LMFDB - Embedded newform 117.4.g.f.55.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [117,4,Mod(55,117)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(117, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 2]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("117.55");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 117 = 3^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	117.g (of order $3$, degree $2$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$6.90322347067$
Analytic rank:	$0$
Dimension:	$16$
Relative dimension:	$8$ over $\Q(\zeta_{3})$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} + 52 x^{14} + 1899 x^{12} + 33440 x^{10} + 424113 x^{8} + 2869882 x^{6} + 13705540 x^{4} + \cdots + 24920064 $ x^16 + 52x^14 + 1899x^12 + 33440x^10 + 424113x^8 + 2869882x^6 + 13705540x^4 + 21016320*x^2 + 24920064
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{4}\cdot 3 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			55.8
Root			$2.69794 + 4.67298i$ of defining polynomial
Character	$\chi$	$=$	117.55
Dual form			117.4.g.f.100.8

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(2.69794 + 4.67298i) q^{2} +(-10.5578 + 18.2867i) q^{4} -13.0421 q^{5} +(3.21247 - 5.56416i) q^{7} -70.7705 q^{8} +O(q^{10})$	\(q+(2.69794 + 4.67298i) q^{2} +(-10.5578 + 18.2867i) q^{4} -13.0421 q^{5} +(3.21247 - 5.56416i) q^{7} -70.7705 q^{8} +(-35.1870 - 60.9457i) q^{10} +(13.1028 + 22.6947i) q^{11} +(43.2448 - 18.0800i) q^{13} +34.6683 q^{14} +(-106.472 - 184.416i) q^{16} +(-61.9384 + 107.280i) q^{17} +(-54.8333 + 94.9741i) q^{19} +(137.697 - 238.497i) q^{20} +(-70.7012 + 122.458i) q^{22} +(-31.7047 - 54.9141i) q^{23} +45.0976 q^{25} +(201.160 + 153.303i) q^{26} +(67.8333 + 117.491i) q^{28} +(112.705 + 195.211i) q^{29} +200.732 q^{31} +(291.431 - 504.774i) q^{32} -668.426 q^{34} +(-41.8975 + 72.5686i) q^{35} +(-126.254 - 218.679i) q^{37} -591.749 q^{38} +922.999 q^{40} +(113.712 + 196.955i) q^{41} +(192.016 - 332.581i) q^{43} -553.348 q^{44} +(171.075 - 296.311i) q^{46} -34.6646 q^{47} +(150.860 + 261.297i) q^{49} +(121.671 + 210.740i) q^{50} +(-125.948 + 981.689i) q^{52} +61.0601 q^{53} +(-170.889 - 295.988i) q^{55} +(-227.348 + 393.778i) q^{56} +(-608.144 + 1053.34i) q^{58} +(-40.2781 + 69.7637i) q^{59} +(13.0692 - 22.6364i) q^{61} +(541.563 + 938.014i) q^{62} +1441.50 q^{64} +(-564.005 + 235.802i) q^{65} +(465.755 + 806.711i) q^{67} +(-1307.87 - 2265.29i) q^{68} -452.149 q^{70} +(213.804 - 370.320i) q^{71} +108.518 q^{73} +(681.255 - 1179.97i) q^{74} +(-1157.84 - 2005.44i) q^{76} +168.369 q^{77} +384.590 q^{79} +(1388.63 + 2405.17i) q^{80} +(-613.576 + 1062.74i) q^{82} +85.9758 q^{83} +(807.810 - 1399.17i) q^{85} +2072.19 q^{86} +(-927.291 - 1606.12i) q^{88} +(-247.952 - 429.465i) q^{89} +(38.3226 - 298.703i) q^{91} +1338.93 q^{92} +(-93.5232 - 161.987i) q^{94} +(715.144 - 1238.67i) q^{95} +(-95.4286 + 165.287i) q^{97} +(-814.024 + 1409.93i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 40 q^{4} + 22 q^{7}+O(q^{10}) $ 16 * q - 40 * q^4 + 22 * q^7	$ 16 q - 40 q^{4} + 22 q^{7} - 36 q^{10} + 36 q^{13} - 204 q^{16} - 244 q^{19} - 136 q^{22} + 708 q^{25} + 452 q^{28} + 484 q^{31} - 2584 q^{34} - 1018 q^{37} + 3400 q^{40} - 74 q^{43} + 896 q^{46} - 298 q^{49} - 1676 q^{52} - 1300 q^{55} - 812 q^{58} - 1148 q^{61} + 7272 q^{64} + 2198 q^{67} + 4400 q^{70} - 4352 q^{73} - 6936 q^{76} + 3724 q^{79} - 5436 q^{82} + 890 q^{85} - 3528 q^{88} - 4754 q^{91} + 3104 q^{94} + 4370 q^{97}+O(q^{100}) $ 16 * q - 40 * q^4 + 22 * q^7 - 36 * q^10 + 36 * q^13 - 204 * q^16 - 244 * q^19 - 136 * q^22 + 708 * q^25 + 452 * q^28 + 484 * q^31 - 2584 * q^34 - 1018 * q^37 + 3400 * q^40 - 74 * q^43 + 896 * q^46 - 298 * q^49 - 1676 * q^52 - 1300 * q^55 - 812 * q^58 - 1148 * q^61 + 7272 * q^64 + 2198 * q^67 + 4400 * q^70 - 4352 * q^73 - 6936 * q^76 + 3724 * q^79 - 5436 * q^82 + 890 * q^85 - 3528 * q^88 - 4754 * q^91 + 3104 * q^94 + 4370 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/117\mathbb{Z}\right)^\times$.

$n$	$28$	$92$
$\chi(n)$	$e\left(\frac{1}{3}\right)$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$	2.69794	+	4.67298i	0.953868	+	1.65215i	0.736939	+	0.675960i	$0.236271\pi$
$2$	2.69794	+	4.67298i	0.953868	+	1.65215i	0.216929	+	0.976187i	$0.430396\pi$
$3$		0			0
$3$		0			0
$4$	−10.5578	+	18.2867i	−1.31973	+	2.28583i
$5$	−13.0421			−1.16653			−0.583263	−	0.812284i	$-0.698224\pi$
$5$	−13.0421			−1.16653			−0.583263	+	0.812284i	$0.698224\pi$
$6$		0			0
$7$	3.21247	−	5.56416i	0.173457	−	0.300437i	−0.766169	−	0.642639i	$-0.777840\pi$
$7$	3.21247	−	5.56416i	0.173457	−	0.300437i	0.939626	+	0.342202i	$0.111173\pi$
$8$	−70.7705			−3.12764
$9$		0			0
$10$	−35.1870	−	60.9457i	−1.11271	−	1.92727i
$11$	13.1028	+	22.6947i	0.359149	+	0.622065i	0.987819	−	0.155608i	$-0.0497336\pi$
$11$	13.1028	+	22.6947i	0.359149	+	0.622065i	−0.628670	+	0.777672i	$0.716400\pi$
$12$		0			0
$13$	43.2448	−	18.0800i	0.922612	−	0.385730i
$13$	43.2448	−	18.0800i	0.922612	−	0.385730i
$14$	34.6683			0.661820
$15$		0			0
$16$	−106.472	−	184.416i	−1.66363	−	2.88149i
$17$	−61.9384	+	107.280i	−0.883663	+	1.53055i	−0.0364243	+	0.999336i	$0.511597\pi$
$17$	−61.9384	+	107.280i	−0.883663	+	1.53055i	−0.847239	+	0.531213i	$0.821737\pi$
$18$		0			0
$19$	−54.8333	+	94.9741i	−0.662086	+	1.14677i	0.317981	+	0.948097i	$0.396995\pi$
$19$	−54.8333	+	94.9741i	−0.662086	+	1.14677i	−0.980067	+	0.198669i	$0.936338\pi$
$20$	137.697	−	238.497i	1.53949	−	2.66648i
$21$		0			0
$22$	−70.7012	+	122.458i	−0.685162	+	1.18673i
$23$	−31.7047	−	54.9141i	−0.287430	−	0.497843i	0.685766	−	0.727822i	$-0.259467\pi$
$23$	−31.7047	−	54.9141i	−0.287430	−	0.497843i	−0.973196	+	0.229979i	$0.926134\pi$
$24$		0			0
$25$	45.0976			0.360781
$26$	201.160	+	153.303i	1.51733	+	1.15635i
$27$		0			0
$28$	67.8333	+	117.491i	0.457832	+	0.792988i
$29$	112.705	+	195.211i	0.721683	+	1.24999i	0.960325	+	0.278884i	$0.0899644\pi$
$29$	112.705	+	195.211i	0.721683	+	1.24999i	−0.238642	+	0.971108i	$0.576702\pi$
$30$		0			0
$31$	200.732			1.16298			0.581491	−	0.813553i	$-0.302470\pi$
$31$	200.732			1.16298			0.581491	+	0.813553i	$0.302470\pi$
$32$	291.431	−	504.774i	1.60995	−	2.78851i
$33$		0			0
$34$	−668.426			−3.37159
$35$	−41.8975	+	72.5686i	−0.202342	+	0.350467i
$36$		0			0
$37$	−126.254	−	218.679i	−0.560976	−	0.971638i	−0.997412	−	0.0719031i	$-0.977093\pi$
$37$	−126.254	−	218.679i	−0.560976	−	0.971638i	0.436436	−	0.899735i	$-0.356241\pi$
$38$	−591.749			−2.52617
$39$		0			0
$40$	922.999			3.64847
$41$	113.712	+	196.955i	0.433141	+	0.750223i	0.997142	−	0.0755517i	$-0.0240718\pi$
$41$	113.712	+	196.955i	0.433141	+	0.750223i	−0.564001	+	0.825774i	$0.690738\pi$
$42$		0			0
$43$	192.016	−	332.581i	0.680980	−	1.17949i	−0.293702	−	0.955897i	$-0.594887\pi$
$43$	192.016	−	332.581i	0.680980	−	1.17949i	0.974682	−	0.223595i	$-0.0717794\pi$
$44$	−553.348			−1.89592
$45$		0			0
$46$	171.075	−	296.311i	0.548340	−	0.949753i
$47$	−34.6646			−0.107582			−0.0537910	−	0.998552i	$-0.517130\pi$
$47$	−34.6646			−0.107582			−0.0537910	+	0.998552i	$0.517130\pi$
$48$		0			0
$49$	150.860	+	261.297i	0.439825	+	0.761800i
$50$	121.671	+	210.740i	0.344137	+	0.596063i
$51$		0			0
$52$	−125.948	+	981.689i	−0.335881	+	2.61800i
$53$	61.0601			0.158250			0.0791251	−	0.996865i	$-0.474787\pi$
$53$	61.0601			0.158250			0.0791251	+	0.996865i	$0.474787\pi$
$54$		0			0
$55$	−170.889	−	295.988i	−0.418957	−	0.725654i
$56$	−227.348	+	393.778i	−0.542512	+	0.939658i
$57$		0			0
$58$	−608.144	+	1053.34i	−1.37678	+	2.38465i
$59$	−40.2781	+	69.7637i	−0.0888774	+	0.153940i	−0.907037	−	0.421051i	$-0.861661\pi$
$59$	−40.2781	+	69.7637i	−0.0888774	+	0.153940i	0.818159	+	0.574991i	$0.194995\pi$
$60$		0			0
$61$	13.0692	−	22.6364i	0.0274317	−	0.0475131i	−0.851984	−	0.523568i	$-0.824600\pi$
$61$	13.0692	−	22.6364i	0.0274317	−	0.0475131i	0.879415	+	0.476055i	$0.157934\pi$
$62$	541.563	+	938.014i	1.10933	+	1.92142i
$63$		0			0
$64$	1441.50			2.81544
$65$	−564.005	+	235.802i	−1.07625	+	0.449964i
$66$		0			0
$67$	465.755	+	806.711i	0.849269	+	1.47098i	0.881861	+	0.471509i	$0.156290\pi$
$67$	465.755	+	806.711i	0.849269	+	1.47098i	−0.0325921	+	0.999469i	$0.510376\pi$
$68$	−1307.87	−	2265.29i	−2.33239	−	4.03981i
$69$		0			0
$70$	−452.149			−0.772030
$71$	213.804	−	370.320i	0.357379	−	0.618998i	−0.630143	−	0.776479i	$-0.717004\pi$
$71$	213.804	−	370.320i	0.357379	−	0.618998i	0.987522	+	0.157481i	$0.0503373\pi$
$72$		0			0
$73$	108.518			0.173987			0.0869935	−	0.996209i	$-0.472274\pi$
$73$	108.518			0.173987			0.0869935	+	0.996209i	$0.472274\pi$
$74$	681.255	−	1179.97i	1.07019	−	1.85363i
$75$		0			0
$76$	−1157.84	−	2005.44i	−1.74754	−	3.02684i
$77$	168.369			0.249188
$78$		0			0
$79$	384.590			0.547718			0.273859	−	0.961770i	$-0.411700\pi$
$79$	384.590			0.547718			0.273859	+	0.961770i	$0.411700\pi$
$80$	1388.63	+	2405.17i	1.94067	+	3.36133i
$81$		0			0
$82$	−613.576	+	1062.74i	−0.826319	+	1.43123i
$83$	85.9758			0.113700			0.0568498	−	0.998383i	$-0.481894\pi$
$83$	85.9758			0.113700			0.0568498	+	0.998383i	$0.481894\pi$
$84$		0			0
$85$	807.810	−	1399.17i	1.03081	−	1.78542i
$86$	2072.19			2.59826
$87$		0			0
$88$	−927.291	−	1606.12i	−1.12329	−	1.94560i
$89$	−247.952	−	429.465i	−0.295313	−	0.511497i	0.679745	−	0.733449i	$-0.262090\pi$
$89$	−247.952	−	429.465i	−0.295313	−	0.511497i	−0.975058	+	0.221952i	$0.928757\pi$
$90$		0			0
$91$	38.3226	−	298.703i	0.0441462	−	0.344094i
$92$	1338.93			1.51732
$93$		0			0
$94$	−93.5232	−	161.987i	−0.102619	−	0.177741i
$95$	715.144	−	1238.67i	0.772340	−	1.33773i
$96$		0			0
$97$	−95.4286	+	165.287i	−0.0998898	+	0.173014i	−0.911639	−	0.410992i	$-0.865182\pi$
$97$	−95.4286	+	165.287i	−0.0998898	+	0.173014i	0.811749	+	0.584006i	$0.198516\pi$
$98$	−814.024	+	1409.93i	−0.839070	+	1.45331i
$99$		0			0
$100$	−476.132	+	824.685i	−0.476132	+	0.824685i
$101$	−425.490	−	736.970i	−0.419186	−	0.726052i	0.576671	−	0.816976i	$-0.304351\pi$
$101$	−425.490	−	736.970i	−0.419186	−	0.726052i	−0.995858	+	0.0909241i	$0.971018\pi$
$102$		0			0
$103$	−1309.50			−1.25270			−0.626352	−	0.779540i	$-0.715453\pi$
$103$	−1309.50			−1.25270			−0.626352	+	0.779540i	$0.715453\pi$
$104$	−3060.46	+	1279.53i	−2.88560	+	1.20643i
$105$		0			0
$106$	164.737	+	285.333i	0.150950	+	0.261452i
$107$	−481.151	−	833.378i	−0.434716	−	0.752951i	0.562556	−	0.826759i	$-0.309818\pi$
$107$	−481.151	−	833.378i	−0.434716	−	0.752951i	−0.997272	+	0.0738085i	$0.976485\pi$
$108$		0			0
$109$	891.973			0.783812			0.391906	−	0.920005i	$-0.371816\pi$
$109$	891.973			0.783812			0.391906	+	0.920005i	$0.371816\pi$
$110$	922.096	−	1597.12i	0.799258	−	1.38436i
$111$		0			0
$112$	−1368.16			−1.15427
$113$	−131.920	+	228.491i	−0.109823	+	0.190218i	−0.915698	−	0.401867i	$-0.868362\pi$
$113$	−131.920	+	228.491i	−0.109823	+	0.190218i	0.805876	+	0.592085i	$0.201695\pi$
$114$		0			0
$115$	413.497	+	716.198i	0.335294	+	0.580746i
$116$	−4759.68			−3.80970
$117$		0			0
$118$	−434.673			−0.339109
$119$	397.951	+	689.271i	0.306555	+	0.530969i
$120$		0			0
$121$	322.133	−	557.952i	0.242024	−	0.419197i
$122$	141.039			0.104665
$123$		0			0
$124$	−2119.29	+	3670.71i	−1.53482	+	2.65838i
$125$	1042.10			0.745665
$126$		0			0
$127$	771.986	+	1337.12i	0.539391	+	0.934252i	0.998937	+	0.0460984i	$0.0146788\pi$
$127$	771.986	+	1337.12i	0.539391	+	0.934252i	−0.459546	+	0.888154i	$0.651988\pi$
$128$	1557.65	+	2697.93i	1.07561	+	1.86301i
$129$		0			0
$130$	−2623.55	−	1999.40i	−1.77001	−	1.34892i
$131$	−776.790			−0.518080			−0.259040	−	0.965867i	$-0.583406\pi$
$131$	−776.790			−0.518080			−0.259040	+	0.965867i	$0.583406\pi$
$132$		0			0
$133$	352.301	+	610.203i	0.229687	+	0.397829i
$134$	−2513.16	+	4352.93i	−1.62018	+	2.80624i
$135$		0			0
$136$	4383.41	−	7592.29i	2.76378	−	4.78701i
$137$	1217.36	−	2108.52i	0.759166	−	1.31491i	−0.184110	−	0.982906i	$-0.558940\pi$
$137$	1217.36	−	2108.52i	0.759166	−	1.31491i	0.943276	−	0.332009i	$-0.107726\pi$
$138$		0			0
$139$	−625.700	+	1083.74i	−0.381807	+	0.661310i	−0.991321	−	0.131466i	$-0.958032\pi$
$139$	−625.700	+	1083.74i	−0.381807	+	0.661310i	0.609513	+	0.792776i	$0.291365\pi$
$140$	−884.692	−	1532.33i	−0.534072	−	0.925041i
$141$		0			0
$142$	2307.33			1.36357
$143$	976.948	+	744.530i	0.571304	+	0.435390i
$144$		0			0
$145$	−1469.92	−	2545.97i	−0.841861	−	1.45815i
$146$	292.775	+	507.101i	0.165961	+	0.287452i
$147$		0			0
$148$	5331.88			2.96134
$149$	391.810	−	678.634i	0.215425	−	0.373127i	−0.737979	−	0.674824i	$-0.764220\pi$
$149$	391.810	−	678.634i	0.215425	−	0.373127i	0.953404	+	0.301697i	$0.0975531\pi$
$150$		0			0
$151$	162.220			0.0874255			0.0437128	−	0.999044i	$-0.486081\pi$
$151$	162.220			0.0874255			0.0437128	+	0.999044i	$0.486081\pi$
$152$	3880.58	−	6721.37i	2.07077	−	3.58668i
$153$		0			0
$154$	454.251	+	786.786i	0.237692	+	0.411695i
$155$	−2617.97			−1.35665
$156$		0			0
$157$	−2355.76			−1.19752			−0.598758	−	0.800930i	$-0.704339\pi$
$157$	−2355.76			−1.19752			−0.598758	+	0.800930i	$0.704339\pi$
$158$	1037.60	+	1797.18i	0.522451	+	0.904911i
$159$		0			0
$160$	−3800.89	+	6583.33i	−1.87804	+	3.25286i
$161$	−407.402			−0.199427
$162$		0			0
$163$	−787.143	+	1363.37i	−0.378244	+	0.655138i	−0.990807	−	0.135284i	$-0.956805\pi$
$163$	−787.143	+	1363.37i	−0.378244	+	0.655138i	0.612563	+	0.790422i	$0.290139\pi$
$164$	−4802.19			−2.28651
$165$		0			0
$166$	231.958	+	401.763i	0.108454	+	0.187849i
$167$	1787.28	+	3095.66i	0.828166	+	1.43443i	0.899475	+	0.436972i	$0.143949\pi$
$167$	1787.28	+	3095.66i	0.828166	+	1.43443i	−0.0713090	+	0.997454i	$0.522718\pi$
$168$		0			0
$169$	1543.23	−	1563.73i	0.702425	−	0.711758i
$170$	8717.70			3.93304
$171$		0			0
$172$	4054.54	+	7022.66i	1.79742	+	3.11322i
$173$	0.720540	−	1.24801i	0.000316657	−	0.000548466i	−0.865867	−	0.500274i	$-0.833233\pi$
$173$	0.720540	−	1.24801i	0.000316657	−	0.000548466i	0.866184	+	0.499726i	$0.166566\pi$
$174$		0			0
$175$	144.875	−	250.930i	0.0625800	−	0.108392i
$176$	2790.17	−	4832.72i	1.19498	−	2.06977i
$177$		0			0
$178$	1337.92	−	2317.35i	0.563378	−	0.975800i
$179$	−1041.14	−	1803.30i	−0.434739	−	0.752990i	0.562535	−	0.826773i	$-0.309826\pi$
$179$	−1041.14	−	1803.30i	−0.434739	−	0.752990i	−0.997274	+	0.0737830i	$0.976493\pi$
$180$		0			0
$181$	−464.500			−0.190751			−0.0953756	−	0.995441i	$-0.530405\pi$
$181$	−464.500			−0.190751			−0.0953756	+	0.995441i	$0.530405\pi$
$182$	1499.22	−	626.802i	0.610603	−	0.255284i
$183$		0			0
$184$	2243.76	+	3886.30i	0.898978	+	1.55708i
$185$	1646.63	+	2852.04i	0.654392	+	1.13344i
$186$		0			0
$187$	−3246.26			−1.26947
$188$	365.982	−	633.900i	0.141979	−	0.245914i
$189$		0			0
$190$	7717.68			2.94684
$191$	−2433.14	+	4214.32i	−0.921758	+	1.59653i	−0.125065	+	0.992149i	$0.539914\pi$
$191$	−2433.14	+	4214.32i	−0.921758	+	1.59653i	−0.796693	+	0.604384i	$0.793419\pi$
$192$		0			0
$193$	428.169	+	741.611i	0.159691	+	0.276592i	0.934757	−	0.355287i	$-0.115617\pi$
$193$	428.169	+	741.611i	0.159691	+	0.276592i	−0.775066	+	0.631880i	$0.782284\pi$
$194$	−1029.84			−0.381127
$195$		0			0
$196$	−6371.01			−2.32180
$197$	−785.965	−	1361.33i	−0.284252	−	0.492339i	0.688175	−	0.725544i	$-0.258412\pi$
$197$	−785.965	−	1361.33i	−0.284252	−	0.492339i	−0.972428	+	0.233205i	$0.925079\pi$
$198$		0			0
$199$	−1563.30	+	2707.72i	−0.556883	+	0.964549i	0.440872	+	0.897570i	$0.354669\pi$
$199$	−1563.30	+	2707.72i	−0.556883	+	0.964549i	−0.997754	+	0.0669791i	$0.978664\pi$
$200$	−3191.58			−1.12839
$201$		0			0
$202$	2295.90	−	3976.61i	0.799697	−	1.38512i
$203$	1448.25			0.500724
$204$		0			0
$205$	−1483.05	−	2568.71i	−0.505270	−	0.875153i
$206$	−3532.95	−	6119.25i	−1.19491	−	2.06965i
$207$		0			0
$208$	−7938.61	−	6050.00i	−2.64636	−	2.01679i
$209$	−2873.88			−0.951150
$210$		0			0
$211$	−61.6959	−	106.860i	−0.0201295	−	0.0348653i	0.855785	−	0.517331i	$-0.173075\pi$
$211$	−61.6959	−	106.860i	−0.0201295	−	0.0348653i	−0.875915	+	0.482466i	$0.839741\pi$
$212$	−644.662	+	1116.59i	−0.208847	+	0.361733i
$213$		0			0
$214$	2596.24	−	4496.82i	0.829323	−	1.43643i
$215$	−2504.30	+	4337.57i	−0.794380	+	1.37591i
$216$		0			0
$217$	644.844	−	1116.90i	0.201728	−	0.349402i
$218$	2406.49	+	4168.17i	0.747653	+	1.29497i
$219$		0			0
$220$	7216.84			2.21163
$221$	−738.884	+	5759.17i	−0.224899	+	1.75296i
$222$		0			0
$223$	1251.84	+	2168.26i	0.375918	+	0.651109i	0.990464	−	0.137772i	$-0.0439941\pi$
$223$	1251.84	+	2168.26i	0.375918	+	0.651109i	−0.614546	+	0.788881i	$0.710661\pi$
$224$	−1872.43	−	3243.14i	−0.558513	−	0.967373i
$225$		0			0
$226$	−1423.65			−0.419025
$227$	289.635	−	501.662i	0.0846861	−	0.146681i	−0.820571	−	0.571544i	$-0.806345\pi$
$227$	289.635	−	501.662i	0.0846861	−	0.146681i	0.905257	+	0.424864i	$0.139678\pi$
$228$		0			0
$229$	−768.922			−0.221886			−0.110943	−	0.993827i	$-0.535387\pi$
$229$	−768.922			−0.221886			−0.110943	+	0.993827i	$0.535387\pi$
$230$	−2231.19	+	3864.53i	−0.639652	+	1.10791i
$231$		0			0
$232$	−7976.19	−	13815.2i	−2.25717	−	3.90953i
$233$	−845.695			−0.237783			−0.118891	−	0.992907i	$-0.537934\pi$
$233$	−845.695			−0.237783			−0.118891	+	0.992907i	$0.537934\pi$
$234$		0			0
$235$	452.101			0.125497
$236$	−850.498	−	1473.11i	−0.234588	−	0.406318i
$237$		0			0
$238$	−2147.30	+	3719.23i	−0.584826	+	1.01295i
$239$	6552.78			1.77349			0.886744	−	0.462260i	$-0.152961\pi$
$239$	6552.78			1.77349			0.886744	+	0.462260i	$0.152961\pi$
$240$		0			0
$241$	−2588.38	+	4483.20i	−0.691834	+	1.19829i	0.279402	+	0.960174i	$0.409864\pi$
$241$	−2588.38	+	4483.20i	−0.691834	+	1.19829i	−0.971236	+	0.238118i	$0.923470\pi$
$242$	3476.39			0.923434
$243$		0			0
$244$	275.963	+	477.983i	0.0724047	+	0.125409i
$245$	−1967.54	−	3407.88i	−0.513067	−	0.888658i
$246$		0			0
$247$	−654.125	+	5098.52i	−0.168506	+	1.31341i
$248$	−14205.9			−3.63739
$249$		0			0
$250$	2811.53	+	4869.70i	0.711266	+	1.23195i
$251$	2148.41	−	3721.15i	0.540264	−	0.935765i	−0.458624	−	0.888630i	$-0.651658\pi$
$251$	2148.41	−	3721.15i	0.540264	−	0.935765i	0.998889	−	0.0471349i	$-0.0150091\pi$
$252$		0			0
$253$	830.840	−	1439.06i	0.206460	−	0.357600i
$254$	−4165.55	+	7214.94i	−1.02901	+	1.78231i
$255$		0			0
$256$	−2638.88	+	4570.68i	−0.644259	+	1.11589i
$257$	−691.970	−	1198.53i	−0.167953	−	0.290903i	0.769747	−	0.638349i	$-0.220382\pi$
$257$	−691.970	−	1198.53i	−0.167953	−	0.290903i	−0.937700	+	0.347446i	$0.887049\pi$
$258$		0			0
$259$	−1622.35			−0.389221
$260$	1642.63	−	12803.3i	0.391813	−	3.05396i
$261$		0			0
$262$	−2095.74	−	3629.92i	−0.494180	−	0.855944i
$263$	−1239.34	−	2146.60i	−0.290574	−	0.503288i	0.683372	−	0.730070i	$-0.260513\pi$
$263$	−1239.34	−	2146.60i	−0.290574	−	0.503288i	−0.973945	+	0.226782i	$0.927179\pi$
$264$		0			0
$265$	−796.355			−0.184603
$266$	−1900.98	+	3292.59i	−0.438182	+	0.758953i
$267$		0			0
$268$	−19669.4			−4.48321
$269$	−1226.32	+	2124.05i	−0.277956	+	0.481433i	−0.970877	−	0.239580i	$-0.922990\pi$
$269$	−1226.32	+	2124.05i	−0.277956	+	0.481433i	0.692921	+	0.721014i	$0.256323\pi$
$270$		0			0
$271$	2990.37	+	5179.47i	0.670302	+	1.16100i	0.977818	+	0.209455i	$0.0671689\pi$
$271$	2990.37	+	5179.47i	0.670302	+	1.16100i	−0.307516	+	0.951543i	$0.599498\pi$
$272$	26378.9			5.88036
$273$		0			0
$274$	13137.4			2.89658
$275$	590.905	+	1023.48i	0.129574	+	0.224429i
$276$		0			0
$277$	3689.00	−	6389.53i	0.800182	−	1.38596i	−0.119314	−	0.992857i	$-0.538070\pi$
$277$	3689.00	−	6389.53i	0.800182	−	1.38596i	0.919496	−	0.393099i	$-0.128597\pi$
$278$	−6752.42			−1.45677
$279$		0			0
$280$	2965.11	−	5135.72i	0.632854	−	1.09613i
$281$	5937.17			1.26043			0.630217	−	0.776419i	$-0.282966\pi$
$281$	5937.17			1.26043			0.630217	+	0.776419i	$0.282966\pi$
$282$		0			0
$283$	−743.259	−	1287.36i	−0.156121	−	0.270409i	0.777346	−	0.629074i	$-0.216566\pi$
$283$	−743.259	−	1287.36i	−0.156121	−	0.270409i	−0.933467	+	0.358664i	$0.883232\pi$
$284$	4514.61	+	7819.53i	0.943284	+	1.63382i
$285$		0			0
$286$	−843.419	+	6573.96i	−0.174379	+	1.35918i
$287$	1461.18			0.300526
$288$		0			0
$289$	−5216.23	−	9034.78i	−1.06172	−	1.83895i
$290$	7931.50	−	13737.8i	1.60605	−	2.78176i
$291$		0			0
$292$	−1145.71	+	1984.43i	−0.229615	+	0.397705i
$293$	1790.02	−	3100.41i	0.356909	−	0.618185i	−0.630534	−	0.776162i	$-0.717164\pi$
$293$	1790.02	−	3100.41i	0.356909	−	0.618185i	0.987443	+	0.157977i	$0.0504972\pi$
$294$		0			0
$295$	525.313	−	909.869i	0.103678	−	0.179575i
$296$	8935.09	+	15476.0i	1.75453	+	3.03894i
$297$		0			0
$298$	4228.32			0.821947
$299$	−2363.91	−	1801.53i	−0.457219	−	0.348446i
$300$		0			0
$301$	−1233.69	−	2136.81i	−0.236242	−	0.409183i
$302$	437.660	+	758.049i	0.0833924	+	0.144440i
$303$		0			0
$304$	23352.9			4.40587
$305$	−170.450	+	295.228i	−0.0319998	+	0.0554252i
$306$		0			0
$307$	−415.013			−0.0771533			−0.0385767	−	0.999256i	$-0.512282\pi$
$307$	−415.013			−0.0771533			−0.0385767	+	0.999256i	$0.512282\pi$
$308$	−1777.61	+	3078.92i	−0.328860	+	0.569602i
$309$		0			0
$310$	−7063.14	−	12233.7i	−1.29406	−	2.24138i
$311$	3009.05			0.548642			0.274321	−	0.961638i	$-0.411547\pi$
$311$	3009.05			0.548642			0.274321	+	0.961638i	$0.411547\pi$
$312$		0			0
$313$	3760.84			0.679154			0.339577	−	0.940578i	$-0.389716\pi$
$313$	3760.84			0.679154			0.339577	+	0.940578i	$0.389716\pi$
$314$	−6355.70	−	11008.4i	−1.14227	−	1.97847i
$315$		0			0
$316$	−4060.43	+	7032.87i	−0.722838	+	1.25199i
$317$	2772.04			0.491146			0.245573	−	0.969378i	$-0.421024\pi$
$317$	2772.04			0.491146			0.245573	+	0.969378i	$0.421024\pi$
$318$		0			0
$319$	−2953.50	+	5115.62i	−0.518384	+	0.897867i
$320$	−18800.3			−3.28428
$321$		0			0
$322$	−1099.15	−	1903.78i	−0.190227	−	0.329483i
$323$	−6792.58	−	11765.1i	−1.17012	−	2.02671i
$324$		0			0
$325$	1950.24	−	815.365i	0.332861	−	0.139164i
$326$	−8494.67			−1.44318
$327$		0			0
$328$	−8047.44	−	13938.6i	−1.35471	−	2.34643i
$329$	−111.359	+	192.879i	−0.0186608	+	0.0323215i
$330$		0			0
$331$	−3279.08	+	5679.53i	−0.544515	+	0.943128i	0.454122	+	0.890939i	$0.349953\pi$
$331$	−3279.08	+	5679.53i	−0.544515	+	0.943128i	−0.998637	+	0.0521882i	$0.983380\pi$
$332$	−907.716	+	1572.21i	−0.150052	+	0.259898i
$333$		0			0
$334$	−9643.96	+	16703.8i	−1.57992	+	2.73650i
$335$	−6074.45	−	10521.2i	−0.990694	−	1.71593i
$336$		0			0
$337$	9509.17			1.53708			0.768542	−	0.639799i	$-0.220983\pi$
$337$	9509.17			1.53708			0.768542	+	0.639799i	$0.220983\pi$
$338$	11470.8	+	2992.60i	1.84595	+	0.481586i
$339$		0			0
$340$	17057.4	+	29544.3i	2.72079	+	4.71254i
$341$	2630.14	+	4555.54i	0.417684	+	0.723450i
$342$		0			0
$343$	4142.29			0.652077
$344$	−13589.1	+	23536.9i	−2.12986	+	3.68903i
$345$		0			0
$346$	7.77591			0.00120820
$347$	−4904.39	+	8494.66i	−0.758736	+	1.31417i	0.184759	+	0.982784i	$0.440850\pi$
$347$	−4904.39	+	8494.66i	−0.758736	+	1.31417i	−0.943495	+	0.331386i	$0.892484\pi$
$348$		0			0
$349$	−397.476	−	688.449i	−0.0609639	−	0.105593i	0.833933	−	0.551866i	$-0.186084\pi$
$349$	−397.476	−	688.449i	−0.0609639	−	0.105593i	−0.894897	+	0.446274i	$0.852751\pi$
$350$	1563.46			0.238772
$351$		0			0
$352$	15274.3			2.31284
$353$	−3557.80	−	6162.29i	−0.536438	−	0.929138i	−0.999092	−	0.0425989i	$-0.986436\pi$
$353$	−3557.80	−	6162.29i	−0.536438	−	0.929138i	0.462654	−	0.886539i	$-0.346897\pi$
$354$		0			0
$355$	−2788.46	+	4829.76i	−0.416891	+	0.722077i
$356$	10471.3			1.55893
$357$		0			0
$358$	5617.87	−	9730.43i	0.829367	−	1.43651i
$359$	6907.19			1.01545			0.507726	−	0.861518i	$-0.330486\pi$
$359$	6907.19			1.01545			0.507726	+	0.861518i	$0.330486\pi$
$360$		0			0
$361$	−2583.89	−	4475.43i	−0.376715	−	0.652490i
$362$	−1253.19	−	2170.60i	−0.181951	−	0.315149i
$363$		0			0
$364$	5057.67	+	3854.44i	0.728280	+	0.555021i
$365$	−1415.31			−0.202960
$366$		0			0
$367$	229.767	+	397.967i	0.0326804	+	0.0566042i	0.881903	−	0.471431i	$-0.156262\pi$
$367$	229.767	+	397.967i	0.0326804	+	0.0566042i	−0.849223	+	0.528035i	$0.822929\pi$
$368$	−6751.35	+	11693.7i	−0.956354	+	1.65645i
$369$		0			0
$370$	−8885.03	+	15389.3i	−1.24841	+	2.16230i
$371$	196.154	−	339.749i	0.0274496	−	0.0475441i
$372$		0			0
$373$	−2679.96	+	4641.83i	−0.372019	+	0.644355i	−0.989876	−	0.141935i	$-0.954668\pi$
$373$	−2679.96	+	4641.83i	−0.372019	+	0.644355i	0.617857	+	0.786290i	$0.288001\pi$
$374$	−8758.24	−	15169.7i	−1.21090	−	2.09735i
$375$		0			0
$376$	2453.23			0.336478
$377$	8403.32	+	6404.15i	1.14799	+	0.874882i
$378$		0			0
$379$	−4297.03	−	7442.68i	−0.582384	−	1.00872i	−0.995196	−	0.0979028i	$-0.968787\pi$
$379$	−4297.03	−	7442.68i	−0.582384	−	1.00872i	0.412812	−	0.910816i	$-0.364547\pi$
$380$	15100.7	+	26155.2i	2.03855	+	3.53088i
$381$		0			0
$382$	−26257.9			−3.51694
$383$	4668.24	−	8085.63i	0.622810	−	1.07874i	−0.366151	−	0.930556i	$-0.619324\pi$
$383$	4668.24	−	8085.63i	0.622810	−	1.07874i	0.988960	−	0.148182i	$-0.0473422\pi$
$384$		0			0
$385$	−2195.90			−0.290684
$386$	−2310.35	+	4001.65i	−0.304648	+	0.527665i
$387$		0			0
$388$	−2015.04	−	3490.14i	−0.263654	−	0.456663i
$389$	−12792.1			−1.66732			−0.833659	−	0.552279i	$-0.813758\pi$
$389$	−12792.1			−1.66732			−0.833659	+	0.552279i	$0.813758\pi$
$390$		0			0
$391$	7854.95			1.01596
$392$	−10676.4	−	18492.1i	−1.37562	−	2.38264i
$393$		0			0
$394$	4240.98	−	7345.59i	0.542278	−	0.939253i
$395$	−5015.88			−0.638927
$396$		0			0
$397$	6692.24	−	11591.3i	0.846030	−	1.46537i	−0.0386932	−	0.999251i	$-0.512319\pi$
$397$	6692.24	−	11591.3i	0.846030	−	1.46537i	0.884723	−	0.466116i	$-0.154347\pi$
$398$	−16870.8			−2.12477
$399$		0			0
$400$	−4801.65	−	8316.70i	−0.600206	−	1.03959i
$401$	−3178.71	−	5505.68i	−0.395853	−	0.685637i	0.597357	−	0.801976i	$-0.296218\pi$
$401$	−3178.71	−	5505.68i	−0.395853	−	0.685637i	−0.993210	+	0.116338i	$0.962884\pi$
$402$		0			0
$403$	8680.60	−	3629.23i	1.07298	−	0.448597i
$404$	17969.0			2.21285
$405$		0			0
$406$	3907.29	+	6767.62i	0.477624	+	0.827270i
$407$	3308.57	−	5730.62i	0.402948	−	0.697926i
$408$		0			0
$409$	2650.81	−	4591.35i	0.320475	−	0.555079i	−0.660111	−	0.751168i	$-0.729491\pi$
$409$	2650.81	−	4591.35i	0.320475	−	0.555079i	0.980586	+	0.196089i	$0.0628241\pi$
$410$	8002.35	−	13860.5i	0.963922	−	1.66956i
$411$		0			0
$412$	13825.4	−	23946.3i	1.65323	−	2.86347i
$413$	258.785	+	448.228i	0.0308328	+	0.0534040i
$414$		0			0
$415$	−1121.31			−0.132633
$416$	3476.58	−	27097.9i	0.409744	−	3.19371i
$417$		0			0
$418$	−7753.57	−	13429.6i	−0.907272	−	1.57144i
$419$	8197.92	+	14199.2i	0.955835	+	1.65555i	0.732447	+	0.680824i	$0.238378\pi$
$419$	8197.92	+	14199.2i	0.955835	+	1.65555i	0.223388	+	0.974730i	$0.428288\pi$
$420$		0			0
$421$	8484.68			0.982227			0.491114	−	0.871095i	$-0.336590\pi$
$421$	8484.68			0.982227			0.491114	+	0.871095i	$0.336590\pi$
$422$	332.904	−	576.607i	0.0384017	−	0.0665137i
$423$		0			0
$424$	−4321.26			−0.494950
$425$	−2793.27	+	4838.09i	−0.318809	+	0.552193i
$426$		0			0
$427$	−83.9685	−	145.438i	−0.00951645	−	0.0164830i
$428$	20319.6			2.29483
$429$		0			0
$430$	−27025.8			−3.03094
$431$	797.370	+	1381.09i	0.0891136	+	0.154349i	0.907137	−	0.420836i	$-0.138263\pi$
$431$	797.370	+	1381.09i	0.0891136	+	0.154349i	−0.818023	+	0.575185i	$0.804930\pi$
$432$		0			0
$433$	1693.79	−	2933.73i	0.187987	−	0.325602i	−0.756592	−	0.653887i	$-0.773137\pi$
$433$	1693.79	−	2933.73i	0.187987	−	0.325602i	0.944579	+	0.328285i	$0.106471\pi$
$434$	6959.02			0.769685
$435$		0			0
$436$	−9417.29	+	16311.2i	−1.03442	+	1.79166i
$437$	6953.90			0.761213
$438$		0			0
$439$	3605.77	+	6245.37i	0.392014	+	0.678987i	0.992715	−	0.120486i	$-0.0384453\pi$
$439$	3605.77	+	6245.37i	0.392014	+	0.678987i	−0.600701	+	0.799473i	$0.705112\pi$
$440$	12093.9	+	20947.2i	1.31035	+	2.26959i
$441$		0			0
$442$	−28905.9	+	12085.1i	−3.11067	+	1.30052i
$443$	7500.66			0.804441			0.402220	−	0.915543i	$-0.368239\pi$
$443$	7500.66			0.804441			0.402220	+	0.915543i	$0.368239\pi$
$444$		0			0
$445$	3233.82	+	5601.15i	0.344490	+	0.596674i
$446$	−6754.82	+	11699.7i	−0.717152	+	1.24214i
$447$		0			0
$448$	4630.79	−	8020.76i	0.488358	−	0.845860i
$449$	−3824.77	+	6624.69i	−0.402009	+	0.696299i	−0.993968	−	0.109668i	$-0.965021\pi$
$449$	−3824.77	+	6624.69i	−0.402009	+	0.696299i	0.591960	+	0.805968i	$0.298354\pi$
$450$		0			0
$451$	−2979.88	+	5161.31i	−0.311125	+	0.538884i
$452$	−2785.56	−	4824.74i	−0.289871	−	0.502072i
$453$		0			0
$454$	3125.67			0.323117
$455$	−499.809	+	3895.72i	−0.0514976	+	0.401394i
$456$		0			0
$457$	4500.90	+	7795.78i	0.460707	+	0.797968i	0.998996	−	0.0447924i	$-0.0142626\pi$
$457$	4500.90	+	7795.78i	0.460707	+	0.797968i	−0.538290	+	0.842760i	$0.680929\pi$
$458$	−2074.51	−	3593.16i	−0.211649	−	0.366588i
$459$		0			0
$460$	−17462.5			−1.76999
$461$	−673.916	+	1167.26i	−0.0680854	+	0.117927i	−0.898059	−	0.439876i	$-0.855022\pi$
$461$	−673.916	+	1167.26i	−0.0680854	+	0.117927i	0.829973	+	0.557803i	$0.188356\pi$
$462$		0			0
$463$	−94.9035			−0.00952600			−0.00476300	−	0.999989i	$-0.501516\pi$
$463$	−94.9035			−0.00952600			−0.00476300	+	0.999989i	$0.501516\pi$
$464$	24000.0	−	41569.1i	2.40123	−	4.15905i
$465$		0			0
$466$	−2281.64	−	3951.92i	−0.226813	−	0.392852i
$467$	−13290.9			−1.31698			−0.658489	−	0.752591i	$-0.728804\pi$
$467$	−13290.9			−1.31698			−0.658489	+	0.752591i	$0.728804\pi$
$468$		0			0
$469$	5984.90			0.589247
$470$	1219.74	+	2112.66i	0.119708	+	0.207340i
$471$		0			0
$472$	2850.50	−	4937.21i	0.277977	−	0.481470i
$473$	10063.8			0.978294
$474$		0			0
$475$	−2472.85	+	4283.10i	−0.238868	+	0.413731i
$476$	−16806.0			−1.61828
$477$		0			0
$478$	17679.0	+	30621.0i	1.69167	+	2.93006i
$479$	−1900.79	−	3292.27i	−0.181314	−	0.314045i	0.761014	−	0.648735i	$-0.224702\pi$
$479$	−1900.79	−	3292.27i	−0.181314	−	0.314045i	−0.942328	+	0.334690i	$0.891368\pi$
$480$		0			0
$481$	−9413.57	−	7174.06i	−0.892353	−	0.680060i
$482$	−27933.2			−2.63967
$483$		0			0
$484$	6802.05	+	11781.5i	0.638810	+	1.10645i
$485$	1244.59	−	2155.70i	0.116524	−	0.201825i
$486$		0			0
$487$	4373.67	−	7575.42i	0.406961	−	0.704877i	−0.587587	−	0.809161i	$-0.699922\pi$
$487$	4373.67	−	7575.42i	0.406961	−	0.704877i	0.994548	+	0.104284i	$0.0332552\pi$
$488$	−924.911	+	1601.99i	−0.0857966	+	0.148604i
$489$		0			0
$490$	10616.6	−	18388.5i	0.978796	−	1.69533i
$491$	7542.69	+	13064.3i	0.693273	+	1.20078i	0.970759	+	0.240054i	$0.0771652\pi$
$491$	7542.69	+	13064.3i	0.693273	+	1.20078i	−0.277487	+	0.960729i	$0.589501\pi$
$492$		0			0
$493$	−27923.1			−2.55090
$494$	−25590.1	+	10698.8i	−2.33067	+	0.974419i
$495$		0			0
$496$	−21372.4	−	37018.0i	−1.93477	−	3.35113i
$497$	−1373.68	−	2379.28i	−0.123980	−	0.214739i
$498$		0			0
$499$	14593.9			1.30925			0.654623	−	0.755955i	$-0.272827\pi$
$499$	14593.9			1.30925			0.654623	+	0.755955i	$0.272827\pi$
$500$	−11002.3	+	19056.5i	−0.984074	+	1.70447i
$501$		0			0
$502$	23185.2			2.06136
$503$	−9102.46	+	15765.9i	−0.806876	+	1.39755i	0.108142	+	0.994135i	$0.465510\pi$
$503$	−9102.46	+	15765.9i	−0.806876	+	1.39755i	−0.915017	+	0.403414i	$0.867823\pi$
$504$		0			0
$505$	5549.30	+	9611.67i	0.488991	+	0.846958i
$506$	8966.25			0.787744
$507$		0			0
$508$	−32601.9			−2.84739
$509$	−3945.96	−	6834.61i	−0.343618	−	0.595164i	0.641483	−	0.767137i	$-0.278319\pi$
$509$	−3945.96	−	6834.61i	−0.343618	−	0.595164i	−0.985102	+	0.171973i	$0.944986\pi$
$510$		0			0
$511$	348.610	−	603.811i	0.0301793	−	0.0522721i
$512$	−3555.88			−0.306932
$513$		0			0
$514$	3733.79	−	6467.12i	0.320410	−	0.554966i
$515$	17078.6			1.46131
$516$		0			0
$517$	−454.203	−	786.703i	−0.0386380	−	0.0669229i
$518$	−4377.02	−	7581.23i	−0.371265	−	0.643050i
$519$		0			0
$520$	39914.9	−	16687.8i	3.36613	−	1.40733i
$521$	−647.902			−0.0544820			−0.0272410	−	0.999629i	$-0.508672\pi$
$521$	−647.902			−0.0544820			−0.0272410	+	0.999629i	$0.508672\pi$
$522$		0			0
$523$	−4634.99	−	8028.04i	−0.387522	−	0.671208i	0.604593	−	0.796534i	$-0.293336\pi$
$523$	−4634.99	−	8028.04i	−0.387522	−	0.671208i	−0.992116	+	0.125326i	$0.960002\pi$
$524$	8201.20	−	14204.9i	0.683724	−	1.18424i
$525$		0			0
$526$	6687.33	−	11582.8i	0.554337	−	0.960141i
$527$	−12433.0	+	21534.6i	−1.02768	+	1.78000i
$528$		0			0
$529$	4073.12	−	7054.86i	0.334768	−	0.579835i
$530$	−2148.52	−	3721.35i	−0.176087	−	0.304991i
$531$		0			0
$532$	−14878.1			−1.21250
$533$	8478.38	+	6461.35i	0.689005	+	0.525089i
$534$		0			0
$535$	6275.24	+	10869.0i	0.507107	+	0.878336i
$536$	−32961.7	−	57091.4i	−2.65621	−	4.60069i
$537$		0			0
$538$	−13234.2			−1.06053
$539$	−3953.38	+	6847.45i	−0.315926	+	0.547200i
$540$		0			0
$541$	3844.83			0.305549			0.152775	−	0.988261i	$-0.451179\pi$
$541$	3844.83			0.305549			0.152775	+	0.988261i	$0.451179\pi$
$542$	−16135.7	+	27947.8i	−1.27876	+	2.21488i
$543$		0			0
$544$	36101.6	+	62529.8i	2.84530	+	4.92820i
$545$	−11633.2			−0.914337
$546$		0			0
$547$	−20245.4			−1.58251			−0.791253	−	0.611489i	$-0.790571\pi$
$547$	−20245.4			−1.58251			−0.791253	+	0.611489i	$0.790571\pi$
$548$	25705.2	+	44522.8i	2.00378	+	3.47065i
$549$		0			0
$550$	−3188.46	+	5522.57i	−0.247193	+	0.428151i
$551$	−24720.0			−1.91126
$552$		0			0
$553$	1235.48	−	2139.92i	0.0950056	−	0.164555i
$554$	39810.8			3.05307
$555$		0			0
$556$	−13212.1	−	22884.0i	−1.00776	−	1.74550i
$557$	−3611.67	−	6255.60i	−0.274742	−	0.475868i	0.695328	−	0.718693i	$-0.255259\pi$
$557$	−3611.67	−	6255.60i	−0.274742	−	0.475868i	−0.970070	+	0.242825i	$0.921926\pi$
$558$		0			0
$559$	2290.62	−	17854.1i	0.173315	−	1.35089i
$560$	17843.7			1.34649
$561$		0			0
$562$	16018.2	+	27744.3i	1.20229	+	2.08242i
$563$	8645.13	−	14973.8i	0.647156	−	1.12091i	−0.336643	−	0.941632i	$-0.609292\pi$
$563$	8645.13	−	14973.8i	0.647156	−	1.12091i	0.983799	−	0.179275i	$-0.0573751\pi$
$564$		0			0
$565$	1720.51	−	2980.02i	0.128111	−	0.221894i
$566$	4010.54	−	6946.47i	0.297837	−	0.515869i
$567$		0			0
$568$	−15131.0	+	26207.7i	−1.11775	+	1.93600i
$569$	−8682.00	−	15037.7i	−0.639663	−	1.10793i	−0.985507	−	0.169637i	$-0.945740\pi$
$569$	−8682.00	−	15037.7i	−0.639663	−	1.10793i	0.345843	−	0.938292i	$-0.387593\pi$
$570$		0			0
$571$	158.149			0.0115908			0.00579540	−	0.999983i	$-0.498155\pi$
$571$	158.149			0.0115908			0.00579540	+	0.999983i	$0.498155\pi$
$572$	−23929.4	+	10004.5i	−1.74919	+	0.731311i
$573$		0			0
$574$	3942.19	+	6828.07i	0.286662	+	0.496513i
$575$	−1429.81	−	2476.50i	−0.103699	−	0.179612i
$576$		0			0
$577$	9627.04			0.694591			0.347295	−	0.937756i	$-0.387100\pi$
$577$	9627.04			0.694591			0.347295	+	0.937756i	$0.387100\pi$
$578$	28146.2	−	48750.7i	2.02548	−	3.50824i
$579$		0			0
$580$	62076.4			4.44411
$581$	276.195	−	478.383i	0.0197220	−	0.0341595i
$582$		0			0
$583$	800.059	+	1385.74i	0.0568354	+	0.0984418i
$584$	−7679.86			−0.544169
$585$		0			0
$586$	19317.6			1.36178
$587$	6660.59	+	11536.5i	0.468334	+	0.811179i	0.999345	−	0.0361864i	$-0.0115210\pi$
$587$	6660.59	+	11536.5i	0.468334	+	0.811179i	−0.531011	+	0.847365i	$0.678188\pi$
$588$		0			0
$589$	−11006.8	+	19064.3i	−0.769994	+	1.33367i
$590$	5669.06			0.395579
$591$		0			0
$592$	−26885.2	+	46566.6i	−1.86651	+	3.23290i
$593$	16723.4			1.15809			0.579044	−	0.815296i	$-0.303426\pi$
$593$	16723.4			1.15809			0.579044	+	0.815296i	$0.303426\pi$
$594$		0			0
$595$	−5190.13	−	8989.57i	−0.357604	−	0.619389i
$596$	8273.31	+	14329.8i	0.568604	+	0.984850i
$597$		0			0
$598$	2040.81	−	15906.9i	0.139557	−	1.08776i
$599$	17160.0			1.17051			0.585256	−	0.810849i	$-0.300994\pi$
$599$	17160.0			1.17051			0.585256	+	0.810849i	$0.300994\pi$
$600$		0			0
$601$	−5524.29	−	9568.34i	−0.374942	−	0.649419i	0.615376	−	0.788234i	$-0.289004\pi$
$601$	−5524.29	−	9568.34i	−0.374942	−	0.649419i	−0.990318	+	0.138814i	$0.955671\pi$
$602$	6656.86	−	11530.0i	0.450687	−	0.780612i
$603$		0			0
$604$	−1712.69	+	2966.46i	−0.115378	+	0.199840i
$605$	−4201.31	+	7276.89i	−0.282327	+	0.489004i
$606$		0			0
$607$	−10559.8	+	18290.1i	−0.706110	+	1.22302i	0.260180	+	0.965560i	$0.416218\pi$
$607$	−10559.8	+	18290.1i	−0.706110	+	1.22302i	−0.966290	+	0.257458i	$0.917115\pi$
$608$	31960.3	+	55356.9i	2.13184	+	3.69246i
$609$		0			0
$610$	−1839.46			−0.122094
$611$	−1499.06	+	626.736i	−0.0992563	+	0.0414976i
$612$		0			0
$613$	3116.91	+	5398.65i	0.205369	+	0.355709i	0.950250	−	0.311488i	$-0.100827\pi$
$613$	3116.91	+	5398.65i	0.205369	+	0.355709i	−0.744881	+	0.667197i	$0.767494\pi$
$614$	−1119.68	−	1939.35i	−0.0735940	−	0.127469i
$615$		0			0
$616$	−11915.6			−0.779371
$617$	2799.18	−	4848.32i	0.182643	−	0.316347i	−0.760137	−	0.649763i	$-0.774868\pi$
$617$	2799.18	−	4848.32i	0.182643	−	0.316347i	0.942780	+	0.333416i	$0.108201\pi$
$618$		0			0
$619$	−15874.3			−1.03076			−0.515382	−	0.856961i	$-0.672350\pi$
$619$	−15874.3			−1.03076			−0.515382	+	0.856961i	$0.672350\pi$
$620$	27640.0	−	47874.0i	1.79040	−	3.10107i
$621$		0			0
$622$	8118.25	+	14061.2i	0.523332	+	0.906437i
$623$	−3186.15			−0.204896
$624$		0			0
$625$	−19228.4			−1.23062
$626$	10146.5	+	17574.3i	0.647823	+	1.12206i
$627$		0			0
$628$	24871.6	−	43079.0i	1.58039	−	2.73732i
$629$	31280.0			1.98285
$630$		0			0
$631$	−5624.51	+	9741.93i	−0.354846	+	0.614612i	−0.987092	−	0.160156i	$-0.948800\pi$
$631$	−5624.51	+	9741.93i	−0.354846	+	0.614612i	0.632245	+	0.774768i	$0.282133\pi$
$632$	−27217.6			−1.71307
$633$		0			0
$634$	7478.81	+	12953.7i	0.468488	+	0.811445i
$635$	−10068.3	−	17438.9i	−0.629213	−	1.08983i
$636$		0			0
$637$	11248.2	+	8572.20i	0.699637	+	0.533192i
$638$	−31873.6			−1.97788
$639$		0			0
$640$	−20315.1	−	35186.8i	−1.25473	−	2.17325i
$641$	−11884.2	+	20584.0i	−0.732287	+	1.26836i	0.223616	+	0.974677i	$0.428214\pi$
$641$	−11884.2	+	20584.0i	−0.732287	+	1.26836i	−0.955903	+	0.293681i	$0.905120\pi$
$642$		0			0
$643$	2184.48	−	3783.63i	0.133977	−	0.232055i	−0.791229	−	0.611520i	$-0.790558\pi$
$643$	2184.48	−	3783.63i	0.133977	−	0.232055i	0.925206	+	0.379465i	$0.123892\pi$
$644$	4301.27	−	7450.02i	0.263189	−	0.455857i
$645$		0			0
$646$	36652.0	−	63483.1i	2.23228	−	3.86642i
$647$	−10652.6	−	18450.8i	−0.647290	−	1.12114i	−0.983767	−	0.179448i	$-0.942569\pi$
$647$	−10652.6	−	18450.8i	−0.647290	−	1.12114i	0.336477	−	0.941692i	$-0.390765\pi$
$648$		0			0
$649$	−2111.02			−0.127681
$650$	9071.81	+	6913.61i	0.547424	+	0.417191i
$651$		0			0
$652$	−16621.0	−	28788.5i	−0.998358	−	1.72921i
$653$	−13979.2	−	24212.6i	−0.837744	−	1.45101i	−0.891777	−	0.452475i	$-0.850541\pi$
$653$	−13979.2	−	24212.6i	−0.837744	−	1.45101i	0.0540335	−	0.998539i	$-0.482792\pi$
$654$		0			0
$655$	10131.0			0.604353
$656$	24214.3	−	41940.4i	1.44117	−	2.49619i
$657$		0			0
$658$	−1201.76			−0.0711999
$659$	4809.73	−	8330.70i	0.284310	−	0.492440i	−0.688131	−	0.725586i	$-0.741569\pi$
$659$	4809.73	−	8330.70i	0.284310	−	0.492440i	0.972442	+	0.233146i	$0.0749021\pi$
$660$		0			0
$661$	−2207.20	−	3822.99i	−0.129879	−	0.224958i	0.793750	−	0.608244i	$-0.208126\pi$
$661$	−2207.20	−	3822.99i	−0.129879	−	0.224958i	−0.923630	+	0.383286i	$0.874792\pi$
$662$	−35387.1			−2.07758
$663$		0			0
$664$	−6084.55			−0.355612
$665$	−4594.76	−	7958.36i	−0.267936	−	0.464078i
$666$		0			0
$667$	7146.56	−	12378.2i	0.414866	−	0.718570i
$668$	−75479.0			−4.37181
$669$		0			0
$670$	32777.0	−	56771.5i	1.88998	−	3.27354i
$671$	684.970			0.0394083
$672$		0			0
$673$	9000.88	+	15590.0i	0.515540	+	0.892942i	0.999837	+	0.0180383i	$0.00574207\pi$
$673$	9000.88	+	15590.0i	0.515540	+	0.892942i	−0.484297	+	0.874904i	$0.660925\pi$
$674$	25655.2	+	44436.1i	1.46617	+	2.53949i
$675$		0			0
$676$	12302.4	+	44730.1i	0.699952	+	2.54495i
$677$	11192.0			0.635369			0.317684	−	0.948197i	$-0.397095\pi$
$677$	11192.0			0.635369			0.317684	+	0.948197i	$0.397095\pi$
$678$		0			0
$679$	613.123	+	1061.96i	0.0346532	+	0.0600211i
$680$	−57169.1	+	99019.8i	−3.22402	+	5.58417i
$681$		0			0
$682$	−14192.0	+	24581.2i	−0.796831	+	1.38015i
$683$	17561.2	−	30416.8i	0.983835	−	1.70405i	0.336833	−	0.941564i	$-0.390644\pi$
$683$	17561.2	−	30416.8i	0.983835	−	1.70405i	0.647002	−	0.762488i	$-0.276022\pi$
$684$		0			0
$685$	−15876.9	+	27499.7i	−0.885586	+	1.53388i
$686$	11175.7	+	19356.8i	0.621996	+	1.07733i
$687$		0			0
$688$	−81777.5			−4.53160
$689$	2640.53	−	1103.97i	0.146003	−	0.0610418i
$690$		0			0
$691$	−14767.3	−	25577.7i	−0.812987	−	1.40813i	−0.910764	−	0.412926i	$-0.864507\pi$
$691$	−14767.3	−	25577.7i	−0.812987	−	1.40813i	0.0977773	−	0.995208i	$-0.468827\pi$
$692$	15.2147	+	26.3526i	0.000835802	+	0.00144765i
$693$		0			0
$694$	−52927.1			−2.89494
$695$	8160.48	−	14134.4i	0.445388	−	0.771434i
$696$		0			0
$697$	−28172.5			−1.53100
$698$	2144.74	−	3714.79i	0.116303	−	0.201443i
$699$		0			0
$700$	3059.12	+	5298.55i	0.165177	+	0.286095i
$701$	−8804.19			−0.474365			−0.237182	−	0.971465i	$-0.576224\pi$
$701$	−8804.19			−0.474365			−0.237182	+	0.971465i	$0.576224\pi$
$702$		0			0
$703$	27691.8			1.48566
$704$	18887.7	+	32714.5i	1.01116	+	1.75138i
$705$		0			0
$706$	19197.5	−	33251.0i	1.02338	−	1.77255i
$707$	−5467.49			−0.290843
$708$		0			0
$709$	−13569.1	+	23502.4i	−0.718758	+	1.24492i	0.242735	+	0.970093i	$0.421956\pi$
$709$	−13569.1	+	23502.4i	−0.718758	+	1.24492i	−0.961492	+	0.274832i	$0.911378\pi$
$710$	−30092.5			−1.59064
$711$		0			0
$712$	17547.7	+	30393.4i	0.923633	+	1.59978i
$713$	−6364.13	−	11023.0i	−0.334276	−	0.578983i
$714$		0			0
$715$	−12741.5	−	9710.27i	−0.666441	−	0.507893i
$716$	43968.6			2.29495
$717$		0			0
$718$	18635.2	+	32277.1i	0.968607	+	1.67768i
$719$	8668.72	−	15014.7i	0.449637	−	0.778794i	−0.548725	−	0.836003i	$-0.684887\pi$
$719$	8668.72	−	15014.7i	0.449637	−	0.778794i	0.998362	+	0.0572088i	$0.0182201\pi$
$720$		0			0
$721$	−4206.72	+	7286.25i	−0.217290	+	0.376358i
$722$	13942.4	−	24148.9i	0.718673	−	1.24478i
$723$		0			0
$724$	4904.10	−	8494.15i	0.251740	−	0.436026i
$725$	5082.73	+	8803.54i	0.260369	+	0.450973i
$726$		0			0
$727$	25771.3			1.31473			0.657363	−	0.753574i	$-0.271672\pi$
$727$	25771.3			1.31473			0.657363	+	0.753574i	$0.271672\pi$
$728$	−2712.11	+	21139.3i	−0.138074	+	1.07620i
$729$		0			0
$730$	−3818.42	−	6613.69i	−0.193597	−	0.335320i
$731$	23786.3	+	41199.1i	1.20351	+	2.08455i
$732$		0			0
$733$	21784.2			1.09771			0.548853	−	0.835919i	$-0.315065\pi$
$733$	21784.2			1.09771			0.548853	+	0.835919i	$0.315065\pi$
$734$	−1239.80	+	2147.39i	−0.0623456	+	0.107986i
$735$		0			0
$736$	−36959.0			−1.85099
$737$	−12205.4	+	21140.4i	−0.610029	+	1.05660i
$738$		0			0
$739$	−11176.9	−	19359.0i	−0.556360	−	0.963643i	−0.997796	−	0.0663507i	$-0.978864\pi$
$739$	−11176.9	−	19359.0i	−0.556360	−	0.963643i	0.441437	−	0.897292i	$-0.354469\pi$
$740$	−69539.2			−3.45448
$741$		0			0
$742$	2116.85			0.104733
$743$	−9191.31	−	15919.8i	−0.453831	−	0.786059i	0.544789	−	0.838573i	$-0.316610\pi$
$743$	−9191.31	−	15919.8i	−0.453831	−	0.786059i	−0.998620	+	0.0525145i	$0.983276\pi$
$744$		0			0
$745$	−5110.04	+	8850.85i	−0.251298	+	0.435262i
$746$	−28921.5			−1.41943
$747$		0			0
$748$	34273.5	−	59363.4i	1.67535	−	2.90179i
$749$	−6182.74			−0.301618
$750$		0			0
$751$	−6891.39	−	11936.2i	−0.334847	−	0.579972i	0.648608	−	0.761122i	$-0.275351\pi$
$751$	−6891.39	−	11936.2i	−0.334847	−	0.579972i	−0.983455	+	0.181150i	$0.942018\pi$
$752$	3690.82	+	6392.69i	0.178977	+	0.309997i
$753$		0			0
$754$	−7254.75	+	56546.6i	−0.350401	+	2.73117i
$755$	−2115.69			−0.101984
$756$		0			0
$757$	−12819.7	−	22204.4i	−0.615510	−	1.06609i	−0.990295	−	0.138983i	$-0.955617\pi$
$757$	−12819.7	−	22204.4i	−0.615510	−	1.06609i	0.374785	−	0.927112i	$-0.377717\pi$
$758$	23186.3	−	40159.9i	1.11104	−	1.92437i
$759$		0			0
$760$	−50611.1	+	87661.0i	−2.41560	+	4.18395i
$761$	21.7379	−	37.6511i	0.00103548	−	0.00179350i	−0.865507	−	0.500896i	$-0.833004\pi$
$761$	21.7379	−	37.6511i	0.00103548	−	0.00179350i	0.866543	+	0.499103i	$0.166337\pi$
$762$		0			0
$763$	2865.44	−	4963.08i	0.135958	−	0.235486i
$764$	−51377.3	−	88988.1i	−2.43294	−	4.21397i
$765$		0			0
$766$	50378.7			2.37631
$767$	−480.491	+	3745.15i	−0.0226200	+	0.176310i
$768$		0			0
$769$	10242.8	+	17741.1i	0.480321	+	0.831940i	0.999745	−	0.0225767i	$-0.00718701\pi$
$769$	10242.8	+	17741.1i	0.480321	+	0.831940i	−0.519425	+	0.854516i	$0.673854\pi$
$770$	−5924.41	−	10261.4i	−0.277274	−	0.480253i
$771$		0			0
$772$	−18082.1			−0.842992
$773$	14725.1	−	25504.7i	0.685157	−	1.18673i	−0.288230	−	0.957561i	$-0.593067\pi$
$773$	14725.1	−	25504.7i	0.685157	−	1.18673i	0.973387	−	0.229166i	$-0.0735998\pi$
$774$		0			0
$775$	9052.51			0.419582
$776$	6753.53	−	11697.5i	0.312420	−	0.541127i
$777$		0			0
$778$	−34512.5	−	59777.4i	−1.59040	−	2.75466i
$779$	−24940.8			−1.14711
$780$		0			0
$781$	11205.7			0.513409
$782$	21192.2	+	36706.0i	0.969095	+	1.67852i
$783$		0			0
$784$	32124.9	−	55641.9i	1.46341	−	2.53471i
$785$	30724.1			1.39693
$786$		0			0
$787$	−4875.55	+	8444.70i	−0.220832	+	0.382492i	−0.955061	−	0.296410i	$-0.904211\pi$
$787$	−4875.55	+	8444.70i	−0.220832	+	0.382492i	0.734229	+	0.678902i	$0.237544\pi$
$788$	33192.3			1.50054
$789$		0			0
$790$	−13532.6	−	23439.1i	−0.609452	−	1.05560i
$791$	847.575	+	1468.04i	0.0380990	+	0.0659894i
$792$		0			0
$793$	155.906	−	1215.20i	0.00698158	−	0.0544174i
$794$	72221.2			3.22800
$795$		0			0
$796$	−33010.1	−	57175.2i	−1.46987	−	2.54588i
$797$	−18875.4	+	32693.1i	−0.838896	+	1.45301i	0.0519216	+	0.998651i	$0.483465\pi$
$797$	−18875.4	+	32693.1i	−0.838896	+	1.45301i	−0.890818	+	0.454360i	$0.849868\pi$
$798$		0			0
$799$	2147.07	−	3718.83i	0.0950661	−	0.164659i
$800$	13142.9	−	22764.1i	0.580837	−	1.00604i
$801$		0			0
$802$	17152.0	−	29708.1i	0.755183	−	1.30801i
$803$	1421.89	+	2462.78i	0.0624873	+	0.108231i
$804$		0			0
$805$	5313.39			0.232637
$806$	40379.1	+	30772.8i	1.76463	+	1.34482i
$807$		0			0
$808$	30112.1	+	52155.7i	1.31107	+	2.27083i
$809$	−3499.72	−	6061.70i	−0.152094	−	0.263434i	0.779903	−	0.625900i	$-0.215268\pi$
$809$	−3499.72	−	6061.70i	−0.152094	−	0.263434i	−0.931997	+	0.362466i	$0.881935\pi$
$810$		0			0
$811$	−35792.1			−1.54973			−0.774864	−	0.632128i	$-0.782182\pi$
$811$	−35792.1			−1.54973			−0.774864	+	0.632128i	$0.782182\pi$
$812$	−15290.3	+	26483.6i	−0.660819	+	1.14457i
$813$		0			0
$814$	35705.4			1.53744
$815$	10266.0	−	17781.3i	0.441231	−	0.764235i
$816$		0			0
$817$	21057.7	+	36473.1i	0.901735	+	1.56185i
$818$	28607.0			1.22276
$819$		0			0
$820$	62630.9			2.66727
$821$	4888.31	+	8466.80i	0.207799	+	0.359919i	0.951021	−	0.309126i	$-0.100037\pi$
$821$	4888.31	+	8466.80i	0.207799	+	0.359919i	−0.743222	+	0.669045i	$0.766703\pi$
$822$		0			0
$823$	−9831.02	+	17027.8i	−0.416389	+	0.721206i	−0.995573	−	0.0939904i	$-0.970038\pi$
$823$	−9831.02	+	17027.8i	−0.416389	+	0.721206i	0.579185	+	0.815196i	$0.303371\pi$
$824$	92673.7			3.91801
$825$		0			0
$826$	−1396.37	+	2418.59i	−0.0588209	+	0.101881i
$827$	−5281.22			−0.222063			−0.111032	−	0.993817i	$-0.535415\pi$
$827$	−5281.22			−0.222063			−0.111032	+	0.993817i	$0.535415\pi$
$828$		0			0
$829$	−10867.7	−	18823.5i	−0.455310	−	0.788621i	0.543396	−	0.839477i	$-0.317138\pi$
$829$	−10867.7	−	18823.5i	−0.455310	−	0.788621i	−0.998706	+	0.0508562i	$0.983805\pi$
$830$	−3025.23	−	5239.85i	−0.126515	−	0.219130i
$831$		0			0
$832$	62337.6	−	26062.4i	2.59756	−	1.08600i
$833$	−37376.1			−1.55463
$834$		0			0
$835$	−23309.9	−	40374.0i	−0.966077	−	1.67329i
$836$	30341.9	−	52553.7i	1.25526	−	2.17417i
$837$		0			0
$838$	−44235.1	+	76617.4i	−1.82348	+	3.15836i
$839$	−14287.4	+	24746.5i	−0.587908	+	1.01829i	0.406598	+	0.913607i	$0.366715\pi$
$839$	−14287.4	+	24746.5i	−0.587908	+	1.01829i	−0.994506	+	0.104679i	$0.966618\pi$
$840$		0			0
$841$	−13210.4	+	22881.0i	−0.541652	+	0.938169i
$842$	22891.2	+	39648.7i	0.936915	+	1.62278i
$843$		0			0
$844$	2605.49			0.106262
$845$	−20127.0	+	20394.4i	−0.819396	+	0.830284i
$846$		0			0
$847$	−2069.69	−	3584.81i	−0.0839614	−	0.145425i
$848$	−6501.22	−	11260.4i	−0.263270	−	0.455997i
$849$		0			0
$850$	−30144.4			−1.21640
$851$	−8005.72	+	13866.3i	−0.322482	+	0.558556i
$852$		0			0
$853$	5251.39			0.210791			0.105395	−	0.994430i	$-0.466389\pi$
$853$	5251.39			0.210791			0.105395	+	0.994430i	$0.466389\pi$
$854$	453.085	−	784.766i	0.0181549	−	0.0314451i
$855$		0			0
$856$	34051.3	+	58978.6i	1.35964	+	2.35496i
$857$	34851.0			1.38913			0.694566	−	0.719429i	$-0.255596\pi$
$857$	34851.0			1.38913			0.694566	+	0.719429i	$0.255596\pi$
$858$		0			0
$859$	41697.5			1.65623			0.828114	−	0.560559i	$-0.189414\pi$
$859$	41697.5			1.65623			0.828114	+	0.560559i	$0.189414\pi$
$860$	−52879.9	−	91590.6i	−2.09673	−	3.63164i
$861$		0			0
$862$	−4302.52	+	7452.19i	−0.170005	+	0.294458i
$863$	−28648.9			−1.13004			−0.565018	−	0.825079i	$-0.691131\pi$
$863$	−28648.9			−1.13004			−0.565018	+	0.825079i	$0.691131\pi$
$864$		0			0
$865$	−9.39739	+	16.2768i	−0.000369388	+	0.000639799i
$866$	18279.0			0.717257
$867$		0			0
$868$	13616.3	+	23584.1i	0.532450	+	0.922231i
$869$	5039.20	+	8728.15i	0.196713	+	0.340716i
$870$		0			0
$871$	34726.8	+	26465.2i	1.35095	+	1.02955i
$872$	−63125.4			−2.45149
$873$		0			0
$874$	18761.2	+	32495.4i	0.726096	+	1.25764i
$875$	3347.71	−	5798.41i	0.129341	−	0.224025i
$876$		0			0
$877$	15758.4	−	27294.3i	0.606754	−	1.05093i	−0.385018	−	0.922909i	$-0.625805\pi$
$877$	15758.4	−	27294.3i	0.606754	−	1.05093i	0.991772	−	0.128020i	$-0.0408621\pi$
$878$	−19456.3	+	33699.4i	−0.747858	+	1.29533i
$879$		0			0
$880$	−36389.8	+	63029.0i	−1.39398	+	2.41444i
$881$	15623.1	+	27059.9i	0.597451	+	1.03482i	0.993196	+	0.116455i	$0.0371531\pi$
$881$	15623.1	+	27059.9i	0.597451	+	1.03482i	−0.395745	+	0.918360i	$0.629514\pi$
$882$		0			0
$883$	−16181.8			−0.616718			−0.308359	−	0.951270i	$-0.599780\pi$
$883$	−16181.8			−0.616718			−0.308359	+	0.951270i	$0.599780\pi$
$884$	−97515.0	−	74316.0i	−3.71016	−	2.82751i
$885$		0			0
$886$	20236.4	+	35050.4i	0.767330	+	1.32905i
$887$	−6644.52	−	11508.6i	−0.251523	−	0.435651i	0.712422	−	0.701751i	$-0.247598\pi$
$887$	−6644.52	−	11508.6i	−0.251523	−	0.435651i	−0.963945	+	0.266100i	$0.914265\pi$
$888$		0			0
$889$	9919.92			0.374245
$890$	−17449.4	+	30223.2i	−0.657195	+	1.13830i
$891$		0			0
$892$	−52867.0			−1.98444
$893$	1900.78	−	3292.24i	0.0712285	−	0.123371i
$894$		0			0
$895$	13578.7	+	23519.0i	0.507134	+	0.878382i
$896$	20015.6			0.746288
$897$		0			0
$898$	−41276.0			−1.53385
$899$	22623.5	+	39185.0i	0.839304	+	1.45372i
$900$		0			0
$901$	−3781.97	+	6550.56i	−0.139840	+	0.242210i
$902$	−32158.2			−1.18709
$903$		0			0
$904$	9336.01	−	16170.4i	0.343486	−	0.594935i
$905$	6058.07			0.222516
$906$		0			0
$907$	−23564.2	−	40814.4i	−0.862664	−	1.49418i	−0.869348	−	0.494200i	$-0.835461\pi$
$907$	−23564.2	−	40814.4i	−0.862664	−	1.49418i	0.00668479	−	0.999978i	$-0.497872\pi$
$908$	6115.82	+	10592.9i	0.223525	+	0.387157i
$909$		0			0
$910$	−19553.1	+	8174.85i	−0.712284	+	0.297795i
$911$	2884.69			0.104911			0.0524556	−	0.998623i	$-0.483295\pi$
$911$	2884.69			0.104911			0.0524556	+	0.998623i	$0.483295\pi$
$912$		0			0
$913$	1126.52	+	1951.20i	0.0408351	+	0.0707285i
$914$	−24286.3	+	42065.2i	−0.878907	+	1.52231i
$915$		0			0
$916$	8118.14	−	14061.0i	0.292828	−	0.507194i
$917$	−2495.41	+	4322.18i	−0.0898646	+	0.155650i
$918$		0			0
$919$	12876.1	−	22302.0i	0.462179	−	0.800518i	−0.536890	−	0.843652i	$-0.680401\pi$
$919$	12876.1	−	22302.0i	0.462179	−	0.800518i	0.999069	+	0.0431343i	$0.0137343\pi$
$920$	−29263.4	−	50685.7i	−1.04868	−	1.81637i
$921$		0			0
$922$	−7272.75			−0.259778
$923$	2550.54	−	19880.0i	0.0909556	−	0.708946i
$924$		0			0
$925$	−5693.77	−	9861.90i	−0.202389	−	0.350548i
$926$	−256.044	−	443.482i	−0.00908655	−	0.0157384i
$927$		0			0
$928$	131383.			4.64748
$929$	−25697.4	+	44509.2i	−0.907539	+	1.57190i	−0.0900658	+	0.995936i	$0.528708\pi$
$929$	−25697.4	+	44509.2i	−0.907539	+	1.57190i	−0.817473	+	0.575967i	$0.804626\pi$
$930$		0			0
$931$	−33088.6			−1.16481
$932$	8928.69	−	15465.0i	0.313808	−	0.543532i
$933$		0			0
$934$	−35858.1	−	62108.0i	−1.25622	−	2.17584i
$935$	42338.3			1.48087
$936$		0			0
$937$	6781.86			0.236450			0.118225	−	0.992987i	$-0.462280\pi$
$937$	6781.86			0.236450			0.118225	+	0.992987i	$0.462280\pi$
$938$	16146.9	+	27967.3i	0.562064	+	0.973523i
$939$		0			0
$940$	−4773.19	+	8267.42i	−0.165622	+	0.286865i
$941$	−22038.2			−0.763470			−0.381735	−	0.924272i	$-0.624673\pi$
$941$	−22038.2			−0.763470			−0.381735	+	0.924272i	$0.624673\pi$
$942$		0			0
$943$	7210.39	−	12488.8i	0.248995	−	0.431273i
$944$	17154.0			0.591436
$945$		0			0
$946$	27151.5	+	47027.8i	0.933163	+	1.61629i
$947$	22765.5	+	39431.0i	0.781182	+	1.35305i	0.931254	+	0.364371i	$0.118716\pi$
$947$	22765.5	+	39431.0i	0.781182	+	1.35305i	−0.150072	+	0.988675i	$0.547951\pi$
$948$		0			0
$949$	4692.83	−	1962.00i	0.160522	−	0.0671120i
$950$	−26686.5			−0.911393
$951$		0			0
$952$	−28163.2	−	48780.0i	−0.958795	−	1.66068i
$953$	13270.6	−	22985.3i	0.451077	−	0.781288i	−0.547376	−	0.836887i	$-0.684373\pi$
$953$	13270.6	−	22985.3i	0.451077	−	0.781288i	0.998453	+	0.0555985i	$0.0177067\pi$
$954$		0			0
$955$	31733.4	−	54963.8i	1.07525	−	1.86240i
$956$	−69183.0	+	119828.i	−2.34052	+	4.05390i
$957$		0			0
$958$	10256.5	−	17764.7i	0.345900	−	0.599116i
$959$	−7821.44	−	13547.1i	−0.263365	−	0.456162i
$960$		0			0
$961$	10502.1			0.352527
$962$	8126.92	−	63344.6i	0.272372	−	2.12299i
$963$		0			0
$964$	−54655.2	−	94665.6i	−1.82606	−	3.16284i
$965$	−5584.25	−	9672.20i	−0.186283	−	0.322652i
$966$		0			0
$967$	48269.6			1.60522			0.802609	−	0.596506i	$-0.203445\pi$
$967$	48269.6			1.60522			0.802609	+	0.596506i	$0.203445\pi$
$968$	−22797.5	+	39486.5i	−0.756964	+	1.31110i
$969$		0			0
$970$	13431.4			0.444594
$971$	−17080.7	+	29584.6i	−0.564515	+	0.977769i	0.432579	+	0.901596i	$0.357604\pi$
$971$	−17080.7	+	29584.6i	−0.564515	+	0.977769i	−0.997095	+	0.0761732i	$0.975730\pi$
$972$		0			0
$973$	4020.09	+	6963.00i	0.132454	+	0.229418i
$974$	47199.7			1.55275
$975$		0			0
$976$	−5566.02			−0.182545
$977$	−12062.1	−	20892.2i	−0.394985	−	0.684135i	0.598114	−	0.801411i	$-0.295917\pi$
$977$	−12062.1	−	20892.2i	−0.394985	−	0.684135i	−0.993099	+	0.117276i	$0.962584\pi$
$978$		0			0
$979$	6497.72	−	11254.4i	0.212123	−	0.367407i
$980$	83091.7			2.70843
$981$		0			0
$982$	−40699.5	+	70493.7i	−1.32258	+	2.29078i
$983$	−19023.4			−0.617246			−0.308623	−	0.951185i	$-0.599868\pi$
$983$	−19023.4			−0.617246			−0.308623	+	0.951185i	$0.599868\pi$
$984$		0			0
$985$	10250.7	+	17754.7i	0.331587	+	0.574326i
$986$	−75334.9	−	130484.i	−2.43322	−	4.21446i
$987$		0			0
$988$	−86328.9	−	65791.0i	−2.77985	−	2.11851i
$989$	−24351.2			−0.782936
$990$		0			0
$991$	114.981	+	199.153i	0.00368566	+	0.00638374i	0.867862	−	0.496805i	$-0.165493\pi$
$991$	114.981	+	199.153i	0.00368566	+	0.00638374i	−0.864177	+	0.503188i	$0.832160\pi$
$992$	58499.4	−	101324.i	1.87234	−	3.24298i
$993$		0			0
$994$	7412.22	−	12838.3i	0.236520	−	0.409665i
$995$	20388.8	−	35314.5i	0.649618	−	1.12517i
$996$		0			0
$997$	−13495.6	+	23375.0i	−0.428694	+	0.742521i	−0.996757	−	0.0804645i	$-0.974360\pi$
$997$	−13495.6	+	23375.0i	−0.428694	+	0.742521i	0.568063	+	0.822985i	$0.307693\pi$
$998$	39373.6	+	68197.1i	1.24885	+	2.16307i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	117.4.g.f.55.8	yes	16
3.2	odd	2	inner	117.4.g.f.55.1	✓	16
13.3	even	3		1521.4.a.bc.1.1		8
13.9	even	3	inner	117.4.g.f.100.8	yes	16
13.10	even	6		1521.4.a.bd.1.8		8
39.23	odd	6		1521.4.a.bd.1.1		8
39.29	odd	6		1521.4.a.bc.1.8		8
39.35	odd	6	inner	117.4.g.f.100.1	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
117.4.g.f.55.1	✓	16	3.2	odd	2	inner
117.4.g.f.55.8	yes	16	1.1	even	1	trivial
117.4.g.f.100.1	yes	16	39.35	odd	6	inner
117.4.g.f.100.8	yes	16	13.9	even	3	inner
1521.4.a.bc.1.1		8	13.3	even	3
1521.4.a.bc.1.8		8	39.29	odd	6
1521.4.a.bd.1.1		8	39.23	odd	6
1521.4.a.bd.1.8		8	13.10	even	6

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 117 = 3^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	117.g (of order \(3\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(2.69794 + 4.67298i) q^{2} +(-10.5578 + 18.2867i) q^{4} -13.0421 q^{5} +(3.21247 - 5.56416i) q^{7} -70.7705 q^{8} +O(q^{10})\)	\(q+(2.69794 + 4.67298i) q^{2} +(-10.5578 + 18.2867i) q^{4} -13.0421 q^{5} +(3.21247 - 5.56416i) q^{7} -70.7705 q^{8} +(-35.1870 - 60.9457i) q^{10} +(13.1028 + 22.6947i) q^{11} +(43.2448 - 18.0800i) q^{13} +34.6683 q^{14} +(-106.472 - 184.416i) q^{16} +(-61.9384 + 107.280i) q^{17} +(-54.8333 + 94.9741i) q^{19} +(137.697 - 238.497i) q^{20} +(-70.7012 + 122.458i) q^{22} +(-31.7047 - 54.9141i) q^{23} +45.0976 q^{25} +(201.160 + 153.303i) q^{26} +(67.8333 + 117.491i) q^{28} +(112.705 + 195.211i) q^{29} +200.732 q^{31} +(291.431 - 504.774i) q^{32} -668.426 q^{34} +(-41.8975 + 72.5686i) q^{35} +(-126.254 - 218.679i) q^{37} -591.749 q^{38} +922.999 q^{40} +(113.712 + 196.955i) q^{41} +(192.016 - 332.581i) q^{43} -553.348 q^{44} +(171.075 - 296.311i) q^{46} -34.6646 q^{47} +(150.860 + 261.297i) q^{49} +(121.671 + 210.740i) q^{50} +(-125.948 + 981.689i) q^{52} +61.0601 q^{53} +(-170.889 - 295.988i) q^{55} +(-227.348 + 393.778i) q^{56} +(-608.144 + 1053.34i) q^{58} +(-40.2781 + 69.7637i) q^{59} +(13.0692 - 22.6364i) q^{61} +(541.563 + 938.014i) q^{62} +1441.50 q^{64} +(-564.005 + 235.802i) q^{65} +(465.755 + 806.711i) q^{67} +(-1307.87 - 2265.29i) q^{68} -452.149 q^{70} +(213.804 - 370.320i) q^{71} +108.518 q^{73} +(681.255 - 1179.97i) q^{74} +(-1157.84 - 2005.44i) q^{76} +168.369 q^{77} +384.590 q^{79} +(1388.63 + 2405.17i) q^{80} +(-613.576 + 1062.74i) q^{82} +85.9758 q^{83} +(807.810 - 1399.17i) q^{85} +2072.19 q^{86} +(-927.291 - 1606.12i) q^{88} +(-247.952 - 429.465i) q^{89} +(38.3226 - 298.703i) q^{91} +1338.93 q^{92} +(-93.5232 - 161.987i) q^{94} +(715.144 - 1238.67i) q^{95} +(-95.4286 + 165.287i) q^{97} +(-814.024 + 1409.93i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 40 q^{4} + 22 q^{7}+O(q^{10}) \) 16 * q - 40 * q^4 + 22 * q^7	\( 16 q - 40 q^{4} + 22 q^{7} - 36 q^{10} + 36 q^{13} - 204 q^{16} - 244 q^{19} - 136 q^{22} + 708 q^{25} + 452 q^{28} + 484 q^{31} - 2584 q^{34} - 1018 q^{37} + 3400 q^{40} - 74 q^{43} + 896 q^{46} - 298 q^{49} - 1676 q^{52} - 1300 q^{55} - 812 q^{58} - 1148 q^{61} + 7272 q^{64} + 2198 q^{67} + 4400 q^{70} - 4352 q^{73} - 6936 q^{76} + 3724 q^{79} - 5436 q^{82} + 890 q^{85} - 3528 q^{88} - 4754 q^{91} + 3104 q^{94} + 4370 q^{97}+O(q^{100}) \) 16 * q - 40 * q^4 + 22 * q^7 - 36 * q^10 + 36 * q^13 - 204 * q^16 - 244 * q^19 - 136 * q^22 + 708 * q^25 + 452 * q^28 + 484 * q^31 - 2584 * q^34 - 1018 * q^37 + 3400 * q^40 - 74 * q^43 + 896 * q^46 - 298 * q^49 - 1676 * q^52 - 1300 * q^55 - 812 * q^58 - 1148 * q^61 + 7272 * q^64 + 2198 * q^67 + 4400 * q^70 - 4352 * q^73 - 6936 * q^76 + 3724 * q^79 - 5436 * q^82 + 890 * q^85 - 3528 * q^88 - 4754 * q^91 + 3104 * q^94 + 4370 * q^97

Introduction

L-functions

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