LMFDB - Embedded newform 117.4.b.d.64.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [117,4,Mod(64,117)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(117, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1]))

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

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

chi := DirichletCharacter("117.64");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 117 = 3^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	117.b (of order $2$, degree $1$, 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:	$4$
Coefficient field:	4.0.5054412.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 29x^{2} + 48 $ x^4 + 29*x^2 + 48
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 39)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			64.1
Root			$-5.21898i$ of defining polynomial
Character	$\chi$	$=$	117.64
Dual form			117.4.b.d.64.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-5.21898i q^{2} -19.2377 q^{4} +5.83936i q^{5} +31.3139i q^{7} +58.6495i q^{8} +O(q^{10})$	\(q-5.21898i q^{2} -19.2377 q^{4} +5.83936i q^{5} +31.3139i q^{7} +58.6495i q^{8} +30.4755 q^{10} -16.2773i q^{11} +(-43.4755 + 17.5181i) q^{13} +163.426 q^{14} +152.189 q^{16} -54.0000 q^{17} +66.3500i q^{19} -112.336i q^{20} -84.9510 q^{22} -182.853 q^{23} +90.9019 q^{25} +(91.4264 + 226.898i) q^{26} -602.408i q^{28} +164.853 q^{29} -58.9055i q^{31} -325.073i q^{32} +281.825i q^{34} -182.853 q^{35} +110.366i q^{37} +346.279 q^{38} -342.475 q^{40} -55.0357i q^{41} +113.147 q^{43} +313.139i q^{44} +954.305i q^{46} +514.089i q^{47} -637.559 q^{49} -474.415i q^{50} +(836.370 - 337.008i) q^{52} -242.559 q^{53} +95.0490 q^{55} -1836.54 q^{56} -860.364i q^{58} -265.036i q^{59} -468.098 q^{61} -307.426 q^{62} -479.042 q^{64} +(-102.294 - 253.869i) q^{65} +852.919i q^{67} +1038.84 q^{68} +954.305i q^{70} -165.619i q^{71} -315.325i q^{73} +576.000 q^{74} -1276.42i q^{76} +509.706 q^{77} +479.608 q^{79} +888.684i q^{80} -287.230 q^{82} -574.235i q^{83} -315.325i q^{85} -590.512i q^{86} +954.657 q^{88} +66.7144i q^{89} +(-548.559 - 1361.39i) q^{91} +3517.68 q^{92} +2683.02 q^{94} -387.441 q^{95} -1438.25i q^{97} +3327.40i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 26 q^{4}+O(q^{10}) $ 4 * q - 26 * q^4	$ 4 q - 26 q^{4} + 20 q^{10} - 72 q^{13} + 348 q^{14} + 354 q^{16} - 216 q^{17} - 136 q^{22} - 120 q^{23} - 44 q^{25} + 60 q^{26} + 48 q^{29} - 120 q^{35} + 468 q^{38} - 1268 q^{40} + 1064 q^{43} - 716 q^{49} + 1766 q^{52} + 864 q^{53} + 584 q^{55} - 3372 q^{56} - 2280 q^{61} - 924 q^{62} - 1050 q^{64} - 1632 q^{65} + 1404 q^{68} + 2304 q^{74} + 816 q^{77} + 288 q^{79} - 28 q^{82} + 2392 q^{88} - 360 q^{91} + 8568 q^{92} + 6656 q^{94} - 3384 q^{95}+O(q^{100}) $ 4 * q - 26 * q^4 + 20 * q^10 - 72 * q^13 + 348 * q^14 + 354 * q^16 - 216 * q^17 - 136 * q^22 - 120 * q^23 - 44 * q^25 + 60 * q^26 + 48 * q^29 - 120 * q^35 + 468 * q^38 - 1268 * q^40 + 1064 * q^43 - 716 * q^49 + 1766 * q^52 + 864 * q^53 + 584 * q^55 - 3372 * q^56 - 2280 * q^61 - 924 * q^62 - 1050 * q^64 - 1632 * q^65 + 1404 * q^68 + 2304 * q^74 + 816 * q^77 + 288 * q^79 - 28 * q^82 + 2392 * q^88 - 360 * q^91 + 8568 * q^92 + 6656 * q^94 - 3384 * q^95

Display 100 coefficients 10 coefficients

Character values

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

$n$	$28$	$92$
$\chi(n)$	$-1$	$1$

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 $

$2$		−	5.21898i		−	1.84519i	−0.385773	−	0.922594i	$-0.626065\pi$
$2$		−	5.21898i		−	1.84519i	0.385773	−	0.922594i	$-0.373935\pi$
$3$		0			0
$3$		0			0
$4$	−19.2377			−2.40472
$5$			5.83936i			0.522288i	0.965300	+	0.261144i	$0.0840997\pi$
$5$			5.83936i			0.522288i	−0.965300	+	0.261144i	$0.915900\pi$
$6$		0			0
$7$			31.3139i			1.69079i	0.534141	+	0.845395i	$0.320635\pi$
$7$			31.3139i			1.69079i	−0.534141	+	0.845395i	$0.679365\pi$
$8$			58.6495i			2.59197i
$9$		0			0
$10$	30.4755			0.963719
$11$		−	16.2773i		−	0.446163i	−0.974800	−	0.223082i	$-0.928388\pi$
$11$		−	16.2773i		−	0.446163i	0.974800	−	0.223082i	$-0.0716116\pi$
$12$		0			0
$13$	−43.4755	+	17.5181i	−0.927533	+	0.373741i
$13$	−43.4755	+	17.5181i	−0.927533	+	0.373741i
$14$	163.426			3.11983
$15$		0			0
$16$	152.189			2.37795
$17$	−54.0000			−0.770407			−0.385204	−	0.922832i	$-0.625869\pi$
$17$	−54.0000			−0.770407			−0.385204	+	0.922832i	$0.625869\pi$
$18$		0			0
$19$			66.3500i			0.801144i	0.916265	+	0.400572i	$0.131189\pi$
$19$			66.3500i			0.801144i	−0.916265	+	0.400572i	$0.868811\pi$
$20$		−	112.336i		−	1.25595i
$21$		0			0
$22$	−84.9510			−0.823255
$23$	−182.853			−1.65772			−0.828858	−	0.559459i	$-0.811009\pi$
$23$	−182.853			−1.65772			−0.828858	+	0.559459i	$0.811009\pi$
$24$		0			0
$25$	90.9019			0.727215
$26$	91.4264	+	226.898i	0.689623	+	1.71147i
$27$		0			0
$28$		−	602.408i		−	4.06587i
$29$	164.853			1.05560			0.527800	−	0.849369i	$-0.323017\pi$
$29$	164.853			1.05560			0.527800	+	0.849369i	$0.323017\pi$
$30$		0			0
$31$		−	58.9055i		−	0.341282i	−0.985333	−	0.170641i	$-0.945416\pi$
$31$		−	58.9055i		−	0.341282i	0.985333	−	0.170641i	$-0.0545838\pi$
$32$		−	325.073i		−	1.79579i
$33$		0			0
$34$			281.825i			1.42155i
$35$	−182.853			−0.883079
$36$		0			0
$37$			110.366i			0.490382i	0.969475	+	0.245191i	$0.0788506\pi$
$37$			110.366i			0.490382i	−0.969475	+	0.245191i	$0.921149\pi$
$38$	346.279			1.47826
$39$		0			0
$40$	−342.475			−1.35375
$41$		−	55.0357i		−	0.209637i	−0.994491	−	0.104819i	$-0.966574\pi$
$41$		−	55.0357i		−	0.209637i	0.994491	−	0.104819i	$-0.0334262\pi$
$42$		0			0
$43$	113.147			0.401274			0.200637	−	0.979666i	$-0.435699\pi$
$43$	113.147			0.401274			0.200637	+	0.979666i	$0.435699\pi$
$44$			313.139i			1.07290i
$45$		0			0
$46$			954.305i			3.05880i
$47$			514.089i			1.59548i	0.603001	+	0.797740i	$0.293971\pi$
$47$			514.089i			1.59548i	−0.603001	+	0.797740i	$0.706029\pi$
$48$		0			0
$49$	−637.559			−1.85877
$50$		−	474.415i		−	1.34185i
$51$		0			0
$52$	836.370	−	337.008i	2.23045	−	0.898742i
$53$	−242.559			−0.628641			−0.314321	−	0.949317i	$-0.601777\pi$
$53$	−242.559			−0.628641			−0.314321	+	0.949317i	$0.601777\pi$
$54$		0			0
$55$	95.0490			0.233026
$56$	−1836.54			−4.38247
$57$		0			0
$58$		−	860.364i		−	1.94778i
$59$		−	265.036i		−	0.584825i	−0.956292	−	0.292413i	$-0.905542\pi$
$59$		−	265.036i		−	0.584825i	0.956292	−	0.292413i	$-0.0944581\pi$
$60$		0			0
$61$	−468.098			−0.982522			−0.491261	−	0.871013i	$-0.663464\pi$
$61$	−468.098			−0.982522			−0.491261	+	0.871013i	$0.663464\pi$
$62$	−307.426			−0.629729
$63$		0			0
$64$	−479.042			−0.935628
$65$	−102.294	−	253.869i	−0.195201	−	0.484439i
$66$		0			0
$67$			852.919i			1.55523i	0.628739	+	0.777617i	$0.283572\pi$
$67$			852.919i			1.55523i	−0.628739	+	0.777617i	$0.716428\pi$
$68$	1038.84			1.85261
$69$		0			0
$70$			954.305i			1.62945i
$71$		−	165.619i		−	0.276836i	−0.990374	−	0.138418i	$-0.955798\pi$
$71$		−	165.619i		−	0.276836i	0.990374	−	0.138418i	$-0.0442018\pi$
$72$		0			0
$73$		−	315.325i		−	0.505562i	−0.967524	−	0.252781i	$-0.918655\pi$
$73$		−	315.325i		−	0.505562i	0.967524	−	0.252781i	$-0.0813452\pi$
$74$	576.000			0.904846
$75$		0			0
$76$		−	1276.42i		−	1.92653i
$77$	509.706			0.754368
$78$		0			0
$79$	479.608			0.683039			0.341519	−	0.939875i	$-0.389058\pi$
$79$	479.608			0.683039			0.341519	+	0.939875i	$0.389058\pi$
$80$			888.684i			1.24197i
$81$		0			0
$82$	−287.230			−0.386820
$83$		−	574.235i		−	0.759404i	−0.925109	−	0.379702i	$-0.876027\pi$
$83$		−	574.235i		−	0.759404i	0.925109	−	0.379702i	$-0.123973\pi$
$84$		0			0
$85$		−	315.325i		−	0.402374i
$86$		−	590.512i		−	0.740426i
$87$		0			0
$88$	954.657			1.15644
$89$			66.7144i			0.0794575i	0.999211	+	0.0397287i	$0.0126494\pi$
$89$			66.7144i			0.0794575i	−0.999211	+	0.0397287i	$0.987351\pi$
$90$		0			0
$91$	−548.559	−	1361.39i	−0.631918	−	1.56826i
$92$	3517.68			3.98634
$93$		0			0
$94$	2683.02			2.94396
$95$	−387.441			−0.418428
$96$		0			0
$97$		−	1438.25i		−	1.50549i	−0.658313	−	0.752744i	$-0.728730\pi$
$97$		−	1438.25i		−	1.50549i	0.658313	−	0.752744i	$-0.271270\pi$
$98$			3327.40i			3.42978i
$99$		0			0
$100$	−1748.75			−1.74875
$101$	−896.264			−0.882986			−0.441493	−	0.897265i	$-0.645551\pi$
$101$	−896.264			−0.882986			−0.441493	+	0.897265i	$0.645551\pi$
$102$		0			0
$103$	22.2644			0.0212988			0.0106494	−	0.999943i	$-0.496610\pi$
$103$	22.2644			0.0212988			0.0106494	+	0.999943i	$0.496610\pi$
$104$	−1027.43	−	2549.82i	−0.968725	−	2.40413i
$105$		0			0
$106$			1265.91i			1.15996i
$107$	351.441			0.317524			0.158762	−	0.987317i	$-0.449250\pi$
$107$	351.441			0.317524			0.158762	+	0.987317i	$0.449250\pi$
$108$		0			0
$109$			967.008i			0.849748i	0.905252	+	0.424874i	$0.139682\pi$
$109$			967.008i			0.849748i	−0.905252	+	0.424874i	$0.860318\pi$
$110$		−	496.059i		−	0.429976i
$111$		0			0
$112$			4765.62i			4.02061i
$113$	−48.2943			−0.0402048			−0.0201024	−	0.999798i	$-0.506399\pi$
$113$	−48.2943			−0.0402048			−0.0201024	+	0.999798i	$0.506399\pi$
$114$		0			0
$115$		−	1067.74i		−	0.865805i
$116$	−3171.40			−2.53842
$117$		0			0
$118$	−1383.22			−1.07911
$119$		−	1690.95i		−	1.30260i
$120$		0			0
$121$	1066.05			0.800938
$122$			2442.99i			1.81294i
$123$		0			0
$124$			1133.21i			0.820686i
$125$			1260.73i			0.902104i
$126$		0			0
$127$	1763.02			1.23183			0.615916	−	0.787812i	$-0.288786\pi$
$127$	1763.02			1.23183			0.615916	+	0.787812i	$0.288786\pi$
$128$		−	100.479i		−	0.0693844i
$129$		0			0
$130$	−1324.94	+	533.872i	−0.893881	+	0.360182i
$131$	−955.970			−0.637584			−0.318792	−	0.947825i	$-0.603277\pi$
$131$	−955.970			−0.637584			−0.318792	+	0.947825i	$0.603277\pi$
$132$		0			0
$133$	−2077.68			−1.35457
$134$	4451.37			2.86970
$135$		0			0
$136$		−	3167.07i		−	1.99687i
$137$			15.9215i			0.00992894i	0.999988	+	0.00496447i	$0.00158025\pi$
$137$			15.9215i			0.00992894i	−0.999988	+	0.00496447i	$0.998420\pi$
$138$		0			0
$139$	2074.26			1.26573			0.632866	−	0.774261i	$-0.281878\pi$
$139$	2074.26			1.26573			0.632866	+	0.774261i	$0.281878\pi$
$140$	3517.68			2.12356
$141$		0			0
$142$	−864.362			−0.510815
$143$	285.147	+	707.664i	0.166750	+	0.413831i
$144$		0			0
$145$			962.635i			0.551327i
$146$	−1645.68			−0.932857
$147$		0			0
$148$		−	2123.20i		−	1.17923i
$149$			2764.08i			1.51975i	0.650071	+	0.759873i	$0.274739\pi$
$149$			2764.08i			1.51975i	−0.650071	+	0.759873i	$0.725261\pi$
$150$		0			0
$151$			1618.46i			0.872239i	0.899889	+	0.436120i	$0.143648\pi$
$151$			1618.46i			0.872239i	−0.899889	+	0.436120i	$0.856352\pi$
$152$	−3891.40			−2.07654
$153$		0			0
$154$		−	2660.14i		−	1.39195i
$155$	343.970			0.178247
$156$		0			0
$157$	−1109.97			−0.564237			−0.282119	−	0.959380i	$-0.591037\pi$
$157$	−1109.97			−0.564237			−0.282119	+	0.959380i	$0.591037\pi$
$158$		−	2503.06i		−	1.26034i
$159$		0			0
$160$	1898.22			0.937921
$161$		−	5725.83i		−	2.80285i
$162$		0			0
$163$		−	233.201i		−	0.112060i	−0.998429	−	0.0560299i	$-0.982156\pi$
$163$		−	233.201i		−	0.112060i	0.998429	−	0.0560299i	$-0.0178442\pi$
$164$			1058.76i			0.504119i
$165$		0			0
$166$	−2996.92			−1.40124
$167$			215.405i			0.0998118i	0.998754	+	0.0499059i	$0.0158921\pi$
$167$			215.405i			0.0998118i	−0.998754	+	0.0499059i	$0.984108\pi$
$168$		0			0
$169$	1583.23	−	1523.21i	0.720635	−	0.693315i
$170$	−1645.68			−0.742456
$171$		0			0
$172$	−2176.69			−0.964950
$173$	−1383.15			−0.607854			−0.303927	−	0.952695i	$-0.598298\pi$
$173$	−1383.15			−0.607854			−0.303927	+	0.952695i	$0.598298\pi$
$174$		0			0
$175$			2846.49i			1.22957i
$176$		−	2477.22i		−	1.06095i
$177$		0			0
$178$	348.181			0.146614
$179$	3642.79			1.52109			0.760545	−	0.649285i	$-0.224932\pi$
$179$	3642.79			1.52109			0.760545	+	0.649285i	$0.224932\pi$
$180$		0			0
$181$	−2621.97			−1.07674			−0.538369	−	0.842709i	$-0.680959\pi$
$181$	−2621.97			−1.07674			−0.538369	+	0.842709i	$0.680959\pi$
$182$	−7105.04	+	2862.92i	−2.89374	+	1.16601i
$183$		0			0
$184$		−	10724.2i		−	4.29674i
$185$	−644.469			−0.256121
$186$		0			0
$187$			878.975i			0.343727i
$188$		−	9889.91i		−	3.83668i
$189$		0			0
$190$			2022.05i			0.772078i
$191$	3419.32			1.29536			0.647679	−	0.761913i	$-0.275740\pi$
$191$	3419.32			1.29536			0.647679	+	0.761913i	$0.275740\pi$
$192$		0			0
$193$			1698.39i			0.633435i	0.948520	+	0.316718i	$0.102581\pi$
$193$			1698.39i			0.633435i	−0.948520	+	0.316718i	$0.897419\pi$
$194$	−7506.20			−2.77791
$195$		0			0
$196$	12265.2			4.46982
$197$			2293.72i			0.829548i	0.909925	+	0.414774i	$0.136139\pi$
$197$			2293.72i			0.829548i	−0.909925	+	0.414774i	$0.863861\pi$
$198$		0			0
$199$	−900.981			−0.320949			−0.160474	−	0.987040i	$-0.551302\pi$
$199$	−900.981			−0.320949			−0.160474	+	0.987040i	$0.551302\pi$
$200$			5331.35i			1.88492i
$201$		0			0
$202$			4677.58i			1.62928i
$203$			5162.18i			1.78480i
$204$		0			0
$205$	321.373			0.109491
$206$		−	116.197i		−	0.0393002i
$207$		0			0
$208$	−6616.48	+	2666.05i	−2.20563	+	0.888738i
$209$	1080.00			0.357441
$210$		0			0
$211$	431.019			0.140628			0.0703142	−	0.997525i	$-0.477600\pi$
$211$	431.019			0.140628			0.0703142	+	0.997525i	$0.477600\pi$
$212$	4666.28			1.51170
$213$		0			0
$214$		−	1834.17i		−	0.585892i
$215$			660.706i			0.209580i
$216$		0			0
$217$	1844.56			0.577036
$218$	5046.79			1.56794
$219$		0			0
$220$	−1828.53			−0.560361
$221$	2347.68	−	945.976i	0.714578	−	0.287933i
$222$		0			0
$223$			4104.30i			1.23249i	0.787556	+	0.616243i	$0.211346\pi$
$223$			4104.30i			1.23249i	−0.787556	+	0.616243i	$0.788654\pi$
$224$	10179.3			3.03631
$225$		0			0
$226$			252.047i			0.0741854i
$227$			1809.11i			0.528963i	0.964391	+	0.264482i	$0.0852009\pi$
$227$			1809.11i			0.528963i	−0.964391	+	0.264482i	$0.914799\pi$
$228$		0			0
$229$		−	5249.33i		−	1.51478i	−0.652961	−	0.757392i	$-0.726473\pi$
$229$		−	5249.33i		−	1.51478i	0.652961	−	0.757392i	$-0.273527\pi$
$230$	−5572.53			−1.59757
$231$		0			0
$232$			9668.54i			2.73608i
$233$	−2808.88			−0.789768			−0.394884	−	0.918731i	$-0.629215\pi$
$233$	−2808.88			−0.789768			−0.394884	+	0.918731i	$0.629215\pi$
$234$		0			0
$235$	−3001.95			−0.833300
$236$			5098.69i			1.40634i
$237$		0			0
$238$	−8825.03			−2.40354
$239$			6712.01i			1.81659i	0.418334	+	0.908293i	$0.362614\pi$
$239$			6712.01i			1.81659i	−0.418334	+	0.908293i	$0.637386\pi$
$240$		0			0
$241$			2519.11i			0.673321i	0.941626	+	0.336661i	$0.109297\pi$
$241$			2519.11i			0.673321i	−0.941626	+	0.336661i	$0.890703\pi$
$242$		−	5563.69i		−	1.47788i
$243$		0			0
$244$	9005.15			2.36269
$245$		−	3722.93i		−	0.970814i
$246$		0			0
$247$	−1162.32	−	2884.60i	−0.299421	−	0.743087i
$248$	3454.78			0.884591
$249$		0			0
$250$	6579.71			1.66455
$251$	828.000			0.208219			0.104109	−	0.994566i	$-0.466801\pi$
$251$	828.000			0.208219			0.104109	+	0.994566i	$0.466801\pi$
$252$		0			0
$253$			2976.35i			0.739612i
$254$		−	9201.16i		−	2.27296i
$255$		0			0
$256$	−4356.73			−1.06366
$257$	−5840.76			−1.41765			−0.708826	−	0.705383i	$-0.750775\pi$
$257$	−5840.76			−1.41765			−0.708826	+	0.705383i	$0.750775\pi$
$258$		0			0
$259$	−3456.00			−0.829133
$260$	1967.91	+	4883.86i	0.469402	+	1.16494i
$261$		0			0
$262$			4989.19i			1.17646i
$263$	4064.06			0.952854			0.476427	−	0.879214i	$-0.341932\pi$
$263$	4064.06			0.952854			0.476427	+	0.879214i	$0.341932\pi$
$264$		0			0
$265$		−	1416.39i		−	0.328332i
$266$			10843.3i			2.49943i
$267$		0			0
$268$		−	16408.2i		−	3.73990i
$269$	−1845.44			−0.418285			−0.209142	−	0.977885i	$-0.567067\pi$
$269$	−1845.44			−0.418285			−0.209142	+	0.977885i	$0.567067\pi$
$270$		0			0
$271$		−	2106.78i		−	0.472242i	−0.971724	−	0.236121i	$-0.924124\pi$
$271$		−	2106.78i		−	0.472242i	0.971724	−	0.236121i	$-0.0758761\pi$
$272$	−8218.19			−1.83199
$273$		0			0
$274$	83.0939			0.0183208
$275$		−	1479.64i		−	0.324457i
$276$		0			0
$277$	4781.94			1.03725			0.518626	−	0.855001i	$-0.326444\pi$
$277$	4781.94			1.03725			0.518626	+	0.855001i	$0.326444\pi$
$278$		−	10825.5i		−	2.33551i
$279$		0			0
$280$		−	10724.2i		−	2.28891i
$281$		−	5865.81i		−	1.24528i	−0.782507	−	0.622642i	$-0.786059\pi$
$281$		−	5865.81i		−	1.24528i	0.782507	−	0.622642i	$-0.213941\pi$
$282$		0			0
$283$	6407.02			1.34579			0.672894	−	0.739739i	$-0.265051\pi$
$283$	6407.02			1.34579			0.672894	+	0.739739i	$0.265051\pi$
$284$			3186.14i			0.665713i
$285$		0			0
$286$	3693.28	−	1488.18i	0.763596	−	0.307684i
$287$	1723.38			0.354453
$288$		0			0
$289$	−1997.00			−0.406473
$290$	5023.97			1.01730
$291$		0			0
$292$			6066.15i			1.21573i
$293$			3010.24i			0.600204i	0.953907	+	0.300102i	$0.0970208\pi$
$293$			3010.24i			0.600204i	−0.953907	+	0.300102i	$0.902979\pi$
$294$		0			0
$295$	1547.64			0.305447
$296$	−6472.94			−1.27105
$297$		0			0
$298$	14425.7			2.80422
$299$	7949.62	−	3203.23i	1.53759	−	0.619557i
$300$		0			0
$301$			3543.07i			0.678470i
$302$	8446.69			1.60944
$303$		0			0
$304$			10097.7i			1.90508i
$305$		−	2733.39i		−	0.513159i
$306$		0			0
$307$		−	3341.84i		−	0.621266i	−0.950530	−	0.310633i	$-0.899459\pi$
$307$		−	3341.84i		−	0.621266i	0.950530	−	0.310633i	$-0.100541\pi$
$308$	−9805.59			−1.81404
$309$		0			0
$310$		−	1795.17i		−	0.328900i
$311$	8755.20			1.59634			0.798170	−	0.602432i	$-0.205801\pi$
$311$	8755.20			1.59634			0.798170	+	0.602432i	$0.205801\pi$
$312$		0			0
$313$	1948.93			0.351949			0.175974	−	0.984395i	$-0.443692\pi$
$313$	1948.93			0.351949			0.175974	+	0.984395i	$0.443692\pi$
$314$			5792.91i			1.04112i
$315$		0			0
$316$	−9226.57			−1.64252
$317$		−	1940.43i		−	0.343802i	−0.985114	−	0.171901i	$-0.945009\pi$
$317$		−	1940.43i		−	0.343802i	0.985114	−	0.171901i	$-0.0549909\pi$
$318$		0			0
$319$		−	2683.36i		−	0.470970i
$320$		−	2797.29i		−	0.488667i
$321$		0			0
$322$	−29883.0			−5.17178
$323$		−	3582.90i		−	0.617207i
$324$		0			0
$325$	−3952.00	+	1592.43i	−0.674516	+	0.271790i
$326$	−1217.07			−0.206771
$327$		0			0
$328$	3227.82			0.543373
$329$	−16098.1			−2.69762
$330$		0			0
$331$			7402.13i			1.22918i	0.788848	+	0.614589i	$0.210678\pi$
$331$			7402.13i			1.22918i	−0.788848	+	0.614589i	$0.789322\pi$
$332$			11047.0i			1.82615i
$333$		0			0
$334$	1124.20			0.184171
$335$	−4980.50			−0.812280
$336$		0			0
$337$	5494.05			0.888071			0.444035	−	0.896009i	$-0.353546\pi$
$337$	5494.05			0.888071			0.444035	+	0.896009i	$0.353546\pi$
$338$	−7949.62	−	8262.87i	−1.27930	−	1.32971i
$339$		0			0
$340$			6066.15i			0.967597i
$341$	−958.823			−0.152267
$342$		0			0
$343$		−	9223.77i		−	1.45200i
$344$			6636.03i			1.04009i
$345$		0			0
$346$			7218.62i			1.12160i
$347$	−3410.76			−0.527664			−0.263832	−	0.964569i	$-0.584986\pi$
$347$	−3410.76			−0.527664			−0.263832	+	0.964569i	$0.584986\pi$
$348$		0			0
$349$		−	12629.8i		−	1.93712i	−0.248773	−	0.968562i	$-0.580027\pi$
$349$		−	12629.8i		−	1.93712i	0.248773	−	0.968562i	$-0.419973\pi$
$350$	14855.8			2.26878
$351$		0			0
$352$	−5291.32			−0.801217
$353$			2981.78i			0.449586i	0.974407	+	0.224793i	$0.0721706\pi$
$353$			2981.78i			0.449586i	−0.974407	+	0.224793i	$0.927829\pi$
$354$		0			0
$355$	967.109			0.144588
$356$		−	1283.43i		−	0.191073i
$357$		0			0
$358$		−	19011.7i		−	2.80670i
$359$		−	8942.30i		−	1.31464i	−0.753611	−	0.657321i	$-0.771690\pi$
$359$		−	8942.30i		−	1.31464i	0.753611	−	0.657321i	$-0.228310\pi$
$360$		0			0
$361$	2456.68			0.358168
$362$			13684.0i			1.98678i
$363$		0			0
$364$	10553.0	+	26190.0i	1.51958	+	3.77123i
$365$	1841.30			0.264049
$366$		0			0
$367$	−4735.26			−0.673511			−0.336756	−	0.941592i	$-0.609330\pi$
$367$	−4735.26			−0.673511			−0.336756	+	0.941592i	$0.609330\pi$
$368$	−27828.1			−3.94196
$369$		0			0
$370$			3363.47i			0.472590i
$371$		−	7595.45i		−	1.06290i
$372$		0			0
$373$	−8304.01			−1.15272			−0.576361	−	0.817195i	$-0.695528\pi$
$373$	−8304.01			−1.15272			−0.576361	+	0.817195i	$0.695528\pi$
$374$	4587.35			0.634241
$375$		0			0
$376$	−30151.1			−4.13543
$377$	−7167.06	+	2887.90i	−0.979104	+	0.394522i
$378$		0			0
$379$		−	4088.11i		−	0.554069i	−0.960860	−	0.277035i	$-0.910648\pi$
$379$		−	4088.11i		−	0.554069i	0.960860	−	0.277035i	$-0.0893517\pi$
$380$	7453.50			1.00620
$381$		0			0
$382$		−	17845.4i		−	2.39018i
$383$			13951.5i			1.86132i	0.365879	+	0.930662i	$0.380768\pi$
$383$			13951.5i			1.86132i	−0.365879	+	0.930662i	$0.619232\pi$
$384$		0			0
$385$			2976.35i			0.393997i
$386$	8863.88			1.16881
$387$		0			0
$388$			27668.7i			3.62027i
$389$	−2804.26			−0.365506			−0.182753	−	0.983159i	$-0.558501\pi$
$389$	−2804.26			−0.365506			−0.182753	+	0.983159i	$0.558501\pi$
$390$		0			0
$391$	9874.06			1.27712
$392$		−	37392.5i		−	4.81787i
$393$		0			0
$394$	11970.9			1.53067
$395$			2800.60i			0.356743i
$396$		0			0
$397$			6556.18i			0.828830i	0.910088	+	0.414415i	$0.136014\pi$
$397$			6556.18i			0.828830i	−0.910088	+	0.414415i	$0.863986\pi$
$398$			4702.20i			0.592211i
$399$		0			0
$400$	13834.2			1.72928
$401$			4730.95i			0.589157i	0.955627	+	0.294579i	$0.0951793\pi$
$401$			4730.95i			0.589157i	−0.955627	+	0.294579i	$0.904821\pi$
$402$		0			0
$403$	1031.91	+	2560.94i	0.127551	+	0.316550i
$404$	17242.1			2.12333
$405$		0			0
$406$	26941.3			3.29329
$407$	1796.47			0.218790
$408$		0			0
$409$			12314.4i			1.48878i	0.667746	+	0.744389i	$0.267259\pi$
$409$			12314.4i			1.48878i	−0.667746	+	0.744389i	$0.732741\pi$
$410$		−	1677.24i		−	0.202032i
$411$		0			0
$412$	−428.316			−0.0512175
$413$	8299.29			0.988817
$414$		0			0
$415$	3353.16			0.396627
$416$	5694.66	+	14132.7i	0.671163	+	1.66566i
$417$		0			0
$418$		−	5636.50i		−	0.659546i
$419$	5499.85			0.641254			0.320627	−	0.947206i	$-0.396106\pi$
$419$	5499.85			0.641254			0.320627	+	0.947206i	$0.396106\pi$
$420$		0			0
$421$		−	12629.7i		−	1.46208i	−0.682336	−	0.731039i	$-0.739036\pi$
$421$		−	12629.7i		−	1.46208i	0.682336	−	0.731039i	$-0.260964\pi$
$422$		−	2249.48i		−	0.259486i
$423$		0			0
$424$		−	14225.9i		−	1.62942i
$425$	−4908.70			−0.560252
$426$		0			0
$427$		−	14658.0i		−	1.66124i
$428$	−6760.94			−0.763557
$429$		0			0
$430$	3448.21			0.386715
$431$			7191.15i			0.803679i	0.915710	+	0.401840i	$0.131629\pi$
$431$			7191.15i			0.803679i	−0.915710	+	0.401840i	$0.868371\pi$
$432$		0			0
$433$	−6062.68			−0.672873			−0.336436	−	0.941706i	$-0.609222\pi$
$433$	−6062.68			−0.672873			−0.336436	+	0.941706i	$0.609222\pi$
$434$		−	9626.71i		−	1.06474i
$435$		0			0
$436$		−	18603.0i		−	2.04340i
$437$		−	12132.3i		−	1.32807i
$438$		0			0
$439$	−11864.3			−1.28986			−0.644932	−	0.764240i	$-0.723114\pi$
$439$	−11864.3			−1.28986			−0.644932	+	0.764240i	$0.723114\pi$
$440$			5574.58i			0.603995i
$441$		0			0
$442$	−4937.03	−	12252.5i	−0.531291	−	1.31853i
$443$	−10560.5			−1.13261			−0.566303	−	0.824197i	$-0.691627\pi$
$443$	−10560.5			−1.13261			−0.566303	+	0.824197i	$0.691627\pi$
$444$		0			0
$445$	−389.569			−0.0414997
$446$	21420.3			2.27417
$447$		0			0
$448$		−	15000.6i		−	1.58195i
$449$			12659.7i			1.33062i	0.746569	+	0.665308i	$0.231700\pi$
$449$			12659.7i			1.33062i	−0.746569	+	0.665308i	$0.768300\pi$
$450$		0			0
$451$	−895.834			−0.0935325
$452$	929.072			0.0966812
$453$		0			0
$454$	9441.69			0.976037
$455$	7949.62	−	3203.23i	0.819085	−	0.330043i
$456$		0			0
$457$			1544.24i			0.158067i	0.996872	+	0.0790336i	$0.0251834\pi$
$457$			1544.24i			0.158067i	−0.996872	+	0.0790336i	$0.974817\pi$
$458$	−27396.1			−2.79506
$459$		0			0
$460$			20541.0i			2.08202i
$461$		−	13196.8i		−	1.33327i	−0.745384	−	0.666635i	$-0.767734\pi$
$461$		−	13196.8i		−	1.33327i	0.745384	−	0.666635i	$-0.232266\pi$
$462$		0			0
$463$			16309.2i			1.63705i	0.574472	+	0.818524i	$0.305208\pi$
$463$			16309.2i			1.63705i	−0.574472	+	0.818524i	$0.694792\pi$
$464$	25088.7			2.51016
$465$		0			0
$466$			14659.5i			1.45727i
$467$	−14260.8			−1.41308			−0.706541	−	0.707672i	$-0.749745\pi$
$467$	−14260.8			−1.41308			−0.706541	+	0.707672i	$0.749745\pi$
$468$		0			0
$469$	−26708.2			−2.62957
$470$			15667.1i			1.53760i
$471$		0			0
$472$	15544.2			1.51585
$473$		−	1841.73i		−	0.179034i
$474$		0			0
$475$			6031.34i			0.582604i
$476$			32530.0i			3.13238i
$477$		0			0
$478$	35029.9			3.35194
$479$		−	18011.5i		−	1.71809i	−0.511899	−	0.859046i	$-0.671058\pi$
$479$		−	18011.5i		−	1.71809i	0.511899	−	0.859046i	$-0.328942\pi$
$480$		0			0
$481$	−1933.41	−	4798.23i	−0.183276	−	0.454845i
$482$	13147.2			1.24240
$483$		0			0
$484$	−20508.4			−1.92603
$485$	8398.46			0.786298
$486$		0			0
$487$			14043.3i			1.30670i	0.757055	+	0.653351i	$0.226637\pi$
$487$			14043.3i			1.30670i	−0.757055	+	0.653351i	$0.773363\pi$
$488$		−	27453.7i		−	2.54666i
$489$		0			0
$490$	−19429.9			−1.79133
$491$	−12966.1			−1.19176			−0.595878	−	0.803075i	$-0.703196\pi$
$491$	−12966.1			−1.19176			−0.595878	+	0.803075i	$0.703196\pi$
$492$		0			0
$493$	−8902.06			−0.813242
$494$	−15054.7	+	6066.15i	−1.37114	+	0.552487i
$495$		0			0
$496$		−	8964.75i		−	0.811551i
$497$	5186.17			0.468072
$498$		0			0
$499$			10215.5i			0.916453i	0.888835	+	0.458227i	$0.151515\pi$
$499$			10215.5i			0.916453i	−0.888835	+	0.458227i	$0.848485\pi$
$500$		−	24253.6i		−	2.16930i
$501$		0			0
$502$		−	4321.31i		−	0.384203i
$503$	16632.0			1.47432			0.737161	−	0.675717i	$-0.236166\pi$
$503$	16632.0			1.47432			0.737161	+	0.675717i	$0.236166\pi$
$504$		0			0
$505$		−	5233.61i		−	0.461173i
$506$	15533.5			1.36472
$507$		0			0
$508$	−33916.5			−2.96221
$509$			15235.3i			1.32671i	0.748306	+	0.663354i	$0.230868\pi$
$509$			15235.3i			1.32671i	−0.748306	+	0.663354i	$0.769132\pi$
$510$		0			0
$511$	9874.06			0.854799
$512$			21933.9i			1.89326i
$513$		0			0
$514$			30482.8i			2.61584i
$515$			130.009i			0.0111241i
$516$		0			0
$517$	8367.99			0.711845
$518$			18036.8i			1.52991i
$519$		0			0
$520$	14889.3	−	5999.51i	1.25565	−	0.505954i
$521$	−2680.23			−0.225380			−0.112690	−	0.993630i	$-0.535947\pi$
$521$	−2680.23			−0.225380			−0.112690	+	0.993630i	$0.535947\pi$
$522$		0			0
$523$	−2410.38			−0.201527			−0.100764	−	0.994910i	$-0.532129\pi$
$523$	−2410.38			−0.201527			−0.100764	+	0.994910i	$0.532129\pi$
$524$	18390.7			1.53321
$525$		0			0
$526$		−	21210.2i		−	1.75819i
$527$			3180.90i			0.262926i
$528$		0			0
$529$	21268.2			1.74802
$530$	−7392.09			−0.605834
$531$		0			0
$532$	39969.8			3.25735
$533$	964.120	+	2392.70i	0.0783502	+	0.194446i
$534$		0			0
$535$			2052.19i			0.165839i
$536$	−50023.3			−4.03111
$537$		0			0
$538$			9631.32i			0.771814i
$539$			10377.7i			0.829315i
$540$		0			0
$541$		−	9969.58i		−	0.792284i	−0.918189	−	0.396142i	$-0.870349\pi$
$541$		−	9969.58i		−	0.792284i	0.918189	−	0.396142i	$-0.129651\pi$
$542$	−10995.2			−0.871375
$543$		0			0
$544$			17554.0i			1.38349i
$545$	−5646.70			−0.443813
$546$		0			0
$547$	16848.8			1.31701			0.658505	−	0.752576i	$-0.271189\pi$
$547$	16848.8			1.31701			0.658505	+	0.752576i	$0.271189\pi$
$548$		−	306.293i		−	0.0238763i
$549$		0			0
$550$	−7722.20			−0.598683
$551$			10938.0i			0.845688i
$552$		0			0
$553$			15018.4i			1.15488i
$554$		−	24956.8i		−	1.91393i
$555$		0			0
$556$	−39904.2			−3.04373
$557$		−	3800.83i		−	0.289132i	−0.989495	−	0.144566i	$-0.953821\pi$
$557$		−	3800.83i		−	0.289132i	0.989495	−	0.144566i	$-0.0461786\pi$
$558$		0			0
$559$	−4919.13	+	1982.12i	−0.372195	+	0.149973i
$560$	−27828.1			−2.09992
$561$		0			0
$562$	−30613.5			−2.29778
$563$	−15750.2			−1.17903			−0.589513	−	0.807759i	$-0.700680\pi$
$563$	−15750.2			−1.17903			−0.589513	+	0.807759i	$0.700680\pi$
$564$		0			0
$565$		−	282.007i		−	0.0209985i
$566$		−	33438.1i		−	2.48323i
$567$		0			0
$568$	9713.48			0.717550
$569$	17753.2			1.30800			0.654002	−	0.756493i	$-0.273089\pi$
$569$	17753.2			1.30800			0.654002	+	0.756493i	$0.273089\pi$
$570$		0			0
$571$	−25293.1			−1.85374			−0.926868	−	0.375388i	$-0.877509\pi$
$571$	−25293.1			−1.85374			−0.926868	+	0.375388i	$0.877509\pi$
$572$	−5485.59	−	13613.9i	−0.400986	−	0.995147i
$573$		0			0
$574$		−	8994.29i		−	0.654032i
$575$	−16621.7			−1.20552
$576$		0			0
$577$		−	18488.8i		−	1.33396i	−0.745073	−	0.666982i	$-0.767586\pi$
$577$		−	18488.8i		−	1.33396i	0.745073	−	0.666982i	$-0.232414\pi$
$578$			10422.3i			0.750018i
$579$		0			0
$580$		−	18518.9i		−	1.32579i
$581$	17981.5			1.28399
$582$		0			0
$583$			3948.20i			0.280477i
$584$	18493.7			1.31040
$585$		0			0
$586$	15710.4			1.10749
$587$			17376.7i			1.22183i	0.791697	+	0.610914i	$0.209198\pi$
$587$			17376.7i			1.22183i	−0.791697	+	0.610914i	$0.790802\pi$
$588$		0			0
$589$	3908.38			0.273416
$590$		−	8077.09i		−	0.563608i
$591$		0			0
$592$			16796.5i			1.16610i
$593$			7991.09i			0.553381i	0.960959	+	0.276690i	$0.0892377\pi$
$593$			7991.09i			0.553381i	−0.960959	+	0.276690i	$0.910762\pi$
$594$		0			0
$595$	9874.06			0.680331
$596$		−	53174.7i		−	3.65456i
$597$		0			0
$598$	−16717.6	−	41488.9i	−1.14320	−	2.83713i
$599$	−10386.5			−0.708480			−0.354240	−	0.935154i	$-0.615260\pi$
$599$	−10386.5			−0.708480			−0.354240	+	0.935154i	$0.615260\pi$
$600$		0			0
$601$	−9241.77			−0.627254			−0.313627	−	0.949546i	$-0.601544\pi$
$601$	−9241.77			−0.627254			−0.313627	+	0.949546i	$0.601544\pi$
$602$	18491.2			1.25190
$603$		0			0
$604$		−	31135.4i		−	2.09749i
$605$			6225.04i			0.418320i
$606$		0			0
$607$	18921.5			1.26524			0.632619	−	0.774463i	$-0.281980\pi$
$607$	18921.5			1.26524			0.632619	+	0.774463i	$0.281980\pi$
$608$	21568.6			1.43869
$609$		0			0
$610$	−14265.5			−0.946875
$611$	−9005.85	−	22350.3i	−0.596297	−	1.47986i
$612$		0			0
$613$			17138.1i			1.12921i	0.825363	+	0.564603i	$0.190971\pi$
$613$			17138.1i			1.12921i	−0.825363	+	0.564603i	$0.809029\pi$
$614$	−17441.0			−1.14635
$615$		0			0
$616$			29894.0i			1.95530i
$617$		−	18825.8i		−	1.22836i	−0.789166	−	0.614180i	$-0.789487\pi$
$617$		−	18825.8i		−	1.22836i	0.789166	−	0.614180i	$-0.210513\pi$
$618$		0			0
$619$		−	1392.83i		−	0.0904404i	−0.998977	−	0.0452202i	$-0.985601\pi$
$619$		−	1392.83i		−	0.0904404i	0.998977	−	0.0452202i	$-0.0143989\pi$
$620$	−6617.21			−0.428635
$621$		0			0
$622$		−	45693.2i		−	2.94555i
$623$	−2089.09			−0.134346
$624$		0			0
$625$	4000.90			0.256057
$626$		−	10171.4i		−	0.649412i
$627$		0			0
$628$	21353.3			1.35683
$629$		−	5959.79i		−	0.377794i
$630$		0			0
$631$		−	25488.4i		−	1.60804i	−0.594599	−	0.804022i	$-0.702689\pi$
$631$		−	25488.4i		−	1.60804i	0.594599	−	0.804022i	$-0.297311\pi$
$632$			28128.8i			1.77041i
$633$		0			0
$634$	−10127.1			−0.634379
$635$			10294.9i			0.643371i
$636$		0			0
$637$	27718.2	−	11168.8i	1.72407	−	0.694700i
$638$	−14004.4			−0.869028
$639$		0			0
$640$	586.735			0.0362387
$641$	−6066.41			−0.373805			−0.186902	−	0.982379i	$-0.559845\pi$
$641$	−6066.41			−0.373805			−0.186902	+	0.982379i	$0.559845\pi$
$642$		0			0
$643$		−	1598.78i		−	0.0980554i	−0.998797	−	0.0490277i	$-0.984388\pi$
$643$		−	1598.78i		−	0.0980554i	0.998797	−	0.0490277i	$-0.0156123\pi$
$644$			110152.i			6.74006i
$645$		0			0
$646$	−18699.1			−1.13886
$647$	23067.2			1.40164			0.700822	−	0.713336i	$-0.252817\pi$
$647$	23067.2			1.40164			0.700822	+	0.713336i	$0.252817\pi$
$648$		0			0
$649$	−4314.07			−0.260928
$650$	8310.84	+	20625.4i	0.501504	+	1.24461i
$651$		0			0
$652$			4486.26i			0.269472i
$653$	23743.9			1.42293			0.711463	−	0.702723i	$-0.248033\pi$
$653$	23743.9			1.42293			0.711463	+	0.702723i	$0.248033\pi$
$654$		0			0
$655$		−	5582.25i		−	0.333002i
$656$		−	8375.81i		−	0.498507i
$657$		0			0
$658$			84015.7i			4.97762i
$659$	7497.65			0.443197			0.221598	−	0.975138i	$-0.428873\pi$
$659$	7497.65			0.443197			0.221598	+	0.975138i	$0.428873\pi$
$660$		0			0
$661$		−	1255.26i		−	0.0738638i	−0.999318	−	0.0369319i	$-0.988242\pi$
$661$		−	1255.26i		−	0.0738638i	0.999318	−	0.0369319i	$-0.0117585\pi$
$662$	38631.5			2.26806
$663$		0			0
$664$	33678.6			1.96835
$665$		−	12132.3i		−	0.707474i
$666$		0			0
$667$	−30143.8			−1.74989
$668$		−	4143.91i		−	0.240019i
$669$		0			0
$670$			25993.1i			1.49881i
$671$			7619.38i			0.438365i
$672$		0			0
$673$	−1505.97			−0.0862569			−0.0431284	−	0.999070i	$-0.513732\pi$
$673$	−1505.97			−0.0862569			−0.0431284	+	0.999070i	$0.513732\pi$
$674$		−	28673.3i		−	1.63866i
$675$		0			0
$676$	−30457.9	+	29303.2i	−1.73292	+	1.66723i
$677$	16201.4			0.919751			0.459876	−	0.887983i	$-0.347894\pi$
$677$	16201.4			0.919751			0.459876	+	0.887983i	$0.347894\pi$
$678$		0			0
$679$	45037.2			2.54546
$680$	18493.7			1.04294
$681$		0			0
$682$			5004.08i			0.280962i
$683$		−	29090.4i		−	1.62974i	−0.579644	−	0.814870i	$-0.696808\pi$
$683$		−	29090.4i		−	1.62974i	0.579644	−	0.814870i	$-0.303192\pi$
$684$		0			0
$685$	−92.9712			−0.00518576
$686$	−48138.7			−2.67922
$687$		0			0
$688$	17219.7			0.954208
$689$	10545.4	−	4249.16i	0.583085	−	0.234949i
$690$		0			0
$691$		−	940.952i		−	0.0518025i	−0.999665	−	0.0259012i	$-0.991754\pi$
$691$		−	940.952i		−	0.0518025i	0.999665	−	0.0259012i	$-0.00824554\pi$
$692$	26608.6			1.46172
$693$		0			0
$694$			17800.7i			0.973639i
$695$			12112.4i			0.661077i
$696$		0			0
$697$			2971.93i			0.161506i
$698$	−65914.5			−3.57436
$699$		0			0
$700$		−	54760.1i		−	2.95676i
$701$	−30713.9			−1.65485			−0.827424	−	0.561578i	$-0.810195\pi$
$701$	−30713.9			−1.65485			−0.827424	+	0.561578i	$0.810195\pi$
$702$		0			0
$703$	−7322.81			−0.392866
$704$			7797.51i			0.417443i
$705$		0			0
$706$	15561.8			0.829571
$707$		−	28065.5i		−	1.49294i
$708$		0			0
$709$			25640.5i			1.35818i	0.734056	+	0.679089i	$0.237625\pi$
$709$			25640.5i			1.35818i	−0.734056	+	0.679089i	$0.762375\pi$
$710$		−	5047.32i		−	0.266792i
$711$		0			0
$712$	−3912.77			−0.205951
$713$			10771.0i			0.565748i
$714$		0			0
$715$	−4132.30	+	1665.08i	−0.216139	+	0.0870913i
$716$	−70079.1			−3.65779
$717$		0			0
$718$	−46669.7			−2.42576
$719$	20842.5			1.08108			0.540538	−	0.841319i	$-0.318221\pi$
$719$	20842.5			1.08108			0.540538	+	0.841319i	$0.318221\pi$
$720$		0			0
$721$			697.183i			0.0360117i
$722$		−	12821.3i		−	0.660888i
$723$		0			0
$724$	50440.8			2.58925
$725$	14985.4			0.767649
$726$		0			0
$727$	263.608			0.0134480			0.00672398	−	0.999977i	$-0.497860\pi$
$727$	263.608			0.0134480			0.00672398	+	0.999977i	$0.497860\pi$
$728$	79844.6	−	32172.7i	4.06489	−	1.63791i
$729$		0			0
$730$		−	9609.69i		−	0.487220i
$731$	−6109.94			−0.309144
$732$		0			0
$733$		−	21835.5i		−	1.10029i	−0.835069	−	0.550146i	$-0.814572\pi$
$733$		−	21835.5i		−	1.10029i	0.835069	−	0.550146i	$-0.185428\pi$
$734$			24713.2i			1.24275i
$735$		0			0
$736$			59440.6i			2.97692i
$737$	13883.2			0.693888
$738$		0			0
$739$		−	19536.4i		−	0.972476i	−0.873826	−	0.486238i	$-0.838369\pi$
$739$		−	19536.4i		−	0.972476i	0.873826	−	0.486238i	$-0.161631\pi$
$740$	12398.1			0.615897
$741$		0			0
$742$	−39640.5			−1.96125
$743$		−	30353.1i		−	1.49872i	−0.662163	−	0.749360i	$-0.730361\pi$
$743$		−	30353.1i		−	1.49872i	0.662163	−	0.749360i	$-0.269639\pi$
$744$		0			0
$745$	−16140.5			−0.793746
$746$			43338.4i			2.12699i
$747$		0			0
$748$		−	16909.5i		−	0.826567i
$749$			11005.0i			0.536867i
$750$		0			0
$751$	3904.20			0.189702			0.0948510	−	0.995491i	$-0.469763\pi$
$751$	3904.20			0.189702			0.0948510	+	0.995491i	$0.469763\pi$
$752$			78238.5i			3.79397i
$753$		0			0
$754$	15071.9	+	37404.7i	0.727966	+	1.80663i
$755$	−9450.74			−0.455560
$756$		0			0
$757$	−2900.52			−0.139262			−0.0696308	−	0.997573i	$-0.522182\pi$
$757$	−2900.52			−0.139262			−0.0696308	+	0.997573i	$0.522182\pi$
$758$	−21335.8			−1.02236
$759$		0			0
$760$		−	22723.3i		−	1.08455i
$761$			33518.7i			1.59665i	0.602227	+	0.798325i	$0.294280\pi$
$761$			33518.7i			1.59665i	−0.602227	+	0.798325i	$0.705720\pi$
$762$		0			0
$763$	−30280.8			−1.43675
$764$	−65780.0			−3.11497
$765$		0			0
$766$	72812.5			3.43449
$767$	4642.91	+	11522.6i	0.218573	+	0.542445i
$768$		0			0
$769$			552.769i			0.0259211i	0.999916	+	0.0129606i	$0.00412559\pi$
$769$			552.769i			0.0259211i	−0.999916	+	0.0129606i	$0.995874\pi$
$770$	15533.5			0.726999
$771$		0			0
$772$		−	32673.3i		−	1.52323i
$773$			36498.8i			1.69828i	0.528166	+	0.849141i	$0.322880\pi$
$773$			36498.8i			1.69828i	−0.528166	+	0.849141i	$0.677120\pi$
$774$		0			0
$775$		−	5354.62i		−	0.248185i
$776$	84352.8			3.90218
$777$		0			0
$778$			14635.4i			0.674427i
$779$	3651.62			0.167950
$780$		0			0
$781$	−2695.83			−0.123514
$782$		−	51532.5i		−	2.35652i
$783$		0			0
$784$	−97029.2			−4.42006
$785$		−	6481.51i		−	0.294694i
$786$		0			0
$787$			4284.79i			0.194074i	0.995281	+	0.0970371i	$0.0309366\pi$
$787$			4284.79i			0.194074i	−0.995281	+	0.0970371i	$0.969063\pi$
$788$

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.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-5.21898i q^{2} -19.2377 q^{4} +5.83936i q^{5} +31.3139i q^{7} +58.6495i q^{8} +O(q^{10})\)	\(q-5.21898i q^{2} -19.2377 q^{4} +5.83936i q^{5} +31.3139i q^{7} +58.6495i q^{8} +30.4755 q^{10} -16.2773i q^{11} +(-43.4755 + 17.5181i) q^{13} +163.426 q^{14} +152.189 q^{16} -54.0000 q^{17} +66.3500i q^{19} -112.336i q^{20} -84.9510 q^{22} -182.853 q^{23} +90.9019 q^{25} +(91.4264 + 226.898i) q^{26} -602.408i q^{28} +164.853 q^{29} -58.9055i q^{31} -325.073i q^{32} +281.825i q^{34} -182.853 q^{35} +110.366i q^{37} +346.279 q^{38} -342.475 q^{40} -55.0357i q^{41} +113.147 q^{43} +313.139i q^{44} +954.305i q^{46} +514.089i q^{47} -637.559 q^{49} -474.415i q^{50} +(836.370 - 337.008i) q^{52} -242.559 q^{53} +95.0490 q^{55} -1836.54 q^{56} -860.364i q^{58} -265.036i q^{59} -468.098 q^{61} -307.426 q^{62} -479.042 q^{64} +(-102.294 - 253.869i) q^{65} +852.919i q^{67} +1038.84 q^{68} +954.305i q^{70} -165.619i q^{71} -315.325i q^{73} +576.000 q^{74} -1276.42i q^{76} +509.706 q^{77} +479.608 q^{79} +888.684i q^{80} -287.230 q^{82} -574.235i q^{83} -315.325i q^{85} -590.512i q^{86} +954.657 q^{88} +66.7144i q^{89} +(-548.559 - 1361.39i) q^{91} +3517.68 q^{92} +2683.02 q^{94} -387.441 q^{95} -1438.25i q^{97} +3327.40i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 26 q^{4}+O(q^{10}) \) 4 * q - 26 * q^4	\( 4 q - 26 q^{4} + 20 q^{10} - 72 q^{13} + 348 q^{14} + 354 q^{16} - 216 q^{17} - 136 q^{22} - 120 q^{23} - 44 q^{25} + 60 q^{26} + 48 q^{29} - 120 q^{35} + 468 q^{38} - 1268 q^{40} + 1064 q^{43} - 716 q^{49} + 1766 q^{52} + 864 q^{53} + 584 q^{55} - 3372 q^{56} - 2280 q^{61} - 924 q^{62} - 1050 q^{64} - 1632 q^{65} + 1404 q^{68} + 2304 q^{74} + 816 q^{77} + 288 q^{79} - 28 q^{82} + 2392 q^{88} - 360 q^{91} + 8568 q^{92} + 6656 q^{94} - 3384 q^{95}+O(q^{100}) \) 4 * q - 26 * q^4 + 20 * q^10 - 72 * q^13 + 348 * q^14 + 354 * q^16 - 216 * q^17 - 136 * q^22 - 120 * q^23 - 44 * q^25 + 60 * q^26 + 48 * q^29 - 120 * q^35 + 468 * q^38 - 1268 * q^40 + 1064 * q^43 - 716 * q^49 + 1766 * q^52 + 864 * q^53 + 584 * q^55 - 3372 * q^56 - 2280 * q^61 - 924 * q^62 - 1050 * q^64 - 1632 * q^65 + 1404 * q^68 + 2304 * q^74 + 816 * q^77 + 288 * q^79 - 28 * q^82 + 2392 * q^88 - 360 * q^91 + 8568 * q^92 + 6656 * q^94 - 3384 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more