LMFDB - Embedded newform 60.5.g.b.41.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [60,5,Mod(41,60)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(60, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 5, names="a")

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

chi := DirichletCharacter("60.41");

S:= CuspForms(chi, 5);

N := Newforms(S);

Level:	$ N $	$=$	$ 60 = 2^{2} \cdot 3 \cdot 5 $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	60.g (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:	$6.20219778503$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{-5}, \sqrt{34})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 58x^{2} + 1521 $ x^4 - 58*x^2 + 1521
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 3\cdot 5 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			41.3
Root			$-5.83095 - 2.23607i$ of defining polynomial
Character	$\chi$	$=$	60.41
Dual form			60.5.g.b.41.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-2.08452 - 8.75527i) q^{3} -11.1803i q^{5} -27.4929 q^{7} +(-72.3095 + 36.5011i) q^{9} +O(q^{10})$	\(q+(-2.08452 - 8.75527i) q^{3} -11.1803i q^{5} -27.4929 q^{7} +(-72.3095 + 36.5011i) q^{9} -29.1008i q^{11} -289.886 q^{13} +(-97.8869 + 23.3057i) q^{15} -125.284i q^{17} +38.9286 q^{19} +(57.3095 + 240.707i) q^{21} -893.038i q^{23} -125.000 q^{25} +(470.308 + 557.002i) q^{27} +438.014i q^{29} +1283.71 q^{31} +(-254.786 + 60.6614i) q^{33} +307.379i q^{35} +139.914 q^{37} +(604.274 + 2538.03i) q^{39} -1998.66i q^{41} +2357.29 q^{43} +(408.095 + 808.445i) q^{45} -1943.67i q^{47} -1645.14 q^{49} +(-1096.89 + 261.157i) q^{51} -4489.58i q^{53} -325.357 q^{55} +(-81.1475 - 340.830i) q^{57} +5225.56i q^{59} -5263.86 q^{61} +(1988.00 - 1003.52i) q^{63} +3241.02i q^{65} -5221.28 q^{67} +(-7818.79 + 1861.56i) q^{69} -5894.88i q^{71} +621.629 q^{73} +(260.566 + 1094.41i) q^{75} +800.065i q^{77} +4994.21 q^{79} +(3896.33 - 5278.76i) q^{81} -3802.29i q^{83} -1400.71 q^{85} +(3834.93 - 913.050i) q^{87} +11054.7i q^{89} +7969.79 q^{91} +(-2675.93 - 11239.3i) q^{93} -435.234i q^{95} +5996.54 q^{97} +(1062.21 + 2104.27i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 20 q^{3} - 40 q^{7} - 56 q^{9}+O(q^{10}) $ 4 * q - 20 * q^3 - 40 * q^7 - 56 * q^9	$ 4 q - 20 q^{3} - 40 q^{7} - 56 q^{9} - 40 q^{13} - 100 q^{15} - 544 q^{19} - 4 q^{21} - 500 q^{25} + 820 q^{27} + 2336 q^{31} + 1080 q^{33} - 280 q^{37} - 3064 q^{39} + 7400 q^{43} - 700 q^{45} - 7980 q^{49} + 1560 q^{51} - 4800 q^{55} + 4760 q^{57} - 64 q^{61} + 4640 q^{63} - 8920 q^{67} - 19380 q^{69} + 18440 q^{73} + 2500 q^{75} - 7312 q^{79} + 2524 q^{81} - 12600 q^{85} + 14640 q^{87} + 19984 q^{91} - 3520 q^{93} - 9880 q^{97} - 23040 q^{99}+O(q^{100}) $ 4 * q - 20 * q^3 - 40 * q^7 - 56 * q^9 - 40 * q^13 - 100 * q^15 - 544 * q^19 - 4 * q^21 - 500 * q^25 + 820 * q^27 + 2336 * q^31 + 1080 * q^33 - 280 * q^37 - 3064 * q^39 + 7400 * q^43 - 700 * q^45 - 7980 * q^49 + 1560 * q^51 - 4800 * q^55 + 4760 * q^57 - 64 * q^61 + 4640 * q^63 - 8920 * q^67 - 19380 * q^69 + 18440 * q^73 + 2500 * q^75 - 7312 * q^79 + 2524 * q^81 - 12600 * q^85 + 14640 * q^87 + 19984 * q^91 - 3520 * q^93 - 9880 * q^97 - 23040 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/60\mathbb{Z}\right)^\times$.

$n$	$31$	$37$	$41$
$\chi(n)$	$1$	$1$	$-1$

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 $

$2$		0			0
$2$		0			0
$3$	−2.08452	−	8.75527i	−0.231614	−	0.972808i
$3$	−2.08452	−	8.75527i	−0.231614	−	0.972808i
$4$		0			0
$5$		−	11.1803i		−	0.447214i
$5$		−	11.1803i		−	0.447214i
$6$		0			0
$7$	−27.4929			−0.561079			−0.280539	−	0.959843i	$-0.590513\pi$
$7$	−27.4929			−0.561079			−0.280539	+	0.959843i	$0.590513\pi$
$8$		0			0
$9$	−72.3095	+	36.5011i	−0.892710	+	0.450631i
$10$		0			0
$11$		−	29.1008i		−	0.240503i	−0.992743	−	0.120251i	$-0.961630\pi$
$11$		−	29.1008i		−	0.240503i	0.992743	−	0.120251i	$-0.0383701\pi$
$12$		0			0
$13$	−289.886			−1.71530			−0.857650	−	0.514234i	$-0.828076\pi$
$13$	−289.886			−1.71530			−0.857650	+	0.514234i	$0.828076\pi$
$14$		0			0
$15$	−97.8869	+	23.3057i	−0.435053	+	0.103581i
$16$		0			0
$17$		−	125.284i		−	0.433508i	−0.976226	−	0.216754i	$-0.930453\pi$
$17$		−	125.284i		−	0.433508i	0.976226	−	0.216754i	$-0.0695469\pi$
$18$		0			0
$19$	38.9286			0.107835			0.0539177	−	0.998545i	$-0.482829\pi$
$19$	38.9286			0.107835			0.0539177	+	0.998545i	$0.482829\pi$
$20$		0			0
$21$	57.3095	+	240.707i	0.129954	+	0.545822i
$22$		0			0
$23$		−	893.038i		−	1.68816i	−0.536216	−	0.844081i	$-0.680147\pi$
$23$		−	893.038i		−	1.68816i	0.536216	−	0.844081i	$-0.319853\pi$
$24$		0			0
$25$	−125.000			−0.200000
$26$		0			0
$27$	470.308	+	557.002i	0.645142	+	0.764063i
$28$		0			0
$29$			438.014i			0.520825i	0.965497	+	0.260412i	$0.0838585\pi$
$29$			438.014i			0.520825i	−0.965497	+	0.260412i	$0.916141\pi$
$30$		0			0
$31$	1283.71			1.33581			0.667905	−	0.744246i	$-0.267191\pi$
$31$	1283.71			1.33581			0.667905	+	0.744246i	$0.267191\pi$
$32$		0			0
$33$	−254.786	+	60.6614i	−0.233963	+	0.0557038i
$34$		0			0
$35$			307.379i			0.250922i
$36$		0			0
$37$	139.914			0.102202			0.0511009	−	0.998693i	$-0.483727\pi$
$37$	139.914			0.102202			0.0511009	+	0.998693i	$0.483727\pi$
$38$		0			0
$39$	604.274	+	2538.03i	0.397287	+	1.66866i
$40$		0			0
$41$		−	1998.66i		−	1.18897i	−0.804106	−	0.594486i	$-0.797356\pi$
$41$		−	1998.66i		−	1.18897i	0.804106	−	0.594486i	$-0.202644\pi$
$42$		0			0
$43$	2357.29			1.27490			0.637451	−	0.770491i	$-0.279989\pi$
$43$	2357.29			1.27490			0.637451	+	0.770491i	$0.279989\pi$
$44$		0			0
$45$	408.095	+	808.445i	0.201528	+	0.399232i
$46$		0			0
$47$		−	1943.67i		−	0.879887i	−0.898025	−	0.439943i	$-0.854998\pi$
$47$		−	1943.67i		−	0.879887i	0.898025	−	0.439943i	$-0.145002\pi$
$48$		0			0
$49$	−1645.14			−0.685191
$50$		0			0
$51$	−1096.89	+	261.157i	−0.421720	+	0.100406i
$52$		0			0
$53$		−	4489.58i		−	1.59828i	−0.601143	−	0.799142i	$-0.705288\pi$
$53$		−	4489.58i		−	1.59828i	0.601143	−	0.799142i	$-0.294712\pi$
$54$		0			0
$55$	−325.357			−0.107556
$56$		0			0
$57$	−81.1475	−	340.830i	−0.0249762	−	0.104903i
$58$		0			0
$59$			5225.56i			1.50117i	0.660776	+	0.750584i	$0.270228\pi$
$59$			5225.56i			1.50117i	−0.660776	+	0.750584i	$0.729772\pi$
$60$		0			0
$61$	−5263.86			−1.41463			−0.707317	−	0.706896i	$-0.750095\pi$
$61$	−5263.86			−1.41463			−0.707317	+	0.706896i	$0.750095\pi$
$62$		0			0
$63$	1988.00	−	1003.52i	0.500881	−	0.252840i
$64$		0			0
$65$			3241.02i			0.767105i
$66$		0			0
$67$	−5221.28			−1.16313			−0.581564	−	0.813501i	$-0.697559\pi$
$67$	−5221.28			−1.16313			−0.581564	+	0.813501i	$0.697559\pi$
$68$		0			0
$69$	−7818.79	+	1861.56i	−1.64226	+	0.391002i
$70$		0			0
$71$		−	5894.88i		−	1.16939i	−0.811254	−	0.584694i	$-0.801215\pi$
$71$		−	5894.88i		−	1.16939i	0.811254	−	0.584694i	$-0.198785\pi$
$72$		0			0
$73$	621.629			0.116650			0.0583251	−	0.998298i	$-0.481424\pi$
$73$	621.629			0.116650			0.0583251	+	0.998298i	$0.481424\pi$
$74$		0			0
$75$	260.566	+	1094.41i	0.0463228	+	0.194562i
$76$		0			0
$77$			800.065i			0.134941i
$78$		0			0
$79$	4994.21			0.800227			0.400113	−	0.916466i	$-0.368971\pi$
$79$	4994.21			0.800227			0.400113	+	0.916466i	$0.368971\pi$
$80$		0			0
$81$	3896.33	−	5278.76i	0.593863	−	0.804566i
$82$		0			0
$83$		−	3802.29i		−	0.551936i	−0.961167	−	0.275968i	$-0.911002\pi$
$83$		−	3802.29i		−	0.551936i	0.961167	−	0.275968i	$-0.0889984\pi$
$84$		0			0
$85$	−1400.71			−0.193871
$86$		0			0
$87$	3834.93	−	913.050i	0.506663	−	0.120630i
$88$		0			0
$89$			11054.7i			1.39562i	0.716281	+	0.697812i	$0.245843\pi$
$89$			11054.7i			1.39562i	−0.716281	+	0.697812i	$0.754157\pi$
$90$		0			0
$91$	7969.79			0.962418
$92$		0			0
$93$	−2675.93	−	11239.3i	−0.309392	−	1.29949i
$94$		0			0
$95$		−	435.234i		−	0.0482254i
$96$		0			0
$97$	5996.54			0.637320			0.318660	−	0.947869i	$-0.396767\pi$
$97$	5996.54			0.637320			0.318660	+	0.947869i	$0.396767\pi$
$98$		0			0
$99$	1062.21	+	2104.27i	0.108378	+	0.214699i
$100$		0			0
$101$			17269.7i			1.69294i	0.532433	+	0.846472i	$0.321278\pi$
$101$			17269.7i			1.69294i	−0.532433	+	0.846472i	$0.678722\pi$
$102$		0			0
$103$	3612.74			0.340535			0.170267	−	0.985398i	$-0.445537\pi$
$103$	3612.74			0.340535			0.170267	+	0.985398i	$0.445537\pi$
$104$		0			0
$105$	2691.19	−	640.740i	0.244099	−	0.0581170i
$106$		0			0
$107$			5257.33i			0.459196i	0.973286	+	0.229598i	$0.0737411\pi$
$107$			5257.33i			0.459196i	−0.973286	+	0.229598i	$0.926259\pi$
$108$		0			0
$109$	5915.57			0.497902			0.248951	−	0.968516i	$-0.419914\pi$
$109$	5915.57			0.497902			0.248951	+	0.968516i	$0.419914\pi$
$110$		0			0
$111$	−291.655	−	1224.99i	−0.0236713	−	0.0994227i
$112$		0			0
$113$			4748.90i			0.371908i	0.982558	+	0.185954i	$0.0595376\pi$
$113$			4748.90i			0.371908i	−0.982558	+	0.185954i	$0.940462\pi$
$114$		0			0
$115$	−9984.46			−0.754969
$116$		0			0
$117$	20961.5	−	10581.2i	1.53127	−	0.772968i
$118$		0			0
$119$			3444.41i			0.243232i
$120$		0			0
$121$	13794.1			0.942158
$122$		0			0
$123$	−17498.8	+	4166.26i	−1.15664	+	0.275382i
$124$		0			0
$125$			1397.54i			0.0894427i
$126$		0			0
$127$	−26280.8			−1.62941			−0.814705	−	0.579875i	$-0.803101\pi$
$127$	−26280.8			−1.62941			−0.814705	+	0.579875i	$0.803101\pi$
$128$		0			0
$129$	−4913.83	−	20638.7i	−0.295285	−	1.24023i
$130$		0			0
$131$		−	13267.9i		−	0.773143i	−0.922260	−	0.386571i	$-0.873659\pi$
$131$		−	13267.9i		−	0.773143i	0.922260	−	0.386571i	$-0.126341\pi$
$132$		0			0
$133$	−1070.26			−0.0605041
$134$		0			0
$135$	6227.47	−	5258.21i	0.341699	−	0.288516i
$136$		0			0
$137$			9032.31i			0.481236i	0.970620	+	0.240618i	$0.0773500\pi$
$137$			9032.31i			0.481236i	−0.970620	+	0.240618i	$0.922650\pi$
$138$		0			0
$139$	2453.22			0.126971			0.0634857	−	0.997983i	$-0.479778\pi$
$139$	2453.22			0.126971			0.0634857	+	0.997983i	$0.479778\pi$
$140$		0			0
$141$	−17017.4	+	4051.63i	−0.855961	+	0.203794i
$142$		0			0
$143$			8435.92i			0.412534i
$144$		0			0
$145$	4897.14			0.232920
$146$		0			0
$147$	3429.34	+	14403.7i	0.158700	+	0.666559i
$148$		0			0
$149$		−	37919.4i		−	1.70800i	−0.520271	−	0.854001i	$-0.674169\pi$
$149$		−	37919.4i		−	1.70800i	0.520271	−	0.854001i	$-0.325831\pi$
$150$		0			0
$151$	−24333.6			−1.06721			−0.533607	−	0.845732i	$-0.679164\pi$
$151$	−24333.6			−1.06721			−0.533607	+	0.845732i	$0.679164\pi$
$152$		0			0
$153$	4573.00	+	9059.20i	0.195352	+	0.386997i
$154$		0			0
$155$		−	14352.4i		−	0.597393i
$156$		0			0
$157$	−42010.9			−1.70436			−0.852182	−	0.523245i	$-0.824721\pi$
$157$	−42010.9			−1.70436			−0.852182	+	0.523245i	$0.824721\pi$
$158$		0			0
$159$	−39307.5	+	9358.63i	−1.55482	+	0.370184i
$160$		0			0
$161$			24552.2i			0.947192i
$162$		0			0
$163$	3347.65			0.125998			0.0629992	−	0.998014i	$-0.479933\pi$
$163$	3347.65			0.125998			0.0629992	+	0.998014i	$0.479933\pi$
$164$		0			0
$165$	678.215	+	2848.59i	0.0249115	+	0.104631i
$166$		0			0
$167$		−	13082.0i		−	0.469075i	−0.972107	−	0.234538i	$-0.924642\pi$
$167$		−	13082.0i		−	0.469075i	0.972107	−	0.234538i	$-0.0753576\pi$
$168$		0			0
$169$	55472.7			1.94225
$170$		0			0
$171$	−2814.91	+	1420.94i	−0.0962657	+	0.0485940i
$172$		0			0
$173$			52099.5i			1.74077i	0.492372	+	0.870385i	$0.336130\pi$
$173$			52099.5i			1.74077i	−0.492372	+	0.870385i	$0.663870\pi$
$174$		0			0
$175$	3436.61			0.112216
$176$		0			0
$177$	45751.2	−	10892.8i	1.46035	−	0.347691i
$178$		0			0
$179$		−	56287.1i		−	1.75672i	−0.477998	−	0.878361i	$-0.658637\pi$
$179$		−	56287.1i		−	1.75672i	0.477998	−	0.878361i	$-0.341363\pi$
$180$		0			0
$181$	−15971.4			−0.487513			−0.243757	−	0.969836i	$-0.578380\pi$
$181$	−15971.4			−0.487513			−0.243757	+	0.969836i	$0.578380\pi$
$182$		0			0
$183$	10972.6	+	46086.5i	0.327649	+	1.37617i
$184$		0			0
$185$		−	1564.29i		−	0.0457060i
$186$		0			0
$187$	−3645.86			−0.104260
$188$		0			0
$189$	−12930.1	−	15313.6i	−0.361975	−	0.428699i
$190$		0			0
$191$		−	30478.8i		−	0.835472i	−0.908568	−	0.417736i	$-0.862824\pi$
$191$		−	30478.8i		−	0.835472i	0.908568	−	0.417736i	$-0.137176\pi$
$192$		0			0
$193$	45607.4			1.22439			0.612196	−	0.790706i	$-0.290286\pi$
$193$	45607.4			1.22439			0.612196	+	0.790706i	$0.290286\pi$
$194$		0			0
$195$	28376.0	−	6755.99i	0.746246	−	0.177672i
$196$		0			0
$197$		−	44648.3i		−	1.15046i	−0.817991	−	0.575232i	$-0.804912\pi$
$197$		−	44648.3i		−	1.15046i	0.817991	−	0.575232i	$-0.195088\pi$
$198$		0			0
$199$	44718.6			1.12923			0.564615	−	0.825354i	$-0.309025\pi$
$199$	44718.6			1.12923			0.564615	+	0.825354i	$0.309025\pi$
$200$		0			0
$201$	10883.9	+	45713.7i	0.269396	+	1.13150i
$202$		0			0
$203$		−	12042.2i		−	0.292224i
$204$		0			0
$205$	−22345.7			−0.531724
$206$		0			0
$207$	32596.9	+	64575.1i	0.760739	+	1.50704i
$208$		0			0
$209$		−	1132.85i		−	0.0259347i
$210$		0			0
$211$	49587.0			1.11379			0.556894	−	0.830583i	$-0.311993\pi$
$211$	49587.0			1.11379			0.556894	+	0.830583i	$0.311993\pi$
$212$		0			0
$213$	−51611.3	+	12288.0i	−1.13759	+	0.270846i
$214$		0			0
$215$		−	26355.3i		−	0.570153i
$216$		0			0
$217$	−35293.0			−0.749495
$218$		0			0
$219$	−1295.80	−	5442.53i	−0.0270178	−	0.113478i
$220$		0			0
$221$			36318.0i			0.743596i
$222$		0			0
$223$	−17030.3			−0.342462			−0.171231	−	0.985231i	$-0.554774\pi$
$223$	−17030.3			−0.342462			−0.171231	+	0.985231i	$0.554774\pi$
$224$		0			0
$225$	9038.69	−	4562.64i	0.178542	−	0.0901263i
$226$		0			0
$227$			90722.9i			1.76062i	0.474400	+	0.880310i	$0.342665\pi$
$227$			90722.9i			1.76062i	−0.474400	+	0.880310i	$0.657335\pi$
$228$		0			0
$229$	−4249.14			−0.0810270			−0.0405135	−	0.999179i	$-0.512899\pi$
$229$	−4249.14			−0.0810270			−0.0405135	+	0.999179i	$0.512899\pi$
$230$		0			0
$231$	7004.79	−	1667.75i	0.131272	−	0.0312542i
$232$		0			0
$233$		−	35838.9i		−	0.660150i	−0.943955	−	0.330075i	$-0.892926\pi$
$233$		−	35838.9i		−	0.660150i	0.943955	−	0.330075i	$-0.107074\pi$
$234$		0			0
$235$	−21730.9			−0.393497
$236$		0			0
$237$	−10410.6	−	43725.7i	−0.185343	−	0.778467i
$238$		0			0
$239$		−	20287.2i		−	0.355162i	−0.984106	−	0.177581i	$-0.943173\pi$
$239$		−	20287.2i		−	0.355162i	0.984106	−	0.177581i	$-0.0568272\pi$
$240$		0			0
$241$	−12132.7			−0.208893			−0.104447	−	0.994531i	$-0.533307\pi$
$241$	−12132.7			−0.208893			−0.104447	+	0.994531i	$0.533307\pi$
$242$		0			0
$243$	−54339.0	−	23109.7i	−0.920235	−	0.391366i
$244$		0			0
$245$			18393.3i			0.306427i
$246$		0			0
$247$	−11284.8			−0.184970
$248$		0			0
$249$	−33290.1	+	7925.96i	−0.536928	+	0.127836i
$250$		0			0
$251$		−	50101.2i		−	0.795245i	−0.917549	−	0.397623i	$-0.869835\pi$
$251$		−	50101.2i		−	0.795245i	0.917549	−	0.397623i	$-0.130165\pi$
$252$		0			0
$253$	−25988.1			−0.406008
$254$		0			0
$255$	2919.82	+	12263.6i	0.0449031	+	0.188599i
$256$		0			0
$257$		−	83918.2i		−	1.27054i	−0.772288	−	0.635272i	$-0.780888\pi$
$257$		−	83918.2i		−	1.27054i	0.772288	−	0.635272i	$-0.219112\pi$
$258$		0			0
$259$	−3846.64			−0.0573433
$260$		0			0
$261$	−15988.0	−	31672.6i	−0.234700	−	0.464946i
$262$		0			0
$263$			18537.8i			0.268008i	0.990981	+	0.134004i	$0.0427834\pi$
$263$			18537.8i			0.268008i	−0.990981	+	0.134004i	$0.957217\pi$
$264$		0			0
$265$	−50195.0			−0.714774
$266$		0			0
$267$	96787.2	−	23043.9i	1.35767	−	0.323246i
$268$		0			0
$269$			21684.1i			0.299666i	0.988711	+	0.149833i	$0.0478735\pi$
$269$			21684.1i			0.299666i	−0.988711	+	0.149833i	$0.952126\pi$
$270$		0			0
$271$	22125.7			0.301272			0.150636	−	0.988589i	$-0.451868\pi$
$271$	22125.7			0.301272			0.150636	+	0.988589i	$0.451868\pi$
$272$		0			0
$273$	−16613.2	−	69777.6i	−0.222909	−	0.936248i
$274$		0			0
$275$			3637.60i			0.0481006i
$276$		0			0
$277$	−87933.5			−1.14603			−0.573013	−	0.819546i	$-0.694226\pi$
$277$	−87933.5			−1.14603			−0.573013	+	0.819546i	$0.694226\pi$
$278$		0			0
$279$	−92824.8	+	46857.0i	−1.19249	+	0.601958i
$280$		0			0
$281$			127665.i			1.61681i	0.588624	+	0.808407i	$0.299670\pi$
$281$			127665.i			1.61681i	−0.588624	+	0.808407i	$0.700330\pi$
$282$		0			0
$283$	−19041.3			−0.237752			−0.118876	−	0.992909i	$-0.537929\pi$
$283$	−19041.3			−0.237752			−0.118876	+	0.992909i	$0.537929\pi$
$284$		0			0
$285$	−3810.60	+	907.257i	−0.0469141	+	0.0111697i
$286$		0			0
$287$			54948.9i			0.667107i
$288$		0			0
$289$	67825.0			0.812071
$290$		0			0
$291$	−12499.9	−	52501.3i	−0.147612	−	0.619990i
$292$		0			0
$293$		−	76347.4i		−	0.889322i	−0.895699	−	0.444661i	$-0.853324\pi$
$293$		−	76347.4i		−	0.889322i	0.895699	−	0.444661i	$-0.146676\pi$
$294$		0			0
$295$	58423.6			0.671342
$296$		0			0
$297$	16209.2	−	13686.4i	0.183759	−	0.155158i
$298$		0			0
$299$			258879.i			2.89570i
$300$		0			0
$301$	−64808.7			−0.715320
$302$		0			0
$303$	151201.	−	35999.2i	1.64691	−	0.392109i
$304$		0			0
$305$			58851.7i			0.632644i
$306$		0			0
$307$	−23870.4			−0.253269			−0.126635	−	0.991949i	$-0.540418\pi$
$307$	−23870.4			−0.253269			−0.126635	+	0.991949i	$0.540418\pi$
$308$		0			0
$309$	−7530.83	−	31630.5i	−0.0788726	−	0.331275i
$310$		0			0
$311$		−	166139.i		−	1.71771i	−0.512217	−	0.858856i	$-0.671176\pi$
$311$		−	166139.i		−	1.71771i	0.512217	−	0.858856i	$-0.328824\pi$
$312$		0			0
$313$	125604.			1.28207			0.641037	−	0.767510i	$-0.278504\pi$
$313$	125604.			1.28207			0.641037	+	0.767510i	$0.278504\pi$
$314$		0			0
$315$	−11219.7	−	22226.5i	−0.113073	−	0.224001i
$316$		0			0
$317$			24175.2i			0.240575i	0.992739	+	0.120288i	$0.0383817\pi$
$317$			24175.2i			0.240575i	−0.992739	+	0.120288i	$0.961618\pi$
$318$		0			0
$319$	12746.6			0.125260
$320$		0			0
$321$	46029.4	−	10959.0i	0.446709	−	0.106356i
$322$		0			0
$323$		−	4877.11i		−	0.0467474i
$324$		0			0
$325$	36235.7			0.343060
$326$		0			0
$327$	−12331.2	−	51792.4i	−0.115321	−	0.484363i
$328$		0			0
$329$			53437.0i			0.493686i
$330$		0			0
$331$	166718.			1.52169			0.760844	−	0.648935i	$-0.224785\pi$
$331$	166718.			1.52169			0.760844	+	0.648935i	$0.224785\pi$
$332$		0			0
$333$	−10117.1	+	5107.03i	−0.0912366	+	0.0460553i
$334$		0			0
$335$			58375.7i			0.520166i
$336$		0			0
$337$	97446.4			0.858037			0.429018	−	0.903296i	$-0.358860\pi$
$337$	97446.4			0.858037			0.429018	+	0.903296i	$0.358860\pi$
$338$		0			0
$339$	41577.9	−	9899.19i	0.361795	−	0.0861391i
$340$		0			0
$341$		−	37357.2i		−	0.321266i
$342$		0			0
$343$	111240.			0.945525
$344$		0			0
$345$	20812.9	+	87416.7i	0.174861	+	0.734440i
$346$		0			0
$347$			26212.4i			0.217695i	0.994058	+	0.108848i	$0.0347160\pi$
$347$			26212.4i			0.217695i	−0.994058	+	0.108848i	$0.965284\pi$
$348$		0			0
$349$	−93634.0			−0.768746			−0.384373	−	0.923178i	$-0.625582\pi$
$349$	−93634.0			−0.768746			−0.384373	+	0.923178i	$0.625582\pi$
$350$		0			0
$351$	−136336.	−	161467.i	−1.10661	−	1.31060i
$352$		0			0
$353$			11144.8i			0.0894383i	0.999000	+	0.0447192i	$0.0142393\pi$
$353$			11144.8i			0.0894383i	−0.999000	+	0.0447192i	$0.985761\pi$
$354$		0			0
$355$	−65906.8			−0.522966
$356$		0			0
$357$	30156.7	−	7179.95i	0.236618	−	0.0563359i
$358$		0			0
$359$			44607.2i			0.346111i	0.984912	+	0.173056i	$0.0553641\pi$
$359$			44607.2i			0.346111i	−0.984912	+	0.173056i	$0.944636\pi$
$360$		0			0
$361$	−128806.			−0.988372
$362$		0			0
$363$	−28754.2	−	120771.i	−0.218217	−	0.916539i
$364$		0			0
$365$		−	6950.02i		−	0.0521676i
$366$		0			0
$367$	−108740.			−0.807341			−0.403670	−	0.914905i	$-0.632266\pi$
$367$	−108740.			−0.807341			−0.403670	+	0.914905i	$0.632266\pi$
$368$		0			0
$369$	72953.4	+	144522.i	0.535788	+	1.06141i
$370$		0			0
$371$			123431.i			0.896763i
$372$		0			0
$373$	76177.7			0.547533			0.273766	−	0.961796i	$-0.411730\pi$
$373$	76177.7			0.547533			0.273766	+	0.961796i	$0.411730\pi$
$374$		0			0
$375$	12235.9	−	2913.21i	0.0870106	−	0.0207162i
$376$		0			0
$377$		−	126974.i		−	0.893371i
$378$		0			0
$379$	−112049.			−0.780064			−0.390032	−	0.920801i	$-0.627536\pi$
$379$	−112049.			−0.780064			−0.390032	+	0.920801i	$0.627536\pi$
$380$		0			0
$381$	54782.9	+	230095.i	0.377394	+	1.58510i
$382$		0			0
$383$		−	121863.i		−	0.830760i	−0.909648	−	0.415380i	$-0.863649\pi$
$383$		−	121863.i		−	0.830760i	0.909648	−	0.415380i	$-0.136351\pi$
$384$		0			0
$385$	8945.00			0.0603474
$386$		0			0
$387$	−170455.	+	86043.9i	−1.13812	+	0.574511i
$388$		0			0
$389$		−	47883.3i		−	0.316435i	−0.987404	−	0.158218i	$-0.949425\pi$
$389$		−	47883.3i		−	0.316435i	0.987404	−	0.158218i	$-0.0505747\pi$
$390$		0			0
$391$	−111883.			−0.731831
$392$		0			0
$393$	−116164.	+	27657.3i	−0.752119	+	0.179070i
$394$		0			0
$395$		−	55837.0i		−	0.357872i
$396$		0			0
$397$	139374.			0.884299			0.442150	−	0.896941i	$-0.354216\pi$
$397$	139374.			0.884299			0.442150	+	0.896941i	$0.354216\pi$
$398$		0			0
$399$	2230.98	+	9370.39i	0.0140136	+	0.0588589i
$400$		0			0
$401$		−	147304.i		−	0.916067i	−0.888935	−	0.458034i	$-0.848554\pi$
$401$		−	147304.i		−	0.916067i	0.888935	−	0.458034i	$-0.151446\pi$
$402$		0			0
$403$	−372130.			−2.29132
$404$		0			0
$405$	−59018.3	−	43562.3i	−0.359813	−	0.265583i
$406$		0			0
$407$		−	4071.62i		−	0.0245798i
$408$		0			0
$409$	−157604.			−0.942149			−0.471074	−	0.882094i	$-0.656134\pi$
$409$	−157604.			−0.942149			−0.471074	+	0.882094i	$0.656134\pi$
$410$		0			0
$411$	79080.3	−	18828.1i	0.468150	−	0.111461i
$412$		0			0
$413$		−	143666.i		−	0.842273i
$414$		0			0
$415$	−42510.9			−0.246833
$416$		0			0
$417$	−5113.79	−	21478.6i	−0.0294083	−	0.123519i
$418$		0			0
$419$		−	59717.3i		−	0.340151i	−0.985431	−	0.170076i	$-0.945599\pi$
$419$		−	59717.3i		−	0.340151i	0.985431	−	0.170076i	$-0.0544012\pi$
$420$		0			0
$421$	198739.			1.12129			0.560645	−	0.828056i	$-0.310553\pi$
$421$	198739.			1.12129			0.560645	+	0.828056i	$0.310553\pi$
$422$		0			0
$423$	70946.2	+	140546.i	0.396505	+	0.785484i
$424$		0			0
$425$			15660.5i			0.0867015i
$426$		0			0
$427$	144718.			0.793722
$428$		0			0
$429$	73858.7	−	17584.9i	0.401317	−	0.0955486i
$430$		0			0
$431$		−	57651.2i		−	0.310351i	−0.987887	−	0.155176i	$-0.950406\pi$
$431$		−	57651.2i		−	0.310351i	0.987887	−	0.155176i	$-0.0495944\pi$
$432$		0			0
$433$	−98353.4			−0.524582			−0.262291	−	0.964989i	$-0.584478\pi$
$433$	−98353.4			−0.524582			−0.262291	+	0.964989i	$0.584478\pi$
$434$		0			0
$435$	−10208.2	−	42875.8i	−0.0539475	−	0.226586i
$436$		0			0
$437$		−	34764.7i		−	0.182043i
$438$		0			0
$439$	112548.			0.583997			0.291999	−	0.956419i	$-0.405680\pi$
$439$	112548.			0.583997			0.291999	+	0.956419i	$0.405680\pi$
$440$		0			0
$441$	118959.	−	60049.6i	0.611677	−	0.308768i
$442$		0			0
$443$			170413.i			0.868349i	0.900829	+	0.434174i	$0.142960\pi$
$443$			170413.i			0.868349i	−0.900829	+	0.434174i	$0.857040\pi$
$444$		0			0
$445$	123596.			0.624142
$446$		0			0
$447$	−331994.	+	79043.8i	−1.66156	+	0.395597i
$448$		0			0
$449$		−	212282.i		−	1.05298i	−0.850181	−	0.526491i	$-0.823507\pi$
$449$		−	212282.i		−	1.05298i	0.850181	−	0.526491i	$-0.176493\pi$
$450$		0			0
$451$	−58162.7			−0.285951
$452$		0			0
$453$	50723.9	+	213047.i	0.247182	+	1.03820i
$454$		0			0
$455$		−	89104.9i		−	0.430407i
$456$		0			0
$457$	207763.			0.994798			0.497399	−	0.867522i	$-0.334288\pi$
$457$	207763.			0.994798			0.497399	+	0.867522i	$0.334288\pi$
$458$		0			0
$459$	69783.3	−	58922.0i	0.331227	−	0.279674i
$460$		0			0
$461$			156089.i			0.734463i	0.930130	+	0.367231i	$0.119694\pi$
$461$			156089.i			0.734463i	−0.930130	+	0.367231i	$0.880306\pi$
$462$		0			0
$463$	−99520.2			−0.464247			−0.232124	−	0.972686i	$-0.574567\pi$
$463$	−99520.2			−0.464247			−0.232124	+	0.972686i	$0.574567\pi$
$464$		0			0
$465$	−125659.	+	29917.8i	−0.581148	+	0.138364i
$466$		0			0
$467$			28473.8i			0.130560i	0.997867	+	0.0652802i	$0.0207941\pi$
$467$			28473.8i			0.130560i	−0.997867	+	0.0652802i	$0.979206\pi$
$468$		0			0
$469$	143548.			0.652606
$470$		0			0
$471$	87572.7	+	367817.i	0.394754	+	1.65802i
$472$		0			0
$473$		−	68599.2i		−	0.306617i
$474$		0			0
$475$	−4866.07			−0.0215671
$476$		0			0
$477$	163875.	+	324639.i	0.720237	+	1.42680i
$478$		0			0
$479$		−	216408.i		−	0.943197i	−0.881813	−	0.471599i	$-0.843677\pi$
$479$		−	216408.i		−	0.943197i	0.881813	−	0.471599i	$-0.156323\pi$
$480$		0			0
$481$	−40559.1			−0.175307
$482$		0			0
$483$	214961.	−	51179.6i	0.921435	−	0.219383i
$484$		0			0
$485$		−	67043.4i		−	0.285018i
$486$		0			0
$487$	221414.			0.933572			0.466786	−	0.884370i	$-0.345412\pi$
$487$	221414.			0.933572			0.466786	+	0.884370i	$0.345412\pi$
$488$		0			0
$489$	−6978.25	−	29309.6i	−0.0291829	−	0.122572i
$490$		0			0
$491$			212401.i			0.881035i	0.897744	+	0.440517i	$0.145205\pi$
$491$			212401.i			0.881035i	−0.897744	+	0.440517i	$0.854795\pi$
$492$		0			0
$493$	54876.0			0.225782
$494$		0			0
$495$	23526.4	−	11875.9i	0.0960164	−	0.0484682i
$496$		0			0
$497$			162067.i			0.656118i
$498$		0			0
$499$	114850.			0.461243			0.230621	−	0.973044i	$-0.425924\pi$
$499$	114850.			0.461243			0.230621	+	0.973044i	$0.425924\pi$
$500$		0			0
$501$	−114537.	+	27269.8i	−0.456320	+	0.108644i
$502$		0			0
$503$			335733.i			1.32696i	0.748194	+	0.663480i	$0.230921\pi$
$503$			335733.i			1.32696i	−0.748194	+	0.663480i	$0.769079\pi$
$504$		0			0
$505$	193081.			0.757108
$506$		0			0
$507$	−115634.	−	485679.i	−0.449853	−	1.88944i
$508$		0			0
$509$		−	471469.i		−	1.81978i	−0.414854	−	0.909888i	$-0.636167\pi$
$509$		−	471469.i		−	1.81978i	0.414854	−	0.909888i	$-0.363833\pi$
$510$		0			0
$511$	−17090.4			−0.0654499
$512$		0			0
$513$	18308.4	+	21683.3i	0.0695691	+	0.0823930i
$514$		0			0
$515$		−	40391.6i		−	0.152292i
$516$		0			0
$517$	−56562.4			−0.211615
$518$		0			0
$519$	456145.	−	108603.i	1.69343	−	0.403186i
$520$		0			0
$521$		−	172558.i		−	0.635711i	−0.948139	−	0.317855i	$-0.897037\pi$
$521$		−	172558.i		−	0.635711i	0.948139	−	0.317855i	$-0.102963\pi$
$522$		0			0
$523$	164847.			0.602669			0.301334	−	0.953519i	$-0.402568\pi$
$523$	164847.			0.602669			0.301334	+	0.953519i	$0.402568\pi$
$524$		0			0
$525$	−7163.69	−	30088.4i	−0.0259907	−	0.109164i
$526$		0			0
$527$		−	160828.i		−	0.579084i
$528$		0			0
$529$	−517675.			−1.84989
$530$		0			0
$531$	−190739.	−	377858.i	−0.676473	−	1.34011i
$532$		0			0
$533$			579383.i			2.03944i
$534$		0			0
$535$	58778.7			0.205359
$536$		0			0
$537$	−492809.	+	117332.i	−1.70895	+	0.406881i
$538$		0			0
$539$			47875.0i			0.164790i
$540$		0			0
$541$	−324235.			−1.10781			−0.553905	−	0.832580i	$-0.686863\pi$
$541$	−324235.			−1.10781			−0.553905	+	0.832580i	$0.686863\pi$
$542$		0			0
$543$	33292.8	+	139834.i	0.112915	+	0.474257i
$544$		0			0
$545$		−	66138.1i		−	0.222668i
$546$		0			0
$547$	236516.			0.790471			0.395235	−	0.918580i	$-0.370663\pi$
$547$	236516.			0.790471			0.395235	+	0.918580i	$0.370663\pi$
$548$		0			0
$549$	380627.	−	192137.i	1.26286	−	0.637479i
$550$		0			0
$551$			17051.2i			0.0561633i
$552$		0			0
$553$	−137305.			−0.448990
$554$		0			0
$555$	−13695.8	+	3260.80i	−0.0444632	+	0.0105861i
$556$		0			0
$557$		−	29656.8i		−	0.0955903i	−0.998857	−	0.0477951i	$-0.984781\pi$
$557$		−	29656.8i		−	0.0955903i	0.998857	−	0.0477951i	$-0.0152195\pi$
$558$		0			0
$559$	−683345.			−2.18684
$560$		0			0
$561$	7599.88	+	31920.5i	0.0241480	+	0.101425i
$562$		0			0
$563$			116004.i			0.365979i	0.983115	+	0.182989i	$0.0585774\pi$
$563$			116004.i			0.365979i	−0.983115	+	0.182989i	$0.941423\pi$
$564$		0			0
$565$	53094.3			0.166322
$566$		0			0
$567$	−107121.	+	145128.i	−0.333204	+	0.451425i
$568$		0			0
$569$			451655.i			1.39503i	0.716572	+	0.697513i	$0.245710\pi$
$569$			451655.i			1.39503i	−0.716572	+	0.697513i	$0.754290\pi$
$570$		0			0
$571$	322102.			0.987920			0.493960	−	0.869485i	$-0.335549\pi$
$571$	322102.			0.987920			0.493960	+	0.869485i	$0.335549\pi$
$572$		0			0
$573$	−266851.	+	63533.9i	−0.812754	+	0.193507i
$574$		0			0
$575$			111630.i			0.337632i
$576$		0			0
$577$	−416558.			−1.25119			−0.625597	−	0.780147i	$-0.715144\pi$
$577$	−416558.			−1.25119			−0.625597	+	0.780147i	$0.715144\pi$
$578$		0			0
$579$	−95069.7	−	399305.i	−0.283586	−	1.19110i
$580$		0			0
$581$			104536.i			0.309680i
$582$		0			0
$583$	−130650.			−0.384391
$584$		0			0
$585$	−118301.	−	234357.i	−0.345682	−	0.684803i
$586$		0			0
$587$			233289.i			0.677045i	0.940958	+	0.338523i	$0.109927\pi$
$587$			233289.i			0.677045i	−0.940958	+	0.338523i	$0.890073\pi$
$588$		0			0
$589$	49973.1			0.144048
$590$		0			0
$591$	−390908.	+	93070.5i	−1.11918	+	0.266463i
$592$		0			0
$593$		−	286788.i		−	0.815551i	−0.913082	−	0.407776i	$-0.866305\pi$
$593$		−	286788.i		−	0.815551i	0.913082	−	0.407776i	$-0.133695\pi$
$594$		0			0
$595$	38509.6			0.108777
$596$		0			0
$597$	−93217.1	−	391524.i	−0.261545	−	1.09852i
$598$		0			0
$599$		−	173144.i		−	0.482562i	−0.970455	−	0.241281i	$-0.922432\pi$
$599$		−	173144.i		−	0.482562i	0.970455	−	0.241281i	$-0.0775675\pi$
$600$		0			0
$601$	−543513.			−1.50474			−0.752369	−	0.658742i	$-0.771089\pi$
$601$	−543513.			−1.50474			−0.752369	+	0.658742i	$0.771089\pi$
$602$		0			0
$603$	377548.	−	190583.i	1.03834	−	0.524142i
$604$		0			0
$605$		−	154223.i		−	0.421346i
$606$		0			0
$607$	435603.			1.18226			0.591131	−	0.806576i	$-0.298682\pi$
$607$	435603.			1.18226			0.591131	+	0.806576i	$0.298682\pi$
$608$		0			0
$609$	−105433.	+	25102.4i	−0.284278	+	0.0676830i
$610$		0			0
$611$			563442.i			1.50927i
$612$		0			0
$613$	179748.			0.478346			0.239173	−	0.970977i	$-0.423124\pi$
$613$	179748.			0.478346			0.239173	+	0.970977i	$0.423124\pi$
$614$		0			0
$615$	46580.2	+	195643.i	0.123155	+	0.517266i
$616$		0			0
$617$			199181.i			0.523213i	0.965175	+	0.261606i	$0.0842522\pi$
$617$			199181.i			0.523213i	−0.965175	+	0.261606i	$0.915748\pi$
$618$		0			0
$619$	371048.			0.968386			0.484193	−	0.874961i	$-0.339113\pi$
$619$	371048.			0.968386			0.484193	+	0.874961i	$0.339113\pi$
$620$		0			0
$621$	497424.	−	420003.i	1.28986	−	1.08910i
$622$		0			0
$623$		−	303926.i		−	0.783055i
$624$		0			0
$625$	15625.0			0.0400000
$626$		0			0
$627$	−9918.44	+	2361.46i	−0.0252295	+	0.00600683i
$628$		0			0
$629$		−	17529.0i		−	0.0443053i
$630$		0			0
$631$	485164.			1.21851			0.609257	−	0.792973i	$-0.291468\pi$
$631$	485164.			1.21851			0.609257	+	0.792973i	$0.291468\pi$
$632$		0			0
$633$	−103365.	−	434148.i	−0.257969	−	1.08350i
$634$		0			0
$635$			293828.i			0.728695i
$636$		0			0
$637$	476903.			1.17531
$638$		0			0
$639$	215170.	+	426256.i	0.526963	+	1.04392i
$640$		0			0
$641$		−	419685.i		−	1.02143i	−0.859751	−	0.510713i	$-0.829381\pi$
$641$		−	419685.i		−	1.02143i	0.859751	−	0.510713i	$-0.170619\pi$
$642$		0			0
$643$	−755857.			−1.82818			−0.914088	−	0.405517i	$-0.867092\pi$
$643$	−755857.			−1.82818			−0.914088	+	0.405517i	$0.867092\pi$
$644$		0			0
$645$	−230748.	+	54938.3i	−0.554650	+	0.132055i
$646$		0			0
$647$		−	183576.i		−	0.438537i	−0.975665	−	0.219268i	$-0.929633\pi$
$647$		−	183576.i		−	0.438537i	0.975665	−	0.219268i	$-0.0703671\pi$
$648$		0			0
$649$	152068.			0.361035
$650$		0			0
$651$	73569.0	+	308999.i	0.173593	+	0.729115i
$652$		0			0
$653$			464810.i			1.09006i	0.838417	+	0.545029i	$0.183481\pi$
$653$			464810.i			1.09006i	−0.838417	+	0.545029i	$0.816519\pi$
$654$		0			0
$655$	−148340.			−0.345760
$656$		0			0
$657$	−44949.7	+	22690.2i	−0.104135	+	0.0525662i
$658$		0			0
$659$		−	5523.27i		−	0.0127182i	−0.999980	−	0.00635909i	$-0.997976\pi$
$659$		−	5523.27i		−	0.0127182i	0.999980	−	0.00635909i	$-0.00202418\pi$
$660$		0			0
$661$	278448.			0.637296			0.318648	−	0.947873i	$-0.396771\pi$
$661$	278448.			0.637296			0.318648	+	0.947873i	$0.396771\pi$
$662$		0			0
$663$	317974.	−	75705.6i	0.723376	−	0.172227i
$664$		0			0
$665$			11965.8i			0.0270583i
$666$		0			0
$667$	391163.			0.879237
$668$		0			0
$669$	35500.0	+	149105.i	0.0793189	+	0.333149i
$670$		0			0
$671$			153183.i			0.340224i
$672$		0			0
$673$	138682.			0.306188			0.153094	−	0.988212i	$-0.451076\pi$
$673$	138682.			0.306188			0.153094	+	0.988212i	$0.451076\pi$
$674$		0			0
$675$	−58788.5	−	69625.2i	−0.129028	−	0.152813i
$676$		0			0
$677$			445028.i			0.970979i	0.874242	+	0.485490i	$0.161359\pi$
$677$			445028.i			0.970979i	−0.874242	+	0.485490i	$0.838641\pi$
$678$		0			0
$679$	−164862.			−0.357587
$680$		0			0
$681$	794304.	−	189114.i	1.71274	−	0.407784i
$682$		0			0
$683$		−	349201.i		−	0.748573i	−0.927313	−	0.374286i	$-0.877888\pi$
$683$		−	349201.i		−	0.748573i	0.927313	−	0.374286i	$-0.122112\pi$
$684$		0			0
$685$	100984.			0.215215
$686$		0			0
$687$	8857.43	+	37202.4i	0.0187670	+	0.0788237i
$688$		0			0
$689$			1.30146e6i			2.74153i
$690$		0			0
$691$	3751.17			0.00785616			0.00392808	−	0.999992i	$-0.498750\pi$
$691$	3751.17			0.00785616			0.00392808	+	0.999992i	$0.498750\pi$
$692$		0			0
$693$	−29203.3	−	57852.3i	−0.0608086	−	0.120463i
$694$		0			0
$695$		−	27427.8i		−	0.0567834i
$696$		0			0
$697$	−250400.			−0.515428
$698$		0			0
$699$	−313779.	+	74707.0i	−0.642199	+	0.152900i
$700$		0			0
$701$		−	371555.i		−	0.756114i	−0.925782	−	0.378057i	$-0.876592\pi$
$701$		−	371555.i		−	0.756114i	0.925782	−	0.378057i	$-0.123408\pi$
$702$		0			0
$703$	5446.66			0.0110210
$704$		0			0
$705$	45298.6	+	190260.i	0.0911394	+	0.382797i
$706$		0			0
$707$		−	474794.i		−	0.949875i
$708$		0			0
$709$	73752.6			0.146718			0.0733592	−	0.997306i	$-0.476628\pi$
$709$	73752.6			0.146718			0.0733592	+	0.997306i	$0.476628\pi$
$710$		0			0
$711$	−361129.	+	182295.i	−0.714370	+	0.360607i
$712$		0			0
$713$		−	1.14641e6i		−	2.25506i
$714$		0			0
$715$	94316.4			0.184491
$716$		0			0
$717$	−177620.	+	42289.2i	−0.345504	+	0.0822604i
$718$		0			0
$719$			145211.i			0.280894i	0.990088	+	0.140447i	$0.0448540\pi$
$719$			145211.i			0.280894i	−0.990088	+	0.140447i	$0.955146\pi$
$720$		0			0
$721$	−99324.4			−0.191067
$722$		0			0
$723$	25290.9	+	106225.i	0.0483825	+	0.203213i
$724$		0			0
$725$		−	54751.7i		−	0.104165i
$726$		0			0
$727$	306651.			0.580197			0.290099	−	0.956997i	$-0.406312\pi$
$727$	306651.			0.580197			0.290099	+	0.956997i	$0.406312\pi$
$728$		0			0
$729$	−89061.2	+	523925.i	−0.167584	+	0.985858i
$730$		0			0
$731$		−	295330.i		−	0.552680i
$732$		0			0
$733$	462843.			0.861441			0.430720	−	0.902485i	$-0.358259\pi$
$733$	462843.			0.861441			0.430720	+	0.902485i	$0.358259\pi$
$734$		0			0
$735$	161038.	−	38341.2i	0.298094	−	0.0709726i
$736$		0			0
$737$			151944.i			0.279735i
$738$		0			0
$739$	−506649.			−0.927723			−0.463861	−	0.885908i	$-0.653536\pi$
$739$	−506649.			−0.927723			−0.463861	+	0.885908i	$0.653536\pi$
$740$		0			0
$741$	23523.5	+	98801.8i	0.0428416	+	0.179940i
$742$		0			0
$743$		−	950761.i		−	1.72224i	−0.508402	−	0.861120i	$-0.669764\pi$
$743$		−	950761.i		−	1.72224i	0.508402	−	0.861120i	$-0.330236\pi$
$744$		0			0
$745$	−423951.			−0.763842
$746$		0			0
$747$	138788.	+	274942.i	0.248720	+	0.492719i
$748$		0			0
$749$		−	144539.i		−	0.257645i
$750$		0			0
$751$	−132721.			−0.235320			−0.117660	−	0.993054i	$-0.537539\pi$
$751$	−132721.			−0.235320			−0.117660	+	0.993054i	$0.537539\pi$
$752$		0			0
$753$	−438650.	+	104437.i	−0.773621	+	0.184190i
$754$		0			0
$755$			272058.i			0.477273i
$756$		0			0
$757$	−604889.			−1.05556			−0.527781	−	0.849381i	$-0.676976\pi$
$757$	−604889.			−1.05556			−0.527781	+	0.849381i	$0.676976\pi$
$758$		0			0
$759$	54172.9	+	227533.i	0.0940369	+	0.394967i
$760$		0			0
$761$		−	632649.i		−	1.09243i	−0.837645	−	0.546215i	$-0.816068\pi$
$761$		−	632649.i		−	1.09243i	0.837645	−	0.546215i	$-0.183932\pi$
$762$		0			0
$763$	−162636.			−0.279362
$764$		0			0
$765$	101285.	−	51127.7i	0.173070	−	0.0873641i
$766$		0			0
$767$		−	1.51482e6i		−	2.57495i
$768$		0			0
$769$	−223721.			−0.378316			−0.189158	−	0.981947i	$-0.560576\pi$
$769$	−223721.			−0.378316			−0.189158	+	0.981947i	$0.560576\pi$
$770$		0			0
$771$	−734727.	+	174930.i	−1.23600	+	0.294276i
$772$		0			0
$773$			683006.i			1.14305i	0.820585	+	0.571525i	$0.193648\pi$
$773$			683006.i			1.14305i	−0.820585	+	0.571525i	$0.806352\pi$
$774$		0			0
$775$	−160464.			−0.267162
$776$		0			0
$777$	8018.42	+	33678.4i	0.0132815	+	0.0557840i
$778$		0			0
$779$		−	77805.0i		−	0.128213i
$780$		0			0
$781$	−171546.			−0.281241
$782$		0			0
$783$	−243974.	+	206001.i	−0.397943	+	0.336006i
$784$		0			0
$785$			469696.i			0.762215i
$786$		0			0
$787$	−833709.			−1.34606			−0.673031	−	0.739615i	$-0.735008\pi$
$787$	−833709.			−1.34606			−0.673031	+	0.739615i	$0.735008\pi$
$788$		0			0
$789$	162304.	−	38642.5i	0.260720	−	0.0620743i
$790$		0			0
$791$		−	130561.i		−	0.208670i
$792$		0			0
$793$	1.52592e6			2.42652
$794$		0			0
$795$	104633.	+	439471.i	0.165551	+	0.695338i
$796$		0			0
$797$			356986.i			0.561998i	0.959708	+	0.280999i	$0.0906658\pi$
$797$			356986.i			0.561998i	−0.959708	+	0.280999i	$0.909334\pi$
$798$		0			0
$799$	−243510.			−0.381438
$800$		0			0
$801$	−403511.	−	799363.i	−0.628912	−	1.24589i
$802$		0			0
$803$		−	18089.9i		−	0.0280547i
$804$		0			0
$805$	274501.			0.423597
$806$		0			0
$807$	189850.	−	45201.0i	0.291517	−	0.0694067i
$808$		0			0
$809$			440631.i			0.673252i	0.941638	+	0.336626i	$0.109286\pi$
$809$			440631.i			0.673252i	−0.941638	+	0.336626i	$0.890714\pi$
$810$		0			0
$811$	−835112.			−1.26970			−0.634852	−	0.772633i	$-0.718939\pi$
$811$	−835112.			−1.26970			−0.634852	+	0.772633i	$0.718939\pi$
$812$		0			0
$813$	−46121.6	−	193717.i	−0.0697787	−	0.293080i
$814$		0			0
$815$		−	37427.9i		−	0.0563482i
$816$		0			0
$817$	91766.0			0.137479
$818$		0			0
$819$	−576291.	+	290906.i	−0.859160	+	0.433696i
$820$		0			0
$821$			33593.3i			0.0498386i	0.999689	+	0.0249193i	$0.00793288\pi$
$821$			33593.3i			0.0498386i	−0.999689	+	0.0249193i	$0.992067\pi$
$822$		0			0
$823$	−627467.			−0.926384			−0.463192	−	0.886258i	$-0.653296\pi$
$823$	−627467.			−0.926384			−0.463192	+	0.886258i	$0.653296\pi$
$824$		0			0
$825$	31848.2	−	7582.67i	0.0467926	−	0.0111408i
$826$		0			0
$827$			629592.i			0.920552i	0.887776	+	0.460276i	$0.152250\pi$
$827$			629592.i			0.920552i	−0.887776	+	0.460276i	$0.847750\pi$
$828$		0			0
$829$	146264.			0.212827			0.106414	−	0.994322i	$-0.466063\pi$
$829$	146264.			0.212827			0.106414	+	0.994322i	$0.466063\pi$
$830$		0			0
$831$	183299.	+	769881.i	0.265435	+	1.11486i
$832$		0			0
$833$			206110.i			0.297035i
$834$		0			0
$835$	−146262.			−0.209777
$836$		0			0
$837$	603741.	+	715031.i	0.861787	+	1.02064i
$838$		0			0
$839$		−	899007.i		−	1.27714i	−0.769563	−	0.638571i	$-0.779526\pi$
$839$		−	899007.i		−	1.27714i	0.769563	−	0.638571i	$-0.220474\pi$
$840$		0			0
$841$	515425.			0.728741
$842$		0			0
$843$	1.11774e6	−	266121.i	1.57285	−	0.374477i
$844$		0			0
$845$		−	620204.i		−	0.868602i
$846$		0			0
$847$	−379240.			−0.528625
$848$		0			0
$849$	39692.1	+	166712.i	0.0550667	+	0.231287i
$850$		0			0
$851$		−	124949.i		−	0.172533i
$852$		0			0
$853$	−222756.			−0.306148			−0.153074	−	0.988215i	$-0.548917\pi$
$853$	−222756.			−0.306148			−0.153074	+	0.988215i	$0.548917\pi$
$854$		0			0
$855$	15886.6	+	31471.6i	0.0217319	+	0.0430513i
$856$		0			0
$857$		−	538201.i		−	0.732795i	−0.930458	−	0.366397i	$-0.880591\pi$
$857$		−	538201.i		−	0.732795i	0.930458	−	0.366397i	$-0.119409\pi$
$858$		0			0
$859$	1.22688e6			1.66270			0.831351	−	0.555748i	$-0.187568\pi$
$859$	1.22688e6			1.66270			0.831351	+	0.555748i	$0.187568\pi$
$860$		0			0
$861$	481093.	−	114542.i	0.648967	−	0.154511i
$862$		0			0
$863$		−	283743.i		−	0.380982i	−0.981689	−	0.190491i	$-0.938992\pi$
$863$		−	283743.i		−	0.380982i	0.981689	−	0.190491i	$-0.0610079\pi$
$864$		0			0
$865$	582490.			0.778496
$866$		0			0
$867$	−141383.	−	593826.i	−0.188087	−	0.789989i
$868$		0			0
$869$		−	145336.i		−	0.192457i
$870$		0			0
$871$	1.51357e6			1.99511
$872$		0			0
$873$	−433607.	+	218881.i	−0.568942	+	0.287196i
$874$		0			0
$875$		−	38422.4i		−	0.0501844i
$876$		0			0
$877$	−447998.			−0.582475			−0.291237	−	0.956651i	$-0.594067\pi$
$877$	−447998.			−0.582475			−0.291237	+	0.956651i	$0.594067\pi$
$878$		0			0
$879$	−668442.	+	159148.i	−0.865139	+	0.205979i
$880$		0			0
$881$			314446.i			0.405130i	0.979269	+	0.202565i	$0.0649277\pi$
$881$			314446.i			0.405130i	−0.979269	+	0.202565i	$0.935072\pi$
$882$		0			0
$883$	839627.			1.07687			0.538437	−	0.842666i	$-0.319015\pi$
$883$	839627.			1.07687			0.538437	+	0.842666i	$0.319015\pi$
$884$		0			0
$885$	−121785.	−	511514.i	−0.155492	−	0.653087i
$886$		0			0
$887$			822318.i			1.04518i	0.852583	+	0.522592i	$0.175035\pi$
$887$			822318.i			1.04518i	−0.852583	+	0.522592i	$0.824965\pi$
$888$		0			0
$889$	722533.			0.914228
$890$		0			0
$891$	−153616.	−	113387.i	−0.193500	−	0.142826i
$892$		0			0
$893$		−	75664.3i		−	0.0948829i
$894$		0			0
$895$	−629309.			−0.785630
$896$		0			0
$897$	2.26655e6	−	539639.i	2.81696	−	0.670685i
$898$		0			0
$899$			562284.i			0.695724i
$900$		0			0
$901$	−562471.			−0.692868
$902$		0			0
$903$	135095.	+	567418.i	0.165678	+	0.695869i
$904$		0			0
$905$			178566.i			0.218023i
$906$		0			0
$907$	−299076.			−0.363552			−0.181776	−	0.983340i	$-0.558185\pi$
$907$	−299076.			−0.363552			−0.181776	+	0.983340i	$0.558185\pi$
$908$		0			0
$909$	−630365.	−	1.24877e6i	−0.762894	−	1.51131i
$910$		0			0
$911$			169371.i			0.204081i	0.994780	+	0.102040i	$0.0325371\pi$
$911$			169371.i			0.204081i	−0.994780	+	0.102040i	$0.967463\pi$
$912$		0			0
$913$	−110650.			−0.132742
$914$		0			0
$915$	515263.	−	122678.i	0.615441	−	0.146529i
$916$		0			0
$917$			364772.i			0.433794i
$918$		0			0
$919$	1.47632e6			1.74803			0.874016	−	0.485897i	$-0.161507\pi$
$919$	1.47632e6			1.74803			0.874016	+	0.485897i	$0.161507\pi$
$920$		0			0
$921$	49758.3	+	208991.i	0.0586606	+	0.246382i
$922$		0			0
$923$			1.70884e6i			2.00585i
$924$		0			0
$925$	−17489.3			−0.0204404
$926$		0			0
$927$	−261235.	+	131869.i	−0.303999	+	0.153456i
$928$		0			0
$929$		−	109784.i		−	0.127205i	−0.997975	−	0.0636027i	$-0.979741\pi$
$929$		−	109784.i		−	0.127205i	0.997975	−	0.0636027i	$-0.0202591\pi$
$930$		0			0
$931$	−64043.0			−0.0738878
$932$		0			0
$933$	−1.45459e6	+	346320.i	−1.67100	+	0.397846i
$934$		0			0
$935$			40762.0i			0.0466264i
$936$		0			0
$937$	−504858.			−0.575029			−0.287515	−	0.957776i	$-0.592829\pi$
$937$	−504858.			−0.575029			−0.287515	+	0.957776i	$0.592829\pi$
$938$		0			0
$939$	−261824.	−	1.09969e6i	−0.296946	−	1.24721i
$940$		0			0
$941$			1.22748e6i			1.38623i	0.720825	+	0.693117i	$0.243763\pi$
$941$			1.22748e6i			1.38623i	−0.720825	+	0.693117i	$0.756237\pi$
$942$		0			0
$943$	−1.78488e6			−2.00718
$944$		0			0
$945$	−171211.	+	144563.i	−0.191720	+	0.161880i
$946$		0			0
$947$		−	890867.i		−	0.993374i	−0.867930	−	0.496687i	$-0.834550\pi$
$947$		−	890867.i		−	0.993374i	0.867930	−	0.496687i	$-0.165450\pi$
$948$		0			0
$949$	−180201.			−0.200090
$950$		0			0
$951$	211660.	−	50393.7i	0.234033	−	0.0557205i
$952$		0			0
$953$			429387.i			0.472784i	0.971658	+	0.236392i	$0.0759650\pi$
$953$			429387.i			0.472784i	−0.971658	+	0.236392i	$0.924035\pi$
$954$		0			0
$955$	−340764.			−0.373634
$956$		0			0
$957$	−26570.5	−	111600.i	−0.0290119	−	0.121854i
$958$		0			0
$959$		−	248324.i		−	0.270011i
$960$		0			0
$961$	724401.			0.784391
$962$		0			0
$963$	−191899.	−	380155.i	−0.206928	−	0.409929i
$964$		0			0
$965$		−	509906.i		−	0.547565i
$966$		0			0
$967$	1.05994e6			1.13352			0.566759	−	0.823884i	$-0.308197\pi$
$967$	1.05994e6			1.13352			0.566759	+	0.823884i	$0.308197\pi$
$968$		0			0
$969$	−42700.5	+	10166.5i	−0.0454763	+	0.0108274i
$970$		0			0
$971$			1.04477e6i			1.10811i	0.832479	+	0.554056i	$0.186921\pi$
$971$			1.04477e6i			1.10811i	−0.832479	+	0.554056i	$0.813079\pi$
$972$		0			0
$973$	−67445.9			−0.0712410
$974$		0			0
$975$	−75534.2	−	317253.i	−0.0794574	−	0.333731i
$976$		0			0
$977$			1.55786e6i			1.63207i	0.578001	+	0.816036i	$0.303833\pi$
$977$			1.55786e6i			1.63207i	−0.578001	+	0.816036i	$0.696167\pi$
$978$		0			0
$979$	321702.			0.335651
$980$		0			0
$981$	−427752.	+	215925.i	−0.444482	+	0.224370i
$982$		0			0
$983$		−	20248.8i		−	0.0209552i	−0.999945	−	0.0104776i	$-0.996665\pi$
$983$		−	20248.8i		−	0.0209552i	0.999945	−	0.0104776i	$-0.00333519\pi$
$984$		0			0
$985$	−499184.			−0.514503
$986$		0			0
$987$	467856.	−	111391.i	0.480261	−	0.114344i
$988$		0			0
$989$		−	2.10515e6i		−	2.15224i
$990$		0			0
$991$	−522200.			−0.531728			−0.265864	−	0.964011i	$-0.585657\pi$
$991$	−522200.			−0.531728			−0.265864	+	0.964011i	$0.585657\pi$
$992$		0			0
$993$	−347527.	−	1.45966e6i	−0.352444	−	1.48031i
$994$		0			0
$995$		−	499970.i		−	0.505007i
$996$		0			0
$997$	−1.77607e6			−1.78678			−0.893389	−	0.449284i	$-0.851679\pi$
$997$	−1.77607e6			−1.78678			−0.893389	+	0.449284i	$0.851679\pi$
$998$		0			0
$999$	65802.8	+	77932.5i	0.0659346	+	0.0780886i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	60.5.g.b.41.3	✓	4
3.2	odd	2	inner	60.5.g.b.41.4	yes	4
4.3	odd	2		240.5.l.c.161.2		4
5.2	odd	4		300.5.b.d.149.2		8
5.3	odd	4		300.5.b.d.149.7		8
5.4	even	2		300.5.g.g.101.2		4
12.11	even	2		240.5.l.c.161.1		4
15.2	even	4		300.5.b.d.149.8		8
15.8	even	4		300.5.b.d.149.1		8
15.14	odd	2		300.5.g.g.101.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
60.5.g.b.41.3	✓	4	1.1	even	1	trivial
60.5.g.b.41.4	yes	4	3.2	odd	2	inner
240.5.l.c.161.1		4	12.11	even	2
240.5.l.c.161.2		4	4.3	odd	2
300.5.b.d.149.1		8	15.8	even	4
300.5.b.d.149.2		8	5.2	odd	4
300.5.b.d.149.7		8	5.3	odd	4
300.5.b.d.149.8		8	15.2	even	4
300.5.g.g.101.1		4	15.14	odd	2
300.5.g.g.101.2		4	5.4	even	2

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:	\( N \)	\(=\)	\( 60 = 2^{2} \cdot 3 \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	60.g (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(-2.08452 - 8.75527i) q^{3} -11.1803i q^{5} -27.4929 q^{7} +(-72.3095 + 36.5011i) q^{9} +O(q^{10})\)	\(q+(-2.08452 - 8.75527i) q^{3} -11.1803i q^{5} -27.4929 q^{7} +(-72.3095 + 36.5011i) q^{9} -29.1008i q^{11} -289.886 q^{13} +(-97.8869 + 23.3057i) q^{15} -125.284i q^{17} +38.9286 q^{19} +(57.3095 + 240.707i) q^{21} -893.038i q^{23} -125.000 q^{25} +(470.308 + 557.002i) q^{27} +438.014i q^{29} +1283.71 q^{31} +(-254.786 + 60.6614i) q^{33} +307.379i q^{35} +139.914 q^{37} +(604.274 + 2538.03i) q^{39} -1998.66i q^{41} +2357.29 q^{43} +(408.095 + 808.445i) q^{45} -1943.67i q^{47} -1645.14 q^{49} +(-1096.89 + 261.157i) q^{51} -4489.58i q^{53} -325.357 q^{55} +(-81.1475 - 340.830i) q^{57} +5225.56i q^{59} -5263.86 q^{61} +(1988.00 - 1003.52i) q^{63} +3241.02i q^{65} -5221.28 q^{67} +(-7818.79 + 1861.56i) q^{69} -5894.88i q^{71} +621.629 q^{73} +(260.566 + 1094.41i) q^{75} +800.065i q^{77} +4994.21 q^{79} +(3896.33 - 5278.76i) q^{81} -3802.29i q^{83} -1400.71 q^{85} +(3834.93 - 913.050i) q^{87} +11054.7i q^{89} +7969.79 q^{91} +(-2675.93 - 11239.3i) q^{93} -435.234i q^{95} +5996.54 q^{97} +(1062.21 + 2104.27i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 20 q^{3} - 40 q^{7} - 56 q^{9}+O(q^{10}) \) 4 * q - 20 * q^3 - 40 * q^7 - 56 * q^9	\( 4 q - 20 q^{3} - 40 q^{7} - 56 q^{9} - 40 q^{13} - 100 q^{15} - 544 q^{19} - 4 q^{21} - 500 q^{25} + 820 q^{27} + 2336 q^{31} + 1080 q^{33} - 280 q^{37} - 3064 q^{39} + 7400 q^{43} - 700 q^{45} - 7980 q^{49} + 1560 q^{51} - 4800 q^{55} + 4760 q^{57} - 64 q^{61} + 4640 q^{63} - 8920 q^{67} - 19380 q^{69} + 18440 q^{73} + 2500 q^{75} - 7312 q^{79} + 2524 q^{81} - 12600 q^{85} + 14640 q^{87} + 19984 q^{91} - 3520 q^{93} - 9880 q^{97} - 23040 q^{99}+O(q^{100}) \) 4 * q - 20 * q^3 - 40 * q^7 - 56 * q^9 - 40 * q^13 - 100 * q^15 - 544 * q^19 - 4 * q^21 - 500 * q^25 + 820 * q^27 + 2336 * q^31 + 1080 * q^33 - 280 * q^37 - 3064 * q^39 + 7400 * q^43 - 700 * q^45 - 7980 * q^49 + 1560 * q^51 - 4800 * q^55 + 4760 * q^57 - 64 * q^61 + 4640 * q^63 - 8920 * q^67 - 19380 * q^69 + 18440 * q^73 + 2500 * q^75 - 7312 * q^79 + 2524 * q^81 - 12600 * q^85 + 14640 * q^87 + 19984 * q^91 - 3520 * q^93 - 9880 * q^97 - 23040 * q^99

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