LMFDB - Embedded newform 4004.2.m.c.2157.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4004,2,Mod(2157,4004)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4004, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("4004.2157");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4004 = 2^{2} \cdot 7 \cdot 11 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4004.m (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:	$31.9721009693$
Analytic rank:	$0$
Dimension:	$36$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2157.5
Character	$\chi$	$=$	4004.2157
Dual form			4004.2.m.c.2157.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.75761 q^{3} +1.98236i q^{5} +1.00000i q^{7} +4.60441 q^{9} +O(q^{10})$	$q-2.75761 q^{3} +1.98236i q^{5} +1.00000i q^{7} +4.60441 q^{9} -1.00000i q^{11} +(-2.54138 + 2.55761i) q^{13} -5.46656i q^{15} -7.39189 q^{17} -4.19788i q^{19} -2.75761i q^{21} +4.61924 q^{23} +1.07027 q^{25} -4.42434 q^{27} +4.07057 q^{29} +3.60497i q^{31} +2.75761i q^{33} -1.98236 q^{35} -9.67276i q^{37} +(7.00813 - 7.05290i) q^{39} -5.43161i q^{41} +12.3904 q^{43} +9.12758i q^{45} -0.284033i q^{47} -1.00000 q^{49} +20.3839 q^{51} -4.46945 q^{53} +1.98236 q^{55} +11.5761i q^{57} -1.96978i q^{59} -2.69145 q^{61} +4.60441i q^{63} +(-5.07010 - 5.03792i) q^{65} +12.8334i q^{67} -12.7381 q^{69} -8.00846i q^{71} -0.542368i q^{73} -2.95137 q^{75} +1.00000 q^{77} -13.5068 q^{79} -1.61263 q^{81} +3.32443i q^{83} -14.6533i q^{85} -11.2251 q^{87} -14.4386i q^{89} +(-2.55761 - 2.54138i) q^{91} -9.94111i q^{93} +8.32168 q^{95} -15.3264i q^{97} -4.60441i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 36 q - 4 q^{3} + 40 q^{9}+O(q^{10}) $ 36 * q - 4 * q^3 + 40 * q^9	$ 36 q - 4 q^{3} + 40 q^{9} - 4 q^{17} + 8 q^{23} - 80 q^{25} + 8 q^{27} + 8 q^{29} - 24 q^{39} + 32 q^{43} - 36 q^{49} - 20 q^{51} + 12 q^{53} + 32 q^{61} - 24 q^{65} + 80 q^{69} - 36 q^{75} + 36 q^{77} + 16 q^{79} + 132 q^{81} + 8 q^{91} + 56 q^{95}+O(q^{100}) $ 36 * q - 4 * q^3 + 40 * q^9 - 4 * q^17 + 8 * q^23 - 80 * q^25 + 8 * q^27 + 8 * q^29 - 24 * q^39 + 32 * q^43 - 36 * q^49 - 20 * q^51 + 12 * q^53 + 32 * q^61 - 24 * q^65 + 80 * q^69 - 36 * q^75 + 36 * q^77 + 16 * q^79 + 132 * q^81 + 8 * q^91 + 56 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4004\mathbb{Z}\right)^\times$.

$n$	$365$	$925$	$2003$	$3433$
$\chi(n)$	$1$	$-1$	$1$	$1$

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
$3$	−2.75761			−1.59211			−0.796053	−	0.605226i	$-0.793083\pi$
$3$	−2.75761			−1.59211			−0.796053	+	0.605226i	$0.793083\pi$
$4$		0			0
$5$			1.98236i			0.886536i	0.896389	+	0.443268i	$0.146181\pi$
$5$			1.98236i			0.886536i	−0.896389	+	0.443268i	$0.853819\pi$
$6$		0			0
$7$			1.00000i			0.377964i
$7$			1.00000i			0.377964i
$8$		0			0
$9$	4.60441			1.53480
$10$		0			0
$11$		−	1.00000i		−	0.301511i
$11$		−	1.00000i		−	0.301511i
$12$		0			0
$13$	−2.54138	+	2.55761i	−0.704852	+	0.709355i
$13$	−2.54138	+	2.55761i	−0.704852	+	0.709355i
$14$		0			0
$15$		−	5.46656i		−	1.41146i
$16$		0			0
$17$	−7.39189			−1.79280			−0.896398	−	0.443250i	$-0.853825\pi$
$17$	−7.39189			−1.79280			−0.896398	+	0.443250i	$0.853825\pi$
$18$		0			0
$19$		−	4.19788i		−	0.963059i	−0.876430	−	0.481529i	$-0.840081\pi$
$19$		−	4.19788i		−	0.963059i	0.876430	−	0.481529i	$-0.159919\pi$
$20$		0			0
$21$		−	2.75761i		−	0.601760i
$22$		0			0
$23$	4.61924			0.963178			0.481589	−	0.876397i	$-0.340060\pi$
$23$	4.61924			0.963178			0.481589	+	0.876397i	$0.340060\pi$
$24$		0			0
$25$	1.07027			0.214053
$26$		0			0
$27$	−4.42434			−0.851465
$28$		0			0
$29$	4.07057			0.755887			0.377943	−	0.925829i	$-0.376631\pi$
$29$	4.07057			0.755887			0.377943	+	0.925829i	$0.376631\pi$
$30$		0			0
$31$			3.60497i			0.647472i	0.946147	+	0.323736i	$0.104939\pi$
$31$			3.60497i			0.647472i	−0.946147	+	0.323736i	$0.895061\pi$
$32$		0			0
$33$			2.75761i			0.480038i
$34$		0			0
$35$	−1.98236			−0.335079
$36$		0			0
$37$		−	9.67276i		−	1.59019i	−0.606484	−	0.795096i	$-0.707420\pi$
$37$		−	9.67276i		−	1.59019i	0.606484	−	0.795096i	$-0.292580\pi$
$38$		0			0
$39$	7.00813	−	7.05290i	1.12220	−	1.12937i
$40$		0			0
$41$		−	5.43161i		−	0.848275i	−0.905598	−	0.424138i	$-0.860577\pi$
$41$		−	5.43161i		−	0.848275i	0.905598	−	0.424138i	$-0.139423\pi$
$42$		0			0
$43$	12.3904			1.88953			0.944763	−	0.327755i	$-0.106292\pi$
$43$	12.3904			1.88953			0.944763	+	0.327755i	$0.106292\pi$
$44$		0			0
$45$			9.12758i			1.36066i
$46$		0			0
$47$		−	0.284033i		−	0.0414305i	−0.999785	−	0.0207153i	$-0.993406\pi$
$47$		−	0.284033i		−	0.0414305i	0.999785	−	0.0207153i	$-0.00659435\pi$
$48$		0			0
$49$	−1.00000			−0.142857
$50$		0			0
$51$	20.3839			2.85432
$52$		0			0
$53$	−4.46945			−0.613926			−0.306963	−	0.951722i	$-0.599313\pi$
$53$	−4.46945			−0.613926			−0.306963	+	0.951722i	$0.599313\pi$
$54$		0			0
$55$	1.98236			0.267301
$56$		0			0
$57$			11.5761i			1.53329i
$58$		0			0
$59$		−	1.96978i		−	0.256444i	−0.991745	−	0.128222i	$-0.959073\pi$
$59$		−	1.96978i		−	0.256444i	0.991745	−	0.128222i	$-0.0409270\pi$
$60$		0			0
$61$	−2.69145			−0.344605			−0.172303	−	0.985044i	$-0.555121\pi$
$61$	−2.69145			−0.344605			−0.172303	+	0.985044i	$0.555121\pi$
$62$		0			0
$63$			4.60441i			0.580101i
$64$		0			0
$65$	−5.07010	−	5.03792i	−0.628869	−	0.624877i
$66$		0			0
$67$			12.8334i			1.56784i	0.620860	+	0.783922i	$0.286784\pi$
$67$			12.8334i			1.56784i	−0.620860	+	0.783922i	$0.713216\pi$
$68$		0			0
$69$	−12.7381			−1.53348
$70$		0			0
$71$		−	8.00846i		−	0.950429i	−0.879870	−	0.475214i	$-0.842370\pi$
$71$		−	8.00846i		−	0.950429i	0.879870	−	0.475214i	$-0.157630\pi$
$72$		0			0
$73$		−	0.542368i		−	0.0634793i	−0.999496	−	0.0317397i	$-0.989895\pi$
$73$		−	0.542368i		−	0.0634793i	0.999496	−	0.0317397i	$-0.0101047\pi$
$74$		0			0
$75$	−2.95137			−0.340795
$76$		0			0
$77$	1.00000			0.113961
$78$		0			0
$79$	−13.5068			−1.51964			−0.759819	−	0.650134i	$-0.774713\pi$
$79$	−13.5068			−1.51964			−0.759819	+	0.650134i	$0.774713\pi$
$80$		0			0
$81$	−1.61263			−0.179181
$82$		0			0
$83$			3.32443i			0.364903i	0.983215	+	0.182452i	$0.0584033\pi$
$83$			3.32443i			0.364903i	−0.983215	+	0.182452i	$0.941597\pi$
$84$		0			0
$85$		−	14.6533i		−	1.58938i
$86$		0			0
$87$	−11.2251			−1.20345
$88$		0			0
$89$		−	14.4386i		−	1.53048i	−0.643742	−	0.765242i	$-0.722619\pi$
$89$		−	14.4386i		−	1.53048i	0.643742	−	0.765242i	$-0.277381\pi$
$90$		0			0
$91$	−2.55761	−	2.54138i	−0.268111	−	0.266409i
$92$		0			0
$93$		−	9.94111i		−	1.03084i
$94$		0			0
$95$	8.32168			0.853787
$96$		0			0
$97$		−	15.3264i		−	1.55616i	−0.628164	−	0.778081i	$-0.716193\pi$
$97$		−	15.3264i		−	1.55616i	0.628164	−	0.778081i	$-0.283807\pi$
$98$		0			0
$99$		−	4.60441i		−	0.462761i
$100$		0			0
$101$	−6.90146			−0.686720			−0.343360	−	0.939204i	$-0.611565\pi$
$101$	−6.90146			−0.686720			−0.343360	+	0.939204i	$0.611565\pi$
$102$		0			0
$103$	13.1336			1.29409			0.647046	−	0.762451i	$-0.276004\pi$
$103$	13.1336			1.29409			0.647046	+	0.762451i	$0.276004\pi$
$104$		0			0
$105$	5.46656			0.533482
$106$		0			0
$107$	16.8985			1.63364			0.816822	−	0.576890i	$-0.195734\pi$
$107$	16.8985			1.63364			0.816822	+	0.576890i	$0.195734\pi$
$108$		0			0
$109$			12.6498i			1.21164i	0.795604	+	0.605818i	$0.207154\pi$
$109$			12.6498i			1.21164i	−0.795604	+	0.605818i	$0.792846\pi$
$110$		0			0
$111$			26.6737i			2.53176i
$112$		0			0
$113$	5.99276			0.563752			0.281876	−	0.959451i	$-0.409043\pi$
$113$	5.99276			0.563752			0.281876	+	0.959451i	$0.409043\pi$
$114$		0			0
$115$			9.15698i			0.853892i
$116$		0			0
$117$	−11.7016	+	11.7763i	−1.08181	+	1.08872i
$118$		0			0
$119$		−	7.39189i		−	0.677613i
$120$		0			0
$121$	−1.00000			−0.0909091
$122$		0			0
$123$			14.9783i			1.35054i
$124$		0			0
$125$			12.0334i			1.07630i
$126$		0			0
$127$	−4.60540			−0.408664			−0.204332	−	0.978902i	$-0.565502\pi$
$127$	−4.60540			−0.408664			−0.204332	+	0.978902i	$0.565502\pi$
$128$		0			0
$129$	−34.1680			−3.00833
$130$		0			0
$131$	−11.5436			−1.00857			−0.504283	−	0.863538i	$-0.668243\pi$
$131$	−11.5436			−1.00857			−0.504283	+	0.863538i	$0.668243\pi$
$132$		0			0
$133$	4.19788			0.364002
$134$		0			0
$135$		−	8.77062i		−	0.754855i
$136$		0			0
$137$			7.68246i			0.656357i	0.944616	+	0.328178i	$0.106435\pi$
$137$			7.68246i			0.656357i	−0.944616	+	0.328178i	$0.893565\pi$
$138$		0			0
$139$	1.55906			0.132237			0.0661187	−	0.997812i	$-0.478938\pi$
$139$	1.55906			0.132237			0.0661187	+	0.997812i	$0.478938\pi$
$140$		0			0
$141$			0.783253i			0.0659618i
$142$		0			0
$143$	2.55761	+	2.54138i	0.213878	+	0.212521i
$144$		0			0
$145$			8.06932i			0.670121i
$146$		0			0
$147$	2.75761			0.227444
$148$		0			0
$149$		−	3.41910i		−	0.280103i	−0.990144	−	0.140052i	$-0.955273\pi$
$149$		−	3.41910i		−	0.280103i	0.990144	−	0.140052i	$-0.0447269\pi$
$150$		0			0
$151$			10.0152i			0.815024i	0.913200	+	0.407512i	$0.133603\pi$
$151$			10.0152i			0.815024i	−0.913200	+	0.407512i	$0.866397\pi$
$152$		0			0
$153$	−34.0353			−2.75159
$154$		0			0
$155$	−7.14634			−0.574008
$156$		0			0
$157$	19.3142			1.54144			0.770719	−	0.637176i	$-0.219897\pi$
$157$	19.3142			1.54144			0.770719	+	0.637176i	$0.219897\pi$
$158$		0			0
$159$	12.3250			0.977435
$160$		0			0
$161$			4.61924i			0.364047i
$162$		0			0
$163$			1.51917i			0.118990i	0.998229	+	0.0594952i	$0.0189491\pi$
$163$			1.51917i			0.118990i	−0.998229	+	0.0594952i	$0.981051\pi$
$164$		0			0
$165$	−5.46656			−0.425571
$166$		0			0
$167$			4.56617i			0.353340i	0.984270	+	0.176670i	$0.0565326\pi$
$167$			4.56617i			0.353340i	−0.984270	+	0.176670i	$0.943467\pi$
$168$		0			0
$169$	−0.0827857	−	12.9997i	−0.00636813	−	0.999980i
$170$		0			0
$171$		−	19.3288i		−	1.47811i
$172$		0			0
$173$	21.0406			1.59969			0.799845	−	0.600207i	$-0.204915\pi$
$173$	21.0406			1.59969			0.799845	+	0.600207i	$0.204915\pi$
$174$		0			0
$175$			1.07027i			0.0809045i
$176$		0			0
$177$			5.43190i			0.408286i
$178$		0			0
$179$	−5.13541			−0.383839			−0.191919	−	0.981411i	$-0.561471\pi$
$179$	−5.13541			−0.383839			−0.191919	+	0.981411i	$0.561471\pi$
$180$		0			0
$181$	0.227231			0.0168899			0.00844497	−	0.999964i	$-0.497312\pi$
$181$	0.227231			0.0168899			0.00844497	+	0.999964i	$0.497312\pi$
$182$		0			0
$183$	7.42198			0.548649
$184$		0			0
$185$	19.1748			1.40976
$186$		0			0
$187$			7.39189i			0.540548i
$188$		0			0
$189$		−	4.42434i		−	0.321824i
$190$		0			0
$191$	15.5458			1.12486			0.562428	−	0.826846i	$-0.309867\pi$
$191$	15.5458			1.12486			0.562428	+	0.826846i	$0.309867\pi$
$192$		0			0
$193$			24.9751i			1.79775i	0.438207	+	0.898874i	$0.355614\pi$
$193$			24.9751i			1.79775i	−0.438207	+	0.898874i	$0.644386\pi$
$194$		0			0
$195$	13.9814	+	13.8926i	1.00123	+	0.994871i
$196$		0			0
$197$			20.4952i			1.46022i	0.683329	+	0.730110i	$0.260531\pi$
$197$			20.4952i			1.46022i	−0.683329	+	0.730110i	$0.739469\pi$
$198$		0			0
$199$	2.18473			0.154872			0.0774358	−	0.996997i	$-0.475327\pi$
$199$	2.18473			0.154872			0.0774358	+	0.996997i	$0.475327\pi$
$200$		0			0
$201$		−	35.3894i		−	2.49617i
$202$		0			0
$203$			4.07057i			0.285698i
$204$		0			0
$205$	10.7674			0.752027
$206$		0			0
$207$	21.2689			1.47829
$208$		0			0
$209$	−4.19788			−0.290373
$210$		0			0
$211$	5.26708			0.362601			0.181300	−	0.983428i	$-0.441969\pi$
$211$	5.26708			0.362601			0.181300	+	0.983428i	$0.441969\pi$
$212$		0			0
$213$			22.0842i			1.51318i
$214$		0			0
$215$			24.5623i			1.67513i
$216$		0			0
$217$	−3.60497			−0.244721
$218$		0			0
$219$			1.49564i			0.101066i
$220$		0			0
$221$	18.7856	−	18.9056i	1.26366	−	1.27173i
$222$		0			0
$223$			10.2976i			0.689578i	0.938680	+	0.344789i	$0.112050\pi$
$223$			10.2976i			0.689578i	−0.938680	+	0.344789i	$0.887950\pi$
$224$		0			0
$225$	4.92794			0.328530
$226$		0			0
$227$			27.0317i			1.79415i	0.441874	+	0.897077i	$0.354314\pi$
$227$			27.0317i			1.79415i	−0.441874	+	0.897077i	$0.645686\pi$
$228$		0			0
$229$		−	2.21512i		−	0.146379i	−0.997318	−	0.0731896i	$-0.976682\pi$
$229$		−	2.21512i		−	0.146379i	0.997318	−	0.0731896i	$-0.0233178\pi$
$230$		0			0
$231$	−2.75761			−0.181437
$232$		0			0
$233$	−21.8938			−1.43431			−0.717154	−	0.696915i	$-0.754556\pi$
$233$	−21.8938			−1.43431			−0.717154	+	0.696915i	$0.754556\pi$
$234$		0			0
$235$	0.563055			0.0367297
$236$		0			0
$237$	37.2466			2.41943
$238$		0			0
$239$		−	4.09702i		−	0.265014i	−0.991182	−	0.132507i	$-0.957697\pi$
$239$		−	4.09702i		−	0.265014i	0.991182	−	0.132507i	$-0.0423027\pi$
$240$		0			0
$241$		−	2.77298i		−	0.178623i	−0.996004	−	0.0893117i	$-0.971533\pi$
$241$		−	2.77298i		−	0.178623i	0.996004	−	0.0893117i	$-0.0284667\pi$
$242$		0			0
$243$	17.7200			1.13674
$244$		0			0
$245$		−	1.98236i		−	0.126648i
$246$		0			0
$247$	10.7365	+	10.6684i	0.683150	+	0.678814i
$248$		0			0
$249$		−	9.16747i		−	0.580965i
$250$		0			0
$251$	20.7097			1.30718			0.653592	−	0.756847i	$-0.273261\pi$
$251$	20.7097			1.30718			0.653592	+	0.756847i	$0.273261\pi$
$252$		0			0
$253$		−	4.61924i		−	0.290409i
$254$		0			0
$255$			40.4082i			2.53046i
$256$		0			0
$257$	1.54096			0.0961227			0.0480613	−	0.998844i	$-0.484696\pi$
$257$	1.54096			0.0961227			0.0480613	+	0.998844i	$0.484696\pi$
$258$		0			0
$259$	9.67276			0.601036
$260$		0			0
$261$	18.7426			1.16014
$262$		0			0
$263$	7.95053			0.490250			0.245125	−	0.969491i	$-0.421171\pi$
$263$	7.95053			0.490250			0.245125	+	0.969491i	$0.421171\pi$
$264$		0			0
$265$		−	8.86003i		−	0.544267i
$266$		0			0
$267$			39.8159i			2.43670i
$268$		0			0
$269$	8.18925			0.499307			0.249654	−	0.968335i	$-0.419683\pi$
$269$	8.18925			0.499307			0.249654	+	0.968335i	$0.419683\pi$
$270$		0			0
$271$		−	11.8033i		−	0.716999i	−0.933530	−	0.358500i	$-0.883288\pi$
$271$		−	11.8033i		−	0.716999i	0.933530	−	0.358500i	$-0.116712\pi$
$272$		0			0
$273$	7.05290	+	7.00813i	0.426861	+	0.424151i
$274$		0			0
$275$		−	1.07027i		−	0.0645394i
$276$		0			0
$277$	−9.29260			−0.558338			−0.279169	−	0.960242i	$-0.590059\pi$
$277$	−9.29260			−0.558338			−0.279169	+	0.960242i	$0.590059\pi$
$278$		0			0
$279$			16.5988i			0.993743i
$280$		0			0
$281$		−	27.3565i		−	1.63195i	−0.578084	−	0.815977i	$-0.696199\pi$
$281$		−	27.3565i		−	1.63195i	0.578084	−	0.815977i	$-0.303801\pi$
$282$		0			0
$283$	19.0094			1.12999			0.564995	−	0.825094i	$-0.308878\pi$
$283$	19.0094			1.12999			0.564995	+	0.825094i	$0.308878\pi$
$284$		0			0
$285$	−22.9480			−1.35932
$286$		0			0
$287$	5.43161			0.320618
$288$		0			0
$289$	37.6400			2.21412
$290$		0			0
$291$			42.2643i			2.47758i
$292$		0			0
$293$			9.11806i			0.532683i	0.963879	+	0.266341i	$0.0858148\pi$
$293$			9.11806i			0.532683i	−0.963879	+	0.266341i	$0.914185\pi$
$294$		0			0
$295$	3.90481			0.227347
$296$		0			0
$297$			4.42434i			0.256726i
$298$		0			0
$299$	−11.7392	+	11.8142i	−0.678898	+	0.683235i
$300$		0			0
$301$			12.3904i			0.714173i
$302$		0			0
$303$	19.0315			1.09333
$304$		0			0
$305$		−	5.33542i		−	0.305505i
$306$		0			0
$307$			17.3379i			0.989524i	0.869028	+	0.494762i	$0.164745\pi$
$307$			17.3379i			0.989524i	−0.869028	+	0.494762i	$0.835255\pi$
$308$		0			0
$309$	−36.2173			−2.06033
$310$		0			0
$311$	2.81808			0.159799			0.0798995	−	0.996803i	$-0.474540\pi$
$311$	2.81808			0.159799			0.0798995	+	0.996803i	$0.474540\pi$
$312$		0			0
$313$	−26.4075			−1.49264			−0.746319	−	0.665588i	$-0.768181\pi$
$313$	−26.4075			−1.49264			−0.746319	+	0.665588i	$0.768181\pi$
$314$		0			0
$315$	−9.12758			−0.514281
$316$		0			0
$317$			3.99221i			0.224225i	0.993696	+	0.112113i	$0.0357617\pi$
$317$			3.99221i			0.224225i	−0.993696	+	0.112113i	$0.964238\pi$
$318$		0			0
$319$		−	4.07057i		−	0.227908i
$320$		0			0
$321$	−46.5996			−2.60094
$322$		0			0
$323$			31.0302i			1.72657i
$324$		0			0
$325$	−2.71995	+	2.73733i	−0.150876	+	0.151840i
$326$		0			0
$327$		−	34.8833i		−	1.92905i
$328$		0			0
$329$	0.284033			0.0156593
$330$		0			0
$331$			21.9021i			1.20385i	0.798554	+	0.601923i	$0.205599\pi$
$331$			21.9021i			1.20385i	−0.798554	+	0.601923i	$0.794401\pi$
$332$		0			0
$333$		−	44.5374i		−	2.44063i
$334$		0			0
$335$	−25.4403			−1.38995
$336$		0			0
$337$	25.2354			1.37466			0.687331	−	0.726344i	$-0.258782\pi$
$337$	25.2354			1.37466			0.687331	+	0.726344i	$0.258782\pi$
$338$		0			0
$339$	−16.5257			−0.897553
$340$		0			0
$341$	3.60497			0.195220
$342$		0			0
$343$		−	1.00000i		−	0.0539949i
$344$		0			0
$345$		−	25.2514i		−	1.35949i
$346$		0			0
$347$	−3.98169			−0.213748			−0.106874	−	0.994273i	$-0.534084\pi$
$347$	−3.98169			−0.213748			−0.106874	+	0.994273i	$0.534084\pi$
$348$		0			0
$349$		−	1.38283i		−	0.0740214i	−0.999315	−	0.0370107i	$-0.988216\pi$
$349$		−	1.38283i		−	0.0740214i	0.999315	−	0.0370107i	$-0.0117836\pi$
$350$		0			0
$351$	11.2439	−	11.3158i	0.600157	−	0.603991i
$352$		0			0
$353$			2.50659i			0.133412i	0.997773	+	0.0667060i	$0.0212490\pi$
$353$			2.50659i			0.133412i	−0.997773	+	0.0667060i	$0.978751\pi$
$354$		0			0
$355$	15.8756			0.842590
$356$		0			0
$357$			20.3839i			1.07883i
$358$		0			0
$359$		−	15.6288i		−	0.824859i	−0.910989	−	0.412430i	$-0.864680\pi$
$359$		−	15.6288i		−	0.824859i	0.910989	−	0.412430i	$-0.135320\pi$
$360$		0			0
$361$	1.37783			0.0725176
$362$		0			0
$363$	2.75761			0.144737
$364$		0			0
$365$	1.07517			0.0562768
$366$		0			0
$367$	19.7136			1.02904			0.514522	−	0.857477i	$-0.327970\pi$
$367$	19.7136			1.02904			0.514522	+	0.857477i	$0.327970\pi$
$368$		0			0
$369$		−	25.0094i		−	1.30194i
$370$		0			0
$371$		−	4.46945i		−	0.232042i
$372$		0			0
$373$	11.5753			0.599346			0.299673	−	0.954042i	$-0.403122\pi$
$373$	11.5753			0.599346			0.299673	+	0.954042i	$0.403122\pi$
$374$		0			0
$375$		−	33.1835i		−	1.71359i
$376$		0			0
$377$	−10.3449	+	10.4110i	−0.532788	+	0.536192i
$378$		0			0
$379$			12.2568i			0.629588i	0.949160	+	0.314794i	$0.101935\pi$
$379$			12.2568i			0.629588i	−0.949160	+	0.314794i	$0.898065\pi$
$380$		0			0
$381$	12.6999			0.650636
$382$		0			0
$383$		−	32.5093i		−	1.66115i	−0.556909	−	0.830574i	$-0.688013\pi$
$383$		−	32.5093i		−	1.66115i	0.556909	−	0.830574i	$-0.311987\pi$
$384$		0			0
$385$			1.98236i			0.101030i
$386$		0			0
$387$	57.0507			2.90005
$388$		0			0
$389$	−7.79700			−0.395324			−0.197662	−	0.980270i	$-0.563335\pi$
$389$	−7.79700			−0.395324			−0.197662	+	0.980270i	$0.563335\pi$
$390$		0			0
$391$	−34.1449			−1.72678
$392$		0			0
$393$	31.8327			1.60575
$394$		0			0
$395$		−	26.7754i		−	1.34722i
$396$		0			0
$397$			19.6911i			0.988269i	0.869386	+	0.494134i	$0.164515\pi$
$397$			19.6911i			0.988269i	−0.869386	+	0.494134i	$0.835485\pi$
$398$		0			0
$399$	−11.5761			−0.579530
$400$		0			0
$401$		−	0.958076i		−	0.0478440i	−0.999714	−	0.0239220i	$-0.992385\pi$
$401$		−	0.958076i		−	0.0478440i	0.999714	−	0.0239220i	$-0.00761534\pi$
$402$		0			0
$403$	−9.22013	−	9.16160i	−0.459287	−	0.456372i
$404$		0			0
$405$		−	3.19680i		−	0.158850i
$406$		0			0
$407$	−9.67276			−0.479461
$408$		0			0
$409$			35.2821i			1.74459i	0.488981	+	0.872294i	$0.337369\pi$
$409$			35.2821i			1.74459i	−0.488981	+	0.872294i	$0.662631\pi$
$410$		0			0
$411$		−	21.1852i		−	1.04499i
$412$		0			0
$413$	1.96978			0.0969268
$414$		0			0
$415$	−6.59019			−0.323500
$416$		0			0
$417$	−4.29927			−0.210536
$418$		0			0
$419$	10.3266			0.504485			0.252243	−	0.967664i	$-0.418832\pi$
$419$	10.3266			0.504485			0.252243	+	0.967664i	$0.418832\pi$
$420$		0			0
$421$		−	13.1500i		−	0.640892i	−0.947267	−	0.320446i	$-0.896167\pi$
$421$		−	13.1500i		−	0.640892i	0.947267	−	0.320446i	$-0.103833\pi$
$422$		0			0
$423$		−	1.30781i		−	0.0635878i
$424$		0			0
$425$	−7.91128			−0.383753
$426$		0			0
$427$		−	2.69145i		−	0.130249i
$428$		0			0
$429$	−7.05290	−	7.00813i	−0.340517	−	0.338356i
$430$		0			0
$431$			5.88317i			0.283382i	0.989911	+	0.141691i	$0.0452540\pi$
$431$			5.88317i			0.283382i	−0.989911	+	0.141691i	$0.954746\pi$
$432$		0			0
$433$	−36.6669			−1.76210			−0.881049	−	0.473025i	$-0.843162\pi$
$433$	−36.6669			−1.76210			−0.881049	+	0.473025i	$0.843162\pi$
$434$		0			0
$435$		−	22.2520i		−	1.06690i
$436$		0			0
$437$		−	19.3910i		−	0.927597i
$438$		0			0
$439$	27.3167			1.30376			0.651878	−	0.758324i	$-0.273981\pi$
$439$	27.3167			1.30376			0.651878	+	0.758324i	$0.273981\pi$
$440$		0			0
$441$	−4.60441			−0.219258
$442$		0			0
$443$	−33.3749			−1.58569			−0.792844	−	0.609424i	$-0.791401\pi$
$443$	−33.3749			−1.58569			−0.792844	+	0.609424i	$0.791401\pi$
$444$		0			0
$445$	28.6224			1.35683
$446$		0			0
$447$			9.42853i			0.445954i
$448$		0			0
$449$			17.2049i			0.811952i	0.913884	+	0.405976i	$0.133068\pi$
$449$			17.2049i			0.811952i	−0.913884	+	0.405976i	$0.866932\pi$
$450$		0			0
$451$	−5.43161			−0.255765
$452$		0			0
$453$		−	27.6179i		−	1.29760i
$454$		0			0
$455$	5.03792	−	5.07010i	0.236181	−	0.237690i
$456$		0			0
$457$			21.5943i			1.01014i	0.863079	+	0.505068i	$0.168533\pi$
$457$			21.5943i			1.01014i	−0.863079	+	0.505068i	$0.831467\pi$
$458$		0			0
$459$	32.7042			1.52650
$460$		0			0
$461$			28.9024i			1.34612i	0.739588	+	0.673060i	$0.235020\pi$
$461$			28.9024i			1.34612i	−0.739588	+	0.673060i	$0.764980\pi$
$462$		0			0
$463$		−	26.3410i		−	1.22417i	−0.790792	−	0.612085i	$-0.790331\pi$
$463$		−	26.3410i		−	1.22417i	0.790792	−	0.612085i	$-0.209669\pi$
$464$		0			0
$465$	19.7068			0.913881
$466$		0			0
$467$	−28.4092			−1.31462			−0.657311	−	0.753619i	$-0.728306\pi$
$467$	−28.4092			−1.31462			−0.657311	+	0.753619i	$0.728306\pi$
$468$		0			0
$469$	−12.8334			−0.592589
$470$		0			0
$471$	−53.2609			−2.45413
$472$		0			0
$473$		−	12.3904i		−	0.569713i
$474$		0			0
$475$		−	4.49284i		−	0.206146i
$476$		0			0
$477$	−20.5792			−0.942256
$478$		0			0
$479$		−	25.4893i		−	1.16464i	−0.812961	−	0.582318i	$-0.802146\pi$
$479$		−	25.4893i		−	1.16464i	0.812961	−	0.582318i	$-0.197854\pi$
$480$		0			0
$481$	24.7392	+	24.5821i	1.12801	+	1.12085i
$482$		0			0
$483$		−	12.7381i		−	0.579602i
$484$		0			0
$485$	30.3824			1.37959
$486$		0			0
$487$		−	33.5089i		−	1.51843i	−0.650839	−	0.759216i	$-0.725583\pi$
$487$		−	33.5089i		−	1.51843i	0.650839	−	0.759216i	$-0.274417\pi$
$488$		0			0
$489$		−	4.18927i		−	0.189445i
$490$		0			0
$491$	3.87148			0.174717			0.0873586	−	0.996177i	$-0.472157\pi$
$491$	3.87148			0.174717			0.0873586	+	0.996177i	$0.472157\pi$
$492$		0			0
$493$	−30.0892			−1.35515
$494$		0			0
$495$	9.12758			0.410254
$496$		0			0
$497$	8.00846			0.359228
$498$		0			0
$499$			16.7283i			0.748862i	0.927255	+	0.374431i	$0.122162\pi$
$499$			16.7283i			0.748862i	−0.927255	+	0.374431i	$0.877838\pi$
$500$		0			0
$501$		−	12.5917i		−	0.562556i
$502$		0			0
$503$	43.5572			1.94212			0.971059	−	0.238839i	$-0.0767669\pi$
$503$	43.5572			1.94212			0.971059	+	0.238839i	$0.0767669\pi$
$504$		0			0
$505$		−	13.6811i		−	0.608803i
$506$		0			0
$507$	0.228291	+	35.8482i	0.0101387	+	1.59207i
$508$		0			0
$509$		−	6.52062i		−	0.289021i	−0.989503	−	0.144511i	$-0.953839\pi$
$509$		−	6.52062i		−	0.289021i	0.989503	−	0.144511i	$-0.0461608\pi$
$510$		0			0
$511$	0.542368			0.0239929
$512$		0			0
$513$			18.5728i			0.820011i
$514$		0			0
$515$			26.0354i			1.14726i
$516$		0			0
$517$	−0.284033			−0.0124918
$518$		0			0
$519$	−58.0218			−2.54688
$520$		0			0
$521$	−3.38853			−0.148454			−0.0742270	−	0.997241i	$-0.523649\pi$
$521$	−3.38853			−0.148454			−0.0742270	+	0.997241i	$0.523649\pi$
$522$		0			0
$523$	36.5564			1.59850			0.799249	−	0.601000i	$-0.205231\pi$
$523$	36.5564			1.59850			0.799249	+	0.601000i	$0.205231\pi$
$524$		0			0
$525$		−	2.95137i		−	0.128809i
$526$		0			0
$527$		−	26.6475i		−	1.16079i
$528$		0			0
$529$	−1.66263			−0.0722881
$530$		0			0
$531$		−	9.06970i		−	0.393591i
$532$		0			0
$533$	13.8920	+	13.8038i	0.601728	+	0.597908i
$534$		0			0
$535$			33.4989i			1.44829i
$536$		0			0
$537$	14.1615			0.611112
$538$		0			0
$539$			1.00000i			0.0430730i
$540$		0			0
$541$			46.0690i			1.98066i	0.138734	+	0.990330i	$0.455697\pi$
$541$			46.0690i			1.98066i	−0.138734	+	0.990330i	$0.544303\pi$
$542$		0			0
$543$	−0.626614			−0.0268906
$544$		0			0
$545$	−25.0765			−1.07416
$546$		0			0
$547$	−11.2186			−0.479674			−0.239837	−	0.970813i	$-0.577094\pi$
$547$	−11.2186			−0.479674			−0.239837	+	0.970813i	$0.577094\pi$
$548$		0			0
$549$	−12.3926			−0.528902
$550$		0			0
$551$		−	17.0878i		−	0.727963i
$552$		0			0
$553$		−	13.5068i		−	0.574370i
$554$		0			0
$555$	−52.8768			−2.24449
$556$		0			0
$557$			15.3169i			0.648998i	0.945886	+	0.324499i	$0.105196\pi$
$557$			15.3169i			0.648998i	−0.945886	+	0.324499i	$0.894804\pi$
$558$		0			0
$559$	−31.4888	+	31.6900i	−1.33184	+	1.34034i
$560$		0			0
$561$		−	20.3839i		−	0.860611i
$562$		0			0
$563$	32.2582			1.35952			0.679760	−	0.733435i	$-0.262084\pi$
$563$	32.2582			1.35952			0.679760	+	0.733435i	$0.262084\pi$
$564$		0			0
$565$			11.8798i			0.499786i
$566$		0			0
$567$		−	1.61263i		−	0.0677239i
$568$		0			0
$569$	18.1639			0.761471			0.380735	−	0.924684i	$-0.375671\pi$
$569$	18.1639			0.761471			0.380735	+	0.924684i	$0.375671\pi$
$570$		0			0
$571$	−26.6686			−1.11605			−0.558023	−	0.829825i	$-0.688440\pi$
$571$	−26.6686			−1.11605			−0.558023	+	0.829825i	$0.688440\pi$
$572$		0			0
$573$	−42.8693			−1.79089
$574$		0			0
$575$	4.94381			0.206171
$576$		0			0
$577$		−	11.9219i		−	0.496315i	−0.968720	−	0.248157i	$-0.920175\pi$
$577$		−	11.9219i		−	0.496315i	0.968720	−	0.248157i	$-0.0798250\pi$
$578$		0			0
$579$		−	68.8716i		−	2.86221i
$580$		0			0
$581$	−3.32443			−0.137920
$582$		0			0
$583$			4.46945i			0.185106i
$584$		0			0
$585$	−23.3448	−	23.1966i	−0.965190	−	0.959063i
$586$		0			0
$587$		−	2.09246i		−	0.0863651i	−0.999067	−	0.0431826i	$-0.986250\pi$
$587$		−	2.09246i		−	0.0863651i	0.999067	−	0.0431826i	$-0.0137497\pi$
$588$		0			0
$589$	15.1332			0.623554
$590$		0			0
$591$		−	56.5177i		−	2.32483i
$592$		0			0
$593$			17.1115i			0.702685i	0.936247	+	0.351343i	$0.114275\pi$
$593$			17.1115i			0.702685i	−0.936247	+	0.351343i	$0.885725\pi$
$594$		0			0
$595$	14.6533			0.600729
$596$		0			0
$597$	−6.02464			−0.246572
$598$		0			0
$599$	43.5056			1.77759			0.888795	−	0.458304i	$-0.151543\pi$
$599$	43.5056			1.77759			0.888795	+	0.458304i	$0.151543\pi$
$600$		0			0
$601$	−12.7825			−0.521411			−0.260705	−	0.965418i	$-0.583955\pi$
$601$	−12.7825			−0.521411			−0.260705	+	0.965418i	$0.583955\pi$
$602$		0			0
$603$			59.0900i			2.40633i
$604$		0			0
$605$		−	1.98236i		−	0.0805942i
$606$		0			0
$607$	23.6720			0.960815			0.480408	−	0.877045i	$-0.340489\pi$
$607$	23.6720			0.960815			0.480408	+	0.877045i	$0.340489\pi$
$608$		0			0
$609$		−	11.2251i		−	0.454862i
$610$		0			0
$611$	0.726448	+	0.721837i	0.0293889	+	0.0292024i
$612$		0			0
$613$		−	18.2292i		−	0.736271i	−0.929772	−	0.368135i	$-0.879996\pi$
$613$		−	18.2292i		−	0.736271i	0.929772	−	0.368135i	$-0.120004\pi$
$614$		0			0
$615$	−29.6922			−1.19731
$616$		0			0
$617$		−	38.0001i		−	1.52983i	−0.644134	−	0.764913i	$-0.722782\pi$
$617$		−	38.0001i		−	1.52983i	0.644134	−	0.764913i	$-0.277218\pi$
$618$		0			0
$619$			14.2348i			0.572147i	0.958208	+	0.286073i	$0.0923501\pi$
$619$			14.2348i			0.572147i	−0.958208	+	0.286073i	$0.907650\pi$
$620$		0			0
$621$	−20.4371			−0.820112
$622$		0			0
$623$	14.4386			0.578469
$624$		0			0
$625$	−18.5032			−0.740128
$626$		0			0
$627$	11.5761			0.462305
$628$		0			0
$629$			71.4999i			2.85089i
$630$		0			0
$631$			4.63927i			0.184687i	0.995727	+	0.0923433i	$0.0294357\pi$
$631$			4.63927i			0.184687i	−0.995727	+	0.0923433i	$0.970564\pi$
$632$		0			0
$633$	−14.5246			−0.577299
$634$		0			0
$635$		−	9.12955i		−	0.362295i
$636$		0			0
$637$	2.54138	−	2.55761i	0.100693	−	0.101336i
$638$		0			0
$639$		−	36.8742i		−	1.45872i
$640$		0			0
$641$	7.60473			0.300369			0.150184	−	0.988658i	$-0.452013\pi$
$641$	7.60473			0.300369			0.150184	+	0.988658i	$0.452013\pi$
$642$		0			0
$643$			25.1261i			0.990875i	0.868643	+	0.495438i	$0.164992\pi$
$643$			25.1261i			0.990875i	−0.868643	+	0.495438i	$0.835008\pi$
$644$		0			0
$645$		−	67.7332i		−	2.66699i
$646$		0			0
$647$	−29.1613			−1.14645			−0.573225	−	0.819398i	$-0.694308\pi$
$647$	−29.1613			−1.14645			−0.573225	+	0.819398i	$0.694308\pi$
$648$		0			0
$649$	−1.96978			−0.0773208
$650$		0			0
$651$	9.94111			0.389623
$652$		0			0
$653$	−21.4154			−0.838048			−0.419024	−	0.907975i	$-0.637628\pi$
$653$	−21.4154			−0.838048			−0.419024	+	0.907975i	$0.637628\pi$
$654$		0			0
$655$		−	22.8835i		−	0.894131i
$656$		0			0
$657$		−	2.49728i		−	0.0974284i
$658$		0			0
$659$	1.27587			0.0497008			0.0248504	−	0.999691i	$-0.492089\pi$
$659$	1.27587			0.0497008			0.0248504	+	0.999691i	$0.492089\pi$
$660$		0			0
$661$		−	17.8826i		−	0.695554i	−0.937577	−	0.347777i	$-0.886937\pi$
$661$		−	17.8826i		−	0.695554i	0.937577	−	0.347777i	$-0.113063\pi$
$662$		0			0
$663$	−51.8033	+	52.1343i	−2.01187	+	2.02473i
$664$		0			0
$665$			8.32168i			0.322701i
$666$		0			0
$667$	18.8030			0.728053
$668$		0			0
$669$		−	28.3968i		−	1.09788i
$670$		0			0
$671$			2.69145i			0.103902i
$672$		0			0
$673$	−10.0259			−0.386470			−0.193235	−	0.981152i	$-0.561898\pi$
$673$	−10.0259			−0.386470			−0.193235	+	0.981152i	$0.561898\pi$
$674$		0			0
$675$	−4.73522			−0.182259
$676$		0			0
$677$	22.6036			0.868728			0.434364	−	0.900737i	$-0.356973\pi$
$677$	22.6036			0.868728			0.434364	+	0.900737i	$0.356973\pi$
$678$		0			0
$679$	15.3264			0.588174
$680$		0			0
$681$		−	74.5428i		−	2.85648i
$682$		0			0
$683$		−	27.3688i		−	1.04724i	−0.851952	−	0.523619i	$-0.824581\pi$
$683$		−	27.3688i		−	1.04724i	0.851952	−	0.523619i	$-0.175419\pi$
$684$		0			0
$685$	−15.2294			−0.581884
$686$		0			0
$687$			6.10844i			0.233051i
$688$		0			0
$689$	11.3586	−	11.4311i	0.432727	−	0.435491i
$690$		0			0
$691$		−	22.4407i		−	0.853683i	−0.904326	−	0.426842i	$-0.859626\pi$
$691$		−	22.4407i		−	0.853683i	0.904326	−	0.426842i	$-0.140374\pi$
$692$		0			0
$693$	4.60441			0.174907
$694$		0			0
$695$			3.09060i			0.117233i
$696$		0			0
$697$			40.1498i			1.52078i
$698$		0			0
$699$	60.3744			2.28357
$700$		0			0
$701$	−37.6066			−1.42038			−0.710191	−	0.704009i	$-0.751391\pi$
$701$	−37.6066			−1.42038			−0.710191	+	0.704009i	$0.751391\pi$
$702$		0			0
$703$	−40.6050			−1.53145
$704$		0			0
$705$	−1.55269			−0.0584776
$706$		0			0
$707$		−	6.90146i		−	0.259556i
$708$		0			0
$709$		−	42.6546i		−	1.60193i	−0.598714	−	0.800963i	$-0.704321\pi$
$709$		−	42.6546i		−	1.60193i	0.598714	−	0.800963i	$-0.295679\pi$
$710$		0			0
$711$	−62.1911			−2.33235
$712$		0			0
$713$			16.6522i			0.623631i
$714$		0			0
$715$	−5.03792	+	5.07010i	−0.188407	+	0.189611i
$716$		0			0
$717$			11.2980i			0.421931i
$718$		0			0
$719$	42.2236			1.57467			0.787337	−	0.616523i	$-0.211459\pi$
$719$	42.2236			1.57467			0.787337	+	0.616523i	$0.211459\pi$
$720$		0			0
$721$			13.1336i			0.489120i
$722$		0			0
$723$			7.64680i			0.284388i
$724$		0			0
$725$	4.35659			0.161800
$726$		0			0
$727$	5.24000			0.194341			0.0971705	−	0.995268i	$-0.469021\pi$
$727$	5.24000			0.194341			0.0971705	+	0.995268i	$0.469021\pi$
$728$		0			0
$729$	−44.0270			−1.63063
$730$		0			0
$731$	−91.5888			−3.38753
$732$		0			0
$733$			24.2788i			0.896757i	0.893844	+	0.448378i	$0.147998\pi$
$733$			24.2788i			0.896757i	−0.893844	+	0.448378i	$0.852002\pi$
$734$		0			0
$735$			5.46656i			0.201637i
$736$		0			0
$737$	12.8334			0.472723
$738$		0			0
$739$			5.37005i			0.197541i	0.995110	+	0.0987703i	$0.0314909\pi$
$739$			5.37005i			0.197541i	−0.995110	+	0.0987703i	$0.968509\pi$
$740$		0			0
$741$	−29.6072	−	29.4193i	−1.08765	−	1.08074i
$742$		0			0
$743$		−	11.4181i		−	0.418890i	−0.977820	−	0.209445i	$-0.932834\pi$
$743$		−	11.4181i		−	0.418890i	0.977820	−	0.209445i	$-0.0671658\pi$
$744$		0			0
$745$	6.77787			0.248322
$746$		0			0
$747$			15.3070i			0.560055i
$748$		0			0
$749$			16.8985i			0.617459i
$750$		0			0
$751$	48.4887			1.76938			0.884689	−	0.466181i	$-0.154371\pi$
$751$	48.4887			1.76938			0.884689	+	0.466181i	$0.154371\pi$
$752$		0			0
$753$	−57.1092			−2.08118
$754$		0			0
$755$	−19.8536			−0.722548
$756$		0			0
$757$	1.79543			0.0652559			0.0326280	−	0.999468i	$-0.489612\pi$
$757$	1.79543			0.0652559			0.0326280	+	0.999468i	$0.489612\pi$
$758$		0			0
$759$			12.7381i			0.462362i
$760$		0			0
$761$		−	8.79075i		−	0.318664i	−0.987225	−	0.159332i	$-0.949066\pi$
$761$		−	8.79075i		−	0.318664i	0.987225	−	0.159332i	$-0.0509341\pi$
$762$		0			0
$763$	−12.6498			−0.457955
$764$		0			0
$765$		−	67.4701i		−	2.43939i
$766$		0			0
$767$	5.03795	+	5.00597i	0.181910	+	0.180755i
$768$		0			0
$769$			44.8648i			1.61787i	0.587901	+	0.808933i	$0.299955\pi$
$769$			44.8648i			1.61787i	−0.587901	+	0.808933i	$0.700045\pi$
$770$		0			0
$771$	−4.24938			−0.153038
$772$		0			0
$773$			3.23482i			0.116348i	0.998306	+	0.0581742i	$0.0185279\pi$
$773$			3.23482i			0.116348i	−0.998306	+	0.0581742i	$0.981472\pi$
$774$		0			0
$775$			3.85828i			0.138593i
$776$		0			0
$777$	−26.6737			−0.956913
$778$		0			0
$779$	−22.8012			−0.816939
$780$		0			0
$781$	−8.00846			−0.286565
$782$		0			0
$783$	−18.0096			−0.643611
$784$		0			0
$785$			38.2875i			1.36654i
$786$		0			0
$787$			52.0427i			1.85512i	0.373673	+	0.927560i	$0.378098\pi$
$787$			52.0427i			1.85512i	−0.373673	+	0.927560i	$0.621902\pi$
$788$		0			0
$789$	−21.9245			−0.780531
$790$		0			0
$791$			5.99276i			0.213078i
$792$		0			0
$793$	6.84001	−	6.88370i	0.242896	−	0.244447i
$794$		0			0
$795$			24.4325i			0.866532i
$796$		0			0
$797$	21.8364			0.773485			0.386743	−	0.922188i	$-0.373600\pi$
$797$	21.8364			0.773485			0.386743	+	0.922188i	$0.373600\pi$
$798$		0			0
$799$			2.09954i			0.0742765i
$800$		0			0
$801$		−	66.4811i		−	2.34899i
$802$		0			0
$803$	−0.542368			−0.0191397
$804$		0			0
$805$	−9.15698			−0.322741
$806$		0			0
$807$	−22.5827			−0.794950
$808$		0			0
$809$	31.2028			1.09703			0.548517	−	0.836140i	$-0.315193\pi$
$809$	31.2028			1.09703			0.548517	+	0.836140i	$0.315193\pi$
$810$		0			0
$811$			3.15331i			0.110728i	0.998466	+	0.0553639i	$0.0176319\pi$
$811$			3.15331i			0.110728i	−0.998466	+	0.0553639i	$0.982368\pi$
$812$		0			0
$813$			32.5489i			1.14154i
$814$		0			0
$815$	−3.01153			−0.105489
$816$		0			0
$817$		−	52.0136i		−	1.81972i
$818$		0			0
$819$	−11.7763	−	11.7016i	−0.411498	−	0.408885i
$820$		0			0
$821$		−	46.9375i		−	1.63813i	−0.573702	−	0.819064i	$-0.694493\pi$
$821$		−	46.9375i		−	1.63813i	0.573702	−	0.819064i	$-0.305507\pi$
$822$		0			0
$823$	−13.4979			−0.470509			−0.235254	−	0.971934i	$-0.575592\pi$
$823$	−13.4979			−0.470509			−0.235254	+	0.971934i	$0.575592\pi$
$824$		0			0
$825$			2.95137i			0.102754i
$826$		0			0
$827$		−	45.5564i		−	1.58415i	−0.610422	−	0.792076i	$-0.709000\pi$
$827$		−	45.5564i		−	1.58415i	0.610422	−	0.792076i	$-0.291000\pi$
$828$		0			0
$829$	37.9162			1.31689			0.658443	−	0.752631i	$-0.271215\pi$
$829$	37.9162			1.31689			0.658443	+	0.752631i	$0.271215\pi$
$830$		0			0
$831$	25.6254			0.888934
$832$		0			0
$833$	7.39189			0.256114
$834$		0			0
$835$	−9.05176			−0.313249
$836$		0			0
$837$		−	15.9496i		−	0.551300i
$838$		0			0
$839$		−	26.2968i		−	0.907865i	−0.891036	−	0.453933i	$-0.850021\pi$
$839$		−	26.2968i		−	0.907865i	0.891036	−	0.453933i	$-0.149979\pi$
$840$		0			0
$841$	−12.4304			−0.428636
$842$		0			0
$843$			75.4387i			2.59825i
$844$		0			0
$845$	25.7701	−	0.164111i	0.886518	−	0.00564558i
$846$		0			0
$847$		−	1.00000i		−	0.0343604i
$848$		0			0
$849$	−52.4204			−1.79907
$850$		0			0
$851$		−	44.6808i		−	1.53164i
$852$		0			0
$853$		−	29.3493i		−	1.00490i	−0.864606	−	0.502450i	$-0.832432\pi$
$853$		−	29.3493i		−	1.00490i	0.864606	−	0.502450i	$-0.167568\pi$
$854$		0			0
$855$	38.3165			1.31040
$856$		0			0
$857$	−43.0706			−1.47126			−0.735632	−	0.677382i	$-0.763115\pi$
$857$	−43.0706			−1.47126			−0.735632	+	0.677382i	$0.763115\pi$
$858$		0			0
$859$	55.4060			1.89043			0.945215	−	0.326450i	$-0.105852\pi$
$859$	55.4060			1.89043			0.945215	+	0.326450i	$0.105852\pi$
$860$		0			0
$861$	−14.9783			−0.510458
$862$		0			0
$863$			38.3524i			1.30553i	0.757560	+	0.652766i	$0.226391\pi$
$863$			38.3524i			1.30553i	−0.757560	+	0.652766i	$0.773609\pi$
$864$		0			0
$865$			41.7100i			1.41818i
$866$		0			0
$867$	−103.796			−3.52511
$868$		0			0
$869$			13.5068i			0.458188i
$870$		0			0
$871$	−32.8228	−	32.6144i	−1.11216	−	1.10510i
$872$		0			0
$873$		−	70.5692i		−	2.38840i
$874$		0			0
$875$	−12.0334			−0.406804
$876$		0			0
$877$		−	0.791681i		−	0.0267332i	−0.999911	−	0.0133666i	$-0.995745\pi$
$877$		−	0.791681i		−	0.0267332i	0.999911	−	0.0133666i	$-0.00425484\pi$
$878$		0			0
$879$		−	25.1440i		−	0.848088i
$880$		0			0
$881$	−11.6101			−0.391153			−0.195577	−	0.980688i	$-0.562658\pi$
$881$	−11.6101			−0.391153			−0.195577	+	0.980688i	$0.562658\pi$
$882$		0			0
$883$	25.4466			0.856347			0.428174	−	0.903696i	$-0.359157\pi$
$883$	25.4466			0.856347			0.428174	+	0.903696i	$0.359157\pi$
$884$		0			0
$885$	−10.7680			−0.361961
$886$		0			0
$887$	19.0189			0.638594			0.319297	−	0.947655i	$-0.396553\pi$
$887$	19.0189			0.638594			0.319297	+	0.947655i	$0.396553\pi$
$888$		0			0
$889$		−	4.60540i		−	0.154460i
$890$		0			0
$891$			1.61263i			0.0540250i
$892$		0			0
$893$	−1.19234			−0.0399000
$894$		0			0
$895$		−	10.1802i		−	0.340287i
$896$		0			0
$897$	32.3722	−	32.5790i	1.08088	−	1.08778i
$898$		0			0
$899$			14.6743i			0.489415i
$900$		0			0
$901$	33.0376			1.10064
$902$		0			0
$903$		−	34.1680i		−	1.13704i
$904$		0			0
$905$			0.450452i			0.0149735i
$906$		0			0
$907$	21.7983			0.723802			0.361901	−	0.932217i	$-0.382128\pi$
$907$	21.7983			0.723802			0.361901	+	0.932217i	$0.382128\pi$
$908$		0			0
$909$	−31.7771			−1.05398
$910$		0			0
$911$	−25.8658			−0.856972			−0.428486	−	0.903548i	$-0.640953\pi$
$911$	−25.8658			−0.856972			−0.428486	+	0.903548i	$0.640953\pi$
$912$		0			0
$913$	3.32443			0.110022
$914$		0			0
$915$			14.7130i			0.486397i
$916$		0			0
$917$		−	11.5436i		−	0.381202i
$918$		0			0
$919$	13.5222			0.446055			0.223028	−	0.974812i	$-0.428406\pi$
$919$	13.5222			0.446055			0.223028	+	0.974812i	$0.428406\pi$
$920$		0			0
$921$		−	47.8111i		−	1.57543i
$922$		0			0
$923$	20.4825	+	20.3525i	0.674191	+	0.669911i
$924$		0			0
$925$		−	10.3524i		−	0.340385i
$926$		0			0
$927$	60.4725			1.98618
$928$		0			0
$929$			51.1065i			1.67675i	0.545095	+	0.838374i	$0.316494\pi$
$929$			51.1065i			1.67675i	−0.545095	+	0.838374i	$0.683506\pi$
$930$		0			0
$931$			4.19788i			0.137580i
$932$		0			0
$933$	−7.77118			−0.254417
$934$		0			0
$935$	−14.6533			−0.479216
$936$		0			0
$937$	−11.0680			−0.361575			−0.180788	−	0.983522i	$-0.557865\pi$
$937$	−11.0680			−0.361575			−0.180788	+	0.983522i	$0.557865\pi$
$938$		0			0
$939$	72.8215			2.37644
$940$		0			0
$941$		−	32.5464i		−	1.06098i	−0.847690	−	0.530492i	$-0.822007\pi$
$941$		−	32.5464i		−	1.06098i	0.847690	−	0.530492i	$-0.177993\pi$
$942$		0			0
$943$		−	25.0899i		−	0.817040i
$944$		0			0
$945$	8.77062			0.285308
$946$		0			0
$947$		−	11.6146i		−	0.377422i	−0.982033	−	0.188711i	$-0.939569\pi$
$947$		−	11.6146i		−	0.377422i	0.982033	−	0.188711i	$-0.0604310\pi$
$948$		0			0
$949$	1.38717	+	1.37836i	0.0450294	+	0.0447435i
$950$		0			0
$951$		−	11.0090i		−	0.356990i
$952$		0			0
$953$	−34.5698			−1.11982			−0.559912	−	0.828552i	$-0.689165\pi$
$953$	−34.5698			−1.11982			−0.559912	+	0.828552i	$0.689165\pi$
$954$		0			0
$955$			30.8173i			0.997226i
$956$		0			0
$957$			11.2251i			0.362854i
$958$		0			0
$959$	−7.68246			−0.248080
$960$		0			0
$961$	18.0042			0.580780
$962$		0			0
$963$	77.8079			2.50732
$964$		0			0
$965$	−49.5096			−1.59377
$966$		0			0
$967$		−	17.4031i		−	0.559646i	−0.960052	−	0.279823i	$-0.909724\pi$
$967$		−	17.4031i		−	0.559646i	0.960052	−	0.279823i	$-0.0902758\pi$
$968$		0			0
$969$		−	85.5693i		−	2.74888i
$970$		0			0
$971$	−24.5169			−0.786785			−0.393393	−	0.919371i	$-0.628699\pi$
$971$	−24.5169			−0.786785			−0.393393	+	0.919371i	$0.628699\pi$
$972$		0			0
$973$			1.55906i			0.0499810i
$974$		0			0
$975$	7.50056	−	7.54848i	0.240210	−	0.241745i
$976$		0			0
$977$		−	6.89582i		−	0.220617i	−0.993897	−	0.110308i	$-0.964816\pi$
$977$		−	6.89582i		−	0.220617i	0.993897	−	0.110308i	$-0.0351839\pi$
$978$		0			0
$979$	−14.4386			−0.461458
$980$		0			0
$981$			58.2451i			1.85962i
$982$		0			0
$983$			4.02126i			0.128258i	0.997942	+	0.0641292i	$0.0204270\pi$
$983$			4.02126i			0.128258i	−0.997942	+	0.0641292i	$0.979573\pi$
$984$		0			0
$985$	−40.6287			−1.29454
$986$		0			0
$987$	−0.783253			−0.0249312
$988$		0			0
$989$	57.2344			1.81995
$990$		0			0
$991$	−43.3571			−1.37728			−0.688642	−	0.725102i	$-0.741793\pi$
$991$	−43.3571			−1.37728			−0.688642	+	0.725102i	$0.741793\pi$
$992$		0			0
$993$		−	60.3973i		−	1.91665i
$994$		0			0
$995$			4.33092i			0.137299i
$996$		0			0
$997$	−14.6297			−0.463328			−0.231664	−	0.972796i	$-0.574417\pi$
$997$	−14.6297			−0.463328			−0.231664	+	0.972796i	$0.574417\pi$
$998$		0			0
$999$			42.7956i			1.35399i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4004.2.m.c.2157.5	✓	36
13.12	even	2	inner	4004.2.m.c.2157.6	yes	36

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4004.2.m.c.2157.5	✓	36	1.1	even	1	trivial
4004.2.m.c.2157.6	yes	36	13.12	even	2	inner

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:	\( N \)	\(=\)	\( 4004 = 2^{2} \cdot 7 \cdot 11 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4004.m (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-2.75761 q^{3} +1.98236i q^{5} +1.00000i q^{7} +4.60441 q^{9} +O(q^{10})\)	\(q-2.75761 q^{3} +1.98236i q^{5} +1.00000i q^{7} +4.60441 q^{9} -1.00000i q^{11} +(-2.54138 + 2.55761i) q^{13} -5.46656i q^{15} -7.39189 q^{17} -4.19788i q^{19} -2.75761i q^{21} +4.61924 q^{23} +1.07027 q^{25} -4.42434 q^{27} +4.07057 q^{29} +3.60497i q^{31} +2.75761i q^{33} -1.98236 q^{35} -9.67276i q^{37} +(7.00813 - 7.05290i) q^{39} -5.43161i q^{41} +12.3904 q^{43} +9.12758i q^{45} -0.284033i q^{47} -1.00000 q^{49} +20.3839 q^{51} -4.46945 q^{53} +1.98236 q^{55} +11.5761i q^{57} -1.96978i q^{59} -2.69145 q^{61} +4.60441i q^{63} +(-5.07010 - 5.03792i) q^{65} +12.8334i q^{67} -12.7381 q^{69} -8.00846i q^{71} -0.542368i q^{73} -2.95137 q^{75} +1.00000 q^{77} -13.5068 q^{79} -1.61263 q^{81} +3.32443i q^{83} -14.6533i q^{85} -11.2251 q^{87} -14.4386i q^{89} +(-2.55761 - 2.54138i) q^{91} -9.94111i q^{93} +8.32168 q^{95} -15.3264i q^{97} -4.60441i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 36 q - 4 q^{3} + 40 q^{9}+O(q^{10}) \) 36 * q - 4 * q^3 + 40 * q^9	\( 36 q - 4 q^{3} + 40 q^{9} - 4 q^{17} + 8 q^{23} - 80 q^{25} + 8 q^{27} + 8 q^{29} - 24 q^{39} + 32 q^{43} - 36 q^{49} - 20 q^{51} + 12 q^{53} + 32 q^{61} - 24 q^{65} + 80 q^{69} - 36 q^{75} + 36 q^{77} + 16 q^{79} + 132 q^{81} + 8 q^{91} + 56 q^{95}+O(q^{100}) \) 36 * q - 4 * q^3 + 40 * q^9 - 4 * q^17 + 8 * q^23 - 80 * q^25 + 8 * q^27 + 8 * q^29 - 24 * q^39 + 32 * q^43 - 36 * q^49 - 20 * q^51 + 12 * q^53 + 32 * q^61 - 24 * q^65 + 80 * q^69 - 36 * q^75 + 36 * q^77 + 16 * q^79 + 132 * q^81 + 8 * q^91 + 56 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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