LMFDB - Embedded newform 416.2.e.c.337.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [416,2,Mod(337,416)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(416, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("416.337");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 416 = 2^{5} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	416.e (of order $2$, degree $1$, not 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:	$3.32177672409$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.4521217600.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + x^{6} - 2x^{4} + 4x^{2} + 16 $ x^8 + x^6 - 2x^4 + 4x^2 + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	no (minimal twist has level 104)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			337.1
Root			$-1.29437 + 0.569745i$ of defining polynomial
Character	$\chi$	$=$	416.337
Dual form			416.2.e.c.337.7

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.94984i q^{3} -1.81616 q^{5} +1.13949i q^{7} -5.70156 q^{9} -4.40490 q^{11} +(-2.58874 - 2.50967i) q^{13} +5.35738i q^{15} +0.701562 q^{17} +5.95005 q^{19} +3.36131 q^{21} -4.00000 q^{23} -1.70156 q^{25} +7.96918i q^{27} +5.01934i q^{29} -8.77585i q^{31} +12.9937i q^{33} -2.06950i q^{35} -3.36131 q^{37} +(-7.40312 + 7.63636i) q^{39} -2.94984i q^{43} +10.3550 q^{45} +1.13949i q^{47} +5.70156 q^{49} -2.06950i q^{51} -11.7994i q^{53} +8.00000 q^{55} -17.5517i q^{57} -5.95005 q^{59} -6.49687i q^{63} +(4.70156 + 4.55796i) q^{65} -7.49521 q^{67} +11.7994i q^{69} -11.8543i q^{71} -8.43579i q^{73} +5.01934i q^{75} -5.01934i q^{77} +10.8062 q^{79} +6.40312 q^{81} -2.85974 q^{83} -1.27415 q^{85} +14.8062 q^{87} +8.43579i q^{89} +(2.85974 - 2.94984i) q^{91} -25.8874 q^{93} -10.8062 q^{95} -12.9937i q^{97} +25.1148 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 20 q^{9} - 20 q^{17} - 32 q^{23} + 12 q^{25} - 8 q^{39} + 20 q^{49} + 64 q^{55} + 12 q^{65} - 16 q^{79} + 16 q^{87} + 16 q^{95}+O(q^{100}) $ 8 * q - 20 * q^9 - 20 * q^17 - 32 * q^23 + 12 * q^25 - 8 * q^39 + 20 * q^49 + 64 * q^55 + 12 * q^65 - 16 * q^79 + 16 * q^87 + 16 * q^95

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/416\mathbb{Z}\right)^\times$.

$n$	$261$	$287$	$353$
$\chi(n)$	$-1$	$1$	$-1$

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
$3$		−	2.94984i		−	1.70309i	−0.524280	−	0.851546i	$-0.675666\pi$
$3$		−	2.94984i		−	1.70309i	0.524280	−	0.851546i	$-0.324334\pi$
$4$		0			0
$5$	−1.81616			−0.812212			−0.406106	−	0.913826i	$-0.633114\pi$
$5$	−1.81616			−0.812212			−0.406106	+	0.913826i	$0.633114\pi$
$6$		0			0
$7$			1.13949i			0.430687i	0.976538	+	0.215343i	$0.0690871\pi$
$7$			1.13949i			0.430687i	−0.976538	+	0.215343i	$0.930913\pi$
$8$		0			0
$9$	−5.70156			−1.90052
$10$		0			0
$11$	−4.40490			−1.32813			−0.664063	−	0.747676i	$-0.731169\pi$
$11$	−4.40490			−1.32813			−0.664063	+	0.747676i	$0.731169\pi$
$12$		0			0
$13$	−2.58874	−	2.50967i	−0.717987	−	0.696057i
$13$	−2.58874	−	2.50967i	−0.717987	−	0.696057i
$14$		0			0
$15$			5.35738i			1.38327i
$16$		0			0
$17$	0.701562			0.170154			0.0850769	−	0.996374i	$-0.472886\pi$
$17$	0.701562			0.170154			0.0850769	+	0.996374i	$0.472886\pi$
$18$		0			0
$19$	5.95005			1.36504			0.682518	−	0.730869i	$-0.260885\pi$
$19$	5.95005			1.36504			0.682518	+	0.730869i	$0.260885\pi$
$20$		0			0
$21$	3.36131			0.733499
$22$		0			0
$23$	−4.00000			−0.834058			−0.417029	−	0.908893i	$-0.636929\pi$
$23$	−4.00000			−0.834058			−0.417029	+	0.908893i	$0.636929\pi$
$24$		0			0
$25$	−1.70156			−0.340312
$26$		0			0
$27$			7.96918i			1.53367i
$28$		0			0
$29$			5.01934i			0.932068i	0.884767	+	0.466034i	$0.154317\pi$
$29$			5.01934i			0.932068i	−0.884767	+	0.466034i	$0.845683\pi$
$30$		0			0
$31$		−	8.77585i		−	1.57619i	−0.615554	−	0.788095i	$-0.711068\pi$
$31$		−	8.77585i		−	1.57619i	0.615554	−	0.788095i	$-0.288932\pi$
$32$		0			0
$33$			12.9937i			2.26192i
$34$		0			0
$35$		−	2.06950i		−	0.349809i
$36$		0			0
$37$	−3.36131			−0.552597			−0.276298	−	0.961072i	$-0.589108\pi$
$37$	−3.36131			−0.552597			−0.276298	+	0.961072i	$0.589108\pi$
$38$		0			0
$39$	−7.40312	+	7.63636i	−1.18545	+	1.22280i
$40$		0			0
$41$		0			0		1.00000			$0$
$41$		0			0		−1.00000			$\pi$
$42$		0			0
$43$		−	2.94984i		−	0.449847i	−0.974376	−	0.224923i	$-0.927787\pi$
$43$		−	2.94984i		−	0.449847i	0.974376	−	0.224923i	$-0.0722131\pi$
$44$		0			0
$45$	10.3550			1.54362
$46$		0			0
$47$			1.13949i			0.166212i	0.996541	+	0.0831059i	$0.0264840\pi$
$47$			1.13949i			0.166212i	−0.996541	+	0.0831059i	$0.973516\pi$
$48$		0			0
$49$	5.70156			0.814509
$50$		0			0
$51$		−	2.06950i		−	0.289788i
$52$		0			0
$53$		−	11.7994i		−	1.62077i	−0.585900	−	0.810384i	$-0.699259\pi$
$53$		−	11.7994i		−	1.62077i	0.585900	−	0.810384i	$-0.300741\pi$
$54$		0			0
$55$	8.00000			1.07872
$56$		0			0
$57$		−	17.5517i		−	2.32478i
$58$		0			0
$59$	−5.95005			−0.774631			−0.387315	−	0.921947i	$-0.626598\pi$
$59$	−5.95005			−0.774631			−0.387315	+	0.921947i	$0.626598\pi$
$60$		0			0
$61$		0			0		1.00000			$0$
$61$		0			0		−1.00000			$\pi$
$62$		0			0
$63$		−	6.49687i		−	0.818529i
$64$		0			0
$65$	4.70156	+	4.55796i	0.583157	+	0.565345i
$66$		0			0
$67$	−7.49521			−0.915685			−0.457843	−	0.889033i	$-0.651378\pi$
$67$	−7.49521			−0.915685			−0.457843	+	0.889033i	$0.651378\pi$
$68$		0			0
$69$			11.7994i			1.42048i
$70$		0			0
$71$		−	11.8543i		−	1.40684i	−0.710774	−	0.703421i	$-0.751655\pi$
$71$		−	11.8543i		−	1.40684i	0.710774	−	0.703421i	$-0.248345\pi$
$72$		0			0
$73$		−	8.43579i		−	0.987334i	−0.869651	−	0.493667i	$-0.835656\pi$
$73$		−	8.43579i		−	0.987334i	0.869651	−	0.493667i	$-0.164344\pi$
$74$		0			0
$75$			5.01934i			0.579583i
$76$		0			0
$77$		−	5.01934i		−	0.572007i
$78$		0			0
$79$	10.8062			1.21580			0.607899	−	0.794014i	$-0.292013\pi$
$79$	10.8062			1.21580			0.607899	+	0.794014i	$0.292013\pi$
$80$		0			0
$81$	6.40312			0.711458
$82$		0			0
$83$	−2.85974			−0.313898			−0.156949	−	0.987607i	$-0.550166\pi$
$83$	−2.85974			−0.313898			−0.156949	+	0.987607i	$0.550166\pi$
$84$		0			0
$85$	−1.27415			−0.138201
$86$		0			0
$87$	14.8062			1.58740
$88$		0			0
$89$			8.43579i			0.894192i	0.894486	+	0.447096i	$0.147542\pi$
$89$			8.43579i			0.894192i	−0.894486	+	0.447096i	$0.852458\pi$
$90$		0			0
$91$	2.85974	−	2.94984i	0.299783	−	0.309227i
$92$		0			0
$93$	−25.8874			−2.68440
$94$		0			0
$95$	−10.8062			−1.10870
$96$		0			0
$97$		−	12.9937i		−	1.31932i	−0.751566	−	0.659658i	$-0.770701\pi$
$97$		−	12.9937i		−	1.31932i	0.751566	−	0.659658i	$-0.229299\pi$
$98$		0			0
$99$	25.1148			2.52413
$100$		0			0
$101$		0			0		1.00000			$0$
$101$		0			0		−1.00000			$\pi$
$102$		0			0
$103$	−4.00000			−0.394132			−0.197066	−	0.980390i	$-0.563141\pi$
$103$	−4.00000			−0.394132			−0.197066	+	0.980390i	$0.563141\pi$
$104$		0			0
$105$	−6.10469			−0.595756
$106$		0			0
$107$		−	0.880344i		−	0.0851061i	−0.999094	−	0.0425530i	$-0.986451\pi$
$107$		−	0.880344i		−	0.0851061i	0.999094	−	0.0425530i	$-0.0135491\pi$
$108$		0			0
$109$	6.99364			0.669869			0.334934	−	0.942241i	$-0.391286\pi$
$109$	6.99364			0.669869			0.334934	+	0.942241i	$0.391286\pi$
$110$		0			0
$111$			9.91534i			0.941123i
$112$		0			0
$113$	−1.40312			−0.131995			−0.0659974	−	0.997820i	$-0.521023\pi$
$113$	−1.40312			−0.131995			−0.0659974	+	0.997820i	$0.521023\pi$
$114$		0			0
$115$	7.26464			0.677431
$116$		0			0
$117$	14.7598	+	14.3090i	1.36455	+	1.32287i
$118$		0			0
$119$			0.799423i			0.0732830i
$120$		0			0
$121$	8.40312			0.763920
$122$		0			0
$123$		0			0
$124$		0			0
$125$	12.1711			1.08862
$126$		0			0
$127$	12.0000			1.06483			0.532414	−	0.846484i	$-0.321285\pi$
$127$	12.0000			1.06483			0.532414	+	0.846484i	$0.321285\pi$
$128$		0			0
$129$	−8.70156			−0.766130
$130$		0			0
$131$		−	1.18915i		−	0.103897i	−0.998650	−	0.0519484i	$-0.983457\pi$
$131$		−	1.18915i		−	0.103897i	0.998650	−	0.0519484i	$-0.0165431\pi$
$132$		0			0
$133$			6.78003i			0.587903i
$134$		0			0
$135$		−	14.4733i		−	1.24566i
$136$		0			0
$137$			4.55796i			0.389413i	0.980862	+	0.194706i	$0.0623754\pi$
$137$			4.55796i			0.389413i	−0.980862	+	0.194706i	$0.937625\pi$
$138$		0			0
$139$			18.8882i			1.60208i	0.598613	+	0.801038i	$0.295719\pi$
$139$			18.8882i			1.60208i	−0.598613	+	0.801038i	$0.704281\pi$
$140$		0			0
$141$	3.36131			0.283074
$142$		0			0
$143$	11.4031	+	11.0548i	0.953577	+	0.924452i
$144$		0			0
$145$		−	9.11592i		−	0.757036i
$146$		0			0
$147$		−	16.8187i		−	1.38718i
$148$		0			0
$149$	−2.08717			−0.170987			−0.0854936	−	0.996339i	$-0.527247\pi$
$149$	−2.08717			−0.170987			−0.0854936	+	0.996339i	$0.527247\pi$
$150$		0			0
$151$			5.69745i			0.463652i	0.972757	+	0.231826i	$0.0744700\pi$
$151$			5.69745i			0.463652i	−0.972757	+	0.231826i	$0.925530\pi$
$152$		0			0
$153$	−4.00000			−0.323381
$154$		0			0
$155$			15.9384i			1.28020i
$156$		0			0
$157$			16.8187i			1.34228i	0.741331	+	0.671139i	$0.234195\pi$
$157$			16.8187i			1.34228i	−0.741331	+	0.671139i	$0.765805\pi$
$158$		0			0
$159$	−34.8062			−2.76032
$160$		0			0
$161$		−	4.55796i		−	0.359218i
$162$		0			0
$163$	−2.85974			−0.223992			−0.111996	−	0.993709i	$-0.535724\pi$
$163$	−2.85974			−0.223992			−0.111996	+	0.993709i	$0.535724\pi$
$164$		0			0
$165$		−	23.5987i		−	1.83716i
$166$		0			0
$167$			4.21789i			0.326390i	0.986594	+	0.163195i	$0.0521800\pi$
$167$			4.21789i			0.326390i	−0.986594	+	0.163195i	$0.947820\pi$
$168$		0			0
$169$	0.403124	+	12.9937i	0.0310096	+	0.999519i
$170$		0			0
$171$	−33.9246			−2.59428
$172$		0			0
$173$		−	11.7994i		−	0.897089i	−0.893760	−	0.448545i	$-0.851943\pi$
$173$		−	11.7994i		−	0.897089i	0.893760	−	0.448545i	$-0.148057\pi$
$174$		0			0
$175$		−	1.93891i		−	0.146568i
$176$		0			0
$177$			17.5517i			1.31927i
$178$		0			0
$179$		−	12.9885i		−	0.970807i	−0.874290	−	0.485404i	$-0.838673\pi$
$179$		−	12.9885i		−	0.970807i	0.874290	−	0.485404i	$-0.161327\pi$
$180$		0			0
$181$		−	6.78003i		−	0.503955i	−0.967733	−	0.251978i	$-0.918919\pi$
$181$		−	6.78003i		−	0.503955i	0.967733	−	0.251978i	$-0.0810810\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	6.10469			0.448825
$186$		0			0
$187$	−3.09031			−0.225986
$188$		0			0
$189$	−9.08080			−0.660531
$190$		0			0
$191$	−22.8062			−1.65020			−0.825101	−	0.564985i	$-0.808882\pi$
$191$	−22.8062			−1.65020			−0.825101	+	0.564985i	$0.808882\pi$
$192$		0			0
$193$		−	8.43579i		−	0.607221i	−0.952796	−	0.303611i	$-0.901808\pi$
$193$		−	8.43579i		−	0.607221i	0.952796	−	0.303611i	$-0.0981922\pi$
$194$		0			0
$195$	13.4453	−	13.8689i	0.962835	−	0.993170i
$196$		0			0
$197$	−3.36131			−0.239484			−0.119742	−	0.992805i	$-0.538207\pi$
$197$	−3.36131			−0.239484			−0.119742	+	0.992805i	$0.538207\pi$
$198$		0			0
$199$		0			0			−	1.00000i	$-0.5\pi$
$199$		0			0				1.00000i	$0.5\pi$
$200$		0			0
$201$			22.1097i			1.55950i
$202$		0			0
$203$	−5.71949			−0.401429
$204$		0			0
$205$		0			0
$206$		0			0
$207$	22.8062			1.58514
$208$		0			0
$209$	−26.2094			−1.81294
$210$		0			0
$211$		−	1.18915i		−	0.0818646i	−0.999162	−	0.0409323i	$-0.986967\pi$
$211$		−	1.18915i		−	0.0818646i	0.999162	−	0.0409323i	$-0.0130328\pi$
$212$		0			0
$213$	−34.9682			−2.39598
$214$		0			0
$215$			5.35738i			0.365371i
$216$		0			0
$217$	10.0000			0.678844
$218$		0			0
$219$	−24.8842			−1.68152
$220$		0			0
$221$	−1.81616	−	1.76069i	−0.122168	−	0.118437i
$222$		0			0
$223$		−	20.9702i		−	1.40427i	−0.712045	−	0.702134i	$-0.752231\pi$
$223$		−	20.9702i		−	1.40427i	0.712045	−	0.702134i	$-0.247769\pi$
$224$		0			0
$225$	9.70156			0.646771
$226$		0			0
$227$	16.3050			1.08220			0.541101	−	0.840958i	$-0.318008\pi$
$227$	16.3050			1.08220			0.541101	+	0.840958i	$0.318008\pi$
$228$		0			0
$229$	18.8937			1.24853			0.624267	−	0.781211i	$-0.285398\pi$
$229$	18.8937			1.24853			0.624267	+	0.781211i	$0.285398\pi$
$230$		0			0
$231$	−14.8062			−0.974180
$232$		0			0
$233$	12.1047			0.793004			0.396502	−	0.918034i	$-0.370224\pi$
$233$	12.1047			0.793004			0.396502	+	0.918034i	$0.370224\pi$
$234$		0			0
$235$		−	2.06950i		−	0.134999i
$236$		0			0
$237$		−	31.8767i		−	2.07062i
$238$		0			0
$239$			9.57528i			0.619373i	0.950839	+	0.309687i	$0.100224\pi$
$239$			9.57528i			0.619373i	−0.950839	+	0.309687i	$0.899776\pi$
$240$		0			0
$241$		−	17.5517i		−	1.13060i	−0.824884	−	0.565302i	$-0.808759\pi$
$241$		−	17.5517i		−	1.13060i	0.824884	−	0.565302i	$-0.191241\pi$
$242$		0			0
$243$			5.01934i			0.321991i
$244$		0			0
$245$	−10.3550			−0.661554
$246$		0			0
$247$	−15.4031	−	14.9327i	−0.980077	−	0.950142i
$248$		0			0
$249$			8.43579i			0.534596i
$250$		0			0
$251$			9.15833i			0.578069i	0.957319	+	0.289034i	$0.0933342\pi$
$251$			9.15833i			0.578069i	−0.957319	+	0.289034i	$0.906666\pi$
$252$		0			0
$253$	17.6196			1.10773
$254$		0			0
$255$			3.75854i			0.235369i
$256$		0			0
$257$	−25.5078			−1.59113			−0.795567	−	0.605866i	$-0.792827\pi$
$257$	−25.5078			−1.59113			−0.795567	+	0.605866i	$0.792827\pi$
$258$		0			0
$259$		−	3.83019i		−	0.237996i
$260$		0			0
$261$		−	28.6181i		−	1.77141i
$262$		0			0
$263$	16.0000			0.986602			0.493301	−	0.869859i	$-0.335790\pi$
$263$	16.0000			0.986602			0.493301	+	0.869859i	$0.335790\pi$
$264$		0			0
$265$			21.4295i			1.31641i
$266$		0			0
$267$	24.8842			1.52289
$268$		0			0
$269$			18.5794i			1.13281i	0.824129	+	0.566403i	$0.191665\pi$
$269$			18.5794i			1.13281i	−0.824129	+	0.566403i	$0.808335\pi$
$270$		0			0
$271$			18.0111i			1.09409i	0.837102	+	0.547047i	$0.184248\pi$
$271$			18.0111i			1.09409i	−0.837102	+	0.547047i	$0.815752\pi$
$272$		0			0
$273$	−8.70156	−	8.43579i	−0.526642	−	0.510557i
$274$		0			0
$275$	7.49521			0.451978
$276$		0			0
$277$			23.5987i			1.41791i	0.705254	+	0.708955i	$0.250833\pi$
$277$			23.5987i			1.41791i	−0.705254	+	0.708955i	$0.749167\pi$
$278$		0			0
$279$			50.0361i			2.99558i
$280$		0			0
$281$			17.5517i			1.04705i	0.852011	+	0.523524i	$0.175383\pi$
$281$			17.5517i			1.04705i	−0.852011	+	0.523524i	$0.824617\pi$
$282$		0			0
$283$		−	10.9190i		−	0.649068i	−0.945874	−	0.324534i	$-0.894792\pi$
$283$		−	10.9190i		−	0.649068i	0.945874	−	0.324534i	$-0.105208\pi$
$284$		0			0
$285$			31.8767i			1.88821i
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−16.5078			−0.971048
$290$		0			0
$291$	−38.3295			−2.24691
$292$		0			0
$293$	23.5292			1.37459			0.687295	−	0.726378i	$-0.258798\pi$
$293$	23.5292			1.37459			0.687295	+	0.726378i	$0.258798\pi$
$294$		0			0
$295$	10.8062			0.629164
$296$		0			0
$297$		−	35.1034i		−	2.03691i
$298$		0			0
$299$	10.3550	+	10.0387i	0.598842	+	0.580552i
$300$		0			0
$301$	3.36131			0.193743
$302$		0			0
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$	8.57923			0.489642			0.244821	−	0.969568i	$-0.421271\pi$
$307$	8.57923			0.489642			0.244821	+	0.969568i	$0.421271\pi$
$308$		0			0
$309$			11.7994i			0.671242i
$310$		0			0
$311$	−12.0000			−0.680458			−0.340229	−	0.940343i	$-0.610505\pi$
$311$	−12.0000			−0.680458			−0.340229	+	0.940343i	$0.610505\pi$
$312$		0			0
$313$	7.50781			0.424367			0.212183	−	0.977230i	$-0.431943\pi$
$313$	7.50781			0.424367			0.212183	+	0.977230i	$0.431943\pi$
$314$		0			0
$315$			11.7994i			0.664819i
$316$		0			0
$317$	−20.1679			−1.13274			−0.566371	−	0.824151i	$-0.691653\pi$
$317$	−20.1679			−1.13274			−0.566371	+	0.824151i	$0.691653\pi$
$318$		0			0
$319$		−	22.1097i		−	1.23790i
$320$		0			0
$321$	−2.59688			−0.144943
$322$		0			0
$323$	4.17433			0.232266
$324$		0			0
$325$	4.40490	+	4.27036i	0.244340	+	0.236877i
$326$		0			0
$327$		−	20.6301i		−	1.14085i
$328$		0			0
$329$	−1.29844			−0.0715852
$330$		0			0
$331$	23.5696			1.29550			0.647752	−	0.761851i	$-0.275709\pi$
$331$	23.5696			1.29550			0.647752	+	0.761851i	$0.275709\pi$
$332$		0			0
$333$	19.1647			1.05022
$334$		0			0
$335$	13.6125			0.743730
$336$		0			0
$337$	20.7016			1.12769			0.563843	−	0.825882i	$-0.309322\pi$
$337$	20.7016			1.12769			0.563843	+	0.825882i	$0.309322\pi$
$338$		0			0
$339$			4.13899i			0.224799i
$340$		0			0
$341$			38.6567i			2.09338i
$342$		0			0
$343$			14.4733i			0.781485i
$344$		0			0
$345$		−	21.4295i		−	1.15373i
$346$		0			0
$347$		−	24.7879i		−	1.33068i	−0.746539	−	0.665342i	$-0.768286\pi$
$347$		−	24.7879i		−	1.33068i	0.746539	−	0.665342i	$-0.231714\pi$
$348$		0			0
$349$	−32.8810			−1.76008			−0.880040	−	0.474899i	$-0.842484\pi$
$349$	−32.8810			−1.76008			−0.880040	+	0.474899i	$0.842484\pi$
$350$		0			0
$351$	20.0000	−	20.6301i	1.06752	−	1.10115i
$352$		0			0
$353$		−	13.6739i		−	0.727787i	−0.931441	−	0.363894i	$-0.881447\pi$
$353$		−	13.6739i		−	0.727787i	0.931441	−	0.363894i	$-0.118553\pi$
$354$		0			0
$355$			21.5292i			1.14265i
$356$		0			0
$357$	2.35817			0.124808
$358$		0			0
$359$			0.340067i			0.0179481i	0.999960	+	0.00897403i	$0.00285656\pi$
$359$			0.340067i			0.0179481i	−0.999960	+	0.00897403i	$0.997143\pi$
$360$		0			0
$361$	16.4031			0.863322
$362$		0			0
$363$		−	24.7879i		−	1.30103i
$364$		0			0
$365$			15.3207i			0.801924i
$366$		0			0
$367$	−8.00000			−0.417597			−0.208798	−	0.977959i	$-0.566955\pi$
$367$	−8.00000			−0.417597			−0.208798	+	0.977959i	$0.566955\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	13.4453			0.698043
$372$		0			0
$373$		−	18.5794i		−	0.962004i	−0.876719	−	0.481002i	$-0.840273\pi$
$373$		−	18.5794i		−	0.962004i	0.876719	−	0.481002i	$-0.159727\pi$
$374$		0			0
$375$		−	35.9028i		−	1.85401i
$376$		0			0
$377$	12.5969	−	12.9937i	0.648772	−	0.669212i
$378$		0			0
$379$	5.95005			0.305634			0.152817	−	0.988255i	$-0.451166\pi$
$379$	5.95005			0.305634			0.152817	+	0.988255i	$0.451166\pi$
$380$		0			0
$381$		−	35.3981i		−	1.81350i
$382$		0			0
$383$			14.1332i			0.722175i	0.932532	+	0.361087i	$0.117594\pi$
$383$			14.1332i			0.722175i	−0.932532	+	0.361087i	$0.882406\pi$
$384$		0			0
$385$			9.11592i			0.464590i
$386$		0			0
$387$			16.8187i			0.854943i
$388$		0			0
$389$		−	26.8574i		−	1.36172i	−0.732412	−	0.680862i	$-0.761606\pi$
$389$		−	26.8574i		−	1.36172i	0.732412	−	0.680862i	$-0.238394\pi$
$390$		0			0
$391$	−2.80625			−0.141918
$392$		0			0
$393$	−3.50781			−0.176946
$394$		0			0
$395$	−19.6259			−0.987485
$396$		0			0
$397$	3.63232			0.182301			0.0911505	−	0.995837i	$-0.470946\pi$
$397$	3.63232			0.182301			0.0911505	+	0.995837i	$0.470946\pi$
$398$		0			0
$399$	20.0000			1.00125
$400$		0			0
$401$			17.5517i			0.876491i	0.898855	+	0.438245i	$0.144400\pi$
$401$			17.5517i			0.876491i	−0.898855	+	0.438245i	$0.855600\pi$
$402$		0			0
$403$	−22.0245	+	22.7184i	−1.09712	+	1.13168i
$404$		0			0
$405$	−11.6291			−0.577855
$406$		0			0
$407$	14.8062			0.733918
$408$		0			0
$409$			13.6739i			0.676130i	0.941123	+	0.338065i	$0.109772\pi$
$409$			13.6739i			0.676130i	−0.941123	+	0.338065i	$0.890228\pi$
$410$		0			0
$411$	13.4453			0.663206
$412$		0			0
$413$		−	6.78003i		−	0.333623i
$414$		0			0
$415$	5.19375			0.254951
$416$		0			0
$417$	55.7172			2.72848
$418$		0			0
$419$		−	26.5486i		−	1.29698i	−0.761222	−	0.648491i	$-0.775400\pi$
$419$		−	26.5486i		−	1.29698i	0.761222	−	0.648491i	$-0.224600\pi$
$420$		0			0
$421$	9.62281			0.468987			0.234494	−	0.972118i	$-0.424657\pi$
$421$	9.62281			0.468987			0.234494	+	0.972118i	$0.424657\pi$
$422$		0			0
$423$		−	6.49687i		−	0.315889i
$424$		0			0
$425$	−1.19375			−0.0579055
$426$		0			0
$427$		0			0
$428$		0			0
$429$	32.6100	−	33.6374i	1.57443	−	1.62403i
$430$		0			0
$431$		−	29.4060i		−	1.41644i	−0.705994	−	0.708218i	$-0.749499\pi$
$431$		−	29.4060i		−	1.41644i	0.705994	−	0.708218i	$-0.250501\pi$
$432$		0			0
$433$	−29.5078			−1.41805			−0.709027	−	0.705181i	$-0.750866\pi$
$433$	−29.5078			−1.41805			−0.709027	+	0.705181i	$0.750866\pi$
$434$		0			0
$435$	−26.8905			−1.28930
$436$		0			0
$437$	−23.8002			−1.13852
$438$		0			0
$439$	−10.8062			−0.515754			−0.257877	−	0.966178i	$-0.583023\pi$
$439$	−10.8062			−0.515754			−0.257877	+	0.966178i	$0.583023\pi$
$440$		0			0
$441$	−32.5078			−1.54799
$442$		0			0
$443$		−	2.94984i		−	0.140151i	−0.997542	−	0.0700756i	$-0.977676\pi$
$443$		−	2.94984i		−	0.140151i	0.997542	−	0.0700756i	$-0.0223240\pi$
$444$		0			0
$445$		−	15.3207i		−	0.726273i
$446$		0			0
$447$			6.15681i			0.291207i
$448$		0			0
$449$		−	38.9812i		−	1.83964i	−0.392342	−	0.919819i	$-0.628335\pi$
$449$		−	38.9812i		−	1.83964i	0.392342	−	0.919819i	$-0.371665\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$	16.8066			0.789642
$454$		0			0
$455$	−5.19375	+	5.35738i	−0.243487	+	0.251158i
$456$		0			0
$457$		−	30.5455i		−	1.42886i	−0.699709	−	0.714428i	$-0.746687\pi$
$457$		−	30.5455i		−	1.42886i	0.699709	−	0.714428i	$-0.253313\pi$
$458$		0			0
$459$			5.59087i			0.260960i
$460$		0			0
$461$	−10.6260			−0.494900			−0.247450	−	0.968901i	$-0.579593\pi$
$461$	−10.6260			−0.494900			−0.247450	+	0.968901i	$0.579593\pi$
$462$		0			0
$463$			17.2116i			0.799893i	0.916539	+	0.399946i	$0.130971\pi$
$463$			17.2116i			0.799893i	−0.916539	+	0.399946i	$0.869029\pi$
$464$		0			0
$465$	47.0156			2.18030
$466$		0			0
$467$			12.6797i			0.586747i	0.955998	+	0.293373i	$0.0947779\pi$
$467$			12.6797i			0.586747i	−0.955998	+	0.293373i	$0.905222\pi$
$468$		0			0
$469$		−	8.54071i		−	0.394374i
$470$		0			0
$471$	49.6125			2.28602
$472$		0			0
$473$			12.9937i			0.597453i
$474$		0			0
$475$	−10.1244			−0.464539
$476$		0			0
$477$			67.2748i			3.08030i
$478$		0			0
$479$		−	2.73834i		−	0.125118i	−0.998041	−	0.0625589i	$-0.980074\pi$
$479$		−	2.73834i		−	0.125118i	0.998041	−	0.0625589i	$-0.0199261\pi$
$480$		0			0
$481$	8.70156	+	8.43579i	0.396757	+	0.384639i
$482$		0			0
$483$	−13.4453			−0.611780
$484$		0			0
$485$			23.5987i			1.07156i
$486$		0			0
$487$		−	25.6474i		−	1.16220i	−0.813834	−	0.581098i	$-0.802623\pi$
$487$		−	25.6474i		−	1.16220i	0.813834	−	0.581098i	$-0.197377\pi$
$488$		0			0
$489$			8.43579i			0.381479i
$490$		0			0
$491$			30.6876i			1.38491i	0.721461	+	0.692455i	$0.243471\pi$
$491$			30.6876i			1.38491i	−0.721461	+	0.692455i	$0.756529\pi$
$492$		0			0
$493$			3.52138i			0.158595i
$494$		0			0
$495$	−45.6125			−2.05013
$496$		0			0
$497$	13.5078			0.605908
$498$		0			0
$499$	1.77572			0.0794922			0.0397461	−	0.999210i	$-0.487345\pi$
$499$	1.77572			0.0794922			0.0397461	+	0.999210i	$0.487345\pi$
$500$		0			0
$501$	12.4421			0.555873
$502$		0			0
$503$	−25.6125			−1.14200			−0.571002	−	0.820948i	$-0.693445\pi$
$503$	−25.6125			−1.14200			−0.571002	+	0.820948i	$0.693445\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	38.3295	−	1.18915i	1.70227	−	0.0528121i
$508$		0			0
$509$	9.81294			0.434951			0.217475	−	0.976066i	$-0.430218\pi$
$509$	9.81294			0.434951			0.217475	+	0.976066i	$0.430218\pi$
$510$		0			0
$511$	9.61250			0.425232
$512$		0			0
$513$			47.4170i			2.09351i
$514$		0			0
$515$	7.26464			0.320118
$516$		0			0
$517$		−	5.01934i		−	0.220750i
$518$		0			0
$519$	−34.8062			−1.52782
$520$		0			0
$521$	−6.10469			−0.267451			−0.133726	−	0.991018i	$-0.542694\pi$
$521$	−6.10469			−0.267451			−0.133726	+	0.991018i	$0.542694\pi$
$522$		0			0
$523$		−	10.9190i		−	0.477455i	−0.971087	−	0.238728i	$-0.923270\pi$
$523$		−	10.9190i		−	0.477455i	0.971087	−	0.238728i	$-0.0767303\pi$
$524$		0			0
$525$	−5.71949			−0.249619
$526$		0			0
$527$		−	6.15681i		−	0.268195i
$528$		0			0
$529$	−7.00000			−0.304348
$530$		0			0
$531$	33.9246			1.47220
$532$		0			0
$533$		0			0
$534$		0			0
$535$			1.59885i			0.0691242i
$536$		0			0
$537$	−38.3141			−1.65337
$538$		0			0
$539$	−25.1148			−1.08177
$540$		0			0
$541$	−10.6260			−0.456846			−0.228423	−	0.973562i	$-0.573357\pi$
$541$	−10.6260			−0.456846			−0.228423	+	0.973562i	$0.573357\pi$
$542$		0			0
$543$	−20.0000			−0.858282
$544$		0			0
$545$	−12.7016			−0.544075
$546$		0			0
$547$		−	6.47122i		−	0.276689i	−0.990384	−	0.138345i	$-0.955822\pi$
$547$		−	6.47122i		−	0.276689i	0.990384	−	0.138345i	$-0.0441782\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$			29.8653i			1.27231i
$552$		0			0
$553$			12.3136i			0.523628i
$554$		0			0
$555$		−	18.0079i		−	0.764391i
$556$		0			0
$557$	24.6132			1.04290			0.521448	−	0.853283i	$-0.325392\pi$
$557$	24.6132			1.04290			0.521448	+	0.853283i	$0.325392\pi$
$558$		0			0
$559$	−7.40312	+	7.63636i	−0.313119	+	0.322984i
$560$		0			0
$561$			9.11592i			0.384875i
$562$		0			0
$563$		−	16.5099i		−	0.695809i	−0.937530	−	0.347905i	$-0.886893\pi$
$563$		−	16.5099i		−	0.695809i	0.937530	−	0.347905i	$-0.113107\pi$
$564$		0			0
$565$	2.54830			0.107208
$566$		0			0
$567$			7.29630i			0.306416i
$568$		0			0
$569$	−13.5078			−0.566277			−0.283138	−	0.959079i	$-0.591376\pi$
$569$	−13.5078			−0.566277			−0.283138	+	0.959079i	$0.591376\pi$
$570$		0			0
$571$		−	1.18915i		−	0.0497645i	−0.999690	−	0.0248822i	$-0.992079\pi$
$571$		−	1.18915i		−	0.0497645i	0.999690	−	0.0248822i	$-0.00792108\pi$
$572$		0			0
$573$			67.2748i			2.81045i
$574$		0			0
$575$	6.80625			0.283840
$576$		0			0
$577$			39.6614i			1.65112i	0.564311	+	0.825562i	$0.309142\pi$
$577$			39.6614i			1.65112i	−0.564311	+	0.825562i	$0.690858\pi$
$578$		0			0
$579$	−24.8842			−1.03415
$580$		0			0
$581$		−	3.25865i		−	0.135192i
$582$		0			0
$583$			51.9750i			2.15258i
$584$		0			0
$585$	−26.8062	−	25.9875i	−1.10830	−	1.07445i
$586$		0			0
$587$	−11.6695			−0.481653			−0.240827	−	0.970568i	$-0.577419\pi$
$587$	−11.6695			−0.481653			−0.240827	+	0.970568i	$0.577419\pi$
$588$		0			0
$589$		−	52.2168i		−	2.15156i
$590$		0			0
$591$			9.91534i			0.407863i
$592$		0			0
$593$		−	25.9875i		−	1.06718i	−0.845744	−	0.533589i	$-0.820843\pi$
$593$		−	25.9875i		−	1.06718i	0.845744	−	0.533589i	$-0.179157\pi$
$594$		0			0
$595$		−	1.45188i		−	0.0595213i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	20.0000			0.817178			0.408589	−	0.912719i	$-0.366021\pi$
$599$	20.0000			0.817178			0.408589	+	0.912719i	$0.366021\pi$
$600$		0			0
$601$	−26.1047			−1.06483			−0.532416	−	0.846483i	$-0.678716\pi$
$601$	−26.1047			−1.06483			−0.532416	+	0.846483i	$0.678716\pi$
$602$		0			0
$603$	42.7344			1.74028
$604$		0			0
$605$	−15.2614			−0.620465
$606$		0			0
$607$	12.0000			0.487065			0.243532	−	0.969893i	$-0.421694\pi$
$607$	12.0000			0.487065			0.243532	+	0.969893i	$0.421694\pi$
$608$		0			0
$609$			16.8716i			0.683671i
$610$		0			0
$611$	2.85974	−	2.94984i	0.115693	−	0.119338i
$612$		0			0
$613$	−37.3263			−1.50760			−0.753798	−	0.657106i	$-0.771781\pi$
$613$	−37.3263			−1.50760			−0.753798	+	0.657106i	$0.771781\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	48.0972i		−	1.93632i	−0.250334	−	0.968159i	$-0.580540\pi$
$617$		−	48.0972i		−	1.93632i	0.250334	−	0.968159i	$-0.419460\pi$
$618$		0			0
$619$	1.77572			0.0713723			0.0356861	−	0.999363i	$-0.488638\pi$
$619$	1.77572			0.0713723			0.0356861	+	0.999363i	$0.488638\pi$
$620$		0			0
$621$		−	31.8767i		−	1.27917i
$622$		0			0
$623$	−9.61250			−0.385117
$624$		0			0
$625$	−13.5969			−0.543875
$626$		0			0
$627$			77.3135i			3.08760i
$628$		0			0
$629$	−2.35817			−0.0940264
$630$		0			0
$631$		−	29.4060i		−	1.17063i	−0.810805	−	0.585317i	$-0.800970\pi$
$631$		−	29.4060i		−	1.17063i	0.810805	−	0.585317i	$-0.199030\pi$
$632$		0			0
$633$	−3.50781			−0.139423
$634$		0			0
$635$	−21.7939			−0.864865
$636$		0			0
$637$	−14.7598	−	14.3090i	−0.584806	−	0.566945i
$638$		0			0
$639$			67.5878i			2.67373i
$640$		0			0
$641$	37.4031			1.47733			0.738667	−	0.674070i	$-0.235455\pi$
$641$	37.4031			1.47733			0.738667	+	0.674070i	$0.235455\pi$
$642$		0			0
$643$	−7.03407			−0.277397			−0.138698	−	0.990335i	$-0.544292\pi$
$643$	−7.03407			−0.277397			−0.138698	+	0.990335i	$0.544292\pi$
$644$		0			0
$645$	15.8034			0.622259
$646$		0			0
$647$	42.8062			1.68289			0.841444	−	0.540345i	$-0.181706\pi$
$647$	42.8062			1.68289			0.841444	+	0.540345i	$0.181706\pi$
$648$		0			0
$649$	26.2094			1.02881
$650$		0			0
$651$		−	29.4984i		−	1.15613i
$652$		0			0
$653$		−	18.5794i		−	0.727068i	−0.931581	−	0.363534i	$-0.881570\pi$
$653$		−	18.5794i		−	0.727068i	0.931581	−	0.363534i	$-0.118430\pi$
$654$		0			0
$655$			2.15969i			0.0843861i
$656$		0			0
$657$			48.0972i			1.87645i
$658$		0			0
$659$		−	34.5177i		−	1.34462i	−0.740269	−	0.672310i	$-0.765302\pi$
$659$		−	34.5177i		−	1.34462i	0.740269	−	0.672310i	$-0.234698\pi$
$660$		0			0
$661$	24.3422			0.946803			0.473401	−	0.880847i	$-0.343026\pi$
$661$	24.3422			0.946803			0.473401	+	0.880847i	$0.343026\pi$
$662$		0			0
$663$	−5.19375	+	5.35738i	−0.201709	+	0.208064i
$664$		0			0
$665$		−	12.3136i		−	0.477501i
$666$		0			0
$667$		−	20.0774i		−	0.777398i
$668$		0			0
$669$	−61.8587			−2.39160
$670$		0			0
$671$		0			0
$672$		0			0
$673$	−44.9109			−1.73119			−0.865595	−	0.500745i	$-0.833059\pi$
$673$	−44.9109			−1.73119			−0.865595	+	0.500745i	$0.833059\pi$
$674$		0			0
$675$		−	13.5601i		−	0.521927i
$676$		0			0
$677$		−	35.3981i		−	1.36046i	−0.732999	−	0.680230i	$-0.761880\pi$
$677$		−	35.3981i		−	1.36046i	0.732999	−	0.680230i	$-0.238120\pi$
$678$		0			0
$679$	14.8062			0.568212
$680$		0			0
$681$		−	48.0972i		−	1.84309i
$682$		0			0
$683$	41.1892			1.57606			0.788031	−	0.615635i	$-0.211101\pi$
$683$	41.1892			1.57606			0.788031	+	0.615635i	$0.211101\pi$
$684$		0			0
$685$		−	8.27799i		−	0.316286i
$686$		0			0
$687$		−	55.7335i		−	2.12637i
$688$		0			0
$689$	−29.6125	+	30.5455i	−1.12815	+	1.16369i
$690$		0			0
$691$	−4.40490			−0.167570			−0.0837851	−	0.996484i	$-0.526701\pi$
$691$	−4.40490			−0.167570			−0.0837851	+	0.996484i	$0.526701\pi$
$692$		0			0
$693$			28.6181i			1.08711i
$694$		0			0
$695$		−	34.3040i		−	1.30122i
$696$		0			0
$697$		0			0
$698$		0			0
$699$		−	35.7069i		−	1.35056i
$700$		0			0
$701$			6.78003i			0.256078i	0.991769	+	0.128039i	$0.0408683\pi$
$701$			6.78003i			0.256078i	−0.991769	+	0.128039i	$0.959132\pi$
$702$		0			0
$703$	−20.0000			−0.754314
$704$		0			0
$705$	−6.10469			−0.229916
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−46.1361			−1.73268			−0.866340	−	0.499455i	$-0.833533\pi$
$709$	−46.1361			−1.73268			−0.866340	+	0.499455i	$0.833533\pi$
$710$		0			0
$711$	−61.6125			−2.31065
$712$		0			0
$713$			35.1034i			1.31463i
$714$		0			0
$715$	−20.7099	−	20.0774i	−0.774506	−	0.750850i
$716$		0			0
$717$	28.2455			1.05485
$718$		0			0
$719$	30.8062			1.14888			0.574440	−	0.818547i	$-0.305220\pi$
$719$	30.8062			1.14888			0.574440	+	0.818547i	$0.305220\pi$
$720$		0			0
$721$		−	4.55796i		−	0.169747i
$722$		0			0
$723$	−51.7748			−1.92552
$724$		0			0
$725$		−	8.54071i		−	0.317194i
$726$		0			0
$727$	−17.1938			−0.637681			−0.318840	−	0.947808i	$-0.603293\pi$
$727$	−17.1938			−0.637681			−0.318840	+	0.947808i	$0.603293\pi$
$728$		0			0
$729$	34.0156			1.25984
$730$		0			0
$731$		−	2.06950i		−	0.0765431i
$732$		0			0
$733$	11.6291			0.429531			0.214765	−	0.976666i	$-0.431101\pi$
$733$	11.6291			0.429531			0.214765	+	0.976666i	$0.431101\pi$
$734$		0			0
$735$			30.5455i			1.12669i
$736$		0			0
$737$	33.0156			1.21615
$738$		0			0
$739$	−45.8247			−1.68569			−0.842844	−	0.538157i	$-0.819121\pi$
$739$	−45.8247			−1.68569			−0.842844	+	0.538157i	$0.819121\pi$
$740$		0			0
$741$	−44.0490	+	45.4368i	−1.61818	+	1.66916i
$742$		0			0
$743$			1.81962i			0.0667555i	0.999443	+	0.0333778i	$0.0106264\pi$
$743$			1.81962i			0.0667555i	−0.999443	+	0.0333778i	$0.989374\pi$
$744$		0			0
$745$	3.79063			0.138878
$746$		0			0
$747$	16.3050			0.596569
$748$		0			0
$749$	1.00314			0.0366541
$750$		0			0
$751$	−44.4187			−1.62086			−0.810432	−	0.585833i	$-0.800767\pi$
$751$	−44.4187			−1.62086			−0.810432	+	0.585833i	$0.800767\pi$
$752$		0			0
$753$	27.0156			0.984504
$754$		0			0
$755$		−	10.3475i		−	0.376583i
$756$		0			0
$757$		−	8.27799i		−	0.300869i	−0.988620	−	0.150434i	$-0.951933\pi$
$757$		−	8.27799i		−	0.300869i	0.988620	−	0.150434i	$-0.0480672\pi$
$758$		0			0
$759$		−	51.9750i		−	1.88657i
$760$		0			0
$761$			47.4170i			1.71887i	0.511248	+	0.859433i	$0.329183\pi$
$761$			47.4170i			1.71887i	−0.511248	+	0.859433i	$0.670817\pi$
$762$		0			0
$763$			7.96918i			0.288504i
$764$		0			0
$765$	7.26464			0.262654
$766$		0			0
$767$	15.4031	+	14.9327i	0.556175	+	0.539187i
$768$		0			0
$769$		−	3.87783i		−	0.139838i	−0.997553	−	0.0699190i	$-0.977726\pi$
$769$		−	3.87783i		−	0.139838i	0.997553	−	0.0699190i	$-0.0222741\pi$
$770$		0			0
$771$			75.2440i			2.70985i
$772$		0			0
$773$	47.9523			1.72472			0.862362	−	0.506292i	$-0.168984\pi$
$773$	47.9523			1.72472			0.862362	+	0.506292i	$0.168984\pi$
$774$		0			0
$775$			14.9327i			0.536397i
$776$		0			0
$777$	−11.2984			−0.405329
$778$		0			0
$779$		0			0
$780$		0			0
$781$			52.2168i			1.86846i
$782$		0			0
$783$	−40.0000			−1.42948
$784$		0			0
$785$		−	30.5455i		−	1.09021i
$786$		0			0
$787$	−43.1955			−1.53975			−0.769877	−	0.638192i	$-0.779682\pi$
$787$	−43.1955			−1.53975			−0.769877	+	0.638192i	$0.779682\pi$
$788$		0			0
$789$		−	47.1975i		−	1.68027i
$790$		0			0
$791$		−	1.59885i		−	0.0568484i
$792$		0			0
$793$		0			0
$794$		0			0
$795$	63.2137			2.24196
$796$		0			0
$797$		−	10.3014i		−	0.364894i	−0.983216	−	0.182447i	$-0.941598\pi$
$797$		−	10.3014i		−	0.364894i	0.983216	−	0.182447i	$-0.0584019\pi$
$798$		0			0
$799$			0.799423i			0.0282816i
$800$		0			0
$801$		−	48.0972i		−	1.69943i
$802$		0			0
$803$			37.1588i			1.31130i
$804$		0			0
$805$			8.27799i			0.291761i
$806$		0			0
$807$	54.8062			1.92927
$808$		0			0
$809$	23.5078			0.826491			0.413245	−	0.910620i	$-0.364395\pi$
$809$	23.5078			0.826491			0.413245	+	0.910620i	$0.364395\pi$
$810$		0			0
$811$	31.2954			1.09893			0.549465	−	0.835516i	$-0.314831\pi$
$811$	31.2954			1.09893			0.549465	+	0.835516i	$0.314831\pi$
$812$		0			0
$813$	53.1298			1.86334
$814$		0			0
$815$	5.19375			0.181929
$816$		0			0
$817$		−	17.5517i		−	0.614057i
$818$		0			0
$819$	−16.3050	+	16.8187i	−0.569743	+	0.587693i
$820$		0			0
$821$	33.4230			1.16647			0.583236	−	0.812303i	$-0.301786\pi$
$821$	33.4230			1.16647			0.583236	+	0.812303i	$0.301786\pi$
$822$		0			0
$823$	−25.6125			−0.892796			−0.446398	−	0.894835i	$-0.647293\pi$
$823$	−25.6125			−0.892796			−0.446398	+	0.894835i	$0.647293\pi$
$824$		0			0
$825$		−	22.1097i		−	0.769760i
$826$		0			0
$827$	−15.2210			−0.529285			−0.264643	−	0.964347i	$-0.585254\pi$
$827$	−15.2210			−0.529285			−0.264643	+	0.964347i	$0.585254\pi$
$828$		0			0
$829$			43.6761i			1.51693i	0.651712	+	0.758466i	$0.274051\pi$
$829$			43.6761i			1.51693i	−0.651712	+	0.758466i	$0.725949\pi$
$830$		0			0
$831$	69.6125			2.41483
$832$		0			0
$833$	4.00000			0.138592
$834$		0			0
$835$		−	7.66037i		−	0.265098i
$836$		0			0
$837$	69.9364			2.41735
$838$		0			0
$839$			0.340067i			0.0117404i	0.999983	+	0.00587021i	$0.00186856\pi$
$839$			0.340067i			0.0117404i	−0.999983	+	0.00587021i	$0.998131\pi$
$840$		0			0
$841$	3.80625			0.131250
$842$		0			0
$843$	51.7748			1.78322
$844$		0			0
$845$	−0.732138	−	23.5987i	−0.0251863	−	0.811821i
$846$		0			0
$847$			9.57528i			0.329010i
$848$		0			0
$849$	−32.2094			−1.10542
$850$		0			0
$851$	13.4453			0.460898
$852$		0			0
$853$	−16.3454			−0.559657			−0.279829	−	0.960050i	$-0.590278\pi$
$853$	−16.3454			−0.559657			−0.279829	+	0.960050i	$0.590278\pi$
$854$		0			0
$855$	61.6125			2.10710
$856$		0			0
$857$	−12.8062			−0.437453			−0.218727	−	0.975786i	$-0.570190\pi$
$857$	−12.8062			−0.437453			−0.218727	+	0.975786i	$0.570190\pi$
$858$		0			0
$859$		−	20.9577i		−	0.715067i	−0.933900	−	0.357534i	$-0.883618\pi$
$859$		−	20.9577i		−	0.715067i	0.933900	−	0.357534i	$-0.116382\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		−	38.5219i		−	1.31130i	−0.755065	−	0.655650i	$-0.772395\pi$
$863$		−	38.5219i		−	1.31130i	0.755065	−	0.655650i	$-0.227605\pi$
$864$		0			0
$865$			21.4295i			0.728626i
$866$		0			0
$867$			48.6954i			1.65378i
$868$		0			0
$869$	−47.6004			−1.61473
$870$		0			0
$871$	19.4031	+	18.8105i	0.657450	+	0.637369i
$872$		0			0
$873$			74.0847i			2.50739i
$874$		0			0
$875$			13.8689i			0.468853i
$876$		0			0
$877$	56.1392			1.89569			0.947843	−	0.318737i	$-0.103259\pi$
$877$	56.1392			1.89569			0.947843	+	0.318737i	$0.103259\pi$
$878$		0			0
$879$		−	69.4074i		−	2.34105i
$880$		0			0
$881$	29.2984			0.987089			0.493545	−	0.869720i	$-0.335701\pi$
$881$	29.2984			0.987089			0.493545	+	0.869720i	$0.335701\pi$
$882$		0			0
$883$			28.9269i			0.973467i	0.873551	+	0.486733i	$0.161812\pi$
$883$			28.9269i			0.973467i	−0.873551	+	0.486733i	$0.838188\pi$
$884$		0			0
$885$		−	31.8767i		−	1.07152i
$886$		0			0
$887$	−51.2250			−1.71997			−0.859983	−	0.510322i	$-0.829526\pi$
$887$	−51.2250			−1.71997			−0.859983	+	0.510322i	$0.829526\pi$
$888$		0			0
$889$			13.6739i			0.458607i
$890$		0			0
$891$	−28.2051			−0.944907
$892$		0			0
$893$			6.78003i			0.226885i
$894$		0			0
$895$			23.5892i			0.788501i
$896$		0			0
$897$	29.6125	−	30.5455i	0.988732	−	1.01988i
$898$		0			0
$899$	44.0490			1.46912
$900$		0			0
$901$		−	8.27799i		−	0.275780i
$902$		0			0
$903$		−	9.91534i		−	0.329962i
$904$		0			0
$905$			12.3136i			0.409318i
$906$		0			0
$907$		−	43.1045i		−	1.43126i	−0.698478	−	0.715631i	$-0.746139\pi$
$907$		−	43.1045i		−	1.43126i	0.698478	−	0.715631i	$-0.253861\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−32.0000			−1.06021			−0.530104	−	0.847933i	$-0.677847\pi$
$911$	−32.0000			−1.06021			−0.530104	+	0.847933i	$0.677847\pi$
$912$		0			0
$913$	12.5969			0.416896
$914$		0			0
$915$		0			0
$916$		0			0
$917$	1.35503			0.0447470
$918$		0			0
$919$	−40.0000			−1.31948			−0.659739	−	0.751495i	$-0.729333\pi$
$919$	−40.0000			−1.31948			−0.659739	+	0.751495i	$0.729333\pi$
$920$		0			0
$921$		−	25.3074i		−	0.833906i
$922$		0			0
$923$	−29.7503	+	30.6876i	−0.979242	+	1.01009i
$924$		0			0
$925$	5.71949			0.188056
$926$		0			0
$927$	22.8062			0.749055
$928$		0			0
$929$		−	38.9812i		−	1.27893i	−0.768819	−	0.639467i	$-0.779155\pi$
$929$		−	38.9812i		−	1.27893i	0.768819	−	0.639467i	$-0.220845\pi$
$930$		0			0
$931$	33.9246			1.11183
$932$		0			0
$933$			35.3981i			1.15888i
$934$		0			0
$935$	5.61250			0.183548
$936$		0			0
$937$	−12.8062			−0.418362			−0.209181	−	0.977877i	$-0.567080\pi$
$937$	−12.8062			−0.418362			−0.209181	+	0.977877i	$0.567080\pi$
$938$		0			0
$939$		−	22.1468i		−	0.722735i
$940$		0			0
$941$	−26.7004			−0.870408			−0.435204	−	0.900332i	$-0.643324\pi$
$941$	−26.7004			−0.870408			−0.435204	+	0.900332i	$0.643324\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$	16.4922			0.536491
$946$		0			0
$947$	−7.49521			−0.243562			−0.121781	−	0.992557i	$-0.538860\pi$
$947$	−7.49521			−0.243562			−0.121781	+	0.992557i	$0.538860\pi$
$948$		0			0
$949$	−21.1710	+	21.8380i	−0.687241	+	0.708893i
$950$		0			0
$951$			59.4921i			1.92916i
$952$		0			0
$953$	44.5234			1.44226			0.721128	−	0.692802i	$-0.243624\pi$
$953$	44.5234			1.44226			0.721128	+	0.692802i	$0.243624\pi$
$954$		0			0
$955$	41.4198			1.34031
$956$		0			0
$957$	−65.2200			−2.10826
$958$		0			0
$959$	−5.19375			−0.167715
$960$		0			0
$961$	−46.0156			−1.48437
$962$		0			0
$963$			5.01934i			0.161746i
$964$		0			0
$965$			15.3207i			0.493192i
$966$		0			0
$967$			6.37758i			0.205089i	0.994728	+	0.102545i	$0.0326985\pi$
$967$			6.37758i			0.205089i	−0.994728	+	0.102545i	$0.967302\pi$
$968$		0			0
$969$		−	12.3136i		−	0.395570i
$970$		0			0
$971$			17.1275i			0.549648i	0.961495	+	0.274824i	$0.0886196\pi$
$971$			17.1275i			0.549648i	−0.961495	+	0.274824i	$0.911380\pi$
$972$		0			0
$973$	−21.5229			−0.689993
$974$		0			0
$975$	12.5969	−	12.9937i	0.403423	−	0.416133i
$976$		0			0
$977$			22.1097i			0.707351i	0.935368	+	0.353675i	$0.115068\pi$
$977$			22.1097i			0.707351i	−0.935368	+	0.353675i	$0.884932\pi$
$978$		0			0
$979$		−	37.1588i		−	1.18760i
$980$		0			0
$981$	−39.8746			−1.27310
$982$		0			0
$983$		−	56.0736i		−	1.78847i	−0.447598	−	0.894235i	$-0.647720\pi$
$983$		−	56.0736i		−	1.78847i	0.447598	−	0.894235i	$-0.352280\pi$
$984$		0			0
$985$	6.10469			0.194511
$986$		0			0
$987$			3.83019i			0.121916i
$988$		0			0
$989$			11.7994i			0.375198i
$990$		0			0
$991$	9.61250			0.305351			0.152676	−	0.988276i	$-0.451211\pi$
$991$	9.61250			0.305351			0.152676	+	0.988276i	$0.451211\pi$
$992$		0			0
$993$		−	69.5267i		−	2.20636i
$994$		0			0
$995$		0			0
$996$		0			0
$997$		−	3.52138i		−	0.111523i	−0.998444	−	0.0557616i	$-0.982241\pi$
$997$		−	3.52138i		−	0.111523i	0.998444	−	0.0557616i	$-0.0177587\pi$
$998$		0			0
$999$		−	26.7869i		−	0.847501i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	416.2.e.c.337.1		8
3.2	odd	2		3744.2.m.g.1585.6		8
4.3	odd	2		104.2.e.c.77.7	yes	8
8.3	odd	2		104.2.e.c.77.1	✓	8
8.5	even	2	inner	416.2.e.c.337.8		8
12.11	even	2		936.2.m.f.181.2		8
13.12	even	2	inner	416.2.e.c.337.2		8
24.5	odd	2		3744.2.m.g.1585.4		8
24.11	even	2		936.2.m.f.181.8		8
39.38	odd	2		3744.2.m.g.1585.3		8
52.51	odd	2		104.2.e.c.77.2	yes	8
104.51	odd	2		104.2.e.c.77.8	yes	8
104.77	even	2	inner	416.2.e.c.337.7		8
156.155	even	2		936.2.m.f.181.7		8
312.77	odd	2		3744.2.m.g.1585.5		8
312.155	even	2		936.2.m.f.181.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
104.2.e.c.77.1	✓	8	8.3	odd	2
104.2.e.c.77.2	yes	8	52.51	odd	2
104.2.e.c.77.7	yes	8	4.3	odd	2
104.2.e.c.77.8	yes	8	104.51	odd	2
416.2.e.c.337.1		8	1.1	even	1	trivial
416.2.e.c.337.2		8	13.12	even	2	inner
416.2.e.c.337.7		8	104.77	even	2	inner
416.2.e.c.337.8		8	8.5	even	2	inner
936.2.m.f.181.1		8	312.155	even	2
936.2.m.f.181.2		8	12.11	even	2
936.2.m.f.181.7		8	156.155	even	2
936.2.m.f.181.8		8	24.11	even	2
3744.2.m.g.1585.3		8	39.38	odd	2
3744.2.m.g.1585.4		8	24.5	odd	2
3744.2.m.g.1585.5		8	312.77	odd	2
3744.2.m.g.1585.6		8	3.2	odd	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:	\( N \)	\(=\)	\( 416 = 2^{5} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	416.e (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-2.94984i q^{3} -1.81616 q^{5} +1.13949i q^{7} -5.70156 q^{9} -4.40490 q^{11} +(-2.58874 - 2.50967i) q^{13} +5.35738i q^{15} +0.701562 q^{17} +5.95005 q^{19} +3.36131 q^{21} -4.00000 q^{23} -1.70156 q^{25} +7.96918i q^{27} +5.01934i q^{29} -8.77585i q^{31} +12.9937i q^{33} -2.06950i q^{35} -3.36131 q^{37} +(-7.40312 + 7.63636i) q^{39} -2.94984i q^{43} +10.3550 q^{45} +1.13949i q^{47} +5.70156 q^{49} -2.06950i q^{51} -11.7994i q^{53} +8.00000 q^{55} -17.5517i q^{57} -5.95005 q^{59} -6.49687i q^{63} +(4.70156 + 4.55796i) q^{65} -7.49521 q^{67} +11.7994i q^{69} -11.8543i q^{71} -8.43579i q^{73} +5.01934i q^{75} -5.01934i q^{77} +10.8062 q^{79} +6.40312 q^{81} -2.85974 q^{83} -1.27415 q^{85} +14.8062 q^{87} +8.43579i q^{89} +(2.85974 - 2.94984i) q^{91} -25.8874 q^{93} -10.8062 q^{95} -12.9937i q^{97} +25.1148 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 20 q^{9} - 20 q^{17} - 32 q^{23} + 12 q^{25} - 8 q^{39} + 20 q^{49} + 64 q^{55} + 12 q^{65} - 16 q^{79} + 16 q^{87} + 16 q^{95}+O(q^{100}) \) 8 * q - 20 * q^9 - 20 * q^17 - 32 * q^23 + 12 * q^25 - 8 * q^39 + 20 * q^49 + 64 * q^55 + 12 * q^65 - 16 * q^79 + 16 * q^87 + 16 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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