LMFDB - Embedded newform 4012.2.b.a.237.13

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4012,2,Mod(237,4012)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4012, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("4012.237");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4012 = 2^{2} \cdot 17 \cdot 59 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4012.b (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$32.0359812909$
Analytic rank:	$0$
Dimension:	$40$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			237.13
Character	$\chi$	$=$	4012.237
Dual form			4012.2.b.a.237.28

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.46162i q^{3} +3.03678i q^{5} -1.90612i q^{7} +0.863667 q^{9} +O(q^{10})$	$q-1.46162i q^{3} +3.03678i q^{5} -1.90612i q^{7} +0.863667 q^{9} +1.80300i q^{11} +1.35513 q^{13} +4.43862 q^{15} +(3.82223 + 1.54614i) q^{17} -8.15232 q^{19} -2.78603 q^{21} -4.71630i q^{23} -4.22205 q^{25} -5.64721i q^{27} -5.60944i q^{29} +6.17313i q^{31} +2.63530 q^{33} +5.78848 q^{35} -5.86607i q^{37} -1.98069i q^{39} -3.86950i q^{41} +7.95523 q^{43} +2.62277i q^{45} +10.1841 q^{47} +3.36669 q^{49} +(2.25987 - 5.58665i) q^{51} +7.47471 q^{53} -5.47531 q^{55} +11.9156i q^{57} -1.00000 q^{59} +12.7651i q^{61} -1.64626i q^{63} +4.11525i q^{65} -3.12076 q^{67} -6.89344 q^{69} +6.11193i q^{71} +0.793064i q^{73} +6.17103i q^{75} +3.43673 q^{77} -5.43651i q^{79} -5.66308 q^{81} +2.89886 q^{83} +(-4.69529 + 11.6073i) q^{85} -8.19888 q^{87} +2.43175 q^{89} -2.58305i q^{91} +9.02278 q^{93} -24.7568i q^{95} -2.53872i q^{97} +1.55719i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 40 q - 36 q^{9}+O(q^{10}) $ 40 * q - 36 * q^9	$ 40 q - 36 q^{9} + 8 q^{13} + 20 q^{15} - 8 q^{17} + 20 q^{19} + 6 q^{21} - 24 q^{25} - 14 q^{33} - 52 q^{35} + 22 q^{43} - 10 q^{47} + 8 q^{49} - 6 q^{51} - 2 q^{53} - 12 q^{55} - 40 q^{59} - 24 q^{67} - 36 q^{69} - 38 q^{77} + 16 q^{81} + 32 q^{83} - 22 q^{85} - 18 q^{87} + 40 q^{89} + 22 q^{93}+O(q^{100}) $ 40 * q - 36 * q^9 + 8 * q^13 + 20 * q^15 - 8 * q^17 + 20 * q^19 + 6 * q^21 - 24 * q^25 - 14 * q^33 - 52 * q^35 + 22 * q^43 - 10 * q^47 + 8 * q^49 - 6 * q^51 - 2 * q^53 - 12 * q^55 - 40 * q^59 - 24 * q^67 - 36 * q^69 - 38 * q^77 + 16 * q^81 + 32 * q^83 - 22 * q^85 - 18 * q^87 + 40 * q^89 + 22 * q^93

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4012\mathbb{Z}\right)^\times$.

$n$	$2007$	$3129$	$3777$
$\chi(n)$	$1$	$1$	$-1$

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
$3$		−	1.46162i		−	0.843867i	−0.906627	−	0.421933i	$-0.861352\pi$
$3$		−	1.46162i		−	0.843867i	0.906627	−	0.421933i	$-0.138648\pi$
$4$		0			0
$5$			3.03678i			1.35809i	0.734097	+	0.679045i	$0.237606\pi$
$5$			3.03678i			1.35809i	−0.734097	+	0.679045i	$0.762394\pi$
$6$		0			0
$7$		−	1.90612i		−	0.720447i	−0.932866	−	0.360224i	$-0.882700\pi$
$7$		−	1.90612i		−	0.720447i	0.932866	−	0.360224i	$-0.117300\pi$
$8$		0			0
$9$	0.863667			0.287889
$10$		0			0
$11$			1.80300i			0.543624i	0.962350	+	0.271812i	$0.0876229\pi$
$11$			1.80300i			0.543624i	−0.962350	+	0.271812i	$0.912377\pi$
$12$		0			0
$13$	1.35513			0.375847			0.187923	−	0.982184i	$-0.439824\pi$
$13$	1.35513			0.375847			0.187923	+	0.982184i	$0.439824\pi$
$14$		0			0
$15$	4.43862			1.14605
$16$		0			0
$17$	3.82223	+	1.54614i	0.927027	+	0.374994i
$17$	3.82223	+	1.54614i	0.927027	+	0.374994i
$18$		0			0
$19$	−8.15232			−1.87027			−0.935136	−	0.354290i	$-0.884723\pi$
$19$	−8.15232			−1.87027			−0.935136	+	0.354290i	$0.884723\pi$
$20$		0			0
$21$	−2.78603			−0.607961
$22$		0			0
$23$		−	4.71630i		−	0.983417i	−0.870760	−	0.491708i	$-0.836373\pi$
$23$		−	4.71630i		−	0.983417i	0.870760	−	0.491708i	$-0.163627\pi$
$24$		0			0
$25$	−4.22205			−0.844409
$26$		0			0
$27$		−	5.64721i		−	1.08681i
$28$		0			0
$29$		−	5.60944i		−	1.04165i	−0.853664	−	0.520824i	$-0.825625\pi$
$29$		−	5.60944i		−	1.04165i	0.853664	−	0.520824i	$-0.174375\pi$
$30$		0			0
$31$			6.17313i			1.10873i	0.832274	+	0.554364i	$0.187038\pi$
$31$			6.17313i			1.10873i	−0.832274	+	0.554364i	$0.812962\pi$
$32$		0			0
$33$	2.63530			0.458746
$34$		0			0
$35$	5.78848			0.978432
$36$		0			0
$37$		−	5.86607i		−	0.964376i	−0.876068	−	0.482188i	$-0.839842\pi$
$37$		−	5.86607i		−	0.964376i	0.876068	−	0.482188i	$-0.160158\pi$
$38$		0			0
$39$		−	1.98069i		−	0.317164i
$40$		0			0
$41$		−	3.86950i		−	0.604314i	−0.953258	−	0.302157i	$-0.902293\pi$
$41$		−	3.86950i		−	0.604314i	0.953258	−	0.302157i	$-0.0977068\pi$
$42$		0			0
$43$	7.95523			1.21316			0.606580	−	0.795022i	$-0.292541\pi$
$43$	7.95523			1.21316			0.606580	+	0.795022i	$0.292541\pi$
$44$		0			0
$45$			2.62277i			0.390979i
$46$		0			0
$47$	10.1841			1.48550			0.742750	−	0.669568i	$-0.233521\pi$
$47$	10.1841			1.48550			0.742750	+	0.669568i	$0.233521\pi$
$48$		0			0
$49$	3.36669			0.480956
$50$		0			0
$51$	2.25987	−	5.58665i	0.316445	−	0.782287i
$52$		0			0
$53$	7.47471			1.02673			0.513365	−	0.858170i	$-0.328399\pi$
$53$	7.47471			1.02673			0.513365	+	0.858170i	$0.328399\pi$
$54$		0			0
$55$	−5.47531			−0.738290
$56$		0			0
$57$			11.9156i			1.57826i
$58$		0			0
$59$	−1.00000			−0.130189
$59$	−1.00000			−0.130189
$60$		0			0
$61$			12.7651i			1.63441i	0.576348	+	0.817204i	$0.304477\pi$
$61$			12.7651i			1.63441i	−0.576348	+	0.817204i	$0.695523\pi$
$62$		0			0
$63$		−	1.64626i		−	0.207409i
$64$		0			0
$65$			4.11525i			0.510434i
$66$		0			0
$67$	−3.12076			−0.381262			−0.190631	−	0.981662i	$-0.561053\pi$
$67$	−3.12076			−0.381262			−0.190631	+	0.981662i	$0.561053\pi$
$68$		0			0
$69$	−6.89344			−0.829873
$70$		0			0
$71$			6.11193i			0.725352i	0.931915	+	0.362676i	$0.118137\pi$
$71$			6.11193i			0.725352i	−0.931915	+	0.362676i	$0.881863\pi$
$72$		0			0
$73$			0.793064i			0.0928211i	0.998922	+	0.0464105i	$0.0147782\pi$
$73$			0.793064i			0.0928211i	−0.998922	+	0.0464105i	$0.985222\pi$
$74$		0			0
$75$			6.17103i			0.712569i
$76$		0			0
$77$	3.43673			0.391652
$78$		0			0
$79$		−	5.43651i		−	0.611655i	−0.952087	−	0.305827i	$-0.901067\pi$
$79$		−	5.43651i		−	0.611655i	0.952087	−	0.305827i	$-0.0989330\pi$
$80$		0			0
$81$	−5.66308			−0.629231
$82$		0			0
$83$	2.89886			0.318191			0.159096	−	0.987263i	$-0.449142\pi$
$83$	2.89886			0.318191			0.159096	+	0.987263i	$0.449142\pi$
$84$		0			0
$85$	−4.69529	+	11.6073i	−0.509276	+	1.25899i
$86$		0			0
$87$	−8.19888			−0.879012
$88$		0			0
$89$	2.43175			0.257765			0.128883	−	0.991660i	$-0.458861\pi$
$89$	2.43175			0.257765			0.128883	+	0.991660i	$0.458861\pi$
$90$		0			0
$91$		−	2.58305i		−	0.270778i
$92$		0			0
$93$	9.02278			0.935618
$94$		0			0
$95$		−	24.7568i		−	2.54000i
$96$		0			0
$97$		−	2.53872i		−	0.257768i	−0.991660	−	0.128884i	$-0.958861\pi$
$97$		−	2.53872i		−	0.257768i	0.991660	−	0.128884i	$-0.0411394\pi$
$98$		0			0
$99$			1.55719i			0.156503i
$100$		0			0
$101$	−10.7158			−1.06626			−0.533132	−	0.846032i	$-0.678985\pi$
$101$	−10.7158			−1.06626			−0.533132	+	0.846032i	$0.678985\pi$
$102$		0			0
$103$	0.799694			0.0787962			0.0393981	−	0.999224i	$-0.487456\pi$
$103$	0.799694			0.0787962			0.0393981	+	0.999224i	$0.487456\pi$
$104$		0			0
$105$		−	8.46056i		−	0.825666i
$106$		0			0
$107$		−	4.18785i		−	0.404854i	−0.979297	−	0.202427i	$-0.935117\pi$
$107$		−	4.18785i		−	0.404854i	0.979297	−	0.202427i	$-0.0648830\pi$
$108$		0			0
$109$			16.9425i			1.62280i	0.584492	+	0.811399i	$0.301294\pi$
$109$			16.9425i			1.62280i	−0.584492	+	0.811399i	$0.698706\pi$
$110$		0			0
$111$	−8.57397			−0.813805
$112$		0			0
$113$			14.0344i			1.32025i	0.751157	+	0.660123i	$0.229496\pi$
$113$			14.0344i			1.32025i	−0.751157	+	0.660123i	$0.770504\pi$
$114$		0			0
$115$	14.3224			1.33557
$116$		0			0
$117$	1.17038			0.108202
$118$		0			0
$119$	2.94714	−	7.28565i	0.270163	−	0.667874i
$120$		0			0
$121$	7.74920			0.704473
$122$		0			0
$123$	−5.65574			−0.509961
$124$		0			0
$125$			2.36248i			0.211306i
$126$		0			0
$127$	17.8882			1.58732			0.793660	−	0.608362i	$-0.208173\pi$
$127$	17.8882			1.58732			0.793660	+	0.608362i	$0.208173\pi$
$128$		0			0
$129$		−	11.6275i		−	1.02375i
$130$		0			0
$131$			2.91841i			0.254983i	0.991840	+	0.127491i	$0.0406925\pi$
$131$			2.91841i			0.254983i	−0.991840	+	0.127491i	$0.959308\pi$
$132$		0			0
$133$			15.5393i			1.34743i
$134$		0			0
$135$	17.1494			1.47598
$136$		0			0
$137$	11.0935			0.947782			0.473891	−	0.880583i	$-0.342849\pi$
$137$	11.0935			0.947782			0.473891	+	0.880583i	$0.342849\pi$
$138$		0			0
$139$		−	10.4831i		−	0.889164i	−0.895738	−	0.444582i	$-0.853352\pi$
$139$		−	10.4831i		−	0.889164i	0.895738	−	0.444582i	$-0.146648\pi$
$140$		0			0
$141$		−	14.8853i		−	1.25356i
$142$		0			0
$143$			2.44330i			0.204319i
$144$		0			0
$145$	17.0347			1.41465
$146$		0			0
$147$		−	4.92082i		−	0.405863i
$148$		0			0
$149$	16.1653			1.32431			0.662157	−	0.749365i	$-0.269641\pi$
$149$	16.1653			1.32431			0.662157	+	0.749365i	$0.269641\pi$
$150$		0			0
$151$	20.0061			1.62807			0.814035	−	0.580816i	$-0.197266\pi$
$151$	20.0061			1.62807			0.814035	+	0.580816i	$0.197266\pi$
$152$		0			0
$153$	3.30113	+	1.33535i	0.266881	+	0.107957i
$154$		0			0
$155$	−18.7465			−1.50575
$156$		0			0
$157$	9.76482			0.779318			0.389659	−	0.920959i	$-0.372593\pi$
$157$	9.76482			0.779318			0.389659	+	0.920959i	$0.372593\pi$
$158$		0			0
$159$		−	10.9252i		−	0.866424i
$160$		0			0
$161$	−8.98985			−0.708500
$162$		0			0
$163$		−	4.02584i		−	0.315328i	−0.987493	−	0.157664i	$-0.949604\pi$
$163$		−	4.02584i		−	0.315328i	0.987493	−	0.157664i	$-0.0503963\pi$
$164$		0			0
$165$			8.00282i			0.623019i
$166$		0			0
$167$		−	15.9967i		−	1.23786i	−0.785446	−	0.618930i	$-0.787566\pi$
$167$		−	15.9967i		−	1.23786i	0.785446	−	0.618930i	$-0.212434\pi$
$168$		0			0
$169$	−11.1636			−0.858739
$170$		0			0
$171$	−7.04089			−0.538430
$172$		0			0
$173$		−	9.35244i		−	0.711053i	−0.934666	−	0.355527i	$-0.884302\pi$
$173$		−	9.35244i		−	0.711053i	0.934666	−	0.355527i	$-0.115698\pi$
$174$		0			0
$175$			8.04774i			0.608352i
$176$		0			0
$177$			1.46162i			0.109862i
$178$		0			0
$179$	10.4186			0.778720			0.389360	−	0.921086i	$-0.372696\pi$
$179$	10.4186			0.778720			0.389360	+	0.921086i	$0.372696\pi$
$180$		0			0
$181$		−	21.1286i		−	1.57047i	−0.619196	−	0.785237i	$-0.712541\pi$
$181$		−	21.1286i		−	1.57047i	0.619196	−	0.785237i	$-0.287459\pi$
$182$		0			0
$183$	18.6578			1.37922
$184$		0			0
$185$	17.8140			1.30971
$186$		0			0
$187$	−2.78769	+	6.89147i	−0.203856	+	0.503954i
$188$		0			0
$189$	−10.7643			−0.782987
$190$		0			0
$191$	−17.2862			−1.25079			−0.625394	−	0.780310i	$-0.715062\pi$
$191$	−17.2862			−1.25079			−0.625394	+	0.780310i	$0.715062\pi$
$192$		0			0
$193$			13.9383i			1.00330i	0.865070	+	0.501652i	$0.167274\pi$
$193$			13.9383i			1.00330i	−0.865070	+	0.501652i	$0.832726\pi$
$194$		0			0
$195$	6.01493			0.430738
$196$		0			0
$197$			15.7982i			1.12558i	0.826601	+	0.562788i	$0.190271\pi$
$197$			15.7982i			1.12558i	−0.826601	+	0.562788i	$0.809729\pi$
$198$		0			0
$199$		−	18.8556i		−	1.33664i	−0.743875	−	0.668319i	$-0.767014\pi$
$199$		−	18.8556i		−	1.33664i	0.743875	−	0.668319i	$-0.232986\pi$
$200$		0			0
$201$			4.56137i			0.321734i
$202$		0			0
$203$	−10.6923			−0.750452
$204$		0			0
$205$	11.7508			0.820714
$206$		0			0
$207$		−	4.07331i		−	0.283115i
$208$		0			0
$209$		−	14.6986i		−	1.01672i
$210$		0			0
$211$			8.47812i			0.583658i	0.956471	+	0.291829i	$0.0942638\pi$
$211$			8.47812i			0.583658i	−0.956471	+	0.291829i	$0.905736\pi$
$212$		0			0
$213$	8.93332			0.612101
$214$		0			0
$215$			24.1583i			1.64758i
$216$		0			0
$217$	11.7668			0.798780
$218$		0			0
$219$	1.15916			0.0783286
$220$		0			0
$221$	5.17963	+	2.09523i	0.348420	+	0.140940i
$222$		0			0
$223$	19.3874			1.29828			0.649139	−	0.760670i	$-0.275129\pi$
$223$	19.3874			1.29828			0.649139	+	0.760670i	$0.275129\pi$
$224$		0			0
$225$	−3.64644			−0.243096
$226$		0			0
$227$		−	22.4428i		−	1.48958i	−0.667298	−	0.744791i	$-0.732549\pi$
$227$		−	22.4428i		−	1.48958i	0.667298	−	0.744791i	$-0.267451\pi$
$228$		0			0
$229$	−11.0879			−0.732711			−0.366355	−	0.930475i	$-0.619395\pi$
$229$	−11.0879			−0.732711			−0.366355	+	0.930475i	$0.619395\pi$
$230$		0			0
$231$		−	5.02320i		−	0.330502i
$232$		0			0
$233$		−	16.8954i		−	1.10685i	−0.832898	−	0.553426i	$-0.813320\pi$
$233$		−	16.8954i		−	1.10685i	0.832898	−	0.553426i	$-0.186680\pi$
$234$		0			0
$235$			30.9268i			2.01744i
$236$		0			0
$237$	−7.94611			−0.516155
$238$		0			0
$239$	7.17946			0.464401			0.232200	−	0.972668i	$-0.425408\pi$
$239$	7.17946			0.464401			0.232200	+	0.972668i	$0.425408\pi$
$240$		0			0
$241$			18.4182i			1.18642i	0.805048	+	0.593210i	$0.202139\pi$
$241$			18.4182i			1.18642i	−0.805048	+	0.593210i	$0.797861\pi$
$242$		0			0
$243$		−	8.66437i		−	0.555819i
$244$		0			0
$245$			10.2239i			0.653182i
$246$		0			0
$247$	−11.0475			−0.702935
$248$		0			0
$249$		−	4.23703i		−	0.268511i
$250$		0			0
$251$	−25.9792			−1.63979			−0.819895	−	0.572514i	$-0.805968\pi$
$251$	−25.9792			−1.63979			−0.819895	+	0.572514i	$0.805968\pi$
$252$		0			0
$253$	8.50347			0.534609
$254$		0			0
$255$	16.9654	+	6.86273i	1.06242	+	0.429761i
$256$		0			0
$257$	2.25098			0.140412			0.0702062	−	0.997532i	$-0.477634\pi$
$257$	2.25098			0.140412			0.0702062	+	0.997532i	$0.477634\pi$
$258$		0			0
$259$	−11.1815			−0.694782
$260$		0			0
$261$		−	4.84469i		−	0.299879i
$262$		0			0
$263$	9.79105			0.603742			0.301871	−	0.953349i	$-0.402389\pi$
$263$	9.79105			0.603742			0.301871	+	0.953349i	$0.402389\pi$
$264$		0			0
$265$			22.6991i			1.39439i
$266$		0			0
$267$		−	3.55430i		−	0.217519i
$268$		0			0
$269$		−	6.73866i		−	0.410863i	−0.978671	−	0.205432i	$-0.934140\pi$
$269$		−	6.73866i		−	0.410863i	0.978671	−	0.205432i	$-0.0658599\pi$
$270$		0			0
$271$	−11.8714			−0.721134			−0.360567	−	0.932733i	$-0.617417\pi$
$271$	−11.8714			−0.721134			−0.360567	+	0.932733i	$0.617417\pi$
$272$		0			0
$273$	−3.77544			−0.228500
$274$		0			0
$275$		−	7.61233i		−	0.459041i
$276$		0			0
$277$		−	16.3961i		−	0.985148i	−0.870270	−	0.492574i	$-0.836056\pi$
$277$		−	16.3961i		−	0.985148i	0.870270	−	0.492574i	$-0.163944\pi$
$278$		0			0
$279$			5.33153i			0.319190i
$280$		0			0
$281$	30.4202			1.81472			0.907359	−	0.420357i	$-0.138095\pi$
$281$	30.4202			1.81472			0.907359	+	0.420357i	$0.138095\pi$
$282$		0			0
$283$			2.06602i			0.122812i	0.998113	+	0.0614061i	$0.0195585\pi$
$283$			2.06602i			0.122812i	−0.998113	+	0.0614061i	$0.980442\pi$
$284$		0			0
$285$	−36.1851			−2.14342
$286$		0			0
$287$	−7.37575			−0.435377
$288$		0			0
$289$	12.2189	+	11.8194i	0.718759	+	0.695260i
$290$		0			0
$291$	−3.71064			−0.217522
$292$		0			0
$293$	−17.3806			−1.01539			−0.507693	−	0.861538i	$-0.669502\pi$
$293$	−17.3806			−1.01539			−0.507693	+	0.861538i	$0.669502\pi$
$294$		0			0
$295$		−	3.03678i		−	0.176808i
$296$		0			0
$297$	10.1819			0.590814
$298$		0			0
$299$		−	6.39122i		−	0.369614i
$300$		0			0
$301$		−	15.1636i		−	0.874018i
$302$		0			0
$303$			15.6624i			0.899784i
$304$		0			0
$305$	−38.7649			−2.21967
$306$		0			0
$307$	33.8692			1.93302			0.966508	−	0.256635i	$-0.0826139\pi$
$307$	33.8692			1.93302			0.966508	+	0.256635i	$0.0826139\pi$
$308$		0			0
$309$		−	1.16885i		−	0.0664935i
$310$		0			0
$311$		−	0.0431294i		−	0.00244564i	−0.999999	−	0.00122282i	$-0.999611\pi$
$311$		−	0.0431294i		−	0.00244564i	0.999999	−	0.00122282i	$-0.000389236\pi$
$312$		0			0
$313$			14.8408i			0.838849i	0.907790	+	0.419425i	$0.137768\pi$
$313$			14.8408i			0.838849i	−0.907790	+	0.419425i	$0.862232\pi$
$314$		0			0
$315$	4.99932			0.281680
$316$		0			0
$317$			12.7306i			0.715022i	0.933909	+	0.357511i	$0.116374\pi$
$317$			12.7306i			0.715022i	−0.933909	+	0.357511i	$0.883626\pi$
$318$		0			0
$319$	10.1138			0.566264
$320$		0			0
$321$	−6.12104			−0.341643
$322$		0			0
$323$	−31.1601	−	12.6046i	−1.73379	−	0.701341i
$324$		0			0
$325$	−5.72144			−0.317368
$326$		0			0
$327$	24.7635			1.36943
$328$		0			0
$329$		−	19.4121i		−	1.07022i
$330$		0			0
$331$	8.50177			0.467300			0.233650	−	0.972321i	$-0.424933\pi$
$331$	8.50177			0.467300			0.233650	+	0.972321i	$0.424933\pi$
$332$		0			0
$333$		−	5.06633i		−	0.277633i
$334$		0			0
$335$		−	9.47707i		−	0.517788i
$336$		0			0
$337$			17.4149i			0.948649i	0.880350	+	0.474324i	$0.157308\pi$
$337$			17.4149i			0.948649i	−0.880350	+	0.474324i	$0.842692\pi$
$338$		0			0
$339$	20.5130			1.11411
$340$		0			0
$341$	−11.1301			−0.602731
$342$		0			0
$343$		−	19.7602i		−	1.06695i
$344$		0			0
$345$		−	20.9339i		−	1.12704i
$346$		0			0
$347$		−	19.1153i		−	1.02616i	−0.858340	−	0.513082i	$-0.828504\pi$
$347$		−	19.1153i		−	1.02616i	0.858340	−	0.513082i	$-0.171496\pi$
$348$		0			0
$349$	−29.1421			−1.55994			−0.779971	−	0.625816i	$-0.784766\pi$
$349$	−29.1421			−1.55994			−0.779971	+	0.625816i	$0.784766\pi$
$350$		0			0
$351$		−	7.65273i		−	0.408473i
$352$		0			0
$353$	−1.58763			−0.0845010			−0.0422505	−	0.999107i	$-0.513453\pi$
$353$	−1.58763			−0.0845010			−0.0422505	+	0.999107i	$0.513453\pi$
$354$		0			0
$355$	−18.5606			−0.985094
$356$		0			0
$357$	−10.6488	−	4.30759i	−0.563597	−	0.227982i
$358$		0			0
$359$	−24.0284			−1.26817			−0.634085	−	0.773263i	$-0.718623\pi$
$359$	−24.0284			−1.26817			−0.634085	+	0.773263i	$0.718623\pi$
$360$		0			0
$361$	47.4604			2.49791
$362$		0			0
$363$		−	11.3264i		−	0.594482i
$364$		0			0
$365$	−2.40836			−0.126059
$366$		0			0
$367$		−	5.18513i		−	0.270661i	−0.990800	−	0.135331i	$-0.956790\pi$
$367$		−	5.18513i		−	0.270661i	0.990800	−	0.135331i	$-0.0432097\pi$
$368$		0			0
$369$		−	3.34196i		−	0.173975i
$370$		0			0
$371$		−	14.2477i		−	0.739705i
$372$		0			0
$373$	15.8814			0.822307			0.411154	−	0.911566i	$-0.365126\pi$
$373$	15.8814			0.822307			0.411154	+	0.911566i	$0.365126\pi$
$374$		0			0
$375$	3.45305			0.178315
$376$		0			0
$377$		−	7.60155i		−	0.391500i
$378$		0			0
$379$		−	6.57352i		−	0.337659i	−0.985645	−	0.168830i	$-0.946001\pi$
$379$		−	6.57352i		−	0.337659i	0.985645	−	0.168830i	$-0.0539988\pi$
$380$		0			0
$381$		−	26.1457i		−	1.33949i
$382$		0			0
$383$	−17.4683			−0.892589			−0.446295	−	0.894886i	$-0.647257\pi$
$383$	−17.4683			−0.892589			−0.446295	+	0.894886i	$0.647257\pi$
$384$		0			0
$385$			10.4366i			0.531899i
$386$		0			0
$387$	6.87066			0.349255
$388$		0			0
$389$	15.2241			0.771894			0.385947	−	0.922521i	$-0.373875\pi$
$389$	15.2241			0.771894			0.385947	+	0.922521i	$0.373875\pi$
$390$		0			0
$391$	7.29207	−	18.0268i	0.368776	−	0.911654i
$392$		0			0
$393$	4.26560			0.215171
$394$		0			0
$395$	16.5095			0.830682
$396$		0			0
$397$		−	4.45900i		−	0.223791i	−0.993720	−	0.111895i	$-0.964308\pi$
$397$		−	4.45900i		−	0.223791i	0.993720	−	0.111895i	$-0.0356922\pi$
$398$		0			0
$399$	22.7126			1.13705
$400$		0			0
$401$			35.5679i			1.77618i	0.459673	+	0.888088i	$0.347967\pi$
$401$			35.5679i			1.77618i	−0.459673	+	0.888088i	$0.652033\pi$
$402$		0			0
$403$			8.36542i			0.416711i
$404$		0			0
$405$		−	17.1975i		−	0.854553i
$406$		0			0
$407$	10.5765			0.524258
$408$		0			0
$409$	12.4282			0.614534			0.307267	−	0.951623i	$-0.400586\pi$
$409$	12.4282			0.614534			0.307267	+	0.951623i	$0.400586\pi$
$410$		0			0
$411$		−	16.2145i		−	0.799802i
$412$		0			0
$413$			1.90612i			0.0937942i
$414$		0			0
$415$			8.80321i			0.432132i
$416$		0			0
$417$	−15.3223			−0.750336
$418$		0			0
$419$		−	22.8622i		−	1.11689i	−0.829540	−	0.558447i	$-0.811397\pi$
$419$		−	22.8622i		−	1.11689i	0.829540	−	0.558447i	$-0.188603\pi$
$420$		0			0
$421$	5.09038			0.248090			0.124045	−	0.992277i	$-0.460413\pi$
$421$	5.09038			0.248090			0.124045	+	0.992277i	$0.460413\pi$
$422$		0			0
$423$	8.79565			0.427659
$424$		0			0
$425$	−16.1376	−	6.52788i	−0.782790	−	0.316649i
$426$		0			0
$427$	24.3319			1.17750
$428$		0			0
$429$	3.57118			0.172418
$430$		0			0
$431$			20.7707i			1.00049i	0.865884	+	0.500245i	$0.166757\pi$
$431$			20.7707i			1.00049i	−0.865884	+	0.500245i	$0.833243\pi$
$432$		0			0
$433$	−16.1958			−0.778322			−0.389161	−	0.921170i	$-0.627235\pi$
$433$	−16.1958			−0.778322			−0.389161	+	0.921170i	$0.627235\pi$
$434$		0			0
$435$		−	24.8982i		−	1.19378i
$436$		0			0
$437$			38.4488i			1.83926i
$438$		0			0
$439$			23.2880i			1.11148i	0.831358	+	0.555738i	$0.187564\pi$
$439$			23.2880i			1.11148i	−0.831358	+	0.555738i	$0.812436\pi$
$440$		0			0
$441$	2.90770			0.138462
$442$		0			0
$443$	−17.5688			−0.834721			−0.417360	−	0.908741i	$-0.637045\pi$
$443$	−17.5688			−0.834721			−0.417360	+	0.908741i	$0.637045\pi$
$444$		0			0
$445$			7.38470i			0.350068i
$446$		0			0
$447$		−	23.6275i		−	1.11754i
$448$		0			0
$449$			21.0161i			0.991812i	0.868376	+	0.495906i	$0.165164\pi$
$449$			21.0161i			0.991812i	−0.868376	+	0.495906i	$0.834836\pi$
$450$		0			0
$451$	6.97669			0.328520
$452$		0			0
$453$		−	29.2412i		−	1.37387i
$454$		0			0
$455$	7.84417			0.367740
$456$		0			0
$457$	−18.8452			−0.881543			−0.440771	−	0.897619i	$-0.645295\pi$
$457$	−18.8452			−0.881543			−0.440771	+	0.897619i	$0.645295\pi$
$458$		0			0
$459$	8.73139	−	21.5850i	0.407546	−	1.00750i
$460$		0			0
$461$	−23.7850			−1.10778			−0.553888	−	0.832591i	$-0.686857\pi$
$461$	−23.7850			−1.10778			−0.553888	+	0.832591i	$0.686857\pi$
$462$		0			0
$463$	−21.3330			−0.991430			−0.495715	−	0.868485i	$-0.665094\pi$
$463$	−21.3330			−0.991430			−0.495715	+	0.868485i	$0.665094\pi$
$464$		0			0
$465$			27.4002i			1.27065i
$466$		0			0
$467$	−7.01123			−0.324441			−0.162221	−	0.986755i	$-0.551866\pi$
$467$	−7.01123			−0.324441			−0.162221	+	0.986755i	$0.551866\pi$
$468$		0			0
$469$			5.94856i			0.274679i
$470$		0			0
$471$		−	14.2725i		−	0.657640i
$472$		0			0
$473$			14.3432i			0.659503i
$474$		0			0
$475$	34.4195			1.57927
$476$		0			0
$477$	6.45566			0.295584
$478$		0			0
$479$		−	9.16555i		−	0.418784i	−0.977832	−	0.209392i	$-0.932851\pi$
$479$		−	9.16555i		−	0.418784i	0.977832	−	0.209392i	$-0.0671485\pi$
$480$		0			0
$481$		−	7.94931i		−	0.362458i
$482$		0			0
$483$			13.1398i			0.597879i
$484$		0			0
$485$	7.70953			0.350072
$486$		0			0
$487$		−	20.3394i		−	0.921666i	−0.887487	−	0.460833i	$-0.847551\pi$
$487$		−	20.3394i		−	0.921666i	0.887487	−	0.460833i	$-0.152449\pi$
$488$		0			0
$489$	−5.88424			−0.266095
$490$		0			0
$491$	−17.0841			−0.770995			−0.385497	−	0.922709i	$-0.625970\pi$
$491$	−17.0841			−0.770995			−0.385497	+	0.922709i	$0.625970\pi$
$492$		0			0
$493$	8.67299	−	21.4406i	0.390612	−	0.965636i
$494$		0			0
$495$	−4.72884			−0.212545
$496$		0			0
$497$	11.6501			0.522578
$498$		0			0
$499$			1.09966i			0.0492274i	0.999697	+	0.0246137i	$0.00783558\pi$
$499$			1.09966i			0.0492274i	−0.999697	+	0.0246137i	$0.992164\pi$
$500$		0			0
$501$	−23.3811			−1.04459
$502$		0			0
$503$		−	6.82460i		−	0.304294i	−0.988358	−	0.152147i	$-0.951381\pi$
$503$		−	6.82460i		−	0.304294i	0.988358	−	0.152147i	$-0.0486187\pi$
$504$		0			0
$505$		−	32.5416i		−	1.44808i
$506$		0			0
$507$			16.3170i			0.724662i
$508$		0			0
$509$	−38.3468			−1.69969			−0.849846	−	0.527032i	$-0.823305\pi$
$509$	−38.3468			−1.69969			−0.849846	+	0.527032i	$0.823305\pi$
$510$		0			0
$511$	1.51168			0.0668727
$512$		0			0
$513$			46.0379i			2.03262i
$514$		0			0
$515$			2.42850i			0.107012i
$516$		0			0
$517$			18.3619i			0.807554i
$518$		0			0
$519$	−13.6697			−0.600034
$520$		0			0
$521$		−	22.0983i		−	0.968144i	−0.875028	−	0.484072i	$-0.839157\pi$
$521$		−	22.0983i		−	0.968144i	0.875028	−	0.484072i	$-0.160843\pi$
$522$		0			0
$523$	−34.8469			−1.52375			−0.761875	−	0.647724i	$-0.775721\pi$
$523$	−34.8469			−1.52375			−0.761875	+	0.647724i	$0.775721\pi$
$524$		0			0
$525$	11.7627			0.513368
$526$		0			0
$527$	−9.54453	+	23.5951i	−0.415766	+	1.02782i
$528$		0			0
$529$	0.756499			0.0328912
$530$		0			0
$531$	−0.863667			−0.0374799
$532$		0			0
$533$		−	5.24369i		−	0.227129i
$534$		0			0
$535$	12.7176			0.549829
$536$		0			0
$537$		−	15.2280i		−	0.657136i
$538$		0			0
$539$			6.07013i			0.261459i
$540$		0			0
$541$		−	31.5884i		−	1.35809i	−0.734095	−	0.679046i	$-0.762394\pi$
$541$		−	31.5884i		−	1.35809i	0.734095	−	0.679046i	$-0.237606\pi$
$542$		0			0
$543$	−30.8819			−1.32527
$544$		0			0
$545$	−51.4507			−2.20391
$546$		0			0
$547$			45.0934i			1.92806i	0.265800	+	0.964028i	$0.414364\pi$
$547$			45.0934i			1.92806i	−0.265800	+	0.964028i	$0.585636\pi$
$548$		0			0
$549$			11.0248i			0.470528i
$550$		0			0
$551$			45.7300i			1.94816i
$552$		0			0
$553$	−10.3627			−0.440665
$554$		0			0
$555$		−	26.0373i		−	1.10522i
$556$		0			0
$557$	−10.0493			−0.425801			−0.212901	−	0.977074i	$-0.568291\pi$
$557$	−10.0493			−0.425801			−0.212901	+	0.977074i	$0.568291\pi$
$558$		0			0
$559$	10.7804			0.455962
$560$		0			0
$561$	10.0727	+	4.07454i	0.425270	+	0.172027i
$562$		0			0
$563$	39.1436			1.64971			0.824854	−	0.565346i	$-0.191258\pi$
$563$	39.1436			1.64971			0.824854	+	0.565346i	$0.191258\pi$
$564$		0			0
$565$	−42.6195			−1.79301
$566$		0			0
$567$			10.7945i			0.453328i
$568$		0			0
$569$	−22.6438			−0.949280			−0.474640	−	0.880180i	$-0.657422\pi$
$569$	−22.6438			−0.949280			−0.474640	+	0.880180i	$0.657422\pi$
$570$		0			0
$571$		−	1.96775i		−	0.0823476i	−0.999152	−	0.0411738i	$-0.986890\pi$
$571$		−	1.96775i		−	0.0823476i	0.999152	−	0.0411738i	$-0.0131097\pi$
$572$		0			0
$573$			25.2659i			1.05550i
$574$		0			0
$575$			19.9124i			0.830406i
$576$		0			0
$577$	17.6364			0.734214			0.367107	−	0.930179i	$-0.380348\pi$
$577$	17.6364			0.734214			0.367107	+	0.930179i	$0.380348\pi$
$578$		0			0
$579$	20.3726			0.846655
$580$		0			0
$581$		−	5.52559i		−	0.229240i
$582$		0			0
$583$			13.4769i			0.558155i
$584$		0			0
$585$			3.55420i			0.146948i
$586$		0			0
$587$	−14.0268			−0.578946			−0.289473	−	0.957186i	$-0.593480\pi$
$587$	−14.0268			−0.578946			−0.289473	+	0.957186i	$0.593480\pi$
$588$		0			0
$589$		−	50.3254i		−	2.07362i
$590$		0			0
$591$	23.0910			0.949836
$592$		0			0
$593$	29.0565			1.19321			0.596604	−	0.802535i	$-0.296516\pi$
$593$	29.0565			1.19321			0.596604	+	0.802535i	$0.296516\pi$
$594$		0			0
$595$	22.1249	+	8.94981i	0.907033	+	0.366906i
$596$		0			0
$597$	−27.5597			−1.12794
$598$		0			0
$599$	−1.27092			−0.0519284			−0.0259642	−	0.999663i	$-0.508266\pi$
$599$	−1.27092			−0.0519284			−0.0259642	+	0.999663i	$0.508266\pi$
$600$		0			0
$601$		−	6.13840i		−	0.250391i	−0.992132	−	0.125195i	$-0.960044\pi$
$601$		−	6.13840i		−	0.250391i	0.992132	−	0.125195i	$-0.0399558\pi$
$602$		0			0
$603$	−2.69530			−0.109761
$604$		0			0
$605$			23.5326i			0.956738i
$606$		0			0
$607$		−	10.5842i		−	0.429598i	−0.976658	−	0.214799i	$-0.931090\pi$
$607$		−	10.5842i		−	0.429598i	0.976658	−	0.214799i	$-0.0689097\pi$
$608$		0			0
$609$			15.6281i			0.633282i
$610$		0			0
$611$	13.8008			0.558320
$612$		0			0
$613$	4.66265			0.188323			0.0941614	−	0.995557i	$-0.469983\pi$
$613$	4.66265			0.188323			0.0941614	+	0.995557i	$0.469983\pi$
$614$		0			0
$615$		−	17.1752i		−	0.692573i
$616$		0			0
$617$		−	41.0436i		−	1.65235i	−0.563413	−	0.826176i	$-0.690512\pi$
$617$		−	41.0436i		−	1.65235i	0.563413	−	0.826176i	$-0.309488\pi$
$618$		0			0
$619$			36.9426i			1.48485i	0.669930	+	0.742424i	$0.266324\pi$
$619$			36.9426i			1.48485i	−0.669930	+	0.742424i	$0.733676\pi$
$620$		0			0
$621$	−26.6340			−1.06878
$622$		0			0
$623$		−	4.63522i		−	0.185706i
$624$		0			0
$625$	−28.2846			−1.13138
$626$		0			0
$627$	−21.4838			−0.857980
$628$		0			0
$629$	9.06977	−	22.4215i	0.361636	−	0.894003i
$630$		0			0
$631$	1.69576			0.0675072			0.0337536	−	0.999430i	$-0.489254\pi$
$631$	1.69576			0.0675072			0.0337536	+	0.999430i	$0.489254\pi$
$632$		0			0
$633$	12.3918			0.492529
$634$		0			0
$635$			54.3225i			2.15572i
$636$		0			0
$637$	4.56232			0.180766
$638$		0			0
$639$			5.27867i			0.208821i
$640$		0			0
$641$		−	13.5492i		−	0.535160i	−0.963536	−	0.267580i	$-0.913776\pi$
$641$		−	13.5492i		−	0.535160i	0.963536	−	0.267580i	$-0.0862240\pi$
$642$		0			0
$643$			18.0247i			0.710824i	0.934710	+	0.355412i	$0.115659\pi$
$643$			18.0247i			0.710824i	−0.934710	+	0.355412i	$0.884341\pi$
$644$		0			0
$645$	35.3102			1.39034
$646$		0			0
$647$	−25.6927			−1.01008			−0.505042	−	0.863095i	$-0.668523\pi$
$647$	−25.6927			−1.01008			−0.505042	+	0.863095i	$0.668523\pi$
$648$		0			0
$649$		−	1.80300i		−	0.0707738i
$650$		0			0
$651$		−	17.1985i		−	0.674063i
$652$		0			0
$653$		−	6.40090i		−	0.250487i	−0.992126	−	0.125243i	$-0.960029\pi$
$653$		−	6.40090i		−	0.250487i	0.992126	−	0.125243i	$-0.0399711\pi$
$654$		0			0
$655$	−8.86257			−0.346289
$656$		0			0
$657$			0.684942i			0.0267222i
$658$		0			0
$659$	20.8896			0.813742			0.406871	−	0.913486i	$-0.366620\pi$
$659$	20.8896			0.813742			0.406871	+	0.913486i	$0.366620\pi$
$660$		0			0
$661$	−47.0328			−1.82937			−0.914683	−	0.404172i	$-0.867560\pi$
$661$	−47.0328			−1.82937			−0.914683	+	0.404172i	$0.867560\pi$
$662$		0			0
$663$	3.06243	−	7.57066i	0.118935	−	0.294020i
$664$		0			0
$665$	−47.1896			−1.82993
$666$		0			0
$667$	−26.4558			−1.02437
$668$		0			0
$669$		−	28.3371i		−	1.09557i
$670$		0			0
$671$	−23.0155			−0.888503
$672$		0			0
$673$		−	27.6201i		−	1.06468i	−0.846531	−	0.532339i	$-0.821313\pi$
$673$		−	27.6201i		−	1.06468i	0.846531	−	0.532339i	$-0.178687\pi$
$674$		0			0
$675$			23.8428i			0.917709i
$676$		0			0
$677$			6.57274i			0.252611i	0.991991	+	0.126305i	$0.0403120\pi$
$677$			6.57274i			0.252611i	−0.991991	+	0.126305i	$0.959688\pi$
$678$		0			0
$679$	−4.83911			−0.185708
$680$		0			0
$681$	−32.8029			−1.25701
$682$		0			0
$683$			40.6780i			1.55650i	0.627955	+	0.778249i	$0.283892\pi$
$683$			40.6780i			1.55650i	−0.627955	+	0.778249i	$0.716108\pi$
$684$		0			0
$685$			33.6886i			1.28717i
$686$		0			0
$687$			16.2063i			0.618310i
$688$		0			0
$689$	10.1292			0.385893
$690$		0			0
$691$		−	16.8806i		−	0.642170i	−0.947050	−	0.321085i	$-0.895952\pi$
$691$		−	16.8806i		−	0.642170i	0.947050	−	0.321085i	$-0.104048\pi$
$692$		0			0
$693$	2.96819			0.112752
$694$		0			0
$695$	31.8348			1.20756
$696$		0			0
$697$	5.98279	−	14.7901i	0.226614	−	0.560216i
$698$		0			0
$699$	−24.6946			−0.934035
$700$		0			0
$701$	15.4307			0.582809			0.291405	−	0.956600i	$-0.405877\pi$
$701$	15.4307			0.582809			0.291405	+	0.956600i	$0.405877\pi$
$702$		0			0
$703$			47.8221i			1.80365i
$704$		0			0
$705$	45.2033			1.70245
$706$		0			0
$707$			20.4257i			0.768186i
$708$		0			0
$709$		−	22.0185i		−	0.826922i	−0.910522	−	0.413461i	$-0.864320\pi$
$709$		−	22.0185i		−	0.826922i	0.910522	−	0.413461i	$-0.135680\pi$
$710$		0			0
$711$		−	4.69533i		−	0.176089i
$712$		0			0
$713$	29.1144			1.09034
$714$		0			0
$715$	−7.41977			−0.277484
$716$		0			0
$717$		−	10.4936i		−	0.391892i
$718$		0			0
$719$			22.9400i			0.855519i	0.903892	+	0.427760i	$0.140697\pi$
$719$			22.9400i			0.855519i	−0.903892	+	0.427760i	$0.859303\pi$
$720$		0			0
$721$		−	1.52432i		−	0.0567685i
$722$		0			0
$723$	26.9204			1.00118
$724$		0			0
$725$			23.6833i			0.879577i
$726$		0			0
$727$	−12.8869			−0.477950			−0.238975	−	0.971026i	$-0.576811\pi$
$727$	−12.8869			−0.477950			−0.238975	+	0.971026i	$0.576811\pi$
$728$		0			0
$729$	−29.6533			−1.09827
$730$		0			0
$731$	30.4067	+	12.2999i	1.12463	+	0.454928i
$732$		0			0
$733$	2.37945			0.0878870			0.0439435	−	0.999034i	$-0.486008\pi$
$733$	2.37945			0.0878870			0.0439435	+	0.999034i	$0.486008\pi$
$734$		0			0
$735$	14.9435			0.551198
$736$		0			0
$737$		−	5.62672i		−	0.207263i
$738$		0			0
$739$	31.9341			1.17471			0.587357	−	0.809328i	$-0.300169\pi$
$739$	31.9341			1.17471			0.587357	+	0.809328i	$0.300169\pi$
$740$		0			0
$741$			16.1472i			0.593183i
$742$		0			0
$743$		−	32.0478i		−	1.17572i	−0.808963	−	0.587860i	$-0.799971\pi$
$743$		−	32.0478i		−	1.17572i	0.808963	−	0.587860i	$-0.200029\pi$
$744$		0			0
$745$			49.0905i			1.79854i
$746$		0			0
$747$	2.50365			0.0916037
$748$		0			0
$749$	−7.98255			−0.291676
$750$		0			0
$751$			29.1027i			1.06197i	0.847381	+	0.530986i	$0.178178\pi$
$751$			29.1027i			1.06197i	−0.847381	+	0.530986i	$0.821822\pi$
$752$		0			0
$753$			37.9717i			1.38376i
$754$		0			0
$755$			60.7540i			2.21107i
$756$		0			0
$757$	−15.1878			−0.552010			−0.276005	−	0.961156i	$-0.589011\pi$
$757$	−15.1878			−0.552010			−0.276005	+	0.961156i	$0.589011\pi$
$758$		0			0
$759$		−	12.4288i		−	0.451139i
$760$		0			0
$761$	1.64185			0.0595169			0.0297584	−	0.999557i	$-0.490526\pi$
$761$	1.64185			0.0595169			0.0297584	+	0.999557i	$0.490526\pi$
$762$		0			0
$763$	32.2945			1.16914
$764$		0			0
$765$	−4.05517	+	10.0248i	−0.146615	+	0.362448i
$766$		0			0
$767$	−1.35513			−0.0489311
$768$		0			0
$769$	37.9399			1.36815			0.684073	−	0.729413i	$-0.260207\pi$
$769$	37.9399			1.36815			0.684073	+	0.729413i	$0.260207\pi$
$770$		0			0
$771$		−	3.29008i		−	0.118489i
$772$		0			0
$773$	−13.7889			−0.495953			−0.247976	−	0.968766i	$-0.579766\pi$
$773$	−13.7889			−0.495953			−0.247976	+	0.968766i	$0.579766\pi$
$774$		0			0
$775$		−	26.0633i		−	0.936220i
$776$		0			0
$777$			16.3430i			0.586304i
$778$		0			0
$779$			31.5454i			1.13023i
$780$		0			0
$781$	−11.0198			−0.394319
$782$		0			0
$783$	−31.6777			−1.13207
$784$		0			0
$785$			29.6536i			1.05838i
$786$		0			0
$787$			34.7927i			1.24023i	0.784512	+	0.620113i	$0.212913\pi$
$787$			34.7927i			1.24023i	−0.784512	+	0.620113i	$0.787087\pi$
$788$		0			0
$789$		−	14.3108i		−	0.509478i
$790$		0			0
$791$	26.7513			0.951168
$792$		0			0
$793$			17.2985i			0.614287i
$794$		0			0
$795$	33.1774			1.17668
$796$		0			0
$797$	−20.5155			−0.726697			−0.363348	−	0.931653i	$-0.618367\pi$
$797$	−20.5155			−0.726697			−0.363348	+	0.931653i	$0.618367\pi$
$798$		0			0
$799$	38.9259	+	15.7460i	1.37710	+	0.557054i
$800$		0			0
$801$	2.10022			0.0742077
$802$		0			0
$803$	−1.42989			−0.0504597
$804$		0			0
$805$		−	27.3002i		−	0.962207i
$806$		0			0
$807$	−9.84936			−0.346714
$808$		0			0
$809$			20.4237i			0.718060i	0.933326	+	0.359030i	$0.116892\pi$
$809$			20.4237i			0.718060i	−0.933326	+	0.359030i	$0.883108\pi$
$810$		0			0
$811$		−	20.2694i		−	0.711754i	−0.934533	−	0.355877i	$-0.884182\pi$
$811$		−	20.2694i		−	0.711754i	0.934533	−	0.355877i	$-0.115818\pi$
$812$		0			0
$813$			17.3514i			0.608541i
$814$		0			0
$815$	12.2256			0.428244
$816$		0			0
$817$	−64.8536			−2.26894
$818$		0			0
$819$		−	2.23090i		−	0.0779538i
$820$		0			0
$821$			24.5522i			0.856879i	0.903571	+	0.428439i	$0.140936\pi$
$821$			24.5522i			0.856879i	−0.903571	+	0.428439i	$0.859064\pi$
$822$		0			0
$823$			7.62928i			0.265940i	0.991120	+	0.132970i	$0.0424514\pi$
$823$			7.62928i			0.265940i	−0.991120	+	0.132970i	$0.957549\pi$
$824$		0			0
$825$	−11.1263			−0.387369
$826$		0			0
$827$			7.44041i			0.258728i	0.991597	+	0.129364i	$0.0412936\pi$
$827$			7.44041i			0.258728i	−0.991597	+	0.129364i	$0.958706\pi$
$828$		0			0
$829$	−37.7634			−1.31158			−0.655788	−	0.754945i	$-0.727663\pi$
$829$	−37.7634			−1.31158			−0.655788	+	0.754945i	$0.727663\pi$
$830$		0			0
$831$	−23.9649			−0.831334
$832$		0			0
$833$	12.8683	+	5.20538i	0.445859	+	0.180356i
$834$		0			0
$835$	48.5784			1.68113
$836$		0			0
$837$	34.8610			1.20497
$838$		0			0
$839$			19.5374i			0.674506i	0.941414	+	0.337253i	$0.109498\pi$
$839$			19.5374i			0.674506i	−0.941414	+	0.337253i	$0.890502\pi$
$840$		0			0
$841$	−2.46586			−0.0850298
$842$		0			0
$843$		−	44.4628i		−	1.53138i
$844$		0			0
$845$		−	33.9015i		−	1.16625i
$846$		0			0
$847$		−	14.7709i		−	0.507536i
$848$		0			0
$849$	3.01974			0.103637
$850$		0			0
$851$	−27.6662			−0.948384
$852$		0			0
$853$			53.7791i			1.84136i	0.390317	+	0.920680i	$0.372365\pi$
$853$			53.7791i			1.84136i	−0.390317	+	0.920680i	$0.627635\pi$
$854$		0			0
$855$		−	21.3816i		−	0.731237i
$856$		0			0
$857$			14.1100i			0.481987i	0.970527	+	0.240994i	$0.0774733\pi$
$857$			14.1100i			0.481987i	−0.970527	+	0.240994i	$0.922527\pi$
$858$		0			0
$859$	44.9103			1.53232			0.766161	−	0.642649i	$-0.222165\pi$
$859$	44.9103			1.53232			0.766161	+	0.642649i	$0.222165\pi$
$860$		0			0
$861$			10.7805i			0.367400i
$862$		0			0
$863$	−27.4510			−0.934443			−0.467222	−	0.884140i	$-0.654745\pi$
$863$	−27.4510			−0.934443			−0.467222	+	0.884140i	$0.654745\pi$
$864$		0			0
$865$	28.4013			0.965674
$866$		0			0
$867$	17.2755	−	17.8594i	0.586706	−	0.606537i
$868$		0			0
$869$	9.80200			0.332510
$870$		0			0
$871$	−4.22905			−0.143296
$872$		0			0
$873$		−	2.19261i		−	0.0742084i
$874$		0			0
$875$	4.50318			0.152235
$876$		0			0
$877$		−	7.91780i		−	0.267365i	−0.991024	−	0.133683i	$-0.957320\pi$
$877$		−	7.91780i		−	0.267365i	0.991024	−	0.133683i	$-0.0426803\pi$
$878$		0			0
$879$			25.4038i			0.856850i
$880$		0			0
$881$			19.1627i			0.645607i	0.946466	+	0.322803i	$0.104625\pi$
$881$			19.1627i			0.645607i	−0.946466	+	0.322803i	$0.895375\pi$
$882$		0			0
$883$	−24.0525			−0.809431			−0.404716	−	0.914443i	$-0.632629\pi$
$883$	−24.0525			−0.809431			−0.404716	+	0.914443i	$0.632629\pi$
$884$		0			0
$885$	−4.43862			−0.149203
$886$		0			0
$887$		−	38.5705i		−	1.29507i	−0.762036	−	0.647535i	$-0.775800\pi$
$887$		−	38.5705i		−	1.29507i	0.762036	−	0.647535i	$-0.224200\pi$
$888$		0			0
$889$		−	34.0971i		−	1.14358i
$890$		0			0
$891$		−	10.2105i		−	0.342065i
$892$		0			0
$893$	−83.0239			−2.77829
$894$		0			0
$895$			31.6389i			1.05757i
$896$		0			0
$897$	−9.34154			−0.311905
$898$		0			0
$899$	34.6278			1.15490
$900$		0			0
$901$	28.5701	+	11.5570i	0.951808	+	0.385018i
$902$		0			0
$903$	−22.1635			−0.737555
$904$		0			0
$905$	64.1628			2.13284
$906$		0			0
$907$			10.0034i			0.332157i	0.986113	+	0.166079i	$0.0531105\pi$
$907$			10.0034i			0.332157i	−0.986113	+	0.166079i	$0.946889\pi$
$908$		0			0
$909$	−9.25489			−0.306965
$910$		0			0
$911$			43.2427i			1.43270i	0.697743	+	0.716348i	$0.254188\pi$
$911$			43.2427i			1.43270i	−0.697743	+	0.716348i	$0.745812\pi$
$912$		0			0
$913$			5.22663i			0.172976i
$914$		0			0
$915$			56.6596i			1.87311i
$916$		0			0
$917$	5.56285			0.183701
$918$		0			0
$919$	−44.4406			−1.46596			−0.732980	−	0.680251i	$-0.761871\pi$
$919$	−44.4406			−1.46596			−0.732980	+	0.680251i	$0.761871\pi$
$920$		0			0
$921$		−	49.5039i		−	1.63121i
$922$		0			0
$923$			8.28248i			0.272621i
$924$		0			0
$925$			24.7668i			0.814328i
$926$		0			0
$927$	0.690669			0.0226845
$928$		0			0
$929$			38.1116i			1.25040i	0.780465	+	0.625200i	$0.214983\pi$
$929$			38.1116i			1.25040i	−0.780465	+	0.625200i	$0.785017\pi$
$930$		0			0
$931$	−27.4464			−0.899518
$932$		0			0
$933$	−0.0630388			−0.00206380
$934$		0			0
$935$	−20.9279	−	8.46559i	−0.684415	−	0.276855i
$936$		0			0
$937$	−0.262990			−0.00859151			−0.00429575	−	0.999991i	$-0.501367\pi$
$937$	−0.262990			−0.00859151			−0.00429575	+	0.999991i	$0.501367\pi$
$938$		0			0
$939$	21.6916			0.707877
$940$		0			0
$941$		−	56.2019i		−	1.83213i	−0.401029	−	0.916066i	$-0.631347\pi$
$941$		−	56.2019i		−	1.83213i	0.401029	−	0.916066i	$-0.368653\pi$
$942$		0			0
$943$	−18.2497			−0.594293
$944$		0			0
$945$		−	32.6888i		−	1.06337i
$946$		0			0
$947$		−	46.3385i		−	1.50580i	−0.658136	−	0.752899i	$-0.728655\pi$
$947$		−	46.3385i		−	1.50580i	0.658136	−	0.752899i	$-0.271345\pi$
$948$		0			0
$949$			1.07471i			0.0348865i
$950$		0			0
$951$	18.6073			0.603383
$952$		0			0
$953$	0.486497			0.0157592			0.00787959	−	0.999969i	$-0.497492\pi$
$953$	0.486497			0.0157592			0.00787959	+	0.999969i	$0.497492\pi$
$954$		0			0
$955$		−	52.4945i		−	1.69868i
$956$		0			0
$957$		−	14.7825i		−	0.477852i
$958$		0			0
$959$		−	21.1456i		−	0.682827i
$960$		0			0
$961$	−7.10758			−0.229277
$962$		0			0
$963$		−	3.61690i		−	0.116553i
$964$		0			0
$965$	−42.3277			−1.36258
$966$		0			0
$967$	−30.6822			−0.986672			−0.493336	−	0.869839i	$-0.664223\pi$
$967$	−30.6822			−0.986672			−0.493336	+	0.869839i	$0.664223\pi$
$968$		0			0
$969$	−18.4232	+	45.5442i	−0.591838	+	1.46309i
$970$		0			0
$971$	−52.4542			−1.68334			−0.841668	−	0.539995i	$-0.818426\pi$
$971$	−52.4542			−1.68334			−0.841668	+	0.539995i	$0.818426\pi$
$972$		0			0
$973$	−19.9821			−0.640595
$974$		0			0
$975$			8.36257i			0.267817i
$976$		0			0
$977$	34.6977			1.11008			0.555039	−	0.831824i	$-0.312703\pi$
$977$	34.6977			1.11008			0.555039	+	0.831824i	$0.312703\pi$
$978$		0			0
$979$			4.38444i			0.140127i
$980$		0			0
$981$			14.6327i			0.467186i
$982$		0			0
$983$			38.7799i			1.23689i	0.785829	+	0.618443i	$0.212236\pi$
$983$			38.7799i			1.23689i	−0.785829	+	0.618443i	$0.787764\pi$
$984$		0			0
$985$	−47.9757			−1.52863
$986$		0			0
$987$	−28.3731			−0.903127
$988$		0			0
$989$		−	37.5193i		−	1.19304i
$990$		0			0
$991$		−	15.3747i		−	0.488394i	−0.969726	−	0.244197i	$-0.921476\pi$
$991$		−	15.3747i		−	0.488394i	0.969726	−	0.244197i	$-0.0785243\pi$
$992$		0			0
$993$		−	12.4264i		−	0.394339i
$994$		0			0
$995$	57.2603			1.81527
$996$		0			0
$997$		−	37.8207i		−	1.19779i	−0.800827	−	0.598896i	$-0.795606\pi$
$997$		−	37.8207i		−	1.19779i	0.800827	−	0.598896i	$-0.204394\pi$
$998$		0			0
$999$	−33.1270			−1.04809

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4012.2.b.a.237.13	✓	40
17.16	even	2	inner	4012.2.b.a.237.28	yes	40

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4012.2.b.a.237.13	✓	40	1.1	even	1	trivial
4012.2.b.a.237.28	yes	40	17.16	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:	\( N \)	\(=\)	\( 4012 = 2^{2} \cdot 17 \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4012.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.46162i q^{3} +3.03678i q^{5} -1.90612i q^{7} +0.863667 q^{9} +O(q^{10})\)	\(q-1.46162i q^{3} +3.03678i q^{5} -1.90612i q^{7} +0.863667 q^{9} +1.80300i q^{11} +1.35513 q^{13} +4.43862 q^{15} +(3.82223 + 1.54614i) q^{17} -8.15232 q^{19} -2.78603 q^{21} -4.71630i q^{23} -4.22205 q^{25} -5.64721i q^{27} -5.60944i q^{29} +6.17313i q^{31} +2.63530 q^{33} +5.78848 q^{35} -5.86607i q^{37} -1.98069i q^{39} -3.86950i q^{41} +7.95523 q^{43} +2.62277i q^{45} +10.1841 q^{47} +3.36669 q^{49} +(2.25987 - 5.58665i) q^{51} +7.47471 q^{53} -5.47531 q^{55} +11.9156i q^{57} -1.00000 q^{59} +12.7651i q^{61} -1.64626i q^{63} +4.11525i q^{65} -3.12076 q^{67} -6.89344 q^{69} +6.11193i q^{71} +0.793064i q^{73} +6.17103i q^{75} +3.43673 q^{77} -5.43651i q^{79} -5.66308 q^{81} +2.89886 q^{83} +(-4.69529 + 11.6073i) q^{85} -8.19888 q^{87} +2.43175 q^{89} -2.58305i q^{91} +9.02278 q^{93} -24.7568i q^{95} -2.53872i q^{97} +1.55719i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q - 36 q^{9}+O(q^{10}) \) 40 * q - 36 * q^9	\( 40 q - 36 q^{9} + 8 q^{13} + 20 q^{15} - 8 q^{17} + 20 q^{19} + 6 q^{21} - 24 q^{25} - 14 q^{33} - 52 q^{35} + 22 q^{43} - 10 q^{47} + 8 q^{49} - 6 q^{51} - 2 q^{53} - 12 q^{55} - 40 q^{59} - 24 q^{67} - 36 q^{69} - 38 q^{77} + 16 q^{81} + 32 q^{83} - 22 q^{85} - 18 q^{87} + 40 q^{89} + 22 q^{93}+O(q^{100}) \) 40 * q - 36 * q^9 + 8 * q^13 + 20 * q^15 - 8 * q^17 + 20 * q^19 + 6 * q^21 - 24 * q^25 - 14 * q^33 - 52 * q^35 + 22 * q^43 - 10 * q^47 + 8 * q^49 - 6 * q^51 - 2 * q^53 - 12 * q^55 - 40 * q^59 - 24 * q^67 - 36 * q^69 - 38 * q^77 + 16 * q^81 + 32 * q^83 - 22 * q^85 - 18 * q^87 + 40 * q^89 + 22 * q^93

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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