LMFDB - Embedded newform 4032.2.v.d.1583.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4032,2,Mod(1583,4032)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4032, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([2, 3, 2, 0]))

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

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

chi := DirichletCharacter("4032.1583");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4032 = 2^{6} \cdot 3^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4032.v (of order $4$, degree $2$, 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:	$32.1956820950$
Analytic rank:	$0$
Dimension:	$36$
Relative dimension:	$18$ over $\Q(i)$
Twist minimal:	no (minimal twist has level 1008)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			1583.3
Character	$\chi$	$=$	4032.1583
Dual form			4032.2.v.d.3599.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-2.28967 - 2.28967i) q^{5} +1.00000 q^{7} +O(q^{10})$	$q+(-2.28967 - 2.28967i) q^{5} +1.00000 q^{7} +(-3.92329 + 3.92329i) q^{11} +(1.38904 + 1.38904i) q^{13} -4.10467i q^{17} +(-3.60592 + 3.60592i) q^{19} -2.68375i q^{23} +5.48518i q^{25} +(6.10020 - 6.10020i) q^{29} +5.45644i q^{31} +(-2.28967 - 2.28967i) q^{35} +(-4.04656 + 4.04656i) q^{37} +7.75676 q^{41} +(-0.0577035 - 0.0577035i) q^{43} +1.70367 q^{47} +1.00000 q^{49} +(0.517297 + 0.517297i) q^{53} +17.9661 q^{55} +(-7.10709 + 7.10709i) q^{59} +(2.19652 + 2.19652i) q^{61} -6.36089i q^{65} +(6.98139 - 6.98139i) q^{67} -0.356496i q^{71} -7.05490i q^{73} +(-3.92329 + 3.92329i) q^{77} +0.163773i q^{79} +(7.33131 + 7.33131i) q^{83} +(-9.39834 + 9.39834i) q^{85} -16.1465 q^{89} +(1.38904 + 1.38904i) q^{91} +16.5127 q^{95} +8.56055 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 36 q + 36 q^{7}+O(q^{10}) $ 36 * q + 36 * q^7	$ 36 q + 36 q^{7} - 16 q^{13} + 16 q^{19} + 20 q^{37} - 36 q^{43} + 36 q^{49} - 32 q^{55} + 112 q^{61} + 36 q^{67} - 96 q^{85} - 16 q^{91} + 56 q^{97}+O(q^{100}) $ 36 * q + 36 * q^7 - 16 * q^13 + 16 * q^19 + 20 * q^37 - 36 * q^43 + 36 * q^49 - 32 * q^55 + 112 * q^61 + 36 * q^67 - 96 * q^85 - 16 * q^91 + 56 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4032\mathbb{Z}\right)^\times$.

$n$	$127$	$577$	$1793$	$3781$
$\chi(n)$	$-1$	$1$	$-1$	$e\left(\frac{3}{4}\right)$

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
$3$		0			0
$3$		0			0
$4$		0			0
$5$	−2.28967	−	2.28967i	−1.02397	−	1.02397i	−0.999706	−	0.0242660i	$-0.992275\pi$
$5$	−2.28967	−	2.28967i	−1.02397	−	1.02397i	−0.0242660	−	0.999706i	$-0.507725\pi$
$6$		0			0
$7$	1.00000			0.377964
$7$	1.00000			0.377964
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−3.92329	+	3.92329i	−1.18292	+	1.18292i	−0.203932	+	0.978985i	$0.565372\pi$
$11$	−3.92329	+	3.92329i	−1.18292	+	1.18292i	−0.978985	+	0.203932i	$0.934628\pi$
$12$		0			0
$13$	1.38904	+	1.38904i	0.385251	+	0.385251i	0.872990	−	0.487739i	$-0.162178\pi$
$13$	1.38904	+	1.38904i	0.385251	+	0.385251i	−0.487739	+	0.872990i	$0.662178\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		−	4.10467i		−	0.995528i	−0.867312	−	0.497764i	$-0.834154\pi$
$17$		−	4.10467i		−	0.995528i	0.867312	−	0.497764i	$-0.165846\pi$
$18$		0			0
$19$	−3.60592	+	3.60592i	−0.827255	+	0.827255i	−0.987136	−	0.159881i	$-0.948889\pi$
$19$	−3.60592	+	3.60592i	−0.827255	+	0.827255i	0.159881	+	0.987136i	$0.448889\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		−	2.68375i		−	0.559600i	−0.960058	−	0.279800i	$-0.909732\pi$
$23$		−	2.68375i		−	0.559600i	0.960058	−	0.279800i	$-0.0902681\pi$
$24$		0			0
$25$			5.48518i			1.09704i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	6.10020	−	6.10020i	1.13278	−	1.13278i	0.143065	−	0.989713i	$-0.454304\pi$
$29$	6.10020	−	6.10020i	1.13278	−	1.13278i	0.989713	−	0.143065i	$-0.0456957\pi$
$30$		0			0
$31$			5.45644i			0.980006i	0.871721	+	0.490003i	$0.163004\pi$
$31$			5.45644i			0.980006i	−0.871721	+	0.490003i	$0.836996\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−2.28967	−	2.28967i	−0.387025	−	0.387025i
$36$		0			0
$37$	−4.04656	+	4.04656i	−0.665250	+	0.665250i	−0.956613	−	0.291362i	$-0.905891\pi$
$37$	−4.04656	+	4.04656i	−0.665250	+	0.665250i	0.291362	+	0.956613i	$0.405891\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	7.75676			1.21140			0.605701	−	0.795692i	$-0.292893\pi$
$41$	7.75676			1.21140			0.605701	+	0.795692i	$0.292893\pi$
$42$		0			0
$43$	−0.0577035	−	0.0577035i	−0.00879970	−	0.00879970i	0.702693	−	0.711493i	$-0.251981\pi$
$43$	−0.0577035	−	0.0577035i	−0.00879970	−	0.00879970i	−0.711493	+	0.702693i	$0.751981\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	1.70367			0.248506			0.124253	−	0.992251i	$-0.460347\pi$
$47$	1.70367			0.248506			0.124253	+	0.992251i	$0.460347\pi$
$48$		0			0
$49$	1.00000			0.142857
$50$		0			0
$51$		0			0
$52$		0			0
$53$	0.517297	+	0.517297i	0.0710562	+	0.0710562i	0.741742	−	0.670686i	$-0.234000\pi$
$53$	0.517297	+	0.517297i	0.0710562	+	0.0710562i	−0.670686	+	0.741742i	$0.734000\pi$
$54$		0			0
$55$	17.9661			2.42255
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−7.10709	+	7.10709i	−0.925265	+	0.925265i	−0.997395	−	0.0721304i	$-0.977020\pi$
$59$	−7.10709	+	7.10709i	−0.925265	+	0.925265i	0.0721304	+	0.997395i	$0.477020\pi$
$60$		0			0
$61$	2.19652	+	2.19652i	0.281236	+	0.281236i	0.833602	−	0.552366i	$-0.186275\pi$
$61$	2.19652	+	2.19652i	0.281236	+	0.281236i	−0.552366	+	0.833602i	$0.686275\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		−	6.36089i		−	0.788972i
$66$		0			0
$67$	6.98139	−	6.98139i	0.852913	−	0.852913i	−0.137578	−	0.990491i	$-0.543932\pi$
$67$	6.98139	−	6.98139i	0.852913	−	0.852913i	0.990491	+	0.137578i	$0.0439318\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	0.356496i		−	0.0423082i	−0.999776	−	0.0211541i	$-0.993266\pi$
$71$		−	0.356496i		−	0.0423082i	0.999776	−	0.0211541i	$-0.00673407\pi$
$72$		0			0
$73$		−	7.05490i		−	0.825713i	−0.910796	−	0.412857i	$-0.864531\pi$
$73$		−	7.05490i		−	0.825713i	0.910796	−	0.412857i	$-0.135469\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−3.92329	+	3.92329i	−0.447101	+	0.447101i
$78$		0			0
$79$			0.163773i			0.0184259i	0.999958	+	0.00921297i	$0.00293262\pi$
$79$			0.163773i			0.0184259i	−0.999958	+	0.00921297i	$0.997067\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	7.33131	+	7.33131i	0.804716	+	0.804716i	0.983829	−	0.179112i	$-0.0573226\pi$
$83$	7.33131	+	7.33131i	0.804716	+	0.804716i	−0.179112	+	0.983829i	$0.557323\pi$
$84$		0			0
$85$	−9.39834	+	9.39834i	−1.01939	+	1.01939i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−16.1465			−1.71153			−0.855764	−	0.517367i	$-0.826912\pi$
$89$	−16.1465			−1.71153			−0.855764	+	0.517367i	$0.826912\pi$
$90$		0			0
$91$	1.38904	+	1.38904i	0.145611	+	0.145611i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	16.5127			1.69417
$96$		0			0
$97$	8.56055			0.869192			0.434596	−	0.900626i	$-0.356891\pi$
$97$	8.56055			0.869192			0.434596	+	0.900626i	$0.356891\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−1.02014	−	1.02014i	−0.101507	−	0.101507i	0.654529	−	0.756037i	$-0.272867\pi$
$101$	−1.02014	−	1.02014i	−0.101507	−	0.101507i	−0.756037	+	0.654529i	$0.772867\pi$
$102$		0			0
$103$	17.7494			1.74890			0.874451	−	0.485113i	$-0.161222\pi$
$103$	17.7494			1.74890			0.874451	+	0.485113i	$0.161222\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	12.0872	−	12.0872i	1.16851	−	1.16851i	0.185951	−	0.982559i	$-0.440464\pi$
$107$	12.0872	−	12.0872i	1.16851	−	1.16851i	0.982559	−	0.185951i	$-0.0595365\pi$
$108$		0			0
$109$	13.8663	+	13.8663i	1.32815	+	1.32815i	0.906986	+	0.421160i	$0.138377\pi$
$109$	13.8663	+	13.8663i	1.32815	+	1.32815i	0.421160	+	0.906986i	$0.361623\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		−	15.0746i		−	1.41810i	−0.705158	−	0.709050i	$-0.749124\pi$
$113$		−	15.0746i		−	1.41810i	0.705158	−	0.709050i	$-0.250876\pi$
$114$		0			0
$115$	−6.14490	+	6.14490i	−0.573014	+	0.573014i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		−	4.10467i		−	0.376274i
$120$		0			0
$121$		−	19.7845i		−	1.79859i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	1.11090	−	1.11090i	0.0993617	−	0.0993617i
$126$		0			0
$127$		−	15.9384i		−	1.41430i	−0.707063	−	0.707151i	$-0.749980\pi$
$127$		−	15.9384i		−	1.41430i	0.707063	−	0.707151i	$-0.250020\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	1.98295	+	1.98295i	0.173251	+	0.173251i	0.788406	−	0.615155i	$-0.210907\pi$
$131$	1.98295	+	1.98295i	0.173251	+	0.173251i	−0.615155	+	0.788406i	$0.710907\pi$
$132$		0			0
$133$	−3.60592	+	3.60592i	−0.312673	+	0.312673i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	4.97010			0.424624			0.212312	−	0.977202i	$-0.431901\pi$
$137$	4.97010			0.424624			0.212312	+	0.977202i	$0.431901\pi$
$138$		0			0
$139$	−6.20608	−	6.20608i	−0.526393	−	0.526393i	0.393102	−	0.919495i	$-0.371402\pi$
$139$	−6.20608	−	6.20608i	−0.526393	−	0.526393i	−0.919495	+	0.393102i	$0.871402\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−10.8992			−0.911440
$144$		0			0
$145$	−27.9349			−2.31987
$146$		0			0
$147$		0			0
$148$		0			0
$149$	10.6879	+	10.6879i	0.875585	+	0.875585i	0.993074	−	0.117489i	$-0.0374846\pi$
$149$	10.6879	+	10.6879i	0.875585	+	0.875585i	−0.117489	+	0.993074i	$0.537485\pi$
$150$		0			0
$151$	−5.12562			−0.417117			−0.208559	−	0.978010i	$-0.566877\pi$
$151$	−5.12562			−0.417117			−0.208559	+	0.978010i	$0.566877\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	12.4935	−	12.4935i	1.00350	−	1.00350i
$156$		0			0
$157$	8.50142	+	8.50142i	0.678487	+	0.678487i	0.959658	−	0.281170i	$-0.0907226\pi$
$157$	8.50142	+	8.50142i	0.678487	+	0.678487i	−0.281170	+	0.959658i	$0.590723\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		−	2.68375i		−	0.211509i
$162$		0			0
$163$	1.05470	−	1.05470i	0.0826108	−	0.0826108i	−0.664594	−	0.747205i	$-0.731395\pi$
$163$	1.05470	−	1.05470i	0.0826108	−	0.0826108i	0.747205	+	0.664594i	$0.231395\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$			5.59381i			0.432862i	0.976298	+	0.216431i	$0.0694417\pi$
$167$			5.59381i			0.432862i	−0.976298	+	0.216431i	$0.930558\pi$
$168$		0			0
$169$		−	9.14113i		−	0.703164i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	14.4457	−	14.4457i	1.09829	−	1.09829i	0.103678	−	0.994611i	$-0.466939\pi$
$173$	14.4457	−	14.4457i	1.09829	−	1.09829i	0.994611	−	0.103678i	$-0.0330611\pi$
$174$		0			0
$175$			5.48518i			0.414640i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−18.8918	−	18.8918i	−1.41204	−	1.41204i	−0.745197	−	0.666845i	$-0.767644\pi$
$179$	−18.8918	−	18.8918i	−1.41204	−	1.41204i	−0.666845	−	0.745197i	$-0.732356\pi$
$180$		0			0
$181$	6.04746	−	6.04746i	0.449504	−	0.449504i	−0.445686	−	0.895190i	$-0.647040\pi$
$181$	6.04746	−	6.04746i	0.449504	−	0.449504i	0.895190	+	0.445686i	$0.147040\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	18.5306			1.36239
$186$		0			0
$187$	16.1038	+	16.1038i	1.17763	+	1.17763i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	21.1862			1.53298			0.766490	−	0.642256i	$-0.222001\pi$
$191$	21.1862			1.53298			0.766490	+	0.642256i	$0.222001\pi$
$192$		0			0
$193$	2.51759			0.181220			0.0906100	−	0.995886i	$-0.471118\pi$
$193$	2.51759			0.181220			0.0906100	+	0.995886i	$0.471118\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−15.3329	−	15.3329i	−1.09242	−	1.09242i	−0.995269	−	0.0971552i	$-0.969026\pi$
$197$	−15.3329	−	15.3329i	−1.09242	−	1.09242i	−0.0971552	−	0.995269i	$-0.530974\pi$
$198$		0			0
$199$	12.9712			0.919505			0.459752	−	0.888047i	$-0.347938\pi$
$199$	12.9712			0.919505			0.459752	+	0.888047i	$0.347938\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	6.10020	−	6.10020i	0.428150	−	0.428150i
$204$		0			0
$205$	−17.7604	−	17.7604i	−1.24044	−	1.24044i
$206$		0			0
$207$		0			0
$208$		0			0
$209$		−	28.2942i		−	1.95715i
$210$		0			0
$211$	−9.79983	+	9.79983i	−0.674648	+	0.674648i	−0.958784	−	0.284136i	$-0.908293\pi$
$211$	−9.79983	+	9.79983i	−0.674648	+	0.674648i	0.284136	+	0.958784i	$0.408293\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$			0.264244i			0.0180213i
$216$		0			0
$217$			5.45644i			0.370407i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	5.70155	−	5.70155i	0.383528	−	0.383528i
$222$		0			0
$223$		−	17.2225i		−	1.15330i	−0.816991	−	0.576651i	$-0.804359\pi$
$223$		−	17.2225i		−	1.15330i	0.816991	−	0.576651i	$-0.195641\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−7.18661	−	7.18661i	−0.476992	−	0.476992i	0.427176	−	0.904168i	$-0.359508\pi$
$227$	−7.18661	−	7.18661i	−0.476992	−	0.476992i	−0.904168	+	0.427176i	$0.859508\pi$
$228$		0			0
$229$	−4.85424	+	4.85424i	−0.320778	+	0.320778i	−0.849065	−	0.528288i	$-0.822834\pi$
$229$	−4.85424	+	4.85424i	−0.320778	+	0.320778i	0.528288	+	0.849065i	$0.322834\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	15.1752			0.994163			0.497081	−	0.867704i	$-0.334405\pi$
$233$	15.1752			0.994163			0.497081	+	0.867704i	$0.334405\pi$
$234$		0			0
$235$	−3.90085	−	3.90085i	−0.254463	−	0.254463i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	4.48591			0.290169			0.145085	−	0.989419i	$-0.453655\pi$
$239$	4.48591			0.290169			0.145085	+	0.989419i	$0.453655\pi$
$240$		0			0
$241$	15.2916			0.985018			0.492509	−	0.870307i	$-0.336080\pi$
$241$	15.2916			0.985018			0.492509	+	0.870307i	$0.336080\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−2.28967	−	2.28967i	−0.146282	−	0.146282i
$246$		0			0
$247$	−10.0175			−0.637401
$248$		0			0
$249$		0			0
$250$		0			0
$251$	21.8987	−	21.8987i	1.38223	−	1.38223i	0.541588	−	0.840644i	$-0.317823\pi$
$251$	21.8987	−	21.8987i	1.38223	−	1.38223i	0.840644	−	0.541588i	$-0.182177\pi$
$252$		0			0
$253$	10.5291	+	10.5291i	0.661960	+	0.661960i
$254$		0			0
$255$		0			0
$256$		0			0
$257$			31.1240i			1.94146i	0.240165	+	0.970732i	$0.422798\pi$
$257$			31.1240i			1.94146i	−0.240165	+	0.970732i	$0.577202\pi$
$258$		0			0
$259$	−4.04656	+	4.04656i	−0.251441	+	0.251441i
$260$		0			0
$261$		0			0
$262$		0			0
$263$			9.83612i			0.606521i	0.952908	+	0.303261i	$0.0980753\pi$
$263$			9.83612i			0.606521i	−0.952908	+	0.303261i	$0.901925\pi$
$264$		0			0
$265$		−	2.36888i		−	0.145519i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−22.7767	+	22.7767i	−1.38872	+	1.38872i	−0.560701	+	0.828018i	$0.689468\pi$
$269$	−22.7767	+	22.7767i	−1.38872	+	1.38872i	−0.828018	+	0.560701i	$0.810532\pi$
$270$		0			0
$271$			6.28014i			0.381492i	0.981639	+	0.190746i	$0.0610906\pi$
$271$			6.28014i			0.381492i	−0.981639	+	0.190746i	$0.938909\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−21.5200	−	21.5200i	−1.29770	−	1.29770i
$276$		0			0
$277$	14.4973	−	14.4973i	0.871058	−	0.871058i	−0.121530	−	0.992588i	$-0.538780\pi$
$277$	14.4973	−	14.4973i	0.871058	−	0.871058i	0.992588	+	0.121530i	$0.0387801\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	23.0904			1.37746			0.688728	−	0.725020i	$-0.258170\pi$
$281$	23.0904			1.37746			0.688728	+	0.725020i	$0.258170\pi$
$282$		0			0
$283$	−4.37723	−	4.37723i	−0.260199	−	0.260199i	0.564936	−	0.825135i	$-0.308901\pi$
$283$	−4.37723	−	4.37723i	−0.260199	−	0.260199i	−0.825135	+	0.564936i	$0.808901\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	7.75676			0.457867
$288$		0			0
$289$	0.151693			0.00892312
$290$		0			0
$291$		0			0
$292$		0			0
$293$	3.34166	+	3.34166i	0.195222	+	0.195222i	0.797948	−	0.602726i	$-0.205919\pi$
$293$	3.34166	+	3.34166i	0.195222	+	0.195222i	−0.602726	+	0.797948i	$0.705919\pi$
$294$		0			0
$295$	32.5458			1.89489
$296$		0			0
$297$		0			0
$298$		0			0
$299$	3.72784	−	3.72784i	0.215586	−	0.215586i
$300$		0			0
$301$	−0.0577035	−	0.0577035i	−0.00332597	−	0.00332597i
$302$		0			0
$303$		0			0
$304$		0			0
$305$		−	10.0586i		−	0.575955i
$306$		0			0
$307$	0.119162	−	0.119162i	0.00680096	−	0.00680096i	−0.703698	−	0.710499i	$-0.748469\pi$
$307$	0.119162	−	0.119162i	0.00680096	−	0.00680096i	0.710499	+	0.703698i	$0.248469\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$			18.6414i			1.05706i	0.848915	+	0.528529i	$0.177256\pi$
$311$			18.6414i			1.05706i	−0.848915	+	0.528529i	$0.822744\pi$
$312$		0			0
$313$		−	1.35618i		−	0.0766556i	−0.999265	−	0.0383278i	$-0.987797\pi$
$313$		−	1.35618i		−	0.0766556i	0.999265	−	0.0383278i	$-0.0122031\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−12.2596	+	12.2596i	−0.688566	+	0.688566i	−0.961915	−	0.273349i	$-0.911869\pi$
$317$	−12.2596	+	12.2596i	−0.688566	+	0.688566i	0.273349	+	0.961915i	$0.411869\pi$
$318$		0			0
$319$			47.8657i			2.67997i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	14.8011	+	14.8011i	0.823556	+	0.823556i
$324$		0			0
$325$	−7.61914	+	7.61914i	−0.422634	+	0.422634i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	1.70367			0.0939265
$330$		0			0
$331$	−11.1478	−	11.1478i	−0.612741	−	0.612741i	0.330918	−	0.943659i	$-0.392642\pi$
$331$	−11.1478	−	11.1478i	−0.612741	−	0.612741i	−0.943659	+	0.330918i	$0.892642\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−31.9702			−1.74672
$336$		0			0
$337$	1.47574			0.0803886			0.0401943	−	0.999192i	$-0.487202\pi$
$337$	1.47574			0.0803886			0.0401943	+	0.999192i	$0.487202\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−21.4072	−	21.4072i	−1.15927	−	1.15927i
$342$		0			0
$343$	1.00000			0.0539949
$344$		0			0
$345$		0			0
$346$		0			0
$347$	6.38522	−	6.38522i	0.342776	−	0.342776i	−0.514634	−	0.857410i	$-0.672072\pi$
$347$	6.38522	−	6.38522i	0.342776	−	0.342776i	0.857410	+	0.514634i	$0.172072\pi$
$348$		0			0
$349$	17.1578	+	17.1578i	0.918435	+	0.918435i	0.996916	−	0.0784806i	$-0.0250069\pi$
$349$	17.1578	+	17.1578i	0.918435	+	0.918435i	−0.0784806	+	0.996916i	$0.525007\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$			5.07308i			0.270013i	0.990845	+	0.135007i	$0.0431055\pi$
$353$			5.07308i			0.270013i	−0.990845	+	0.135007i	$0.956894\pi$
$354$		0			0
$355$	−0.816257	+	0.816257i	−0.0433224	+	0.0433224i
$356$		0			0
$357$		0			0
$358$		0			0
$359$			1.13760i			0.0600402i	0.999549	+	0.0300201i	$0.00955712\pi$
$359$			1.13760i			0.0600402i	−0.999549	+	0.0300201i	$0.990443\pi$
$360$		0			0
$361$		−	7.00533i		−	0.368701i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−16.1534	+	16.1534i	−0.845507	+	0.845507i
$366$		0			0
$367$		−	2.78975i		−	0.145624i	−0.997346	−	0.0728119i	$-0.976803\pi$
$367$		−	2.78975i		−	0.145624i	0.997346	−	0.0728119i	$-0.0231973\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	0.517297	+	0.517297i	0.0268567	+	0.0268567i
$372$		0			0
$373$	20.7939	−	20.7939i	1.07667	−	1.07667i	0.0798639	−	0.996806i	$-0.474551\pi$
$373$	20.7939	−	20.7939i	1.07667	−	1.07667i	0.996806	−	0.0798639i	$-0.0254486\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	16.9469			0.872807
$378$		0			0
$379$	23.7712	+	23.7712i	1.22105	+	1.22105i	0.967261	+	0.253785i	$0.0816756\pi$
$379$	23.7712	+	23.7712i	1.22105	+	1.22105i	0.253785	+	0.967261i	$0.418324\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−7.74341			−0.395670			−0.197835	−	0.980235i	$-0.563391\pi$
$383$	−7.74341			−0.395670			−0.197835	+	0.980235i	$0.563391\pi$
$384$		0			0
$385$	17.9661			0.915637
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−14.2477	−	14.2477i	−0.722385	−	0.722385i	0.246705	−	0.969091i	$-0.420652\pi$
$389$	−14.2477	−	14.2477i	−0.722385	−	0.722385i	−0.969091	+	0.246705i	$0.920652\pi$
$390$		0			0
$391$	−11.0159			−0.557098
$392$		0			0
$393$		0			0
$394$		0			0
$395$	0.374987	−	0.374987i	0.0188676	−	0.0188676i
$396$		0			0
$397$	21.6073	+	21.6073i	1.08444	+	1.08444i	0.996089	+	0.0883525i	$0.0281602\pi$
$397$	21.6073	+	21.6073i	1.08444	+	1.08444i	0.0883525	+	0.996089i	$0.471840\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$			32.1405i			1.60502i	0.596638	+	0.802510i	$0.296503\pi$
$401$			32.1405i			1.60502i	−0.596638	+	0.802510i	$0.703497\pi$
$402$		0			0
$403$	−7.57922	+	7.57922i	−0.377548	+	0.377548i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		−	31.7517i		−	1.57387i
$408$		0			0
$409$			25.4416i			1.25801i	0.777402	+	0.629004i	$0.216537\pi$
$409$			25.4416i			1.25801i	−0.777402	+	0.629004i	$0.783463\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−7.10709	+	7.10709i	−0.349717	+	0.349717i
$414$		0			0
$415$		−	33.5726i		−	1.64801i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	5.68536	+	5.68536i	0.277748	+	0.277748i	0.832209	−	0.554461i	$-0.187076\pi$
$419$	5.68536	+	5.68536i	0.277748	+	0.277748i	−0.554461	+	0.832209i	$0.687076\pi$
$420$		0			0
$421$	1.82028	−	1.82028i	0.0887151	−	0.0887151i	−0.661357	−	0.750072i	$-0.730019\pi$
$421$	1.82028	−	1.82028i	0.0887151	−	0.0887151i	0.750072	+	0.661357i	$0.230019\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	22.5148			1.09213
$426$		0			0
$427$	2.19652	+	2.19652i	0.106297	+	0.106297i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	16.7699			0.807779			0.403889	−	0.914808i	$-0.367658\pi$
$431$	16.7699			0.807779			0.403889	+	0.914808i	$0.367658\pi$
$432$		0			0
$433$	−28.6891			−1.37871			−0.689355	−	0.724424i	$-0.742106\pi$
$433$	−28.6891			−1.37871			−0.689355	+	0.724424i	$0.742106\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	9.67738	+	9.67738i	0.462932	+	0.462932i
$438$		0			0
$439$	25.4747			1.21584			0.607920	−	0.793998i	$-0.292004\pi$
$439$	25.4747			1.21584			0.607920	+	0.793998i	$0.292004\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−11.0293	+	11.0293i	−0.524020	+	0.524020i	−0.918783	−	0.394763i	$-0.870827\pi$
$443$	−11.0293	+	11.0293i	−0.524020	+	0.524020i	0.394763	+	0.918783i	$0.370827\pi$
$444$		0			0
$445$	36.9702	+	36.9702i	1.75256	+	1.75256i
$446$		0			0
$447$		0			0
$448$		0			0
$449$			25.9697i			1.22558i	0.790244	+	0.612792i	$0.209954\pi$
$449$			25.9697i			1.22558i	−0.790244	+	0.612792i	$0.790046\pi$
$450$		0			0
$451$	−30.4320	+	30.4320i	−1.43299	+	1.43299i
$452$		0			0
$453$		0			0
$454$		0			0
$455$		−	6.36089i		−	0.298203i
$456$		0			0
$457$		−	21.9383i		−	1.02623i	−0.858320	−	0.513115i	$-0.828492\pi$
$457$		−	21.9383i		−	1.02623i	0.858320	−	0.513115i	$-0.171508\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	5.54081	−	5.54081i	0.258061	−	0.258061i	−0.566204	−	0.824265i	$-0.691589\pi$
$461$	5.54081	−	5.54081i	0.258061	−	0.258061i	0.824265	+	0.566204i	$0.191589\pi$
$462$		0			0
$463$		−	3.14263i		−	0.146050i	−0.997330	−	0.0730252i	$-0.976735\pi$
$463$		−	3.14263i		−	0.146050i	0.997330	−	0.0730252i	$-0.0232653\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−11.0517	−	11.0517i	−0.511411	−	0.511411i	0.403548	−	0.914959i	$-0.367777\pi$
$467$	−11.0517	−	11.0517i	−0.511411	−	0.511411i	−0.914959	+	0.403548i	$0.867777\pi$
$468$		0			0
$469$	6.98139	−	6.98139i	0.322371	−	0.322371i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	0.452775			0.0208186
$474$		0			0
$475$	−19.7791	−	19.7791i	−0.907528	−	0.907528i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−29.5140			−1.34853			−0.674264	−	0.738490i	$-0.735539\pi$
$479$	−29.5140			−1.34853			−0.674264	+	0.738490i	$0.735539\pi$
$480$		0			0
$481$	−11.2417			−0.512576
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−19.6008	−	19.6008i	−0.890028	−	0.890028i
$486$		0			0
$487$	−9.24906			−0.419115			−0.209557	−	0.977796i	$-0.567202\pi$
$487$	−9.24906			−0.419115			−0.209557	+	0.977796i	$0.567202\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−1.07931	+	1.07931i	−0.0487085	+	0.0487085i	−0.731042	−	0.682333i	$-0.760965\pi$
$491$	−1.07931	+	1.07931i	−0.0487085	+	0.0487085i	0.682333	+	0.731042i	$0.260965\pi$
$492$		0			0
$493$	−25.0393	−	25.0393i	−1.12771	−	1.12771i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		−	0.356496i		−	0.0159910i
$498$		0			0
$499$	−20.5136	+	20.5136i	−0.918314	+	0.918314i	−0.996907	−	0.0785932i	$-0.974957\pi$
$499$	−20.5136	+	20.5136i	−0.918314	+	0.918314i	0.0785932	+	0.996907i	$0.474957\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$			13.4273i			0.598694i	0.954144	+	0.299347i	$0.0967689\pi$
$503$			13.4273i			0.598694i	−0.954144	+	0.299347i	$0.903231\pi$
$504$		0			0
$505$			4.67155i			0.207881i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−4.73152	+	4.73152i	−0.209721	+	0.209721i	−0.804149	−	0.594428i	$-0.797379\pi$
$509$	−4.73152	+	4.73152i	−0.209721	+	0.209721i	0.594428	+	0.804149i	$0.297379\pi$
$510$		0			0
$511$		−	7.05490i		−	0.312090i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−40.6403	−	40.6403i	−1.79083	−	1.79083i
$516$		0			0
$517$	−6.68401	+	6.68401i	−0.293962	+	0.293962i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−26.7681			−1.17273			−0.586365	−	0.810047i	$-0.699442\pi$
$521$	−26.7681			−1.17273			−0.586365	+	0.810047i	$0.699442\pi$
$522$		0			0
$523$	−11.3476	−	11.3476i	−0.496194	−	0.496194i	0.414057	−	0.910251i	$-0.364112\pi$
$523$	−11.3476	−	11.3476i	−0.496194	−	0.496194i	−0.910251	+	0.414057i	$0.864112\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	22.3969			0.975624
$528$		0			0
$529$	15.7975			0.686848
$530$		0			0
$531$		0			0
$532$		0			0
$533$	10.7745	+	10.7745i	0.466694	+	0.466694i
$534$		0			0
$535$	−55.3512			−2.39304
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−3.92329	+	3.92329i	−0.168988	+	0.168988i
$540$		0			0
$541$	23.3649	+	23.3649i	1.00454	+	1.00454i	0.999990	+	0.00454629i	$0.00144714\pi$
$541$	23.3649	+	23.3649i	1.00454	+	1.00454i	0.00454629	+	0.999990i	$0.498553\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		−	63.4983i		−	2.71997i
$546$		0			0
$547$	15.0340	−	15.0340i	0.642807	−	0.642807i	−0.308438	−	0.951245i	$-0.599806\pi$
$547$	15.0340	−	15.0340i	0.642807	−	0.642807i	0.951245	+	0.308438i	$0.0998061\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$			43.9937i			1.87419i
$552$		0			0
$553$			0.163773i			0.00696435i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	28.5441	−	28.5441i	1.20945	−	1.20945i	0.238246	−	0.971205i	$-0.423427\pi$
$557$	28.5441	−	28.5441i	1.20945	−	1.20945i	0.971205	−	0.238246i	$-0.0765726\pi$
$558$		0			0
$559$		−	0.160305i		−	0.00678018i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	12.1713	+	12.1713i	0.512959	+	0.512959i	0.915432	−	0.402473i	$-0.131849\pi$
$563$	12.1713	+	12.1713i	0.512959	+	0.512959i	−0.402473	+	0.915432i	$0.631849\pi$
$564$		0			0
$565$	−34.5159	+	34.5159i	−1.45209	+	1.45209i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−1.31985			−0.0553309			−0.0276654	−	0.999617i	$-0.508807\pi$
$569$	−1.31985			−0.0553309			−0.0276654	+	0.999617i	$0.508807\pi$
$570$		0			0
$571$	−20.4342	−	20.4342i	−0.855143	−	0.855143i	0.135618	−	0.990761i	$-0.456698\pi$
$571$	−20.4342	−	20.4342i	−0.855143	−	0.855143i	−0.990761	+	0.135618i	$0.956698\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	14.7208			0.613901
$576$		0			0
$577$	−11.0034			−0.458076			−0.229038	−	0.973417i	$-0.573558\pi$
$577$	−11.0034			−0.458076			−0.229038	+	0.973417i	$0.573558\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	7.33131	+	7.33131i	0.304154	+	0.304154i
$582$		0			0
$583$	−4.05902			−0.168107
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−11.6048	+	11.6048i	−0.478983	+	0.478983i	−0.904806	−	0.425824i	$-0.859984\pi$
$587$	−11.6048	+	11.6048i	−0.478983	+	0.478983i	0.425824	+	0.904806i	$0.359984\pi$
$588$		0			0
$589$	−19.6755	−	19.6755i	−0.810715	−	0.810715i
$590$		0			0
$591$		0			0
$592$		0			0
$593$		−	29.8971i		−	1.22773i	−0.789412	−	0.613863i	$-0.789615\pi$
$593$		−	29.8971i		−	1.22773i	0.789412	−	0.613863i	$-0.210385\pi$
$594$		0			0
$595$	−9.39834	+	9.39834i	−0.385294	+	0.385294i
$596$		0			0
$597$		0			0
$598$		0			0
$599$			17.3069i			0.707142i	0.935408	+	0.353571i	$0.115033\pi$
$599$			17.3069i			0.707142i	−0.935408	+	0.353571i	$0.884967\pi$
$600$		0			0
$601$			20.4350i			0.833561i	0.909007	+	0.416781i	$0.136842\pi$
$601$			20.4350i			0.833561i	−0.909007	+	0.416781i	$0.863158\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−45.2999	+	45.2999i	−1.84170	+	1.84170i
$606$		0			0
$607$		−	26.2358i		−	1.06488i	−0.846468	−	0.532440i	$-0.821275\pi$
$607$		−	26.2358i		−	1.06488i	0.846468	−	0.532440i	$-0.178725\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	2.36647	+	2.36647i	0.0957372	+	0.0957372i
$612$		0			0
$613$	−2.40107	+	2.40107i	−0.0969784	+	0.0969784i	−0.753931	−	0.656953i	$-0.771845\pi$
$613$	−2.40107	+	2.40107i	−0.0969784	+	0.0969784i	0.656953	+	0.753931i	$0.271845\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−6.65661			−0.267985			−0.133992	−	0.990982i	$-0.542780\pi$
$617$	−6.65661			−0.267985			−0.133992	+	0.990982i	$0.542780\pi$
$618$		0			0
$619$	−18.6833	−	18.6833i	−0.750946	−	0.750946i	0.223710	−	0.974656i	$-0.428183\pi$
$619$	−18.6833	−	18.6833i	−0.750946	−	0.750946i	−0.974656	+	0.223710i	$0.928183\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−16.1465			−0.646897
$624$		0			0
$625$	22.3387			0.893549
$626$		0			0
$627$		0			0
$628$		0			0
$629$	16.6098	+	16.6098i	0.662276	+	0.662276i
$630$		0			0
$631$	15.4855			0.616468			0.308234	−	0.951311i	$-0.400262\pi$
$631$	15.4855			0.616468			0.308234	+	0.951311i	$0.400262\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−36.4936	+	36.4936i	−1.44820	+	1.44820i
$636$		0			0
$637$	1.38904	+	1.38904i	0.0550358	+	0.0550358i
$638$		0			0
$639$		0			0
$640$		0			0
$641$			48.4867i			1.91511i	0.288248	+	0.957556i	$0.406927\pi$
$641$			48.4867i			1.91511i	−0.288248	+	0.957556i	$0.593073\pi$
$642$		0			0
$643$	−35.0610	+	35.0610i	−1.38267	+	1.38267i	−0.542829	+	0.839843i	$0.682647\pi$
$643$	−35.0610	+	35.0610i	−1.38267	+	1.38267i	−0.839843	+	0.542829i	$0.817353\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$			17.4142i			0.684625i	0.939586	+	0.342312i	$0.111210\pi$
$647$			17.4142i			0.684625i	−0.939586	+	0.342312i	$0.888790\pi$
$648$		0			0
$649$		−	55.7664i		−	2.18902i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−3.53254	+	3.53254i	−0.138239	+	0.138239i	−0.772840	−	0.634601i	$-0.781164\pi$
$653$	−3.53254	+	3.53254i	−0.138239	+	0.138239i	0.634601	+	0.772840i	$0.281164\pi$
$654$		0			0
$655$		−	9.08060i		−	0.354809i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−11.8364	−	11.8364i	−0.461080	−	0.461080i	0.437929	−	0.899010i	$-0.355712\pi$
$659$	−11.8364	−	11.8364i	−0.461080	−	0.461080i	−0.899010	+	0.437929i	$0.855712\pi$
$660$		0			0
$661$	5.27298	−	5.27298i	0.205095	−	0.205095i	−0.597084	−	0.802179i	$-0.703674\pi$
$661$	5.27298	−	5.27298i	0.205095	−	0.205095i	0.802179	+	0.597084i	$0.203674\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	16.5127			0.640336
$666$		0			0
$667$	−16.3714	−	16.3714i	−0.633903	−	0.633903i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−17.2352			−0.665357
$672$		0			0
$673$	11.3408			0.437154			0.218577	−	0.975820i	$-0.429858\pi$
$673$	11.3408			0.437154			0.218577	+	0.975820i	$0.429858\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−10.5290	−	10.5290i	−0.404664	−	0.404664i	0.475209	−	0.879873i	$-0.342372\pi$
$677$	−10.5290	−	10.5290i	−0.404664	−	0.404664i	−0.879873	+	0.475209i	$0.842372\pi$
$678$		0			0
$679$	8.56055			0.328524
$680$		0			0
$681$		0			0
$682$		0			0
$683$	33.4933	−	33.4933i	1.28159	−	1.28159i	0.341821	−	0.939765i	$-0.388956\pi$
$683$	33.4933	−	33.4933i	1.28159	−	1.28159i	0.939765	−	0.341821i	$-0.111044\pi$
$684$		0			0
$685$	−11.3799	−	11.3799i	−0.434803	−	0.434803i
$686$		0			0
$687$		0			0
$688$		0			0
$689$			1.43709i			0.0547489i
$690$		0			0
$691$	34.1173	−	34.1173i	1.29788	−	1.29788i	0.368094	−	0.929789i	$-0.380010\pi$
$691$	34.1173	−	34.1173i	1.29788	−	1.29788i	0.929789	−	0.368094i	$-0.119990\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$			28.4197i			1.07802i
$696$		0			0
$697$		−	31.8389i		−	1.20599i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	18.5268	−	18.5268i	0.699746	−	0.699746i	−0.264610	−	0.964356i	$-0.585243\pi$
$701$	18.5268	−	18.5268i	0.699746	−	0.699746i	0.964356	+	0.264610i	$0.0852431\pi$
$702$		0			0
$703$		−	29.1831i		−	1.10066i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−1.02014	−	1.02014i	−0.0383662	−	0.0383662i
$708$		0			0
$709$	−13.9116	+	13.9116i	−0.522462	+	0.522462i	−0.918314	−	0.395852i	$-0.870449\pi$
$709$	−13.9116	+	13.9116i	−0.522462	+	0.522462i	0.395852	+	0.918314i	$0.370449\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	14.6437			0.548411
$714$		0			0
$715$	24.9556	+	24.9556i	0.933288	+	0.933288i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	29.8916			1.11477			0.557384	−	0.830255i	$-0.311805\pi$
$719$	29.8916			1.11477			0.557384	+	0.830255i	$0.311805\pi$
$720$		0			0
$721$	17.7494			0.661023
$722$		0			0
$723$		0			0
$724$		0			0
$725$	33.4607	+	33.4607i	1.24270	+	1.24270i
$726$		0			0
$727$	9.42362			0.349503			0.174751	−	0.984613i	$-0.444088\pi$
$727$	9.42362			0.349503			0.174751	+	0.984613i	$0.444088\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−0.236854	+	0.236854i	−0.00876035	+	0.00876035i
$732$		0			0
$733$	−31.8531	−	31.8531i	−1.17652	−	1.17652i	−0.980624	−	0.195897i	$-0.937238\pi$
$733$	−31.8531	−	31.8531i	−1.17652	−	1.17652i	−0.195897	−	0.980624i	$-0.562762\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$			54.7801i			2.01785i
$738$		0			0
$739$	1.55441	−	1.55441i	0.0571797	−	0.0571797i	−0.677939	−	0.735118i	$-0.737127\pi$
$739$	1.55441	−	1.55441i	0.0571797	−	0.0571797i	0.735118	+	0.677939i	$0.237127\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$			5.57392i			0.204487i	0.994759	+	0.102244i	$0.0326021\pi$
$743$			5.57392i			0.204487i	−0.994759	+	0.102244i	$0.967398\pi$
$744$		0			0
$745$		−	48.9434i		−	1.79315i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	12.0872	−	12.0872i	0.441655	−	0.441655i
$750$		0			0
$751$			22.0891i			0.806043i	0.915190	+	0.403022i	$0.132040\pi$
$751$			22.0891i			0.806043i	−0.915190	+	0.403022i	$0.867960\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	11.7360	+	11.7360i	0.427116	+	0.427116i
$756$		0			0
$757$	−11.9027	+	11.9027i	−0.432612	+	0.432612i	−0.889516	−	0.456904i	$-0.848958\pi$
$757$	−11.9027	+	11.9027i	−0.432612	+	0.432612i	0.456904	+	0.889516i	$0.348958\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−26.6151			−0.964798			−0.482399	−	0.875952i	$-0.660234\pi$
$761$	−26.6151			−0.964798			−0.482399	+	0.875952i	$0.660234\pi$
$762$		0			0
$763$	13.8663	+	13.8663i	0.501992	+	0.501992i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−19.7441			−0.712918
$768$		0			0
$769$	−6.30839			−0.227486			−0.113743	−	0.993510i	$-0.536284\pi$
$769$	−6.30839			−0.227486			−0.113743	+	0.993510i	$0.536284\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	22.5947	+	22.5947i	0.812676	+	0.812676i	0.985034	−	0.172359i	$-0.0551388\pi$
$773$	22.5947	+	22.5947i	0.812676	+	0.812676i	−0.172359	+	0.985034i	$0.555139\pi$
$774$		0			0
$775$	−29.9296			−1.07510
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−27.9703	+	27.9703i	−1.00214	+	1.00214i
$780$		0			0
$781$	1.39864	+	1.39864i	0.0500472	+	0.0500472i
$782$		0			0
$783$		0			0
$784$		0			0
$785$		−	38.9309i		−	1.38950i
$786$		0			0
$787$	15.1599	−	15.1599i	0.540393	−	0.540393i	−0.383251	−	0.923644i	$-0.625196\pi$
$787$	15.1599	−	15.1599i	0.540393	−	0.540393i	0.923644	+	0.383251i	$0.125196\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		−	15.0746i		−	0.535991i
$792$		0			0
$793$			6.10212i			0.216692i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	0.154906	−	0.154906i	0.00548705	−	0.00548705i	−0.704358	−	0.709845i	$-0.748765\pi$
$797$	0.154906	−	0.154906i	0.00548705	−	0.00548705i	0.709845	+	0.704358i	$0.248765\pi$
$798$		0			0
$799$		−	6.99301i		−	0.247395i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	27.6784	+	27.6784i	0.976751	+	0.976751i
$804$		0			0
$805$	−6.14490	+	6.14490i	−0.216579	+	0.216579i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−20.6800			−0.727071			−0.363536	−	0.931580i	$-0.618430\pi$
$809$	−20.6800			−0.727071			−0.363536	+	0.931580i	$0.618430\pi$
$810$		0			0
$811$	8.15298	+	8.15298i	0.286290	+	0.286290i	0.835611	−	0.549321i	$-0.185114\pi$
$811$	8.15298	+	8.15298i	0.286290	+	0.286290i	−0.549321	+	0.835611i	$0.685114\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−4.82985			−0.169182
$816$		0			0
$817$	0.416148			0.0145592
$818$		0			0
$819$		0			0
$820$		0			0
$821$	7.36475	+	7.36475i	0.257031	+	0.257031i	0.823846	−	0.566814i	$-0.191824\pi$
$821$	7.36475	+	7.36475i	0.257031	+	0.257031i	−0.566814	+	0.823846i	$0.691824\pi$
$822$		0			0
$823$	32.9745			1.14942			0.574710	−	0.818357i	$-0.305115\pi$
$823$	32.9745			1.14942			0.574710	+	0.818357i	$0.305115\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	10.7733	−	10.7733i	0.374623	−	0.374623i	−0.494535	−	0.869158i	$-0.664661\pi$
$827$	10.7733	−	10.7733i	0.374623	−	0.374623i	0.869158	+	0.494535i	$0.164661\pi$
$828$		0			0
$829$	2.50209	+	2.50209i	0.0869011	+	0.0869011i	0.749221	−	0.662320i	$-0.230428\pi$
$829$	2.50209	+	2.50209i	0.0869011	+	0.0869011i	−0.662320	+	0.749221i	$0.730428\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		−	4.10467i		−	0.142218i
$834$		0			0
$835$	12.8080	−	12.8080i	0.443239	−	0.443239i
$836$		0			0
$837$		0			0
$838$		0			0
$839$		−	40.5205i		−	1.39892i	−0.714671	−	0.699461i	$-0.753423\pi$
$839$		−	40.5205i		−	1.39892i	0.714671	−	0.699461i	$-0.246577\pi$
$840$		0			0
$841$		−	45.4248i		−	1.56637i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−20.9302	+	20.9302i	−0.720020	+	0.720020i
$846$		0			0
$847$		−	19.7845i		−	0.679802i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	10.8599	+	10.8599i	0.372274	+	0.372274i
$852$		0			0
$853$	20.2785	−	20.2785i	0.694323	−	0.694323i	−0.268857	−	0.963180i	$-0.586646\pi$
$853$	20.2785	−	20.2785i	0.694323	−	0.694323i	0.963180	+	0.268857i	$0.0866460\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−37.6034			−1.28451			−0.642253	−	0.766492i	$-0.722000\pi$
$857$	−37.6034			−1.28451			−0.642253	+	0.766492i	$0.722000\pi$
$858$		0			0
$859$	10.7814	+	10.7814i	0.367856	+	0.367856i	0.866695	−	0.498839i	$-0.166240\pi$
$859$	10.7814	+	10.7814i	0.367856	+	0.367856i	−0.498839	+	0.866695i	$0.666240\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	1.09947			0.0374264			0.0187132	−	0.999825i	$-0.494043\pi$
$863$	1.09947			0.0374264			0.0187132	+	0.999825i	$0.494043\pi$
$864$		0			0
$865$	−66.1519			−2.24923
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−0.642531	−	0.642531i	−0.0217964	−	0.0217964i
$870$		0			0
$871$	19.3949			0.657171
$872$		0			0
$873$		0			0
$874$		0			0
$875$	1.11090	−	1.11090i	0.0375552	−	0.0375552i
$876$		0			0
$877$	−2.46654	−	2.46654i	−0.0832892	−	0.0832892i	0.664235	−	0.747524i	$-0.268758\pi$
$877$	−2.46654	−	2.46654i	−0.0832892	−	0.0832892i	−0.747524	+	0.664235i	$0.768758\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		−	3.16526i		−	0.106640i	−0.998577	−	0.0533201i	$-0.983020\pi$
$881$		−	3.16526i		−	0.106640i	0.998577	−	0.0533201i	$-0.0169804\pi$
$882$		0			0
$883$	−24.5299	+	24.5299i	−0.825498	+	0.825498i	−0.986890	−	0.161392i	$-0.948402\pi$
$883$	−24.5299	+	24.5299i	−0.825498	+	0.825498i	0.161392	+	0.986890i	$0.448402\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$			28.9642i			0.972521i	0.873814	+	0.486260i	$0.161639\pi$
$887$			28.9642i			0.972521i	−0.873814	+	0.486260i	$0.838361\pi$
$888$		0			0
$889$		−	15.9384i		−	0.534556i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−6.14331	+	6.14331i	−0.205578	+	0.205578i
$894$		0			0
$895$			86.5121i			2.89178i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	33.2854	+	33.2854i	1.11013	+	1.11013i
$900$		0			0
$901$	2.12333	−	2.12333i	0.0707385	−	0.0707385i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−27.6934			−0.920558
$906$		0			0
$907$	−32.1593	−	32.1593i	−1.06783	−	1.06783i	−0.997525	−	0.0703065i	$-0.977602\pi$
$907$	−32.1593	−	32.1593i	−1.06783	−	1.06783i	−0.0703065	−	0.997525i	$-0.522398\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	25.3226			0.838974			0.419487	−	0.907761i	$-0.362210\pi$
$911$	25.3226			0.838974			0.419487	+	0.907761i	$0.362210\pi$
$912$		0			0
$913$	−57.5258			−1.90383
$914$		0			0
$915$		0			0
$916$		0			0
$917$	1.98295	+	1.98295i	0.0654828	+	0.0654828i
$918$		0			0
$919$	−1.44624			−0.0477071			−0.0238535	−	0.999715i	$-0.507594\pi$
$919$	−1.44624			−0.0477071			−0.0238535	+	0.999715i	$0.507594\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	0.495187	−	0.495187i	0.0162993	−	0.0162993i
$924$		0			0
$925$	−22.1961	−	22.1961i	−0.729803	−	0.729803i
$926$		0			0
$927$		0			0
$928$		0			0
$929$			9.88117i			0.324191i	0.986775	+	0.162095i	$0.0518252\pi$
$929$			9.88117i			0.324191i	−0.986775	+	0.162095i	$0.948175\pi$
$930$		0			0
$931$	−3.60592	+	3.60592i	−0.118179	+	0.118179i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		−	73.7449i		−	2.41171i
$936$		0			0
$937$			16.0952i			0.525807i	0.964822	+	0.262904i	$0.0846801\pi$
$937$			16.0952i			0.525807i	−0.964822	+	0.262904i	$0.915320\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−15.7553	+	15.7553i	−0.513608	+	0.513608i	−0.915630	−	0.402022i	$-0.868307\pi$
$941$	−15.7553	+	15.7553i	−0.513608	+	0.513608i	0.402022	+	0.915630i	$0.368307\pi$
$942$		0			0
$943$		−	20.8172i		−	0.677901i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	9.07070	+	9.07070i	0.294758	+	0.294758i	0.838957	−	0.544198i	$-0.183166\pi$
$947$	9.07070	+	9.07070i	0.294758	+	0.294758i	−0.544198	+	0.838957i	$0.683166\pi$
$948$		0			0
$949$	9.79954	−	9.79954i	0.318107	−	0.318107i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−7.94742			−0.257442			−0.128721	−	0.991681i	$-0.541087\pi$
$953$	−7.94742			−0.257442			−0.128721	+	0.991681i	$0.541087\pi$
$954$		0			0
$955$	−48.5094	−	48.5094i	−1.56973	−	1.56973i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	4.97010			0.160493
$960$		0			0
$961$	1.22724			0.0395883
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−5.76445	−	5.76445i	−0.185564	−	0.185564i
$966$		0			0
$967$	16.3273			0.525051			0.262525	−	0.964925i	$-0.415445\pi$
$967$	16.3273			0.525051			0.262525	+	0.964925i	$0.415445\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−18.6065	+	18.6065i	−0.597111	+	0.597111i	−0.939543	−	0.342432i	$-0.888749\pi$
$971$	−18.6065	+	18.6065i	−0.597111	+	0.597111i	0.342432	+	0.939543i	$0.388749\pi$
$972$		0			0
$973$	−6.20608	−	6.20608i	−0.198958	−	0.198958i
$974$		0			0
$975$		0			0
$976$		0			0
$977$		−	20.8881i		−	0.668271i	−0.942525	−	0.334135i	$-0.891556\pi$
$977$		−	20.8881i		−	0.668271i	0.942525	−	0.334135i	$-0.108444\pi$
$978$		0			0
$979$	63.3475	−	63.3475i	2.02460	−	2.02460i
$980$		0			0
$981$		0			0
$982$		0			0
$983$		−	0.996126i		−	0.0317715i	−0.999874	−	0.0158857i	$-0.994943\pi$
$983$		−	0.996126i		−	0.0317715i	0.999874	−	0.0158857i	$-0.00505680\pi$
$984$		0			0
$985$			70.2146i			2.23722i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−0.154862	+	0.154862i	−0.00492431	+	0.00492431i
$990$		0			0
$991$			35.3570i			1.12315i	0.827425	+	0.561576i	$0.189805\pi$
$991$			35.3570i			1.12315i	−0.827425	+	0.561576i	$0.810195\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−29.6998	−	29.6998i	−0.941546	−	0.941546i
$996$		0			0
$997$	10.7796	−	10.7796i	0.341392	−	0.341392i	−0.515498	−	0.856891i	$-0.672393\pi$
$997$	10.7796	−	10.7796i	0.341392	−	0.341392i	0.856891	+	0.515498i	$0.172393\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4032.2.v.d.1583.3		36
3.2	odd	2	inner	4032.2.v.d.1583.16		36
4.3	odd	2		1008.2.v.d.323.14	yes	36
12.11	even	2		1008.2.v.d.323.5	✓	36
16.5	even	4		1008.2.v.d.827.5	yes	36
16.11	odd	4	inner	4032.2.v.d.3599.16		36
48.5	odd	4		1008.2.v.d.827.14	yes	36
48.11	even	4	inner	4032.2.v.d.3599.3		36

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1008.2.v.d.323.5	✓	36	12.11	even	2
1008.2.v.d.323.14	yes	36	4.3	odd	2
1008.2.v.d.827.5	yes	36	16.5	even	4
1008.2.v.d.827.14	yes	36	48.5	odd	4
4032.2.v.d.1583.3		36	1.1	even	1	trivial
4032.2.v.d.1583.16		36	3.2	odd	2	inner
4032.2.v.d.3599.3		36	48.11	even	4	inner
4032.2.v.d.3599.16		36	16.11	odd	4	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:	\( N \)	\(=\)	\( 4032 = 2^{6} \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4032.v (of order \(4\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-2.28967 - 2.28967i) q^{5} +1.00000 q^{7} +O(q^{10})\)	\(q+(-2.28967 - 2.28967i) q^{5} +1.00000 q^{7} +(-3.92329 + 3.92329i) q^{11} +(1.38904 + 1.38904i) q^{13} -4.10467i q^{17} +(-3.60592 + 3.60592i) q^{19} -2.68375i q^{23} +5.48518i q^{25} +(6.10020 - 6.10020i) q^{29} +5.45644i q^{31} +(-2.28967 - 2.28967i) q^{35} +(-4.04656 + 4.04656i) q^{37} +7.75676 q^{41} +(-0.0577035 - 0.0577035i) q^{43} +1.70367 q^{47} +1.00000 q^{49} +(0.517297 + 0.517297i) q^{53} +17.9661 q^{55} +(-7.10709 + 7.10709i) q^{59} +(2.19652 + 2.19652i) q^{61} -6.36089i q^{65} +(6.98139 - 6.98139i) q^{67} -0.356496i q^{71} -7.05490i q^{73} +(-3.92329 + 3.92329i) q^{77} +0.163773i q^{79} +(7.33131 + 7.33131i) q^{83} +(-9.39834 + 9.39834i) q^{85} -16.1465 q^{89} +(1.38904 + 1.38904i) q^{91} +16.5127 q^{95} +8.56055 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 36 q + 36 q^{7}+O(q^{10}) \) 36 * q + 36 * q^7	\( 36 q + 36 q^{7} - 16 q^{13} + 16 q^{19} + 20 q^{37} - 36 q^{43} + 36 q^{49} - 32 q^{55} + 112 q^{61} + 36 q^{67} - 96 q^{85} - 16 q^{91} + 56 q^{97}+O(q^{100}) \) 36 * q + 36 * q^7 - 16 * q^13 + 16 * q^19 + 20 * q^37 - 36 * q^43 + 36 * q^49 - 32 * q^55 + 112 * q^61 + 36 * q^67 - 96 * q^85 - 16 * q^91 + 56 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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