LMFDB - Embedded newform 1008.4.h.b.575.21

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1008,4,Mod(575,1008)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1008, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("1008.575");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1008 = 2^{4} \cdot 3^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1008.h (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:	$59.4739252858$
Analytic rank:	$0$
Dimension:	$24$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			575.21
Character	$\chi$	$=$	1008.575
Dual form			1008.4.h.b.575.22

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-16.1444i q^{5} +7.00000i q^{7} +O(q^{10})$	$q-16.1444i q^{5} +7.00000i q^{7} +41.4793 q^{11} -24.1086 q^{13} +68.2665i q^{17} +88.4124i q^{19} +52.4307 q^{23} -135.640 q^{25} +159.744i q^{29} -16.7397i q^{31} +113.010 q^{35} -150.032 q^{37} +309.274i q^{41} -29.1603i q^{43} -17.2598 q^{47} -49.0000 q^{49} -97.8779i q^{53} -669.657i q^{55} +247.584 q^{59} +851.995 q^{61} +389.218i q^{65} +897.685i q^{67} -454.426 q^{71} +342.498 q^{73} +290.355i q^{77} -487.979i q^{79} -17.8662 q^{83} +1102.12 q^{85} -905.907i q^{89} -168.761i q^{91} +1427.36 q^{95} +586.705 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q+O(q^{10}) $ 24 * q	$ 24 q - 192 q^{13} - 984 q^{25} + 720 q^{37} - 1176 q^{49} + 2736 q^{61} + 3408 q^{85} + 1152 q^{97}+O(q^{100}) $ 24 * q - 192 * q^13 - 984 * q^25 + 720 * q^37 - 1176 * q^49 + 2736 * q^61 + 3408 * q^85 + 1152 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1008\mathbb{Z}\right)^\times$.

$n$	$127$	$577$	$757$	$785$
$\chi(n)$	$-1$	$1$	$1$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	0	0
$3$	0	0
$4$	0	0
$5$	− 16.1444i	− 1.44399i	−0.691896	−	0.721997i	$-0.743224\pi$
$5$	− 16.1444i	− 1.44399i	0.691896	−	0.721997i	$-0.256776\pi$
$6$	0	0
$7$	7.00000i	0.377964i
$7$	7.00000i	0.377964i
$8$	0	0
$9$	0	0
$10$	0	0
$11$	41.4793	1.13695	0.568477	−	0.822699i	$-0.307533\pi$
$11$	41.4793	1.13695	0.568477	+	0.822699i	$0.307533\pi$
$12$	0	0
$13$	−24.1086	−0.514349	−0.257174	−	0.966365i	$-0.582791\pi$
$13$	−24.1086	−0.514349	−0.257174	+	0.966365i	$0.582791\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	68.2665i	0.973945i	0.873417	+	0.486972i	$0.161899\pi$
$17$	68.2665i	0.973945i	−0.873417	+	0.486972i	$0.838101\pi$
$18$	0	0
$19$	88.4124i	1.06754i	0.845631	+	0.533768i	$0.179225\pi$
$19$	88.4124i	1.06754i	−0.845631	+	0.533768i	$0.820775\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	52.4307	0.475329	0.237664	−	0.971347i	$-0.423618\pi$
$23$	52.4307	0.475329	0.237664	+	0.971347i	$0.423618\pi$
$24$	0	0
$25$	−135.640	−1.08512
$26$	0	0
$27$	0	0
$28$	0	0
$29$	159.744i	1.02289i	0.859317	+	0.511443i	$0.170889\pi$
$29$	159.744i	1.02289i	−0.859317	+	0.511443i	$0.829111\pi$
$30$	0	0
$31$	− 16.7397i	− 0.0969852i	−0.998824	−	0.0484926i	$-0.984558\pi$
$31$	− 16.7397i	− 0.0969852i	0.998824	−	0.0484926i	$-0.0154417\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	113.010	0.545779
$36$	0	0
$37$	−150.032	−0.666625	−0.333313	−	0.942816i	$-0.608166\pi$
$37$	−150.032	−0.666625	−0.333313	+	0.942816i	$0.608166\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	309.274i	1.17806i	0.808111	+	0.589030i	$0.200490\pi$
$41$	309.274i	1.17806i	−0.808111	+	0.589030i	$0.799510\pi$
$42$	0	0
$43$	− 29.1603i	− 0.103416i	−0.998662	−	0.0517081i	$-0.983533\pi$
$43$	− 29.1603i	− 0.103416i	0.998662	−	0.0517081i	$-0.0164665\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−17.2598	−0.0535658	−0.0267829	−	0.999641i	$-0.508526\pi$
$47$	−17.2598	−0.0535658	−0.0267829	+	0.999641i	$0.508526\pi$
$48$	0	0
$49$	−49.0000	−0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	− 97.8779i	− 0.253671i	−0.991924	−	0.126836i	$-0.959518\pi$
$53$	− 97.8779i	− 0.253671i	0.991924	−	0.126836i	$-0.0404820\pi$
$54$	0	0
$55$	− 669.657i	− 1.64176i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	247.584	0.546317	0.273158	−	0.961969i	$-0.411932\pi$
$59$	247.584	0.546317	0.273158	+	0.961969i	$0.411932\pi$
$60$	0	0
$61$	851.995	1.78831	0.894154	−	0.447760i	$-0.147778\pi$
$61$	851.995	1.78831	0.894154	+	0.447760i	$0.147778\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	389.218i	0.742717i
$66$	0	0
$67$	897.685i	1.63686i	0.574605	+	0.818431i	$0.305156\pi$
$67$	897.685i	1.63686i	−0.574605	+	0.818431i	$0.694844\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−454.426	−0.759584	−0.379792	−	0.925072i	$-0.624004\pi$
$71$	−454.426	−0.759584	−0.379792	+	0.925072i	$0.624004\pi$
$72$	0	0
$73$	342.498	0.549129	0.274564	−	0.961569i	$-0.411466\pi$
$73$	342.498	0.549129	0.274564	+	0.961569i	$0.411466\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	290.355i	0.429728i
$78$	0	0
$79$	− 487.979i	− 0.694962i	−0.937687	−	0.347481i	$-0.887037\pi$
$79$	− 487.979i	− 0.694962i	0.937687	−	0.347481i	$-0.112963\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−17.8662	−0.0236274	−0.0118137	−	0.999930i	$-0.503761\pi$
$83$	−17.8662	−0.0236274	−0.0118137	+	0.999930i	$0.503761\pi$
$84$	0	0
$85$	1102.12	1.40637
$86$	0	0
$87$	0	0
$88$	0	0
$89$	− 905.907i	− 1.07894i	−0.842004	−	0.539472i	$-0.818624\pi$
$89$	− 905.907i	− 1.07894i	0.842004	−	0.539472i	$-0.181376\pi$
$90$	0	0
$91$	− 168.761i	− 0.194406i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	1427.36	1.54152
$96$	0	0
$97$	586.705	0.614132	0.307066	−	0.951688i	$-0.400653\pi$
$97$	586.705	0.614132	0.307066	+	0.951688i	$0.400653\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	1439.22i	1.41790i	0.705259	+	0.708950i	$0.250831\pi$
$101$	1439.22i	1.41790i	−0.705259	+	0.708950i	$0.749169\pi$
$102$	0	0
$103$	− 948.479i	− 0.907344i	−0.891169	−	0.453672i	$-0.850114\pi$
$103$	− 948.479i	− 0.907344i	0.891169	−	0.453672i	$-0.149886\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	407.626	0.368287	0.184143	−	0.982899i	$-0.441049\pi$
$107$	407.626	0.368287	0.184143	+	0.982899i	$0.441049\pi$
$108$	0	0
$109$	1603.23	1.40882	0.704411	−	0.709793i	$-0.251211\pi$
$109$	1603.23	1.40882	0.704411	+	0.709793i	$0.251211\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	1035.89i	0.862372i	0.902263	+	0.431186i	$0.141905\pi$
$113$	1035.89i	0.862372i	−0.902263	+	0.431186i	$0.858095\pi$
$114$	0	0
$115$	− 846.460i	− 0.686372i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−477.866	−0.368117
$120$	0	0
$121$	389.536	0.292664
$122$	0	0
$123$	0	0
$124$	0	0
$125$	171.778i	0.122914i
$126$	0	0
$127$	− 2426.17i	− 1.69518i	−0.530653	−	0.847589i	$-0.678053\pi$
$127$	− 2426.17i	− 1.69518i	0.530653	−	0.847589i	$-0.321947\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	2185.13	1.45737	0.728687	−	0.684847i	$-0.240131\pi$
$131$	2185.13	1.45737	0.728687	+	0.684847i	$0.240131\pi$
$132$	0	0
$133$	−618.887	−0.403491
$134$	0	0
$135$	0	0
$136$	0	0
$137$	1730.64i	1.07926i	0.841903	+	0.539628i	$0.181435\pi$
$137$	1730.64i	1.07926i	−0.841903	+	0.539628i	$0.818565\pi$
$138$	0	0
$139$	− 2031.27i	− 1.23950i	−0.784800	−	0.619748i	$-0.787235\pi$
$139$	− 2031.27i	− 1.23950i	0.784800	−	0.619748i	$-0.212765\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−1000.01	−0.584791
$144$	0	0
$145$	2578.96	1.47704
$146$	0	0
$147$	0	0
$148$	0	0
$149$	2344.17i	1.28887i	0.764659	+	0.644435i	$0.222908\pi$
$149$	2344.17i	1.28887i	−0.764659	+	0.644435i	$0.777092\pi$
$150$	0	0
$151$	2460.09i	1.32582i	0.748698	+	0.662911i	$0.230679\pi$
$151$	2460.09i	1.32582i	−0.748698	+	0.662911i	$0.769321\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−270.252	−0.140046
$156$	0	0
$157$	−2506.58	−1.27418	−0.637092	−	0.770787i	$-0.719863\pi$
$157$	−2506.58	−1.27418	−0.637092	+	0.770787i	$0.719863\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	367.015i	0.179657i
$162$	0	0
$163$	3926.85i	1.88696i	0.331426	+	0.943481i	$0.392470\pi$
$163$	3926.85i	1.88696i	−0.331426	+	0.943481i	$0.607530\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	1007.10	0.466658	0.233329	−	0.972398i	$-0.425038\pi$
$167$	1007.10	0.466658	0.233329	+	0.972398i	$0.425038\pi$
$168$	0	0
$169$	−1615.77	−0.735445
$170$	0	0
$171$	0	0
$172$	0	0
$173$	3403.43i	1.49571i	0.663862	+	0.747855i	$0.268916\pi$
$173$	3403.43i	1.49571i	−0.663862	+	0.747855i	$0.731084\pi$
$174$	0	0
$175$	− 949.481i	− 0.410137i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	4581.41	1.91302	0.956511	−	0.291697i	$-0.0942200\pi$
$179$	4581.41	1.91302	0.956511	+	0.291697i	$0.0942200\pi$
$180$	0	0
$181$	−2440.16	−1.00208	−0.501038	−	0.865425i	$-0.667048\pi$
$181$	−2440.16	−1.00208	−0.501038	+	0.865425i	$0.667048\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	2422.17i	0.962603i
$186$	0	0
$187$	2831.65i	1.10733i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	3744.42	1.41852	0.709258	−	0.704949i	$-0.249030\pi$
$191$	3744.42	1.41852	0.709258	+	0.704949i	$0.249030\pi$
$192$	0	0
$193$	3439.80	1.28291	0.641456	−	0.767160i	$-0.278331\pi$
$193$	3439.80	1.28291	0.641456	+	0.767160i	$0.278331\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	795.600i	0.287737i	0.989597	+	0.143868i	$0.0459542\pi$
$197$	795.600i	0.287737i	−0.989597	+	0.143868i	$0.954046\pi$
$198$	0	0
$199$	− 4252.33i	− 1.51477i	−0.652968	−	0.757385i	$-0.726476\pi$
$199$	− 4252.33i	− 1.51477i	0.652968	−	0.757385i	$-0.273524\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−1118.21	−0.386615
$204$	0	0
$205$	4993.02	1.70111
$206$	0	0
$207$	0	0
$208$	0	0
$209$	3667.29i	1.21374i
$210$	0	0
$211$	2147.50i	0.700665i	0.936625	+	0.350332i	$0.113931\pi$
$211$	2147.50i	0.700665i	−0.936625	+	0.350332i	$0.886069\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−470.773	−0.149332
$216$	0	0
$217$	117.178	0.0366570
$218$	0	0
$219$	0	0
$220$	0	0
$221$	− 1645.81i	− 0.500947i
$222$	0	0
$223$	− 3472.14i	− 1.04265i	−0.853357	−	0.521327i	$-0.825437\pi$
$223$	− 3472.14i	− 1.04265i	0.853357	−	0.521327i	$-0.174563\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−1552.54	−0.453947	−0.226973	−	0.973901i	$-0.572883\pi$
$227$	−1552.54	−0.453947	−0.226973	+	0.973901i	$0.572883\pi$
$228$	0	0
$229$	−1206.84	−0.348255	−0.174128	−	0.984723i	$-0.555711\pi$
$229$	−1206.84	−0.348255	−0.174128	+	0.984723i	$0.555711\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	− 2049.30i	− 0.576198i	−0.957601	−	0.288099i	$-0.906977\pi$
$233$	− 2049.30i	− 0.576198i	0.957601	−	0.288099i	$-0.0930233\pi$
$234$	0	0
$235$	278.648i	0.0773488i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	1084.17	0.293427	0.146713	−	0.989179i	$-0.453131\pi$
$239$	1084.17	0.293427	0.146713	+	0.989179i	$0.453131\pi$
$240$	0	0
$241$	1195.88	0.319642	0.159821	−	0.987146i	$-0.448908\pi$
$241$	1195.88	0.319642	0.159821	+	0.987146i	$0.448908\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	791.073i	0.206285i
$246$	0	0
$247$	− 2131.50i	− 0.549086i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−5775.20	−1.45230	−0.726150	−	0.687536i	$-0.758692\pi$
$251$	−5775.20	−1.45230	−0.726150	+	0.687536i	$0.758692\pi$
$252$	0	0
$253$	2174.79	0.540427
$254$	0	0
$255$	0	0
$256$	0	0
$257$	627.339i	0.152266i	0.997098	+	0.0761330i	$0.0242574\pi$
$257$	627.339i	0.152266i	−0.997098	+	0.0761330i	$0.975743\pi$
$258$	0	0
$259$	− 1050.22i	− 0.251961i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	5423.45	1.27158	0.635788	−	0.771864i	$-0.280675\pi$
$263$	5423.45	1.27158	0.635788	+	0.771864i	$0.280675\pi$
$264$	0	0
$265$	−1580.18	−0.366300
$266$	0	0
$267$	0	0
$268$	0	0
$269$	− 3362.75i	− 0.762195i	−0.924535	−	0.381098i	$-0.875546\pi$
$269$	− 3362.75i	− 0.762195i	0.924535	−	0.381098i	$-0.124454\pi$
$270$	0	0
$271$	5591.87i	1.25344i	0.779245	+	0.626719i	$0.215603\pi$
$271$	5591.87i	1.25344i	−0.779245	+	0.626719i	$0.784397\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−5626.26	−1.23373
$276$	0	0
$277$	3165.70	0.686674	0.343337	−	0.939212i	$-0.388443\pi$
$277$	3165.70	0.686674	0.343337	+	0.939212i	$0.388443\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	5943.44i	1.26177i	0.775878	+	0.630883i	$0.217307\pi$
$281$	5943.44i	1.26177i	−0.775878	+	0.630883i	$0.782693\pi$
$282$	0	0
$283$	− 7295.25i	− 1.53236i	−0.642626	−	0.766180i	$-0.722155\pi$
$283$	− 7295.25i	− 1.53236i	0.642626	−	0.766180i	$-0.277845\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−2164.92	−0.445265
$288$	0	0
$289$	252.683	0.0514316
$290$	0	0
$291$	0	0
$292$	0	0
$293$	− 2180.26i	− 0.434716i	−0.976092	−	0.217358i	$-0.930256\pi$
$293$	− 2180.26i	− 0.434716i	0.976092	−	0.217358i	$-0.0697440\pi$
$294$	0	0
$295$	− 3997.08i	− 0.788878i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−1264.03	−0.244485
$300$	0	0
$301$	204.122	0.0390876
$302$	0	0
$303$	0	0
$304$	0	0
$305$	− 13754.9i	− 2.58231i
$306$	0	0
$307$	− 6209.36i	− 1.15436i	−0.816619	−	0.577178i	$-0.804154\pi$
$307$	− 6209.36i	− 1.15436i	0.816619	−	0.577178i	$-0.195846\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	4126.41	0.752371	0.376185	−	0.926544i	$-0.377236\pi$
$311$	4126.41	0.752371	0.376185	+	0.926544i	$0.377236\pi$
$312$	0	0
$313$	768.261	0.138737	0.0693685	−	0.997591i	$-0.477902\pi$
$313$	768.261	0.138737	0.0693685	+	0.997591i	$0.477902\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 4556.50i	− 0.807314i	−0.914910	−	0.403657i	$-0.867739\pi$
$317$	− 4556.50i	− 0.807314i	0.914910	−	0.403657i	$-0.132261\pi$
$318$	0	0
$319$	6626.08i	1.16298i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−6035.61	−1.03972
$324$	0	0
$325$	3270.10	0.558131
$326$	0	0
$327$	0	0
$328$	0	0
$329$	− 120.818i	− 0.0202460i
$330$	0	0
$331$	1727.87i	0.286925i	0.989656	+	0.143462i	$0.0458236\pi$
$331$	1727.87i	0.286925i	−0.989656	+	0.143462i	$0.954176\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	14492.5	2.36362
$336$	0	0
$337$	−7804.93	−1.26161	−0.630803	−	0.775943i	$-0.717274\pi$
$337$	−7804.93	−1.26161	−0.630803	+	0.775943i	$0.717274\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	− 694.353i	− 0.110268i
$342$	0	0
$343$	− 343.000i	− 0.0539949i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−7200.19	−1.11391	−0.556955	−	0.830543i	$-0.688030\pi$
$347$	−7200.19	−1.11391	−0.556955	+	0.830543i	$0.688030\pi$
$348$	0	0
$349$	−2267.47	−0.347779	−0.173889	−	0.984765i	$-0.555634\pi$
$349$	−2267.47	−0.347779	−0.173889	+	0.984765i	$0.555634\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	− 3053.33i	− 0.460374i	−0.973146	−	0.230187i	$-0.926066\pi$
$353$	− 3053.33i	− 0.460374i	0.973146	−	0.230187i	$-0.0739339\pi$
$354$	0	0
$355$	7336.41i	1.09683i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−8210.61	−1.20707	−0.603537	−	0.797335i	$-0.706242\pi$
$359$	−8210.61	−1.20707	−0.603537	+	0.797335i	$0.706242\pi$
$360$	0	0
$361$	−957.755	−0.139635
$362$	0	0
$363$	0	0
$364$	0	0
$365$	− 5529.42i	− 0.792939i
$366$	0	0
$367$	7607.04i	1.08197i	0.841031	+	0.540987i	$0.181949\pi$
$367$	7607.04i	1.08197i	−0.841031	+	0.540987i	$0.818051\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	685.145	0.0958786
$372$	0	0
$373$	−210.602	−0.0292348	−0.0146174	−	0.999893i	$-0.504653\pi$
$373$	−210.602	−0.0292348	−0.0146174	+	0.999893i	$0.504653\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 3851.21i	− 0.526121i
$378$	0	0
$379$	2578.23i	0.349432i	0.984619	+	0.174716i	$0.0559007\pi$
$379$	2578.23i	0.349432i	−0.984619	+	0.174716i	$0.944099\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	4950.71	0.660494	0.330247	−	0.943895i	$-0.392868\pi$
$383$	4950.71	0.660494	0.330247	+	0.943895i	$0.392868\pi$
$384$	0	0
$385$	4687.60	0.620525
$386$	0	0
$387$	0	0
$388$	0	0
$389$	1655.28i	0.215749i	0.994165	+	0.107874i	$0.0344044\pi$
$389$	1655.28i	0.215749i	−0.994165	+	0.107874i	$0.965596\pi$
$390$	0	0
$391$	3579.26i	0.462944i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−7878.11	−1.00352
$396$	0	0
$397$	−11158.9	−1.41070	−0.705349	−	0.708860i	$-0.749210\pi$
$397$	−11158.9	−1.41070	−0.705349	+	0.708860i	$0.749210\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	11463.1i	1.42754i	0.700382	+	0.713768i	$0.253013\pi$
$401$	11463.1i	1.42754i	−0.700382	+	0.713768i	$0.746987\pi$
$402$	0	0
$403$	403.572i	0.0498842i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−6223.23	−0.757922
$408$	0	0
$409$	−7783.37	−0.940985	−0.470493	−	0.882404i	$-0.655924\pi$
$409$	−7783.37	−0.940985	−0.470493	+	0.882404i	$0.655924\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	1733.09i	0.206488i
$414$	0	0
$415$	288.439i	0.0341178i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−1474.42	−0.171909	−0.0859545	−	0.996299i	$-0.527394\pi$
$419$	−1474.42	−0.171909	−0.0859545	+	0.996299i	$0.527394\pi$
$420$	0	0
$421$	−2772.73	−0.320985	−0.160492	−	0.987037i	$-0.551308\pi$
$421$	−2772.73	−0.320985	−0.160492	+	0.987037i	$0.551308\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	− 9259.68i	− 1.05685i
$426$	0	0
$427$	5963.97i	0.675917i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	9891.19	1.10543	0.552717	−	0.833369i	$-0.313591\pi$
$431$	9891.19	1.10543	0.552717	+	0.833369i	$0.313591\pi$
$432$	0	0
$433$	−12757.4	−1.41589	−0.707947	−	0.706266i	$-0.750378\pi$
$433$	−12757.4	−1.41589	−0.707947	+	0.706266i	$0.750378\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	4635.53i	0.507431i
$438$	0	0
$439$	− 6922.35i	− 0.752587i	−0.926501	−	0.376293i	$-0.877198\pi$
$439$	− 6922.35i	− 0.752587i	0.926501	−	0.376293i	$-0.122802\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−9905.24	−1.06233	−0.531165	−	0.847269i	$-0.678245\pi$
$443$	−9905.24	−1.06233	−0.531165	+	0.847269i	$0.678245\pi$
$444$	0	0
$445$	−14625.3	−1.55799
$446$	0	0
$447$	0	0
$448$	0	0
$449$	1900.26i	0.199730i	0.995001	+	0.0998652i	$0.0318412\pi$
$449$	1900.26i	0.199730i	−0.995001	+	0.0998652i	$0.968159\pi$
$450$	0	0
$451$	12828.5i	1.33940i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−2724.53	−0.280721
$456$	0	0
$457$	−16444.8	−1.68328	−0.841638	−	0.540042i	$-0.818408\pi$
$457$	−16444.8	−1.68328	−0.841638	+	0.540042i	$0.818408\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	− 16906.9i	− 1.70810i	−0.520194	−	0.854048i	$-0.674140\pi$
$461$	− 16906.9i	− 1.70810i	0.520194	−	0.854048i	$-0.325860\pi$
$462$	0	0
$463$	13404.7i	1.34551i	0.739866	+	0.672754i	$0.234889\pi$
$463$	13404.7i	1.34551i	−0.739866	+	0.672754i	$0.765111\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−12967.9	−1.28497	−0.642487	−	0.766296i	$-0.722098\pi$
$467$	−12967.9	−1.28497	−0.642487	+	0.766296i	$0.722098\pi$
$468$	0	0
$469$	−6283.80	−0.618675
$470$	0	0
$471$	0	0
$472$	0	0
$473$	− 1209.55i	− 0.117579i
$474$	0	0
$475$	− 11992.3i	− 1.15841i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	8462.68	0.807244	0.403622	−	0.914926i	$-0.367751\pi$
$479$	8462.68	0.807244	0.403622	+	0.914926i	$0.367751\pi$
$480$	0	0
$481$	3617.07	0.342878
$482$	0	0
$483$	0	0
$484$	0	0
$485$	− 9471.97i	− 0.886804i
$486$	0	0
$487$	− 19498.6i	− 1.81430i	−0.420806	−	0.907151i	$-0.638253\pi$
$487$	− 19498.6i	− 1.81430i	0.420806	−	0.907151i	$-0.361747\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−7202.09	−0.661967	−0.330983	−	0.943637i	$-0.607380\pi$
$491$	−7202.09	−0.661967	−0.330983	+	0.943637i	$0.607380\pi$
$492$	0	0
$493$	−10905.2	−0.996235
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 3180.98i	− 0.287096i
$498$	0	0
$499$	19102.4i	1.71371i	0.515558	+	0.856855i	$0.327585\pi$
$499$	19102.4i	1.71371i	−0.515558	+	0.856855i	$0.672415\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	12700.1	1.12578	0.562891	−	0.826531i	$-0.309689\pi$
$503$	12700.1	1.12578	0.562891	+	0.826531i	$0.309689\pi$
$504$	0	0
$505$	23235.3	2.04744
$506$	0	0
$507$	0	0
$508$	0	0
$509$	3916.62i	0.341063i	0.985352	+	0.170532i	$0.0545485\pi$
$509$	3916.62i	0.341063i	−0.985352	+	0.170532i	$0.945451\pi$
$510$	0	0
$511$	2397.49i	0.207551i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−15312.6	−1.31020
$516$	0	0
$517$	−715.923	−0.0609019
$518$	0	0
$519$	0	0
$520$	0	0
$521$	− 23066.7i	− 1.93967i	−0.243753	−	0.969837i	$-0.578379\pi$
$521$	− 23066.7i	− 1.93967i	0.243753	−	0.969837i	$-0.421621\pi$
$522$	0	0
$523$	− 4136.47i	− 0.345842i	−0.984936	−	0.172921i	$-0.944679\pi$
$523$	− 4136.47i	− 0.345842i	0.984936	−	0.172921i	$-0.0553205\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	1142.76	0.0944583
$528$	0	0
$529$	−9418.02	−0.774063
$530$	0	0
$531$	0	0
$532$	0	0
$533$	− 7456.17i	− 0.605933i
$534$	0	0
$535$	− 6580.86i	− 0.531804i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−2032.49	−0.162422
$540$	0	0
$541$	18652.5	1.48232	0.741159	−	0.671329i	$-0.234276\pi$
$541$	18652.5	1.48232	0.741159	+	0.671329i	$0.234276\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	− 25883.1i	− 2.03433i
$546$	0	0
$547$	23638.0i	1.84769i	0.382766	+	0.923845i	$0.374971\pi$
$547$	23638.0i	1.84769i	−0.382766	+	0.923845i	$0.625029\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−14123.4	−1.09197
$552$	0	0
$553$	3415.86	0.262671
$554$	0	0
$555$	0	0
$556$	0	0
$557$	13513.1i	1.02795i	0.857805	+	0.513975i	$0.171827\pi$
$557$	13513.1i	1.02795i	−0.857805	+	0.513975i	$0.828173\pi$
$558$	0	0
$559$	703.014i	0.0531920i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	6604.68	0.494412	0.247206	−	0.968963i	$-0.420488\pi$
$563$	6604.68	0.494412	0.247206	+	0.968963i	$0.420488\pi$
$564$	0	0
$565$	16723.7	1.24526
$566$	0	0
$567$	0	0
$568$	0	0
$569$	8889.76i	0.654970i	0.944856	+	0.327485i	$0.106201\pi$
$569$	8889.76i	0.654970i	−0.944856	+	0.327485i	$0.893799\pi$
$570$	0	0
$571$	− 17097.7i	− 1.25309i	−0.779385	−	0.626545i	$-0.784468\pi$
$571$	− 17097.7i	− 1.25309i	0.779385	−	0.626545i	$-0.215532\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−7111.71	−0.515789
$576$	0	0
$577$	12882.1	0.929444	0.464722	−	0.885457i	$-0.346154\pi$
$577$	12882.1	0.929444	0.464722	+	0.885457i	$0.346154\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	− 125.064i	− 0.00893032i
$582$	0	0
$583$	− 4059.91i	− 0.288412i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	22536.1	1.58460	0.792302	−	0.610129i	$-0.208882\pi$
$587$	22536.1	1.58460	0.792302	+	0.610129i	$0.208882\pi$
$588$	0	0
$589$	1480.00	0.103535
$590$	0	0
$591$	0	0
$592$	0	0
$593$	− 1068.41i	− 0.0739870i	−0.999316	−	0.0369935i	$-0.988222\pi$
$593$	− 1068.41i	− 0.0739870i	0.999316	−	0.0369935i	$-0.0117781\pi$
$594$	0	0
$595$	7714.83i	0.531558i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	944.445	0.0644223	0.0322112	−	0.999481i	$-0.489745\pi$
$599$	944.445	0.0644223	0.0322112	+	0.999481i	$0.489745\pi$
$600$	0	0
$601$	−211.060	−0.0143250	−0.00716249	−	0.999974i	$-0.502280\pi$
$601$	−211.060	−0.0143250	−0.00716249	+	0.999974i	$0.502280\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	− 6288.80i	− 0.422605i
$606$	0	0
$607$	− 17899.8i	− 1.19692i	−0.801152	−	0.598462i	$-0.795779\pi$
$607$	− 17899.8i	− 1.19692i	0.801152	−	0.598462i	$-0.204221\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	416.109	0.0275515
$612$	0	0
$613$	−17300.3	−1.13989	−0.569945	−	0.821683i	$-0.693036\pi$
$613$	−17300.3	−1.13989	−0.569945	+	0.821683i	$0.693036\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 14150.3i	− 0.923291i	−0.887064	−	0.461646i	$-0.847259\pi$
$617$	− 14150.3i	− 0.923291i	0.887064	−	0.461646i	$-0.152741\pi$
$618$	0	0
$619$	5913.91i	0.384007i	0.981394	+	0.192003i	$0.0614984\pi$
$619$	5913.91i	0.384007i	−0.981394	+	0.192003i	$0.938502\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	6341.35	0.407802
$624$	0	0
$625$	−14181.8	−0.907634
$626$	0	0
$627$	0	0
$628$	0	0
$629$	− 10242.2i	− 0.649256i
$630$	0	0
$631$	− 8117.82i	− 0.512148i	−0.966657	−	0.256074i	$-0.917571\pi$
$631$	− 8117.82i	− 0.512148i	0.966657	−	0.256074i	$-0.0824290\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−39168.9	−2.44783
$636$	0	0
$637$	1181.32	0.0734784
$638$	0	0
$639$	0	0
$640$	0	0
$641$	5934.93i	0.365703i	0.983141	+	0.182851i	$0.0585328\pi$
$641$	5934.93i	0.365703i	−0.983141	+	0.182851i	$0.941467\pi$
$642$	0	0
$643$	15441.9i	0.947077i	0.880773	+	0.473539i	$0.157024\pi$
$643$	15441.9i	0.947077i	−0.880773	+	0.473539i	$0.842976\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	9615.02	0.584244	0.292122	−	0.956381i	$-0.405639\pi$
$647$	9615.02	0.584244	0.292122	+	0.956381i	$0.405639\pi$
$648$	0	0
$649$	10269.6	0.621137
$650$	0	0
$651$	0	0
$652$	0	0
$653$	1447.78i	0.0867626i	0.999059	+	0.0433813i	$0.0138130\pi$
$653$	1447.78i	0.0867626i	−0.999059	+	0.0433813i	$0.986187\pi$
$654$	0	0
$655$	− 35277.5i	− 2.10444i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−25749.4	−1.52208	−0.761042	−	0.648703i	$-0.775312\pi$
$659$	−25749.4	−1.52208	−0.761042	+	0.648703i	$0.775312\pi$
$660$	0	0
$661$	−17045.5	−1.00302	−0.501508	−	0.865153i	$-0.667221\pi$
$661$	−17045.5	−1.00302	−0.501508	+	0.865153i	$0.667221\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	9991.53i	0.582639i
$666$	0	0
$667$	8375.49i	0.486208i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	35340.2	2.03322
$672$	0	0
$673$	−478.114	−0.0273848	−0.0136924	−	0.999906i	$-0.504359\pi$
$673$	−478.114	−0.0273848	−0.0136924	+	0.999906i	$0.504359\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	8729.54i	0.495574i	0.968815	+	0.247787i	$0.0797033\pi$
$677$	8729.54i	0.495574i	−0.968815	+	0.247787i	$0.920297\pi$
$678$	0	0
$679$	4106.93i	0.232120i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	16364.6	0.916798	0.458399	−	0.888746i	$-0.348423\pi$
$683$	16364.6	0.916798	0.458399	+	0.888746i	$0.348423\pi$
$684$	0	0
$685$	27940.0	1.55844
$686$	0	0
$687$	0	0
$688$	0	0
$689$	2359.70i	0.130475i
$690$	0	0
$691$	8623.96i	0.474777i	0.971415	+	0.237389i	$0.0762914\pi$
$691$	8623.96i	0.474777i	−0.971415	+	0.237389i	$0.923709\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−32793.5	−1.78983
$696$	0	0
$697$	−21113.0	−1.14736
$698$	0	0
$699$	0	0
$700$	0	0
$701$	29824.6i	1.60693i	0.595352	+	0.803465i	$0.297013\pi$
$701$	29824.6i	1.60693i	−0.595352	+	0.803465i	$0.702987\pi$
$702$	0	0
$703$	− 13264.7i	− 0.711647i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−10074.6	−0.535916
$708$	0	0
$709$	26635.3	1.41087	0.705436	−	0.708774i	$-0.250751\pi$
$709$	26635.3	1.41087	0.705436	+	0.708774i	$0.250751\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	− 877.676i	− 0.0460999i
$714$	0	0
$715$	16144.5i	0.844435i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	6659.89	0.345441	0.172720	−	0.984971i	$-0.444744\pi$
$719$	6659.89	0.345441	0.172720	+	0.984971i	$0.444744\pi$
$720$	0	0
$721$	6639.35	0.342944
$722$	0	0
$723$	0	0
$724$	0	0
$725$	− 21667.7i	− 1.10996i
$726$	0	0
$727$	− 16638.0i	− 0.848788i	−0.905478	−	0.424394i	$-0.860487\pi$
$727$	− 16638.0i	− 0.848788i	0.905478	−	0.424394i	$-0.139513\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	1990.67	0.100722
$732$	0	0
$733$	−29196.7	−1.47122	−0.735610	−	0.677405i	$-0.763104\pi$
$733$	−29196.7	−1.47122	−0.735610	+	0.677405i	$0.763104\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	37235.4i	1.86104i
$738$	0	0
$739$	− 6913.07i	− 0.344116i	−0.985087	−	0.172058i	$-0.944958\pi$
$739$	− 6913.07i	− 0.344116i	0.985087	−	0.172058i	$-0.0550416\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	24329.2	1.20128	0.600640	−	0.799519i	$-0.294912\pi$
$743$	24329.2	1.20128	0.600640	+	0.799519i	$0.294912\pi$
$744$	0	0
$745$	37845.1	1.86112
$746$	0	0
$747$	0	0
$748$	0	0
$749$	2853.38i	0.139199i
$750$	0	0
$751$	15688.1i	0.762272i	0.924519	+	0.381136i	$0.124467\pi$
$751$	15688.1i	0.762272i	−0.924519	+	0.381136i	$0.875533\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	39716.6	1.91448
$756$	0	0
$757$	18725.4	0.899058	0.449529	−	0.893266i	$-0.351592\pi$
$757$	18725.4	0.899058	0.449529	+	0.893266i	$0.351592\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	26482.9i	1.26151i	0.775984	+	0.630753i	$0.217254\pi$
$761$	26482.9i	1.26151i	−0.775984	+	0.630753i	$0.782746\pi$
$762$	0	0
$763$	11222.6i	0.532484i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−5968.91	−0.280997
$768$	0	0
$769$	−33520.6	−1.57189	−0.785945	−	0.618297i	$-0.787823\pi$
$769$	−33520.6	−1.57189	−0.785945	+	0.618297i	$0.787823\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	32397.7i	1.50746i	0.657186	+	0.753729i	$0.271747\pi$
$773$	32397.7i	1.50746i	−0.657186	+	0.753729i	$0.728253\pi$
$774$	0	0
$775$	2270.58i	0.105241i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−27343.6	−1.25762
$780$	0	0
$781$	−18849.3	−0.863612
$782$	0	0
$783$	0	0
$784$	0	0
$785$	40467.1i	1.83992i
$786$	0	0
$787$	36145.7i	1.63717i	0.574385	+	0.818586i	$0.305241\pi$
$787$	36145.7i	1.63717i	−0.574385	+	0.818586i	$0.694759\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−7251.21	−0.325946
$792$	0	0
$793$	−20540.4	−0.919814
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 18153.0i	− 0.806790i	−0.915026	−	0.403395i	$-0.867830\pi$
$797$	− 18153.0i	− 0.806790i	0.915026	−	0.403395i	$-0.132170\pi$
$798$	0	0
$799$	− 1178.26i	− 0.0521702i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	14206.6	0.624334
$804$	0	0
$805$	5925.22	0.259424
$806$	0	0
$807$	0	0
$808$	0	0
$809$	11687.6i	0.507931i	0.967213	+	0.253965i	$0.0817349\pi$
$809$	11687.6i	0.507931i	−0.967213	+	0.253965i	$0.918265\pi$
$810$	0	0
$811$	− 13261.9i	− 0.574216i	−0.957898	−	0.287108i	$-0.907306\pi$
$811$	− 13261.9i	− 0.574216i	0.957898	−	0.287108i	$-0.0926938\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	63396.5	2.72476
$816$	0	0
$817$	2578.13	0.110401
$818$	0	0
$819$	0	0
$820$	0	0
$821$	6121.32i	0.260214i	0.991500	+	0.130107i	$0.0415320\pi$
$821$	6121.32i	0.260214i	−0.991500	+	0.130107i	$0.958468\pi$
$822$	0	0
$823$	− 9675.04i	− 0.409782i	−0.978785	−	0.204891i	$-0.934316\pi$
$823$	− 9675.04i	− 0.409782i	0.978785	−	0.204891i	$-0.0656840\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	34831.5	1.46458	0.732292	−	0.680991i	$-0.238451\pi$
$827$	34831.5	1.46458	0.732292	+	0.680991i	$0.238451\pi$
$828$	0	0
$829$	993.122	0.0416074	0.0208037	−	0.999784i	$-0.493377\pi$
$829$	993.122	0.0416074	0.0208037	+	0.999784i	$0.493377\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	− 3345.06i	− 0.139135i
$834$	0	0
$835$	− 16259.0i	− 0.673851i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	42717.0	1.75775	0.878877	−	0.477048i	$-0.158293\pi$
$839$	42717.0	1.75775	0.878877	+	0.477048i	$0.158293\pi$
$840$	0	0
$841$	−1129.15	−0.0462975
$842$	0	0
$843$	0	0
$844$	0	0
$845$	26085.6i	1.06198i
$846$	0	0
$847$	2726.75i	0.110617i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−7866.29	−0.316866
$852$	0	0
$853$	−6139.04	−0.246421	−0.123210	−	0.992381i	$-0.539319\pi$
$853$	−6139.04	−0.246421	−0.123210	+	0.992381i	$0.539319\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 37289.8i	− 1.48634i	−0.669101	−	0.743171i	$-0.733321\pi$
$857$	− 37289.8i	− 1.48634i	0.669101	−	0.743171i	$-0.266679\pi$
$858$	0	0
$859$	23219.9i	0.922296i	0.887323	+	0.461148i	$0.152562\pi$
$859$	23219.9i	0.922296i	−0.887323	+	0.461148i	$0.847438\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−18013.4	−0.710526	−0.355263	−	0.934766i	$-0.615609\pi$
$863$	−18013.4	−0.710526	−0.355263	+	0.934766i	$0.615609\pi$
$864$	0	0
$865$	54946.1	2.15980
$866$	0	0
$867$	0	0
$868$	0	0
$869$	− 20241.1i	− 0.790139i
$870$	0	0
$871$	− 21642.0i	− 0.841918i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−1202.44	−0.0464571
$876$	0	0
$877$	36769.1	1.41574	0.707870	−	0.706343i	$-0.249656\pi$
$877$	36769.1	1.41574	0.707870	+	0.706343i	$0.249656\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	46521.3i	1.77905i	0.456886	+	0.889525i	$0.348965\pi$
$881$	46521.3i	1.77905i	−0.456886	+	0.889525i	$0.651035\pi$
$882$	0	0
$883$	6524.30i	0.248653i	0.992241	+	0.124326i	$0.0396770\pi$
$883$	6524.30i	0.248653i	−0.992241	+	0.124326i	$0.960323\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−9695.31	−0.367009	−0.183504	−	0.983019i	$-0.558744\pi$
$887$	−9695.31	−0.367009	−0.183504	+	0.983019i	$0.558744\pi$
$888$	0	0
$889$	16983.2	0.640717
$890$	0	0
$891$	0	0
$892$	0	0
$893$	− 1525.98i	− 0.0571835i
$894$	0	0
$895$	− 73963.9i	− 2.76239i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	2674.07	0.0992049
$900$	0	0
$901$	6681.78	0.247062
$902$	0	0
$903$	0	0
$904$	0	0
$905$	39394.8i	1.44699i
$906$	0	0
$907$	− 5806.45i	− 0.212569i	−0.994336	−	0.106284i	$-0.966105\pi$
$907$	− 5806.45i	− 0.212569i	0.994336	−	0.106284i	$-0.0338954\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−43989.8	−1.59983	−0.799916	−	0.600112i	$-0.795123\pi$
$911$	−43989.8	−1.59983	−0.799916	+	0.600112i	$0.795123\pi$
$912$	0	0
$913$	−741.080	−0.0268633
$914$	0	0
$915$	0	0
$916$	0	0
$917$	15295.9i	0.550835i
$918$	0	0
$919$	− 2457.41i	− 0.0882073i	−0.999027	−	0.0441036i	$-0.985957\pi$
$919$	− 2457.41i	− 0.0882073i	0.999027	−	0.0441036i	$-0.0140432\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	10955.6	0.390691
$924$	0	0
$925$	20350.4	0.723369
$926$	0	0
$927$	0	0
$928$	0	0
$929$	− 12769.4i	− 0.450968i	−0.974247	−	0.225484i	$-0.927604\pi$
$929$	− 12769.4i	− 0.450968i	0.974247	−	0.225484i	$-0.0723963\pi$
$930$	0	0
$931$	− 4332.21i	− 0.152505i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	45715.2	1.59898
$936$	0	0
$937$	−8882.69	−0.309696	−0.154848	−	0.987938i	$-0.549489\pi$
$937$	−8882.69	−0.309696	−0.154848	+	0.987938i	$0.549489\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	30403.8i	1.05328i	0.850089	+	0.526640i	$0.176548\pi$
$941$	30403.8i	1.05328i	−0.850089	+	0.526640i	$0.823452\pi$
$942$	0	0
$943$	16215.4i	0.559965i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−16195.0	−0.555720	−0.277860	−	0.960622i	$-0.589625\pi$
$947$	−16195.0	−0.555720	−0.277860	+	0.960622i	$0.589625\pi$
$948$	0	0
$949$	−8257.18	−0.282444
$950$	0	0
$951$	0	0
$952$	0	0
$953$	20013.1i	0.680261i	0.940378	+	0.340131i	$0.110471\pi$
$953$	20013.1i	0.680261i	−0.940378	+	0.340131i	$0.889529\pi$
$954$	0	0
$955$	− 60451.2i	− 2.04833i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−12114.5	−0.407921
$960$	0	0
$961$	29510.8	0.990594
$962$	0	0
$963$	0	0
$964$	0	0
$965$	− 55533.3i	− 1.85252i
$966$	0	0
$967$	36134.7i	1.20167i	0.799374	+	0.600834i	$0.205165\pi$
$967$	36134.7i	1.20167i	−0.799374	+	0.600834i	$0.794835\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	5659.91	0.187060	0.0935300	−	0.995616i	$-0.470185\pi$
$971$	5659.91	0.187060	0.0935300	+	0.995616i	$0.470185\pi$
$972$	0	0
$973$	14218.9	0.468486
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 38648.6i	− 1.26559i	−0.774321	−	0.632793i	$-0.781908\pi$
$977$	− 38648.6i	− 1.26559i	0.774321	−	0.632793i	$-0.218092\pi$
$978$	0	0
$979$	− 37576.4i	− 1.22671i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−41325.6	−1.34088	−0.670439	−	0.741965i	$-0.733894\pi$
$983$	−41325.6	−1.34088	−0.670439	+	0.741965i	$0.733894\pi$
$984$	0	0
$985$	12844.5	0.415491
$986$	0	0
$987$	0	0
$988$	0	0
$989$	− 1528.89i	− 0.0491567i
$990$	0	0
$991$	− 42445.1i	− 1.36056i	−0.732953	−	0.680279i	$-0.761859\pi$
$991$	− 42445.1i	− 1.36056i	0.732953	−	0.680279i	$-0.238141\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−68651.0	−2.18732
$996$	0	0
$997$	−685.611	−0.0217789	−0.0108894	−	0.999941i	$-0.503466\pi$
$997$	−685.611	−0.0217789	−0.0108894	+	0.999941i	$0.503466\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1008.4.h.b.575.21	yes	24
3.2	odd	2	inner	1008.4.h.b.575.4	yes	24
4.3	odd	2	inner	1008.4.h.b.575.3	✓	24
12.11	even	2	inner	1008.4.h.b.575.22	yes	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1008.4.h.b.575.3	✓	24	4.3	odd	2	inner
1008.4.h.b.575.4	yes	24	3.2	odd	2	inner
1008.4.h.b.575.21	yes	24	1.1	even	1	trivial
1008.4.h.b.575.22	yes	24	12.11	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:	\( N \)	\(=\)	\( 1008 = 2^{4} \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1008.h (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-16.1444i q^{5} +7.00000i q^{7} +O(q^{10})\)	\(q-16.1444i q^{5} +7.00000i q^{7} +41.4793 q^{11} -24.1086 q^{13} +68.2665i q^{17} +88.4124i q^{19} +52.4307 q^{23} -135.640 q^{25} +159.744i q^{29} -16.7397i q^{31} +113.010 q^{35} -150.032 q^{37} +309.274i q^{41} -29.1603i q^{43} -17.2598 q^{47} -49.0000 q^{49} -97.8779i q^{53} -669.657i q^{55} +247.584 q^{59} +851.995 q^{61} +389.218i q^{65} +897.685i q^{67} -454.426 q^{71} +342.498 q^{73} +290.355i q^{77} -487.979i q^{79} -17.8662 q^{83} +1102.12 q^{85} -905.907i q^{89} -168.761i q^{91} +1427.36 q^{95} +586.705 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q+O(q^{10}) \) 24 * q	\( 24 q - 192 q^{13} - 984 q^{25} + 720 q^{37} - 1176 q^{49} + 2736 q^{61} + 3408 q^{85} + 1152 q^{97}+O(q^{100}) \) 24 * q - 192 * q^13 - 984 * q^25 + 720 * q^37 - 1176 * q^49 + 2736 * q^61 + 3408 * q^85 + 1152 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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