LMFDB - Embedded newform 2496.4.a.x.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2496,4,Mod(1,2496)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2496.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2496, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 2496 = 2^{6} \cdot 3 \cdot 13 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2496.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,0,-6,0,4,0,-20,0,18,0,-20] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	yes
Analytic conductor:	$147.268767374$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{55}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 55 $ x^2 - 55
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 312)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$7.41620$ of defining polynomial
Character	$\chi$	$=$	2496.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-3.00000 q^{3} +16.8324 q^{5} +4.83240 q^{7} +9.00000 q^{9} -39.6648 q^{11} +13.0000 q^{13} -50.4972 q^{15} -4.33521 q^{17} +28.8324 q^{19} -14.4972 q^{21} -0.670412 q^{23} +158.330 q^{25} -27.0000 q^{27} +6.00000 q^{29} -274.497 q^{31} +118.994 q^{33} +81.3408 q^{35} -84.3352 q^{37} -39.0000 q^{39} +209.156 q^{41} +364.324 q^{43} +151.492 q^{45} -239.341 q^{47} -319.648 q^{49} +13.0056 q^{51} +415.318 q^{53} -667.654 q^{55} -86.4972 q^{57} +78.0000 q^{59} +813.955 q^{61} +43.4916 q^{63} +218.821 q^{65} +858.452 q^{67} +2.01124 q^{69} +213.006 q^{71} +568.313 q^{73} -474.989 q^{75} -191.676 q^{77} -583.352 q^{79} +81.0000 q^{81} +162.648 q^{83} -72.9719 q^{85} -18.0000 q^{87} +1417.41 q^{89} +62.8212 q^{91} +823.492 q^{93} +485.318 q^{95} -798.949 q^{97} -356.983 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 6 q^{3} + 4 q^{5} - 20 q^{7} + 18 q^{9} - 20 q^{11} + 26 q^{13} - 12 q^{15} - 68 q^{17} + 28 q^{19} + 60 q^{21} - 120 q^{23} + 198 q^{25} - 54 q^{27} + 12 q^{29} - 460 q^{31} + 60 q^{33} + 400 q^{35}+ \cdots - 180 q^{99}+O(q^{100}) $ 2 * q - 6 * q^3 + 4 * q^5 - 20 * q^7 + 18 * q^9 - 20 * q^11 + 26 * q^13 - 12 * q^15 - 68 * q^17 + 28 * q^19 + 60 * q^21 - 120 * q^23 + 198 * q^25 - 54 * q^27 + 12 * q^29 - 460 * q^31 + 60 * q^33 + 400 * q^35 - 228 * q^37 - 78 * q^39 + 92 * q^41 + 432 * q^43 + 36 * q^45 - 716 * q^47 - 46 * q^49 + 204 * q^51 + 356 * q^53 - 920 * q^55 - 84 * q^57 + 156 * q^59 + 204 * q^61 - 180 * q^63 + 52 * q^65 + 204 * q^67 + 360 * q^69 + 604 * q^71 + 484 * q^73 - 594 * q^75 - 680 * q^77 - 1760 * q^79 + 162 * q^81 - 268 * q^83 + 744 * q^85 - 36 * q^87 + 76 * q^89 - 260 * q^91 + 1380 * q^93 + 496 * q^95 + 4 * q^97 - 180 * q^99

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
$3$	−3.00000	−0.577350
$3$	−3.00000	−0.577350
$4$	0	0
$5$	16.8324	1.50554	0.752768	−	0.658286i	$-0.228718\pi$
$5$	16.8324	1.50554	0.752768	+	0.658286i	$0.228718\pi$
$6$	0	0
$7$	4.83240	0.260925	0.130462	−	0.991453i	$-0.458354\pi$
$7$	4.83240	0.260925	0.130462	+	0.991453i	$0.458354\pi$
$8$	0	0
$9$	9.00000	0.333333
$10$	0	0
$11$	−39.6648	−1.08722	−0.543608	−	0.839339i	$-0.682942\pi$
$11$	−39.6648	−1.08722	−0.543608	+	0.839339i	$0.682942\pi$
$12$	0	0
$13$	13.0000	0.277350
$13$	13.0000	0.277350
$14$	0	0
$15$	−50.4972	−0.869221
$16$	0	0
$17$	−4.33521	−0.0618495	−0.0309248	−	0.999522i	$-0.509845\pi$
$17$	−4.33521	−0.0618495	−0.0309248	+	0.999522i	$0.509845\pi$
$18$	0	0
$19$	28.8324	0.348137	0.174069	−	0.984734i	$-0.444309\pi$
$19$	28.8324	0.348137	0.174069	+	0.984734i	$0.444309\pi$
$20$	0	0
$21$	−14.4972	−0.150645
$22$	0	0
$23$	−0.670412	−0.00607785	−0.00303893	−	0.999995i	$-0.500967\pi$
$23$	−0.670412	−0.00607785	−0.00303893	+	0.999995i	$0.500967\pi$
$24$	0	0
$25$	158.330	1.26664
$26$	0	0
$27$	−27.0000	−0.192450
$28$	0	0
$29$	6.00000	0.0384197	0.0192099	−	0.999815i	$-0.493885\pi$
$29$	6.00000	0.0384197	0.0192099	+	0.999815i	$0.493885\pi$
$30$	0	0
$31$	−274.497	−1.59036	−0.795180	−	0.606374i	$-0.792624\pi$
$31$	−274.497	−1.59036	−0.795180	+	0.606374i	$0.792624\pi$
$32$	0	0
$33$	118.994	0.627705
$34$	0	0
$35$	81.3408	0.392832
$36$	0	0
$37$	−84.3352	−0.374720	−0.187360	−	0.982291i	$-0.559993\pi$
$37$	−84.3352	−0.374720	−0.187360	+	0.982291i	$0.559993\pi$
$38$	0	0
$39$	−39.0000	−0.160128
$40$	0	0
$41$	209.156	0.796701	0.398350	−	0.917233i	$-0.369583\pi$
$41$	209.156	0.796701	0.398350	+	0.917233i	$0.369583\pi$
$42$	0	0
$43$	364.324	1.29207	0.646034	−	0.763309i	$-0.276427\pi$
$43$	364.324	1.29207	0.646034	+	0.763309i	$0.276427\pi$
$44$	0	0
$45$	151.492	0.501845
$46$	0	0
$47$	−239.341	−0.742797	−0.371398	−	0.928474i	$-0.621122\pi$
$47$	−239.341	−0.742797	−0.371398	+	0.928474i	$0.621122\pi$
$48$	0	0
$49$	−319.648	−0.931918
$50$	0	0
$51$	13.0056	0.0357088
$52$	0	0
$53$	415.318	1.07638	0.538192	−	0.842822i	$-0.319107\pi$
$53$	415.318	1.07638	0.538192	+	0.842822i	$0.319107\pi$
$54$	0	0
$55$	−667.654	−1.63684
$56$	0	0
$57$	−86.4972	−0.200997
$58$	0	0
$59$	78.0000	0.172114	0.0860571	−	0.996290i	$-0.472573\pi$
$59$	78.0000	0.172114	0.0860571	+	0.996290i	$0.472573\pi$
$60$	0	0
$61$	813.955	1.70846	0.854232	−	0.519893i	$-0.174028\pi$
$61$	813.955	1.70846	0.854232	+	0.519893i	$0.174028\pi$
$62$	0	0
$63$	43.4916	0.0869750
$64$	0	0
$65$	218.821	0.417560
$66$	0	0
$67$	858.452	1.56532	0.782661	−	0.622448i	$-0.213862\pi$
$67$	858.452	1.56532	0.782661	+	0.622448i	$0.213862\pi$
$68$	0	0
$69$	2.01124	0.00350905
$70$	0	0
$71$	213.006	0.356044	0.178022	−	0.984027i	$-0.443030\pi$
$71$	213.006	0.356044	0.178022	+	0.984027i	$0.443030\pi$
$72$	0	0
$73$	568.313	0.911178	0.455589	−	0.890190i	$-0.349429\pi$
$73$	568.313	0.911178	0.455589	+	0.890190i	$0.349429\pi$
$74$	0	0
$75$	−474.989	−0.731293
$76$	0	0
$77$	−191.676	−0.283682
$78$	0	0
$79$	−583.352	−0.830788	−0.415394	−	0.909642i	$-0.636356\pi$
$79$	−583.352	−0.830788	−0.415394	+	0.909642i	$0.636356\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	162.648	0.215096	0.107548	−	0.994200i	$-0.465700\pi$
$83$	162.648	0.215096	0.107548	+	0.994200i	$0.465700\pi$
$84$	0	0
$85$	−72.9719	−0.0931166
$86$	0	0
$87$	−18.0000	−0.0221816
$88$	0	0
$89$	1417.41	1.68815	0.844076	−	0.536224i	$-0.180150\pi$
$89$	1417.41	1.68815	0.844076	+	0.536224i	$0.180150\pi$
$90$	0	0
$91$	62.8212	0.0723675
$92$	0	0
$93$	823.492	0.918195
$94$	0	0
$95$	485.318	0.524133
$96$	0	0
$97$	−798.949	−0.836299	−0.418150	−	0.908378i	$-0.637321\pi$
$97$	−798.949	−0.836299	−0.418150	+	0.908378i	$0.637321\pi$
$98$	0	0
$99$	−356.983	−0.362406
$100$	0	0
$101$	1629.63	1.60549	0.802744	−	0.596323i	$-0.203372\pi$
$101$	1629.63	1.60549	0.802744	+	0.596323i	$0.203372\pi$
$102$	0	0
$103$	384.927	0.368233	0.184116	−	0.982904i	$-0.441058\pi$
$103$	384.927	0.368233	0.184116	+	0.982904i	$0.441058\pi$
$104$	0	0
$105$	−244.022	−0.226801
$106$	0	0
$107$	−336.045	−0.303614	−0.151807	−	0.988410i	$-0.548509\pi$
$107$	−336.045	−0.303614	−0.151807	+	0.988410i	$0.548509\pi$
$108$	0	0
$109$	−777.307	−0.683051	−0.341525	−	0.939873i	$-0.610944\pi$
$109$	−777.307	−0.683051	−0.341525	+	0.939873i	$0.610944\pi$
$110$	0	0
$111$	253.006	0.216344
$112$	0	0
$113$	−338.648	−0.281923	−0.140962	−	0.990015i	$-0.545019\pi$
$113$	−338.648	−0.281923	−0.140962	+	0.990015i	$0.545019\pi$
$114$	0	0
$115$	−11.2846	−0.00915042
$116$	0	0
$117$	117.000	0.0924500
$118$	0	0
$119$	−20.9494	−0.0161381
$120$	0	0
$121$	242.296	0.182040
$122$	0	0
$123$	−627.469	−0.459975
$124$	0	0
$125$	561.017	0.401431
$126$	0	0
$127$	1389.60	0.970920	0.485460	−	0.874259i	$-0.338652\pi$
$127$	1389.60	0.970920	0.485460	+	0.874259i	$0.338652\pi$
$128$	0	0
$129$	−1092.97	−0.745975
$130$	0	0
$131$	−701.966	−0.468176	−0.234088	−	0.972215i	$-0.575210\pi$
$131$	−701.966	−0.468176	−0.234088	+	0.972215i	$0.575210\pi$
$132$	0	0
$133$	139.330	0.0908377
$134$	0	0
$135$	−454.475	−0.289740
$136$	0	0
$137$	−1044.21	−0.651187	−0.325593	−	0.945510i	$-0.605564\pi$
$137$	−1044.21	−0.651187	−0.325593	+	0.945510i	$0.605564\pi$
$138$	0	0
$139$	1935.33	1.18095	0.590477	−	0.807055i	$-0.298940\pi$
$139$	1935.33	1.18095	0.590477	+	0.807055i	$0.298940\pi$
$140$	0	0
$141$	718.022	0.428854
$142$	0	0
$143$	−515.642	−0.301540
$144$	0	0
$145$	100.994	0.0578423
$146$	0	0
$147$	958.944	0.538043
$148$	0	0
$149$	2252.37	1.23840	0.619200	−	0.785233i	$-0.287457\pi$
$149$	2252.37	1.23840	0.619200	+	0.785233i	$0.287457\pi$
$150$	0	0
$151$	−626.128	−0.337441	−0.168721	−	0.985664i	$-0.553964\pi$
$151$	−626.128	−0.337441	−0.168721	+	0.985664i	$0.553964\pi$
$152$	0	0
$153$	−39.0169	−0.0206165
$154$	0	0
$155$	−4620.45	−2.39434
$156$	0	0
$157$	−1102.56	−0.560470	−0.280235	−	0.959931i	$-0.590412\pi$
$157$	−1102.56	−0.560470	−0.280235	+	0.959931i	$0.590412\pi$
$158$	0	0
$159$	−1245.96	−0.621451
$160$	0	0
$161$	−3.23970	−0.00158586
$162$	0	0
$163$	695.145	0.334037	0.167018	−	0.985954i	$-0.446586\pi$
$163$	695.145	0.334037	0.167018	+	0.985954i	$0.446586\pi$
$164$	0	0
$165$	2002.96	0.945032
$166$	0	0
$167$	−1984.50	−0.919553	−0.459777	−	0.888035i	$-0.652070\pi$
$167$	−1984.50	−0.919553	−0.459777	+	0.888035i	$0.652070\pi$
$168$	0	0
$169$	169.000	0.0769231
$170$	0	0
$171$	259.492	0.116046
$172$	0	0
$173$	1100.85	0.483792	0.241896	−	0.970302i	$-0.422231\pi$
$173$	1100.85	0.483792	0.241896	+	0.970302i	$0.422231\pi$
$174$	0	0
$175$	765.111	0.330497
$176$	0	0
$177$	−234.000	−0.0993702
$178$	0	0
$179$	−3792.63	−1.58365	−0.791827	−	0.610745i	$-0.790870\pi$
$179$	−3792.63	−1.58365	−0.791827	+	0.610745i	$0.790870\pi$
$180$	0	0
$181$	−1926.47	−0.791123	−0.395561	−	0.918440i	$-0.629450\pi$
$181$	−1926.47	−0.791123	−0.395561	+	0.918440i	$0.629450\pi$
$182$	0	0
$183$	−2441.87	−0.986382
$184$	0	0
$185$	−1419.56	−0.564153
$186$	0	0
$187$	171.955	0.0672438
$188$	0	0
$189$	−130.475	−0.0502150
$190$	0	0
$191$	−2048.51	−0.776048	−0.388024	−	0.921649i	$-0.626842\pi$
$191$	−2048.51	−0.776048	−0.388024	+	0.921649i	$0.626842\pi$
$192$	0	0
$193$	4134.67	1.54207	0.771036	−	0.636791i	$-0.219739\pi$
$193$	4134.67	1.54207	0.771036	+	0.636791i	$0.219739\pi$
$194$	0	0
$195$	−656.463	−0.241079
$196$	0	0
$197$	−1856.98	−0.671594	−0.335797	−	0.941934i	$-0.609006\pi$
$197$	−1856.98	−0.671594	−0.335797	+	0.941934i	$0.609006\pi$
$198$	0	0
$199$	−5515.56	−1.96476	−0.982382	−	0.186885i	$-0.940161\pi$
$199$	−5515.56	−1.96476	−0.982382	+	0.186885i	$0.940161\pi$
$200$	0	0
$201$	−2575.36	−0.903740
$202$	0	0
$203$	28.9944	0.0100247
$204$	0	0
$205$	3520.60	1.19946
$206$	0	0
$207$	−6.03371	−0.00202595
$208$	0	0
$209$	−1143.63	−0.378501
$210$	0	0
$211$	4035.73	1.31674	0.658368	−	0.752697i	$-0.271247\pi$
$211$	4035.73	1.31674	0.658368	+	0.752697i	$0.271247\pi$
$212$	0	0
$213$	−639.017	−0.205562
$214$	0	0
$215$	6132.45	1.94525
$216$	0	0
$217$	−1326.48	−0.414964
$218$	0	0
$219$	−1704.94	−0.526069
$220$	0	0
$221$	−56.3577	−0.0171540
$222$	0	0
$223$	4452.72	1.33711	0.668556	−	0.743661i	$-0.266912\pi$
$223$	4452.72	1.33711	0.668556	+	0.743661i	$0.266912\pi$
$224$	0	0
$225$	1424.97	0.422212
$226$	0	0
$227$	5482.68	1.60308	0.801538	−	0.597944i	$-0.204015\pi$
$227$	5482.68	1.60308	0.801538	+	0.597944i	$0.204015\pi$
$228$	0	0
$229$	1406.74	0.405938	0.202969	−	0.979185i	$-0.434941\pi$
$229$	1406.74	0.405938	0.202969	+	0.979185i	$0.434941\pi$
$230$	0	0
$231$	575.028	0.163784
$232$	0	0
$233$	6420.16	1.80514	0.902572	−	0.430540i	$-0.141677\pi$
$233$	6420.16	1.80514	0.902572	+	0.430540i	$0.141677\pi$
$234$	0	0
$235$	−4028.68	−1.11831
$236$	0	0
$237$	1750.06	0.479656
$238$	0	0
$239$	4899.44	1.32602	0.663009	−	0.748611i	$-0.269279\pi$
$239$	4899.44	1.32602	0.663009	+	0.748611i	$0.269279\pi$
$240$	0	0
$241$	259.129	0.0692613	0.0346307	−	0.999400i	$-0.488975\pi$
$241$	259.129	0.0692613	0.0346307	+	0.999400i	$0.488975\pi$
$242$	0	0
$243$	−243.000	−0.0641500
$244$	0	0
$245$	−5380.44	−1.40304
$246$	0	0
$247$	374.821	0.0965559
$248$	0	0
$249$	−487.944	−0.124185
$250$	0	0
$251$	884.760	0.222492	0.111246	−	0.993793i	$-0.464516\pi$
$251$	884.760	0.222492	0.111246	+	0.993793i	$0.464516\pi$
$252$	0	0
$253$	26.5918	0.00660794
$254$	0	0
$255$	218.916	0.0537609
$256$	0	0
$257$	2913.01	0.707036	0.353518	−	0.935428i	$-0.384985\pi$
$257$	2913.01	0.707036	0.353518	+	0.935428i	$0.384985\pi$
$258$	0	0
$259$	−407.541	−0.0977737
$260$	0	0
$261$	54.0000	0.0128066
$262$	0	0
$263$	127.330	0.0298535	0.0149268	−	0.999889i	$-0.495248\pi$
$263$	127.330	0.0298535	0.0149268	+	0.999889i	$0.495248\pi$
$264$	0	0
$265$	6990.80	1.62053
$266$	0	0
$267$	−4252.24	−0.974655
$268$	0	0
$269$	−8177.51	−1.85350	−0.926750	−	0.375679i	$-0.877409\pi$
$269$	−8177.51	−1.85350	−0.926750	+	0.375679i	$0.877409\pi$
$270$	0	0
$271$	4113.84	0.922132	0.461066	−	0.887366i	$-0.347467\pi$
$271$	4113.84	0.922132	0.461066	+	0.887366i	$0.347467\pi$
$272$	0	0
$273$	−188.463	−0.0417814
$274$	0	0
$275$	−6280.11	−1.37711
$276$	0	0
$277$	−587.543	−0.127444	−0.0637221	−	0.997968i	$-0.520297\pi$
$277$	−587.543	−0.127444	−0.0637221	+	0.997968i	$0.520297\pi$
$278$	0	0
$279$	−2470.47	−0.530120
$280$	0	0
$281$	3925.13	0.833288	0.416644	−	0.909070i	$-0.363206\pi$
$281$	3925.13	0.833288	0.416644	+	0.909070i	$0.363206\pi$
$282$	0	0
$283$	6461.26	1.35718	0.678590	−	0.734517i	$-0.262591\pi$
$283$	6461.26	1.35718	0.678590	+	0.734517i	$0.262591\pi$
$284$	0	0
$285$	−1455.96	−0.302608
$286$	0	0
$287$	1010.73	0.207879
$288$	0	0
$289$	−4894.21	−0.996175
$290$	0	0
$291$	2396.85	0.482838
$292$	0	0
$293$	−9603.82	−1.91489	−0.957443	−	0.288623i	$-0.906802\pi$
$293$	−9603.82	−1.91489	−0.957443	+	0.288623i	$0.906802\pi$
$294$	0	0
$295$	1312.93	0.259124
$296$	0	0
$297$	1070.95	0.209235
$298$	0	0
$299$	−8.71536	−0.00168569
$300$	0	0
$301$	1760.56	0.337132
$302$	0	0
$303$	−4888.89	−0.926929
$304$	0	0
$305$	13700.8	2.57215
$306$	0	0
$307$	5313.21	0.987756	0.493878	−	0.869531i	$-0.335579\pi$
$307$	5313.21	0.987756	0.493878	+	0.869531i	$0.335579\pi$
$308$	0	0
$309$	−1154.78	−0.212599
$310$	0	0
$311$	−6662.88	−1.21485	−0.607423	−	0.794378i	$-0.707797\pi$
$311$	−6662.88	−1.21485	−0.607423	+	0.794378i	$0.707797\pi$
$312$	0	0
$313$	1556.70	0.281119	0.140559	−	0.990072i	$-0.455110\pi$
$313$	1556.70	0.281119	0.140559	+	0.990072i	$0.455110\pi$
$314$	0	0
$315$	732.067	0.130944
$316$	0	0
$317$	662.664	0.117410	0.0587049	−	0.998275i	$-0.481303\pi$
$317$	662.664	0.117410	0.0587049	+	0.998275i	$0.481303\pi$
$318$	0	0
$319$	−237.989	−0.0417706
$320$	0	0
$321$	1008.13	0.175292
$322$	0	0
$323$	−124.994	−0.0215321
$324$	0	0
$325$	2058.28	0.351302
$326$	0	0
$327$	2331.92	0.394359
$328$	0	0
$329$	−1156.59	−0.193814
$330$	0	0
$331$	1378.83	0.228965	0.114482	−	0.993425i	$-0.463479\pi$
$331$	1378.83	0.228965	0.114482	+	0.993425i	$0.463479\pi$
$332$	0	0
$333$	−759.017	−0.124907
$334$	0	0
$335$	14449.8	2.35665
$336$	0	0
$337$	8180.86	1.32237	0.661187	−	0.750222i	$-0.270053\pi$
$337$	8180.86	1.32237	0.661187	+	0.750222i	$0.270053\pi$
$338$	0	0
$339$	1015.94	0.162768
$340$	0	0
$341$	10887.9	1.72907
$342$	0	0
$343$	−3202.18	−0.504086
$344$	0	0
$345$	33.8539	0.00528300
$346$	0	0
$347$	2712.49	0.419637	0.209819	−	0.977740i	$-0.432713\pi$
$347$	2712.49	0.419637	0.209819	+	0.977740i	$0.432713\pi$
$348$	0	0
$349$	6789.20	1.04131	0.520656	−	0.853767i	$-0.325687\pi$
$349$	6789.20	1.04131	0.520656	+	0.853767i	$0.325687\pi$
$350$	0	0
$351$	−351.000	−0.0533761
$352$	0	0
$353$	228.018	0.0343801	0.0171900	−	0.999852i	$-0.494528\pi$
$353$	228.018	0.0343801	0.0171900	+	0.999852i	$0.494528\pi$
$354$	0	0
$355$	3585.40	0.536037
$356$	0	0
$357$	62.8483	0.00931733
$358$	0	0
$359$	507.161	0.0745597	0.0372799	−	0.999305i	$-0.488131\pi$
$359$	507.161	0.0745597	0.0372799	+	0.999305i	$0.488131\pi$
$360$	0	0
$361$	−6027.69	−0.878801
$362$	0	0
$363$	−726.888	−0.105101
$364$	0	0
$365$	9566.07	1.37181
$366$	0	0
$367$	9627.62	1.36937	0.684683	−	0.728841i	$-0.259941\pi$
$367$	9627.62	1.36937	0.684683	+	0.728841i	$0.259941\pi$
$368$	0	0
$369$	1882.41	0.265567
$370$	0	0
$371$	2006.98	0.280855
$372$	0	0
$373$	7342.00	1.01918	0.509590	−	0.860417i	$-0.329797\pi$
$373$	7342.00	1.01918	0.509590	+	0.860417i	$0.329797\pi$
$374$	0	0
$375$	−1683.05	−0.231766
$376$	0	0
$377$	78.0000	0.0106557
$378$	0	0
$379$	−11735.7	−1.59055	−0.795277	−	0.606246i	$-0.792675\pi$
$379$	−11735.7	−1.59055	−0.795277	+	0.606246i	$0.792675\pi$
$380$	0	0
$381$	−4168.79	−0.560561
$382$	0	0
$383$	−6739.67	−0.899168	−0.449584	−	0.893238i	$-0.648428\pi$
$383$	−6739.67	−0.899168	−0.449584	+	0.893238i	$0.648428\pi$
$384$	0	0
$385$	−3226.37	−0.427093
$386$	0	0
$387$	3278.92	0.430689
$388$	0	0
$389$	13234.7	1.72500	0.862498	−	0.506060i	$-0.168898\pi$
$389$	13234.7	1.72500	0.862498	+	0.506060i	$0.168898\pi$
$390$	0	0
$391$	2.90637	0.000375912 0
$392$	0	0
$393$	2105.90	0.270302
$394$	0	0
$395$	−9819.21	−1.25078
$396$	0	0
$397$	14292.8	1.80689	0.903446	−	0.428701i	$-0.141029\pi$
$397$	14292.8	1.80689	0.903446	+	0.428701i	$0.141029\pi$
$398$	0	0
$399$	−417.989	−0.0524451
$400$	0	0
$401$	11779.5	1.46693	0.733466	−	0.679726i	$-0.237901\pi$
$401$	11779.5	1.46693	0.733466	+	0.679726i	$0.237901\pi$
$402$	0	0
$403$	−3568.46	−0.441086
$404$	0	0
$405$	1363.42	0.167282
$406$	0	0
$407$	3345.14	0.407401
$408$	0	0
$409$	1039.22	0.125638	0.0628189	−	0.998025i	$-0.479991\pi$
$409$	1039.22	0.125638	0.0628189	+	0.998025i	$0.479991\pi$
$410$	0	0
$411$	3132.62	0.375963
$412$	0	0
$413$	376.927	0.0449089
$414$	0	0
$415$	2737.75	0.323834
$416$	0	0
$417$	−5805.99	−0.681824
$418$	0	0
$419$	6085.74	0.709565	0.354783	−	0.934949i	$-0.384555\pi$
$419$	6085.74	0.709565	0.354783	+	0.934949i	$0.384555\pi$
$420$	0	0
$421$	−8309.91	−0.961995	−0.480998	−	0.876722i	$-0.659725\pi$
$421$	−8309.91	−0.961995	−0.480998	+	0.876722i	$0.659725\pi$
$422$	0	0
$423$	−2154.07	−0.247599
$424$	0	0
$425$	−686.391	−0.0783409
$426$	0	0
$427$	3933.35	0.445781
$428$	0	0
$429$	1546.93	0.174094
$430$	0	0
$431$	15601.0	1.74356	0.871778	−	0.489902i	$-0.162967\pi$
$431$	15601.0	1.74356	0.871778	+	0.489902i	$0.162967\pi$
$432$	0	0
$433$	7887.07	0.875354	0.437677	−	0.899132i	$-0.355801\pi$
$433$	7887.07	0.875354	0.437677	+	0.899132i	$0.355801\pi$
$434$	0	0
$435$	−302.983	−0.0333952
$436$	0	0
$437$	−19.3296	−0.00211593
$438$	0	0
$439$	−2105.13	−0.228866	−0.114433	−	0.993431i	$-0.536505\pi$
$439$	−2105.13	−0.228866	−0.114433	+	0.993431i	$0.536505\pi$
$440$	0	0
$441$	−2876.83	−0.310639
$442$	0	0
$443$	−1387.00	−0.148754	−0.0743772	−	0.997230i	$-0.523697\pi$
$443$	−1387.00	−0.148754	−0.0743772	+	0.997230i	$0.523697\pi$
$444$	0	0
$445$	23858.5	2.54157
$446$	0	0
$447$	−6757.12	−0.714991
$448$	0	0
$449$	12563.5	1.32050	0.660252	−	0.751044i	$-0.270449\pi$
$449$	12563.5	1.32050	0.660252	+	0.751044i	$0.270449\pi$
$450$	0	0
$451$	−8296.14	−0.866187
$452$	0	0
$453$	1878.38	0.194822
$454$	0	0
$455$	1057.43	0.108952
$456$	0	0
$457$	−12331.4	−1.26222	−0.631112	−	0.775692i	$-0.717401\pi$
$457$	−12331.4	−1.26222	−0.631112	+	0.775692i	$0.717401\pi$
$458$	0	0
$459$	117.051	0.0119029
$460$	0	0
$461$	−1825.26	−0.184406	−0.0922029	−	0.995740i	$-0.529391\pi$
$461$	−1825.26	−0.184406	−0.0922029	+	0.995740i	$0.529391\pi$
$462$	0	0
$463$	−8172.00	−0.820270	−0.410135	−	0.912025i	$-0.634518\pi$
$463$	−8172.00	−0.820270	−0.410135	+	0.912025i	$0.634518\pi$
$464$	0	0
$465$	13861.3	1.38237
$466$	0	0
$467$	9959.22	0.986847	0.493424	−	0.869789i	$-0.335745\pi$
$467$	9959.22	0.986847	0.493424	+	0.869789i	$0.335745\pi$
$468$	0	0
$469$	4148.38	0.408432
$470$	0	0
$471$	3307.67	0.323587
$472$	0	0
$473$	−14450.8	−1.40476
$474$	0	0
$475$	4565.02	0.440963
$476$	0	0
$477$	3737.87	0.358795
$478$	0	0
$479$	2793.32	0.266451	0.133226	−	0.991086i	$-0.457467\pi$
$479$	2793.32	0.266451	0.133226	+	0.991086i	$0.457467\pi$
$480$	0	0
$481$	−1096.36	−0.103928
$482$	0	0
$483$	9.71909	0.000915598 0
$484$	0	0
$485$	−13448.2	−1.25908
$486$	0	0
$487$	3870.17	0.360111	0.180056	−	0.983656i	$-0.442372\pi$
$487$	3870.17	0.360111	0.180056	+	0.983656i	$0.442372\pi$
$488$	0	0
$489$	−2085.44	−0.192856
$490$	0	0
$491$	−1888.65	−0.173592	−0.0867958	−	0.996226i	$-0.527663\pi$
$491$	−1888.65	−0.173592	−0.0867958	+	0.996226i	$0.527663\pi$
$492$	0	0
$493$	−26.0112	−0.00237624
$494$	0	0
$495$	−6008.88	−0.545614
$496$	0	0
$497$	1029.33	0.0929007
$498$	0	0
$499$	−15879.0	−1.42453	−0.712266	−	0.701910i	$-0.752331\pi$
$499$	−15879.0	−1.42453	−0.712266	+	0.701910i	$0.752331\pi$
$500$	0	0
$501$	5953.51	0.530904
$502$	0	0
$503$	−16338.7	−1.44833	−0.724164	−	0.689628i	$-0.757774\pi$
$503$	−16338.7	−1.44833	−0.724164	+	0.689628i	$0.757774\pi$
$504$	0	0
$505$	27430.6	2.41712
$506$	0	0
$507$	−507.000	−0.0444116
$508$	0	0
$509$	11875.7	1.03415	0.517076	−	0.855940i	$-0.327021\pi$
$509$	11875.7	1.03415	0.517076	+	0.855940i	$0.327021\pi$
$510$	0	0
$511$	2746.31	0.237749
$512$	0	0
$513$	−778.475	−0.0669990
$514$	0	0
$515$	6479.24	0.554388
$516$	0	0
$517$	9493.40	0.807581
$518$	0	0
$519$	−3302.54	−0.279317
$520$	0	0
$521$	4971.92	0.418088	0.209044	−	0.977906i	$-0.432965\pi$
$521$	4971.92	0.418088	0.209044	+	0.977906i	$0.432965\pi$
$522$	0	0
$523$	1131.76	0.0946238	0.0473119	−	0.998880i	$-0.484935\pi$
$523$	1131.76	0.0946238	0.0473119	+	0.998880i	$0.484935\pi$
$524$	0	0
$525$	−2295.33	−0.190813
$526$	0	0
$527$	1190.00	0.0983630
$528$	0	0
$529$	−12166.6	−0.999963
$530$	0	0
$531$	702.000	0.0573714
$532$	0	0
$533$	2719.03	0.220965
$534$	0	0
$535$	−5656.44	−0.457102
$536$	0	0
$537$	11377.9	0.914323
$538$	0	0
$539$	12678.8	1.01320
$540$	0	0
$541$	10514.6	0.835593	0.417797	−	0.908541i	$-0.362802\pi$
$541$	10514.6	0.835593	0.417797	+	0.908541i	$0.362802\pi$
$542$	0	0
$543$	5779.40	0.456755
$544$	0	0
$545$	−13083.9	−1.02836
$546$	0	0
$547$	−21257.4	−1.66161	−0.830806	−	0.556562i	$-0.812120\pi$
$547$	−21257.4	−1.66161	−0.830806	+	0.556562i	$0.812120\pi$
$548$	0	0
$549$	7325.60	0.569488
$550$	0	0
$551$	172.994	0.0133753
$552$	0	0
$553$	−2818.99	−0.216773
$554$	0	0
$555$	4258.69	0.325714
$556$	0	0
$557$	−2350.31	−0.178789	−0.0893947	−	0.995996i	$-0.528493\pi$
$557$	−2350.31	−0.178789	−0.0893947	+	0.995996i	$0.528493\pi$
$558$	0	0
$559$	4736.21	0.358355
$560$	0	0
$561$	−515.865	−0.0388233
$562$	0	0
$563$	19787.9	1.48128	0.740639	−	0.671904i	$-0.234523\pi$
$563$	19787.9	1.48128	0.740639	+	0.671904i	$0.234523\pi$
$564$	0	0
$565$	−5700.26	−0.424445
$566$	0	0
$567$	391.424	0.0289917
$568$	0	0
$569$	−20580.0	−1.51627	−0.758136	−	0.652097i	$-0.773890\pi$
$569$	−20580.0	−1.51627	−0.758136	+	0.652097i	$0.773890\pi$
$570$	0	0
$571$	−13301.9	−0.974897	−0.487449	−	0.873152i	$-0.662072\pi$
$571$	−13301.9	−0.974897	−0.487449	+	0.873152i	$0.662072\pi$
$572$	0	0
$573$	6145.54	0.448052
$574$	0	0
$575$	−106.146	−0.00769843
$576$	0	0
$577$	−6423.54	−0.463458	−0.231729	−	0.972780i	$-0.574438\pi$
$577$	−6423.54	−0.463458	−0.231729	+	0.972780i	$0.574438\pi$
$578$	0	0
$579$	−12404.0	−0.890316
$580$	0	0
$581$	785.979	0.0561238
$582$	0	0
$583$	−16473.5	−1.17026
$584$	0	0
$585$	1969.39	0.139187
$586$	0	0
$587$	18600.5	1.30788	0.653941	−	0.756546i	$-0.273115\pi$
$587$	18600.5	1.30788	0.653941	+	0.756546i	$0.273115\pi$
$588$	0	0
$589$	−7914.41	−0.553663
$590$	0	0
$591$	5570.93	0.387745
$592$	0	0
$593$	11942.3	0.827001	0.413501	−	0.910504i	$-0.364306\pi$
$593$	11942.3	0.827001	0.413501	+	0.910504i	$0.364306\pi$
$594$	0	0
$595$	−352.629	−0.0242965
$596$	0	0
$597$	16546.7	1.13436
$598$	0	0
$599$	−746.614	−0.0509279	−0.0254640	−	0.999676i	$-0.508106\pi$
$599$	−746.614	−0.0509279	−0.0254640	+	0.999676i	$0.508106\pi$
$600$	0	0
$601$	11287.4	0.766096	0.383048	−	0.923728i	$-0.374874\pi$
$601$	11287.4	0.766096	0.383048	+	0.923728i	$0.374874\pi$
$602$	0	0
$603$	7726.07	0.521774
$604$	0	0
$605$	4078.42	0.274068
$606$	0	0
$607$	−7090.08	−0.474098	−0.237049	−	0.971498i	$-0.576180\pi$
$607$	−7090.08	−0.474098	−0.237049	+	0.971498i	$0.576180\pi$
$608$	0	0
$609$	−86.9831	−0.00578774
$610$	0	0
$611$	−3111.43	−0.206015
$612$	0	0
$613$	−21842.3	−1.43915	−0.719577	−	0.694413i	$-0.755664\pi$
$613$	−21842.3	−1.43915	−0.719577	+	0.694413i	$0.755664\pi$
$614$	0	0
$615$	−10561.8	−0.692509
$616$	0	0
$617$	7845.18	0.511888	0.255944	−	0.966692i	$-0.417614\pi$
$617$	7845.18	0.511888	0.255944	+	0.966692i	$0.417614\pi$
$618$	0	0
$619$	29667.4	1.92639	0.963193	−	0.268809i	$-0.0866302\pi$
$619$	29667.4	1.92639	0.963193	+	0.268809i	$0.0866302\pi$
$620$	0	0
$621$	18.1011	0.00116968
$622$	0	0
$623$	6849.50	0.440481
$624$	0	0
$625$	−10347.9	−0.662268
$626$	0	0
$627$	3430.89	0.218527
$628$	0	0
$629$	365.610	0.0231762
$630$	0	0
$631$	−24321.2	−1.53441	−0.767204	−	0.641404i	$-0.778352\pi$
$631$	−24321.2	−1.53441	−0.767204	+	0.641404i	$0.778352\pi$
$632$	0	0
$633$	−12107.2	−0.760217
$634$	0	0
$635$	23390.3	1.46175
$636$	0	0
$637$	−4155.42	−0.258468
$638$	0	0
$639$	1917.05	0.118681
$640$	0	0
$641$	16751.4	1.03220	0.516099	−	0.856529i	$-0.327384\pi$
$641$	16751.4	1.03220	0.516099	+	0.856529i	$0.327384\pi$
$642$	0	0
$643$	−27260.5	−1.67193	−0.835964	−	0.548784i	$-0.815091\pi$
$643$	−27260.5	−1.67193	−0.835964	+	0.548784i	$0.815091\pi$
$644$	0	0
$645$	−18397.3	−1.12309
$646$	0	0
$647$	9550.09	0.580298	0.290149	−	0.956981i	$-0.406295\pi$
$647$	9550.09	0.580298	0.290149	+	0.956981i	$0.406295\pi$
$648$	0	0
$649$	−3093.85	−0.187125
$650$	0	0
$651$	3979.44	0.239580
$652$	0	0
$653$	1093.89	0.0655548	0.0327774	−	0.999463i	$-0.489565\pi$
$653$	1093.89	0.0655548	0.0327774	+	0.999463i	$0.489565\pi$
$654$	0	0
$655$	−11815.8	−0.704856
$656$	0	0
$657$	5114.81	0.303726
$658$	0	0
$659$	−9129.47	−0.539657	−0.269828	−	0.962908i	$-0.586967\pi$
$659$	−9129.47	−0.539657	−0.269828	+	0.962908i	$0.586967\pi$
$660$	0	0
$661$	−10726.5	−0.631181	−0.315591	−	0.948895i	$-0.602203\pi$
$661$	−10726.5	−0.631181	−0.315591	+	0.948895i	$0.602203\pi$
$662$	0	0
$663$	169.073	0.00990385
$664$	0	0
$665$	2345.25	0.136759
$666$	0	0
$667$	−4.02247	−0.000233509 0
$668$	0	0
$669$	−13358.2	−0.771983
$670$	0	0
$671$	−32285.4	−1.85747
$672$	0	0
$673$	−6797.04	−0.389311	−0.194656	−	0.980872i	$-0.562359\pi$
$673$	−6797.04	−0.389311	−0.194656	+	0.980872i	$0.562359\pi$
$674$	0	0
$675$	−4274.90	−0.243764
$676$	0	0
$677$	22167.0	1.25841	0.629207	−	0.777238i	$-0.283380\pi$
$677$	22167.0	1.25841	0.629207	+	0.777238i	$0.283380\pi$
$678$	0	0
$679$	−3860.84	−0.218211
$680$	0	0
$681$	−16448.0	−0.925536
$682$	0	0
$683$	16756.8	0.938773	0.469387	−	0.882993i	$-0.344475\pi$
$683$	16756.8	0.938773	0.469387	+	0.882993i	$0.344475\pi$
$684$	0	0
$685$	−17576.5	−0.980385
$686$	0	0
$687$	−4220.21	−0.234369
$688$	0	0
$689$	5399.14	0.298535
$690$	0	0
$691$	−1807.58	−0.0995132	−0.0497566	−	0.998761i	$-0.515845\pi$
$691$	−1807.58	−0.0995132	−0.0497566	+	0.998761i	$0.515845\pi$
$692$	0	0
$693$	−1725.08	−0.0945607
$694$	0	0
$695$	32576.2	1.77797
$696$	0	0
$697$	−906.736	−0.0492756
$698$	0	0
$699$	−19260.5	−1.04220
$700$	0	0
$701$	26699.0	1.43853	0.719264	−	0.694737i	$-0.244479\pi$
$701$	26699.0	1.43853	0.719264	+	0.694737i	$0.244479\pi$
$702$	0	0
$703$	−2431.59	−0.130454
$704$	0	0
$705$	12086.0	0.645655
$706$	0	0
$707$	7875.02	0.418912
$708$	0	0
$709$	−19226.1	−1.01841	−0.509204	−	0.860646i	$-0.670060\pi$
$709$	−19226.1	−1.01841	−0.509204	+	0.860646i	$0.670060\pi$
$710$	0	0
$711$	−5250.17	−0.276929
$712$	0	0
$713$	184.026	0.00966597
$714$	0	0
$715$	−8679.50	−0.453979
$716$	0	0
$717$	−14698.3	−0.765577
$718$	0	0
$719$	14372.2	0.745470	0.372735	−	0.927938i	$-0.378420\pi$
$719$	14372.2	0.745470	0.372735	+	0.927938i	$0.378420\pi$
$720$	0	0
$721$	1860.12	0.0960811
$722$	0	0
$723$	−777.388	−0.0399880
$724$	0	0
$725$	949.978	0.0486638
$726$	0	0
$727$	−26347.2	−1.34410	−0.672052	−	0.740504i	$-0.734587\pi$
$727$	−26347.2	−1.34410	−0.672052	+	0.740504i	$0.734587\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	0	0
$731$	−1579.42	−0.0799137
$732$	0	0
$733$	−27597.8	−1.39065	−0.695326	−	0.718694i	$-0.744740\pi$
$733$	−27597.8	−1.39065	−0.695326	+	0.718694i	$0.744740\pi$
$734$	0	0
$735$	16141.3	0.810043
$736$	0	0
$737$	−34050.3	−1.70185
$738$	0	0
$739$	9546.08	0.475180	0.237590	−	0.971365i	$-0.423642\pi$
$739$	9546.08	0.475180	0.237590	+	0.971365i	$0.423642\pi$
$740$	0	0
$741$	−1124.46	−0.0557466
$742$	0	0
$743$	32976.0	1.62823	0.814114	−	0.580705i	$-0.197223\pi$
$743$	32976.0	1.62823	0.814114	+	0.580705i	$0.197223\pi$
$744$	0	0
$745$	37912.8	1.86446
$746$	0	0
$747$	1463.83	0.0716985
$748$	0	0
$749$	−1623.90	−0.0792204
$750$	0	0
$751$	9781.24	0.475263	0.237631	−	0.971355i	$-0.423629\pi$
$751$	9781.24	0.475263	0.237631	+	0.971355i	$0.423629\pi$
$752$	0	0
$753$	−2654.28	−0.128456
$754$	0	0
$755$	−10539.2	−0.508029
$756$	0	0
$757$	−4414.55	−0.211955	−0.105977	−	0.994369i	$-0.533797\pi$
$757$	−4414.55	−0.211955	−0.105977	+	0.994369i	$0.533797\pi$
$758$	0	0
$759$	−79.7753	−0.00381510
$760$	0	0
$761$	−33676.2	−1.60415	−0.802077	−	0.597220i	$-0.796272\pi$
$761$	−33676.2	−1.60415	−0.802077	+	0.597220i	$0.796272\pi$
$762$	0	0
$763$	−3756.26	−0.178225
$764$	0	0
$765$	−656.747	−0.0310389
$766$	0	0
$767$	1014.00	0.0477359
$768$	0	0
$769$	−34795.0	−1.63165	−0.815827	−	0.578297i	$-0.803718\pi$
$769$	−34795.0	−1.63165	−0.815827	+	0.578297i	$0.803718\pi$
$770$	0	0
$771$	−8739.02	−0.408208
$772$	0	0
$773$	−28152.1	−1.30991	−0.654954	−	0.755668i	$-0.727312\pi$
$773$	−28152.1	−1.30991	−0.654954	+	0.755668i	$0.727312\pi$
$774$	0	0
$775$	−43461.0	−2.01441
$776$	0	0
$777$	1222.62	0.0564496
$778$	0	0
$779$	6030.48	0.277361
$780$	0	0
$781$	−8448.82	−0.387097
$782$	0	0
$783$	−162.000	−0.00739388
$784$	0	0
$785$	−18558.7	−0.843807
$786$	0	0
$787$	2659.15	0.120443	0.0602215	−	0.998185i	$-0.480819\pi$
$787$	2659.15	0.120443	0.0602215	+	0.998185i	$0.480819\pi$
$788$	0	0
$789$	−381.989	−0.0172359
$790$	0	0
$791$	−1636.48	−0.0735608
$792$	0	0
$793$	10581.4	0.473842
$794$	0	0
$795$	−20972.4	−0.935616
$796$	0	0
$797$	−33193.5	−1.47525	−0.737625	−	0.675210i	$-0.764053\pi$
$797$	−33193.5	−1.47525	−0.737625	+	0.675210i	$0.764053\pi$
$798$	0	0
$799$	1037.59	0.0459416
$800$	0	0
$801$	12756.7	0.562717
$802$	0	0
$803$	−22542.0	−0.990648
$804$	0	0
$805$	−54.5319	−0.00238757
$806$	0	0
$807$	24532.5	1.07012
$808$	0	0
$809$	9198.73	0.399765	0.199883	−	0.979820i	$-0.435944\pi$
$809$	9198.73	0.399765	0.199883	+	0.979820i	$0.435944\pi$
$810$	0	0
$811$	2435.11	0.105436	0.0527179	−	0.998609i	$-0.483212\pi$
$811$	2435.11	0.105436	0.0527179	+	0.998609i	$0.483212\pi$
$812$	0	0
$813$	−12341.5	−0.532393
$814$	0	0
$815$	11701.0	0.502904
$816$	0	0
$817$	10504.3	0.449817
$818$	0	0
$819$	565.390	0.0241225
$820$	0	0
$821$	−17767.2	−0.755275	−0.377638	−	0.925954i	$-0.623263\pi$
$821$	−17767.2	−0.755275	−0.377638	+	0.925954i	$0.623263\pi$
$822$	0	0
$823$	15112.0	0.640060	0.320030	−	0.947407i	$-0.396307\pi$
$823$	15112.0	0.640060	0.320030	+	0.947407i	$0.396307\pi$
$824$	0	0
$825$	18840.3	0.795074
$826$	0	0
$827$	32606.2	1.37102	0.685508	−	0.728065i	$-0.259580\pi$
$827$	32606.2	1.37102	0.685508	+	0.728065i	$0.259580\pi$
$828$	0	0
$829$	26701.9	1.11869	0.559347	−	0.828934i	$-0.311052\pi$
$829$	26701.9	1.11869	0.559347	+	0.828934i	$0.311052\pi$
$830$	0	0
$831$	1762.63	0.0735799
$832$	0	0
$833$	1385.74	0.0576387
$834$	0	0
$835$	−33403.9	−1.38442
$836$	0	0
$837$	7411.42	0.306065
$838$	0	0
$839$	20747.5	0.853736	0.426868	−	0.904314i	$-0.359617\pi$
$839$	20747.5	0.853736	0.426868	+	0.904314i	$0.359617\pi$
$840$	0	0
$841$	−24353.0	−0.998524
$842$	0	0
$843$	−11775.4	−0.481099
$844$	0	0
$845$	2844.68	0.115810
$846$	0	0
$847$	1170.87	0.0474989
$848$	0	0
$849$	−19383.8	−0.783568
$850$	0	0
$851$	56.5393	0.00227749
$852$	0	0
$853$	−4041.50	−0.162225	−0.0811127	−	0.996705i	$-0.525847\pi$
$853$	−4041.50	−0.162225	−0.0811127	+	0.996705i	$0.525847\pi$
$854$	0	0
$855$	4367.87	0.174711
$856$	0	0
$857$	−32067.4	−1.27818	−0.639091	−	0.769131i	$-0.720689\pi$
$857$	−32067.4	−1.27818	−0.639091	+	0.769131i	$0.720689\pi$
$858$	0	0
$859$	−3739.80	−0.148545	−0.0742726	−	0.997238i	$-0.523664\pi$
$859$	−3739.80	−0.148545	−0.0742726	+	0.997238i	$0.523664\pi$
$860$	0	0
$861$	−3032.18	−0.120019
$862$	0	0
$863$	−38258.1	−1.50906	−0.754531	−	0.656265i	$-0.772135\pi$
$863$	−38258.1	−1.50906	−0.754531	+	0.656265i	$0.772135\pi$
$864$	0	0
$865$	18529.9	0.728365
$866$	0	0
$867$	14682.6	0.575142
$868$	0	0
$869$	23138.5	0.903246
$870$	0	0
$871$	11159.9	0.434142
$872$	0	0
$873$	−7190.54	−0.278766
$874$	0	0
$875$	2711.06	0.104743
$876$	0	0
$877$	21528.7	0.828932	0.414466	−	0.910065i	$-0.363968\pi$
$877$	21528.7	0.828932	0.414466	+	0.910065i	$0.363968\pi$
$878$	0	0
$879$	28811.5	1.10556
$880$	0	0
$881$	25995.3	0.994101	0.497051	−	0.867722i	$-0.334416\pi$
$881$	25995.3	0.994101	0.497051	+	0.867722i	$0.334416\pi$
$882$	0	0
$883$	9586.18	0.365346	0.182673	−	0.983174i	$-0.441525\pi$
$883$	9586.18	0.365346	0.182673	+	0.983174i	$0.441525\pi$
$884$	0	0
$885$	−3938.78	−0.149605
$886$	0	0
$887$	−20879.1	−0.790361	−0.395180	−	0.918604i	$-0.629318\pi$
$887$	−20879.1	−0.790361	−0.395180	+	0.918604i	$0.629318\pi$
$888$	0	0
$889$	6715.09	0.253337
$890$	0	0
$891$	−3212.85	−0.120802
$892$	0	0
$893$	−6900.77	−0.258595
$894$	0	0
$895$	−63839.0	−2.38425
$896$	0	0
$897$	26.1461	0.000973235 0
$898$	0	0
$899$	−1646.98	−0.0611012
$900$	0	0
$901$	−1800.49	−0.0665739
$902$	0	0
$903$	−5281.67	−0.194644
$904$	0	0
$905$	−32427.1	−1.19106
$906$	0	0
$907$	1094.00	0.0400503	0.0200251	−	0.999799i	$-0.493625\pi$
$907$	1094.00	0.0400503	0.0200251	+	0.999799i	$0.493625\pi$
$908$	0	0
$909$	14666.7	0.535163
$910$	0	0
$911$	29816.7	1.08438	0.542190	−	0.840256i	$-0.317595\pi$
$911$	29816.7	1.08438	0.542190	+	0.840256i	$0.317595\pi$
$912$	0	0
$913$	−6451.40	−0.233856
$914$	0	0
$915$	−41102.4	−1.48503
$916$	0	0
$917$	−3392.18	−0.122159
$918$	0	0
$919$	40744.8	1.46251	0.731255	−	0.682104i	$-0.238935\pi$
$919$	40744.8	1.46251	0.731255	+	0.682104i	$0.238935\pi$
$920$	0	0
$921$	−15939.6	−0.570281
$922$	0	0
$923$	2769.07	0.0987488
$924$	0	0
$925$	−13352.8	−0.474634
$926$	0	0
$927$	3464.34	0.122744
$928$	0	0
$929$	−40223.7	−1.42056	−0.710278	−	0.703921i	$-0.751431\pi$
$929$	−40223.7	−1.42056	−0.710278	+	0.703921i	$0.751431\pi$
$930$	0	0
$931$	−9216.22	−0.324435
$932$	0	0
$933$	19988.6	0.701392
$934$	0	0
$935$	2894.42	0.101238
$936$	0	0
$937$	−511.273	−0.0178256	−0.00891279	−	0.999960i	$-0.502837\pi$
$937$	−511.273	−0.0178256	−0.00891279	+	0.999960i	$0.502837\pi$
$938$	0	0
$939$	−4670.11	−0.162304
$940$	0	0
$941$	−24696.7	−0.855569	−0.427785	−	0.903881i	$-0.640706\pi$
$941$	−24696.7	−0.855569	−0.427785	+	0.903881i	$0.640706\pi$
$942$	0	0
$943$	−140.221	−0.00484223
$944$	0	0
$945$	−2196.20	−0.0756005
$946$	0	0
$947$	−11177.7	−0.383555	−0.191778	−	0.981438i	$-0.561425\pi$
$947$	−11177.7	−0.383555	−0.191778	+	0.981438i	$0.561425\pi$
$948$	0	0
$949$	7388.07	0.252715
$950$	0	0
$951$	−1987.99	−0.0677866
$952$	0	0
$953$	31891.5	1.08402	0.542008	−	0.840374i	$-0.317664\pi$
$953$	31891.5	1.08402	0.542008	+	0.840374i	$0.317664\pi$
$954$	0	0
$955$	−34481.4	−1.16837
$956$	0	0
$957$	713.966	0.0241163
$958$	0	0
$959$	−5046.02	−0.169911
$960$	0	0
$961$	45557.7	1.52924
$962$	0	0
$963$	−3024.40	−0.101205
$964$	0	0
$965$	69596.4	2.32165
$966$	0	0
$967$	46487.2	1.54594	0.772971	−	0.634441i	$-0.218770\pi$
$967$	46487.2	1.54594	0.772971	+	0.634441i	$0.218770\pi$
$968$	0	0
$969$	374.983	0.0124316
$970$	0	0
$971$	45959.6	1.51897	0.759483	−	0.650528i	$-0.225452\pi$
$971$	45959.6	1.51897	0.759483	+	0.650528i	$0.225452\pi$
$972$	0	0
$973$	9352.28	0.308140
$974$	0	0
$975$	−6174.85	−0.202824
$976$	0	0
$977$	3859.76	0.126392	0.0631959	−	0.998001i	$-0.479871\pi$
$977$	3859.76	0.126392	0.0631959	+	0.998001i	$0.479871\pi$
$978$	0	0
$979$	−56221.4	−1.83539
$980$	0	0
$981$	−6995.76	−0.227684
$982$	0	0
$983$	12285.9	0.398637	0.199318	−	0.979935i	$-0.436127\pi$
$983$	12285.9	0.398637	0.199318	+	0.979935i	$0.436127\pi$
$984$	0	0
$985$	−31257.4	−1.01111
$986$	0	0
$987$	3469.77	0.111899
$988$	0	0
$989$	−244.247	−0.00785299
$990$	0	0
$991$	−5559.34	−0.178202	−0.0891011	−	0.996023i	$-0.528399\pi$
$991$	−5559.34	−0.178202	−0.0891011	+	0.996023i	$0.528399\pi$
$992$	0	0
$993$	−4136.49	−0.132193
$994$	0	0
$995$	−92840.2	−2.95802
$996$	0	0
$997$	−35491.1	−1.12740	−0.563698	−	0.825981i	$-0.690622\pi$
$997$	−35491.1	−1.12740	−0.563698	+	0.825981i	$0.690622\pi$
$998$	0	0
$999$	2277.05	0.0721148

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2496.4.a.x.1.2		2
4.3	odd	2		2496.4.a.bi.1.2		2
8.3	odd	2		312.4.a.b.1.1	✓	2
8.5	even	2		624.4.a.n.1.1		2
24.5	odd	2		1872.4.a.bd.1.2		2
24.11	even	2		936.4.a.h.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
312.4.a.b.1.1	✓	2	8.3	odd	2
624.4.a.n.1.1		2	8.5	even	2
936.4.a.h.1.2		2	24.11	even	2
1872.4.a.bd.1.2		2	24.5	odd	2
2496.4.a.x.1.2		2	1.1	even	1	trivial
2496.4.a.bi.1.2		2	4.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 2496 = 2^{6} \cdot 3 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2496.a (trivial)

\(f(q)\)	\(=\)	\(q-3.00000 q^{3} +16.8324 q^{5} +4.83240 q^{7} +9.00000 q^{9} -39.6648 q^{11} +13.0000 q^{13} -50.4972 q^{15} -4.33521 q^{17} +28.8324 q^{19} -14.4972 q^{21} -0.670412 q^{23} +158.330 q^{25} -27.0000 q^{27} +6.00000 q^{29} -274.497 q^{31} +118.994 q^{33} +81.3408 q^{35} -84.3352 q^{37} -39.0000 q^{39} +209.156 q^{41} +364.324 q^{43} +151.492 q^{45} -239.341 q^{47} -319.648 q^{49} +13.0056 q^{51} +415.318 q^{53} -667.654 q^{55} -86.4972 q^{57} +78.0000 q^{59} +813.955 q^{61} +43.4916 q^{63} +218.821 q^{65} +858.452 q^{67} +2.01124 q^{69} +213.006 q^{71} +568.313 q^{73} -474.989 q^{75} -191.676 q^{77} -583.352 q^{79} +81.0000 q^{81} +162.648 q^{83} -72.9719 q^{85} -18.0000 q^{87} +1417.41 q^{89} +62.8212 q^{91} +823.492 q^{93} +485.318 q^{95} -798.949 q^{97} -356.983 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 6 q^{3} + 4 q^{5} - 20 q^{7} + 18 q^{9} - 20 q^{11} + 26 q^{13} - 12 q^{15} - 68 q^{17} + 28 q^{19} + 60 q^{21} - 120 q^{23} + 198 q^{25} - 54 q^{27} + 12 q^{29} - 460 q^{31} + 60 q^{33} + 400 q^{35}+ \cdots - 180 q^{99}+O(q^{100}) \) 2 * q - 6 * q^3 + 4 * q^5 - 20 * q^7 + 18 * q^9 - 20 * q^11 + 26 * q^13 - 12 * q^15 - 68 * q^17 + 28 * q^19 + 60 * q^21 - 120 * q^23 + 198 * q^25 - 54 * q^27 + 12 * q^29 - 460 * q^31 + 60 * q^33 + 400 * q^35 - 228 * q^37 - 78 * q^39 + 92 * q^41 + 432 * q^43 + 36 * q^45 - 716 * q^47 - 46 * q^49 + 204 * q^51 + 356 * q^53 - 920 * q^55 - 84 * q^57 + 156 * q^59 + 204 * q^61 - 180 * q^63 + 52 * q^65 + 204 * q^67 + 360 * q^69 + 604 * q^71 + 484 * q^73 - 594 * q^75 - 680 * q^77 - 1760 * q^79 + 162 * q^81 - 268 * q^83 + 744 * q^85 - 36 * q^87 + 76 * q^89 - 260 * q^91 + 1380 * q^93 + 496 * q^95 + 4 * q^97 - 180 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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