LMFDB - Embedded newform 912.4.a.d.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [912,4,Mod(1,912)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(912, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("912.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 912 = 2^{4} \cdot 3 \cdot 19 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	912.a (trivial)

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:	yes
Analytic conductor:	$53.8097419252$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 114)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	912.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-3.00000 q^{3} +12.0000 q^{5} -4.00000 q^{7} +9.00000 q^{9} +O(q^{10})$

$q-3.00000 q^{3} +12.0000 q^{5} -4.00000 q^{7} +9.00000 q^{9} -8.00000 q^{11} -24.0000 q^{13} -36.0000 q^{15} +62.0000 q^{17} -19.0000 q^{19} +12.0000 q^{21} -194.000 q^{23} +19.0000 q^{25} -27.0000 q^{27} +102.000 q^{29} -18.0000 q^{31} +24.0000 q^{33} -48.0000 q^{35} -296.000 q^{37} +72.0000 q^{39} +134.000 q^{41} +60.0000 q^{43} +108.000 q^{45} +226.000 q^{47} -327.000 q^{49} -186.000 q^{51} -362.000 q^{53} -96.0000 q^{55} +57.0000 q^{57} +316.000 q^{59} +134.000 q^{61} -36.0000 q^{63} -288.000 q^{65} +240.000 q^{67} +582.000 q^{69} +800.000 q^{71} -578.000 q^{73} -57.0000 q^{75} +32.0000 q^{77} -1078.00 q^{79} +81.0000 q^{81} -940.000 q^{83} +744.000 q^{85} -306.000 q^{87} +170.000 q^{89} +96.0000 q^{91} +54.0000 q^{93} -228.000 q^{95} +206.000 q^{97} -72.0000 q^{99} +O(q^{100})$

Display 100 coefficients 10 coefficients

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
$5$	12.0000	1.07331	0.536656	−	0.843801i	$-0.319687\pi$
$5$	12.0000	1.07331	0.536656	+	0.843801i	$0.319687\pi$
$6$	0	0
$7$	−4.00000	−0.215980	−0.107990	−	0.994152i	$-0.534441\pi$
$7$	−4.00000	−0.215980	−0.107990	+	0.994152i	$0.534441\pi$
$8$	0	0
$9$	9.00000	0.333333
$10$	0	0
$11$	−8.00000	−0.219281	−0.109640	−	0.993971i	$-0.534970\pi$
$11$	−8.00000	−0.219281	−0.109640	+	0.993971i	$0.534970\pi$
$12$	0	0
$13$	−24.0000	−0.512031	−0.256015	−	0.966673i	$-0.582410\pi$
$13$	−24.0000	−0.512031	−0.256015	+	0.966673i	$0.582410\pi$
$14$	0	0
$15$	−36.0000	−0.619677
$16$	0	0
$17$	62.0000	0.884542	0.442271	−	0.896882i	$-0.354173\pi$
$17$	62.0000	0.884542	0.442271	+	0.896882i	$0.354173\pi$
$18$	0	0
$19$	−19.0000	−0.229416
$19$	−19.0000	−0.229416
$20$	0	0
$21$	12.0000	0.124696
$22$	0	0
$23$	−194.000	−1.75877	−0.879387	−	0.476108i	$-0.842047\pi$
$23$	−194.000	−1.75877	−0.879387	+	0.476108i	$0.842047\pi$
$24$	0	0
$25$	19.0000	0.152000
$26$	0	0
$27$	−27.0000	−0.192450
$28$	0	0
$29$	102.000	0.653135	0.326568	−	0.945174i	$-0.394108\pi$
$29$	102.000	0.653135	0.326568	+	0.945174i	$0.394108\pi$
$30$	0	0
$31$	−18.0000	−0.104287	−0.0521435	−	0.998640i	$-0.516605\pi$
$31$	−18.0000	−0.104287	−0.0521435	+	0.998640i	$0.516605\pi$
$32$	0	0
$33$	24.0000	0.126602
$34$	0	0
$35$	−48.0000	−0.231814
$36$	0	0
$37$	−296.000	−1.31519	−0.657596	−	0.753371i	$-0.728427\pi$
$37$	−296.000	−1.31519	−0.657596	+	0.753371i	$0.728427\pi$
$38$	0	0
$39$	72.0000	0.295621
$40$	0	0
$41$	134.000	0.510422	0.255211	−	0.966885i	$-0.417855\pi$
$41$	134.000	0.510422	0.255211	+	0.966885i	$0.417855\pi$
$42$	0	0
$43$	60.0000	0.212789	0.106394	−	0.994324i	$-0.466069\pi$
$43$	60.0000	0.212789	0.106394	+	0.994324i	$0.466069\pi$
$44$	0	0
$45$	108.000	0.357771
$46$	0	0
$47$	226.000	0.701393	0.350697	−	0.936489i	$-0.385945\pi$
$47$	226.000	0.701393	0.350697	+	0.936489i	$0.385945\pi$
$48$	0	0
$49$	−327.000	−0.953353
$50$	0	0
$51$	−186.000	−0.510690
$52$	0	0
$53$	−362.000	−0.938199	−0.469099	−	0.883145i	$-0.655421\pi$
$53$	−362.000	−0.938199	−0.469099	+	0.883145i	$0.655421\pi$
$54$	0	0
$55$	−96.0000	−0.235357
$56$	0	0
$57$	57.0000	0.132453
$58$	0	0
$59$	316.000	0.697283	0.348641	−	0.937256i	$-0.386643\pi$
$59$	316.000	0.697283	0.348641	+	0.937256i	$0.386643\pi$
$60$	0	0
$61$	134.000	0.281261	0.140631	−	0.990062i	$-0.455087\pi$
$61$	134.000	0.281261	0.140631	+	0.990062i	$0.455087\pi$
$62$	0	0
$63$	−36.0000	−0.0719932
$64$	0	0
$65$	−288.000	−0.549569
$66$	0	0
$67$	240.000	0.437622	0.218811	−	0.975767i	$-0.429782\pi$
$67$	240.000	0.437622	0.218811	+	0.975767i	$0.429782\pi$
$68$	0	0
$69$	582.000	1.01543
$70$	0	0
$71$	800.000	1.33722	0.668609	−	0.743614i	$-0.266890\pi$
$71$	800.000	1.33722	0.668609	+	0.743614i	$0.266890\pi$
$72$	0	0
$73$	−578.000	−0.926709	−0.463355	−	0.886173i	$-0.653354\pi$
$73$	−578.000	−0.926709	−0.463355	+	0.886173i	$0.653354\pi$
$74$	0	0
$75$	−57.0000	−0.0877572
$76$	0	0
$77$	32.0000	0.0473602
$78$	0	0
$79$	−1078.00	−1.53525	−0.767623	−	0.640901i	$-0.778561\pi$
$79$	−1078.00	−1.53525	−0.767623	+	0.640901i	$0.778561\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	−940.000	−1.24311	−0.621557	−	0.783369i	$-0.713499\pi$
$83$	−940.000	−1.24311	−0.621557	+	0.783369i	$0.713499\pi$
$84$	0	0
$85$	744.000	0.949390
$86$	0	0
$87$	−306.000	−0.377088
$88$	0	0
$89$	170.000	0.202472	0.101236	−	0.994862i	$-0.467720\pi$
$89$	170.000	0.202472	0.101236	+	0.994862i	$0.467720\pi$
$90$	0	0
$91$	96.0000	0.110588
$92$	0	0
$93$	54.0000	0.0602101
$94$	0	0
$95$	−228.000	−0.246235
$96$	0	0
$97$	206.000	0.215630	0.107815	−	0.994171i	$-0.465615\pi$
$97$	206.000	0.215630	0.107815	+	0.994171i	$0.465615\pi$
$98$	0	0
$99$	−72.0000	−0.0730937
$100$	0	0
$101$	−884.000	−0.870904	−0.435452	−	0.900212i	$-0.643411\pi$
$101$	−884.000	−0.870904	−0.435452	+	0.900212i	$0.643411\pi$
$102$	0	0
$103$	1146.00	1.09630	0.548149	−	0.836381i	$-0.315333\pi$
$103$	1146.00	1.09630	0.548149	+	0.836381i	$0.315333\pi$
$104$	0	0
$105$	144.000	0.133838
$106$	0	0
$107$	−940.000	−0.849283	−0.424641	−	0.905362i	$-0.639600\pi$
$107$	−940.000	−0.849283	−0.424641	+	0.905362i	$0.639600\pi$
$108$	0	0
$109$	−892.000	−0.783836	−0.391918	−	0.920000i	$-0.628188\pi$
$109$	−892.000	−0.783836	−0.391918	+	0.920000i	$0.628188\pi$
$110$	0	0
$111$	888.000	0.759326
$112$	0	0
$113$	−1858.00	−1.54678	−0.773389	−	0.633932i	$-0.781440\pi$
$113$	−1858.00	−1.54678	−0.773389	+	0.633932i	$0.781440\pi$
$114$	0	0
$115$	−2328.00	−1.88771
$116$	0	0
$117$	−216.000	−0.170677
$118$	0	0
$119$	−248.000	−0.191043
$120$	0	0
$121$	−1267.00	−0.951916
$122$	0	0
$123$	−402.000	−0.294692
$124$	0	0
$125$	−1272.00	−0.910169
$126$	0	0
$127$	−322.000	−0.224983	−0.112492	−	0.993653i	$-0.535883\pi$
$127$	−322.000	−0.224983	−0.112492	+	0.993653i	$0.535883\pi$
$128$	0	0
$129$	−180.000	−0.122854
$130$	0	0
$131$	1500.00	1.00042	0.500212	−	0.865903i	$-0.333255\pi$
$131$	1500.00	1.00042	0.500212	+	0.865903i	$0.333255\pi$
$132$	0	0
$133$	76.0000	0.0495491
$134$	0	0
$135$	−324.000	−0.206559
$136$	0	0
$137$	−1146.00	−0.714667	−0.357334	−	0.933977i	$-0.616314\pi$
$137$	−1146.00	−0.714667	−0.357334	+	0.933977i	$0.616314\pi$
$138$	0	0
$139$	−1364.00	−0.832324	−0.416162	−	0.909291i	$-0.636625\pi$
$139$	−1364.00	−0.832324	−0.416162	+	0.909291i	$0.636625\pi$
$140$	0	0
$141$	−678.000	−0.404950
$142$	0	0
$143$	192.000	0.112279
$144$	0	0
$145$	1224.00	0.701018
$146$	0	0
$147$	981.000	0.550418
$148$	0	0
$149$	1168.00	0.642190	0.321095	−	0.947047i	$-0.395949\pi$
$149$	1168.00	0.642190	0.321095	+	0.947047i	$0.395949\pi$
$150$	0	0
$151$	−1270.00	−0.684445	−0.342222	−	0.939619i	$-0.611180\pi$
$151$	−1270.00	−0.684445	−0.342222	+	0.939619i	$0.611180\pi$
$152$	0	0
$153$	558.000	0.294847
$154$	0	0
$155$	−216.000	−0.111933
$156$	0	0
$157$	−1182.00	−0.600853	−0.300426	−	0.953805i	$-0.597129\pi$
$157$	−1182.00	−0.600853	−0.300426	+	0.953805i	$0.597129\pi$
$158$	0	0
$159$	1086.00	0.541669
$160$	0	0
$161$	776.000	0.379859
$162$	0	0
$163$	−764.000	−0.367123	−0.183562	−	0.983008i	$-0.558763\pi$
$163$	−764.000	−0.367123	−0.183562	+	0.983008i	$0.558763\pi$
$164$	0	0
$165$	288.000	0.135883
$166$	0	0
$167$	−1284.00	−0.594963	−0.297482	−	0.954728i	$-0.596147\pi$
$167$	−1284.00	−0.594963	−0.297482	+	0.954728i	$0.596147\pi$
$168$	0	0
$169$	−1621.00	−0.737824
$170$	0	0
$171$	−171.000	−0.0764719
$172$	0	0
$173$	−4134.00	−1.81678	−0.908388	−	0.418129i	$-0.862686\pi$
$173$	−4134.00	−1.81678	−0.908388	+	0.418129i	$0.862686\pi$
$174$	0	0
$175$	−76.0000	−0.0328289
$176$	0	0
$177$	−948.000	−0.402577
$178$	0	0
$179$	−2660.00	−1.11071	−0.555357	−	0.831612i	$-0.687418\pi$
$179$	−2660.00	−1.11071	−0.555357	+	0.831612i	$0.687418\pi$
$180$	0	0
$181$	2340.00	0.960944	0.480472	−	0.877010i	$-0.340465\pi$
$181$	2340.00	0.960944	0.480472	+	0.877010i	$0.340465\pi$
$182$	0	0
$183$	−402.000	−0.162386
$184$	0	0
$185$	−3552.00	−1.41161
$186$	0	0
$187$	−496.000	−0.193963
$188$	0	0
$189$	108.000	0.0415653
$190$	0	0
$191$	−3818.00	−1.44639	−0.723196	−	0.690643i	$-0.757328\pi$
$191$	−3818.00	−1.44639	−0.723196	+	0.690643i	$0.757328\pi$
$192$	0	0
$193$	−1330.00	−0.496039	−0.248019	−	0.968755i	$-0.579780\pi$
$193$	−1330.00	−0.496039	−0.248019	+	0.968755i	$0.579780\pi$
$194$	0	0
$195$	864.000	0.317294
$196$	0	0
$197$	2432.00	0.879557	0.439779	−	0.898106i	$-0.355057\pi$
$197$	2432.00	0.879557	0.439779	+	0.898106i	$0.355057\pi$
$198$	0	0
$199$	−3788.00	−1.34937	−0.674684	−	0.738107i	$-0.735720\pi$
$199$	−3788.00	−1.34937	−0.674684	+	0.738107i	$0.735720\pi$
$200$	0	0
$201$	−720.000	−0.252661
$202$	0	0
$203$	−408.000	−0.141064
$204$	0	0
$205$	1608.00	0.547842
$206$	0	0
$207$	−1746.00	−0.586258
$208$	0	0
$209$	152.000	0.0503065
$210$	0	0
$211$	1868.00	0.609471	0.304736	−	0.952437i	$-0.401432\pi$
$211$	1868.00	0.609471	0.304736	+	0.952437i	$0.401432\pi$
$212$	0	0
$213$	−2400.00	−0.772044
$214$	0	0
$215$	720.000	0.228389
$216$	0	0
$217$	72.0000	0.0225239
$218$	0	0
$219$	1734.00	0.535036
$220$	0	0
$221$	−1488.00	−0.452913
$222$	0	0
$223$	3990.00	1.19816	0.599081	−	0.800688i	$-0.295533\pi$
$223$	3990.00	1.19816	0.599081	+	0.800688i	$0.295533\pi$
$224$	0	0
$225$	171.000	0.0506667
$226$	0	0
$227$	3596.00	1.05143	0.525716	−	0.850660i	$-0.323798\pi$
$227$	3596.00	1.05143	0.525716	+	0.850660i	$0.323798\pi$
$228$	0	0
$229$	5518.00	1.59231	0.796157	−	0.605091i	$-0.206863\pi$
$229$	5518.00	1.59231	0.796157	+	0.605091i	$0.206863\pi$
$230$	0	0
$231$	−96.0000	−0.0273434
$232$	0	0
$233$	138.000	0.0388012	0.0194006	−	0.999812i	$-0.493824\pi$
$233$	138.000	0.0388012	0.0194006	+	0.999812i	$0.493824\pi$
$234$	0	0
$235$	2712.00	0.752814
$236$	0	0
$237$	3234.00	0.886375
$238$	0	0
$239$	−6870.00	−1.85934	−0.929672	−	0.368388i	$-0.879910\pi$
$239$	−6870.00	−1.85934	−0.929672	+	0.368388i	$0.879910\pi$
$240$	0	0
$241$	5830.00	1.55827	0.779136	−	0.626856i	$-0.215658\pi$
$241$	5830.00	1.55827	0.779136	+	0.626856i	$0.215658\pi$
$242$	0	0
$243$	−243.000	−0.0641500
$244$	0	0
$245$	−3924.00	−1.02325
$246$	0	0
$247$	456.000	0.117468
$248$	0	0
$249$	2820.00	0.717712
$250$	0	0
$251$	−564.000	−0.141830	−0.0709151	−	0.997482i	$-0.522592\pi$
$251$	−564.000	−0.141830	−0.0709151	+	0.997482i	$0.522592\pi$
$252$	0	0
$253$	1552.00	0.385666
$254$	0	0
$255$	−2232.00	−0.548130
$256$	0	0
$257$	1842.00	0.447085	0.223542	−	0.974694i	$-0.428238\pi$
$257$	1842.00	0.447085	0.223542	+	0.974694i	$0.428238\pi$
$258$	0	0
$259$	1184.00	0.284055
$260$	0	0
$261$	918.000	0.217712
$262$	0	0
$263$	−3702.00	−0.867966	−0.433983	−	0.900921i	$-0.642892\pi$
$263$	−3702.00	−0.867966	−0.433983	+	0.900921i	$0.642892\pi$
$264$	0	0
$265$	−4344.00	−1.00698
$266$	0	0
$267$	−510.000	−0.116897
$268$	0	0
$269$	6130.00	1.38942	0.694708	−	0.719292i	$-0.255534\pi$
$269$	6130.00	1.38942	0.694708	+	0.719292i	$0.255534\pi$
$270$	0	0
$271$	7300.00	1.63632	0.818161	−	0.574989i	$-0.194994\pi$
$271$	7300.00	1.63632	0.818161	+	0.574989i	$0.194994\pi$
$272$	0	0
$273$	−288.000	−0.0638482
$274$	0	0
$275$	−152.000	−0.0333307
$276$	0	0
$277$	7822.00	1.69667	0.848337	−	0.529457i	$-0.177604\pi$
$277$	7822.00	1.69667	0.848337	+	0.529457i	$0.177604\pi$
$278$	0	0
$279$	−162.000	−0.0347623
$280$	0	0
$281$	3278.00	0.695904	0.347952	−	0.937512i	$-0.386877\pi$
$281$	3278.00	0.695904	0.347952	+	0.937512i	$0.386877\pi$
$282$	0	0
$283$	5636.00	1.18384	0.591918	−	0.805998i	$-0.298371\pi$
$283$	5636.00	1.18384	0.591918	+	0.805998i	$0.298371\pi$
$284$	0	0
$285$	684.000	0.142164
$286$	0	0
$287$	−536.000	−0.110241
$288$	0	0
$289$	−1069.00	−0.217586
$290$	0	0
$291$	−618.000	−0.124494
$292$	0	0
$293$	−7310.00	−1.45752	−0.728762	−	0.684767i	$-0.759904\pi$
$293$	−7310.00	−1.45752	−0.728762	+	0.684767i	$0.759904\pi$
$294$	0	0
$295$	3792.00	0.748403
$296$	0	0
$297$	216.000	0.0422006
$298$	0	0
$299$	4656.00	0.900547
$300$	0	0
$301$	−240.000	−0.0459580
$302$	0	0
$303$	2652.00	0.502817
$304$	0	0
$305$	1608.00	0.301881
$306$	0	0
$307$	4008.00	0.745110	0.372555	−	0.928010i	$-0.378482\pi$
$307$	4008.00	0.745110	0.372555	+	0.928010i	$0.378482\pi$
$308$	0	0
$309$	−3438.00	−0.632948
$310$	0	0
$311$	−6182.00	−1.12717	−0.563584	−	0.826059i	$-0.690578\pi$
$311$	−6182.00	−1.12717	−0.563584	+	0.826059i	$0.690578\pi$
$312$	0	0
$313$	−9594.00	−1.73254	−0.866270	−	0.499576i	$-0.833489\pi$
$313$	−9594.00	−1.73254	−0.866270	+	0.499576i	$0.833489\pi$
$314$	0	0
$315$	−432.000	−0.0772712
$316$	0	0
$317$	2874.00	0.509211	0.254606	−	0.967045i	$-0.418054\pi$
$317$	2874.00	0.509211	0.254606	+	0.967045i	$0.418054\pi$
$318$	0	0
$319$	−816.000	−0.143220
$320$	0	0
$321$	2820.00	0.490333
$322$	0	0
$323$	−1178.00	−0.202928
$324$	0	0
$325$	−456.000	−0.0778287
$326$	0	0
$327$	2676.00	0.452548
$328$	0	0
$329$	−904.000	−0.151487
$330$	0	0
$331$	6240.00	1.03620	0.518099	−	0.855321i	$-0.326640\pi$
$331$	6240.00	1.03620	0.518099	+	0.855321i	$0.326640\pi$
$332$	0	0
$333$	−2664.00	−0.438397
$334$	0	0
$335$	2880.00	0.469705
$336$	0	0
$337$	−10126.0	−1.63679	−0.818395	−	0.574656i	$-0.805136\pi$
$337$	−10126.0	−1.63679	−0.818395	+	0.574656i	$0.805136\pi$
$338$	0	0
$339$	5574.00	0.893033
$340$	0	0
$341$	144.000	0.0228681
$342$	0	0
$343$	2680.00	0.421885
$344$	0	0
$345$	6984.00	1.08987
$346$	0	0
$347$	3376.00	0.522286	0.261143	−	0.965300i	$-0.415901\pi$
$347$	3376.00	0.522286	0.261143	+	0.965300i	$0.415901\pi$
$348$	0	0
$349$	−9806.00	−1.50402	−0.752010	−	0.659151i	$-0.770916\pi$
$349$	−9806.00	−1.50402	−0.752010	+	0.659151i	$0.770916\pi$
$350$	0	0
$351$	648.000	0.0985404
$352$	0	0
$353$	9402.00	1.41761	0.708807	−	0.705402i	$-0.249234\pi$
$353$	9402.00	1.41761	0.708807	+	0.705402i	$0.249234\pi$
$354$	0	0
$355$	9600.00	1.43525
$356$	0	0
$357$	744.000	0.110299
$358$	0	0
$359$	−11010.0	−1.61862	−0.809311	−	0.587380i	$-0.800159\pi$
$359$	−11010.0	−1.61862	−0.809311	+	0.587380i	$0.800159\pi$
$360$	0	0
$361$	361.000	0.0526316
$362$	0	0
$363$	3801.00	0.549589
$364$	0	0
$365$	−6936.00	−0.994649
$366$	0	0
$367$	−9956.00	−1.41607	−0.708037	−	0.706176i	$-0.750419\pi$
$367$	−9956.00	−1.41607	−0.708037	+	0.706176i	$0.750419\pi$
$368$	0	0
$369$	1206.00	0.170141
$370$	0	0
$371$	1448.00	0.202632
$372$	0	0
$373$	2024.00	0.280962	0.140481	−	0.990083i	$-0.455135\pi$
$373$	2024.00	0.280962	0.140481	+	0.990083i	$0.455135\pi$
$374$	0	0
$375$	3816.00	0.525486
$376$	0	0
$377$	−2448.00	−0.334426
$378$	0	0
$379$	8560.00	1.16015	0.580076	−	0.814562i	$-0.303023\pi$
$379$	8560.00	1.16015	0.580076	+	0.814562i	$0.303023\pi$
$380$	0	0
$381$	966.000	0.129894
$382$	0	0
$383$	3944.00	0.526185	0.263093	−	0.964771i	$-0.415258\pi$
$383$	3944.00	0.526185	0.263093	+	0.964771i	$0.415258\pi$
$384$	0	0
$385$	384.000	0.0508323
$386$	0	0
$387$	540.000	0.0709296
$388$	0	0
$389$	−1964.00	−0.255987	−0.127993	−	0.991775i	$-0.540854\pi$
$389$	−1964.00	−0.255987	−0.127993	+	0.991775i	$0.540854\pi$
$390$	0	0
$391$	−12028.0	−1.55571
$392$	0	0
$393$	−4500.00	−0.577595
$394$	0	0
$395$	−12936.0	−1.64780
$396$	0	0
$397$	7434.00	0.939803	0.469901	−	0.882719i	$-0.344289\pi$
$397$	7434.00	0.939803	0.469901	+	0.882719i	$0.344289\pi$
$398$	0	0
$399$	−228.000	−0.0286072
$400$	0	0
$401$	−7026.00	−0.874967	−0.437483	−	0.899226i	$-0.644130\pi$
$401$	−7026.00	−0.874967	−0.437483	+	0.899226i	$0.644130\pi$
$402$	0	0
$403$	432.000	0.0533981
$404$	0	0
$405$	972.000	0.119257
$406$	0	0
$407$	2368.00	0.288397
$408$	0	0
$409$	8454.00	1.02206	0.511031	−	0.859562i	$-0.329264\pi$
$409$	8454.00	1.02206	0.511031	+	0.859562i	$0.329264\pi$
$410$	0	0
$411$	3438.00	0.412613
$412$	0	0
$413$	−1264.00	−0.150599
$414$	0	0
$415$	−11280.0	−1.33425
$416$	0	0
$417$	4092.00	0.480542
$418$	0	0
$419$	12552.0	1.46350	0.731749	−	0.681575i	$-0.238705\pi$
$419$	12552.0	1.46350	0.731749	+	0.681575i	$0.238705\pi$
$420$	0	0
$421$	13120.0	1.51884	0.759418	−	0.650603i	$-0.225484\pi$
$421$	13120.0	1.51884	0.759418	+	0.650603i	$0.225484\pi$
$422$	0	0
$423$	2034.00	0.233798
$424$	0	0
$425$	1178.00	0.134450
$426$	0	0
$427$	−536.000	−0.0607467
$428$	0	0
$429$	−576.000	−0.0648241
$430$	0	0
$431$	−12388.0	−1.38448	−0.692238	−	0.721669i	$-0.743375\pi$
$431$	−12388.0	−1.38448	−0.692238	+	0.721669i	$0.743375\pi$
$432$	0	0
$433$	7182.00	0.797101	0.398551	−	0.917146i	$-0.369513\pi$
$433$	7182.00	0.797101	0.398551	+	0.917146i	$0.369513\pi$
$434$	0	0
$435$	−3672.00	−0.404733
$436$	0	0
$437$	3686.00	0.403490
$438$	0	0
$439$	−16090.0	−1.74928	−0.874640	−	0.484773i	$-0.838902\pi$
$439$	−16090.0	−1.74928	−0.874640	+	0.484773i	$0.838902\pi$
$440$	0	0
$441$	−2943.00	−0.317784
$442$	0	0
$443$	2544.00	0.272842	0.136421	−	0.990651i	$-0.456440\pi$
$443$	2544.00	0.272842	0.136421	+	0.990651i	$0.456440\pi$
$444$	0	0
$445$	2040.00	0.217315
$446$	0	0
$447$	−3504.00	−0.370768
$448$	0	0
$449$	4338.00	0.455953	0.227976	−	0.973667i	$-0.426789\pi$
$449$	4338.00	0.455953	0.227976	+	0.973667i	$0.426789\pi$
$450$	0	0
$451$	−1072.00	−0.111926
$452$	0	0
$453$	3810.00	0.395164
$454$	0	0
$455$	1152.00	0.118696
$456$	0	0
$457$	10274.0	1.05164	0.525818	−	0.850597i	$-0.323759\pi$
$457$	10274.0	1.05164	0.525818	+	0.850597i	$0.323759\pi$
$458$	0	0
$459$	−1674.00	−0.170230
$460$	0	0
$461$	768.000	0.0775907	0.0387954	−	0.999247i	$-0.487648\pi$
$461$	768.000	0.0775907	0.0387954	+	0.999247i	$0.487648\pi$
$462$	0	0
$463$	11148.0	1.11899	0.559494	−	0.828834i	$-0.310995\pi$
$463$	11148.0	1.11899	0.559494	+	0.828834i	$0.310995\pi$
$464$	0	0
$465$	648.000	0.0646243
$466$	0	0
$467$	−18532.0	−1.83631	−0.918157	−	0.396217i	$-0.870323\pi$
$467$	−18532.0	−1.83631	−0.918157	+	0.396217i	$0.870323\pi$
$468$	0	0
$469$	−960.000	−0.0945174
$470$	0	0
$471$	3546.00	0.346902
$472$	0	0
$473$	−480.000	−0.0466605
$474$	0	0
$475$	−361.000	−0.0348712
$476$	0	0
$477$	−3258.00	−0.312733
$478$	0	0
$479$	6942.00	0.662188	0.331094	−	0.943598i	$-0.392582\pi$
$479$	6942.00	0.662188	0.331094	+	0.943598i	$0.392582\pi$
$480$	0	0
$481$	7104.00	0.673419
$482$	0	0
$483$	−2328.00	−0.219312
$484$	0	0
$485$	2472.00	0.231439
$486$	0	0
$487$	9370.00	0.871859	0.435929	−	0.899981i	$-0.356420\pi$
$487$	9370.00	0.871859	0.435929	+	0.899981i	$0.356420\pi$
$488$	0	0
$489$	2292.00	0.211959
$490$	0	0
$491$	13572.0	1.24745	0.623723	−	0.781646i	$-0.285619\pi$
$491$	13572.0	1.24745	0.623723	+	0.781646i	$0.285619\pi$
$492$	0	0
$493$	6324.00	0.577725
$494$	0	0
$495$	−864.000	−0.0784523
$496$	0	0
$497$	−3200.00	−0.288812
$498$	0	0
$499$	124.000	0.0111243	0.00556213	−	0.999985i	$-0.498230\pi$
$499$	124.000	0.0111243	0.00556213	+	0.999985i	$0.498230\pi$
$500$	0	0
$501$	3852.00	0.343502
$502$	0	0
$503$	−638.000	−0.0565547	−0.0282773	−	0.999600i	$-0.509002\pi$
$503$	−638.000	−0.0565547	−0.0282773	+	0.999600i	$0.509002\pi$
$504$	0	0
$505$	−10608.0	−0.934752
$506$	0	0
$507$	4863.00	0.425983
$508$	0	0
$509$	−9758.00	−0.849736	−0.424868	−	0.905255i	$-0.639680\pi$
$509$	−9758.00	−0.849736	−0.424868	+	0.905255i	$0.639680\pi$
$510$	0	0
$511$	2312.00	0.200150
$512$	0	0
$513$	513.000	0.0441511
$514$	0	0
$515$	13752.0	1.17667
$516$	0	0
$517$	−1808.00	−0.153802
$518$	0	0
$519$	12402.0	1.04892
$520$	0	0
$521$	−4750.00	−0.399427	−0.199713	−	0.979854i	$-0.564001\pi$
$521$	−4750.00	−0.399427	−0.199713	+	0.979854i	$0.564001\pi$
$522$	0	0
$523$	18080.0	1.51163	0.755816	−	0.654784i	$-0.227241\pi$
$523$	18080.0	1.51163	0.755816	+	0.654784i	$0.227241\pi$
$524$	0	0
$525$	228.000	0.0189538
$526$	0	0
$527$	−1116.00	−0.0922462
$528$	0	0
$529$	25469.0	2.09329
$530$	0	0
$531$	2844.00	0.232428
$532$	0	0
$533$	−3216.00	−0.261352
$534$	0	0
$535$	−11280.0	−0.911546
$536$	0	0
$537$	7980.00	0.641271
$538$	0	0
$539$	2616.00	0.209052
$540$	0	0
$541$	−3478.00	−0.276397	−0.138199	−	0.990405i	$-0.544131\pi$
$541$	−3478.00	−0.276397	−0.138199	+	0.990405i	$0.544131\pi$
$542$	0	0
$543$	−7020.00	−0.554801
$544$	0	0
$545$	−10704.0	−0.841301
$546$	0	0
$547$	−13832.0	−1.08119	−0.540597	−	0.841281i	$-0.681802\pi$
$547$	−13832.0	−1.08119	−0.540597	+	0.841281i	$0.681802\pi$
$548$	0	0
$549$	1206.00	0.0937538
$550$	0	0
$551$	−1938.00	−0.149840
$552$	0	0
$553$	4312.00	0.331582
$554$	0	0
$555$	10656.0	0.814995
$556$	0	0
$557$	−17296.0	−1.31572	−0.657859	−	0.753141i	$-0.728538\pi$
$557$	−17296.0	−1.31572	−0.657859	+	0.753141i	$0.728538\pi$
$558$	0	0
$559$	−1440.00	−0.108954
$560$	0	0
$561$	1488.00	0.111985
$562$	0	0
$563$	2196.00	0.164388	0.0821939	−	0.996616i	$-0.473807\pi$
$563$	2196.00	0.164388	0.0821939	+	0.996616i	$0.473807\pi$
$564$	0	0
$565$	−22296.0	−1.66018
$566$	0	0
$567$	−324.000	−0.0239977
$568$	0	0
$569$	25386.0	1.87036	0.935181	−	0.354169i	$-0.115236\pi$
$569$	25386.0	1.87036	0.935181	+	0.354169i	$0.115236\pi$
$570$	0	0
$571$	−1196.00	−0.0876551	−0.0438275	−	0.999039i	$-0.513955\pi$
$571$	−1196.00	−0.0876551	−0.0438275	+	0.999039i	$0.513955\pi$
$572$	0	0
$573$	11454.0	0.835075
$574$	0	0
$575$	−3686.00	−0.267334
$576$	0	0
$577$	−7934.00	−0.572438	−0.286219	−	0.958164i	$-0.592398\pi$
$577$	−7934.00	−0.572438	−0.286219	+	0.958164i	$0.592398\pi$
$578$	0	0
$579$	3990.00	0.286388
$580$	0	0
$581$	3760.00	0.268487
$582$	0	0
$583$	2896.00	0.205729
$584$	0	0
$585$	−2592.00	−0.183190
$586$	0	0
$587$	22120.0	1.55535	0.777675	−	0.628667i	$-0.216399\pi$
$587$	22120.0	1.55535	0.777675	+	0.628667i	$0.216399\pi$
$588$	0	0
$589$	342.000	0.0239251
$590$	0	0
$591$	−7296.00	−0.507813
$592$	0	0
$593$	9722.00	0.673246	0.336623	−	0.941640i	$-0.390715\pi$
$593$	9722.00	0.673246	0.336623	+	0.941640i	$0.390715\pi$
$594$	0	0
$595$	−2976.00	−0.205049
$596$	0	0
$597$	11364.0	0.779058
$598$	0	0
$599$	9980.00	0.680754	0.340377	−	0.940289i	$-0.389445\pi$
$599$	9980.00	0.680754	0.340377	+	0.940289i	$0.389445\pi$
$600$	0	0
$601$	6138.00	0.416596	0.208298	−	0.978065i	$-0.433208\pi$
$601$	6138.00	0.416596	0.208298	+	0.978065i	$0.433208\pi$
$602$	0	0
$603$	2160.00	0.145874
$604$	0	0
$605$	−15204.0	−1.02170
$606$	0	0
$607$	−10286.0	−0.687802	−0.343901	−	0.939006i	$-0.611748\pi$
$607$	−10286.0	−0.687802	−0.343901	+	0.939006i	$0.611748\pi$
$608$	0	0
$609$	1224.00	0.0814433
$610$	0	0
$611$	−5424.00	−0.359135
$612$	0	0
$613$	4178.00	0.275282	0.137641	−	0.990482i	$-0.456048\pi$
$613$	4178.00	0.275282	0.137641	+	0.990482i	$0.456048\pi$
$614$	0	0
$615$	−4824.00	−0.316297
$616$	0	0
$617$	−19086.0	−1.24534	−0.622669	−	0.782485i	$-0.713952\pi$
$617$	−19086.0	−1.24534	−0.622669	+	0.782485i	$0.713952\pi$
$618$	0	0
$619$	11084.0	0.719715	0.359857	−	0.933007i	$-0.382825\pi$
$619$	11084.0	0.719715	0.359857	+	0.933007i	$0.382825\pi$
$620$	0	0
$621$	5238.00	0.338476
$622$	0	0
$623$	−680.000	−0.0437297
$624$	0	0
$625$	−17639.0	−1.12890
$626$	0	0
$627$	−456.000	−0.0290445
$628$	0	0
$629$	−18352.0	−1.16334
$630$	0	0
$631$	9160.00	0.577898	0.288949	−	0.957344i	$-0.406694\pi$
$631$	9160.00	0.577898	0.288949	+	0.957344i	$0.406694\pi$
$632$	0	0
$633$	−5604.00	−0.351878
$634$	0	0
$635$	−3864.00	−0.241477
$636$	0	0
$637$	7848.00	0.488146
$638$	0	0
$639$	7200.00	0.445740
$640$	0	0
$641$	13082.0	0.806097	0.403048	−	0.915179i	$-0.367951\pi$
$641$	13082.0	0.806097	0.403048	+	0.915179i	$0.367951\pi$
$642$	0	0
$643$	−7852.00	−0.481575	−0.240787	−	0.970578i	$-0.577406\pi$
$643$	−7852.00	−0.481575	−0.240787	+	0.970578i	$0.577406\pi$
$644$	0	0
$645$	−2160.00	−0.131860
$646$	0	0
$647$	−10530.0	−0.639841	−0.319920	−	0.947444i	$-0.603656\pi$
$647$	−10530.0	−0.639841	−0.319920	+	0.947444i	$0.603656\pi$
$648$	0	0
$649$	−2528.00	−0.152901
$650$	0	0
$651$	−216.000	−0.0130042
$652$	0	0
$653$	−17880.0	−1.07151	−0.535757	−	0.844372i	$-0.679974\pi$
$653$	−17880.0	−1.07151	−0.535757	+	0.844372i	$0.679974\pi$
$654$	0	0
$655$	18000.0	1.07377
$656$	0	0
$657$	−5202.00	−0.308903
$658$	0	0
$659$	−27404.0	−1.61989	−0.809946	−	0.586505i	$-0.800503\pi$
$659$	−27404.0	−1.61989	−0.809946	+	0.586505i	$0.800503\pi$
$660$	0	0
$661$	22936.0	1.34963	0.674816	−	0.737986i	$-0.264223\pi$
$661$	22936.0	1.34963	0.674816	+	0.737986i	$0.264223\pi$
$662$	0	0
$663$	4464.00	0.261489
$664$	0	0
$665$	912.000	0.0531817
$666$	0	0
$667$	−19788.0	−1.14872
$668$	0	0
$669$	−11970.0	−0.691759
$670$	0	0
$671$	−1072.00	−0.0616753
$672$	0	0
$673$	−25762.0	−1.47556	−0.737780	−	0.675041i	$-0.764126\pi$
$673$	−25762.0	−1.47556	−0.737780	+	0.675041i	$0.764126\pi$
$674$	0	0
$675$	−513.000	−0.0292524
$676$	0	0
$677$	11214.0	0.636616	0.318308	−	0.947987i	$-0.396885\pi$
$677$	11214.0	0.636616	0.318308	+	0.947987i	$0.396885\pi$
$678$	0	0
$679$	−824.000	−0.0465717
$680$	0	0
$681$	−10788.0	−0.607044
$682$	0	0
$683$	−15516.0	−0.869258	−0.434629	−	0.900610i	$-0.643121\pi$
$683$	−15516.0	−0.869258	−0.434629	+	0.900610i	$0.643121\pi$
$684$	0	0
$685$	−13752.0	−0.767061
$686$	0	0
$687$	−16554.0	−0.919322
$688$	0	0
$689$	8688.00	0.480387
$690$	0	0
$691$	13028.0	0.717234	0.358617	−	0.933485i	$-0.383248\pi$
$691$	13028.0	0.717234	0.358617	+	0.933485i	$0.383248\pi$
$692$	0	0
$693$	288.000	0.0157867
$694$	0	0
$695$	−16368.0	−0.893343
$696$	0	0
$697$	8308.00	0.451489
$698$	0	0
$699$	−414.000	−0.0224019
$700$	0	0
$701$	−14404.0	−0.776079	−0.388040	−	0.921643i	$-0.626848\pi$
$701$	−14404.0	−0.776079	−0.388040	+	0.921643i	$0.626848\pi$
$702$	0	0
$703$	5624.00	0.301726
$704$	0	0
$705$	−8136.00	−0.434638
$706$	0	0
$707$	3536.00	0.188098
$708$	0	0
$709$	13070.0	0.692319	0.346159	−	0.938176i	$-0.387486\pi$
$709$	13070.0	0.692319	0.346159	+	0.938176i	$0.387486\pi$
$710$	0	0
$711$	−9702.00	−0.511749
$712$	0	0
$713$	3492.00	0.183417
$714$	0	0
$715$	2304.00	0.120510
$716$	0	0
$717$	20610.0	1.07349
$718$	0	0
$719$	7450.00	0.386423	0.193211	−	0.981157i	$-0.438110\pi$
$719$	7450.00	0.386423	0.193211	+	0.981157i	$0.438110\pi$
$720$	0	0
$721$	−4584.00	−0.236778
$722$	0	0
$723$	−17490.0	−0.899668
$724$	0	0
$725$	1938.00	0.0992766
$726$	0	0
$727$	−23792.0	−1.21375	−0.606875	−	0.794797i	$-0.707577\pi$
$727$	−23792.0	−1.21375	−0.606875	+	0.794797i	$0.707577\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	0	0
$731$	3720.00	0.188220
$732$	0	0
$733$	−29390.0	−1.48096	−0.740481	−	0.672078i	$-0.765402\pi$
$733$	−29390.0	−1.48096	−0.740481	+	0.672078i	$0.765402\pi$
$734$	0	0
$735$	11772.0	0.590771
$736$	0	0
$737$	−1920.00	−0.0959622
$738$	0	0
$739$	22580.0	1.12398	0.561988	−	0.827145i	$-0.310037\pi$
$739$	22580.0	1.12398	0.561988	+	0.827145i	$0.310037\pi$
$740$	0	0
$741$	−1368.00	−0.0678202
$742$	0	0
$743$	−21732.0	−1.07304	−0.536521	−	0.843887i	$-0.680262\pi$
$743$	−21732.0	−1.07304	−0.536521	+	0.843887i	$0.680262\pi$
$744$	0	0
$745$	14016.0	0.689270
$746$	0	0
$747$	−8460.00	−0.414371
$748$	0	0
$749$	3760.00	0.183428
$750$	0	0
$751$	19754.0	0.959832	0.479916	−	0.877314i	$-0.340667\pi$
$751$	19754.0	0.959832	0.479916	+	0.877314i	$0.340667\pi$
$752$	0	0
$753$	1692.00	0.0818857
$754$	0	0
$755$	−15240.0	−0.734623
$756$	0	0
$757$	−5662.00	−0.271848	−0.135924	−	0.990719i	$-0.543400\pi$
$757$	−5662.00	−0.271848	−0.135924	+	0.990719i	$0.543400\pi$
$758$	0	0
$759$	−4656.00	−0.222664
$760$	0	0
$761$	−6818.00	−0.324773	−0.162387	−	0.986727i	$-0.551919\pi$
$761$	−6818.00	−0.324773	−0.162387	+	0.986727i	$0.551919\pi$
$762$	0	0
$763$	3568.00	0.169293
$764$	0	0
$765$	6696.00	0.316463
$766$	0	0
$767$	−7584.00	−0.357030
$768$	0	0
$769$	−7986.00	−0.374490	−0.187245	−	0.982313i	$-0.559956\pi$
$769$	−7986.00	−0.374490	−0.187245	+	0.982313i	$0.559956\pi$
$770$	0	0
$771$	−5526.00	−0.258125
$772$	0	0
$773$	21666.0	1.00811	0.504057	−	0.863670i	$-0.331840\pi$
$773$	21666.0	1.00811	0.504057	+	0.863670i	$0.331840\pi$
$774$	0	0
$775$	−342.000	−0.0158516
$776$	0	0
$777$	−3552.00	−0.163999
$778$	0	0
$779$	−2546.00	−0.117099
$780$	0	0
$781$	−6400.00	−0.293227
$782$	0	0
$783$	−2754.00	−0.125696
$784$	0	0
$785$	−14184.0	−0.644903
$786$	0	0
$787$	10628.0	0.481382	0.240691	−	0.970602i	$-0.422626\pi$
$787$	10628.0	0.481382	0.240691	+	0.970602i	$0.422626\pi$
$788$	0	0
$789$	11106.0	0.501120
$790$	0	0
$791$	7432.00	0.334073
$792$	0	0
$793$	−3216.00	−0.144015
$794$	0	0
$795$	13032.0	0.581380
$796$	0	0
$797$	1450.00	0.0644437	0.0322219	−	0.999481i	$-0.489742\pi$
$797$	1450.00	0.0644437	0.0322219	+	0.999481i	$0.489742\pi$
$798$	0	0
$799$	14012.0	0.620412
$800$	0	0
$801$	1530.00	0.0674905
$802$	0	0
$803$	4624.00	0.203210
$804$	0	0
$805$	9312.00	0.407708
$806$	0	0
$807$	−18390.0	−0.802179
$808$	0	0
$809$	−32850.0	−1.42762	−0.713810	−	0.700339i	$-0.753032\pi$
$809$	−32850.0	−1.42762	−0.713810	+	0.700339i	$0.753032\pi$
$810$	0	0
$811$	2744.00	0.118810	0.0594050	−	0.998234i	$-0.481080\pi$
$811$	2744.00	0.118810	0.0594050	+	0.998234i	$0.481080\pi$
$812$	0	0
$813$	−21900.0	−0.944731
$814$	0	0
$815$	−9168.00	−0.394038
$816$	0	0
$817$	−1140.00	−0.0488171
$818$	0	0
$819$	864.000	0.0368628
$820$	0	0
$821$	−13884.0	−0.590201	−0.295100	−	0.955466i	$-0.595353\pi$
$821$	−13884.0	−0.590201	−0.295100	+	0.955466i	$0.595353\pi$
$822$	0	0
$823$	−20968.0	−0.888091	−0.444045	−	0.896004i	$-0.646457\pi$
$823$	−20968.0	−0.888091	−0.444045	+	0.896004i	$0.646457\pi$
$824$	0	0
$825$	456.000	0.0192435
$826$	0	0
$827$	12516.0	0.526268	0.263134	−	0.964759i	$-0.415244\pi$
$827$	12516.0	0.526268	0.263134	+	0.964759i	$0.415244\pi$
$828$	0	0
$829$	22484.0	0.941980	0.470990	−	0.882138i	$-0.343897\pi$
$829$	22484.0	0.941980	0.470990	+	0.882138i	$0.343897\pi$
$830$	0	0
$831$	−23466.0	−0.979575
$832$	0	0
$833$	−20274.0	−0.843280
$834$	0	0
$835$	−15408.0	−0.638582
$836$	0	0
$837$	486.000	0.0200700
$838$	0	0
$839$	−8436.00	−0.347131	−0.173566	−	0.984822i	$-0.555529\pi$
$839$	−8436.00	−0.347131	−0.173566	+	0.984822i	$0.555529\pi$
$840$	0	0
$841$	−13985.0	−0.573414
$842$	0	0
$843$	−9834.00	−0.401780
$844$	0	0
$845$	−19452.0	−0.791916
$846$	0	0
$847$	5068.00	0.205594
$848$	0	0
$849$	−16908.0	−0.683488
$850$	0	0
$851$	57424.0	2.31312
$852$	0	0
$853$	−4818.00	−0.193394	−0.0966971	−	0.995314i	$-0.530828\pi$
$853$	−4818.00	−0.193394	−0.0966971	+	0.995314i	$0.530828\pi$
$854$	0	0
$855$	−2052.00	−0.0820783
$856$	0	0
$857$	29346.0	1.16971	0.584854	−	0.811138i	$-0.301152\pi$
$857$	29346.0	1.16971	0.584854	+	0.811138i	$0.301152\pi$
$858$	0	0
$859$	−45532.0	−1.80854	−0.904268	−	0.426966i	$-0.859582\pi$
$859$	−45532.0	−1.80854	−0.904268	+	0.426966i	$0.859582\pi$
$860$	0	0
$861$	1608.00	0.0636475
$862$	0	0
$863$	−18124.0	−0.714888	−0.357444	−	0.933935i	$-0.616352\pi$
$863$	−18124.0	−0.714888	−0.357444	+	0.933935i	$0.616352\pi$
$864$	0	0
$865$	−49608.0	−1.94997
$866$	0	0
$867$	3207.00	0.125623
$868$	0	0
$869$	8624.00	0.336650
$870$	0	0
$871$	−5760.00	−0.224076
$872$	0	0
$873$	1854.00	0.0718767
$874$	0	0
$875$	5088.00	0.196578
$876$	0	0
$877$	−2824.00	−0.108734	−0.0543670	−	0.998521i	$-0.517314\pi$
$877$	−2824.00	−0.108734	−0.0543670	+	0.998521i	$0.517314\pi$
$878$	0	0
$879$	21930.0	0.841502
$880$	0	0
$881$	45906.0	1.75552	0.877760	−	0.479101i	$-0.159037\pi$
$881$	45906.0	1.75552	0.877760	+	0.479101i	$0.159037\pi$
$882$	0	0
$883$	32292.0	1.23070	0.615352	−	0.788252i	$-0.289014\pi$
$883$	32292.0	1.23070	0.615352	+	0.788252i	$0.289014\pi$
$884$	0	0
$885$	−11376.0	−0.432090
$886$	0	0
$887$	−2364.00	−0.0894874	−0.0447437	−	0.998998i	$-0.514247\pi$
$887$	−2364.00	−0.0894874	−0.0447437	+	0.998998i	$0.514247\pi$
$888$	0	0
$889$	1288.00	0.0485918
$890$	0	0
$891$	−648.000	−0.0243646
$892$	0	0
$893$	−4294.00	−0.160911
$894$	0	0
$895$	−31920.0	−1.19214
$896$	0	0
$897$	−13968.0	−0.519931
$898$	0	0
$899$	−1836.00	−0.0681135
$900$	0	0
$901$	−22444.0	−0.829876
$902$	0	0
$903$	720.000	0.0265339
$904$	0	0
$905$	28080.0	1.03139
$906$	0	0
$907$	4304.00	0.157566	0.0787828	−	0.996892i	$-0.474897\pi$
$907$	4304.00	0.157566	0.0787828	+	0.996892i	$0.474897\pi$
$908$	0	0
$909$	−7956.00	−0.290301
$910$	0	0
$911$	15060.0	0.547706	0.273853	−	0.961772i	$-0.411702\pi$
$911$	15060.0	0.547706	0.273853	+	0.961772i	$0.411702\pi$
$912$	0	0
$913$	7520.00	0.272591
$914$	0	0
$915$	−4824.00	−0.174291
$916$	0	0
$917$	−6000.00	−0.216071
$918$	0	0
$919$	−39160.0	−1.40562	−0.702812	−	0.711375i	$-0.748073\pi$
$919$	−39160.0	−1.40562	−0.702812	+	0.711375i	$0.748073\pi$
$920$	0	0
$921$	−12024.0	−0.430189
$922$	0	0
$923$	−19200.0	−0.684697
$924$	0	0
$925$	−5624.00	−0.199909
$926$	0	0
$927$	10314.0	0.365433
$928$	0	0
$929$	−33810.0	−1.19405	−0.597024	−	0.802224i	$-0.703650\pi$
$929$	−33810.0	−1.19405	−0.597024	+	0.802224i	$0.703650\pi$
$930$	0	0
$931$	6213.00	0.218714
$932$	0	0
$933$	18546.0	0.650770
$934$	0	0
$935$	−5952.00	−0.208183
$936$	0	0
$937$	54614.0	1.90412	0.952061	−	0.305909i	$-0.0989605\pi$
$937$	54614.0	1.90412	0.952061	+	0.305909i	$0.0989605\pi$
$938$	0	0
$939$	28782.0	1.00028
$940$	0	0
$941$	11594.0	0.401651	0.200826	−	0.979627i	$-0.435638\pi$
$941$	11594.0	0.401651	0.200826	+	0.979627i	$0.435638\pi$
$942$	0	0
$943$	−25996.0	−0.897716
$944$	0	0
$945$	1296.00	0.0446126
$946$	0	0
$947$	−9452.00	−0.324339	−0.162169	−	0.986763i	$-0.551849\pi$
$947$	−9452.00	−0.324339	−0.162169	+	0.986763i	$0.551849\pi$
$948$	0	0
$949$	13872.0	0.474504
$950$	0	0
$951$	−8622.00	−0.293993
$952$	0	0
$953$	36966.0	1.25650	0.628251	−	0.778011i	$-0.283771\pi$
$953$	36966.0	1.25650	0.628251	+	0.778011i	$0.283771\pi$
$954$	0	0
$955$	−45816.0	−1.55243
$956$	0	0
$957$	2448.00	0.0826882
$958$	0	0
$959$	4584.00	0.154354
$960$	0	0
$961$	−29467.0	−0.989124
$962$	0	0
$963$	−8460.00	−0.283094
$964$	0	0
$965$	−15960.0	−0.532405
$966$	0	0
$967$	324.000	0.0107747	0.00538735	−	0.999985i	$-0.498285\pi$
$967$	324.000	0.0107747	0.00538735	+	0.999985i	$0.498285\pi$
$968$	0	0
$969$	3534.00	0.117160
$970$	0	0
$971$	−9084.00	−0.300226	−0.150113	−	0.988669i	$-0.547964\pi$
$971$	−9084.00	−0.300226	−0.150113	+	0.988669i	$0.547964\pi$
$972$	0	0
$973$	5456.00	0.179765
$974$	0	0
$975$	1368.00	0.0449344
$976$	0	0
$977$	38142.0	1.24900	0.624499	−	0.781026i	$-0.285303\pi$
$977$	38142.0	1.24900	0.624499	+	0.781026i	$0.285303\pi$
$978$	0	0
$979$	−1360.00	−0.0443981
$980$	0	0
$981$	−8028.00	−0.261279
$982$	0	0
$983$	−19248.0	−0.624533	−0.312266	−	0.949995i	$-0.601088\pi$
$983$	−19248.0	−0.624533	−0.312266	+	0.949995i	$0.601088\pi$
$984$	0	0
$985$	29184.0	0.944040
$986$	0	0
$987$	2712.00	0.0874609
$988$	0	0
$989$	−11640.0	−0.374247
$990$	0	0
$991$	34070.0	1.09210	0.546049	−	0.837753i	$-0.316131\pi$
$991$	34070.0	1.09210	0.546049	+	0.837753i	$0.316131\pi$
$992$	0	0
$993$	−18720.0	−0.598249
$994$	0	0
$995$	−45456.0	−1.44829
$996$	0	0
$997$	26282.0	0.834864	0.417432	−	0.908708i	$-0.362930\pi$
$997$	26282.0	0.834864	0.417432	+	0.908708i	$0.362930\pi$
$998$	0	0
$999$	7992.00	0.253109

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	912.4.a.d.1.1		1
4.3	odd	2		114.4.a.c.1.1	✓	1
12.11	even	2		342.4.a.b.1.1		1
76.75	even	2		2166.4.a.g.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
114.4.a.c.1.1	✓	1	4.3	odd	2
342.4.a.b.1.1		1	12.11	even	2
912.4.a.d.1.1		1	1.1	even	1	trivial
2166.4.a.g.1.1		1	76.75	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 912 = 2^{4} \cdot 3 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	912.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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