LMFDB - Embedded newform 8400.2.a.ch.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8400,2,Mod(1,8400)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8400, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("8400.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8400 = 2^{4} \cdot 3 \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8400.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:	$67.0743376979$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 105)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	8400.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+1.00000 q^{3} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})$

$q+1.00000 q^{3} -1.00000 q^{7} +1.00000 q^{9} +6.00000 q^{11} +2.00000 q^{13} -4.00000 q^{17} +6.00000 q^{19} -1.00000 q^{21} +1.00000 q^{27} -2.00000 q^{29} +10.0000 q^{31} +6.00000 q^{33} +4.00000 q^{37} +2.00000 q^{39} +2.00000 q^{41} -4.00000 q^{43} +1.00000 q^{49} -4.00000 q^{51} -6.00000 q^{53} +6.00000 q^{57} +8.00000 q^{59} -2.00000 q^{61} -1.00000 q^{63} -16.0000 q^{67} -10.0000 q^{71} +6.00000 q^{73} -6.00000 q^{77} -4.00000 q^{79} +1.00000 q^{81} +8.00000 q^{83} -2.00000 q^{87} +6.00000 q^{89} -2.00000 q^{91} +10.0000 q^{93} +2.00000 q^{97} +6.00000 q^{99} +O(q^{100})$

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
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	6.00000	1.80907	0.904534	−	0.426401i	$-0.140219\pi$
$11$	6.00000	1.80907	0.904534	+	0.426401i	$0.140219\pi$
$12$	0	0
$13$	2.00000	0.554700	0.277350	−	0.960769i	$-0.410544\pi$
$13$	2.00000	0.554700	0.277350	+	0.960769i	$0.410544\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−4.00000	−0.970143	−0.485071	−	0.874475i	$-0.661206\pi$
$17$	−4.00000	−0.970143	−0.485071	+	0.874475i	$0.661206\pi$
$18$	0	0
$19$	6.00000	1.37649	0.688247	−	0.725476i	$-0.258380\pi$
$19$	6.00000	1.37649	0.688247	+	0.725476i	$0.258380\pi$
$20$	0	0
$21$	−1.00000	−0.218218
$22$	0	0
$23$	0	0		−	1.00000i	$-0.5\pi$
$23$	0	0			1.00000i	$0.5\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	−2.00000	−0.371391	−0.185695	−	0.982607i	$-0.559454\pi$
$29$	−2.00000	−0.371391	−0.185695	+	0.982607i	$0.559454\pi$
$30$	0	0
$31$	10.0000	1.79605	0.898027	−	0.439941i	$-0.145001\pi$
$31$	10.0000	1.79605	0.898027	+	0.439941i	$0.145001\pi$
$32$	0	0
$33$	6.00000	1.04447
$34$	0	0
$35$	0	0
$36$	0	0
$37$	4.00000	0.657596	0.328798	−	0.944400i	$-0.393356\pi$
$37$	4.00000	0.657596	0.328798	+	0.944400i	$0.393356\pi$
$38$	0	0
$39$	2.00000	0.320256
$40$	0	0
$41$	2.00000	0.312348	0.156174	−	0.987730i	$-0.450084\pi$
$41$	2.00000	0.312348	0.156174	+	0.987730i	$0.450084\pi$
$42$	0	0
$43$	−4.00000	−0.609994	−0.304997	−	0.952353i	$-0.598656\pi$
$43$	−4.00000	−0.609994	−0.304997	+	0.952353i	$0.598656\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	0	0		−	1.00000i	$-0.5\pi$
$47$	0	0			1.00000i	$0.5\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−4.00000	−0.560112
$52$	0	0
$53$	−6.00000	−0.824163	−0.412082	−	0.911147i	$-0.635198\pi$
$53$	−6.00000	−0.824163	−0.412082	+	0.911147i	$0.635198\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	6.00000	0.794719
$58$	0	0
$59$	8.00000	1.04151	0.520756	−	0.853706i	$-0.325650\pi$
$59$	8.00000	1.04151	0.520756	+	0.853706i	$0.325650\pi$
$60$	0	0
$61$	−2.00000	−0.256074	−0.128037	−	0.991769i	$-0.540868\pi$
$61$	−2.00000	−0.256074	−0.128037	+	0.991769i	$0.540868\pi$
$62$	0	0
$63$	−1.00000	−0.125988
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−16.0000	−1.95471	−0.977356	−	0.211604i	$-0.932131\pi$
$67$	−16.0000	−1.95471	−0.977356	+	0.211604i	$0.932131\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−10.0000	−1.18678	−0.593391	−	0.804914i	$-0.702211\pi$
$71$	−10.0000	−1.18678	−0.593391	+	0.804914i	$0.702211\pi$
$72$	0	0
$73$	6.00000	0.702247	0.351123	−	0.936329i	$-0.385800\pi$
$73$	6.00000	0.702247	0.351123	+	0.936329i	$0.385800\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−6.00000	−0.683763
$78$	0	0
$79$	−4.00000	−0.450035	−0.225018	−	0.974355i	$-0.572244\pi$
$79$	−4.00000	−0.450035	−0.225018	+	0.974355i	$0.572244\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	8.00000	0.878114	0.439057	−	0.898459i	$-0.355313\pi$
$83$	8.00000	0.878114	0.439057	+	0.898459i	$0.355313\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−2.00000	−0.214423
$88$	0	0
$89$	6.00000	0.635999	0.317999	−	0.948091i	$-0.396989\pi$
$89$	6.00000	0.635999	0.317999	+	0.948091i	$0.396989\pi$
$90$	0	0
$91$	−2.00000	−0.209657
$92$	0	0
$93$	10.0000	1.03695
$94$	0	0
$95$	0	0
$96$	0	0
$97$	2.00000	0.203069	0.101535	−	0.994832i	$-0.467625\pi$
$97$	2.00000	0.203069	0.101535	+	0.994832i	$0.467625\pi$
$98$	0	0
$99$	6.00000	0.603023
$100$	0	0
$101$	−6.00000	−0.597022	−0.298511	−	0.954406i	$-0.596490\pi$
$101$	−6.00000	−0.597022	−0.298511	+	0.954406i	$0.596490\pi$
$102$	0	0
$103$	8.00000	0.788263	0.394132	−	0.919054i	$-0.371045\pi$
$103$	8.00000	0.788263	0.394132	+	0.919054i	$0.371045\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	4.00000	0.386695	0.193347	−	0.981130i	$-0.438066\pi$
$107$	4.00000	0.386695	0.193347	+	0.981130i	$0.438066\pi$
$108$	0	0
$109$	2.00000	0.191565	0.0957826	−	0.995402i	$-0.469465\pi$
$109$	2.00000	0.191565	0.0957826	+	0.995402i	$0.469465\pi$
$110$	0	0
$111$	4.00000	0.379663
$112$	0	0
$113$	−6.00000	−0.564433	−0.282216	−	0.959351i	$-0.591070\pi$
$113$	−6.00000	−0.564433	−0.282216	+	0.959351i	$0.591070\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	2.00000	0.184900
$118$	0	0
$119$	4.00000	0.366679
$120$	0	0
$121$	25.0000	2.27273
$122$	0	0
$123$	2.00000	0.180334
$124$	0	0
$125$	0	0
$126$	0	0
$127$	20.0000	1.77471	0.887357	−	0.461084i	$-0.152539\pi$
$127$	20.0000	1.77471	0.887357	+	0.461084i	$0.152539\pi$
$128$	0	0
$129$	−4.00000	−0.352180
$130$	0	0
$131$	−4.00000	−0.349482	−0.174741	−	0.984614i	$-0.555909\pi$
$131$	−4.00000	−0.349482	−0.174741	+	0.984614i	$0.555909\pi$
$132$	0	0
$133$	−6.00000	−0.520266
$134$	0	0
$135$	0	0
$136$	0	0
$137$	6.00000	0.512615	0.256307	−	0.966595i	$-0.417494\pi$
$137$	6.00000	0.512615	0.256307	+	0.966595i	$0.417494\pi$
$138$	0	0
$139$	−2.00000	−0.169638	−0.0848189	−	0.996396i	$-0.527031\pi$
$139$	−2.00000	−0.169638	−0.0848189	+	0.996396i	$0.527031\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	12.0000	1.00349
$144$	0	0
$145$	0	0
$146$	0	0
$147$	1.00000	0.0824786
$148$	0	0
$149$	−14.0000	−1.14692	−0.573462	−	0.819232i	$-0.694400\pi$
$149$	−14.0000	−1.14692	−0.573462	+	0.819232i	$0.694400\pi$
$150$	0	0
$151$	−8.00000	−0.651031	−0.325515	−	0.945537i	$-0.605538\pi$
$151$	−8.00000	−0.651031	−0.325515	+	0.945537i	$0.605538\pi$
$152$	0	0
$153$	−4.00000	−0.323381
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−18.0000	−1.43656	−0.718278	−	0.695756i	$-0.755069\pi$
$157$	−18.0000	−1.43656	−0.718278	+	0.695756i	$0.755069\pi$
$158$	0	0
$159$	−6.00000	−0.475831
$160$	0	0
$161$	0	0
$162$	0	0
$163$	4.00000	0.313304	0.156652	−	0.987654i	$-0.449930\pi$
$163$	4.00000	0.313304	0.156652	+	0.987654i	$0.449930\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	12.0000	0.928588	0.464294	−	0.885681i	$-0.346308\pi$
$167$	12.0000	0.928588	0.464294	+	0.885681i	$0.346308\pi$
$168$	0	0
$169$	−9.00000	−0.692308
$170$	0	0
$171$	6.00000	0.458831
$172$	0	0
$173$	0	0		−	1.00000i	$-0.5\pi$
$173$	0	0			1.00000i	$0.5\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	8.00000	0.601317
$178$	0	0
$179$	14.0000	1.04641	0.523205	−	0.852207i	$-0.324736\pi$
$179$	14.0000	1.04641	0.523205	+	0.852207i	$0.324736\pi$
$180$	0	0
$181$	−6.00000	−0.445976	−0.222988	−	0.974821i	$-0.571581\pi$
$181$	−6.00000	−0.445976	−0.222988	+	0.974821i	$0.571581\pi$
$182$	0	0
$183$	−2.00000	−0.147844
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−24.0000	−1.75505
$188$	0	0
$189$	−1.00000	−0.0727393
$190$	0	0
$191$	18.0000	1.30243	0.651217	−	0.758891i	$-0.274259\pi$
$191$	18.0000	1.30243	0.651217	+	0.758891i	$0.274259\pi$
$192$	0	0
$193$	8.00000	0.575853	0.287926	−	0.957653i	$-0.407034\pi$
$193$	8.00000	0.575853	0.287926	+	0.957653i	$0.407034\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−2.00000	−0.142494	−0.0712470	−	0.997459i	$-0.522698\pi$
$197$	−2.00000	−0.142494	−0.0712470	+	0.997459i	$0.522698\pi$
$198$	0	0
$199$	−14.0000	−0.992434	−0.496217	−	0.868199i	$-0.665278\pi$
$199$	−14.0000	−0.992434	−0.496217	+	0.868199i	$0.665278\pi$
$200$	0	0
$201$	−16.0000	−1.12855
$202$	0	0
$203$	2.00000	0.140372
$204$	0	0
$205$	0	0
$206$	0	0
$207$	0	0
$208$	0	0
$209$	36.0000	2.49017
$210$	0	0
$211$	16.0000	1.10149	0.550743	−	0.834675i	$-0.314345\pi$
$211$	16.0000	1.10149	0.550743	+	0.834675i	$0.314345\pi$
$212$	0	0
$213$	−10.0000	−0.685189
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−10.0000	−0.678844
$218$	0	0
$219$	6.00000	0.405442
$220$	0	0
$221$	−8.00000	−0.538138
$222$	0	0
$223$	−24.0000	−1.60716	−0.803579	−	0.595198i	$-0.797074\pi$
$223$	−24.0000	−1.60716	−0.803579	+	0.595198i	$0.797074\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	8.00000	0.530979	0.265489	−	0.964114i	$-0.414466\pi$
$227$	8.00000	0.530979	0.265489	+	0.964114i	$0.414466\pi$
$228$	0	0
$229$	−10.0000	−0.660819	−0.330409	−	0.943838i	$-0.607187\pi$
$229$	−10.0000	−0.660819	−0.330409	+	0.943838i	$0.607187\pi$
$230$	0	0
$231$	−6.00000	−0.394771
$232$	0	0
$233$	26.0000	1.70332	0.851658	−	0.524097i	$-0.175597\pi$
$233$	26.0000	1.70332	0.851658	+	0.524097i	$0.175597\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−4.00000	−0.259828
$238$	0	0
$239$	6.00000	0.388108	0.194054	−	0.980991i	$-0.437836\pi$
$239$	6.00000	0.388108	0.194054	+	0.980991i	$0.437836\pi$
$240$	0	0
$241$	22.0000	1.41714	0.708572	−	0.705638i	$-0.249340\pi$
$241$	22.0000	1.41714	0.708572	+	0.705638i	$0.249340\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	0	0
$246$	0	0
$247$	12.0000	0.763542
$248$	0	0
$249$	8.00000	0.506979
$250$	0	0
$251$	0	0		−	1.00000i	$-0.5\pi$
$251$	0	0			1.00000i	$0.5\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−16.0000	−0.998053	−0.499026	−	0.866587i	$-0.666309\pi$
$257$	−16.0000	−0.998053	−0.499026	+	0.866587i	$0.666309\pi$
$258$	0	0
$259$	−4.00000	−0.248548
$260$	0	0
$261$	−2.00000	−0.123797
$262$	0	0
$263$	−24.0000	−1.47990	−0.739952	−	0.672660i	$-0.765152\pi$
$263$	−24.0000	−1.47990	−0.739952	+	0.672660i	$0.765152\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	6.00000	0.367194
$268$	0	0
$269$	−14.0000	−0.853595	−0.426798	−	0.904347i	$-0.640358\pi$
$269$	−14.0000	−0.853595	−0.426798	+	0.904347i	$0.640358\pi$
$270$	0	0
$271$	14.0000	0.850439	0.425220	−	0.905090i	$-0.360197\pi$
$271$	14.0000	0.850439	0.425220	+	0.905090i	$0.360197\pi$
$272$	0	0
$273$	−2.00000	−0.121046
$274$	0	0
$275$	0	0
$276$	0	0
$277$	28.0000	1.68236	0.841178	−	0.540758i	$-0.181862\pi$
$277$	28.0000	1.68236	0.841178	+	0.540758i	$0.181862\pi$
$278$	0	0
$279$	10.0000	0.598684
$280$	0	0
$281$	−2.00000	−0.119310	−0.0596550	−	0.998219i	$-0.519000\pi$
$281$	−2.00000	−0.119310	−0.0596550	+	0.998219i	$0.519000\pi$
$282$	0	0
$283$	20.0000	1.18888	0.594438	−	0.804141i	$-0.297374\pi$
$283$	20.0000	1.18888	0.594438	+	0.804141i	$0.297374\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−2.00000	−0.118056
$288$	0	0
$289$	−1.00000	−0.0588235
$290$	0	0
$291$	2.00000	0.117242
$292$	0	0
$293$	−24.0000	−1.40209	−0.701047	−	0.713115i	$-0.747284\pi$
$293$	−24.0000	−1.40209	−0.701047	+	0.713115i	$0.747284\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	6.00000	0.348155
$298$	0	0
$299$	0	0
$300$	0	0
$301$	4.00000	0.230556
$302$	0	0
$303$	−6.00000	−0.344691
$304$	0	0
$305$	0	0
$306$	0	0
$307$	4.00000	0.228292	0.114146	−	0.993464i	$-0.463587\pi$
$307$	4.00000	0.228292	0.114146	+	0.993464i	$0.463587\pi$
$308$	0	0
$309$	8.00000	0.455104
$310$	0	0
$311$	24.0000	1.36092	0.680458	−	0.732787i	$-0.261781\pi$
$311$	24.0000	1.36092	0.680458	+	0.732787i	$0.261781\pi$
$312$	0	0
$313$	−6.00000	−0.339140	−0.169570	−	0.985518i	$-0.554238\pi$
$313$	−6.00000	−0.339140	−0.169570	+	0.985518i	$0.554238\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	18.0000	1.01098	0.505490	−	0.862832i	$-0.331312\pi$
$317$	18.0000	1.01098	0.505490	+	0.862832i	$0.331312\pi$
$318$	0	0
$319$	−12.0000	−0.671871
$320$	0	0
$321$	4.00000	0.223258
$322$	0	0
$323$	−24.0000	−1.33540
$324$	0	0
$325$	0	0
$326$	0	0
$327$	2.00000	0.110600
$328$	0	0
$329$	0	0
$330$	0	0
$331$	24.0000	1.31916	0.659580	−	0.751635i	$-0.270734\pi$
$331$	24.0000	1.31916	0.659580	+	0.751635i	$0.270734\pi$
$332$	0	0
$333$	4.00000	0.219199
$334$	0	0
$335$	0	0
$336$	0	0
$337$	24.0000	1.30736	0.653682	−	0.756770i	$-0.273224\pi$
$337$	24.0000	1.30736	0.653682	+	0.756770i	$0.273224\pi$
$338$	0	0
$339$	−6.00000	−0.325875
$340$	0	0
$341$	60.0000	3.24918
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	0	0
$346$	0	0
$347$	12.0000	0.644194	0.322097	−	0.946707i	$-0.395612\pi$
$347$	12.0000	0.644194	0.322097	+	0.946707i	$0.395612\pi$
$348$	0	0
$349$	−2.00000	−0.107058	−0.0535288	−	0.998566i	$-0.517047\pi$
$349$	−2.00000	−0.107058	−0.0535288	+	0.998566i	$0.517047\pi$
$350$	0	0
$351$	2.00000	0.106752
$352$	0	0
$353$	−20.0000	−1.06449	−0.532246	−	0.846590i	$-0.678652\pi$
$353$	−20.0000	−1.06449	−0.532246	+	0.846590i	$0.678652\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	4.00000	0.211702
$358$	0	0
$359$	−22.0000	−1.16112	−0.580558	−	0.814219i	$-0.697165\pi$
$359$	−22.0000	−1.16112	−0.580558	+	0.814219i	$0.697165\pi$
$360$	0	0
$361$	17.0000	0.894737
$362$	0	0
$363$	25.0000	1.31216
$364$	0	0
$365$	0	0
$366$	0	0
$367$	0	0		−	1.00000i	$-0.5\pi$
$367$	0	0			1.00000i	$0.5\pi$
$368$	0	0
$369$	2.00000	0.104116
$370$	0	0
$371$	6.00000	0.311504
$372$	0	0
$373$	0	0		−	1.00000i	$-0.5\pi$
$373$	0	0			1.00000i	$0.5\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−4.00000	−0.206010
$378$	0	0
$379$	28.0000	1.43826	0.719132	−	0.694874i	$-0.244540\pi$
$379$	28.0000	1.43826	0.719132	+	0.694874i	$0.244540\pi$
$380$	0	0
$381$	20.0000	1.02463
$382$	0	0
$383$	−20.0000	−1.02195	−0.510976	−	0.859595i	$-0.670716\pi$
$383$	−20.0000	−1.02195	−0.510976	+	0.859595i	$0.670716\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−4.00000	−0.203331
$388$	0	0
$389$	26.0000	1.31825	0.659126	−	0.752032i	$-0.270926\pi$
$389$	26.0000	1.31825	0.659126	+	0.752032i	$0.270926\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	−4.00000	−0.201773
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−22.0000	−1.10415	−0.552074	−	0.833795i	$-0.686163\pi$
$397$	−22.0000	−1.10415	−0.552074	+	0.833795i	$0.686163\pi$
$398$	0	0
$399$	−6.00000	−0.300376
$400$	0	0
$401$	−30.0000	−1.49813	−0.749064	−	0.662497i	$-0.769497\pi$
$401$	−30.0000	−1.49813	−0.749064	+	0.662497i	$0.769497\pi$
$402$	0	0
$403$	20.0000	0.996271
$404$	0	0
$405$	0	0
$406$	0	0
$407$	24.0000	1.18964
$408$	0	0
$409$	−22.0000	−1.08783	−0.543915	−	0.839140i	$-0.683059\pi$
$409$	−22.0000	−1.08783	−0.543915	+	0.839140i	$0.683059\pi$
$410$	0	0
$411$	6.00000	0.295958
$412$	0	0
$413$	−8.00000	−0.393654
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−2.00000	−0.0979404
$418$	0	0
$419$	12.0000	0.586238	0.293119	−	0.956076i	$-0.405307\pi$
$419$	12.0000	0.586238	0.293119	+	0.956076i	$0.405307\pi$
$420$	0	0
$421$	18.0000	0.877266	0.438633	−	0.898666i	$-0.355463\pi$
$421$	18.0000	0.877266	0.438633	+	0.898666i	$0.355463\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	0	0
$426$	0	0
$427$	2.00000	0.0967868
$428$	0	0
$429$	12.0000	0.579365
$430$	0	0
$431$	14.0000	0.674356	0.337178	−	0.941441i	$-0.390528\pi$
$431$	14.0000	0.674356	0.337178	+	0.941441i	$0.390528\pi$
$432$	0	0
$433$	34.0000	1.63394	0.816968	−	0.576683i	$-0.195653\pi$
$433$	34.0000	1.63394	0.816968	+	0.576683i	$0.195653\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	0	0
$438$	0	0
$439$	−6.00000	−0.286364	−0.143182	−	0.989696i	$-0.545733\pi$
$439$	−6.00000	−0.286364	−0.143182	+	0.989696i	$0.545733\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	0	0
$443$	−4.00000	−0.190046	−0.0950229	−	0.995475i	$-0.530292\pi$
$443$	−4.00000	−0.190046	−0.0950229	+	0.995475i	$0.530292\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−14.0000	−0.662177
$448$	0	0
$449$	10.0000	0.471929	0.235965	−	0.971762i	$-0.424175\pi$
$449$	10.0000	0.471929	0.235965	+	0.971762i	$0.424175\pi$
$450$	0	0
$451$	12.0000	0.565058
$452$	0	0
$453$	−8.00000	−0.375873
$454$	0	0
$455$	0	0
$456$	0	0
$457$	20.0000	0.935561	0.467780	−	0.883845i	$-0.345054\pi$
$457$	20.0000	0.935561	0.467780	+	0.883845i	$0.345054\pi$
$458$	0	0
$459$	−4.00000	−0.186704
$460$	0	0
$461$	30.0000	1.39724	0.698620	−	0.715493i	$-0.253798\pi$
$461$	30.0000	1.39724	0.698620	+	0.715493i	$0.253798\pi$
$462$	0	0
$463$	36.0000	1.67306	0.836531	−	0.547920i	$-0.184580\pi$
$463$	36.0000	1.67306	0.836531	+	0.547920i	$0.184580\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−24.0000	−1.11059	−0.555294	−	0.831654i	$-0.687394\pi$
$467$	−24.0000	−1.11059	−0.555294	+	0.831654i	$0.687394\pi$
$468$	0	0
$469$	16.0000	0.738811
$470$	0	0
$471$	−18.0000	−0.829396
$472$	0	0
$473$	−24.0000	−1.10352
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−6.00000	−0.274721
$478$	0	0
$479$	−24.0000	−1.09659	−0.548294	−	0.836286i	$-0.684723\pi$
$479$	−24.0000	−1.09659	−0.548294	+	0.836286i	$0.684723\pi$
$480$	0	0
$481$	8.00000	0.364769
$482$	0	0
$483$	0	0
$484$	0	0
$485$	0	0
$486$	0	0
$487$	12.0000	0.543772	0.271886	−	0.962329i	$-0.412353\pi$
$487$	12.0000	0.543772	0.271886	+	0.962329i	$0.412353\pi$
$488$	0	0
$489$	4.00000	0.180886
$490$	0	0
$491$	−10.0000	−0.451294	−0.225647	−	0.974209i	$-0.572450\pi$
$491$	−10.0000	−0.451294	−0.225647	+	0.974209i	$0.572450\pi$
$492$	0	0
$493$	8.00000	0.360302
$494$	0	0
$495$	0	0
$496$	0	0
$497$	10.0000	0.448561
$498$	0	0
$499$	−4.00000	−0.179065	−0.0895323	−	0.995984i	$-0.528537\pi$
$499$	−4.00000	−0.179065	−0.0895323	+	0.995984i	$0.528537\pi$
$500$	0	0
$501$	12.0000	0.536120
$502$	0	0
$503$	−36.0000	−1.60516	−0.802580	−	0.596544i	$-0.796540\pi$
$503$	−36.0000	−1.60516	−0.802580	+	0.596544i	$0.796540\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−9.00000	−0.399704
$508$	0	0
$509$	30.0000	1.32973	0.664863	−	0.746965i	$-0.268490\pi$
$509$	30.0000	1.32973	0.664863	+	0.746965i	$0.268490\pi$
$510$	0	0
$511$	−6.00000	−0.265424
$512$	0	0
$513$	6.00000	0.264906
$514$	0	0
$515$	0	0
$516$	0	0
$517$	0	0
$518$	0	0
$519$	0	0
$520$	0	0
$521$	38.0000	1.66481	0.832405	−	0.554168i	$-0.186963\pi$
$521$	38.0000	1.66481	0.832405	+	0.554168i	$0.186963\pi$
$522$	0	0
$523$	20.0000	0.874539	0.437269	−	0.899331i	$-0.355946\pi$
$523$	20.0000	0.874539	0.437269	+	0.899331i	$0.355946\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−40.0000	−1.74243
$528$	0	0
$529$	−23.0000	−1.00000
$530$	0	0
$531$	8.00000	0.347170
$532$	0	0
$533$	4.00000	0.173259
$534$	0	0
$535$	0	0
$536$	0	0
$537$	14.0000	0.604145
$538$	0	0
$539$	6.00000	0.258438
$540$	0	0
$541$	6.00000	0.257960	0.128980	−	0.991647i	$-0.458830\pi$
$541$	6.00000	0.257960	0.128980	+	0.991647i	$0.458830\pi$
$542$	0	0
$543$	−6.00000	−0.257485
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−16.0000	−0.684111	−0.342055	−	0.939680i	$-0.611123\pi$
$547$	−16.0000	−0.684111	−0.342055	+	0.939680i	$0.611123\pi$
$548$	0	0
$549$	−2.00000	−0.0853579
$550$	0	0
$551$	−12.0000	−0.511217
$552$	0	0
$553$	4.00000	0.170097
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−38.0000	−1.61011	−0.805056	−	0.593199i	$-0.797865\pi$
$557$	−38.0000	−1.61011	−0.805056	+	0.593199i	$0.797865\pi$
$558$	0	0
$559$	−8.00000	−0.338364
$560$	0	0
$561$	−24.0000	−1.01328
$562$	0	0
$563$	−36.0000	−1.51722	−0.758610	−	0.651546i	$-0.774121\pi$
$563$	−36.0000	−1.51722	−0.758610	+	0.651546i	$0.774121\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−1.00000	−0.0419961
$568$	0	0
$569$	−30.0000	−1.25767	−0.628833	−	0.777541i	$-0.716467\pi$
$569$	−30.0000	−1.25767	−0.628833	+	0.777541i	$0.716467\pi$
$570$	0	0
$571$	−36.0000	−1.50655	−0.753277	−	0.657704i	$-0.771528\pi$
$571$	−36.0000	−1.50655	−0.753277	+	0.657704i	$0.771528\pi$
$572$	0	0
$573$	18.0000	0.751961
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−14.0000	−0.582828	−0.291414	−	0.956597i	$-0.594126\pi$
$577$	−14.0000	−0.582828	−0.291414	+	0.956597i	$0.594126\pi$
$578$	0	0
$579$	8.00000	0.332469
$580$	0	0
$581$	−8.00000	−0.331896
$582$	0	0
$583$	−36.0000	−1.49097
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−12.0000	−0.495293	−0.247647	−	0.968850i	$-0.579657\pi$
$587$	−12.0000	−0.495293	−0.247647	+	0.968850i	$0.579657\pi$
$588$	0	0
$589$	60.0000	2.47226
$590$	0	0
$591$	−2.00000	−0.0822690
$592$	0	0
$593$	−44.0000	−1.80686	−0.903432	−	0.428732i	$-0.858960\pi$
$593$	−44.0000	−1.80686	−0.903432	+	0.428732i	$0.858960\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−14.0000	−0.572982
$598$	0	0
$599$	2.00000	0.0817178	0.0408589	−	0.999165i	$-0.486991\pi$
$599$	2.00000	0.0817178	0.0408589	+	0.999165i	$0.486991\pi$
$600$	0	0
$601$	−26.0000	−1.06056	−0.530281	−	0.847822i	$-0.677914\pi$
$601$	−26.0000	−1.06056	−0.530281	+	0.847822i	$0.677914\pi$
$602$	0	0
$603$	−16.0000	−0.651570
$604$	0	0
$605$	0	0
$606$	0	0
$607$	24.0000	0.974130	0.487065	−	0.873366i	$-0.338067\pi$
$607$	24.0000	0.974130	0.487065	+	0.873366i	$0.338067\pi$
$608$	0	0
$609$	2.00000	0.0810441
$610$	0	0
$611$	0	0
$612$	0	0
$613$	−4.00000	−0.161558	−0.0807792	−	0.996732i	$-0.525741\pi$
$613$	−4.00000	−0.161558	−0.0807792	+	0.996732i	$0.525741\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	26.0000	1.04672	0.523360	−	0.852111i	$-0.324678\pi$
$617$	26.0000	1.04672	0.523360	+	0.852111i	$0.324678\pi$
$618$	0	0
$619$	−10.0000	−0.401934	−0.200967	−	0.979598i	$-0.564408\pi$
$619$	−10.0000	−0.401934	−0.200967	+	0.979598i	$0.564408\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−6.00000	−0.240385
$624$	0	0
$625$	0	0
$626$	0	0
$627$	36.0000	1.43770
$628$	0	0
$629$	−16.0000	−0.637962
$630$	0	0
$631$	20.0000	0.796187	0.398094	−	0.917345i	$-0.369672\pi$
$631$	20.0000	0.796187	0.398094	+	0.917345i	$0.369672\pi$
$632$	0	0
$633$	16.0000	0.635943
$634$	0	0
$635$	0	0
$636$	0	0
$637$	2.00000	0.0792429
$638$	0	0
$639$	−10.0000	−0.395594
$640$	0	0
$641$	−2.00000	−0.0789953	−0.0394976	−	0.999220i	$-0.512576\pi$
$641$	−2.00000	−0.0789953	−0.0394976	+	0.999220i	$0.512576\pi$
$642$	0	0
$643$	−20.0000	−0.788723	−0.394362	−	0.918955i	$-0.629034\pi$
$643$	−20.0000	−0.788723	−0.394362	+	0.918955i	$0.629034\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−20.0000	−0.786281	−0.393141	−	0.919478i	$-0.628611\pi$
$647$	−20.0000	−0.786281	−0.393141	+	0.919478i	$0.628611\pi$
$648$	0	0
$649$	48.0000	1.88416
$650$	0	0
$651$	−10.0000	−0.391931
$652$	0	0
$653$	−2.00000	−0.0782660	−0.0391330	−	0.999234i	$-0.512460\pi$
$653$	−2.00000	−0.0782660	−0.0391330	+	0.999234i	$0.512460\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	6.00000	0.234082
$658$	0	0
$659$	18.0000	0.701180	0.350590	−	0.936529i	$-0.385981\pi$
$659$	18.0000	0.701180	0.350590	+	0.936529i	$0.385981\pi$
$660$	0	0
$661$	−18.0000	−0.700119	−0.350059	−	0.936727i	$-0.613839\pi$
$661$	−18.0000	−0.700119	−0.350059	+	0.936727i	$0.613839\pi$
$662$	0	0
$663$	−8.00000	−0.310694
$664$	0	0
$665$	0	0
$666$	0	0
$667$	0	0
$668$	0	0
$669$	−24.0000	−0.927894
$670$	0	0
$671$	−12.0000	−0.463255
$672$	0	0
$673$	36.0000	1.38770	0.693849	−	0.720121i	$-0.255914\pi$
$673$	36.0000	1.38770	0.693849	+	0.720121i	$0.255914\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−32.0000	−1.22986	−0.614930	−	0.788582i	$-0.710816\pi$
$677$	−32.0000	−1.22986	−0.614930	+	0.788582i	$0.710816\pi$
$678$	0	0
$679$	−2.00000	−0.0767530
$680$	0	0
$681$	8.00000	0.306561
$682$	0	0
$683$	28.0000	1.07139	0.535695	−	0.844411i	$-0.320050\pi$
$683$	28.0000	1.07139	0.535695	+	0.844411i	$0.320050\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−10.0000	−0.381524
$688$	0	0
$689$	−12.0000	−0.457164
$690$	0	0
$691$	50.0000	1.90209	0.951045	−	0.309053i	$-0.100012\pi$
$691$	50.0000	1.90209	0.951045	+	0.309053i	$0.100012\pi$
$692$	0	0
$693$	−6.00000	−0.227921
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−8.00000	−0.303022
$698$	0	0
$699$	26.0000	0.983410
$700$	0	0
$701$	−10.0000	−0.377695	−0.188847	−	0.982006i	$-0.560475\pi$
$701$	−10.0000	−0.377695	−0.188847	+	0.982006i	$0.560475\pi$
$702$	0	0
$703$	24.0000	0.905177
$704$	0	0
$705$	0	0
$706$	0	0
$707$	6.00000	0.225653
$708$	0	0
$709$	26.0000	0.976450	0.488225	−	0.872718i	$-0.337644\pi$
$709$	26.0000	0.976450	0.488225	+	0.872718i	$0.337644\pi$
$710$	0	0
$711$	−4.00000	−0.150012
$712$	0	0
$713$	0	0
$714$	0	0
$715$	0	0
$716$	0	0
$717$	6.00000	0.224074
$718$	0	0
$719$	−12.0000	−0.447524	−0.223762	−	0.974644i	$-0.571834\pi$
$719$	−12.0000	−0.447524	−0.223762	+	0.974644i	$0.571834\pi$
$720$	0	0
$721$	−8.00000	−0.297936
$722$	0	0
$723$	22.0000	0.818189
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−40.0000	−1.48352	−0.741759	−	0.670667i	$-0.766008\pi$
$727$	−40.0000	−1.48352	−0.741759	+	0.670667i	$0.766008\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	16.0000	0.591781
$732$	0	0
$733$	−22.0000	−0.812589	−0.406294	−	0.913742i	$-0.633179\pi$
$733$	−22.0000	−0.812589	−0.406294	+	0.913742i	$0.633179\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−96.0000	−3.53621
$738$	0	0
$739$	44.0000	1.61857	0.809283	−	0.587419i	$-0.199856\pi$
$739$	44.0000	1.61857	0.809283	+	0.587419i	$0.199856\pi$
$740$	0	0
$741$	12.0000	0.440831
$742$	0	0
$743$	40.0000	1.46746	0.733729	−	0.679442i	$-0.237778\pi$
$743$	40.0000	1.46746	0.733729	+	0.679442i	$0.237778\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	8.00000	0.292705
$748$	0	0
$749$	−4.00000	−0.146157
$750$	0	0
$751$	12.0000	0.437886	0.218943	−	0.975738i	$-0.429739\pi$
$751$	12.0000	0.437886	0.218943	+	0.975738i	$0.429739\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	40.0000	1.45382	0.726912	−	0.686730i	$-0.240955\pi$
$757$	40.0000	1.45382	0.726912	+	0.686730i	$0.240955\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−22.0000	−0.797499	−0.398750	−	0.917060i	$-0.630556\pi$
$761$	−22.0000	−0.797499	−0.398750	+	0.917060i	$0.630556\pi$
$762$	0	0
$763$	−2.00000	−0.0724049
$764$	0	0
$765$	0	0
$766$	0	0
$767$	16.0000	0.577727
$768$	0	0
$769$	−18.0000	−0.649097	−0.324548	−	0.945869i	$-0.605212\pi$
$769$	−18.0000	−0.649097	−0.324548	+	0.945869i	$0.605212\pi$
$770$	0	0
$771$	−16.0000	−0.576226
$772$	0	0
$773$	−24.0000	−0.863220	−0.431610	−	0.902060i	$-0.642054\pi$
$773$	−24.0000	−0.863220	−0.431610	+	0.902060i	$0.642054\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−4.00000	−0.143499
$778$	0	0
$779$	12.0000	0.429945
$780$	0	0
$781$	−60.0000	−2.14697
$782$	0	0
$783$	−2.00000	−0.0714742
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−4.00000	−0.142585	−0.0712923	−	0.997455i	$-0.522712\pi$
$787$	−4.00000	−0.142585	−0.0712923	+	0.997455i	$0.522712\pi$
$788$	0	0
$789$	−24.0000	−0.854423
$790$	0	0
$791$	6.00000	0.213335
$792$	0	0
$793$	−4.00000	−0.142044
$794$	0	0
$795$	0	0
$796$	0	0
$797$	16.0000	0.566749	0.283375	−	0.959009i	$-0.408546\pi$
$797$	16.0000	0.566749	0.283375	+	0.959009i	$0.408546\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	6.00000	0.212000
$802$	0	0
$803$	36.0000	1.27041
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−14.0000	−0.492823
$808$	0	0
$809$	−18.0000	−0.632846	−0.316423	−	0.948618i	$-0.602482\pi$
$809$	−18.0000	−0.632846	−0.316423	+	0.948618i	$0.602482\pi$
$810$	0	0
$811$	−34.0000	−1.19390	−0.596951	−	0.802278i	$-0.703621\pi$
$811$	−34.0000	−1.19390	−0.596951	+	0.802278i	$0.703621\pi$
$812$	0	0
$813$	14.0000	0.491001
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−24.0000	−0.839654
$818$	0	0
$819$	−2.00000	−0.0698857
$820$	0	0
$821$	−2.00000	−0.0698005	−0.0349002	−	0.999391i	$-0.511111\pi$
$821$	−2.00000	−0.0698005	−0.0349002	+	0.999391i	$0.511111\pi$
$822$	0	0
$823$	44.0000	1.53374	0.766872	−	0.641800i	$-0.221812\pi$
$823$	44.0000	1.53374	0.766872	+	0.641800i	$0.221812\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−12.0000	−0.417281	−0.208640	−	0.977992i	$-0.566904\pi$
$827$	−12.0000	−0.417281	−0.208640	+	0.977992i	$0.566904\pi$
$828$	0	0
$829$	−6.00000	−0.208389	−0.104194	−	0.994557i	$-0.533226\pi$
$829$	−6.00000	−0.208389	−0.104194	+	0.994557i	$0.533226\pi$
$830$	0	0
$831$	28.0000	0.971309
$832$	0	0
$833$	−4.00000	−0.138592
$834$	0	0
$835$	0	0
$836$	0	0
$837$	10.0000	0.345651
$838$	0	0
$839$	−12.0000	−0.414286	−0.207143	−	0.978311i	$-0.566417\pi$
$839$	−12.0000	−0.414286	−0.207143	+	0.978311i	$0.566417\pi$
$840$	0	0
$841$	−25.0000	−0.862069
$842$	0	0
$843$	−2.00000	−0.0688837
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−25.0000	−0.859010
$848$	0	0
$849$	20.0000	0.686398
$850$	0	0
$851$	0	0
$852$	0	0
$853$	−30.0000	−1.02718	−0.513590	−	0.858036i	$-0.671685\pi$
$853$	−30.0000	−1.02718	−0.513590	+	0.858036i	$0.671685\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	24.0000	0.819824	0.409912	−	0.912125i	$-0.365559\pi$
$857$	24.0000	0.819824	0.409912	+	0.912125i	$0.365559\pi$
$858$	0	0
$859$	26.0000	0.887109	0.443554	−	0.896248i	$-0.353717\pi$
$859$	26.0000	0.887109	0.443554	+	0.896248i	$0.353717\pi$
$860$	0	0
$861$	−2.00000	−0.0681598
$862$	0	0
$863$	−24.0000	−0.816970	−0.408485	−	0.912765i	$-0.633943\pi$
$863$	−24.0000	−0.816970	−0.408485	+	0.912765i	$0.633943\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−1.00000	−0.0339618
$868$	0	0
$869$	−24.0000	−0.814144
$870$	0	0
$871$	−32.0000	−1.08428
$872$	0	0
$873$	2.00000	0.0676897
$874$	0	0
$875$	0	0
$876$	0	0
$877$	4.00000	0.135070	0.0675352	−	0.997717i	$-0.478487\pi$
$877$	4.00000	0.135070	0.0675352	+	0.997717i	$0.478487\pi$
$878$	0	0
$879$	−24.0000	−0.809500
$880$	0	0
$881$	14.0000	0.471672	0.235836	−	0.971793i	$-0.424217\pi$
$881$	14.0000	0.471672	0.235836	+	0.971793i	$0.424217\pi$
$882$	0	0
$883$	−48.0000	−1.61533	−0.807664	−	0.589643i	$-0.799269\pi$
$883$	−48.0000	−1.61533	−0.807664	+	0.589643i	$0.799269\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	8.00000	0.268614	0.134307	−	0.990940i	$-0.457119\pi$
$887$	8.00000	0.268614	0.134307	+	0.990940i	$0.457119\pi$
$888$	0	0
$889$	−20.0000	−0.670778
$890$	0	0
$891$	6.00000	0.201008
$892$	0	0
$893$	0	0
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−20.0000	−0.667037
$900$	0	0
$901$	24.0000	0.799556
$902$	0	0
$903$	4.00000	0.133112
$904$	0	0
$905$	0	0
$906$	0	0
$907$	16.0000	0.531271	0.265636	−	0.964073i	$-0.414418\pi$
$907$	16.0000	0.531271	0.265636	+	0.964073i	$0.414418\pi$
$908$	0	0
$909$	−6.00000	−0.199007
$910$	0	0
$911$	−6.00000	−0.198789	−0.0993944	−	0.995048i	$-0.531691\pi$
$911$	−6.00000	−0.198789	−0.0993944	+	0.995048i	$0.531691\pi$
$912$	0	0
$913$	48.0000	1.58857
$914$	0	0
$915$	0	0
$916$	0	0
$917$	4.00000	0.132092
$918$	0	0
$919$	−28.0000	−0.923635	−0.461817	−	0.886975i	$-0.652802\pi$
$919$	−28.0000	−0.923635	−0.461817	+	0.886975i	$0.652802\pi$
$920$	0	0
$921$	4.00000	0.131804
$922$	0	0
$923$	−20.0000	−0.658308
$924$	0	0
$925$	0	0
$926$	0	0
$927$	8.00000	0.262754
$928$	0	0
$929$	−14.0000	−0.459325	−0.229663	−	0.973270i	$-0.573762\pi$
$929$	−14.0000	−0.459325	−0.229663	+	0.973270i	$0.573762\pi$
$930$	0	0
$931$	6.00000	0.196642
$932$	0	0
$933$	24.0000	0.785725
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−2.00000	−0.0653372	−0.0326686	−	0.999466i	$-0.510401\pi$
$937$	−2.00000	−0.0653372	−0.0326686	+	0.999466i	$0.510401\pi$
$938$	0	0
$939$	−6.00000	−0.195803
$940$	0	0
$941$	−42.0000	−1.36916	−0.684580	−	0.728937i	$-0.740015\pi$
$941$	−42.0000	−1.36916	−0.684580	+	0.728937i	$0.740015\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−36.0000	−1.16984	−0.584921	−	0.811090i	$-0.698875\pi$
$947$	−36.0000	−1.16984	−0.584921	+	0.811090i	$0.698875\pi$
$948$	0	0
$949$	12.0000	0.389536
$950$	0	0
$951$	18.0000	0.583690
$952$	0	0
$953$	−6.00000	−0.194359	−0.0971795	−	0.995267i	$-0.530982\pi$
$953$	−6.00000	−0.194359	−0.0971795	+	0.995267i	$0.530982\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−12.0000	−0.387905
$958$	0	0
$959$	−6.00000	−0.193750
$960$	0	0
$961$	69.0000	2.22581
$962$	0	0
$963$	4.00000	0.128898
$964$	0	0
$965$	0	0
$966$	0	0
$967$	40.0000	1.28631	0.643157	−	0.765735i	$-0.277624\pi$
$967$	40.0000	1.28631	0.643157	+	0.765735i	$0.277624\pi$
$968$	0	0
$969$	−24.0000	−0.770991
$970$	0	0
$971$	8.00000	0.256732	0.128366	−	0.991727i	$-0.459027\pi$
$971$	8.00000	0.256732	0.128366	+	0.991727i	$0.459027\pi$
$972$	0	0
$973$	2.00000	0.0641171
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−30.0000	−0.959785	−0.479893	−	0.877327i	$-0.659324\pi$
$977$	−30.0000	−0.959785	−0.479893	+	0.877327i	$0.659324\pi$
$978$	0	0
$979$	36.0000	1.15056
$980$	0	0
$981$	2.00000	0.0638551
$982$	0	0
$983$	24.0000	0.765481	0.382741	−	0.923856i	$-0.374980\pi$
$983$	24.0000	0.765481	0.382741	+	0.923856i	$0.374980\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	0	0
$990$	0	0
$991$	−4.00000	−0.127064	−0.0635321	−	0.997980i	$-0.520237\pi$
$991$	−4.00000	−0.127064	−0.0635321	+	0.997980i	$0.520237\pi$
$992$	0	0
$993$	24.0000	0.761617
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−14.0000	−0.443384	−0.221692	−	0.975117i	$-0.571158\pi$
$997$	−14.0000	−0.443384	−0.221692	+	0.975117i	$0.571158\pi$
$998$	0	0
$999$	4.00000	0.126554

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8400.2.a.ch.1.1		1
4.3	odd	2		525.2.a.c.1.1		1
5.2	odd	4		1680.2.t.f.1009.1		2
5.3	odd	4		1680.2.t.f.1009.2		2
5.4	even	2		8400.2.a.bj.1.1		1
12.11	even	2		1575.2.a.e.1.1		1
15.2	even	4		5040.2.t.e.1009.2		2
15.8	even	4		5040.2.t.e.1009.1		2
20.3	even	4		105.2.d.a.64.1	✓	2
20.7	even	4		105.2.d.a.64.2	yes	2
20.19	odd	2		525.2.a.b.1.1		1
28.27	even	2		3675.2.a.l.1.1		1
60.23	odd	4		315.2.d.c.64.2		2
60.47	odd	4		315.2.d.c.64.1		2
60.59	even	2		1575.2.a.i.1.1		1
140.3	odd	12		735.2.q.b.79.2		4
140.23	even	12		735.2.q.a.214.1		4
140.27	odd	4		735.2.d.a.589.2		2
140.47	odd	12		735.2.q.b.214.2		4
140.67	even	12		735.2.q.a.79.1		4
140.83	odd	4		735.2.d.a.589.1		2
140.87	odd	12		735.2.q.b.79.1		4
140.103	odd	12		735.2.q.b.214.1		4
140.107	even	12		735.2.q.a.214.2		4
140.123	even	12		735.2.q.a.79.2		4
140.139	even	2		3675.2.a.d.1.1		1
420.83	even	4		2205.2.d.f.1324.2		2
420.167	even	4		2205.2.d.f.1324.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
105.2.d.a.64.1	✓	2	20.3	even	4
105.2.d.a.64.2	yes	2	20.7	even	4
315.2.d.c.64.1		2	60.47	odd	4
315.2.d.c.64.2		2	60.23	odd	4
525.2.a.b.1.1		1	20.19	odd	2
525.2.a.c.1.1		1	4.3	odd	2
735.2.d.a.589.1		2	140.83	odd	4
735.2.d.a.589.2		2	140.27	odd	4
735.2.q.a.79.1		4	140.67	even	12
735.2.q.a.79.2		4	140.123	even	12
735.2.q.a.214.1		4	140.23	even	12
735.2.q.a.214.2		4	140.107	even	12
735.2.q.b.79.1		4	140.87	odd	12
735.2.q.b.79.2		4	140.3	odd	12
735.2.q.b.214.1		4	140.103	odd	12
735.2.q.b.214.2		4	140.47	odd	12
1575.2.a.e.1.1		1	12.11	even	2
1575.2.a.i.1.1		1	60.59	even	2
1680.2.t.f.1009.1		2	5.2	odd	4
1680.2.t.f.1009.2		2	5.3	odd	4
2205.2.d.f.1324.1		2	420.167	even	4
2205.2.d.f.1324.2		2	420.83	even	4
3675.2.a.d.1.1		1	140.139	even	2
3675.2.a.l.1.1		1	28.27	even	2
5040.2.t.e.1009.1		2	15.8	even	4
5040.2.t.e.1009.2		2	15.2	even	4
8400.2.a.bj.1.1		1	5.4	even	2
8400.2.a.ch.1.1		1	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 8400 = 2^{4} \cdot 3 \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8400.a (trivial)

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