LMFDB - Embedded newform 9300.2.g.q.3349.6

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [9300,2,Mod(3349,9300)] 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("9300.3349"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(9300, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 9300 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 31 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9300.g (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [6,0,0,0,0,0,0,0,-6,0,4,0,0,0,0,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(19)] == traces)

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

Self dual:	no
Analytic conductor:	$74.2608738798$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.2611456.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{3} + 16x^{2} - 8x + 2 $ x^6 - 2x^3 + 16x^2 - 8*x + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 1860)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			3349.6
Root			$1.26704 - 1.26704i$ of defining polynomial
Character	$\chi$	$=$	9300.3349
Dual form			9300.2.g.q.3349.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{3} +3.74483i q^{7} -1.00000 q^{9} -5.48965 q^{11} -1.32331i q^{13} -6.95558i q^{17} -5.06814 q^{19} -3.74483 q^{21} +2.11256i q^{23} -1.00000i q^{27} -3.63227 q^{29} +1.00000 q^{31} -5.48965i q^{33} -8.39145i q^{37} +1.32331 q^{39} +11.4897 q^{41} -0.421512i q^{43} -1.46593i q^{47} -7.02372 q^{49} +6.95558 q^{51} +4.53407i q^{53} -5.06814i q^{57} -4.78924 q^{59} +6.84302 q^{61} -3.74483i q^{63} +4.39145i q^{67} -2.11256 q^{69} -7.34704 q^{71} -4.39145i q^{73} -20.5578i q^{77} +14.0237 q^{79} +1.00000 q^{81} +5.04442i q^{83} -3.63227i q^{87} +13.3470 q^{89} +4.95558 q^{91} +1.00000i q^{93} +3.48965i q^{97} +5.48965 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 6 q^{9} + 4 q^{11} - 4 q^{21} - 12 q^{29} + 6 q^{31} - 4 q^{39} + 32 q^{41} + 10 q^{49} + 20 q^{51} - 32 q^{59} + 28 q^{61} - 4 q^{69} + 20 q^{71} + 32 q^{79} + 6 q^{81} + 16 q^{89} + 8 q^{91} - 4 q^{99}+O(q^{100}) $ 6 * q - 6 * q^9 + 4 * q^11 - 4 * q^21 - 12 * q^29 + 6 * q^31 - 4 * q^39 + 32 * q^41 + 10 * q^49 + 20 * q^51 - 32 * q^59 + 28 * q^61 - 4 * q^69 + 20 * q^71 + 32 * q^79 + 6 * q^81 + 16 * q^89 + 8 * q^91 - 4 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/9300\mathbb{Z}\right)^\times$.

$n$	$1801$	$2977$	$3101$	$4651$
$\chi(n)$	$1$	$-1$	$1$	$1$

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	1.00000i	0.577350i
$3$	1.00000i	0.577350i
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	3.74483i	1.41541i	0.706507	+	0.707706i	$0.250270\pi$
$7$	3.74483i	1.41541i	−0.706507	+	0.707706i	$0.749730\pi$
$8$	0	0
$9$	−1.00000	−0.333333
$10$	0	0
$11$	−5.48965	−1.65519	−0.827596	−	0.561324i	$-0.810292\pi$
$11$	−5.48965	−1.65519	−0.827596	+	0.561324i	$0.810292\pi$
$12$	0	0
$13$	− 1.32331i	− 0.367021i	−0.983018	−	0.183511i	$-0.941254\pi$
$13$	− 1.32331i	− 0.367021i	0.983018	−	0.183511i	$-0.0587462\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 6.95558i	− 1.68698i	−0.537148	−	0.843488i	$-0.680498\pi$
$17$	− 6.95558i	− 1.68698i	0.537148	−	0.843488i	$-0.319502\pi$
$18$	0	0
$19$	−5.06814	−1.16271	−0.581356	−	0.813650i	$-0.697477\pi$
$19$	−5.06814	−1.16271	−0.581356	+	0.813650i	$0.697477\pi$
$20$	0	0
$21$	−3.74483	−0.817188
$22$	0	0
$23$	2.11256i	0.440499i	0.975444	+	0.220249i	$0.0706871\pi$
$23$	2.11256i	0.440499i	−0.975444	+	0.220249i	$0.929313\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	− 1.00000i	− 0.192450i
$28$	0	0
$29$	−3.63227	−0.674495	−0.337248	−	0.941416i	$-0.609496\pi$
$29$	−3.63227	−0.674495	−0.337248	+	0.941416i	$0.609496\pi$
$30$	0	0
$31$	1.00000	0.179605
$31$	1.00000	0.179605
$32$	0	0
$33$	− 5.48965i	− 0.955626i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	− 8.39145i	− 1.37955i	−0.724025	−	0.689773i	$-0.757710\pi$
$37$	− 8.39145i	− 1.37955i	0.724025	−	0.689773i	$-0.242290\pi$
$38$	0	0
$39$	1.32331	0.211900
$40$	0	0
$41$	11.4897	1.79438	0.897191	−	0.441643i	$-0.145604\pi$
$41$	11.4897	1.79438	0.897191	+	0.441643i	$0.145604\pi$
$42$	0	0
$43$	− 0.421512i	− 0.0642799i	−0.999483	−	0.0321400i	$-0.989768\pi$
$43$	− 0.421512i	− 0.0642799i	0.999483	−	0.0321400i	$-0.0102322\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 1.46593i	− 0.213828i	−0.994268	−	0.106914i	$-0.965903\pi$
$47$	− 1.46593i	− 0.213828i	0.994268	−	0.106914i	$-0.0340969\pi$
$48$	0	0
$49$	−7.02372	−1.00339
$50$	0	0
$51$	6.95558	0.973976
$52$	0	0
$53$	4.53407i	0.622802i	0.950279	+	0.311401i	$0.100798\pi$
$53$	4.53407i	0.622802i	−0.950279	+	0.311401i	$0.899202\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	− 5.06814i	− 0.671292i
$58$	0	0
$59$	−4.78924	−0.623506	−0.311753	−	0.950163i	$-0.600916\pi$
$59$	−4.78924	−0.623506	−0.311753	+	0.950163i	$0.600916\pi$
$60$	0	0
$61$	6.84302	0.876159	0.438080	−	0.898936i	$-0.355659\pi$
$61$	6.84302	0.876159	0.438080	+	0.898936i	$0.355659\pi$
$62$	0	0
$63$	− 3.74483i	− 0.471804i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	4.39145i	0.536502i	0.963349	+	0.268251i	$0.0864456\pi$
$67$	4.39145i	0.536502i	−0.963349	+	0.268251i	$0.913554\pi$
$68$	0	0
$69$	−2.11256	−0.254322
$70$	0	0
$71$	−7.34704	−0.871933	−0.435966	−	0.899963i	$-0.643593\pi$
$71$	−7.34704	−0.871933	−0.435966	+	0.899963i	$0.643593\pi$
$72$	0	0
$73$	− 4.39145i	− 0.513981i	−0.966414	−	0.256990i	$-0.917269\pi$
$73$	− 4.39145i	− 0.513981i	0.966414	−	0.256990i	$-0.0827309\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 20.5578i	− 2.34278i
$78$	0	0
$79$	14.0237	1.57779	0.788896	−	0.614527i	$-0.210653\pi$
$79$	14.0237	1.57779	0.788896	+	0.614527i	$0.210653\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	5.04442i	0.553697i	0.960914	+	0.276848i	$0.0892900\pi$
$83$	5.04442i	0.553697i	−0.960914	+	0.276848i	$0.910710\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	− 3.63227i	− 0.389420i
$88$	0	0
$89$	13.3470	1.41478	0.707392	−	0.706822i	$-0.249872\pi$
$89$	13.3470	1.41478	0.707392	+	0.706822i	$0.249872\pi$
$90$	0	0
$91$	4.95558	0.519486
$92$	0	0
$93$	1.00000i	0.103695i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	3.48965i	0.354320i	0.984182	+	0.177160i	$0.0566911\pi$
$97$	3.48965i	0.354320i	−0.984182	+	0.177160i	$0.943309\pi$
$98$	0	0
$99$	5.48965	0.551731
$100$	0	0
$101$	−13.4008	−1.33343	−0.666716	−	0.745312i	$-0.732300\pi$
$101$	−13.4008	−1.33343	−0.666716	+	0.745312i	$0.732300\pi$
$102$	0	0
$103$	15.1456i	1.49234i	0.665753	+	0.746172i	$0.268110\pi$
$103$	15.1456i	1.49234i	−0.665753	+	0.746172i	$0.731890\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 12.9556i	− 1.25246i	−0.779637	−	0.626232i	$-0.784596\pi$
$107$	− 12.9556i	− 1.25246i	0.779637	−	0.626232i	$-0.215404\pi$
$108$	0	0
$109$	13.3771	1.28129	0.640647	−	0.767836i	$-0.278666\pi$
$109$	13.3771	1.28129	0.640647	+	0.767836i	$0.278666\pi$
$110$	0	0
$111$	8.39145	0.796482
$112$	0	0
$113$	− 1.26454i	− 0.118957i	−0.998230	−	0.0594787i	$-0.981056\pi$
$113$	− 1.26454i	− 0.118957i	0.998230	−	0.0594787i	$-0.0189439\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	1.32331i	0.122340i
$118$	0	0
$119$	26.0474	2.38777
$120$	0	0
$121$	19.1363	1.73966
$122$	0	0
$123$	11.4897i	1.03599i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	20.3327i	1.80423i	0.431492	+	0.902117i	$0.357987\pi$
$127$	20.3327i	1.80423i	−0.431492	+	0.902117i	$0.642013\pi$
$128$	0	0
$129$	0.421512	0.0371120
$130$	0	0
$131$	19.9937	1.74685	0.873427	−	0.486955i	$-0.161892\pi$
$131$	19.9937	1.74685	0.873427	+	0.486955i	$0.161892\pi$
$132$	0	0
$133$	− 18.9793i	− 1.64571i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 11.7986i	− 1.00802i	−0.863697	−	0.504011i	$-0.831857\pi$
$137$	− 11.7986i	− 1.00802i	0.863697	−	0.504011i	$-0.168143\pi$
$138$	0	0
$139$	−2.64663	−0.224484	−0.112242	−	0.993681i	$-0.535803\pi$
$139$	−2.64663	−0.224484	−0.112242	+	0.993681i	$0.535803\pi$
$140$	0	0
$141$	1.46593	0.123454
$142$	0	0
$143$	7.26454i	0.607491i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	− 7.02372i	− 0.579307i
$148$	0	0
$149$	0.510348	0.0418093	0.0209047	−	0.999781i	$-0.493345\pi$
$149$	0.510348	0.0418093	0.0209047	+	0.999781i	$0.493345\pi$
$150$	0	0
$151$	−1.24081	−0.100976	−0.0504880	−	0.998725i	$-0.516078\pi$
$151$	−1.24081	−0.100976	−0.0504880	+	0.998725i	$0.516078\pi$
$152$	0	0
$153$	6.95558i	0.562325i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	7.71477i	0.615706i	0.951434	+	0.307853i	$0.0996104\pi$
$157$	7.71477i	0.615706i	−0.951434	+	0.307853i	$0.900390\pi$
$158$	0	0
$159$	−4.53407	−0.359575
$160$	0	0
$161$	−7.91116	−0.623487
$162$	0	0
$163$	− 1.18703i	− 0.0929756i	−0.998919	−	0.0464878i	$-0.985197\pi$
$163$	− 1.18703i	− 0.0929756i	0.998919	−	0.0464878i	$-0.0148029\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 8.55779i	− 0.662222i	−0.943592	−	0.331111i	$-0.892577\pi$
$167$	− 8.55779i	− 0.662222i	0.943592	−	0.331111i	$-0.107423\pi$
$168$	0	0
$169$	11.2488	0.865295
$170$	0	0
$171$	5.06814	0.387570
$172$	0	0
$173$	2.55779i	0.194465i	0.995262	+	0.0972327i	$0.0309991\pi$
$173$	2.55779i	0.194465i	−0.995262	+	0.0972327i	$0.969001\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	− 4.78924i	− 0.359982i
$178$	0	0
$179$	−24.8905	−1.86040	−0.930200	−	0.367052i	$-0.880367\pi$
$179$	−24.8905	−1.86040	−0.930200	+	0.367052i	$0.880367\pi$
$180$	0	0
$181$	17.8223	1.32472	0.662362	−	0.749184i	$-0.269554\pi$
$181$	17.8223	1.32472	0.662362	+	0.749184i	$0.269554\pi$
$182$	0	0
$183$	6.84302i	0.505851i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	38.1837i	2.79227i
$188$	0	0
$189$	3.74483	0.272396
$190$	0	0
$191$	−10.9255	−0.790543	−0.395272	−	0.918564i	$-0.629350\pi$
$191$	−10.9255	−0.790543	−0.395272	+	0.918564i	$0.629350\pi$
$192$	0	0
$193$	15.2645i	1.09877i	0.835571	+	0.549383i	$0.185137\pi$
$193$	15.2645i	1.09877i	−0.835571	+	0.549383i	$0.814863\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 6.33768i	− 0.451541i	−0.974181	−	0.225770i	$-0.927510\pi$
$197$	− 6.33768i	− 0.451541i	0.974181	−	0.225770i	$-0.0724899\pi$
$198$	0	0
$199$	3.55477	0.251991	0.125995	−	0.992031i	$-0.459788\pi$
$199$	3.55477	0.251991	0.125995	+	0.992031i	$0.459788\pi$
$200$	0	0
$201$	−4.39145	−0.309749
$202$	0	0
$203$	− 13.6022i	− 0.954688i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	− 2.11256i	− 0.146833i
$208$	0	0
$209$	27.8223	1.92451
$210$	0	0
$211$	−12.5578	−0.864514	−0.432257	−	0.901750i	$-0.642283\pi$
$211$	−12.5578	−0.864514	−0.432257	+	0.901750i	$0.642283\pi$
$212$	0	0
$213$	− 7.34704i	− 0.503411i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	3.74483i	0.254215i
$218$	0	0
$219$	4.39145	0.296747
$220$	0	0
$221$	−9.20442	−0.619156
$222$	0	0
$223$	17.2044i	1.15209i	0.817417	+	0.576047i	$0.195405\pi$
$223$	17.2044i	1.15209i	−0.817417	+	0.576047i	$0.804595\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 19.0919i	− 1.26717i	−0.773673	−	0.633586i	$-0.781582\pi$
$227$	− 19.0919i	− 1.26717i	0.773673	−	0.633586i	$-0.218418\pi$
$228$	0	0
$229$	−12.4690	−0.823972	−0.411986	−	0.911190i	$-0.635165\pi$
$229$	−12.4690	−0.823972	−0.411986	+	0.911190i	$0.635165\pi$
$230$	0	0
$231$	20.5578	1.35260
$232$	0	0
$233$	− 2.75919i	− 0.180760i	−0.995907	−	0.0903802i	$-0.971192\pi$
$233$	− 2.75919i	− 0.180760i	0.995907	−	0.0903802i	$-0.0288082\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	14.0237i	0.910939i
$238$	0	0
$239$	6.61791	0.428077	0.214038	−	0.976825i	$-0.431338\pi$
$239$	6.61791	0.428077	0.214038	+	0.976825i	$0.431338\pi$
$240$	0	0
$241$	−5.26454	−0.339119	−0.169559	−	0.985520i	$-0.554234\pi$
$241$	−5.26454	−0.339119	−0.169559	+	0.985520i	$0.554234\pi$
$242$	0	0
$243$	1.00000i	0.0641500i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	6.70674i	0.426740i
$248$	0	0
$249$	−5.04442	−0.319677
$250$	0	0
$251$	20.0474	1.26538	0.632692	−	0.774404i	$-0.281950\pi$
$251$	20.0474	1.26538	0.632692	+	0.774404i	$0.281950\pi$
$252$	0	0
$253$	− 11.5972i	− 0.729110i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 25.7385i	− 1.60552i	−0.596300	−	0.802761i	$-0.703363\pi$
$257$	− 25.7385i	− 1.60552i	0.596300	−	0.802761i	$-0.296637\pi$
$258$	0	0
$259$	31.4245	1.95263
$260$	0	0
$261$	3.63227	0.224832
$262$	0	0
$263$	16.3327i	1.00712i	0.863961	+	0.503558i	$0.167976\pi$
$263$	16.3327i	1.00712i	−0.863961	+	0.503558i	$0.832024\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	13.3470i	0.816825i
$268$	0	0
$269$	−19.4546	−1.18617	−0.593084	−	0.805141i	$-0.702090\pi$
$269$	−19.4546	−1.18617	−0.593084	+	0.805141i	$0.702090\pi$
$270$	0	0
$271$	−28.7829	−1.74844	−0.874219	−	0.485533i	$-0.838626\pi$
$271$	−28.7829	−1.74844	−0.874219	+	0.485533i	$0.838626\pi$
$272$	0	0
$273$	4.95558i	0.299925i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	− 15.0381i	− 0.903551i	−0.892132	−	0.451775i	$-0.850791\pi$
$277$	− 15.0381i	− 0.903551i	0.892132	−	0.451775i	$-0.149209\pi$
$278$	0	0
$279$	−1.00000	−0.0598684
$280$	0	0
$281$	18.7829	1.12049	0.560247	−	0.828325i	$-0.310706\pi$
$281$	18.7829	1.12049	0.560247	+	0.828325i	$0.310706\pi$
$282$	0	0
$283$	− 23.7923i	− 1.41430i	−0.707062	−	0.707152i	$-0.749980\pi$
$283$	− 23.7923i	− 1.41430i	0.707062	−	0.707152i	$-0.250020\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	43.0267i	2.53979i
$288$	0	0
$289$	−31.3801	−1.84589
$290$	0	0
$291$	−3.48965	−0.204567
$292$	0	0
$293$	18.3090i	1.06962i	0.844972	+	0.534810i	$0.179617\pi$
$293$	18.3090i	1.06962i	−0.844972	+	0.534810i	$0.820383\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	5.48965i	0.318542i
$298$	0	0
$299$	2.79558	0.161672
$300$	0	0
$301$	1.57849	0.0909825
$302$	0	0
$303$	− 13.4008i	− 0.769857i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	− 6.72413i	− 0.383766i	−0.981418	−	0.191883i	$-0.938541\pi$
$307$	− 6.72413i	− 0.383766i	0.981418	−	0.191883i	$-0.0614595\pi$
$308$	0	0
$309$	−15.1456	−0.861605
$310$	0	0
$311$	24.8080	1.40673	0.703365	−	0.710828i	$-0.251680\pi$
$311$	24.8080	1.40673	0.703365	+	0.710828i	$0.251680\pi$
$312$	0	0
$313$	29.5959i	1.67286i	0.548075	+	0.836429i	$0.315361\pi$
$313$	29.5959i	1.67286i	−0.548075	+	0.836429i	$0.684639\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	13.7385i	0.771631i	0.922576	+	0.385815i	$0.126080\pi$
$317$	13.7385i	0.771631i	−0.922576	+	0.385815i	$0.873920\pi$
$318$	0	0
$319$	19.9399	1.11642
$320$	0	0
$321$	12.9556	0.723110
$322$	0	0
$323$	35.2519i	1.96147i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	13.3771i	0.739755i
$328$	0	0
$329$	5.48965	0.302654
$330$	0	0
$331$	0.908137	0.0499157	0.0249579	−	0.999689i	$-0.492055\pi$
$331$	0.908137	0.0499157	0.0249579	+	0.999689i	$0.492055\pi$
$332$	0	0
$333$	8.39145i	0.459849i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	− 13.2345i	− 0.720928i	−0.932773	−	0.360464i	$-0.882618\pi$
$337$	− 13.2345i	− 0.720928i	0.932773	−	0.360464i	$-0.117382\pi$
$338$	0	0
$339$	1.26454	0.0686801
$340$	0	0
$341$	−5.48965	−0.297281
$342$	0	0
$343$	− 0.0888361i	− 0.00479670i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 1.12825i	− 0.0605679i	−0.999541	−	0.0302839i	$-0.990359\pi$
$347$	− 1.12825i	− 0.0605679i	0.999541	−	0.0302839i	$-0.00964115\pi$
$348$	0	0
$349$	−6.67035	−0.357056	−0.178528	−	0.983935i	$-0.557133\pi$
$349$	−6.67035	−0.357056	−0.178528	+	0.983935i	$0.557133\pi$
$350$	0	0
$351$	−1.32331	−0.0706333
$352$	0	0
$353$	− 0.201395i	− 0.0107191i	−0.999986	−	0.00535957i	$-0.998294\pi$
$353$	− 0.201395i	− 0.0107191i	0.999986	−	0.00535957i	$-0.00170601\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	26.0474i	1.37858i
$358$	0	0
$359$	−14.2375	−0.751427	−0.375713	−	0.926736i	$-0.622602\pi$
$359$	−14.2375	−0.751427	−0.375713	+	0.926736i	$0.622602\pi$
$360$	0	0
$361$	6.68605	0.351897
$362$	0	0
$363$	19.1363i	1.00439i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	16.0000i	0.835193i	0.908633	+	0.417597i	$0.137127\pi$
$367$	16.0000i	0.835193i	−0.908633	+	0.417597i	$0.862873\pi$
$368$	0	0
$369$	−11.4897	−0.598127
$370$	0	0
$371$	−16.9793	−0.881522
$372$	0	0
$373$	24.8717i	1.28781i	0.765105	+	0.643905i	$0.222687\pi$
$373$	24.8717i	1.28781i	−0.765105	+	0.643905i	$0.777313\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	4.80663i	0.247554i
$378$	0	0
$379$	31.2044	1.60286	0.801432	−	0.598086i	$-0.204072\pi$
$379$	31.2044	1.60286	0.801432	+	0.598086i	$0.204072\pi$
$380$	0	0
$381$	−20.3327	−1.04167
$382$	0	0
$383$	− 28.9142i	− 1.47745i	−0.674009	−	0.738723i	$-0.735429\pi$
$383$	− 28.9142i	− 1.47745i	0.674009	−	0.738723i	$-0.264571\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0.421512i	0.0214266i
$388$	0	0
$389$	33.1219	1.67935	0.839674	−	0.543091i	$-0.182746\pi$
$389$	33.1219	1.67935	0.839674	+	0.543091i	$0.182746\pi$
$390$	0	0
$391$	14.6941	0.743111
$392$	0	0
$393$	19.9937i	1.00855i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	− 13.5972i	− 0.682424i	−0.939986	−	0.341212i	$-0.889163\pi$
$397$	− 13.5972i	− 0.682424i	0.939986	−	0.341212i	$-0.110837\pi$
$398$	0	0
$399$	18.9793	0.950154
$400$	0	0
$401$	1.34704	0.0672678	0.0336339	−	0.999434i	$-0.489292\pi$
$401$	1.34704	0.0672678	0.0336339	+	0.999434i	$0.489292\pi$
$402$	0	0
$403$	− 1.32331i	− 0.0659190i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	46.0662i	2.28342i
$408$	0	0
$409$	36.8016	1.81972	0.909862	−	0.414911i	$-0.136187\pi$
$409$	36.8016	1.81972	0.909862	+	0.414911i	$0.136187\pi$
$410$	0	0
$411$	11.7986	0.581982
$412$	0	0
$413$	− 17.9349i	− 0.882518i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	− 2.64663i	− 0.129606i
$418$	0	0
$419$	−14.1012	−0.688890	−0.344445	−	0.938807i	$-0.611933\pi$
$419$	−14.1012	−0.688890	−0.344445	+	0.938807i	$0.611933\pi$
$420$	0	0
$421$	−29.0919	−1.41785	−0.708925	−	0.705284i	$-0.750820\pi$
$421$	−29.0919	−1.41785	−0.708925	+	0.705284i	$0.750820\pi$
$422$	0	0
$423$	1.46593i	0.0712759i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	25.6259i	1.24013i
$428$	0	0
$429$	−7.26454	−0.350735
$430$	0	0
$431$	−21.3658	−1.02915	−0.514576	−	0.857445i	$-0.672051\pi$
$431$	−21.3658	−1.02915	−0.514576	+	0.857445i	$0.672051\pi$
$432$	0	0
$433$	− 10.5277i	− 0.505931i	−0.967475	−	0.252965i	$-0.918594\pi$
$433$	− 10.5277i	− 0.505931i	0.967475	−	0.252965i	$-0.0814059\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 10.7067i	− 0.512173i
$438$	0	0
$439$	−24.0949	−1.14999	−0.574993	−	0.818158i	$-0.694995\pi$
$439$	−24.0949	−1.14999	−0.574993	+	0.818158i	$0.694995\pi$
$440$	0	0
$441$	7.02372	0.334463
$442$	0	0
$443$	− 33.8935i	− 1.61033i	−0.593052	−	0.805164i	$-0.702077\pi$
$443$	− 33.8935i	− 1.61033i	0.593052	−	0.805164i	$-0.297923\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	0.510348i	0.0241386i
$448$	0	0
$449$	33.9048	1.60007	0.800034	−	0.599955i	$-0.204815\pi$
$449$	33.9048	1.60007	0.800034	+	0.599955i	$0.204815\pi$
$450$	0	0
$451$	−63.0742	−2.97005
$452$	0	0
$453$	− 1.24081i	− 0.0582985i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 2.36273i	− 0.110524i	−0.998472	−	0.0552620i	$-0.982401\pi$
$457$	− 2.36273i	− 0.110524i	0.998472	−	0.0552620i	$-0.0175994\pi$
$458$	0	0
$459$	−6.95558	−0.324659
$460$	0	0
$461$	−1.10320	−0.0513810	−0.0256905	−	0.999670i	$-0.508178\pi$
$461$	−1.10320	−0.0513810	−0.0256905	+	0.999670i	$0.508178\pi$
$462$	0	0
$463$	− 16.6466i	− 0.773634i	−0.922156	−	0.386817i	$-0.873574\pi$
$463$	− 16.6466i	− 0.773634i	0.922156	−	0.386817i	$-0.126426\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	1.46593i	0.0678351i	0.999425	+	0.0339176i	$0.0107984\pi$
$467$	1.46593i	0.0678351i	−0.999425	+	0.0339176i	$0.989202\pi$
$468$	0	0
$469$	−16.4452	−0.759370
$470$	0	0
$471$	−7.71477	−0.355478
$472$	0	0
$473$	2.31395i	0.106396i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	− 4.53407i	− 0.207601i
$478$	0	0
$479$	−1.99367	−0.0910929	−0.0455464	−	0.998962i	$-0.514503\pi$
$479$	−1.99367	−0.0910929	−0.0455464	+	0.998962i	$0.514503\pi$
$480$	0	0
$481$	−11.1045	−0.506323
$482$	0	0
$483$	− 7.91116i	− 0.359970i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	− 38.5164i	− 1.74534i	−0.488306	−	0.872672i	$-0.662385\pi$
$487$	− 38.5164i	− 1.74534i	0.488306	−	0.872672i	$-0.337615\pi$
$488$	0	0
$489$	1.18703	0.0536795
$490$	0	0
$491$	7.26454	0.327844	0.163922	−	0.986473i	$-0.447585\pi$
$491$	7.26454	0.327844	0.163922	+	0.986473i	$0.447585\pi$
$492$	0	0
$493$	25.2645i	1.13786i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 27.5134i	− 1.23414i
$498$	0	0
$499$	39.2519	1.75715	0.878577	−	0.477600i	$-0.158493\pi$
$499$	39.2519	1.75715	0.878577	+	0.477600i	$0.158493\pi$
$500$	0	0
$501$	8.55779	0.382334
$502$	0	0
$503$	− 5.69105i	− 0.253751i	−0.991919	−	0.126876i	$-0.959505\pi$
$503$	− 5.69105i	− 0.253751i	0.991919	−	0.126876i	$-0.0404949\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	11.2488i	0.499578i
$508$	0	0
$509$	28.3865	1.25821	0.629104	−	0.777321i	$-0.283422\pi$
$509$	28.3865	1.25821	0.629104	+	0.777321i	$0.283422\pi$
$510$	0	0
$511$	16.4452	0.727494
$512$	0	0
$513$	5.06814i	0.223764i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	8.04744i	0.353926i
$518$	0	0
$519$	−2.55779	−0.112275
$520$	0	0
$521$	12.7829	0.560029	0.280015	−	0.959996i	$-0.409661\pi$
$521$	12.7829	0.560029	0.280015	+	0.959996i	$0.409661\pi$
$522$	0	0
$523$	− 8.73546i	− 0.381975i	−0.981592	−	0.190988i	$-0.938831\pi$
$523$	− 8.73546i	− 0.381975i	0.981592	−	0.190988i	$-0.0611690\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 6.95558i	− 0.302990i
$528$	0	0
$529$	18.5371	0.805961
$530$	0	0
$531$	4.78924	0.207835
$532$	0	0
$533$	− 15.2044i	− 0.658577i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	− 24.8905i	− 1.07410i
$538$	0	0
$539$	38.5578	1.66080
$540$	0	0
$541$	4.65163	0.199989	0.0999946	−	0.994988i	$-0.468117\pi$
$541$	4.65163	0.199989	0.0999946	+	0.994988i	$0.468117\pi$
$542$	0	0
$543$	17.8223i	0.764829i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 38.9079i	− 1.66358i	−0.555091	−	0.831790i	$-0.687316\pi$
$547$	− 38.9079i	− 1.66358i	0.555091	−	0.831790i	$-0.312684\pi$
$548$	0	0
$549$	−6.84302	−0.292053
$550$	0	0
$551$	18.4088	0.784243
$552$	0	0
$553$	52.5164i	2.23322i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 19.6309i	− 0.831789i	−0.909413	−	0.415895i	$-0.863469\pi$
$557$	− 19.6309i	− 0.831789i	0.909413	−	0.415895i	$-0.136531\pi$
$558$	0	0
$559$	−0.557793	−0.0235921
$560$	0	0
$561$	−38.1837	−1.61212
$562$	0	0
$563$	− 16.7779i	− 0.707105i	−0.935415	−	0.353552i	$-0.884974\pi$
$563$	− 16.7779i	− 0.707105i	0.935415	−	0.353552i	$-0.115026\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	3.74483i	0.157268i
$568$	0	0
$569$	24.7004	1.03549	0.517747	−	0.855533i	$-0.326771\pi$
$569$	24.7004	1.03549	0.517747	+	0.855533i	$0.326771\pi$
$570$	0	0
$571$	9.80361	0.410268	0.205134	−	0.978734i	$-0.434237\pi$
$571$	9.80361	0.410268	0.205134	+	0.978734i	$0.434237\pi$
$572$	0	0
$573$	− 10.9255i	− 0.456420i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	2.70674i	0.112683i	0.998412	+	0.0563416i	$0.0179436\pi$
$577$	2.70674i	0.112683i	−0.998412	+	0.0563416i	$0.982056\pi$
$578$	0	0
$579$	−15.2645	−0.634372
$580$	0	0
$581$	−18.8905	−0.783709
$582$	0	0
$583$	− 24.8905i	− 1.03086i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 4.33268i	− 0.178829i	−0.995995	−	0.0894143i	$-0.971500\pi$
$587$	− 4.33268i	− 0.178829i	0.995995	−	0.0894143i	$-0.0284995\pi$
$588$	0	0
$589$	−5.06814	−0.208829
$590$	0	0
$591$	6.33768	0.260697
$592$	0	0
$593$	15.1757i	0.623191i	0.950215	+	0.311596i	$0.100863\pi$
$593$	15.1757i	0.623191i	−0.950215	+	0.311596i	$0.899137\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	3.55477i	0.145487i
$598$	0	0
$599$	34.0411	1.39088	0.695441	−	0.718583i	$-0.255209\pi$
$599$	34.0411	1.39088	0.695441	+	0.718583i	$0.255209\pi$
$600$	0	0
$601$	11.1757	0.455866	0.227933	−	0.973677i	$-0.426803\pi$
$601$	11.1757	0.455866	0.227933	+	0.973677i	$0.426803\pi$
$602$	0	0
$603$	− 4.39145i	− 0.178834i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	37.9699i	1.54115i	0.637348	+	0.770576i	$0.280031\pi$
$607$	37.9699i	1.54115i	−0.637348	+	0.770576i	$0.719969\pi$
$608$	0	0
$609$	13.6022	0.551189
$610$	0	0
$611$	−1.93989	−0.0784794
$612$	0	0
$613$	− 44.7302i	− 1.80664i	−0.428972	−	0.903318i	$-0.641124\pi$
$613$	− 44.7302i	− 1.80664i	0.428972	−	0.903318i	$-0.358876\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 14.1777i	− 0.570772i	−0.958413	−	0.285386i	$-0.907878\pi$
$617$	− 14.1777i	− 0.570772i	0.958413	−	0.285386i	$-0.0921217\pi$
$618$	0	0
$619$	−11.5421	−0.463916	−0.231958	−	0.972726i	$-0.574513\pi$
$619$	−11.5421	−0.463916	−0.231958	+	0.972726i	$0.574513\pi$
$620$	0	0
$621$	2.11256	0.0847740
$622$	0	0
$623$	49.9823i	2.00250i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	27.8223i	1.11112i
$628$	0	0
$629$	−58.3675	−2.32726
$630$	0	0
$631$	19.2519	0.766405	0.383202	−	0.923664i	$-0.374821\pi$
$631$	19.2519	0.766405	0.383202	+	0.923664i	$0.374821\pi$
$632$	0	0
$633$	− 12.5578i	− 0.499127i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	9.29459i	0.368265i
$638$	0	0
$639$	7.34704	0.290644
$640$	0	0
$641$	27.3945	1.08202	0.541008	−	0.841017i	$-0.318043\pi$
$641$	27.3945	1.08202	0.541008	+	0.841017i	$0.318043\pi$
$642$	0	0
$643$	− 11.2645i	− 0.444230i	−0.975020	−	0.222115i	$-0.928704\pi$
$643$	− 11.2645i	− 0.444230i	0.975020	−	0.222115i	$-0.0712960\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 37.1630i	− 1.46103i	−0.682897	−	0.730515i	$-0.739280\pi$
$647$	− 37.1630i	− 1.46103i	0.682897	−	0.730515i	$-0.260720\pi$
$648$	0	0
$649$	26.2913	1.03202
$650$	0	0
$651$	−3.74483	−0.146771
$652$	0	0
$653$	1.06314i	0.0416039i	0.999784	+	0.0208020i	$0.00662195\pi$
$653$	1.06314i	0.0416039i	−0.999784	+	0.0208020i	$0.993378\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	4.39145i	0.171327i
$658$	0	0
$659$	14.3965	0.560806	0.280403	−	0.959882i	$-0.409532\pi$
$659$	14.3965	0.560806	0.280403	+	0.959882i	$0.409532\pi$
$660$	0	0
$661$	9.26454	0.360349	0.180174	−	0.983635i	$-0.442334\pi$
$661$	9.26454	0.360349	0.180174	+	0.983635i	$0.442334\pi$
$662$	0	0
$663$	− 9.20442i	− 0.357470i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 7.67338i	− 0.297114i
$668$	0	0
$669$	−17.2044	−0.665161
$670$	0	0
$671$	−37.5658	−1.45021
$672$	0	0
$673$	− 6.84169i	− 0.263728i	−0.991268	−	0.131864i	$-0.957904\pi$
$673$	− 6.84169i	− 0.263728i	0.991268	−	0.131864i	$-0.0420962\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	18.7178i	0.719383i	0.933071	+	0.359692i	$0.117118\pi$
$677$	18.7178i	0.719383i	−0.933071	+	0.359692i	$0.882882\pi$
$678$	0	0
$679$	−13.0681	−0.501509
$680$	0	0
$681$	19.0919	0.731602
$682$	0	0
$683$	35.5895i	1.36180i	0.732378	+	0.680898i	$0.238410\pi$
$683$	35.5895i	1.36180i	−0.732378	+	0.680898i	$0.761590\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	− 12.4690i	− 0.475720i
$688$	0	0
$689$	6.00000	0.228582
$690$	0	0
$691$	7.21709	0.274551	0.137276	−	0.990533i	$-0.456165\pi$
$691$	7.21709	0.274551	0.137276	+	0.990533i	$0.456165\pi$
$692$	0	0
$693$	20.5578i	0.780926i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	− 79.9172i	− 3.02708i
$698$	0	0
$699$	2.75919	0.104362
$700$	0	0
$701$	−32.7228	−1.23592	−0.617961	−	0.786208i	$-0.712041\pi$
$701$	−32.7228	−1.23592	−0.617961	+	0.786208i	$0.712041\pi$
$702$	0	0
$703$	42.5291i	1.60401i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 50.1837i	− 1.88735i
$708$	0	0
$709$	25.8223	0.969778	0.484889	−	0.874576i	$-0.338860\pi$
$709$	25.8223	0.969778	0.484889	+	0.874576i	$0.338860\pi$
$710$	0	0
$711$	−14.0237	−0.525931
$712$	0	0
$713$	2.11256i	0.0791159i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	6.61791i	0.247150i
$718$	0	0
$719$	40.4690	1.50924	0.754619	−	0.656164i	$-0.227822\pi$
$719$	40.4690	1.50924	0.754619	+	0.656164i	$0.227822\pi$
$720$	0	0
$721$	−56.7178	−2.11228
$722$	0	0
$723$	− 5.26454i	− 0.195790i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	20.2138i	0.749688i	0.927088	+	0.374844i	$0.122304\pi$
$727$	20.2138i	0.749688i	−0.927088	+	0.374844i	$0.877696\pi$
$728$	0	0
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	−2.93186	−0.108439
$732$	0	0
$733$	16.3140i	0.602570i	0.953534	+	0.301285i	$0.0974155\pi$
$733$	16.3140i	0.602570i	−0.953534	+	0.301285i	$0.902585\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 24.1076i	− 0.888013i
$738$	0	0
$739$	−23.7799	−0.874757	−0.437379	−	0.899277i	$-0.644093\pi$
$739$	−23.7799	−0.874757	−0.437379	+	0.899277i	$0.644093\pi$
$740$	0	0
$741$	−6.70674	−0.246378
$742$	0	0
$743$	− 45.7208i	− 1.67733i	−0.544644	−	0.838667i	$-0.683335\pi$
$743$	− 45.7208i	− 1.67733i	0.544644	−	0.838667i	$-0.316665\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	− 5.04442i	− 0.184566i
$748$	0	0
$749$	48.5164	1.77275
$750$	0	0
$751$	−2.81430	−0.102695	−0.0513477	−	0.998681i	$-0.516352\pi$
$751$	−2.81430	−0.102695	−0.0513477	+	0.998681i	$0.516352\pi$
$752$	0	0
$753$	20.0474i	0.730569i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	48.1724i	1.75086i	0.483349	+	0.875428i	$0.339420\pi$
$757$	48.1724i	1.75086i	−0.483349	+	0.875428i	$0.660580\pi$
$758$	0	0
$759$	11.5972	0.420952
$760$	0	0
$761$	46.4212	1.68277	0.841384	−	0.540438i	$-0.181741\pi$
$761$	46.4212	1.68277	0.841384	+	0.540438i	$0.181741\pi$
$762$	0	0
$763$	50.0949i	1.81356i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	6.33768i	0.228840i
$768$	0	0
$769$	51.1580	1.84481	0.922403	−	0.386229i	$-0.126223\pi$
$769$	51.1580	1.84481	0.922403	+	0.386229i	$0.126223\pi$
$770$	0	0
$771$	25.7385	0.926949
$772$	0	0
$773$	52.1661i	1.87628i	0.346253	+	0.938141i	$0.387454\pi$
$773$	52.1661i	1.87628i	−0.346253	+	0.938141i	$0.612546\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	31.4245i	1.12735i
$778$	0	0
$779$	−58.2312	−2.08635
$780$	0	0
$781$	40.3327	1.44322
$782$	0	0
$783$	3.63227i	0.129807i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	36.8016i	1.31184i	0.754832	+	0.655918i	$0.227718\pi$
$787$	36.8016i	1.31184i	−0.754832	+	0.655918i	$0.772282\pi$
$788$	0	0
$789$	−16.3327	−0.581459
$790$	0	0
$791$	4.73546	0.168374
$792$	0	0
$793$	− 9.05547i	− 0.321569i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 24.8193i	− 0.879145i	−0.898207	−	0.439572i	$-0.855130\pi$
$797$	− 24.8193i	− 0.879145i	0.898207	−	0.439572i	$-0.144870\pi$
$798$	0	0
$799$	−10.1964	−0.360723
$800$	0	0
$801$	−13.3470	−0.471594
$802$	0	0
$803$	24.1076i	0.850737i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	− 19.4546i	− 0.684834i
$808$	0	0
$809$	29.9523	1.05307	0.526533	−	0.850155i	$-0.323492\pi$
$809$	29.9523	1.05307	0.526533	+	0.850155i	$0.323492\pi$
$810$	0	0
$811$	13.3534	0.468900	0.234450	−	0.972128i	$-0.424671\pi$
$811$	13.3534	0.468900	0.234450	+	0.972128i	$0.424671\pi$
$812$	0	0
$813$	− 28.7829i	− 1.00946i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	2.13628i	0.0747390i
$818$	0	0
$819$	−4.95558	−0.173162
$820$	0	0
$821$	31.9937	1.11659	0.558293	−	0.829644i	$-0.311456\pi$
$821$	31.9937	1.11659	0.558293	+	0.829644i	$0.311456\pi$
$822$	0	0
$823$	− 9.48965i	− 0.330788i	−0.986228	−	0.165394i	$-0.947110\pi$
$823$	− 9.48965i	− 0.330788i	0.986228	−	0.165394i	$-0.0528897\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	10.7779i	0.374785i	0.982285	+	0.187392i	$0.0600036\pi$
$827$	10.7779i	0.374785i	−0.982285	+	0.187392i	$0.939996\pi$
$828$	0	0
$829$	−14.1076	−0.489976	−0.244988	−	0.969526i	$-0.578784\pi$
$829$	−14.1076	−0.489976	−0.244988	+	0.969526i	$0.578784\pi$
$830$	0	0
$831$	15.0381	0.521665
$832$	0	0
$833$	48.8541i	1.69269i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	− 1.00000i	− 0.0345651i
$838$	0	0
$839$	33.2295	1.14721	0.573605	−	0.819132i	$-0.305545\pi$
$839$	33.2295	1.14721	0.573605	+	0.819132i	$0.305545\pi$
$840$	0	0
$841$	−15.8066	−0.545056
$842$	0	0
$843$	18.7829i	0.646918i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	71.6620i	2.46234i
$848$	0	0
$849$	23.7923	0.816549
$850$	0	0
$851$	17.7274	0.607689
$852$	0	0
$853$	− 5.63860i	− 0.193062i	−0.995330	−	0.0965310i	$-0.969225\pi$
$853$	− 5.63860i	− 0.193062i	0.995330	−	0.0965310i	$-0.0307747\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 2.55779i	− 0.0873725i	−0.999045	−	0.0436863i	$-0.986090\pi$
$857$	− 2.55779i	− 0.0873725i	0.999045	−	0.0436863i	$-0.0139102\pi$
$858$	0	0
$859$	46.8965	1.60009	0.800044	−	0.599941i	$-0.204809\pi$
$859$	46.8965	1.60009	0.800044	+	0.599941i	$0.204809\pi$
$860$	0	0
$861$	−43.0267	−1.46635
$862$	0	0
$863$	− 8.39279i	− 0.285694i	−0.989745	−	0.142847i	$-0.954374\pi$
$863$	− 8.39279i	− 0.285694i	0.989745	−	0.142847i	$-0.0456257\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	− 31.3801i	− 1.06572i
$868$	0	0
$869$	−76.9854	−2.61155
$870$	0	0
$871$	5.81127	0.196908
$872$	0	0
$873$	− 3.48965i	− 0.118107i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 44.4877i	− 1.50224i	−0.660164	−	0.751121i	$-0.729513\pi$
$877$	− 44.4877i	− 1.50224i	0.660164	−	0.751121i	$-0.270487\pi$
$878$	0	0
$879$	−18.3090	−0.617546
$880$	0	0
$881$	−28.5515	−0.961923	−0.480962	−	0.876742i	$-0.659712\pi$
$881$	−28.5515	−0.961923	−0.480962	+	0.876742i	$0.659712\pi$
$882$	0	0
$883$	33.4295i	1.12499i	0.826799	+	0.562497i	$0.190159\pi$
$883$	33.4295i	1.12499i	−0.826799	+	0.562497i	$0.809841\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	12.0424i	0.404346i	0.979350	+	0.202173i	$0.0648003\pi$
$887$	12.0424i	0.404346i	−0.979350	+	0.202173i	$0.935200\pi$
$888$	0	0
$889$	−76.1423	−2.55373
$890$	0	0
$891$	−5.48965	−0.183910
$892$	0	0
$893$	7.42954i	0.248620i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	2.79558i	0.0933417i
$898$	0	0
$899$	−3.63227	−0.121143
$900$	0	0
$901$	31.5371	1.05065
$902$	0	0
$903$	1.57849i	0.0525288i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	− 49.7509i	− 1.65195i	−0.563706	−	0.825975i	$-0.690625\pi$
$907$	− 49.7509i	− 1.65195i	0.563706	−	0.825975i	$-0.309375\pi$
$908$	0	0
$909$	13.4008	0.444477
$910$	0	0
$911$	−27.1757	−0.900371	−0.450186	−	0.892935i	$-0.648642\pi$
$911$	−27.1757	−0.900371	−0.450186	+	0.892935i	$0.648642\pi$
$912$	0	0
$913$	− 27.6921i	− 0.916475i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	74.8728i	2.47252i
$918$	0	0
$919$	−17.6860	−0.583409	−0.291704	−	0.956509i	$-0.594222\pi$
$919$	−17.6860	−0.583409	−0.291704	+	0.956509i	$0.594222\pi$
$920$	0	0
$921$	6.72413	0.221568
$922$	0	0
$923$	9.72244i	0.320018i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	− 15.1456i	− 0.497448i
$928$	0	0
$929$	38.5040	1.26328	0.631638	−	0.775264i	$-0.282383\pi$
$929$	38.5040	1.26328	0.631638	+	0.775264i	$0.282383\pi$
$930$	0	0
$931$	35.5972	1.16665
$932$	0	0
$933$	24.8080i	0.812176i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	− 4.96058i	− 0.162055i	−0.996712	−	0.0810276i	$-0.974180\pi$
$937$	− 4.96058i	− 0.162055i	0.996712	−	0.0810276i	$-0.0258202\pi$
$938$	0	0
$939$	−29.5959	−0.965825
$940$	0	0
$941$	8.88413	0.289614	0.144807	−	0.989460i	$-0.453744\pi$
$941$	8.88413	0.289614	0.144807	+	0.989460i	$0.453744\pi$
$942$	0	0
$943$	24.2726i	0.790423i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	6.06511i	0.197090i	0.995133	+	0.0985449i	$0.0314188\pi$
$947$	6.06511i	0.197090i	−0.995133	+	0.0985449i	$0.968581\pi$
$948$	0	0
$949$	−5.81127	−0.188642
$950$	0	0
$951$	−13.7385	−0.445501
$952$	0	0
$953$	− 28.6416i	− 0.927793i	−0.885889	−	0.463897i	$-0.846451\pi$
$953$	− 28.6416i	− 0.927793i	0.885889	−	0.463897i	$-0.153549\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	19.9399i	0.644565i
$958$	0	0
$959$	44.1837	1.42677
$960$	0	0
$961$	1.00000	0.0322581
$962$	0	0
$963$	12.9556i	0.417488i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	− 36.5765i	− 1.17622i	−0.808780	−	0.588111i	$-0.799872\pi$
$967$	− 36.5765i	− 1.17622i	0.808780	−	0.588111i	$-0.200128\pi$
$968$	0	0
$969$	−35.2519	−1.13245
$970$	0	0
$971$	−6.16134	−0.197727	−0.0988634	−	0.995101i	$-0.531521\pi$
$971$	−6.16134	−0.197727	−0.0988634	+	0.995101i	$0.531521\pi$
$972$	0	0
$973$	− 9.91116i	− 0.317737i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	21.8223i	0.698158i	0.937093	+	0.349079i	$0.113506\pi$
$977$	21.8223i	0.698158i	−0.937093	+	0.349079i	$0.886494\pi$
$978$	0	0
$979$	−73.2706	−2.34174
$980$	0	0
$981$	−13.3771	−0.427098
$982$	0	0
$983$	9.06814i	0.289229i	0.989488	+	0.144614i	$0.0461942\pi$
$983$	9.06814i	0.289229i	−0.989488	+	0.144614i	$0.953806\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	5.48965i	0.174738i
$988$	0	0
$989$	0.890468	0.0283152
$990$	0	0
$991$	−0.155004	−0.00492385	−0.00246192	−	0.999997i	$-0.500784\pi$
$991$	−0.155004	−0.00492385	−0.00246192	+	0.999997i	$0.500784\pi$
$992$	0	0
$993$	0.908137i	0.0288189i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 44.2438i	− 1.40122i	−0.713546	−	0.700608i	$-0.752912\pi$
$997$	− 44.2438i	− 1.40122i	0.713546	−	0.700608i	$-0.247088\pi$
$998$	0	0
$999$	−8.39145	−0.265494

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9300.2.g.q.3349.6		6
5.2	odd	4		1860.2.a.g.1.1	✓	3
5.3	odd	4		9300.2.a.u.1.3		3
5.4	even	2	inner	9300.2.g.q.3349.1		6
15.2	even	4		5580.2.a.j.1.1		3
20.7	even	4		7440.2.a.bn.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1860.2.a.g.1.1	✓	3	5.2	odd	4
5580.2.a.j.1.1		3	15.2	even	4
7440.2.a.bn.1.3		3	20.7	even	4
9300.2.a.u.1.3		3	5.3	odd	4
9300.2.g.q.3349.1		6	5.4	even	2	inner
9300.2.g.q.3349.6		6	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 9300 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9300.g (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{3} +3.74483i q^{7} -1.00000 q^{9} -5.48965 q^{11} -1.32331i q^{13} -6.95558i q^{17} -5.06814 q^{19} -3.74483 q^{21} +2.11256i q^{23} -1.00000i q^{27} -3.63227 q^{29} +1.00000 q^{31} -5.48965i q^{33} -8.39145i q^{37} +1.32331 q^{39} +11.4897 q^{41} -0.421512i q^{43} -1.46593i q^{47} -7.02372 q^{49} +6.95558 q^{51} +4.53407i q^{53} -5.06814i q^{57} -4.78924 q^{59} +6.84302 q^{61} -3.74483i q^{63} +4.39145i q^{67} -2.11256 q^{69} -7.34704 q^{71} -4.39145i q^{73} -20.5578i q^{77} +14.0237 q^{79} +1.00000 q^{81} +5.04442i q^{83} -3.63227i q^{87} +13.3470 q^{89} +4.95558 q^{91} +1.00000i q^{93} +3.48965i q^{97} +5.48965 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 6 q^{9} + 4 q^{11} - 4 q^{21} - 12 q^{29} + 6 q^{31} - 4 q^{39} + 32 q^{41} + 10 q^{49} + 20 q^{51} - 32 q^{59} + 28 q^{61} - 4 q^{69} + 20 q^{71} + 32 q^{79} + 6 q^{81} + 16 q^{89} + 8 q^{91} - 4 q^{99}+O(q^{100}) \) 6 * q - 6 * q^9 + 4 * q^11 - 4 * q^21 - 12 * q^29 + 6 * q^31 - 4 * q^39 + 32 * q^41 + 10 * q^49 + 20 * q^51 - 32 * q^59 + 28 * q^61 - 4 * q^69 + 20 * q^71 + 32 * q^79 + 6 * q^81 + 16 * q^89 + 8 * q^91 - 4 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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