LMFDB - Embedded newform 8100.2.a.x.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [8100,2,Mod(1,8100)] 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("8100.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(8100, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 8100 = 2^{2} \cdot 3^{4} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8100.a (trivial)

Newform invariants

comment:select newform

sage:traces = [4,0,0,0,0,0,-1,0,0,0,-3,0,2,0,0,0,-9,0,-4,0,0,0,3] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(23)] == traces)

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

Self dual:	yes
Analytic conductor:	$64.6788256372$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.3981.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 4x^{2} + 2x + 1 $ x^4 - x^3 - 4x^2 + 2x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 3^{2} $
Twist minimal:	no (minimal twist has level 900)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.28400$ of defining polynomial
Character	$\chi$	$=$	8100.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-0.0864793 q^{7} -0.913521 q^{11} -2.62499 q^{13} -2.08648 q^{17} +4.93847 q^{19} -8.47698 q^{23} +2.39798 q^{29} +3.62499 q^{31} +5.85199 q^{37} +6.64994 q^{41} +8.24997 q^{43} -2.68850 q^{47} -6.99252 q^{49} +5.73642 q^{53} +12.3384 q^{59} -6.33645 q^{61} -6.16548 q^{67} -12.3905 q^{71} -5.31349 q^{73} +0.0790006 q^{77} -13.4479 q^{79} +6.07900 q^{83} -8.13440 q^{89} +0.227007 q^{91} -11.1020 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - q^{7} - 3 q^{11} + 2 q^{13} - 9 q^{17} - 4 q^{19} + 3 q^{23} + 9 q^{29} + 2 q^{31} - q^{37} - 9 q^{41} + 8 q^{43} - 12 q^{47} + 9 q^{49} - 12 q^{53} + 15 q^{59} - q^{61} + 11 q^{67} - 12 q^{71}+ \cdots + 5 q^{97}+O(q^{100}) $ 4 * q - q^7 - 3 * q^11 + 2 * q^13 - 9 * q^17 - 4 * q^19 + 3 * q^23 + 9 * q^29 + 2 * q^31 - q^37 - 9 * q^41 + 8 * q^43 - 12 * q^47 + 9 * q^49 - 12 * q^53 + 15 * q^59 - q^61 + 11 * q^67 - 12 * q^71 - 10 * q^73 - 36 * q^77 - 7 * q^79 - 12 * q^83 + 3 * q^89 - 11 * q^91 + 5 * q^97

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	0	0
$3$	0	0
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−0.0864793	−0.0326861	−0.0163431	−	0.999866i	$-0.505202\pi$
$7$	−0.0864793	−0.0326861	−0.0163431	+	0.999866i	$0.505202\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−0.913521	−0.275437	−0.137718	−	0.990471i	$-0.543977\pi$
$11$	−0.913521	−0.275437	−0.137718	+	0.990471i	$0.543977\pi$
$12$	0	0
$13$	−2.62499	−0.728040	−0.364020	−	0.931391i	$-0.618596\pi$
$13$	−2.62499	−0.728040	−0.364020	+	0.931391i	$0.618596\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−2.08648	−0.506046	−0.253023	−	0.967460i	$-0.581425\pi$
$17$	−2.08648	−0.506046	−0.253023	+	0.967460i	$0.581425\pi$
$18$	0	0
$19$	4.93847	1.13296	0.566482	−	0.824074i	$-0.308304\pi$
$19$	4.93847	1.13296	0.566482	+	0.824074i	$0.308304\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−8.47698	−1.76757	−0.883786	−	0.467891i	$-0.845014\pi$
$23$	−8.47698	−1.76757	−0.883786	+	0.467891i	$0.845014\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	0	0
$28$	0	0
$29$	2.39798	0.445294	0.222647	−	0.974899i	$-0.428530\pi$
$29$	2.39798	0.445294	0.222647	+	0.974899i	$0.428530\pi$
$30$	0	0
$31$	3.62499	0.651067	0.325533	−	0.945531i	$-0.394456\pi$
$31$	3.62499	0.651067	0.325533	+	0.945531i	$0.394456\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	5.85199	0.962062	0.481031	−	0.876704i	$-0.340262\pi$
$37$	5.85199	0.962062	0.481031	+	0.876704i	$0.340262\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	6.64994	1.03855	0.519273	−	0.854608i	$-0.326203\pi$
$41$	6.64994	1.03855	0.519273	+	0.854608i	$0.326203\pi$
$42$	0	0
$43$	8.24997	1.25811	0.629055	−	0.777361i	$-0.283442\pi$
$43$	8.24997	1.25811	0.629055	+	0.777361i	$0.283442\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−2.68850	−0.392158	−0.196079	−	0.980588i	$-0.562821\pi$
$47$	−2.68850	−0.392158	−0.196079	+	0.980588i	$0.562821\pi$
$48$	0	0
$49$	−6.99252	−0.998932
$50$	0	0
$51$	0	0
$52$	0	0
$53$	5.73642	0.787958	0.393979	−	0.919120i	$-0.371098\pi$
$53$	5.73642	0.787958	0.393979	+	0.919120i	$0.371098\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	0	0
$59$	12.3384	1.60633	0.803164	−	0.595758i	$-0.203148\pi$
$59$	12.3384	1.60633	0.803164	+	0.595758i	$0.203148\pi$
$60$	0	0
$61$	−6.33645	−0.811300	−0.405650	−	0.914029i	$-0.632955\pi$
$61$	−6.33645	−0.811300	−0.405650	+	0.914029i	$0.632955\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−6.16548	−0.753233	−0.376617	−	0.926369i	$-0.622913\pi$
$67$	−6.16548	−0.753233	−0.376617	+	0.926369i	$0.622913\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−12.3905	−1.47048	−0.735241	−	0.677806i	$-0.762931\pi$
$71$	−12.3905	−1.47048	−0.735241	+	0.677806i	$0.762931\pi$
$72$	0	0
$73$	−5.31349	−0.621897	−0.310948	−	0.950427i	$-0.600647\pi$
$73$	−5.31349	−0.621897	−0.310948	+	0.950427i	$0.600647\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0.0790006	0.00900296
$78$	0	0
$79$	−13.4479	−1.51301	−0.756503	−	0.653991i	$-0.773094\pi$
$79$	−13.4479	−1.51301	−0.756503	+	0.653991i	$0.773094\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	6.07900	0.667257	0.333629	−	0.942705i	$-0.391727\pi$
$83$	6.07900	0.667257	0.333629	+	0.942705i	$0.391727\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−8.13440	−0.862244	−0.431122	−	0.902294i	$-0.641882\pi$
$89$	−8.13440	−0.862244	−0.431122	+	0.902294i	$0.641882\pi$
$90$	0	0
$91$	0.227007	0.0237968
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−11.1020	−1.12723	−0.563617	−	0.826036i	$-0.690591\pi$
$97$	−11.1020	−1.12723	−0.563617	+	0.826036i	$0.690591\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−12.0790	−1.20191	−0.600953	−	0.799285i	$-0.705212\pi$
$101$	−12.0790	−1.20191	−0.600953	+	0.799285i	$0.705212\pi$
$102$	0	0
$103$	4.36753	0.430346	0.215173	−	0.976576i	$-0.430968\pi$
$103$	4.36753	0.430346	0.215173	+	0.976576i	$0.430968\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−18.9614	−1.83307	−0.916536	−	0.399953i	$-0.869027\pi$
$107$	−18.9614	−1.83307	−0.916536	+	0.399953i	$0.869027\pi$
$108$	0	0
$109$	15.1904	1.45498	0.727490	−	0.686119i	$-0.240687\pi$
$109$	15.1904	1.45498	0.727490	+	0.686119i	$0.240687\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−6.60202	−0.621066	−0.310533	−	0.950563i	$-0.600507\pi$
$113$	−6.60202	−0.621066	−0.310533	+	0.950563i	$0.600507\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	0	0
$118$	0	0
$119$	0.180437	0.0165407
$120$	0	0
$121$	−10.1655	−0.924135
$122$	0	0
$123$	0	0
$124$	0	0
$125$	0	0
$126$	0	0
$127$	7.42492	0.658855	0.329427	−	0.944181i	$-0.393144\pi$
$127$	7.42492	0.658855	0.329427	+	0.944181i	$0.393144\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−15.9614	−1.39456	−0.697279	−	0.716800i	$-0.745606\pi$
$131$	−15.9614	−1.39456	−0.697279	+	0.716800i	$0.745606\pi$
$132$	0	0
$133$	−0.427076	−0.0370322
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−18.3905	−1.57121	−0.785603	−	0.618731i	$-0.787647\pi$
$137$	−18.3905	−1.57121	−0.785603	+	0.618731i	$0.787647\pi$
$138$	0	0
$139$	5.56147	0.471718	0.235859	−	0.971787i	$-0.424210\pi$
$139$	5.56147	0.471718	0.235859	+	0.971787i	$0.424210\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	2.39798	0.200529
$144$	0	0
$145$	0	0
$146$	0	0
$147$	0	0
$148$	0	0
$149$	23.5560	1.92978	0.964891	−	0.262652i	$-0.0845972\pi$
$149$	23.5560	1.92978	0.964891	+	0.262652i	$0.0845972\pi$
$150$	0	0
$151$	0.0229661	0.00186896	0.000934479	−	1.00000i	$-0.499703\pi$
$151$	0.0229661	0.00186896	0.000934479		1.00000i	$0.499703\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	12.1593	0.970422	0.485211	−	0.874397i	$-0.338743\pi$
$157$	12.1593	0.970422	0.485211	+	0.874397i	$0.338743\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	0.733083	0.0577751
$162$	0	0
$163$	−19.0654	−1.49332	−0.746658	−	0.665208i	$-0.768343\pi$
$163$	−19.0654	−1.49332	−0.746658	+	0.665208i	$0.768343\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	1.82290	0.141060	0.0705300	−	0.997510i	$-0.477531\pi$
$167$	1.82290	0.141060	0.0705300	+	0.997510i	$0.477531\pi$
$168$	0	0
$169$	−6.10945	−0.469957
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−20.7885	−1.58052	−0.790259	−	0.612772i	$-0.790054\pi$
$173$	−20.7885	−1.58052	−0.790259	+	0.612772i	$0.790054\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−3.12692	−0.233717	−0.116858	−	0.993149i	$-0.537282\pi$
$179$	−3.12692	−0.233717	−0.116858	+	0.993149i	$0.537282\pi$
$180$	0	0
$181$	22.9249	1.70399	0.851996	−	0.523549i	$-0.175392\pi$
$181$	22.9249	1.70399	0.851996	+	0.523549i	$0.175392\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	0	0
$186$	0	0
$187$	1.90604	0.139384
$188$	0	0
$189$	0	0
$190$	0	0
$191$	7.95396	0.575528	0.287764	−	0.957701i	$-0.407088\pi$
$191$	7.95396	0.575528	0.287764	+	0.957701i	$0.407088\pi$
$192$	0	0
$193$	5.18845	0.373473	0.186736	−	0.982410i	$-0.440209\pi$
$193$	5.18845	0.373473	0.186736	+	0.982410i	$0.440209\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−21.3384	−1.52030	−0.760150	−	0.649747i	$-0.774875\pi$
$197$	−21.3384	−1.52030	−0.760150	+	0.649747i	$0.774875\pi$
$198$	0	0
$199$	9.50608	0.673868	0.336934	−	0.941528i	$-0.390610\pi$
$199$	9.50608	0.673868	0.336934	+	0.941528i	$0.390610\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−0.207376	−0.0145549
$204$	0	0
$205$	0	0
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−4.51140	−0.312060
$210$	0	0
$211$	−0.590570	−0.0406565	−0.0203282	−	0.999793i	$-0.506471\pi$
$211$	−0.590570	−0.0406565	−0.0203282	+	0.999793i	$0.506471\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−0.313486	−0.0212808
$218$	0	0
$219$	0	0
$220$	0	0
$221$	5.47698	0.368422
$222$	0	0
$223$	−5.76551	−0.386087	−0.193044	−	0.981190i	$-0.561836\pi$
$223$	−5.76551	−0.386087	−0.193044	+	0.981190i	$0.561836\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−13.2561	−0.879838	−0.439919	−	0.898037i	$-0.644993\pi$
$227$	−13.2561	−0.879838	−0.439919	+	0.898037i	$0.644993\pi$
$228$	0	0
$229$	7.85001	0.518743	0.259372	−	0.965778i	$-0.416485\pi$
$229$	7.85001	0.518743	0.259372	+	0.965778i	$0.416485\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−15.9094	−1.04226	−0.521129	−	0.853478i	$-0.674489\pi$
$233$	−15.9094	−1.04226	−0.521129	+	0.853478i	$0.674489\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	0	0
$238$	0	0
$239$	14.4770	0.936438	0.468219	−	0.883612i	$-0.344896\pi$
$239$	14.4770	0.936438	0.468219	+	0.883612i	$0.344896\pi$
$240$	0	0
$241$	6.65607	0.428755	0.214378	−	0.976751i	$-0.431228\pi$
$241$	6.65607	0.428755	0.214378	+	0.976751i	$0.431228\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−12.9634	−0.824843
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−14.6885	−0.927130	−0.463565	−	0.886063i	$-0.653430\pi$
$251$	−14.6885	−0.927130	−0.463565	+	0.886063i	$0.653430\pi$
$252$	0	0
$253$	7.74390	0.486855
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−17.0865	−1.06583	−0.532913	−	0.846170i	$-0.678903\pi$
$257$	−17.0865	−1.06583	−0.532913	+	0.846170i	$0.678903\pi$
$258$	0	0
$259$	−0.506076	−0.0314461
$260$	0	0
$261$	0	0
$262$	0	0
$263$	3.34258	0.206112	0.103056	−	0.994676i	$-0.467138\pi$
$263$	3.34258	0.206112	0.103056	+	0.994676i	$0.467138\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	0	0
$269$	1.82704	0.111397	0.0556983	−	0.998448i	$-0.482261\pi$
$269$	1.82704	0.111397	0.0556983	+	0.998448i	$0.482261\pi$
$270$	0	0
$271$	−24.0634	−1.46175	−0.730874	−	0.682513i	$-0.760887\pi$
$271$	−24.0634	−1.46175	−0.730874	+	0.682513i	$0.760887\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−13.9310	−0.837032	−0.418516	−	0.908209i	$-0.637450\pi$
$277$	−13.9310	−0.837032	−0.418516	+	0.908209i	$0.637450\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−19.2478	−1.14823	−0.574114	−	0.818775i	$-0.694653\pi$
$281$	−19.2478	−1.14823	−0.574114	+	0.818775i	$0.694653\pi$
$282$	0	0
$283$	22.2749	1.32411	0.662053	−	0.749457i	$-0.269685\pi$
$283$	22.2749	1.32411	0.662053	+	0.749457i	$0.269685\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−0.575082	−0.0339460
$288$	0	0
$289$	−12.6466	−0.743918
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−2.09062	−0.122136	−0.0610678	−	0.998134i	$-0.519451\pi$
$293$	−2.09062	−0.122136	−0.0610678	+	0.998134i	$0.519451\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0	0
$298$	0	0
$299$	22.2520	1.28686
$300$	0	0
$301$	−0.713452	−0.0411227
$302$	0	0
$303$	0	0
$304$	0	0
$305$	0	0
$306$	0	0
$307$	1.74056	0.0993391	0.0496696	−	0.998766i	$-0.484183\pi$
$307$	1.74056	0.0993391	0.0496696	+	0.998766i	$0.484183\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−23.7364	−1.34597	−0.672984	−	0.739657i	$-0.734988\pi$
$311$	−23.7364	−1.34597	−0.672984	+	0.739657i	$0.734988\pi$
$312$	0	0
$313$	10.8811	0.615036	0.307518	−	0.951542i	$-0.400502\pi$
$313$	10.8811	0.615036	0.307518	+	0.951542i	$0.400502\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−8.00748	−0.449745	−0.224872	−	0.974388i	$-0.572197\pi$
$317$	−8.00748	−0.449745	−0.224872	+	0.974388i	$0.572197\pi$
$318$	0	0
$319$	−2.19060	−0.122650
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−10.3040	−0.573331
$324$	0	0
$325$	0	0
$326$	0	0
$327$	0	0
$328$	0	0
$329$	0.232500	0.0128181
$330$	0	0
$331$	−4.21953	−0.231926	−0.115963	−	0.993254i	$-0.536995\pi$
$331$	−4.21953	−0.231926	−0.115963	+	0.993254i	$0.536995\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	0	0
$336$	0	0
$337$	6.39663	0.348447	0.174223	−	0.984706i	$-0.444259\pi$
$337$	6.39663	0.348447	0.174223	+	0.984706i	$0.444259\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−3.31150	−0.179328
$342$	0	0
$343$	1.21006	0.0653373
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−13.5155	−0.725552	−0.362776	−	0.931876i	$-0.618171\pi$
$347$	−13.5155	−0.725552	−0.362776	+	0.931876i	$0.618171\pi$
$348$	0	0
$349$	0.817577	0.0437639	0.0218819	−	0.999761i	$-0.493034\pi$
$349$	0.817577	0.0437639	0.0218819	+	0.999761i	$0.493034\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−18.9405	−1.00810	−0.504049	−	0.863675i	$-0.668157\pi$
$353$	−18.9405	−1.00810	−0.504049	+	0.863675i	$0.668157\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	36.4863	1.92568	0.962838	−	0.270081i	$-0.0870505\pi$
$359$	36.4863	1.92568	0.962838	+	0.270081i	$0.0870505\pi$
$360$	0	0
$361$	5.38851	0.283606
$362$	0	0
$363$	0	0
$364$	0	0
$365$	0	0
$366$	0	0
$367$	19.4249	1.01397	0.506986	−	0.861954i	$-0.330759\pi$
$367$	19.4249	1.01397	0.506986	+	0.861954i	$0.330759\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−0.496081	−0.0257553
$372$	0	0
$373$	−29.1768	−1.51072	−0.755359	−	0.655311i	$-0.772538\pi$
$373$	−29.1768	−1.51072	−0.755359	+	0.655311i	$0.772538\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−6.29466	−0.324192
$378$	0	0
$379$	−21.8060	−1.12010	−0.560048	−	0.828460i	$-0.689217\pi$
$379$	−21.8060	−1.12010	−0.560048	+	0.828460i	$0.689217\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−9.67092	−0.494161	−0.247080	−	0.968995i	$-0.579471\pi$
$383$	−9.67092	−0.494161	−0.247080	+	0.968995i	$0.579471\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	15.7289	0.797489	0.398744	−	0.917062i	$-0.369446\pi$
$389$	15.7289	0.797489	0.398744	+	0.917062i	$0.369446\pi$
$390$	0	0
$391$	17.6870	0.894472
$392$	0	0
$393$	0	0
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−22.8999	−1.14931	−0.574657	−	0.818394i	$-0.694864\pi$
$397$	−22.8999	−1.14931	−0.574657	+	0.818394i	$0.694864\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	6.09998	0.304618	0.152309	−	0.988333i	$-0.451329\pi$
$401$	6.09998	0.304618	0.152309	+	0.988333i	$0.451329\pi$
$402$	0	0
$403$	−9.51554	−0.474003
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−5.34592	−0.264987
$408$	0	0
$409$	−28.8688	−1.42747	−0.713736	−	0.700415i	$-0.752998\pi$
$409$	−28.8688	−1.42747	−0.713736	+	0.700415i	$0.752998\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−1.06702	−0.0525046
$414$	0	0
$415$	0	0
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−17.9423	−0.876541	−0.438270	−	0.898843i	$-0.644409\pi$
$419$	−17.9423	−0.876541	−0.438270	+	0.898843i	$0.644409\pi$
$420$	0	0
$421$	−0.723089	−0.0352412	−0.0176206	−	0.999845i	$-0.505609\pi$
$421$	−0.723089	−0.0352412	−0.0176206	+	0.999845i	$0.505609\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	0	0
$426$	0	0
$427$	0.547972	0.0265182
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−8.19656	−0.394815	−0.197407	−	0.980322i	$-0.563252\pi$
$431$	−8.19656	−0.394815	−0.197407	+	0.980322i	$0.563252\pi$
$432$	0	0
$433$	35.7464	1.71786	0.858931	−	0.512091i	$-0.171129\pi$
$433$	35.7464	1.71786	0.858931	+	0.512091i	$0.171129\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−41.8633	−2.00259
$438$	0	0
$439$	18.2688	0.871922	0.435961	−	0.899966i	$-0.356409\pi$
$439$	18.2688	0.871922	0.435961	+	0.899966i	$0.356409\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	29.8323	1.41737	0.708687	−	0.705523i	$-0.249288\pi$
$443$	29.8323	1.41737	0.708687	+	0.705523i	$0.249288\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	0	0
$448$	0	0
$449$	38.9423	1.83780	0.918901	−	0.394488i	$-0.129078\pi$
$449$	38.9423	1.83780	0.918901	+	0.394488i	$0.129078\pi$
$450$	0	0
$451$	−6.07486	−0.286054
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	20.4905	0.958504	0.479252	−	0.877677i	$-0.340908\pi$
$457$	20.4905	0.958504	0.479252	+	0.877677i	$0.340908\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−28.1715	−1.31208	−0.656039	−	0.754727i	$-0.727769\pi$
$461$	−28.1715	−1.31208	−0.656039	+	0.754727i	$0.727769\pi$
$462$	0	0
$463$	−34.3195	−1.59496	−0.797481	−	0.603344i	$-0.793835\pi$
$463$	−34.3195	−1.59496	−0.797481	+	0.603344i	$0.793835\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	4.33096	0.200413	0.100206	−	0.994967i	$-0.468050\pi$
$467$	4.33096	0.200413	0.100206	+	0.994967i	$0.468050\pi$
$468$	0	0
$469$	0.533186	0.0246203
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−7.53652	−0.346530
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0	0
$478$	0	0
$479$	28.9850	1.32436	0.662180	−	0.749345i	$-0.269631\pi$
$479$	28.9850	1.32436	0.662180	+	0.749345i	$0.269631\pi$
$480$	0	0
$481$	−15.3614	−0.700420
$482$	0	0
$483$	0	0
$484$	0	0
$485$	0	0
$486$	0	0
$487$	11.2189	0.508376	0.254188	−	0.967155i	$-0.418192\pi$
$487$	11.2189	0.508376	0.254188	+	0.967155i	$0.418192\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−25.5635	−1.15366	−0.576831	−	0.816863i	$-0.695711\pi$
$491$	−25.5635	−1.15366	−0.576831	+	0.816863i	$0.695711\pi$
$492$	0	0
$493$	−5.00333	−0.225339
$494$	0	0
$495$	0	0
$496$	0	0
$497$	1.07152	0.0480643
$498$	0	0
$499$	18.9371	0.847742	0.423871	−	0.905723i	$-0.360671\pi$
$499$	18.9371	0.847742	0.423871	+	0.905723i	$0.360671\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	21.1790	0.944324	0.472162	−	0.881512i	$-0.343474\pi$
$503$	21.1790	0.944324	0.472162	+	0.881512i	$0.343474\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−32.6560	−1.44745	−0.723725	−	0.690088i	$-0.757572\pi$
$509$	−32.6560	−1.44745	−0.723725	+	0.690088i	$0.757572\pi$
$510$	0	0
$511$	0.459507	0.0203274
$512$	0	0
$513$	0	0
$514$	0	0
$515$	0	0
$516$	0	0
$517$	2.45600	0.108015
$518$	0	0
$519$	0	0
$520$	0	0
$521$	2.06550	0.0904912	0.0452456	−	0.998976i	$-0.485593\pi$
$521$	2.06550	0.0904912	0.0452456	+	0.998976i	$0.485593\pi$
$522$	0	0
$523$	8.34790	0.365028	0.182514	−	0.983203i	$-0.441576\pi$
$523$	8.34790	0.365028	0.182514	+	0.983203i	$0.441576\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−7.56346	−0.329469
$528$	0	0
$529$	48.8592	2.12431
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−17.4560	−0.756103
$534$	0	0
$535$	0	0
$536$	0	0
$537$	0	0
$538$	0	0
$539$	6.38781	0.275143
$540$	0	0
$541$	−12.2175	−0.525273	−0.262637	−	0.964895i	$-0.584592\pi$
$541$	−12.2175	−0.525273	−0.262637	+	0.964895i	$0.584592\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	0	0
$546$	0	0
$547$	9.50193	0.406273	0.203137	−	0.979150i	$-0.434886\pi$
$547$	9.50193	0.406273	0.203137	+	0.979150i	$0.434886\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	11.8424	0.504501
$552$	0	0
$553$	1.16296	0.0494542
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−19.1999	−0.813526	−0.406763	−	0.913534i	$-0.633342\pi$
$557$	−19.1999	−0.813526	−0.406763	+	0.913534i	$0.633342\pi$
$558$	0	0
$559$	−21.6561	−0.915954
$560$	0	0
$561$	0	0
$562$	0	0
$563$	16.2209	0.683628	0.341814	−	0.939768i	$-0.388959\pi$
$563$	16.2209	0.683628	0.341814	+	0.939768i	$0.388959\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−28.0573	−1.17622	−0.588111	−	0.808780i	$-0.700128\pi$
$569$	−28.0573	−1.17622	−0.588111	+	0.808780i	$0.700128\pi$
$570$	0	0
$571$	41.5444	1.73858	0.869289	−	0.494305i	$-0.164577\pi$
$571$	41.5444	1.73858	0.869289	+	0.494305i	$0.164577\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−18.4113	−0.766473	−0.383236	−	0.923650i	$-0.625191\pi$
$577$	−18.4113	−0.766473	−0.383236	+	0.923650i	$0.625191\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−0.525708	−0.0218100
$582$	0	0
$583$	−5.24034	−0.217033
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−35.8843	−1.48110	−0.740552	−	0.671999i	$-0.765436\pi$
$587$	−35.8843	−1.48110	−0.740552	+	0.671999i	$0.765436\pi$
$588$	0	0
$589$	17.9019	0.737635
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−47.6039	−1.95486	−0.977429	−	0.211265i	$-0.932242\pi$
$593$	−47.6039	−1.95486	−0.977429	+	0.211265i	$0.932242\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−24.8559	−1.01558	−0.507791	−	0.861480i	$-0.669538\pi$
$599$	−24.8559	−1.01558	−0.507791	+	0.861480i	$0.669538\pi$
$600$	0	0
$601$	23.0965	0.942125	0.471062	−	0.882100i	$-0.343871\pi$
$601$	23.0965	0.942125	0.471062	+	0.882100i	$0.343871\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	0	0
$606$	0	0
$607$	4.62697	0.187803	0.0939015	−	0.995581i	$-0.470066\pi$
$607$	4.62697	0.187803	0.0939015	+	0.995581i	$0.470066\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	7.05728	0.285507
$612$	0	0
$613$	−12.1791	−0.491909	−0.245954	−	0.969281i	$-0.579101\pi$
$613$	−12.1791	−0.491909	−0.245954	+	0.969281i	$0.579101\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−46.5369	−1.87350	−0.936752	−	0.349994i	$-0.886184\pi$
$617$	−46.5369	−1.87350	−0.936752	+	0.349994i	$0.886184\pi$
$618$	0	0
$619$	15.8040	0.635215	0.317608	−	0.948222i	$-0.397121\pi$
$619$	15.8040	0.635215	0.317608	+	0.948222i	$0.397121\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	0.703457	0.0281834
$624$	0	0
$625$	0	0
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−12.2101	−0.486847
$630$	0	0
$631$	−24.6749	−0.982292	−0.491146	−	0.871077i	$-0.663422\pi$
$631$	−24.6749	−0.982292	−0.491146	+	0.871077i	$0.663422\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	0	0
$636$	0	0
$637$	18.3553	0.727262
$638$	0	0
$639$	0	0
$640$	0	0
$641$	1.51554	0.0598603	0.0299301	−	0.999552i	$-0.490472\pi$
$641$	1.51554	0.0598603	0.0299301	+	0.999552i	$0.490472\pi$
$642$	0	0
$643$	24.3999	0.962236	0.481118	−	0.876656i	$-0.340231\pi$
$643$	24.3999	0.962236	0.481118	+	0.876656i	$0.340231\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	26.9195	1.05832	0.529158	−	0.848523i	$-0.322508\pi$
$647$	26.9195	1.05832	0.529158	+	0.848523i	$0.322508\pi$
$648$	0	0
$649$	−11.2714	−0.442442
$650$	0	0
$651$	0	0
$652$	0	0
$653$	8.18232	0.320199	0.160099	−	0.987101i	$-0.448819\pi$
$653$	8.18232	0.320199	0.160099	+	0.987101i	$0.448819\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	0	0
$658$	0	0
$659$	15.1101	0.588605	0.294303	−	0.955712i	$-0.404913\pi$
$659$	15.1101	0.588605	0.294303	+	0.955712i	$0.404913\pi$
$660$	0	0
$661$	−34.0601	−1.32478	−0.662392	−	0.749158i	$-0.730458\pi$
$661$	−34.0601	−1.32478	−0.662392	+	0.749158i	$0.730458\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−20.3276	−0.787089
$668$	0	0
$669$	0	0
$670$	0	0
$671$	5.78848	0.223462
$672$	0	0
$673$	−29.4135	−1.13381	−0.566903	−	0.823785i	$-0.691859\pi$
$673$	−29.4135	−1.13381	−0.566903	+	0.823785i	$0.691859\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	33.9038	1.30303	0.651514	−	0.758637i	$-0.274134\pi$
$677$	33.9038	1.30303	0.651514	+	0.758637i	$0.274134\pi$
$678$	0	0
$679$	0.960090	0.0368449
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−29.1078	−1.11378	−0.556890	−	0.830586i	$-0.688005\pi$
$683$	−29.1078	−1.11378	−0.556890	+	0.830586i	$0.688005\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−15.0580	−0.573665
$690$	0	0
$691$	−10.4216	−0.396456	−0.198228	−	0.980156i	$-0.563519\pi$
$691$	−10.4216	−0.396456	−0.198228	+	0.980156i	$0.563519\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−13.8750	−0.525552
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−9.75406	−0.368406	−0.184203	−	0.982888i	$-0.558970\pi$
$701$	−9.75406	−0.368406	−0.184203	+	0.982888i	$0.558970\pi$
$702$	0	0
$703$	28.8999	1.08998
$704$	0	0
$705$	0	0
$706$	0	0
$707$	1.04458	0.0392856
$708$	0	0
$709$	−32.3079	−1.21335	−0.606674	−	0.794951i	$-0.707497\pi$
$709$	−32.3079	−1.21335	−0.606674	+	0.794951i	$0.707497\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−30.7289	−1.15081
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	0	0
$719$	18.4905	0.689579	0.344789	−	0.938680i	$-0.387950\pi$
$719$	18.4905	0.689579	0.344789	+	0.938680i	$0.387950\pi$
$720$	0	0
$721$	−0.377701	−0.0140663
$722$	0	0
$723$	0	0
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−51.4714	−1.90897	−0.954484	−	0.298262i	$-0.903593\pi$
$727$	−51.4714	−1.90897	−0.954484	+	0.298262i	$0.903593\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−17.2134	−0.636661
$732$	0	0
$733$	−6.64058	−0.245275	−0.122638	−	0.992452i	$-0.539135\pi$
$733$	−6.64058	−0.245275	−0.122638	+	0.992452i	$0.539135\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	5.63229	0.207468
$738$	0	0
$739$	−38.6058	−1.42014	−0.710068	−	0.704133i	$-0.751336\pi$
$739$	−38.6058	−1.42014	−0.710068	+	0.704133i	$0.751336\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	25.5904	0.938821	0.469410	−	0.882980i	$-0.344467\pi$
$743$	25.5904	0.938821	0.469410	+	0.882980i	$0.344467\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	0	0
$748$	0	0
$749$	1.63977	0.0599160
$750$	0	0
$751$	15.1114	0.551424	0.275712	−	0.961240i	$-0.411086\pi$
$751$	15.1114	0.551424	0.275712	+	0.961240i	$0.411086\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−13.2710	−0.482341	−0.241170	−	0.970483i	$-0.577531\pi$
$757$	−13.2710	−0.482341	−0.241170	+	0.970483i	$0.577531\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−18.5174	−0.671256	−0.335628	−	0.941995i	$-0.608948\pi$
$761$	−18.5174	−0.671256	−0.335628	+	0.941995i	$0.608948\pi$
$762$	0	0
$763$	−1.31366	−0.0475576
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−32.3882	−1.16947
$768$	0	0
$769$	−2.01882	−0.0728006	−0.0364003	−	0.999337i	$-0.511589\pi$
$769$	−2.01882	−0.0728006	−0.0364003	+	0.999337i	$0.511589\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−5.76154	−0.207228	−0.103614	−	0.994618i	$-0.533041\pi$
$773$	−5.76154	−0.207228	−0.103614	+	0.994618i	$0.533041\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0	0
$778$	0	0
$779$	32.8405	1.17663
$780$	0	0
$781$	11.3190	0.405025
$782$	0	0
$783$	0	0
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−7.33043	−0.261302	−0.130651	−	0.991428i	$-0.541707\pi$
$787$	−7.33043	−0.261302	−0.130651	+	0.991428i	$0.541707\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	0.570938	0.0203002
$792$	0	0
$793$	16.6331	0.590659
$794$	0	0
$795$	0	0
$796$	0	0
$797$	7.12358	0.252330	0.126165	−	0.992009i	$-0.459733\pi$
$797$	7.12358	0.252330	0.126165	+	0.992009i	$0.459733\pi$
$798$	0	0
$799$	5.60950	0.198450
$800$	0	0
$801$	0	0
$802$	0	0
$803$	4.85398	0.171293
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	29.0809	1.02243	0.511215	−	0.859453i	$-0.329196\pi$
$809$	29.0809	1.02243	0.511215	+	0.859453i	$0.329196\pi$
$810$	0	0
$811$	34.3214	1.20519	0.602593	−	0.798048i	$-0.294134\pi$
$811$	34.3214	1.20519	0.602593	+	0.798048i	$0.294134\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	0	0
$816$	0	0
$817$	40.7423	1.42539
$818$	0	0
$819$	0	0
$820$	0	0
$821$	42.8768	1.49641	0.748206	−	0.663466i	$-0.230915\pi$
$821$	42.8768	1.49641	0.748206	+	0.663466i	$0.230915\pi$
$822$	0	0
$823$	46.2309	1.61151	0.805753	−	0.592251i	$-0.201761\pi$
$823$	46.2309	1.61151	0.805753	+	0.592251i	$0.201761\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−23.6350	−0.821869	−0.410934	−	0.911665i	$-0.634797\pi$
$827$	−23.6350	−0.821869	−0.410934	+	0.911665i	$0.634797\pi$
$828$	0	0
$829$	−18.8979	−0.656352	−0.328176	−	0.944617i	$-0.606434\pi$
$829$	−18.8979	−0.656352	−0.328176	+	0.944617i	$0.606434\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	14.5898	0.505505
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0	0
$838$	0	0
$839$	10.4309	0.360116	0.180058	−	0.983656i	$-0.442371\pi$
$839$	10.4309	0.360116	0.180058	+	0.983656i	$0.442371\pi$
$840$	0	0
$841$	−23.2497	−0.801714
$842$	0	0
$843$	0	0
$844$	0	0
$845$	0	0
$846$	0	0
$847$	0.879104	0.0302064
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−49.6072	−1.70051
$852$	0	0
$853$	39.8809	1.36550	0.682748	−	0.730654i	$-0.260785\pi$
$853$	39.8809	1.36550	0.682748	+	0.730654i	$0.260785\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	18.1269	0.619204	0.309602	−	0.950866i	$-0.399804\pi$
$857$	18.1269	0.619204	0.309602	+	0.950866i	$0.399804\pi$
$858$	0	0
$859$	4.02081	0.137188	0.0685941	−	0.997645i	$-0.478149\pi$
$859$	4.02081	0.137188	0.0685941	+	0.997645i	$0.478149\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	43.9540	1.49621	0.748105	−	0.663580i	$-0.230964\pi$
$863$	43.9540	1.49621	0.748105	+	0.663580i	$0.230964\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	0	0
$868$	0	0
$869$	12.2849	0.416737
$870$	0	0
$871$	16.1843	0.548384
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	49.5728	1.67396	0.836978	−	0.547237i	$-0.184320\pi$
$877$	49.5728	1.67396	0.836978	+	0.547237i	$0.184320\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	51.0775	1.72085	0.860423	−	0.509580i	$-0.170199\pi$
$881$	51.0775	1.72085	0.860423	+	0.509580i	$0.170199\pi$
$882$	0	0
$883$	−51.1113	−1.72003	−0.860017	−	0.510266i	$-0.829547\pi$
$883$	−51.1113	−1.72003	−0.860017	+	0.510266i	$0.829547\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	33.9573	1.14017	0.570087	−	0.821584i	$-0.306909\pi$
$887$	33.9573	1.14017	0.570087	+	0.821584i	$0.306909\pi$
$888$	0	0
$889$	−0.642102	−0.0215354
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−13.2771	−0.444301
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	0	0
$899$	8.69264	0.289916
$900$	0	0
$901$	−11.9689	−0.398742
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−5.80397	−0.192718	−0.0963588	−	0.995347i	$-0.530720\pi$
$907$	−5.80397	−0.192718	−0.0963588	+	0.995347i	$0.530720\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−4.19990	−0.139149	−0.0695744	−	0.997577i	$-0.522164\pi$
$911$	−4.19990	−0.139149	−0.0695744	+	0.997577i	$0.522164\pi$
$912$	0	0
$913$	−5.55329	−0.183787
$914$	0	0
$915$	0	0
$916$	0	0
$917$	1.38033	0.0455827
$918$	0	0
$919$	25.2655	0.833431	0.416715	−	0.909037i	$-0.363181\pi$
$919$	25.2655	0.833431	0.416715	+	0.909037i	$0.363181\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	32.5249	1.07057
$924$	0	0
$925$	0	0
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−36.3864	−1.19380	−0.596899	−	0.802317i	$-0.703601\pi$
$929$	−36.3864	−1.19380	−0.596899	+	0.802317i	$0.703601\pi$
$930$	0	0
$931$	−34.5324	−1.13175
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−17.5425	−0.573088	−0.286544	−	0.958067i	$-0.592506\pi$
$937$	−17.5425	−0.573088	−0.286544	+	0.958067i	$0.592506\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	24.3384	0.793410	0.396705	−	0.917946i	$-0.370154\pi$
$941$	24.3384	0.793410	0.396705	+	0.917946i	$0.370154\pi$
$942$	0	0
$943$	−56.3714	−1.83571
$944$	0	0
$945$	0	0
$946$	0	0
$947$	3.05206	0.0991787	0.0495893	−	0.998770i	$-0.484209\pi$
$947$	3.05206	0.0991787	0.0495893	+	0.998770i	$0.484209\pi$
$948$	0	0
$949$	13.9478	0.452766
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−11.6054	−0.375934	−0.187967	−	0.982175i	$-0.560190\pi$
$953$	−11.6054	−0.375934	−0.187967	+	0.982175i	$0.560190\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	1.59040	0.0513566
$960$	0	0
$961$	−17.8595	−0.576112
$962$	0	0
$963$	0	0
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−16.8384	−0.541486	−0.270743	−	0.962652i	$-0.587269\pi$
$967$	−16.8384	−0.541486	−0.270743	+	0.962652i	$0.587269\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−18.3864	−0.590046	−0.295023	−	0.955490i	$-0.595327\pi$
$971$	−18.3864	−0.590046	−0.295023	+	0.955490i	$0.595327\pi$
$972$	0	0
$973$	−0.480952	−0.0154186
$974$	0	0
$975$	0	0
$976$	0	0
$977$	29.4687	0.942787	0.471393	−	0.881923i	$-0.343751\pi$
$977$	29.4687	0.942787	0.471393	+	0.881923i	$0.343751\pi$
$978$	0	0
$979$	7.43094	0.237494
$980$	0	0
$981$	0	0
$982$	0	0
$983$	21.5541	0.687469	0.343735	−	0.939067i	$-0.388308\pi$
$983$	21.5541	0.687469	0.343735	+	0.939067i	$0.388308\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−69.9349	−2.22380
$990$	0	0
$991$	9.01027	0.286221	0.143110	−	0.989707i	$-0.454290\pi$
$991$	9.01027	0.286221	0.143110	+	0.989707i	$0.454290\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−30.4998	−0.965938	−0.482969	−	0.875638i	$-0.660442\pi$
$997$	−30.4998	−0.965938	−0.482969	+	0.875638i	$0.660442\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8100.2.a.x.1.2		4
3.2	odd	2		8100.2.a.y.1.2		4
5.2	odd	4		8100.2.d.q.649.4		8
5.3	odd	4		8100.2.d.q.649.5		8
5.4	even	2		8100.2.a.z.1.3		4
9.2	odd	6		2700.2.i.e.901.3		8
9.4	even	3		900.2.i.d.601.4	yes	8
9.5	odd	6		2700.2.i.e.1801.3		8
9.7	even	3		900.2.i.d.301.4	✓	8
15.2	even	4		8100.2.d.s.649.4		8
15.8	even	4		8100.2.d.s.649.5		8
15.14	odd	2		8100.2.a.ba.1.3		4
45.2	even	12		2700.2.s.d.1549.4		16
45.4	even	6		900.2.i.e.601.1	yes	8
45.7	odd	12		900.2.s.d.49.3		16
45.13	odd	12		900.2.s.d.349.3		16
45.14	odd	6		2700.2.i.d.1801.2		8
45.22	odd	12		900.2.s.d.349.6		16
45.23	even	12		2700.2.s.d.2449.4		16
45.29	odd	6		2700.2.i.d.901.2		8
45.32	even	12		2700.2.s.d.2449.5		16
45.34	even	6		900.2.i.e.301.1	yes	8
45.38	even	12		2700.2.s.d.1549.5		16
45.43	odd	12		900.2.s.d.49.6		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
900.2.i.d.301.4	✓	8	9.7	even	3
900.2.i.d.601.4	yes	8	9.4	even	3
900.2.i.e.301.1	yes	8	45.34	even	6
900.2.i.e.601.1	yes	8	45.4	even	6
900.2.s.d.49.3		16	45.7	odd	12
900.2.s.d.49.6		16	45.43	odd	12
900.2.s.d.349.3		16	45.13	odd	12
900.2.s.d.349.6		16	45.22	odd	12
2700.2.i.d.901.2		8	45.29	odd	6
2700.2.i.d.1801.2		8	45.14	odd	6
2700.2.i.e.901.3		8	9.2	odd	6
2700.2.i.e.1801.3		8	9.5	odd	6
2700.2.s.d.1549.4		16	45.2	even	12
2700.2.s.d.1549.5		16	45.38	even	12
2700.2.s.d.2449.4		16	45.23	even	12
2700.2.s.d.2449.5		16	45.32	even	12
8100.2.a.x.1.2		4	1.1	even	1	trivial
8100.2.a.y.1.2		4	3.2	odd	2
8100.2.a.z.1.3		4	5.4	even	2
8100.2.a.ba.1.3		4	15.14	odd	2
8100.2.d.q.649.4		8	5.2	odd	4
8100.2.d.q.649.5		8	5.3	odd	4
8100.2.d.s.649.4		8	15.2	even	4
8100.2.d.s.649.5		8	15.8	even	4

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:	\( N \)	\(=\)	\( 8100 = 2^{2} \cdot 3^{4} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8100.a (trivial)

\(f(q)\)	\(=\)	\(q-0.0864793 q^{7} -0.913521 q^{11} -2.62499 q^{13} -2.08648 q^{17} +4.93847 q^{19} -8.47698 q^{23} +2.39798 q^{29} +3.62499 q^{31} +5.85199 q^{37} +6.64994 q^{41} +8.24997 q^{43} -2.68850 q^{47} -6.99252 q^{49} +5.73642 q^{53} +12.3384 q^{59} -6.33645 q^{61} -6.16548 q^{67} -12.3905 q^{71} -5.31349 q^{73} +0.0790006 q^{77} -13.4479 q^{79} +6.07900 q^{83} -8.13440 q^{89} +0.227007 q^{91} -11.1020 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - q^{7} - 3 q^{11} + 2 q^{13} - 9 q^{17} - 4 q^{19} + 3 q^{23} + 9 q^{29} + 2 q^{31} - q^{37} - 9 q^{41} + 8 q^{43} - 12 q^{47} + 9 q^{49} - 12 q^{53} + 15 q^{59} - q^{61} + 11 q^{67} - 12 q^{71}+ \cdots + 5 q^{97}+O(q^{100}) \) 4 * q - q^7 - 3 * q^11 + 2 * q^13 - 9 * q^17 - 4 * q^19 + 3 * q^23 + 9 * q^29 + 2 * q^31 - q^37 - 9 * q^41 + 8 * q^43 - 12 * q^47 + 9 * q^49 - 12 * q^53 + 15 * q^59 - q^61 + 11 * q^67 - 12 * q^71 - 10 * q^73 - 36 * q^77 - 7 * q^79 - 12 * q^83 + 3 * q^89 - 11 * q^91 + 5 * q^97

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