LMFDB - Embedded newform 8100.2.d.s.649.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [8100,2,Mod(649,8100)] 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("8100.649"); 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, 1])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 8100 = 2^{2} \cdot 3^{4} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8100.d (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [8,0,0,0,0,0,0,0,0,0,6,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,0,18] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(29)] == traces)

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

Self dual:	no
Analytic conductor:	$64.6788256372$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.4057180416.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 9x^{6} + 22x^{4} + 12x^{2} + 1 $ x^8 + 9x^6 + 22x^4 + 12*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 3^{4} $
Twist minimal:	no (minimal twist has level 900)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			649.4
Root			$-2.28400i$ of defining polynomial
Character	$\chi$	$=$	8100.649
Dual form			8100.2.d.s.649.5

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-0.0864793i q^{7} +0.913521 q^{11} +2.62499i q^{13} +2.08648i q^{17} -4.93847 q^{19} -8.47698i q^{23} +2.39798 q^{29} +3.62499 q^{31} +5.85199i q^{37} -6.64994 q^{41} -8.24997i q^{43} +2.68850i q^{47} +6.99252 q^{49} +5.73642i q^{53} +12.3384 q^{59} -6.33645 q^{61} -6.16548i q^{67} +12.3905 q^{71} +5.31349i q^{73} -0.0790006i q^{77} +13.4479 q^{79} +6.07900i q^{83} -8.13440 q^{89} +0.227007 q^{91} -11.1020i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 6 q^{11} + 8 q^{19} + 18 q^{29} + 4 q^{31} + 18 q^{41} - 18 q^{49} + 30 q^{59} - 2 q^{61} + 24 q^{71} + 14 q^{79} + 6 q^{89} - 22 q^{91}+O(q^{100}) $ 8 * q + 6 * q^11 + 8 * q^19 + 18 * q^29 + 4 * q^31 + 18 * q^41 - 18 * q^49 + 30 * q^59 - 2 * q^61 + 24 * q^71 + 14 * q^79 + 6 * q^89 - 22 * q^91

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/8100\mathbb{Z}\right)^\times$.

$n$	$4051$	$6401$	$7777$
$\chi(n)$	$1$	$1$	$-1$

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
$3$	0	0
$3$	0	0
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 0.0864793i	− 0.0326861i	−0.999866	−	0.0163431i	$-0.994798\pi$
$7$	− 0.0864793i	− 0.0326861i	0.999866	−	0.0163431i	$-0.00520239\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	0.913521	0.275437	0.137718	−	0.990471i	$-0.456023\pi$
$11$	0.913521	0.275437	0.137718	+	0.990471i	$0.456023\pi$
$12$	0	0
$13$	2.62499i	0.728040i	0.931391	+	0.364020i	$0.118596\pi$
$13$	2.62499i	0.728040i	−0.931391	+	0.364020i	$0.881404\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	2.08648i	0.506046i	0.967460	+	0.253023i	$0.0814248\pi$
$17$	2.08648i	0.506046i	−0.967460	+	0.253023i	$0.918575\pi$
$18$	0	0
$19$	−4.93847	−1.13296	−0.566482	−	0.824074i	$-0.691696\pi$
$19$	−4.93847	−1.13296	−0.566482	+	0.824074i	$0.691696\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	− 8.47698i	− 1.76757i	−0.467891	−	0.883786i	$-0.654986\pi$
$23$	− 8.47698i	− 1.76757i	0.467891	−	0.883786i	$-0.345014\pi$
$24$	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
$37$	5.85199i	0.962062i	0.876704	+	0.481031i	$0.159738\pi$
$37$	5.85199i	0.962062i	−0.876704	+	0.481031i	$0.840262\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−6.64994	−1.03855	−0.519273	−	0.854608i	$-0.673797\pi$
$41$	−6.64994	−1.03855	−0.519273	+	0.854608i	$0.673797\pi$
$42$	0	0
$43$	− 8.24997i	− 1.25811i	−0.777361	−	0.629055i	$-0.783442\pi$
$43$	− 8.24997i	− 1.25811i	0.777361	−	0.629055i	$-0.216558\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	2.68850i	0.392158i	0.980588	+	0.196079i	$0.0628209\pi$
$47$	2.68850i	0.392158i	−0.980588	+	0.196079i	$0.937179\pi$
$48$	0	0
$49$	6.99252	0.998932
$50$	0	0
$51$	0	0
$52$	0	0
$53$	5.73642i	0.787958i	0.919120	+	0.393979i	$0.128902\pi$
$53$	5.73642i	0.787958i	−0.919120	+	0.393979i	$0.871098\pi$
$54$	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
$67$	− 6.16548i	− 0.753233i	−0.926369	−	0.376617i	$-0.877087\pi$
$67$	− 6.16548i	− 0.753233i	0.926369	−	0.376617i	$-0.122913\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	12.3905	1.47048	0.735241	−	0.677806i	$-0.237069\pi$
$71$	12.3905	1.47048	0.735241	+	0.677806i	$0.237069\pi$
$72$	0	0
$73$	5.31349i	0.621897i	0.950427	+	0.310948i	$0.100647\pi$
$73$	5.31349i	0.621897i	−0.950427	+	0.310948i	$0.899353\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 0.0790006i	− 0.00900296i
$78$	0	0
$79$	13.4479	1.51301	0.756503	−	0.653991i	$-0.226906\pi$
$79$	13.4479	1.51301	0.756503	+	0.653991i	$0.226906\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	6.07900i	0.667257i	0.942705	+	0.333629i	$0.108273\pi$
$83$	6.07900i	0.667257i	−0.942705	+	0.333629i	$0.891727\pi$
$84$	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
$97$	− 11.1020i	− 1.12723i	−0.826036	−	0.563617i	$-0.809409\pi$
$97$	− 11.1020i	− 1.12723i	0.826036	−	0.563617i	$-0.190591\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	12.0790	1.20191	0.600953	−	0.799285i	$-0.294788\pi$
$101$	12.0790	1.20191	0.600953	+	0.799285i	$0.294788\pi$
$102$	0	0
$103$	− 4.36753i	− 0.430346i	−0.976576	−	0.215173i	$-0.930968\pi$
$103$	− 4.36753i	− 0.430346i	0.976576	−	0.215173i	$-0.0690315\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	18.9614i	1.83307i	0.399953	+	0.916536i	$0.369027\pi$
$107$	18.9614i	1.83307i	−0.399953	+	0.916536i	$0.630973\pi$
$108$	0	0
$109$	−15.1904	−1.45498	−0.727490	−	0.686119i	$-0.759313\pi$
$109$	−15.1904	−1.45498	−0.727490	+	0.686119i	$0.759313\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	− 6.60202i	− 0.621066i	−0.950563	−	0.310533i	$-0.899493\pi$
$113$	− 6.60202i	− 0.621066i	0.950563	−	0.310533i	$-0.100507\pi$
$114$	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
$127$	7.42492i	0.658855i	0.944181	+	0.329427i	$0.106856\pi$
$127$	7.42492i	0.658855i	−0.944181	+	0.329427i	$0.893144\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	15.9614	1.39456	0.697279	−	0.716800i	$-0.254394\pi$
$131$	15.9614	1.39456	0.697279	+	0.716800i	$0.254394\pi$
$132$	0	0
$133$	0.427076i	0.0370322i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	18.3905i	1.57121i	0.618731	+	0.785603i	$0.287647\pi$
$137$	18.3905i	1.57121i	−0.618731	+	0.785603i	$0.712353\pi$
$138$	0	0
$139$	−5.56147	−0.471718	−0.235859	−	0.971787i	$-0.575790\pi$
$139$	−5.56147	−0.471718	−0.235859	+	0.971787i	$0.575790\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	2.39798i	0.200529i
$144$	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
$157$	12.1593i	0.970422i	0.874397	+	0.485211i	$0.161257\pi$
$157$	12.1593i	0.970422i	−0.874397	+	0.485211i	$0.838743\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−0.733083	−0.0577751
$162$	0	0
$163$	19.0654i	1.49332i	0.665208	+	0.746658i	$0.268343\pi$
$163$	19.0654i	1.49332i	−0.665208	+	0.746658i	$0.731657\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 1.82290i	− 0.141060i	−0.997510	−	0.0705300i	$-0.977531\pi$
$167$	− 1.82290i	− 0.141060i	0.997510	−	0.0705300i	$-0.0224691\pi$
$168$	0	0
$169$	6.10945	0.469957
$170$	0	0
$171$	0	0
$172$	0	0
$173$	− 20.7885i	− 1.58052i	−0.612772	−	0.790259i	$-0.709946\pi$
$173$	− 20.7885i	− 1.58052i	0.612772	−	0.790259i	$-0.290054\pi$
$174$	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
$187$	1.90604i	0.139384i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−7.95396	−0.575528	−0.287764	−	0.957701i	$-0.592912\pi$
$191$	−7.95396	−0.575528	−0.287764	+	0.957701i	$0.592912\pi$
$192$	0	0
$193$	− 5.18845i	− 0.373473i	−0.982410	−	0.186736i	$-0.940209\pi$
$193$	− 5.18845i	− 0.373473i	0.982410	−	0.186736i	$-0.0597910\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	21.3384i	1.52030i	0.649747	+	0.760150i	$0.274875\pi$
$197$	21.3384i	1.52030i	−0.649747	+	0.760150i	$0.725125\pi$
$198$	0	0
$199$	−9.50608	−0.673868	−0.336934	−	0.941528i	$-0.609390\pi$
$199$	−9.50608	−0.673868	−0.336934	+	0.941528i	$0.609390\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	− 0.207376i	− 0.0145549i
$204$	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
$217$	− 0.313486i	− 0.0212808i
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−5.47698	−0.368422
$222$	0	0
$223$	5.76551i	0.386087i	0.981190	+	0.193044i	$0.0618359\pi$
$223$	5.76551i	0.386087i	−0.981190	+	0.193044i	$0.938164\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	13.2561i	0.879838i	0.898037	+	0.439919i	$0.144993\pi$
$227$	13.2561i	0.879838i	−0.898037	+	0.439919i	$0.855007\pi$
$228$	0	0
$229$	−7.85001	−0.518743	−0.259372	−	0.965778i	$-0.583515\pi$
$229$	−7.85001	−0.518743	−0.259372	+	0.965778i	$0.583515\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	− 15.9094i	− 1.04226i	−0.853478	−	0.521129i	$-0.825511\pi$
$233$	− 15.9094i	− 1.04226i	0.853478	−	0.521129i	$-0.174489\pi$
$234$	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
$247$	− 12.9634i	− 0.824843i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	14.6885	0.927130	0.463565	−	0.886063i	$-0.346570\pi$
$251$	14.6885	0.927130	0.463565	+	0.886063i	$0.346570\pi$
$252$	0	0
$253$	− 7.74390i	− 0.486855i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	17.0865i	1.06583i	0.846170	+	0.532913i	$0.178903\pi$
$257$	17.0865i	1.06583i	−0.846170	+	0.532913i	$0.821097\pi$
$258$	0	0
$259$	0.506076	0.0314461
$260$	0	0
$261$	0	0
$262$	0	0
$263$	3.34258i	0.206112i	0.994676	+	0.103056i	$0.0328622\pi$
$263$	3.34258i	0.206112i	−0.994676	+	0.103056i	$0.967138\pi$
$264$	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
$277$	− 13.9310i	− 0.837032i	−0.908209	−	0.418516i	$-0.862550\pi$
$277$	− 13.9310i	− 0.837032i	0.908209	−	0.418516i	$-0.137450\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	19.2478	1.14823	0.574114	−	0.818775i	$-0.305347\pi$
$281$	19.2478	1.14823	0.574114	+	0.818775i	$0.305347\pi$
$282$	0	0
$283$	− 22.2749i	− 1.32411i	−0.749457	−	0.662053i	$-0.769685\pi$
$283$	− 22.2749i	− 1.32411i	0.749457	−	0.662053i	$-0.230315\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0.575082i	0.0339460i
$288$	0	0
$289$	12.6466	0.743918
$290$	0	0
$291$	0	0
$292$	0	0
$293$	− 2.09062i	− 0.122136i	−0.998134	−	0.0610678i	$-0.980549\pi$
$293$	− 2.09062i	− 0.122136i	0.998134	−	0.0610678i	$-0.0194506\pi$
$294$	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
$307$	1.74056i	0.0993391i	0.998766	+	0.0496696i	$0.0158168\pi$
$307$	1.74056i	0.0993391i	−0.998766	+	0.0496696i	$0.984183\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	23.7364	1.34597	0.672984	−	0.739657i	$-0.265012\pi$
$311$	23.7364	1.34597	0.672984	+	0.739657i	$0.265012\pi$
$312$	0	0
$313$	− 10.8811i	− 0.615036i	−0.951542	−	0.307518i	$-0.900502\pi$
$313$	− 10.8811i	− 0.615036i	0.951542	−	0.307518i	$-0.0994984\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	8.00748i	0.449745i	0.974388	+	0.224872i	$0.0721965\pi$
$317$	8.00748i	0.449745i	−0.974388	+	0.224872i	$0.927803\pi$
$318$	0	0
$319$	2.19060	0.122650
$320$	0	0
$321$	0	0
$322$	0	0
$323$	− 10.3040i	− 0.573331i
$324$	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
$337$	6.39663i	0.348447i	0.984706	+	0.174223i	$0.0557415\pi$
$337$	6.39663i	0.348447i	−0.984706	+	0.174223i	$0.944259\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	3.31150	0.179328
$342$	0	0
$343$	− 1.21006i	− 0.0653373i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	13.5155i	0.725552i	0.931876	+	0.362776i	$0.118171\pi$
$347$	13.5155i	0.725552i	−0.931876	+	0.362776i	$0.881829\pi$
$348$	0	0
$349$	−0.817577	−0.0437639	−0.0218819	−	0.999761i	$-0.506966\pi$
$349$	−0.817577	−0.0437639	−0.0218819	+	0.999761i	$0.506966\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	− 18.9405i	− 1.00810i	−0.863675	−	0.504049i	$-0.831843\pi$
$353$	− 18.9405i	− 1.00810i	0.863675	−	0.504049i	$-0.168157\pi$
$354$	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
$367$	19.4249i	1.01397i	0.861954	+	0.506986i	$0.169241\pi$
$367$	19.4249i	1.01397i	−0.861954	+	0.506986i	$0.830759\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	0.496081	0.0257553
$372$	0	0
$373$	29.1768i	1.51072i	0.655311	+	0.755359i	$0.272538\pi$
$373$	29.1768i	1.51072i	−0.655311	+	0.755359i	$0.727462\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	6.29466i	0.324192i
$378$	0	0
$379$	21.8060	1.12010	0.560048	−	0.828460i	$-0.310783\pi$
$379$	21.8060	1.12010	0.560048	+	0.828460i	$0.310783\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	− 9.67092i	− 0.494161i	−0.968995	−	0.247080i	$-0.920529\pi$
$383$	− 9.67092i	− 0.494161i	0.968995	−	0.247080i	$-0.0794712\pi$
$384$	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
$397$	− 22.8999i	− 1.14931i	−0.818394	−	0.574657i	$-0.805136\pi$
$397$	− 22.8999i	− 1.14931i	0.818394	−	0.574657i	$-0.194864\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−6.09998	−0.304618	−0.152309	−	0.988333i	$-0.548671\pi$
$401$	−6.09998	−0.304618	−0.152309	+	0.988333i	$0.548671\pi$
$402$	0	0
$403$	9.51554i	0.474003i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	5.34592i	0.264987i
$408$	0	0
$409$	28.8688	1.42747	0.713736	−	0.700415i	$-0.247002\pi$
$409$	28.8688	1.42747	0.713736	+	0.700415i	$0.247002\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	− 1.06702i	− 0.0525046i
$414$	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
$427$	0.547972i	0.0265182i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	8.19656	0.394815	0.197407	−	0.980322i	$-0.436748\pi$
$431$	8.19656	0.394815	0.197407	+	0.980322i	$0.436748\pi$
$432$	0	0
$433$	− 35.7464i	− 1.71786i	−0.512091	−	0.858931i	$-0.671129\pi$
$433$	− 35.7464i	− 1.71786i	0.512091	−	0.858931i	$-0.328871\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	41.8633i	2.00259i
$438$	0	0
$439$	−18.2688	−0.871922	−0.435961	−	0.899966i	$-0.643591\pi$
$439$	−18.2688	−0.871922	−0.435961	+	0.899966i	$0.643591\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	29.8323i	1.41737i	0.705523	+	0.708687i	$0.250712\pi$
$443$	29.8323i	1.41737i	−0.705523	+	0.708687i	$0.749288\pi$
$444$	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
$457$	20.4905i	0.958504i	0.877677	+	0.479252i	$0.159092\pi$
$457$	20.4905i	0.958504i	−0.877677	+	0.479252i	$0.840908\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	28.1715	1.31208	0.656039	−	0.754727i	$-0.272231\pi$
$461$	28.1715	1.31208	0.656039	+	0.754727i	$0.272231\pi$
$462$	0	0
$463$	34.3195i	1.59496i	0.603344	+	0.797481i	$0.293835\pi$
$463$	34.3195i	1.59496i	−0.603344	+	0.797481i	$0.706165\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 4.33096i	− 0.200413i	−0.994967	−	0.100206i	$-0.968050\pi$
$467$	− 4.33096i	− 0.200413i	0.994967	−	0.100206i	$-0.0319503\pi$
$468$	0	0
$469$	−0.533186	−0.0246203
$470$	0	0
$471$	0	0
$472$	0	0
$473$	− 7.53652i	− 0.346530i
$474$	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
$487$	11.2189i	0.508376i	0.967155	+	0.254188i	$0.0818083\pi$
$487$	11.2189i	0.508376i	−0.967155	+	0.254188i	$0.918192\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	25.5635	1.15366	0.576831	−	0.816863i	$-0.304289\pi$
$491$	25.5635	1.15366	0.576831	+	0.816863i	$0.304289\pi$
$492$	0	0
$493$	5.00333i	0.225339i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 1.07152i	− 0.0480643i
$498$	0	0
$499$	−18.9371	−0.847742	−0.423871	−	0.905723i	$-0.639329\pi$
$499$	−18.9371	−0.847742	−0.423871	+	0.905723i	$0.639329\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	21.1790i	0.944324i	0.881512	+	0.472162i	$0.156526\pi$
$503$	21.1790i	0.944324i	−0.881512	+	0.472162i	$0.843474\pi$
$504$	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
$517$	2.45600i	0.108015i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−2.06550	−0.0904912	−0.0452456	−	0.998976i	$-0.514407\pi$
$521$	−2.06550	−0.0904912	−0.0452456	+	0.998976i	$0.514407\pi$
$522$	0	0
$523$	− 8.34790i	− 0.365028i	−0.983203	−	0.182514i	$-0.941576\pi$
$523$	− 8.34790i	− 0.365028i	0.983203	−	0.182514i	$-0.0584235\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	7.56346i	0.329469i
$528$	0	0
$529$	−48.8592	−2.12431
$530$	0	0
$531$	0	0
$532$	0	0
$533$	− 17.4560i	− 0.756103i
$534$	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
$547$	9.50193i	0.406273i	0.979150	+	0.203137i	$0.0651136\pi$
$547$	9.50193i	0.406273i	−0.979150	+	0.203137i	$0.934886\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−11.8424	−0.504501
$552$	0	0
$553$	− 1.16296i	− 0.0494542i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	19.1999i	0.813526i	0.913534	+	0.406763i	$0.133342\pi$
$557$	19.1999i	0.813526i	−0.913534	+	0.406763i	$0.866658\pi$
$558$	0	0
$559$	21.6561	0.915954
$560$	0	0
$561$	0	0
$562$	0	0
$563$	16.2209i	0.683628i	0.939768	+	0.341814i	$0.111041\pi$
$563$	16.2209i	0.683628i	−0.939768	+	0.341814i	$0.888959\pi$
$564$	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
$577$	− 18.4113i	− 0.766473i	−0.923650	−	0.383236i	$-0.874809\pi$
$577$	− 18.4113i	− 0.766473i	0.923650	−	0.383236i	$-0.125191\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	0.525708	0.0218100
$582$	0	0
$583$	5.24034i	0.217033i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	35.8843i	1.48110i	0.671999	+	0.740552i	$0.265436\pi$
$587$	35.8843i	1.48110i	−0.671999	+	0.740552i	$0.734564\pi$
$588$	0	0
$589$	−17.9019	−0.737635
$590$	0	0
$591$	0	0
$592$	0	0
$593$	− 47.6039i	− 1.95486i	−0.211265	−	0.977429i	$-0.567758\pi$
$593$	− 47.6039i	− 1.95486i	0.211265	−	0.977429i	$-0.432242\pi$
$594$	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
$607$	4.62697i	0.187803i	0.995581	+	0.0939015i	$0.0299339\pi$
$607$	4.62697i	0.187803i	−0.995581	+	0.0939015i	$0.970066\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−7.05728	−0.285507
$612$	0	0
$613$	12.1791i	0.491909i	0.969281	+	0.245954i	$0.0791013\pi$
$613$	12.1791i	0.491909i	−0.969281	+	0.245954i	$0.920899\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	46.5369i	1.87350i	0.349994	+	0.936752i	$0.386184\pi$
$617$	46.5369i	1.87350i	−0.349994	+	0.936752i	$0.613816\pi$
$618$	0	0
$619$	−15.8040	−0.635215	−0.317608	−	0.948222i	$-0.602879\pi$
$619$	−15.8040	−0.635215	−0.317608	+	0.948222i	$0.602879\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	0.703457i	0.0281834i
$624$	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
$637$	18.3553i	0.727262i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−1.51554	−0.0598603	−0.0299301	−	0.999552i	$-0.509528\pi$
$641$	−1.51554	−0.0598603	−0.0299301	+	0.999552i	$0.509528\pi$
$642$	0	0
$643$	− 24.3999i	− 0.962236i	−0.876656	−	0.481118i	$-0.840231\pi$
$643$	− 24.3999i	− 0.962236i	0.876656	−	0.481118i	$-0.159769\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 26.9195i	− 1.05832i	−0.848523	−	0.529158i	$-0.822508\pi$
$647$	− 26.9195i	− 1.05832i	0.848523	−	0.529158i	$-0.177492\pi$
$648$	0	0
$649$	11.2714	0.442442
$650$	0	0
$651$	0	0
$652$	0	0
$653$	8.18232i	0.320199i	0.987101	+	0.160099i	$0.0511815\pi$
$653$	8.18232i	0.320199i	−0.987101	+	0.160099i	$0.948819\pi$
$654$	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
$667$	− 20.3276i	− 0.787089i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−5.78848	−0.223462
$672$	0	0
$673$	29.4135i	1.13381i	0.823785	+	0.566903i	$0.191859\pi$
$673$	29.4135i	1.13381i	−0.823785	+	0.566903i	$0.808141\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 33.9038i	− 1.30303i	−0.758637	−	0.651514i	$-0.774134\pi$
$677$	− 33.9038i	− 1.30303i	0.758637	−	0.651514i	$-0.225866\pi$
$678$	0	0
$679$	−0.960090	−0.0368449
$680$	0	0
$681$	0	0
$682$	0	0
$683$	− 29.1078i	− 1.11378i	−0.830586	−	0.556890i	$-0.811995\pi$
$683$	− 29.1078i	− 1.11378i	0.830586	−	0.556890i	$-0.188005\pi$
$684$	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
$697$	− 13.8750i	− 0.525552i
$698$	0	0
$699$	0	0
$700$	0	0
$701$	9.75406	0.368406	0.184203	−	0.982888i	$-0.441030\pi$
$701$	9.75406	0.368406	0.184203	+	0.982888i	$0.441030\pi$
$702$	0	0
$703$	− 28.8999i	− 1.08998i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 1.04458i	− 0.0392856i
$708$	0	0
$709$	32.3079	1.21335	0.606674	−	0.794951i	$-0.292503\pi$
$709$	32.3079	1.21335	0.606674	+	0.794951i	$0.292503\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	− 30.7289i	− 1.15081i
$714$	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
$727$	− 51.4714i	− 1.90897i	−0.298262	−	0.954484i	$-0.596407\pi$
$727$	− 51.4714i	− 1.90897i	0.298262	−	0.954484i	$-0.403593\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	17.2134	0.636661
$732$	0	0
$733$	6.64058i	0.245275i	0.992452	+	0.122638i	$0.0391353\pi$
$733$	6.64058i	0.245275i	−0.992452	+	0.122638i	$0.960865\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 5.63229i	− 0.207468i
$738$	0	0
$739$	38.6058	1.42014	0.710068	−	0.704133i	$-0.248664\pi$
$739$	38.6058	1.42014	0.710068	+	0.704133i	$0.248664\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	25.5904i	0.938821i	0.882980	+	0.469410i	$0.155533\pi$
$743$	25.5904i	0.938821i	−0.882980	+	0.469410i	$0.844467\pi$
$744$	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
$757$	− 13.2710i	− 0.482341i	−0.970483	−	0.241170i	$-0.922469\pi$
$757$	− 13.2710i	− 0.482341i	0.970483	−	0.241170i	$-0.0775313\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	18.5174	0.671256	0.335628	−	0.941995i	$-0.391052\pi$
$761$	18.5174	0.671256	0.335628	+	0.941995i	$0.391052\pi$
$762$	0	0
$763$	1.31366i	0.0475576i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	32.3882i	1.16947i
$768$	0	0
$769$	2.01882	0.0728006	0.0364003	−	0.999337i	$-0.488411\pi$
$769$	2.01882	0.0728006	0.0364003	+	0.999337i	$0.488411\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	− 5.76154i	− 0.207228i	−0.994618	−	0.103614i	$-0.966959\pi$
$773$	− 5.76154i	− 0.207228i	0.994618	−	0.103614i	$-0.0330407\pi$
$774$	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
$787$	− 7.33043i	− 0.261302i	−0.991428	−	0.130651i	$-0.958293\pi$
$787$	− 7.33043i	− 0.261302i	0.991428	−	0.130651i	$-0.0417067\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−0.570938	−0.0203002
$792$	0	0
$793$	− 16.6331i	− 0.590659i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 7.12358i	− 0.252330i	−0.992009	−	0.126165i	$-0.959733\pi$
$797$	− 7.12358i	− 0.252330i	0.992009	−	0.126165i	$-0.0402669\pi$
$798$	0	0
$799$	−5.60950	−0.198450
$800$	0	0
$801$	0	0
$802$	0	0
$803$	4.85398i	0.171293i
$804$	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
$817$	40.7423i	1.42539i
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−42.8768	−1.49641	−0.748206	−	0.663466i	$-0.769085\pi$
$821$	−42.8768	−1.49641	−0.748206	+	0.663466i	$0.769085\pi$
$822$	0	0
$823$	− 46.2309i	− 1.61151i	−0.592251	−	0.805753i	$-0.701761\pi$
$823$	− 46.2309i	− 1.61151i	0.592251	−	0.805753i	$-0.298239\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	23.6350i	0.821869i	0.911665	+	0.410934i	$0.134797\pi$
$827$	23.6350i	0.821869i	−0.911665	+	0.410934i	$0.865203\pi$
$828$	0	0
$829$	18.8979	0.656352	0.328176	−	0.944617i	$-0.393566\pi$
$829$	18.8979	0.656352	0.328176	+	0.944617i	$0.393566\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	14.5898i	0.505505i
$834$	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
$847$	0.879104i	0.0302064i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	49.6072	1.70051
$852$	0	0
$853$	− 39.8809i	− 1.36550i	−0.730654	−	0.682748i	$-0.760785\pi$
$853$	− 39.8809i	− 1.36550i	0.730654	−	0.682748i	$-0.239215\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 18.1269i	− 0.619204i	−0.950866	−	0.309602i	$-0.899804\pi$
$857$	− 18.1269i	− 0.619204i	0.950866	−	0.309602i	$-0.100196\pi$
$858$	0	0
$859$	−4.02081	−0.137188	−0.0685941	−	0.997645i	$-0.521851\pi$
$859$	−4.02081	−0.137188	−0.0685941	+	0.997645i	$0.521851\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	43.9540i	1.49621i	0.663580	+	0.748105i	$0.269036\pi$
$863$	43.9540i	1.49621i	−0.663580	+	0.748105i	$0.730964\pi$
$864$	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
$877$	49.5728i	1.67396i	0.547237	+	0.836978i	$0.315680\pi$
$877$	49.5728i	1.67396i	−0.547237	+	0.836978i	$0.684320\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−51.0775	−1.72085	−0.860423	−	0.509580i	$-0.829801\pi$
$881$	−51.0775	−1.72085	−0.860423	+	0.509580i	$0.829801\pi$
$882$	0	0
$883$	51.1113i	1.72003i	0.510266	+	0.860017i	$0.329547\pi$
$883$	51.1113i	1.72003i	−0.510266	+	0.860017i	$0.670453\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 33.9573i	− 1.14017i	−0.821584	−	0.570087i	$-0.806909\pi$
$887$	− 33.9573i	− 1.14017i	0.821584	−	0.570087i	$-0.193091\pi$
$888$	0	0
$889$	0.642102	0.0215354
$890$	0	0
$891$	0	0
$892$	0	0
$893$	− 13.2771i	− 0.444301i
$894$	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
$907$	− 5.80397i	− 0.192718i	−0.995347	−	0.0963588i	$-0.969280\pi$
$907$	− 5.80397i	− 0.192718i	0.995347	−	0.0963588i	$-0.0307196\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	4.19990	0.139149	0.0695744	−	0.997577i	$-0.477836\pi$
$911$	4.19990	0.139149	0.0695744	+	0.997577i	$0.477836\pi$
$912$	0	0
$913$	5.55329i	0.183787i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 1.38033i	− 0.0455827i
$918$	0	0
$919$	−25.2655	−0.833431	−0.416715	−	0.909037i	$-0.636819\pi$
$919$	−25.2655	−0.833431	−0.416715	+	0.909037i	$0.636819\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	32.5249i	1.07057i
$924$	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
$937$	− 17.5425i	− 0.573088i	−0.958067	−	0.286544i	$-0.907494\pi$
$937$	− 17.5425i	− 0.573088i	0.958067	−	0.286544i	$-0.0925064\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−24.3384	−0.793410	−0.396705	−	0.917946i	$-0.629846\pi$
$941$	−24.3384	−0.793410	−0.396705	+	0.917946i	$0.629846\pi$
$942$	0	0
$943$	56.3714i	1.83571i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 3.05206i	− 0.0991787i	−0.998770	−	0.0495893i	$-0.984209\pi$
$947$	− 3.05206i	− 0.0991787i	0.998770	−	0.0495893i	$-0.0157913\pi$
$948$	0	0
$949$	−13.9478	−0.452766
$950$	0	0
$951$	0	0
$952$	0	0
$953$	− 11.6054i	− 0.375934i	−0.982175	−	0.187967i	$-0.939810\pi$
$953$	− 11.6054i	− 0.375934i	0.982175	−	0.187967i	$-0.0601899\pi$
$954$	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
$967$	− 16.8384i	− 0.541486i	−0.962652	−	0.270743i	$-0.912731\pi$
$967$	− 16.8384i	− 0.541486i	0.962652	−	0.270743i	$-0.0872694\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	18.3864	0.590046	0.295023	−	0.955490i	$-0.404673\pi$
$971$	18.3864	0.590046	0.295023	+	0.955490i	$0.404673\pi$
$972$	0	0
$973$	0.480952i	0.0154186i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 29.4687i	− 0.942787i	−0.881923	−	0.471393i	$-0.843751\pi$
$977$	− 29.4687i	− 0.942787i	0.881923	−	0.471393i	$-0.156249\pi$
$978$	0	0
$979$	−7.43094	−0.237494
$980$	0	0
$981$	0	0
$982$	0	0
$983$	21.5541i	0.687469i	0.939067	+	0.343735i	$0.111692\pi$
$983$	21.5541i	0.687469i	−0.939067	+	0.343735i	$0.888308\pi$
$984$	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
$997$	− 30.4998i	− 0.965938i	−0.875638	−	0.482969i	$-0.839558\pi$
$997$	− 30.4998i	− 0.965938i	0.875638	−	0.482969i	$-0.160442\pi$
$998$	0	0
$999$	0	0

Twists

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

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

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.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-0.0864793i q^{7} +0.913521 q^{11} +2.62499i q^{13} +2.08648i q^{17} -4.93847 q^{19} -8.47698i q^{23} +2.39798 q^{29} +3.62499 q^{31} +5.85199i q^{37} -6.64994 q^{41} -8.24997i q^{43} +2.68850i q^{47} +6.99252 q^{49} +5.73642i q^{53} +12.3384 q^{59} -6.33645 q^{61} -6.16548i q^{67} +12.3905 q^{71} +5.31349i q^{73} -0.0790006i q^{77} +13.4479 q^{79} +6.07900i q^{83} -8.13440 q^{89} +0.227007 q^{91} -11.1020i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 6 q^{11} + 8 q^{19} + 18 q^{29} + 4 q^{31} + 18 q^{41} - 18 q^{49} + 30 q^{59} - 2 q^{61} + 24 q^{71} + 14 q^{79} + 6 q^{89} - 22 q^{91}+O(q^{100}) \) 8 * q + 6 * q^11 + 8 * q^19 + 18 * q^29 + 4 * q^31 + 18 * q^41 - 18 * q^49 + 30 * q^59 - 2 * q^61 + 24 * q^71 + 14 * q^79 + 6 * q^89 - 22 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more