LMFDB - Embedded newform 784.4.a.y.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [784,4,Mod(1,784)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(784, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("784.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 784 = 2^{4} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	784.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$46.2574974445$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 98)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.41421$ of defining polynomial
Character	$\chi$	$=$	784.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-7.07107 q^{3} +19.7990 q^{5} +23.0000 q^{9} +O(q^{10})$	$q-7.07107 q^{3} +19.7990 q^{5} +23.0000 q^{9} +14.0000 q^{11} -50.9117 q^{13} -140.000 q^{15} -1.41421 q^{17} -1.41421 q^{19} -140.000 q^{23} +267.000 q^{25} +28.2843 q^{27} -286.000 q^{29} -93.3381 q^{31} -98.9949 q^{33} -38.0000 q^{37} +360.000 q^{39} +125.865 q^{41} +34.0000 q^{43} +455.377 q^{45} +523.259 q^{47} +10.0000 q^{51} -74.0000 q^{53} +277.186 q^{55} +10.0000 q^{57} +434.164 q^{59} -14.1421 q^{61} -1008.00 q^{65} -684.000 q^{67} +989.949 q^{69} -588.000 q^{71} +270.115 q^{73} -1887.98 q^{75} -1220.00 q^{79} -821.000 q^{81} +422.850 q^{83} -28.0000 q^{85} +2022.33 q^{87} -618.011 q^{89} +660.000 q^{93} -28.0000 q^{95} -1483.51 q^{97} +322.000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 46 q^{9}+O(q^{10}) $ 2 * q + 46 * q^9	$ 2 q + 46 q^{9} + 28 q^{11} - 280 q^{15} - 280 q^{23} + 534 q^{25} - 572 q^{29} - 76 q^{37} + 720 q^{39} + 68 q^{43} + 20 q^{51} - 148 q^{53} + 20 q^{57} - 2016 q^{65} - 1368 q^{67} - 1176 q^{71} - 2440 q^{79} - 1642 q^{81} - 56 q^{85} + 1320 q^{93} - 56 q^{95} + 644 q^{99}+O(q^{100}) $ 2 * q + 46 * q^9 + 28 * q^11 - 280 * q^15 - 280 * q^23 + 534 * q^25 - 572 * q^29 - 76 * q^37 + 720 * q^39 + 68 * q^43 + 20 * q^51 - 148 * q^53 + 20 * q^57 - 2016 * q^65 - 1368 * q^67 - 1176 * q^71 - 2440 * q^79 - 1642 * q^81 - 56 * q^85 + 1320 * q^93 - 56 * q^95 + 644 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	−7.07107	−1.36083	−0.680414	−	0.732828i	$-0.738200\pi$
$3$	−7.07107	−1.36083	−0.680414	+	0.732828i	$0.738200\pi$
$4$	0	0
$5$	19.7990	1.77088	0.885438	−	0.464758i	$-0.153859\pi$
$5$	19.7990	1.77088	0.885438	+	0.464758i	$0.153859\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	23.0000	0.851852
$10$	0	0
$11$	14.0000	0.383742	0.191871	−	0.981420i	$-0.438545\pi$
$11$	14.0000	0.383742	0.191871	+	0.981420i	$0.438545\pi$
$12$	0	0
$13$	−50.9117	−1.08618	−0.543091	−	0.839674i	$-0.682746\pi$
$13$	−50.9117	−1.08618	−0.543091	+	0.839674i	$0.682746\pi$
$14$	0	0
$15$	−140.000	−2.40986
$16$	0	0
$17$	−1.41421	−0.0201763	−0.0100882	−	0.999949i	$-0.503211\pi$
$17$	−1.41421	−0.0201763	−0.0100882	+	0.999949i	$0.503211\pi$
$18$	0	0
$19$	−1.41421	−0.0170759	−0.00853797	−	0.999964i	$-0.502718\pi$
$19$	−1.41421	−0.0170759	−0.00853797	+	0.999964i	$0.502718\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−140.000	−1.26922	−0.634609	−	0.772833i	$-0.718839\pi$
$23$	−140.000	−1.26922	−0.634609	+	0.772833i	$0.718839\pi$
$24$	0	0
$25$	267.000	2.13600
$26$	0	0
$27$	28.2843	0.201604
$28$	0	0
$29$	−286.000	−1.83134	−0.915670	−	0.401931i	$-0.868339\pi$
$29$	−286.000	−1.83134	−0.915670	+	0.401931i	$0.868339\pi$
$30$	0	0
$31$	−93.3381	−0.540775	−0.270387	−	0.962752i	$-0.587152\pi$
$31$	−93.3381	−0.540775	−0.270387	+	0.962752i	$0.587152\pi$
$32$	0	0
$33$	−98.9949	−0.522206
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−38.0000	−0.168842	−0.0844211	−	0.996430i	$-0.526904\pi$
$37$	−38.0000	−0.168842	−0.0844211	+	0.996430i	$0.526904\pi$
$38$	0	0
$39$	360.000	1.47811
$40$	0	0
$41$	125.865	0.479434	0.239717	−	0.970843i	$-0.422945\pi$
$41$	125.865	0.479434	0.239717	+	0.970843i	$0.422945\pi$
$42$	0	0
$43$	34.0000	0.120580	0.0602901	−	0.998181i	$-0.480797\pi$
$43$	34.0000	0.120580	0.0602901	+	0.998181i	$0.480797\pi$
$44$	0	0
$45$	455.377	1.50852
$46$	0	0
$47$	523.259	1.62394	0.811970	−	0.583699i	$-0.198395\pi$
$47$	523.259	1.62394	0.811970	+	0.583699i	$0.198395\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	10.0000	0.0274565
$52$	0	0
$53$	−74.0000	−0.191786	−0.0958932	−	0.995392i	$-0.530571\pi$
$53$	−74.0000	−0.191786	−0.0958932	+	0.995392i	$0.530571\pi$
$54$	0	0
$55$	277.186	0.679559
$56$	0	0
$57$	10.0000	0.0232374
$58$	0	0
$59$	434.164	0.958022	0.479011	−	0.877809i	$-0.340995\pi$
$59$	434.164	0.958022	0.479011	+	0.877809i	$0.340995\pi$
$60$	0	0
$61$	−14.1421	−0.0296839	−0.0148419	−	0.999890i	$-0.504725\pi$
$61$	−14.1421	−0.0296839	−0.0148419	+	0.999890i	$0.504725\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−1008.00	−1.92349
$66$	0	0
$67$	−684.000	−1.24722	−0.623611	−	0.781735i	$-0.714335\pi$
$67$	−684.000	−1.24722	−0.623611	+	0.781735i	$0.714335\pi$
$68$	0	0
$69$	989.949	1.72719
$70$	0	0
$71$	−588.000	−0.982856	−0.491428	−	0.870918i	$-0.663525\pi$
$71$	−588.000	−0.982856	−0.491428	+	0.870918i	$0.663525\pi$
$72$	0	0
$73$	270.115	0.433076	0.216538	−	0.976274i	$-0.430523\pi$
$73$	270.115	0.433076	0.216538	+	0.976274i	$0.430523\pi$
$74$	0	0
$75$	−1887.98	−2.90673
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−1220.00	−1.73748	−0.868739	−	0.495271i	$-0.835069\pi$
$79$	−1220.00	−1.73748	−0.868739	+	0.495271i	$0.835069\pi$
$80$	0	0
$81$	−821.000	−1.12620
$82$	0	0
$83$	422.850	0.559202	0.279601	−	0.960116i	$-0.409798\pi$
$83$	422.850	0.559202	0.279601	+	0.960116i	$0.409798\pi$
$84$	0	0
$85$	−28.0000	−0.0357297
$86$	0	0
$87$	2022.33	2.49214
$88$	0	0
$89$	−618.011	−0.736057	−0.368028	−	0.929815i	$-0.619967\pi$
$89$	−618.011	−0.736057	−0.368028	+	0.929815i	$0.619967\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	660.000	0.735901
$94$	0	0
$95$	−28.0000	−0.0302394
$96$	0	0
$97$	−1483.51	−1.55286	−0.776431	−	0.630202i	$-0.782972\pi$
$97$	−1483.51	−1.55286	−0.776431	+	0.630202i	$0.782972\pi$
$98$	0	0
$99$	322.000	0.326891
$100$	0	0
$101$	−1128.54	−1.11182	−0.555912	−	0.831241i	$-0.687631\pi$
$101$	−1128.54	−1.11182	−0.555912	+	0.831241i	$0.687631\pi$
$102$	0	0
$103$	−868.327	−0.830668	−0.415334	−	0.909669i	$-0.636335\pi$
$103$	−868.327	−0.830668	−0.415334	+	0.909669i	$0.636335\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	1684.00	1.52148	0.760740	−	0.649056i	$-0.224836\pi$
$107$	1684.00	1.52148	0.760740	+	0.649056i	$0.224836\pi$
$108$	0	0
$109$	−818.000	−0.718809	−0.359405	−	0.933182i	$-0.617020\pi$
$109$	−818.000	−0.718809	−0.359405	+	0.933182i	$0.617020\pi$
$110$	0	0
$111$	268.701	0.229765
$112$	0	0
$113$	−540.000	−0.449548	−0.224774	−	0.974411i	$-0.572164\pi$
$113$	−540.000	−0.449548	−0.224774	+	0.974411i	$0.572164\pi$
$114$	0	0
$115$	−2771.86	−2.24763
$116$	0	0
$117$	−1170.97	−0.925266
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−1135.00	−0.852742
$122$	0	0
$123$	−890.000	−0.652428
$124$	0	0
$125$	2811.46	2.01171
$126$	0	0
$127$	−1720.00	−1.20177	−0.600887	−	0.799334i	$-0.705186\pi$
$127$	−1720.00	−1.20177	−0.600887	+	0.799334i	$0.705186\pi$
$128$	0	0
$129$	−240.416	−0.164089
$130$	0	0
$131$	−1735.24	−1.15732	−0.578659	−	0.815570i	$-0.696424\pi$
$131$	−1735.24	−1.15732	−0.578659	+	0.815570i	$0.696424\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	560.000	0.357016
$136$	0	0
$137$	828.000	0.516356	0.258178	−	0.966097i	$-0.416878\pi$
$137$	828.000	0.516356	0.258178	+	0.966097i	$0.416878\pi$
$138$	0	0
$139$	−425.678	−0.259752	−0.129876	−	0.991530i	$-0.541458\pi$
$139$	−425.678	−0.259752	−0.129876	+	0.991530i	$0.541458\pi$
$140$	0	0
$141$	−3700.00	−2.20990
$142$	0	0
$143$	−712.764	−0.416813
$144$	0	0
$145$	−5662.51	−3.24308
$146$	0	0
$147$	0	0
$148$	0	0
$149$	2050.00	1.12713	0.563566	−	0.826071i	$-0.309429\pi$
$149$	2050.00	1.12713	0.563566	+	0.826071i	$0.309429\pi$
$150$	0	0
$151$	472.000	0.254376	0.127188	−	0.991879i	$-0.459405\pi$
$151$	472.000	0.254376	0.127188	+	0.991879i	$0.459405\pi$
$152$	0	0
$153$	−32.5269	−0.0171872
$154$	0	0
$155$	−1848.00	−0.957645
$156$	0	0
$157$	2211.83	1.12435	0.562176	−	0.827018i	$-0.309964\pi$
$157$	2211.83	1.12435	0.562176	+	0.827018i	$0.309964\pi$
$158$	0	0
$159$	523.259	0.260988
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−3286.00	−1.57901	−0.789507	−	0.613741i	$-0.789664\pi$
$163$	−3286.00	−1.57901	−0.789507	+	0.613741i	$0.789664\pi$
$164$	0	0
$165$	−1960.00	−0.924762
$166$	0	0
$167$	−1490.58	−0.690686	−0.345343	−	0.938476i	$-0.612237\pi$
$167$	−1490.58	−0.690686	−0.345343	+	0.938476i	$0.612237\pi$
$168$	0	0
$169$	395.000	0.179791
$170$	0	0
$171$	−32.5269	−0.0145462
$172$	0	0
$173$	2070.41	0.909886	0.454943	−	0.890521i	$-0.349660\pi$
$173$	2070.41	0.909886	0.454943	+	0.890521i	$0.349660\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−3070.00	−1.30370
$178$	0	0
$179$	−540.000	−0.225483	−0.112742	−	0.993624i	$-0.535963\pi$
$179$	−540.000	−0.225483	−0.112742	+	0.993624i	$0.535963\pi$
$180$	0	0
$181$	−3784.44	−1.55412	−0.777058	−	0.629429i	$-0.783289\pi$
$181$	−3784.44	−1.55412	−0.777058	+	0.629429i	$0.783289\pi$
$182$	0	0
$183$	100.000	0.0403946
$184$	0	0
$185$	−752.362	−0.298999
$186$	0	0
$187$	−19.7990	−0.00774249
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−1028.00	−0.389442	−0.194721	−	0.980859i	$-0.562380\pi$
$191$	−1028.00	−0.389442	−0.194721	+	0.980859i	$0.562380\pi$
$192$	0	0
$193$	4592.00	1.71264	0.856320	−	0.516446i	$-0.172745\pi$
$193$	4592.00	1.71264	0.856320	+	0.516446i	$0.172745\pi$
$194$	0	0
$195$	7127.64	2.61754
$196$	0	0
$197$	794.000	0.287158	0.143579	−	0.989639i	$-0.454139\pi$
$197$	794.000	0.287158	0.143579	+	0.989639i	$0.454139\pi$
$198$	0	0
$199$	2486.19	0.885634	0.442817	−	0.896612i	$-0.353979\pi$
$199$	2486.19	0.885634	0.442817	+	0.896612i	$0.353979\pi$
$200$	0	0
$201$	4836.61	1.69725
$202$	0	0
$203$	0	0
$204$	0	0
$205$	2492.00	0.849019
$206$	0	0
$207$	−3220.00	−1.08119
$208$	0	0
$209$	−19.7990	−0.00655275
$210$	0	0
$211$	2748.00	0.896588	0.448294	−	0.893886i	$-0.352032\pi$
$211$	2748.00	0.896588	0.448294	+	0.893886i	$0.352032\pi$
$212$	0	0
$213$	4157.79	1.33750
$214$	0	0
$215$	673.166	0.213533
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−1910.00	−0.589342
$220$	0	0
$221$	72.0000	0.0219151
$222$	0	0
$223$	−3428.05	−1.02941	−0.514707	−	0.857366i	$-0.672099\pi$
$223$	−3428.05	−1.02941	−0.514707	+	0.857366i	$0.672099\pi$
$224$	0	0
$225$	6141.00	1.81956
$226$	0	0
$227$	5290.57	1.54691	0.773453	−	0.633854i	$-0.218528\pi$
$227$	5290.57	1.54691	0.773453	+	0.633854i	$0.218528\pi$
$228$	0	0
$229$	2749.23	0.793338	0.396669	−	0.917962i	$-0.370166\pi$
$229$	2749.23	0.793338	0.396669	+	0.917962i	$0.370166\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	72.0000	0.0202441	0.0101221	−	0.999949i	$-0.496778\pi$
$233$	72.0000	0.0202441	0.0101221	+	0.999949i	$0.496778\pi$
$234$	0	0
$235$	10360.0	2.87580
$236$	0	0
$237$	8626.70	2.36441
$238$	0	0
$239$	−4308.00	−1.16595	−0.582974	−	0.812491i	$-0.698111\pi$
$239$	−4308.00	−1.16595	−0.582974	+	0.812491i	$0.698111\pi$
$240$	0	0
$241$	1540.08	0.411640	0.205820	−	0.978590i	$-0.434014\pi$
$241$	1540.08	0.411640	0.205820	+	0.978590i	$0.434014\pi$
$242$	0	0
$243$	5041.67	1.33096
$244$	0	0
$245$	0	0
$246$	0	0
$247$	72.0000	0.0185476
$248$	0	0
$249$	−2990.00	−0.760978
$250$	0	0
$251$	931.967	0.234363	0.117182	−	0.993110i	$-0.462614\pi$
$251$	931.967	0.234363	0.117182	+	0.993110i	$0.462614\pi$
$252$	0	0
$253$	−1960.00	−0.487052
$254$	0	0
$255$	197.990	0.0486220
$256$	0	0
$257$	−937.624	−0.227577	−0.113789	−	0.993505i	$-0.536299\pi$
$257$	−937.624	−0.227577	−0.113789	+	0.993505i	$0.536299\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−6578.00	−1.56003
$262$	0	0
$263$	−7140.00	−1.67404	−0.837018	−	0.547176i	$-0.815703\pi$
$263$	−7140.00	−1.67404	−0.837018	+	0.547176i	$0.815703\pi$
$264$	0	0
$265$	−1465.13	−0.339630
$266$	0	0
$267$	4370.00	1.00165
$268$	0	0
$269$	−4610.34	−1.04497	−0.522485	−	0.852648i	$-0.674995\pi$
$269$	−4610.34	−1.04497	−0.522485	+	0.852648i	$0.674995\pi$
$270$	0	0
$271$	−2364.57	−0.530026	−0.265013	−	0.964245i	$-0.585376\pi$
$271$	−2364.57	−0.530026	−0.265013	+	0.964245i	$0.585376\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	3738.00	0.819672
$276$	0	0
$277$	4006.00	0.868943	0.434472	−	0.900686i	$-0.356935\pi$
$277$	4006.00	0.868943	0.434472	+	0.900686i	$0.356935\pi$
$278$	0	0
$279$	−2146.78	−0.460660
$280$	0	0
$281$	−5984.00	−1.27038	−0.635188	−	0.772358i	$-0.719077\pi$
$281$	−5984.00	−1.27038	−0.635188	+	0.772358i	$0.719077\pi$
$282$	0	0
$283$	−4928.53	−1.03523	−0.517617	−	0.855613i	$-0.673181\pi$
$283$	−4928.53	−1.03523	−0.517617	+	0.855613i	$0.673181\pi$
$284$	0	0
$285$	197.990	0.0411506
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−4911.00	−0.999593
$290$	0	0
$291$	10490.0	2.11318
$292$	0	0
$293$	−1971.41	−0.393076	−0.196538	−	0.980496i	$-0.562970\pi$
$293$	−1971.41	−0.393076	−0.196538	+	0.980496i	$0.562970\pi$
$294$	0	0
$295$	8596.00	1.69654
$296$	0	0
$297$	395.980	0.0773639
$298$	0	0
$299$	7127.64	1.37860
$300$	0	0
$301$	0	0
$302$	0	0
$303$	7980.00	1.51300
$304$	0	0
$305$	−280.000	−0.0525664
$306$	0	0
$307$	−4767.31	−0.886270	−0.443135	−	0.896455i	$-0.646134\pi$
$307$	−4767.31	−0.886270	−0.443135	+	0.896455i	$0.646134\pi$
$308$	0	0
$309$	6140.00	1.13040
$310$	0	0
$311$	−6776.91	−1.23564	−0.617819	−	0.786320i	$-0.711984\pi$
$311$	−6776.91	−1.23564	−0.617819	+	0.786320i	$0.711984\pi$
$312$	0	0
$313$	6190.01	1.11783	0.558914	−	0.829226i	$-0.311218\pi$
$313$	6190.01	1.11783	0.558914	+	0.829226i	$0.311218\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	9826.00	1.74096	0.870478	−	0.492207i	$-0.163810\pi$
$317$	9826.00	1.74096	0.870478	+	0.492207i	$0.163810\pi$
$318$	0	0
$319$	−4004.00	−0.702762
$320$	0	0
$321$	−11907.7	−2.07047
$322$	0	0
$323$	2.00000	0.000344529 0
$324$	0	0
$325$	−13593.4	−2.32008
$326$	0	0
$327$	5784.13	0.978175
$328$	0	0
$329$	0	0
$330$	0	0
$331$	5738.00	0.952837	0.476418	−	0.879219i	$-0.341935\pi$
$331$	5738.00	0.952837	0.476418	+	0.879219i	$0.341935\pi$
$332$	0	0
$333$	−874.000	−0.143829
$334$	0	0
$335$	−13542.5	−2.20868
$336$	0	0
$337$	−2254.00	−0.364342	−0.182171	−	0.983267i	$-0.558312\pi$
$337$	−2254.00	−0.364342	−0.182171	+	0.983267i	$0.558312\pi$
$338$	0	0
$339$	3818.38	0.611757
$340$	0	0
$341$	−1306.73	−0.207518
$342$	0	0
$343$	0	0
$344$	0	0
$345$	19600.0	3.05863
$346$	0	0
$347$	1986.00	0.307245	0.153623	−	0.988130i	$-0.450906\pi$
$347$	1986.00	0.307245	0.153623	+	0.988130i	$0.450906\pi$
$348$	0	0
$349$	−6771.25	−1.03856	−0.519279	−	0.854605i	$-0.673800\pi$
$349$	−6771.25	−1.03856	−0.519279	+	0.854605i	$0.673800\pi$
$350$	0	0
$351$	−1440.00	−0.218979
$352$	0	0
$353$	−6993.29	−1.05443	−0.527217	−	0.849731i	$-0.676764\pi$
$353$	−6993.29	−1.05443	−0.527217	+	0.849731i	$0.676764\pi$
$354$	0	0
$355$	−11641.8	−1.74052
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−5944.00	−0.873850	−0.436925	−	0.899498i	$-0.643933\pi$
$359$	−5944.00	−0.873850	−0.436925	+	0.899498i	$0.643933\pi$
$360$	0	0
$361$	−6857.00	−0.999708
$362$	0	0
$363$	8025.66	1.16044
$364$	0	0
$365$	5348.00	0.766924
$366$	0	0
$367$	−842.871	−0.119884	−0.0599421	−	0.998202i	$-0.519092\pi$
$367$	−842.871	−0.119884	−0.0599421	+	0.998202i	$0.519092\pi$
$368$	0	0
$369$	2894.90	0.408407
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−5726.00	−0.794855	−0.397428	−	0.917634i	$-0.630097\pi$
$373$	−5726.00	−0.794855	−0.397428	+	0.917634i	$0.630097\pi$
$374$	0	0
$375$	−19880.0	−2.73760
$376$	0	0
$377$	14560.7	1.98917
$378$	0	0
$379$	−10330.0	−1.40004	−0.700022	−	0.714122i	$-0.746826\pi$
$379$	−10330.0	−1.40004	−0.700022	+	0.714122i	$0.746826\pi$
$380$	0	0
$381$	12162.2	1.63541
$382$	0	0
$383$	1004.09	0.133960	0.0669800	−	0.997754i	$-0.478664\pi$
$383$	1004.09	0.133960	0.0669800	+	0.997754i	$0.478664\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	782.000	0.102717
$388$	0	0
$389$	5210.00	0.679068	0.339534	−	0.940594i	$-0.389731\pi$
$389$	5210.00	0.679068	0.339534	+	0.940594i	$0.389731\pi$
$390$	0	0
$391$	197.990	0.0256081
$392$	0	0
$393$	12270.0	1.57491
$394$	0	0
$395$	−24154.8	−3.07686
$396$	0	0
$397$	73.5391	0.00929678	0.00464839	−	0.999989i	$-0.498520\pi$
$397$	73.5391	0.00929678	0.00464839	+	0.999989i	$0.498520\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−498.000	−0.0620173	−0.0310086	−	0.999519i	$-0.509872\pi$
$401$	−498.000	−0.0620173	−0.0310086	+	0.999519i	$0.509872\pi$
$402$	0	0
$403$	4752.00	0.587380
$404$	0	0
$405$	−16255.0	−1.99436
$406$	0	0
$407$	−532.000	−0.0647918
$408$	0	0
$409$	−3355.93	−0.405721	−0.202861	−	0.979208i	$-0.565024\pi$
$409$	−3355.93	−0.405721	−0.202861	+	0.979208i	$0.565024\pi$
$410$	0	0
$411$	−5854.84	−0.702672
$412$	0	0
$413$	0	0
$414$	0	0
$415$	8372.00	0.990278
$416$	0	0
$417$	3010.00	0.353478
$418$	0	0
$419$	14545.2	1.69589	0.847946	−	0.530082i	$-0.177839\pi$
$419$	14545.2	1.69589	0.847946	+	0.530082i	$0.177839\pi$
$420$	0	0
$421$	10854.0	1.25651	0.628256	−	0.778007i	$-0.283769\pi$
$421$	10854.0	1.25651	0.628256	+	0.778007i	$0.283769\pi$
$422$	0	0
$423$	12035.0	1.38336
$424$	0	0
$425$	−377.595	−0.0430966
$426$	0	0
$427$	0	0
$428$	0	0
$429$	5040.00	0.567211
$430$	0	0
$431$	5364.00	0.599477	0.299739	−	0.954021i	$-0.403100\pi$
$431$	5364.00	0.599477	0.299739	+	0.954021i	$0.403100\pi$
$432$	0	0
$433$	6487.00	0.719966	0.359983	−	0.932959i	$-0.382783\pi$
$433$	6487.00	0.719966	0.359983	+	0.932959i	$0.382783\pi$
$434$	0	0
$435$	40040.0	4.41327
$436$	0	0
$437$	197.990	0.0216731
$438$	0	0
$439$	13932.8	1.51476	0.757378	−	0.652977i	$-0.226480\pi$
$439$	13932.8	1.51476	0.757378	+	0.652977i	$0.226480\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−5996.00	−0.643067	−0.321533	−	0.946898i	$-0.604198\pi$
$443$	−5996.00	−0.643067	−0.321533	+	0.946898i	$0.604198\pi$
$444$	0	0
$445$	−12236.0	−1.30347
$446$	0	0
$447$	−14495.7	−1.53383
$448$	0	0
$449$	2622.00	0.275590	0.137795	−	0.990461i	$-0.455999\pi$
$449$	2622.00	0.275590	0.137795	+	0.990461i	$0.455999\pi$
$450$	0	0
$451$	1762.11	0.183979
$452$	0	0
$453$	−3337.54	−0.346162
$454$	0	0
$455$	0	0
$456$	0	0
$457$	11208.0	1.14724	0.573619	−	0.819122i	$-0.305539\pi$
$457$	11208.0	1.14724	0.573619	+	0.819122i	$0.305539\pi$
$458$	0	0
$459$	−40.0000	−0.00406763
$460$	0	0
$461$	−9786.36	−0.988712	−0.494356	−	0.869260i	$-0.664596\pi$
$461$	−9786.36	−0.988712	−0.494356	+	0.869260i	$0.664596\pi$
$462$	0	0
$463$	−3952.00	−0.396685	−0.198342	−	0.980133i	$-0.563556\pi$
$463$	−3952.00	−0.396685	−0.198342	+	0.980133i	$0.563556\pi$
$464$	0	0
$465$	13067.3	1.30319
$466$	0	0
$467$	17506.5	1.73470	0.867352	−	0.497696i	$-0.165820\pi$
$467$	17506.5	1.73470	0.867352	+	0.497696i	$0.165820\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−15640.0	−1.53005
$472$	0	0
$473$	476.000	0.0462717
$474$	0	0
$475$	−377.595	−0.0364742
$476$	0	0
$477$	−1702.00	−0.163374
$478$	0	0
$479$	2288.20	0.218268	0.109134	−	0.994027i	$-0.465192\pi$
$479$	2288.20	0.218268	0.109134	+	0.994027i	$0.465192\pi$
$480$	0	0
$481$	1934.64	0.183393
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−29372.0	−2.74993
$486$	0	0
$487$	−972.000	−0.0904426	−0.0452213	−	0.998977i	$-0.514399\pi$
$487$	−972.000	−0.0904426	−0.0452213	+	0.998977i	$0.514399\pi$
$488$	0	0
$489$	23235.5	2.14877
$490$	0	0
$491$	7404.00	0.680525	0.340263	−	0.940330i	$-0.389484\pi$
$491$	7404.00	0.680525	0.340263	+	0.940330i	$0.389484\pi$
$492$	0	0
$493$	404.465	0.0369497
$494$	0	0
$495$	6375.27	0.578883
$496$	0	0
$497$	0	0
$498$	0	0
$499$	12244.0	1.09843	0.549215	−	0.835681i	$-0.314927\pi$
$499$	12244.0	1.09843	0.549215	+	0.835681i	$0.314927\pi$
$500$	0	0
$501$	10540.0	0.939905
$502$	0	0
$503$	−2415.48	−0.214117	−0.107058	−	0.994253i	$-0.534143\pi$
$503$	−2415.48	−0.214117	−0.107058	+	0.994253i	$0.534143\pi$
$504$	0	0
$505$	−22344.0	−1.96890
$506$	0	0
$507$	−2793.07	−0.244664
$508$	0	0
$509$	5707.77	0.497038	0.248519	−	0.968627i	$-0.420056\pi$
$509$	5707.77	0.497038	0.248519	+	0.968627i	$0.420056\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−40.0000	−0.00344258
$514$	0	0
$515$	−17192.0	−1.47101
$516$	0	0
$517$	7325.63	0.623173
$518$	0	0
$519$	−14640.0	−1.23820
$520$	0	0
$521$	−1.41421	−0.000118921 0	−5.94605e−5	−	1.00000i	$-0.500019\pi$
$521$	−1.41421	−0.000118921 0	−5.94605e−5		1.00000i	$0.500019\pi$
$522$	0	0
$523$	−12257.0	−1.02478	−0.512391	−	0.858752i	$-0.671240\pi$
$523$	−12257.0	−1.02478	−0.512391	+	0.858752i	$0.671240\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	132.000	0.0109108
$528$	0	0
$529$	7433.00	0.610915
$530$	0	0
$531$	9985.76	0.816093
$532$	0	0
$533$	−6408.00	−0.520753
$534$	0	0
$535$	33341.5	2.69435
$536$	0	0
$537$	3818.38	0.306844
$538$	0	0
$539$	0	0
$540$	0	0
$541$	2050.00	0.162914	0.0814569	−	0.996677i	$-0.474043\pi$
$541$	2050.00	0.162914	0.0814569	+	0.996677i	$0.474043\pi$
$542$	0	0
$543$	26760.0	2.11488
$544$	0	0
$545$	−16195.6	−1.27292
$546$	0	0
$547$	−14554.0	−1.13763	−0.568815	−	0.822465i	$-0.692598\pi$
$547$	−14554.0	−1.13763	−0.568815	+	0.822465i	$0.692598\pi$
$548$	0	0
$549$	−325.269	−0.0252862
$550$	0	0
$551$	404.465	0.0312719
$552$	0	0
$553$	0	0
$554$	0	0
$555$	5320.00	0.406885
$556$	0	0
$557$	6954.00	0.528995	0.264498	−	0.964386i	$-0.414794\pi$
$557$	6954.00	0.528995	0.264498	+	0.964386i	$0.414794\pi$
$558$	0	0
$559$	−1731.00	−0.130972
$560$	0	0
$561$	140.000	0.0105362
$562$	0	0
$563$	−1636.25	−0.122486	−0.0612429	−	0.998123i	$-0.519506\pi$
$563$	−1636.25	−0.122486	−0.0612429	+	0.998123i	$0.519506\pi$
$564$	0	0
$565$	−10691.5	−0.796094
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−7142.00	−0.526201	−0.263100	−	0.964768i	$-0.584745\pi$
$569$	−7142.00	−0.526201	−0.263100	+	0.964768i	$0.584745\pi$
$570$	0	0
$571$	20606.0	1.51022	0.755109	−	0.655599i	$-0.227584\pi$
$571$	20606.0	1.51022	0.755109	+	0.655599i	$0.227584\pi$
$572$	0	0
$573$	7269.06	0.529964
$574$	0	0
$575$	−37380.0	−2.71105
$576$	0	0
$577$	8803.48	0.635171	0.317585	−	0.948230i	$-0.397128\pi$
$577$	8803.48	0.635171	0.317585	+	0.948230i	$0.397128\pi$
$578$	0	0
$579$	−32470.3	−2.33061
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−1036.00	−0.0735965
$584$	0	0
$585$	−23184.0	−1.63853
$586$	0	0
$587$	6503.97	0.457321	0.228661	−	0.973506i	$-0.426565\pi$
$587$	6503.97	0.457321	0.228661	+	0.973506i	$0.426565\pi$
$588$	0	0
$589$	132.000	0.00923424
$590$	0	0
$591$	−5614.43	−0.390773
$592$	0	0
$593$	23140.8	1.60249	0.801246	−	0.598335i	$-0.204171\pi$
$593$	23140.8	1.60249	0.801246	+	0.598335i	$0.204171\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−17580.0	−1.20520
$598$	0	0
$599$	11296.0	0.770521	0.385260	−	0.922808i	$-0.374112\pi$
$599$	11296.0	0.770521	0.385260	+	0.922808i	$0.374112\pi$
$600$	0	0
$601$	−8727.11	−0.592323	−0.296162	−	0.955138i	$-0.595707\pi$
$601$	−8727.11	−0.592323	−0.296162	+	0.955138i	$0.595707\pi$
$602$	0	0
$603$	−15732.0	−1.06245
$604$	0	0
$605$	−22471.9	−1.51010
$606$	0	0
$607$	19736.8	1.31975	0.659877	−	0.751374i	$-0.270608\pi$
$607$	19736.8	1.31975	0.659877	+	0.751374i	$0.270608\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−26640.0	−1.76389
$612$	0	0
$613$	16962.0	1.11760	0.558800	−	0.829302i	$-0.311262\pi$
$613$	16962.0	1.11760	0.558800	+	0.829302i	$0.311262\pi$
$614$	0	0
$615$	−17621.1	−1.15537
$616$	0	0
$617$	−19034.0	−1.24194	−0.620972	−	0.783832i	$-0.713262\pi$
$617$	−19034.0	−1.24194	−0.620972	+	0.783832i	$0.713262\pi$
$618$	0	0
$619$	−18677.5	−1.21278	−0.606392	−	0.795166i	$-0.707384\pi$
$619$	−18677.5	−1.21278	−0.606392	+	0.795166i	$0.707384\pi$
$620$	0	0
$621$	−3959.80	−0.255880
$622$	0	0
$623$	0	0
$624$	0	0
$625$	22289.0	1.42650
$626$	0	0
$627$	140.000	0.00891716
$628$	0	0
$629$	53.7401	0.00340661
$630$	0	0
$631$	14716.0	0.928423	0.464211	−	0.885724i	$-0.346338\pi$
$631$	14716.0	0.928423	0.464211	+	0.885724i	$0.346338\pi$
$632$	0	0
$633$	−19431.3	−1.22010
$634$	0	0
$635$	−34054.3	−2.12819
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−13524.0	−0.837248
$640$	0	0
$641$	4730.00	0.291457	0.145728	−	0.989325i	$-0.453447\pi$
$641$	4730.00	0.291457	0.145728	+	0.989325i	$0.453447\pi$
$642$	0	0
$643$	19056.5	1.16877	0.584383	−	0.811478i	$-0.301337\pi$
$643$	19056.5	1.16877	0.584383	+	0.811478i	$0.301337\pi$
$644$	0	0
$645$	−4760.00	−0.290581
$646$	0	0
$647$	9342.29	0.567672	0.283836	−	0.958873i	$-0.408393\pi$
$647$	9342.29	0.567672	0.283836	+	0.958873i	$0.408393\pi$
$648$	0	0
$649$	6078.29	0.367633
$650$	0	0
$651$	0	0
$652$	0	0
$653$	3774.00	0.226169	0.113084	−	0.993585i	$-0.463927\pi$
$653$	3774.00	0.226169	0.113084	+	0.993585i	$0.463927\pi$
$654$	0	0
$655$	−34356.0	−2.04947
$656$	0	0
$657$	6212.64	0.368917
$658$	0	0
$659$	21150.0	1.25021	0.625104	−	0.780541i	$-0.285057\pi$
$659$	21150.0	1.25021	0.625104	+	0.780541i	$0.285057\pi$
$660$	0	0
$661$	10377.5	0.610647	0.305324	−	0.952249i	$-0.401235\pi$
$661$	10377.5	0.610647	0.305324	+	0.952249i	$0.401235\pi$
$662$	0	0
$663$	−509.117	−0.0298227
$664$	0	0
$665$	0	0
$666$	0	0
$667$	40040.0	2.32437
$668$	0	0
$669$	24240.0	1.40086
$670$	0	0
$671$	−197.990	−0.0113909
$672$	0	0
$673$	−1164.00	−0.0666700	−0.0333350	−	0.999444i	$-0.510613\pi$
$673$	−1164.00	−0.0666700	−0.0333350	+	0.999444i	$0.510613\pi$
$674$	0	0
$675$	7551.90	0.430626
$676$	0	0
$677$	−27152.9	−1.54146	−0.770732	−	0.637160i	$-0.780109\pi$
$677$	−27152.9	−1.54146	−0.770732	+	0.637160i	$0.780109\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−37410.0	−2.10507
$682$	0	0
$683$	16596.0	0.929763	0.464882	−	0.885373i	$-0.346097\pi$
$683$	16596.0	0.929763	0.464882	+	0.885373i	$0.346097\pi$
$684$	0	0
$685$	16393.6	0.914403
$686$	0	0
$687$	−19440.0	−1.07960
$688$	0	0
$689$	3767.46	0.208315
$690$	0	0
$691$	11298.2	0.622000	0.311000	−	0.950410i	$-0.399336\pi$
$691$	11298.2	0.622000	0.311000	+	0.950410i	$0.399336\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−8428.00	−0.459989
$696$	0	0
$697$	−178.000	−0.00967321
$698$	0	0
$699$	−509.117	−0.0275487
$700$	0	0
$701$	−2754.00	−0.148384	−0.0741920	−	0.997244i	$-0.523638\pi$
$701$	−2754.00	−0.148384	−0.0741920	+	0.997244i	$0.523638\pi$
$702$	0	0
$703$	53.7401	0.00288314
$704$	0	0
$705$	−73256.3	−3.91346
$706$	0	0
$707$	0	0
$708$	0	0
$709$	29434.0	1.55912	0.779561	−	0.626327i	$-0.215442\pi$
$709$	29434.0	1.55912	0.779561	+	0.626327i	$0.215442\pi$
$710$	0	0
$711$	−28060.0	−1.48007
$712$	0	0
$713$	13067.3	0.686361
$714$	0	0
$715$	−14112.0	−0.738124
$716$	0	0
$717$	30462.2	1.58665
$718$	0	0
$719$	−17669.2	−0.916480	−0.458240	−	0.888828i	$-0.651520\pi$
$719$	−17669.2	−0.916480	−0.458240	+	0.888828i	$0.651520\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	−10890.0	−0.560171
$724$	0	0
$725$	−76362.0	−3.91174
$726$	0	0
$727$	28445.5	1.45115	0.725574	−	0.688144i	$-0.241574\pi$
$727$	28445.5	1.45115	0.725574	+	0.688144i	$0.241574\pi$
$728$	0	0
$729$	−13483.0	−0.685007
$730$	0	0
$731$	−48.0833	−0.00243286
$732$	0	0
$733$	−22341.7	−1.12580	−0.562900	−	0.826525i	$-0.690314\pi$
$733$	−22341.7	−1.12580	−0.562900	+	0.826525i	$0.690314\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−9576.00	−0.478611
$738$	0	0
$739$	−20670.0	−1.02890	−0.514451	−	0.857520i	$-0.672004\pi$
$739$	−20670.0	−1.02890	−0.514451	+	0.857520i	$0.672004\pi$
$740$	0	0
$741$	−509.117	−0.0252400
$742$	0	0
$743$	25400.0	1.25415	0.627076	−	0.778958i	$-0.284251\pi$
$743$	25400.0	1.25415	0.627076	+	0.778958i	$0.284251\pi$
$744$	0	0
$745$	40587.9	1.99601
$746$	0	0
$747$	9725.55	0.476358
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−29180.0	−1.41783	−0.708917	−	0.705292i	$-0.750816\pi$
$751$	−29180.0	−1.41783	−0.708917	+	0.705292i	$0.750816\pi$
$752$	0	0
$753$	−6590.00	−0.318928
$754$	0	0
$755$	9345.12	0.450469
$756$	0	0
$757$	−26206.0	−1.25822	−0.629110	−	0.777316i	$-0.716581\pi$
$757$	−26206.0	−1.25822	−0.629110	+	0.777316i	$0.716581\pi$
$758$	0	0
$759$	13859.3	0.662794
$760$	0	0
$761$	−6863.18	−0.326925	−0.163463	−	0.986550i	$-0.552266\pi$
$761$	−6863.18	−0.326925	−0.163463	+	0.986550i	$0.552266\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	−644.000	−0.0304364
$766$	0	0
$767$	−22104.0	−1.04059
$768$	0	0
$769$	9058.04	0.424761	0.212380	−	0.977187i	$-0.431878\pi$
$769$	9058.04	0.424761	0.212380	+	0.977187i	$0.431878\pi$
$770$	0	0
$771$	6630.00	0.309693
$772$	0	0
$773$	132.936	0.00618548	0.00309274	−	0.999995i	$-0.499016\pi$
$773$	132.936	0.00618548	0.00309274	+	0.999995i	$0.499016\pi$
$774$	0	0
$775$	−24921.3	−1.15509
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−178.000	−0.00818679
$780$	0	0
$781$	−8232.00	−0.377163
$782$	0	0
$783$	−8089.30	−0.369206
$784$	0	0
$785$	43792.0	1.99109
$786$	0	0
$787$	−8729.94	−0.395411	−0.197706	−	0.980261i	$-0.563349\pi$
$787$	−8729.94	−0.395411	−0.197706	+	0.980261i	$0.563349\pi$
$788$	0	0
$789$	50487.4	2.27807
$790$	0	0
$791$	0	0
$792$	0	0
$793$	720.000	0.0322421
$794$	0	0
$795$	10360.0	0.462178
$796$	0	0
$797$	−7517.96	−0.334128	−0.167064	−	0.985946i	$-0.553429\pi$
$797$	−7517.96	−0.334128	−0.167064	+	0.985946i	$0.553429\pi$
$798$	0	0
$799$	−740.000	−0.0327651
$800$	0	0
$801$	−14214.3	−0.627011
$802$	0	0
$803$	3781.61	0.166189
$804$	0	0
$805$	0	0
$806$	0	0
$807$	32600.0	1.42203
$808$	0	0
$809$	3776.00	0.164100	0.0820501	−	0.996628i	$-0.473853\pi$
$809$	3776.00	0.164100	0.0820501	+	0.996628i	$0.473853\pi$
$810$	0	0
$811$	−36227.9	−1.56860	−0.784300	−	0.620382i	$-0.786977\pi$
$811$	−36227.9	−1.56860	−0.784300	+	0.620382i	$0.786977\pi$
$812$	0	0
$813$	16720.0	0.721274
$814$	0	0
$815$	−65059.5	−2.79624
$816$	0	0
$817$	−48.0833	−0.00205902
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−16410.0	−0.697580	−0.348790	−	0.937201i	$-0.613407\pi$
$821$	−16410.0	−0.697580	−0.348790	+	0.937201i	$0.613407\pi$
$822$	0	0
$823$	−22072.0	−0.934850	−0.467425	−	0.884033i	$-0.654818\pi$
$823$	−22072.0	−0.934850	−0.467425	+	0.884033i	$0.654818\pi$
$824$	0	0
$825$	−26431.7	−1.11543
$826$	0	0
$827$	11628.0	0.488930	0.244465	−	0.969658i	$-0.421388\pi$
$827$	11628.0	0.488930	0.244465	+	0.969658i	$0.421388\pi$
$828$	0	0
$829$	30906.2	1.29483	0.647417	−	0.762136i	$-0.275849\pi$
$829$	30906.2	1.29483	0.647417	+	0.762136i	$0.275849\pi$
$830$	0	0
$831$	−28326.7	−1.18248
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−29512.0	−1.22312
$836$	0	0
$837$	−2640.00	−0.109022
$838$	0	0
$839$	17884.1	0.735911	0.367955	−	0.929843i	$-0.380058\pi$
$839$	17884.1	0.735911	0.367955	+	0.929843i	$0.380058\pi$
$840$	0	0
$841$	57407.0	2.35381
$842$	0	0
$843$	42313.3	1.72876
$844$	0	0
$845$	7820.60	0.318387
$846$	0	0
$847$	0	0
$848$	0	0
$849$	34850.0	1.40877
$850$	0	0
$851$	5320.00	0.214298
$852$	0	0
$853$	20755.0	0.833104	0.416552	−	0.909112i	$-0.363238\pi$
$853$	20755.0	0.833104	0.416552	+	0.909112i	$0.363238\pi$
$854$	0	0
$855$	−644.000	−0.0257595
$856$	0	0
$857$	−44919.7	−1.79046	−0.895231	−	0.445602i	$-0.852990\pi$
$857$	−44919.7	−1.79046	−0.895231	+	0.445602i	$0.852990\pi$
$858$	0	0
$859$	69.2965	0.00275246	0.00137623	−	0.999999i	$-0.499562\pi$
$859$	69.2965	0.00275246	0.00137623	+	0.999999i	$0.499562\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	5452.00	0.215050	0.107525	−	0.994202i	$-0.465707\pi$
$863$	5452.00	0.215050	0.107525	+	0.994202i	$0.465707\pi$
$864$	0	0
$865$	40992.0	1.61129
$866$	0	0
$867$	34726.0	1.36027
$868$	0	0
$869$	−17080.0	−0.666743
$870$	0	0
$871$	34823.6	1.35471
$872$	0	0
$873$	−34120.7	−1.32281
$874$	0	0
$875$	0	0
$876$	0	0
$877$	31106.0	1.19769	0.598845	−	0.800865i	$-0.295626\pi$
$877$	31106.0	1.19769	0.598845	+	0.800865i	$0.295626\pi$
$878$	0	0
$879$	13940.0	0.534908
$880$	0	0
$881$	−5943.94	−0.227306	−0.113653	−	0.993521i	$-0.536255\pi$
$881$	−5943.94	−0.227306	−0.113653	+	0.993521i	$0.536255\pi$
$882$	0	0
$883$	−34796.0	−1.32614	−0.663068	−	0.748559i	$-0.730746\pi$
$883$	−34796.0	−1.32614	−0.663068	+	0.748559i	$0.730746\pi$
$884$	0	0
$885$	−60782.9	−2.30869
$886$	0	0
$887$	9964.55	0.377200	0.188600	−	0.982054i	$-0.439605\pi$
$887$	9964.55	0.377200	0.188600	+	0.982054i	$0.439605\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−11494.0	−0.432170
$892$	0	0
$893$	−740.000	−0.0277303
$894$	0	0
$895$	−10691.5	−0.399303
$896$	0	0
$897$	−50400.0	−1.87604
$898$	0	0
$899$	26694.7	0.990343
$900$	0	0
$901$	104.652	0.00386954
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−74928.0	−2.75214
$906$	0	0
$907$	29756.0	1.08934	0.544670	−	0.838650i	$-0.316655\pi$
$907$	29756.0	1.08934	0.544670	+	0.838650i	$0.316655\pi$
$908$	0	0
$909$	−25956.5	−0.947109
$910$	0	0
$911$	−21440.0	−0.779735	−0.389868	−	0.920871i	$-0.627479\pi$
$911$	−21440.0	−0.779735	−0.389868	+	0.920871i	$0.627479\pi$
$912$	0	0
$913$	5919.90	0.214589
$914$	0	0
$915$	1979.90	0.0715338
$916$	0	0
$917$	0	0
$918$	0	0
$919$	8288.00	0.297493	0.148746	−	0.988875i	$-0.452476\pi$
$919$	8288.00	0.297493	0.148746	+	0.988875i	$0.452476\pi$
$920$	0	0
$921$	33710.0	1.20606
$922$	0	0
$923$	29936.1	1.06756
$924$	0	0
$925$	−10146.0	−0.360647
$926$	0	0
$927$	−19971.5	−0.707606
$928$	0	0
$929$	45581.5	1.60978	0.804888	−	0.593427i	$-0.202226\pi$
$929$	45581.5	1.60978	0.804888	+	0.593427i	$0.202226\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	47920.0	1.68149
$934$	0	0
$935$	−392.000	−0.0137110
$936$	0	0
$937$	−11665.8	−0.406731	−0.203365	−	0.979103i	$-0.565188\pi$
$937$	−11665.8	−0.406731	−0.203365	+	0.979103i	$0.565188\pi$
$938$	0	0
$939$	−43770.0	−1.52117
$940$	0	0
$941$	−14.1421	−0.000489926 0	−0.000244963	−	1.00000i	$-0.500078\pi$
$941$	−14.1421	−0.000489926 0	−0.000244963		1.00000i	$0.500078\pi$
$942$	0	0
$943$	−17621.1	−0.608507
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−14034.0	−0.481567	−0.240783	−	0.970579i	$-0.577404\pi$
$947$	−14034.0	−0.481567	−0.240783	+	0.970579i	$0.577404\pi$
$948$	0	0
$949$	−13752.0	−0.470399
$950$	0	0
$951$	−69480.3	−2.36914
$952$	0	0
$953$	−42698.0	−1.45134	−0.725668	−	0.688045i	$-0.758469\pi$
$953$	−42698.0	−1.45134	−0.725668	+	0.688045i	$0.758469\pi$
$954$	0	0
$955$	−20353.4	−0.689654
$956$	0	0
$957$	28312.6	0.956337
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−21079.0	−0.707563
$962$	0	0
$963$	38732.0	1.29608
$964$	0	0
$965$	90917.0	3.03287
$966$	0	0
$967$	48492.0	1.61261	0.806307	−	0.591497i	$-0.201463\pi$
$967$	48492.0	1.61261	0.806307	+	0.591497i	$0.201463\pi$
$968$	0	0
$969$	−14.1421	−0.000468845 0
$970$	0	0
$971$	52669.6	1.74073	0.870364	−	0.492409i	$-0.163884\pi$
$971$	52669.6	1.74073	0.870364	+	0.492409i	$0.163884\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	96120.0	3.15723
$976$	0	0
$977$	−55380.0	−1.81347	−0.906737	−	0.421698i	$-0.861434\pi$
$977$	−55380.0	−1.81347	−0.906737	+	0.421698i	$0.861434\pi$
$978$	0	0
$979$	−8652.16	−0.282456
$980$	0	0
$981$	−18814.0	−0.612319
$982$	0	0
$983$	−50535.5	−1.63971	−0.819854	−	0.572573i	$-0.805945\pi$
$983$	−50535.5	−1.63971	−0.819854	+	0.572573i	$0.805945\pi$
$984$	0	0
$985$	15720.4	0.508521
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−4760.00	−0.153043
$990$	0	0
$991$	39712.0	1.27295	0.636475	−	0.771297i	$-0.280392\pi$
$991$	39712.0	1.27295	0.636475	+	0.771297i	$0.280392\pi$
$992$	0	0
$993$	−40573.8	−1.29665
$994$	0	0
$995$	49224.0	1.56835
$996$	0	0
$997$	−2186.37	−0.0694515	−0.0347258	−	0.999397i	$-0.511056\pi$
$997$	−2186.37	−0.0694515	−0.0347258	+	0.999397i	$0.511056\pi$
$998$	0	0
$999$	−1074.80	−0.0340393

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	784.4.a.y.1.1		2
4.3	odd	2		98.4.a.g.1.2	yes	2
7.6	odd	2	inner	784.4.a.y.1.2		2
12.11	even	2		882.4.a.bg.1.1		2
20.19	odd	2		2450.4.a.bx.1.1		2
28.3	even	6		98.4.c.h.79.2		4
28.11	odd	6		98.4.c.h.79.1		4
28.19	even	6		98.4.c.h.67.2		4
28.23	odd	6		98.4.c.h.67.1		4
28.27	even	2		98.4.a.g.1.1	✓	2
84.11	even	6		882.4.g.ba.667.2		4
84.23	even	6		882.4.g.ba.361.2		4
84.47	odd	6		882.4.g.ba.361.1		4
84.59	odd	6		882.4.g.ba.667.1		4
84.83	odd	2		882.4.a.bg.1.2		2
140.139	even	2		2450.4.a.bx.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
98.4.a.g.1.1	✓	2	28.27	even	2
98.4.a.g.1.2	yes	2	4.3	odd	2
98.4.c.h.67.1		4	28.23	odd	6
98.4.c.h.67.2		4	28.19	even	6
98.4.c.h.79.1		4	28.11	odd	6
98.4.c.h.79.2		4	28.3	even	6
784.4.a.y.1.1		2	1.1	even	1	trivial
784.4.a.y.1.2		2	7.6	odd	2	inner
882.4.a.bg.1.1		2	12.11	even	2
882.4.a.bg.1.2		2	84.83	odd	2
882.4.g.ba.361.1		4	84.47	odd	6
882.4.g.ba.361.2		4	84.23	even	6
882.4.g.ba.667.1		4	84.59	odd	6
882.4.g.ba.667.2		4	84.11	even	6
2450.4.a.bx.1.1		2	20.19	odd	2
2450.4.a.bx.1.2		2	140.139	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 784 = 2^{4} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	784.a (trivial)

\(f(q)\)	\(=\)	\(q-7.07107 q^{3} +19.7990 q^{5} +23.0000 q^{9} +O(q^{10})\)	\(q-7.07107 q^{3} +19.7990 q^{5} +23.0000 q^{9} +14.0000 q^{11} -50.9117 q^{13} -140.000 q^{15} -1.41421 q^{17} -1.41421 q^{19} -140.000 q^{23} +267.000 q^{25} +28.2843 q^{27} -286.000 q^{29} -93.3381 q^{31} -98.9949 q^{33} -38.0000 q^{37} +360.000 q^{39} +125.865 q^{41} +34.0000 q^{43} +455.377 q^{45} +523.259 q^{47} +10.0000 q^{51} -74.0000 q^{53} +277.186 q^{55} +10.0000 q^{57} +434.164 q^{59} -14.1421 q^{61} -1008.00 q^{65} -684.000 q^{67} +989.949 q^{69} -588.000 q^{71} +270.115 q^{73} -1887.98 q^{75} -1220.00 q^{79} -821.000 q^{81} +422.850 q^{83} -28.0000 q^{85} +2022.33 q^{87} -618.011 q^{89} +660.000 q^{93} -28.0000 q^{95} -1483.51 q^{97} +322.000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 46 q^{9}+O(q^{10}) \) 2 * q + 46 * q^9	\( 2 q + 46 q^{9} + 28 q^{11} - 280 q^{15} - 280 q^{23} + 534 q^{25} - 572 q^{29} - 76 q^{37} + 720 q^{39} + 68 q^{43} + 20 q^{51} - 148 q^{53} + 20 q^{57} - 2016 q^{65} - 1368 q^{67} - 1176 q^{71} - 2440 q^{79} - 1642 q^{81} - 56 q^{85} + 1320 q^{93} - 56 q^{95} + 644 q^{99}+O(q^{100}) \) 2 * q + 46 * q^9 + 28 * q^11 - 280 * q^15 - 280 * q^23 + 534 * q^25 - 572 * q^29 - 76 * q^37 + 720 * q^39 + 68 * q^43 + 20 * q^51 - 148 * q^53 + 20 * q^57 - 2016 * q^65 - 1368 * q^67 - 1176 * q^71 - 2440 * q^79 - 1642 * q^81 - 56 * q^85 + 1320 * q^93 - 56 * q^95 + 644 * q^99

Introduction

L-functions

Modular forms

Varieties

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Representations

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