LMFDB - Embedded newform 784.4.a.bg.1.2

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:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{2}, \sqrt{65})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 35x^{2} + 36x + 194 $ x^4 - 2x^3 - 35x^2 + 36*x + 194
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 392)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$3.11692$ 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)

$f(q)$	$=$	$q-3.57956 q^{3} -2.16534 q^{5} -14.1868 q^{9} +O(q^{10})$	$q-3.57956 q^{3} -2.16534 q^{5} -14.1868 q^{9} -9.31129 q^{11} -56.2576 q^{13} +7.75097 q^{15} -7.42325 q^{17} -9.23641 q^{19} +129.245 q^{23} -120.311 q^{25} +147.430 q^{27} +9.56032 q^{29} -180.221 q^{31} +33.3303 q^{33} +142.805 q^{37} +201.377 q^{39} -385.904 q^{41} +474.307 q^{43} +30.7192 q^{45} -205.314 q^{47} +26.5719 q^{51} -558.864 q^{53} +20.1621 q^{55} +33.0623 q^{57} -539.385 q^{59} +752.040 q^{61} +121.817 q^{65} +540.879 q^{67} -462.640 q^{69} +800.864 q^{71} +203.019 q^{73} +430.661 q^{75} +405.875 q^{79} -144.693 q^{81} +297.290 q^{83} +16.0739 q^{85} -34.2217 q^{87} +915.348 q^{89} +645.113 q^{93} +20.0000 q^{95} -1095.36 q^{97} +132.097 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 40 q^{9}+O(q^{10}) $ 4 * q + 40 * q^9	$ 4 q + 40 q^{9} + 124 q^{11} + 160 q^{15} - 128 q^{23} - 320 q^{25} - 252 q^{29} - 364 q^{37} + 1128 q^{39} + 1220 q^{43} + 1364 q^{51} - 752 q^{53} + 100 q^{57} + 1100 q^{65} + 2744 q^{67} + 1720 q^{71} + 1688 q^{79} - 1256 q^{81} + 1580 q^{85} + 968 q^{93} + 80 q^{95} + 5140 q^{99}+O(q^{100}) $ 4 * q + 40 * q^9 + 124 * q^11 + 160 * q^15 - 128 * q^23 - 320 * q^25 - 252 * q^29 - 364 * q^37 + 1128 * q^39 + 1220 * q^43 + 1364 * q^51 - 752 * q^53 + 100 * q^57 + 1100 * q^65 + 2744 * q^67 + 1720 * q^71 + 1688 * q^79 - 1256 * q^81 + 1580 * q^85 + 968 * q^93 + 80 * q^95 + 5140 * 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
$3$	−3.57956	−0.688886	−0.344443	−	0.938807i	$-0.611932\pi$
$3$	−3.57956	−0.688886	−0.344443	+	0.938807i	$0.611932\pi$
$4$	0	0
$5$	−2.16534	−0.193674	−0.0968371	−	0.995300i	$-0.530873\pi$
$5$	−2.16534	−0.193674	−0.0968371	+	0.995300i	$0.530873\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	−14.1868	−0.525436
$10$	0	0
$11$	−9.31129	−0.255224	−0.127612	−	0.991824i	$-0.540731\pi$
$11$	−9.31129	−0.255224	−0.127612	+	0.991824i	$0.540731\pi$
$12$	0	0
$13$	−56.2576	−1.20024	−0.600118	−	0.799912i	$-0.704880\pi$
$13$	−56.2576	−1.20024	−0.600118	+	0.799912i	$0.704880\pi$
$14$	0	0
$15$	7.75097	0.133419
$16$	0	0
$17$	−7.42325	−0.105906	−0.0529530	−	0.998597i	$-0.516863\pi$
$17$	−7.42325	−0.105906	−0.0529530	+	0.998597i	$0.516863\pi$
$18$	0	0
$19$	−9.23641	−0.111525	−0.0557626	−	0.998444i	$-0.517759\pi$
$19$	−9.23641	−0.111525	−0.0557626	+	0.998444i	$0.517759\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	129.245	1.17172	0.585858	−	0.810414i	$-0.300758\pi$
$23$	129.245	1.17172	0.585858	+	0.810414i	$0.300758\pi$
$24$	0	0
$25$	−120.311	−0.962490
$26$	0	0
$27$	147.430	1.05085
$28$	0	0
$29$	9.56032	0.0612175	0.0306087	−	0.999531i	$-0.490255\pi$
$29$	9.56032	0.0612175	0.0306087	+	0.999531i	$0.490255\pi$
$30$	0	0
$31$	−180.221	−1.04415	−0.522076	−	0.852899i	$-0.674842\pi$
$31$	−180.221	−1.04415	−0.522076	+	0.852899i	$0.674842\pi$
$32$	0	0
$33$	33.3303	0.175820
$34$	0	0
$35$	0	0
$36$	0	0
$37$	142.805	0.634516	0.317258	−	0.948339i	$-0.397238\pi$
$37$	142.805	0.634516	0.317258	+	0.948339i	$0.397238\pi$
$38$	0	0
$39$	201.377	0.826826
$40$	0	0
$41$	−385.904	−1.46995	−0.734977	−	0.678092i	$-0.762807\pi$
$41$	−385.904	−1.46995	−0.734977	+	0.678092i	$0.762807\pi$
$42$	0	0
$43$	474.307	1.68212	0.841060	−	0.540941i	$-0.181932\pi$
$43$	474.307	1.68212	0.841060	+	0.540941i	$0.181932\pi$
$44$	0	0
$45$	30.7192	0.101763
$46$	0	0
$47$	−205.314	−0.637195	−0.318597	−	0.947890i	$-0.603212\pi$
$47$	−205.314	−0.637195	−0.318597	+	0.947890i	$0.603212\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	26.5719	0.0729572
$52$	0	0
$53$	−558.864	−1.44841	−0.724206	−	0.689584i	$-0.757794\pi$
$53$	−558.864	−1.44841	−0.724206	+	0.689584i	$0.757794\pi$
$54$	0	0
$55$	20.1621	0.0494302
$56$	0	0
$57$	33.0623	0.0768281
$58$	0	0
$59$	−539.385	−1.19020	−0.595101	−	0.803651i	$-0.702888\pi$
$59$	−539.385	−1.19020	−0.595101	+	0.803651i	$0.702888\pi$
$60$	0	0
$61$	752.040	1.57851	0.789253	−	0.614069i	$-0.210468\pi$
$61$	752.040	1.57851	0.789253	+	0.614069i	$0.210468\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	121.817	0.232455
$66$	0	0
$67$	540.879	0.986253	0.493126	−	0.869958i	$-0.335854\pi$
$67$	540.879	0.986253	0.493126	+	0.869958i	$0.335854\pi$
$68$	0	0
$69$	−462.640	−0.807179
$70$	0	0
$71$	800.864	1.33866	0.669331	−	0.742964i	$-0.266581\pi$
$71$	800.864	1.33866	0.669331	+	0.742964i	$0.266581\pi$
$72$	0	0
$73$	203.019	0.325502	0.162751	−	0.986667i	$-0.447963\pi$
$73$	203.019	0.325502	0.162751	+	0.986667i	$0.447963\pi$
$74$	0	0
$75$	430.661	0.663046
$76$	0	0
$77$	0	0
$78$	0	0
$79$	405.875	0.578032	0.289016	−	0.957324i	$-0.406672\pi$
$79$	405.875	0.578032	0.289016	+	0.957324i	$0.406672\pi$
$80$	0	0
$81$	−144.693	−0.198481
$82$	0	0
$83$	297.290	0.393155	0.196577	−	0.980488i	$-0.437017\pi$
$83$	297.290	0.393155	0.196577	+	0.980488i	$0.437017\pi$
$84$	0	0
$85$	16.0739	0.0205113
$86$	0	0
$87$	−34.2217	−0.0421719
$88$	0	0
$89$	915.348	1.09019	0.545094	−	0.838375i	$-0.316494\pi$
$89$	915.348	1.09019	0.545094	+	0.838375i	$0.316494\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	645.113	0.719302
$94$	0	0
$95$	20.0000	0.0215995
$96$	0	0
$97$	−1095.36	−1.14657	−0.573284	−	0.819357i	$-0.694331\pi$
$97$	−1095.36	−1.14657	−0.573284	+	0.819357i	$0.694331\pi$
$98$	0	0
$99$	132.097	0.134104
$100$	0	0
$101$	−38.1589	−0.0375936	−0.0187968	−	0.999823i	$-0.505984\pi$
$101$	−38.1589	−0.0375936	−0.0187968	+	0.999823i	$0.505984\pi$
$102$	0	0
$103$	1654.53	1.58277	0.791385	−	0.611319i	$-0.209361\pi$
$103$	1654.53	1.58277	0.791385	+	0.611319i	$0.209361\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	862.109	0.778909	0.389454	−	0.921046i	$-0.372664\pi$
$107$	862.109	0.778909	0.389454	+	0.921046i	$0.372664\pi$
$108$	0	0
$109$	926.409	0.814072	0.407036	−	0.913412i	$-0.366562\pi$
$109$	926.409	0.814072	0.407036	+	0.913412i	$0.366562\pi$
$110$	0	0
$111$	−511.180	−0.437109
$112$	0	0
$113$	1805.40	1.50299	0.751496	−	0.659737i	$-0.229332\pi$
$113$	1805.40	1.50299	0.751496	+	0.659737i	$0.229332\pi$
$114$	0	0
$115$	−279.860	−0.226931
$116$	0	0
$117$	798.114	0.630647
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−1244.30	−0.934861
$122$	0	0
$123$	1381.37	1.01263
$124$	0	0
$125$	531.183	0.380084
$126$	0	0
$127$	413.128	0.288655	0.144328	−	0.989530i	$-0.453898\pi$
$127$	413.128	0.288655	0.144328	+	0.989530i	$0.453898\pi$
$128$	0	0
$129$	−1697.81	−1.15879
$130$	0	0
$131$	1157.91	0.772266	0.386133	−	0.922443i	$-0.373811\pi$
$131$	1157.91	0.772266	0.386133	+	0.922443i	$0.373811\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−319.237	−0.203523
$136$	0	0
$137$	2453.32	1.52993	0.764967	−	0.644070i	$-0.222755\pi$
$137$	2453.32	1.52993	0.764967	+	0.644070i	$0.222755\pi$
$138$	0	0
$139$	−1258.16	−0.767738	−0.383869	−	0.923388i	$-0.625409\pi$
$139$	−1258.16	−0.767738	−0.383869	+	0.923388i	$0.625409\pi$
$140$	0	0
$141$	734.934	0.438954
$142$	0	0
$143$	523.831	0.306328
$144$	0	0
$145$	−20.7014	−0.0118562
$146$	0	0
$147$	0	0
$148$	0	0
$149$	1930.97	1.06168	0.530842	−	0.847471i	$-0.321876\pi$
$149$	1930.97	1.06168	0.530842	+	0.847471i	$0.321876\pi$
$150$	0	0
$151$	−2507.84	−1.35156	−0.675778	−	0.737106i	$-0.736192\pi$
$151$	−2507.84	−1.35156	−0.675778	+	0.737106i	$0.736192\pi$
$152$	0	0
$153$	105.312	0.0556469
$154$	0	0
$155$	390.241	0.202225
$156$	0	0
$157$	−424.597	−0.215838	−0.107919	−	0.994160i	$-0.534419\pi$
$157$	−424.597	−0.215838	−0.107919	+	0.994160i	$0.534419\pi$
$158$	0	0
$159$	2000.48	0.997791
$160$	0	0
$161$	0	0
$162$	0	0
$163$	459.335	0.220723	0.110362	−	0.993892i	$-0.464799\pi$
$163$	459.335	0.220723	0.110362	+	0.993892i	$0.464799\pi$
$164$	0	0
$165$	−72.1715	−0.0340518
$166$	0	0
$167$	−3642.19	−1.68767	−0.843837	−	0.536600i	$-0.819708\pi$
$167$	−3642.19	−1.68767	−0.843837	+	0.536600i	$0.819708\pi$
$168$	0	0
$169$	967.922	0.440565
$170$	0	0
$171$	131.035	0.0585993
$172$	0	0
$173$	1935.84	0.850747	0.425374	−	0.905018i	$-0.360143\pi$
$173$	1935.84	0.850747	0.425374	+	0.905018i	$0.360143\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	1930.76	0.819914
$178$	0	0
$179$	−1305.56	−0.545153	−0.272577	−	0.962134i	$-0.587876\pi$
$179$	−1305.56	−0.545153	−0.272577	+	0.962134i	$0.587876\pi$
$180$	0	0
$181$	4015.80	1.64913	0.824564	−	0.565768i	$-0.191420\pi$
$181$	4015.80	1.64913	0.824564	+	0.565768i	$0.191420\pi$
$182$	0	0
$183$	−2691.97	−1.08741
$184$	0	0
$185$	−309.223	−0.122889
$186$	0	0
$187$	69.1200	0.0270297
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−552.366	−0.209255	−0.104628	−	0.994511i	$-0.533365\pi$
$191$	−552.366	−0.209255	−0.104628	+	0.994511i	$0.533365\pi$
$192$	0	0
$193$	234.346	0.0874021	0.0437011	−	0.999045i	$-0.486085\pi$
$193$	234.346	0.0874021	0.0437011	+	0.999045i	$0.486085\pi$
$194$	0	0
$195$	−436.051	−0.160135
$196$	0	0
$197$	996.903	0.360540	0.180270	−	0.983617i	$-0.442303\pi$
$197$	996.903	0.360540	0.180270	+	0.983617i	$0.442303\pi$
$198$	0	0
$199$	513.437	0.182897	0.0914486	−	0.995810i	$-0.470850\pi$
$199$	513.437	0.182897	0.0914486	+	0.995810i	$0.470850\pi$
$200$	0	0
$201$	−1936.11	−0.679416
$202$	0	0
$203$	0	0
$204$	0	0
$205$	835.615	0.284692
$206$	0	0
$207$	−1833.57	−0.615662
$208$	0	0
$209$	86.0029	0.0284638
$210$	0	0
$211$	4713.74	1.53795	0.768974	−	0.639280i	$-0.220768\pi$
$211$	4713.74	1.53795	0.768974	+	0.639280i	$0.220768\pi$
$212$	0	0
$213$	−2866.74	−0.922186
$214$	0	0
$215$	−1027.04	−0.325783
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−726.720	−0.224234
$220$	0	0
$221$	417.615	0.127112
$222$	0	0
$223$	−4436.55	−1.33226	−0.666128	−	0.745837i	$-0.732050\pi$
$223$	−4436.55	−1.33226	−0.666128	+	0.745837i	$0.732050\pi$
$224$	0	0
$225$	1706.83	0.505727
$226$	0	0
$227$	1229.48	0.359486	0.179743	−	0.983714i	$-0.442473\pi$
$227$	1229.48	0.359486	0.179743	+	0.983714i	$0.442473\pi$
$228$	0	0
$229$	−3714.71	−1.07194	−0.535971	−	0.844236i	$-0.680054\pi$
$229$	−3714.71	−1.07194	−0.535971	+	0.844236i	$0.680054\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−2039.64	−0.573482	−0.286741	−	0.958008i	$-0.592572\pi$
$233$	−2039.64	−0.573482	−0.286741	+	0.958008i	$0.592572\pi$
$234$	0	0
$235$	444.576	0.123408
$236$	0	0
$237$	−1452.85	−0.398198
$238$	0	0
$239$	−3219.77	−0.871420	−0.435710	−	0.900087i	$-0.643503\pi$
$239$	−3219.77	−0.871420	−0.435710	+	0.900087i	$0.643503\pi$
$240$	0	0
$241$	899.550	0.240436	0.120218	−	0.992748i	$-0.461641\pi$
$241$	899.550	0.240436	0.120218	+	0.992748i	$0.461641\pi$
$242$	0	0
$243$	−3462.69	−0.914121
$244$	0	0
$245$	0	0
$246$	0	0
$247$	519.619	0.133856
$248$	0	0
$249$	−1064.17	−0.270839
$250$	0	0
$251$	7400.76	1.86108	0.930541	−	0.366187i	$-0.119337\pi$
$251$	7400.76	1.86108	0.930541	+	0.366187i	$0.119337\pi$
$252$	0	0
$253$	−1203.44	−0.299050
$254$	0	0
$255$	−57.5374	−0.0141299
$256$	0	0
$257$	−1519.88	−0.368902	−0.184451	−	0.982842i	$-0.559051\pi$
$257$	−1519.88	−0.368902	−0.184451	+	0.982842i	$0.559051\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−135.630	−0.0321659
$262$	0	0
$263$	−2846.05	−0.667281	−0.333640	−	0.942700i	$-0.608277\pi$
$263$	−2846.05	−0.667281	−0.333640	+	0.942700i	$0.608277\pi$
$264$	0	0
$265$	1210.13	0.280520
$266$	0	0
$267$	−3276.54	−0.751015
$268$	0	0
$269$	3243.49	0.735165	0.367582	−	0.929991i	$-0.380186\pi$
$269$	3243.49	0.735165	0.367582	+	0.929991i	$0.380186\pi$
$270$	0	0
$271$	1875.42	0.420383	0.210192	−	0.977660i	$-0.432591\pi$
$271$	1875.42	0.420383	0.210192	+	0.977660i	$0.432591\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	1120.25	0.245650
$276$	0	0
$277$	−2700.82	−0.585835	−0.292918	−	0.956138i	$-0.594626\pi$
$277$	−2700.82	−0.585835	−0.292918	+	0.956138i	$0.594626\pi$
$278$	0	0
$279$	2556.76	0.548635
$280$	0	0
$281$	−2504.44	−0.531680	−0.265840	−	0.964017i	$-0.585649\pi$
$281$	−2504.44	−0.531680	−0.265840	+	0.964017i	$0.585649\pi$
$282$	0	0
$283$	−6672.48	−1.40155	−0.700774	−	0.713384i	$-0.747162\pi$
$283$	−6672.48	−1.40155	−0.700774	+	0.713384i	$0.747162\pi$
$284$	0	0
$285$	−71.5911	−0.0148796
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−4857.90	−0.988784
$290$	0	0
$291$	3920.91	0.789854
$292$	0	0
$293$	−2338.55	−0.466278	−0.233139	−	0.972443i	$-0.574900\pi$
$293$	−2338.55	−0.466278	−0.233139	+	0.972443i	$0.574900\pi$
$294$	0	0
$295$	1167.95	0.230511
$296$	0	0
$297$	−1372.77	−0.268202
$298$	0	0
$299$	−7271.03	−1.40634
$300$	0	0
$301$	0	0
$302$	0	0
$303$	136.592	0.0258977
$304$	0	0
$305$	−1628.42	−0.305716
$306$	0	0
$307$	8862.17	1.64753	0.823763	−	0.566934i	$-0.191870\pi$
$307$	8862.17	1.64753	0.823763	+	0.566934i	$0.191870\pi$
$308$	0	0
$309$	−5922.47	−1.09035
$310$	0	0
$311$	10760.8	1.96202	0.981012	−	0.193945i	$-0.0621284\pi$
$311$	10760.8	1.96202	0.981012	+	0.193945i	$0.0621284\pi$
$312$	0	0
$313$	−2834.31	−0.511836	−0.255918	−	0.966699i	$-0.582378\pi$
$313$	−2834.31	−0.511836	−0.255918	+	0.966699i	$0.582378\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−1386.01	−0.245571	−0.122785	−	0.992433i	$-0.539183\pi$
$317$	−1386.01	−0.245571	−0.122785	+	0.992433i	$0.539183\pi$
$318$	0	0
$319$	−89.0189	−0.0156241
$320$	0	0
$321$	−3085.97	−0.536579
$322$	0	0
$323$	68.5642	0.0118112
$324$	0	0
$325$	6768.43	1.15522
$326$	0	0
$327$	−3316.13	−0.560803
$328$	0	0
$329$	0	0
$330$	0	0
$331$	26.6340	0.00442276	0.00221138	−	0.999998i	$-0.499296\pi$
$331$	26.6340	0.00442276	0.00221138	+	0.999998i	$0.499296\pi$
$332$	0	0
$333$	−2025.95	−0.333397
$334$	0	0
$335$	−1171.19	−0.191012
$336$	0	0
$337$	−11211.1	−1.81219	−0.906094	−	0.423077i	$-0.860950\pi$
$337$	−11211.1	−1.81219	−0.906094	+	0.423077i	$0.860950\pi$
$338$	0	0
$339$	−6462.55	−1.03539
$340$	0	0
$341$	1678.09	0.266492
$342$	0	0
$343$	0	0
$344$	0	0
$345$	1001.78	0.156330
$346$	0	0
$347$	12203.4	1.88794	0.943968	−	0.330038i	$-0.107062\pi$
$347$	12203.4	1.88794	0.943968	+	0.330038i	$0.107062\pi$
$348$	0	0
$349$	−1014.94	−0.155668	−0.0778341	−	0.996966i	$-0.524800\pi$
$349$	−1014.94	−0.155668	−0.0778341	+	0.996966i	$0.524800\pi$
$350$	0	0
$351$	−8294.09	−1.26127
$352$	0	0
$353$	9084.97	1.36981	0.684907	−	0.728630i	$-0.259843\pi$
$353$	9084.97	1.36981	0.684907	+	0.728630i	$0.259843\pi$
$354$	0	0
$355$	−1734.15	−0.259264
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−10062.6	−1.47934	−0.739669	−	0.672971i	$-0.765018\pi$
$359$	−10062.6	−1.47934	−0.739669	+	0.672971i	$0.765018\pi$
$360$	0	0
$361$	−6773.69	−0.987562
$362$	0	0
$363$	4454.04	0.644013
$364$	0	0
$365$	−439.607	−0.0630413
$366$	0	0
$367$	1472.08	0.209379	0.104689	−	0.994505i	$-0.466615\pi$
$367$	1472.08	0.209379	0.104689	+	0.994505i	$0.466615\pi$
$368$	0	0
$369$	5474.74	0.772367
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−12534.9	−1.74003	−0.870017	−	0.493022i	$-0.835892\pi$
$373$	−12534.9	−1.74003	−0.870017	+	0.493022i	$0.835892\pi$
$374$	0	0
$375$	−1901.40	−0.261834
$376$	0	0
$377$	−537.841	−0.0734754
$378$	0	0
$379$	5322.24	0.721332	0.360666	−	0.932695i	$-0.382549\pi$
$379$	5322.24	0.721332	0.360666	+	0.932695i	$0.382549\pi$
$380$	0	0
$381$	−1478.82	−0.198851
$382$	0	0
$383$	−6918.47	−0.923022	−0.461511	−	0.887135i	$-0.652692\pi$
$383$	−6918.47	−0.923022	−0.461511	+	0.887135i	$0.652692\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−6728.89	−0.883847
$388$	0	0
$389$	794.821	0.103596	0.0517982	−	0.998658i	$-0.483505\pi$
$389$	794.821	0.103596	0.0517982	+	0.998658i	$0.483505\pi$
$390$	0	0
$391$	−959.419	−0.124092
$392$	0	0
$393$	−4144.80	−0.532004
$394$	0	0
$395$	−878.860	−0.111950
$396$	0	0
$397$	−12050.4	−1.52340	−0.761700	−	0.647930i	$-0.775635\pi$
$397$	−12050.4	−1.52340	−0.761700	+	0.647930i	$0.775635\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	10131.3	1.26168	0.630842	−	0.775911i	$-0.282710\pi$
$401$	10131.3	1.26168	0.630842	+	0.775911i	$0.282710\pi$
$402$	0	0
$403$	10138.8	1.25323
$404$	0	0
$405$	313.309	0.0384406
$406$	0	0
$407$	−1329.70	−0.161943
$408$	0	0
$409$	3587.46	0.433712	0.216856	−	0.976204i	$-0.430420\pi$
$409$	3587.46	0.433712	0.216856	+	0.976204i	$0.430420\pi$
$410$	0	0
$411$	−8781.78	−1.05395
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−643.735	−0.0761439
$416$	0	0
$417$	4503.65	0.528884
$418$	0	0
$419$	15156.6	1.76718	0.883590	−	0.468262i	$-0.155120\pi$
$419$	15156.6	1.76718	0.883590	+	0.468262i	$0.155120\pi$
$420$	0	0
$421$	−8933.19	−1.03415	−0.517075	−	0.855940i	$-0.672979\pi$
$421$	−8933.19	−1.03415	−0.517075	+	0.855940i	$0.672979\pi$
$422$	0	0
$423$	2912.75	0.334805
$424$	0	0
$425$	893.101	0.101934
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−1875.08	−0.211025
$430$	0	0
$431$	−6929.62	−0.774450	−0.387225	−	0.921985i	$-0.626566\pi$
$431$	−6929.62	−0.774450	−0.387225	+	0.921985i	$0.626566\pi$
$432$	0	0
$433$	14812.9	1.64403	0.822013	−	0.569469i	$-0.192851\pi$
$433$	14812.9	1.64403	0.822013	+	0.569469i	$0.192851\pi$
$434$	0	0
$435$	74.1017	0.00816760
$436$	0	0
$437$	−1193.76	−0.130676
$438$	0	0
$439$	15671.5	1.70378	0.851890	−	0.523721i	$-0.175456\pi$
$439$	15671.5	1.70378	0.851890	+	0.523721i	$0.175456\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	8829.22	0.946928	0.473464	−	0.880813i	$-0.343003\pi$
$443$	8829.22	0.946928	0.473464	+	0.880813i	$0.343003\pi$
$444$	0	0
$445$	−1982.04	−0.211141
$446$	0	0
$447$	−6912.00	−0.731379
$448$	0	0
$449$	5523.45	0.580551	0.290276	−	0.956943i	$-0.406253\pi$
$449$	5523.45	0.580551	0.290276	+	0.956943i	$0.406253\pi$
$450$	0	0
$451$	3593.27	0.375167
$452$	0	0
$453$	8976.94	0.931068
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−7242.67	−0.741351	−0.370676	−	0.928762i	$-0.620874\pi$
$457$	−7242.67	−0.741351	−0.370676	+	0.928762i	$0.620874\pi$
$458$	0	0
$459$	−1094.41	−0.111292
$460$	0	0
$461$	−4312.58	−0.435699	−0.217849	−	0.975982i	$-0.569904\pi$
$461$	−4312.58	−0.435699	−0.217849	+	0.975982i	$0.569904\pi$
$462$	0	0
$463$	−9641.82	−0.967805	−0.483902	−	0.875122i	$-0.660781\pi$
$463$	−9641.82	−0.967805	−0.483902	+	0.875122i	$0.660781\pi$
$464$	0	0
$465$	−1396.89	−0.139310
$466$	0	0
$467$	−8462.50	−0.838539	−0.419270	−	0.907862i	$-0.637714\pi$
$467$	−8462.50	−0.838539	−0.419270	+	0.907862i	$0.637714\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	1519.87	0.148688
$472$	0	0
$473$	−4416.41	−0.429317
$474$	0	0
$475$	1111.24	0.107342
$476$	0	0
$477$	7928.47	0.761048
$478$	0	0
$479$	3176.83	0.303034	0.151517	−	0.988455i	$-0.451584\pi$
$479$	3176.83	0.303034	0.151517	+	0.988455i	$0.451584\pi$
$480$	0	0
$481$	−8033.90	−0.761568
$482$	0	0
$483$	0	0
$484$	0	0
$485$	2371.83	0.222061
$486$	0	0
$487$	−16213.7	−1.50865	−0.754323	−	0.656503i	$-0.772035\pi$
$487$	−16213.7	−1.50865	−0.754323	+	0.656503i	$0.772035\pi$
$488$	0	0
$489$	−1644.21	−0.152053
$490$	0	0
$491$	−861.456	−0.0791791	−0.0395896	−	0.999216i	$-0.512605\pi$
$491$	−861.456	−0.0791791	−0.0395896	+	0.999216i	$0.512605\pi$
$492$	0	0
$493$	−70.9687	−0.00648330
$494$	0	0
$495$	−286.036	−0.0259724
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−1970.65	−0.176791	−0.0883954	−	0.996085i	$-0.528174\pi$
$499$	−1970.65	−0.176791	−0.0883954	+	0.996085i	$0.528174\pi$
$500$	0	0
$501$	13037.4	1.16261
$502$	0	0
$503$	−17603.1	−1.56041	−0.780203	−	0.625526i	$-0.784884\pi$
$503$	−17603.1	−1.56041	−0.780203	+	0.625526i	$0.784884\pi$
$504$	0	0
$505$	82.6272	0.00728091
$506$	0	0
$507$	−3464.73	−0.303499
$508$	0	0
$509$	−17758.1	−1.54639	−0.773195	−	0.634168i	$-0.781343\pi$
$509$	−17758.1	−1.54639	−0.773195	+	0.634168i	$0.781343\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−1361.73	−0.117196
$514$	0	0
$515$	−3582.62	−0.306542
$516$	0	0
$517$	1911.74	0.162627
$518$	0	0
$519$	−6929.45	−0.586068
$520$	0	0
$521$	−11492.4	−0.966394	−0.483197	−	0.875512i	$-0.660525\pi$
$521$	−11492.4	−0.966394	−0.483197	+	0.875512i	$0.660525\pi$
$522$	0	0
$523$	12472.4	1.04279	0.521394	−	0.853316i	$-0.325412\pi$
$523$	12472.4	1.04279	0.521394	+	0.853316i	$0.325412\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	1337.83	0.110582
$528$	0	0
$529$	4537.31	0.372919
$530$	0	0
$531$	7652.13	0.625375
$532$	0	0
$533$	21710.1	1.76429
$534$	0	0
$535$	−1866.76	−0.150854
$536$	0	0
$537$	4673.34	0.375549
$538$	0	0
$539$	0	0
$540$	0	0
$541$	608.460	0.0483544	0.0241772	−	0.999708i	$-0.492303\pi$
$541$	608.460	0.0483544	0.0241772	+	0.999708i	$0.492303\pi$
$542$	0	0
$543$	−14374.8	−1.13606
$544$	0	0
$545$	−2005.99	−0.157665
$546$	0	0
$547$	23512.2	1.83786	0.918930	−	0.394420i	$-0.129054\pi$
$547$	23512.2	1.83786	0.918930	+	0.394420i	$0.129054\pi$
$548$	0	0
$549$	−10669.0	−0.829404
$550$	0	0
$551$	−88.3030	−0.00682729
$552$	0	0
$553$	0	0
$554$	0	0
$555$	1106.88	0.0846567
$556$	0	0
$557$	7986.80	0.607561	0.303781	−	0.952742i	$-0.401751\pi$
$557$	7986.80	0.607561	0.303781	+	0.952742i	$0.401751\pi$
$558$	0	0
$559$	−26683.4	−2.01894
$560$	0	0
$561$	−247.419	−0.0186204
$562$	0	0
$563$	15127.5	1.13241	0.566207	−	0.824263i	$-0.308410\pi$
$563$	15127.5	1.13241	0.566207	+	0.824263i	$0.308410\pi$
$564$	0	0
$565$	−3909.32	−0.291091
$566$	0	0
$567$	0	0
$568$	0	0
$569$	18892.4	1.39194	0.695968	−	0.718073i	$-0.254976\pi$
$569$	18892.4	1.39194	0.695968	+	0.718073i	$0.254976\pi$
$570$	0	0
$571$	6065.41	0.444535	0.222268	−	0.974986i	$-0.428654\pi$
$571$	6065.41	0.444535	0.222268	+	0.974986i	$0.428654\pi$
$572$	0	0
$573$	1977.22	0.144153
$574$	0	0
$575$	−15549.7	−1.12777
$576$	0	0
$577$	−3639.65	−0.262600	−0.131300	−	0.991343i	$-0.541915\pi$
$577$	−3639.65	−0.262600	−0.131300	+	0.991343i	$0.541915\pi$
$578$	0	0
$579$	−838.855	−0.0602101
$580$	0	0
$581$	0	0
$582$	0	0
$583$	5203.74	0.369669
$584$	0	0
$585$	−1728.19	−0.122140
$586$	0	0
$587$	−10126.4	−0.712033	−0.356016	−	0.934480i	$-0.615865\pi$
$587$	−10126.4	−0.712033	−0.356016	+	0.934480i	$0.615865\pi$
$588$	0	0
$589$	1664.60	0.116449
$590$	0	0
$591$	−3568.47	−0.248371
$592$	0	0
$593$	4423.27	0.306310	0.153155	−	0.988202i	$-0.451057\pi$
$593$	4423.27	0.306310	0.153155	+	0.988202i	$0.451057\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−1837.88	−0.125995
$598$	0	0
$599$	−10289.9	−0.701892	−0.350946	−	0.936396i	$-0.614140\pi$
$599$	−10289.9	−0.701892	−0.350946	+	0.936396i	$0.614140\pi$
$600$	0	0
$601$	22325.5	1.51527	0.757633	−	0.652681i	$-0.226356\pi$
$601$	22325.5	1.51527	0.757633	+	0.652681i	$0.226356\pi$
$602$	0	0
$603$	−7673.33	−0.518213
$604$	0	0
$605$	2694.34	0.181058
$606$	0	0
$607$	22725.9	1.51963	0.759814	−	0.650140i	$-0.225290\pi$
$607$	22725.9	1.51963	0.759814	+	0.650140i	$0.225290\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	11550.5	0.764784
$612$	0	0
$613$	−14824.1	−0.976734	−0.488367	−	0.872638i	$-0.662407\pi$
$613$	−14824.1	−0.976734	−0.488367	+	0.872638i	$0.662407\pi$
$614$	0	0
$615$	−2991.13	−0.196120
$616$	0	0
$617$	14919.0	0.973444	0.486722	−	0.873557i	$-0.338192\pi$
$617$	14919.0	0.973444	0.486722	+	0.873557i	$0.338192\pi$
$618$	0	0
$619$	−27539.2	−1.78820	−0.894099	−	0.447869i	$-0.852183\pi$
$619$	−27539.2	−1.78820	−0.894099	+	0.447869i	$0.852183\pi$
$620$	0	0
$621$	19054.7	1.23130
$622$	0	0
$623$	0	0
$624$	0	0
$625$	13888.7	0.888878
$626$	0	0
$627$	−307.852	−0.0196083
$628$	0	0
$629$	−1060.08	−0.0671990
$630$	0	0
$631$	19605.3	1.23688	0.618442	−	0.785831i	$-0.287764\pi$
$631$	19605.3	1.23688	0.618442	+	0.785831i	$0.287764\pi$
$632$	0	0
$633$	−16873.1	−1.05947
$634$	0	0
$635$	−894.565	−0.0559051
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−11361.7	−0.703382
$640$	0	0
$641$	14428.5	0.889065	0.444533	−	0.895763i	$-0.353370\pi$
$641$	14428.5	0.889065	0.444533	+	0.895763i	$0.353370\pi$
$642$	0	0
$643$	−4525.97	−0.277584	−0.138792	−	0.990322i	$-0.544322\pi$
$643$	−4525.97	−0.277584	−0.138792	+	0.990322i	$0.544322\pi$
$644$	0	0
$645$	3676.34	0.224428
$646$	0	0
$647$	22755.2	1.38269	0.691343	−	0.722526i	$-0.257019\pi$
$647$	22755.2	1.38269	0.691343	+	0.722526i	$0.257019\pi$
$648$	0	0
$649$	5022.37	0.303768
$650$	0	0
$651$	0	0
$652$	0	0
$653$	16265.5	0.974759	0.487380	−	0.873190i	$-0.337953\pi$
$653$	16265.5	0.974759	0.487380	+	0.873190i	$0.337953\pi$
$654$	0	0
$655$	−2507.27	−0.149568
$656$	0	0
$657$	−2880.19	−0.171030
$658$	0	0
$659$	12929.6	0.764290	0.382145	−	0.924102i	$-0.375186\pi$
$659$	12929.6	0.764290	0.382145	+	0.924102i	$0.375186\pi$
$660$	0	0
$661$	−21920.6	−1.28989	−0.644943	−	0.764231i	$-0.723119\pi$
$661$	−21920.6	−1.28989	−0.644943	+	0.764231i	$0.723119\pi$
$662$	0	0
$663$	−1494.88	−0.0875658
$664$	0	0
$665$	0	0
$666$	0	0
$667$	1235.63	0.0717295
$668$	0	0
$669$	15880.9	0.917773
$670$	0	0
$671$	−7002.46	−0.402872
$672$	0	0
$673$	−1803.76	−0.103313	−0.0516566	−	0.998665i	$-0.516450\pi$
$673$	−1803.76	−0.103313	−0.0516566	+	0.998665i	$0.516450\pi$
$674$	0	0
$675$	−17737.5	−1.01143
$676$	0	0
$677$	16075.8	0.912620	0.456310	−	0.889821i	$-0.349171\pi$
$677$	16075.8	0.912620	0.456310	+	0.889821i	$0.349171\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−4400.98	−0.247645
$682$	0	0
$683$	32434.7	1.81710	0.908550	−	0.417776i	$-0.137190\pi$
$683$	32434.7	1.81710	0.908550	+	0.417776i	$0.137190\pi$
$684$	0	0
$685$	−5312.27	−0.296309
$686$	0	0
$687$	13297.0	0.738446
$688$	0	0
$689$	31440.4	1.73844
$690$	0	0
$691$	18812.7	1.03570	0.517851	−	0.855471i	$-0.326732\pi$
$691$	18812.7	1.03570	0.517851	+	0.855471i	$0.326732\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	2724.34	0.148691
$696$	0	0
$697$	2864.66	0.155677
$698$	0	0
$699$	7301.02	0.395064
$700$	0	0
$701$	−938.122	−0.0505455	−0.0252727	−	0.999681i	$-0.508045\pi$
$701$	−938.122	−0.0505455	−0.0252727	+	0.999681i	$0.508045\pi$
$702$	0	0
$703$	−1319.01	−0.0707644
$704$	0	0
$705$	−1591.38	−0.0850141
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−5141.30	−0.272335	−0.136168	−	0.990686i	$-0.543479\pi$
$709$	−5141.30	−0.272335	−0.136168	+	0.990686i	$0.543479\pi$
$710$	0	0
$711$	−5758.06	−0.303719
$712$	0	0
$713$	−23292.7	−1.22345
$714$	0	0
$715$	−1134.27	−0.0593279
$716$	0	0
$717$	11525.3	0.600309
$718$	0	0
$719$	6223.01	0.322780	0.161390	−	0.986891i	$-0.448402\pi$
$719$	6223.01	0.322780	0.161390	+	0.986891i	$0.448402\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	−3219.99	−0.165633
$724$	0	0
$725$	−1150.21	−0.0589212
$726$	0	0
$727$	−1599.67	−0.0816075	−0.0408037	−	0.999167i	$-0.512992\pi$
$727$	−1599.67	−0.0816075	−0.0408037	+	0.999167i	$0.512992\pi$
$728$	0	0
$729$	16301.6	0.828206
$730$	0	0
$731$	−3520.90	−0.178147
$732$	0	0
$733$	8767.06	0.441772	0.220886	−	0.975300i	$-0.429105\pi$
$733$	8767.06	0.441772	0.220886	+	0.975300i	$0.429105\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−5036.28	−0.251715
$738$	0	0
$739$	1477.28	0.0735353	0.0367677	−	0.999324i	$-0.488294\pi$
$739$	1477.28	0.0735353	0.0367677	+	0.999324i	$0.488294\pi$
$740$	0	0
$741$	−1860.00	−0.0922118
$742$	0	0
$743$	7873.64	0.388770	0.194385	−	0.980925i	$-0.437729\pi$
$743$	7873.64	0.388770	0.194385	+	0.980925i	$0.437729\pi$
$744$	0	0
$745$	−4181.20	−0.205621
$746$	0	0
$747$	−4217.59	−0.206578
$748$	0	0
$749$	0	0
$750$	0	0
$751$	24140.9	1.17299	0.586493	−	0.809954i	$-0.300508\pi$
$751$	24140.9	1.17299	0.586493	+	0.809954i	$0.300508\pi$
$752$	0	0
$753$	−26491.4	−1.28207
$754$	0	0
$755$	5430.33	0.261761
$756$	0	0
$757$	−3269.51	−0.156978	−0.0784889	−	0.996915i	$-0.525010\pi$
$757$	−3269.51	−0.156978	−0.0784889	+	0.996915i	$0.525010\pi$
$758$	0	0
$759$	4307.78	0.206011
$760$	0	0
$761$	31648.1	1.50755	0.753773	−	0.657135i	$-0.228232\pi$
$761$	31648.1	1.50755	0.753773	+	0.657135i	$0.228232\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	−228.037	−0.0107774
$766$	0	0
$767$	30344.5	1.42852
$768$	0	0
$769$	−35490.4	−1.66426	−0.832131	−	0.554580i	$-0.812879\pi$
$769$	−35490.4	−1.66426	−0.832131	+	0.554580i	$0.812879\pi$
$770$	0	0
$771$	5440.51	0.254131
$772$	0	0
$773$	2262.96	0.105295	0.0526474	−	0.998613i	$-0.483234\pi$
$773$	2262.96	0.105295	0.0526474	+	0.998613i	$0.483234\pi$
$774$	0	0
$775$	21682.7	1.00499
$776$	0	0
$777$	0	0
$778$	0	0
$779$	3564.37	0.163937
$780$	0	0
$781$	−7457.07	−0.341658
$782$	0	0
$783$	1409.48	0.0643305
$784$	0	0
$785$	919.398	0.0418022
$786$	0	0
$787$	−9474.33	−0.429128	−0.214564	−	0.976710i	$-0.568833\pi$
$787$	−9474.33	−0.429128	−0.214564	+	0.976710i	$0.568833\pi$
$788$	0	0
$789$	10187.6	0.459680
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−42308.0	−1.89458
$794$	0	0
$795$	−4331.74	−0.193246
$796$	0	0
$797$	−20909.1	−0.929283	−0.464642	−	0.885499i	$-0.653817\pi$
$797$	−20909.1	−0.929283	−0.464642	+	0.885499i	$0.653817\pi$
$798$	0	0
$799$	1524.10	0.0674828
$800$	0	0
$801$	−12985.8	−0.572824
$802$	0	0
$803$	−1890.37	−0.0830757
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−11610.3	−0.506445
$808$	0	0
$809$	23121.1	1.00481	0.502407	−	0.864631i	$-0.332448\pi$
$809$	23121.1	1.00481	0.502407	+	0.864631i	$0.332448\pi$
$810$	0	0
$811$	33868.1	1.46642	0.733212	−	0.680000i	$-0.238020\pi$
$811$	33868.1	1.46642	0.733212	+	0.680000i	$0.238020\pi$
$812$	0	0
$813$	−6713.18	−0.289596
$814$	0	0
$815$	−994.617	−0.0427484
$816$	0	0
$817$	−4380.90	−0.187599
$818$	0	0
$819$	0	0
$820$	0	0
$821$	22436.6	0.953765	0.476883	−	0.878967i	$-0.341767\pi$
$821$	22436.6	0.953765	0.476883	+	0.878967i	$0.341767\pi$
$822$	0	0
$823$	9990.16	0.423129	0.211565	−	0.977364i	$-0.432144\pi$
$823$	9990.16	0.423129	0.211565	+	0.977364i	$0.432144\pi$
$824$	0	0
$825$	−4010.01	−0.169225
$826$	0	0
$827$	10633.3	0.447105	0.223552	−	0.974692i	$-0.428235\pi$
$827$	10633.3	0.447105	0.223552	+	0.974692i	$0.428235\pi$
$828$	0	0
$829$	32926.4	1.37947	0.689734	−	0.724062i	$-0.257727\pi$
$829$	32926.4	1.37947	0.689734	+	0.724062i	$0.257727\pi$
$830$	0	0
$831$	9667.73	0.403574
$832$	0	0
$833$	0	0
$834$	0	0
$835$	7886.60	0.326859
$836$	0	0
$837$	−26570.1	−1.09725
$838$	0	0
$839$	27144.2	1.11695	0.558475	−	0.829521i	$-0.311387\pi$
$839$	27144.2	1.11695	0.558475	+	0.829521i	$0.311387\pi$
$840$	0	0
$841$	−24297.6	−0.996252
$842$	0	0
$843$	8964.77	0.366267
$844$	0	0
$845$	−2095.88	−0.0853261
$846$	0	0
$847$	0	0
$848$	0	0
$849$	23884.5	0.965507
$850$	0	0
$851$	18456.9	0.743472
$852$	0	0
$853$	12586.3	0.505214	0.252607	−	0.967569i	$-0.418712\pi$
$853$	12586.3	0.505214	0.252607	+	0.967569i	$0.418712\pi$
$854$	0	0
$855$	−283.735	−0.0113492
$856$	0	0
$857$	1645.17	0.0655754	0.0327877	−	0.999462i	$-0.489561\pi$
$857$	1645.17	0.0655754	0.0327877	+	0.999462i	$0.489561\pi$
$858$	0	0
$859$	−1873.36	−0.0744102	−0.0372051	−	0.999308i	$-0.511845\pi$
$859$	−1873.36	−0.0744102	−0.0372051	+	0.999308i	$0.511845\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	30065.5	1.18591	0.592957	−	0.805234i	$-0.297961\pi$
$863$	30065.5	1.18591	0.592957	+	0.805234i	$0.297961\pi$
$864$	0	0
$865$	−4191.76	−0.164768
$866$	0	0
$867$	17389.1	0.681159
$868$	0	0
$869$	−3779.22	−0.147527
$870$	0	0
$871$	−30428.6	−1.18374
$872$	0	0
$873$	15539.6	0.602448
$874$	0	0
$875$	0	0
$876$	0	0
$877$	3716.92	0.143115	0.0715573	−	0.997436i	$-0.477203\pi$
$877$	3716.92	0.143115	0.0715573	+	0.997436i	$0.477203\pi$
$878$	0	0
$879$	8370.97	0.321213
$880$	0	0
$881$	−9281.72	−0.354948	−0.177474	−	0.984125i	$-0.556793\pi$
$881$	−9281.72	−0.354948	−0.177474	+	0.984125i	$0.556793\pi$
$882$	0	0
$883$	12050.0	0.459247	0.229623	−	0.973280i	$-0.426251\pi$
$883$	12050.0	0.459247	0.229623	+	0.973280i	$0.426251\pi$
$884$	0	0
$885$	−4180.76	−0.158796
$886$	0	0
$887$	−31051.5	−1.17543	−0.587715	−	0.809068i	$-0.699972\pi$
$887$	−31051.5	−1.17543	−0.587715	+	0.809068i	$0.699972\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	1347.27	0.0506570
$892$	0	0
$893$	1896.37	0.0710632
$894$	0	0
$895$	2827.00	0.105582
$896$	0	0
$897$	26027.1	0.968805
$898$	0	0
$899$	−1722.97	−0.0639204
$900$	0	0
$901$	4148.59	0.153396
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−8695.59	−0.319394
$906$	0	0
$907$	−8764.36	−0.320856	−0.160428	−	0.987048i	$-0.551287\pi$
$907$	−8764.36	−0.320856	−0.160428	+	0.987048i	$0.551287\pi$
$908$	0	0
$909$	541.352	0.0197530
$910$	0	0
$911$	−13226.3	−0.481017	−0.240508	−	0.970647i	$-0.577314\pi$
$911$	−13226.3	−0.481017	−0.240508	+	0.970647i	$0.577314\pi$
$912$	0	0
$913$	−2768.16	−0.100342
$914$	0	0
$915$	5829.04	0.210603
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−42149.5	−1.51293	−0.756465	−	0.654034i	$-0.773075\pi$
$919$	−42149.5	−1.51293	−0.756465	+	0.654034i	$0.773075\pi$
$920$	0	0
$921$	−31722.6	−1.13496
$922$	0	0
$923$	−45054.7	−1.60671
$924$	0	0
$925$	−17181.1	−0.610715
$926$	0	0
$927$	−23472.4	−0.831644
$928$	0	0
$929$	−46058.3	−1.62661	−0.813307	−	0.581835i	$-0.802335\pi$
$929$	−46058.3	−1.62661	−0.813307	+	0.581835i	$0.802335\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−38518.9	−1.35161
$934$	0	0
$935$	−149.669	−0.00523496
$936$	0	0
$937$	6074.37	0.211783	0.105892	−	0.994378i	$-0.466230\pi$
$937$	6074.37	0.211783	0.105892	+	0.994378i	$0.466230\pi$
$938$	0	0
$939$	10145.6	0.352597
$940$	0	0
$941$	14674.1	0.508354	0.254177	−	0.967158i	$-0.418195\pi$
$941$	14674.1	0.508354	0.254177	+	0.967158i	$0.418195\pi$
$942$	0	0
$943$	−49876.2	−1.72237
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−8956.63	−0.307340	−0.153670	−	0.988122i	$-0.549109\pi$
$947$	−8956.63	−0.307340	−0.153670	+	0.988122i	$0.549109\pi$
$948$	0	0
$949$	−11421.4	−0.390679
$950$	0	0
$951$	4961.30	0.169170
$952$	0	0
$953$	−27435.3	−0.932546	−0.466273	−	0.884641i	$-0.654404\pi$
$953$	−27435.3	−0.932546	−0.466273	+	0.884641i	$0.654404\pi$
$954$	0	0
$955$	1196.06	0.0405274
$956$	0	0
$957$	318.648	0.0107633
$958$	0	0
$959$	0	0
$960$	0	0
$961$	2688.77	0.0902544
$962$	0	0
$963$	−12230.5	−0.409267
$964$	0	0
$965$	−507.440	−0.0169275
$966$	0	0
$967$	−3972.12	−0.132094	−0.0660470	−	0.997817i	$-0.521039\pi$
$967$	−3972.12	−0.132094	−0.0660470	+	0.997817i	$0.521039\pi$
$968$	0	0
$969$	−245.429	−0.00813656
$970$	0	0
$971$	−32157.7	−1.06281	−0.531406	−	0.847117i	$-0.678336\pi$
$971$	−32157.7	−1.06281	−0.531406	+	0.847117i	$0.678336\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	−24228.0	−0.795812
$976$	0	0
$977$	30632.9	1.00311	0.501553	−	0.865127i	$-0.332762\pi$
$977$	30632.9	1.00311	0.501553	+	0.865127i	$0.332762\pi$
$978$	0	0
$979$	−8523.07	−0.278242
$980$	0	0
$981$	−13142.7	−0.427743
$982$	0	0
$983$	39735.2	1.28927	0.644637	−	0.764489i	$-0.277008\pi$
$983$	39735.2	1.28927	0.644637	+	0.764489i	$0.277008\pi$
$984$	0	0
$985$	−2158.64	−0.0698273
$986$	0	0
$987$	0	0
$988$	0	0
$989$	61301.9	1.97097
$990$	0	0
$991$	9289.70	0.297777	0.148888	−	0.988854i	$-0.452430\pi$
$991$	9289.70	0.297777	0.148888	+	0.988854i	$0.452430\pi$
$992$	0	0
$993$	−95.3378	−0.00304678
$994$	0	0
$995$	−1111.77	−0.0354225
$996$	0	0
$997$	−55261.0	−1.75540	−0.877700	−	0.479210i	$-0.840923\pi$
$997$	−55261.0	−1.75540	−0.877700	+	0.479210i	$0.840923\pi$
$998$	0	0
$999$	21053.9	0.666782

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	784.4.a.bg.1.2		4
4.3	odd	2		392.4.a.m.1.3	yes	4
7.6	odd	2	inner	784.4.a.bg.1.3		4
28.3	even	6		392.4.i.p.177.3		8
28.11	odd	6		392.4.i.p.177.2		8
28.19	even	6		392.4.i.p.361.3		8
28.23	odd	6		392.4.i.p.361.2		8
28.27	even	2		392.4.a.m.1.2	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
392.4.a.m.1.2	✓	4	28.27	even	2
392.4.a.m.1.3	yes	4	4.3	odd	2
392.4.i.p.177.2		8	28.11	odd	6
392.4.i.p.177.3		8	28.3	even	6
392.4.i.p.361.2		8	28.23	odd	6
392.4.i.p.361.3		8	28.19	even	6
784.4.a.bg.1.2		4	1.1	even	1	trivial
784.4.a.bg.1.3		4	7.6	odd	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 $\F_{q}$

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

\(f(q)\)	\(=\)	\(q-3.57956 q^{3} -2.16534 q^{5} -14.1868 q^{9} +O(q^{10})\)	\(q-3.57956 q^{3} -2.16534 q^{5} -14.1868 q^{9} -9.31129 q^{11} -56.2576 q^{13} +7.75097 q^{15} -7.42325 q^{17} -9.23641 q^{19} +129.245 q^{23} -120.311 q^{25} +147.430 q^{27} +9.56032 q^{29} -180.221 q^{31} +33.3303 q^{33} +142.805 q^{37} +201.377 q^{39} -385.904 q^{41} +474.307 q^{43} +30.7192 q^{45} -205.314 q^{47} +26.5719 q^{51} -558.864 q^{53} +20.1621 q^{55} +33.0623 q^{57} -539.385 q^{59} +752.040 q^{61} +121.817 q^{65} +540.879 q^{67} -462.640 q^{69} +800.864 q^{71} +203.019 q^{73} +430.661 q^{75} +405.875 q^{79} -144.693 q^{81} +297.290 q^{83} +16.0739 q^{85} -34.2217 q^{87} +915.348 q^{89} +645.113 q^{93} +20.0000 q^{95} -1095.36 q^{97} +132.097 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 40 q^{9}+O(q^{10}) \) 4 * q + 40 * q^9	\( 4 q + 40 q^{9} + 124 q^{11} + 160 q^{15} - 128 q^{23} - 320 q^{25} - 252 q^{29} - 364 q^{37} + 1128 q^{39} + 1220 q^{43} + 1364 q^{51} - 752 q^{53} + 100 q^{57} + 1100 q^{65} + 2744 q^{67} + 1720 q^{71} + 1688 q^{79} - 1256 q^{81} + 1580 q^{85} + 968 q^{93} + 80 q^{95} + 5140 q^{99}+O(q^{100}) \) 4 * q + 40 * q^9 + 124 * q^11 + 160 * q^15 - 128 * q^23 - 320 * q^25 - 252 * q^29 - 364 * q^37 + 1128 * q^39 + 1220 * q^43 + 1364 * q^51 - 752 * q^53 + 100 * q^57 + 1100 * q^65 + 2744 * q^67 + 1720 * q^71 + 1688 * q^79 - 1256 * q^81 + 1580 * q^85 + 968 * q^93 + 80 * q^95 + 5140 * q^99

Introduction

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