LMFDB - Embedded newform 784.6.a.bg.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [784,6,Mod(1,784)] 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("784.1"); S:= CuspForms(chi, 6); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(784, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 6, names="a")

Level:	$ N $	$=$	$ 784 = 2^{4} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	784.a (trivial)

Newform invariants

comment:select newform

sage:traces = [4,0,0,0,0,0,0,0,712] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(9)] == traces)

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

Self dual:	yes
Analytic conductor:	$125.740914733$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{2}, \sqrt{793})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 399x^{2} + 400x + 38414 $ x^4 - 2x^3 - 399x^2 + 400*x + 38414
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2\cdot 7^{2} $
Twist minimal:	no (minimal twist has level 98)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$13.1659$ 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)

$f(q)$	$=$	$q-14.9626 q^{3} +34.9882 q^{5} -19.1218 q^{9} -748.365 q^{11} +12.4696 q^{13} -523.513 q^{15} -2213.03 q^{17} -1356.76 q^{19} -3021.46 q^{23} -1900.83 q^{25} +3922.01 q^{27} +5024.37 q^{29} +4554.07 q^{31} +11197.5 q^{33} -1041.10 q^{37} -186.577 q^{39} -3248.18 q^{41} +2863.29 q^{43} -669.037 q^{45} -20556.2 q^{47} +33112.5 q^{51} -22083.9 q^{53} -26183.9 q^{55} +20300.6 q^{57} +18875.7 q^{59} -56289.1 q^{61} +436.288 q^{65} -14070.6 q^{67} +45208.8 q^{69} +74485.3 q^{71} -39606.3 q^{73} +28441.2 q^{75} +39198.5 q^{79} -54036.8 q^{81} -13646.9 q^{83} -77429.8 q^{85} -75177.4 q^{87} +67288.1 q^{89} -68140.5 q^{93} -47470.6 q^{95} -88714.1 q^{97} +14310.1 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 712 q^{9} - 628 q^{11} - 5248 q^{15} - 2624 q^{23} + 4224 q^{25} + 17732 q^{29} + 2932 q^{37} + 41832 q^{39} - 9836 q^{43} + 51236 q^{51} + 1552 q^{53} - 104092 q^{57} - 142548 q^{65} - 13704 q^{67}+ \cdots + 354500 q^{99}+O(q^{100}) $ 4 * q + 712 * q^9 - 628 * q^11 - 5248 * q^15 - 2624 * q^23 + 4224 * q^25 + 17732 * q^29 + 2932 * q^37 + 41832 * q^39 - 9836 * q^43 + 51236 * q^51 + 1552 * q^53 - 104092 * q^57 - 142548 * q^65 - 13704 * q^67 + 56664 * q^71 + 218296 * q^79 - 127048 * q^81 - 103924 * q^85 - 88056 * q^93 + 396752 * q^95 + 354500 * q^99

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
$3$	−14.9626	−0.959849	−0.479924	−	0.877310i	$-0.659336\pi$
$3$	−14.9626	−0.959849	−0.479924	+	0.877310i	$0.659336\pi$
$4$	0	0
$5$	34.9882	0.625888	0.312944	−	0.949772i	$-0.398685\pi$
$5$	34.9882	0.625888	0.312944	+	0.949772i	$0.398685\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	−19.1218	−0.0786905
$10$	0	0
$11$	−748.365	−1.86480	−0.932399	−	0.361430i	$-0.882289\pi$
$11$	−748.365	−1.86480	−0.932399	+	0.361430i	$0.882289\pi$
$12$	0	0
$13$	12.4696	0.0204641	0.0102321	−	0.999948i	$-0.496743\pi$
$13$	12.4696	0.0204641	0.0102321	+	0.999948i	$0.496743\pi$
$14$	0	0
$15$	−523.513	−0.600757
$16$	0	0
$17$	−2213.03	−1.85722	−0.928612	−	0.371052i	$-0.878997\pi$
$17$	−2213.03	−1.85722	−0.928612	+	0.371052i	$0.878997\pi$
$18$	0	0
$19$	−1356.76	−0.862223	−0.431111	−	0.902299i	$-0.641878\pi$
$19$	−1356.76	−0.862223	−0.431111	+	0.902299i	$0.641878\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−3021.46	−1.19096	−0.595480	−	0.803370i	$-0.703038\pi$
$23$	−3021.46	−1.19096	−0.595480	+	0.803370i	$0.703038\pi$
$24$	0	0
$25$	−1900.83	−0.608265
$26$	0	0
$27$	3922.01	1.03538
$28$	0	0
$29$	5024.37	1.10940	0.554698	−	0.832052i	$-0.312834\pi$
$29$	5024.37	1.10940	0.554698	+	0.832052i	$0.312834\pi$
$30$	0	0
$31$	4554.07	0.851129	0.425564	−	0.904928i	$-0.360076\pi$
$31$	4554.07	0.851129	0.425564	+	0.904928i	$0.360076\pi$
$32$	0	0
$33$	11197.5	1.78992
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−1041.10	−0.125022	−0.0625110	−	0.998044i	$-0.519911\pi$
$37$	−1041.10	−0.125022	−0.0625110	+	0.998044i	$0.519911\pi$
$38$	0	0
$39$	−186.577	−0.0196425
$40$	0	0
$41$	−3248.18	−0.301773	−0.150886	−	0.988551i	$-0.548213\pi$
$41$	−3248.18	−0.301773	−0.150886	+	0.988551i	$0.548213\pi$
$42$	0	0
$43$	2863.29	0.236153	0.118077	−	0.993004i	$-0.462327\pi$
$43$	2863.29	0.236153	0.118077	+	0.993004i	$0.462327\pi$
$44$	0	0
$45$	−669.037	−0.0492514
$46$	0	0
$47$	−20556.2	−1.35737	−0.678684	−	0.734431i	$-0.737449\pi$
$47$	−20556.2	−1.35737	−0.678684	+	0.734431i	$0.737449\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	33112.5	1.78265
$52$	0	0
$53$	−22083.9	−1.07991	−0.539953	−	0.841695i	$-0.681558\pi$
$53$	−22083.9	−1.07991	−0.539953	+	0.841695i	$0.681558\pi$
$54$	0	0
$55$	−26183.9	−1.16715
$56$	0	0
$57$	20300.6	0.827603
$58$	0	0
$59$	18875.7	0.705948	0.352974	−	0.935633i	$-0.385170\pi$
$59$	18875.7	0.705948	0.352974	+	0.935633i	$0.385170\pi$
$60$	0	0
$61$	−56289.1	−1.93687	−0.968434	−	0.249270i	$-0.919809\pi$
$61$	−56289.1	−1.93687	−0.968434	+	0.249270i	$0.919809\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	436.288	0.0128082
$66$	0	0
$67$	−14070.6	−0.382935	−0.191467	−	0.981499i	$-0.561325\pi$
$67$	−14070.6	−0.382935	−0.191467	+	0.981499i	$0.561325\pi$
$68$	0	0
$69$	45208.8	1.14314
$70$	0	0
$71$	74485.3	1.75358	0.876788	−	0.480877i	$-0.159682\pi$
$71$	74485.3	1.75358	0.876788	+	0.480877i	$0.159682\pi$
$72$	0	0
$73$	−39606.3	−0.869875	−0.434937	−	0.900461i	$-0.643230\pi$
$73$	−39606.3	−0.869875	−0.434937	+	0.900461i	$0.643230\pi$
$74$	0	0
$75$	28441.2	0.583842
$76$	0	0
$77$	0	0
$78$	0	0
$79$	39198.5	0.706646	0.353323	−	0.935501i	$-0.385052\pi$
$79$	39198.5	0.706646	0.353323	+	0.935501i	$0.385052\pi$
$80$	0	0
$81$	−54036.8	−0.915117
$82$	0	0
$83$	−13646.9	−0.217440	−0.108720	−	0.994072i	$-0.534675\pi$
$83$	−13646.9	−0.217440	−0.108720	+	0.994072i	$0.534675\pi$
$84$	0	0
$85$	−77429.8	−1.16241
$86$	0	0
$87$	−75177.4	−1.06485
$88$	0	0
$89$	67288.1	0.900457	0.450229	−	0.892913i	$-0.351343\pi$
$89$	67288.1	0.900457	0.450229	+	0.892913i	$0.351343\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−68140.5	−0.816955
$94$	0	0
$95$	−47470.6	−0.539655
$96$	0	0
$97$	−88714.1	−0.957334	−0.478667	−	0.877997i	$-0.658880\pi$
$97$	−88714.1	−0.957334	−0.478667	+	0.877997i	$0.658880\pi$
$98$	0	0
$99$	14310.1	0.146742
$100$	0	0
$101$	−89154.5	−0.869640	−0.434820	−	0.900517i	$-0.643188\pi$
$101$	−89154.5	−0.869640	−0.434820	+	0.900517i	$0.643188\pi$
$102$	0	0
$103$	2762.86	0.0256605	0.0128303	−	0.999918i	$-0.495916\pi$
$103$	2762.86	0.0256605	0.0128303	+	0.999918i	$0.495916\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−29402.0	−0.248266	−0.124133	−	0.992266i	$-0.539615\pi$
$107$	−29402.0	−0.248266	−0.124133	+	0.992266i	$0.539615\pi$
$108$	0	0
$109$	−1239.90	−0.00999588	−0.00499794	−	0.999988i	$-0.501591\pi$
$109$	−1239.90	−0.00999588	−0.00499794	+	0.999988i	$0.501591\pi$
$110$	0	0
$111$	15577.5	0.120002
$112$	0	0
$113$	209535.	1.54369	0.771846	−	0.635810i	$-0.219334\pi$
$113$	209535.	1.54369	0.771846	+	0.635810i	$0.219334\pi$
$114$	0	0
$115$	−105715.	−0.745408
$116$	0	0
$117$	−238.440	−0.00161033
$118$	0	0
$119$	0	0
$120$	0	0
$121$	399000.	2.47747
$122$	0	0
$123$	48601.0	0.289656
$124$	0	0
$125$	−175845.	−1.00659
$126$	0	0
$127$	−181882.	−1.00065	−0.500323	−	0.865839i	$-0.666786\pi$
$127$	−181882.	−1.00065	−0.500323	+	0.865839i	$0.666786\pi$
$128$	0	0
$129$	−42842.1	−0.226671
$130$	0	0
$131$	−100024.	−0.509244	−0.254622	−	0.967041i	$-0.581951\pi$
$131$	−100024.	−0.509244	−0.254622	+	0.967041i	$0.581951\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	137224.	0.648031
$136$	0	0
$137$	179675.	0.817874	0.408937	−	0.912563i	$-0.365900\pi$
$137$	179675.	0.817874	0.408937	+	0.912563i	$0.365900\pi$
$138$	0	0
$139$	−171096.	−0.751108	−0.375554	−	0.926801i	$-0.622547\pi$
$139$	−171096.	−0.751108	−0.375554	+	0.926801i	$0.622547\pi$
$140$	0	0
$141$	307573.	1.30287
$142$	0	0
$143$	−9331.79	−0.0381615
$144$	0	0
$145$	175793.	0.694357
$146$	0	0
$147$	0	0
$148$	0	0
$149$	143150.	0.528234	0.264117	−	0.964491i	$-0.414920\pi$
$149$	143150.	0.528234	0.264117	+	0.964491i	$0.414920\pi$
$150$	0	0
$151$	−164944.	−0.588699	−0.294350	−	0.955698i	$-0.595103\pi$
$151$	−164944.	−0.588699	−0.294350	+	0.955698i	$0.595103\pi$
$152$	0	0
$153$	42317.0	0.146146
$154$	0	0
$155$	159339.	0.532711
$156$	0	0
$157$	418001.	1.35341	0.676703	−	0.736256i	$-0.263408\pi$
$157$	418001.	1.35341	0.676703	+	0.736256i	$0.263408\pi$
$158$	0	0
$159$	330431.	1.03655
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−298894.	−0.881147	−0.440573	−	0.897717i	$-0.645225\pi$
$163$	−298894.	−0.881147	−0.440573	+	0.897717i	$0.645225\pi$
$164$	0	0
$165$	391779.	1.12029
$166$	0	0
$167$	119265.	0.330920	0.165460	−	0.986216i	$-0.447089\pi$
$167$	119265.	0.330920	0.165460	+	0.986216i	$0.447089\pi$
$168$	0	0
$169$	−371138.	−0.999581
$170$	0	0
$171$	25943.7	0.0678487
$172$	0	0
$173$	423841.	1.07668	0.538341	−	0.842727i	$-0.319051\pi$
$173$	423841.	1.07668	0.538341	+	0.842727i	$0.319051\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−282429.	−0.677603
$178$	0	0
$179$	668814.	1.56017	0.780086	−	0.625672i	$-0.215175\pi$
$179$	668814.	1.56017	0.780086	+	0.625672i	$0.215175\pi$
$180$	0	0
$181$	−566620.	−1.28557	−0.642785	−	0.766047i	$-0.722221\pi$
$181$	−566620.	−1.28557	−0.642785	+	0.766047i	$0.722221\pi$
$182$	0	0
$183$	842230.	1.85910
$184$	0	0
$185$	−36426.1	−0.0782497
$186$	0	0
$187$	1.65615e6	3.46335
$188$	0	0
$189$	0	0
$190$	0	0
$191$	215834.	0.428091	0.214045	−	0.976824i	$-0.431336\pi$
$191$	215834.	0.428091	0.214045	+	0.976824i	$0.431336\pi$
$192$	0	0
$193$	−181256.	−0.350268	−0.175134	−	0.984545i	$-0.556036\pi$
$193$	−181256.	−0.350268	−0.175134	+	0.984545i	$0.556036\pi$
$194$	0	0
$195$	−6527.98	−0.0122940
$196$	0	0
$197$	440619.	0.808906	0.404453	−	0.914559i	$-0.367462\pi$
$197$	440619.	0.808906	0.404453	+	0.914559i	$0.367462\pi$
$198$	0	0
$199$	−757541.	−1.35604	−0.678022	−	0.735042i	$-0.737162\pi$
$199$	−757541.	−1.35604	−0.678022	+	0.735042i	$0.737162\pi$
$200$	0	0
$201$	210532.	0.367560
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−113648.	−0.188876
$206$	0	0
$207$	57775.8	0.0937173
$208$	0	0
$209$	1.01535e6	1.60787
$210$	0	0
$211$	290551.	0.449278	0.224639	−	0.974442i	$-0.427880\pi$
$211$	290551.	0.449278	0.224639	+	0.974442i	$0.427880\pi$
$212$	0	0
$213$	−1.11449e6	−1.68317
$214$	0	0
$215$	100181.	0.147805
$216$	0	0
$217$	0	0
$218$	0	0
$219$	592611.	0.834948
$220$	0	0
$221$	−27595.5	−0.0380065
$222$	0	0
$223$	−942010.	−1.26851	−0.634254	−	0.773124i	$-0.718693\pi$
$223$	−942010.	−1.26851	−0.634254	+	0.773124i	$0.718693\pi$
$224$	0	0
$225$	36347.2	0.0478646
$226$	0	0
$227$	507698.	0.653944	0.326972	−	0.945034i	$-0.393972\pi$
$227$	507698.	0.653944	0.326972	+	0.945034i	$0.393972\pi$
$228$	0	0
$229$	911540.	1.14865	0.574324	−	0.818628i	$-0.305265\pi$
$229$	911540.	1.14865	0.574324	+	0.818628i	$0.305265\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	57291.9	0.0691359	0.0345679	−	0.999402i	$-0.488994\pi$
$233$	57291.9	0.0691359	0.0345679	+	0.999402i	$0.488994\pi$
$234$	0	0
$235$	−719223.	−0.849560
$236$	0	0
$237$	−586510.	−0.678273
$238$	0	0
$239$	662765.	0.750524	0.375262	−	0.926919i	$-0.377553\pi$
$239$	662765.	0.750524	0.375262	+	0.926919i	$0.377553\pi$
$240$	0	0
$241$	1.27757e6	1.41691	0.708454	−	0.705757i	$-0.249393\pi$
$241$	1.27757e6	1.41691	0.708454	+	0.705757i	$0.249393\pi$
$242$	0	0
$243$	−144521.	−0.157006
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−16918.2	−0.0176446
$248$	0	0
$249$	204193.	0.208710
$250$	0	0
$251$	−422281.	−0.423074	−0.211537	−	0.977370i	$-0.567847\pi$
$251$	−422281.	−0.423074	−0.211537	+	0.977370i	$0.567847\pi$
$252$	0	0
$253$	2.26116e6	2.22090
$254$	0	0
$255$	1.15855e6	1.11574
$256$	0	0
$257$	1.66228e6	1.56990	0.784950	−	0.619559i	$-0.212689\pi$
$257$	1.66228e6	1.56990	0.784950	+	0.619559i	$0.212689\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−96074.9	−0.0872988
$262$	0	0
$263$	1.98006e6	1.76518	0.882588	−	0.470147i	$-0.155799\pi$
$263$	1.98006e6	1.76518	0.882588	+	0.470147i	$0.155799\pi$
$264$	0	0
$265$	−772675.	−0.675900
$266$	0	0
$267$	−1.00680e6	−0.864303
$268$	0	0
$269$	−1.17111e6	−0.986774	−0.493387	−	0.869810i	$-0.664241\pi$
$269$	−1.17111e6	−0.986774	−0.493387	+	0.869810i	$0.664241\pi$
$270$	0	0
$271$	−131162.	−0.108489	−0.0542445	−	0.998528i	$-0.517275\pi$
$271$	−131162.	−0.108489	−0.0542445	+	0.998528i	$0.517275\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	1.42251e6	1.13429
$276$	0	0
$277$	1.07740e6	0.843676	0.421838	−	0.906671i	$-0.361385\pi$
$277$	1.07740e6	0.843676	0.421838	+	0.906671i	$0.361385\pi$
$278$	0	0
$279$	−87081.9	−0.0669757
$280$	0	0
$281$	−193756.	−0.146382	−0.0731912	−	0.997318i	$-0.523318\pi$
$281$	−193756.	−0.146382	−0.0731912	+	0.997318i	$0.523318\pi$
$282$	0	0
$283$	250024.	0.185573	0.0927867	−	0.995686i	$-0.470423\pi$
$283$	250024.	0.185573	0.0927867	+	0.995686i	$0.470423\pi$
$284$	0	0
$285$	710282.	0.517987
$286$	0	0
$287$	0	0
$288$	0	0
$289$	3.47763e6	2.44928
$290$	0	0
$291$	1.32739e6	0.918896
$292$	0	0
$293$	−2.13317e6	−1.45163	−0.725816	−	0.687889i	$-0.758537\pi$
$293$	−2.13317e6	−1.45163	−0.725816	+	0.687889i	$0.758537\pi$
$294$	0	0
$295$	660426.	0.441844
$296$	0	0
$297$	−2.93510e6	−1.93077
$298$	0	0
$299$	−37676.3	−0.0243720
$300$	0	0
$301$	0	0
$302$	0	0
$303$	1.33398e6	0.834723
$304$	0	0
$305$	−1.96945e6	−1.21226
$306$	0	0
$307$	389090.	0.235616	0.117808	−	0.993036i	$-0.462413\pi$
$307$	389090.	0.235616	0.117808	+	0.993036i	$0.462413\pi$
$308$	0	0
$309$	−41339.4	−0.0246302
$310$	0	0
$311$	−2.41308e6	−1.41472	−0.707360	−	0.706854i	$-0.750114\pi$
$311$	−2.41308e6	−1.41472	−0.707360	+	0.706854i	$0.750114\pi$
$312$	0	0
$313$	857374.	0.494663	0.247332	−	0.968931i	$-0.420446\pi$
$313$	857374.	0.494663	0.247332	+	0.968931i	$0.420446\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−2.18422e6	−1.22081	−0.610406	−	0.792088i	$-0.708994\pi$
$317$	−2.18422e6	−1.22081	−0.610406	+	0.792088i	$0.708994\pi$
$318$	0	0
$319$	−3.76006e6	−2.06880
$320$	0	0
$321$	439930.	0.238298
$322$	0	0
$323$	3.00255e6	1.60134
$324$	0	0
$325$	−23702.5	−0.0124476
$326$	0	0
$327$	18552.1	0.00959453
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−1.67770e6	−0.841675	−0.420838	−	0.907136i	$-0.638264\pi$
$331$	−1.67770e6	−0.841675	−0.420838	+	0.907136i	$0.638264\pi$
$332$	0	0
$333$	19907.6	0.00983804
$334$	0	0
$335$	−492304.	−0.239674
$336$	0	0
$337$	1.03458e6	0.496238	0.248119	−	0.968730i	$-0.420188\pi$
$337$	1.03458e6	0.496238	0.248119	+	0.968730i	$0.420188\pi$
$338$	0	0
$339$	−3.13518e6	−1.48171
$340$	0	0
$341$	−3.40811e6	−1.58718
$342$	0	0
$343$	0	0
$344$	0	0
$345$	1.58177e6	0.715479
$346$	0	0
$347$	−2.93179e6	−1.30710	−0.653550	−	0.756884i	$-0.726721\pi$
$347$	−2.93179e6	−1.30710	−0.653550	+	0.756884i	$0.726721\pi$
$348$	0	0
$349$	1.58391e6	0.696093	0.348046	−	0.937477i	$-0.386845\pi$
$349$	1.58391e6	0.696093	0.348046	+	0.937477i	$0.386845\pi$
$350$	0	0
$351$	48905.8	0.0211881
$352$	0	0
$353$	−693771.	−0.296332	−0.148166	−	0.988962i	$-0.547337\pi$
$353$	−693771.	−0.296332	−0.148166	+	0.988962i	$0.547337\pi$
$354$	0	0
$355$	2.60610e6	1.09754
$356$	0	0
$357$	0	0
$358$	0	0
$359$	1.64948e6	0.675477	0.337739	−	0.941240i	$-0.390338\pi$
$359$	1.64948e6	0.675477	0.337739	+	0.941240i	$0.390338\pi$
$360$	0	0
$361$	−635298.	−0.256572
$362$	0	0
$363$	−5.97006e6	−2.37800
$364$	0	0
$365$	−1.38575e6	−0.544444
$366$	0	0
$367$	−1.41980e6	−0.550252	−0.275126	−	0.961408i	$-0.588720\pi$
$367$	−1.41980e6	−0.550252	−0.275126	+	0.961408i	$0.588720\pi$
$368$	0	0
$369$	62111.0	0.0237466
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−425803.	−0.158466	−0.0792332	−	0.996856i	$-0.525247\pi$
$373$	−425803.	−0.158466	−0.0792332	+	0.996856i	$0.525247\pi$
$374$	0	0
$375$	2.63108e6	0.966177
$376$	0	0
$377$	62651.7	0.0227028
$378$	0	0
$379$	−176909.	−0.0632633	−0.0316316	−	0.999500i	$-0.510070\pi$
$379$	−176909.	−0.0632633	−0.0316316	+	0.999500i	$0.510070\pi$
$380$	0	0
$381$	2.72142e6	0.960470
$382$	0	0
$383$	2.64656e6	0.921902	0.460951	−	0.887426i	$-0.347508\pi$
$383$	2.64656e6	0.921902	0.460951	+	0.887426i	$0.347508\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−54751.2	−0.0185830
$388$	0	0
$389$	−1.14149e6	−0.382470	−0.191235	−	0.981544i	$-0.561249\pi$
$389$	−1.14149e6	−0.382470	−0.191235	+	0.981544i	$0.561249\pi$
$390$	0	0
$391$	6.68657e6	2.21188
$392$	0	0
$393$	1.49662e6	0.488798
$394$	0	0
$395$	1.37148e6	0.442281
$396$	0	0
$397$	−5.33798e6	−1.69981	−0.849906	−	0.526934i	$-0.823341\pi$
$397$	−5.33798e6	−1.69981	−0.849906	+	0.526934i	$0.823341\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	1.43598e6	0.445950	0.222975	−	0.974824i	$-0.428423\pi$
$401$	1.43598e6	0.445950	0.222975	+	0.974824i	$0.428423\pi$
$402$	0	0
$403$	56787.2	0.0174176
$404$	0	0
$405$	−1.89065e6	−0.572761
$406$	0	0
$407$	779120.	0.233141
$408$	0	0
$409$	3.74760e6	1.10776	0.553878	−	0.832598i	$-0.313147\pi$
$409$	3.74760e6	1.10776	0.553878	+	0.832598i	$0.313147\pi$
$410$	0	0
$411$	−2.68840e6	−0.785035
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−477481.	−0.136093
$416$	0	0
$417$	2.56003e6	0.720950
$418$	0	0
$419$	1.51473e6	0.421503	0.210752	−	0.977540i	$-0.432409\pi$
$419$	1.51473e6	0.421503	0.210752	+	0.977540i	$0.432409\pi$
$420$	0	0
$421$	515241.	0.141679	0.0708395	−	0.997488i	$-0.477432\pi$
$421$	515241.	0.141679	0.0708395	+	0.997488i	$0.477432\pi$
$422$	0	0
$423$	393071.	0.106812
$424$	0	0
$425$	4.20658e6	1.12968
$426$	0	0
$427$	0	0
$428$	0	0
$429$	139628.	0.0366292
$430$	0	0
$431$	3.28901e6	0.852849	0.426424	−	0.904523i	$-0.359773\pi$
$431$	3.28901e6	0.852849	0.426424	+	0.904523i	$0.359773\pi$
$432$	0	0
$433$	2.40962e6	0.617631	0.308816	−	0.951122i	$-0.400067\pi$
$433$	2.40962e6	0.617631	0.308816	+	0.951122i	$0.400067\pi$
$434$	0	0
$435$	−2.63032e6	−0.666477
$436$	0	0
$437$	4.09940e6	1.02687
$438$	0	0
$439$	−3.05894e6	−0.757547	−0.378773	−	0.925489i	$-0.623654\pi$
$439$	−3.05894e6	−0.757547	−0.378773	+	0.925489i	$0.623654\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−1.51135e6	−0.365895	−0.182948	−	0.983123i	$-0.558564\pi$
$443$	−1.51135e6	−0.365895	−0.182948	+	0.983123i	$0.558564\pi$
$444$	0	0
$445$	2.35429e6	0.563585
$446$	0	0
$447$	−2.14189e6	−0.507024
$448$	0	0
$449$	−238029.	−0.0557204	−0.0278602	−	0.999612i	$-0.508869\pi$
$449$	−238029.	−0.0557204	−0.0278602	+	0.999612i	$0.508869\pi$
$450$	0	0
$451$	2.43082e6	0.562745
$452$	0	0
$453$	2.46798e6	0.565062
$454$	0	0
$455$	0	0
$456$	0	0
$457$	2.28636e6	0.512100	0.256050	−	0.966664i	$-0.417579\pi$
$457$	2.28636e6	0.512100	0.256050	+	0.966664i	$0.417579\pi$
$458$	0	0
$459$	−8.67952e6	−1.92293
$460$	0	0
$461$	−7.83709e6	−1.71752	−0.858761	−	0.512376i	$-0.828765\pi$
$461$	−7.83709e6	−1.71752	−0.858761	+	0.512376i	$0.828765\pi$
$462$	0	0
$463$	187039.	0.0405489	0.0202744	−	0.999794i	$-0.493546\pi$
$463$	187039.	0.0405489	0.0202744	+	0.999794i	$0.493546\pi$
$464$	0	0
$465$	−2.38411e6	−0.511322
$466$	0	0
$467$	−5.59905e6	−1.18801	−0.594007	−	0.804460i	$-0.702455\pi$
$467$	−5.59905e6	−1.18801	−0.594007	+	0.804460i	$0.702455\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−6.25437e6	−1.29907
$472$	0	0
$473$	−2.14279e6	−0.440378
$474$	0	0
$475$	2.57897e6	0.524459
$476$	0	0
$477$	422283.	0.0849783
$478$	0	0
$479$	5.28367e6	1.05220	0.526098	−	0.850424i	$-0.323654\pi$
$479$	5.28367e6	1.05220	0.526098	+	0.850424i	$0.323654\pi$
$480$	0	0
$481$	−12982.0	−0.00255847
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−3.10395e6	−0.599184
$486$	0	0
$487$	3.90472e6	0.746049	0.373024	−	0.927822i	$-0.378321\pi$
$487$	3.90472e6	0.746049	0.373024	+	0.927822i	$0.378321\pi$
$488$	0	0
$489$	4.47222e6	0.845768
$490$	0	0
$491$	−3.55252e6	−0.665018	−0.332509	−	0.943100i	$-0.607895\pi$
$491$	−3.55252e6	−0.665018	−0.332509	+	0.943100i	$0.607895\pi$
$492$	0	0
$493$	−1.11191e7	−2.06040
$494$	0	0
$495$	500684.	0.0918440
$496$	0	0
$497$	0	0
$498$	0	0
$499$	2.52545e6	0.454032	0.227016	−	0.973891i	$-0.427103\pi$
$499$	2.52545e6	0.454032	0.227016	+	0.973891i	$0.427103\pi$
$500$	0	0
$501$	−1.78452e6	−0.317634
$502$	0	0
$503$	5.89904e6	1.03959	0.519795	−	0.854291i	$-0.326009\pi$
$503$	5.89904e6	1.03959	0.519795	+	0.854291i	$0.326009\pi$
$504$	0	0
$505$	−3.11935e6	−0.544297
$506$	0	0
$507$	5.55317e6	0.959447
$508$	0	0
$509$	2.36970e6	0.405413	0.202707	−	0.979240i	$-0.435026\pi$
$509$	2.36970e6	0.405413	0.202707	+	0.979240i	$0.435026\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−5.32124e6	−0.892728
$514$	0	0
$515$	96667.4	0.0160606
$516$	0	0
$517$	1.53835e7	2.53122
$518$	0	0
$519$	−6.34174e6	−1.03345
$520$	0	0
$521$	8.20591e6	1.32444	0.662220	−	0.749309i	$-0.269614\pi$
$521$	8.20591e6	1.32444	0.662220	+	0.749309i	$0.269614\pi$
$522$	0	0
$523$	5.59590e6	0.894572	0.447286	−	0.894391i	$-0.352391\pi$
$523$	5.59590e6	0.894572	0.447286	+	0.894391i	$0.352391\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−1.00783e7	−1.58074
$528$	0	0
$529$	2.69289e6	0.418388
$530$	0	0
$531$	−360937.	−0.0555514
$532$	0	0
$533$	−40503.4	−0.00617551
$534$	0	0
$535$	−1.02872e6	−0.155387
$536$	0	0
$537$	−1.00072e7	−1.49753
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−1.11429e7	−1.63683	−0.818416	−	0.574626i	$-0.805148\pi$
$541$	−1.11429e7	−1.63683	−0.818416	+	0.574626i	$0.805148\pi$
$542$	0	0
$543$	8.47809e6	1.23395
$544$	0	0
$545$	−43381.9	−0.00625630
$546$	0	0
$547$	−1.03208e7	−1.47483	−0.737417	−	0.675438i	$-0.763955\pi$
$547$	−1.03208e7	−1.47483	−0.737417	+	0.675438i	$0.763955\pi$
$548$	0	0
$549$	1.07635e6	0.152413
$550$	0	0
$551$	−6.81686e6	−0.956545
$552$	0	0
$553$	0	0
$554$	0	0
$555$	545027.	0.0751079
$556$	0	0
$557$	3.20238e6	0.437356	0.218678	−	0.975797i	$-0.429826\pi$
$557$	3.20238e6	0.437356	0.218678	+	0.975797i	$0.429826\pi$
$558$	0	0
$559$	35704.0	0.00483267
$560$	0	0
$561$	−2.47803e7	−3.32429
$562$	0	0
$563$	−1.60241e6	−0.213060	−0.106530	−	0.994309i	$-0.533974\pi$
$563$	−1.60241e6	−0.213060	−0.106530	+	0.994309i	$0.533974\pi$
$564$	0	0
$565$	7.33125e6	0.966178
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−1.47395e7	−1.90855	−0.954273	−	0.298936i	$-0.903368\pi$
$569$	−1.47395e7	−1.90855	−0.954273	+	0.298936i	$0.903368\pi$
$570$	0	0
$571$	5.87638e6	0.754257	0.377129	−	0.926161i	$-0.376911\pi$
$571$	5.87638e6	0.754257	0.377129	+	0.926161i	$0.376911\pi$
$572$	0	0
$573$	−3.22943e6	−0.410903
$574$	0	0
$575$	5.74328e6	0.724419
$576$	0	0
$577$	−244876.	−0.0306200	−0.0153100	−	0.999883i	$-0.504874\pi$
$577$	−244876.	−0.0306200	−0.0153100	+	0.999883i	$0.504874\pi$
$578$	0	0
$579$	2.71206e6	0.336204
$580$	0	0
$581$	0	0
$582$	0	0
$583$	1.65268e7	2.01381
$584$	0	0
$585$	−8342.60	−0.00100789
$586$	0	0
$587$	−6.63168e6	−0.794380	−0.397190	−	0.917736i	$-0.630015\pi$
$587$	−6.63168e6	−0.794380	−0.397190	+	0.917736i	$0.630015\pi$
$588$	0	0
$589$	−6.17878e6	−0.733862
$590$	0	0
$591$	−6.59279e6	−0.776427
$592$	0	0
$593$	−1.03581e7	−1.20961	−0.604803	−	0.796375i	$-0.706748\pi$
$593$	−1.03581e7	−1.20961	−0.604803	+	0.796375i	$0.706748\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	1.13348e7	1.30160
$598$	0	0
$599$	7.13439e6	0.812437	0.406218	−	0.913776i	$-0.366847\pi$
$599$	7.13439e6	0.812437	0.406218	+	0.913776i	$0.366847\pi$
$600$	0	0
$601$	1.64350e6	0.185602	0.0928010	−	0.995685i	$-0.470418\pi$
$601$	1.64350e6	0.185602	0.0928010	+	0.995685i	$0.470418\pi$
$602$	0	0
$603$	269055.	0.0301333
$604$	0	0
$605$	1.39603e7	1.55062
$606$	0	0
$607$	−1.09809e7	−1.20967	−0.604835	−	0.796351i	$-0.706761\pi$
$607$	−1.09809e7	−1.20967	−0.604835	+	0.796351i	$0.706761\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−256326.	−0.0277773
$612$	0	0
$613$	−5.74628e6	−0.617640	−0.308820	−	0.951120i	$-0.599934\pi$
$613$	−5.74628e6	−0.617640	−0.308820	+	0.951120i	$0.599934\pi$
$614$	0	0
$615$	1.70046e6	0.181292
$616$	0	0
$617$	6.02625e6	0.637286	0.318643	−	0.947875i	$-0.396773\pi$
$617$	6.02625e6	0.637286	0.318643	+	0.947875i	$0.396773\pi$
$618$	0	0
$619$	1.72334e7	1.80777	0.903887	−	0.427771i	$-0.140701\pi$
$619$	1.72334e7	1.80777	0.903887	+	0.427771i	$0.140701\pi$
$620$	0	0
$621$	−1.18502e7	−1.23310
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−212398.	−0.0217496
$626$	0	0
$627$	−1.51923e7	−1.54331
$628$	0	0
$629$	2.30397e6	0.232194
$630$	0	0
$631$	2.84236e6	0.284188	0.142094	−	0.989853i	$-0.454616\pi$
$631$	2.84236e6	0.284188	0.142094	+	0.989853i	$0.454616\pi$
$632$	0	0
$633$	−4.34738e6	−0.431239
$634$	0	0
$635$	−6.36373e6	−0.626293
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−1.42429e6	−0.137990
$640$	0	0
$641$	−1.33572e7	−1.28402	−0.642008	−	0.766698i	$-0.721899\pi$
$641$	−1.33572e7	−1.28402	−0.642008	+	0.766698i	$0.721899\pi$
$642$	0	0
$643$	−1.66914e7	−1.59208	−0.796042	−	0.605242i	$-0.793076\pi$
$643$	−1.66914e7	−1.59208	−0.796042	+	0.605242i	$0.793076\pi$
$644$	0	0
$645$	−1.49897e6	−0.141871
$646$	0	0
$647$	3.43361e6	0.322471	0.161236	−	0.986916i	$-0.448452\pi$
$647$	3.43361e6	0.322471	0.161236	+	0.986916i	$0.448452\pi$
$648$	0	0
$649$	−1.41259e7	−1.31645
$650$	0	0
$651$	0	0
$652$	0	0
$653$	1.76145e7	1.61655	0.808274	−	0.588807i	$-0.200402\pi$
$653$	1.76145e7	1.61655	0.808274	+	0.588807i	$0.200402\pi$
$654$	0	0
$655$	−3.49966e6	−0.318730
$656$	0	0
$657$	757343.	0.0684509
$658$	0	0
$659$	−8.15115e6	−0.731148	−0.365574	−	0.930782i	$-0.619127\pi$
$659$	−8.15115e6	−0.731148	−0.365574	+	0.930782i	$0.619127\pi$
$660$	0	0
$661$	−3.06437e6	−0.272796	−0.136398	−	0.990654i	$-0.543553\pi$
$661$	−3.06437e6	−0.272796	−0.136398	+	0.990654i	$0.543553\pi$
$662$	0	0
$663$	412899.	0.0364804
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−1.51809e7	−1.32125
$668$	0	0
$669$	1.40949e7	1.21758
$670$	0	0
$671$	4.21248e7	3.61187
$672$	0	0
$673$	1.00272e7	0.853383	0.426691	−	0.904397i	$-0.359679\pi$
$673$	1.00272e7	0.853383	0.426691	+	0.904397i	$0.359679\pi$
$674$	0	0
$675$	−7.45507e6	−0.629785
$676$	0	0
$677$	−3.24118e6	−0.271789	−0.135894	−	0.990723i	$-0.543391\pi$
$677$	−3.24118e6	−0.271789	−0.135894	+	0.990723i	$0.543391\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−7.59646e6	−0.627687
$682$	0	0
$683$	−1.27107e7	−1.04260	−0.521301	−	0.853373i	$-0.674553\pi$
$683$	−1.27107e7	−1.04260	−0.521301	+	0.853373i	$0.674553\pi$
$684$	0	0
$685$	6.28650e6	0.511897
$686$	0	0
$687$	−1.36390e7	−1.10253
$688$	0	0
$689$	−275376.	−0.0220993
$690$	0	0
$691$	−1.10794e7	−0.882715	−0.441358	−	0.897331i	$-0.645503\pi$
$691$	−1.10794e7	−0.882715	−0.441358	+	0.897331i	$0.645503\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−5.98633e6	−0.470109
$696$	0	0
$697$	7.18830e6	0.560460
$698$	0	0
$699$	−857234.	−0.0663600
$700$	0	0
$701$	−1.02838e7	−0.790425	−0.395212	−	0.918590i	$-0.629329\pi$
$701$	−1.02838e7	−0.790425	−0.395212	+	0.918590i	$0.629329\pi$
$702$	0	0
$703$	1.41252e6	0.107797
$704$	0	0
$705$	1.07614e7	0.815449
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−1.01764e7	−0.760289	−0.380144	−	0.924927i	$-0.624126\pi$
$709$	−1.01764e7	−0.760289	−0.380144	+	0.924927i	$0.624126\pi$
$710$	0	0
$711$	−749545.	−0.0556063
$712$	0	0
$713$	−1.37599e7	−1.01366
$714$	0	0
$715$	−326502.	−0.0238848
$716$	0	0
$717$	−9.91666e6	−0.720390
$718$	0	0
$719$	−1.03436e7	−0.746187	−0.373093	−	0.927794i	$-0.621703\pi$
$719$	−1.03436e7	−0.746187	−0.373093	+	0.927794i	$0.621703\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	−1.91157e7	−1.36002
$724$	0	0
$725$	−9.55045e6	−0.674806
$726$	0	0
$727$	−3.18206e6	−0.223291	−0.111646	−	0.993748i	$-0.535612\pi$
$727$	−3.18206e6	−0.223291	−0.111646	+	0.993748i	$0.535612\pi$
$728$	0	0
$729$	1.52933e7	1.06582
$730$	0	0
$731$	−6.33653e6	−0.438590
$732$	0	0
$733$	1.11021e7	0.763210	0.381605	−	0.924326i	$-0.375371\pi$
$733$	1.11021e7	0.763210	0.381605	+	0.924326i	$0.375371\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	1.05299e7	0.714096
$738$	0	0
$739$	−1.95846e7	−1.31918	−0.659591	−	0.751625i	$-0.729270\pi$
$739$	−1.95846e7	−1.31918	−0.659591	+	0.751625i	$0.729270\pi$
$740$	0	0
$741$	253140.	0.0169362
$742$	0	0
$743$	4.83653e6	0.321412	0.160706	−	0.987002i	$-0.448623\pi$
$743$	4.83653e6	0.321412	0.160706	+	0.987002i	$0.448623\pi$
$744$	0	0
$745$	5.00856e6	0.330615
$746$	0	0
$747$	260954.	0.0171105
$748$	0	0
$749$	0	0
$750$	0	0
$751$	1.86920e6	0.120936	0.0604681	−	0.998170i	$-0.480741\pi$
$751$	1.86920e6	0.120936	0.0604681	+	0.998170i	$0.480741\pi$
$752$	0	0
$753$	6.31840e6	0.406087
$754$	0	0
$755$	−5.77108e6	−0.368460
$756$	0	0
$757$	−3.61268e6	−0.229134	−0.114567	−	0.993416i	$-0.536548\pi$
$757$	−3.61268e6	−0.229134	−0.114567	+	0.993416i	$0.536548\pi$
$758$	0	0
$759$	−3.38327e7	−2.13173
$760$	0	0
$761$	1.51147e6	0.0946099	0.0473049	−	0.998880i	$-0.484937\pi$
$761$	1.51147e6	0.0946099	0.0473049	+	0.998880i	$0.484937\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	1.48060e6	0.0914709
$766$	0	0
$767$	235372.	0.0144466
$768$	0	0
$769$	−3.96989e6	−0.242082	−0.121041	−	0.992648i	$-0.538623\pi$
$769$	−3.96989e6	−0.242082	−0.121041	+	0.992648i	$0.538623\pi$
$770$	0	0
$771$	−2.48720e7	−1.50687
$772$	0	0
$773$	2.35033e7	1.41475	0.707377	−	0.706836i	$-0.249878\pi$
$773$	2.35033e7	1.41475	0.707377	+	0.706836i	$0.249878\pi$
$774$	0	0
$775$	−8.65649e6	−0.517711
$776$	0	0
$777$	0	0
$778$	0	0
$779$	4.40700e6	0.260195
$780$	0	0
$781$	−5.57422e7	−3.27007
$782$	0	0
$783$	1.97056e7	1.14865
$784$	0	0
$785$	1.46251e7	0.847081
$786$	0	0
$787$	2.75631e7	1.58632	0.793161	−	0.609012i	$-0.208434\pi$
$787$	2.75631e7	1.58632	0.793161	+	0.609012i	$0.208434\pi$
$788$	0	0
$789$	−2.96267e7	−1.69430
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−701901.	−0.0396363
$794$	0	0
$795$	1.15612e7	0.648761
$796$	0	0
$797$	1.97282e7	1.10012	0.550061	−	0.835124i	$-0.314604\pi$
$797$	1.97282e7	1.10012	0.550061	+	0.835124i	$0.314604\pi$
$798$	0	0
$799$	4.54913e7	2.52094
$800$	0	0
$801$	−1.28667e6	−0.0708574
$802$	0	0
$803$	2.96400e7	1.62214
$804$	0	0
$805$	0	0
$806$	0	0
$807$	1.75228e7	0.947153
$808$	0	0
$809$	1.69706e7	0.911647	0.455824	−	0.890070i	$-0.349345\pi$
$809$	1.69706e7	0.911647	0.455824	+	0.890070i	$0.349345\pi$
$810$	0	0
$811$	5.79743e6	0.309516	0.154758	−	0.987952i	$-0.450540\pi$
$811$	5.79743e6	0.309516	0.154758	+	0.987952i	$0.450540\pi$
$812$	0	0
$813$	1.96252e6	0.104133
$814$	0	0
$815$	−1.04578e7	−0.551499
$816$	0	0
$817$	−3.88480e6	−0.203617
$818$	0	0
$819$	0	0
$820$	0	0
$821$	2.52205e7	1.30586	0.652928	−	0.757420i	$-0.273540\pi$
$821$	2.52205e7	1.30586	0.652928	+	0.757420i	$0.273540\pi$
$822$	0	0
$823$	−4.50651e6	−0.231921	−0.115961	−	0.993254i	$-0.536995\pi$
$823$	−4.50651e6	−0.231921	−0.115961	+	0.993254i	$0.536995\pi$
$824$	0	0
$825$	−2.12844e7	−1.08875
$826$	0	0
$827$	1.86065e7	0.946022	0.473011	−	0.881057i	$-0.343167\pi$
$827$	1.86065e7	0.946022	0.473011	+	0.881057i	$0.343167\pi$
$828$	0	0
$829$	3.74194e6	0.189108	0.0945540	−	0.995520i	$-0.469857\pi$
$829$	3.74194e6	0.189108	0.0945540	+	0.995520i	$0.469857\pi$
$830$	0	0
$831$	−1.61206e7	−0.809801
$832$	0	0
$833$	0	0
$834$	0	0
$835$	4.17288e6	0.207119
$836$	0	0
$837$	1.78611e7	0.881241
$838$	0	0
$839$	−3.60871e7	−1.76989	−0.884945	−	0.465695i	$-0.845804\pi$
$839$	−3.60871e7	−1.76989	−0.884945	+	0.465695i	$0.845804\pi$
$840$	0	0
$841$	4.73310e6	0.230757
$842$	0	0
$843$	2.89908e6	0.140505
$844$	0	0
$845$	−1.29854e7	−0.625626
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−3.74100e6	−0.178122
$850$	0	0
$851$	3.14563e6	0.148896
$852$	0	0
$853$	1.45992e7	0.687001	0.343501	−	0.939152i	$-0.388387\pi$
$853$	1.45992e7	0.687001	0.343501	+	0.939152i	$0.388387\pi$
$854$	0	0
$855$	907723.	0.0424657
$856$	0	0
$857$	−2.05674e7	−0.956594	−0.478297	−	0.878198i	$-0.658746\pi$
$857$	−2.05674e7	−0.956594	−0.478297	+	0.878198i	$0.658746\pi$
$858$	0	0
$859$	3.41480e7	1.57900	0.789500	−	0.613751i	$-0.210340\pi$
$859$	3.41480e7	1.57900	0.789500	+	0.613751i	$0.210340\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	2.33245e7	1.06607	0.533034	−	0.846094i	$-0.321052\pi$
$863$	2.33245e7	1.06607	0.533034	+	0.846094i	$0.321052\pi$
$864$	0	0
$865$	1.48294e7	0.673882
$866$	0	0
$867$	−5.20342e7	−2.35094
$868$	0	0
$869$	−2.93348e7	−1.31775
$870$	0	0
$871$	−175454.	−0.00783642
$872$	0	0
$873$	1.69637e6	0.0753331
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−2.79697e7	−1.22797	−0.613986	−	0.789317i	$-0.710435\pi$
$877$	−2.79697e7	−1.22797	−0.613986	+	0.789317i	$0.710435\pi$
$878$	0	0
$879$	3.19177e7	1.39335
$880$	0	0
$881$	1.48700e6	0.0645464	0.0322732	−	0.999479i	$-0.489725\pi$
$881$	1.48700e6	0.0645464	0.0322732	+	0.999479i	$0.489725\pi$
$882$	0	0
$883$	−2.11130e7	−0.911274	−0.455637	−	0.890166i	$-0.650588\pi$
$883$	−2.11130e7	−0.911274	−0.455637	+	0.890166i	$0.650588\pi$
$884$	0	0
$885$	−9.88166e6	−0.424104
$886$	0	0
$887$	−4.19979e6	−0.179233	−0.0896165	−	0.995976i	$-0.528564\pi$
$887$	−4.19979e6	−0.179233	−0.0896165	+	0.995976i	$0.528564\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	4.04392e7	1.70651
$892$	0	0
$893$	2.78898e7	1.17035
$894$	0	0
$895$	2.34006e7	0.976493
$896$	0	0
$897$	563734.	0.0233934
$898$	0	0
$899$	2.28813e7	0.944238
$900$	0	0
$901$	4.88722e7	2.00563
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−1.98250e7	−0.804622
$906$	0	0
$907$	−3.19107e7	−1.28801	−0.644003	−	0.765023i	$-0.722727\pi$
$907$	−3.19107e7	−1.28801	−0.644003	+	0.765023i	$0.722727\pi$
$908$	0	0
$909$	1.70479e6	0.0684324
$910$	0	0
$911$	4.56611e7	1.82285	0.911424	−	0.411469i	$-0.134984\pi$
$911$	4.56611e7	1.82285	0.911424	+	0.411469i	$0.134984\pi$
$912$	0	0
$913$	1.02129e7	0.405482
$914$	0	0
$915$	2.94681e7	1.16359
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−4.13031e7	−1.61322	−0.806610	−	0.591083i	$-0.798700\pi$
$919$	−4.13031e7	−1.61322	−0.806610	+	0.591083i	$0.798700\pi$
$920$	0	0
$921$	−5.82178e6	−0.226155
$922$	0	0
$923$	928799.	0.0358854
$924$	0	0
$925$	1.97894e6	0.0760465
$926$	0	0
$927$	−52830.8	−0.00201924
$928$	0	0
$929$	1.44541e7	0.549482	0.274741	−	0.961518i	$-0.411408\pi$
$929$	1.44541e7	0.549482	0.274741	+	0.961518i	$0.411408\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	3.61058e7	1.35792
$934$	0	0
$935$	5.79458e7	2.16767
$936$	0	0
$937$	−3.53477e7	−1.31526	−0.657631	−	0.753340i	$-0.728441\pi$
$937$	−3.53477e7	−1.31526	−0.657631	+	0.753340i	$0.728441\pi$
$938$	0	0
$939$	−1.28285e7	−0.474802
$940$	0	0
$941$	−3.68406e6	−0.135629	−0.0678144	−	0.997698i	$-0.521603\pi$
$941$	−3.68406e6	−0.135629	−0.0678144	+	0.997698i	$0.521603\pi$
$942$	0	0
$943$	9.81424e6	0.359400
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−1.21957e6	−0.0441908	−0.0220954	−	0.999756i	$-0.507034\pi$
$947$	−1.21957e6	−0.0441908	−0.0220954	+	0.999756i	$0.507034\pi$
$948$	0	0
$949$	−493873.	−0.0178012
$950$	0	0
$951$	3.26816e7	1.17180
$952$	0	0
$953$	−1.22041e7	−0.435286	−0.217643	−	0.976028i	$-0.569837\pi$
$953$	−1.22041e7	−0.435286	−0.217643	+	0.976028i	$0.569837\pi$
$954$	0	0
$955$	7.55163e6	0.267937
$956$	0	0
$957$	5.62601e7	1.98573
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−7.88963e6	−0.275580
$962$	0	0
$963$	562220.	0.0195362
$964$	0	0
$965$	−6.34184e6	−0.219228
$966$	0	0
$967$	−4.11920e7	−1.41660	−0.708299	−	0.705913i	$-0.750537\pi$
$967$	−4.11920e7	−1.41660	−0.708299	+	0.705913i	$0.750537\pi$
$968$	0	0
$969$	−4.49258e7	−1.53704
$970$	0	0
$971$	3.91352e6	0.133205	0.0666024	−	0.997780i	$-0.478784\pi$
$971$	3.91352e6	0.133205	0.0666024	+	0.997780i	$0.478784\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	354650.	0.0119478
$976$	0	0
$977$	−2.76330e7	−0.926171	−0.463086	−	0.886314i	$-0.653258\pi$
$977$	−2.76330e7	−0.926171	−0.463086	+	0.886314i	$0.653258\pi$
$978$	0	0
$979$	−5.03561e7	−1.67917
$980$	0	0
$981$	23709.1	0.000786581 0
$982$	0	0
$983$	−2.09731e7	−0.692276	−0.346138	−	0.938184i	$-0.612507\pi$
$983$	−2.09731e7	−0.692276	−0.346138	+	0.938184i	$0.612507\pi$
$984$	0	0
$985$	1.54165e7	0.506284
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−8.65132e6	−0.281249
$990$	0	0
$991$	3.91689e7	1.26694	0.633471	−	0.773766i	$-0.281630\pi$
$991$	3.91689e7	1.26694	0.633471	+	0.773766i	$0.281630\pi$
$992$	0	0
$993$	2.51027e7	0.807881
$994$	0	0
$995$	−2.65050e7	−0.848731
$996$	0	0
$997$	−4.21788e7	−1.34387	−0.671934	−	0.740611i	$-0.734536\pi$
$997$	−4.21788e7	−1.34387	−0.671934	+	0.740611i	$0.734536\pi$
$998$	0	0
$999$	−4.08319e6	−0.129445

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	784.6.a.bg.1.2		4
4.3	odd	2		98.6.a.i.1.3	yes	4
7.6	odd	2	inner	784.6.a.bg.1.3		4
12.11	even	2		882.6.a.bv.1.2		4
28.3	even	6		98.6.c.i.79.3		8
28.11	odd	6		98.6.c.i.79.2		8
28.19	even	6		98.6.c.i.67.3		8
28.23	odd	6		98.6.c.i.67.2		8
28.27	even	2		98.6.a.i.1.2	✓	4
84.83	odd	2		882.6.a.bv.1.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
98.6.a.i.1.2	✓	4	28.27	even	2
98.6.a.i.1.3	yes	4	4.3	odd	2
98.6.c.i.67.2		8	28.23	odd	6
98.6.c.i.67.3		8	28.19	even	6
98.6.c.i.79.2		8	28.11	odd	6
98.6.c.i.79.3		8	28.3	even	6
784.6.a.bg.1.2		4	1.1	even	1	trivial
784.6.a.bg.1.3		4	7.6	odd	2	inner
882.6.a.bv.1.2		4	12.11	even	2
882.6.a.bv.1.3		4	84.83	odd	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 784 = 2^{4} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	784.a (trivial)

\(f(q)\)	\(=\)	\(q-14.9626 q^{3} +34.9882 q^{5} -19.1218 q^{9} -748.365 q^{11} +12.4696 q^{13} -523.513 q^{15} -2213.03 q^{17} -1356.76 q^{19} -3021.46 q^{23} -1900.83 q^{25} +3922.01 q^{27} +5024.37 q^{29} +4554.07 q^{31} +11197.5 q^{33} -1041.10 q^{37} -186.577 q^{39} -3248.18 q^{41} +2863.29 q^{43} -669.037 q^{45} -20556.2 q^{47} +33112.5 q^{51} -22083.9 q^{53} -26183.9 q^{55} +20300.6 q^{57} +18875.7 q^{59} -56289.1 q^{61} +436.288 q^{65} -14070.6 q^{67} +45208.8 q^{69} +74485.3 q^{71} -39606.3 q^{73} +28441.2 q^{75} +39198.5 q^{79} -54036.8 q^{81} -13646.9 q^{83} -77429.8 q^{85} -75177.4 q^{87} +67288.1 q^{89} -68140.5 q^{93} -47470.6 q^{95} -88714.1 q^{97} +14310.1 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 712 q^{9} - 628 q^{11} - 5248 q^{15} - 2624 q^{23} + 4224 q^{25} + 17732 q^{29} + 2932 q^{37} + 41832 q^{39} - 9836 q^{43} + 51236 q^{51} + 1552 q^{53} - 104092 q^{57} - 142548 q^{65} - 13704 q^{67}+ \cdots + 354500 q^{99}+O(q^{100}) \) 4 * q + 712 * q^9 - 628 * q^11 - 5248 * q^15 - 2624 * q^23 + 4224 * q^25 + 17732 * q^29 + 2932 * q^37 + 41832 * q^39 - 9836 * q^43 + 51236 * q^51 + 1552 * q^53 - 104092 * q^57 - 142548 * q^65 - 13704 * q^67 + 56664 * q^71 + 218296 * q^79 - 127048 * q^81 - 103924 * q^85 - 88056 * q^93 + 396752 * q^95 + 354500 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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