LMFDB - Embedded newform 784.6.a.bl.1.1

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 = [5,0,5,0,81,0,0,0,390] 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:	$5$
Coefficient field:	$\mathbb{Q}[x]/(x^{5} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 200x^{3} - 99x^{2} + 5803x - 3615 $ x^5 - 200x^3 - 99x^2 + 5803*x - 3615
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{9}\cdot 7 $
Twist minimal:	no (minimal twist has level 56)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-12.3209$ 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)

$f(q)$	$=$	$q-23.6419 q^{3} -20.0456 q^{5} +315.939 q^{9} -58.7531 q^{11} +691.185 q^{13} +473.916 q^{15} +526.776 q^{17} +127.568 q^{19} -2268.61 q^{23} -2723.17 q^{25} -1724.41 q^{27} +8636.30 q^{29} +10290.8 q^{31} +1389.03 q^{33} -4723.11 q^{37} -16340.9 q^{39} +6676.64 q^{41} -22926.7 q^{43} -6333.18 q^{45} -24112.0 q^{47} -12454.0 q^{51} -8242.96 q^{53} +1177.74 q^{55} -3015.94 q^{57} -36134.3 q^{59} +15654.8 q^{61} -13855.2 q^{65} -29371.7 q^{67} +53634.1 q^{69} +9025.02 q^{71} +4435.42 q^{73} +64381.0 q^{75} -8201.62 q^{79} -36004.8 q^{81} +54575.9 q^{83} -10559.5 q^{85} -204178. q^{87} -86933.9 q^{89} -243294. q^{93} -2557.17 q^{95} +157783. q^{97} -18562.4 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + 5 q^{3} + 81 q^{5} + 390 q^{9} - 361 q^{11} + 342 q^{13} + 1049 q^{15} + 1809 q^{17} - 1277 q^{19} - 911 q^{23} + 3940 q^{25} + 4751 q^{27} + 5442 q^{29} - 2187 q^{31} - 5553 q^{33} - 8181 q^{37}+ \cdots - 249798 q^{99}+O(q^{100}) $ 5 * q + 5 * q^3 + 81 * q^5 + 390 * q^9 - 361 * q^11 + 342 * q^13 + 1049 * q^15 + 1809 * q^17 - 1277 * q^19 - 911 * q^23 + 3940 * q^25 + 4751 * q^27 + 5442 * q^29 - 2187 * q^31 - 5553 * q^33 - 8181 * q^37 + 3422 * q^39 + 16578 * q^41 - 6332 * q^43 + 41310 * q^45 - 16101 * q^47 + 67865 * q^51 + 16047 * q^53 - 45629 * q^55 + 22347 * q^57 - 71027 * q^59 + 31093 * q^61 - 64370 * q^65 - 47981 * q^67 + 137249 * q^69 - 22512 * q^71 + 123333 * q^73 + 45460 * q^75 + 212481 * q^79 + 52917 * q^81 - 87460 * q^83 + 222141 * q^85 - 318070 * q^87 + 129045 * q^89 - 252835 * q^93 - 300417 * q^95 + 328274 * q^97 - 249798 * 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
$3$	−23.6419	−1.51663	−0.758314	−	0.651890i	$-0.773977\pi$
$3$	−23.6419	−1.51663	−0.758314	+	0.651890i	$0.773977\pi$
$4$	0	0
$5$	−20.0456	−0.358587	−0.179293	−	0.983796i	$-0.557381\pi$
$5$	−20.0456	−0.358587	−0.179293	+	0.983796i	$0.557381\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	315.939	1.30016
$10$	0	0
$11$	−58.7531	−0.146403	−0.0732014	−	0.997317i	$-0.523322\pi$
$11$	−58.7531	−0.146403	−0.0732014	+	0.997317i	$0.523322\pi$
$12$	0	0
$13$	691.185	1.13432	0.567160	−	0.823607i	$-0.308042\pi$
$13$	691.185	1.13432	0.567160	+	0.823607i	$0.308042\pi$
$14$	0	0
$15$	473.916	0.543842
$16$	0	0
$17$	526.776	0.442083	0.221041	−	0.975264i	$-0.429054\pi$
$17$	526.776	0.442083	0.221041	+	0.975264i	$0.429054\pi$
$18$	0	0
$19$	127.568	0.0810692	0.0405346	−	0.999178i	$-0.487094\pi$
$19$	127.568	0.0810692	0.0405346	+	0.999178i	$0.487094\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−2268.61	−0.894210	−0.447105	−	0.894482i	$-0.647545\pi$
$23$	−2268.61	−0.894210	−0.447105	+	0.894482i	$0.647545\pi$
$24$	0	0
$25$	−2723.17	−0.871416
$26$	0	0
$27$	−1724.41	−0.455230
$28$	0	0
$29$	8636.30	1.90692	0.953461	−	0.301518i	$-0.0974933\pi$
$29$	8636.30	1.90692	0.953461	+	0.301518i	$0.0974933\pi$
$30$	0	0
$31$	10290.8	1.92330	0.961648	−	0.274288i	$-0.0884422\pi$
$31$	10290.8	1.92330	0.961648	+	0.274288i	$0.0884422\pi$
$32$	0	0
$33$	1389.03	0.222039
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−4723.11	−0.567183	−0.283592	−	0.958945i	$-0.591526\pi$
$37$	−4723.11	−0.567183	−0.283592	+	0.958945i	$0.591526\pi$
$38$	0	0
$39$	−16340.9	−1.72034
$40$	0	0
$41$	6676.64	0.620295	0.310147	−	0.950688i	$-0.399622\pi$
$41$	6676.64	0.620295	0.310147	+	0.950688i	$0.399622\pi$
$42$	0	0
$43$	−22926.7	−1.89091	−0.945455	−	0.325754i	$-0.894382\pi$
$43$	−22926.7	−1.89091	−0.945455	+	0.325754i	$0.894382\pi$
$44$	0	0
$45$	−6333.18	−0.466220
$46$	0	0
$47$	−24112.0	−1.59217	−0.796085	−	0.605185i	$-0.793099\pi$
$47$	−24112.0	−1.59217	−0.796085	+	0.605185i	$0.793099\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	−12454.0	−0.670475
$52$	0	0
$53$	−8242.96	−0.403082	−0.201541	−	0.979480i	$-0.564595\pi$
$53$	−8242.96	−0.403082	−0.201541	+	0.979480i	$0.564595\pi$
$54$	0	0
$55$	1177.74	0.0524981
$56$	0	0
$57$	−3015.94	−0.122952
$58$	0	0
$59$	−36134.3	−1.35142	−0.675709	−	0.737169i	$-0.736162\pi$
$59$	−36134.3	−1.35142	−0.675709	+	0.737169i	$0.736162\pi$
$60$	0	0
$61$	15654.8	0.538672	0.269336	−	0.963046i	$-0.413196\pi$
$61$	15654.8	0.538672	0.269336	+	0.963046i	$0.413196\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−13855.2	−0.406752
$66$	0	0
$67$	−29371.7	−0.799361	−0.399680	−	0.916655i	$-0.630879\pi$
$67$	−29371.7	−0.799361	−0.399680	+	0.916655i	$0.630879\pi$
$68$	0	0
$69$	53634.1	1.35618
$70$	0	0
$71$	9025.02	0.212472	0.106236	−	0.994341i	$-0.466120\pi$
$71$	9025.02	0.212472	0.106236	+	0.994341i	$0.466120\pi$
$72$	0	0
$73$	4435.42	0.0974154	0.0487077	−	0.998813i	$-0.484490\pi$
$73$	4435.42	0.0974154	0.0487077	+	0.998813i	$0.484490\pi$
$74$	0	0
$75$	64381.0	1.32161
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−8201.62	−0.147854	−0.0739268	−	0.997264i	$-0.523553\pi$
$79$	−8201.62	−0.147854	−0.0739268	+	0.997264i	$0.523553\pi$
$80$	0	0
$81$	−36004.8	−0.609745
$82$	0	0
$83$	54575.9	0.869572	0.434786	−	0.900534i	$-0.356824\pi$
$83$	54575.9	0.869572	0.434786	+	0.900534i	$0.356824\pi$
$84$	0	0
$85$	−10559.5	−0.158525
$86$	0	0
$87$	−204178.	−2.89209
$88$	0	0
$89$	−86933.9	−1.16336	−0.581680	−	0.813418i	$-0.697604\pi$
$89$	−86933.9	−1.16336	−0.581680	+	0.813418i	$0.697604\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−243294.	−2.91692
$94$	0	0
$95$	−2557.17	−0.0290703
$96$	0	0
$97$	157783.	1.70268	0.851338	−	0.524617i	$-0.175792\pi$
$97$	157783.	1.70268	0.851338	+	0.524617i	$0.175792\pi$
$98$	0	0
$99$	−18562.4	−0.190347
$100$	0	0
$101$	77781.9	0.758709	0.379354	−	0.925251i	$-0.376146\pi$
$101$	77781.9	0.758709	0.379354	+	0.925251i	$0.376146\pi$
$102$	0	0
$103$	−107006.	−0.993840	−0.496920	−	0.867796i	$-0.665536\pi$
$103$	−107006.	−0.993840	−0.496920	+	0.867796i	$0.665536\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	160396.	1.35436	0.677182	−	0.735816i	$-0.263201\pi$
$107$	160396.	1.35436	0.677182	+	0.735816i	$0.263201\pi$
$108$	0	0
$109$	80176.6	0.646371	0.323185	−	0.946336i	$-0.395246\pi$
$109$	80176.6	0.646371	0.323185	+	0.946336i	$0.395246\pi$
$110$	0	0
$111$	111663.	0.860206
$112$	0	0
$113$	−99679.0	−0.734358	−0.367179	−	0.930150i	$-0.619676\pi$
$113$	−99679.0	−0.734358	−0.367179	+	0.930150i	$0.619676\pi$
$114$	0	0
$115$	45475.6	0.320652
$116$	0	0
$117$	218372.	1.47480
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−157599.	−0.978566
$122$	0	0
$123$	−157848.	−0.940756
$124$	0	0
$125$	117230.	0.671065
$126$	0	0
$127$	120594.	0.663463	0.331732	−	0.943374i	$-0.392367\pi$
$127$	120594.	0.663463	0.331732	+	0.943374i	$0.392367\pi$
$128$	0	0
$129$	542031.	2.86781
$130$	0	0
$131$	50227.8	0.255721	0.127860	−	0.991792i	$-0.459189\pi$
$131$	50227.8	0.255721	0.127860	+	0.991792i	$0.459189\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	34566.8	0.163239
$136$	0	0
$137$	−208225.	−0.947834	−0.473917	−	0.880569i	$-0.657160\pi$
$137$	−208225.	−0.947834	−0.473917	+	0.880569i	$0.657160\pi$
$138$	0	0
$139$	−215032.	−0.943988	−0.471994	−	0.881602i	$-0.656466\pi$
$139$	−215032.	−0.943988	−0.471994	+	0.881602i	$0.656466\pi$
$140$	0	0
$141$	570054.	2.41473
$142$	0	0
$143$	−40609.3	−0.166068
$144$	0	0
$145$	−173120.	−0.683796
$146$	0	0
$147$	0	0
$148$	0	0
$149$	383526.	1.41524	0.707618	−	0.706595i	$-0.249770\pi$
$149$	383526.	1.41524	0.707618	+	0.706595i	$0.249770\pi$
$150$	0	0
$151$	−410918.	−1.46660	−0.733302	−	0.679903i	$-0.762022\pi$
$151$	−410918.	−1.46660	−0.733302	+	0.679903i	$0.762022\pi$
$152$	0	0
$153$	166429.	0.574778
$154$	0	0
$155$	−206286.	−0.689668
$156$	0	0
$157$	−329710.	−1.06754	−0.533769	−	0.845630i	$-0.679225\pi$
$157$	−329710.	−1.06754	−0.533769	+	0.845630i	$0.679225\pi$
$158$	0	0
$159$	194879.	0.611326
$160$	0	0
$161$	0	0
$162$	0	0
$163$	381832.	1.12565	0.562824	−	0.826577i	$-0.309715\pi$
$163$	381832.	1.12565	0.562824	+	0.826577i	$0.309715\pi$
$164$	0	0
$165$	−27844.0	−0.0796200
$166$	0	0
$167$	−195870.	−0.543471	−0.271735	−	0.962372i	$-0.587598\pi$
$167$	−195870.	−0.543471	−0.271735	+	0.962372i	$0.587598\pi$
$168$	0	0
$169$	106443.	0.286683
$170$	0	0
$171$	40303.5	0.105403
$172$	0	0
$173$	531987.	1.35141	0.675703	−	0.737174i	$-0.263840\pi$
$173$	531987.	1.35141	0.675703	+	0.737174i	$0.263840\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	854283.	2.04960
$178$	0	0
$179$	374611.	0.873873	0.436937	−	0.899492i	$-0.356063\pi$
$179$	374611.	0.873873	0.436937	+	0.899492i	$0.356063\pi$
$180$	0	0
$181$	148562.	0.337063	0.168532	−	0.985696i	$-0.446097\pi$
$181$	148562.	0.337063	0.168532	+	0.985696i	$0.446097\pi$
$182$	0	0
$183$	−370110.	−0.816965
$184$	0	0
$185$	94677.5	0.203384
$186$	0	0
$187$	−30949.7	−0.0647222
$188$	0	0
$189$	0	0
$190$	0	0
$191$	616890.	1.22356	0.611779	−	0.791029i	$-0.290454\pi$
$191$	616890.	1.22356	0.611779	+	0.791029i	$0.290454\pi$
$192$	0	0
$193$	69951.0	0.135176	0.0675882	−	0.997713i	$-0.478470\pi$
$193$	69951.0	0.135176	0.0675882	+	0.997713i	$0.478470\pi$
$194$	0	0
$195$	327563.	0.616892
$196$	0	0
$197$	440150.	0.808045	0.404022	−	0.914749i	$-0.367612\pi$
$197$	440150.	0.808045	0.404022	+	0.914749i	$0.367612\pi$
$198$	0	0
$199$	350409.	0.627253	0.313627	−	0.949546i	$-0.398456\pi$
$199$	350409.	0.627253	0.313627	+	0.949546i	$0.398456\pi$
$200$	0	0
$201$	694403.	1.21233
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−133837.	−0.222429
$206$	0	0
$207$	−716740.	−1.16262
$208$	0	0
$209$	−7494.99	−0.0118688
$210$	0	0
$211$	637860.	0.986323	0.493161	−	0.869938i	$-0.335841\pi$
$211$	637860.	0.986323	0.493161	+	0.869938i	$0.335841\pi$
$212$	0	0
$213$	−213369.	−0.322241
$214$	0	0
$215$	459580.	0.678055
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−104862.	−0.147743
$220$	0	0
$221$	364100.	0.501464
$222$	0	0
$223$	56418.4	0.0759729	0.0379864	−	0.999278i	$-0.487906\pi$
$223$	56418.4	0.0759729	0.0379864	+	0.999278i	$0.487906\pi$
$224$	0	0
$225$	−860356.	−1.13298
$226$	0	0
$227$	293499.	0.378043	0.189022	−	0.981973i	$-0.439468\pi$
$227$	293499.	0.378043	0.189022	+	0.981973i	$0.439468\pi$
$228$	0	0
$229$	620574.	0.781996	0.390998	−	0.920391i	$-0.372130\pi$
$229$	620574.	0.781996	0.390998	+	0.920391i	$0.372130\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−724323.	−0.874062	−0.437031	−	0.899446i	$-0.643970\pi$
$233$	−724323.	−0.874062	−0.437031	+	0.899446i	$0.643970\pi$
$234$	0	0
$235$	483340.	0.570931
$236$	0	0
$237$	193902.	0.224239
$238$	0	0
$239$	272208.	0.308252	0.154126	−	0.988051i	$-0.450744\pi$
$239$	272208.	0.308252	0.154126	+	0.988051i	$0.450744\pi$
$240$	0	0
$241$	850014.	0.942722	0.471361	−	0.881940i	$-0.343763\pi$
$241$	850014.	0.942722	0.471361	+	0.881940i	$0.343763\pi$
$242$	0	0
$243$	1.27025e6	1.37999
$244$	0	0
$245$	0	0
$246$	0	0
$247$	88172.7	0.0919585
$248$	0	0
$249$	−1.29028e6	−1.31882
$250$	0	0
$251$	1.62295e6	1.62601	0.813003	−	0.582260i	$-0.197831\pi$
$251$	1.62295e6	1.62601	0.813003	+	0.582260i	$0.197831\pi$
$252$	0	0
$253$	133288.	0.130915
$254$	0	0
$255$	249648.	0.240423
$256$	0	0
$257$	−1.29876e6	−1.22658	−0.613292	−	0.789857i	$-0.710155\pi$
$257$	−1.29876e6	−1.22658	−0.613292	+	0.789857i	$0.710155\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	2.72854e6	2.47930
$262$	0	0
$263$	1.58814e6	1.41579	0.707895	−	0.706318i	$-0.249645\pi$
$263$	1.58814e6	1.41579	0.707895	+	0.706318i	$0.249645\pi$
$264$	0	0
$265$	165235.	0.144540
$266$	0	0
$267$	2.05528e6	1.76438
$268$	0	0
$269$	−850606.	−0.716717	−0.358358	−	0.933584i	$-0.616663\pi$
$269$	−850606.	−0.716717	−0.358358	+	0.933584i	$0.616663\pi$
$270$	0	0
$271$	1.75962e6	1.45544	0.727722	−	0.685872i	$-0.240579\pi$
$271$	1.75962e6	1.45544	0.727722	+	0.685872i	$0.240579\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	159995.	0.127578
$276$	0	0
$277$	−1.74579e6	−1.36707	−0.683537	−	0.729916i	$-0.739559\pi$
$277$	−1.74579e6	−1.36707	−0.683537	+	0.729916i	$0.739559\pi$
$278$	0	0
$279$	3.25127e6	2.50059
$280$	0	0
$281$	−2.07171e6	−1.56518	−0.782588	−	0.622541i	$-0.786101\pi$
$281$	−2.07171e6	−1.56518	−0.782588	+	0.622541i	$0.786101\pi$
$282$	0	0
$283$	314936.	0.233752	0.116876	−	0.993146i	$-0.462712\pi$
$283$	314936.	0.233752	0.116876	+	0.993146i	$0.462712\pi$
$284$	0	0
$285$	60456.3	0.0440889
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−1.14236e6	−0.804563
$290$	0	0
$291$	−3.73030e6	−2.58233
$292$	0	0
$293$	2.07017e6	1.40876	0.704381	−	0.709822i	$-0.251225\pi$
$293$	2.07017e6	1.40876	0.704381	+	0.709822i	$0.251225\pi$
$294$	0	0
$295$	724334.	0.484600
$296$	0	0
$297$	101314.	0.0666470
$298$	0	0
$299$	−1.56803e6	−1.01432
$300$	0	0
$301$	0	0
$302$	0	0
$303$	−1.83891e6	−1.15068
$304$	0	0
$305$	−313811.	−0.193161
$306$	0	0
$307$	−1.16092e6	−0.703000	−0.351500	−	0.936188i	$-0.614328\pi$
$307$	−1.16092e6	−0.703000	−0.351500	+	0.936188i	$0.614328\pi$
$308$	0	0
$309$	2.52983e6	1.50728
$310$	0	0
$311$	663375.	0.388918	0.194459	−	0.980911i	$-0.437705\pi$
$311$	663375.	0.388918	0.194459	+	0.980911i	$0.437705\pi$
$312$	0	0
$313$	620692.	0.358109	0.179054	−	0.983839i	$-0.442696\pi$
$313$	620692.	0.358109	0.179054	+	0.983839i	$0.442696\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	1.82339e6	1.01914	0.509568	−	0.860431i	$-0.329805\pi$
$317$	1.82339e6	1.01914	0.509568	+	0.860431i	$0.329805\pi$
$318$	0	0
$319$	−507410.	−0.279179
$320$	0	0
$321$	−3.79207e6	−2.05407
$322$	0	0
$323$	67199.5	0.0358393
$324$	0	0
$325$	−1.88222e6	−0.988465
$326$	0	0
$327$	−1.89553e6	−0.980304
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−1.34113e6	−0.672822	−0.336411	−	0.941715i	$-0.609213\pi$
$331$	−1.34113e6	−0.672822	−0.336411	+	0.941715i	$0.609213\pi$
$332$	0	0
$333$	−1.49221e6	−0.737429
$334$	0	0
$335$	588774.	0.286640
$336$	0	0
$337$	320075.	0.153524	0.0767621	−	0.997049i	$-0.475542\pi$
$337$	320075.	0.153524	0.0767621	+	0.997049i	$0.475542\pi$
$338$	0	0
$339$	2.35660e6	1.11375
$340$	0	0
$341$	−604618.	−0.281576
$342$	0	0
$343$	0	0
$344$	0	0
$345$	−1.07513e6	−0.486309
$346$	0	0
$347$	−697566.	−0.311001	−0.155500	−	0.987836i	$-0.549699\pi$
$347$	−697566.	−0.311001	−0.155500	+	0.987836i	$0.549699\pi$
$348$	0	0
$349$	2.08445e6	0.916068	0.458034	−	0.888935i	$-0.348554\pi$
$349$	2.08445e6	0.916068	0.458034	+	0.888935i	$0.348554\pi$
$350$	0	0
$351$	−1.19189e6	−0.516377
$352$	0	0
$353$	−3.13719e6	−1.34000	−0.669998	−	0.742363i	$-0.733705\pi$
$353$	−3.13719e6	−1.34000	−0.669998	+	0.742363i	$0.733705\pi$
$354$	0	0
$355$	−180912.	−0.0761897
$356$	0	0
$357$	0	0
$358$	0	0
$359$	1.80529e6	0.739284	0.369642	−	0.929174i	$-0.379480\pi$
$359$	1.80529e6	0.739284	0.369642	+	0.929174i	$0.379480\pi$
$360$	0	0
$361$	−2.45983e6	−0.993428
$362$	0	0
$363$	3.72594e6	1.48412
$364$	0	0
$365$	−88910.6	−0.0349318
$366$	0	0
$367$	1.58136e6	0.612864	0.306432	−	0.951893i	$-0.400865\pi$
$367$	1.58136e6	0.612864	0.306432	+	0.951893i	$0.400865\pi$
$368$	0	0
$369$	2.10941e6	0.806482
$370$	0	0
$371$	0	0
$372$	0	0
$373$	1.70096e6	0.633028	0.316514	−	0.948588i	$-0.397487\pi$
$373$	1.70096e6	0.633028	0.316514	+	0.948588i	$0.397487\pi$
$374$	0	0
$375$	−2.77154e6	−1.01776
$376$	0	0
$377$	5.96928e6	2.16306
$378$	0	0
$379$	−4.93679e6	−1.76541	−0.882707	−	0.469923i	$-0.844282\pi$
$379$	−4.93679e6	−1.76541	−0.882707	+	0.469923i	$0.844282\pi$
$380$	0	0
$381$	−2.85107e6	−1.00623
$382$	0	0
$383$	448450.	0.156213	0.0781065	−	0.996945i	$-0.475113\pi$
$383$	448450.	0.156213	0.0781065	+	0.996945i	$0.475113\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−7.24344e6	−2.45848
$388$	0	0
$389$	245076.	0.0821157	0.0410579	−	0.999157i	$-0.486927\pi$
$389$	245076.	0.0821157	0.0410579	+	0.999157i	$0.486927\pi$
$390$	0	0
$391$	−1.19505e6	−0.395315
$392$	0	0
$393$	−1.18748e6	−0.387833
$394$	0	0
$395$	164406.	0.0530183
$396$	0	0
$397$	−3.06740e6	−0.976774	−0.488387	−	0.872627i	$-0.662414\pi$
$397$	−3.06740e6	−0.976774	−0.488387	+	0.872627i	$0.662414\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−4.51686e6	−1.40274	−0.701368	−	0.712800i	$-0.747427\pi$
$401$	−4.51686e6	−1.40274	−0.701368	+	0.712800i	$0.747427\pi$
$402$	0	0
$403$	7.11286e6	2.18163
$404$	0	0
$405$	721738.	0.218646
$406$	0	0
$407$	277497.	0.0830372
$408$	0	0
$409$	5.08298e6	1.50249	0.751243	−	0.660026i	$-0.229455\pi$
$409$	5.08298e6	1.50249	0.751243	+	0.660026i	$0.229455\pi$
$410$	0	0
$411$	4.92284e6	1.43751
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−1.09401e6	−0.311817
$416$	0	0
$417$	5.08377e6	1.43168
$418$	0	0
$419$	2.14263e6	0.596227	0.298114	−	0.954530i	$-0.403643\pi$
$419$	2.14263e6	0.596227	0.298114	+	0.954530i	$0.403643\pi$
$420$	0	0
$421$	3.63121e6	0.998497	0.499248	−	0.866459i	$-0.333609\pi$
$421$	3.63121e6	0.998497	0.499248	+	0.866459i	$0.333609\pi$
$422$	0	0
$423$	−7.61793e6	−2.07007
$424$	0	0
$425$	−1.43450e6	−0.385238
$426$	0	0
$427$	0	0
$428$	0	0
$429$	960080.	0.251863
$430$	0	0
$431$	−1.95928e6	−0.508046	−0.254023	−	0.967198i	$-0.581754\pi$
$431$	−1.95928e6	−0.508046	−0.254023	+	0.967198i	$0.581754\pi$
$432$	0	0
$433$	−2.49301e6	−0.639005	−0.319502	−	0.947585i	$-0.603516\pi$
$433$	−2.49301e6	−0.639005	−0.319502	+	0.947585i	$0.603516\pi$
$434$	0	0
$435$	4.09288e6	1.03706
$436$	0	0
$437$	−289400.	−0.0724929
$438$	0	0
$439$	−4.02098e6	−0.995796	−0.497898	−	0.867236i	$-0.665895\pi$
$439$	−4.02098e6	−0.995796	−0.497898	+	0.867236i	$0.665895\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	4.46554e6	1.08110	0.540548	−	0.841313i	$-0.318217\pi$
$443$	4.46554e6	1.08110	0.540548	+	0.841313i	$0.318217\pi$
$444$	0	0
$445$	1.74264e6	0.417165
$446$	0	0
$447$	−9.06727e6	−2.14639
$448$	0	0
$449$	−3.68564e6	−0.862774	−0.431387	−	0.902167i	$-0.641976\pi$
$449$	−3.68564e6	−0.862774	−0.431387	+	0.902167i	$0.641976\pi$
$450$	0	0
$451$	−392273.	−0.0908129
$452$	0	0
$453$	9.71488e6	2.22429
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−2.13435e6	−0.478051	−0.239025	−	0.971013i	$-0.576828\pi$
$457$	−2.13435e6	−0.478051	−0.239025	+	0.971013i	$0.576828\pi$
$458$	0	0
$459$	−908378.	−0.201250
$460$	0	0
$461$	−3.11136e6	−0.681865	−0.340932	−	0.940088i	$-0.610743\pi$
$461$	−3.11136e6	−0.681865	−0.340932	+	0.940088i	$0.610743\pi$
$462$	0	0
$463$	4.03151e6	0.874009	0.437004	−	0.899459i	$-0.356039\pi$
$463$	4.03151e6	0.874009	0.437004	+	0.899459i	$0.356039\pi$
$464$	0	0
$465$	4.87698e6	1.04597
$466$	0	0
$467$	−465000.	−0.0986644	−0.0493322	−	0.998782i	$-0.515709\pi$
$467$	−465000.	−0.0986644	−0.0493322	+	0.998782i	$0.515709\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	7.79498e6	1.61906
$472$	0	0
$473$	1.34702e6	0.276834
$474$	0	0
$475$	−347389.	−0.0706450
$476$	0	0
$477$	−2.60427e6	−0.524071
$478$	0	0
$479$	−3.88192e6	−0.773050	−0.386525	−	0.922279i	$-0.626325\pi$
$479$	−3.88192e6	−0.773050	−0.386525	+	0.922279i	$0.626325\pi$
$480$	0	0
$481$	−3.26454e6	−0.643368
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−3.16286e6	−0.610557
$486$	0	0
$487$	3.77230e6	0.720750	0.360375	−	0.932808i	$-0.382649\pi$
$487$	3.77230e6	0.720750	0.360375	+	0.932808i	$0.382649\pi$
$488$	0	0
$489$	−9.02722e6	−1.70719
$490$	0	0
$491$	−3.97153e6	−0.743455	−0.371727	−	0.928342i	$-0.621234\pi$
$491$	−3.97153e6	−0.743455	−0.371727	+	0.928342i	$0.621234\pi$
$492$	0	0
$493$	4.54940e6	0.843017
$494$	0	0
$495$	372094.	0.0682559
$496$	0	0
$497$	0	0
$498$	0	0
$499$	9.30366e6	1.67264	0.836320	−	0.548241i	$-0.184702\pi$
$499$	9.30366e6	1.67264	0.836320	+	0.548241i	$0.184702\pi$
$500$	0	0
$501$	4.63073e6	0.824243
$502$	0	0
$503$	−2.66706e6	−0.470016	−0.235008	−	0.971993i	$-0.575512\pi$
$503$	−2.66706e6	−0.470016	−0.235008	+	0.971993i	$0.575512\pi$
$504$	0	0
$505$	−1.55918e6	−0.272063
$506$	0	0
$507$	−2.51652e6	−0.434792
$508$	0	0
$509$	−5.93994e6	−1.01622	−0.508110	−	0.861292i	$-0.669656\pi$
$509$	−5.93994e6	−1.01622	−0.508110	+	0.861292i	$0.669656\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−219979.	−0.0369052
$514$	0	0
$515$	2.14501e6	0.356378
$516$	0	0
$517$	1.41666e6	0.233098
$518$	0	0
$519$	−1.25772e7	−2.04958
$520$	0	0
$521$	6.49423e6	1.04817	0.524087	−	0.851665i	$-0.324407\pi$
$521$	6.49423e6	1.04817	0.524087	+	0.851665i	$0.324407\pi$
$522$	0	0
$523$	−5.26803e6	−0.842158	−0.421079	−	0.907024i	$-0.638349\pi$
$523$	−5.26803e6	−0.842158	−0.421079	+	0.907024i	$0.638349\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	5.42096e6	0.850256
$528$	0	0
$529$	−1.28977e6	−0.200389
$530$	0	0
$531$	−1.14162e7	−1.75706
$532$	0	0
$533$	4.61479e6	0.703613
$534$	0	0
$535$	−3.21524e6	−0.485657
$536$	0	0
$537$	−8.85652e6	−1.32534
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−3.94194e6	−0.579052	−0.289526	−	0.957170i	$-0.593498\pi$
$541$	−3.94194e6	−0.579052	−0.289526	+	0.957170i	$0.593498\pi$
$542$	0	0
$543$	−3.51229e6	−0.511200
$544$	0	0
$545$	−1.60719e6	−0.231780
$546$	0	0
$547$	1.04260e7	1.48987	0.744934	−	0.667138i	$-0.232481\pi$
$547$	1.04260e7	1.48987	0.744934	+	0.667138i	$0.232481\pi$
$548$	0	0
$549$	4.94597e6	0.700359
$550$	0	0
$551$	1.10171e6	0.154593
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−2.23835e6	−0.308458
$556$	0	0
$557$	9.10886e6	1.24402	0.622008	−	0.783011i	$-0.286317\pi$
$557$	9.10886e6	1.24402	0.622008	+	0.783011i	$0.286317\pi$
$558$	0	0
$559$	−1.58466e7	−2.14490
$560$	0	0
$561$	731710.	0.0981594
$562$	0	0
$563$	9.52612e6	1.26662	0.633308	−	0.773900i	$-0.281697\pi$
$563$	9.52612e6	1.26662	0.633308	+	0.773900i	$0.281697\pi$
$564$	0	0
$565$	1.99813e6	0.263331
$566$	0	0
$567$	0	0
$568$	0	0
$569$	1.26911e7	1.64331	0.821656	−	0.569983i	$-0.193050\pi$
$569$	1.26911e7	1.64331	0.821656	+	0.569983i	$0.193050\pi$
$570$	0	0
$571$	4.00866e6	0.514528	0.257264	−	0.966341i	$-0.417179\pi$
$571$	4.00866e6	0.514528	0.257264	+	0.966341i	$0.417179\pi$
$572$	0	0
$573$	−1.45845e7	−1.85568
$574$	0	0
$575$	6.17781e6	0.779228
$576$	0	0
$577$	−1.44306e6	−0.180445	−0.0902224	−	0.995922i	$-0.528758\pi$
$577$	−1.44306e6	−0.180445	−0.0902224	+	0.995922i	$0.528758\pi$
$578$	0	0
$579$	−1.65377e6	−0.205012
$580$	0	0
$581$	0	0
$582$	0	0
$583$	484300.	0.0590124
$584$	0	0
$585$	−4.37740e6	−0.528843
$586$	0	0
$587$	−3.38345e6	−0.405289	−0.202645	−	0.979252i	$-0.564954\pi$
$587$	−3.38345e6	−0.405289	−0.202645	+	0.979252i	$0.564954\pi$
$588$	0	0
$589$	1.31278e6	0.155920
$590$	0	0
$591$	−1.04060e7	−1.22550
$592$	0	0
$593$	4.94666e6	0.577664	0.288832	−	0.957380i	$-0.406733\pi$
$593$	4.94666e6	0.577664	0.288832	+	0.957380i	$0.406733\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−8.28434e6	−0.951310
$598$	0	0
$599$	1.29641e7	1.47631	0.738153	−	0.674633i	$-0.235698\pi$
$599$	1.29641e7	1.47631	0.738153	+	0.674633i	$0.235698\pi$
$600$	0	0
$601$	1.52347e7	1.72047	0.860235	−	0.509897i	$-0.170317\pi$
$601$	1.52347e7	1.72047	0.860235	+	0.509897i	$0.170317\pi$
$602$	0	0
$603$	−9.27967e6	−1.03930
$604$	0	0
$605$	3.15917e6	0.350901
$606$	0	0
$607$	−6.50606e6	−0.716714	−0.358357	−	0.933585i	$-0.616663\pi$
$607$	−6.50606e6	−0.716714	−0.358357	+	0.933585i	$0.616663\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−1.66659e7	−1.80603
$612$	0	0
$613$	7.28274e6	0.782787	0.391394	−	0.920223i	$-0.371993\pi$
$613$	7.28274e6	0.782787	0.391394	+	0.920223i	$0.371993\pi$
$614$	0	0
$615$	3.16416e6	0.337343
$616$	0	0
$617$	1.34924e7	1.42684	0.713422	−	0.700734i	$-0.247144\pi$
$617$	1.34924e7	1.42684	0.713422	+	0.700734i	$0.247144\pi$
$618$	0	0
$619$	7.05648e6	0.740221	0.370111	−	0.928988i	$-0.379320\pi$
$619$	7.05648e6	0.740221	0.370111	+	0.928988i	$0.379320\pi$
$620$	0	0
$621$	3.91201e6	0.407071
$622$	0	0
$623$	0	0
$624$	0	0
$625$	6.15997e6	0.630781
$626$	0	0
$627$	177196.	0.0180005
$628$	0	0
$629$	−2.48802e6	−0.250742
$630$	0	0
$631$	−1.46502e7	−1.46477	−0.732384	−	0.680891i	$-0.761593\pi$
$631$	−1.46502e7	−1.46477	−0.732384	+	0.680891i	$0.761593\pi$
$632$	0	0
$633$	−1.50802e7	−1.49588
$634$	0	0
$635$	−2.41738e6	−0.237909
$636$	0	0
$637$	0	0
$638$	0	0
$639$	2.85135e6	0.276248
$640$	0	0
$641$	1.99039e7	1.91335	0.956673	−	0.291166i	$-0.0940432\pi$
$641$	1.99039e7	1.91335	0.956673	+	0.291166i	$0.0940432\pi$
$642$	0	0
$643$	5.52786e6	0.527266	0.263633	−	0.964623i	$-0.415079\pi$
$643$	5.52786e6	0.527266	0.263633	+	0.964623i	$0.415079\pi$
$644$	0	0
$645$	−1.08653e7	−1.02836
$646$	0	0
$647$	1.81508e6	0.170465	0.0852323	−	0.996361i	$-0.472837\pi$
$647$	1.81508e6	0.170465	0.0852323	+	0.996361i	$0.472837\pi$
$648$	0	0
$649$	2.12300e6	0.197851
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−1.59251e7	−1.46151	−0.730753	−	0.682642i	$-0.760831\pi$
$653$	−1.59251e7	−1.46151	−0.730753	+	0.682642i	$0.760831\pi$
$654$	0	0
$655$	−1.00685e6	−0.0916980
$656$	0	0
$657$	1.40132e6	0.126656
$658$	0	0
$659$	−38652.4	−0.00346707	−0.00173353	−	0.999998i	$-0.500552\pi$
$659$	−38652.4	−0.00346707	−0.00173353	+	0.999998i	$0.500552\pi$
$660$	0	0
$661$	−1.52544e7	−1.35797	−0.678986	−	0.734151i	$-0.737580\pi$
$661$	−1.52544e7	−1.35797	−0.678986	+	0.734151i	$0.737580\pi$
$662$	0	0
$663$	−8.60800e6	−0.760534
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−1.95924e7	−1.70519
$668$	0	0
$669$	−1.33384e6	−0.115223
$670$	0	0
$671$	−919771.	−0.0788631
$672$	0	0
$673$	2.71207e6	0.230815	0.115407	−	0.993318i	$-0.463183\pi$
$673$	2.71207e6	0.230815	0.115407	+	0.993318i	$0.463183\pi$
$674$	0	0
$675$	4.69587e6	0.396695
$676$	0	0
$677$	−1.27201e7	−1.06664	−0.533319	−	0.845914i	$-0.679056\pi$
$677$	−1.27201e7	−1.06664	−0.533319	+	0.845914i	$0.679056\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−6.93886e6	−0.573351
$682$	0	0
$683$	4.90908e6	0.402669	0.201335	−	0.979523i	$-0.435472\pi$
$683$	4.90908e6	0.402669	0.201335	+	0.979523i	$0.435472\pi$
$684$	0	0
$685$	4.17401e6	0.339881
$686$	0	0
$687$	−1.46715e7	−1.18600
$688$	0	0
$689$	−5.69741e6	−0.457224
$690$	0	0
$691$	−7.59865e6	−0.605399	−0.302699	−	0.953086i	$-0.597888\pi$
$691$	−7.59865e6	−0.605399	−0.302699	+	0.953086i	$0.597888\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	4.31045e6	0.338502
$696$	0	0
$697$	3.51709e6	0.274222
$698$	0	0
$699$	1.71244e7	1.32563
$700$	0	0
$701$	−3.72699e6	−0.286460	−0.143230	−	0.989689i	$-0.545749\pi$
$701$	−3.72699e6	−0.286460	−0.143230	+	0.989689i	$0.545749\pi$
$702$	0	0
$703$	−602515.	−0.0459811
$704$	0	0
$705$	−1.14271e7	−0.865889
$706$	0	0
$707$	0	0
$708$	0	0
$709$	94232.7	0.00704022	0.00352011	−	0.999994i	$-0.498880\pi$
$709$	94232.7	0.00704022	0.00352011	+	0.999994i	$0.498880\pi$
$710$	0	0
$711$	−2.59121e6	−0.192233
$712$	0	0
$713$	−2.33458e7	−1.71983
$714$	0	0
$715$	814037.	0.0595497
$716$	0	0
$717$	−6.43550e6	−0.467503
$718$	0	0
$719$	1.52391e7	1.09935	0.549675	−	0.835379i	$-0.314752\pi$
$719$	1.52391e7	1.09935	0.549675	+	0.835379i	$0.314752\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	−2.00959e7	−1.42976
$724$	0	0
$725$	−2.35181e7	−1.66172
$726$	0	0
$727$	3.09483e6	0.217171	0.108585	−	0.994087i	$-0.465368\pi$
$727$	3.09483e6	0.217171	0.108585	+	0.994087i	$0.465368\pi$
$728$	0	0
$729$	−2.12820e7	−1.48318
$730$	0	0
$731$	−1.20772e7	−0.835939
$732$	0	0
$733$	6.11165e6	0.420144	0.210072	−	0.977686i	$-0.432630\pi$
$733$	6.11165e6	0.420144	0.210072	+	0.977686i	$0.432630\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	1.72568e6	0.117029
$738$	0	0
$739$	3.67820e6	0.247756	0.123878	−	0.992297i	$-0.460467\pi$
$739$	3.67820e6	0.247756	0.123878	+	0.992297i	$0.460467\pi$
$740$	0	0
$741$	−2.08457e6	−0.139467
$742$	0	0
$743$	2.68224e7	1.78249	0.891243	−	0.453526i	$-0.149834\pi$
$743$	2.68224e7	1.78249	0.891243	+	0.453526i	$0.149834\pi$
$744$	0	0
$745$	−7.68801e6	−0.507485
$746$	0	0
$747$	1.72426e7	1.13058
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−1.57747e7	−1.02061	−0.510306	−	0.859993i	$-0.670468\pi$
$751$	−1.57747e7	−1.02061	−0.510306	+	0.859993i	$0.670468\pi$
$752$	0	0
$753$	−3.83697e7	−2.46604
$754$	0	0
$755$	8.23710e6	0.525905
$756$	0	0
$757$	−2.33178e7	−1.47893	−0.739466	−	0.673194i	$-0.764922\pi$
$757$	−2.33178e7	−1.47893	−0.739466	+	0.673194i	$0.764922\pi$
$758$	0	0
$759$	−3.15117e6	−0.198549
$760$	0	0
$761$	5.84571e6	0.365911	0.182956	−	0.983121i	$-0.441434\pi$
$761$	5.84571e6	0.365911	0.182956	+	0.983121i	$0.441434\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	−3.33617e6	−0.206108
$766$	0	0
$767$	−2.49755e7	−1.53294
$768$	0	0
$769$	1.79261e7	1.09313	0.546563	−	0.837418i	$-0.315936\pi$
$769$	1.79261e7	1.09313	0.546563	+	0.837418i	$0.315936\pi$
$770$	0	0
$771$	3.07052e7	1.86027
$772$	0	0
$773$	2.15535e7	1.29739	0.648694	−	0.761049i	$-0.275315\pi$
$773$	2.15535e7	1.29739	0.648694	+	0.761049i	$0.275315\pi$
$774$	0	0
$775$	−2.80237e7	−1.67599
$776$	0	0
$777$	0	0
$778$	0	0
$779$	851722.	0.0502868
$780$	0	0
$781$	−530248.	−0.0311065
$782$	0	0
$783$	−1.48925e7	−0.868088
$784$	0	0
$785$	6.60924e6	0.382805
$786$	0	0
$787$	2.56889e6	0.147846	0.0739228	−	0.997264i	$-0.476448\pi$
$787$	2.56889e6	0.147846	0.0739228	+	0.997264i	$0.476448\pi$
$788$	0	0
$789$	−3.75466e7	−2.14723
$790$	0	0
$791$	0	0
$792$	0	0
$793$	1.08204e7	0.611027
$794$	0	0
$795$	−3.90647e6	−0.219213
$796$	0	0
$797$	−2.63767e7	−1.47087	−0.735437	−	0.677593i	$-0.763023\pi$
$797$	−2.63767e7	−1.47087	−0.735437	+	0.677593i	$0.763023\pi$
$798$	0	0
$799$	−1.27016e7	−0.703871
$800$	0	0
$801$	−2.74658e7	−1.51255
$802$	0	0
$803$	−260595.	−0.0142619
$804$	0	0
$805$	0	0
$806$	0	0
$807$	2.01099e7	1.08699
$808$	0	0
$809$	2.28856e7	1.22939	0.614696	−	0.788764i	$-0.289279\pi$
$809$	2.28856e7	1.22939	0.614696	+	0.788764i	$0.289279\pi$
$810$	0	0
$811$	1.52953e6	0.0816592	0.0408296	−	0.999166i	$-0.487000\pi$
$811$	1.52953e6	0.0816592	0.0408296	+	0.999166i	$0.487000\pi$
$812$	0	0
$813$	−4.16007e7	−2.20737
$814$	0	0
$815$	−7.65404e6	−0.403642
$816$	0	0
$817$	−2.92470e6	−0.153295
$818$	0	0
$819$	0	0
$820$	0	0
$821$	1.58749e7	0.821963	0.410982	−	0.911644i	$-0.365186\pi$
$821$	1.58749e7	0.821963	0.410982	+	0.911644i	$0.365186\pi$
$822$	0	0
$823$	3.10952e7	1.60027	0.800136	−	0.599819i	$-0.204761\pi$
$823$	3.10952e7	1.60027	0.800136	+	0.599819i	$0.204761\pi$
$824$	0	0
$825$	−3.78258e6	−0.193488
$826$	0	0
$827$	4.81422e6	0.244772	0.122386	−	0.992483i	$-0.460945\pi$
$827$	4.81422e6	0.244772	0.122386	+	0.992483i	$0.460945\pi$
$828$	0	0
$829$	2.54355e7	1.28545	0.642724	−	0.766098i	$-0.277804\pi$
$829$	2.54355e7	1.28545	0.642724	+	0.766098i	$0.277804\pi$
$830$	0	0
$831$	4.12737e7	2.07334
$832$	0	0
$833$	0	0
$834$	0	0
$835$	3.92633e6	0.194881
$836$	0	0
$837$	−1.77456e7	−0.875542
$838$	0	0
$839$	2.74688e7	1.34721	0.673605	−	0.739091i	$-0.264745\pi$
$839$	2.74688e7	1.34721	0.673605	+	0.739091i	$0.264745\pi$
$840$	0	0
$841$	5.40745e7	2.63635
$842$	0	0
$843$	4.89791e7	2.37379
$844$	0	0
$845$	−2.13372e6	−0.102801
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−7.44568e6	−0.354515
$850$	0	0
$851$	1.07149e7	0.507181
$852$	0	0
$853$	5.98613e6	0.281691	0.140846	−	0.990032i	$-0.455018\pi$
$853$	5.98613e6	0.281691	0.140846	+	0.990032i	$0.455018\pi$
$854$	0	0
$855$	−807909.	−0.0377961
$856$	0	0
$857$	−1.30595e7	−0.607397	−0.303699	−	0.952768i	$-0.598222\pi$
$857$	−1.30595e7	−0.607397	−0.303699	+	0.952768i	$0.598222\pi$
$858$	0	0
$859$	−7.51349e6	−0.347423	−0.173712	−	0.984797i	$-0.555576\pi$
$859$	−7.51349e6	−0.347423	−0.173712	+	0.984797i	$0.555576\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	834999.	0.0381644	0.0190822	−	0.999818i	$-0.493926\pi$
$863$	834999.	0.0381644	0.0190822	+	0.999818i	$0.493926\pi$
$864$	0	0
$865$	−1.06640e7	−0.484596
$866$	0	0
$867$	2.70076e7	1.22022
$868$	0	0
$869$	481871.	0.0216462
$870$	0	0
$871$	−2.03013e7	−0.906731
$872$	0	0
$873$	4.98499e7	2.21375
$874$	0	0
$875$	0	0
$876$	0	0
$877$	1.35679e7	0.595679	0.297840	−	0.954616i	$-0.403734\pi$
$877$	1.35679e7	0.595679	0.297840	+	0.954616i	$0.403734\pi$
$878$	0	0
$879$	−4.89428e7	−2.13657
$880$	0	0
$881$	1.02028e7	0.442873	0.221436	−	0.975175i	$-0.428926\pi$
$881$	1.02028e7	0.442873	0.221436	+	0.975175i	$0.428926\pi$
$882$	0	0
$883$	2.57631e7	1.11198	0.555989	−	0.831190i	$-0.312340\pi$
$883$	2.57631e7	1.11198	0.555989	+	0.831190i	$0.312340\pi$
$884$	0	0
$885$	−1.71246e7	−0.734958
$886$	0	0
$887$	−1.71056e7	−0.730012	−0.365006	−	0.931005i	$-0.618933\pi$
$887$	−1.71056e7	−0.730012	−0.365006	+	0.931005i	$0.618933\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	2.11540e6	0.0892683
$892$	0	0
$893$	−3.07591e6	−0.129076
$894$	0	0
$895$	−7.50931e6	−0.313359
$896$	0	0
$897$	3.70711e7	1.53835
$898$	0	0
$899$	8.88746e7	3.66757
$900$	0	0
$901$	−4.34220e6	−0.178196
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−2.97802e6	−0.120866
$906$	0	0
$907$	2.41290e6	0.0973915	0.0486958	−	0.998814i	$-0.484494\pi$
$907$	2.41290e6	0.0973915	0.0486958	+	0.998814i	$0.484494\pi$
$908$	0	0
$909$	2.45743e7	0.986442
$910$	0	0
$911$	−2.36633e7	−0.944668	−0.472334	−	0.881420i	$-0.656588\pi$
$911$	−2.36633e7	−0.944668	−0.472334	+	0.881420i	$0.656588\pi$
$912$	0	0
$913$	−3.20650e6	−0.127308
$914$	0	0
$915$	7.41908e6	0.292953
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−3.62159e6	−0.141452	−0.0707262	−	0.997496i	$-0.522532\pi$
$919$	−3.62159e6	−0.141452	−0.0707262	+	0.997496i	$0.522532\pi$
$920$	0	0
$921$	2.74463e7	1.06619
$922$	0	0
$923$	6.23796e6	0.241012
$924$	0	0
$925$	1.28618e7	0.494252
$926$	0	0
$927$	−3.38074e7	−1.29215
$928$	0	0
$929$	3.21340e7	1.22159	0.610796	−	0.791788i	$-0.290850\pi$
$929$	3.21340e7	1.22159	0.610796	+	0.791788i	$0.290850\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−1.56834e7	−0.589844
$934$	0	0
$935$	620406.	0.0232085
$936$	0	0
$937$	−1.50936e7	−0.561621	−0.280811	−	0.959763i	$-0.590603\pi$
$937$	−1.50936e7	−0.561621	−0.280811	+	0.959763i	$0.590603\pi$
$938$	0	0
$939$	−1.46743e7	−0.543118
$940$	0	0
$941$	3.88634e7	1.43076	0.715379	−	0.698736i	$-0.246254\pi$
$941$	3.88634e7	1.43076	0.715379	+	0.698736i	$0.246254\pi$
$942$	0	0
$943$	−1.51467e7	−0.554674
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−1.94261e7	−0.703898	−0.351949	−	0.936019i	$-0.614481\pi$
$947$	−1.94261e7	−0.703898	−0.351949	+	0.936019i	$0.614481\pi$
$948$	0	0
$949$	3.06569e6	0.110500
$950$	0	0
$951$	−4.31084e7	−1.54565
$952$	0	0
$953$	−3.32921e7	−1.18743	−0.593717	−	0.804674i	$-0.702340\pi$
$953$	−3.32921e7	−1.18743	−0.593717	+	0.804674i	$0.702340\pi$
$954$	0	0
$955$	−1.23659e7	−0.438752
$956$	0	0
$957$	1.19961e7	0.423410
$958$	0	0
$959$	0	0
$960$	0	0
$961$	7.72719e7	2.69906
$962$	0	0
$963$	5.06755e7	1.76089
$964$	0	0
$965$	−1.40221e6	−0.0484724
$966$	0	0
$967$	−1.99707e7	−0.686796	−0.343398	−	0.939190i	$-0.611578\pi$
$967$	−1.99707e7	−0.686796	−0.343398	+	0.939190i	$0.611578\pi$
$968$	0	0
$969$	−1.58872e6	−0.0543549
$970$	0	0
$971$	4.92737e7	1.67713	0.838567	−	0.544799i	$-0.183394\pi$
$971$	4.92737e7	1.67713	0.838567	+	0.544799i	$0.183394\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	4.44991e7	1.49913
$976$	0	0
$977$	4.75393e7	1.59337	0.796684	−	0.604396i	$-0.206586\pi$
$977$	4.75393e7	1.59337	0.796684	+	0.604396i	$0.206586\pi$
$978$	0	0
$979$	5.10764e6	0.170319
$980$	0	0
$981$	2.53309e7	0.840385
$982$	0	0
$983$	−1.82946e7	−0.603863	−0.301932	−	0.953330i	$-0.597631\pi$
$983$	−1.82946e7	−0.603863	−0.301932	+	0.953330i	$0.597631\pi$
$984$	0	0
$985$	−8.82308e6	−0.289754
$986$	0	0
$987$	0	0
$988$	0	0
$989$	5.20117e7	1.69087
$990$	0	0
$991$	−5.61760e6	−0.181705	−0.0908524	−	0.995864i	$-0.528959\pi$
$991$	−5.61760e6	−0.181705	−0.0908524	+	0.995864i	$0.528959\pi$
$992$	0	0
$993$	3.17068e7	1.02042
$994$	0	0
$995$	−7.02416e6	−0.224925
$996$	0	0
$997$	1.97151e7	0.628146	0.314073	−	0.949399i	$-0.398306\pi$
$997$	1.97151e7	0.628146	0.314073	+	0.949399i	$0.398306\pi$
$998$	0	0
$999$	8.14457e6	0.258199

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	784.6.a.bl.1.1		5
4.3	odd	2		392.6.a.j.1.5		5
7.3	odd	6		112.6.i.f.65.1		10
7.5	odd	6		112.6.i.f.81.1		10
7.6	odd	2		784.6.a.bk.1.5		5
28.3	even	6		56.6.i.b.9.5	✓	10
28.11	odd	6		392.6.i.o.177.1		10
28.19	even	6		56.6.i.b.25.5	yes	10
28.23	odd	6		392.6.i.o.361.1		10
28.27	even	2		392.6.a.k.1.1		5
84.47	odd	6		504.6.s.b.361.3		10
84.59	odd	6		504.6.s.b.289.3		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
56.6.i.b.9.5	✓	10	28.3	even	6
56.6.i.b.25.5	yes	10	28.19	even	6
112.6.i.f.65.1		10	7.3	odd	6
112.6.i.f.81.1		10	7.5	odd	6
392.6.a.j.1.5		5	4.3	odd	2
392.6.a.k.1.1		5	28.27	even	2
392.6.i.o.177.1		10	28.11	odd	6
392.6.i.o.361.1		10	28.23	odd	6
504.6.s.b.289.3		10	84.59	odd	6
504.6.s.b.361.3		10	84.47	odd	6
784.6.a.bk.1.5		5	7.6	odd	2
784.6.a.bl.1.1		5	1.1	even	1	trivial

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)

\(f(q)\)	\(=\)	\(q-23.6419 q^{3} -20.0456 q^{5} +315.939 q^{9} -58.7531 q^{11} +691.185 q^{13} +473.916 q^{15} +526.776 q^{17} +127.568 q^{19} -2268.61 q^{23} -2723.17 q^{25} -1724.41 q^{27} +8636.30 q^{29} +10290.8 q^{31} +1389.03 q^{33} -4723.11 q^{37} -16340.9 q^{39} +6676.64 q^{41} -22926.7 q^{43} -6333.18 q^{45} -24112.0 q^{47} -12454.0 q^{51} -8242.96 q^{53} +1177.74 q^{55} -3015.94 q^{57} -36134.3 q^{59} +15654.8 q^{61} -13855.2 q^{65} -29371.7 q^{67} +53634.1 q^{69} +9025.02 q^{71} +4435.42 q^{73} +64381.0 q^{75} -8201.62 q^{79} -36004.8 q^{81} +54575.9 q^{83} -10559.5 q^{85} -204178. q^{87} -86933.9 q^{89} -243294. q^{93} -2557.17 q^{95} +157783. q^{97} -18562.4 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 5 q^{3} + 81 q^{5} + 390 q^{9} - 361 q^{11} + 342 q^{13} + 1049 q^{15} + 1809 q^{17} - 1277 q^{19} - 911 q^{23} + 3940 q^{25} + 4751 q^{27} + 5442 q^{29} - 2187 q^{31} - 5553 q^{33} - 8181 q^{37}+ \cdots - 249798 q^{99}+O(q^{100}) \) 5 * q + 5 * q^3 + 81 * q^5 + 390 * q^9 - 361 * q^11 + 342 * q^13 + 1049 * q^15 + 1809 * q^17 - 1277 * q^19 - 911 * q^23 + 3940 * q^25 + 4751 * q^27 + 5442 * q^29 - 2187 * q^31 - 5553 * q^33 - 8181 * q^37 + 3422 * q^39 + 16578 * q^41 - 6332 * q^43 + 41310 * q^45 - 16101 * q^47 + 67865 * q^51 + 16047 * q^53 - 45629 * q^55 + 22347 * q^57 - 71027 * q^59 + 31093 * q^61 - 64370 * q^65 - 47981 * q^67 + 137249 * q^69 - 22512 * q^71 + 123333 * q^73 + 45460 * q^75 + 212481 * q^79 + 52917 * q^81 - 87460 * q^83 + 222141 * q^85 - 318070 * q^87 + 129045 * q^89 - 252835 * q^93 - 300417 * q^95 + 328274 * q^97 - 249798 * q^99

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