LMFDB - Embedded newform 784.6.f.e.783.14

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [784,6,Mod(783,784)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(784, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0, 1])) N = Newforms(chi, 6, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("784.783"); S:= CuspForms(chi, 6); N := Newforms(S);

Level:	$ N $	$=$	$ 784 = 2^{4} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	784.f (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [16,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

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

Self dual:	no
Analytic conductor:	$125.740914733$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 232 x^{14} + 23092 x^{12} - 1258560 x^{10} + 40996255 x^{8} - 775625040 x^{6} + \cdots + 20552376321 $ x^16 - 232x^14 + 23092x^12 - 1258560x^10 + 40996255x^8 - 775625040x^6 + 7073588484x^4 - 15906881088*x^2 + 20552376321
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{34}\cdot 7^{18} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			783.14
Root			$1.27628 + 0.597902i$ of defining polynomial
Character	$\chi$	$=$	784.783
Dual form			784.6.f.e.783.13

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+22.3313 q^{3} +90.0418i q^{5} +255.687 q^{9} +729.115i q^{11} -874.475i q^{13} +2010.75i q^{15} -1440.53i q^{17} -2448.40 q^{19} +629.236i q^{23} -4982.52 q^{25} +283.307 q^{27} -5885.40 q^{29} -6148.39 q^{31} +16282.1i q^{33} +2888.22 q^{37} -19528.2i q^{39} +4102.36i q^{41} +2070.55i q^{43} +23022.5i q^{45} +13580.5 q^{47} -32168.9i q^{51} -12085.8 q^{53} -65650.8 q^{55} -54675.9 q^{57} +34784.0 q^{59} +12060.1i q^{61} +78739.3 q^{65} -18133.0i q^{67} +14051.7i q^{69} +67372.9i q^{71} -6284.55i q^{73} -111266. q^{75} -15185.8i q^{79} -55805.2 q^{81} -90365.2 q^{83} +129708. q^{85} -131428. q^{87} -101934. i q^{89} -137302. q^{93} -220458. i q^{95} -91420.7i q^{97} +186425. i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 1712 q^{9} - 496 q^{25} - 17024 q^{29} + 85664 q^{53} - 95424 q^{57} + 233344 q^{65} - 511216 q^{81} + 697952 q^{85} - 857024 q^{93}+O(q^{100}) $ 16 * q + 1712 * q^9 - 496 * q^25 - 17024 * q^29 + 85664 * q^53 - 95424 * q^57 + 233344 * q^65 - 511216 * q^81 + 697952 * q^85 - 857024 * q^93

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/784\mathbb{Z}\right)^\times$.

$n$	$197$	$687$	$689$
$\chi(n)$	$1$	$-1$	$-1$

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	22.3313	1.43255	0.716276	−	0.697817i	$-0.245845\pi$
$3$	22.3313	1.43255	0.716276	+	0.697817i	$0.245845\pi$
$4$	0	0
$5$	90.0418i	1.61072i	0.592789	+	0.805358i	$0.298027\pi$
$5$	90.0418i	1.61072i	−0.592789	+	0.805358i	$0.701973\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	255.687	1.05221
$10$	0	0
$11$	729.115i	1.81683i	0.418069	+	0.908415i	$0.362707\pi$
$11$	729.115i	1.81683i	−0.418069	+	0.908415i	$0.637293\pi$
$12$	0	0
$13$	− 874.475i	− 1.43512i	−0.696495	−	0.717562i	$-0.745258\pi$
$13$	− 874.475i	− 1.43512i	0.696495	−	0.717562i	$-0.254742\pi$
$14$	0	0
$15$	2010.75i	2.30744i
$16$	0	0
$17$	− 1440.53i	− 1.20893i	−0.796633	−	0.604463i	$-0.793388\pi$
$17$	− 1440.53i	− 1.20893i	0.796633	−	0.604463i	$-0.206612\pi$
$18$	0	0
$19$	−2448.40	−1.55596	−0.777979	−	0.628290i	$-0.783755\pi$
$19$	−2448.40	−1.55596	−0.777979	+	0.628290i	$0.783755\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	629.236i	0.248024i	0.992281	+	0.124012i	$0.0395762\pi$
$23$	629.236i	0.248024i	−0.992281	+	0.124012i	$0.960424\pi$
$24$	0	0
$25$	−4982.52	−1.59441
$26$	0	0
$27$	283.307	0.0747907
$28$	0	0
$29$	−5885.40	−1.29951	−0.649757	−	0.760142i	$-0.725129\pi$
$29$	−5885.40	−1.29951	−0.649757	+	0.760142i	$0.725129\pi$
$30$	0	0
$31$	−6148.39	−1.14910	−0.574549	−	0.818470i	$-0.694823\pi$
$31$	−6148.39	−1.14910	−0.574549	+	0.818470i	$0.694823\pi$
$32$	0	0
$33$	16282.1i	2.60271i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	2888.22	0.346837	0.173419	−	0.984848i	$-0.444519\pi$
$37$	2888.22	0.346837	0.173419	+	0.984848i	$0.444519\pi$
$38$	0	0
$39$	− 19528.2i	− 2.05589i
$40$	0	0
$41$	4102.36i	0.381131i	0.981674	+	0.190565i	$0.0610321\pi$
$41$	4102.36i	0.381131i	−0.981674	+	0.190565i	$0.938968\pi$
$42$	0	0
$43$	2070.55i	0.170771i	0.996348	+	0.0853854i	$0.0272122\pi$
$43$	2070.55i	0.170771i	−0.996348	+	0.0853854i	$0.972788\pi$
$44$	0	0
$45$	23022.5i	1.69481i
$46$	0	0
$47$	13580.5	0.896748	0.448374	−	0.893846i	$-0.352003\pi$
$47$	13580.5	0.896748	0.448374	+	0.893846i	$0.352003\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	− 32168.9i	− 1.73185i
$52$	0	0
$53$	−12085.8	−0.590996	−0.295498	−	0.955343i	$-0.595486\pi$
$53$	−12085.8	−0.590996	−0.295498	+	0.955343i	$0.595486\pi$
$54$	0	0
$55$	−65650.8	−2.92640
$56$	0	0
$57$	−54675.9	−2.22899
$58$	0	0
$59$	34784.0	1.30092	0.650458	−	0.759542i	$-0.274577\pi$
$59$	34784.0	1.30092	0.650458	+	0.759542i	$0.274577\pi$
$60$	0	0
$61$	12060.1i	0.414980i	0.978237	+	0.207490i	$0.0665295\pi$
$61$	12060.1i	0.414980i	−0.978237	+	0.207490i	$0.933470\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	78739.3	2.31158
$66$	0	0
$67$	− 18133.0i	− 0.493496i	−0.969080	−	0.246748i	$-0.920638\pi$
$67$	− 18133.0i	− 0.493496i	0.969080	−	0.246748i	$-0.0793620\pi$
$68$	0	0
$69$	14051.7i	0.355308i
$70$	0	0
$71$	67372.9i	1.58613i	0.609136	+	0.793066i	$0.291516\pi$
$71$	67372.9i	1.58613i	−0.609136	+	0.793066i	$0.708484\pi$
$72$	0	0
$73$	− 6284.55i	− 0.138028i	−0.997616	−	0.0690139i	$-0.978015\pi$
$73$	− 6284.55i	− 0.138028i	0.997616	−	0.0690139i	$-0.0219853\pi$
$74$	0	0
$75$	−111266.	−2.28407
$76$	0	0
$77$	0	0
$78$	0	0
$79$	− 15185.8i	− 0.273761i	−0.990588	−	0.136880i	$-0.956292\pi$
$79$	− 15185.8i	− 0.273761i	0.990588	−	0.136880i	$-0.0437076\pi$
$80$	0	0
$81$	−55805.2	−0.945066
$82$	0	0
$83$	−90365.2	−1.43981	−0.719906	−	0.694071i	$-0.755815\pi$
$83$	−90365.2	−1.43981	−0.719906	+	0.694071i	$0.755815\pi$
$84$	0	0
$85$	129708.	1.94724
$86$	0	0
$87$	−131428.	−1.86162
$88$	0	0
$89$	− 101934.i	− 1.36410i	−0.731307	−	0.682049i	$-0.761089\pi$
$89$	− 101934.i	− 1.36410i	0.731307	−	0.682049i	$-0.238911\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−137302.	−1.64614
$94$	0	0
$95$	− 220458.i	− 2.50621i
$96$	0	0
$97$	− 91420.7i	− 0.986541i	−0.869876	−	0.493270i	$-0.835801\pi$
$97$	− 91420.7i	− 0.986541i	0.869876	−	0.493270i	$-0.164199\pi$
$98$	0	0
$99$	186425.i	1.91168i
$100$	0	0
$101$	28916.9i	0.282065i	0.990005	+	0.141032i	$0.0450422\pi$
$101$	28916.9i	0.282065i	−0.990005	+	0.141032i	$0.954958\pi$
$102$	0	0
$103$	94969.3	0.882044	0.441022	−	0.897496i	$-0.354616\pi$
$103$	94969.3	0.882044	0.441022	+	0.897496i	$0.354616\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 125423.i	− 1.05905i	−0.848295	−	0.529525i	$-0.822370\pi$
$107$	− 125423.i	− 1.05905i	0.848295	−	0.529525i	$-0.177630\pi$
$108$	0	0
$109$	−9636.18	−0.0776852	−0.0388426	−	0.999245i	$-0.512367\pi$
$109$	−9636.18	−0.0776852	−0.0388426	+	0.999245i	$0.512367\pi$
$110$	0	0
$111$	64497.6	0.496863
$112$	0	0
$113$	−127124.	−0.936555	−0.468277	−	0.883582i	$-0.655125\pi$
$113$	−127124.	−0.936555	−0.468277	+	0.883582i	$0.655125\pi$
$114$	0	0
$115$	−56657.5	−0.399496
$116$	0	0
$117$	− 223592.i	− 1.51005i
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−370558.	−2.30087
$122$	0	0
$123$	91611.0i	0.545990i
$124$	0	0
$125$	− 167254.i	− 0.957420i
$126$	0	0
$127$	− 131727.i	− 0.724712i	−0.932040	−	0.362356i	$-0.881972\pi$
$127$	− 131727.i	− 0.724712i	0.932040	−	0.362356i	$-0.118028\pi$
$128$	0	0
$129$	46238.0i	0.244638i
$130$	0	0
$131$	−7301.55	−0.0371738	−0.0185869	−	0.999827i	$-0.505917\pi$
$131$	−7301.55	−0.0371738	−0.0185869	+	0.999827i	$0.505917\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	25509.4i	0.120467i
$136$	0	0
$137$	171441.	0.780394	0.390197	−	0.920731i	$-0.372407\pi$
$137$	171441.	0.780394	0.390197	+	0.920731i	$0.372407\pi$
$138$	0	0
$139$	−434869.	−1.90907	−0.954535	−	0.298100i	$-0.903647\pi$
$139$	−434869.	−1.90907	−0.954535	+	0.298100i	$0.903647\pi$
$140$	0	0
$141$	303270.	1.28464
$142$	0	0
$143$	637593.	2.60738
$144$	0	0
$145$	− 529931.i	− 2.09315i
$146$	0	0
$147$	0	0
$148$	0	0
$149$	150774.	0.556365	0.278183	−	0.960528i	$-0.410268\pi$
$149$	150774.	0.556365	0.278183	+	0.960528i	$0.410268\pi$
$150$	0	0
$151$	261301.i	0.932607i	0.884625	+	0.466304i	$0.154415\pi$
$151$	261301.i	0.932607i	−0.884625	+	0.466304i	$0.845585\pi$
$152$	0	0
$153$	− 368324.i	− 1.27204i
$154$	0	0
$155$	− 553612.i	− 1.85087i
$156$	0	0
$157$	442817.i	1.43376i	0.697199	+	0.716878i	$0.254429\pi$
$157$	442817.i	1.43376i	−0.697199	+	0.716878i	$0.745571\pi$
$158$	0	0
$159$	−269891.	−0.846633
$160$	0	0
$161$	0	0
$162$	0	0
$163$	365529.i	1.07759i	0.842438	+	0.538794i	$0.181120\pi$
$163$	365529.i	1.07759i	−0.842438	+	0.538794i	$0.818880\pi$
$164$	0	0
$165$	−1.46607e6	−4.19222
$166$	0	0
$167$	706892.	1.96138	0.980690	−	0.195569i	$-0.0626555\pi$
$167$	706892.	1.96138	0.980690	+	0.195569i	$0.0626555\pi$
$168$	0	0
$169$	−393414.	−1.05958
$170$	0	0
$171$	−626022.	−1.63719
$172$	0	0
$173$	24407.4i	0.0620020i	0.999519	+	0.0310010i	$0.00986950\pi$
$173$	24407.4i	0.0620020i	−0.999519	+	0.0310010i	$0.990130\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	776771.	1.86363
$178$	0	0
$179$	403220.i	0.940609i	0.882504	+	0.470305i	$0.155856\pi$
$179$	403220.i	0.940609i	−0.882504	+	0.470305i	$0.844144\pi$
$180$	0	0
$181$	804214.i	1.82463i	0.409486	+	0.912316i	$0.365708\pi$
$181$	804214.i	1.82463i	−0.409486	+	0.912316i	$0.634292\pi$
$182$	0	0
$183$	269318.i	0.594482i
$184$	0	0
$185$	260060.i	0.558656i
$186$	0	0
$187$	1.05031e6	2.19641
$188$	0	0
$189$	0	0
$190$	0	0
$191$	324997.i	0.644609i	0.946636	+	0.322305i	$0.104458\pi$
$191$	324997.i	0.644609i	−0.946636	+	0.322305i	$0.895542\pi$
$192$	0	0
$193$	−229824.	−0.444122	−0.222061	−	0.975033i	$-0.571278\pi$
$193$	−229824.	−0.444122	−0.222061	+	0.975033i	$0.571278\pi$
$194$	0	0
$195$	1.75835e6	3.31146
$196$	0	0
$197$	−862809.	−1.58398	−0.791989	−	0.610535i	$-0.790954\pi$
$197$	−862809.	−1.58398	−0.791989	+	0.610535i	$0.790954\pi$
$198$	0	0
$199$	−41015.0	−0.0734193	−0.0367097	−	0.999326i	$-0.511688\pi$
$199$	−41015.0	−0.0734193	−0.0367097	+	0.999326i	$0.511688\pi$
$200$	0	0
$201$	− 404934.i	− 0.706959i
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−369384.	−0.613894
$206$	0	0
$207$	160887.i	0.260973i
$208$	0	0
$209$	− 1.78516e6i	− 2.82691i
$210$	0	0
$211$	− 478410.i	− 0.739765i	−0.929079	−	0.369882i	$-0.879398\pi$
$211$	− 478410.i	− 0.739765i	0.929079	−	0.369882i	$-0.120602\pi$
$212$	0	0
$213$	1.50452e6i	2.27222i
$214$	0	0
$215$	−186436.	−0.275063
$216$	0	0
$217$	0	0
$218$	0	0
$219$	− 140342.i	− 0.197732i
$220$	0	0
$221$	−1.25971e6	−1.73496
$222$	0	0
$223$	−152367.	−0.205177	−0.102588	−	0.994724i	$-0.532712\pi$
$223$	−152367.	−0.205177	−0.102588	+	0.994724i	$0.532712\pi$
$224$	0	0
$225$	−1.27396e6	−1.67765
$226$	0	0
$227$	−458558.	−0.590649	−0.295324	−	0.955397i	$-0.595428\pi$
$227$	−458558.	−0.590649	−0.295324	+	0.955397i	$0.595428\pi$
$228$	0	0
$229$	534765.i	0.673868i	0.941528	+	0.336934i	$0.109390\pi$
$229$	534765.i	0.673868i	−0.941528	+	0.336934i	$0.890610\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	6568.36	0.00792624	0.00396312	−	0.999992i	$-0.498738\pi$
$233$	6568.36	0.00792624	0.00396312	+	0.999992i	$0.498738\pi$
$234$	0	0
$235$	1.22281e6i	1.44441i
$236$	0	0
$237$	− 339119.i	− 0.392177i
$238$	0	0
$239$	− 186056.i	− 0.210692i	−0.994436	−	0.105346i	$-0.966405\pi$
$239$	− 186056.i	− 0.210692i	0.994436	−	0.105346i	$-0.0335951\pi$
$240$	0	0
$241$	1.52949e6i	1.69631i	0.529750	+	0.848154i	$0.322286\pi$
$241$	1.52949e6i	1.69631i	−0.529750	+	0.848154i	$0.677714\pi$
$242$	0	0
$243$	−1.31505e6	−1.42865
$244$	0	0
$245$	0	0
$246$	0	0
$247$	2.14106e6i	2.23299i
$248$	0	0
$249$	−2.01797e6	−2.06261
$250$	0	0
$251$	−803905.	−0.805416	−0.402708	−	0.915328i	$-0.631931\pi$
$251$	−803905.	−0.805416	−0.402708	+	0.915328i	$0.631931\pi$
$252$	0	0
$253$	−458785.	−0.450618
$254$	0	0
$255$	2.89654e6	2.78952
$256$	0	0
$257$	− 1.03205e6i	− 0.974695i	−0.873208	−	0.487347i	$-0.837965\pi$
$257$	− 1.03205e6i	− 0.974695i	0.873208	−	0.487347i	$-0.162035\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−1.50482e6	−1.36736
$262$	0	0
$263$	78896.0i	0.0703340i	0.999381	+	0.0351670i	$0.0111963\pi$
$263$	78896.0i	0.0703340i	−0.999381	+	0.0351670i	$0.988804\pi$
$264$	0	0
$265$	− 1.08822e6i	− 0.951926i
$266$	0	0
$267$	− 2.27632e6i	− 1.95414i
$268$	0	0
$269$	− 1.62611e6i	− 1.37015i	−0.728471	−	0.685077i	$-0.759769\pi$
$269$	− 1.62611e6i	− 1.37015i	0.728471	−	0.685077i	$-0.240231\pi$
$270$	0	0
$271$	1.48721e6	1.23013	0.615064	−	0.788477i	$-0.289130\pi$
$271$	1.48721e6	1.23013	0.615064	+	0.788477i	$0.289130\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	− 3.63283e6i	− 2.89677i
$276$	0	0
$277$	−664834.	−0.520612	−0.260306	−	0.965526i	$-0.583823\pi$
$277$	−664834.	−0.520612	−0.260306	+	0.965526i	$0.583823\pi$
$278$	0	0
$279$	−1.57206e6	−1.20909
$280$	0	0
$281$	2.15082e6	1.62494	0.812472	−	0.583000i	$-0.198121\pi$
$281$	2.15082e6	1.62494	0.812472	+	0.583000i	$0.198121\pi$
$282$	0	0
$283$	−1.24079e6	−0.920944	−0.460472	−	0.887674i	$-0.652320\pi$
$283$	−1.24079e6	−0.920944	−0.460472	+	0.887674i	$0.652320\pi$
$284$	0	0
$285$	− 4.92311e6i	− 3.59027i
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−655268.	−0.461503
$290$	0	0
$291$	− 2.04154e6i	− 1.41327i
$292$	0	0
$293$	410751.i	0.279518i	0.990186	+	0.139759i	$0.0446328\pi$
$293$	410751.i	0.279518i	−0.990186	+	0.139759i	$0.955367\pi$
$294$	0	0
$295$	3.13201e6i	2.09541i
$296$	0	0
$297$	206563.i	0.135882i
$298$	0	0
$299$	550251.	0.355945
$300$	0	0
$301$	0	0
$302$	0	0
$303$	645753.i	0.404073i
$304$	0	0
$305$	−1.08592e6	−0.668416
$306$	0	0
$307$	−394121.	−0.238662	−0.119331	−	0.992855i	$-0.538075\pi$
$307$	−394121.	−0.238662	−0.119331	+	0.992855i	$0.538075\pi$
$308$	0	0
$309$	2.12079e6	1.26357
$310$	0	0
$311$	−2.03583e6	−1.19355	−0.596774	−	0.802410i	$-0.703551\pi$
$311$	−2.03583e6	−1.19355	−0.596774	+	0.802410i	$0.703551\pi$
$312$	0	0
$313$	− 1.03944e6i	− 0.599709i	−0.953985	−	0.299854i	$-0.903062\pi$
$313$	− 1.03944e6i	− 0.599709i	0.953985	−	0.299854i	$-0.0969381\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	1.29032e6	0.721191	0.360595	−	0.932722i	$-0.382573\pi$
$317$	1.29032e6	0.721191	0.360595	+	0.932722i	$0.382573\pi$
$318$	0	0
$319$	− 4.29113e6i	− 2.36099i
$320$	0	0
$321$	− 2.80085e6i	− 1.51714i
$322$	0	0
$323$	3.52699e6i	1.88104i
$324$	0	0
$325$	4.35709e6i	2.28817i
$326$	0	0
$327$	−215188.	−0.111288
$328$	0	0
$329$	0	0
$330$	0	0
$331$	− 2.06363e6i	− 1.03529i	−0.855595	−	0.517645i	$-0.826809\pi$
$331$	− 2.06363e6i	− 1.03529i	0.855595	−	0.517645i	$-0.173191\pi$
$332$	0	0
$333$	738478.	0.364945
$334$	0	0
$335$	1.63273e6	0.794882
$336$	0	0
$337$	−2.94918e6	−1.41458	−0.707289	−	0.706924i	$-0.750082\pi$
$337$	−2.94918e6	−1.41458	−0.707289	+	0.706924i	$0.750082\pi$
$338$	0	0
$339$	−2.83885e6	−1.34166
$340$	0	0
$341$	− 4.48289e6i	− 2.08772i
$342$	0	0
$343$	0	0
$344$	0	0
$345$	−1.26524e6	−0.572300
$346$	0	0
$347$	2.25144e6i	1.00377i	0.864933	+	0.501887i	$0.167361\pi$
$347$	2.25144e6i	1.00377i	−0.864933	+	0.501887i	$0.832639\pi$
$348$	0	0
$349$	− 307448.i	− 0.135116i	−0.997715	−	0.0675582i	$-0.978479\pi$
$349$	− 307448.i	− 0.135116i	0.997715	−	0.0675582i	$-0.0215208\pi$
$350$	0	0
$351$	− 247745.i	− 0.107334i
$352$	0	0
$353$	33430.2i	0.0142792i	0.999975	+	0.00713958i	$0.00227262\pi$
$353$	33430.2i	0.0142792i	−0.999975	+	0.00713958i	$0.997727\pi$
$354$	0	0
$355$	−6.06637e6	−2.55481
$356$	0	0
$357$	0	0
$358$	0	0
$359$	1.49877e6i	0.613761i	0.951748	+	0.306881i	$0.0992852\pi$
$359$	1.49877e6i	0.613761i	−0.951748	+	0.306881i	$0.900715\pi$
$360$	0	0
$361$	3.51855e6	1.42101
$362$	0	0
$363$	−8.27503e6	−3.29612
$364$	0	0
$365$	565872.	0.222324
$366$	0	0
$367$	2.64339e6	1.02446	0.512232	−	0.858847i	$-0.328819\pi$
$367$	2.64339e6	1.02446	0.512232	+	0.858847i	$0.328819\pi$
$368$	0	0
$369$	1.04892e6i	0.401029i
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−3.24492e6	−1.20762	−0.603812	−	0.797126i	$-0.706352\pi$
$373$	−3.24492e6	−1.20762	−0.603812	+	0.797126i	$0.706352\pi$
$374$	0	0
$375$	− 3.73501e6i	− 1.37155i
$376$	0	0
$377$	5.14663e6i	1.86496i
$378$	0	0
$379$	2.83313e6i	1.01314i	0.862199	+	0.506570i	$0.169087\pi$
$379$	2.83313e6i	1.01314i	−0.862199	+	0.506570i	$0.830913\pi$
$380$	0	0
$381$	− 2.94163e6i	− 1.03819i
$382$	0	0
$383$	−1.06238e6	−0.370068	−0.185034	−	0.982732i	$-0.559240\pi$
$383$	−1.06238e6	−0.370068	−0.185034	+	0.982732i	$0.559240\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	529411.i	0.179686i
$388$	0	0
$389$	−4.09561e6	−1.37228	−0.686142	−	0.727468i	$-0.740697\pi$
$389$	−4.09561e6	−1.37228	−0.686142	+	0.727468i	$0.740697\pi$
$390$	0	0
$391$	906433.	0.299843
$392$	0	0
$393$	−163053.	−0.0532534
$394$	0	0
$395$	1.36736e6	0.440951
$396$	0	0
$397$	5.91148e6i	1.88244i	0.337799	+	0.941218i	$0.390318\pi$
$397$	5.91148e6i	1.88244i	−0.337799	+	0.941218i	$0.609682\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−1.14124e6	−0.354419	−0.177209	−	0.984173i	$-0.556707\pi$
$401$	−1.14124e6	−0.354419	−0.177209	+	0.984173i	$0.556707\pi$
$402$	0	0
$403$	5.37662e6i	1.64910i
$404$	0	0
$405$	− 5.02480e6i	− 1.52223i
$406$	0	0
$407$	2.10584e6i	0.630144i
$408$	0	0
$409$	3.45697e6i	1.02185i	0.859625	+	0.510925i	$0.170697\pi$
$409$	3.45697e6i	1.02185i	−0.859625	+	0.510925i	$0.829303\pi$
$410$	0	0
$411$	3.82850e6	1.11796
$412$	0	0
$413$	0	0
$414$	0	0
$415$	− 8.13664e6i	− 2.31913i
$416$	0	0
$417$	−9.71119e6	−2.73484
$418$	0	0
$419$	−2.18354e6	−0.607611	−0.303805	−	0.952734i	$-0.598257\pi$
$419$	−2.18354e6	−0.607611	−0.303805	+	0.952734i	$0.598257\pi$
$420$	0	0
$421$	−5.95861e6	−1.63848	−0.819238	−	0.573454i	$-0.805603\pi$
$421$	−5.95861e6	−1.63848	−0.819238	+	0.573454i	$0.805603\pi$
$422$	0	0
$423$	3.47235e6	0.943565
$424$	0	0
$425$	7.17747e6i	1.92752i
$426$	0	0
$427$	0	0
$428$	0	0
$429$	1.42383e7	3.73520
$430$	0	0
$431$	− 5.01453e6i	− 1.30028i	−0.759814	−	0.650140i	$-0.774710\pi$
$431$	− 5.01453e6i	− 1.30028i	0.759814	−	0.650140i	$-0.225290\pi$
$432$	0	0
$433$	647482.i	0.165962i	0.996551	+	0.0829809i	$0.0264440\pi$
$433$	647482.i	0.165962i	−0.996551	+	0.0829809i	$0.973556\pi$
$434$	0	0
$435$	− 1.18341e7i	− 2.99854i
$436$	0	0
$437$	− 1.54062e6i	− 0.385915i
$438$	0	0
$439$	7.05251e6	1.74655	0.873277	−	0.487223i	$-0.161990\pi$
$439$	7.05251e6	1.74655	0.873277	+	0.487223i	$0.161990\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	3.41951e6i	0.827856i	0.910310	+	0.413928i	$0.135843\pi$
$443$	3.41951e6i	0.827856i	−0.910310	+	0.413928i	$0.864157\pi$
$444$	0	0
$445$	9.17835e6	2.19717
$446$	0	0
$447$	3.36697e6	0.797023
$448$	0	0
$449$	3.98846e6	0.933662	0.466831	−	0.884346i	$-0.345396\pi$
$449$	3.98846e6	0.933662	0.466831	+	0.884346i	$0.345396\pi$
$450$	0	0
$451$	−2.99109e6	−0.692450
$452$	0	0
$453$	5.83519e6i	1.33601i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−7.26224e6	−1.62660	−0.813299	−	0.581847i	$-0.802330\pi$
$457$	−7.26224e6	−1.62660	−0.813299	+	0.581847i	$0.802330\pi$
$458$	0	0
$459$	− 408112.i	− 0.0904165i
$460$	0	0
$461$	− 678936.i	− 0.148791i	−0.997229	−	0.0743955i	$-0.976297\pi$
$461$	− 678936.i	− 0.148791i	0.997229	−	0.0743955i	$-0.0237027\pi$
$462$	0	0
$463$	− 1.99460e6i	− 0.432418i	−0.976347	−	0.216209i	$-0.930631\pi$
$463$	− 1.99460e6i	− 0.432418i	0.976347	−	0.216209i	$-0.0693692\pi$
$464$	0	0
$465$	− 1.23629e7i	− 2.65147i
$466$	0	0
$467$	6.32117e6	1.34124	0.670618	−	0.741803i	$-0.266029\pi$
$467$	6.32117e6	1.34124	0.670618	+	0.741803i	$0.266029\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	9.88867e6i	2.05393i
$472$	0	0
$473$	−1.50967e6	−0.310262
$474$	0	0
$475$	1.21992e7	2.48083
$476$	0	0
$477$	−3.09017e6	−0.621851
$478$	0	0
$479$	−781569.	−0.155643	−0.0778214	−	0.996967i	$-0.524796\pi$
$479$	−781569.	−0.155643	−0.0778214	+	0.996967i	$0.524796\pi$
$480$	0	0
$481$	− 2.52568e6i	− 0.497754i
$482$	0	0
$483$	0	0
$484$	0	0
$485$	8.23168e6	1.58904
$486$	0	0
$487$	5.23084e6i	0.999423i	0.866192	+	0.499711i	$0.166561\pi$
$487$	5.23084e6i	0.999423i	−0.866192	+	0.499711i	$0.833439\pi$
$488$	0	0
$489$	8.16273e6i	1.54370i
$490$	0	0
$491$	2.68974e6i	0.503508i	0.967791	+	0.251754i	$0.0810074\pi$
$491$	2.68974e6i	0.503508i	−0.967791	+	0.251754i	$0.918993\pi$
$492$	0	0
$493$	8.47809e6i	1.57102i
$494$	0	0
$495$	−1.67860e7	−3.07918
$496$	0	0
$497$	0	0
$498$	0	0
$499$	− 2.76539e6i	− 0.497170i	−0.968610	−	0.248585i	$-0.920034\pi$
$499$	− 2.76539e6i	− 0.497170i	0.968610	−	0.248585i	$-0.0799656\pi$
$500$	0	0
$501$	1.57858e7	2.80978
$502$	0	0
$503$	865965.	0.152609	0.0763046	−	0.997085i	$-0.475688\pi$
$503$	865965.	0.152609	0.0763046	+	0.997085i	$0.475688\pi$
$504$	0	0
$505$	−2.60373e6	−0.454326
$506$	0	0
$507$	−8.78545e6	−1.51790
$508$	0	0
$509$	1.91520e6i	0.327658i	0.986489	+	0.163829i	$0.0523845\pi$
$509$	1.91520e6i	0.327658i	−0.986489	+	0.163829i	$0.947615\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−693648.	−0.116371
$514$	0	0
$515$	8.55120e6i	1.42072i
$516$	0	0
$517$	9.90173e6i	1.62924i
$518$	0	0
$519$	545048.i	0.0888211i
$520$	0	0
$521$	− 3.45794e6i	− 0.558114i	−0.960275	−	0.279057i	$-0.909978\pi$
$521$	− 3.45794e6i	− 0.558114i	0.960275	−	0.279057i	$-0.0900218\pi$
$522$	0	0
$523$	−3.02386e6	−0.483400	−0.241700	−	0.970351i	$-0.577705\pi$
$523$	−3.02386e6	−0.483400	−0.241700	+	0.970351i	$0.577705\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	8.85694e6i	1.38918i
$528$	0	0
$529$	6.04041e6	0.938484
$530$	0	0
$531$	8.89379e6	1.36883
$532$	0	0
$533$	3.58741e6	0.546970
$534$	0	0
$535$	1.12933e7	1.70583
$536$	0	0
$537$	9.00442e6i	1.34747i
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−4.06347e6	−0.596904	−0.298452	−	0.954425i	$-0.596470\pi$
$541$	−4.06347e6	−0.596904	−0.298452	+	0.954425i	$0.596470\pi$
$542$	0	0
$543$	1.79591e7i	2.61388i
$544$	0	0
$545$	− 867658.i	− 0.125129i
$546$	0	0
$547$	− 1.82099e6i	− 0.260219i	−0.991500	−	0.130110i	$-0.958467\pi$
$547$	− 1.82099e6i	− 0.260219i	0.991500	−	0.130110i	$-0.0415329\pi$
$548$	0	0
$549$	3.08361e6i	0.436646i
$550$	0	0
$551$	1.44098e7	2.02199
$552$	0	0
$553$	0	0
$554$	0	0
$555$	5.80748e6i	0.800305i
$556$	0	0
$557$	2.58038e6	0.352407	0.176204	−	0.984354i	$-0.443618\pi$
$557$	2.58038e6	0.352407	0.176204	+	0.984354i	$0.443618\pi$
$558$	0	0
$559$	1.81064e6	0.245077
$560$	0	0
$561$	2.34548e7	3.14648
$562$	0	0
$563$	1.24406e7	1.65414	0.827069	−	0.562100i	$-0.190006\pi$
$563$	1.24406e7	1.65414	0.827069	+	0.562100i	$0.190006\pi$
$564$	0	0
$565$	− 1.14465e7i	− 1.50852i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−9.52025e6	−1.23273	−0.616365	−	0.787461i	$-0.711395\pi$
$569$	−9.52025e6	−1.23273	−0.616365	+	0.787461i	$0.711395\pi$
$570$	0	0
$571$	− 4.85113e6i	− 0.622662i	−0.950302	−	0.311331i	$-0.899225\pi$
$571$	− 4.85113e6i	− 0.622662i	0.950302	−	0.311331i	$-0.100775\pi$
$572$	0	0
$573$	7.25761e6i	0.923437i
$574$	0	0
$575$	− 3.13518e6i	− 0.395451i
$576$	0	0
$577$	− 1.19031e7i	− 1.48840i	−0.667958	−	0.744199i	$-0.732831\pi$
$577$	− 1.19031e7i	− 1.48840i	0.667958	−	0.744199i	$-0.267169\pi$
$578$	0	0
$579$	−5.13227e6	−0.636228
$580$	0	0
$581$	0	0
$582$	0	0
$583$	− 8.81191e6i	− 1.07374i
$584$	0	0
$585$	2.01326e7	2.43226
$586$	0	0
$587$	4.57010e6	0.547433	0.273716	−	0.961810i	$-0.411747\pi$
$587$	4.57010e6	0.547433	0.273716	+	0.961810i	$0.411747\pi$
$588$	0	0
$589$	1.50537e7	1.78795
$590$	0	0
$591$	−1.92676e7	−2.26913
$592$	0	0
$593$	3.70366e6i	0.432509i	0.976337	+	0.216254i	$0.0693841\pi$
$593$	3.70366e6i	0.432509i	−0.976337	+	0.216254i	$0.930616\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−915918.	−0.105177
$598$	0	0
$599$	4.87963e6i	0.555674i	0.960628	+	0.277837i	$0.0896174\pi$
$599$	4.87963e6i	0.555674i	−0.960628	+	0.277837i	$0.910383\pi$
$600$	0	0
$601$	− 4.80619e6i	− 0.542769i	−0.962471	−	0.271384i	$-0.912519\pi$
$601$	− 4.80619e6i	− 0.542769i	0.962471	−	0.271384i	$-0.0874815\pi$
$602$	0	0
$603$	− 4.63638e6i	− 0.519261i
$604$	0	0
$605$	− 3.33657e7i	− 3.70605i
$606$	0	0
$607$	1.29902e7	1.43101	0.715506	−	0.698607i	$-0.246196\pi$
$607$	1.29902e7	1.43101	0.715506	+	0.698607i	$0.246196\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	− 1.18758e7i	− 1.28694i
$612$	0	0
$613$	1.32895e7	1.42843	0.714213	−	0.699928i	$-0.246785\pi$
$613$	1.32895e7	1.42843	0.714213	+	0.699928i	$0.246785\pi$
$614$	0	0
$615$	−8.24882e6	−0.879435
$616$	0	0
$617$	−1.40996e7	−1.49106	−0.745528	−	0.666474i	$-0.767803\pi$
$617$	−1.40996e7	−1.49106	−0.745528	+	0.666474i	$0.767803\pi$
$618$	0	0
$619$	9.01931e6	0.946121	0.473060	−	0.881030i	$-0.343149\pi$
$619$	9.01931e6	0.946121	0.473060	+	0.881030i	$0.343149\pi$
$620$	0	0
$621$	178267.i	0.0185499i
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−510496.	−0.0522748
$626$	0	0
$627$	− 3.98650e7i	− 4.04970i
$628$	0	0
$629$	− 4.16056e6i	− 0.419301i
$630$	0	0
$631$	− 1.41945e7i	− 1.41921i	−0.704599	−	0.709606i	$-0.748873\pi$
$631$	− 1.41945e7i	− 1.41921i	0.704599	−	0.709606i	$-0.251127\pi$
$632$	0	0
$633$	− 1.06835e7i	− 1.05975i
$634$	0	0
$635$	1.18609e7	1.16731
$636$	0	0
$637$	0	0
$638$	0	0
$639$	1.72263e7i	1.66894i
$640$	0	0
$641$	−3.48764e6	−0.335264	−0.167632	−	0.985850i	$-0.553612\pi$
$641$	−3.48764e6	−0.335264	−0.167632	+	0.985850i	$0.553612\pi$
$642$	0	0
$643$	−1.28177e7	−1.22259	−0.611296	−	0.791402i	$-0.709352\pi$
$643$	−1.28177e7	−1.22259	−0.611296	+	0.791402i	$0.709352\pi$
$644$	0	0
$645$	−4.16335e6	−0.394043
$646$	0	0
$647$	−94387.8	−0.00886452	−0.00443226	−	0.999990i	$-0.501411\pi$
$647$	−94387.8	−0.00886452	−0.00443226	+	0.999990i	$0.501411\pi$
$648$	0	0
$649$	2.53615e7i	2.36354i
$650$	0	0
$651$	0	0
$652$	0	0
$653$	1.25042e7	1.14755	0.573777	−	0.819012i	$-0.305478\pi$
$653$	1.25042e7	1.14755	0.573777	+	0.819012i	$0.305478\pi$
$654$	0	0
$655$	− 657444.i	− 0.0598764i
$656$	0	0
$657$	− 1.60687e6i	− 0.145234i
$658$	0	0
$659$	523557.i	0.0469624i	0.999724	+	0.0234812i	$0.00747499\pi$
$659$	523557.i	0.0469624i	−0.999724	+	0.0234812i	$0.992525\pi$
$660$	0	0
$661$	1.15819e7i	1.03104i	0.856877	+	0.515521i	$0.172402\pi$
$661$	1.15819e7i	1.03104i	−0.856877	+	0.515521i	$0.827598\pi$
$662$	0	0
$663$	−2.81309e7	−2.48542
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 3.70330e6i	− 0.322311i
$668$	0	0
$669$	−3.40254e6	−0.293926
$670$	0	0
$671$	−8.79323e6	−0.753949
$672$	0	0
$673$	7.72861e6	0.657755	0.328877	−	0.944373i	$-0.393330\pi$
$673$	7.72861e6	0.657755	0.328877	+	0.944373i	$0.393330\pi$
$674$	0	0
$675$	−1.41158e6	−0.119247
$676$	0	0
$677$	− 7.28721e6i	− 0.611068i	−0.952181	−	0.305534i	$-0.901165\pi$
$677$	− 7.28721e6i	− 0.611068i	0.952181	−	0.305534i	$-0.0988349\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−1.02402e7	−0.846136
$682$	0	0
$683$	1.45815e7i	1.19606i	0.801475	+	0.598028i	$0.204049\pi$
$683$	1.45815e7i	1.19606i	−0.801475	+	0.598028i	$0.795951\pi$
$684$	0	0
$685$	1.54369e7i	1.25699i
$686$	0	0
$687$	1.19420e7i	0.965351i
$688$	0	0
$689$	1.05687e7i	0.848152i
$690$	0	0
$691$	468176.	0.0373004	0.0186502	−	0.999826i	$-0.494063\pi$
$691$	468176.	0.0373004	0.0186502	+	0.999826i	$0.494063\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	− 3.91564e7i	− 3.07497i
$696$	0	0
$697$	5.90957e6	0.460759
$698$	0	0
$699$	146680.	0.0113548
$700$	0	0
$701$	6.19052e6	0.475808	0.237904	−	0.971289i	$-0.423540\pi$
$701$	6.19052e6	0.475808	0.237904	+	0.971289i	$0.423540\pi$
$702$	0	0
$703$	−7.07151e6	−0.539664
$704$	0	0
$705$	2.73069e7i	2.06919i
$706$	0	0
$707$	0	0
$708$	0	0
$709$	275001.	0.0205456	0.0102728	−	0.999947i	$-0.496730\pi$
$709$	275001.	0.0205456	0.0102728	+	0.999947i	$0.496730\pi$
$710$	0	0
$711$	− 3.88281e6i	− 0.288053i
$712$	0	0
$713$	− 3.86879e6i	− 0.285004i
$714$	0	0
$715$	5.74100e7i	4.19974i
$716$	0	0
$717$	− 4.15487e6i	− 0.301828i
$718$	0	0
$719$	−1.47456e7	−1.06375	−0.531874	−	0.846823i	$-0.678512\pi$
$719$	−1.47456e7	−1.06375	−0.531874	+	0.846823i	$0.678512\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	3.41555e7i	2.43005i
$724$	0	0
$725$	2.93241e7	2.07195
$726$	0	0
$727$	−1.10507e7	−0.775449	−0.387724	−	0.921775i	$-0.626739\pi$
$727$	−1.10507e7	−0.775449	−0.387724	+	0.921775i	$0.626739\pi$
$728$	0	0
$729$	−1.58060e7	−1.10155
$730$	0	0
$731$	2.98268e6	0.206449
$732$	0	0
$733$	1.40998e7i	0.969289i	0.874711	+	0.484645i	$0.161051\pi$
$733$	1.40998e7i	0.969289i	−0.874711	+	0.484645i	$0.838949\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	1.32211e7	0.896599
$738$	0	0
$739$	9.70671e6i	0.653824i	0.945055	+	0.326912i	$0.106008\pi$
$739$	9.70671e6i	0.653824i	−0.945055	+	0.326912i	$0.893992\pi$
$740$	0	0
$741$	4.78127e7i	3.19888i
$742$	0	0
$743$	2.38861e7i	1.58735i	0.608342	+	0.793675i	$0.291835\pi$
$743$	2.38861e7i	1.58735i	−0.608342	+	0.793675i	$0.708165\pi$
$744$	0	0
$745$	1.35759e7i	0.896146i
$746$	0	0
$747$	−2.31052e7	−1.51498
$748$	0	0
$749$	0	0
$750$	0	0
$751$	− 1.59689e7i	− 1.03317i	−0.856234	−	0.516587i	$-0.827202\pi$
$751$	− 1.59689e7i	− 1.03317i	0.856234	−	0.516587i	$-0.172798\pi$
$752$	0	0
$753$	−1.79522e7	−1.15380
$754$	0	0
$755$	−2.35280e7	−1.50217
$756$	0	0
$757$	−1.59670e7	−1.01271	−0.506353	−	0.862326i	$-0.669007\pi$
$757$	−1.59670e7	−1.01271	−0.506353	+	0.862326i	$0.669007\pi$
$758$	0	0
$759$	−1.02453e7	−0.645534
$760$	0	0
$761$	994670.i	0.0622612i	0.999515	+	0.0311306i	$0.00991077\pi$
$761$	994670.i	0.0622612i	−0.999515	+	0.0311306i	$0.990089\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	3.31645e7	2.04890
$766$	0	0
$767$	− 3.04177e7i	− 1.86697i
$768$	0	0
$769$	− 1.94076e7i	− 1.18346i	−0.806135	−	0.591732i	$-0.798444\pi$
$769$	− 1.94076e7i	− 1.18346i	0.806135	−	0.591732i	$-0.201556\pi$
$770$	0	0
$771$	− 2.30470e7i	− 1.39630i
$772$	0	0
$773$	1.91046e7i	1.14997i	0.818162	+	0.574987i	$0.194993\pi$
$773$	1.91046e7i	1.14997i	−0.818162	+	0.574987i	$0.805007\pi$
$774$	0	0
$775$	3.06345e7	1.83213
$776$	0	0
$777$	0	0
$778$	0	0
$779$	− 1.00442e7i	− 0.593024i
$780$	0	0
$781$	−4.91226e7	−2.88173
$782$	0	0
$783$	−1.66737e6	−0.0971915
$784$	0	0
$785$	−3.98720e7	−2.30937
$786$	0	0
$787$	2.59070e6	0.149101	0.0745504	−	0.997217i	$-0.476248\pi$
$787$	2.59070e6	0.149101	0.0745504	+	0.997217i	$0.476248\pi$
$788$	0	0
$789$	1.76185e6i	0.100757i
$790$	0	0
$791$	0	0
$792$	0	0
$793$	1.05463e7	0.595548
$794$	0	0
$795$	− 2.43014e7i	− 1.36369i
$796$	0	0
$797$	2.66061e7i	1.48366i	0.670587	+	0.741831i	$0.266043\pi$
$797$	2.66061e7i	1.48366i	−0.670587	+	0.741831i	$0.733957\pi$
$798$	0	0
$799$	− 1.95631e7i	− 1.08410i
$800$	0	0
$801$	− 2.60632e7i	− 1.43531i
$802$	0	0
$803$	4.58216e6	0.250773
$804$	0	0
$805$	0	0
$806$	0	0
$807$	− 3.63131e7i	− 1.96282i
$808$	0	0
$809$	2.13273e7	1.14568	0.572842	−	0.819666i	$-0.305841\pi$
$809$	2.13273e7	1.14568	0.572842	+	0.819666i	$0.305841\pi$
$810$	0	0
$811$	−447782.	−0.0239064	−0.0119532	−	0.999929i	$-0.503805\pi$
$811$	−447782.	−0.0239064	−0.0119532	+	0.999929i	$0.503805\pi$
$812$	0	0
$813$	3.32114e7	1.76222
$814$	0	0
$815$	−3.29129e7	−1.73569
$816$	0	0
$817$	− 5.06952e6i	− 0.265712i
$818$	0	0
$819$	0	0
$820$	0	0
$821$	1.30413e6	0.0675246	0.0337623	−	0.999430i	$-0.489251\pi$
$821$	1.30413e6	0.0675246	0.0337623	+	0.999430i	$0.489251\pi$
$822$	0	0
$823$	− 8.60253e6i	− 0.442717i	−0.975192	−	0.221359i	$-0.928951\pi$
$823$	− 8.60253e6i	− 0.442717i	0.975192	−	0.221359i	$-0.0710492\pi$
$824$	0	0
$825$	− 8.11258e7i	− 4.14977i
$826$	0	0
$827$	− 1.47150e6i	− 0.0748162i	−0.999300	−	0.0374081i	$-0.988090\pi$
$827$	− 1.47150e6i	− 0.0748162i	0.999300	−	0.0374081i	$-0.0119101\pi$
$828$	0	0
$829$	− 2.69544e6i	− 0.136221i	−0.997678	−	0.0681103i	$-0.978303\pi$
$829$	− 2.69544e6i	− 0.136221i	0.997678	−	0.0681103i	$-0.0216970\pi$
$830$	0	0
$831$	−1.48466e7	−0.745804
$832$	0	0
$833$	0	0
$834$	0	0
$835$	6.36498e7i	3.15923i
$836$	0	0
$837$	−1.74188e6	−0.0859419
$838$	0	0
$839$	2.98409e7	1.46355	0.731775	−	0.681546i	$-0.238692\pi$
$839$	2.98409e7	1.46355	0.731775	+	0.681546i	$0.238692\pi$
$840$	0	0
$841$	1.41267e7	0.688735
$842$	0	0
$843$	4.80306e7	2.32782
$844$	0	0
$845$	− 3.54237e7i	− 1.70668i
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−2.77085e7	−1.31930
$850$	0	0
$851$	1.81737e6i	0.0860240i
$852$	0	0
$853$	− 679437.i	− 0.0319725i	−0.999872	−	0.0159863i	$-0.994911\pi$
$853$	− 679437.i	− 0.0319725i	0.999872	−	0.0159863i	$-0.00508880\pi$
$854$	0	0
$855$	− 5.63682e7i	− 2.63705i
$856$	0	0
$857$	− 1.60609e7i	− 0.746995i	−0.927631	−	0.373497i	$-0.878159\pi$
$857$	− 1.60609e7i	− 0.746995i	0.927631	−	0.373497i	$-0.121841\pi$
$858$	0	0
$859$	3.51618e7	1.62588	0.812938	−	0.582350i	$-0.197866\pi$
$859$	3.51618e7	1.62588	0.812938	+	0.582350i	$0.197866\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	− 4.36890e6i	− 0.199685i	−0.995003	−	0.0998424i	$-0.968166\pi$
$863$	− 4.36890e6i	− 0.199685i	0.995003	−	0.0998424i	$-0.0318339\pi$
$864$	0	0
$865$	−2.19768e6	−0.0998676
$866$	0	0
$867$	−1.46330e7	−0.661127
$868$	0	0
$869$	1.10722e7	0.497377
$870$	0	0
$871$	−1.58569e7	−0.708228
$872$	0	0
$873$	− 2.33750e7i	− 1.03805i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	5.61623e6	0.246573	0.123287	−	0.992371i	$-0.460657\pi$
$877$	5.61623e6	0.246573	0.123287	+	0.992371i	$0.460657\pi$
$878$	0	0
$879$	9.17261e6i	0.400424i
$880$	0	0
$881$	− 1.53408e6i	− 0.0665898i	−0.999446	−	0.0332949i	$-0.989400\pi$
$881$	− 1.53408e6i	− 0.0665898i	0.999446	−	0.0332949i	$-0.0106001\pi$
$882$	0	0
$883$	2.56997e7i	1.10924i	0.832103	+	0.554620i	$0.187137\pi$
$883$	2.56997e7i	1.10924i	−0.832103	+	0.554620i	$0.812863\pi$
$884$	0	0
$885$	6.99418e7i	3.00178i
$886$	0	0
$887$	9.88941e6	0.422047	0.211024	−	0.977481i	$-0.432320\pi$
$887$	9.88941e6	0.422047	0.211024	+	0.977481i	$0.432320\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	− 4.06884e7i	− 1.71702i
$892$	0	0
$893$	−3.32504e7	−1.39530
$894$	0	0
$895$	−3.63066e7	−1.51505
$896$	0	0
$897$	1.22878e7	0.509910
$898$	0	0
$899$	3.61857e7	1.49327
$900$	0	0
$901$	1.74099e7i	0.714470i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−7.24129e7	−2.93897
$906$	0	0
$907$	4.59501e7i	1.85468i	0.374225	+	0.927338i	$0.377909\pi$
$907$	4.59501e7i	1.85468i	−0.374225	+	0.927338i	$0.622091\pi$
$908$	0	0
$909$	7.39367e6i	0.296791i
$910$	0	0
$911$	6.43441e6i	0.256870i	0.991718	+	0.128435i	$0.0409953\pi$
$911$	6.43441e6i	0.256870i	−0.991718	+	0.128435i	$0.959005\pi$
$912$	0	0
$913$	− 6.58866e7i	− 2.61590i
$914$	0	0
$915$	−2.42499e7	−0.957541
$916$	0	0
$917$	0	0
$918$	0	0
$919$	7.80466e6i	0.304835i	0.988316	+	0.152418i	$0.0487059\pi$
$919$	7.80466e6i	0.304835i	−0.988316	+	0.152418i	$0.951294\pi$
$920$	0	0
$921$	−8.80123e6	−0.341896
$922$	0	0
$923$	5.89159e7	2.27630
$924$	0	0
$925$	−1.43906e7	−0.552999
$926$	0	0
$927$	2.42824e7	0.928094
$928$	0	0
$929$	− 1.65944e7i	− 0.630845i	−0.948951	−	0.315422i	$-0.897854\pi$
$929$	− 1.65944e7i	− 0.630845i	0.948951	−	0.315422i	$-0.102146\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−4.54626e7	−1.70982
$934$	0	0
$935$	9.45719e7i	3.53780i
$936$	0	0
$937$	− 1.62143e6i	− 0.0603322i	−0.999545	−	0.0301661i	$-0.990396\pi$
$937$	− 1.62143e6i	− 0.0603322i	0.999545	−	0.0301661i	$-0.00960363\pi$
$938$	0	0
$939$	− 2.32121e7i	− 0.859114i
$940$	0	0
$941$	3.39935e7i	1.25147i	0.780034	+	0.625737i	$0.215202\pi$
$941$	3.39935e7i	1.25147i	−0.780034	+	0.625737i	$0.784798\pi$
$942$	0	0
$943$	−2.58135e6	−0.0945297
$944$	0	0
$945$	0	0
$946$	0	0
$947$	3.93465e7i	1.42571i	0.701311	+	0.712855i	$0.252598\pi$
$947$	3.93465e7i	1.42571i	−0.701311	+	0.712855i	$0.747402\pi$
$948$	0	0
$949$	−5.49568e6	−0.198087
$950$	0	0
$951$	2.88146e7	1.03314
$952$	0	0
$953$	−2.11074e7	−0.752841	−0.376420	−	0.926449i	$-0.622845\pi$
$953$	−2.11074e7	−0.752841	−0.376420	+	0.926449i	$0.622845\pi$
$954$	0	0
$955$	−2.92633e7	−1.03828
$956$	0	0
$957$	− 9.58265e7i	− 3.38225i
$958$	0	0
$959$	0	0
$960$	0	0
$961$	9.17358e6	0.320428
$962$	0	0
$963$	− 3.20689e7i	− 1.11434i
$964$	0	0
$965$	− 2.06938e7i	− 0.715355i
$966$	0	0
$967$	4.54428e7i	1.56278i	0.624040	+	0.781392i	$0.285490\pi$
$967$	4.54428e7i	1.56278i	−0.624040	+	0.781392i	$0.714510\pi$
$968$	0	0
$969$	7.87622e7i	2.69469i
$970$	0	0
$971$	−5.00919e6	−0.170498	−0.0852490	−	0.996360i	$-0.527169\pi$
$971$	−5.00919e6	−0.170498	−0.0852490	+	0.996360i	$0.527169\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	9.72995e7i	3.27792i
$976$	0	0
$977$	−2.16804e7	−0.726660	−0.363330	−	0.931661i	$-0.618360\pi$
$977$	−2.16804e7	−0.726660	−0.363330	+	0.931661i	$0.618360\pi$
$978$	0	0
$979$	7.43218e7	2.47833
$980$	0	0
$981$	−2.46384e6	−0.0817410
$982$	0	0
$983$	−1.56013e7	−0.514963	−0.257481	−	0.966283i	$-0.582893\pi$
$983$	−1.56013e7	−0.514963	−0.257481	+	0.966283i	$0.582893\pi$
$984$	0	0
$985$	− 7.76888e7i	− 2.55134i
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−1.30286e6	−0.0423553
$990$	0	0
$991$	4.03862e7i	1.30632i	0.757222	+	0.653158i	$0.226556\pi$
$991$	4.03862e7i	1.30632i	−0.757222	+	0.653158i	$0.773444\pi$
$992$	0	0
$993$	− 4.60836e7i	− 1.48311i
$994$	0	0
$995$	− 3.69307e6i	− 0.118258i
$996$	0	0
$997$	2.30603e6i	0.0734730i	0.999325	+	0.0367365i	$0.0116962\pi$
$997$	2.30603e6i	0.0734730i	−0.999325	+	0.0367365i	$0.988304\pi$
$998$	0	0
$999$	818252.	0.0259402

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	784.6.f.e.783.14	yes	16
4.3	odd	2	inner	784.6.f.e.783.4	yes	16
7.6	odd	2	inner	784.6.f.e.783.3	✓	16
28.27	even	2	inner	784.6.f.e.783.13	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
784.6.f.e.783.3	✓	16	7.6	odd	2	inner
784.6.f.e.783.4	yes	16	4.3	odd	2	inner
784.6.f.e.783.13	yes	16	28.27	even	2	inner
784.6.f.e.783.14	yes	16	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.f (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+22.3313 q^{3} +90.0418i q^{5} +255.687 q^{9} +729.115i q^{11} -874.475i q^{13} +2010.75i q^{15} -1440.53i q^{17} -2448.40 q^{19} +629.236i q^{23} -4982.52 q^{25} +283.307 q^{27} -5885.40 q^{29} -6148.39 q^{31} +16282.1i q^{33} +2888.22 q^{37} -19528.2i q^{39} +4102.36i q^{41} +2070.55i q^{43} +23022.5i q^{45} +13580.5 q^{47} -32168.9i q^{51} -12085.8 q^{53} -65650.8 q^{55} -54675.9 q^{57} +34784.0 q^{59} +12060.1i q^{61} +78739.3 q^{65} -18133.0i q^{67} +14051.7i q^{69} +67372.9i q^{71} -6284.55i q^{73} -111266. q^{75} -15185.8i q^{79} -55805.2 q^{81} -90365.2 q^{83} +129708. q^{85} -131428. q^{87} -101934. i q^{89} -137302. q^{93} -220458. i q^{95} -91420.7i q^{97} +186425. i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 1712 q^{9} - 496 q^{25} - 17024 q^{29} + 85664 q^{53} - 95424 q^{57} + 233344 q^{65} - 511216 q^{81} + 697952 q^{85} - 857024 q^{93}+O(q^{100}) \) 16 * q + 1712 * q^9 - 496 * q^25 - 17024 * q^29 + 85664 * q^53 - 95424 * q^57 + 233344 * q^65 - 511216 * q^81 + 697952 * q^85 - 857024 * q^93

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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