LMFDB - Embedded newform 400.8.c.r.49.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [400,8,Mod(49,400)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(400, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1])) N = Newforms(chi, 8, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("400.49"); S:= CuspForms(chi, 8); N := Newforms(S);

Level:	$ N $	$=$	$ 400 = 2^{4} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	400.c (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [4,0,0,0,0,0,0,0,-2936] 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:	no
Analytic conductor:	$124.954010194$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{2521})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 1261x^{2} + 396900 $ x^4 + 1261*x^2 + 396900
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2}\cdot 5^{2} $
Twist minimal:	no (minimal twist has level 100)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			49.4
Root			$-24.6048i$ of defining polynomial
Character	$\chi$	$=$	400.49
Dual form			400.8.c.r.49.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+70.2096i q^{3} -441.257i q^{7} -2742.38 q^{9} -3399.43 q^{11} +10512.7i q^{13} +36255.5i q^{17} +54919.5 q^{19} +30980.5 q^{21} -13459.2i q^{23} -38993.2i q^{27} +214891. q^{29} -99804.6 q^{31} -238673. i q^{33} +242589. i q^{37} -738091. q^{39} -191194. q^{41} +762206. i q^{43} -756534. i q^{47} +628835. q^{49} -2.54548e6 q^{51} +932053. i q^{53} +3.85587e6i q^{57} +1.55254e6 q^{59} -568835. q^{61} +1.21010e6i q^{63} -1.20006e6i q^{67} +944966. q^{69} -3.45325e6 q^{71} +1.31744e6i q^{73} +1.50002e6i q^{77} -4.35327e6 q^{79} -3.25990e6 q^{81} +3.64469e6i q^{83} +1.50874e7i q^{87} +4.59832e6 q^{89} +4.63880e6 q^{91} -7.00723e6i q^{93} -3.13274e6i q^{97} +9.32254e6 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 2936 q^{9} - 4560 q^{11} + 41936 q^{19} + 71704 q^{21} + 461904 q^{29} - 429344 q^{31} - 1787504 q^{39} + 898164 q^{41} + 2852748 q^{49} - 5521488 q^{51} - 1887648 q^{59} + 2532728 q^{61} + 3370152 q^{69}+ \cdots + 21498240 q^{99}+O(q^{100}) $ 4 * q - 2936 * q^9 - 4560 * q^11 + 41936 * q^19 + 71704 * q^21 + 461904 * q^29 - 429344 * q^31 - 1787504 * q^39 + 898164 * q^41 + 2852748 * q^49 - 5521488 * q^51 - 1887648 * q^59 + 2532728 * q^61 + 3370152 * q^69 - 7233552 * q^71 - 6079792 * q^79 - 7263484 * q^81 - 6478524 * q^89 + 10939424 * q^91 + 21498240 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/400\mathbb{Z}\right)^\times$.

$n$	$101$	$177$	$351$
$\chi(n)$	$1$	$-1$	$1$

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
$3$	70.2096i	1.50132i	0.660691	+	0.750658i	$0.270263\pi$
$3$	70.2096i	1.50132i	−0.660691	+	0.750658i	$0.729737\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 441.257i	− 0.486238i	−0.969996	−	0.243119i	$-0.921829\pi$
$7$	− 441.257i	− 0.486238i	0.969996	−	0.243119i	$-0.0781705\pi$
$8$	0	0
$9$	−2742.38	−1.25395
$10$	0	0
$11$	−3399.43	−0.770073	−0.385036	−	0.922901i	$-0.625811\pi$
$11$	−3399.43	−0.770073	−0.385036	+	0.922901i	$0.625811\pi$
$12$	0	0
$13$	10512.7i	1.32713i	0.748120	+	0.663563i	$0.230957\pi$
$13$	10512.7i	1.32713i	−0.748120	+	0.663563i	$0.769043\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	36255.5i	1.78979i	0.446273	+	0.894897i	$0.352751\pi$
$17$	36255.5i	1.78979i	−0.446273	+	0.894897i	$0.647249\pi$
$18$	0	0
$19$	54919.5	1.83691	0.918457	−	0.395522i	$-0.129436\pi$
$19$	54919.5	1.83691	0.918457	+	0.395522i	$0.129436\pi$
$20$	0	0
$21$	30980.5	0.729996
$22$	0	0
$23$	− 13459.2i	− 0.230660i	−0.993327	−	0.115330i	$-0.963207\pi$
$23$	− 13459.2i	− 0.230660i	0.993327	−	0.115330i	$-0.0367926\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	− 38993.2i	− 0.381255i
$28$	0	0
$29$	214891.	1.63616	0.818079	−	0.575106i	$-0.195039\pi$
$29$	214891.	1.63616	0.818079	+	0.575106i	$0.195039\pi$
$30$	0	0
$31$	−99804.6	−0.601706	−0.300853	−	0.953670i	$-0.597271\pi$
$31$	−99804.6	−0.601706	−0.300853	+	0.953670i	$0.597271\pi$
$32$	0	0
$33$	− 238673.i	− 1.15612i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	242589.i	0.787346i	0.919250	+	0.393673i	$0.128796\pi$
$37$	242589.i	0.787346i	−0.919250	+	0.393673i	$0.871204\pi$
$38$	0	0
$39$	−738091.	−1.99243
$40$	0	0
$41$	−191194.	−0.433243	−0.216621	−	0.976256i	$-0.569504\pi$
$41$	−191194.	−0.433243	−0.216621	+	0.976256i	$0.569504\pi$
$42$	0	0
$43$	762206.i	1.46195i	0.682404	+	0.730975i	$0.260935\pi$
$43$	762206.i	1.46195i	−0.682404	+	0.730975i	$0.739065\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 756534.i	− 1.06288i	−0.847094	−	0.531442i	$-0.821650\pi$
$47$	− 756534.i	− 1.06288i	0.847094	−	0.531442i	$-0.178350\pi$
$48$	0	0
$49$	628835.	0.763573
$50$	0	0
$51$	−2.54548e6	−2.68704
$52$	0	0
$53$	932053.i	0.859954i	0.902840	+	0.429977i	$0.141478\pi$
$53$	932053.i	0.859954i	−0.902840	+	0.429977i	$0.858522\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	3.85587e6i	2.75779i
$58$	0	0
$59$	1.55254e6	0.984147	0.492074	−	0.870554i	$-0.336239\pi$
$59$	1.55254e6	0.984147	0.492074	+	0.870554i	$0.336239\pi$
$60$	0	0
$61$	−568835.	−0.320872	−0.160436	−	0.987046i	$-0.551290\pi$
$61$	−568835.	−0.320872	−0.160436	+	0.987046i	$0.551290\pi$
$62$	0	0
$63$	1.21010e6i	0.609717i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	− 1.20006e6i	− 0.487463i	−0.969843	−	0.243732i	$-0.921628\pi$
$67$	− 1.20006e6i	− 0.487463i	0.969843	−	0.243732i	$-0.0783716\pi$
$68$	0	0
$69$	944966.	0.346293
$70$	0	0
$71$	−3.45325e6	−1.14505	−0.572525	−	0.819887i	$-0.694036\pi$
$71$	−3.45325e6	−1.14505	−0.572525	+	0.819887i	$0.694036\pi$
$72$	0	0
$73$	1.31744e6i	0.396370i	0.980165	+	0.198185i	$0.0635047\pi$
$73$	1.31744e6i	0.396370i	−0.980165	+	0.198185i	$0.936495\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	1.50002e6i	0.374439i
$78$	0	0
$79$	−4.35327e6	−0.993393	−0.496697	−	0.867924i	$-0.665454\pi$
$79$	−4.35327e6	−0.993393	−0.496697	+	0.867924i	$0.665454\pi$
$80$	0	0
$81$	−3.25990e6	−0.681564
$82$	0	0
$83$	3.64469e6i	0.699660i	0.936813	+	0.349830i	$0.113761\pi$
$83$	3.64469e6i	0.699660i	−0.936813	+	0.349830i	$0.886239\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	1.50874e7i	2.45639i
$88$	0	0
$89$	4.59832e6	0.691408	0.345704	−	0.938344i	$-0.387640\pi$
$89$	4.59832e6	0.691408	0.345704	+	0.938344i	$0.387640\pi$
$90$	0	0
$91$	4.63880e6	0.645299
$92$	0	0
$93$	− 7.00723e6i	− 0.903351i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	− 3.13274e6i	− 0.348517i	−0.984700	−	0.174258i	$-0.944247\pi$
$97$	− 3.13274e6i	− 0.348517i	0.984700	−	0.174258i	$-0.0557528\pi$
$98$	0	0
$99$	9.32254e6	0.965630
$100$	0	0
$101$	−1.70809e7	−1.64963	−0.824815	−	0.565402i	$-0.808721\pi$
$101$	−1.70809e7	−1.64963	−0.824815	+	0.565402i	$0.808721\pi$
$102$	0	0
$103$	− 1.76460e7i	− 1.59117i	−0.605844	−	0.795584i	$-0.707164\pi$
$103$	− 1.76460e7i	− 1.59117i	0.605844	−	0.795584i	$-0.292836\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	858442.i	0.0677435i	0.999426	+	0.0338718i	$0.0107838\pi$
$107$	858442.i	0.0677435i	−0.999426	+	0.0338718i	$0.989216\pi$
$108$	0	0
$109$	−1.91116e7	−1.41352	−0.706762	−	0.707451i	$-0.749845\pi$
$109$	−1.91116e7	−1.41352	−0.706762	+	0.707451i	$0.749845\pi$
$110$	0	0
$111$	−1.70321e7	−1.18206
$112$	0	0
$113$	1.50866e7i	0.983593i	0.870710	+	0.491797i	$0.163660\pi$
$113$	1.50866e7i	0.983593i	−0.870710	+	0.491797i	$0.836340\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	− 2.88298e7i	− 1.66415i
$118$	0	0
$119$	1.59980e7	0.870266
$120$	0	0
$121$	−7.93105e6	−0.406988
$122$	0	0
$123$	− 1.34237e7i	− 0.650434i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	− 1.06922e7i	− 0.463185i	−0.972813	−	0.231592i	$-0.925606\pi$
$127$	− 1.06922e7i	− 0.463185i	0.972813	−	0.231592i	$-0.0743936\pi$
$128$	0	0
$129$	−5.35141e7	−2.19485
$130$	0	0
$131$	3.95488e7	1.53704	0.768518	−	0.639828i	$-0.220994\pi$
$131$	3.95488e7	1.53704	0.768518	+	0.639828i	$0.220994\pi$
$132$	0	0
$133$	− 2.42336e7i	− 0.893177i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	5.16362e6i	0.171566i	0.996314	+	0.0857832i	$0.0273392\pi$
$137$	5.16362e6i	0.171566i	−0.996314	+	0.0857832i	$0.972661\pi$
$138$	0	0
$139$	−5.76675e7	−1.82129	−0.910645	−	0.413189i	$-0.864415\pi$
$139$	−5.76675e7	−1.82129	−0.910645	+	0.413189i	$0.864415\pi$
$140$	0	0
$141$	5.31159e7	1.59572
$142$	0	0
$143$	− 3.57372e7i	− 1.02198i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	4.41502e7i	1.14636i
$148$	0	0
$149$	6.03696e7	1.49509	0.747544	−	0.664212i	$-0.231233\pi$
$149$	6.03696e7	1.49509	0.747544	+	0.664212i	$0.231233\pi$
$150$	0	0
$151$	5.16114e7	1.21990	0.609952	−	0.792438i	$-0.291189\pi$
$151$	5.16114e7	1.21990	0.609952	+	0.792438i	$0.291189\pi$
$152$	0	0
$153$	− 9.94265e7i	− 2.24431i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	1.39259e7i	0.287193i	0.989636	+	0.143596i	$0.0458667\pi$
$157$	1.39259e7i	0.287193i	−0.989636	+	0.143596i	$0.954133\pi$
$158$	0	0
$159$	−6.54391e7	−1.29106
$160$	0	0
$161$	−5.93898e6	−0.112156
$162$	0	0
$163$	− 5.05328e7i	− 0.913938i	−0.889483	−	0.456969i	$-0.848935\pi$
$163$	− 5.05328e7i	− 0.913938i	0.889483	−	0.456969i	$-0.151065\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 6.11316e7i	− 1.01568i	−0.861451	−	0.507841i	$-0.830444\pi$
$167$	− 6.11316e7i	− 1.01568i	0.861451	−	0.507841i	$-0.169556\pi$
$168$	0	0
$169$	−4.77682e7	−0.761264
$170$	0	0
$171$	−1.50610e8	−2.30339
$172$	0	0
$173$	− 1.22519e8i	− 1.79905i	−0.436869	−	0.899525i	$-0.643913\pi$
$173$	− 1.22519e8i	− 1.79905i	0.436869	−	0.899525i	$-0.356087\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	1.09003e8i	1.47752i
$178$	0	0
$179$	−3.21175e7	−0.418559	−0.209280	−	0.977856i	$-0.567112\pi$
$179$	−3.21175e7	−0.418559	−0.209280	+	0.977856i	$0.567112\pi$
$180$	0	0
$181$	−2.65874e7	−0.333273	−0.166637	−	0.986018i	$-0.553291\pi$
$181$	−2.65874e7	−0.333273	−0.166637	+	0.986018i	$0.553291\pi$
$182$	0	0
$183$	− 3.99376e7i	− 0.481730i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	− 1.23248e8i	− 1.37827i
$188$	0	0
$189$	−1.72060e7	−0.185380
$190$	0	0
$191$	−6.15505e7	−0.639168	−0.319584	−	0.947558i	$-0.603543\pi$
$191$	−6.15505e7	−0.639168	−0.319584	+	0.947558i	$0.603543\pi$
$192$	0	0
$193$	1.27483e8i	1.27645i	0.769852	+	0.638223i	$0.220330\pi$
$193$	1.27483e8i	1.27645i	−0.769852	+	0.638223i	$0.779670\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	9.77703e7i	0.911119i	0.890205	+	0.455560i	$0.150561\pi$
$197$	9.77703e7i	0.911119i	−0.890205	+	0.455560i	$0.849439\pi$
$198$	0	0
$199$	−2.64808e7	−0.238202	−0.119101	−	0.992882i	$-0.538001\pi$
$199$	−2.64808e7	−0.238202	−0.119101	+	0.992882i	$0.538001\pi$
$200$	0	0
$201$	8.42558e7	0.731836
$202$	0	0
$203$	− 9.48222e7i	− 0.795562i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	3.69103e7i	0.289235i
$208$	0	0
$209$	−1.86695e8	−1.41456
$210$	0	0
$211$	−1.28704e8	−0.943199	−0.471600	−	0.881813i	$-0.656323\pi$
$211$	−1.28704e8	−0.943199	−0.471600	+	0.881813i	$0.656323\pi$
$212$	0	0
$213$	− 2.42451e8i	− 1.71908i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	4.40395e7i	0.292572i
$218$	0	0
$219$	−9.24969e7	−0.595076
$220$	0	0
$221$	−3.81143e8	−2.37528
$222$	0	0
$223$	8.69653e7i	0.525145i	0.964912	+	0.262572i	$0.0845708\pi$
$223$	8.69653e7i	0.525145i	−0.964912	+	0.262572i	$0.915429\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	1.88984e8i	1.07234i	0.844109	+	0.536172i	$0.180130\pi$
$227$	1.88984e8i	1.07234i	−0.844109	+	0.536172i	$0.819870\pi$
$228$	0	0
$229$	3.32292e8	1.82850	0.914252	−	0.405145i	$-0.132779\pi$
$229$	3.32292e8	1.82850	0.914252	+	0.405145i	$0.132779\pi$
$230$	0	0
$231$	−1.05316e8	−0.562150
$232$	0	0
$233$	− 6.91582e7i	− 0.358177i	−0.983833	−	0.179089i	$-0.942685\pi$
$233$	− 6.91582e7i	− 0.358177i	0.983833	−	0.179089i	$-0.0573149\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	− 3.05641e8i	− 1.49140i
$238$	0	0
$239$	−3.82039e6	−0.0181015	−0.00905077	−	0.999959i	$-0.502881\pi$
$239$	−3.82039e6	−0.0181015	−0.00905077	+	0.999959i	$0.502881\pi$
$240$	0	0
$241$	−2.11029e8	−0.971140	−0.485570	−	0.874198i	$-0.661388\pi$
$241$	−2.11029e8	−0.971140	−0.485570	+	0.874198i	$0.661388\pi$
$242$	0	0
$243$	− 3.14154e8i	− 1.40450i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	5.77351e8i	2.43782i
$248$	0	0
$249$	−2.55892e8	−1.05041
$250$	0	0
$251$	−2.01112e7	−0.0802751	−0.0401375	−	0.999194i	$-0.512780\pi$
$251$	−2.01112e7	−0.0802751	−0.0401375	+	0.999194i	$0.512780\pi$
$252$	0	0
$253$	4.57537e7i	0.177625i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	4.41797e8i	1.62352i	0.583994	+	0.811758i	$0.301489\pi$
$257$	4.41797e8i	1.62352i	−0.583994	+	0.811758i	$0.698511\pi$
$258$	0	0
$259$	1.07044e8	0.382838
$260$	0	0
$261$	−5.89313e8	−2.05165
$262$	0	0
$263$	1.24901e8i	0.423369i	0.977338	+	0.211685i	$0.0678950\pi$
$263$	1.24901e8i	0.423369i	−0.977338	+	0.211685i	$0.932105\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	3.22846e8i	1.03802i
$268$	0	0
$269$	−3.10603e8	−0.972908	−0.486454	−	0.873706i	$-0.661710\pi$
$269$	−3.10603e8	−0.972908	−0.486454	+	0.873706i	$0.661710\pi$
$270$	0	0
$271$	−2.55577e8	−0.780063	−0.390031	−	0.920802i	$-0.627536\pi$
$271$	−2.55577e8	−0.780063	−0.390031	+	0.920802i	$0.627536\pi$
$272$	0	0
$273$	3.25688e8i	0.968797i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	− 6.49357e7i	− 0.183571i	−0.995779	−	0.0917856i	$-0.970743\pi$
$277$	− 6.49357e7i	− 0.183571i	0.995779	−	0.0917856i	$-0.0292574\pi$
$278$	0	0
$279$	2.73702e8	0.754508
$280$	0	0
$281$	2.74082e8	0.736901	0.368450	−	0.929647i	$-0.379888\pi$
$281$	2.74082e8	0.736901	0.368450	+	0.929647i	$0.379888\pi$
$282$	0	0
$283$	− 1.94443e8i	− 0.509965i	−0.966946	−	0.254982i	$-0.917930\pi$
$283$	− 1.94443e8i	− 0.509965i	0.966946	−	0.254982i	$-0.0820697\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	8.43658e7i	0.210659i
$288$	0	0
$289$	−9.04125e8	−2.20336
$290$	0	0
$291$	2.19949e8	0.523234
$292$	0	0
$293$	− 2.29968e8i	− 0.534110i	−0.963681	−	0.267055i	$-0.913949\pi$
$293$	− 2.29968e8i	− 0.534110i	0.963681	−	0.267055i	$-0.0860506\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	1.32555e8i	0.293594i
$298$	0	0
$299$	1.41493e8	0.306115
$300$	0	0
$301$	3.36329e8	0.710856
$302$	0	0
$303$	− 1.19924e9i	− 2.47662i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	− 6.12658e8i	− 1.20847i	−0.796808	−	0.604233i	$-0.793480\pi$
$307$	− 6.12658e8i	− 1.20847i	0.796808	−	0.604233i	$-0.206520\pi$
$308$	0	0
$309$	1.23892e9	2.38884
$310$	0	0
$311$	2.83965e8	0.535308	0.267654	−	0.963515i	$-0.413752\pi$
$311$	2.83965e8	0.535308	0.267654	+	0.963515i	$0.413752\pi$
$312$	0	0
$313$	− 2.19538e8i	− 0.404674i	−0.979316	−	0.202337i	$-0.935146\pi$
$313$	− 2.19538e8i	− 0.404674i	0.979316	−	0.202337i	$-0.0648537\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 1.04184e9i	− 1.83693i	−0.395504	−	0.918464i	$-0.629430\pi$
$317$	− 1.04184e9i	− 1.83693i	0.395504	−	0.918464i	$-0.370570\pi$
$318$	0	0
$319$	−7.30507e8	−1.25996
$320$	0	0
$321$	−6.02708e7	−0.101704
$322$	0	0
$323$	1.99113e9i	3.28770i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	− 1.34181e9i	− 2.12214i
$328$	0	0
$329$	−3.33826e8	−0.516815
$330$	0	0
$331$	−1.76873e8	−0.268079	−0.134040	−	0.990976i	$-0.542795\pi$
$331$	−1.76873e8	−0.268079	−0.134040	+	0.990976i	$0.542795\pi$
$332$	0	0
$333$	− 6.65273e8i	− 0.987291i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	− 3.07250e8i	− 0.437307i	−0.975803	−	0.218654i	$-0.929833\pi$
$337$	− 3.07250e8i	− 0.437307i	0.975803	−	0.218654i	$-0.0701665\pi$
$338$	0	0
$339$	−1.05922e9	−1.47668
$340$	0	0
$341$	3.39279e8	0.463358
$342$	0	0
$343$	− 6.40872e8i	− 0.857516i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 2.16301e8i	− 0.277911i	−0.990299	−	0.138956i	$-0.955625\pi$
$347$	− 2.16301e8i	− 0.277911i	0.990299	−	0.138956i	$-0.0443745\pi$
$348$	0	0
$349$	1.03009e9	1.29713	0.648567	−	0.761157i	$-0.275369\pi$
$349$	1.03009e9	1.29713	0.648567	+	0.761157i	$0.275369\pi$
$350$	0	0
$351$	4.09923e8	0.505973
$352$	0	0
$353$	− 2.36545e8i	− 0.286222i	−0.989707	−	0.143111i	$-0.954289\pi$
$353$	− 2.36545e8i	− 0.286222i	0.989707	−	0.143111i	$-0.0457106\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	1.12321e9i	1.30654i
$358$	0	0
$359$	2.52947e8	0.288536	0.144268	−	0.989539i	$-0.453917\pi$
$359$	2.52947e8	0.288536	0.144268	+	0.989539i	$0.453917\pi$
$360$	0	0
$361$	2.12228e9	2.37425
$362$	0	0
$363$	− 5.56835e8i	− 0.611017i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	1.01344e9i	1.07021i	0.844787	+	0.535103i	$0.179727\pi$
$367$	1.01344e9i	1.07021i	−0.844787	+	0.535103i	$0.820273\pi$
$368$	0	0
$369$	5.24328e8	0.543263
$370$	0	0
$371$	4.11275e8	0.418142
$372$	0	0
$373$	− 5.02965e8i	− 0.501830i	−0.968009	−	0.250915i	$-0.919268\pi$
$373$	− 5.02965e8i	− 0.501830i	0.968009	−	0.250915i	$-0.0807315\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	2.25908e9i	2.17139i
$378$	0	0
$379$	2.99749e8	0.282827	0.141413	−	0.989951i	$-0.454835\pi$
$379$	2.99749e8	0.282827	0.141413	+	0.989951i	$0.454835\pi$
$380$	0	0
$381$	7.50695e8	0.695387
$382$	0	0
$383$	1.40661e9i	1.27932i	0.768659	+	0.639659i	$0.220924\pi$
$383$	1.40661e9i	1.27932i	−0.768659	+	0.639659i	$0.779076\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	− 2.09026e9i	− 1.83321i
$388$	0	0
$389$	−1.50654e9	−1.29765	−0.648824	−	0.760938i	$-0.724739\pi$
$389$	−1.50654e9	−1.29765	−0.648824	+	0.760938i	$0.724739\pi$
$390$	0	0
$391$	4.87971e8	0.412834
$392$	0	0
$393$	2.77671e9i	2.30758i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	− 6.48209e8i	− 0.519934i	−0.965617	−	0.259967i	$-0.916288\pi$
$397$	− 6.48209e8i	− 0.519934i	0.965617	−	0.259967i	$-0.0837117\pi$
$398$	0	0
$399$	1.70143e9	1.34094
$400$	0	0
$401$	−5.27771e8	−0.408733	−0.204367	−	0.978894i	$-0.565513\pi$
$401$	−5.27771e8	−0.408733	−0.204367	+	0.978894i	$0.565513\pi$
$402$	0	0
$403$	− 1.04921e9i	− 0.798540i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 8.24666e8i	− 0.606314i
$408$	0	0
$409$	−5.10465e7	−0.0368922	−0.0184461	−	0.999830i	$-0.505872\pi$
$409$	−5.10465e7	−0.0368922	−0.0184461	+	0.999830i	$0.505872\pi$
$410$	0	0
$411$	−3.62535e8	−0.257575
$412$	0	0
$413$	− 6.85069e8i	− 0.478530i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	− 4.04881e9i	− 2.73433i
$418$	0	0
$419$	7.59761e8	0.504577	0.252289	−	0.967652i	$-0.418817\pi$
$419$	7.59761e8	0.504577	0.252289	+	0.967652i	$0.418817\pi$
$420$	0	0
$421$	1.31573e9	0.859370	0.429685	−	0.902979i	$-0.358625\pi$
$421$	1.31573e9	0.859370	0.429685	+	0.902979i	$0.358625\pi$
$422$	0	0
$423$	2.07471e9i	1.33280i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	2.51003e8i	0.156020i
$428$	0	0
$429$	2.50909e9	1.53432
$430$	0	0
$431$	1.89700e9	1.14129	0.570645	−	0.821197i	$-0.306693\pi$
$431$	1.89700e9	1.14129	0.570645	+	0.821197i	$0.306693\pi$
$432$	0	0
$433$	2.35825e9i	1.39599i	0.716103	+	0.697995i	$0.245924\pi$
$433$	2.35825e9i	1.39599i	−0.716103	+	0.697995i	$0.754076\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 7.39173e8i	− 0.423702i
$438$	0	0
$439$	−9.29569e8	−0.524392	−0.262196	−	0.965015i	$-0.584447\pi$
$439$	−9.29569e8	−0.524392	−0.262196	+	0.965015i	$0.584447\pi$
$440$	0	0
$441$	−1.72451e9	−0.957480
$442$	0	0
$443$	2.20109e9i	1.20289i	0.798916	+	0.601443i	$0.205407\pi$
$443$	2.20109e9i	1.20289i	−0.798916	+	0.601443i	$0.794593\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	4.23853e9i	2.24460i
$448$	0	0
$449$	6.13244e6	0.00319721	0.00159860	−	0.999999i	$-0.499491\pi$
$449$	6.13244e6	0.00319721	0.00159860	+	0.999999i	$0.499491\pi$
$450$	0	0
$451$	6.49951e8	0.333628
$452$	0	0
$453$	3.62361e9i	1.83146i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	7.94414e8i	0.389350i	0.980868	+	0.194675i	$0.0623652\pi$
$457$	7.94414e8i	0.389350i	−0.980868	+	0.194675i	$0.937635\pi$
$458$	0	0
$459$	1.41372e9	0.682367
$460$	0	0
$461$	−4.10368e9	−1.95083	−0.975417	−	0.220366i	$-0.929275\pi$
$461$	−4.10368e9	−1.95083	−0.975417	+	0.220366i	$0.929275\pi$
$462$	0	0
$463$	2.96925e9i	1.39031i	0.718858	+	0.695157i	$0.244665\pi$
$463$	2.96925e9i	1.39031i	−0.718858	+	0.695157i	$0.755335\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	1.59415e9i	0.724303i	0.932119	+	0.362151i	$0.117958\pi$
$467$	1.59415e9i	0.724303i	−0.932119	+	0.362151i	$0.882042\pi$
$468$	0	0
$469$	−5.29536e8	−0.237023
$470$	0	0
$471$	−9.77730e8	−0.431167
$472$	0	0
$473$	− 2.59107e9i	− 1.12581i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	− 2.55605e9i	− 1.07834i
$478$	0	0
$479$	3.56500e9	1.48213	0.741064	−	0.671435i	$-0.234322\pi$
$479$	3.56500e9	1.48213	0.741064	+	0.671435i	$0.234322\pi$
$480$	0	0
$481$	−2.55027e9	−1.04491
$482$	0	0
$483$	− 4.16973e8i	− 0.168381i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	2.08588e9i	0.818349i	0.912456	+	0.409174i	$0.134183\pi$
$487$	2.08588e9i	0.818349i	−0.912456	+	0.409174i	$0.865817\pi$
$488$	0	0
$489$	3.54789e9	1.37211
$490$	0	0
$491$	−1.60239e9	−0.610918	−0.305459	−	0.952205i	$-0.598810\pi$
$491$	−1.60239e9	−0.610918	−0.305459	+	0.952205i	$0.598810\pi$
$492$	0	0
$493$	7.79098e9i	2.92838i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	1.52377e9i	0.556766i
$498$	0	0
$499$	−8.43362e8	−0.303852	−0.151926	−	0.988392i	$-0.548548\pi$
$499$	−8.43362e8	−0.303852	−0.151926	+	0.988392i	$0.548548\pi$
$500$	0	0
$501$	4.29202e9	1.52486
$502$	0	0
$503$	− 1.82551e9i	− 0.639581i	−0.947488	−	0.319790i	$-0.896388\pi$
$503$	− 1.82551e9i	− 0.639581i	0.947488	−	0.319790i	$-0.103612\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	− 3.35378e9i	− 1.14290i
$508$	0	0
$509$	−4.63020e9	−1.55628	−0.778140	−	0.628091i	$-0.783837\pi$
$509$	−4.63020e9	−1.55628	−0.778140	+	0.628091i	$0.783837\pi$
$510$	0	0
$511$	5.81330e8	0.192730
$512$	0	0
$513$	− 2.14148e9i	− 0.700332i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	2.57179e9i	0.818498i
$518$	0	0
$519$	8.60203e9	2.70094
$520$	0	0
$521$	3.67003e8	0.113694	0.0568471	−	0.998383i	$-0.481895\pi$
$521$	3.67003e8	0.113694	0.0568471	+	0.998383i	$0.481895\pi$
$522$	0	0
$523$	1.39584e9i	0.426657i	0.976981	+	0.213328i	$0.0684304\pi$
$523$	1.39584e9i	0.426657i	−0.976981	+	0.213328i	$0.931570\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 3.61847e9i	− 1.07693i
$528$	0	0
$529$	3.22368e9	0.946796
$530$	0	0
$531$	−4.25765e9	−1.23407
$532$	0	0
$533$	− 2.00997e9i	− 0.574968i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	− 2.25496e9i	− 0.628389i
$538$	0	0
$539$	−2.13768e9	−0.588006
$540$	0	0
$541$	7.69994e8	0.209072	0.104536	−	0.994521i	$-0.466664\pi$
$541$	7.69994e8	0.209072	0.104536	+	0.994521i	$0.466664\pi$
$542$	0	0
$543$	− 1.86669e9i	− 0.500348i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 1.00580e9i	− 0.262757i	−0.991332	−	0.131379i	$-0.958060\pi$
$547$	− 1.00580e9i	− 0.262757i	0.991332	−	0.131379i	$-0.0419404\pi$
$548$	0	0
$549$	1.55996e9	0.402357
$550$	0	0
$551$	1.18017e10	3.00548
$552$	0	0
$553$	1.92091e9i	0.483025i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	7.69979e9i	1.88793i	0.330046	+	0.943965i	$0.392936\pi$
$557$	7.69979e9i	1.88793i	−0.330046	+	0.943965i	$0.607064\pi$
$558$	0	0
$559$	−8.01284e9	−1.94019
$560$	0	0
$561$	8.65320e9	2.06922
$562$	0	0
$563$	7.19305e9i	1.69877i	0.527777	+	0.849383i	$0.323026\pi$
$563$	7.19305e9i	1.69877i	−0.527777	+	0.849383i	$0.676974\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	1.43845e9i	0.331402i
$568$	0	0
$569$	4.07740e9	0.927876	0.463938	−	0.885868i	$-0.346436\pi$
$569$	4.07740e9	0.927876	0.463938	+	0.885868i	$0.346436\pi$
$570$	0	0
$571$	−3.94726e9	−0.887298	−0.443649	−	0.896201i	$-0.646316\pi$
$571$	−3.94726e9	−0.887298	−0.443649	+	0.896201i	$0.646316\pi$
$572$	0	0
$573$	− 4.32143e9i	− 0.959592i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	− 2.06436e9i	− 0.447372i	−0.974661	−	0.223686i	$-0.928191\pi$
$577$	− 2.06436e9i	− 0.447372i	0.974661	−	0.223686i	$-0.0718091\pi$
$578$	0	0
$579$	−8.95054e9	−1.91635
$580$	0	0
$581$	1.60825e9	0.340201
$582$	0	0
$583$	− 3.16845e9i	− 0.662227i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 5.42776e9i	− 1.10761i	−0.832646	−	0.553806i	$-0.813175\pi$
$587$	− 5.42776e9i	− 1.10761i	0.832646	−	0.553806i	$-0.186825\pi$
$588$	0	0
$589$	−5.48121e9	−1.10528
$590$	0	0
$591$	−6.86441e9	−1.36788
$592$	0	0
$593$	− 5.36795e9i	− 1.05710i	−0.848902	−	0.528551i	$-0.822736\pi$
$593$	− 5.36795e9i	− 1.05710i	0.848902	−	0.528551i	$-0.177264\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	− 1.85921e9i	− 0.357616i
$598$	0	0
$599$	−3.96320e9	−0.753446	−0.376723	−	0.926326i	$-0.622949\pi$
$599$	−3.96320e9	−0.753446	−0.376723	+	0.926326i	$0.622949\pi$
$600$	0	0
$601$	−3.57751e9	−0.672233	−0.336117	−	0.941820i	$-0.609114\pi$
$601$	−3.57751e9	−0.672233	−0.336117	+	0.941820i	$0.609114\pi$
$602$	0	0
$603$	3.29103e9i	0.611253i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	1.79748e7i	0.00326215i	0.999999	+	0.00163108i	$0.000519188\pi$
$607$	1.79748e7i	0.00326215i	−0.999999	+	0.00163108i	$0.999481\pi$
$608$	0	0
$609$	6.65743e9	1.19439
$610$	0	0
$611$	7.95321e9	1.41058
$612$	0	0
$613$	9.92247e8i	0.173984i	0.996209	+	0.0869918i	$0.0277254\pi$
$613$	9.92247e8i	0.173984i	−0.996209	+	0.0869918i	$0.972275\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 5.79730e9i	− 0.993637i	−0.867854	−	0.496819i	$-0.834501\pi$
$617$	− 5.79730e9i	− 0.993637i	0.867854	−	0.496819i	$-0.165499\pi$
$618$	0	0
$619$	−1.23127e9	−0.208658	−0.104329	−	0.994543i	$-0.533269\pi$
$619$	−1.23127e9	−0.208658	−0.104329	+	0.994543i	$0.533269\pi$
$620$	0	0
$621$	−5.24817e8	−0.0879402
$622$	0	0
$623$	− 2.02904e9i	− 0.336189i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	− 1.31078e10i	− 2.12370i
$628$	0	0
$629$	−8.79521e9	−1.40919
$630$	0	0
$631$	−4.58423e9	−0.726379	−0.363189	−	0.931715i	$-0.618312\pi$
$631$	−4.58423e9	−0.726379	−0.363189	+	0.931715i	$0.618312\pi$
$632$	0	0
$633$	− 9.03625e9i	− 1.41604i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	6.61075e9i	1.01336i
$638$	0	0
$639$	9.47014e9	1.43583
$640$	0	0
$641$	1.09681e9	0.164486	0.0822431	−	0.996612i	$-0.473792\pi$
$641$	1.09681e9	0.164486	0.0822431	+	0.996612i	$0.473792\pi$
$642$	0	0
$643$	− 1.12025e10i	− 1.66179i	−0.556430	−	0.830894i	$-0.687829\pi$
$643$	− 1.12025e10i	− 1.66179i	0.556430	−	0.830894i	$-0.312171\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	7.13092e9i	1.03510i	0.855654	+	0.517548i	$0.173155\pi$
$647$	7.13092e9i	1.03510i	−0.855654	+	0.517548i	$0.826845\pi$
$648$	0	0
$649$	−5.27774e9	−0.757865
$650$	0	0
$651$	−3.09199e9	−0.439243
$652$	0	0
$653$	3.81333e9i	0.535930i	0.963429	+	0.267965i	$0.0863511\pi$
$653$	3.81333e9i	0.535930i	−0.963429	+	0.267965i	$0.913649\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	− 3.61292e9i	− 0.497027i
$658$	0	0
$659$	7.02323e7	0.00955956	0.00477978	−	0.999989i	$-0.498479\pi$
$659$	7.02323e7	0.00955956	0.00477978	+	0.999989i	$0.498479\pi$
$660$	0	0
$661$	6.27708e9	0.845381	0.422691	−	0.906274i	$-0.361086\pi$
$661$	6.27708e9	0.845381	0.422691	+	0.906274i	$0.361086\pi$
$662$	0	0
$663$	− 2.67599e10i	− 3.56605i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 2.89226e9i	− 0.377396i
$668$	0	0
$669$	−6.10580e9	−0.788408
$670$	0	0
$671$	1.93371e9	0.247095
$672$	0	0
$673$	− 7.13029e9i	− 0.901684i	−0.892604	−	0.450842i	$-0.851124\pi$
$673$	− 7.13029e9i	− 0.901684i	0.892604	−	0.450842i	$-0.148876\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	4.73732e9i	0.586776i	0.955993	+	0.293388i	$0.0947829\pi$
$677$	4.73732e9i	0.586776i	−0.955993	+	0.293388i	$0.905217\pi$
$678$	0	0
$679$	−1.38235e9	−0.169462
$680$	0	0
$681$	−1.32685e10	−1.60992
$682$	0	0
$683$	5.84588e9i	0.702066i	0.936363	+	0.351033i	$0.114169\pi$
$683$	5.84588e9i	0.702066i	−0.936363	+	0.351033i	$0.885831\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	2.33301e10i	2.74516i
$688$	0	0
$689$	−9.79839e9	−1.14127
$690$	0	0
$691$	9.53623e9	1.09952	0.549761	−	0.835322i	$-0.314719\pi$
$691$	9.53623e9	1.09952	0.549761	+	0.835322i	$0.314719\pi$
$692$	0	0
$693$	− 4.11364e9i	− 0.469526i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	− 6.93185e9i	− 0.775415i
$698$	0	0
$699$	4.85557e9	0.537737
$700$	0	0
$701$	6.30244e9	0.691029	0.345514	−	0.938414i	$-0.387704\pi$
$701$	6.30244e9	0.691029	0.345514	+	0.938414i	$0.387704\pi$
$702$	0	0
$703$	1.33229e10i	1.44629i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	7.53709e9i	0.802113i
$708$	0	0
$709$	1.61363e9	0.170036	0.0850182	−	0.996379i	$-0.472905\pi$
$709$	1.61363e9	0.170036	0.0850182	+	0.996379i	$0.472905\pi$
$710$	0	0
$711$	1.19383e10	1.24566
$712$	0	0
$713$	1.34329e9i	0.138790i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	− 2.68228e8i	− 0.0271761i
$718$	0	0
$719$	1.87867e9	0.188495	0.0942474	−	0.995549i	$-0.469956\pi$
$719$	1.87867e9	0.188495	0.0942474	+	0.995549i	$0.469956\pi$
$720$	0	0
$721$	−7.78642e9	−0.773686
$722$	0	0
$723$	− 1.48162e10i	− 1.45799i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	1.76641e10i	1.70499i	0.522734	+	0.852496i	$0.324912\pi$
$727$	1.76641e10i	1.70499i	−0.522734	+	0.852496i	$0.675088\pi$
$728$	0	0
$729$	1.49272e10	1.42703
$730$	0	0
$731$	−2.76342e10	−2.61659
$732$	0	0
$733$	3.51351e9i	0.329516i	0.986334	+	0.164758i	$0.0526844\pi$
$733$	3.51351e9i	0.329516i	−0.986334	+	0.164758i	$0.947316\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	4.07953e9i	0.375382i
$738$	0	0
$739$	−1.28441e10	−1.17070	−0.585351	−	0.810780i	$-0.699043\pi$
$739$	−1.28441e10	−1.17070	−0.585351	+	0.810780i	$0.699043\pi$
$740$	0	0
$741$	−4.05356e10	−3.65993
$742$	0	0
$743$	6.83686e9i	0.611499i	0.952112	+	0.305749i	$0.0989070\pi$
$743$	6.83686e9i	0.611499i	−0.952112	+	0.305749i	$0.901093\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	− 9.99513e9i	− 0.877336i
$748$	0	0
$749$	3.78794e8	0.0329395
$750$	0	0
$751$	8.09190e8	0.0697125	0.0348563	−	0.999392i	$-0.488903\pi$
$751$	8.09190e8	0.0697125	0.0348563	+	0.999392i	$0.488903\pi$
$752$	0	0
$753$	− 1.41200e9i	− 0.120518i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	− 1.94075e10i	− 1.62605i	−0.582231	−	0.813023i	$-0.697820\pi$
$757$	− 1.94075e10i	− 1.62605i	0.582231	−	0.813023i	$-0.302180\pi$
$758$	0	0
$759$	−3.21234e9	−0.266671
$760$	0	0
$761$	−1.05361e10	−0.866633	−0.433316	−	0.901242i	$-0.642657\pi$
$761$	−1.05361e10	−0.866633	−0.433316	+	0.901242i	$0.642657\pi$
$762$	0	0
$763$	8.43311e9i	0.687309i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	1.63213e10i	1.30609i
$768$	0	0
$769$	−1.43764e10	−1.14001	−0.570005	−	0.821641i	$-0.693059\pi$
$769$	−1.43764e10	−1.14001	−0.570005	+	0.821641i	$0.693059\pi$
$770$	0	0
$771$	−3.10184e10	−2.43741
$772$	0	0
$773$	4.52876e9i	0.352655i	0.984332	+	0.176328i	$0.0564219\pi$
$773$	4.52876e9i	0.352655i	−0.984332	+	0.176328i	$0.943578\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	7.51554e9i	0.574760i
$778$	0	0
$779$	−1.05003e10	−0.795829
$780$	0	0
$781$	1.17391e10	0.881771
$782$	0	0
$783$	− 8.37928e9i	− 0.623793i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	− 1.16498e10i	− 0.851939i	−0.904737	−	0.425970i	$-0.859933\pi$
$787$	− 1.16498e10i	− 0.851939i	0.904737	−	0.425970i	$-0.140067\pi$
$788$	0	0
$789$	−8.76922e9	−0.635611
$790$	0	0
$791$	6.65705e9	0.478260
$792$	0	0
$793$	− 5.97999e9i	− 0.425838i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	2.52156e10i	1.76427i	0.470998	+	0.882134i	$0.343894\pi$
$797$	2.52156e10i	1.76427i	−0.470998	+	0.882134i	$0.656106\pi$
$798$	0	0
$799$	2.74286e10	1.90234
$800$	0	0
$801$	−1.26104e10	−0.866989
$802$	0	0
$803$	− 4.47855e9i	− 0.305234i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	− 2.18073e10i	− 1.46064i
$808$	0	0
$809$	−7.02887e9	−0.466730	−0.233365	−	0.972389i	$-0.574974\pi$
$809$	−7.02887e9	−0.466730	−0.233365	+	0.972389i	$0.574974\pi$
$810$	0	0
$811$	−2.36366e10	−1.55601	−0.778003	−	0.628261i	$-0.783767\pi$
$811$	−2.36366e10	−1.55601	−0.778003	+	0.628261i	$0.783767\pi$
$812$	0	0
$813$	− 1.79440e10i	− 1.17112i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	4.18599e10i	2.68548i
$818$	0	0
$819$	−1.27214e10	−0.809171
$820$	0	0
$821$	1.76032e10	1.11017	0.555085	−	0.831794i	$-0.312686\pi$
$821$	1.76032e10	1.11017	0.555085	+	0.831794i	$0.312686\pi$
$822$	0	0
$823$	− 1.69875e10i	− 1.06225i	−0.847292	−	0.531127i	$-0.821769\pi$
$823$	− 1.69875e10i	− 1.06225i	0.847292	−	0.531127i	$-0.178231\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	1.92153e10i	1.18135i	0.806911	+	0.590673i	$0.201138\pi$
$827$	1.92153e10i	1.18135i	−0.806911	+	0.590673i	$0.798862\pi$
$828$	0	0
$829$	4.89038e9	0.298128	0.149064	−	0.988828i	$-0.452374\pi$
$829$	4.89038e9	0.298128	0.149064	+	0.988828i	$0.452374\pi$
$830$	0	0
$831$	4.55911e9	0.275598
$832$	0	0
$833$	2.27987e10i	1.36664i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	3.89170e9i	0.229403i
$838$	0	0
$839$	−4.84102e9	−0.282989	−0.141495	−	0.989939i	$-0.545191\pi$
$839$	−4.84102e9	−0.282989	−0.141495	+	0.989939i	$0.545191\pi$
$840$	0	0
$841$	2.89282e10	1.67701
$842$	0	0
$843$	1.92432e10i	1.10632i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	3.49963e9i	0.197893i
$848$	0	0
$849$	1.36518e10	0.765618
$850$	0	0
$851$	3.26506e9	0.181609
$852$	0	0
$853$	− 1.44504e10i	− 0.797182i	−0.917129	−	0.398591i	$-0.869499\pi$
$853$	− 1.44504e10i	− 0.797182i	0.917129	−	0.398591i	$-0.130501\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 1.34760e10i	− 0.731355i	−0.930742	−	0.365678i	$-0.880837\pi$
$857$	− 1.34760e10i	− 0.731355i	0.930742	−	0.365678i	$-0.119163\pi$
$858$	0	0
$859$	−2.71763e10	−1.46290	−0.731449	−	0.681896i	$-0.761156\pi$
$859$	−2.71763e10	−1.46290	−0.731449	+	0.681896i	$0.761156\pi$
$860$	0	0
$861$	−5.92329e9	−0.316266
$862$	0	0
$863$	− 1.14514e10i	− 0.606486i	−0.952913	−	0.303243i	$-0.901931\pi$
$863$	− 1.14514e10i	− 0.606486i	0.952913	−	0.303243i	$-0.0980694\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	− 6.34782e10i	− 3.30794i
$868$	0	0
$869$	1.47986e10	0.764985
$870$	0	0
$871$	1.26159e10	0.646925
$872$	0	0
$873$	8.59118e9i	0.437022i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 1.94305e10i	− 0.972714i	−0.873760	−	0.486357i	$-0.838325\pi$
$877$	− 1.94305e10i	− 0.972714i	0.873760	−	0.486357i	$-0.161675\pi$
$878$	0	0
$879$	1.61460e10	0.801868
$880$	0	0
$881$	1.67671e10	0.826118	0.413059	−	0.910704i	$-0.364460\pi$
$881$	1.67671e10	0.826118	0.413059	+	0.910704i	$0.364460\pi$
$882$	0	0
$883$	2.67210e10i	1.30614i	0.757297	+	0.653071i	$0.226520\pi$
$883$	2.67210e10i	1.30614i	−0.757297	+	0.653071i	$0.773480\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	2.70750e9i	0.130267i	0.997877	+	0.0651336i	$0.0207474\pi$
$887$	2.70750e9i	0.130267i	−0.997877	+	0.0651336i	$0.979253\pi$
$888$	0	0
$889$	−4.71802e9	−0.225218
$890$	0	0
$891$	1.10818e10	0.524854
$892$	0	0
$893$	− 4.15485e10i	− 1.95243i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	9.93413e9i	0.459575i
$898$	0	0
$899$	−2.14471e10	−0.984486
$900$	0	0
$901$	−3.37921e10	−1.53914
$902$	0	0
$903$	2.36135e10i	1.06722i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	1.50146e10i	0.668171i	0.942543	+	0.334085i	$0.108427\pi$
$907$	1.50146e10i	0.668171i	−0.942543	+	0.334085i	$0.891573\pi$
$908$	0	0
$909$	4.68425e10	2.06855
$910$	0	0
$911$	−2.93619e10	−1.28668	−0.643339	−	0.765581i	$-0.722452\pi$
$911$	−2.93619e10	−1.28668	−0.643339	+	0.765581i	$0.722452\pi$
$912$	0	0
$913$	− 1.23899e10i	− 0.538789i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 1.74512e10i	− 0.747365i
$918$	0	0
$919$	2.07262e10	0.880879	0.440439	−	0.897782i	$-0.354823\pi$
$919$	2.07262e10	0.880879	0.440439	+	0.897782i	$0.354823\pi$
$920$	0	0
$921$	4.30145e10	1.81429
$922$	0	0
$923$	− 3.63030e10i	− 1.51962i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	4.83921e10i	1.99524i
$928$	0	0
$929$	3.07919e10	1.26003	0.630015	−	0.776583i	$-0.283049\pi$
$929$	3.07919e10	1.26003	0.630015	+	0.776583i	$0.283049\pi$
$930$	0	0
$931$	3.45353e10	1.40262
$932$	0	0
$933$	1.99371e10i	0.803666i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	1.12790e10i	0.447901i	0.974601	+	0.223951i	$0.0718954\pi$
$937$	1.12790e10i	0.447901i	−0.974601	+	0.223951i	$0.928105\pi$
$938$	0	0
$939$	1.54137e10	0.607543
$940$	0	0
$941$	1.74971e10	0.684546	0.342273	−	0.939600i	$-0.388803\pi$
$941$	1.74971e10	0.684546	0.342273	+	0.939600i	$0.388803\pi$
$942$	0	0
$943$	2.57332e9i	0.0999317i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 4.00359e10i	− 1.53188i	−0.642912	−	0.765940i	$-0.722274\pi$
$947$	− 4.00359e10i	− 1.53188i	0.642912	−	0.765940i	$-0.277726\pi$
$948$	0	0
$949$	−1.38498e10	−0.526033
$950$	0	0
$951$	7.31469e10	2.75781
$952$	0	0
$953$	3.20525e10i	1.19960i	0.800149	+	0.599801i	$0.204754\pi$
$953$	3.20525e10i	1.19960i	−0.800149	+	0.599801i	$0.795246\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	− 5.12886e10i	− 1.89160i
$958$	0	0
$959$	2.27849e9	0.0834221
$960$	0	0
$961$	−1.75517e10	−0.637950
$962$	0	0
$963$	− 2.35418e9i	− 0.0849468i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	− 2.74626e10i	− 0.976671i	−0.872656	−	0.488336i	$-0.837604\pi$
$967$	− 2.74626e10i	− 0.976671i	0.872656	−	0.488336i	$-0.162396\pi$
$968$	0	0
$969$	−1.39797e11	−4.93587
$970$	0	0
$971$	−1.74172e10	−0.610537	−0.305268	−	0.952266i	$-0.598746\pi$
$971$	−1.74172e10	−0.610537	−0.305268	+	0.952266i	$0.598746\pi$
$972$	0	0
$973$	2.54462e10i	0.885581i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 2.15528e10i	− 0.739390i	−0.929153	−	0.369695i	$-0.879462\pi$
$977$	− 2.15528e10i	− 0.739390i	0.929153	−	0.369695i	$-0.120538\pi$
$978$	0	0
$979$	−1.56317e10	−0.532434
$980$	0	0
$981$	5.24112e10	1.77248
$982$	0	0
$983$	− 3.94252e10i	− 1.32384i	−0.749573	−	0.661922i	$-0.769741\pi$
$983$	− 3.94252e10i	− 1.32384i	0.749573	−	0.661922i	$-0.230259\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	− 2.34378e10i	− 0.775902i
$988$	0	0
$989$	1.02587e10	0.337214
$990$	0	0
$991$	9.69036e9	0.316287	0.158144	−	0.987416i	$-0.449449\pi$
$991$	9.69036e9	0.316287	0.158144	+	0.987416i	$0.449449\pi$
$992$	0	0
$993$	− 1.24182e10i	− 0.402471i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	1.17694e9i	0.0376115i	0.999823	+	0.0188057i	$0.00598640\pi$
$997$	1.17694e9i	0.0376115i	−0.999823	+	0.0188057i	$0.994014\pi$
$998$	0	0
$999$	9.45933e9	0.300180

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	400.8.c.r.49.4		4
4.3	odd	2		100.8.c.c.49.1		4
5.2	odd	4		400.8.a.be.1.2		2
5.3	odd	4		400.8.a.u.1.1		2
5.4	even	2	inner	400.8.c.r.49.1		4
20.3	even	4		100.8.a.d.1.2	yes	2
20.7	even	4		100.8.a.b.1.1	✓	2
20.19	odd	2		100.8.c.c.49.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
100.8.a.b.1.1	✓	2	20.7	even	4
100.8.a.d.1.2	yes	2	20.3	even	4
100.8.c.c.49.1		4	4.3	odd	2
100.8.c.c.49.4		4	20.19	odd	2
400.8.a.u.1.1		2	5.3	odd	4
400.8.a.be.1.2		2	5.2	odd	4
400.8.c.r.49.1		4	5.4	even	2	inner
400.8.c.r.49.4		4	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:	\( N \)	\(=\)	\( 400 = 2^{4} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	400.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+70.2096i q^{3} -441.257i q^{7} -2742.38 q^{9} -3399.43 q^{11} +10512.7i q^{13} +36255.5i q^{17} +54919.5 q^{19} +30980.5 q^{21} -13459.2i q^{23} -38993.2i q^{27} +214891. q^{29} -99804.6 q^{31} -238673. i q^{33} +242589. i q^{37} -738091. q^{39} -191194. q^{41} +762206. i q^{43} -756534. i q^{47} +628835. q^{49} -2.54548e6 q^{51} +932053. i q^{53} +3.85587e6i q^{57} +1.55254e6 q^{59} -568835. q^{61} +1.21010e6i q^{63} -1.20006e6i q^{67} +944966. q^{69} -3.45325e6 q^{71} +1.31744e6i q^{73} +1.50002e6i q^{77} -4.35327e6 q^{79} -3.25990e6 q^{81} +3.64469e6i q^{83} +1.50874e7i q^{87} +4.59832e6 q^{89} +4.63880e6 q^{91} -7.00723e6i q^{93} -3.13274e6i q^{97} +9.32254e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 2936 q^{9} - 4560 q^{11} + 41936 q^{19} + 71704 q^{21} + 461904 q^{29} - 429344 q^{31} - 1787504 q^{39} + 898164 q^{41} + 2852748 q^{49} - 5521488 q^{51} - 1887648 q^{59} + 2532728 q^{61} + 3370152 q^{69}+ \cdots + 21498240 q^{99}+O(q^{100}) \) 4 * q - 2936 * q^9 - 4560 * q^11 + 41936 * q^19 + 71704 * q^21 + 461904 * q^29 - 429344 * q^31 - 1787504 * q^39 + 898164 * q^41 + 2852748 * q^49 - 5521488 * q^51 - 1887648 * q^59 + 2532728 * q^61 + 3370152 * q^69 - 7233552 * q^71 - 6079792 * q^79 - 7263484 * q^81 - 6478524 * q^89 + 10939424 * q^91 + 21498240 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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