LMFDB - Embedded newform 350.6.c.i.99.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [350,6,Mod(99,350)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(350, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0]))

N = Newforms(chi, 6, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("350.99");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 350 = 2 \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	350.c (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$56.1343369345$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{79})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 39x^{2} + 400 $ x^4 - 39*x^2 + 400
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			99.4
Root			$-4.44410 - 0.500000i$ of defining polynomial
Character	$\chi$	$=$	350.99
Dual form			350.6.c.i.99.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+4.00000i q^{2} +27.7764i q^{3} -16.0000 q^{4} -111.106 q^{6} +49.0000i q^{7} -64.0000i q^{8} -528.528 q^{9} +O(q^{10})$	\(q+4.00000i q^{2} +27.7764i q^{3} -16.0000 q^{4} -111.106 q^{6} +49.0000i q^{7} -64.0000i q^{8} -528.528 q^{9} -481.671 q^{11} -444.422i q^{12} +883.515i q^{13} -196.000 q^{14} +256.000 q^{16} +1840.82i q^{17} -2114.11i q^{18} -2440.51 q^{19} -1361.04 q^{21} -1926.68i q^{22} +169.274i q^{23} +1777.69 q^{24} -3534.06 q^{26} -7930.93i q^{27} -784.000i q^{28} +554.156 q^{29} +7734.52 q^{31} +1024.00i q^{32} -13379.1i q^{33} -7363.28 q^{34} +8456.44 q^{36} -11425.8i q^{37} -9762.03i q^{38} -24540.9 q^{39} +10893.6 q^{41} -5444.17i q^{42} +15335.5i q^{43} +7706.73 q^{44} -677.094 q^{46} -9705.85i q^{47} +7110.76i q^{48} -2401.00 q^{49} -51131.3 q^{51} -14136.2i q^{52} +8829.12i q^{53} +31723.7 q^{54} +3136.00 q^{56} -67788.5i q^{57} +2216.62i q^{58} +40580.0 q^{59} -1675.50 q^{61} +30938.1i q^{62} -25897.9i q^{63} -4096.00 q^{64} +53516.3 q^{66} -9885.31i q^{67} -29453.1i q^{68} -4701.81 q^{69} -51350.0 q^{71} +33825.8i q^{72} +26718.3i q^{73} +45703.2 q^{74} +39048.1 q^{76} -23601.9i q^{77} -98163.5i q^{78} +70237.0 q^{79} +91860.4 q^{81} +43574.3i q^{82} +58622.5i q^{83} +21776.7 q^{84} -61342.0 q^{86} +15392.4i q^{87} +30826.9i q^{88} -28370.7 q^{89} -43292.2 q^{91} -2708.38i q^{92} +214837. i q^{93} +38823.4 q^{94} -28443.0 q^{96} +117548. i q^{97} -9604.00i q^{98} +254576. q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 64 q^{4} - 160 q^{6} - 692 q^{9}+O(q^{10}) $ 4 * q - 64 * q^4 - 160 * q^6 - 692 * q^9	$ 4 q - 64 q^{4} - 160 q^{6} - 692 q^{9} - 2140 q^{11} - 784 q^{14} + 1024 q^{16} - 1656 q^{19} - 1960 q^{21} + 2560 q^{24} - 5888 q^{26} + 4492 q^{29} + 576 q^{31} - 15232 q^{34} + 11072 q^{36} - 51376 q^{39} + 22456 q^{41} + 34240 q^{44} - 13232 q^{46} - 9604 q^{49} - 101280 q^{51} + 89920 q^{54} + 12544 q^{56} + 185856 q^{59} + 30344 q^{61} - 16384 q^{64} + 70432 q^{66} + 13688 q^{69} - 41644 q^{71} + 71888 q^{74} + 26496 q^{76} + 144212 q^{79} + 220964 q^{81} + 31360 q^{84} - 27216 q^{86} + 19840 q^{89} - 72128 q^{91} + 124576 q^{94} - 40960 q^{96} + 294380 q^{99}+O(q^{100}) $ 4 * q - 64 * q^4 - 160 * q^6 - 692 * q^9 - 2140 * q^11 - 784 * q^14 + 1024 * q^16 - 1656 * q^19 - 1960 * q^21 + 2560 * q^24 - 5888 * q^26 + 4492 * q^29 + 576 * q^31 - 15232 * q^34 + 11072 * q^36 - 51376 * q^39 + 22456 * q^41 + 34240 * q^44 - 13232 * q^46 - 9604 * q^49 - 101280 * q^51 + 89920 * q^54 + 12544 * q^56 + 185856 * q^59 + 30344 * q^61 - 16384 * q^64 + 70432 * q^66 + 13688 * q^69 - 41644 * q^71 + 71888 * q^74 + 26496 * q^76 + 144212 * q^79 + 220964 * q^81 + 31360 * q^84 - 27216 * q^86 + 19840 * q^89 - 72128 * q^91 + 124576 * q^94 - 40960 * q^96 + 294380 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/350\mathbb{Z}\right)^\times$.

$n$	$101$	$127$
$\chi(n)$	$1$	$-1$

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 $

$2$	4.00000i	0.707107i
$2$	4.00000i	0.707107i
$3$	27.7764i	1.78186i	0.454144	+	0.890928i	$0.349945\pi$
$3$	27.7764i	1.78186i	−0.454144	+	0.890928i	$0.650055\pi$
$4$	−16.0000	−0.500000
$5$	0	0
$5$	0	0
$6$	−111.106	−1.25996
$7$	49.0000i	0.377964i
$7$	49.0000i	0.377964i
$8$	− 64.0000i	− 0.353553i
$9$	−528.528	−2.17501
$10$	0	0
$11$	−481.671	−1.20024	−0.600121	−	0.799909i	$-0.704881\pi$
$11$	−481.671	−1.20024	−0.600121	+	0.799909i	$0.704881\pi$
$12$	− 444.422i	− 0.890928i
$13$	883.515i	1.44996i	0.688771	+	0.724979i	$0.258151\pi$
$13$	883.515i	1.44996i	−0.688771	+	0.724979i	$0.741849\pi$
$14$	−196.000	−0.267261
$15$	0	0
$16$	256.000	0.250000
$17$	1840.82i	1.54486i	0.635100	+	0.772430i	$0.280959\pi$
$17$	1840.82i	1.54486i	−0.635100	+	0.772430i	$0.719041\pi$
$18$	− 2114.11i	− 1.53797i
$19$	−2440.51	−1.55094	−0.775472	−	0.631382i	$-0.782488\pi$
$19$	−2440.51	−1.55094	−0.775472	+	0.631382i	$0.782488\pi$
$20$	0	0
$21$	−1361.04	−0.673478
$22$	− 1926.68i	− 0.848699i
$23$	169.274i	0.0667221i	0.999443	+	0.0333610i	$0.0106211\pi$
$23$	169.274i	0.0667221i	−0.999443	+	0.0333610i	$0.989379\pi$
$24$	1777.69	0.629981
$25$	0	0
$26$	−3534.06	−1.02528
$27$	− 7930.93i	− 2.09370i
$28$	− 784.000i	− 0.188982i
$29$	554.156	0.122359	0.0611796	−	0.998127i	$-0.480514\pi$
$29$	554.156	0.122359	0.0611796	+	0.998127i	$0.480514\pi$
$30$	0	0
$31$	7734.52	1.44554	0.722768	−	0.691091i	$-0.242869\pi$
$31$	7734.52	1.44554	0.722768	+	0.691091i	$0.242869\pi$
$32$	1024.00i	0.176777i
$33$	− 13379.1i	− 2.13866i
$34$	−7363.28	−1.09238
$35$	0	0
$36$	8456.44	1.08751
$37$	− 11425.8i	− 1.37209i	−0.727560	−	0.686044i	$-0.759346\pi$
$37$	− 11425.8i	− 1.37209i	0.727560	−	0.686044i	$-0.240654\pi$
$38$	− 9762.03i	− 1.09668i
$39$	−24540.9	−2.58362
$40$	0	0
$41$	10893.6	1.01207	0.506036	−	0.862512i	$-0.331110\pi$
$41$	10893.6	1.01207	0.506036	+	0.862512i	$0.331110\pi$
$42$	− 5444.17i	− 0.476221i
$43$	15335.5i	1.26481i	0.774636	+	0.632407i	$0.217933\pi$
$43$	15335.5i	1.26481i	−0.774636	+	0.632407i	$0.782067\pi$
$44$	7706.73	0.600121
$45$	0	0
$46$	−677.094	−0.0471796
$47$	− 9705.85i	− 0.640898i	−0.947266	−	0.320449i	$-0.896166\pi$
$47$	− 9705.85i	− 0.640898i	0.947266	−	0.320449i	$-0.103834\pi$
$48$	7110.76i	0.445464i
$49$	−2401.00	−0.142857
$50$	0	0
$51$	−51131.3	−2.75272
$52$	− 14136.2i	− 0.724979i
$53$	8829.12i	0.431746i	0.976422	+	0.215873i	$0.0692596\pi$
$53$	8829.12i	0.431746i	−0.976422	+	0.215873i	$0.930740\pi$
$54$	31723.7	1.48047
$55$	0	0
$56$	3136.00	0.133631
$57$	− 67788.5i	− 2.76356i
$58$	2216.62i	0.0865210i
$59$	40580.0	1.51769	0.758843	−	0.651273i	$-0.225765\pi$
$59$	40580.0	1.51769	0.758843	+	0.651273i	$0.225765\pi$
$60$	0	0
$61$	−1675.50	−0.0576527	−0.0288263	−	0.999584i	$-0.509177\pi$
$61$	−1675.50	−0.0576527	−0.0288263	+	0.999584i	$0.509177\pi$
$62$	30938.1i	1.02215i
$63$	− 25897.9i	− 0.822077i
$64$	−4096.00	−0.125000
$65$	0	0
$66$	53516.3	1.51226
$67$	− 9885.31i	− 0.269032i	−0.990911	−	0.134516i	$-0.957052\pi$
$67$	− 9885.31i	− 0.269032i	0.990911	−	0.134516i	$-0.0429479\pi$
$68$	− 29453.1i	− 0.772430i
$69$	−4701.81	−0.118889
$70$	0	0
$71$	−51350.0	−1.20891	−0.604456	−	0.796638i	$-0.706610\pi$
$71$	−51350.0	−1.20891	−0.604456	+	0.796638i	$0.706610\pi$
$72$	33825.8i	0.768983i
$73$	26718.3i	0.586816i	0.955987	+	0.293408i	$0.0947894\pi$
$73$	26718.3i	0.586816i	−0.955987	+	0.293408i	$0.905211\pi$
$74$	45703.2	0.970213
$75$	0	0
$76$	39048.1	0.775472
$77$	− 23601.9i	− 0.453649i
$78$	− 98163.5i	− 1.82689i
$79$	70237.0	1.26619	0.633094	−	0.774075i	$-0.281785\pi$
$79$	70237.0	1.26619	0.633094	+	0.774075i	$0.281785\pi$
$80$	0	0
$81$	91860.4	1.55566
$82$	43574.3i	0.715643i
$83$	58622.5i	0.934047i	0.884245	+	0.467024i	$0.154674\pi$
$83$	58622.5i	0.934047i	−0.884245	+	0.467024i	$0.845326\pi$
$84$	21776.7	0.336739
$85$	0	0
$86$	−61342.0	−0.894358
$87$	15392.4i	0.218027i
$88$	30826.9i	0.424349i
$89$	−28370.7	−0.379661	−0.189830	−	0.981817i	$-0.560794\pi$
$89$	−28370.7	−0.379661	−0.189830	+	0.981817i	$0.560794\pi$
$90$	0	0
$91$	−43292.2	−0.548033
$92$	− 2708.38i	− 0.0333610i
$93$	214837.i	2.57574i
$94$	38823.4	0.453183
$95$	0	0
$96$	−28443.0	−0.314991
$97$	117548.i	1.26849i	0.773134	+	0.634243i	$0.218688\pi$
$97$	117548.i	1.26849i	−0.773134	+	0.634243i	$0.781312\pi$
$98$	− 9604.00i	− 0.101015i
$99$	254576.	2.61054
$100$	0	0
$101$	−25228.5	−0.246087	−0.123043	−	0.992401i	$-0.539265\pi$
$101$	−25228.5	−0.246087	−0.123043	+	0.992401i	$0.539265\pi$
$102$	− 204525.i	− 1.94646i
$103$	99166.7i	0.921028i	0.887653	+	0.460514i	$0.152335\pi$
$103$	99166.7i	0.921028i	−0.887653	+	0.460514i	$0.847665\pi$
$104$	56545.0	0.512638
$105$	0	0
$106$	−35316.5	−0.305290
$107$	− 127493.i	− 1.07653i	−0.842776	−	0.538264i	$-0.819080\pi$
$107$	− 127493.i	− 1.07653i	0.842776	−	0.538264i	$-0.180920\pi$
$108$	126895.i	1.04685i
$109$	−147420.	−1.18848	−0.594238	−	0.804289i	$-0.702546\pi$
$109$	−147420.	−1.18848	−0.594238	+	0.804289i	$0.702546\pi$
$110$	0	0
$111$	317367.	2.44486
$112$	12544.0i	0.0944911i
$113$	− 88159.3i	− 0.649489i	−0.945802	−	0.324745i	$-0.894722\pi$
$113$	− 88159.3i	− 0.649489i	0.945802	−	0.324745i	$-0.105278\pi$
$114$	271154.	1.95413
$115$	0	0
$116$	−8866.49	−0.0611796
$117$	− 466962.i	− 3.15368i
$118$	162320.i	1.07317i
$119$	−90200.2	−0.583902
$120$	0	0
$121$	70955.8	0.440580
$122$	− 6701.99i	− 0.0407666i
$123$	302585.i	1.80337i
$124$	−123752.	−0.722768
$125$	0	0
$126$	103591.	0.581296
$127$	− 89172.9i	− 0.490595i	−0.969448	−	0.245298i	$-0.921114\pi$
$127$	− 89172.9i	− 0.490595i	0.969448	−	0.245298i	$-0.0788857\pi$
$128$	− 16384.0i	− 0.0883883i
$129$	−425965.	−2.25372
$130$	0	0
$131$	298138.	1.51789	0.758944	−	0.651156i	$-0.225716\pi$
$131$	298138.	1.51789	0.758944	+	0.651156i	$0.225716\pi$
$132$	214065.i	1.06933i
$133$	− 119585.i	− 0.586202i
$134$	39541.3	0.190234
$135$	0	0
$136$	117812.	0.546190
$137$	23992.0i	0.109211i	0.998508	+	0.0546053i	$0.0173901\pi$
$137$	23992.0i	0.109211i	−0.998508	+	0.0546053i	$0.982610\pi$
$138$	− 18807.2i	− 0.0840673i
$139$	376423.	1.65249	0.826245	−	0.563311i	$-0.190473\pi$
$139$	376423.	1.65249	0.826245	+	0.563311i	$0.190473\pi$
$140$	0	0
$141$	269593.	1.14199
$142$	− 205400.i	− 0.854830i
$143$	− 425564.i	− 1.74030i
$144$	−135303.	−0.543753
$145$	0	0
$146$	−106873.	−0.414942
$147$	− 66691.1i	− 0.254551i
$148$	182813.i	0.686044i
$149$	8717.80	0.0321693	0.0160846	−	0.999871i	$-0.494880\pi$
$149$	8717.80	0.0321693	0.0160846	+	0.999871i	$0.494880\pi$
$150$	0	0
$151$	−447551.	−1.59735	−0.798676	−	0.601762i	$-0.794466\pi$
$151$	−447551.	−1.59735	−0.798676	+	0.601762i	$0.794466\pi$
$152$	156193.i	0.548342i
$153$	− 972924.i	− 3.36009i
$154$	94407.5	0.320778
$155$	0	0
$156$	392654.	1.29181
$157$	− 63881.6i	− 0.206836i	−0.994638	−	0.103418i	$-0.967022\pi$
$157$	− 63881.6i	− 0.206836i	0.994638	−	0.103418i	$-0.0329780\pi$
$158$	280948.i	0.895330i
$159$	−245241.	−0.769308
$160$	0	0
$161$	−8294.41	−0.0252186
$162$	367441.i	1.10002i
$163$	− 270208.i	− 0.796581i	−0.917259	−	0.398290i	$-0.869604\pi$
$163$	− 270208.i	− 0.796581i	0.917259	−	0.398290i	$-0.130396\pi$
$164$	−174297.	−0.506036
$165$	0	0
$166$	−234490.	−0.660471
$167$	− 575770.i	− 1.59756i	−0.601622	−	0.798781i	$-0.705479\pi$
$167$	− 575770.i	− 1.59756i	0.601622	−	0.798781i	$-0.294521\pi$
$168$	87106.8i	0.238111i
$169$	−409306.	−1.10238
$170$	0	0
$171$	1.28988e6	3.37332
$172$	− 245368.i	− 0.632407i
$173$	408907.i	1.03875i	0.854548	+	0.519373i	$0.173835\pi$
$173$	408907.i	1.03875i	−0.854548	+	0.519373i	$0.826165\pi$
$174$	−61569.8	−0.154168
$175$	0	0
$176$	−123308.	−0.300060
$177$	1.12717e6i	2.70430i
$178$	− 113483.i	− 0.268461i
$179$	−701240.	−1.63581	−0.817907	−	0.575350i	$-0.804866\pi$
$179$	−701240.	−1.63581	−0.817907	+	0.575350i	$0.804866\pi$
$180$	0	0
$181$	−653879.	−1.48355	−0.741773	−	0.670652i	$-0.766015\pi$
$181$	−653879.	−1.48355	−0.741773	+	0.670652i	$0.766015\pi$
$182$	− 173169.i	− 0.387518i
$183$	− 46539.3i	− 0.102729i
$184$	10833.5	0.0235898
$185$	0	0
$186$	−859348.	−1.82132
$187$	− 886669.i	− 1.85420i
$188$	155294.i	0.320449i
$189$	388616.	0.791345
$190$	0	0
$191$	−205677.	−0.407946	−0.203973	−	0.978977i	$-0.565385\pi$
$191$	−205677.	−0.407946	−0.203973	+	0.978977i	$0.565385\pi$
$192$	− 113772.i	− 0.222732i
$193$	163999.i	0.316918i	0.987366	+	0.158459i	$0.0506526\pi$
$193$	163999.i	0.316918i	−0.987366	+	0.158459i	$0.949347\pi$
$194$	−470191.	−0.896954
$195$	0	0
$196$	38416.0	0.0714286
$197$	77300.2i	0.141911i	0.997479	+	0.0709553i	$0.0226048\pi$
$197$	77300.2i	0.141911i	−0.997479	+	0.0709553i	$0.977395\pi$
$198$	1.01831e6i	1.84593i
$199$	466915.	0.835806	0.417903	−	0.908492i	$-0.362765\pi$
$199$	466915.	0.835806	0.417903	+	0.908492i	$0.362765\pi$
$200$	0	0
$201$	274578.	0.479376
$202$	− 100914.i	− 0.174010i
$203$	27153.6i	0.0462474i
$204$	818101.	1.37636
$205$	0	0
$206$	−396667.	−0.651265
$207$	− 89465.8i	− 0.145121i
$208$	226180.i	0.362490i
$209$	1.17552e6	1.86151
$210$	0	0
$211$	1.09847e6	1.69857	0.849284	−	0.527935i	$-0.177034\pi$
$211$	1.09847e6	1.69857	0.849284	+	0.527935i	$0.177034\pi$
$212$	− 141266.i	− 0.215873i
$213$	− 1.42632e6i	− 2.15411i
$214$	509970.	0.761221
$215$	0	0
$216$	−507580.	−0.740235
$217$	378991.i	0.546361i
$218$	− 589680.i	− 0.840379i
$219$	−742138.	−1.04562
$220$	0	0
$221$	−1.62639e6	−2.23998
$222$	1.26947e6i	1.72878i
$223$	− 1.33836e6i	− 1.80224i	−0.433573	−	0.901119i	$-0.642747\pi$
$223$	− 1.33836e6i	− 1.80224i	0.433573	−	0.901119i	$-0.357253\pi$
$224$	−50176.0	−0.0668153
$225$	0	0
$226$	352637.	0.459258
$227$	1.06702e6i	1.37438i	0.726479	+	0.687189i	$0.241156\pi$
$227$	1.06702e6i	1.37438i	−0.726479	+	0.687189i	$0.758844\pi$
$228$	1.08462e6i	1.38178i
$229$	−395172.	−0.497964	−0.248982	−	0.968508i	$-0.580096\pi$
$229$	−395172.	−0.497964	−0.248982	+	0.968508i	$0.580096\pi$
$230$	0	0
$231$	655575.	0.808337
$232$	− 35466.0i	− 0.0432605i
$233$	1.40396e6i	1.69420i	0.531436	+	0.847099i	$0.321653\pi$
$233$	1.40396e6i	1.69420i	−0.531436	+	0.847099i	$0.678347\pi$
$234$	1.86785e6	2.22999
$235$	0	0
$236$	−649280.	−0.758843
$237$	1.95093e6i	2.25616i
$238$	− 360801.i	− 0.412881i
$239$	617173.	0.698896	0.349448	−	0.936956i	$-0.386369\pi$
$239$	617173.	0.698896	0.349448	+	0.936956i	$0.386369\pi$
$240$	0	0
$241$	−1.16204e6	−1.28878	−0.644390	−	0.764697i	$-0.722889\pi$
$241$	−1.16204e6	−1.28878	−0.644390	+	0.764697i	$0.722889\pi$
$242$	283823.i	0.311537i
$243$	624333.i	0.678267i
$244$	26808.0	0.0288263
$245$	0	0
$246$	−1.21034e6	−1.27517
$247$	− 2.15623e6i	− 2.24881i
$248$	− 495009.i	− 0.511074i
$249$	−1.62832e6	−1.66434
$250$	0	0
$251$	−1.39683e6	−1.39946	−0.699730	−	0.714407i	$-0.746697\pi$
$251$	−1.39683e6	−1.39946	−0.699730	+	0.714407i	$0.746697\pi$
$252$	414366.i	0.411039i
$253$	− 81534.2i	− 0.0800826i
$254$	356692.	0.346903
$255$	0	0
$256$	65536.0	0.0625000
$257$	566476.i	0.534994i	0.963559	+	0.267497i	$0.0861966\pi$
$257$	566476.i	0.534994i	−0.963559	+	0.267497i	$0.913803\pi$
$258$	− 1.70386e6i	− 1.59362i
$259$	559864.	0.518601
$260$	0	0
$261$	−292887.	−0.266133
$262$	1.19255e6i	1.07331i
$263$	670643.i	0.597864i	0.954274	+	0.298932i	$0.0966303\pi$
$263$	670643.i	0.597864i	−0.954274	+	0.298932i	$0.903370\pi$
$264$	−856261.	−0.756130
$265$	0	0
$266$	478340.	0.414507
$267$	− 788036.i	− 0.676501i
$268$	158165.i	0.134516i
$269$	−19288.8	−0.0162526	−0.00812632	−	0.999967i	$-0.502587\pi$
$269$	−19288.8	−0.0162526	−0.00812632	+	0.999967i	$0.502587\pi$
$270$	0	0
$271$	−1.92342e6	−1.59093	−0.795466	−	0.605999i	$-0.792774\pi$
$271$	−1.92342e6	−1.59093	−0.795466	+	0.605999i	$0.792774\pi$
$272$	471250.i	0.386215i
$273$	− 1.20250e6i	− 0.976516i
$274$	−95968.0	−0.0772236
$275$	0	0
$276$	75229.0	0.0594446
$277$	− 1.07324e6i	− 0.840423i	−0.907426	−	0.420212i	$-0.861956\pi$
$277$	− 1.07324e6i	− 0.840423i	0.907426	−	0.420212i	$-0.138044\pi$
$278$	1.50569e6i	1.16849i
$279$	−4.08791e6	−3.14406
$280$	0	0
$281$	1.05444e6	0.796627	0.398313	−	0.917249i	$-0.369596\pi$
$281$	1.05444e6	0.796627	0.398313	+	0.917249i	$0.369596\pi$
$282$	1.07837e6i	0.807508i
$283$	2.29016e6i	1.69981i	0.526937	+	0.849904i	$0.323340\pi$
$283$	2.29016e6i	1.69981i	−0.526937	+	0.849904i	$0.676660\pi$
$284$	821600.	0.604456
$285$	0	0
$286$	1.70225e6	1.23058
$287$	533786.i	0.382527i
$288$	− 541212.i	− 0.384491i
$289$	−1.96876e6	−1.38659
$290$	0	0
$291$	−3.26506e6	−2.26026
$292$	− 427493.i	− 0.293408i
$293$	− 2.41541e6i	− 1.64370i	−0.569707	−	0.821848i	$-0.692944\pi$
$293$	− 2.41541e6i	− 1.64370i	0.569707	−	0.821848i	$-0.307056\pi$
$294$	266764.	0.179995
$295$	0	0
$296$	−731251.	−0.485106
$297$	3.82010e6i	2.51295i
$298$	34871.2i	0.0227471i
$299$	−149556.	−0.0967443
$300$	0	0
$301$	−751439.	−0.478055
$302$	− 1.79021e6i	− 1.12950i
$303$	− 700757.i	− 0.438491i
$304$	−624770.	−0.387736
$305$	0	0
$306$	3.89170e6	2.37594
$307$	810075.i	0.490545i	0.969454	+	0.245273i	$0.0788775\pi$
$307$	810075.i	0.490545i	−0.969454	+	0.245273i	$0.921123\pi$
$308$	377630.i	0.226824i
$309$	−2.75449e6	−1.64114
$310$	0	0
$311$	2.00540e6	1.17571	0.587856	−	0.808966i	$-0.299972\pi$
$311$	2.00540e6	1.17571	0.587856	+	0.808966i	$0.299972\pi$
$312$	1.57062e6i	0.913447i
$313$	− 657277.i	− 0.379217i	−0.981860	−	0.189608i	$-0.939278\pi$
$313$	− 657277.i	− 0.379217i	0.981860	−	0.189608i	$-0.0607218\pi$
$314$	255526.	0.146255
$315$	0	0
$316$	−1.12379e6	−0.633094
$317$	− 312965.i	− 0.174923i	−0.996168	−	0.0874616i	$-0.972124\pi$
$317$	− 312965.i	− 0.174923i	0.996168	−	0.0874616i	$-0.0278755\pi$
$318$	− 980965.i	− 0.543983i
$319$	−266921.	−0.146861
$320$	0	0
$321$	3.54128e6	1.91822
$322$	− 33177.6i	− 0.0178322i
$323$	− 4.49254e6i	− 2.39599i
$324$	−1.46977e6	−0.777832
$325$	0	0
$326$	1.08083e6	0.563267
$327$	− 4.09480e6i	− 2.11769i
$328$	− 697190.i	− 0.357821i
$329$	475587.	0.242237
$330$	0	0
$331$	−252813.	−0.126832	−0.0634162	−	0.997987i	$-0.520200\pi$
$331$	−252813.	−0.126832	−0.0634162	+	0.997987i	$0.520200\pi$
$332$	− 937959.i	− 0.467024i
$333$	6.03885e6i	2.98431i
$334$	2.30308e6	1.12965
$335$	0	0
$336$	−348427.	−0.168370
$337$	748958.i	0.359238i	0.983736	+	0.179619i	$0.0574865\pi$
$337$	748958.i	0.359238i	−0.983736	+	0.179619i	$0.942513\pi$
$338$	− 1.63722e6i	− 0.779501i
$339$	2.44875e6	1.15730
$340$	0	0
$341$	−3.72549e6	−1.73499
$342$	5.15951e6i	2.38530i
$343$	− 117649.i	− 0.0539949i
$344$	981471.	0.447179
$345$	0	0
$346$	−1.63563e6	−0.734505
$347$	772587.i	0.344448i	0.985058	+	0.172224i	$0.0550953\pi$
$347$	772587.i	0.344448i	−0.985058	+	0.172224i	$0.944905\pi$
$348$	− 246279.i	− 0.109013i
$349$	−2.00308e6	−0.880309	−0.440155	−	0.897922i	$-0.645076\pi$
$349$	−2.00308e6	−0.880309	−0.440155	+	0.897922i	$0.645076\pi$
$350$	0	0
$351$	7.00710e6	3.03578
$352$	− 493231.i	− 0.212175i
$353$	− 1.10664e6i	− 0.472684i	−0.971670	−	0.236342i	$-0.924052\pi$
$353$	− 1.10664e6i	− 0.472684i	0.971670	−	0.236342i	$-0.0759485\pi$
$354$	−4.50867e6	−1.91223
$355$	0	0
$356$	453932.	0.189830
$357$	− 2.50543e6i	− 1.04043i
$358$	− 2.80496e6i	− 1.15670i
$359$	4.22306e6	1.72938	0.864692	−	0.502302i	$-0.167514\pi$
$359$	4.22306e6	1.72938	0.864692	+	0.502302i	$0.167514\pi$
$360$	0	0
$361$	3.47998e6	1.40543
$362$	− 2.61551e6i	− 1.04902i
$363$	1.97090e6i	0.785050i
$364$	692676.	0.274016
$365$	0	0
$366$	186157.	0.0726402
$367$	− 721326.i	− 0.279555i	−0.990183	−	0.139777i	$-0.955361\pi$
$367$	− 721326.i	− 0.279555i	0.990183	−	0.139777i	$-0.0446387\pi$
$368$	43334.0i	0.0166805i
$369$	−5.75756e6	−2.20127
$370$	0	0
$371$	−432627.	−0.163184
$372$	− 3.43739e6i	− 1.28787i
$373$	− 2.91702e6i	− 1.08559i	−0.839864	−	0.542797i	$-0.817365\pi$
$373$	− 2.91702e6i	− 1.08559i	0.839864	−	0.542797i	$-0.182635\pi$
$374$	3.54668e6	1.31112
$375$	0	0
$376$	−621174.	−0.226592
$377$	489605.i	0.177416i
$378$	1.55446e6i	0.559565i
$379$	75909.9	0.0271457	0.0135728	−	0.999908i	$-0.495679\pi$
$379$	75909.9	0.0271457	0.0135728	+	0.999908i	$0.495679\pi$
$380$	0	0
$381$	2.47690e6	0.874170
$382$	− 822708.i	− 0.288461i
$383$	− 1.48698e6i	− 0.517974i	−0.965881	−	0.258987i	$-0.916611\pi$
$383$	− 1.48698e6i	− 0.517974i	0.965881	−	0.258987i	$-0.0833886\pi$
$384$	455088.	0.157495
$385$	0	0
$386$	−655995.	−0.224095
$387$	− 8.10523e6i	− 2.75098i
$388$	− 1.88077e6i	− 0.634243i
$389$	−2.38812e6	−0.800169	−0.400084	−	0.916478i	$-0.631019\pi$
$389$	−2.38812e6	−0.800169	−0.400084	+	0.916478i	$0.631019\pi$
$390$	0	0
$391$	−311602.	−0.103076
$392$	153664.i	0.0505076i
$393$	8.28121e6i	2.70466i
$394$	−309201.	−0.100346
$395$	0	0
$396$	−4.07322e6	−1.30527
$397$	931473.i	0.296616i	0.988941	+	0.148308i	$0.0473827\pi$
$397$	931473.i	0.296616i	−0.988941	+	0.148308i	$0.952617\pi$
$398$	1.86766e6i	0.591004i
$399$	3.32164e6	1.04453
$400$	0	0
$401$	−502048.	−0.155914	−0.0779568	−	0.996957i	$-0.524840\pi$
$401$	−502048.	−0.155914	−0.0779568	+	0.996957i	$0.524840\pi$
$402$	1.09831e6i	0.338970i
$403$	6.83356e6i	2.09597i
$404$	403656.	0.123043
$405$	0	0
$406$	−108614.	−0.0327019
$407$	5.50347e6i	1.64684i
$408$	3.27240e6i	0.973232i
$409$	−2.46221e6	−0.727807	−0.363903	−	0.931437i	$-0.618556\pi$
$409$	−2.46221e6	−0.727807	−0.363903	+	0.931437i	$0.618556\pi$
$410$	0	0
$411$	−666411.	−0.194598
$412$	− 1.58667e6i	− 0.460514i
$413$	1.98842e6i	0.573632i
$414$	357863.	0.102616
$415$	0	0
$416$	−904720.	−0.256319
$417$	1.04557e7i	2.94450i
$418$	4.70209e6i	1.31628i
$419$	1.72560e6	0.480182	0.240091	−	0.970750i	$-0.422823\pi$
$419$	1.72560e6	0.480182	0.240091	+	0.970750i	$0.422823\pi$
$420$	0	0
$421$	−4.78043e6	−1.31450	−0.657251	−	0.753671i	$-0.728281\pi$
$421$	−4.78043e6	−1.31450	−0.657251	+	0.753671i	$0.728281\pi$
$422$	4.39389e6i	1.20107i
$423$	5.12981e6i	1.39396i
$424$	565064.	0.152645
$425$	0	0
$426$	5.70527e6	1.52318
$427$	− 82099.4i	− 0.0217907i
$428$	2.03988e6i	0.538264i
$429$	1.18206e7	3.10097
$430$	0	0
$431$	−3.18383e6	−0.825574	−0.412787	−	0.910828i	$-0.635445\pi$
$431$	−3.18383e6	−0.825574	−0.412787	+	0.910828i	$0.635445\pi$
$432$	− 2.03032e6i	− 0.523425i
$433$	− 6.03313e6i	− 1.54640i	−0.634159	−	0.773202i	$-0.718654\pi$
$433$	− 6.03313e6i	− 1.54640i	0.634159	−	0.773202i	$-0.281346\pi$
$434$	−1.51597e6	−0.386336
$435$	0	0
$436$	2.35872e6	0.594238
$437$	− 413114.i	− 0.103482i
$438$	− 2.96855e6i	− 0.739366i
$439$	5.13976e6	1.27286	0.636431	−	0.771334i	$-0.280410\pi$
$439$	5.13976e6	1.27286	0.636431	+	0.771334i	$0.280410\pi$
$440$	0	0
$441$	1.26900e6	0.310716
$442$	− 6.50557e6i	− 1.58391i
$443$	1.03796e6i	0.251287i	0.992075	+	0.125644i	$0.0400996\pi$
$443$	1.03796e6i	0.251287i	−0.992075	+	0.125644i	$0.959900\pi$
$444$	−5.07788e6	−1.22243
$445$	0	0
$446$	5.35345e6	1.27437
$447$	242149.i	0.0573210i
$448$	− 200704.i	− 0.0472456i
$449$	7.38979e6	1.72988	0.864940	−	0.501875i	$-0.167356\pi$
$449$	7.38979e6	1.72988	0.864940	+	0.501875i	$0.167356\pi$
$450$	0	0
$451$	−5.24712e6	−1.21473
$452$	1.41055e6i	0.324745i
$453$	− 1.24314e7i	− 2.84625i
$454$	−4.26806e6	−0.971832
$455$	0	0
$456$	−4.33846e6	−0.977066
$457$	− 2.17988e6i	− 0.488250i	−0.969744	−	0.244125i	$-0.921499\pi$
$457$	− 2.17988e6i	− 0.488250i	0.969744	−	0.244125i	$-0.0785006\pi$
$458$	− 1.58069e6i	− 0.352113i
$459$	1.45994e7	3.23447
$460$	0	0
$461$	−7.41009e6	−1.62394	−0.811972	−	0.583696i	$-0.801606\pi$
$461$	−7.41009e6	−1.62394	−0.811972	+	0.583696i	$0.801606\pi$
$462$	2.62230e6i	0.571580i
$463$	− 1.73426e6i	− 0.375978i	−0.982171	−	0.187989i	$-0.939803\pi$
$463$	− 1.73426e6i	− 0.375978i	0.982171	−	0.187989i	$-0.0601970\pi$
$464$	141864.	0.0305898
$465$	0	0
$466$	−5.61583e6	−1.19798
$467$	− 1.41704e6i	− 0.300670i	−0.988635	−	0.150335i	$-0.951965\pi$
$467$	− 1.41704e6i	− 0.300670i	0.988635	−	0.150335i	$-0.0480352\pi$
$468$	7.47140e6i	1.57684i
$469$	484380.	0.101684
$470$	0	0
$471$	1.77440e6	0.368553
$472$	− 2.59712e6i	− 0.536583i
$473$	− 7.38666e6i	− 1.51808i
$474$	−7.80372e6	−1.59535
$475$	0	0
$476$	1.44320e6	0.291951
$477$	− 4.66644e6i	− 0.939051i
$478$	2.46869e6i	0.494194i
$479$	3.71353e6	0.739517	0.369758	−	0.929128i	$-0.379440\pi$
$479$	3.71353e6	0.739517	0.369758	+	0.929128i	$0.379440\pi$
$480$	0	0
$481$	1.00949e7	1.98947
$482$	− 4.64817e6i	− 0.911306i
$483$	− 230389.i	− 0.0449359i
$484$	−1.13529e6	−0.220290
$485$	0	0
$486$	−2.49733e6	−0.479607
$487$	− 6.53152e6i	− 1.24794i	−0.781450	−	0.623968i	$-0.785520\pi$
$487$	− 6.53152e6i	− 1.24794i	0.781450	−	0.623968i	$-0.214480\pi$
$488$	107232.i	0.0203833i
$489$	7.50541e6	1.41939
$490$	0	0
$491$	1.03633e7	1.93997	0.969986	−	0.243159i	$-0.0781836\pi$
$491$	1.03633e7	1.93997	0.969986	+	0.243159i	$0.0781836\pi$
$492$	− 4.84135e6i	− 0.901683i
$493$	1.02010e6i	0.189028i
$494$	8.62491e6	1.59015
$495$	0	0
$496$	1.98004e6	0.361384
$497$	− 2.51615e6i	− 0.456926i
$498$	− 6.51328e6i	− 1.17686i
$499$	1.34074e6	0.241043	0.120521	−	0.992711i	$-0.461543\pi$
$499$	1.34074e6	0.241043	0.120521	+	0.992711i	$0.461543\pi$
$500$	0	0
$501$	1.59928e7	2.84663
$502$	− 5.58734e6i	− 0.989568i
$503$	1.07226e6i	0.188964i	0.995527	+	0.0944820i	$0.0301195\pi$
$503$	1.07226e6i	0.188964i	−0.995527	+	0.0944820i	$0.969881\pi$
$504$	−1.65746e6	−0.290648
$505$	0	0
$506$	326137.	0.0566270
$507$	− 1.13690e7i	− 1.96428i
$508$	1.42677e6i	0.245298i
$509$	−7.26315e6	−1.24260	−0.621299	−	0.783574i	$-0.713395\pi$
$509$	−7.26315e6	−1.24260	−0.621299	+	0.783574i	$0.713395\pi$
$510$	0	0
$511$	−1.30920e6	−0.221796
$512$	262144.i	0.0441942i
$513$	1.93555e7i	3.24721i
$514$	−2.26591e6	−0.378298
$515$	0	0
$516$	6.81543e6	1.12686
$517$	4.67502e6i	0.769232i
$518$	2.23946e6i	0.366706i
$519$	−1.13580e7	−1.85090
$520$	0	0
$521$	−3.31930e6	−0.535738	−0.267869	−	0.963455i	$-0.586320\pi$
$521$	−3.31930e6	−0.535738	−0.267869	+	0.963455i	$0.586320\pi$
$522$	− 1.17155e6i	− 0.188184i
$523$	− 4.01229e6i	− 0.641414i	−0.947178	−	0.320707i	$-0.896080\pi$
$523$	− 4.01229e6i	− 0.641414i	0.947178	−	0.320707i	$-0.103920\pi$
$524$	−4.77021e6	−0.758944
$525$	0	0
$526$	−2.68257e6	−0.422754
$527$	1.42379e7i	2.23315i
$528$	− 3.42504e6i	− 0.534664i
$529$	6.40769e6	0.995548
$530$	0	0
$531$	−2.14477e7	−3.30099
$532$	1.91336e6i	0.293101i
$533$	9.62465e6i	1.46746i
$534$	3.15215e6	0.478358
$535$	0	0
$536$	−632660.	−0.0951171
$537$	− 1.94779e7i	− 2.91479i
$538$	− 77155.1i	− 0.0114923i
$539$	1.15649e6	0.171463
$540$	0	0
$541$	−1.20893e7	−1.77586	−0.887929	−	0.459981i	$-0.847856\pi$
$541$	−1.20893e7	−1.77586	−0.887929	+	0.459981i	$0.847856\pi$
$542$	− 7.69369e6i	− 1.12496i
$543$	− 1.81624e7i	− 2.64346i
$544$	−1.88500e6	−0.273095
$545$	0	0
$546$	4.81001e6	0.690501
$547$	4.43431e6i	0.633663i	0.948482	+	0.316831i	$0.102619\pi$
$547$	4.43431e6i	0.633663i	−0.948482	+	0.316831i	$0.897381\pi$
$548$	− 383872.i	− 0.0546053i
$549$	885548.	0.125395
$550$	0	0
$551$	−1.35242e6	−0.189772
$552$	300916.i	0.0420337i
$553$	3.44161e6i	0.478574i
$554$	4.29297e6	0.594269
$555$	0	0
$556$	−6.02276e6	−0.826245
$557$	390925.i	0.0533895i	0.999644	+	0.0266947i	$0.00849821\pi$
$557$	390925.i	0.0533895i	−0.999644	+	0.0266947i	$0.991502\pi$
$558$	− 1.63516e7i	− 2.22318i
$559$	−1.35491e7	−1.83393
$560$	0	0
$561$	2.46285e7	3.30392
$562$	4.21775e6i	0.563300i
$563$	− 2.64029e6i	− 0.351059i	−0.984474	−	0.175530i	$-0.943836\pi$
$563$	− 2.64029e6i	− 0.351059i	0.984474	−	0.175530i	$-0.0561638\pi$
$564$	−4.31350e6	−0.570994
$565$	0	0
$566$	−9.16065e6	−1.20195
$567$	4.50116e6i	0.587985i
$568$	3.28640e6i	0.427415i
$569$	228685.	0.0296112	0.0148056	−	0.999890i	$-0.495287\pi$
$569$	228685.	0.0296112	0.0148056	+	0.999890i	$0.495287\pi$
$570$	0	0
$571$	5.96584e6	0.765740	0.382870	−	0.923802i	$-0.374936\pi$
$571$	5.96584e6	0.765740	0.382870	+	0.923802i	$0.374936\pi$
$572$	6.80902e6i	0.870150i
$573$	− 5.71296e6i	− 0.726900i
$574$	−2.13514e6	−0.270488
$575$	0	0
$576$	2.16485e6	0.271876
$577$	7.86646e6i	0.983648i	0.870695	+	0.491824i	$0.163670\pi$
$577$	7.86646e6i	0.983648i	−0.870695	+	0.491824i	$0.836330\pi$
$578$	− 7.87504e6i	− 0.980467i
$579$	−4.55530e6	−0.564703
$580$	0	0
$581$	−2.87250e6	−0.353037
$582$	− 1.30602e7i	− 1.59824i
$583$	− 4.25273e6i	− 0.518199i
$584$	1.70997e6	0.207471
$585$	0	0
$586$	9.66163e6	1.16227
$587$	− 1.19474e7i	− 1.43113i	−0.698546	−	0.715565i	$-0.746169\pi$
$587$	− 1.19474e7i	− 1.43113i	0.698546	−	0.715565i	$-0.253831\pi$
$588$	1.06706e6i	0.127275i
$589$	−1.88762e7	−2.24195
$590$	0	0
$591$	−2.14712e6	−0.252864
$592$	− 2.92500e6i	− 0.343022i
$593$	7.64277e6i	0.892512i	0.894905	+	0.446256i	$0.147243\pi$
$593$	7.64277e6i	0.892512i	−0.894905	+	0.446256i	$0.852757\pi$
$594$	−1.52804e7	−1.77692
$595$	0	0
$596$	−139485.	−0.0160846
$597$	1.29692e7i	1.48929i
$598$	− 598223.i	− 0.0684085i
$599$	−615573.	−0.0700991	−0.0350496	−	0.999386i	$-0.511159\pi$
$599$	−615573.	−0.0700991	−0.0350496	+	0.999386i	$0.511159\pi$
$600$	0	0
$601$	−6.68100e6	−0.754493	−0.377247	−	0.926113i	$-0.623129\pi$
$601$	−6.68100e6	−0.754493	−0.377247	+	0.926113i	$0.623129\pi$
$602$	− 3.00576e6i	− 0.338036i
$603$	5.22466e6i	0.585147i
$604$	7.16082e6	0.798676
$605$	0	0
$606$	2.80303e6	0.310060
$607$	810442.i	0.0892792i	0.999003	+	0.0446396i	$0.0142139\pi$
$607$	810442.i	0.0892792i	−0.999003	+	0.0446396i	$0.985786\pi$
$608$	− 2.49908e6i	− 0.274171i
$609$	−754230.	−0.0824063
$610$	0	0
$611$	8.57527e6	0.929276
$612$	1.55668e7i	1.68004i
$613$	− 4.96233e6i	− 0.533377i	−0.963783	−	0.266688i	$-0.914071\pi$
$613$	− 4.96233e6i	− 0.533377i	0.963783	−	0.266688i	$-0.0859294\pi$
$614$	−3.24030e6	−0.346868
$615$	0	0
$616$	−1.51052e6	−0.160389
$617$	− 2.35809e6i	− 0.249371i	−0.992196	−	0.124686i	$-0.960208\pi$
$617$	− 2.35809e6i	− 0.249371i	0.992196	−	0.124686i	$-0.0397923\pi$
$618$	− 1.10180e7i	− 1.16046i
$619$	−7.68885e6	−0.806556	−0.403278	−	0.915077i	$-0.632129\pi$
$619$	−7.68885e6	−0.806556	−0.403278	+	0.915077i	$0.632129\pi$
$620$	0	0
$621$	1.34250e6	0.139696
$622$	8.02162e6i	0.831354i
$623$	− 1.39017e6i	− 0.143498i
$624$	−6.28246e6	−0.645905
$625$	0	0
$626$	2.62911e6	0.268147
$627$	3.26517e7i	3.31694i
$628$	1.02211e6i	0.103418i
$629$	2.10328e7	2.11968
$630$	0	0
$631$	−1.65595e7	−1.65567	−0.827835	−	0.560971i	$-0.810428\pi$
$631$	−1.65595e7	−1.65567	−0.827835	+	0.560971i	$0.810428\pi$
$632$	− 4.49517e6i	− 0.447665i
$633$	3.05116e7i	3.02661i
$634$	1.25186e6	0.123689
$635$	0	0
$636$	3.92386e6	0.384654
$637$	− 2.12132e6i	− 0.207137i
$638$	− 1.06768e6i	− 0.103846i
$639$	2.71399e7	2.62940
$640$	0	0
$641$	3.81994e6	0.367207	0.183604	−	0.983000i	$-0.441224\pi$
$641$	3.81994e6	0.367207	0.183604	+	0.983000i	$0.441224\pi$
$642$	1.41651e7i	1.35639i
$643$	1.54152e7i	1.47035i	0.677877	+	0.735176i	$0.262900\pi$
$643$	1.54152e7i	1.47035i	−0.677877	+	0.735176i	$0.737100\pi$
$644$	132711.	0.0126093
$645$	0	0
$646$	1.79701e7	1.69422
$647$	1.24057e7i	1.16509i	0.812798	+	0.582546i	$0.197943\pi$
$647$	1.24057e7i	1.16509i	−0.812798	+	0.582546i	$0.802057\pi$
$648$	− 5.87906e6i	− 0.550010i
$649$	−1.95462e7	−1.82159
$650$	0	0
$651$	−1.05270e7	−0.973537
$652$	4.32333e6i	0.398290i
$653$	9.76650e6i	0.896306i	0.893957	+	0.448153i	$0.147918\pi$
$653$	9.76650e6i	0.896306i	−0.893957	+	0.448153i	$0.852082\pi$
$654$	1.63792e7	1.49744
$655$	0	0
$656$	2.78876e6	0.253018
$657$	− 1.41214e7i	− 1.27633i
$658$	1.90235e6i	0.171287i
$659$	−1.51635e7	−1.36014	−0.680072	−	0.733146i	$-0.738051\pi$
$659$	−1.51635e7	−1.36014	−0.680072	+	0.733146i	$0.738051\pi$
$660$	0	0
$661$	1.84762e6	0.164479	0.0822394	−	0.996613i	$-0.473793\pi$
$661$	1.84762e6	0.164479	0.0822394	+	0.996613i	$0.473793\pi$
$662$	− 1.01125e6i	− 0.0896840i
$663$	− 4.51753e7i	− 3.99133i
$664$	3.75184e6	0.330236
$665$	0	0
$666$	−2.41554e7	−2.11022
$667$	93803.9i	0.00816406i
$668$	9.21232e6i	0.798781i
$669$	3.71749e7	3.21133
$670$	0	0
$671$	807039.	0.0691971
$672$	− 1.39371e6i	− 0.119055i
$673$	1.70205e7i	1.44855i	0.689511	+	0.724275i	$0.257825\pi$
$673$	1.70205e7i	1.44855i	−0.689511	+	0.724275i	$0.742175\pi$
$674$	−2.99583e6	−0.254020
$675$	0	0
$676$	6.54890e6	0.551190
$677$	5.61056e6i	0.470472i	0.971938	+	0.235236i	$0.0755864\pi$
$677$	5.61056e6i	0.470472i	−0.971938	+	0.235236i	$0.924414\pi$
$678$	9.79499e6i	0.818332i
$679$	−5.75985e6	−0.479442
$680$	0	0
$681$	−2.96378e7	−2.44894
$682$	− 1.49020e7i	− 1.22683i
$683$	1.73009e7i	1.41911i	0.704649	+	0.709556i	$0.251105\pi$
$683$	1.73009e7i	1.41911i	−0.704649	+	0.709556i	$0.748895\pi$
$684$	−2.06380e7	−1.68666
$685$	0	0
$686$	470596.	0.0381802
$687$	− 1.09765e7i	− 0.887299i
$688$	3.92589e6i	0.316203i
$689$	−7.80066e6	−0.626013
$690$	0	0
$691$	−9.30689e6	−0.741497	−0.370748	−	0.928733i	$-0.620899\pi$
$691$	−9.30689e6	−0.741497	−0.370748	+	0.928733i	$0.620899\pi$
$692$	− 6.54251e6i	− 0.519373i
$693$	1.24742e7i	0.986691i
$694$	−3.09035e6	−0.243562
$695$	0	0
$696$	985116.	0.0770840
$697$	2.00531e7i	1.56351i
$698$	− 8.01233e6i	− 0.622473i
$699$	−3.89969e7	−3.01882
$700$	0	0
$701$	1.44180e7	1.10818	0.554090	−	0.832457i	$-0.313066\pi$
$701$	1.44180e7	1.10818	0.554090	+	0.832457i	$0.313066\pi$
$702$	2.80284e7i	2.14662i
$703$	2.78847e7i	2.12803i
$704$	1.97292e6	0.150030
$705$	0	0
$706$	4.42657e6	0.334238
$707$	− 1.23620e6i	− 0.0930120i
$708$	− 1.80347e7i	− 1.35215i
$709$	1.00341e7	0.749656	0.374828	−	0.927094i	$-0.377702\pi$
$709$	1.00341e7	0.749656	0.374828	+	0.927094i	$0.377702\pi$
$710$	0	0
$711$	−3.71222e7	−2.75397
$712$	1.81573e6i	0.134230i
$713$	1.30925e6i	0.0964492i
$714$	1.00217e7	0.735695
$715$	0	0
$716$	1.12198e7	0.817907
$717$	1.71428e7i	1.24533i
$718$	1.68923e7i	1.22286i
$719$	1.43364e7	1.03423	0.517114	−	0.855916i	$-0.327006\pi$
$719$	1.43364e7	1.03423	0.517114	+	0.855916i	$0.327006\pi$
$720$	0	0
$721$	−4.85917e6	−0.348116
$722$	1.39199e7i	0.993789i
$723$	− 3.22773e7i	− 2.29642i
$724$	1.04621e7	0.741773
$725$	0	0
$726$	−7.88358e6	−0.555114
$727$	− 577386.i	− 0.0405164i	−0.999795	−	0.0202582i	$-0.993551\pi$
$727$	− 577386.i	− 0.0405164i	0.999795	−	0.0202582i	$-0.00644882\pi$
$728$	2.77070e6i	0.193759i
$729$	4.98035e6	0.347089
$730$	0	0
$731$	−2.82299e7	−1.95396
$732$	744629.i	0.0513644i
$733$	2.12449e7i	1.46048i	0.683191	+	0.730240i	$0.260592\pi$
$733$	2.12449e7i	1.46048i	−0.683191	+	0.730240i	$0.739408\pi$
$734$	2.88530e6	0.197675
$735$	0	0
$736$	−173336.	−0.0117949
$737$	4.76147e6i	0.322903i
$738$	− 2.30303e7i	− 1.55653i
$739$	−1.09365e7	−0.736662	−0.368331	−	0.929695i	$-0.620071\pi$
$739$	−1.09365e7	−0.736662	−0.368331	+	0.929695i	$0.620071\pi$
$740$	0	0
$741$	5.98922e7	4.00705
$742$	− 1.73051e6i	− 0.115389i
$743$	− 2.42027e7i	− 1.60839i	−0.594366	−	0.804195i	$-0.702597\pi$
$743$	− 2.42027e7i	− 1.60839i	0.594366	−	0.804195i	$-0.297403\pi$
$744$	1.37496e7	0.910661
$745$	0	0
$746$	1.16681e7	0.767631
$747$	− 3.09836e7i	− 2.03156i
$748$	1.41867e7i	0.927102i
$749$	6.24714e6	0.406890
$750$	0	0
$751$	−1.22083e7	−0.789871	−0.394936	−	0.918709i	$-0.629233\pi$
$751$	−1.22083e7	−0.789871	−0.394936	+	0.918709i	$0.629233\pi$
$752$	− 2.48470e6i	− 0.160225i
$753$	− 3.87990e7i	− 2.49364i
$754$	−1.95842e6	−0.125452
$755$	0	0
$756$	−6.21785e6	−0.395672
$757$	6.73453e6i	0.427138i	0.976928	+	0.213569i	$0.0685088\pi$
$757$	6.73453e6i	0.427138i	−0.976928	+	0.213569i	$0.931491\pi$
$758$	303640.i	0.0191949i
$759$	2.26472e6	0.142696
$760$	0	0
$761$	1.28864e7	0.806625	0.403312	−	0.915062i	$-0.367859\pi$
$761$	1.28864e7	0.806625	0.403312	+	0.915062i	$0.367859\pi$
$762$	9.90760e6i	0.618132i
$763$	− 7.22358e6i	− 0.449202i
$764$	3.29083e6	0.203973
$765$	0	0
$766$	5.94791e6	0.366263
$767$	3.58531e7i	2.20058i
$768$	1.82035e6i	0.111366i
$769$	−1.11806e7	−0.681787	−0.340893	−	0.940102i	$-0.610730\pi$
$769$	−1.11806e7	−0.681787	−0.340893	+	0.940102i	$0.610730\pi$
$770$	0	0
$771$	−1.57347e7	−0.953283
$772$	− 2.62398e6i	− 0.158459i
$773$	2.70973e6i	0.163109i	0.996669	+	0.0815543i	$0.0259884\pi$
$773$	2.70973e6i	0.163109i	−0.996669	+	0.0815543i	$0.974012\pi$
$774$	3.24209e7	1.94524
$775$	0	0
$776$	7.52306e6	0.448477
$777$	1.55510e7i	0.924072i
$778$	− 9.55247e6i	− 0.565805i
$779$	−2.65859e7	−1.56967
$780$	0	0
$781$	2.47338e7	1.45099
$782$	− 1.24641e6i	− 0.0728859i
$783$	− 4.39497e6i	− 0.256184i
$784$	−614656.	−0.0357143
$785$	0	0
$786$	−3.31248e7	−1.91248
$787$	7.17969e6i	0.413208i	0.978425	+	0.206604i	$0.0662412\pi$
$787$	7.17969e6i	0.413208i	−0.978425	+	0.206604i	$0.933759\pi$
$788$	− 1.23680e6i	− 0.0709553i
$789$	−1.86281e7	−1.06531
$790$	0	0
$791$	4.31981e6	0.245484
$792$	− 1.62929e7i	− 0.922965i
$793$	− 1.48033e6i	− 0.0835940i
$794$	−3.72589e6	−0.209739
$795$	0	0
$796$	−7.47064e6	−0.417903
$797$	2.09215e7i	1.16667i	0.812232	+	0.583335i	$0.198252\pi$
$797$	2.09215e7i	1.16667i	−0.812232	+	0.583335i	$0.801748\pi$
$798$	1.32865e7i	0.738593i
$799$	1.78667e7	0.990097
$800$	0	0
$801$	1.49947e7	0.825766
$802$	− 2.00819e6i	− 0.110248i
$803$	− 1.28694e7i	− 0.704321i
$804$	−4.39325e6	−0.239688
$805$	0	0
$806$	−2.73343e7	−1.48207
$807$	− 535772.i	− 0.0289599i
$808$	1.61462e6i	0.0870048i
$809$	8.14429e6	0.437504	0.218752	−	0.975780i	$-0.429801\pi$
$809$	8.14429e6	0.437504	0.218752	+	0.975780i	$0.429801\pi$
$810$	0	0
$811$	−1.99391e7	−1.06452	−0.532260	−	0.846581i	$-0.678657\pi$
$811$	−1.99391e7	−1.06452	−0.532260	+	0.846581i	$0.678657\pi$
$812$	− 434458.i	− 0.0231237i
$813$	− 5.34257e7i	− 2.83481i
$814$	−2.20139e7	−1.16449
$815$	0	0
$816$	−1.30896e7	−0.688179
$817$	− 3.74264e7i	− 1.96166i
$818$	− 9.84882e6i	− 0.514637i
$819$	2.28812e7	1.19198
$820$	0	0
$821$	3.57285e7	1.84994	0.924968	−	0.380046i	$-0.124092\pi$
$821$	3.57285e7	1.84994	0.924968	+	0.380046i	$0.124092\pi$
$822$	− 2.66564e6i	− 0.137601i
$823$	1.29131e7i	0.664556i	0.943182	+	0.332278i	$0.107817\pi$
$823$	1.29131e7i	0.664556i	−0.943182	+	0.332278i	$0.892183\pi$
$824$	6.34667e6	0.325632
$825$	0	0
$826$	−7.95368e6	−0.405619
$827$	− 365916.i	− 0.0186045i	−0.999957	−	0.00930225i	$-0.997039\pi$
$827$	− 365916.i	− 0.0186045i	0.999957	−	0.00930225i	$-0.00296104\pi$
$828$	1.43145e6i	0.0725607i
$829$	−2.46028e7	−1.24336	−0.621682	−	0.783270i	$-0.713550\pi$
$829$	−2.46028e7	−1.24336	−0.621682	+	0.783270i	$0.713550\pi$
$830$	0	0
$831$	2.98108e7	1.49751
$832$	− 3.61888e6i	− 0.181245i
$833$	− 4.41981e6i	− 0.220694i
$834$	−4.18227e7	−2.08208
$835$	0	0
$836$	−1.88083e7	−0.930754
$837$	− 6.13419e7i	− 3.02652i
$838$	6.90241e6i	0.339540i
$839$	1.60455e7	0.786950	0.393475	−	0.919335i	$-0.371273\pi$
$839$	1.60455e7	0.786950	0.393475	+	0.919335i	$0.371273\pi$
$840$	0	0
$841$	−2.02041e7	−0.985028
$842$	− 1.91217e7i	− 0.929494i
$843$	2.92885e7i	1.41947i
$844$	−1.75756e7	−0.849284
$845$	0	0
$846$	−2.05192e7	−0.985679
$847$	3.47683e6i	0.166523i
$848$	2.26026e6i	0.107936i
$849$	−6.36124e7	−3.02881
$850$	0	0
$851$	1.93409e6	0.0915486
$852$	2.28211e7i	1.07705i
$853$	9.50336e6i	0.447203i	0.974681	+	0.223601i	$0.0717814\pi$
$853$	9.50336e6i	0.447203i	−0.974681	+	0.223601i	$0.928219\pi$
$854$	328398.	0.0154083
$855$	0	0
$856$	−8.15953e6	−0.380610
$857$	8.55821e6i	0.398044i	0.979995	+	0.199022i	$0.0637765\pi$
$857$	8.55821e6i	0.398044i	−0.979995	+	0.199022i	$0.936223\pi$
$858$	4.72825e7i	2.19271i
$859$	−3.73000e7	−1.72475	−0.862376	−	0.506269i	$-0.831024\pi$
$859$	−3.73000e7	−1.72475	−0.862376	+	0.506269i	$0.831024\pi$
$860$	0	0
$861$	−1.48266e7	−0.681608
$862$	− 1.27353e7i	− 0.583769i
$863$	− 1.41708e7i	− 0.647690i	−0.946110	−	0.323845i	$-0.895024\pi$
$863$	− 1.41708e7i	− 0.647690i	0.946110	−	0.323845i	$-0.104976\pi$
$864$	8.12127e6	0.370118
$865$	0	0
$866$	2.41325e7	1.09347
$867$	− 5.46850e7i	− 2.47070i
$868$	− 6.06386e6i	− 0.273181i
$869$	−3.38311e7	−1.51973
$870$	0	0
$871$	8.73382e6	0.390085
$872$	9.43488e6i	0.420190i
$873$	− 6.21273e7i	− 2.75897i
$874$	1.65245e6	0.0731730
$875$	0	0
$876$	1.18742e7	0.522811
$877$	− 2.66505e7i	− 1.17006i	−0.811013	−	0.585028i	$-0.801083\pi$
$877$	− 2.66505e7i	− 1.17006i	0.811013	−	0.585028i	$-0.198917\pi$
$878$	2.05590e7i	0.900049i
$879$	6.70913e7	2.92883
$880$	0	0
$881$	2.56753e7	1.11449	0.557245	−	0.830348i	$-0.311859\pi$
$881$	2.56753e7	1.11449	0.557245	+	0.830348i	$0.311859\pi$
$882$	5.07598e6i	0.219709i
$883$	− 9.63276e6i	− 0.415766i	−0.978154	−	0.207883i	$-0.933343\pi$
$883$	− 9.63276e6i	− 0.415766i	0.978154	−	0.207883i	$-0.0666573\pi$
$884$	2.60223e7	1.11999
$885$	0	0
$886$	−4.15183e6	−0.177687
$887$	6.02461e6i	0.257110i	0.991702	+	0.128555i	$0.0410340\pi$
$887$	6.02461e6i	0.257110i	−0.991702	+	0.128555i	$0.958966\pi$
$888$	− 2.03115e7i	− 0.864390i
$889$	4.36947e6	0.185428
$890$	0	0
$891$	−4.42465e7	−1.86717
$892$	2.14138e7i	0.901119i
$893$	2.36872e7i	0.993997i
$894$	−968596.	−0.0405321
$895$	0	0
$896$	802816.	0.0334077
$897$	− 4.15412e6i	− 0.172384i
$898$	2.95592e7i	1.22321i
$899$	4.28613e6	0.176875
$900$	0	0
$901$	−1.62528e7	−0.666986
$902$	− 2.09885e7i	− 0.858944i
$903$	− 2.08723e7i	− 0.851825i
$904$	−5.64219e6	−0.229629
$905$	0	0
$906$	4.97254e7	2.01260
$907$	− 1.62666e7i	− 0.656566i	−0.944579	−	0.328283i	$-0.893530\pi$
$907$	− 1.62666e7i	− 0.656566i	0.944579	−	0.328283i	$-0.106470\pi$
$908$	− 1.70723e7i	− 0.687189i
$909$	1.33340e7	0.535241
$910$	0	0
$911$	2.90438e7	1.15946	0.579732	−	0.814807i	$-0.303157\pi$
$911$	2.90438e7	1.15946	0.579732	+	0.814807i	$0.303157\pi$
$912$	− 1.73539e7i	− 0.690890i
$913$	− 2.82367e7i	− 1.12108i
$914$	8.71952e6	0.345245
$915$	0	0
$916$	6.32275e6	0.248982
$917$	1.46088e7i	0.573708i
$918$	5.83976e7i	2.28712i
$919$	4.01091e7	1.56659	0.783293	−	0.621652i	$-0.213538\pi$
$919$	4.01091e7	1.56659	0.783293	+	0.621652i	$0.213538\pi$
$920$	0	0
$921$	−2.25009e7	−0.874081
$922$	− 2.96404e7i	− 1.14830i
$923$	− 4.53685e7i	− 1.75287i
$924$	−1.04892e7	−0.404168
$925$	0	0
$926$	6.93705e6	0.265857
$927$	− 5.24123e7i	− 2.00325i
$928$	567455.i	0.0216303i
$929$	−1.54551e7	−0.587533	−0.293766	−	0.955877i	$-0.594909\pi$
$929$	−1.54551e7	−0.587533	−0.293766	+	0.955877i	$0.594909\pi$
$930$	0	0
$931$	5.85966e6	0.221564
$932$	− 2.24633e7i	− 0.847099i
$933$	5.57029e7i	2.09495i
$934$	5.66816e6	0.212606
$935$	0	0
$936$	−2.98856e7	−1.11499
$937$	− 4.63024e7i	− 1.72288i	−0.507861	−	0.861439i	$-0.669564\pi$
$937$	− 4.63024e7i	− 1.72288i	0.507861	−	0.861439i	$-0.330436\pi$
$938$	1.93752e6i	0.0719017i
$939$	1.82568e7	0.675710
$940$	0	0
$941$	1.24284e7	0.457553	0.228777	−	0.973479i	$-0.426527\pi$
$941$	1.24284e7	0.457553	0.228777	+	0.973479i	$0.426527\pi$
$942$	7.09760e6i	0.260606i
$943$	1.84400e6i	0.0675275i
$944$	1.03885e7	0.379422
$945$	0	0
$946$	2.95466e7	1.07345
$947$	3.59653e7i	1.30319i	0.758565	+	0.651597i	$0.225901\pi$
$947$	3.59653e7i	1.30319i	−0.758565	+	0.651597i	$0.774099\pi$
$948$	− 3.12149e7i	− 1.12808i
$949$	−2.36060e7	−0.850859
$950$	0	0
$951$	8.69303e6	0.311688
$952$	5.77281e6i	0.206440i
$953$	1.07512e7i	0.383465i	0.981447	+	0.191733i	$0.0614106\pi$
$953$	1.07512e7i	0.383465i	−0.981447	+	0.191733i	$0.938589\pi$
$954$	1.86657e7	0.664010
$955$	0	0
$956$	−9.87477e6	−0.349448
$957$	− 7.41409e6i	− 0.261684i
$958$	1.48541e7i	0.522917i
$959$	−1.17561e6	−0.0412777
$960$	0	0
$961$	3.11936e7	1.08958
$962$	4.03794e7i	1.40677i
$963$	6.73834e7i	2.34146i
$964$	1.85927e7	0.644390
$965$	0	0
$966$	921555.	0.0317745
$967$	1.57963e7i	0.543235i	0.962405	+	0.271618i	$0.0875586\pi$
$967$	1.57963e7i	0.543235i	−0.962405	+	0.271618i	$0.912441\pi$
$968$	− 4.54117e6i	− 0.155768i
$969$	1.24786e8	4.26931
$970$	0	0
$971$	2.98682e7	1.01662	0.508312	−	0.861173i	$-0.330270\pi$
$971$	2.98682e7	1.01662	0.508312	+	0.861173i	$0.330270\pi$
$972$	− 9.98933e6i	− 0.339133i
$973$	1.84447e7i	0.624582i
$974$	2.61261e7	0.882424
$975$	0	0
$976$	−428928.	−0.0144132
$977$	− 8.43017e6i	− 0.282553i	−0.989970	−	0.141276i	$-0.954879\pi$
$977$	− 8.43017e6i	− 0.282553i	0.989970	−	0.141276i	$-0.0451206\pi$
$978$	3.00216e7i	1.00366i
$979$	1.36654e7	0.455684
$980$	0	0
$981$	7.79156e7	2.58495
$982$	4.14533e7i	1.37177i
$983$	− 4.42301e7i	− 1.45994i	−0.683480	−	0.729969i	$-0.739535\pi$
$983$	− 4.42301e7i	− 1.45994i	0.683480	−	0.729969i	$-0.260465\pi$
$984$	1.93654e7	0.637586
$985$	0	0
$986$	−4.08040e6	−0.133663
$987$	1.32101e7i	0.431631i
$988$	3.44996e7i	1.12440i
$989$	−2.59589e6	−0.0843910
$990$	0	0
$991$	−7.69521e6	−0.248907	−0.124453	−	0.992225i	$-0.539718\pi$
$991$	−7.69521e6	−0.248907	−0.124453	+	0.992225i	$0.539718\pi$
$992$	7.92015e6i	0.255537i
$993$	− 7.02224e6i	− 0.225997i
$994$	1.00646e7	0.323095
$995$	0	0
$996$	2.60531e7	0.832169
$997$	− 4.71077e6i	− 0.150091i	−0.997180	−	0.0750453i	$-0.976090\pi$
$997$	− 4.71077e6i	− 0.150091i	0.997180	−	0.0750453i	$-0.0239101\pi$
$998$	5.36297e6i	0.170443i
$999$	−9.06172e7	−2.87274

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	350.6.c.i.99.4		4
5.2	odd	4		350.6.a.r.1.2	✓	2
5.3	odd	4		350.6.a.s.1.1	yes	2
5.4	even	2	inner	350.6.c.i.99.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
350.6.a.r.1.2	✓	2	5.2	odd	4
350.6.a.s.1.1	yes	2	5.3	odd	4
350.6.c.i.99.1		4	5.4	even	2	inner
350.6.c.i.99.4		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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 350 = 2 \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	350.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+4.00000i q^{2} +27.7764i q^{3} -16.0000 q^{4} -111.106 q^{6} +49.0000i q^{7} -64.0000i q^{8} -528.528 q^{9} +O(q^{10})\)	\(q+4.00000i q^{2} +27.7764i q^{3} -16.0000 q^{4} -111.106 q^{6} +49.0000i q^{7} -64.0000i q^{8} -528.528 q^{9} -481.671 q^{11} -444.422i q^{12} +883.515i q^{13} -196.000 q^{14} +256.000 q^{16} +1840.82i q^{17} -2114.11i q^{18} -2440.51 q^{19} -1361.04 q^{21} -1926.68i q^{22} +169.274i q^{23} +1777.69 q^{24} -3534.06 q^{26} -7930.93i q^{27} -784.000i q^{28} +554.156 q^{29} +7734.52 q^{31} +1024.00i q^{32} -13379.1i q^{33} -7363.28 q^{34} +8456.44 q^{36} -11425.8i q^{37} -9762.03i q^{38} -24540.9 q^{39} +10893.6 q^{41} -5444.17i q^{42} +15335.5i q^{43} +7706.73 q^{44} -677.094 q^{46} -9705.85i q^{47} +7110.76i q^{48} -2401.00 q^{49} -51131.3 q^{51} -14136.2i q^{52} +8829.12i q^{53} +31723.7 q^{54} +3136.00 q^{56} -67788.5i q^{57} +2216.62i q^{58} +40580.0 q^{59} -1675.50 q^{61} +30938.1i q^{62} -25897.9i q^{63} -4096.00 q^{64} +53516.3 q^{66} -9885.31i q^{67} -29453.1i q^{68} -4701.81 q^{69} -51350.0 q^{71} +33825.8i q^{72} +26718.3i q^{73} +45703.2 q^{74} +39048.1 q^{76} -23601.9i q^{77} -98163.5i q^{78} +70237.0 q^{79} +91860.4 q^{81} +43574.3i q^{82} +58622.5i q^{83} +21776.7 q^{84} -61342.0 q^{86} +15392.4i q^{87} +30826.9i q^{88} -28370.7 q^{89} -43292.2 q^{91} -2708.38i q^{92} +214837. i q^{93} +38823.4 q^{94} -28443.0 q^{96} +117548. i q^{97} -9604.00i q^{98} +254576. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 64 q^{4} - 160 q^{6} - 692 q^{9}+O(q^{10}) \) 4 * q - 64 * q^4 - 160 * q^6 - 692 * q^9	\( 4 q - 64 q^{4} - 160 q^{6} - 692 q^{9} - 2140 q^{11} - 784 q^{14} + 1024 q^{16} - 1656 q^{19} - 1960 q^{21} + 2560 q^{24} - 5888 q^{26} + 4492 q^{29} + 576 q^{31} - 15232 q^{34} + 11072 q^{36} - 51376 q^{39} + 22456 q^{41} + 34240 q^{44} - 13232 q^{46} - 9604 q^{49} - 101280 q^{51} + 89920 q^{54} + 12544 q^{56} + 185856 q^{59} + 30344 q^{61} - 16384 q^{64} + 70432 q^{66} + 13688 q^{69} - 41644 q^{71} + 71888 q^{74} + 26496 q^{76} + 144212 q^{79} + 220964 q^{81} + 31360 q^{84} - 27216 q^{86} + 19840 q^{89} - 72128 q^{91} + 124576 q^{94} - 40960 q^{96} + 294380 q^{99}+O(q^{100}) \) 4 * q - 64 * q^4 - 160 * q^6 - 692 * q^9 - 2140 * q^11 - 784 * q^14 + 1024 * q^16 - 1656 * q^19 - 1960 * q^21 + 2560 * q^24 - 5888 * q^26 + 4492 * q^29 + 576 * q^31 - 15232 * q^34 + 11072 * q^36 - 51376 * q^39 + 22456 * q^41 + 34240 * q^44 - 13232 * q^46 - 9604 * q^49 - 101280 * q^51 + 89920 * q^54 + 12544 * q^56 + 185856 * q^59 + 30344 * q^61 - 16384 * q^64 + 70432 * q^66 + 13688 * q^69 - 41644 * q^71 + 71888 * q^74 + 26496 * q^76 + 144212 * q^79 + 220964 * q^81 + 31360 * q^84 - 27216 * q^86 + 19840 * q^89 - 72128 * q^91 + 124576 * q^94 - 40960 * q^96 + 294380 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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