LMFDB - Embedded newform 784.5.c.a.97.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [784,5,Mod(97,784)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(784, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 5, names="a")

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

chi := DirichletCharacter("784.97");

S:= CuspForms(chi, 5);

N := Newforms(S);

Level:	$ N $	$=$	$ 784 = 2^{4} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	784.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:	$81.0420510577$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.0.2048.2
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 4x^{2} + 2 $ x^4 + 4*x^2 + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 98)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			97.3
Root			$-1.84776i$ of defining polynomial
Character	$\chi$	$=$	784.97
Dual form			784.5.c.a.97.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.03347i q^{3} -0.710974i q^{5} +76.8650 q^{9} +O(q^{10})$	$q+2.03347i q^{3} -0.710974i q^{5} +76.8650 q^{9} -151.664 q^{11} -260.864i q^{13} +1.44574 q^{15} +385.598i q^{17} +390.140i q^{19} +177.647 q^{23} +624.495 q^{25} +321.013i q^{27} -320.887 q^{29} -1346.37i q^{31} -308.404i q^{33} -797.088 q^{37} +530.459 q^{39} -815.856i q^{41} +2167.70 q^{43} -54.6490i q^{45} -4285.61i q^{47} -784.102 q^{51} -3171.57 q^{53} +107.829i q^{55} -793.337 q^{57} +4706.39i q^{59} +2534.78i q^{61} -185.468 q^{65} +4092.43 q^{67} +361.239i q^{69} +2255.28 q^{71} +4653.19i q^{73} +1269.89i q^{75} +4193.74 q^{79} +5573.29 q^{81} -7799.12i q^{83} +274.150 q^{85} -652.514i q^{87} -9469.11i q^{89} +2737.81 q^{93} +277.379 q^{95} -9945.39i q^{97} -11657.7 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 196 q^{9}+O(q^{10}) $ 4 * q - 196 * q^9	$ 4 q - 196 q^{9} - 24 q^{11} - 888 q^{15} - 104 q^{23} + 948 q^{25} - 1408 q^{29} - 3392 q^{37} - 2200 q^{39} + 2024 q^{43} - 18936 q^{51} - 16680 q^{53} - 6064 q^{57} - 6048 q^{65} + 20816 q^{67} + 1984 q^{71} + 29616 q^{79} + 30852 q^{81} - 29688 q^{85} + 18192 q^{93} - 7240 q^{95} - 72160 q^{99}+O(q^{100}) $ 4 * q - 196 * q^9 - 24 * q^11 - 888 * q^15 - 104 * q^23 + 948 * q^25 - 1408 * q^29 - 3392 * q^37 - 2200 * q^39 + 2024 * q^43 - 18936 * q^51 - 16680 * q^53 - 6064 * q^57 - 6048 * q^65 + 20816 * q^67 + 1984 * q^71 + 29616 * q^79 + 30852 * q^81 - 29688 * q^85 + 18192 * q^93 - 7240 * q^95 - 72160 * q^99

Display 100 coefficients 10 coefficients

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
$3$	2.03347i	0.225941i	0.993598	+	0.112970i	$0.0360365\pi$
$3$	2.03347i	0.225941i	−0.993598	+	0.112970i	$0.963963\pi$
$4$	0	0
$5$	− 0.710974i	− 0.0284390i	−0.999899	−	0.0142195i	$-0.995474\pi$
$5$	− 0.710974i	− 0.0284390i	0.999899	−	0.0142195i	$-0.00452635\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	76.8650	0.948951
$10$	0	0
$11$	−151.664	−1.25342	−0.626711	−	0.779252i	$-0.715599\pi$
$11$	−151.664	−1.25342	−0.626711	+	0.779252i	$0.715599\pi$
$12$	0	0
$13$	− 260.864i	− 1.54358i	−0.635880	−	0.771788i	$-0.719363\pi$
$13$	− 260.864i	− 1.54358i	0.635880	−	0.771788i	$-0.280637\pi$
$14$	0	0
$15$	1.44574	0.00642552
$16$	0	0
$17$	385.598i	1.33425i	0.744946	+	0.667125i	$0.232475\pi$
$17$	385.598i	1.33425i	−0.744946	+	0.667125i	$0.767525\pi$
$18$	0	0
$19$	390.140i	1.08072i	0.841434	+	0.540360i	$0.181712\pi$
$19$	390.140i	1.08072i	−0.841434	+	0.540360i	$0.818288\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	177.647	0.335816	0.167908	−	0.985803i	$-0.446299\pi$
$23$	177.647	0.335816	0.167908	+	0.985803i	$0.446299\pi$
$24$	0	0
$25$	624.495	0.999191
$26$	0	0
$27$	321.013i	0.440348i
$28$	0	0
$29$	−320.887	−0.381554	−0.190777	−	0.981633i	$-0.561101\pi$
$29$	−320.887	−0.381554	−0.190777	+	0.981633i	$0.561101\pi$
$30$	0	0
$31$	− 1346.37i	− 1.40101i	−0.713646	−	0.700506i	$-0.752958\pi$
$31$	− 1346.37i	− 1.40101i	0.713646	−	0.700506i	$-0.247042\pi$
$32$	0	0
$33$	− 308.404i	− 0.283199i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−797.088	−0.582241	−0.291121	−	0.956686i	$-0.594028\pi$
$37$	−797.088	−0.582241	−0.291121	+	0.956686i	$0.594028\pi$
$38$	0	0
$39$	530.459	0.348757
$40$	0	0
$41$	− 815.856i	− 0.485340i	−0.970109	−	0.242670i	$-0.921977\pi$
$41$	− 815.856i	− 0.485340i	0.970109	−	0.242670i	$-0.0780232\pi$
$42$	0	0
$43$	2167.70	1.17236	0.586182	−	0.810179i	$-0.300630\pi$
$43$	2167.70	1.17236	0.586182	+	0.810179i	$0.300630\pi$
$44$	0	0
$45$	− 54.6490i	− 0.0269872i
$46$	0	0
$47$	− 4285.61i	− 1.94007i	−0.242970	−	0.970034i	$-0.578122\pi$
$47$	− 4285.61i	− 1.94007i	0.242970	−	0.970034i	$-0.421878\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	−784.102	−0.301462
$52$	0	0
$53$	−3171.57	−1.12907	−0.564536	−	0.825408i	$-0.690945\pi$
$53$	−3171.57	−1.12907	−0.564536	+	0.825408i	$0.690945\pi$
$54$	0	0
$55$	107.829i	0.0356460i
$56$	0	0
$57$	−793.337	−0.244179
$58$	0	0
$59$	4706.39i	1.35202i	0.736891	+	0.676012i	$0.236293\pi$
$59$	4706.39i	1.35202i	−0.736891	+	0.676012i	$0.763707\pi$
$60$	0	0
$61$	2534.78i	0.681209i	0.940207	+	0.340604i	$0.110632\pi$
$61$	2534.78i	0.681209i	−0.940207	+	0.340604i	$0.889368\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−185.468	−0.0438977
$66$	0	0
$67$	4092.43	0.911657	0.455828	−	0.890068i	$-0.349343\pi$
$67$	4092.43	0.911657	0.455828	+	0.890068i	$0.349343\pi$
$68$	0	0
$69$	361.239i	0.0758746i
$70$	0	0
$71$	2255.28	0.447388	0.223694	−	0.974659i	$-0.428188\pi$
$71$	2255.28	0.447388	0.223694	+	0.974659i	$0.428188\pi$
$72$	0	0
$73$	4653.19i	0.873183i	0.899660	+	0.436591i	$0.143814\pi$
$73$	4653.19i	0.873183i	−0.899660	+	0.436591i	$0.856186\pi$
$74$	0	0
$75$	1269.89i	0.225758i
$76$	0	0
$77$	0	0
$78$	0	0
$79$	4193.74	0.671965	0.335983	−	0.941868i	$-0.390932\pi$
$79$	4193.74	0.671965	0.335983	+	0.941868i	$0.390932\pi$
$80$	0	0
$81$	5573.29	0.849458
$82$	0	0
$83$	− 7799.12i	− 1.13211i	−0.824367	−	0.566056i	$-0.808468\pi$
$83$	− 7799.12i	− 1.13211i	0.824367	−	0.566056i	$-0.191532\pi$
$84$	0	0
$85$	274.150	0.0379447
$86$	0	0
$87$	− 652.514i	− 0.0862088i
$88$	0	0
$89$	− 9469.11i	− 1.19544i	−0.801704	−	0.597722i	$-0.796073\pi$
$89$	− 9469.11i	− 1.19544i	0.801704	−	0.597722i	$-0.203927\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	2737.81	0.316546
$94$	0	0
$95$	277.379	0.0307345
$96$	0	0
$97$	− 9945.39i	− 1.05701i	−0.848931	−	0.528504i	$-0.822753\pi$
$97$	− 9945.39i	− 1.05701i	0.848931	−	0.528504i	$-0.177247\pi$
$98$	0	0
$99$	−11657.7	−1.18944
$100$	0	0
$101$	− 2068.32i	− 0.202756i	−0.994848	−	0.101378i	$-0.967675\pi$
$101$	− 2068.32i	− 0.202756i	0.994848	−	0.101378i	$-0.0323252\pi$
$102$	0	0
$103$	2504.76i	0.236098i	0.993008	+	0.118049i	$0.0376639\pi$
$103$	2504.76i	0.236098i	−0.993008	+	0.118049i	$0.962336\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−6443.39	−0.562791	−0.281395	−	0.959592i	$-0.590797\pi$
$107$	−6443.39	−0.562791	−0.281395	+	0.959592i	$0.590797\pi$
$108$	0	0
$109$	23546.0	1.98182	0.990912	−	0.134515i	$-0.0429477\pi$
$109$	23546.0	1.98182	0.990912	+	0.134515i	$0.0429477\pi$
$110$	0	0
$111$	− 1620.85i	− 0.131552i
$112$	0	0
$113$	19692.2	1.54219	0.771096	−	0.636719i	$-0.219709\pi$
$113$	19692.2	1.54219	0.771096	+	0.636719i	$0.219709\pi$
$114$	0	0
$115$	− 126.302i	− 0.00955026i
$116$	0	0
$117$	− 20051.3i	− 1.46478i
$118$	0	0
$119$	0	0
$120$	0	0
$121$	8360.97	0.571065
$122$	0	0
$123$	1659.02	0.109658
$124$	0	0
$125$	− 888.358i	− 0.0568549i
$126$	0	0
$127$	12550.5	0.778135	0.389067	−	0.921209i	$-0.372797\pi$
$127$	12550.5	0.778135	0.389067	+	0.921209i	$0.372797\pi$
$128$	0	0
$129$	4407.95i	0.264885i
$130$	0	0
$131$	− 19323.3i	− 1.12600i	−0.826457	−	0.562999i	$-0.809647\pi$
$131$	− 19323.3i	− 1.12600i	0.826457	−	0.562999i	$-0.190353\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	228.232	0.0125230
$136$	0	0
$137$	3708.59	0.197591	0.0987955	−	0.995108i	$-0.468501\pi$
$137$	3708.59	0.197591	0.0987955	+	0.995108i	$0.468501\pi$
$138$	0	0
$139$	1096.09i	0.0567307i	0.999598	+	0.0283653i	$0.00903018\pi$
$139$	1096.09i	0.0567307i	−0.999598	+	0.0283653i	$0.990970\pi$
$140$	0	0
$141$	8714.65	0.438341
$142$	0	0
$143$	39563.7i	1.93475i
$144$	0	0
$145$	228.143i	0.0108510i
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−1233.82	−0.0555748	−0.0277874	−	0.999614i	$-0.508846\pi$
$149$	−1233.82	−0.0555748	−0.0277874	+	0.999614i	$0.508846\pi$
$150$	0	0
$151$	42672.8	1.87153	0.935766	−	0.352622i	$-0.114710\pi$
$151$	42672.8	1.87153	0.935766	+	0.352622i	$0.114710\pi$
$152$	0	0
$153$	29639.0i	1.26614i
$154$	0	0
$155$	−957.236	−0.0398433
$156$	0	0
$157$	29884.9i	1.21242i	0.795305	+	0.606209i	$0.207311\pi$
$157$	29884.9i	1.21242i	−0.795305	+	0.606209i	$0.792689\pi$
$158$	0	0
$159$	− 6449.28i	− 0.255104i
$160$	0	0
$161$	0	0
$162$	0	0
$163$	26205.2	0.986308	0.493154	−	0.869942i	$-0.335844\pi$
$163$	26205.2	0.986308	0.493154	+	0.869942i	$0.335844\pi$
$164$	0	0
$165$	−219.267	−0.00805389
$166$	0	0
$167$	− 35887.5i	− 1.28680i	−0.765532	−	0.643398i	$-0.777524\pi$
$167$	− 35887.5i	− 1.28680i	0.765532	−	0.643398i	$-0.222476\pi$
$168$	0	0
$169$	−39489.2	−1.38263
$170$	0	0
$171$	29988.1i	1.02555i
$172$	0	0
$173$	− 10972.7i	− 0.366625i	−0.983055	−	0.183312i	$-0.941318\pi$
$173$	− 10972.7i	− 0.366625i	0.983055	−	0.183312i	$-0.0586820\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−9570.30	−0.305477
$178$	0	0
$179$	33844.1	1.05628	0.528138	−	0.849159i	$-0.322891\pi$
$179$	33844.1	1.05628	0.528138	+	0.849159i	$0.322891\pi$
$180$	0	0
$181$	− 50094.5i	− 1.52909i	−0.644571	−	0.764545i	$-0.722964\pi$
$181$	− 50094.5i	− 1.52909i	0.644571	−	0.764545i	$-0.277036\pi$
$182$	0	0
$183$	−5154.39	−0.153913
$184$	0	0
$185$	566.709i	0.0165583i
$186$	0	0
$187$	− 58481.4i	− 1.67238i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−17612.9	−0.482798	−0.241399	−	0.970426i	$-0.577606\pi$
$191$	−17612.9	−0.482798	−0.241399	+	0.970426i	$0.577606\pi$
$192$	0	0
$193$	13264.2	0.356097	0.178048	−	0.984022i	$-0.443022\pi$
$193$	13264.2	0.356097	0.178048	+	0.984022i	$0.443022\pi$
$194$	0	0
$195$	− 377.143i	− 0.00991828i
$196$	0	0
$197$	5362.20	0.138169	0.0690844	−	0.997611i	$-0.477992\pi$
$197$	5362.20	0.138169	0.0690844	+	0.997611i	$0.477992\pi$
$198$	0	0
$199$	− 42230.6i	− 1.06640i	−0.845988	−	0.533202i	$-0.820989\pi$
$199$	− 42230.6i	− 1.06640i	0.845988	−	0.533202i	$-0.179011\pi$
$200$	0	0
$201$	8321.82i	0.205981i
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−580.053	−0.0138026
$206$	0	0
$207$	13654.8	0.318673
$208$	0	0
$209$	− 59170.2i	− 1.35460i
$210$	0	0
$211$	77049.0	1.73062	0.865311	−	0.501236i	$-0.167121\pi$
$211$	77049.0	1.73062	0.865311	+	0.501236i	$0.167121\pi$
$212$	0	0
$213$	4586.04i	0.101083i
$214$	0	0
$215$	− 1541.18i	− 0.0333408i
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−9462.11	−0.197288
$220$	0	0
$221$	100589.	2.05951
$222$	0	0
$223$	3702.60i	0.0744555i	0.999307	+	0.0372278i	$0.0118527\pi$
$223$	3702.60i	0.0744555i	−0.999307	+	0.0372278i	$0.988147\pi$
$224$	0	0
$225$	48001.8	0.948183
$226$	0	0
$227$	− 12018.8i	− 0.233243i	−0.993176	−	0.116621i	$-0.962794\pi$
$227$	− 12018.8i	− 0.233243i	0.993176	−	0.116621i	$-0.0372064\pi$
$228$	0	0
$229$	− 33360.5i	− 0.636153i	−0.948065	−	0.318076i	$-0.896963\pi$
$229$	− 33360.5i	− 0.636153i	0.948065	−	0.318076i	$-0.103037\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	54138.9	0.997235	0.498618	−	0.866822i	$-0.333841\pi$
$233$	54138.9	0.997235	0.498618	+	0.866822i	$0.333841\pi$
$234$	0	0
$235$	−3046.96	−0.0551735
$236$	0	0
$237$	8527.83i	0.151824i
$238$	0	0
$239$	−29165.4	−0.510590	−0.255295	−	0.966863i	$-0.582173\pi$
$239$	−29165.4	−0.510590	−0.255295	+	0.966863i	$0.582173\pi$
$240$	0	0
$241$	100130.i	1.72398i	0.506929	+	0.861988i	$0.330781\pi$
$241$	100130.i	1.72398i	−0.506929	+	0.861988i	$0.669219\pi$
$242$	0	0
$243$	37335.2i	0.632275i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	101774.	1.66817
$248$	0	0
$249$	15859.3	0.255791
$250$	0	0
$251$	81988.2i	1.30138i	0.759344	+	0.650690i	$0.225520\pi$
$251$	81988.2i	1.30138i	−0.759344	+	0.650690i	$0.774480\pi$
$252$	0	0
$253$	−26942.6	−0.420919
$254$	0	0
$255$	557.476i	0.00857325i
$256$	0	0
$257$	− 73774.8i	− 1.11697i	−0.829515	−	0.558485i	$-0.811383\pi$
$257$	− 73774.8i	− 1.11697i	0.829515	−	0.558485i	$-0.188617\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−24665.0	−0.362076
$262$	0	0
$263$	−39243.6	−0.567358	−0.283679	−	0.958919i	$-0.591555\pi$
$263$	−39243.6	−0.567358	−0.283679	+	0.958919i	$0.591555\pi$
$264$	0	0
$265$	2254.90i	0.0321096i
$266$	0	0
$267$	19255.1	0.270100
$268$	0	0
$269$	10311.6i	0.142502i	0.997458	+	0.0712509i	$0.0226991\pi$
$269$	10311.6i	0.142502i	−0.997458	+	0.0712509i	$0.977301\pi$
$270$	0	0
$271$	− 36969.3i	− 0.503388i	−0.967807	−	0.251694i	$-0.919012\pi$
$271$	− 36969.3i	− 0.503388i	0.967807	−	0.251694i	$-0.0809876\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−94713.3	−1.25241
$276$	0	0
$277$	−92405.8	−1.20431	−0.602157	−	0.798378i	$-0.705692\pi$
$277$	−92405.8	−1.20431	−0.602157	+	0.798378i	$0.705692\pi$
$278$	0	0
$279$	− 103489.i	− 1.32949i
$280$	0	0
$281$	21171.6	0.268128	0.134064	−	0.990973i	$-0.457197\pi$
$281$	21171.6	0.268128	0.134064	+	0.990973i	$0.457197\pi$
$282$	0	0
$283$	48065.2i	0.600147i	0.953916	+	0.300074i	$0.0970112\pi$
$283$	48065.2i	0.600147i	−0.953916	+	0.300074i	$0.902989\pi$
$284$	0	0
$285$	564.042i	0.00694419i
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−65164.9	−0.780222
$290$	0	0
$291$	20223.6	0.238821
$292$	0	0
$293$	26781.1i	0.311956i	0.987761	+	0.155978i	$0.0498528\pi$
$293$	26781.1i	0.311956i	−0.987761	+	0.155978i	$0.950147\pi$
$294$	0	0
$295$	3346.12	0.0384501
$296$	0	0
$297$	− 48686.2i	− 0.551941i
$298$	0	0
$299$	− 46341.7i	− 0.518358i
$300$	0	0
$301$	0	0
$302$	0	0
$303$	4205.85	0.0458109
$304$	0	0
$305$	1802.16	0.0193729
$306$	0	0
$307$	− 69996.0i	− 0.742671i	−0.928499	−	0.371336i	$-0.878900\pi$
$307$	− 69996.0i	− 0.742671i	0.928499	−	0.371336i	$-0.121100\pi$
$308$	0	0
$309$	−5093.35	−0.0533441
$310$	0	0
$311$	86961.6i	0.899097i	0.893256	+	0.449549i	$0.148415\pi$
$311$	86961.6i	0.899097i	−0.893256	+	0.449549i	$0.851585\pi$
$312$	0	0
$313$	33163.2i	0.338507i	0.985573	+	0.169253i	$0.0541357\pi$
$313$	33163.2i	0.338507i	−0.985573	+	0.169253i	$0.945864\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	29206.0	0.290639	0.145319	−	0.989385i	$-0.453579\pi$
$317$	29206.0	0.290639	0.145319	+	0.989385i	$0.453579\pi$
$318$	0	0
$319$	48667.1	0.478249
$320$	0	0
$321$	− 13102.4i	− 0.127157i
$322$	0	0
$323$	−150437.	−1.44195
$324$	0	0
$325$	− 162908.i	− 1.54233i
$326$	0	0
$327$	47880.1i	0.447775i
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−166993.	−1.52420	−0.762102	−	0.647457i	$-0.775832\pi$
$331$	−166993.	−1.52420	−0.762102	+	0.647457i	$0.775832\pi$
$332$	0	0
$333$	−61268.2	−0.552518
$334$	0	0
$335$	− 2909.61i	− 0.0259266i
$336$	0	0
$337$	16126.0	0.141993	0.0709964	−	0.997477i	$-0.477382\pi$
$337$	16126.0	0.141993	0.0709964	+	0.997477i	$0.477382\pi$
$338$	0	0
$339$	40043.5i	0.348444i
$340$	0	0
$341$	204196.i	1.75606i
$342$	0	0
$343$	0	0
$344$	0	0
$345$	256.832	0.00215779
$346$	0	0
$347$	−10071.3	−0.0836428	−0.0418214	−	0.999125i	$-0.513316\pi$
$347$	−10071.3	−0.0836428	−0.0418214	+	0.999125i	$0.513316\pi$
$348$	0	0
$349$	97674.1i	0.801915i	0.916097	+	0.400958i	$0.131323\pi$
$349$	97674.1i	0.801915i	−0.916097	+	0.400958i	$0.868677\pi$
$350$	0	0
$351$	83740.9	0.679710
$352$	0	0
$353$	− 85255.3i	− 0.684182i	−0.939667	−	0.342091i	$-0.888865\pi$
$353$	− 85255.3i	− 0.684182i	0.939667	−	0.342091i	$-0.111135\pi$
$354$	0	0
$355$	− 1603.45i	− 0.0127232i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	142240.	1.10365	0.551826	−	0.833959i	$-0.313931\pi$
$359$	142240.	1.10365	0.551826	+	0.833959i	$0.313931\pi$
$360$	0	0
$361$	−21888.1	−0.167955
$362$	0	0
$363$	17001.8i	0.129027i
$364$	0	0
$365$	3308.30	0.0248324
$366$	0	0
$367$	− 177841.i	− 1.32038i	−0.751098	−	0.660191i	$-0.770475\pi$
$367$	− 177841.i	− 1.32038i	0.751098	−	0.660191i	$-0.229525\pi$
$368$	0	0
$369$	− 62710.8i	− 0.460564i
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−21670.8	−0.155761	−0.0778803	−	0.996963i	$-0.524815\pi$
$373$	−21670.8	−0.155761	−0.0778803	+	0.996963i	$0.524815\pi$
$374$	0	0
$375$	1806.45	0.0128459
$376$	0	0
$377$	83708.0i	0.588958i
$378$	0	0
$379$	−125199.	−0.871608	−0.435804	−	0.900042i	$-0.643536\pi$
$379$	−125199.	−0.871608	−0.435804	+	0.900042i	$0.643536\pi$
$380$	0	0
$381$	25521.1i	0.175812i
$382$	0	0
$383$	211410.i	1.44121i	0.693345	+	0.720605i	$0.256136\pi$
$383$	211410.i	1.44121i	−0.693345	+	0.720605i	$0.743864\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	166620.	1.11252
$388$	0	0
$389$	−39160.8	−0.258793	−0.129396	−	0.991593i	$-0.541304\pi$
$389$	−39160.8	−0.258793	−0.129396	+	0.991593i	$0.541304\pi$
$390$	0	0
$391$	68500.3i	0.448063i
$392$	0	0
$393$	39293.2	0.254409
$394$	0	0
$395$	− 2981.64i	− 0.0191100i
$396$	0	0
$397$	− 196525.i	− 1.24691i	−0.781858	−	0.623456i	$-0.785728\pi$
$397$	− 196525.i	− 1.24691i	0.781858	−	0.623456i	$-0.214272\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	213650.	1.32866	0.664330	−	0.747439i	$-0.268717\pi$
$401$	213650.	1.32866	0.664330	+	0.747439i	$0.268717\pi$
$402$	0	0
$403$	−351221.	−2.16257
$404$	0	0
$405$	− 3962.47i	− 0.0241577i
$406$	0	0
$407$	120890.	0.729794
$408$	0	0
$409$	35604.1i	0.212840i	0.994321	+	0.106420i	$0.0339388\pi$
$409$	35604.1i	0.212840i	−0.994321	+	0.106420i	$0.966061\pi$
$410$	0	0
$411$	7541.29i	0.0446439i
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−5544.97	−0.0321961
$416$	0	0
$417$	−2228.87	−0.0128178
$418$	0	0
$419$	254947.i	1.45219i	0.687597	+	0.726093i	$0.258666\pi$
$419$	254947.i	1.45219i	−0.687597	+	0.726093i	$0.741334\pi$
$420$	0	0
$421$	107186.	0.604748	0.302374	−	0.953189i	$-0.402221\pi$
$421$	107186.	0.604748	0.302374	+	0.953189i	$0.402221\pi$
$422$	0	0
$423$	− 329413.i	− 1.84103i
$424$	0	0
$425$	240804.i	1.33317i
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−80451.6	−0.437139
$430$	0	0
$431$	−119958.	−0.645768	−0.322884	−	0.946439i	$-0.604652\pi$
$431$	−119958.	−0.645768	−0.322884	+	0.946439i	$0.604652\pi$
$432$	0	0
$433$	110159.i	0.587547i	0.955875	+	0.293773i	$0.0949111\pi$
$433$	110159.i	0.587547i	−0.955875	+	0.293773i	$0.905089\pi$
$434$	0	0
$435$	−463.921	−0.00245169
$436$	0	0
$437$	69307.1i	0.362923i
$438$	0	0
$439$	352295.i	1.82801i	0.405708	+	0.914003i	$0.367025\pi$
$439$	352295.i	1.82801i	−0.405708	+	0.914003i	$0.632975\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−285732.	−1.45597	−0.727984	−	0.685594i	$-0.759543\pi$
$443$	−285732.	−1.45597	−0.727984	+	0.685594i	$0.759543\pi$
$444$	0	0
$445$	−6732.29	−0.0339972
$446$	0	0
$447$	− 2508.92i	− 0.0125566i
$448$	0	0
$449$	−342184.	−1.69733	−0.848667	−	0.528927i	$-0.822594\pi$
$449$	−342184.	−1.69733	−0.848667	+	0.528927i	$0.822594\pi$
$450$	0	0
$451$	123736.i	0.608335i
$452$	0	0
$453$	86773.8i	0.422856i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−277624.	−1.32931	−0.664653	−	0.747152i	$-0.731421\pi$
$457$	−277624.	−1.32931	−0.664653	+	0.747152i	$0.731421\pi$
$458$	0	0
$459$	−123782.	−0.587534
$460$	0	0
$461$	− 361934.i	− 1.70305i	−0.524315	−	0.851524i	$-0.675678\pi$
$461$	− 361934.i	− 1.70305i	0.524315	−	0.851524i	$-0.324322\pi$
$462$	0	0
$463$	−146876.	−0.685153	−0.342577	−	0.939490i	$-0.611300\pi$
$463$	−146876.	−0.685153	−0.342577	+	0.939490i	$0.611300\pi$
$464$	0	0
$465$	− 1946.51i	− 0.00900224i
$466$	0	0
$467$	− 268410.i	− 1.23073i	−0.788241	−	0.615367i	$-0.789008\pi$
$467$	− 268410.i	− 1.23073i	0.788241	−	0.615367i	$-0.210992\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−60770.0	−0.273935
$472$	0	0
$473$	−328762.	−1.46947
$474$	0	0
$475$	243640.i	1.07985i
$476$	0	0
$477$	−243782.	−1.07143
$478$	0	0
$479$	− 155285.i	− 0.676797i	−0.941003	−	0.338399i	$-0.890115\pi$
$479$	− 155285.i	− 0.676797i	0.941003	−	0.338399i	$-0.109885\pi$
$480$	0	0
$481$	207932.i	0.898733i
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−7070.91	−0.0300602
$486$	0	0
$487$	155628.	0.656190	0.328095	−	0.944645i	$-0.393593\pi$
$487$	155628.	0.656190	0.328095	+	0.944645i	$0.393593\pi$
$488$	0	0
$489$	53287.5i	0.222847i
$490$	0	0
$491$	−37550.7	−0.155760	−0.0778798	−	0.996963i	$-0.524815\pi$
$491$	−37550.7	−0.155760	−0.0778798	+	0.996963i	$0.524815\pi$
$492$	0	0
$493$	− 123734.i	− 0.509089i
$494$	0	0
$495$	8288.29i	0.0338263i
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−167820.	−0.673975	−0.336987	−	0.941509i	$-0.609408\pi$
$499$	−167820.	−0.673975	−0.336987	+	0.941509i	$0.609408\pi$
$500$	0	0
$501$	72976.0	0.290740
$502$	0	0
$503$	− 133035.i	− 0.525810i	−0.964822	−	0.262905i	$-0.915319\pi$
$503$	− 133035.i	− 0.525810i	0.964822	−	0.262905i	$-0.0846806\pi$
$504$	0	0
$505$	−1470.52	−0.00576617
$506$	0	0
$507$	− 80299.9i	− 0.312392i
$508$	0	0
$509$	227994.i	0.880011i	0.897995	+	0.440005i	$0.145024\pi$
$509$	227994.i	0.880011i	−0.897995	+	0.440005i	$0.854976\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−125240.	−0.475892
$514$	0	0
$515$	1780.82	0.00671437
$516$	0	0
$517$	649973.i	2.43172i
$518$	0	0
$519$	22312.7	0.0828355
$520$	0	0
$521$	− 341411.i	− 1.25777i	−0.777498	−	0.628885i	$-0.783511\pi$
$521$	− 341411.i	− 1.25777i	0.777498	−	0.628885i	$-0.216489\pi$
$522$	0	0
$523$	62912.0i	0.230001i	0.993365	+	0.115001i	$0.0366870\pi$
$523$	62912.0i	0.230001i	−0.993365	+	0.115001i	$0.963313\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	519159.	1.86930
$528$	0	0
$529$	−248283.	−0.887228
$530$	0	0
$531$	361757.i	1.28300i
$532$	0	0
$533$	−212828.	−0.749159
$534$	0	0
$535$	4581.08i	0.0160052i
$536$	0	0
$537$	68820.9i	0.238656i
$538$	0	0
$539$	0	0
$540$	0	0
$541$	251203.	0.858282	0.429141	−	0.903237i	$-0.358816\pi$
$541$	251203.	0.858282	0.429141	+	0.903237i	$0.358816\pi$
$542$	0	0
$543$	101866.	0.345484
$544$	0	0
$545$	− 16740.6i	− 0.0563610i
$546$	0	0
$547$	153431.	0.512789	0.256395	−	0.966572i	$-0.417465\pi$
$547$	153431.	0.512789	0.256395	+	0.966572i	$0.417465\pi$
$548$	0	0
$549$	194836.i	0.646434i
$550$	0	0
$551$	− 125191.i	− 0.412353i
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−1152.38	−0.00374121
$556$	0	0
$557$	−475151.	−1.53152	−0.765758	−	0.643129i	$-0.777636\pi$
$557$	−475151.	−1.53152	−0.765758	+	0.643129i	$0.777636\pi$
$558$	0	0
$559$	− 565476.i	− 1.80963i
$560$	0	0
$561$	118920.	0.377858
$562$	0	0
$563$	− 39487.0i	− 0.124577i	−0.998058	−	0.0622884i	$-0.980160\pi$
$563$	− 39487.0i	− 0.124577i	0.998058	−	0.0622884i	$-0.0198398\pi$
$564$	0	0
$565$	− 14000.7i	− 0.0438583i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−85307.7	−0.263490	−0.131745	−	0.991284i	$-0.542058\pi$
$569$	−85307.7	−0.263490	−0.131745	+	0.991284i	$0.542058\pi$
$570$	0	0
$571$	−89099.0	−0.273276	−0.136638	−	0.990621i	$-0.543630\pi$
$571$	−89099.0	−0.273276	−0.136638	+	0.990621i	$0.543630\pi$
$572$	0	0
$573$	− 35815.4i	− 0.109084i
$574$	0	0
$575$	110939.	0.335545
$576$	0	0
$577$	− 311076.i	− 0.934362i	−0.884162	−	0.467181i	$-0.845270\pi$
$577$	− 311076.i	− 0.934362i	0.884162	−	0.467181i	$-0.154730\pi$
$578$	0	0
$579$	26972.4i	0.0804568i
$580$	0	0
$581$	0	0
$582$	0	0
$583$	481012.	1.41520
$584$	0	0
$585$	−14256.0	−0.0416567
$586$	0	0
$587$	62466.3i	0.181288i	0.995883	+	0.0906440i	$0.0288926\pi$
$587$	62466.3i	0.181288i	−0.995883	+	0.0906440i	$0.971107\pi$
$588$	0	0
$589$	525274.	1.51410
$590$	0	0
$591$	10903.9i	0.0312180i
$592$	0	0
$593$	47567.3i	0.135269i	0.997710	+	0.0676346i	$0.0215452\pi$
$593$	47567.3i	0.135269i	−0.997710	+	0.0676346i	$0.978455\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	85874.6	0.240944
$598$	0	0
$599$	−564035.	−1.57200	−0.785999	−	0.618227i	$-0.787851\pi$
$599$	−564035.	−1.57200	−0.785999	+	0.618227i	$0.787851\pi$
$600$	0	0
$601$	− 255637.i	− 0.707740i	−0.935295	−	0.353870i	$-0.884865\pi$
$601$	− 255637.i	− 0.707740i	0.935295	−	0.353870i	$-0.115135\pi$
$602$	0	0
$603$	314565.	0.865118
$604$	0	0
$605$	− 5944.43i	− 0.0162405i
$606$	0	0
$607$	572699.i	1.55435i	0.629284	+	0.777176i	$0.283348\pi$
$607$	572699.i	1.55435i	−0.629284	+	0.777176i	$0.716652\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−1.11796e6	−2.99464
$612$	0	0
$613$	283002.	0.753127	0.376563	−	0.926391i	$-0.377106\pi$
$613$	283002.	0.753127	0.376563	+	0.926391i	$0.377106\pi$
$614$	0	0
$615$	− 1179.52i	− 0.00311856i
$616$	0	0
$617$	248170.	0.651896	0.325948	−	0.945388i	$-0.394317\pi$
$617$	248170.	0.651896	0.325948	+	0.945388i	$0.394317\pi$
$618$	0	0
$619$	56588.1i	0.147688i	0.997270	+	0.0738438i	$0.0235266\pi$
$619$	56588.1i	0.147688i	−0.997270	+	0.0738438i	$0.976473\pi$
$620$	0	0
$621$	57027.0i	0.147876i
$622$	0	0
$623$	0	0
$624$	0	0
$625$	389677.	0.997574
$626$	0	0
$627$	120321.	0.306059
$628$	0	0
$629$	− 307356.i	− 0.776855i
$630$	0	0
$631$	−681448.	−1.71149	−0.855744	−	0.517399i	$-0.826900\pi$
$631$	−681448.	−1.71149	−0.855744	+	0.517399i	$0.826900\pi$
$632$	0	0
$633$	156677.i	0.391018i
$634$	0	0
$635$	− 8923.10i	− 0.0221293i
$636$	0	0
$637$	0	0
$638$	0	0
$639$	173352.	0.424549
$640$	0	0
$641$	678114.	1.65039	0.825195	−	0.564848i	$-0.191065\pi$
$641$	678114.	1.65039	0.825195	+	0.564848i	$0.191065\pi$
$642$	0	0
$643$	345520.i	0.835701i	0.908516	+	0.417851i	$0.137216\pi$
$643$	345520.i	0.835701i	−0.908516	+	0.417851i	$0.862784\pi$
$644$	0	0
$645$	3133.94	0.00753305
$646$	0	0
$647$	− 7757.50i	− 0.0185316i	−0.999957	−	0.00926581i	$-0.997051\pi$
$647$	− 7757.50i	− 0.0185316i	0.999957	−	0.00926581i	$-0.00294944\pi$
$648$	0	0
$649$	− 713791.i	− 1.69466i
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−448473.	−1.05174	−0.525872	−	0.850564i	$-0.676261\pi$
$653$	−448473.	−1.05174	−0.525872	+	0.850564i	$0.676261\pi$
$654$	0	0
$655$	−13738.3	−0.0320222
$656$	0	0
$657$	357667.i	0.828607i
$658$	0	0
$659$	−583660.	−1.34397	−0.671984	−	0.740565i	$-0.734558\pi$
$659$	−583660.	−1.34397	−0.671984	+	0.740565i	$0.734558\pi$
$660$	0	0
$661$	− 382886.i	− 0.876327i	−0.898895	−	0.438164i	$-0.855629\pi$
$661$	− 382886.i	− 0.876327i	0.898895	−	0.438164i	$-0.144371\pi$
$662$	0	0
$663$	204544.i	0.465329i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−57004.6	−0.128132
$668$	0	0
$669$	−7529.12	−0.0168226
$670$	0	0
$671$	− 384435.i	− 0.853842i
$672$	0	0
$673$	−323802.	−0.714907	−0.357453	−	0.933931i	$-0.616355\pi$
$673$	−323802.	−0.714907	−0.357453	+	0.933931i	$0.616355\pi$
$674$	0	0
$675$	200471.i	0.439992i
$676$	0	0
$677$	− 374803.i	− 0.817760i	−0.912588	−	0.408880i	$-0.865919\pi$
$677$	− 374803.i	− 0.817760i	0.912588	−	0.408880i	$-0.134081\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	24439.8	0.0526990
$682$	0	0
$683$	−97364.5	−0.208718	−0.104359	−	0.994540i	$-0.533279\pi$
$683$	−97364.5	−0.208718	−0.104359	+	0.994540i	$0.533279\pi$
$684$	0	0
$685$	− 2636.71i	− 0.00561928i
$686$	0	0
$687$	67837.5	0.143733
$688$	0	0
$689$	827348.i	1.74281i
$690$	0	0
$691$	− 787116.i	− 1.64848i	−0.566244	−	0.824238i	$-0.691604\pi$
$691$	− 787116.i	− 1.64848i	0.566244	−	0.824238i	$-0.308396\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	779.294	0.00161336
$696$	0	0
$697$	314593.	0.647564
$698$	0	0
$699$	110090.i	0.225316i
$700$	0	0
$701$	−232522.	−0.473182	−0.236591	−	0.971609i	$-0.576030\pi$
$701$	−232522.	−0.473182	−0.236591	+	0.971609i	$0.576030\pi$
$702$	0	0
$703$	− 310976.i	− 0.629240i
$704$	0	0
$705$	− 6195.89i	− 0.0124659i
$706$	0	0
$707$	0	0
$708$	0	0
$709$	151049.	0.300487	0.150243	−	0.988649i	$-0.451994\pi$
$709$	151049.	0.300487	0.150243	+	0.988649i	$0.451994\pi$
$710$	0	0
$711$	322351.	0.637662
$712$	0	0
$713$	− 239179.i	− 0.470483i
$714$	0	0
$715$	28128.8	0.0550223
$716$	0	0
$717$	− 59306.9i	− 0.115363i
$718$	0	0
$719$	712110.i	1.37749i	0.725002	+	0.688746i	$0.241839\pi$
$719$	712110.i	1.37749i	−0.725002	+	0.688746i	$0.758161\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	−203612.	−0.389517
$724$	0	0
$725$	−200392.	−0.381246
$726$	0	0
$727$	− 693160.i	− 1.31149i	−0.754983	−	0.655744i	$-0.772355\pi$
$727$	− 693160.i	− 1.31149i	0.754983	−	0.655744i	$-0.227645\pi$
$728$	0	0
$729$	375517.	0.706601
$730$	0	0
$731$	835861.i	1.56423i
$732$	0	0
$733$	269107.i	0.500861i	0.968135	+	0.250431i	$0.0805722\pi$
$733$	269107.i	0.500861i	−0.968135	+	0.250431i	$0.919428\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−620674.	−1.14269
$738$	0	0
$739$	−86185.1	−0.157813	−0.0789066	−	0.996882i	$-0.525143\pi$
$739$	−86185.1	−0.157813	−0.0789066	+	0.996882i	$0.525143\pi$
$740$	0	0
$741$	206953.i	0.376908i
$742$	0	0
$743$	457439.	0.828620	0.414310	−	0.910136i	$-0.364023\pi$
$743$	457439.	0.828620	0.414310	+	0.910136i	$0.364023\pi$
$744$	0	0
$745$	877.210i	0.00158049i
$746$	0	0
$747$	− 599480.i	− 1.07432i
$748$	0	0
$749$	0	0
$750$	0	0
$751$	581442.	1.03092	0.515462	−	0.856913i	$-0.327620\pi$
$751$	581442.	1.03092	0.515462	+	0.856913i	$0.327620\pi$
$752$	0	0
$753$	−166720.	−0.294035
$754$	0	0
$755$	− 30339.2i	− 0.0532244i
$756$	0	0
$757$	398071.	0.694655	0.347327	−	0.937744i	$-0.387089\pi$
$757$	398071.	0.694655	0.347327	+	0.937744i	$0.387089\pi$
$758$	0	0
$759$	− 54787.0i	− 0.0951029i
$760$	0	0
$761$	− 323965.i	− 0.559408i	−0.960086	−	0.279704i	$-0.909764\pi$
$761$	− 323965.i	− 0.559408i	0.960086	−	0.279704i	$-0.0902363\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	21072.6	0.0360076
$766$	0	0
$767$	1.22773e6	2.08695
$768$	0	0
$769$	− 318602.i	− 0.538760i	−0.963034	−	0.269380i	$-0.913181\pi$
$769$	− 318602.i	− 0.538760i	0.963034	−	0.269380i	$-0.0868187\pi$
$770$	0	0
$771$	150019.	0.252369
$772$	0	0
$773$	512015.i	0.856887i	0.903569	+	0.428443i	$0.140938\pi$
$773$	512015.i	0.856887i	−0.903569	+	0.428443i	$0.859062\pi$
$774$	0	0
$775$	− 840803.i	− 1.39988i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	318298.	0.524516
$780$	0	0
$781$	−342045.	−0.560765
$782$	0	0
$783$	− 103009.i	− 0.168017i
$784$	0	0
$785$	21247.4	0.0344799
$786$	0	0
$787$	644919.i	1.04125i	0.853785	+	0.520626i	$0.174301\pi$
$787$	644919.i	1.04125i	−0.853785	+	0.520626i	$0.825699\pi$
$788$	0	0
$789$	− 79800.5i	− 0.128189i
$790$	0	0
$791$	0	0
$792$	0	0
$793$	661233.	1.05150
$794$	0	0
$795$	−4585.27	−0.00725488
$796$	0	0
$797$	278844.i	0.438980i	0.975615	+	0.219490i	$0.0704393\pi$
$797$	278844.i	0.438980i	−0.975615	+	0.219490i	$0.929561\pi$
$798$	0	0
$799$	1.65252e6	2.58853
$800$	0	0
$801$	− 727843.i	− 1.13442i
$802$	0	0
$803$	− 705721.i	− 1.09447i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−20968.2	−0.0321970
$808$	0	0
$809$	662520.	1.01228	0.506142	−	0.862450i	$-0.331071\pi$
$809$	662520.	1.01228	0.506142	+	0.862450i	$0.331071\pi$
$810$	0	0
$811$	586078.i	0.891073i	0.895264	+	0.445537i	$0.146987\pi$
$811$	586078.i	0.891073i	−0.895264	+	0.445537i	$0.853013\pi$
$812$	0	0
$813$	75175.9	0.113736
$814$	0	0
$815$	− 18631.2i	− 0.0280496i
$816$	0	0
$817$	845706.i	1.26700i
$818$	0	0
$819$	0	0
$820$	0	0
$821$	790534.	1.17283	0.586414	−	0.810012i	$-0.300539\pi$
$821$	790534.	1.17283	0.586414	+	0.810012i	$0.300539\pi$
$822$	0	0
$823$	1.15645e6	1.70737	0.853684	−	0.520791i	$-0.174363\pi$
$823$	1.15645e6	1.70737	0.853684	+	0.520791i	$0.174363\pi$
$824$	0	0
$825$	− 192597.i	− 0.282970i
$826$	0	0
$827$	−390408.	−0.570832	−0.285416	−	0.958404i	$-0.592132\pi$
$827$	−390408.	−0.570832	−0.285416	+	0.958404i	$0.592132\pi$
$828$	0	0
$829$	531226.i	0.772984i	0.922293	+	0.386492i	$0.126313\pi$
$829$	531226.i	0.772984i	−0.922293	+	0.386492i	$0.873687\pi$
$830$	0	0
$831$	− 187904.i	− 0.272104i
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−25515.1	−0.0365951
$836$	0	0
$837$	432204.	0.616933
$838$	0	0
$839$	− 263842.i	− 0.374817i	−0.982282	−	0.187409i	$-0.939991\pi$
$839$	− 263842.i	− 0.374817i	0.982282	−	0.187409i	$-0.0600089\pi$
$840$	0	0
$841$	−604312.	−0.854416
$842$	0	0
$843$	43051.9i	0.0605811i
$844$	0	0
$845$	28075.8i	0.0393204i
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−97739.0	−0.135598
$850$	0	0
$851$	−141600.	−0.195526
$852$	0	0
$853$	− 1.17089e6i	− 1.60923i	−0.593798	−	0.804614i	$-0.702372\pi$
$853$	− 1.17089e6i	− 1.60923i	0.593798	−	0.804614i	$-0.297628\pi$
$854$	0	0
$855$	21320.8	0.0291656
$856$	0	0
$857$	424840.i	0.578447i	0.957262	+	0.289223i	$0.0933970\pi$
$857$	424840.i	0.578447i	−0.957262	+	0.289223i	$0.906603\pi$
$858$	0	0
$859$	− 408175.i	− 0.553172i	−0.960989	−	0.276586i	$-0.910797\pi$
$859$	− 408175.i	− 0.553172i	0.960989	−	0.276586i	$-0.0892030\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	882741.	1.18525	0.592627	−	0.805477i	$-0.298091\pi$
$863$	882741.	1.18525	0.592627	+	0.805477i	$0.298091\pi$
$864$	0	0
$865$	−7801.31	−0.0104264
$866$	0	0
$867$	− 132511.i	− 0.176284i
$868$	0	0
$869$	−636039.	−0.842256
$870$	0	0
$871$	− 1.06757e6i	− 1.40721i
$872$	0	0
$873$	− 764452.i	− 1.00305i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−595392.	−0.774112	−0.387056	−	0.922056i	$-0.626508\pi$
$877$	−595392.	−0.774112	−0.387056	+	0.922056i	$0.626508\pi$
$878$	0	0
$879$	−54458.5	−0.0704835
$880$	0	0
$881$	− 238444.i	− 0.307209i	−0.988132	−	0.153605i	$-0.950912\pi$
$881$	− 238444.i	− 0.307209i	0.988132	−	0.153605i	$-0.0490882\pi$
$882$	0	0
$883$	−189523.	−0.243076	−0.121538	−	0.992587i	$-0.538783\pi$
$883$	−189523.	−0.243076	−0.121538	+	0.992587i	$0.538783\pi$
$884$	0	0
$885$	6804.24i	0.00868746i
$886$	0	0
$887$	819553.i	1.04167i	0.853658	+	0.520834i	$0.174379\pi$
$887$	819553.i	1.04167i	−0.853658	+	0.520834i	$0.825621\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−845268.	−1.06473
$892$	0	0
$893$	1.67199e6	2.09667
$894$	0	0
$895$	− 24062.3i	− 0.0300394i
$896$	0	0
$897$	94234.3	0.117118
$898$	0	0
$899$	432034.i	0.534563i
$900$	0	0
$901$	− 1.22295e6i	− 1.50646i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−35615.9	−0.0434857
$906$	0	0
$907$	657619.	0.799392	0.399696	−	0.916648i	$-0.369116\pi$
$907$	657619.	0.799392	0.399696	+	0.916648i	$0.369116\pi$
$908$	0	0
$909$	− 158981.i	− 0.192406i
$910$	0	0
$911$	−1.32632e6	−1.59812	−0.799061	−	0.601250i	$-0.794670\pi$
$911$	−1.32632e6	−1.59812	−0.799061	+	0.601250i	$0.794670\pi$
$912$	0	0
$913$	1.18285e6i	1.41901i
$914$	0	0
$915$	3664.64i	0.00437712i
$916$	0	0
$917$	0	0
$918$	0	0
$919$	236310.	0.279803	0.139901	−	0.990165i	$-0.455321\pi$
$919$	236310.	0.279803	0.139901	+	0.990165i	$0.455321\pi$
$920$	0	0
$921$	142335.	0.167800
$922$	0	0
$923$	− 588322.i	− 0.690577i
$924$	0	0
$925$	−497777.	−0.581770
$926$	0	0
$927$	192528.i	0.224045i
$928$	0	0
$929$	− 352762.i	− 0.408743i	−0.978893	−	0.204372i	$-0.934485\pi$
$929$	− 352762.i	− 0.408743i	0.978893	−	0.204372i	$-0.0655151\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−176834.	−0.203143
$934$	0	0
$935$	−41578.7	−0.0475607
$936$	0	0
$937$	113445.i	0.129213i	0.997911	+	0.0646066i	$0.0205793\pi$
$937$	113445.i	0.129213i	−0.997911	+	0.0646066i	$0.979421\pi$
$938$	0	0
$939$	−67436.3	−0.0764826
$940$	0	0
$941$	1.17351e6i	1.32528i	0.748937	+	0.662642i	$0.230565\pi$
$941$	1.17351e6i	1.32528i	−0.748937	+	0.662642i	$0.769435\pi$
$942$	0	0
$943$	− 144934.i	− 0.162985i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−271958.	−0.303250	−0.151625	−	0.988438i	$-0.548451\pi$
$947$	−271958.	−0.303250	−0.151625	+	0.988438i	$0.548451\pi$
$948$	0	0
$949$	1.21385e6	1.34782
$950$	0	0
$951$	59389.5i	0.0656672i
$952$	0	0
$953$	465476.	0.512520	0.256260	−	0.966608i	$-0.417510\pi$
$953$	465476.	0.512520	0.256260	+	0.966608i	$0.417510\pi$
$954$	0	0
$955$	12522.3i	0.0137303i
$956$	0	0
$957$	98962.9i	0.108056i
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−889199.	−0.962836
$962$	0	0
$963$	−495271.	−0.534061
$964$	0	0
$965$	− 9430.53i	− 0.0101270i
$966$	0	0
$967$	1.49573e6	1.59956	0.799782	−	0.600291i	$-0.204949\pi$
$967$	1.49573e6	1.59956	0.799782	+	0.600291i	$0.204949\pi$
$968$	0	0
$969$	− 305909.i	− 0.325795i
$970$	0	0
$971$	404524.i	0.429048i	0.976719	+	0.214524i	$0.0688201\pi$
$971$	404524.i	0.429048i	−0.976719	+	0.214524i	$0.931180\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	331269.	0.348475
$976$	0	0
$977$	−1.65194e6	−1.73063	−0.865315	−	0.501229i	$-0.832882\pi$
$977$	−1.65194e6	−1.73063	−0.865315	+	0.501229i	$0.832882\pi$
$978$	0	0
$979$	1.43612e6i	1.49839i
$980$	0	0
$981$	1.80987e6	1.88065
$982$	0	0
$983$	978562.i	1.01270i	0.862328	+	0.506350i	$0.169006\pi$
$983$	978562.i	1.01270i	−0.862328	+	0.506350i	$0.830994\pi$
$984$	0	0
$985$	− 3812.38i	− 0.00392938i
$986$	0	0
$987$	0	0
$988$	0	0
$989$	385085.	0.393699
$990$	0	0
$991$	−1.26854e6	−1.29168	−0.645841	−	0.763472i	$-0.723493\pi$
$991$	−1.26854e6	−1.29168	−0.645841	+	0.763472i	$0.723493\pi$
$992$	0	0
$993$	− 339575.i	− 0.344380i
$994$	0	0
$995$	−30024.9	−0.0303274
$996$	0	0
$997$	− 2071.33i	− 0.00208382i	−0.999999	−	0.00104191i	$-0.999668\pi$
$997$	− 2071.33i	− 0.00208382i	0.999999	−	0.00104191i	$-0.000331650\pi$
$998$	0	0
$999$	− 255876.i	− 0.256389i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	784.5.c.a.97.3		4
4.3	odd	2		98.5.b.a.97.3	✓	4
7.6	odd	2	inner	784.5.c.a.97.2		4
12.11	even	2		882.5.c.c.685.2		4
28.3	even	6		98.5.d.c.19.2		8
28.11	odd	6		98.5.d.c.19.1		8
28.19	even	6		98.5.d.c.31.1		8
28.23	odd	6		98.5.d.c.31.2		8
28.27	even	2		98.5.b.a.97.4	yes	4
84.83	odd	2		882.5.c.c.685.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
98.5.b.a.97.3	✓	4	4.3	odd	2
98.5.b.a.97.4	yes	4	28.27	even	2
98.5.d.c.19.1		8	28.11	odd	6
98.5.d.c.19.2		8	28.3	even	6
98.5.d.c.31.1		8	28.19	even	6
98.5.d.c.31.2		8	28.23	odd	6
784.5.c.a.97.2		4	7.6	odd	2	inner
784.5.c.a.97.3		4	1.1	even	1	trivial
882.5.c.c.685.1		4	84.83	odd	2
882.5.c.c.685.2		4	12.11	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 784 = 2^{4} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	784.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+2.03347i q^{3} -0.710974i q^{5} +76.8650 q^{9} +O(q^{10})\)	\(q+2.03347i q^{3} -0.710974i q^{5} +76.8650 q^{9} -151.664 q^{11} -260.864i q^{13} +1.44574 q^{15} +385.598i q^{17} +390.140i q^{19} +177.647 q^{23} +624.495 q^{25} +321.013i q^{27} -320.887 q^{29} -1346.37i q^{31} -308.404i q^{33} -797.088 q^{37} +530.459 q^{39} -815.856i q^{41} +2167.70 q^{43} -54.6490i q^{45} -4285.61i q^{47} -784.102 q^{51} -3171.57 q^{53} +107.829i q^{55} -793.337 q^{57} +4706.39i q^{59} +2534.78i q^{61} -185.468 q^{65} +4092.43 q^{67} +361.239i q^{69} +2255.28 q^{71} +4653.19i q^{73} +1269.89i q^{75} +4193.74 q^{79} +5573.29 q^{81} -7799.12i q^{83} +274.150 q^{85} -652.514i q^{87} -9469.11i q^{89} +2737.81 q^{93} +277.379 q^{95} -9945.39i q^{97} -11657.7 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 196 q^{9}+O(q^{10}) \) 4 * q - 196 * q^9	\( 4 q - 196 q^{9} - 24 q^{11} - 888 q^{15} - 104 q^{23} + 948 q^{25} - 1408 q^{29} - 3392 q^{37} - 2200 q^{39} + 2024 q^{43} - 18936 q^{51} - 16680 q^{53} - 6064 q^{57} - 6048 q^{65} + 20816 q^{67} + 1984 q^{71} + 29616 q^{79} + 30852 q^{81} - 29688 q^{85} + 18192 q^{93} - 7240 q^{95} - 72160 q^{99}+O(q^{100}) \) 4 * q - 196 * q^9 - 24 * q^11 - 888 * q^15 - 104 * q^23 + 948 * q^25 - 1408 * q^29 - 3392 * q^37 - 2200 * q^39 + 2024 * q^43 - 18936 * q^51 - 16680 * q^53 - 6064 * q^57 - 6048 * q^65 + 20816 * q^67 + 1984 * q^71 + 29616 * q^79 + 30852 * q^81 - 29688 * q^85 + 18192 * q^93 - 7240 * q^95 - 72160 * q^99

Introduction

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Modular forms

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