LMFDB - Embedded newform 784.5.c.g.97.8

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:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 4x^{7} - 66x^{6} + 212x^{5} + 2021x^{4} - 4400x^{3} - 25028x^{2} + 27264x + 127778 $ x^8 - 4x^7 - 66x^6 + 212x^5 + 2021x^4 - 4400x^3 - 25028x^2 + 27264*x + 127778
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2\cdot 7^{6} $
Twist minimal:	no (minimal twist has level 49)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			97.8
Root			$3.71722 - 0.765367i$ of defining polynomial
Character	$\chi$	$=$	784.97
Dual form			784.5.c.g.97.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+16.1337i q^{3} +7.83726i q^{5} -179.298 q^{9} +O(q^{10})$	$q+16.1337i q^{3} +7.83726i q^{5} -179.298 q^{9} -30.5126 q^{11} -162.301i q^{13} -126.444 q^{15} -345.619i q^{17} +263.892i q^{19} -213.179 q^{23} +563.577 q^{25} -1585.91i q^{27} -1011.68 q^{29} -273.802i q^{31} -492.283i q^{33} +1546.00 q^{37} +2618.52 q^{39} -531.628i q^{41} +1470.93 q^{43} -1405.20i q^{45} -2383.59i q^{47} +5576.13 q^{51} +3206.62 q^{53} -239.135i q^{55} -4257.56 q^{57} +3118.51i q^{59} -3944.63i q^{61} +1272.00 q^{65} +6919.27 q^{67} -3439.38i q^{69} -8580.05 q^{71} -359.285i q^{73} +9092.62i q^{75} -2087.16 q^{79} +11063.6 q^{81} -10594.2i q^{83} +2708.71 q^{85} -16322.2i q^{87} +10431.1i q^{89} +4417.46 q^{93} -2068.19 q^{95} -1403.85i q^{97} +5470.85 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 352 q^{9}+O(q^{10}) $ 8 * q - 352 * q^9	$ 8 q - 352 q^{9} - 120 q^{11} + 632 q^{15} - 1752 q^{23} - 2192 q^{25} + 1248 q^{29} + 2368 q^{37} + 7672 q^{39} + 8552 q^{43} + 11976 q^{51} + 5496 q^{53} - 9200 q^{57} + 30240 q^{65} + 7440 q^{67} - 9984 q^{71} + 14096 q^{79} + 3432 q^{81} + 11912 q^{85} + 9584 q^{93} - 22488 q^{95} - 30144 q^{99}+O(q^{100}) $ 8 * q - 352 * q^9 - 120 * q^11 + 632 * q^15 - 1752 * q^23 - 2192 * q^25 + 1248 * q^29 + 2368 * q^37 + 7672 * q^39 + 8552 * q^43 + 11976 * q^51 + 5496 * q^53 - 9200 * q^57 + 30240 * q^65 + 7440 * q^67 - 9984 * q^71 + 14096 * q^79 + 3432 * q^81 + 11912 * q^85 + 9584 * q^93 - 22488 * q^95 - 30144 * 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
$3$	16.1337i	1.79264i	0.443409	+	0.896319i	$0.353769\pi$
$3$	16.1337i	1.79264i	−0.443409	+	0.896319i	$0.646231\pi$
$4$	0	0
$5$	7.83726i	0.313490i	0.987639	+	0.156745i	$0.0501001\pi$
$5$	7.83726i	0.313490i	−0.987639	+	0.156745i	$0.949900\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	−179.298	−2.21355
$10$	0	0
$11$	−30.5126	−0.252170	−0.126085	−	0.992019i	$-0.540241\pi$
$11$	−30.5126	−0.252170	−0.126085	+	0.992019i	$0.540241\pi$
$12$	0	0
$13$	− 162.301i	− 0.960361i	−0.877170	−	0.480181i	$-0.840571\pi$
$13$	− 162.301i	− 0.960361i	0.877170	−	0.480181i	$-0.159429\pi$
$14$	0	0
$15$	−126.444	−0.561975
$16$	0	0
$17$	− 345.619i	− 1.19591i	−0.801528	−	0.597957i	$-0.795979\pi$
$17$	− 345.619i	− 1.19591i	0.801528	−	0.597957i	$-0.204021\pi$
$18$	0	0
$19$	263.892i	0.731002i	0.930811	+	0.365501i	$0.119102\pi$
$19$	263.892i	0.731002i	−0.930811	+	0.365501i	$0.880898\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−213.179	−0.402985	−0.201492	−	0.979490i	$-0.564579\pi$
$23$	−213.179	−0.402985	−0.201492	+	0.979490i	$0.564579\pi$
$24$	0	0
$25$	563.577	0.901724
$26$	0	0
$27$	− 1585.91i	− 2.17546i
$28$	0	0
$29$	−1011.68	−1.20295	−0.601474	−	0.798893i	$-0.705419\pi$
$29$	−1011.68	−1.20295	−0.601474	+	0.798893i	$0.705419\pi$
$30$	0	0
$31$	− 273.802i	− 0.284914i	−0.989801	−	0.142457i	$-0.954500\pi$
$31$	− 273.802i	− 0.284914i	0.989801	−	0.142457i	$-0.0455002\pi$
$32$	0	0
$33$	− 492.283i	− 0.452050i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	1546.00	1.12929	0.564646	−	0.825333i	$-0.309013\pi$
$37$	1546.00	1.12929	0.564646	+	0.825333i	$0.309013\pi$
$38$	0	0
$39$	2618.52	1.72158
$40$	0	0
$41$	− 531.628i	− 0.316257i	−0.987419	−	0.158128i	$-0.949454\pi$
$41$	− 531.628i	− 0.316257i	0.987419	−	0.158128i	$-0.0505460\pi$
$42$	0	0
$43$	1470.93	0.795526	0.397763	−	0.917488i	$-0.369787\pi$
$43$	1470.93	0.795526	0.397763	+	0.917488i	$0.369787\pi$
$44$	0	0
$45$	− 1405.20i	− 0.693928i
$46$	0	0
$47$	− 2383.59i	− 1.07903i	−0.841975	−	0.539517i	$-0.818607\pi$
$47$	− 2383.59i	− 1.07903i	0.841975	−	0.539517i	$-0.181393\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	5576.13	2.14384
$52$	0	0
$53$	3206.62	1.14155	0.570777	−	0.821105i	$-0.306642\pi$
$53$	3206.62	1.14155	0.570777	+	0.821105i	$0.306642\pi$
$54$	0	0
$55$	− 239.135i	− 0.0790530i
$56$	0	0
$57$	−4257.56	−1.31042
$58$	0	0
$59$	3118.51i	0.895865i	0.894067	+	0.447932i	$0.147840\pi$
$59$	3118.51i	0.895865i	−0.894067	+	0.447932i	$0.852160\pi$
$60$	0	0
$61$	− 3944.63i	− 1.06010i	−0.847966	−	0.530050i	$-0.822173\pi$
$61$	− 3944.63i	− 1.06010i	0.847966	−	0.530050i	$-0.177827\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	1272.00	0.301064
$66$	0	0
$67$	6919.27	1.54138	0.770692	−	0.637208i	$-0.219911\pi$
$67$	6919.27	1.54138	0.770692	+	0.637208i	$0.219911\pi$
$68$	0	0
$69$	− 3439.38i	− 0.722406i
$70$	0	0
$71$	−8580.05	−1.70205	−0.851027	−	0.525122i	$-0.824020\pi$
$71$	−8580.05	−1.70205	−0.851027	+	0.525122i	$0.824020\pi$
$72$	0	0
$73$	− 359.285i	− 0.0674207i	−0.999432	−	0.0337104i	$-0.989268\pi$
$73$	− 359.285i	− 0.0674207i	0.999432	−	0.0337104i	$-0.0107324\pi$
$74$	0	0
$75$	9092.62i	1.61647i
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−2087.16	−0.334427	−0.167214	−	0.985921i	$-0.553477\pi$
$79$	−2087.16	−0.334427	−0.167214	+	0.985921i	$0.553477\pi$
$80$	0	0
$81$	11063.6	1.68627
$82$	0	0
$83$	− 10594.2i	− 1.53784i	−0.639344	−	0.768921i	$-0.720794\pi$
$83$	− 10594.2i	− 1.53784i	0.639344	−	0.768921i	$-0.279206\pi$
$84$	0	0
$85$	2708.71	0.374908
$86$	0	0
$87$	− 16322.2i	− 2.15645i
$88$	0	0
$89$	10431.1i	1.31689i	0.752628	+	0.658446i	$0.228786\pi$
$89$	10431.1i	1.31689i	−0.752628	+	0.658446i	$0.771214\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	4417.46	0.510748
$94$	0	0
$95$	−2068.19	−0.229162
$96$	0	0
$97$	− 1403.85i	− 0.149203i	−0.997213	−	0.0746016i	$-0.976231\pi$
$97$	− 1403.85i	− 0.149203i	0.997213	−	0.0746016i	$-0.0237685\pi$
$98$	0	0
$99$	5470.85	0.558193
$100$	0	0
$101$	− 14051.5i	− 1.37747i	−0.725014	−	0.688734i	$-0.758167\pi$
$101$	− 14051.5i	− 1.37747i	0.725014	−	0.688734i	$-0.241833\pi$
$102$	0	0
$103$	− 1748.62i	− 0.164824i	−0.996598	−	0.0824121i	$-0.973738\pi$
$103$	− 1748.62i	− 0.164824i	0.996598	−	0.0824121i	$-0.0262624\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−8463.37	−0.739224	−0.369612	−	0.929186i	$-0.620509\pi$
$107$	−8463.37	−0.739224	−0.369612	+	0.929186i	$0.620509\pi$
$108$	0	0
$109$	−10952.5	−0.921852	−0.460926	−	0.887439i	$-0.652483\pi$
$109$	−10952.5	−0.921852	−0.460926	+	0.887439i	$0.652483\pi$
$110$	0	0
$111$	24942.8i	2.02441i
$112$	0	0
$113$	5884.58	0.460849	0.230424	−	0.973090i	$-0.425989\pi$
$113$	5884.58	0.460849	0.230424	+	0.973090i	$0.425989\pi$
$114$	0	0
$115$	− 1670.74i	− 0.126332i
$116$	0	0
$117$	29100.2i	2.12581i
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−13710.0	−0.936410
$122$	0	0
$123$	8577.15	0.566934
$124$	0	0
$125$	9315.19i	0.596172i
$126$	0	0
$127$	8451.12	0.523971	0.261985	−	0.965072i	$-0.415623\pi$
$127$	8451.12	0.523971	0.261985	+	0.965072i	$0.415623\pi$
$128$	0	0
$129$	23731.6i	1.42609i
$130$	0	0
$131$	− 801.065i	− 0.0466794i	−0.999728	−	0.0233397i	$-0.992570\pi$
$131$	− 801.065i	− 0.0466794i	0.999728	−	0.0233397i	$-0.00742993\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	12429.2	0.681987
$136$	0	0
$137$	22808.9	1.21524	0.607621	−	0.794227i	$-0.292124\pi$
$137$	22808.9	1.21524	0.607621	+	0.794227i	$0.292124\pi$
$138$	0	0
$139$	32895.2i	1.70256i	0.524709	+	0.851281i	$0.324174\pi$
$139$	32895.2i	1.70256i	−0.524709	+	0.851281i	$0.675826\pi$
$140$	0	0
$141$	38456.2	1.93432
$142$	0	0
$143$	4952.23i	0.242175i
$144$	0	0
$145$	− 7928.79i	− 0.377112i
$146$	0	0
$147$	0	0
$148$	0	0
$149$	29961.9	1.34957	0.674786	−	0.738013i	$-0.264236\pi$
$149$	29961.9	1.34957	0.674786	+	0.738013i	$0.264236\pi$
$150$	0	0
$151$	4517.62	0.198132	0.0990662	−	0.995081i	$-0.468414\pi$
$151$	4517.62	0.198132	0.0990662	+	0.995081i	$0.468414\pi$
$152$	0	0
$153$	61968.8i	2.64722i
$154$	0	0
$155$	2145.86	0.0893178
$156$	0	0
$157$	− 31715.2i	− 1.28667i	−0.765583	−	0.643337i	$-0.777550\pi$
$157$	− 31715.2i	− 1.28667i	0.765583	−	0.643337i	$-0.222450\pi$
$158$	0	0
$159$	51734.9i	2.04639i
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−13505.2	−0.508306	−0.254153	−	0.967164i	$-0.581797\pi$
$163$	−13505.2	−0.508306	−0.254153	+	0.967164i	$0.581797\pi$
$164$	0	0
$165$	3858.15	0.141713
$166$	0	0
$167$	24965.3i	0.895168i	0.894242	+	0.447584i	$0.147715\pi$
$167$	24965.3i	0.895168i	−0.894242	+	0.447584i	$0.852285\pi$
$168$	0	0
$169$	2219.36	0.0777060
$170$	0	0
$171$	− 47315.2i	− 1.61811i
$172$	0	0
$173$	− 25428.2i	− 0.849618i	−0.905283	−	0.424809i	$-0.860341\pi$
$173$	− 25428.2i	− 0.849618i	0.905283	−	0.424809i	$-0.139659\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−50313.2	−1.60596
$178$	0	0
$179$	3612.57	0.112748	0.0563742	−	0.998410i	$-0.482046\pi$
$179$	3612.57	0.112748	0.0563742	+	0.998410i	$0.482046\pi$
$180$	0	0
$181$	15458.2i	0.471847i	0.971772	+	0.235923i	$0.0758114\pi$
$181$	15458.2i	0.471847i	−0.971772	+	0.235923i	$0.924189\pi$
$182$	0	0
$183$	63641.8	1.90038
$184$	0	0
$185$	12116.4i	0.354022i
$186$	0	0
$187$	10545.7i	0.301574i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−13375.6	−0.366646	−0.183323	−	0.983053i	$-0.558685\pi$
$191$	−13375.6	−0.366646	−0.183323	+	0.983053i	$0.558685\pi$
$192$	0	0
$193$	13763.6	0.369501	0.184751	−	0.982785i	$-0.440852\pi$
$193$	13763.6	0.369501	0.184751	+	0.982785i	$0.440852\pi$
$194$	0	0
$195$	20522.1i	0.539699i
$196$	0	0
$197$	25150.4	0.648057	0.324028	−	0.946047i	$-0.394963\pi$
$197$	25150.4	0.648057	0.324028	+	0.946047i	$0.394963\pi$
$198$	0	0
$199$	42790.6i	1.08054i	0.841491	+	0.540271i	$0.181678\pi$
$199$	42790.6i	1.08054i	−0.841491	+	0.540271i	$0.818322\pi$
$200$	0	0
$201$	111634.i	2.76314i
$202$	0	0
$203$	0	0
$204$	0	0
$205$	4166.50	0.0991435
$206$	0	0
$207$	38222.5	0.892028
$208$	0	0
$209$	− 8052.02i	− 0.184337i
$210$	0	0
$211$	3516.62	0.0789879	0.0394940	−	0.999220i	$-0.487425\pi$
$211$	3516.62	0.0789879	0.0394940	+	0.999220i	$0.487425\pi$
$212$	0	0
$213$	− 138428.i	− 3.05117i
$214$	0	0
$215$	11528.0i	0.249390i
$216$	0	0
$217$	0	0
$218$	0	0
$219$	5796.61	0.120861
$220$	0	0
$221$	−56094.4	−1.14851
$222$	0	0
$223$	− 67281.8i	− 1.35297i	−0.736457	−	0.676485i	$-0.763503\pi$
$223$	− 67281.8i	− 1.35297i	0.736457	−	0.676485i	$-0.236497\pi$
$224$	0	0
$225$	−101048.	−1.99601
$226$	0	0
$227$	− 74331.3i	− 1.44251i	−0.692667	−	0.721257i	$-0.743565\pi$
$227$	− 74331.3i	− 1.44251i	0.692667	−	0.721257i	$-0.256435\pi$
$228$	0	0
$229$	27965.0i	0.533266i	0.963798	+	0.266633i	$0.0859112\pi$
$229$	27965.0i	0.533266i	−0.963798	+	0.266633i	$0.914089\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−39880.8	−0.734602	−0.367301	−	0.930102i	$-0.619718\pi$
$233$	−39880.8	−0.734602	−0.367301	+	0.930102i	$0.619718\pi$
$234$	0	0
$235$	18680.8	0.338267
$236$	0	0
$237$	− 33673.7i	− 0.599508i
$238$	0	0
$239$	29726.5	0.520413	0.260206	−	0.965553i	$-0.416209\pi$
$239$	29726.5	0.520413	0.260206	+	0.965553i	$0.416209\pi$
$240$	0	0
$241$	62828.5i	1.08174i	0.841106	+	0.540870i	$0.181905\pi$
$241$	62828.5i	1.08174i	−0.841106	+	0.540870i	$0.818095\pi$
$242$	0	0
$243$	50038.4i	0.847405i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	42829.9	0.702026
$248$	0	0
$249$	170924.	2.75679
$250$	0	0
$251$	− 62181.9i	− 0.986999i	−0.869746	−	0.493499i	$-0.835718\pi$
$251$	− 62181.9i	− 0.986999i	0.869746	−	0.493499i	$-0.164282\pi$
$252$	0	0
$253$	6504.64	0.101621
$254$	0	0
$255$	43701.6i	0.672074i
$256$	0	0
$257$	− 114684.i	− 1.73634i	−0.496264	−	0.868172i	$-0.665295\pi$
$257$	− 114684.i	− 1.73634i	0.496264	−	0.868172i	$-0.334705\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	181392.	2.66279
$262$	0	0
$263$	83591.6	1.20851	0.604256	−	0.796790i	$-0.293470\pi$
$263$	83591.6	1.20851	0.604256	+	0.796790i	$0.293470\pi$
$264$	0	0
$265$	25131.1i	0.357866i
$266$	0	0
$267$	−168293.	−2.36071
$268$	0	0
$269$	6858.72i	0.0947848i	0.998876	+	0.0473924i	$0.0150911\pi$
$269$	6858.72i	0.0947848i	−0.998876	+	0.0473924i	$0.984909\pi$
$270$	0	0
$271$	− 50356.0i	− 0.685666i	−0.939396	−	0.342833i	$-0.888613\pi$
$271$	− 50356.0i	− 0.685666i	0.939396	−	0.342833i	$-0.111387\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−17196.2	−0.227388
$276$	0	0
$277$	84305.6	1.09875	0.549373	−	0.835577i	$-0.314867\pi$
$277$	84305.6	1.09875	0.549373	+	0.835577i	$0.314867\pi$
$278$	0	0
$279$	49092.2i	0.630672i
$280$	0	0
$281$	9852.07	0.124771	0.0623857	−	0.998052i	$-0.480129\pi$
$281$	9852.07	0.124771	0.0623857	+	0.998052i	$0.480129\pi$
$282$	0	0
$283$	31733.2i	0.396225i	0.980179	+	0.198112i	$0.0634811\pi$
$283$	31733.2i	0.396225i	−0.980179	+	0.198112i	$0.936519\pi$
$284$	0	0
$285$	− 33367.6i	− 0.410805i
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−35931.6	−0.430211
$290$	0	0
$291$	22649.4	0.267468
$292$	0	0
$293$	− 17750.9i	− 0.206769i	−0.994641	−	0.103384i	$-0.967033\pi$
$293$	− 17750.9i	− 0.206769i	0.994641	−	0.103384i	$-0.0329672\pi$
$294$	0	0
$295$	−24440.5	−0.280845
$296$	0	0
$297$	48390.3i	0.548587i
$298$	0	0
$299$	34599.2i	0.387011i
$300$	0	0
$301$	0	0
$302$	0	0
$303$	226704.	2.46930
$304$	0	0
$305$	30915.1	0.332331
$306$	0	0
$307$	− 5360.84i	− 0.0568795i	−0.999596	−	0.0284398i	$-0.990946\pi$
$307$	− 5360.84i	− 0.0568795i	0.999596	−	0.0284398i	$-0.00905388\pi$
$308$	0	0
$309$	28211.8	0.295470
$310$	0	0
$311$	− 138382.i	− 1.43073i	−0.698751	−	0.715365i	$-0.746260\pi$
$311$	− 138382.i	− 1.43073i	0.698751	−	0.715365i	$-0.253740\pi$
$312$	0	0
$313$	45760.7i	0.467094i	0.972346	+	0.233547i	$0.0750333\pi$
$313$	45760.7i	0.467094i	−0.972346	+	0.233547i	$0.924967\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	39438.7	0.392468	0.196234	−	0.980557i	$-0.437129\pi$
$317$	39438.7	0.392468	0.196234	+	0.980557i	$0.437129\pi$
$318$	0	0
$319$	30868.9	0.303348
$320$	0	0
$321$	− 136546.i	− 1.32516i
$322$	0	0
$323$	91206.0	0.874215
$324$	0	0
$325$	− 91469.2i	− 0.865981i
$326$	0	0
$327$	− 176705.i	− 1.65255i
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−39807.2	−0.363333	−0.181667	−	0.983360i	$-0.558149\pi$
$331$	−39807.2	−0.363333	−0.181667	+	0.983360i	$0.558149\pi$
$332$	0	0
$333$	−277195.	−2.49975
$334$	0	0
$335$	54228.1i	0.483209i
$336$	0	0
$337$	−71094.3	−0.626001	−0.313001	−	0.949753i	$-0.601334\pi$
$337$	−71094.3	−0.626001	−0.313001	+	0.949753i	$0.601334\pi$
$338$	0	0
$339$	94940.3i	0.826136i
$340$	0	0
$341$	8354.42i	0.0718468i
$342$	0	0
$343$	0	0
$344$	0	0
$345$	26955.3	0.226467
$346$	0	0
$347$	−66258.0	−0.550275	−0.275137	−	0.961405i	$-0.588723\pi$
$347$	−66258.0	−0.550275	−0.275137	+	0.961405i	$0.588723\pi$
$348$	0	0
$349$	− 225684.i	− 1.85289i	−0.376429	−	0.926445i	$-0.622848\pi$
$349$	− 225684.i	− 1.85289i	0.376429	−	0.926445i	$-0.377152\pi$
$350$	0	0
$351$	−257395.	−2.08923
$352$	0	0
$353$	132999.i	1.06733i	0.845697	+	0.533664i	$0.179185\pi$
$353$	132999.i	1.06733i	−0.845697	+	0.533664i	$0.820815\pi$
$354$	0	0
$355$	− 67244.1i	− 0.533578i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−232730.	−1.80578	−0.902888	−	0.429875i	$-0.858557\pi$
$359$	−232730.	−1.80578	−0.902888	+	0.429875i	$0.858557\pi$
$360$	0	0
$361$	60682.2	0.465636
$362$	0	0
$363$	− 221193.i	− 1.67865i
$364$	0	0
$365$	2815.81	0.0211357
$366$	0	0
$367$	− 108571.i	− 0.806084i	−0.915181	−	0.403042i	$-0.867953\pi$
$367$	− 108571.i	− 0.806084i	0.915181	−	0.403042i	$-0.132047\pi$
$368$	0	0
$369$	95319.7i	0.700052i
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−169883.	−1.22105	−0.610523	−	0.791999i	$-0.709041\pi$
$373$	−169883.	−1.22105	−0.610523	+	0.791999i	$0.709041\pi$
$374$	0	0
$375$	−150289.	−1.06872
$376$	0	0
$377$	164197.i	1.15526i
$378$	0	0
$379$	224839.	1.56528	0.782641	−	0.622474i	$-0.213872\pi$
$379$	224839.	1.56528	0.782641	+	0.622474i	$0.213872\pi$
$380$	0	0
$381$	136348.i	0.939290i
$382$	0	0
$383$	− 24577.7i	− 0.167550i	−0.996485	−	0.0837750i	$-0.973302\pi$
$383$	− 24577.7i	− 0.167550i	0.996485	−	0.0837750i	$-0.0266977\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−263734.	−1.76094
$388$	0	0
$389$	144136.	0.952520	0.476260	−	0.879305i	$-0.341992\pi$
$389$	144136.	0.952520	0.476260	+	0.879305i	$0.341992\pi$
$390$	0	0
$391$	73678.7i	0.481935i
$392$	0	0
$393$	12924.2	0.0836793
$394$	0	0
$395$	− 16357.6i	− 0.104840i
$396$	0	0
$397$	− 153247.i	− 0.972327i	−0.873868	−	0.486164i	$-0.838396\pi$
$397$	− 153247.i	− 0.972327i	0.873868	−	0.486164i	$-0.161604\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−45426.3	−0.282500	−0.141250	−	0.989974i	$-0.545112\pi$
$401$	−45426.3	−0.282500	−0.141250	+	0.989974i	$0.545112\pi$
$402$	0	0
$403$	−44438.4	−0.273620
$404$	0	0
$405$	86708.3i	0.528629i
$406$	0	0
$407$	−47172.5	−0.284774
$408$	0	0
$409$	− 10421.7i	− 0.0623003i	−0.999515	−	0.0311502i	$-0.990083\pi$
$409$	− 10421.7i	− 0.0623003i	0.999515	−	0.0311502i	$-0.00991701\pi$
$410$	0	0
$411$	367993.i	2.17849i
$412$	0	0
$413$	0	0
$414$	0	0
$415$	83029.4	0.482099
$416$	0	0
$417$	−530723.	−3.05208
$418$	0	0
$419$	− 148730.i	− 0.847167i	−0.905857	−	0.423584i	$-0.860772\pi$
$419$	− 148730.i	− 0.847167i	0.905857	−	0.423584i	$-0.139228\pi$
$420$	0	0
$421$	238275.	1.34436	0.672178	−	0.740389i	$-0.265359\pi$
$421$	238275.	1.34436	0.672178	+	0.740389i	$0.265359\pi$
$422$	0	0
$423$	427372.i	2.38850i
$424$	0	0
$425$	− 194783.i	− 1.07838i
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−79898.0	−0.434132
$430$	0	0
$431$	−280105.	−1.50788	−0.753938	−	0.656945i	$-0.771848\pi$
$431$	−280105.	−1.50788	−0.753938	+	0.656945i	$0.771848\pi$
$432$	0	0
$433$	78101.0i	0.416563i	0.978069	+	0.208281i	$0.0667870\pi$
$433$	78101.0i	0.416563i	−0.978069	+	0.208281i	$0.933213\pi$
$434$	0	0
$435$	127921.	0.676026
$436$	0	0
$437$	− 56256.1i	− 0.294583i
$438$	0	0
$439$	− 310618.i	− 1.61175i	−0.592087	−	0.805874i	$-0.701696\pi$
$439$	− 310618.i	− 1.61175i	0.592087	−	0.805874i	$-0.298304\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	5382.84	0.0274286	0.0137143	−	0.999906i	$-0.495634\pi$
$443$	5382.84	0.0274286	0.0137143	+	0.999906i	$0.495634\pi$
$444$	0	0
$445$	−81751.3	−0.412833
$446$	0	0
$447$	483397.i	2.41930i
$448$	0	0
$449$	167992.	0.833292	0.416646	−	0.909069i	$-0.363206\pi$
$449$	167992.	0.833292	0.416646	+	0.909069i	$0.363206\pi$
$450$	0	0
$451$	16221.3i	0.0797506i
$452$	0	0
$453$	72886.1i	0.355180i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−195647.	−0.936787	−0.468394	−	0.883520i	$-0.655167\pi$
$457$	−195647.	−0.936787	−0.468394	+	0.883520i	$0.655167\pi$
$458$	0	0
$459$	−548122.	−2.60167
$460$	0	0
$461$	355715.i	1.67379i	0.547365	+	0.836894i	$0.315631\pi$
$461$	355715.i	1.67379i	−0.547365	+	0.836894i	$0.684369\pi$
$462$	0	0
$463$	103798.	0.484205	0.242102	−	0.970251i	$-0.422163\pi$
$463$	103798.	0.484205	0.242102	+	0.970251i	$0.422163\pi$
$464$	0	0
$465$	34620.8i	0.160114i
$466$	0	0
$467$	− 294112.i	− 1.34859i	−0.738463	−	0.674295i	$-0.764448\pi$
$467$	− 294112.i	− 1.34859i	0.738463	−	0.674295i	$-0.235552\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	511685.	2.30654
$472$	0	0
$473$	−44881.8	−0.200608
$474$	0	0
$475$	148723.i	0.659162i
$476$	0	0
$477$	−574941.	−2.52689
$478$	0	0
$479$	367747.i	1.60279i	0.598133	+	0.801397i	$0.295909\pi$
$479$	367747.i	1.60279i	−0.598133	+	0.801397i	$0.704091\pi$
$480$	0	0
$481$	− 250918.i	− 1.08453i
$482$	0	0
$483$	0	0
$484$	0	0
$485$	11002.4	0.0467738
$486$	0	0
$487$	342314.	1.44333	0.721667	−	0.692240i	$-0.243376\pi$
$487$	342314.	1.44333	0.721667	+	0.692240i	$0.243376\pi$
$488$	0	0
$489$	− 217889.i	− 0.911209i
$490$	0	0
$491$	−41579.7	−0.172472	−0.0862360	−	0.996275i	$-0.527484\pi$
$491$	−41579.7	−0.172472	−0.0862360	+	0.996275i	$0.527484\pi$
$492$	0	0
$493$	349655.i	1.43862i
$494$	0	0
$495$	42876.4i	0.174988i
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−355975.	−1.42961	−0.714807	−	0.699322i	$-0.753485\pi$
$499$	−355975.	−1.42961	−0.714807	+	0.699322i	$0.753485\pi$
$500$	0	0
$501$	−402785.	−1.60471
$502$	0	0
$503$	− 196448.i	− 0.776446i	−0.921566	−	0.388223i	$-0.873089\pi$
$503$	− 196448.i	− 0.776446i	0.921566	−	0.388223i	$-0.126911\pi$
$504$	0	0
$505$	110126.	0.431823
$506$	0	0
$507$	35806.6i	0.139299i
$508$	0	0
$509$	− 133569.i	− 0.515551i	−0.966205	−	0.257776i	$-0.917010\pi$
$509$	− 133569.i	− 0.515551i	0.966205	−	0.257776i	$-0.0829895\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	418509.	1.59027
$514$	0	0
$515$	13704.4	0.0516708
$516$	0	0
$517$	72729.4i	0.272100i
$518$	0	0
$519$	410253.	1.52306
$520$	0	0
$521$	− 359413.i	− 1.32409i	−0.749463	−	0.662046i	$-0.769688\pi$
$521$	− 359413.i	− 1.32409i	0.749463	−	0.662046i	$-0.230312\pi$
$522$	0	0
$523$	149926.i	0.548119i	0.961713	+	0.274060i	$0.0883665\pi$
$523$	149926.i	0.548119i	−0.961713	+	0.274060i	$0.911633\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−94631.3	−0.340733
$528$	0	0
$529$	−234396.	−0.837603
$530$	0	0
$531$	− 559142.i	− 1.98305i
$532$	0	0
$533$	−86283.7	−0.303721
$534$	0	0
$535$	− 66329.6i	− 0.231739i
$536$	0	0
$537$	58284.3i	0.202117i
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−168242.	−0.574830	−0.287415	−	0.957806i	$-0.592796\pi$
$541$	−168242.	−0.574830	−0.287415	+	0.957806i	$0.592796\pi$
$542$	0	0
$543$	−249398.	−0.845851
$544$	0	0
$545$	− 85837.7i	− 0.288992i
$546$	0	0
$547$	−297422.	−0.994027	−0.497013	−	0.867743i	$-0.665570\pi$
$547$	−297422.	−0.994027	−0.497013	+	0.867743i	$0.665570\pi$
$548$	0	0
$549$	707265.i	2.34659i
$550$	0	0
$551$	− 266974.i	− 0.879357i
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−195483.	−0.634634
$556$	0	0
$557$	451666.	1.45582	0.727909	−	0.685674i	$-0.240492\pi$
$557$	451666.	1.45582	0.727909	+	0.685674i	$0.240492\pi$
$558$	0	0
$559$	− 238733.i	− 0.763992i
$560$	0	0
$561$	−170142.	−0.540613
$562$	0	0
$563$	− 550994.i	− 1.73832i	−0.494531	−	0.869160i	$-0.664660\pi$
$563$	− 550994.i	− 1.73832i	0.494531	−	0.869160i	$-0.335340\pi$
$564$	0	0
$565$	46119.0i	0.144472i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	143742.	0.443975	0.221988	−	0.975049i	$-0.428746\pi$
$569$	143742.	0.443975	0.221988	+	0.975049i	$0.428746\pi$
$570$	0	0
$571$	−118245.	−0.362668	−0.181334	−	0.983422i	$-0.558042\pi$
$571$	−118245.	−0.362668	−0.181334	+	0.983422i	$0.558042\pi$
$572$	0	0
$573$	− 215799.i	− 0.657264i
$574$	0	0
$575$	−120143.	−0.363381
$576$	0	0
$577$	− 451941.i	− 1.35747i	−0.734383	−	0.678735i	$-0.762529\pi$
$577$	− 451941.i	− 1.35747i	0.734383	−	0.678735i	$-0.237471\pi$
$578$	0	0
$579$	222058.i	0.662383i
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−97842.4	−0.287866
$584$	0	0
$585$	−228066.	−0.666422
$586$	0	0
$587$	59990.7i	0.174104i	0.996204	+	0.0870518i	$0.0277446\pi$
$587$	59990.7i	0.174104i	−0.996204	+	0.0870518i	$0.972255\pi$
$588$	0	0
$589$	72254.1	0.208273
$590$	0	0
$591$	405771.i	1.16173i
$592$	0	0
$593$	− 77447.2i	− 0.220240i	−0.993918	−	0.110120i	$-0.964876\pi$
$593$	− 77447.2i	− 0.220240i	0.993918	−	0.110120i	$-0.0351235\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−690372.	−1.93702
$598$	0	0
$599$	407051.	1.13448	0.567238	−	0.823554i	$-0.308012\pi$
$599$	407051.	1.13448	0.567238	+	0.823554i	$0.308012\pi$
$600$	0	0
$601$	− 681741.i	− 1.88743i	−0.330761	−	0.943715i	$-0.607305\pi$
$601$	− 681741.i	− 1.88743i	0.330761	−	0.943715i	$-0.392695\pi$
$602$	0	0
$603$	−1.24061e6	−3.41193
$604$	0	0
$605$	− 107449.i	− 0.293556i
$606$	0	0
$607$	96785.6i	0.262684i	0.991337	+	0.131342i	$0.0419286\pi$
$607$	96785.6i	0.262684i	−0.991337	+	0.131342i	$0.958071\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−386858.	−1.03626
$612$	0	0
$613$	478095.	1.27231	0.636156	−	0.771561i	$-0.280524\pi$
$613$	478095.	1.27231	0.636156	+	0.771561i	$0.280524\pi$
$614$	0	0
$615$	67221.3i	0.177728i
$616$	0	0
$617$	−55323.0	−0.145323	−0.0726617	−	0.997357i	$-0.523149\pi$
$617$	−55323.0	−0.145323	−0.0726617	+	0.997357i	$0.523149\pi$
$618$	0	0
$619$	142719.i	0.372479i	0.982504	+	0.186239i	$0.0596300\pi$
$619$	142719.i	0.372479i	−0.982504	+	0.186239i	$0.940370\pi$
$620$	0	0
$621$	338083.i	0.876679i
$622$	0	0
$623$	0	0
$624$	0	0
$625$	279230.	0.714830
$626$	0	0
$627$	129909.	0.330450
$628$	0	0
$629$	− 534328.i	− 1.35054i
$630$	0	0
$631$	667428.	1.67628	0.838138	−	0.545459i	$-0.183644\pi$
$631$	667428.	1.67628	0.838138	+	0.545459i	$0.183644\pi$
$632$	0	0
$633$	56736.3i	0.141597i
$634$	0	0
$635$	66233.6i	0.164260i
$636$	0	0
$637$	0	0
$638$	0	0
$639$	1.53839e6	3.76759
$640$	0	0
$641$	11099.0	0.0270127	0.0135063	−	0.999909i	$-0.495701\pi$
$641$	11099.0	0.0270127	0.0135063	+	0.999909i	$0.495701\pi$
$642$	0	0
$643$	− 136149.i	− 0.329300i	−0.986352	−	0.164650i	$-0.947351\pi$
$643$	− 136149.i	− 0.329300i	0.986352	−	0.164650i	$-0.0526494\pi$
$644$	0	0
$645$	−185991.	−0.447066
$646$	0	0
$647$	− 110825.i	− 0.264746i	−0.991200	−	0.132373i	$-0.957740\pi$
$647$	− 110825.i	− 0.264746i	0.991200	−	0.132373i	$-0.0422597\pi$
$648$	0	0
$649$	− 95153.7i	− 0.225911i
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−88318.2	−0.207121	−0.103560	−	0.994623i	$-0.533024\pi$
$653$	−88318.2	−0.207121	−0.103560	+	0.994623i	$0.533024\pi$
$654$	0	0
$655$	6278.16	0.0146335
$656$	0	0
$657$	64419.0i	0.149239i
$658$	0	0
$659$	−136014.	−0.313194	−0.156597	−	0.987663i	$-0.550052\pi$
$659$	−136014.	−0.313194	−0.156597	+	0.987663i	$0.550052\pi$
$660$	0	0
$661$	138133.i	0.316150i	0.987427	+	0.158075i	$0.0505288\pi$
$661$	138133.i	0.316150i	−0.987427	+	0.158075i	$0.949471\pi$
$662$	0	0
$663$	− 905012.i	− 2.05886i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	215669.	0.484769
$668$	0	0
$669$	1.08551e6	2.42539
$670$	0	0
$671$	120361.i	0.267326i
$672$	0	0
$673$	752500.	1.66141	0.830704	−	0.556714i	$-0.187938\pi$
$673$	752500.	1.66141	0.830704	+	0.556714i	$0.187938\pi$
$674$	0	0
$675$	− 893785.i	− 1.96167i
$676$	0	0
$677$	− 774498.i	− 1.68983i	−0.534901	−	0.844914i	$-0.679651\pi$
$677$	− 774498.i	− 1.68983i	0.534901	−	0.844914i	$-0.320349\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	1.19924e6	2.58591
$682$	0	0
$683$	330088.	0.707600	0.353800	−	0.935321i	$-0.384889\pi$
$683$	330088.	0.707600	0.353800	+	0.935321i	$0.384889\pi$
$684$	0	0
$685$	178759.i	0.380967i
$686$	0	0
$687$	−451181.	−0.955954
$688$	0	0
$689$	− 520438.i	− 1.09630i
$690$	0	0
$691$	− 705962.i	− 1.47851i	−0.673424	−	0.739257i	$-0.735177\pi$
$691$	− 705962.i	− 1.47851i	0.673424	−	0.739257i	$-0.264823\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−257808.	−0.533737
$696$	0	0
$697$	−183741.	−0.378216
$698$	0	0
$699$	− 643427.i	− 1.31688i
$700$	0	0
$701$	−387191.	−0.787933	−0.393966	−	0.919125i	$-0.628897\pi$
$701$	−387191.	−0.787933	−0.393966	+	0.919125i	$0.628897\pi$
$702$	0	0
$703$	407977.i	0.825515i
$704$	0	0
$705$	301391.i	0.606390i
$706$	0	0
$707$	0	0
$708$	0	0
$709$	650735.	1.29453	0.647264	−	0.762266i	$-0.275913\pi$
$709$	650735.	1.29453	0.647264	+	0.762266i	$0.275913\pi$
$710$	0	0
$711$	374224.	0.740273
$712$	0	0
$713$	58368.9i	0.114816i
$714$	0	0
$715$	−38811.9	−0.0759194
$716$	0	0
$717$	479600.i	0.932912i
$718$	0	0
$719$	84700.1i	0.163842i	0.996639	+	0.0819212i	$0.0261056\pi$
$719$	84700.1i	0.163842i	−0.996639	+	0.0819212i	$0.973894\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	−1.01366e6	−1.93917
$724$	0	0
$725$	−570159.	−1.08473
$726$	0	0
$727$	602305.i	1.13959i	0.821788	+	0.569794i	$0.192977\pi$
$727$	602305.i	1.13959i	−0.821788	+	0.569794i	$0.807023\pi$
$728$	0	0
$729$	88844.8	0.167177
$730$	0	0
$731$	− 508381.i	− 0.951381i
$732$	0	0
$733$	471815.i	0.878140i	0.898453	+	0.439070i	$0.144692\pi$
$733$	471815.i	0.878140i	−0.898453	+	0.439070i	$0.855308\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−211125.	−0.388691
$738$	0	0
$739$	328102.	0.600786	0.300393	−	0.953815i	$-0.402882\pi$
$739$	328102.	0.600786	0.300393	+	0.953815i	$0.402882\pi$
$740$	0	0
$741$	691007.i	1.25848i
$742$	0	0
$743$	237585.	0.430370	0.215185	−	0.976573i	$-0.430965\pi$
$743$	237585.	0.430370	0.215185	+	0.976573i	$0.430965\pi$
$744$	0	0
$745$	234819.i	0.423078i
$746$	0	0
$747$	1.89952e6i	3.40410i
$748$	0	0
$749$	0	0
$750$	0	0
$751$	364285.	0.645894	0.322947	−	0.946417i	$-0.395326\pi$
$751$	364285.	0.645894	0.322947	+	0.946417i	$0.395326\pi$
$752$	0	0
$753$	1.00323e6	1.76933
$754$	0	0
$755$	35405.7i	0.0621126i
$756$	0	0
$757$	−590661.	−1.03073	−0.515367	−	0.856969i	$-0.672345\pi$
$757$	−590661.	−1.03073	−0.515367	+	0.856969i	$0.672345\pi$
$758$	0	0
$759$	104944.i	0.182169i
$760$	0	0
$761$	− 272715.i	− 0.470912i	−0.971885	−	0.235456i	$-0.924342\pi$
$761$	− 272715.i	− 0.470912i	0.971885	−	0.235456i	$-0.0756584\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	−485665.	−0.829878
$766$	0	0
$767$	506137.	0.860354
$768$	0	0
$769$	725929.i	1.22756i	0.789478	+	0.613778i	$0.210351\pi$
$769$	725929.i	1.22756i	−0.789478	+	0.613778i	$0.789649\pi$
$770$	0	0
$771$	1.85028e6	3.11264
$772$	0	0
$773$	392620.i	0.657073i	0.944491	+	0.328536i	$0.106555\pi$
$773$	392620.i	0.657073i	−0.944491	+	0.328536i	$0.893445\pi$
$774$	0	0
$775$	− 154309.i	− 0.256914i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	140292.	0.231184
$780$	0	0
$781$	261800.	0.429207
$782$	0	0
$783$	1.60443e6i	2.61697i
$784$	0	0
$785$	248560.	0.403360
$786$	0	0
$787$	324279.i	0.523563i	0.965127	+	0.261782i	$0.0843100\pi$
$787$	324279.i	0.523563i	−0.965127	+	0.261782i	$0.915690\pi$
$788$	0	0
$789$	1.34865e6i	2.16643i
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−640218.	−1.01808
$794$	0	0
$795$	−405459.	−0.641524
$796$	0	0
$797$	179577.i	0.282706i	0.989959	+	0.141353i	$0.0451453\pi$
$797$	179577.i	0.282706i	−0.989959	+	0.141353i	$0.954855\pi$
$798$	0	0
$799$	−823813.	−1.29043
$800$	0	0
$801$	− 1.87027e6i	− 2.91501i
$802$	0	0
$803$	10962.7i	0.0170015i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−110657.	−0.169915
$808$	0	0
$809$	−982618.	−1.50137	−0.750685	−	0.660660i	$-0.770276\pi$
$809$	−982618.	−1.50137	−0.750685	+	0.660660i	$0.770276\pi$
$810$	0	0
$811$	− 994497.i	− 1.51203i	−0.654552	−	0.756017i	$-0.727143\pi$
$811$	− 994497.i	− 1.51203i	0.654552	−	0.756017i	$-0.272857\pi$
$812$	0	0
$813$	812431.	1.22915
$814$	0	0
$815$	− 105844.i	− 0.159349i
$816$	0	0
$817$	388166.i	0.581531i
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−1.02275e6	−1.51734	−0.758668	−	0.651477i	$-0.774150\pi$
$821$	−1.02275e6	−1.51734	−0.758668	+	0.651477i	$0.774150\pi$
$822$	0	0
$823$	−317711.	−0.469065	−0.234532	−	0.972108i	$-0.575356\pi$
$823$	−317711.	−0.469065	−0.234532	+	0.972108i	$0.575356\pi$
$824$	0	0
$825$	− 277439.i	− 0.407624i
$826$	0	0
$827$	50540.8	0.0738977	0.0369489	−	0.999317i	$-0.488236\pi$
$827$	50540.8	0.0738977	0.0369489	+	0.999317i	$0.488236\pi$
$828$	0	0
$829$	− 1.07529e6i	− 1.56465i	−0.622868	−	0.782327i	$-0.714033\pi$
$829$	− 1.07529e6i	− 1.56465i	0.622868	−	0.782327i	$-0.285967\pi$
$830$	0	0
$831$	1.36017e6i	1.96965i
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−195660.	−0.280627
$836$	0	0
$837$	−434227.	−0.619820
$838$	0	0
$839$	966211.i	1.37261i	0.727312	+	0.686307i	$0.240769\pi$
$839$	966211.i	1.37261i	−0.727312	+	0.686307i	$0.759231\pi$
$840$	0	0
$841$	316212.	0.447082
$842$	0	0
$843$	158951.i	0.223670i
$844$	0	0
$845$	17393.7i	0.0243601i
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−511976.	−0.710288
$850$	0	0
$851$	−329575.	−0.455088
$852$	0	0
$853$	− 395709.i	− 0.543848i	−0.962319	−	0.271924i	$-0.912340\pi$
$853$	− 395709.i	− 0.543848i	0.962319	−	0.271924i	$-0.0876600\pi$
$854$	0	0
$855$	370822.	0.507263
$856$	0	0
$857$	787067.i	1.07164i	0.844331	+	0.535822i	$0.179998\pi$
$857$	787067.i	1.07164i	−0.844331	+	0.535822i	$0.820002\pi$
$858$	0	0
$859$	1.32116e6i	1.79047i	0.445592	+	0.895236i	$0.352993\pi$
$859$	1.32116e6i	1.79047i	−0.445592	+	0.895236i	$0.647007\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	648112.	0.870219	0.435110	−	0.900377i	$-0.356710\pi$
$863$	648112.	0.870219	0.435110	+	0.900377i	$0.356710\pi$
$864$	0	0
$865$	199288.	0.266347
$866$	0	0
$867$	− 579712.i	− 0.771212i
$868$	0	0
$869$	63684.7	0.0843327
$870$	0	0
$871$	− 1.12300e6i	− 1.48028i
$872$	0	0
$873$	251708.i	0.330270i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	35839.1	0.0465969	0.0232985	−	0.999729i	$-0.492583\pi$
$877$	35839.1	0.0465969	0.0232985	+	0.999729i	$0.492583\pi$
$878$	0	0
$879$	286389.	0.370662
$880$	0	0
$881$	830960.i	1.07060i	0.844661	+	0.535302i	$0.179802\pi$
$881$	830960.i	1.07060i	−0.844661	+	0.535302i	$0.820198\pi$
$882$	0	0
$883$	1.54585e6	1.98265	0.991327	−	0.131415i	$-0.0419522\pi$
$883$	1.54585e6	1.98265	0.991327	+	0.131415i	$0.0419522\pi$
$884$	0	0
$885$	− 394318.i	− 0.503454i
$886$	0	0
$887$	− 853705.i	− 1.08508i	−0.840031	−	0.542539i	$-0.817463\pi$
$887$	− 853705.i	− 1.08508i	0.840031	−	0.542539i	$-0.182537\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−337579.	−0.425227
$892$	0	0
$893$	629008.	0.788776
$894$	0	0
$895$	28312.7i	0.0353455i
$896$	0	0
$897$	−558214.	−0.693771
$898$	0	0
$899$	277000.i	0.342736i
$900$	0	0
$901$	− 1.10827e6i	− 1.36520i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−121150.	−0.147919
$906$	0	0
$907$	−194393.	−0.236302	−0.118151	−	0.992996i	$-0.537697\pi$
$907$	−194393.	−0.236302	−0.118151	+	0.992996i	$0.537697\pi$
$908$	0	0
$909$	2.51941e6i	3.04910i
$910$	0	0
$911$	−433279.	−0.522072	−0.261036	−	0.965329i	$-0.584064\pi$
$911$	−433279.	−0.522072	−0.261036	+	0.965329i	$0.584064\pi$
$912$	0	0
$913$	323256.i	0.387798i
$914$	0	0
$915$	498777.i	0.595750i
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−935259.	−1.10739	−0.553695	−	0.832719i	$-0.686783\pi$
$919$	−935259.	−1.10739	−0.553695	+	0.832719i	$0.686783\pi$
$920$	0	0
$921$	86490.4	0.101964
$922$	0	0
$923$	1.39255e6i	1.63459i
$924$	0	0
$925$	871291.	1.01831
$926$	0	0
$927$	313524.i	0.364847i
$928$	0	0
$929$	1.16877e6i	1.35425i	0.735867	+	0.677126i	$0.236775\pi$
$929$	1.16877e6i	1.35425i	−0.735867	+	0.677126i	$0.763225\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	2.23262e6	2.56478
$934$	0	0
$935$	−82649.7	−0.0945405
$936$	0	0
$937$	695595.i	0.792277i	0.918191	+	0.396139i	$0.129650\pi$
$937$	695595.i	0.792277i	−0.918191	+	0.396139i	$0.870350\pi$
$938$	0	0
$939$	−738292.	−0.837331
$940$	0	0
$941$	− 1.55308e6i	− 1.75394i	−0.480543	−	0.876971i	$-0.659560\pi$
$941$	− 1.55308e6i	− 1.75394i	0.480543	−	0.876971i	$-0.340440\pi$
$942$	0	0
$943$	113332.i	0.127447i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−1.04501e6	−1.16525	−0.582627	−	0.812740i	$-0.697975\pi$
$947$	−1.04501e6	−1.16525	−0.582627	+	0.812740i	$0.697975\pi$
$948$	0	0
$949$	−58312.3	−0.0647482
$950$	0	0
$951$	636294.i	0.703553i
$952$	0	0
$953$	−855805.	−0.942300	−0.471150	−	0.882053i	$-0.656161\pi$
$953$	−855805.	−0.942300	−0.471150	+	0.882053i	$0.656161\pi$
$954$	0	0
$955$	− 104828.i	− 0.114940i
$956$	0	0
$957$	498032.i	0.543793i
$958$	0	0
$959$	0	0
$960$	0	0
$961$	848553.	0.918824
$962$	0	0
$963$	1.51746e6	1.63631
$964$	0	0
$965$	107869.i	0.115835i
$966$	0	0
$967$	1.73940e6	1.86015	0.930074	−	0.367371i	$-0.119742\pi$
$967$	1.73940e6	1.86015	0.930074	+	0.367371i	$0.119742\pi$
$968$	0	0
$969$	1.47150e6i	1.56715i
$970$	0	0
$971$	− 93336.5i	− 0.0989950i	−0.998774	−	0.0494975i	$-0.984238\pi$
$971$	− 93336.5i	− 0.0989950i	0.998774	−	0.0494975i	$-0.0157620\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	1.47574e6	1.55239
$976$	0	0
$977$	1.05190e6	1.10201	0.551004	−	0.834503i	$-0.314245\pi$
$977$	1.05190e6	1.10201	0.551004	+	0.834503i	$0.314245\pi$
$978$	0	0
$979$	− 318280.i	− 0.332081i
$980$	0	0
$981$	1.96376e6	2.04057
$982$	0	0
$983$	32971.7i	0.0341219i	0.999854	+	0.0170610i	$0.00543094\pi$
$983$	32971.7i	0.0341219i	−0.999854	+	0.0170610i	$0.994569\pi$
$984$	0	0
$985$	197110.i	0.203159i
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−313571.	−0.320585
$990$	0	0
$991$	−638020.	−0.649661	−0.324831	−	0.945772i	$-0.605307\pi$
$991$	−638020.	−0.649661	−0.324831	+	0.945772i	$0.605307\pi$
$992$	0	0
$993$	− 642239.i	− 0.651325i
$994$	0	0
$995$	−335361.	−0.338740
$996$	0	0
$997$	1.72164e6i	1.73202i	0.500026	+	0.866010i	$0.333324\pi$
$997$	1.72164e6i	1.73202i	−0.500026	+	0.866010i	$0.666676\pi$
$998$	0	0
$999$	− 2.45182e6i	− 2.45674i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	784.5.c.g.97.8		8
4.3	odd	2		49.5.b.b.48.5	✓	8
7.6	odd	2	inner	784.5.c.g.97.1		8
12.11	even	2		441.5.d.g.244.3		8
28.3	even	6		49.5.d.c.19.4		16
28.11	odd	6		49.5.d.c.19.3		16
28.19	even	6		49.5.d.c.31.3		16
28.23	odd	6		49.5.d.c.31.4		16
28.27	even	2		49.5.b.b.48.6	yes	8
84.83	odd	2		441.5.d.g.244.4		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
49.5.b.b.48.5	✓	8	4.3	odd	2
49.5.b.b.48.6	yes	8	28.27	even	2
49.5.d.c.19.3		16	28.11	odd	6
49.5.d.c.19.4		16	28.3	even	6
49.5.d.c.31.3		16	28.19	even	6
49.5.d.c.31.4		16	28.23	odd	6
441.5.d.g.244.3		8	12.11	even	2
441.5.d.g.244.4		8	84.83	odd	2
784.5.c.g.97.1		8	7.6	odd	2	inner
784.5.c.g.97.8		8	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:	\( N \)	\(=\)	\( 784 = 2^{4} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	784.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+16.1337i q^{3} +7.83726i q^{5} -179.298 q^{9} +O(q^{10})\)	\(q+16.1337i q^{3} +7.83726i q^{5} -179.298 q^{9} -30.5126 q^{11} -162.301i q^{13} -126.444 q^{15} -345.619i q^{17} +263.892i q^{19} -213.179 q^{23} +563.577 q^{25} -1585.91i q^{27} -1011.68 q^{29} -273.802i q^{31} -492.283i q^{33} +1546.00 q^{37} +2618.52 q^{39} -531.628i q^{41} +1470.93 q^{43} -1405.20i q^{45} -2383.59i q^{47} +5576.13 q^{51} +3206.62 q^{53} -239.135i q^{55} -4257.56 q^{57} +3118.51i q^{59} -3944.63i q^{61} +1272.00 q^{65} +6919.27 q^{67} -3439.38i q^{69} -8580.05 q^{71} -359.285i q^{73} +9092.62i q^{75} -2087.16 q^{79} +11063.6 q^{81} -10594.2i q^{83} +2708.71 q^{85} -16322.2i q^{87} +10431.1i q^{89} +4417.46 q^{93} -2068.19 q^{95} -1403.85i q^{97} +5470.85 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 352 q^{9}+O(q^{10}) \) 8 * q - 352 * q^9	\( 8 q - 352 q^{9} - 120 q^{11} + 632 q^{15} - 1752 q^{23} - 2192 q^{25} + 1248 q^{29} + 2368 q^{37} + 7672 q^{39} + 8552 q^{43} + 11976 q^{51} + 5496 q^{53} - 9200 q^{57} + 30240 q^{65} + 7440 q^{67} - 9984 q^{71} + 14096 q^{79} + 3432 q^{81} + 11912 q^{85} + 9584 q^{93} - 22488 q^{95} - 30144 q^{99}+O(q^{100}) \) 8 * q - 352 * q^9 - 120 * q^11 + 632 * q^15 - 1752 * q^23 - 2192 * q^25 + 1248 * q^29 + 2368 * q^37 + 7672 * q^39 + 8552 * q^43 + 11976 * q^51 + 5496 * q^53 - 9200 * q^57 + 30240 * q^65 + 7440 * q^67 - 9984 * q^71 + 14096 * q^79 + 3432 * q^81 + 11912 * q^85 + 9584 * q^93 - 22488 * q^95 - 30144 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more