LMFDB - Embedded newform 441.5.d.g.244.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [441,5,Mod(244,441)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(441, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("441.244");

S:= CuspForms(chi, 5);

N := Newforms(S);

Level:	$ N $	$=$	$ 441 = 3^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	441.d (of order $2$, degree $1$, 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:	$45.5861537200$
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:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 7^{6} $
Twist minimal:	no (minimal twist has level 49)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			244.4
Root			$3.71722 - 0.765367i$ of defining polynomial
Character	$\chi$	$=$	441.244
Dual form			441.5.d.g.244.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.71722 q^{2} -13.0512 q^{4} +7.83726i q^{5} +49.8872 q^{8} +O(q^{10})$	$q-1.71722 q^{2} -13.0512 q^{4} +7.83726i q^{5} +49.8872 q^{8} -13.4583i q^{10} -30.5126 q^{11} +162.301i q^{13} +123.151 q^{16} -345.619i q^{17} +263.892i q^{19} -102.285i q^{20} +52.3969 q^{22} -213.179 q^{23} +563.577 q^{25} -278.707i q^{26} +1011.68 q^{29} -273.802i q^{31} -1009.67 q^{32} +593.504i q^{34} +1546.00 q^{37} -453.160i q^{38} +390.979i q^{40} -531.628i q^{41} -1470.93 q^{43} +398.225 q^{44} +366.075 q^{46} +2383.59i q^{47} -967.786 q^{50} -2118.22i q^{52} -3206.62 q^{53} -239.135i q^{55} -1737.27 q^{58} -3118.51i q^{59} +3944.63i q^{61} +470.179i q^{62} -236.587 q^{64} -1272.00 q^{65} -6919.27 q^{67} +4510.73i q^{68} -8580.05 q^{71} +359.285i q^{73} -2654.82 q^{74} -3444.09i q^{76} +2087.16 q^{79} +965.167i q^{80} +912.922i q^{82} +10594.2i q^{83} +2708.71 q^{85} +2525.91 q^{86} -1522.19 q^{88} +10431.1i q^{89} +2782.23 q^{92} -4093.14i q^{94} -2068.19 q^{95} +1403.85i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 12 q^{2} + 52 q^{4} + 372 q^{8}+O(q^{10}) $ 8 * q + 12 * q^2 + 52 * q^4 + 372 * q^8	$ 8 q + 12 q^{2} + 52 q^{4} + 372 q^{8} - 120 q^{11} - 300 q^{16} + 1952 q^{22} - 1752 q^{23} - 2192 q^{25} - 1248 q^{29} - 3156 q^{32} + 2368 q^{37} - 8552 q^{43} - 264 q^{44} + 7208 q^{46} + 5556 q^{50} - 5496 q^{53} - 17496 q^{58} + 3980 q^{64} - 30240 q^{65} - 7440 q^{67} - 9984 q^{71} + 1080 q^{74} - 14096 q^{79} + 11912 q^{85} - 44496 q^{86} - 44464 q^{88} + 32232 q^{92} - 22488 q^{95}+O(q^{100}) $ 8 * q + 12 * q^2 + 52 * q^4 + 372 * q^8 - 120 * q^11 - 300 * q^16 + 1952 * q^22 - 1752 * q^23 - 2192 * q^25 - 1248 * q^29 - 3156 * q^32 + 2368 * q^37 - 8552 * q^43 - 264 * q^44 + 7208 * q^46 + 5556 * q^50 - 5496 * q^53 - 17496 * q^58 + 3980 * q^64 - 30240 * q^65 - 7440 * q^67 - 9984 * q^71 + 1080 * q^74 - 14096 * q^79 + 11912 * q^85 - 44496 * q^86 - 44464 * q^88 + 32232 * q^92 - 22488 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/441\mathbb{Z}\right)^\times$.

$n$	$199$	$344$
$\chi(n)$	$-1$	$1$

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 $

$2$	−1.71722	−0.429305	−0.214653	−	0.976690i	$-0.568862\pi$
$2$	−1.71722	−0.429305	−0.214653	+	0.976690i	$0.568862\pi$
$3$	0	0
$3$	0	0
$4$	−13.0512	−0.815697
$5$	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
$8$	49.8872	0.779488
$9$	0	0
$10$	− 13.4583i	− 0.134583i
$11$	−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	$0.159429\pi$
$13$	162.301i	0.960361i	−0.877170	+	0.480181i	$0.840571\pi$
$14$	0	0
$15$	0	0
$16$	123.151	0.481059
$17$	− 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	$0.880898\pi$
$20$	− 102.285i	− 0.255713i
$21$	0	0
$22$	52.3969	0.108258
$23$	−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
$26$	− 278.707i	− 0.412288i
$27$	0	0
$28$	0	0
$29$	1011.68	1.20295	0.601474	−	0.798893i	$-0.294581\pi$
$29$	1011.68	1.20295	0.601474	+	0.798893i	$0.294581\pi$
$30$	0	0
$31$	− 273.802i	− 0.284914i	−0.989801	−	0.142457i	$-0.954500\pi$
$31$	− 273.802i	− 0.284914i	0.989801	−	0.142457i	$-0.0455002\pi$
$32$	−1009.67	−0.986009
$33$	0	0
$34$	593.504i	0.513412i
$35$	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	$0.309013\pi$
$38$	− 453.160i	− 0.313823i
$39$	0	0
$40$	390.979i	0.244362i
$41$	− 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	$-0.630213\pi$
$43$	−1470.93	−0.795526	−0.397763	+	0.917488i	$0.630213\pi$
$44$	398.225	0.205695
$45$	0	0
$46$	366.075	0.173003
$47$	2383.59i	1.07903i	0.841975	+	0.539517i	$0.181393\pi$
$47$	2383.59i	1.07903i	−0.841975	+	0.539517i	$0.818607\pi$
$48$	0	0
$49$	0	0
$50$	−967.786	−0.387115
$51$	0	0
$52$	− 2118.22i	− 0.783364i
$53$	−3206.62	−1.14155	−0.570777	−	0.821105i	$-0.693358\pi$
$53$	−3206.62	−1.14155	−0.570777	+	0.821105i	$0.693358\pi$
$54$	0	0
$55$	− 239.135i	− 0.0790530i
$56$	0	0
$57$	0	0
$58$	−1737.27	−0.516431
$59$	− 3118.51i	− 0.895865i	−0.894067	−	0.447932i	$-0.852160\pi$
$59$	− 3118.51i	− 0.895865i	0.894067	−	0.447932i	$-0.147840\pi$
$60$	0	0
$61$	3944.63i	1.06010i	0.847966	+	0.530050i	$0.177827\pi$
$61$	3944.63i	1.06010i	−0.847966	+	0.530050i	$0.822173\pi$
$62$	470.179i	0.122315i
$63$	0	0
$64$	−236.587	−0.0577605
$65$	−1272.00	−0.301064
$66$	0	0
$67$	−6919.27	−1.54138	−0.770692	−	0.637208i	$-0.780089\pi$
$67$	−6919.27	−1.54138	−0.770692	+	0.637208i	$0.780089\pi$
$68$	4510.73i	0.975504i
$69$	0	0
$70$	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	$0.0107324\pi$
$73$	359.285i	0.0674207i	−0.999432	+	0.0337104i	$0.989268\pi$
$74$	−2654.82	−0.484811
$75$	0	0
$76$	− 3444.09i	− 0.596276i
$77$	0	0
$78$	0	0
$79$	2087.16	0.334427	0.167214	−	0.985921i	$-0.446523\pi$
$79$	2087.16	0.334427	0.167214	+	0.985921i	$0.446523\pi$
$80$	965.167i	0.150807i
$81$	0	0
$82$	912.922i	0.135771i
$83$	10594.2i	1.53784i	0.639344	+	0.768921i	$0.279206\pi$
$83$	10594.2i	1.53784i	−0.639344	+	0.768921i	$0.720794\pi$
$84$	0	0
$85$	2708.71	0.374908
$86$	2525.91	0.341523
$87$	0	0
$88$	−1522.19	−0.196564
$89$	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
$92$	2782.23	0.328714
$93$	0	0
$94$	− 4093.14i	− 0.463235i
$95$	−2068.19	−0.229162
$96$	0	0
$97$	1403.85i	0.149203i	0.997213	+	0.0746016i	$0.0237685\pi$
$97$	1403.85i	0.149203i	−0.997213	+	0.0746016i	$0.976231\pi$
$98$	0	0
$99$	0	0
$100$	−7355.34	−0.735534
$101$	− 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	$-0.0262624\pi$
$104$	8096.75i	0.748590i
$105$	0	0
$106$	5506.48	0.490075
$107$	−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	$0.652483\pi$
$110$	410.648i	0.0339378i
$111$	0	0
$112$	0	0
$113$	−5884.58	−0.460849	−0.230424	−	0.973090i	$-0.574011\pi$
$113$	−5884.58	−0.460849	−0.230424	+	0.973090i	$0.574011\pi$
$114$	0	0
$115$	− 1670.74i	− 0.126332i
$116$	−13203.6	−0.981241
$117$	0	0
$118$	5355.16i	0.384599i
$119$	0	0
$120$	0	0
$121$	−13710.0	−0.936410
$122$	− 6773.81i	− 0.455107i
$123$	0	0
$124$	3573.44i	0.232403i
$125$	9315.19i	0.596172i
$126$	0	0
$127$	−8451.12	−0.523971	−0.261985	−	0.965072i	$-0.584377\pi$
$127$	−8451.12	−0.523971	−0.261985	+	0.965072i	$0.584377\pi$
$128$	16561.0	1.01081
$129$	0	0
$130$	2184.30	0.129248
$131$	801.065i	0.0466794i	0.999728	+	0.0233397i	$0.00742993\pi$
$131$	801.065i	0.0466794i	−0.999728	+	0.0233397i	$0.992570\pi$
$132$	0	0
$133$	0	0
$134$	11881.9	0.661723
$135$	0	0
$136$	− 17242.0i	− 0.932201i
$137$	−22808.9	−1.21524	−0.607621	−	0.794227i	$-0.707876\pi$
$137$	−22808.9	−1.21524	−0.607621	+	0.794227i	$0.707876\pi$
$138$	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
$141$	0	0
$142$	14733.8	0.730700
$143$	− 4952.23i	− 0.242175i
$144$	0	0
$145$	7928.79i	0.377112i
$146$	− 616.971i	− 0.0289440i
$147$	0	0
$148$	−20177.1	−0.921161
$149$	−29961.9	−1.34957	−0.674786	−	0.738013i	$-0.735764\pi$
$149$	−29961.9	−1.34957	−0.674786	+	0.738013i	$0.735764\pi$
$150$	0	0
$151$	−4517.62	−0.198132	−0.0990662	−	0.995081i	$-0.531586\pi$
$151$	−4517.62	−0.198132	−0.0990662	+	0.995081i	$0.531586\pi$
$152$	13164.8i	0.569807i
$153$	0	0
$154$	0	0
$155$	2145.86	0.0893178
$156$	0	0
$157$	31715.2i	1.28667i	0.765583	+	0.643337i	$0.222450\pi$
$157$	31715.2i	1.28667i	−0.765583	+	0.643337i	$0.777550\pi$
$158$	−3584.12	−0.143571
$159$	0	0
$160$	− 7913.07i	− 0.309104i
$161$	0	0
$162$	0	0
$163$	13505.2	0.508306	0.254153	−	0.967164i	$-0.418203\pi$
$163$	13505.2	0.508306	0.254153	+	0.967164i	$0.418203\pi$
$164$	6938.36i	0.257970i
$165$	0	0
$166$	− 18192.6i	− 0.660203i
$167$	− 24965.3i	− 0.895168i	−0.894242	−	0.447584i	$-0.852285\pi$
$167$	− 24965.3i	− 0.895168i	0.894242	−	0.447584i	$-0.147715\pi$
$168$	0	0
$169$	2219.36	0.0777060
$170$	−4651.45	−0.160950
$171$	0	0
$172$	19197.3	0.648908
$173$	− 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
$176$	−3757.66	−0.121309
$177$	0	0
$178$	− 17912.5i	− 0.565348i
$179$	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	$-0.924189\pi$
$181$	− 15458.2i	− 0.471847i	0.971772	−	0.235923i	$-0.0758114\pi$
$182$	0	0
$183$	0	0
$184$	−10634.9	−0.314122
$185$	12116.4i	0.354022i
$186$	0	0
$187$	10545.7i	0.301574i
$188$	− 31108.5i	− 0.880165i
$189$	0	0
$190$	3551.53	0.0983804
$191$	−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	$0.440852\pi$
$194$	− 2410.73i	− 0.0640537i
$195$	0	0
$196$	0	0
$197$	−25150.4	−0.648057	−0.324028	−	0.946047i	$-0.605037\pi$
$197$	−25150.4	−0.648057	−0.324028	+	0.946047i	$0.605037\pi$
$198$	0	0
$199$	42790.6i	1.08054i	0.841491	+	0.540271i	$0.181678\pi$
$199$	42790.6i	1.08054i	−0.841491	+	0.540271i	$0.818322\pi$
$200$	28115.3	0.702883
$201$	0	0
$202$	24129.6i	0.591354i
$203$	0	0
$204$	0	0
$205$	4166.50	0.0991435
$206$	3002.77i	0.0707599i
$207$	0	0
$208$	19987.6i	0.461991i
$209$	− 8052.02i	− 0.184337i
$210$	0	0
$211$	−3516.62	−0.0789879	−0.0394940	−	0.999220i	$-0.512575\pi$
$211$	−3516.62	−0.0789879	−0.0394940	+	0.999220i	$0.512575\pi$
$212$	41850.1	0.931162
$213$	0	0
$214$	14533.5	0.317352
$215$	− 11528.0i	− 0.249390i
$216$	0	0
$217$	0	0
$218$	18807.9	0.395756
$219$	0	0
$220$	3120.99i	0.0644833i
$221$	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
$225$	0	0
$226$	10105.1	0.197845
$227$	74331.3i	1.44251i	0.692667	+	0.721257i	$0.256435\pi$
$227$	74331.3i	1.44251i	−0.692667	+	0.721257i	$0.743565\pi$
$228$	0	0
$229$	− 27965.0i	− 0.533266i	−0.963798	−	0.266633i	$-0.914089\pi$
$229$	− 27965.0i	− 0.533266i	0.963798	−	0.266633i	$-0.0859112\pi$
$230$	2869.03i	0.0542349i
$231$	0	0
$232$	50469.8	0.937683
$233$	39880.8	0.734602	0.367301	−	0.930102i	$-0.380282\pi$
$233$	39880.8	0.734602	0.367301	+	0.930102i	$0.380282\pi$
$234$	0	0
$235$	−18680.8	−0.338267
$236$	40700.1i	0.730755i
$237$	0	0
$238$	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	$-0.818095\pi$
$241$	− 62828.5i	− 1.08174i	0.841106	−	0.540870i	$-0.181905\pi$
$242$	23543.1	0.402006
$243$	0	0
$244$	− 51482.0i	− 0.864721i
$245$	0	0
$246$	0	0
$247$	−42829.9	−0.702026
$248$	− 13659.2i	− 0.222087i
$249$	0	0
$250$	− 15996.2i	− 0.255940i
$251$	62181.9i	0.986999i	0.869746	+	0.493499i	$0.164282\pi$
$251$	62181.9i	0.986999i	−0.869746	+	0.493499i	$0.835718\pi$
$252$	0	0
$253$	6504.64	0.101621
$254$	14512.4	0.224943
$255$	0	0
$256$	−24653.6	−0.376184
$257$	− 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
$260$	16601.0	0.245577
$261$	0	0
$262$	− 1375.61i	− 0.0200397i
$263$	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
$267$	0	0
$268$	90304.4	1.25730
$269$	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	$-0.111387\pi$
$272$	− 42563.4i	− 0.575305i
$273$	0	0
$274$	39167.9	0.521710
$275$	−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	$0.314867\pi$
$278$	− 56488.3i	− 0.730919i
$279$	0	0
$280$	0	0
$281$	−9852.07	−0.124771	−0.0623857	−	0.998052i	$-0.519871\pi$
$281$	−9852.07	−0.124771	−0.0623857	+	0.998052i	$0.519871\pi$
$282$	0	0
$283$	31733.2i	0.396225i	0.980179	+	0.198112i	$0.0634811\pi$
$283$	31733.2i	0.396225i	−0.980179	+	0.198112i	$0.936519\pi$
$284$	111980.	1.38836
$285$	0	0
$286$	8504.07i	0.103967i
$287$	0	0
$288$	0	0
$289$	−35931.6	−0.430211
$290$	− 13615.5i	− 0.161896i
$291$	0	0
$292$	− 4689.08i	− 0.0549949i
$293$	− 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
$296$	77125.7	0.880270
$297$	0	0
$298$	51451.1	0.579378
$299$	− 34599.2i	− 0.387011i
$300$	0	0
$301$	0	0
$302$	7757.74	0.0850592
$303$	0	0
$304$	32498.6i	0.351655i
$305$	−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
$309$	0	0
$310$	−3684.91	−0.0383446
$311$	138382.i	1.43073i	0.698751	+	0.715365i	$0.253740\pi$
$311$	138382.i	1.43073i	−0.698751	+	0.715365i	$0.746260\pi$
$312$	0	0
$313$	− 45760.7i	− 0.467094i	−0.972346	−	0.233547i	$-0.924967\pi$
$313$	− 45760.7i	− 0.467094i	0.972346	−	0.233547i	$-0.0750333\pi$
$314$	− 54462.0i	− 0.552375i
$315$	0	0
$316$	−27239.9	−0.272792
$317$	−39438.7	−0.392468	−0.196234	−	0.980557i	$-0.562871\pi$
$317$	−39438.7	−0.392468	−0.196234	+	0.980557i	$0.562871\pi$
$318$	0	0
$319$	−30868.9	−0.303348
$320$	− 1854.19i	− 0.0181074i
$321$	0	0
$322$	0	0
$323$	91206.0	0.874215
$324$	0	0
$325$	91469.2i	0.865981i
$326$	−23191.4	−0.218218
$327$	0	0
$328$	− 26521.4i	− 0.246518i
$329$	0	0
$330$	0	0
$331$	39807.2	0.363333	0.181667	−	0.983360i	$-0.441851\pi$
$331$	39807.2	0.363333	0.181667	+	0.983360i	$0.441851\pi$
$332$	− 138266.i	− 1.25441i
$333$	0	0
$334$	42871.0i	0.384300i
$335$	− 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	$0.601334\pi$
$338$	−3811.13	−0.0333596
$339$	0	0
$340$	−35351.8	−0.305811
$341$	8354.42i	0.0718468i
$342$	0	0
$343$	0	0
$344$	−73380.5	−0.620103
$345$	0	0
$346$	43665.8i	0.364745i
$347$	−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	$0.377152\pi$
$349$	225684.i	1.85289i	−0.376429	+	0.926445i	$0.622848\pi$
$350$	0	0
$351$	0	0
$352$	30807.8	0.248642
$353$	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
$356$	− 136138.i	− 1.07419i
$357$	0	0
$358$	−6203.58	−0.0484034
$359$	−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
$362$	26545.1i	0.202566i
$363$	0	0
$364$	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	$-0.132047\pi$
$368$	−26253.2	−0.193859
$369$	0	0
$370$	− 20806.5i	− 0.151984i
$371$	0	0
$372$	0	0
$373$	−169883.	−1.22105	−0.610523	−	0.791999i	$-0.709041\pi$
$373$	−169883.	−1.22105	−0.610523	+	0.791999i	$0.709041\pi$
$374$	− 18109.4i	− 0.129467i
$375$	0	0
$376$	118910.i	0.841094i
$377$	164197.i	1.15526i
$378$	0	0
$379$	−224839.	−1.56528	−0.782641	−	0.622474i	$-0.786128\pi$
$379$	−224839.	−1.56528	−0.782641	+	0.622474i	$0.786128\pi$
$380$	26992.2	0.186927
$381$	0	0
$382$	22968.9	0.157403
$383$	24577.7i	0.167550i	0.996485	+	0.0837750i	$0.0266977\pi$
$383$	24577.7i	0.167550i	−0.996485	+	0.0837750i	$0.973302\pi$
$384$	0	0
$385$	0	0
$386$	−23635.1	−0.158629
$387$	0	0
$388$	− 18321.9i	− 0.121705i
$389$	−144136.	−0.952520	−0.476260	−	0.879305i	$-0.658008\pi$
$389$	−144136.	−0.952520	−0.476260	+	0.879305i	$0.658008\pi$
$390$	0	0
$391$	73678.7i	0.481935i
$392$	0	0
$393$	0	0
$394$	43188.8	0.278214
$395$	16357.6i	0.104840i
$396$	0	0
$397$	153247.i	0.972327i	0.873868	+	0.486164i	$0.161604\pi$
$397$	153247.i	0.972327i	−0.873868	+	0.486164i	$0.838396\pi$
$398$	− 73480.8i	− 0.463882i
$399$	0	0
$400$	69405.2	0.433782
$401$	45426.3	0.282500	0.141250	−	0.989974i	$-0.454888\pi$
$401$	45426.3	0.282500	0.141250	+	0.989974i	$0.454888\pi$
$402$	0	0
$403$	44438.4	0.273620
$404$	183389.i	1.12360i
$405$	0	0
$406$	0	0
$407$	−47172.5	−0.284774
$408$	0	0
$409$	10421.7i	0.0623003i	0.999515	+	0.0311502i	$0.00991701\pi$
$409$	10421.7i	0.0623003i	−0.999515	+	0.0311502i	$0.990083\pi$
$410$	−7154.80	−0.0425628
$411$	0	0
$412$	22821.5i	0.134447i
$413$	0	0
$414$	0	0
$415$	−83029.4	−0.482099
$416$	− 163871.i	− 0.946925i
$417$	0	0
$418$	13827.1i	0.0791368i
$419$	148730.i	0.847167i	0.905857	+	0.423584i	$0.139228\pi$
$419$	148730.i	0.847167i	−0.905857	+	0.423584i	$0.860772\pi$
$420$	0	0
$421$	238275.	1.34436	0.672178	−	0.740389i	$-0.265359\pi$
$421$	238275.	1.34436	0.672178	+	0.740389i	$0.265359\pi$
$422$	6038.81	0.0339099
$423$	0	0
$424$	−159970.	−0.889827
$425$	− 194783.i	− 1.07838i
$426$	0	0
$427$	0	0
$428$	110457.	0.602983
$429$	0	0
$430$	19796.2i	0.107064i
$431$	−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	$-0.933213\pi$
$433$	− 78101.0i	− 0.416563i	0.978069	−	0.208281i	$-0.0667870\pi$
$434$	0	0
$435$	0	0
$436$	142943.	0.751952
$437$	− 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	$-0.298304\pi$
$440$	− 11929.8i	− 0.0616208i
$441$	0	0
$442$	−96326.4	−0.493061
$443$	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
$446$	115538.i	0.580836i
$447$	0	0
$448$	0	0
$449$	−167992.	−0.833292	−0.416646	−	0.909069i	$-0.636794\pi$
$449$	−167992.	−0.833292	−0.416646	+	0.909069i	$0.636794\pi$
$450$	0	0
$451$	16221.3i	0.0797506i
$452$	76800.6	0.375913
$453$	0	0
$454$	− 127643.i	− 0.619279i
$455$	0	0
$456$	0	0
$457$	−195647.	−0.936787	−0.468394	−	0.883520i	$-0.655167\pi$
$457$	−195647.	−0.936787	−0.468394	+	0.883520i	$0.655167\pi$
$458$	48022.1i	0.228934i
$459$	0	0
$460$	21805.1i	0.103049i
$461$	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	$-0.577837\pi$
$463$	−103798.	−0.484205	−0.242102	+	0.970251i	$0.577837\pi$
$464$	124589.	0.578689
$465$	0	0
$466$	−68484.1	−0.315368
$467$	294112.i	1.34859i	0.738463	+	0.674295i	$0.235552\pi$
$467$	294112.i	1.34859i	−0.738463	+	0.674295i	$0.764448\pi$
$468$	0	0
$469$	0	0
$470$	32079.0	0.145220
$471$	0	0
$472$	− 155574.i	− 0.698316i
$473$	44881.8	0.200608
$474$	0	0
$475$	148723.i	0.659162i
$476$	0	0
$477$	0	0
$478$	−51046.9	−0.223416
$479$	− 367747.i	− 1.60279i	−0.598133	−	0.801397i	$-0.704091\pi$
$479$	− 367747.i	− 1.60279i	0.598133	−	0.801397i	$-0.295909\pi$
$480$	0	0
$481$	250918.i	1.08453i
$482$	107890.i	0.464396i
$483$	0	0
$484$	178931.	0.763827
$485$	−11002.4	−0.0467738
$486$	0	0
$487$	−342314.	−1.44333	−0.721667	−	0.692240i	$-0.756624\pi$
$487$	−342314.	−1.44333	−0.721667	+	0.692240i	$0.756624\pi$
$488$	196787.i	0.826336i
$489$	0	0
$490$	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
$494$	73548.4	0.301383
$495$	0	0
$496$	− 33719.1i	− 0.137060i
$497$	0	0
$498$	0	0
$499$	355975.	1.42961	0.714807	−	0.699322i	$-0.246515\pi$
$499$	355975.	1.42961	0.714807	+	0.699322i	$0.246515\pi$
$500$	− 121574.i	− 0.486296i
$501$	0	0
$502$	− 106780.i	− 0.423723i
$503$	196448.i	0.776446i	0.921566	+	0.388223i	$0.126911\pi$
$503$	196448.i	0.776446i	−0.921566	+	0.388223i	$0.873089\pi$
$504$	0	0
$505$	110126.	0.431823
$506$	−11169.9	−0.0436263
$507$	0	0
$508$	110297.	0.427401
$509$	− 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
$512$	−222641.	−0.849308
$513$	0	0
$514$	196937.i	0.745421i
$515$	13704.4	0.0516708
$516$	0	0
$517$	− 72729.4i	− 0.272100i
$518$	0	0
$519$	0	0
$520$	−63456.3	−0.234676
$521$	− 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	$0.911633\pi$
$524$	− 10454.8i	− 0.0380763i
$525$	0	0
$526$	−143545.	−0.518821
$527$	−94631.3	−0.340733
$528$	0	0
$529$	−234396.	−0.837603
$530$	43155.7i	0.153634i
$531$	0	0
$532$	0	0
$533$	86283.7	0.303721
$534$	0	0
$535$	− 66329.6i	− 0.231739i
$536$	−345183.	−1.20149
$537$	0	0
$538$	− 11777.9i	− 0.0406916i
$539$	0	0
$540$	0	0
$541$	−168242.	−0.574830	−0.287415	−	0.957806i	$-0.592796\pi$
$541$	−168242.	−0.574830	−0.287415	+	0.957806i	$0.592796\pi$
$542$	86472.4i	0.294360i
$543$	0	0
$544$	348962.i	1.17918i
$545$	− 85837.7i	− 0.288992i
$546$	0	0
$547$	297422.	0.994027	0.497013	−	0.867743i	$-0.334430\pi$
$547$	297422.	0.994027	0.497013	+	0.867743i	$0.334430\pi$
$548$	297682.	0.991270
$549$	0	0
$550$	29529.7	0.0976188
$551$	266974.i	0.879357i
$552$	0	0
$553$	0	0
$554$	−144771.	−0.471697
$555$	0	0
$556$	− 429321.i	− 1.38878i
$557$	−451666.	−1.45582	−0.727909	−	0.685674i	$-0.759508\pi$
$557$	−451666.	−1.45582	−0.727909	+	0.685674i	$0.759508\pi$
$558$	0	0
$559$	− 238733.i	− 0.763992i
$560$	0	0
$561$	0	0
$562$	16918.2	0.0535650
$563$	550994.i	1.73832i	0.494531	+	0.869160i	$0.335340\pi$
$563$	550994.i	1.73832i	−0.494531	+	0.869160i	$0.664660\pi$
$564$	0	0
$565$	− 46119.0i	− 0.144472i
$566$	− 54493.0i	− 0.170101i
$567$	0	0
$568$	−428035.	−1.32673
$569$	−143742.	−0.443975	−0.221988	−	0.975049i	$-0.571254\pi$
$569$	−143742.	−0.443975	−0.221988	+	0.975049i	$0.571254\pi$
$570$	0	0
$571$	118245.	0.362668	0.181334	−	0.983422i	$-0.441958\pi$
$571$	118245.	0.362668	0.181334	+	0.983422i	$0.441958\pi$
$572$	64632.3i	0.197541i
$573$	0	0
$574$	0	0
$575$	−120143.	−0.363381
$576$	0	0
$577$	451941.i	1.35747i	0.734383	+	0.678735i	$0.237471\pi$
$577$	451941.i	1.35747i	−0.734383	+	0.678735i	$0.762529\pi$
$578$	61702.5	0.184692
$579$	0	0
$580$	− 103480.i	− 0.307609i
$581$	0	0
$582$	0	0
$583$	97842.4	0.287866
$584$	17923.7i	0.0525536i
$585$	0	0
$586$	30482.2i	0.0887670i
$587$	− 59990.7i	− 0.174104i	−0.996204	−	0.0870518i	$-0.972255\pi$
$587$	− 59990.7i	− 0.174104i	0.996204	−	0.0870518i	$-0.0277446\pi$
$588$	0	0
$589$	72254.1	0.208273
$590$	−41969.8	−0.120568
$591$	0	0
$592$	190392.	0.543256
$593$	− 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
$596$	391037.	1.10084
$597$	0	0
$598$	59414.4i	0.166146i
$599$	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	$0.392695\pi$
$601$	681741.i	1.88743i	−0.330761	+	0.943715i	$0.607305\pi$
$602$	0	0
$603$	0	0
$604$	58960.1	0.161616
$605$	− 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	$0.958071\pi$
$608$	− 266444.i	− 0.720774i
$609$	0	0
$610$	53088.1	0.142672
$611$	−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	$0.280524\pi$
$614$	9205.74i	0.0244187i
$615$	0	0
$616$	0	0
$617$	55323.0	0.145323	0.0726617	−	0.997357i	$-0.476851\pi$
$617$	55323.0	0.145323	0.0726617	+	0.997357i	$0.476851\pi$
$618$	0	0
$619$	142719.i	0.372479i	0.982504	+	0.186239i	$0.0596300\pi$
$619$	142719.i	0.372479i	−0.982504	+	0.186239i	$0.940370\pi$
$620$	−28005.9	−0.0728562
$621$	0	0
$622$	− 237632.i	− 0.614220i
$623$	0	0
$624$	0	0
$625$	279230.	0.714830
$626$	78581.2i	0.200526i
$627$	0	0
$628$	− 413920.i	− 1.04954i
$629$	− 534328.i	− 1.35054i
$630$	0	0
$631$	−667428.	−1.67628	−0.838138	−	0.545459i	$-0.816356\pi$
$631$	−667428.	−1.67628	−0.838138	+	0.545459i	$0.816356\pi$
$632$	104123.	0.260682
$633$	0	0
$634$	67724.9	0.168488
$635$	− 66233.6i	− 0.164260i
$636$	0	0
$637$	0	0
$638$	53008.8	0.130229
$639$	0	0
$640$	129793.i	0.316878i
$641$	−11099.0	−0.0270127	−0.0135063	−	0.999909i	$-0.504299\pi$
$641$	−11099.0	−0.0270127	−0.0135063	+	0.999909i	$0.504299\pi$
$642$	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
$645$	0	0
$646$	−156621.	−0.375305
$647$	110825.i	0.264746i	0.991200	+	0.132373i	$0.0422597\pi$
$647$	110825.i	0.264746i	−0.991200	+	0.132373i	$0.957740\pi$
$648$	0	0
$649$	95153.7i	0.225911i
$650$	− 157073.i	− 0.371770i
$651$	0	0
$652$	−176258.	−0.414624
$653$	88318.2	0.207121	0.103560	−	0.994623i	$-0.466976\pi$
$653$	88318.2	0.207121	0.103560	+	0.994623i	$0.466976\pi$
$654$	0	0
$655$	−6278.16	−0.0146335
$656$	− 65470.6i	− 0.152138i
$657$	0	0
$658$	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	$-0.949471\pi$
$661$	− 138133.i	− 0.316150i	0.987427	−	0.158075i	$-0.0505288\pi$
$662$	−68357.7	−0.155981
$663$	0	0
$664$	528515.i	1.19873i
$665$	0	0
$666$	0	0
$667$	−215669.	−0.484769
$668$	325827.i	0.730186i
$669$	0	0
$670$	93121.6i	0.207444i
$671$	− 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	$0.187938\pi$
$674$	122085.	0.268745
$675$	0	0
$676$	−28965.2	−0.0633846
$677$	− 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
$680$	135130.	0.292236
$681$	0	0
$682$	− 14346.4i	− 0.0308442i
$683$	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
$687$	0	0
$688$	−181146.	−0.382695
$689$	− 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	$-0.264823\pi$
$692$	331868.i	0.693031i
$693$	0	0
$694$	113780.	0.236236
$695$	−257808.	−0.533737
$696$	0	0
$697$	−183741.	−0.378216
$698$	− 387549.i	− 0.795455i
$699$	0	0
$700$	0	0
$701$	387191.	0.787933	0.393966	−	0.919125i	$-0.371103\pi$
$701$	387191.	0.787933	0.393966	+	0.919125i	$0.371103\pi$
$702$	0	0
$703$	407977.i	0.825515i
$704$	7218.89	0.0145655
$705$	0	0
$706$	− 228388.i	− 0.458209i
$707$	0	0
$708$	0	0
$709$	650735.	1.29453	0.647264	−	0.762266i	$-0.275913\pi$
$709$	650735.	1.29453	0.647264	+	0.762266i	$0.275913\pi$
$710$	115473.i	0.229068i
$711$	0	0
$712$	520379.i	1.02650i
$713$	58368.9i	0.114816i
$714$	0	0
$715$	38811.9	0.0759194
$716$	−47148.2	−0.0919685
$717$	0	0
$718$	399649.	0.775229
$719$	− 84700.1i	− 0.163842i	−0.996639	−	0.0819212i	$-0.973894\pi$
$719$	− 84700.1i	− 0.163842i	0.996639	−	0.0819212i	$-0.0261056\pi$
$720$	0	0
$721$	0	0
$722$	−104205.	−0.199900
$723$	0	0
$724$	201747.i	0.384884i
$725$	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
$729$	0	0
$730$	4835.36	0.00907368
$731$	508381.i	0.951381i
$732$	0	0
$733$	− 471815.i	− 0.878140i	−0.898453	−	0.439070i	$-0.855308\pi$
$733$	− 471815.i	− 0.878140i	0.898453	−	0.439070i	$-0.144692\pi$
$734$	186440.i	0.346056i
$735$	0	0
$736$	215241.	0.397347
$737$	211125.	0.388691
$738$	0	0
$739$	−328102.	−0.600786	−0.300393	−	0.953815i	$-0.597118\pi$
$739$	−328102.	−0.600786	−0.300393	+	0.953815i	$0.597118\pi$
$740$	− 158133.i	− 0.288775i
$741$	0	0
$742$	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
$746$	291726.	0.524201
$747$	0	0
$748$	− 137634.i	− 0.245993i
$749$	0	0
$750$	0	0
$751$	−364285.	−0.645894	−0.322947	−	0.946417i	$-0.604674\pi$
$751$	−364285.	−0.645894	−0.322947	+	0.946417i	$0.604674\pi$
$752$	293541.i	0.519079i
$753$	0	0
$754$	− 281962.i	− 0.495961i
$755$	− 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	$0.672345\pi$
$758$	386097.	0.671983
$759$	0	0
$760$	−103176.	−0.178629
$761$	− 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
$764$	174567.	0.299072
$765$	0	0
$766$	− 42205.4i	− 0.0719301i
$767$	506137.	0.860354
$768$	0	0
$769$	− 725929.i	− 1.22756i	−0.789478	−	0.613778i	$-0.789649\pi$
$769$	− 725929.i	− 1.22756i	0.789478	−	0.613778i	$-0.210351\pi$
$770$	0	0
$771$	0	0
$772$	−179630.	−0.301401
$773$	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
$776$	70034.4i	0.116302i
$777$	0	0
$778$	247514.	0.408922
$779$	140292.	0.231184
$780$	0	0
$781$	261800.	0.429207
$782$	− 126523.i	− 0.206897i
$783$	0	0
$784$	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	$0.915690\pi$
$788$	328242.	0.528618
$789$	0	0
$790$	− 28089.6i	− 0.0450082i
$791$	0	0
$792$	0	0
$793$	−640218.	−1.01808
$794$	− 263160.i	− 0.417425i
$795$	0	0
$796$	− 558466.i	− 0.881396i
$797$	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
$800$	−569029.	−0.889108
$801$	0	0
$802$	−78006.9	−0.121279
$803$	− 10962.7i	− 0.0170015i
$804$	0	0
$805$	0	0
$806$	−76310.5	−0.117467
$807$	0	0
$808$	− 700993.i	− 1.07372i
$809$	982618.	1.50137	0.750685	−	0.660660i	$-0.229724\pi$
$809$	982618.	1.50137	0.750685	+	0.660660i	$0.229724\pi$
$810$	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
$813$	0	0
$814$	81005.6	0.122255
$815$	105844.i	0.159349i
$816$	0	0
$817$	− 388166.i	− 0.581531i
$818$	− 17896.3i	− 0.0267459i
$819$	0	0
$820$	−54377.7	−0.0808710
$821$	1.02275e6	1.51734	0.758668	−	0.651477i	$-0.225850\pi$
$821$	1.02275e6	1.51734	0.758668	+	0.651477i	$0.225850\pi$
$822$	0	0
$823$	317711.	0.469065	0.234532	−	0.972108i	$-0.424644\pi$
$823$	317711.	0.469065	0.234532	+	0.972108i	$0.424644\pi$
$824$	− 87233.8i	− 0.128478i
$825$	0	0
$826$	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	$0.285967\pi$
$829$	1.07529e6i	1.56465i	−0.622868	+	0.782327i	$0.714033\pi$
$830$	142580.	0.206967
$831$	0	0
$832$	− 38398.3i	− 0.0554710i
$833$	0	0
$834$	0	0
$835$	195660.	0.280627
$836$	105088.i	0.150363i
$837$	0	0
$838$	− 255401.i	− 0.363693i
$839$	− 966211.i	− 1.37261i	−0.727312	−	0.686307i	$-0.759231\pi$
$839$	− 966211.i	− 1.37261i	0.727312	−	0.686307i	$-0.240769\pi$
$840$	0	0
$841$	316212.	0.447082
$842$	−409171.	−0.577139
$843$	0	0
$844$	45896.0	0.0644302
$845$	17393.7i	0.0243601i
$846$	0	0
$847$	0	0
$848$	−394899.	−0.549155
$849$	0	0
$850$	334486.i	0.462956i
$851$	−329575.	−0.455088
$852$	0	0
$853$	395709.i	0.543848i	0.962319	+	0.271924i	$0.0876600\pi$
$853$	395709.i	0.543848i	−0.962319	+	0.271924i	$0.912340\pi$
$854$	0	0
$855$	0	0
$856$	−422214.	−0.576216
$857$	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	$0.647007\pi$
$860$	150454.i	0.203426i
$861$	0	0
$862$	481001.	0.647339
$863$	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
$866$	134117.i	0.178833i
$867$	0	0
$868$	0	0
$869$	−63684.7	−0.0843327
$870$	0	0
$871$	− 1.12300e6i	− 1.48028i
$872$	−546391.	−0.718572
$873$	0	0
$874$	96604.2i	0.126466i
$875$	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	$0.492583\pi$
$878$	533399.i	0.691931i
$879$	0	0
$880$	− 29449.8i	− 0.0380291i
$881$	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	$-0.958048\pi$
$883$	−1.54585e6	−1.98265	−0.991327	+	0.131415i	$0.958048\pi$
$884$	−732096.	−0.936836
$885$	0	0
$886$	−9243.52	−0.0117752
$887$	853705.i	1.08508i	0.840031	+	0.542539i	$0.182537\pi$
$887$	853705.i	1.08508i	−0.840031	+	0.542539i	$0.817463\pi$
$888$	0	0
$889$	0	0
$890$	140385.	0.177231
$891$	0	0
$892$	878105.i	1.10361i
$893$	−629008.	−0.788776
$894$	0	0
$895$	28312.7i	0.0353455i
$896$	0	0
$897$	0	0
$898$	288480.	0.357736
$899$	− 277000.i	− 0.342736i
$900$	0	0
$901$	1.10827e6i	1.36520i
$902$	− 27855.6i	− 0.0342373i
$903$	0	0
$904$	−293565.	−0.359226
$905$	121150.	0.147919
$906$	0	0
$907$	194393.	0.236302	0.118151	−	0.992996i	$-0.462303\pi$
$907$	194393.	0.236302	0.118151	+	0.992996i	$0.462303\pi$
$908$	− 970110.i	− 1.17665i
$909$	0	0
$910$	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
$914$	335969.	0.402168
$915$	0	0
$916$	364976.i	0.434984i
$917$	0	0
$918$	0	0
$919$	935259.	1.10739	0.553695	−	0.832719i	$-0.313217\pi$
$919$	935259.	1.10739	0.553695	+	0.832719i	$0.313217\pi$
$920$	− 83348.5i	− 0.0984741i
$921$	0	0
$922$	− 610841.i	− 0.718566i
$923$	− 1.39255e6i	− 1.63459i
$924$	0	0
$925$	871291.	1.01831
$926$	178245.	0.207872
$927$	0	0
$928$	−1.02146e6	−1.18612
$929$	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
$932$	−520491.	−0.599213
$933$	0	0
$934$	− 505056.i	− 0.578956i
$935$	−82649.7	−0.0945405
$936$	0	0
$937$	− 695595.i	− 0.792277i	−0.918191	−	0.396139i	$-0.870350\pi$
$937$	− 695595.i	− 0.792277i	0.918191	−	0.396139i	$-0.129650\pi$
$938$	0	0
$939$	0	0
$940$	243806.	0.275923
$941$	− 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
$944$	− 384048.i	− 0.430964i
$945$	0	0
$946$	−77072.0	−0.0861220
$947$	−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
$950$	− 255391.i	− 0.282981i
$951$	0	0
$952$	0	0
$953$	855805.	0.942300	0.471150	−	0.882053i	$-0.343839\pi$
$953$	855805.	0.942300	0.471150	+	0.882053i	$0.343839\pi$
$954$	0	0
$955$	− 104828.i	− 0.114940i
$956$	−387965.	−0.424499
$957$	0	0
$958$	631502.i	0.688087i
$959$	0	0
$960$	0	0
$961$	848553.	0.918824
$962$	− 430881.i	− 0.465594i
$963$	0	0
$964$	819985.i	0.882372i
$965$	107869.i	0.115835i
$966$	0	0
$967$	−1.73940e6	−1.86015	−0.930074	−	0.367371i	$-0.880258\pi$
$967$	−1.73940e6	−1.86015	−0.930074	+	0.367371i	$0.880258\pi$
$968$	−683953.	−0.729920
$969$	0	0
$970$	18893.5	0.0200802
$971$	93336.5i	0.0989950i	0.998774	+	0.0494975i	$0.0157620\pi$
$971$	93336.5i	0.0989950i	−0.998774	+	0.0494975i	$0.984238\pi$
$972$	0	0
$973$	0	0
$974$	587829.	0.619630
$975$	0	0
$976$	485786.i	0.509971i
$977$	−1.05190e6	−1.10201	−0.551004	−	0.834503i	$-0.685755\pi$
$977$	−1.05190e6	−1.10201	−0.551004	+	0.834503i	$0.685755\pi$
$978$	0	0
$979$	− 318280.i	− 0.332081i
$980$	0	0
$981$	0	0
$982$	71401.5	0.0740431
$983$	− 32971.7i	− 0.0341219i	−0.999854	−	0.0170610i	$-0.994569\pi$
$983$	− 32971.7i	− 0.0341219i	0.999854	−	0.0170610i	$-0.00543094\pi$
$984$	0	0
$985$	− 197110.i	− 0.203159i
$986$	600435.i	0.617607i
$987$	0	0
$988$	558980.	0.572641
$989$	313571.	0.320585
$990$	0	0
$991$	638020.	0.649661	0.324831	−	0.945772i	$-0.394693\pi$
$991$	638020.	0.649661	0.324831	+	0.945772i	$0.394693\pi$
$992$	276451.i	0.280928i
$993$	0	0
$994$	0	0
$995$	−335361.	−0.338740
$996$	0	0
$997$	− 1.72164e6i	− 1.73202i	−0.500026	−	0.866010i	$-0.666676\pi$
$997$	− 1.72164e6i	− 1.73202i	0.500026	−	0.866010i	$-0.333324\pi$
$998$	−611288.	−0.613740
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	441.5.d.g.244.4		8
3.2	odd	2		49.5.b.b.48.6	yes	8
7.6	odd	2	inner	441.5.d.g.244.3		8
12.11	even	2		784.5.c.g.97.1		8
21.2	odd	6		49.5.d.c.31.3		16
21.5	even	6		49.5.d.c.31.4		16
21.11	odd	6		49.5.d.c.19.4		16
21.17	even	6		49.5.d.c.19.3		16
21.20	even	2		49.5.b.b.48.5	✓	8
84.83	odd	2		784.5.c.g.97.8		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
49.5.b.b.48.5	✓	8	21.20	even	2
49.5.b.b.48.6	yes	8	3.2	odd	2
49.5.d.c.19.3		16	21.17	even	6
49.5.d.c.19.4		16	21.11	odd	6
49.5.d.c.31.3		16	21.2	odd	6
49.5.d.c.31.4		16	21.5	even	6
441.5.d.g.244.3		8	7.6	odd	2	inner
441.5.d.g.244.4		8	1.1	even	1	trivial
784.5.c.g.97.1		8	12.11	even	2
784.5.c.g.97.8		8	84.83	odd	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:	\( N \)	\(=\)	\( 441 = 3^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	441.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.71722 q^{2} -13.0512 q^{4} +7.83726i q^{5} +49.8872 q^{8} +O(q^{10})\)	\(q-1.71722 q^{2} -13.0512 q^{4} +7.83726i q^{5} +49.8872 q^{8} -13.4583i q^{10} -30.5126 q^{11} +162.301i q^{13} +123.151 q^{16} -345.619i q^{17} +263.892i q^{19} -102.285i q^{20} +52.3969 q^{22} -213.179 q^{23} +563.577 q^{25} -278.707i q^{26} +1011.68 q^{29} -273.802i q^{31} -1009.67 q^{32} +593.504i q^{34} +1546.00 q^{37} -453.160i q^{38} +390.979i q^{40} -531.628i q^{41} -1470.93 q^{43} +398.225 q^{44} +366.075 q^{46} +2383.59i q^{47} -967.786 q^{50} -2118.22i q^{52} -3206.62 q^{53} -239.135i q^{55} -1737.27 q^{58} -3118.51i q^{59} +3944.63i q^{61} +470.179i q^{62} -236.587 q^{64} -1272.00 q^{65} -6919.27 q^{67} +4510.73i q^{68} -8580.05 q^{71} +359.285i q^{73} -2654.82 q^{74} -3444.09i q^{76} +2087.16 q^{79} +965.167i q^{80} +912.922i q^{82} +10594.2i q^{83} +2708.71 q^{85} +2525.91 q^{86} -1522.19 q^{88} +10431.1i q^{89} +2782.23 q^{92} -4093.14i q^{94} -2068.19 q^{95} +1403.85i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 12 q^{2} + 52 q^{4} + 372 q^{8}+O(q^{10}) \) 8 * q + 12 * q^2 + 52 * q^4 + 372 * q^8	\( 8 q + 12 q^{2} + 52 q^{4} + 372 q^{8} - 120 q^{11} - 300 q^{16} + 1952 q^{22} - 1752 q^{23} - 2192 q^{25} - 1248 q^{29} - 3156 q^{32} + 2368 q^{37} - 8552 q^{43} - 264 q^{44} + 7208 q^{46} + 5556 q^{50} - 5496 q^{53} - 17496 q^{58} + 3980 q^{64} - 30240 q^{65} - 7440 q^{67} - 9984 q^{71} + 1080 q^{74} - 14096 q^{79} + 11912 q^{85} - 44496 q^{86} - 44464 q^{88} + 32232 q^{92} - 22488 q^{95}+O(q^{100}) \) 8 * q + 12 * q^2 + 52 * q^4 + 372 * q^8 - 120 * q^11 - 300 * q^16 + 1952 * q^22 - 1752 * q^23 - 2192 * q^25 - 1248 * q^29 - 3156 * q^32 + 2368 * q^37 - 8552 * q^43 - 264 * q^44 + 7208 * q^46 + 5556 * q^50 - 5496 * q^53 - 17496 * q^58 + 3980 * q^64 - 30240 * q^65 - 7440 * q^67 - 9984 * q^71 + 1080 * q^74 - 14096 * q^79 + 11912 * q^85 - 44496 * q^86 - 44464 * q^88 + 32232 * q^92 - 22488 * q^95

Introduction

L-functions

Modular forms

Varieties

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Database

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