LMFDB - Embedded newform 819.2.c.d.64.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [819,2,Mod(64,819)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(819, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("819.64");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$N$	$=$	$819 = 3^{2} \cdot 7 \cdot 13$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	819.c (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:	$6.53974792554$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.265727878144.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{8} + 15x^{6} + 67x^{4} + 77x^{2} + 4$ x^8 + 15x^6 + 67x^4 + 77*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$2$
Twist minimal:	no (minimal twist has level 273)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			64.1
Root			$-2.60520i$ of defining polynomial
Character	$\chi$	$=$	819.64
Dual form			819.2.c.d.64.8

$q$ -expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.60520i q^{2} -4.78706 q^{4} +3.78706i q^{5} -1.00000i q^{7} +7.26084i q^{8} +9.86604 q^{10} -2.55475i q^{11} +(0.0504439 - 3.60520i) q^{13} -2.60520 q^{14} +9.34181 q^{16} +3.21040 q^{17} -8.44270i q^{19} -18.1289i q^{20} -6.65564 q^{22} -3.97809 q^{23} -9.34181 q^{25} +(-9.39226 - 0.131416i) q^{26} +4.78706i q^{28} +2.86858 q^{29} +0.868584i q^{31} -9.81560i q^{32} -8.36372i q^{34} +3.78706 q^{35} -5.10951i q^{37} -21.9949 q^{38} -27.4972 q^{40} +3.34436i q^{41} +4.86858 q^{43} +12.2298i q^{44} +10.3637i q^{46} -11.0983i q^{47} -1.00000 q^{49} +24.3373i q^{50} +(-0.241478 + 17.2583i) q^{52} -1.33319 q^{53} +9.67501 q^{55} +7.26084 q^{56} -7.47323i q^{58} -10.6556i q^{59} -4.10089 q^{61} +2.26283 q^{62} -6.88795 q^{64} +(13.6531 + 0.191034i) q^{65} -10.8467i q^{67} -15.3684 q^{68} -9.86604i q^{70} +2.55475i q^{71} -6.76770i q^{73} -13.3113 q^{74} +40.4157i q^{76} -2.55475 q^{77} -1.23230 q^{79} +35.3780i q^{80} +8.71272 q^{82} +11.0983i q^{83} +12.1580i q^{85} -12.6836i q^{86} +18.5497 q^{88} -5.52423i q^{89} +(-3.60520 - 0.0504439i) q^{91} +19.0434 q^{92} -28.9134 q^{94} +31.9730 q^{95} +5.65819i q^{97} +2.60520i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$8 q - 14 q^{4} + 4 q^{10} - 6 q^{13} + 2 q^{14} + 34 q^{16} - 20 q^{17} - 24 q^{22} + 6 q^{23} - 34 q^{25} - 28 q^{26} + 18 q^{29} + 6 q^{35} - 36 q^{38} - 8 q^{40} + 34 q^{43} - 8 q^{49} + 18 q^{52} + 10 q^{53}+ \cdots + 78 q^{95}+O(q^{100})$ 8 * q - 14 * q^4 + 4 * q^10 - 6 * q^13 + 2 * q^14 + 34 * q^16 - 20 * q^17 - 24 * q^22 + 6 * q^23 - 34 * q^25 - 28 * q^26 + 18 * q^29 + 6 * q^35 - 36 * q^38 - 8 * q^40 + 34 * q^43 - 8 * q^49 + 18 * q^52 + 10 * q^53 + 16 * q^55 + 6 * q^56 - 20 * q^61 + 28 * q^62 - 18 * q^64 + 10 * q^65 + 24 * q^68 - 48 * q^74 - 4 * q^77 - 2 * q^79 - 48 * q^82 - 8 * q^88 - 6 * q^91 - 56 * q^92 - 72 * q^94 + 78 * q^95

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/819\mathbb{Z}\right)^\times$ .

$n$	$92$	$379$	$703$
$\chi(n)$	$1$	$-1$	$1$

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

)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$\alpha_n$			$\theta_n$
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$\alpha_p$			$\theta_p$

$2$		−	2.60520i		−	1.84215i	−0.389381	−	0.921077i	$-0.627311\pi$
$2$		−	2.60520i		−	1.84215i	0.389381	−	0.921077i	$-0.372689\pi$
$3$		0			0
$3$		0			0
$4$	−4.78706			−2.39353
$5$			3.78706i			1.69362i	0.531892	+	0.846812i	$0.321481\pi$
$5$			3.78706i			1.69362i	−0.531892	+	0.846812i	$0.678519\pi$
$6$		0			0
$7$		−	1.00000i		−	0.377964i
$7$		−	1.00000i		−	0.377964i
$8$			7.26084i			2.56709i
$9$		0			0
$10$	9.86604			3.11992
$11$		−	2.55475i		−	0.770287i	−0.922857	−	0.385144i	$-0.874152\pi$
$11$		−	2.55475i		−	0.770287i	0.922857	−	0.385144i	$-0.125848\pi$
$12$		0			0
$13$	0.0504439	−	3.60520i	0.0139906	−	0.999902i
$13$	0.0504439	−	3.60520i	0.0139906	−	0.999902i
$14$	−2.60520			−0.696269
$15$		0			0
$16$	9.34181			2.33545
$17$	3.21040			0.778636			0.389318	−	0.921103i	$-0.372711\pi$
$17$	3.21040			0.778636			0.389318	+	0.921103i	$0.372711\pi$
$18$		0			0
$19$		−	8.44270i		−	1.93689i	−0.249230	−	0.968444i	$-0.580178\pi$
$19$		−	8.44270i		−	1.93689i	0.249230	−	0.968444i	$-0.419822\pi$
$20$		−	18.1289i		−	4.05374i
$21$		0			0
$22$	−6.65564			−1.41899
$23$	−3.97809			−0.829490			−0.414745	−	0.909938i	$-0.636129\pi$
$23$	−3.97809			−0.829490			−0.414745	+	0.909938i	$0.636129\pi$
$24$		0			0
$25$	−9.34181			−1.86836
$26$	−9.39226	−	0.131416i	−1.84197	−	0.0257729i
$27$		0			0
$28$			4.78706i			0.904669i
$29$	2.86858			0.532683			0.266341	−	0.963879i	$-0.414185\pi$
$29$	2.86858			0.532683			0.266341	+	0.963879i	$0.414185\pi$
$30$		0			0
$31$			0.868584i			0.156002i	0.996953	+	0.0780011i	$0.0248538\pi$
$31$			0.868584i			0.156002i	−0.996953	+	0.0780011i	$0.975146\pi$
$32$		−	9.81560i		−	1.73517i
$33$		0			0
$34$		−	8.36372i		−	1.43437i
$35$	3.78706			0.640130
$36$		0			0
$37$		−	5.10951i		−	0.839998i	−0.907525	−	0.419999i	$-0.862030\pi$
$37$		−	5.10951i		−	0.839998i	0.907525	−	0.419999i	$-0.137970\pi$
$38$	−21.9949			−3.56805
$39$		0			0
$40$	−27.4972			−4.34769
$41$			3.34436i			0.522301i	0.965298	+	0.261150i	$0.0841019\pi$
$41$			3.34436i			0.522301i	−0.965298	+	0.261150i	$0.915898\pi$
$42$		0			0
$43$	4.86858			0.742452			0.371226	−	0.928543i	$-0.378938\pi$
$43$	4.86858			0.742452			0.371226	+	0.928543i	$0.378938\pi$
$44$			12.2298i			1.84371i
$45$		0			0
$46$			10.3637i			1.52805i
$47$		−	11.0983i		−	1.61886i	−0.587217	−	0.809430i	$-0.699776\pi$
$47$		−	11.0983i		−	1.61886i	0.587217	−	0.809430i	$-0.300224\pi$
$48$		0			0
$49$	−1.00000			−0.142857
$50$			24.3373i			3.44181i
$51$		0			0
$52$	−0.241478	+	17.2583i	−0.0334870	+	2.39330i
$53$	−1.33319			−0.183128			−0.0915640	−	0.995799i	$-0.529187\pi$
$53$	−1.33319			−0.183128			−0.0915640	+	0.995799i	$0.529187\pi$
$54$		0			0
$55$	9.67501			1.30458
$56$	7.26084			0.970271
$57$		0			0
$58$		−	7.47323i		−	0.981283i
$59$		−	10.6556i		−	1.38725i	−0.720338	−	0.693623i	$-0.756013\pi$
$59$		−	10.6556i		−	1.38725i	0.720338	−	0.693623i	$-0.243987\pi$
$60$		0			0
$61$	−4.10089			−0.525065			−0.262532	−	0.964923i	$-0.584558\pi$
$61$	−4.10089			−0.525065			−0.262532	+	0.964923i	$0.584558\pi$
$62$	2.26283			0.287380
$63$		0			0
$64$	−6.88795			−0.860993
$65$	13.6531	+	0.191034i	1.69346	+	0.0236949i
$66$		0			0
$67$		−	10.8467i		−	1.32513i	−0.749003	−	0.662566i	$-0.769467\pi$
$67$		−	10.8467i		−	1.32513i	0.749003	−	0.662566i	$-0.230533\pi$
$68$	−15.3684			−1.86369
$69$		0			0
$70$		−	9.86604i		−	1.17922i
$71$			2.55475i			0.303194i	0.988442	+	0.151597i	$0.0484415\pi$
$71$			2.55475i			0.303194i	−0.988442	+	0.151597i	$0.951558\pi$
$72$		0			0
$73$		−	6.76770i		−	0.792099i	−0.918229	−	0.396049i	$-0.870381\pi$
$73$		−	6.76770i		−	0.792099i	0.918229	−	0.396049i	$-0.129619\pi$
$74$	−13.3113			−1.54741
$75$		0			0
$76$			40.4157i			4.63600i
$77$	−2.55475			−0.291141
$78$		0			0
$79$	−1.23230			−0.138645			−0.0693225	−	0.997594i	$-0.522084\pi$
$79$	−1.23230			−0.138645			−0.0693225	+	0.997594i	$0.522084\pi$
$80$			35.3780i			3.95538i
$81$		0			0
$82$	8.71272			0.962158
$83$			11.0983i			1.21820i	0.793093	+	0.609101i	$0.208469\pi$
$83$			11.0983i			1.21820i	−0.793093	+	0.609101i	$0.791531\pi$
$84$		0			0
$85$			12.1580i			1.31872i
$86$		−	12.6836i		−	1.36771i
$87$		0			0
$88$	18.5497			1.97740
$89$		−	5.52423i		−	0.585567i	−0.956179	−	0.292783i	$-0.905419\pi$
$89$		−	5.52423i		−	0.585567i	0.956179	−	0.292783i	$-0.0945815\pi$
$90$		0			0
$91$	−3.60520	−	0.0504439i	−0.377927	−	0.00528796i
$92$	19.0434			1.98541
$93$		0			0
$94$	−28.9134			−2.98219
$95$	31.9730			3.28036
$96$		0			0
$97$			5.65819i			0.574502i	0.957855	+	0.287251i	$0.0927414\pi$
$97$			5.65819i			0.574502i	−0.957855	+	0.287251i	$0.907259\pi$
$98$			2.60520i			0.263165i
$99$		0			0
$100$	44.7198			4.47198
$101$	8.94756			0.890316			0.445158	−	0.895452i	$-0.353148\pi$
$101$	8.94756			0.890316			0.445158	+	0.895452i	$0.353148\pi$
$102$		0			0
$103$	−1.67501			−0.165043			−0.0825216	−	0.996589i	$-0.526297\pi$
$103$	−1.67501			−0.165043			−0.0825216	+	0.996589i	$0.526297\pi$
$104$	26.1768	+	0.366265i	2.56684	+	0.0359152i
$105$		0			0
$106$			3.47323i			0.337350i
$107$	0.846676			0.0818513			0.0409256	−	0.999162i	$-0.486969\pi$
$107$	0.846676			0.0818513			0.0409256	+	0.999162i	$0.486969\pi$
$108$		0			0
$109$			17.9949i			1.72360i	0.507248	+	0.861800i	$0.330663\pi$
$109$			17.9949i			1.72360i	−0.507248	+	0.861800i	$0.669337\pi$
$110$		−	25.2053i		−	2.40323i
$111$		0			0
$112$		−	9.34181i		−	0.882718i
$113$	−2.86858			−0.269854			−0.134927	−	0.990856i	$-0.543080\pi$
$113$	−2.86858			−0.269854			−0.134927	+	0.990856i	$0.543080\pi$
$114$		0			0
$115$		−	15.0653i		−	1.40484i
$116$	−13.7321			−1.27499
$117$		0			0
$118$	−27.7601			−2.55552
$119$		−	3.21040i		−	0.294297i
$120$		0			0
$121$	4.47323			0.406657
$122$			10.6836i			0.967250i
$123$		0			0
$124$		−	4.15796i		−	0.373396i
$125$		−	16.4427i		−	1.47068i
$126$		0			0
$127$	17.2053			1.52672			0.763362	−	0.645971i	$-0.223547\pi$
$127$	17.2053			1.52672			0.763362	+	0.645971i	$0.223547\pi$
$128$		−	1.68672i		−	0.149087i
$129$		0			0
$130$	0.497681	−	35.5690i	0.0436496	−	3.11961i
$131$	−16.6836			−1.45766			−0.728828	−	0.684697i	$-0.759934\pi$
$131$	−16.6836			−1.45766			−0.728828	+	0.684697i	$0.759934\pi$
$132$		0			0
$133$	−8.44270			−0.732075
$134$	−28.2577			−2.44110
$135$		0			0
$136$			23.3102i			1.99883i
$137$			18.9134i			1.61588i	0.589265	+	0.807940i	$0.299417\pi$
$137$			18.9134i			1.61588i	−0.589265	+	0.807940i	$0.700583\pi$
$138$		0			0
$139$	5.67501			0.481348			0.240674	−	0.970606i	$-0.422632\pi$
$139$	5.67501			0.481348			0.240674	+	0.970606i	$0.422632\pi$
$140$	−18.1289			−1.53217
$141$		0			0
$142$	6.65564			0.558529
$143$	−9.21040	−	0.128872i	−0.770212	−	0.0107768i
$144$		0			0
$145$			10.8635i			0.902164i
$146$	−17.6312			−1.45917
$147$		0			0
$148$			24.4595i			2.01056i
$149$		−	8.49259i		−	0.695740i	−0.937543	−	0.347870i	$-0.886905\pi$
$149$		−	8.49259i		−	0.695740i	0.937543	−	0.347870i	$-0.113095\pi$
$150$		0			0
$151$			14.2190i			1.15713i	0.815637	+	0.578564i	$0.196387\pi$
$151$			14.2190i			1.15713i	−0.815637	+	0.578564i	$0.803613\pi$
$152$	61.3011			4.97218
$153$		0			0
$154$			6.65564i			0.536327i
$155$	−3.28938			−0.264209
$156$		0			0
$157$	8.42079			0.672052			0.336026	−	0.941853i	$-0.390917\pi$
$157$	8.42079			0.672052			0.336026	+	0.941853i	$0.390917\pi$
$158$			3.21040i			0.255405i
$159$		0			0
$160$	37.1722			2.93872
$161$			3.97809i			0.313518i
$162$		0			0
$163$		−	3.77589i		−	0.295751i	−0.989006	−	0.147875i	$-0.952757\pi$
$163$		−	3.77589i		−	0.295751i	0.989006	−	0.147875i	$-0.0472435\pi$
$164$		−	16.0096i		−	1.25014i
$165$		0			0
$166$	28.9134			2.24411
$167$		−	8.83551i		−	0.683712i	−0.939752	−	0.341856i	$-0.888944\pi$
$167$		−	8.83551i		−	0.683712i	0.939752	−	0.341856i	$-0.111056\pi$
$168$		0			0
$169$	−12.9949	−	0.363721i	−0.999609	−	0.0279785i
$170$	31.6739			2.42928
$171$		0			0
$172$	−23.3062			−1.77708
$173$	−17.6750			−1.34381			−0.671903	−	0.740639i	$-0.734523\pi$
$173$	−17.6750			−1.34381			−0.671903	+	0.740639i	$0.734523\pi$
$174$		0			0
$175$			9.34181i			0.706175i
$176$		−	23.8660i		−	1.79897i
$177$		0			0
$178$	−14.3917			−1.07870
$179$	−16.9073			−1.26371			−0.631856	−	0.775086i	$-0.717707\pi$
$179$	−16.9073			−1.26371			−0.631856	+	0.775086i	$0.717707\pi$
$180$		0			0
$181$	7.73717			0.575099			0.287550	−	0.957766i	$-0.407159\pi$
$181$	7.73717			0.575099			0.287550	+	0.957766i	$0.407159\pi$
$182$	−0.131416	+	9.39226i	−0.00974123	+	0.696200i
$183$		0			0
$184$		−	28.8843i		−	2.12938i
$185$	19.3500			1.42264
$186$		0			0
$187$		−	8.20178i		−	0.599773i
$188$			53.1284i			3.87479i
$189$		0			0
$190$		−	83.2960i		−	6.04293i
$191$	−5.57412			−0.403329			−0.201664	−	0.979455i	$-0.564635\pi$
$191$	−5.57412			−0.403329			−0.201664	+	0.979455i	$0.564635\pi$
$192$		0			0
$193$		−	7.57412i		−	0.545197i	−0.962128	−	0.272598i	$-0.912117\pi$
$193$		−	7.57412i		−	0.545197i	0.962128	−	0.272598i	$-0.0878830\pi$
$194$	14.7407			1.05832
$195$		0			0
$196$	4.78706			0.341933
$197$			13.3444i			0.950746i	0.879784	+	0.475373i	$0.157687\pi$
$197$			13.3444i			0.950746i	−0.879784	+	0.475373i	$0.842313\pi$
$198$		0			0
$199$	−16.8854			−1.19697			−0.598487	−	0.801132i	$-0.704231\pi$
$199$	−16.8854			−1.19697			−0.598487	+	0.801132i	$0.704231\pi$
$200$		−	67.8294i		−	4.79626i
$201$		0			0
$202$		−	23.3102i		−	1.64010i
$203$		−	2.86858i		−	0.201335i
$204$		0			0
$205$	−12.6653			−0.884581
$206$			4.36372i			0.304035i
$207$		0			0
$208$	0.471237	−	33.6791i	0.0326744	−	2.33522i
$209$	−21.5690			−1.49196
$210$		0			0
$211$	11.6969			0.805249			0.402624	−	0.915365i	$-0.368098\pi$
$211$	11.6969			0.805249			0.402624	+	0.915365i	$0.368098\pi$
$212$	6.38207			0.438322
$213$		0			0
$214$		−	2.20576i		−	0.150783i
$215$			18.4376i			1.25743i
$216$		0			0
$217$	0.868584			0.0589633
$218$	46.8803			3.17514
$219$		0			0
$220$	−46.3148			−3.12254
$221$	0.161945	−	11.5741i	0.0108936	−	0.778559i
$222$		0			0
$223$			4.24093i			0.283993i	0.989867	+	0.141997i	$0.0453522\pi$
$223$			4.24093i			0.283993i	−0.989867	+	0.141997i	$0.954648\pi$
$224$	−9.81560			−0.655832
$225$		0			0
$226$			7.47323i			0.497112i
$227$			1.72643i			0.114587i	0.998357	+	0.0572934i	$0.0182471\pi$
$227$			1.72643i			0.114587i	−0.998357	+	0.0572934i	$0.981753\pi$
$228$		0			0
$229$			6.82833i			0.451229i	0.974217	+	0.225614i	$0.0724389\pi$
$229$			6.82833i			0.451229i	−0.974217	+	0.225614i	$0.927561\pi$
$230$	−39.2480			−2.58794
$231$		0			0
$232$			20.8283i			1.36745i
$233$	1.33319			0.0873403			0.0436702	−	0.999046i	$-0.486095\pi$
$233$	1.33319			0.0873403			0.0436702	+	0.999046i	$0.486095\pi$
$234$		0			0
$235$	42.0301			2.74174
$236$			51.0092i			3.32042i
$237$		0			0
$238$	−8.36372			−0.542139
$239$		−	3.90985i		−	0.252907i	−0.991973	−	0.126454i	$-0.959640\pi$
$239$		−	3.90985i		−	0.252907i	0.991973	−	0.126454i	$-0.0403595\pi$
$240$		0			0
$241$		−	25.3903i		−	1.63553i	−0.575552	−	0.817765i	$-0.695213\pi$
$241$		−	25.3903i		−	1.63553i	0.575552	−	0.817765i	$-0.304787\pi$
$242$		−	11.6537i		−	0.749125i
$243$		0			0
$244$	19.6312			1.25676
$245$		−	3.78706i		−	0.241946i
$246$		0			0
$247$	−30.4376	−	0.425883i	−1.93670	−	0.0270983i
$248$	−6.30665			−0.400473
$249$		0			0
$250$	−42.8365			−2.70922
$251$	25.7708			1.62664			0.813319	−	0.581818i	$-0.197658\pi$
$251$	25.7708			1.62664			0.813319	+	0.581818i	$0.197658\pi$
$252$		0			0
$253$			10.1631i			0.638945i
$254$		−	44.8232i		−	2.81246i
$255$		0			0
$256$	−18.1701			−1.13563
$257$	−5.87678			−0.366584			−0.183292	−	0.983059i	$-0.558675\pi$
$257$	−5.87678			−0.366584			−0.183292	+	0.983059i	$0.558675\pi$
$258$		0			0
$259$	−5.10951			−0.317489
$260$	−65.3582	−	0.914491i	−4.05334	−	0.0567143i
$261$		0			0
$262$			43.4642i			2.68522i
$263$	10.2409			0.631483			0.315741	−	0.948845i	$-0.397747\pi$
$263$	10.2409			0.631483			0.315741	+	0.948845i	$0.397747\pi$
$264$		0			0
$265$		−	5.04888i		−	0.310150i
$266$			21.9949i			1.34859i
$267$		0			0
$268$			51.9237i			3.17174i
$269$	16.1396			0.984050			0.492025	−	0.870581i	$-0.336257\pi$
$269$	16.1396			0.984050			0.492025	+	0.870581i	$0.336257\pi$
$270$		0			0
$271$		−	12.2577i		−	0.744605i	−0.928111	−	0.372302i	$-0.878568\pi$
$271$		−	12.2577i		−	0.744605i	0.928111	−	0.372302i	$-0.121432\pi$
$272$	29.9909			1.81847
$273$		0			0
$274$	49.2731			2.97670
$275$			23.8660i			1.43918i
$276$		0			0
$277$	9.81504			0.589729			0.294864	−	0.955539i	$-0.404726\pi$
$277$	9.81504			0.589729			0.294864	+	0.955539i	$0.404726\pi$
$278$		−	14.7845i		−	0.886716i
$279$		0			0
$280$			27.4972i			1.64327i
$281$		−	16.1910i		−	0.965876i	−0.875655	−	0.482938i	$-0.839570\pi$
$281$		−	16.1910i		−	0.965876i	0.875655	−	0.482938i	$-0.160430\pi$
$282$		0			0
$283$	19.1482			1.13824			0.569122	−	0.822253i	$-0.307283\pi$
$283$	19.1482			1.13824			0.569122	+	0.822253i	$0.307283\pi$
$284$		−	12.2298i		−	0.725703i
$285$		0			0
$286$	−0.335737	+	23.9949i	−0.0198525	+	1.41885i
$287$	3.34436			0.197411
$288$		0			0
$289$	−6.69335			−0.393727
$290$	28.3016			1.66192
$291$		0			0
$292$			32.3974i			1.89591i
$293$			24.6725i			1.44138i	0.693257	+	0.720690i	$0.256175\pi$
$293$			24.6725i			1.44138i	−0.693257	+	0.720690i	$0.743825\pi$
$294$		0			0
$295$	40.3535			2.34947
$296$	37.0993			2.15635
$297$		0			0
$298$	−22.1249			−1.28166
$299$	−0.200670	+	14.3418i	−0.0116051	+	0.829408i
$300$		0			0
$301$		−	4.86858i		−	0.280620i
$302$	37.0434			2.13161
$303$		0			0
$304$		−	78.8701i		−	4.52351i
$305$		−	15.5303i		−	0.889263i
$306$		0			0
$307$		−	13.7540i		−	0.784981i	−0.919756	−	0.392491i	$-0.871614\pi$
$307$		−	13.7540i		−	0.784981i	0.919756	−	0.392491i	$-0.128386\pi$
$308$	12.2298			0.696855
$309$		0			0
$310$			8.56948i			0.486714i
$311$	−12.1580			−0.689415			−0.344707	−	0.938710i	$-0.612022\pi$
$311$	−12.1580			−0.689415			−0.344707	+	0.938710i	$0.612022\pi$
$312$		0			0
$313$	21.6699			1.22486			0.612428	−	0.790526i	$-0.290193\pi$
$313$	21.6699			1.22486			0.612428	+	0.790526i	$0.290193\pi$
$314$		−	21.9378i		−	1.23802i
$315$		0			0
$316$	5.89911			0.331851
$317$			29.3780i			1.65003i	0.565109	+	0.825016i	$0.308834\pi$
$317$			29.3780i			1.65003i	−0.565109	+	0.825016i	$0.691166\pi$
$318$		0			0
$319$		−	7.32853i		−	0.410319i
$320$		−	26.0851i		−	1.45820i
$321$		0			0
$322$	10.3637			0.577548
$323$		−	27.1044i		−	1.50813i
$324$		0			0
$325$	−0.471237	+	33.6791i	−0.0261396	+	1.86818i
$326$	−9.83695			−0.544818
$327$		0			0
$328$	−24.2828			−1.34080
$329$	−11.0983			−0.611871
$330$		0			0
$331$			23.9124i			1.31434i	0.753741	+	0.657171i	$0.228247\pi$
$331$			23.9124i			1.31434i	−0.753741	+	0.657171i	$0.771753\pi$
$332$		−	53.1284i		−	2.91580i
$333$		0			0
$334$	−23.0183			−1.25950
$335$	41.0770			2.24428
$336$		0			0
$337$	−15.5706			−0.848182			−0.424091	−	0.905620i	$-0.639406\pi$
$337$	−15.5706			−0.848182			−0.424091	+	0.905620i	$0.639406\pi$
$338$	−0.947564	+	33.8543i	−0.0515407	+	1.84143i
$339$		0			0
$340$		−	58.2009i		−	3.15639i
$341$	2.21902			0.120167
$342$		0			0
$343$			1.00000i			0.0539949i
$344$			35.3500i			1.90594i
$345$		0			0
$346$			46.0469i			2.47550i
$347$	−1.04845			−0.0562838			−0.0281419	−	0.999604i	$-0.508959\pi$
$347$	−1.04845			−0.0562838			−0.0281419	+	0.999604i	$0.508959\pi$
$348$		0			0
$349$			13.8721i			0.742557i	0.928521	+	0.371279i	$0.121081\pi$
$349$			13.8721i			0.742557i	−0.928521	+	0.371279i	$0.878919\pi$
$350$	24.3373			1.30088
$351$		0			0
$352$	−25.0764			−1.33658
$353$			9.96693i			0.530486i	0.964182	+	0.265243i	$0.0854522\pi$
$353$			9.96693i			0.530486i	−0.964182	+	0.265243i	$0.914548\pi$
$354$		0			0
$355$	−9.67501			−0.513496
$356$			26.4448i			1.40157i
$357$		0			0
$358$			44.0469i			2.32795i
$359$			18.9705i			1.00122i	0.865672	+	0.500611i	$0.166891\pi$
$359$			18.9705i			1.00122i	−0.865672	+	0.500611i	$0.833109\pi$
$360$		0			0
$361$	−52.2792			−2.75154
$362$		−	20.1569i		−	1.05942i
$363$		0			0
$364$	17.2583	+	0.241478i	0.904581	+	0.0126569i
$365$	25.6297			1.34152
$366$		0			0
$367$	22.3586			1.16711			0.583556	−	0.812073i	$-0.301661\pi$
$367$	22.3586			1.16711			0.583556	+	0.812073i	$0.301661\pi$
$368$	−37.1626			−1.93723
$369$		0			0
$370$		−	50.4106i		−	2.62072i
$371$			1.33319i			0.0692159i
$372$		0			0
$373$	26.0958			1.35119			0.675595	−	0.737273i	$-0.263887\pi$
$373$	26.0958			1.35119			0.675595	+	0.737273i	$0.263887\pi$
$374$	−21.3673			−1.10487
$375$		0			0
$376$	80.5833			4.15577
$377$	0.144703	−	10.3418i	0.00745256	−	0.532630i
$378$		0			0
$379$		−	12.0000i		−	0.616399i	−0.951322	−	0.308199i	$-0.900274\pi$
$379$		−	12.0000i		−	0.616399i	0.951322	−	0.308199i	$-0.0997264\pi$
$380$	−153.057			−7.85164
$381$		0			0
$382$			14.5217i			0.742994i
$383$			4.96229i			0.253561i	0.991931	+	0.126781i	$0.0404644\pi$
$383$			4.96229i			0.253561i	−0.991931	+	0.126781i	$0.959536\pi$
$384$		0			0
$385$		−	9.67501i		−	0.493084i
$386$	−19.7321			−1.00434
$387$		0			0
$388$		−	27.0861i		−	1.37509i
$389$	−17.8757			−0.906333			−0.453166	−	0.891426i	$-0.649706\pi$
$389$	−17.8757			−0.906333			−0.453166	+	0.891426i	$0.649706\pi$
$390$		0			0
$391$	−12.7713			−0.645870
$392$		−	7.26084i		−	0.366728i
$393$		0			0
$394$	34.7647			1.75142
$395$		−	4.66681i		−	0.234813i
$396$		0			0
$397$			7.06925i			0.354796i	0.984139	+	0.177398i	$0.0567679\pi$
$397$			7.06925i			0.354796i	−0.984139	+	0.177398i	$0.943232\pi$
$398$			43.9898i			2.20501i
$399$		0			0
$400$	−87.2695			−4.36347
$401$			9.12025i			0.455444i	0.973726	+	0.227722i	$0.0731277\pi$
$401$			9.12025i			0.455444i	−0.973726	+	0.227722i	$0.926872\pi$
$402$		0			0
$403$	3.13142	+	0.0438147i	0.155987	+	0.00218257i
$404$	−42.8325			−2.13100
$405$		0			0
$406$	−7.47323			−0.370890
$407$	−13.0535			−0.647040
$408$		0			0
$409$		−	18.2542i		−	0.902611i	−0.892369	−	0.451306i	$-0.850958\pi$
$409$		−	18.2542i		−	0.902611i	0.892369	−	0.451306i	$-0.149042\pi$
$410$			32.9956i			1.62953i
$411$		0			0
$412$	8.01835			0.395036
$413$	−10.6556			−0.524330
$414$		0			0
$415$	−42.0301			−2.06318
$416$	−35.3872	−	0.495137i	−1.73500	−	0.0242761i
$417$		0			0
$418$			56.1916i			2.74842i
$419$	−13.6934			−0.668964			−0.334482	−	0.942402i	$-0.608561\pi$
$419$	−13.6934			−0.668964			−0.334482	+	0.942402i	$0.608561\pi$
$420$		0			0
$421$		−	33.8146i		−	1.64802i	−0.566573	−	0.824012i	$-0.691731\pi$
$421$		−	33.8146i		−	1.64802i	0.566573	−	0.824012i	$-0.308269\pi$
$422$		−	30.4728i		−	1.48339i
$423$		0			0
$424$		−	9.68009i		−	0.470107i
$425$	−29.9909			−1.45477
$426$		0			0
$427$			4.10089i			0.198456i
$428$	−4.05309			−0.195913
$429$		0			0
$430$	48.0336			2.31639
$431$		−	20.3256i		−	0.979048i	−0.871990	−	0.489524i	$-0.837171\pi$
$431$		−	20.3256i		−	0.979048i	0.871990	−	0.489524i	$-0.162829\pi$
$432$		0			0
$433$	−17.1106			−0.822284			−0.411142	−	0.911571i	$-0.634870\pi$
$433$	−17.1106			−0.822284			−0.411142	+	0.911571i	$0.634870\pi$
$434$		−	2.26283i		−	0.108619i
$435$		0			0
$436$		−	86.1427i		−	4.12549i
$437$			33.5858i			1.60663i
$438$		0			0
$439$	−1.67501			−0.0799436			−0.0399718	−	0.999201i	$-0.512727\pi$
$439$	−1.67501			−0.0799436			−0.0399718	+	0.999201i	$0.512727\pi$
$440$			70.2487i			3.34897i
$441$		0			0
$442$	−30.1529	−	0.421899i	−1.43423	−	0.0200677i
$443$	12.9073			0.613245			0.306622	−	0.951831i	$-0.400801\pi$
$443$	12.9073			0.613245			0.306622	+	0.951831i	$0.400801\pi$
$444$		0			0
$445$	20.9206			0.991730
$446$	11.0485			0.523159
$447$		0			0
$448$			6.88795i			0.325425i
$449$		−	21.1762i		−	0.999368i	−0.866208	−	0.499684i	$-0.833450\pi$
$449$		−	21.1762i		−	0.999368i	0.866208	−	0.499684i	$-0.166550\pi$
$450$		0			0
$451$	8.54401			0.402322
$452$	13.7321			0.645903
$453$		0			0
$454$	4.49768			0.211087
$455$	0.191034	−	13.6531i	0.00895581	−	0.640067i
$456$		0			0
$457$		−	8.41570i		−	0.393670i	−0.980437	−	0.196835i	$-0.936934\pi$
$457$		−	8.41570i		−	0.393670i	0.980437	−	0.196835i	$-0.0630663\pi$
$458$	17.7892			0.831232
$459$		0			0
$460$			72.1183i			3.36253i
$461$		−	1.76515i		−	0.0822113i	−0.999155	−	0.0411056i	$-0.986912\pi$
$461$		−	1.76515i		−	0.0822113i	0.999155	−	0.0411056i	$-0.0130880\pi$
$462$		0			0
$463$			38.9241i			1.80896i	0.426519	+	0.904479i	$0.359740\pi$
$463$			38.9241i			1.80896i	−0.426519	+	0.904479i	$0.640260\pi$
$464$	26.7978			1.24406
$465$		0			0
$466$		−	3.47323i		−	0.160894i
$467$	30.4544			1.40926			0.704631	−	0.709573i	$-0.251112\pi$
$467$	30.4544			1.40926			0.704631	+	0.709573i	$0.251112\pi$
$468$		0			0
$469$	−10.8467			−0.500853
$470$		−	109.497i		−	5.05071i
$471$		0			0
$472$	77.3689			3.56119
$473$		−	12.4380i		−	0.571902i
$474$		0			0
$475$			78.8701i			3.61881i
$476$			15.3684i			0.704408i
$477$		0			0
$478$	−10.1859			−0.465894
$479$			3.54656i			0.162046i	0.996712	+	0.0810232i	$0.0258188\pi$
$479$			3.54656i			0.162046i	−0.996712	+	0.0810232i	$0.974181\pi$
$480$		0			0
$481$	−18.4208	−	0.257744i	−0.839916	−	0.0117521i
$482$	−66.1467			−3.01290
$483$		0			0
$484$	−21.4136			−0.973346
$485$	−21.4279			−0.972990
$486$		0			0
$487$			39.9898i			1.81211i	0.423158	+	0.906056i	$0.360922\pi$
$487$			39.9898i			1.81211i	−0.423158	+	0.906056i	$0.639078\pi$
$488$		−	29.7759i		−	1.34789i
$489$		0			0
$490$	−9.86604			−0.445702
$491$	9.69844			0.437685			0.218842	−	0.975760i	$-0.429772\pi$
$491$	9.69844			0.437685			0.218842	+	0.975760i	$0.429772\pi$
$492$		0			0
$493$	9.20929			0.414766
$494$	−1.10951	+	79.2960i	−0.0499192	+	3.56770i
$495$		0			0
$496$			8.11415i			0.364336i
$497$	2.55475			0.114596
$498$		0			0
$499$			24.0336i			1.07589i	0.842979	+	0.537947i	$0.180800\pi$
$499$			24.0336i			1.07589i	−0.842979	+	0.537947i	$0.819200\pi$
$500$			78.7122i			3.52012i
$501$		0			0
$502$		−	67.1381i		−	2.99652i
$503$	−3.67390			−0.163811			−0.0819056	−	0.996640i	$-0.526101\pi$
$503$	−3.67390			−0.163811			−0.0819056	+	0.996640i	$0.526101\pi$
$504$		0			0
$505$			33.8849i			1.50786i
$506$	26.4768			1.17704
$507$		0			0
$508$	−82.3628			−3.65426
$509$		−	19.3612i		−	0.858169i	−0.903264	−	0.429085i	$-0.858836\pi$
$509$		−	19.3612i		−	0.858169i	0.903264	−	0.429085i	$-0.141164\pi$
$510$		0			0
$511$	−6.76770			−0.299385
$512$			43.9634i			1.94293i
$513$		0			0
$514$			15.3102i			0.675303i
$515$		−	6.34334i		−	0.279521i
$516$		0			0
$517$	−28.3535			−1.24699
$518$			13.3113i			0.584864i
$519$		0			0
$520$	−1.38707	+	99.1330i	−0.0608269	+	4.34727i
$521$	−34.1130			−1.49452			−0.747260	−	0.664532i	$-0.768631\pi$
$521$	−34.1130			−1.49452			−0.747260	+	0.664532i	$0.768631\pi$
$522$		0			0
$523$	−33.0250			−1.44408			−0.722042	−	0.691850i	$-0.756796\pi$
$523$	−33.0250			−1.44408			−0.722042	+	0.691850i	$0.756796\pi$
$524$	79.8655			3.48894
$525$		0			0
$526$		−	26.6796i		−	1.16329i
$527$			2.78850i			0.121469i
$528$		0			0
$529$	−7.17478			−0.311947
$530$	−13.1533			−0.571344
$531$		0			0
$532$	40.4157			1.75224
$533$	12.0571	+	0.168702i	0.522250	+	0.00730731i
$534$		0			0
$535$			3.20641i			0.138625i
$536$	78.7560			3.40174
$537$		0			0
$538$		−	42.0469i		−	1.81277i
$539$			2.55475i			0.110041i
$540$		0			0
$541$		−	24.8416i		−	1.06802i	−0.845477	−	0.534012i	$-0.820684\pi$
$541$		−	24.8416i		−	1.06802i	0.845477	−	0.534012i	$-0.179316\pi$
$542$	−31.9339			−1.37168
$543$		0			0
$544$		−	31.5120i		−	1.35106i
$545$	−68.1478			−2.91913
$546$		0			0
$547$	8.14114			0.348090			0.174045	−	0.984738i	$-0.444316\pi$
$547$	8.14114			0.348090			0.174045	+	0.984738i	$0.444316\pi$
$548$		−	90.5395i		−	3.86766i
$549$		0			0
$550$	62.1758			2.65118
$551$		−	24.2186i		−	1.03175i
$552$		0			0
$553$			1.23230i			0.0524029i
$554$		−	25.5701i		−	1.08637i
$555$		0			0
$556$	−27.1666			−1.15212
$557$		−	8.11052i		−	0.343654i	−0.985127	−	0.171827i	$-0.945033\pi$
$557$		−	8.11052i		−	0.343654i	0.985127	−	0.171827i	$-0.0549670\pi$
$558$		0			0
$559$	0.245590	−	17.5522i	0.0103874	−	0.742379i
$560$	35.3780			1.49499
$561$		0			0
$562$	−42.1809			−1.77929
$563$	−11.1044			−0.467995			−0.233998	−	0.972237i	$-0.575181\pi$
$563$	−11.1044			−0.467995			−0.233998	+	0.972237i	$0.575181\pi$
$564$		0			0
$565$		−	10.8635i		−	0.457031i
$566$		−	49.8849i		−	2.09682i
$567$		0			0
$568$	−18.5497			−0.778327
$569$	−2.86858			−0.120257			−0.0601286	−	0.998191i	$-0.519151\pi$
$569$	−2.86858			−0.120257			−0.0601286	+	0.998191i	$0.519151\pi$
$570$		0			0
$571$	32.4947			1.35986			0.679930	−	0.733277i	$-0.262010\pi$
$571$	32.4947			1.35986			0.679930	+	0.733277i	$0.262010\pi$
$572$	44.0907	+	0.616917i	1.84353	+	0.0257946i
$573$		0			0
$574$		−	8.71272i		−	0.363662i
$575$	37.1626			1.54979
$576$		0			0
$577$		−	24.3759i		−	1.01478i	−0.861716	−	0.507390i	$-0.830610\pi$
$577$		−	24.3759i		−	1.01478i	0.861716	−	0.507390i	$-0.169390\pi$
$578$			17.4375i			0.725305i
$579$		0			0
$580$		−	52.0042i		−	2.15936i
$581$	11.0983			0.460437
$582$		0			0
$583$			3.40598i			0.141061i
$584$	49.1392			2.03339
$585$		0			0
$586$	64.2767			2.65524
$587$			11.7483i			0.484906i	0.970163	+	0.242453i	$0.0779520\pi$
$587$			11.7483i			0.484906i	−0.970163	+	0.242453i	$0.922048\pi$
$588$		0			0
$589$	7.33319			0.302159
$590$		−	105.129i		−	4.32809i
$591$		0			0
$592$		−	47.7321i		−	1.96178i
$593$		−	46.6459i		−	1.91552i	−0.287574	−	0.957759i	$-0.592849\pi$
$593$		−	46.6459i		−	1.91552i	0.287574	−	0.957759i	$-0.407151\pi$
$594$		0			0
$595$	12.1580			0.498428
$596$			40.6545i			1.66527i
$597$		0			0
$598$	37.3633	+	0.522786i	1.52790	+	0.0213783i
$599$	20.8197			0.850669			0.425335	−	0.905036i	$-0.360156\pi$
$599$	20.8197			0.850669			0.425335	+	0.905036i	$0.360156\pi$
$600$		0			0
$601$	24.2588			0.989539			0.494770	−	0.869024i	$-0.335252\pi$
$601$	24.2588			0.989539			0.494770	+	0.869024i	$0.335252\pi$
$602$	−12.6836			−0.516946
$603$		0			0
$604$		−	68.0673i		−	2.76962i
$605$			16.9404i			0.688724i
$606$		0			0
$607$	6.20067			0.251677			0.125839	−	0.992051i	$-0.459838\pi$
$607$	6.20067			0.251677			0.125839	+	0.992051i	$0.459838\pi$
$608$	−82.8701			−3.36083
$609$		0			0
$610$	−40.4595			−1.63816
$611$	−40.0117	−	0.559844i	−1.61870	−	0.0226489i
$612$		0			0
$613$		−	14.9026i		−	0.601912i	−0.953638	−	0.300956i	$-0.902694\pi$
$613$		−	14.9026i		−	0.601912i	0.953638	−	0.300956i	$-0.0973058\pi$
$614$	−35.8319			−1.44606
$615$		0			0
$616$		−	18.5497i		−	0.747387i
$617$			24.1472i			0.972130i	0.873923	+	0.486065i	$0.161568\pi$
$617$			24.1472i			0.972130i	−0.873923	+	0.486065i	$0.838432\pi$
$618$		0			0
$619$			43.9339i			1.76585i	0.469513	+	0.882925i	$0.344429\pi$
$619$			43.9339i			1.76585i	−0.469513	+	0.882925i	$0.655571\pi$
$620$	15.7464			0.632392
$621$		0			0
$622$			31.6739i			1.27001i
$623$	−5.52423			−0.221323
$624$		0			0
$625$	15.5604			0.622416
$626$		−	56.4544i		−	2.25637i
$627$		0			0
$628$	−40.3108			−1.60858
$629$		−	16.4036i		−	0.654052i
$630$		0			0
$631$		−	25.9511i		−	1.03310i	−0.856258	−	0.516548i	$-0.827217\pi$
$631$		−	25.9511i		−	1.03310i	0.856258	−	0.516548i	$-0.172783\pi$
$632$		−	8.94756i		−	0.355915i
$633$		0			0
$634$	76.5355			3.03961
$635$			65.1575i			2.58570i
$636$		0			0
$637$	−0.0504439	+	3.60520i	−0.00199866	+	0.142843i
$638$	−19.0923			−0.755870
$639$		0			0
$640$	6.38772			0.252497
$641$	39.5858			1.56355			0.781773	−	0.623563i	$-0.214315\pi$
$641$	39.5858			1.56355			0.781773	+	0.623563i	$0.214315\pi$
$642$		0			0
$643$		−	17.0434i		−	0.672125i	−0.941840	−	0.336062i	$-0.890905\pi$
$643$		−	17.0434i		−	0.672125i	0.941840	−	0.336062i	$-0.109095\pi$
$644$		−	19.0434i		−	0.750414i
$645$		0			0
$646$	−70.6124			−2.77821
$647$	27.7544			1.09114			0.545569	−	0.838066i	$-0.316313\pi$
$647$	27.7544			1.09114			0.545569	+	0.838066i	$0.316313\pi$
$648$		0			0
$649$	−27.2225			−1.06858
$650$	87.7407	+	1.22767i	3.44147	+	0.0481531i
$651$		0			0
$652$			18.0754i			0.707888i
$653$	7.77080			0.304095			0.152048	−	0.988373i	$-0.451413\pi$
$653$	7.77080			0.304095			0.152048	+	0.988373i	$0.451413\pi$
$654$		0			0
$655$		−	63.1819i		−	2.46872i
$656$			31.2424i			1.21981i
$657$		0			0
$658$			28.9134i			1.12716i
$659$	4.74935			0.185008			0.0925042	−	0.995712i	$-0.470513\pi$
$659$	4.74935			0.185008			0.0925042	+	0.995712i	$0.470513\pi$
$660$		0			0
$661$		−	38.0301i		−	1.47920i	−0.673047	−	0.739599i	$-0.735015\pi$
$661$		−	38.0301i		−	1.47920i	0.673047	−	0.739599i	$-0.264985\pi$
$662$	62.2965			2.42122
$663$		0			0
$664$	−80.5833			−3.12724
$665$		−	31.9730i		−	1.23986i
$666$		0			0
$667$	−11.4115			−0.441855
$668$			42.2961i			1.63649i
$669$		0			0
$670$		−	107.014i		−	4.13430i
$671$			10.4768i			0.404451i
$672$		0			0
$673$	23.0602			0.888905			0.444452	−	0.895803i	$-0.353398\pi$
$673$	23.0602			0.888905			0.444452	+	0.895803i	$0.353398\pi$
$674$			40.5644i			1.56248i
$675$		0			0
$676$	62.2074	+	1.74115i	2.39259	+	0.0669674i
$677$	44.3759			1.70550			0.852752	−	0.522317i	$-0.174932\pi$
$677$	44.3759			1.70550			0.852752	+	0.522317i	$0.174932\pi$
$678$		0			0
$679$	5.65819			0.217141
$680$	−88.2770			−3.38527
$681$		0			0
$682$		−	5.78098i		−	0.221365i
$683$		−	5.91494i		−	0.226329i	−0.993576	−	0.113165i	$-0.963901\pi$
$683$		−	5.91494i		−	0.226329i	0.993576	−	0.113165i	$-0.0360987\pi$
$684$		0			0
$685$	−71.6261			−2.73669
$686$	2.60520			0.0994669
$687$		0			0
$688$	45.4814			1.73396
$689$	−0.0672514	+	4.80642i	−0.00256207	+	0.183110i
$690$		0			0
$691$		−	12.6883i		−	0.482685i	−0.970440	−	0.241343i	$-0.922412\pi$
$691$		−	12.6883i		−	0.482685i	0.970440	−	0.241343i	$-0.0775878\pi$
$692$	84.6113			3.21644
$693$		0			0
$694$			2.73143i			0.103683i
$695$			21.4916i			0.815222i
$696$		0			0
$697$			10.7367i			0.406682i
$698$	36.1396			1.36790
$699$		0			0
$700$		−	44.7198i		−	1.69025i
$701$	−25.8487			−0.976291			−0.488146	−	0.872762i	$-0.662326\pi$
$701$	−25.8487			−0.976291			−0.488146	+	0.872762i	$0.662326\pi$
$702$		0			0
$703$	−43.1381			−1.62698
$704$			17.5970i			0.663212i
$705$		0			0
$706$	25.9658			0.977237
$707$		−	8.94756i		−	0.336508i
$708$		0			0
$709$			18.8803i			0.709065i	0.935044	+	0.354533i	$0.115360\pi$
$709$			18.8803i			0.709065i	−0.935044	+	0.354533i	$0.884640\pi$
$710$			25.2053i			0.945938i
$711$		0			0
$712$	40.1105			1.50321
$713$		−	3.45531i		−	0.129402i
$714$		0			0
$715$	0.488045	−	34.8803i	0.0182518	−	1.30445i
$716$	80.9363			3.02473
$717$		0			0
$718$	49.4218			1.84441
$719$	40.1580			1.49764			0.748820	−	0.662774i	$-0.230621\pi$
$719$	40.1580			1.49764			0.748820	+	0.662774i	$0.230621\pi$
$720$		0			0
$721$			1.67501i			0.0623804i
$722$			136.198i			5.06875i
$723$		0			0
$724$	−37.0383			−1.37652
$725$	−26.7978			−0.995244
$726$		0			0
$727$	−32.0336			−1.18806			−0.594031	−	0.804442i	$-0.702464\pi$
$727$	−32.0336			−1.18806			−0.594031	+	0.804442i	$0.702464\pi$
$728$	0.366265	−	26.1768i	0.0135747	−	0.970176i
$729$		0			0
$730$		−	66.7704i		−	2.47128i
$731$	15.6301			0.578100
$732$		0			0
$733$			48.3265i			1.78498i	0.451066	+	0.892491i	$0.351044\pi$
$733$			48.3265i			1.78498i	−0.451066	+	0.892491i	$0.648956\pi$
$734$		−	58.2487i		−	2.15000i
$735$		0			0
$736$			39.0473i			1.43930i
$737$	−27.7106			−1.02073
$738$		0			0
$739$			35.7321i			1.31443i	0.753705	+	0.657213i	$0.228265\pi$
$739$			35.7321i			1.31443i	−0.753705	+	0.657213i	$0.771735\pi$
$740$	−92.6296			−3.40513
$741$		0			0
$742$	3.47323			0.127506
$743$			45.0092i			1.65123i	0.564236	+	0.825613i	$0.309171\pi$
$743$			45.0092i			1.65123i	−0.564236	+	0.825613i	$0.690829\pi$
$744$		0			0
$745$	32.1619			1.17832
$746$		−	67.9847i		−	2.48910i
$747$		0			0
$748$			39.2624i			1.43557i
$749$		−	0.846676i		−	0.0309369i
$750$		0			0
$751$	5.39425			0.196839			0.0984195	−	0.995145i	$-0.468621\pi$
$751$	5.39425			0.196839			0.0984195	+	0.995145i	$0.468621\pi$
$752$		−	103.679i		−	3.78077i
$753$		0			0
$754$	−26.9425	−	0.376979i	−0.981187	−	0.0137288i
$755$	−53.8483			−1.95974
$756$		0			0
$757$	13.7896			0.501191			0.250595	−	0.968092i	$-0.419374\pi$
$757$	13.7896			0.501191			0.250595	+	0.968092i	$0.419374\pi$
$758$	−31.2624			−1.13550
$759$		0			0
$760$			232.151i			8.42100i
$761$		−	30.6674i		−	1.11169i	−0.831286	−	0.555846i	$-0.812395\pi$
$761$		−	30.6674i		−	1.11169i	0.831286	−	0.555846i	$-0.187605\pi$
$762$		0			0
$763$	17.9949			0.651460
$764$	26.6836			0.965380
$765$		0			0
$766$	12.9277			0.467099
$767$	−38.4157	−	0.537512i	−1.38711	−	0.0194084i
$768$		0			0
$769$		−	48.6160i		−	1.75314i	−0.481278	−	0.876568i	$-0.659827\pi$
$769$		−	48.6160i		−	1.75314i	0.481278	−	0.876568i	$-0.340173\pi$
$770$	−25.2053			−0.908336
$771$		0			0
$772$			36.2577i			1.30494i
$773$		−	18.6729i		−	0.671617i	−0.941930	−	0.335808i	$-0.890991\pi$
$773$		−	18.6729i		−	0.671617i	0.941930	−	0.335808i	$-0.109009\pi$
$774$		0			0
$775$		−	8.11415i		−	0.291469i
$776$	−41.0832			−1.47480
$777$		0			0
$778$			46.5697i			1.66960i
$779$	28.2354			1.01164
$780$		0			0
$781$	6.52677			0.233546
$782$			33.2717i			1.18979i
$783$		0			0
$784$	−9.34181			−0.333636
$785$			31.8900i			1.13820i
$786$		0			0
$787$		−	9.61751i		−	0.342827i	−0.985199	−	0.171414i	$-0.945167\pi$
$787$		−	9.61751i		−	0.342827i	0.985199	−	0.171414i	$-0.0548334\pi$
$788$		−	63.8802i		−	2.27564i
$789$		0			0
$790$	−12.1580			−0.432561
$791$			2.86858i			0.101995i
$792$		0			0
$793$	−0.206865	+	14.7845i	−0.00734598	+	0.525013i
$794$	18.4168			0.653588
$795$		0			0
$796$	80.8314			2.86499
$797$	3.73606			0.132338			0.0661691	−	0.997808i	$-0.478922\pi$
$797$	3.73606			0.132338			0.0661691	+	0.997808i	$0.478922\pi$
$798$		0			0
$799$		−	35.6301i		−	1.26050i
$800$			91.6955i			3.24192i
$801$		0			0
$802$	23.7601			0.838997
$803$	−17.2898			−0.610144
$804$		0			0
$805$	−15.0653			−0.530981
$806$	0.114146	−	8.15796i	0.00402062	−	0.287352i
$807$		0			0
$808$			64.9668i			2.28553i
$809$	10.1850			0.358084			0.179042	−	0.983841i	$-0.442700\pi$
$809$	10.1850			0.358084			0.179042	+	0.983841i	$0.442700\pi$
$810$		0			0
$811$			6.21902i			0.218379i	0.994021	+	0.109190i	$0.0348256\pi$
$811$			6.21902i			0.218379i	−0.994021	+	0.109190i	$0.965174\pi$
$812$			13.7321i			0.481901i
$813$		0			0
$814$			34.0071i			1.19195i
$815$	14.2995			0.500891
$816$		0			0
$817$		−	41.1040i		−	1.43805i
$818$	−47.5558			−1.66275
$819$		0			0
$820$	60.6294			2.11727
$821$			55.4534i			1.93534i	0.252224	+	0.967669i	$0.418838\pi$
$821$			55.4534i			1.93534i	−0.252224	+	0.967669i	$0.581162\pi$
$822$		0			0
$823$	−20.4380			−0.712425			−0.356213	−	0.934405i	$-0.615932\pi$
$823$	−20.4380			−0.712425			−0.356213	+	0.934405i	$0.615932\pi$
$824$		−	12.1619i		−	0.423681i
$825$		0			0
$826$			27.7601i			0.965896i
$827$			14.1289i			0.491309i	0.969357	+	0.245655i	$0.0790029\pi$
$827$			14.1289i			0.491309i	−0.969357	+	0.245655i	$0.920997\pi$
$828$		0			0
$829$	34.9159			1.21268			0.606340	−	0.795206i	$-0.292637\pi$
$829$	34.9159			1.21268			0.606340	+	0.795206i	$0.292637\pi$
$830$			109.497i			3.80069i
$831$		0			0
$832$	−0.347455	+	24.8324i	−0.0120458	+	0.860909i
$833$	−3.21040			−0.111234
$834$		0			0
$835$	33.4606			1.15795
$836$	103.252			3.57105
$837$		0			0
$838$			35.6739i			1.23233i
$839$		−	46.4437i		−	1.60341i	−0.597717	−	0.801707i	$-0.703925\pi$
$839$		−	46.4437i		−	1.60341i	0.597717	−	0.801707i	$-0.296075\pi$
$840$		0			0
$841$	−20.7712			−0.716249
$842$	−88.0938			−3.03591
$843$		0			0
$844$	−55.9938			−1.92739
$845$	1.37743	−	49.2125i	0.0473851	−	1.69296i
$846$		0			0
$847$		−	4.47323i		−	0.153702i
$848$	−12.4544			−0.427687
$849$		0			0
$850$			78.1323i			2.67992i
$851$			20.3261i			0.696770i
$852$		0			0
$853$			41.4462i			1.41909i	0.704659	+	0.709546i	$0.251100\pi$
$853$			41.4462i			1.41909i	−0.704659	+	0.709546i	$0.748900\pi$
$854$	10.6836			0.365586
$855$		0			0
$856$			6.14758i			0.210120i
$857$	−33.8666			−1.15686			−0.578431	−	0.815732i	$-0.696335\pi$
$857$	−33.8666			−1.15686			−0.578431	+	0.815732i	$0.696335\pi$
$858$		0			0
$859$	−37.2890			−1.27228			−0.636141	−	0.771573i	$-0.719470\pi$
$859$	−37.2890			−1.27228			−0.636141	+	0.771573i	$0.719470\pi$
$860$		−	88.2619i		−	3.00971i
$861$		0			0
$862$	−52.9521			−1.80356
$863$		−	38.7464i		−	1.31894i	−0.751730	−	0.659471i	$-0.770781\pi$
$863$		−	38.7464i		−	1.31894i	0.751730	−	0.659471i	$-0.229219\pi$
$864$		0			0
$865$		−	66.9363i		−	2.27590i
$866$			44.5765i			1.51477i
$867$		0			0
$868$	−4.15796			−0.141130
$869$			3.14823i			0.106797i
$870$		0			0
$871$	−39.1044	−	0.547149i	−1.32500	−	0.0185394i
$872$	−130.658			−4.42464
$873$		0			0
$874$	87.4978			2.95966
$875$	−16.4427			−0.555865
$876$		0			0
$877$			51.8319i			1.75024i	0.483908	+	0.875119i	$0.339217\pi$
$877$			51.8319i			1.75024i	−0.483908	+	0.875119i	$0.660783\pi$
$878$			4.36372i			0.147268i
$879$		0			0
$880$	90.3821			3.04678
$881$	−20.7896			−0.700420			−0.350210	−	0.936671i	$-0.613890\pi$
$881$	−20.7896			−0.700420			−0.350210	+	0.936671i	$0.613890\pi$
$882$		0			0
$883$	17.5292			0.589904			0.294952	−	0.955512i	$-0.404696\pi$
$883$	17.5292			0.589904			0.294952	+	0.955512i	$0.404696\pi$
$884$	−0.775240	+	55.4060i	−0.0260741	+	1.86350i
$885$		0			0
$886$		−	33.6261i		−	1.12969i
$887$	−9.53539			−0.320167			−0.160084	−	0.987103i	$-0.551176\pi$
$887$	−9.53539			−0.320167			−0.160084	+	0.987103i	$0.551176\pi$
$888$		0			0
$889$		−	17.2053i		−	0.577047i
$890$		−	54.5022i		−	1.82692i
$891$		0			0
$892$		−	20.3016i		−	0.679746i
$893$	−93.7000			−3.13555
$894$		0			0
$895$		−	64.0290i		−	2.14025i
$896$	−1.68672			−0.0563495
$897$		0			0
$898$	−55.1683			−1.84099
$899$			2.49160i			0.0830997i
$900$		0			0
$901$	−4.28008			−0.142590
$902$		−	22.2588i		−	0.741139i
$903$		0			0
$904$		−	20.8283i		−	0.692740i
$905$			29.3011i			0.974002i
$906$		0			0
$907$	27.3270			0.907378			0.453689	−	0.891160i	$-0.350108\pi$
$907$	27.3270			0.907378			0.453689	+	0.891160i	$0.350108\pi$
$908$		−	8.26450i		−	0.274267i
$909$		0			0
$910$	−35.5690	−	0.497681i	−1.17910	−	0.0164980i
$911$	38.9582			1.29074			0.645371	−	0.763869i	$-0.276703\pi$
$911$	38.9582			1.29074			0.645371	+	0.763869i	$0.276703\pi$
$912$		0			0
$913$	28.3535			0.938365
$914$	−21.9246			−0.725201
$915$		0			0
$916$		−	32.6876i		−	1.08003i
$917$			16.6836i			0.550942i
$918$		0			0
$919$	9.94293			0.327987			0.163993	−	0.986461i	$-0.447562\pi$
$919$	9.94293			0.327987			0.163993	+	0.986461i	$0.447562\pi$
$920$	109.387			3.60637
$921$		0			0
$922$	−4.59857			−0.151446
$923$	9.21040	+	0.128872i	0.303164	+	0.00424187i
$924$		0			0
$925$			47.7321i			1.56942i
$926$	101.405			3.33238
$927$		0			0
$928$		−	28.1569i		−	0.924294i
$929$			7.80854i			0.256190i	0.991762	+	0.128095i	$0.0408862\pi$
$929$			7.80854i			0.256190i	−0.991762	+	0.128095i	$0.959114\pi$
$930$		0			0
$931$			8.44270i			0.276698i
$932$	−6.38207			−0.209052
$933$		0			0
$934$		−	79.3398i		−	2.59608i
$935$	31.0606			1.01579
$936$		0			0
$937$	−9.59246			−0.313372			−0.156686	−	0.987648i	$-0.550081\pi$
$937$	−9.59246			−0.313372			−0.156686	+	0.987648i	$0.550081\pi$
$938$			28.2577i			0.922648i
$939$		0			0
$940$	−201.200			−6.56243
$941$		−	28.5043i		−	0.929214i	−0.885517	−	0.464607i	$-0.846196\pi$
$941$		−	28.5043i		−	0.929214i	0.885517	−	0.464607i	$-0.153804\pi$
$942$		0			0
$943$		−	13.3042i		−	0.433243i
$944$		−	99.5430i		−	3.23985i
$945$		0			0
$946$	−32.4036			−1.05353
$947$		−	47.8997i		−	1.55653i	−0.627936	−	0.778265i	$-0.716100\pi$
$947$		−	47.8997i		−	1.55653i	0.627936	−	0.778265i	$-0.283900\pi$
$948$		0			0
$949$	−24.3989	−	0.341389i	−0.792021	−	0.0110820i
$950$	205.472			6.66640
$951$		0			0
$952$	23.3102			0.755487
$953$	−32.5181			−1.05337			−0.526683	−	0.850062i	$-0.676564\pi$
$953$	−32.5181			−1.05337			−0.526683	+	0.850062i	$0.676564\pi$
$954$		0			0
$955$		−	21.1095i		−	0.683088i
$956$			18.7167i			0.605341i
$957$		0			0
$958$	9.23949			0.298514
$959$	18.9134			0.610745
$960$		0			0
$961$	30.2456			0.975663
$962$	−0.671473	+	47.9898i	−0.0216492	+	1.54725i
$963$		0			0
$964$			121.545i			3.91469i
$965$	28.6836			0.923359
$966$		0			0
$967$			18.0387i			0.580086i	0.957014	+	0.290043i	$0.0936697\pi$
$967$			18.0387i			0.580086i	−0.957014	+	0.290043i	$0.906330\pi$
$968$			32.4794i			1.04393i
$969$		0			0
$970$			55.8239i			1.79240i
$971$	−30.2190			−0.969774			−0.484887	−	0.874577i	$-0.661139\pi$
$971$	−30.2190			−0.969774			−0.484887	+	0.874577i	$0.661139\pi$
$972$		0			0
$973$		−	5.67501i		−	0.181932i
$974$	104.181			3.33819
$975$		0			0
$976$	−38.3097			−1.22626
$977$			38.4652i			1.23061i	0.788289	+	0.615305i	$0.210967\pi$
$977$			38.4652i			1.23061i	−0.788289	+	0.615305i	$0.789033\pi$
$978$		0			0
$979$	−14.1130			−0.451055
$980$			18.1289i			0.579106i
$981$		0			0
$982$		−	25.2664i		−	0.806282i
$983$		−	12.1742i		−	0.388297i	−0.980972	−	0.194149i	$-0.937806\pi$
$983$		−	12.1742i		−	0.388297i	0.980972	−	0.194149i	$-0.0621944\pi$
$984$		0			0
$985$	−50.5359			−1.61021
$986$		−	23.9920i		−	0.764062i
$987$		0			0
$988$	145.707	+	2.03873i	4.63555	+	0.0648605i
$989$	−19.3677			−0.615856
$990$		0			0
$991$	14.4982			0.460552			0.230276	−	0.973125i	$-0.426037\pi$
$991$	14.4982			0.460552			0.230276	+	0.973125i	$0.426037\pi$
$992$	8.52567			0.270690
$993$		0			0
$994$		−	6.65564i		−	0.211104i
$995$		−	63.9460i		−	2.02722i
$996$		0			0
$997$	−9.35086			−0.296145			−0.148072	−	0.988977i	$-0.547307\pi$
$997$	−9.35086			−0.296145			−0.148072	+	0.988977i	$0.547307\pi$
$998$	62.6124			1.98196
$999$		0			0

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

)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	819.2.c.d.64.1		8
3.2	odd	2		273.2.c.c.64.8	yes	8
12.11	even	2		4368.2.h.q.337.1		8
13.12	even	2	inner	819.2.c.d.64.8		8
21.20	even	2		1911.2.c.l.883.8		8
39.5	even	4		3549.2.a.v.1.4		4
39.8	even	4		3549.2.a.x.1.1		4
39.38	odd	2		273.2.c.c.64.1	✓	8
156.155	even	2		4368.2.h.q.337.8		8
273.272	even	2		1911.2.c.l.883.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
273.2.c.c.64.1	✓	8	39.38	odd	2
273.2.c.c.64.8	yes	8	3.2	odd	2
819.2.c.d.64.1		8	1.1	even	1	trivial
819.2.c.d.64.8		8	13.12	even	2	inner
1911.2.c.l.883.1		8	273.272	even	2
1911.2.c.l.883.8		8	21.20	even	2
3549.2.a.v.1.4		4	39.5	even	4
3549.2.a.x.1.1		4	39.8	even	4
4368.2.h.q.337.1		8	12.11	even	2
4368.2.h.q.337.8		8	156.155	even	2

Overview	Random
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Classical	Maass
Hilbert	Bianchi

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

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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Newspace parameters

Newform invariants

Embedding invariants

$q$ -expansion

Character values

Coefficient data

Twists

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	819.2.c.d.64.1

Level	$819$
Weight	$2$
Character	819.64
Analytic conductor	$6.540$
Analytic rank	$0$
Dimension	$8$
Inner twists	$2$