LMFDB - Embedded newform 425.2.c.c.424.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [425,2,Mod(424,425)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(425, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("425.424");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$N$	$=$	$425 = 5^{2} \cdot 17$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	425.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:	$3.39364208590$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{12} + 16 x^{10} - 4 x^{9} + 111 x^{8} + 4 x^{7} + 394 x^{6} + 236 x^{5} + 581 x^{4} + 1000 x^{3} + \cdots + 845$ x^12 + 16x^10 - 4x^9 + 111x^8 + 4x^7 + 394x^6 + 236x^5 + 581x^4 + 1000x^3 + 962x^2 + 1504x + 845
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$2^{4}$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			424.5
Root			$1.10716 - 1.97403i$ of defining polynomial
Character	$\chi$	$=$	425.424
Dual form			425.2.c.c.424.7

$q$ -expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.311108i q^{2} -1.12261 q^{3} +1.90321 q^{4} +0.349253i q^{6} -3.60843 q^{7} -1.21432i q^{8} -1.73975 q^{9} -5.74499i q^{11} -2.13656 q^{12} -2.59210i q^{13} +1.12261i q^{14} +3.42864 q^{16} +(3.25917 - 2.52543i) q^{17} +0.541249i q^{18} -4.28100 q^{19} +4.05086 q^{21} -1.78731 q^{22} +1.47186 q^{23} +1.36321i q^{24} -0.806424 q^{26} +5.32089 q^{27} -6.86760 q^{28} +5.50439i q^{29} -8.33946i q^{31} -3.49532i q^{32} +6.44938i q^{33} +(-0.785680 - 1.01395i) q^{34} -3.31111 q^{36} -6.20290 q^{37} +1.33185i q^{38} +2.90992i q^{39} +3.95768i q^{41} -1.26025i q^{42} -9.76049i q^{43} -10.9339i q^{44} -0.457908i q^{46} +6.62222i q^{47} -3.84902 q^{48} +6.02074 q^{49} +(-3.65878 + 2.83507i) q^{51} -4.93332i q^{52} -0.658781i q^{53} -1.65537i q^{54} +4.38178i q^{56} +4.80589 q^{57} +1.71246 q^{58} -5.95407 q^{59} +8.76357i q^{61} -2.59447 q^{62} +6.27775 q^{63} +5.76986 q^{64} +2.00645 q^{66} -0.428639i q^{67} +(6.20290 - 4.80642i) q^{68} -1.65233 q^{69} -1.36321i q^{71} +2.11261i q^{72} +3.25917 q^{73} +1.92977i q^{74} -8.14764 q^{76} +20.7304i q^{77} +0.905299 q^{78} -1.89597i q^{79} -0.754037 q^{81} +1.23127 q^{82} +16.6637i q^{83} +7.70964 q^{84} -3.03657 q^{86} -6.17929i q^{87} -6.97626 q^{88} +3.52543 q^{89} +9.35342i q^{91} +2.80127 q^{92} +9.36196i q^{93} +2.06022 q^{94} +3.92388i q^{96} +11.7073 q^{97} -1.87310i q^{98} +9.99483i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$12 q - 4 q^{4} + 32 q^{9} - 12 q^{16} - 24 q^{19} - 4 q^{21} + 44 q^{26} - 36 q^{34} - 40 q^{36} - 8 q^{49} - 16 q^{51} + 8 q^{59} + 44 q^{64} - 32 q^{66} - 48 q^{69} - 72 q^{76} + 68 q^{81} + 12 q^{84}+ \cdots + 80 q^{94}+O(q^{100})$ 12 * q - 4 * q^4 + 32 * q^9 - 12 * q^16 - 24 * q^19 - 4 * q^21 + 44 * q^26 - 36 * q^34 - 40 * q^36 - 8 * q^49 - 16 * q^51 + 8 * q^59 + 44 * q^64 - 32 * q^66 - 48 * q^69 - 72 * q^76 + 68 * q^81 + 12 * q^84 - 8 * q^86 + 16 * q^89 + 80 * q^94

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/425\mathbb{Z}\right)^\times$ .

$n$	$52$	$326$
$\chi(n)$	$-1$	$-1$

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

)

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

$2$		−	0.311108i		−	0.219986i	−0.993932	−	0.109993i	$-0.964917\pi$
$2$		−	0.311108i		−	0.219986i	0.993932	−	0.109993i	$-0.0350829\pi$
$3$	−1.12261			−0.648139			−0.324070	−	0.946033i	$-0.605051\pi$
$3$	−1.12261			−0.648139			−0.324070	+	0.946033i	$0.605051\pi$
$4$	1.90321			0.951606
$5$		0			0
$5$		0			0
$6$			0.349253i			0.142582i
$7$	−3.60843			−1.36386			−0.681929	−	0.731419i	$-0.738859\pi$
$7$	−3.60843			−1.36386			−0.681929	+	0.731419i	$0.738859\pi$
$8$		−	1.21432i		−	0.429327i
$9$	−1.73975			−0.579916
$10$		0			0
$11$		−	5.74499i		−	1.73218i	−0.499888	−	0.866090i	$-0.666626\pi$
$11$		−	5.74499i		−	1.73218i	0.499888	−	0.866090i	$-0.333374\pi$
$12$	−2.13656			−0.616773
$13$		−	2.59210i		−	0.718920i	−0.933160	−	0.359460i	$-0.882961\pi$
$13$		−	2.59210i		−	0.718920i	0.933160	−	0.359460i	$-0.117039\pi$
$14$			1.12261i			0.300030i
$15$		0			0
$16$	3.42864			0.857160
$17$	3.25917	−	2.52543i	0.790466	−	0.612506i
$17$	3.25917	−	2.52543i	0.790466	−	0.612506i
$18$			0.541249i			0.127574i
$19$	−4.28100			−0.982128			−0.491064	−	0.871124i	$-0.663392\pi$
$19$	−4.28100			−0.982128			−0.491064	+	0.871124i	$0.663392\pi$
$20$		0			0
$21$	4.05086			0.883969
$22$	−1.78731			−0.381056
$23$	1.47186			0.306905			0.153452	−	0.988156i	$-0.450961\pi$
$23$	1.47186			0.306905			0.153452	+	0.988156i	$0.450961\pi$
$24$			1.36321i			0.278264i
$25$		0			0
$26$	−0.806424			−0.158153
$27$	5.32089			1.02401
$28$	−6.86760			−1.29785
$29$			5.50439i			1.02214i	0.859539	+	0.511070i	$0.170751\pi$
$29$			5.50439i			1.02214i	−0.859539	+	0.511070i	$0.829249\pi$
$30$		0			0
$31$		−	8.33946i		−	1.49781i	−0.662676	−	0.748906i	$-0.730579\pi$
$31$		−	8.33946i		−	1.49781i	0.662676	−	0.748906i	$-0.269421\pi$
$32$		−	3.49532i		−	0.617890i
$33$			6.44938i			1.12269i
$34$	−0.785680	−	1.01395i	−0.134743	−	0.173892i
$35$		0			0
$36$	−3.31111			−0.551851
$37$	−6.20290			−1.01975			−0.509875	−	0.860248i	$-0.670308\pi$
$37$	−6.20290			−1.01975			−0.509875	+	0.860248i	$0.670308\pi$
$38$			1.33185i			0.216055i
$39$			2.90992i			0.465960i
$40$		0			0
$41$			3.95768i			0.618086i	0.951048	+	0.309043i	$0.100009\pi$
$41$			3.95768i			0.618086i	−0.951048	+	0.309043i	$0.899991\pi$
$42$		−	1.26025i		−	0.194461i
$43$		−	9.76049i		−	1.48846i	−0.667923	−	0.744230i	$-0.732816\pi$
$43$		−	9.76049i		−	1.48846i	0.667923	−	0.744230i	$-0.267184\pi$
$44$		−	10.9339i		−	1.64835i
$45$		0			0
$46$		−	0.457908i		−	0.0675148i
$47$			6.62222i			0.965949i	0.875634	+	0.482975i	$0.160444\pi$
$47$			6.62222i			0.965949i	−0.875634	+	0.482975i	$0.839556\pi$
$48$	−3.84902			−0.555559
$49$	6.02074			0.860106
$50$		0			0
$51$	−3.65878	+	2.83507i	−0.512332	+	0.396989i
$52$		−	4.93332i		−	0.684129i
$53$		−	0.658781i		−	0.0904905i	−0.998976	−	0.0452452i	$-0.985593\pi$
$53$		−	0.658781i		−	0.0904905i	0.998976	−	0.0452452i	$-0.0144069\pi$
$54$		−	1.65537i		−	0.225267i
$55$		0			0
$56$			4.38178i			0.585540i
$57$	4.80589			0.636555
$58$	1.71246			0.224857
$59$	−5.95407			−0.775154			−0.387577	−	0.921837i	$-0.626688\pi$
$59$	−5.95407			−0.775154			−0.387577	+	0.921837i	$0.626688\pi$
$60$		0			0
$61$			8.76357i			1.12206i	0.827796	+	0.561030i	$0.189595\pi$
$61$			8.76357i			1.12206i	−0.827796	+	0.561030i	$0.810405\pi$
$62$	−2.59447			−0.329498
$63$	6.27775			0.790922
$64$	5.76986			0.721232
$65$		0			0
$66$	2.00645			0.246977
$67$		−	0.428639i		−	0.0523666i	−0.999657	−	0.0261833i	$-0.991665\pi$
$67$		−	0.428639i		−	0.0523666i	0.999657	−	0.0261833i	$-0.00833536\pi$
$68$	6.20290	−	4.80642i	0.752212	−	0.582865i
$69$	−1.65233			−0.198917
$70$		0			0
$71$		−	1.36321i		−	0.161783i	−0.996723	−	0.0808915i	$-0.974223\pi$
$71$		−	1.36321i		−	0.161783i	0.996723	−	0.0808915i	$-0.0257767\pi$
$72$			2.11261i			0.248973i
$73$	3.25917			0.381457			0.190729	−	0.981643i	$-0.438915\pi$
$73$	3.25917			0.381457			0.190729	+	0.981643i	$0.438915\pi$
$74$			1.92977i			0.224331i
$75$		0			0
$76$	−8.14764			−0.934599
$77$			20.7304i			2.36245i
$78$	0.905299			0.102505
$79$		−	1.89597i		−	0.213313i	−0.994296	−	0.106656i	$-0.965985\pi$
$79$		−	1.89597i		−	0.213313i	0.994296	−	0.106656i	$-0.0340145\pi$
$80$		0			0
$81$	−0.754037			−0.0837819
$82$	1.23127			0.135970
$83$			16.6637i			1.82908i	0.404497	+	0.914540i	$0.367447\pi$
$83$			16.6637i			1.82908i	−0.404497	+	0.914540i	$0.632553\pi$
$84$	7.70964			0.841190
$85$		0			0
$86$	−3.03657			−0.327441
$87$		−	6.17929i		−	0.662489i
$88$	−6.97626			−0.743671
$89$	3.52543			0.373695			0.186847	−	0.982389i	$-0.440173\pi$
$89$	3.52543			0.373695			0.186847	+	0.982389i	$0.440173\pi$
$90$		0			0
$91$			9.35342i			0.980505i
$92$	2.80127			0.292052
$93$			9.36196i			0.970790i
$94$	2.06022			0.212496
$95$		0			0
$96$			3.92388i			0.400479i
$97$	11.7073			1.18870			0.594348	−	0.804208i	$-0.297410\pi$
$97$	11.7073			1.18870			0.594348	+	0.804208i	$0.297410\pi$
$98$		−	1.87310i		−	0.189212i
$99$			9.99483i			1.00452i
$100$		0			0
$101$	3.96989			0.395019			0.197509	−	0.980301i	$-0.436715\pi$
$101$	3.96989			0.395019			0.197509	+	0.980301i	$0.436715\pi$
$102$	0.882012	+	1.13828i	0.0873322	+	0.112706i
$103$			4.23506i			0.417293i	0.977991	+	0.208647i	$0.0669058\pi$
$103$			4.23506i			0.417293i	−0.977991	+	0.208647i	$0.933094\pi$
$104$	−3.14764			−0.308652
$105$		0			0
$106$	−0.204952			−0.0199067
$107$	11.8392			1.14454			0.572271	−	0.820065i	$-0.306063\pi$
$107$	11.8392			1.14454			0.572271	+	0.820065i	$0.306063\pi$
$108$	10.1268			0.974449
$109$		−	12.0227i		−	1.15157i	−0.817601	−	0.575785i	$-0.804697\pi$
$109$		−	12.0227i		−	1.15157i	0.817601	−	0.575785i	$-0.195303\pi$
$110$		0			0
$111$	6.96343			0.660940
$112$	−12.3720			−1.16904
$113$	10.7915			1.01518			0.507588	−	0.861600i	$-0.330537\pi$
$113$	10.7915			1.01518			0.507588	+	0.861600i	$0.330537\pi$
$114$		−	1.49515i		−	0.140034i
$115$		0			0
$116$			10.4760i			0.972675i
$117$			4.50961i			0.416913i
$118$			1.85236i			0.170523i
$119$	−11.7605	+	9.11282i	−1.07808	+	0.835371i
$120$		0			0
$121$	−22.0049			−2.00045
$122$	2.72641			0.246838
$123$		−	4.44293i		−	0.400605i
$124$		−	15.8718i		−	1.42533i
$125$		0			0
$126$		−	1.95306i		−	0.173992i
$127$		−	17.1383i		−	1.52078i	−0.649469	−	0.760388i	$-0.725009\pi$
$127$		−	17.1383i		−	1.52078i	0.649469	−	0.760388i	$-0.274991\pi$
$128$		−	8.78568i		−	0.776552i
$129$			10.9572i			0.964730i
$130$		0			0
$131$		−	8.89551i		−	0.777204i	−0.921406	−	0.388602i	$-0.872958\pi$
$131$		−	8.89551i		−	0.777204i	0.921406	−	0.388602i	$-0.127042\pi$
$132$			12.2745i			1.06836i
$133$	15.4477			1.33948
$134$	−0.133353			−0.0115200
$135$		0			0
$136$	−3.06668	−	3.95768i	−0.262965	−	0.339368i
$137$		−	14.0716i		−	1.20222i	−0.799167	−	0.601109i	$-0.794726\pi$
$137$		−	14.0716i		−	1.20222i	0.799167	−	0.601109i	$-0.205274\pi$
$138$			0.514052i			0.0437590i
$139$		−	16.7876i		−	1.42390i	−0.702228	−	0.711952i	$-0.747811\pi$
$139$		−	16.7876i		−	1.42390i	0.702228	−	0.711952i	$-0.252189\pi$
$140$		0			0
$141$		−	7.43416i		−	0.626070i
$142$	−0.424104			−0.0355901
$143$	−14.8916			−1.24530
$144$	−5.96497			−0.497081
$145$		0			0
$146$		−	1.01395i		−	0.0839155i
$147$	−6.75895			−0.557468
$148$	−11.8054			−0.970400
$149$	4.76494			0.390359			0.195179	−	0.980768i	$-0.437471\pi$
$149$	4.76494			0.390359			0.195179	+	0.980768i	$0.437471\pi$
$150$		0			0
$151$	2.99063			0.243374			0.121687	−	0.992569i	$-0.461170\pi$
$151$	2.99063			0.243374			0.121687	+	0.992569i	$0.461170\pi$
$152$			5.19850i			0.421654i
$153$	−5.67014	+	4.39361i	−0.458404	+	0.355202i
$154$	6.44938			0.519706
$155$		0			0
$156$			5.53820i			0.443411i
$157$			13.9447i			1.11291i	0.830878	+	0.556454i	$0.187838\pi$
$157$			13.9447i			1.11291i	−0.830878	+	0.556454i	$0.812162\pi$
$158$	−0.589850			−0.0469260
$159$			0.739554i			0.0586504i
$160$		0			0
$161$	−5.31111			−0.418574
$162$			0.234587i			0.0184309i
$163$	−8.65491			−0.677905			−0.338953	−	0.940803i	$-0.610073\pi$
$163$	−8.65491			−0.677905			−0.338953	+	0.940803i	$0.610073\pi$
$164$			7.53230i			0.588174i
$165$		0			0
$166$	5.18421			0.402373
$167$	0.773357			0.0598442			0.0299221	−	0.999552i	$-0.490474\pi$
$167$	0.773357			0.0598442			0.0299221	+	0.999552i	$0.490474\pi$
$168$		−	4.91903i		−	0.379512i
$169$	6.28100			0.483154
$170$		0			0
$171$	7.44785			0.569551
$172$		−	18.5763i		−	1.41643i
$173$	2.41097			0.183302			0.0916511	−	0.995791i	$-0.470786\pi$
$173$	2.41097			0.183302			0.0916511	+	0.995791i	$0.470786\pi$
$174$	−1.92242			−0.145739
$175$		0			0
$176$		−	19.6975i		−	1.48476i
$177$	6.68409			0.502407
$178$		−	1.09679i		−	0.0822077i
$179$	18.4558			1.37945			0.689727	−	0.724070i	$-0.257731\pi$
$179$	18.4558			1.37945			0.689727	+	0.724070i	$0.257731\pi$
$180$		0			0
$181$			20.6366i			1.53391i	0.641703	+	0.766953i	$0.278228\pi$
$181$			20.6366i			1.53391i	−0.641703	+	0.766953i	$0.721772\pi$
$182$	2.90992			0.215698
$183$		−	9.83807i		−	0.727251i
$184$		−	1.78731i		−	0.131762i
$185$		0			0
$186$	2.91258			0.213561
$187$	−14.5086	−	18.7239i	−1.06097	−	1.36923i
$188$			12.6035i			0.919203i
$189$	−19.2000			−1.39660
$190$		0			0
$191$	22.8113			1.65057			0.825286	−	0.564716i	$-0.191014\pi$
$191$	22.8113			1.65057			0.825286	+	0.564716i	$0.191014\pi$
$192$	−6.47730			−0.467459
$193$	−19.7208			−1.41953			−0.709767	−	0.704437i	$-0.751200\pi$
$193$	−19.7208			−1.41953			−0.709767	+	0.704437i	$0.751200\pi$
$194$		−	3.64223i		−	0.261497i
$195$		0			0
$196$	11.4588			0.818482
$197$	−12.8870			−0.918160			−0.459080	−	0.888395i	$-0.651821\pi$
$197$	−12.8870			−0.918160			−0.459080	+	0.888395i	$0.651821\pi$
$198$	3.10947			0.220980
$199$			6.44350i			0.456767i	0.973571	+	0.228384i	$0.0733441\pi$
$199$			6.44350i			0.456767i	−0.973571	+	0.228384i	$0.926656\pi$
$200$		0			0
$201$			0.481195i			0.0339409i
$202$		−	1.23506i		−	0.0868988i
$203$		−	19.8622i		−	1.39405i
$204$	−6.96343	+	5.39574i	−0.487538	+	0.377777i
$205$		0			0
$206$	1.31756			0.0917988
$207$	−2.56067			−0.177979
$208$		−	8.88739i		−	0.616230i
$209$			24.5943i			1.70122i
$210$		0			0
$211$		−	11.3580i		−	0.781920i	−0.920408	−	0.390960i	$-0.872143\pi$
$211$		−	11.3580i		−	0.781920i	0.920408	−	0.390960i	$-0.127857\pi$
$212$		−	1.25380i		−	0.0861113i
$213$			1.53035i			0.104858i
$214$		−	3.68328i		−	0.251784i
$215$		0			0
$216$		−	6.46126i		−	0.439633i
$217$			30.0923i			2.04280i
$218$	−3.74037			−0.253330
$219$	−3.65878			−0.247237
$220$		0			0
$221$	−6.54617	−	8.44812i	−0.440343	−	0.568282i
$222$		−	2.16638i		−	0.145398i
$223$			0.888922i			0.0595266i	0.999557	+	0.0297633i	$0.00947535\pi$
$223$			0.888922i			0.0595266i	−0.999557	+	0.0297633i	$0.990525\pi$
$224$			12.6126i			0.842714i
$225$		0			0
$226$		−	3.35731i		−	0.223325i
$227$	−12.9280			−0.858064			−0.429032	−	0.903289i	$-0.641145\pi$
$227$	−12.9280			−0.858064			−0.429032	+	0.903289i	$0.641145\pi$
$228$	9.14662			0.605750
$229$	−2.83161			−0.187118			−0.0935591	−	0.995614i	$-0.529824\pi$
$229$	−2.83161			−0.187118			−0.0935591	+	0.995614i	$0.529824\pi$
$230$		0			0
$231$		−	23.2721i		−	1.53119i
$232$	6.68409			0.438832
$233$	19.9381			1.30619			0.653094	−	0.757277i	$-0.273471\pi$
$233$	19.9381			1.30619			0.653094	+	0.757277i	$0.273471\pi$
$234$	1.40297			0.0917153
$235$		0			0
$236$	−11.3319			−0.737641
$237$			2.12843i			0.138256i
$238$	2.83507	+	3.65878i	0.183770	+	0.237164i
$239$	−27.2859			−1.76498			−0.882490	−	0.470332i	$-0.844134\pi$
$239$	−27.2859			−1.76498			−0.882490	+	0.470332i	$0.844134\pi$
$240$		0			0
$241$			3.57462i			0.230262i	0.993350	+	0.115131i	$0.0367287\pi$
$241$			3.57462i			0.230262i	−0.993350	+	0.115131i	$0.963271\pi$
$242$			6.84590i			0.440071i
$243$	−15.1162			−0.969703
$244$			16.6789i			1.06776i
$245$		0			0
$246$	−1.38223			−0.0881278
$247$			11.0968i			0.706072i
$248$	−10.1268			−0.643051
$249$		−	18.7068i		−	1.18550i
$250$		0			0
$251$	17.6128			1.11171			0.555857	−	0.831278i	$-0.312390\pi$
$251$	17.6128			1.11171			0.555857	+	0.831278i	$0.312390\pi$
$252$	11.9479			0.752646
$253$		−	8.45584i		−	0.531614i
$254$	−5.33185			−0.334550
$255$		0			0
$256$	8.80642			0.550401
$257$		−	20.5462i		−	1.28163i	−0.767693	−	0.640817i	$-0.778596\pi$
$257$		−	20.5462i		−	1.28163i	0.767693	−	0.640817i	$-0.221404\pi$
$258$	3.40888			0.212227
$259$	22.3827			1.39079
$260$		0			0
$261$		−	9.57625i		−	0.592755i
$262$	−2.76746			−0.170974
$263$			8.38715i			0.517174i	0.965988	+	0.258587i	$0.0832569\pi$
$263$			8.38715i			0.517174i	−0.965988	+	0.258587i	$0.916743\pi$
$264$	7.83161			0.482002
$265$		0			0
$266$		−	4.80589i		−	0.294668i
$267$	−3.95768			−0.242206
$268$		−	0.815792i		−	0.0498324i
$269$		−	12.4058i		−	0.756395i	−0.925725	−	0.378197i	$-0.876544\pi$
$269$		−	12.4058i		−	0.756395i	0.925725	−	0.378197i	$-0.123456\pi$
$270$		0			0
$271$	−7.13828			−0.433619			−0.216810	−	0.976214i	$-0.569565\pi$
$271$	−7.13828			−0.433619			−0.216810	+	0.976214i	$0.569565\pi$
$272$	11.1745	−	8.65878i	0.677556	−	0.525016i
$273$		−	10.5002i		−	0.635503i
$274$	−4.37778			−0.264472
$275$		0			0
$276$	−3.14473			−0.189290
$277$	−20.4193			−1.22688			−0.613438	−	0.789743i	$-0.710214\pi$
$277$	−20.4193			−1.22688			−0.613438	+	0.789743i	$0.710214\pi$
$278$	−5.22275			−0.313240
$279$			14.5086i			0.868605i
$280$		0			0
$281$	29.1432			1.73854			0.869269	−	0.494340i	$-0.164590\pi$
$281$	29.1432			1.73854			0.869269	+	0.494340i	$0.164590\pi$
$282$	−2.31283			−0.137727
$283$	−9.16991			−0.545095			−0.272547	−	0.962142i	$-0.587866\pi$
$283$	−9.16991			−0.545095			−0.272547	+	0.962142i	$0.587866\pi$
$284$		−	2.59447i		−	0.153954i
$285$		0			0
$286$			4.63290i			0.273949i
$287$		−	14.2810i		−	0.842981i
$288$			6.08097i			0.358324i
$289$	4.24443	−	16.4616i	0.249672	−	0.968330i
$290$		0			0
$291$	−13.1427			−0.770440
$292$	6.20290			0.362997
$293$		−	16.2351i		−	0.948463i	−0.880400	−	0.474231i	$-0.842726\pi$
$293$		−	16.2351i		−	0.948463i	0.880400	−	0.474231i	$-0.157274\pi$
$294$			2.10276i			0.122636i
$295$		0			0
$296$			7.53230i			0.437806i
$297$		−	30.5684i		−	1.77376i
$298$		−	1.48241i		−	0.0858737i
$299$		−	3.81522i		−	0.220640i
$300$		0			0
$301$			35.2200i			2.03005i
$302$		−	0.930409i		−	0.0535390i
$303$	−4.45664			−0.256027
$304$	−14.6780			−0.841841
$305$		0			0
$306$	1.36689	+	1.76402i	0.0781396	+	0.100843i
$307$		−	3.37778i		−	0.192780i	−0.995344	−	0.0963902i	$-0.969270\pi$
$307$		−	3.37778i		−	0.192780i	0.995344	−	0.0963902i	$-0.0307297\pi$
$308$			39.4543i			2.24812i
$309$		−	4.75432i		−	0.270464i
$310$		0			0
$311$		−	7.95641i		−	0.451166i	−0.974224	−	0.225583i	$-0.927571\pi$
$311$		−	7.95641i		−	0.451166i	0.974224	−	0.225583i	$-0.0724288\pi$
$312$	3.53358			0.200049
$313$	6.83380			0.386269			0.193135	−	0.981172i	$-0.438135\pi$
$313$	6.83380			0.386269			0.193135	+	0.981172i	$0.438135\pi$
$314$	4.33830			0.244825
$315$		0			0
$316$		−	3.60843i		−	0.202990i
$317$	27.1550			1.52517			0.762587	−	0.646886i	$-0.223929\pi$
$317$	27.1550			1.52517			0.762587	+	0.646886i	$0.223929\pi$
$318$	0.230081			0.0129023
$319$	31.6227			1.77053
$320$		0			0
$321$	−13.2908			−0.741822
$322$			1.65233i			0.0920806i
$323$	−13.9525	+	10.8113i	−0.776339	+	0.601559i
$324$	−1.43509			−0.0797274
$325$		0			0
$326$			2.69261i			0.149130i
$327$			13.4968i			0.746377i
$328$	4.80589			0.265361
$329$		−	23.8958i		−	1.31742i
$330$		0			0
$331$	−20.3783			−1.12009			−0.560045	−	0.828462i	$-0.689216\pi$
$331$	−20.3783			−1.12009			−0.560045	+	0.828462i	$0.689216\pi$
$332$			31.7146i			1.74056i
$333$	10.7915			0.591369
$334$		−	0.240597i		−	0.0131649i
$335$		0			0
$336$	13.8889			0.757703
$337$	−4.70775			−0.256447			−0.128224	−	0.991745i	$-0.540928\pi$
$337$	−4.70775			−0.256447			−0.128224	+	0.991745i	$0.540928\pi$
$338$		−	1.95407i		−	0.106287i
$339$	−12.1146			−0.657976
$340$		0			0
$341$	−47.9101			−2.59448
$342$		−	2.31708i		−	0.125294i
$343$	3.53358			0.190795
$344$	−11.8524			−0.639036
$345$		0			0
$346$		−	0.750070i		−	0.0403240i
$347$	−35.3520			−1.89779			−0.948896	−	0.315588i	$-0.897798\pi$
$347$	−35.3520			−1.89779			−0.948896	+	0.315588i	$0.897798\pi$
$348$		−	11.7605i		−	0.630428i
$349$	25.0558			1.34121			0.670603	−	0.741817i	$-0.266036\pi$
$349$	25.0558			1.34121			0.670603	+	0.741817i	$0.266036\pi$
$350$		0			0
$351$		−	13.7923i		−	0.736178i
$352$	−20.0806			−1.07030
$353$			13.6953i			0.728930i	0.931217	+	0.364465i	$0.118748\pi$
$353$			13.6953i			0.728930i	−0.931217	+	0.364465i	$0.881252\pi$
$354$		−	2.07947i		−	0.110523i
$355$		0			0
$356$	6.70964			0.355610
$357$	13.2024	−	10.2301i	0.698747	−	0.541436i
$358$		−	5.74176i		−	0.303461i
$359$	−2.53480			−0.133781			−0.0668907	−	0.997760i	$-0.521308\pi$
$359$	−2.53480			−0.133781			−0.0668907	+	0.997760i	$0.521308\pi$
$360$		0			0
$361$	−0.673071			−0.0354248
$362$	6.42021			0.337439
$363$	24.7029			1.29657
$364$			17.8015i			0.933054i
$365$		0			0
$366$	−3.06070			−0.159985
$367$	36.3426			1.89707			0.948535	−	0.316673i	$-0.102566\pi$
$367$	36.3426			1.89707			0.948535	+	0.316673i	$0.102566\pi$
$368$	5.04649			0.263066
$369$		−	6.88536i		−	0.358438i
$370$		0			0
$371$			2.37716i			0.123416i
$372$			17.8178i			0.923810i
$373$		−	12.7714i		−	0.661278i	−0.943757	−	0.330639i	$-0.892736\pi$
$373$		−	12.7714i		−	0.661278i	0.943757	−	0.330639i	$-0.107264\pi$
$374$	−5.82516	+	4.51373i	−0.301212	+	0.233399i
$375$		0			0
$376$	8.04149			0.414708
$377$	14.2680			0.734837
$378$			5.97328i			0.307232i
$379$		−	22.1495i		−	1.13774i	−0.822426	−	0.568872i	$-0.807380\pi$
$379$		−	22.1495i		−	1.13774i	0.822426	−	0.568872i	$-0.192620\pi$
$380$		0			0
$381$			19.2396i			0.985674i
$382$		−	7.09679i		−	0.363103i
$383$		−	1.95407i		−	0.0998482i	−0.998753	−	0.0499241i	$-0.984102\pi$
$383$		−	1.95407i		−	0.0998482i	0.998753	−	0.0499241i	$-0.0158979\pi$
$384$			9.86289i			0.503314i
$385$		0			0
$386$			6.13529i			0.312278i
$387$			16.9808i			0.863182i
$388$	22.2815			1.13117
$389$	7.91258			0.401184			0.200592	−	0.979675i	$-0.435713\pi$
$389$	7.91258			0.401184			0.200592	+	0.979675i	$0.435713\pi$
$390$		0			0
$391$	4.79706	−	3.71708i	0.242598	−	0.187981i
$392$		−	7.31111i		−	0.369267i
$393$			9.98619i			0.503736i
$394$			4.00924i			0.201983i
$395$		0			0
$396$			19.0223i			0.955906i
$397$	9.24476			0.463981			0.231991	−	0.972718i	$-0.425476\pi$
$397$	9.24476			0.463981			0.231991	+	0.972718i	$0.425476\pi$
$398$	2.00462			0.100483
$399$	−17.3417			−0.868171
$400$		0			0
$401$			21.4848i			1.07290i	0.843932	+	0.536450i	$0.180235\pi$
$401$			21.4848i			1.07290i	−0.843932	+	0.536450i	$0.819765\pi$
$402$	0.149703			0.00746653
$403$	−21.6168			−1.07681
$404$	7.55554			0.375902
$405$		0			0
$406$	−6.17929			−0.306673
$407$			35.6356i			1.76639i
$408$	3.44268	+	4.44293i	0.170438	+	0.219958i
$409$	−18.1669			−0.898293			−0.449147	−	0.893458i	$-0.648272\pi$
$409$	−18.1669			−0.898293			−0.449147	+	0.893458i	$0.648272\pi$
$410$		0			0
$411$			15.7969i			0.779204i
$412$			8.06022i			0.397099i
$413$	21.4848			1.05720
$414$			0.796644i			0.0391529i
$415$		0			0
$416$	−9.06022			−0.444214
$417$			18.8459i			0.922888i
$418$	7.65147			0.374246
$419$			14.3589i			0.701476i	0.936474	+	0.350738i	$0.114069\pi$
$419$			14.3589i			0.701476i	−0.936474	+	0.350738i	$0.885931\pi$
$420$		0			0
$421$	3.96989			0.193481			0.0967403	−	0.995310i	$-0.469158\pi$
$421$	3.96989			0.193481			0.0967403	+	0.995310i	$0.469158\pi$
$422$	−3.53358			−0.172012
$423$		−	11.5210i		−	0.560169i
$424$	−0.799970			−0.0388500
$425$		0			0
$426$	0.476104			0.0230673
$427$		−	31.6227i		−	1.53033i
$428$	22.5326			1.08915
$429$	16.7175			0.807127
$430$		0			0
$431$			32.9337i			1.58636i	0.608985	+	0.793181i	$0.291577\pi$
$431$			32.9337i			1.58636i	−0.608985	+	0.793181i	$0.708423\pi$
$432$	18.2434			0.877736
$433$			38.2306i			1.83725i	0.395135	+	0.918623i	$0.370698\pi$
$433$			38.2306i			1.83725i	−0.395135	+	0.918623i	$0.629302\pi$
$434$	9.36196			0.449389
$435$		0			0
$436$		−	22.8818i		−	1.09584i
$437$	−6.30104			−0.301420
$438$			1.13828i			0.0543889i
$439$		−	0.664702i		−	0.0317245i	−0.999874	−	0.0158622i	$-0.994951\pi$
$439$		−	0.664702i		−	0.0317245i	0.999874	−	0.0158622i	$-0.00504932\pi$
$440$		0			0
$441$	−10.4746			−0.498789
$442$	−2.62828	+	2.03657i	−0.125014	+	0.0968695i
$443$		−	27.5669i		−	1.30974i	−0.755740	−	0.654872i	$-0.772723\pi$
$443$		−	27.5669i		−	1.30974i	0.755740	−	0.654872i	$-0.227277\pi$
$444$	13.2529			0.628954
$445$		0			0
$446$	0.276551			0.0130950
$447$	−5.34916			−0.253007
$448$	−20.8201			−0.983658
$449$		−	17.4756i		−	0.824723i	−0.911020	−	0.412362i	$-0.864704\pi$
$449$		−	17.4756i		−	0.824723i	0.911020	−	0.412362i	$-0.135296\pi$
$450$		0			0
$451$	22.7368			1.07064
$452$	20.5385			0.966048
$453$	−3.35731			−0.157740
$454$			4.02201i			0.188762i
$455$		0			0
$456$		−	5.83589i		−	0.273290i
$457$			13.0350i			0.609753i	0.952392	+	0.304877i	$0.0986152\pi$
$457$			13.0350i			0.609753i	−0.952392	+	0.304877i	$0.901385\pi$
$458$			0.880937i			0.0411635i
$459$	17.3417	−	13.4375i	0.809441	−	0.627209i
$460$		0			0
$461$	−10.9491			−0.509953			−0.254976	−	0.966947i	$-0.582068\pi$
$461$	−10.9491			−0.509953			−0.254976	+	0.966947i	$0.582068\pi$
$462$	−7.24014			−0.336842
$463$			17.2400i			0.801210i	0.916251	+	0.400605i	$0.131200\pi$
$463$			17.2400i			0.801210i	−0.916251	+	0.400605i	$0.868800\pi$
$464$			18.8726i			0.876138i
$465$		0			0
$466$		−	6.20290i		−	0.287344i
$467$			0.0414872i			0.00191980i	1.00000		0.000959899i	$0.000305545\pi$
$467$			0.0414872i			0.00191980i	−1.00000		0.000959899i	$0.999694\pi$
$468$			8.58274i			0.396737i
$469$			1.54671i			0.0714206i
$470$		0			0
$471$		−	15.6545i		−	0.721319i
$472$			7.23014i			0.332794i
$473$	−56.0739			−2.57828
$474$	0.662171			0.0304145
$475$		0			0
$476$	−22.3827	+	17.3436i	−1.02591	+	0.794944i
$477$			1.14611i			0.0524769i
$478$			8.48886i			0.388272i
$479$			35.4606i			1.62024i	0.586266	+	0.810118i	$0.300597\pi$
$479$			35.4606i			1.62024i	−0.586266	+	0.810118i	$0.699403\pi$
$480$		0			0
$481$			16.0786i			0.733119i
$482$	1.11209			0.0506545
$483$	5.96230			0.271294
$484$	−41.8800			−1.90364
$485$		0			0
$486$			4.70276i			0.213321i
$487$	−24.6691			−1.11787			−0.558933	−	0.829213i	$-0.688789\pi$
$487$	−24.6691			−1.11787			−0.558933	+	0.829213i	$0.688789\pi$
$488$	10.6418			0.481730
$489$	9.71609			0.439377
$490$		0			0
$491$	8.58073			0.387243			0.193621	−	0.981076i	$-0.437977\pi$
$491$	8.58073			0.387243			0.193621	+	0.981076i	$0.437977\pi$
$492$		−	8.45584i		−	0.381219i
$493$	13.9009	+	17.9398i	0.626067	+	0.807967i
$494$	3.45230			0.155326
$495$		0			0
$496$		−	28.5930i		−	1.28386i
$497$			4.91903i			0.220649i
$498$	−5.81984			−0.260793
$499$		−	8.48917i		−	0.380027i	−0.981781	−	0.190014i	$-0.939147\pi$
$499$		−	8.48917i		−	0.380027i	0.981781	−	0.190014i	$-0.0608532\pi$
$500$		0			0
$501$	−0.868178			−0.0387873
$502$		−	5.47949i		−	0.244562i
$503$	24.1879			1.07849			0.539244	−	0.842150i	$-0.318710\pi$
$503$	24.1879			1.07849			0.539244	+	0.842150i	$0.318710\pi$
$504$		−	7.62320i		−	0.339564i
$505$		0			0
$506$	−2.63068			−0.116948
$507$	−7.05111			−0.313151
$508$		−	32.6178i		−	1.44718i
$509$	25.8671			1.14654			0.573270	−	0.819366i	$-0.305675\pi$
$509$	25.8671			1.14654			0.573270	+	0.819366i	$0.305675\pi$
$510$		0			0
$511$	−11.7605			−0.520253
$512$		−	20.3111i		−	0.897633i
$513$	−22.7787			−1.00570
$514$	−6.39207			−0.281942
$515$		0			0
$516$			20.8539i			0.918042i
$517$	38.0446			1.67320
$518$		−	6.96343i		−	0.305956i
$519$	−2.70657			−0.118805
$520$		0			0
$521$			11.4900i			0.503385i	0.967807	+	0.251693i	$0.0809872\pi$
$521$			11.4900i			0.503385i	−0.967807	+	0.251693i	$0.919013\pi$
$522$	−2.97925			−0.130398
$523$		−	33.0509i		−	1.44521i	−0.691260	−	0.722606i	$-0.742944\pi$
$523$		−	33.0509i		−	1.44521i	0.691260	−	0.722606i	$-0.257056\pi$
$524$		−	16.9300i		−	0.739592i
$525$		0			0
$526$	2.60931			0.113771
$527$	−21.0607	−	27.1798i	−0.917419	−	1.18397i
$528$			22.1126i			0.962328i
$529$	−20.8336			−0.905810
$530$		0			0
$531$	10.3586			0.449524
$532$	29.4002			1.27466
$533$	10.2587			0.444354
$534$			1.23127i			0.0532820i
$535$		0			0
$536$	−0.520505			−0.0224824
$537$	−20.7187			−0.894078
$538$	−3.85954			−0.166397
$539$		−	34.5891i		−	1.48986i
$540$		0			0
$541$		−	19.7884i		−	0.850770i	−0.905013	−	0.425385i	$-0.860139\pi$
$541$		−	19.7884i		−	0.850770i	0.905013	−	0.425385i	$-0.139861\pi$
$542$			2.22077i			0.0953904i
$543$		−	23.1669i		−	0.994185i
$544$	−8.82717	−	11.3918i	−0.378462	−	0.488421i
$545$		0			0
$546$	−3.26671			−0.139802
$547$	−15.4815			−0.661940			−0.330970	−	0.943641i	$-0.607376\pi$
$547$	−15.4815			−0.661940			−0.330970	+	0.943641i	$0.607376\pi$
$548$		−	26.7812i		−	1.14404i
$549$		−	15.2464i		−	0.650700i
$550$		0			0
$551$		−	23.5643i		−	1.00387i
$552$			2.00645i			0.0854003i
$553$			6.84146i			0.290928i
$554$			6.35260i			0.269896i
$555$		0			0
$556$		−	31.9503i		−	1.35500i
$557$		−	15.3477i		−	0.650302i	−0.945662	−	0.325151i	$-0.894585\pi$
$557$		−	15.3477i		−	0.650302i	0.945662	−	0.325151i	$-0.105415\pi$
$558$	4.51373			0.191081
$559$	−25.3002			−1.07008
$560$		0			0
$561$	16.2874	+	21.0197i	0.687657	+	0.887451i
$562$		−	9.06668i		−	0.382455i
$563$		−	14.3827i		−	0.606159i	−0.952965	−	0.303079i	$-0.901985\pi$
$563$		−	14.3827i		−	0.606159i	0.952965	−	0.303079i	$-0.0980147\pi$
$564$		−	14.1488i		−	0.595772i
$565$		0			0
$566$			2.85283i			0.119913i
$567$	2.72089			0.114267
$568$	−1.65537			−0.0694578
$569$	12.1432			0.509069			0.254535	−	0.967064i	$-0.418078\pi$
$569$	12.1432			0.509069			0.254535	+	0.967064i	$0.418078\pi$
$570$		0			0
$571$			2.88663i			0.120802i	0.998174	+	0.0604009i	$0.0192379\pi$
$571$			2.88663i			0.120802i	−0.998174	+	0.0604009i	$0.980762\pi$
$572$	−28.3419			−1.18503
$573$	−25.6082			−1.06980
$574$	−4.44293			−0.185444
$575$		0			0
$576$	−10.0381			−0.418254
$577$		−	3.10970i		−	0.129458i	−0.997903	−	0.0647291i	$-0.979382\pi$
$577$		−	3.10970i		−	0.129458i	0.997903	−	0.0647291i	$-0.0206183\pi$
$578$	−5.12134	−	1.32048i	−0.213020	−	0.0549246i
$579$	22.1388			0.920055
$580$		0			0
$581$		−	60.1298i		−	2.49460i
$582$			4.08880i			0.169486i
$583$	−3.78469			−0.156746
$584$		−	3.95768i		−	0.163770i
$585$		0			0
$586$	−5.05086			−0.208649
$587$			10.2034i			0.421140i	0.977579	+	0.210570i	$0.0675320\pi$
$587$			10.2034i			0.421140i	−0.977579	+	0.210570i	$0.932468\pi$
$588$	−12.8637			−0.530490
$589$			35.7012i			1.47104i
$590$		0			0
$591$	14.4671			0.595095
$592$	−21.2675			−0.874089
$593$			16.2301i			0.666492i	0.942840	+	0.333246i	$0.108144\pi$
$593$			16.2301i			0.666492i	−0.942840	+	0.333246i	$0.891856\pi$
$594$	−9.51008			−0.390203
$595$		0			0
$596$	9.06868			0.371468
$597$		−	7.23353i		−	0.296049i
$598$	−1.18694			−0.0485378
$599$	8.43309			0.344567			0.172283	−	0.985047i	$-0.444886\pi$
$599$	8.43309			0.344567			0.172283	+	0.985047i	$0.444886\pi$
$600$		0			0
$601$		−	20.2535i		−	0.826160i	−0.910695	−	0.413080i	$-0.864453\pi$
$601$		−	20.2535i		−	0.826160i	0.910695	−	0.413080i	$-0.135547\pi$
$602$	10.9572			0.446583
$603$			0.745724i			0.0303682i
$604$	5.69181			0.231596
$605$		0			0
$606$			1.38649i			0.0563225i
$607$	22.7732			0.924334			0.462167	−	0.886793i	$-0.347072\pi$
$607$	22.7732			0.924334			0.462167	+	0.886793i	$0.347072\pi$
$608$			14.9634i			0.606847i
$609$			22.2975i			0.903540i
$610$		0			0
$611$	17.1655			0.694441
$612$	−10.7915	+	8.36196i	−0.436220	+	0.338012i
$613$			8.40636i			0.339530i	0.985485	+	0.169765i	$0.0543008\pi$
$613$			8.40636i			0.339530i	−0.985485	+	0.169765i	$0.945699\pi$
$614$	−1.05086			−0.0424091
$615$		0			0
$616$	25.1733			1.01426
$617$	14.8168			0.596500			0.298250	−	0.954488i	$-0.403597\pi$
$617$	14.8168			0.596500			0.298250	+	0.954488i	$0.403597\pi$
$618$	−1.47911			−0.0594984
$619$			31.6221i			1.27100i	0.772101	+	0.635500i	$0.219206\pi$
$619$			31.6221i			1.27100i	−0.772101	+	0.635500i	$0.780794\pi$
$620$		0			0
$621$	7.83161			0.314272
$622$	−2.47530			−0.0992505
$623$	−12.7212			−0.509666
$624$			9.97707i			0.399403i
$625$		0			0
$626$		−	2.12605i		−	0.0849740i
$627$		−	27.6098i		−	1.10263i
$628$			26.5397i			1.05905i
$629$	−20.2163	+	15.6650i	−0.806078	+	0.624603i
$630$		0			0
$631$	−17.3733			−0.691622			−0.345811	−	0.938304i	$-0.612396\pi$
$631$	−17.3733			−0.691622			−0.345811	+	0.938304i	$0.612396\pi$
$632$	−2.30231			−0.0915810
$633$			12.7506i			0.506793i
$634$		−	8.44812i		−	0.335518i
$635$		0			0
$636$			1.40753i			0.0558121i
$637$		−	15.6064i		−	0.618348i
$638$		−	9.83807i		−	0.389493i
$639$			2.37164i			0.0938205i
$640$		0			0
$641$			7.26842i			0.287085i	0.989644	+	0.143543i	$0.0458494\pi$
$641$			7.26842i			0.287085i	−0.989644	+	0.143543i	$0.954151\pi$
$642$			4.13488i			0.163191i
$643$	−4.40507			−0.173719			−0.0868595	−	0.996221i	$-0.527683\pi$
$643$	−4.40507			−0.173719			−0.0868595	+	0.996221i	$0.527683\pi$
$644$	−10.1082			−0.398317
$645$		0			0
$646$	3.36349	+	4.34074i	0.132335	+	0.170784i
$647$			0.198022i			0.00778504i	0.999992	+	0.00389252i	$0.00123903\pi$
$647$			0.198022i			0.00778504i	−0.999992	+	0.00389252i	$0.998761\pi$
$648$			0.915643i			0.0359698i
$649$			34.2061i			1.34271i
$650$		0			0
$651$		−	33.7820i		−	1.32402i
$652$	−16.4721			−0.645098
$653$	−9.92723			−0.388482			−0.194241	−	0.980954i	$-0.562224\pi$
$653$	−9.92723			−0.388482			−0.194241	+	0.980954i	$0.562224\pi$
$654$	4.19897			0.164193
$655$		0			0
$656$			13.5695i			0.529798i
$657$	−5.67014			−0.221213
$658$	−7.43416			−0.289814
$659$	−3.00492			−0.117055			−0.0585276	−	0.998286i	$-0.518641\pi$
$659$	−3.00492			−0.117055			−0.0585276	+	0.998286i	$0.518641\pi$
$660$		0			0
$661$	23.6923			0.921523			0.460762	−	0.887524i	$-0.347576\pi$
$661$	23.6923			0.921523			0.460762	+	0.887524i	$0.347576\pi$
$662$			6.33984i			0.246405i
$663$	7.34880	+	9.48394i	0.285404	+	0.368326i
$664$	20.2351			0.785273
$665$		0			0
$666$		−	3.35731i		−	0.130093i
$667$			8.10171i			0.313699i
$668$	1.47186			0.0569481
$669$		−	0.997912i		−	0.0385815i
$670$		0			0
$671$	50.3466			1.94361
$672$		−	14.1590i		−	0.546196i
$673$	30.1968			1.16400			0.582001	−	0.813188i	$-0.302270\pi$
$673$	30.1968			1.16400			0.582001	+	0.813188i	$0.302270\pi$
$674$			1.46462i			0.0564150i
$675$		0			0
$676$	11.9541			0.459772
$677$	−47.9413			−1.84253			−0.921266	−	0.388933i	$-0.872844\pi$
$677$	−47.9413			−1.84253			−0.921266	+	0.388933i	$0.872844\pi$
$678$			3.76895i			0.144746i
$679$	−42.2449			−1.62121
$680$		0			0
$681$	14.5131			0.556145
$682$			14.9052i			0.570750i
$683$	−1.41477			−0.0541347			−0.0270674	−	0.999634i	$-0.508617\pi$
$683$	−1.41477			−0.0541347			−0.0270674	+	0.999634i	$0.508617\pi$
$684$	14.1748			0.541989
$685$		0			0
$686$		−	1.09932i		−	0.0419723i
$687$	3.17880			0.121279
$688$		−	33.4652i		−	1.27585i
$689$	−1.70763			−0.0650554
$690$		0			0
$691$			1.36321i			0.0518588i	0.999664	+	0.0259294i	$0.00825452\pi$
$691$			1.36321i			0.0518588i	−0.999664	+	0.0259294i	$0.991745\pi$
$692$	4.58858			0.174432
$693$		−	36.0656i		−	1.37002i
$694$			10.9983i			0.417489i
$695$		0			0
$696$	−7.50363			−0.284424
$697$	9.99483	+	12.8988i	0.378581	+	0.488576i
$698$		−	7.79505i		−	0.295047i
$699$	−22.3827			−0.846592
$700$		0			0
$701$	−40.1990			−1.51829			−0.759147	−	0.650919i	$-0.774384\pi$
$701$	−40.1990			−1.51829			−0.759147	+	0.650919i	$0.774384\pi$
$702$	−4.29089			−0.161949
$703$	26.5546			1.00153
$704$		−	33.1478i		−	1.24930i
$705$		0			0
$706$	4.26073			0.160355
$707$	−14.3251			−0.538749
$708$	12.7212			0.478094
$709$		−	19.4729i		−	0.731322i	−0.930748	−	0.365661i	$-0.880843\pi$
$709$		−	19.4729i		−	0.731322i	0.930748	−	0.365661i	$-0.119157\pi$
$710$		0			0
$711$			3.29850i			0.123704i
$712$		−	4.28100i		−	0.160437i
$713$		−	12.2745i		−	0.459685i
$714$	−3.18268	−	4.10738i	−0.119109	−	0.153715i
$715$		0			0
$716$	35.1254			1.31270
$717$	30.6314			1.14395
$718$			0.788595i			0.0294301i
$719$		−	21.9765i		−	0.819586i	−0.912179	−	0.409793i	$-0.865601\pi$
$719$		−	21.9765i		−	0.819586i	0.912179	−	0.409793i	$-0.134399\pi$
$720$		0			0
$721$		−	15.2819i		−	0.569128i
$722$			0.209398i			0.00779297i
$723$		−	4.01291i		−	0.149242i
$724$			39.2758i			1.45967i
$725$		0			0
$726$		−	7.68528i		−	0.285227i
$727$			36.7007i			1.36116i	0.732676	+	0.680578i	$0.238271\pi$
$727$			36.7007i			1.36116i	−0.732676	+	0.680578i	$0.761729\pi$
$728$	11.3580			0.420957
$729$	19.2317			0.712284
$730$		0			0
$731$	−24.6494	−	31.8111i	−0.911691	−	1.17658i
$732$		−	18.7239i		−	0.692056i
$733$		−	24.8666i		−	0.918471i	−0.888315	−	0.459235i	$-0.848123\pi$
$733$		−	24.8666i		−	0.918471i	0.888315	−	0.459235i	$-0.151877\pi$
$734$		−	11.3065i		−	0.417330i
$735$		0			0
$736$		−	5.14462i		−	0.189633i
$737$	−2.46253			−0.0907085
$738$	−2.14209			−0.0788514
$739$	32.7368			1.20424			0.602122	−	0.798404i	$-0.294322\pi$
$739$	32.7368			1.20424			0.602122	+	0.798404i	$0.294322\pi$
$740$		0			0
$741$		−	12.4574i		−	0.457633i
$742$	0.739554			0.0271499
$743$	−21.3762			−0.784215			−0.392108	−	0.919919i	$-0.628254\pi$
$743$	−21.3762			−0.784215			−0.392108	+	0.919919i	$0.628254\pi$
$744$	11.3684			0.416786
$745$		0			0
$746$	−3.97328			−0.145472
$747$		−	28.9906i		−	1.06071i
$748$	−27.6129	−	35.6356i	−1.00963	−	1.30297i
$749$	−42.7210			−1.56099
$750$		0			0
$751$			44.5557i			1.62586i	0.582362	+	0.812930i	$0.302129\pi$
$751$			44.5557i			1.62586i	−0.582362	+	0.812930i	$0.697871\pi$
$752$			22.7052i			0.827973i
$753$	−19.7724			−0.720545
$754$		−	4.43887i		−	0.161654i
$755$		0			0
$756$	−36.5417			−1.32901
$757$			12.1793i			0.442664i	0.975199	+	0.221332i	$0.0710404\pi$
$757$			12.1793i			0.442664i	−0.975199	+	0.221332i	$0.928960\pi$
$758$	−6.89089			−0.250288
$759$			9.49260i			0.344560i
$760$		0			0
$761$	7.18712			0.260533			0.130266	−	0.991479i	$-0.458417\pi$
$761$	7.18712			0.260533			0.130266	+	0.991479i	$0.458417\pi$
$762$	5.98559			0.216835
$763$			43.3832i			1.57058i
$764$	43.4148			1.57069
$765$		0			0
$766$	−0.607926			−0.0219652
$767$			15.4336i			0.557274i
$768$	−9.88618			−0.356737
$769$	−14.9748			−0.540005			−0.270003	−	0.962860i	$-0.587025\pi$
$769$	−14.9748			−0.540005			−0.270003	+	0.962860i	$0.587025\pi$
$770$		0			0
$771$			23.0653i			0.830678i
$772$	−37.5328			−1.35084
$773$			26.2938i			0.945721i	0.881137	+	0.472860i	$0.156778\pi$
$773$			26.2938i			0.945721i	−0.881137	+	0.472860i	$0.843222\pi$
$774$	5.28286			0.189888
$775$		0			0
$776$		−	14.2164i		−	0.510339i
$777$	−25.1270			−0.901428
$778$		−	2.46167i		−	0.0882550i
$779$		−	16.9428i		−	0.607039i
$780$		0			0
$781$	−7.83161			−0.280237
$782$	−1.15641	−	1.49240i	−0.0413533	−	0.0533682i
$783$			29.2883i			1.04668i
$784$	20.6430			0.737249
$785$		0			0
$786$	3.10678			0.110815
$787$	−9.35342			−0.333413			−0.166707	−	0.986007i	$-0.553313\pi$
$787$	−9.35342			−0.333413			−0.166707	+	0.986007i	$0.553313\pi$
$788$	−24.5267			−0.873727
$789$		−	9.41550i		−	0.335201i
$790$		0			0
$791$	−38.9403			−1.38456
$792$	12.1369			0.431267
$793$	22.7161			0.806672
$794$		−	2.87612i		−	0.102070i
$795$		0			0
$796$			12.2633i			0.434663i
$797$		−	29.4800i		−	1.04423i	−0.852874	−	0.522117i	$-0.825142\pi$
$797$		−	29.4800i		−	1.04423i	0.852874	−	0.522117i	$-0.174858\pi$
$798$			5.39514i			0.190986i
$799$	16.7239	+	21.5830i	0.591650	+	0.763550i
$800$		0			0
$801$	−6.13335			−0.216711
$802$	6.68409			0.236024
$803$		−	18.7239i		−	0.660753i
$804$			0.915816i			0.0322983i
$805$		0			0
$806$			6.72514i			0.236883i
$807$			13.9269i			0.490249i
$808$		−	4.82071i		−	0.169592i
$809$		−	35.1219i		−	1.23482i	−0.786642	−	0.617410i	$-0.788182\pi$
$809$		−	35.1219i		−	1.23482i	0.786642	−	0.617410i	$-0.211818\pi$
$810$		0			0
$811$			17.5682i			0.616902i	0.951240	+	0.308451i	$0.0998106\pi$
$811$			17.5682i			0.616902i	−0.951240	+	0.308451i	$0.900189\pi$
$812$		−	37.8020i		−	1.32659i
$813$	8.01350			0.281046
$814$	11.0865			0.388582
$815$		0			0
$816$	−12.5446	+	9.72043i	−0.439150	+	0.340283i
$817$			41.7846i			1.46186i
$818$			5.65185i			0.197612i
$819$		−	16.2726i		−	0.568610i
$820$		0			0
$821$			13.1043i			0.457343i	0.973504	+	0.228672i	$0.0734382\pi$
$821$			13.1043i			0.457343i	−0.973504	+	0.228672i	$0.926562\pi$
$822$	4.91454			0.171414
$823$	23.7638			0.828355			0.414178	−	0.910196i	$-0.364069\pi$
$823$	23.7638			0.828355			0.414178	+	0.910196i	$0.364069\pi$
$824$	5.14272			0.179155
$825$		0			0
$826$		−	6.68409i		−	0.232569i
$827$	28.9423			1.00642			0.503211	−	0.864164i	$-0.332152\pi$
$827$	28.9423			1.00642			0.503211	+	0.864164i	$0.332152\pi$
$828$	−4.87350			−0.169366
$829$	−48.3131			−1.67798			−0.838992	−	0.544144i	$-0.816855\pi$
$829$	−48.3131			−1.67798			−0.838992	+	0.544144i	$0.816855\pi$
$830$		0			0
$831$	22.9229			0.795187
$832$		−	14.9561i		−	0.518509i
$833$	19.6227	−	15.2050i	0.679885	−	0.526820i
$834$	5.86311			0.203023
$835$		0			0
$836$			46.8081i			1.61889i
$837$		−	44.3733i		−	1.53377i
$838$	4.46715			0.154315
$839$			36.8238i			1.27130i	0.771978	+	0.635650i	$0.219268\pi$
$839$			36.8238i			1.27130i	−0.771978	+	0.635650i	$0.780732\pi$
$840$		0			0
$841$	−1.29835			−0.0447707
$842$		−	1.23506i		−	0.0425631i
$843$	−32.7164			−1.12681
$844$		−	21.6168i		−	0.744079i
$845$		0			0
$846$	−3.58427			−0.123230
$847$	79.4031			2.72832
$848$		−	2.25872i		−	0.0775648i
$849$	10.2942			0.353297
$850$		0			0
$851$	−9.12981			−0.312966
$852$			2.91258i			0.0997833i
$853$	29.8138			1.02080			0.510402	−	0.859936i	$-0.329497\pi$
$853$	29.8138			1.02080			0.510402	+	0.859936i	$0.329497\pi$
$854$	−9.83807			−0.336652
$855$		0			0
$856$		−	14.3766i		−	0.491383i
$857$	1.71246			0.0584965			0.0292483	−	0.999572i	$-0.490689\pi$
$857$	1.71246			0.0584965			0.0292483	+	0.999572i	$0.490689\pi$
$858$		−	5.20094i		−	0.177557i
$859$	−22.5749			−0.770246			−0.385123	−	0.922865i	$-0.625841\pi$
$859$	−22.5749			−0.770246			−0.385123	+	0.922865i	$0.625841\pi$
$860$		0			0
$861$			16.0320i			0.546369i
$862$	10.2459			0.348978
$863$			54.2578i			1.84696i	0.383650	+	0.923479i	$0.374667\pi$
$863$			54.2578i			1.84696i	−0.383650	+	0.923479i	$0.625333\pi$
$864$		−	18.5982i		−	0.632723i
$865$		0			0
$866$	11.8938			0.404169
$867$	−4.76484	+	18.4800i	−0.161822	+	0.627613i
$868$			57.2721i			1.94394i
$869$	−10.8923			−0.369496
$870$		0			0
$871$	−1.11108			−0.0376474
$872$	−14.5995			−0.494400
$873$	−20.3677			−0.689343
$874$			1.96030i			0.0663082i
$875$		0			0
$876$	−6.96343			−0.235273
$877$	14.8989			0.503099			0.251549	−	0.967844i	$-0.419060\pi$
$877$	14.8989			0.503099			0.251549	+	0.967844i	$0.419060\pi$
$878$	−0.206794			−0.00697896
$879$			18.2256i			0.614736i
$880$		0			0
$881$			22.3330i			0.752419i	0.926535	+	0.376209i	$0.122773\pi$
$881$			22.3330i			0.752419i	−0.926535	+	0.376209i	$0.877227\pi$
$882$			3.25872i			0.109727i
$883$			25.3319i			0.852485i	0.904609	+	0.426242i	$0.140163\pi$
$883$			25.3319i			0.852485i	−0.904609	+	0.426242i	$0.859837\pi$
$884$	−12.4588	−	16.0786i	−0.419033	−	0.540780i
$885$		0			0
$886$	−8.57628			−0.288126
$887$	6.35813			0.213485			0.106743	−	0.994287i	$-0.465958\pi$
$887$	6.35813			0.213485			0.106743	+	0.994287i	$0.465958\pi$
$888$		−	8.45584i		−	0.283759i
$889$			61.8422i			2.07412i
$890$		0			0
$891$			4.33194i			0.145125i
$892$			1.69181i			0.0566459i
$893$		−	28.3497i		−	0.948686i
$894$			1.66417i			0.0556581i
$895$		0			0
$896$			31.7025i			1.05911i
$897$			4.28300i			0.143005i
$898$	−5.43679			−0.181428
$899$	45.9037			1.53097
$900$		0			0
$901$	−1.66370	−	2.14708i	−0.0554260	−	0.0715296i
$902$		−	7.07361i		−	0.235525i
$903$		−	39.5383i		−	1.31575i
$904$		−	13.1043i		−	0.435843i
$905$		0			0
$906$			1.04449i			0.0347007i
$907$	13.0295			0.432636			0.216318	−	0.976323i	$-0.430595\pi$
$907$	13.0295			0.432636			0.216318	+	0.976323i	$0.430595\pi$
$908$	−24.6048			−0.816539
$909$	−6.90660			−0.229078
$910$		0			0
$911$		−	20.7685i		−	0.688093i	−0.938953	−	0.344046i	$-0.888202\pi$
$911$		−	20.7685i		−	0.688093i	0.938953	−	0.344046i	$-0.111798\pi$
$912$	16.4777			0.545630
$913$	95.7328			3.16829
$914$	4.05530			0.134137
$915$		0			0
$916$	−5.38916			−0.178063
$917$			32.0988i			1.06000i
$918$	−4.18052	−	5.39514i	−0.137978	−	0.178066i
$919$	5.10970			0.168553			0.0842766	−	0.996442i	$-0.473142\pi$
$919$	5.10970			0.168553			0.0842766	+	0.996442i	$0.473142\pi$
$920$		0			0
$921$			3.79193i			0.124948i
$922$			3.40636i			0.112183i
$923$	−3.53358			−0.116309
$924$		−	44.2918i		−	1.45709i
$925$		0			0
$926$	5.36349			0.176255
$927$		−	7.36794i		−	0.241995i
$928$	19.2396			0.631571
$929$		−	39.6073i		−	1.29947i	−0.760159	−	0.649737i	$-0.774879\pi$
$929$		−	39.6073i		−	1.29947i	0.760159	−	0.649737i	$-0.225121\pi$
$930$		0			0
$931$	−25.7748			−0.844734
$932$	37.9464			1.24298
$933$			8.93194i			0.292419i
$934$	0.0129070			0.000422330
$935$		0			0
$936$	5.47610			0.178992
$937$			39.5575i			1.29229i	0.763215	+	0.646144i	$0.223620\pi$
$937$			39.5575i			1.29229i	−0.763215	+	0.646144i	$0.776380\pi$
$938$	0.481195			0.0157116
$939$	−7.67169			−0.250356
$940$		0			0
$941$		−	53.2283i		−	1.73519i	−0.497268	−	0.867597i	$-0.665663\pi$
$941$		−	53.2283i		−	1.73519i	0.497268	−	0.867597i	$-0.334337\pi$
$942$	−4.87022			−0.158680
$943$			5.82516i			0.189693i
$944$	−20.4143			−0.664430
$945$		0			0
$946$			17.4450i			0.567187i
$947$	0.773357			0.0251307			0.0125654	−	0.999921i	$-0.496000\pi$
$947$	0.773357			0.0251307			0.0125654	+	0.999921i	$0.496000\pi$
$948$			4.05086i			0.131566i
$949$		−	8.44812i		−	0.274238i
$950$		0			0
$951$	−30.4844			−0.988525
$952$	11.0659	+	14.2810i	0.358647	+	0.462850i
$953$		−	36.8731i		−	1.19444i	−0.802079	−	0.597218i	$-0.796273\pi$
$953$		−	36.8731i		−	1.19444i	0.802079	−	0.597218i	$-0.203727\pi$
$954$	0.356564			0.0115442
$955$		0			0
$956$	−51.9309			−1.67956
$957$	−35.4999			−1.14755
$958$	11.0321			0.356430
$959$			50.7763i			1.63965i
$960$		0			0
$961$	−38.5466			−1.24344
$962$	5.00217			0.161276
$963$	−20.5973			−0.663738
$964$			6.80327i			0.219118i
$965$		0			0
$966$		−	1.85492i		−	0.0596810i
$967$			17.8306i			0.573392i	0.958022	+	0.286696i	$0.0925570\pi$
$967$			17.8306i			0.573392i	−0.958022	+	0.286696i	$0.907443\pi$
$968$			26.7210i			0.858846i
$969$	15.6632	−	12.1369i	0.503175	−	0.389894i
$970$		0			0
$971$	−20.3783			−0.653970			−0.326985	−	0.945030i	$-0.606033\pi$
$971$	−20.3783			−0.653970			−0.326985	+	0.945030i	$0.606033\pi$
$972$	−28.7693			−0.922775
$973$			60.5768i			1.94200i
$974$			7.67476i			0.245915i
$975$		0			0
$976$			30.0471i			0.961785i
$977$		−	8.64587i		−	0.276606i	−0.990390	−	0.138303i	$-0.955835\pi$
$977$		−	8.64587i		−	0.276606i	0.990390	−	0.138303i	$-0.0441648\pi$
$978$		−	3.02275i		−	0.0966569i
$979$		−	20.2535i		−	0.647306i
$980$		0			0
$981$			20.9165i			0.667813i
$982$		−	2.66953i		−	0.0851882i
$983$	61.1615			1.95075			0.975374	−	0.220558i	$-0.0707877\pi$
$983$	61.1615			1.95075			0.975374	+	0.220558i	$0.0707877\pi$
$984$	−5.39514			−0.171991
$985$		0			0
$986$	5.58120	−	4.32469i	0.177742	−	0.137726i
$987$			26.8256i			0.853869i
$988$			21.1195i			0.671902i
$989$		−	14.3661i		−	0.456815i
$990$		0			0
$991$			27.0979i			0.860792i	0.902640	+	0.430396i	$0.141626\pi$
$991$			27.0979i			0.860792i	−0.902640	+	0.430396i	$0.858374\pi$
$992$	−29.1491			−0.925484
$993$	22.8768			0.725974
$994$	1.53035			0.0485397
$995$		0			0
$996$		−	35.6031i		−	1.12813i
$997$	48.3759			1.53208			0.766040	−	0.642793i	$-0.222225\pi$
$997$	48.3759			1.53208			0.766040	+	0.642793i	$0.222225\pi$
$998$	−2.64105			−0.0836009
$999$	−33.0049			−1.04423

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	425.2.c.c.424.5		12
5.2	odd	4		425.2.d.b.101.4	yes	6
5.3	odd	4		425.2.d.a.101.3	✓	6
5.4	even	2	inner	425.2.c.c.424.8		12
17.16	even	2	inner	425.2.c.c.424.6		12
85.13	odd	4		7225.2.a.bg.1.4		6
85.33	odd	4		425.2.d.a.101.4	yes	6
85.38	odd	4		7225.2.a.bg.1.3		6
85.47	odd	4		7225.2.a.ba.1.3		6
85.67	odd	4		425.2.d.b.101.3	yes	6
85.72	odd	4		7225.2.a.ba.1.4		6
85.84	even	2	inner	425.2.c.c.424.7		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
425.2.c.c.424.5		12	1.1	even	1	trivial
425.2.c.c.424.6		12	17.16	even	2	inner
425.2.c.c.424.7		12	85.84	even	2	inner
425.2.c.c.424.8		12	5.4	even	2	inner
425.2.d.a.101.3	✓	6	5.3	odd	4
425.2.d.a.101.4	yes	6	85.33	odd	4
425.2.d.b.101.3	yes	6	85.67	odd	4
425.2.d.b.101.4	yes	6	5.2	odd	4
7225.2.a.ba.1.3		6	85.47	odd	4
7225.2.a.ba.1.4		6	85.72	odd	4
7225.2.a.bg.1.3		6	85.38	odd	4
7225.2.a.bg.1.4		6	85.13	odd	4

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 $\F_{q}$

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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	425.2.c.c.424.5

Level	$425$
Weight	$2$
Character	425.424
Analytic conductor	$3.394$
Analytic rank	$0$
Dimension	$12$
Inner twists	$4$