LMFDB - Embedded newform 1305.2.f.l.289.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1305,2,Mod(289,1305)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1305, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("1305.289");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$N$	$=$	$1305 = 3^{2} \cdot 5 \cdot 29$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	1305.f (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:	$10.4204774638$
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:	$x^{12} - x^{11} - 11 x^{9} + 55 x^{8} - 66 x^{7} + 328 x^{6} - 214 x^{5} + 207 x^{4} + 383 x^{3} + \cdots + 209$ x^12 - x^11 - 11x^9 + 55x^8 - 66x^7 + 328x^6 - 214x^5 + 207x^4 + 383x^3 + 16x^2 - 107*x + 209
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$2^{6}$
Twist minimal:	no (minimal twist has level 435)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			289.8
Root			$-2.21342 - 2.00212i$ of defining polynomial
Character	$\chi$	$=$	1305.289
Dual form			1305.2.f.l.289.7

$q$ -expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.334522 q^{2} -1.88809 q^{4} +(-1.95387 + 1.08739i) q^{5} +0.275019i q^{7} -1.30066 q^{8} +(-0.653612 + 0.363756i) q^{10} +0.541705i q^{11} -5.03697i q^{13} +0.0920000i q^{14} +3.34109 q^{16} -3.32386 q^{17} +6.31271i q^{19} +(3.68908 - 2.05309i) q^{20} +0.181212i q^{22} +6.93673i q^{23} +(2.63518 - 4.24922i) q^{25} -1.68498i q^{26} -0.519262i q^{28} +(-2.25292 - 4.89125i) q^{29} -10.1057i q^{31} +3.71898 q^{32} -1.11191 q^{34} +(-0.299052 - 0.537350i) q^{35} +4.07094 q^{37} +2.11174i q^{38} +(2.54130 - 1.41432i) q^{40} -8.49843i q^{41} +5.97627 q^{43} -1.02279i q^{44} +2.32049i q^{46} +12.4310 q^{47} +6.92436 q^{49} +(0.881526 - 1.42146i) q^{50} +9.51028i q^{52} -3.98569i q^{53} +(-0.589043 - 1.05842i) q^{55} -0.357705i q^{56} +(-0.753651 - 1.63623i) q^{58} +8.01554 q^{59} -4.84022i q^{61} -3.38058i q^{62} -5.43810 q^{64} +(5.47714 + 9.84156i) q^{65} +12.1664i q^{67} +6.27576 q^{68} +(-0.100040 - 0.179756i) q^{70} +7.20182 q^{71} +1.23869 q^{73} +1.36182 q^{74} -11.9190i q^{76} -0.148979 q^{77} -2.36972i q^{79} +(-6.52804 + 3.63306i) q^{80} -2.84292i q^{82} -10.3724i q^{83} +(6.49437 - 3.61432i) q^{85} +1.99920 q^{86} -0.704571i q^{88} -2.50630i q^{89} +1.38526 q^{91} -13.0972i q^{92} +4.15845 q^{94} +(-6.86436 - 12.3342i) q^{95} -17.8627 q^{97} +2.31636 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$12 q + 16 q^{4} + 6 q^{5} + 2 q^{10} + 32 q^{16} + 8 q^{17} + 20 q^{20} + 12 q^{25} - 8 q^{29} + 40 q^{32} - 52 q^{34} - 14 q^{35} + 20 q^{37} - 22 q^{40} + 44 q^{43} + 32 q^{47} - 4 q^{49} - 24 q^{50}+ \cdots + 112 q^{98}+O(q^{100})$ 12 * q + 16 * q^4 + 6 * q^5 + 2 * q^10 + 32 * q^16 + 8 * q^17 + 20 * q^20 + 12 * q^25 - 8 * q^29 + 40 * q^32 - 52 * q^34 - 14 * q^35 + 20 * q^37 - 22 * q^40 + 44 * q^43 + 32 * q^47 - 4 * q^49 - 24 * q^50 + 42 * q^55 - 24 * q^58 + 28 * q^59 - 4 * q^64 - 22 * q^65 - 16 * q^68 - 26 * q^70 + 16 * q^71 - 36 * q^73 + 28 * q^74 + 22 * q^80 + 30 * q^85 + 16 * q^86 + 24 * q^91 - 28 * q^94 - 16 * q^95 - 40 * q^97 + 112 * q^98

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1305\mathbb{Z}\right)^\times$ .

$n$	$146$	$262$	$901$
$\chi(n)$	$1$	$-1$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$ -expansion $a_n$ , the Satake parameters $\alpha_p$ , and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$ .

Display

a_p

with

p

up to: 50 250 1000 (See

a_n

instead) (See

a_n

instead) (See

a_n

instead) Display

a_n

with

n

up to: 50 250 1000 (See only

a_p

) (See only

a_p

) (See only

a_p

)

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

$2$	0.334522			0.236543			0.118272	−	0.992981i	$-0.462265\pi$
$2$	0.334522			0.236543			0.118272	+	0.992981i	$0.462265\pi$
$3$		0			0
$3$		0			0
$4$	−1.88809			−0.944047
$5$	−1.95387	+	1.08739i	−0.873795	+	0.486294i
$5$	−1.95387	+	1.08739i	−0.873795	+	0.486294i
$6$		0			0
$7$			0.275019i			0.103947i	0.998648	+	0.0519737i	$0.0165512\pi$
$7$			0.275019i			0.103947i	−0.998648	+	0.0519737i	$0.983449\pi$
$8$	−1.30066			−0.459851
$9$		0			0
$10$	−0.653612	+	0.363756i	−0.206690	+	0.115030i
$11$			0.541705i			0.163330i	0.996660	+	0.0816650i	$0.0260238\pi$
$11$			0.541705i			0.163330i	−0.996660	+	0.0816650i	$0.973976\pi$
$12$		0			0
$13$		−	5.03697i		−	1.39700i	−0.715608	−	0.698502i	$-0.753850\pi$
$13$		−	5.03697i		−	1.39700i	0.715608	−	0.698502i	$-0.246150\pi$
$14$			0.0920000i			0.0245880i
$15$		0			0
$16$	3.34109			0.835273
$17$	−3.32386			−0.806154			−0.403077	−	0.915166i	$-0.632059\pi$
$17$	−3.32386			−0.806154			−0.403077	+	0.915166i	$0.632059\pi$
$18$		0			0
$19$			6.31271i			1.44824i	0.689676	+	0.724118i	$0.257753\pi$
$19$			6.31271i			1.44824i	−0.689676	+	0.724118i	$0.742247\pi$
$20$	3.68908	−	2.05309i	0.824904	−	0.459085i
$21$		0			0
$22$			0.181212i			0.0386346i
$23$			6.93673i			1.44641i	0.690635	+	0.723204i	$0.257331\pi$
$23$			6.93673i			1.44641i	−0.690635	+	0.723204i	$0.742669\pi$
$24$		0			0
$25$	2.63518	−	4.24922i	0.527036	−	0.849843i
$26$		−	1.68498i		−	0.330452i
$27$		0			0
$28$		−	0.519262i		−	0.0981312i
$29$	−2.25292	−	4.89125i	−0.418356	−	0.908283i
$29$	−2.25292	−	4.89125i	−0.418356	−	0.908283i
$30$		0			0
$31$		−	10.1057i		−	1.81504i	−0.420012	−	0.907518i	$-0.637974\pi$
$31$		−	10.1057i		−	1.81504i	0.420012	−	0.907518i	$-0.362026\pi$
$32$	3.71898			0.657429
$33$		0			0
$34$	−1.11191			−0.190690
$35$	−0.299052	−	0.537350i	−0.0505490	−	0.0908287i
$36$		0			0
$37$	4.07094			0.669259			0.334629	−	0.942350i	$-0.391389\pi$
$37$	4.07094			0.669259			0.334629	+	0.942350i	$0.391389\pi$
$38$			2.11174i			0.342570i
$39$		0			0
$40$	2.54130	−	1.41432i	0.401816	−	0.223623i
$41$		−	8.49843i		−	1.32723i	−0.748073	−	0.663616i	$-0.769021\pi$
$41$		−	8.49843i		−	1.32723i	0.748073	−	0.663616i	$-0.230979\pi$
$42$		0			0
$43$	5.97627			0.911372			0.455686	−	0.890140i	$-0.349394\pi$
$43$	5.97627			0.911372			0.455686	+	0.890140i	$0.349394\pi$
$44$		−	1.02279i		−	0.154191i
$45$		0			0
$46$			2.32049i			0.342138i
$47$	12.4310			1.81325			0.906624	−	0.421939i	$-0.138651\pi$
$47$	12.4310			1.81325			0.906624	+	0.421939i	$0.138651\pi$
$48$		0			0
$49$	6.92436			0.989195
$50$	0.881526	−	1.42146i	0.124667	−	0.201025i
$51$		0			0
$52$			9.51028i			1.31884i
$53$		−	3.98569i		−	0.547477i	−0.961804	−	0.273739i	$-0.911740\pi$
$53$		−	3.98569i		−	0.547477i	0.961804	−	0.273739i	$-0.0882603\pi$
$54$		0			0
$55$	−0.589043	−	1.05842i	−0.0794265	−	0.142717i
$56$		−	0.357705i		−	0.0478003i
$57$		0			0
$58$	−0.753651	−	1.63623i	−0.0989593	−	0.214848i
$59$	8.01554			1.04353			0.521767	−	0.853088i	$-0.325273\pi$
$59$	8.01554			1.04353			0.521767	+	0.853088i	$0.325273\pi$
$60$		0			0
$61$		−	4.84022i		−	0.619726i	−0.950781	−	0.309863i	$-0.899717\pi$
$61$		−	4.84022i		−	0.619726i	0.950781	−	0.309863i	$-0.100283\pi$
$62$		−	3.38058i		−	0.429334i
$63$		0			0
$64$	−5.43810			−0.679762
$65$	5.47714	+	9.84156i	0.679355	+	1.22070i
$66$		0			0
$67$			12.1664i			1.48636i	0.669090	+	0.743182i	$0.266684\pi$
$67$			12.1664i			1.48636i	−0.669090	+	0.743182i	$0.733316\pi$
$68$	6.27576			0.761048
$69$		0			0
$70$	−0.100040	−	0.179756i	−0.0119570	−	0.0214849i
$71$	7.20182			0.854698			0.427349	−	0.904087i	$-0.359447\pi$
$71$	7.20182			0.854698			0.427349	+	0.904087i	$0.359447\pi$
$72$		0			0
$73$	1.23869			0.144977			0.0724886	−	0.997369i	$-0.476906\pi$
$73$	1.23869			0.144977			0.0724886	+	0.997369i	$0.476906\pi$
$74$	1.36182			0.158309
$75$		0			0
$76$		−	11.9190i		−	1.36720i
$77$	−0.148979			−0.0169777
$78$		0			0
$79$		−	2.36972i		−	0.266614i	−0.991075	−	0.133307i	$-0.957440\pi$
$79$		−	2.36972i		−	0.266614i	0.991075	−	0.133307i	$-0.0425597\pi$
$80$	−6.52804	+	3.63306i	−0.729857	+	0.406188i
$81$		0			0
$82$		−	2.84292i		−	0.313948i
$83$		−	10.3724i		−	1.13852i	−0.822159	−	0.569258i	$-0.807230\pi$
$83$		−	10.3724i		−	1.13852i	0.822159	−	0.569258i	$-0.192770\pi$
$84$		0			0
$85$	6.49437	−	3.61432i	0.704413	−	0.392028i
$86$	1.99920			0.215579
$87$		0			0
$88$		−	0.704571i		−	0.0751075i
$89$		−	2.50630i		−	0.265667i	−0.991138	−	0.132833i	$-0.957592\pi$
$89$		−	2.50630i		−	0.265667i	0.991138	−	0.132833i	$-0.0424076\pi$
$90$		0			0
$91$	1.38526			0.145215
$92$		−	13.0972i		−	1.36548i
$93$		0			0
$94$	4.15845			0.428911
$95$	−6.86436	−	12.3342i	−0.704269	−	1.26546i
$96$		0			0
$97$	−17.8627			−1.81368			−0.906840	−	0.421476i	$-0.861512\pi$
$97$	−17.8627			−1.81368			−0.906840	+	0.421476i	$0.861512\pi$
$98$	2.31636			0.233987
$99$		0			0
$100$	−4.97546	+	8.02292i	−0.497546	+	0.802292i
$101$		−	13.6767i		−	1.36088i	−0.732803	−	0.680441i	$-0.761788\pi$
$101$		−	13.6767i		−	1.36088i	0.732803	−	0.680441i	$-0.238212\pi$
$102$		0			0
$103$			10.6870i			1.05302i	0.850168	+	0.526511i	$0.176500\pi$
$103$			10.6870i			1.05302i	−0.850168	+	0.526511i	$0.823500\pi$
$104$			6.55136i			0.642414i
$105$		0			0
$106$		−	1.33330i		−	0.129502i
$107$			4.59224i			0.443948i	0.975053	+	0.221974i	$0.0712501\pi$
$107$			4.59224i			0.443948i	−0.975053	+	0.221974i	$0.928750\pi$
$108$		0			0
$109$	−13.1473			−1.25928			−0.629639	−	0.776888i	$-0.716797\pi$
$109$	−13.1473			−1.25928			−0.629639	+	0.776888i	$0.716797\pi$
$110$	−0.197048	−	0.354065i	−0.0187878	−	0.0337587i
$111$		0			0
$112$			0.918863i			0.0868244i
$113$	12.4441			1.17065			0.585323	−	0.810800i	$-0.300968\pi$
$113$	12.4441			1.17065			0.585323	+	0.810800i	$0.300968\pi$
$114$		0			0
$115$	−7.54291	−	13.5534i	−0.703380	−	1.26386i
$116$	4.25372	+	9.23515i	0.394948	+	0.857462i
$117$		0			0
$118$	2.68138			0.246841
$119$		−	0.914124i		−	0.0837976i
$120$		0			0
$121$	10.7066			0.973323
$122$		−	1.61916i		−	0.146592i
$123$		0			0
$124$			19.0805i			1.71348i
$125$	−0.528238	+	11.1679i	−0.0472470	+	0.998883i
$126$		0			0
$127$	−10.3846			−0.921487			−0.460744	−	0.887533i	$-0.652417\pi$
$127$	−10.3846			−0.921487			−0.460744	+	0.887533i	$0.652417\pi$
$128$	−9.25713			−0.818222
$129$		0			0
$130$	1.83223	+	3.29222i	0.160697	+	0.288747i
$131$		−	3.80784i		−	0.332693i	−0.986067	−	0.166346i	$-0.946803\pi$
$131$		−	3.80784i		−	0.332693i	0.986067	−	0.166346i	$-0.0531970\pi$
$132$		0			0
$133$	−1.73611			−0.150540
$134$			4.06994i			0.351589i
$135$		0			0
$136$	4.32319			0.370711
$137$	−9.05968			−0.774021			−0.387010	−	0.922075i	$-0.626492\pi$
$137$	−9.05968			−0.774021			−0.387010	+	0.922075i	$0.626492\pi$
$138$		0			0
$139$	1.41853			0.120318			0.0601591	−	0.998189i	$-0.480839\pi$
$139$	1.41853			0.120318			0.0601591	+	0.998189i	$0.480839\pi$
$140$	0.564638	+	1.01457i	0.0477207	+	0.0857466i
$141$		0			0
$142$	2.40917			0.202173
$143$	2.72855			0.228173
$144$		0			0
$145$	9.72058	+	7.10706i	0.807251	+	0.590209i
$146$	0.414368			0.0342933
$147$		0			0
$148$	−7.68632			−0.631812
$149$	−4.70591			−0.385523			−0.192762	−	0.981246i	$-0.561744\pi$
$149$	−4.70591			−0.385523			−0.192762	+	0.981246i	$0.561744\pi$
$150$		0			0
$151$	−5.99181			−0.487606			−0.243803	−	0.969825i	$-0.578395\pi$
$151$	−5.99181			−0.487606			−0.243803	+	0.969825i	$0.578395\pi$
$152$		−	8.21066i		−	0.665973i
$153$		0			0
$154$	−0.0498368			−0.00401596
$155$	10.9888	+	19.7452i	0.882642	+	1.58597i
$156$		0			0
$157$	11.2728			0.899664			0.449832	−	0.893113i	$-0.351484\pi$
$157$	11.2728			0.899664			0.449832	+	0.893113i	$0.351484\pi$
$158$		−	0.792725i		−	0.0630658i
$159$		0			0
$160$	−7.26639	+	4.04397i	−0.574458	+	0.319704i
$161$	−1.90773			−0.150350
$162$		0			0
$163$	−11.2859			−0.883980			−0.441990	−	0.897020i	$-0.645727\pi$
$163$	−11.2859			−0.883980			−0.441990	+	0.897020i	$0.645727\pi$
$164$			16.0458i			1.25297i
$165$		0			0
$166$		−	3.46980i		−	0.269308i
$167$		−	9.11150i		−	0.705069i	−0.935799	−	0.352535i	$-0.885320\pi$
$167$		−	9.11150i		−	0.705069i	0.935799	−	0.352535i	$-0.114680\pi$
$168$		0			0
$169$	−12.3711			−0.951621
$170$	2.17251	−	1.20907i	0.166624	−	0.0927316i
$171$		0			0
$172$	−11.2838			−0.860379
$173$			6.68099i			0.507946i	0.967211	+	0.253973i	$0.0817374\pi$
$173$			6.68099i			0.507946i	−0.967211	+	0.253973i	$0.918263\pi$
$174$		0			0
$175$	1.16861	+	0.724723i	0.0883389	+	0.0547839i
$176$			1.80988i			0.136425i
$177$		0			0
$178$		−	0.838413i		−	0.0628417i
$179$	−17.6127			−1.31644			−0.658218	−	0.752827i	$-0.728690\pi$
$179$	−17.6127			−1.31644			−0.658218	+	0.752827i	$0.728690\pi$
$180$		0			0
$181$	0.799278			0.0594099			0.0297049	−	0.999559i	$-0.490543\pi$
$181$	0.799278			0.0594099			0.0297049	+	0.999559i	$0.490543\pi$
$182$	0.463401			0.0343496
$183$		0			0
$184$		−	9.02229i		−	0.665132i
$185$	−7.95407	+	4.42669i	−0.584795	+	0.325457i
$186$		0			0
$187$		−	1.80055i		−	0.131669i
$188$	−23.4709			−1.71179
$189$		0			0
$190$	−2.29628	−	4.12606i	−0.166590	−	0.299336i
$191$		−	14.0344i		−	1.01549i	−0.861506	−	0.507747i	$-0.830478\pi$
$191$		−	14.0344i		−	1.01549i	0.861506	−	0.507747i	$-0.169522\pi$
$192$		0			0
$193$	22.3693			1.61018			0.805089	−	0.593154i	$-0.202117\pi$
$193$	22.3693			1.61018			0.805089	+	0.593154i	$0.202117\pi$
$194$	−5.97546			−0.429013
$195$		0			0
$196$	−13.0739			−0.933847
$197$		−	4.66274i		−	0.332207i	−0.986108	−	0.166103i	$-0.946881\pi$
$197$		−	4.66274i		−	0.332207i	0.986108	−	0.166103i	$-0.0531186\pi$
$198$		0			0
$199$	14.9769			1.06169			0.530843	−	0.847470i	$-0.321876\pi$
$199$	14.9769			1.06169			0.530843	+	0.847470i	$0.321876\pi$
$200$	−3.42746	+	5.52676i	−0.242358	+	0.390801i
$201$		0			0
$202$		−	4.57516i		−	0.321907i
$203$	1.34519	−	0.619595i	0.0944136	−	0.0434870i
$204$		0			0
$205$	9.24109	+	16.6048i	0.645426	+	1.15973i
$206$			3.57504i			0.249085i
$207$		0			0
$208$		−	16.8290i		−	1.16688i
$209$	−3.41963			−0.236540
$210$		0			0
$211$		−	15.8960i		−	1.09432i	−0.837027	−	0.547162i	$-0.815708\pi$
$211$		−	15.8960i		−	1.09432i	0.837027	−	0.547162i	$-0.184292\pi$
$212$			7.52537i			0.516844i
$213$		0			0
$214$			1.53621i			0.105013i
$215$	−11.6768	+	6.49852i	−0.796353	+	0.443195i
$216$		0			0
$217$	2.77926			0.188668
$218$	−4.39805			−0.297874
$219$		0			0
$220$	1.11217	+	1.99839i	0.0749824	+	0.134732i
$221$			16.7422i			1.12620i
$222$		0			0
$223$			7.51910i			0.503516i	0.967790	+	0.251758i	$0.0810087\pi$
$223$			7.51910i			0.503516i	−0.967790	+	0.251758i	$0.918991\pi$
$224$			1.02279i			0.0683380i
$225$		0			0
$226$	4.16284			0.276908
$227$			14.9639i			0.993187i	0.867983	+	0.496594i	$0.165416\pi$
$227$			14.9639i			0.993187i	−0.867983	+	0.496594i	$0.834584\pi$
$228$		0			0
$229$		−	12.3067i		−	0.813247i	−0.913596	−	0.406623i	$-0.866706\pi$
$229$		−	12.3067i		−	0.813247i	0.913596	−	0.406623i	$-0.133294\pi$
$230$	−2.52327	−	4.53393i	−0.166380	−	0.298958i
$231$		0			0
$232$	2.93027	+	6.36183i	0.192382	+	0.417675i
$233$			10.7500i			0.704258i	0.935951	+	0.352129i	$0.114542\pi$
$233$			10.7500i			0.704258i	−0.935951	+	0.352129i	$0.885458\pi$
$234$		0			0
$235$	−24.2885	+	13.5173i	−1.58441	+	0.881772i
$236$	−15.1341			−0.985146
$237$		0			0
$238$		−	0.305795i		−	0.0198217i
$239$	6.04481			0.391006			0.195503	−	0.980703i	$-0.437366\pi$
$239$	6.04481			0.391006			0.195503	+	0.980703i	$0.437366\pi$
$240$		0			0
$241$	24.1998			1.55885			0.779423	−	0.626498i	$-0.215512\pi$
$241$	24.1998			1.55885			0.779423	+	0.626498i	$0.215512\pi$
$242$	3.58158			0.230233
$243$		0			0
$244$			9.13879i			0.585051i
$245$	−13.5293	+	7.52947i	−0.864354	+	0.481040i
$246$		0			0
$247$	31.7969			2.02319
$248$			13.1440i			0.834647i
$249$		0			0
$250$	−0.176707	+	3.73590i	−0.0111760	+	0.236279i
$251$		−	26.0151i		−	1.64206i	−0.570886	−	0.821030i	$-0.693400\pi$
$251$		−	26.0151i		−	1.64206i	0.570886	−	0.821030i	$-0.306600\pi$
$252$		0			0
$253$	−3.75766			−0.236242
$254$	−3.47389			−0.217971
$255$		0			0
$256$	7.77948			0.486218
$257$			3.58709i			0.223757i	0.993722	+	0.111878i	$0.0356867\pi$
$257$			3.58709i			0.223757i	−0.993722	+	0.111878i	$0.964313\pi$
$258$		0			0
$259$			1.11959i			0.0695677i
$260$	−10.3414	−	18.5818i	−0.641344	−	1.15239i
$261$		0			0
$262$		−	1.27381i		−	0.0786962i
$263$	−3.92407			−0.241969			−0.120984	−	0.992654i	$-0.538605\pi$
$263$	−3.92407			−0.241969			−0.120984	+	0.992654i	$0.538605\pi$
$264$		0			0
$265$	4.33399	+	7.78751i	0.266235	+	0.478383i
$266$	−0.580769			−0.0356093
$267$		0			0
$268$		−	22.9713i		−	1.40320i
$269$			13.8826i			0.846437i	0.906028	+	0.423218i	$0.139100\pi$
$269$			13.8826i			0.846437i	−0.906028	+	0.423218i	$0.860900\pi$
$270$		0			0
$271$		−	13.3880i		−	0.813265i	−0.913592	−	0.406632i	$-0.866703\pi$
$271$		−	13.3880i		−	0.813265i	0.913592	−	0.406632i	$-0.133297\pi$
$272$	−11.1053			−0.673358
$273$		0			0
$274$	−3.03067			−0.183089
$275$	2.30182	+	1.42749i	0.138805	+	0.0860808i
$276$		0			0
$277$		−	9.91356i		−	0.595648i	−0.954621	−	0.297824i	$-0.903739\pi$
$277$		−	9.91356i		−	0.595648i	0.954621	−	0.297824i	$-0.0962609\pi$
$278$	0.474530			0.0284604
$279$		0			0
$280$	0.388963	+	0.698907i	0.0232450	+	0.0417677i
$281$	18.7167			1.11654			0.558271	−	0.829658i	$-0.311465\pi$
$281$	18.7167			1.11654			0.558271	+	0.829658i	$0.311465\pi$
$282$		0			0
$283$		−	18.9462i		−	1.12624i	−0.826377	−	0.563118i	$-0.809602\pi$
$283$		−	18.9462i		−	1.12624i	0.826377	−	0.563118i	$-0.190398\pi$
$284$	−13.5977			−0.806876
$285$		0			0
$286$	0.912761			0.0539727
$287$	2.33723			0.137962
$288$		0			0
$289$	−5.95197			−0.350116
$290$	3.25175	+	2.37747i	0.190950	+	0.139610i
$291$		0			0
$292$	−2.33875			−0.136865
$293$	7.13932			0.417083			0.208542	−	0.978013i	$-0.433128\pi$
$293$	7.13932			0.417083			0.208542	+	0.978013i	$0.433128\pi$
$294$		0			0
$295$	−15.6613	+	8.71600i	−0.911835	+	0.507465i
$296$	−5.29489			−0.307759
$297$		0			0
$298$	−1.57423			−0.0911929
$299$	34.9401			2.02064
$300$		0			0
$301$			1.64359i			0.0947347i
$302$	−2.00439			−0.115340
$303$		0			0
$304$			21.0913i			1.20967i
$305$	5.26319	+	9.45713i	0.301369	+	0.541514i
$306$		0			0
$307$	10.0637			0.574368			0.287184	−	0.957875i	$-0.407281\pi$
$307$	10.0637			0.574368			0.287184	+	0.957875i	$0.407281\pi$
$308$	0.281286			0.0160278
$309$		0			0
$310$	3.67600	+	6.60520i	0.208783	+	0.375150i
$311$			16.2493i			0.921412i	0.887553	+	0.460706i	$0.152404\pi$
$311$			16.2493i			0.921412i	−0.887553	+	0.460706i	$0.847596\pi$
$312$		0			0
$313$			21.4838i			1.21434i	0.794573	+	0.607168i	$0.207695\pi$
$313$			21.4838i			1.21434i	−0.794573	+	0.607168i	$0.792305\pi$
$314$	3.77099			0.212809
$315$		0			0
$316$			4.47426i			0.251697i
$317$	24.4204			1.37159			0.685794	−	0.727796i	$-0.259455\pi$
$317$	24.4204			1.37159			0.685794	+	0.727796i	$0.259455\pi$
$318$		0			0
$319$	2.64961	−	1.22042i	0.148350	−	0.0683302i
$320$	10.6253	−	5.91332i	0.593973	−	0.330565i
$321$		0			0
$322$	−0.638179			−0.0355643
$323$		−	20.9826i		−	1.16750i
$324$		0			0
$325$	−21.4032	−	13.2733i	−1.18723	−	0.736271i
$326$	−3.77539			−0.209099
$327$		0			0
$328$			11.0535i			0.610329i
$329$			3.41876i			0.188482i
$330$		0			0
$331$			0.627675i			0.0345001i	0.999851	+	0.0172501i	$0.00549114\pi$
$331$			0.627675i			0.0345001i	−0.999851	+	0.0172501i	$0.994509\pi$
$332$			19.5840i			1.07481i
$333$		0			0
$334$		−	3.04800i		−	0.166779i
$335$	−13.2296	−	23.7715i	−0.722810	−	1.29878i
$336$		0			0
$337$	−27.2068			−1.48205			−0.741024	−	0.671479i	$-0.765659\pi$
$337$	−27.2068			−1.48205			−0.741024	+	0.671479i	$0.765659\pi$
$338$	−4.13840			−0.225099
$339$		0			0
$340$	−12.2620	+	6.82418i	−0.665000	+	0.370093i
$341$	5.47430			0.296450
$342$		0			0
$343$			3.82946i			0.206772i
$344$	−7.77306			−0.419096
$345$		0			0
$346$			2.23494i			0.120151i
$347$		−	13.0161i		−	0.698739i	−0.936985	−	0.349369i	$-0.886396\pi$
$347$		−	13.0161i		−	0.698739i	0.936985	−	0.349369i	$-0.113604\pi$
$348$		0			0
$349$	23.2480			1.24444			0.622218	−	0.782844i	$-0.286232\pi$
$349$	23.2480			1.24444			0.622218	+	0.782844i	$0.286232\pi$
$350$	0.390928	+	0.242436i	0.0208960	+	0.0129588i
$351$		0			0
$352$			2.01459i			0.107378i
$353$		−	33.0268i		−	1.75784i	−0.476969	−	0.878920i	$-0.658265\pi$
$353$		−	33.0268i		−	1.75784i	0.476969	−	0.878920i	$-0.341735\pi$
$354$		0			0
$355$	−14.0714	+	7.83116i	−0.746831	+	0.415635i
$356$			4.73213i			0.250802i
$357$		0			0
$358$	−5.89185			−0.311394
$359$		−	30.0802i		−	1.58757i	−0.608195	−	0.793787i	$-0.708106\pi$
$359$		−	30.0802i		−	1.58757i	0.608195	−	0.793787i	$-0.291894\pi$
$360$		0			0
$361$	−20.8503			−1.09739
$362$	0.267376			0.0140530
$363$		0			0
$364$	−2.61551			−0.137090
$365$	−2.42022	+	1.34693i	−0.126680	+	0.0705016i
$366$		0			0
$367$	−3.89299			−0.203213			−0.101606	−	0.994825i	$-0.532398\pi$
$367$	−3.89299			−0.203213			−0.101606	+	0.994825i	$0.532398\pi$
$368$			23.1762i			1.20814i
$369$		0			0
$370$	−2.66082	+	1.48083i	−0.138329	+	0.0769845i
$371$	1.09614			0.0569088
$372$		0			0
$373$		−	1.78697i		−	0.0925260i	−0.998929	−	0.0462630i	$-0.985269\pi$
$373$		−	1.78697i		−	0.0925260i	0.998929	−	0.0462630i	$-0.0147312\pi$
$374$		−	0.602324i		−	0.0311454i
$375$		0			0
$376$	−16.1684			−0.833824
$377$	−24.6371	+	11.3479i	−1.26888	+	0.584445i
$378$		0			0
$379$			18.4165i			0.945993i	0.881064	+	0.472996i	$0.156828\pi$
$379$			18.4165i			0.945993i	−0.881064	+	0.472996i	$0.843172\pi$
$380$	12.9606	+	23.2881i	0.664863	+	1.19466i
$381$		0			0
$382$		−	4.69482i		−	0.240208i
$383$		−	28.3971i		−	1.45102i	−0.688210	−	0.725512i	$-0.741603\pi$
$383$		−	28.3971i		−	1.45102i	0.688210	−	0.725512i	$-0.258397\pi$
$384$		0			0
$385$	0.291085	−	0.161998i	0.0148351	−	0.00825617i
$386$	7.48304			0.380877
$387$		0			0
$388$	33.7264			1.71220
$389$		−	8.49677i		−	0.430803i	−0.976526	−	0.215402i	$-0.930894\pi$
$389$		−	8.49677i		−	0.430803i	0.976526	−	0.215402i	$-0.0691061\pi$
$390$		0			0
$391$		−	23.0567i		−	1.16603i
$392$	−9.00621			−0.454882
$393$		0			0
$394$		−	1.55979i		−	0.0785812i
$395$	2.57680	+	4.63011i	0.129653	+	0.232966i
$396$		0			0
$397$			1.07708i			0.0540571i	0.999635	+	0.0270286i	$0.00860451\pi$
$397$			1.07708i			0.0540571i	−0.999635	+	0.0270286i	$0.991395\pi$
$398$	5.01011			0.251134
$399$		0			0
$400$	8.80437	−	14.1970i	0.440218	−	0.709851i
$401$	−18.2009			−0.908910			−0.454455	−	0.890770i	$-0.650166\pi$
$401$	−18.2009			−0.908910			−0.454455	+	0.890770i	$0.650166\pi$
$402$		0			0
$403$	−50.9021			−2.53561
$404$			25.8229i			1.28474i
$405$		0			0
$406$	0.449995	−	0.207268i	0.0223329	−	0.0102866i
$407$			2.20525i			0.109310i
$408$		0			0
$409$		−	29.4035i		−	1.45391i	−0.686684	−	0.726956i	$-0.740934\pi$
$409$		−	29.4035i		−	1.45391i	0.686684	−	0.726956i	$-0.259066\pi$
$410$	3.09135	+	5.55468i	0.152671	+	0.274326i
$411$		0			0
$412$		−	20.1781i		−	0.994103i
$413$			2.20442i			0.108473i
$414$		0			0
$415$	11.2788	+	20.2662i	0.553654	+	0.994830i
$416$		−	18.7324i		−	0.918431i
$417$		0			0
$418$	−1.14394			−0.0559520
$419$	7.33375			0.358277			0.179139	−	0.983824i	$-0.442669\pi$
$419$	7.33375			0.358277			0.179139	+	0.983824i	$0.442669\pi$
$420$		0			0
$421$			21.0761i			1.02718i	0.858034	+	0.513592i	$0.171686\pi$
$421$			21.0761i			1.02718i	−0.858034	+	0.513592i	$0.828314\pi$
$422$		−	5.31756i		−	0.258855i
$423$		0			0
$424$			5.18401i			0.251758i
$425$	−8.75896	+	14.1238i	−0.424872	+	0.685105i
$426$		0			0
$427$	1.33115			0.0644189
$428$		−	8.67058i		−	0.419108i
$429$		0			0
$430$	−3.90616	+	2.17390i	−0.188372	+	0.104835i
$431$	10.6334			0.512195			0.256098	−	0.966651i	$-0.417563\pi$
$431$	10.6334			0.512195			0.256098	+	0.966651i	$0.417563\pi$
$432$		0			0
$433$	−12.7281			−0.611674			−0.305837	−	0.952084i	$-0.598936\pi$
$433$	−12.7281			−0.611674			−0.305837	+	0.952084i	$0.598936\pi$
$434$	0.929724			0.0446282
$435$		0			0
$436$	24.8233			1.18882
$437$	−43.7896			−2.09474
$438$		0			0
$439$	−4.48366			−0.213994			−0.106997	−	0.994259i	$-0.534123\pi$
$439$	−4.48366			−0.213994			−0.106997	+	0.994259i	$0.534123\pi$
$440$	0.766141	+	1.37664i	0.0365244	+	0.0656286i
$441$		0			0
$442$			5.60063i			0.266395i
$443$	12.8049			0.608379			0.304189	−	0.952612i	$-0.401614\pi$
$443$	12.8049			0.608379			0.304189	+	0.952612i	$0.401614\pi$
$444$		0			0
$445$	2.72532	+	4.89697i	0.129192	+	0.232138i
$446$			2.51531i			0.119103i
$447$		0			0
$448$		−	1.49558i		−	0.0706595i
$449$			22.1787i			1.04668i	0.852124	+	0.523340i	$0.175314\pi$
$449$			22.1787i			1.04668i	−0.852124	+	0.523340i	$0.824686\pi$
$450$		0			0
$451$	4.60364			0.216777
$452$	−23.4957			−1.10515
$453$		0			0
$454$			5.00575i			0.234932i
$455$	−2.70661	+	1.50632i	−0.126888	+	0.0706172i
$456$		0			0
$457$		−	12.2239i		−	0.571808i	−0.958258	−	0.285904i	$-0.907706\pi$
$457$		−	12.2239i		−	0.571808i	0.958258	−	0.285904i	$-0.0922939\pi$
$458$		−	4.11685i		−	0.192368i
$459$		0			0
$460$	14.2417	+	25.5902i	0.664024	+	1.19315i
$461$			26.1495i			1.21790i	0.793207	+	0.608952i	$0.208410\pi$
$461$			26.1495i			1.21790i	−0.793207	+	0.608952i	$0.791590\pi$
$462$		0			0
$463$			23.5519i			1.09455i	0.836953	+	0.547275i	$0.184335\pi$
$463$			23.5519i			1.09455i	−0.836953	+	0.547275i	$0.815665\pi$
$464$	−7.52720	−	16.3421i	−0.349442	−	0.758664i
$465$		0			0
$466$			3.59613i			0.166587i
$467$	12.7060			0.587963			0.293981	−	0.955811i	$-0.405020\pi$
$467$	12.7060			0.587963			0.293981	+	0.955811i	$0.405020\pi$
$468$		0			0
$469$	−3.34599			−0.154503
$470$	−8.12505	+	4.52185i	−0.374781	+	0.208577i
$471$		0			0
$472$	−10.4255			−0.479870
$473$			3.23737i			0.148855i
$474$		0			0
$475$	26.8241	+	16.6351i	1.23077	+	0.763272i
$476$			1.72595i			0.0791089i
$477$		0			0
$478$	2.02212			0.0924898
$479$		−	14.2867i		−	0.652775i	−0.945236	−	0.326388i	$-0.894169\pi$
$479$		−	14.2867i		−	0.652775i	0.945236	−	0.326388i	$-0.105831\pi$
$480$		0			0
$481$		−	20.5052i		−	0.934957i
$482$	8.09538			0.368734
$483$		0			0
$484$	−20.2150			−0.918863
$485$	34.9012	−	19.4236i	1.58478	−	0.881982i
$486$		0			0
$487$		−	34.4239i		−	1.55990i	−0.625844	−	0.779948i	$-0.715246\pi$
$487$		−	34.4239i		−	1.55990i	0.625844	−	0.779948i	$-0.284754\pi$
$488$			6.29545i			0.284982i
$489$		0			0
$490$	−4.52585	+	2.51878i	−0.204457	+	0.113787i
$491$			20.5793i			0.928733i	0.885643	+	0.464366i	$0.153718\pi$
$491$			20.5793i			0.928733i	−0.885643	+	0.464366i	$0.846282\pi$
$492$		0			0
$493$	7.48838	+	16.2578i	0.337260	+	0.732216i
$494$	10.6368			0.478572
$495$		0			0
$496$		−	33.7641i		−	1.51605i
$497$			1.98064i			0.0888436i
$498$		0			0
$499$	−19.5569			−0.875487			−0.437743	−	0.899100i	$-0.644222\pi$
$499$	−19.5569			−0.875487			−0.437743	+	0.899100i	$0.644222\pi$
$500$	0.997363	−	21.0860i	0.0446034	−	0.942993i
$501$		0			0
$502$		−	8.70264i		−	0.388418i
$503$	38.5345			1.71817			0.859084	−	0.511835i	$-0.171034\pi$
$503$	38.5345			1.71817			0.859084	+	0.511835i	$0.171034\pi$
$504$		0			0
$505$	14.8719	+	26.7224i	0.661790	+	1.18913i
$506$	−1.25702			−0.0558814
$507$		0			0
$508$	19.6072			0.869928
$509$	−9.66065			−0.428201			−0.214100	−	0.976812i	$-0.568682\pi$
$509$	−9.66065			−0.428201			−0.214100	+	0.976812i	$0.568682\pi$
$510$		0			0
$511$			0.340662i			0.0150700i
$512$	21.1167			0.933234
$513$		0			0
$514$			1.19996i			0.0529281i
$515$	−11.6209	−	20.8810i	−0.512079	−	0.920126i
$516$		0			0
$517$			6.73393i			0.296158i
$518$			0.374527i			0.0164557i
$519$		0			0
$520$	−7.12387	−	12.8005i	−0.312402	−	0.561338i
$521$	−20.3242			−0.890421			−0.445211	−	0.895426i	$-0.646871\pi$
$521$	−20.3242			−0.890421			−0.445211	+	0.895426i	$0.646871\pi$
$522$		0			0
$523$			40.6870i			1.77912i	0.456820	+	0.889559i	$0.348988\pi$
$523$			40.6870i			1.77912i	−0.456820	+	0.889559i	$0.651012\pi$
$524$			7.18957i			0.314078i
$525$		0			0
$526$	−1.31269			−0.0572360
$527$			33.5899i			1.46320i
$528$		0			0
$529$	−25.1182			−1.09209
$530$	1.44982	+	2.60510i	0.0629761	+	0.113158i
$531$		0			0
$532$	3.27795			0.142117
$533$	−42.8064			−1.85415
$534$		0			0
$535$	−4.99354	−	8.97261i	−0.215890	−	0.387920i
$536$		−	15.8243i		−	0.683506i
$537$		0			0
$538$			4.64404i			0.200219i
$539$			3.75096i			0.161565i
$540$		0			0
$541$		−	5.02402i		−	0.216000i	−0.994151	−	0.108000i	$-0.965555\pi$
$541$		−	5.02402i		−	0.216000i	0.994151	−	0.108000i	$-0.0344446\pi$
$542$		−	4.47860i		−	0.192372i
$543$		0			0
$544$	−12.3614			−0.529989
$545$	25.6880	−	14.2962i	1.10035	−	0.612380i
$546$		0			0
$547$			8.61344i			0.368284i	0.982900	+	0.184142i	$0.0589506\pi$
$547$			8.61344i			0.368284i	−0.982900	+	0.184142i	$0.941049\pi$
$548$	17.1055			0.730712
$549$		0			0
$550$	0.770011	+	0.477527i	0.0328334	+	0.0203618i
$551$	30.8771	−	14.2220i	1.31541	−	0.605878i
$552$		0			0
$553$	0.651718			0.0277139
$554$		−	3.31631i		−	0.140896i
$555$		0			0
$556$	−2.67832			−0.113586
$557$			0.455853i			0.0193151i	0.999953	+	0.00965755i	$0.00307414\pi$
$557$			0.455853i			0.0193151i	−0.999953	+	0.00965755i	$0.996926\pi$
$558$		0			0
$559$		−	30.1023i		−	1.27319i
$560$	−0.999160	−	1.79533i	−0.0422222	−	0.0758667i
$561$		0			0
$562$	6.26114			0.264110
$563$	−0.120515			−0.00507911			−0.00253956	−	0.999997i	$-0.500808\pi$
$563$	−0.120515			−0.00507911			−0.00253956	+	0.999997i	$0.500808\pi$
$564$		0			0
$565$	−24.3142	+	13.5316i	−1.02290	+	0.569279i
$566$		−	6.33794i		−	0.266403i
$567$		0			0
$568$	−9.36708			−0.393034
$569$		−	22.7315i		−	0.952954i	−0.879187	−	0.476477i	$-0.841914\pi$
$569$		−	22.7315i		−	0.952954i	0.879187	−	0.476477i	$-0.158086\pi$
$570$		0			0
$571$	26.7167			1.11806			0.559030	−	0.829147i	$-0.311174\pi$
$571$	26.7167			1.11806			0.559030	+	0.829147i	$0.311174\pi$
$572$	−5.15176			−0.215406
$573$		0			0
$574$	0.781856			0.0326340
$575$	29.4756	+	18.2795i	1.22922	+	0.762308i
$576$		0			0
$577$	15.3024			0.637049			0.318524	−	0.947915i	$-0.396813\pi$
$577$	15.3024			0.637049			0.318524	+	0.947915i	$0.396813\pi$
$578$	−1.99107			−0.0828175
$579$		0			0
$580$	−18.3534	−	13.4188i	−0.762083	−	0.557185i
$581$	2.85260			0.118346
$582$		0			0
$583$	2.15907			0.0894195
$584$	−1.61110			−0.0666679
$585$		0			0
$586$	2.38826			0.0986582
$587$			41.6229i			1.71796i	0.512009	+	0.858980i	$0.328902\pi$
$587$			41.6229i			1.71796i	−0.512009	+	0.858980i	$0.671098\pi$
$588$		0			0
$589$	63.7944			2.62860
$590$	−5.23905	+	2.91570i	−0.215688	+	0.120037i
$591$		0			0
$592$	13.6014			0.559013
$593$		−	11.7493i		−	0.482487i	−0.970465	−	0.241244i	$-0.922445\pi$
$593$		−	11.7493i		−	0.482487i	0.970465	−	0.241244i	$-0.0775553\pi$
$594$		0			0
$595$	0.994006	+	1.78607i	0.0407503	+	0.0732219i
$596$	8.88521			0.363952
$597$		0			0
$598$	11.6882			0.477968
$599$		−	8.86844i		−	0.362354i	−0.983450	−	0.181177i	$-0.942009\pi$
$599$		−	8.86844i		−	0.362354i	0.983450	−	0.181177i	$-0.0579908\pi$
$600$		0			0
$601$		−	0.339050i		−	0.0138301i	−0.999976	−	0.00691506i	$-0.997799\pi$
$601$		−	0.339050i		−	0.0138301i	0.999976	−	0.00691506i	$-0.00220115\pi$
$602$			0.549817i			0.0224089i
$603$		0			0
$604$	11.3131			0.460324
$605$	−20.9192	+	11.6422i	−0.850485	+	0.473322i
$606$		0			0
$607$	−8.27856			−0.336017			−0.168008	−	0.985786i	$-0.553734\pi$
$607$	−8.27856			−0.336017			−0.168008	+	0.985786i	$0.553734\pi$
$608$			23.4769i			0.952112i
$609$		0			0
$610$	1.76066	+	3.16362i	0.0712869	+	0.128091i
$611$		−	62.6146i		−	2.53312i
$612$		0			0
$613$		−	42.0873i		−	1.69989i	−0.526870	−	0.849946i	$-0.676635\pi$
$613$		−	42.0873i		−	1.69989i	0.526870	−	0.849946i	$-0.323365\pi$
$614$	3.36655			0.135863
$615$		0			0
$616$	0.193770			0.00780723
$617$	28.7087			1.15577			0.577884	−	0.816119i	$-0.303879\pi$
$617$	28.7087			1.15577			0.577884	+	0.816119i	$0.303879\pi$
$618$		0			0
$619$			8.90716i			0.358009i	0.983848	+	0.179005i	$0.0572877\pi$
$619$			8.90716i			0.358009i	−0.983848	+	0.179005i	$0.942712\pi$
$620$	−20.7479	−	37.2807i	−0.833256	−	1.49723i
$621$		0			0
$622$			5.43575i			0.217954i
$623$	0.689279			0.0276154
$624$		0			0
$625$	−11.1117	−	22.3949i	−0.444467	−	0.895795i
$626$			7.18681i			0.287243i
$627$		0			0
$628$	−21.2840			−0.849325
$629$	−13.5312			−0.539525
$630$		0			0
$631$	17.9351			0.713986			0.356993	−	0.934107i	$-0.383802\pi$
$631$	17.9351			0.713986			0.356993	+	0.934107i	$0.383802\pi$
$632$			3.08219i			0.122603i
$633$		0			0
$634$	8.16918			0.324440
$635$	20.2902	−	11.2921i	0.805191	−	0.448114i
$636$		0			0
$637$		−	34.8778i		−	1.38191i
$638$	0.886356	−	0.408256i	0.0350912	−	0.0161630i
$639$		0			0
$640$	18.0872	−	10.0661i	0.714958	−	0.397897i
$641$		−	0.220741i		−	0.00871875i	−0.999990	−	0.00435938i	$-0.998612\pi$
$641$		−	0.220741i		−	0.00871875i	0.999990	−	0.00435938i	$-0.00138764\pi$
$642$		0			0
$643$			6.25440i			0.246650i	0.992366	+	0.123325i	$0.0393557\pi$
$643$			6.25440i			0.246650i	−0.992366	+	0.123325i	$0.960644\pi$
$644$	3.60198			0.141938
$645$		0			0
$646$		−	7.01914i		−	0.276164i
$647$			24.3697i			0.958071i	0.877796	+	0.479035i	$0.159013\pi$
$647$			24.3697i			0.958071i	−0.877796	+	0.479035i	$0.840987\pi$
$648$		0			0
$649$			4.34205i			0.170441i
$650$	−7.15984	−	4.44022i	−0.280832	−	0.174160i
$651$		0			0
$652$	21.3088			0.834519
$653$	−41.1834			−1.61163			−0.805815	−	0.592167i	$-0.798273\pi$
$653$	−41.1834			−1.61163			−0.805815	+	0.592167i	$0.798273\pi$
$654$		0			0
$655$	4.14060	+	7.44001i	0.161787	+	0.290705i
$656$		−	28.3940i		−	1.10860i
$657$		0			0
$658$			1.14365i			0.0445842i
$659$		−	4.18021i		−	0.162838i	−0.996680	−	0.0814190i	$-0.974055\pi$
$659$		−	4.18021i		−	0.162838i	0.996680	−	0.0814190i	$-0.0259452\pi$
$660$		0			0
$661$	−8.11965			−0.315818			−0.157909	−	0.987454i	$-0.550475\pi$
$661$	−8.11965			−0.315818			−0.157909	+	0.987454i	$0.550475\pi$
$662$			0.209971i			0.00816077i
$663$		0			0
$664$			13.4909i			0.523548i
$665$	3.39213	−	1.88783i	0.131541	−	0.0732069i
$666$		0			0
$667$	33.9293	−	15.6279i	1.31375	−	0.605113i
$668$			17.2034i			0.665619i
$669$		0			0
$670$	−4.42560	−	7.95211i	−0.170976	−	0.307217i
$671$	2.62197			0.101220
$672$		0			0
$673$		−	33.1749i		−	1.27880i	−0.768875	−	0.639399i	$-0.779183\pi$
$673$		−	33.1749i		−	1.27880i	0.768875	−	0.639399i	$-0.220817\pi$
$674$	−9.10128			−0.350568
$675$		0			0
$676$	23.3577			0.898375
$677$	14.7844			0.568212			0.284106	−	0.958793i	$-0.408303\pi$
$677$	14.7844			0.568212			0.284106	+	0.958793i	$0.408303\pi$
$678$		0			0
$679$		−	4.91257i		−	0.188527i
$680$	−8.44694	+	4.70099i	−0.323925	+	0.180275i
$681$		0			0
$682$	1.83128			0.0701232
$683$			33.5573i			1.28404i	0.766690	+	0.642018i	$0.221902\pi$
$683$			33.5573i			1.28404i	−0.766690	+	0.642018i	$0.778098\pi$
$684$		0			0
$685$	17.7014	−	9.85138i	0.676336	−	0.376402i
$686$			1.28104i			0.0489104i
$687$		0			0
$688$	19.9673			0.761245
$689$	−20.0758			−0.764828
$690$		0			0
$691$	−11.4767			−0.436596			−0.218298	−	0.975882i	$-0.570050\pi$
$691$	−11.4767			−0.436596			−0.218298	+	0.975882i	$0.570050\pi$
$692$		−	12.6143i		−	0.479525i
$693$		0			0
$694$		−	4.35417i		−	0.165282i
$695$	−2.77162	+	1.54249i	−0.105133	+	0.0585101i
$696$		0			0
$697$			28.2476i			1.06995i
$698$	7.77697			0.294363
$699$		0			0
$700$	−2.20645	−	1.36835i	−0.0833962	−	0.0517186i
$701$	−11.0065			−0.415711			−0.207855	−	0.978160i	$-0.566648\pi$
$701$	−11.0065			−0.415711			−0.207855	+	0.978160i	$0.566648\pi$
$702$		0			0
$703$			25.6987i			0.969244i
$704$		−	2.94584i		−	0.111026i
$705$		0			0
$706$		−	11.0482i		−	0.415805i
$707$	3.76135			0.141460
$708$		0			0
$709$	21.4248			0.804624			0.402312	−	0.915503i	$-0.368207\pi$
$709$	21.4248			0.804624			0.402312	+	0.915503i	$0.368207\pi$
$710$	−4.70719	+	2.61970i	−0.176658	+	0.0983156i
$711$		0			0
$712$			3.25983i			0.122167i
$713$	70.1005			2.62528
$714$		0			0
$715$	−5.33122	+	2.96699i	−0.199376	+	0.110959i
$716$	33.2545			1.24278
$717$		0			0
$718$		−	10.0625i		−	0.375530i
$719$	21.8797			0.815977			0.407988	−	0.912987i	$-0.366230\pi$
$719$	21.8797			0.815977			0.407988	+	0.912987i	$0.366230\pi$
$720$		0			0
$721$	−2.93913			−0.109459
$722$	−6.97491			−0.259579
$723$		0			0
$724$	−1.50911			−0.0560857
$725$	−26.7208	−	3.31619i	−0.992387	−	0.123160i
$726$		0			0
$727$	36.6219			1.35823			0.679115	−	0.734032i	$-0.262364\pi$
$727$	36.6219			1.35823			0.679115	+	0.734032i	$0.262364\pi$
$728$	−1.80175			−0.0667772
$729$		0			0
$730$	−0.809619	+	0.450579i	−0.0299654	+	0.0166767i
$731$	−19.8643			−0.734707
$732$		0			0
$733$	−7.68719			−0.283933			−0.141966	−	0.989871i	$-0.545342\pi$
$733$	−7.68719			−0.283933			−0.141966	+	0.989871i	$0.545342\pi$
$734$	−1.30229			−0.0480686
$735$		0			0
$736$			25.7975i			0.950910i
$737$	−6.59060			−0.242768
$738$		0			0
$739$		−	26.2160i		−	0.964371i	−0.876069	−	0.482186i	$-0.839843\pi$
$739$		−	26.2160i		−	0.964371i	0.876069	−	0.482186i	$-0.160157\pi$
$740$	15.0180	−	8.35801i	0.552074	−	0.307247i
$741$		0			0
$742$	0.366684			0.0134614
$743$	−7.62063			−0.279574			−0.139787	−	0.990182i	$-0.544642\pi$
$743$	−7.62063			−0.279574			−0.139787	+	0.990182i	$0.544642\pi$
$744$		0			0
$745$	9.19472	−	5.11715i	0.336868	−	0.187478i
$746$		−	0.597783i		−	0.0218864i
$747$		0			0
$748$			3.39961i			0.124302i
$749$	−1.26295			−0.0461472
$750$		0			0
$751$		−	28.5397i		−	1.04143i	−0.853731	−	0.520714i	$-0.825666\pi$
$751$		−	28.5397i		−	1.04143i	0.853731	−	0.520714i	$-0.174334\pi$
$752$	41.5331			1.51456
$753$		0			0
$754$	−8.24166	+	3.79612i	−0.300144	+	0.138247i
$755$	11.7072	−	6.51542i	0.426068	−	0.237120i
$756$		0			0
$757$	−19.8761			−0.722408			−0.361204	−	0.932487i	$-0.617634\pi$
$757$	−19.8761			−0.722408			−0.361204	+	0.932487i	$0.617634\pi$
$758$			6.16074i			0.223768i
$759$		0			0
$760$	8.92817	+	16.0425i	0.323859	+	0.581924i
$761$	−33.8887			−1.22847			−0.614233	−	0.789125i	$-0.710535\pi$
$761$	−33.8887			−1.22847			−0.614233	+	0.789125i	$0.710535\pi$
$762$		0			0
$763$		−	3.61574i		−	0.130899i
$764$			26.4983i			0.958674i
$765$		0			0
$766$		−	9.49947i		−	0.343230i
$767$		−	40.3740i		−	1.45782i
$768$		0			0
$769$			37.6604i			1.35807i	0.734107	+	0.679034i	$0.237601\pi$
$769$			37.6604i			1.35807i	−0.734107	+	0.679034i	$0.762399\pi$
$770$	0.0973744	−	0.0541919i	0.00350913	−	0.00195294i
$771$		0			0
$772$	−42.2354			−1.52008
$773$	−23.1391			−0.832255			−0.416127	−	0.909306i	$-0.636613\pi$
$773$	−23.1391			−0.832255			−0.416127	+	0.909306i	$0.636613\pi$
$774$		0			0
$775$	−42.9413	−	26.6303i	−1.54250	−	0.956589i
$776$	23.2332			0.834022
$777$		0			0
$778$		−	2.84236i		−	0.101904i
$779$	53.6482			1.92214
$780$		0			0
$781$			3.90126i			0.139598i
$782$		−	7.71298i		−	0.275816i
$783$		0			0
$784$	23.1349			0.826248
$785$	−22.0254	+	12.2579i	−0.786122	+	0.437502i
$786$		0			0
$787$		−	11.0598i		−	0.394238i	−0.980380	−	0.197119i	$-0.936841\pi$
$787$		−	11.0598i		−	0.394238i	0.980380	−	0.197119i	$-0.0631586\pi$
$788$			8.80370i			0.313619i
$789$		0			0
$790$	0.861999	+	1.54888i	0.0306685	+	0.0551066i
$791$			3.42237i			0.121686i
$792$		0			0
$793$	−24.3800			−0.865760
$794$			0.360308i			0.0127868i
$795$		0			0
$796$	−28.2778			−1.00228
$797$	11.5108			0.407733			0.203866	−	0.978999i	$-0.434649\pi$
$797$	11.5108			0.407733			0.203866	+	0.978999i	$0.434649\pi$
$798$		0			0
$799$	−41.3189			−1.46176
$800$	9.80017	−	15.8028i	0.346488	−	0.558712i
$801$		0			0
$802$	−6.08861			−0.214996
$803$			0.671001i			0.0236791i
$804$		0			0
$805$	3.72745	−	2.07444i	0.131375	−	0.0731145i
$806$	−17.0279			−0.599782
$807$		0			0
$808$			17.7887i			0.625803i
$809$			26.4897i			0.931330i	0.884961	+	0.465665i	$0.154185\pi$
$809$			26.4897i			0.931330i	−0.884961	+	0.465665i	$0.845815\pi$
$810$		0			0
$811$	−40.5686			−1.42455			−0.712277	−	0.701898i	$-0.752336\pi$
$811$	−40.5686			−1.42455			−0.712277	+	0.701898i	$0.752336\pi$
$812$	−2.53984	+	1.16985i	−0.0891309	+	0.0410538i
$813$		0			0
$814$			0.737705i			0.0258565i
$815$	22.0511	−	12.2721i	0.772417	−	0.429874i
$816$		0			0
$817$			37.7265i			1.31988i
$818$		−	9.83615i		−	0.343913i
$819$		0			0
$820$	−17.4480	−	31.3514i	−0.609312	−	1.09484i
$821$	34.6649			1.20981			0.604906	−	0.796297i	$-0.293211\pi$
$821$	34.6649			1.20981			0.604906	+	0.796297i	$0.293211\pi$
$822$		0			0
$823$	37.6231			1.31146			0.655729	−	0.754996i	$-0.272361\pi$
$823$	37.6231			1.31146			0.655729	+	0.754996i	$0.272361\pi$
$824$		−	13.9001i		−	0.484233i
$825$		0			0
$826$			0.737429i			0.0256585i
$827$	−45.0238			−1.56563			−0.782816	−	0.622253i	$-0.786217\pi$
$827$	−45.0238			−1.56563			−0.782816	+	0.622253i	$0.786217\pi$
$828$		0			0
$829$			12.5148i			0.434656i	0.976099	+	0.217328i	$0.0697341\pi$
$829$			12.5148i			0.434656i	−0.976099	+	0.217328i	$0.930266\pi$
$830$	3.77301	+	6.77951i	0.130963	+	0.235320i
$831$		0			0
$832$			27.3915i			0.949631i
$833$	−23.0156			−0.797443
$834$		0			0
$835$	9.90773	+	17.8026i	0.342871	+	0.616086i
$836$	6.45658			0.223305
$837$		0			0
$838$	2.45330			0.0847480
$839$		−	35.4253i		−	1.22302i	−0.791238	−	0.611508i	$-0.790563\pi$
$839$		−	35.4253i		−	1.22302i	0.791238	−	0.611508i	$-0.209437\pi$
$840$		0			0
$841$	−18.8487	+	22.0392i	−0.649956	+	0.759972i
$842$			7.05042i			0.242973i
$843$		0			0
$844$			30.0131i			1.03309i
$845$	24.1714	−	13.4521i	0.831521	−	0.462768i
$846$		0			0
$847$			2.94450i			0.101174i
$848$		−	13.3166i		−	0.457293i
$849$		0			0
$850$	−2.93007	+	4.72473i	−0.100501	+	0.162057i
$851$			28.2390i			0.968021i
$852$		0			0
$853$	−4.63924			−0.158844			−0.0794222	−	0.996841i	$-0.525308\pi$
$853$	−4.63924			−0.158844			−0.0794222	+	0.996841i	$0.525308\pi$
$854$	0.445300			0.0152379
$855$		0			0
$856$		−	5.97292i		−	0.204150i
$857$		−	52.9487i		−	1.80869i	−0.426800	−	0.904346i	$-0.640359\pi$
$857$		−	52.9487i		−	1.80869i	0.426800	−	0.904346i	$-0.359641\pi$
$858$		0			0
$859$		−	37.3538i		−	1.27450i	−0.770658	−	0.637248i	$-0.780073\pi$
$859$		−	37.3538i		−	1.27450i	0.770658	−	0.637248i	$-0.219927\pi$
$860$	22.0469	−	12.2698i	0.751795	−	0.418397i
$861$		0			0
$862$	3.55713			0.121156
$863$			10.2002i			0.347219i	0.984815	+	0.173609i	$0.0555431\pi$
$863$			10.2002i			0.347219i	−0.984815	+	0.173609i	$0.944457\pi$
$864$		0			0
$865$	−7.26482	−	13.0537i	−0.247011	−	0.443841i
$866$	−4.25784			−0.144687
$867$		0			0
$868$	−5.24750			−0.178112
$869$	1.28369			0.0435461
$870$		0			0
$871$	61.2818			2.07646
$872$	17.1000			0.579081
$873$		0			0
$874$	−14.6486			−0.495496
$875$	−3.07137	−	0.145275i	−0.103831	−	0.00491120i
$876$		0			0
$877$		−	8.25649i		−	0.278802i	−0.990236	−	0.139401i	$-0.955482\pi$
$877$		−	8.25649i		−	0.278802i	0.990236	−	0.139401i	$-0.0445177\pi$
$878$	−1.49989			−0.0506187
$879$		0			0
$880$	−1.96805	−	3.53627i	−0.0663428	−	0.119208i
$881$			4.94367i			0.166556i	0.996526	+	0.0832782i	$0.0265390\pi$
$881$			4.94367i			0.166556i	−0.996526	+	0.0832782i	$0.973461\pi$
$882$		0			0
$883$		−	35.6378i		−	1.19931i	−0.800259	−	0.599654i	$-0.795305\pi$
$883$		−	35.6378i		−	1.19931i	0.800259	−	0.599654i	$-0.204695\pi$
$884$		−	31.6108i		−	1.06319i
$885$		0			0
$886$	4.28352			0.143908
$887$	−2.14284			−0.0719497			−0.0359748	−	0.999353i	$-0.511454\pi$
$887$	−2.14284			−0.0719497			−0.0359748	+	0.999353i	$0.511454\pi$
$888$		0			0
$889$		−	2.85597i		−	0.0957862i
$890$	0.911679	+	1.63815i	0.0305596	+	0.0549108i
$891$		0			0
$892$		−	14.1968i		−	0.475343i
$893$			78.4733i			2.62601i
$894$		0			0
$895$	34.4129	−	19.1519i	1.15030	−	0.640176i
$896$		−	2.54588i		−	0.0850520i
$897$		0			0
$898$			7.41929i			0.247585i
$899$	−49.4295	+	22.7673i	−1.64857	+	0.759332i
$900$		0			0
$901$			13.2479i			0.441351i
$902$	1.54002			0.0512771
$903$		0			0
$904$	−16.1855			−0.538323
$905$	−1.56168	+	0.869125i	−0.0519120	+	0.0288907i
$906$		0			0
$907$	−27.7834			−0.922533			−0.461266	−	0.887262i	$-0.652605\pi$
$907$	−27.7834			−0.922533			−0.461266	+	0.887262i	$0.652605\pi$
$908$		−	28.2532i		−	0.937616i
$909$		0			0
$910$	−0.905423	+	0.503897i	−0.0300145	+	0.0167040i
$911$		−	7.92701i		−	0.262634i	−0.991340	−	0.131317i	$-0.958080\pi$
$911$		−	7.92701i		−	0.262634i	0.991340	−	0.131317i	$-0.0419205\pi$
$912$		0			0
$913$	5.61877			0.185954
$914$		−	4.08916i		−	0.135257i
$915$		0			0
$916$			23.2361i			0.767743i
$917$	1.04723			0.0345825
$918$		0			0
$919$	24.5480			0.809764			0.404882	−	0.914369i	$-0.367313\pi$
$919$	24.5480			0.809764			0.404882	+	0.914369i	$0.367313\pi$
$920$	9.81072	+	17.6283i	0.323450	+	0.581189i
$921$		0			0
$922$			8.74760i			0.288087i
$923$		−	36.2753i		−	1.19402i
$924$		0			0
$925$	10.7277	−	17.2983i	0.352723	−	0.568765i
$926$			7.87864i			0.258908i
$927$		0			0
$928$	−8.37855	−	18.1905i	−0.275040	−	0.597132i
$929$	−2.45572			−0.0805695			−0.0402847	−	0.999188i	$-0.512827\pi$
$929$	−2.45572			−0.0805695			−0.0402847	+	0.999188i	$0.512827\pi$
$930$		0			0
$931$			43.7115i			1.43259i
$932$		−	20.2971i		−	0.664853i
$933$		0			0
$934$	4.25044			0.139079
$935$	1.95789	+	3.51803i	0.0640300	+	0.115052i
$936$		0			0
$937$			39.7814i			1.29960i	0.760104	+	0.649801i	$0.225148\pi$
$937$			39.7814i			1.29960i	−0.760104	+	0.649801i	$0.774852\pi$
$938$	−1.11931			−0.0365467
$939$		0			0
$940$	45.8590	−	25.5220i	1.49576	−	0.832435i
$941$	−33.7621			−1.10061			−0.550306	−	0.834963i	$-0.685489\pi$
$941$	−33.7621			−1.10061			−0.550306	+	0.834963i	$0.685489\pi$
$942$		0			0
$943$	58.9513			1.91972
$944$	26.7806			0.871636
$945$		0			0
$946$			1.08297i			0.0352105i
$947$	1.32780			0.0431476			0.0215738	−	0.999767i	$-0.493132\pi$
$947$	1.32780			0.0431476			0.0215738	+	0.999767i	$0.493132\pi$
$948$		0			0
$949$		−	6.23922i		−	0.202534i
$950$	8.97326	+	5.56482i	0.291131	+	0.180547i
$951$		0			0
$952$			1.18896i			0.0385344i
$953$		−	21.2315i		−	0.687755i	−0.939014	−	0.343878i	$-0.888259\pi$
$953$		−	21.2315i		−	0.687755i	0.939014	−	0.343878i	$-0.111741\pi$
$954$		0			0
$955$	15.2608	+	27.4213i	0.493829	+	0.887333i
$956$	−11.4132			−0.369128
$957$		0			0
$958$		−	4.77922i		−	0.154409i
$959$		−	2.49158i		−	0.0804574i
$960$		0			0
$961$	−71.1251			−2.29436
$962$		−	6.85945i		−	0.221158i
$963$		0			0
$964$	−45.6915			−1.47162
$965$	−43.7066	+	24.3241i	−1.40697	+	0.783021i
$966$		0			0
$967$	−8.54151			−0.274677			−0.137338	−	0.990524i	$-0.543855\pi$
$967$	−8.54151			−0.274677			−0.137338	+	0.990524i	$0.543855\pi$
$968$	−13.9255			−0.447584
$969$		0			0
$970$	11.6753	−	6.49764i	0.374870	−	0.208627i
$971$			18.0454i			0.579104i	0.957162	+	0.289552i	$0.0935063\pi$
$971$			18.0454i			0.579104i	−0.957162	+	0.289552i	$0.906494\pi$
$972$		0			0
$973$			0.390123i			0.0125068i
$974$		−	11.5156i		−	0.368983i
$975$		0			0
$976$		−	16.1716i		−	0.517641i
$977$		−	15.2948i		−	0.489324i	−0.969608	−	0.244662i	$-0.921323\pi$
$977$		−	15.2948i		−	0.489324i	0.969608	−	0.244662i	$-0.0786770\pi$
$978$		0			0
$979$	1.35767			0.0433914
$980$	25.5446	−	14.2163i	0.815991	−	0.454124i
$981$		0			0
$982$			6.88425i			0.219685i
$983$	38.2643			1.22044			0.610221	−	0.792231i	$-0.291081\pi$
$983$	38.2643			1.22044			0.610221	+	0.792231i	$0.291081\pi$
$984$		0			0
$985$	5.07021	+	9.11037i	0.161550	+	0.290281i
$986$	2.50503	+	5.43861i	0.0797764	+	0.173201i
$987$		0			0
$988$	−60.0356			−1.90999
$989$			41.4557i			1.31822i
$990$		0			0
$991$	22.5768			0.717174			0.358587	−	0.933496i	$-0.383259\pi$
$991$	22.5768			0.717174			0.358587	+	0.933496i	$0.383259\pi$
$992$		−	37.5829i		−	1.19326i
$993$		0			0
$994$			0.662567i			0.0210153i
$995$	−29.2629	+	16.2857i	−0.927695	+	0.516292i
$996$		0			0
$997$	−28.4370			−0.900608			−0.450304	−	0.892875i	$-0.648684\pi$
$997$	−28.4370			−0.900608			−0.450304	+	0.892875i	$0.648684\pi$
$998$	−6.54222			−0.207090
$999$		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	1305.2.f.l.289.8		12
3.2	odd	2		435.2.f.f.289.5	yes	12
5.4	even	2		1305.2.f.k.289.5		12
15.2	even	4		2175.2.d.j.376.13		24
15.8	even	4		2175.2.d.j.376.12		24
15.14	odd	2		435.2.f.e.289.8	yes	12
29.28	even	2		1305.2.f.k.289.6		12
87.86	odd	2		435.2.f.e.289.7	✓	12
145.144	even	2	inner	1305.2.f.l.289.7		12
435.173	even	4		2175.2.d.j.376.11		24
435.347	even	4		2175.2.d.j.376.14		24
435.434	odd	2		435.2.f.f.289.6	yes	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
435.2.f.e.289.7	✓	12	87.86	odd	2
435.2.f.e.289.8	yes	12	15.14	odd	2
435.2.f.f.289.5	yes	12	3.2	odd	2
435.2.f.f.289.6	yes	12	435.434	odd	2
1305.2.f.k.289.5		12	5.4	even	2
1305.2.f.k.289.6		12	29.28	even	2
1305.2.f.l.289.7		12	145.144	even	2	inner
1305.2.f.l.289.8		12	1.1	even	1	trivial
2175.2.d.j.376.11		24	435.173	even	4
2175.2.d.j.376.12		24	15.8	even	4
2175.2.d.j.376.13		24	15.2	even	4
2175.2.d.j.376.14		24	435.347	even	4

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Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
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Newspace parameters

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$q$ -expansion

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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	1305.2.f.l.289.8

Level	$1305$
Weight	$2$
Character	1305.289
Analytic conductor	$10.420$
Analytic rank	$0$
Dimension	$12$
Inner twists	$2$