LMFDB - Embedded newform 425.2.c.a.424.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [425,2,Mod(424,425)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage: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")

magma://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:traces = [6,0,0,-2,0,0,-4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$3.39364208590$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.350464.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$x^{6} - 2x^{5} + 2x^{4} + 2x^{3} + 4x^{2} - 4x + 2$ x^6 - 2x^5 + 2x^4 + 2x^3 + 4x^2 - 4*x + 2
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$2$
Twist minimal:	no (minimal twist has level 85)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			424.4
Root			$1.45161 - 1.45161i$ of defining polynomial
Character	$\chi$	$=$	425.424
Dual form			425.2.c.a.424.3

$q$ -expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+0.311108i q^{2} -2.21432 q^{3} +1.90321 q^{4} -0.688892i q^{6} +1.59210 q^{7} +1.21432i q^{8} +1.90321 q^{9} -1.31111i q^{11} -4.21432 q^{12} +3.52543i q^{13} +0.495316i q^{14} +3.42864 q^{16} +(-0.214320 - 4.11753i) q^{17} +0.592104i q^{18} +4.42864 q^{19} -3.52543 q^{21} +0.407896 q^{22} +4.96989 q^{23} -2.68889i q^{24} -1.09679 q^{26} +2.42864 q^{27} +3.03011 q^{28} +8.42864i q^{29} +7.73975i q^{31} +3.49532i q^{32} +2.90321i q^{33} +(1.28100 - 0.0666765i) q^{34} +3.62222 q^{36} +7.05086 q^{37} +1.37778i q^{38} -7.80642i q^{39} -3.67307i q^{41} -1.09679i q^{42} +2.47457i q^{43} -2.49532i q^{44} +1.54617i q^{46} -3.33185i q^{47} -7.59210 q^{48} -4.46520 q^{49} +(0.474572 + 9.11753i) q^{51} +6.70964i q^{52} -9.18421i q^{53} +0.755569i q^{54} +1.93332i q^{56} -9.80642 q^{57} -2.62222 q^{58} -1.37778 q^{59} -15.4193i q^{61} -2.40790 q^{62} +3.03011 q^{63} +5.76986 q^{64} -0.903212 q^{66} +9.13828i q^{67} +(-0.407896 - 7.83654i) q^{68} -11.0049 q^{69} -10.5970i q^{71} +2.31111i q^{72} -5.57136 q^{73} +2.19358i q^{74} +8.42864 q^{76} -2.08742i q^{77} +2.42864 q^{78} +7.87310i q^{79} -11.0874 q^{81} +1.14272 q^{82} +7.19850i q^{83} -6.70964 q^{84} -0.769859 q^{86} -18.6637i q^{87} +1.59210 q^{88} -11.6271 q^{89} +5.61285i q^{91} +9.45875 q^{92} -17.1383i q^{93} +1.03657 q^{94} -7.73975i q^{96} -15.4795 q^{97} -1.38916i q^{98} -2.49532i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$6 q - 2 q^{4} - 4 q^{7} - 2 q^{9} - 12 q^{12} - 6 q^{16} + 12 q^{17} - 8 q^{21} + 16 q^{22} + 16 q^{23} - 20 q^{26} - 12 q^{27} + 32 q^{28} - 6 q^{34} + 22 q^{36} + 16 q^{37} - 32 q^{48} + 14 q^{49} + 16 q^{51}+ \cdots - 40 q^{97}+O(q^{100})$ 6 * q - 2 * q^4 - 4 * q^7 - 2 * q^9 - 12 * q^12 - 6 * q^16 + 12 * q^17 - 8 * q^21 + 16 * q^22 + 16 * q^23 - 20 * q^26 - 12 * q^27 + 32 * q^28 - 6 * q^34 + 22 * q^36 + 16 * q^37 - 32 * q^48 + 14 * q^49 + 16 * q^51 - 32 * q^57 - 16 * q^58 - 8 * q^59 - 28 * q^62 + 32 * q^63 + 22 * q^64 + 8 * q^66 - 16 * q^68 - 60 * q^73 + 24 * q^76 - 12 * q^78 - 26 * q^81 + 60 * q^82 + 8 * q^86 - 4 * q^88 - 4 * q^89 + 44 * q^92 - 8 * q^94 - 40 * q^97

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	$0.0350829\pi$
$2$			0.311108i			0.219986i	−0.993932	+	0.109993i	$0.964917\pi$
$3$	−2.21432			−1.27844			−0.639219	−	0.769025i	$-0.720742\pi$
$3$	−2.21432			−1.27844			−0.639219	+	0.769025i	$0.720742\pi$
$4$	1.90321			0.951606
$5$		0			0
$5$		0			0
$6$		−	0.688892i		−	0.281239i
$7$	1.59210			0.601759			0.300879	−	0.953662i	$-0.402720\pi$
$7$	1.59210			0.601759			0.300879	+	0.953662i	$0.402720\pi$
$8$			1.21432i			0.429327i
$9$	1.90321			0.634404
$10$		0			0
$11$		−	1.31111i		−	0.395314i	−0.980271	−	0.197657i	$-0.936667\pi$
$11$		−	1.31111i		−	0.395314i	0.980271	−	0.197657i	$-0.0633332\pi$
$12$	−4.21432			−1.21657
$13$			3.52543i			0.977778i	0.872346	+	0.488889i	$0.162598\pi$
$13$			3.52543i			0.977778i	−0.872346	+	0.488889i	$0.837402\pi$
$14$			0.495316i			0.132379i
$15$		0			0
$16$	3.42864			0.857160
$17$	−0.214320	−	4.11753i	−0.0519802	−	0.998648i
$17$	−0.214320	−	4.11753i	−0.0519802	−	0.998648i
$18$			0.592104i			0.139560i
$19$	4.42864			1.01600			0.508000	−	0.861357i	$-0.330385\pi$
$19$	4.42864			1.01600			0.508000	+	0.861357i	$0.330385\pi$
$20$		0			0
$21$	−3.52543			−0.769311
$22$	0.407896			0.0869637
$23$	4.96989			1.03629			0.518147	−	0.855292i	$-0.326622\pi$
$23$	4.96989			1.03629			0.518147	+	0.855292i	$0.326622\pi$
$24$		−	2.68889i		−	0.548868i
$25$		0			0
$26$	−1.09679			−0.215098
$27$	2.42864			0.467392
$28$	3.03011			0.572637
$29$			8.42864i			1.56516i	0.622551	+	0.782580i	$0.286096\pi$
$29$			8.42864i			1.56516i	−0.622551	+	0.782580i	$0.713904\pi$
$30$		0			0
$31$			7.73975i			1.39010i	0.718962	+	0.695050i	$0.244618\pi$
$31$			7.73975i			1.39010i	−0.718962	+	0.695050i	$0.755382\pi$
$32$			3.49532i			0.617890i
$33$			2.90321i			0.505384i
$34$	1.28100	−	0.0666765i	0.219689	−	0.0114349i
$35$		0			0
$36$	3.62222			0.603703
$37$	7.05086			1.15915			0.579577	−	0.814918i	$-0.303218\pi$
$37$	7.05086			1.15915			0.579577	+	0.814918i	$0.303218\pi$
$38$			1.37778i			0.223506i
$39$		−	7.80642i		−	1.25003i
$40$		0			0
$41$		−	3.67307i		−	0.573637i	−0.957985	−	0.286819i	$-0.907402\pi$
$41$		−	3.67307i		−	0.573637i	0.957985	−	0.286819i	$-0.0925977\pi$
$42$		−	1.09679i		−	0.169238i
$43$			2.47457i			0.377369i	0.982038	+	0.188684i	$0.0604223\pi$
$43$			2.47457i			0.377369i	−0.982038	+	0.188684i	$0.939578\pi$
$44$		−	2.49532i		−	0.376183i
$45$		0			0
$46$			1.54617i			0.227970i
$47$		−	3.33185i		−	0.486000i	−0.970026	−	0.243000i	$-0.921868\pi$
$47$		−	3.33185i		−	0.486000i	0.970026	−	0.243000i	$-0.0781316\pi$
$48$	−7.59210			−1.09583
$49$	−4.46520			−0.637886
$50$		0			0
$51$	0.474572	+	9.11753i	0.0664534	+	1.27671i
$52$			6.70964i			0.930459i
$53$		−	9.18421i		−	1.26155i	−0.775967	−	0.630774i	$-0.782737\pi$
$53$		−	9.18421i		−	1.26155i	0.775967	−	0.630774i	$-0.217263\pi$
$54$			0.755569i			0.102820i
$55$		0			0
$56$			1.93332i			0.258351i
$57$	−9.80642			−1.29889
$58$	−2.62222			−0.344314
$59$	−1.37778			−0.179372			−0.0896861	−	0.995970i	$-0.528586\pi$
$59$	−1.37778			−0.179372			−0.0896861	+	0.995970i	$0.528586\pi$
$60$		0			0
$61$		−	15.4193i		−	1.97424i	−0.159996	−	0.987118i	$-0.551148\pi$
$61$		−	15.4193i		−	1.97424i	0.159996	−	0.987118i	$-0.448852\pi$
$62$	−2.40790			−0.305803
$63$	3.03011			0.381758
$64$	5.76986			0.721232
$65$		0			0
$66$	−0.903212			−0.111178
$67$			9.13828i			1.11642i	0.829700	+	0.558209i	$0.188511\pi$
$67$			9.13828i			1.11642i	−0.829700	+	0.558209i	$0.811489\pi$
$68$	−0.407896	−	7.83654i	−0.0494646	−	0.950320i
$69$	−11.0049			−1.32484
$70$		0			0
$71$		−	10.5970i		−	1.25764i	−0.777553	−	0.628818i	$-0.783539\pi$
$71$		−	10.5970i		−	1.25764i	0.777553	−	0.628818i	$-0.216461\pi$
$72$			2.31111i			0.272367i
$73$	−5.57136			−0.652078			−0.326039	−	0.945356i	$-0.605714\pi$
$73$	−5.57136			−0.652078			−0.326039	+	0.945356i	$0.605714\pi$
$74$			2.19358i			0.254998i
$75$		0			0
$76$	8.42864			0.966831
$77$		−	2.08742i		−	0.237884i
$78$	2.42864			0.274989
$79$			7.87310i			0.885793i	0.896573	+	0.442897i	$0.146049\pi$
$79$			7.87310i			0.885793i	−0.896573	+	0.442897i	$0.853951\pi$
$80$		0			0
$81$	−11.0874			−1.23194
$82$	1.14272			0.126192
$83$			7.19850i			0.790138i	0.918651	+	0.395069i	$0.129279\pi$
$83$			7.19850i			0.790138i	−0.918651	+	0.395069i	$0.870721\pi$
$84$	−6.70964			−0.732081
$85$		0			0
$86$	−0.769859			−0.0830160
$87$		−	18.6637i		−	2.00096i
$88$	1.59210			0.169719
$89$	−11.6271			−1.23247			−0.616237	−	0.787561i	$-0.711344\pi$
$89$	−11.6271			−1.23247			−0.616237	+	0.787561i	$0.711344\pi$
$90$		0			0
$91$			5.61285i			0.588386i
$92$	9.45875			0.986143
$93$		−	17.1383i		−	1.77716i
$94$	1.03657			0.106914
$95$		0			0
$96$		−	7.73975i		−	0.789935i
$97$	−15.4795			−1.57170			−0.785852	−	0.618414i	$-0.787775\pi$
$97$	−15.4795			−1.57170			−0.785852	+	0.618414i	$0.787775\pi$
$98$		−	1.38916i		−	0.140326i
$99$		−	2.49532i		−	0.250789i
$100$		0			0
$101$	−14.3827			−1.43113			−0.715566	−	0.698545i	$-0.753831\pi$
$101$	−14.3827			−1.43113			−0.715566	+	0.698545i	$0.753831\pi$
$102$	−2.83654	+	0.147643i	−0.280859	+	0.0146189i
$103$		−	9.39207i		−	0.925429i	−0.886507	−	0.462714i	$-0.846876\pi$
$103$		−	9.39207i		−	0.925429i	0.886507	−	0.462714i	$-0.153124\pi$
$104$	−4.28100			−0.419786
$105$		0			0
$106$	2.85728			0.277523
$107$	6.77631			0.655091			0.327545	−	0.944835i	$-0.393779\pi$
$107$	6.77631			0.655091			0.327545	+	0.944835i	$0.393779\pi$
$108$	4.62222			0.444773
$109$		−	2.85728i		−	0.273678i	−0.990593	−	0.136839i	$-0.956306\pi$
$109$		−	2.85728i		−	0.273678i	0.990593	−	0.136839i	$-0.0436942\pi$
$110$		0			0
$111$	−15.6128			−1.48191
$112$	5.45875			0.515803
$113$	−5.86665			−0.551888			−0.275944	−	0.961174i	$-0.588990\pi$
$113$	−5.86665			−0.551888			−0.275944	+	0.961174i	$0.588990\pi$
$114$		−	3.05086i		−	0.285739i
$115$		0			0
$116$			16.0415i			1.48941i
$117$			6.70964i			0.620306i
$118$		−	0.428639i		−	0.0394595i
$119$	−0.341219	−	6.55554i	−0.0312795	−	0.600945i
$120$		0			0
$121$	9.28100			0.843727
$122$	4.79706			0.434305
$123$			8.13335i			0.733360i
$124$			14.7304i			1.32283i
$125$		0			0
$126$			0.942691i			0.0839816i
$127$		−	0.280996i		−	0.0249344i	−0.999922	−	0.0124672i	$-0.996031\pi$
$127$		−	0.280996i		−	0.0249344i	0.999922	−	0.0124672i	$-0.00396853\pi$
$128$			8.78568i			0.776552i
$129$		−	5.47949i		−	0.482443i
$130$		0			0
$131$			13.4128i			1.17188i	0.810353	+	0.585942i	$0.199275\pi$
$131$			13.4128i			1.17188i	−0.810353	+	0.585942i	$0.800725\pi$
$132$			5.52543i			0.480927i
$133$	7.05086			0.611387
$134$	−2.84299			−0.245597
$135$		0			0
$136$	5.00000	−	0.260253i	0.428746	−	0.0223165i
$137$		−	10.2810i		−	0.878365i	−0.898398	−	0.439182i	$-0.855268\pi$
$137$		−	10.2810i		−	0.878365i	0.898398	−	0.439182i	$-0.144732\pi$
$138$		−	3.42372i		−	0.291446i
$139$		−	18.3017i		−	1.55233i	−0.630528	−	0.776167i	$-0.717162\pi$
$139$		−	18.3017i		−	1.55233i	0.630528	−	0.776167i	$-0.282838\pi$
$140$		0			0
$141$			7.37778i			0.621322i
$142$	3.29682			0.276663
$143$	4.62222			0.386529
$144$	6.52543			0.543786
$145$		0			0
$146$		−	1.73329i		−	0.143448i
$147$	9.88739			0.815498
$148$	13.4193			1.10306
$149$	4.91750			0.402857			0.201429	−	0.979503i	$-0.435442\pi$
$149$	4.91750			0.402857			0.201429	+	0.979503i	$0.435442\pi$
$150$		0			0
$151$	12.7239			1.03546			0.517729	−	0.855545i	$-0.326777\pi$
$151$	12.7239			1.03546			0.517729	+	0.855545i	$0.326777\pi$
$152$			5.37778i			0.436196i
$153$	−0.407896	−	7.83654i	−0.0329764	−	0.633546i
$154$	0.649413			0.0523312
$155$		0			0
$156$		−	14.8573i		−	1.18953i
$157$			9.18421i			0.732980i	0.930422	+	0.366490i	$0.119441\pi$
$157$			9.18421i			0.732980i	−0.930422	+	0.366490i	$0.880559\pi$
$158$	−2.44938			−0.194862
$159$			20.3368i			1.61281i
$160$		0			0
$161$	7.91258			0.623599
$162$		−	3.44938i		−	0.271009i
$163$	−5.07160			−0.397238			−0.198619	−	0.980077i	$-0.563646\pi$
$163$	−5.07160			−0.397238			−0.198619	+	0.980077i	$0.563646\pi$
$164$		−	6.99063i		−	0.545877i
$165$		0			0
$166$	−2.23951			−0.173820
$167$	−19.8272			−1.53427			−0.767136	−	0.641484i	$-0.778319\pi$
$167$	−19.8272			−1.53427			−0.767136	+	0.641484i	$0.778319\pi$
$168$		−	4.28100i		−	0.330286i
$169$	0.571361			0.0439508
$170$		0			0
$171$	8.42864			0.644554
$172$			4.70964i			0.359106i
$173$	2.48886			0.189225			0.0946124	−	0.995514i	$-0.469839\pi$
$173$	2.48886			0.189225			0.0946124	+	0.995514i	$0.469839\pi$
$174$	5.80642			0.440184
$175$		0			0
$176$		−	4.49532i		−	0.338847i
$177$	3.05086			0.229316
$178$		−	3.61729i		−	0.271128i
$179$	11.6128			0.867985			0.433992	−	0.900916i	$-0.357104\pi$
$179$	11.6128			0.867985			0.433992	+	0.900916i	$0.357104\pi$
$180$		0			0
$181$			3.86665i			0.287406i	0.989621	+	0.143703i	$0.0459009\pi$
$181$			3.86665i			0.287406i	−0.989621	+	0.143703i	$0.954099\pi$
$182$	−1.74620			−0.129437
$183$			34.1432i			2.52394i
$184$			6.03503i			0.444909i
$185$		0			0
$186$	5.33185			0.390950
$187$	−5.39853	+	0.280996i	−0.394779	+	0.0205485i
$188$		−	6.34122i		−	0.462481i
$189$	3.86665			0.281257
$190$		0			0
$191$	−14.9175			−1.07939			−0.539696	−	0.841860i	$-0.681461\pi$
$191$	−14.9175			−1.07939			−0.539696	+	0.841860i	$0.681461\pi$
$192$	−12.7763			−0.922051
$193$	−17.7462			−1.27740			−0.638700	−	0.769456i	$-0.720527\pi$
$193$	−17.7462			−1.27740			−0.638700	+	0.769456i	$0.720527\pi$
$194$		−	4.81579i		−	0.345754i
$195$		0			0
$196$	−8.49823			−0.607016
$197$	9.41927			0.671095			0.335548	−	0.942023i	$-0.391079\pi$
$197$	9.41927			0.671095			0.335548	+	0.942023i	$0.391079\pi$
$198$	0.776312			0.0551701
$199$			7.21924i			0.511758i	0.966709	+	0.255879i	$0.0823649\pi$
$199$			7.21924i			0.511758i	−0.966709	+	0.255879i	$0.917635\pi$
$200$		0			0
$201$		−	20.2351i		−	1.42727i
$202$		−	4.47457i		−	0.314830i
$203$			13.4193i			0.941848i
$204$	0.903212	+	17.3526i	0.0632375	+	1.21492i
$205$		0			0
$206$	2.92195			0.203582
$207$	9.45875			0.657429
$208$			12.0874i			0.838112i
$209$		−	5.80642i		−	0.401639i
$210$		0			0
$211$		−	14.8825i		−	1.02455i	−0.858821	−	0.512276i	$-0.828803\pi$
$211$		−	14.8825i		−	1.02455i	0.858821	−	0.512276i	$-0.171197\pi$
$212$		−	17.4795i		−	1.20050i
$213$			23.4652i			1.60781i
$214$			2.10816i			0.144111i
$215$		0			0
$216$			2.94914i			0.200664i
$217$			12.3225i			0.836505i
$218$	0.888922			0.0602054
$219$	12.3368			0.833642
$220$		0			0
$221$	14.5161	−	0.755569i	0.976456	−	0.0508251i
$222$		−	4.85728i		−	0.325999i
$223$		−	23.4652i		−	1.57135i	−0.618642	−	0.785673i	$-0.712317\pi$
$223$		−	23.4652i		−	1.57135i	0.618642	−	0.785673i	$-0.287683\pi$
$224$			5.56491i			0.371821i
$225$		0			0
$226$		−	1.82516i		−	0.121408i
$227$	−4.34767			−0.288565			−0.144283	−	0.989537i	$-0.546087\pi$
$227$	−4.34767			−0.288565			−0.144283	+	0.989537i	$0.546087\pi$
$228$	−18.6637			−1.23603
$229$	−6.47457			−0.427852			−0.213926	−	0.976850i	$-0.568625\pi$
$229$	−6.47457			−0.427852			−0.213926	+	0.976850i	$0.568625\pi$
$230$		0			0
$231$			4.62222i			0.304119i
$232$	−10.2351			−0.671965
$233$	−18.5303			−1.21396			−0.606982	−	0.794716i	$-0.707620\pi$
$233$	−18.5303			−1.21396			−0.606982	+	0.794716i	$0.707620\pi$
$234$	−2.08742			−0.136459
$235$		0			0
$236$	−2.62222			−0.170692
$237$		−	17.4336i		−	1.13243i
$238$	2.03948	−	0.106156i	0.132200	−	0.00688107i
$239$	21.4193			1.38550			0.692749	−	0.721179i	$-0.256399\pi$
$239$	21.4193			1.38550			0.692749	+	0.721179i	$0.256399\pi$
$240$		0			0
$241$			12.2351i			0.788130i	0.919083	+	0.394065i	$0.128931\pi$
$241$			12.2351i			0.788130i	−0.919083	+	0.394065i	$0.871069\pi$
$242$			2.88739i			0.185608i
$243$	17.2652			1.10756
$244$		−	29.3461i		−	1.87869i
$245$		0			0
$246$	−2.53035			−0.161329
$247$			15.6128i			0.993422i
$248$	−9.39853			−0.596807
$249$		−	15.9398i		−	1.01014i
$250$		0			0
$251$	−4.52051			−0.285332			−0.142666	−	0.989771i	$-0.545567\pi$
$251$	−4.52051			−0.285332			−0.142666	+	0.989771i	$0.545567\pi$
$252$	5.76694			0.363283
$253$		−	6.51606i		−	0.409661i
$254$	0.0874201			0.00548523
$255$		0			0
$256$	8.80642			0.550401
$257$			6.18913i			0.386067i	0.981192	+	0.193034i	$0.0618327\pi$
$257$			6.18913i			0.386067i	−0.981192	+	0.193034i	$0.938167\pi$
$258$	1.70471			0.106131
$259$	11.2257			0.697531
$260$		0			0
$261$			16.0415i			0.992943i
$262$	−4.17283			−0.257798
$263$			8.18913i			0.504963i	0.967602	+	0.252482i	$0.0812468\pi$
$263$			8.18913i			0.504963i	−0.967602	+	0.252482i	$0.918753\pi$
$264$	−3.52543			−0.216975
$265$		0			0
$266$			2.19358i			0.134497i
$267$	25.7462			1.57564
$268$			17.3921i			1.06239i
$269$			3.89829i			0.237683i	0.992913	+	0.118841i	$0.0379180\pi$
$269$			3.89829i			0.237683i	−0.992913	+	0.118841i	$0.962082\pi$
$270$		0			0
$271$	8.85728			0.538041			0.269021	−	0.963134i	$-0.413300\pi$
$271$	8.85728			0.538041			0.269021	+	0.963134i	$0.413300\pi$
$272$	−0.734825	−	14.1175i	−0.0445553	−	0.856001i
$273$		−	12.4286i		−	0.752215i
$274$	3.19850			0.193228
$275$		0			0
$276$	−20.9447			−1.26072
$277$	1.14272			0.0686595			0.0343297	−	0.999411i	$-0.489070\pi$
$277$	1.14272			0.0686595			0.0343297	+	0.999411i	$0.489070\pi$
$278$	5.69381			0.341492
$279$			14.7304i			0.881885i
$280$		0			0
$281$	−14.6953			−0.876651			−0.438325	−	0.898816i	$-0.644428\pi$
$281$	−14.6953			−0.876651			−0.438325	+	0.898816i	$0.644428\pi$
$282$	−2.29529			−0.136682
$283$	1.97926			0.117655			0.0588273	−	0.998268i	$-0.481264\pi$
$283$	1.97926			0.117655			0.0588273	+	0.998268i	$0.481264\pi$
$284$		−	20.1684i		−	1.19677i
$285$		0			0
$286$			1.43801i			0.0850312i
$287$		−	5.84791i		−	0.345191i
$288$			6.65233i			0.391992i
$289$	−16.9081	+	1.76494i	−0.994596	+	0.103820i
$290$		0			0
$291$	34.2766			2.00933
$292$	−10.6035			−0.620522
$293$			2.94914i			0.172291i	0.996283	+	0.0861454i	$0.0274550\pi$
$293$			2.94914i			0.172291i	−0.996283	+	0.0861454i	$0.972545\pi$
$294$			3.07604i			0.179399i
$295$		0			0
$296$			8.56199i			0.497656i
$297$		−	3.18421i		−	0.184767i
$298$			1.52987i			0.0886232i
$299$			17.5210i			1.01326i
$300$		0			0
$301$			3.93978i			0.227085i
$302$			3.95851i			0.227787i
$303$	31.8479			1.82961
$304$	15.1842			0.870874
$305$		0			0
$306$	2.43801	−	0.126900i	0.139372	−	0.00725437i
$307$			5.68736i			0.324595i	0.986742	+	0.162297i	$0.0518904\pi$
$307$			5.68736i			0.324595i	−0.986742	+	0.162297i	$0.948110\pi$
$308$		−	3.97280i		−	0.226371i
$309$			20.7971i			1.18310i
$310$		0			0
$311$		−	23.7210i		−	1.34510i	−0.740054	−	0.672548i	$-0.765200\pi$
$311$		−	23.7210i		−	1.34510i	0.740054	−	0.672548i	$-0.234800\pi$
$312$	9.47949			0.536671
$313$	30.8988			1.74650			0.873251	−	0.487271i	$-0.162008\pi$
$313$	30.8988			1.74650			0.873251	+	0.487271i	$0.162008\pi$
$314$	−2.85728			−0.161246
$315$		0			0
$316$			14.9842i			0.842926i
$317$	13.7047			0.769733			0.384867	−	0.922972i	$-0.374247\pi$
$317$	13.7047			0.769733			0.384867	+	0.922972i	$0.374247\pi$
$318$	−6.32693			−0.354797
$319$	11.0509			0.618729
$320$		0			0
$321$	−15.0049			−0.837493
$322$			2.46167i			0.137183i
$323$	−0.949145	−	18.2351i	−0.0528118	−	1.01463i
$324$	−21.1017			−1.17232
$325$		0			0
$326$		−	1.57781i		−	0.0873870i
$327$			6.32693i			0.349880i
$328$	4.46028			0.246278
$329$		−	5.30465i		−	0.292455i
$330$		0			0
$331$	24.8988			1.36856			0.684280	−	0.729219i	$-0.260117\pi$
$331$	24.8988			1.36856			0.684280	+	0.729219i	$0.260117\pi$
$332$			13.7003i			0.751900i
$333$	13.4193			0.735372
$334$		−	6.16839i		−	0.337519i
$335$		0			0
$336$	−12.0874			−0.659423
$337$	5.28592			0.287942			0.143971	−	0.989582i	$-0.454013\pi$
$337$	5.28592			0.287942			0.143971	+	0.989582i	$0.454013\pi$
$338$			0.177755i			0.00966858i
$339$	12.9906			0.705554
$340$		0			0
$341$	10.1476			0.549526
$342$			2.62222i			0.141793i
$343$	−18.2538			−0.985613
$344$	−3.00492			−0.162015
$345$		0			0
$346$			0.774305i			0.0416269i
$347$	−31.6019			−1.69648			−0.848241	−	0.529611i	$-0.822338\pi$
$347$	−31.6019			−1.69648			−0.848241	+	0.529611i	$0.822338\pi$
$348$		−	35.5210i		−	1.90412i
$349$	−26.5116			−1.41913			−0.709567	−	0.704638i	$-0.751109\pi$
$349$	−26.5116			−1.41913			−0.709567	+	0.704638i	$0.751109\pi$
$350$		0			0
$351$			8.56199i			0.457005i
$352$	4.58274			0.244261
$353$		−	3.71456i		−	0.197706i	−0.995102	−	0.0988530i	$-0.968483\pi$
$353$		−	3.71456i		−	0.197706i	0.995102	−	0.0988530i	$-0.0315174\pi$
$354$			0.949145i			0.0504465i
$355$		0			0
$356$	−22.1289			−1.17283
$357$	0.755569	+	14.5161i	0.0399889	+	0.768271i
$358$			3.61285i			0.190945i
$359$	−10.6637			−0.562809			−0.281404	−	0.959589i	$-0.590800\pi$
$359$	−10.6637			−0.562809			−0.281404	+	0.959589i	$0.590800\pi$
$360$		0			0
$361$	0.612848			0.0322551
$362$	−1.20294			−0.0632253
$363$	−20.5511			−1.07865
$364$			10.6824i			0.559912i
$365$		0			0
$366$	−10.6222			−0.555232
$367$	−10.8780			−0.567828			−0.283914	−	0.958850i	$-0.591633\pi$
$367$	−10.8780			−0.567828			−0.283914	+	0.958850i	$0.591633\pi$
$368$	17.0400			0.888269
$369$		−	6.99063i		−	0.363918i
$370$		0			0
$371$		−	14.6222i		−	0.759148i
$372$		−	32.6178i		−	1.69115i
$373$		−	16.5575i		−	0.857317i	−0.903467	−	0.428659i	$-0.858986\pi$
$373$		−	16.5575i		−	0.857317i	0.903467	−	0.428659i	$-0.141014\pi$
$374$	−0.0874201	−	1.67952i	−0.00452039	−	0.0868461i
$375$		0			0
$376$	4.04593			0.208653
$377$	−29.7146			−1.53038
$378$			1.20294i			0.0618728i
$379$		−	8.16839i		−	0.419582i	−0.977746	−	0.209791i	$-0.932722\pi$
$379$		−	8.16839i		−	0.419582i	0.977746	−	0.209791i	$-0.0672783\pi$
$380$		0			0
$381$			0.622216i			0.0318771i
$382$		−	4.64095i		−	0.237452i
$383$			36.7926i			1.88001i	0.341154	+	0.940007i	$0.389182\pi$
$383$			36.7926i			1.88001i	−0.341154	+	0.940007i	$0.610818\pi$
$384$		−	19.4543i		−	0.992773i
$385$		0			0
$386$		−	5.52098i		−	0.281011i
$387$			4.70964i			0.239404i
$388$	−29.4608			−1.49564
$389$	−12.5348			−0.635539			−0.317770	−	0.948168i	$-0.602934\pi$
$389$	−12.5348			−0.635539			−0.317770	+	0.948168i	$0.602934\pi$
$390$		0			0
$391$	−1.06515	−	20.4637i	−0.0538667	−	1.03489i
$392$		−	5.42219i		−	0.273862i
$393$		−	29.7003i		−	1.49818i
$394$			2.93041i			0.147632i
$395$		0			0
$396$		−	4.74912i		−	0.238652i
$397$	32.1017			1.61114			0.805569	−	0.592502i	$-0.201860\pi$
$397$	32.1017			1.61114			0.805569	+	0.592502i	$0.201860\pi$
$398$	−2.24596			−0.112580
$399$	−15.6128			−0.781620
$400$		0			0
$401$			13.0321i			0.650793i	0.945578	+	0.325396i	$0.105498\pi$
$401$			13.0321i			0.650793i	−0.945578	+	0.325396i	$0.894502\pi$
$402$	6.29529			0.313980
$403$	−27.2859			−1.35921
$404$	−27.3733			−1.36187
$405$		0			0
$406$	−4.17484			−0.207194
$407$		−	9.24443i		−	0.458229i
$408$	−11.0716	+	0.576283i	−0.548126	+	0.0285302i
$409$	−3.12399			−0.154471			−0.0772356	−	0.997013i	$-0.524609\pi$
$409$	−3.12399			−0.154471			−0.0772356	+	0.997013i	$0.524609\pi$
$410$		0			0
$411$			22.7654i			1.12294i
$412$		−	17.8751i		−	0.880643i
$413$	−2.19358			−0.107939
$414$			2.94269i			0.144625i
$415$		0			0
$416$	−12.3225			−0.604159
$417$			40.5259i			1.98456i
$418$	1.80642			0.0883551
$419$			13.1590i			0.642860i	0.946933	+	0.321430i	$0.104164\pi$
$419$			13.1590i			0.642860i	−0.946933	+	0.321430i	$0.895836\pi$
$420$		0			0
$421$	−27.6686			−1.34849			−0.674243	−	0.738509i	$-0.735530\pi$
$421$	−27.6686			−1.34849			−0.674243	+	0.738509i	$0.735530\pi$
$422$	4.63005			0.225387
$423$		−	6.34122i		−	0.308321i
$424$	11.1526			0.541616
$425$		0			0
$426$	−7.30021			−0.353696
$427$		−	24.5491i		−	1.18801i
$428$	12.8968			0.623388
$429$	−10.2351			−0.494154
$430$		0			0
$431$		−	17.6795i		−	0.851593i	−0.904819	−	0.425796i	$-0.859994\pi$
$431$		−	17.6795i		−	0.851593i	0.904819	−	0.425796i	$-0.140006\pi$
$432$	8.32693			0.400630
$433$		−	24.9447i		−	1.19877i	−0.800462	−	0.599383i	$-0.795413\pi$
$433$		−	24.9447i		−	1.19877i	0.800462	−	0.599383i	$-0.204587\pi$
$434$	−3.83362			−0.184020
$435$		0			0
$436$		−	5.43801i		−	0.260433i
$437$	22.0098			1.05287
$438$			3.83807i			0.183390i
$439$		−	11.9748i		−	0.571527i	−0.958300	−	0.285763i	$-0.907753\pi$
$439$		−	11.9748i		−	0.571527i	0.958300	−	0.285763i	$-0.0922471\pi$
$440$		0			0
$441$	−8.49823			−0.404678
$442$	0.235063	+	4.51606i	0.0111808	+	0.214807i
$443$		−	30.6909i		−	1.45817i	−0.684424	−	0.729084i	$-0.739946\pi$
$443$		−	30.6909i		−	1.45817i	0.684424	−	0.729084i	$-0.260054\pi$
$444$	−29.7146			−1.41019
$445$		0			0
$446$	7.30021			0.345675
$447$	−10.8889			−0.515028
$448$	9.18622			0.434008
$449$		−	13.6543i		−	0.644388i	−0.946674	−	0.322194i	$-0.895580\pi$
$449$		−	13.6543i		−	0.644388i	0.946674	−	0.322194i	$-0.104420\pi$
$450$		0			0
$451$	−4.81579			−0.226767
$452$	−11.1655			−0.525180
$453$	−28.1748			−1.32377
$454$		−	1.35260i		−	0.0634804i
$455$		0			0
$456$		−	11.9081i		−	0.557649i
$457$			8.20787i			0.383948i	0.981400	+	0.191974i	$0.0614889\pi$
$457$			8.20787i			0.383948i	−0.981400	+	0.191974i	$0.938511\pi$
$458$		−	2.01429i		−	0.0941216i
$459$	−0.520505	−	10.0000i	−0.0242951	−	0.466760i
$460$		0			0
$461$	5.18421			0.241453			0.120726	−	0.992686i	$-0.461478\pi$
$461$	5.18421			0.241453			0.120726	+	0.992686i	$0.461478\pi$
$462$	−1.43801			−0.0669022
$463$			23.1985i			1.07813i	0.842266	+	0.539063i	$0.181221\pi$
$463$			23.1985i			1.07813i	−0.842266	+	0.539063i	$0.818779\pi$
$464$			28.8988i			1.34159i
$465$		0			0
$466$		−	5.76494i		−	0.267056i
$467$			35.2400i			1.63071i	0.578960	+	0.815356i	$0.303459\pi$
$467$			35.2400i			1.63071i	−0.578960	+	0.815356i	$0.696541\pi$
$468$			12.7699i			0.590287i
$469$			14.5491i			0.671814i
$470$		0			0
$471$		−	20.3368i		−	0.937069i
$472$		−	1.67307i		−	0.0770093i
$473$	3.24443			0.149179
$474$	5.42372			0.249120
$475$		0			0
$476$	−0.649413	−	12.4766i	−0.0297658	−	0.571863i
$477$		−	17.4795i		−	0.800331i
$478$			6.66370i			0.304791i
$479$		−	11.4445i		−	0.522911i	−0.965216	−	0.261455i	$-0.915798\pi$
$479$		−	11.4445i		−	0.522911i	0.965216	−	0.261455i	$-0.0842024\pi$
$480$		0			0
$481$			24.8573i			1.13339i
$482$	−3.80642			−0.173378
$483$	−17.5210			−0.797232
$484$	17.6637			0.802896
$485$		0			0
$486$			5.37133i			0.243649i
$487$	−29.7540			−1.34828			−0.674142	−	0.738602i	$-0.735486\pi$
$487$	−29.7540			−1.34828			−0.674142	+	0.738602i	$0.735486\pi$
$488$	18.7239			0.847592
$489$	11.2301			0.507845
$490$		0			0
$491$	25.4193			1.14716			0.573578	−	0.819151i	$-0.305555\pi$
$491$	25.4193			1.14716			0.573578	+	0.819151i	$0.305555\pi$
$492$			15.4795i			0.697870i
$493$	34.7052	−	1.80642i	1.56304	−	0.0813572i
$494$	−4.85728			−0.218539
$495$		0			0
$496$			26.5368i			1.19154i
$497$		−	16.8716i		−	0.756793i
$498$	4.95899			0.222218
$499$		−	14.8035i		−	0.662696i	−0.943509	−	0.331348i	$-0.892497\pi$
$499$		−	14.8035i		−	0.662696i	0.943509	−	0.331348i	$-0.107503\pi$
$500$		0			0
$501$	43.9037			1.96147
$502$		−	1.40636i		−	0.0627691i
$503$	−4.34767			−0.193853			−0.0969266	−	0.995292i	$-0.530901\pi$
$503$	−4.34767			−0.193853			−0.0969266	+	0.995292i	$0.530901\pi$
$504$			3.67952i			0.163899i
$505$		0			0
$506$	2.02720			0.0901199
$507$	−1.26517			−0.0561884
$508$		−	0.534795i		−	0.0237277i
$509$	15.9813			0.708357			0.354179	−	0.935178i	$-0.384761\pi$
$509$	15.9813			0.708357			0.354179	+	0.935178i	$0.384761\pi$
$510$		0			0
$511$	−8.87019			−0.392394
$512$			20.3111i			0.897633i
$513$	10.7556			0.474870
$514$	−1.92549			−0.0849296
$515$		0			0
$516$		−	10.4286i		−	0.459095i
$517$	−4.36842			−0.192123
$518$			3.49240i			0.153447i
$519$	−5.51114			−0.241912
$520$		0			0
$521$			27.9081i			1.22268i	0.791369	+	0.611339i	$0.209369\pi$
$521$			27.9081i			1.22268i	−0.791369	+	0.611339i	$0.790631\pi$
$522$	−4.99063			−0.218434
$523$			15.8938i			0.694989i	0.937682	+	0.347495i	$0.112968\pi$
$523$			15.8938i			0.694989i	−0.937682	+	0.347495i	$0.887032\pi$
$524$			25.5274i			1.11517i
$525$		0			0
$526$	−2.54770			−0.111085
$527$	31.8687	−	1.65878i	1.38822	−	0.0722576i
$528$			9.95407i			0.433195i
$529$	1.69979			0.0739040
$530$		0			0
$531$	−2.62222			−0.113794
$532$	13.4193			0.581799
$533$	12.9491			0.560890
$534$			8.00984i			0.346620i
$535$		0			0
$536$	−11.0968			−0.479308
$537$	−25.7146			−1.10967
$538$	−1.21279			−0.0522870
$539$			5.85436i			0.252165i
$540$		0			0
$541$		−	24.2449i		−	1.04237i	−0.853444	−	0.521185i	$-0.825490\pi$
$541$		−	24.2449i		−	1.04237i	0.853444	−	0.521185i	$-0.174510\pi$
$542$			2.75557i			0.118362i
$543$		−	8.56199i		−	0.367430i
$544$	14.3921	−	0.749115i	0.617055	−	0.0321181i
$545$		0			0
$546$	3.86665			0.165477
$547$	11.7255			0.501344			0.250672	−	0.968072i	$-0.419348\pi$
$547$	11.7255			0.501344			0.250672	+	0.968072i	$0.419348\pi$
$548$		−	19.5669i		−	0.835857i
$549$		−	29.3461i		−	1.25246i
$550$		0			0
$551$			37.3274i			1.59020i
$552$		−	13.3635i		−	0.568788i
$553$			12.5348i			0.533034i
$554$			0.355509i			0.0151041i
$555$		0			0
$556$		−	34.8321i		−	1.47721i
$557$			31.4336i			1.33188i	0.746004	+	0.665941i	$0.231970\pi$
$557$			31.4336i			1.33188i	−0.746004	+	0.665941i	$0.768030\pi$
$558$	−4.58274			−0.194003
$559$	−8.72393			−0.368983
$560$		0			0
$561$	11.9541	−	0.622216i	0.504701	−	0.0262700i
$562$		−	4.57184i		−	0.192851i
$563$			11.9353i			0.503014i	0.967855	+	0.251507i	$0.0809262\pi$
$563$			11.9353i			0.503014i	−0.967855	+	0.251507i	$0.919074\pi$
$564$			14.0415i			0.591253i
$565$		0			0
$566$			0.615762i			0.0258824i
$567$	−17.6523			−0.741328
$568$	12.8682			0.539937
$569$	−1.14272			−0.0479054			−0.0239527	−	0.999713i	$-0.507625\pi$
$569$	−1.14272			−0.0479054			−0.0239527	+	0.999713i	$0.507625\pi$
$570$		0			0
$571$			26.7402i			1.11904i	0.828816	+	0.559522i	$0.189015\pi$
$571$			26.7402i			1.11904i	−0.828816	+	0.559522i	$0.810985\pi$
$572$	8.79706			0.367823
$573$	33.0321			1.37994
$574$	1.81933			0.0759374
$575$		0			0
$576$	10.9813			0.457553
$577$		−	27.9956i		−	1.16547i	−0.812662	−	0.582735i	$-0.801983\pi$
$577$		−	27.9956i		−	1.16547i	0.812662	−	0.582735i	$-0.198017\pi$
$578$	−0.549086	−	5.26025i	−0.0228389	−	0.218798i
$579$	39.2958			1.63308
$580$		0			0
$581$			11.4608i			0.475472i
$582$			10.6637i			0.442025i
$583$	−12.0415			−0.498707
$584$		−	6.76541i		−	0.279955i
$585$		0			0
$586$	−0.917502			−0.0379017
$587$		−	6.60793i		−	0.272738i	−0.990658	−	0.136369i	$-0.956457\pi$
$587$		−	6.60793i		−	0.272738i	0.990658	−	0.136369i	$-0.0435433\pi$
$588$	18.8178			0.776033
$589$			34.2766i			1.41234i
$590$		0			0
$591$	−20.8573			−0.857954
$592$	24.1748			0.993580
$593$		−	4.82564i		−	0.198165i	−0.995079	−	0.0990826i	$-0.968409\pi$
$593$		−	4.82564i		−	0.198165i	0.995079	−	0.0990826i	$-0.0315908\pi$
$594$	0.990632			0.0406461
$595$		0			0
$596$	9.35905			0.383362
$597$		−	15.9857i		−	0.654252i
$598$	−5.45091			−0.222904
$599$	39.0005			1.59352			0.796758	−	0.604298i	$-0.206546\pi$
$599$	39.0005			1.59352			0.796758	+	0.604298i	$0.206546\pi$
$600$		0			0
$601$		−	7.19405i		−	0.293452i	−0.989177	−	0.146726i	$-0.953127\pi$
$601$		−	7.19405i		−	0.293452i	0.989177	−	0.146726i	$-0.0468735\pi$
$602$	−1.22570			−0.0499556
$603$			17.3921i			0.708260i
$604$	24.2163			0.985348
$605$		0			0
$606$			9.90813i			0.402490i
$607$	7.23353			0.293600			0.146800	−	0.989166i	$-0.453103\pi$
$607$	7.23353			0.293600			0.146800	+	0.989166i	$0.453103\pi$
$608$			15.4795i			0.627776i
$609$		−	29.7146i		−	1.20409i
$610$		0			0
$611$	11.7462			0.475200
$612$	−0.776312	−	14.9146i	−0.0313806	−	0.602886i
$613$			14.7654i			0.596369i	0.954508	+	0.298185i	$0.0963811\pi$
$613$			14.7654i			0.596369i	−0.954508	+	0.298185i	$0.903619\pi$
$614$	−1.76938			−0.0714065
$615$		0			0
$616$	2.53480			0.102130
$617$	10.5205			0.423540			0.211770	−	0.977320i	$-0.432077\pi$
$617$	10.5205			0.423540			0.211770	+	0.977320i	$0.432077\pi$
$618$	−6.47013			−0.260267
$619$			7.94623i			0.319386i	0.987167	+	0.159693i	$0.0510504\pi$
$619$			7.94623i			0.319386i	−0.987167	+	0.159693i	$0.948950\pi$
$620$		0			0
$621$	12.0701			0.484355
$622$	7.37979			0.295903
$623$	−18.5116			−0.741652
$624$		−	26.7654i		−	1.07147i
$625$		0			0
$626$			9.61285i			0.384207i
$627$			12.8573i			0.513470i
$628$			17.4795i			0.697508i
$629$	−1.51114	−	29.0321i	−0.0602530	−	1.15759i
$630$		0			0
$631$	−14.2636			−0.567827			−0.283913	−	0.958850i	$-0.591633\pi$
$631$	−14.2636			−0.567827			−0.283913	+	0.958850i	$0.591633\pi$
$632$	−9.56046			−0.380295
$633$			32.9545i			1.30983i
$634$			4.26364i			0.169331i
$635$		0			0
$636$			38.7052i			1.53476i
$637$		−	15.7418i		−	0.623711i
$638$			3.43801i			0.136112i
$639$		−	20.1684i		−	0.797849i
$640$		0			0
$641$		−	25.4509i		−	1.00525i	−0.864504	−	0.502625i	$-0.832368\pi$
$641$		−	25.4509i		−	1.00525i	0.864504	−	0.502625i	$-0.167632\pi$
$642$		−	4.66815i		−	0.184237i
$643$	−16.1126			−0.635419			−0.317710	−	0.948188i	$-0.602914\pi$
$643$	−16.1126			−0.635419			−0.317710	+	0.948188i	$0.602914\pi$
$644$	15.0593			0.593420
$645$		0			0
$646$	5.67307	−	0.295286i	0.223204	−	0.0116179i
$647$		−	40.4558i		−	1.59048i	−0.606293	−	0.795242i	$-0.707344\pi$
$647$		−	40.4558i		−	1.59048i	0.606293	−	0.795242i	$-0.292656\pi$
$648$		−	13.4637i		−	0.528903i
$649$			1.80642i			0.0709083i
$650$		0			0
$651$		−	27.2859i		−	1.06942i
$652$	−9.65233			−0.378014
$653$	−6.87601			−0.269079			−0.134540	−	0.990908i	$-0.542956\pi$
$653$	−6.87601			−0.269079			−0.134540	+	0.990908i	$0.542956\pi$
$654$	−1.96836			−0.0769689
$655$		0			0
$656$		−	12.5936i		−	0.491699i
$657$	−10.6035			−0.413681
$658$	1.65032			0.0643361
$659$	−16.0286			−0.624385			−0.312192	−	0.950019i	$-0.601063\pi$
$659$	−16.0286			−0.624385			−0.312192	+	0.950019i	$0.601063\pi$
$660$		0			0
$661$	−31.1655			−1.21220			−0.606098	−	0.795390i	$-0.707266\pi$
$661$	−31.1655			−1.21220			−0.606098	+	0.795390i	$0.707266\pi$
$662$			7.74620i			0.301065i
$663$	−32.1432	+	1.67307i	−1.24834	+	0.0649767i
$664$	−8.74128			−0.339227
$665$		0			0
$666$			4.17484i			0.161772i
$667$			41.8894i			1.62196i
$668$	−37.7353			−1.46002
$669$			51.9595i			2.00887i
$670$		0			0
$671$	−20.2163			−0.780443
$672$		−	12.3225i		−	0.475350i
$673$	−19.8479			−0.765081			−0.382540	−	0.923939i	$-0.624951\pi$
$673$	−19.8479			−0.765081			−0.382540	+	0.923939i	$0.624951\pi$
$674$			1.64449i			0.0633434i
$675$		0			0
$676$	1.08742			0.0418239
$677$	28.8385			1.10836			0.554178	−	0.832398i	$-0.313033\pi$
$677$	28.8385			1.10836			0.554178	+	0.832398i	$0.313033\pi$
$678$			4.04149i			0.155212i
$679$	−24.6450			−0.945787
$680$		0			0
$681$	9.62714			0.368913
$682$			3.15701i			0.120888i
$683$	19.2050			0.734857			0.367429	−	0.930052i	$-0.380238\pi$
$683$	19.2050			0.734857			0.367429	+	0.930052i	$0.380238\pi$
$684$	16.0415			0.613362
$685$		0			0
$686$		−	5.67890i		−	0.216821i
$687$	14.3368			0.546982
$688$			8.48442i			0.323465i
$689$	32.3783			1.23351
$690$		0			0
$691$		−	34.5555i		−	1.31455i	−0.753649	−	0.657277i	$-0.771708\pi$
$691$		−	34.5555i		−	1.31455i	0.753649	−	0.657277i	$-0.228292\pi$
$692$	4.73683			0.180067
$693$		−	3.97280i		−	0.150914i
$694$		−	9.83161i		−	0.373203i
$695$		0			0
$696$	22.6637			0.859065
$697$	−15.1240	+	0.787212i	−0.572862	+	0.0298178i
$698$		−	8.24797i		−	0.312190i
$699$	41.0321			1.55198
$700$		0			0
$701$	−20.9131			−0.789875			−0.394938	−	0.918708i	$-0.629234\pi$
$701$	−20.9131			−0.789875			−0.394938	+	0.918708i	$0.629234\pi$
$702$	−2.66370			−0.100535
$703$	31.2257			1.17770
$704$		−	7.56491i		−	0.285113i
$705$		0			0
$706$	1.15563			0.0434926
$707$	−22.8988			−0.861197
$708$	5.80642			0.218219
$709$			32.9906i			1.23899i	0.785001	+	0.619495i	$0.212662\pi$
$709$			32.9906i			1.23899i	−0.785001	+	0.619495i	$0.787338\pi$
$710$		0			0
$711$			14.9842i			0.561951i
$712$		−	14.1191i		−	0.529134i
$713$			38.4657i			1.44055i
$714$	−4.51606	+	0.235063i	−0.169009	+	0.00879702i
$715$		0			0
$716$	22.1017			0.825980
$717$	−47.4291			−1.77127
$718$		−	3.31756i		−	0.123810i
$719$			10.5141i			0.392108i	0.980593	+	0.196054i	$0.0628128\pi$
$719$			10.5141i			0.392108i	−0.980593	+	0.196054i	$0.937187\pi$
$720$		0			0
$721$		−	14.9532i		−	0.556885i
$722$			0.190662i			0.00709569i
$723$		−	27.0923i		−	1.00758i
$724$			7.35905i			0.273497i
$725$		0			0
$726$		−	6.39361i		−	0.237289i
$727$		−	43.2815i		−	1.60522i	−0.596503	−	0.802610i	$-0.703444\pi$
$727$		−	43.2815i		−	1.60522i	0.596503	−	0.802610i	$-0.296556\pi$
$728$	−6.81579			−0.252610
$729$	−4.96836			−0.184013
$730$		0			0
$731$	10.1891	−	0.530350i	0.376859	−	0.0196157i
$732$			64.9817i			2.40179i
$733$		−	22.8859i		−	0.845308i	−0.906291	−	0.422654i	$-0.861098\pi$
$733$		−	22.8859i		−	0.845308i	0.906291	−	0.422654i	$-0.138902\pi$
$734$		−	3.38424i		−	0.124914i
$735$		0			0
$736$			17.3713i			0.640316i
$737$	11.9813			0.441336
$738$	2.17484			0.0800570
$739$	−0.815792			−0.0300094			−0.0150047	−	0.999887i	$-0.504776\pi$
$739$	−0.815792			−0.0300094			−0.0150047	+	0.999887i	$0.504776\pi$
$740$		0			0
$741$		−	34.5718i		−	1.27003i
$742$	4.54909			0.167002
$743$	−30.1639			−1.10661			−0.553304	−	0.832980i	$-0.686633\pi$
$743$	−30.1639			−1.10661			−0.553304	+	0.832980i	$0.686633\pi$
$744$	20.8113			0.762981
$745$		0			0
$746$	5.15118			0.188598
$747$			13.7003i			0.501267i
$748$	−10.2745	+	0.534795i	−0.375674	+	0.0195541i
$749$	10.7886			0.394207
$750$		0			0
$751$			2.17728i			0.0794500i	0.999211	+	0.0397250i	$0.0126482\pi$
$751$			2.17728i			0.0794500i	−0.999211	+	0.0397250i	$0.987352\pi$
$752$		−	11.4237i		−	0.416580i
$753$	10.0098			0.364779
$754$		−	9.24443i		−	0.336662i
$755$		0			0
$756$	7.35905			0.267646
$757$		−	29.3230i		−	1.06576i	−0.846190	−	0.532881i	$-0.821110\pi$
$757$		−	29.3230i		−	1.06576i	0.846190	−	0.532881i	$-0.178890\pi$
$758$	2.54125			0.0923023
$759$			14.4286i			0.523726i
$760$		0			0
$761$	49.2212			1.78427			0.892134	−	0.451770i	$-0.149207\pi$
$761$	49.2212			1.78427			0.892134	+	0.451770i	$0.149207\pi$
$762$	−0.193576			−0.00701252
$763$		−	4.54909i		−	0.164688i
$764$	−28.3912			−1.02716
$765$		0			0
$766$	−11.4465			−0.413578
$767$		−	4.85728i		−	0.175386i
$768$	−19.5002			−0.703654
$769$	−37.0178			−1.33490			−0.667449	−	0.744656i	$-0.732614\pi$
$769$	−37.0178			−1.33490			−0.667449	+	0.744656i	$0.732614\pi$
$770$		0			0
$771$		−	13.7047i		−	0.493563i
$772$	−33.7748			−1.21558
$773$		−	4.20787i		−	0.151346i	−0.997133	−	0.0756732i	$-0.975889\pi$
$773$		−	4.20787i		−	0.151346i	0.997133	−	0.0756732i	$-0.0241106\pi$
$774$	−1.46520			−0.0526657
$775$		0			0
$776$		−	18.7971i		−	0.674775i
$777$	−24.8573			−0.891750
$778$		−	3.89967i		−	0.139810i
$779$		−	16.2667i		−	0.582815i
$780$		0			0
$781$	−13.8938			−0.497161
$782$	6.36641	−	0.331375i	0.227662	−	0.0118499i
$783$			20.4701i			0.731543i
$784$	−15.3096			−0.546771
$785$		0			0
$786$	9.23999			0.329579
$787$	23.8557			0.850366			0.425183	−	0.905108i	$-0.360210\pi$
$787$	23.8557			0.850366			0.425183	+	0.905108i	$0.360210\pi$
$788$	17.9269			0.638618
$789$		−	18.1334i		−	0.645564i
$790$		0			0
$791$	−9.34031			−0.332103
$792$	3.03011			0.107670
$793$	54.3595			1.93036
$794$			9.98709i			0.354429i
$795$		0			0
$796$			13.7397i			0.486992i
$797$			34.4415i			1.21998i	0.792408	+	0.609991i	$0.208827\pi$
$797$			34.4415i			1.21998i	−0.792408	+	0.609991i	$0.791173\pi$
$798$		−	4.85728i		−	0.171946i
$799$	−13.7190	+	0.714082i	−0.485343	+	0.0252624i
$800$		0			0
$801$	−22.1289			−0.781886
$802$	−4.05439			−0.143166
$803$			7.30465i			0.257776i
$804$		−	38.5116i		−	1.35820i
$805$		0			0
$806$		−	8.48886i		−	0.299007i
$807$		−	8.63206i		−	0.303863i
$808$		−	17.4652i		−	0.614424i
$809$			40.2578i			1.41539i	0.706518	+	0.707695i	$0.250265\pi$
$809$			40.2578i			1.41539i	−0.706518	+	0.707695i	$0.749735\pi$
$810$		0			0
$811$			20.4953i			0.719688i	0.933013	+	0.359844i	$0.117170\pi$
$811$			20.4953i			0.719688i	−0.933013	+	0.359844i	$0.882830\pi$
$812$			25.5397i			0.896268i
$813$	−19.6128			−0.687853
$814$	2.87601			0.100804
$815$		0			0
$816$	1.62714	+	31.2607i	0.0569612	+	1.09434i
$817$			10.9590i			0.383407i
$818$		−	0.971896i		−	0.0339816i
$819$			10.6824i			0.373275i
$820$		0			0
$821$			33.6958i			1.17599i	0.808864	+	0.587996i	$0.200083\pi$
$821$			33.6958i			1.17599i	−0.808864	+	0.587996i	$0.799917\pi$
$822$	−7.08250			−0.247030
$823$	16.6113			0.579034			0.289517	−	0.957173i	$-0.406505\pi$
$823$	16.6113			0.579034			0.289517	+	0.957173i	$0.406505\pi$
$824$	11.4050			0.397311
$825$		0			0
$826$		−	0.682439i		−	0.0237451i
$827$	0.316030			0.0109894			0.00549472	−	0.999985i	$-0.498251\pi$
$827$	0.316030			0.0109894			0.00549472	+	0.999985i	$0.498251\pi$
$828$	18.0020			0.625613
$829$	−18.2034			−0.632231			−0.316115	−	0.948721i	$-0.602379\pi$
$829$	−18.2034			−0.632231			−0.316115	+	0.948721i	$0.602379\pi$
$830$		0			0
$831$	−2.53035			−0.0877769
$832$			20.3412i			0.705205i
$833$	0.956981	+	18.3856i	0.0331574	+	0.637024i
$834$	−12.6079			−0.436577
$835$		0			0
$836$		−	11.0509i		−	0.382202i
$837$			18.7971i			0.649721i
$838$	−4.09387			−0.141421
$839$		−	2.63851i		−	0.0910916i	−0.998962	−	0.0455458i	$-0.985497\pi$
$839$		−	2.63851i		−	0.0910916i	0.998962	−	0.0455458i	$-0.0145027\pi$
$840$		0			0
$841$	−42.0420			−1.44972
$842$		−	8.60793i		−	0.296649i
$843$	32.5402			1.12074
$844$		−	28.3245i		−	0.974969i
$845$		0			0
$846$	1.97280			0.0678264
$847$	14.7763			0.507720
$848$		−	31.4893i		−	1.08135i
$849$	−4.38271			−0.150414
$850$		0			0
$851$	35.0420			1.20122
$852$			44.6593i			1.53000i
$853$	4.68244			0.160324			0.0801618	−	0.996782i	$-0.474456\pi$
$853$	4.68244			0.160324			0.0801618	+	0.996782i	$0.474456\pi$
$854$	7.63741			0.261347
$855$		0			0
$856$			8.22861i			0.281248i
$857$	47.3689			1.61809			0.809045	−	0.587746i	$-0.199985\pi$
$857$	47.3689			1.61809			0.809045	+	0.587746i	$0.199985\pi$
$858$		−	3.18421i		−	0.108707i
$859$	−16.1748			−0.551878			−0.275939	−	0.961175i	$-0.588989\pi$
$859$	−16.1748			−0.551878			−0.275939	+	0.961175i	$0.588989\pi$
$860$		0			0
$861$			12.9491i			0.441306i
$862$	5.50024			0.187339
$863$		−	32.5674i		−	1.10861i	−0.832315	−	0.554303i	$-0.812985\pi$
$863$		−	32.5674i		−	1.10861i	0.832315	−	0.554303i	$-0.187015\pi$
$864$			8.48886i			0.288797i
$865$		0			0
$866$	7.76049			0.263712
$867$	37.4400	−	3.90813i	1.27153	−	0.132727i
$868$			23.4523i			0.796023i
$869$	10.3225			0.350166
$870$		0			0
$871$	−32.2163			−1.09161
$872$	3.46965			0.117497
$873$	−29.4608			−0.997096
$874$			6.84743i			0.231618i
$875$		0			0
$876$	23.4795			0.793299
$877$	13.2543			0.447565			0.223783	−	0.974639i	$-0.428159\pi$
$877$	13.2543			0.447565			0.223783	+	0.974639i	$0.428159\pi$
$878$	3.72546			0.125728
$879$		−	6.53035i		−	0.220263i
$880$		0			0
$881$			2.78721i			0.0939035i	0.998897	+	0.0469518i	$0.0149507\pi$
$881$			2.78721i			0.0939035i	−0.998897	+	0.0469518i	$0.985049\pi$
$882$		−	2.64387i		−	0.0890236i
$883$		−	27.1985i		−	0.915302i	−0.889132	−	0.457651i	$-0.848691\pi$
$883$		−	27.1985i		−	0.915302i	0.889132	−	0.457651i	$-0.151309\pi$
$884$	27.6271	−	1.43801i	0.929201	−	0.0483654i
$885$		0			0
$886$	9.54818			0.320777
$887$	48.2657			1.62060			0.810301	−	0.586014i	$-0.199304\pi$
$887$	48.2657			1.62060			0.810301	+	0.586014i	$0.199304\pi$
$888$		−	18.9590i		−	0.636222i
$889$		−	0.447375i		−	0.0150045i
$890$		0			0
$891$			14.5368i			0.487001i
$892$		−	44.6593i		−	1.49530i
$893$		−	14.7556i		−	0.493776i
$894$		−	3.38763i		−	0.113299i
$895$		0			0
$896$			13.9877i			0.467297i
$897$		−	38.7971i		−	1.29540i
$898$	4.24797			0.141757
$899$	−65.2355			−2.17573
$900$		0			0
$901$	−37.8163	+	1.96836i	−1.25984	+	0.0655755i
$902$		−	1.49823i		−	0.0498856i
$903$		−	8.72393i		−	0.290314i
$904$		−	7.12399i		−	0.236940i
$905$		0			0
$906$		−	8.76541i		−	0.291211i
$907$	−8.93825			−0.296790			−0.148395	−	0.988928i	$-0.547411\pi$
$907$	−8.93825			−0.296790			−0.148395	+	0.988928i	$0.547411\pi$
$908$	−8.27454			−0.274600
$909$	−27.3733			−0.907916
$910$		0			0
$911$			51.6795i			1.71222i	0.516794	+	0.856110i	$0.327125\pi$
$911$			51.6795i			1.71222i	−0.516794	+	0.856110i	$0.672875\pi$
$912$	−33.6227			−1.11336
$913$	9.43801			0.312352
$914$	−2.55353			−0.0844633
$915$		0			0
$916$	−12.3225			−0.407146
$917$			21.3546i			0.705191i
$918$	3.11108	−	0.161933i	0.102681	−	0.00534460i
$919$	−18.9719			−0.625825			−0.312913	−	0.949782i	$-0.601305\pi$
$919$	−18.9719			−0.625825			−0.312913	+	0.949782i	$0.601305\pi$
$920$		0			0
$921$		−	12.5936i		−	0.414974i
$922$			1.61285i			0.0531163i
$923$	37.3590			1.22969
$924$			8.79706i			0.289402i
$925$		0			0
$926$	−7.21723			−0.237173
$927$		−	17.8751i		−	0.587096i
$928$	−29.4608			−0.967097
$929$		−	50.3912i		−	1.65328i	−0.562731	−	0.826640i	$-0.690249\pi$
$929$		−	50.3912i		−	1.65328i	0.562731	−	0.826640i	$-0.309751\pi$
$930$		0			0
$931$	−19.7748			−0.648092
$932$	−35.2672			−1.15521
$933$			52.5259i			1.71962i
$934$	−10.9634			−0.358735
$935$		0			0
$936$	−8.14764			−0.266314
$937$		−	58.1530i		−	1.89978i	−0.312589	−	0.949889i	$-0.601196\pi$
$937$		−	58.1530i		−	1.89978i	0.312589	−	0.949889i	$-0.398804\pi$
$938$	−4.52633			−0.147790
$939$	−68.4197			−2.23279
$940$		0			0
$941$		−	9.25734i		−	0.301781i	−0.988551	−	0.150890i	$-0.951786\pi$
$941$		−	9.25734i		−	0.301781i	0.988551	−	0.150890i	$-0.0482140\pi$
$942$	6.32693			0.206142
$943$		−	18.2548i		−	0.594457i
$944$	−4.72393			−0.153751
$945$		0			0
$946$			1.00937i			0.0328174i
$947$	−43.9516			−1.42824			−0.714118	−	0.700025i	$-0.753172\pi$
$947$	−43.9516			−1.42824			−0.714118	+	0.700025i	$0.753172\pi$
$948$		−	33.1798i		−	1.07763i
$949$		−	19.6414i		−	0.637588i
$950$		0			0
$951$	−30.3466			−0.984057
$952$	7.96052	−	0.414349i	0.258002	−	0.0134291i
$953$		−	38.1891i		−	1.23707i	−0.785758	−	0.618534i	$-0.787727\pi$
$953$		−	38.1891i		−	1.23707i	0.785758	−	0.618534i	$-0.212273\pi$
$954$	5.43801			0.176062
$955$		0			0
$956$	40.7654			1.31845
$957$	−24.4701			−0.791007
$958$	3.56046			0.115033
$959$		−	16.3684i		−	0.528564i
$960$		0			0
$961$	−28.9037			−0.932377
$962$	−7.73329			−0.249331
$963$	12.8968			0.415592
$964$			23.2859i			0.749989i
$965$		0			0
$966$		−	5.45091i		−	0.175380i
$967$			7.36043i			0.236696i	0.992972	+	0.118348i	$0.0377598\pi$
$967$			7.36043i			0.236696i	−0.992972	+	0.118348i	$0.962240\pi$
$968$			11.2701i			0.362235i
$969$	2.10171	+	40.3783i	0.0675167	+	1.29714i
$970$		0			0
$971$	−35.2257			−1.13045			−0.565223	−	0.824938i	$-0.691210\pi$
$971$	−35.2257			−1.13045			−0.565223	+	0.824938i	$0.691210\pi$
$972$	32.8593			1.05396
$973$		−	29.1383i		−	0.934130i
$974$		−	9.25671i		−	0.296604i
$975$		0			0
$976$		−	52.8671i		−	1.69224i
$977$			46.9273i			1.50134i	0.660678	+	0.750669i	$0.270269\pi$
$977$			46.9273i			1.50134i	−0.660678	+	0.750669i	$0.729731\pi$
$978$			3.49378i			0.111719i
$979$			15.2444i			0.487214i
$980$		0			0
$981$		−	5.43801i		−	0.173622i
$982$			7.90813i			0.252359i
$983$	8.21432			0.261996			0.130998	−	0.991383i	$-0.458182\pi$
$983$	8.21432			0.261996			0.130998	+	0.991383i	$0.458182\pi$
$984$	−9.87649			−0.314851
$985$		0			0
$986$	0.561993	+	10.7971i	0.0178975	+	0.343848i
$987$			11.7462i			0.373886i
$988$			29.7146i			0.945346i
$989$			12.2983i			0.391065i
$990$		0			0
$991$		−	17.7683i		−	0.564430i	−0.959351	−	0.282215i	$-0.908931\pi$
$991$		−	17.7683i		−	0.564430i	0.959351	−	0.282215i	$-0.0910691\pi$
$992$	−27.0529			−0.858929
$993$	−55.1338			−1.74962
$994$	5.24888			0.166484
$995$		0			0
$996$		−	30.3368i		−	0.961257i
$997$	11.6958			0.370410			0.185205	−	0.982700i	$-0.440705\pi$
$997$	11.6958			0.370410			0.185205	+	0.982700i	$0.440705\pi$
$998$	4.60549			0.145784
$999$	17.1240			0.541779

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.a.424.4		6
5.2	odd	4		85.2.d.a.16.4	yes	6
5.3	odd	4		425.2.d.c.101.3		6
5.4	even	2		425.2.c.b.424.3		6
15.2	even	4		765.2.g.b.271.3		6
17.16	even	2		425.2.c.b.424.4		6
20.7	even	4		1360.2.c.f.1121.1		6
85.13	odd	4		7225.2.a.q.1.2		3
85.33	odd	4		425.2.d.c.101.4		6
85.38	odd	4		7225.2.a.r.1.2		3
85.47	odd	4		1445.2.a.j.1.2		3
85.67	odd	4		85.2.d.a.16.3	✓	6
85.72	odd	4		1445.2.a.k.1.2		3
85.84	even	2	inner	425.2.c.a.424.3		6
255.152	even	4		765.2.g.b.271.4		6
340.67	even	4		1360.2.c.f.1121.6		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
85.2.d.a.16.3	✓	6	85.67	odd	4
85.2.d.a.16.4	yes	6	5.2	odd	4
425.2.c.a.424.3		6	85.84	even	2	inner
425.2.c.a.424.4		6	1.1	even	1	trivial
425.2.c.b.424.3		6	5.4	even	2
425.2.c.b.424.4		6	17.16	even	2
425.2.d.c.101.3		6	5.3	odd	4
425.2.d.c.101.4		6	85.33	odd	4
765.2.g.b.271.3		6	15.2	even	4
765.2.g.b.271.4		6	255.152	even	4
1360.2.c.f.1121.1		6	20.7	even	4
1360.2.c.f.1121.6		6	340.67	even	4
1445.2.a.j.1.2		3	85.47	odd	4
1445.2.a.k.1.2		3	85.72	odd	4
7225.2.a.q.1.2		3	85.13	odd	4
7225.2.a.r.1.2		3	85.38	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 $\F_{q}$
Belyi maps

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Newspace parameters

Newform invariants

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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	425.2.c.a.424.4

Level	$425$
Weight	$2$
Character	425.424
Analytic conductor	$3.394$
Analytic rank	$0$
Dimension	$6$
Inner twists	$2$