LMFDB - Embedded newform 9075.2.a.dy.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9075,2,Mod(1,9075)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9075, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("9075.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 9075 = 3 \cdot 5^{2} \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9075.a (trivial)

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:	yes
Analytic conductor:	$72.4642398343$
Analytic rank:	$1$
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} - 4 x^{11} - 10 x^{10} + 52 x^{9} + 18 x^{8} - 220 x^{7} + 61 x^{6} + 330 x^{5} - 145 x^{4} + \cdots - 5 $ x^12 - 4x^11 - 10x^10 + 52x^9 + 18x^8 - 220x^7 + 61x^6 + 330x^5 - 145x^4 - 148x^3 + 71x^2 + 10*x - 5
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 165)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$0.803366$ of defining polynomial
Character	$\chi$	$=$	9075.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.803366 q^{2} +1.00000 q^{3} -1.35460 q^{4} -0.803366 q^{6} +0.508505 q^{7} +2.69497 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-0.803366 q^{2} +1.00000 q^{3} -1.35460 q^{4} -0.803366 q^{6} +0.508505 q^{7} +2.69497 q^{8} +1.00000 q^{9} -1.35460 q^{12} +5.13798 q^{13} -0.408516 q^{14} +0.544158 q^{16} -3.34542 q^{17} -0.803366 q^{18} -1.17303 q^{19} +0.508505 q^{21} -2.91459 q^{23} +2.69497 q^{24} -4.12768 q^{26} +1.00000 q^{27} -0.688823 q^{28} -0.392594 q^{29} -6.35529 q^{31} -5.82710 q^{32} +2.68759 q^{34} -1.35460 q^{36} -4.45218 q^{37} +0.942375 q^{38} +5.13798 q^{39} -2.79867 q^{41} -0.408516 q^{42} +3.05350 q^{43} +2.34148 q^{46} +11.4923 q^{47} +0.544158 q^{48} -6.74142 q^{49} -3.34542 q^{51} -6.95993 q^{52} +9.80747 q^{53} -0.803366 q^{54} +1.37041 q^{56} -1.17303 q^{57} +0.315396 q^{58} -3.02854 q^{59} +1.05512 q^{61} +5.10562 q^{62} +0.508505 q^{63} +3.59298 q^{64} -5.31327 q^{67} +4.53171 q^{68} -2.91459 q^{69} +4.09976 q^{71} +2.69497 q^{72} -13.1083 q^{73} +3.57673 q^{74} +1.58900 q^{76} -4.12768 q^{78} -16.1403 q^{79} +1.00000 q^{81} +2.24836 q^{82} -15.3587 q^{83} -0.688823 q^{84} -2.45308 q^{86} -0.392594 q^{87} -9.84603 q^{89} +2.61269 q^{91} +3.94812 q^{92} -6.35529 q^{93} -9.23252 q^{94} -5.82710 q^{96} -15.1692 q^{97} +5.41583 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 4 q^{2} + 12 q^{3} + 12 q^{4} - 4 q^{6} - 8 q^{7} - 12 q^{8} + 12 q^{9}+O(q^{10}) $ 12 * q - 4 * q^2 + 12 * q^3 + 12 * q^4 - 4 * q^6 - 8 * q^7 - 12 * q^8 + 12 * q^9	$ 12 q - 4 q^{2} + 12 q^{3} + 12 q^{4} - 4 q^{6} - 8 q^{7} - 12 q^{8} + 12 q^{9} + 12 q^{12} - 18 q^{13} + 6 q^{14} + 24 q^{16} - 18 q^{17} - 4 q^{18} - 16 q^{19} - 8 q^{21} - 12 q^{24} + 16 q^{26} + 12 q^{27} - 30 q^{28} - 28 q^{32} + 6 q^{34} + 12 q^{36} - 28 q^{38} - 18 q^{39} + 6 q^{42} - 32 q^{43} - 28 q^{46} - 4 q^{47} + 24 q^{48} + 12 q^{49} - 18 q^{51} - 48 q^{52} - 12 q^{53} - 4 q^{54} + 6 q^{56} - 16 q^{57} - 10 q^{58} + 20 q^{59} - 20 q^{61} - 20 q^{62} - 8 q^{63} - 6 q^{64} + 10 q^{67} - 26 q^{68} + 32 q^{71} - 12 q^{72} - 26 q^{73} + 68 q^{74} - 34 q^{76} + 16 q^{78} - 32 q^{79} + 12 q^{81} - 62 q^{82} - 26 q^{83} - 30 q^{84} - 36 q^{86} - 10 q^{89} + 8 q^{92} + 2 q^{94} - 28 q^{96} - 22 q^{97} - 60 q^{98}+O(q^{100}) $ 12 * q - 4 * q^2 + 12 * q^3 + 12 * q^4 - 4 * q^6 - 8 * q^7 - 12 * q^8 + 12 * q^9 + 12 * q^12 - 18 * q^13 + 6 * q^14 + 24 * q^16 - 18 * q^17 - 4 * q^18 - 16 * q^19 - 8 * q^21 - 12 * q^24 + 16 * q^26 + 12 * q^27 - 30 * q^28 - 28 * q^32 + 6 * q^34 + 12 * q^36 - 28 * q^38 - 18 * q^39 + 6 * q^42 - 32 * q^43 - 28 * q^46 - 4 * q^47 + 24 * q^48 + 12 * q^49 - 18 * q^51 - 48 * q^52 - 12 * q^53 - 4 * q^54 + 6 * q^56 - 16 * q^57 - 10 * q^58 + 20 * q^59 - 20 * q^61 - 20 * q^62 - 8 * q^63 - 6 * q^64 + 10 * q^67 - 26 * q^68 + 32 * q^71 - 12 * q^72 - 26 * q^73 + 68 * q^74 - 34 * q^76 + 16 * q^78 - 32 * q^79 + 12 * q^81 - 62 * q^82 - 26 * q^83 - 30 * q^84 - 36 * q^86 - 10 * q^89 + 8 * q^92 + 2 * q^94 - 28 * q^96 - 22 * q^97 - 60 * q^98

Display 100 coefficients 10 coefficients

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 $

$2$	−0.803366	−0.568065	−0.284033	−	0.958815i	$-0.591672\pi$
$2$	−0.803366	−0.568065	−0.284033	+	0.958815i	$0.591672\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	−1.35460	−0.677302
$5$	0	0
$5$	0	0
$6$	−0.803366	−0.327973
$7$	0.508505	0.192197	0.0960984	−	0.995372i	$-0.469364\pi$
$7$	0.508505	0.192197	0.0960984	+	0.995372i	$0.469364\pi$
$8$	2.69497	0.952817
$9$	1.00000	0.333333
$10$	0	0
$11$	0	0
$11$	0	0
$12$	−1.35460	−0.391040
$13$	5.13798	1.42502	0.712509	−	0.701662i	$-0.247559\pi$
$13$	5.13798	1.42502	0.712509	+	0.701662i	$0.247559\pi$
$14$	−0.408516	−0.109180
$15$	0	0
$16$	0.544158	0.136040
$17$	−3.34542	−0.811383	−0.405691	−	0.914010i	$-0.632969\pi$
$17$	−3.34542	−0.811383	−0.405691	+	0.914010i	$0.632969\pi$
$18$	−0.803366	−0.189355
$19$	−1.17303	−0.269113	−0.134556	−	0.990906i	$-0.542961\pi$
$19$	−1.17303	−0.269113	−0.134556	+	0.990906i	$0.542961\pi$
$20$	0	0
$21$	0.508505	0.110965
$22$	0	0
$23$	−2.91459	−0.607734	−0.303867	−	0.952714i	$-0.598278\pi$
$23$	−2.91459	−0.607734	−0.303867	+	0.952714i	$0.598278\pi$
$24$	2.69497	0.550109
$25$	0	0
$26$	−4.12768	−0.809504
$27$	1.00000	0.192450
$28$	−0.688823	−0.130175
$29$	−0.392594	−0.0729028	−0.0364514	−	0.999335i	$-0.511605\pi$
$29$	−0.392594	−0.0729028	−0.0364514	+	0.999335i	$0.511605\pi$
$30$	0	0
$31$	−6.35529	−1.14144	−0.570722	−	0.821143i	$-0.693337\pi$
$31$	−6.35529	−1.14144	−0.570722	+	0.821143i	$0.693337\pi$
$32$	−5.82710	−1.03010
$33$	0	0
$34$	2.68759	0.460918
$35$	0	0
$36$	−1.35460	−0.225767
$37$	−4.45218	−0.731934	−0.365967	−	0.930628i	$-0.619262\pi$
$37$	−4.45218	−0.731934	−0.365967	+	0.930628i	$0.619262\pi$
$38$	0.942375	0.152873
$39$	5.13798	0.822735
$40$	0	0
$41$	−2.79867	−0.437079	−0.218540	−	0.975828i	$-0.570129\pi$
$41$	−2.79867	−0.437079	−0.218540	+	0.975828i	$0.570129\pi$
$42$	−0.408516	−0.0630353
$43$	3.05350	0.465654	0.232827	−	0.972518i	$-0.425202\pi$
$43$	3.05350	0.465654	0.232827	+	0.972518i	$0.425202\pi$
$44$	0	0
$45$	0	0
$46$	2.34148	0.345233
$47$	11.4923	1.67633	0.838163	−	0.545421i	$-0.183630\pi$
$47$	11.4923	1.67633	0.838163	+	0.545421i	$0.183630\pi$
$48$	0.544158	0.0785425
$49$	−6.74142	−0.963060
$50$	0	0
$51$	−3.34542	−0.468452
$52$	−6.95993	−0.965168
$53$	9.80747	1.34716	0.673580	−	0.739114i	$-0.264756\pi$
$53$	9.80747	1.34716	0.673580	+	0.739114i	$0.264756\pi$
$54$	−0.803366	−0.109324
$55$	0	0
$56$	1.37041	0.183128
$57$	−1.17303	−0.155372
$58$	0.315396	0.0414136
$59$	−3.02854	−0.394282	−0.197141	−	0.980375i	$-0.563166\pi$
$59$	−3.02854	−0.394282	−0.197141	+	0.980375i	$0.563166\pi$
$60$	0	0
$61$	1.05512	0.135095	0.0675473	−	0.997716i	$-0.478483\pi$
$61$	1.05512	0.135095	0.0675473	+	0.997716i	$0.478483\pi$
$62$	5.10562	0.648415
$63$	0.508505	0.0640656
$64$	3.59298	0.449122
$65$	0	0
$66$	0	0
$67$	−5.31327	−0.649120	−0.324560	−	0.945865i	$-0.605216\pi$
$67$	−5.31327	−0.649120	−0.324560	+	0.945865i	$0.605216\pi$
$68$	4.53171	0.549551
$69$	−2.91459	−0.350876
$70$	0	0
$71$	4.09976	0.486552	0.243276	−	0.969957i	$-0.421778\pi$
$71$	4.09976	0.486552	0.243276	+	0.969957i	$0.421778\pi$
$72$	2.69497	0.317606
$73$	−13.1083	−1.53422	−0.767108	−	0.641518i	$-0.778305\pi$
$73$	−13.1083	−1.53422	−0.767108	+	0.641518i	$0.778305\pi$
$74$	3.57673	0.415787
$75$	0	0
$76$	1.58900	0.182270
$77$	0	0
$78$	−4.12768	−0.467367
$79$	−16.1403	−1.81593	−0.907963	−	0.419051i	$-0.862363\pi$
$79$	−16.1403	−1.81593	−0.907963	+	0.419051i	$0.862363\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	2.24836	0.248290
$83$	−15.3587	−1.68584	−0.842920	−	0.538039i	$-0.819165\pi$
$83$	−15.3587	−1.68584	−0.842920	+	0.538039i	$0.819165\pi$
$84$	−0.688823	−0.0751567
$85$	0	0
$86$	−2.45308	−0.264522
$87$	−0.392594	−0.0420905
$88$	0	0
$89$	−9.84603	−1.04368	−0.521838	−	0.853044i	$-0.674754\pi$
$89$	−9.84603	−1.04368	−0.521838	+	0.853044i	$0.674754\pi$
$90$	0	0
$91$	2.61269	0.273884
$92$	3.94812	0.411620
$93$	−6.35529	−0.659013
$94$	−9.23252	−0.952262
$95$	0	0
$96$	−5.82710	−0.594726
$97$	−15.1692	−1.54020	−0.770098	−	0.637926i	$-0.779793\pi$
$97$	−15.1692	−1.54020	−0.770098	+	0.637926i	$0.779793\pi$
$98$	5.41583	0.547081
$99$	0	0
$100$	0	0
$101$	−16.5472	−1.64650	−0.823252	−	0.567677i	$-0.807842\pi$
$101$	−16.5472	−1.64650	−0.823252	+	0.567677i	$0.807842\pi$
$102$	2.68759	0.266111
$103$	14.3340	1.41237	0.706187	−	0.708025i	$-0.250414\pi$
$103$	14.3340	1.41237	0.706187	+	0.708025i	$0.250414\pi$
$104$	13.8467	1.35778
$105$	0	0
$106$	−7.87899	−0.765275
$107$	6.87095	0.664239	0.332120	−	0.943237i	$-0.392236\pi$
$107$	6.87095	0.664239	0.332120	+	0.943237i	$0.392236\pi$
$108$	−1.35460	−0.130347
$109$	16.9516	1.62367	0.811833	−	0.583890i	$-0.198470\pi$
$109$	16.9516	1.62367	0.811833	+	0.583890i	$0.198470\pi$
$110$	0	0
$111$	−4.45218	−0.422583
$112$	0.276707	0.0261464
$113$	−4.50913	−0.424184	−0.212092	−	0.977250i	$-0.568028\pi$
$113$	−4.50913	−0.424184	−0.212092	+	0.977250i	$0.568028\pi$
$114$	0.942375	0.0882615
$115$	0	0
$116$	0.531809	0.0493772
$117$	5.13798	0.475006
$118$	2.43303	0.223978
$119$	−1.70116	−0.155945
$120$	0	0
$121$	0	0
$122$	−0.847649	−0.0767425
$123$	−2.79867	−0.252348
$124$	8.60890	0.773102
$125$	0	0
$126$	−0.408516	−0.0363935
$127$	−10.6946	−0.948990	−0.474495	−	0.880258i	$-0.657369\pi$
$127$	−10.6946	−0.948990	−0.474495	+	0.880258i	$0.657369\pi$
$128$	8.76773	0.774966
$129$	3.05350	0.268846
$130$	0	0
$131$	−7.00557	−0.612079	−0.306040	−	0.952019i	$-0.599004\pi$
$131$	−7.00557	−0.612079	−0.306040	+	0.952019i	$0.599004\pi$
$132$	0	0
$133$	−0.596494	−0.0517226
$134$	4.26850	0.368742
$135$	0	0
$136$	−9.01581	−0.773099
$137$	11.0079	0.940465	0.470233	−	0.882542i	$-0.344170\pi$
$137$	11.0079	0.940465	0.470233	+	0.882542i	$0.344170\pi$
$138$	2.34148	0.199320
$139$	−14.2028	−1.20467	−0.602334	−	0.798244i	$-0.705763\pi$
$139$	−14.2028	−1.20467	−0.602334	+	0.798244i	$0.705763\pi$
$140$	0	0
$141$	11.4923	0.967827
$142$	−3.29361	−0.276393
$143$	0	0
$144$	0.544158	0.0453465
$145$	0	0
$146$	10.5308	0.871535
$147$	−6.74142	−0.556023
$148$	6.03094	0.495741
$149$	13.0575	1.06971	0.534854	−	0.844944i	$-0.320367\pi$
$149$	13.0575	1.06971	0.534854	+	0.844944i	$0.320367\pi$
$150$	0	0
$151$	18.3513	1.49340	0.746702	−	0.665158i	$-0.231636\pi$
$151$	18.3513	1.49340	0.746702	+	0.665158i	$0.231636\pi$
$152$	−3.16130	−0.256415
$153$	−3.34542	−0.270461
$154$	0	0
$155$	0	0
$156$	−6.95993	−0.557240
$157$	10.9402	0.873127	0.436563	−	0.899673i	$-0.356195\pi$
$157$	10.9402	0.873127	0.436563	+	0.899673i	$0.356195\pi$
$158$	12.9666	1.03156
$159$	9.80747	0.777783
$160$	0	0
$161$	−1.48208	−0.116805
$162$	−0.803366	−0.0631184
$163$	−23.0823	−1.80795	−0.903973	−	0.427589i	$-0.859363\pi$
$163$	−23.0823	−1.80795	−0.903973	+	0.427589i	$0.859363\pi$
$164$	3.79109	0.296035
$165$	0	0
$166$	12.3387	0.957667
$167$	−4.17425	−0.323013	−0.161507	−	0.986872i	$-0.551635\pi$
$167$	−4.17425	−0.323013	−0.161507	+	0.986872i	$0.551635\pi$
$168$	1.37041	0.105729
$169$	13.3988	1.03068
$170$	0	0
$171$	−1.17303	−0.0897042
$172$	−4.13628	−0.315388
$173$	−3.71693	−0.282593	−0.141297	−	0.989967i	$-0.545127\pi$
$173$	−3.71693	−0.282593	−0.141297	+	0.989967i	$0.545127\pi$
$174$	0.315396	0.0239101
$175$	0	0
$176$	0	0
$177$	−3.02854	−0.227639
$178$	7.90996	0.592876
$179$	17.8921	1.33732	0.668661	−	0.743568i	$-0.266868\pi$
$179$	17.8921	1.33732	0.668661	+	0.743568i	$0.266868\pi$
$180$	0	0
$181$	15.1181	1.12372	0.561859	−	0.827233i	$-0.310086\pi$
$181$	15.1181	1.12372	0.561859	+	0.827233i	$0.310086\pi$
$182$	−2.09894	−0.155584
$183$	1.05512	0.0779969
$184$	−7.85475	−0.579060
$185$	0	0
$186$	5.10562	0.374362
$187$	0	0
$188$	−15.5675	−1.13538
$189$	0.508505	0.0369883
$190$	0	0
$191$	0.745573	0.0539478	0.0269739	−	0.999636i	$-0.491413\pi$
$191$	0.745573	0.0539478	0.0269739	+	0.999636i	$0.491413\pi$
$192$	3.59298	0.259301
$193$	−8.87896	−0.639121	−0.319561	−	0.947566i	$-0.603535\pi$
$193$	−8.87896	−0.639121	−0.319561	+	0.947566i	$0.603535\pi$
$194$	12.1864	0.874931
$195$	0	0
$196$	9.13196	0.652283
$197$	−1.05280	−0.0750089	−0.0375044	−	0.999296i	$-0.511941\pi$
$197$	−1.05280	−0.0750089	−0.0375044	+	0.999296i	$0.511941\pi$
$198$	0	0
$199$	−2.28171	−0.161746	−0.0808730	−	0.996724i	$-0.525771\pi$
$199$	−2.28171	−0.161746	−0.0808730	+	0.996724i	$0.525771\pi$
$200$	0	0
$201$	−5.31327	−0.374769
$202$	13.2934	0.935321
$203$	−0.199636	−0.0140117
$204$	4.53171	0.317283
$205$	0	0
$206$	−11.5155	−0.802321
$207$	−2.91459	−0.202578
$208$	2.79587	0.193859
$209$	0	0
$210$	0	0
$211$	−9.56919	−0.658771	−0.329385	−	0.944196i	$-0.606841\pi$
$211$	−9.56919	−0.658771	−0.329385	+	0.944196i	$0.606841\pi$
$212$	−13.2852	−0.912434
$213$	4.09976	0.280911
$214$	−5.51988	−0.377331
$215$	0	0
$216$	2.69497	0.183370
$217$	−3.23170	−0.219382
$218$	−13.6183	−0.922348
$219$	−13.1083	−0.885780
$220$	0	0
$221$	−17.1887	−1.15624
$222$	3.57673	0.240054
$223$	8.81925	0.590581	0.295290	−	0.955408i	$-0.404584\pi$
$223$	8.81925	0.590581	0.295290	+	0.955408i	$0.404584\pi$
$224$	−2.96311	−0.197981
$225$	0	0
$226$	3.62248	0.240964
$227$	5.38764	0.357590	0.178795	−	0.983886i	$-0.442780\pi$
$227$	5.38764	0.357590	0.178795	+	0.983886i	$0.442780\pi$
$228$	1.58900	0.105234
$229$	24.3817	1.61119	0.805595	−	0.592466i	$-0.201846\pi$
$229$	24.3817	1.61119	0.805595	+	0.592466i	$0.201846\pi$
$230$	0	0
$231$	0	0
$232$	−1.05803	−0.0694631
$233$	−4.61992	−0.302661	−0.151331	−	0.988483i	$-0.548356\pi$
$233$	−4.61992	−0.302661	−0.151331	+	0.988483i	$0.548356\pi$
$234$	−4.12768	−0.269835
$235$	0	0
$236$	4.10247	0.267048
$237$	−16.1403	−1.04843
$238$	1.36665	0.0885870
$239$	4.81862	0.311691	0.155845	−	0.987781i	$-0.450190\pi$
$239$	4.81862	0.311691	0.155845	+	0.987781i	$0.450190\pi$
$240$	0	0
$241$	0.424606	0.0273513	0.0136757	−	0.999906i	$-0.495647\pi$
$241$	0.424606	0.0273513	0.0136757	+	0.999906i	$0.495647\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	−1.42927	−0.0914998
$245$	0	0
$246$	2.24836	0.143350
$247$	−6.02703	−0.383490
$248$	−17.1273	−1.08759
$249$	−15.3587	−0.973320
$250$	0	0
$251$	15.3627	0.969688	0.484844	−	0.874601i	$-0.338876\pi$
$251$	15.3627	0.969688	0.484844	+	0.874601i	$0.338876\pi$
$252$	−0.688823	−0.0433918
$253$	0	0
$254$	8.59165	0.539088
$255$	0	0
$256$	−14.2297	−0.889353
$257$	7.41474	0.462519	0.231259	−	0.972892i	$-0.425715\pi$
$257$	7.41474	0.462519	0.231259	+	0.972892i	$0.425715\pi$
$258$	−2.45308	−0.152722
$259$	−2.26396	−0.140676
$260$	0	0
$261$	−0.392594	−0.0243009
$262$	5.62803	0.347701
$263$	−30.6870	−1.89224	−0.946120	−	0.323815i	$-0.895034\pi$
$263$	−30.6870	−1.89224	−0.946120	+	0.323815i	$0.895034\pi$
$264$	0	0
$265$	0	0
$266$	0.479203	0.0293818
$267$	−9.84603	−0.602567
$268$	7.19738	0.439650
$269$	18.2413	1.11219	0.556095	−	0.831119i	$-0.312299\pi$
$269$	18.2413	1.11219	0.556095	+	0.831119i	$0.312299\pi$
$270$	0	0
$271$	13.7411	0.834714	0.417357	−	0.908743i	$-0.362956\pi$
$271$	13.7411	0.834714	0.417357	+	0.908743i	$0.362956\pi$
$272$	−1.82044	−0.110380
$273$	2.61269	0.158127
$274$	−8.84334	−0.534246
$275$	0	0
$276$	3.94812	0.237649
$277$	−1.37752	−0.0827674	−0.0413837	−	0.999143i	$-0.513177\pi$
$277$	−1.37752	−0.0827674	−0.0413837	+	0.999143i	$0.513177\pi$
$278$	11.4101	0.684330
$279$	−6.35529	−0.380481
$280$	0	0
$281$	−19.3025	−1.15149	−0.575746	−	0.817629i	$-0.695288\pi$
$281$	−19.3025	−1.15149	−0.575746	+	0.817629i	$0.695288\pi$
$282$	−9.23252	−0.549789
$283$	−2.19620	−0.130551	−0.0652754	−	0.997867i	$-0.520793\pi$
$283$	−2.19620	−0.130551	−0.0652754	+	0.997867i	$0.520793\pi$
$284$	−5.55355	−0.329543
$285$	0	0
$286$	0	0
$287$	−1.42314	−0.0840053
$288$	−5.82710	−0.343365
$289$	−5.80819	−0.341658
$290$	0	0
$291$	−15.1692	−0.889232
$292$	17.7566	1.03913
$293$	−27.3180	−1.59594	−0.797968	−	0.602700i	$-0.794091\pi$
$293$	−27.3180	−1.59594	−0.797968	+	0.602700i	$0.794091\pi$
$294$	5.41583	0.315857
$295$	0	0
$296$	−11.9985	−0.697400
$297$	0	0
$298$	−10.4899	−0.607664
$299$	−14.9751	−0.866033
$300$	0	0
$301$	1.55272	0.0894973
$302$	−14.7428	−0.848351
$303$	−16.5472	−0.950609
$304$	−0.638317	−0.0366100
$305$	0	0
$306$	2.68759	0.153639
$307$	26.9287	1.53690	0.768451	−	0.639908i	$-0.221028\pi$
$307$	26.9287	1.53690	0.768451	+	0.639908i	$0.221028\pi$
$308$	0	0
$309$	14.3340	0.815434
$310$	0	0
$311$	11.9344	0.676740	0.338370	−	0.941013i	$-0.390124\pi$
$311$	11.9344	0.676740	0.338370	+	0.941013i	$0.390124\pi$
$312$	13.8467	0.783916
$313$	14.9612	0.845658	0.422829	−	0.906209i	$-0.361037\pi$
$313$	14.9612	0.845658	0.422829	+	0.906209i	$0.361037\pi$
$314$	−8.78902	−0.495993
$315$	0	0
$316$	21.8637	1.22993
$317$	9.78534	0.549600	0.274800	−	0.961501i	$-0.411388\pi$
$317$	9.78534	0.549600	0.274800	+	0.961501i	$0.411388\pi$
$318$	−7.87899	−0.441832
$319$	0	0
$320$	0	0
$321$	6.87095	0.383499
$322$	1.19066	0.0663527
$323$	3.92429	0.218353
$324$	−1.35460	−0.0752558
$325$	0	0
$326$	18.5435	1.02703
$327$	16.9516	0.937424
$328$	−7.54235	−0.416456
$329$	5.84390	0.322184
$330$	0	0
$331$	5.14693	0.282901	0.141450	−	0.989945i	$-0.454823\pi$
$331$	5.14693	0.282901	0.141450	+	0.989945i	$0.454823\pi$
$332$	20.8050	1.14182
$333$	−4.45218	−0.243978
$334$	3.35345	0.183492
$335$	0	0
$336$	0.276707	0.0150956
$337$	−29.3514	−1.59887	−0.799437	−	0.600750i	$-0.794869\pi$
$337$	−29.3514	−1.59887	−0.799437	+	0.600750i	$0.794869\pi$
$338$	−10.7642	−0.585493
$339$	−4.50913	−0.244902
$340$	0	0
$341$	0	0
$342$	0.942375	0.0509578
$343$	−6.98758	−0.377294
$344$	8.22910	0.443683
$345$	0	0
$346$	2.98606	0.160531
$347$	−7.33591	−0.393812	−0.196906	−	0.980422i	$-0.563089\pi$
$347$	−7.33591	−0.393812	−0.196906	+	0.980422i	$0.563089\pi$
$348$	0.531809	0.0285080
$349$	−28.5268	−1.52701	−0.763503	−	0.645804i	$-0.776522\pi$
$349$	−28.5268	−1.52701	−0.763503	+	0.645804i	$0.776522\pi$
$350$	0	0
$351$	5.13798	0.274245
$352$	0	0
$353$	−16.1818	−0.861271	−0.430636	−	0.902526i	$-0.641711\pi$
$353$	−16.1818	−0.861271	−0.430636	+	0.902526i	$0.641711\pi$
$354$	2.43303	0.129314
$355$	0	0
$356$	13.3375	0.706884
$357$	−1.70116	−0.0900350
$358$	−14.3739	−0.759686
$359$	16.9324	0.893659	0.446829	−	0.894619i	$-0.352553\pi$
$359$	16.9324	0.893659	0.446829	+	0.894619i	$0.352553\pi$
$360$	0	0
$361$	−17.6240	−0.927578
$362$	−12.1453	−0.638345
$363$	0	0
$364$	−3.53916	−0.185502
$365$	0	0
$366$	−0.847649	−0.0443073
$367$	−4.02281	−0.209989	−0.104994	−	0.994473i	$-0.533482\pi$
$367$	−4.02281	−0.209989	−0.104994	+	0.994473i	$0.533482\pi$
$368$	−1.58600	−0.0826759
$369$	−2.79867	−0.145693
$370$	0	0
$371$	4.98715	0.258920
$372$	8.60890	0.446351
$373$	−36.2624	−1.87760	−0.938798	−	0.344469i	$-0.888059\pi$
$373$	−36.2624	−1.87760	−0.938798	+	0.344469i	$0.888059\pi$
$374$	0	0
$375$	0	0
$376$	30.9715	1.59723
$377$	−2.01714	−0.103888
$378$	−0.408516	−0.0210118
$379$	−19.8668	−1.02049	−0.510243	−	0.860030i	$-0.670445\pi$
$379$	−19.8668	−1.02049	−0.510243	+	0.860030i	$0.670445\pi$
$380$	0	0
$381$	−10.6946	−0.547899
$382$	−0.598968	−0.0306459
$383$	−24.4491	−1.24929	−0.624646	−	0.780908i	$-0.714757\pi$
$383$	−24.4491	−1.24929	−0.624646	+	0.780908i	$0.714757\pi$
$384$	8.76773	0.447427
$385$	0	0
$386$	7.13305	0.363063
$387$	3.05350	0.155218
$388$	20.5482	1.04318
$389$	−12.0876	−0.612866	−0.306433	−	0.951892i	$-0.599135\pi$
$389$	−12.0876	−0.612866	−0.306433	+	0.951892i	$0.599135\pi$
$390$	0	0
$391$	9.75052	0.493105
$392$	−18.1680	−0.917620
$393$	−7.00557	−0.353384
$394$	0.845783	0.0426099
$395$	0	0
$396$	0	0
$397$	−22.7158	−1.14007	−0.570037	−	0.821619i	$-0.693071\pi$
$397$	−22.7158	−1.14007	−0.570037	+	0.821619i	$0.693071\pi$
$398$	1.83305	0.0918823
$399$	−0.596494	−0.0298620
$400$	0	0
$401$	−2.35795	−0.117750	−0.0588751	−	0.998265i	$-0.518751\pi$
$401$	−2.35795	−0.117750	−0.0588751	+	0.998265i	$0.518751\pi$
$402$	4.26850	0.212893
$403$	−32.6533	−1.62658
$404$	22.4148	1.11518
$405$	0	0
$406$	0.160381	0.00795956
$407$	0	0
$408$	−9.01581	−0.446349
$409$	−2.74377	−0.135671	−0.0678353	−	0.997697i	$-0.521609\pi$
$409$	−2.74377	−0.135671	−0.0678353	+	0.997697i	$0.521609\pi$
$410$	0	0
$411$	11.0079	0.542978
$412$	−19.4169	−0.956603
$413$	−1.54003	−0.0757799
$414$	2.34148	0.115078
$415$	0	0
$416$	−29.9395	−1.46791
$417$	−14.2028	−0.695516
$418$	0	0
$419$	−13.9861	−0.683266	−0.341633	−	0.939833i	$-0.610980\pi$
$419$	−13.9861	−0.683266	−0.341633	+	0.939833i	$0.610980\pi$
$420$	0	0
$421$	−17.7744	−0.866273	−0.433137	−	0.901328i	$-0.642593\pi$
$421$	−17.7744	−0.866273	−0.433137	+	0.901328i	$0.642593\pi$
$422$	7.68756	0.374225
$423$	11.4923	0.558775
$424$	26.4309	1.28360
$425$	0	0
$426$	−3.29361	−0.159576
$427$	0.536535	0.0259647
$428$	−9.30741	−0.449891
$429$	0	0
$430$	0	0
$431$	7.11619	0.342775	0.171387	−	0.985204i	$-0.445175\pi$
$431$	7.11619	0.342775	0.171387	+	0.985204i	$0.445175\pi$
$432$	0.544158	0.0261808
$433$	−11.3817	−0.546970	−0.273485	−	0.961876i	$-0.588176\pi$
$433$	−11.3817	−0.546970	−0.273485	+	0.961876i	$0.588176\pi$
$434$	2.59623	0.124623
$435$	0	0
$436$	−22.9626	−1.09971
$437$	3.41892	0.163549
$438$	10.5308	0.503181
$439$	−16.1917	−0.772788	−0.386394	−	0.922334i	$-0.626279\pi$
$439$	−16.1917	−0.772788	−0.386394	+	0.922334i	$0.626279\pi$
$440$	0	0
$441$	−6.74142	−0.321020
$442$	13.8088	0.656817
$443$	23.7522	1.12850	0.564250	−	0.825604i	$-0.309166\pi$
$443$	23.7522	1.12850	0.564250	+	0.825604i	$0.309166\pi$
$444$	6.03094	0.286216
$445$	0	0
$446$	−7.08508	−0.335488
$447$	13.0575	0.617596
$448$	1.82705	0.0863199
$449$	−20.6589	−0.974954	−0.487477	−	0.873136i	$-0.662083\pi$
$449$	−20.6589	−0.974954	−0.487477	+	0.873136i	$0.662083\pi$
$450$	0	0
$451$	0	0
$452$	6.10809	0.287300
$453$	18.3513	0.862218
$454$	−4.32824	−0.203135
$455$	0	0
$456$	−3.16130	−0.148041
$457$	−7.72035	−0.361143	−0.180571	−	0.983562i	$-0.557795\pi$
$457$	−7.72035	−0.361143	−0.180571	+	0.983562i	$0.557795\pi$
$458$	−19.5874	−0.915261
$459$	−3.34542	−0.156151
$460$	0	0
$461$	41.3034	1.92369	0.961845	−	0.273595i	$-0.0882128\pi$
$461$	41.3034	1.92369	0.961845	+	0.273595i	$0.0882128\pi$
$462$	0	0
$463$	−19.5424	−0.908214	−0.454107	−	0.890947i	$-0.650042\pi$
$463$	−19.5424	−0.908214	−0.454107	+	0.890947i	$0.650042\pi$
$464$	−0.213633	−0.00991767
$465$	0	0
$466$	3.71149	0.171931
$467$	21.3613	0.988483	0.494242	−	0.869325i	$-0.335446\pi$
$467$	21.3613	0.988483	0.494242	+	0.869325i	$0.335446\pi$
$468$	−6.95993	−0.321723
$469$	−2.70183	−0.124759
$470$	0	0
$471$	10.9402	0.504100
$472$	−8.16184	−0.375679
$473$	0	0
$474$	12.9666	0.595574
$475$	0	0
$476$	2.30440	0.105622
$477$	9.80747	0.449053
$478$	−3.87111	−0.177061
$479$	−15.4162	−0.704383	−0.352191	−	0.935928i	$-0.614563\pi$
$479$	−15.4162	−0.704383	−0.352191	+	0.935928i	$0.614563\pi$
$480$	0	0
$481$	−22.8752	−1.04302
$482$	−0.341114	−0.0155373
$483$	−1.48208	−0.0674372
$484$	0	0
$485$	0	0
$486$	−0.803366	−0.0364414
$487$	−25.1124	−1.13795	−0.568976	−	0.822354i	$-0.692660\pi$
$487$	−25.1124	−1.13795	−0.568976	+	0.822354i	$0.692660\pi$
$488$	2.84353	0.128720
$489$	−23.0823	−1.04382
$490$	0	0
$491$	1.10386	0.0498165	0.0249083	−	0.999690i	$-0.492071\pi$
$491$	1.10386	0.0498165	0.0249083	+	0.999690i	$0.492071\pi$
$492$	3.79109	0.170916
$493$	1.31339	0.0591521
$494$	4.84190	0.217848
$495$	0	0
$496$	−3.45829	−0.155282
$497$	2.08475	0.0935138
$498$	12.3387	0.552909
$499$	2.29718	0.102836	0.0514179	−	0.998677i	$-0.483626\pi$
$499$	2.29718	0.102836	0.0514179	+	0.998677i	$0.483626\pi$
$500$	0	0
$501$	−4.17425	−0.186492
$502$	−12.3419	−0.550846
$503$	−6.85667	−0.305724	−0.152862	−	0.988248i	$-0.548849\pi$
$503$	−6.85667	−0.305724	−0.152862	+	0.988248i	$0.548849\pi$
$504$	1.37041	0.0610428
$505$	0	0
$506$	0	0
$507$	13.3988	0.595063
$508$	14.4869	0.642752
$509$	25.8834	1.14726	0.573630	−	0.819114i	$-0.305535\pi$
$509$	25.8834	1.14726	0.573630	+	0.819114i	$0.305535\pi$
$510$	0	0
$511$	−6.66566	−0.294871
$512$	−6.10385	−0.269755
$513$	−1.17303	−0.0517907
$514$	−5.95675	−0.262741
$515$	0	0
$516$	−4.13628	−0.182090
$517$	0	0
$518$	1.81879	0.0799129
$519$	−3.71693	−0.163155
$520$	0	0
$521$	−28.7809	−1.26091	−0.630456	−	0.776225i	$-0.717132\pi$
$521$	−28.7809	−1.26091	−0.630456	+	0.776225i	$0.717132\pi$
$522$	0.315396	0.0138045
$523$	−29.9930	−1.31150	−0.655752	−	0.754976i	$-0.727648\pi$
$523$	−29.9930	−1.31150	−0.655752	+	0.754976i	$0.727648\pi$
$524$	9.48977	0.414563
$525$	0	0
$526$	24.6529	1.07492
$527$	21.2611	0.926148
$528$	0	0
$529$	−14.5052	−0.630659
$530$	0	0
$531$	−3.02854	−0.131427
$532$	0.808013	0.0350318
$533$	−14.3795	−0.622846
$534$	7.90996	0.342297
$535$	0	0
$536$	−14.3191	−0.618492
$537$	17.8921	0.772103
$538$	−14.6544	−0.631796
$539$	0	0
$540$	0	0
$541$	2.65449	0.114126	0.0570628	−	0.998371i	$-0.481826\pi$
$541$	2.65449	0.114126	0.0570628	+	0.998371i	$0.481826\pi$
$542$	−11.0392	−0.474172
$543$	15.1181	0.648779
$544$	19.4941	0.835802
$545$	0	0
$546$	−2.09894	−0.0898265
$547$	−3.36255	−0.143772	−0.0718861	−	0.997413i	$-0.522902\pi$
$547$	−3.36255	−0.143772	−0.0718861	+	0.997413i	$0.522902\pi$
$548$	−14.9113	−0.636979
$549$	1.05512	0.0450315
$550$	0	0
$551$	0.460526	0.0196191
$552$	−7.85475	−0.334320
$553$	−8.20743	−0.349015
$554$	1.10666	0.0470173
$555$	0	0
$556$	19.2392	0.815924
$557$	−24.5074	−1.03841	−0.519206	−	0.854649i	$-0.673772\pi$
$557$	−24.5074	−1.03841	−0.519206	+	0.854649i	$0.673772\pi$
$558$	5.10562	0.216138
$559$	15.6888	0.663566
$560$	0	0
$561$	0	0
$562$	15.5070	0.654123
$563$	8.01826	0.337929	0.168965	−	0.985622i	$-0.445958\pi$
$563$	8.01826	0.337929	0.168965	+	0.985622i	$0.445958\pi$
$564$	−15.5675	−0.655511
$565$	0	0
$566$	1.76435	0.0741613
$567$	0.508505	0.0213552
$568$	11.0487	0.463595
$569$	9.51091	0.398718	0.199359	−	0.979927i	$-0.436114\pi$
$569$	9.51091	0.398718	0.199359	+	0.979927i	$0.436114\pi$
$570$	0	0
$571$	−11.6232	−0.486415	−0.243208	−	0.969974i	$-0.578200\pi$
$571$	−11.6232	−0.486415	−0.243208	+	0.969974i	$0.578200\pi$
$572$	0	0
$573$	0.745573	0.0311468
$574$	1.14330	0.0477205
$575$	0	0
$576$	3.59298	0.149707
$577$	−9.73590	−0.405311	−0.202655	−	0.979250i	$-0.564957\pi$
$577$	−9.73590	−0.405311	−0.202655	+	0.979250i	$0.564957\pi$
$578$	4.66610	0.194084
$579$	−8.87896	−0.368997
$580$	0	0
$581$	−7.80999	−0.324013
$582$	12.1864	0.505142
$583$	0	0
$584$	−35.3266	−1.46183
$585$	0	0
$586$	21.9463	0.906595
$587$	−6.35480	−0.262291	−0.131145	−	0.991363i	$-0.541865\pi$
$587$	−6.35480	−0.262291	−0.131145	+	0.991363i	$0.541865\pi$
$588$	9.13196	0.376596
$589$	7.45497	0.307177
$590$	0	0
$591$	−1.05280	−0.0433064
$592$	−2.42269	−0.0995721
$593$	−4.58134	−0.188133	−0.0940666	−	0.995566i	$-0.529987\pi$
$593$	−4.58134	−0.188133	−0.0940666	+	0.995566i	$0.529987\pi$
$594$	0	0
$595$	0	0
$596$	−17.6877	−0.724516
$597$	−2.28171	−0.0933841
$598$	12.0305	0.491963
$599$	−9.26327	−0.378487	−0.189244	−	0.981930i	$-0.560604\pi$
$599$	−9.26327	−0.378487	−0.189244	+	0.981930i	$0.560604\pi$
$600$	0	0
$601$	−10.4939	−0.428054	−0.214027	−	0.976828i	$-0.568658\pi$
$601$	−10.4939	−0.428054	−0.214027	+	0.976828i	$0.568658\pi$
$602$	−1.24740	−0.0508403
$603$	−5.31327	−0.216373
$604$	−24.8587	−1.01149
$605$	0	0
$606$	13.2934	0.540008
$607$	−20.8000	−0.844244	−0.422122	−	0.906539i	$-0.638715\pi$
$607$	−20.8000	−0.844244	−0.422122	+	0.906539i	$0.638715\pi$
$608$	6.83539	0.277212
$609$	−0.199636	−0.00808966
$610$	0	0
$611$	59.0472	2.38879
$612$	4.53171	0.183184
$613$	−24.7575	−0.999945	−0.499973	−	0.866041i	$-0.666657\pi$
$613$	−24.7575	−0.999945	−0.499973	+	0.866041i	$0.666657\pi$
$614$	−21.6336	−0.873061
$615$	0	0
$616$	0	0
$617$	−28.2943	−1.13909	−0.569543	−	0.821962i	$-0.692880\pi$
$617$	−28.2943	−1.13909	−0.569543	+	0.821962i	$0.692880\pi$
$618$	−11.5155	−0.463220
$619$	−26.6041	−1.06931	−0.534654	−	0.845071i	$-0.679558\pi$
$619$	−26.6041	−1.06931	−0.534654	+	0.845071i	$0.679558\pi$
$620$	0	0
$621$	−2.91459	−0.116959
$622$	−9.58772	−0.384432
$623$	−5.00675	−0.200591
$624$	2.79587	0.111925
$625$	0	0
$626$	−12.0193	−0.480389
$627$	0	0
$628$	−14.8197	−0.591370
$629$	14.8944	0.593879
$630$	0	0
$631$	16.4724	0.655757	0.327879	−	0.944720i	$-0.393666\pi$
$631$	16.4724	0.655757	0.327879	+	0.944720i	$0.393666\pi$
$632$	−43.4977	−1.73024
$633$	−9.56919	−0.380341
$634$	−7.86121	−0.312208
$635$	0	0
$636$	−13.2852	−0.526794
$637$	−34.6373	−1.37238
$638$	0	0
$639$	4.09976	0.162184
$640$	0	0
$641$	7.15197	0.282486	0.141243	−	0.989975i	$-0.454890\pi$
$641$	7.15197	0.282486	0.141243	+	0.989975i	$0.454890\pi$
$642$	−5.51988	−0.217852
$643$	−17.7623	−0.700477	−0.350238	−	0.936661i	$-0.613899\pi$
$643$	−17.7623	−0.700477	−0.350238	+	0.936661i	$0.613899\pi$
$644$	2.00764	0.0791120
$645$	0	0
$646$	−3.15264	−0.124039
$647$	−1.43164	−0.0562836	−0.0281418	−	0.999604i	$-0.508959\pi$
$647$	−1.43164	−0.0562836	−0.0281418	+	0.999604i	$0.508959\pi$
$648$	2.69497	0.105869
$649$	0	0
$650$	0	0
$651$	−3.23170	−0.126660
$652$	31.2674	1.22453
$653$	29.2874	1.14611	0.573053	−	0.819518i	$-0.305759\pi$
$653$	29.2874	1.14611	0.573053	+	0.819518i	$0.305759\pi$
$654$	−13.6183	−0.532518
$655$	0	0
$656$	−1.52292	−0.0594601
$657$	−13.1083	−0.511405
$658$	−4.69479	−0.183022
$659$	21.2056	0.826054	0.413027	−	0.910719i	$-0.364471\pi$
$659$	21.2056	0.826054	0.413027	+	0.910719i	$0.364471\pi$
$660$	0	0
$661$	−16.5453	−0.643537	−0.321768	−	0.946818i	$-0.604277\pi$
$661$	−16.5453	−0.643537	−0.321768	+	0.946818i	$0.604277\pi$
$662$	−4.13486	−0.160706
$663$	−17.1887	−0.667553
$664$	−41.3914	−1.60630
$665$	0	0
$666$	3.57673	0.138596
$667$	1.14425	0.0443056
$668$	5.65445	0.218777
$669$	8.81925	0.340972
$670$	0	0
$671$	0	0
$672$	−2.96311	−0.114305
$673$	−18.9533	−0.730597	−0.365299	−	0.930890i	$-0.619033\pi$
$673$	−18.9533	−0.730597	−0.365299	+	0.930890i	$0.619033\pi$
$674$	23.5799	0.908265
$675$	0	0
$676$	−18.1501	−0.698081
$677$	−33.0448	−1.27002	−0.635008	−	0.772506i	$-0.719003\pi$
$677$	−33.0448	−1.27002	−0.635008	+	0.772506i	$0.719003\pi$
$678$	3.62248	0.139121
$679$	−7.71360	−0.296021
$680$	0	0
$681$	5.38764	0.206455
$682$	0	0
$683$	26.1102	0.999079	0.499540	−	0.866291i	$-0.333503\pi$
$683$	26.1102	0.999079	0.499540	+	0.866291i	$0.333503\pi$
$684$	1.58900	0.0607568
$685$	0	0
$686$	5.61358	0.214328
$687$	24.3817	0.930221
$688$	1.66159	0.0633474
$689$	50.3906	1.91973
$690$	0	0
$691$	−23.8340	−0.906690	−0.453345	−	0.891335i	$-0.649769\pi$
$691$	−23.8340	−0.906690	−0.453345	+	0.891335i	$0.649769\pi$
$692$	5.03497	0.191401
$693$	0	0
$694$	5.89342	0.223711
$695$	0	0
$696$	−1.05803	−0.0401045
$697$	9.36272	0.354638
$698$	22.9175	0.867439
$699$	−4.61992	−0.174741
$700$	0	0
$701$	16.3561	0.617761	0.308880	−	0.951101i	$-0.400046\pi$
$701$	16.3561	0.617761	0.308880	+	0.951101i	$0.400046\pi$
$702$	−4.12768	−0.155789
$703$	5.22256	0.196973
$704$	0	0
$705$	0	0
$706$	12.9999	0.489258
$707$	−8.41431	−0.316453
$708$	4.10247	0.154180
$709$	15.0833	0.566464	0.283232	−	0.959051i	$-0.408593\pi$
$709$	15.0833	0.566464	0.283232	+	0.959051i	$0.408593\pi$
$710$	0	0
$711$	−16.1403	−0.605309
$712$	−26.5348	−0.994433
$713$	18.5231	0.693695
$714$	1.36665	0.0511458
$715$	0	0
$716$	−24.2368	−0.905770
$717$	4.81862	0.179955
$718$	−13.6029	−0.507657
$719$	22.6232	0.843702	0.421851	−	0.906665i	$-0.361381\pi$
$719$	22.6232	0.843702	0.421851	+	0.906665i	$0.361381\pi$
$720$	0	0
$721$	7.28893	0.271454
$722$	14.1585	0.526925
$723$	0.424606	0.0157913
$724$	−20.4790	−0.761096
$725$	0	0
$726$	0	0
$727$	20.2062	0.749408	0.374704	−	0.927144i	$-0.377744\pi$
$727$	20.2062	0.749408	0.374704	+	0.927144i	$0.377744\pi$
$728$	7.04113	0.260961
$729$	1.00000	0.0370370
$730$	0	0
$731$	−10.2152	−0.377824
$732$	−1.42927	−0.0528274
$733$	1.50517	0.0555947	0.0277974	−	0.999614i	$-0.491151\pi$
$733$	1.50517	0.0555947	0.0277974	+	0.999614i	$0.491151\pi$
$734$	3.23178	0.119287
$735$	0	0
$736$	16.9836	0.626025
$737$	0	0
$738$	2.24836	0.0827632
$739$	38.7837	1.42668	0.713340	−	0.700818i	$-0.247182\pi$
$739$	38.7837	1.42668	0.713340	+	0.700818i	$0.247182\pi$
$740$	0	0
$741$	−6.02703	−0.221408
$742$	−4.00651	−0.147083
$743$	−26.8606	−0.985419	−0.492710	−	0.870194i	$-0.663993\pi$
$743$	−26.8606	−0.985419	−0.492710	+	0.870194i	$0.663993\pi$
$744$	−17.1273	−0.627919
$745$	0	0
$746$	29.1320	1.06660
$747$	−15.3587	−0.561947
$748$	0	0
$749$	3.49391	0.127665
$750$	0	0
$751$	26.0025	0.948845	0.474423	−	0.880297i	$-0.342657\pi$
$751$	26.0025	0.948845	0.474423	+	0.880297i	$0.342657\pi$
$752$	6.25364	0.228047
$753$	15.3627	0.559850
$754$	1.62050	0.0590151
$755$	0	0
$756$	−0.688823	−0.0250522
$757$	−29.5909	−1.07550	−0.537749	−	0.843105i	$-0.680725\pi$
$757$	−29.5909	−1.07550	−0.537749	+	0.843105i	$0.680725\pi$
$758$	15.9603	0.579703
$759$	0	0
$760$	0	0
$761$	33.0705	1.19880	0.599401	−	0.800449i	$-0.295405\pi$
$761$	33.0705	1.19880	0.599401	+	0.800449i	$0.295405\pi$
$762$	8.59165	0.311243
$763$	8.61996	0.312063
$764$	−1.00996	−0.0365389
$765$	0	0
$766$	19.6416	0.709679
$767$	−15.5606	−0.561860
$768$	−14.2297	−0.513468
$769$	3.37489	0.121702	0.0608508	−	0.998147i	$-0.480619\pi$
$769$	3.37489	0.121702	0.0608508	+	0.998147i	$0.480619\pi$
$770$	0	0
$771$	7.41474	0.267035
$772$	12.0275	0.432878
$773$	−23.1887	−0.834041	−0.417020	−	0.908897i	$-0.636926\pi$
$773$	−23.1887	−0.834041	−0.417020	+	0.908897i	$0.636926\pi$
$774$	−2.45308	−0.0881740
$775$	0	0
$776$	−40.8805	−1.46752
$777$	−2.26396	−0.0812190
$778$	9.71076	0.348148
$779$	3.28294	0.117623
$780$	0	0
$781$	0	0
$782$	−7.83323	−0.280116
$783$	−0.392594	−0.0140302
$784$	−3.66840	−0.131014
$785$	0	0
$786$	5.62803	0.200745
$787$	41.0681	1.46392	0.731959	−	0.681348i	$-0.238606\pi$
$787$	41.0681	1.46392	0.731959	+	0.681348i	$0.238606\pi$
$788$	1.42613	0.0508036
$789$	−30.6870	−1.09249
$790$	0	0
$791$	−2.29292	−0.0815267
$792$	0	0
$793$	5.42119	0.192512
$794$	18.2491	0.647637
$795$	0	0
$796$	3.09081	0.109551
$797$	48.2620	1.70953	0.854765	−	0.519016i	$-0.173701\pi$
$797$	48.2620	1.70953	0.854765	+	0.519016i	$0.173701\pi$
$798$	0.479203	0.0169636
$799$	−38.4465	−1.36014
$800$	0	0
$801$	−9.84603	−0.347892
$802$	1.89429	0.0668898
$803$	0	0
$804$	7.19738	0.253832
$805$	0	0
$806$	26.2326	0.924003
$807$	18.2413	0.642123
$808$	−44.5941	−1.56882
$809$	−43.8433	−1.54145	−0.770724	−	0.637170i	$-0.780105\pi$
$809$	−43.8433	−1.54145	−0.770724	+	0.637170i	$0.780105\pi$
$810$	0	0
$811$	−41.3933	−1.45351	−0.726757	−	0.686895i	$-0.758973\pi$
$811$	−41.3933	−1.45351	−0.726757	+	0.686895i	$0.758973\pi$
$812$	0.270428	0.00949015
$813$	13.7411	0.481922
$814$	0	0
$815$	0	0
$816$	−1.82044	−0.0637280
$817$	−3.58186	−0.125313
$818$	2.20425	0.0770698
$819$	2.61269	0.0912947
$820$	0	0
$821$	42.9590	1.49928	0.749639	−	0.661847i	$-0.230227\pi$
$821$	42.9590	1.49928	0.749639	+	0.661847i	$0.230227\pi$
$822$	−8.84334	−0.308447
$823$	−14.0263	−0.488925	−0.244462	−	0.969659i	$-0.578611\pi$
$823$	−14.0263	−0.488925	−0.244462	+	0.969659i	$0.578611\pi$
$824$	38.6298	1.34573
$825$	0	0
$826$	1.23721	0.0430479
$827$	−30.9045	−1.07465	−0.537327	−	0.843374i	$-0.680566\pi$
$827$	−30.9045	−1.07465	−0.537327	+	0.843374i	$0.680566\pi$
$828$	3.94812	0.137207
$829$	36.9798	1.28436	0.642181	−	0.766553i	$-0.278030\pi$
$829$	36.9798	1.28436	0.642181	+	0.766553i	$0.278030\pi$
$830$	0	0
$831$	−1.37752	−0.0477858
$832$	18.4606	0.640008
$833$	22.5529	0.781410
$834$	11.4101	0.395098
$835$	0	0
$836$	0	0
$837$	−6.35529	−0.219671
$838$	11.2360	0.388140
$839$	2.20030	0.0759626	0.0379813	−	0.999278i	$-0.487907\pi$
$839$	2.20030	0.0759626	0.0379813	+	0.999278i	$0.487907\pi$
$840$	0	0
$841$	−28.8459	−0.994685
$842$	14.2794	0.492100
$843$	−19.3025	−0.664814
$844$	12.9625	0.446186
$845$	0	0
$846$	−9.23252	−0.317421
$847$	0	0
$848$	5.33682	0.183267
$849$	−2.19620	−0.0753735
$850$	0	0
$851$	12.9763	0.444822
$852$	−5.55355	−0.190262
$853$	33.1729	1.13582	0.567909	−	0.823092i	$-0.307753\pi$
$853$	33.1729	1.13582	0.567909	+	0.823092i	$0.307753\pi$
$854$	−0.431034	−0.0147497
$855$	0	0
$856$	18.5170	0.632899
$857$	−28.5419	−0.974972	−0.487486	−	0.873131i	$-0.662086\pi$
$857$	−28.5419	−0.974972	−0.487486	+	0.873131i	$0.662086\pi$
$858$	0	0
$859$	53.6600	1.83085	0.915427	−	0.402484i	$-0.131853\pi$
$859$	53.6600	1.83085	0.915427	+	0.402484i	$0.131853\pi$
$860$	0	0
$861$	−1.42314	−0.0485005
$862$	−5.71690	−0.194719
$863$	19.0942	0.649974	0.324987	−	0.945719i	$-0.394640\pi$
$863$	19.0942	0.649974	0.324987	+	0.945719i	$0.394640\pi$
$864$	−5.82710	−0.198242
$865$	0	0
$866$	9.14368	0.310715
$867$	−5.80819	−0.197257
$868$	4.37767	0.148588
$869$	0	0
$870$	0	0
$871$	−27.2995	−0.925008
$872$	45.6840	1.54706
$873$	−15.1692	−0.513398
$874$	−2.74664	−0.0929065
$875$	0	0
$876$	17.7566	0.599940
$877$	13.8148	0.466494	0.233247	−	0.972418i	$-0.425065\pi$
$877$	13.8148	0.466494	0.233247	+	0.972418i	$0.425065\pi$
$878$	13.0079	0.438994
$879$	−27.3180	−0.921414
$880$	0	0
$881$	−14.7416	−0.496658	−0.248329	−	0.968676i	$-0.579881\pi$
$881$	−14.7416	−0.496658	−0.248329	+	0.968676i	$0.579881\pi$
$882$	5.41583	0.182360
$883$	−0.165486	−0.00556904	−0.00278452	−	0.999996i	$-0.500886\pi$
$883$	−0.165486	−0.00556904	−0.00278452	+	0.999996i	$0.500886\pi$
$884$	23.2838	0.783120
$885$	0	0
$886$	−19.0817	−0.641061
$887$	13.8749	0.465874	0.232937	−	0.972492i	$-0.425166\pi$
$887$	13.8749	0.465874	0.232937	+	0.972492i	$0.425166\pi$
$888$	−11.9985	−0.402644
$889$	−5.43824	−0.182393
$890$	0	0
$891$	0	0
$892$	−11.9466	−0.400001
$893$	−13.4809	−0.451120
$894$	−10.4899	−0.350835
$895$	0	0
$896$	4.45844	0.148946
$897$	−14.9751	−0.500004
$898$	16.5967	0.553838
$899$	2.49505	0.0832145
$900$	0	0
$901$	−32.8101	−1.09306
$902$	0	0
$903$	1.55272	0.0516713
$904$	−12.1520	−0.404169
$905$	0	0
$906$	−14.7428	−0.489796
$907$	9.10404	0.302295	0.151147	−	0.988511i	$-0.451703\pi$
$907$	9.10404	0.302295	0.151147	+	0.988511i	$0.451703\pi$
$908$	−7.29811	−0.242196
$909$	−16.5472	−0.548834
$910$	0	0
$911$	−45.2779	−1.50012	−0.750062	−	0.661367i	$-0.769977\pi$
$911$	−45.2779	−1.50012	−0.750062	+	0.661367i	$0.769977\pi$
$912$	−0.638317	−0.0211368
$913$	0	0
$914$	6.20227	0.205153
$915$	0	0
$916$	−33.0276	−1.09126
$917$	−3.56237	−0.117640
$918$	2.68759	0.0887038
$919$	−14.8633	−0.490296	−0.245148	−	0.969486i	$-0.578837\pi$
$919$	−14.8633	−0.490296	−0.245148	+	0.969486i	$0.578837\pi$
$920$	0	0
$921$	26.9287	0.887331
$922$	−33.1817	−1.09278
$923$	21.0645	0.693346
$924$	0	0
$925$	0	0
$926$	15.6997	0.515925
$927$	14.3340	0.470791
$928$	2.28769	0.0750969
$929$	3.29372	0.108063	0.0540317	−	0.998539i	$-0.482793\pi$
$929$	3.29372	0.108063	0.0540317	+	0.998539i	$0.482793\pi$
$930$	0	0
$931$	7.90792	0.259172
$932$	6.25816	0.204993
$933$	11.9344	0.390716
$934$	−17.1609	−0.561523
$935$	0	0
$936$	13.8467	0.452594
$937$	0.798303	0.0260794	0.0130397	−	0.999915i	$-0.495849\pi$
$937$	0.798303	0.0260794	0.0130397	+	0.999915i	$0.495849\pi$
$938$	2.17055	0.0708711
$939$	14.9612	0.488241
$940$	0	0
$941$	−25.2852	−0.824275	−0.412138	−	0.911122i	$-0.635218\pi$
$941$	−25.2852	−0.824275	−0.412138	+	0.911122i	$0.635218\pi$
$942$	−8.78902	−0.286362
$943$	8.15699	0.265628
$944$	−1.64801	−0.0536380
$945$	0	0
$946$	0	0
$947$	49.6668	1.61396	0.806978	−	0.590582i	$-0.201102\pi$
$947$	49.6668	1.61396	0.806978	+	0.590582i	$0.201102\pi$
$948$	21.8637	0.710100
$949$	−67.3504	−2.18629
$950$	0	0
$951$	9.78534	0.317311
$952$	−4.58458	−0.148587
$953$	27.7742	0.899694	0.449847	−	0.893106i	$-0.351479\pi$
$953$	27.7742	0.899694	0.449847	+	0.893106i	$0.351479\pi$
$954$	−7.87899	−0.255092
$955$	0	0
$956$	−6.52732	−0.211109
$957$	0	0
$958$	12.3848	0.400135
$959$	5.59756	0.180754
$960$	0	0
$961$	9.38972	0.302894
$962$	18.3772	0.592504
$963$	6.87095	0.221413
$964$	−0.575173	−0.0185251
$965$	0	0
$966$	1.19066	0.0383087
$967$	−11.0951	−0.356793	−0.178396	−	0.983959i	$-0.557091\pi$
$967$	−11.0951	−0.356793	−0.178396	+	0.983959i	$0.557091\pi$
$968$	0	0
$969$	3.92429	0.126066
$970$	0	0
$971$	−31.9217	−1.02442	−0.512208	−	0.858861i	$-0.671172\pi$
$971$	−31.9217	−1.02442	−0.512208	+	0.858861i	$0.671172\pi$
$972$	−1.35460	−0.0434489
$973$	−7.22221	−0.231534
$974$	20.1745	0.646431
$975$	0	0
$976$	0.574154	0.0183782
$977$	−10.4056	−0.332904	−0.166452	−	0.986050i	$-0.553231\pi$
$977$	−10.4056	−0.332904	−0.166452	+	0.986050i	$0.553231\pi$
$978$	18.5435	0.592957
$979$	0	0
$980$	0	0
$981$	16.9516	0.541222
$982$	−0.886804	−0.0282991
$983$	41.4334	1.32152	0.660760	−	0.750597i	$-0.270234\pi$
$983$	41.4334	1.32152	0.660760	+	0.750597i	$0.270234\pi$
$984$	−7.54235	−0.240441
$985$	0	0
$986$	−1.05513	−0.0336022
$987$	5.84390	0.186013
$988$	8.16423	0.259739
$989$	−8.89970	−0.282994
$990$	0	0
$991$	−2.17000	−0.0689322	−0.0344661	−	0.999406i	$-0.510973\pi$
$991$	−2.17000	−0.0689322	−0.0344661	+	0.999406i	$0.510973\pi$
$992$	37.0329	1.17580
$993$	5.14693	0.163333
$994$	−1.67482	−0.0531219
$995$	0	0
$996$	20.8050	0.659231
$997$	0.808994	0.0256211	0.0128105	−	0.999918i	$-0.495922\pi$
$997$	0.808994	0.0256211	0.0128105	+	0.999918i	$0.495922\pi$
$998$	−1.84547	−0.0584174
$999$	−4.45218	−0.140861

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9075.2.a.dy.1.5		12
5.2	odd	4		1815.2.c.j.364.8		24
5.3	odd	4		1815.2.c.j.364.17		24
5.4	even	2		9075.2.a.dz.1.8		12
11.5	even	5		825.2.n.p.751.4		24
11.9	even	5		825.2.n.p.301.4		24
11.10	odd	2		9075.2.a.ea.1.8		12
55.9	even	10		825.2.n.o.301.3		24
55.27	odd	20		165.2.s.a.124.9	yes	48
55.32	even	4		1815.2.c.k.364.17		24
55.38	odd	20		165.2.s.a.124.4	yes	48
55.42	odd	20		165.2.s.a.4.4	✓	48
55.43	even	4		1815.2.c.k.364.8		24
55.49	even	10		825.2.n.o.751.3		24
55.53	odd	20		165.2.s.a.4.9	yes	48
55.54	odd	2		9075.2.a.dx.1.5		12
165.38	even	20		495.2.ba.c.289.9		48
165.53	even	20		495.2.ba.c.334.4		48
165.137	even	20		495.2.ba.c.289.4		48
165.152	even	20		495.2.ba.c.334.9		48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
165.2.s.a.4.4	✓	48	55.42	odd	20
165.2.s.a.4.9	yes	48	55.53	odd	20
165.2.s.a.124.4	yes	48	55.38	odd	20
165.2.s.a.124.9	yes	48	55.27	odd	20
495.2.ba.c.289.4		48	165.137	even	20
495.2.ba.c.289.9		48	165.38	even	20
495.2.ba.c.334.4		48	165.53	even	20
495.2.ba.c.334.9		48	165.152	even	20
825.2.n.o.301.3		24	55.9	even	10
825.2.n.o.751.3		24	55.49	even	10
825.2.n.p.301.4		24	11.9	even	5
825.2.n.p.751.4		24	11.5	even	5
1815.2.c.j.364.8		24	5.2	odd	4
1815.2.c.j.364.17		24	5.3	odd	4
1815.2.c.k.364.8		24	55.43	even	4
1815.2.c.k.364.17		24	55.32	even	4
9075.2.a.dx.1.5		12	55.54	odd	2
9075.2.a.dy.1.5		12	1.1	even	1	trivial
9075.2.a.dz.1.8		12	5.4	even	2
9075.2.a.ea.1.8		12	11.10	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 9075 = 3 \cdot 5^{2} \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9075.a (trivial)

\(f(q)\)	\(=\)	\(q-0.803366 q^{2} +1.00000 q^{3} -1.35460 q^{4} -0.803366 q^{6} +0.508505 q^{7} +2.69497 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-0.803366 q^{2} +1.00000 q^{3} -1.35460 q^{4} -0.803366 q^{6} +0.508505 q^{7} +2.69497 q^{8} +1.00000 q^{9} -1.35460 q^{12} +5.13798 q^{13} -0.408516 q^{14} +0.544158 q^{16} -3.34542 q^{17} -0.803366 q^{18} -1.17303 q^{19} +0.508505 q^{21} -2.91459 q^{23} +2.69497 q^{24} -4.12768 q^{26} +1.00000 q^{27} -0.688823 q^{28} -0.392594 q^{29} -6.35529 q^{31} -5.82710 q^{32} +2.68759 q^{34} -1.35460 q^{36} -4.45218 q^{37} +0.942375 q^{38} +5.13798 q^{39} -2.79867 q^{41} -0.408516 q^{42} +3.05350 q^{43} +2.34148 q^{46} +11.4923 q^{47} +0.544158 q^{48} -6.74142 q^{49} -3.34542 q^{51} -6.95993 q^{52} +9.80747 q^{53} -0.803366 q^{54} +1.37041 q^{56} -1.17303 q^{57} +0.315396 q^{58} -3.02854 q^{59} +1.05512 q^{61} +5.10562 q^{62} +0.508505 q^{63} +3.59298 q^{64} -5.31327 q^{67} +4.53171 q^{68} -2.91459 q^{69} +4.09976 q^{71} +2.69497 q^{72} -13.1083 q^{73} +3.57673 q^{74} +1.58900 q^{76} -4.12768 q^{78} -16.1403 q^{79} +1.00000 q^{81} +2.24836 q^{82} -15.3587 q^{83} -0.688823 q^{84} -2.45308 q^{86} -0.392594 q^{87} -9.84603 q^{89} +2.61269 q^{91} +3.94812 q^{92} -6.35529 q^{93} -9.23252 q^{94} -5.82710 q^{96} -15.1692 q^{97} +5.41583 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 4 q^{2} + 12 q^{3} + 12 q^{4} - 4 q^{6} - 8 q^{7} - 12 q^{8} + 12 q^{9}+O(q^{10}) \) 12 * q - 4 * q^2 + 12 * q^3 + 12 * q^4 - 4 * q^6 - 8 * q^7 - 12 * q^8 + 12 * q^9	\( 12 q - 4 q^{2} + 12 q^{3} + 12 q^{4} - 4 q^{6} - 8 q^{7} - 12 q^{8} + 12 q^{9} + 12 q^{12} - 18 q^{13} + 6 q^{14} + 24 q^{16} - 18 q^{17} - 4 q^{18} - 16 q^{19} - 8 q^{21} - 12 q^{24} + 16 q^{26} + 12 q^{27} - 30 q^{28} - 28 q^{32} + 6 q^{34} + 12 q^{36} - 28 q^{38} - 18 q^{39} + 6 q^{42} - 32 q^{43} - 28 q^{46} - 4 q^{47} + 24 q^{48} + 12 q^{49} - 18 q^{51} - 48 q^{52} - 12 q^{53} - 4 q^{54} + 6 q^{56} - 16 q^{57} - 10 q^{58} + 20 q^{59} - 20 q^{61} - 20 q^{62} - 8 q^{63} - 6 q^{64} + 10 q^{67} - 26 q^{68} + 32 q^{71} - 12 q^{72} - 26 q^{73} + 68 q^{74} - 34 q^{76} + 16 q^{78} - 32 q^{79} + 12 q^{81} - 62 q^{82} - 26 q^{83} - 30 q^{84} - 36 q^{86} - 10 q^{89} + 8 q^{92} + 2 q^{94} - 28 q^{96} - 22 q^{97} - 60 q^{98}+O(q^{100}) \) 12 * q - 4 * q^2 + 12 * q^3 + 12 * q^4 - 4 * q^6 - 8 * q^7 - 12 * q^8 + 12 * q^9 + 12 * q^12 - 18 * q^13 + 6 * q^14 + 24 * q^16 - 18 * q^17 - 4 * q^18 - 16 * q^19 - 8 * q^21 - 12 * q^24 + 16 * q^26 + 12 * q^27 - 30 * q^28 - 28 * q^32 + 6 * q^34 + 12 * q^36 - 28 * q^38 - 18 * q^39 + 6 * q^42 - 32 * q^43 - 28 * q^46 - 4 * q^47 + 24 * q^48 + 12 * q^49 - 18 * q^51 - 48 * q^52 - 12 * q^53 - 4 * q^54 + 6 * q^56 - 16 * q^57 - 10 * q^58 + 20 * q^59 - 20 * q^61 - 20 * q^62 - 8 * q^63 - 6 * q^64 + 10 * q^67 - 26 * q^68 + 32 * q^71 - 12 * q^72 - 26 * q^73 + 68 * q^74 - 34 * q^76 + 16 * q^78 - 32 * q^79 + 12 * q^81 - 62 * q^82 - 26 * q^83 - 30 * q^84 - 36 * q^86 - 10 * q^89 + 8 * q^92 + 2 * q^94 - 28 * q^96 - 22 * q^97 - 60 * q^98

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