LMFDB - Embedded newform 9075.2.a.cg.1.2

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:	$0$
Dimension:	$3$
Coefficient field:	3.3.148.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 3x + 1 $ x^3 - x^2 - 3*x + 1
Coefficient ring:	$\Z[a_1, a_2]$
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			1.2
Root			$0.311108$ 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)

$f(q)$	$=$	$q-0.311108 q^{2} +1.00000 q^{3} -1.90321 q^{4} -0.311108 q^{6} +0.903212 q^{7} +1.21432 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-0.311108 q^{2} +1.00000 q^{3} -1.90321 q^{4} -0.311108 q^{6} +0.903212 q^{7} +1.21432 q^{8} +1.00000 q^{9} -1.90321 q^{12} -2.90321 q^{13} -0.280996 q^{14} +3.42864 q^{16} -2.28100 q^{17} -0.311108 q^{18} -2.42864 q^{19} +0.903212 q^{21} +4.00000 q^{23} +1.21432 q^{24} +0.903212 q^{26} +1.00000 q^{27} -1.71900 q^{28} -7.05086 q^{29} -2.62222 q^{31} -3.49532 q^{32} +0.709636 q^{34} -1.90321 q^{36} +5.80642 q^{37} +0.755569 q^{38} -2.90321 q^{39} +10.6637 q^{41} -0.280996 q^{42} -10.7096 q^{43} -1.24443 q^{46} +0.949145 q^{47} +3.42864 q^{48} -6.18421 q^{49} -2.28100 q^{51} +5.52543 q^{52} +0.815792 q^{53} -0.311108 q^{54} +1.09679 q^{56} -2.42864 q^{57} +2.19358 q^{58} -1.67307 q^{59} +7.24443 q^{61} +0.815792 q^{62} +0.903212 q^{63} -5.76986 q^{64} +12.8573 q^{67} +4.34122 q^{68} +4.00000 q^{69} +9.28592 q^{71} +1.21432 q^{72} +5.65878 q^{73} -1.80642 q^{74} +4.62222 q^{76} +0.903212 q^{78} +16.5303 q^{79} +1.00000 q^{81} -3.31756 q^{82} -7.76049 q^{83} -1.71900 q^{84} +3.33185 q^{86} -7.05086 q^{87} +6.13335 q^{89} -2.62222 q^{91} -7.61285 q^{92} -2.62222 q^{93} -0.295286 q^{94} -3.49532 q^{96} -12.4701 q^{97} +1.92396 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - q^{2} + 3 q^{3} + q^{4} - q^{6} - 4 q^{7} - 3 q^{8} + 3 q^{9}+O(q^{10}) $ 3 * q - q^2 + 3 * q^3 + q^4 - q^6 - 4 * q^7 - 3 * q^8 + 3 * q^9	$ 3 q - q^{2} + 3 q^{3} + q^{4} - q^{6} - 4 q^{7} - 3 q^{8} + 3 q^{9} + q^{12} - 2 q^{13} + 6 q^{14} - 3 q^{16} - q^{18} + 6 q^{19} - 4 q^{21} + 12 q^{23} - 3 q^{24} - 4 q^{26} + 3 q^{27} - 12 q^{28} - 8 q^{29} - 8 q^{31} + 3 q^{32} - 18 q^{34} + q^{36} + 4 q^{37} + 2 q^{38} - 2 q^{39} - 8 q^{41} + 6 q^{42} - 12 q^{43} - 4 q^{46} + 16 q^{47} - 3 q^{48} - 5 q^{49} + 10 q^{52} + 16 q^{53} - q^{54} + 10 q^{56} + 6 q^{57} + 20 q^{58} + 8 q^{59} + 22 q^{61} + 16 q^{62} - 4 q^{63} - 11 q^{64} + 12 q^{67} + 20 q^{68} + 12 q^{69} - 12 q^{71} - 3 q^{72} + 10 q^{73} + 8 q^{74} + 14 q^{76} - 4 q^{78} + 10 q^{79} + 3 q^{81} + 4 q^{82} + 10 q^{83} - 12 q^{84} - 10 q^{86} - 8 q^{87} + 18 q^{89} - 8 q^{91} + 4 q^{92} - 8 q^{93} + 12 q^{94} + 3 q^{96} + 16 q^{97} - 21 q^{98}+O(q^{100}) $ 3 * q - q^2 + 3 * q^3 + q^4 - q^6 - 4 * q^7 - 3 * q^8 + 3 * q^9 + q^12 - 2 * q^13 + 6 * q^14 - 3 * q^16 - q^18 + 6 * q^19 - 4 * q^21 + 12 * q^23 - 3 * q^24 - 4 * q^26 + 3 * q^27 - 12 * q^28 - 8 * q^29 - 8 * q^31 + 3 * q^32 - 18 * q^34 + q^36 + 4 * q^37 + 2 * q^38 - 2 * q^39 - 8 * q^41 + 6 * q^42 - 12 * q^43 - 4 * q^46 + 16 * q^47 - 3 * q^48 - 5 * q^49 + 10 * q^52 + 16 * q^53 - q^54 + 10 * q^56 + 6 * q^57 + 20 * q^58 + 8 * q^59 + 22 * q^61 + 16 * q^62 - 4 * q^63 - 11 * q^64 + 12 * q^67 + 20 * q^68 + 12 * q^69 - 12 * q^71 - 3 * q^72 + 10 * q^73 + 8 * q^74 + 14 * q^76 - 4 * q^78 + 10 * q^79 + 3 * q^81 + 4 * q^82 + 10 * q^83 - 12 * q^84 - 10 * q^86 - 8 * q^87 + 18 * q^89 - 8 * q^91 + 4 * q^92 - 8 * q^93 + 12 * q^94 + 3 * q^96 + 16 * q^97 - 21 * 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 $

$2$	−0.311108	−0.219986	−0.109993	−	0.993932i	$-0.535083\pi$
$2$	−0.311108	−0.219986	−0.109993	+	0.993932i	$0.535083\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	−1.90321	−0.951606
$5$	0	0
$5$	0	0
$6$	−0.311108	−0.127009
$7$	0.903212	0.341382	0.170691	−	0.985325i	$-0.445400\pi$
$7$	0.903212	0.341382	0.170691	+	0.985325i	$0.445400\pi$
$8$	1.21432	0.429327
$9$	1.00000	0.333333
$10$	0	0
$11$	0	0
$11$	0	0
$12$	−1.90321	−0.549410
$13$	−2.90321	−0.805206	−0.402603	−	0.915375i	$-0.631894\pi$
$13$	−2.90321	−0.805206	−0.402603	+	0.915375i	$0.631894\pi$
$14$	−0.280996	−0.0750994
$15$	0	0
$16$	3.42864	0.857160
$17$	−2.28100	−0.553223	−0.276611	−	0.960982i	$-0.589211\pi$
$17$	−2.28100	−0.553223	−0.276611	+	0.960982i	$0.589211\pi$
$18$	−0.311108	−0.0733288
$19$	−2.42864	−0.557168	−0.278584	−	0.960412i	$-0.589865\pi$
$19$	−2.42864	−0.557168	−0.278584	+	0.960412i	$0.589865\pi$
$20$	0	0
$21$	0.903212	0.197097
$22$	0	0
$23$	4.00000	0.834058	0.417029	−	0.908893i	$-0.363071\pi$
$23$	4.00000	0.834058	0.417029	+	0.908893i	$0.363071\pi$
$24$	1.21432	0.247872
$25$	0	0
$26$	0.903212	0.177134
$27$	1.00000	0.192450
$28$	−1.71900	−0.324861
$29$	−7.05086	−1.30931	−0.654655	−	0.755927i	$-0.727186\pi$
$29$	−7.05086	−1.30931	−0.654655	+	0.755927i	$0.727186\pi$
$30$	0	0
$31$	−2.62222	−0.470964	−0.235482	−	0.971879i	$-0.575667\pi$
$31$	−2.62222	−0.470964	−0.235482	+	0.971879i	$0.575667\pi$
$32$	−3.49532	−0.617890
$33$	0	0
$34$	0.709636	0.121702
$35$	0	0
$36$	−1.90321	−0.317202
$37$	5.80642	0.954570	0.477285	−	0.878749i	$-0.341621\pi$
$37$	5.80642	0.954570	0.477285	+	0.878749i	$0.341621\pi$
$38$	0.755569	0.122569
$39$	−2.90321	−0.464886
$40$	0	0
$41$	10.6637	1.66539	0.832695	−	0.553731i	$-0.186797\pi$
$41$	10.6637	1.66539	0.832695	+	0.553731i	$0.186797\pi$
$42$	−0.280996	−0.0433587
$43$	−10.7096	−1.63320	−0.816602	−	0.577201i	$-0.804145\pi$
$43$	−10.7096	−1.63320	−0.816602	+	0.577201i	$0.804145\pi$
$44$	0	0
$45$	0	0
$46$	−1.24443	−0.183481
$47$	0.949145	0.138447	0.0692235	−	0.997601i	$-0.477948\pi$
$47$	0.949145	0.138447	0.0692235	+	0.997601i	$0.477948\pi$
$48$	3.42864	0.494881
$49$	−6.18421	−0.883458
$50$	0	0
$51$	−2.28100	−0.319403
$52$	5.52543	0.766239
$53$	0.815792	0.112058	0.0560288	−	0.998429i	$-0.482156\pi$
$53$	0.815792	0.112058	0.0560288	+	0.998429i	$0.482156\pi$
$54$	−0.311108	−0.0423364
$55$	0	0
$56$	1.09679	0.146564
$57$	−2.42864	−0.321681
$58$	2.19358	0.288031
$59$	−1.67307	−0.217815	−0.108908	−	0.994052i	$-0.534735\pi$
$59$	−1.67307	−0.217815	−0.108908	+	0.994052i	$0.534735\pi$
$60$	0	0
$61$	7.24443	0.927554	0.463777	−	0.885952i	$-0.346494\pi$
$61$	7.24443	0.927554	0.463777	+	0.885952i	$0.346494\pi$
$62$	0.815792	0.103606
$63$	0.903212	0.113794
$64$	−5.76986	−0.721232
$65$	0	0
$66$	0	0
$67$	12.8573	1.57077	0.785383	−	0.619010i	$-0.212466\pi$
$67$	12.8573	1.57077	0.785383	+	0.619010i	$0.212466\pi$
$68$	4.34122	0.526450
$69$	4.00000	0.481543
$70$	0	0
$71$	9.28592	1.10204	0.551018	−	0.834493i	$-0.314240\pi$
$71$	9.28592	1.10204	0.551018	+	0.834493i	$0.314240\pi$
$72$	1.21432	0.143109
$73$	5.65878	0.662310	0.331155	−	0.943576i	$-0.392562\pi$
$73$	5.65878	0.662310	0.331155	+	0.943576i	$0.392562\pi$
$74$	−1.80642	−0.209993
$75$	0	0
$76$	4.62222	0.530204
$77$	0	0
$78$	0.903212	0.102269
$79$	16.5303	1.85981	0.929905	−	0.367800i	$-0.119889\pi$
$79$	16.5303	1.85981	0.929905	+	0.367800i	$0.119889\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	−3.31756	−0.366363
$83$	−7.76049	−0.851825	−0.425912	−	0.904764i	$-0.640047\pi$
$83$	−7.76049	−0.851825	−0.425912	+	0.904764i	$0.640047\pi$
$84$	−1.71900	−0.187559
$85$	0	0
$86$	3.33185	0.359283
$87$	−7.05086	−0.755931
$88$	0	0
$89$	6.13335	0.650134	0.325067	−	0.945691i	$-0.394613\pi$
$89$	6.13335	0.650134	0.325067	+	0.945691i	$0.394613\pi$
$90$	0	0
$91$	−2.62222	−0.274883
$92$	−7.61285	−0.793694
$93$	−2.62222	−0.271911
$94$	−0.295286	−0.0304565
$95$	0	0
$96$	−3.49532	−0.356739
$97$	−12.4701	−1.26615	−0.633075	−	0.774091i	$-0.718207\pi$
$97$	−12.4701	−1.26615	−0.633075	+	0.774091i	$0.718207\pi$
$98$	1.92396	0.194349
$99$	0	0
$100$	0	0
$101$	−16.1748	−1.60946	−0.804728	−	0.593643i	$-0.797689\pi$
$101$	−16.1748	−1.60946	−0.804728	+	0.593643i	$0.797689\pi$
$102$	0.709636	0.0702644
$103$	−17.1526	−1.69009	−0.845046	−	0.534693i	$-0.820427\pi$
$103$	−17.1526	−1.69009	−0.845046	+	0.534693i	$0.820427\pi$
$104$	−3.52543	−0.345697
$105$	0	0
$106$	−0.253799	−0.0246512
$107$	13.5669	1.31156	0.655782	−	0.754951i	$-0.272339\pi$
$107$	13.5669	1.31156	0.655782	+	0.754951i	$0.272339\pi$
$108$	−1.90321	−0.183137
$109$	10.0000	0.957826	0.478913	−	0.877862i	$-0.341031\pi$
$109$	10.0000	0.957826	0.478913	+	0.877862i	$0.341031\pi$
$110$	0	0
$111$	5.80642	0.551121
$112$	3.09679	0.292619
$113$	14.2351	1.33912	0.669561	−	0.742757i	$-0.266482\pi$
$113$	14.2351	1.33912	0.669561	+	0.742757i	$0.266482\pi$
$114$	0.755569	0.0707655
$115$	0	0
$116$	13.4193	1.24595
$117$	−2.90321	−0.268402
$118$	0.520505	0.0479164
$119$	−2.06022	−0.188860
$120$	0	0
$121$	0	0
$122$	−2.25380	−0.204049
$123$	10.6637	0.961514
$124$	4.99063	0.448172
$125$	0	0
$126$	−0.280996	−0.0250331
$127$	−11.0049	−0.976529	−0.488264	−	0.872696i	$-0.662370\pi$
$127$	−11.0049	−0.976529	−0.488264	+	0.872696i	$0.662370\pi$
$128$	8.78568	0.776552
$129$	−10.7096	−0.942931
$130$	0	0
$131$	−1.24443	−0.108726	−0.0543632	−	0.998521i	$-0.517313\pi$
$131$	−1.24443	−0.108726	−0.0543632	+	0.998521i	$0.517313\pi$
$132$	0	0
$133$	−2.19358	−0.190207
$134$	−4.00000	−0.345547
$135$	0	0
$136$	−2.76986	−0.237513
$137$	−4.42864	−0.378364	−0.189182	−	0.981942i	$-0.560584\pi$
$137$	−4.42864	−0.378364	−0.189182	+	0.981942i	$0.560584\pi$
$138$	−1.24443	−0.105933
$139$	−0.917502	−0.0778215	−0.0389108	−	0.999243i	$-0.512389\pi$
$139$	−0.917502	−0.0778215	−0.0389108	+	0.999243i	$0.512389\pi$
$140$	0	0
$141$	0.949145	0.0799324
$142$	−2.88892	−0.242433
$143$	0	0
$144$	3.42864	0.285720
$145$	0	0
$146$	−1.76049	−0.145699
$147$	−6.18421	−0.510065
$148$	−11.0509	−0.908375
$149$	−2.19358	−0.179705	−0.0898524	−	0.995955i	$-0.528640\pi$
$149$	−2.19358	−0.179705	−0.0898524	+	0.995955i	$0.528640\pi$
$150$	0	0
$151$	10.4286	0.848671	0.424335	−	0.905505i	$-0.360508\pi$
$151$	10.4286	0.848671	0.424335	+	0.905505i	$0.360508\pi$
$152$	−2.94914	−0.239207
$153$	−2.28100	−0.184408
$154$	0	0
$155$	0	0
$156$	5.52543	0.442388
$157$	5.80642	0.463403	0.231702	−	0.972787i	$-0.425571\pi$
$157$	5.80642	0.463403	0.231702	+	0.972787i	$0.425571\pi$
$158$	−5.14272	−0.409133
$159$	0.815792	0.0646965
$160$	0	0
$161$	3.61285	0.284732
$162$	−0.311108	−0.0244429
$163$	11.0509	0.865570	0.432785	−	0.901497i	$-0.357531\pi$
$163$	11.0509	0.865570	0.432785	+	0.901497i	$0.357531\pi$
$164$	−20.2953	−1.58480
$165$	0	0
$166$	2.41435	0.187390
$167$	−13.9541	−1.07980	−0.539899	−	0.841730i	$-0.681538\pi$
$167$	−13.9541	−1.07980	−0.539899	+	0.841730i	$0.681538\pi$
$168$	1.09679	0.0846190
$169$	−4.57136	−0.351643
$170$	0	0
$171$	−2.42864	−0.185723
$172$	20.3827	1.55417
$173$	18.8430	1.43261	0.716303	−	0.697789i	$-0.245833\pi$
$173$	18.8430	1.43261	0.716303	+	0.697789i	$0.245833\pi$
$174$	2.19358	0.166295
$175$	0	0
$176$	0	0
$177$	−1.67307	−0.125756
$178$	−1.90813	−0.143021
$179$	4.85728	0.363050	0.181525	−	0.983386i	$-0.441897\pi$
$179$	4.85728	0.363050	0.181525	+	0.983386i	$0.441897\pi$
$180$	0	0
$181$	−16.7971	−1.24852	−0.624258	−	0.781219i	$-0.714598\pi$
$181$	−16.7971	−1.24852	−0.624258	+	0.781219i	$0.714598\pi$
$182$	0.815792	0.0604705
$183$	7.24443	0.535524
$184$	4.85728	0.358083
$185$	0	0
$186$	0.815792	0.0598168
$187$	0	0
$188$	−1.80642	−0.131747
$189$	0.903212	0.0656990
$190$	0	0
$191$	16.8573	1.21975	0.609875	−	0.792498i	$-0.291220\pi$
$191$	16.8573	1.21975	0.609875	+	0.792498i	$0.291220\pi$
$192$	−5.76986	−0.416404
$193$	24.6178	1.77203	0.886013	−	0.463661i	$-0.153464\pi$
$193$	24.6178	1.77203	0.886013	+	0.463661i	$0.153464\pi$
$194$	3.87955	0.278536
$195$	0	0
$196$	11.7699	0.840704
$197$	−14.8716	−1.05956	−0.529778	−	0.848137i	$-0.677725\pi$
$197$	−14.8716	−1.05956	−0.529778	+	0.848137i	$0.677725\pi$
$198$	0	0
$199$	−11.2257	−0.795768	−0.397884	−	0.917436i	$-0.630255\pi$
$199$	−11.2257	−0.795768	−0.397884	+	0.917436i	$0.630255\pi$
$200$	0	0
$201$	12.8573	0.906883
$202$	5.03212	0.354059
$203$	−6.36842	−0.446975
$204$	4.34122	0.303946
$205$	0	0
$206$	5.33630	0.371797
$207$	4.00000	0.278019
$208$	−9.95407	−0.690190
$209$	0	0
$210$	0	0
$211$	11.9398	0.821968	0.410984	−	0.911643i	$-0.365185\pi$
$211$	11.9398	0.821968	0.410984	+	0.911643i	$0.365185\pi$
$212$	−1.55262	−0.106635
$213$	9.28592	0.636261
$214$	−4.22077	−0.288526
$215$	0	0
$216$	1.21432	0.0826240
$217$	−2.36842	−0.160779
$218$	−3.11108	−0.210709
$219$	5.65878	0.382385
$220$	0	0
$221$	6.62222	0.445458
$222$	−1.80642	−0.121239
$223$	21.8064	1.46027	0.730133	−	0.683305i	$-0.239458\pi$
$223$	21.8064	1.46027	0.730133	+	0.683305i	$0.239458\pi$
$224$	−3.15701	−0.210937
$225$	0	0
$226$	−4.42864	−0.294589
$227$	−3.19850	−0.212292	−0.106146	−	0.994351i	$-0.533851\pi$
$227$	−3.19850	−0.212292	−0.106146	+	0.994351i	$0.533851\pi$
$228$	4.62222	0.306114
$229$	7.12399	0.470766	0.235383	−	0.971903i	$-0.424366\pi$
$229$	7.12399	0.470766	0.235383	+	0.971903i	$0.424366\pi$
$230$	0	0
$231$	0	0
$232$	−8.56199	−0.562122
$233$	19.5254	1.27915	0.639577	−	0.768727i	$-0.279110\pi$
$233$	19.5254	1.27915	0.639577	+	0.768727i	$0.279110\pi$
$234$	0.903212	0.0590448
$235$	0	0
$236$	3.18421	0.207274
$237$	16.5303	1.07376
$238$	0.640951	0.0415467
$239$	21.9813	1.42185	0.710925	−	0.703268i	$-0.248277\pi$
$239$	21.9813	1.42185	0.710925	+	0.703268i	$0.248277\pi$
$240$	0	0
$241$	−5.34614	−0.344375	−0.172188	−	0.985064i	$-0.555084\pi$
$241$	−5.34614	−0.344375	−0.172188	+	0.985064i	$0.555084\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	−13.7877	−0.882666
$245$	0	0
$246$	−3.31756	−0.211520
$247$	7.05086	0.448635
$248$	−3.18421	−0.202197
$249$	−7.76049	−0.491801
$250$	0	0
$251$	−23.7748	−1.50065	−0.750325	−	0.661069i	$-0.770103\pi$
$251$	−23.7748	−1.50065	−0.750325	+	0.661069i	$0.770103\pi$
$252$	−1.71900	−0.108287
$253$	0	0
$254$	3.42372	0.214823
$255$	0	0
$256$	8.80642	0.550401
$257$	−8.13335	−0.507345	−0.253672	−	0.967290i	$-0.581638\pi$
$257$	−8.13335	−0.507345	−0.253672	+	0.967290i	$0.581638\pi$
$258$	3.33185	0.207432
$259$	5.24443	0.325873
$260$	0	0
$261$	−7.05086	−0.436437
$262$	0.387152	0.0239183
$263$	−22.9032	−1.41227	−0.706136	−	0.708076i	$-0.749563\pi$
$263$	−22.9032	−1.41227	−0.706136	+	0.708076i	$0.749563\pi$
$264$	0	0
$265$	0	0
$266$	0.682439	0.0418430
$267$	6.13335	0.375355
$268$	−24.4701	−1.49475
$269$	11.8350	0.721593	0.360796	−	0.932645i	$-0.382505\pi$
$269$	11.8350	0.721593	0.360796	+	0.932645i	$0.382505\pi$
$270$	0	0
$271$	−14.8988	−0.905036	−0.452518	−	0.891755i	$-0.649474\pi$
$271$	−14.8988	−0.905036	−0.452518	+	0.891755i	$0.649474\pi$
$272$	−7.82071	−0.474200
$273$	−2.62222	−0.158704
$274$	1.37778	0.0832350
$275$	0	0
$276$	−7.61285	−0.458240
$277$	27.6686	1.66245	0.831223	−	0.555939i	$-0.187641\pi$
$277$	27.6686	1.66245	0.831223	+	0.555939i	$0.187641\pi$
$278$	0.285442	0.0171197
$279$	−2.62222	−0.156988
$280$	0	0
$281$	−9.80642	−0.585002	−0.292501	−	0.956265i	$-0.594488\pi$
$281$	−9.80642	−0.585002	−0.292501	+	0.956265i	$0.594488\pi$
$282$	−0.295286	−0.0175840
$283$	19.0049	1.12973	0.564863	−	0.825185i	$-0.308929\pi$
$283$	19.0049	1.12973	0.564863	+	0.825185i	$0.308929\pi$
$284$	−17.6731	−1.04870
$285$	0	0
$286$	0	0
$287$	9.63158	0.568534
$288$	−3.49532	−0.205963
$289$	−11.7971	−0.693944
$290$	0	0
$291$	−12.4701	−0.731012
$292$	−10.7699	−0.630258
$293$	30.7511	1.79650	0.898250	−	0.439485i	$-0.144839\pi$
$293$	30.7511	1.79650	0.898250	+	0.439485i	$0.144839\pi$
$294$	1.92396	0.112207
$295$	0	0
$296$	7.05086	0.409823
$297$	0	0
$298$	0.682439	0.0395326
$299$	−11.6128	−0.671588
$300$	0	0
$301$	−9.67307	−0.557547
$302$	−3.24443	−0.186696
$303$	−16.1748	−0.929220
$304$	−8.32693	−0.477582
$305$	0	0
$306$	0.709636	0.0405672
$307$	13.4938	0.770131	0.385065	−	0.922889i	$-0.374179\pi$
$307$	13.4938	0.770131	0.385065	+	0.922889i	$0.374179\pi$
$308$	0	0
$309$	−17.1526	−0.975775
$310$	0	0
$311$	−17.5526	−0.995318	−0.497659	−	0.867373i	$-0.665807\pi$
$311$	−17.5526	−0.995318	−0.497659	+	0.867373i	$0.665807\pi$
$312$	−3.52543	−0.199588
$313$	14.3970	0.813766	0.406883	−	0.913480i	$-0.366616\pi$
$313$	14.3970	0.813766	0.406883	+	0.913480i	$0.366616\pi$
$314$	−1.80642	−0.101942
$315$	0	0
$316$	−31.4608	−1.76981
$317$	29.4608	1.65468	0.827341	−	0.561701i	$-0.189853\pi$
$317$	29.4608	1.65468	0.827341	+	0.561701i	$0.189853\pi$
$318$	−0.253799	−0.0142324
$319$	0	0
$320$	0	0
$321$	13.5669	0.757231
$322$	−1.12399	−0.0626372
$323$	5.53972	0.308238
$324$	−1.90321	−0.105734
$325$	0	0
$326$	−3.43801	−0.190414
$327$	10.0000	0.553001
$328$	12.9491	0.714997
$329$	0.857279	0.0472633
$330$	0	0
$331$	−2.62222	−0.144130	−0.0720650	−	0.997400i	$-0.522959\pi$
$331$	−2.62222	−0.144130	−0.0720650	+	0.997400i	$0.522959\pi$
$332$	14.7699	0.810601
$333$	5.80642	0.318190
$334$	4.34122	0.237541
$335$	0	0
$336$	3.09679	0.168944
$337$	5.00492	0.272635	0.136318	−	0.990665i	$-0.456473\pi$
$337$	5.00492	0.272635	0.136318	+	0.990665i	$0.456473\pi$
$338$	1.42219	0.0773567
$339$	14.2351	0.773143
$340$	0	0
$341$	0	0
$342$	0.755569	0.0408565
$343$	−11.9081	−0.642979
$344$	−13.0049	−0.701178
$345$	0	0
$346$	−5.86220	−0.315154
$347$	22.8113	1.22458	0.612289	−	0.790634i	$-0.290249\pi$
$347$	22.8113	1.22458	0.612289	+	0.790634i	$0.290249\pi$
$348$	13.4193	0.719348
$349$	−21.2257	−1.13619	−0.568093	−	0.822965i	$-0.692319\pi$
$349$	−21.2257	−1.13619	−0.568093	+	0.822965i	$0.692319\pi$
$350$	0	0
$351$	−2.90321	−0.154962
$352$	0	0
$353$	7.18421	0.382377	0.191188	−	0.981553i	$-0.438766\pi$
$353$	7.18421	0.382377	0.191188	+	0.981553i	$0.438766\pi$
$354$	0.520505	0.0276645
$355$	0	0
$356$	−11.6731	−0.618672
$357$	−2.06022	−0.109039
$358$	−1.51114	−0.0798661
$359$	14.1017	0.744260	0.372130	−	0.928181i	$-0.378628\pi$
$359$	14.1017	0.744260	0.372130	+	0.928181i	$0.378628\pi$
$360$	0	0
$361$	−13.1017	−0.689564
$362$	5.22570	0.274656
$363$	0	0
$364$	4.99063	0.261580
$365$	0	0
$366$	−2.25380	−0.117808
$367$	−3.90813	−0.204003	−0.102001	−	0.994784i	$-0.532525\pi$
$367$	−3.90813	−0.204003	−0.102001	+	0.994784i	$0.532525\pi$
$368$	13.7146	0.714921
$369$	10.6637	0.555130
$370$	0	0
$371$	0.736833	0.0382545
$372$	4.99063	0.258752
$373$	−12.9763	−0.671890	−0.335945	−	0.941882i	$-0.609056\pi$
$373$	−12.9763	−0.671890	−0.335945	+	0.941882i	$0.609056\pi$
$374$	0	0
$375$	0	0
$376$	1.15257	0.0594390
$377$	20.4701	1.05427
$378$	−0.280996	−0.0144529
$379$	36.0830	1.85346	0.926729	−	0.375731i	$-0.122608\pi$
$379$	36.0830	1.85346	0.926729	+	0.375731i	$0.122608\pi$
$380$	0	0
$381$	−11.0049	−0.563799
$382$	−5.24443	−0.268328
$383$	20.2953	1.03704	0.518520	−	0.855065i	$-0.326483\pi$
$383$	20.2953	1.03704	0.518520	+	0.855065i	$0.326483\pi$
$384$	8.78568	0.448342
$385$	0	0
$386$	−7.65878	−0.389822
$387$	−10.7096	−0.544401
$388$	23.7333	1.20488
$389$	30.4701	1.54490	0.772448	−	0.635078i	$-0.219032\pi$
$389$	30.4701	1.54490	0.772448	+	0.635078i	$0.219032\pi$
$390$	0	0
$391$	−9.12399	−0.461420
$392$	−7.50961	−0.379292
$393$	−1.24443	−0.0627733
$394$	4.62666	0.233088
$395$	0	0
$396$	0	0
$397$	4.97773	0.249825	0.124912	−	0.992168i	$-0.460135\pi$
$397$	4.97773	0.249825	0.124912	+	0.992168i	$0.460135\pi$
$398$	3.49240	0.175058
$399$	−2.19358	−0.109816
$400$	0	0
$401$	−1.86665	−0.0932159	−0.0466079	−	0.998913i	$-0.514841\pi$
$401$	−1.86665	−0.0932159	−0.0466079	+	0.998913i	$0.514841\pi$
$402$	−4.00000	−0.199502
$403$	7.61285	0.379223
$404$	30.7841	1.53157
$405$	0	0
$406$	1.98126	0.0983285
$407$	0	0
$408$	−2.76986	−0.137128
$409$	3.63158	0.179570	0.0897851	−	0.995961i	$-0.471382\pi$
$409$	3.63158	0.179570	0.0897851	+	0.995961i	$0.471382\pi$
$410$	0	0
$411$	−4.42864	−0.218449
$412$	32.6450	1.60830
$413$	−1.51114	−0.0743582
$414$	−1.24443	−0.0611605
$415$	0	0
$416$	10.1476	0.497529
$417$	−0.917502	−0.0449303
$418$	0	0
$419$	4.85728	0.237294	0.118647	−	0.992937i	$-0.462144\pi$
$419$	4.85728	0.237294	0.118647	+	0.992937i	$0.462144\pi$
$420$	0	0
$421$	22.6321	1.10302	0.551510	−	0.834169i	$-0.314052\pi$
$421$	22.6321	1.10302	0.551510	+	0.834169i	$0.314052\pi$
$422$	−3.71456	−0.180822
$423$	0.949145	0.0461490
$424$	0.990632	0.0481093
$425$	0	0
$426$	−2.88892	−0.139969
$427$	6.54326	0.316650
$428$	−25.8207	−1.24809
$429$	0	0
$430$	0	0
$431$	−1.24443	−0.0599421	−0.0299711	−	0.999551i	$-0.509542\pi$
$431$	−1.24443	−0.0599421	−0.0299711	+	0.999551i	$0.509542\pi$
$432$	3.42864	0.164960
$433$	−16.0000	−0.768911	−0.384455	−	0.923144i	$-0.625611\pi$
$433$	−16.0000	−0.768911	−0.384455	+	0.923144i	$0.625611\pi$
$434$	0.736833	0.0353691
$435$	0	0
$436$	−19.0321	−0.911473
$437$	−9.71456	−0.464710
$438$	−1.76049	−0.0841195
$439$	2.42864	0.115913	0.0579563	−	0.998319i	$-0.481542\pi$
$439$	2.42864	0.115913	0.0579563	+	0.998319i	$0.481542\pi$
$440$	0	0
$441$	−6.18421	−0.294486
$442$	−2.06022	−0.0979948
$443$	31.0509	1.47527	0.737635	−	0.675199i	$-0.235942\pi$
$443$	31.0509	1.47527	0.737635	+	0.675199i	$0.235942\pi$
$444$	−11.0509	−0.524450
$445$	0	0
$446$	−6.78415	−0.321239
$447$	−2.19358	−0.103753
$448$	−5.21141	−0.246216
$449$	−37.3590	−1.76308	−0.881541	−	0.472107i	$-0.843494\pi$
$449$	−37.3590	−1.76308	−0.881541	+	0.472107i	$0.843494\pi$
$450$	0	0
$451$	0	0
$452$	−27.0923	−1.27432
$453$	10.4286	0.489980
$454$	0.995078	0.0467013
$455$	0	0
$456$	−2.94914	−0.138106
$457$	−8.73822	−0.408757	−0.204378	−	0.978892i	$-0.565517\pi$
$457$	−8.73822	−0.408757	−0.204378	+	0.978892i	$0.565517\pi$
$458$	−2.21633	−0.103562
$459$	−2.28100	−0.106468
$460$	0	0
$461$	−31.7877	−1.48050	−0.740250	−	0.672332i	$-0.765293\pi$
$461$	−31.7877	−1.48050	−0.740250	+	0.672332i	$0.765293\pi$
$462$	0	0
$463$	12.0919	0.561957	0.280978	−	0.959714i	$-0.409341\pi$
$463$	12.0919	0.561957	0.280978	+	0.959714i	$0.409341\pi$
$464$	−24.1748	−1.12229
$465$	0	0
$466$	−6.07451	−0.281396
$467$	−15.3461	−0.710135	−0.355067	−	0.934841i	$-0.615542\pi$
$467$	−15.3461	−0.710135	−0.355067	+	0.934841i	$0.615542\pi$
$468$	5.52543	0.255413
$469$	11.6128	0.536231
$470$	0	0
$471$	5.80642	0.267546
$472$	−2.03164	−0.0935139
$473$	0	0
$474$	−5.14272	−0.236213
$475$	0	0
$476$	3.92104	0.179721
$477$	0.815792	0.0373525
$478$	−6.83854	−0.312788
$479$	−5.89829	−0.269500	−0.134750	−	0.990880i	$-0.543023\pi$
$479$	−5.89829	−0.269500	−0.134750	+	0.990880i	$0.543023\pi$
$480$	0	0
$481$	−16.8573	−0.768626
$482$	1.66323	0.0757579
$483$	3.61285	0.164390
$484$	0	0
$485$	0	0
$486$	−0.311108	−0.0141121
$487$	31.3461	1.42043	0.710215	−	0.703985i	$-0.248598\pi$
$487$	31.3461	1.42043	0.710215	+	0.703985i	$0.248598\pi$
$488$	8.79706	0.398224
$489$	11.0509	0.499737
$490$	0	0
$491$	8.00000	0.361035	0.180517	−	0.983572i	$-0.442223\pi$
$491$	8.00000	0.361035	0.180517	+	0.983572i	$0.442223\pi$
$492$	−20.2953	−0.914982
$493$	16.0830	0.724341
$494$	−2.19358	−0.0986937
$495$	0	0
$496$	−8.99063	−0.403691
$497$	8.38715	0.376215
$498$	2.41435	0.108190
$499$	15.1427	0.677881	0.338941	−	0.940808i	$-0.389931\pi$
$499$	15.1427	0.677881	0.338941	+	0.940808i	$0.389931\pi$
$500$	0	0
$501$	−13.9541	−0.623422
$502$	7.39652	0.330123
$503$	−26.0370	−1.16093	−0.580467	−	0.814284i	$-0.697130\pi$
$503$	−26.0370	−1.16093	−0.580467	+	0.814284i	$0.697130\pi$
$504$	1.09679	0.0488548
$505$	0	0
$506$	0	0
$507$	−4.57136	−0.203021
$508$	20.9447	0.929271
$509$	−24.5718	−1.08913	−0.544564	−	0.838719i	$-0.683305\pi$
$509$	−24.5718	−1.08913	−0.544564	+	0.838719i	$0.683305\pi$
$510$	0	0
$511$	5.11108	0.226101
$512$	−20.3111	−0.897633
$513$	−2.42864	−0.107227
$514$	2.53035	0.111609
$515$	0	0
$516$	20.3827	0.897299
$517$	0	0
$518$	−1.63158	−0.0716877
$519$	18.8430	0.827115
$520$	0	0
$521$	−4.88892	−0.214188	−0.107094	−	0.994249i	$-0.534155\pi$
$521$	−4.88892	−0.214188	−0.107094	+	0.994249i	$0.534155\pi$
$522$	2.19358	0.0960102
$523$	4.22077	0.184562	0.0922808	−	0.995733i	$-0.470584\pi$
$523$	4.22077	0.184562	0.0922808	+	0.995733i	$0.470584\pi$
$524$	2.36842	0.103465
$525$	0	0
$526$	7.12537	0.310681
$527$	5.98126	0.260548
$528$	0	0
$529$	−7.00000	−0.304348
$530$	0	0
$531$	−1.67307	−0.0726051
$532$	4.17484	0.181002
$533$	−30.9590	−1.34098
$534$	−1.90813	−0.0825730
$535$	0	0
$536$	15.6128	0.674372
$537$	4.85728	0.209607
$538$	−3.68196	−0.158741
$539$	0	0
$540$	0	0
$541$	13.6128	0.585262	0.292631	−	0.956225i	$-0.405469\pi$
$541$	13.6128	0.585262	0.292631	+	0.956225i	$0.405469\pi$
$542$	4.63512	0.199096
$543$	−16.7971	−0.720831
$544$	7.97280	0.341831
$545$	0	0
$546$	0.815792	0.0349127
$547$	7.48394	0.319990	0.159995	−	0.987118i	$-0.448852\pi$
$547$	7.48394	0.319990	0.159995	+	0.987118i	$0.448852\pi$
$548$	8.42864	0.360054
$549$	7.24443	0.309185
$550$	0	0
$551$	17.1240	0.729506
$552$	4.85728	0.206740
$553$	14.9304	0.634906
$554$	−8.60793	−0.365716
$555$	0	0
$556$	1.74620	0.0740554
$557$	32.2908	1.36821	0.684103	−	0.729385i	$-0.260194\pi$
$557$	32.2908	1.36821	0.684103	+	0.729385i	$0.260194\pi$
$558$	0.815792	0.0345352
$559$	31.0923	1.31507
$560$	0	0
$561$	0	0
$562$	3.05086	0.128693
$563$	7.49378	0.315825	0.157913	−	0.987453i	$-0.449524\pi$
$563$	7.49378	0.315825	0.157913	+	0.987453i	$0.449524\pi$
$564$	−1.80642	−0.0760642
$565$	0	0
$566$	−5.91258	−0.248524
$567$	0.903212	0.0379313
$568$	11.2761	0.473134
$569$	12.9491	0.542856	0.271428	−	0.962459i	$-0.412504\pi$
$569$	12.9491	0.542856	0.271428	+	0.962459i	$0.412504\pi$
$570$	0	0
$571$	15.2859	0.639696	0.319848	−	0.947469i	$-0.396368\pi$
$571$	15.2859	0.639696	0.319848	+	0.947469i	$0.396368\pi$
$572$	0	0
$573$	16.8573	0.704223
$574$	−2.99646	−0.125070
$575$	0	0
$576$	−5.76986	−0.240411
$577$	−28.4415	−1.18404	−0.592019	−	0.805924i	$-0.701669\pi$
$577$	−28.4415	−1.18404	−0.592019	+	0.805924i	$0.701669\pi$
$578$	3.67016	0.152658
$579$	24.6178	1.02308
$580$	0	0
$581$	−7.00937	−0.290798
$582$	3.87955	0.160813
$583$	0	0
$584$	6.87157	0.284348
$585$	0	0
$586$	−9.56691	−0.395206
$587$	8.47013	0.349600	0.174800	−	0.984604i	$-0.444072\pi$
$587$	8.47013	0.349600	0.174800	+	0.984604i	$0.444072\pi$
$588$	11.7699	0.485381
$589$	6.36842	0.262406
$590$	0	0
$591$	−14.8716	−0.611735
$592$	19.9081	0.818219
$593$	26.5763	1.09136	0.545679	−	0.837995i	$-0.316272\pi$
$593$	26.5763	1.09136	0.545679	+	0.837995i	$0.316272\pi$
$594$	0	0
$595$	0	0
$596$	4.17484	0.171008
$597$	−11.2257	−0.459437
$598$	3.61285	0.147740
$599$	−8.77430	−0.358508	−0.179254	−	0.983803i	$-0.557368\pi$
$599$	−8.77430	−0.358508	−0.179254	+	0.983803i	$0.557368\pi$
$600$	0	0
$601$	41.8163	1.70572	0.852861	−	0.522139i	$-0.174866\pi$
$601$	41.8163	1.70572	0.852861	+	0.522139i	$0.174866\pi$
$602$	3.00937	0.122653
$603$	12.8573	0.523589
$604$	−19.8479	−0.807600
$605$	0	0
$606$	5.03212	0.204416
$607$	29.9353	1.21504	0.607519	−	0.794305i	$-0.292165\pi$
$607$	29.9353	1.21504	0.607519	+	0.794305i	$0.292165\pi$
$608$	8.48886	0.344269
$609$	−6.36842	−0.258061
$610$	0	0
$611$	−2.75557	−0.111478
$612$	4.34122	0.175483
$613$	26.1289	1.05534	0.527668	−	0.849450i	$-0.323066\pi$
$613$	26.1289	1.05534	0.527668	+	0.849450i	$0.323066\pi$
$614$	−4.19802	−0.169418
$615$	0	0
$616$	0	0
$617$	3.66323	0.147476	0.0737380	−	0.997278i	$-0.476507\pi$
$617$	3.66323	0.147476	0.0737380	+	0.997278i	$0.476507\pi$
$618$	5.33630	0.214657
$619$	43.2958	1.74020	0.870102	−	0.492872i	$-0.164053\pi$
$619$	43.2958	1.74020	0.870102	+	0.492872i	$0.164053\pi$
$620$	0	0
$621$	4.00000	0.160514
$622$	5.46076	0.218956
$623$	5.53972	0.221944
$624$	−9.95407	−0.398482
$625$	0	0
$626$	−4.47902	−0.179018
$627$	0	0
$628$	−11.0509	−0.440977
$629$	−13.2444	−0.528090
$630$	0	0
$631$	8.97773	0.357398	0.178699	−	0.983904i	$-0.442811\pi$
$631$	8.97773	0.357398	0.178699	+	0.983904i	$0.442811\pi$
$632$	20.0731	0.798466
$633$	11.9398	0.474564
$634$	−9.16547	−0.364007
$635$	0	0
$636$	−1.55262	−0.0615656
$637$	17.9541	0.711366
$638$	0	0
$639$	9.28592	0.367345
$640$	0	0
$641$	9.21279	0.363883	0.181942	−	0.983309i	$-0.441762\pi$
$641$	9.21279	0.363883	0.181942	+	0.983309i	$0.441762\pi$
$642$	−4.22077	−0.166581
$643$	16.3783	0.645896	0.322948	−	0.946417i	$-0.395326\pi$
$643$	16.3783	0.645896	0.322948	+	0.946417i	$0.395326\pi$
$644$	−6.87601	−0.270953
$645$	0	0
$646$	−1.72345	−0.0678082
$647$	−9.80642	−0.385530	−0.192765	−	0.981245i	$-0.561746\pi$
$647$	−9.80642	−0.385530	−0.192765	+	0.981245i	$0.561746\pi$
$648$	1.21432	0.0477030
$649$	0	0
$650$	0	0
$651$	−2.36842	−0.0928256
$652$	−21.0321	−0.823681
$653$	−33.0736	−1.29427	−0.647135	−	0.762375i	$-0.724033\pi$
$653$	−33.0736	−1.29427	−0.647135	+	0.762375i	$0.724033\pi$
$654$	−3.11108	−0.121653
$655$	0	0
$656$	36.5620	1.42751
$657$	5.65878	0.220770
$658$	−0.266706	−0.0103973
$659$	−34.1017	−1.32841	−0.664207	−	0.747549i	$-0.731231\pi$
$659$	−34.1017	−1.32841	−0.664207	+	0.747549i	$0.731231\pi$
$660$	0	0
$661$	−5.40943	−0.210402	−0.105201	−	0.994451i	$-0.533549\pi$
$661$	−5.40943	−0.210402	−0.105201	+	0.994451i	$0.533549\pi$
$662$	0.815792	0.0317066
$663$	6.62222	0.257186
$664$	−9.42372	−0.365711
$665$	0	0
$666$	−1.80642	−0.0699975
$667$	−28.2034	−1.09204
$668$	26.5575	1.02754
$669$	21.8064	0.843085
$670$	0	0
$671$	0	0
$672$	−3.15701	−0.121784
$673$	24.1476	0.930823	0.465412	−	0.885094i	$-0.345906\pi$
$673$	24.1476	0.930823	0.465412	+	0.885094i	$0.345906\pi$
$674$	−1.55707	−0.0599761
$675$	0	0
$676$	8.70027	0.334626
$677$	26.2810	1.01006	0.505030	−	0.863102i	$-0.331481\pi$
$677$	26.2810	1.01006	0.505030	+	0.863102i	$0.331481\pi$
$678$	−4.42864	−0.170081
$679$	−11.2632	−0.432241
$680$	0	0
$681$	−3.19850	−0.122567
$682$	0	0
$683$	−15.3176	−0.586110	−0.293055	−	0.956096i	$-0.594672\pi$
$683$	−15.3176	−0.586110	−0.293055	+	0.956096i	$0.594672\pi$
$684$	4.62222	0.176735
$685$	0	0
$686$	3.70471	0.141447
$687$	7.12399	0.271797
$688$	−36.7195	−1.39992
$689$	−2.36842	−0.0902295
$690$	0	0
$691$	15.0223	0.571474	0.285737	−	0.958308i	$-0.407762\pi$
$691$	15.0223	0.571474	0.285737	+	0.958308i	$0.407762\pi$
$692$	−35.8622	−1.36328
$693$	0	0
$694$	−7.09679	−0.269390
$695$	0	0
$696$	−8.56199	−0.324541
$697$	−24.3239	−0.921332
$698$	6.60348	0.249945
$699$	19.5254	0.738519
$700$	0	0
$701$	19.9081	0.751920	0.375960	−	0.926636i	$-0.377313\pi$
$701$	19.9081	0.751920	0.375960	+	0.926636i	$0.377313\pi$
$702$	0.903212	0.0340895
$703$	−14.1017	−0.531856
$704$	0	0
$705$	0	0
$706$	−2.23506	−0.0841177
$707$	−14.6093	−0.549440
$708$	3.18421	0.119670
$709$	−13.5081	−0.507306	−0.253653	−	0.967295i	$-0.581632\pi$
$709$	−13.5081	−0.507306	−0.253653	+	0.967295i	$0.581632\pi$
$710$	0	0
$711$	16.5303	0.619937
$712$	7.44785	0.279120
$713$	−10.4889	−0.392811
$714$	0.640951	0.0239870
$715$	0	0
$716$	−9.24443	−0.345481
$717$	21.9813	0.820905
$718$	−4.38715	−0.163727
$719$	−16.0830	−0.599794	−0.299897	−	0.953972i	$-0.596952\pi$
$719$	−16.0830	−0.599794	−0.299897	+	0.953972i	$0.596952\pi$
$720$	0	0
$721$	−15.4924	−0.576967
$722$	4.07604	0.151695
$723$	−5.34614	−0.198825
$724$	31.9684	1.18809
$725$	0	0
$726$	0	0
$727$	23.6128	0.875752	0.437876	−	0.899035i	$-0.355731\pi$
$727$	23.6128	0.875752	0.437876	+	0.899035i	$0.355731\pi$
$728$	−3.18421	−0.118015
$729$	1.00000	0.0370370
$730$	0	0
$731$	24.4286	0.903526
$732$	−13.7877	−0.509608
$733$	30.0459	1.10977	0.554886	−	0.831926i	$-0.312762\pi$
$733$	30.0459	1.10977	0.554886	+	0.831926i	$0.312762\pi$
$734$	1.21585	0.0448779
$735$	0	0
$736$	−13.9813	−0.515356
$737$	0	0
$738$	−3.31756	−0.122121
$739$	24.4099	0.897933	0.448966	−	0.893549i	$-0.351792\pi$
$739$	24.4099	0.897933	0.448966	+	0.893549i	$0.351792\pi$
$740$	0	0
$741$	7.05086	0.259020
$742$	−0.229234	−0.00841546
$743$	33.1798	1.21725	0.608624	−	0.793459i	$-0.291722\pi$
$743$	33.1798	1.21725	0.608624	+	0.793459i	$0.291722\pi$
$744$	−3.18421	−0.116739
$745$	0	0
$746$	4.03704	0.147807
$747$	−7.76049	−0.283942
$748$	0	0
$749$	12.2538	0.447744
$750$	0	0
$751$	−22.5718	−0.823658	−0.411829	−	0.911261i	$-0.635110\pi$
$751$	−22.5718	−0.823658	−0.411829	+	0.911261i	$0.635110\pi$
$752$	3.25428	0.118671
$753$	−23.7748	−0.866401
$754$	−6.36842	−0.231924
$755$	0	0
$756$	−1.71900	−0.0625196
$757$	−4.94914	−0.179880	−0.0899399	−	0.995947i	$-0.528667\pi$
$757$	−4.94914	−0.179880	−0.0899399	+	0.995947i	$0.528667\pi$
$758$	−11.2257	−0.407736
$759$	0	0
$760$	0	0
$761$	−14.6637	−0.531559	−0.265779	−	0.964034i	$-0.585629\pi$
$761$	−14.6637	−0.531559	−0.265779	+	0.964034i	$0.585629\pi$
$762$	3.42372	0.124028
$763$	9.03212	0.326985
$764$	−32.0830	−1.16072
$765$	0	0
$766$	−6.31402	−0.228135
$767$	4.85728	0.175386
$768$	8.80642	0.317774
$769$	44.5718	1.60730	0.803651	−	0.595101i	$-0.202888\pi$
$769$	44.5718	1.60730	0.803651	+	0.595101i	$0.202888\pi$
$770$	0	0
$771$	−8.13335	−0.292916
$772$	−46.8528	−1.68627
$773$	−17.3145	−0.622759	−0.311380	−	0.950286i	$-0.600791\pi$
$773$	−17.3145	−0.622759	−0.311380	+	0.950286i	$0.600791\pi$
$774$	3.33185	0.119761
$775$	0	0
$776$	−15.1427	−0.543592
$777$	5.24443	0.188143
$778$	−9.47949	−0.339856
$779$	−25.8983	−0.927903
$780$	0	0
$781$	0	0
$782$	2.83854	0.101506
$783$	−7.05086	−0.251977
$784$	−21.2034	−0.757265
$785$	0	0
$786$	0.387152	0.0138093
$787$	36.5161	1.30166	0.650828	−	0.759225i	$-0.274422\pi$
$787$	36.5161	1.30166	0.650828	+	0.759225i	$0.274422\pi$
$788$	28.3037	1.00828
$789$	−22.9032	−0.815376
$790$	0	0
$791$	12.8573	0.457152
$792$	0	0
$793$	−21.0321	−0.746872
$794$	−1.54861	−0.0549581
$795$	0	0
$796$	21.3649	0.757258
$797$	14.3180	0.507171	0.253585	−	0.967313i	$-0.418390\pi$
$797$	14.3180	0.507171	0.253585	+	0.967313i	$0.418390\pi$
$798$	0.682439	0.0241581
$799$	−2.16500	−0.0765921
$800$	0	0
$801$	6.13335	0.216711
$802$	0.580728	0.0205062
$803$	0	0
$804$	−24.4701	−0.862995
$805$	0	0
$806$	−2.36842	−0.0834239
$807$	11.8350	0.416612
$808$	−19.6414	−0.690983
$809$	−32.0544	−1.12697	−0.563486	−	0.826125i	$-0.690540\pi$
$809$	−32.0544	−1.12697	−0.563486	+	0.826125i	$0.690540\pi$
$810$	0	0
$811$	8.44738	0.296627	0.148314	−	0.988940i	$-0.452615\pi$
$811$	8.44738	0.296627	0.148314	+	0.988940i	$0.452615\pi$
$812$	12.1204	0.425344
$813$	−14.8988	−0.522523
$814$	0	0
$815$	0	0
$816$	−7.82071	−0.273780
$817$	26.0098	0.909969
$818$	−1.12981	−0.0395030
$819$	−2.62222	−0.0916276
$820$	0	0
$821$	−17.2159	−0.600837	−0.300419	−	0.953807i	$-0.597126\pi$
$821$	−17.2159	−0.600837	−0.300419	+	0.953807i	$0.597126\pi$
$822$	1.37778	0.0480557
$823$	−12.7654	−0.444974	−0.222487	−	0.974936i	$-0.571418\pi$
$823$	−12.7654	−0.444974	−0.222487	+	0.974936i	$0.571418\pi$
$824$	−20.8287	−0.725602
$825$	0	0
$826$	0.470127	0.0163578
$827$	8.70964	0.302864	0.151432	−	0.988468i	$-0.451612\pi$
$827$	8.70964	0.302864	0.151432	+	0.988468i	$0.451612\pi$
$828$	−7.61285	−0.264565
$829$	−8.32693	−0.289206	−0.144603	−	0.989490i	$-0.546191\pi$
$829$	−8.32693	−0.289206	−0.144603	+	0.989490i	$0.546191\pi$
$830$	0	0
$831$	27.6686	0.959814
$832$	16.7511	0.580741
$833$	14.1062	0.488749
$834$	0.285442	0.00988405
$835$	0	0
$836$	0	0
$837$	−2.62222	−0.0906370
$838$	−1.51114	−0.0522014
$839$	12.8988	0.445315	0.222657	−	0.974897i	$-0.428527\pi$
$839$	12.8988	0.445315	0.222657	+	0.974897i	$0.428527\pi$
$840$	0	0
$841$	20.7146	0.714295
$842$	−7.04101	−0.242649
$843$	−9.80642	−0.337751
$844$	−22.7239	−0.782190
$845$	0	0
$846$	−0.295286	−0.0101522
$847$	0	0
$848$	2.79706	0.0960513
$849$	19.0049	0.652247
$850$	0	0
$851$	23.2257	0.796167
$852$	−17.6731	−0.605469
$853$	−19.6686	−0.673441	−0.336720	−	0.941605i	$-0.609318\pi$
$853$	−19.6686	−0.673441	−0.336720	+	0.941605i	$0.609318\pi$
$854$	−2.03566	−0.0696588
$855$	0	0
$856$	16.4746	0.563089
$857$	31.8207	1.08697	0.543487	−	0.839417i	$-0.317104\pi$
$857$	31.8207	1.08697	0.543487	+	0.839417i	$0.317104\pi$
$858$	0	0
$859$	27.8292	0.949519	0.474760	−	0.880116i	$-0.342535\pi$
$859$	27.8292	0.949519	0.474760	+	0.880116i	$0.342535\pi$
$860$	0	0
$861$	9.63158	0.328243
$862$	0.387152	0.0131865
$863$	4.82870	0.164371	0.0821854	−	0.996617i	$-0.473810\pi$
$863$	4.82870	0.164371	0.0821854	+	0.996617i	$0.473810\pi$
$864$	−3.49532	−0.118913
$865$	0	0
$866$	4.97773	0.169150
$867$	−11.7971	−0.400649
$868$	4.50760	0.152998
$869$	0	0
$870$	0	0
$871$	−37.3274	−1.26479
$872$	12.1432	0.411221
$873$	−12.4701	−0.422050
$874$	3.02227	0.102230
$875$	0	0
$876$	−10.7699	−0.363880
$877$	21.9826	0.742301	0.371151	−	0.928573i	$-0.378963\pi$
$877$	21.9826	0.742301	0.371151	+	0.928573i	$0.378963\pi$
$878$	−0.755569	−0.0254992
$879$	30.7511	1.03721
$880$	0	0
$881$	−12.1017	−0.407717	−0.203858	−	0.979000i	$-0.565348\pi$
$881$	−12.1017	−0.407717	−0.203858	+	0.979000i	$0.565348\pi$
$882$	1.92396	0.0647830
$883$	−8.73683	−0.294018	−0.147009	−	0.989135i	$-0.546965\pi$
$883$	−8.73683	−0.294018	−0.147009	+	0.989135i	$0.546965\pi$
$884$	−12.6035	−0.423901
$885$	0	0
$886$	−9.66016	−0.324540
$887$	−19.8524	−0.666577	−0.333288	−	0.942825i	$-0.608158\pi$
$887$	−19.8524	−0.666577	−0.333288	+	0.942825i	$0.608158\pi$
$888$	7.05086	0.236611
$889$	−9.93978	−0.333369
$890$	0	0
$891$	0	0
$892$	−41.5022	−1.38960
$893$	−2.30513	−0.0771383
$894$	0.682439	0.0228242
$895$	0	0
$896$	7.93533	0.265101
$897$	−11.6128	−0.387742
$898$	11.6227	0.387854
$899$	18.4889	0.616638
$900$	0	0
$901$	−1.86082	−0.0619928
$902$	0	0
$903$	−9.67307	−0.321900
$904$	17.2859	0.574921
$905$	0	0
$906$	−3.24443	−0.107789
$907$	−32.8287	−1.09006	−0.545030	−	0.838417i	$-0.683482\pi$
$907$	−32.8287	−1.09006	−0.545030	+	0.838417i	$0.683482\pi$
$908$	6.08742	0.202018
$909$	−16.1748	−0.536486
$910$	0	0
$911$	−16.3497	−0.541689	−0.270845	−	0.962623i	$-0.587303\pi$
$911$	−16.3497	−0.541689	−0.270845	+	0.962623i	$0.587303\pi$
$912$	−8.32693	−0.275732
$913$	0	0
$914$	2.71853	0.0899209
$915$	0	0
$916$	−13.5585	−0.447984
$917$	−1.12399	−0.0371173
$918$	0.709636	0.0234215
$919$	−20.0228	−0.660490	−0.330245	−	0.943895i	$-0.607131\pi$
$919$	−20.0228	−0.660490	−0.330245	+	0.943895i	$0.607131\pi$
$920$	0	0
$921$	13.4938	0.444635
$922$	9.88940	0.325690
$923$	−26.9590	−0.887366
$924$	0	0
$925$	0	0
$926$	−3.76187	−0.123623
$927$	−17.1526	−0.563364
$928$	24.6450	0.809011
$929$	−43.5308	−1.42820	−0.714100	−	0.700044i	$-0.753164\pi$
$929$	−43.5308	−1.42820	−0.714100	+	0.700044i	$0.753164\pi$
$930$	0	0
$931$	15.0192	0.492235
$932$	−37.1610	−1.21725
$933$	−17.5526	−0.574647
$934$	4.77430	0.156220
$935$	0	0
$936$	−3.52543	−0.115232
$937$	−43.4563	−1.41966	−0.709828	−	0.704375i	$-0.751227\pi$
$937$	−43.4563	−1.41966	−0.709828	+	0.704375i	$0.751227\pi$
$938$	−3.61285	−0.117964
$939$	14.3970	0.469828
$940$	0	0
$941$	23.7244	0.773393	0.386697	−	0.922207i	$-0.373616\pi$
$941$	23.7244	0.773393	0.386697	+	0.922207i	$0.373616\pi$
$942$	−1.80642	−0.0588565
$943$	42.6548	1.38903
$944$	−5.73636	−0.186703
$945$	0	0
$946$	0	0
$947$	11.7047	0.380352	0.190176	−	0.981750i	$-0.439094\pi$
$947$	11.7047	0.380352	0.190176	+	0.981750i	$0.439094\pi$
$948$	−31.4608	−1.02180
$949$	−16.4286	−0.533296
$950$	0	0
$951$	29.4608	0.955331
$952$	−2.50177	−0.0810828
$953$	−46.1258	−1.49416	−0.747081	−	0.664733i	$-0.768545\pi$
$953$	−46.1258	−1.49416	−0.747081	+	0.664733i	$0.768545\pi$
$954$	−0.253799	−0.00821705
$955$	0	0
$956$	−41.8350	−1.35304
$957$	0	0
$958$	1.83500	0.0592863
$959$	−4.00000	−0.129167
$960$	0	0
$961$	−24.1240	−0.778193
$962$	5.24443	0.169087
$963$	13.5669	0.437188
$964$	10.1748	0.327710
$965$	0	0
$966$	−1.12399	−0.0361636
$967$	−17.0495	−0.548274	−0.274137	−	0.961691i	$-0.588392\pi$
$967$	−17.0495	−0.548274	−0.274137	+	0.961691i	$0.588392\pi$
$968$	0	0
$969$	5.53972	0.177961
$970$	0	0
$971$	−58.1847	−1.86724	−0.933618	−	0.358271i	$-0.883366\pi$
$971$	−58.1847	−1.86724	−0.933618	+	0.358271i	$0.883366\pi$
$972$	−1.90321	−0.0610456
$973$	−0.828699	−0.0265669
$974$	−9.75203	−0.312475
$975$	0	0
$976$	24.8385	0.795062
$977$	−51.7373	−1.65522	−0.827612	−	0.561301i	$-0.810301\pi$
$977$	−51.7373	−1.65522	−0.827612	+	0.561301i	$0.810301\pi$
$978$	−3.43801	−0.109935
$979$	0	0
$980$	0	0
$981$	10.0000	0.319275
$982$	−2.48886	−0.0794228
$983$	−26.3970	−0.841933	−0.420967	−	0.907076i	$-0.638309\pi$
$983$	−26.3970	−0.841933	−0.420967	+	0.907076i	$0.638309\pi$
$984$	12.9491	0.412804
$985$	0	0
$986$	−5.00354	−0.159345
$987$	0.857279	0.0272875
$988$	−13.4193	−0.426924
$989$	−42.8385	−1.36219
$990$	0	0
$991$	−23.0923	−0.733552	−0.366776	−	0.930309i	$-0.619539\pi$
$991$	−23.0923	−0.733552	−0.366776	+	0.930309i	$0.619539\pi$
$992$	9.16547	0.291004
$993$	−2.62222	−0.0832135
$994$	−2.60931	−0.0827622
$995$	0	0
$996$	14.7699	0.468001
$997$	−12.9131	−0.408961	−0.204480	−	0.978871i	$-0.565550\pi$
$997$	−12.9131	−0.408961	−0.204480	+	0.978871i	$0.565550\pi$
$998$	−4.71102	−0.149125
$999$	5.80642	0.183707

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9075.2.a.cg.1.2		3
5.2	odd	4		1815.2.c.e.364.3		6
5.3	odd	4		1815.2.c.e.364.4		6
5.4	even	2		9075.2.a.ch.1.2		3
11.10	odd	2		825.2.a.l.1.2		3
33.32	even	2		2475.2.a.ba.1.2		3
55.32	even	4		165.2.c.b.34.4	yes	6
55.43	even	4		165.2.c.b.34.3	✓	6
55.54	odd	2		825.2.a.j.1.2		3
165.32	odd	4		495.2.c.e.199.3		6
165.98	odd	4		495.2.c.e.199.4		6
165.164	even	2		2475.2.a.bc.1.2		3
220.43	odd	4		2640.2.d.h.529.2		6
220.87	odd	4		2640.2.d.h.529.5		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
165.2.c.b.34.3	✓	6	55.43	even	4
165.2.c.b.34.4	yes	6	55.32	even	4
495.2.c.e.199.3		6	165.32	odd	4
495.2.c.e.199.4		6	165.98	odd	4
825.2.a.j.1.2		3	55.54	odd	2
825.2.a.l.1.2		3	11.10	odd	2
1815.2.c.e.364.3		6	5.2	odd	4
1815.2.c.e.364.4		6	5.3	odd	4
2475.2.a.ba.1.2		3	33.32	even	2
2475.2.a.bc.1.2		3	165.164	even	2
2640.2.d.h.529.2		6	220.43	odd	4
2640.2.d.h.529.5		6	220.87	odd	4
9075.2.a.cg.1.2		3	1.1	even	1	trivial
9075.2.a.ch.1.2		3	5.4	even	2

Overview	Random
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Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q-0.311108 q^{2} +1.00000 q^{3} -1.90321 q^{4} -0.311108 q^{6} +0.903212 q^{7} +1.21432 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-0.311108 q^{2} +1.00000 q^{3} -1.90321 q^{4} -0.311108 q^{6} +0.903212 q^{7} +1.21432 q^{8} +1.00000 q^{9} -1.90321 q^{12} -2.90321 q^{13} -0.280996 q^{14} +3.42864 q^{16} -2.28100 q^{17} -0.311108 q^{18} -2.42864 q^{19} +0.903212 q^{21} +4.00000 q^{23} +1.21432 q^{24} +0.903212 q^{26} +1.00000 q^{27} -1.71900 q^{28} -7.05086 q^{29} -2.62222 q^{31} -3.49532 q^{32} +0.709636 q^{34} -1.90321 q^{36} +5.80642 q^{37} +0.755569 q^{38} -2.90321 q^{39} +10.6637 q^{41} -0.280996 q^{42} -10.7096 q^{43} -1.24443 q^{46} +0.949145 q^{47} +3.42864 q^{48} -6.18421 q^{49} -2.28100 q^{51} +5.52543 q^{52} +0.815792 q^{53} -0.311108 q^{54} +1.09679 q^{56} -2.42864 q^{57} +2.19358 q^{58} -1.67307 q^{59} +7.24443 q^{61} +0.815792 q^{62} +0.903212 q^{63} -5.76986 q^{64} +12.8573 q^{67} +4.34122 q^{68} +4.00000 q^{69} +9.28592 q^{71} +1.21432 q^{72} +5.65878 q^{73} -1.80642 q^{74} +4.62222 q^{76} +0.903212 q^{78} +16.5303 q^{79} +1.00000 q^{81} -3.31756 q^{82} -7.76049 q^{83} -1.71900 q^{84} +3.33185 q^{86} -7.05086 q^{87} +6.13335 q^{89} -2.62222 q^{91} -7.61285 q^{92} -2.62222 q^{93} -0.295286 q^{94} -3.49532 q^{96} -12.4701 q^{97} +1.92396 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - q^{2} + 3 q^{3} + q^{4} - q^{6} - 4 q^{7} - 3 q^{8} + 3 q^{9}+O(q^{10}) \) 3 * q - q^2 + 3 * q^3 + q^4 - q^6 - 4 * q^7 - 3 * q^8 + 3 * q^9	\( 3 q - q^{2} + 3 q^{3} + q^{4} - q^{6} - 4 q^{7} - 3 q^{8} + 3 q^{9} + q^{12} - 2 q^{13} + 6 q^{14} - 3 q^{16} - q^{18} + 6 q^{19} - 4 q^{21} + 12 q^{23} - 3 q^{24} - 4 q^{26} + 3 q^{27} - 12 q^{28} - 8 q^{29} - 8 q^{31} + 3 q^{32} - 18 q^{34} + q^{36} + 4 q^{37} + 2 q^{38} - 2 q^{39} - 8 q^{41} + 6 q^{42} - 12 q^{43} - 4 q^{46} + 16 q^{47} - 3 q^{48} - 5 q^{49} + 10 q^{52} + 16 q^{53} - q^{54} + 10 q^{56} + 6 q^{57} + 20 q^{58} + 8 q^{59} + 22 q^{61} + 16 q^{62} - 4 q^{63} - 11 q^{64} + 12 q^{67} + 20 q^{68} + 12 q^{69} - 12 q^{71} - 3 q^{72} + 10 q^{73} + 8 q^{74} + 14 q^{76} - 4 q^{78} + 10 q^{79} + 3 q^{81} + 4 q^{82} + 10 q^{83} - 12 q^{84} - 10 q^{86} - 8 q^{87} + 18 q^{89} - 8 q^{91} + 4 q^{92} - 8 q^{93} + 12 q^{94} + 3 q^{96} + 16 q^{97} - 21 q^{98}+O(q^{100}) \) 3 * q - q^2 + 3 * q^3 + q^4 - q^6 - 4 * q^7 - 3 * q^8 + 3 * q^9 + q^12 - 2 * q^13 + 6 * q^14 - 3 * q^16 - q^18 + 6 * q^19 - 4 * q^21 + 12 * q^23 - 3 * q^24 - 4 * q^26 + 3 * q^27 - 12 * q^28 - 8 * q^29 - 8 * q^31 + 3 * q^32 - 18 * q^34 + q^36 + 4 * q^37 + 2 * q^38 - 2 * q^39 - 8 * q^41 + 6 * q^42 - 12 * q^43 - 4 * q^46 + 16 * q^47 - 3 * q^48 - 5 * q^49 + 10 * q^52 + 16 * q^53 - q^54 + 10 * q^56 + 6 * q^57 + 20 * q^58 + 8 * q^59 + 22 * q^61 + 16 * q^62 - 4 * q^63 - 11 * q^64 + 12 * q^67 + 20 * q^68 + 12 * q^69 - 12 * q^71 - 3 * q^72 + 10 * q^73 + 8 * q^74 + 14 * q^76 - 4 * q^78 + 10 * q^79 + 3 * q^81 + 4 * q^82 + 10 * q^83 - 12 * q^84 - 10 * q^86 - 8 * q^87 + 18 * q^89 - 8 * q^91 + 4 * q^92 - 8 * q^93 + 12 * q^94 + 3 * q^96 + 16 * q^97 - 21 * q^98

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