LMFDB - Embedded newform 9075.2.a.cd.1.3

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:	3.3.568.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 6x - 2 $ x^3 - x^2 - 6*x - 2
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 825)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$3.12489$ 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)

$f(q)$	$=$	$q+2.12489 q^{2} +1.00000 q^{3} +2.51514 q^{4} +2.12489 q^{6} +3.64002 q^{7} +1.09461 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+2.12489 q^{2} +1.00000 q^{3} +2.51514 q^{4} +2.12489 q^{6} +3.64002 q^{7} +1.09461 q^{8} +1.00000 q^{9} +2.51514 q^{12} +1.51514 q^{13} +7.73463 q^{14} -2.70436 q^{16} +1.15516 q^{17} +2.12489 q^{18} -2.60975 q^{19} +3.64002 q^{21} +5.73463 q^{23} +1.09461 q^{24} +3.21949 q^{26} +1.00000 q^{27} +9.15516 q^{28} -6.24977 q^{29} +5.51514 q^{31} -7.93567 q^{32} +2.45459 q^{34} +2.51514 q^{36} -0.454586 q^{37} -5.54541 q^{38} +1.51514 q^{39} -4.12489 q^{41} +7.73463 q^{42} +11.7044 q^{43} +12.1854 q^{46} +3.48486 q^{47} -2.70436 q^{48} +6.24977 q^{49} +1.15516 q^{51} +3.81078 q^{52} +12.5601 q^{53} +2.12489 q^{54} +3.98440 q^{56} -2.60975 q^{57} -13.2800 q^{58} -7.73463 q^{59} +12.0147 q^{61} +11.7190 q^{62} +3.64002 q^{63} -11.4537 q^{64} +14.2645 q^{67} +2.90539 q^{68} +5.73463 q^{69} +8.51514 q^{71} +1.09461 q^{72} -9.21949 q^{73} -0.965943 q^{74} -6.56387 q^{76} +3.21949 q^{78} -5.09461 q^{79} +1.00000 q^{81} -8.76491 q^{82} -14.7493 q^{83} +9.15516 q^{84} +24.8704 q^{86} -6.24977 q^{87} -10.4995 q^{89} +5.51514 q^{91} +14.4234 q^{92} +5.51514 q^{93} +7.40493 q^{94} -7.93567 q^{96} +6.77959 q^{97} +13.2800 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 2 q^{2} + 3 q^{3} + 8 q^{4} - 2 q^{6} + 3 q^{7} - 6 q^{8} + 3 q^{9}+O(q^{10}) $ 3 * q - 2 * q^2 + 3 * q^3 + 8 * q^4 - 2 * q^6 + 3 * q^7 - 6 * q^8 + 3 * q^9	$ 3 q - 2 q^{2} + 3 q^{3} + 8 q^{4} - 2 q^{6} + 3 q^{7} - 6 q^{8} + 3 q^{9} + 8 q^{12} + 5 q^{13} + 6 q^{14} + 10 q^{16} - 4 q^{17} - 2 q^{18} + q^{19} + 3 q^{21} - 6 q^{24} - 8 q^{26} + 3 q^{27} + 20 q^{28} - 2 q^{29} + 17 q^{31} - 34 q^{32} + 6 q^{34} + 8 q^{36} - 18 q^{38} + 5 q^{39} - 4 q^{41} + 6 q^{42} + 17 q^{43} + 30 q^{46} + 10 q^{47} + 10 q^{48} + 2 q^{49} - 4 q^{51} + 30 q^{52} + 6 q^{53} - 2 q^{54} - 22 q^{56} + q^{57} - 24 q^{58} - 6 q^{59} + 3 q^{61} - 16 q^{62} + 3 q^{63} + 34 q^{64} - 7 q^{67} + 18 q^{68} + 26 q^{71} - 6 q^{72} - 10 q^{73} - 14 q^{74} + 24 q^{76} - 8 q^{78} - 6 q^{79} + 3 q^{81} - 10 q^{82} + 6 q^{83} + 20 q^{84} + 28 q^{86} - 2 q^{87} + 2 q^{89} + 17 q^{91} - 26 q^{92} + 17 q^{93} - 2 q^{94} - 34 q^{96} - 29 q^{97} + 24 q^{98}+O(q^{100}) $ 3 * q - 2 * q^2 + 3 * q^3 + 8 * q^4 - 2 * q^6 + 3 * q^7 - 6 * q^8 + 3 * q^9 + 8 * q^12 + 5 * q^13 + 6 * q^14 + 10 * q^16 - 4 * q^17 - 2 * q^18 + q^19 + 3 * q^21 - 6 * q^24 - 8 * q^26 + 3 * q^27 + 20 * q^28 - 2 * q^29 + 17 * q^31 - 34 * q^32 + 6 * q^34 + 8 * q^36 - 18 * q^38 + 5 * q^39 - 4 * q^41 + 6 * q^42 + 17 * q^43 + 30 * q^46 + 10 * q^47 + 10 * q^48 + 2 * q^49 - 4 * q^51 + 30 * q^52 + 6 * q^53 - 2 * q^54 - 22 * q^56 + q^57 - 24 * q^58 - 6 * q^59 + 3 * q^61 - 16 * q^62 + 3 * q^63 + 34 * q^64 - 7 * q^67 + 18 * q^68 + 26 * q^71 - 6 * q^72 - 10 * q^73 - 14 * q^74 + 24 * q^76 - 8 * q^78 - 6 * q^79 + 3 * q^81 - 10 * q^82 + 6 * q^83 + 20 * q^84 + 28 * q^86 - 2 * q^87 + 2 * q^89 + 17 * q^91 - 26 * q^92 + 17 * q^93 - 2 * q^94 - 34 * q^96 - 29 * q^97 + 24 * 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 $

$2$	2.12489	1.50252	0.751260	−	0.660006i	$-0.229446\pi$
$2$	2.12489	1.50252	0.751260	+	0.660006i	$0.229446\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	2.51514	1.25757
$5$	0	0
$5$	0	0
$6$	2.12489	0.867481
$7$	3.64002	1.37580	0.687900	−	0.725806i	$-0.258533\pi$
$7$	3.64002	1.37580	0.687900	+	0.725806i	$0.258533\pi$
$8$	1.09461	0.387003
$9$	1.00000	0.333333
$10$	0	0
$11$	0	0
$11$	0	0
$12$	2.51514	0.726058
$13$	1.51514	0.420224	0.210112	−	0.977677i	$-0.432617\pi$
$13$	1.51514	0.420224	0.210112	+	0.977677i	$0.432617\pi$
$14$	7.73463	2.06717
$15$	0	0
$16$	−2.70436	−0.676089
$17$	1.15516	0.280168	0.140084	−	0.990140i	$-0.455263\pi$
$17$	1.15516	0.280168	0.140084	+	0.990140i	$0.455263\pi$
$18$	2.12489	0.500840
$19$	−2.60975	−0.598717	−0.299359	−	0.954141i	$-0.596773\pi$
$19$	−2.60975	−0.598717	−0.299359	+	0.954141i	$0.596773\pi$
$20$	0	0
$21$	3.64002	0.794318
$22$	0	0
$23$	5.73463	1.19575	0.597877	−	0.801588i	$-0.296011\pi$
$23$	5.73463	1.19575	0.597877	+	0.801588i	$0.296011\pi$
$24$	1.09461	0.223436
$25$	0	0
$26$	3.21949	0.631395
$27$	1.00000	0.192450
$28$	9.15516	1.73016
$29$	−6.24977	−1.16055	−0.580277	−	0.814419i	$-0.697056\pi$
$29$	−6.24977	−1.16055	−0.580277	+	0.814419i	$0.697056\pi$
$30$	0	0
$31$	5.51514	0.990548	0.495274	−	0.868737i	$-0.335068\pi$
$31$	5.51514	0.990548	0.495274	+	0.868737i	$0.335068\pi$
$32$	−7.93567	−1.40284
$33$	0	0
$34$	2.45459	0.420958
$35$	0	0
$36$	2.51514	0.419190
$37$	−0.454586	−0.0747335	−0.0373667	−	0.999302i	$-0.511897\pi$
$37$	−0.454586	−0.0747335	−0.0373667	+	0.999302i	$0.511897\pi$
$38$	−5.54541	−0.899585
$39$	1.51514	0.242616
$40$	0	0
$41$	−4.12489	−0.644199	−0.322099	−	0.946706i	$-0.604389\pi$
$41$	−4.12489	−0.644199	−0.322099	+	0.946706i	$0.604389\pi$
$42$	7.73463	1.19348
$43$	11.7044	1.78490	0.892449	−	0.451149i	$-0.148986\pi$
$43$	11.7044	1.78490	0.892449	+	0.451149i	$0.148986\pi$
$44$	0	0
$45$	0	0
$46$	12.1854	1.79664
$47$	3.48486	0.508319	0.254160	−	0.967162i	$-0.418201\pi$
$47$	3.48486	0.508319	0.254160	+	0.967162i	$0.418201\pi$
$48$	−2.70436	−0.390340
$49$	6.24977	0.892824
$50$	0	0
$51$	1.15516	0.161755
$52$	3.81078	0.528460
$53$	12.5601	1.72526	0.862631	−	0.505834i	$-0.168815\pi$
$53$	12.5601	1.72526	0.862631	+	0.505834i	$0.168815\pi$
$54$	2.12489	0.289160
$55$	0	0
$56$	3.98440	0.532438
$57$	−2.60975	−0.345669
$58$	−13.2800	−1.74376
$59$	−7.73463	−1.00696	−0.503482	−	0.864006i	$-0.667948\pi$
$59$	−7.73463	−1.00696	−0.503482	+	0.864006i	$0.667948\pi$
$60$	0	0
$61$	12.0147	1.53832	0.769161	−	0.639055i	$-0.220674\pi$
$61$	12.0147	1.53832	0.769161	+	0.639055i	$0.220674\pi$
$62$	11.7190	1.48832
$63$	3.64002	0.458600
$64$	−11.4537	−1.43171
$65$	0	0
$66$	0	0
$67$	14.2645	1.74268	0.871340	−	0.490680i	$-0.163251\pi$
$67$	14.2645	1.74268	0.871340	+	0.490680i	$0.163251\pi$
$68$	2.90539	0.352330
$69$	5.73463	0.690369
$70$	0	0
$71$	8.51514	1.01056	0.505280	−	0.862955i	$-0.331389\pi$
$71$	8.51514	1.01056	0.505280	+	0.862955i	$0.331389\pi$
$72$	1.09461	0.129001
$73$	−9.21949	−1.07906	−0.539530	−	0.841966i	$-0.681398\pi$
$73$	−9.21949	−1.07906	−0.539530	+	0.841966i	$0.681398\pi$
$74$	−0.965943	−0.112289
$75$	0	0
$76$	−6.56387	−0.752928
$77$	0	0
$78$	3.21949	0.364536
$79$	−5.09461	−0.573188	−0.286594	−	0.958052i	$-0.592523\pi$
$79$	−5.09461	−0.573188	−0.286594	+	0.958052i	$0.592523\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	−8.76491	−0.967922
$83$	−14.7493	−1.61895	−0.809474	−	0.587156i	$-0.800247\pi$
$83$	−14.7493	−1.61895	−0.809474	+	0.587156i	$0.800247\pi$
$84$	9.15516	0.998910
$85$	0	0
$86$	24.8704	2.68185
$87$	−6.24977	−0.670046
$88$	0	0
$89$	−10.4995	−1.11295	−0.556475	−	0.830865i	$-0.687846\pi$
$89$	−10.4995	−1.11295	−0.556475	+	0.830865i	$0.687846\pi$
$90$	0	0
$91$	5.51514	0.578144
$92$	14.4234	1.50374
$93$	5.51514	0.571893
$94$	7.40493	0.763760
$95$	0	0
$96$	−7.93567	−0.809931
$97$	6.77959	0.688363	0.344181	−	0.938903i	$-0.388156\pi$
$97$	6.77959	0.688363	0.344181	+	0.938903i	$0.388156\pi$
$98$	13.2800	1.34149
$99$	0	0
$100$	0	0
$101$	−7.40493	−0.736818	−0.368409	−	0.929664i	$-0.620097\pi$
$101$	−7.40493	−0.736818	−0.368409	+	0.929664i	$0.620097\pi$
$102$	2.45459	0.243040
$103$	16.4995	1.62575	0.812874	−	0.582439i	$-0.197902\pi$
$103$	16.4995	1.62575	0.812874	+	0.582439i	$0.197902\pi$
$104$	1.65848	0.162628
$105$	0	0
$106$	26.6888	2.59224
$107$	3.93945	0.380841	0.190420	−	0.981703i	$-0.439015\pi$
$107$	3.93945	0.380841	0.190420	+	0.981703i	$0.439015\pi$
$108$	2.51514	0.242019
$109$	−6.73463	−0.645061	−0.322530	−	0.946559i	$-0.604533\pi$
$109$	−6.73463	−0.645061	−0.322530	+	0.946559i	$0.604533\pi$
$110$	0	0
$111$	−0.454586	−0.0431474
$112$	−9.84392	−0.930163
$113$	6.00000	0.564433	0.282216	−	0.959351i	$-0.408930\pi$
$113$	6.00000	0.564433	0.282216	+	0.959351i	$0.408930\pi$
$114$	−5.54541	−0.519376
$115$	0	0
$116$	−15.7190	−1.45948
$117$	1.51514	0.140075
$118$	−16.4352	−1.51298
$119$	4.20482	0.385455
$120$	0	0
$121$	0	0
$122$	25.5298	2.31136
$123$	−4.12489	−0.371928
$124$	13.8713	1.24568
$125$	0	0
$126$	7.73463	0.689056
$127$	8.06433	0.715594	0.357797	−	0.933799i	$-0.383528\pi$
$127$	8.06433	0.715594	0.357797	+	0.933799i	$0.383528\pi$
$128$	−8.46640	−0.748331
$129$	11.7044	1.03051
$130$	0	0
$131$	12.8099	1.11920	0.559602	−	0.828762i	$-0.310954\pi$
$131$	12.8099	1.11920	0.559602	+	0.828762i	$0.310954\pi$
$132$	0	0
$133$	−9.49954	−0.823715
$134$	30.3103	2.61841
$135$	0	0
$136$	1.26445	0.108426
$137$	−22.8099	−1.94878	−0.974389	−	0.224868i	$-0.927805\pi$
$137$	−22.8099	−1.94878	−0.974389	+	0.224868i	$0.927805\pi$
$138$	12.1854	1.03729
$139$	−7.59037	−0.643807	−0.321903	−	0.946773i	$-0.604323\pi$
$139$	−7.59037	−0.643807	−0.321903	+	0.946773i	$0.604323\pi$
$140$	0	0
$141$	3.48486	0.293478
$142$	18.0937	1.51839
$143$	0	0
$144$	−2.70436	−0.225363
$145$	0	0
$146$	−19.5904	−1.62131
$147$	6.24977	0.515472
$148$	−1.14335	−0.0939825
$149$	1.81456	0.148655	0.0743274	−	0.997234i	$-0.476319\pi$
$149$	1.81456	0.148655	0.0743274	+	0.997234i	$0.476319\pi$
$150$	0	0
$151$	−24.3250	−1.97954	−0.989770	−	0.142670i	$-0.954431\pi$
$151$	−24.3250	−1.97954	−0.989770	+	0.142670i	$0.954431\pi$
$152$	−2.85665	−0.231705
$153$	1.15516	0.0933893
$154$	0	0
$155$	0	0
$156$	3.81078	0.305107
$157$	−9.76491	−0.779325	−0.389662	−	0.920958i	$-0.627408\pi$
$157$	−9.76491	−0.779325	−0.389662	+	0.920958i	$0.627408\pi$
$158$	−10.8255	−0.861227
$159$	12.5601	0.996080
$160$	0	0
$161$	20.8742	1.64512
$162$	2.12489	0.166947
$163$	6.98440	0.547061	0.273530	−	0.961863i	$-0.411809\pi$
$163$	6.98440	0.547061	0.273530	+	0.961863i	$0.411809\pi$
$164$	−10.3747	−0.810125
$165$	0	0
$166$	−31.3406	−2.43250
$167$	−6.31032	−0.488307	−0.244154	−	0.969737i	$-0.578510\pi$
$167$	−6.31032	−0.488307	−0.244154	+	0.969737i	$0.578510\pi$
$168$	3.98440	0.307403
$169$	−10.7044	−0.823412
$170$	0	0
$171$	−2.60975	−0.199572
$172$	29.4381	2.24463
$173$	−12.8448	−0.976575	−0.488287	−	0.872683i	$-0.662378\pi$
$173$	−12.8448	−0.976575	−0.488287	+	0.872683i	$0.662378\pi$
$174$	−13.2800	−1.00676
$175$	0	0
$176$	0	0
$177$	−7.73463	−0.581371
$178$	−22.3103	−1.67223
$179$	13.4849	1.00791	0.503953	−	0.863731i	$-0.331878\pi$
$179$	13.4849	1.00791	0.503953	+	0.863731i	$0.331878\pi$
$180$	0	0
$181$	−23.0899	−1.71626	−0.858130	−	0.513433i	$-0.828374\pi$
$181$	−23.0899	−1.71626	−0.858130	+	0.513433i	$0.828374\pi$
$182$	11.7190	0.868673
$183$	12.0147	0.888151
$184$	6.27718	0.462760
$185$	0	0
$186$	11.7190	0.859281
$187$	0	0
$188$	8.76491	0.639247
$189$	3.64002	0.264773
$190$	0	0
$191$	−7.98440	−0.577731	−0.288866	−	0.957370i	$-0.593278\pi$
$191$	−7.98440	−0.577731	−0.288866	+	0.957370i	$0.593278\pi$
$192$	−11.4537	−0.826597
$193$	11.7649	0.846857	0.423428	−	0.905930i	$-0.360827\pi$
$193$	11.7649	0.846857	0.423428	+	0.905930i	$0.360827\pi$
$194$	14.4058	1.03428
$195$	0	0
$196$	15.7190	1.12279
$197$	−3.81456	−0.271776	−0.135888	−	0.990724i	$-0.543389\pi$
$197$	−3.81456	−0.271776	−0.135888	+	0.990724i	$0.543389\pi$
$198$	0	0
$199$	−12.0752	−0.855990	−0.427995	−	0.903781i	$-0.640780\pi$
$199$	−12.0752	−0.855990	−0.427995	+	0.903781i	$0.640780\pi$
$200$	0	0
$201$	14.2645	1.00614
$202$	−15.7346	−1.10708
$203$	−22.7493	−1.59669
$204$	2.90539	0.203418
$205$	0	0
$206$	35.0596	2.44272
$207$	5.73463	0.398585
$208$	−4.09747	−0.284109
$209$	0	0
$210$	0	0
$211$	−10.2645	−0.706634	−0.353317	−	0.935504i	$-0.614946\pi$
$211$	−10.2645	−0.706634	−0.353317	+	0.935504i	$0.614946\pi$
$212$	31.5904	2.16964
$213$	8.51514	0.583448
$214$	8.37088	0.572221
$215$	0	0
$216$	1.09461	0.0744787
$217$	20.0752	1.36280
$218$	−14.3103	−0.969217
$219$	−9.21949	−0.622996
$220$	0	0
$221$	1.75023	0.117733
$222$	−0.965943	−0.0648298
$223$	−12.9239	−0.865445	−0.432723	−	0.901527i	$-0.642447\pi$
$223$	−12.9239	−0.865445	−0.432723	+	0.901527i	$0.642447\pi$
$224$	−28.8860	−1.93003
$225$	0	0
$226$	12.7493	0.848072
$227$	22.8099	1.51394	0.756972	−	0.653447i	$-0.226678\pi$
$227$	22.8099	1.51394	0.756972	+	0.653447i	$0.226678\pi$
$228$	−6.56387	−0.434703
$229$	14.7796	0.976663	0.488331	−	0.872658i	$-0.337606\pi$
$229$	14.7796	0.976663	0.488331	+	0.872658i	$0.337606\pi$
$230$	0	0
$231$	0	0
$232$	−6.84106	−0.449137
$233$	4.96594	0.325330	0.162665	−	0.986681i	$-0.447991\pi$
$233$	4.96594	0.325330	0.162665	+	0.986681i	$0.447991\pi$
$234$	3.21949	0.210465
$235$	0	0
$236$	−19.4537	−1.26633
$237$	−5.09461	−0.330930
$238$	8.93475	0.579154
$239$	14.9991	0.970210	0.485105	−	0.874456i	$-0.338781\pi$
$239$	14.9991	0.970210	0.485105	+	0.874456i	$0.338781\pi$
$240$	0	0
$241$	−5.04496	−0.324974	−0.162487	−	0.986711i	$-0.551952\pi$
$241$	−5.04496	−0.324974	−0.162487	+	0.986711i	$0.551952\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	30.2186	1.93455
$245$	0	0
$246$	−8.76491	−0.558830
$247$	−3.95413	−0.251595
$248$	6.03692	0.383345
$249$	−14.7493	−0.934700
$250$	0	0
$251$	−3.03028	−0.191269	−0.0956347	−	0.995417i	$-0.530488\pi$
$251$	−3.03028	−0.191269	−0.0956347	+	0.995417i	$0.530488\pi$
$252$	9.15516	0.576721
$253$	0	0
$254$	17.1358	1.07519
$255$	0	0
$256$	4.91721	0.307325
$257$	13.6509	0.851521	0.425761	−	0.904836i	$-0.360007\pi$
$257$	13.6509	0.851521	0.425761	+	0.904836i	$0.360007\pi$
$258$	24.8704	1.54836
$259$	−1.65470	−0.102818
$260$	0	0
$261$	−6.24977	−0.386851
$262$	27.2195	1.68163
$263$	12.5601	0.774489	0.387244	−	0.921977i	$-0.373427\pi$
$263$	12.5601	0.774489	0.387244	+	0.921977i	$0.373427\pi$
$264$	0	0
$265$	0	0
$266$	−20.1854	−1.23765
$267$	−10.4995	−0.642562
$268$	35.8771	2.19154
$269$	−24.6888	−1.50530	−0.752650	−	0.658421i	$-0.771225\pi$
$269$	−24.6888	−1.50530	−0.752650	+	0.658421i	$0.771225\pi$
$270$	0	0
$271$	−7.56479	−0.459528	−0.229764	−	0.973246i	$-0.573795\pi$
$271$	−7.56479	−0.459528	−0.229764	+	0.973246i	$0.573795\pi$
$272$	−3.12397	−0.189418
$273$	5.51514	0.333791
$274$	−48.4683	−2.92808
$275$	0	0
$276$	14.4234	0.868186
$277$	1.92477	0.115648	0.0578241	−	0.998327i	$-0.481584\pi$
$277$	1.92477	0.115648	0.0578241	+	0.998327i	$0.481584\pi$
$278$	−16.1287	−0.967333
$279$	5.51514	0.330183
$280$	0	0
$281$	1.87511	0.111860	0.0559300	−	0.998435i	$-0.482188\pi$
$281$	1.87511	0.111860	0.0559300	+	0.998435i	$0.482188\pi$
$282$	7.40493	0.440957
$283$	30.1396	1.79161	0.895806	−	0.444446i	$-0.146599\pi$
$283$	30.1396	1.79161	0.895806	+	0.444446i	$0.146599\pi$
$284$	21.4167	1.27085
$285$	0	0
$286$	0	0
$287$	−15.0147	−0.886289
$288$	−7.93567	−0.467614
$289$	−15.6656	−0.921506
$290$	0	0
$291$	6.77959	0.397427
$292$	−23.1883	−1.35699
$293$	−29.1552	−1.70326	−0.851631	−	0.524141i	$-0.824386\pi$
$293$	−29.1552	−1.70326	−0.851631	+	0.524141i	$0.824386\pi$
$294$	13.2800	0.774508
$295$	0	0
$296$	−0.497594	−0.0289221
$297$	0	0
$298$	3.85574	0.223357
$299$	8.68876	0.502484
$300$	0	0
$301$	42.6041	2.45566
$302$	−51.6878	−2.97430
$303$	−7.40493	−0.425402
$304$	7.05769	0.404786
$305$	0	0
$306$	2.45459	0.140319
$307$	27.8548	1.58976	0.794879	−	0.606768i	$-0.207534\pi$
$307$	27.8548	1.58976	0.794879	+	0.606768i	$0.207534\pi$
$308$	0	0
$309$	16.4995	0.938626
$310$	0	0
$311$	23.9083	1.35571	0.677856	−	0.735194i	$-0.262909\pi$
$311$	23.9083	1.35571	0.677856	+	0.735194i	$0.262909\pi$
$312$	1.65848	0.0938932
$313$	−28.3094	−1.60014	−0.800071	−	0.599905i	$-0.795205\pi$
$313$	−28.3094	−1.60014	−0.800071	+	0.599905i	$0.795205\pi$
$314$	−20.7493	−1.17095
$315$	0	0
$316$	−12.8136	−0.720824
$317$	8.80986	0.494811	0.247406	−	0.968912i	$-0.420422\pi$
$317$	8.80986	0.494811	0.247406	+	0.968912i	$0.420422\pi$
$318$	26.6888	1.49663
$319$	0	0
$320$	0	0
$321$	3.93945	0.219879
$322$	44.3553	2.47182
$323$	−3.01468	−0.167741
$324$	2.51514	0.139730
$325$	0	0
$326$	14.8411	0.821970
$327$	−6.73463	−0.372426
$328$	−4.51514	−0.249307
$329$	12.6850	0.699346
$330$	0	0
$331$	32.2498	1.77261	0.886304	−	0.463104i	$-0.153264\pi$
$331$	32.2498	1.77261	0.886304	+	0.463104i	$0.153264\pi$
$332$	−37.0966	−2.03594
$333$	−0.454586	−0.0249112
$334$	−13.4087	−0.733692
$335$	0	0
$336$	−9.84392	−0.537030
$337$	28.9844	1.57888	0.789441	−	0.613827i	$-0.210371\pi$
$337$	28.9844	1.57888	0.789441	+	0.613827i	$0.210371\pi$
$338$	−22.7455	−1.23719
$339$	6.00000	0.325875
$340$	0	0
$341$	0	0
$342$	−5.54541	−0.299862
$343$	−2.73085	−0.147452
$344$	12.8117	0.690760
$345$	0	0
$346$	−27.2938	−1.46732
$347$	−35.7190	−1.91750	−0.958749	−	0.284253i	$-0.908254\pi$
$347$	−35.7190	−1.91750	−0.958749	+	0.284253i	$0.908254\pi$
$348$	−15.7190	−0.842629
$349$	23.2800	1.24615	0.623076	−	0.782161i	$-0.285883\pi$
$349$	23.2800	1.24615	0.623076	+	0.782161i	$0.285883\pi$
$350$	0	0
$351$	1.51514	0.0808721
$352$	0	0
$353$	9.75023	0.518952	0.259476	−	0.965750i	$-0.416450\pi$
$353$	9.75023	0.518952	0.259476	+	0.965750i	$0.416450\pi$
$354$	−16.4352	−0.873521
$355$	0	0
$356$	−26.4078	−1.39961
$357$	4.20482	0.222542
$358$	28.6538	1.51440
$359$	−33.9007	−1.78921	−0.894605	−	0.446858i	$-0.852543\pi$
$359$	−33.9007	−1.78921	−0.894605	+	0.446858i	$0.852543\pi$
$360$	0	0
$361$	−12.1892	−0.641538
$362$	−49.0634	−2.57872
$363$	0	0
$364$	13.8713	0.727055
$365$	0	0
$366$	25.5298	1.33446
$367$	−1.88601	−0.0984491	−0.0492245	−	0.998788i	$-0.515675\pi$
$367$	−1.88601	−0.0984491	−0.0492245	+	0.998788i	$0.515675\pi$
$368$	−15.5085	−0.808436
$369$	−4.12489	−0.214733
$370$	0	0
$371$	45.7190	2.37361
$372$	13.8713	0.719195
$373$	−16.3250	−0.845277	−0.422638	−	0.906298i	$-0.638896\pi$
$373$	−16.3250	−0.845277	−0.422638	+	0.906298i	$0.638896\pi$
$374$	0	0
$375$	0	0
$376$	3.81456	0.196721
$377$	−9.46927	−0.487692
$378$	7.73463	0.397827
$379$	−26.0440	−1.33779	−0.668896	−	0.743356i	$-0.733233\pi$
$379$	−26.0440	−1.33779	−0.668896	+	0.743356i	$0.733233\pi$
$380$	0	0
$381$	8.06433	0.413148
$382$	−16.9659	−0.868053
$383$	−12.4702	−0.637197	−0.318598	−	0.947890i	$-0.603212\pi$
$383$	−12.4702	−0.637197	−0.318598	+	0.947890i	$0.603212\pi$
$384$	−8.46640	−0.432049
$385$	0	0
$386$	24.9991	1.27242
$387$	11.7044	0.594966
$388$	17.0516	0.865664
$389$	−18.0899	−0.917195	−0.458597	−	0.888644i	$-0.651648\pi$
$389$	−18.0899	−0.917195	−0.458597	+	0.888644i	$0.651648\pi$
$390$	0	0
$391$	6.62443	0.335012
$392$	6.84106	0.345526
$393$	12.8099	0.646172
$394$	−8.10551	−0.408350
$395$	0	0
$396$	0	0
$397$	15.2342	0.764581	0.382291	−	0.924042i	$-0.375135\pi$
$397$	15.2342	0.764581	0.382291	+	0.924042i	$0.375135\pi$
$398$	−25.6585	−1.28614
$399$	−9.49954	−0.475572
$400$	0	0
$401$	2.74931	0.137294	0.0686471	−	0.997641i	$-0.478132\pi$
$401$	2.74931	0.137294	0.0686471	+	0.997641i	$0.478132\pi$
$402$	30.3103	1.51174
$403$	8.35620	0.416252
$404$	−18.6244	−0.926600
$405$	0	0
$406$	−48.3397	−2.39906
$407$	0	0
$408$	1.26445	0.0625996
$409$	3.98532	0.197061	0.0985307	−	0.995134i	$-0.468586\pi$
$409$	3.98532	0.197061	0.0985307	+	0.995134i	$0.468586\pi$
$410$	0	0
$411$	−22.8099	−1.12513
$412$	41.4986	2.04449
$413$	−28.1542	−1.38538
$414$	12.1854	0.598882
$415$	0	0
$416$	−12.0236	−0.589507
$417$	−7.59037	−0.371702
$418$	0	0
$419$	−5.13578	−0.250899	−0.125450	−	0.992100i	$-0.540037\pi$
$419$	−5.13578	−0.250899	−0.125450	+	0.992100i	$0.540037\pi$
$420$	0	0
$421$	−8.94657	−0.436029	−0.218014	−	0.975946i	$-0.569958\pi$
$421$	−8.94657	−0.436029	−0.218014	+	0.975946i	$0.569958\pi$
$422$	−21.8108	−1.06173
$423$	3.48486	0.169440
$424$	13.7484	0.667681
$425$	0	0
$426$	18.0937	0.876642
$427$	43.7337	2.11642
$428$	9.90826	0.478934
$429$	0	0
$430$	0	0
$431$	−22.7493	−1.09580	−0.547898	−	0.836545i	$-0.684572\pi$
$431$	−22.7493	−1.09580	−0.547898	+	0.836545i	$0.684572\pi$
$432$	−2.70436	−0.130113
$433$	−7.58325	−0.364428	−0.182214	−	0.983259i	$-0.558326\pi$
$433$	−7.58325	−0.364428	−0.182214	+	0.983259i	$0.558326\pi$
$434$	42.6576	2.04763
$435$	0	0
$436$	−16.9385	−0.811209
$437$	−14.9659	−0.715918
$438$	−19.5904	−0.936064
$439$	−17.3903	−0.829991	−0.414996	−	0.909823i	$-0.636217\pi$
$439$	−17.3903	−0.829991	−0.414996	+	0.909823i	$0.636217\pi$
$440$	0	0
$441$	6.24977	0.297608
$442$	3.71904	0.176897
$443$	−10.6438	−0.505702	−0.252851	−	0.967505i	$-0.581368\pi$
$443$	−10.6438	−0.505702	−0.252851	+	0.967505i	$0.581368\pi$
$444$	−1.14335	−0.0542608
$445$	0	0
$446$	−27.4617	−1.30035
$447$	1.81456	0.0858259
$448$	−41.6916	−1.96974
$449$	−26.9310	−1.27095	−0.635476	−	0.772121i	$-0.719196\pi$
$449$	−26.9310	−1.27095	−0.635476	+	0.772121i	$0.719196\pi$
$450$	0	0
$451$	0	0
$452$	15.0908	0.709813
$453$	−24.3250	−1.14289
$454$	48.4683	2.27473
$455$	0	0
$456$	−2.85665	−0.133775
$457$	−15.7796	−0.738138	−0.369069	−	0.929402i	$-0.620323\pi$
$457$	−15.7796	−0.738138	−0.369069	+	0.929402i	$0.620323\pi$
$458$	31.4049	1.46746
$459$	1.15516	0.0539183
$460$	0	0
$461$	−8.18922	−0.381410	−0.190705	−	0.981647i	$-0.561077\pi$
$461$	−8.18922	−0.381410	−0.190705	+	0.981647i	$0.561077\pi$
$462$	0	0
$463$	−16.0899	−0.747762	−0.373881	−	0.927477i	$-0.621973\pi$
$463$	−16.0899	−0.747762	−0.373881	+	0.927477i	$0.621973\pi$
$464$	16.9016	0.784638
$465$	0	0
$466$	10.5521	0.488815
$467$	29.4693	1.36367	0.681837	−	0.731504i	$-0.261181\pi$
$467$	29.4693	1.36367	0.681837	+	0.731504i	$0.261181\pi$
$468$	3.81078	0.176153
$469$	51.9229	2.39758
$470$	0	0
$471$	−9.76491	−0.449943
$472$	−8.46640	−0.389698
$473$	0	0
$474$	−10.8255	−0.497230
$475$	0	0
$476$	10.5757	0.484736
$477$	12.5601	0.575087
$478$	31.8713	1.45776
$479$	−32.2186	−1.47210	−0.736052	−	0.676925i	$-0.763312\pi$
$479$	−32.2186	−1.47210	−0.736052	+	0.676925i	$0.763312\pi$
$480$	0	0
$481$	−0.688760	−0.0314048
$482$	−10.7200	−0.488280
$483$	20.8742	0.949809
$484$	0	0
$485$	0	0
$486$	2.12489	0.0963868
$487$	35.8936	1.62649	0.813247	−	0.581919i	$-0.197698\pi$
$487$	35.8936	1.62649	0.813247	+	0.581919i	$0.197698\pi$
$488$	13.1514	0.595335
$489$	6.98440	0.315846
$490$	0	0
$491$	7.15894	0.323079	0.161539	−	0.986866i	$-0.448354\pi$
$491$	7.15894	0.323079	0.161539	+	0.986866i	$0.448354\pi$
$492$	−10.3747	−0.467726
$493$	−7.21949	−0.325150
$494$	−8.40207	−0.378027
$495$	0	0
$496$	−14.9149	−0.669699
$497$	30.9953	1.39033
$498$	−31.3406	−1.40441
$499$	27.0743	1.21201	0.606006	−	0.795460i	$-0.292771\pi$
$499$	27.0743	1.21201	0.606006	+	0.795460i	$0.292771\pi$
$500$	0	0
$501$	−6.31032	−0.281924
$502$	−6.43899	−0.287386
$503$	26.9991	1.20383	0.601915	−	0.798560i	$-0.294405\pi$
$503$	26.9991	1.20383	0.601915	+	0.798560i	$0.294405\pi$
$504$	3.98440	0.177479
$505$	0	0
$506$	0	0
$507$	−10.7044	−0.475397
$508$	20.2829	0.899909
$509$	15.5904	0.691031	0.345515	−	0.938413i	$-0.387704\pi$
$509$	15.5904	0.691031	0.345515	+	0.938413i	$0.387704\pi$
$510$	0	0
$511$	−33.5592	−1.48457
$512$	27.3813	1.21009
$513$	−2.60975	−0.115223
$514$	29.0066	1.27943
$515$	0	0
$516$	29.4381	1.29594
$517$	0	0
$518$	−3.51605	−0.154487
$519$	−12.8448	−0.563826
$520$	0	0
$521$	11.1589	0.488882	0.244441	−	0.969664i	$-0.421396\pi$
$521$	11.1589	0.488882	0.244441	+	0.969664i	$0.421396\pi$
$522$	−13.2800	−0.581252
$523$	−10.5786	−0.462568	−0.231284	−	0.972886i	$-0.574293\pi$
$523$	−10.5786	−0.462568	−0.231284	+	0.972886i	$0.574293\pi$
$524$	32.2186	1.40748
$525$	0	0
$526$	26.6888	1.16369
$527$	6.37088	0.277520
$528$	0	0
$529$	9.88601	0.429827
$530$	0	0
$531$	−7.73463	−0.335654
$532$	−23.8927	−1.03588
$533$	−6.24977	−0.270708
$534$	−22.3103	−0.965462
$535$	0	0
$536$	15.6140	0.674422
$537$	13.4849	0.581915
$538$	−52.4608	−2.26175
$539$	0	0
$540$	0	0
$541$	11.2947	0.485598	0.242799	−	0.970077i	$-0.421934\pi$
$541$	11.2947	0.485598	0.242799	+	0.970077i	$0.421934\pi$
$542$	−16.0743	−0.690451
$543$	−23.0899	−0.990883
$544$	−9.16698	−0.393031
$545$	0	0
$546$	11.7190	0.501528
$547$	6.09369	0.260547	0.130274	−	0.991478i	$-0.458414\pi$
$547$	6.09369	0.260547	0.130274	+	0.991478i	$0.458414\pi$
$548$	−57.3700	−2.45072
$549$	12.0147	0.512774
$550$	0	0
$551$	16.3103	0.694843
$552$	6.27718	0.267175
$553$	−18.5445	−0.788592
$554$	4.08991	0.173764
$555$	0	0
$556$	−19.0908	−0.809631
$557$	−5.90069	−0.250020	−0.125010	−	0.992155i	$-0.539896\pi$
$557$	−5.90069	−0.250020	−0.125010	+	0.992155i	$0.539896\pi$
$558$	11.7190	0.496106
$559$	17.7337	0.750056
$560$	0	0
$561$	0	0
$562$	3.98440	0.168072
$563$	3.03028	0.127711	0.0638555	−	0.997959i	$-0.479660\pi$
$563$	3.03028	0.127711	0.0638555	+	0.997959i	$0.479660\pi$
$564$	8.76491	0.369069
$565$	0	0
$566$	64.0431	2.69193
$567$	3.64002	0.152867
$568$	9.32075	0.391090
$569$	−13.4049	−0.561964	−0.280982	−	0.959713i	$-0.590660\pi$
$569$	−13.4049	−0.561964	−0.280982	+	0.959713i	$0.590660\pi$
$570$	0	0
$571$	−26.8851	−1.12511	−0.562553	−	0.826761i	$-0.690181\pi$
$571$	−26.8851	−1.12511	−0.562553	+	0.826761i	$0.690181\pi$
$572$	0	0
$573$	−7.98440	−0.333553
$574$	−31.9045	−1.33167
$575$	0	0
$576$	−11.4537	−0.477236
$577$	−2.03028	−0.0845215	−0.0422607	−	0.999107i	$-0.513456\pi$
$577$	−2.03028	−0.0845215	−0.0422607	+	0.999107i	$0.513456\pi$
$578$	−33.2876	−1.38458
$579$	11.7649	0.488933
$580$	0	0
$581$	−53.6878	−2.22735
$582$	14.4058	0.597142
$583$	0	0
$584$	−10.0917	−0.417599
$585$	0	0
$586$	−61.9514	−2.55919
$587$	−21.8245	−0.900795	−0.450398	−	0.892828i	$-0.648718\pi$
$587$	−21.8245	−0.900795	−0.450398	+	0.892828i	$0.648718\pi$
$588$	15.7190	0.648242
$589$	−14.3931	−0.593058
$590$	0	0
$591$	−3.81456	−0.156910
$592$	1.22936	0.0505265
$593$	8.06811	0.331318	0.165659	−	0.986183i	$-0.447025\pi$
$593$	8.06811	0.331318	0.165659	+	0.986183i	$0.447025\pi$
$594$	0	0
$595$	0	0
$596$	4.56387	0.186944
$597$	−12.0752	−0.494206
$598$	18.4626	0.754993
$599$	7.61353	0.311080	0.155540	−	0.987830i	$-0.450288\pi$
$599$	7.61353	0.311080	0.155540	+	0.987830i	$0.450288\pi$
$600$	0	0
$601$	−3.57569	−0.145855	−0.0729277	−	0.997337i	$-0.523234\pi$
$601$	−3.57569	−0.145855	−0.0729277	+	0.997337i	$0.523234\pi$
$602$	90.5289	3.68968
$603$	14.2645	0.580893
$604$	−61.1807	−2.48941
$605$	0	0
$606$	−15.7346	−0.639176
$607$	17.5298	0.711513	0.355757	−	0.934579i	$-0.384223\pi$
$607$	17.5298	0.711513	0.355757	+	0.934579i	$0.384223\pi$
$608$	20.7101	0.839905
$609$	−22.7493	−0.921849
$610$	0	0
$611$	5.28005	0.213608
$612$	2.90539	0.117443
$613$	12.5601	0.507297	0.253649	−	0.967296i	$-0.418369\pi$
$613$	12.5601	0.507297	0.253649	+	0.967296i	$0.418369\pi$
$614$	59.1883	2.38865
$615$	0	0
$616$	0	0
$617$	−15.9612	−0.642576	−0.321288	−	0.946982i	$-0.604116\pi$
$617$	−15.9612	−0.642576	−0.321288	+	0.946982i	$0.604116\pi$
$618$	35.0596	1.41031
$619$	−9.23417	−0.371153	−0.185576	−	0.982630i	$-0.559415\pi$
$619$	−9.23417	−0.371153	−0.185576	+	0.982630i	$0.559415\pi$
$620$	0	0
$621$	5.73463	0.230123
$622$	50.8023	2.03699
$623$	−38.2186	−1.53119
$624$	−4.09747	−0.164030
$625$	0	0
$626$	−60.1542	−2.40425
$627$	0	0
$628$	−24.5601	−0.980054
$629$	−0.525120	−0.0209379
$630$	0	0
$631$	29.2342	1.16379	0.581897	−	0.813262i	$-0.302311\pi$
$631$	29.2342	1.16379	0.581897	+	0.813262i	$0.302311\pi$
$632$	−5.57661	−0.221826
$633$	−10.2645	−0.407975
$634$	18.7200	0.743464
$635$	0	0
$636$	31.5904	1.25264
$637$	9.46927	0.375186
$638$	0	0
$639$	8.51514	0.336854
$640$	0	0
$641$	13.9612	0.551436	0.275718	−	0.961239i	$-0.411084\pi$
$641$	13.9612	0.551436	0.275718	+	0.961239i	$0.411084\pi$
$642$	8.37088	0.330372
$643$	−12.6206	−0.497710	−0.248855	−	0.968541i	$-0.580054\pi$
$643$	−12.6206	−0.497710	−0.248855	+	0.968541i	$0.580054\pi$
$644$	52.5015	2.06885
$645$	0	0
$646$	−6.40585	−0.252035
$647$	−29.6429	−1.16538	−0.582691	−	0.812694i	$-0.698000\pi$
$647$	−29.6429	−1.16538	−0.582691	+	0.812694i	$0.698000\pi$
$648$	1.09461	0.0430003
$649$	0	0
$650$	0	0
$651$	20.0752	0.786810
$652$	17.5667	0.687967
$653$	−9.90069	−0.387444	−0.193722	−	0.981056i	$-0.562056\pi$
$653$	−9.90069	−0.387444	−0.193722	+	0.981056i	$0.562056\pi$
$654$	−14.3103	−0.559578
$655$	0	0
$656$	11.1552	0.435536
$657$	−9.21949	−0.359687
$658$	26.9541	1.05078
$659$	5.28005	0.205681	0.102841	−	0.994698i	$-0.467207\pi$
$659$	5.28005	0.205681	0.102841	+	0.994698i	$0.467207\pi$
$660$	0	0
$661$	−26.8548	−1.04453	−0.522266	−	0.852783i	$-0.674913\pi$
$661$	−26.8548	−1.04453	−0.522266	+	0.852783i	$0.674913\pi$
$662$	68.5271	2.66338
$663$	1.75023	0.0679733
$664$	−16.1447	−0.626537
$665$	0	0
$666$	−0.965943	−0.0374295
$667$	−35.8401	−1.38774
$668$	−15.8713	−0.614080
$669$	−12.9239	−0.499665
$670$	0	0
$671$	0	0
$672$	−28.8860	−1.11430
$673$	−3.81834	−0.147186	−0.0735932	−	0.997288i	$-0.523447\pi$
$673$	−3.81834	−0.147186	−0.0735932	+	0.997288i	$0.523447\pi$
$674$	61.5885	2.37230
$675$	0	0
$676$	−26.9229	−1.03550
$677$	−15.6897	−0.603003	−0.301502	−	0.953466i	$-0.597488\pi$
$677$	−15.6897	−0.603003	−0.301502	+	0.953466i	$0.597488\pi$
$678$	12.7493	0.489634
$679$	24.6779	0.947049
$680$	0	0
$681$	22.8099	0.874076
$682$	0	0
$683$	15.6353	0.598269	0.299135	−	0.954211i	$-0.403302\pi$
$683$	15.6353	0.598269	0.299135	+	0.954211i	$0.403302\pi$
$684$	−6.56387	−0.250976
$685$	0	0
$686$	−5.80275	−0.221550
$687$	14.7796	0.563876
$688$	−31.6528	−1.20675
$689$	19.0303	0.724996
$690$	0	0
$691$	−31.4305	−1.19567	−0.597836	−	0.801618i	$-0.703973\pi$
$691$	−31.4305	−1.19567	−0.597836	+	0.801618i	$0.703973\pi$
$692$	−32.3065	−1.22811
$693$	0	0
$694$	−75.8989	−2.88108
$695$	0	0
$696$	−6.84106	−0.259310
$697$	−4.76491	−0.180484
$698$	49.4674	1.87237
$699$	4.96594	0.187829
$700$	0	0
$701$	−3.24507	−0.122565	−0.0612824	−	0.998120i	$-0.519519\pi$
$701$	−3.24507	−0.122565	−0.0612824	+	0.998120i	$0.519519\pi$
$702$	3.21949	0.121512
$703$	1.18635	0.0447442
$704$	0	0
$705$	0	0
$706$	20.7181	0.779737
$707$	−26.9541	−1.01371
$708$	−19.4537	−0.731114
$709$	−26.7190	−1.00345	−0.501727	−	0.865026i	$-0.667302\pi$
$709$	−26.7190	−1.00345	−0.501727	+	0.865026i	$0.667302\pi$
$710$	0	0
$711$	−5.09461	−0.191063
$712$	−11.4929	−0.430714
$713$	31.6273	1.18445
$714$	8.93475	0.334375
$715$	0	0
$716$	33.9163	1.26751
$717$	14.9991	0.560151
$718$	−72.0351	−2.68833
$719$	−6.78807	−0.253152	−0.126576	−	0.991957i	$-0.540399\pi$
$719$	−6.78807	−0.253152	−0.126576	+	0.991957i	$0.540399\pi$
$720$	0	0
$721$	60.0587	2.23670
$722$	−25.9007	−0.963924
$723$	−5.04496	−0.187624
$724$	−58.0743	−2.15831
$725$	0	0
$726$	0	0
$727$	19.9154	0.738620	0.369310	−	0.929306i	$-0.379594\pi$
$727$	19.9154	0.738620	0.369310	+	0.929306i	$0.379594\pi$
$728$	6.03692	0.223743
$729$	1.00000	0.0370370
$730$	0	0
$731$	13.5204	0.500071
$732$	30.2186	1.11691
$733$	−31.3388	−1.15752	−0.578762	−	0.815497i	$-0.696464\pi$
$733$	−31.3388	−1.15752	−0.578762	+	0.815497i	$0.696464\pi$
$734$	−4.00756	−0.147922
$735$	0	0
$736$	−45.5081	−1.67745
$737$	0	0
$738$	−8.76491	−0.322641
$739$	2.25355	0.0828982	0.0414491	−	0.999141i	$-0.486803\pi$
$739$	2.25355	0.0828982	0.0414491	+	0.999141i	$0.486803\pi$
$740$	0	0
$741$	−3.95413	−0.145259
$742$	97.1477	3.56640
$743$	−6.74931	−0.247608	−0.123804	−	0.992307i	$-0.539509\pi$
$743$	−6.74931	−0.247608	−0.123804	+	0.992307i	$0.539509\pi$
$744$	6.03692	0.221324
$745$	0	0
$746$	−34.6888	−1.27005
$747$	−14.7493	−0.539649
$748$	0	0
$749$	14.3397	0.523961
$750$	0	0
$751$	22.4390	0.818810	0.409405	−	0.912353i	$-0.365736\pi$
$751$	22.4390	0.818810	0.409405	+	0.912353i	$0.365736\pi$
$752$	−9.42431	−0.343669
$753$	−3.03028	−0.110429
$754$	−20.1211	−0.732767
$755$	0	0
$756$	9.15516	0.332970
$757$	−25.4158	−0.923754	−0.461877	−	0.886944i	$-0.652824\pi$
$757$	−25.4158	−0.923754	−0.461877	+	0.886944i	$0.652824\pi$
$758$	−55.3406	−2.01006
$759$	0	0
$760$	0	0
$761$	30.7493	1.11466	0.557331	−	0.830291i	$-0.311826\pi$
$761$	30.7493	1.11466	0.557331	+	0.830291i	$0.311826\pi$
$762$	17.1358	0.620764
$763$	−24.5142	−0.887474
$764$	−20.0819	−0.726537
$765$	0	0
$766$	−26.4977	−0.957401
$767$	−11.7190	−0.423150
$768$	4.91721	0.177434
$769$	16.2956	0.587636	0.293818	−	0.955861i	$-0.405074\pi$
$769$	16.2956	0.587636	0.293818	+	0.955861i	$0.405074\pi$
$770$	0	0
$771$	13.6509	0.491626
$772$	29.5904	1.06498
$773$	−48.7787	−1.75445	−0.877223	−	0.480082i	$-0.840607\pi$
$773$	−48.7787	−1.75445	−0.877223	+	0.480082i	$0.840607\pi$
$774$	24.8704	0.893949
$775$	0	0
$776$	7.42100	0.266398
$777$	−1.65470	−0.0593621
$778$	−38.4390	−1.37810
$779$	10.7649	0.385693
$780$	0	0
$781$	0	0
$782$	14.0761	0.503362
$783$	−6.24977	−0.223349
$784$	−16.9016	−0.603629
$785$	0	0
$786$	27.2195	0.970887
$787$	−46.2001	−1.64686	−0.823428	−	0.567421i	$-0.807941\pi$
$787$	−46.2001	−1.64686	−0.823428	+	0.567421i	$0.807941\pi$
$788$	−9.59415	−0.341777
$789$	12.5601	0.447151
$790$	0	0
$791$	21.8401	0.776546
$792$	0	0
$793$	18.2039	0.646439
$794$	32.3709	1.14880
$795$	0	0
$796$	−30.3709	−1.07647
$797$	−36.3784	−1.28859	−0.644295	−	0.764777i	$-0.722849\pi$
$797$	−36.3784	−1.28859	−0.644295	+	0.764777i	$0.722849\pi$
$798$	−20.1854	−0.714557
$799$	4.02558	0.142415
$800$	0	0
$801$	−10.4995	−0.370983
$802$	5.84197	0.206287
$803$	0	0
$804$	35.8771	1.26529
$805$	0	0
$806$	17.7560	0.625427
$807$	−24.6888	−0.869086
$808$	−8.10551	−0.285151
$809$	−11.3737	−0.399879	−0.199940	−	0.979808i	$-0.564075\pi$
$809$	−11.3737	−0.399879	−0.199940	+	0.979808i	$0.564075\pi$
$810$	0	0
$811$	13.3903	0.470195	0.235098	−	0.971972i	$-0.424459\pi$
$811$	13.3903	0.470195	0.235098	+	0.971972i	$0.424459\pi$
$812$	−57.2177	−2.00795
$813$	−7.56479	−0.265309
$814$	0	0
$815$	0	0
$816$	−3.12397	−0.109361
$817$	−30.5454	−1.06865
$818$	8.46835	0.296089
$819$	5.51514	0.192715
$820$	0	0
$821$	−32.0975	−1.12021	−0.560105	−	0.828422i	$-0.689239\pi$
$821$	−32.0975	−1.12021	−0.560105	+	0.828422i	$0.689239\pi$
$822$	−48.4683	−1.69053
$823$	−16.7952	−0.585443	−0.292722	−	0.956198i	$-0.594561\pi$
$823$	−16.7952	−0.585443	−0.292722	+	0.956198i	$0.594561\pi$
$824$	18.0606	0.629169
$825$	0	0
$826$	−59.8245	−2.08156
$827$	−45.5904	−1.58533	−0.792666	−	0.609656i	$-0.791308\pi$
$827$	−45.5904	−1.58533	−0.792666	+	0.609656i	$0.791308\pi$
$828$	14.4234	0.501248
$829$	12.9385	0.449374	0.224687	−	0.974431i	$-0.427864\pi$
$829$	12.9385	0.449374	0.224687	+	0.974431i	$0.427864\pi$
$830$	0	0
$831$	1.92477	0.0667695
$832$	−17.3539	−0.601638
$833$	7.21949	0.250141
$834$	−16.1287	−0.558490
$835$	0	0
$836$	0	0
$837$	5.51514	0.190631
$838$	−10.9130	−0.376982
$839$	−1.59037	−0.0549057	−0.0274528	−	0.999623i	$-0.508740\pi$
$839$	−1.59037	−0.0549057	−0.0274528	+	0.999623i	$0.508740\pi$
$840$	0	0
$841$	10.0596	0.346884
$842$	−19.0104	−0.655143
$843$	1.87511	0.0645824
$844$	−25.8165	−0.888641
$845$	0	0
$846$	7.40493	0.254587
$847$	0	0
$848$	−33.9670	−1.16643
$849$	30.1396	1.03439
$850$	0	0
$851$	−2.60688	−0.0893628
$852$	21.4167	0.733726
$853$	10.5161	0.360063	0.180031	−	0.983661i	$-0.442380\pi$
$853$	10.5161	0.360063	0.180031	+	0.983661i	$0.442380\pi$
$854$	92.9291	3.17997
$855$	0	0
$856$	4.31216	0.147386
$857$	−57.4637	−1.96292	−0.981460	−	0.191665i	$-0.938611\pi$
$857$	−57.4637	−1.96292	−0.981460	+	0.191665i	$0.938611\pi$
$858$	0	0
$859$	32.7181	1.11633	0.558164	−	0.829731i	$-0.311506\pi$
$859$	32.7181	1.11633	0.558164	+	0.829731i	$0.311506\pi$
$860$	0	0
$861$	−15.0147	−0.511699
$862$	−48.3397	−1.64646
$863$	43.1807	1.46989	0.734945	−	0.678127i	$-0.237208\pi$
$863$	43.1807	1.46989	0.734945	+	0.678127i	$0.237208\pi$
$864$	−7.93567	−0.269977
$865$	0	0
$866$	−16.1135	−0.547560
$867$	−15.6656	−0.532032
$868$	50.4920	1.71381
$869$	0	0
$870$	0	0
$871$	21.6126	0.732315
$872$	−7.37179	−0.249640
$873$	6.77959	0.229454
$874$	−31.8009	−1.07568
$875$	0	0
$876$	−23.1883	−0.783460
$877$	−16.0752	−0.542822	−0.271411	−	0.962464i	$-0.587490\pi$
$877$	−16.0752	−0.542822	−0.271411	+	0.962464i	$0.587490\pi$
$878$	−36.9523	−1.24708
$879$	−29.1552	−0.983379
$880$	0	0
$881$	31.2876	1.05411	0.527053	−	0.849832i	$-0.323297\pi$
$881$	31.2876	1.05411	0.527053	+	0.849832i	$0.323297\pi$
$882$	13.2800	0.447162
$883$	−24.7640	−0.833375	−0.416687	−	0.909050i	$-0.636809\pi$
$883$	−24.7640	−0.833375	−0.416687	+	0.909050i	$0.636809\pi$
$884$	4.40207	0.148058
$885$	0	0
$886$	−22.6169	−0.759828
$887$	−26.3085	−0.883353	−0.441676	−	0.897174i	$-0.645616\pi$
$887$	−26.3085	−0.883353	−0.441676	+	0.897174i	$0.645616\pi$
$888$	−0.497594	−0.0166982
$889$	29.3544	0.984514
$890$	0	0
$891$	0	0
$892$	−32.5053	−1.08836
$893$	−9.09461	−0.304339
$894$	3.85574	0.128955
$895$	0	0
$896$	−30.8179	−1.02955
$897$	8.68876	0.290109
$898$	−57.2252	−1.90963
$899$	−34.4683	−1.14958
$900$	0	0
$901$	14.5089	0.483363
$902$	0	0
$903$	42.6041	1.41778
$904$	6.56766	0.218437
$905$	0	0
$906$	−51.6878	−1.71721
$907$	55.9301	1.85713	0.928563	−	0.371174i	$-0.121045\pi$
$907$	55.9301	1.85713	0.928563	+	0.371174i	$0.121045\pi$
$908$	57.3700	1.90389
$909$	−7.40493	−0.245606
$910$	0	0
$911$	−15.3931	−0.509997	−0.254998	−	0.966941i	$-0.582075\pi$
$911$	−15.3931	−0.509997	−0.254998	+	0.966941i	$0.582075\pi$
$912$	7.05769	0.233703
$913$	0	0
$914$	−33.5298	−1.10907
$915$	0	0
$916$	37.1727	1.22822
$917$	46.6282	1.53980
$918$	2.45459	0.0810134
$919$	51.2598	1.69090	0.845452	−	0.534052i	$-0.179331\pi$
$919$	51.2598	1.69090	0.845452	+	0.534052i	$0.179331\pi$
$920$	0	0
$921$	27.8548	0.917848
$922$	−17.4012	−0.573076
$923$	12.9016	0.424662
$924$	0	0
$925$	0	0
$926$	−34.1892	−1.12353
$927$	16.4995	0.541916
$928$	49.5961	1.62807
$929$	14.8099	0.485896	0.242948	−	0.970039i	$-0.421886\pi$
$929$	14.8099	0.485896	0.242948	+	0.970039i	$0.421886\pi$
$930$	0	0
$931$	−16.3103	−0.534549
$932$	12.4900	0.409125
$933$	23.9083	0.782721
$934$	62.6188	2.04895
$935$	0	0
$936$	1.65848	0.0542093
$937$	27.2654	0.890721	0.445360	−	0.895351i	$-0.353076\pi$
$937$	27.2654	0.890721	0.445360	+	0.895351i	$0.353076\pi$
$938$	110.330	3.60241
$939$	−28.3094	−0.923843
$940$	0	0
$941$	49.5630	1.61571	0.807853	−	0.589384i	$-0.200629\pi$
$941$	49.5630	1.61571	0.807853	+	0.589384i	$0.200629\pi$
$942$	−20.7493	−0.676049
$943$	−23.6547	−0.770303
$944$	20.9172	0.680797
$945$	0	0
$946$	0	0
$947$	13.9844	0.454432	0.227216	−	0.973844i	$-0.427038\pi$
$947$	13.9844	0.454432	0.227216	+	0.973844i	$0.427038\pi$
$948$	−12.8136	−0.416168
$949$	−13.9688	−0.453447
$950$	0	0
$951$	8.80986	0.285679
$952$	4.60263	0.149172
$953$	17.1240	0.554700	0.277350	−	0.960769i	$-0.410544\pi$
$953$	17.1240	0.554700	0.277350	+	0.960769i	$0.410544\pi$
$954$	26.6888	0.864081
$955$	0	0
$956$	37.7248	1.22011
$957$	0	0
$958$	−68.4608	−2.21187
$959$	−83.0284	−2.68113
$960$	0	0
$961$	−0.583252	−0.0188146
$962$	−1.46354	−0.0471863
$963$	3.93945	0.126947
$964$	−12.6888	−0.408677
$965$	0	0
$966$	44.3553	1.42711
$967$	1.90826	0.0613653	0.0306827	−	0.999529i	$-0.490232\pi$
$967$	1.90826	0.0613653	0.0306827	+	0.999529i	$0.490232\pi$
$968$	0	0
$969$	−3.01468	−0.0968455
$970$	0	0
$971$	31.3856	1.00721	0.503605	−	0.863934i	$-0.332007\pi$
$971$	31.3856	1.00721	0.503605	+	0.863934i	$0.332007\pi$
$972$	2.51514	0.0806731
$973$	−27.6291	−0.885749
$974$	76.2697	2.44384
$975$	0	0
$976$	−32.4920	−1.04004
$977$	−41.4693	−1.32672	−0.663360	−	0.748301i	$-0.730870\pi$
$977$	−41.4693	−1.32672	−0.663360	+	0.748301i	$0.730870\pi$
$978$	14.8411	0.474565
$979$	0	0
$980$	0	0
$981$	−6.73463	−0.215020
$982$	15.2119	0.485432
$983$	6.23601	0.198898	0.0994489	−	0.995043i	$-0.468292\pi$
$983$	6.23601	0.198898	0.0994489	+	0.995043i	$0.468292\pi$
$984$	−4.51514	−0.143937
$985$	0	0
$986$	−15.3406	−0.488544
$987$	12.6850	0.403767
$988$	−9.94518	−0.316398
$989$	67.1202	2.13430
$990$	0	0
$991$	27.2048	0.864189	0.432095	−	0.901828i	$-0.357775\pi$
$991$	27.2048	0.864189	0.432095	+	0.901828i	$0.357775\pi$
$992$	−43.7663	−1.38958
$993$	32.2498	1.02342
$994$	65.8615	2.08900
$995$	0	0
$996$	−37.0966	−1.17545
$997$	28.0606	0.888687	0.444343	−	0.895857i	$-0.353437\pi$
$997$	28.0606	0.888687	0.444343	+	0.895857i	$0.353437\pi$
$998$	57.5298	1.82107
$999$	−0.454586	−0.0143825

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9075.2.a.cd.1.3		3
5.4	even	2		9075.2.a.cj.1.1		3
11.10	odd	2		825.2.a.m.1.1	yes	3
33.32	even	2		2475.2.a.z.1.3		3
55.32	even	4		825.2.c.f.199.2		6
55.43	even	4		825.2.c.f.199.5		6
55.54	odd	2		825.2.a.i.1.3	✓	3
165.32	odd	4		2475.2.c.q.199.5		6
165.98	odd	4		2475.2.c.q.199.2		6
165.164	even	2		2475.2.a.bd.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
825.2.a.i.1.3	✓	3	55.54	odd	2
825.2.a.m.1.1	yes	3	11.10	odd	2
825.2.c.f.199.2		6	55.32	even	4
825.2.c.f.199.5		6	55.43	even	4
2475.2.a.z.1.3		3	33.32	even	2
2475.2.a.bd.1.1		3	165.164	even	2
2475.2.c.q.199.2		6	165.98	odd	4
2475.2.c.q.199.5		6	165.32	odd	4
9075.2.a.cd.1.3		3	1.1	even	1	trivial
9075.2.a.cj.1.1		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)

\(f(q)\)	\(=\)	\(q+2.12489 q^{2} +1.00000 q^{3} +2.51514 q^{4} +2.12489 q^{6} +3.64002 q^{7} +1.09461 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+2.12489 q^{2} +1.00000 q^{3} +2.51514 q^{4} +2.12489 q^{6} +3.64002 q^{7} +1.09461 q^{8} +1.00000 q^{9} +2.51514 q^{12} +1.51514 q^{13} +7.73463 q^{14} -2.70436 q^{16} +1.15516 q^{17} +2.12489 q^{18} -2.60975 q^{19} +3.64002 q^{21} +5.73463 q^{23} +1.09461 q^{24} +3.21949 q^{26} +1.00000 q^{27} +9.15516 q^{28} -6.24977 q^{29} +5.51514 q^{31} -7.93567 q^{32} +2.45459 q^{34} +2.51514 q^{36} -0.454586 q^{37} -5.54541 q^{38} +1.51514 q^{39} -4.12489 q^{41} +7.73463 q^{42} +11.7044 q^{43} +12.1854 q^{46} +3.48486 q^{47} -2.70436 q^{48} +6.24977 q^{49} +1.15516 q^{51} +3.81078 q^{52} +12.5601 q^{53} +2.12489 q^{54} +3.98440 q^{56} -2.60975 q^{57} -13.2800 q^{58} -7.73463 q^{59} +12.0147 q^{61} +11.7190 q^{62} +3.64002 q^{63} -11.4537 q^{64} +14.2645 q^{67} +2.90539 q^{68} +5.73463 q^{69} +8.51514 q^{71} +1.09461 q^{72} -9.21949 q^{73} -0.965943 q^{74} -6.56387 q^{76} +3.21949 q^{78} -5.09461 q^{79} +1.00000 q^{81} -8.76491 q^{82} -14.7493 q^{83} +9.15516 q^{84} +24.8704 q^{86} -6.24977 q^{87} -10.4995 q^{89} +5.51514 q^{91} +14.4234 q^{92} +5.51514 q^{93} +7.40493 q^{94} -7.93567 q^{96} +6.77959 q^{97} +13.2800 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 2 q^{2} + 3 q^{3} + 8 q^{4} - 2 q^{6} + 3 q^{7} - 6 q^{8} + 3 q^{9}+O(q^{10}) \) 3 * q - 2 * q^2 + 3 * q^3 + 8 * q^4 - 2 * q^6 + 3 * q^7 - 6 * q^8 + 3 * q^9	\( 3 q - 2 q^{2} + 3 q^{3} + 8 q^{4} - 2 q^{6} + 3 q^{7} - 6 q^{8} + 3 q^{9} + 8 q^{12} + 5 q^{13} + 6 q^{14} + 10 q^{16} - 4 q^{17} - 2 q^{18} + q^{19} + 3 q^{21} - 6 q^{24} - 8 q^{26} + 3 q^{27} + 20 q^{28} - 2 q^{29} + 17 q^{31} - 34 q^{32} + 6 q^{34} + 8 q^{36} - 18 q^{38} + 5 q^{39} - 4 q^{41} + 6 q^{42} + 17 q^{43} + 30 q^{46} + 10 q^{47} + 10 q^{48} + 2 q^{49} - 4 q^{51} + 30 q^{52} + 6 q^{53} - 2 q^{54} - 22 q^{56} + q^{57} - 24 q^{58} - 6 q^{59} + 3 q^{61} - 16 q^{62} + 3 q^{63} + 34 q^{64} - 7 q^{67} + 18 q^{68} + 26 q^{71} - 6 q^{72} - 10 q^{73} - 14 q^{74} + 24 q^{76} - 8 q^{78} - 6 q^{79} + 3 q^{81} - 10 q^{82} + 6 q^{83} + 20 q^{84} + 28 q^{86} - 2 q^{87} + 2 q^{89} + 17 q^{91} - 26 q^{92} + 17 q^{93} - 2 q^{94} - 34 q^{96} - 29 q^{97} + 24 q^{98}+O(q^{100}) \) 3 * q - 2 * q^2 + 3 * q^3 + 8 * q^4 - 2 * q^6 + 3 * q^7 - 6 * q^8 + 3 * q^9 + 8 * q^12 + 5 * q^13 + 6 * q^14 + 10 * q^16 - 4 * q^17 - 2 * q^18 + q^19 + 3 * q^21 - 6 * q^24 - 8 * q^26 + 3 * q^27 + 20 * q^28 - 2 * q^29 + 17 * q^31 - 34 * q^32 + 6 * q^34 + 8 * q^36 - 18 * q^38 + 5 * q^39 - 4 * q^41 + 6 * q^42 + 17 * q^43 + 30 * q^46 + 10 * q^47 + 10 * q^48 + 2 * q^49 - 4 * q^51 + 30 * q^52 + 6 * q^53 - 2 * q^54 - 22 * q^56 + q^57 - 24 * q^58 - 6 * q^59 + 3 * q^61 - 16 * q^62 + 3 * q^63 + 34 * q^64 - 7 * q^67 + 18 * q^68 + 26 * q^71 - 6 * q^72 - 10 * q^73 - 14 * q^74 + 24 * q^76 - 8 * q^78 - 6 * q^79 + 3 * q^81 - 10 * q^82 + 6 * q^83 + 20 * q^84 + 28 * q^86 - 2 * q^87 + 2 * q^89 + 17 * q^91 - 26 * q^92 + 17 * q^93 - 2 * q^94 - 34 * q^96 - 29 * q^97 + 24 * q^98

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