LMFDB - Embedded newform 9075.2.a.ea.1.10

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:	$12$
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			1.10
Root			$2.10651$ 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)

$f(q)$	$=$	\(q+2.10651 q^{2} +1.00000 q^{3} +2.43738 q^{4} +2.10651 q^{6} -3.31633 q^{7} +0.921348 q^{8} +1.00000 q^{9} +2.43738 q^{12} +2.49889 q^{13} -6.98587 q^{14} -2.93393 q^{16} +6.81019 q^{17} +2.10651 q^{18} +1.14769 q^{19} -3.31633 q^{21} +5.89026 q^{23} +0.921348 q^{24} +5.26393 q^{26} +1.00000 q^{27} -8.08315 q^{28} +1.12257 q^{29} -1.09673 q^{31} -8.02306 q^{32} +14.3457 q^{34} +2.43738 q^{36} -11.5389 q^{37} +2.41763 q^{38} +2.49889 q^{39} +0.932568 q^{41} -6.98587 q^{42} -0.552188 q^{43} +12.4079 q^{46} -2.13427 q^{47} -2.93393 q^{48} +3.99802 q^{49} +6.81019 q^{51} +6.09074 q^{52} +11.6356 q^{53} +2.10651 q^{54} -3.05549 q^{56} +1.14769 q^{57} +2.36469 q^{58} +8.39837 q^{59} +8.21313 q^{61} -2.31027 q^{62} -3.31633 q^{63} -11.0328 q^{64} -4.15419 q^{67} +16.5990 q^{68} +5.89026 q^{69} +12.5340 q^{71} +0.921348 q^{72} +5.73761 q^{73} -24.3067 q^{74} +2.79736 q^{76} +5.26393 q^{78} +3.86754 q^{79} +1.00000 q^{81} +1.96446 q^{82} +6.08055 q^{83} -8.08315 q^{84} -1.16319 q^{86} +1.12257 q^{87} +12.5950 q^{89} -8.28712 q^{91} +14.3568 q^{92} -1.09673 q^{93} -4.49586 q^{94} -8.02306 q^{96} -2.86873 q^{97} +8.42187 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 4 q^{2} + 12 q^{3} + 12 q^{4} + 4 q^{6} + 8 q^{7} + 12 q^{8} + 12 q^{9} + 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}+ \cdots + 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

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 $

$2$	2.10651	1.48953	0.744764	−	0.667328i	$-0.232562\pi$
$2$	2.10651	1.48953	0.744764	+	0.667328i	$0.232562\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	2.43738	1.21869
$5$	0	0
$5$	0	0
$6$	2.10651	0.859979
$7$	−3.31633	−1.25345	−0.626727	−	0.779239i	$-0.715606\pi$
$7$	−3.31633	−1.25345	−0.626727	+	0.779239i	$0.715606\pi$
$8$	0.921348	0.325746
$9$	1.00000	0.333333
$10$	0	0
$11$	0	0
$11$	0	0
$12$	2.43738	0.703611
$13$	2.49889	0.693066	0.346533	−	0.938038i	$-0.387359\pi$
$13$	2.49889	0.693066	0.346533	+	0.938038i	$0.387359\pi$
$14$	−6.98587	−1.86705
$15$	0	0
$16$	−2.93393	−0.733484
$17$	6.81019	1.65171	0.825857	−	0.563880i	$-0.190692\pi$
$17$	6.81019	1.65171	0.825857	+	0.563880i	$0.190692\pi$
$18$	2.10651	0.496509
$19$	1.14769	0.263299	0.131649	−	0.991296i	$-0.457973\pi$
$19$	1.14769	0.263299	0.131649	+	0.991296i	$0.457973\pi$
$20$	0	0
$21$	−3.31633	−0.723682
$22$	0	0
$23$	5.89026	1.22820	0.614102	−	0.789226i	$-0.289518\pi$
$23$	5.89026	1.22820	0.614102	+	0.789226i	$0.289518\pi$
$24$	0.921348	0.188069
$25$	0	0
$26$	5.26393	1.03234
$27$	1.00000	0.192450
$28$	−8.08315	−1.52757
$29$	1.12257	0.208455	0.104228	−	0.994553i	$-0.466763\pi$
$29$	1.12257	0.208455	0.104228	+	0.994553i	$0.466763\pi$
$30$	0	0
$31$	−1.09673	−0.196978	−0.0984892	−	0.995138i	$-0.531401\pi$
$31$	−1.09673	−0.196978	−0.0984892	+	0.995138i	$0.531401\pi$
$32$	−8.02306	−1.41829
$33$	0	0
$34$	14.3457	2.46027
$35$	0	0
$36$	2.43738	0.406230
$37$	−11.5389	−1.89698	−0.948489	−	0.316810i	$-0.897388\pi$
$37$	−11.5389	−1.89698	−0.948489	+	0.316810i	$0.897388\pi$
$38$	2.41763	0.392191
$39$	2.49889	0.400142
$40$	0	0
$41$	0.932568	0.145643	0.0728213	−	0.997345i	$-0.476800\pi$
$41$	0.932568	0.145643	0.0728213	+	0.997345i	$0.476800\pi$
$42$	−6.98587	−1.07794
$43$	−0.552188	−0.0842079	−0.0421040	−	0.999113i	$-0.513406\pi$
$43$	−0.552188	−0.0842079	−0.0421040	+	0.999113i	$0.513406\pi$
$44$	0	0
$45$	0	0
$46$	12.4079	1.82944
$47$	−2.13427	−0.311316	−0.155658	−	0.987811i	$-0.549750\pi$
$47$	−2.13427	−0.311316	−0.155658	+	0.987811i	$0.549750\pi$
$48$	−2.93393	−0.423477
$49$	3.99802	0.571146
$50$	0	0
$51$	6.81019	0.953617
$52$	6.09074	0.844633
$53$	11.6356	1.59827	0.799136	−	0.601150i	$-0.205291\pi$
$53$	11.6356	1.59827	0.799136	+	0.601150i	$0.205291\pi$
$54$	2.10651	0.286660
$55$	0	0
$56$	−3.05549	−0.408307
$57$	1.14769	0.152016
$58$	2.36469	0.310500
$59$	8.39837	1.09337	0.546687	−	0.837337i	$-0.315889\pi$
$59$	8.39837	1.09337	0.546687	+	0.837337i	$0.315889\pi$
$60$	0	0
$61$	8.21313	1.05158	0.525792	−	0.850613i	$-0.323769\pi$
$61$	8.21313	1.05158	0.525792	+	0.850613i	$0.323769\pi$
$62$	−2.31027	−0.293405
$63$	−3.31633	−0.417818
$64$	−11.0328	−1.37910
$65$	0	0
$66$	0	0
$67$	−4.15419	−0.507515	−0.253757	−	0.967268i	$-0.581666\pi$
$67$	−4.15419	−0.507515	−0.253757	+	0.967268i	$0.581666\pi$
$68$	16.5990	2.01293
$69$	5.89026	0.709104
$70$	0	0
$71$	12.5340	1.48752	0.743759	−	0.668448i	$-0.233041\pi$
$71$	12.5340	1.48752	0.743759	+	0.668448i	$0.233041\pi$
$72$	0.921348	0.108582
$73$	5.73761	0.671537	0.335768	−	0.941945i	$-0.391004\pi$
$73$	5.73761	0.671537	0.335768	+	0.941945i	$0.391004\pi$
$74$	−24.3067	−2.82560
$75$	0	0
$76$	2.79736	0.320880
$77$	0	0
$78$	5.26393	0.596022
$79$	3.86754	0.435132	0.217566	−	0.976046i	$-0.430188\pi$
$79$	3.86754	0.435132	0.217566	+	0.976046i	$0.430188\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	1.96446	0.216939
$83$	6.08055	0.667427	0.333714	−	0.942675i	$-0.391698\pi$
$83$	6.08055	0.667427	0.333714	+	0.942675i	$0.391698\pi$
$84$	−8.08315	−0.881944
$85$	0	0
$86$	−1.16319	−0.125430
$87$	1.12257	0.120352
$88$	0	0
$89$	12.5950	1.33506	0.667531	−	0.744582i	$-0.267351\pi$
$89$	12.5950	1.33506	0.667531	+	0.744582i	$0.267351\pi$
$90$	0	0
$91$	−8.28712	−0.868726
$92$	14.3568	1.49680
$93$	−1.09673	−0.113726
$94$	−4.49586	−0.463713
$95$	0	0
$96$	−8.02306	−0.818850
$97$	−2.86873	−0.291275	−0.145638	−	0.989338i	$-0.546523\pi$
$97$	−2.86873	−0.291275	−0.145638	+	0.989338i	$0.546523\pi$
$98$	8.42187	0.850737
$99$	0	0
$100$	0	0
$101$	−9.87001	−0.982103	−0.491052	−	0.871131i	$-0.663387\pi$
$101$	−9.87001	−0.982103	−0.491052	+	0.871131i	$0.663387\pi$
$102$	14.3457	1.42044
$103$	7.71953	0.760628	0.380314	−	0.924857i	$-0.375816\pi$
$103$	7.71953	0.760628	0.380314	+	0.924857i	$0.375816\pi$
$104$	2.30234	0.225763
$105$	0	0
$106$	24.5105	2.38067
$107$	15.5114	1.49954	0.749771	−	0.661697i	$-0.230163\pi$
$107$	15.5114	1.49954	0.749771	+	0.661697i	$0.230163\pi$
$108$	2.43738	0.234537
$109$	−6.79834	−0.651163	−0.325581	−	0.945514i	$-0.605560\pi$
$109$	−6.79834	−0.651163	−0.325581	+	0.945514i	$0.605560\pi$
$110$	0	0
$111$	−11.5389	−1.09522
$112$	9.72988	0.919388
$113$	−0.301489	−0.0283617	−0.0141808	−	0.999899i	$-0.504514\pi$
$113$	−0.301489	−0.0283617	−0.0141808	+	0.999899i	$0.504514\pi$
$114$	2.41763	0.226431
$115$	0	0
$116$	2.73612	0.254042
$117$	2.49889	0.231022
$118$	17.6912	1.62861
$119$	−22.5848	−2.07035
$120$	0	0
$121$	0	0
$122$	17.3010	1.56636
$123$	0.932568	0.0840868
$124$	−2.67315	−0.240056
$125$	0	0
$126$	−6.98587	−0.622351
$127$	6.70539	0.595007	0.297504	−	0.954721i	$-0.403846\pi$
$127$	6.70539	0.595007	0.297504	+	0.954721i	$0.403846\pi$
$128$	−7.19453	−0.635913
$129$	−0.552188	−0.0486175
$130$	0	0
$131$	8.07729	0.705716	0.352858	−	0.935677i	$-0.385210\pi$
$131$	8.07729	0.705716	0.352858	+	0.935677i	$0.385210\pi$
$132$	0	0
$133$	−3.80612	−0.330033
$134$	−8.75084	−0.755957
$135$	0	0
$136$	6.27456	0.538039
$137$	−6.43781	−0.550020	−0.275010	−	0.961441i	$-0.588681\pi$
$137$	−6.43781	−0.550020	−0.275010	+	0.961441i	$0.588681\pi$
$138$	12.4079	1.05623
$139$	−6.97567	−0.591669	−0.295834	−	0.955239i	$-0.595598\pi$
$139$	−6.97567	−0.591669	−0.295834	+	0.955239i	$0.595598\pi$
$140$	0	0
$141$	−2.13427	−0.179738
$142$	26.4031	2.21570
$143$	0	0
$144$	−2.93393	−0.244495
$145$	0	0
$146$	12.0863	1.00027
$147$	3.99802	0.329751
$148$	−28.1246	−2.31183
$149$	19.0205	1.55822	0.779110	−	0.626887i	$-0.215671\pi$
$149$	19.0205	1.55822	0.779110	+	0.626887i	$0.215671\pi$
$150$	0	0
$151$	−3.90081	−0.317444	−0.158722	−	0.987323i	$-0.550737\pi$
$151$	−3.90081	−0.317444	−0.158722	+	0.987323i	$0.550737\pi$
$152$	1.05742	0.0857685
$153$	6.81019	0.550571
$154$	0	0
$155$	0	0
$156$	6.09074	0.487649
$157$	−18.3843	−1.46723	−0.733613	−	0.679568i	$-0.762167\pi$
$157$	−18.3843	−1.46723	−0.733613	+	0.679568i	$0.762167\pi$
$158$	8.14701	0.648141
$159$	11.6356	0.922763
$160$	0	0
$161$	−19.5340	−1.53950
$162$	2.10651	0.165503
$163$	−17.5720	−1.37635	−0.688173	−	0.725547i	$-0.741587\pi$
$163$	−17.5720	−1.37635	−0.688173	+	0.725547i	$0.741587\pi$
$164$	2.27302	0.177493
$165$	0	0
$166$	12.8087	0.994151
$167$	12.0777	0.934598	0.467299	−	0.884099i	$-0.345227\pi$
$167$	12.0777	0.934598	0.467299	+	0.884099i	$0.345227\pi$
$168$	−3.05549	−0.235736
$169$	−6.75557	−0.519659
$170$	0	0
$171$	1.14769	0.0877662
$172$	−1.34589	−0.102623
$173$	24.2436	1.84321	0.921603	−	0.388133i	$-0.126880\pi$
$173$	24.2436	1.84321	0.921603	+	0.388133i	$0.126880\pi$
$174$	2.36469	0.179267
$175$	0	0
$176$	0	0
$177$	8.39837	0.631260
$178$	26.5314	1.98861
$179$	−19.9390	−1.49031	−0.745155	−	0.666891i	$-0.767625\pi$
$179$	−19.9390	−1.49031	−0.745155	+	0.666891i	$0.767625\pi$
$180$	0	0
$181$	−5.13290	−0.381525	−0.190763	−	0.981636i	$-0.561096\pi$
$181$	−5.13290	−0.381525	−0.190763	+	0.981636i	$0.561096\pi$
$182$	−17.4569	−1.29399
$183$	8.21313	0.607132
$184$	5.42698	0.400082
$185$	0	0
$186$	−2.31027	−0.169397
$187$	0	0
$188$	−5.20204	−0.379397
$189$	−3.31633	−0.241227
$190$	0	0
$191$	−2.81336	−0.203568	−0.101784	−	0.994807i	$-0.532455\pi$
$191$	−2.81336	−0.203568	−0.101784	+	0.994807i	$0.532455\pi$
$192$	−11.0328	−0.796222
$193$	19.3032	1.38948	0.694738	−	0.719263i	$-0.255520\pi$
$193$	19.3032	1.38948	0.694738	+	0.719263i	$0.255520\pi$
$194$	−6.04300	−0.433862
$195$	0	0
$196$	9.74470	0.696050
$197$	−3.57956	−0.255033	−0.127516	−	0.991836i	$-0.540701\pi$
$197$	−3.57956	−0.255033	−0.127516	+	0.991836i	$0.540701\pi$
$198$	0	0
$199$	12.7087	0.900893	0.450447	−	0.892803i	$-0.351265\pi$
$199$	12.7087	0.900893	0.450447	+	0.892803i	$0.351265\pi$
$200$	0	0
$201$	−4.15419	−0.293014
$202$	−20.7913	−1.46287
$203$	−3.72279	−0.261289
$204$	16.5990	1.16216
$205$	0	0
$206$	16.2613	1.13298
$207$	5.89026	0.409401
$208$	−7.33157	−0.508353
$209$	0	0
$210$	0	0
$211$	3.49518	0.240618	0.120309	−	0.992736i	$-0.461611\pi$
$211$	3.49518	0.240618	0.120309	+	0.992736i	$0.461611\pi$
$212$	28.3604	1.94780
$213$	12.5340	0.858819
$214$	32.6749	2.23361
$215$	0	0
$216$	0.921348	0.0626898
$217$	3.63711	0.246903
$218$	−14.3208	−0.969925
$219$	5.73761	0.387712
$220$	0	0
$221$	17.0179	1.14475
$222$	−24.3067	−1.63136
$223$	−5.20558	−0.348591	−0.174296	−	0.984693i	$-0.555765\pi$
$223$	−5.20558	−0.348591	−0.174296	+	0.984693i	$0.555765\pi$
$224$	26.6071	1.77776
$225$	0	0
$226$	−0.635089	−0.0422455
$227$	9.70244	0.643973	0.321987	−	0.946744i	$-0.395649\pi$
$227$	9.70244	0.643973	0.321987	+	0.946744i	$0.395649\pi$
$228$	2.79736	0.185260
$229$	18.5011	1.22259	0.611293	−	0.791404i	$-0.290650\pi$
$229$	18.5011	1.22259	0.611293	+	0.791404i	$0.290650\pi$
$230$	0	0
$231$	0	0
$232$	1.03427	0.0679034
$233$	−14.9543	−0.979692	−0.489846	−	0.871809i	$-0.662947\pi$
$233$	−14.9543	−0.979692	−0.489846	+	0.871809i	$0.662947\pi$
$234$	5.26393	0.344114
$235$	0	0
$236$	20.4700	1.33249
$237$	3.86754	0.251224
$238$	−47.5751	−3.08384
$239$	1.50716	0.0974901	0.0487451	−	0.998811i	$-0.484478\pi$
$239$	1.50716	0.0974901	0.0487451	+	0.998811i	$0.484478\pi$
$240$	0	0
$241$	7.09571	0.457075	0.228537	−	0.973535i	$-0.426606\pi$
$241$	7.09571	0.457075	0.228537	+	0.973535i	$0.426606\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	20.0185	1.28156
$245$	0	0
$246$	1.96446	0.125250
$247$	2.86795	0.182483
$248$	−1.01047	−0.0641649
$249$	6.08055	0.385339
$250$	0	0
$251$	−13.6786	−0.863386	−0.431693	−	0.902021i	$-0.642084\pi$
$251$	−13.6786	−0.863386	−0.431693	+	0.902021i	$0.642084\pi$
$252$	−8.08315	−0.509191
$253$	0	0
$254$	14.1250	0.886279
$255$	0	0
$256$	6.91021	0.431888
$257$	8.30411	0.517996	0.258998	−	0.965878i	$-0.416608\pi$
$257$	8.30411	0.517996	0.258998	+	0.965878i	$0.416608\pi$
$258$	−1.16319	−0.0724171
$259$	38.2667	2.37777
$260$	0	0
$261$	1.12257	0.0694851
$262$	17.0149	1.05118
$263$	11.1345	0.686582	0.343291	−	0.939229i	$-0.388458\pi$
$263$	11.1345	0.686582	0.343291	+	0.939229i	$0.388458\pi$
$264$	0	0
$265$	0	0
$266$	−8.01763	−0.491593
$267$	12.5950	0.770799
$268$	−10.1253	−0.618504
$269$	22.7563	1.38748	0.693739	−	0.720227i	$-0.255962\pi$
$269$	22.7563	1.38748	0.693739	+	0.720227i	$0.255962\pi$
$270$	0	0
$271$	−7.79448	−0.473481	−0.236740	−	0.971573i	$-0.576079\pi$
$271$	−7.79448	−0.473481	−0.236740	+	0.971573i	$0.576079\pi$
$272$	−19.9806	−1.21150
$273$	−8.28712	−0.501559
$274$	−13.5613	−0.819269
$275$	0	0
$276$	14.3568	0.864179
$277$	9.46719	0.568828	0.284414	−	0.958702i	$-0.408201\pi$
$277$	9.46719	0.568828	0.284414	+	0.958702i	$0.408201\pi$
$278$	−14.6943	−0.881307
$279$	−1.09673	−0.0656595
$280$	0	0
$281$	−31.3016	−1.86729	−0.933647	−	0.358194i	$-0.883393\pi$
$281$	−31.3016	−1.86729	−0.933647	+	0.358194i	$0.883393\pi$
$282$	−4.49586	−0.267725
$283$	−24.4123	−1.45116	−0.725581	−	0.688136i	$-0.758429\pi$
$283$	−24.4123	−1.45116	−0.725581	+	0.688136i	$0.758429\pi$
$284$	30.5503	1.81282
$285$	0	0
$286$	0	0
$287$	−3.09270	−0.182556
$288$	−8.02306	−0.472763
$289$	29.3787	1.72816
$290$	0	0
$291$	−2.86873	−0.168168
$292$	13.9847	0.818395
$293$	11.8197	0.690512	0.345256	−	0.938509i	$-0.387792\pi$
$293$	11.8197	0.690512	0.345256	+	0.938509i	$0.387792\pi$
$294$	8.42187	0.491173
$295$	0	0
$296$	−10.6313	−0.617933
$297$	0	0
$298$	40.0669	2.32101
$299$	14.7191	0.851227
$300$	0	0
$301$	1.83124	0.105551
$302$	−8.21710	−0.472841
$303$	−9.87001	−0.567018
$304$	−3.36725	−0.193125
$305$	0	0
$306$	14.3457	0.820091
$307$	22.3217	1.27396	0.636982	−	0.770878i	$-0.280182\pi$
$307$	22.3217	1.27396	0.636982	+	0.770878i	$0.280182\pi$
$308$	0	0
$309$	7.71953	0.439149
$310$	0	0
$311$	−19.4834	−1.10480	−0.552401	−	0.833578i	$-0.686288\pi$
$311$	−19.4834	−1.10480	−0.552401	+	0.833578i	$0.686288\pi$
$312$	2.30234	0.130345
$313$	−17.3527	−0.980833	−0.490416	−	0.871488i	$-0.663155\pi$
$313$	−17.3527	−0.980833	−0.490416	+	0.871488i	$0.663155\pi$
$314$	−38.7267	−2.18547
$315$	0	0
$316$	9.42667	0.530292
$317$	−0.830717	−0.0466577	−0.0233289	−	0.999728i	$-0.507426\pi$
$317$	−0.830717	−0.0466577	−0.0233289	+	0.999728i	$0.507426\pi$
$318$	24.5105	1.37448
$319$	0	0
$320$	0	0
$321$	15.5114	0.865762
$322$	−41.1486	−2.29312
$323$	7.81600	0.434894
$324$	2.43738	0.135410
$325$	0	0
$326$	−37.0156	−2.05010
$327$	−6.79834	−0.375949
$328$	0.859219	0.0474425
$329$	7.07794	0.390220
$330$	0	0
$331$	−31.5311	−1.73311	−0.866554	−	0.499084i	$-0.833670\pi$
$331$	−31.5311	−1.73311	−0.866554	+	0.499084i	$0.833670\pi$
$332$	14.8206	0.813387
$333$	−11.5389	−0.632326
$334$	25.4417	1.39211
$335$	0	0
$336$	9.72988	0.530809
$337$	−16.2339	−0.884319	−0.442160	−	0.896936i	$-0.645788\pi$
$337$	−16.2339	−0.884319	−0.442160	+	0.896936i	$0.645788\pi$
$338$	−14.2307	−0.774047
$339$	−0.301489	−0.0163746
$340$	0	0
$341$	0	0
$342$	2.41763	0.130730
$343$	9.95555	0.537549
$344$	−0.508758	−0.0274304
$345$	0	0
$346$	51.0694	2.74551
$347$	−12.4500	−0.668350	−0.334175	−	0.942511i	$-0.608458\pi$
$347$	−12.4500	−0.668350	−0.334175	+	0.942511i	$0.608458\pi$
$348$	2.73612	0.146671
$349$	−32.6297	−1.74663	−0.873314	−	0.487157i	$-0.838034\pi$
$349$	−32.6297	−1.74663	−0.873314	+	0.487157i	$0.838034\pi$
$350$	0	0
$351$	2.49889	0.133381
$352$	0	0
$353$	15.7921	0.840530	0.420265	−	0.907402i	$-0.361937\pi$
$353$	15.7921	0.840530	0.420265	+	0.907402i	$0.361937\pi$
$354$	17.6912	0.940279
$355$	0	0
$356$	30.6987	1.62703
$357$	−22.5848	−1.19531
$358$	−42.0017	−2.21986
$359$	−18.1622	−0.958563	−0.479282	−	0.877661i	$-0.659103\pi$
$359$	−18.1622	−0.958563	−0.479282	+	0.877661i	$0.659103\pi$
$360$	0	0
$361$	−17.6828	−0.930674
$362$	−10.8125	−0.568292
$363$	0	0
$364$	−20.1989	−1.05871
$365$	0	0
$366$	17.3010	0.904340
$367$	0.472046	0.0246406	0.0123203	−	0.999924i	$-0.496078\pi$
$367$	0.472046	0.0246406	0.0123203	+	0.999924i	$0.496078\pi$
$368$	−17.2816	−0.900868
$369$	0.932568	0.0485475
$370$	0	0
$371$	−38.5874	−2.00336
$372$	−2.67315	−0.138596
$373$	15.4102	0.797910	0.398955	−	0.916970i	$-0.369373\pi$
$373$	15.4102	0.797910	0.398955	+	0.916970i	$0.369373\pi$
$374$	0	0
$375$	0	0
$376$	−1.96641	−0.101410
$377$	2.80516	0.144473
$378$	−6.98587	−0.359314
$379$	−19.2746	−0.990068	−0.495034	−	0.868874i	$-0.664844\pi$
$379$	−19.2746	−0.990068	−0.495034	+	0.868874i	$0.664844\pi$
$380$	0	0
$381$	6.70539	0.343527
$382$	−5.92637	−0.303220
$383$	7.44824	0.380587	0.190294	−	0.981727i	$-0.439056\pi$
$383$	7.44824	0.380587	0.190294	+	0.981727i	$0.439056\pi$
$384$	−7.19453	−0.367144
$385$	0	0
$386$	40.6624	2.06966
$387$	−0.552188	−0.0280693
$388$	−6.99218	−0.354974
$389$	−0.713819	−0.0361921	−0.0180960	−	0.999836i	$-0.505760\pi$
$389$	−0.713819	−0.0361921	−0.0180960	+	0.999836i	$0.505760\pi$
$390$	0	0
$391$	40.1138	2.02864
$392$	3.68357	0.186048
$393$	8.07729	0.407446
$394$	−7.54037	−0.379878
$395$	0	0
$396$	0	0
$397$	0.764171	0.0383526	0.0191763	−	0.999816i	$-0.493896\pi$
$397$	0.764171	0.0383526	0.0191763	+	0.999816i	$0.493896\pi$
$398$	26.7709	1.34191
$399$	−3.80612	−0.190544
$400$	0	0
$401$	−15.7544	−0.786740	−0.393370	−	0.919380i	$-0.628691\pi$
$401$	−15.7544	−0.786740	−0.393370	+	0.919380i	$0.628691\pi$
$402$	−8.75084	−0.436452
$403$	−2.74060	−0.136519
$404$	−24.0570	−1.19688
$405$	0	0
$406$	−7.84210	−0.389197
$407$	0	0
$408$	6.27456	0.310637
$409$	2.62100	0.129600	0.0648001	−	0.997898i	$-0.479359\pi$
$409$	2.62100	0.129600	0.0648001	+	0.997898i	$0.479359\pi$
$410$	0	0
$411$	−6.43781	−0.317554
$412$	18.8154	0.926970
$413$	−27.8517	−1.37049
$414$	12.4079	0.609815
$415$	0	0
$416$	−20.0487	−0.982968
$417$	−6.97567	−0.341600
$418$	0	0
$419$	−18.0078	−0.879737	−0.439868	−	0.898062i	$-0.644975\pi$
$419$	−18.0078	−0.879737	−0.439868	+	0.898062i	$0.644975\pi$
$420$	0	0
$421$	1.17117	0.0570792	0.0285396	−	0.999593i	$-0.490914\pi$
$421$	1.17117	0.0570792	0.0285396	+	0.999593i	$0.490914\pi$
$422$	7.36264	0.358408
$423$	−2.13427	−0.103772
$424$	10.7204	0.520630
$425$	0	0
$426$	26.4031	1.27923
$427$	−27.2374	−1.31811
$428$	37.8072	1.82748
$429$	0	0
$430$	0	0
$431$	−35.0921	−1.69033	−0.845163	−	0.534508i	$-0.820497\pi$
$431$	−35.0921	−1.69033	−0.845163	+	0.534508i	$0.820497\pi$
$432$	−2.93393	−0.141159
$433$	−18.2260	−0.875888	−0.437944	−	0.899002i	$-0.644293\pi$
$433$	−18.2260	−0.875888	−0.437944	+	0.899002i	$0.644293\pi$
$434$	7.66161	0.367769
$435$	0	0
$436$	−16.5701	−0.793566
$437$	6.76021	0.323385
$438$	12.0863	0.577507
$439$	19.8155	0.945743	0.472871	−	0.881131i	$-0.343218\pi$
$439$	19.8155	0.945743	0.472871	+	0.881131i	$0.343218\pi$
$440$	0	0
$441$	3.99802	0.190382
$442$	35.8483	1.70513
$443$	16.6884	0.792892	0.396446	−	0.918058i	$-0.370243\pi$
$443$	16.6884	0.792892	0.396446	+	0.918058i	$0.370243\pi$
$444$	−28.1246	−1.33474
$445$	0	0
$446$	−10.9656	−0.519236
$447$	19.0205	0.899639
$448$	36.5883	1.72863
$449$	−13.1971	−0.622808	−0.311404	−	0.950278i	$-0.600799\pi$
$449$	−13.1971	−0.622808	−0.311404	+	0.950278i	$0.600799\pi$
$450$	0	0
$451$	0	0
$452$	−0.734843	−0.0345641
$453$	−3.90081	−0.183276
$454$	20.4383	0.959216
$455$	0	0
$456$	1.05742	0.0495184
$457$	−21.2673	−0.994840	−0.497420	−	0.867510i	$-0.665719\pi$
$457$	−21.2673	−0.994840	−0.497420	+	0.867510i	$0.665719\pi$
$458$	38.9727	1.82107
$459$	6.81019	0.317872
$460$	0	0
$461$	20.0772	0.935086	0.467543	−	0.883970i	$-0.345139\pi$
$461$	20.0772	0.935086	0.467543	+	0.883970i	$0.345139\pi$
$462$	0	0
$463$	−0.962526	−0.0447324	−0.0223662	−	0.999750i	$-0.507120\pi$
$463$	−0.962526	−0.0447324	−0.0223662	+	0.999750i	$0.507120\pi$
$464$	−3.29353	−0.152898
$465$	0	0
$466$	−31.5015	−1.45928
$467$	−14.9130	−0.690092	−0.345046	−	0.938586i	$-0.612137\pi$
$467$	−14.9130	−0.690092	−0.345046	+	0.938586i	$0.612137\pi$
$468$	6.09074	0.281544
$469$	13.7766	0.636146
$470$	0	0
$471$	−18.3843	−0.847103
$472$	7.73783	0.356162
$473$	0	0
$474$	8.14701	0.374204
$475$	0	0
$476$	−55.0478	−2.52311
$477$	11.6356	0.532757
$478$	3.17485	0.145214
$479$	0.475527	0.0217274	0.0108637	−	0.999941i	$-0.496542\pi$
$479$	0.475527	0.0217274	0.0108637	+	0.999941i	$0.496542\pi$
$480$	0	0
$481$	−28.8343	−1.31473
$482$	14.9472	0.680825
$483$	−19.5340	−0.888829
$484$	0	0
$485$	0	0
$486$	2.10651	0.0955532
$487$	−35.3529	−1.60199	−0.800997	−	0.598668i	$-0.795697\pi$
$487$	−35.3529	−1.60199	−0.800997	+	0.598668i	$0.795697\pi$
$488$	7.56716	0.342549
$489$	−17.5720	−0.794634
$490$	0	0
$491$	30.1328	1.35987	0.679937	−	0.733271i	$-0.262007\pi$
$491$	30.1328	1.35987	0.679937	+	0.733271i	$0.262007\pi$
$492$	2.27302	0.102476
$493$	7.64488	0.344308
$494$	6.04137	0.271814
$495$	0	0
$496$	3.21773	0.144480
$497$	−41.5670	−1.86453
$498$	12.8087	0.573973
$499$	−36.4797	−1.63306	−0.816528	−	0.577305i	$-0.804104\pi$
$499$	−36.4797	−1.63306	−0.816528	+	0.577305i	$0.804104\pi$
$500$	0	0
$501$	12.0777	0.539590
$502$	−28.8141	−1.28604
$503$	−15.4251	−0.687772	−0.343886	−	0.939011i	$-0.611743\pi$
$503$	−15.4251	−0.687772	−0.343886	+	0.939011i	$0.611743\pi$
$504$	−3.05549	−0.136102
$505$	0	0
$506$	0	0
$507$	−6.75557	−0.300025
$508$	16.3436	0.725130
$509$	12.6374	0.560142	0.280071	−	0.959979i	$-0.409642\pi$
$509$	12.6374	0.560142	0.280071	+	0.959979i	$0.409642\pi$
$510$	0	0
$511$	−19.0278	−0.841740
$512$	28.9455	1.27922
$513$	1.14769	0.0506719
$514$	17.4927	0.771570
$515$	0	0
$516$	−1.34589	−0.0592497
$517$	0	0
$518$	80.6091	3.54176
$519$	24.2436	1.06418
$520$	0	0
$521$	−9.18439	−0.402376	−0.201188	−	0.979553i	$-0.564480\pi$
$521$	−9.18439	−0.402376	−0.201188	+	0.979553i	$0.564480\pi$
$522$	2.36469	0.103500
$523$	−33.5841	−1.46853	−0.734265	−	0.678863i	$-0.762473\pi$
$523$	−33.5841	−1.46853	−0.734265	+	0.678863i	$0.762473\pi$
$524$	19.6874	0.860050
$525$	0	0
$526$	23.4549	1.02268
$527$	−7.46893	−0.325352
$528$	0	0
$529$	11.6952	0.508486
$530$	0	0
$531$	8.39837	0.364458
$532$	−9.27697	−0.402208
$533$	2.33038	0.100940
$534$	26.5314	1.14813
$535$	0	0
$536$	−3.82745	−0.165321
$537$	−19.9390	−0.860431
$538$	47.9364	2.06669
$539$	0	0
$540$	0	0
$541$	−44.6024	−1.91761	−0.958803	−	0.284072i	$-0.908315\pi$
$541$	−44.6024	−1.91761	−0.958803	+	0.284072i	$0.908315\pi$
$542$	−16.4191	−0.705262
$543$	−5.13290	−0.220274
$544$	−54.6385	−2.34261
$545$	0	0
$546$	−17.4569	−0.747086
$547$	−19.2310	−0.822256	−0.411128	−	0.911578i	$-0.634865\pi$
$547$	−19.2310	−0.822256	−0.411128	+	0.911578i	$0.634865\pi$
$548$	−15.6914	−0.670304
$549$	8.21313	0.350528
$550$	0	0
$551$	1.28836	0.0548860
$552$	5.42698	0.230988
$553$	−12.8260	−0.545418
$554$	19.9427	0.847285
$555$	0	0
$556$	−17.0024	−0.721061
$557$	33.3669	1.41380	0.706900	−	0.707314i	$-0.250093\pi$
$557$	33.3669	1.41380	0.706900	+	0.707314i	$0.250093\pi$
$558$	−2.31027	−0.0978016
$559$	−1.37986	−0.0583617
$560$	0	0
$561$	0	0
$562$	−65.9370	−2.78139
$563$	2.09215	0.0881735	0.0440868	−	0.999028i	$-0.485962\pi$
$563$	2.09215	0.0881735	0.0440868	+	0.999028i	$0.485962\pi$
$564$	−5.20204	−0.219045
$565$	0	0
$566$	−51.4248	−2.16155
$567$	−3.31633	−0.139273
$568$	11.5482	0.484553
$569$	9.31615	0.390553	0.195277	−	0.980748i	$-0.437439\pi$
$569$	9.31615	0.390553	0.195277	+	0.980748i	$0.437439\pi$
$570$	0	0
$571$	−2.32989	−0.0975029	−0.0487514	−	0.998811i	$-0.515524\pi$
$571$	−2.32989	−0.0975029	−0.0487514	+	0.998811i	$0.515524\pi$
$572$	0	0
$573$	−2.81336	−0.117530
$574$	−6.51480	−0.271922
$575$	0	0
$576$	−11.0328	−0.459699
$577$	29.4627	1.22655	0.613275	−	0.789869i	$-0.289852\pi$
$577$	29.4627	1.22655	0.613275	+	0.789869i	$0.289852\pi$
$578$	61.8864	2.57414
$579$	19.3032	0.802214
$580$	0	0
$581$	−20.1651	−0.836589
$582$	−6.04300	−0.250491
$583$	0	0
$584$	5.28634	0.218750
$585$	0	0
$586$	24.8982	1.02854
$587$	41.5167	1.71358	0.856789	−	0.515667i	$-0.172456\pi$
$587$	41.5167	1.71358	0.856789	+	0.515667i	$0.172456\pi$
$588$	9.74470	0.401865
$589$	−1.25871	−0.0518642
$590$	0	0
$591$	−3.57956	−0.147243
$592$	33.8543	1.39140
$593$	−9.36964	−0.384765	−0.192383	−	0.981320i	$-0.561621\pi$
$593$	−9.36964	−0.384765	−0.192383	+	0.981320i	$0.561621\pi$
$594$	0	0
$595$	0	0
$596$	46.3602	1.89899
$597$	12.7087	0.520131
$598$	31.0059	1.26793
$599$	1.31702	0.0538122	0.0269061	−	0.999638i	$-0.491434\pi$
$599$	1.31702	0.0538122	0.0269061	+	0.999638i	$0.491434\pi$
$600$	0	0
$601$	−15.6267	−0.637425	−0.318713	−	0.947851i	$-0.603250\pi$
$601$	−15.6267	−0.637425	−0.318713	+	0.947851i	$0.603250\pi$
$602$	3.85752	0.157221
$603$	−4.15419	−0.169172
$604$	−9.50777	−0.386866
$605$	0	0
$606$	−20.7913	−0.844588
$607$	24.2123	0.982746	0.491373	−	0.870949i	$-0.336495\pi$
$607$	24.2123	0.982746	0.491373	+	0.870949i	$0.336495\pi$
$608$	−9.20800	−0.373434
$609$	−3.72279	−0.150855
$610$	0	0
$611$	−5.33330	−0.215762
$612$	16.5990	0.670976
$613$	4.08630	0.165044	0.0825221	−	0.996589i	$-0.473702\pi$
$613$	4.08630	0.165044	0.0825221	+	0.996589i	$0.473702\pi$
$614$	47.0208	1.89760
$615$	0	0
$616$	0	0
$617$	−16.9835	−0.683731	−0.341866	−	0.939749i	$-0.611059\pi$
$617$	−16.9835	−0.683731	−0.341866	+	0.939749i	$0.611059\pi$
$618$	16.2613	0.654124
$619$	−32.1306	−1.29144	−0.645720	−	0.763575i	$-0.723442\pi$
$619$	−32.1306	−1.29144	−0.645720	+	0.763575i	$0.723442\pi$
$620$	0	0
$621$	5.89026	0.236368
$622$	−41.0420	−1.64563
$623$	−41.7690	−1.67344
$624$	−7.33157	−0.293498
$625$	0	0
$626$	−36.5536	−1.46098
$627$	0	0
$628$	−44.8095	−1.78809
$629$	−78.5819	−3.13326
$630$	0	0
$631$	−6.31566	−0.251422	−0.125711	−	0.992067i	$-0.540121\pi$
$631$	−6.31566	−0.251422	−0.125711	+	0.992067i	$0.540121\pi$
$632$	3.56335	0.141742
$633$	3.49518	0.138921
$634$	−1.74991	−0.0694980
$635$	0	0
$636$	28.3604	1.12456
$637$	9.99059	0.395842
$638$	0	0
$639$	12.5340	0.495839
$640$	0	0
$641$	14.6862	0.580072	0.290036	−	0.957016i	$-0.406333\pi$
$641$	14.6862	0.580072	0.290036	+	0.957016i	$0.406333\pi$
$642$	32.6749	1.28958
$643$	16.1510	0.636933	0.318467	−	0.947934i	$-0.396832\pi$
$643$	16.1510	0.636933	0.318467	+	0.947934i	$0.396832\pi$
$644$	−47.6119	−1.87617
$645$	0	0
$646$	16.4645	0.647786
$647$	−27.7874	−1.09243	−0.546217	−	0.837644i	$-0.683933\pi$
$647$	−27.7874	−1.09243	−0.546217	+	0.837644i	$0.683933\pi$
$648$	0.921348	0.0361940
$649$	0	0
$650$	0	0
$651$	3.63711	0.142550
$652$	−42.8297	−1.67734
$653$	34.0655	1.33309	0.666543	−	0.745467i	$-0.267773\pi$
$653$	34.0655	1.33309	0.666543	+	0.745467i	$0.267773\pi$
$654$	−14.3208	−0.559986
$655$	0	0
$656$	−2.73609	−0.106826
$657$	5.73761	0.223846
$658$	14.9098	0.581243
$659$	−14.4196	−0.561709	−0.280855	−	0.959750i	$-0.590618\pi$
$659$	−14.4196	−0.561709	−0.280855	+	0.959750i	$0.590618\pi$
$660$	0	0
$661$	26.4275	1.02791	0.513956	−	0.857817i	$-0.328179\pi$
$661$	26.4275	1.02791	0.513956	+	0.857817i	$0.328179\pi$
$662$	−66.4206	−2.58151
$663$	17.0179	0.660920
$664$	5.60231	0.217412
$665$	0	0
$666$	−24.3067	−0.941867
$667$	6.61220	0.256026
$668$	29.4379	1.13899
$669$	−5.20558	−0.201259
$670$	0	0
$671$	0	0
$672$	26.6071	1.02639
$673$	36.3130	1.39976	0.699881	−	0.714260i	$-0.253236\pi$
$673$	36.3130	1.39976	0.699881	+	0.714260i	$0.253236\pi$
$674$	−34.1970	−1.31722
$675$	0	0
$676$	−16.4659	−0.633304
$677$	11.3442	0.435994	0.217997	−	0.975949i	$-0.430048\pi$
$677$	11.3442	0.435994	0.217997	+	0.975949i	$0.430048\pi$
$678$	−0.635089	−0.0243904
$679$	9.51364	0.365100
$680$	0	0
$681$	9.70244	0.371798
$682$	0	0
$683$	−39.4269	−1.50863	−0.754314	−	0.656514i	$-0.772030\pi$
$683$	−39.4269	−1.50863	−0.754314	+	0.656514i	$0.772030\pi$
$684$	2.79736	0.106960
$685$	0	0
$686$	20.9715	0.800694
$687$	18.5011	0.705860
$688$	1.62008	0.0617651
$689$	29.0760	1.10771
$690$	0	0
$691$	14.7763	0.562118	0.281059	−	0.959691i	$-0.409314\pi$
$691$	14.7763	0.562118	0.281059	+	0.959691i	$0.409314\pi$
$692$	59.0909	2.24630
$693$	0	0
$694$	−26.2260	−0.995526
$695$	0	0
$696$	1.03427	0.0392040
$697$	6.35096	0.240560
$698$	−68.7348	−2.60165
$699$	−14.9543	−0.565625
$700$	0	0
$701$	7.12932	0.269271	0.134635	−	0.990895i	$-0.457014\pi$
$701$	7.12932	0.269271	0.134635	+	0.990895i	$0.457014\pi$
$702$	5.26393	0.198674
$703$	−13.2431	−0.499472
$704$	0	0
$705$	0	0
$706$	33.2662	1.25199
$707$	32.7322	1.23102
$708$	20.4700	0.769311
$709$	−5.35482	−0.201104	−0.100552	−	0.994932i	$-0.532061\pi$
$709$	−5.35482	−0.201104	−0.100552	+	0.994932i	$0.532061\pi$
$710$	0	0
$711$	3.86754	0.145044
$712$	11.6043	0.434891
$713$	−6.46002	−0.241930
$714$	−47.5751	−1.78045
$715$	0	0
$716$	−48.5989	−1.81623
$717$	1.50716	0.0562860
$718$	−38.2588	−1.42781
$719$	−40.7829	−1.52095	−0.760473	−	0.649370i	$-0.775033\pi$
$719$	−40.7829	−1.52095	−0.760473	+	0.649370i	$0.775033\pi$
$720$	0	0
$721$	−25.6005	−0.953412
$722$	−37.2490	−1.38626
$723$	7.09571	0.263892
$724$	−12.5108	−0.464961
$725$	0	0
$726$	0	0
$727$	28.6101	1.06109	0.530546	−	0.847656i	$-0.321987\pi$
$727$	28.6101	1.06109	0.530546	+	0.847656i	$0.321987\pi$
$728$	−7.63532	−0.282984
$729$	1.00000	0.0370370
$730$	0	0
$731$	−3.76051	−0.139087
$732$	20.0185	0.739907
$733$	4.39612	0.162374	0.0811871	−	0.996699i	$-0.474129\pi$
$733$	4.39612	0.162374	0.0811871	+	0.996699i	$0.474129\pi$
$734$	0.994370	0.0367029
$735$	0	0
$736$	−47.2579	−1.74195
$737$	0	0
$738$	1.96446	0.0723129
$739$	10.5172	0.386882	0.193441	−	0.981112i	$-0.438035\pi$
$739$	10.5172	0.386882	0.193441	+	0.981112i	$0.438035\pi$
$740$	0	0
$741$	2.86795	0.105357
$742$	−81.2848	−2.98406
$743$	−34.4901	−1.26532	−0.632659	−	0.774431i	$-0.718036\pi$
$743$	−34.4901	−1.26532	−0.632659	+	0.774431i	$0.718036\pi$
$744$	−1.01047	−0.0370456
$745$	0	0
$746$	32.4617	1.18851
$747$	6.08055	0.222476
$748$	0	0
$749$	−51.4408	−1.87961
$750$	0	0
$751$	−2.38104	−0.0868854	−0.0434427	−	0.999056i	$-0.513833\pi$
$751$	−2.38104	−0.0868854	−0.0434427	+	0.999056i	$0.513833\pi$
$752$	6.26181	0.228345
$753$	−13.6786	−0.498476
$754$	5.90910	0.215197
$755$	0	0
$756$	−8.08315	−0.293981
$757$	8.39549	0.305139	0.152570	−	0.988293i	$-0.451245\pi$
$757$	8.39549	0.305139	0.152570	+	0.988293i	$0.451245\pi$
$758$	−40.6021	−1.47473
$759$	0	0
$760$	0	0
$761$	−10.4681	−0.379467	−0.189734	−	0.981836i	$-0.560762\pi$
$761$	−10.4681	−0.379467	−0.189734	+	0.981836i	$0.560762\pi$
$762$	14.1250	0.511693
$763$	22.5455	0.816202
$764$	−6.85724	−0.248086
$765$	0	0
$766$	15.6898	0.566895
$767$	20.9866	0.757781
$768$	6.91021	0.249351
$769$	9.70799	0.350079	0.175040	−	0.984561i	$-0.443995\pi$
$769$	9.70799	0.350079	0.175040	+	0.984561i	$0.443995\pi$
$770$	0	0
$771$	8.30411	0.299065
$772$	47.0493	1.69334
$773$	−31.9598	−1.14952	−0.574758	−	0.818324i	$-0.694904\pi$
$773$	−31.9598	−1.14952	−0.574758	+	0.818324i	$0.694904\pi$
$774$	−1.16319	−0.0418100
$775$	0	0
$776$	−2.64310	−0.0948817
$777$	38.2667	1.37281
$778$	−1.50367	−0.0539090
$779$	1.07030	0.0383475
$780$	0	0
$781$	0	0
$782$	84.5001	3.02172
$783$	1.12257	0.0401172
$784$	−11.7299	−0.418926
$785$	0	0
$786$	17.0149	0.606901
$787$	−46.4715	−1.65653	−0.828266	−	0.560336i	$-0.810672\pi$
$787$	−46.4715	−1.65653	−0.828266	+	0.560336i	$0.810672\pi$
$788$	−8.72475	−0.310806
$789$	11.1345	0.396398
$790$	0	0
$791$	0.999835	0.0355500
$792$	0	0
$793$	20.5237	0.728817
$794$	1.60973	0.0571273
$795$	0	0
$796$	30.9759	1.09791
$797$	32.2353	1.14183	0.570917	−	0.821008i	$-0.306588\pi$
$797$	32.2353	1.14183	0.570917	+	0.821008i	$0.306588\pi$
$798$	−8.01763	−0.283821
$799$	−14.5348	−0.514204
$800$	0	0
$801$	12.5950	0.445021
$802$	−33.1869	−1.17187
$803$	0	0
$804$	−10.1253	−0.357093
$805$	0	0
$806$	−5.77310	−0.203349
$807$	22.7563	0.801061
$808$	−9.09372	−0.319916
$809$	−21.1977	−0.745273	−0.372637	−	0.927977i	$-0.621546\pi$
$809$	−21.1977	−0.745273	−0.372637	+	0.927977i	$0.621546\pi$
$810$	0	0
$811$	23.6049	0.828879	0.414440	−	0.910077i	$-0.363978\pi$
$811$	23.6049	0.828879	0.414440	+	0.910077i	$0.363978\pi$
$812$	−9.07387	−0.318430
$813$	−7.79448	−0.273364
$814$	0	0
$815$	0	0
$816$	−19.9806	−0.699463
$817$	−0.633743	−0.0221718
$818$	5.52117	0.193043
$819$	−8.28712	−0.289575
$820$	0	0
$821$	44.5174	1.55367	0.776834	−	0.629706i	$-0.216825\pi$
$821$	44.5174	1.55367	0.776834	+	0.629706i	$0.216825\pi$
$822$	−13.5613	−0.473005
$823$	0.136429	0.00475560	0.00237780	−	0.999997i	$-0.499243\pi$
$823$	0.136429	0.00475560	0.00237780	+	0.999997i	$0.499243\pi$
$824$	7.11238	0.247771
$825$	0	0
$826$	−58.6700	−2.04139
$827$	11.7712	0.409325	0.204662	−	0.978833i	$-0.434390\pi$
$827$	11.7712	0.409325	0.204662	+	0.978833i	$0.434390\pi$
$828$	14.3568	0.498934
$829$	11.0290	0.383052	0.191526	−	0.981488i	$-0.438656\pi$
$829$	11.0290	0.383052	0.191526	+	0.981488i	$0.438656\pi$
$830$	0	0
$831$	9.46719	0.328413
$832$	−27.5696	−0.955805
$833$	27.2273	0.943369
$834$	−14.6943	−0.508823
$835$	0	0
$836$	0	0
$837$	−1.09673	−0.0379085
$838$	−37.9335	−1.31039
$839$	−10.5903	−0.365616	−0.182808	−	0.983149i	$-0.558519\pi$
$839$	−10.5903	−0.365616	−0.182808	+	0.983149i	$0.558519\pi$
$840$	0	0
$841$	−27.7398	−0.956546
$842$	2.46708	0.0850211
$843$	−31.3016	−1.07808
$844$	8.51910	0.293239
$845$	0	0
$846$	−4.49586	−0.154571
$847$	0	0
$848$	−34.1381	−1.17231
$849$	−24.4123	−0.837829
$850$	0	0
$851$	−67.9670	−2.32988
$852$	30.5503	1.04663
$853$	44.5811	1.52643	0.763215	−	0.646145i	$-0.223620\pi$
$853$	44.5811	1.52643	0.763215	+	0.646145i	$0.223620\pi$
$854$	−57.3759	−1.96336
$855$	0	0
$856$	14.2914	0.488470
$857$	24.3105	0.830430	0.415215	−	0.909723i	$-0.363706\pi$
$857$	24.3105	0.830430	0.415215	+	0.909723i	$0.363706\pi$
$858$	0	0
$859$	−35.5045	−1.21140	−0.605699	−	0.795694i	$-0.707107\pi$
$859$	−35.5045	−1.21140	−0.605699	+	0.795694i	$0.707107\pi$
$860$	0	0
$861$	−3.09270	−0.105399
$862$	−73.9218	−2.51779
$863$	11.6720	0.397319	0.198659	−	0.980069i	$-0.436341\pi$
$863$	11.6720	0.397319	0.198659	+	0.980069i	$0.436341\pi$
$864$	−8.02306	−0.272950
$865$	0	0
$866$	−38.3933	−1.30466
$867$	29.3787	0.997752
$868$	8.86503	0.300899
$869$	0	0
$870$	0	0
$871$	−10.3808	−0.351741
$872$	−6.26364	−0.212114
$873$	−2.86873	−0.0970917
$874$	14.2404	0.481690
$875$	0	0
$876$	13.9847	0.472501
$877$	26.4819	0.894230	0.447115	−	0.894476i	$-0.352451\pi$
$877$	26.4819	0.894230	0.447115	+	0.894476i	$0.352451\pi$
$878$	41.7416	1.40871
$879$	11.8197	0.398667
$880$	0	0
$881$	10.3488	0.348658	0.174329	−	0.984687i	$-0.444224\pi$
$881$	10.3488	0.348658	0.174329	+	0.984687i	$0.444224\pi$
$882$	8.42187	0.283579
$883$	−2.71148	−0.0912487	−0.0456243	−	0.998959i	$-0.514528\pi$
$883$	−2.71148	−0.0912487	−0.0456243	+	0.998959i	$0.514528\pi$
$884$	41.4791	1.39509
$885$	0	0
$886$	35.1543	1.18103
$887$	−0.122673	−0.00411896	−0.00205948	−	0.999998i	$-0.500656\pi$
$887$	−0.122673	−0.00411896	−0.00205948	+	0.999998i	$0.500656\pi$
$888$	−10.6313	−0.356764
$889$	−22.2373	−0.745814
$890$	0	0
$891$	0	0
$892$	−12.6880	−0.424825
$893$	−2.44949	−0.0819690
$894$	40.0669	1.34004
$895$	0	0
$896$	23.8594	0.797087
$897$	14.7191	0.491456
$898$	−27.7997	−0.927689
$899$	−1.23115	−0.0410612
$900$	0	0
$901$	79.2406	2.63989
$902$	0	0
$903$	1.83124	0.0609398
$904$	−0.277776	−0.00923870
$905$	0	0
$906$	−8.21710	−0.272995
$907$	−13.8188	−0.458844	−0.229422	−	0.973327i	$-0.573684\pi$
$907$	−13.8188	−0.458844	−0.229422	+	0.973327i	$0.573684\pi$
$908$	23.6485	0.784804
$909$	−9.87001	−0.327368
$910$	0	0
$911$	23.2558	0.770501	0.385250	−	0.922812i	$-0.374115\pi$
$911$	23.2558	0.770501	0.385250	+	0.922812i	$0.374115\pi$
$912$	−3.36725	−0.111501
$913$	0	0
$914$	−44.7997	−1.48184
$915$	0	0
$916$	45.0942	1.48995
$917$	−26.7869	−0.884583
$918$	14.3457	0.473480
$919$	−26.4278	−0.871772	−0.435886	−	0.900002i	$-0.643565\pi$
$919$	−26.4278	−0.871772	−0.435886	+	0.900002i	$0.643565\pi$
$920$	0	0
$921$	22.3217	0.735524
$922$	42.2927	1.39284
$923$	31.3212	1.03095
$924$	0	0
$925$	0	0
$926$	−2.02757	−0.0666301
$927$	7.71953	0.253543
$928$	−9.00641	−0.295650
$929$	−8.03463	−0.263608	−0.131804	−	0.991276i	$-0.542077\pi$
$929$	−8.03463	−0.263608	−0.131804	+	0.991276i	$0.542077\pi$
$930$	0	0
$931$	4.58850	0.150382
$932$	−36.4494	−1.19394
$933$	−19.4834	−0.637858
$934$	−31.4144	−1.02791
$935$	0	0
$936$	2.30234	0.0752545
$937$	−13.4913	−0.440740	−0.220370	−	0.975416i	$-0.570726\pi$
$937$	−13.4913	−0.440740	−0.220370	+	0.975416i	$0.570726\pi$
$938$	29.0206	0.947557
$939$	−17.3527	−0.566284
$940$	0	0
$941$	31.9040	1.04004	0.520021	−	0.854154i	$-0.325924\pi$
$941$	31.9040	1.04004	0.520021	+	0.854154i	$0.325924\pi$
$942$	−38.7267	−1.26178
$943$	5.49307	0.178879
$944$	−24.6403	−0.801973
$945$	0	0
$946$	0	0
$947$	−10.2719	−0.333793	−0.166896	−	0.985974i	$-0.553375\pi$
$947$	−10.2719	−0.333793	−0.166896	+	0.985974i	$0.553375\pi$
$948$	9.42667	0.306164
$949$	14.3376	0.465419
$950$	0	0
$951$	−0.830717	−0.0269379
$952$	−20.8085	−0.674407
$953$	9.83426	0.318563	0.159281	−	0.987233i	$-0.449082\pi$
$953$	9.83426	0.318563	0.159281	+	0.987233i	$0.449082\pi$
$954$	24.5105	0.793557
$955$	0	0
$956$	3.67353	0.118810
$957$	0	0
$958$	1.00170	0.0323635
$959$	21.3499	0.689424
$960$	0	0
$961$	−29.7972	−0.961199
$962$	−60.7397	−1.95833
$963$	15.5114	0.499848
$964$	17.2949	0.557033
$965$	0	0
$966$	−41.1486	−1.32393
$967$	−3.85001	−0.123808	−0.0619041	−	0.998082i	$-0.519717\pi$
$967$	−3.85001	−0.123808	−0.0619041	+	0.998082i	$0.519717\pi$
$968$	0	0
$969$	7.81600	0.251086
$970$	0	0
$971$	21.3920	0.686502	0.343251	−	0.939244i	$-0.388472\pi$
$971$	21.3920	0.686502	0.343251	+	0.939244i	$0.388472\pi$
$972$	2.43738	0.0781790
$973$	23.1336	0.741630
$974$	−74.4713	−2.38621
$975$	0	0
$976$	−24.0968	−0.771320
$977$	16.6681	0.533258	0.266629	−	0.963799i	$-0.414090\pi$
$977$	16.6681	0.533258	0.266629	+	0.963799i	$0.414090\pi$
$978$	−37.0156	−1.18363
$979$	0	0
$980$	0	0
$981$	−6.79834	−0.217054
$982$	63.4750	2.02557
$983$	−51.8570	−1.65398	−0.826991	−	0.562216i	$-0.809949\pi$
$983$	−51.8570	−1.65398	−0.826991	+	0.562216i	$0.809949\pi$
$984$	0.859219	0.0273909
$985$	0	0
$986$	16.1040	0.512856
$987$	7.07794	0.225293
$988$	6.99029	0.222391
$989$	−3.25253	−0.103425
$990$	0	0
$991$	19.8195	0.629588	0.314794	−	0.949160i	$-0.398065\pi$
$991$	19.8195	0.629588	0.314794	+	0.949160i	$0.398065\pi$
$992$	8.79912	0.279372
$993$	−31.5311	−1.00061
$994$	−87.5613	−2.77727
$995$	0	0
$996$	14.8206	0.469609
$997$	−51.6594	−1.63607	−0.818036	−	0.575168i	$-0.804937\pi$
$997$	−51.6594	−1.63607	−0.818036	+	0.575168i	$0.804937\pi$
$998$	−76.8449	−2.43248
$999$	−11.5389	−0.365074

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9075.2.a.ea.1.10		12
5.2	odd	4		1815.2.c.k.364.20		24
5.3	odd	4		1815.2.c.k.364.5		24
5.4	even	2		9075.2.a.dx.1.3		12
11.7	odd	10		825.2.n.p.676.2		24
11.8	odd	10		825.2.n.p.526.2		24
11.10	odd	2		9075.2.a.dy.1.3		12
55.7	even	20		165.2.s.a.49.10	yes	48
55.8	even	20		165.2.s.a.64.10	yes	48
55.18	even	20		165.2.s.a.49.3	✓	48
55.19	odd	10		825.2.n.o.526.5		24
55.29	odd	10		825.2.n.o.676.5		24
55.32	even	4		1815.2.c.j.364.5		24
55.43	even	4		1815.2.c.j.364.20		24
55.52	even	20		165.2.s.a.64.3	yes	48
55.54	odd	2		9075.2.a.dz.1.10		12
165.8	odd	20		495.2.ba.c.64.3		48
165.62	odd	20		495.2.ba.c.379.3		48
165.107	odd	20		495.2.ba.c.64.10		48
165.128	odd	20		495.2.ba.c.379.10		48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
165.2.s.a.49.3	✓	48	55.18	even	20
165.2.s.a.49.10	yes	48	55.7	even	20
165.2.s.a.64.3	yes	48	55.52	even	20
165.2.s.a.64.10	yes	48	55.8	even	20
495.2.ba.c.64.3		48	165.8	odd	20
495.2.ba.c.64.10		48	165.107	odd	20
495.2.ba.c.379.3		48	165.62	odd	20
495.2.ba.c.379.10		48	165.128	odd	20
825.2.n.o.526.5		24	55.19	odd	10
825.2.n.o.676.5		24	55.29	odd	10
825.2.n.p.526.2		24	11.8	odd	10
825.2.n.p.676.2		24	11.7	odd	10
1815.2.c.j.364.5		24	55.32	even	4
1815.2.c.j.364.20		24	55.43	even	4
1815.2.c.k.364.5		24	5.3	odd	4
1815.2.c.k.364.20		24	5.2	odd	4
9075.2.a.dx.1.3		12	5.4	even	2
9075.2.a.dy.1.3		12	11.10	odd	2
9075.2.a.dz.1.10		12	55.54	odd	2
9075.2.a.ea.1.10		12	1.1	even	1	trivial

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)

\(f(q)\)	\(=\)	\(q+2.10651 q^{2} +1.00000 q^{3} +2.43738 q^{4} +2.10651 q^{6} -3.31633 q^{7} +0.921348 q^{8} +1.00000 q^{9} +2.43738 q^{12} +2.49889 q^{13} -6.98587 q^{14} -2.93393 q^{16} +6.81019 q^{17} +2.10651 q^{18} +1.14769 q^{19} -3.31633 q^{21} +5.89026 q^{23} +0.921348 q^{24} +5.26393 q^{26} +1.00000 q^{27} -8.08315 q^{28} +1.12257 q^{29} -1.09673 q^{31} -8.02306 q^{32} +14.3457 q^{34} +2.43738 q^{36} -11.5389 q^{37} +2.41763 q^{38} +2.49889 q^{39} +0.932568 q^{41} -6.98587 q^{42} -0.552188 q^{43} +12.4079 q^{46} -2.13427 q^{47} -2.93393 q^{48} +3.99802 q^{49} +6.81019 q^{51} +6.09074 q^{52} +11.6356 q^{53} +2.10651 q^{54} -3.05549 q^{56} +1.14769 q^{57} +2.36469 q^{58} +8.39837 q^{59} +8.21313 q^{61} -2.31027 q^{62} -3.31633 q^{63} -11.0328 q^{64} -4.15419 q^{67} +16.5990 q^{68} +5.89026 q^{69} +12.5340 q^{71} +0.921348 q^{72} +5.73761 q^{73} -24.3067 q^{74} +2.79736 q^{76} +5.26393 q^{78} +3.86754 q^{79} +1.00000 q^{81} +1.96446 q^{82} +6.08055 q^{83} -8.08315 q^{84} -1.16319 q^{86} +1.12257 q^{87} +12.5950 q^{89} -8.28712 q^{91} +14.3568 q^{92} -1.09673 q^{93} -4.49586 q^{94} -8.02306 q^{96} -2.86873 q^{97} +8.42187 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 4 q^{2} + 12 q^{3} + 12 q^{4} + 4 q^{6} + 8 q^{7} + 12 q^{8} + 12 q^{9} + 12 q^{12} + 18 q^{13} + 6 q^{14} + 24 q^{16} + 18 q^{17} + 4 q^{18} + 16 q^{19} + 8 q^{21} + 12 q^{24} + 16 q^{26} + 12 q^{27}+ \cdots + 60 q^{98}+O(q^{100}) \) 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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