LMFDB - Embedded newform 9075.2.a.cv.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

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

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

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

chi := DirichletCharacter("9075.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

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

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$72.4642398343$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{3}, \sqrt{11})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 7x^{2} + 4 $ x^4 - 7*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 363)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-2.52434$ 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)

$f(q)$	$=$	\(q-2.52434 q^{2} -1.00000 q^{3} +4.37228 q^{4} +2.52434 q^{6} -0.792287 q^{7} -5.98844 q^{8} +1.00000 q^{9} -4.37228 q^{12} +4.10891 q^{13} +2.00000 q^{14} +6.37228 q^{16} +5.98844 q^{17} -2.52434 q^{18} +4.25639 q^{19} +0.792287 q^{21} -2.00000 q^{23} +5.98844 q^{24} -10.3723 q^{26} -1.00000 q^{27} -3.46410 q^{28} -2.52434 q^{29} +7.37228 q^{31} -4.10891 q^{32} -15.1168 q^{34} +4.37228 q^{36} -5.00000 q^{37} -10.7446 q^{38} -4.10891 q^{39} -5.69349 q^{41} -2.00000 q^{42} +6.63325 q^{43} +5.04868 q^{46} +1.25544 q^{47} -6.37228 q^{48} -6.37228 q^{49} -5.98844 q^{51} +17.9653 q^{52} -13.1168 q^{53} +2.52434 q^{54} +4.74456 q^{56} -4.25639 q^{57} +6.37228 q^{58} -6.00000 q^{59} -2.67181 q^{61} -18.6101 q^{62} -0.792287 q^{63} -2.37228 q^{64} -16.1168 q^{67} +26.1831 q^{68} +2.00000 q^{69} +0.744563 q^{71} -5.98844 q^{72} -7.42554 q^{73} +12.6217 q^{74} +18.6101 q^{76} +10.3723 q^{78} +5.84096 q^{79} +1.00000 q^{81} +14.3723 q^{82} -8.51278 q^{83} +3.46410 q^{84} -16.7446 q^{86} +2.52434 q^{87} -6.37228 q^{89} -3.25544 q^{91} -8.74456 q^{92} -7.37228 q^{93} -3.16915 q^{94} +4.10891 q^{96} +12.4891 q^{97} +16.0858 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{3} + 6 q^{4} + 4 q^{9} - 6 q^{12} + 8 q^{14} + 14 q^{16} - 8 q^{23} - 30 q^{26} - 4 q^{27} + 18 q^{31} - 26 q^{34} + 6 q^{36} - 20 q^{37} - 20 q^{38} - 8 q^{42} + 28 q^{47} - 14 q^{48} - 14 q^{49}+ \cdots + 4 q^{97}+O(q^{100}) $ 4 * q - 4 * q^3 + 6 * q^4 + 4 * q^9 - 6 * q^12 + 8 * q^14 + 14 * q^16 - 8 * q^23 - 30 * q^26 - 4 * q^27 + 18 * q^31 - 26 * q^34 + 6 * q^36 - 20 * q^37 - 20 * q^38 - 8 * q^42 + 28 * q^47 - 14 * q^48 - 14 * q^49 - 18 * q^53 - 4 * q^56 + 14 * q^58 - 24 * q^59 + 2 * q^64 - 30 * q^67 + 8 * q^69 - 20 * q^71 + 30 * q^78 + 4 * q^81 + 46 * q^82 - 44 * q^86 - 14 * q^89 - 36 * q^91 - 12 * q^92 - 18 * q^93 + 4 * q^97

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 $

$2$	−2.52434	−1.78498	−0.892488	−	0.451071i	$-0.851042\pi$
$2$	−2.52434	−1.78498	−0.892488	+	0.451071i	$0.851042\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	4.37228	2.18614
$5$	0	0
$5$	0	0
$6$	2.52434	1.03056
$7$	−0.792287	−0.299456	−0.149728	−	0.988727i	$-0.547840\pi$
$7$	−0.792287	−0.299456	−0.149728	+	0.988727i	$0.547840\pi$
$8$	−5.98844	−2.11723
$9$	1.00000	0.333333
$10$	0	0
$11$	0	0
$11$	0	0
$12$	−4.37228	−1.26217
$13$	4.10891	1.13961	0.569804	−	0.821781i	$-0.307019\pi$
$13$	4.10891	1.13961	0.569804	+	0.821781i	$0.307019\pi$
$14$	2.00000	0.534522
$15$	0	0
$16$	6.37228	1.59307
$17$	5.98844	1.45241	0.726205	−	0.687478i	$-0.241282\pi$
$17$	5.98844	1.45241	0.726205	+	0.687478i	$0.241282\pi$
$18$	−2.52434	−0.594992
$19$	4.25639	0.976483	0.488241	−	0.872709i	$-0.337639\pi$
$19$	4.25639	0.976483	0.488241	+	0.872709i	$0.337639\pi$
$20$	0	0
$21$	0.792287	0.172891
$22$	0	0
$23$	−2.00000	−0.417029	−0.208514	−	0.978019i	$-0.566863\pi$
$23$	−2.00000	−0.417029	−0.208514	+	0.978019i	$0.566863\pi$
$24$	5.98844	1.22239
$25$	0	0
$26$	−10.3723	−2.03417
$27$	−1.00000	−0.192450
$28$	−3.46410	−0.654654
$29$	−2.52434	−0.468758	−0.234379	−	0.972145i	$-0.575306\pi$
$29$	−2.52434	−0.468758	−0.234379	+	0.972145i	$0.575306\pi$
$30$	0	0
$31$	7.37228	1.32410	0.662050	−	0.749459i	$-0.269686\pi$
$31$	7.37228	1.32410	0.662050	+	0.749459i	$0.269686\pi$
$32$	−4.10891	−0.726360
$33$	0	0
$34$	−15.1168	−2.59252
$35$	0	0
$36$	4.37228	0.728714
$37$	−5.00000	−0.821995	−0.410997	−	0.911636i	$-0.634819\pi$
$37$	−5.00000	−0.821995	−0.410997	+	0.911636i	$0.634819\pi$
$38$	−10.7446	−1.74300
$39$	−4.10891	−0.657952
$40$	0	0
$41$	−5.69349	−0.889173	−0.444587	−	0.895736i	$-0.646649\pi$
$41$	−5.69349	−0.889173	−0.444587	+	0.895736i	$0.646649\pi$
$42$	−2.00000	−0.308607
$43$	6.63325	1.01156	0.505781	−	0.862662i	$-0.331205\pi$
$43$	6.63325	1.01156	0.505781	+	0.862662i	$0.331205\pi$
$44$	0	0
$45$	0	0
$46$	5.04868	0.744387
$47$	1.25544	0.183124	0.0915622	−	0.995799i	$-0.470814\pi$
$47$	1.25544	0.183124	0.0915622	+	0.995799i	$0.470814\pi$
$48$	−6.37228	−0.919760
$49$	−6.37228	−0.910326
$50$	0	0
$51$	−5.98844	−0.838549
$52$	17.9653	2.49134
$53$	−13.1168	−1.80174	−0.900869	−	0.434092i	$-0.857069\pi$
$53$	−13.1168	−1.80174	−0.900869	+	0.434092i	$0.857069\pi$
$54$	2.52434	0.343519
$55$	0	0
$56$	4.74456	0.634019
$57$	−4.25639	−0.563772
$58$	6.37228	0.836722
$59$	−6.00000	−0.781133	−0.390567	−	0.920575i	$-0.627721\pi$
$59$	−6.00000	−0.781133	−0.390567	+	0.920575i	$0.627721\pi$
$60$	0	0
$61$	−2.67181	−0.342091	−0.171045	−	0.985263i	$-0.554714\pi$
$61$	−2.67181	−0.342091	−0.171045	+	0.985263i	$0.554714\pi$
$62$	−18.6101	−2.36349
$63$	−0.792287	−0.0998188
$64$	−2.37228	−0.296535
$65$	0	0
$66$	0	0
$67$	−16.1168	−1.96899	−0.984493	−	0.175424i	$-0.943870\pi$
$67$	−16.1168	−1.96899	−0.984493	+	0.175424i	$0.943870\pi$
$68$	26.1831	3.17517
$69$	2.00000	0.240772
$70$	0	0
$71$	0.744563	0.0883633	0.0441817	−	0.999024i	$-0.485932\pi$
$71$	0.744563	0.0883633	0.0441817	+	0.999024i	$0.485932\pi$
$72$	−5.98844	−0.705744
$73$	−7.42554	−0.869093	−0.434547	−	0.900649i	$-0.643091\pi$
$73$	−7.42554	−0.869093	−0.434547	+	0.900649i	$0.643091\pi$
$74$	12.6217	1.46724
$75$	0	0
$76$	18.6101	2.13473
$77$	0	0
$78$	10.3723	1.17443
$79$	5.84096	0.657160	0.328580	−	0.944476i	$-0.393430\pi$
$79$	5.84096	0.657160	0.328580	+	0.944476i	$0.393430\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	14.3723	1.58715
$83$	−8.51278	−0.934399	−0.467199	−	0.884152i	$-0.654737\pi$
$83$	−8.51278	−0.934399	−0.467199	+	0.884152i	$0.654737\pi$
$84$	3.46410	0.377964
$85$	0	0
$86$	−16.7446	−1.80561
$87$	2.52434	0.270637
$88$	0	0
$89$	−6.37228	−0.675460	−0.337730	−	0.941243i	$-0.609659\pi$
$89$	−6.37228	−0.675460	−0.337730	+	0.941243i	$0.609659\pi$
$90$	0	0
$91$	−3.25544	−0.341263
$92$	−8.74456	−0.911684
$93$	−7.37228	−0.764470
$94$	−3.16915	−0.326873
$95$	0	0
$96$	4.10891	0.419364
$97$	12.4891	1.26808	0.634039	−	0.773301i	$-0.281396\pi$
$97$	12.4891	1.26808	0.634039	+	0.773301i	$0.281396\pi$
$98$	16.0858	1.62491
$99$	0	0
$100$	0	0
$101$	−5.34363	−0.531711	−0.265855	−	0.964013i	$-0.585654\pi$
$101$	−5.34363	−0.531711	−0.265855	+	0.964013i	$0.585654\pi$
$102$	15.1168	1.49679
$103$	−8.11684	−0.799776	−0.399888	−	0.916564i	$-0.630951\pi$
$103$	−8.11684	−0.799776	−0.399888	+	0.916564i	$0.630951\pi$
$104$	−24.6060	−2.41281
$105$	0	0
$106$	33.1113	3.21606
$107$	0.294954	0.0285142	0.0142571	−	0.999898i	$-0.495462\pi$
$107$	0.294954	0.0285142	0.0142571	+	0.999898i	$0.495462\pi$
$108$	−4.37228	−0.420723
$109$	9.94987	0.953025	0.476513	−	0.879168i	$-0.341901\pi$
$109$	9.94987	0.953025	0.476513	+	0.879168i	$0.341901\pi$
$110$	0	0
$111$	5.00000	0.474579
$112$	−5.04868	−0.477055
$113$	4.37228	0.411310	0.205655	−	0.978625i	$-0.434068\pi$
$113$	4.37228	0.411310	0.205655	+	0.978625i	$0.434068\pi$
$114$	10.7446	1.00632
$115$	0	0
$116$	−11.0371	−1.02477
$117$	4.10891	0.379869
$118$	15.1460	1.39430
$119$	−4.74456	−0.434933
$120$	0	0
$121$	0	0
$122$	6.74456	0.610624
$123$	5.69349	0.513364
$124$	32.2337	2.89467
$125$	0	0
$126$	2.00000	0.178174
$127$	−19.6974	−1.74786	−0.873929	−	0.486053i	$-0.838436\pi$
$127$	−19.6974	−1.74786	−0.873929	+	0.486053i	$0.838436\pi$
$128$	14.2063	1.25567
$129$	−6.63325	−0.584025
$130$	0	0
$131$	6.63325	0.579550	0.289775	−	0.957095i	$-0.406420\pi$
$131$	6.63325	0.579550	0.289775	+	0.957095i	$0.406420\pi$
$132$	0	0
$133$	−3.37228	−0.292414
$134$	40.6844	3.51459
$135$	0	0
$136$	−35.8614	−3.07509
$137$	−0.744563	−0.0636123	−0.0318061	−	0.999494i	$-0.510126\pi$
$137$	−0.744563	−0.0636123	−0.0318061	+	0.999494i	$0.510126\pi$
$138$	−5.04868	−0.429772
$139$	−10.3923	−0.881464	−0.440732	−	0.897639i	$-0.645281\pi$
$139$	−10.3923	−0.881464	−0.440732	+	0.897639i	$0.645281\pi$
$140$	0	0
$141$	−1.25544	−0.105727
$142$	−1.87953	−0.157726
$143$	0	0
$144$	6.37228	0.531023
$145$	0	0
$146$	18.7446	1.55131
$147$	6.37228	0.525577
$148$	−21.8614	−1.79700
$149$	−5.98844	−0.490592	−0.245296	−	0.969448i	$-0.578885\pi$
$149$	−5.98844	−0.490592	−0.245296	+	0.969448i	$0.578885\pi$
$150$	0	0
$151$	2.17448	0.176957	0.0884784	−	0.996078i	$-0.471800\pi$
$151$	2.17448	0.176957	0.0884784	+	0.996078i	$0.471800\pi$
$152$	−25.4891	−2.06744
$153$	5.98844	0.484137
$154$	0	0
$155$	0	0
$156$	−17.9653	−1.43838
$157$	−6.62772	−0.528950	−0.264475	−	0.964393i	$-0.585199\pi$
$157$	−6.62772	−0.528950	−0.264475	+	0.964393i	$0.585199\pi$
$158$	−14.7446	−1.17301
$159$	13.1168	1.04023
$160$	0	0
$161$	1.58457	0.124882
$162$	−2.52434	−0.198331
$163$	9.37228	0.734094	0.367047	−	0.930202i	$-0.380369\pi$
$163$	9.37228	0.734094	0.367047	+	0.930202i	$0.380369\pi$
$164$	−24.8935	−1.94386
$165$	0	0
$166$	21.4891	1.66788
$167$	2.87419	0.222412	0.111206	−	0.993797i	$-0.464529\pi$
$167$	2.87419	0.222412	0.111206	+	0.993797i	$0.464529\pi$
$168$	−4.74456	−0.366051
$169$	3.88316	0.298704
$170$	0	0
$171$	4.25639	0.325494
$172$	29.0024	2.21141
$173$	6.92820	0.526742	0.263371	−	0.964695i	$-0.415166\pi$
$173$	6.92820	0.526742	0.263371	+	0.964695i	$0.415166\pi$
$174$	−6.37228	−0.483081
$175$	0	0
$176$	0	0
$177$	6.00000	0.450988
$178$	16.0858	1.20568
$179$	−24.2337	−1.81131	−0.905655	−	0.424014i	$-0.860621\pi$
$179$	−24.2337	−1.81131	−0.905655	+	0.424014i	$0.860621\pi$
$180$	0	0
$181$	17.0000	1.26360	0.631800	−	0.775131i	$-0.282316\pi$
$181$	17.0000	1.26360	0.631800	+	0.775131i	$0.282316\pi$
$182$	8.21782	0.609146
$183$	2.67181	0.197506
$184$	11.9769	0.882947
$185$	0	0
$186$	18.6101	1.36456
$187$	0	0
$188$	5.48913	0.400336
$189$	0.792287	0.0576304
$190$	0	0
$191$	−17.4891	−1.26547	−0.632734	−	0.774369i	$-0.718067\pi$
$191$	−17.4891	−1.26547	−0.632734	+	0.774369i	$0.718067\pi$
$192$	2.37228	0.171205
$193$	17.1730	1.23614	0.618071	−	0.786122i	$-0.287914\pi$
$193$	17.1730	1.23614	0.618071	+	0.786122i	$0.287914\pi$
$194$	−31.5268	−2.26349
$195$	0	0
$196$	−27.8614	−1.99010
$197$	−21.1345	−1.50577	−0.752884	−	0.658153i	$-0.771338\pi$
$197$	−21.1345	−1.50577	−0.752884	+	0.658153i	$0.771338\pi$
$198$	0	0
$199$	−6.86141	−0.486392	−0.243196	−	0.969977i	$-0.578196\pi$
$199$	−6.86141	−0.486392	−0.243196	+	0.969977i	$0.578196\pi$
$200$	0	0
$201$	16.1168	1.13679
$202$	13.4891	0.949092
$203$	2.00000	0.140372
$204$	−26.1831	−1.83319
$205$	0	0
$206$	20.4897	1.42758
$207$	−2.00000	−0.139010
$208$	26.1831	1.81547
$209$	0	0
$210$	0	0
$211$	12.4742	0.858760	0.429380	−	0.903124i	$-0.358732\pi$
$211$	12.4742	0.858760	0.429380	+	0.903124i	$0.358732\pi$
$212$	−57.3505	−3.93885
$213$	−0.744563	−0.0510166
$214$	−0.744563	−0.0508973
$215$	0	0
$216$	5.98844	0.407462
$217$	−5.84096	−0.396510
$218$	−25.1168	−1.70113
$219$	7.42554	0.501771
$220$	0	0
$221$	24.6060	1.65518
$222$	−12.6217	−0.847112
$223$	4.86141	0.325544	0.162772	−	0.986664i	$-0.447957\pi$
$223$	4.86141	0.325544	0.162772	+	0.986664i	$0.447957\pi$
$224$	3.25544	0.217513
$225$	0	0
$226$	−11.0371	−0.734178
$227$	−25.2434	−1.67546	−0.837731	−	0.546083i	$-0.816118\pi$
$227$	−25.2434	−1.67546	−0.837731	+	0.546083i	$0.816118\pi$
$228$	−18.6101	−1.23249
$229$	−3.62772	−0.239726	−0.119863	−	0.992790i	$-0.538246\pi$
$229$	−3.62772	−0.239726	−0.119863	+	0.992790i	$0.538246\pi$
$230$	0	0
$231$	0	0
$232$	15.1168	0.992469
$233$	−2.52434	−0.165375	−0.0826874	−	0.996576i	$-0.526350\pi$
$233$	−2.52434	−0.165375	−0.0826874	+	0.996576i	$0.526350\pi$
$234$	−10.3723	−0.678057
$235$	0	0
$236$	−26.2337	−1.70767
$237$	−5.84096	−0.379411
$238$	11.9769	0.776346
$239$	−29.2974	−1.89509	−0.947545	−	0.319622i	$-0.896444\pi$
$239$	−29.2974	−1.89509	−0.947545	+	0.319622i	$0.896444\pi$
$240$	0	0
$241$	−13.2665	−0.854570	−0.427285	−	0.904117i	$-0.640530\pi$
$241$	−13.2665	−0.854570	−0.427285	+	0.904117i	$0.640530\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	−11.6819	−0.747859
$245$	0	0
$246$	−14.3723	−0.916343
$247$	17.4891	1.11281
$248$	−44.1485	−2.80343
$249$	8.51278	0.539475
$250$	0	0
$251$	−0.510875	−0.0322461	−0.0161231	−	0.999870i	$-0.505132\pi$
$251$	−0.510875	−0.0322461	−0.0161231	+	0.999870i	$0.505132\pi$
$252$	−3.46410	−0.218218
$253$	0	0
$254$	49.7228	3.11989
$255$	0	0
$256$	−31.1168	−1.94480
$257$	−24.3723	−1.52030	−0.760151	−	0.649747i	$-0.774875\pi$
$257$	−24.3723	−1.52030	−0.760151	+	0.649747i	$0.774875\pi$
$258$	16.7446	1.04247
$259$	3.96143	0.246152
$260$	0	0
$261$	−2.52434	−0.156253
$262$	−16.7446	−1.03448
$263$	22.3692	1.37934	0.689671	−	0.724122i	$-0.257755\pi$
$263$	22.3692	1.37934	0.689671	+	0.724122i	$0.257755\pi$
$264$	0	0
$265$	0	0
$266$	8.51278	0.521952
$267$	6.37228	0.389977
$268$	−70.4674	−4.30448
$269$	9.11684	0.555864	0.277932	−	0.960601i	$-0.410351\pi$
$269$	9.11684	0.555864	0.277932	+	0.960601i	$0.410351\pi$
$270$	0	0
$271$	−10.3923	−0.631288	−0.315644	−	0.948878i	$-0.602220\pi$
$271$	−10.3923	−0.631288	−0.315644	+	0.948878i	$0.602220\pi$
$272$	38.1600	2.31379
$273$	3.25544	0.197028
$274$	1.87953	0.113546
$275$	0	0
$276$	8.74456	0.526361
$277$	8.66025	0.520344	0.260172	−	0.965562i	$-0.416221\pi$
$277$	8.66025	0.520344	0.260172	+	0.965562i	$0.416221\pi$
$278$	26.2337	1.57339
$279$	7.37228	0.441367
$280$	0	0
$281$	−0.589907	−0.0351909	−0.0175955	−	0.999845i	$-0.505601\pi$
$281$	−0.589907	−0.0351909	−0.0175955	+	0.999845i	$0.505601\pi$
$282$	3.16915	0.188720
$283$	2.37686	0.141290	0.0706449	−	0.997502i	$-0.477494\pi$
$283$	2.37686	0.141290	0.0706449	+	0.997502i	$0.477494\pi$
$284$	3.25544	0.193175
$285$	0	0
$286$	0	0
$287$	4.51087	0.266269
$288$	−4.10891	−0.242120
$289$	18.8614	1.10949
$290$	0	0
$291$	−12.4891	−0.732125
$292$	−32.4665	−1.89996
$293$	−6.57835	−0.384311	−0.192156	−	0.981364i	$-0.561548\pi$
$293$	−6.57835	−0.384311	−0.192156	+	0.981364i	$0.561548\pi$
$294$	−16.0858	−0.938142
$295$	0	0
$296$	29.9422	1.74035
$297$	0	0
$298$	15.1168	0.875695
$299$	−8.21782	−0.475249
$300$	0	0
$301$	−5.25544	−0.302918
$302$	−5.48913	−0.315864
$303$	5.34363	0.306983
$304$	27.1229	1.55561
$305$	0	0
$306$	−15.1168	−0.864172
$307$	17.8178	1.01692	0.508459	−	0.861086i	$-0.330215\pi$
$307$	17.8178	1.01692	0.508459	+	0.861086i	$0.330215\pi$
$308$	0	0
$309$	8.11684	0.461751
$310$	0	0
$311$	−6.51087	−0.369198	−0.184599	−	0.982814i	$-0.559099\pi$
$311$	−6.51087	−0.369198	−0.184599	+	0.982814i	$0.559099\pi$
$312$	24.6060	1.39304
$313$	−9.11684	−0.515314	−0.257657	−	0.966236i	$-0.582951\pi$
$313$	−9.11684	−0.515314	−0.257657	+	0.966236i	$0.582951\pi$
$314$	16.7306	0.944162
$315$	0	0
$316$	25.5383	1.43664
$317$	−15.2554	−0.856831	−0.428415	−	0.903582i	$-0.640928\pi$
$317$	−15.2554	−0.856831	−0.428415	+	0.903582i	$0.640928\pi$
$318$	−33.1113	−1.85679
$319$	0	0
$320$	0	0
$321$	−0.294954	−0.0164627
$322$	−4.00000	−0.222911
$323$	25.4891	1.41825
$324$	4.37228	0.242905
$325$	0	0
$326$	−23.6588	−1.31034
$327$	−9.94987	−0.550229
$328$	34.0951	1.88259
$329$	−0.994667	−0.0548377
$330$	0	0
$331$	−0.627719	−0.0345025	−0.0172513	−	0.999851i	$-0.505492\pi$
$331$	−0.627719	−0.0345025	−0.0172513	+	0.999851i	$0.505492\pi$
$332$	−37.2203	−2.04273
$333$	−5.00000	−0.273998
$334$	−7.25544	−0.397000
$335$	0	0
$336$	5.04868	0.275428
$337$	−15.2935	−0.833090	−0.416545	−	0.909115i	$-0.636759\pi$
$337$	−15.2935	−0.833090	−0.416545	+	0.909115i	$0.636759\pi$
$338$	−9.80240	−0.533180
$339$	−4.37228	−0.237470
$340$	0	0
$341$	0	0
$342$	−10.7446	−0.580999
$343$	10.5947	0.572059
$344$	−39.7228	−2.14171
$345$	0	0
$346$	−17.4891	−0.940221
$347$	18.0202	0.967376	0.483688	−	0.875240i	$-0.339297\pi$
$347$	18.0202	0.967376	0.483688	+	0.875240i	$0.339297\pi$
$348$	11.0371	0.591651
$349$	−25.9808	−1.39072	−0.695359	−	0.718662i	$-0.744755\pi$
$349$	−25.9808	−1.39072	−0.695359	+	0.718662i	$0.744755\pi$
$350$	0	0
$351$	−4.10891	−0.219317
$352$	0	0
$353$	29.3505	1.56217	0.781086	−	0.624424i	$-0.214666\pi$
$353$	29.3505	1.56217	0.781086	+	0.624424i	$0.214666\pi$
$354$	−15.1460	−0.805002
$355$	0	0
$356$	−27.8614	−1.47665
$357$	4.74456	0.251109
$358$	61.1740	3.23315
$359$	−1.28962	−0.0680636	−0.0340318	−	0.999421i	$-0.510835\pi$
$359$	−1.28962	−0.0680636	−0.0340318	+	0.999421i	$0.510835\pi$
$360$	0	0
$361$	−0.883156	−0.0464819
$362$	−42.9137	−2.25550
$363$	0	0
$364$	−14.2337	−0.746048
$365$	0	0
$366$	−6.74456	−0.352544
$367$	23.7228	1.23832	0.619160	−	0.785265i	$-0.287473\pi$
$367$	23.7228	1.23832	0.619160	+	0.785265i	$0.287473\pi$
$368$	−12.7446	−0.664356
$369$	−5.69349	−0.296391
$370$	0	0
$371$	10.3923	0.539542
$372$	−32.2337	−1.67124
$373$	24.4511	1.26603	0.633015	−	0.774140i	$-0.281817\pi$
$373$	24.4511	1.26603	0.633015	+	0.774140i	$0.281817\pi$
$374$	0	0
$375$	0	0
$376$	−7.51811	−0.387717
$377$	−10.3723	−0.534200
$378$	−2.00000	−0.102869
$379$	23.7228	1.21856	0.609280	−	0.792956i	$-0.291459\pi$
$379$	23.7228	1.21856	0.609280	+	0.792956i	$0.291459\pi$
$380$	0	0
$381$	19.6974	1.00913
$382$	44.1485	2.25883
$383$	23.4891	1.20024	0.600119	−	0.799911i	$-0.295120\pi$
$383$	23.4891	1.20024	0.600119	+	0.799911i	$0.295120\pi$
$384$	−14.2063	−0.724960
$385$	0	0
$386$	−43.3505	−2.20648
$387$	6.63325	0.337187
$388$	54.6060	2.77220
$389$	−9.11684	−0.462242	−0.231121	−	0.972925i	$-0.574239\pi$
$389$	−9.11684	−0.462242	−0.231121	+	0.972925i	$0.574239\pi$
$390$	0	0
$391$	−11.9769	−0.605697
$392$	38.1600	1.92737
$393$	−6.63325	−0.334603
$394$	53.3505	2.68776
$395$	0	0
$396$	0	0
$397$	−24.4891	−1.22907	−0.614537	−	0.788888i	$-0.710657\pi$
$397$	−24.4891	−1.22907	−0.614537	+	0.788888i	$0.710657\pi$
$398$	17.3205	0.868199
$399$	3.37228	0.168825
$400$	0	0
$401$	−19.1168	−0.954650	−0.477325	−	0.878727i	$-0.658393\pi$
$401$	−19.1168	−0.954650	−0.477325	+	0.878727i	$0.658393\pi$
$402$	−40.6844	−2.02915
$403$	30.2921	1.50895
$404$	−23.3639	−1.16240
$405$	0	0
$406$	−5.04868	−0.250562
$407$	0	0
$408$	35.8614	1.77540
$409$	0.442430	0.0218768	0.0109384	−	0.999940i	$-0.496518\pi$
$409$	0.442430	0.0218768	0.0109384	+	0.999940i	$0.496518\pi$
$410$	0	0
$411$	0.744563	0.0367266
$412$	−35.4891	−1.74842
$413$	4.75372	0.233915
$414$	5.04868	0.248129
$415$	0	0
$416$	−16.8832	−0.827765
$417$	10.3923	0.508913
$418$	0	0
$419$	−8.51087	−0.415783	−0.207892	−	0.978152i	$-0.566660\pi$
$419$	−8.51087	−0.415783	−0.207892	+	0.978152i	$0.566660\pi$
$420$	0	0
$421$	6.13859	0.299177	0.149588	−	0.988748i	$-0.452205\pi$
$421$	6.13859	0.299177	0.149588	+	0.988748i	$0.452205\pi$
$422$	−31.4891	−1.53287
$423$	1.25544	0.0610415
$424$	78.5494	3.81470
$425$	0	0
$426$	1.87953	0.0910634
$427$	2.11684	0.102441
$428$	1.28962	0.0623362
$429$	0	0
$430$	0	0
$431$	23.6588	1.13960	0.569802	−	0.821782i	$-0.307020\pi$
$431$	23.6588	1.13960	0.569802	+	0.821782i	$0.307020\pi$
$432$	−6.37228	−0.306587
$433$	23.2337	1.11654	0.558270	−	0.829659i	$-0.311465\pi$
$433$	23.2337	1.11654	0.558270	+	0.829659i	$0.311465\pi$
$434$	14.7446	0.707762
$435$	0	0
$436$	43.5036	2.08345
$437$	−8.51278	−0.407221
$438$	−18.7446	−0.895650
$439$	24.4511	1.16699	0.583493	−	0.812118i	$-0.301685\pi$
$439$	24.4511	1.16699	0.583493	+	0.812118i	$0.301685\pi$
$440$	0	0
$441$	−6.37228	−0.303442
$442$	−62.1138	−2.95445
$443$	−20.7446	−0.985604	−0.492802	−	0.870142i	$-0.664027\pi$
$443$	−20.7446	−0.985604	−0.492802	+	0.870142i	$0.664027\pi$
$444$	21.8614	1.03750
$445$	0	0
$446$	−12.2718	−0.581088
$447$	5.98844	0.283243
$448$	1.87953	0.0887993
$449$	24.6060	1.16123	0.580614	−	0.814179i	$-0.302813\pi$
$449$	24.6060	1.16123	0.580614	+	0.814179i	$0.302813\pi$
$450$	0	0
$451$	0	0
$452$	19.1168	0.899181
$453$	−2.17448	−0.102166
$454$	63.7228	2.99066
$455$	0	0
$456$	25.4891	1.19364
$457$	8.45787	0.395643	0.197821	−	0.980238i	$-0.436613\pi$
$457$	8.45787	0.395643	0.197821	+	0.980238i	$0.436613\pi$
$458$	9.15759	0.427906
$459$	−5.98844	−0.279516
$460$	0	0
$461$	14.5012	0.675389	0.337694	−	0.941256i	$-0.390353\pi$
$461$	14.5012	0.675389	0.337694	+	0.941256i	$0.390353\pi$
$462$	0	0
$463$	3.25544	0.151293	0.0756465	−	0.997135i	$-0.475898\pi$
$463$	3.25544	0.151293	0.0756465	+	0.997135i	$0.475898\pi$
$464$	−16.0858	−0.746764
$465$	0	0
$466$	6.37228	0.295190
$467$	22.2337	1.02885	0.514426	−	0.857535i	$-0.328005\pi$
$467$	22.2337	1.02885	0.514426	+	0.857535i	$0.328005\pi$
$468$	17.9653	0.830447
$469$	12.7692	0.589625
$470$	0	0
$471$	6.62772	0.305389
$472$	35.9306	1.65384
$473$	0	0
$474$	14.7446	0.677240
$475$	0	0
$476$	−20.7446	−0.950825
$477$	−13.1168	−0.600579
$478$	73.9565	3.38269
$479$	37.5152	1.71411	0.857057	−	0.515222i	$-0.172290\pi$
$479$	37.5152	1.71411	0.857057	+	0.515222i	$0.172290\pi$
$480$	0	0
$481$	−20.5446	−0.936751
$482$	33.4891	1.52539
$483$	−1.58457	−0.0721006
$484$	0	0
$485$	0	0
$486$	2.52434	0.114506
$487$	12.7446	0.577511	0.288756	−	0.957403i	$-0.406758\pi$
$487$	12.7446	0.577511	0.288756	+	0.957403i	$0.406758\pi$
$488$	16.0000	0.724286
$489$	−9.37228	−0.423829
$490$	0	0
$491$	18.6101	0.839863	0.419932	−	0.907556i	$-0.362054\pi$
$491$	18.6101	0.839863	0.419932	+	0.907556i	$0.362054\pi$
$492$	24.8935	1.12229
$493$	−15.1168	−0.680828
$494$	−44.1485	−1.98633
$495$	0	0
$496$	46.9783	2.10939
$497$	−0.589907	−0.0264610
$498$	−21.4891	−0.962951
$499$	−37.8397	−1.69394	−0.846968	−	0.531644i	$-0.821574\pi$
$499$	−37.8397	−1.69394	−0.846968	+	0.531644i	$0.821574\pi$
$500$	0	0
$501$	−2.87419	−0.128410
$502$	1.28962	0.0575586
$503$	−22.0742	−0.984241	−0.492121	−	0.870527i	$-0.663778\pi$
$503$	−22.0742	−0.984241	−0.492121	+	0.870527i	$0.663778\pi$
$504$	4.74456	0.211340
$505$	0	0
$506$	0	0
$507$	−3.88316	−0.172457
$508$	−86.1224	−3.82107
$509$	−40.9783	−1.81633	−0.908165	−	0.418613i	$-0.862516\pi$
$509$	−40.9783	−1.81633	−0.908165	+	0.418613i	$0.862516\pi$
$510$	0	0
$511$	5.88316	0.260256
$512$	50.1369	2.21576
$513$	−4.25639	−0.187924
$514$	61.5239	2.71370
$515$	0	0
$516$	−29.0024	−1.27676
$517$	0	0
$518$	−10.0000	−0.439375
$519$	−6.92820	−0.304114
$520$	0	0
$521$	16.9783	0.743831	0.371915	−	0.928267i	$-0.378701\pi$
$521$	16.9783	0.743831	0.371915	+	0.928267i	$0.378701\pi$
$522$	6.37228	0.278907
$523$	6.83563	0.298901	0.149451	−	0.988769i	$-0.452250\pi$
$523$	6.83563	0.298901	0.149451	+	0.988769i	$0.452250\pi$
$524$	29.0024	1.26698
$525$	0	0
$526$	−56.4674	−2.46209
$527$	44.1485	1.92314
$528$	0	0
$529$	−19.0000	−0.826087
$530$	0	0
$531$	−6.00000	−0.260378
$532$	−14.7446	−0.639258
$533$	−23.3940	−1.01331
$534$	−16.0858	−0.696100
$535$	0	0
$536$	96.5147	4.16880
$537$	24.2337	1.04576
$538$	−23.0140	−0.992204
$539$	0	0
$540$	0	0
$541$	−27.7128	−1.19147	−0.595733	−	0.803182i	$-0.703139\pi$
$541$	−27.7128	−1.19147	−0.595733	+	0.803182i	$0.703139\pi$
$542$	26.2337	1.12683
$543$	−17.0000	−0.729540
$544$	−24.6060	−1.05497
$545$	0	0
$546$	−8.21782	−0.351690
$547$	−41.9740	−1.79468	−0.897339	−	0.441342i	$-0.854502\pi$
$547$	−41.9740	−1.79468	−0.897339	+	0.441342i	$0.854502\pi$
$548$	−3.25544	−0.139065
$549$	−2.67181	−0.114030
$550$	0	0
$551$	−10.7446	−0.457734
$552$	−11.9769	−0.509770
$553$	−4.62772	−0.196791
$554$	−21.8614	−0.928802
$555$	0	0
$556$	−45.4381	−1.92700
$557$	−9.50744	−0.402843	−0.201422	−	0.979505i	$-0.564556\pi$
$557$	−9.50744	−0.402843	−0.201422	+	0.979505i	$0.564556\pi$
$558$	−18.6101	−0.787830
$559$	27.2554	1.15278
$560$	0	0
$561$	0	0
$562$	1.48913	0.0628150
$563$	−14.5561	−0.613467	−0.306734	−	0.951795i	$-0.599236\pi$
$563$	−14.5561	−0.613467	−0.306734	+	0.951795i	$0.599236\pi$
$564$	−5.48913	−0.231134
$565$	0	0
$566$	−6.00000	−0.252199
$567$	−0.792287	−0.0332729
$568$	−4.45877	−0.187086
$569$	24.5437	1.02892	0.514462	−	0.857513i	$-0.327992\pi$
$569$	24.5437	1.02892	0.514462	+	0.857513i	$0.327992\pi$
$570$	0	0
$571$	4.55134	0.190468	0.0952339	−	0.995455i	$-0.469640\pi$
$571$	4.55134	0.190468	0.0952339	+	0.995455i	$0.469640\pi$
$572$	0	0
$573$	17.4891	0.730619
$574$	−11.3870	−0.475283
$575$	0	0
$576$	−2.37228	−0.0988451
$577$	16.4891	0.686451	0.343226	−	0.939253i	$-0.388480\pi$
$577$	16.4891	0.686451	0.343226	+	0.939253i	$0.388480\pi$
$578$	−47.6126	−1.98042
$579$	−17.1730	−0.713687
$580$	0	0
$581$	6.74456	0.279812
$582$	31.5268	1.30683
$583$	0	0
$584$	44.4674	1.84007
$585$	0	0
$586$	16.6060	0.685986
$587$	−38.9783	−1.60880	−0.804402	−	0.594085i	$-0.797514\pi$
$587$	−38.9783	−1.60880	−0.804402	+	0.594085i	$0.797514\pi$
$588$	27.8614	1.14899
$589$	31.3793	1.29296
$590$	0	0
$591$	21.1345	0.869356
$592$	−31.8614	−1.30950
$593$	−27.7677	−1.14028	−0.570142	−	0.821546i	$-0.693112\pi$
$593$	−27.7677	−1.14028	−0.570142	+	0.821546i	$0.693112\pi$
$594$	0	0
$595$	0	0
$596$	−26.1831	−1.07250
$597$	6.86141	0.280819
$598$	20.7446	0.848308
$599$	40.4674	1.65345	0.826726	−	0.562605i	$-0.190201\pi$
$599$	40.4674	1.65345	0.826726	+	0.562605i	$0.190201\pi$
$600$	0	0
$601$	−40.2419	−1.64150	−0.820751	−	0.571286i	$-0.806445\pi$
$601$	−40.2419	−1.64150	−0.820751	+	0.571286i	$0.806445\pi$
$602$	13.2665	0.540702
$603$	−16.1168	−0.656329
$604$	9.50744	0.386852
$605$	0	0
$606$	−13.4891	−0.547958
$607$	−26.1282	−1.06051	−0.530256	−	0.847837i	$-0.677904\pi$
$607$	−26.1282	−1.06051	−0.530256	+	0.847837i	$0.677904\pi$
$608$	−17.4891	−0.709278
$609$	−2.00000	−0.0810441
$610$	0	0
$611$	5.15848	0.208690
$612$	26.1831	1.05839
$613$	43.4111	1.75336	0.876678	−	0.481077i	$-0.159754\pi$
$613$	43.4111	1.75336	0.876678	+	0.481077i	$0.159754\pi$
$614$	−44.9783	−1.81517
$615$	0	0
$616$	0	0
$617$	−43.8614	−1.76579	−0.882897	−	0.469567i	$-0.844410\pi$
$617$	−43.8614	−1.76579	−0.882897	+	0.469567i	$0.844410\pi$
$618$	−20.4897	−0.824215
$619$	10.2337	0.411327	0.205663	−	0.978623i	$-0.434065\pi$
$619$	10.2337	0.411327	0.205663	+	0.978623i	$0.434065\pi$
$620$	0	0
$621$	2.00000	0.0802572
$622$	16.4356	0.659009
$623$	5.04868	0.202271
$624$	−26.1831	−1.04816
$625$	0	0
$626$	23.0140	0.919824
$627$	0	0
$628$	−28.9783	−1.15636
$629$	−29.9422	−1.19387
$630$	0	0
$631$	22.2337	0.885109	0.442555	−	0.896742i	$-0.354072\pi$
$631$	22.2337	0.885109	0.442555	+	0.896742i	$0.354072\pi$
$632$	−34.9783	−1.39136
$633$	−12.4742	−0.495805
$634$	38.5099	1.52942
$635$	0	0
$636$	57.3505	2.27410
$637$	−26.1831	−1.03741
$638$	0	0
$639$	0.744563	0.0294544
$640$	0	0
$641$	7.11684	0.281098	0.140549	−	0.990074i	$-0.455113\pi$
$641$	7.11684	0.281098	0.140549	+	0.990074i	$0.455113\pi$
$642$	0.744563	0.0293855
$643$	−21.0951	−0.831909	−0.415955	−	0.909385i	$-0.636553\pi$
$643$	−21.0951	−0.831909	−0.415955	+	0.909385i	$0.636553\pi$
$644$	6.92820	0.273009
$645$	0	0
$646$	−64.3432	−2.53155
$647$	12.0000	0.471769	0.235884	−	0.971781i	$-0.424201\pi$
$647$	12.0000	0.471769	0.235884	+	0.971781i	$0.424201\pi$
$648$	−5.98844	−0.235248
$649$	0	0
$650$	0	0
$651$	5.84096	0.228925
$652$	40.9783	1.60483
$653$	−4.51087	−0.176524	−0.0882621	−	0.996097i	$-0.528131\pi$
$653$	−4.51087	−0.176524	−0.0882621	+	0.996097i	$0.528131\pi$
$654$	25.1168	0.982146
$655$	0	0
$656$	−36.2805	−1.41652
$657$	−7.42554	−0.289698
$658$	2.51087	0.0978841
$659$	22.3692	0.871380	0.435690	−	0.900097i	$-0.356504\pi$
$659$	22.3692	0.871380	0.435690	+	0.900097i	$0.356504\pi$
$660$	0	0
$661$	−26.7228	−1.03940	−0.519698	−	0.854350i	$-0.673956\pi$
$661$	−26.7228	−1.03940	−0.519698	+	0.854350i	$0.673956\pi$
$662$	1.58457	0.0615862
$663$	−24.6060	−0.955617
$664$	50.9783	1.97834
$665$	0	0
$666$	12.6217	0.489081
$667$	5.04868	0.195485
$668$	12.5668	0.486224
$669$	−4.86141	−0.187953
$670$	0	0
$671$	0	0
$672$	−3.25544	−0.125581
$673$	−41.4766	−1.59881	−0.799404	−	0.600794i	$-0.794851\pi$
$673$	−41.4766	−1.59881	−0.799404	+	0.600794i	$0.794851\pi$
$674$	38.6060	1.48705
$675$	0	0
$676$	16.9783	0.653010
$677$	−36.2805	−1.39437	−0.697186	−	0.716890i	$-0.745565\pi$
$677$	−36.2805	−1.39437	−0.697186	+	0.716890i	$0.745565\pi$
$678$	11.0371	0.423878
$679$	−9.89497	−0.379734
$680$	0	0
$681$	25.2434	0.967328
$682$	0	0
$683$	−2.00000	−0.0765279	−0.0382639	−	0.999268i	$-0.512183\pi$
$683$	−2.00000	−0.0765279	−0.0382639	+	0.999268i	$0.512183\pi$
$684$	18.6101	0.711576
$685$	0	0
$686$	−26.7446	−1.02111
$687$	3.62772	0.138406
$688$	42.2689	1.61149
$689$	−53.8960	−2.05327
$690$	0	0
$691$	−4.11684	−0.156612	−0.0783061	−	0.996929i	$-0.524951\pi$
$691$	−4.11684	−0.156612	−0.0783061	+	0.996929i	$0.524951\pi$
$692$	30.2921	1.15153
$693$	0	0
$694$	−45.4891	−1.72674
$695$	0	0
$696$	−15.1168	−0.573002
$697$	−34.0951	−1.29144
$698$	65.5842	2.48240
$699$	2.52434	0.0954792
$700$	0	0
$701$	10.0424	0.379298	0.189649	−	0.981852i	$-0.439265\pi$
$701$	10.0424	0.379298	0.189649	+	0.981852i	$0.439265\pi$
$702$	10.3723	0.391477
$703$	−21.2819	−0.802664
$704$	0	0
$705$	0	0
$706$	−74.0907	−2.78844
$707$	4.23369	0.159224
$708$	26.2337	0.985922
$709$	6.00000	0.225335	0.112667	−	0.993633i	$-0.464061\pi$
$709$	6.00000	0.225335	0.112667	+	0.993633i	$0.464061\pi$
$710$	0	0
$711$	5.84096	0.219053
$712$	38.1600	1.43011
$713$	−14.7446	−0.552188
$714$	−11.9769	−0.448223
$715$	0	0
$716$	−105.957	−3.95978
$717$	29.2974	1.09413
$718$	3.25544	0.121492
$719$	−6.00000	−0.223762	−0.111881	−	0.993722i	$-0.535688\pi$
$719$	−6.00000	−0.223762	−0.111881	+	0.993722i	$0.535688\pi$
$720$	0	0
$721$	6.43087	0.239498
$722$	2.22938	0.0829691
$723$	13.2665	0.493386
$724$	74.3288	2.76241
$725$	0	0
$726$	0	0
$727$	−7.25544	−0.269089	−0.134545	−	0.990908i	$-0.542957\pi$
$727$	−7.25544	−0.269089	−0.134545	+	0.990908i	$0.542957\pi$
$728$	19.4950	0.722532
$729$	1.00000	0.0370370
$730$	0	0
$731$	39.7228	1.46920
$732$	11.6819	0.431776
$733$	9.15759	0.338243	0.169122	−	0.985595i	$-0.445907\pi$
$733$	9.15759	0.338243	0.169122	+	0.985595i	$0.445907\pi$
$734$	−59.8844	−2.21037
$735$	0	0
$736$	8.21782	0.302913
$737$	0	0
$738$	14.3723	0.529051
$739$	−38.8974	−1.43086	−0.715432	−	0.698682i	$-0.753770\pi$
$739$	−38.8974	−1.43086	−0.715432	+	0.698682i	$0.753770\pi$
$740$	0	0
$741$	−17.4891	−0.642479
$742$	−26.2337	−0.963069
$743$	24.6535	0.904448	0.452224	−	0.891904i	$-0.350631\pi$
$743$	24.6535	0.904448	0.452224	+	0.891904i	$0.350631\pi$
$744$	44.1485	1.61856
$745$	0	0
$746$	−61.7228	−2.25983
$747$	−8.51278	−0.311466
$748$	0	0
$749$	−0.233688	−0.00853877
$750$	0	0
$751$	−16.3505	−0.596639	−0.298320	−	0.954466i	$-0.596426\pi$
$751$	−16.3505	−0.596639	−0.298320	+	0.954466i	$0.596426\pi$
$752$	8.00000	0.291730
$753$	0.510875	0.0186173
$754$	26.1831	0.953534
$755$	0	0
$756$	3.46410	0.125988
$757$	−9.51087	−0.345679	−0.172839	−	0.984950i	$-0.555294\pi$
$757$	−9.51087	−0.345679	−0.172839	+	0.984950i	$0.555294\pi$
$758$	−59.8844	−2.17510
$759$	0	0
$760$	0	0
$761$	40.6295	1.47282	0.736408	−	0.676537i	$-0.236520\pi$
$761$	40.6295	1.47282	0.736408	+	0.676537i	$0.236520\pi$
$762$	−49.7228	−1.80127
$763$	−7.88316	−0.285389
$764$	−76.4674	−2.76649
$765$	0	0
$766$	−59.2945	−2.14240
$767$	−24.6535	−0.890185
$768$	31.1168	1.12283
$769$	34.7885	1.25451	0.627253	−	0.778816i	$-0.284179\pi$
$769$	34.7885	1.25451	0.627253	+	0.778816i	$0.284179\pi$
$770$	0	0
$771$	24.3723	0.877746
$772$	75.0853	2.70238
$773$	−33.2119	−1.19455	−0.597275	−	0.802036i	$-0.703750\pi$
$773$	−33.2119	−1.19455	−0.597275	+	0.802036i	$0.703750\pi$
$774$	−16.7446	−0.601871
$775$	0	0
$776$	−74.7904	−2.68482
$777$	−3.96143	−0.142116
$778$	23.0140	0.825092
$779$	−24.2337	−0.868262
$780$	0	0
$781$	0	0
$782$	30.2337	1.08115
$783$	2.52434	0.0902125
$784$	−40.6060	−1.45021
$785$	0	0
$786$	16.7446	0.597259
$787$	4.75372	0.169452	0.0847259	−	0.996404i	$-0.472999\pi$
$787$	4.75372	0.169452	0.0847259	+	0.996404i	$0.472999\pi$
$788$	−92.4058	−3.29182
$789$	−22.3692	−0.796364
$790$	0	0
$791$	−3.46410	−0.123169
$792$	0	0
$793$	−10.9783	−0.389849
$794$	61.8188	2.19387
$795$	0	0
$796$	−30.0000	−1.06332
$797$	−6.00000	−0.212531	−0.106265	−	0.994338i	$-0.533889\pi$
$797$	−6.00000	−0.212531	−0.106265	+	0.994338i	$0.533889\pi$
$798$	−8.51278	−0.301349
$799$	7.51811	0.265972
$800$	0	0
$801$	−6.37228	−0.225153
$802$	48.2574	1.70403
$803$	0	0
$804$	70.4674	2.48519
$805$	0	0
$806$	−76.4674	−2.69345
$807$	−9.11684	−0.320928
$808$	32.0000	1.12576
$809$	−53.6559	−1.88644	−0.943221	−	0.332167i	$-0.892220\pi$
$809$	−53.6559	−1.88644	−0.943221	+	0.332167i	$0.892220\pi$
$810$	0	0
$811$	44.0559	1.54701	0.773506	−	0.633789i	$-0.218501\pi$
$811$	44.0559	1.54701	0.773506	+	0.633789i	$0.218501\pi$
$812$	8.74456	0.306874
$813$	10.3923	0.364474
$814$	0	0
$815$	0	0
$816$	−38.1600	−1.33587
$817$	28.2337	0.987772
$818$	−1.11684	−0.0390495
$819$	−3.25544	−0.113754
$820$	0	0
$821$	−12.8617	−0.448878	−0.224439	−	0.974488i	$-0.572055\pi$
$821$	−12.8617	−0.448878	−0.224439	+	0.974488i	$0.572055\pi$
$822$	−1.87953	−0.0655561
$823$	32.3505	1.12767	0.563834	−	0.825888i	$-0.309326\pi$
$823$	32.3505	1.12767	0.563834	+	0.825888i	$0.309326\pi$
$824$	48.6072	1.69331
$825$	0	0
$826$	−12.0000	−0.417533
$827$	16.1407	0.561267	0.280633	−	0.959815i	$-0.409455\pi$
$827$	16.1407	0.561267	0.280633	+	0.959815i	$0.409455\pi$
$828$	−8.74456	−0.303895
$829$	−22.2554	−0.772963	−0.386482	−	0.922297i	$-0.626310\pi$
$829$	−22.2554	−0.772963	−0.386482	+	0.922297i	$0.626310\pi$
$830$	0	0
$831$	−8.66025	−0.300421
$832$	−9.74749	−0.337934
$833$	−38.1600	−1.32217
$834$	−26.2337	−0.908398
$835$	0	0
$836$	0	0
$837$	−7.37228	−0.254823
$838$	21.4843	0.742164
$839$	26.9783	0.931393	0.465696	−	0.884945i	$-0.345804\pi$
$839$	26.9783	0.931393	0.465696	+	0.884945i	$0.345804\pi$
$840$	0	0
$841$	−22.6277	−0.780266
$842$	−15.4959	−0.534023
$843$	0.589907	0.0203175
$844$	54.5408	1.87737
$845$	0	0
$846$	−3.16915	−0.108958
$847$	0	0
$848$	−83.5842	−2.87029
$849$	−2.37686	−0.0815737
$850$	0	0
$851$	10.0000	0.342796
$852$	−3.25544	−0.111529
$853$	−31.7292	−1.08639	−0.543193	−	0.839608i	$-0.682785\pi$
$853$	−31.7292	−1.08639	−0.543193	+	0.839608i	$0.682785\pi$
$854$	−5.34363	−0.182855
$855$	0	0
$856$	−1.76631	−0.0603713
$857$	0	0		−	1.00000i	$-0.5\pi$
$857$	0	0			1.00000i	$0.5\pi$
$858$	0	0
$859$	−12.1168	−0.413421	−0.206711	−	0.978402i	$-0.566276\pi$
$859$	−12.1168	−0.413421	−0.206711	+	0.978402i	$0.566276\pi$
$860$	0	0
$861$	−4.51087	−0.153730
$862$	−59.7228	−2.03417
$863$	−16.5109	−0.562037	−0.281018	−	0.959702i	$-0.590672\pi$
$863$	−16.5109	−0.562037	−0.281018	+	0.959702i	$0.590672\pi$
$864$	4.10891	0.139788
$865$	0	0
$866$	−58.6497	−1.99300
$867$	−18.8614	−0.640567
$868$	−25.5383	−0.866827
$869$	0	0
$870$	0	0
$871$	−66.2227	−2.24387
$872$	−59.5842	−2.01778
$873$	12.4891	0.422693
$874$	21.4891	0.726881
$875$	0	0
$876$	32.4665	1.09694
$877$	−25.3909	−0.857388	−0.428694	−	0.903450i	$-0.641026\pi$
$877$	−25.3909	−0.857388	−0.428694	+	0.903450i	$0.641026\pi$
$878$	−61.7228	−2.08304
$879$	6.57835	0.221882
$880$	0	0
$881$	−5.39403	−0.181730	−0.0908648	−	0.995863i	$-0.528963\pi$
$881$	−5.39403	−0.181730	−0.0908648	+	0.995863i	$0.528963\pi$
$882$	16.0858	0.541637
$883$	4.86141	0.163599	0.0817997	−	0.996649i	$-0.473933\pi$
$883$	4.86141	0.163599	0.0817997	+	0.996649i	$0.473933\pi$
$884$	107.584	3.61845
$885$	0	0
$886$	52.3663	1.75928
$887$	−29.4072	−0.987397	−0.493698	−	0.869633i	$-0.664355\pi$
$887$	−29.4072	−0.987397	−0.493698	+	0.869633i	$0.664355\pi$
$888$	−29.9422	−1.00479
$889$	15.6060	0.523407
$890$	0	0
$891$	0	0
$892$	21.2554	0.711685
$893$	5.34363	0.178818
$894$	−15.1168	−0.505583
$895$	0	0
$896$	−11.2554	−0.376018
$897$	8.21782	0.274385
$898$	−62.1138	−2.07276
$899$	−18.6101	−0.620683
$900$	0	0
$901$	−78.5494	−2.61686
$902$	0	0
$903$	5.25544	0.174890
$904$	−26.1831	−0.870838
$905$	0	0
$906$	5.48913	0.182364
$907$	36.3505	1.20700	0.603500	−	0.797363i	$-0.293772\pi$
$907$	36.3505	1.20700	0.603500	+	0.797363i	$0.293772\pi$
$908$	−110.371	−3.66280
$909$	−5.34363	−0.177237
$910$	0	0
$911$	38.2337	1.26674	0.633369	−	0.773850i	$-0.281671\pi$
$911$	38.2337	1.26674	0.633369	+	0.773850i	$0.281671\pi$
$912$	−27.1229	−0.898129
$913$	0	0
$914$	−21.3505	−0.706213
$915$	0	0
$916$	−15.8614	−0.524076
$917$	−5.25544	−0.173550
$918$	15.1168	0.498930
$919$	−31.6742	−1.04484	−0.522419	−	0.852689i	$-0.674970\pi$
$919$	−31.6742	−1.04484	−0.522419	+	0.852689i	$0.674970\pi$
$920$	0	0
$921$	−17.8178	−0.587118
$922$	−36.6060	−1.20555
$923$	3.05934	0.100699
$924$	0	0
$925$	0	0
$926$	−8.21782	−0.270054
$927$	−8.11684	−0.266592
$928$	10.3723	0.340487
$929$	−34.8832	−1.14448	−0.572240	−	0.820086i	$-0.693925\pi$
$929$	−34.8832	−1.14448	−0.572240	+	0.820086i	$0.693925\pi$
$930$	0	0
$931$	−27.1229	−0.888917
$932$	−11.0371	−0.361533
$933$	6.51087	0.213156
$934$	−56.1253	−1.83648
$935$	0	0
$936$	−24.6060	−0.804271
$937$	−40.2419	−1.31465	−0.657323	−	0.753609i	$-0.728311\pi$
$937$	−40.2419	−1.31465	−0.657323	+	0.753609i	$0.728311\pi$
$938$	−32.2337	−1.05247
$939$	9.11684	0.297517
$940$	0	0
$941$	16.3807	0.533997	0.266999	−	0.963697i	$-0.413968\pi$
$941$	16.3807	0.533997	0.266999	+	0.963697i	$0.413968\pi$
$942$	−16.7306	−0.545112
$943$	11.3870	0.370811
$944$	−38.2337	−1.24440
$945$	0	0
$946$	0	0
$947$	31.4891	1.02326	0.511630	−	0.859206i	$-0.329042\pi$
$947$	31.4891	1.02326	0.511630	+	0.859206i	$0.329042\pi$
$948$	−25.5383	−0.829446
$949$	−30.5109	−0.990425
$950$	0	0
$951$	15.2554	0.494691
$952$	28.4125	0.920855
$953$	47.2627	1.53099	0.765495	−	0.643442i	$-0.222494\pi$
$953$	47.2627	1.53099	0.765495	+	0.643442i	$0.222494\pi$
$954$	33.1113	1.07202
$955$	0	0
$956$	−128.096	−4.14293
$957$	0	0
$958$	−94.7011	−3.05965
$959$	0.589907	0.0190491
$960$	0	0
$961$	23.3505	0.753243
$962$	51.8614	1.67208
$963$	0.294954	0.00950475
$964$	−58.0049	−1.86821
$965$	0	0
$966$	4.00000	0.128698
$967$	−44.3508	−1.42623	−0.713113	−	0.701049i	$-0.752716\pi$
$967$	−44.3508	−1.42623	−0.713113	+	0.701049i	$0.752716\pi$
$968$	0	0
$969$	−25.4891	−0.818829
$970$	0	0
$971$	59.4891	1.90910	0.954548	−	0.298056i	$-0.0963382\pi$
$971$	59.4891	1.90910	0.954548	+	0.298056i	$0.0963382\pi$
$972$	−4.37228	−0.140241
$973$	8.23369	0.263960
$974$	−32.1716	−1.03084
$975$	0	0
$976$	−17.0256	−0.544975
$977$	−13.1168	−0.419645	−0.209823	−	0.977739i	$-0.567289\pi$
$977$	−13.1168	−0.419645	−0.209823	+	0.977739i	$0.567289\pi$
$978$	23.6588	0.756525
$979$	0	0
$980$	0	0
$981$	9.94987	0.317675
$982$	−46.9783	−1.49914
$983$	56.2337	1.79358	0.896788	−	0.442460i	$-0.145894\pi$
$983$	56.2337	1.79358	0.896788	+	0.442460i	$0.145894\pi$
$984$	−34.0951	−1.08691
$985$	0	0
$986$	38.1600	1.21526
$987$	0.994667	0.0316606
$988$	76.4674	2.43275
$989$	−13.2665	−0.421850
$990$	0	0
$991$	34.5109	1.09627	0.548137	−	0.836389i	$-0.315337\pi$
$991$	34.5109	1.09627	0.548137	+	0.836389i	$0.315337\pi$
$992$	−30.2921	−0.961774
$993$	0.627719	0.0199201
$994$	1.48913	0.0472322
$995$	0	0
$996$	37.2203	1.17937
$997$	−48.4598	−1.53474	−0.767368	−	0.641207i	$-0.778434\pi$
$997$	−48.4598	−1.53474	−0.767368	+	0.641207i	$0.778434\pi$
$998$	95.5201	3.02364
$999$	5.00000	0.158193

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9075.2.a.cv.1.1		4
5.4	even	2		363.2.a.j.1.4	yes	4
11.10	odd	2	inner	9075.2.a.cv.1.4		4
15.14	odd	2		1089.2.a.u.1.1		4
20.19	odd	2		5808.2.a.ck.1.1		4
55.4	even	10		363.2.e.n.148.4		16
55.9	even	10		363.2.e.n.202.1		16
55.14	even	10		363.2.e.n.130.4		16
55.19	odd	10		363.2.e.n.130.1		16
55.24	odd	10		363.2.e.n.202.4		16
55.29	odd	10		363.2.e.n.148.1		16
55.39	odd	10		363.2.e.n.124.4		16
55.49	even	10		363.2.e.n.124.1		16
55.54	odd	2		363.2.a.j.1.1	✓	4
165.164	even	2		1089.2.a.u.1.4		4
220.219	even	2		5808.2.a.ck.1.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
363.2.a.j.1.1	✓	4	55.54	odd	2
363.2.a.j.1.4	yes	4	5.4	even	2
363.2.e.n.124.1		16	55.49	even	10
363.2.e.n.124.4		16	55.39	odd	10
363.2.e.n.130.1		16	55.19	odd	10
363.2.e.n.130.4		16	55.14	even	10
363.2.e.n.148.1		16	55.29	odd	10
363.2.e.n.148.4		16	55.4	even	10
363.2.e.n.202.1		16	55.9	even	10
363.2.e.n.202.4		16	55.24	odd	10
1089.2.a.u.1.1		4	15.14	odd	2
1089.2.a.u.1.4		4	165.164	even	2
5808.2.a.ck.1.1		4	20.19	odd	2
5808.2.a.ck.1.2		4	220.219	even	2
9075.2.a.cv.1.1		4	1.1	even	1	trivial
9075.2.a.cv.1.4		4	11.10	odd	2	inner

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)

\(f(q)\)	\(=\)	\(q-2.52434 q^{2} -1.00000 q^{3} +4.37228 q^{4} +2.52434 q^{6} -0.792287 q^{7} -5.98844 q^{8} +1.00000 q^{9} -4.37228 q^{12} +4.10891 q^{13} +2.00000 q^{14} +6.37228 q^{16} +5.98844 q^{17} -2.52434 q^{18} +4.25639 q^{19} +0.792287 q^{21} -2.00000 q^{23} +5.98844 q^{24} -10.3723 q^{26} -1.00000 q^{27} -3.46410 q^{28} -2.52434 q^{29} +7.37228 q^{31} -4.10891 q^{32} -15.1168 q^{34} +4.37228 q^{36} -5.00000 q^{37} -10.7446 q^{38} -4.10891 q^{39} -5.69349 q^{41} -2.00000 q^{42} +6.63325 q^{43} +5.04868 q^{46} +1.25544 q^{47} -6.37228 q^{48} -6.37228 q^{49} -5.98844 q^{51} +17.9653 q^{52} -13.1168 q^{53} +2.52434 q^{54} +4.74456 q^{56} -4.25639 q^{57} +6.37228 q^{58} -6.00000 q^{59} -2.67181 q^{61} -18.6101 q^{62} -0.792287 q^{63} -2.37228 q^{64} -16.1168 q^{67} +26.1831 q^{68} +2.00000 q^{69} +0.744563 q^{71} -5.98844 q^{72} -7.42554 q^{73} +12.6217 q^{74} +18.6101 q^{76} +10.3723 q^{78} +5.84096 q^{79} +1.00000 q^{81} +14.3723 q^{82} -8.51278 q^{83} +3.46410 q^{84} -16.7446 q^{86} +2.52434 q^{87} -6.37228 q^{89} -3.25544 q^{91} -8.74456 q^{92} -7.37228 q^{93} -3.16915 q^{94} +4.10891 q^{96} +12.4891 q^{97} +16.0858 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{3} + 6 q^{4} + 4 q^{9} - 6 q^{12} + 8 q^{14} + 14 q^{16} - 8 q^{23} - 30 q^{26} - 4 q^{27} + 18 q^{31} - 26 q^{34} + 6 q^{36} - 20 q^{37} - 20 q^{38} - 8 q^{42} + 28 q^{47} - 14 q^{48} - 14 q^{49}+ \cdots + 4 q^{97}+O(q^{100}) \) 4 * q - 4 * q^3 + 6 * q^4 + 4 * q^9 - 6 * q^12 + 8 * q^14 + 14 * q^16 - 8 * q^23 - 30 * q^26 - 4 * q^27 + 18 * q^31 - 26 * q^34 + 6 * q^36 - 20 * q^37 - 20 * q^38 - 8 * q^42 + 28 * q^47 - 14 * q^48 - 14 * q^49 - 18 * q^53 - 4 * q^56 + 14 * q^58 - 24 * q^59 + 2 * q^64 - 30 * q^67 + 8 * q^69 - 20 * q^71 + 30 * q^78 + 4 * q^81 + 46 * q^82 - 44 * q^86 - 14 * q^89 - 36 * q^91 - 12 * q^92 - 18 * q^93 + 4 * q^97

Introduction

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