LMFDB - Embedded newform 9075.2.a.db.1.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [9075,2,Mod(1,9075)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://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);

sage: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")

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:traces = [4,0,4,6,0,0,0,0,4,0,0,6,0,8,0,14,0,0,0,0,0,0,-32] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(23)] == traces)

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

Self dual:	yes
Analytic conductor:	$72.4642398343$
Analytic rank:	$0$
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:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
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)

$f(q)$	$=$	\(q+2.52434 q^{2} +1.00000 q^{3} +4.37228 q^{4} +2.52434 q^{6} +0.792287 q^{7} +5.98844 q^{8} +1.00000 q^{9} +4.37228 q^{12} -0.147477 q^{13} +2.00000 q^{14} +6.37228 q^{16} +6.63325 q^{17} +2.52434 q^{18} -4.40387 q^{19} +0.792287 q^{21} -8.00000 q^{23} +5.98844 q^{24} -0.372281 q^{26} +1.00000 q^{27} +3.46410 q^{28} +10.0974 q^{29} +2.37228 q^{31} +4.10891 q^{32} +16.7446 q^{34} +4.37228 q^{36} -5.00000 q^{37} -11.1168 q^{38} -0.147477 q^{39} +6.92820 q^{41} +2.00000 q^{42} +9.94987 q^{43} -20.1947 q^{46} +8.74456 q^{47} +6.37228 q^{48} -6.37228 q^{49} +6.63325 q^{51} -0.644810 q^{52} +1.25544 q^{53} +2.52434 q^{54} +4.74456 q^{56} -4.40387 q^{57} +25.4891 q^{58} +4.00000 q^{59} +5.98844 q^{61} +5.98844 q^{62} +0.792287 q^{63} -2.37228 q^{64} +11.1168 q^{67} +29.0024 q^{68} -8.00000 q^{69} +10.7446 q^{71} +5.98844 q^{72} -5.19615 q^{73} -12.6217 q^{74} -19.2549 q^{76} -0.372281 q^{78} -6.78073 q^{79} +1.00000 q^{81} +17.4891 q^{82} +8.51278 q^{83} +3.46410 q^{84} +25.1168 q^{86} +10.0974 q^{87} +5.48913 q^{89} -0.116844 q^{91} -34.9783 q^{92} +2.37228 q^{93} +22.0742 q^{94} +4.10891 q^{96} +9.37228 q^{97} -16.0858 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{3} + 6 q^{4} + 4 q^{9} + 6 q^{12} + 8 q^{14} + 14 q^{16} - 32 q^{23} + 10 q^{26} + 4 q^{27} - 2 q^{31} + 44 q^{34} + 6 q^{36} - 20 q^{37} - 10 q^{38} + 8 q^{42} + 12 q^{47} + 14 q^{48} - 14 q^{49}+ \cdots + 26 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 - 32 * q^23 + 10 * q^26 + 4 * q^27 - 2 * q^31 + 44 * q^34 + 6 * q^36 - 20 * q^37 - 10 * q^38 + 8 * q^42 + 12 * q^47 + 14 * q^48 - 14 * q^49 + 28 * q^53 - 4 * q^56 + 56 * q^58 + 16 * q^59 + 2 * q^64 + 10 * q^67 - 32 * q^69 + 20 * q^71 + 10 * q^78 + 4 * q^81 + 24 * q^82 + 66 * q^86 - 24 * q^89 + 34 * q^91 - 48 * q^92 - 2 * q^93 + 26 * 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	$-0.148958\pi$
$2$	2.52434	1.78498	0.892488	+	0.451071i	$0.148958\pi$
$3$	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	$-0.452160\pi$
$7$	0.792287	0.299456	0.149728	+	0.988727i	$0.452160\pi$
$8$	5.98844	2.11723
$9$	1.00000	0.333333
$10$	0	0
$11$	0	0
$11$	0	0
$12$	4.37228	1.26217
$13$	−0.147477	−0.0409027	−0.0204514	−	0.999791i	$-0.506510\pi$
$13$	−0.147477	−0.0409027	−0.0204514	+	0.999791i	$0.506510\pi$
$14$	2.00000	0.534522
$15$	0	0
$16$	6.37228	1.59307
$17$	6.63325	1.60880	0.804400	−	0.594089i	$-0.202487\pi$
$17$	6.63325	1.60880	0.804400	+	0.594089i	$0.202487\pi$
$18$	2.52434	0.594992
$19$	−4.40387	−1.01032	−0.505158	−	0.863027i	$-0.668566\pi$
$19$	−4.40387	−1.01032	−0.505158	+	0.863027i	$0.668566\pi$
$20$	0	0
$21$	0.792287	0.172891
$22$	0	0
$23$	−8.00000	−1.66812	−0.834058	−	0.551677i	$-0.813988\pi$
$23$	−8.00000	−1.66812	−0.834058	+	0.551677i	$0.813988\pi$
$24$	5.98844	1.22239
$25$	0	0
$26$	−0.372281	−0.0730104
$27$	1.00000	0.192450
$28$	3.46410	0.654654
$29$	10.0974	1.87503	0.937516	−	0.347943i	$-0.113120\pi$
$29$	10.0974	1.87503	0.937516	+	0.347943i	$0.113120\pi$
$30$	0	0
$31$	2.37228	0.426074	0.213037	−	0.977044i	$-0.431664\pi$
$31$	2.37228	0.426074	0.213037	+	0.977044i	$0.431664\pi$
$32$	4.10891	0.726360
$33$	0	0
$34$	16.7446	2.87167
$35$	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	$0.634819\pi$
$38$	−11.1168	−1.80339
$39$	−0.147477	−0.0236152
$40$	0	0
$41$	6.92820	1.08200	0.541002	−	0.841021i	$-0.318045\pi$
$41$	6.92820	1.08200	0.541002	+	0.841021i	$0.318045\pi$
$42$	2.00000	0.308607
$43$	9.94987	1.51734	0.758671	−	0.651474i	$-0.225849\pi$
$43$	9.94987	1.51734	0.758671	+	0.651474i	$0.225849\pi$
$44$	0	0
$45$	0	0
$46$	−20.1947	−2.97755
$47$	8.74456	1.27553	0.637763	−	0.770233i	$-0.279860\pi$
$47$	8.74456	1.27553	0.637763	+	0.770233i	$0.279860\pi$
$48$	6.37228	0.919760
$49$	−6.37228	−0.910326
$50$	0	0
$51$	6.63325	0.928841
$52$	−0.644810	−0.0894191
$53$	1.25544	0.172448	0.0862238	−	0.996276i	$-0.472520\pi$
$53$	1.25544	0.172448	0.0862238	+	0.996276i	$0.472520\pi$
$54$	2.52434	0.343519
$55$	0	0
$56$	4.74456	0.634019
$57$	−4.40387	−0.583306
$58$	25.4891	3.34689
$59$	4.00000	0.520756	0.260378	−	0.965507i	$-0.416153\pi$
$59$	4.00000	0.520756	0.260378	+	0.965507i	$0.416153\pi$
$60$	0	0
$61$	5.98844	0.766741	0.383371	−	0.923595i	$-0.374763\pi$
$61$	5.98844	0.766741	0.383371	+	0.923595i	$0.374763\pi$
$62$	5.98844	0.760533
$63$	0.792287	0.0998188
$64$	−2.37228	−0.296535
$65$	0	0
$66$	0	0
$67$	11.1168	1.35814	0.679069	−	0.734074i	$-0.262384\pi$
$67$	11.1168	1.35814	0.679069	+	0.734074i	$0.262384\pi$
$68$	29.0024	3.51706
$69$	−8.00000	−0.963087
$70$	0	0
$71$	10.7446	1.27514	0.637572	−	0.770390i	$-0.279939\pi$
$71$	10.7446	1.27514	0.637572	+	0.770390i	$0.279939\pi$
$72$	5.98844	0.705744
$73$	−5.19615	−0.608164	−0.304082	−	0.952646i	$-0.598350\pi$
$73$	−5.19615	−0.608164	−0.304082	+	0.952646i	$0.598350\pi$
$74$	−12.6217	−1.46724
$75$	0	0
$76$	−19.2549	−2.20869
$77$	0	0
$78$	−0.372281	−0.0421526
$79$	−6.78073	−0.762891	−0.381446	−	0.924391i	$-0.624574\pi$
$79$	−6.78073	−0.762891	−0.381446	+	0.924391i	$0.624574\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	17.4891	1.93135
$83$	8.51278	0.934399	0.467199	−	0.884152i	$-0.345263\pi$
$83$	8.51278	0.934399	0.467199	+	0.884152i	$0.345263\pi$
$84$	3.46410	0.377964
$85$	0	0
$86$	25.1168	2.70842
$87$	10.0974	1.08255
$88$	0	0
$89$	5.48913	0.581846	0.290923	−	0.956746i	$-0.406038\pi$
$89$	5.48913	0.581846	0.290923	+	0.956746i	$0.406038\pi$
$90$	0	0
$91$	−0.116844	−0.0122486
$92$	−34.9783	−3.64673
$93$	2.37228	0.245994
$94$	22.0742	2.27678
$95$	0	0
$96$	4.10891	0.419364
$97$	9.37228	0.951611	0.475805	−	0.879551i	$-0.342157\pi$
$97$	9.37228	0.951611	0.475805	+	0.879551i	$0.342157\pi$
$98$	−16.0858	−1.62491
$99$	0	0
$100$	0	0
$101$	−13.2665	−1.32007	−0.660033	−	0.751237i	$-0.729458\pi$
$101$	−13.2665	−1.32007	−0.660033	+	0.751237i	$0.729458\pi$
$102$	16.7446	1.65796
$103$	−3.74456	−0.368963	−0.184481	−	0.982836i	$-0.559061\pi$
$103$	−3.74456	−0.368963	−0.184481	+	0.982836i	$0.559061\pi$
$104$	−0.883156	−0.0866006
$105$	0	0
$106$	3.16915	0.307815
$107$	−8.21782	−0.794447	−0.397223	−	0.917722i	$-0.630026\pi$
$107$	−8.21782	−0.794447	−0.397223	+	0.917722i	$0.630026\pi$
$108$	4.37228	0.420723
$109$	−2.67181	−0.255913	−0.127957	−	0.991780i	$-0.540842\pi$
$109$	−2.67181	−0.255913	−0.127957	+	0.991780i	$0.540842\pi$
$110$	0	0
$111$	−5.00000	−0.474579
$112$	5.04868	0.477055
$113$	−16.2337	−1.52714	−0.763568	−	0.645727i	$-0.776554\pi$
$113$	−16.2337	−1.52714	−0.763568	+	0.645727i	$0.776554\pi$
$114$	−11.1168	−1.04119
$115$	0	0
$116$	44.1485	4.09908
$117$	−0.147477	−0.0136342
$118$	10.0974	0.929537
$119$	5.25544	0.481765
$120$	0	0
$121$	0	0
$122$	15.1168	1.36861
$123$	6.92820	0.624695
$124$	10.3723	0.931458
$125$	0	0
$126$	2.00000	0.178174
$127$	−10.2448	−0.909081	−0.454541	−	0.890726i	$-0.650197\pi$
$127$	−10.2448	−0.909081	−0.454541	+	0.890726i	$0.650197\pi$
$128$	−14.2063	−1.25567
$129$	9.94987	0.876038
$130$	0	0
$131$	−18.6101	−1.62597	−0.812987	−	0.582282i	$-0.802160\pi$
$131$	−18.6101	−1.62597	−0.812987	+	0.582282i	$0.802160\pi$
$132$	0	0
$133$	−3.48913	−0.302546
$134$	28.0627	2.42425
$135$	0	0
$136$	39.7228	3.40620
$137$	0.744563	0.0636123	0.0318061	−	0.999494i	$-0.489874\pi$
$137$	0.744563	0.0636123	0.0318061	+	0.999494i	$0.489874\pi$
$138$	−20.1947	−1.71909
$139$	−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
$141$	8.74456	0.736425
$142$	27.1229	2.27610
$143$	0	0
$144$	6.37228	0.531023
$145$	0	0
$146$	−13.1168	−1.08556
$147$	−6.37228	−0.525577
$148$	−21.8614	−1.79700
$149$	−1.28962	−0.105650	−0.0528249	−	0.998604i	$-0.516823\pi$
$149$	−1.28962	−0.105650	−0.0528249	+	0.998604i	$0.516823\pi$
$150$	0	0
$151$	−23.8063	−1.93733	−0.968664	−	0.248375i	$-0.920103\pi$
$151$	−23.8063	−1.93733	−0.968664	+	0.248375i	$0.920103\pi$
$152$	−26.3723	−2.13907
$153$	6.63325	0.536266
$154$	0	0
$155$	0	0
$156$	−0.644810	−0.0516261
$157$	−8.37228	−0.668181	−0.334090	−	0.942541i	$-0.608429\pi$
$157$	−8.37228	−0.668181	−0.334090	+	0.942541i	$0.608429\pi$
$158$	−17.1168	−1.36174
$159$	1.25544	0.0995627
$160$	0	0
$161$	−6.33830	−0.499528
$162$	2.52434	0.198331
$163$	−14.3723	−1.12572	−0.562862	−	0.826551i	$-0.690300\pi$
$163$	−14.3723	−1.12572	−0.562862	+	0.826551i	$0.690300\pi$
$164$	30.2921	2.36541
$165$	0	0
$166$	21.4891	1.66788
$167$	5.04868	0.390678	0.195339	−	0.980736i	$-0.437419\pi$
$167$	5.04868	0.390678	0.195339	+	0.980736i	$0.437419\pi$
$168$	4.74456	0.366051
$169$	−12.9783	−0.998327
$170$	0	0
$171$	−4.40387	−0.336772
$172$	43.5036	3.31712
$173$	−6.92820	−0.526742	−0.263371	−	0.964695i	$-0.584834\pi$
$173$	−6.92820	−0.526742	−0.263371	+	0.964695i	$0.584834\pi$
$174$	25.4891	1.93233
$175$	0	0
$176$	0	0
$177$	4.00000	0.300658
$178$	13.8564	1.03858
$179$	19.4891	1.45669	0.728343	−	0.685213i	$-0.240291\pi$
$179$	19.4891	1.45669	0.728343	+	0.685213i	$0.240291\pi$
$180$	0	0
$181$	13.8614	1.03031	0.515155	−	0.857097i	$-0.327734\pi$
$181$	13.8614	1.03031	0.515155	+	0.857097i	$0.327734\pi$
$182$	−0.294954	−0.0218634
$183$	5.98844	0.442678
$184$	−47.9075	−3.53179
$185$	0	0
$186$	5.98844	0.439094
$187$	0	0
$188$	38.2337	2.78848
$189$	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
$193$	−21.1345	−1.52129	−0.760646	−	0.649167i	$-0.775118\pi$
$193$	−21.1345	−1.52129	−0.760646	+	0.649167i	$0.775118\pi$
$194$	23.6588	1.69860
$195$	0	0
$196$	−27.8614	−1.99010
$197$	8.51278	0.606510	0.303255	−	0.952909i	$-0.401927\pi$
$197$	8.51278	0.606510	0.303255	+	0.952909i	$0.401927\pi$
$198$	0	0
$199$	16.8614	1.19527	0.597637	−	0.801767i	$-0.296107\pi$
$199$	16.8614	1.19527	0.597637	+	0.801767i	$0.296107\pi$
$200$	0	0
$201$	11.1168	0.784122
$202$	−33.4891	−2.35629
$203$	8.00000	0.561490
$204$	29.0024	2.03058
$205$	0	0
$206$	−9.45254	−0.658590
$207$	−8.00000	−0.556038
$208$	−0.939764	−0.0651609
$209$	0	0
$210$	0	0
$211$	4.55134	0.313327	0.156664	−	0.987652i	$-0.449926\pi$
$211$	4.55134	0.313327	0.156664	+	0.987652i	$0.449926\pi$
$212$	5.48913	0.376995
$213$	10.7446	0.736205
$214$	−20.7446	−1.41807
$215$	0	0
$216$	5.98844	0.407462
$217$	1.87953	0.127591
$218$	−6.74456	−0.456799
$219$	−5.19615	−0.351123
$220$	0	0
$221$	−0.978251	−0.0658043
$222$	−12.6217	−0.847112
$223$	23.8614	1.59788	0.798939	−	0.601412i	$-0.205395\pi$
$223$	23.8614	1.59788	0.798939	+	0.601412i	$0.205395\pi$
$224$	3.25544	0.217513
$225$	0	0
$226$	−40.9793	−2.72590
$227$	−17.3205	−1.14960	−0.574801	−	0.818293i	$-0.694921\pi$
$227$	−17.3205	−1.14960	−0.574801	+	0.818293i	$0.694921\pi$
$228$	−19.2549	−1.27519
$229$	−10.4891	−0.693141	−0.346570	−	0.938024i	$-0.612654\pi$
$229$	−10.4891	−0.693141	−0.346570	+	0.938024i	$0.612654\pi$
$230$	0	0
$231$	0	0
$232$	60.4674	3.96988
$233$	15.1460	0.992249	0.496125	−	0.868251i	$-0.334756\pi$
$233$	15.1460	0.992249	0.496125	+	0.868251i	$0.334756\pi$
$234$	−0.372281	−0.0243368
$235$	0	0
$236$	17.4891	1.13845
$237$	−6.78073	−0.440456
$238$	13.2665	0.859939
$239$	13.2665	0.858138	0.429069	−	0.903272i	$-0.358842\pi$
$239$	13.2665	0.858138	0.429069	+	0.903272i	$0.358842\pi$
$240$	0	0
$241$	−14.0039	−0.902069	−0.451035	−	0.892506i	$-0.648945\pi$
$241$	−14.0039	−0.902069	−0.451035	+	0.892506i	$0.648945\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	26.1831	1.67620
$245$	0	0
$246$	17.4891	1.11507
$247$	0.649468	0.0413247
$248$	14.2063	0.902099
$249$	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
$254$	−25.8614	−1.62269
$255$	0	0
$256$	−31.1168	−1.94480
$257$	−17.4891	−1.09094	−0.545471	−	0.838130i	$-0.683649\pi$
$257$	−17.4891	−1.09094	−0.545471	+	0.838130i	$0.683649\pi$
$258$	25.1168	1.56371
$259$	−3.96143	−0.246152
$260$	0	0
$261$	10.0974	0.625010
$262$	−46.9783	−2.90233
$263$	−5.04868	−0.311315	−0.155657	−	0.987811i	$-0.549750\pi$
$263$	−5.04868	−0.311315	−0.155657	+	0.987811i	$0.549750\pi$
$264$	0	0
$265$	0	0
$266$	−8.80773	−0.540037
$267$	5.48913	0.335929
$268$	48.6060	2.96908
$269$	−2.74456	−0.167339	−0.0836695	−	0.996494i	$-0.526664\pi$
$269$	−2.74456	−0.167339	−0.0836695	+	0.996494i	$0.526664\pi$
$270$	0	0
$271$	−10.3923	−0.631288	−0.315644	−	0.948878i	$-0.602220\pi$
$271$	−10.3923	−0.631288	−0.315644	+	0.948878i	$0.602220\pi$
$272$	42.2689	2.56293
$273$	−0.116844	−0.00707172
$274$	1.87953	0.113546
$275$	0	0
$276$	−34.9783	−2.10544
$277$	20.5446	1.23440	0.617201	−	0.786805i	$-0.288266\pi$
$277$	20.5446	1.23440	0.617201	+	0.786805i	$0.288266\pi$
$278$	−26.2337	−1.57339
$279$	2.37228	0.142025
$280$	0	0
$281$	−8.51278	−0.507830	−0.253915	−	0.967227i	$-0.581718\pi$
$281$	−8.51278	−0.507830	−0.253915	+	0.967227i	$0.581718\pi$
$282$	22.0742	1.31450
$283$	−28.3576	−1.68569	−0.842843	−	0.538160i	$-0.819120\pi$
$283$	−28.3576	−1.68569	−0.842843	+	0.538160i	$0.819120\pi$
$284$	46.9783	2.78765
$285$	0	0
$286$	0	0
$287$	5.48913	0.324013
$288$	4.10891	0.242120
$289$	27.0000	1.58824
$290$	0	0
$291$	9.37228	0.549413
$292$	−22.7190	−1.32953
$293$	−32.7615	−1.91395	−0.956973	−	0.290176i	$-0.906286\pi$
$293$	−32.7615	−1.91395	−0.956973	+	0.290176i	$0.906286\pi$
$294$	−16.0858	−0.938142
$295$	0	0
$296$	−29.9422	−1.74035
$297$	0	0
$298$	−3.25544	−0.188582
$299$	1.17981	0.0682304
$300$	0	0
$301$	7.88316	0.454378
$302$	−60.0951	−3.45808
$303$	−13.2665	−0.762140
$304$	−28.0627	−1.60950
$305$	0	0
$306$	16.7446	0.957223
$307$	−26.4781	−1.51118	−0.755592	−	0.655042i	$-0.772651\pi$
$307$	−26.4781	−1.51118	−0.755592	+	0.655042i	$0.772651\pi$
$308$	0	0
$309$	−3.74456	−0.213021
$310$	0	0
$311$	−10.2337	−0.580299	−0.290150	−	0.956981i	$-0.593705\pi$
$311$	−10.2337	−0.580299	−0.290150	+	0.956981i	$0.593705\pi$
$312$	−0.883156	−0.0499989
$313$	25.9783	1.46838	0.734189	−	0.678945i	$-0.237563\pi$
$313$	25.9783	1.46838	0.734189	+	0.678945i	$0.237563\pi$
$314$	−21.1345	−1.19269
$315$	0	0
$316$	−29.6472	−1.66779
$317$	18.9783	1.06592	0.532962	−	0.846139i	$-0.321079\pi$
$317$	18.9783	1.06592	0.532962	+	0.846139i	$0.321079\pi$
$318$	3.16915	0.177717
$319$	0	0
$320$	0	0
$321$	−8.21782	−0.458674
$322$	−16.0000	−0.891645
$323$	−29.2119	−1.62540
$324$	4.37228	0.242905
$325$	0	0
$326$	−36.2805	−2.00939
$327$	−2.67181	−0.147752
$328$	41.4891	2.29085
$329$	6.92820	0.381964
$330$	0	0
$331$	1.23369	0.0678096	0.0339048	−	0.999425i	$-0.489206\pi$
$331$	1.23369	0.0678096	0.0339048	+	0.999425i	$0.489206\pi$
$332$	37.2203	2.04273
$333$	−5.00000	−0.273998
$334$	12.7446	0.697351
$335$	0	0
$336$	5.04868	0.275428
$337$	10.5947	0.577129	0.288565	−	0.957460i	$-0.406822\pi$
$337$	10.5947	0.577129	0.288565	+	0.957460i	$0.406822\pi$
$338$	−32.7615	−1.78199
$339$	−16.2337	−0.881693
$340$	0	0
$341$	0	0
$342$	−11.1168	−0.601130
$343$	−10.5947	−0.572059
$344$	59.5842	3.21257
$345$	0	0
$346$	−17.4891	−0.940221
$347$	−2.17448	−0.116732	−0.0583661	−	0.998295i	$-0.518589\pi$
$347$	−2.17448	−0.116732	−0.0583661	+	0.998295i	$0.518589\pi$
$348$	44.1485	2.36661
$349$	8.66025	0.463573	0.231786	−	0.972767i	$-0.425543\pi$
$349$	8.66025	0.463573	0.231786	+	0.972767i	$0.425543\pi$
$350$	0	0
$351$	−0.147477	−0.00787173
$352$	0	0
$353$	16.2337	0.864032	0.432016	−	0.901866i	$-0.357802\pi$
$353$	16.2337	0.864032	0.432016	+	0.901866i	$0.357802\pi$
$354$	10.0974	0.536668
$355$	0	0
$356$	24.0000	1.27200
$357$	5.25544	0.278147
$358$	49.1971	2.60015
$359$	6.63325	0.350090	0.175045	−	0.984560i	$-0.443993\pi$
$359$	6.63325	0.350090	0.175045	+	0.984560i	$0.443993\pi$
$360$	0	0
$361$	0.394031	0.0207385
$362$	34.9909	1.83908
$363$	0	0
$364$	−0.510875	−0.0267771
$365$	0	0
$366$	15.1168	0.790170
$367$	15.0000	0.782994	0.391497	−	0.920179i	$-0.371957\pi$
$367$	15.0000	0.782994	0.391497	+	0.920179i	$0.371957\pi$
$368$	−50.9783	−2.65743
$369$	6.92820	0.360668
$370$	0	0
$371$	0.994667	0.0516405
$372$	10.3723	0.537778
$373$	26.0357	1.34808	0.674038	−	0.738697i	$-0.264558\pi$
$373$	26.0357	1.34808	0.674038	+	0.738697i	$0.264558\pi$
$374$	0	0
$375$	0	0
$376$	52.3663	2.70058
$377$	−1.48913	−0.0766939
$378$	2.00000	0.102869
$379$	−5.00000	−0.256833	−0.128416	−	0.991720i	$-0.540989\pi$
$379$	−5.00000	−0.256833	−0.128416	+	0.991720i	$0.540989\pi$
$380$	0	0
$381$	−10.2448	−0.524858
$382$	−44.1485	−2.25883
$383$	10.2337	0.522917	0.261459	−	0.965215i	$-0.415797\pi$
$383$	10.2337	0.522917	0.261459	+	0.965215i	$0.415797\pi$
$384$	−14.2063	−0.724960
$385$	0	0
$386$	−53.3505	−2.71547
$387$	9.94987	0.505781
$388$	40.9783	2.08036
$389$	32.7446	1.66022	0.830108	−	0.557603i	$-0.188279\pi$
$389$	32.7446	1.66022	0.830108	+	0.557603i	$0.188279\pi$
$390$	0	0
$391$	−53.0660	−2.68366
$392$	−38.1600	−1.92737
$393$	−18.6101	−0.938757
$394$	21.4891	1.08261
$395$	0	0
$396$	0	0
$397$	2.62772	0.131881	0.0659407	−	0.997824i	$-0.478995\pi$
$397$	2.62772	0.131881	0.0659407	+	0.997824i	$0.478995\pi$
$398$	42.5639	2.13353
$399$	−3.48913	−0.174675
$400$	0	0
$401$	−30.9783	−1.54698	−0.773490	−	0.633808i	$-0.781491\pi$
$401$	−30.9783	−1.54698	−0.773490	+	0.633808i	$0.781491\pi$
$402$	28.0627	1.39964
$403$	−0.349857	−0.0174276
$404$	−58.0049	−2.88585
$405$	0	0
$406$	20.1947	1.00225
$407$	0	0
$408$	39.7228	1.96657
$409$	38.3075	1.89418	0.947092	−	0.320962i	$-0.104006\pi$
$409$	38.3075	1.89418	0.947092	+	0.320962i	$0.104006\pi$
$410$	0	0
$411$	0.744563	0.0367266
$412$	−16.3723	−0.806604
$413$	3.16915	0.155944
$414$	−20.1947	−0.992515
$415$	0	0
$416$	−0.605969	−0.0297101
$417$	−10.3923	−0.508913
$418$	0	0
$419$	−2.23369	−0.109123	−0.0545614	−	0.998510i	$-0.517376\pi$
$419$	−2.23369	−0.109123	−0.0545614	+	0.998510i	$0.517376\pi$
$420$	0	0
$421$	18.0000	0.877266	0.438633	−	0.898666i	$-0.355463\pi$
$421$	18.0000	0.877266	0.438633	+	0.898666i	$0.355463\pi$
$422$	11.4891	0.559282
$423$	8.74456	0.425175
$424$	7.51811	0.365112
$425$	0	0
$426$	27.1229	1.31411
$427$	4.74456	0.229605
$428$	−35.9306	−1.73677
$429$	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
$433$	−40.0951	−1.92685	−0.963424	−	0.267983i	$-0.913643\pi$
$433$	−40.0951	−1.92685	−0.963424	+	0.267983i	$0.913643\pi$
$434$	4.74456	0.227746
$435$	0	0
$436$	−11.6819	−0.559463
$437$	35.2309	1.68532
$438$	−13.1168	−0.626747
$439$	7.13058	0.340324	0.170162	−	0.985416i	$-0.445571\pi$
$439$	7.13058	0.340324	0.170162	+	0.985416i	$0.445571\pi$
$440$	0	0
$441$	−6.37228	−0.303442
$442$	−2.46943	−0.117459
$443$	−9.25544	−0.439739	−0.219870	−	0.975529i	$-0.570563\pi$
$443$	−9.25544	−0.439739	−0.219870	+	0.975529i	$0.570563\pi$
$444$	−21.8614	−1.03750
$445$	0	0
$446$	60.2343	2.85217
$447$	−1.28962	−0.0609969
$448$	−1.87953	−0.0887993
$449$	12.7446	0.601453	0.300727	−	0.953710i	$-0.402771\pi$
$449$	12.7446	0.601453	0.300727	+	0.953710i	$0.402771\pi$
$450$	0	0
$451$	0	0
$452$	−70.9783	−3.33854
$453$	−23.8063	−1.11852
$454$	−43.7228	−2.05201
$455$	0	0
$456$	−26.3723	−1.23500
$457$	−19.6048	−0.917074	−0.458537	−	0.888675i	$-0.651626\pi$
$457$	−19.6048	−0.917074	−0.458537	+	0.888675i	$0.651626\pi$
$458$	−26.4781	−1.23724
$459$	6.63325	0.309614
$460$	0	0
$461$	1.87953	0.0875383	0.0437692	−	0.999042i	$-0.486063\pi$
$461$	1.87953	0.0875383	0.0437692	+	0.999042i	$0.486063\pi$
$462$	0	0
$463$	−26.9783	−1.25379	−0.626893	−	0.779106i	$-0.715674\pi$
$463$	−26.9783	−1.25379	−0.626893	+	0.779106i	$0.715674\pi$
$464$	64.3432	2.98706
$465$	0	0
$466$	38.2337	1.77114
$467$	1.48913	0.0689085	0.0344543	−	0.999406i	$-0.489031\pi$
$467$	1.48913	0.0689085	0.0344543	+	0.999406i	$0.489031\pi$
$468$	−0.644810	−0.0298064
$469$	8.80773	0.406703
$470$	0	0
$471$	−8.37228	−0.385774
$472$	23.9538	1.10256
$473$	0	0
$474$	−17.1168	−0.786203
$475$	0	0
$476$	22.9783	1.05321
$477$	1.25544	0.0574825
$478$	33.4891	1.53176
$479$	2.87419	0.131325	0.0656626	−	0.997842i	$-0.479084\pi$
$479$	2.87419	0.131325	0.0656626	+	0.997842i	$0.479084\pi$
$480$	0	0
$481$	0.737384	0.0336218
$482$	−35.3505	−1.61017
$483$	−6.33830	−0.288402
$484$	0	0
$485$	0	0
$486$	2.52434	0.114506
$487$	35.9783	1.63033	0.815165	−	0.579229i	$-0.196646\pi$
$487$	35.9783	1.63033	0.815165	+	0.579229i	$0.196646\pi$
$488$	35.8614	1.62337
$489$	−14.3723	−0.649937
$490$	0	0
$491$	−6.63325	−0.299354	−0.149677	−	0.988735i	$-0.547823\pi$
$491$	−6.63325	−0.299354	−0.149677	+	0.988735i	$0.547823\pi$
$492$	30.2921	1.36567
$493$	66.9783	3.01655
$494$	1.63948	0.0737636
$495$	0	0
$496$	15.1168	0.678766
$497$	8.51278	0.381850
$498$	21.4891	0.962951
$499$	−14.1168	−0.631957	−0.315978	−	0.948766i	$-0.602333\pi$
$499$	−14.1168	−0.631957	−0.315978	+	0.948766i	$0.602333\pi$
$500$	0	0
$501$	5.04868	0.225558
$502$	−1.28962	−0.0575586
$503$	−3.16915	−0.141305	−0.0706527	−	0.997501i	$-0.522508\pi$
$503$	−3.16915	−0.141305	−0.0706527	+	0.997501i	$0.522508\pi$
$504$	4.74456	0.211340
$505$	0	0
$506$	0	0
$507$	−12.9783	−0.576384
$508$	−44.7933	−1.98738
$509$	−30.9783	−1.37309	−0.686543	−	0.727089i	$-0.740873\pi$
$509$	−30.9783	−1.37309	−0.686543	+	0.727089i	$0.740873\pi$
$510$	0	0
$511$	−4.11684	−0.182118
$512$	−50.1369	−2.21576
$513$	−4.40387	−0.194435
$514$	−44.1485	−1.94731
$515$	0	0
$516$	43.5036	1.91514
$517$	0	0
$518$	−10.0000	−0.439375
$519$	−6.92820	−0.304114
$520$	0	0
$521$	−36.7446	−1.60981	−0.804904	−	0.593405i	$-0.797783\pi$
$521$	−36.7446	−1.60981	−0.804904	+	0.593405i	$0.797783\pi$
$522$	25.4891	1.11563
$523$	27.0680	1.18360	0.591801	−	0.806084i	$-0.298417\pi$
$523$	27.0680	1.18360	0.591801	+	0.806084i	$0.298417\pi$
$524$	−81.3687	−3.55461
$525$	0	0
$526$	−12.7446	−0.555689
$527$	15.7359	0.685468
$528$	0	0
$529$	41.0000	1.78261
$530$	0	0
$531$	4.00000	0.173585
$532$	−15.2554	−0.661407
$533$	−1.02175	−0.0442569
$534$	13.8564	0.599625
$535$	0	0
$536$	66.5725	2.87550
$537$	19.4891	0.841018
$538$	−6.92820	−0.298696
$539$	0	0
$540$	0	0
$541$	23.5113	1.01083	0.505415	−	0.862876i	$-0.331339\pi$
$541$	23.5113	1.01083	0.505415	+	0.862876i	$0.331339\pi$
$542$	−26.2337	−1.12683
$543$	13.8614	0.594850
$544$	27.2554	1.16857
$545$	0	0
$546$	−0.294954	−0.0126229
$547$	0.884861	0.0378339	0.0189170	−	0.999821i	$-0.493978\pi$
$547$	0.884861	0.0378339	0.0189170	+	0.999821i	$0.493978\pi$
$548$	3.25544	0.139065
$549$	5.98844	0.255580
$550$	0	0
$551$	−44.4674	−1.89437
$552$	−47.9075	−2.03908
$553$	−5.37228	−0.228453
$554$	51.8614	2.20338
$555$	0	0
$556$	−45.4381	−1.92700
$557$	−6.33830	−0.268562	−0.134281	−	0.990943i	$-0.542873\pi$
$557$	−6.33830	−0.268562	−0.134281	+	0.990943i	$0.542873\pi$
$558$	5.98844	0.253511
$559$	−1.46738	−0.0620634
$560$	0	0
$561$	0	0
$562$	−21.4891	−0.906464
$563$	−43.8535	−1.84820	−0.924102	−	0.382145i	$-0.875186\pi$
$563$	−43.8535	−1.84820	−0.924102	+	0.382145i	$0.875186\pi$
$564$	38.2337	1.60993
$565$	0	0
$566$	−71.5842	−3.00891
$567$	0.792287	0.0332729
$568$	64.3432	2.69978
$569$	23.0689	0.967098	0.483549	−	0.875317i	$-0.339347\pi$
$569$	23.0689	0.967098	0.483549	+	0.875317i	$0.339347\pi$
$570$	0	0
$571$	−3.37153	−0.141094	−0.0705470	−	0.997508i	$-0.522474\pi$
$571$	−3.37153	−0.141094	−0.0705470	+	0.997508i	$0.522474\pi$
$572$	0	0
$573$	−17.4891	−0.730619
$574$	13.8564	0.578355
$575$	0	0
$576$	−2.37228	−0.0988451
$577$	24.0951	1.00309	0.501546	−	0.865131i	$-0.332765\pi$
$577$	24.0951	1.00309	0.501546	+	0.865131i	$0.332765\pi$
$578$	68.1571	2.83496
$579$	−21.1345	−0.878318
$580$	0	0
$581$	6.74456	0.279812
$582$	23.6588	0.980689
$583$	0	0
$584$	−31.1168	−1.28762
$585$	0	0
$586$	−82.7011	−3.41635
$587$	−34.7446	−1.43406	−0.717031	−	0.697041i	$-0.754499\pi$
$587$	−34.7446	−1.43406	−0.717031	+	0.697041i	$0.754499\pi$
$588$	−27.8614	−1.14899
$589$	−10.4472	−0.430470
$590$	0	0
$591$	8.51278	0.350169
$592$	−31.8614	−1.30950
$593$	24.5437	1.00789	0.503944	−	0.863736i	$-0.331882\pi$
$593$	24.5437	1.00789	0.503944	+	0.863736i	$0.331882\pi$
$594$	0	0
$595$	0	0
$596$	−5.63858	−0.230965
$597$	16.8614	0.690091
$598$	2.97825	0.121790
$599$	−33.2554	−1.35878	−0.679390	−	0.733777i	$-0.737756\pi$
$599$	−33.2554	−1.35878	−0.679390	+	0.733777i	$0.737756\pi$
$600$	0	0
$601$	−11.0371	−0.450213	−0.225107	−	0.974334i	$-0.572273\pi$
$601$	−11.0371	−0.450213	−0.225107	+	0.974334i	$0.572273\pi$
$602$	19.8997	0.811053
$603$	11.1168	0.452713
$604$	−104.088	−4.23527
$605$	0	0
$606$	−33.4891	−1.36040
$607$	−33.7562	−1.37012	−0.685060	−	0.728487i	$-0.740224\pi$
$607$	−33.7562	−1.37012	−0.685060	+	0.728487i	$0.740224\pi$
$608$	−18.0951	−0.733853
$609$	8.00000	0.324176
$610$	0	0
$611$	−1.28962	−0.0521725
$612$	29.0024	1.17235
$613$	14.9985	0.605786	0.302893	−	0.953025i	$-0.402048\pi$
$613$	14.9985	0.605786	0.302893	+	0.953025i	$0.402048\pi$
$614$	−66.8397	−2.69743
$615$	0	0
$616$	0	0
$617$	2.00000	0.0805170	0.0402585	−	0.999189i	$-0.487182\pi$
$617$	2.00000	0.0805170	0.0402585	+	0.999189i	$0.487182\pi$
$618$	−9.45254	−0.380237
$619$	−48.4891	−1.94894	−0.974471	−	0.224512i	$-0.927921\pi$
$619$	−48.4891	−1.94894	−0.974471	+	0.224512i	$0.927921\pi$
$620$	0	0
$621$	−8.00000	−0.321029
$622$	−25.8333	−1.03582
$623$	4.34896	0.174238
$624$	−0.939764	−0.0376207
$625$	0	0
$626$	65.5779	2.62102
$627$	0	0
$628$	−36.6060	−1.46074
$629$	−33.1662	−1.32242
$630$	0	0
$631$	−36.4891	−1.45261	−0.726305	−	0.687373i	$-0.758764\pi$
$631$	−36.4891	−1.45261	−0.726305	+	0.687373i	$0.758764\pi$
$632$	−40.6060	−1.61522
$633$	4.55134	0.180900
$634$	47.9075	1.90265
$635$	0	0
$636$	5.48913	0.217658
$637$	0.939764	0.0372348
$638$	0	0
$639$	10.7446	0.425048
$640$	0	0
$641$	28.9783	1.14457	0.572286	−	0.820054i	$-0.306057\pi$
$641$	28.9783	1.14457	0.572286	+	0.820054i	$0.306057\pi$
$642$	−20.7446	−0.818723
$643$	9.23369	0.364141	0.182071	−	0.983285i	$-0.441720\pi$
$643$	9.23369	0.364141	0.182071	+	0.983285i	$0.441720\pi$
$644$	−27.7128	−1.09204
$645$	0	0
$646$	−73.7408	−2.90129
$647$	−2.00000	−0.0786281	−0.0393141	−	0.999227i	$-0.512517\pi$
$647$	−2.00000	−0.0786281	−0.0393141	+	0.999227i	$0.512517\pi$
$648$	5.98844	0.235248
$649$	0	0
$650$	0	0
$651$	1.87953	0.0736645
$652$	−62.8397	−2.46099
$653$	−45.4891	−1.78013	−0.890064	−	0.455837i	$-0.849340\pi$
$653$	−45.4891	−1.78013	−0.890064	+	0.455837i	$0.849340\pi$
$654$	−6.74456	−0.263733
$655$	0	0
$656$	44.1485	1.72371
$657$	−5.19615	−0.202721
$658$	17.4891	0.681797
$659$	−12.2718	−0.478043	−0.239021	−	0.971014i	$-0.576827\pi$
$659$	−12.2718	−0.478043	−0.239021	+	0.971014i	$0.576827\pi$
$660$	0	0
$661$	−3.00000	−0.116686	−0.0583432	−	0.998297i	$-0.518582\pi$
$661$	−3.00000	−0.116686	−0.0583432	+	0.998297i	$0.518582\pi$
$662$	3.11425	0.121039
$663$	−0.978251	−0.0379921
$664$	50.9783	1.97834
$665$	0	0
$666$	−12.6217	−0.489081
$667$	−80.7788	−3.12777
$668$	22.0742	0.854078
$669$	23.8614	0.922535
$670$	0	0
$671$	0	0
$672$	3.25544	0.125581
$673$	28.8550	1.11228	0.556138	−	0.831090i	$-0.312282\pi$
$673$	28.8550	1.11228	0.556138	+	0.831090i	$0.312282\pi$
$674$	26.7446	1.03016
$675$	0	0
$676$	−56.7446	−2.18248
$677$	−10.9822	−0.422081	−0.211040	−	0.977477i	$-0.567685\pi$
$677$	−10.9822	−0.422081	−0.211040	+	0.977477i	$0.567685\pi$
$678$	−40.9793	−1.57380
$679$	7.42554	0.284966
$680$	0	0
$681$	−17.3205	−0.663723
$682$	0	0
$683$	−18.0000	−0.688751	−0.344375	−	0.938832i	$-0.611909\pi$
$683$	−18.0000	−0.688751	−0.344375	+	0.938832i	$0.611909\pi$
$684$	−19.2549	−0.736231
$685$	0	0
$686$	−26.7446	−1.02111
$687$	−10.4891	−0.400185
$688$	63.4034	2.41723
$689$	−0.185148	−0.00705357
$690$	0	0
$691$	−2.25544	−0.0858009	−0.0429004	−	0.999079i	$-0.513660\pi$
$691$	−2.25544	−0.0858009	−0.0429004	+	0.999079i	$0.513660\pi$
$692$	−30.2921	−1.15153
$693$	0	0
$694$	−5.48913	−0.208364
$695$	0	0
$696$	60.4674	2.29201
$697$	45.9565	1.74073
$698$	21.8614	0.827466
$699$	15.1460	0.572875
$700$	0	0
$701$	−4.05401	−0.153118	−0.0765589	−	0.997065i	$-0.524393\pi$
$701$	−4.05401	−0.153118	−0.0765589	+	0.997065i	$0.524393\pi$
$702$	−0.372281	−0.0140509
$703$	22.0193	0.830475
$704$	0	0
$705$	0	0
$706$	40.9793	1.54228
$707$	−10.5109	−0.395302
$708$	17.4891	0.657282
$709$	31.0000	1.16423	0.582115	−	0.813107i	$-0.302225\pi$
$709$	31.0000	1.16423	0.582115	+	0.813107i	$0.302225\pi$
$710$	0	0
$711$	−6.78073	−0.254297
$712$	32.8713	1.23190
$713$	−18.9783	−0.710741
$714$	13.2665	0.496486
$715$	0	0
$716$	85.2119	3.18452
$717$	13.2665	0.495446
$718$	16.7446	0.624902
$719$	14.0000	0.522112	0.261056	−	0.965324i	$-0.415929\pi$
$719$	14.0000	0.522112	0.261056	+	0.965324i	$0.415929\pi$
$720$	0	0
$721$	−2.96677	−0.110488
$722$	0.994667	0.0370177
$723$	−14.0039	−0.520810
$724$	60.6060	2.25240
$725$	0	0
$726$	0	0
$727$	−21.4674	−0.796181	−0.398090	−	0.917346i	$-0.630327\pi$
$727$	−21.4674	−0.796181	−0.398090	+	0.917346i	$0.630327\pi$
$728$	−0.699713	−0.0259331
$729$	1.00000	0.0370370
$730$	0	0
$731$	66.0000	2.44110
$732$	26.1831	0.967757
$733$	−13.8564	−0.511798	−0.255899	−	0.966704i	$-0.582371\pi$
$733$	−13.8564	−0.511798	−0.255899	+	0.966704i	$0.582371\pi$
$734$	37.8651	1.39763
$735$	0	0
$736$	−32.8713	−1.21165
$737$	0	0
$738$	17.4891	0.643784
$739$	33.6087	1.23632	0.618158	−	0.786054i	$-0.287879\pi$
$739$	33.6087	1.23632	0.618158	+	0.786054i	$0.287879\pi$
$740$	0	0
$741$	0.649468	0.0238588
$742$	2.51087	0.0921771
$743$	−41.9740	−1.53988	−0.769938	−	0.638119i	$-0.779713\pi$
$743$	−41.9740	−1.53988	−0.769938	+	0.638119i	$0.779713\pi$
$744$	14.2063	0.520827
$745$	0	0
$746$	65.7228	2.40628
$747$	8.51278	0.311466
$748$	0	0
$749$	−6.51087	−0.237902
$750$	0	0
$751$	29.2337	1.06675	0.533376	−	0.845878i	$-0.320923\pi$
$751$	29.2337	1.06675	0.533376	+	0.845878i	$0.320923\pi$
$752$	55.7228	2.03200
$753$	−0.510875	−0.0186173
$754$	−3.75906	−0.136897
$755$	0	0
$756$	3.46410	0.125988
$757$	−32.3505	−1.17580	−0.587900	−	0.808934i	$-0.700045\pi$
$757$	−32.3505	−1.17580	−0.587900	+	0.808934i	$0.700045\pi$
$758$	−12.6217	−0.458440
$759$	0	0
$760$	0	0
$761$	18.6101	0.674617	0.337308	−	0.941394i	$-0.390484\pi$
$761$	18.6101	0.674617	0.337308	+	0.941394i	$0.390484\pi$
$762$	−25.8614	−0.936860
$763$	−2.11684	−0.0766349
$764$	−76.4674	−2.76649
$765$	0	0
$766$	25.8333	0.933395
$767$	−0.589907	−0.0213003
$768$	−31.1168	−1.12283
$769$	29.3523	1.05847	0.529235	−	0.848475i	$-0.322479\pi$
$769$	29.3523	1.05847	0.529235	+	0.848475i	$0.322479\pi$
$770$	0	0
$771$	−17.4891	−0.629855
$772$	−92.4058	−3.32576
$773$	−0.510875	−0.0183749	−0.00918744	−	0.999958i	$-0.502924\pi$
$773$	−0.510875	−0.0183749	−0.00918744	+	0.999958i	$0.502924\pi$
$774$	25.1168	0.902806
$775$	0	0
$776$	56.1253	2.01478
$777$	−3.96143	−0.142116
$778$	82.6583	2.96344
$779$	−30.5109	−1.09317
$780$	0	0
$781$	0	0
$782$	−133.957	−4.79027
$783$	10.0974	0.360850
$784$	−40.6060	−1.45021
$785$	0	0
$786$	−46.9783	−1.67566
$787$	−13.4140	−0.478157	−0.239078	−	0.971000i	$-0.576845\pi$
$787$	−13.4140	−0.478157	−0.239078	+	0.971000i	$0.576845\pi$
$788$	37.2203	1.32592
$789$	−5.04868	−0.179738
$790$	0	0
$791$	−12.8617	−0.457311
$792$	0	0
$793$	−0.883156	−0.0313618
$794$	6.63325	0.235405
$795$	0	0
$796$	73.7228	2.61304
$797$	29.7228	1.05284	0.526418	−	0.850226i	$-0.323535\pi$
$797$	29.7228	1.05284	0.526418	+	0.850226i	$0.323535\pi$
$798$	−8.80773	−0.311790
$799$	58.0049	2.05206
$800$	0	0
$801$	5.48913	0.193949
$802$	−78.1996	−2.76132
$803$	0	0
$804$	48.6060	1.71420
$805$	0	0
$806$	−0.883156	−0.0311078
$807$	−2.74456	−0.0966132
$808$	−79.4456	−2.79489
$809$	−20.4897	−0.720378	−0.360189	−	0.932879i	$-0.617288\pi$
$809$	−20.4897	−0.720378	−0.360189	+	0.932879i	$0.617288\pi$
$810$	0	0
$811$	−4.95610	−0.174032	−0.0870161	−	0.996207i	$-0.527733\pi$
$811$	−4.95610	−0.174032	−0.0870161	+	0.996207i	$0.527733\pi$
$812$	34.9783	1.22750
$813$	−10.3923	−0.364474
$814$	0	0
$815$	0	0
$816$	42.2689	1.47971
$817$	−43.8179	−1.53299
$818$	96.7011	3.38107
$819$	−0.116844	−0.00408286
$820$	0	0
$821$	−46.0280	−1.60639	−0.803194	−	0.595718i	$-0.796868\pi$
$821$	−46.0280	−1.60639	−0.803194	+	0.595718i	$0.796868\pi$
$822$	1.87953	0.0655561
$823$	−2.35053	−0.0819344	−0.0409672	−	0.999160i	$-0.513044\pi$
$823$	−2.35053	−0.0819344	−0.0409672	+	0.999160i	$0.513044\pi$
$824$	−22.4241	−0.781180
$825$	0	0
$826$	8.00000	0.278356
$827$	−16.1407	−0.561267	−0.280633	−	0.959815i	$-0.590545\pi$
$827$	−16.1407	−0.561267	−0.280633	+	0.959815i	$0.590545\pi$
$828$	−34.9783	−1.21558
$829$	41.4674	1.44022	0.720111	−	0.693859i	$-0.244091\pi$
$829$	41.4674	1.44022	0.720111	+	0.693859i	$0.244091\pi$
$830$	0	0
$831$	20.5446	0.712683
$832$	0.349857	0.0121291
$833$	−42.2689	−1.46453
$834$	−26.2337	−0.908398
$835$	0	0
$836$	0	0
$837$	2.37228	0.0819980
$838$	−5.63858	−0.194782
$839$	23.2554	0.802867	0.401433	−	0.915888i	$-0.368512\pi$
$839$	23.2554	0.802867	0.401433	+	0.915888i	$0.368512\pi$
$840$	0	0
$841$	72.9565	2.51574
$842$	45.4381	1.56590
$843$	−8.51278	−0.293196
$844$	19.8997	0.684978
$845$	0	0
$846$	22.0742	0.758928
$847$	0	0
$848$	8.00000	0.274721
$849$	−28.3576	−0.973231
$850$	0	0
$851$	40.0000	1.37118
$852$	46.9783	1.60945
$853$	10.4472	0.357706	0.178853	−	0.983876i	$-0.442761\pi$
$853$	10.4472	0.357706	0.178853	+	0.983876i	$0.442761\pi$
$854$	11.9769	0.409840
$855$	0	0
$856$	−49.2119	−1.68203
$857$	−42.5639	−1.45395	−0.726977	−	0.686661i	$-0.759075\pi$
$857$	−42.5639	−1.45395	−0.726977	+	0.686661i	$0.759075\pi$
$858$	0	0
$859$	−10.2554	−0.349911	−0.174956	−	0.984576i	$-0.555978\pi$
$859$	−10.2554	−0.349911	−0.174956	+	0.984576i	$0.555978\pi$
$860$	0	0
$861$	5.48913	0.187069
$862$	59.7228	2.03417
$863$	20.2337	0.688763	0.344381	−	0.938830i	$-0.388089\pi$
$863$	20.2337	0.688763	0.344381	+	0.938830i	$0.388089\pi$
$864$	4.10891	0.139788
$865$	0	0
$866$	−101.214	−3.43938
$867$	27.0000	0.916968
$868$	8.21782	0.278931
$869$	0	0
$870$	0	0
$871$	−1.63948	−0.0555516
$872$	−16.0000	−0.541828
$873$	9.37228	0.317204
$874$	88.9348	3.00826
$875$	0	0
$876$	−22.7190	−0.767605
$877$	−46.3778	−1.56607	−0.783034	−	0.621979i	$-0.786329\pi$
$877$	−46.3778	−1.56607	−0.783034	+	0.621979i	$0.786329\pi$
$878$	18.0000	0.607471
$879$	−32.7615	−1.10502
$880$	0	0
$881$	46.4674	1.56553	0.782763	−	0.622320i	$-0.213810\pi$
$881$	46.4674	1.56553	0.782763	+	0.622320i	$0.213810\pi$
$882$	−16.0858	−0.541637
$883$	−11.1386	−0.374844	−0.187422	−	0.982280i	$-0.560013\pi$
$883$	−11.1386	−0.374844	−0.187422	+	0.982280i	$0.560013\pi$
$884$	−4.27719	−0.143857
$885$	0	0
$886$	−23.3639	−0.784924
$887$	21.4843	0.721373	0.360686	−	0.932687i	$-0.382542\pi$
$887$	21.4843	0.721373	0.360686	+	0.932687i	$0.382542\pi$
$888$	−29.9422	−1.00479
$889$	−8.11684	−0.272230
$890$	0	0
$891$	0	0
$892$	104.329	3.49319
$893$	−38.5099	−1.28868
$894$	−3.25544	−0.108878
$895$	0	0
$896$	−11.2554	−0.376018
$897$	1.17981	0.0393929
$898$	32.1716	1.07358
$899$	23.9538	0.798903
$900$	0	0
$901$	8.32763	0.277434
$902$	0	0
$903$	7.88316	0.262335
$904$	−97.2145	−3.23330
$905$	0	0
$906$	−60.0951	−1.99653
$907$	19.2337	0.638644	0.319322	−	0.947646i	$-0.396545\pi$
$907$	19.2337	0.638644	0.319322	+	0.947646i	$0.396545\pi$
$908$	−75.7301	−2.51319
$909$	−13.2665	−0.440022
$910$	0	0
$911$	4.51087	0.149452	0.0747260	−	0.997204i	$-0.476192\pi$
$911$	4.51087	0.149452	0.0747260	+	0.997204i	$0.476192\pi$
$912$	−28.0627	−0.929248
$913$	0	0
$914$	−49.4891	−1.63695
$915$	0	0
$916$	−45.8614	−1.51530
$917$	−14.7446	−0.486908
$918$	16.7446	0.552653
$919$	10.8896	0.359216	0.179608	−	0.983738i	$-0.442517\pi$
$919$	10.8896	0.359216	0.179608	+	0.983738i	$0.442517\pi$
$920$	0	0
$921$	−26.4781	−0.872483
$922$	4.74456	0.156254
$923$	−1.58457	−0.0521569
$924$	0	0
$925$	0	0
$926$	−68.1022	−2.23798
$927$	−3.74456	−0.122988
$928$	41.4891	1.36195
$929$	13.2554	0.434897	0.217448	−	0.976072i	$-0.430227\pi$
$929$	13.2554	0.434897	0.217448	+	0.976072i	$0.430227\pi$
$930$	0	0
$931$	28.0627	0.919717
$932$	66.2227	2.16920
$933$	−10.2337	−0.335036
$934$	3.75906	0.123000
$935$	0	0
$936$	−0.883156	−0.0288669
$937$	−14.9436	−0.488188	−0.244094	−	0.969752i	$-0.578490\pi$
$937$	−14.9436	−0.488188	−0.244094	+	0.969752i	$0.578490\pi$
$938$	22.2337	0.725956
$939$	25.9783	0.847768
$940$	0	0
$941$	3.75906	0.122542	0.0612708	−	0.998121i	$-0.480485\pi$
$941$	3.75906	0.122542	0.0612708	+	0.998121i	$0.480485\pi$
$942$	−21.1345	−0.688598
$943$	−55.4256	−1.80491
$944$	25.4891	0.829600
$945$	0	0
$946$	0	0
$947$	−21.4891	−0.698303	−0.349151	−	0.937066i	$-0.613530\pi$
$947$	−21.4891	−0.698303	−0.349151	+	0.937066i	$0.613530\pi$
$948$	−29.6472	−0.962898
$949$	0.766312	0.0248755
$950$	0	0
$951$	18.9783	0.615412
$952$	31.4719	1.02001
$953$	−25.2434	−0.817713	−0.408857	−	0.912599i	$-0.634072\pi$
$953$	−25.2434	−0.817713	−0.408857	+	0.912599i	$0.634072\pi$
$954$	3.16915	0.102605
$955$	0	0
$956$	58.0049	1.87601
$957$	0	0
$958$	7.25544	0.234413
$959$	0.589907	0.0190491
$960$	0	0
$961$	−25.3723	−0.818461
$962$	1.86141	0.0600142
$963$	−8.21782	−0.264816
$964$	−61.2289	−1.97205
$965$	0	0
$966$	−16.0000	−0.514792
$967$	7.96054	0.255994	0.127997	−	0.991775i	$-0.459145\pi$
$967$	7.96054	0.255994	0.127997	+	0.991775i	$0.459145\pi$
$968$	0	0
$969$	−29.2119	−0.938423
$970$	0	0
$971$	49.4891	1.58818	0.794091	−	0.607799i	$-0.207947\pi$
$971$	49.4891	1.58818	0.794091	+	0.607799i	$0.207947\pi$
$972$	4.37228	0.140241
$973$	−8.23369	−0.263960
$974$	90.8213	2.91010
$975$	0	0
$976$	38.1600	1.22147
$977$	−38.7446	−1.23955	−0.619774	−	0.784780i	$-0.712776\pi$
$977$	−38.7446	−1.23955	−0.619774	+	0.784780i	$0.712776\pi$
$978$	−36.2805	−1.16012
$979$	0	0
$980$	0	0
$981$	−2.67181	−0.0853045
$982$	−16.7446	−0.534340
$983$	37.4891	1.19572	0.597859	−	0.801602i	$-0.296018\pi$
$983$	37.4891	1.19572	0.597859	+	0.801602i	$0.296018\pi$
$984$	41.4891	1.32263
$985$	0	0
$986$	169.076	5.38447
$987$	6.92820	0.220527
$988$	2.83966	0.0903415
$989$	−79.5990	−2.53110
$990$	0	0
$991$	46.9565	1.49162	0.745811	−	0.666157i	$-0.232062\pi$
$991$	46.9565	1.49162	0.745811	+	0.666157i	$0.232062\pi$
$992$	9.74749	0.309483
$993$	1.23369	0.0391499
$994$	21.4891	0.681594
$995$	0	0
$996$	37.2203	1.17937
$997$	−0.552236	−0.0174895	−0.00874475	−	0.999962i	$-0.502784\pi$
$997$	−0.552236	−0.0174895	−0.00874475	+	0.999962i	$0.502784\pi$
$998$	−35.6357	−1.12803
$999$	−5.00000	−0.158193

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9075.2.a.db.1.4	yes	4
5.4	even	2		9075.2.a.cw.1.1	✓	4
11.10	odd	2	inner	9075.2.a.db.1.1	yes	4
55.54	odd	2		9075.2.a.cw.1.4	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
9075.2.a.cw.1.1	✓	4	5.4	even	2
9075.2.a.cw.1.4	yes	4	55.54	odd	2
9075.2.a.db.1.1	yes	4	11.10	odd	2	inner
9075.2.a.db.1.4	yes	4	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 $\F_{q}$
Belyi maps

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

\(f(q)\)	\(=\)	\(q+2.52434 q^{2} +1.00000 q^{3} +4.37228 q^{4} +2.52434 q^{6} +0.792287 q^{7} +5.98844 q^{8} +1.00000 q^{9} +4.37228 q^{12} -0.147477 q^{13} +2.00000 q^{14} +6.37228 q^{16} +6.63325 q^{17} +2.52434 q^{18} -4.40387 q^{19} +0.792287 q^{21} -8.00000 q^{23} +5.98844 q^{24} -0.372281 q^{26} +1.00000 q^{27} +3.46410 q^{28} +10.0974 q^{29} +2.37228 q^{31} +4.10891 q^{32} +16.7446 q^{34} +4.37228 q^{36} -5.00000 q^{37} -11.1168 q^{38} -0.147477 q^{39} +6.92820 q^{41} +2.00000 q^{42} +9.94987 q^{43} -20.1947 q^{46} +8.74456 q^{47} +6.37228 q^{48} -6.37228 q^{49} +6.63325 q^{51} -0.644810 q^{52} +1.25544 q^{53} +2.52434 q^{54} +4.74456 q^{56} -4.40387 q^{57} +25.4891 q^{58} +4.00000 q^{59} +5.98844 q^{61} +5.98844 q^{62} +0.792287 q^{63} -2.37228 q^{64} +11.1168 q^{67} +29.0024 q^{68} -8.00000 q^{69} +10.7446 q^{71} +5.98844 q^{72} -5.19615 q^{73} -12.6217 q^{74} -19.2549 q^{76} -0.372281 q^{78} -6.78073 q^{79} +1.00000 q^{81} +17.4891 q^{82} +8.51278 q^{83} +3.46410 q^{84} +25.1168 q^{86} +10.0974 q^{87} +5.48913 q^{89} -0.116844 q^{91} -34.9783 q^{92} +2.37228 q^{93} +22.0742 q^{94} +4.10891 q^{96} +9.37228 q^{97} -16.0858 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{3} + 6 q^{4} + 4 q^{9} + 6 q^{12} + 8 q^{14} + 14 q^{16} - 32 q^{23} + 10 q^{26} + 4 q^{27} - 2 q^{31} + 44 q^{34} + 6 q^{36} - 20 q^{37} - 10 q^{38} + 8 q^{42} + 12 q^{47} + 14 q^{48} - 14 q^{49}+ \cdots + 26 q^{97}+O(q^{100}) \) 4 * q + 4 * q^3 + 6 * q^4 + 4 * q^9 + 6 * q^12 + 8 * q^14 + 14 * q^16 - 32 * q^23 + 10 * q^26 + 4 * q^27 - 2 * q^31 + 44 * q^34 + 6 * q^36 - 20 * q^37 - 10 * q^38 + 8 * q^42 + 12 * q^47 + 14 * q^48 - 14 * q^49 + 28 * q^53 - 4 * q^56 + 56 * q^58 + 16 * q^59 + 2 * q^64 + 10 * q^67 - 32 * q^69 + 20 * q^71 + 10 * q^78 + 4 * q^81 + 24 * q^82 + 66 * q^86 - 24 * q^89 + 34 * q^91 - 48 * q^92 - 2 * q^93 + 26 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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