LMFDB - Embedded newform 9450.2.a.eb.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9450,2,Mod(1,9450)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9450, 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("9450.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 9450 = 2 \cdot 3^{3} \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9450.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:	$75.4586299101$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{12})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 3 $ x^2 - 3
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1890)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.73205$ of defining polynomial
Character	$\chi$	$=$	9450.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{7} -1.00000 q^{8} +O(q^{10})$	$q-1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{7} -1.00000 q^{8} -1.73205 q^{11} +5.46410 q^{13} +1.00000 q^{14} +1.00000 q^{16} -4.00000 q^{17} +5.73205 q^{19} +1.73205 q^{22} -2.46410 q^{23} -5.46410 q^{26} -1.00000 q^{28} -1.46410 q^{29} -0.267949 q^{31} -1.00000 q^{32} +4.00000 q^{34} -3.19615 q^{37} -5.73205 q^{38} +3.92820 q^{41} -4.53590 q^{43} -1.73205 q^{44} +2.46410 q^{46} -11.4641 q^{47} +1.00000 q^{49} +5.46410 q^{52} +13.8564 q^{53} +1.00000 q^{56} +1.46410 q^{58} -4.92820 q^{59} -13.8564 q^{61} +0.267949 q^{62} +1.00000 q^{64} -12.3923 q^{67} -4.00000 q^{68} -4.26795 q^{71} +16.9282 q^{73} +3.19615 q^{74} +5.73205 q^{76} +1.73205 q^{77} +8.92820 q^{79} -3.92820 q^{82} -9.46410 q^{83} +4.53590 q^{86} +1.73205 q^{88} -1.00000 q^{89} -5.46410 q^{91} -2.46410 q^{92} +11.4641 q^{94} -6.92820 q^{97} -1.00000 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} + 2 q^{4} - 2 q^{7} - 2 q^{8}+O(q^{10}) $ 2 * q - 2 * q^2 + 2 * q^4 - 2 * q^7 - 2 * q^8	$ 2 q - 2 q^{2} + 2 q^{4} - 2 q^{7} - 2 q^{8} + 4 q^{13} + 2 q^{14} + 2 q^{16} - 8 q^{17} + 8 q^{19} + 2 q^{23} - 4 q^{26} - 2 q^{28} + 4 q^{29} - 4 q^{31} - 2 q^{32} + 8 q^{34} + 4 q^{37} - 8 q^{38} - 6 q^{41} - 16 q^{43} - 2 q^{46} - 16 q^{47} + 2 q^{49} + 4 q^{52} + 2 q^{56} - 4 q^{58} + 4 q^{59} + 4 q^{62} + 2 q^{64} - 4 q^{67} - 8 q^{68} - 12 q^{71} + 20 q^{73} - 4 q^{74} + 8 q^{76} + 4 q^{79} + 6 q^{82} - 12 q^{83} + 16 q^{86} - 2 q^{89} - 4 q^{91} + 2 q^{92} + 16 q^{94} - 2 q^{98}+O(q^{100}) $ 2 * q - 2 * q^2 + 2 * q^4 - 2 * q^7 - 2 * q^8 + 4 * q^13 + 2 * q^14 + 2 * q^16 - 8 * q^17 + 8 * q^19 + 2 * q^23 - 4 * q^26 - 2 * q^28 + 4 * q^29 - 4 * q^31 - 2 * q^32 + 8 * q^34 + 4 * q^37 - 8 * q^38 - 6 * q^41 - 16 * q^43 - 2 * q^46 - 16 * q^47 + 2 * q^49 + 4 * q^52 + 2 * q^56 - 4 * q^58 + 4 * q^59 + 4 * q^62 + 2 * q^64 - 4 * q^67 - 8 * q^68 - 12 * q^71 + 20 * q^73 - 4 * q^74 + 8 * q^76 + 4 * q^79 + 6 * q^82 - 12 * q^83 + 16 * q^86 - 2 * q^89 - 4 * q^91 + 2 * q^92 + 16 * q^94 - 2 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	−1.00000	−0.353553
$9$	0	0
$10$	0	0
$11$	−1.73205	−0.522233	−0.261116	−	0.965307i	$-0.584091\pi$
$11$	−1.73205	−0.522233	−0.261116	+	0.965307i	$0.584091\pi$
$12$	0	0
$13$	5.46410	1.51547	0.757735	−	0.652563i	$-0.226306\pi$
$13$	5.46410	1.51547	0.757735	+	0.652563i	$0.226306\pi$
$14$	1.00000	0.267261
$15$	0	0
$16$	1.00000	0.250000
$17$	−4.00000	−0.970143	−0.485071	−	0.874475i	$-0.661206\pi$
$17$	−4.00000	−0.970143	−0.485071	+	0.874475i	$0.661206\pi$
$18$	0	0
$19$	5.73205	1.31502	0.657511	−	0.753445i	$-0.271609\pi$
$19$	5.73205	1.31502	0.657511	+	0.753445i	$0.271609\pi$
$20$	0	0
$21$	0	0
$22$	1.73205	0.369274
$23$	−2.46410	−0.513801	−0.256900	−	0.966438i	$-0.582701\pi$
$23$	−2.46410	−0.513801	−0.256900	+	0.966438i	$0.582701\pi$
$24$	0	0
$25$	0	0
$26$	−5.46410	−1.07160
$27$	0	0
$28$	−1.00000	−0.188982
$29$	−1.46410	−0.271877	−0.135938	−	0.990717i	$-0.543405\pi$
$29$	−1.46410	−0.271877	−0.135938	+	0.990717i	$0.543405\pi$
$30$	0	0
$31$	−0.267949	−0.0481251	−0.0240625	−	0.999710i	$-0.507660\pi$
$31$	−0.267949	−0.0481251	−0.0240625	+	0.999710i	$0.507660\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	4.00000	0.685994
$35$	0	0
$36$	0	0
$37$	−3.19615	−0.525444	−0.262722	−	0.964872i	$-0.584620\pi$
$37$	−3.19615	−0.525444	−0.262722	+	0.964872i	$0.584620\pi$
$38$	−5.73205	−0.929861
$39$	0	0
$40$	0	0
$41$	3.92820	0.613482	0.306741	−	0.951793i	$-0.400761\pi$
$41$	3.92820	0.613482	0.306741	+	0.951793i	$0.400761\pi$
$42$	0	0
$43$	−4.53590	−0.691718	−0.345859	−	0.938286i	$-0.612412\pi$
$43$	−4.53590	−0.691718	−0.345859	+	0.938286i	$0.612412\pi$
$44$	−1.73205	−0.261116
$45$	0	0
$46$	2.46410	0.363312
$47$	−11.4641	−1.67221	−0.836106	−	0.548569i	$-0.815173\pi$
$47$	−11.4641	−1.67221	−0.836106	+	0.548569i	$0.815173\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	0	0
$52$	5.46410	0.757735
$53$	13.8564	1.90332	0.951662	−	0.307148i	$-0.0993745\pi$
$53$	13.8564	1.90332	0.951662	+	0.307148i	$0.0993745\pi$
$54$	0	0
$55$	0	0
$56$	1.00000	0.133631
$57$	0	0
$58$	1.46410	0.192246
$59$	−4.92820	−0.641597	−0.320799	−	0.947147i	$-0.603951\pi$
$59$	−4.92820	−0.641597	−0.320799	+	0.947147i	$0.603951\pi$
$60$	0	0
$61$	−13.8564	−1.77413	−0.887066	−	0.461644i	$-0.847260\pi$
$61$	−13.8564	−1.77413	−0.887066	+	0.461644i	$0.847260\pi$
$62$	0.267949	0.0340296
$63$	0	0
$64$	1.00000	0.125000
$65$	0	0
$66$	0	0
$67$	−12.3923	−1.51396	−0.756980	−	0.653437i	$-0.773326\pi$
$67$	−12.3923	−1.51396	−0.756980	+	0.653437i	$0.773326\pi$
$68$	−4.00000	−0.485071
$69$	0	0
$70$	0	0
$71$	−4.26795	−0.506512	−0.253256	−	0.967399i	$-0.581502\pi$
$71$	−4.26795	−0.506512	−0.253256	+	0.967399i	$0.581502\pi$
$72$	0	0
$73$	16.9282	1.98130	0.990648	−	0.136441i	$-0.0435665\pi$
$73$	16.9282	1.98130	0.990648	+	0.136441i	$0.0435665\pi$
$74$	3.19615	0.371545
$75$	0	0
$76$	5.73205	0.657511
$77$	1.73205	0.197386
$78$	0	0
$79$	8.92820	1.00450	0.502251	−	0.864722i	$-0.332505\pi$
$79$	8.92820	1.00450	0.502251	+	0.864722i	$0.332505\pi$
$80$	0	0
$81$	0	0
$82$	−3.92820	−0.433797
$83$	−9.46410	−1.03882	−0.519410	−	0.854525i	$-0.673848\pi$
$83$	−9.46410	−1.03882	−0.519410	+	0.854525i	$0.673848\pi$
$84$	0	0
$85$	0	0
$86$	4.53590	0.489119
$87$	0	0
$88$	1.73205	0.184637
$89$	−1.00000	−0.106000	−0.0529999	−	0.998595i	$-0.516878\pi$
$89$	−1.00000	−0.106000	−0.0529999	+	0.998595i	$0.516878\pi$
$90$	0	0
$91$	−5.46410	−0.572793
$92$	−2.46410	−0.256900
$93$	0	0
$94$	11.4641	1.18243
$95$	0	0
$96$	0	0
$97$	−6.92820	−0.703452	−0.351726	−	0.936103i	$-0.614405\pi$
$97$	−6.92820	−0.703452	−0.351726	+	0.936103i	$0.614405\pi$
$98$	−1.00000	−0.101015
$99$	0	0
$100$	0	0
$101$	12.9282	1.28640	0.643202	−	0.765696i	$-0.277605\pi$
$101$	12.9282	1.28640	0.643202	+	0.765696i	$0.277605\pi$
$102$	0	0
$103$	−10.4641	−1.03106	−0.515529	−	0.856872i	$-0.672405\pi$
$103$	−10.4641	−1.03106	−0.515529	+	0.856872i	$0.672405\pi$
$104$	−5.46410	−0.535799
$105$	0	0
$106$	−13.8564	−1.34585
$107$	4.00000	0.386695	0.193347	−	0.981130i	$-0.438066\pi$
$107$	4.00000	0.386695	0.193347	+	0.981130i	$0.438066\pi$
$108$	0	0
$109$	7.53590	0.721808	0.360904	−	0.932603i	$-0.382468\pi$
$109$	7.53590	0.721808	0.360904	+	0.932603i	$0.382468\pi$
$110$	0	0
$111$	0	0
$112$	−1.00000	−0.0944911
$113$	−4.53590	−0.426701	−0.213351	−	0.976976i	$-0.568438\pi$
$113$	−4.53590	−0.426701	−0.213351	+	0.976976i	$0.568438\pi$
$114$	0	0
$115$	0	0
$116$	−1.46410	−0.135938
$117$	0	0
$118$	4.92820	0.453678
$119$	4.00000	0.366679
$120$	0	0
$121$	−8.00000	−0.727273
$122$	13.8564	1.25450
$123$	0	0
$124$	−0.267949	−0.0240625
$125$	0	0
$126$	0	0
$127$	5.46410	0.484861	0.242430	−	0.970169i	$-0.422055\pi$
$127$	5.46410	0.484861	0.242430	+	0.970169i	$0.422055\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	0	0
$131$	20.9282	1.82851	0.914253	−	0.405144i	$-0.132779\pi$
$131$	20.9282	1.82851	0.914253	+	0.405144i	$0.132779\pi$
$132$	0	0
$133$	−5.73205	−0.497032
$134$	12.3923	1.07053
$135$	0	0
$136$	4.00000	0.342997
$137$	−13.4641	−1.15032	−0.575158	−	0.818042i	$-0.695059\pi$
$137$	−13.4641	−1.15032	−0.575158	+	0.818042i	$0.695059\pi$
$138$	0	0
$139$	0.535898	0.0454543	0.0227272	−	0.999742i	$-0.492765\pi$
$139$	0.535898	0.0454543	0.0227272	+	0.999742i	$0.492765\pi$
$140$	0	0
$141$	0	0
$142$	4.26795	0.358158
$143$	−9.46410	−0.791428
$144$	0	0
$145$	0	0
$146$	−16.9282	−1.40099
$147$	0	0
$148$	−3.19615	−0.262722
$149$	12.9282	1.05912	0.529560	−	0.848273i	$-0.322357\pi$
$149$	12.9282	1.05912	0.529560	+	0.848273i	$0.322357\pi$
$150$	0	0
$151$	17.3205	1.40952	0.704761	−	0.709444i	$-0.251054\pi$
$151$	17.3205	1.40952	0.704761	+	0.709444i	$0.251054\pi$
$152$	−5.73205	−0.464931
$153$	0	0
$154$	−1.73205	−0.139573
$155$	0	0
$156$	0	0
$157$	2.53590	0.202387	0.101193	−	0.994867i	$-0.467734\pi$
$157$	2.53590	0.202387	0.101193	+	0.994867i	$0.467734\pi$
$158$	−8.92820	−0.710290
$159$	0	0
$160$	0	0
$161$	2.46410	0.194198
$162$	0	0
$163$	2.53590	0.198627	0.0993134	−	0.995056i	$-0.468335\pi$
$163$	2.53590	0.198627	0.0993134	+	0.995056i	$0.468335\pi$
$164$	3.92820	0.306741
$165$	0	0
$166$	9.46410	0.734557
$167$	−12.9282	−1.00041	−0.500207	−	0.865906i	$-0.666743\pi$
$167$	−12.9282	−1.00041	−0.500207	+	0.865906i	$0.666743\pi$
$168$	0	0
$169$	16.8564	1.29665
$170$	0	0
$171$	0	0
$172$	−4.53590	−0.345859
$173$	−18.1244	−1.37797	−0.688985	−	0.724776i	$-0.741943\pi$
$173$	−18.1244	−1.37797	−0.688985	+	0.724776i	$0.741943\pi$
$174$	0	0
$175$	0	0
$176$	−1.73205	−0.130558
$177$	0	0
$178$	1.00000	0.0749532
$179$	−4.53590	−0.339029	−0.169514	−	0.985528i	$-0.554220\pi$
$179$	−4.53590	−0.339029	−0.169514	+	0.985528i	$0.554220\pi$
$180$	0	0
$181$	3.85641	0.286644	0.143322	−	0.989676i	$-0.454221\pi$
$181$	3.85641	0.286644	0.143322	+	0.989676i	$0.454221\pi$
$182$	5.46410	0.405026
$183$	0	0
$184$	2.46410	0.181656
$185$	0	0
$186$	0	0
$187$	6.92820	0.506640
$188$	−11.4641	−0.836106
$189$	0	0
$190$	0	0
$191$	−8.80385	−0.637024	−0.318512	−	0.947919i	$-0.603183\pi$
$191$	−8.80385	−0.637024	−0.318512	+	0.947919i	$0.603183\pi$
$192$	0	0
$193$	17.8564	1.28533	0.642666	−	0.766146i	$-0.277828\pi$
$193$	17.8564	1.28533	0.642666	+	0.766146i	$0.277828\pi$
$194$	6.92820	0.497416
$195$	0	0
$196$	1.00000	0.0714286
$197$	−19.4641	−1.38676	−0.693380	−	0.720572i	$-0.743879\pi$
$197$	−19.4641	−1.38676	−0.693380	+	0.720572i	$0.743879\pi$
$198$	0	0
$199$	−13.0526	−0.925271	−0.462636	−	0.886548i	$-0.653096\pi$
$199$	−13.0526	−0.925271	−0.462636	+	0.886548i	$0.653096\pi$
$200$	0	0
$201$	0	0
$202$	−12.9282	−0.909625
$203$	1.46410	0.102760
$204$	0	0
$205$	0	0
$206$	10.4641	0.729069
$207$	0	0
$208$	5.46410	0.378867
$209$	−9.92820	−0.686748
$210$	0	0
$211$	−10.3923	−0.715436	−0.357718	−	0.933830i	$-0.616445\pi$
$211$	−10.3923	−0.715436	−0.357718	+	0.933830i	$0.616445\pi$
$212$	13.8564	0.951662
$213$	0	0
$214$	−4.00000	−0.273434
$215$	0	0
$216$	0	0
$217$	0.267949	0.0181896
$218$	−7.53590	−0.510395
$219$	0	0
$220$	0	0
$221$	−21.8564	−1.47022
$222$	0	0
$223$	−17.3923	−1.16467	−0.582337	−	0.812947i	$-0.697862\pi$
$223$	−17.3923	−1.16467	−0.582337	+	0.812947i	$0.697862\pi$
$224$	1.00000	0.0668153
$225$	0	0
$226$	4.53590	0.301723
$227$	26.7846	1.77776	0.888878	−	0.458143i	$-0.151485\pi$
$227$	26.7846	1.77776	0.888878	+	0.458143i	$0.151485\pi$
$228$	0	0
$229$	10.9282	0.722156	0.361078	−	0.932536i	$-0.382409\pi$
$229$	10.9282	0.722156	0.361078	+	0.932536i	$0.382409\pi$
$230$	0	0
$231$	0	0
$232$	1.46410	0.0961230
$233$	−12.5359	−0.821254	−0.410627	−	0.911803i	$-0.634690\pi$
$233$	−12.5359	−0.821254	−0.410627	+	0.911803i	$0.634690\pi$
$234$	0	0
$235$	0	0
$236$	−4.92820	−0.320799
$237$	0	0
$238$	−4.00000	−0.259281
$239$	11.4641	0.741551	0.370776	−	0.928723i	$-0.379092\pi$
$239$	11.4641	0.741551	0.370776	+	0.928723i	$0.379092\pi$
$240$	0	0
$241$	−10.3923	−0.669427	−0.334714	−	0.942320i	$-0.608640\pi$
$241$	−10.3923	−0.669427	−0.334714	+	0.942320i	$0.608640\pi$
$242$	8.00000	0.514259
$243$	0	0
$244$	−13.8564	−0.887066
$245$	0	0
$246$	0	0
$247$	31.3205	1.99288
$248$	0.267949	0.0170148
$249$	0	0
$250$	0	0
$251$	−5.85641	−0.369653	−0.184827	−	0.982771i	$-0.559172\pi$
$251$	−5.85641	−0.369653	−0.184827	+	0.982771i	$0.559172\pi$
$252$	0	0
$253$	4.26795	0.268324
$254$	−5.46410	−0.342848
$255$	0	0
$256$	1.00000	0.0625000
$257$	−27.5885	−1.72092	−0.860460	−	0.509517i	$-0.829824\pi$
$257$	−27.5885	−1.72092	−0.860460	+	0.509517i	$0.829824\pi$
$258$	0	0
$259$	3.19615	0.198599
$260$	0	0
$261$	0	0
$262$	−20.9282	−1.29295
$263$	−17.5359	−1.08131	−0.540655	−	0.841244i	$-0.681824\pi$
$263$	−17.5359	−1.08131	−0.540655	+	0.841244i	$0.681824\pi$
$264$	0	0
$265$	0	0
$266$	5.73205	0.351455
$267$	0	0
$268$	−12.3923	−0.756980
$269$	−18.3205	−1.11702	−0.558511	−	0.829497i	$-0.688627\pi$
$269$	−18.3205	−1.11702	−0.558511	+	0.829497i	$0.688627\pi$
$270$	0	0
$271$	10.3923	0.631288	0.315644	−	0.948878i	$-0.397780\pi$
$271$	10.3923	0.631288	0.315644	+	0.948878i	$0.397780\pi$
$272$	−4.00000	−0.242536
$273$	0	0
$274$	13.4641	0.813396
$275$	0	0
$276$	0	0
$277$	9.05256	0.543916	0.271958	−	0.962309i	$-0.412329\pi$
$277$	9.05256	0.543916	0.271958	+	0.962309i	$0.412329\pi$
$278$	−0.535898	−0.0321410
$279$	0	0
$280$	0	0
$281$	10.3923	0.619953	0.309976	−	0.950744i	$-0.399679\pi$
$281$	10.3923	0.619953	0.309976	+	0.950744i	$0.399679\pi$
$282$	0	0
$283$	9.07180	0.539262	0.269631	−	0.962964i	$-0.413098\pi$
$283$	9.07180	0.539262	0.269631	+	0.962964i	$0.413098\pi$
$284$	−4.26795	−0.253256
$285$	0	0
$286$	9.46410	0.559624
$287$	−3.92820	−0.231875
$288$	0	0
$289$	−1.00000	−0.0588235
$290$	0	0
$291$	0	0
$292$	16.9282	0.990648
$293$	−5.07180	−0.296298	−0.148149	−	0.988965i	$-0.547331\pi$
$293$	−5.07180	−0.296298	−0.148149	+	0.988965i	$0.547331\pi$
$294$	0	0
$295$	0	0
$296$	3.19615	0.185773
$297$	0	0
$298$	−12.9282	−0.748911
$299$	−13.4641	−0.778649
$300$	0	0
$301$	4.53590	0.261445
$302$	−17.3205	−0.996683
$303$	0	0
$304$	5.73205	0.328756
$305$	0	0
$306$	0	0
$307$	−23.9282	−1.36565	−0.682827	−	0.730580i	$-0.739250\pi$
$307$	−23.9282	−1.36565	−0.682827	+	0.730580i	$0.739250\pi$
$308$	1.73205	0.0986928
$309$	0	0
$310$	0	0
$311$	−4.39230	−0.249065	−0.124532	−	0.992216i	$-0.539743\pi$
$311$	−4.39230	−0.249065	−0.124532	+	0.992216i	$0.539743\pi$
$312$	0	0
$313$	−31.3205	−1.77034	−0.885170	−	0.465268i	$-0.845958\pi$
$313$	−31.3205	−1.77034	−0.885170	+	0.465268i	$0.845958\pi$
$314$	−2.53590	−0.143109
$315$	0	0
$316$	8.92820	0.502251
$317$	−15.0718	−0.846516	−0.423258	−	0.906009i	$-0.639114\pi$
$317$	−15.0718	−0.846516	−0.423258	+	0.906009i	$0.639114\pi$
$318$	0	0
$319$	2.53590	0.141983
$320$	0	0
$321$	0	0
$322$	−2.46410	−0.137319
$323$	−22.9282	−1.27576
$324$	0	0
$325$	0	0
$326$	−2.53590	−0.140450
$327$	0	0
$328$	−3.92820	−0.216899
$329$	11.4641	0.632036
$330$	0	0
$331$	−12.7846	−0.702706	−0.351353	−	0.936243i	$-0.614278\pi$
$331$	−12.7846	−0.702706	−0.351353	+	0.936243i	$0.614278\pi$
$332$	−9.46410	−0.519410
$333$	0	0
$334$	12.9282	0.707400
$335$	0	0
$336$	0	0
$337$	21.7321	1.18382	0.591910	−	0.806004i	$-0.298374\pi$
$337$	21.7321	1.18382	0.591910	+	0.806004i	$0.298374\pi$
$338$	−16.8564	−0.916868
$339$	0	0
$340$	0	0
$341$	0.464102	0.0251325
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	4.53590	0.244559
$345$	0	0
$346$	18.1244	0.974371
$347$	21.9282	1.17717	0.588584	−	0.808436i	$-0.299686\pi$
$347$	21.9282	1.17717	0.588584	+	0.808436i	$0.299686\pi$
$348$	0	0
$349$	4.14359	0.221801	0.110901	−	0.993831i	$-0.464626\pi$
$349$	4.14359	0.221801	0.110901	+	0.993831i	$0.464626\pi$
$350$	0	0
$351$	0	0
$352$	1.73205	0.0923186
$353$	−8.66025	−0.460939	−0.230469	−	0.973080i	$-0.574026\pi$
$353$	−8.66025	−0.460939	−0.230469	+	0.973080i	$0.574026\pi$
$354$	0	0
$355$	0	0
$356$	−1.00000	−0.0529999
$357$	0	0
$358$	4.53590	0.239730
$359$	−28.5359	−1.50607	−0.753034	−	0.657982i	$-0.771410\pi$
$359$	−28.5359	−1.50607	−0.753034	+	0.657982i	$0.771410\pi$
$360$	0	0
$361$	13.8564	0.729285
$362$	−3.85641	−0.202688
$363$	0	0
$364$	−5.46410	−0.286397
$365$	0	0
$366$	0	0
$367$	−23.3923	−1.22107	−0.610534	−	0.791990i	$-0.709045\pi$
$367$	−23.3923	−1.22107	−0.610534	+	0.791990i	$0.709045\pi$
$368$	−2.46410	−0.128450
$369$	0	0
$370$	0	0
$371$	−13.8564	−0.719389
$372$	0	0
$373$	−7.19615	−0.372603	−0.186301	−	0.982493i	$-0.559650\pi$
$373$	−7.19615	−0.372603	−0.186301	+	0.982493i	$0.559650\pi$
$374$	−6.92820	−0.358249
$375$	0	0
$376$	11.4641	0.591216
$377$	−8.00000	−0.412021
$378$	0	0
$379$	−14.0000	−0.719132	−0.359566	−	0.933120i	$-0.617075\pi$
$379$	−14.0000	−0.719132	−0.359566	+	0.933120i	$0.617075\pi$
$380$	0	0
$381$	0	0
$382$	8.80385	0.450444
$383$	−11.0718	−0.565742	−0.282871	−	0.959158i	$-0.591287\pi$
$383$	−11.0718	−0.565742	−0.282871	+	0.959158i	$0.591287\pi$
$384$	0	0
$385$	0	0
$386$	−17.8564	−0.908867
$387$	0	0
$388$	−6.92820	−0.351726
$389$	−6.92820	−0.351274	−0.175637	−	0.984455i	$-0.556198\pi$
$389$	−6.92820	−0.351274	−0.175637	+	0.984455i	$0.556198\pi$
$390$	0	0
$391$	9.85641	0.498460
$392$	−1.00000	−0.0505076
$393$	0	0
$394$	19.4641	0.980587
$395$	0	0
$396$	0	0
$397$	−5.60770	−0.281442	−0.140721	−	0.990049i	$-0.544942\pi$
$397$	−5.60770	−0.281442	−0.140721	+	0.990049i	$0.544942\pi$
$398$	13.0526	0.654266
$399$	0	0
$400$	0	0
$401$	−7.60770	−0.379910	−0.189955	−	0.981793i	$-0.560834\pi$
$401$	−7.60770	−0.379910	−0.189955	+	0.981793i	$0.560834\pi$
$402$	0	0
$403$	−1.46410	−0.0729321
$404$	12.9282	0.643202
$405$	0	0
$406$	−1.46410	−0.0726621
$407$	5.53590	0.274404
$408$	0	0
$409$	−37.3205	−1.84538	−0.922690	−	0.385542i	$-0.874014\pi$
$409$	−37.3205	−1.84538	−0.922690	+	0.385542i	$0.874014\pi$
$410$	0	0
$411$	0	0
$412$	−10.4641	−0.515529
$413$	4.92820	0.242501
$414$	0	0
$415$	0	0
$416$	−5.46410	−0.267900
$417$	0	0
$418$	9.92820	0.485604
$419$	23.4641	1.14630	0.573148	−	0.819452i	$-0.305722\pi$
$419$	23.4641	1.14630	0.573148	+	0.819452i	$0.305722\pi$
$420$	0	0
$421$	4.46410	0.217567	0.108784	−	0.994065i	$-0.465304\pi$
$421$	4.46410	0.217567	0.108784	+	0.994065i	$0.465304\pi$
$422$	10.3923	0.505889
$423$	0	0
$424$	−13.8564	−0.672927
$425$	0	0
$426$	0	0
$427$	13.8564	0.670559
$428$	4.00000	0.193347
$429$	0	0
$430$	0	0
$431$	24.5167	1.18093	0.590463	−	0.807065i	$-0.298945\pi$
$431$	24.5167	1.18093	0.590463	+	0.807065i	$0.298945\pi$
$432$	0	0
$433$	38.2487	1.83812	0.919058	−	0.394123i	$-0.128951\pi$
$433$	38.2487	1.83812	0.919058	+	0.394123i	$0.128951\pi$
$434$	−0.267949	−0.0128620
$435$	0	0
$436$	7.53590	0.360904
$437$	−14.1244	−0.675660
$438$	0	0
$439$	−17.6077	−0.840369	−0.420185	−	0.907439i	$-0.638035\pi$
$439$	−17.6077	−0.840369	−0.420185	+	0.907439i	$0.638035\pi$
$440$	0	0
$441$	0	0
$442$	21.8564	1.03960
$443$	10.8564	0.515803	0.257902	−	0.966171i	$-0.416969\pi$
$443$	10.8564	0.515803	0.257902	+	0.966171i	$0.416969\pi$
$444$	0	0
$445$	0	0
$446$	17.3923	0.823550
$447$	0	0
$448$	−1.00000	−0.0472456
$449$	−10.0000	−0.471929	−0.235965	−	0.971762i	$-0.575825\pi$
$449$	−10.0000	−0.471929	−0.235965	+	0.971762i	$0.575825\pi$
$450$	0	0
$451$	−6.80385	−0.320381
$452$	−4.53590	−0.213351
$453$	0	0
$454$	−26.7846	−1.25706
$455$	0	0
$456$	0	0
$457$	14.5167	0.679061	0.339530	−	0.940595i	$-0.389732\pi$
$457$	14.5167	0.679061	0.339530	+	0.940595i	$0.389732\pi$
$458$	−10.9282	−0.510641
$459$	0	0
$460$	0	0
$461$	5.39230	0.251145	0.125572	−	0.992084i	$-0.459923\pi$
$461$	5.39230	0.251145	0.125572	+	0.992084i	$0.459923\pi$
$462$	0	0
$463$	0.143594	0.00667336	0.00333668	−	0.999994i	$-0.498938\pi$
$463$	0.143594	0.00667336	0.00333668	+	0.999994i	$0.498938\pi$
$464$	−1.46410	−0.0679692
$465$	0	0
$466$	12.5359	0.580714
$467$	20.2487	0.936999	0.468499	−	0.883464i	$-0.344795\pi$
$467$	20.2487	0.936999	0.468499	+	0.883464i	$0.344795\pi$
$468$	0	0
$469$	12.3923	0.572223
$470$	0	0
$471$	0	0
$472$	4.92820	0.226839
$473$	7.85641	0.361238
$474$	0	0
$475$	0	0
$476$	4.00000	0.183340
$477$	0	0
$478$	−11.4641	−0.524356
$479$	20.7846	0.949673	0.474837	−	0.880074i	$-0.342507\pi$
$479$	20.7846	0.949673	0.474837	+	0.880074i	$0.342507\pi$
$480$	0	0
$481$	−17.4641	−0.796294
$482$	10.3923	0.473357
$483$	0	0
$484$	−8.00000	−0.363636
$485$	0	0
$486$	0	0
$487$	−26.3923	−1.19595	−0.597975	−	0.801515i	$-0.704028\pi$
$487$	−26.3923	−1.19595	−0.597975	+	0.801515i	$0.704028\pi$
$488$	13.8564	0.627250
$489$	0	0
$490$	0	0
$491$	−41.4449	−1.87038	−0.935190	−	0.354146i	$-0.884772\pi$
$491$	−41.4449	−1.87038	−0.935190	+	0.354146i	$0.884772\pi$
$492$	0	0
$493$	5.85641	0.263759
$494$	−31.3205	−1.40918
$495$	0	0
$496$	−0.267949	−0.0120313
$497$	4.26795	0.191444
$498$	0	0
$499$	−7.07180	−0.316577	−0.158289	−	0.987393i	$-0.550598\pi$
$499$	−7.07180	−0.316577	−0.158289	+	0.987393i	$0.550598\pi$
$500$	0	0
$501$	0	0
$502$	5.85641	0.261384
$503$	−8.39230	−0.374194	−0.187097	−	0.982341i	$-0.559908\pi$
$503$	−8.39230	−0.374194	−0.187097	+	0.982341i	$0.559908\pi$
$504$	0	0
$505$	0	0
$506$	−4.26795	−0.189733
$507$	0	0
$508$	5.46410	0.242430
$509$	−13.7128	−0.607810	−0.303905	−	0.952702i	$-0.598291\pi$
$509$	−13.7128	−0.607810	−0.303905	+	0.952702i	$0.598291\pi$
$510$	0	0
$511$	−16.9282	−0.748860
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	27.5885	1.21687
$515$	0	0
$516$	0	0
$517$	19.8564	0.873284
$518$	−3.19615	−0.140431
$519$	0	0
$520$	0	0
$521$	41.7846	1.83062	0.915308	−	0.402753i	$-0.131947\pi$
$521$	41.7846	1.83062	0.915308	+	0.402753i	$0.131947\pi$
$522$	0	0
$523$	−18.0718	−0.790224	−0.395112	−	0.918633i	$-0.629294\pi$
$523$	−18.0718	−0.790224	−0.395112	+	0.918633i	$0.629294\pi$
$524$	20.9282	0.914253
$525$	0	0
$526$	17.5359	0.764602
$527$	1.07180	0.0466882
$528$	0	0
$529$	−16.9282	−0.736009
$530$	0	0
$531$	0	0
$532$	−5.73205	−0.248516
$533$	21.4641	0.929713
$534$	0	0
$535$	0	0
$536$	12.3923	0.535266
$537$	0	0
$538$	18.3205	0.789853
$539$	−1.73205	−0.0746047
$540$	0	0
$541$	−4.46410	−0.191927	−0.0959634	−	0.995385i	$-0.530593\pi$
$541$	−4.46410	−0.191927	−0.0959634	+	0.995385i	$0.530593\pi$
$542$	−10.3923	−0.446388
$543$	0	0
$544$	4.00000	0.171499
$545$	0	0
$546$	0	0
$547$	20.0000	0.855138	0.427569	−	0.903983i	$-0.359370\pi$
$547$	20.0000	0.855138	0.427569	+	0.903983i	$0.359370\pi$
$548$	−13.4641	−0.575158
$549$	0	0
$550$	0	0
$551$	−8.39230	−0.357524
$552$	0	0
$553$	−8.92820	−0.379666
$554$	−9.05256	−0.384606
$555$	0	0
$556$	0.535898	0.0227272
$557$	5.60770	0.237606	0.118803	−	0.992918i	$-0.462094\pi$
$557$	5.60770	0.237606	0.118803	+	0.992918i	$0.462094\pi$
$558$	0	0
$559$	−24.7846	−1.04828
$560$	0	0
$561$	0	0
$562$	−10.3923	−0.438373
$563$	−24.3923	−1.02801	−0.514007	−	0.857786i	$-0.671839\pi$
$563$	−24.3923	−1.02801	−0.514007	+	0.857786i	$0.671839\pi$
$564$	0	0
$565$	0	0
$566$	−9.07180	−0.381316
$567$	0	0
$568$	4.26795	0.179079
$569$	−33.7128	−1.41331	−0.706657	−	0.707556i	$-0.749798\pi$
$569$	−33.7128	−1.41331	−0.706657	+	0.707556i	$0.749798\pi$
$570$	0	0
$571$	−31.4641	−1.31673	−0.658366	−	0.752698i	$-0.728752\pi$
$571$	−31.4641	−1.31673	−0.658366	+	0.752698i	$0.728752\pi$
$572$	−9.46410	−0.395714
$573$	0	0
$574$	3.92820	0.163960
$575$	0	0
$576$	0	0
$577$	33.3205	1.38715	0.693575	−	0.720384i	$-0.256034\pi$
$577$	33.3205	1.38715	0.693575	+	0.720384i	$0.256034\pi$
$578$	1.00000	0.0415945
$579$	0	0
$580$	0	0
$581$	9.46410	0.392637
$582$	0	0
$583$	−24.0000	−0.993978
$584$	−16.9282	−0.700494
$585$	0	0
$586$	5.07180	0.209514
$587$	−4.92820	−0.203409	−0.101704	−	0.994815i	$-0.532430\pi$
$587$	−4.92820	−0.203409	−0.101704	+	0.994815i	$0.532430\pi$
$588$	0	0
$589$	−1.53590	−0.0632856
$590$	0	0
$591$	0	0
$592$	−3.19615	−0.131361
$593$	−43.3013	−1.77817	−0.889085	−	0.457742i	$-0.848658\pi$
$593$	−43.3013	−1.77817	−0.889085	+	0.457742i	$0.848658\pi$
$594$	0	0
$595$	0	0
$596$	12.9282	0.529560
$597$	0	0
$598$	13.4641	0.550588
$599$	42.1244	1.72115	0.860577	−	0.509320i	$-0.170103\pi$
$599$	42.1244	1.72115	0.860577	+	0.509320i	$0.170103\pi$
$600$	0	0
$601$	0.392305	0.0160024	0.00800122	−	0.999968i	$-0.497453\pi$
$601$	0.392305	0.0160024	0.00800122	+	0.999968i	$0.497453\pi$
$602$	−4.53590	−0.184869
$603$	0	0
$604$	17.3205	0.704761
$605$	0	0
$606$	0	0
$607$	−28.7846	−1.16833	−0.584166	−	0.811634i	$-0.698578\pi$
$607$	−28.7846	−1.16833	−0.584166	+	0.811634i	$0.698578\pi$
$608$	−5.73205	−0.232465
$609$	0	0
$610$	0	0
$611$	−62.6410	−2.53418
$612$	0	0
$613$	10.9474	0.442163	0.221081	−	0.975255i	$-0.429041\pi$
$613$	10.9474	0.442163	0.221081	+	0.975255i	$0.429041\pi$
$614$	23.9282	0.965664
$615$	0	0
$616$	−1.73205	−0.0697863
$617$	4.92820	0.198402	0.0992010	−	0.995067i	$-0.468371\pi$
$617$	4.92820	0.198402	0.0992010	+	0.995067i	$0.468371\pi$
$618$	0	0
$619$	−11.3397	−0.455783	−0.227891	−	0.973687i	$-0.573183\pi$
$619$	−11.3397	−0.455783	−0.227891	+	0.973687i	$0.573183\pi$
$620$	0	0
$621$	0	0
$622$	4.39230	0.176115
$623$	1.00000	0.0400642
$624$	0	0
$625$	0	0
$626$	31.3205	1.25182
$627$	0	0
$628$	2.53590	0.101193
$629$	12.7846	0.509756
$630$	0	0
$631$	−7.85641	−0.312759	−0.156379	−	0.987697i	$-0.549982\pi$
$631$	−7.85641	−0.312759	−0.156379	+	0.987697i	$0.549982\pi$
$632$	−8.92820	−0.355145
$633$	0	0
$634$	15.0718	0.598578
$635$	0	0
$636$	0	0
$637$	5.46410	0.216496
$638$	−2.53590	−0.100397
$639$	0	0
$640$	0	0
$641$	3.07180	0.121329	0.0606643	−	0.998158i	$-0.480678\pi$
$641$	3.07180	0.121329	0.0606643	+	0.998158i	$0.480678\pi$
$642$	0	0
$643$	16.0718	0.633810	0.316905	−	0.948457i	$-0.397356\pi$
$643$	16.0718	0.633810	0.316905	+	0.948457i	$0.397356\pi$
$644$	2.46410	0.0970992
$645$	0	0
$646$	22.9282	0.902098
$647$	−20.5359	−0.807349	−0.403675	−	0.914903i	$-0.632267\pi$
$647$	−20.5359	−0.807349	−0.403675	+	0.914903i	$0.632267\pi$
$648$	0	0
$649$	8.53590	0.335063
$650$	0	0
$651$	0	0
$652$	2.53590	0.0993134
$653$	7.85641	0.307445	0.153722	−	0.988114i	$-0.450874\pi$
$653$	7.85641	0.307445	0.153722	+	0.988114i	$0.450874\pi$
$654$	0	0
$655$	0	0
$656$	3.92820	0.153371
$657$	0	0
$658$	−11.4641	−0.446917
$659$	22.5167	0.877125	0.438562	−	0.898701i	$-0.355488\pi$
$659$	22.5167	0.877125	0.438562	+	0.898701i	$0.355488\pi$
$660$	0	0
$661$	22.2487	0.865375	0.432687	−	0.901544i	$-0.357565\pi$
$661$	22.2487	0.865375	0.432687	+	0.901544i	$0.357565\pi$
$662$	12.7846	0.496888
$663$	0	0
$664$	9.46410	0.367278
$665$	0	0
$666$	0	0
$667$	3.60770	0.139691
$668$	−12.9282	−0.500207
$669$	0	0
$670$	0	0
$671$	24.0000	0.926510
$672$	0	0
$673$	−9.85641	−0.379937	−0.189968	−	0.981790i	$-0.560839\pi$
$673$	−9.85641	−0.379937	−0.189968	+	0.981790i	$0.560839\pi$
$674$	−21.7321	−0.837087
$675$	0	0
$676$	16.8564	0.648323
$677$	46.9090	1.80286	0.901429	−	0.432927i	$-0.142519\pi$
$677$	46.9090	1.80286	0.901429	+	0.432927i	$0.142519\pi$
$678$	0	0
$679$	6.92820	0.265880
$680$	0	0
$681$	0	0
$682$	−0.464102	−0.0177714
$683$	−20.8564	−0.798048	−0.399024	−	0.916940i	$-0.630651\pi$
$683$	−20.8564	−0.798048	−0.399024	+	0.916940i	$0.630651\pi$
$684$	0	0
$685$	0	0
$686$	1.00000	0.0381802
$687$	0	0
$688$	−4.53590	−0.172930
$689$	75.7128	2.88443
$690$	0	0
$691$	16.2487	0.618130	0.309065	−	0.951041i	$-0.399984\pi$
$691$	16.2487	0.618130	0.309065	+	0.951041i	$0.399984\pi$
$692$	−18.1244	−0.688985
$693$	0	0
$694$	−21.9282	−0.832383
$695$	0	0
$696$	0	0
$697$	−15.7128	−0.595165
$698$	−4.14359	−0.156837
$699$	0	0
$700$	0	0
$701$	12.3923	0.468051	0.234025	−	0.972230i	$-0.424810\pi$
$701$	12.3923	0.468051	0.234025	+	0.972230i	$0.424810\pi$
$702$	0	0
$703$	−18.3205	−0.690971
$704$	−1.73205	−0.0652791
$705$	0	0
$706$	8.66025	0.325933
$707$	−12.9282	−0.486215
$708$	0	0
$709$	−5.53590	−0.207905	−0.103953	−	0.994582i	$-0.533149\pi$
$709$	−5.53590	−0.207905	−0.103953	+	0.994582i	$0.533149\pi$
$710$	0	0
$711$	0	0
$712$	1.00000	0.0374766
$713$	0.660254	0.0247267
$714$	0	0
$715$	0	0
$716$	−4.53590	−0.169514
$717$	0	0
$718$	28.5359	1.06495
$719$	−47.8564	−1.78474	−0.892371	−	0.451302i	$-0.850960\pi$
$719$	−47.8564	−1.78474	−0.892371	+	0.451302i	$0.850960\pi$
$720$	0	0
$721$	10.4641	0.389704
$722$	−13.8564	−0.515682
$723$	0	0
$724$	3.85641	0.143322
$725$	0	0
$726$	0	0
$727$	−48.7846	−1.80932	−0.904661	−	0.426133i	$-0.859876\pi$
$727$	−48.7846	−1.80932	−0.904661	+	0.426133i	$0.859876\pi$
$728$	5.46410	0.202513
$729$	0	0
$730$	0	0
$731$	18.1436	0.671065
$732$	0	0
$733$	−27.8564	−1.02890	−0.514450	−	0.857520i	$-0.672004\pi$
$733$	−27.8564	−1.02890	−0.514450	+	0.857520i	$0.672004\pi$
$734$	23.3923	0.863426
$735$	0	0
$736$	2.46410	0.0908280
$737$	21.4641	0.790640
$738$	0	0
$739$	−40.5359	−1.49114	−0.745569	−	0.666429i	$-0.767822\pi$
$739$	−40.5359	−1.49114	−0.745569	+	0.666429i	$0.767822\pi$
$740$	0	0
$741$	0	0
$742$	13.8564	0.508685
$743$	−22.6077	−0.829396	−0.414698	−	0.909959i	$-0.636113\pi$
$743$	−22.6077	−0.829396	−0.414698	+	0.909959i	$0.636113\pi$
$744$	0	0
$745$	0	0
$746$	7.19615	0.263470
$747$	0	0
$748$	6.92820	0.253320
$749$	−4.00000	−0.146157
$750$	0	0
$751$	−15.8564	−0.578608	−0.289304	−	0.957237i	$-0.593424\pi$
$751$	−15.8564	−0.578608	−0.289304	+	0.957237i	$0.593424\pi$
$752$	−11.4641	−0.418053
$753$	0	0
$754$	8.00000	0.291343
$755$	0	0
$756$	0	0
$757$	−13.8564	−0.503620	−0.251810	−	0.967777i	$-0.581026\pi$
$757$	−13.8564	−0.503620	−0.251810	+	0.967777i	$0.581026\pi$
$758$	14.0000	0.508503
$759$	0	0
$760$	0	0
$761$	49.7128	1.80209	0.901044	−	0.433728i	$-0.142802\pi$
$761$	49.7128	1.80209	0.901044	+	0.433728i	$0.142802\pi$
$762$	0	0
$763$	−7.53590	−0.272818
$764$	−8.80385	−0.318512
$765$	0	0
$766$	11.0718	0.400040
$767$	−26.9282	−0.972321
$768$	0	0
$769$	−53.9615	−1.94590	−0.972951	−	0.231011i	$-0.925797\pi$
$769$	−53.9615	−1.94590	−0.972951	+	0.231011i	$0.925797\pi$
$770$	0	0
$771$	0	0
$772$	17.8564	0.642666
$773$	30.3731	1.09244	0.546222	−	0.837641i	$-0.316066\pi$
$773$	30.3731	1.09244	0.546222	+	0.837641i	$0.316066\pi$
$774$	0	0
$775$	0	0
$776$	6.92820	0.248708
$777$	0	0
$778$	6.92820	0.248388
$779$	22.5167	0.806743
$780$	0	0
$781$	7.39230	0.264517
$782$	−9.85641	−0.352464
$783$	0	0
$784$	1.00000	0.0357143
$785$	0	0
$786$	0	0
$787$	13.8564	0.493928	0.246964	−	0.969025i	$-0.420567\pi$
$787$	13.8564	0.493928	0.246964	+	0.969025i	$0.420567\pi$
$788$	−19.4641	−0.693380
$789$	0	0
$790$	0	0
$791$	4.53590	0.161278
$792$	0	0
$793$	−75.7128	−2.68864
$794$	5.60770	0.199010
$795$	0	0
$796$	−13.0526	−0.462636
$797$	−20.5167	−0.726737	−0.363369	−	0.931645i	$-0.618373\pi$
$797$	−20.5167	−0.726737	−0.363369	+	0.931645i	$0.618373\pi$
$798$	0	0
$799$	45.8564	1.62228
$800$	0	0
$801$	0	0
$802$	7.60770	0.268637
$803$	−29.3205	−1.03470
$804$	0	0
$805$	0	0
$806$	1.46410	0.0515708
$807$	0	0
$808$	−12.9282	−0.454813
$809$	−29.4641	−1.03590	−0.517951	−	0.855410i	$-0.673305\pi$
$809$	−29.4641	−1.03590	−0.517951	+	0.855410i	$0.673305\pi$
$810$	0	0
$811$	−34.2679	−1.20331	−0.601655	−	0.798756i	$-0.705492\pi$
$811$	−34.2679	−1.20331	−0.601655	+	0.798756i	$0.705492\pi$
$812$	1.46410	0.0513799
$813$	0	0
$814$	−5.53590	−0.194033
$815$	0	0
$816$	0	0
$817$	−26.0000	−0.909625
$818$	37.3205	1.30488
$819$	0	0
$820$	0	0
$821$	−18.6795	−0.651919	−0.325959	−	0.945384i	$-0.605687\pi$
$821$	−18.6795	−0.651919	−0.325959	+	0.945384i	$0.605687\pi$
$822$	0	0
$823$	42.4974	1.48137	0.740684	−	0.671854i	$-0.234502\pi$
$823$	42.4974	1.48137	0.740684	+	0.671854i	$0.234502\pi$
$824$	10.4641	0.364534
$825$	0	0
$826$	−4.92820	−0.171474
$827$	20.8564	0.725248	0.362624	−	0.931935i	$-0.381881\pi$
$827$	20.8564	0.725248	0.362624	+	0.931935i	$0.381881\pi$
$828$	0	0
$829$	43.4641	1.50957	0.754785	−	0.655972i	$-0.227741\pi$
$829$	43.4641	1.50957	0.754785	+	0.655972i	$0.227741\pi$
$830$	0	0
$831$	0	0
$832$	5.46410	0.189434
$833$	−4.00000	−0.138592
$834$	0	0
$835$	0	0
$836$	−9.92820	−0.343374
$837$	0	0
$838$	−23.4641	−0.810554
$839$	10.5359	0.363740	0.181870	−	0.983323i	$-0.441785\pi$
$839$	10.5359	0.363740	0.181870	+	0.983323i	$0.441785\pi$
$840$	0	0
$841$	−26.8564	−0.926083
$842$	−4.46410	−0.153843
$843$	0	0
$844$	−10.3923	−0.357718
$845$	0	0
$846$	0	0
$847$	8.00000	0.274883
$848$	13.8564	0.475831
$849$	0	0
$850$	0	0
$851$	7.87564	0.269974
$852$	0	0
$853$	40.4974	1.38661	0.693303	−	0.720647i	$-0.256155\pi$
$853$	40.4974	1.38661	0.693303	+	0.720647i	$0.256155\pi$
$854$	−13.8564	−0.474156
$855$	0	0
$856$	−4.00000	−0.136717
$857$	5.73205	0.195803	0.0979016	−	0.995196i	$-0.468787\pi$
$857$	5.73205	0.195803	0.0979016	+	0.995196i	$0.468787\pi$
$858$	0	0
$859$	0.124356	0.00424296	0.00212148	−	0.999998i	$-0.499325\pi$
$859$	0.124356	0.00424296	0.00212148	+	0.999998i	$0.499325\pi$
$860$	0	0
$861$	0	0
$862$	−24.5167	−0.835041
$863$	−48.7846	−1.66065	−0.830324	−	0.557281i	$-0.811844\pi$
$863$	−48.7846	−1.66065	−0.830324	+	0.557281i	$0.811844\pi$
$864$	0	0
$865$	0	0
$866$	−38.2487	−1.29974
$867$	0	0
$868$	0.267949	0.00909479
$869$	−15.4641	−0.524584
$870$	0	0
$871$	−67.7128	−2.29436
$872$	−7.53590	−0.255198
$873$	0	0
$874$	14.1244	0.477763
$875$	0	0
$876$	0	0
$877$	−0.784610	−0.0264944	−0.0132472	−	0.999912i	$-0.504217\pi$
$877$	−0.784610	−0.0264944	−0.0132472	+	0.999912i	$0.504217\pi$
$878$	17.6077	0.594231
$879$	0	0
$880$	0	0
$881$	−52.8564	−1.78078	−0.890389	−	0.455201i	$-0.849567\pi$
$881$	−52.8564	−1.78078	−0.890389	+	0.455201i	$0.849567\pi$
$882$	0	0
$883$	50.9282	1.71387	0.856935	−	0.515424i	$-0.172366\pi$
$883$	50.9282	1.71387	0.856935	+	0.515424i	$0.172366\pi$
$884$	−21.8564	−0.735111
$885$	0	0
$886$	−10.8564	−0.364728
$887$	25.1769	0.845358	0.422679	−	0.906279i	$-0.361090\pi$
$887$	25.1769	0.845358	0.422679	+	0.906279i	$0.361090\pi$
$888$	0	0
$889$	−5.46410	−0.183260
$890$	0	0
$891$	0	0
$892$	−17.3923	−0.582337
$893$	−65.7128	−2.19900
$894$	0	0
$895$	0	0
$896$	1.00000	0.0334077
$897$	0	0
$898$	10.0000	0.333704
$899$	0.392305	0.0130841
$900$	0	0
$901$	−55.4256	−1.84650
$902$	6.80385	0.226543
$903$	0	0
$904$	4.53590	0.150862
$905$	0	0
$906$	0	0
$907$	−31.0718	−1.03172	−0.515861	−	0.856672i	$-0.672528\pi$
$907$	−31.0718	−1.03172	−0.515861	+	0.856672i	$0.672528\pi$
$908$	26.7846	0.888878
$909$	0	0
$910$	0	0
$911$	32.2487	1.06845	0.534224	−	0.845343i	$-0.320604\pi$
$911$	32.2487	1.06845	0.534224	+	0.845343i	$0.320604\pi$
$912$	0	0
$913$	16.3923	0.542506
$914$	−14.5167	−0.480168
$915$	0	0
$916$	10.9282	0.361078
$917$	−20.9282	−0.691110
$918$	0	0
$919$	−14.1436	−0.466554	−0.233277	−	0.972410i	$-0.574945\pi$
$919$	−14.1436	−0.466554	−0.233277	+	0.972410i	$0.574945\pi$
$920$	0	0
$921$	0	0
$922$	−5.39230	−0.177586
$923$	−23.3205	−0.767604
$924$	0	0
$925$	0	0
$926$	−0.143594	−0.00471878
$927$	0	0
$928$	1.46410	0.0480615
$929$	−16.1436	−0.529654	−0.264827	−	0.964296i	$-0.585315\pi$
$929$	−16.1436	−0.529654	−0.264827	+	0.964296i	$0.585315\pi$
$930$	0	0
$931$	5.73205	0.187860
$932$	−12.5359	−0.410627
$933$	0	0
$934$	−20.2487	−0.662558
$935$	0	0
$936$	0	0
$937$	−26.6410	−0.870324	−0.435162	−	0.900352i	$-0.643309\pi$
$937$	−26.6410	−0.870324	−0.435162	+	0.900352i	$0.643309\pi$
$938$	−12.3923	−0.404623
$939$	0	0
$940$	0	0
$941$	−21.1051	−0.688007	−0.344004	−	0.938968i	$-0.611783\pi$
$941$	−21.1051	−0.688007	−0.344004	+	0.938968i	$0.611783\pi$
$942$	0	0
$943$	−9.67949	−0.315208
$944$	−4.92820	−0.160399
$945$	0	0
$946$	−7.85641	−0.255434
$947$	−32.5692	−1.05836	−0.529179	−	0.848510i	$-0.677500\pi$
$947$	−32.5692	−1.05836	−0.529179	+	0.848510i	$0.677500\pi$
$948$	0	0
$949$	92.4974	3.00259
$950$	0	0
$951$	0	0
$952$	−4.00000	−0.129641
$953$	2.53590	0.0821458	0.0410729	−	0.999156i	$-0.486922\pi$
$953$	2.53590	0.0821458	0.0410729	+	0.999156i	$0.486922\pi$
$954$	0	0
$955$	0	0
$956$	11.4641	0.370776
$957$	0	0
$958$	−20.7846	−0.671520
$959$	13.4641	0.434779
$960$	0	0
$961$	−30.9282	−0.997684
$962$	17.4641	0.563065
$963$	0	0
$964$	−10.3923	−0.334714
$965$	0	0
$966$	0	0
$967$	35.8564	1.15306	0.576532	−	0.817074i	$-0.304406\pi$
$967$	35.8564	1.15306	0.576532	+	0.817074i	$0.304406\pi$
$968$	8.00000	0.257130
$969$	0	0
$970$	0	0
$971$	0.679492	0.0218059	0.0109030	−	0.999941i	$-0.496529\pi$
$971$	0.679492	0.0218059	0.0109030	+	0.999941i	$0.496529\pi$
$972$	0	0
$973$	−0.535898	−0.0171801
$974$	26.3923	0.845664
$975$	0	0
$976$	−13.8564	−0.443533
$977$	−41.5692	−1.32992	−0.664959	−	0.746880i	$-0.731551\pi$
$977$	−41.5692	−1.32992	−0.664959	+	0.746880i	$0.731551\pi$
$978$	0	0
$979$	1.73205	0.0553566
$980$	0	0
$981$	0	0
$982$	41.4449	1.32256
$983$	−17.6077	−0.561598	−0.280799	−	0.959767i	$-0.590599\pi$
$983$	−17.6077	−0.561598	−0.280799	+	0.959767i	$0.590599\pi$
$984$	0	0
$985$	0	0
$986$	−5.85641	−0.186506
$987$	0	0
$988$	31.3205	0.996438
$989$	11.1769	0.355405
$990$	0	0
$991$	−36.3923	−1.15604	−0.578020	−	0.816023i	$-0.696174\pi$
$991$	−36.3923	−1.15604	−0.578020	+	0.816023i	$0.696174\pi$
$992$	0.267949	0.00850740
$993$	0	0
$994$	−4.26795	−0.135371
$995$	0	0
$996$	0	0
$997$	−55.9615	−1.77232	−0.886160	−	0.463380i	$-0.846636\pi$
$997$	−55.9615	−1.77232	−0.886160	+	0.463380i	$0.846636\pi$
$998$	7.07180	0.223854
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9450.2.a.eb.1.1		2
3.2	odd	2		9450.2.a.er.1.2		2
5.2	odd	4		1890.2.g.n.379.2	✓	4
5.3	odd	4		1890.2.g.n.379.4	yes	4
5.4	even	2		9450.2.a.eu.1.1		2
15.2	even	4		1890.2.g.q.379.3	yes	4
15.8	even	4		1890.2.g.q.379.1	yes	4
15.14	odd	2		9450.2.a.ei.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1890.2.g.n.379.2	✓	4	5.2	odd	4
1890.2.g.n.379.4	yes	4	5.3	odd	4
1890.2.g.q.379.1	yes	4	15.8	even	4
1890.2.g.q.379.3	yes	4	15.2	even	4
9450.2.a.eb.1.1		2	1.1	even	1	trivial
9450.2.a.ei.1.2		2	15.14	odd	2
9450.2.a.er.1.2		2	3.2	odd	2
9450.2.a.eu.1.1		2	5.4	even	2

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:	\( N \)	\(=\)	\( 9450 = 2 \cdot 3^{3} \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9450.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{7} -1.00000 q^{8} +O(q^{10})\)	\(q-1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{7} -1.00000 q^{8} -1.73205 q^{11} +5.46410 q^{13} +1.00000 q^{14} +1.00000 q^{16} -4.00000 q^{17} +5.73205 q^{19} +1.73205 q^{22} -2.46410 q^{23} -5.46410 q^{26} -1.00000 q^{28} -1.46410 q^{29} -0.267949 q^{31} -1.00000 q^{32} +4.00000 q^{34} -3.19615 q^{37} -5.73205 q^{38} +3.92820 q^{41} -4.53590 q^{43} -1.73205 q^{44} +2.46410 q^{46} -11.4641 q^{47} +1.00000 q^{49} +5.46410 q^{52} +13.8564 q^{53} +1.00000 q^{56} +1.46410 q^{58} -4.92820 q^{59} -13.8564 q^{61} +0.267949 q^{62} +1.00000 q^{64} -12.3923 q^{67} -4.00000 q^{68} -4.26795 q^{71} +16.9282 q^{73} +3.19615 q^{74} +5.73205 q^{76} +1.73205 q^{77} +8.92820 q^{79} -3.92820 q^{82} -9.46410 q^{83} +4.53590 q^{86} +1.73205 q^{88} -1.00000 q^{89} -5.46410 q^{91} -2.46410 q^{92} +11.4641 q^{94} -6.92820 q^{97} -1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} + 2 q^{4} - 2 q^{7} - 2 q^{8}+O(q^{10}) \) 2 * q - 2 * q^2 + 2 * q^4 - 2 * q^7 - 2 * q^8	\( 2 q - 2 q^{2} + 2 q^{4} - 2 q^{7} - 2 q^{8} + 4 q^{13} + 2 q^{14} + 2 q^{16} - 8 q^{17} + 8 q^{19} + 2 q^{23} - 4 q^{26} - 2 q^{28} + 4 q^{29} - 4 q^{31} - 2 q^{32} + 8 q^{34} + 4 q^{37} - 8 q^{38} - 6 q^{41} - 16 q^{43} - 2 q^{46} - 16 q^{47} + 2 q^{49} + 4 q^{52} + 2 q^{56} - 4 q^{58} + 4 q^{59} + 4 q^{62} + 2 q^{64} - 4 q^{67} - 8 q^{68} - 12 q^{71} + 20 q^{73} - 4 q^{74} + 8 q^{76} + 4 q^{79} + 6 q^{82} - 12 q^{83} + 16 q^{86} - 2 q^{89} - 4 q^{91} + 2 q^{92} + 16 q^{94} - 2 q^{98}+O(q^{100}) \) 2 * q - 2 * q^2 + 2 * q^4 - 2 * q^7 - 2 * q^8 + 4 * q^13 + 2 * q^14 + 2 * q^16 - 8 * q^17 + 8 * q^19 + 2 * q^23 - 4 * q^26 - 2 * q^28 + 4 * q^29 - 4 * q^31 - 2 * q^32 + 8 * q^34 + 4 * q^37 - 8 * q^38 - 6 * q^41 - 16 * q^43 - 2 * q^46 - 16 * q^47 + 2 * q^49 + 4 * q^52 + 2 * q^56 - 4 * q^58 + 4 * q^59 + 4 * q^62 + 2 * q^64 - 4 * q^67 - 8 * q^68 - 12 * q^71 + 20 * q^73 - 4 * q^74 + 8 * q^76 + 4 * q^79 + 6 * q^82 - 12 * q^83 + 16 * q^86 - 2 * q^89 - 4 * q^91 + 2 * q^92 + 16 * q^94 - 2 * q^98

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