LMFDB - Embedded newform 4950.2.a.bw.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4950,2,Mod(1,4950)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4950, 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("4950.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4950 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4950.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:	$39.5259490005$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{33}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 8 $ x^2 - x - 8
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 110)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-2.37228$ of defining polynomial
Character	$\chi$	$=$	4950.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{4} +2.37228 q^{7} -1.00000 q^{8} +1.00000 q^{11} -2.00000 q^{13} -2.37228 q^{14} +1.00000 q^{16} -4.37228 q^{17} +6.37228 q^{19} -1.00000 q^{22} -8.74456 q^{23} +2.00000 q^{26} +2.37228 q^{28} +4.37228 q^{29} -2.37228 q^{31} -1.00000 q^{32} +4.37228 q^{34} -3.62772 q^{37} -6.37228 q^{38} -11.4891 q^{41} +4.00000 q^{43} +1.00000 q^{44} +8.74456 q^{46} -8.74456 q^{47} -1.37228 q^{49} -2.00000 q^{52} +13.1168 q^{53} -2.37228 q^{56} -4.37228 q^{58} -8.74456 q^{59} +0.372281 q^{61} +2.37228 q^{62} +1.00000 q^{64} -8.00000 q^{67} -4.37228 q^{68} +7.11684 q^{71} -7.48913 q^{73} +3.62772 q^{74} +6.37228 q^{76} +2.37228 q^{77} -12.7446 q^{79} +11.4891 q^{82} +8.74456 q^{83} -4.00000 q^{86} -1.00000 q^{88} -4.37228 q^{89} -4.74456 q^{91} -8.74456 q^{92} +8.74456 q^{94} +1.25544 q^{97} +1.37228 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} + 2 q^{4} - q^{7} - 2 q^{8} + 2 q^{11} - 4 q^{13} + q^{14} + 2 q^{16} - 3 q^{17} + 7 q^{19} - 2 q^{22} - 6 q^{23} + 4 q^{26} - q^{28} + 3 q^{29} + q^{31} - 2 q^{32} + 3 q^{34} - 13 q^{37}+ \cdots - 3 q^{98}+O(q^{100}) $ 2 * q - 2 * q^2 + 2 * q^4 - q^7 - 2 * q^8 + 2 * q^11 - 4 * q^13 + q^14 + 2 * q^16 - 3 * q^17 + 7 * q^19 - 2 * q^22 - 6 * q^23 + 4 * q^26 - q^28 + 3 * q^29 + q^31 - 2 * q^32 + 3 * q^34 - 13 * q^37 - 7 * q^38 + 8 * q^43 + 2 * q^44 + 6 * q^46 - 6 * q^47 + 3 * q^49 - 4 * q^52 + 9 * q^53 + q^56 - 3 * q^58 - 6 * q^59 - 5 * q^61 - q^62 + 2 * q^64 - 16 * q^67 - 3 * q^68 - 3 * q^71 + 8 * q^73 + 13 * q^74 + 7 * q^76 - q^77 - 14 * q^79 + 6 * q^83 - 8 * q^86 - 2 * q^88 - 3 * q^89 + 2 * q^91 - 6 * q^92 + 6 * q^94 + 14 * q^97 - 3 * q^98

Coefficient data

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

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

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

$2$	−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
$7$	2.37228	0.896638	0.448319	−	0.893874i	$-0.352023\pi$
$7$	2.37228	0.896638	0.448319	+	0.893874i	$0.352023\pi$
$8$	−1.00000	−0.353553
$9$	0	0
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	−2.00000	−0.554700	−0.277350	−	0.960769i	$-0.589456\pi$
$13$	−2.00000	−0.554700	−0.277350	+	0.960769i	$0.589456\pi$
$14$	−2.37228	−0.634019
$15$	0	0
$16$	1.00000	0.250000
$17$	−4.37228	−1.06043	−0.530217	−	0.847862i	$-0.677890\pi$
$17$	−4.37228	−1.06043	−0.530217	+	0.847862i	$0.677890\pi$
$18$	0	0
$19$	6.37228	1.46190	0.730951	−	0.682430i	$-0.239077\pi$
$19$	6.37228	1.46190	0.730951	+	0.682430i	$0.239077\pi$
$20$	0	0
$21$	0	0
$22$	−1.00000	−0.213201
$23$	−8.74456	−1.82337	−0.911684	−	0.410893i	$-0.865217\pi$
$23$	−8.74456	−1.82337	−0.911684	+	0.410893i	$0.865217\pi$
$24$	0	0
$25$	0	0
$26$	2.00000	0.392232
$27$	0	0
$28$	2.37228	0.448319
$29$	4.37228	0.811912	0.405956	−	0.913893i	$-0.366939\pi$
$29$	4.37228	0.811912	0.405956	+	0.913893i	$0.366939\pi$
$30$	0	0
$31$	−2.37228	−0.426074	−0.213037	−	0.977044i	$-0.568336\pi$
$31$	−2.37228	−0.426074	−0.213037	+	0.977044i	$0.568336\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	4.37228	0.749840
$35$	0	0
$36$	0	0
$37$	−3.62772	−0.596393	−0.298197	−	0.954504i	$-0.596385\pi$
$37$	−3.62772	−0.596393	−0.298197	+	0.954504i	$0.596385\pi$
$38$	−6.37228	−1.03372
$39$	0	0
$40$	0	0
$41$	−11.4891	−1.79430	−0.897150	−	0.441726i	$-0.854366\pi$
$41$	−11.4891	−1.79430	−0.897150	+	0.441726i	$0.854366\pi$
$42$	0	0
$43$	4.00000	0.609994	0.304997	−	0.952353i	$-0.401344\pi$
$43$	4.00000	0.609994	0.304997	+	0.952353i	$0.401344\pi$
$44$	1.00000	0.150756
$45$	0	0
$46$	8.74456	1.28932
$47$	−8.74456	−1.27553	−0.637763	−	0.770233i	$-0.720140\pi$
$47$	−8.74456	−1.27553	−0.637763	+	0.770233i	$0.720140\pi$
$48$	0	0
$49$	−1.37228	−0.196040
$50$	0	0
$51$	0	0
$52$	−2.00000	−0.277350
$53$	13.1168	1.80174	0.900869	−	0.434092i	$-0.142931\pi$
$53$	13.1168	1.80174	0.900869	+	0.434092i	$0.142931\pi$
$54$	0	0
$55$	0	0
$56$	−2.37228	−0.317009
$57$	0	0
$58$	−4.37228	−0.574109
$59$	−8.74456	−1.13845	−0.569223	−	0.822183i	$-0.692756\pi$
$59$	−8.74456	−1.13845	−0.569223	+	0.822183i	$0.692756\pi$
$60$	0	0
$61$	0.372281	0.0476657	0.0238329	−	0.999716i	$-0.492413\pi$
$61$	0.372281	0.0476657	0.0238329	+	0.999716i	$0.492413\pi$
$62$	2.37228	0.301280
$63$	0	0
$64$	1.00000	0.125000
$65$	0	0
$66$	0	0
$67$	−8.00000	−0.977356	−0.488678	−	0.872464i	$-0.662521\pi$
$67$	−8.00000	−0.977356	−0.488678	+	0.872464i	$0.662521\pi$
$68$	−4.37228	−0.530217
$69$	0	0
$70$	0	0
$71$	7.11684	0.844614	0.422307	−	0.906453i	$-0.361220\pi$
$71$	7.11684	0.844614	0.422307	+	0.906453i	$0.361220\pi$
$72$	0	0
$73$	−7.48913	−0.876536	−0.438268	−	0.898844i	$-0.644408\pi$
$73$	−7.48913	−0.876536	−0.438268	+	0.898844i	$0.644408\pi$
$74$	3.62772	0.421714
$75$	0	0
$76$	6.37228	0.730951
$77$	2.37228	0.270347
$78$	0	0
$79$	−12.7446	−1.43388	−0.716938	−	0.697137i	$-0.754457\pi$
$79$	−12.7446	−1.43388	−0.716938	+	0.697137i	$0.754457\pi$
$80$	0	0
$81$	0	0
$82$	11.4891	1.26876
$83$	8.74456	0.959840	0.479920	−	0.877312i	$-0.340666\pi$
$83$	8.74456	0.959840	0.479920	+	0.877312i	$0.340666\pi$
$84$	0	0
$85$	0	0
$86$	−4.00000	−0.431331
$87$	0	0
$88$	−1.00000	−0.106600
$89$	−4.37228	−0.463461	−0.231730	−	0.972780i	$-0.574439\pi$
$89$	−4.37228	−0.463461	−0.231730	+	0.972780i	$0.574439\pi$
$90$	0	0
$91$	−4.74456	−0.497365
$92$	−8.74456	−0.911684
$93$	0	0
$94$	8.74456	0.901933
$95$	0	0
$96$	0	0
$97$	1.25544	0.127470	0.0637352	−	0.997967i	$-0.479699\pi$
$97$	1.25544	0.127470	0.0637352	+	0.997967i	$0.479699\pi$
$98$	1.37228	0.138621
$99$	0	0
$100$	0	0
$101$	−6.00000	−0.597022	−0.298511	−	0.954406i	$-0.596490\pi$
$101$	−6.00000	−0.597022	−0.298511	+	0.954406i	$0.596490\pi$
$102$	0	0
$103$	−13.4891	−1.32912	−0.664562	−	0.747234i	$-0.731382\pi$
$103$	−13.4891	−1.32912	−0.664562	+	0.747234i	$0.731382\pi$
$104$	2.00000	0.196116
$105$	0	0
$106$	−13.1168	−1.27402
$107$	−12.0000	−1.16008	−0.580042	−	0.814587i	$-0.696964\pi$
$107$	−12.0000	−1.16008	−0.580042	+	0.814587i	$0.696964\pi$
$108$	0	0
$109$	7.48913	0.717328	0.358664	−	0.933467i	$-0.383232\pi$
$109$	7.48913	0.717328	0.358664	+	0.933467i	$0.383232\pi$
$110$	0	0
$111$	0	0
$112$	2.37228	0.224160
$113$	14.7446	1.38705	0.693526	−	0.720432i	$-0.256056\pi$
$113$	14.7446	1.38705	0.693526	+	0.720432i	$0.256056\pi$
$114$	0	0
$115$	0	0
$116$	4.37228	0.405956
$117$	0	0
$118$	8.74456	0.805002
$119$	−10.3723	−0.950825
$120$	0	0
$121$	1.00000	0.0909091
$122$	−0.372281	−0.0337048
$123$	0	0
$124$	−2.37228	−0.213037
$125$	0	0
$126$	0	0
$127$	−8.00000	−0.709885	−0.354943	−	0.934888i	$-0.615500\pi$
$127$	−8.00000	−0.709885	−0.354943	+	0.934888i	$0.615500\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	0	0
$131$	−4.88316	−0.426643	−0.213322	−	0.976982i	$-0.568428\pi$
$131$	−4.88316	−0.426643	−0.213322	+	0.976982i	$0.568428\pi$
$132$	0	0
$133$	15.1168	1.31080
$134$	8.00000	0.691095
$135$	0	0
$136$	4.37228	0.374920
$137$	−2.74456	−0.234484	−0.117242	−	0.993103i	$-0.537405\pi$
$137$	−2.74456	−0.234484	−0.117242	+	0.993103i	$0.537405\pi$
$138$	0	0
$139$	−4.00000	−0.339276	−0.169638	−	0.985506i	$-0.554260\pi$
$139$	−4.00000	−0.339276	−0.169638	+	0.985506i	$0.554260\pi$
$140$	0	0
$141$	0	0
$142$	−7.11684	−0.597232
$143$	−2.00000	−0.167248
$144$	0	0
$145$	0	0
$146$	7.48913	0.619804
$147$	0	0
$148$	−3.62772	−0.298197
$149$	18.6060	1.52426	0.762130	−	0.647424i	$-0.224154\pi$
$149$	18.6060	1.52426	0.762130	+	0.647424i	$0.224154\pi$
$150$	0	0
$151$	22.2337	1.80935	0.904676	−	0.426100i	$-0.140113\pi$
$151$	22.2337	1.80935	0.904676	+	0.426100i	$0.140113\pi$
$152$	−6.37228	−0.516860
$153$	0	0
$154$	−2.37228	−0.191164
$155$	0	0
$156$	0	0
$157$	−3.62772	−0.289523	−0.144762	−	0.989467i	$-0.546242\pi$
$157$	−3.62772	−0.289523	−0.144762	+	0.989467i	$0.546242\pi$
$158$	12.7446	1.01390
$159$	0	0
$160$	0	0
$161$	−20.7446	−1.63490
$162$	0	0
$163$	23.1168	1.81065	0.905325	−	0.424718i	$-0.139627\pi$
$163$	23.1168	1.81065	0.905325	+	0.424718i	$0.139627\pi$
$164$	−11.4891	−0.897150
$165$	0	0
$166$	−8.74456	−0.678710
$167$	−10.3723	−0.802631	−0.401316	−	0.915940i	$-0.631447\pi$
$167$	−10.3723	−0.802631	−0.401316	+	0.915940i	$0.631447\pi$
$168$	0	0
$169$	−9.00000	−0.692308
$170$	0	0
$171$	0	0
$172$	4.00000	0.304997
$173$	−6.00000	−0.456172	−0.228086	−	0.973641i	$-0.573247\pi$
$173$	−6.00000	−0.456172	−0.228086	+	0.973641i	$0.573247\pi$
$174$	0	0
$175$	0	0
$176$	1.00000	0.0753778
$177$	0	0
$178$	4.37228	0.327716
$179$	12.0000	0.896922	0.448461	−	0.893802i	$-0.351972\pi$
$179$	12.0000	0.896922	0.448461	+	0.893802i	$0.351972\pi$
$180$	0	0
$181$	−10.0000	−0.743294	−0.371647	−	0.928374i	$-0.621207\pi$
$181$	−10.0000	−0.743294	−0.371647	+	0.928374i	$0.621207\pi$
$182$	4.74456	0.351690
$183$	0	0
$184$	8.74456	0.644658
$185$	0	0
$186$	0	0
$187$	−4.37228	−0.319733
$188$	−8.74456	−0.637763
$189$	0	0
$190$	0	0
$191$	−17.4891	−1.26547	−0.632734	−	0.774369i	$-0.718067\pi$
$191$	−17.4891	−1.26547	−0.632734	+	0.774369i	$0.718067\pi$
$192$	0	0
$193$	13.8614	0.997766	0.498883	−	0.866669i	$-0.333744\pi$
$193$	13.8614	0.997766	0.498883	+	0.866669i	$0.333744\pi$
$194$	−1.25544	−0.0901351
$195$	0	0
$196$	−1.37228	−0.0980201
$197$	−9.25544	−0.659423	−0.329711	−	0.944082i	$-0.606951\pi$
$197$	−9.25544	−0.659423	−0.329711	+	0.944082i	$0.606951\pi$
$198$	0	0
$199$	0.883156	0.0626053	0.0313026	−	0.999510i	$-0.490034\pi$
$199$	0.883156	0.0626053	0.0313026	+	0.999510i	$0.490034\pi$
$200$	0	0
$201$	0	0
$202$	6.00000	0.422159
$203$	10.3723	0.727991
$204$	0	0
$205$	0	0
$206$	13.4891	0.939832
$207$	0	0
$208$	−2.00000	−0.138675
$209$	6.37228	0.440780
$210$	0	0
$211$	−11.1168	−0.765315	−0.382658	−	0.923890i	$-0.624991\pi$
$211$	−11.1168	−0.765315	−0.382658	+	0.923890i	$0.624991\pi$
$212$	13.1168	0.900869
$213$	0	0
$214$	12.0000	0.820303
$215$	0	0
$216$	0	0
$217$	−5.62772	−0.382034
$218$	−7.48913	−0.507228
$219$	0	0
$220$	0	0
$221$	8.74456	0.588223
$222$	0	0
$223$	7.25544	0.485860	0.242930	−	0.970044i	$-0.421891\pi$
$223$	7.25544	0.485860	0.242930	+	0.970044i	$0.421891\pi$
$224$	−2.37228	−0.158505
$225$	0	0
$226$	−14.7446	−0.980794
$227$	8.74456	0.580397	0.290199	−	0.956966i	$-0.406279\pi$
$227$	8.74456	0.580397	0.290199	+	0.956966i	$0.406279\pi$
$228$	0	0
$229$	−10.0000	−0.660819	−0.330409	−	0.943838i	$-0.607187\pi$
$229$	−10.0000	−0.660819	−0.330409	+	0.943838i	$0.607187\pi$
$230$	0	0
$231$	0	0
$232$	−4.37228	−0.287054
$233$	−4.37228	−0.286438	−0.143219	−	0.989691i	$-0.545745\pi$
$233$	−4.37228	−0.286438	−0.143219	+	0.989691i	$0.545745\pi$
$234$	0	0
$235$	0	0
$236$	−8.74456	−0.569223
$237$	0	0
$238$	10.3723	0.672335
$239$	3.25544	0.210577	0.105288	−	0.994442i	$-0.466423\pi$
$239$	3.25544	0.210577	0.105288	+	0.994442i	$0.466423\pi$
$240$	0	0
$241$	−22.0000	−1.41714	−0.708572	−	0.705638i	$-0.750660\pi$
$241$	−22.0000	−1.41714	−0.708572	+	0.705638i	$0.750660\pi$
$242$	−1.00000	−0.0642824
$243$	0	0
$244$	0.372281	0.0238329
$245$	0	0
$246$	0	0
$247$	−12.7446	−0.810917
$248$	2.37228	0.150640
$249$	0	0
$250$	0	0
$251$	8.74456	0.551952	0.275976	−	0.961165i	$-0.410999\pi$
$251$	8.74456	0.551952	0.275976	+	0.961165i	$0.410999\pi$
$252$	0	0
$253$	−8.74456	−0.549766
$254$	8.00000	0.501965
$255$	0	0
$256$	1.00000	0.0625000
$257$	18.0000	1.12281	0.561405	−	0.827541i	$-0.310261\pi$
$257$	18.0000	1.12281	0.561405	+	0.827541i	$0.310261\pi$
$258$	0	0
$259$	−8.60597	−0.534749
$260$	0	0
$261$	0	0
$262$	4.88316	0.301682
$263$	3.86141	0.238105	0.119052	−	0.992888i	$-0.462014\pi$
$263$	3.86141	0.238105	0.119052	+	0.992888i	$0.462014\pi$
$264$	0	0
$265$	0	0
$266$	−15.1168	−0.926873
$267$	0	0
$268$	−8.00000	−0.488678
$269$	−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
$271$	−16.0000	−0.971931	−0.485965	−	0.873978i	$-0.661532\pi$
$271$	−16.0000	−0.971931	−0.485965	+	0.873978i	$0.661532\pi$
$272$	−4.37228	−0.265108
$273$	0	0
$274$	2.74456	0.165805
$275$	0	0
$276$	0	0
$277$	1.25544	0.0754319	0.0377160	−	0.999289i	$-0.487992\pi$
$277$	1.25544	0.0754319	0.0377160	+	0.999289i	$0.487992\pi$
$278$	4.00000	0.239904
$279$	0	0
$280$	0	0
$281$	−18.0000	−1.07379	−0.536895	−	0.843649i	$-0.680403\pi$
$281$	−18.0000	−1.07379	−0.536895	+	0.843649i	$0.680403\pi$
$282$	0	0
$283$	−16.7446	−0.995361	−0.497680	−	0.867360i	$-0.665815\pi$
$283$	−16.7446	−0.995361	−0.497680	+	0.867360i	$0.665815\pi$
$284$	7.11684	0.422307
$285$	0	0
$286$	2.00000	0.118262
$287$	−27.2554	−1.60884
$288$	0	0
$289$	2.11684	0.124520
$290$	0	0
$291$	0	0
$292$	−7.48913	−0.438268
$293$	0.510875	0.0298456	0.0149228	−	0.999889i	$-0.495250\pi$
$293$	0.510875	0.0298456	0.0149228	+	0.999889i	$0.495250\pi$
$294$	0	0
$295$	0	0
$296$	3.62772	0.210857
$297$	0	0
$298$	−18.6060	−1.07781
$299$	17.4891	1.01142
$300$	0	0
$301$	9.48913	0.546944
$302$	−22.2337	−1.27940
$303$	0	0
$304$	6.37228	0.365475
$305$	0	0
$306$	0	0
$307$	−16.7446	−0.955663	−0.477831	−	0.878452i	$-0.658577\pi$
$307$	−16.7446	−0.955663	−0.477831	+	0.878452i	$0.658577\pi$
$308$	2.37228	0.135173
$309$	0	0
$310$	0	0
$311$	−13.6277	−0.772757	−0.386379	−	0.922340i	$-0.626274\pi$
$311$	−13.6277	−0.772757	−0.386379	+	0.922340i	$0.626274\pi$
$312$	0	0
$313$	22.0000	1.24351	0.621757	−	0.783210i	$-0.286419\pi$
$313$	22.0000	1.24351	0.621757	+	0.783210i	$0.286419\pi$
$314$	3.62772	0.204724
$315$	0	0
$316$	−12.7446	−0.716938
$317$	27.3505	1.53616	0.768079	−	0.640355i	$-0.221213\pi$
$317$	27.3505	1.53616	0.768079	+	0.640355i	$0.221213\pi$
$318$	0	0
$319$	4.37228	0.244801
$320$	0	0
$321$	0	0
$322$	20.7446	1.15605
$323$	−27.8614	−1.55025
$324$	0	0
$325$	0	0
$326$	−23.1168	−1.28032
$327$	0	0
$328$	11.4891	0.634381
$329$	−20.7446	−1.14368
$330$	0	0
$331$	−14.9783	−0.823279	−0.411640	−	0.911347i	$-0.635044\pi$
$331$	−14.9783	−0.823279	−0.411640	+	0.911347i	$0.635044\pi$
$332$	8.74456	0.479920
$333$	0	0
$334$	10.3723	0.567546
$335$	0	0
$336$	0	0
$337$	−6.88316	−0.374949	−0.187475	−	0.982269i	$-0.560030\pi$
$337$	−6.88316	−0.374949	−0.187475	+	0.982269i	$0.560030\pi$
$338$	9.00000	0.489535
$339$	0	0
$340$	0	0
$341$	−2.37228	−0.128466
$342$	0	0
$343$	−19.8614	−1.07242
$344$	−4.00000	−0.215666
$345$	0	0
$346$	6.00000	0.322562
$347$	2.23369	0.119911	0.0599553	−	0.998201i	$-0.480904\pi$
$347$	2.23369	0.119911	0.0599553	+	0.998201i	$0.480904\pi$
$348$	0	0
$349$	−3.48913	−0.186769	−0.0933843	−	0.995630i	$-0.529769\pi$
$349$	−3.48913	−0.186769	−0.0933843	+	0.995630i	$0.529769\pi$
$350$	0	0
$351$	0	0
$352$	−1.00000	−0.0533002
$353$	−23.4891	−1.25020	−0.625100	−	0.780545i	$-0.714942\pi$
$353$	−23.4891	−1.25020	−0.625100	+	0.780545i	$0.714942\pi$
$354$	0	0
$355$	0	0
$356$	−4.37228	−0.231730
$357$	0	0
$358$	−12.0000	−0.634220
$359$	0	0		−	1.00000i	$-0.5\pi$
$359$	0	0			1.00000i	$0.5\pi$
$360$	0	0
$361$	21.6060	1.13716
$362$	10.0000	0.525588
$363$	0	0
$364$	−4.74456	−0.248683
$365$	0	0
$366$	0	0
$367$	4.00000	0.208798	0.104399	−	0.994535i	$-0.466708\pi$
$367$	4.00000	0.208798	0.104399	+	0.994535i	$0.466708\pi$
$368$	−8.74456	−0.455842
$369$	0	0
$370$	0	0
$371$	31.1168	1.61551
$372$	0	0
$373$	−8.51087	−0.440676	−0.220338	−	0.975424i	$-0.570716\pi$
$373$	−8.51087	−0.440676	−0.220338	+	0.975424i	$0.570716\pi$
$374$	4.37228	0.226085
$375$	0	0
$376$	8.74456	0.450966
$377$	−8.74456	−0.450368
$378$	0	0
$379$	34.2337	1.75847	0.879233	−	0.476392i	$-0.158056\pi$
$379$	34.2337	1.75847	0.879233	+	0.476392i	$0.158056\pi$
$380$	0	0
$381$	0	0
$382$	17.4891	0.894821
$383$	−2.23369	−0.114136	−0.0570681	−	0.998370i	$-0.518175\pi$
$383$	−2.23369	−0.114136	−0.0570681	+	0.998370i	$0.518175\pi$
$384$	0	0
$385$	0	0
$386$	−13.8614	−0.705527
$387$	0	0
$388$	1.25544	0.0637352
$389$	−6.00000	−0.304212	−0.152106	−	0.988364i	$-0.548606\pi$
$389$	−6.00000	−0.304212	−0.152106	+	0.988364i	$0.548606\pi$
$390$	0	0
$391$	38.2337	1.93356
$392$	1.37228	0.0693107
$393$	0	0
$394$	9.25544	0.466282
$395$	0	0
$396$	0	0
$397$	20.9783	1.05287	0.526434	−	0.850216i	$-0.323529\pi$
$397$	20.9783	1.05287	0.526434	+	0.850216i	$0.323529\pi$
$398$	−0.883156	−0.0442686
$399$	0	0
$400$	0	0
$401$	−7.62772	−0.380910	−0.190455	−	0.981696i	$-0.560996\pi$
$401$	−7.62772	−0.380910	−0.190455	+	0.981696i	$0.560996\pi$
$402$	0	0
$403$	4.74456	0.236343
$404$	−6.00000	−0.298511
$405$	0	0
$406$	−10.3723	−0.514768
$407$	−3.62772	−0.179819
$408$	0	0
$409$	−36.2337	−1.79164	−0.895820	−	0.444417i	$-0.853411\pi$
$409$	−36.2337	−1.79164	−0.895820	+	0.444417i	$0.853411\pi$
$410$	0	0
$411$	0	0
$412$	−13.4891	−0.664562
$413$	−20.7446	−1.02077
$414$	0	0
$415$	0	0
$416$	2.00000	0.0980581
$417$	0	0
$418$	−6.37228	−0.311678
$419$	12.0000	0.586238	0.293119	−	0.956076i	$-0.405307\pi$
$419$	12.0000	0.586238	0.293119	+	0.956076i	$0.405307\pi$
$420$	0	0
$421$	−24.2337	−1.18108	−0.590539	−	0.807009i	$-0.701085\pi$
$421$	−24.2337	−1.18108	−0.590539	+	0.807009i	$0.701085\pi$
$422$	11.1168	0.541159
$423$	0	0
$424$	−13.1168	−0.637010
$425$	0	0
$426$	0	0
$427$	0.883156	0.0427389
$428$	−12.0000	−0.580042
$429$	0	0
$430$	0	0
$431$	−10.9783	−0.528804	−0.264402	−	0.964413i	$-0.585175\pi$
$431$	−10.9783	−0.528804	−0.264402	+	0.964413i	$0.585175\pi$
$432$	0	0
$433$	29.7228	1.42839	0.714194	−	0.699948i	$-0.246794\pi$
$433$	29.7228	1.42839	0.714194	+	0.699948i	$0.246794\pi$
$434$	5.62772	0.270139
$435$	0	0
$436$	7.48913	0.358664
$437$	−55.7228	−2.66558
$438$	0	0
$439$	−26.9783	−1.28760	−0.643801	−	0.765193i	$-0.722643\pi$
$439$	−26.9783	−1.28760	−0.643801	+	0.765193i	$0.722643\pi$
$440$	0	0
$441$	0	0
$442$	−8.74456	−0.415936
$443$	6.51087	0.309341	0.154670	−	0.987966i	$-0.450568\pi$
$443$	6.51087	0.309341	0.154670	+	0.987966i	$0.450568\pi$
$444$	0	0
$445$	0	0
$446$	−7.25544	−0.343555
$447$	0	0
$448$	2.37228	0.112080
$449$	16.9783	0.801253	0.400627	−	0.916241i	$-0.368792\pi$
$449$	16.9783	0.801253	0.400627	+	0.916241i	$0.368792\pi$
$450$	0	0
$451$	−11.4891	−0.541002
$452$	14.7446	0.693526
$453$	0	0
$454$	−8.74456	−0.410403
$455$	0	0
$456$	0	0
$457$	−35.3505	−1.65363	−0.826814	−	0.562475i	$-0.809849\pi$
$457$	−35.3505	−1.65363	−0.826814	+	0.562475i	$0.809849\pi$
$458$	10.0000	0.467269
$459$	0	0
$460$	0	0
$461$	1.11684	0.0520166	0.0260083	−	0.999662i	$-0.491720\pi$
$461$	1.11684	0.0520166	0.0260083	+	0.999662i	$0.491720\pi$
$462$	0	0
$463$	−34.2337	−1.59097	−0.795487	−	0.605970i	$-0.792785\pi$
$463$	−34.2337	−1.59097	−0.795487	+	0.605970i	$0.792785\pi$
$464$	4.37228	0.202978
$465$	0	0
$466$	4.37228	0.202542
$467$	−13.6277	−0.630616	−0.315308	−	0.948989i	$-0.602108\pi$
$467$	−13.6277	−0.630616	−0.315308	+	0.948989i	$0.602108\pi$
$468$	0	0
$469$	−18.9783	−0.876334
$470$	0	0
$471$	0	0
$472$	8.74456	0.402501
$473$	4.00000	0.183920
$474$	0	0
$475$	0	0
$476$	−10.3723	−0.475413
$477$	0	0
$478$	−3.25544	−0.148900
$479$	17.4891	0.799099	0.399549	−	0.916712i	$-0.369167\pi$
$479$	17.4891	0.799099	0.399549	+	0.916712i	$0.369167\pi$
$480$	0	0
$481$	7.25544	0.330819
$482$	22.0000	1.00207
$483$	0	0
$484$	1.00000	0.0454545
$485$	0	0
$486$	0	0
$487$	−20.0000	−0.906287	−0.453143	−	0.891438i	$-0.649697\pi$
$487$	−20.0000	−0.906287	−0.453143	+	0.891438i	$0.649697\pi$
$488$	−0.372281	−0.0168524
$489$	0	0
$490$	0	0
$491$	1.62772	0.0734579	0.0367290	−	0.999325i	$-0.488306\pi$
$491$	1.62772	0.0734579	0.0367290	+	0.999325i	$0.488306\pi$
$492$	0	0
$493$	−19.1168	−0.860979
$494$	12.7446	0.573405
$495$	0	0
$496$	−2.37228	−0.106519
$497$	16.8832	0.757313
$498$	0	0
$499$	−10.5109	−0.470531	−0.235266	−	0.971931i	$-0.575596\pi$
$499$	−10.5109	−0.470531	−0.235266	+	0.971931i	$0.575596\pi$
$500$	0	0
$501$	0	0
$502$	−8.74456	−0.390289
$503$	10.9783	0.489496	0.244748	−	0.969587i	$-0.421295\pi$
$503$	10.9783	0.489496	0.244748	+	0.969587i	$0.421295\pi$
$504$	0	0
$505$	0	0
$506$	8.74456	0.388743
$507$	0	0
$508$	−8.00000	−0.354943
$509$	−44.2337	−1.96062	−0.980312	−	0.197455i	$-0.936732\pi$
$509$	−44.2337	−1.96062	−0.980312	+	0.197455i	$0.936732\pi$
$510$	0	0
$511$	−17.7663	−0.785935
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	−18.0000	−0.793946
$515$	0	0
$516$	0	0
$517$	−8.74456	−0.384585
$518$	8.60597	0.378125
$519$	0	0
$520$	0	0
$521$	−35.4891	−1.55481	−0.777403	−	0.629002i	$-0.783464\pi$
$521$	−35.4891	−1.55481	−0.777403	+	0.629002i	$0.783464\pi$
$522$	0	0
$523$	14.9783	0.654953	0.327477	−	0.944859i	$-0.393802\pi$
$523$	14.9783	0.654953	0.327477	+	0.944859i	$0.393802\pi$
$524$	−4.88316	−0.213322
$525$	0	0
$526$	−3.86141	−0.168365
$527$	10.3723	0.451824
$528$	0	0
$529$	53.4674	2.32467
$530$	0	0
$531$	0	0
$532$	15.1168	0.655398
$533$	22.9783	0.995299
$534$	0	0
$535$	0	0
$536$	8.00000	0.345547
$537$	0	0
$538$	2.74456	0.118326
$539$	−1.37228	−0.0591083
$540$	0	0
$541$	−2.88316	−0.123957	−0.0619783	−	0.998077i	$-0.519741\pi$
$541$	−2.88316	−0.123957	−0.0619783	+	0.998077i	$0.519741\pi$
$542$	16.0000	0.687259
$543$	0	0
$544$	4.37228	0.187460
$545$	0	0
$546$	0	0
$547$	−20.0000	−0.855138	−0.427569	−	0.903983i	$-0.640630\pi$
$547$	−20.0000	−0.855138	−0.427569	+	0.903983i	$0.640630\pi$
$548$	−2.74456	−0.117242
$549$	0	0
$550$	0	0
$551$	27.8614	1.18694
$552$	0	0
$553$	−30.2337	−1.28567
$554$	−1.25544	−0.0533384
$555$	0	0
$556$	−4.00000	−0.169638
$557$	−40.9783	−1.73630	−0.868152	−	0.496298i	$-0.834692\pi$
$557$	−40.9783	−1.73630	−0.868152	+	0.496298i	$0.834692\pi$
$558$	0	0
$559$	−8.00000	−0.338364
$560$	0	0
$561$	0	0
$562$	18.0000	0.759284
$563$	26.2337	1.10562	0.552809	−	0.833308i	$-0.313556\pi$
$563$	26.2337	1.10562	0.552809	+	0.833308i	$0.313556\pi$
$564$	0	0
$565$	0	0
$566$	16.7446	0.703826
$567$	0	0
$568$	−7.11684	−0.298616
$569$	26.7446	1.12119	0.560595	−	0.828090i	$-0.310572\pi$
$569$	26.7446	1.12119	0.560595	+	0.828090i	$0.310572\pi$
$570$	0	0
$571$	9.62772	0.402907	0.201454	−	0.979498i	$-0.435433\pi$
$571$	9.62772	0.402907	0.201454	+	0.979498i	$0.435433\pi$
$572$	−2.00000	−0.0836242
$573$	0	0
$574$	27.2554	1.13762
$575$	0	0
$576$	0	0
$577$	8.97825	0.373769	0.186885	−	0.982382i	$-0.440161\pi$
$577$	8.97825	0.373769	0.186885	+	0.982382i	$0.440161\pi$
$578$	−2.11684	−0.0880491
$579$	0	0
$580$	0	0
$581$	20.7446	0.860629
$582$	0	0
$583$	13.1168	0.543244
$584$	7.48913	0.309902
$585$	0	0
$586$	−0.510875	−0.0211040
$587$	−3.86141	−0.159377	−0.0796887	−	0.996820i	$-0.525393\pi$
$587$	−3.86141	−0.159377	−0.0796887	+	0.996820i	$0.525393\pi$
$588$	0	0
$589$	−15.1168	−0.622879
$590$	0	0
$591$	0	0
$592$	−3.62772	−0.149098
$593$	−35.4891	−1.45736	−0.728682	−	0.684852i	$-0.759867\pi$
$593$	−35.4891	−1.45736	−0.728682	+	0.684852i	$0.759867\pi$
$594$	0	0
$595$	0	0
$596$	18.6060	0.762130
$597$	0	0
$598$	−17.4891	−0.715184
$599$	−0.605969	−0.0247592	−0.0123796	−	0.999923i	$-0.503941\pi$
$599$	−0.605969	−0.0247592	−0.0123796	+	0.999923i	$0.503941\pi$
$600$	0	0
$601$	−39.4891	−1.61080	−0.805398	−	0.592735i	$-0.798048\pi$
$601$	−39.4891	−1.61080	−0.805398	+	0.592735i	$0.798048\pi$
$602$	−9.48913	−0.386748
$603$	0	0
$604$	22.2337	0.904676
$605$	0	0
$606$	0	0
$607$	23.1168	0.938284	0.469142	−	0.883123i	$-0.344563\pi$
$607$	23.1168	0.938284	0.469142	+	0.883123i	$0.344563\pi$
$608$	−6.37228	−0.258430
$609$	0	0
$610$	0	0
$611$	17.4891	0.707534
$612$	0	0
$613$	−43.4891	−1.75651	−0.878255	−	0.478193i	$-0.841292\pi$
$613$	−43.4891	−1.75651	−0.878255	+	0.478193i	$0.841292\pi$
$614$	16.7446	0.675756
$615$	0	0
$616$	−2.37228	−0.0955819
$617$	32.2337	1.29768	0.648840	−	0.760925i	$-0.275255\pi$
$617$	32.2337	1.29768	0.648840	+	0.760925i	$0.275255\pi$
$618$	0	0
$619$	24.4674	0.983427	0.491713	−	0.870757i	$-0.336371\pi$
$619$	24.4674	0.983427	0.491713	+	0.870757i	$0.336371\pi$
$620$	0	0
$621$	0	0
$622$	13.6277	0.546422
$623$	−10.3723	−0.415557
$624$	0	0
$625$	0	0
$626$	−22.0000	−0.879297
$627$	0	0
$628$	−3.62772	−0.144762
$629$	15.8614	0.632436
$630$	0	0
$631$	24.8832	0.990583	0.495291	−	0.868727i	$-0.335061\pi$
$631$	24.8832	0.990583	0.495291	+	0.868727i	$0.335061\pi$
$632$	12.7446	0.506951
$633$	0	0
$634$	−27.3505	−1.08623
$635$	0	0
$636$	0	0
$637$	2.74456	0.108744
$638$	−4.37228	−0.173100
$639$	0	0
$640$	0	0
$641$	−36.0951	−1.42567	−0.712835	−	0.701332i	$-0.752589\pi$
$641$	−36.0951	−1.42567	−0.712835	+	0.701332i	$0.752589\pi$
$642$	0	0
$643$	23.1168	0.911639	0.455820	−	0.890072i	$-0.349346\pi$
$643$	23.1168	0.911639	0.455820	+	0.890072i	$0.349346\pi$
$644$	−20.7446	−0.817450
$645$	0	0
$646$	27.8614	1.09619
$647$	19.7228	0.775384	0.387692	−	0.921789i	$-0.373272\pi$
$647$	19.7228	0.775384	0.387692	+	0.921789i	$0.373272\pi$
$648$	0	0
$649$	−8.74456	−0.343254
$650$	0	0
$651$	0	0
$652$	23.1168	0.905325
$653$	16.3723	0.640697	0.320348	−	0.947300i	$-0.396200\pi$
$653$	16.3723	0.640697	0.320348	+	0.947300i	$0.396200\pi$
$654$	0	0
$655$	0	0
$656$	−11.4891	−0.448575
$657$	0	0
$658$	20.7446	0.808707
$659$	15.8614	0.617873	0.308936	−	0.951083i	$-0.400027\pi$
$659$	15.8614	0.617873	0.308936	+	0.951083i	$0.400027\pi$
$660$	0	0
$661$	31.4891	1.22479	0.612393	−	0.790554i	$-0.290207\pi$
$661$	31.4891	1.22479	0.612393	+	0.790554i	$0.290207\pi$
$662$	14.9783	0.582146
$663$	0	0
$664$	−8.74456	−0.339355
$665$	0	0
$666$	0	0
$667$	−38.2337	−1.48041
$668$	−10.3723	−0.401316
$669$	0	0
$670$	0	0
$671$	0.372281	0.0143718
$672$	0	0
$673$	13.8614	0.534318	0.267159	−	0.963652i	$-0.413915\pi$
$673$	13.8614	0.534318	0.267159	+	0.963652i	$0.413915\pi$
$674$	6.88316	0.265129
$675$	0	0
$676$	−9.00000	−0.346154
$677$	14.7446	0.566680	0.283340	−	0.959020i	$-0.408558\pi$
$677$	14.7446	0.566680	0.283340	+	0.959020i	$0.408558\pi$
$678$	0	0
$679$	2.97825	0.114295
$680$	0	0
$681$	0	0
$682$	2.37228	0.0908393
$683$	−34.3723	−1.31522	−0.657609	−	0.753359i	$-0.728432\pi$
$683$	−34.3723	−1.31522	−0.657609	+	0.753359i	$0.728432\pi$
$684$	0	0
$685$	0	0
$686$	19.8614	0.758312
$687$	0	0
$688$	4.00000	0.152499
$689$	−26.2337	−0.999424
$690$	0	0
$691$	5.76631	0.219361	0.109680	−	0.993967i	$-0.465017\pi$
$691$	5.76631	0.219361	0.109680	+	0.993967i	$0.465017\pi$
$692$	−6.00000	−0.228086
$693$	0	0
$694$	−2.23369	−0.0847896
$695$	0	0
$696$	0	0
$697$	50.2337	1.90274
$698$	3.48913	0.132065
$699$	0	0
$700$	0	0
$701$	31.6277	1.19456	0.597281	−	0.802032i	$-0.296248\pi$
$701$	31.6277	1.19456	0.597281	+	0.802032i	$0.296248\pi$
$702$	0	0
$703$	−23.1168	−0.871868
$704$	1.00000	0.0376889
$705$	0	0
$706$	23.4891	0.884025
$707$	−14.2337	−0.535313
$708$	0	0
$709$	−10.0000	−0.375558	−0.187779	−	0.982211i	$-0.560129\pi$
$709$	−10.0000	−0.375558	−0.187779	+	0.982211i	$0.560129\pi$
$710$	0	0
$711$	0	0
$712$	4.37228	0.163858
$713$	20.7446	0.776890
$714$	0	0
$715$	0	0
$716$	12.0000	0.448461
$717$	0	0
$718$	0	0
$719$	−31.1168	−1.16046	−0.580231	−	0.814452i	$-0.697038\pi$
$719$	−31.1168	−1.16046	−0.580231	+	0.814452i	$0.697038\pi$
$720$	0	0
$721$	−32.0000	−1.19174
$722$	−21.6060	−0.804091
$723$	0	0
$724$	−10.0000	−0.371647
$725$	0	0
$726$	0	0
$727$	−10.2337	−0.379546	−0.189773	−	0.981828i	$-0.560775\pi$
$727$	−10.2337	−0.379546	−0.189773	+	0.981828i	$0.560775\pi$
$728$	4.74456	0.175845
$729$	0	0
$730$	0	0
$731$	−17.4891	−0.646859
$732$	0	0
$733$	−11.7663	−0.434599	−0.217299	−	0.976105i	$-0.569725\pi$
$733$	−11.7663	−0.434599	−0.217299	+	0.976105i	$0.569725\pi$
$734$	−4.00000	−0.147643
$735$	0	0
$736$	8.74456	0.322329
$737$	−8.00000	−0.294684
$738$	0	0
$739$	48.4674	1.78290	0.891451	−	0.453118i	$-0.149688\pi$
$739$	48.4674	1.78290	0.891451	+	0.453118i	$0.149688\pi$
$740$	0	0
$741$	0	0
$742$	−31.1168	−1.14234
$743$	−10.3723	−0.380522	−0.190261	−	0.981734i	$-0.560933\pi$
$743$	−10.3723	−0.380522	−0.190261	+	0.981734i	$0.560933\pi$
$744$	0	0
$745$	0	0
$746$	8.51087	0.311605
$747$	0	0
$748$	−4.37228	−0.159866
$749$	−28.4674	−1.04018
$750$	0	0
$751$	−19.8614	−0.724753	−0.362377	−	0.932032i	$-0.618035\pi$
$751$	−19.8614	−0.724753	−0.362377	+	0.932032i	$0.618035\pi$
$752$	−8.74456	−0.318881
$753$	0	0
$754$	8.74456	0.318458
$755$	0	0
$756$	0	0
$757$	−24.9783	−0.907850	−0.453925	−	0.891040i	$-0.649977\pi$
$757$	−24.9783	−0.907850	−0.453925	+	0.891040i	$0.649977\pi$
$758$	−34.2337	−1.24342
$759$	0	0
$760$	0	0
$761$	40.9783	1.48546	0.742730	−	0.669591i	$-0.233531\pi$
$761$	40.9783	1.48546	0.742730	+	0.669591i	$0.233531\pi$
$762$	0	0
$763$	17.7663	0.643184
$764$	−17.4891	−0.632734
$765$	0	0
$766$	2.23369	0.0807064
$767$	17.4891	0.631496
$768$	0	0
$769$	−22.0000	−0.793340	−0.396670	−	0.917961i	$-0.629834\pi$
$769$	−22.0000	−0.793340	−0.396670	+	0.917961i	$0.629834\pi$
$770$	0	0
$771$	0	0
$772$	13.8614	0.498883
$773$	6.60597	0.237600	0.118800	−	0.992918i	$-0.462095\pi$
$773$	6.60597	0.237600	0.118800	+	0.992918i	$0.462095\pi$
$774$	0	0
$775$	0	0
$776$	−1.25544	−0.0450676
$777$	0	0
$778$	6.00000	0.215110
$779$	−73.2119	−2.62309
$780$	0	0
$781$	7.11684	0.254661
$782$	−38.2337	−1.36723
$783$	0	0
$784$	−1.37228	−0.0490100
$785$	0	0
$786$	0	0
$787$	−24.4674	−0.872168	−0.436084	−	0.899906i	$-0.643635\pi$
$787$	−24.4674	−0.872168	−0.436084	+	0.899906i	$0.643635\pi$
$788$	−9.25544	−0.329711
$789$	0	0
$790$	0	0
$791$	34.9783	1.24368
$792$	0	0
$793$	−0.744563	−0.0264402
$794$	−20.9783	−0.744490
$795$	0	0
$796$	0.883156	0.0313026
$797$	−11.4891	−0.406966	−0.203483	−	0.979079i	$-0.565226\pi$
$797$	−11.4891	−0.406966	−0.203483	+	0.979079i	$0.565226\pi$
$798$	0	0
$799$	38.2337	1.35261
$800$	0	0
$801$	0	0
$802$	7.62772	0.269344
$803$	−7.48913	−0.264285
$804$	0	0
$805$	0	0
$806$	−4.74456	−0.167120
$807$	0	0
$808$	6.00000	0.211079
$809$	−18.0000	−0.632846	−0.316423	−	0.948618i	$-0.602482\pi$
$809$	−18.0000	−0.632846	−0.316423	+	0.948618i	$0.602482\pi$
$810$	0	0
$811$	16.1386	0.566703	0.283351	−	0.959016i	$-0.408554\pi$
$811$	16.1386	0.566703	0.283351	+	0.959016i	$0.408554\pi$
$812$	10.3723	0.363996
$813$	0	0
$814$	3.62772	0.127151
$815$	0	0
$816$	0	0
$817$	25.4891	0.891752
$818$	36.2337	1.26688
$819$	0	0
$820$	0	0
$821$	11.4891	0.400973	0.200487	−	0.979696i	$-0.435748\pi$
$821$	11.4891	0.400973	0.200487	+	0.979696i	$0.435748\pi$
$822$	0	0
$823$	28.0000	0.976019	0.488009	−	0.872838i	$-0.337723\pi$
$823$	28.0000	0.976019	0.488009	+	0.872838i	$0.337723\pi$
$824$	13.4891	0.469916
$825$	0	0
$826$	20.7446	0.721796
$827$	1.02175	0.0355297	0.0177649	−	0.999842i	$-0.494345\pi$
$827$	1.02175	0.0355297	0.0177649	+	0.999842i	$0.494345\pi$
$828$	0	0
$829$	−13.2554	−0.460380	−0.230190	−	0.973146i	$-0.573935\pi$
$829$	−13.2554	−0.460380	−0.230190	+	0.973146i	$0.573935\pi$
$830$	0	0
$831$	0	0
$832$	−2.00000	−0.0693375
$833$	6.00000	0.207888
$834$	0	0
$835$	0	0
$836$	6.37228	0.220390
$837$	0	0
$838$	−12.0000	−0.414533
$839$	−34.9783	−1.20758	−0.603792	−	0.797142i	$-0.706344\pi$
$839$	−34.9783	−1.20758	−0.603792	+	0.797142i	$0.706344\pi$
$840$	0	0
$841$	−9.88316	−0.340798
$842$	24.2337	0.835148
$843$	0	0
$844$	−11.1168	−0.382658
$845$	0	0
$846$	0	0
$847$	2.37228	0.0815126
$848$	13.1168	0.450434
$849$	0	0
$850$	0	0
$851$	31.7228	1.08744
$852$	0	0
$853$	−30.4674	−1.04318	−0.521592	−	0.853195i	$-0.674661\pi$
$853$	−30.4674	−1.04318	−0.521592	+	0.853195i	$0.674661\pi$
$854$	−0.883156	−0.0302210
$855$	0	0
$856$	12.0000	0.410152
$857$	−15.3505	−0.524364	−0.262182	−	0.965018i	$-0.584442\pi$
$857$	−15.3505	−0.524364	−0.262182	+	0.965018i	$0.584442\pi$
$858$	0	0
$859$	−31.2554	−1.06642	−0.533211	−	0.845982i	$-0.679015\pi$
$859$	−31.2554	−1.06642	−0.533211	+	0.845982i	$0.679015\pi$
$860$	0	0
$861$	0	0
$862$	10.9783	0.373921
$863$	32.7446	1.11464	0.557319	−	0.830299i	$-0.311830\pi$
$863$	32.7446	1.11464	0.557319	+	0.830299i	$0.311830\pi$
$864$	0	0
$865$	0	0
$866$	−29.7228	−1.01002
$867$	0	0
$868$	−5.62772	−0.191017
$869$	−12.7446	−0.432330
$870$	0	0
$871$	16.0000	0.542139
$872$	−7.48913	−0.253614
$873$	0	0
$874$	55.7228	1.88485
$875$	0	0
$876$	0	0
$877$	8.97825	0.303174	0.151587	−	0.988444i	$-0.451562\pi$
$877$	8.97825	0.303174	0.151587	+	0.988444i	$0.451562\pi$
$878$	26.9783	0.910472
$879$	0	0
$880$	0	0
$881$	−18.0000	−0.606435	−0.303218	−	0.952921i	$-0.598061\pi$
$881$	−18.0000	−0.606435	−0.303218	+	0.952921i	$0.598061\pi$
$882$	0	0
$883$	2.37228	0.0798336	0.0399168	−	0.999203i	$-0.487291\pi$
$883$	2.37228	0.0798336	0.0399168	+	0.999203i	$0.487291\pi$
$884$	8.74456	0.294111
$885$	0	0
$886$	−6.51087	−0.218737
$887$	34.9783	1.17445	0.587227	−	0.809422i	$-0.300219\pi$
$887$	34.9783	1.17445	0.587227	+	0.809422i	$0.300219\pi$
$888$	0	0
$889$	−18.9783	−0.636510
$890$	0	0
$891$	0	0
$892$	7.25544	0.242930
$893$	−55.7228	−1.86469
$894$	0	0
$895$	0	0
$896$	−2.37228	−0.0792524
$897$	0	0
$898$	−16.9783	−0.566572
$899$	−10.3723	−0.345935
$900$	0	0
$901$	−57.3505	−1.91062
$902$	11.4891	0.382546
$903$	0	0
$904$	−14.7446	−0.490397
$905$	0	0
$906$	0	0
$907$	40.6060	1.34830	0.674150	−	0.738595i	$-0.264510\pi$
$907$	40.6060	1.34830	0.674150	+	0.738595i	$0.264510\pi$
$908$	8.74456	0.290199
$909$	0	0
$910$	0	0
$911$	48.6060	1.61039	0.805194	−	0.593012i	$-0.202061\pi$
$911$	48.6060	1.61039	0.805194	+	0.593012i	$0.202061\pi$
$912$	0	0
$913$	8.74456	0.289403
$914$	35.3505	1.16929
$915$	0	0
$916$	−10.0000	−0.330409
$917$	−11.5842	−0.382545
$918$	0	0
$919$	−26.9783	−0.889930	−0.444965	−	0.895548i	$-0.646784\pi$
$919$	−26.9783	−0.889930	−0.444965	+	0.895548i	$0.646784\pi$
$920$	0	0
$921$	0	0
$922$	−1.11684	−0.0367813
$923$	−14.2337	−0.468508
$924$	0	0
$925$	0	0
$926$	34.2337	1.12499
$927$	0	0
$928$	−4.37228	−0.143527
$929$	27.3505	0.897342	0.448671	−	0.893697i	$-0.351898\pi$
$929$	27.3505	0.897342	0.448671	+	0.893697i	$0.351898\pi$
$930$	0	0
$931$	−8.74456	−0.286591
$932$	−4.37228	−0.143219
$933$	0	0
$934$	13.6277	0.445913
$935$	0	0
$936$	0	0
$937$	51.4891	1.68208	0.841038	−	0.540976i	$-0.181945\pi$
$937$	51.4891	1.68208	0.841038	+	0.540976i	$0.181945\pi$
$938$	18.9783	0.619662
$939$	0	0
$940$	0	0
$941$	−48.0951	−1.56786	−0.783928	−	0.620852i	$-0.786787\pi$
$941$	−48.0951	−1.56786	−0.783928	+	0.620852i	$0.786787\pi$
$942$	0	0
$943$	100.467	3.27167
$944$	−8.74456	−0.284611
$945$	0	0
$946$	−4.00000	−0.130051
$947$	20.1386	0.654416	0.327208	−	0.944952i	$-0.393892\pi$
$947$	20.1386	0.654416	0.327208	+	0.944952i	$0.393892\pi$
$948$	0	0
$949$	14.9783	0.486215
$950$	0	0
$951$	0	0
$952$	10.3723	0.336168
$953$	22.8832	0.741258	0.370629	−	0.928781i	$-0.379142\pi$
$953$	22.8832	0.741258	0.370629	+	0.928781i	$0.379142\pi$
$954$	0	0
$955$	0	0
$956$	3.25544	0.105288
$957$	0	0
$958$	−17.4891	−0.565048
$959$	−6.51087	−0.210247
$960$	0	0
$961$	−25.3723	−0.818461
$962$	−7.25544	−0.233925
$963$	0	0
$964$	−22.0000	−0.708572
$965$	0	0
$966$	0	0
$967$	−7.39403	−0.237776	−0.118888	−	0.992908i	$-0.537933\pi$
$967$	−7.39403	−0.237776	−0.118888	+	0.992908i	$0.537933\pi$
$968$	−1.00000	−0.0321412
$969$	0	0
$970$	0	0
$971$	46.9783	1.50760	0.753802	−	0.657102i	$-0.228218\pi$
$971$	46.9783	1.50760	0.753802	+	0.657102i	$0.228218\pi$
$972$	0	0
$973$	−9.48913	−0.304207
$974$	20.0000	0.640841
$975$	0	0
$976$	0.372281	0.0119164
$977$	−20.2337	−0.647333	−0.323667	−	0.946171i	$-0.604916\pi$
$977$	−20.2337	−0.647333	−0.323667	+	0.946171i	$0.604916\pi$
$978$	0	0
$979$	−4.37228	−0.139739
$980$	0	0
$981$	0	0
$982$	−1.62772	−0.0519426
$983$	43.7228	1.39454	0.697271	−	0.716808i	$-0.254398\pi$
$983$	43.7228	1.39454	0.697271	+	0.716808i	$0.254398\pi$
$984$	0	0
$985$	0	0
$986$	19.1168	0.608804
$987$	0	0
$988$	−12.7446	−0.405459
$989$	−34.9783	−1.11224
$990$	0	0
$991$	8.00000	0.254128	0.127064	−	0.991894i	$-0.459445\pi$
$991$	8.00000	0.254128	0.127064	+	0.991894i	$0.459445\pi$
$992$	2.37228	0.0753200
$993$	0	0
$994$	−16.8832	−0.535501
$995$	0	0
$996$	0	0
$997$	12.2337	0.387445	0.193722	−	0.981056i	$-0.437944\pi$
$997$	12.2337	0.387445	0.193722	+	0.981056i	$0.437944\pi$
$998$	10.5109	0.332716
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4950.2.a.bw.1.2		2
3.2	odd	2		550.2.a.n.1.1		2
5.2	odd	4		4950.2.c.bc.199.2		4
5.3	odd	4		4950.2.c.bc.199.3		4
5.4	even	2		990.2.a.m.1.1		2
12.11	even	2		4400.2.a.bl.1.2		2
15.2	even	4		550.2.b.f.199.4		4
15.8	even	4		550.2.b.f.199.1		4
15.14	odd	2		110.2.a.d.1.2	✓	2
20.19	odd	2		7920.2.a.bq.1.2		2
33.32	even	2		6050.2.a.cb.1.1		2
60.23	odd	4		4400.2.b.p.4049.3		4
60.47	odd	4		4400.2.b.p.4049.2		4
60.59	even	2		880.2.a.n.1.1		2
105.104	even	2		5390.2.a.bp.1.1		2
120.29	odd	2		3520.2.a.bq.1.1		2
120.59	even	2		3520.2.a.bj.1.2		2
165.164	even	2		1210.2.a.r.1.2		2
660.659	odd	2		9680.2.a.bt.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
110.2.a.d.1.2	✓	2	15.14	odd	2
550.2.a.n.1.1		2	3.2	odd	2
550.2.b.f.199.1		4	15.8	even	4
550.2.b.f.199.4		4	15.2	even	4
880.2.a.n.1.1		2	60.59	even	2
990.2.a.m.1.1		2	5.4	even	2
1210.2.a.r.1.2		2	165.164	even	2
3520.2.a.bj.1.2		2	120.59	even	2
3520.2.a.bq.1.1		2	120.29	odd	2
4400.2.a.bl.1.2		2	12.11	even	2
4400.2.b.p.4049.2		4	60.47	odd	4
4400.2.b.p.4049.3		4	60.23	odd	4
4950.2.a.bw.1.2		2	1.1	even	1	trivial
4950.2.c.bc.199.2		4	5.2	odd	4
4950.2.c.bc.199.3		4	5.3	odd	4
5390.2.a.bp.1.1		2	105.104	even	2
6050.2.a.cb.1.1		2	33.32	even	2
7920.2.a.bq.1.2		2	20.19	odd	2
9680.2.a.bt.1.1		2	660.659	odd	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:	\( N \)	\(=\)	\( 4950 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4950.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{4} +2.37228 q^{7} -1.00000 q^{8} +1.00000 q^{11} -2.00000 q^{13} -2.37228 q^{14} +1.00000 q^{16} -4.37228 q^{17} +6.37228 q^{19} -1.00000 q^{22} -8.74456 q^{23} +2.00000 q^{26} +2.37228 q^{28} +4.37228 q^{29} -2.37228 q^{31} -1.00000 q^{32} +4.37228 q^{34} -3.62772 q^{37} -6.37228 q^{38} -11.4891 q^{41} +4.00000 q^{43} +1.00000 q^{44} +8.74456 q^{46} -8.74456 q^{47} -1.37228 q^{49} -2.00000 q^{52} +13.1168 q^{53} -2.37228 q^{56} -4.37228 q^{58} -8.74456 q^{59} +0.372281 q^{61} +2.37228 q^{62} +1.00000 q^{64} -8.00000 q^{67} -4.37228 q^{68} +7.11684 q^{71} -7.48913 q^{73} +3.62772 q^{74} +6.37228 q^{76} +2.37228 q^{77} -12.7446 q^{79} +11.4891 q^{82} +8.74456 q^{83} -4.00000 q^{86} -1.00000 q^{88} -4.37228 q^{89} -4.74456 q^{91} -8.74456 q^{92} +8.74456 q^{94} +1.25544 q^{97} +1.37228 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} + 2 q^{4} - q^{7} - 2 q^{8} + 2 q^{11} - 4 q^{13} + q^{14} + 2 q^{16} - 3 q^{17} + 7 q^{19} - 2 q^{22} - 6 q^{23} + 4 q^{26} - q^{28} + 3 q^{29} + q^{31} - 2 q^{32} + 3 q^{34} - 13 q^{37}+ \cdots - 3 q^{98}+O(q^{100}) \) 2 * q - 2 * q^2 + 2 * q^4 - q^7 - 2 * q^8 + 2 * q^11 - 4 * q^13 + q^14 + 2 * q^16 - 3 * q^17 + 7 * q^19 - 2 * q^22 - 6 * q^23 + 4 * q^26 - q^28 + 3 * q^29 + q^31 - 2 * q^32 + 3 * q^34 - 13 * q^37 - 7 * q^38 + 8 * q^43 + 2 * q^44 + 6 * q^46 - 6 * q^47 + 3 * q^49 - 4 * q^52 + 9 * q^53 + q^56 - 3 * q^58 - 6 * q^59 - 5 * q^61 - q^62 + 2 * q^64 - 16 * q^67 - 3 * q^68 - 3 * q^71 + 8 * q^73 + 13 * q^74 + 7 * q^76 - q^77 - 14 * q^79 + 6 * q^83 - 8 * q^86 - 2 * q^88 - 3 * q^89 + 2 * q^91 - 6 * q^92 + 6 * q^94 + 14 * q^97 - 3 * q^98

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