LMFDB - Embedded newform 9800.2.a.db.1.9

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9800,2,Mod(1,9800)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9800, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("9800.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 9800 = 2^{3} \cdot 5^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9800.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:	$78.2533939809$
Analytic rank:	$1$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - 22x^{8} - 16x^{7} + 146x^{6} + 200x^{5} - 206x^{4} - 440x^{3} - 124x^{2} + 72x + 28 $ x^10 - 22x^8 - 16x^7 + 146x^6 + 200x^5 - 206x^4 - 440x^3 - 124x^2 + 72x + 28
Coefficient ring:	$\Z[a_1, \ldots, a_{41}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	no (minimal twist has level 1960)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.9
Root			$3.39389$ of defining polynomial
Character	$\chi$	$=$	9800.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.69859 q^{3} +4.28239 q^{9} +O(q^{10})$	$q+2.69859 q^{3} +4.28239 q^{9} +0.919787 q^{11} -5.24470 q^{13} -4.25577 q^{17} -1.83949 q^{19} -8.78042 q^{23} +3.46065 q^{27} +6.71298 q^{29} +4.64200 q^{31} +2.48213 q^{33} -1.42175 q^{37} -14.1533 q^{39} +7.35699 q^{41} -9.80292 q^{43} +6.80187 q^{47} -11.4846 q^{51} -11.2567 q^{53} -4.96404 q^{57} -11.3331 q^{59} +9.64670 q^{61} -2.78505 q^{67} -23.6948 q^{69} +11.8349 q^{71} -5.11486 q^{73} +0.727336 q^{79} -3.50830 q^{81} +4.37545 q^{83} +18.1156 q^{87} +0.963998 q^{89} +12.5269 q^{93} -13.5588 q^{97} +3.93889 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q + 14 q^{9}+O(q^{10}) $ 10 * q + 14 * q^9	$ 10 q + 14 q^{9} - 12 q^{11} - 16 q^{23} + 12 q^{29} - 36 q^{37} - 20 q^{39} - 24 q^{43} - 36 q^{51} - 8 q^{53} - 16 q^{57} - 40 q^{67} - 8 q^{71} - 4 q^{79} + 50 q^{81} - 48 q^{93} - 72 q^{99}+O(q^{100}) $ 10 * q + 14 * q^9 - 12 * q^11 - 16 * q^23 + 12 * q^29 - 36 * q^37 - 20 * q^39 - 24 * q^43 - 36 * q^51 - 8 * q^53 - 16 * q^57 - 40 * q^67 - 8 * q^71 - 4 * q^79 + 50 * q^81 - 48 * q^93 - 72 * q^99

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 $

$2$	0	0
$2$	0	0
$3$	2.69859	1.55803	0.779016	−	0.627004i	$-0.215719\pi$
$3$	2.69859	1.55803	0.779016	+	0.627004i	$0.215719\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	4.28239	1.42746
$10$	0	0
$11$	0.919787	0.277326	0.138663	−	0.990340i	$-0.455719\pi$
$11$	0.919787	0.277326	0.138663	+	0.990340i	$0.455719\pi$
$12$	0	0
$13$	−5.24470	−1.45462	−0.727309	−	0.686310i	$-0.759229\pi$
$13$	−5.24470	−1.45462	−0.727309	+	0.686310i	$0.759229\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−4.25577	−1.03217	−0.516087	−	0.856536i	$-0.672612\pi$
$17$	−4.25577	−1.03217	−0.516087	+	0.856536i	$0.672612\pi$
$18$	0	0
$19$	−1.83949	−0.422009	−0.211004	−	0.977485i	$-0.567673\pi$
$19$	−1.83949	−0.422009	−0.211004	+	0.977485i	$0.567673\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−8.78042	−1.83085	−0.915423	−	0.402494i	$-0.868143\pi$
$23$	−8.78042	−1.83085	−0.915423	+	0.402494i	$0.868143\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	3.46065	0.666002
$28$	0	0
$29$	6.71298	1.24657	0.623284	−	0.781995i	$-0.285798\pi$
$29$	6.71298	1.24657	0.623284	+	0.781995i	$0.285798\pi$
$30$	0	0
$31$	4.64200	0.833728	0.416864	−	0.908969i	$-0.363129\pi$
$31$	4.64200	0.833728	0.416864	+	0.908969i	$0.363129\pi$
$32$	0	0
$33$	2.48213	0.432083
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−1.42175	−0.233735	−0.116867	−	0.993148i	$-0.537285\pi$
$37$	−1.42175	−0.233735	−0.116867	+	0.993148i	$0.537285\pi$
$38$	0	0
$39$	−14.1533	−2.26634
$40$	0	0
$41$	7.35699	1.14897	0.574484	−	0.818515i	$-0.305203\pi$
$41$	7.35699	1.14897	0.574484	+	0.818515i	$0.305203\pi$
$42$	0	0
$43$	−9.80292	−1.49493	−0.747466	−	0.664301i	$-0.768730\pi$
$43$	−9.80292	−1.49493	−0.747466	+	0.664301i	$0.768730\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	6.80187	0.992155	0.496078	−	0.868278i	$-0.334773\pi$
$47$	6.80187	0.992155	0.496078	+	0.868278i	$0.334773\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	−11.4846	−1.60816
$52$	0	0
$53$	−11.2567	−1.54623	−0.773113	−	0.634268i	$-0.781302\pi$
$53$	−11.2567	−1.54623	−0.773113	+	0.634268i	$0.781302\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−4.96404	−0.657503
$58$	0	0
$59$	−11.3331	−1.47544	−0.737721	−	0.675106i	$-0.764098\pi$
$59$	−11.3331	−1.47544	−0.737721	+	0.675106i	$0.764098\pi$
$60$	0	0
$61$	9.64670	1.23513	0.617567	−	0.786519i	$-0.288119\pi$
$61$	9.64670	1.23513	0.617567	+	0.786519i	$0.288119\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−2.78505	−0.340248	−0.170124	−	0.985423i	$-0.554417\pi$
$67$	−2.78505	−0.340248	−0.170124	+	0.985423i	$0.554417\pi$
$68$	0	0
$69$	−23.6948	−2.85252
$70$	0	0
$71$	11.8349	1.40455	0.702275	−	0.711906i	$-0.252168\pi$
$71$	11.8349	1.40455	0.702275	+	0.711906i	$0.252168\pi$
$72$	0	0
$73$	−5.11486	−0.598649	−0.299325	−	0.954151i	$-0.596761\pi$
$73$	−5.11486	−0.598649	−0.299325	+	0.954151i	$0.596761\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0	0
$78$	0	0
$79$	0.727336	0.0818317	0.0409159	−	0.999163i	$-0.486972\pi$
$79$	0.727336	0.0818317	0.0409159	+	0.999163i	$0.486972\pi$
$80$	0	0
$81$	−3.50830	−0.389811
$82$	0	0
$83$	4.37545	0.480268	0.240134	−	0.970740i	$-0.422809\pi$
$83$	4.37545	0.480268	0.240134	+	0.970740i	$0.422809\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	18.1156	1.94219
$88$	0	0
$89$	0.963998	0.102184	0.0510918	−	0.998694i	$-0.483730\pi$
$89$	0.963998	0.102184	0.0510918	+	0.998694i	$0.483730\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	12.5269	1.29898
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−13.5588	−1.37669	−0.688344	−	0.725384i	$-0.741662\pi$
$97$	−13.5588	−1.37669	−0.688344	+	0.725384i	$0.741662\pi$
$98$	0	0
$99$	3.93889	0.395873
$100$	0	0
$101$	−19.2029	−1.91076	−0.955379	−	0.295384i	$-0.904552\pi$
$101$	−19.2029	−1.91076	−0.955379	+	0.295384i	$0.904552\pi$
$102$	0	0
$103$	1.67686	0.165226	0.0826130	−	0.996582i	$-0.473673\pi$
$103$	1.67686	0.165226	0.0826130	+	0.996582i	$0.473673\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−18.1090	−1.75067	−0.875334	−	0.483520i	$-0.839358\pi$
$107$	−18.1090	−1.75067	−0.875334	+	0.483520i	$0.839358\pi$
$108$	0	0
$109$	−2.07628	−0.198871	−0.0994357	−	0.995044i	$-0.531704\pi$
$109$	−2.07628	−0.198871	−0.0994357	+	0.995044i	$0.531704\pi$
$110$	0	0
$111$	−3.83673	−0.364167
$112$	0	0
$113$	−1.45378	−0.136760	−0.0683800	−	0.997659i	$-0.521783\pi$
$113$	−1.45378	−0.136760	−0.0683800	+	0.997659i	$0.521783\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−22.4599	−2.07641
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−10.1540	−0.923090
$122$	0	0
$123$	19.8535	1.79013
$124$	0	0
$125$	0	0
$126$	0	0
$127$	7.25670	0.643928	0.321964	−	0.946752i	$-0.395657\pi$
$127$	7.25670	0.643928	0.321964	+	0.946752i	$0.395657\pi$
$128$	0	0
$129$	−26.4541	−2.32915
$130$	0	0
$131$	−4.04070	−0.353038	−0.176519	−	0.984297i	$-0.556484\pi$
$131$	−4.04070	−0.353038	−0.176519	+	0.984297i	$0.556484\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−4.35288	−0.371891	−0.185946	−	0.982560i	$-0.559535\pi$
$137$	−4.35288	−0.371891	−0.185946	+	0.982560i	$0.559535\pi$
$138$	0	0
$139$	−14.9602	−1.26891	−0.634455	−	0.772960i	$-0.718775\pi$
$139$	−14.9602	−1.26891	−0.634455	+	0.772960i	$0.718775\pi$
$140$	0	0
$141$	18.3555	1.54581
$142$	0	0
$143$	−4.82401	−0.403404
$144$	0	0
$145$	0	0
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−9.90223	−0.811223	−0.405611	−	0.914046i	$-0.632941\pi$
$149$	−9.90223	−0.811223	−0.405611	+	0.914046i	$0.632941\pi$
$150$	0	0
$151$	0.794625	0.0646657	0.0323328	−	0.999477i	$-0.489706\pi$
$151$	0.794625	0.0646657	0.0323328	+	0.999477i	$0.489706\pi$
$152$	0	0
$153$	−18.2249	−1.47339
$154$	0	0
$155$	0	0
$156$	0	0
$157$	23.1524	1.84777	0.923883	−	0.382675i	$-0.124997\pi$
$157$	23.1524	1.84777	0.923883	+	0.382675i	$0.124997\pi$
$158$	0	0
$159$	−30.3772	−2.40907
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−7.04479	−0.551791	−0.275895	−	0.961188i	$-0.588974\pi$
$163$	−7.04479	−0.551791	−0.275895	+	0.961188i	$0.588974\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−7.39931	−0.572576	−0.286288	−	0.958144i	$-0.592421\pi$
$167$	−7.39931	−0.572576	−0.286288	+	0.958144i	$0.592421\pi$
$168$	0	0
$169$	14.5069	1.11591
$170$	0	0
$171$	−7.87743	−0.602402
$172$	0	0
$173$	−9.11776	−0.693211	−0.346605	−	0.938011i	$-0.612666\pi$
$173$	−9.11776	−0.693211	−0.346605	+	0.938011i	$0.612666\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−30.5833	−2.29878
$178$	0	0
$179$	−13.8396	−1.03442	−0.517209	−	0.855859i	$-0.673029\pi$
$179$	−13.8396	−1.03442	−0.517209	+	0.855859i	$0.673029\pi$
$180$	0	0
$181$	0.0875186	0.00650521	0.00325260	−	0.999995i	$-0.498965\pi$
$181$	0.0875186	0.00650521	0.00325260	+	0.999995i	$0.498965\pi$
$182$	0	0
$183$	26.0325	1.92438
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−3.91440	−0.286249
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−12.6296	−0.913844	−0.456922	−	0.889507i	$-0.651048\pi$
$191$	−12.6296	−0.913844	−0.456922	+	0.889507i	$0.651048\pi$
$192$	0	0
$193$	−20.5968	−1.48259	−0.741295	−	0.671179i	$-0.765788\pi$
$193$	−20.5968	−1.48259	−0.741295	+	0.671179i	$0.765788\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−10.4851	−0.747033	−0.373517	−	0.927624i	$-0.621848\pi$
$197$	−10.4851	−0.747033	−0.373517	+	0.927624i	$0.621848\pi$
$198$	0	0
$199$	−11.8092	−0.837133	−0.418567	−	0.908186i	$-0.637467\pi$
$199$	−11.8092	−0.837133	−0.418567	+	0.908186i	$0.637467\pi$
$200$	0	0
$201$	−7.51572	−0.530118
$202$	0	0
$203$	0	0
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−37.6012	−2.61346
$208$	0	0
$209$	−1.69194	−0.117034
$210$	0	0
$211$	−11.3594	−0.782014	−0.391007	−	0.920388i	$-0.627873\pi$
$211$	−11.3594	−0.782014	−0.391007	+	0.920388i	$0.627873\pi$
$212$	0	0
$213$	31.9377	2.18833
$214$	0	0
$215$	0	0
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−13.8029	−0.932715
$220$	0	0
$221$	22.3202	1.50142
$222$	0	0
$223$	26.9321	1.80351	0.901753	−	0.432252i	$-0.142281\pi$
$223$	26.9321	1.80351	0.901753	+	0.432252i	$0.142281\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	17.9510	1.19145	0.595724	−	0.803190i	$-0.296865\pi$
$227$	17.9510	1.19145	0.595724	+	0.803190i	$0.296865\pi$
$228$	0	0
$229$	−16.0850	−1.06293	−0.531463	−	0.847082i	$-0.678357\pi$
$229$	−16.0850	−1.06293	−0.531463	+	0.847082i	$0.678357\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	19.5915	1.28348	0.641740	−	0.766922i	$-0.278213\pi$
$233$	19.5915	1.28348	0.641740	+	0.766922i	$0.278213\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	1.96278	0.127496
$238$	0	0
$239$	26.0767	1.68676	0.843382	−	0.537314i	$-0.180561\pi$
$239$	26.0767	1.68676	0.843382	+	0.537314i	$0.180561\pi$
$240$	0	0
$241$	23.2436	1.49725	0.748625	−	0.662994i	$-0.230714\pi$
$241$	23.2436	1.49725	0.748625	+	0.662994i	$0.230714\pi$
$242$	0	0
$243$	−19.8494	−1.27334
$244$	0	0
$245$	0	0
$246$	0	0
$247$	9.64759	0.613862
$248$	0	0
$249$	11.8075	0.748273
$250$	0	0
$251$	6.75366	0.426287	0.213144	−	0.977021i	$-0.431630\pi$
$251$	6.75366	0.426287	0.213144	+	0.977021i	$0.431630\pi$
$252$	0	0
$253$	−8.07612	−0.507742
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−18.4392	−1.15021	−0.575104	−	0.818080i	$-0.695038\pi$
$257$	−18.4392	−1.15021	−0.575104	+	0.818080i	$0.695038\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	28.7476	1.77943
$262$	0	0
$263$	9.92808	0.612192	0.306096	−	0.952001i	$-0.400977\pi$
$263$	9.92808	0.612192	0.306096	+	0.952001i	$0.400977\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	2.60144	0.159205
$268$	0	0
$269$	14.8533	0.905624	0.452812	−	0.891606i	$-0.350421\pi$
$269$	14.8533	0.905624	0.452812	+	0.891606i	$0.350421\pi$
$270$	0	0
$271$	−3.34123	−0.202965	−0.101483	−	0.994837i	$-0.532359\pi$
$271$	−3.34123	−0.202965	−0.101483	+	0.994837i	$0.532359\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	17.5428	1.05404	0.527021	−	0.849852i	$-0.323309\pi$
$277$	17.5428	1.05404	0.527021	+	0.849852i	$0.323309\pi$
$278$	0	0
$279$	19.8789	1.19012
$280$	0	0
$281$	1.54676	0.0922720	0.0461360	−	0.998935i	$-0.485309\pi$
$281$	1.54676	0.0922720	0.0461360	+	0.998935i	$0.485309\pi$
$282$	0	0
$283$	6.37758	0.379108	0.189554	−	0.981870i	$-0.439296\pi$
$283$	6.37758	0.379108	0.189554	+	0.981870i	$0.439296\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	1.11154	0.0653848
$290$	0	0
$291$	−36.5897	−2.14492
$292$	0	0
$293$	2.65690	0.155218	0.0776090	−	0.996984i	$-0.475271\pi$
$293$	2.65690	0.155218	0.0776090	+	0.996984i	$0.475271\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	3.18306	0.184700
$298$	0	0
$299$	46.0507	2.66318
$300$	0	0
$301$	0	0
$302$	0	0
$303$	−51.8207	−2.97702
$304$	0	0
$305$	0	0
$306$	0	0
$307$	11.0822	0.632493	0.316246	−	0.948677i	$-0.397577\pi$
$307$	11.0822	0.632493	0.316246	+	0.948677i	$0.397577\pi$
$308$	0	0
$309$	4.52516	0.257427
$310$	0	0
$311$	−22.9133	−1.29930	−0.649648	−	0.760235i	$-0.725084\pi$
$311$	−22.9133	−1.29930	−0.649648	+	0.760235i	$0.725084\pi$
$312$	0	0
$313$	−20.7070	−1.17043	−0.585214	−	0.810879i	$-0.698989\pi$
$313$	−20.7070	−1.17043	−0.585214	+	0.810879i	$0.698989\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	16.4624	0.924620	0.462310	−	0.886718i	$-0.347021\pi$
$317$	16.4624	0.924620	0.462310	+	0.886718i	$0.347021\pi$
$318$	0	0
$319$	6.17451	0.345706
$320$	0	0
$321$	−48.8689	−2.72760
$322$	0	0
$323$	7.82846	0.435587
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−5.60303	−0.309848
$328$	0	0
$329$	0	0
$330$	0	0
$331$	18.5231	1.01812	0.509062	−	0.860730i	$-0.329993\pi$
$331$	18.5231	1.01812	0.509062	+	0.860730i	$0.329993\pi$
$332$	0	0
$333$	−6.08851	−0.333648
$334$	0	0
$335$	0	0
$336$	0	0
$337$	9.84378	0.536225	0.268112	−	0.963388i	$-0.413600\pi$
$337$	9.84378	0.536225	0.268112	+	0.963388i	$0.413600\pi$
$338$	0	0
$339$	−3.92316	−0.213077
$340$	0	0
$341$	4.26965	0.231215
$342$	0	0
$343$	0	0
$344$	0	0
$345$	0	0
$346$	0	0
$347$	26.6878	1.43267	0.716337	−	0.697755i	$-0.245817\pi$
$347$	26.6878	1.43267	0.716337	+	0.697755i	$0.245817\pi$
$348$	0	0
$349$	8.56272	0.458352	0.229176	−	0.973385i	$-0.426397\pi$
$349$	8.56272	0.458352	0.229176	+	0.973385i	$0.426397\pi$
$350$	0	0
$351$	−18.1501	−0.968778
$352$	0	0
$353$	13.6743	0.727808	0.363904	−	0.931436i	$-0.381444\pi$
$353$	13.6743	0.727808	0.363904	+	0.931436i	$0.381444\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	5.99861	0.316594	0.158297	−	0.987392i	$-0.449400\pi$
$359$	5.99861	0.316594	0.158297	+	0.987392i	$0.449400\pi$
$360$	0	0
$361$	−15.6163	−0.821909
$362$	0	0
$363$	−27.4015	−1.43820
$364$	0	0
$365$	0	0
$366$	0	0
$367$	3.14847	0.164349	0.0821745	−	0.996618i	$-0.473814\pi$
$367$	3.14847	0.164349	0.0821745	+	0.996618i	$0.473814\pi$
$368$	0	0
$369$	31.5055	1.64011
$370$	0	0
$371$	0	0
$372$	0	0
$373$	3.95038	0.204543	0.102271	−	0.994757i	$-0.467389\pi$
$373$	3.95038	0.204543	0.102271	+	0.994757i	$0.467389\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−35.2076	−1.81328
$378$	0	0
$379$	−29.9555	−1.53871	−0.769355	−	0.638822i	$-0.779422\pi$
$379$	−29.9555	−1.53871	−0.769355	+	0.638822i	$0.779422\pi$
$380$	0	0
$381$	19.5829	1.00326
$382$	0	0
$383$	14.0242	0.716602	0.358301	−	0.933606i	$-0.383356\pi$
$383$	14.0242	0.716602	0.358301	+	0.933606i	$0.383356\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−41.9799	−2.13396
$388$	0	0
$389$	−10.3188	−0.523185	−0.261592	−	0.965178i	$-0.584248\pi$
$389$	−10.3188	−0.523185	−0.261592	+	0.965178i	$0.584248\pi$
$390$	0	0
$391$	37.3674	1.88975
$392$	0	0
$393$	−10.9042	−0.550044
$394$	0	0
$395$	0	0
$396$	0	0
$397$	9.68961	0.486308	0.243154	−	0.969988i	$-0.421818\pi$
$397$	9.68961	0.486308	0.243154	+	0.969988i	$0.421818\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−19.2731	−0.962454	−0.481227	−	0.876596i	$-0.659809\pi$
$401$	−19.2731	−0.962454	−0.481227	+	0.876596i	$0.659809\pi$
$402$	0	0
$403$	−24.3459	−1.21276
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−1.30771	−0.0648208
$408$	0	0
$409$	22.8951	1.13209	0.566044	−	0.824375i	$-0.308473\pi$
$409$	22.8951	1.13209	0.566044	+	0.824375i	$0.308473\pi$
$410$	0	0
$411$	−11.7466	−0.579419
$412$	0	0
$413$	0	0
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−40.3715	−1.97700
$418$	0	0
$419$	30.7282	1.50117	0.750585	−	0.660774i	$-0.229772\pi$
$419$	30.7282	1.50117	0.750585	+	0.660774i	$0.229772\pi$
$420$	0	0
$421$	−19.2411	−0.937754	−0.468877	−	0.883263i	$-0.655341\pi$
$421$	−19.2411	−0.937754	−0.468877	+	0.883263i	$0.655341\pi$
$422$	0	0
$423$	29.1283	1.41627
$424$	0	0
$425$	0	0
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−13.0180	−0.628516
$430$	0	0
$431$	14.0860	0.678499	0.339250	−	0.940696i	$-0.389827\pi$
$431$	14.0860	0.678499	0.339250	+	0.940696i	$0.389827\pi$
$432$	0	0
$433$	−1.91785	−0.0921658	−0.0460829	−	0.998938i	$-0.514674\pi$
$433$	−1.91785	−0.0921658	−0.0460829	+	0.998938i	$0.514674\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	16.1515	0.772633
$438$	0	0
$439$	12.6532	0.603906	0.301953	−	0.953323i	$-0.402361\pi$
$439$	12.6532	0.603906	0.301953	+	0.953323i	$0.402361\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−21.6954	−1.03078	−0.515390	−	0.856956i	$-0.672353\pi$
$443$	−21.6954	−1.03078	−0.515390	+	0.856956i	$0.672353\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−26.7221	−1.26391
$448$	0	0
$449$	23.9706	1.13124	0.565621	−	0.824666i	$-0.308637\pi$
$449$	23.9706	1.13124	0.565621	+	0.824666i	$0.308637\pi$
$450$	0	0
$451$	6.76687	0.318639
$452$	0	0
$453$	2.14437	0.100751
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−7.29836	−0.341403	−0.170701	−	0.985323i	$-0.554603\pi$
$457$	−7.29836	−0.341403	−0.170701	+	0.985323i	$0.554603\pi$
$458$	0	0
$459$	−14.7277	−0.687431
$460$	0	0
$461$	−19.2232	−0.895312	−0.447656	−	0.894206i	$-0.647741\pi$
$461$	−19.2232	−0.895312	−0.447656	+	0.894206i	$0.647741\pi$
$462$	0	0
$463$	−1.89096	−0.0878803	−0.0439401	−	0.999034i	$-0.513991\pi$
$463$	−1.89096	−0.0878803	−0.0439401	+	0.999034i	$0.513991\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−7.15284	−0.330994	−0.165497	−	0.986210i	$-0.552923\pi$
$467$	−7.15284	−0.330994	−0.165497	+	0.986210i	$0.552923\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	62.4790	2.87888
$472$	0	0
$473$	−9.01660	−0.414584
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−48.2056	−2.20718
$478$	0	0
$479$	7.46388	0.341033	0.170517	−	0.985355i	$-0.445456\pi$
$479$	7.46388	0.341033	0.170517	+	0.985355i	$0.445456\pi$
$480$	0	0
$481$	7.45667	0.339995
$482$	0	0
$483$	0	0
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−38.1373	−1.72817	−0.864084	−	0.503348i	$-0.832101\pi$
$487$	−38.1373	−1.72817	−0.864084	+	0.503348i	$0.832101\pi$
$488$	0	0
$489$	−19.0110	−0.859708
$490$	0	0
$491$	−4.25569	−0.192057	−0.0960283	−	0.995379i	$-0.530614\pi$
$491$	−4.25569	−0.192057	−0.0960283	+	0.995379i	$0.530614\pi$
$492$	0	0
$493$	−28.5689	−1.28668
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−36.0796	−1.61515	−0.807573	−	0.589768i	$-0.799219\pi$
$499$	−36.0796	−1.61515	−0.807573	+	0.589768i	$0.799219\pi$
$500$	0	0
$501$	−19.9677	−0.892091
$502$	0	0
$503$	10.0442	0.447848	0.223924	−	0.974607i	$-0.428113\pi$
$503$	10.0442	0.447848	0.223924	+	0.974607i	$0.428113\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	39.1481	1.73863
$508$	0	0
$509$	22.2612	0.986710	0.493355	−	0.869828i	$-0.335770\pi$
$509$	22.2612	0.986710	0.493355	+	0.869828i	$0.335770\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−6.36584	−0.281059
$514$	0	0
$515$	0	0
$516$	0	0
$517$	6.25628	0.275151
$518$	0	0
$519$	−24.6051	−1.08004
$520$	0	0
$521$	−4.18727	−0.183447	−0.0917237	−	0.995784i	$-0.529238\pi$
$521$	−4.18727	−0.183447	−0.0917237	+	0.995784i	$0.529238\pi$
$522$	0	0
$523$	6.49224	0.283886	0.141943	−	0.989875i	$-0.454665\pi$
$523$	6.49224	0.283886	0.141943	+	0.989875i	$0.454665\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−19.7553	−0.860553
$528$	0	0
$529$	54.0958	2.35199
$530$	0	0
$531$	−48.5327	−2.10614
$532$	0	0
$533$	−38.5852	−1.67131
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−37.3473	−1.61166
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−6.42739	−0.276335	−0.138168	−	0.990409i	$-0.544121\pi$
$541$	−6.42739	−0.276335	−0.138168	+	0.990409i	$0.544121\pi$
$542$	0	0
$543$	0.236177	0.0101353
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−0.626973	−0.0268074	−0.0134037	−	0.999910i	$-0.504267\pi$
$547$	−0.626973	−0.0268074	−0.0134037	+	0.999910i	$0.504267\pi$
$548$	0	0
$549$	41.3109	1.76311
$550$	0	0
$551$	−12.3485	−0.526063
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−24.6950	−1.04636	−0.523181	−	0.852221i	$-0.675255\pi$
$557$	−24.6950	−1.04636	−0.523181	+	0.852221i	$0.675255\pi$
$558$	0	0
$559$	51.4134	2.17455
$560$	0	0
$561$	−10.5634	−0.445985
$562$	0	0
$563$	−8.88596	−0.374498	−0.187249	−	0.982312i	$-0.559957\pi$
$563$	−8.88596	−0.374498	−0.187249	+	0.982312i	$0.559957\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	0	0
$568$	0	0
$569$	11.1464	0.467281	0.233641	−	0.972323i	$-0.424936\pi$
$569$	11.1464	0.467281	0.233641	+	0.972323i	$0.424936\pi$
$570$	0	0
$571$	21.2210	0.888072	0.444036	−	0.896009i	$-0.353546\pi$
$571$	21.2210	0.888072	0.444036	+	0.896009i	$0.353546\pi$
$572$	0	0
$573$	−34.0820	−1.42380
$574$	0	0
$575$	0	0
$576$	0	0
$577$	10.2379	0.426209	0.213104	−	0.977029i	$-0.431643\pi$
$577$	10.2379	0.426209	0.213104	+	0.977029i	$0.431643\pi$
$578$	0	0
$579$	−55.5824	−2.30992
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−10.3538	−0.428809
$584$	0	0
$585$	0	0
$586$	0	0
$587$	13.6732	0.564354	0.282177	−	0.959362i	$-0.408943\pi$
$587$	13.6732	0.564354	0.282177	+	0.959362i	$0.408943\pi$
$588$	0	0
$589$	−8.53893	−0.351841
$590$	0	0
$591$	−28.2950	−1.16390
$592$	0	0
$593$	18.0203	0.740004	0.370002	−	0.929031i	$-0.379357\pi$
$593$	18.0203	0.740004	0.370002	+	0.929031i	$0.379357\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−31.8682	−1.30428
$598$	0	0
$599$	−2.02798	−0.0828609	−0.0414304	−	0.999141i	$-0.513191\pi$
$599$	−2.02798	−0.0828609	−0.0414304	+	0.999141i	$0.513191\pi$
$600$	0	0
$601$	31.5443	1.28672	0.643359	−	0.765565i	$-0.277540\pi$
$601$	31.5443	1.28672	0.643359	+	0.765565i	$0.277540\pi$
$602$	0	0
$603$	−11.9267	−0.485692
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−36.5276	−1.48261	−0.741305	−	0.671169i	$-0.765793\pi$
$607$	−36.5276	−1.48261	−0.741305	+	0.671169i	$0.765793\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−35.6738	−1.44321
$612$	0	0
$613$	27.1153	1.09518	0.547588	−	0.836748i	$-0.315546\pi$
$613$	27.1153	1.09518	0.547588	+	0.836748i	$0.315546\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	33.9078	1.36508	0.682539	−	0.730849i	$-0.260876\pi$
$617$	33.9078	1.36508	0.682539	+	0.730849i	$0.260876\pi$
$618$	0	0
$619$	−1.51850	−0.0610337	−0.0305168	−	0.999534i	$-0.509715\pi$
$619$	−1.51850	−0.0610337	−0.0305168	+	0.999534i	$0.509715\pi$
$620$	0	0
$621$	−30.3860	−1.21935
$622$	0	0
$623$	0	0
$624$	0	0
$625$	0	0
$626$	0	0
$627$	−4.56586	−0.182343
$628$	0	0
$629$	6.05065	0.241255
$630$	0	0
$631$	27.7605	1.10513	0.552565	−	0.833470i	$-0.313649\pi$
$631$	27.7605	1.10513	0.552565	+	0.833470i	$0.313649\pi$
$632$	0	0
$633$	−30.6544	−1.21840
$634$	0	0
$635$	0	0
$636$	0	0
$637$	0	0
$638$	0	0
$639$	50.6819	2.00494
$640$	0	0
$641$	21.4853	0.848619	0.424309	−	0.905517i	$-0.360517\pi$
$641$	21.4853	0.848619	0.424309	+	0.905517i	$0.360517\pi$
$642$	0	0
$643$	29.7709	1.17405	0.587025	−	0.809569i	$-0.300299\pi$
$643$	29.7709	1.17405	0.587025	+	0.809569i	$0.300299\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	18.2683	0.718202	0.359101	−	0.933299i	$-0.383083\pi$
$647$	18.2683	0.718202	0.359101	+	0.933299i	$0.383083\pi$
$648$	0	0
$649$	−10.4240	−0.409179
$650$	0	0
$651$	0	0
$652$	0	0
$653$	4.69936	0.183900	0.0919501	−	0.995764i	$-0.470690\pi$
$653$	4.69936	0.183900	0.0919501	+	0.995764i	$0.470690\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−21.9038	−0.854550
$658$	0	0
$659$	−46.9732	−1.82982	−0.914908	−	0.403663i	$-0.867737\pi$
$659$	−46.9732	−1.82982	−0.914908	+	0.403663i	$0.867737\pi$
$660$	0	0
$661$	8.23347	0.320245	0.160122	−	0.987097i	$-0.448811\pi$
$661$	8.23347	0.320245	0.160122	+	0.987097i	$0.448811\pi$
$662$	0	0
$663$	60.2331	2.33926
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−58.9428	−2.28227
$668$	0	0
$669$	72.6787	2.80992
$670$	0	0
$671$	8.87291	0.342535
$672$	0	0
$673$	30.8570	1.18945	0.594725	−	0.803929i	$-0.297261\pi$
$673$	30.8570	1.18945	0.594725	+	0.803929i	$0.297261\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	17.8391	0.685612	0.342806	−	0.939406i	$-0.388623\pi$
$677$	17.8391	0.685612	0.342806	+	0.939406i	$0.388623\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	48.4423	1.85631
$682$	0	0
$683$	−2.86296	−0.109548	−0.0547741	−	0.998499i	$-0.517444\pi$
$683$	−2.86296	−0.109548	−0.0547741	+	0.998499i	$0.517444\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−43.4068	−1.65607
$688$	0	0
$689$	59.0380	2.24917
$690$	0	0
$691$	39.3220	1.49588	0.747940	−	0.663767i	$-0.231043\pi$
$691$	39.3220	1.49588	0.747940	+	0.663767i	$0.231043\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−31.3096	−1.18594
$698$	0	0
$699$	52.8694	1.99970
$700$	0	0
$701$	−8.95691	−0.338298	−0.169149	−	0.985591i	$-0.554102\pi$
$701$	−8.95691	−0.338298	−0.169149	+	0.985591i	$0.554102\pi$
$702$	0	0
$703$	2.61531	0.0986382
$704$	0	0
$705$	0	0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	13.1893	0.495333	0.247666	−	0.968845i	$-0.420336\pi$
$709$	13.1893	0.495333	0.247666	+	0.968845i	$0.420336\pi$
$710$	0	0
$711$	3.11474	0.116812
$712$	0	0
$713$	−40.7587	−1.52643
$714$	0	0
$715$	0	0
$716$	0	0
$717$	70.3705	2.62803
$718$	0	0
$719$	−20.9300	−0.780556	−0.390278	−	0.920697i	$-0.627621\pi$
$719$	−20.9300	−0.780556	−0.390278	+	0.920697i	$0.627621\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	62.7249	2.33276
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−34.8251	−1.29159	−0.645797	−	0.763510i	$-0.723475\pi$
$727$	−34.8251	−1.29159	−0.645797	+	0.763510i	$0.723475\pi$
$728$	0	0
$729$	−43.0405	−1.59409
$730$	0	0
$731$	41.7189	1.54303
$732$	0	0
$733$	−22.6429	−0.836335	−0.418167	−	0.908370i	$-0.637327\pi$
$733$	−22.6429	−0.836335	−0.418167	+	0.908370i	$0.637327\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−2.56166	−0.0943599
$738$	0	0
$739$	−5.40246	−0.198733	−0.0993663	−	0.995051i	$-0.531682\pi$
$739$	−5.40246	−0.198733	−0.0993663	+	0.995051i	$0.531682\pi$
$740$	0	0
$741$	26.0349	0.956416
$742$	0	0
$743$	13.6717	0.501566	0.250783	−	0.968043i	$-0.419312\pi$
$743$	13.6717	0.501566	0.250783	+	0.968043i	$0.419312\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	18.7374	0.685565
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−2.94579	−0.107493	−0.0537467	−	0.998555i	$-0.517116\pi$
$751$	−2.94579	−0.107493	−0.0537467	+	0.998555i	$0.517116\pi$
$752$	0	0
$753$	18.2254	0.664169
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−41.7761	−1.51838	−0.759188	−	0.650871i	$-0.774404\pi$
$757$	−41.7761	−1.51838	−0.759188	+	0.650871i	$0.774404\pi$
$758$	0	0
$759$	−21.7941	−0.791077
$760$	0	0
$761$	−31.6836	−1.14853	−0.574264	−	0.818670i	$-0.694712\pi$
$761$	−31.6836	−1.14853	−0.574264	+	0.818670i	$0.694712\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	0	0
$766$	0	0
$767$	59.4386	2.14620
$768$	0	0
$769$	−33.7916	−1.21856	−0.609278	−	0.792957i	$-0.708541\pi$
$769$	−33.7916	−1.21856	−0.609278	+	0.792957i	$0.708541\pi$
$770$	0	0
$771$	−49.7599	−1.79206
$772$	0	0
$773$	29.3136	1.05434	0.527168	−	0.849761i	$-0.323254\pi$
$773$	29.3136	1.05434	0.527168	+	0.849761i	$0.323254\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−13.5331	−0.484875
$780$	0	0
$781$	10.8856	0.389519
$782$	0	0
$783$	23.2313	0.830217
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−39.6786	−1.41439	−0.707195	−	0.707019i	$-0.750040\pi$
$787$	−39.6786	−1.41439	−0.707195	+	0.707019i	$0.750040\pi$
$788$	0	0
$789$	26.7918	0.953814
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−50.5940	−1.79665
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−6.82244	−0.241663	−0.120832	−	0.992673i	$-0.538556\pi$
$797$	−6.82244	−0.241663	−0.120832	+	0.992673i	$0.538556\pi$
$798$	0	0
$799$	−28.9472	−1.02408
$800$	0	0
$801$	4.12822	0.145863
$802$	0	0
$803$	−4.70459	−0.166021
$804$	0	0
$805$	0	0
$806$	0	0
$807$	40.0831	1.41099
$808$	0	0
$809$	−9.70620	−0.341252	−0.170626	−	0.985336i	$-0.554579\pi$
$809$	−9.70620	−0.341252	−0.170626	+	0.985336i	$0.554579\pi$
$810$	0	0
$811$	23.8942	0.839037	0.419519	−	0.907747i	$-0.362199\pi$
$811$	23.8942	0.839037	0.419519	+	0.907747i	$0.362199\pi$
$812$	0	0
$813$	−9.01660	−0.316226
$814$	0	0
$815$	0	0
$816$	0	0
$817$	18.0324	0.630874
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−48.5915	−1.69586	−0.847928	−	0.530111i	$-0.822150\pi$
$821$	−48.5915	−1.69586	−0.847928	+	0.530111i	$0.822150\pi$
$822$	0	0
$823$	−0.0389997	−0.00135944	−0.000679722	−	1.00000i	$-0.500216\pi$
$823$	−0.0389997	−0.00135944	−0.000679722		1.00000i	$0.500216\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	26.7489	0.930149	0.465075	−	0.885271i	$-0.346027\pi$
$827$	26.7489	0.930149	0.465075	+	0.885271i	$0.346027\pi$
$828$	0	0
$829$	−10.9119	−0.378986	−0.189493	−	0.981882i	$-0.560684\pi$
$829$	−10.9119	−0.378986	−0.189493	+	0.981882i	$0.560684\pi$
$830$	0	0
$831$	47.3407	1.64223
$832$	0	0
$833$	0	0
$834$	0	0
$835$	0	0
$836$	0	0
$837$	16.0643	0.555265
$838$	0	0
$839$	−3.86425	−0.133409	−0.0667043	−	0.997773i	$-0.521248\pi$
$839$	−3.86425	−0.133409	−0.0667043	+	0.997773i	$0.521248\pi$
$840$	0	0
$841$	16.0641	0.553934
$842$	0	0
$843$	4.17407	0.143763
$844$	0	0
$845$	0	0
$846$	0	0
$847$	0	0
$848$	0	0
$849$	17.2105	0.590662
$850$	0	0
$851$	12.4836	0.427932
$852$	0	0
$853$	39.3320	1.34670	0.673351	−	0.739323i	$-0.264854\pi$
$853$	39.3320	1.34670	0.673351	+	0.739323i	$0.264854\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−31.1984	−1.06572	−0.532858	−	0.846205i	$-0.678882\pi$
$857$	−31.1984	−1.06572	−0.532858	+	0.846205i	$0.678882\pi$
$858$	0	0
$859$	−7.83869	−0.267453	−0.133726	−	0.991018i	$-0.542694\pi$
$859$	−7.83869	−0.267453	−0.133726	+	0.991018i	$0.542694\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	6.16750	0.209944	0.104972	−	0.994475i	$-0.466525\pi$
$863$	6.16750	0.209944	0.104972	+	0.994475i	$0.466525\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	2.99960	0.101872
$868$	0	0
$869$	0.668995	0.0226941
$870$	0	0
$871$	14.6068	0.494931
$872$	0	0
$873$	−58.0641	−1.96517
$874$	0	0
$875$	0	0
$876$	0	0
$877$	25.0332	0.845311	0.422655	−	0.906290i	$-0.361098\pi$
$877$	25.0332	0.845311	0.422655	+	0.906290i	$0.361098\pi$
$878$	0	0
$879$	7.16990	0.241835
$880$	0	0
$881$	30.9200	1.04172	0.520861	−	0.853641i	$-0.325611\pi$
$881$	30.9200	1.04172	0.520861	+	0.853641i	$0.325611\pi$
$882$	0	0
$883$	−40.5853	−1.36580	−0.682902	−	0.730510i	$-0.739283\pi$
$883$	−40.5853	−1.36580	−0.682902	+	0.730510i	$0.739283\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−30.8084	−1.03444	−0.517222	−	0.855851i	$-0.673034\pi$
$887$	−30.8084	−1.03444	−0.517222	+	0.855851i	$0.673034\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−3.22689	−0.108105
$892$	0	0
$893$	−12.5120	−0.418698
$894$	0	0
$895$	0	0
$896$	0	0
$897$	124.272	4.14932
$898$	0	0
$899$	31.1617	1.03930
$900$	0	0
$901$	47.9059	1.59598
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	11.2998	0.375205	0.187603	−	0.982245i	$-0.439928\pi$
$907$	11.2998	0.375205	0.187603	+	0.982245i	$0.439928\pi$
$908$	0	0
$909$	−82.2342	−2.72754
$910$	0	0
$911$	−31.4425	−1.04174	−0.520869	−	0.853637i	$-0.674392\pi$
$911$	−31.4425	−1.04174	−0.520869	+	0.853637i	$0.674392\pi$
$912$	0	0
$913$	4.02448	0.133191
$914$	0	0
$915$	0	0
$916$	0	0
$917$	0	0
$918$	0	0
$919$	6.88776	0.227206	0.113603	−	0.993526i	$-0.463761\pi$
$919$	6.88776	0.227206	0.113603	+	0.993526i	$0.463761\pi$
$920$	0	0
$921$	29.9062	0.985444
$922$	0	0
$923$	−62.0707	−2.04308
$924$	0	0
$925$	0	0
$926$	0	0
$927$	7.18097	0.235854
$928$	0	0
$929$	8.77677	0.287956	0.143978	−	0.989581i	$-0.454010\pi$
$929$	8.77677	0.287956	0.143978	+	0.989581i	$0.454010\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−61.8337	−2.02435
$934$	0	0
$935$	0	0
$936$	0	0
$937$	49.7847	1.62639	0.813197	−	0.581988i	$-0.197725\pi$
$937$	49.7847	1.62639	0.813197	+	0.581988i	$0.197725\pi$
$938$	0	0
$939$	−55.8797	−1.82356
$940$	0	0
$941$	37.6116	1.22610	0.613052	−	0.790042i	$-0.289941\pi$
$941$	37.6116	1.22610	0.613052	+	0.790042i	$0.289941\pi$
$942$	0	0
$943$	−64.5975	−2.10358
$944$	0	0
$945$	0	0
$946$	0	0
$947$	7.33224	0.238266	0.119133	−	0.992878i	$-0.461989\pi$
$947$	7.33224	0.238266	0.119133	+	0.992878i	$0.461989\pi$
$948$	0	0
$949$	26.8259	0.870806
$950$	0	0
$951$	44.4252	1.44059
$952$	0	0
$953$	50.4457	1.63410	0.817048	−	0.576570i	$-0.195609\pi$
$953$	50.4457	1.63410	0.817048	+	0.576570i	$0.195609\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	16.6625	0.538622
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−9.45182	−0.304897
$962$	0	0
$963$	−77.5500	−2.49901
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−20.1685	−0.648574	−0.324287	−	0.945959i	$-0.605124\pi$
$967$	−20.1685	−0.648574	−0.324287	+	0.945959i	$0.605124\pi$
$968$	0	0
$969$	21.1258	0.678658
$970$	0	0
$971$	−61.7398	−1.98133	−0.990663	−	0.136334i	$-0.956468\pi$
$971$	−61.7398	−1.98133	−0.990663	+	0.136334i	$0.956468\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−34.0613	−1.08972	−0.544858	−	0.838528i	$-0.683417\pi$
$977$	−34.0613	−1.08972	−0.544858	+	0.838528i	$0.683417\pi$
$978$	0	0
$979$	0.886674	0.0283382
$980$	0	0
$981$	−8.89144	−0.283882
$982$	0	0
$983$	52.8543	1.68579	0.842895	−	0.538078i	$-0.180849\pi$
$983$	52.8543	1.68579	0.842895	+	0.538078i	$0.180849\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	86.0738	2.73699
$990$	0	0
$991$	−30.0746	−0.955352	−0.477676	−	0.878536i	$-0.658521\pi$
$991$	−30.0746	−0.955352	−0.477676	+	0.878536i	$0.658521\pi$
$992$	0	0
$993$	49.9863	1.58627
$994$	0	0
$995$	0	0
$996$	0	0
$997$	19.0097	0.602044	0.301022	−	0.953617i	$-0.402672\pi$
$997$	19.0097	0.602044	0.301022	+	0.953617i	$0.402672\pi$
$998$	0	0
$999$	−4.92019	−0.155668

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9800.2.a.db.1.9		10
5.2	odd	4		1960.2.g.g.1569.4	yes	20
5.3	odd	4		1960.2.g.g.1569.18	yes	20
5.4	even	2		9800.2.a.dc.1.2		10
7.6	odd	2	inner	9800.2.a.db.1.2		10
35.13	even	4		1960.2.g.g.1569.3	✓	20
35.27	even	4		1960.2.g.g.1569.17	yes	20
35.34	odd	2		9800.2.a.dc.1.9		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1960.2.g.g.1569.3	✓	20	35.13	even	4
1960.2.g.g.1569.4	yes	20	5.2	odd	4
1960.2.g.g.1569.17	yes	20	35.27	even	4
1960.2.g.g.1569.18	yes	20	5.3	odd	4
9800.2.a.db.1.2		10	7.6	odd	2	inner
9800.2.a.db.1.9		10	1.1	even	1	trivial
9800.2.a.dc.1.2		10	5.4	even	2
9800.2.a.dc.1.9		10	35.34	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:	\( N \)	\(=\)	\( 9800 = 2^{3} \cdot 5^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9800.a (trivial)

\(f(q)\)	\(=\)	\(q+2.69859 q^{3} +4.28239 q^{9} +O(q^{10})\)	\(q+2.69859 q^{3} +4.28239 q^{9} +0.919787 q^{11} -5.24470 q^{13} -4.25577 q^{17} -1.83949 q^{19} -8.78042 q^{23} +3.46065 q^{27} +6.71298 q^{29} +4.64200 q^{31} +2.48213 q^{33} -1.42175 q^{37} -14.1533 q^{39} +7.35699 q^{41} -9.80292 q^{43} +6.80187 q^{47} -11.4846 q^{51} -11.2567 q^{53} -4.96404 q^{57} -11.3331 q^{59} +9.64670 q^{61} -2.78505 q^{67} -23.6948 q^{69} +11.8349 q^{71} -5.11486 q^{73} +0.727336 q^{79} -3.50830 q^{81} +4.37545 q^{83} +18.1156 q^{87} +0.963998 q^{89} +12.5269 q^{93} -13.5588 q^{97} +3.93889 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q + 14 q^{9}+O(q^{10}) \) 10 * q + 14 * q^9	\( 10 q + 14 q^{9} - 12 q^{11} - 16 q^{23} + 12 q^{29} - 36 q^{37} - 20 q^{39} - 24 q^{43} - 36 q^{51} - 8 q^{53} - 16 q^{57} - 40 q^{67} - 8 q^{71} - 4 q^{79} + 50 q^{81} - 48 q^{93} - 72 q^{99}+O(q^{100}) \) 10 * q + 14 * q^9 - 12 * q^11 - 16 * q^23 + 12 * q^29 - 36 * q^37 - 20 * q^39 - 24 * q^43 - 36 * q^51 - 8 * q^53 - 16 * q^57 - 40 * q^67 - 8 * q^71 - 4 * q^79 + 50 * q^81 - 48 * q^93 - 72 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more