LMFDB - Embedded newform 4900.2.a.bh.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [4900,2,Mod(1,4900)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("4900.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(4900, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 4900 = 2^{2} \cdot 5^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4900.a (trivial)

Newform invariants

comment:select newform

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

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

Self dual:	yes
Analytic conductor:	$39.1266969904$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{24})^+$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 4x^{2} + 1 $ x^4 - 4*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{23}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 980)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$0.517638$ of defining polynomial
Character	$\chi$	$=$	4900.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-1.73205 q^{3} -1.00000 q^{11} -1.73205 q^{13} +5.19615 q^{17} -2.82843 q^{19} +2.44949 q^{23} +5.19615 q^{27} -7.00000 q^{29} +7.07107 q^{31} +1.73205 q^{33} -7.34847 q^{37} +3.00000 q^{39} +7.07107 q^{41} -9.79796 q^{43} +12.1244 q^{47} -9.00000 q^{51} +12.2474 q^{53} +4.89898 q^{57} -7.07107 q^{59} -14.1421 q^{61} +12.2474 q^{67} -4.24264 q^{69} -10.0000 q^{71} +3.00000 q^{79} -9.00000 q^{81} +12.1244 q^{87} -12.2474 q^{93} -5.19615 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{11} - 28 q^{29} + 12 q^{39} - 36 q^{51} - 40 q^{71} + 12 q^{79} - 36 q^{81}+O(q^{100}) $ 4 * q - 4 * q^11 - 28 * q^29 + 12 * q^39 - 36 * q^51 - 40 * q^71 + 12 * q^79 - 36 * q^81

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
$3$	−1.73205	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$3$	−1.73205	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−1.00000	−0.301511	−0.150756	−	0.988571i	$-0.548171\pi$
$11$	−1.00000	−0.301511	−0.150756	+	0.988571i	$0.548171\pi$
$12$	0	0
$13$	−1.73205	−0.480384	−0.240192	−	0.970725i	$-0.577210\pi$
$13$	−1.73205	−0.480384	−0.240192	+	0.970725i	$0.577210\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	5.19615	1.26025	0.630126	−	0.776493i	$-0.283003\pi$
$17$	5.19615	1.26025	0.630126	+	0.776493i	$0.283003\pi$
$18$	0	0
$19$	−2.82843	−0.648886	−0.324443	−	0.945905i	$-0.605177\pi$
$19$	−2.82843	−0.648886	−0.324443	+	0.945905i	$0.605177\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	2.44949	0.510754	0.255377	−	0.966842i	$-0.417800\pi$
$23$	2.44949	0.510754	0.255377	+	0.966842i	$0.417800\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	5.19615	1.00000
$28$	0	0
$29$	−7.00000	−1.29987	−0.649934	−	0.759991i	$-0.725203\pi$
$29$	−7.00000	−1.29987	−0.649934	+	0.759991i	$0.725203\pi$
$30$	0	0
$31$	7.07107	1.27000	0.635001	−	0.772512i	$-0.281000\pi$
$31$	7.07107	1.27000	0.635001	+	0.772512i	$0.281000\pi$
$32$	0	0
$33$	1.73205	0.301511
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−7.34847	−1.20808	−0.604040	−	0.796954i	$-0.706443\pi$
$37$	−7.34847	−1.20808	−0.604040	+	0.796954i	$0.706443\pi$
$38$	0	0
$39$	3.00000	0.480384
$40$	0	0
$41$	7.07107	1.10432	0.552158	−	0.833740i	$-0.313805\pi$
$41$	7.07107	1.10432	0.552158	+	0.833740i	$0.313805\pi$
$42$	0	0
$43$	−9.79796	−1.49417	−0.747087	−	0.664726i	$-0.768548\pi$
$43$	−9.79796	−1.49417	−0.747087	+	0.664726i	$0.768548\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	12.1244	1.76852	0.884260	−	0.466996i	$-0.154664\pi$
$47$	12.1244	1.76852	0.884260	+	0.466996i	$0.154664\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	−9.00000	−1.26025
$52$	0	0
$53$	12.2474	1.68232	0.841158	−	0.540789i	$-0.181874\pi$
$53$	12.2474	1.68232	0.841158	+	0.540789i	$0.181874\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	4.89898	0.648886
$58$	0	0
$59$	−7.07107	−0.920575	−0.460287	−	0.887770i	$-0.652254\pi$
$59$	−7.07107	−0.920575	−0.460287	+	0.887770i	$0.652254\pi$
$60$	0	0
$61$	−14.1421	−1.81071	−0.905357	−	0.424650i	$-0.860397\pi$
$61$	−14.1421	−1.81071	−0.905357	+	0.424650i	$0.860397\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0	0
$66$	0	0
$67$	12.2474	1.49626	0.748132	−	0.663550i	$-0.230951\pi$
$67$	12.2474	1.49626	0.748132	+	0.663550i	$0.230951\pi$
$68$	0	0
$69$	−4.24264	−0.510754
$70$	0	0
$71$	−10.0000	−1.18678	−0.593391	−	0.804914i	$-0.702211\pi$
$71$	−10.0000	−1.18678	−0.593391	+	0.804914i	$0.702211\pi$
$72$	0	0
$73$	0	0		−	1.00000i	$-0.5\pi$
$73$	0	0			1.00000i	$0.5\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0	0
$78$	0	0
$79$	3.00000	0.337526	0.168763	−	0.985657i	$-0.446023\pi$
$79$	3.00000	0.337526	0.168763	+	0.985657i	$0.446023\pi$
$80$	0	0
$81$	−9.00000	−1.00000
$82$	0	0
$83$	0	0		−	1.00000i	$-0.5\pi$
$83$	0	0			1.00000i	$0.5\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	12.1244	1.29987
$88$	0	0
$89$	0	0		−	1.00000i	$-0.5\pi$
$89$	0	0			1.00000i	$0.5\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−12.2474	−1.27000
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−5.19615	−0.527589	−0.263795	−	0.964579i	$-0.584974\pi$
$97$	−5.19615	−0.527589	−0.263795	+	0.964579i	$0.584974\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	5.65685	0.562878	0.281439	−	0.959579i	$-0.409188\pi$
$101$	5.65685	0.562878	0.281439	+	0.959579i	$0.409188\pi$
$102$	0	0
$103$	1.73205	0.170664	0.0853320	−	0.996353i	$-0.472805\pi$
$103$	1.73205	0.170664	0.0853320	+	0.996353i	$0.472805\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−4.89898	−0.473602	−0.236801	−	0.971558i	$-0.576099\pi$
$107$	−4.89898	−0.473602	−0.236801	+	0.971558i	$0.576099\pi$
$108$	0	0
$109$	−3.00000	−0.287348	−0.143674	−	0.989625i	$-0.545892\pi$
$109$	−3.00000	−0.287348	−0.143674	+	0.989625i	$0.545892\pi$
$110$	0	0
$111$	12.7279	1.20808
$112$	0	0
$113$	2.44949	0.230429	0.115214	−	0.993341i	$-0.463245\pi$
$113$	2.44949	0.230429	0.115214	+	0.993341i	$0.463245\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	0	0
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−10.0000	−0.909091
$122$	0	0
$123$	−12.2474	−1.10432
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−12.2474	−1.08679	−0.543393	−	0.839479i	$-0.682861\pi$
$127$	−12.2474	−1.08679	−0.543393	+	0.839479i	$0.682861\pi$
$128$	0	0
$129$	16.9706	1.49417
$130$	0	0
$131$	−12.7279	−1.11204	−0.556022	−	0.831168i	$-0.687673\pi$
$131$	−12.7279	−1.11204	−0.556022	+	0.831168i	$0.687673\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	0	0
$136$	0	0
$137$	0	0		−	1.00000i	$-0.5\pi$
$137$	0	0			1.00000i	$0.5\pi$
$138$	0	0
$139$	18.3848	1.55938	0.779688	−	0.626168i	$-0.215378\pi$
$139$	18.3848	1.55938	0.779688	+	0.626168i	$0.215378\pi$
$140$	0	0
$141$	−21.0000	−1.76852
$142$	0	0
$143$	1.73205	0.144841
$144$	0	0
$145$	0	0
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−20.0000	−1.63846	−0.819232	−	0.573462i	$-0.805600\pi$
$149$	−20.0000	−1.63846	−0.819232	+	0.573462i	$0.805600\pi$
$150$	0	0
$151$	11.0000	0.895167	0.447584	−	0.894242i	$-0.352285\pi$
$151$	11.0000	0.895167	0.447584	+	0.894242i	$0.352285\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−17.3205	−1.38233	−0.691164	−	0.722698i	$-0.742902\pi$
$157$	−17.3205	−1.38233	−0.691164	+	0.722698i	$0.742902\pi$
$158$	0	0
$159$	−21.2132	−1.68232
$160$	0	0
$161$	0	0
$162$	0	0
$163$	22.0454	1.72673	0.863365	−	0.504580i	$-0.168353\pi$
$163$	22.0454	1.72673	0.863365	+	0.504580i	$0.168353\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	5.19615	0.402090	0.201045	−	0.979582i	$-0.435566\pi$
$167$	5.19615	0.402090	0.201045	+	0.979582i	$0.435566\pi$
$168$	0	0
$169$	−10.0000	−0.769231
$170$	0	0
$171$	0	0
$172$	0	0
$173$	1.73205	0.131685	0.0658427	−	0.997830i	$-0.479026\pi$
$173$	1.73205	0.131685	0.0658427	+	0.997830i	$0.479026\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	12.2474	0.920575
$178$	0	0
$179$	−10.0000	−0.747435	−0.373718	−	0.927543i	$-0.621917\pi$
$179$	−10.0000	−0.747435	−0.373718	+	0.927543i	$0.621917\pi$
$180$	0	0
$181$	−7.07107	−0.525588	−0.262794	−	0.964852i	$-0.584644\pi$
$181$	−7.07107	−0.525588	−0.262794	+	0.964852i	$0.584644\pi$
$182$	0	0
$183$	24.4949	1.81071
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−5.19615	−0.379980
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−1.00000	−0.0723575	−0.0361787	−	0.999345i	$-0.511519\pi$
$191$	−1.00000	−0.0723575	−0.0361787	+	0.999345i	$0.511519\pi$
$192$	0	0
$193$	0	0		−	1.00000i	$-0.5\pi$
$193$	0	0			1.00000i	$0.5\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−12.2474	−0.872595	−0.436297	−	0.899803i	$-0.643710\pi$
$197$	−12.2474	−0.872595	−0.436297	+	0.899803i	$0.643710\pi$
$198$	0	0
$199$	7.07107	0.501255	0.250627	−	0.968084i	$-0.419363\pi$
$199$	7.07107	0.501255	0.250627	+	0.968084i	$0.419363\pi$
$200$	0	0
$201$	−21.2132	−1.49626
$202$	0	0
$203$	0	0
$204$	0	0
$205$	0	0
$206$	0	0
$207$	0	0
$208$	0	0
$209$	2.82843	0.195646
$210$	0	0
$211$	−21.0000	−1.44570	−0.722850	−	0.691005i	$-0.757168\pi$
$211$	−21.0000	−1.44570	−0.722850	+	0.691005i	$0.757168\pi$
$212$	0	0
$213$	17.3205	1.18678
$214$	0	0
$215$	0	0
$216$	0	0
$217$	0	0
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−9.00000	−0.605406
$222$	0	0
$223$	19.0526	1.27585	0.637927	−	0.770097i	$-0.279792\pi$
$223$	19.0526	1.27585	0.637927	+	0.770097i	$0.279792\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−12.1244	−0.804722	−0.402361	−	0.915481i	$-0.631810\pi$
$227$	−12.1244	−0.804722	−0.402361	+	0.915481i	$0.631810\pi$
$228$	0	0
$229$	7.07107	0.467269	0.233635	−	0.972324i	$-0.424938\pi$
$229$	7.07107	0.467269	0.233635	+	0.972324i	$0.424938\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−24.4949	−1.60471	−0.802357	−	0.596844i	$-0.796421\pi$
$233$	−24.4949	−1.60471	−0.802357	+	0.596844i	$0.796421\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−5.19615	−0.337526
$238$	0	0
$239$	13.0000	0.840900	0.420450	−	0.907316i	$-0.361872\pi$
$239$	13.0000	0.840900	0.420450	+	0.907316i	$0.361872\pi$
$240$	0	0
$241$	1.41421	0.0910975	0.0455488	−	0.998962i	$-0.485496\pi$
$241$	1.41421	0.0910975	0.0455488	+	0.998962i	$0.485496\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	0	0
$246$	0	0
$247$	4.89898	0.311715
$248$	0	0
$249$	0	0
$250$	0	0
$251$	15.5563	0.981908	0.490954	−	0.871185i	$-0.336648\pi$
$251$	15.5563	0.981908	0.490954	+	0.871185i	$0.336648\pi$
$252$	0	0
$253$	−2.44949	−0.153998
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−17.3205	−1.08042	−0.540212	−	0.841529i	$-0.681656\pi$
$257$	−17.3205	−1.08042	−0.540212	+	0.841529i	$0.681656\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−9.79796	−0.604168	−0.302084	−	0.953281i	$-0.597682\pi$
$263$	−9.79796	−0.604168	−0.302084	+	0.953281i	$0.597682\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	0	0
$269$	4.24264	0.258678	0.129339	−	0.991600i	$-0.458714\pi$
$269$	4.24264	0.258678	0.129339	+	0.991600i	$0.458714\pi$
$270$	0	0
$271$	−28.2843	−1.71815	−0.859074	−	0.511852i	$-0.828960\pi$
$271$	−28.2843	−1.71815	−0.859074	+	0.511852i	$0.828960\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−7.34847	−0.441527	−0.220763	−	0.975327i	$-0.570855\pi$
$277$	−7.34847	−0.441527	−0.220763	+	0.975327i	$0.570855\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−29.0000	−1.72999	−0.864997	−	0.501776i	$-0.832680\pi$
$281$	−29.0000	−1.72999	−0.864997	+	0.501776i	$0.832680\pi$
$282$	0	0
$283$	−1.73205	−0.102960	−0.0514799	−	0.998674i	$-0.516394\pi$
$283$	−1.73205	−0.102960	−0.0514799	+	0.998674i	$0.516394\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	10.0000	0.588235
$290$	0	0
$291$	9.00000	0.527589
$292$	0	0
$293$	−19.0526	−1.11306	−0.556531	−	0.830827i	$-0.687868\pi$
$293$	−19.0526	−1.11306	−0.556531	+	0.830827i	$0.687868\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−5.19615	−0.301511
$298$	0	0
$299$	−4.24264	−0.245358
$300$	0	0
$301$	0	0
$302$	0	0
$303$	−9.79796	−0.562878
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−29.4449	−1.68051	−0.840254	−	0.542194i	$-0.817594\pi$
$307$	−29.4449	−1.68051	−0.840254	+	0.542194i	$0.817594\pi$
$308$	0	0
$309$	−3.00000	−0.170664
$310$	0	0
$311$	8.48528	0.481156	0.240578	−	0.970630i	$-0.422663\pi$
$311$	8.48528	0.481156	0.240578	+	0.970630i	$0.422663\pi$
$312$	0	0
$313$	−1.73205	−0.0979013	−0.0489506	−	0.998801i	$-0.515588\pi$
$313$	−1.73205	−0.0979013	−0.0489506	+	0.998801i	$0.515588\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	19.5959	1.10062	0.550308	−	0.834962i	$-0.314510\pi$
$317$	19.5959	1.10062	0.550308	+	0.834962i	$0.314510\pi$
$318$	0	0
$319$	7.00000	0.391925
$320$	0	0
$321$	8.48528	0.473602
$322$	0	0
$323$	−14.6969	−0.817760
$324$	0	0
$325$	0	0
$326$	0	0
$327$	5.19615	0.287348
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−30.0000	−1.64895	−0.824475	−	0.565899i	$-0.808529\pi$
$331$	−30.0000	−1.64895	−0.824475	+	0.565899i	$0.808529\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	0	0
$336$	0	0
$337$	31.8434	1.73462	0.867309	−	0.497770i	$-0.165847\pi$
$337$	31.8434	1.73462	0.867309	+	0.497770i	$0.165847\pi$
$338$	0	0
$339$	−4.24264	−0.230429
$340$	0	0
$341$	−7.07107	−0.382920
$342$	0	0
$343$	0	0
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−17.1464	−0.920468	−0.460234	−	0.887798i	$-0.652235\pi$
$347$	−17.1464	−0.920468	−0.460234	+	0.887798i	$0.652235\pi$
$348$	0	0
$349$	−2.82843	−0.151402	−0.0757011	−	0.997131i	$-0.524119\pi$
$349$	−2.82843	−0.151402	−0.0757011	+	0.997131i	$0.524119\pi$
$350$	0	0
$351$	−9.00000	−0.480384
$352$	0	0
$353$	−15.5885	−0.829690	−0.414845	−	0.909892i	$-0.636164\pi$
$353$	−15.5885	−0.829690	−0.414845	+	0.909892i	$0.636164\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−20.0000	−1.05556	−0.527780	−	0.849381i	$-0.676975\pi$
$359$	−20.0000	−1.05556	−0.527780	+	0.849381i	$0.676975\pi$
$360$	0	0
$361$	−11.0000	−0.578947
$362$	0	0
$363$	17.3205	0.909091
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−5.19615	−0.271237	−0.135618	−	0.990761i	$-0.543302\pi$
$367$	−5.19615	−0.271237	−0.135618	+	0.990761i	$0.543302\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	0	0
$372$	0	0
$373$	24.4949	1.26830	0.634149	−	0.773211i	$-0.281351\pi$
$373$	24.4949	1.26830	0.634149	+	0.773211i	$0.281351\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	12.1244	0.624436
$378$	0	0
$379$	−10.0000	−0.513665	−0.256833	−	0.966456i	$-0.582679\pi$
$379$	−10.0000	−0.513665	−0.256833	+	0.966456i	$0.582679\pi$
$380$	0	0
$381$	21.2132	1.08679
$382$	0	0
$383$	−20.7846	−1.06204	−0.531022	−	0.847358i	$-0.678192\pi$
$383$	−20.7846	−1.06204	−0.531022	+	0.847358i	$0.678192\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−17.0000	−0.861934	−0.430967	−	0.902368i	$-0.641828\pi$
$389$	−17.0000	−0.861934	−0.430967	+	0.902368i	$0.641828\pi$
$390$	0	0
$391$	12.7279	0.643679
$392$	0	0
$393$	22.0454	1.11204
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−29.4449	−1.47780	−0.738898	−	0.673818i	$-0.764653\pi$
$397$	−29.4449	−1.47780	−0.738898	+	0.673818i	$0.764653\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−11.0000	−0.549314	−0.274657	−	0.961542i	$-0.588564\pi$
$401$	−11.0000	−0.549314	−0.274657	+	0.961542i	$0.588564\pi$
$402$	0	0
$403$	−12.2474	−0.610089
$404$	0	0
$405$	0	0
$406$	0	0
$407$	7.34847	0.364250
$408$	0	0
$409$	11.3137	0.559427	0.279713	−	0.960084i	$-0.409761\pi$
$409$	11.3137	0.559427	0.279713	+	0.960084i	$0.409761\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	0	0
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−31.8434	−1.55938
$418$	0	0
$419$	21.2132	1.03633	0.518166	−	0.855280i	$-0.326615\pi$
$419$	21.2132	1.03633	0.518166	+	0.855280i	$0.326615\pi$
$420$	0	0
$421$	21.0000	1.02348	0.511739	−	0.859141i	$-0.329002\pi$
$421$	21.0000	1.02348	0.511739	+	0.859141i	$0.329002\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	0	0
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−3.00000	−0.144841
$430$	0	0
$431$	−11.0000	−0.529851	−0.264926	−	0.964269i	$-0.585347\pi$
$431$	−11.0000	−0.529851	−0.264926	+	0.964269i	$0.585347\pi$
$432$	0	0
$433$	−17.3205	−0.832370	−0.416185	−	0.909280i	$-0.636633\pi$
$433$	−17.3205	−0.832370	−0.416185	+	0.909280i	$0.636633\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−6.92820	−0.331421
$438$	0	0
$439$	−18.3848	−0.877457	−0.438729	−	0.898620i	$-0.644571\pi$
$439$	−18.3848	−0.877457	−0.438729	+	0.898620i	$0.644571\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	34.2929	1.62930	0.814651	−	0.579951i	$-0.196928\pi$
$443$	34.2929	1.62930	0.814651	+	0.579951i	$0.196928\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	34.6410	1.63846
$448$	0	0
$449$	−23.0000	−1.08544	−0.542719	−	0.839915i	$-0.682605\pi$
$449$	−23.0000	−1.08544	−0.542719	+	0.839915i	$0.682605\pi$
$450$	0	0
$451$	−7.07107	−0.332964
$452$	0	0
$453$	−19.0526	−0.895167
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−36.7423	−1.71873	−0.859367	−	0.511359i	$-0.829142\pi$
$457$	−36.7423	−1.71873	−0.859367	+	0.511359i	$0.829142\pi$
$458$	0	0
$459$	27.0000	1.26025
$460$	0	0
$461$	0	0		−	1.00000i	$-0.5\pi$
$461$	0	0			1.00000i	$0.5\pi$
$462$	0	0
$463$	−24.4949	−1.13837	−0.569187	−	0.822208i	$-0.692742\pi$
$463$	−24.4949	−1.13837	−0.569187	+	0.822208i	$0.692742\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−12.1244	−0.561048	−0.280524	−	0.959847i	$-0.590508\pi$
$467$	−12.1244	−0.561048	−0.280524	+	0.959847i	$0.590508\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	30.0000	1.38233
$472$	0	0
$473$	9.79796	0.450511
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−38.1838	−1.74466	−0.872330	−	0.488917i	$-0.837392\pi$
$479$	−38.1838	−1.74466	−0.872330	+	0.488917i	$0.837392\pi$
$480$	0	0
$481$	12.7279	0.580343
$482$	0	0
$483$	0	0
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−17.1464	−0.776979	−0.388489	−	0.921453i	$-0.627003\pi$
$487$	−17.1464	−0.776979	−0.388489	+	0.921453i	$0.627003\pi$
$488$	0	0
$489$	−38.1838	−1.72673
$490$	0	0
$491$	29.0000	1.30875	0.654376	−	0.756169i	$-0.272931\pi$
$491$	29.0000	1.30875	0.654376	+	0.756169i	$0.272931\pi$
$492$	0	0
$493$	−36.3731	−1.63816
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	37.0000	1.65635	0.828174	−	0.560471i	$-0.189380\pi$
$499$	37.0000	1.65635	0.828174	+	0.560471i	$0.189380\pi$
$500$	0	0
$501$	−9.00000	−0.402090
$502$	0	0
$503$	−32.9090	−1.46734	−0.733669	−	0.679507i	$-0.762194\pi$
$503$	−32.9090	−1.46734	−0.733669	+	0.679507i	$0.762194\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	17.3205	0.769231
$508$	0	0
$509$	4.24264	0.188052	0.0940259	−	0.995570i	$-0.470026\pi$
$509$	4.24264	0.188052	0.0940259	+	0.995570i	$0.470026\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−14.6969	−0.648886
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−12.1244	−0.533229
$518$	0	0
$519$	−3.00000	−0.131685
$520$	0	0
$521$	−8.48528	−0.371747	−0.185873	−	0.982574i	$-0.559511\pi$
$521$	−8.48528	−0.371747	−0.185873	+	0.982574i	$0.559511\pi$
$522$	0	0
$523$	34.6410	1.51475	0.757373	−	0.652983i	$-0.226483\pi$
$523$	34.6410	1.51475	0.757373	+	0.652983i	$0.226483\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	36.7423	1.60052
$528$	0	0
$529$	−17.0000	−0.739130
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−12.2474	−0.530496
$534$	0	0
$535$	0	0
$536$	0	0
$537$	17.3205	0.747435
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−9.00000	−0.386940	−0.193470	−	0.981106i	$-0.561974\pi$
$541$	−9.00000	−0.386940	−0.193470	+	0.981106i	$0.561974\pi$
$542$	0	0
$543$	12.2474	0.525588
$544$	0	0
$545$	0	0
$546$	0	0
$547$	24.4949	1.04733	0.523663	−	0.851925i	$-0.324565\pi$
$547$	24.4949	1.04733	0.523663	+	0.851925i	$0.324565\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	19.7990	0.843465
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0	0
$556$	0	0
$557$	24.4949	1.03788	0.518941	−	0.854810i	$-0.326326\pi$
$557$	24.4949	1.03788	0.518941	+	0.854810i	$0.326326\pi$
$558$	0	0
$559$	16.9706	0.717778
$560$	0	0
$561$	9.00000	0.379980
$562$	0	0
$563$	−17.3205	−0.729972	−0.364986	−	0.931013i	$-0.618926\pi$
$563$	−17.3205	−0.729972	−0.364986	+	0.931013i	$0.618926\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−20.0000	−0.838444	−0.419222	−	0.907884i	$-0.637697\pi$
$569$	−20.0000	−0.838444	−0.419222	+	0.907884i	$0.637697\pi$
$570$	0	0
$571$	10.0000	0.418487	0.209243	−	0.977864i	$-0.432900\pi$
$571$	10.0000	0.418487	0.209243	+	0.977864i	$0.432900\pi$
$572$	0	0
$573$	1.73205	0.0723575
$574$	0	0
$575$	0	0
$576$	0	0
$577$	46.7654	1.94687	0.973434	−	0.228968i	$-0.0735351\pi$
$577$	46.7654	1.94687	0.973434	+	0.228968i	$0.0735351\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−12.2474	−0.507237
$584$	0	0
$585$	0	0
$586$	0	0
$587$	6.92820	0.285958	0.142979	−	0.989726i	$-0.454332\pi$
$587$	6.92820	0.285958	0.142979	+	0.989726i	$0.454332\pi$
$588$	0	0
$589$	−20.0000	−0.824086
$590$	0	0
$591$	21.2132	0.872595
$592$	0	0
$593$	−1.73205	−0.0711268	−0.0355634	−	0.999367i	$-0.511323\pi$
$593$	−1.73205	−0.0711268	−0.0355634	+	0.999367i	$0.511323\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−12.2474	−0.501255
$598$	0	0
$599$	43.0000	1.75693	0.878466	−	0.477805i	$-0.158567\pi$
$599$	43.0000	1.75693	0.878466	+	0.477805i	$0.158567\pi$
$600$	0	0
$601$	−19.7990	−0.807618	−0.403809	−	0.914843i	$-0.632314\pi$
$601$	−19.7990	−0.807618	−0.403809	+	0.914843i	$0.632314\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	0	0
$606$	0	0
$607$	5.19615	0.210905	0.105453	−	0.994424i	$-0.466371\pi$
$607$	5.19615	0.210905	0.105453	+	0.994424i	$0.466371\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−21.0000	−0.849569
$612$	0	0
$613$	9.79796	0.395736	0.197868	−	0.980229i	$-0.436598\pi$
$613$	9.79796	0.395736	0.197868	+	0.980229i	$0.436598\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−7.34847	−0.295838	−0.147919	−	0.988999i	$-0.547258\pi$
$617$	−7.34847	−0.295838	−0.147919	+	0.988999i	$0.547258\pi$
$618$	0	0
$619$	9.89949	0.397894	0.198947	−	0.980010i	$-0.436248\pi$
$619$	9.89949	0.397894	0.198947	+	0.980010i	$0.436248\pi$
$620$	0	0
$621$	12.7279	0.510754
$622$	0	0
$623$	0	0
$624$	0	0
$625$	0	0
$626$	0	0
$627$	−4.89898	−0.195646
$628$	0	0
$629$	−38.1838	−1.52249
$630$	0	0
$631$	9.00000	0.358284	0.179142	−	0.983823i	$-0.442668\pi$
$631$	9.00000	0.358284	0.179142	+	0.983823i	$0.442668\pi$
$632$	0	0
$633$	36.3731	1.44570
$634$	0	0
$635$	0	0
$636$	0	0
$637$	0	0
$638$	0	0
$639$	0	0
$640$	0	0
$641$	8.00000	0.315981	0.157991	−	0.987441i	$-0.449498\pi$
$641$	8.00000	0.315981	0.157991	+	0.987441i	$0.449498\pi$
$642$	0	0
$643$	36.3731	1.43441	0.717207	−	0.696860i	$-0.245420\pi$
$643$	36.3731	1.43441	0.717207	+	0.696860i	$0.245420\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	0	0		−	1.00000i	$-0.5\pi$
$647$	0	0			1.00000i	$0.5\pi$
$648$	0	0
$649$	7.07107	0.277564
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−9.79796	−0.383424	−0.191712	−	0.981451i	$-0.561404\pi$
$653$	−9.79796	−0.383424	−0.191712	+	0.981451i	$0.561404\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	0	0
$658$	0	0
$659$	17.0000	0.662226	0.331113	−	0.943591i	$-0.392576\pi$
$659$	17.0000	0.662226	0.331113	+	0.943591i	$0.392576\pi$
$660$	0	0
$661$	14.1421	0.550065	0.275033	−	0.961435i	$-0.411311\pi$
$661$	14.1421	0.550065	0.275033	+	0.961435i	$0.411311\pi$
$662$	0	0
$663$	15.5885	0.605406
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−17.1464	−0.663912
$668$	0	0
$669$	−33.0000	−1.27585
$670$	0	0
$671$	14.1421	0.545951
$672$	0	0
$673$	−24.4949	−0.944209	−0.472104	−	0.881543i	$-0.656505\pi$
$673$	−24.4949	−0.944209	−0.472104	+	0.881543i	$0.656505\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	46.7654	1.79734	0.898670	−	0.438626i	$-0.144535\pi$
$677$	46.7654	1.79734	0.898670	+	0.438626i	$0.144535\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	21.0000	0.804722
$682$	0	0
$683$	24.4949	0.937271	0.468636	−	0.883392i	$-0.344746\pi$
$683$	24.4949	0.937271	0.468636	+	0.883392i	$0.344746\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−12.2474	−0.467269
$688$	0	0
$689$	−21.2132	−0.808159
$690$	0	0
$691$	−14.1421	−0.537992	−0.268996	−	0.963141i	$-0.586692\pi$
$691$	−14.1421	−0.537992	−0.268996	+	0.963141i	$0.586692\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	36.7423	1.39172
$698$	0	0
$699$	42.4264	1.60471
$700$	0	0
$701$	−11.0000	−0.415464	−0.207732	−	0.978186i	$-0.566608\pi$
$701$	−11.0000	−0.415464	−0.207732	+	0.978186i	$0.566608\pi$
$702$	0	0
$703$	20.7846	0.783906
$704$	0	0
$705$	0	0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	37.0000	1.38956	0.694782	−	0.719220i	$-0.255501\pi$
$709$	37.0000	1.38956	0.694782	+	0.719220i	$0.255501\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	17.3205	0.648658
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−22.5167	−0.840900
$718$	0	0
$719$	24.0416	0.896602	0.448301	−	0.893883i	$-0.352029\pi$
$719$	24.0416	0.896602	0.448301	+	0.893883i	$0.352029\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	−2.44949	−0.0910975
$724$	0	0
$725$	0	0
$726$	0	0
$727$	17.3205	0.642382	0.321191	−	0.947014i	$-0.395917\pi$
$727$	17.3205	0.642382	0.321191	+	0.947014i	$0.395917\pi$
$728$	0	0
$729$	27.0000	1.00000
$730$	0	0
$731$	−50.9117	−1.88304
$732$	0	0
$733$	−19.0526	−0.703722	−0.351861	−	0.936052i	$-0.614451\pi$
$733$	−19.0526	−0.703722	−0.351861	+	0.936052i	$0.614451\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−12.2474	−0.451141
$738$	0	0
$739$	23.0000	0.846069	0.423034	−	0.906114i	$-0.360965\pi$
$739$	23.0000	0.846069	0.423034	+	0.906114i	$0.360965\pi$
$740$	0	0
$741$	−8.48528	−0.311715
$742$	0	0
$743$	−12.2474	−0.449315	−0.224658	−	0.974438i	$-0.572126\pi$
$743$	−12.2474	−0.449315	−0.224658	+	0.974438i	$0.572126\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	0	0
$748$	0	0
$749$	0	0
$750$	0	0
$751$	51.0000	1.86102	0.930508	−	0.366271i	$-0.119366\pi$
$751$	51.0000	1.86102	0.930508	+	0.366271i	$0.119366\pi$
$752$	0	0
$753$	−26.9444	−0.981908
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−24.4949	−0.890282	−0.445141	−	0.895460i	$-0.646846\pi$
$757$	−24.4949	−0.890282	−0.445141	+	0.895460i	$0.646846\pi$
$758$	0	0
$759$	4.24264	0.153998
$760$	0	0
$761$	29.6985	1.07657	0.538285	−	0.842763i	$-0.319073\pi$
$761$	29.6985	1.07657	0.538285	+	0.842763i	$0.319073\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	0	0
$766$	0	0
$767$	12.2474	0.442230
$768$	0	0
$769$	−39.5980	−1.42794	−0.713970	−	0.700176i	$-0.753105\pi$
$769$	−39.5980	−1.42794	−0.713970	+	0.700176i	$0.753105\pi$
$770$	0	0
$771$	30.0000	1.08042
$772$	0	0
$773$	−15.5885	−0.560678	−0.280339	−	0.959901i	$-0.590447\pi$
$773$	−15.5885	−0.560678	−0.280339	+	0.959901i	$0.590447\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−20.0000	−0.716574
$780$	0	0
$781$	10.0000	0.357828
$782$	0	0
$783$	−36.3731	−1.29987
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−22.5167	−0.802632	−0.401316	−	0.915940i	$-0.631447\pi$
$787$	−22.5167	−0.802632	−0.401316	+	0.915940i	$0.631447\pi$
$788$	0	0
$789$	16.9706	0.604168
$790$	0	0
$791$	0	0
$792$	0	0
$793$	24.4949	0.869839
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−39.8372	−1.41110	−0.705552	−	0.708658i	$-0.749301\pi$
$797$	−39.8372	−1.41110	−0.705552	+	0.708658i	$0.749301\pi$
$798$	0	0
$799$	63.0000	2.22878
$800$	0	0
$801$	0	0
$802$	0	0
$803$	0	0
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−7.34847	−0.258678
$808$	0	0
$809$	−53.0000	−1.86338	−0.931690	−	0.363253i	$-0.881666\pi$
$809$	−53.0000	−1.86338	−0.931690	+	0.363253i	$0.881666\pi$
$810$	0	0
$811$	−43.8406	−1.53945	−0.769726	−	0.638374i	$-0.779607\pi$
$811$	−43.8406	−1.53945	−0.769726	+	0.638374i	$0.779607\pi$
$812$	0	0
$813$	48.9898	1.71815
$814$	0	0
$815$	0	0
$816$	0	0
$817$	27.7128	0.969549
$818$	0	0
$819$	0	0
$820$	0	0
$821$	1.00000	0.0349002	0.0174501	−	0.999848i	$-0.494445\pi$
$821$	1.00000	0.0349002	0.0174501	+	0.999848i	$0.494445\pi$
$822$	0	0
$823$	12.2474	0.426919	0.213460	−	0.976952i	$-0.431527\pi$
$823$	12.2474	0.426919	0.213460	+	0.976952i	$0.431527\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−17.1464	−0.596240	−0.298120	−	0.954528i	$-0.596360\pi$
$827$	−17.1464	−0.596240	−0.298120	+	0.954528i	$0.596360\pi$
$828$	0	0
$829$	49.4975	1.71912	0.859559	−	0.511036i	$-0.170738\pi$
$829$	49.4975	1.71912	0.859559	+	0.511036i	$0.170738\pi$
$830$	0	0
$831$	12.7279	0.441527
$832$	0	0
$833$	0	0
$834$	0	0
$835$	0	0
$836$	0	0
$837$	36.7423	1.27000
$838$	0	0
$839$	18.3848	0.634713	0.317356	−	0.948306i	$-0.397205\pi$
$839$	18.3848	0.634713	0.317356	+	0.948306i	$0.397205\pi$
$840$	0	0
$841$	20.0000	0.689655
$842$	0	0
$843$	50.2295	1.72999
$844$	0	0
$845$	0	0
$846$	0	0
$847$	0	0
$848$	0	0
$849$	3.00000	0.102960
$850$	0	0
$851$	−18.0000	−0.617032
$852$	0	0
$853$	3.46410	0.118609	0.0593043	−	0.998240i	$-0.481112\pi$
$853$	3.46410	0.118609	0.0593043	+	0.998240i	$0.481112\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	17.3205	0.591657	0.295829	−	0.955241i	$-0.404404\pi$
$857$	17.3205	0.591657	0.295829	+	0.955241i	$0.404404\pi$
$858$	0	0
$859$	14.1421	0.482523	0.241262	−	0.970460i	$-0.422439\pi$
$859$	14.1421	0.482523	0.241262	+	0.970460i	$0.422439\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−36.7423	−1.25072	−0.625362	−	0.780335i	$-0.715049\pi$
$863$	−36.7423	−1.25072	−0.625362	+	0.780335i	$0.715049\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−17.3205	−0.588235
$868$	0	0
$869$	−3.00000	−0.101768
$870$	0	0
$871$	−21.2132	−0.718782
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	19.5959	0.661707	0.330854	−	0.943682i	$-0.392663\pi$
$877$	19.5959	0.661707	0.330854	+	0.943682i	$0.392663\pi$
$878$	0	0
$879$	33.0000	1.11306
$880$	0	0
$881$	−28.2843	−0.952921	−0.476461	−	0.879196i	$-0.658081\pi$
$881$	−28.2843	−0.952921	−0.476461	+	0.879196i	$0.658081\pi$
$882$	0	0
$883$	24.4949	0.824319	0.412159	−	0.911112i	$-0.364775\pi$
$883$	24.4949	0.824319	0.412159	+	0.911112i	$0.364775\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−17.3205	−0.581566	−0.290783	−	0.956789i	$-0.593916\pi$
$887$	−17.3205	−0.581566	−0.290783	+	0.956789i	$0.593916\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	9.00000	0.301511
$892$	0	0
$893$	−34.2929	−1.14757
$894$	0	0
$895$	0	0
$896$	0	0
$897$	7.34847	0.245358
$898$	0	0
$899$	−49.4975	−1.65083
$900$	0	0
$901$	63.6396	2.12014
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	41.6413	1.38268	0.691339	−	0.722531i	$-0.257021\pi$
$907$	41.6413	1.38268	0.691339	+	0.722531i	$0.257021\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	20.0000	0.662630	0.331315	−	0.943520i	$-0.392508\pi$
$911$	20.0000	0.662630	0.331315	+	0.943520i	$0.392508\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	0	0
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−3.00000	−0.0989609	−0.0494804	−	0.998775i	$-0.515757\pi$
$919$	−3.00000	−0.0989609	−0.0494804	+	0.998775i	$0.515757\pi$
$920$	0	0
$921$	51.0000	1.68051
$922$	0	0
$923$	17.3205	0.570111
$924$	0	0
$925$	0	0
$926$	0	0
$927$	0	0
$928$	0	0
$929$	21.2132	0.695983	0.347991	−	0.937498i	$-0.386864\pi$
$929$	21.2132	0.695983	0.347991	+	0.937498i	$0.386864\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−14.6969	−0.481156
$934$	0	0
$935$	0	0
$936$	0	0
$937$	39.8372	1.30142	0.650712	−	0.759325i	$-0.274471\pi$
$937$	39.8372	1.30142	0.650712	+	0.759325i	$0.274471\pi$
$938$	0	0
$939$	3.00000	0.0979013
$940$	0	0
$941$	−56.5685	−1.84408	−0.922041	−	0.387092i	$-0.873479\pi$
$941$	−56.5685	−1.84408	−0.922041	+	0.387092i	$0.873479\pi$
$942$	0	0
$943$	17.3205	0.564033
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−36.7423	−1.19397	−0.596983	−	0.802254i	$-0.703634\pi$
$947$	−36.7423	−1.19397	−0.596983	+	0.802254i	$0.703634\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	−33.9411	−1.10062
$952$	0	0
$953$	36.7423	1.19020	0.595101	−	0.803651i	$-0.297112\pi$
$953$	36.7423	1.19020	0.595101	+	0.803651i	$0.297112\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−12.1244	−0.391925
$958$	0	0
$959$	0	0
$960$	0	0
$961$	19.0000	0.612903
$962$	0	0
$963$	0	0
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−44.0908	−1.41787	−0.708933	−	0.705276i	$-0.750823\pi$
$967$	−44.0908	−1.41787	−0.708933	+	0.705276i	$0.750823\pi$
$968$	0	0
$969$	25.4558	0.817760
$970$	0	0
$971$	49.4975	1.58845	0.794225	−	0.607624i	$-0.207877\pi$
$971$	49.4975	1.58845	0.794225	+	0.607624i	$0.207877\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−4.89898	−0.156732	−0.0783661	−	0.996925i	$-0.524970\pi$
$977$	−4.89898	−0.156732	−0.0783661	+	0.996925i	$0.524970\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−32.9090	−1.04963	−0.524816	−	0.851215i	$-0.675866\pi$
$983$	−32.9090	−1.04963	−0.524816	+	0.851215i	$0.675866\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−24.0000	−0.763156
$990$	0	0
$991$	30.0000	0.952981	0.476491	−	0.879180i	$-0.341909\pi$
$991$	30.0000	0.952981	0.476491	+	0.879180i	$0.341909\pi$
$992$	0	0
$993$	51.9615	1.64895
$994$	0	0
$995$	0	0
$996$	0	0
$997$	5.19615	0.164564	0.0822819	−	0.996609i	$-0.473779\pi$
$997$	5.19615	0.164564	0.0822819	+	0.996609i	$0.473779\pi$
$998$	0	0
$999$	−38.1838	−1.20808

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4900.2.a.bh.1.1		4
5.2	odd	4		980.2.e.e.589.4	yes	4
5.3	odd	4		980.2.e.e.589.2	yes	4
5.4	even	2	inner	4900.2.a.bh.1.3		4
7.6	odd	2	inner	4900.2.a.bh.1.4		4
35.2	odd	12		980.2.q.a.949.1		4
35.3	even	12		980.2.q.h.569.2		4
35.12	even	12		980.2.q.h.949.2		4
35.13	even	4		980.2.e.e.589.3	yes	4
35.17	even	12		980.2.q.a.569.2		4
35.18	odd	12		980.2.q.a.569.1		4
35.23	odd	12		980.2.q.h.949.1		4
35.27	even	4		980.2.e.e.589.1	✓	4
35.32	odd	12		980.2.q.h.569.1		4
35.33	even	12		980.2.q.a.949.2		4
35.34	odd	2	inner	4900.2.a.bh.1.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
980.2.e.e.589.1	✓	4	35.27	even	4
980.2.e.e.589.2	yes	4	5.3	odd	4
980.2.e.e.589.3	yes	4	35.13	even	4
980.2.e.e.589.4	yes	4	5.2	odd	4
980.2.q.a.569.1		4	35.18	odd	12
980.2.q.a.569.2		4	35.17	even	12
980.2.q.a.949.1		4	35.2	odd	12
980.2.q.a.949.2		4	35.33	even	12
980.2.q.h.569.1		4	35.32	odd	12
980.2.q.h.569.2		4	35.3	even	12
980.2.q.h.949.1		4	35.23	odd	12
980.2.q.h.949.2		4	35.12	even	12
4900.2.a.bh.1.1		4	1.1	even	1	trivial
4900.2.a.bh.1.2		4	35.34	odd	2	inner
4900.2.a.bh.1.3		4	5.4	even	2	inner
4900.2.a.bh.1.4		4	7.6	odd	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 4900 = 2^{2} \cdot 5^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4900.a (trivial)

\(f(q)\)	\(=\)	\(q-1.73205 q^{3} -1.00000 q^{11} -1.73205 q^{13} +5.19615 q^{17} -2.82843 q^{19} +2.44949 q^{23} +5.19615 q^{27} -7.00000 q^{29} +7.07107 q^{31} +1.73205 q^{33} -7.34847 q^{37} +3.00000 q^{39} +7.07107 q^{41} -9.79796 q^{43} +12.1244 q^{47} -9.00000 q^{51} +12.2474 q^{53} +4.89898 q^{57} -7.07107 q^{59} -14.1421 q^{61} +12.2474 q^{67} -4.24264 q^{69} -10.0000 q^{71} +3.00000 q^{79} -9.00000 q^{81} +12.1244 q^{87} -12.2474 q^{93} -5.19615 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{11} - 28 q^{29} + 12 q^{39} - 36 q^{51} - 40 q^{71} + 12 q^{79} - 36 q^{81}+O(q^{100}) \) 4 * q - 4 * q^11 - 28 * q^29 + 12 * q^39 - 36 * q^51 - 40 * q^71 + 12 * q^79 - 36 * q^81

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