LMFDB - Embedded newform 2601.2.a.bb.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2601,2,Mod(1,2601)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2601, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("2601.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2601 = 3^{2} \cdot 17^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2601.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:	$20.7690895657$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{16})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 4x^{2} + 2 $ x^4 - 4*x^2 + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 17)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.84776$ of defining polynomial
Character	$\chi$	$=$	2601.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.41421 q^{2} +3.82843 q^{4} -0.765367 q^{5} +2.61313 q^{7} -4.41421 q^{8} +O(q^{10})$	$q-2.41421 q^{2} +3.82843 q^{4} -0.765367 q^{5} +2.61313 q^{7} -4.41421 q^{8} +1.84776 q^{10} +2.61313 q^{11} +1.41421 q^{13} -6.30864 q^{14} +3.00000 q^{16} -0.828427 q^{19} -2.93015 q^{20} -6.30864 q^{22} -4.77791 q^{23} -4.41421 q^{25} -3.41421 q^{26} +10.0042 q^{28} +0.317025 q^{29} -7.83938 q^{31} +1.58579 q^{32} -2.00000 q^{35} -9.23880 q^{37} +2.00000 q^{38} +3.37849 q^{40} +1.21371 q^{41} +0.828427 q^{43} +10.0042 q^{44} +11.5349 q^{46} -5.17157 q^{47} -0.171573 q^{49} +10.6569 q^{50} +5.41421 q^{52} -1.41421 q^{53} -2.00000 q^{55} -11.5349 q^{56} -0.765367 q^{58} -6.00000 q^{59} -3.82683 q^{61} +18.9259 q^{62} -9.82843 q^{64} -1.08239 q^{65} -1.17157 q^{67} +4.82843 q^{70} -5.41196 q^{71} +12.9343 q^{73} +22.3044 q^{74} -3.17157 q^{76} +6.82843 q^{77} +4.77791 q^{79} -2.29610 q^{80} -2.93015 q^{82} -11.6569 q^{83} -2.00000 q^{86} -11.5349 q^{88} -6.58579 q^{89} +3.69552 q^{91} -18.2919 q^{92} +12.4853 q^{94} +0.634051 q^{95} +10.3212 q^{97} +0.414214 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{2} + 4 q^{4} - 12 q^{8}+O(q^{10}) $ 4 * q - 4 * q^2 + 4 * q^4 - 12 * q^8	$ 4 q - 4 q^{2} + 4 q^{4} - 12 q^{8} + 12 q^{16} + 8 q^{19} - 12 q^{25} - 8 q^{26} + 12 q^{32} - 8 q^{35} + 8 q^{38} - 8 q^{43} - 32 q^{47} - 12 q^{49} + 20 q^{50} + 16 q^{52} - 8 q^{55} - 24 q^{59} - 28 q^{64} - 16 q^{67} + 8 q^{70} - 24 q^{76} + 16 q^{77} - 24 q^{83} - 8 q^{86} - 32 q^{89} + 16 q^{94} - 4 q^{98}+O(q^{100}) $ 4 * q - 4 * q^2 + 4 * q^4 - 12 * q^8 + 12 * q^16 + 8 * q^19 - 12 * q^25 - 8 * q^26 + 12 * q^32 - 8 * q^35 + 8 * q^38 - 8 * q^43 - 32 * q^47 - 12 * q^49 + 20 * q^50 + 16 * q^52 - 8 * q^55 - 24 * q^59 - 28 * q^64 - 16 * q^67 + 8 * q^70 - 24 * q^76 + 16 * q^77 - 24 * q^83 - 8 * q^86 - 32 * q^89 + 16 * q^94 - 4 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−2.41421	−1.70711	−0.853553	−	0.521005i	$-0.825557\pi$
$2$	−2.41421	−1.70711	−0.853553	+	0.521005i	$0.825557\pi$
$3$	0	0
$3$	0	0
$4$	3.82843	1.91421
$5$	−0.765367	−0.342282	−0.171141	−	0.985247i	$-0.554745\pi$
$5$	−0.765367	−0.342282	−0.171141	+	0.985247i	$0.554745\pi$
$6$	0	0
$7$	2.61313	0.987669	0.493834	−	0.869556i	$-0.335595\pi$
$7$	2.61313	0.987669	0.493834	+	0.869556i	$0.335595\pi$
$8$	−4.41421	−1.56066
$9$	0	0
$10$	1.84776	0.584313
$11$	2.61313	0.787887	0.393944	−	0.919135i	$-0.371111\pi$
$11$	2.61313	0.787887	0.393944	+	0.919135i	$0.371111\pi$
$12$	0	0
$13$	1.41421	0.392232	0.196116	−	0.980581i	$-0.437167\pi$
$13$	1.41421	0.392232	0.196116	+	0.980581i	$0.437167\pi$
$14$	−6.30864	−1.68606
$15$	0	0
$16$	3.00000	0.750000
$17$	0	0
$17$	0	0
$18$	0	0
$19$	−0.828427	−0.190054	−0.0950271	−	0.995475i	$-0.530294\pi$
$19$	−0.828427	−0.190054	−0.0950271	+	0.995475i	$0.530294\pi$
$20$	−2.93015	−0.655202
$21$	0	0
$22$	−6.30864	−1.34501
$23$	−4.77791	−0.996263	−0.498132	−	0.867101i	$-0.665980\pi$
$23$	−4.77791	−0.996263	−0.498132	+	0.867101i	$0.665980\pi$
$24$	0	0
$25$	−4.41421	−0.882843
$26$	−3.41421	−0.669582
$27$	0	0
$28$	10.0042	1.89061
$29$	0.317025	0.0588701	0.0294351	−	0.999567i	$-0.490629\pi$
$29$	0.317025	0.0588701	0.0294351	+	0.999567i	$0.490629\pi$
$30$	0	0
$31$	−7.83938	−1.40799	−0.703997	−	0.710203i	$-0.748603\pi$
$31$	−7.83938	−1.40799	−0.703997	+	0.710203i	$0.748603\pi$
$32$	1.58579	0.280330
$33$	0	0
$34$	0	0
$35$	−2.00000	−0.338062
$36$	0	0
$37$	−9.23880	−1.51885	−0.759424	−	0.650596i	$-0.774519\pi$
$37$	−9.23880	−1.51885	−0.759424	+	0.650596i	$0.774519\pi$
$38$	2.00000	0.324443
$39$	0	0
$40$	3.37849	0.534187
$41$	1.21371	0.189549	0.0947747	−	0.995499i	$-0.469787\pi$
$41$	1.21371	0.189549	0.0947747	+	0.995499i	$0.469787\pi$
$42$	0	0
$43$	0.828427	0.126334	0.0631670	−	0.998003i	$-0.479880\pi$
$43$	0.828427	0.126334	0.0631670	+	0.998003i	$0.479880\pi$
$44$	10.0042	1.50818
$45$	0	0
$46$	11.5349	1.70073
$47$	−5.17157	−0.754351	−0.377176	−	0.926142i	$-0.623105\pi$
$47$	−5.17157	−0.754351	−0.377176	+	0.926142i	$0.623105\pi$
$48$	0	0
$49$	−0.171573	−0.0245104
$50$	10.6569	1.50711
$51$	0	0
$52$	5.41421	0.750816
$53$	−1.41421	−0.194257	−0.0971286	−	0.995272i	$-0.530966\pi$
$53$	−1.41421	−0.194257	−0.0971286	+	0.995272i	$0.530966\pi$
$54$	0	0
$55$	−2.00000	−0.269680
$56$	−11.5349	−1.54142
$57$	0	0
$58$	−0.765367	−0.100498
$59$	−6.00000	−0.781133	−0.390567	−	0.920575i	$-0.627721\pi$
$59$	−6.00000	−0.781133	−0.390567	+	0.920575i	$0.627721\pi$
$60$	0	0
$61$	−3.82683	−0.489976	−0.244988	−	0.969526i	$-0.578784\pi$
$61$	−3.82683	−0.489976	−0.244988	+	0.969526i	$0.578784\pi$
$62$	18.9259	2.40360
$63$	0	0
$64$	−9.82843	−1.22855
$65$	−1.08239	−0.134254
$66$	0	0
$67$	−1.17157	−0.143130	−0.0715652	−	0.997436i	$-0.522799\pi$
$67$	−1.17157	−0.143130	−0.0715652	+	0.997436i	$0.522799\pi$
$68$	0	0
$69$	0	0
$70$	4.82843	0.577107
$71$	−5.41196	−0.642282	−0.321141	−	0.947031i	$-0.604066\pi$
$71$	−5.41196	−0.642282	−0.321141	+	0.947031i	$0.604066\pi$
$72$	0	0
$73$	12.9343	1.51385	0.756923	−	0.653504i	$-0.226702\pi$
$73$	12.9343	1.51385	0.756923	+	0.653504i	$0.226702\pi$
$74$	22.3044	2.59284
$75$	0	0
$76$	−3.17157	−0.363804
$77$	6.82843	0.778171
$78$	0	0
$79$	4.77791	0.537557	0.268778	−	0.963202i	$-0.413380\pi$
$79$	4.77791	0.537557	0.268778	+	0.963202i	$0.413380\pi$
$80$	−2.29610	−0.256712
$81$	0	0
$82$	−2.93015	−0.323581
$83$	−11.6569	−1.27951	−0.639753	−	0.768581i	$-0.720963\pi$
$83$	−11.6569	−1.27951	−0.639753	+	0.768581i	$0.720963\pi$
$84$	0	0
$85$	0	0
$86$	−2.00000	−0.215666
$87$	0	0
$88$	−11.5349	−1.22962
$89$	−6.58579	−0.698092	−0.349046	−	0.937106i	$-0.613494\pi$
$89$	−6.58579	−0.698092	−0.349046	+	0.937106i	$0.613494\pi$
$90$	0	0
$91$	3.69552	0.387396
$92$	−18.2919	−1.90706
$93$	0	0
$94$	12.4853	1.28776
$95$	0.634051	0.0650522
$96$	0	0
$97$	10.3212	1.04796	0.523979	−	0.851731i	$-0.324447\pi$
$97$	10.3212	1.04796	0.523979	+	0.851731i	$0.324447\pi$
$98$	0.414214	0.0418419
$99$	0	0
$100$	−16.8995	−1.68995
$101$	−10.5858	−1.05333	−0.526663	−	0.850074i	$-0.676557\pi$
$101$	−10.5858	−1.05333	−0.526663	+	0.850074i	$0.676557\pi$
$102$	0	0
$103$	12.4853	1.23021	0.615106	−	0.788445i	$-0.289113\pi$
$103$	12.4853	1.23021	0.615106	+	0.788445i	$0.289113\pi$
$104$	−6.24264	−0.612141
$105$	0	0
$106$	3.41421	0.331618
$107$	0.448342	0.0433428	0.0216714	−	0.999765i	$-0.493101\pi$
$107$	0.448342	0.0433428	0.0216714	+	0.999765i	$0.493101\pi$
$108$	0	0
$109$	−15.5474	−1.48917	−0.744587	−	0.667525i	$-0.767354\pi$
$109$	−15.5474	−1.48917	−0.744587	+	0.667525i	$0.767354\pi$
$110$	4.82843	0.460372
$111$	0	0
$112$	7.83938	0.740752
$113$	−13.1969	−1.24146	−0.620732	−	0.784023i	$-0.713165\pi$
$113$	−13.1969	−1.24146	−0.620732	+	0.784023i	$0.713165\pi$
$114$	0	0
$115$	3.65685	0.341003
$116$	1.21371	0.112690
$117$	0	0
$118$	14.4853	1.33348
$119$	0	0
$120$	0	0
$121$	−4.17157	−0.379234
$122$	9.23880	0.836441
$123$	0	0
$124$	−30.0125	−2.69520
$125$	7.20533	0.644464
$126$	0	0
$127$	−5.31371	−0.471515	−0.235758	−	0.971812i	$-0.575757\pi$
$127$	−5.31371	−0.471515	−0.235758	+	0.971812i	$0.575757\pi$
$128$	20.5563	1.81694
$129$	0	0
$130$	2.61313	0.229186
$131$	15.2304	1.33069	0.665344	−	0.746537i	$-0.268285\pi$
$131$	15.2304	1.33069	0.665344	+	0.746537i	$0.268285\pi$
$132$	0	0
$133$	−2.16478	−0.187711
$134$	2.82843	0.244339
$135$	0	0
$136$	0	0
$137$	16.7279	1.42916	0.714581	−	0.699552i	$-0.246617\pi$
$137$	16.7279	1.42916	0.714581	+	0.699552i	$0.246617\pi$
$138$	0	0
$139$	21.3533	1.81117	0.905584	−	0.424168i	$-0.139433\pi$
$139$	21.3533	1.81117	0.905584	+	0.424168i	$0.139433\pi$
$140$	−7.65685	−0.647122
$141$	0	0
$142$	13.0656	1.09644
$143$	3.69552	0.309035
$144$	0	0
$145$	−0.242641	−0.0201502
$146$	−31.2262	−2.58430
$147$	0	0
$148$	−35.3701	−2.90740
$149$	16.9706	1.39028	0.695141	−	0.718873i	$-0.255342\pi$
$149$	16.9706	1.39028	0.695141	+	0.718873i	$0.255342\pi$
$150$	0	0
$151$	7.17157	0.583614	0.291807	−	0.956477i	$-0.405743\pi$
$151$	7.17157	0.583614	0.291807	+	0.956477i	$0.405743\pi$
$152$	3.65685	0.296610
$153$	0	0
$154$	−16.4853	−1.32842
$155$	6.00000	0.481932
$156$	0	0
$157$	−9.65685	−0.770701	−0.385350	−	0.922770i	$-0.625919\pi$
$157$	−9.65685	−0.770701	−0.385350	+	0.922770i	$0.625919\pi$
$158$	−11.5349	−0.917667
$159$	0	0
$160$	−1.21371	−0.0959521
$161$	−12.4853	−0.983978
$162$	0	0
$163$	−8.47343	−0.663690	−0.331845	−	0.943334i	$-0.607671\pi$
$163$	−8.47343	−0.663690	−0.331845	+	0.943334i	$0.607671\pi$
$164$	4.64659	0.362838
$165$	0	0
$166$	28.1421	2.18425
$167$	−1.97908	−0.153145	−0.0765727	−	0.997064i	$-0.524398\pi$
$167$	−1.97908	−0.153145	−0.0765727	+	0.997064i	$0.524398\pi$
$168$	0	0
$169$	−11.0000	−0.846154
$170$	0	0
$171$	0	0
$172$	3.17157	0.241830
$173$	2.93015	0.222775	0.111388	−	0.993777i	$-0.464470\pi$
$173$	2.93015	0.222775	0.111388	+	0.993777i	$0.464470\pi$
$174$	0	0
$175$	−11.5349	−0.871956
$176$	7.83938	0.590915
$177$	0	0
$178$	15.8995	1.19172
$179$	−6.00000	−0.448461	−0.224231	−	0.974536i	$-0.571987\pi$
$179$	−6.00000	−0.448461	−0.224231	+	0.974536i	$0.571987\pi$
$180$	0	0
$181$	−11.6662	−0.867143	−0.433571	−	0.901119i	$-0.642747\pi$
$181$	−11.6662	−0.867143	−0.433571	+	0.901119i	$0.642747\pi$
$182$	−8.92177	−0.661326
$183$	0	0
$184$	21.0907	1.55483
$185$	7.07107	0.519875
$186$	0	0
$187$	0	0
$188$	−19.7990	−1.44399
$189$	0	0
$190$	−1.53073	−0.111051
$191$	−20.0000	−1.44715	−0.723575	−	0.690246i	$-0.757502\pi$
$191$	−20.0000	−1.44715	−0.723575	+	0.690246i	$0.757502\pi$
$192$	0	0
$193$	−2.29610	−0.165277	−0.0826385	−	0.996580i	$-0.526335\pi$
$193$	−2.29610	−0.165277	−0.0826385	+	0.996580i	$0.526335\pi$
$194$	−24.9176	−1.78898
$195$	0	0
$196$	−0.656854	−0.0469182
$197$	4.64659	0.331056	0.165528	−	0.986205i	$-0.447067\pi$
$197$	4.64659	0.331056	0.165528	+	0.986205i	$0.447067\pi$
$198$	0	0
$199$	11.5349	0.817687	0.408844	−	0.912604i	$-0.365932\pi$
$199$	11.5349	0.817687	0.408844	+	0.912604i	$0.365932\pi$
$200$	19.4853	1.37782
$201$	0	0
$202$	25.5563	1.79814
$203$	0.828427	0.0581442
$204$	0	0
$205$	−0.928932	−0.0648794
$206$	−30.1421	−2.10010
$207$	0	0
$208$	4.24264	0.294174
$209$	−2.16478	−0.149741
$210$	0	0
$211$	21.3533	1.47003	0.735013	−	0.678053i	$-0.237176\pi$
$211$	21.3533	1.47003	0.735013	+	0.678053i	$0.237176\pi$
$212$	−5.41421	−0.371850
$213$	0	0
$214$	−1.08239	−0.0739908
$215$	−0.634051	−0.0432419
$216$	0	0
$217$	−20.4853	−1.39063
$218$	37.5348	2.54218
$219$	0	0
$220$	−7.65685	−0.516225
$221$	0	0
$222$	0	0
$223$	−4.82843	−0.323335	−0.161668	−	0.986845i	$-0.551687\pi$
$223$	−4.82843	−0.323335	−0.161668	+	0.986845i	$0.551687\pi$
$224$	4.14386	0.276873
$225$	0	0
$226$	31.8602	2.11931
$227$	17.3952	1.15456	0.577280	−	0.816546i	$-0.304114\pi$
$227$	17.3952	1.15456	0.577280	+	0.816546i	$0.304114\pi$
$228$	0	0
$229$	−17.1716	−1.13473	−0.567365	−	0.823467i	$-0.692037\pi$
$229$	−17.1716	−1.13473	−0.567365	+	0.823467i	$0.692037\pi$
$230$	−8.82843	−0.582129
$231$	0	0
$232$	−1.39942	−0.0918763
$233$	−8.79045	−0.575882	−0.287941	−	0.957648i	$-0.592971\pi$
$233$	−8.79045	−0.575882	−0.287941	+	0.957648i	$0.592971\pi$
$234$	0	0
$235$	3.95815	0.258201
$236$	−22.9706	−1.49526
$237$	0	0
$238$	0	0
$239$	−14.8284	−0.959171	−0.479586	−	0.877495i	$-0.659213\pi$
$239$	−14.8284	−0.959171	−0.479586	+	0.877495i	$0.659213\pi$
$240$	0	0
$241$	3.56420	0.229590	0.114795	−	0.993389i	$-0.463379\pi$
$241$	3.56420	0.229590	0.114795	+	0.993389i	$0.463379\pi$
$242$	10.0711	0.647393
$243$	0	0
$244$	−14.6508	−0.937919
$245$	0.131316	0.00838948
$246$	0	0
$247$	−1.17157	−0.0745454
$248$	34.6047	2.19740
$249$	0	0
$250$	−17.3952	−1.10017
$251$	−20.4853	−1.29302	−0.646510	−	0.762906i	$-0.723772\pi$
$251$	−20.4853	−1.29302	−0.646510	+	0.762906i	$0.723772\pi$
$252$	0	0
$253$	−12.4853	−0.784943
$254$	12.8284	0.804927
$255$	0	0
$256$	−29.9706	−1.87316
$257$	−6.14214	−0.383136	−0.191568	−	0.981479i	$-0.561357\pi$
$257$	−6.14214	−0.383136	−0.191568	+	0.981479i	$0.561357\pi$
$258$	0	0
$259$	−24.1421	−1.50012
$260$	−4.14386	−0.256991
$261$	0	0
$262$	−36.7695	−2.27163
$263$	10.4853	0.646550	0.323275	−	0.946305i	$-0.395216\pi$
$263$	10.4853	0.646550	0.323275	+	0.946305i	$0.395216\pi$
$264$	0	0
$265$	1.08239	0.0664908
$266$	5.22625	0.320442
$267$	0	0
$268$	−4.48528	−0.273982
$269$	−26.4483	−1.61258	−0.806290	−	0.591520i	$-0.798528\pi$
$269$	−26.4483	−1.61258	−0.806290	+	0.591520i	$0.798528\pi$
$270$	0	0
$271$	−22.1421	−1.34504	−0.672519	−	0.740079i	$-0.734788\pi$
$271$	−22.1421	−1.34504	−0.672519	+	0.740079i	$0.734788\pi$
$272$	0	0
$273$	0	0
$274$	−40.3848	−2.43973
$275$	−11.5349	−0.695580
$276$	0	0
$277$	19.9539	1.19892	0.599458	−	0.800406i	$-0.295383\pi$
$277$	19.9539	1.19892	0.599458	+	0.800406i	$0.295383\pi$
$278$	−51.5515	−3.09186
$279$	0	0
$280$	8.82843	0.527599
$281$	−1.89949	−0.113314	−0.0566572	−	0.998394i	$-0.518044\pi$
$281$	−1.89949	−0.113314	−0.0566572	+	0.998394i	$0.518044\pi$
$282$	0	0
$283$	−18.6633	−1.10942	−0.554709	−	0.832044i	$-0.687170\pi$
$283$	−18.6633	−1.10942	−0.554709	+	0.832044i	$0.687170\pi$
$284$	−20.7193	−1.22946
$285$	0	0
$286$	−8.92177	−0.527555
$287$	3.17157	0.187212
$288$	0	0
$289$	0	0
$290$	0.585786	0.0343986
$291$	0	0
$292$	49.5181	2.89783
$293$	12.3431	0.721094	0.360547	−	0.932741i	$-0.382590\pi$
$293$	12.3431	0.721094	0.360547	+	0.932741i	$0.382590\pi$
$294$	0	0
$295$	4.59220	0.267368
$296$	40.7820	2.37041
$297$	0	0
$298$	−40.9706	−2.37336
$299$	−6.75699	−0.390767
$300$	0	0
$301$	2.16478	0.124776
$302$	−17.3137	−0.996292
$303$	0	0
$304$	−2.48528	−0.142541
$305$	2.92893	0.167710
$306$	0	0
$307$	−26.1421	−1.49201	−0.746005	−	0.665940i	$-0.768031\pi$
$307$	−26.1421	−1.49201	−0.746005	+	0.665940i	$0.768031\pi$
$308$	26.1421	1.48959
$309$	0	0
$310$	−14.4853	−0.822709
$311$	25.6829	1.45634	0.728172	−	0.685394i	$-0.240370\pi$
$311$	25.6829	1.45634	0.728172	+	0.685394i	$0.240370\pi$
$312$	0	0
$313$	9.87285	0.558046	0.279023	−	0.960284i	$-0.409989\pi$
$313$	9.87285	0.558046	0.279023	+	0.960284i	$0.409989\pi$
$314$	23.3137	1.31567
$315$	0	0
$316$	18.2919	1.02900
$317$	−19.2430	−1.08079	−0.540396	−	0.841411i	$-0.681726\pi$
$317$	−19.2430	−1.08079	−0.540396	+	0.841411i	$0.681726\pi$
$318$	0	0
$319$	0.828427	0.0463830
$320$	7.52235	0.420512
$321$	0	0
$322$	30.1421	1.67976
$323$	0	0
$324$	0	0
$325$	−6.24264	−0.346279
$326$	20.4567	1.13299
$327$	0	0
$328$	−5.35757	−0.295822
$329$	−13.5140	−0.745049
$330$	0	0
$331$	−21.7990	−1.19818	−0.599090	−	0.800681i	$-0.704471\pi$
$331$	−21.7990	−1.19818	−0.599090	+	0.800681i	$0.704471\pi$
$332$	−44.6274	−2.44925
$333$	0	0
$334$	4.77791	0.261436
$335$	0.896683	0.0489910
$336$	0	0
$337$	−5.62020	−0.306152	−0.153076	−	0.988214i	$-0.548918\pi$
$337$	−5.62020	−0.306152	−0.153076	+	0.988214i	$0.548918\pi$
$338$	26.5563	1.44447
$339$	0	0
$340$	0	0
$341$	−20.4853	−1.10934
$342$	0	0
$343$	−18.7402	−1.01188
$344$	−3.65685	−0.197164
$345$	0	0
$346$	−7.07401	−0.380301
$347$	16.7611	0.899786	0.449893	−	0.893083i	$-0.351462\pi$
$347$	16.7611	0.899786	0.449893	+	0.893083i	$0.351462\pi$
$348$	0	0
$349$	−4.24264	−0.227103	−0.113552	−	0.993532i	$-0.536223\pi$
$349$	−4.24264	−0.227103	−0.113552	+	0.993532i	$0.536223\pi$
$350$	27.8477	1.48852
$351$	0	0
$352$	4.14386	0.220868
$353$	14.0000	0.745145	0.372572	−	0.928003i	$-0.378476\pi$
$353$	14.0000	0.745145	0.372572	+	0.928003i	$0.378476\pi$
$354$	0	0
$355$	4.14214	0.219842
$356$	−25.2132	−1.33630
$357$	0	0
$358$	14.4853	0.765571
$359$	−28.8284	−1.52151	−0.760753	−	0.649041i	$-0.775170\pi$
$359$	−28.8284	−1.52151	−0.760753	+	0.649041i	$0.775170\pi$
$360$	0	0
$361$	−18.3137	−0.963879
$362$	28.1647	1.48031
$363$	0	0
$364$	14.1480	0.741558
$365$	−9.89949	−0.518163
$366$	0	0
$367$	−4.40649	−0.230017	−0.115009	−	0.993365i	$-0.536690\pi$
$367$	−4.40649	−0.230017	−0.115009	+	0.993365i	$0.536690\pi$
$368$	−14.3337	−0.747197
$369$	0	0
$370$	−17.0711	−0.887483
$371$	−3.69552	−0.191862
$372$	0	0
$373$	−11.5563	−0.598365	−0.299183	−	0.954196i	$-0.596714\pi$
$373$	−11.5563	−0.598365	−0.299183	+	0.954196i	$0.596714\pi$
$374$	0	0
$375$	0	0
$376$	22.8284	1.17729
$377$	0.448342	0.0230908
$378$	0	0
$379$	2.61313	0.134227	0.0671136	−	0.997745i	$-0.478621\pi$
$379$	2.61313	0.134227	0.0671136	+	0.997745i	$0.478621\pi$
$380$	2.42742	0.124524
$381$	0	0
$382$	48.2843	2.47044
$383$	−22.4853	−1.14894	−0.574472	−	0.818524i	$-0.694793\pi$
$383$	−22.4853	−1.14894	−0.574472	+	0.818524i	$0.694793\pi$
$384$	0	0
$385$	−5.22625	−0.266354
$386$	5.54328	0.282145
$387$	0	0
$388$	39.5139	2.00602
$389$	12.1421	0.615631	0.307815	−	0.951446i	$-0.400402\pi$
$389$	12.1421	0.615631	0.307815	+	0.951446i	$0.400402\pi$
$390$	0	0
$391$	0	0
$392$	0.757359	0.0382524
$393$	0	0
$394$	−11.2179	−0.565148
$395$	−3.65685	−0.183996
$396$	0	0
$397$	−17.7891	−0.892812	−0.446406	−	0.894831i	$-0.647296\pi$
$397$	−17.7891	−0.892812	−0.446406	+	0.894831i	$0.647296\pi$
$398$	−27.8477	−1.39588
$399$	0	0
$400$	−13.2426	−0.662132
$401$	0.579658	0.0289467	0.0144734	−	0.999895i	$-0.495393\pi$
$401$	0.579658	0.0289467	0.0144734	+	0.999895i	$0.495393\pi$
$402$	0	0
$403$	−11.0866	−0.552261
$404$	−40.5269	−2.01629
$405$	0	0
$406$	−2.00000	−0.0992583
$407$	−24.1421	−1.19668
$408$	0	0
$409$	−3.31371	−0.163852	−0.0819262	−	0.996638i	$-0.526107\pi$
$409$	−3.31371	−0.163852	−0.0819262	+	0.996638i	$0.526107\pi$
$410$	2.24264	0.110756
$411$	0	0
$412$	47.7990	2.35489
$413$	−15.6788	−0.771501
$414$	0	0
$415$	8.92177	0.437952
$416$	2.24264	0.109955
$417$	0	0
$418$	5.22625	0.255624
$419$	13.3283	0.651128	0.325564	−	0.945520i	$-0.394446\pi$
$419$	13.3283	0.651128	0.325564	+	0.945520i	$0.394446\pi$
$420$	0	0
$421$	14.5858	0.710868	0.355434	−	0.934701i	$-0.384333\pi$
$421$	14.5858	0.710868	0.355434	+	0.934701i	$0.384333\pi$
$422$	−51.5515	−2.50949
$423$	0	0
$424$	6.24264	0.303169
$425$	0	0
$426$	0	0
$427$	−10.0000	−0.483934
$428$	1.71644	0.0829674
$429$	0	0
$430$	1.53073	0.0738185
$431$	7.31411	0.352308	0.176154	−	0.984363i	$-0.443634\pi$
$431$	7.31411	0.352308	0.176154	+	0.984363i	$0.443634\pi$
$432$	0	0
$433$	20.8284	1.00095	0.500475	−	0.865751i	$-0.333159\pi$
$433$	20.8284	1.00095	0.500475	+	0.865751i	$0.333159\pi$
$434$	49.4558	2.37396
$435$	0	0
$436$	−59.5222	−2.85060
$437$	3.95815	0.189344
$438$	0	0
$439$	−10.6382	−0.507734	−0.253867	−	0.967239i	$-0.581703\pi$
$439$	−10.6382	−0.507734	−0.253867	+	0.967239i	$0.581703\pi$
$440$	8.82843	0.420879
$441$	0	0
$442$	0	0
$443$	23.7990	1.13072	0.565362	−	0.824843i	$-0.308736\pi$
$443$	23.7990	1.13072	0.565362	+	0.824843i	$0.308736\pi$
$444$	0	0
$445$	5.04054	0.238945
$446$	11.6569	0.551968
$447$	0	0
$448$	−25.6829	−1.21340
$449$	12.1146	0.571721	0.285861	−	0.958271i	$-0.407721\pi$
$449$	12.1146	0.571721	0.285861	+	0.958271i	$0.407721\pi$
$450$	0	0
$451$	3.17157	0.149344
$452$	−50.5235	−2.37643
$453$	0	0
$454$	−41.9957	−1.97096
$455$	−2.82843	−0.132599
$456$	0	0
$457$	13.1716	0.616140	0.308070	−	0.951364i	$-0.400317\pi$
$457$	13.1716	0.616140	0.308070	+	0.951364i	$0.400317\pi$
$458$	41.4558	1.93710
$459$	0	0
$460$	14.0000	0.652753
$461$	24.0416	1.11973	0.559865	−	0.828584i	$-0.310853\pi$
$461$	24.0416	1.11973	0.559865	+	0.828584i	$0.310853\pi$
$462$	0	0
$463$	14.6274	0.679794	0.339897	−	0.940463i	$-0.389608\pi$
$463$	14.6274	0.679794	0.339897	+	0.940463i	$0.389608\pi$
$464$	0.951076	0.0441526
$465$	0	0
$466$	21.2220	0.983092
$467$	−32.6274	−1.50982	−0.754908	−	0.655830i	$-0.772319\pi$
$467$	−32.6274	−1.50982	−0.754908	+	0.655830i	$0.772319\pi$
$468$	0	0
$469$	−3.06147	−0.141365
$470$	−9.55582	−0.440777
$471$	0	0
$472$	26.4853	1.21908
$473$	2.16478	0.0995369
$474$	0	0
$475$	3.65685	0.167788
$476$	0	0
$477$	0	0
$478$	35.7990	1.63741
$479$	5.14933	0.235279	0.117639	−	0.993056i	$-0.462467\pi$
$479$	5.14933	0.235279	0.117639	+	0.993056i	$0.462467\pi$
$480$	0	0
$481$	−13.0656	−0.595741
$482$	−8.60474	−0.391935
$483$	0	0
$484$	−15.9706	−0.725935
$485$	−7.89949	−0.358698
$486$	0	0
$487$	26.3170	1.19254	0.596268	−	0.802786i	$-0.296650\pi$
$487$	26.3170	1.19254	0.596268	+	0.802786i	$0.296650\pi$
$488$	16.8925	0.764686
$489$	0	0
$490$	−0.317025	−0.0143217
$491$	−37.1127	−1.67487	−0.837436	−	0.546535i	$-0.815947\pi$
$491$	−37.1127	−1.67487	−0.837436	+	0.546535i	$0.815947\pi$
$492$	0	0
$493$	0	0
$494$	2.82843	0.127257
$495$	0	0
$496$	−23.5181	−1.05600
$497$	−14.1421	−0.634361
$498$	0	0
$499$	21.4621	0.960777	0.480389	−	0.877056i	$-0.340496\pi$
$499$	21.4621	0.960777	0.480389	+	0.877056i	$0.340496\pi$
$500$	27.5851	1.23364
$501$	0	0
$502$	49.4558	2.20732
$503$	21.3533	0.952099	0.476049	−	0.879419i	$-0.342068\pi$
$503$	21.3533	0.952099	0.476049	+	0.879419i	$0.342068\pi$
$504$	0	0
$505$	8.10201	0.360535
$506$	30.1421	1.33998
$507$	0	0
$508$	−20.3431	−0.902581
$509$	−36.9706	−1.63869	−0.819346	−	0.573300i	$-0.805663\pi$
$509$	−36.9706	−1.63869	−0.819346	+	0.573300i	$0.805663\pi$
$510$	0	0
$511$	33.7990	1.49518
$512$	31.2426	1.38074
$513$	0	0
$514$	14.8284	0.654054
$515$	−9.55582	−0.421080
$516$	0	0
$517$	−13.5140	−0.594344
$518$	58.2843	2.56086
$519$	0	0
$520$	4.77791	0.209525
$521$	−18.6089	−0.815271	−0.407636	−	0.913145i	$-0.633647\pi$
$521$	−18.6089	−0.815271	−0.407636	+	0.913145i	$0.633647\pi$
$522$	0	0
$523$	1.17157	0.0512293	0.0256147	−	0.999672i	$-0.491846\pi$
$523$	1.17157	0.0512293	0.0256147	+	0.999672i	$0.491846\pi$
$524$	58.3085	2.54722
$525$	0	0
$526$	−25.3137	−1.10373
$527$	0	0
$528$	0	0
$529$	−0.171573	−0.00745969
$530$	−2.61313	−0.113507
$531$	0	0
$532$	−8.28772	−0.359318
$533$	1.71644	0.0743474
$534$	0	0
$535$	−0.343146	−0.0148355
$536$	5.17157	0.223378
$537$	0	0
$538$	63.8518	2.75285
$539$	−0.448342	−0.0193114
$540$	0	0
$541$	−18.4232	−0.792075	−0.396038	−	0.918234i	$-0.629615\pi$
$541$	−18.4232	−0.792075	−0.396038	+	0.918234i	$0.629615\pi$
$542$	53.4558	2.29613
$543$	0	0
$544$	0	0
$545$	11.8995	0.509718
$546$	0	0
$547$	−8.10201	−0.346417	−0.173208	−	0.984885i	$-0.555413\pi$
$547$	−8.10201	−0.346417	−0.173208	+	0.984885i	$0.555413\pi$
$548$	64.0416	2.73572
$549$	0	0
$550$	27.8477	1.18743
$551$	−0.262632	−0.0111885
$552$	0	0
$553$	12.4853	0.530928
$554$	−48.1731	−2.04668
$555$	0	0
$556$	81.7497	3.46696
$557$	19.7574	0.837146	0.418573	−	0.908183i	$-0.362530\pi$
$557$	19.7574	0.837146	0.418573	+	0.908183i	$0.362530\pi$
$558$	0	0
$559$	1.17157	0.0495523
$560$	−6.00000	−0.253546
$561$	0	0
$562$	4.58579	0.193440
$563$	34.7696	1.46536	0.732681	−	0.680572i	$-0.238269\pi$
$563$	34.7696	1.46536	0.732681	+	0.680572i	$0.238269\pi$
$564$	0	0
$565$	10.1005	0.424931
$566$	45.0572	1.89390
$567$	0	0
$568$	23.8896	1.00238
$569$	−12.0416	−0.504811	−0.252406	−	0.967621i	$-0.581222\pi$
$569$	−12.0416	−0.504811	−0.252406	+	0.967621i	$0.581222\pi$
$570$	0	0
$571$	−4.25265	−0.177968	−0.0889838	−	0.996033i	$-0.528362\pi$
$571$	−4.25265	−0.177968	−0.0889838	+	0.996033i	$0.528362\pi$
$572$	14.1480	0.591559
$573$	0	0
$574$	−7.65685	−0.319591
$575$	21.0907	0.879544
$576$	0	0
$577$	27.0711	1.12698	0.563492	−	0.826122i	$-0.309458\pi$
$577$	27.0711	1.12698	0.563492	+	0.826122i	$0.309458\pi$
$578$	0	0
$579$	0	0
$580$	−0.928932	−0.0385718
$581$	−30.4608	−1.26373
$582$	0	0
$583$	−3.69552	−0.153053
$584$	−57.0948	−2.36260
$585$	0	0
$586$	−29.7990	−1.23098
$587$	45.3137	1.87030	0.935148	−	0.354256i	$-0.115266\pi$
$587$	45.3137	1.87030	0.935148	+	0.354256i	$0.115266\pi$
$588$	0	0
$589$	6.49435	0.267595
$590$	−11.0866	−0.456426
$591$	0	0
$592$	−27.7164	−1.13914
$593$	−12.9289	−0.530928	−0.265464	−	0.964121i	$-0.585525\pi$
$593$	−12.9289	−0.530928	−0.265464	+	0.964121i	$0.585525\pi$
$594$	0	0
$595$	0	0
$596$	64.9706	2.66130
$597$	0	0
$598$	16.3128	0.667080
$599$	10.6274	0.434224	0.217112	−	0.976147i	$-0.430336\pi$
$599$	10.6274	0.434224	0.217112	+	0.976147i	$0.430336\pi$
$600$	0	0
$601$	8.41904	0.343420	0.171710	−	0.985148i	$-0.445071\pi$
$601$	8.41904	0.343420	0.171710	+	0.985148i	$0.445071\pi$
$602$	−5.22625	−0.213006
$603$	0	0
$604$	27.4558	1.11716
$605$	3.19278	0.129805
$606$	0	0
$607$	16.3897	0.665239	0.332619	−	0.943061i	$-0.392068\pi$
$607$	16.3897	0.665239	0.332619	+	0.943061i	$0.392068\pi$
$608$	−1.31371	−0.0532779
$609$	0	0
$610$	−7.07107	−0.286299
$611$	−7.31371	−0.295881
$612$	0	0
$613$	5.31371	0.214619	0.107309	−	0.994226i	$-0.465776\pi$
$613$	5.31371	0.214619	0.107309	+	0.994226i	$0.465776\pi$
$614$	63.1127	2.54702
$615$	0	0
$616$	−30.1421	−1.21446
$617$	−2.93015	−0.117963	−0.0589817	−	0.998259i	$-0.518785\pi$
$617$	−2.93015	−0.117963	−0.0589817	+	0.998259i	$0.518785\pi$
$618$	0	0
$619$	−28.4818	−1.14478	−0.572389	−	0.819982i	$-0.693983\pi$
$619$	−28.4818	−1.14478	−0.572389	+	0.819982i	$0.693983\pi$
$620$	22.9706	0.922520
$621$	0	0
$622$	−62.0040	−2.48614
$623$	−17.2095	−0.689484
$624$	0	0
$625$	16.5563	0.662254
$626$	−23.8352	−0.952645
$627$	0	0
$628$	−36.9706	−1.47529
$629$	0	0
$630$	0	0
$631$	29.3137	1.16696	0.583480	−	0.812127i	$-0.301691\pi$
$631$	29.3137	1.16696	0.583480	+	0.812127i	$0.301691\pi$
$632$	−21.0907	−0.838944
$633$	0	0
$634$	46.4566	1.84503
$635$	4.06694	0.161391
$636$	0	0
$637$	−0.242641	−0.00961377
$638$	−2.00000	−0.0791808
$639$	0	0
$640$	−15.7331	−0.621907
$641$	41.4161	1.63584	0.817918	−	0.575335i	$-0.195128\pi$
$641$	41.4161	1.63584	0.817918	+	0.575335i	$0.195128\pi$
$642$	0	0
$643$	28.8532	1.13786	0.568929	−	0.822387i	$-0.307358\pi$
$643$	28.8532	1.13786	0.568929	+	0.822387i	$0.307358\pi$
$644$	−47.7990	−1.88354
$645$	0	0
$646$	0	0
$647$	2.82843	0.111197	0.0555985	−	0.998453i	$-0.482293\pi$
$647$	2.82843	0.111197	0.0555985	+	0.998453i	$0.482293\pi$
$648$	0	0
$649$	−15.6788	−0.615445
$650$	15.0711	0.591136
$651$	0	0
$652$	−32.4399	−1.27044
$653$	9.50143	0.371820	0.185910	−	0.982567i	$-0.440477\pi$
$653$	9.50143	0.371820	0.185910	+	0.982567i	$0.440477\pi$
$654$	0	0
$655$	−11.6569	−0.455471
$656$	3.64113	0.142162
$657$	0	0
$658$	32.6256	1.27188
$659$	8.48528	0.330540	0.165270	−	0.986248i	$-0.447151\pi$
$659$	8.48528	0.330540	0.165270	+	0.986248i	$0.447151\pi$
$660$	0	0
$661$	1.21320	0.0471881	0.0235941	−	0.999722i	$-0.492489\pi$
$661$	1.21320	0.0471881	0.0235941	+	0.999722i	$0.492489\pi$
$662$	52.6274	2.04542
$663$	0	0
$664$	51.4558	1.99687
$665$	1.65685	0.0642501
$666$	0	0
$667$	−1.51472	−0.0586501
$668$	−7.57675	−0.293153
$669$	0	0
$670$	−2.16478	−0.0836329
$671$	−10.0000	−0.386046
$672$	0	0
$673$	4.46088	0.171954	0.0859772	−	0.996297i	$-0.472599\pi$
$673$	4.46088	0.171954	0.0859772	+	0.996297i	$0.472599\pi$
$674$	13.5684	0.522634
$675$	0	0
$676$	−42.1127	−1.61972
$677$	38.0920	1.46399	0.731997	−	0.681308i	$-0.238589\pi$
$677$	38.0920	1.46399	0.731997	+	0.681308i	$0.238589\pi$
$678$	0	0
$679$	26.9706	1.03504
$680$	0	0
$681$	0	0
$682$	49.4558	1.89376
$683$	−23.7808	−0.909946	−0.454973	−	0.890505i	$-0.650351\pi$
$683$	−23.7808	−0.909946	−0.454973	+	0.890505i	$0.650351\pi$
$684$	0	0
$685$	−12.8030	−0.489177
$686$	45.2429	1.72738
$687$	0	0
$688$	2.48528	0.0947505
$689$	−2.00000	−0.0761939
$690$	0	0
$691$	−19.9314	−0.758226	−0.379113	−	0.925350i	$-0.623771\pi$
$691$	−19.9314	−0.758226	−0.379113	+	0.925350i	$0.623771\pi$
$692$	11.2179	0.426439
$693$	0	0
$694$	−40.4650	−1.53603
$695$	−16.3431	−0.619931
$696$	0	0
$697$	0	0
$698$	10.2426	0.387690
$699$	0	0
$700$	−44.1605	−1.66911
$701$	−37.6985	−1.42385	−0.711926	−	0.702254i	$-0.752177\pi$
$701$	−37.6985	−1.42385	−0.711926	+	0.702254i	$0.752177\pi$
$702$	0	0
$703$	7.65367	0.288664
$704$	−25.6829	−0.967961
$705$	0	0
$706$	−33.7990	−1.27204
$707$	−27.6620	−1.04034
$708$	0	0
$709$	24.2835	0.911986	0.455993	−	0.889983i	$-0.349284\pi$
$709$	24.2835	0.911986	0.455993	+	0.889983i	$0.349284\pi$
$710$	−10.0000	−0.375293
$711$	0	0
$712$	29.0711	1.08948
$713$	37.4558	1.40273
$714$	0	0
$715$	−2.82843	−0.105777
$716$	−22.9706	−0.858450
$717$	0	0
$718$	69.5980	2.59737
$719$	−33.9706	−1.26689	−0.633445	−	0.773787i	$-0.718360\pi$
$719$	−33.9706	−1.26689	−0.633445	+	0.773787i	$0.718360\pi$
$720$	0	0
$721$	32.6256	1.21504
$722$	44.2132	1.64545
$723$	0	0
$724$	−44.6632	−1.65990
$725$	−1.39942	−0.0519731
$726$	0	0
$727$	−43.1127	−1.59896	−0.799481	−	0.600692i	$-0.794892\pi$
$727$	−43.1127	−1.59896	−0.799481	+	0.600692i	$0.794892\pi$
$728$	−16.3128	−0.604593
$729$	0	0
$730$	23.8995	0.884560
$731$	0	0
$732$	0	0
$733$	36.0416	1.33123	0.665614	−	0.746296i	$-0.268170\pi$
$733$	36.0416	1.33123	0.665614	+	0.746296i	$0.268170\pi$
$734$	10.6382	0.392664
$735$	0	0
$736$	−7.57675	−0.279283
$737$	−3.06147	−0.112771
$738$	0	0
$739$	22.2843	0.819740	0.409870	−	0.912144i	$-0.365574\pi$
$739$	22.2843	0.819740	0.409870	+	0.912144i	$0.365574\pi$
$740$	27.0711	0.995152
$741$	0	0
$742$	8.92177	0.327528
$743$	−51.0263	−1.87197	−0.935986	−	0.352036i	$-0.885489\pi$
$743$	−51.0263	−1.87197	−0.935986	+	0.352036i	$0.885489\pi$
$744$	0	0
$745$	−12.9887	−0.475869
$746$	27.8995	1.02147
$747$	0	0
$748$	0	0
$749$	1.17157	0.0428083
$750$	0	0
$751$	−47.5934	−1.73671	−0.868354	−	0.495945i	$-0.834822\pi$
$751$	−47.5934	−1.73671	−0.868354	+	0.495945i	$0.834822\pi$
$752$	−15.5147	−0.565764
$753$	0	0
$754$	−1.08239	−0.0394184
$755$	−5.48888	−0.199761
$756$	0	0
$757$	2.54416	0.0924689	0.0462345	−	0.998931i	$-0.485278\pi$
$757$	2.54416	0.0924689	0.0462345	+	0.998931i	$0.485278\pi$
$758$	−6.30864	−0.229140
$759$	0	0
$760$	−2.79884	−0.101524
$761$	−37.6985	−1.36657	−0.683285	−	0.730152i	$-0.739449\pi$
$761$	−37.6985	−1.36657	−0.683285	+	0.730152i	$0.739449\pi$
$762$	0	0
$763$	−40.6274	−1.47081
$764$	−76.5685	−2.77015
$765$	0	0
$766$	54.2843	1.96137
$767$	−8.48528	−0.306386
$768$	0	0
$769$	12.7279	0.458981	0.229490	−	0.973311i	$-0.426294\pi$
$769$	12.7279	0.458981	0.229490	+	0.973311i	$0.426294\pi$
$770$	12.6173	0.454696
$771$	0	0
$772$	−8.79045	−0.316375
$773$	0.828427	0.0297965	0.0148982	−	0.999889i	$-0.495258\pi$
$773$	0.828427	0.0297965	0.0148982	+	0.999889i	$0.495258\pi$
$774$	0	0
$775$	34.6047	1.24304
$776$	−45.5599	−1.63551
$777$	0	0
$778$	−29.3137	−1.05095
$779$	−1.00547	−0.0360247
$780$	0	0
$781$	−14.1421	−0.506045
$782$	0	0
$783$	0	0
$784$	−0.514719	−0.0183828
$785$	7.39104	0.263797
$786$	0	0
$787$	20.5655	0.733079	0.366540	−	0.930402i	$-0.380542\pi$
$787$	20.5655	0.733079	0.366540	+	0.930402i	$0.380542\pi$
$788$	17.7891	0.633712
$789$	0	0
$790$	8.82843	0.314101
$791$	−34.4853	−1.22616
$792$	0	0
$793$	−5.41196	−0.192184
$794$	42.9468	1.52412
$795$	0	0
$796$	44.1605	1.56523
$797$	25.2132	0.893097	0.446549	−	0.894759i	$-0.352653\pi$
$797$	25.2132	0.893097	0.446549	+	0.894759i	$0.352653\pi$
$798$	0	0
$799$	0	0
$800$	−7.00000	−0.247487
$801$	0	0
$802$	−1.39942	−0.0494152
$803$	33.7990	1.19274
$804$	0	0
$805$	9.55582	0.336798
$806$	26.7653	0.942768
$807$	0	0
$808$	46.7279	1.64388
$809$	35.2931	1.24084	0.620420	−	0.784270i	$-0.286962\pi$
$809$	35.2931	1.24084	0.620420	+	0.784270i	$0.286962\pi$
$810$	0	0
$811$	−55.1383	−1.93617	−0.968083	−	0.250628i	$-0.919363\pi$
$811$	−55.1383	−1.93617	−0.968083	+	0.250628i	$0.919363\pi$
$812$	3.17157	0.111300
$813$	0	0
$814$	58.2843	2.04286
$815$	6.48528	0.227169
$816$	0	0
$817$	−0.686292	−0.0240103
$818$	8.00000	0.279713
$819$	0	0
$820$	−3.55635	−0.124193
$821$	38.4315	1.34127	0.670635	−	0.741788i	$-0.266022\pi$
$821$	38.4315	1.34127	0.670635	+	0.741788i	$0.266022\pi$
$822$	0	0
$823$	−9.74153	−0.339568	−0.169784	−	0.985481i	$-0.554307\pi$
$823$	−9.74153	−0.339568	−0.169784	+	0.985481i	$0.554307\pi$
$824$	−55.1127	−1.91994
$825$	0	0
$826$	37.8519	1.31703
$827$	46.8506	1.62915	0.814577	−	0.580056i	$-0.196969\pi$
$827$	46.8506	1.62915	0.814577	+	0.580056i	$0.196969\pi$
$828$	0	0
$829$	53.9411	1.87345	0.936726	−	0.350062i	$-0.113840\pi$
$829$	53.9411	1.87345	0.936726	+	0.350062i	$0.113840\pi$
$830$	−21.5391	−0.747632
$831$	0	0
$832$	−13.8995	−0.481878
$833$	0	0
$834$	0	0
$835$	1.51472	0.0524190
$836$	−8.28772	−0.286637
$837$	0	0
$838$	−32.1773	−1.11155
$839$	−16.7611	−0.578659	−0.289330	−	0.957230i	$-0.593432\pi$
$839$	−16.7611	−0.578659	−0.289330	+	0.957230i	$0.593432\pi$
$840$	0	0
$841$	−28.8995	−0.996534
$842$	−35.2132	−1.21353
$843$	0	0
$844$	81.7497	2.81394
$845$	8.41904	0.289624
$846$	0	0
$847$	−10.9008	−0.374557
$848$	−4.24264	−0.145693
$849$	0	0
$850$	0	0
$851$	44.1421	1.51317
$852$	0	0
$853$	19.1660	0.656233	0.328116	−	0.944637i	$-0.393586\pi$
$853$	19.1660	0.656233	0.328116	+	0.944637i	$0.393586\pi$
$854$	24.1421	0.826127
$855$	0	0
$856$	−1.97908	−0.0676434
$857$	−9.23880	−0.315591	−0.157796	−	0.987472i	$-0.550439\pi$
$857$	−9.23880	−0.315591	−0.157796	+	0.987472i	$0.550439\pi$
$858$	0	0
$859$	−34.9706	−1.19318	−0.596590	−	0.802546i	$-0.703478\pi$
$859$	−34.9706	−1.19318	−0.596590	+	0.802546i	$0.703478\pi$
$860$	−2.42742	−0.0827742
$861$	0	0
$862$	−17.6578	−0.601428
$863$	10.6274	0.361761	0.180881	−	0.983505i	$-0.442105\pi$
$863$	10.6274	0.361761	0.180881	+	0.983505i	$0.442105\pi$
$864$	0	0
$865$	−2.24264	−0.0762521
$866$	−50.2843	−1.70873
$867$	0	0
$868$	−78.4264	−2.66197
$869$	12.4853	0.423534
$870$	0	0
$871$	−1.65685	−0.0561404
$872$	68.6297	2.32410
$873$	0	0
$874$	−9.55582	−0.323230
$875$	18.8284	0.636517
$876$	0	0
$877$	50.2291	1.69611	0.848057	−	0.529905i	$-0.177772\pi$
$877$	50.2291	1.69611	0.848057	+	0.529905i	$0.177772\pi$
$878$	25.6829	0.866756
$879$	0	0
$880$	−6.00000	−0.202260
$881$	−33.6536	−1.13382	−0.566909	−	0.823780i	$-0.691861\pi$
$881$	−33.6536	−1.13382	−0.566909	+	0.823780i	$0.691861\pi$
$882$	0	0
$883$	−8.00000	−0.269221	−0.134611	−	0.990899i	$-0.542978\pi$
$883$	−8.00000	−0.269221	−0.134611	+	0.990899i	$0.542978\pi$
$884$	0	0
$885$	0	0
$886$	−57.4558	−1.93027
$887$	44.0517	1.47911	0.739556	−	0.673095i	$-0.235035\pi$
$887$	44.0517	1.47911	0.739556	+	0.673095i	$0.235035\pi$
$888$	0	0
$889$	−13.8854	−0.465701
$890$	−12.1689	−0.407904
$891$	0	0
$892$	−18.4853	−0.618933
$893$	4.28427	0.143368
$894$	0	0
$895$	4.59220	0.153500
$896$	53.7163	1.79454
$897$	0	0
$898$	−29.2471	−0.975989
$899$	−2.48528	−0.0828888
$900$	0	0
$901$	0	0
$902$	−7.65685	−0.254945
$903$	0	0
$904$	58.2541	1.93750
$905$	8.92893	0.296808
$906$	0	0
$907$	−13.4370	−0.446170	−0.223085	−	0.974799i	$-0.571613\pi$
$907$	−13.4370	−0.446170	−0.223085	+	0.974799i	$0.571613\pi$
$908$	66.5962	2.21007
$909$	0	0
$910$	6.82843	0.226360
$911$	−5.67459	−0.188008	−0.0940038	−	0.995572i	$-0.529967\pi$
$911$	−5.67459	−0.188008	−0.0940038	+	0.995572i	$0.529967\pi$
$912$	0	0
$913$	−30.4608	−1.00811
$914$	−31.7990	−1.05182
$915$	0	0
$916$	−65.7401	−2.17211
$917$	39.7990	1.31428
$918$	0	0
$919$	19.3137	0.637100	0.318550	−	0.947906i	$-0.396804\pi$
$919$	19.3137	0.637100	0.318550	+	0.947906i	$0.396804\pi$
$920$	−16.1421	−0.532190
$921$	0	0
$922$	−58.0416	−1.91150
$923$	−7.65367	−0.251924
$924$	0	0
$925$	40.7820	1.34090
$926$	−35.3137	−1.16048
$927$	0	0
$928$	0.502734	0.0165031
$929$	−17.3408	−0.568933	−0.284467	−	0.958686i	$-0.591817\pi$
$929$	−17.3408	−0.568933	−0.284467	+	0.958686i	$0.591817\pi$
$930$	0	0
$931$	0.142136	0.00465831
$932$	−33.6536	−1.10236
$933$	0	0
$934$	78.7696	2.57742
$935$	0	0
$936$	0	0
$937$	−27.5563	−0.900227	−0.450113	−	0.892971i	$-0.648616\pi$
$937$	−27.5563	−0.900227	−0.450113	+	0.892971i	$0.648616\pi$
$938$	7.39104	0.241326
$939$	0	0
$940$	15.1535	0.494252
$941$	−39.8853	−1.30022	−0.650112	−	0.759838i	$-0.725278\pi$
$941$	−39.8853	−1.30022	−0.650112	+	0.759838i	$0.725278\pi$
$942$	0	0
$943$	−5.79899	−0.188841
$944$	−18.0000	−0.585850
$945$	0	0
$946$	−5.22625	−0.169920
$947$	−30.6465	−0.995879	−0.497939	−	0.867212i	$-0.665910\pi$
$947$	−30.6465	−0.995879	−0.497939	+	0.867212i	$0.665910\pi$
$948$	0	0
$949$	18.2919	0.593780
$950$	−8.82843	−0.286432
$951$	0	0
$952$	0	0
$953$	−49.6985	−1.60989	−0.804946	−	0.593348i	$-0.797806\pi$
$953$	−49.6985	−1.60989	−0.804946	+	0.593348i	$0.797806\pi$
$954$	0	0
$955$	15.3073	0.495334
$956$	−56.7696	−1.83606
$957$	0	0
$958$	−12.4316	−0.401646
$959$	43.7122	1.41154
$960$	0	0
$961$	30.4558	0.982447
$962$	31.5432	1.01699
$963$	0	0
$964$	13.6453	0.439485
$965$	1.75736	0.0565714
$966$	0	0
$967$	−43.6569	−1.40391	−0.701955	−	0.712221i	$-0.747689\pi$
$967$	−43.6569	−1.40391	−0.701955	+	0.712221i	$0.747689\pi$
$968$	18.4142	0.591855
$969$	0	0
$970$	19.0711	0.612335
$971$	−51.7401	−1.66042	−0.830210	−	0.557451i	$-0.811779\pi$
$971$	−51.7401	−1.66042	−0.830210	+	0.557451i	$0.811779\pi$
$972$	0	0
$973$	55.7990	1.78883
$974$	−63.5348	−2.03579
$975$	0	0
$976$	−11.4805	−0.367482
$977$	38.3848	1.22804	0.614019	−	0.789291i	$-0.289552\pi$
$977$	38.3848	1.22804	0.614019	+	0.789291i	$0.289552\pi$
$978$	0	0
$979$	−17.2095	−0.550018
$980$	0.502734	0.0160593
$981$	0	0
$982$	89.5980	2.85919
$983$	−9.74153	−0.310707	−0.155353	−	0.987859i	$-0.549652\pi$
$983$	−9.74153	−0.310707	−0.155353	+	0.987859i	$0.549652\pi$
$984$	0	0
$985$	−3.55635	−0.113315
$986$	0	0
$987$	0	0
$988$	−4.48528	−0.142696
$989$	−3.95815	−0.125862
$990$	0	0
$991$	−40.8364	−1.29721	−0.648606	−	0.761125i	$-0.724648\pi$
$991$	−40.8364	−1.29721	−0.648606	+	0.761125i	$0.724648\pi$
$992$	−12.4316	−0.394703
$993$	0	0
$994$	34.1421	1.08292
$995$	−8.82843	−0.279880
$996$	0	0
$997$	−7.52235	−0.238235	−0.119118	−	0.992880i	$-0.538007\pi$
$997$	−7.52235	−0.238235	−0.119118	+	0.992880i	$0.538007\pi$
$998$	−51.8142	−1.64015
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2601.2.a.bb.1.1		4
3.2	odd	2		289.2.a.f.1.3		4
12.11	even	2		4624.2.a.bp.1.3		4
15.14	odd	2		7225.2.a.u.1.2		4
17.5	odd	16		153.2.l.c.127.1		4
17.7	odd	16		153.2.l.c.100.1		4
17.16	even	2	inner	2601.2.a.bb.1.2		4
51.2	odd	8		289.2.c.c.38.3		8
51.5	even	16		17.2.d.a.8.1	✓	4
51.8	odd	8		289.2.c.c.251.2		8
51.11	even	16		289.2.d.c.155.1		4
51.14	even	16		289.2.d.c.179.1		4
51.20	even	16		289.2.d.b.179.1		4
51.23	even	16		289.2.d.b.155.1		4
51.26	odd	8		289.2.c.c.251.1		8
51.29	even	16		289.2.d.a.110.1		4
51.32	odd	8		289.2.c.c.38.4		8
51.38	odd	4		289.2.b.b.288.2		4
51.41	even	16		17.2.d.a.15.1	yes	4
51.44	even	16		289.2.d.a.134.1		4
51.47	odd	4		289.2.b.b.288.1		4
51.50	odd	2		289.2.a.f.1.4		4
204.107	odd	16		272.2.v.d.161.1		4
204.143	odd	16		272.2.v.d.49.1		4
204.203	even	2		4624.2.a.bp.1.2		4
255.92	odd	16		425.2.n.b.49.1		4
255.107	odd	16		425.2.n.a.399.1		4
255.143	odd	16		425.2.n.a.49.1		4
255.158	odd	16		425.2.n.b.399.1		4
255.194	even	16		425.2.m.a.151.1		4
255.209	even	16		425.2.m.a.76.1		4
255.254	odd	2		7225.2.a.u.1.1		4
357.5	odd	48		833.2.v.a.263.1		8
357.41	odd	16		833.2.l.a.491.1		4
357.107	even	48		833.2.v.b.263.1		8
357.143	odd	48		833.2.v.a.814.1		8
357.158	even	48		833.2.v.b.569.1		8
357.194	odd	48		833.2.v.a.508.1		8
357.209	odd	16		833.2.l.a.246.1		4
357.296	even	48		833.2.v.b.508.1		8
357.311	odd	48		833.2.v.a.569.1		8
357.347	even	48		833.2.v.b.814.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
17.2.d.a.8.1	✓	4	51.5	even	16
17.2.d.a.15.1	yes	4	51.41	even	16
153.2.l.c.100.1		4	17.7	odd	16
153.2.l.c.127.1		4	17.5	odd	16
272.2.v.d.49.1		4	204.143	odd	16
272.2.v.d.161.1		4	204.107	odd	16
289.2.a.f.1.3		4	3.2	odd	2
289.2.a.f.1.4		4	51.50	odd	2
289.2.b.b.288.1		4	51.47	odd	4
289.2.b.b.288.2		4	51.38	odd	4
289.2.c.c.38.3		8	51.2	odd	8
289.2.c.c.38.4		8	51.32	odd	8
289.2.c.c.251.1		8	51.26	odd	8
289.2.c.c.251.2		8	51.8	odd	8
289.2.d.a.110.1		4	51.29	even	16
289.2.d.a.134.1		4	51.44	even	16
289.2.d.b.155.1		4	51.23	even	16
289.2.d.b.179.1		4	51.20	even	16
289.2.d.c.155.1		4	51.11	even	16
289.2.d.c.179.1		4	51.14	even	16
425.2.m.a.76.1		4	255.209	even	16
425.2.m.a.151.1		4	255.194	even	16
425.2.n.a.49.1		4	255.143	odd	16
425.2.n.a.399.1		4	255.107	odd	16
425.2.n.b.49.1		4	255.92	odd	16
425.2.n.b.399.1		4	255.158	odd	16
833.2.l.a.246.1		4	357.209	odd	16
833.2.l.a.491.1		4	357.41	odd	16
833.2.v.a.263.1		8	357.5	odd	48
833.2.v.a.508.1		8	357.194	odd	48
833.2.v.a.569.1		8	357.311	odd	48
833.2.v.a.814.1		8	357.143	odd	48
833.2.v.b.263.1		8	357.107	even	48
833.2.v.b.508.1		8	357.296	even	48
833.2.v.b.569.1		8	357.158	even	48
833.2.v.b.814.1		8	357.347	even	48
2601.2.a.bb.1.1		4	1.1	even	1	trivial
2601.2.a.bb.1.2		4	17.16	even	2	inner
4624.2.a.bp.1.2		4	204.203	even	2
4624.2.a.bp.1.3		4	12.11	even	2
7225.2.a.u.1.1		4	255.254	odd	2
7225.2.a.u.1.2		4	15.14	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:	\( N \)	\(=\)	\( 2601 = 3^{2} \cdot 17^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2601.a (trivial)

\(f(q)\)	\(=\)	\(q-2.41421 q^{2} +3.82843 q^{4} -0.765367 q^{5} +2.61313 q^{7} -4.41421 q^{8} +O(q^{10})\)	\(q-2.41421 q^{2} +3.82843 q^{4} -0.765367 q^{5} +2.61313 q^{7} -4.41421 q^{8} +1.84776 q^{10} +2.61313 q^{11} +1.41421 q^{13} -6.30864 q^{14} +3.00000 q^{16} -0.828427 q^{19} -2.93015 q^{20} -6.30864 q^{22} -4.77791 q^{23} -4.41421 q^{25} -3.41421 q^{26} +10.0042 q^{28} +0.317025 q^{29} -7.83938 q^{31} +1.58579 q^{32} -2.00000 q^{35} -9.23880 q^{37} +2.00000 q^{38} +3.37849 q^{40} +1.21371 q^{41} +0.828427 q^{43} +10.0042 q^{44} +11.5349 q^{46} -5.17157 q^{47} -0.171573 q^{49} +10.6569 q^{50} +5.41421 q^{52} -1.41421 q^{53} -2.00000 q^{55} -11.5349 q^{56} -0.765367 q^{58} -6.00000 q^{59} -3.82683 q^{61} +18.9259 q^{62} -9.82843 q^{64} -1.08239 q^{65} -1.17157 q^{67} +4.82843 q^{70} -5.41196 q^{71} +12.9343 q^{73} +22.3044 q^{74} -3.17157 q^{76} +6.82843 q^{77} +4.77791 q^{79} -2.29610 q^{80} -2.93015 q^{82} -11.6569 q^{83} -2.00000 q^{86} -11.5349 q^{88} -6.58579 q^{89} +3.69552 q^{91} -18.2919 q^{92} +12.4853 q^{94} +0.634051 q^{95} +10.3212 q^{97} +0.414214 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{2} + 4 q^{4} - 12 q^{8}+O(q^{10}) \) 4 * q - 4 * q^2 + 4 * q^4 - 12 * q^8	\( 4 q - 4 q^{2} + 4 q^{4} - 12 q^{8} + 12 q^{16} + 8 q^{19} - 12 q^{25} - 8 q^{26} + 12 q^{32} - 8 q^{35} + 8 q^{38} - 8 q^{43} - 32 q^{47} - 12 q^{49} + 20 q^{50} + 16 q^{52} - 8 q^{55} - 24 q^{59} - 28 q^{64} - 16 q^{67} + 8 q^{70} - 24 q^{76} + 16 q^{77} - 24 q^{83} - 8 q^{86} - 32 q^{89} + 16 q^{94} - 4 q^{98}+O(q^{100}) \) 4 * q - 4 * q^2 + 4 * q^4 - 12 * q^8 + 12 * q^16 + 8 * q^19 - 12 * q^25 - 8 * q^26 + 12 * q^32 - 8 * q^35 + 8 * q^38 - 8 * q^43 - 32 * q^47 - 12 * q^49 + 20 * q^50 + 16 * q^52 - 8 * q^55 - 24 * q^59 - 28 * q^64 - 16 * q^67 + 8 * q^70 - 24 * q^76 + 16 * q^77 - 24 * q^83 - 8 * q^86 - 32 * q^89 + 16 * q^94 - 4 * q^98

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