LMFDB - Embedded newform 5239.2.a.l.1.9

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5239,2,Mod(1,5239)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5239, 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("5239.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5239 = 13^{2} \cdot 31 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5239.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:	$41.8336256189$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 4 x^{15} - 17 x^{14} + 80 x^{13} + 98 x^{12} - 628 x^{11} - 158 x^{10} + 2458 x^{9} + \cdots - 147 $ x^16 - 4x^15 - 17x^14 + 80x^13 + 98x^12 - 628x^11 - 158x^10 + 2458x^9 - 510x^8 - 5030x^7 + 2318x^6 + 5112x^5 - 3154x^4 - 2086x^3 + 1542x^2 + 124*x - 147
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 403)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.9
Root			$0.624841$ of defining polynomial
Character	$\chi$	$=$	5239.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.624841 q^{2} -2.78388 q^{3} -1.60957 q^{4} +0.619834 q^{5} -1.73948 q^{6} +2.35216 q^{7} -2.25541 q^{8} +4.74999 q^{9} +O(q^{10})$	\(q+0.624841 q^{2} -2.78388 q^{3} -1.60957 q^{4} +0.619834 q^{5} -1.73948 q^{6} +2.35216 q^{7} -2.25541 q^{8} +4.74999 q^{9} +0.387298 q^{10} +1.21518 q^{11} +4.48086 q^{12} +1.46972 q^{14} -1.72554 q^{15} +1.80988 q^{16} -4.00546 q^{17} +2.96799 q^{18} +4.95017 q^{19} -0.997669 q^{20} -6.54812 q^{21} +0.759295 q^{22} -2.11759 q^{23} +6.27879 q^{24} -4.61581 q^{25} -4.87176 q^{27} -3.78597 q^{28} +10.4829 q^{29} -1.07819 q^{30} -1.00000 q^{31} +5.64170 q^{32} -3.38292 q^{33} -2.50277 q^{34} +1.45795 q^{35} -7.64546 q^{36} +10.5475 q^{37} +3.09307 q^{38} -1.39798 q^{40} +1.88041 q^{41} -4.09153 q^{42} +5.95396 q^{43} -1.95592 q^{44} +2.94420 q^{45} -1.32315 q^{46} -12.5015 q^{47} -5.03848 q^{48} -1.46736 q^{49} -2.88414 q^{50} +11.1507 q^{51} -2.74137 q^{53} -3.04407 q^{54} +0.753211 q^{55} -5.30507 q^{56} -13.7807 q^{57} +6.55011 q^{58} +7.46822 q^{59} +2.77739 q^{60} -6.44716 q^{61} -0.624841 q^{62} +11.1727 q^{63} -0.0945869 q^{64} -2.11379 q^{66} -13.2892 q^{67} +6.44708 q^{68} +5.89511 q^{69} +0.910984 q^{70} -2.71975 q^{71} -10.7132 q^{72} -7.81310 q^{73} +6.59050 q^{74} +12.8498 q^{75} -7.96766 q^{76} +2.85830 q^{77} +2.11825 q^{79} +1.12182 q^{80} -0.687580 q^{81} +1.17495 q^{82} -11.0061 q^{83} +10.5397 q^{84} -2.48272 q^{85} +3.72027 q^{86} -29.1830 q^{87} -2.74073 q^{88} -2.01694 q^{89} +1.83966 q^{90} +3.40841 q^{92} +2.78388 q^{93} -7.81144 q^{94} +3.06828 q^{95} -15.7058 q^{96} -5.43723 q^{97} -0.916869 q^{98} +5.77210 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 4 q^{2} - 2 q^{3} + 18 q^{4} + 4 q^{5} + 6 q^{6} + 2 q^{7} + 12 q^{8} + 10 q^{9}+O(q^{10}) $ 16 * q + 4 * q^2 - 2 * q^3 + 18 * q^4 + 4 * q^5 + 6 * q^6 + 2 * q^7 + 12 * q^8 + 10 * q^9	$ 16 q + 4 q^{2} - 2 q^{3} + 18 q^{4} + 4 q^{5} + 6 q^{6} + 2 q^{7} + 12 q^{8} + 10 q^{9} - 2 q^{10} + 14 q^{11} + 8 q^{12} - 8 q^{14} + 14 q^{16} + 4 q^{17} - 28 q^{18} + 22 q^{19} + 28 q^{20} + 12 q^{21} - 8 q^{22} + 4 q^{23} - 8 q^{24} - 2 q^{25} + 10 q^{27} + 16 q^{28} - 8 q^{29} - 20 q^{30} - 16 q^{31} + 48 q^{32} + 10 q^{33} + 8 q^{34} - 2 q^{35} + 22 q^{36} + 16 q^{37} - 6 q^{38} + 14 q^{40} + 44 q^{41} + 14 q^{42} + 16 q^{43} + 4 q^{44} + 56 q^{45} + 10 q^{47} + 32 q^{49} + 2 q^{50} - 6 q^{53} + 24 q^{54} + 22 q^{55} - 4 q^{56} - 8 q^{57} + 74 q^{58} + 2 q^{59} + 40 q^{60} + 8 q^{61} - 4 q^{62} + 56 q^{63} + 38 q^{64} - 34 q^{66} - 8 q^{67} + 32 q^{68} - 10 q^{69} - 108 q^{70} + 50 q^{71} - 44 q^{72} + 14 q^{73} + 8 q^{74} + 44 q^{76} + 16 q^{77} + 32 q^{79} + 68 q^{80} - 8 q^{81} - 6 q^{82} - 20 q^{83} + 136 q^{84} - 32 q^{85} + 8 q^{86} - 36 q^{87} - 40 q^{88} + 52 q^{89} - 34 q^{90} + 14 q^{92} + 2 q^{93} + 44 q^{94} - 2 q^{95} - 80 q^{96} + 18 q^{97} + 12 q^{98} + 38 q^{99}+O(q^{100}) $ 16 * q + 4 * q^2 - 2 * q^3 + 18 * q^4 + 4 * q^5 + 6 * q^6 + 2 * q^7 + 12 * q^8 + 10 * q^9 - 2 * q^10 + 14 * q^11 + 8 * q^12 - 8 * q^14 + 14 * q^16 + 4 * q^17 - 28 * q^18 + 22 * q^19 + 28 * q^20 + 12 * q^21 - 8 * q^22 + 4 * q^23 - 8 * q^24 - 2 * q^25 + 10 * q^27 + 16 * q^28 - 8 * q^29 - 20 * q^30 - 16 * q^31 + 48 * q^32 + 10 * q^33 + 8 * q^34 - 2 * q^35 + 22 * q^36 + 16 * q^37 - 6 * q^38 + 14 * q^40 + 44 * q^41 + 14 * q^42 + 16 * q^43 + 4 * q^44 + 56 * q^45 + 10 * q^47 + 32 * q^49 + 2 * q^50 - 6 * q^53 + 24 * q^54 + 22 * q^55 - 4 * q^56 - 8 * q^57 + 74 * q^58 + 2 * q^59 + 40 * q^60 + 8 * q^61 - 4 * q^62 + 56 * q^63 + 38 * q^64 - 34 * q^66 - 8 * q^67 + 32 * q^68 - 10 * q^69 - 108 * q^70 + 50 * q^71 - 44 * q^72 + 14 * q^73 + 8 * q^74 + 44 * q^76 + 16 * q^77 + 32 * q^79 + 68 * q^80 - 8 * q^81 - 6 * q^82 - 20 * q^83 + 136 * q^84 - 32 * q^85 + 8 * q^86 - 36 * q^87 - 40 * q^88 + 52 * q^89 - 34 * q^90 + 14 * q^92 + 2 * q^93 + 44 * q^94 - 2 * q^95 - 80 * q^96 + 18 * q^97 + 12 * q^98 + 38 * 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 $

$2$	0.624841	0.441829	0.220915	−	0.975293i	$-0.429096\pi$
$2$	0.624841	0.441829	0.220915	+	0.975293i	$0.429096\pi$
$3$	−2.78388	−1.60727	−0.803637	−	0.595120i	$-0.797105\pi$
$3$	−2.78388	−1.60727	−0.803637	+	0.595120i	$0.797105\pi$
$4$	−1.60957	−0.804787
$5$	0.619834	0.277198	0.138599	−	0.990349i	$-0.455740\pi$
$5$	0.619834	0.277198	0.138599	+	0.990349i	$0.455740\pi$
$6$	−1.73948	−0.710140
$7$	2.35216	0.889031	0.444516	−	0.895771i	$-0.353376\pi$
$7$	2.35216	0.889031	0.444516	+	0.895771i	$0.353376\pi$
$8$	−2.25541	−0.797408
$9$	4.74999	1.58333
$10$	0.387298	0.122474
$11$	1.21518	0.366391	0.183196	−	0.983077i	$-0.441356\pi$
$11$	1.21518	0.366391	0.183196	+	0.983077i	$0.441356\pi$
$12$	4.48086	1.29351
$13$	0	0
$13$	0	0
$14$	1.46972	0.392800
$15$	−1.72554	−0.445533
$16$	1.80988	0.452469
$17$	−4.00546	−0.971466	−0.485733	−	0.874107i	$-0.661447\pi$
$17$	−4.00546	−0.971466	−0.485733	+	0.874107i	$0.661447\pi$
$18$	2.96799	0.699561
$19$	4.95017	1.13565	0.567823	−	0.823151i	$-0.307786\pi$
$19$	4.95017	1.13565	0.567823	+	0.823151i	$0.307786\pi$
$20$	−0.997669	−0.223086
$21$	−6.54812	−1.42892
$22$	0.759295	0.161882
$23$	−2.11759	−0.441547	−0.220774	−	0.975325i	$-0.570858\pi$
$23$	−2.11759	−0.441547	−0.220774	+	0.975325i	$0.570858\pi$
$24$	6.27879	1.28165
$25$	−4.61581	−0.923161
$26$	0	0
$27$	−4.87176	−0.937570
$28$	−3.78597	−0.715481
$29$	10.4829	1.94662	0.973309	−	0.229500i	$-0.0737092\pi$
$29$	10.4829	1.94662	0.973309	+	0.229500i	$0.0737092\pi$
$30$	−1.07819	−0.196850
$31$	−1.00000	−0.179605
$31$	−1.00000	−0.179605
$32$	5.64170	0.997322
$33$	−3.38292	−0.588891
$34$	−2.50277	−0.429222
$35$	1.45795	0.246438
$36$	−7.64546	−1.27424
$37$	10.5475	1.73400	0.866999	−	0.498310i	$-0.166046\pi$
$37$	10.5475	1.73400	0.866999	+	0.498310i	$0.166046\pi$
$38$	3.09307	0.501761
$39$	0	0
$40$	−1.39798	−0.221040
$41$	1.88041	0.293670	0.146835	−	0.989161i	$-0.453091\pi$
$41$	1.88041	0.293670	0.146835	+	0.989161i	$0.453091\pi$
$42$	−4.09153	−0.631337
$43$	5.95396	0.907970	0.453985	−	0.891009i	$-0.350002\pi$
$43$	5.95396	0.907970	0.453985	+	0.891009i	$0.350002\pi$
$44$	−1.95592	−0.294867
$45$	2.94420	0.438896
$46$	−1.32315	−0.195089
$47$	−12.5015	−1.82353	−0.911765	−	0.410712i	$-0.865280\pi$
$47$	−12.5015	−1.82353	−0.911765	+	0.410712i	$0.865280\pi$
$48$	−5.03848	−0.727242
$49$	−1.46736	−0.209623
$50$	−2.88414	−0.407880
$51$	11.1507	1.56141
$52$	0	0
$53$	−2.74137	−0.376557	−0.188278	−	0.982116i	$-0.560291\pi$
$53$	−2.74137	−0.376557	−0.188278	+	0.982116i	$0.560291\pi$
$54$	−3.04407	−0.414246
$55$	0.753211	0.101563
$56$	−5.30507	−0.708920
$57$	−13.7807	−1.82529
$58$	6.55011	0.860072
$59$	7.46822	0.972280	0.486140	−	0.873881i	$-0.338405\pi$
$59$	7.46822	0.972280	0.486140	+	0.873881i	$0.338405\pi$
$60$	2.77739	0.358560
$61$	−6.44716	−0.825474	−0.412737	−	0.910850i	$-0.635427\pi$
$61$	−6.44716	−0.825474	−0.412737	+	0.910850i	$0.635427\pi$
$62$	−0.624841	−0.0793549
$63$	11.1727	1.40763
$64$	−0.0945869	−0.0118234
$65$	0	0
$66$	−2.11379	−0.260189
$67$	−13.2892	−1.62353	−0.811764	−	0.583985i	$-0.801493\pi$
$67$	−13.2892	−1.62353	−0.811764	+	0.583985i	$0.801493\pi$
$68$	6.44708	0.781823
$69$	5.89511	0.709688
$70$	0.910984	0.108883
$71$	−2.71975	−0.322775	−0.161388	−	0.986891i	$-0.551597\pi$
$71$	−2.71975	−0.322775	−0.161388	+	0.986891i	$0.551597\pi$
$72$	−10.7132	−1.26256
$73$	−7.81310	−0.914455	−0.457227	−	0.889350i	$-0.651157\pi$
$73$	−7.81310	−0.914455	−0.457227	+	0.889350i	$0.651157\pi$
$74$	6.59050	0.766131
$75$	12.8498	1.48377
$76$	−7.96766	−0.913953
$77$	2.85830	0.325733
$78$	0	0
$79$	2.11825	0.238322	0.119161	−	0.992875i	$-0.461980\pi$
$79$	2.11825	0.238322	0.119161	+	0.992875i	$0.461980\pi$
$80$	1.12182	0.125424
$81$	−0.687580	−0.0763978
$82$	1.17495	0.129752
$83$	−11.0061	−1.20808	−0.604039	−	0.796955i	$-0.706443\pi$
$83$	−11.0061	−1.20808	−0.604039	+	0.796955i	$0.706443\pi$
$84$	10.5397	1.14997
$85$	−2.48272	−0.269289
$86$	3.72027	0.401168
$87$	−29.1830	−3.12875
$88$	−2.74073	−0.292163
$89$	−2.01694	−0.213795	−0.106898	−	0.994270i	$-0.534092\pi$
$89$	−2.01694	−0.213795	−0.106898	+	0.994270i	$0.534092\pi$
$90$	1.83966	0.193917
$91$	0	0
$92$	3.40841	0.355352
$93$	2.78388	0.288675
$94$	−7.81144	−0.805689
$95$	3.06828	0.314799
$96$	−15.7058	−1.60297
$97$	−5.43723	−0.552067	−0.276034	−	0.961148i	$-0.589020\pi$
$97$	−5.43723	−0.552067	−0.276034	+	0.961148i	$0.589020\pi$
$98$	−0.916869	−0.0926178
$99$	5.77210	0.580118
$100$	7.42948	0.742948
$101$	12.4640	1.24022	0.620108	−	0.784516i	$-0.287089\pi$
$101$	12.4640	1.24022	0.620108	+	0.784516i	$0.287089\pi$
$102$	6.96742	0.689877
$103$	12.3489	1.21678	0.608388	−	0.793640i	$-0.291816\pi$
$103$	12.3489	1.21678	0.608388	+	0.793640i	$0.291816\pi$
$104$	0	0
$105$	−4.05875	−0.396093
$106$	−1.71292	−0.166374
$107$	−2.92701	−0.282965	−0.141482	−	0.989941i	$-0.545187\pi$
$107$	−2.92701	−0.282965	−0.141482	+	0.989941i	$0.545187\pi$
$108$	7.84145	0.754544
$109$	9.65896	0.925160	0.462580	−	0.886577i	$-0.346924\pi$
$109$	9.65896	0.925160	0.462580	+	0.886577i	$0.346924\pi$
$110$	0.470637	0.0448735
$111$	−29.3630	−2.78701
$112$	4.25711	0.402259
$113$	11.8792	1.11750	0.558748	−	0.829337i	$-0.311282\pi$
$113$	11.8792	1.11750	0.558748	+	0.829337i	$0.311282\pi$
$114$	−8.61072	−0.806468
$115$	−1.31255	−0.122396
$116$	−16.8729	−1.56661
$117$	0	0
$118$	4.66645	0.429582
$119$	−9.42146	−0.863664
$120$	3.89181	0.355272
$121$	−9.52333	−0.865758
$122$	−4.02845	−0.364718
$123$	−5.23482	−0.472008
$124$	1.60957	0.144544
$125$	−5.96020	−0.533097
$126$	6.98116	0.621932
$127$	8.62348	0.765210	0.382605	−	0.923912i	$-0.375027\pi$
$127$	8.62348	0.765210	0.382605	+	0.923912i	$0.375027\pi$
$128$	−11.3425	−1.00255
$129$	−16.5751	−1.45936
$130$	0	0
$131$	11.3070	0.987898	0.493949	−	0.869491i	$-0.335553\pi$
$131$	11.3070	0.987898	0.493949	+	0.869491i	$0.335553\pi$
$132$	5.44506	0.473932
$133$	11.6436	1.00962
$134$	−8.30360	−0.717322
$135$	−3.01968	−0.259893
$136$	9.03394	0.774654
$137$	−6.79159	−0.580244	−0.290122	−	0.956990i	$-0.593696\pi$
$137$	−6.79159	−0.580244	−0.290122	+	0.956990i	$0.593696\pi$
$138$	3.68350	0.313561
$139$	4.80698	0.407723	0.203862	−	0.979000i	$-0.434651\pi$
$139$	4.80698	0.407723	0.203862	+	0.979000i	$0.434651\pi$
$140$	−2.34667	−0.198330
$141$	34.8027	2.93091
$142$	−1.69941	−0.142611
$143$	0	0
$144$	8.59689	0.716408
$145$	6.49763	0.539599
$146$	−4.88195	−0.404033
$147$	4.08497	0.336922
$148$	−16.9770	−1.39550
$149$	10.3179	0.845277	0.422639	−	0.906298i	$-0.361104\pi$
$149$	10.3179	0.845277	0.422639	+	0.906298i	$0.361104\pi$
$150$	8.02911	0.655574
$151$	−7.91075	−0.643768	−0.321884	−	0.946779i	$-0.604316\pi$
$151$	−7.91075	−0.643768	−0.321884	+	0.946779i	$0.604316\pi$
$152$	−11.1646	−0.905573
$153$	−19.0259	−1.53815
$154$	1.78598	0.143918
$155$	−0.619834	−0.0497863
$156$	0	0
$157$	−4.63624	−0.370012	−0.185006	−	0.982737i	$-0.559230\pi$
$157$	−4.63624	−0.370012	−0.185006	+	0.982737i	$0.559230\pi$
$158$	1.32357	0.105298
$159$	7.63166	0.605230
$160$	3.49692	0.276456
$161$	−4.98089	−0.392549
$162$	−0.429628	−0.0337548
$163$	−7.44256	−0.582946	−0.291473	−	0.956579i	$-0.594145\pi$
$163$	−7.44256	−0.582946	−0.291473	+	0.956579i	$0.594145\pi$
$164$	−3.02665	−0.236342
$165$	−2.09685	−0.163239
$166$	−6.87707	−0.533764
$167$	6.43784	0.498175	0.249087	−	0.968481i	$-0.419869\pi$
$167$	6.43784	0.498175	0.249087	+	0.968481i	$0.419869\pi$
$168$	14.7687	1.13943
$169$	0	0
$170$	−1.55130	−0.118980
$171$	23.5132	1.79810
$172$	−9.58333	−0.730722
$173$	10.0469	0.763854	0.381927	−	0.924192i	$-0.375261\pi$
$173$	10.0469	0.763854	0.381927	+	0.924192i	$0.375261\pi$
$174$	−18.2347	−1.38237
$175$	−10.8571	−0.820719
$176$	2.19933	0.165781
$177$	−20.7906	−1.56272
$178$	−1.26027	−0.0944610
$179$	19.8830	1.48612	0.743061	−	0.669223i	$-0.233373\pi$
$179$	19.8830	1.48612	0.743061	+	0.669223i	$0.233373\pi$
$180$	−4.73891	−0.353218
$181$	−6.79278	−0.504903	−0.252452	−	0.967609i	$-0.581237\pi$
$181$	−6.79278	−0.504903	−0.252452	+	0.967609i	$0.581237\pi$
$182$	0	0
$183$	17.9481	1.32676
$184$	4.77603	0.352093
$185$	6.53770	0.480661
$186$	1.73948	0.127545
$187$	−4.86736	−0.355936
$188$	20.1221	1.46755
$189$	−11.4591	−0.833529
$190$	1.91719	0.139087
$191$	10.4860	0.758738	0.379369	−	0.925245i	$-0.376141\pi$
$191$	10.4860	0.758738	0.379369	+	0.925245i	$0.376141\pi$
$192$	0.263319	0.0190034
$193$	12.9006	0.928604	0.464302	−	0.885677i	$-0.346305\pi$
$193$	12.9006	0.928604	0.464302	+	0.885677i	$0.346305\pi$
$194$	−3.39740	−0.243919
$195$	0	0
$196$	2.36183	0.168702
$197$	20.5640	1.46513	0.732563	−	0.680699i	$-0.238324\pi$
$197$	20.5640	1.46513	0.732563	+	0.680699i	$0.238324\pi$
$198$	3.60664	0.256313
$199$	1.44890	0.102710	0.0513549	−	0.998680i	$-0.483646\pi$
$199$	1.44890	0.102710	0.0513549	+	0.998680i	$0.483646\pi$
$200$	10.4105	0.736136
$201$	36.9954	2.60945
$202$	7.78803	0.547964
$203$	24.6573	1.73060
$204$	−17.9479	−1.25660
$205$	1.16554	0.0814048
$206$	7.71612	0.537607
$207$	−10.0585	−0.699115
$208$	0	0
$209$	6.01535	0.416090
$210$	−2.53607	−0.175006
$211$	−7.11123	−0.489557	−0.244779	−	0.969579i	$-0.578715\pi$
$211$	−7.11123	−0.489557	−0.244779	+	0.969579i	$0.578715\pi$
$212$	4.41244	0.303048
$213$	7.57146	0.518788
$214$	−1.82892	−0.125022
$215$	3.69046	0.251688
$216$	10.9878	0.747625
$217$	−2.35216	−0.159675
$218$	6.03531	0.408763
$219$	21.7507	1.46978
$220$	−1.21235	−0.0817365
$221$	0	0
$222$	−18.3472	−1.23138
$223$	8.70644	0.583026	0.291513	−	0.956567i	$-0.405841\pi$
$223$	8.70644	0.583026	0.291513	+	0.956567i	$0.405841\pi$
$224$	13.2702	0.886650
$225$	−21.9250	−1.46167
$226$	7.42258	0.493743
$227$	0.260146	0.0172665	0.00863326	−	0.999963i	$-0.497252\pi$
$227$	0.260146	0.0172665	0.00863326	+	0.999963i	$0.497252\pi$
$228$	22.1810	1.46897
$229$	−1.38191	−0.0913190	−0.0456595	−	0.998957i	$-0.514539\pi$
$229$	−1.38191	−0.0913190	−0.0456595	+	0.998957i	$0.514539\pi$
$230$	−0.820136	−0.0540782
$231$	−7.95715	−0.523542
$232$	−23.6431	−1.55225
$233$	−14.6663	−0.960823	−0.480412	−	0.877043i	$-0.659513\pi$
$233$	−14.6663	−0.960823	−0.480412	+	0.877043i	$0.659513\pi$
$234$	0	0
$235$	−7.74885	−0.505479
$236$	−12.0207	−0.782478
$237$	−5.89696	−0.383048
$238$	−5.88691	−0.381592
$239$	27.1469	1.75599	0.877995	−	0.478670i	$-0.158881\pi$
$239$	27.1469	1.75599	0.877995	+	0.478670i	$0.158881\pi$
$240$	−3.12302	−0.201590
$241$	22.9384	1.47759	0.738796	−	0.673929i	$-0.235395\pi$
$241$	22.9384	1.47759	0.738796	+	0.673929i	$0.235395\pi$
$242$	−5.95057	−0.382517
$243$	16.5294	1.06036
$244$	10.3772	0.664331
$245$	−0.909522	−0.0581072
$246$	−3.27093	−0.208547
$247$	0	0
$248$	2.25541	0.143219
$249$	30.6397	1.94171
$250$	−3.72418	−0.235538
$251$	5.40311	0.341041	0.170521	−	0.985354i	$-0.445455\pi$
$251$	5.40311	0.341041	0.170521	+	0.985354i	$0.445455\pi$
$252$	−17.9833	−1.13284
$253$	−2.57325	−0.161779
$254$	5.38830	0.338092
$255$	6.91159	0.432821
$256$	−6.89809	−0.431130
$257$	21.4190	1.33608	0.668042	−	0.744124i	$-0.267133\pi$
$257$	21.4190	1.33608	0.668042	+	0.744124i	$0.267133\pi$
$258$	−10.3568	−0.644786
$259$	24.8093	1.54158
$260$	0	0
$261$	49.7934	3.08214
$262$	7.06508	0.436482
$263$	6.53589	0.403020	0.201510	−	0.979486i	$-0.435415\pi$
$263$	6.53589	0.403020	0.201510	+	0.979486i	$0.435415\pi$
$264$	7.62987	0.469586
$265$	−1.69920	−0.104381
$266$	7.27537	0.446082
$267$	5.61492	0.343628
$268$	21.3899	1.30659
$269$	−18.8323	−1.14822	−0.574112	−	0.818777i	$-0.694653\pi$
$269$	−18.8323	−1.14822	−0.574112	+	0.818777i	$0.694653\pi$
$270$	−1.88682	−0.114828
$271$	18.1725	1.10390	0.551950	−	0.833877i	$-0.313884\pi$
$271$	18.1725	1.10390	0.551950	+	0.833877i	$0.313884\pi$
$272$	−7.24938	−0.439558
$273$	0	0
$274$	−4.24366	−0.256369
$275$	−5.60904	−0.338238
$276$	−9.48861	−0.571147
$277$	31.5861	1.89782	0.948912	−	0.315541i	$-0.102186\pi$
$277$	31.5861	1.89782	0.948912	+	0.315541i	$0.102186\pi$
$278$	3.00360	0.180144
$279$	−4.74999	−0.284374
$280$	−3.28826	−0.196511
$281$	27.0399	1.61306	0.806531	−	0.591191i	$-0.201342\pi$
$281$	27.0399	1.61306	0.806531	+	0.591191i	$0.201342\pi$
$282$	21.7461	1.29496
$283$	−32.3052	−1.92035	−0.960173	−	0.279405i	$-0.909863\pi$
$283$	−32.3052	−1.92035	−0.960173	+	0.279405i	$0.909863\pi$
$284$	4.37764	0.259765
$285$	−8.54173	−0.505968
$286$	0	0
$287$	4.42301	0.261082
$288$	26.7980	1.57909
$289$	−0.956311	−0.0562536
$290$	4.05998	0.238410
$291$	15.1366	0.887323
$292$	12.5758	0.735941
$293$	−18.6860	−1.09165	−0.545825	−	0.837899i	$-0.683784\pi$
$293$	−18.6860	−1.09165	−0.545825	+	0.837899i	$0.683784\pi$
$294$	2.55245	0.148862
$295$	4.62906	0.269514
$296$	−23.7889	−1.38270
$297$	−5.92007	−0.343517
$298$	6.44706	0.373468
$299$	0	0
$300$	−20.6828	−1.19412
$301$	14.0046	0.807213
$302$	−4.94296	−0.284435
$303$	−34.6983	−1.99337
$304$	8.95919	0.513845
$305$	−3.99617	−0.228820
$306$	−11.8881	−0.679600
$307$	−21.9423	−1.25231	−0.626156	−	0.779698i	$-0.715373\pi$
$307$	−21.9423	−1.25231	−0.626156	+	0.779698i	$0.715373\pi$
$308$	−4.60064	−0.262146
$309$	−34.3779	−1.95569
$310$	−0.387298	−0.0219970
$311$	17.9051	1.01531	0.507653	−	0.861562i	$-0.330513\pi$
$311$	17.9051	1.01531	0.507653	+	0.861562i	$0.330513\pi$
$312$	0	0
$313$	−4.73057	−0.267388	−0.133694	−	0.991023i	$-0.542684\pi$
$313$	−4.73057	−0.267388	−0.133694	+	0.991023i	$0.542684\pi$
$314$	−2.89691	−0.163482
$315$	6.92523	0.390192
$316$	−3.40948	−0.191798
$317$	19.0145	1.06796	0.533981	−	0.845496i	$-0.320695\pi$
$317$	19.0145	1.06796	0.533981	+	0.845496i	$0.320695\pi$
$318$	4.76857	0.267408
$319$	12.7386	0.713223
$320$	−0.0586282	−0.00327742
$321$	8.14845	0.454802
$322$	−3.11227	−0.173440
$323$	−19.8277	−1.10324
$324$	1.10671	0.0614840
$325$	0	0
$326$	−4.65042	−0.257563
$327$	−26.8894	−1.48699
$328$	−4.24108	−0.234175
$329$	−29.4055	−1.62118
$330$	−1.31020	−0.0721239
$331$	24.5020	1.34675	0.673375	−	0.739301i	$-0.264844\pi$
$331$	24.5020	1.34675	0.673375	+	0.739301i	$0.264844\pi$
$332$	17.7152	0.972245
$333$	50.1005	2.74549
$334$	4.02262	0.220108
$335$	−8.23707	−0.450039
$336$	−11.8513	−0.646541
$337$	−6.91654	−0.376768	−0.188384	−	0.982095i	$-0.560325\pi$
$337$	−6.91654	−0.376768	−0.188384	+	0.982095i	$0.560325\pi$
$338$	0	0
$339$	−33.0701	−1.79612
$340$	3.99612	0.216720
$341$	−1.21518	−0.0658058
$342$	14.6920	0.794454
$343$	−19.9166	−1.07539
$344$	−13.4286	−0.724022
$345$	3.65399	0.196724
$346$	6.27773	0.337493
$347$	−17.8295	−0.957137	−0.478569	−	0.878050i	$-0.658844\pi$
$347$	−17.8295	−0.957137	−0.478569	+	0.878050i	$0.658844\pi$
$348$	46.9722	2.51797
$349$	7.98868	0.427624	0.213812	−	0.976875i	$-0.431412\pi$
$349$	7.98868	0.427624	0.213812	+	0.976875i	$0.431412\pi$
$350$	−6.78395	−0.362618
$351$	0	0
$352$	6.85569	0.365410
$353$	4.94127	0.262997	0.131499	−	0.991316i	$-0.458021\pi$
$353$	4.94127	0.262997	0.131499	+	0.991316i	$0.458021\pi$
$354$	−12.9908	−0.690455
$355$	−1.68579	−0.0894727
$356$	3.24642	0.172060
$357$	26.2282	1.38814
$358$	12.4237	0.656612
$359$	9.47701	0.500178	0.250089	−	0.968223i	$-0.419540\pi$
$359$	9.47701	0.500178	0.250089	+	0.968223i	$0.419540\pi$
$360$	−6.64038	−0.349979
$361$	5.50414	0.289691
$362$	−4.24441	−0.223081
$363$	26.5118	1.39151
$364$	0	0
$365$	−4.84283	−0.253485
$366$	11.2147	0.586202
$367$	−11.4595	−0.598183	−0.299091	−	0.954225i	$-0.596684\pi$
$367$	−11.4595	−0.598183	−0.299091	+	0.954225i	$0.596684\pi$
$368$	−3.83257	−0.199787
$369$	8.93190	0.464976
$370$	4.08502	0.212370
$371$	−6.44814	−0.334771
$372$	−4.48086	−0.232322
$373$	34.3146	1.77674	0.888372	−	0.459124i	$-0.151836\pi$
$373$	34.3146	1.77674	0.888372	+	0.459124i	$0.151836\pi$
$374$	−3.04132	−0.157263
$375$	16.5925	0.856833
$376$	28.1960	1.45410
$377$	0	0
$378$	−7.16013	−0.368277
$379$	−29.0334	−1.49135	−0.745674	−	0.666311i	$-0.767872\pi$
$379$	−29.0334	−1.49135	−0.745674	+	0.666311i	$0.767872\pi$
$380$	−4.93863	−0.253346
$381$	−24.0067	−1.22990
$382$	6.55206	0.335233
$383$	19.8755	1.01559	0.507794	−	0.861478i	$-0.330461\pi$
$383$	19.8755	1.01559	0.507794	+	0.861478i	$0.330461\pi$
$384$	31.5762	1.61137
$385$	1.77167	0.0902926
$386$	8.06081	0.410284
$387$	28.2812	1.43762
$388$	8.75163	0.444297
$389$	25.2071	1.27805	0.639024	−	0.769187i	$-0.279338\pi$
$389$	25.2071	1.27805	0.639024	+	0.769187i	$0.279338\pi$
$390$	0	0
$391$	8.48190	0.428948
$392$	3.30951	0.167155
$393$	−31.4774	−1.58782
$394$	12.8492	0.647335
$395$	1.31296	0.0660624
$396$	−9.29062	−0.466871
$397$	−7.74047	−0.388483	−0.194241	−	0.980954i	$-0.562225\pi$
$397$	−7.74047	−0.388483	−0.194241	+	0.980954i	$0.562225\pi$
$398$	0.905332	0.0453802
$399$	−32.4143	−1.62274
$400$	−8.35404	−0.417702
$401$	−7.36568	−0.367825	−0.183912	−	0.982943i	$-0.558876\pi$
$401$	−7.36568	−0.367825	−0.183912	+	0.982943i	$0.558876\pi$
$402$	23.1162	1.15293
$403$	0	0
$404$	−20.0618	−0.998110
$405$	−0.426186	−0.0211773
$406$	15.4069	0.764631
$407$	12.8171	0.635321
$408$	−25.1494	−1.24508
$409$	−8.29760	−0.410290	−0.205145	−	0.978732i	$-0.565767\pi$
$409$	−8.29760	−0.410290	−0.205145	+	0.978732i	$0.565767\pi$
$410$	0.728277	0.0359670
$411$	18.9070	0.932612
$412$	−19.8765	−0.979246
$413$	17.5664	0.864387
$414$	−6.28497	−0.308889
$415$	−6.82196	−0.334877
$416$	0	0
$417$	−13.3821	−0.655323
$418$	3.75864	0.183841
$419$	6.23536	0.304617	0.152309	−	0.988333i	$-0.451329\pi$
$419$	6.23536	0.304617	0.152309	+	0.988333i	$0.451329\pi$
$420$	6.53285	0.318771
$421$	10.9675	0.534525	0.267262	−	0.963624i	$-0.413881\pi$
$421$	10.9675	0.534525	0.267262	+	0.963624i	$0.413881\pi$
$422$	−4.44339	−0.216301
$423$	−59.3819	−2.88725
$424$	6.18292	0.300269
$425$	18.4884	0.896820
$426$	4.73096	0.229216
$427$	−15.1647	−0.733872
$428$	4.71124	0.227726
$429$	0	0
$430$	2.30595	0.111203
$431$	7.82125	0.376736	0.188368	−	0.982098i	$-0.439680\pi$
$431$	7.82125	0.376736	0.188368	+	0.982098i	$0.439680\pi$
$432$	−8.81728	−0.424221
$433$	33.6131	1.61534	0.807671	−	0.589633i	$-0.200728\pi$
$433$	33.6131	1.61534	0.807671	+	0.589633i	$0.200728\pi$
$434$	−1.46972	−0.0705489
$435$	−18.0886	−0.867283
$436$	−15.5468	−0.744557
$437$	−10.4824	−0.501441
$438$	13.5908	0.649391
$439$	−17.3442	−0.827793	−0.413896	−	0.910324i	$-0.635832\pi$
$439$	−17.3442	−0.827793	−0.413896	+	0.910324i	$0.635832\pi$
$440$	−1.69880	−0.0809870
$441$	−6.96996	−0.331903
$442$	0	0
$443$	−10.2049	−0.484849	−0.242425	−	0.970170i	$-0.577943\pi$
$443$	−10.2049	−0.484849	−0.242425	+	0.970170i	$0.577943\pi$
$444$	47.2619	2.24295
$445$	−1.25017	−0.0592637
$446$	5.44014	0.257598
$447$	−28.7239	−1.35859
$448$	−0.222483	−0.0105113
$449$	3.32509	0.156921	0.0784604	−	0.996917i	$-0.475000\pi$
$449$	3.32509	0.156921	0.0784604	+	0.996917i	$0.475000\pi$
$450$	−13.6996	−0.645808
$451$	2.28503	0.107598
$452$	−19.1204	−0.899347
$453$	22.0226	1.03471
$454$	0.162550	0.00762885
$455$	0	0
$456$	31.0810	1.45550
$457$	29.0747	1.36006	0.680029	−	0.733185i	$-0.261967\pi$
$457$	29.0747	1.36006	0.680029	+	0.733185i	$0.261967\pi$
$458$	−0.863472	−0.0403474
$459$	19.5136	0.910817
$460$	2.11265	0.0985028
$461$	−28.7557	−1.33929	−0.669643	−	0.742684i	$-0.733553\pi$
$461$	−28.7557	−1.33929	−0.669643	+	0.742684i	$0.733553\pi$
$462$	−4.97195	−0.231316
$463$	−34.6020	−1.60809	−0.804045	−	0.594568i	$-0.797323\pi$
$463$	−34.6020	−1.60809	−0.804045	+	0.594568i	$0.797323\pi$
$464$	18.9727	0.880784
$465$	1.72554	0.0800202
$466$	−9.16412	−0.424520
$467$	−33.1318	−1.53316	−0.766578	−	0.642152i	$-0.778042\pi$
$467$	−33.1318	−1.53316	−0.766578	+	0.642152i	$0.778042\pi$
$468$	0	0
$469$	−31.2582	−1.44337
$470$	−4.84180	−0.223335
$471$	12.9067	0.594711
$472$	−16.8439	−0.775303
$473$	7.23514	0.332672
$474$	−3.68466	−0.169242
$475$	−22.8490	−1.04838
$476$	15.1645	0.695065
$477$	−13.0215	−0.596213
$478$	16.9625	0.775847
$479$	−28.5444	−1.30423	−0.652113	−	0.758122i	$-0.726117\pi$
$479$	−28.5444	−1.30423	−0.652113	+	0.758122i	$0.726117\pi$
$480$	−9.73500	−0.444340
$481$	0	0
$482$	14.3328	0.652843
$483$	13.8662	0.630934
$484$	15.3285	0.696750
$485$	−3.37018	−0.153032
$486$	10.3282	0.468499
$487$	−22.2944	−1.01026	−0.505129	−	0.863044i	$-0.668555\pi$
$487$	−22.2944	−1.01026	−0.505129	+	0.863044i	$0.668555\pi$
$488$	14.5410	0.658239
$489$	20.7192	0.936955
$490$	−0.568307	−0.0256735
$491$	2.28882	0.103293	0.0516465	−	0.998665i	$-0.483553\pi$
$491$	2.28882	0.103293	0.0516465	+	0.998665i	$0.483553\pi$
$492$	8.42584	0.379866
$493$	−41.9886	−1.89107
$494$	0	0
$495$	3.57774	0.160808
$496$	−1.80988	−0.0812659
$497$	−6.39728	−0.286957
$498$	19.1449	0.857905
$499$	10.7617	0.481758	0.240879	−	0.970555i	$-0.422564\pi$
$499$	10.7617	0.481758	0.240879	+	0.970555i	$0.422564\pi$
$500$	9.59339	0.429029
$501$	−17.9222	−0.800703
$502$	3.37608	0.150682
$503$	15.7196	0.700901	0.350451	−	0.936581i	$-0.386028\pi$
$503$	15.7196	0.700901	0.350451	+	0.936581i	$0.386028\pi$
$504$	−25.1990	−1.12245
$505$	7.72562	0.343786
$506$	−1.60787	−0.0714787
$507$	0	0
$508$	−13.8801	−0.615831
$509$	−9.05022	−0.401144	−0.200572	−	0.979679i	$-0.564280\pi$
$509$	−9.05022	−0.401144	−0.200572	+	0.979679i	$0.564280\pi$
$510$	4.31864	0.191233
$511$	−18.3776	−0.812979
$512$	18.3748	0.812059
$513$	−24.1160	−1.06475
$514$	13.3835	0.590321
$515$	7.65429	0.337288
$516$	26.6788	1.17447
$517$	−15.1916	−0.668125
$518$	15.5019	0.681114
$519$	−27.9694	−1.22772
$520$	0	0
$521$	−14.5767	−0.638615	−0.319308	−	0.947651i	$-0.603450\pi$
$521$	−14.5767	−0.638615	−0.319308	+	0.947651i	$0.603450\pi$
$522$	31.1130	1.36178
$523$	21.1296	0.923935	0.461967	−	0.886897i	$-0.347144\pi$
$523$	21.1296	0.923935	0.461967	+	0.886897i	$0.347144\pi$
$524$	−18.1995	−0.795048
$525$	30.2248	1.31912
$526$	4.08389	0.178066
$527$	4.00546	0.174480
$528$	−6.12267	−0.266455
$529$	−18.5158	−0.805036
$530$	−1.06173	−0.0461185
$531$	35.4740	1.53944
$532$	−18.7412	−0.812533
$533$	0	0
$534$	3.50843	0.151825
$535$	−1.81426	−0.0784373
$536$	29.9725	1.29461
$537$	−55.3518	−2.38861
$538$	−11.7672	−0.507319
$539$	−1.78311	−0.0768042
$540$	4.86040	0.209158
$541$	16.1842	0.695811	0.347906	−	0.937530i	$-0.386893\pi$
$541$	16.1842	0.695811	0.347906	+	0.937530i	$0.386893\pi$
$542$	11.3549	0.487736
$543$	18.9103	0.811518
$544$	−22.5976	−0.968864
$545$	5.98695	0.256453
$546$	0	0
$547$	−21.4532	−0.917274	−0.458637	−	0.888624i	$-0.651662\pi$
$547$	−21.4532	−0.917274	−0.458637	+	0.888624i	$0.651662\pi$
$548$	10.9316	0.466973
$549$	−30.6239	−1.30700
$550$	−3.50476	−0.149443
$551$	51.8919	2.21067
$552$	−13.2959	−0.565910
$553$	4.98246	0.211876
$554$	19.7363	0.838514
$555$	−18.2002	−0.772554
$556$	−7.73720	−0.328130
$557$	24.1842	1.02472	0.512359	−	0.858772i	$-0.328772\pi$
$557$	24.1842	1.02472	0.512359	+	0.858772i	$0.328772\pi$
$558$	−2.96799	−0.125645
$559$	0	0
$560$	2.63870	0.111506
$561$	13.5501	0.572087
$562$	16.8956	0.712698
$563$	29.9506	1.26227	0.631133	−	0.775675i	$-0.282590\pi$
$563$	29.9506	1.26227	0.631133	+	0.775675i	$0.282590\pi$
$564$	−56.0174	−2.35876
$565$	7.36311	0.309768
$566$	−20.1856	−0.848465
$567$	−1.61730	−0.0679200
$568$	6.13415	0.257383
$569$	−7.78865	−0.326517	−0.163259	−	0.986583i	$-0.552200\pi$
$569$	−7.78865	−0.326517	−0.163259	+	0.986583i	$0.552200\pi$
$570$	−5.33722	−0.223551
$571$	−20.2796	−0.848676	−0.424338	−	0.905504i	$-0.639493\pi$
$571$	−20.2796	−0.848676	−0.424338	+	0.905504i	$0.639493\pi$
$572$	0	0
$573$	−29.1917	−1.21950
$574$	2.76367	0.115354
$575$	9.77437	0.407619
$576$	−0.449287	−0.0187203
$577$	16.5988	0.691016	0.345508	−	0.938416i	$-0.387707\pi$
$577$	16.5988	0.691016	0.345508	+	0.938416i	$0.387707\pi$
$578$	−0.597542	−0.0248545
$579$	−35.9137	−1.49252
$580$	−10.4584	−0.434262
$581$	−25.8881	−1.07402
$582$	9.45797	0.392045
$583$	−3.33127	−0.137967
$584$	17.6217	0.729193
$585$	0	0
$586$	−11.6758	−0.482323
$587$	−12.9010	−0.532482	−0.266241	−	0.963907i	$-0.585782\pi$
$587$	−12.9010	−0.532482	−0.266241	+	0.963907i	$0.585782\pi$
$588$	−6.57505	−0.271151
$589$	−4.95017	−0.203968
$590$	2.89243	0.119079
$591$	−57.2478	−2.35486
$592$	19.0897	0.784580
$593$	44.5857	1.83091	0.915457	−	0.402415i	$-0.131829\pi$
$593$	44.5857	1.83091	0.915457	+	0.402415i	$0.131829\pi$
$594$	−3.69910	−0.151776
$595$	−5.83974	−0.239406
$596$	−16.6075	−0.680268
$597$	−4.03356	−0.165083
$598$	0	0
$599$	−7.39164	−0.302014	−0.151007	−	0.988533i	$-0.548252\pi$
$599$	−7.39164	−0.302014	−0.151007	+	0.988533i	$0.548252\pi$
$600$	−28.9817	−1.18317
$601$	20.6091	0.840662	0.420331	−	0.907371i	$-0.361914\pi$
$601$	20.6091	0.840662	0.420331	+	0.907371i	$0.361914\pi$
$602$	8.75066	0.356650
$603$	−63.1233	−2.57058
$604$	12.7329	0.518096
$605$	−5.90289	−0.239986
$606$	−21.6809	−0.880728
$607$	32.5879	1.32270	0.661350	−	0.750077i	$-0.269984\pi$
$607$	32.5879	1.32270	0.661350	+	0.750077i	$0.269984\pi$
$608$	27.9274	1.13260
$609$	−68.6430	−2.78155
$610$	−2.49697	−0.101099
$611$	0	0
$612$	30.6236	1.23788
$613$	−2.62416	−0.105989	−0.0529943	−	0.998595i	$-0.516877\pi$
$613$	−2.62416	−0.105989	−0.0529943	+	0.998595i	$0.516877\pi$
$614$	−13.7104	−0.553308
$615$	−3.24472	−0.130840
$616$	−6.44663	−0.259742
$617$	−2.69130	−0.108348	−0.0541738	−	0.998532i	$-0.517252\pi$
$617$	−2.69130	−0.108348	−0.0541738	+	0.998532i	$0.517252\pi$
$618$	−21.4807	−0.864082
$619$	−27.2021	−1.09334	−0.546672	−	0.837347i	$-0.684105\pi$
$619$	−27.2021	−1.09334	−0.546672	+	0.837347i	$0.684105\pi$
$620$	0.997669	0.0400673
$621$	10.3164	0.413982
$622$	11.1878	0.448592
$623$	−4.74416	−0.190071
$624$	0	0
$625$	19.3847	0.775388
$626$	−2.95585	−0.118140
$627$	−16.7460	−0.668771
$628$	7.46237	0.297781
$629$	−42.2475	−1.68452
$630$	4.32716	0.172398
$631$	26.9529	1.07298	0.536490	−	0.843907i	$-0.319750\pi$
$631$	26.9529	1.07298	0.536490	+	0.843907i	$0.319750\pi$
$632$	−4.77752	−0.190040
$633$	19.7968	0.786853
$634$	11.8811	0.471857
$635$	5.34512	0.212115
$636$	−12.2837	−0.487081
$637$	0	0
$638$	7.95958	0.315123
$639$	−12.9188	−0.511059
$640$	−7.03047	−0.277904
$641$	11.3030	0.446441	0.223220	−	0.974768i	$-0.428343\pi$
$641$	11.3030	0.446441	0.223220	+	0.974768i	$0.428343\pi$
$642$	5.09148	0.200945
$643$	13.9895	0.551693	0.275847	−	0.961202i	$-0.411042\pi$
$643$	13.9895	0.551693	0.275847	+	0.961202i	$0.411042\pi$
$644$	8.01712	0.315919
$645$	−10.2738	−0.404531
$646$	−12.3891	−0.487444
$647$	22.5333	0.885877	0.442938	−	0.896552i	$-0.353936\pi$
$647$	22.5333	0.885877	0.442938	+	0.896552i	$0.353936\pi$
$648$	1.55077	0.0609202
$649$	9.07525	0.356235
$650$	0	0
$651$	6.54812	0.256641
$652$	11.9794	0.469148
$653$	−10.7751	−0.421663	−0.210831	−	0.977522i	$-0.567617\pi$
$653$	−10.7751	−0.421663	−0.210831	+	0.977522i	$0.567617\pi$
$654$	−16.8016	−0.656994
$655$	7.00847	0.273844
$656$	3.40330	0.132877
$657$	−37.1121	−1.44788
$658$	−18.3737	−0.716282
$659$	−39.5160	−1.53932	−0.769662	−	0.638452i	$-0.779575\pi$
$659$	−39.5160	−1.53932	−0.769662	+	0.638452i	$0.779575\pi$
$660$	3.37503	0.131373
$661$	40.9839	1.59409	0.797044	−	0.603921i	$-0.206396\pi$
$661$	40.9839	1.59409	0.797044	+	0.603921i	$0.206396\pi$
$662$	15.3098	0.595033
$663$	0	0
$664$	24.8233	0.963330
$665$	7.21707	0.279866
$666$	31.3048	1.21304
$667$	−22.1984	−0.859524
$668$	−10.3622	−0.400925
$669$	−24.2377	−0.937083
$670$	−5.14686	−0.198840
$671$	−7.83446	−0.302446
$672$	−36.9425	−1.42509
$673$	16.4585	0.634430	0.317215	−	0.948354i	$-0.397252\pi$
$673$	16.4585	0.634430	0.317215	+	0.948354i	$0.397252\pi$
$674$	−4.32174	−0.166467
$675$	22.4871	0.865528
$676$	0	0
$677$	31.8404	1.22372	0.611862	−	0.790964i	$-0.290421\pi$
$677$	31.8404	1.22372	0.611862	+	0.790964i	$0.290421\pi$
$678$	−20.6636	−0.793580
$679$	−12.7892	−0.490805
$680$	5.59955	0.214733
$681$	−0.724216	−0.0277520
$682$	−0.759295	−0.0290749
$683$	−3.58966	−0.137354	−0.0686772	−	0.997639i	$-0.521878\pi$
$683$	−3.58966	−0.137354	−0.0686772	+	0.997639i	$0.521878\pi$
$684$	−37.8463	−1.44709
$685$	−4.20966	−0.160843
$686$	−12.4447	−0.475140
$687$	3.84706	0.146775
$688$	10.7759	0.410828
$689$	0	0
$690$	2.28316	0.0869185
$691$	−30.1287	−1.14615	−0.573075	−	0.819503i	$-0.694250\pi$
$691$	−30.1287	−1.14615	−0.573075	+	0.819503i	$0.694250\pi$
$692$	−16.1713	−0.614740
$693$	13.5769	0.515743
$694$	−11.1406	−0.422891
$695$	2.97953	0.113020
$696$	65.8196	2.49489
$697$	−7.53188	−0.285290
$698$	4.99165	0.188937
$699$	40.8293	1.54431
$700$	17.4753	0.660504
$701$	11.0470	0.417238	0.208619	−	0.977997i	$-0.433103\pi$
$701$	11.0470	0.417238	0.208619	+	0.977997i	$0.433103\pi$
$702$	0	0
$703$	52.2118	1.96921
$704$	−0.114940	−0.00433197
$705$	21.5719	0.812444
$706$	3.08751	0.116200
$707$	29.3173	1.10259
$708$	33.4641	1.25766
$709$	41.5907	1.56197	0.780986	−	0.624549i	$-0.214717\pi$
$709$	41.5907	1.56197	0.780986	+	0.624549i	$0.214717\pi$
$710$	−1.05335	−0.0395316
$711$	10.0617	0.377342
$712$	4.54903	0.170482
$713$	2.11759	0.0793043
$714$	16.3885	0.613323
$715$	0	0
$716$	−32.0031	−1.19601
$717$	−75.5738	−2.82236
$718$	5.92163	0.220993
$719$	38.3515	1.43027	0.715134	−	0.698987i	$-0.246365\pi$
$719$	38.3515	1.43027	0.715134	+	0.698987i	$0.246365\pi$
$720$	5.32865	0.198587
$721$	29.0466	1.08175
$722$	3.43921	0.127994
$723$	−63.8578	−2.37490
$724$	10.9335	0.406340
$725$	−48.3868	−1.79704
$726$	16.5657	0.614810
$727$	−49.9648	−1.85309	−0.926546	−	0.376182i	$-0.877237\pi$
$727$	−49.9648	−1.85309	−0.926546	+	0.376182i	$0.877237\pi$
$728$	0	0
$729$	−43.9531	−1.62789
$730$	−3.02600	−0.111997
$731$	−23.8483	−0.882062
$732$	−28.8888	−1.06776
$733$	32.9576	1.21732	0.608658	−	0.793433i	$-0.291708\pi$
$733$	32.9576	1.21732	0.608658	+	0.793433i	$0.291708\pi$
$734$	−7.16038	−0.264294
$735$	2.53200	0.0933943
$736$	−11.9468	−0.440365
$737$	−16.1487	−0.594846
$738$	5.58102	0.205440
$739$	19.5295	0.718406	0.359203	−	0.933259i	$-0.383049\pi$
$739$	19.5295	0.718406	0.359203	+	0.933259i	$0.383049\pi$
$740$	−10.5229	−0.386830
$741$	0	0
$742$	−4.02906	−0.147911
$743$	19.0224	0.697864	0.348932	−	0.937148i	$-0.386544\pi$
$743$	19.0224	0.697864	0.348932	+	0.937148i	$0.386544\pi$
$744$	−6.27879	−0.230192
$745$	6.39540	0.234309
$746$	21.4412	0.785018
$747$	−52.2789	−1.91278
$748$	7.83437	0.286453
$749$	−6.88478	−0.251565
$750$	10.3677	0.378574
$751$	26.8083	0.978248	0.489124	−	0.872214i	$-0.337317\pi$
$751$	26.8083	0.978248	0.489124	+	0.872214i	$0.337317\pi$
$752$	−22.6262	−0.825091
$753$	−15.0416	−0.548146
$754$	0	0
$755$	−4.90335	−0.178451
$756$	18.4443	0.670813
$757$	−17.2202	−0.625880	−0.312940	−	0.949773i	$-0.601314\pi$
$757$	−17.2202	−0.625880	−0.312940	+	0.949773i	$0.601314\pi$
$758$	−18.1413	−0.658921
$759$	7.16363	0.260023
$760$	−6.92023	−0.251023
$761$	−43.3306	−1.57073	−0.785366	−	0.619032i	$-0.787525\pi$
$761$	−43.3306	−1.57073	−0.785366	+	0.619032i	$0.787525\pi$
$762$	−15.0004	−0.543406
$763$	22.7194	0.822496
$764$	−16.8779	−0.610622
$765$	−11.7929	−0.426373
$766$	12.4190	0.448716
$767$	0	0
$768$	19.2034	0.692945
$769$	−29.5903	−1.06705	−0.533527	−	0.845783i	$-0.679134\pi$
$769$	−29.5903	−1.06705	−0.533527	+	0.845783i	$0.679134\pi$
$770$	1.10701	0.0398939
$771$	−59.6280	−2.14745
$772$	−20.7644	−0.747328
$773$	47.4918	1.70816	0.854080	−	0.520142i	$-0.174121\pi$
$773$	47.4918	1.70816	0.854080	+	0.520142i	$0.174121\pi$
$774$	17.6713	0.635180
$775$	4.61581	0.165805
$776$	12.2632	0.440223
$777$	−69.0662	−2.47774
$778$	15.7504	0.564679
$779$	9.30832	0.333505
$780$	0	0
$781$	−3.30499	−0.118262
$782$	5.29984	0.189522
$783$	−51.0699	−1.82509
$784$	−2.65575	−0.0948481
$785$	−2.87370	−0.102567
$786$	−19.6683	−0.701546
$787$	37.6005	1.34031	0.670157	−	0.742220i	$-0.266227\pi$
$787$	37.6005	1.34031	0.670157	+	0.742220i	$0.266227\pi$
$788$	−33.0993	−1.17911
$789$	−18.1951	−0.647764
$790$	0.820393	0.0291883
$791$	27.9416	0.993490
$792$	−13.0184	−0.462590
$793$	0	0
$794$	−4.83656	−0.171643
$795$	4.73036	0.167769
$796$	−2.33211	−0.0826595
$797$	22.2917	0.789611	0.394806	−	0.918765i	$-0.370812\pi$
$797$	22.2917	0.789611	0.394806	+	0.918765i	$0.370812\pi$
$798$	−20.2538	−0.716975
$799$	50.0742	1.77150
$800$	−26.0410	−0.920689
$801$	−9.58045	−0.338508
$802$	−4.60238	−0.162516
$803$	−9.49434	−0.335048
$804$	−59.5468	−2.10006
$805$	−3.08733	−0.108814
$806$	0	0
$807$	52.4268	1.84551
$808$	−28.1115	−0.988958
$809$	16.2551	0.571498	0.285749	−	0.958305i	$-0.407758\pi$
$809$	16.2551	0.571498	0.285749	+	0.958305i	$0.407758\pi$
$810$	−0.266298	−0.00935676
$811$	28.5796	1.00356	0.501782	−	0.864994i	$-0.332678\pi$
$811$	28.5796	1.00356	0.501782	+	0.864994i	$0.332678\pi$
$812$	−39.6878	−1.39277
$813$	−50.5901	−1.77427
$814$	8.00866	0.280703
$815$	−4.61315	−0.161592
$816$	20.1814	0.706491
$817$	29.4731	1.03113
$818$	−5.18468	−0.181278
$819$	0	0
$820$	−1.87602	−0.0655135
$821$	−15.3714	−0.536464	−0.268232	−	0.963354i	$-0.586439\pi$
$821$	−15.3714	−0.536464	−0.268232	+	0.963354i	$0.586439\pi$
$822$	11.8138	0.412055
$823$	−47.1895	−1.64492	−0.822462	−	0.568820i	$-0.807400\pi$
$823$	−47.1895	−1.64492	−0.822462	+	0.568820i	$0.807400\pi$
$824$	−27.8519	−0.970267
$825$	15.6149	0.543641
$826$	10.9762	0.381912
$827$	−10.7778	−0.374781	−0.187390	−	0.982286i	$-0.560003\pi$
$827$	−10.7778	−0.374781	−0.187390	+	0.982286i	$0.560003\pi$
$828$	16.1899	0.562639
$829$	30.4045	1.05599	0.527997	−	0.849246i	$-0.322943\pi$
$829$	30.4045	1.05599	0.527997	+	0.849246i	$0.322943\pi$
$830$	−4.26264	−0.147958
$831$	−87.9319	−3.05032
$832$	0	0
$833$	5.87746	0.203642
$834$	−8.36166	−0.289541
$835$	3.99039	0.138093
$836$	−9.68215	−0.334864
$837$	4.87176	0.168393
$838$	3.89611	0.134589
$839$	41.7161	1.44020	0.720099	−	0.693871i	$-0.244096\pi$
$839$	41.7161	1.44020	0.720099	+	0.693871i	$0.244096\pi$
$840$	9.15413	0.315848
$841$	80.8902	2.78932
$842$	6.85296	0.236169
$843$	−75.2757	−2.59263
$844$	11.4461	0.393989
$845$	0	0
$846$	−37.1043	−1.27567
$847$	−22.4004	−0.769686
$848$	−4.96155	−0.170380
$849$	89.9339	3.08652
$850$	11.5523	0.396241
$851$	−22.3352	−0.765642
$852$	−12.1868	−0.417514
$853$	−36.5309	−1.25080	−0.625398	−	0.780306i	$-0.715063\pi$
$853$	−36.5309	−1.25080	−0.625398	+	0.780306i	$0.715063\pi$
$854$	−9.47553	−0.324246
$855$	14.5743	0.498430
$856$	6.60161	0.225638
$857$	37.5045	1.28113	0.640565	−	0.767904i	$-0.278700\pi$
$857$	37.5045	1.28113	0.640565	+	0.767904i	$0.278700\pi$
$858$	0	0
$859$	56.8532	1.93981	0.969903	−	0.243492i	$-0.0782931\pi$
$859$	56.8532	1.93981	0.969903	+	0.243492i	$0.0782931\pi$
$860$	−5.94008	−0.202555
$861$	−12.3131	−0.419630
$862$	4.88704	0.166453
$863$	15.5581	0.529602	0.264801	−	0.964303i	$-0.414694\pi$
$863$	15.5581	0.529602	0.264801	+	0.964303i	$0.414694\pi$
$864$	−27.4850	−0.935059
$865$	6.22743	0.211739
$866$	21.0028	0.713705
$867$	2.66226	0.0904150
$868$	3.78597	0.128504
$869$	2.57406	0.0873190
$870$	−11.3025	−0.383191
$871$	0	0
$872$	−21.7849	−0.737730
$873$	−25.8268	−0.874104
$874$	−6.54983	−0.221551
$875$	−14.0193	−0.473940
$876$	−35.0094	−1.18286
$877$	−3.07559	−0.103855	−0.0519277	−	0.998651i	$-0.516537\pi$
$877$	−3.07559	−0.103855	−0.0519277	+	0.998651i	$0.516537\pi$
$878$	−10.8374	−0.365743
$879$	52.0197	1.75458
$880$	1.36322	0.0459541
$881$	33.1770	1.11776	0.558880	−	0.829248i	$-0.311231\pi$
$881$	33.1770	1.11776	0.558880	+	0.829248i	$0.311231\pi$
$882$	−4.35512	−0.146644
$883$	34.0077	1.14445	0.572225	−	0.820097i	$-0.306081\pi$
$883$	34.0077	1.14445	0.572225	+	0.820097i	$0.306081\pi$
$884$	0	0
$885$	−12.8867	−0.433183
$886$	−6.37643	−0.214220
$887$	−49.0181	−1.64587	−0.822934	−	0.568137i	$-0.807664\pi$
$887$	−49.0181	−1.64587	−0.822934	+	0.568137i	$0.807664\pi$
$888$	66.2255	2.22238
$889$	20.2838	0.680295
$890$	−0.781156	−0.0261844
$891$	−0.835535	−0.0279915
$892$	−14.0137	−0.469212
$893$	−61.8845	−2.07088
$894$	−17.9478	−0.600265
$895$	12.3241	0.411950
$896$	−26.6793	−0.891294
$897$	0	0
$898$	2.07765	0.0693322
$899$	−10.4829	−0.349623
$900$	35.2899	1.17633
$901$	10.9805	0.365812
$902$	1.42778	0.0475400
$903$	−38.9872	−1.29741
$904$	−26.7924	−0.891100
$905$	−4.21040	−0.139958
$906$	13.7606	0.457166
$907$	−27.1941	−0.902965	−0.451483	−	0.892280i	$-0.649105\pi$
$907$	−27.1941	−0.902965	−0.451483	+	0.892280i	$0.649105\pi$
$908$	−0.418725	−0.0138959
$909$	59.2039	1.96367
$910$	0	0
$911$	−51.0536	−1.69148	−0.845741	−	0.533594i	$-0.820841\pi$
$911$	−51.0536	−1.69148	−0.845741	+	0.533594i	$0.820841\pi$
$912$	−24.9413	−0.825889
$913$	−13.3744	−0.442629
$914$	18.1671	0.600913
$915$	11.1248	0.367776
$916$	2.22428	0.0734923
$917$	26.5958	0.878272
$918$	12.1929	0.402426
$919$	−16.1289	−0.532045	−0.266022	−	0.963967i	$-0.585710\pi$
$919$	−16.1289	−0.532045	−0.266022	+	0.963967i	$0.585710\pi$
$920$	2.96034	0.0975996
$921$	61.0846	2.01281
$922$	−17.9677	−0.591735
$923$	0	0
$924$	12.8076	0.421340
$925$	−48.6852	−1.60076
$926$	−21.6207	−0.710501
$927$	58.6573	1.92656
$928$	59.1411	1.94140
$929$	1.10502	0.0362546	0.0181273	−	0.999836i	$-0.494230\pi$
$929$	1.10502	0.0362546	0.0181273	+	0.999836i	$0.494230\pi$
$930$	1.07819	0.0353552
$931$	−7.26370	−0.238058
$932$	23.6065	0.773258
$933$	−49.8457	−1.63188
$934$	−20.7021	−0.677393
$935$	−3.01695	−0.0986649
$936$	0	0
$937$	7.77910	0.254132	0.127066	−	0.991894i	$-0.459444\pi$
$937$	7.77910	0.254132	0.127066	+	0.991894i	$0.459444\pi$
$938$	−19.5314	−0.637722
$939$	13.1693	0.429765
$940$	12.4723	0.406803
$941$	57.1242	1.86219	0.931097	−	0.364771i	$-0.118853\pi$
$941$	57.1242	1.86219	0.931097	+	0.364771i	$0.118853\pi$
$942$	8.06465	0.262761
$943$	−3.98192	−0.129669
$944$	13.5166	0.439927
$945$	−7.10276	−0.231053
$946$	4.52081	0.146984
$947$	−17.7962	−0.578299	−0.289150	−	0.957284i	$-0.593373\pi$
$947$	−17.7962	−0.578299	−0.289150	+	0.957284i	$0.593373\pi$
$948$	9.49159	0.308272
$949$	0	0
$950$	−14.2770	−0.463207
$951$	−52.9342	−1.71651
$952$	21.2492	0.688692
$953$	−36.4920	−1.18209	−0.591045	−	0.806638i	$-0.701285\pi$
$953$	−36.4920	−1.18209	−0.591045	+	0.806638i	$0.701285\pi$
$954$	−8.13636	−0.263424
$955$	6.49956	0.210321
$956$	−43.6950	−1.41320
$957$	−35.4627	−1.14634
$958$	−17.8357	−0.576245
$959$	−15.9749	−0.515855
$960$	0.163214	0.00526770
$961$	1.00000	0.0322581
$962$	0	0
$963$	−13.9033	−0.448026
$964$	−36.9211	−1.18915
$965$	7.99622	0.257407
$966$	8.66417	0.278765
$967$	0.254947	0.00819854	0.00409927	−	0.999992i	$-0.498695\pi$
$967$	0.254947	0.00819854	0.00409927	+	0.999992i	$0.498695\pi$
$968$	21.4790	0.690362
$969$	55.1979	1.77321
$970$	−2.10583	−0.0676140
$971$	−33.0115	−1.05939	−0.529695	−	0.848188i	$-0.677694\pi$
$971$	−33.0115	−1.05939	−0.529695	+	0.848188i	$0.677694\pi$
$972$	−26.6053	−0.853366
$973$	11.3068	0.362479
$974$	−13.9305	−0.446361
$975$	0	0
$976$	−11.6686	−0.373501
$977$	−20.0922	−0.642806	−0.321403	−	0.946943i	$-0.604154\pi$
$977$	−20.0922	−0.642806	−0.321403	+	0.946943i	$0.604154\pi$
$978$	12.9462	0.413974
$979$	−2.45095	−0.0783327
$980$	1.46394	0.0467640
$981$	45.8799	1.46483
$982$	1.43015	0.0456378
$983$	38.2966	1.22147	0.610736	−	0.791835i	$-0.290874\pi$
$983$	38.2966	1.22147	0.610736	+	0.791835i	$0.290874\pi$
$984$	11.8067	0.376383
$985$	12.7463	0.406130
$986$	−26.2362	−0.835531
$987$	81.8613	2.60567
$988$	0	0
$989$	−12.6080	−0.400912
$990$	2.23552	0.0710495
$991$	−53.4115	−1.69667	−0.848337	−	0.529457i	$-0.822396\pi$
$991$	−53.4115	−1.69667	−0.848337	+	0.529457i	$0.822396\pi$
$992$	−5.64170	−0.179124
$993$	−68.2105	−2.16460
$994$	−3.99728	−0.126786
$995$	0.898077	0.0284710
$996$	−49.3169	−1.56266
$997$	−7.78304	−0.246491	−0.123246	−	0.992376i	$-0.539330\pi$
$997$	−7.78304	−0.246491	−0.123246	+	0.992376i	$0.539330\pi$
$998$	6.72433	0.212855
$999$	−51.3848	−1.62574

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5239.2.a.l.1.9		16
13.5	odd	4		403.2.c.b.311.14	✓	32
13.8	odd	4		403.2.c.b.311.19	yes	32
13.12	even	2		5239.2.a.k.1.8		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
403.2.c.b.311.14	✓	32	13.5	odd	4
403.2.c.b.311.19	yes	32	13.8	odd	4
5239.2.a.k.1.8		16	13.12	even	2
5239.2.a.l.1.9		16	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 5239 = 13^{2} \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5239.a (trivial)

\(f(q)\)	\(=\)	\(q+0.624841 q^{2} -2.78388 q^{3} -1.60957 q^{4} +0.619834 q^{5} -1.73948 q^{6} +2.35216 q^{7} -2.25541 q^{8} +4.74999 q^{9} +O(q^{10})\)	\(q+0.624841 q^{2} -2.78388 q^{3} -1.60957 q^{4} +0.619834 q^{5} -1.73948 q^{6} +2.35216 q^{7} -2.25541 q^{8} +4.74999 q^{9} +0.387298 q^{10} +1.21518 q^{11} +4.48086 q^{12} +1.46972 q^{14} -1.72554 q^{15} +1.80988 q^{16} -4.00546 q^{17} +2.96799 q^{18} +4.95017 q^{19} -0.997669 q^{20} -6.54812 q^{21} +0.759295 q^{22} -2.11759 q^{23} +6.27879 q^{24} -4.61581 q^{25} -4.87176 q^{27} -3.78597 q^{28} +10.4829 q^{29} -1.07819 q^{30} -1.00000 q^{31} +5.64170 q^{32} -3.38292 q^{33} -2.50277 q^{34} +1.45795 q^{35} -7.64546 q^{36} +10.5475 q^{37} +3.09307 q^{38} -1.39798 q^{40} +1.88041 q^{41} -4.09153 q^{42} +5.95396 q^{43} -1.95592 q^{44} +2.94420 q^{45} -1.32315 q^{46} -12.5015 q^{47} -5.03848 q^{48} -1.46736 q^{49} -2.88414 q^{50} +11.1507 q^{51} -2.74137 q^{53} -3.04407 q^{54} +0.753211 q^{55} -5.30507 q^{56} -13.7807 q^{57} +6.55011 q^{58} +7.46822 q^{59} +2.77739 q^{60} -6.44716 q^{61} -0.624841 q^{62} +11.1727 q^{63} -0.0945869 q^{64} -2.11379 q^{66} -13.2892 q^{67} +6.44708 q^{68} +5.89511 q^{69} +0.910984 q^{70} -2.71975 q^{71} -10.7132 q^{72} -7.81310 q^{73} +6.59050 q^{74} +12.8498 q^{75} -7.96766 q^{76} +2.85830 q^{77} +2.11825 q^{79} +1.12182 q^{80} -0.687580 q^{81} +1.17495 q^{82} -11.0061 q^{83} +10.5397 q^{84} -2.48272 q^{85} +3.72027 q^{86} -29.1830 q^{87} -2.74073 q^{88} -2.01694 q^{89} +1.83966 q^{90} +3.40841 q^{92} +2.78388 q^{93} -7.81144 q^{94} +3.06828 q^{95} -15.7058 q^{96} -5.43723 q^{97} -0.916869 q^{98} +5.77210 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 4 q^{2} - 2 q^{3} + 18 q^{4} + 4 q^{5} + 6 q^{6} + 2 q^{7} + 12 q^{8} + 10 q^{9}+O(q^{10}) \) 16 * q + 4 * q^2 - 2 * q^3 + 18 * q^4 + 4 * q^5 + 6 * q^6 + 2 * q^7 + 12 * q^8 + 10 * q^9	\( 16 q + 4 q^{2} - 2 q^{3} + 18 q^{4} + 4 q^{5} + 6 q^{6} + 2 q^{7} + 12 q^{8} + 10 q^{9} - 2 q^{10} + 14 q^{11} + 8 q^{12} - 8 q^{14} + 14 q^{16} + 4 q^{17} - 28 q^{18} + 22 q^{19} + 28 q^{20} + 12 q^{21} - 8 q^{22} + 4 q^{23} - 8 q^{24} - 2 q^{25} + 10 q^{27} + 16 q^{28} - 8 q^{29} - 20 q^{30} - 16 q^{31} + 48 q^{32} + 10 q^{33} + 8 q^{34} - 2 q^{35} + 22 q^{36} + 16 q^{37} - 6 q^{38} + 14 q^{40} + 44 q^{41} + 14 q^{42} + 16 q^{43} + 4 q^{44} + 56 q^{45} + 10 q^{47} + 32 q^{49} + 2 q^{50} - 6 q^{53} + 24 q^{54} + 22 q^{55} - 4 q^{56} - 8 q^{57} + 74 q^{58} + 2 q^{59} + 40 q^{60} + 8 q^{61} - 4 q^{62} + 56 q^{63} + 38 q^{64} - 34 q^{66} - 8 q^{67} + 32 q^{68} - 10 q^{69} - 108 q^{70} + 50 q^{71} - 44 q^{72} + 14 q^{73} + 8 q^{74} + 44 q^{76} + 16 q^{77} + 32 q^{79} + 68 q^{80} - 8 q^{81} - 6 q^{82} - 20 q^{83} + 136 q^{84} - 32 q^{85} + 8 q^{86} - 36 q^{87} - 40 q^{88} + 52 q^{89} - 34 q^{90} + 14 q^{92} + 2 q^{93} + 44 q^{94} - 2 q^{95} - 80 q^{96} + 18 q^{97} + 12 q^{98} + 38 q^{99}+O(q^{100}) \) 16 * q + 4 * q^2 - 2 * q^3 + 18 * q^4 + 4 * q^5 + 6 * q^6 + 2 * q^7 + 12 * q^8 + 10 * q^9 - 2 * q^10 + 14 * q^11 + 8 * q^12 - 8 * q^14 + 14 * q^16 + 4 * q^17 - 28 * q^18 + 22 * q^19 + 28 * q^20 + 12 * q^21 - 8 * q^22 + 4 * q^23 - 8 * q^24 - 2 * q^25 + 10 * q^27 + 16 * q^28 - 8 * q^29 - 20 * q^30 - 16 * q^31 + 48 * q^32 + 10 * q^33 + 8 * q^34 - 2 * q^35 + 22 * q^36 + 16 * q^37 - 6 * q^38 + 14 * q^40 + 44 * q^41 + 14 * q^42 + 16 * q^43 + 4 * q^44 + 56 * q^45 + 10 * q^47 + 32 * q^49 + 2 * q^50 - 6 * q^53 + 24 * q^54 + 22 * q^55 - 4 * q^56 - 8 * q^57 + 74 * q^58 + 2 * q^59 + 40 * q^60 + 8 * q^61 - 4 * q^62 + 56 * q^63 + 38 * q^64 - 34 * q^66 - 8 * q^67 + 32 * q^68 - 10 * q^69 - 108 * q^70 + 50 * q^71 - 44 * q^72 + 14 * q^73 + 8 * q^74 + 44 * q^76 + 16 * q^77 + 32 * q^79 + 68 * q^80 - 8 * q^81 - 6 * q^82 - 20 * q^83 + 136 * q^84 - 32 * q^85 + 8 * q^86 - 36 * q^87 - 40 * q^88 + 52 * q^89 - 34 * q^90 + 14 * q^92 + 2 * q^93 + 44 * q^94 - 2 * q^95 - 80 * q^96 + 18 * q^97 + 12 * q^98 + 38 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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