LMFDB - Embedded newform 5239.2.a.k.1.8

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:	$1$
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			1.8
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	$-0.570904\pi$
$2$	−0.624841	−0.441829	−0.220915	+	0.975293i	$0.570904\pi$
$3$	−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	$-0.544260\pi$
$5$	−0.619834	−0.277198	−0.138599	+	0.990349i	$0.544260\pi$
$6$	1.73948	0.710140
$7$	−2.35216	−0.889031	−0.444516	−	0.895771i	$-0.646624\pi$
$7$	−2.35216	−0.889031	−0.444516	+	0.895771i	$0.646624\pi$
$8$	2.25541	0.797408
$9$	4.74999	1.58333
$10$	0.387298	0.122474
$11$	−1.21518	−0.366391	−0.183196	−	0.983077i	$-0.558644\pi$
$11$	−1.21518	−0.366391	−0.183196	+	0.983077i	$0.558644\pi$
$12$	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	$-0.692214\pi$
$19$	−4.95017	−1.13565	−0.567823	+	0.823151i	$0.692214\pi$
$20$	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	$-0.833954\pi$
$37$	−10.5475	−1.73400	−0.866999	+	0.498310i	$0.833954\pi$
$38$	3.09307	0.501761
$39$	0	0
$40$	−1.39798	−0.221040
$41$	−1.88041	−0.293670	−0.146835	−	0.989161i	$-0.546909\pi$
$41$	−1.88041	−0.293670	−0.146835	+	0.989161i	$0.546909\pi$
$42$	−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	$-0.134720\pi$
$47$	12.5015	1.82353	0.911765	+	0.410712i	$0.134720\pi$
$48$	−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	$-0.661595\pi$
$59$	−7.46822	−0.972280	−0.486140	+	0.873881i	$0.661595\pi$
$60$	−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	$-0.198507\pi$
$67$	13.2892	1.62353	0.811764	+	0.583985i	$0.198507\pi$
$68$	6.44708	0.781823
$69$	5.89511	0.709688
$70$	−0.910984	−0.108883
$71$	2.71975	0.322775	0.161388	−	0.986891i	$-0.448403\pi$
$71$	2.71975	0.322775	0.161388	+	0.986891i	$0.448403\pi$
$72$	10.7132	1.26256
$73$	7.81310	0.914455	0.457227	−	0.889350i	$-0.348843\pi$
$73$	7.81310	0.914455	0.457227	+	0.889350i	$0.348843\pi$
$74$	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	$-0.293557\pi$
$83$	11.0061	1.20808	0.604039	+	0.796955i	$0.293557\pi$
$84$	−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	$-0.465908\pi$
$89$	2.01694	0.213795	0.106898	+	0.994270i	$0.465908\pi$
$90$	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	$-0.410980\pi$
$97$	5.43723	0.552067	0.276034	+	0.961148i	$0.410980\pi$
$98$	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	$-0.653076\pi$
$109$	−9.65896	−0.925160	−0.462580	+	0.886577i	$0.653076\pi$
$110$	−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	$-0.406304\pi$
$137$	6.79159	0.580244	0.290122	+	0.956990i	$0.406304\pi$
$138$	−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	$-0.638896\pi$
$149$	−10.3179	−0.845277	−0.422639	+	0.906298i	$0.638896\pi$
$150$	−8.02911	−0.655574
$151$	7.91075	0.643768	0.321884	−	0.946779i	$-0.395684\pi$
$151$	7.91075	0.643768	0.321884	+	0.946779i	$0.395684\pi$
$152$	−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	$-0.405855\pi$
$163$	7.44256	0.582946	0.291473	+	0.956579i	$0.405855\pi$
$164$	3.02665	0.236342
$165$	−2.09685	−0.163239
$166$	−6.87707	−0.533764
$167$	−6.43784	−0.498175	−0.249087	−	0.968481i	$-0.580131\pi$
$167$	−6.43784	−0.498175	−0.249087	+	0.968481i	$0.580131\pi$
$168$	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	$-0.653695\pi$
$193$	−12.9006	−0.928604	−0.464302	+	0.885677i	$0.653695\pi$
$194$	−3.39740	−0.243919
$195$	0	0
$196$	2.36183	0.168702
$197$	−20.5640	−1.46513	−0.732563	−	0.680699i	$-0.761676\pi$
$197$	−20.5640	−1.46513	−0.732563	+	0.680699i	$0.761676\pi$
$198$	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	$-0.594159\pi$
$223$	−8.70644	−0.583026	−0.291513	+	0.956567i	$0.594159\pi$
$224$	13.2702	0.886650
$225$	−21.9250	−1.46167
$226$	−7.42258	−0.493743
$227$	−0.260146	−0.0172665	−0.00863326	−	0.999963i	$-0.502748\pi$
$227$	−0.260146	−0.0172665	−0.00863326	+	0.999963i	$0.502748\pi$
$228$	−22.1810	−1.46897
$229$	1.38191	0.0913190	0.0456595	−	0.998957i	$-0.485461\pi$
$229$	1.38191	0.0913190	0.0456595	+	0.998957i	$0.485461\pi$
$230$	−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	$-0.841119\pi$
$239$	−27.1469	−1.75599	−0.877995	+	0.478670i	$0.841119\pi$
$240$	3.12302	0.201590
$241$	−22.9384	−1.47759	−0.738796	−	0.673929i	$-0.764605\pi$
$241$	−22.9384	−1.47759	−0.738796	+	0.673929i	$0.764605\pi$
$242$	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	$-0.686116\pi$
$271$	−18.1725	−1.10390	−0.551950	+	0.833877i	$0.686116\pi$
$272$	−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	$-0.798658\pi$
$281$	−27.0399	−1.61306	−0.806531	+	0.591191i	$0.798658\pi$
$282$	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	$-0.316216\pi$
$293$	18.6860	1.09165	0.545825	+	0.837899i	$0.316216\pi$
$294$	−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	$-0.284627\pi$
$307$	21.9423	1.25231	0.626156	+	0.779698i	$0.284627\pi$
$308$	−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	$-0.679305\pi$
$317$	−19.0145	−1.06796	−0.533981	+	0.845496i	$0.679305\pi$
$318$	−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	$-0.735156\pi$
$331$	−24.5020	−1.34675	−0.673375	+	0.739301i	$0.735156\pi$
$332$	−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	$-0.568588\pi$
$349$	−7.98868	−0.427624	−0.213812	+	0.976875i	$0.568588\pi$
$350$	−6.78395	−0.362618
$351$	0	0
$352$	6.85569	0.365410
$353$	−4.94127	−0.262997	−0.131499	−	0.991316i	$-0.541979\pi$
$353$	−4.94127	−0.262997	−0.131499	+	0.991316i	$0.541979\pi$
$354$	−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	$-0.580460\pi$
$359$	−9.47701	−0.500178	−0.250089	+	0.968223i	$0.580460\pi$
$360$	−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	$-0.232128\pi$
$379$	29.0334	1.49135	0.745674	+	0.666311i	$0.232128\pi$
$380$	−4.93863	−0.253346
$381$	−24.0067	−1.22990
$382$	−6.55206	−0.335233
$383$	−19.8755	−1.01559	−0.507794	−	0.861478i	$-0.669539\pi$
$383$	−19.8755	−1.01559	−0.507794	+	0.861478i	$0.669539\pi$
$384$	−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	$-0.437775\pi$
$397$	7.74047	0.388483	0.194241	+	0.980954i	$0.437775\pi$
$398$	−0.905332	−0.0453802
$399$	−32.4143	−1.62274
$400$	−8.35404	−0.417702
$401$	7.36568	0.367825	0.183912	−	0.982943i	$-0.441124\pi$
$401$	7.36568	0.367825	0.183912	+	0.982943i	$0.441124\pi$
$402$	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	$-0.434233\pi$
$409$	8.29760	0.410290	0.205145	+	0.978732i	$0.434233\pi$
$410$	−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	$-0.586119\pi$
$421$	−10.9675	−0.534525	−0.267262	+	0.963624i	$0.586119\pi$
$422$	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	$-0.560320\pi$
$431$	−7.82125	−0.376736	−0.188368	+	0.982098i	$0.560320\pi$
$432$	−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	$-0.525000\pi$
$449$	−3.32509	−0.156921	−0.0784604	+	0.996917i	$0.525000\pi$
$450$	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	$-0.738033\pi$
$457$	−29.0747	−1.36006	−0.680029	+	0.733185i	$0.738033\pi$
$458$	−0.863472	−0.0403474
$459$	19.5136	0.910817
$460$	−2.11265	−0.0985028
$461$	28.7557	1.33929	0.669643	−	0.742684i	$-0.266447\pi$
$461$	28.7557	1.33929	0.669643	+	0.742684i	$0.266447\pi$
$462$	4.97195	0.231316
$463$	34.6020	1.60809	0.804045	−	0.594568i	$-0.202677\pi$
$463$	34.6020	1.60809	0.804045	+	0.594568i	$0.202677\pi$
$464$	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	$-0.273883\pi$
$479$	28.5444	1.30423	0.652113	+	0.758122i	$0.273883\pi$
$480$	−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	$-0.331445\pi$
$487$	22.2944	1.01026	0.505129	+	0.863044i	$0.331445\pi$
$488$	−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	$-0.577436\pi$
$499$	−10.7617	−0.481758	−0.240879	+	0.970555i	$0.577436\pi$
$500$	−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	$-0.435720\pi$
$509$	9.05022	0.401144	0.200572	+	0.979679i	$0.435720\pi$
$510$	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	$-0.613107\pi$
$541$	−16.1842	−0.695811	−0.347906	+	0.937530i	$0.613107\pi$
$542$	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	$-0.671228\pi$
$557$	−24.1842	−1.02472	−0.512359	+	0.858772i	$0.671228\pi$
$558$	−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	$-0.612293\pi$
$577$	−16.5988	−0.691016	−0.345508	+	0.938416i	$0.612293\pi$
$578$	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	$-0.414218\pi$
$587$	12.9010	0.532482	0.266241	+	0.963907i	$0.414218\pi$
$588$	−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	$-0.868171\pi$
$593$	−44.5857	−1.83091	−0.915457	+	0.402415i	$0.868171\pi$
$594$	−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	$-0.483123\pi$
$613$	2.62416	0.105989	0.0529943	+	0.998595i	$0.483123\pi$
$614$	−13.7104	−0.553308
$615$	−3.24472	−0.130840
$616$	6.44663	0.259742
$617$	2.69130	0.108348	0.0541738	−	0.998532i	$-0.482748\pi$
$617$	2.69130	0.108348	0.0541738	+	0.998532i	$0.482748\pi$
$618$	21.4807	0.864082
$619$	27.2021	1.09334	0.546672	−	0.837347i	$-0.315895\pi$
$619$	27.2021	1.09334	0.546672	+	0.837347i	$0.315895\pi$
$620$	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	$-0.680250\pi$
$631$	−26.9529	−1.07298	−0.536490	+	0.843907i	$0.680250\pi$
$632$	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	$-0.588958\pi$
$643$	−13.9895	−0.551693	−0.275847	+	0.961202i	$0.588958\pi$
$644$	−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	$-0.793604\pi$
$661$	−40.9839	−1.59409	−0.797044	+	0.603921i	$0.793604\pi$
$662$	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	$-0.478122\pi$
$683$	3.58966	0.137354	0.0686772	+	0.997639i	$0.478122\pi$
$684$	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	$-0.305750\pi$
$691$	30.1287	1.14615	0.573075	+	0.819503i	$0.305750\pi$
$692$	−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	$-0.785283\pi$
$709$	−41.5907	−1.56197	−0.780986	+	0.624549i	$0.785283\pi$
$710$	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	$-0.708292\pi$
$733$	−32.9576	−1.21732	−0.608658	+	0.793433i	$0.708292\pi$
$734$	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	$-0.616951\pi$
$739$	−19.5295	−0.718406	−0.359203	+	0.933259i	$0.616951\pi$
$740$	−10.5229	−0.386830
$741$	0	0
$742$	−4.02906	−0.147911
$743$	−19.0224	−0.697864	−0.348932	−	0.937148i	$-0.613456\pi$
$743$	−19.0224	−0.697864	−0.348932	+	0.937148i	$0.613456\pi$
$744$	−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	$-0.212475\pi$
$761$	43.3306	1.57073	0.785366	+	0.619032i	$0.212475\pi$
$762$	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	$-0.320866\pi$
$769$	29.5903	1.06705	0.533527	+	0.845783i	$0.320866\pi$
$770$	1.10701	0.0398939
$771$	−59.6280	−2.14745
$772$	20.7644	0.747328
$773$	−47.4918	−1.70816	−0.854080	−	0.520142i	$-0.825879\pi$
$773$	−47.4918	−1.70816	−0.854080	+	0.520142i	$0.825879\pi$
$774$	−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	$-0.733773\pi$
$787$	−37.6005	−1.34031	−0.670157	+	0.742220i	$0.733773\pi$
$788$	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	$-0.667322\pi$
$811$	−28.5796	−1.00356	−0.501782	+	0.864994i	$0.667322\pi$
$812$	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	$-0.413561\pi$
$821$	15.3714	0.536464	0.268232	+	0.963354i	$0.413561\pi$
$822$	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	$-0.439997\pi$
$827$	10.7778	0.374781	0.187390	+	0.982286i	$0.439997\pi$
$828$	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	$-0.755904\pi$
$839$	−41.7161	−1.44020	−0.720099	+	0.693871i	$0.755904\pi$
$840$	−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	$-0.284937\pi$
$853$	36.5309	1.25080	0.625398	+	0.780306i	$0.284937\pi$
$854$	−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	$-0.585306\pi$
$863$	−15.5581	−0.529602	−0.264801	+	0.964303i	$0.585306\pi$
$864$	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	$-0.483463\pi$
$877$	3.07559	0.103855	0.0519277	+	0.998651i	$0.483463\pi$
$878$	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	$-0.505770\pi$
$929$	−1.10502	−0.0362546	−0.0181273	+	0.999836i	$0.505770\pi$
$930$	−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	$-0.881147\pi$
$941$	−57.1242	−1.86219	−0.931097	+	0.364771i	$0.881147\pi$
$942$	−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	$-0.406627\pi$
$947$	17.7962	0.578299	0.289150	+	0.957284i	$0.406627\pi$
$948$	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	$-0.501305\pi$
$967$	−0.254947	−0.00819854	−0.00409927	+	0.999992i	$0.501305\pi$
$968$	−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	$-0.395846\pi$
$977$	20.0922	0.642806	0.321403	+	0.946943i	$0.395846\pi$
$978$	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	$-0.709126\pi$
$983$	−38.2966	−1.22147	−0.610736	+	0.791835i	$0.709126\pi$
$984$	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.k.1.8		16
13.5	odd	4		403.2.c.b.311.19	yes	32
13.8	odd	4		403.2.c.b.311.14	✓	32
13.12	even	2		5239.2.a.l.1.9		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
403.2.c.b.311.14	✓	32	13.8	odd	4
403.2.c.b.311.19	yes	32	13.5	odd	4
5239.2.a.k.1.8		16	1.1	even	1	trivial
5239.2.a.l.1.9		16	13.12	even	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:	\( N \)	\(=\)	\( 5239 = 13^{2} \cdot 31 \)
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

Related objects

Downloads

Learn more