LMFDB - Embedded newform 5239.2.a.l.1.12

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			1.12
Root			$1.48947$ 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)

$f(q)$	$=$	$q+1.48947 q^{2} +2.99747 q^{3} +0.218531 q^{4} -0.692562 q^{5} +4.46466 q^{6} +1.15260 q^{7} -2.65345 q^{8} +5.98485 q^{9} +O(q^{10})$	\(q+1.48947 q^{2} +2.99747 q^{3} +0.218531 q^{4} -0.692562 q^{5} +4.46466 q^{6} +1.15260 q^{7} -2.65345 q^{8} +5.98485 q^{9} -1.03155 q^{10} +2.69554 q^{11} +0.655040 q^{12} +1.71676 q^{14} -2.07594 q^{15} -4.38931 q^{16} +4.04263 q^{17} +8.91428 q^{18} +5.06961 q^{19} -0.151346 q^{20} +3.45488 q^{21} +4.01494 q^{22} +1.01940 q^{23} -7.95365 q^{24} -4.52036 q^{25} +8.94703 q^{27} +0.251878 q^{28} -8.56509 q^{29} -3.09205 q^{30} -1.00000 q^{31} -1.23085 q^{32} +8.07982 q^{33} +6.02139 q^{34} -0.798246 q^{35} +1.30787 q^{36} +5.88104 q^{37} +7.55104 q^{38} +1.83768 q^{40} +11.8895 q^{41} +5.14596 q^{42} +3.63935 q^{43} +0.589058 q^{44} -4.14488 q^{45} +1.51836 q^{46} -0.848777 q^{47} -13.1568 q^{48} -5.67152 q^{49} -6.73295 q^{50} +12.1177 q^{51} +1.67652 q^{53} +13.3264 q^{54} -1.86683 q^{55} -3.05836 q^{56} +15.1960 q^{57} -12.7575 q^{58} +1.68042 q^{59} -0.453656 q^{60} +2.00772 q^{61} -1.48947 q^{62} +6.89813 q^{63} +6.94529 q^{64} +12.0347 q^{66} +8.32497 q^{67} +0.883439 q^{68} +3.05562 q^{69} -1.18897 q^{70} +5.15229 q^{71} -15.8805 q^{72} +14.8210 q^{73} +8.75965 q^{74} -13.5497 q^{75} +1.10786 q^{76} +3.10687 q^{77} +0.873238 q^{79} +3.03987 q^{80} +8.86392 q^{81} +17.7091 q^{82} -10.8692 q^{83} +0.754998 q^{84} -2.79977 q^{85} +5.42072 q^{86} -25.6736 q^{87} -7.15249 q^{88} -3.90690 q^{89} -6.17370 q^{90} +0.222769 q^{92} -2.99747 q^{93} -1.26423 q^{94} -3.51102 q^{95} -3.68945 q^{96} -10.3729 q^{97} -8.44757 q^{98} +16.1324 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	1.48947	1.05322	0.526608	−	0.850108i	$-0.323463\pi$
$2$	1.48947	1.05322	0.526608	+	0.850108i	$0.323463\pi$
$3$	2.99747	1.73059	0.865296	−	0.501261i	$-0.167130\pi$
$3$	2.99747	1.73059	0.865296	+	0.501261i	$0.167130\pi$
$4$	0.218531	0.109265
$5$	−0.692562	−0.309723	−0.154862	−	0.987936i	$-0.549493\pi$
$5$	−0.692562	−0.309723	−0.154862	+	0.987936i	$0.549493\pi$
$6$	4.46466	1.82269
$7$	1.15260	0.435641	0.217821	−	0.975989i	$-0.430105\pi$
$7$	1.15260	0.435641	0.217821	+	0.975989i	$0.430105\pi$
$8$	−2.65345	−0.938137
$9$	5.98485	1.99495
$10$	−1.03155	−0.326206
$11$	2.69554	0.812736	0.406368	−	0.913709i	$-0.366795\pi$
$11$	2.69554	0.812736	0.406368	+	0.913709i	$0.366795\pi$
$12$	0.655040	0.189094
$13$	0	0
$13$	0	0
$14$	1.71676	0.458824
$15$	−2.07594	−0.536005
$16$	−4.38931	−1.09733
$17$	4.04263	0.980482	0.490241	−	0.871587i	$-0.336909\pi$
$17$	4.04263	0.980482	0.490241	+	0.871587i	$0.336909\pi$
$18$	8.91428	2.10112
$19$	5.06961	1.16305	0.581524	−	0.813530i	$-0.302457\pi$
$19$	5.06961	1.16305	0.581524	+	0.813530i	$0.302457\pi$
$20$	−0.151346	−0.0338420
$21$	3.45488	0.753917
$22$	4.01494	0.855987
$23$	1.01940	0.212559	0.106279	−	0.994336i	$-0.466106\pi$
$23$	1.01940	0.212559	0.106279	+	0.994336i	$0.466106\pi$
$24$	−7.95365	−1.62353
$25$	−4.52036	−0.904071
$26$	0	0
$27$	8.94703	1.72186
$28$	0.251878	0.0476005
$29$	−8.56509	−1.59050	−0.795249	−	0.606283i	$-0.792660\pi$
$29$	−8.56509	−1.59050	−0.795249	+	0.606283i	$0.792660\pi$
$30$	−3.09205	−0.564529
$31$	−1.00000	−0.179605
$31$	−1.00000	−0.179605
$32$	−1.23085	−0.217586
$33$	8.07982	1.40652
$34$	6.02139	1.03266
$35$	−0.798246	−0.134928
$36$	1.30787	0.217979
$37$	5.88104	0.966837	0.483418	−	0.875389i	$-0.339395\pi$
$37$	5.88104	0.966837	0.483418	+	0.875389i	$0.339395\pi$
$38$	7.55104	1.22494
$39$	0	0
$40$	1.83768	0.290563
$41$	11.8895	1.85683	0.928413	−	0.371551i	$-0.121174\pi$
$41$	11.8895	1.85683	0.928413	+	0.371551i	$0.121174\pi$
$42$	5.14596	0.794038
$43$	3.63935	0.554996	0.277498	−	0.960726i	$-0.410495\pi$
$43$	3.63935	0.554996	0.277498	+	0.960726i	$0.410495\pi$
$44$	0.589058	0.0888039
$45$	−4.14488	−0.617883
$46$	1.51836	0.223871
$47$	−0.848777	−0.123807	−0.0619034	−	0.998082i	$-0.519717\pi$
$47$	−0.848777	−0.123807	−0.0619034	+	0.998082i	$0.519717\pi$
$48$	−13.1568	−1.89903
$49$	−5.67152	−0.810217
$50$	−6.73295	−0.952183
$51$	12.1177	1.69681
$52$	0	0
$53$	1.67652	0.230287	0.115144	−	0.993349i	$-0.463267\pi$
$53$	1.67652	0.230287	0.115144	+	0.993349i	$0.463267\pi$
$54$	13.3264	1.81349
$55$	−1.86683	−0.251723
$56$	−3.05836	−0.408691
$57$	15.1960	2.01276
$58$	−12.7575	−1.67514
$59$	1.68042	0.218772	0.109386	−	0.993999i	$-0.465111\pi$
$59$	1.68042	0.218772	0.109386	+	0.993999i	$0.465111\pi$
$60$	−0.453656	−0.0585668
$61$	2.00772	0.257062	0.128531	−	0.991705i	$-0.458974\pi$
$61$	2.00772	0.257062	0.128531	+	0.991705i	$0.458974\pi$
$62$	−1.48947	−0.189163
$63$	6.89813	0.869083
$64$	6.94529	0.868161
$65$	0	0
$66$	12.0347	1.48137
$67$	8.32497	1.01706	0.508529	−	0.861045i	$-0.330190\pi$
$67$	8.32497	1.01706	0.508529	+	0.861045i	$0.330190\pi$
$68$	0.883439	0.107133
$69$	3.05562	0.367853
$70$	−1.18897	−0.142109
$71$	5.15229	0.611465	0.305732	−	0.952117i	$-0.401099\pi$
$71$	5.15229	0.611465	0.305732	+	0.952117i	$0.401099\pi$
$72$	−15.8805	−1.87154
$73$	14.8210	1.73466	0.867332	−	0.497730i	$-0.165833\pi$
$73$	14.8210	1.73466	0.867332	+	0.497730i	$0.165833\pi$
$74$	8.75965	1.01829
$75$	−13.5497	−1.56458
$76$	1.10786	0.127081
$77$	3.10687	0.354061
$78$	0	0
$79$	0.873238	0.0982470	0.0491235	−	0.998793i	$-0.484357\pi$
$79$	0.873238	0.0982470	0.0491235	+	0.998793i	$0.484357\pi$
$80$	3.03987	0.339868
$81$	8.86392	0.984880
$82$	17.7091	1.95564
$83$	−10.8692	−1.19305	−0.596523	−	0.802596i	$-0.703452\pi$
$83$	−10.8692	−1.19305	−0.596523	+	0.802596i	$0.703452\pi$
$84$	0.754998	0.0823770
$85$	−2.79977	−0.303678
$86$	5.42072	0.584531
$87$	−25.6736	−2.75250
$88$	−7.15249	−0.762457
$89$	−3.90690	−0.414131	−0.207065	−	0.978327i	$-0.566391\pi$
$89$	−3.90690	−0.414131	−0.207065	+	0.978327i	$0.566391\pi$
$90$	−6.17370	−0.650765
$91$	0	0
$92$	0.222769	0.0232253
$93$	−2.99747	−0.310824
$94$	−1.26423	−0.130395
$95$	−3.51102	−0.360223
$96$	−3.68945	−0.376553
$97$	−10.3729	−1.05321	−0.526606	−	0.850110i	$-0.676536\pi$
$97$	−10.3729	−1.05321	−0.526606	+	0.850110i	$0.676536\pi$
$98$	−8.44757	−0.853334
$99$	16.1324	1.62137
$100$	−0.987837	−0.0987837
$101$	1.55946	0.155172	0.0775862	−	0.996986i	$-0.475279\pi$
$101$	1.55946	0.155172	0.0775862	+	0.996986i	$0.475279\pi$
$102$	18.0490	1.78711
$103$	−18.4902	−1.82189	−0.910947	−	0.412523i	$-0.864648\pi$
$103$	−18.4902	−1.82189	−0.910947	+	0.412523i	$0.864648\pi$
$104$	0	0
$105$	−2.39272	−0.233506
$106$	2.49713	0.242543
$107$	15.4750	1.49602	0.748010	−	0.663687i	$-0.231009\pi$
$107$	15.4750	1.49602	0.748010	+	0.663687i	$0.231009\pi$
$108$	1.95520	0.188139
$109$	−11.0776	−1.06104	−0.530518	−	0.847673i	$-0.678003\pi$
$109$	−11.0776	−1.06104	−0.530518	+	0.847673i	$0.678003\pi$
$110$	−2.78059	−0.265119
$111$	17.6283	1.67320
$112$	−5.05910	−0.478040
$113$	1.83667	0.172780	0.0863898	−	0.996261i	$-0.472467\pi$
$113$	1.83667	0.172780	0.0863898	+	0.996261i	$0.472467\pi$
$114$	22.6341	2.11987
$115$	−0.705996	−0.0658344
$116$	−1.87174	−0.173786
$117$	0	0
$118$	2.50294	0.230415
$119$	4.65953	0.427138
$120$	5.50840	0.502846
$121$	−3.73406	−0.339460
$122$	2.99045	0.270743
$123$	35.6384	3.21341
$124$	−0.218531	−0.0196246
$125$	6.59344	0.589735
$126$	10.2746	0.915333
$127$	−14.4471	−1.28198	−0.640988	−	0.767551i	$-0.721475\pi$
$127$	−14.4471	−1.28198	−0.640988	+	0.767551i	$0.721475\pi$
$128$	12.8065	1.13195
$129$	10.9089	0.960472
$130$	0	0
$131$	−6.01302	−0.525360	−0.262680	−	0.964883i	$-0.584606\pi$
$131$	−6.01302	−0.525360	−0.262680	+	0.964883i	$0.584606\pi$
$132$	1.76569	0.153683
$133$	5.84322	0.506671
$134$	12.3998	1.07118
$135$	−6.19637	−0.533299
$136$	−10.7269	−0.919826
$137$	−16.9039	−1.44420	−0.722098	−	0.691791i	$-0.756822\pi$
$137$	−16.9039	−1.44420	−0.722098	+	0.691791i	$0.756822\pi$
$138$	4.55126	0.387429
$139$	−0.0379050	−0.00321506	−0.00160753	−	0.999999i	$-0.500512\pi$
$139$	−0.0379050	−0.00321506	−0.00160753	+	0.999999i	$0.500512\pi$
$140$	−0.174441	−0.0147430
$141$	−2.54419	−0.214259
$142$	7.67421	0.644005
$143$	0	0
$144$	−26.2694	−2.18911
$145$	5.93186	0.492614
$146$	22.0754	1.82698
$147$	−17.0002	−1.40216
$148$	1.28519	0.105642
$149$	−16.6078	−1.36056	−0.680281	−	0.732952i	$-0.738142\pi$
$149$	−16.6078	−1.36056	−0.680281	+	0.732952i	$0.738142\pi$
$150$	−20.1819	−1.64784
$151$	3.90890	0.318102	0.159051	−	0.987270i	$-0.449157\pi$
$151$	3.90890	0.318102	0.159051	+	0.987270i	$0.449157\pi$
$152$	−13.4519	−1.09110
$153$	24.1945	1.95601
$154$	4.62761	0.372903
$155$	0.692562	0.0556279
$156$	0	0
$157$	4.63903	0.370235	0.185117	−	0.982716i	$-0.440733\pi$
$157$	4.63903	0.370235	0.185117	+	0.982716i	$0.440733\pi$
$158$	1.30066	0.103475
$159$	5.02532	0.398534
$160$	0.852441	0.0673914
$161$	1.17495	0.0925994
$162$	13.2026	1.03729
$163$	14.1214	1.10607	0.553035	−	0.833158i	$-0.313469\pi$
$163$	14.1214	1.10607	0.553035	+	0.833158i	$0.313469\pi$
$164$	2.59822	0.202887
$165$	−5.59578	−0.435631
$166$	−16.1893	−1.25654
$167$	−10.5161	−0.813759	−0.406879	−	0.913482i	$-0.633383\pi$
$167$	−10.5161	−0.813759	−0.406879	+	0.913482i	$0.633383\pi$
$168$	−9.16736	−0.707277
$169$	0	0
$170$	−4.17019	−0.319839
$171$	30.3409	2.32022
$172$	0.795310	0.0606418
$173$	−20.5134	−1.55960	−0.779801	−	0.626028i	$-0.784680\pi$
$173$	−20.5134	−1.55960	−0.779801	+	0.626028i	$0.784680\pi$
$174$	−38.2402	−2.89898
$175$	−5.21015	−0.393851
$176$	−11.8316	−0.891837
$177$	5.03702	0.378606
$178$	−5.81923	−0.436170
$179$	10.7116	0.800620	0.400310	−	0.916380i	$-0.368902\pi$
$179$	10.7116	0.800620	0.400310	+	0.916380i	$0.368902\pi$
$180$	−0.905785	−0.0675132
$181$	23.9491	1.78012	0.890061	−	0.455842i	$-0.150662\pi$
$181$	23.9491	1.78012	0.890061	+	0.455842i	$0.150662\pi$
$182$	0	0
$183$	6.01810	0.444871
$184$	−2.70492	−0.199409
$185$	−4.07299	−0.299452
$186$	−4.46466	−0.327365
$187$	10.8971	0.796873
$188$	−0.185484	−0.0135278
$189$	10.3123	0.750111
$190$	−5.22957	−0.379393
$191$	−3.30915	−0.239442	−0.119721	−	0.992808i	$-0.538200\pi$
$191$	−3.30915	−0.239442	−0.119721	+	0.992808i	$0.538200\pi$
$192$	20.8183	1.50243
$193$	−9.66509	−0.695709	−0.347854	−	0.937549i	$-0.613090\pi$
$193$	−9.66509	−0.695709	−0.347854	+	0.937549i	$0.613090\pi$
$194$	−15.4502	−1.10926
$195$	0	0
$196$	−1.23940	−0.0885286
$197$	26.5454	1.89128	0.945641	−	0.325212i	$-0.105436\pi$
$197$	26.5454	1.89128	0.945641	+	0.325212i	$0.105436\pi$
$198$	24.0288	1.70765
$199$	6.60253	0.468041	0.234021	−	0.972232i	$-0.424812\pi$
$199$	6.60253	0.468041	0.234021	+	0.972232i	$0.424812\pi$
$200$	11.9945	0.848143
$201$	24.9539	1.76011
$202$	2.32278	0.163430
$203$	−9.87211	−0.692886
$204$	2.64808	0.185403
$205$	−8.23421	−0.575102
$206$	−27.5407	−1.91885
$207$	6.10094	0.424045
$208$	0	0
$209$	13.6653	0.945250
$210$	−3.56390	−0.245932
$211$	−18.1733	−1.25110	−0.625551	−	0.780183i	$-0.715126\pi$
$211$	−18.1733	−1.25110	−0.625551	+	0.780183i	$0.715126\pi$
$212$	0.366371	0.0251624
$213$	15.4439	1.05820
$214$	23.0495	1.57563
$215$	−2.52048	−0.171895
$216$	−23.7405	−1.61534
$217$	−1.15260	−0.0782434
$218$	−16.4997	−1.11750
$219$	44.4255	3.00200
$220$	−0.407960	−0.0275046
$221$	0	0
$222$	26.2568	1.76224
$223$	3.90568	0.261544	0.130772	−	0.991412i	$-0.458254\pi$
$223$	3.90568	0.261544	0.130772	+	0.991412i	$0.458254\pi$
$224$	−1.41868	−0.0947894
$225$	−27.0537	−1.80358
$226$	2.73568	0.181974
$227$	16.9202	1.12303	0.561515	−	0.827466i	$-0.310219\pi$
$227$	16.9202	1.12303	0.561515	+	0.827466i	$0.310219\pi$
$228$	3.32080	0.219925
$229$	−11.5304	−0.761949	−0.380975	−	0.924585i	$-0.624411\pi$
$229$	−11.5304	−0.761949	−0.380975	+	0.924585i	$0.624411\pi$
$230$	−1.05156	−0.0693379
$231$	9.31278	0.612736
$232$	22.7271	1.49210
$233$	−0.108649	−0.00711784	−0.00355892	−	0.999994i	$-0.501133\pi$
$233$	−0.108649	−0.00711784	−0.00355892	+	0.999994i	$0.501133\pi$
$234$	0	0
$235$	0.587831	0.0383459
$236$	0.367224	0.0239042
$237$	2.61751	0.170026
$238$	6.94024	0.449869
$239$	−15.8424	−1.02476	−0.512378	−	0.858760i	$-0.671235\pi$
$239$	−15.8424	−1.02476	−0.512378	+	0.858760i	$0.671235\pi$
$240$	9.11193	0.588172
$241$	−4.44336	−0.286222	−0.143111	−	0.989707i	$-0.545711\pi$
$241$	−4.44336	−0.286222	−0.143111	+	0.989707i	$0.545711\pi$
$242$	−5.56178	−0.357525
$243$	−0.271696	−0.0174293
$244$	0.438749	0.0280880
$245$	3.92788	0.250943
$246$	53.0825	3.38442
$247$	0	0
$248$	2.65345	0.168494
$249$	−32.5801	−2.06468
$250$	9.82075	0.621119
$251$	−8.24869	−0.520653	−0.260326	−	0.965521i	$-0.583830\pi$
$251$	−8.24869	−0.520653	−0.260326	+	0.965521i	$0.583830\pi$
$252$	1.50745	0.0949606
$253$	2.74783	0.172754
$254$	−21.5186	−1.35020
$255$	−8.39225	−0.525543
$256$	5.18440	0.324025
$257$	8.29303	0.517305	0.258652	−	0.965970i	$-0.416722\pi$
$257$	8.29303	0.517305	0.258652	+	0.965970i	$0.416722\pi$
$258$	16.2485	1.01159
$259$	6.77847	0.421194
$260$	0	0
$261$	−51.2608	−3.17297
$262$	−8.95624	−0.553318
$263$	−19.7412	−1.21729	−0.608646	−	0.793442i	$-0.708287\pi$
$263$	−19.7412	−1.21729	−0.608646	+	0.793442i	$0.708287\pi$
$264$	−21.4394	−1.31950
$265$	−1.16109	−0.0713254
$266$	8.70331	0.533634
$267$	−11.7108	−0.716692
$268$	1.81926	0.111129
$269$	−28.8863	−1.76123	−0.880614	−	0.473834i	$-0.842870\pi$
$269$	−28.8863	−1.76123	−0.880614	+	0.473834i	$0.842870\pi$
$270$	−9.22933	−0.561679
$271$	0.993095	0.0603262	0.0301631	−	0.999545i	$-0.490397\pi$
$271$	0.993095	0.0603262	0.0301631	+	0.999545i	$0.490397\pi$
$272$	−17.7443	−1.07591
$273$	0	0
$274$	−25.1779	−1.52105
$275$	−12.1848	−0.734772
$276$	0.667746	0.0401936
$277$	−1.99748	−0.120017	−0.0600085	−	0.998198i	$-0.519113\pi$
$277$	−1.99748	−0.120017	−0.0600085	+	0.998198i	$0.519113\pi$
$278$	−0.0564585	−0.00338616
$279$	−5.98485	−0.358304
$280$	2.11811	0.126581
$281$	−9.22146	−0.550106	−0.275053	−	0.961429i	$-0.588695\pi$
$281$	−9.22146	−0.550106	−0.275053	+	0.961429i	$0.588695\pi$
$282$	−3.78950	−0.225661
$283$	6.66812	0.396379	0.198189	−	0.980164i	$-0.436494\pi$
$283$	6.66812	0.396379	0.198189	+	0.980164i	$0.436494\pi$
$284$	1.12593	0.0668119
$285$	−10.5242	−0.623399
$286$	0	0
$287$	13.7038	0.808909
$288$	−7.36647	−0.434073
$289$	−0.657150	−0.0386559
$290$	8.83535	0.518829
$291$	−31.0926	−1.82268
$292$	3.23884	0.189539
$293$	20.6409	1.20585	0.602926	−	0.797797i	$-0.294001\pi$
$293$	20.6409	1.20585	0.602926	+	0.797797i	$0.294001\pi$
$294$	−25.3214	−1.47677
$295$	−1.16380	−0.0677589
$296$	−15.6050	−0.907025
$297$	24.1171	1.39941
$298$	−24.7368	−1.43297
$299$	0	0
$300$	−2.96102	−0.170954
$301$	4.19471	0.241779
$302$	5.82220	0.335030
$303$	4.67445	0.268540
$304$	−22.2520	−1.27624
$305$	−1.39047	−0.0796182
$306$	36.0371	2.06011
$307$	3.05469	0.174340	0.0871702	−	0.996193i	$-0.472218\pi$
$307$	3.05469	0.174340	0.0871702	+	0.996193i	$0.472218\pi$
$308$	0.678947	0.0386866
$309$	−55.4239	−3.15296
$310$	1.03155	0.0585883
$311$	22.9204	1.29969	0.649847	−	0.760065i	$-0.274833\pi$
$311$	22.9204	1.29969	0.649847	+	0.760065i	$0.274833\pi$
$312$	0	0
$313$	10.3566	0.585391	0.292695	−	0.956206i	$-0.405448\pi$
$313$	10.3566	0.585391	0.292695	+	0.956206i	$0.405448\pi$
$314$	6.90971	0.389937
$315$	−4.77739	−0.269175
$316$	0.190829	0.0107350
$317$	9.04755	0.508161	0.254080	−	0.967183i	$-0.418227\pi$
$317$	9.04755	0.508161	0.254080	+	0.967183i	$0.418227\pi$
$318$	7.48508	0.419742
$319$	−23.0876	−1.29265
$320$	−4.81005	−0.268890
$321$	46.3858	2.58900
$322$	1.75006	0.0975272
$323$	20.4945	1.14035
$324$	1.93704	0.107613
$325$	0	0
$326$	21.0334	1.16493
$327$	−33.2047	−1.83622
$328$	−31.5482	−1.74196
$329$	−0.978299	−0.0539354
$330$	−8.33476	−0.458813
$331$	32.1476	1.76699	0.883496	−	0.468438i	$-0.155183\pi$
$331$	32.1476	1.76699	0.883496	+	0.468438i	$0.155183\pi$
$332$	−2.37525	−0.130359
$333$	35.1972	1.92879
$334$	−15.6634	−0.857064
$335$	−5.76556	−0.315006
$336$	−15.1645	−0.827293
$337$	−31.7351	−1.72872	−0.864361	−	0.502871i	$-0.832277\pi$
$337$	−31.7351	−1.72872	−0.864361	+	0.502871i	$0.832277\pi$
$338$	0	0
$339$	5.50538	0.299011
$340$	−0.611836	−0.0331815
$341$	−2.69554	−0.145972
$342$	45.1919	2.44370
$343$	−14.6052	−0.788605
$344$	−9.65684	−0.520662
$345$	−2.11620	−0.113933
$346$	−30.5541	−1.64260
$347$	−1.57708	−0.0846622	−0.0423311	−	0.999104i	$-0.513478\pi$
$347$	−1.57708	−0.0846622	−0.0423311	+	0.999104i	$0.513478\pi$
$348$	−5.61048	−0.300753
$349$	−33.0608	−1.76970	−0.884852	−	0.465872i	$-0.845741\pi$
$349$	−33.0608	−1.76970	−0.884852	+	0.465872i	$0.845741\pi$
$350$	−7.76039	−0.414810
$351$	0	0
$352$	−3.31781	−0.176840
$353$	−4.70845	−0.250605	−0.125303	−	0.992119i	$-0.539990\pi$
$353$	−4.70845	−0.250605	−0.125303	+	0.992119i	$0.539990\pi$
$354$	7.50251	0.398754
$355$	−3.56829	−0.189385
$356$	−0.853778	−0.0452502
$357$	13.9668	0.739202
$358$	15.9546	0.843226
$359$	27.2147	1.43634	0.718169	−	0.695868i	$-0.244980\pi$
$359$	27.2147	1.43634	0.718169	+	0.695868i	$0.244980\pi$
$360$	10.9982	0.579659
$361$	6.70090	0.352679
$362$	35.6715	1.87485
$363$	−11.1928	−0.587467
$364$	0	0
$365$	−10.2644	−0.537266
$366$	8.96379	0.468545
$367$	4.82539	0.251883	0.125942	−	0.992038i	$-0.459805\pi$
$367$	4.82539	0.251883	0.125942	+	0.992038i	$0.459805\pi$
$368$	−4.47444	−0.233246
$369$	71.1568	3.70428
$370$	−6.06660	−0.315388
$371$	1.93235	0.100323
$372$	−0.655040	−0.0339623
$373$	17.5308	0.907710	0.453855	−	0.891076i	$-0.350048\pi$
$373$	17.5308	0.907710	0.453855	+	0.891076i	$0.350048\pi$
$374$	16.2309	0.839280
$375$	19.7637	1.02059
$376$	2.25219	0.116148
$377$	0	0
$378$	15.3599	0.790030
$379$	−33.9284	−1.74279	−0.871393	−	0.490586i	$-0.836783\pi$
$379$	−33.9284	−1.74279	−0.871393	+	0.490586i	$0.836783\pi$
$380$	−0.767265	−0.0393599
$381$	−43.3049	−2.21858
$382$	−4.92889	−0.252184
$383$	6.07830	0.310587	0.155293	−	0.987868i	$-0.450368\pi$
$383$	6.07830	0.310587	0.155293	+	0.987868i	$0.450368\pi$
$384$	38.3872	1.95894
$385$	−2.15170	−0.109661
$386$	−14.3959	−0.732732
$387$	21.7810	1.10719
$388$	−2.26680	−0.115080
$389$	−15.9014	−0.806232	−0.403116	−	0.915149i	$-0.632073\pi$
$389$	−15.9014	−0.806232	−0.403116	+	0.915149i	$0.632073\pi$
$390$	0	0
$391$	4.12104	0.208410
$392$	15.0491	0.760094
$393$	−18.0239	−0.909185
$394$	39.5387	1.99193
$395$	−0.604772	−0.0304294
$396$	3.52543	0.177159
$397$	−26.8597	−1.34805	−0.674024	−	0.738709i	$-0.735436\pi$
$397$	−26.8597	−1.34805	−0.674024	+	0.738709i	$0.735436\pi$
$398$	9.83430	0.492949
$399$	17.5149	0.876841
$400$	19.8412	0.992062
$401$	−29.4467	−1.47050	−0.735249	−	0.677797i	$-0.762935\pi$
$401$	−29.4467	−1.47050	−0.735249	+	0.677797i	$0.762935\pi$
$402$	37.1682	1.85378
$403$	0	0
$404$	0.340791	0.0169550
$405$	−6.13882	−0.305040
$406$	−14.7042	−0.729759
$407$	15.8526	0.785783
$408$	−32.1537	−1.59184
$409$	38.4627	1.90186	0.950930	−	0.309408i	$-0.100131\pi$
$409$	38.4627	1.90186	0.950930	+	0.309408i	$0.100131\pi$
$410$	−12.2646	−0.605707
$411$	−50.6690	−2.49932
$412$	−4.04068	−0.199070
$413$	1.93685	0.0953062
$414$	9.08719	0.446611
$415$	7.52757	0.369514
$416$	0	0
$417$	−0.113619	−0.00556396
$418$	20.3541	0.995554
$419$	6.73204	0.328882	0.164441	−	0.986387i	$-0.447418\pi$
$419$	6.73204	0.328882	0.164441	+	0.986387i	$0.447418\pi$
$420$	−0.522883	−0.0255141
$421$	0.815884	0.0397637	0.0198819	−	0.999802i	$-0.493671\pi$
$421$	0.815884	0.0397637	0.0198819	+	0.999802i	$0.493671\pi$
$422$	−27.0686	−1.31768
$423$	−5.07981	−0.246989
$424$	−4.44856	−0.216041
$425$	−18.2741	−0.886425
$426$	23.0032	1.11451
$427$	2.31410	0.111987
$428$	3.38175	0.163463
$429$	0	0
$430$	−3.75418	−0.181043
$431$	−35.7637	−1.72268	−0.861338	−	0.508032i	$-0.830373\pi$
$431$	−35.7637	−1.72268	−0.861338	+	0.508032i	$0.830373\pi$
$432$	−39.2712	−1.88944
$433$	25.8780	1.24362	0.621808	−	0.783170i	$-0.286399\pi$
$433$	25.8780	1.24362	0.621808	+	0.783170i	$0.286399\pi$
$434$	−1.71676	−0.0824073
$435$	17.7806	0.852515
$436$	−2.42078	−0.115935
$437$	5.16794	0.247216
$438$	66.1706	3.16175
$439$	24.4490	1.16689	0.583444	−	0.812153i	$-0.301705\pi$
$439$	24.4490	1.16689	0.583444	+	0.812153i	$0.301705\pi$
$440$	4.95354	0.236151
$441$	−33.9432	−1.61634
$442$	0	0
$443$	15.4138	0.732330	0.366165	−	0.930550i	$-0.380671\pi$
$443$	15.4138	0.732330	0.366165	+	0.930550i	$0.380671\pi$
$444$	3.85232	0.182823
$445$	2.70577	0.128266
$446$	5.81741	0.275462
$447$	−49.7814	−2.35458
$448$	8.00513	0.378207
$449$	1.20092	0.0566750	0.0283375	−	0.999598i	$-0.490979\pi$
$449$	1.20092	0.0566750	0.0283375	+	0.999598i	$0.490979\pi$
$450$	−40.2957	−1.89956
$451$	32.0486	1.50911
$452$	0.401369	0.0188788
$453$	11.7168	0.550504
$454$	25.2021	1.18279
$455$	0	0
$456$	−40.3219	−1.88824
$457$	−4.20576	−0.196737	−0.0983686	−	0.995150i	$-0.531362\pi$
$457$	−4.20576	−0.196737	−0.0983686	+	0.995150i	$0.531362\pi$
$458$	−17.1742	−0.802498
$459$	36.1695	1.68825
$460$	−0.154282	−0.00719342
$461$	−14.1239	−0.657814	−0.328907	−	0.944362i	$-0.606680\pi$
$461$	−14.1239	−0.657814	−0.328907	+	0.944362i	$0.606680\pi$
$462$	13.8711	0.645344
$463$	21.9558	1.02037	0.510186	−	0.860064i	$-0.329577\pi$
$463$	21.9558	1.02037	0.510186	+	0.860064i	$0.329577\pi$
$464$	37.5948	1.74530
$465$	2.07594	0.0962693
$466$	−0.161830	−0.00749663
$467$	0.918022	0.0424810	0.0212405	−	0.999774i	$-0.493238\pi$
$467$	0.918022	0.0424810	0.0212405	+	0.999774i	$0.493238\pi$
$468$	0	0
$469$	9.59535	0.443072
$470$	0.875559	0.0403865
$471$	13.9054	0.640726
$472$	−4.45892	−0.205238
$473$	9.81002	0.451065
$474$	3.89871	0.179074
$475$	−22.9164	−1.05148
$476$	1.01825	0.0466714
$477$	10.0337	0.459412
$478$	−23.5968	−1.07929
$479$	4.85495	0.221828	0.110914	−	0.993830i	$-0.464622\pi$
$479$	4.85495	0.221828	0.110914	+	0.993830i	$0.464622\pi$
$480$	2.55517	0.116627
$481$	0	0
$482$	−6.61826	−0.301454
$483$	3.52190	0.160252
$484$	−0.816007	−0.0370912
$485$	7.18390	0.326204
$486$	−0.404684	−0.0183568
$487$	−39.7612	−1.80175	−0.900877	−	0.434075i	$-0.857075\pi$
$487$	−39.7612	−1.80175	−0.900877	+	0.434075i	$0.857075\pi$
$488$	−5.32739	−0.241160
$489$	42.3284	1.91416
$490$	5.85047	0.264297
$491$	−6.08825	−0.274759	−0.137379	−	0.990519i	$-0.543868\pi$
$491$	−6.08825	−0.274759	−0.137379	+	0.990519i	$0.543868\pi$
$492$	7.78809	0.351114
$493$	−34.6255	−1.55945
$494$	0	0
$495$	−11.1727	−0.502176
$496$	4.38931	0.197086
$497$	5.93852	0.266379
$498$	−48.5271	−2.17455
$499$	−26.0135	−1.16453	−0.582263	−	0.813001i	$-0.697832\pi$
$499$	−26.0135	−1.16453	−0.582263	+	0.813001i	$0.697832\pi$
$500$	1.44087	0.0644376
$501$	−31.5217	−1.40829
$502$	−12.2862	−0.548360
$503$	−26.6423	−1.18792	−0.593962	−	0.804493i	$-0.702437\pi$
$503$	−26.6423	−1.18792	−0.593962	+	0.804493i	$0.702437\pi$
$504$	−18.3039	−0.815318
$505$	−1.08003	−0.0480605
$506$	4.09281	0.181948
$507$	0	0
$508$	−3.15714	−0.140075
$509$	33.0607	1.46539	0.732694	−	0.680558i	$-0.238262\pi$
$509$	33.0607	1.46539	0.732694	+	0.680558i	$0.238262\pi$
$510$	−12.5000	−0.553511
$511$	17.0826	0.755691
$512$	−17.8910	−0.790679
$513$	45.3579	2.00260
$514$	12.3522	0.544834
$515$	12.8056	0.564283
$516$	2.38392	0.104946
$517$	−2.28791	−0.100622
$518$	10.0964	0.443608
$519$	−61.4883	−2.69904
$520$	0	0
$521$	25.8957	1.13451	0.567255	−	0.823543i	$-0.308006\pi$
$521$	25.8957	1.13451	0.567255	+	0.823543i	$0.308006\pi$
$522$	−76.3516	−3.34182
$523$	−25.9541	−1.13489	−0.567447	−	0.823410i	$-0.692069\pi$
$523$	−25.9541	−1.13489	−0.567447	+	0.823410i	$0.692069\pi$
$524$	−1.31403	−0.0574037
$525$	−15.6173	−0.681595
$526$	−29.4039	−1.28207
$527$	−4.04263	−0.176100
$528$	−35.4648	−1.54341
$529$	−21.9608	−0.954819
$530$	−1.72942	−0.0751211
$531$	10.0571	0.436440
$532$	1.27692	0.0553616
$533$	0	0
$534$	−17.4430	−0.754832
$535$	−10.7174	−0.463352
$536$	−22.0899	−0.954139
$537$	32.1077	1.38555
$538$	−43.0254	−1.85496
$539$	−15.2878	−0.658493
$540$	−1.35410	−0.0582711
$541$	−28.6632	−1.23233	−0.616164	−	0.787618i	$-0.711314\pi$
$541$	−28.6632	−1.23233	−0.616164	+	0.787618i	$0.711314\pi$
$542$	1.47919	0.0635366
$543$	71.7868	3.08067
$544$	−4.97588	−0.213339
$545$	7.67189	0.328628
$546$	0	0
$547$	12.0729	0.516198	0.258099	−	0.966119i	$-0.416904\pi$
$547$	12.0729	0.516198	0.258099	+	0.966119i	$0.416904\pi$
$548$	−3.69402	−0.157801
$549$	12.0159	0.512827
$550$	−18.1489	−0.773874
$551$	−43.4216	−1.84982
$552$	−8.10793	−0.345096
$553$	1.00649	0.0428004
$554$	−2.97520	−0.126404
$555$	−12.2087	−0.518229
$556$	−0.00828341	−0.000351295 0
$557$	−2.10575	−0.0892234	−0.0446117	−	0.999004i	$-0.514205\pi$
$557$	−2.10575	−0.0892234	−0.0446117	+	0.999004i	$0.514205\pi$
$558$	−8.91428	−0.377372
$559$	0	0
$560$	3.50374	0.148060
$561$	32.6637	1.37906
$562$	−13.7351	−0.579381
$563$	−22.0349	−0.928662	−0.464331	−	0.885662i	$-0.653705\pi$
$563$	−22.0349	−0.928662	−0.464331	+	0.885662i	$0.653705\pi$
$564$	−0.555983	−0.0234111
$565$	−1.27201	−0.0535139
$566$	9.93199	0.417473
$567$	10.2165	0.429054
$568$	−13.6714	−0.573638
$569$	−36.0563	−1.51156	−0.755780	−	0.654826i	$-0.772742\pi$
$569$	−36.0563	−1.51156	−0.755780	+	0.654826i	$0.772742\pi$
$570$	−15.6755	−0.656574
$571$	−34.7416	−1.45389	−0.726945	−	0.686695i	$-0.759061\pi$
$571$	−34.7416	−1.45389	−0.726945	+	0.686695i	$0.759061\pi$
$572$	0	0
$573$	−9.91909	−0.414376
$574$	20.4114	0.851957
$575$	−4.60804	−0.192168
$576$	41.5666	1.73194
$577$	−11.2505	−0.468364	−0.234182	−	0.972193i	$-0.575241\pi$
$577$	−11.2505	−0.468364	−0.234182	+	0.972193i	$0.575241\pi$
$578$	−0.978807	−0.0407130
$579$	−28.9709	−1.20399
$580$	1.29629	0.0538257
$581$	−12.5278	−0.519740
$582$	−46.3116	−1.91968
$583$	4.51912	0.187163
$584$	−39.3267	−1.62735
$585$	0	0
$586$	30.7440	1.27002
$587$	16.1947	0.668426	0.334213	−	0.942498i	$-0.391529\pi$
$587$	16.1947	0.668426	0.334213	+	0.942498i	$0.391529\pi$
$588$	−3.71507	−0.153207
$589$	−5.06961	−0.208889
$590$	−1.73344	−0.0713648
$591$	79.5692	3.27304
$592$	−25.8137	−1.06094
$593$	23.3449	0.958660	0.479330	−	0.877635i	$-0.340880\pi$
$593$	23.3449	0.958660	0.479330	+	0.877635i	$0.340880\pi$
$594$	35.9217	1.47389
$595$	−3.22701	−0.132295
$596$	−3.62931	−0.148662
$597$	19.7909	0.809989
$598$	0	0
$599$	−40.5953	−1.65868	−0.829340	−	0.558744i	$-0.811284\pi$
$599$	−40.5953	−1.65868	−0.829340	+	0.558744i	$0.811284\pi$
$600$	35.9534	1.46779
$601$	−39.5067	−1.61151	−0.805755	−	0.592249i	$-0.798240\pi$
$601$	−39.5067	−1.61151	−0.805755	+	0.592249i	$0.798240\pi$
$602$	6.24791	0.254646
$603$	49.8237	2.02898
$604$	0.854214	0.0347575
$605$	2.58607	0.105139
$606$	6.96247	0.282831
$607$	1.07768	0.0437419	0.0218709	−	0.999761i	$-0.493038\pi$
$607$	1.07768	0.0437419	0.0218709	+	0.999761i	$0.493038\pi$
$608$	−6.23993	−0.253063
$609$	−29.5914	−1.19910
$610$	−2.07107	−0.0838553
$611$	0	0
$612$	5.28725	0.213724
$613$	−19.5448	−0.789406	−0.394703	−	0.918809i	$-0.629152\pi$
$613$	−19.5448	−0.789406	−0.394703	+	0.918809i	$0.629152\pi$
$614$	4.54988	0.183618
$615$	−24.6818	−0.995267
$616$	−8.24394	−0.332158
$617$	−25.7573	−1.03695	−0.518474	−	0.855093i	$-0.673500\pi$
$617$	−25.7573	−1.03695	−0.518474	+	0.855093i	$0.673500\pi$
$618$	−82.5525	−3.32075
$619$	23.1573	0.930770	0.465385	−	0.885108i	$-0.345916\pi$
$619$	23.1573	0.930770	0.465385	+	0.885108i	$0.345916\pi$
$620$	0.151346	0.00607821
$621$	9.12057	0.365996
$622$	34.1393	1.36886
$623$	−4.50309	−0.180412
$624$	0	0
$625$	18.0354	0.721417
$626$	15.4259	0.616543
$627$	40.9615	1.63584
$628$	1.01377	0.0404538
$629$	23.7749	0.947966
$630$	−7.11579	−0.283500
$631$	−42.2030	−1.68008	−0.840038	−	0.542528i	$-0.817467\pi$
$631$	−42.2030	−1.68008	−0.840038	+	0.542528i	$0.817467\pi$
$632$	−2.31709	−0.0921691
$633$	−54.4740	−2.16515
$634$	13.4761	0.535203
$635$	10.0055	0.397058
$636$	1.09819	0.0435459
$637$	0	0
$638$	−34.3883	−1.36145
$639$	30.8357	1.21984
$640$	−8.86932	−0.350591
$641$	−35.9647	−1.42052	−0.710260	−	0.703939i	$-0.751423\pi$
$641$	−35.9647	−1.42052	−0.710260	+	0.703939i	$0.751423\pi$
$642$	69.0904	2.72678
$643$	−11.9104	−0.469701	−0.234850	−	0.972032i	$-0.575460\pi$
$643$	−11.9104	−0.469701	−0.234850	+	0.972032i	$0.575460\pi$
$644$	0.256764	0.0101179
$645$	−7.55507	−0.297481
$646$	30.5261	1.20103
$647$	−3.70864	−0.145802	−0.0729009	−	0.997339i	$-0.523226\pi$
$647$	−3.70864	−0.145802	−0.0729009	+	0.997339i	$0.523226\pi$
$648$	−23.5200	−0.923952
$649$	4.52965	0.177804
$650$	0	0
$651$	−3.45488	−0.135408
$652$	3.08595	0.120855
$653$	−13.2111	−0.516991	−0.258496	−	0.966012i	$-0.583227\pi$
$653$	−13.2111	−0.516991	−0.258496	+	0.966012i	$0.583227\pi$
$654$	−49.4575	−1.93394
$655$	4.16439	0.162716
$656$	−52.1866	−2.03754
$657$	88.7014	3.46057
$658$	−1.45715	−0.0568056
$659$	−37.0706	−1.44407	−0.722033	−	0.691858i	$-0.756792\pi$
$659$	−37.0706	−1.44407	−0.722033	+	0.691858i	$0.756792\pi$
$660$	−1.22285	−0.0475993
$661$	−1.86574	−0.0725687	−0.0362844	−	0.999342i	$-0.511552\pi$
$661$	−1.86574	−0.0725687	−0.0362844	+	0.999342i	$0.511552\pi$
$662$	47.8830	1.86103
$663$	0	0
$664$	28.8408	1.11924
$665$	−4.04679	−0.156928
$666$	52.4252	2.03144
$667$	−8.73123	−0.338074
$668$	−2.29809	−0.0889156
$669$	11.7072	0.452626
$670$	−8.58765	−0.331770
$671$	5.41190	0.208924
$672$	−4.25245	−0.164042
$673$	−38.8121	−1.49610	−0.748048	−	0.663645i	$-0.769009\pi$
$673$	−38.8121	−1.49610	−0.748048	+	0.663645i	$0.769009\pi$
$674$	−47.2686	−1.82072
$675$	−40.4438	−1.55668
$676$	0	0
$677$	−18.1456	−0.697394	−0.348697	−	0.937236i	$-0.613376\pi$
$677$	−18.1456	−0.697394	−0.348697	+	0.937236i	$0.613376\pi$
$678$	8.20012	0.314924
$679$	−11.9558	−0.458822
$680$	7.42906	0.284891
$681$	50.7178	1.94351
$682$	−4.01494	−0.153740
$683$	23.3612	0.893891	0.446946	−	0.894561i	$-0.352512\pi$
$683$	23.3612	0.893891	0.446946	+	0.894561i	$0.352512\pi$
$684$	6.63041	0.253520
$685$	11.7070	0.447301
$686$	−21.7540	−0.830572
$687$	−34.5620	−1.31862
$688$	−15.9742	−0.609012
$689$	0	0
$690$	−3.15203	−0.119996
$691$	9.52166	0.362221	0.181111	−	0.983463i	$-0.442031\pi$
$691$	9.52166	0.362221	0.181111	+	0.983463i	$0.442031\pi$
$692$	−4.48280	−0.170410
$693$	18.5942	0.706335
$694$	−2.34902	−0.0891676
$695$	0.0262516	0.000995780 0
$696$	68.1238	2.58222
$697$	48.0648	1.82058
$698$	−49.2432	−1.86388
$699$	−0.325673	−0.0123181
$700$	−1.13858	−0.0430342
$701$	−7.92968	−0.299500	−0.149750	−	0.988724i	$-0.547847\pi$
$701$	−7.92968	−0.299500	−0.149750	+	0.988724i	$0.547847\pi$
$702$	0	0
$703$	29.8145	1.12448
$704$	18.7213	0.705586
$705$	1.76201	0.0663611
$706$	−7.01310	−0.263942
$707$	1.79743	0.0675994
$708$	1.10074	0.0413685
$709$	−0.516551	−0.0193995	−0.00969973	−	0.999953i	$-0.503088\pi$
$709$	−0.516551	−0.0193995	−0.00969973	+	0.999953i	$0.503088\pi$
$710$	−5.31487	−0.199463
$711$	5.22620	0.195998
$712$	10.3668	0.388511
$713$	−1.01940	−0.0381767
$714$	20.8032	0.778540
$715$	0	0
$716$	2.34081	0.0874800
$717$	−47.4871	−1.77344
$718$	40.5356	1.51278
$719$	25.0988	0.936026	0.468013	−	0.883722i	$-0.344970\pi$
$719$	25.0988	0.936026	0.468013	+	0.883722i	$0.344970\pi$
$720$	18.1932	0.678019
$721$	−21.3118	−0.793692
$722$	9.98081	0.371447
$723$	−13.3188	−0.495333
$724$	5.23361	0.194506
$725$	38.7173	1.43792
$726$	−16.6713	−0.618730
$727$	−24.2348	−0.898819	−0.449409	−	0.893326i	$-0.648366\pi$
$727$	−24.2348	−0.898819	−0.449409	+	0.893326i	$0.648366\pi$
$728$	0	0
$729$	−27.4062	−1.01504
$730$	−15.2886	−0.565857
$731$	14.7126	0.544163
$732$	1.31514	0.0486089
$733$	47.9992	1.77289	0.886445	−	0.462834i	$-0.153167\pi$
$733$	47.9992	1.77289	0.886445	+	0.462834i	$0.153167\pi$
$734$	7.18729	0.265288
$735$	11.7737	0.434280
$736$	−1.25473	−0.0462498
$737$	22.4403	0.826599
$738$	105.986	3.90141
$739$	13.4514	0.494817	0.247409	−	0.968911i	$-0.420421\pi$
$739$	13.4514	0.494817	0.247409	+	0.968911i	$0.420421\pi$
$740$	−0.890072	−0.0327197
$741$	0	0
$742$	2.87818	0.105661
$743$	17.9897	0.659977	0.329989	−	0.943985i	$-0.392955\pi$
$743$	17.9897	0.659977	0.329989	+	0.943985i	$0.392955\pi$
$744$	7.95365	0.291595
$745$	11.5019	0.421397
$746$	26.1116	0.956015
$747$	−65.0504	−2.38007
$748$	2.38134	0.0870706
$749$	17.8364	0.651728
$750$	29.4375	1.07490
$751$	−5.17026	−0.188666	−0.0943328	−	0.995541i	$-0.530072\pi$
$751$	−5.17026	−0.188666	−0.0943328	+	0.995541i	$0.530072\pi$
$752$	3.72554	0.135857
$753$	−24.7252	−0.901038
$754$	0	0
$755$	−2.70716	−0.0985235
$756$	2.25356	0.0819612
$757$	16.2563	0.590845	0.295422	−	0.955367i	$-0.404540\pi$
$757$	16.2563	0.590845	0.295422	+	0.955367i	$0.404540\pi$
$758$	−50.5355	−1.83553
$759$	8.23654	0.298967
$760$	9.31631	0.337938
$761$	26.5204	0.961365	0.480682	−	0.876895i	$-0.340389\pi$
$761$	26.5204	0.961365	0.480682	+	0.876895i	$0.340389\pi$
$762$	−64.5015	−2.33664
$763$	−12.7680	−0.462231
$764$	−0.723150	−0.0261627
$765$	−16.7562	−0.605823
$766$	9.05347	0.327115
$767$	0	0
$768$	15.5401	0.560755
$769$	−0.983871	−0.0354793	−0.0177397	−	0.999843i	$-0.505647\pi$
$769$	−0.983871	−0.0354793	−0.0177397	+	0.999843i	$0.505647\pi$
$770$	−3.20491	−0.115497
$771$	24.8581	0.895244
$772$	−2.11212	−0.0760168
$773$	−27.6879	−0.995866	−0.497933	−	0.867215i	$-0.665907\pi$
$773$	−27.6879	−0.995866	−0.497933	+	0.867215i	$0.665907\pi$
$774$	32.4422	1.16611
$775$	4.52036	0.162376
$776$	27.5241	0.988056
$777$	20.3183	0.728915
$778$	−23.6847	−0.849137
$779$	60.2750	2.15958
$780$	0	0
$781$	13.8882	0.496960
$782$	6.13818	0.219501
$783$	−76.6321	−2.73861
$784$	24.8940	0.889072
$785$	−3.21282	−0.114670
$786$	−26.8461	−0.957569
$787$	−50.0529	−1.78419	−0.892096	−	0.451846i	$-0.850766\pi$
$787$	−50.0529	−1.78419	−0.892096	+	0.451846i	$0.850766\pi$
$788$	5.80099	0.206652
$789$	−59.1736	−2.10664
$790$	−0.900791	−0.0320487
$791$	2.11695	0.0752699
$792$	−42.8066	−1.52107
$793$	0	0
$794$	−40.0068	−1.41979
$795$	−3.48035	−0.123435
$796$	1.44286	0.0511407
$797$	33.6343	1.19139	0.595694	−	0.803212i	$-0.296877\pi$
$797$	33.6343	1.19139	0.595694	+	0.803212i	$0.296877\pi$
$798$	26.0880	0.923504
$799$	−3.43129	−0.121390
$800$	5.56389	0.196713
$801$	−23.3823	−0.826171
$802$	−43.8601	−1.54875
$803$	39.9505	1.40982
$804$	5.45319	0.192319
$805$	−0.813729	−0.0286802
$806$	0	0
$807$	−86.5859	−3.04797
$808$	−4.13796	−0.145573
$809$	20.5333	0.721913	0.360956	−	0.932583i	$-0.382450\pi$
$809$	20.5333	0.721913	0.360956	+	0.932583i	$0.382450\pi$
$810$	−9.14361	−0.321274
$811$	−21.2908	−0.747622	−0.373811	−	0.927505i	$-0.621949\pi$
$811$	−21.2908	−0.747622	−0.373811	+	0.927505i	$0.621949\pi$
$812$	−2.15736	−0.0757084
$813$	2.97678	0.104400
$814$	23.6120	0.827600
$815$	−9.77993	−0.342576
$816$	−53.1882	−1.86196
$817$	18.4501	0.645487
$818$	57.2892	2.00307
$819$	0	0
$820$	−1.79943	−0.0628387
$821$	18.4733	0.644723	0.322362	−	0.946617i	$-0.395523\pi$
$821$	18.4733	0.644723	0.322362	+	0.946617i	$0.395523\pi$
$822$	−75.4701	−2.63232
$823$	−19.9508	−0.695442	−0.347721	−	0.937598i	$-0.613044\pi$
$823$	−19.9508	−0.695442	−0.347721	+	0.937598i	$0.613044\pi$
$824$	49.0629	1.70919
$825$	−36.5237	−1.27159
$826$	2.88489	0.100378
$827$	44.1949	1.53681	0.768403	−	0.639966i	$-0.221052\pi$
$827$	44.1949	1.53681	0.768403	+	0.639966i	$0.221052\pi$
$828$	1.33324	0.0463334
$829$	8.44679	0.293369	0.146685	−	0.989183i	$-0.453140\pi$
$829$	8.44679	0.293369	0.146685	+	0.989183i	$0.453140\pi$
$830$	11.2121	0.389178
$831$	−5.98740	−0.207701
$832$	0	0
$833$	−22.9278	−0.794403
$834$	−0.169233	−0.00586006
$835$	7.28304	0.252040
$836$	2.98629	0.103283
$837$	−8.94703	−0.309254
$838$	10.0272	0.346384
$839$	39.8660	1.37633	0.688163	−	0.725556i	$-0.258417\pi$
$839$	39.8660	1.37633	0.688163	+	0.725556i	$0.258417\pi$
$840$	6.34897	0.219060
$841$	44.3608	1.52968
$842$	1.21524	0.0418798
$843$	−27.6411	−0.952010
$844$	−3.97142	−0.136702
$845$	0	0
$846$	−7.56624	−0.260133
$847$	−4.30387	−0.147883
$848$	−7.35875	−0.252700
$849$	19.9875	0.685970
$850$	−27.2188	−0.933598
$851$	5.99511	0.205510
$852$	3.37496	0.115624
$853$	9.26849	0.317347	0.158674	−	0.987331i	$-0.449278\pi$
$853$	9.26849	0.317347	0.158674	+	0.987331i	$0.449278\pi$
$854$	3.44678	0.117947
$855$	−21.0129	−0.718627
$856$	−41.0620	−1.40347
$857$	37.4028	1.27766	0.638828	−	0.769350i	$-0.279420\pi$
$857$	37.4028	1.27766	0.638828	+	0.769350i	$0.279420\pi$
$858$	0	0
$859$	−23.2534	−0.793395	−0.396698	−	0.917949i	$-0.629844\pi$
$859$	−23.2534	−0.793395	−0.396698	+	0.917949i	$0.629844\pi$
$860$	−0.550802	−0.0187822
$861$	41.0768	1.39989
$862$	−53.2691	−1.81435
$863$	−22.9063	−0.779741	−0.389871	−	0.920870i	$-0.627480\pi$
$863$	−22.9063	−0.779741	−0.389871	+	0.920870i	$0.627480\pi$
$864$	−11.0125	−0.374652
$865$	14.2068	0.483045
$866$	38.5445	1.30980
$867$	−1.96979	−0.0668976
$868$	−0.251878	−0.00854930
$869$	2.35385	0.0798489
$870$	26.4837	0.897883
$871$	0	0
$872$	29.3937	0.995398
$873$	−62.0805	−2.10111
$874$	7.69751	0.260372
$875$	7.59959	0.256913
$876$	9.70834	0.328014
$877$	−20.8719	−0.704795	−0.352397	−	0.935850i	$-0.614633\pi$
$877$	−20.8719	−0.704795	−0.352397	+	0.935850i	$0.614633\pi$
$878$	36.4162	1.22899
$879$	61.8705	2.08684
$880$	8.19409	0.276223
$881$	24.7872	0.835102	0.417551	−	0.908653i	$-0.362888\pi$
$881$	24.7872	0.835102	0.417551	+	0.908653i	$0.362888\pi$
$882$	−50.5575	−1.70236
$883$	42.9772	1.44630	0.723148	−	0.690693i	$-0.242694\pi$
$883$	42.9772	1.44630	0.723148	+	0.690693i	$0.242694\pi$
$884$	0	0
$885$	−3.48845	−0.117263
$886$	22.9584	0.771302
$887$	20.4768	0.687544	0.343772	−	0.939053i	$-0.388295\pi$
$887$	20.4768	0.687544	0.343772	+	0.939053i	$0.388295\pi$
$888$	−46.7757	−1.56969
$889$	−16.6517	−0.558481
$890$	4.03018	0.135092
$891$	23.8931	0.800448
$892$	0.853512	0.0285777
$893$	−4.30297	−0.143993
$894$	−74.1480	−2.47988
$895$	−7.41843	−0.247971
$896$	14.7608	0.493123
$897$	0	0
$898$	1.78874	0.0596910
$899$	8.56509	0.285662
$900$	−5.91206	−0.197069
$901$	6.77754	0.225793
$902$	47.7355	1.58942
$903$	12.5735	0.418421
$904$	−4.87352	−0.162091
$905$	−16.5862	−0.551345
$906$	17.4519	0.579800
$907$	38.8180	1.28893	0.644466	−	0.764633i	$-0.277080\pi$
$907$	38.8180	1.28893	0.644466	+	0.764633i	$0.277080\pi$
$908$	3.69757	0.122708
$909$	9.33316	0.309561
$910$	0	0
$911$	14.2402	0.471799	0.235900	−	0.971777i	$-0.424196\pi$
$911$	14.2402	0.471799	0.235900	+	0.971777i	$0.424196\pi$
$912$	−66.7000	−2.20866
$913$	−29.2983	−0.969632
$914$	−6.26437	−0.207207
$915$	−4.16791	−0.137787
$916$	−2.51974	−0.0832546
$917$	−6.93060	−0.228869
$918$	53.8735	1.77809
$919$	53.7283	1.77233	0.886166	−	0.463367i	$-0.153359\pi$
$919$	53.7283	1.77233	0.886166	+	0.463367i	$0.153359\pi$
$920$	1.87332	0.0617617
$921$	9.15636	0.301712
$922$	−21.0371	−0.692821
$923$	0	0
$924$	2.03513	0.0669508
$925$	−26.5844	−0.874090
$926$	32.7025	1.07467
$927$	−110.661	−3.63459
$928$	10.5424	0.346070
$929$	29.9480	0.982562	0.491281	−	0.871001i	$-0.336529\pi$
$929$	29.9480	0.982562	0.491281	+	0.871001i	$0.336529\pi$
$930$	3.09205	0.101392
$931$	−28.7524	−0.942320
$932$	−0.0237432	−0.000777734 0
$933$	68.7032	2.24924
$934$	1.36737	0.0447417
$935$	−7.54690	−0.246810
$936$	0	0
$937$	−32.2927	−1.05496	−0.527479	−	0.849568i	$-0.676863\pi$
$937$	−32.2927	−1.05496	−0.527479	+	0.849568i	$0.676863\pi$
$938$	14.2920	0.466651
$939$	31.0437	1.01307
$940$	0.128459	0.00418987
$941$	−10.7946	−0.351894	−0.175947	−	0.984400i	$-0.556299\pi$
$941$	−10.7946	−0.351894	−0.175947	+	0.984400i	$0.556299\pi$
$942$	20.7117	0.674823
$943$	12.1201	0.394685
$944$	−7.37589	−0.240065
$945$	−7.14193	−0.232327
$946$	14.6118	0.475069
$947$	55.7380	1.81124	0.905620	−	0.424090i	$-0.139406\pi$
$947$	55.7380	1.81124	0.905620	+	0.424090i	$0.139406\pi$
$948$	0.572006	0.0185779
$949$	0	0
$950$	−34.1334	−1.10743
$951$	27.1198	0.879419
$952$	−12.3638	−0.400714
$953$	−42.0978	−1.36368	−0.681841	−	0.731500i	$-0.738820\pi$
$953$	−42.0978	−1.36368	−0.681841	+	0.731500i	$0.738820\pi$
$954$	14.9449	0.483861
$955$	2.29179	0.0741606
$956$	−3.46204	−0.111970
$957$	−69.2044	−2.23706
$958$	7.23131	0.233633
$959$	−19.4834	−0.629151
$960$	−14.4180	−0.465339
$961$	1.00000	0.0322581
$962$	0	0
$963$	92.6153	2.98449
$964$	−0.971010	−0.0312741
$965$	6.69368	0.215477
$966$	5.24577	0.168780
$967$	46.8530	1.50669	0.753345	−	0.657625i	$-0.228439\pi$
$967$	46.8530	1.50669	0.753345	+	0.657625i	$0.228439\pi$
$968$	9.90814	0.318460
$969$	61.4318	1.97348
$970$	10.7002	0.343564
$971$	43.0627	1.38195	0.690975	−	0.722879i	$-0.257182\pi$
$971$	43.0627	1.38195	0.690975	+	0.722879i	$0.257182\pi$
$972$	−0.0593739	−0.00190442
$973$	−0.0436892	−0.00140061
$974$	−59.2233	−1.89764
$975$	0	0
$976$	−8.81251	−0.282081
$977$	−11.4541	−0.366448	−0.183224	−	0.983071i	$-0.558653\pi$
$977$	−11.4541	−0.366448	−0.183224	+	0.983071i	$0.558653\pi$
$978$	63.0471	2.01602
$979$	−10.5312	−0.336579
$980$	0.858362	0.0274194
$981$	−66.2975	−2.11672
$982$	−9.06828	−0.289380
$983$	7.93525	0.253095	0.126548	−	0.991961i	$-0.459610\pi$
$983$	7.93525	0.253095	0.126548	+	0.991961i	$0.459610\pi$
$984$	−94.5648	−3.01462
$985$	−18.3843	−0.585774
$986$	−51.5737	−1.64244
$987$	−2.93243	−0.0933401
$988$	0	0
$989$	3.70994	0.117969
$990$	−16.6414	−0.528900
$991$	20.2465	0.643150	0.321575	−	0.946884i	$-0.395788\pi$
$991$	20.2465	0.643150	0.321575	+	0.946884i	$0.395788\pi$
$992$	1.23085	0.0390796
$993$	96.3617	3.05795
$994$	8.84527	0.280555
$995$	−4.57266	−0.144963
$996$	−7.11974	−0.225598
$997$	−18.4051	−0.582896	−0.291448	−	0.956587i	$-0.594137\pi$
$997$	−18.4051	−0.582896	−0.291448	+	0.956587i	$0.594137\pi$
$998$	−38.7465	−1.22650
$999$	52.6178	1.66475

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5239.2.a.l.1.12		16
13.5	odd	4		403.2.c.b.311.8	✓	32
13.8	odd	4		403.2.c.b.311.25	yes	32
13.12	even	2		5239.2.a.k.1.5		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
403.2.c.b.311.8	✓	32	13.5	odd	4
403.2.c.b.311.25	yes	32	13.8	odd	4
5239.2.a.k.1.5		16	13.12	even	2
5239.2.a.l.1.12		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)

\(f(q)\)	\(=\)	\(q+1.48947 q^{2} +2.99747 q^{3} +0.218531 q^{4} -0.692562 q^{5} +4.46466 q^{6} +1.15260 q^{7} -2.65345 q^{8} +5.98485 q^{9} +O(q^{10})\)	\(q+1.48947 q^{2} +2.99747 q^{3} +0.218531 q^{4} -0.692562 q^{5} +4.46466 q^{6} +1.15260 q^{7} -2.65345 q^{8} +5.98485 q^{9} -1.03155 q^{10} +2.69554 q^{11} +0.655040 q^{12} +1.71676 q^{14} -2.07594 q^{15} -4.38931 q^{16} +4.04263 q^{17} +8.91428 q^{18} +5.06961 q^{19} -0.151346 q^{20} +3.45488 q^{21} +4.01494 q^{22} +1.01940 q^{23} -7.95365 q^{24} -4.52036 q^{25} +8.94703 q^{27} +0.251878 q^{28} -8.56509 q^{29} -3.09205 q^{30} -1.00000 q^{31} -1.23085 q^{32} +8.07982 q^{33} +6.02139 q^{34} -0.798246 q^{35} +1.30787 q^{36} +5.88104 q^{37} +7.55104 q^{38} +1.83768 q^{40} +11.8895 q^{41} +5.14596 q^{42} +3.63935 q^{43} +0.589058 q^{44} -4.14488 q^{45} +1.51836 q^{46} -0.848777 q^{47} -13.1568 q^{48} -5.67152 q^{49} -6.73295 q^{50} +12.1177 q^{51} +1.67652 q^{53} +13.3264 q^{54} -1.86683 q^{55} -3.05836 q^{56} +15.1960 q^{57} -12.7575 q^{58} +1.68042 q^{59} -0.453656 q^{60} +2.00772 q^{61} -1.48947 q^{62} +6.89813 q^{63} +6.94529 q^{64} +12.0347 q^{66} +8.32497 q^{67} +0.883439 q^{68} +3.05562 q^{69} -1.18897 q^{70} +5.15229 q^{71} -15.8805 q^{72} +14.8210 q^{73} +8.75965 q^{74} -13.5497 q^{75} +1.10786 q^{76} +3.10687 q^{77} +0.873238 q^{79} +3.03987 q^{80} +8.86392 q^{81} +17.7091 q^{82} -10.8692 q^{83} +0.754998 q^{84} -2.79977 q^{85} +5.42072 q^{86} -25.6736 q^{87} -7.15249 q^{88} -3.90690 q^{89} -6.17370 q^{90} +0.222769 q^{92} -2.99747 q^{93} -1.26423 q^{94} -3.51102 q^{95} -3.68945 q^{96} -10.3729 q^{97} -8.44757 q^{98} +16.1324 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 4 q^{2} - 2 q^{3} + 18 q^{4} + 4 q^{5} + 6 q^{6} + 2 q^{7} + 12 q^{8} + 10 q^{9}+O(q^{10}) \) 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

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