LMFDB - Embedded newform 5239.2.a.l.1.13

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			1.13
Root			$1.88981$ 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)

$f(q)$	$=$	$q+1.88981 q^{2} -0.327950 q^{3} +1.57140 q^{4} -1.75790 q^{5} -0.619765 q^{6} -0.652358 q^{7} -0.809976 q^{8} -2.89245 q^{9} +O(q^{10})$	\(q+1.88981 q^{2} -0.327950 q^{3} +1.57140 q^{4} -1.75790 q^{5} -0.619765 q^{6} -0.652358 q^{7} -0.809976 q^{8} -2.89245 q^{9} -3.32211 q^{10} -4.04168 q^{11} -0.515341 q^{12} -1.23284 q^{14} +0.576505 q^{15} -4.67350 q^{16} +2.05056 q^{17} -5.46619 q^{18} +7.52193 q^{19} -2.76237 q^{20} +0.213941 q^{21} -7.63802 q^{22} +1.17786 q^{23} +0.265632 q^{24} -1.90978 q^{25} +1.93243 q^{27} -1.02511 q^{28} +6.42057 q^{29} +1.08949 q^{30} -1.00000 q^{31} -7.21210 q^{32} +1.32547 q^{33} +3.87518 q^{34} +1.14678 q^{35} -4.54519 q^{36} +4.69077 q^{37} +14.2150 q^{38} +1.42386 q^{40} -5.09819 q^{41} +0.404309 q^{42} -2.44144 q^{43} -6.35109 q^{44} +5.08465 q^{45} +2.22594 q^{46} +1.20893 q^{47} +1.53268 q^{48} -6.57443 q^{49} -3.60912 q^{50} -0.672482 q^{51} +0.419540 q^{53} +3.65193 q^{54} +7.10488 q^{55} +0.528394 q^{56} -2.46682 q^{57} +12.1337 q^{58} +10.2580 q^{59} +0.905919 q^{60} -12.7823 q^{61} -1.88981 q^{62} +1.88691 q^{63} -4.28253 q^{64} +2.50489 q^{66} +11.5333 q^{67} +3.22225 q^{68} -0.386280 q^{69} +2.16721 q^{70} -6.53421 q^{71} +2.34282 q^{72} -0.170013 q^{73} +8.86469 q^{74} +0.626311 q^{75} +11.8200 q^{76} +2.63662 q^{77} +8.39155 q^{79} +8.21557 q^{80} +8.04360 q^{81} -9.63464 q^{82} +13.1610 q^{83} +0.336186 q^{84} -3.60469 q^{85} -4.61387 q^{86} -2.10563 q^{87} +3.27366 q^{88} +7.61808 q^{89} +9.60904 q^{90} +1.85089 q^{92} +0.327950 q^{93} +2.28465 q^{94} -13.2228 q^{95} +2.36521 q^{96} +18.9945 q^{97} -12.4245 q^{98} +11.6903 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	1.88981	1.33630	0.668150	−	0.744026i	$-0.267086\pi$
$2$	1.88981	1.33630	0.668150	+	0.744026i	$0.267086\pi$
$3$	−0.327950	−0.189342	−0.0946711	−	0.995509i	$-0.530180\pi$
$3$	−0.327950	−0.189342	−0.0946711	+	0.995509i	$0.530180\pi$
$4$	1.57140	0.785699
$5$	−1.75790	−0.786158	−0.393079	−	0.919505i	$-0.628590\pi$
$5$	−1.75790	−0.786158	−0.393079	+	0.919505i	$0.628590\pi$
$6$	−0.619765	−0.253018
$7$	−0.652358	−0.246568	−0.123284	−	0.992371i	$-0.539343\pi$
$7$	−0.652358	−0.246568	−0.123284	+	0.992371i	$0.539343\pi$
$8$	−0.809976	−0.286370
$9$	−2.89245	−0.964150
$10$	−3.32211	−1.05054
$11$	−4.04168	−1.21861	−0.609306	−	0.792935i	$-0.708552\pi$
$11$	−4.04168	−1.21861	−0.609306	+	0.792935i	$0.708552\pi$
$12$	−0.515341	−0.148766
$13$	0	0
$13$	0	0
$14$	−1.23284	−0.329489
$15$	0.576505	0.148853
$16$	−4.67350	−1.16838
$17$	2.05056	0.497334	0.248667	−	0.968589i	$-0.420008\pi$
$17$	2.05056	0.497334	0.248667	+	0.968589i	$0.420008\pi$
$18$	−5.46619	−1.28839
$19$	7.52193	1.72565	0.862824	−	0.505504i	$-0.168693\pi$
$19$	7.52193	1.72565	0.862824	+	0.505504i	$0.168693\pi$
$20$	−2.76237	−0.617684
$21$	0.213941	0.0466857
$22$	−7.63802	−1.62843
$23$	1.17786	0.245601	0.122801	−	0.992431i	$-0.460812\pi$
$23$	1.17786	0.245601	0.122801	+	0.992431i	$0.460812\pi$
$24$	0.265632	0.0542219
$25$	−1.90978	−0.381955
$26$	0	0
$27$	1.93243	0.371896
$28$	−1.02511	−0.193728
$29$	6.42057	1.19227	0.596135	−	0.802884i	$-0.296702\pi$
$29$	6.42057	1.19227	0.596135	+	0.802884i	$0.296702\pi$
$30$	1.08949	0.198912
$31$	−1.00000	−0.179605
$31$	−1.00000	−0.179605
$32$	−7.21210	−1.27493
$33$	1.32547	0.230735
$34$	3.87518	0.664588
$35$	1.14678	0.193842
$36$	−4.54519	−0.757532
$37$	4.69077	0.771158	0.385579	−	0.922675i	$-0.374002\pi$
$37$	4.69077	0.771158	0.385579	+	0.922675i	$0.374002\pi$
$38$	14.2150	2.30599
$39$	0	0
$40$	1.42386	0.225132
$41$	−5.09819	−0.796204	−0.398102	−	0.917341i	$-0.630331\pi$
$41$	−5.09819	−0.796204	−0.398102	+	0.917341i	$0.630331\pi$
$42$	0.404309	0.0623862
$43$	−2.44144	−0.372317	−0.186158	−	0.982520i	$-0.559604\pi$
$43$	−2.44144	−0.372317	−0.186158	+	0.982520i	$0.559604\pi$
$44$	−6.35109	−0.957463
$45$	5.08465	0.757974
$46$	2.22594	0.328197
$47$	1.20893	0.176341	0.0881703	−	0.996105i	$-0.471898\pi$
$47$	1.20893	0.176341	0.0881703	+	0.996105i	$0.471898\pi$
$48$	1.53268	0.221223
$49$	−6.57443	−0.939204
$50$	−3.60912	−0.510407
$51$	−0.672482	−0.0941663
$52$	0	0
$53$	0.419540	0.0576282	0.0288141	−	0.999585i	$-0.490827\pi$
$53$	0.419540	0.0576282	0.0288141	+	0.999585i	$0.490827\pi$
$54$	3.65193	0.496965
$55$	7.10488	0.958022
$56$	0.528394	0.0706097
$57$	−2.46682	−0.326738
$58$	12.1337	1.59323
$59$	10.2580	1.33548	0.667738	−	0.744397i	$-0.267263\pi$
$59$	10.2580	1.33548	0.667738	+	0.744397i	$0.267263\pi$
$60$	0.905919	0.116954
$61$	−12.7823	−1.63661	−0.818303	−	0.574788i	$-0.805085\pi$
$61$	−12.7823	−1.63661	−0.818303	+	0.574788i	$0.805085\pi$
$62$	−1.88981	−0.240007
$63$	1.88691	0.237728
$64$	−4.28253	−0.535316
$65$	0	0
$66$	2.50489	0.308331
$67$	11.5333	1.40902	0.704511	−	0.709693i	$-0.251166\pi$
$67$	11.5333	1.40902	0.704511	+	0.709693i	$0.251166\pi$
$68$	3.22225	0.390755
$69$	−0.386280	−0.0465027
$70$	2.16721	0.259031
$71$	−6.53421	−0.775468	−0.387734	−	0.921771i	$-0.626742\pi$
$71$	−6.53421	−0.775468	−0.387734	+	0.921771i	$0.626742\pi$
$72$	2.34282	0.276103
$73$	−0.170013	−0.0198985	−0.00994926	−	0.999951i	$-0.503167\pi$
$73$	−0.170013	−0.0198985	−0.00994926	+	0.999951i	$0.503167\pi$
$74$	8.86469	1.03050
$75$	0.626311	0.0723202
$76$	11.8200	1.35584
$77$	2.63662	0.300471
$78$	0	0
$79$	8.39155	0.944124	0.472062	−	0.881565i	$-0.343510\pi$
$79$	8.39155	0.944124	0.472062	+	0.881565i	$0.343510\pi$
$80$	8.21557	0.918528
$81$	8.04360	0.893734
$82$	−9.63464	−1.06397
$83$	13.1610	1.44461	0.722304	−	0.691576i	$-0.243083\pi$
$83$	13.1610	1.44461	0.722304	+	0.691576i	$0.243083\pi$
$84$	0.336186	0.0366810
$85$	−3.60469	−0.390983
$86$	−4.61387	−0.497527
$87$	−2.10563	−0.225747
$88$	3.27366	0.348974
$89$	7.61808	0.807515	0.403758	−	0.914866i	$-0.367704\pi$
$89$	7.61808	0.807515	0.403758	+	0.914866i	$0.367704\pi$
$90$	9.60904	1.01288
$91$	0	0
$92$	1.85089	0.192969
$93$	0.327950	0.0340069
$94$	2.28465	0.235644
$95$	−13.2228	−1.35663
$96$	2.36521	0.241398
$97$	18.9945	1.92860	0.964300	−	0.264812i	$-0.0853100\pi$
$97$	18.9945	1.92860	0.964300	+	0.264812i	$0.0853100\pi$
$98$	−12.4245	−1.25506
$99$	11.6903	1.17492
$100$	−3.00102	−0.300102
$101$	12.9992	1.29347	0.646733	−	0.762717i	$-0.276135\pi$
$101$	12.9992	1.29347	0.646733	+	0.762717i	$0.276135\pi$
$102$	−1.27087	−0.125834
$103$	7.58987	0.747852	0.373926	−	0.927458i	$-0.378011\pi$
$103$	7.58987	0.747852	0.373926	+	0.927458i	$0.378011\pi$
$104$	0	0
$105$	−0.376087	−0.0367024
$106$	0.792853	0.0770087
$107$	14.5830	1.40979	0.704895	−	0.709312i	$-0.250994\pi$
$107$	14.5830	1.40979	0.704895	+	0.709312i	$0.250994\pi$
$108$	3.03662	0.292199
$109$	4.51756	0.432703	0.216352	−	0.976315i	$-0.430584\pi$
$109$	4.51756	0.432703	0.216352	+	0.976315i	$0.430584\pi$
$110$	13.4269	1.28021
$111$	−1.53834	−0.146013
$112$	3.04880	0.288084
$113$	−12.3097	−1.15800	−0.578998	−	0.815329i	$-0.696556\pi$
$113$	−12.3097	−1.15800	−0.578998	+	0.815329i	$0.696556\pi$
$114$	−4.66183	−0.436620
$115$	−2.07057	−0.193082
$116$	10.0893	0.936766
$117$	0	0
$118$	19.3857	1.78460
$119$	−1.33770	−0.122627
$120$	−0.466955	−0.0426270
$121$	5.33516	0.485015
$122$	−24.1562	−2.18700
$123$	1.67195	0.150755
$124$	−1.57140	−0.141116
$125$	12.1467	1.08644
$126$	3.56591	0.317677
$127$	−20.8813	−1.85292	−0.926459	−	0.376396i	$-0.877163\pi$
$127$	−20.8813	−1.85292	−0.926459	+	0.376396i	$0.877163\pi$
$128$	6.33102	0.559588
$129$	0.800672	0.0704952
$130$	0	0
$131$	8.64964	0.755723	0.377861	−	0.925862i	$-0.376660\pi$
$131$	8.64964	0.755723	0.377861	+	0.925862i	$0.376660\pi$
$132$	2.08284	0.181288
$133$	−4.90699	−0.425490
$134$	21.7959	1.88288
$135$	−3.39703	−0.292369
$136$	−1.66091	−0.142421
$137$	−14.7125	−1.25697	−0.628486	−	0.777821i	$-0.716325\pi$
$137$	−14.7125	−1.25697	−0.628486	+	0.777821i	$0.716325\pi$
$138$	−0.729998	−0.0621416
$139$	15.5107	1.31560	0.657801	−	0.753192i	$-0.271487\pi$
$139$	15.5107	1.31560	0.657801	+	0.753192i	$0.271487\pi$
$140$	1.80205	0.152301
$141$	−0.396469	−0.0333887
$142$	−12.3484	−1.03626
$143$	0	0
$144$	13.5179	1.12649
$145$	−11.2867	−0.937313
$146$	−0.321293	−0.0265904
$147$	2.15609	0.177831
$148$	7.37108	0.605899
$149$	−1.96195	−0.160729	−0.0803645	−	0.996766i	$-0.525608\pi$
$149$	−1.96195	−0.160729	−0.0803645	+	0.996766i	$0.525608\pi$
$150$	1.18361	0.0966415
$151$	−3.06613	−0.249518	−0.124759	−	0.992187i	$-0.539816\pi$
$151$	−3.06613	−0.249518	−0.124759	+	0.992187i	$0.539816\pi$
$152$	−6.09258	−0.494174
$153$	−5.93114	−0.479504
$154$	4.98272	0.401519
$155$	1.75790	0.141198
$156$	0	0
$157$	9.92590	0.792173	0.396086	−	0.918213i	$-0.370368\pi$
$157$	9.92590	0.792173	0.396086	+	0.918213i	$0.370368\pi$
$158$	15.8585	1.26163
$159$	−0.137588	−0.0109115
$160$	12.6782	1.00230
$161$	−0.768388	−0.0605575
$162$	15.2009	1.19430
$163$	−9.23484	−0.723328	−0.361664	−	0.932308i	$-0.617791\pi$
$163$	−9.23484	−0.723328	−0.361664	+	0.932308i	$0.617791\pi$
$164$	−8.01129	−0.625577
$165$	−2.33005	−0.181394
$166$	24.8718	1.93043
$167$	18.2427	1.41166	0.705832	−	0.708379i	$-0.250573\pi$
$167$	18.2427	1.41166	0.705832	+	0.708379i	$0.250573\pi$
$168$	−0.173287	−0.0133694
$169$	0	0
$170$	−6.81219	−0.522471
$171$	−21.7568	−1.66378
$172$	−3.83648	−0.292529
$173$	0.0577833	0.00439318	0.00219659	−	0.999998i	$-0.499301\pi$
$173$	0.0577833	0.00439318	0.00219659	+	0.999998i	$0.499301\pi$
$174$	−3.97925	−0.301666
$175$	1.24586	0.0941779
$176$	18.8888	1.42380
$177$	−3.36411	−0.252862
$178$	14.3968	1.07908
$179$	−12.4809	−0.932867	−0.466433	−	0.884556i	$-0.654461\pi$
$179$	−12.4809	−0.932867	−0.466433	+	0.884556i	$0.654461\pi$
$180$	7.99001	0.595540
$181$	−9.94653	−0.739319	−0.369660	−	0.929167i	$-0.620526\pi$
$181$	−9.94653	−0.739319	−0.369660	+	0.929167i	$0.620526\pi$
$182$	0	0
$183$	4.19196	0.309878
$184$	−0.954041	−0.0703329
$185$	−8.24593	−0.606253
$186$	0.619765	0.0454434
$187$	−8.28771	−0.606057
$188$	1.89971	0.138551
$189$	−1.26064	−0.0916978
$190$	−24.9887	−1.81287
$191$	−26.3529	−1.90683	−0.953413	−	0.301669i	$-0.902456\pi$
$191$	−26.3529	−1.90683	−0.953413	+	0.301669i	$0.902456\pi$
$192$	1.40446	0.101358
$193$	0.942300	0.0678282	0.0339141	−	0.999425i	$-0.489203\pi$
$193$	0.942300	0.0678282	0.0339141	+	0.999425i	$0.489203\pi$
$194$	35.8961	2.57719
$195$	0	0
$196$	−10.3311	−0.737932
$197$	10.5101	0.748812	0.374406	−	0.927265i	$-0.377847\pi$
$197$	10.5101	0.748812	0.374406	+	0.927265i	$0.377847\pi$
$198$	22.0926	1.57005
$199$	−10.5659	−0.748994	−0.374497	−	0.927228i	$-0.622185\pi$
$199$	−10.5659	−0.748994	−0.374497	+	0.927228i	$0.622185\pi$
$200$	1.54687	0.109380
$201$	−3.78236	−0.266787
$202$	24.5660	1.72846
$203$	−4.18851	−0.293976
$204$	−1.05674	−0.0739864
$205$	8.96213	0.625942
$206$	14.3435	0.999356
$207$	−3.40691	−0.236796
$208$	0	0
$209$	−30.4012	−2.10290
$210$	−0.710736	−0.0490454
$211$	−0.214908	−0.0147949	−0.00739745	−	0.999973i	$-0.502355\pi$
$211$	−0.214908	−0.0147949	−0.00739745	+	0.999973i	$0.502355\pi$
$212$	0.659265	0.0452785
$213$	2.14290	0.146829
$214$	27.5591	1.88390
$215$	4.29182	0.292700
$216$	−1.56522	−0.106500
$217$	0.652358	0.0442849
$218$	8.53734	0.578222
$219$	0.0557558	0.00376763
$220$	11.1646	0.752717
$221$	0	0
$222$	−2.90718	−0.195117
$223$	7.54869	0.505497	0.252749	−	0.967532i	$-0.418665\pi$
$223$	7.54869	0.505497	0.252749	+	0.967532i	$0.418665\pi$
$224$	4.70487	0.314357
$225$	5.52393	0.368262
$226$	−23.2630	−1.54743
$227$	−17.5811	−1.16690	−0.583450	−	0.812149i	$-0.698298\pi$
$227$	−17.5811	−1.16690	−0.583450	+	0.812149i	$0.698298\pi$
$228$	−3.87636	−0.256718
$229$	2.48342	0.164109	0.0820544	−	0.996628i	$-0.473852\pi$
$229$	2.48342	0.164109	0.0820544	+	0.996628i	$0.473852\pi$
$230$	−3.91299	−0.258015
$231$	−0.864680	−0.0568918
$232$	−5.20051	−0.341430
$233$	1.73165	0.113444	0.0567220	−	0.998390i	$-0.481935\pi$
$233$	1.73165	0.113444	0.0567220	+	0.998390i	$0.481935\pi$
$234$	0	0
$235$	−2.12518	−0.138632
$236$	16.1194	1.04928
$237$	−2.75201	−0.178762
$238$	−2.52800	−0.163866
$239$	1.40954	0.0911755	0.0455878	−	0.998960i	$-0.485484\pi$
$239$	1.40954	0.0911755	0.0455878	+	0.998960i	$0.485484\pi$
$240$	−2.69430	−0.173916
$241$	−0.348571	−0.0224534	−0.0112267	−	0.999937i	$-0.503574\pi$
$241$	−0.348571	−0.0224534	−0.0112267	+	0.999937i	$0.503574\pi$
$242$	10.0825	0.648125
$243$	−8.43519	−0.541118
$244$	−20.0861	−1.28588
$245$	11.5572	0.738363
$246$	3.15968	0.201454
$247$	0	0
$248$	0.809976	0.0514336
$249$	−4.31615	−0.273525
$250$	22.9550	1.45180
$251$	7.54455	0.476208	0.238104	−	0.971240i	$-0.423474\pi$
$251$	7.54455	0.476208	0.238104	+	0.971240i	$0.423474\pi$
$252$	2.96509	0.186783
$253$	−4.76054	−0.299293
$254$	−39.4618	−2.47606
$255$	1.18216	0.0740296
$256$	20.5295	1.28309
$257$	22.3905	1.39668	0.698341	−	0.715765i	$-0.253922\pi$
$257$	22.3905	1.39668	0.698341	+	0.715765i	$0.253922\pi$
$258$	1.51312	0.0942028
$259$	−3.06006	−0.190143
$260$	0	0
$261$	−18.5712	−1.14953
$262$	16.3462	1.00987
$263$	19.5006	1.20246	0.601230	−	0.799076i	$-0.294678\pi$
$263$	19.5006	1.20246	0.601230	+	0.799076i	$0.294678\pi$
$264$	−1.07360	−0.0660755
$265$	−0.737511	−0.0453049
$266$	−9.27330	−0.568582
$267$	−2.49835	−0.152897
$268$	18.1235	1.10707
$269$	22.4446	1.36847	0.684236	−	0.729261i	$-0.260136\pi$
$269$	22.4446	1.36847	0.684236	+	0.729261i	$0.260136\pi$
$270$	−6.41975	−0.390693
$271$	−19.0371	−1.15642	−0.578211	−	0.815887i	$-0.696249\pi$
$271$	−19.0371	−1.15642	−0.578211	+	0.815887i	$0.696249\pi$
$272$	−9.58330	−0.581073
$273$	0	0
$274$	−27.8039	−1.67969
$275$	7.71870	0.465455
$276$	−0.607001	−0.0365372
$277$	−3.01871	−0.181377	−0.0906883	−	0.995879i	$-0.528907\pi$
$277$	−3.01871	−0.181377	−0.0906883	+	0.995879i	$0.528907\pi$
$278$	29.3124	1.75804
$279$	2.89245	0.173166
$280$	−0.928866	−0.0555104
$281$	−28.7820	−1.71699	−0.858496	−	0.512820i	$-0.828601\pi$
$281$	−28.7820	−1.71699	−0.858496	+	0.512820i	$0.828601\pi$
$282$	−0.749253	−0.0446174
$283$	1.28804	0.0765662	0.0382831	−	0.999267i	$-0.487811\pi$
$283$	1.28804	0.0765662	0.0382831	+	0.999267i	$0.487811\pi$
$284$	−10.2678	−0.609285
$285$	4.33643	0.256868
$286$	0	0
$287$	3.32584	0.196318
$288$	20.8606	1.22922
$289$	−12.7952	−0.752659
$290$	−21.3298	−1.25253
$291$	−6.22925	−0.365165
$292$	−0.267158	−0.0156343
$293$	−21.1020	−1.23279	−0.616397	−	0.787436i	$-0.711408\pi$
$293$	−21.1020	−1.23279	−0.616397	+	0.787436i	$0.711408\pi$
$294$	4.07460	0.237636
$295$	−18.0325	−1.04990
$296$	−3.79942	−0.220837
$297$	−7.81026	−0.453197
$298$	−3.70772	−0.214782
$299$	0	0
$300$	0.984185	0.0568219
$301$	1.59269	0.0918014
$302$	−5.79442	−0.333431
$303$	−4.26308	−0.244908
$304$	−35.1538	−2.01621
$305$	22.4700	1.28663
$306$	−11.2088	−0.640762
$307$	2.44457	0.139519	0.0697595	−	0.997564i	$-0.477777\pi$
$307$	2.44457	0.139519	0.0697595	+	0.997564i	$0.477777\pi$
$308$	4.14318	0.236080
$309$	−2.48910	−0.141600
$310$	3.32211	0.188683
$311$	−19.4882	−1.10508	−0.552538	−	0.833487i	$-0.686341\pi$
$311$	−19.4882	−1.10508	−0.552538	+	0.833487i	$0.686341\pi$
$312$	0	0
$313$	17.7760	1.00476	0.502380	−	0.864647i	$-0.332458\pi$
$313$	17.7760	1.00476	0.502380	+	0.864647i	$0.332458\pi$
$314$	18.7581	1.05858
$315$	−3.31701	−0.186892
$316$	13.1865	0.741797
$317$	−2.03343	−0.114209	−0.0571044	−	0.998368i	$-0.518187\pi$
$317$	−2.03343	−0.114209	−0.0571044	+	0.998368i	$0.518187\pi$
$318$	−0.260016	−0.0145810
$319$	−25.9499	−1.45291
$320$	7.52827	0.420843
$321$	−4.78249	−0.266933
$322$	−1.45211	−0.0809230
$323$	15.4242	0.858224
$324$	12.6397	0.702206
$325$	0	0
$326$	−17.4521	−0.966584
$327$	−1.48153	−0.0819290
$328$	4.12941	0.228009
$329$	−0.788655	−0.0434799
$330$	−4.40336	−0.242397
$331$	−18.5709	−1.02075	−0.510376	−	0.859952i	$-0.670494\pi$
$331$	−18.5709	−1.02075	−0.510376	+	0.859952i	$0.670494\pi$
$332$	20.6812	1.13503
$333$	−13.5678	−0.743512
$334$	34.4754	1.88641
$335$	−20.2745	−1.10771
$336$	−0.999853	−0.0545465
$337$	−33.2525	−1.81138	−0.905689	−	0.423942i	$-0.860646\pi$
$337$	−33.2525	−1.81138	−0.905689	+	0.423942i	$0.860646\pi$
$338$	0	0
$339$	4.03696	0.219258
$340$	−5.66440	−0.307195
$341$	4.04168	0.218869
$342$	−41.1163	−2.22331
$343$	8.85538	0.478146
$344$	1.97751	0.106620
$345$	0.679044	0.0365585
$346$	0.109200	0.00587061
$347$	20.9692	1.12569	0.562843	−	0.826564i	$-0.309708\pi$
$347$	20.9692	1.12569	0.562843	+	0.826564i	$0.309708\pi$
$348$	−3.30878	−0.177369
$349$	30.4835	1.63175	0.815873	−	0.578231i	$-0.196257\pi$
$349$	30.4835	1.63175	0.815873	+	0.578231i	$0.196257\pi$
$350$	2.35444	0.125850
$351$	0	0
$352$	29.1490	1.55365
$353$	10.5824	0.563243	0.281621	−	0.959526i	$-0.409128\pi$
$353$	10.5824	0.563243	0.281621	+	0.959526i	$0.409128\pi$
$354$	−6.35754	−0.337899
$355$	11.4865	0.609640
$356$	11.9710	0.634464
$357$	0.438699	0.0232184
$358$	−23.5866	−1.24659
$359$	21.8192	1.15157	0.575786	−	0.817600i	$-0.304696\pi$
$359$	21.8192	1.15157	0.575786	+	0.817600i	$0.304696\pi$
$360$	−4.11844	−0.217061
$361$	37.5794	1.97786
$362$	−18.7971	−0.987953
$363$	−1.74967	−0.0918337
$364$	0	0
$365$	0.298866	0.0156434
$366$	7.92202	0.414091
$367$	−20.1551	−1.05209	−0.526044	−	0.850457i	$-0.676325\pi$
$367$	−20.1551	−1.05209	−0.526044	+	0.850457i	$0.676325\pi$
$368$	−5.50475	−0.286955
$369$	14.7463	0.767659
$370$	−15.5833	−0.810136
$371$	−0.273690	−0.0142093
$372$	0.515341	0.0267192
$373$	−29.0660	−1.50498	−0.752491	−	0.658602i	$-0.771148\pi$
$373$	−29.0660	−1.50498	−0.752491	+	0.658602i	$0.771148\pi$
$374$	−15.6622	−0.809874
$375$	−3.98352	−0.205708
$376$	−0.979205	−0.0504986
$377$	0	0
$378$	−2.38237	−0.122536
$379$	12.4985	0.642003	0.321002	−	0.947079i	$-0.395981\pi$
$379$	12.4985	0.642003	0.321002	+	0.947079i	$0.395981\pi$
$380$	−20.7783	−1.06591
$381$	6.84804	0.350835
$382$	−49.8020	−2.54809
$383$	5.69126	0.290810	0.145405	−	0.989372i	$-0.453551\pi$
$383$	5.69126	0.290810	0.145405	+	0.989372i	$0.453551\pi$
$384$	−2.07626	−0.105954
$385$	−4.63492	−0.236218
$386$	1.78077	0.0906389
$387$	7.06175	0.358969
$388$	29.8479	1.51530
$389$	−23.5136	−1.19219	−0.596093	−	0.802915i	$-0.703281\pi$
$389$	−23.5136	−1.19219	−0.596093	+	0.802915i	$0.703281\pi$
$390$	0	0
$391$	2.41528	0.122146
$392$	5.32513	0.268960
$393$	−2.83665	−0.143090
$394$	19.8621	1.00064
$395$	−14.7515	−0.742231
$396$	18.3702	0.923137
$397$	8.32867	0.418004	0.209002	−	0.977915i	$-0.432979\pi$
$397$	8.32867	0.418004	0.209002	+	0.977915i	$0.432979\pi$
$398$	−19.9675	−1.00088
$399$	1.60925	0.0805632
$400$	8.92534	0.446267
$401$	32.6105	1.62849	0.814246	−	0.580520i	$-0.197151\pi$
$401$	32.6105	1.62849	0.814246	+	0.580520i	$0.197151\pi$
$402$	−7.14797	−0.356508
$403$	0	0
$404$	20.4269	1.01628
$405$	−14.1399	−0.702616
$406$	−7.91550	−0.392840
$407$	−18.9586	−0.939743
$408$	0.544694	0.0269664
$409$	−22.2951	−1.10242	−0.551211	−	0.834366i	$-0.685834\pi$
$409$	−22.2951	−1.10242	−0.551211	+	0.834366i	$0.685834\pi$
$410$	16.9368	0.836447
$411$	4.82496	0.237998
$412$	11.9267	0.587587
$413$	−6.69187	−0.329286
$414$	−6.43842	−0.316431
$415$	−23.1358	−1.13569
$416$	0	0
$417$	−5.08674	−0.249099
$418$	−57.4527	−2.81010
$419$	−30.9395	−1.51149	−0.755747	−	0.654864i	$-0.772726\pi$
$419$	−30.9395	−1.51149	−0.755747	+	0.654864i	$0.772726\pi$
$420$	−0.590983	−0.0288370
$421$	5.10402	0.248754	0.124377	−	0.992235i	$-0.460307\pi$
$421$	5.10402	0.248754	0.124377	+	0.992235i	$0.460307\pi$
$422$	−0.406137	−0.0197704
$423$	−3.49677	−0.170019
$424$	−0.339817	−0.0165030
$425$	−3.91611	−0.189959
$426$	4.04967	0.196207
$427$	8.33863	0.403534
$428$	22.9157	1.10767
$429$	0	0
$430$	8.11075	0.391135
$431$	24.6933	1.18944	0.594718	−	0.803935i	$-0.297264\pi$
$431$	24.6933	1.18944	0.594718	+	0.803935i	$0.297264\pi$
$432$	−9.03122	−0.434515
$433$	4.59601	0.220870	0.110435	−	0.993883i	$-0.464776\pi$
$433$	4.59601	0.220870	0.110435	+	0.993883i	$0.464776\pi$
$434$	1.23284	0.0591780
$435$	3.70149	0.177473
$436$	7.09888	0.339975
$437$	8.85980	0.423822
$438$	0.105368	0.00503468
$439$	22.3774	1.06801	0.534007	−	0.845480i	$-0.320686\pi$
$439$	22.3774	1.06801	0.534007	+	0.845480i	$0.320686\pi$
$440$	−5.75479	−0.274349
$441$	19.0162	0.905533
$442$	0	0
$443$	27.0807	1.28664	0.643322	−	0.765596i	$-0.277556\pi$
$443$	27.0807	1.28664	0.643322	+	0.765596i	$0.277556\pi$
$444$	−2.41735	−0.114722
$445$	−13.3919	−0.634835
$446$	14.2656	0.675497
$447$	0.643422	0.0304328
$448$	2.79374	0.131992
$449$	33.4114	1.57678	0.788391	−	0.615175i	$-0.210915\pi$
$449$	33.4114	1.57678	0.788391	+	0.615175i	$0.210915\pi$
$450$	10.4392	0.492108
$451$	20.6052	0.970263
$452$	−19.3434	−0.909837
$453$	1.00554	0.0472443
$454$	−33.2251	−1.55933
$455$	0	0
$456$	1.99806	0.0935680
$457$	25.7519	1.20462	0.602311	−	0.798262i	$-0.294247\pi$
$457$	25.7519	1.20462	0.602311	+	0.798262i	$0.294247\pi$
$458$	4.69320	0.219299
$459$	3.96256	0.184957
$460$	−3.25369	−0.151704
$461$	7.17831	0.334327	0.167164	−	0.985929i	$-0.446539\pi$
$461$	7.17831	0.334327	0.167164	+	0.985929i	$0.446539\pi$
$462$	−1.63409	−0.0760245
$463$	2.14846	0.0998475	0.0499238	−	0.998753i	$-0.484102\pi$
$463$	2.14846	0.0998475	0.0499238	+	0.998753i	$0.484102\pi$
$464$	−30.0065	−1.39302
$465$	−0.576505	−0.0267348
$466$	3.27249	0.151595
$467$	7.19142	0.332779	0.166390	−	0.986060i	$-0.446789\pi$
$467$	7.19142	0.332779	0.166390	+	0.986060i	$0.446789\pi$
$468$	0	0
$469$	−7.52387	−0.347420
$470$	−4.01620	−0.185254
$471$	−3.25520	−0.149992
$472$	−8.30872	−0.382440
$473$	9.86753	0.453709
$474$	−5.20079	−0.238880
$475$	−14.3652	−0.659120
$476$	−2.10206	−0.0963477
$477$	−1.21350	−0.0555622
$478$	2.66377	0.121838
$479$	25.8734	1.18219	0.591094	−	0.806603i	$-0.298696\pi$
$479$	25.8734	1.18219	0.591094	+	0.806603i	$0.298696\pi$
$480$	−4.15781	−0.189777
$481$	0	0
$482$	−0.658734	−0.0300045
$483$	0.251993	0.0114661
$484$	8.38367	0.381076
$485$	−33.3905	−1.51618
$486$	−15.9410	−0.723096
$487$	1.91760	0.0868946	0.0434473	−	0.999056i	$-0.486166\pi$
$487$	1.91760	0.0868946	0.0434473	+	0.999056i	$0.486166\pi$
$488$	10.3534	0.468674
$489$	3.02857	0.136957
$490$	21.8410	0.986675
$491$	−5.36083	−0.241931	−0.120965	−	0.992657i	$-0.538599\pi$
$491$	−5.36083	−0.241931	−0.120965	+	0.992657i	$0.538599\pi$
$492$	2.62731	0.118448
$493$	13.1658	0.592956
$494$	0	0
$495$	−20.5505	−0.923676
$496$	4.67350	0.209846
$497$	4.26264	0.191206
$498$	−8.15673	−0.365512
$499$	22.3837	1.00203	0.501016	−	0.865438i	$-0.332960\pi$
$499$	22.3837	1.00203	0.501016	+	0.865438i	$0.332960\pi$
$500$	19.0873	0.853612
$501$	−5.98271	−0.267288
$502$	14.2578	0.636357
$503$	1.51578	0.0675852	0.0337926	−	0.999429i	$-0.489241\pi$
$503$	1.51578	0.0675852	0.0337926	+	0.999429i	$0.489241\pi$
$504$	−1.52835	−0.0680783
$505$	−22.8513	−1.01687
$506$	−8.99654	−0.399945
$507$	0	0
$508$	−32.8129	−1.45584
$509$	−15.8794	−0.703842	−0.351921	−	0.936030i	$-0.614471\pi$
$509$	−15.8794	−0.703842	−0.351921	+	0.936030i	$0.614471\pi$
$510$	2.23406	0.0989258
$511$	0.110909	0.00490634
$512$	26.1349	1.15501
$513$	14.5356	0.641762
$514$	42.3139	1.86639
$515$	−13.3423	−0.587930
$516$	1.25817	0.0553881
$517$	−4.88611	−0.214891
$518$	−5.78295	−0.254088
$519$	−0.0189500	−0.000831815 0
$520$	0	0
$521$	−15.5542	−0.681440	−0.340720	−	0.940165i	$-0.610671\pi$
$521$	−15.5542	−0.681440	−0.340720	+	0.940165i	$0.610671\pi$
$522$	−35.0961	−1.53611
$523$	−33.1558	−1.44980	−0.724901	−	0.688853i	$-0.758115\pi$
$523$	−33.1558	−1.44980	−0.724901	+	0.688853i	$0.758115\pi$
$524$	13.5920	0.593771
$525$	−0.408579	−0.0178319
$526$	36.8525	1.60685
$527$	−2.05056	−0.0893238
$528$	−6.19459	−0.269585
$529$	−21.6126	−0.939680
$530$	−1.39376	−0.0605410
$531$	−29.6707	−1.28760
$532$	−7.71084	−0.334307
$533$	0	0
$534$	−4.72142	−0.204316
$535$	−25.6355	−1.10832
$536$	−9.34174	−0.403502
$537$	4.09312	0.176631
$538$	42.4161	1.82869
$539$	26.5717	1.14453
$540$	−5.33808	−0.229715
$541$	38.6097	1.65996	0.829981	−	0.557792i	$-0.188351\pi$
$541$	38.6097	1.65996	0.829981	+	0.557792i	$0.188351\pi$
$542$	−35.9766	−1.54533
$543$	3.26197	0.139984
$544$	−14.7889	−0.634067
$545$	−7.94143	−0.340173
$546$	0	0
$547$	−9.56630	−0.409026	−0.204513	−	0.978864i	$-0.565561\pi$
$547$	−9.56630	−0.409026	−0.204513	+	0.978864i	$0.565561\pi$
$548$	−23.1192	−0.987602
$549$	36.9721	1.57793
$550$	14.5869	0.621988
$551$	48.2951	2.05744
$552$	0.312878	0.0133170
$553$	−5.47430	−0.232791
$554$	−5.70480	−0.242374
$555$	2.70425	0.114789
$556$	24.3735	1.03367
$557$	6.45936	0.273692	0.136846	−	0.990592i	$-0.456303\pi$
$557$	6.45936	0.273692	0.136846	+	0.990592i	$0.456303\pi$
$558$	5.46619	0.231402
$559$	0	0
$560$	−5.35949	−0.226480
$561$	2.71796	0.114752
$562$	−54.3927	−2.29442
$563$	−3.68120	−0.155144	−0.0775721	−	0.996987i	$-0.524717\pi$
$563$	−3.68120	−0.155144	−0.0775721	+	0.996987i	$0.524717\pi$
$564$	−0.623011	−0.0262335
$565$	21.6392	0.910368
$566$	2.43416	0.102315
$567$	−5.24731	−0.220366
$568$	5.29255	0.222071
$569$	27.0122	1.13241	0.566205	−	0.824265i	$-0.308411\pi$
$569$	27.0122	1.13241	0.566205	+	0.824265i	$0.308411\pi$
$570$	8.19505	0.343253
$571$	−26.6613	−1.11574	−0.557870	−	0.829928i	$-0.688381\pi$
$571$	−26.6613	−1.11574	−0.557870	+	0.829928i	$0.688381\pi$
$572$	0	0
$573$	8.64242	0.361042
$574$	6.28523	0.262340
$575$	−2.24945	−0.0938087
$576$	12.3870	0.516125
$577$	6.72496	0.279964	0.139982	−	0.990154i	$-0.455296\pi$
$577$	6.72496	0.279964	0.139982	+	0.990154i	$0.455296\pi$
$578$	−24.1806	−1.00578
$579$	−0.309027	−0.0128427
$580$	−17.7360	−0.736446
$581$	−8.58568	−0.356194
$582$	−11.7721	−0.487971
$583$	−1.69565	−0.0702265
$584$	0.137707	0.00569834
$585$	0	0
$586$	−39.8789	−1.64738
$587$	28.7519	1.18672	0.593359	−	0.804938i	$-0.297801\pi$
$587$	28.7519	1.18672	0.593359	+	0.804938i	$0.297801\pi$
$588$	3.38807	0.139722
$589$	−7.52193	−0.309936
$590$	−34.0782	−1.40298
$591$	−3.44678	−0.141782
$592$	−21.9223	−0.901003
$593$	1.97573	0.0811335	0.0405668	−	0.999177i	$-0.487084\pi$
$593$	1.97573	0.0811335	0.0405668	+	0.999177i	$0.487084\pi$
$594$	−14.7599	−0.605608
$595$	2.35155	0.0964040
$596$	−3.08300	−0.126285
$597$	3.46508	0.141816
$598$	0	0
$599$	7.67011	0.313392	0.156696	−	0.987647i	$-0.449916\pi$
$599$	7.67011	0.313392	0.156696	+	0.987647i	$0.449916\pi$
$600$	−0.507297	−0.0207103
$601$	−34.0529	−1.38905	−0.694523	−	0.719470i	$-0.744385\pi$
$601$	−34.0529	−1.38905	−0.694523	+	0.719470i	$0.744385\pi$
$602$	3.00990	0.122674
$603$	−33.3596	−1.35851
$604$	−4.81811	−0.196046
$605$	−9.37870	−0.381298
$606$	−8.05643	−0.327270
$607$	32.5479	1.32108	0.660540	−	0.750791i	$-0.270328\pi$
$607$	32.5479	1.32108	0.660540	+	0.750791i	$0.270328\pi$
$608$	−54.2489	−2.20008
$609$	1.37362	0.0556620
$610$	42.4642	1.71933
$611$	0	0
$612$	−9.32019	−0.376746
$613$	−17.8597	−0.721348	−0.360674	−	0.932692i	$-0.617453\pi$
$613$	−17.8597	−0.721348	−0.360674	+	0.932692i	$0.617453\pi$
$614$	4.61978	0.186439
$615$	−2.93913	−0.118517
$616$	−2.13560	−0.0860458
$617$	−14.9164	−0.600511	−0.300255	−	0.953859i	$-0.597072\pi$
$617$	−14.9164	−0.600511	−0.300255	+	0.953859i	$0.597072\pi$
$618$	−4.70394	−0.189220
$619$	12.1473	0.488242	0.244121	−	0.969745i	$-0.421501\pi$
$619$	12.1473	0.488242	0.244121	+	0.969745i	$0.421501\pi$
$620$	2.76237	0.110939
$621$	2.27614	0.0913383
$622$	−36.8292	−1.47671
$623$	−4.96971	−0.199107
$624$	0	0
$625$	−11.8039	−0.472155
$626$	33.5934	1.34266
$627$	9.97009	0.398167
$628$	15.5975	0.622410
$629$	9.61872	0.383523
$630$	−6.26853	−0.249744
$631$	18.8097	0.748800	0.374400	−	0.927267i	$-0.377849\pi$
$631$	18.8097	0.748800	0.374400	+	0.927267i	$0.377849\pi$
$632$	−6.79696	−0.270369
$633$	0.0704793	0.00280130
$634$	−3.84281	−0.152617
$635$	36.7073	1.45669
$636$	−0.216206	−0.00857313
$637$	0	0
$638$	−49.0404	−1.94153
$639$	18.8999	0.747667
$640$	−11.1293	−0.439925
$641$	2.35909	0.0931783	0.0465892	−	0.998914i	$-0.485165\pi$
$641$	2.35909	0.0931783	0.0465892	+	0.998914i	$0.485165\pi$
$642$	−9.03802	−0.356702
$643$	−17.9916	−0.709518	−0.354759	−	0.934958i	$-0.615437\pi$
$643$	−17.9916	−0.709518	−0.354759	+	0.934958i	$0.615437\pi$
$644$	−1.20744	−0.0475800
$645$	−1.40750	−0.0554204
$646$	29.1488	1.14684
$647$	43.8760	1.72494	0.862472	−	0.506105i	$-0.168915\pi$
$647$	43.8760	1.72494	0.862472	+	0.506105i	$0.168915\pi$
$648$	−6.51513	−0.255938
$649$	−41.4595	−1.62743
$650$	0	0
$651$	−0.213941	−0.00838500
$652$	−14.5116	−0.568319
$653$	−0.256698	−0.0100454	−0.00502269	−	0.999987i	$-0.501599\pi$
$653$	−0.256698	−0.0100454	−0.00502269	+	0.999987i	$0.501599\pi$
$654$	−2.79982	−0.109482
$655$	−15.2052	−0.594118
$656$	23.8264	0.930265
$657$	0.491754	0.0191851
$658$	−1.49041	−0.0581023
$659$	8.42299	0.328113	0.164056	−	0.986451i	$-0.447542\pi$
$659$	8.42299	0.328113	0.164056	+	0.986451i	$0.447542\pi$
$660$	−3.66143	−0.142521
$661$	11.7394	0.456609	0.228304	−	0.973590i	$-0.426682\pi$
$661$	11.7394	0.456609	0.228304	+	0.973590i	$0.426682\pi$
$662$	−35.0956	−1.36403
$663$	0	0
$664$	−10.6601	−0.413692
$665$	8.62601	0.334502
$666$	−25.6407	−0.993556
$667$	7.56255	0.292823
$668$	28.6666	1.10914
$669$	−2.47559	−0.0957120
$670$	−38.3151	−1.48024
$671$	51.6619	1.99439
$672$	−1.54296	−0.0595211
$673$	17.2708	0.665742	0.332871	−	0.942972i	$-0.391983\pi$
$673$	17.2708	0.665742	0.332871	+	0.942972i	$0.391983\pi$
$674$	−62.8410	−2.42055
$675$	−3.69051	−0.142048
$676$	0	0
$677$	14.1990	0.545713	0.272857	−	0.962055i	$-0.412032\pi$
$677$	14.1990	0.545713	0.272857	+	0.962055i	$0.412032\pi$
$678$	7.62910	0.292994
$679$	−12.3912	−0.475531
$680$	2.91971	0.111966
$681$	5.76574	0.220943
$682$	7.63802	0.292475
$683$	10.1038	0.386611	0.193305	−	0.981139i	$-0.438079\pi$
$683$	10.1038	0.386611	0.193305	+	0.981139i	$0.438079\pi$
$684$	−34.1886	−1.30723
$685$	25.8631	0.988179
$686$	16.7350	0.638947
$687$	−0.814437	−0.0310727
$688$	11.4101	0.435006
$689$	0	0
$690$	1.28327	0.0488531
$691$	−9.41120	−0.358019	−0.179009	−	0.983847i	$-0.557289\pi$
$691$	−9.41120	−0.358019	−0.179009	+	0.983847i	$0.557289\pi$
$692$	0.0908006	0.00345172
$693$	−7.62629	−0.289699
$694$	39.6279	1.50426
$695$	−27.2663	−1.03427
$696$	1.70551	0.0646471
$697$	−10.4542	−0.395979
$698$	57.6082	2.18050
$699$	−0.567894	−0.0214797
$700$	1.95774	0.0739955
$701$	22.3912	0.845703	0.422852	−	0.906199i	$-0.361029\pi$
$701$	22.3912	0.845703	0.422852	+	0.906199i	$0.361029\pi$
$702$	0	0
$703$	35.2837	1.33075
$704$	17.3086	0.652342
$705$	0.696954	0.0262488
$706$	19.9987	0.752662
$707$	−8.48011	−0.318927
$708$	−5.28636	−0.198673
$709$	20.3716	0.765072	0.382536	−	0.923941i	$-0.375051\pi$
$709$	20.3716	0.765072	0.382536	+	0.923941i	$0.375051\pi$
$710$	21.7074	0.814663
$711$	−24.2721	−0.910276
$712$	−6.17047	−0.231248
$713$	−1.17786	−0.0441113
$714$	0.829059	0.0310268
$715$	0	0
$716$	−19.6125	−0.732953
$717$	−0.462259	−0.0172634
$718$	41.2342	1.53885
$719$	−14.9478	−0.557459	−0.278730	−	0.960370i	$-0.589913\pi$
$719$	−14.9478	−0.557459	−0.278730	+	0.960370i	$0.589913\pi$
$720$	−23.7631	−0.885599
$721$	−4.95131	−0.184396
$722$	71.0181	2.64302
$723$	0.114314	0.00425138
$724$	−15.6300	−0.580883
$725$	−12.2618	−0.455393
$726$	−3.30655	−0.122717
$727$	−27.0908	−1.00474	−0.502372	−	0.864652i	$-0.667539\pi$
$727$	−27.0908	−1.00474	−0.502372	+	0.864652i	$0.667539\pi$
$728$	0	0
$729$	−21.3645	−0.791277
$730$	0.564802	0.0209043
$731$	−5.00633	−0.185166
$732$	6.58724	0.243471
$733$	−16.9857	−0.627382	−0.313691	−	0.949525i	$-0.601566\pi$
$733$	−16.9857	−0.627382	−0.313691	+	0.949525i	$0.601566\pi$
$734$	−38.0894	−1.40591
$735$	−3.79019	−0.139803
$736$	−8.49487	−0.313125
$737$	−46.6141	−1.71705
$738$	27.8677	1.02582
$739$	4.03178	0.148311	0.0741557	−	0.997247i	$-0.476374\pi$
$739$	4.03178	0.148311	0.0741557	+	0.997247i	$0.476374\pi$
$740$	−12.9576	−0.476332
$741$	0	0
$742$	−0.517224	−0.0189879
$743$	5.56680	0.204226	0.102113	−	0.994773i	$-0.467440\pi$
$743$	5.56680	0.204226	0.102113	+	0.994773i	$0.467440\pi$
$744$	−0.265632	−0.00973854
$745$	3.44892	0.126359
$746$	−54.9294	−2.01111
$747$	−38.0675	−1.39282
$748$	−13.0233	−0.476179
$749$	−9.51332	−0.347609
$750$	−7.52811	−0.274888
$751$	2.33856	0.0853354	0.0426677	−	0.999089i	$-0.486414\pi$
$751$	2.33856	0.0853354	0.0426677	+	0.999089i	$0.486414\pi$
$752$	−5.64994	−0.206032
$753$	−2.47424	−0.0901662
$754$	0	0
$755$	5.38996	0.196161
$756$	−1.98096	−0.0720469
$757$	8.27850	0.300887	0.150444	−	0.988619i	$-0.451930\pi$
$757$	8.27850	0.300887	0.150444	+	0.988619i	$0.451930\pi$
$758$	23.6198	0.857909
$759$	1.56122	0.0566687
$760$	10.7102	0.388499
$761$	26.7917	0.971199	0.485600	−	0.874181i	$-0.338601\pi$
$761$	26.7917	0.971199	0.485600	+	0.874181i	$0.338601\pi$
$762$	12.9415	0.468822
$763$	−2.94706	−0.106691
$764$	−41.4108	−1.49819
$765$	10.4264	0.376966
$766$	10.7554	0.388610
$767$	0	0
$768$	−6.73266	−0.242944
$769$	−43.5055	−1.56885	−0.784425	−	0.620224i	$-0.787042\pi$
$769$	−43.5055	−1.56885	−0.784425	+	0.620224i	$0.787042\pi$
$770$	−8.75915	−0.315658
$771$	−7.34298	−0.264451
$772$	1.48073	0.0532926
$773$	34.3969	1.23717	0.618585	−	0.785718i	$-0.287706\pi$
$773$	34.3969	1.23717	0.618585	+	0.785718i	$0.287706\pi$
$774$	13.3454	0.479690
$775$	1.90978	0.0686012
$776$	−15.3851	−0.552293
$777$	1.00355	0.0360021
$778$	−44.4363	−1.59312
$779$	−38.3482	−1.37397
$780$	0	0
$781$	26.4092	0.944994
$782$	4.56443	0.163224
$783$	12.4073	0.443401
$784$	30.7256	1.09734
$785$	−17.4488	−0.622773
$786$	−5.36075	−0.191212
$787$	−21.6617	−0.772155	−0.386077	−	0.922466i	$-0.626170\pi$
$787$	−21.6617	−0.772155	−0.386077	+	0.922466i	$0.626170\pi$
$788$	16.5155	0.588341
$789$	−6.39523	−0.227676
$790$	−27.8777	−0.991843
$791$	8.03031	0.285525
$792$	−9.46890	−0.336463
$793$	0	0
$794$	15.7396	0.558579
$795$	0.241867	0.00857813
$796$	−16.6032	−0.588484
$797$	−52.2693	−1.85147	−0.925736	−	0.378170i	$-0.876553\pi$
$797$	−52.2693	−1.85147	−0.925736	+	0.378170i	$0.876553\pi$
$798$	3.04118	0.107657
$799$	2.47898	0.0877002
$800$	13.7735	0.486967
$801$	−22.0349	−0.778565
$802$	61.6278	2.17615
$803$	0.687138	0.0242486
$804$	−5.94360	−0.209615
$805$	1.35075	0.0476078
$806$	0	0
$807$	−7.36071	−0.259109
$808$	−10.5290	−0.370410
$809$	29.6179	1.04131	0.520654	−	0.853768i	$-0.325688\pi$
$809$	29.6179	1.04131	0.520654	+	0.853768i	$0.325688\pi$
$810$	−26.7218	−0.938907
$811$	−0.0929122	−0.00326259	−0.00163129	−	0.999999i	$-0.500519\pi$
$811$	−0.0929122	−0.00326259	−0.00163129	+	0.999999i	$0.500519\pi$
$812$	−6.58182	−0.230976
$813$	6.24323	0.218960
$814$	−35.8282	−1.25578
$815$	16.2339	0.568650
$816$	3.14285	0.110022
$817$	−18.3644	−0.642488
$818$	−42.1336	−1.47317
$819$	0	0
$820$	14.0831	0.491802
$821$	−22.6436	−0.790267	−0.395133	−	0.918624i	$-0.629302\pi$
$821$	−22.6436	−0.790267	−0.395133	+	0.918624i	$0.629302\pi$
$822$	9.11828	0.318037
$823$	43.4443	1.51437	0.757187	−	0.653199i	$-0.226573\pi$
$823$	43.4443	1.51437	0.757187	+	0.653199i	$0.226573\pi$
$824$	−6.14762	−0.214162
$825$	−2.53135	−0.0881302
$826$	−12.6464	−0.440025
$827$	32.1031	1.11633	0.558167	−	0.829728i	$-0.311505\pi$
$827$	32.1031	1.11633	0.558167	+	0.829728i	$0.311505\pi$
$828$	−5.35361	−0.186051
$829$	−48.9477	−1.70002	−0.850011	−	0.526765i	$-0.823405\pi$
$829$	−48.9477	−1.70002	−0.850011	+	0.526765i	$0.823405\pi$
$830$	−43.7223	−1.51762
$831$	0.989986	0.0343422
$832$	0	0
$833$	−13.4813	−0.467098
$834$	−9.61300	−0.332871
$835$	−32.0690	−1.10979
$836$	−47.7724	−1.65224
$837$	−1.93243	−0.0667946
$838$	−58.4699	−2.01981
$839$	−26.2951	−0.907808	−0.453904	−	0.891051i	$-0.649969\pi$
$839$	−26.2951	−0.907808	−0.453904	+	0.891051i	$0.649969\pi$
$840$	0.304622	0.0105105
$841$	12.2237	0.421507
$842$	9.64564	0.332411
$843$	9.43908	0.325099
$844$	−0.337707	−0.0116244
$845$	0	0
$846$	−6.60824	−0.227196
$847$	−3.48043	−0.119589
$848$	−1.96072	−0.0673314
$849$	−0.422414	−0.0144972
$850$	−7.40072	−0.253843
$851$	5.52509	0.189398
$852$	3.36734	0.115363
$853$	40.0545	1.37144	0.685720	−	0.727865i	$-0.259487\pi$
$853$	40.0545	1.37144	0.685720	+	0.727865i	$0.259487\pi$
$854$	15.7585	0.539243
$855$	38.2463	1.30800
$856$	−11.8119	−0.403721
$857$	42.5354	1.45298	0.726491	−	0.687176i	$-0.241150\pi$
$857$	42.5354	1.45298	0.726491	+	0.687176i	$0.241150\pi$
$858$	0	0
$859$	−26.3544	−0.899201	−0.449600	−	0.893230i	$-0.648434\pi$
$859$	−26.3544	−0.899201	−0.449600	+	0.893230i	$0.648434\pi$
$860$	6.74416	0.229974
$861$	−1.09071	−0.0371714
$862$	46.6658	1.58944
$863$	−15.7514	−0.536183	−0.268091	−	0.963394i	$-0.586393\pi$
$863$	−15.7514	−0.536183	−0.268091	+	0.963394i	$0.586393\pi$
$864$	−13.9369	−0.474142
$865$	−0.101577	−0.00345374
$866$	8.68560	0.295149
$867$	4.19619	0.142510
$868$	1.02511	0.0347946
$869$	−33.9160	−1.15052
$870$	6.99513	0.237157
$871$	0	0
$872$	−3.65911	−0.123913
$873$	−54.9406	−1.85946
$874$	16.7434	0.566353
$875$	−7.92401	−0.267880
$876$	0.0876146	0.00296022
$877$	33.1442	1.11920	0.559601	−	0.828762i	$-0.310955\pi$
$877$	33.1442	1.11920	0.559601	+	0.828762i	$0.310955\pi$
$878$	42.2891	1.42719
$879$	6.92041	0.233420
$880$	−33.2047	−1.11933
$881$	−2.63745	−0.0888580	−0.0444290	−	0.999013i	$-0.514147\pi$
$881$	−2.63745	−0.0888580	−0.0444290	+	0.999013i	$0.514147\pi$
$882$	35.9371	1.21006
$883$	22.2303	0.748109	0.374055	−	0.927407i	$-0.377967\pi$
$883$	22.2303	0.748109	0.374055	+	0.927407i	$0.377967\pi$
$884$	0	0
$885$	5.91378	0.198789
$886$	51.1776	1.71934
$887$	−11.9865	−0.402466	−0.201233	−	0.979543i	$-0.564495\pi$
$887$	−11.9865	−0.402466	−0.201233	+	0.979543i	$0.564495\pi$
$888$	1.24602	0.0418137
$889$	13.6221	0.456870
$890$	−25.3081	−0.848330
$891$	−32.5097	−1.08911
$892$	11.8620	0.397169
$893$	9.09349	0.304302
$894$	1.21595	0.0406674
$895$	21.9402	0.733381
$896$	−4.13009	−0.137977
$897$	0	0
$898$	63.1413	2.10705
$899$	−6.42057	−0.214138
$900$	8.68029	0.289343
$901$	0.860292	0.0286605
$902$	38.9401	1.29656
$903$	−0.522324	−0.0173819
$904$	9.97054	0.331615
$905$	17.4850	0.581222
$906$	1.90028	0.0631326
$907$	−29.8644	−0.991630	−0.495815	−	0.868428i	$-0.665131\pi$
$907$	−29.8644	−0.991630	−0.495815	+	0.868428i	$0.665131\pi$
$908$	−27.6270	−0.916833
$909$	−37.5994	−1.24709
$910$	0	0
$911$	46.2206	1.53136	0.765678	−	0.643224i	$-0.222404\pi$
$911$	46.2206	1.53136	0.765678	+	0.643224i	$0.222404\pi$
$912$	11.5287	0.381753
$913$	−53.1925	−1.76042
$914$	48.6663	1.60974
$915$	−7.36906	−0.243613
$916$	3.90244	0.128940
$917$	−5.64266	−0.186337
$918$	7.48851	0.247158
$919$	−20.0899	−0.662705	−0.331352	−	0.943507i	$-0.607505\pi$
$919$	−20.0899	−0.662705	−0.331352	+	0.943507i	$0.607505\pi$
$920$	1.67711	0.0552928
$921$	−0.801698	−0.0264168
$922$	13.5657	0.446762
$923$	0	0
$924$	−1.35876	−0.0446998
$925$	−8.95832	−0.294548
$926$	4.06020	0.133426
$927$	−21.9533	−0.721042
$928$	−46.3058	−1.52006
$929$	−25.3137	−0.830516	−0.415258	−	0.909704i	$-0.636309\pi$
$929$	−25.3137	−0.830516	−0.415258	+	0.909704i	$0.636309\pi$
$930$	−1.08949	−0.0357257
$931$	−49.4524	−1.62074
$932$	2.72111	0.0891328
$933$	6.39117	0.209238
$934$	13.5904	0.444693
$935$	14.5690	0.476457
$936$	0	0
$937$	33.0728	1.08044	0.540221	−	0.841523i	$-0.318341\pi$
$937$	33.0728	1.08044	0.540221	+	0.841523i	$0.318341\pi$
$938$	−14.2187	−0.464257
$939$	−5.82965	−0.190243
$940$	−3.33951	−0.108923
$941$	33.7982	1.10179	0.550895	−	0.834574i	$-0.314286\pi$
$941$	33.7982	1.10179	0.550895	+	0.834574i	$0.314286\pi$
$942$	−6.15173	−0.200434
$943$	−6.00497	−0.195549
$944$	−47.9407	−1.56034
$945$	2.21608	0.0720890
$946$	18.6478	0.606292
$947$	−30.3168	−0.985163	−0.492581	−	0.870266i	$-0.663947\pi$
$947$	−30.3168	−0.985163	−0.492581	+	0.870266i	$0.663947\pi$
$948$	−4.32451	−0.140454
$949$	0	0
$950$	−27.1475	−0.880783
$951$	0.666864	0.0216246
$952$	1.08350	0.0351166
$953$	−19.2346	−0.623068	−0.311534	−	0.950235i	$-0.600843\pi$
$953$	−19.2346	−0.623068	−0.311534	+	0.950235i	$0.600843\pi$
$954$	−2.29329	−0.0742479
$955$	46.3258	1.49907
$956$	2.21495	0.0716365
$957$	8.51027	0.275098
$958$	48.8960	1.57976
$959$	9.59780	0.309929
$960$	−2.46890	−0.0796834
$961$	1.00000	0.0322581
$962$	0	0
$963$	−42.1805	−1.35925
$964$	−0.547744	−0.0176416
$965$	−1.65647	−0.0533237
$966$	0.476220	0.0153221
$967$	42.6693	1.37215	0.686076	−	0.727530i	$-0.259332\pi$
$967$	42.6693	1.37215	0.686076	+	0.727530i	$0.259332\pi$
$968$	−4.32135	−0.138894
$969$	−5.05836	−0.162498
$970$	−63.1019	−2.02608
$971$	50.8279	1.63114	0.815572	−	0.578656i	$-0.196423\pi$
$971$	50.8279	1.63114	0.815572	+	0.578656i	$0.196423\pi$
$972$	−13.2551	−0.425156
$973$	−10.1185	−0.324385
$974$	3.62390	0.116117
$975$	0	0
$976$	59.7381	1.91217
$977$	−12.8260	−0.410340	−0.205170	−	0.978726i	$-0.565775\pi$
$977$	−12.8260	−0.410340	−0.205170	+	0.978726i	$0.565775\pi$
$978$	5.72343	0.183015
$979$	−30.7898	−0.984047
$980$	18.1610	0.580132
$981$	−13.0668	−0.417191
$982$	−10.1310	−0.323293
$983$	33.5024	1.06856	0.534280	−	0.845307i	$-0.320583\pi$
$983$	33.5024	1.06856	0.534280	+	0.845307i	$0.320583\pi$
$984$	−1.35424	−0.0431717
$985$	−18.4757	−0.588685
$986$	24.8809	0.792368
$987$	0.258640	0.00823259
$988$	0	0
$989$	−2.87569	−0.0914415
$990$	−38.8366	−1.23431
$991$	42.9474	1.36427	0.682134	−	0.731227i	$-0.261052\pi$
$991$	42.9474	1.36427	0.682134	+	0.731227i	$0.261052\pi$
$992$	7.21210	0.228984
$993$	6.09035	0.193271
$994$	8.05560	0.255508
$995$	18.5738	0.588828
$996$	−6.78240	−0.214909
$997$	38.7880	1.22843	0.614214	−	0.789139i	$-0.289473\pi$
$997$	38.7880	1.22843	0.614214	+	0.789139i	$0.289473\pi$
$998$	42.3010	1.33902
$999$	9.06459	0.286791

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5239.2.a.l.1.13		16
13.5	odd	4		403.2.c.b.311.7	✓	32
13.8	odd	4		403.2.c.b.311.26	yes	32
13.12	even	2		5239.2.a.k.1.4		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
403.2.c.b.311.7	✓	32	13.5	odd	4
403.2.c.b.311.26	yes	32	13.8	odd	4
5239.2.a.k.1.4		16	13.12	even	2
5239.2.a.l.1.13		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)

\(f(q)\)	\(=\)	\(q+1.88981 q^{2} -0.327950 q^{3} +1.57140 q^{4} -1.75790 q^{5} -0.619765 q^{6} -0.652358 q^{7} -0.809976 q^{8} -2.89245 q^{9} +O(q^{10})\)	\(q+1.88981 q^{2} -0.327950 q^{3} +1.57140 q^{4} -1.75790 q^{5} -0.619765 q^{6} -0.652358 q^{7} -0.809976 q^{8} -2.89245 q^{9} -3.32211 q^{10} -4.04168 q^{11} -0.515341 q^{12} -1.23284 q^{14} +0.576505 q^{15} -4.67350 q^{16} +2.05056 q^{17} -5.46619 q^{18} +7.52193 q^{19} -2.76237 q^{20} +0.213941 q^{21} -7.63802 q^{22} +1.17786 q^{23} +0.265632 q^{24} -1.90978 q^{25} +1.93243 q^{27} -1.02511 q^{28} +6.42057 q^{29} +1.08949 q^{30} -1.00000 q^{31} -7.21210 q^{32} +1.32547 q^{33} +3.87518 q^{34} +1.14678 q^{35} -4.54519 q^{36} +4.69077 q^{37} +14.2150 q^{38} +1.42386 q^{40} -5.09819 q^{41} +0.404309 q^{42} -2.44144 q^{43} -6.35109 q^{44} +5.08465 q^{45} +2.22594 q^{46} +1.20893 q^{47} +1.53268 q^{48} -6.57443 q^{49} -3.60912 q^{50} -0.672482 q^{51} +0.419540 q^{53} +3.65193 q^{54} +7.10488 q^{55} +0.528394 q^{56} -2.46682 q^{57} +12.1337 q^{58} +10.2580 q^{59} +0.905919 q^{60} -12.7823 q^{61} -1.88981 q^{62} +1.88691 q^{63} -4.28253 q^{64} +2.50489 q^{66} +11.5333 q^{67} +3.22225 q^{68} -0.386280 q^{69} +2.16721 q^{70} -6.53421 q^{71} +2.34282 q^{72} -0.170013 q^{73} +8.86469 q^{74} +0.626311 q^{75} +11.8200 q^{76} +2.63662 q^{77} +8.39155 q^{79} +8.21557 q^{80} +8.04360 q^{81} -9.63464 q^{82} +13.1610 q^{83} +0.336186 q^{84} -3.60469 q^{85} -4.61387 q^{86} -2.10563 q^{87} +3.27366 q^{88} +7.61808 q^{89} +9.60904 q^{90} +1.85089 q^{92} +0.327950 q^{93} +2.28465 q^{94} -13.2228 q^{95} +2.36521 q^{96} +18.9945 q^{97} -12.4245 q^{98} +11.6903 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 4 q^{2} - 2 q^{3} + 18 q^{4} + 4 q^{5} + 6 q^{6} + 2 q^{7} + 12 q^{8} + 10 q^{9}+O(q^{10}) \) 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

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