LMFDB - Embedded newform 8003.2.a.c.1.10

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8003,2,Mod(1,8003)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8003, 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("8003.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8003 = 53 \cdot 151 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8003.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:	$63.9042767376$
Analytic rank:	$0$
Dimension:	$172$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.10
Character	$\chi$	$=$	8003.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.56961 q^{2} -1.95716 q^{3} +4.60287 q^{4} +1.92681 q^{5} +5.02912 q^{6} +3.09633 q^{7} -6.68835 q^{8} +0.830463 q^{9} +O(q^{10})$	\(q-2.56961 q^{2} -1.95716 q^{3} +4.60287 q^{4} +1.92681 q^{5} +5.02912 q^{6} +3.09633 q^{7} -6.68835 q^{8} +0.830463 q^{9} -4.95115 q^{10} +6.15017 q^{11} -9.00854 q^{12} +6.36427 q^{13} -7.95633 q^{14} -3.77108 q^{15} +7.98068 q^{16} -2.90133 q^{17} -2.13396 q^{18} -0.236083 q^{19} +8.86888 q^{20} -6.05999 q^{21} -15.8035 q^{22} +7.62505 q^{23} +13.0902 q^{24} -1.28739 q^{25} -16.3537 q^{26} +4.24612 q^{27} +14.2520 q^{28} -2.66639 q^{29} +9.69018 q^{30} +10.1255 q^{31} -7.13049 q^{32} -12.0369 q^{33} +7.45528 q^{34} +5.96604 q^{35} +3.82251 q^{36} -10.8285 q^{37} +0.606640 q^{38} -12.4559 q^{39} -12.8872 q^{40} -1.58063 q^{41} +15.5718 q^{42} +4.88250 q^{43} +28.3084 q^{44} +1.60015 q^{45} -19.5934 q^{46} -8.88615 q^{47} -15.6194 q^{48} +2.58723 q^{49} +3.30808 q^{50} +5.67836 q^{51} +29.2939 q^{52} -1.00000 q^{53} -10.9109 q^{54} +11.8502 q^{55} -20.7093 q^{56} +0.462051 q^{57} +6.85158 q^{58} +5.79894 q^{59} -17.3578 q^{60} -9.67044 q^{61} -26.0186 q^{62} +2.57138 q^{63} +2.36119 q^{64} +12.2628 q^{65} +30.9300 q^{66} +10.5985 q^{67} -13.3545 q^{68} -14.9234 q^{69} -15.3304 q^{70} -2.11613 q^{71} -5.55443 q^{72} +5.85458 q^{73} +27.8251 q^{74} +2.51962 q^{75} -1.08666 q^{76} +19.0429 q^{77} +32.0067 q^{78} -4.48344 q^{79} +15.3773 q^{80} -10.8017 q^{81} +4.06159 q^{82} +10.5266 q^{83} -27.8934 q^{84} -5.59033 q^{85} -12.5461 q^{86} +5.21855 q^{87} -41.1345 q^{88} +0.534793 q^{89} -4.11175 q^{90} +19.7059 q^{91} +35.0971 q^{92} -19.8173 q^{93} +22.8339 q^{94} -0.454888 q^{95} +13.9555 q^{96} -8.93829 q^{97} -6.64816 q^{98} +5.10749 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 172 q + 8 q^{2} + 25 q^{3} + 188 q^{4} + 27 q^{5} + 10 q^{6} + 31 q^{7} + 21 q^{8} + 179 q^{9}+O(q^{10}) $ 172 * q + 8 * q^2 + 25 * q^3 + 188 * q^4 + 27 * q^5 + 10 * q^6 + 31 * q^7 + 21 * q^8 + 179 * q^9	\( 172 q + 8 q^{2} + 25 q^{3} + 188 q^{4} + 27 q^{5} + 10 q^{6} + 31 q^{7} + 21 q^{8} + 179 q^{9} + 20 q^{10} - 3 q^{11} + 66 q^{12} + 121 q^{13} + 12 q^{14} + 30 q^{15} + 212 q^{16} + 8 q^{17} + 40 q^{18} + 41 q^{19} + 64 q^{20} + 56 q^{21} + 50 q^{22} + 28 q^{23} + 30 q^{24} + 231 q^{25} + 38 q^{26} + 100 q^{27} + 80 q^{28} + 26 q^{29} + 55 q^{30} + 66 q^{31} + 65 q^{32} + 99 q^{33} + 81 q^{34} + 36 q^{35} + 212 q^{36} + 153 q^{37} + q^{38} + 20 q^{39} + 59 q^{40} + 40 q^{41} + 50 q^{42} + 39 q^{43} - 51 q^{44} + 123 q^{45} + 59 q^{46} + 29 q^{47} + 128 q^{48} + 245 q^{49} + 19 q^{50} + 36 q^{51} + 215 q^{52} - 172 q^{53} + 40 q^{54} + 40 q^{55} + 15 q^{56} + 54 q^{57} + 44 q^{58} - 54 q^{60} + 100 q^{61} - 29 q^{62} + 92 q^{63} + 253 q^{64} + 77 q^{65} + 14 q^{66} + 126 q^{67} - 27 q^{68} + 47 q^{69} + 72 q^{70} + 38 q^{71} + 65 q^{72} + 185 q^{73} + 48 q^{74} + 75 q^{75} + 38 q^{76} + 120 q^{77} + 75 q^{78} + 79 q^{79} + 43 q^{80} + 232 q^{81} + 110 q^{82} + 90 q^{83} + 158 q^{84} + 115 q^{85} + 68 q^{86} + 61 q^{87} + 15 q^{88} - 36 q^{89} - 6 q^{90} + 33 q^{91} + 139 q^{92} + 103 q^{93} - 24 q^{94} - 45 q^{95} + 34 q^{96} + 159 q^{97} - 36 q^{98} + 27 q^{99}+O(q^{100}) \) 172 * q + 8 * q^2 + 25 * q^3 + 188 * q^4 + 27 * q^5 + 10 * q^6 + 31 * q^7 + 21 * q^8 + 179 * q^9 + 20 * q^10 - 3 * q^11 + 66 * q^12 + 121 * q^13 + 12 * q^14 + 30 * q^15 + 212 * q^16 + 8 * q^17 + 40 * q^18 + 41 * q^19 + 64 * q^20 + 56 * q^21 + 50 * q^22 + 28 * q^23 + 30 * q^24 + 231 * q^25 + 38 * q^26 + 100 * q^27 + 80 * q^28 + 26 * q^29 + 55 * q^30 + 66 * q^31 + 65 * q^32 + 99 * q^33 + 81 * q^34 + 36 * q^35 + 212 * q^36 + 153 * q^37 + q^38 + 20 * q^39 + 59 * q^40 + 40 * q^41 + 50 * q^42 + 39 * q^43 - 51 * q^44 + 123 * q^45 + 59 * q^46 + 29 * q^47 + 128 * q^48 + 245 * q^49 + 19 * q^50 + 36 * q^51 + 215 * q^52 - 172 * q^53 + 40 * q^54 + 40 * q^55 + 15 * q^56 + 54 * q^57 + 44 * q^58 - 54 * q^60 + 100 * q^61 - 29 * q^62 + 92 * q^63 + 253 * q^64 + 77 * q^65 + 14 * q^66 + 126 * q^67 - 27 * q^68 + 47 * q^69 + 72 * q^70 + 38 * q^71 + 65 * q^72 + 185 * q^73 + 48 * q^74 + 75 * q^75 + 38 * q^76 + 120 * q^77 + 75 * q^78 + 79 * q^79 + 43 * q^80 + 232 * q^81 + 110 * q^82 + 90 * q^83 + 158 * q^84 + 115 * q^85 + 68 * q^86 + 61 * q^87 + 15 * q^88 - 36 * q^89 - 6 * q^90 + 33 * q^91 + 139 * q^92 + 103 * q^93 - 24 * q^94 - 45 * q^95 + 34 * q^96 + 159 * q^97 - 36 * q^98 + 27 * 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 $

$2$	−2.56961	−1.81699	−0.908493	−	0.417901i	$-0.862766\pi$
$2$	−2.56961	−1.81699	−0.908493	+	0.417901i	$0.862766\pi$
$3$	−1.95716	−1.12997	−0.564983	−	0.825103i	$-0.691117\pi$
$3$	−1.95716	−1.12997	−0.564983	+	0.825103i	$0.691117\pi$
$4$	4.60287	2.30144
$5$	1.92681	0.861697	0.430849	−	0.902424i	$-0.358214\pi$
$5$	1.92681	0.861697	0.430849	+	0.902424i	$0.358214\pi$
$6$	5.02912	2.05313
$7$	3.09633	1.17030	0.585150	−	0.810925i	$-0.301035\pi$
$7$	3.09633	1.17030	0.585150	+	0.810925i	$0.301035\pi$
$8$	−6.68835	−2.36469
$9$	0.830463	0.276821
$10$	−4.95115	−1.56569
$11$	6.15017	1.85435	0.927173	−	0.374633i	$-0.122231\pi$
$11$	6.15017	1.85435	0.927173	+	0.374633i	$0.122231\pi$
$12$	−9.00854	−2.60054
$13$	6.36427	1.76513	0.882566	−	0.470189i	$-0.155814\pi$
$13$	6.36427	1.76513	0.882566	+	0.470189i	$0.155814\pi$
$14$	−7.95633	−2.12642
$15$	−3.77108	−0.973688
$16$	7.98068	1.99517
$17$	−2.90133	−0.703677	−0.351838	−	0.936061i	$-0.614443\pi$
$17$	−2.90133	−0.703677	−0.351838	+	0.936061i	$0.614443\pi$
$18$	−2.13396	−0.502980
$19$	−0.236083	−0.0541611	−0.0270806	−	0.999633i	$-0.508621\pi$
$19$	−0.236083	−0.0541611	−0.0270806	+	0.999633i	$0.508621\pi$
$20$	8.86888	1.98314
$21$	−6.05999	−1.32240
$22$	−15.8035	−3.36932
$23$	7.62505	1.58993	0.794967	−	0.606653i	$-0.207488\pi$
$23$	7.62505	1.58993	0.794967	+	0.606653i	$0.207488\pi$
$24$	13.0902	2.67202
$25$	−1.28739	−0.257478
$26$	−16.3537	−3.20722
$27$	4.24612	0.817167
$28$	14.2520	2.69337
$29$	−2.66639	−0.495137	−0.247568	−	0.968870i	$-0.579631\pi$
$29$	−2.66639	−0.495137	−0.247568	+	0.968870i	$0.579631\pi$
$30$	9.69018	1.76918
$31$	10.1255	1.81860	0.909300	−	0.416142i	$-0.136618\pi$
$31$	10.1255	1.81860	0.909300	+	0.416142i	$0.136618\pi$
$32$	−7.13049	−1.26050
$33$	−12.0369	−2.09535
$34$	7.45528	1.27857
$35$	5.96604	1.00845
$36$	3.82251	0.637086
$37$	−10.8285	−1.78020	−0.890101	−	0.455764i	$-0.849366\pi$
$37$	−10.8285	−1.78020	−0.890101	+	0.455764i	$0.849366\pi$
$38$	0.606640	0.0984100
$39$	−12.4559	−1.99454
$40$	−12.8872	−2.03765
$41$	−1.58063	−0.246852	−0.123426	−	0.992354i	$-0.539388\pi$
$41$	−1.58063	−0.246852	−0.123426	+	0.992354i	$0.539388\pi$
$42$	15.5718	2.40278
$43$	4.88250	0.744574	0.372287	−	0.928118i	$-0.378574\pi$
$43$	4.88250	0.744574	0.372287	+	0.928118i	$0.378574\pi$
$44$	28.3084	4.26766
$45$	1.60015	0.238536
$46$	−19.5934	−2.88889
$47$	−8.88615	−1.29618	−0.648089	−	0.761565i	$-0.724431\pi$
$47$	−8.88615	−1.29618	−0.648089	+	0.761565i	$0.724431\pi$
$48$	−15.6194	−2.25447
$49$	2.58723	0.369604
$50$	3.30808	0.467833
$51$	5.67836	0.795130
$52$	29.2939	4.06234
$53$	−1.00000	−0.137361
$53$	−1.00000	−0.137361
$54$	−10.9109	−1.48478
$55$	11.8502	1.59789
$56$	−20.7093	−2.76740
$57$	0.462051	0.0612002
$58$	6.85158	0.899656
$59$	5.79894	0.754958	0.377479	−	0.926018i	$-0.376791\pi$
$59$	5.79894	0.754958	0.377479	+	0.926018i	$0.376791\pi$
$60$	−17.3578	−2.24088
$61$	−9.67044	−1.23817	−0.619086	−	0.785323i	$-0.712497\pi$
$61$	−9.67044	−1.23817	−0.619086	+	0.785323i	$0.712497\pi$
$62$	−26.0186	−3.30437
$63$	2.57138	0.323964
$64$	2.36119	0.295149
$65$	12.2628	1.52101
$66$	30.9300	3.80721
$67$	10.5985	1.29481	0.647406	−	0.762146i	$-0.275854\pi$
$67$	10.5985	1.29481	0.647406	+	0.762146i	$0.275854\pi$
$68$	−13.3545	−1.61947
$69$	−14.9234	−1.79657
$70$	−15.3304	−1.83233
$71$	−2.11613	−0.251138	−0.125569	−	0.992085i	$-0.540076\pi$
$71$	−2.11613	−0.251138	−0.125569	+	0.992085i	$0.540076\pi$
$72$	−5.55443	−0.654596
$73$	5.85458	0.685227	0.342614	−	0.939476i	$-0.388688\pi$
$73$	5.85458	0.685227	0.342614	+	0.939476i	$0.388688\pi$
$74$	27.8251	3.23460
$75$	2.51962	0.290941
$76$	−1.08666	−0.124648
$77$	19.0429	2.17014
$78$	32.0067	3.62404
$79$	−4.48344	−0.504426	−0.252213	−	0.967672i	$-0.581158\pi$
$79$	−4.48344	−0.504426	−0.252213	+	0.967672i	$0.581158\pi$
$80$	15.3773	1.71923
$81$	−10.8017	−1.20019
$82$	4.06159	0.448527
$83$	10.5266	1.15544	0.577722	−	0.816233i	$-0.303942\pi$
$83$	10.5266	1.15544	0.577722	+	0.816233i	$0.303942\pi$
$84$	−27.8934	−3.04342
$85$	−5.59033	−0.606356
$86$	−12.5461	−1.35288
$87$	5.21855	0.559487
$88$	−41.1345	−4.38495
$89$	0.534793	0.0566879	0.0283440	−	0.999598i	$-0.490977\pi$
$89$	0.534793	0.0566879	0.0283440	+	0.999598i	$0.490977\pi$
$90$	−4.11175	−0.433416
$91$	19.7059	2.06574
$92$	35.0971	3.65913
$93$	−19.8173	−2.05495
$94$	22.8339	2.35514
$95$	−0.454888	−0.0466705
$96$	13.9555	1.42433
$97$	−8.93829	−0.907546	−0.453773	−	0.891117i	$-0.649922\pi$
$97$	−8.93829	−0.907546	−0.453773	+	0.891117i	$0.649922\pi$
$98$	−6.64816	−0.671565
$99$	5.10749	0.513322
$100$	−5.92568	−0.592568
$101$	−7.12335	−0.708799	−0.354400	−	0.935094i	$-0.615315\pi$
$101$	−7.12335	−0.708799	−0.354400	+	0.935094i	$0.615315\pi$
$102$	−14.5912	−1.44474
$103$	6.75786	0.665871	0.332936	−	0.942950i	$-0.391961\pi$
$103$	6.75786	0.665871	0.332936	+	0.942950i	$0.391961\pi$
$104$	−42.5665	−4.17399
$105$	−11.6765	−1.13951
$106$	2.56961	0.249582
$107$	−7.60615	−0.735315	−0.367657	−	0.929961i	$-0.619840\pi$
$107$	−7.60615	−0.735315	−0.367657	+	0.929961i	$0.619840\pi$
$108$	19.5444	1.88066
$109$	17.6098	1.68671	0.843357	−	0.537354i	$-0.180576\pi$
$109$	17.6098	1.68671	0.843357	+	0.537354i	$0.180576\pi$
$110$	−30.4504	−2.90333
$111$	21.1932	2.01157
$112$	24.7108	2.33495
$113$	12.0941	1.13772	0.568858	−	0.822435i	$-0.307385\pi$
$113$	12.0941	1.13772	0.568858	+	0.822435i	$0.307385\pi$
$114$	−1.18729	−0.111200
$115$	14.6921	1.37004
$116$	−12.2731	−1.13953
$117$	5.28529	0.488626
$118$	−14.9010	−1.37175
$119$	−8.98347	−0.823513
$120$	25.2223	2.30247
$121$	26.8246	2.43860
$122$	24.8492	2.24974
$123$	3.09354	0.278935
$124$	46.6065	4.18539
$125$	−12.1146	−1.08357
$126$	−6.60744	−0.588638
$127$	14.0469	1.24646	0.623229	−	0.782040i	$-0.285821\pi$
$127$	14.0469	1.24646	0.623229	+	0.782040i	$0.285821\pi$
$128$	8.19365	0.724223
$129$	−9.55582	−0.841343
$130$	−31.5105	−2.76365
$131$	−15.3049	−1.33719	−0.668597	−	0.743625i	$-0.733105\pi$
$131$	−15.3049	−1.33719	−0.668597	+	0.743625i	$0.733105\pi$
$132$	−55.4041	−4.82231
$133$	−0.730989	−0.0633848
$134$	−27.2339	−2.35265
$135$	8.18149	0.704151
$136$	19.4051	1.66398
$137$	3.90928	0.333992	0.166996	−	0.985958i	$-0.446593\pi$
$137$	3.90928	0.333992	0.166996	+	0.985958i	$0.446593\pi$
$138$	38.3473	3.26434
$139$	17.3281	1.46975	0.734873	−	0.678205i	$-0.237242\pi$
$139$	17.3281	1.46975	0.734873	+	0.678205i	$0.237242\pi$
$140$	27.4609	2.32087
$141$	17.3916	1.46464
$142$	5.43761	0.456314
$143$	39.1414	3.27317
$144$	6.62766	0.552305
$145$	−5.13764	−0.426658
$146$	−15.0440	−1.24505
$147$	−5.06361	−0.417640
$148$	−49.8424	−4.09702
$149$	−17.1782	−1.40729	−0.703646	−	0.710550i	$-0.748446\pi$
$149$	−17.1782	−1.40729	−0.703646	+	0.710550i	$0.748446\pi$
$150$	−6.47443	−0.528635
$151$	1.00000	0.0813788
$151$	1.00000	0.0813788
$152$	1.57901	0.128074
$153$	−2.40945	−0.194793
$154$	−48.9328	−3.94312
$155$	19.5100	1.56708
$156$	−57.3328	−4.59030
$157$	−9.59457	−0.765730	−0.382865	−	0.923804i	$-0.625063\pi$
$157$	−9.59457	−0.765730	−0.382865	+	0.923804i	$0.625063\pi$
$158$	11.5207	0.916535
$159$	1.95716	0.155213
$160$	−13.7391	−1.08617
$161$	23.6096	1.86070
$162$	27.7562	2.18073
$163$	13.3034	1.04200	0.521002	−	0.853556i	$-0.325559\pi$
$163$	13.3034	1.04200	0.521002	+	0.853556i	$0.325559\pi$
$164$	−7.27542	−0.568115
$165$	−23.1928	−1.80555
$166$	−27.0492	−2.09943
$167$	10.6748	0.826041	0.413021	−	0.910722i	$-0.364474\pi$
$167$	10.6748	0.826041	0.413021	+	0.910722i	$0.364474\pi$
$168$	40.5314	3.12706
$169$	27.5040	2.11569
$170$	14.3649	1.10174
$171$	−0.196058	−0.0149929
$172$	22.4735	1.71359
$173$	1.63788	0.124526	0.0622628	−	0.998060i	$-0.480168\pi$
$173$	1.63788	0.124526	0.0622628	+	0.998060i	$0.480168\pi$
$174$	−13.4096	−1.01658
$175$	−3.98617	−0.301326
$176$	49.0825	3.69974
$177$	−11.3494	−0.853076
$178$	−1.37421	−0.103001
$179$	−5.15232	−0.385102	−0.192551	−	0.981287i	$-0.561676\pi$
$179$	−5.15232	−0.385102	−0.192551	+	0.981287i	$0.561676\pi$
$180$	7.36527	0.548975
$181$	19.3628	1.43923	0.719614	−	0.694374i	$-0.244319\pi$
$181$	19.3628	1.43923	0.719614	+	0.694374i	$0.244319\pi$
$182$	−50.6363	−3.75341
$183$	18.9266	1.39909
$184$	−50.9990	−3.75970
$185$	−20.8646	−1.53399
$186$	50.9225	3.73382
$187$	−17.8437	−1.30486
$188$	−40.9018	−2.98307
$189$	13.1474	0.956331
$190$	1.16888	0.0847996
$191$	−5.56546	−0.402702	−0.201351	−	0.979519i	$-0.564533\pi$
$191$	−5.56546	−0.402702	−0.201351	+	0.979519i	$0.564533\pi$
$192$	−4.62122	−0.333508
$193$	−2.83189	−0.203844	−0.101922	−	0.994792i	$-0.532499\pi$
$193$	−2.83189	−0.203844	−0.101922	+	0.994792i	$0.532499\pi$
$194$	22.9679	1.64900
$195$	−24.0002	−1.71869
$196$	11.9087	0.850620
$197$	17.9925	1.28191	0.640957	−	0.767576i	$-0.278538\pi$
$197$	17.9925	1.28191	0.640957	+	0.767576i	$0.278538\pi$
$198$	−13.1242	−0.932699
$199$	−2.94895	−0.209045	−0.104523	−	0.994523i	$-0.533331\pi$
$199$	−2.94895	−0.209045	−0.104523	+	0.994523i	$0.533331\pi$
$200$	8.61050	0.608855
$201$	−20.7429	−1.46309
$202$	18.3042	1.28788
$203$	−8.25602	−0.579459
$204$	26.1368	1.82994
$205$	−3.04557	−0.212712
$206$	−17.3650	−1.20988
$207$	6.33232	0.440127
$208$	50.7912	3.52174
$209$	−1.45195	−0.100434
$210$	30.0039	2.07047
$211$	25.0992	1.72790	0.863949	−	0.503580i	$-0.167984\pi$
$211$	25.0992	1.72790	0.863949	+	0.503580i	$0.167984\pi$
$212$	−4.60287	−0.316126
$213$	4.14159	0.283777
$214$	19.5448	1.33606
$215$	9.40767	0.641598
$216$	−28.3996	−1.93235
$217$	31.3519	2.12831
$218$	−45.2502	−3.06473
$219$	−11.4583	−0.774283
$220$	54.5451	3.67743
$221$	−18.4649	−1.24208
$222$	−54.4580	−3.65498
$223$	16.0509	1.07485	0.537425	−	0.843312i	$-0.319397\pi$
$223$	16.0509	1.07485	0.537425	+	0.843312i	$0.319397\pi$
$224$	−22.0783	−1.47517
$225$	−1.06913	−0.0712752
$226$	−31.0771	−2.06721
$227$	1.84432	0.122412	0.0612059	−	0.998125i	$-0.480505\pi$
$227$	1.84432	0.122412	0.0612059	+	0.998125i	$0.480505\pi$
$228$	2.12676	0.140848
$229$	−3.96237	−0.261841	−0.130920	−	0.991393i	$-0.541793\pi$
$229$	−3.96237	−0.261841	−0.130920	+	0.991393i	$0.541793\pi$
$230$	−37.7528	−2.48934
$231$	−37.2700	−2.45219
$232$	17.8338	1.17084
$233$	−18.6650	−1.22278	−0.611392	−	0.791328i	$-0.709390\pi$
$233$	−18.6650	−1.22278	−0.611392	+	0.791328i	$0.709390\pi$
$234$	−13.5811	−0.887825
$235$	−17.1220	−1.11691
$236$	26.6918	1.73749
$237$	8.77479	0.569984
$238$	23.0840	1.49631
$239$	−29.0092	−1.87645	−0.938225	−	0.346025i	$-0.887531\pi$
$239$	−29.0092	−1.87645	−0.938225	+	0.346025i	$0.887531\pi$
$240$	−30.0958	−1.94267
$241$	1.00207	0.0645493	0.0322746	−	0.999479i	$-0.489725\pi$
$241$	1.00207	0.0645493	0.0322746	+	0.999479i	$0.489725\pi$
$242$	−68.9287	−4.43090
$243$	8.40229	0.539007
$244$	−44.5118	−2.84957
$245$	4.98511	0.318487
$246$	−7.94916	−0.506820
$247$	−1.50250	−0.0956015
$248$	−67.7231	−4.30042
$249$	−20.6022	−1.30561
$250$	31.1298	1.96882
$251$	−21.1688	−1.33616	−0.668081	−	0.744089i	$-0.732884\pi$
$251$	−21.1688	−1.33616	−0.668081	+	0.744089i	$0.732884\pi$
$252$	11.8357	0.745582
$253$	46.8954	2.94829
$254$	−36.0949	−2.26479
$255$	10.9412	0.685161
$256$	−25.7768	−1.61105
$257$	1.63829	0.102194	0.0510969	−	0.998694i	$-0.483728\pi$
$257$	1.63829	0.102194	0.0510969	+	0.998694i	$0.483728\pi$
$258$	24.5547	1.52871
$259$	−33.5287	−2.08337
$260$	56.4439	3.50050
$261$	−2.21434	−0.137064
$262$	39.3275	2.42966
$263$	−14.6672	−0.904421	−0.452210	−	0.891911i	$-0.649364\pi$
$263$	−14.6672	−0.904421	−0.452210	+	0.891911i	$0.649364\pi$
$264$	80.5067	4.95484
$265$	−1.92681	−0.118363
$266$	1.87835	0.115169
$267$	−1.04667	−0.0640554
$268$	48.7835	2.97992
$269$	11.1480	0.679707	0.339853	−	0.940478i	$-0.389623\pi$
$269$	11.1480	0.679707	0.339853	+	0.940478i	$0.389623\pi$
$270$	−21.0232	−1.27943
$271$	−6.90273	−0.419311	−0.209655	−	0.977775i	$-0.567234\pi$
$271$	−6.90273	−0.419311	−0.209655	+	0.977775i	$0.567234\pi$
$272$	−23.1546	−1.40395
$273$	−38.5675	−2.33421
$274$	−10.0453	−0.606859
$275$	−7.91766	−0.477453
$276$	−68.6906	−4.13469
$277$	10.1010	0.606909	0.303454	−	0.952846i	$-0.401860\pi$
$277$	10.1010	0.606909	0.303454	+	0.952846i	$0.401860\pi$
$278$	−44.5263	−2.67051
$279$	8.40888	0.503427
$280$	−39.9030	−2.38466
$281$	14.5577	0.868438	0.434219	−	0.900807i	$-0.357024\pi$
$281$	14.5577	0.868438	0.434219	+	0.900807i	$0.357024\pi$
$282$	−44.6895	−2.66122
$283$	−11.3298	−0.673489	−0.336744	−	0.941596i	$-0.609326\pi$
$283$	−11.3298	−0.673489	−0.336744	+	0.941596i	$0.609326\pi$
$284$	−9.74026	−0.577978
$285$	0.890287	0.0527360
$286$	−100.578	−5.94729
$287$	−4.89414	−0.288892
$288$	−5.92161	−0.348934
$289$	−8.58227	−0.504839
$290$	13.2017	0.775231
$291$	17.4936	1.02550
$292$	26.9479	1.57701
$293$	26.6184	1.55506	0.777532	−	0.628844i	$-0.216472\pi$
$293$	26.6184	1.55506	0.777532	+	0.628844i	$0.216472\pi$
$294$	13.0115	0.758845
$295$	11.1735	0.650546
$296$	72.4251	4.20962
$297$	26.1144	1.51531
$298$	44.1412	2.55703
$299$	48.5279	2.80644
$300$	11.5975	0.669581
$301$	15.1178	0.871376
$302$	−2.56961	−0.147864
$303$	13.9415	0.800919
$304$	−1.88410	−0.108061
$305$	−18.6331	−1.06693
$306$	6.19134	0.353935
$307$	−28.8719	−1.64781	−0.823903	−	0.566731i	$-0.808208\pi$
$307$	−28.8719	−1.64781	−0.823903	+	0.566731i	$0.808208\pi$
$308$	87.6522	4.99444
$309$	−13.2262	−0.752411
$310$	−50.1330	−2.84737
$311$	14.4098	0.817106	0.408553	−	0.912735i	$-0.366034\pi$
$311$	14.4098	0.817106	0.408553	+	0.912735i	$0.366034\pi$
$312$	83.3093	4.71646
$313$	0.0941437	0.00532132	0.00266066	−	0.999996i	$-0.499153\pi$
$313$	0.0941437	0.00532132	0.00266066	+	0.999996i	$0.499153\pi$
$314$	24.6543	1.39132
$315$	4.95458	0.279159
$316$	−20.6367	−1.16090
$317$	7.16773	0.402580	0.201290	−	0.979532i	$-0.435487\pi$
$317$	7.16773	0.402580	0.201290	+	0.979532i	$0.435487\pi$
$318$	−5.02912	−0.282019
$319$	−16.3988	−0.918155
$320$	4.54958	0.254329
$321$	14.8864	0.830880
$322$	−60.6675	−3.38086
$323$	0.684955	0.0381119
$324$	−49.7189	−2.76216
$325$	−8.19329	−0.454482
$326$	−34.1845	−1.89330
$327$	−34.4652	−1.90593
$328$	10.5718	0.583729
$329$	−27.5144	−1.51692
$330$	59.5963	3.28067
$331$	−21.3369	−1.17278	−0.586390	−	0.810029i	$-0.699451\pi$
$331$	−21.3369	−1.17278	−0.586390	+	0.810029i	$0.699451\pi$
$332$	48.4526	2.65918
$333$	−8.99270	−0.492797
$334$	−27.4300	−1.50090
$335$	20.4213	1.11574
$336$	−48.3629	−2.63841
$337$	−20.7162	−1.12848	−0.564240	−	0.825611i	$-0.690831\pi$
$337$	−20.7162	−1.12848	−0.564240	+	0.825611i	$0.690831\pi$
$338$	−70.6743	−3.84418
$339$	−23.6700	−1.28558
$340$	−25.7316	−1.39549
$341$	62.2738	3.37231
$342$	0.503792	0.0272420
$343$	−13.6634	−0.737753
$344$	−32.6559	−1.76069
$345$	−28.7547	−1.54810
$346$	−4.20870	−0.226261
$347$	−26.0475	−1.39830	−0.699152	−	0.714973i	$-0.746439\pi$
$347$	−26.0475	−1.39830	−0.699152	+	0.714973i	$0.746439\pi$
$348$	24.0203	1.28762
$349$	16.9724	0.908514	0.454257	−	0.890871i	$-0.349905\pi$
$349$	16.9724	0.908514	0.454257	+	0.890871i	$0.349905\pi$
$350$	10.2429	0.547505
$351$	27.0235	1.44241
$352$	−43.8537	−2.33741
$353$	−10.7292	−0.571057	−0.285528	−	0.958370i	$-0.592169\pi$
$353$	−10.7292	−0.571057	−0.285528	+	0.958370i	$0.592169\pi$
$354$	29.1636	1.55003
$355$	−4.07738	−0.216405
$356$	2.46158	0.130464
$357$	17.5821	0.930541
$358$	13.2394	0.699725
$359$	−6.92824	−0.365659	−0.182829	−	0.983145i	$-0.558526\pi$
$359$	−6.92824	−0.365659	−0.182829	+	0.983145i	$0.558526\pi$
$360$	−10.7024	−0.564063
$361$	−18.9443	−0.997067
$362$	−49.7548	−2.61506
$363$	−52.5000	−2.75553
$364$	90.7035	4.75416
$365$	11.2807	0.590459
$366$	−48.6338	−2.54213
$367$	7.39768	0.386156	0.193078	−	0.981183i	$-0.438153\pi$
$367$	7.39768	0.386156	0.193078	+	0.981183i	$0.438153\pi$
$368$	60.8531	3.17219
$369$	−1.31265	−0.0683340
$370$	53.6137	2.78725
$371$	−3.09633	−0.160753
$372$	−91.2163	−4.72934
$373$	13.1800	0.682437	0.341218	−	0.939984i	$-0.389160\pi$
$373$	13.1800	0.682437	0.341218	+	0.939984i	$0.389160\pi$
$374$	45.8513	2.37091
$375$	23.7102	1.22439
$376$	59.4337	3.06506
$377$	−16.9697	−0.873981
$378$	−33.7836	−1.73764
$379$	−14.0861	−0.723552	−0.361776	−	0.932265i	$-0.617829\pi$
$379$	−14.0861	−0.723552	−0.361776	+	0.932265i	$0.617829\pi$
$380$	−2.09379	−0.107409
$381$	−27.4919	−1.40845
$382$	14.3010	0.731704
$383$	28.1796	1.43991	0.719955	−	0.694021i	$-0.244162\pi$
$383$	28.1796	1.43991	0.719955	+	0.694021i	$0.244162\pi$
$384$	−16.0363	−0.818347
$385$	36.6922	1.87001
$386$	7.27684	0.370381
$387$	4.05474	0.206114
$388$	−41.1418	−2.08866
$389$	−21.4230	−1.08619	−0.543095	−	0.839671i	$-0.682748\pi$
$389$	−21.4230	−1.08619	−0.543095	+	0.839671i	$0.682748\pi$
$390$	61.6709	3.12283
$391$	−22.1228	−1.11880
$392$	−17.3043	−0.873999
$393$	29.9541	1.51098
$394$	−46.2337	−2.32922
$395$	−8.63875	−0.434663
$396$	23.5091	1.18138
$397$	−5.93437	−0.297838	−0.148919	−	0.988849i	$-0.547579\pi$
$397$	−5.93437	−0.297838	−0.148919	+	0.988849i	$0.547579\pi$
$398$	7.57763	0.379832
$399$	1.43066	0.0716226
$400$	−10.2742	−0.513712
$401$	−1.51787	−0.0757988	−0.0378994	−	0.999282i	$-0.512067\pi$
$401$	−1.51787	−0.0757988	−0.0378994	+	0.999282i	$0.512067\pi$
$402$	53.3011	2.65842
$403$	64.4416	3.21007
$404$	−32.7878	−1.63126
$405$	−20.8129	−1.03420
$406$	21.2147	1.05287
$407$	−66.5974	−3.30111
$408$	−37.9789	−1.88024
$409$	−4.02674	−0.199109	−0.0995547	−	0.995032i	$-0.531742\pi$
$409$	−4.02674	−0.199109	−0.0995547	+	0.995032i	$0.531742\pi$
$410$	7.82592	0.386495
$411$	−7.65107	−0.377400
$412$	31.1055	1.53246
$413$	17.9554	0.883528
$414$	−16.2716	−0.799704
$415$	20.2828	0.995643
$416$	−45.3804	−2.22496
$417$	−33.9137	−1.66076
$418$	3.73094	0.182486
$419$	−15.3243	−0.748641	−0.374320	−	0.927299i	$-0.622124\pi$
$419$	−15.3243	−0.748641	−0.374320	+	0.927299i	$0.622124\pi$
$420$	−53.7453	−2.62250
$421$	−16.1333	−0.786290	−0.393145	−	0.919477i	$-0.628613\pi$
$421$	−16.1333	−0.786290	−0.393145	+	0.919477i	$0.628613\pi$
$422$	−64.4949	−3.13956
$423$	−7.37962	−0.358809
$424$	6.68835	0.324815
$425$	3.73514	0.181181
$426$	−10.6423	−0.515619
$427$	−29.9428	−1.44903
$428$	−35.0101	−1.69228
$429$	−76.6058	−3.69856
$430$	−24.1740	−1.16577
$431$	−28.7911	−1.38682	−0.693410	−	0.720543i	$-0.743893\pi$
$431$	−28.7911	−1.38682	−0.693410	+	0.720543i	$0.743893\pi$
$432$	33.8869	1.63039
$433$	−23.7864	−1.14310	−0.571552	−	0.820566i	$-0.693658\pi$
$433$	−23.7864	−1.14310	−0.571552	+	0.820566i	$0.693658\pi$
$434$	−80.5621	−3.86710
$435$	10.0552	0.482109
$436$	81.0557	3.88186
$437$	−1.80014	−0.0861126
$438$	29.4434	1.40686
$439$	1.00425	0.0479301	0.0239650	−	0.999713i	$-0.492371\pi$
$439$	1.00425	0.0479301	0.0239650	+	0.999713i	$0.492371\pi$
$440$	−79.2585	−3.77850
$441$	2.14860	0.102314
$442$	47.4474	2.25684
$443$	27.3989	1.30176	0.650881	−	0.759180i	$-0.274400\pi$
$443$	27.3989	1.30176	0.650881	+	0.759180i	$0.274400\pi$
$444$	97.5493	4.62949
$445$	1.03045	0.0488478
$446$	−41.2446	−1.95299
$447$	33.6204	1.59019
$448$	7.31102	0.345413
$449$	−33.3591	−1.57431	−0.787156	−	0.616754i	$-0.788447\pi$
$449$	−33.3591	−1.57431	−0.787156	+	0.616754i	$0.788447\pi$
$450$	2.74724	0.129506
$451$	−9.72113	−0.457750
$452$	55.6676	2.61838
$453$	−1.95716	−0.0919553
$454$	−4.73917	−0.222420
$455$	37.9695	1.78004
$456$	−3.09036	−0.144719
$457$	−1.82851	−0.0855339	−0.0427669	−	0.999085i	$-0.513617\pi$
$457$	−1.82851	−0.0855339	−0.0427669	+	0.999085i	$0.513617\pi$
$458$	10.1817	0.475761
$459$	−12.3194	−0.575021
$460$	67.6256	3.15306
$461$	−20.1952	−0.940584	−0.470292	−	0.882511i	$-0.655851\pi$
$461$	−20.1952	−0.940584	−0.470292	+	0.882511i	$0.655851\pi$
$462$	95.7692	4.45559
$463$	39.4200	1.83200	0.916002	−	0.401174i	$-0.131398\pi$
$463$	39.4200	1.83200	0.916002	+	0.401174i	$0.131398\pi$
$464$	−21.2796	−0.987882
$465$	−38.1842	−1.77075
$466$	47.9617	2.22178
$467$	0.938522	0.0434296	0.0217148	−	0.999764i	$-0.493087\pi$
$467$	0.938522	0.0434296	0.0217148	+	0.999764i	$0.493087\pi$
$468$	24.3275	1.12454
$469$	32.8164	1.51532
$470$	43.9967	2.02941
$471$	18.7781	0.865248
$472$	−38.7854	−1.78524
$473$	30.0282	1.38070
$474$	−22.5478	−1.03565
$475$	0.303930	0.0139453
$476$	−41.3498	−1.89526
$477$	−0.830463	−0.0380243
$478$	74.5422	3.40948
$479$	35.8664	1.63878	0.819390	−	0.573237i	$-0.194313\pi$
$479$	35.8664	1.63878	0.819390	+	0.573237i	$0.194313\pi$
$480$	26.8896	1.22734
$481$	−68.9158	−3.14229
$482$	−2.57494	−0.117285
$483$	−46.2078	−2.10253
$484$	123.470	5.61228
$485$	−17.2224	−0.782030
$486$	−21.5906	−0.979368
$487$	34.2292	1.55107	0.775537	−	0.631302i	$-0.217479\pi$
$487$	34.2292	1.55107	0.775537	+	0.631302i	$0.217479\pi$
$488$	64.6793	2.92789
$489$	−26.0369	−1.17743
$490$	−12.8098	−0.578686
$491$	5.62962	0.254061	0.127031	−	0.991899i	$-0.459455\pi$
$491$	5.62962	0.254061	0.127031	+	0.991899i	$0.459455\pi$
$492$	14.2391	0.641950
$493$	7.73609	0.348416
$494$	3.86082	0.173707
$495$	9.84118	0.442328
$496$	80.8086	3.62841
$497$	−6.55222	−0.293907
$498$	52.9395	2.37228
$499$	−17.9921	−0.805436	−0.402718	−	0.915324i	$-0.631935\pi$
$499$	−17.9921	−0.805436	−0.402718	+	0.915324i	$0.631935\pi$
$500$	−55.7621	−2.49376
$501$	−20.8923	−0.933397
$502$	54.3954	2.42779
$503$	−21.8662	−0.974965	−0.487483	−	0.873133i	$-0.662085\pi$
$503$	−21.8662	−0.974965	−0.487483	+	0.873133i	$0.662085\pi$
$504$	−17.1983	−0.766074
$505$	−13.7254	−0.610771
$506$	−120.503	−5.35699
$507$	−53.8296	−2.39066
$508$	64.6559	2.86864
$509$	7.98458	0.353910	0.176955	−	0.984219i	$-0.443375\pi$
$509$	7.98458	0.353910	0.176955	+	0.984219i	$0.443375\pi$
$510$	−28.1144	−1.24493
$511$	18.1277	0.801922
$512$	49.8490	2.20303
$513$	−1.00244	−0.0442587
$514$	−4.20976	−0.185685
$515$	13.0211	0.573780
$516$	−43.9842	−1.93630
$517$	−54.6513	−2.40356
$518$	86.1555	3.78545
$519$	−3.20559	−0.140710
$520$	−82.0177	−3.59671
$521$	−10.3063	−0.451527	−0.225763	−	0.974182i	$-0.572488\pi$
$521$	−10.3063	−0.451527	−0.225763	+	0.974182i	$0.572488\pi$
$522$	5.68998	0.249044
$523$	5.41422	0.236748	0.118374	−	0.992969i	$-0.462232\pi$
$523$	5.41422	0.236748	0.118374	+	0.992969i	$0.462232\pi$
$524$	−70.4464	−3.07747
$525$	7.80157	0.340488
$526$	37.6890	1.64332
$527$	−29.3775	−1.27971
$528$	−96.0622	−4.18057
$529$	35.1414	1.52789
$530$	4.95115	0.215064
$531$	4.81581	0.208988
$532$	−3.36465	−0.145876
$533$	−10.0595	−0.435727
$534$	2.68954	0.116388
$535$	−14.6556	−0.633619
$536$	−70.8864	−3.06183
$537$	10.0839	0.435152
$538$	−28.6460	−1.23502
$539$	15.9119	0.685374
$540$	37.6583	1.62056
$541$	6.20288	0.266682	0.133341	−	0.991070i	$-0.457429\pi$
$541$	6.20288	0.266682	0.133341	+	0.991070i	$0.457429\pi$
$542$	17.7373	0.761882
$543$	−37.8961	−1.62628
$544$	20.6879	0.886988
$545$	33.9308	1.45344
$546$	99.1031	4.24122
$547$	−24.9925	−1.06860	−0.534301	−	0.845294i	$-0.679425\pi$
$547$	−24.9925	−1.06860	−0.534301	+	0.845294i	$0.679425\pi$
$548$	17.9939	0.768662
$549$	−8.03094	−0.342752
$550$	20.3453	0.867525
$551$	0.629490	0.0268172
$552$	99.8131	4.24833
$553$	−13.8822	−0.590330
$554$	−25.9555	−1.10274
$555$	40.8353	1.73336
$556$	79.7588	3.38253
$557$	−1.40811	−0.0596636	−0.0298318	−	0.999555i	$-0.509497\pi$
$557$	−1.40811	−0.0596636	−0.0298318	+	0.999555i	$0.509497\pi$
$558$	−21.6075	−0.914719
$559$	31.0736	1.31427
$560$	47.6131	2.01202
$561$	34.9229	1.47445
$562$	−37.4075	−1.57794
$563$	−20.7359	−0.873913	−0.436956	−	0.899483i	$-0.643944\pi$
$563$	−20.7359	−0.873913	−0.436956	+	0.899483i	$0.643944\pi$
$564$	80.0512	3.37076
$565$	23.3031	0.980368
$566$	29.1132	1.22372
$567$	−33.4456	−1.40458
$568$	14.1534	0.593863
$569$	32.3798	1.35743	0.678716	−	0.734400i	$-0.262537\pi$
$569$	32.3798	1.35743	0.678716	+	0.734400i	$0.262537\pi$
$570$	−2.28769	−0.0958206
$571$	−2.25737	−0.0944682	−0.0472341	−	0.998884i	$-0.515041\pi$
$571$	−2.25737	−0.0944682	−0.0472341	+	0.998884i	$0.515041\pi$
$572$	180.163	7.53298
$573$	10.8925	0.455040
$574$	12.5760	0.524912
$575$	−9.81640	−0.409372
$576$	1.96088	0.0817034
$577$	46.0912	1.91880	0.959402	−	0.282044i	$-0.0910123\pi$
$577$	46.0912	1.91880	0.959402	+	0.282044i	$0.0910123\pi$
$578$	22.0530	0.917285
$579$	5.54245	0.230336
$580$	−23.6479	−0.981926
$581$	32.5938	1.35222
$582$	−44.9518	−1.86331
$583$	−6.15017	−0.254714
$584$	−39.1575	−1.62035
$585$	10.1838	0.421047
$586$	−68.3988	−2.82553
$587$	−32.3043	−1.33334	−0.666670	−	0.745353i	$-0.732281\pi$
$587$	−32.3043	−1.33334	−0.666670	+	0.745353i	$0.732281\pi$
$588$	−23.3072	−0.961171
$589$	−2.39047	−0.0984974
$590$	−28.7115	−1.18203
$591$	−35.2142	−1.44852
$592$	−86.4191	−3.55180
$593$	−6.44386	−0.264618	−0.132309	−	0.991209i	$-0.542239\pi$
$593$	−6.44386	−0.264618	−0.132309	+	0.991209i	$0.542239\pi$
$594$	−67.1037	−2.75330
$595$	−17.3095	−0.709619
$596$	−79.0690	−3.23879
$597$	5.77155	0.236214
$598$	−124.698	−5.09926
$599$	−1.11296	−0.0454741	−0.0227371	−	0.999741i	$-0.507238\pi$
$599$	−1.11296	−0.0454741	−0.0227371	+	0.999741i	$0.507238\pi$
$600$	−16.8521	−0.687984
$601$	−21.9545	−0.895542	−0.447771	−	0.894148i	$-0.647782\pi$
$601$	−21.9545	−0.895542	−0.447771	+	0.894148i	$0.647782\pi$
$602$	−38.8468	−1.58328
$603$	8.80165	0.358431
$604$	4.60287	0.187288
$605$	51.6860	2.10134
$606$	−35.8242	−1.45526
$607$	2.54798	0.103419	0.0517097	−	0.998662i	$-0.483533\pi$
$607$	2.54798	0.103419	0.0517097	+	0.998662i	$0.483533\pi$
$608$	1.68339	0.0682704
$609$	16.1583	0.654768
$610$	47.8798	1.93860
$611$	−56.5539	−2.28792
$612$	−11.0904	−0.448302
$613$	12.3377	0.498316	0.249158	−	0.968463i	$-0.419846\pi$
$613$	12.3377	0.498316	0.249158	+	0.968463i	$0.419846\pi$
$614$	74.1893	2.99404
$615$	5.96067	0.240357
$616$	−127.366	−5.13171
$617$	13.5676	0.546212	0.273106	−	0.961984i	$-0.411949\pi$
$617$	13.5676	0.546212	0.273106	+	0.961984i	$0.411949\pi$
$618$	33.9861	1.36712
$619$	0.945242	0.0379925	0.0189962	−	0.999820i	$-0.493953\pi$
$619$	0.945242	0.0379925	0.0189962	+	0.999820i	$0.493953\pi$
$620$	89.8021	3.60654
$621$	32.3769	1.29924
$622$	−37.0275	−1.48467
$623$	1.65589	0.0663419
$624$	−99.4064	−3.97944
$625$	−16.9057	−0.676228
$626$	−0.241912	−0.00966875
$627$	2.84169	0.113486
$628$	−44.1626	−1.76228
$629$	31.4172	1.25269
$630$	−12.7313	−0.507228
$631$	−24.8837	−0.990605	−0.495303	−	0.868721i	$-0.664943\pi$
$631$	−24.8837	−0.990605	−0.495303	+	0.868721i	$0.664943\pi$
$632$	29.9868	1.19281
$633$	−49.1230	−1.95246
$634$	−18.4182	−0.731482
$635$	27.0657	1.07407
$636$	9.00854	0.357212
$637$	16.4658	0.652400
$638$	42.1384	1.66827
$639$	−1.75737	−0.0695203
$640$	15.7876	0.624061
$641$	18.5341	0.732054	0.366027	−	0.930604i	$-0.380718\pi$
$641$	18.5341	0.732054	0.366027	+	0.930604i	$0.380718\pi$
$642$	−38.2523	−1.50970
$643$	13.5133	0.532911	0.266455	−	0.963847i	$-0.414147\pi$
$643$	13.5133	0.532911	0.266455	+	0.963847i	$0.414147\pi$
$644$	108.672	4.28228
$645$	−18.4123	−0.724983
$646$	−1.76006	−0.0692488
$647$	−7.52570	−0.295866	−0.147933	−	0.988997i	$-0.547262\pi$
$647$	−7.52570	−0.295866	−0.147933	+	0.988997i	$0.547262\pi$
$648$	72.2457	2.83808
$649$	35.6645	1.39995
$650$	21.0535	0.825787
$651$	−61.3607	−2.40491
$652$	61.2339	2.39810
$653$	6.81586	0.266725	0.133363	−	0.991067i	$-0.457423\pi$
$653$	6.81586	0.266725	0.133363	+	0.991067i	$0.457423\pi$
$654$	88.5618	3.46304
$655$	−29.4897	−1.15226
$656$	−12.6145	−0.492513
$657$	4.86202	0.189685
$658$	70.7011	2.75622
$659$	1.03706	0.0403982	0.0201991	−	0.999796i	$-0.493570\pi$
$659$	1.03706	0.0403982	0.0201991	+	0.999796i	$0.493570\pi$
$660$	−106.753	−4.15537
$661$	−18.4538	−0.717770	−0.358885	−	0.933382i	$-0.616843\pi$
$661$	−18.4538	−0.717770	−0.358885	+	0.933382i	$0.616843\pi$
$662$	54.8273	2.13092
$663$	36.1387	1.40351
$664$	−70.4056	−2.73227
$665$	−1.40848	−0.0546185
$666$	23.1077	0.895405
$667$	−20.3314	−0.787234
$668$	49.1347	1.90108
$669$	−31.4142	−1.21454
$670$	−52.4747	−2.02727
$671$	−59.4749	−2.29600
$672$	43.2107	1.66689
$673$	47.3958	1.82697	0.913487	−	0.406868i	$-0.133379\pi$
$673$	47.3958	1.82697	0.913487	+	0.406868i	$0.133379\pi$
$674$	53.2323	2.05043
$675$	−5.46641	−0.210402
$676$	126.597	4.86912
$677$	−30.2052	−1.16088	−0.580441	−	0.814303i	$-0.697120\pi$
$677$	−30.2052	−1.16088	−0.580441	+	0.814303i	$0.697120\pi$
$678$	60.8227	2.33588
$679$	−27.6759	−1.06210
$680$	37.3901	1.43384
$681$	−3.60962	−0.138321
$682$	−160.019	−6.12744
$683$	7.78364	0.297833	0.148916	−	0.988850i	$-0.452421\pi$
$683$	7.78364	0.297833	0.148916	+	0.988850i	$0.452421\pi$
$684$	−0.902430	−0.0345053
$685$	7.53245	0.287800
$686$	35.1095	1.34049
$687$	7.75498	0.295871
$688$	38.9657	1.48555
$689$	−6.36427	−0.242459
$690$	73.8881	2.81287
$691$	25.3481	0.964287	0.482144	−	0.876092i	$-0.339858\pi$
$691$	25.3481	0.964287	0.482144	+	0.876092i	$0.339858\pi$
$692$	7.53895	0.286588
$693$	15.8145	0.600741
$694$	66.9318	2.54070
$695$	33.3879	1.26648
$696$	−34.9035	−1.32301
$697$	4.58593	0.173704
$698$	−43.6125	−1.65076
$699$	36.5303	1.38170
$700$	−18.3478	−0.693483
$701$	−30.1439	−1.13852	−0.569260	−	0.822158i	$-0.692770\pi$
$701$	−30.1439	−1.13852	−0.569260	+	0.822158i	$0.692770\pi$
$702$	−69.4397	−2.62083
$703$	2.55643	0.0964177
$704$	14.5217	0.547308
$705$	33.5103	1.26207
$706$	27.5698	1.03760
$707$	−22.0562	−0.829509
$708$	−52.2400	−1.96330
$709$	−42.6602	−1.60214	−0.801069	−	0.598572i	$-0.795735\pi$
$709$	−42.6602	−1.60214	−0.801069	+	0.598572i	$0.795735\pi$
$710$	10.4773	0.393205
$711$	−3.72333	−0.139636
$712$	−3.57688	−0.134049
$713$	77.2077	2.89145
$714$	−45.1790	−1.69078
$715$	75.4181	2.82048
$716$	−23.7154	−0.886288
$717$	56.7756	2.12032
$718$	17.8028	0.664396
$719$	38.5686	1.43837	0.719183	−	0.694820i	$-0.244516\pi$
$719$	38.5686	1.43837	0.719183	+	0.694820i	$0.244516\pi$
$720$	12.7703	0.475920
$721$	20.9245	0.779270
$722$	48.6793	1.81166
$723$	−1.96122	−0.0729384
$724$	89.1246	3.31229
$725$	3.43268	0.127487
$726$	134.904	5.00676
$727$	22.9505	0.851189	0.425594	−	0.904914i	$-0.360065\pi$
$727$	22.9505	0.851189	0.425594	+	0.904914i	$0.360065\pi$
$728$	−131.800	−4.88482
$729$	15.9606	0.591132
$730$	−28.9869	−1.07285
$731$	−14.1658	−0.523940
$732$	87.1165	3.21992
$733$	−20.8831	−0.771336	−0.385668	−	0.922638i	$-0.626029\pi$
$733$	−20.8831	−0.771336	−0.385668	+	0.922638i	$0.626029\pi$
$734$	−19.0091	−0.701639
$735$	−9.75664	−0.359879
$736$	−54.3704	−2.00412
$737$	65.1825	2.40103
$738$	3.37300	0.124162
$739$	22.4207	0.824760	0.412380	−	0.911012i	$-0.364698\pi$
$739$	22.4207	0.824760	0.412380	+	0.911012i	$0.364698\pi$
$740$	−96.0370	−3.53039
$741$	2.94062	0.108026
$742$	7.95633	0.292086
$743$	−6.87486	−0.252214	−0.126107	−	0.992017i	$-0.540248\pi$
$743$	−6.87486	−0.252214	−0.126107	+	0.992017i	$0.540248\pi$
$744$	132.545	4.85933
$745$	−33.0992	−1.21266
$746$	−33.8675	−1.23998
$747$	8.74195	0.319851
$748$	−82.1322	−3.00305
$749$	−23.5511	−0.860539
$750$	−60.9259	−2.22470
$751$	−3.98480	−0.145408	−0.0727038	−	0.997354i	$-0.523163\pi$
$751$	−3.98480	−0.145408	−0.0727038	+	0.997354i	$0.523163\pi$
$752$	−70.9175	−2.58609
$753$	41.4306	1.50982
$754$	43.6053	1.58801
$755$	1.92681	0.0701239
$756$	60.5157	2.20093
$757$	26.9694	0.980219	0.490110	−	0.871661i	$-0.336957\pi$
$757$	26.9694	0.980219	0.490110	+	0.871661i	$0.336957\pi$
$758$	36.1956	1.31468
$759$	−91.7816	−3.33146
$760$	3.04245	0.110361
$761$	36.7973	1.33390	0.666950	−	0.745103i	$-0.267600\pi$
$761$	36.7973	1.33390	0.666950	+	0.745103i	$0.267600\pi$
$762$	70.6434	2.55914
$763$	54.5257	1.97396
$764$	−25.6171	−0.926794
$765$	−4.64256	−0.167852
$766$	−72.4104	−2.61629
$767$	36.9061	1.33260
$768$	50.4493	1.82043
$769$	0.0864354	0.00311694	0.00155847	−	0.999999i	$-0.499504\pi$
$769$	0.0864354	0.00311694	0.00155847	+	0.999999i	$0.499504\pi$
$770$	−94.2844	−3.39777
$771$	−3.20639	−0.115475
$772$	−13.0348	−0.469133
$773$	−44.3466	−1.59504	−0.797518	−	0.603295i	$-0.793854\pi$
$773$	−44.3466	−1.59504	−0.797518	+	0.603295i	$0.793854\pi$
$774$	−10.4191	−0.374506
$775$	−13.0355	−0.468249
$776$	59.7824	2.14606
$777$	65.6209	2.35414
$778$	55.0487	1.97359
$779$	0.373159	0.0133698
$780$	−110.470	−3.95545
$781$	−13.0145	−0.465697
$782$	56.8469	2.03284
$783$	−11.3218	−0.404609
$784$	20.6478	0.737423
$785$	−18.4870	−0.659828
$786$	−76.9701	−2.74543
$787$	27.7727	0.989989	0.494995	−	0.868896i	$-0.335170\pi$
$787$	27.7727	0.989989	0.494995	+	0.868896i	$0.335170\pi$
$788$	82.8173	2.95024
$789$	28.7061	1.02196
$790$	22.1982	0.789776
$791$	37.4473	1.33147
$792$	−34.1607	−1.21385
$793$	−61.5453	−2.18554
$794$	15.2490	0.541167
$795$	3.77108	0.133746
$796$	−13.5736	−0.481104
$797$	11.8945	0.421323	0.210662	−	0.977559i	$-0.432438\pi$
$797$	11.8945	0.421323	0.210662	+	0.977559i	$0.432438\pi$
$798$	−3.67623	−0.130137
$799$	25.7817	0.912090
$800$	9.17971	0.324552
$801$	0.444126	0.0156924
$802$	3.90033	0.137725
$803$	36.0067	1.27065
$804$	−95.4769	−3.36721
$805$	45.4914	1.60336
$806$	−165.590	−5.83264
$807$	−21.8184	−0.768045
$808$	47.6434	1.67609
$809$	4.14679	0.145794	0.0728968	−	0.997339i	$-0.476776\pi$
$809$	4.14679	0.145794	0.0728968	+	0.997339i	$0.476776\pi$
$810$	53.4809	1.87913
$811$	17.1220	0.601235	0.300617	−	0.953745i	$-0.402807\pi$
$811$	17.1220	0.601235	0.300617	+	0.953745i	$0.402807\pi$
$812$	−38.0014	−1.33359
$813$	13.5097	0.473807
$814$	171.129	5.99807
$815$	25.6332	0.897891
$816$	45.3172	1.58642
$817$	−1.15267	−0.0403270
$818$	10.3471	0.361779
$819$	16.3650	0.571839
$820$	−14.0184	−0.489543
$821$	−7.96579	−0.278008	−0.139004	−	0.990292i	$-0.544390\pi$
$821$	−7.96579	−0.278008	−0.139004	+	0.990292i	$0.544390\pi$
$822$	19.6602	0.685729
$823$	7.28353	0.253888	0.126944	−	0.991910i	$-0.459483\pi$
$823$	7.28353	0.253888	0.126944	+	0.991910i	$0.459483\pi$
$824$	−45.1989	−1.57458
$825$	15.4961	0.539505
$826$	−46.1383	−1.60536
$827$	40.7437	1.41680	0.708399	−	0.705812i	$-0.249418\pi$
$827$	40.7437	1.41680	0.708399	+	0.705812i	$0.249418\pi$
$828$	29.1469	1.01292
$829$	−18.6694	−0.648414	−0.324207	−	0.945986i	$-0.605097\pi$
$829$	−18.6694	−0.648414	−0.324207	+	0.945986i	$0.605097\pi$
$830$	−52.1188	−1.80907
$831$	−19.7692	−0.685786
$832$	15.0273	0.520977
$833$	−7.50642	−0.260082
$834$	87.1449	3.01758
$835$	20.5684	0.711797
$836$	−6.68314	−0.231141
$837$	42.9943	1.48610
$838$	39.3774	1.36027
$839$	10.6419	0.367399	0.183700	−	0.982982i	$-0.441193\pi$
$839$	10.6419	0.367399	0.183700	+	0.982982i	$0.441193\pi$
$840$	78.0964	2.69458
$841$	−21.8903	−0.754840
$842$	41.4563	1.42868
$843$	−28.4917	−0.981305
$844$	115.528	3.97664
$845$	52.9950	1.82308
$846$	18.9627	0.651951
$847$	83.0577	2.85390
$848$	−7.98068	−0.274058
$849$	22.1743	0.761019
$850$	−9.59784	−0.329203
$851$	−82.5682	−2.83040
$852$	19.0632	0.653095
$853$	16.8532	0.577041	0.288520	−	0.957474i	$-0.406837\pi$
$853$	16.8532	0.577041	0.288520	+	0.957474i	$0.406837\pi$
$854$	76.9412	2.63287
$855$	−0.377768	−0.0129194
$856$	50.8726	1.73879
$857$	23.0230	0.786449	0.393225	−	0.919442i	$-0.371360\pi$
$857$	23.0230	0.786449	0.393225	+	0.919442i	$0.371360\pi$
$858$	196.847	6.72023
$859$	43.1268	1.47147	0.735733	−	0.677271i	$-0.236838\pi$
$859$	43.1268	1.47147	0.735733	+	0.677271i	$0.236838\pi$
$860$	43.3023	1.47660
$861$	9.57859	0.326438
$862$	73.9819	2.51983
$863$	39.5391	1.34593	0.672963	−	0.739676i	$-0.265021\pi$
$863$	39.5391	1.34593	0.672963	+	0.739676i	$0.265021\pi$
$864$	−30.2770	−1.03004
$865$	3.15589	0.107303
$866$	61.1218	2.07700
$867$	16.7968	0.570451
$868$	144.309	4.89816
$869$	−27.5739	−0.935381
$870$	−25.8378	−0.875984
$871$	67.4516	2.28551
$872$	−117.781	−3.98855
$873$	−7.42292	−0.251228
$874$	4.62566	0.156465
$875$	−37.5108	−1.26810
$876$	−52.7413	−1.78196
$877$	−46.6303	−1.57459	−0.787297	−	0.616574i	$-0.788520\pi$
$877$	−46.6303	−1.57459	−0.787297	+	0.616574i	$0.788520\pi$
$878$	−2.58052	−0.0870882
$879$	−52.0964	−1.75717
$880$	94.5729	3.18805
$881$	10.0041	0.337045	0.168523	−	0.985698i	$-0.446100\pi$
$881$	10.0041	0.337045	0.168523	+	0.985698i	$0.446100\pi$
$882$	−5.52105	−0.185903
$883$	−0.433467	−0.0145873	−0.00729366	−	0.999973i	$-0.502322\pi$
$883$	−0.433467	−0.0145873	−0.00729366	+	0.999973i	$0.502322\pi$
$884$	−84.9914	−2.85857
$885$	−21.8683	−0.735094
$886$	−70.4044	−2.36528
$887$	37.3796	1.25508	0.627541	−	0.778583i	$-0.284061\pi$
$887$	37.3796	1.25508	0.627541	+	0.778583i	$0.284061\pi$
$888$	−141.747	−4.75673
$889$	43.4936	1.45873
$890$	−2.64784	−0.0887558
$891$	−66.4324	−2.22557
$892$	73.8804	2.47370
$893$	2.09787	0.0702025
$894$	−86.3912	−2.88935
$895$	−9.92755	−0.331841
$896$	25.3702	0.847559
$897$	−94.9767	−3.17118
$898$	85.7196	2.86050
$899$	−26.9986	−0.900455
$900$	−4.92106	−0.164035
$901$	2.90133	0.0966574
$902$	24.9795	0.831725
$903$	−29.5879	−0.984624
$904$	−80.8896	−2.69035
$905$	37.3086	1.24018
$906$	5.02912	0.167081
$907$	−22.6439	−0.751879	−0.375940	−	0.926644i	$-0.622680\pi$
$907$	−22.6439	−0.751879	−0.375940	+	0.926644i	$0.622680\pi$
$908$	8.48916	0.281723
$909$	−5.91568	−0.196211
$910$	−97.5667	−3.23430
$911$	31.0427	1.02849	0.514246	−	0.857643i	$-0.328072\pi$
$911$	31.0427	1.02849	0.514246	+	0.857643i	$0.328072\pi$
$912$	3.68748	0.122105
$913$	64.7404	2.14259
$914$	4.69854	0.155414
$915$	36.4680	1.20559
$916$	−18.2383	−0.602610
$917$	−47.3889	−1.56492
$918$	31.6560	1.04481
$919$	53.4816	1.76420	0.882098	−	0.471066i	$-0.156130\pi$
$919$	53.4816	1.76420	0.882098	+	0.471066i	$0.156130\pi$
$920$	−98.2656	−3.23972
$921$	56.5068	1.86196
$922$	51.8937	1.70903
$923$	−13.4676	−0.443292
$924$	−171.549	−5.64355
$925$	13.9405	0.458362
$926$	−101.294	−3.32872
$927$	5.61215	0.184327
$928$	19.0127	0.624122
$929$	−43.2319	−1.41839	−0.709196	−	0.705011i	$-0.750942\pi$
$929$	−43.2319	−1.41839	−0.709196	+	0.705011i	$0.750942\pi$
$930$	98.1182	3.21742
$931$	−0.610801	−0.0200182
$932$	−85.9125	−2.81416
$933$	−28.2023	−0.923301
$934$	−2.41163	−0.0789110
$935$	−34.3815	−1.12439
$936$	−35.3499	−1.15545
$937$	8.45574	0.276237	0.138119	−	0.990416i	$-0.455895\pi$
$937$	8.45574	0.276237	0.138119	+	0.990416i	$0.455895\pi$
$938$	−84.3251	−2.75331
$939$	−0.184254	−0.00601290
$940$	−78.8101	−2.57050
$941$	−27.3643	−0.892050	−0.446025	−	0.895020i	$-0.647161\pi$
$941$	−27.3643	−0.892050	−0.446025	+	0.895020i	$0.647161\pi$
$942$	−48.2522	−1.57214
$943$	−12.0524	−0.392479
$944$	46.2795	1.50627
$945$	25.3326	0.824068
$946$	−77.1607	−2.50871
$947$	−24.4766	−0.795382	−0.397691	−	0.917520i	$-0.630188\pi$
$947$	−24.4766	−0.795382	−0.397691	+	0.917520i	$0.630188\pi$
$948$	40.3892	1.31178
$949$	37.2602	1.20952
$950$	−0.780981	−0.0253384
$951$	−14.0284	−0.454901
$952$	60.0846	1.94735
$953$	19.2937	0.624984	0.312492	−	0.949920i	$-0.398836\pi$
$953$	19.2937	0.624984	0.312492	+	0.949920i	$0.398836\pi$
$954$	2.13396	0.0690896
$955$	−10.7236	−0.347008
$956$	−133.526	−4.31853
$957$	32.0950	1.03748
$958$	−92.1626	−2.97764
$959$	12.1044	0.390871
$960$	−8.90424	−0.287383
$961$	71.5264	2.30730
$962$	177.086	5.70949
$963$	−6.31663	−0.203551
$964$	4.61242	0.148556
$965$	−5.45652	−0.175652
$966$	118.736	3.82026
$967$	−2.54289	−0.0817739	−0.0408870	−	0.999164i	$-0.513018\pi$
$967$	−2.54289	−0.0817739	−0.0408870	+	0.999164i	$0.513018\pi$
$968$	−179.412	−5.76653
$969$	−1.34056	−0.0430651
$970$	44.2548	1.42094
$971$	−48.9246	−1.57006	−0.785032	−	0.619455i	$-0.787353\pi$
$971$	−48.9246	−1.57006	−0.785032	+	0.619455i	$0.787353\pi$
$972$	38.6747	1.24049
$973$	53.6533	1.72005
$974$	−87.9556	−2.81828
$975$	16.0356	0.513549
$976$	−77.1767	−2.47036
$977$	2.22069	0.0710461	0.0355230	−	0.999369i	$-0.488690\pi$
$977$	2.22069	0.0710461	0.0355230	+	0.999369i	$0.488690\pi$
$978$	66.9044	2.13937
$979$	3.28907	0.105119
$980$	22.9458	0.732977
$981$	14.6243	0.466918
$982$	−14.4659	−0.461625
$983$	−0.529805	−0.0168981	−0.00844907	−	0.999964i	$-0.502689\pi$
$983$	−0.529805	−0.0168981	−0.00844907	+	0.999964i	$0.502689\pi$
$984$	−20.6906	−0.659594
$985$	34.6683	1.10462
$986$	−19.8787	−0.633067
$987$	53.8500	1.71406
$988$	−6.91579	−0.220021
$989$	37.2293	1.18382
$990$	−25.2880	−0.803704
$991$	−24.2643	−0.770780	−0.385390	−	0.922754i	$-0.625933\pi$
$991$	−24.2643	−0.770780	−0.385390	+	0.922754i	$0.625933\pi$
$992$	−72.2000	−2.29235
$993$	41.7596	1.32520
$994$	16.8366	0.534025
$995$	−5.68207	−0.180134
$996$	−94.8293	−3.00478
$997$	48.2040	1.52664	0.763318	−	0.646023i	$-0.223569\pi$
$997$	48.2040	1.52664	0.763318	+	0.646023i	$0.223569\pi$
$998$	46.2326	1.46347
$999$	−45.9793	−1.45472

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8003.2.a.c.1.10	✓	172

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8003.2.a.c.1.10	✓	172	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:	\( N \)	\(=\)	\( 8003 = 53 \cdot 151 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8003.a (trivial)

\(f(q)\)	\(=\)	\(q-2.56961 q^{2} -1.95716 q^{3} +4.60287 q^{4} +1.92681 q^{5} +5.02912 q^{6} +3.09633 q^{7} -6.68835 q^{8} +0.830463 q^{9} +O(q^{10})\)	\(q-2.56961 q^{2} -1.95716 q^{3} +4.60287 q^{4} +1.92681 q^{5} +5.02912 q^{6} +3.09633 q^{7} -6.68835 q^{8} +0.830463 q^{9} -4.95115 q^{10} +6.15017 q^{11} -9.00854 q^{12} +6.36427 q^{13} -7.95633 q^{14} -3.77108 q^{15} +7.98068 q^{16} -2.90133 q^{17} -2.13396 q^{18} -0.236083 q^{19} +8.86888 q^{20} -6.05999 q^{21} -15.8035 q^{22} +7.62505 q^{23} +13.0902 q^{24} -1.28739 q^{25} -16.3537 q^{26} +4.24612 q^{27} +14.2520 q^{28} -2.66639 q^{29} +9.69018 q^{30} +10.1255 q^{31} -7.13049 q^{32} -12.0369 q^{33} +7.45528 q^{34} +5.96604 q^{35} +3.82251 q^{36} -10.8285 q^{37} +0.606640 q^{38} -12.4559 q^{39} -12.8872 q^{40} -1.58063 q^{41} +15.5718 q^{42} +4.88250 q^{43} +28.3084 q^{44} +1.60015 q^{45} -19.5934 q^{46} -8.88615 q^{47} -15.6194 q^{48} +2.58723 q^{49} +3.30808 q^{50} +5.67836 q^{51} +29.2939 q^{52} -1.00000 q^{53} -10.9109 q^{54} +11.8502 q^{55} -20.7093 q^{56} +0.462051 q^{57} +6.85158 q^{58} +5.79894 q^{59} -17.3578 q^{60} -9.67044 q^{61} -26.0186 q^{62} +2.57138 q^{63} +2.36119 q^{64} +12.2628 q^{65} +30.9300 q^{66} +10.5985 q^{67} -13.3545 q^{68} -14.9234 q^{69} -15.3304 q^{70} -2.11613 q^{71} -5.55443 q^{72} +5.85458 q^{73} +27.8251 q^{74} +2.51962 q^{75} -1.08666 q^{76} +19.0429 q^{77} +32.0067 q^{78} -4.48344 q^{79} +15.3773 q^{80} -10.8017 q^{81} +4.06159 q^{82} +10.5266 q^{83} -27.8934 q^{84} -5.59033 q^{85} -12.5461 q^{86} +5.21855 q^{87} -41.1345 q^{88} +0.534793 q^{89} -4.11175 q^{90} +19.7059 q^{91} +35.0971 q^{92} -19.8173 q^{93} +22.8339 q^{94} -0.454888 q^{95} +13.9555 q^{96} -8.93829 q^{97} -6.64816 q^{98} +5.10749 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 172 q + 8 q^{2} + 25 q^{3} + 188 q^{4} + 27 q^{5} + 10 q^{6} + 31 q^{7} + 21 q^{8} + 179 q^{9}+O(q^{10}) \) 172 * q + 8 * q^2 + 25 * q^3 + 188 * q^4 + 27 * q^5 + 10 * q^6 + 31 * q^7 + 21 * q^8 + 179 * q^9	\( 172 q + 8 q^{2} + 25 q^{3} + 188 q^{4} + 27 q^{5} + 10 q^{6} + 31 q^{7} + 21 q^{8} + 179 q^{9} + 20 q^{10} - 3 q^{11} + 66 q^{12} + 121 q^{13} + 12 q^{14} + 30 q^{15} + 212 q^{16} + 8 q^{17} + 40 q^{18} + 41 q^{19} + 64 q^{20} + 56 q^{21} + 50 q^{22} + 28 q^{23} + 30 q^{24} + 231 q^{25} + 38 q^{26} + 100 q^{27} + 80 q^{28} + 26 q^{29} + 55 q^{30} + 66 q^{31} + 65 q^{32} + 99 q^{33} + 81 q^{34} + 36 q^{35} + 212 q^{36} + 153 q^{37} + q^{38} + 20 q^{39} + 59 q^{40} + 40 q^{41} + 50 q^{42} + 39 q^{43} - 51 q^{44} + 123 q^{45} + 59 q^{46} + 29 q^{47} + 128 q^{48} + 245 q^{49} + 19 q^{50} + 36 q^{51} + 215 q^{52} - 172 q^{53} + 40 q^{54} + 40 q^{55} + 15 q^{56} + 54 q^{57} + 44 q^{58} - 54 q^{60} + 100 q^{61} - 29 q^{62} + 92 q^{63} + 253 q^{64} + 77 q^{65} + 14 q^{66} + 126 q^{67} - 27 q^{68} + 47 q^{69} + 72 q^{70} + 38 q^{71} + 65 q^{72} + 185 q^{73} + 48 q^{74} + 75 q^{75} + 38 q^{76} + 120 q^{77} + 75 q^{78} + 79 q^{79} + 43 q^{80} + 232 q^{81} + 110 q^{82} + 90 q^{83} + 158 q^{84} + 115 q^{85} + 68 q^{86} + 61 q^{87} + 15 q^{88} - 36 q^{89} - 6 q^{90} + 33 q^{91} + 139 q^{92} + 103 q^{93} - 24 q^{94} - 45 q^{95} + 34 q^{96} + 159 q^{97} - 36 q^{98} + 27 q^{99}+O(q^{100}) \) 172 * q + 8 * q^2 + 25 * q^3 + 188 * q^4 + 27 * q^5 + 10 * q^6 + 31 * q^7 + 21 * q^8 + 179 * q^9 + 20 * q^10 - 3 * q^11 + 66 * q^12 + 121 * q^13 + 12 * q^14 + 30 * q^15 + 212 * q^16 + 8 * q^17 + 40 * q^18 + 41 * q^19 + 64 * q^20 + 56 * q^21 + 50 * q^22 + 28 * q^23 + 30 * q^24 + 231 * q^25 + 38 * q^26 + 100 * q^27 + 80 * q^28 + 26 * q^29 + 55 * q^30 + 66 * q^31 + 65 * q^32 + 99 * q^33 + 81 * q^34 + 36 * q^35 + 212 * q^36 + 153 * q^37 + q^38 + 20 * q^39 + 59 * q^40 + 40 * q^41 + 50 * q^42 + 39 * q^43 - 51 * q^44 + 123 * q^45 + 59 * q^46 + 29 * q^47 + 128 * q^48 + 245 * q^49 + 19 * q^50 + 36 * q^51 + 215 * q^52 - 172 * q^53 + 40 * q^54 + 40 * q^55 + 15 * q^56 + 54 * q^57 + 44 * q^58 - 54 * q^60 + 100 * q^61 - 29 * q^62 + 92 * q^63 + 253 * q^64 + 77 * q^65 + 14 * q^66 + 126 * q^67 - 27 * q^68 + 47 * q^69 + 72 * q^70 + 38 * q^71 + 65 * q^72 + 185 * q^73 + 48 * q^74 + 75 * q^75 + 38 * q^76 + 120 * q^77 + 75 * q^78 + 79 * q^79 + 43 * q^80 + 232 * q^81 + 110 * q^82 + 90 * q^83 + 158 * q^84 + 115 * q^85 + 68 * q^86 + 61 * q^87 + 15 * q^88 - 36 * q^89 - 6 * q^90 + 33 * q^91 + 139 * q^92 + 103 * q^93 - 24 * q^94 - 45 * q^95 + 34 * q^96 + 159 * q^97 - 36 * q^98 + 27 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more