LMFDB - Embedded newform 8003.2.a.c.1.3

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			1.3
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)

$f(q)$	$=$	$q-2.76129 q^{2} +2.33703 q^{3} +5.62470 q^{4} -2.36515 q^{5} -6.45322 q^{6} +1.79490 q^{7} -10.0088 q^{8} +2.46173 q^{9} +O(q^{10})$	\(q-2.76129 q^{2} +2.33703 q^{3} +5.62470 q^{4} -2.36515 q^{5} -6.45322 q^{6} +1.79490 q^{7} -10.0088 q^{8} +2.46173 q^{9} +6.53087 q^{10} -3.92879 q^{11} +13.1451 q^{12} +7.17497 q^{13} -4.95624 q^{14} -5.52745 q^{15} +16.3879 q^{16} -5.22520 q^{17} -6.79754 q^{18} +5.03522 q^{19} -13.3033 q^{20} +4.19475 q^{21} +10.8485 q^{22} +2.81115 q^{23} -23.3910 q^{24} +0.593953 q^{25} -19.8121 q^{26} -1.25795 q^{27} +10.0958 q^{28} +5.34023 q^{29} +15.2629 q^{30} -4.38643 q^{31} -25.2339 q^{32} -9.18172 q^{33} +14.4283 q^{34} -4.24522 q^{35} +13.8465 q^{36} -8.62816 q^{37} -13.9037 q^{38} +16.7682 q^{39} +23.6724 q^{40} +5.19348 q^{41} -11.5829 q^{42} -7.09294 q^{43} -22.0983 q^{44} -5.82237 q^{45} -7.76240 q^{46} +9.20357 q^{47} +38.2990 q^{48} -3.77832 q^{49} -1.64007 q^{50} -12.2115 q^{51} +40.3571 q^{52} -1.00000 q^{53} +3.47357 q^{54} +9.29219 q^{55} -17.9649 q^{56} +11.7675 q^{57} -14.7459 q^{58} +11.8676 q^{59} -31.0902 q^{60} -11.3443 q^{61} +12.1122 q^{62} +4.41857 q^{63} +36.9023 q^{64} -16.9699 q^{65} +25.3534 q^{66} +3.73957 q^{67} -29.3902 q^{68} +6.56976 q^{69} +11.7223 q^{70} +5.58484 q^{71} -24.6391 q^{72} +16.4782 q^{73} +23.8248 q^{74} +1.38809 q^{75} +28.3216 q^{76} -7.05180 q^{77} -46.3017 q^{78} -3.34689 q^{79} -38.7598 q^{80} -10.3251 q^{81} -14.3407 q^{82} +11.2123 q^{83} +23.5942 q^{84} +12.3584 q^{85} +19.5856 q^{86} +12.4803 q^{87} +39.3226 q^{88} -8.89396 q^{89} +16.0772 q^{90} +12.8784 q^{91} +15.8119 q^{92} -10.2512 q^{93} -25.4137 q^{94} -11.9091 q^{95} -58.9725 q^{96} +11.4072 q^{97} +10.4330 q^{98} -9.67163 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−2.76129	−1.95252	−0.976262	−	0.216593i	$-0.930506\pi$
$2$	−2.76129	−1.95252	−0.976262	+	0.216593i	$0.930506\pi$
$3$	2.33703	1.34929	0.674644	−	0.738143i	$-0.264297\pi$
$3$	2.33703	1.34929	0.674644	+	0.738143i	$0.264297\pi$
$4$	5.62470	2.81235
$5$	−2.36515	−1.05773	−0.528864	−	0.848706i	$-0.677382\pi$
$5$	−2.36515	−1.05773	−0.528864	+	0.848706i	$0.677382\pi$
$6$	−6.45322	−2.63452
$7$	1.79490	0.678410	0.339205	−	0.940713i	$-0.389842\pi$
$7$	1.79490	0.678410	0.339205	+	0.940713i	$0.389842\pi$
$8$	−10.0088	−3.53866
$9$	2.46173	0.820577
$10$	6.53087	2.06524
$11$	−3.92879	−1.18458	−0.592288	−	0.805727i	$-0.701775\pi$
$11$	−3.92879	−1.18458	−0.592288	+	0.805727i	$0.701775\pi$
$12$	13.1451	3.79467
$13$	7.17497	1.98998	0.994989	−	0.0999822i	$-0.0318786\pi$
$13$	7.17497	1.98998	0.994989	+	0.0999822i	$0.0318786\pi$
$14$	−4.95624	−1.32461
$15$	−5.52745	−1.42718
$16$	16.3879	4.09697
$17$	−5.22520	−1.26730	−0.633649	−	0.773621i	$-0.718443\pi$
$17$	−5.22520	−1.26730	−0.633649	+	0.773621i	$0.718443\pi$
$18$	−6.79754	−1.60220
$19$	5.03522	1.15516	0.577579	−	0.816335i	$-0.303998\pi$
$19$	5.03522	1.15516	0.577579	+	0.816335i	$0.303998\pi$
$20$	−13.3033	−2.97470
$21$	4.19475	0.915370
$22$	10.8485	2.31291
$23$	2.81115	0.586166	0.293083	−	0.956087i	$-0.405319\pi$
$23$	2.81115	0.586166	0.293083	+	0.956087i	$0.405319\pi$
$24$	−23.3910	−4.77467
$25$	0.593953	0.118791
$26$	−19.8121	−3.88548
$27$	−1.25795	−0.242093
$28$	10.0958	1.90793
$29$	5.34023	0.991656	0.495828	−	0.868421i	$-0.334865\pi$
$29$	5.34023	0.991656	0.495828	+	0.868421i	$0.334865\pi$
$30$	15.2629	2.78660
$31$	−4.38643	−0.787826	−0.393913	−	0.919148i	$-0.628879\pi$
$31$	−4.38643	−0.787826	−0.393913	+	0.919148i	$0.628879\pi$
$32$	−25.2339	−4.46077
$33$	−9.18172	−1.59833
$34$	14.4283	2.47443
$35$	−4.24522	−0.717574
$36$	13.8465	2.30775
$37$	−8.62816	−1.41846	−0.709230	−	0.704977i	$-0.750957\pi$
$37$	−8.62816	−1.41846	−0.709230	+	0.704977i	$0.750957\pi$
$38$	−13.9037	−2.25547
$39$	16.7682	2.68505
$40$	23.6724	3.74294
$41$	5.19348	0.811085	0.405542	−	0.914076i	$-0.367083\pi$
$41$	5.19348	0.811085	0.405542	+	0.914076i	$0.367083\pi$
$42$	−11.5829	−1.78728
$43$	−7.09294	−1.08166	−0.540831	−	0.841131i	$-0.681890\pi$
$43$	−7.09294	−1.08166	−0.540831	+	0.841131i	$0.681890\pi$
$44$	−22.0983	−3.33144
$45$	−5.82237	−0.867948
$46$	−7.76240	−1.14450
$47$	9.20357	1.34248	0.671239	−	0.741241i	$-0.265762\pi$
$47$	9.20357	1.34248	0.671239	+	0.741241i	$0.265762\pi$
$48$	38.2990	5.52798
$49$	−3.77832	−0.539760
$50$	−1.64007	−0.231941
$51$	−12.2115	−1.70995
$52$	40.3571	5.59652
$53$	−1.00000	−0.137361
$53$	−1.00000	−0.137361
$54$	3.47357	0.472693
$55$	9.29219	1.25296
$56$	−17.9649	−2.40066
$57$	11.7675	1.55864
$58$	−14.7459	−1.93623
$59$	11.8676	1.54503	0.772516	−	0.634996i	$-0.218998\pi$
$59$	11.8676	1.54503	0.772516	+	0.634996i	$0.218998\pi$
$60$	−31.0902	−4.01373
$61$	−11.3443	−1.45249	−0.726247	−	0.687434i	$-0.758737\pi$
$61$	−11.3443	−1.45249	−0.726247	+	0.687434i	$0.758737\pi$
$62$	12.1122	1.53825
$63$	4.41857	0.556687
$64$	36.9023	4.61279
$65$	−16.9699	−2.10486
$66$	25.3534	3.12078
$67$	3.73957	0.456861	0.228430	−	0.973560i	$-0.426641\pi$
$67$	3.73957	0.456861	0.228430	+	0.973560i	$0.426641\pi$
$68$	−29.3902	−3.56408
$69$	6.56976	0.790906
$70$	11.7223	1.40108
$71$	5.58484	0.662799	0.331400	−	0.943491i	$-0.392479\pi$
$71$	5.58484	0.662799	0.331400	+	0.943491i	$0.392479\pi$
$72$	−24.6391	−2.90374
$73$	16.4782	1.92863	0.964316	−	0.264752i	$-0.0852902\pi$
$73$	16.4782	1.92863	0.964316	+	0.264752i	$0.0852902\pi$
$74$	23.8248	2.76958
$75$	1.38809	0.160283
$76$	28.3216	3.24871
$77$	−7.05180	−0.803627
$78$	−46.3017	−5.24263
$79$	−3.34689	−0.376555	−0.188278	−	0.982116i	$-0.560290\pi$
$79$	−3.34689	−0.376555	−0.188278	+	0.982116i	$0.560290\pi$
$80$	−38.7598	−4.33348
$81$	−10.3251	−1.14723
$82$	−14.3407	−1.58366
$83$	11.2123	1.23071	0.615357	−	0.788249i	$-0.289012\pi$
$83$	11.2123	1.23071	0.615357	+	0.788249i	$0.289012\pi$
$84$	23.5942	2.57434
$85$	12.3584	1.34046
$86$	19.5856	2.11197
$87$	12.4803	1.33803
$88$	39.3226	4.19181
$89$	−8.89396	−0.942758	−0.471379	−	0.881931i	$-0.656244\pi$
$89$	−8.89396	−0.942758	−0.471379	+	0.881931i	$0.656244\pi$
$90$	16.0772	1.69469
$91$	12.8784	1.35002
$92$	15.8119	1.64850
$93$	−10.2512	−1.06300
$94$	−25.4137	−2.62122
$95$	−11.9091	−1.22184
$96$	−58.9725	−6.01886
$97$	11.4072	1.15823	0.579113	−	0.815247i	$-0.303399\pi$
$97$	11.4072	1.15823	0.579113	+	0.815247i	$0.303399\pi$
$98$	10.4330	1.05390
$99$	−9.67163	−0.972035
$100$	3.34081	0.334081
$101$	−1.50754	−0.150006	−0.0750031	−	0.997183i	$-0.523897\pi$
$101$	−1.50754	−0.150006	−0.0750031	+	0.997183i	$0.523897\pi$
$102$	33.7194	3.33872
$103$	14.3509	1.41404	0.707020	−	0.707193i	$-0.250039\pi$
$103$	14.3509	1.41404	0.707020	+	0.707193i	$0.250039\pi$
$104$	−71.8131	−7.04185
$105$	−9.92123	−0.968213
$106$	2.76129	0.268200
$107$	8.91867	0.862200	0.431100	−	0.902304i	$-0.358126\pi$
$107$	8.91867	0.862200	0.431100	+	0.902304i	$0.358126\pi$
$108$	−7.07561	−0.680851
$109$	7.78466	0.745635	0.372817	−	0.927905i	$-0.378392\pi$
$109$	7.78466	0.745635	0.372817	+	0.927905i	$0.378392\pi$
$110$	−25.6584	−2.44643
$111$	−20.1643	−1.91391
$112$	29.4146	2.77942
$113$	−9.48131	−0.891926	−0.445963	−	0.895051i	$-0.647139\pi$
$113$	−9.48131	−0.891926	−0.445963	+	0.895051i	$0.647139\pi$
$114$	−32.4934	−3.04328
$115$	−6.64881	−0.620004
$116$	30.0372	2.78888
$117$	17.6628	1.63293
$118$	−32.7699	−3.01671
$119$	−9.37873	−0.859747
$120$	55.3233	5.05030
$121$	4.43540	0.403218
$122$	31.3250	2.83603
$123$	12.1373	1.09439
$124$	−24.6723	−2.21564
$125$	10.4210	0.932081
$126$	−12.2009	−1.08695
$127$	4.86594	0.431782	0.215891	−	0.976417i	$-0.430734\pi$
$127$	4.86594	0.431782	0.215891	+	0.976417i	$0.430734\pi$
$128$	−51.4300	−4.54581
$129$	−16.5764	−1.45947
$130$	46.8588	4.10979
$131$	6.64485	0.580564	0.290282	−	0.956941i	$-0.406251\pi$
$131$	6.64485	0.580564	0.290282	+	0.956941i	$0.406251\pi$
$132$	−51.6444	−4.49507
$133$	9.03773	0.783670
$134$	−10.3260	−0.892032
$135$	2.97525	0.256069
$136$	52.2982	4.48453
$137$	−6.76189	−0.577708	−0.288854	−	0.957373i	$-0.593274\pi$
$137$	−6.76189	−0.577708	−0.288854	+	0.957373i	$0.593274\pi$
$138$	−18.1410	−1.54426
$139$	6.35654	0.539155	0.269578	−	0.962979i	$-0.413116\pi$
$139$	6.35654	0.539155	0.269578	+	0.962979i	$0.413116\pi$
$140$	−23.8781	−2.01807
$141$	21.5091	1.81139
$142$	−15.4214	−1.29413
$143$	−28.1890	−2.35728
$144$	40.3425	3.36188
$145$	−12.6305	−1.04890
$146$	−45.5012	−3.76570
$147$	−8.83007	−0.728292
$148$	−48.5308	−3.98921
$149$	−0.962559	−0.0788559	−0.0394279	−	0.999222i	$-0.512554\pi$
$149$	−0.962559	−0.0788559	−0.0394279	+	0.999222i	$0.512554\pi$
$150$	−3.83291	−0.312956
$151$	1.00000	0.0813788
$151$	1.00000	0.0813788
$152$	−50.3967	−4.08771
$153$	−12.8630	−1.03991
$154$	19.4720	1.56910
$155$	10.3746	0.833306
$156$	94.3158	7.55131
$157$	−23.1858	−1.85043	−0.925215	−	0.379444i	$-0.876115\pi$
$157$	−23.1858	−1.85043	−0.925215	+	0.379444i	$0.876115\pi$
$158$	9.24173	0.735233
$159$	−2.33703	−0.185339
$160$	59.6821	4.71828
$161$	5.04575	0.397660
$162$	28.5105	2.24000
$163$	−10.5937	−0.829762	−0.414881	−	0.909876i	$-0.636177\pi$
$163$	−10.5937	−0.829762	−0.414881	+	0.909876i	$0.636177\pi$
$164$	29.2118	2.28105
$165$	21.7162	1.69060
$166$	−30.9605	−2.40300
$167$	−7.58214	−0.586724	−0.293362	−	0.956001i	$-0.594774\pi$
$167$	−7.58214	−0.586724	−0.293362	+	0.956001i	$0.594774\pi$
$168$	−41.9846	−3.23918
$169$	38.4802	2.96001
$170$	−34.1251	−2.61727
$171$	12.3953	0.947896
$172$	−39.8957	−3.04201
$173$	−3.45146	−0.262410	−0.131205	−	0.991355i	$-0.541885\pi$
$173$	−3.45146	−0.262410	−0.131205	+	0.991355i	$0.541885\pi$
$174$	−34.4617	−2.61253
$175$	1.06609	0.0805887
$176$	−64.3845	−4.85316
$177$	27.7350	2.08469
$178$	24.5588	1.84076
$179$	12.8909	0.963512	0.481756	−	0.876305i	$-0.339999\pi$
$179$	12.8909	0.963512	0.481756	+	0.876305i	$0.339999\pi$
$180$	−32.7491	−2.44097
$181$	−18.1025	−1.34555	−0.672775	−	0.739847i	$-0.734898\pi$
$181$	−18.1025	−1.34555	−0.672775	+	0.739847i	$0.734898\pi$
$182$	−35.5609	−2.63595
$183$	−26.5121	−1.95983
$184$	−28.1364	−2.07424
$185$	20.4069	1.50035
$186$	28.3066	2.07554
$187$	20.5287	1.50121
$188$	51.7673	3.77552
$189$	−2.25790	−0.164238
$190$	32.8843	2.38568
$191$	22.9101	1.65771	0.828857	−	0.559461i	$-0.188992\pi$
$191$	22.9101	1.65771	0.828857	+	0.559461i	$0.188992\pi$
$192$	86.2420	6.22398
$193$	−12.6503	−0.910591	−0.455296	−	0.890340i	$-0.650466\pi$
$193$	−12.6503	−0.910591	−0.455296	+	0.890340i	$0.650466\pi$
$194$	−31.4985	−2.26146
$195$	−39.6593	−2.84006
$196$	−21.2519	−1.51800
$197$	−11.2199	−0.799383	−0.399692	−	0.916650i	$-0.630883\pi$
$197$	−11.2199	−0.799383	−0.399692	+	0.916650i	$0.630883\pi$
$198$	26.7061	1.89792
$199$	−11.7382	−0.832102	−0.416051	−	0.909341i	$-0.636586\pi$
$199$	−11.7382	−0.832102	−0.416051	+	0.909341i	$0.636586\pi$
$200$	−5.94478	−0.420359
$201$	8.73950	0.616437
$202$	4.16276	0.292891
$203$	9.58520	0.672749
$204$	−68.6859	−4.80897
$205$	−12.2834	−0.857908
$206$	−39.6271	−2.76095
$207$	6.92030	0.480994
$208$	117.582	8.15287
$209$	−19.7823	−1.36837
$210$	27.3954	1.89046
$211$	10.5115	0.723644	0.361822	−	0.932247i	$-0.382155\pi$
$211$	10.5115	0.723644	0.361822	+	0.932247i	$0.382155\pi$
$212$	−5.62470	−0.386306
$213$	13.0520	0.894306
$214$	−24.6270	−1.68347
$215$	16.7759	1.14411
$216$	12.5907	0.856685
$217$	−7.87321	−0.534469
$218$	−21.4957	−1.45587
$219$	38.5102	2.60228
$220$	52.2658	3.52376
$221$	−37.4906	−2.52189
$222$	55.6794	3.73696
$223$	8.76755	0.587118	0.293559	−	0.955941i	$-0.405160\pi$
$223$	8.76755	0.587118	0.293559	+	0.955941i	$0.405160\pi$
$224$	−45.2924	−3.02623
$225$	1.46215	0.0974768
$226$	26.1806	1.74151
$227$	11.7274	0.778376	0.389188	−	0.921158i	$-0.372756\pi$
$227$	11.7274	0.778376	0.389188	+	0.921158i	$0.372756\pi$
$228$	66.1885	4.38344
$229$	7.14519	0.472167	0.236084	−	0.971733i	$-0.424136\pi$
$229$	7.14519	0.472167	0.236084	+	0.971733i	$0.424136\pi$
$230$	18.3593	1.21057
$231$	−16.4803	−1.08432
$232$	−53.4495	−3.50913
$233$	−4.78774	−0.313655	−0.156828	−	0.987626i	$-0.550127\pi$
$233$	−4.78774	−0.313655	−0.156828	+	0.987626i	$0.550127\pi$
$234$	−48.7722	−3.18834
$235$	−21.7679	−1.41998
$236$	66.7518	4.34517
$237$	−7.82181	−0.508081
$238$	25.8974	1.67868
$239$	−4.87259	−0.315182	−0.157591	−	0.987504i	$-0.550373\pi$
$239$	−4.87259	−0.315182	−0.157591	+	0.987504i	$0.550373\pi$
$240$	−90.5830	−5.84711
$241$	−15.8055	−1.01812	−0.509062	−	0.860730i	$-0.670008\pi$
$241$	−15.8055	−1.01812	−0.509062	+	0.860730i	$0.670008\pi$
$242$	−12.2474	−0.787293
$243$	−20.3562	−1.30585
$244$	−63.8085	−4.08492
$245$	8.93631	0.570920
$246$	−33.5146	−2.13682
$247$	36.1275	2.29874
$248$	43.9030	2.78785
$249$	26.2036	1.66059
$250$	−28.7753	−1.81991
$251$	−7.34957	−0.463901	−0.231950	−	0.972728i	$-0.574511\pi$
$251$	−7.34957	−0.463901	−0.231950	+	0.972728i	$0.574511\pi$
$252$	24.8531	1.56560
$253$	−11.0444	−0.694357
$254$	−13.4362	−0.843065
$255$	28.8820	1.80866
$256$	68.2084	4.26302
$257$	18.8184	1.17386	0.586929	−	0.809638i	$-0.300337\pi$
$257$	18.8184	1.17386	0.586929	+	0.809638i	$0.300337\pi$
$258$	45.7723	2.84966
$259$	−15.4867	−0.962298
$260$	−95.4507	−5.91960
$261$	13.1462	0.813730
$262$	−18.3483	−1.13356
$263$	16.3997	1.01125	0.505623	−	0.862755i	$-0.331263\pi$
$263$	16.3997	1.01125	0.505623	+	0.862755i	$0.331263\pi$
$264$	91.8983	5.65595
$265$	2.36515	0.145290
$266$	−24.9557	−1.53014
$267$	−20.7855	−1.27205
$268$	21.0340	1.28485
$269$	26.1132	1.59215	0.796073	−	0.605200i	$-0.206907\pi$
$269$	26.1132	1.59215	0.796073	+	0.605200i	$0.206907\pi$
$270$	−8.21553	−0.499981
$271$	29.6185	1.79919	0.899596	−	0.436722i	$-0.143861\pi$
$271$	29.6185	1.79919	0.899596	+	0.436722i	$0.143861\pi$
$272$	−85.6299	−5.19207
$273$	30.0972	1.82157
$274$	18.6715	1.12799
$275$	−2.33352	−0.140716
$276$	36.9529	2.22431
$277$	8.80288	0.528914	0.264457	−	0.964398i	$-0.414807\pi$
$277$	8.80288	0.528914	0.264457	+	0.964398i	$0.414807\pi$
$278$	−17.5522	−1.05271
$279$	−10.7982	−0.646472
$280$	42.4897	2.53925
$281$	−6.38749	−0.381046	−0.190523	−	0.981683i	$-0.561018\pi$
$281$	−6.38749	−0.381046	−0.190523	+	0.981683i	$0.561018\pi$
$282$	−59.3927	−3.53678
$283$	7.47971	0.444623	0.222311	−	0.974976i	$-0.428640\pi$
$283$	7.47971	0.444623	0.222311	+	0.974976i	$0.428640\pi$
$284$	31.4131	1.86402
$285$	−27.8319	−1.64862
$286$	77.8378	4.60264
$287$	9.32179	0.550248
$288$	−62.1191	−3.66040
$289$	10.3027	0.606042
$290$	34.8763	2.04801
$291$	26.6590	1.56278
$292$	92.6852	5.42399
$293$	11.4604	0.669523	0.334762	−	0.942303i	$-0.391344\pi$
$293$	11.4604	0.669523	0.334762	+	0.942303i	$0.391344\pi$
$294$	24.3824	1.42201
$295$	−28.0687	−1.63422
$296$	86.3579	5.01945
$297$	4.94224	0.286778
$298$	2.65790	0.153968
$299$	20.1699	1.16646
$300$	7.80758	0.450771
$301$	−12.7311	−0.733810
$302$	−2.76129	−0.158894
$303$	−3.52318	−0.202401
$304$	82.5164	4.73264
$305$	26.8311	1.53635
$306$	35.5185	2.03046
$307$	11.7556	0.670929	0.335464	−	0.942053i	$-0.391107\pi$
$307$	11.7556	0.670929	0.335464	+	0.942053i	$0.391107\pi$
$308$	−39.6643	−2.26008
$309$	33.5387	1.90795
$310$	−28.6472	−1.62705
$311$	−5.05379	−0.286574	−0.143287	−	0.989681i	$-0.545767\pi$
$311$	−5.05379	−0.286574	−0.143287	+	0.989681i	$0.545767\pi$
$312$	−167.830	−9.50149
$313$	29.5189	1.66851	0.834253	−	0.551382i	$-0.185899\pi$
$313$	29.5189	1.66851	0.834253	+	0.551382i	$0.185899\pi$
$314$	64.0227	3.61301
$315$	−10.4506	−0.588824
$316$	−18.8253	−1.05900
$317$	−34.7865	−1.95380	−0.976901	−	0.213692i	$-0.931451\pi$
$317$	−34.7865	−1.95380	−0.976901	+	0.213692i	$0.931451\pi$
$318$	6.45322	0.361879
$319$	−20.9807	−1.17469
$320$	−87.2796	−4.87908
$321$	20.8432	1.16336
$322$	−13.9327	−0.776442
$323$	−26.3100	−1.46393
$324$	−58.0755	−3.22641
$325$	4.26159	0.236391
$326$	29.2522	1.62013
$327$	18.1930	1.00608
$328$	−51.9807	−2.87015
$329$	16.5195	0.910750
$330$	−59.9646	−3.30094
$331$	20.1989	1.11023	0.555116	−	0.831773i	$-0.312674\pi$
$331$	20.1989	1.11023	0.555116	+	0.831773i	$0.312674\pi$
$332$	63.0660	3.46120
$333$	−21.2402	−1.16396
$334$	20.9365	1.14559
$335$	−8.84465	−0.483235
$336$	68.7430	3.75024
$337$	17.4053	0.948129	0.474065	−	0.880490i	$-0.342786\pi$
$337$	17.4053	0.948129	0.474065	+	0.880490i	$0.342786\pi$
$338$	−106.255	−5.77950
$339$	−22.1581	−1.20347
$340$	69.5123	3.76983
$341$	17.2334	0.933239
$342$	−34.2271	−1.85079
$343$	−19.3460	−1.04459
$344$	70.9921	3.82764
$345$	−15.5385	−0.836564
$346$	9.53046	0.512361
$347$	23.1663	1.24363	0.621816	−	0.783164i	$-0.286395\pi$
$347$	23.1663	1.24363	0.621816	+	0.783164i	$0.286395\pi$
$348$	70.1980	3.76301
$349$	−7.44983	−0.398781	−0.199390	−	0.979920i	$-0.563896\pi$
$349$	−7.44983	−0.398781	−0.199390	+	0.979920i	$0.563896\pi$
$350$	−2.94377	−0.157351
$351$	−9.02578	−0.481760
$352$	99.1387	5.28411
$353$	−25.3894	−1.35134	−0.675670	−	0.737205i	$-0.736145\pi$
$353$	−25.3894	−1.35134	−0.675670	+	0.737205i	$0.736145\pi$
$354$	−76.5843	−4.07041
$355$	−13.2090	−0.701062
$356$	−50.0259	−2.65137
$357$	−21.9184	−1.16005
$358$	−35.5955	−1.88128
$359$	−5.21783	−0.275387	−0.137693	−	0.990475i	$-0.543969\pi$
$359$	−5.21783	−0.275387	−0.137693	+	0.990475i	$0.543969\pi$
$360$	58.2752	3.07137
$361$	6.35341	0.334390
$362$	49.9862	2.62722
$363$	10.3657	0.544057
$364$	72.4370	3.79673
$365$	−38.9736	−2.03997
$366$	73.2076	3.82662
$367$	−15.5521	−0.811811	−0.405906	−	0.913915i	$-0.633044\pi$
$367$	−15.5521	−0.811811	−0.405906	+	0.913915i	$0.633044\pi$
$368$	46.0688	2.40150
$369$	12.7849	0.665557
$370$	−56.3494	−2.92946
$371$	−1.79490	−0.0931867
$372$	−57.6601	−2.98954
$373$	3.17049	0.164162	0.0820809	−	0.996626i	$-0.473843\pi$
$373$	3.17049	0.164162	0.0820809	+	0.996626i	$0.473843\pi$
$374$	−56.6857	−2.93115
$375$	24.3542	1.25764
$376$	−92.1170	−4.75057
$377$	38.3160	1.97337
$378$	6.23472	0.320679
$379$	28.1192	1.44439	0.722194	−	0.691691i	$-0.243134\pi$
$379$	28.1192	1.44439	0.722194	+	0.691691i	$0.243134\pi$
$380$	−66.9849	−3.43625
$381$	11.3719	0.582598
$382$	−63.2612	−3.23673
$383$	1.02985	0.0526231	0.0263115	−	0.999654i	$-0.491624\pi$
$383$	1.02985	0.0526231	0.0263115	+	0.999654i	$0.491624\pi$
$384$	−120.194	−6.13361
$385$	16.6786	0.850020
$386$	34.9312	1.77795
$387$	−17.4609	−0.887588
$388$	64.1621	3.25734
$389$	−6.63149	−0.336230	−0.168115	−	0.985767i	$-0.553768\pi$
$389$	−6.63149	−0.336230	−0.168115	+	0.985767i	$0.553768\pi$
$390$	109.511	5.54528
$391$	−14.6888	−0.742846
$392$	37.8166	1.91003
$393$	15.5293	0.783347
$394$	30.9813	1.56082
$395$	7.91592	0.398293
$396$	−54.4000	−2.73370
$397$	25.3921	1.27439	0.637195	−	0.770702i	$-0.280094\pi$
$397$	25.3921	1.27439	0.637195	+	0.770702i	$0.280094\pi$
$398$	32.4127	1.62470
$399$	21.1215	1.05740
$400$	9.73362	0.486681
$401$	−8.75988	−0.437448	−0.218724	−	0.975787i	$-0.570189\pi$
$401$	−8.75988	−0.437448	−0.218724	+	0.975787i	$0.570189\pi$
$402$	−24.1323	−1.20361
$403$	−31.4725	−1.56776
$404$	−8.47948	−0.421870
$405$	24.4204	1.21346
$406$	−26.4675	−1.31356
$407$	33.8982	1.68027
$408$	122.223	6.05092
$409$	−2.45200	−0.121244	−0.0606218	−	0.998161i	$-0.519308\pi$
$409$	−2.45200	−0.121244	−0.0606218	+	0.998161i	$0.519308\pi$
$410$	33.9179	1.67509
$411$	−15.8028	−0.779494
$412$	80.7198	3.97678
$413$	21.3012	1.04816
$414$	−19.1089	−0.939153
$415$	−26.5189	−1.30176
$416$	−181.052	−8.87683
$417$	14.8555	0.727475
$418$	54.6246	2.67178
$419$	15.4740	0.755955	0.377977	−	0.925815i	$-0.376620\pi$
$419$	15.4740	0.755955	0.377977	+	0.925815i	$0.376620\pi$
$420$	−55.8040	−2.72295
$421$	10.9346	0.532920	0.266460	−	0.963846i	$-0.414146\pi$
$421$	10.9346	0.532920	0.266460	+	0.963846i	$0.414146\pi$
$422$	−29.0254	−1.41293
$423$	22.6567	1.10161
$424$	10.0088	0.486072
$425$	−3.10352	−0.150543
$426$	−36.0402	−1.74616
$427$	−20.3620	−0.985386
$428$	50.1649	2.42481
$429$	−65.8786	−3.18065
$430$	−46.3230	−2.23389
$431$	−21.5242	−1.03678	−0.518392	−	0.855143i	$-0.673469\pi$
$431$	−21.5242	−1.03678	−0.518392	+	0.855143i	$0.673469\pi$
$432$	−20.6152	−0.991848
$433$	5.38467	0.258771	0.129385	−	0.991594i	$-0.458700\pi$
$433$	5.38467	0.258771	0.129385	+	0.991594i	$0.458700\pi$
$434$	21.7402	1.04356
$435$	−29.5178	−1.41527
$436$	43.7864	2.09699
$437$	14.1548	0.677114
$438$	−106.338	−5.08102
$439$	8.33818	0.397960	0.198980	−	0.980004i	$-0.436237\pi$
$439$	8.33818	0.397960	0.198980	+	0.980004i	$0.436237\pi$
$440$	−93.0041	−4.43380
$441$	−9.30121	−0.442915
$442$	103.522	4.92406
$443$	16.9782	0.806658	0.403329	−	0.915055i	$-0.367853\pi$
$443$	16.9782	0.806658	0.403329	+	0.915055i	$0.367853\pi$
$444$	−113.418	−5.38259
$445$	21.0356	0.997182
$446$	−24.2097	−1.14636
$447$	−2.24953	−0.106399
$448$	66.2361	3.12936
$449$	13.6749	0.645360	0.322680	−	0.946508i	$-0.395416\pi$
$449$	13.6749	0.645360	0.322680	+	0.946508i	$0.395416\pi$
$450$	−4.03742	−0.190326
$451$	−20.4041	−0.960791
$452$	−53.3295	−2.50841
$453$	2.33703	0.109803
$454$	−32.3828	−1.51980
$455$	−30.4593	−1.42796
$456$	−117.779	−5.51550
$457$	31.2714	1.46282	0.731408	−	0.681940i	$-0.238864\pi$
$457$	31.2714	1.46282	0.731408	+	0.681940i	$0.238864\pi$
$458$	−19.7299	−0.921918
$459$	6.57306	0.306804
$460$	−37.3976	−1.74367
$461$	−1.92748	−0.0897717	−0.0448858	−	0.998992i	$-0.514292\pi$
$461$	−1.92748	−0.0897717	−0.0448858	+	0.998992i	$0.514292\pi$
$462$	45.5068	2.11717
$463$	10.4675	0.486464	0.243232	−	0.969968i	$-0.421792\pi$
$463$	10.4675	0.486464	0.243232	+	0.969968i	$0.421792\pi$
$464$	87.5150	4.06278
$465$	24.2457	1.12437
$466$	13.2203	0.612420
$467$	−34.4614	−1.59469	−0.797343	−	0.603527i	$-0.793762\pi$
$467$	−34.4614	−1.59469	−0.797343	+	0.603527i	$0.793762\pi$
$468$	99.3482	4.59237
$469$	6.71216	0.309939
$470$	60.1073	2.77254
$471$	−54.1861	−2.49676
$472$	−118.781	−5.46734
$473$	27.8667	1.28131
$474$	21.5983	0.992041
$475$	2.99068	0.137222
$476$	−52.7525	−2.41791
$477$	−2.46173	−0.112715
$478$	13.4546	0.615400
$479$	9.19771	0.420254	0.210127	−	0.977674i	$-0.432612\pi$
$479$	9.19771	0.420254	0.210127	+	0.977674i	$0.432612\pi$
$480$	139.479	6.36632
$481$	−61.9068	−2.82271
$482$	43.6436	1.98791
$483$	11.7921	0.536558
$484$	24.9478	1.13399
$485$	−26.9798	−1.22509
$486$	56.2093	2.54970
$487$	−23.0931	−1.04645	−0.523224	−	0.852195i	$-0.675271\pi$
$487$	−23.0931	−1.04645	−0.523224	+	0.852195i	$0.675271\pi$
$488$	113.544	5.13988
$489$	−24.7578	−1.11959
$490$	−24.6757	−1.11474
$491$	1.29987	0.0586623	0.0293311	−	0.999570i	$-0.490662\pi$
$491$	1.29987	0.0586623	0.0293311	+	0.999570i	$0.490662\pi$
$492$	68.2689	3.07780
$493$	−27.9038	−1.25672
$494$	−99.7584	−4.48834
$495$	22.8749	1.02815
$496$	−71.8842	−3.22770
$497$	10.0243	0.449649
$498$	−72.3557	−3.24234
$499$	−35.0347	−1.56837	−0.784184	−	0.620528i	$-0.786918\pi$
$499$	−35.0347	−1.56837	−0.784184	+	0.620528i	$0.786918\pi$
$500$	58.6149	2.62134
$501$	−17.7197	−0.791659
$502$	20.2943	0.905777
$503$	−0.114212	−0.00509244	−0.00254622	−	0.999997i	$-0.500810\pi$
$503$	−0.114212	−0.00509244	−0.00254622	+	0.999997i	$0.500810\pi$
$504$	−44.2247	−1.96993
$505$	3.56557	0.158666
$506$	30.4968	1.35575
$507$	89.9295	3.99391
$508$	27.3694	1.21432
$509$	11.0101	0.488014	0.244007	−	0.969773i	$-0.421538\pi$
$509$	11.0101	0.488014	0.244007	+	0.969773i	$0.421538\pi$
$510$	−79.7515	−3.53146
$511$	29.5769	1.30840
$512$	−85.4828	−3.77784
$513$	−6.33407	−0.279656
$514$	−51.9629	−2.29199
$515$	−33.9422	−1.49567
$516$	−93.2375	−4.10455
$517$	−36.1589	−1.59027
$518$	42.7632	1.87891
$519$	−8.06618	−0.354066
$520$	169.849	7.44837
$521$	6.63159	0.290535	0.145268	−	0.989392i	$-0.453596\pi$
$521$	6.63159	0.290535	0.145268	+	0.989392i	$0.453596\pi$
$522$	−36.3005	−1.58883
$523$	−27.2968	−1.19361	−0.596804	−	0.802387i	$-0.703563\pi$
$523$	−27.2968	−1.19361	−0.596804	+	0.802387i	$0.703563\pi$
$524$	37.3753	1.63275
$525$	2.49148	0.108737
$526$	−45.2841	−1.97448
$527$	22.9200	0.998409
$528$	−150.469	−6.54831
$529$	−15.0974	−0.656410
$530$	−6.53087	−0.283683
$531$	29.2149	1.26782
$532$	50.8345	2.20396
$533$	37.2630	1.61404
$534$	57.3947	2.48371
$535$	−21.0940	−0.911974
$536$	−37.4287	−1.61667
$537$	30.1265	1.30005
$538$	−72.1059	−3.10871
$539$	14.8442	0.639387
$540$	16.7349	0.720156
$541$	30.0273	1.29097	0.645486	−	0.763772i	$-0.276655\pi$
$541$	30.0273	1.29097	0.645486	+	0.763772i	$0.276655\pi$
$542$	−81.7850	−3.51297
$543$	−42.3062	−1.81553
$544$	131.852	5.65312
$545$	−18.4119	−0.788679
$546$	−83.1070	−3.55665
$547$	4.82485	0.206296	0.103148	−	0.994666i	$-0.467108\pi$
$547$	4.82485	0.206296	0.103148	+	0.994666i	$0.467108\pi$
$548$	−38.0336	−1.62472
$549$	−27.9267	−1.19188
$550$	6.44351	0.274752
$551$	26.8892	1.14552
$552$	−65.7557	−2.79875
$553$	−6.00735	−0.255459
$554$	−24.3073	−1.03272
$555$	47.6917	2.02440
$556$	35.7537	1.51629
$557$	34.4899	1.46138	0.730692	−	0.682707i	$-0.239198\pi$
$557$	34.4899	1.46138	0.730692	+	0.682707i	$0.239198\pi$
$558$	29.8169	1.26225
$559$	−50.8916	−2.15249
$560$	−69.5701	−2.93987
$561$	47.9763	2.02556
$562$	17.6377	0.744001
$563$	−26.7370	−1.12683	−0.563415	−	0.826174i	$-0.690513\pi$
$563$	−26.7370	−1.12683	−0.563415	+	0.826174i	$0.690513\pi$
$564$	120.982	5.09426
$565$	22.4247	0.943416
$566$	−20.6536	−0.868136
$567$	−18.5325	−0.778292
$568$	−55.8978	−2.34542
$569$	7.15467	0.299940	0.149970	−	0.988691i	$-0.452082\pi$
$569$	7.15467	0.299940	0.149970	+	0.988691i	$0.452082\pi$
$570$	76.8518	3.21897
$571$	−42.0543	−1.75992	−0.879959	−	0.475049i	$-0.842430\pi$
$571$	−42.0543	−1.75992	−0.879959	+	0.475049i	$0.842430\pi$
$572$	−158.554	−6.62949
$573$	53.5416	2.23673
$574$	−25.7401	−1.07437
$575$	1.66969	0.0696310
$576$	90.8435	3.78515
$577$	22.4821	0.935943	0.467971	−	0.883744i	$-0.344985\pi$
$577$	22.4821	0.935943	0.467971	+	0.883744i	$0.344985\pi$
$578$	−28.4487	−1.18331
$579$	−29.5643	−1.22865
$580$	−71.0426	−2.94988
$581$	20.1251	0.834928
$582$	−73.6132	−3.05136
$583$	3.92879	0.162714
$584$	−164.928	−6.82477
$585$	−41.7753	−1.72720
$586$	−31.6454	−1.30726
$587$	15.9550	0.658535	0.329267	−	0.944237i	$-0.393198\pi$
$587$	15.9550	0.658535	0.329267	+	0.944237i	$0.393198\pi$
$588$	−49.6665	−2.04821
$589$	−22.0866	−0.910063
$590$	77.5058	3.19086
$591$	−26.2212	−1.07860
$592$	−141.397	−5.81139
$593$	6.71413	0.275716	0.137858	−	0.990452i	$-0.455978\pi$
$593$	6.71413	0.275716	0.137858	+	0.990452i	$0.455978\pi$
$594$	−13.6469	−0.559940
$595$	22.1821	0.909379
$596$	−5.41410	−0.221770
$597$	−27.4327	−1.12275
$598$	−55.6949	−2.27754
$599$	−36.7167	−1.50020	−0.750101	−	0.661323i	$-0.769995\pi$
$599$	−36.7167	−1.50020	−0.750101	+	0.661323i	$0.769995\pi$
$600$	−13.8931	−0.567185
$601$	8.78601	0.358389	0.179194	−	0.983814i	$-0.442651\pi$
$601$	8.78601	0.358389	0.179194	+	0.983814i	$0.442651\pi$
$602$	35.1543	1.43278
$603$	9.20581	0.374890
$604$	5.62470	0.228866
$605$	−10.4904	−0.426495
$606$	9.72851	0.395194
$607$	−34.6363	−1.40585	−0.702923	−	0.711266i	$-0.748122\pi$
$607$	−34.6363	−1.40585	−0.702923	+	0.711266i	$0.748122\pi$
$608$	−127.058	−5.15289
$609$	22.4009	0.907732
$610$	−74.0884	−2.99975
$611$	66.0353	2.67150
$612$	−72.3507	−2.92461
$613$	22.6588	0.915180	0.457590	−	0.889163i	$-0.348713\pi$
$613$	22.6588	0.915180	0.457590	+	0.889163i	$0.348713\pi$
$614$	−32.4606	−1.31000
$615$	−28.7067	−1.15756
$616$	70.5803	2.84376
$617$	36.3186	1.46213	0.731065	−	0.682307i	$-0.239023\pi$
$617$	36.3186	1.46213	0.731065	+	0.682307i	$0.239023\pi$
$618$	−92.6098	−3.72531
$619$	19.2407	0.773348	0.386674	−	0.922216i	$-0.373624\pi$
$619$	19.2407	0.773348	0.386674	+	0.922216i	$0.373624\pi$
$620$	58.3539	2.34355
$621$	−3.53630	−0.141907
$622$	13.9550	0.559543
$623$	−15.9638	−0.639576
$624$	274.794	11.0006
$625$	−27.6170	−1.10468
$626$	−81.5101	−3.25780
$627$	−46.2320	−1.84633
$628$	−130.413	−5.20406
$629$	45.0839	1.79761
$630$	28.8571	1.14969
$631$	27.9064	1.11094	0.555468	−	0.831538i	$-0.312539\pi$
$631$	27.9064	1.11094	0.555468	+	0.831538i	$0.312539\pi$
$632$	33.4985	1.33250
$633$	24.5658	0.976404
$634$	96.0554	3.81485
$635$	−11.5087	−0.456708
$636$	−13.1451	−0.521238
$637$	−27.1093	−1.07411
$638$	57.9336	2.29361
$639$	13.7484	0.543878
$640$	121.640	4.80824
$641$	26.8581	1.06083	0.530416	−	0.847737i	$-0.322036\pi$
$641$	26.8581	1.06083	0.530416	+	0.847737i	$0.322036\pi$
$642$	−57.5542	−2.27148
$643$	0.470811	0.0185670	0.00928348	−	0.999957i	$-0.497045\pi$
$643$	0.470811	0.0185670	0.00928348	+	0.999957i	$0.497045\pi$
$644$	28.3808	1.11836
$645$	39.2058	1.54373
$646$	72.6495	2.85836
$647$	39.1129	1.53769	0.768843	−	0.639437i	$-0.220833\pi$
$647$	39.1129	1.53769	0.768843	+	0.639437i	$0.220833\pi$
$648$	103.342	4.05966
$649$	−46.6254	−1.83021
$650$	−11.7675	−0.461558
$651$	−18.4000	−0.721152
$652$	−59.5864	−2.33358
$653$	−50.1561	−1.96276	−0.981380	−	0.192078i	$-0.938477\pi$
$653$	−50.1561	−1.96276	−0.981380	+	0.192078i	$0.938477\pi$
$654$	−50.2361	−1.96439
$655$	−15.7161	−0.614079
$656$	85.1100	3.32299
$657$	40.5650	1.58259
$658$	−45.6151	−1.77826
$659$	38.8976	1.51524	0.757618	−	0.652698i	$-0.226363\pi$
$659$	38.8976	1.51524	0.757618	+	0.652698i	$0.226363\pi$
$660$	122.147	4.75457
$661$	30.8760	1.20094	0.600469	−	0.799648i	$-0.294980\pi$
$661$	30.8760	1.20094	0.600469	+	0.799648i	$0.294980\pi$
$662$	−55.7750	−2.16776
$663$	−87.6169	−3.40276
$664$	−112.222	−4.35507
$665$	−21.3756	−0.828911
$666$	58.6503	2.27265
$667$	15.0122	0.581275
$668$	−42.6473	−1.65007
$669$	20.4901	0.792192
$670$	24.4226	0.943528
$671$	44.5695	1.72059
$672$	−105.850	−4.08325
$673$	−31.8844	−1.22905	−0.614526	−	0.788896i	$-0.710653\pi$
$673$	−31.8844	−1.22905	−0.614526	+	0.788896i	$0.710653\pi$
$674$	−48.0611	−1.85125
$675$	−0.747165	−0.0287584
$676$	216.440	8.32460
$677$	15.1325	0.581587	0.290794	−	0.956786i	$-0.406081\pi$
$677$	15.1325	0.581587	0.290794	+	0.956786i	$0.406081\pi$
$678$	61.1850	2.34979
$679$	20.4748	0.785751
$680$	−123.693	−4.74342
$681$	27.4074	1.05025
$682$	−47.5862	−1.82217
$683$	7.29998	0.279326	0.139663	−	0.990199i	$-0.455398\pi$
$683$	7.29998	0.279326	0.139663	+	0.990199i	$0.455398\pi$
$684$	69.7201	2.66582
$685$	15.9929	0.611058
$686$	53.4200	2.03958
$687$	16.6986	0.637090
$688$	−116.238	−4.43154
$689$	−7.17497	−0.273345
$690$	42.9062	1.63341
$691$	21.4149	0.814661	0.407330	−	0.913281i	$-0.366460\pi$
$691$	21.4149	0.814661	0.407330	+	0.913281i	$0.366460\pi$
$692$	−19.4134	−0.737988
$693$	−17.3596	−0.659438
$694$	−63.9688	−2.42822
$695$	−15.0342	−0.570280
$696$	−124.913	−4.73483
$697$	−27.1369	−1.02789
$698$	20.5711	0.778629
$699$	−11.1891	−0.423211
$700$	5.99642	0.226644
$701$	13.1337	0.496054	0.248027	−	0.968753i	$-0.420218\pi$
$701$	13.1337	0.496054	0.248027	+	0.968753i	$0.420218\pi$
$702$	24.9228	0.940649
$703$	−43.4447	−1.63855
$704$	−144.981	−5.46419
$705$	−50.8722	−1.91596
$706$	70.1073	2.63852
$707$	−2.70589	−0.101766
$708$	156.001	5.86288
$709$	16.7147	0.627735	0.313868	−	0.949467i	$-0.398375\pi$
$709$	16.7147	0.627735	0.313868	+	0.949467i	$0.398375\pi$
$710$	36.4739	1.36884
$711$	−8.23915	−0.308992
$712$	89.0182	3.33610
$713$	−12.3309	−0.461796
$714$	60.5230	2.26502
$715$	66.6712	2.49336
$716$	72.5075	2.70973
$717$	−11.3874	−0.425271
$718$	14.4079	0.537699
$719$	−8.02992	−0.299466	−0.149733	−	0.988726i	$-0.547841\pi$
$719$	−8.02992	−0.299466	−0.149733	+	0.988726i	$0.547841\pi$
$720$	−95.4162	−3.55595
$721$	25.7586	0.959299
$722$	−17.5436	−0.652904
$723$	−36.9381	−1.37374
$724$	−101.821	−3.78416
$725$	3.17185	0.117799
$726$	−28.6226	−1.06228
$727$	−17.1457	−0.635898	−0.317949	−	0.948108i	$-0.602994\pi$
$727$	−17.1457	−0.635898	−0.317949	+	0.948108i	$0.602994\pi$
$728$	−128.898	−4.77726
$729$	−16.5979	−0.614737
$730$	107.617	3.98309
$731$	37.0620	1.37079
$732$	−149.123	−5.51174
$733$	−35.5926	−1.31464	−0.657321	−	0.753610i	$-0.728311\pi$
$733$	−35.5926	−1.31464	−0.657321	+	0.753610i	$0.728311\pi$
$734$	42.9437	1.58508
$735$	20.8845	0.770335
$736$	−70.9363	−2.61475
$737$	−14.6920	−0.541186
$738$	−35.3029	−1.29952
$739$	29.2693	1.07669	0.538344	−	0.842725i	$-0.319050\pi$
$739$	29.2693	1.07669	0.538344	+	0.842725i	$0.319050\pi$
$740$	114.783	4.21950
$741$	84.4313	3.10166
$742$	4.95624	0.181949
$743$	40.4073	1.48240	0.741200	−	0.671284i	$-0.234257\pi$
$743$	40.4073	1.48240	0.741200	+	0.671284i	$0.234257\pi$
$744$	102.603	3.76161
$745$	2.27660	0.0834081
$746$	−8.75464	−0.320530
$747$	27.6017	1.00990
$748$	115.468	4.22192
$749$	16.0081	0.584925
$750$	−67.2489	−2.45558
$751$	5.98975	0.218569	0.109285	−	0.994011i	$-0.465144\pi$
$751$	5.98975	0.218569	0.109285	+	0.994011i	$0.465144\pi$
$752$	150.827	5.50009
$753$	−17.1762	−0.625935
$754$	−105.801	−3.85306
$755$	−2.36515	−0.0860768
$756$	−12.7000	−0.461896
$757$	5.32614	0.193582	0.0967909	−	0.995305i	$-0.469142\pi$
$757$	5.32614	0.193582	0.0967909	+	0.995305i	$0.469142\pi$
$758$	−77.6453	−2.82020
$759$	−25.8112	−0.936888
$760$	119.196	4.32369
$761$	−28.4337	−1.03072	−0.515360	−	0.856974i	$-0.672342\pi$
$761$	−28.4337	−1.03072	−0.515360	+	0.856974i	$0.672342\pi$
$762$	−31.4010	−1.13754
$763$	13.9727	0.505846
$764$	128.862	4.66207
$765$	30.4231	1.09995
$766$	−2.84372	−0.102748
$767$	85.1498	3.07458
$768$	159.405	5.75204
$769$	−18.4136	−0.664012	−0.332006	−	0.943277i	$-0.607725\pi$
$769$	−18.4136	−0.664012	−0.332006	+	0.943277i	$0.607725\pi$
$770$	−46.0544	−1.65968
$771$	43.9792	1.58387
$772$	−71.1544	−2.56090
$773$	−4.33310	−0.155851	−0.0779255	−	0.996959i	$-0.524830\pi$
$773$	−4.33310	−0.155851	−0.0779255	+	0.996959i	$0.524830\pi$
$774$	48.2145	1.73304
$775$	−2.60533	−0.0935863
$776$	−114.173	−4.09857
$777$	−36.1930	−1.29842
$778$	18.3114	0.656497
$779$	26.1503	0.936931
$780$	−223.071	−7.98724
$781$	−21.9417	−0.785135
$782$	40.5601	1.45043
$783$	−6.71776	−0.240073
$784$	−61.9186	−2.21138
$785$	54.8380	1.95725
$786$	−42.8807	−1.52950
$787$	37.5204	1.33746	0.668728	−	0.743507i	$-0.266839\pi$
$787$	37.5204	1.33746	0.668728	+	0.743507i	$0.266839\pi$
$788$	−63.1085	−2.24815
$789$	38.3266	1.36446
$790$	−21.8581	−0.777677
$791$	−17.0180	−0.605091
$792$	96.8017	3.43970
$793$	−81.3953	−2.89043
$794$	−70.1147	−2.48828
$795$	5.52745	0.196038
$796$	−66.0241	−2.34016
$797$	−19.0892	−0.676172	−0.338086	−	0.941115i	$-0.609780\pi$
$797$	−19.0892	−0.676172	−0.338086	+	0.941115i	$0.609780\pi$
$798$	−58.3225	−2.06459
$799$	−48.0905	−1.70132
$800$	−14.9877	−0.529897
$801$	−21.8945	−0.773605
$802$	24.1885	0.854127
$803$	−64.7396	−2.28461
$804$	49.1571	1.73364
$805$	−11.9340	−0.420617
$806$	86.9045	3.06108
$807$	61.0273	2.14826
$808$	15.0888	0.530821
$809$	−32.7474	−1.15134	−0.575669	−	0.817683i	$-0.695258\pi$
$809$	−32.7474	−1.15134	−0.575669	+	0.817683i	$0.695258\pi$
$810$	−67.4317	−2.36931
$811$	−11.2508	−0.395070	−0.197535	−	0.980296i	$-0.563294\pi$
$811$	−11.2508	−0.395070	−0.197535	+	0.980296i	$0.563294\pi$
$812$	53.9139	1.89201
$813$	69.2194	2.42763
$814$	−93.6027	−3.28077
$815$	25.0557	0.877664
$816$	−200.120	−7.00560
$817$	−35.7145	−1.24949
$818$	6.77067	0.236731
$819$	31.7031	1.10780
$820$	−69.0903	−2.41274
$821$	15.8790	0.554180	0.277090	−	0.960844i	$-0.410630\pi$
$821$	15.8790	0.554180	0.277090	+	0.960844i	$0.410630\pi$
$822$	43.6360	1.52198
$823$	−27.1915	−0.947837	−0.473918	−	0.880569i	$-0.657161\pi$
$823$	−27.1915	−0.947837	−0.473918	+	0.880569i	$0.657161\pi$
$824$	−143.636	−5.00381
$825$	−5.45351	−0.189867
$826$	−58.8188	−2.04657
$827$	−31.0013	−1.07802	−0.539011	−	0.842299i	$-0.681202\pi$
$827$	−31.0013	−1.07802	−0.539011	+	0.842299i	$0.681202\pi$
$828$	38.9246	1.35272
$829$	22.2474	0.772684	0.386342	−	0.922356i	$-0.373739\pi$
$829$	22.2474	0.772684	0.386342	+	0.922356i	$0.373739\pi$
$830$	73.2263	2.54172
$831$	20.5726	0.713656
$832$	264.773	9.17935
$833$	19.7425	0.684037
$834$	−41.0202	−1.42041
$835$	17.9329	0.620595
$836$	−111.270	−3.84834
$837$	5.51792	0.190727
$838$	−42.7282	−1.47602
$839$	24.5429	0.847314	0.423657	−	0.905823i	$-0.360746\pi$
$839$	24.5429	0.847314	0.423657	+	0.905823i	$0.360746\pi$
$840$	99.3000	3.42618
$841$	−0.481922	−0.0166180
$842$	−30.1936	−1.04054
$843$	−14.9278	−0.514140
$844$	59.1243	2.03514
$845$	−91.0116	−3.13089
$846$	−62.5616	−2.15091
$847$	7.96111	0.273547
$848$	−16.3879	−0.562762
$849$	17.4803	0.599924
$850$	8.56971	0.293939
$851$	−24.2551	−0.831453
$852$	73.4135	2.51510
$853$	−47.2614	−1.61820	−0.809100	−	0.587672i	$-0.800045\pi$
$853$	−47.2614	−1.61820	−0.809100	+	0.587672i	$0.800045\pi$
$854$	56.2253	1.92399
$855$	−29.3169	−1.00262
$856$	−89.2655	−3.05103
$857$	30.9901	1.05860	0.529300	−	0.848435i	$-0.322455\pi$
$857$	30.9901	1.05860	0.529300	+	0.848435i	$0.322455\pi$
$858$	181.910	6.21029
$859$	13.6456	0.465582	0.232791	−	0.972527i	$-0.425214\pi$
$859$	13.6456	0.465582	0.232791	+	0.972527i	$0.425214\pi$
$860$	94.3594	3.21763
$861$	21.7853	0.742442
$862$	59.4345	2.02435
$863$	29.3074	0.997635	0.498818	−	0.866707i	$-0.333768\pi$
$863$	29.3074	0.997635	0.498818	+	0.866707i	$0.333768\pi$
$864$	31.7431	1.07992
$865$	8.16323	0.277558
$866$	−14.8686	−0.505256
$867$	24.0778	0.817725
$868$	−44.2845	−1.50311
$869$	13.1492	0.446058
$870$	81.5072	2.76335
$871$	26.8313	0.909143
$872$	−77.9154	−2.63855
$873$	28.0815	0.950413
$874$	−39.0853	−1.32208
$875$	18.7046	0.632333
$876$	216.609	7.31853
$877$	49.0892	1.65763	0.828813	−	0.559526i	$-0.189017\pi$
$877$	49.0892	1.65763	0.828813	+	0.559526i	$0.189017\pi$
$878$	−23.0241	−0.777026
$879$	26.7833	0.903379
$880$	152.279	5.13333
$881$	22.6078	0.761675	0.380838	−	0.924642i	$-0.375636\pi$
$881$	22.6078	0.761675	0.380838	+	0.924642i	$0.375636\pi$
$882$	25.6833	0.864802
$883$	49.0082	1.64926	0.824628	−	0.565675i	$-0.191384\pi$
$883$	49.0082	1.64926	0.824628	+	0.565675i	$0.191384\pi$
$884$	−210.874	−7.09245
$885$	−65.5976	−2.20504
$886$	−46.8816	−1.57502
$887$	−19.3206	−0.648724	−0.324362	−	0.945933i	$-0.605150\pi$
$887$	−19.3206	−0.648724	−0.324362	+	0.945933i	$0.605150\pi$
$888$	201.821	6.77268
$889$	8.73388	0.292925
$890$	−58.0853	−1.94702
$891$	40.5651	1.35898
$892$	49.3148	1.65118
$893$	46.3420	1.55077
$894$	6.21160	0.207747
$895$	−30.4890	−1.01913
$896$	−92.3119	−3.08392
$897$	47.1378	1.57389
$898$	−37.7604	−1.26008
$899$	−23.4245	−0.781252
$900$	8.22417	0.274139
$901$	5.22520	0.174077
$902$	56.3415	1.87597
$903$	−29.7531	−0.990121
$904$	94.8969	3.15622
$905$	42.8152	1.42323
$906$	−6.45322	−0.214394
$907$	19.2189	0.638153	0.319076	−	0.947729i	$-0.396627\pi$
$907$	19.2189	0.638153	0.319076	+	0.947729i	$0.396627\pi$
$908$	65.9632	2.18907
$909$	−3.71117	−0.123092
$910$	84.1070	2.78812
$911$	35.0012	1.15964	0.579821	−	0.814744i	$-0.303122\pi$
$911$	35.0012	1.15964	0.579821	+	0.814744i	$0.303122\pi$
$912$	192.844	6.38570
$913$	−44.0509	−1.45787
$914$	−86.3493	−2.85618
$915$	62.7052	2.07297
$916$	40.1895	1.32790
$917$	11.9269	0.393860
$918$	−18.1501	−0.599042
$919$	48.5539	1.60165	0.800823	−	0.598901i	$-0.204396\pi$
$919$	48.5539	1.60165	0.800823	+	0.598901i	$0.204396\pi$
$920$	66.5468	2.19398
$921$	27.4733	0.905276
$922$	5.32232	0.175281
$923$	40.0711	1.31896
$924$	−92.6968	−3.04950
$925$	−5.12472	−0.168500
$926$	−28.9037	−0.949833
$927$	35.3282	1.16033
$928$	−134.755	−4.42355
$929$	−9.62734	−0.315863	−0.157931	−	0.987450i	$-0.550482\pi$
$929$	−9.62734	−0.315863	−0.157931	+	0.987450i	$0.550482\pi$
$930$	−66.9494	−2.19536
$931$	−19.0247	−0.623508
$932$	−26.9296	−0.882109
$933$	−11.8109	−0.386671
$934$	95.1579	3.11366
$935$	−48.5536	−1.58787
$936$	−176.785	−5.77838
$937$	15.3328	0.500902	0.250451	−	0.968129i	$-0.419421\pi$
$937$	15.3328	0.500902	0.250451	+	0.968129i	$0.419421\pi$
$938$	−18.5342	−0.605163
$939$	68.9866	2.25129
$940$	−122.438	−3.99348
$941$	−31.6146	−1.03061	−0.515303	−	0.857008i	$-0.672321\pi$
$941$	−31.6146	−1.03061	−0.515303	+	0.857008i	$0.672321\pi$
$942$	149.623	4.87499
$943$	14.5997	0.475430
$944$	194.485	6.32994
$945$	5.34029	0.173720
$946$	−76.9478	−2.50179
$947$	54.4648	1.76987	0.884934	−	0.465717i	$-0.154203\pi$
$947$	54.4648	1.76987	0.884934	+	0.465717i	$0.154203\pi$
$948$	−43.9953	−1.42890
$949$	118.231	3.83794
$950$	−8.25813	−0.267929
$951$	−81.2972	−2.63624
$952$	93.8702	3.04235
$953$	−46.5677	−1.50848	−0.754238	−	0.656601i	$-0.771993\pi$
$953$	−46.5677	−1.50848	−0.754238	+	0.656601i	$0.771993\pi$
$954$	6.79754	0.220079
$955$	−54.1858	−1.75341
$956$	−27.4069	−0.886401
$957$	−49.0325	−1.58500
$958$	−25.3975	−0.820556
$959$	−12.1369	−0.391922
$960$	−203.976	−6.58328
$961$	−11.7592	−0.379331
$962$	170.942	5.51140
$963$	21.9554	0.707502
$964$	−88.9015	−2.86332
$965$	29.9200	0.963159
$966$	−32.5613	−1.04764
$967$	35.9410	1.15578	0.577892	−	0.816113i	$-0.303876\pi$
$967$	35.9410	1.15578	0.577892	+	0.816113i	$0.303876\pi$
$968$	−44.3932	−1.42685
$969$	−61.4874	−1.97526
$970$	74.4989	2.39202
$971$	45.0837	1.44681	0.723403	−	0.690426i	$-0.242577\pi$
$971$	45.0837	1.44681	0.723403	+	0.690426i	$0.242577\pi$
$972$	−114.498	−3.67251
$973$	11.4094	0.365768
$974$	63.7667	2.04322
$975$	9.95949	0.318959
$976$	−185.910	−5.95082
$977$	−9.74460	−0.311757	−0.155879	−	0.987776i	$-0.549821\pi$
$977$	−9.74460	−0.311757	−0.155879	+	0.987776i	$0.549821\pi$
$978$	68.3635	2.18602
$979$	34.9425	1.11677
$980$	50.2641	1.60563
$981$	19.1637	0.611851
$982$	−3.58931	−0.114540
$983$	27.2791	0.870069	0.435034	−	0.900414i	$-0.356736\pi$
$983$	27.2791	0.870069	0.435034	+	0.900414i	$0.356736\pi$
$984$	−121.481	−3.87266
$985$	26.5367	0.845531
$986$	77.0503	2.45378
$987$	38.6067	1.22886
$988$	203.207	6.46486
$989$	−19.9393	−0.634034
$990$	−63.1641	−2.00749
$991$	−17.3512	−0.551180	−0.275590	−	0.961275i	$-0.588873\pi$
$991$	−17.3512	−0.551180	−0.275590	+	0.961275i	$0.588873\pi$
$992$	110.687	3.51431
$993$	47.2056	1.49802
$994$	−27.6798	−0.877951
$995$	27.7628	0.880139
$996$	147.387	4.67015
$997$	−60.2153	−1.90704	−0.953518	−	0.301335i	$-0.902568\pi$
$997$	−60.2153	−1.90704	−0.953518	+	0.301335i	$0.902568\pi$
$998$	96.7408	3.06228
$999$	10.8538	0.343400

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8003.2.a.c.1.3	✓	172

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8003.2.a.c.1.3	✓	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)

\(f(q)\)	\(=\)	\(q-2.76129 q^{2} +2.33703 q^{3} +5.62470 q^{4} -2.36515 q^{5} -6.45322 q^{6} +1.79490 q^{7} -10.0088 q^{8} +2.46173 q^{9} +O(q^{10})\)	\(q-2.76129 q^{2} +2.33703 q^{3} +5.62470 q^{4} -2.36515 q^{5} -6.45322 q^{6} +1.79490 q^{7} -10.0088 q^{8} +2.46173 q^{9} +6.53087 q^{10} -3.92879 q^{11} +13.1451 q^{12} +7.17497 q^{13} -4.95624 q^{14} -5.52745 q^{15} +16.3879 q^{16} -5.22520 q^{17} -6.79754 q^{18} +5.03522 q^{19} -13.3033 q^{20} +4.19475 q^{21} +10.8485 q^{22} +2.81115 q^{23} -23.3910 q^{24} +0.593953 q^{25} -19.8121 q^{26} -1.25795 q^{27} +10.0958 q^{28} +5.34023 q^{29} +15.2629 q^{30} -4.38643 q^{31} -25.2339 q^{32} -9.18172 q^{33} +14.4283 q^{34} -4.24522 q^{35} +13.8465 q^{36} -8.62816 q^{37} -13.9037 q^{38} +16.7682 q^{39} +23.6724 q^{40} +5.19348 q^{41} -11.5829 q^{42} -7.09294 q^{43} -22.0983 q^{44} -5.82237 q^{45} -7.76240 q^{46} +9.20357 q^{47} +38.2990 q^{48} -3.77832 q^{49} -1.64007 q^{50} -12.2115 q^{51} +40.3571 q^{52} -1.00000 q^{53} +3.47357 q^{54} +9.29219 q^{55} -17.9649 q^{56} +11.7675 q^{57} -14.7459 q^{58} +11.8676 q^{59} -31.0902 q^{60} -11.3443 q^{61} +12.1122 q^{62} +4.41857 q^{63} +36.9023 q^{64} -16.9699 q^{65} +25.3534 q^{66} +3.73957 q^{67} -29.3902 q^{68} +6.56976 q^{69} +11.7223 q^{70} +5.58484 q^{71} -24.6391 q^{72} +16.4782 q^{73} +23.8248 q^{74} +1.38809 q^{75} +28.3216 q^{76} -7.05180 q^{77} -46.3017 q^{78} -3.34689 q^{79} -38.7598 q^{80} -10.3251 q^{81} -14.3407 q^{82} +11.2123 q^{83} +23.5942 q^{84} +12.3584 q^{85} +19.5856 q^{86} +12.4803 q^{87} +39.3226 q^{88} -8.89396 q^{89} +16.0772 q^{90} +12.8784 q^{91} +15.8119 q^{92} -10.2512 q^{93} -25.4137 q^{94} -11.9091 q^{95} -58.9725 q^{96} +11.4072 q^{97} +10.4330 q^{98} -9.67163 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 172 q + 8 q^{2} + 25 q^{3} + 188 q^{4} + 27 q^{5} + 10 q^{6} + 31 q^{7} + 21 q^{8} + 179 q^{9}+O(q^{10}) \) 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