LMFDB - Embedded newform 6002.2.a.d.1.12

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6002,2,Mod(1,6002)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6002, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("6002.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6002 = 2 \cdot 3001 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6002.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:	$47.9262112932$
Analytic rank:	$0$
Dimension:	$79$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.12
Character	$\chi$	$=$	6002.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} -2.46943 q^{3} +1.00000 q^{4} +4.39243 q^{5} -2.46943 q^{6} -4.67132 q^{7} +1.00000 q^{8} +3.09808 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} -2.46943 q^{3} +1.00000 q^{4} +4.39243 q^{5} -2.46943 q^{6} -4.67132 q^{7} +1.00000 q^{8} +3.09808 q^{9} +4.39243 q^{10} -0.564487 q^{11} -2.46943 q^{12} -5.67880 q^{13} -4.67132 q^{14} -10.8468 q^{15} +1.00000 q^{16} +5.55146 q^{17} +3.09808 q^{18} -2.00989 q^{19} +4.39243 q^{20} +11.5355 q^{21} -0.564487 q^{22} -3.26966 q^{23} -2.46943 q^{24} +14.2934 q^{25} -5.67880 q^{26} -0.242205 q^{27} -4.67132 q^{28} -6.52544 q^{29} -10.8468 q^{30} -1.65210 q^{31} +1.00000 q^{32} +1.39396 q^{33} +5.55146 q^{34} -20.5185 q^{35} +3.09808 q^{36} +7.15165 q^{37} -2.00989 q^{38} +14.0234 q^{39} +4.39243 q^{40} -0.483297 q^{41} +11.5355 q^{42} -0.613072 q^{43} -0.564487 q^{44} +13.6081 q^{45} -3.26966 q^{46} +0.363074 q^{47} -2.46943 q^{48} +14.8213 q^{49} +14.2934 q^{50} -13.7089 q^{51} -5.67880 q^{52} -10.6595 q^{53} -0.242205 q^{54} -2.47947 q^{55} -4.67132 q^{56} +4.96329 q^{57} -6.52544 q^{58} +10.9057 q^{59} -10.8468 q^{60} +1.54368 q^{61} -1.65210 q^{62} -14.4721 q^{63} +1.00000 q^{64} -24.9437 q^{65} +1.39396 q^{66} +13.7382 q^{67} +5.55146 q^{68} +8.07421 q^{69} -20.5185 q^{70} -4.98170 q^{71} +3.09808 q^{72} +7.67787 q^{73} +7.15165 q^{74} -35.2966 q^{75} -2.00989 q^{76} +2.63690 q^{77} +14.0234 q^{78} +8.17935 q^{79} +4.39243 q^{80} -8.69614 q^{81} -0.483297 q^{82} +11.8862 q^{83} +11.5355 q^{84} +24.3844 q^{85} -0.613072 q^{86} +16.1141 q^{87} -0.564487 q^{88} +12.1011 q^{89} +13.6081 q^{90} +26.5275 q^{91} -3.26966 q^{92} +4.07974 q^{93} +0.363074 q^{94} -8.82832 q^{95} -2.46943 q^{96} +11.6394 q^{97} +14.8213 q^{98} -1.74883 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 79 q + 79 q^{2} + 17 q^{3} + 79 q^{4} + 18 q^{5} + 17 q^{6} + 19 q^{7} + 79 q^{8} + 118 q^{9}+O(q^{10}) $ 79 * q + 79 * q^2 + 17 * q^3 + 79 * q^4 + 18 * q^5 + 17 * q^6 + 19 * q^7 + 79 * q^8 + 118 * q^9	\( 79 q + 79 q^{2} + 17 q^{3} + 79 q^{4} + 18 q^{5} + 17 q^{6} + 19 q^{7} + 79 q^{8} + 118 q^{9} + 18 q^{10} + 28 q^{11} + 17 q^{12} + 47 q^{13} + 19 q^{14} + 14 q^{15} + 79 q^{16} + 36 q^{17} + 118 q^{18} + 29 q^{19} + 18 q^{20} + 45 q^{21} + 28 q^{22} + 23 q^{23} + 17 q^{24} + 161 q^{25} + 47 q^{26} + 50 q^{27} + 19 q^{28} + 53 q^{29} + 14 q^{30} + 29 q^{31} + 79 q^{32} + 34 q^{33} + 36 q^{34} + 33 q^{35} + 118 q^{36} + 89 q^{37} + 29 q^{38} - 7 q^{39} + 18 q^{40} + 58 q^{41} + 45 q^{42} + 88 q^{43} + 28 q^{44} + 45 q^{45} + 23 q^{46} + 3 q^{47} + 17 q^{48} + 162 q^{49} + 161 q^{50} + 29 q^{51} + 47 q^{52} + 88 q^{53} + 50 q^{54} + 37 q^{55} + 19 q^{56} + 54 q^{57} + 53 q^{58} + 37 q^{59} + 14 q^{60} + 55 q^{61} + 29 q^{62} + 21 q^{63} + 79 q^{64} + 55 q^{65} + 34 q^{66} + 107 q^{67} + 36 q^{68} + 39 q^{69} + 33 q^{70} - 5 q^{71} + 118 q^{72} + 71 q^{73} + 89 q^{74} + 37 q^{75} + 29 q^{76} + 61 q^{77} - 7 q^{78} + 29 q^{79} + 18 q^{80} + 215 q^{81} + 58 q^{82} + 42 q^{83} + 45 q^{84} + 84 q^{85} + 88 q^{86} + 15 q^{87} + 28 q^{88} + 72 q^{89} + 45 q^{90} + 70 q^{91} + 23 q^{92} + 97 q^{93} + 3 q^{94} - 18 q^{95} + 17 q^{96} + 93 q^{97} + 162 q^{98} + 49 q^{99}+O(q^{100}) \) 79 * q + 79 * q^2 + 17 * q^3 + 79 * q^4 + 18 * q^5 + 17 * q^6 + 19 * q^7 + 79 * q^8 + 118 * q^9 + 18 * q^10 + 28 * q^11 + 17 * q^12 + 47 * q^13 + 19 * q^14 + 14 * q^15 + 79 * q^16 + 36 * q^17 + 118 * q^18 + 29 * q^19 + 18 * q^20 + 45 * q^21 + 28 * q^22 + 23 * q^23 + 17 * q^24 + 161 * q^25 + 47 * q^26 + 50 * q^27 + 19 * q^28 + 53 * q^29 + 14 * q^30 + 29 * q^31 + 79 * q^32 + 34 * q^33 + 36 * q^34 + 33 * q^35 + 118 * q^36 + 89 * q^37 + 29 * q^38 - 7 * q^39 + 18 * q^40 + 58 * q^41 + 45 * q^42 + 88 * q^43 + 28 * q^44 + 45 * q^45 + 23 * q^46 + 3 * q^47 + 17 * q^48 + 162 * q^49 + 161 * q^50 + 29 * q^51 + 47 * q^52 + 88 * q^53 + 50 * q^54 + 37 * q^55 + 19 * q^56 + 54 * q^57 + 53 * q^58 + 37 * q^59 + 14 * q^60 + 55 * q^61 + 29 * q^62 + 21 * q^63 + 79 * q^64 + 55 * q^65 + 34 * q^66 + 107 * q^67 + 36 * q^68 + 39 * q^69 + 33 * q^70 - 5 * q^71 + 118 * q^72 + 71 * q^73 + 89 * q^74 + 37 * q^75 + 29 * q^76 + 61 * q^77 - 7 * q^78 + 29 * q^79 + 18 * q^80 + 215 * q^81 + 58 * q^82 + 42 * q^83 + 45 * q^84 + 84 * q^85 + 88 * q^86 + 15 * q^87 + 28 * q^88 + 72 * q^89 + 45 * q^90 + 70 * q^91 + 23 * q^92 + 97 * q^93 + 3 * q^94 - 18 * q^95 + 17 * q^96 + 93 * q^97 + 162 * q^98 + 49 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	−2.46943	−1.42573	−0.712863	−	0.701303i	$-0.752602\pi$
$3$	−2.46943	−1.42573	−0.712863	+	0.701303i	$0.752602\pi$
$4$	1.00000	0.500000
$5$	4.39243	1.96435	0.982177	−	0.187958i	$-0.0601869\pi$
$5$	4.39243	1.96435	0.982177	+	0.187958i	$0.0601869\pi$
$6$	−2.46943	−1.00814
$7$	−4.67132	−1.76559	−0.882797	−	0.469754i	$-0.844343\pi$
$7$	−4.67132	−1.76559	−0.882797	+	0.469754i	$0.844343\pi$
$8$	1.00000	0.353553
$9$	3.09808	1.03269
$10$	4.39243	1.38901
$11$	−0.564487	−0.170199	−0.0850996	−	0.996372i	$-0.527121\pi$
$11$	−0.564487	−0.170199	−0.0850996	+	0.996372i	$0.527121\pi$
$12$	−2.46943	−0.712863
$13$	−5.67880	−1.57502	−0.787508	−	0.616304i	$-0.788629\pi$
$13$	−5.67880	−1.57502	−0.787508	+	0.616304i	$0.788629\pi$
$14$	−4.67132	−1.24846
$15$	−10.8468	−2.80063
$16$	1.00000	0.250000
$17$	5.55146	1.34643	0.673213	−	0.739448i	$-0.264913\pi$
$17$	5.55146	1.34643	0.673213	+	0.739448i	$0.264913\pi$
$18$	3.09808	0.730225
$19$	−2.00989	−0.461101	−0.230551	−	0.973060i	$-0.574053\pi$
$19$	−2.00989	−0.461101	−0.230551	+	0.973060i	$0.574053\pi$
$20$	4.39243	0.982177
$21$	11.5355	2.51725
$22$	−0.564487	−0.120349
$23$	−3.26966	−0.681772	−0.340886	−	0.940105i	$-0.610727\pi$
$23$	−3.26966	−0.681772	−0.340886	+	0.940105i	$0.610727\pi$
$24$	−2.46943	−0.504070
$25$	14.2934	2.85869
$26$	−5.67880	−1.11370
$27$	−0.242205	−0.0466124
$28$	−4.67132	−0.882797
$29$	−6.52544	−1.21174	−0.605872	−	0.795562i	$-0.707176\pi$
$29$	−6.52544	−1.21174	−0.605872	+	0.795562i	$0.707176\pi$
$30$	−10.8468	−1.98034
$31$	−1.65210	−0.296726	−0.148363	−	0.988933i	$-0.547400\pi$
$31$	−1.65210	−0.296726	−0.148363	+	0.988933i	$0.547400\pi$
$32$	1.00000	0.176777
$33$	1.39396	0.242657
$34$	5.55146	0.952068
$35$	−20.5185	−3.46825
$36$	3.09808	0.516347
$37$	7.15165	1.17572	0.587862	−	0.808961i	$-0.299970\pi$
$37$	7.15165	1.17572	0.587862	+	0.808961i	$0.299970\pi$
$38$	−2.00989	−0.326048
$39$	14.0234	2.24554
$40$	4.39243	0.694504
$41$	−0.483297	−0.0754782	−0.0377391	−	0.999288i	$-0.512016\pi$
$41$	−0.483297	−0.0754782	−0.0377391	+	0.999288i	$0.512016\pi$
$42$	11.5355	1.77997
$43$	−0.613072	−0.0934925	−0.0467463	−	0.998907i	$-0.514885\pi$
$43$	−0.613072	−0.0934925	−0.0467463	+	0.998907i	$0.514885\pi$
$44$	−0.564487	−0.0850996
$45$	13.6081	2.02858
$46$	−3.26966	−0.482086
$47$	0.363074	0.0529598	0.0264799	−	0.999649i	$-0.491570\pi$
$47$	0.363074	0.0529598	0.0264799	+	0.999649i	$0.491570\pi$
$48$	−2.46943	−0.356431
$49$	14.8213	2.11732
$50$	14.2934	2.02140
$51$	−13.7089	−1.91964
$52$	−5.67880	−0.787508
$53$	−10.6595	−1.46419	−0.732096	−	0.681201i	$-0.761458\pi$
$53$	−10.6595	−1.46419	−0.732096	+	0.681201i	$0.761458\pi$
$54$	−0.242205	−0.0329600
$55$	−2.47947	−0.334332
$56$	−4.67132	−0.624232
$57$	4.96329	0.657404
$58$	−6.52544	−0.856833
$59$	10.9057	1.41981	0.709903	−	0.704299i	$-0.248739\pi$
$59$	10.9057	1.41981	0.709903	+	0.704299i	$0.248739\pi$
$60$	−10.8468	−1.40032
$61$	1.54368	0.197648	0.0988239	−	0.995105i	$-0.468492\pi$
$61$	1.54368	0.197648	0.0988239	+	0.995105i	$0.468492\pi$
$62$	−1.65210	−0.209817
$63$	−14.4721	−1.82332
$64$	1.00000	0.125000
$65$	−24.9437	−3.09389
$66$	1.39396	0.171585
$67$	13.7382	1.67839	0.839197	−	0.543828i	$-0.183025\pi$
$67$	13.7382	1.67839	0.839197	+	0.543828i	$0.183025\pi$
$68$	5.55146	0.673213
$69$	8.07421	0.972020
$70$	−20.5185	−2.45243
$71$	−4.98170	−0.591219	−0.295610	−	0.955309i	$-0.595523\pi$
$71$	−4.98170	−0.591219	−0.295610	+	0.955309i	$0.595523\pi$
$72$	3.09808	0.365112
$73$	7.67787	0.898627	0.449313	−	0.893374i	$-0.351669\pi$
$73$	7.67787	0.898627	0.449313	+	0.893374i	$0.351669\pi$
$74$	7.15165	0.831362
$75$	−35.2966	−4.07570
$76$	−2.00989	−0.230551
$77$	2.63690	0.300503
$78$	14.0234	1.58784
$79$	8.17935	0.920249	0.460125	−	0.887854i	$-0.347805\pi$
$79$	8.17935	0.920249	0.460125	+	0.887854i	$0.347805\pi$
$80$	4.39243	0.491089
$81$	−8.69614	−0.966237
$82$	−0.483297	−0.0533712
$83$	11.8862	1.30468	0.652341	−	0.757926i	$-0.273787\pi$
$83$	11.8862	1.30468	0.652341	+	0.757926i	$0.273787\pi$
$84$	11.5355	1.25863
$85$	24.3844	2.64486
$86$	−0.613072	−0.0661092
$87$	16.1141	1.72762
$88$	−0.564487	−0.0601745
$89$	12.1011	1.28272	0.641359	−	0.767241i	$-0.278371\pi$
$89$	12.1011	1.28272	0.641359	+	0.767241i	$0.278371\pi$
$90$	13.6081	1.43442
$91$	26.5275	2.78084
$92$	−3.26966	−0.340886
$93$	4.07974	0.423049
$94$	0.363074	0.0374482
$95$	−8.82832	−0.905766
$96$	−2.46943	−0.252035
$97$	11.6394	1.18180	0.590901	−	0.806744i	$-0.298772\pi$
$97$	11.6394	1.18180	0.590901	+	0.806744i	$0.298772\pi$
$98$	14.8213	1.49717
$99$	−1.74883	−0.175764
$100$	14.2934	1.42934
$101$	−2.00586	−0.199590	−0.0997950	−	0.995008i	$-0.531819\pi$
$101$	−2.00586	−0.199590	−0.0997950	+	0.995008i	$0.531819\pi$
$102$	−13.7089	−1.35739
$103$	5.76412	0.567956	0.283978	−	0.958831i	$-0.408346\pi$
$103$	5.76412	0.567956	0.283978	+	0.958831i	$0.408346\pi$
$104$	−5.67880	−0.556852
$105$	50.6689	4.94478
$106$	−10.6595	−1.03534
$107$	5.25937	0.508443	0.254221	−	0.967146i	$-0.418181\pi$
$107$	5.25937	0.508443	0.254221	+	0.967146i	$0.418181\pi$
$108$	−0.242205	−0.0233062
$109$	−3.29153	−0.315271	−0.157635	−	0.987497i	$-0.550387\pi$
$109$	−3.29153	−0.315271	−0.157635	+	0.987497i	$0.550387\pi$
$110$	−2.47947	−0.236408
$111$	−17.6605	−1.67626
$112$	−4.67132	−0.441399
$113$	−12.2954	−1.15666	−0.578329	−	0.815804i	$-0.696295\pi$
$113$	−12.2954	−1.15666	−0.578329	+	0.815804i	$0.696295\pi$
$114$	4.96329	0.464855
$115$	−14.3618	−1.33924
$116$	−6.52544	−0.605872
$117$	−17.5934	−1.62651
$118$	10.9057	1.00395
$119$	−25.9327	−2.37724
$120$	−10.8468	−0.990172
$121$	−10.6814	−0.971032
$122$	1.54368	0.139758
$123$	1.19347	0.107611
$124$	−1.65210	−0.148363
$125$	40.8208	3.65112
$126$	−14.4721	−1.28928
$127$	12.9134	1.14588	0.572939	−	0.819598i	$-0.305803\pi$
$127$	12.9134	1.14588	0.572939	+	0.819598i	$0.305803\pi$
$128$	1.00000	0.0883883
$129$	1.51394	0.133295
$130$	−24.9437	−2.18771
$131$	−5.69318	−0.497415	−0.248708	−	0.968579i	$-0.580006\pi$
$131$	−5.69318	−0.497415	−0.248708	+	0.968579i	$0.580006\pi$
$132$	1.39396	0.121329
$133$	9.38886	0.814118
$134$	13.7382	1.18680
$135$	−1.06387	−0.0915633
$136$	5.55146	0.476034
$137$	17.0429	1.45607	0.728035	−	0.685540i	$-0.240434\pi$
$137$	17.0429	1.45607	0.728035	+	0.685540i	$0.240434\pi$
$138$	8.07421	0.687322
$139$	−11.1872	−0.948888	−0.474444	−	0.880286i	$-0.657351\pi$
$139$	−11.1872	−0.948888	−0.474444	+	0.880286i	$0.657351\pi$
$140$	−20.5185	−1.73413
$141$	−0.896586	−0.0755061
$142$	−4.98170	−0.418055
$143$	3.20561	0.268067
$144$	3.09808	0.258173
$145$	−28.6625	−2.38029
$146$	7.67787	0.635425
$147$	−36.6001	−3.01872
$148$	7.15165	0.587862
$149$	5.91008	0.484172	0.242086	−	0.970255i	$-0.422168\pi$
$149$	5.91008	0.484172	0.242086	+	0.970255i	$0.422168\pi$
$150$	−35.2966	−2.88196
$151$	−18.0453	−1.46850	−0.734251	−	0.678878i	$-0.762466\pi$
$151$	−18.0453	−1.46850	−0.734251	+	0.678878i	$0.762466\pi$
$152$	−2.00989	−0.163024
$153$	17.1989	1.39045
$154$	2.63690	0.212488
$155$	−7.25672	−0.582874
$156$	14.0234	1.12277
$157$	−6.44995	−0.514762	−0.257381	−	0.966310i	$-0.582860\pi$
$157$	−6.44995	−0.514762	−0.257381	+	0.966310i	$0.582860\pi$
$158$	8.17935	0.650714
$159$	26.3228	2.08754
$160$	4.39243	0.347252
$161$	15.2737	1.20373
$162$	−8.69614	−0.683233
$163$	−6.77425	−0.530600	−0.265300	−	0.964166i	$-0.585471\pi$
$163$	−6.77425	−0.530600	−0.265300	+	0.964166i	$0.585471\pi$
$164$	−0.483297	−0.0377391
$165$	6.12287	0.476665
$166$	11.8862	0.922550
$167$	−8.94840	−0.692448	−0.346224	−	0.938152i	$-0.612536\pi$
$167$	−8.94840	−0.692448	−0.346224	+	0.938152i	$0.612536\pi$
$168$	11.5355	0.889984
$169$	19.2488	1.48068
$170$	24.3844	1.87020
$171$	−6.22681	−0.476176
$172$	−0.613072	−0.0467463
$173$	10.4812	0.796868	0.398434	−	0.917197i	$-0.369554\pi$
$173$	10.4812	0.796868	0.398434	+	0.917197i	$0.369554\pi$
$174$	16.1141	1.22161
$175$	−66.7693	−5.04728
$176$	−0.564487	−0.0425498
$177$	−26.9310	−2.02425
$178$	12.1011	0.907019
$179$	5.93971	0.443955	0.221978	−	0.975052i	$-0.428749\pi$
$179$	5.93971	0.443955	0.221978	+	0.975052i	$0.428749\pi$
$180$	13.6081	1.01429
$181$	6.44359	0.478948	0.239474	−	0.970903i	$-0.423025\pi$
$181$	6.44359	0.478948	0.239474	+	0.970903i	$0.423025\pi$
$182$	26.5275	1.96635
$183$	−3.81201	−0.281792
$184$	−3.26966	−0.241043
$185$	31.4131	2.30954
$186$	4.07974	0.299141
$187$	−3.13373	−0.229161
$188$	0.363074	0.0264799
$189$	1.13142	0.0822987
$190$	−8.82832	−0.640473
$191$	−13.2681	−0.960048	−0.480024	−	0.877255i	$-0.659372\pi$
$191$	−13.2681	−0.960048	−0.480024	+	0.877255i	$0.659372\pi$
$192$	−2.46943	−0.178216
$193$	22.7173	1.63523	0.817613	−	0.575769i	$-0.195297\pi$
$193$	22.7173	1.63523	0.817613	+	0.575769i	$0.195297\pi$
$194$	11.6394	0.835660
$195$	61.5968	4.41104
$196$	14.8213	1.05866
$197$	26.7242	1.90402	0.952010	−	0.306068i	$-0.0990136\pi$
$197$	26.7242	1.90402	0.952010	+	0.306068i	$0.0990136\pi$
$198$	−1.74883	−0.124284
$199$	−19.5722	−1.38743	−0.693717	−	0.720248i	$-0.744028\pi$
$199$	−19.5722	−1.38743	−0.693717	+	0.720248i	$0.744028\pi$
$200$	14.2934	1.01070
$201$	−33.9256	−2.39293
$202$	−2.00586	−0.141132
$203$	30.4825	2.13945
$204$	−13.7089	−0.959818
$205$	−2.12285	−0.148266
$206$	5.76412	0.401605
$207$	−10.1297	−0.704062
$208$	−5.67880	−0.393754
$209$	1.13456	0.0784791
$210$	50.6689	3.49649
$211$	12.5465	0.863735	0.431867	−	0.901937i	$-0.357855\pi$
$211$	12.5465	0.863735	0.431867	+	0.901937i	$0.357855\pi$
$212$	−10.6595	−0.732096
$213$	12.3020	0.842917
$214$	5.25937	0.359523
$215$	−2.69287	−0.183652
$216$	−0.242205	−0.0164800
$217$	7.71748	0.523897
$218$	−3.29153	−0.222930
$219$	−18.9600	−1.28120
$220$	−2.47947	−0.167166
$221$	−31.5256	−2.12064
$222$	−17.6605	−1.18529
$223$	24.3760	1.63234	0.816169	−	0.577814i	$-0.196094\pi$
$223$	24.3760	1.63234	0.816169	+	0.577814i	$0.196094\pi$
$224$	−4.67132	−0.312116
$225$	44.2822	2.95215
$226$	−12.2954	−0.817881
$227$	−15.2227	−1.01036	−0.505182	−	0.863013i	$-0.668575\pi$
$227$	−15.2227	−1.01036	−0.505182	+	0.863013i	$0.668575\pi$
$228$	4.96329	0.328702
$229$	25.5897	1.69101	0.845506	−	0.533965i	$-0.179299\pi$
$229$	25.5897	1.69101	0.845506	+	0.533965i	$0.179299\pi$
$230$	−14.3618	−0.946987
$231$	−6.51164	−0.428435
$232$	−6.52544	−0.428416
$233$	−14.7509	−0.966365	−0.483183	−	0.875520i	$-0.660519\pi$
$233$	−14.7509	−0.966365	−0.483183	+	0.875520i	$0.660519\pi$
$234$	−17.5934	−1.15012
$235$	1.59478	0.104032
$236$	10.9057	0.709903
$237$	−20.1983	−1.31202
$238$	−25.9327	−1.68097
$239$	−2.62908	−0.170061	−0.0850306	−	0.996378i	$-0.527099\pi$
$239$	−2.62908	−0.170061	−0.0850306	+	0.996378i	$0.527099\pi$
$240$	−10.8468	−0.700158
$241$	−22.5965	−1.45557	−0.727786	−	0.685805i	$-0.759450\pi$
$241$	−22.5965	−1.45557	−0.727786	+	0.685805i	$0.759450\pi$
$242$	−10.6814	−0.686623
$243$	22.2011	1.42420
$244$	1.54368	0.0988239
$245$	65.1014	4.15917
$246$	1.19347	0.0760927
$247$	11.4138	0.726242
$248$	−1.65210	−0.104908
$249$	−29.3522	−1.86012
$250$	40.8208	2.58173
$251$	6.25962	0.395104	0.197552	−	0.980292i	$-0.436701\pi$
$251$	6.25962	0.395104	0.197552	+	0.980292i	$0.436701\pi$
$252$	−14.4721	−0.911659
$253$	1.84568	0.116037
$254$	12.9134	0.810258
$255$	−60.2156	−3.77084
$256$	1.00000	0.0625000
$257$	−22.1055	−1.37891	−0.689453	−	0.724330i	$-0.742149\pi$
$257$	−22.1055	−1.37891	−0.689453	+	0.724330i	$0.742149\pi$
$258$	1.51394	0.0942536
$259$	−33.4077	−2.07585
$260$	−24.9437	−1.54695
$261$	−20.2164	−1.25136
$262$	−5.69318	−0.351726
$263$	11.7966	0.727407	0.363704	−	0.931515i	$-0.381512\pi$
$263$	11.7966	0.727407	0.363704	+	0.931515i	$0.381512\pi$
$264$	1.39396	0.0857923
$265$	−46.8210	−2.87619
$266$	9.38886	0.575668
$267$	−29.8829	−1.82881
$268$	13.7382	0.839197
$269$	1.87525	0.114336	0.0571680	−	0.998365i	$-0.481793\pi$
$269$	1.87525	0.114336	0.0571680	+	0.998365i	$0.481793\pi$
$270$	−1.06387	−0.0647450
$271$	−26.0348	−1.58150	−0.790751	−	0.612138i	$-0.790310\pi$
$271$	−26.0348	−1.58150	−0.790751	+	0.612138i	$0.790310\pi$
$272$	5.55146	0.336607
$273$	−65.5079	−3.96472
$274$	17.0429	1.02960
$275$	−8.06846	−0.486546
$276$	8.07421	0.486010
$277$	−18.6556	−1.12091	−0.560453	−	0.828187i	$-0.689373\pi$
$277$	−18.6556	−1.12091	−0.560453	+	0.828187i	$0.689373\pi$
$278$	−11.1872	−0.670965
$279$	−5.11833	−0.306427
$280$	−20.5185	−1.22621
$281$	27.5656	1.64443	0.822214	−	0.569178i	$-0.192739\pi$
$281$	27.5656	1.64443	0.822214	+	0.569178i	$0.192739\pi$
$282$	−0.896586	−0.0533909
$283$	1.53352	0.0911584	0.0455792	−	0.998961i	$-0.485487\pi$
$283$	1.53352	0.0911584	0.0455792	+	0.998961i	$0.485487\pi$
$284$	−4.98170	−0.295610
$285$	21.8009	1.29137
$286$	3.20561	0.189552
$287$	2.25764	0.133264
$288$	3.09808	0.182556
$289$	13.8187	0.812866
$290$	−28.6625	−1.68312
$291$	−28.7427	−1.68492
$292$	7.67787	0.449313
$293$	1.44638	0.0844983	0.0422492	−	0.999107i	$-0.486548\pi$
$293$	1.44638	0.0844983	0.0422492	+	0.999107i	$0.486548\pi$
$294$	−36.6001	−2.13456
$295$	47.9027	2.78900
$296$	7.15165	0.415681
$297$	0.136722	0.00793340
$298$	5.91008	0.342362
$299$	18.5678	1.07380
$300$	−35.2966	−2.03785
$301$	2.86386	0.165070
$302$	−18.0453	−1.03839
$303$	4.95332	0.284561
$304$	−2.00989	−0.115275
$305$	6.78050	0.388250
$306$	17.1989	0.983194
$307$	23.9928	1.36934	0.684671	−	0.728852i	$-0.259946\pi$
$307$	23.9928	1.36934	0.684671	+	0.728852i	$0.259946\pi$
$308$	2.63690	0.150251
$309$	−14.2341	−0.809749
$310$	−7.25672	−0.412154
$311$	1.76254	0.0999444	0.0499722	−	0.998751i	$-0.484087\pi$
$311$	1.76254	0.0999444	0.0499722	+	0.998751i	$0.484087\pi$
$312$	14.0234	0.793919
$313$	24.2797	1.37237	0.686184	−	0.727428i	$-0.259285\pi$
$313$	24.2797	1.37237	0.686184	+	0.727428i	$0.259285\pi$
$314$	−6.44995	−0.363992
$315$	−63.5679	−3.58164
$316$	8.17935	0.460125
$317$	24.8129	1.39363	0.696815	−	0.717251i	$-0.254600\pi$
$317$	24.8129	1.39363	0.696815	+	0.717251i	$0.254600\pi$
$318$	26.3228	1.47611
$319$	3.68353	0.206238
$320$	4.39243	0.245544
$321$	−12.9876	−0.724900
$322$	15.2737	0.851168
$323$	−11.1578	−0.620839
$324$	−8.69614	−0.483119
$325$	−81.1696	−4.50248
$326$	−6.77425	−0.375191
$327$	8.12819	0.449490
$328$	−0.483297	−0.0266856
$329$	−1.69604	−0.0935055
$330$	6.12287	0.337053
$331$	24.3934	1.34078	0.670391	−	0.742008i	$-0.266126\pi$
$331$	24.3934	1.34078	0.670391	+	0.742008i	$0.266126\pi$
$332$	11.8862	0.652341
$333$	22.1564	1.21416
$334$	−8.94840	−0.489635
$335$	60.3443	3.29696
$336$	11.5355	0.629313
$337$	10.4448	0.568966	0.284483	−	0.958681i	$-0.408178\pi$
$337$	10.4448	0.568966	0.284483	+	0.958681i	$0.408178\pi$
$338$	19.2488	1.04700
$339$	30.3627	1.64908
$340$	24.3844	1.32243
$341$	0.932588	0.0505025
$342$	−6.22681	−0.336708
$343$	−36.5357	−1.97274
$344$	−0.613072	−0.0330546
$345$	35.4654	1.90939
$346$	10.4812	0.563471
$347$	21.6490	1.16218	0.581091	−	0.813839i	$-0.302626\pi$
$347$	21.6490	1.16218	0.581091	+	0.813839i	$0.302626\pi$
$348$	16.1141	0.863808
$349$	−27.9168	−1.49435	−0.747175	−	0.664627i	$-0.768590\pi$
$349$	−27.9168	−1.49435	−0.747175	+	0.664627i	$0.768590\pi$
$350$	−66.7693	−3.56897
$351$	1.37544	0.0734153
$352$	−0.564487	−0.0300873
$353$	−6.30504	−0.335584	−0.167792	−	0.985822i	$-0.553664\pi$
$353$	−6.30504	−0.335584	−0.167792	+	0.985822i	$0.553664\pi$
$354$	−26.9310	−1.43136
$355$	−21.8818	−1.16136
$356$	12.1011	0.641359
$357$	64.0389	3.38930
$358$	5.93971	0.313924
$359$	13.3201	0.703007	0.351503	−	0.936187i	$-0.385671\pi$
$359$	13.3201	0.703007	0.351503	+	0.936187i	$0.385671\pi$
$360$	13.6081	0.717210
$361$	−14.9603	−0.787386
$362$	6.44359	0.338668
$363$	26.3769	1.38443
$364$	26.5275	1.39042
$365$	33.7245	1.76522
$366$	−3.81201	−0.199257
$367$	35.8170	1.86963	0.934817	−	0.355129i	$-0.115563\pi$
$367$	35.8170	1.86963	0.934817	+	0.355129i	$0.115563\pi$
$368$	−3.26966	−0.170443
$369$	−1.49729	−0.0779459
$370$	31.4131	1.63309
$371$	49.7939	2.58517
$372$	4.07974	0.211525
$373$	21.5396	1.11528	0.557638	−	0.830084i	$-0.311708\pi$
$373$	21.5396	1.11528	0.557638	+	0.830084i	$0.311708\pi$
$374$	−3.13373	−0.162041
$375$	−100.804	−5.20550
$376$	0.363074	0.0187241
$377$	37.0567	1.90852
$378$	1.13142	0.0581939
$379$	−14.5186	−0.745769	−0.372885	−	0.927878i	$-0.621631\pi$
$379$	−14.5186	−0.745769	−0.372885	+	0.927878i	$0.621631\pi$
$380$	−8.82832	−0.452883
$381$	−31.8887	−1.63371
$382$	−13.2681	−0.678857
$383$	5.82831	0.297813	0.148906	−	0.988851i	$-0.452425\pi$
$383$	5.82831	0.297813	0.148906	+	0.988851i	$0.452425\pi$
$384$	−2.46943	−0.126018
$385$	11.5824	0.590294
$386$	22.7173	1.15628
$387$	−1.89935	−0.0965492
$388$	11.6394	0.590901
$389$	12.3203	0.624664	0.312332	−	0.949973i	$-0.398890\pi$
$389$	12.3203	0.624664	0.312332	+	0.949973i	$0.398890\pi$
$390$	61.5968	3.11908
$391$	−18.1514	−0.917956
$392$	14.8213	0.748587
$393$	14.0589	0.709178
$394$	26.7242	1.34634
$395$	35.9272	1.80770
$396$	−1.74883	−0.0878818
$397$	−29.4480	−1.47795	−0.738976	−	0.673731i	$-0.764691\pi$
$397$	−29.4480	−1.47795	−0.738976	+	0.673731i	$0.764691\pi$
$398$	−19.5722	−0.981063
$399$	−23.1851	−1.16071
$400$	14.2934	0.714672
$401$	−2.22165	−0.110944	−0.0554720	−	0.998460i	$-0.517666\pi$
$401$	−2.22165	−0.110944	−0.0554720	+	0.998460i	$0.517666\pi$
$402$	−33.9256	−1.69206
$403$	9.38194	0.467348
$404$	−2.00586	−0.0997950
$405$	−38.1972	−1.89803
$406$	30.4825	1.51282
$407$	−4.03701	−0.200107
$408$	−13.7089	−0.678694
$409$	10.1106	0.499938	0.249969	−	0.968254i	$-0.419580\pi$
$409$	10.1106	0.499938	0.249969	+	0.968254i	$0.419580\pi$
$410$	−2.12285	−0.104840
$411$	−42.0861	−2.07596
$412$	5.76412	0.283978
$413$	−50.9443	−2.50680
$414$	−10.1297	−0.497847
$415$	52.2094	2.56286
$416$	−5.67880	−0.278426
$417$	27.6260	1.35285
$418$	1.13456	0.0554931
$419$	−32.0186	−1.56421	−0.782106	−	0.623146i	$-0.785854\pi$
$419$	−32.0186	−1.56421	−0.782106	+	0.623146i	$0.785854\pi$
$420$	50.6689	2.47239
$421$	5.61392	0.273606	0.136803	−	0.990598i	$-0.456317\pi$
$421$	5.61392	0.273606	0.136803	+	0.990598i	$0.456317\pi$
$422$	12.5465	0.610753
$423$	1.12483	0.0546912
$424$	−10.6595	−0.517670
$425$	79.3494	3.84901
$426$	12.3020	0.596032
$427$	−7.21103	−0.348966
$428$	5.25937	0.254221
$429$	−7.91603	−0.382189
$430$	−2.69287	−0.129862
$431$	−40.5533	−1.95339	−0.976693	−	0.214641i	$-0.931142\pi$
$431$	−40.5533	−1.95339	−0.976693	+	0.214641i	$0.931142\pi$
$432$	−0.242205	−0.0116531
$433$	−11.2499	−0.540635	−0.270318	−	0.962771i	$-0.587129\pi$
$433$	−11.2499	−0.540635	−0.270318	+	0.962771i	$0.587129\pi$
$434$	7.71748	0.370451
$435$	70.7801	3.39365
$436$	−3.29153	−0.157635
$437$	6.57168	0.314366
$438$	−18.9600	−0.905942
$439$	12.2065	0.582583	0.291291	−	0.956634i	$-0.405915\pi$
$439$	12.2065	0.582583	0.291291	+	0.956634i	$0.405915\pi$
$440$	−2.47947	−0.118204
$441$	45.9175	2.18655
$442$	−31.5256	−1.49952
$443$	15.6656	0.744295	0.372147	−	0.928174i	$-0.378622\pi$
$443$	15.6656	0.744295	0.372147	+	0.928174i	$0.378622\pi$
$444$	−17.6605	−0.838130
$445$	53.1534	2.51971
$446$	24.3760	1.15424
$447$	−14.5945	−0.690297
$448$	−4.67132	−0.220699
$449$	12.2753	0.579306	0.289653	−	0.957132i	$-0.406460\pi$
$449$	12.2753	0.579306	0.289653	+	0.957132i	$0.406460\pi$
$450$	44.2822	2.08748
$451$	0.272815	0.0128463
$452$	−12.2954	−0.578329
$453$	44.5615	2.09368
$454$	−15.2227	−0.714436
$455$	116.520	5.46256
$456$	4.96329	0.232427
$457$	−5.78124	−0.270435	−0.135217	−	0.990816i	$-0.543173\pi$
$457$	−5.78124	−0.270435	−0.135217	+	0.990816i	$0.543173\pi$
$458$	25.5897	1.19573
$459$	−1.34459	−0.0627602
$460$	−14.3618	−0.669621
$461$	26.0209	1.21191	0.605956	−	0.795498i	$-0.292791\pi$
$461$	26.0209	1.21191	0.605956	+	0.795498i	$0.292791\pi$
$462$	−6.51164	−0.302949
$463$	−7.26276	−0.337529	−0.168765	−	0.985656i	$-0.553978\pi$
$463$	−7.26276	−0.337529	−0.168765	+	0.985656i	$0.553978\pi$
$464$	−6.52544	−0.302936
$465$	17.9200	0.831018
$466$	−14.7509	−0.683324
$467$	16.0621	0.743267	0.371634	−	0.928379i	$-0.378798\pi$
$467$	16.0621	0.743267	0.371634	+	0.928379i	$0.378798\pi$
$468$	−17.5934	−0.813255
$469$	−64.1758	−2.96336
$470$	1.59478	0.0735616
$471$	15.9277	0.733910
$472$	10.9057	0.501977
$473$	0.346071	0.0159124
$474$	−20.1983	−0.927740
$475$	−28.7283	−1.31814
$476$	−25.9327	−1.18862
$477$	−33.0240	−1.51206
$478$	−2.62908	−0.120251
$479$	4.51193	0.206155	0.103078	−	0.994673i	$-0.467131\pi$
$479$	4.51193	0.206155	0.103078	+	0.994673i	$0.467131\pi$
$480$	−10.8468	−0.495086
$481$	−40.6128	−1.85178
$482$	−22.5965	−1.02924
$483$	−37.7172	−1.71619
$484$	−10.6814	−0.485516
$485$	51.1252	2.32148
$486$	22.2011	1.00706
$487$	38.9960	1.76708	0.883538	−	0.468360i	$-0.155155\pi$
$487$	38.9960	1.76708	0.883538	+	0.468360i	$0.155155\pi$
$488$	1.54368	0.0698791
$489$	16.7285	0.756491
$490$	65.1014	2.94098
$491$	28.3121	1.27771	0.638854	−	0.769328i	$-0.279409\pi$
$491$	28.3121	1.27771	0.638854	+	0.769328i	$0.279409\pi$
$492$	1.19347	0.0538056
$493$	−36.2257	−1.63153
$494$	11.4138	0.513531
$495$	−7.68160	−0.345262
$496$	−1.65210	−0.0741814
$497$	23.2712	1.04385
$498$	−29.3522	−1.31530
$499$	37.5311	1.68012	0.840062	−	0.542491i	$-0.182519\pi$
$499$	37.5311	1.68012	0.840062	+	0.542491i	$0.182519\pi$
$500$	40.8208	1.82556
$501$	22.0975	0.987241
$502$	6.25962	0.279381
$503$	−8.25033	−0.367864	−0.183932	−	0.982939i	$-0.558883\pi$
$503$	−8.25033	−0.367864	−0.183932	+	0.982939i	$0.558883\pi$
$504$	−14.4721	−0.644641
$505$	−8.81058	−0.392066
$506$	1.84568	0.0820506
$507$	−47.5335	−2.11104
$508$	12.9134	0.572939
$509$	−41.0588	−1.81990	−0.909950	−	0.414719i	$-0.863880\pi$
$509$	−41.0588	−1.81990	−0.909950	+	0.414719i	$0.863880\pi$
$510$	−60.2156	−2.66639
$511$	−35.8658	−1.58661
$512$	1.00000	0.0441942
$513$	0.486807	0.0214930
$514$	−22.1055	−0.975034
$515$	25.3185	1.11567
$516$	1.51394	0.0666474
$517$	−0.204951	−0.00901371
$518$	−33.4077	−1.46785
$519$	−25.8825	−1.13611
$520$	−24.9437	−1.09386
$521$	33.1000	1.45014	0.725068	−	0.688677i	$-0.241808\pi$
$521$	33.1000	1.45014	0.725068	+	0.688677i	$0.241808\pi$
$522$	−20.2164	−0.884846
$523$	2.80144	0.122499	0.0612493	−	0.998122i	$-0.480492\pi$
$523$	2.80144	0.122499	0.0612493	+	0.998122i	$0.480492\pi$
$524$	−5.69318	−0.248708
$525$	164.882	7.19604
$526$	11.7966	0.514354
$527$	−9.17156	−0.399519
$528$	1.39396	0.0606644
$529$	−12.3093	−0.535187
$530$	−46.8210	−2.03378
$531$	33.7869	1.46623
$532$	9.38886	0.407059
$533$	2.74455	0.118879
$534$	−29.8829	−1.29316
$535$	23.1014	0.998762
$536$	13.7382	0.593402
$537$	−14.6677	−0.632958
$538$	1.87525	0.0808477
$539$	−8.36641	−0.360367
$540$	−1.06387	−0.0457817
$541$	−10.4140	−0.447734	−0.223867	−	0.974620i	$-0.571868\pi$
$541$	−10.4140	−0.447734	−0.223867	+	0.974620i	$0.571868\pi$
$542$	−26.0348	−1.11829
$543$	−15.9120	−0.682849
$544$	5.55146	0.238017
$545$	−14.4578	−0.619304
$546$	−65.5079	−2.80348
$547$	−9.81017	−0.419452	−0.209726	−	0.977760i	$-0.567257\pi$
$547$	−9.81017	−0.419452	−0.209726	+	0.977760i	$0.567257\pi$
$548$	17.0429	0.728035
$549$	4.78244	0.204110
$550$	−8.06846	−0.344040
$551$	13.1154	0.558737
$552$	8.07421	0.343661
$553$	−38.2084	−1.62479
$554$	−18.6556	−0.792600
$555$	−77.5724	−3.29277
$556$	−11.1872	−0.474444
$557$	−43.7020	−1.85171	−0.925856	−	0.377877i	$-0.876654\pi$
$557$	−43.7020	−1.85171	−0.925856	+	0.377877i	$0.876654\pi$
$558$	−5.11833	−0.216676
$559$	3.48151	0.147252
$560$	−20.5185	−0.867063
$561$	7.73852	0.326720
$562$	27.5656	1.16279
$563$	−0.560696	−0.0236305	−0.0118153	−	0.999930i	$-0.503761\pi$
$563$	−0.560696	−0.0236305	−0.0118153	+	0.999930i	$0.503761\pi$
$564$	−0.896586	−0.0377531
$565$	−54.0069	−2.27209
$566$	1.53352	0.0644587
$567$	40.6225	1.70598
$568$	−4.98170	−0.209028
$569$	28.4185	1.19137	0.595684	−	0.803219i	$-0.296881\pi$
$569$	28.4185	1.19137	0.595684	+	0.803219i	$0.296881\pi$
$570$	21.8009	0.913139
$571$	−15.8359	−0.662711	−0.331356	−	0.943506i	$-0.607506\pi$
$571$	−15.8359	−0.662711	−0.331356	+	0.943506i	$0.607506\pi$
$572$	3.20561	0.134033
$573$	32.7647	1.36877
$574$	2.25764	0.0942319
$575$	−46.7347	−1.94897
$576$	3.09808	0.129087
$577$	−15.6398	−0.651095	−0.325547	−	0.945526i	$-0.605549\pi$
$577$	−15.6398	−0.651095	−0.325547	+	0.945526i	$0.605549\pi$
$578$	13.8187	0.574783
$579$	−56.0987	−2.33138
$580$	−28.6625	−1.19015
$581$	−55.5244	−2.30354
$582$	−28.7427	−1.19142
$583$	6.01714	0.249204
$584$	7.67787	0.317713
$585$	−77.2777	−3.19504
$586$	1.44638	0.0597493
$587$	29.3547	1.21160	0.605799	−	0.795618i	$-0.292853\pi$
$587$	29.3547	1.21160	0.605799	+	0.795618i	$0.292853\pi$
$588$	−36.6001	−1.50936
$589$	3.32054	0.136820
$590$	47.9027	1.97212
$591$	−65.9935	−2.71461
$592$	7.15165	0.293931
$593$	11.5104	0.472674	0.236337	−	0.971671i	$-0.424053\pi$
$593$	11.5104	0.472674	0.236337	+	0.971671i	$0.424053\pi$
$594$	0.136722	0.00560976
$595$	−113.907	−4.66975
$596$	5.91008	0.242086
$597$	48.3320	1.97810
$598$	18.5678	0.759293
$599$	−26.7707	−1.09382	−0.546911	−	0.837191i	$-0.684196\pi$
$599$	−26.7707	−1.09382	−0.546911	+	0.837191i	$0.684196\pi$
$600$	−35.2966	−1.44098
$601$	36.4017	1.48486	0.742429	−	0.669925i	$-0.233674\pi$
$601$	36.4017	1.48486	0.742429	+	0.669925i	$0.233674\pi$
$602$	2.86386	0.116722
$603$	42.5622	1.73327
$604$	−18.0453	−0.734251
$605$	−46.9171	−1.90745
$606$	4.95332	0.201215
$607$	−8.35565	−0.339146	−0.169573	−	0.985518i	$-0.554239\pi$
$607$	−8.35565	−0.339146	−0.169573	+	0.985518i	$0.554239\pi$
$608$	−2.00989	−0.0815119
$609$	−75.2743	−3.05027
$610$	6.78050	0.274535
$611$	−2.06183	−0.0834125
$612$	17.1989	0.695223
$613$	2.89728	0.117020	0.0585100	−	0.998287i	$-0.481365\pi$
$613$	2.89728	0.117020	0.0585100	+	0.998287i	$0.481365\pi$
$614$	23.9928	0.968271
$615$	5.24222	0.211387
$616$	2.63690	0.106244
$617$	−20.4143	−0.821847	−0.410923	−	0.911670i	$-0.634794\pi$
$617$	−20.4143	−0.821847	−0.410923	+	0.911670i	$0.634794\pi$
$618$	−14.2341	−0.572579
$619$	−8.85405	−0.355874	−0.177937	−	0.984042i	$-0.556942\pi$
$619$	−8.85405	−0.355874	−0.177937	+	0.984042i	$0.556942\pi$
$620$	−7.25672	−0.291437
$621$	0.791930	0.0317791
$622$	1.76254	0.0706714
$623$	−56.5284	−2.26476
$624$	14.0234	0.561385
$625$	107.835	4.31341
$626$	24.2797	0.970410
$627$	−2.80171	−0.111890
$628$	−6.44995	−0.257381
$629$	39.7021	1.58303
$630$	−63.5679	−2.53260
$631$	43.6517	1.73775	0.868873	−	0.495036i	$-0.164845\pi$
$631$	43.6517	1.73775	0.868873	+	0.495036i	$0.164845\pi$
$632$	8.17935	0.325357
$633$	−30.9826	−1.23145
$634$	24.8129	0.985445
$635$	56.7212	2.25091
$636$	26.3228	1.04377
$637$	−84.1671	−3.33482
$638$	3.68353	0.145832
$639$	−15.4337	−0.610549
$640$	4.39243	0.173626
$641$	−28.6621	−1.13209	−0.566043	−	0.824376i	$-0.691526\pi$
$641$	−28.6621	−1.13209	−0.566043	+	0.824376i	$0.691526\pi$
$642$	−12.9876	−0.512582
$643$	14.7420	0.581368	0.290684	−	0.956819i	$-0.406117\pi$
$643$	14.7420	0.581368	0.290684	+	0.956819i	$0.406117\pi$
$644$	15.2737	0.601867
$645$	6.64986	0.261838
$646$	−11.1578	−0.439000
$647$	−11.6609	−0.458437	−0.229218	−	0.973375i	$-0.573617\pi$
$647$	−11.6609	−0.458437	−0.229218	+	0.973375i	$0.573617\pi$
$648$	−8.69614	−0.341616
$649$	−6.15615	−0.241650
$650$	−81.1696	−3.18373
$651$	−19.0578	−0.746933
$652$	−6.77425	−0.265300
$653$	−26.9151	−1.05327	−0.526634	−	0.850092i	$-0.676546\pi$
$653$	−26.9151	−1.05327	−0.526634	+	0.850092i	$0.676546\pi$
$654$	8.12819	0.317837
$655$	−25.0069	−0.977100
$656$	−0.483297	−0.0188696
$657$	23.7867	0.928007
$658$	−1.69604	−0.0661184
$659$	9.39462	0.365962	0.182981	−	0.983116i	$-0.441425\pi$
$659$	9.39462	0.365962	0.182981	+	0.983116i	$0.441425\pi$
$660$	6.12287	0.238333
$661$	26.5562	1.03292	0.516459	−	0.856312i	$-0.327250\pi$
$661$	26.5562	1.03292	0.516459	+	0.856312i	$0.327250\pi$
$662$	24.3934	0.948076
$663$	77.8504	3.02346
$664$	11.8862	0.461275
$665$	41.2399	1.59922
$666$	22.1564	0.858543
$667$	21.3360	0.826133
$668$	−8.94840	−0.346224
$669$	−60.1948	−2.32727
$670$	60.3443	2.33130
$671$	−0.871387	−0.0336395
$672$	11.5355	0.444992
$673$	−12.2832	−0.473482	−0.236741	−	0.971573i	$-0.576079\pi$
$673$	−12.2832	−0.473482	−0.236741	+	0.971573i	$0.576079\pi$
$674$	10.4448	0.402319
$675$	−3.46195	−0.133250
$676$	19.2488	0.740338
$677$	−25.3616	−0.974724	−0.487362	−	0.873200i	$-0.662041\pi$
$677$	−25.3616	−0.974724	−0.487362	+	0.873200i	$0.662041\pi$
$678$	30.3627	1.16607
$679$	−54.3714	−2.08658
$680$	24.3844	0.935099
$681$	37.5913	1.44050
$682$	0.932588	0.0357106
$683$	−18.6572	−0.713897	−0.356949	−	0.934124i	$-0.616183\pi$
$683$	−18.6572	−0.713897	−0.356949	+	0.934124i	$0.616183\pi$
$684$	−6.22681	−0.238088
$685$	74.8595	2.86024
$686$	−36.5357	−1.39494
$687$	−63.1919	−2.41092
$688$	−0.613072	−0.0233731
$689$	60.5331	2.30613
$690$	35.4654	1.35014
$691$	−29.9983	−1.14119	−0.570595	−	0.821232i	$-0.693287\pi$
$691$	−29.9983	−1.14119	−0.570595	+	0.821232i	$0.693287\pi$
$692$	10.4812	0.398434
$693$	8.16934	0.310327
$694$	21.6490	0.821786
$695$	−49.1391	−1.86395
$696$	16.1141	0.610804
$697$	−2.68300	−0.101626
$698$	−27.9168	−1.05667
$699$	36.4264	1.37777
$700$	−66.7693	−2.52364
$701$	−43.4525	−1.64118	−0.820590	−	0.571517i	$-0.806355\pi$
$701$	−43.4525	−1.64118	−0.820590	+	0.571517i	$0.806355\pi$
$702$	1.37544	0.0519125
$703$	−14.3740	−0.542128
$704$	−0.564487	−0.0212749
$705$	−3.93819	−0.148321
$706$	−6.30504	−0.237293
$707$	9.37000	0.352395
$708$	−26.9310	−1.01213
$709$	−5.09739	−0.191436	−0.0957182	−	0.995408i	$-0.530515\pi$
$709$	−5.09739	−0.191436	−0.0957182	+	0.995408i	$0.530515\pi$
$710$	−21.8818	−0.821209
$711$	25.3403	0.950336
$712$	12.1011	0.453510
$713$	5.40180	0.202299
$714$	64.0389	2.39660
$715$	14.0804	0.526578
$716$	5.93971	0.221978
$717$	6.49233	0.242461
$718$	13.3201	0.497101
$719$	−15.5870	−0.581299	−0.290649	−	0.956830i	$-0.593871\pi$
$719$	−15.5870	−0.581299	−0.290649	+	0.956830i	$0.593871\pi$
$720$	13.6081	0.507144
$721$	−26.9261	−1.00278
$722$	−14.9603	−0.556766
$723$	55.8006	2.07525
$724$	6.44359	0.239474
$725$	−93.2710	−3.46400
$726$	26.3769	0.978937
$727$	−1.98215	−0.0735139	−0.0367569	−	0.999324i	$-0.511703\pi$
$727$	−1.98215	−0.0735139	−0.0367569	+	0.999324i	$0.511703\pi$
$728$	26.5275	0.983176
$729$	−28.7357	−1.06428
$730$	33.7245	1.24820
$731$	−3.40344	−0.125881
$732$	−3.81201	−0.140896
$733$	−17.7353	−0.655068	−0.327534	−	0.944839i	$-0.606218\pi$
$733$	−17.7353	−0.655068	−0.327534	+	0.944839i	$0.606218\pi$
$734$	35.8170	1.32203
$735$	−160.763	−5.92984
$736$	−3.26966	−0.120521
$737$	−7.75506	−0.285661
$738$	−1.49729	−0.0551161
$739$	−1.83451	−0.0674835	−0.0337418	−	0.999431i	$-0.510742\pi$
$739$	−1.83451	−0.0674835	−0.0337418	+	0.999431i	$0.510742\pi$
$740$	31.4131	1.15477
$741$	−28.1855	−1.03542
$742$	49.7939	1.82799
$743$	−10.2826	−0.377231	−0.188615	−	0.982051i	$-0.560400\pi$
$743$	−10.2826	−0.377231	−0.188615	+	0.982051i	$0.560400\pi$
$744$	4.07974	0.149570
$745$	25.9596	0.951086
$746$	21.5396	0.788620
$747$	36.8245	1.34734
$748$	−3.13373	−0.114580
$749$	−24.5682	−0.897704
$750$	−100.804	−3.68084
$751$	22.1528	0.808369	0.404184	−	0.914677i	$-0.367555\pi$
$751$	22.1528	0.808369	0.404184	+	0.914677i	$0.367555\pi$
$752$	0.363074	0.0132399
$753$	−15.4577	−0.563310
$754$	37.0567	1.34953
$755$	−79.2625	−2.88466
$756$	1.13142	0.0411493
$757$	2.56099	0.0930809	0.0465405	−	0.998916i	$-0.485180\pi$
$757$	2.56099	0.0930809	0.0465405	+	0.998916i	$0.485180\pi$
$758$	−14.5186	−0.527339
$759$	−4.55778	−0.165437
$760$	−8.82832	−0.320237
$761$	−12.2194	−0.442953	−0.221476	−	0.975166i	$-0.571088\pi$
$761$	−12.2194	−0.442953	−0.221476	+	0.975166i	$0.571088\pi$
$762$	−31.8887	−1.15521
$763$	15.3758	0.556641
$764$	−13.2681	−0.480024
$765$	75.5449	2.73133
$766$	5.82831	0.210585
$767$	−61.9316	−2.23622
$768$	−2.46943	−0.0891079
$769$	−10.1976	−0.367736	−0.183868	−	0.982951i	$-0.558862\pi$
$769$	−10.1976	−0.367736	−0.183868	+	0.982951i	$0.558862\pi$
$770$	11.5824	0.417401
$771$	54.5881	1.96594
$772$	22.7173	0.817613
$773$	−18.3313	−0.659330	−0.329665	−	0.944098i	$-0.606936\pi$
$773$	−18.3313	−0.659330	−0.329665	+	0.944098i	$0.606936\pi$
$774$	−1.89935	−0.0682706
$775$	−23.6142	−0.848245
$776$	11.6394	0.417830
$777$	82.4979	2.95959
$778$	12.3203	0.441704
$779$	0.971375	0.0348031
$780$	61.5968	2.20552
$781$	2.81211	0.100625
$782$	−18.1514	−0.649093
$783$	1.58050	0.0564823
$784$	14.8213	0.529331
$785$	−28.3310	−1.01118
$786$	14.0589	0.501464
$787$	39.2354	1.39859	0.699296	−	0.714833i	$-0.253497\pi$
$787$	39.2354	1.39859	0.699296	+	0.714833i	$0.253497\pi$
$788$	26.7242	0.952010
$789$	−29.1308	−1.03708
$790$	35.9272	1.27823
$791$	57.4360	2.04219
$792$	−1.74883	−0.0621418
$793$	−8.76625	−0.311299
$794$	−29.4480	−1.04507
$795$	115.621	4.10066
$796$	−19.5722	−0.693717
$797$	−46.2734	−1.63909	−0.819544	−	0.573017i	$-0.805773\pi$
$797$	−46.2734	−1.63909	−0.819544	+	0.573017i	$0.805773\pi$
$798$	−23.1851	−0.820745
$799$	2.01559	0.0713065
$800$	14.2934	0.505349
$801$	37.4903	1.32466
$802$	−2.22165	−0.0784493
$803$	−4.33406	−0.152946
$804$	−33.9256	−1.19646
$805$	67.0885	2.36456
$806$	9.38194	0.330465
$807$	−4.63080	−0.163012
$808$	−2.00586	−0.0705658
$809$	33.9773	1.19458	0.597290	−	0.802026i	$-0.296244\pi$
$809$	33.9773	1.19458	0.597290	+	0.802026i	$0.296244\pi$
$810$	−38.1972	−1.34211
$811$	49.4695	1.73711	0.868554	−	0.495594i	$-0.165050\pi$
$811$	49.4695	1.73711	0.868554	+	0.495594i	$0.165050\pi$
$812$	30.4825	1.06972
$813$	64.2911	2.25479
$814$	−4.03701	−0.141497
$815$	−29.7554	−1.04229
$816$	−13.7089	−0.479909
$817$	1.23221	0.0431095
$818$	10.1106	0.353510
$819$	82.1844	2.87176
$820$	−2.12285	−0.0741330
$821$	−2.39100	−0.0834464	−0.0417232	−	0.999129i	$-0.513285\pi$
$821$	−2.39100	−0.0834464	−0.0417232	+	0.999129i	$0.513285\pi$
$822$	−42.0861	−1.46792
$823$	−35.9091	−1.25171	−0.625856	−	0.779938i	$-0.715250\pi$
$823$	−35.9091	−1.25171	−0.625856	+	0.779938i	$0.715250\pi$
$824$	5.76412	0.200803
$825$	19.9245	0.693682
$826$	−50.9443	−1.77258
$827$	−1.87043	−0.0650414	−0.0325207	−	0.999471i	$-0.510353\pi$
$827$	−1.87043	−0.0650414	−0.0325207	+	0.999471i	$0.510353\pi$
$828$	−10.1297	−0.352031
$829$	−28.7207	−0.997513	−0.498756	−	0.866742i	$-0.666210\pi$
$829$	−28.7207	−0.997513	−0.498756	+	0.866742i	$0.666210\pi$
$830$	52.2094	1.81221
$831$	46.0686	1.59810
$832$	−5.67880	−0.196877
$833$	82.2797	2.85082
$834$	27.6260	0.956612
$835$	−39.3052	−1.36021
$836$	1.13456	0.0392395
$837$	0.400147	0.0138311
$838$	−32.0186	−1.10606
$839$	14.5409	0.502006	0.251003	−	0.967986i	$-0.419240\pi$
$839$	14.5409	0.502006	0.251003	+	0.967986i	$0.419240\pi$
$840$	50.6689	1.74824
$841$	13.5814	0.468324
$842$	5.61392	0.193469
$843$	−68.0714	−2.34450
$844$	12.5465	0.431867
$845$	84.5490	2.90857
$846$	1.12483	0.0386725
$847$	49.8961	1.71445
$848$	−10.6595	−0.366048
$849$	−3.78692	−0.129967
$850$	79.3494	2.72166
$851$	−23.3835	−0.801576
$852$	12.3020	0.421458
$853$	−5.25799	−0.180030	−0.0900150	−	0.995940i	$-0.528692\pi$
$853$	−5.25799	−0.180030	−0.0900150	+	0.995940i	$0.528692\pi$
$854$	−7.21103	−0.246756
$855$	−27.3508	−0.935379
$856$	5.25937	0.179762
$857$	−40.3689	−1.37897	−0.689487	−	0.724298i	$-0.742164\pi$
$857$	−40.3689	−1.37897	−0.689487	+	0.724298i	$0.742164\pi$
$858$	−7.91603	−0.270249
$859$	−20.6301	−0.703889	−0.351945	−	0.936021i	$-0.614479\pi$
$859$	−20.6301	−0.703889	−0.351945	+	0.936021i	$0.614479\pi$
$860$	−2.69287	−0.0918262
$861$	−5.57507	−0.189998
$862$	−40.5533	−1.38125
$863$	−0.495229	−0.0168578	−0.00842889	−	0.999964i	$-0.502683\pi$
$863$	−0.495229	−0.0168578	−0.00842889	+	0.999964i	$0.502683\pi$
$864$	−0.242205	−0.00823999
$865$	46.0377	1.56533
$866$	−11.2499	−0.382287
$867$	−34.1243	−1.15892
$868$	7.71748	0.261948
$869$	−4.61714	−0.156626
$870$	70.7801	2.39967
$871$	−78.0168	−2.64350
$872$	−3.29153	−0.111465
$873$	36.0598	1.22044
$874$	6.57168	0.222290
$875$	−190.687	−6.44640
$876$	−18.9600	−0.640598
$877$	−38.0589	−1.28516	−0.642579	−	0.766219i	$-0.722136\pi$
$877$	−38.0589	−1.28516	−0.642579	+	0.766219i	$0.722136\pi$
$878$	12.2065	0.411948
$879$	−3.57173	−0.120471
$880$	−2.47947	−0.0835829
$881$	−23.2952	−0.784834	−0.392417	−	0.919787i	$-0.628361\pi$
$881$	−23.2952	−0.784834	−0.392417	+	0.919787i	$0.628361\pi$
$882$	45.9175	1.54612
$883$	−33.7521	−1.13585	−0.567925	−	0.823081i	$-0.692253\pi$
$883$	−33.7521	−1.13585	−0.567925	+	0.823081i	$0.692253\pi$
$884$	−31.5256	−1.06032
$885$	−118.292	−3.97635
$886$	15.6656	0.526296
$887$	21.2151	0.712335	0.356167	−	0.934422i	$-0.384083\pi$
$887$	21.2151	0.712335	0.356167	+	0.934422i	$0.384083\pi$
$888$	−17.6605	−0.592647
$889$	−60.3226	−2.02316
$890$	53.1534	1.78171
$891$	4.90886	0.164453
$892$	24.3760	0.816169
$893$	−0.729740	−0.0244198
$894$	−14.5945	−0.488114
$895$	26.0898	0.872085
$896$	−4.67132	−0.156058
$897$	−45.8518	−1.53095
$898$	12.2753	0.409632
$899$	10.7807	0.359555
$900$	44.2822	1.47607
$901$	−59.1757	−1.97143
$902$	0.272815	0.00908373
$903$	−7.07209	−0.235344
$904$	−12.2954	−0.408940
$905$	28.3030	0.940824
$906$	44.5615	1.48046
$907$	57.3102	1.90296	0.951478	−	0.307718i	$-0.0995652\pi$
$907$	57.3102	1.90296	0.951478	+	0.307718i	$0.0995652\pi$
$908$	−15.2227	−0.505182
$909$	−6.21430	−0.206115
$910$	116.520	3.86261
$911$	53.4659	1.77141	0.885703	−	0.464253i	$-0.153677\pi$
$911$	53.4659	1.77141	0.885703	+	0.464253i	$0.153677\pi$
$912$	4.96329	0.164351
$913$	−6.70962	−0.222056
$914$	−5.78124	−0.191226
$915$	−16.7440	−0.553539
$916$	25.5897	0.845506
$917$	26.5947	0.878234
$918$	−1.34459	−0.0443782
$919$	−9.26304	−0.305560	−0.152780	−	0.988260i	$-0.548823\pi$
$919$	−9.26304	−0.305560	−0.152780	+	0.988260i	$0.548823\pi$
$920$	−14.3618	−0.473494
$921$	−59.2486	−1.95231
$922$	26.0209	0.856951
$923$	28.2901	0.931180
$924$	−6.51164	−0.214217
$925$	102.222	3.36103
$926$	−7.26276	−0.238669
$927$	17.8577	0.586524
$928$	−6.52544	−0.214208
$929$	−16.4878	−0.540948	−0.270474	−	0.962727i	$-0.587180\pi$
$929$	−16.4878	−0.540948	−0.270474	+	0.962727i	$0.587180\pi$
$930$	17.9200	0.587619
$931$	−29.7892	−0.976301
$932$	−14.7509	−0.483183
$933$	−4.35246	−0.142493
$934$	16.0621	0.525569
$935$	−13.7647	−0.450153
$936$	−17.5934	−0.575058
$937$	13.0654	0.426828	0.213414	−	0.976962i	$-0.431542\pi$
$937$	13.0654	0.426828	0.213414	+	0.976962i	$0.431542\pi$
$938$	−64.1758	−2.09541
$939$	−59.9569	−1.95662
$940$	1.59478	0.0520159
$941$	60.4526	1.97070	0.985349	−	0.170548i	$-0.0545537\pi$
$941$	60.4526	1.97070	0.985349	+	0.170548i	$0.0545537\pi$
$942$	15.9277	0.518953
$943$	1.58022	0.0514590
$944$	10.9057	0.354952
$945$	4.96968	0.161664
$946$	0.346071	0.0112517
$947$	45.5942	1.48161	0.740806	−	0.671719i	$-0.234444\pi$
$947$	45.5942	1.48161	0.740806	+	0.671719i	$0.234444\pi$
$948$	−20.1983	−0.656011
$949$	−43.6011	−1.41535
$950$	−28.7283	−0.932069
$951$	−61.2737	−1.98693
$952$	−25.9327	−0.840483
$953$	36.1556	1.17119	0.585597	−	0.810602i	$-0.300860\pi$
$953$	36.1556	1.17119	0.585597	+	0.810602i	$0.300860\pi$
$954$	−33.0240	−1.06919
$955$	−58.2793	−1.88588
$956$	−2.62908	−0.0850306
$957$	−9.09621	−0.294039
$958$	4.51193	0.145774
$959$	−79.6127	−2.57083
$960$	−10.8468	−0.350079
$961$	−28.2706	−0.911954
$962$	−40.6128	−1.30941
$963$	16.2940	0.525066
$964$	−22.5965	−0.727786
$965$	99.7840	3.21216
$966$	−37.7172	−1.21353
$967$	−44.2800	−1.42395	−0.711975	−	0.702205i	$-0.752199\pi$
$967$	−44.2800	−1.42395	−0.711975	+	0.702205i	$0.752199\pi$
$968$	−10.6814	−0.343312
$969$	27.5535	0.885146
$970$	51.1252	1.64153
$971$	−1.17922	−0.0378430	−0.0189215	−	0.999821i	$-0.506023\pi$
$971$	−1.17922	−0.0378430	−0.0189215	+	0.999821i	$0.506023\pi$
$972$	22.2011	0.712101
$973$	52.2591	1.67535
$974$	38.9960	1.24951
$975$	200.443	6.41930
$976$	1.54368	0.0494120
$977$	−2.54445	−0.0814041	−0.0407021	−	0.999171i	$-0.512959\pi$
$977$	−2.54445	−0.0814041	−0.0407021	+	0.999171i	$0.512959\pi$
$978$	16.7285	0.534920
$979$	−6.83094	−0.218318
$980$	65.1014	2.07959
$981$	−10.1974	−0.325578
$982$	28.3121	0.903476
$983$	−10.3398	−0.329788	−0.164894	−	0.986311i	$-0.552728\pi$
$983$	−10.3398	−0.329788	−0.164894	+	0.986311i	$0.552728\pi$
$984$	1.19347	0.0380463
$985$	117.384	3.74017
$986$	−36.2257	−1.15366
$987$	4.18824	0.133313
$988$	11.4138	0.363121
$989$	2.00454	0.0637406
$990$	−7.68160	−0.244137
$991$	17.1341	0.544284	0.272142	−	0.962257i	$-0.412268\pi$
$991$	17.1341	0.544284	0.272142	+	0.962257i	$0.412268\pi$
$992$	−1.65210	−0.0524542
$993$	−60.2378	−1.91159
$994$	23.2712	0.738116
$995$	−85.9693	−2.72541
$996$	−29.3522	−0.930059
$997$	−20.6905	−0.655276	−0.327638	−	0.944803i	$-0.606253\pi$
$997$	−20.6905	−0.655276	−0.327638	+	0.944803i	$0.606253\pi$
$998$	37.5311	1.18803
$999$	−1.73217	−0.0548033

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6002.2.a.d.1.12	✓	79

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6002.2.a.d.1.12	✓	79	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:	\( N \)	\(=\)	\( 6002 = 2 \cdot 3001 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6002.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -2.46943 q^{3} +1.00000 q^{4} +4.39243 q^{5} -2.46943 q^{6} -4.67132 q^{7} +1.00000 q^{8} +3.09808 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} -2.46943 q^{3} +1.00000 q^{4} +4.39243 q^{5} -2.46943 q^{6} -4.67132 q^{7} +1.00000 q^{8} +3.09808 q^{9} +4.39243 q^{10} -0.564487 q^{11} -2.46943 q^{12} -5.67880 q^{13} -4.67132 q^{14} -10.8468 q^{15} +1.00000 q^{16} +5.55146 q^{17} +3.09808 q^{18} -2.00989 q^{19} +4.39243 q^{20} +11.5355 q^{21} -0.564487 q^{22} -3.26966 q^{23} -2.46943 q^{24} +14.2934 q^{25} -5.67880 q^{26} -0.242205 q^{27} -4.67132 q^{28} -6.52544 q^{29} -10.8468 q^{30} -1.65210 q^{31} +1.00000 q^{32} +1.39396 q^{33} +5.55146 q^{34} -20.5185 q^{35} +3.09808 q^{36} +7.15165 q^{37} -2.00989 q^{38} +14.0234 q^{39} +4.39243 q^{40} -0.483297 q^{41} +11.5355 q^{42} -0.613072 q^{43} -0.564487 q^{44} +13.6081 q^{45} -3.26966 q^{46} +0.363074 q^{47} -2.46943 q^{48} +14.8213 q^{49} +14.2934 q^{50} -13.7089 q^{51} -5.67880 q^{52} -10.6595 q^{53} -0.242205 q^{54} -2.47947 q^{55} -4.67132 q^{56} +4.96329 q^{57} -6.52544 q^{58} +10.9057 q^{59} -10.8468 q^{60} +1.54368 q^{61} -1.65210 q^{62} -14.4721 q^{63} +1.00000 q^{64} -24.9437 q^{65} +1.39396 q^{66} +13.7382 q^{67} +5.55146 q^{68} +8.07421 q^{69} -20.5185 q^{70} -4.98170 q^{71} +3.09808 q^{72} +7.67787 q^{73} +7.15165 q^{74} -35.2966 q^{75} -2.00989 q^{76} +2.63690 q^{77} +14.0234 q^{78} +8.17935 q^{79} +4.39243 q^{80} -8.69614 q^{81} -0.483297 q^{82} +11.8862 q^{83} +11.5355 q^{84} +24.3844 q^{85} -0.613072 q^{86} +16.1141 q^{87} -0.564487 q^{88} +12.1011 q^{89} +13.6081 q^{90} +26.5275 q^{91} -3.26966 q^{92} +4.07974 q^{93} +0.363074 q^{94} -8.82832 q^{95} -2.46943 q^{96} +11.6394 q^{97} +14.8213 q^{98} -1.74883 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 79 q + 79 q^{2} + 17 q^{3} + 79 q^{4} + 18 q^{5} + 17 q^{6} + 19 q^{7} + 79 q^{8} + 118 q^{9}+O(q^{10}) \) 79 * q + 79 * q^2 + 17 * q^3 + 79 * q^4 + 18 * q^5 + 17 * q^6 + 19 * q^7 + 79 * q^8 + 118 * q^9	\( 79 q + 79 q^{2} + 17 q^{3} + 79 q^{4} + 18 q^{5} + 17 q^{6} + 19 q^{7} + 79 q^{8} + 118 q^{9} + 18 q^{10} + 28 q^{11} + 17 q^{12} + 47 q^{13} + 19 q^{14} + 14 q^{15} + 79 q^{16} + 36 q^{17} + 118 q^{18} + 29 q^{19} + 18 q^{20} + 45 q^{21} + 28 q^{22} + 23 q^{23} + 17 q^{24} + 161 q^{25} + 47 q^{26} + 50 q^{27} + 19 q^{28} + 53 q^{29} + 14 q^{30} + 29 q^{31} + 79 q^{32} + 34 q^{33} + 36 q^{34} + 33 q^{35} + 118 q^{36} + 89 q^{37} + 29 q^{38} - 7 q^{39} + 18 q^{40} + 58 q^{41} + 45 q^{42} + 88 q^{43} + 28 q^{44} + 45 q^{45} + 23 q^{46} + 3 q^{47} + 17 q^{48} + 162 q^{49} + 161 q^{50} + 29 q^{51} + 47 q^{52} + 88 q^{53} + 50 q^{54} + 37 q^{55} + 19 q^{56} + 54 q^{57} + 53 q^{58} + 37 q^{59} + 14 q^{60} + 55 q^{61} + 29 q^{62} + 21 q^{63} + 79 q^{64} + 55 q^{65} + 34 q^{66} + 107 q^{67} + 36 q^{68} + 39 q^{69} + 33 q^{70} - 5 q^{71} + 118 q^{72} + 71 q^{73} + 89 q^{74} + 37 q^{75} + 29 q^{76} + 61 q^{77} - 7 q^{78} + 29 q^{79} + 18 q^{80} + 215 q^{81} + 58 q^{82} + 42 q^{83} + 45 q^{84} + 84 q^{85} + 88 q^{86} + 15 q^{87} + 28 q^{88} + 72 q^{89} + 45 q^{90} + 70 q^{91} + 23 q^{92} + 97 q^{93} + 3 q^{94} - 18 q^{95} + 17 q^{96} + 93 q^{97} + 162 q^{98} + 49 q^{99}+O(q^{100}) \) 79 * q + 79 * q^2 + 17 * q^3 + 79 * q^4 + 18 * q^5 + 17 * q^6 + 19 * q^7 + 79 * q^8 + 118 * q^9 + 18 * q^10 + 28 * q^11 + 17 * q^12 + 47 * q^13 + 19 * q^14 + 14 * q^15 + 79 * q^16 + 36 * q^17 + 118 * q^18 + 29 * q^19 + 18 * q^20 + 45 * q^21 + 28 * q^22 + 23 * q^23 + 17 * q^24 + 161 * q^25 + 47 * q^26 + 50 * q^27 + 19 * q^28 + 53 * q^29 + 14 * q^30 + 29 * q^31 + 79 * q^32 + 34 * q^33 + 36 * q^34 + 33 * q^35 + 118 * q^36 + 89 * q^37 + 29 * q^38 - 7 * q^39 + 18 * q^40 + 58 * q^41 + 45 * q^42 + 88 * q^43 + 28 * q^44 + 45 * q^45 + 23 * q^46 + 3 * q^47 + 17 * q^48 + 162 * q^49 + 161 * q^50 + 29 * q^51 + 47 * q^52 + 88 * q^53 + 50 * q^54 + 37 * q^55 + 19 * q^56 + 54 * q^57 + 53 * q^58 + 37 * q^59 + 14 * q^60 + 55 * q^61 + 29 * q^62 + 21 * q^63 + 79 * q^64 + 55 * q^65 + 34 * q^66 + 107 * q^67 + 36 * q^68 + 39 * q^69 + 33 * q^70 - 5 * q^71 + 118 * q^72 + 71 * q^73 + 89 * q^74 + 37 * q^75 + 29 * q^76 + 61 * q^77 - 7 * q^78 + 29 * q^79 + 18 * q^80 + 215 * q^81 + 58 * q^82 + 42 * q^83 + 45 * q^84 + 84 * q^85 + 88 * q^86 + 15 * q^87 + 28 * q^88 + 72 * q^89 + 45 * q^90 + 70 * q^91 + 23 * q^92 + 97 * q^93 + 3 * q^94 - 18 * q^95 + 17 * q^96 + 93 * q^97 + 162 * q^98 + 49 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more