LMFDB - Embedded newform 6009.2.a.c.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6009,2,Mod(1,6009)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6009, 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("6009.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6009 = 3 \cdot 2003 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6009.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:	$47.9821065746$
Analytic rank:	$0$
Dimension:	$92$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	6009.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.78023 q^{2} +1.00000 q^{3} +5.72970 q^{4} -1.51058 q^{5} -2.78023 q^{6} -3.00947 q^{7} -10.3695 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-2.78023 q^{2} +1.00000 q^{3} +5.72970 q^{4} -1.51058 q^{5} -2.78023 q^{6} -3.00947 q^{7} -10.3695 q^{8} +1.00000 q^{9} +4.19978 q^{10} +3.47752 q^{11} +5.72970 q^{12} +3.37085 q^{13} +8.36703 q^{14} -1.51058 q^{15} +17.3701 q^{16} +7.54917 q^{17} -2.78023 q^{18} -6.37443 q^{19} -8.65520 q^{20} -3.00947 q^{21} -9.66833 q^{22} +4.06788 q^{23} -10.3695 q^{24} -2.71814 q^{25} -9.37176 q^{26} +1.00000 q^{27} -17.2434 q^{28} -3.05312 q^{29} +4.19978 q^{30} -4.83112 q^{31} -27.5540 q^{32} +3.47752 q^{33} -20.9885 q^{34} +4.54605 q^{35} +5.72970 q^{36} -4.41563 q^{37} +17.7224 q^{38} +3.37085 q^{39} +15.6639 q^{40} +11.3428 q^{41} +8.36703 q^{42} +6.23723 q^{43} +19.9252 q^{44} -1.51058 q^{45} -11.3096 q^{46} -1.41599 q^{47} +17.3701 q^{48} +2.05690 q^{49} +7.55706 q^{50} +7.54917 q^{51} +19.3140 q^{52} +7.07246 q^{53} -2.78023 q^{54} -5.25309 q^{55} +31.2065 q^{56} -6.37443 q^{57} +8.48839 q^{58} +2.40583 q^{59} -8.65520 q^{60} -8.00620 q^{61} +13.4316 q^{62} -3.00947 q^{63} +41.8665 q^{64} -5.09195 q^{65} -9.66833 q^{66} -4.52087 q^{67} +43.2545 q^{68} +4.06788 q^{69} -12.6391 q^{70} +15.2576 q^{71} -10.3695 q^{72} +7.53074 q^{73} +12.2765 q^{74} -2.71814 q^{75} -36.5236 q^{76} -10.4655 q^{77} -9.37176 q^{78} +14.5425 q^{79} -26.2390 q^{80} +1.00000 q^{81} -31.5357 q^{82} -17.2162 q^{83} -17.2434 q^{84} -11.4037 q^{85} -17.3410 q^{86} -3.05312 q^{87} -36.0600 q^{88} +4.05133 q^{89} +4.19978 q^{90} -10.1445 q^{91} +23.3077 q^{92} -4.83112 q^{93} +3.93678 q^{94} +9.62912 q^{95} -27.5540 q^{96} -9.82592 q^{97} -5.71867 q^{98} +3.47752 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 92 q + 17 q^{2} + 92 q^{3} + 107 q^{4} + 34 q^{5} + 17 q^{6} + 22 q^{7} + 51 q^{8} + 92 q^{9}+O(q^{10}) $ 92 * q + 17 * q^2 + 92 * q^3 + 107 * q^4 + 34 * q^5 + 17 * q^6 + 22 * q^7 + 51 * q^8 + 92 * q^9	\( 92 q + 17 q^{2} + 92 q^{3} + 107 q^{4} + 34 q^{5} + 17 q^{6} + 22 q^{7} + 51 q^{8} + 92 q^{9} + 13 q^{10} + 40 q^{11} + 107 q^{12} + 6 q^{13} + 37 q^{14} + 34 q^{15} + 133 q^{16} + 77 q^{17} + 17 q^{18} + 34 q^{19} + 55 q^{20} + 22 q^{21} + 8 q^{22} + 83 q^{23} + 51 q^{24} + 110 q^{25} + 22 q^{26} + 92 q^{27} + 32 q^{28} + 97 q^{29} + 13 q^{30} + 44 q^{31} + 104 q^{32} + 40 q^{33} + 20 q^{34} + 80 q^{35} + 107 q^{36} + 12 q^{37} + 54 q^{38} + 6 q^{39} + 23 q^{40} + 67 q^{41} + 37 q^{42} + 30 q^{43} + 87 q^{44} + 34 q^{45} + 33 q^{46} + 69 q^{47} + 133 q^{48} + 112 q^{49} + 58 q^{50} + 77 q^{51} - 3 q^{52} + 113 q^{53} + 17 q^{54} + 42 q^{55} + 92 q^{56} + 34 q^{57} - 30 q^{58} + 72 q^{59} + 55 q^{60} + 19 q^{61} + 60 q^{62} + 22 q^{63} + 147 q^{64} + 74 q^{65} + 8 q^{66} + 26 q^{67} + 171 q^{68} + 83 q^{69} - 35 q^{70} + 134 q^{71} + 51 q^{72} - 17 q^{73} + 95 q^{74} + 110 q^{75} + 27 q^{76} + 108 q^{77} + 22 q^{78} + 159 q^{79} + 79 q^{80} + 92 q^{81} - 64 q^{82} + 73 q^{83} + 32 q^{84} - 4 q^{85} + 22 q^{86} + 97 q^{87} - 16 q^{88} + 50 q^{89} + 13 q^{90} + 17 q^{91} + 154 q^{92} + 44 q^{93} + 8 q^{94} + 155 q^{95} + 104 q^{96} - 20 q^{97} + 63 q^{98} + 40 q^{99}+O(q^{100}) \) 92 * q + 17 * q^2 + 92 * q^3 + 107 * q^4 + 34 * q^5 + 17 * q^6 + 22 * q^7 + 51 * q^8 + 92 * q^9 + 13 * q^10 + 40 * q^11 + 107 * q^12 + 6 * q^13 + 37 * q^14 + 34 * q^15 + 133 * q^16 + 77 * q^17 + 17 * q^18 + 34 * q^19 + 55 * q^20 + 22 * q^21 + 8 * q^22 + 83 * q^23 + 51 * q^24 + 110 * q^25 + 22 * q^26 + 92 * q^27 + 32 * q^28 + 97 * q^29 + 13 * q^30 + 44 * q^31 + 104 * q^32 + 40 * q^33 + 20 * q^34 + 80 * q^35 + 107 * q^36 + 12 * q^37 + 54 * q^38 + 6 * q^39 + 23 * q^40 + 67 * q^41 + 37 * q^42 + 30 * q^43 + 87 * q^44 + 34 * q^45 + 33 * q^46 + 69 * q^47 + 133 * q^48 + 112 * q^49 + 58 * q^50 + 77 * q^51 - 3 * q^52 + 113 * q^53 + 17 * q^54 + 42 * q^55 + 92 * q^56 + 34 * q^57 - 30 * q^58 + 72 * q^59 + 55 * q^60 + 19 * q^61 + 60 * q^62 + 22 * q^63 + 147 * q^64 + 74 * q^65 + 8 * q^66 + 26 * q^67 + 171 * q^68 + 83 * q^69 - 35 * q^70 + 134 * q^71 + 51 * q^72 - 17 * q^73 + 95 * q^74 + 110 * q^75 + 27 * q^76 + 108 * q^77 + 22 * q^78 + 159 * q^79 + 79 * q^80 + 92 * q^81 - 64 * q^82 + 73 * q^83 + 32 * q^84 - 4 * q^85 + 22 * q^86 + 97 * q^87 - 16 * q^88 + 50 * q^89 + 13 * q^90 + 17 * q^91 + 154 * q^92 + 44 * q^93 + 8 * q^94 + 155 * q^95 + 104 * q^96 - 20 * q^97 + 63 * q^98 + 40 * 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 $

$2$	−2.78023	−1.96592	−0.982961	−	0.183813i	$-0.941156\pi$
$2$	−2.78023	−1.96592	−0.982961	+	0.183813i	$0.941156\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	5.72970	2.86485
$5$	−1.51058	−0.675554	−0.337777	−	0.941226i	$-0.609675\pi$
$5$	−1.51058	−0.675554	−0.337777	+	0.941226i	$0.609675\pi$
$6$	−2.78023	−1.13503
$7$	−3.00947	−1.13747	−0.568736	−	0.822520i	$-0.692567\pi$
$7$	−3.00947	−1.13747	−0.568736	+	0.822520i	$0.692567\pi$
$8$	−10.3695	−3.66615
$9$	1.00000	0.333333
$10$	4.19978	1.32809
$11$	3.47752	1.04851	0.524257	−	0.851560i	$-0.324343\pi$
$11$	3.47752	1.04851	0.524257	+	0.851560i	$0.324343\pi$
$12$	5.72970	1.65402
$13$	3.37085	0.934906	0.467453	−	0.884018i	$-0.345172\pi$
$13$	3.37085	0.934906	0.467453	+	0.884018i	$0.345172\pi$
$14$	8.36703	2.23618
$15$	−1.51058	−0.390031
$16$	17.3701	4.34252
$17$	7.54917	1.83094	0.915472	−	0.402382i	$-0.131818\pi$
$17$	7.54917	1.83094	0.915472	+	0.402382i	$0.131818\pi$
$18$	−2.78023	−0.655308
$19$	−6.37443	−1.46240	−0.731198	−	0.682166i	$-0.761038\pi$
$19$	−6.37443	−1.46240	−0.731198	+	0.682166i	$0.761038\pi$
$20$	−8.65520	−1.93536
$21$	−3.00947	−0.656720
$22$	−9.66833	−2.06130
$23$	4.06788	0.848211	0.424105	−	0.905613i	$-0.360589\pi$
$23$	4.06788	0.848211	0.424105	+	0.905613i	$0.360589\pi$
$24$	−10.3695	−2.11666
$25$	−2.71814	−0.543627
$26$	−9.37176	−1.83795
$27$	1.00000	0.192450
$28$	−17.2434	−3.25869
$29$	−3.05312	−0.566950	−0.283475	−	0.958980i	$-0.591487\pi$
$29$	−3.05312	−0.566950	−0.283475	+	0.958980i	$0.591487\pi$
$30$	4.19978	0.766771
$31$	−4.83112	−0.867694	−0.433847	−	0.900987i	$-0.642844\pi$
$31$	−4.83112	−0.867694	−0.433847	+	0.900987i	$0.642844\pi$
$32$	−27.5540	−4.87091
$33$	3.47752	0.605359
$34$	−20.9885	−3.59949
$35$	4.54605	0.768424
$36$	5.72970	0.954951
$37$	−4.41563	−0.725925	−0.362962	−	0.931804i	$-0.618235\pi$
$37$	−4.41563	−0.725925	−0.362962	+	0.931804i	$0.618235\pi$
$38$	17.7224	2.87496
$39$	3.37085	0.539768
$40$	15.6639	2.47668
$41$	11.3428	1.77145	0.885725	−	0.464211i	$-0.153662\pi$
$41$	11.3428	1.77145	0.885725	+	0.464211i	$0.153662\pi$
$42$	8.36703	1.29106
$43$	6.23723	0.951168	0.475584	−	0.879670i	$-0.342237\pi$
$43$	6.23723	0.951168	0.475584	+	0.879670i	$0.342237\pi$
$44$	19.9252	3.00383
$45$	−1.51058	−0.225185
$46$	−11.3096	−1.66752
$47$	−1.41599	−0.206543	−0.103271	−	0.994653i	$-0.532931\pi$
$47$	−1.41599	−0.206543	−0.103271	+	0.994653i	$0.532931\pi$
$48$	17.3701	2.50716
$49$	2.05690	0.293843
$50$	7.55706	1.06873
$51$	7.54917	1.05710
$52$	19.3140	2.67837
$53$	7.07246	0.971477	0.485738	−	0.874104i	$-0.338551\pi$
$53$	7.07246	0.971477	0.485738	+	0.874104i	$0.338551\pi$
$54$	−2.78023	−0.378342
$55$	−5.25309	−0.708327
$56$	31.2065	4.17015
$57$	−6.37443	−0.844314
$58$	8.48839	1.11458
$59$	2.40583	0.313213	0.156606	−	0.987661i	$-0.449945\pi$
$59$	2.40583	0.313213	0.156606	+	0.987661i	$0.449945\pi$
$60$	−8.65520	−1.11738
$61$	−8.00620	−1.02509	−0.512545	−	0.858661i	$-0.671297\pi$
$61$	−8.00620	−1.02509	−0.512545	+	0.858661i	$0.671297\pi$
$62$	13.4316	1.70582
$63$	−3.00947	−0.379157
$64$	41.8665	5.23331
$65$	−5.09195	−0.631579
$66$	−9.66833	−1.19009
$67$	−4.52087	−0.552313	−0.276156	−	0.961113i	$-0.589061\pi$
$67$	−4.52087	−0.552313	−0.276156	+	0.961113i	$0.589061\pi$
$68$	43.2545	5.24538
$69$	4.06788	0.489715
$70$	−12.6391	−1.51066
$71$	15.2576	1.81074	0.905370	−	0.424623i	$-0.139594\pi$
$71$	15.2576	1.81074	0.905370	+	0.424623i	$0.139594\pi$
$72$	−10.3695	−1.22205
$73$	7.53074	0.881407	0.440703	−	0.897653i	$-0.354729\pi$
$73$	7.53074	0.881407	0.440703	+	0.897653i	$0.354729\pi$
$74$	12.2765	1.42711
$75$	−2.71814	−0.313863
$76$	−36.5236	−4.18955
$77$	−10.4655	−1.19265
$78$	−9.37176	−1.06114
$79$	14.5425	1.63616	0.818079	−	0.575106i	$-0.195039\pi$
$79$	14.5425	1.63616	0.818079	+	0.575106i	$0.195039\pi$
$80$	−26.2390	−2.93361
$81$	1.00000	0.111111
$82$	−31.5357	−3.48253
$83$	−17.2162	−1.88973	−0.944863	−	0.327467i	$-0.893805\pi$
$83$	−17.2162	−1.88973	−0.944863	+	0.327467i	$0.893805\pi$
$84$	−17.2434	−1.88141
$85$	−11.4037	−1.23690
$86$	−17.3410	−1.86992
$87$	−3.05312	−0.327329
$88$	−36.0600	−3.84401
$89$	4.05133	0.429440	0.214720	−	0.976676i	$-0.431116\pi$
$89$	4.05133	0.429440	0.214720	+	0.976676i	$0.431116\pi$
$90$	4.19978	0.442695
$91$	−10.1445	−1.06343
$92$	23.3077	2.43000
$93$	−4.83112	−0.500963
$94$	3.93678	0.406047
$95$	9.62912	0.987927
$96$	−27.5540	−2.81222
$97$	−9.82592	−0.997671	−0.498836	−	0.866697i	$-0.666239\pi$
$97$	−9.82592	−0.997671	−0.498836	+	0.866697i	$0.666239\pi$
$98$	−5.71867	−0.577673
$99$	3.47752	0.349504
$100$	−15.5741	−1.55741
$101$	7.68986	0.765170	0.382585	−	0.923920i	$-0.375034\pi$
$101$	7.68986	0.765170	0.382585	+	0.923920i	$0.375034\pi$
$102$	−20.9885	−2.07817
$103$	0.0787749	0.00776192	0.00388096	−	0.999992i	$-0.498765\pi$
$103$	0.0787749	0.00776192	0.00388096	+	0.999992i	$0.498765\pi$
$104$	−34.9539	−3.42751
$105$	4.54605	0.443650
$106$	−19.6631	−1.90985
$107$	−10.4326	−1.00856	−0.504280	−	0.863540i	$-0.668242\pi$
$107$	−10.4326	−1.00856	−0.504280	+	0.863540i	$0.668242\pi$
$108$	5.72970	0.551341
$109$	12.2632	1.17460	0.587301	−	0.809369i	$-0.300191\pi$
$109$	12.2632	1.17460	0.587301	+	0.809369i	$0.300191\pi$
$110$	14.6048	1.39252
$111$	−4.41563	−0.419113
$112$	−52.2748	−4.93950
$113$	−10.7672	−1.01289	−0.506447	−	0.862271i	$-0.669041\pi$
$113$	−10.7672	−1.01289	−0.506447	+	0.862271i	$0.669041\pi$
$114$	17.7224	1.65986
$115$	−6.14487	−0.573012
$116$	−17.4935	−1.62423
$117$	3.37085	0.311635
$118$	−6.68878	−0.615752
$119$	−22.7190	−2.08265
$120$	15.6639	1.42991
$121$	1.09318	0.0993797
$122$	22.2591	2.01525
$123$	11.3428	1.02275
$124$	−27.6809	−2.48581
$125$	11.6589	1.04280
$126$	8.36703	0.745394
$127$	16.9947	1.50804	0.754018	−	0.656854i	$-0.228113\pi$
$127$	16.9947	1.50804	0.754018	+	0.656854i	$0.228113\pi$
$128$	−61.2906	−5.41738
$129$	6.23723	0.549157
$130$	14.1568	1.24164
$131$	1.24502	0.108778	0.0543890	−	0.998520i	$-0.482679\pi$
$131$	1.24502	0.108778	0.0543890	+	0.998520i	$0.482679\pi$
$132$	19.9252	1.73426
$133$	19.1837	1.66343
$134$	12.5691	1.08580
$135$	−1.51058	−0.130010
$136$	−78.2808	−6.71252
$137$	−14.7943	−1.26396	−0.631981	−	0.774984i	$-0.717758\pi$
$137$	−14.7943	−1.26396	−0.631981	+	0.774984i	$0.717758\pi$
$138$	−11.3096	−0.962741
$139$	−6.69804	−0.568120	−0.284060	−	0.958806i	$-0.591682\pi$
$139$	−6.69804	−0.568120	−0.284060	+	0.958806i	$0.591682\pi$
$140$	26.0475	2.20142
$141$	−1.41599	−0.119248
$142$	−42.4196	−3.55977
$143$	11.7222	0.980261
$144$	17.3701	1.44751
$145$	4.61200	0.383005
$146$	−20.9372	−1.73278
$147$	2.05690	0.169651
$148$	−25.3002	−2.07967
$149$	−2.61804	−0.214478	−0.107239	−	0.994233i	$-0.534201\pi$
$149$	−2.61804	−0.214478	−0.107239	+	0.994233i	$0.534201\pi$
$150$	7.55706	0.617031
$151$	21.7092	1.76667	0.883333	−	0.468746i	$-0.155294\pi$
$151$	21.7092	1.76667	0.883333	+	0.468746i	$0.155294\pi$
$152$	66.0994	5.36137
$153$	7.54917	0.610314
$154$	29.0965	2.34467
$155$	7.29780	0.586174
$156$	19.3140	1.54636
$157$	1.07237	0.0855841	0.0427920	−	0.999084i	$-0.486375\pi$
$157$	1.07237	0.0855841	0.0427920	+	0.999084i	$0.486375\pi$
$158$	−40.4315	−3.21656
$159$	7.07246	0.560882
$160$	41.6227	3.29056
$161$	−12.2421	−0.964816
$162$	−2.78023	−0.218436
$163$	9.48499	0.742921	0.371461	−	0.928449i	$-0.378857\pi$
$163$	9.48499	0.742921	0.371461	+	0.928449i	$0.378857\pi$
$164$	64.9909	5.07494
$165$	−5.25309	−0.408953
$166$	47.8651	3.71505
$167$	−15.3780	−1.18998	−0.594991	−	0.803732i	$-0.702844\pi$
$167$	−15.3780	−1.18998	−0.594991	+	0.803732i	$0.702844\pi$
$168$	31.2065	2.40764
$169$	−1.63736	−0.125951
$170$	31.7048	2.43165
$171$	−6.37443	−0.487465
$172$	35.7375	2.72496
$173$	−19.9633	−1.51778	−0.758892	−	0.651216i	$-0.774259\pi$
$173$	−19.9633	−1.51778	−0.758892	+	0.651216i	$0.774259\pi$
$174$	8.48839	0.643503
$175$	8.18015	0.618361
$176$	60.4049	4.55319
$177$	2.40583	0.180834
$178$	−11.2636	−0.844245
$179$	6.42839	0.480481	0.240240	−	0.970713i	$-0.422774\pi$
$179$	6.42839	0.480481	0.240240	+	0.970713i	$0.422774\pi$
$180$	−8.65520	−0.645120
$181$	−22.6389	−1.68274	−0.841369	−	0.540461i	$-0.818250\pi$
$181$	−22.6389	−1.68274	−0.841369	+	0.540461i	$0.818250\pi$
$182$	28.2040	2.09062
$183$	−8.00620	−0.591836
$184$	−42.1816	−3.10967
$185$	6.67017	0.490401
$186$	13.4316	0.984855
$187$	26.2524	1.91977
$188$	−8.11319	−0.591715
$189$	−3.00947	−0.218907
$190$	−26.7712	−1.94219
$191$	−0.884621	−0.0640090	−0.0320045	−	0.999488i	$-0.510189\pi$
$191$	−0.884621	−0.0640090	−0.0320045	+	0.999488i	$0.510189\pi$
$192$	41.8665	3.02145
$193$	−12.0992	−0.870916	−0.435458	−	0.900209i	$-0.643414\pi$
$193$	−12.0992	−0.870916	−0.435458	+	0.900209i	$0.643414\pi$
$194$	27.3184	1.96134
$195$	−5.09195	−0.364642
$196$	11.7854	0.841818
$197$	10.3552	0.737778	0.368889	−	0.929473i	$-0.379738\pi$
$197$	10.3552	0.737778	0.368889	+	0.929473i	$0.379738\pi$
$198$	−9.66833	−0.687099
$199$	−4.65662	−0.330099	−0.165050	−	0.986285i	$-0.552778\pi$
$199$	−4.65662	−0.330099	−0.165050	+	0.986285i	$0.552778\pi$
$200$	28.1856	1.99302
$201$	−4.52087	−0.318878
$202$	−21.3796	−1.50426
$203$	9.18827	0.644890
$204$	43.2545	3.02842
$205$	−17.1343	−1.19671
$206$	−0.219013	−0.0152593
$207$	4.06788	0.282737
$208$	58.5520	4.05985
$209$	−22.1673	−1.53334
$210$	−12.6391	−0.872181
$211$	−8.70631	−0.599367	−0.299683	−	0.954039i	$-0.596881\pi$
$211$	−8.70631	−0.599367	−0.299683	+	0.954039i	$0.596881\pi$
$212$	40.5231	2.78314
$213$	15.2576	1.04543
$214$	29.0051	1.98275
$215$	−9.42185	−0.642565
$216$	−10.3695	−0.705552
$217$	14.5391	0.986978
$218$	−34.0946	−2.30918
$219$	7.53074	0.508880
$220$	−30.0987	−2.02925
$221$	25.4471	1.71176
$222$	12.2765	0.823943
$223$	−1.44816	−0.0969761	−0.0484880	−	0.998824i	$-0.515440\pi$
$223$	−1.44816	−0.0969761	−0.0484880	+	0.998824i	$0.515440\pi$
$224$	82.9230	5.54053
$225$	−2.71814	−0.181209
$226$	29.9354	1.99127
$227$	−13.1542	−0.873073	−0.436536	−	0.899687i	$-0.643795\pi$
$227$	−13.1542	−0.873073	−0.436536	+	0.899687i	$0.643795\pi$
$228$	−36.5236	−2.41884
$229$	−17.5412	−1.15915	−0.579576	−	0.814918i	$-0.696782\pi$
$229$	−17.5412	−1.15915	−0.579576	+	0.814918i	$0.696782\pi$
$230$	17.0842	1.12650
$231$	−10.4655	−0.688579
$232$	31.6592	2.07853
$233$	−22.9591	−1.50410	−0.752050	−	0.659106i	$-0.770935\pi$
$233$	−22.9591	−1.50410	−0.752050	+	0.659106i	$0.770935\pi$
$234$	−9.37176	−0.612651
$235$	2.13897	0.139531
$236$	13.7847	0.897309
$237$	14.5425	0.944636
$238$	63.1641	4.09432
$239$	17.2723	1.11725	0.558625	−	0.829420i	$-0.311329\pi$
$239$	17.2723	1.11725	0.558625	+	0.829420i	$0.311329\pi$
$240$	−26.2390	−1.69372
$241$	18.7096	1.20519	0.602595	−	0.798047i	$-0.294133\pi$
$241$	18.7096	1.20519	0.602595	+	0.798047i	$0.294133\pi$
$242$	−3.03929	−0.195373
$243$	1.00000	0.0641500
$244$	−45.8732	−2.93673
$245$	−3.10712	−0.198507
$246$	−31.5357	−2.01064
$247$	−21.4873	−1.36720
$248$	50.0960	3.18110
$249$	−17.2162	−1.09103
$250$	−32.4145	−2.05007
$251$	16.2558	1.02606	0.513028	−	0.858372i	$-0.328524\pi$
$251$	16.2558	1.02606	0.513028	+	0.858372i	$0.328524\pi$
$252$	−17.2434	−1.08623
$253$	14.1461	0.889360
$254$	−47.2492	−2.96468
$255$	−11.4037	−0.714125
$256$	86.6693	5.41683
$257$	5.31616	0.331613	0.165807	−	0.986158i	$-0.446977\pi$
$257$	5.31616	0.331613	0.165807	+	0.986158i	$0.446977\pi$
$258$	−17.3410	−1.07960
$259$	13.2887	0.825719
$260$	−29.1754	−1.80938
$261$	−3.05312	−0.188983
$262$	−3.46145	−0.213849
$263$	13.4500	0.829362	0.414681	−	0.909967i	$-0.363893\pi$
$263$	13.4500	0.829362	0.414681	+	0.909967i	$0.363893\pi$
$264$	−36.0600	−2.21934
$265$	−10.6835	−0.656285
$266$	−53.3351	−3.27018
$267$	4.05133	0.247937
$268$	−25.9033	−1.58229
$269$	13.6762	0.833856	0.416928	−	0.908940i	$-0.363107\pi$
$269$	13.6762	0.833856	0.416928	+	0.908940i	$0.363107\pi$
$270$	4.19978	0.255590
$271$	25.4733	1.54739	0.773696	−	0.633557i	$-0.218406\pi$
$271$	25.4733	1.54739	0.773696	+	0.633557i	$0.218406\pi$
$272$	131.130	7.95092
$273$	−10.1445	−0.613971
$274$	41.1316	2.48485
$275$	−9.45239	−0.570000
$276$	23.3077	1.40296
$277$	4.76078	0.286047	0.143024	−	0.989719i	$-0.454317\pi$
$277$	4.76078	0.286047	0.143024	+	0.989719i	$0.454317\pi$
$278$	18.6221	1.11688
$279$	−4.83112	−0.289231
$280$	−47.1401	−2.81716
$281$	8.16192	0.486899	0.243450	−	0.969914i	$-0.421721\pi$
$281$	8.16192	0.486899	0.243450	+	0.969914i	$0.421721\pi$
$282$	3.93678	0.234432
$283$	14.8541	0.882986	0.441493	−	0.897265i	$-0.354449\pi$
$283$	14.8541	0.882986	0.441493	+	0.897265i	$0.354449\pi$
$284$	87.4213	5.18750
$285$	9.62912	0.570380
$286$	−32.5905	−1.92712
$287$	−34.1358	−2.01497
$288$	−27.5540	−1.62364
$289$	39.9900	2.35235
$290$	−12.8224	−0.752959
$291$	−9.82592	−0.576006
$292$	43.1489	2.52510
$293$	16.7083	0.976110	0.488055	−	0.872813i	$-0.337707\pi$
$293$	16.7083	0.976110	0.488055	+	0.872813i	$0.337707\pi$
$294$	−5.71867	−0.333520
$295$	−3.63421	−0.211592
$296$	45.7876	2.66135
$297$	3.47752	0.201786
$298$	7.27877	0.421648
$299$	13.7122	0.792997
$300$	−15.5741	−0.899172
$301$	−18.7707	−1.08193
$302$	−60.3565	−3.47313
$303$	7.68986	0.441771
$304$	−110.725	−6.35049
$305$	12.0940	0.692503
$306$	−20.9885	−1.19983
$307$	−23.8549	−1.36147	−0.680737	−	0.732528i	$-0.738340\pi$
$307$	−23.8549	−1.36147	−0.680737	+	0.732528i	$0.738340\pi$
$308$	−59.9642	−3.41678
$309$	0.0787749	0.00448135
$310$	−20.2896	−1.15237
$311$	12.8878	0.730798	0.365399	−	0.930851i	$-0.380933\pi$
$311$	12.8878	0.730798	0.365399	+	0.930851i	$0.380933\pi$
$312$	−34.9539	−1.97887
$313$	−3.44156	−0.194528	−0.0972641	−	0.995259i	$-0.531009\pi$
$313$	−3.44156	−0.194528	−0.0972641	+	0.995259i	$0.531009\pi$
$314$	−2.98143	−0.168252
$315$	4.54605	0.256141
$316$	83.3241	4.68735
$317$	0.106297	0.00597024	0.00298512	−	0.999996i	$-0.499050\pi$
$317$	0.106297	0.00597024	0.00298512	+	0.999996i	$0.499050\pi$
$318$	−19.6631	−1.10265
$319$	−10.6173	−0.594455
$320$	−63.2429	−3.53538
$321$	−10.4326	−0.582292
$322$	34.0360	1.89675
$323$	−48.1217	−2.67756
$324$	5.72970	0.318317
$325$	−9.16244	−0.508241
$326$	−26.3705	−1.46053
$327$	12.2632	0.678156
$328$	−117.619	−6.49441
$329$	4.26137	0.234937
$330$	14.6048	0.803969
$331$	−6.97269	−0.383254	−0.191627	−	0.981468i	$-0.561376\pi$
$331$	−6.97269	−0.383254	−0.191627	+	0.981468i	$0.561376\pi$
$332$	−98.6438	−5.41378
$333$	−4.41563	−0.241975
$334$	42.7543	2.33941
$335$	6.82916	0.373117
$336$	−52.2748	−2.85182
$337$	−10.5746	−0.576036	−0.288018	−	0.957625i	$-0.592996\pi$
$337$	−10.5746	−0.576036	−0.288018	+	0.957625i	$0.592996\pi$
$338$	4.55224	0.247609
$339$	−10.7672	−0.584795
$340$	−65.3396	−3.54354
$341$	−16.8003	−0.909788
$342$	17.7224	0.958319
$343$	14.8761	0.803234
$344$	−64.6766	−3.48713
$345$	−6.14487	−0.330829
$346$	55.5028	2.98385
$347$	2.88573	0.154914	0.0774570	−	0.996996i	$-0.475320\pi$
$347$	2.88573	0.154914	0.0774570	+	0.996996i	$0.475320\pi$
$348$	−17.4935	−0.937749
$349$	−7.05471	−0.377630	−0.188815	−	0.982013i	$-0.560465\pi$
$349$	−7.05471	−0.377630	−0.188815	+	0.982013i	$0.560465\pi$
$350$	−22.7427	−1.21565
$351$	3.37085	0.179923
$352$	−95.8199	−5.10722
$353$	−30.5963	−1.62848	−0.814238	−	0.580531i	$-0.802845\pi$
$353$	−30.5963	−1.62848	−0.814238	+	0.580531i	$0.802845\pi$
$354$	−6.68878	−0.355505
$355$	−23.0478	−1.22325
$356$	23.2129	1.23028
$357$	−22.7190	−1.20242
$358$	−17.8724	−0.944588
$359$	37.4135	1.97461	0.987305	−	0.158838i	$-0.0507749\pi$
$359$	37.4135	1.97461	0.987305	+	0.158838i	$0.0507749\pi$
$360$	15.6639	0.825561
$361$	21.6334	1.13860
$362$	62.9415	3.30813
$363$	1.09318	0.0573769
$364$	−58.1248	−3.04657
$365$	−11.3758	−0.595437
$366$	22.2591	1.16350
$367$	22.9693	1.19899	0.599495	−	0.800379i	$-0.295368\pi$
$367$	22.9693	1.19899	0.599495	+	0.800379i	$0.295368\pi$
$368$	70.6594	3.68338
$369$	11.3428	0.590483
$370$	−18.5446	−0.964090
$371$	−21.2843	−1.10503
$372$	−27.6809	−1.43519
$373$	33.0410	1.71080	0.855398	−	0.517971i	$-0.173313\pi$
$373$	33.0410	1.71080	0.855398	+	0.517971i	$0.173313\pi$
$374$	−72.9879	−3.77412
$375$	11.6589	0.602063
$376$	14.6830	0.757218
$377$	−10.2916	−0.530045
$378$	8.36703	0.430354
$379$	−6.26554	−0.321839	−0.160920	−	0.986968i	$-0.551446\pi$
$379$	−6.26554	−0.321839	−0.160920	+	0.986968i	$0.551446\pi$
$380$	55.1720	2.83026
$381$	16.9947	0.870665
$382$	2.45945	0.125837
$383$	23.2869	1.18990	0.594951	−	0.803762i	$-0.297171\pi$
$383$	23.2869	1.18990	0.594951	+	0.803762i	$0.297171\pi$
$384$	−61.2906	−3.12772
$385$	15.8090	0.805702
$386$	33.6385	1.71215
$387$	6.23723	0.317056
$388$	−56.2996	−2.85818
$389$	−5.48204	−0.277950	−0.138975	−	0.990296i	$-0.544381\pi$
$389$	−5.48204	−0.277950	−0.138975	+	0.990296i	$0.544381\pi$
$390$	14.1568	0.716859
$391$	30.7091	1.55303
$392$	−21.3290	−1.07728
$393$	1.24502	0.0628030
$394$	−28.7899	−1.45041
$395$	−21.9676	−1.10531
$396$	19.9252	1.00128
$397$	−10.8471	−0.544401	−0.272201	−	0.962241i	$-0.587751\pi$
$397$	−10.8471	−0.544401	−0.272201	+	0.962241i	$0.587751\pi$
$398$	12.9465	0.648949
$399$	19.1837	0.960384
$400$	−47.2143	−2.36072
$401$	27.1130	1.35396	0.676979	−	0.736003i	$-0.263289\pi$
$401$	27.1130	1.35396	0.676979	+	0.736003i	$0.263289\pi$
$402$	12.5691	0.626889
$403$	−16.2850	−0.811212
$404$	44.0606	2.19210
$405$	−1.51058	−0.0750615
$406$	−25.5456	−1.26780
$407$	−15.3555	−0.761141
$408$	−78.2808	−3.87548
$409$	−12.1067	−0.598637	−0.299319	−	0.954153i	$-0.596759\pi$
$409$	−12.1067	−0.598637	−0.299319	+	0.954153i	$0.596759\pi$
$410$	47.6373	2.35264
$411$	−14.7943	−0.729748
$412$	0.451357	0.0222368
$413$	−7.24028	−0.356271
$414$	−11.3096	−0.555839
$415$	26.0065	1.27661
$416$	−92.8806	−4.55385
$417$	−6.69804	−0.328004
$418$	61.6302	3.01443
$419$	−33.6580	−1.64430	−0.822150	−	0.569270i	$-0.807226\pi$
$419$	−33.6580	−1.64430	−0.822150	+	0.569270i	$0.807226\pi$
$420$	26.0475	1.27099
$421$	−18.8647	−0.919411	−0.459705	−	0.888071i	$-0.652045\pi$
$421$	−18.8647	−0.919411	−0.459705	+	0.888071i	$0.652045\pi$
$422$	24.2056	1.17831
$423$	−1.41599	−0.0688477
$424$	−73.3375	−3.56158
$425$	−20.5197	−0.995351
$426$	−42.4196	−2.05524
$427$	24.0944	1.16601
$428$	−59.7758	−2.88937
$429$	11.7222	0.565954
$430$	26.1950	1.26323
$431$	22.6460	1.09082	0.545409	−	0.838170i	$-0.316374\pi$
$431$	22.6460	1.09082	0.545409	+	0.838170i	$0.316374\pi$
$432$	17.3701	0.835719
$433$	14.8490	0.713598	0.356799	−	0.934181i	$-0.383868\pi$
$433$	14.8490	0.713598	0.356799	+	0.934181i	$0.383868\pi$
$434$	−40.4221	−1.94032
$435$	4.61200	0.221128
$436$	70.2645	3.36506
$437$	−25.9304	−1.24042
$438$	−20.9372	−1.00042
$439$	36.6382	1.74865	0.874324	−	0.485343i	$-0.161305\pi$
$439$	36.6382	1.74865	0.874324	+	0.485343i	$0.161305\pi$
$440$	54.4717	2.59684
$441$	2.05690	0.0979478
$442$	−70.7490	−3.36519
$443$	34.7600	1.65150	0.825749	−	0.564038i	$-0.190753\pi$
$443$	34.7600	1.65150	0.825749	+	0.564038i	$0.190753\pi$
$444$	−25.3002	−1.20070
$445$	−6.11987	−0.290110
$446$	4.02623	0.190647
$447$	−2.61804	−0.123829
$448$	−125.996	−5.95275
$449$	39.2748	1.85349	0.926746	−	0.375688i	$-0.122593\pi$
$449$	39.2748	1.85349	0.926746	+	0.375688i	$0.122593\pi$
$450$	7.55706	0.356243
$451$	39.4449	1.85739
$452$	−61.6929	−2.90179
$453$	21.7092	1.01999
$454$	36.5717	1.71639
$455$	15.3241	0.718404
$456$	66.0994	3.09539
$457$	−4.97635	−0.232784	−0.116392	−	0.993203i	$-0.537133\pi$
$457$	−4.97635	−0.232784	−0.116392	+	0.993203i	$0.537133\pi$
$458$	48.7685	2.27880
$459$	7.54917	0.352365
$460$	−35.2083	−1.64159
$461$	−0.650173	−0.0302816	−0.0151408	−	0.999885i	$-0.504820\pi$
$461$	−0.650173	−0.0302816	−0.0151408	+	0.999885i	$0.504820\pi$
$462$	29.0965	1.35369
$463$	31.7169	1.47401	0.737004	−	0.675888i	$-0.236240\pi$
$463$	31.7169	1.47401	0.737004	+	0.675888i	$0.236240\pi$
$464$	−53.0330	−2.46200
$465$	7.29780	0.338428
$466$	63.8317	2.95695
$467$	21.6420	1.00147	0.500736	−	0.865600i	$-0.333063\pi$
$467$	21.6420	1.00147	0.500736	+	0.865600i	$0.333063\pi$
$468$	19.3140	0.892789
$469$	13.6054	0.628240
$470$	−5.94683	−0.274307
$471$	1.07237	0.0494120
$472$	−24.9472	−1.14829
$473$	21.6901	0.997312
$474$	−40.4315	−1.85708
$475$	17.3266	0.794998
$476$	−130.173	−5.96648
$477$	7.07246	0.323826
$478$	−48.0210	−2.19643
$479$	15.6345	0.714359	0.357179	−	0.934036i	$-0.383738\pi$
$479$	15.6345	0.714359	0.357179	+	0.934036i	$0.383738\pi$
$480$	41.6227	1.89981
$481$	−14.8844	−0.678671
$482$	−52.0170	−2.36931
$483$	−12.2421	−0.557037
$484$	6.26358	0.284708
$485$	14.8429	0.673980
$486$	−2.78023	−0.126114
$487$	27.9131	1.26486	0.632432	−	0.774616i	$-0.282057\pi$
$487$	27.9131	1.26486	0.632432	+	0.774616i	$0.282057\pi$
$488$	83.0199	3.75814
$489$	9.48499	0.428926
$490$	8.63853	0.390249
$491$	2.44489	0.110336	0.0551681	−	0.998477i	$-0.482431\pi$
$491$	2.44489	0.110336	0.0551681	+	0.998477i	$0.482431\pi$
$492$	64.9909	2.93002
$493$	−23.0485	−1.03805
$494$	59.7397	2.68781
$495$	−5.25309	−0.236109
$496$	−83.9169	−3.76798
$497$	−45.9172	−2.05967
$498$	47.8651	2.14489
$499$	3.07512	0.137661	0.0688307	−	0.997628i	$-0.478073\pi$
$499$	3.07512	0.137661	0.0688307	+	0.997628i	$0.478073\pi$
$500$	66.8020	2.98748
$501$	−15.3780	−0.687036
$502$	−45.1949	−2.01715
$503$	26.9236	1.20046	0.600231	−	0.799827i	$-0.295075\pi$
$503$	26.9236	1.20046	0.600231	+	0.799827i	$0.295075\pi$
$504$	31.2065	1.39005
$505$	−11.6162	−0.516913
$506$	−39.3296	−1.74841
$507$	−1.63736	−0.0727176
$508$	97.3746	4.32030
$509$	−31.5036	−1.39637	−0.698186	−	0.715917i	$-0.746009\pi$
$509$	−31.5036	−1.39637	−0.698186	+	0.715917i	$0.746009\pi$
$510$	31.7048	1.40391
$511$	−22.6635	−1.00258
$512$	−118.380	−5.23169
$513$	−6.37443	−0.281438
$514$	−14.7802	−0.651926
$515$	−0.118996	−0.00524360
$516$	35.7375	1.57325
$517$	−4.92413	−0.216563
$518$	−36.9457	−1.62330
$519$	−19.9633	−0.876293
$520$	52.8008	2.31547
$521$	33.2062	1.45479	0.727395	−	0.686219i	$-0.240731\pi$
$521$	33.2062	1.45479	0.727395	+	0.686219i	$0.240731\pi$
$522$	8.48839	0.371527
$523$	13.0063	0.568727	0.284363	−	0.958717i	$-0.408218\pi$
$523$	13.0063	0.568727	0.284363	+	0.958717i	$0.408218\pi$
$524$	7.13361	0.311633
$525$	8.18015	0.357011
$526$	−37.3941	−1.63046
$527$	−36.4709	−1.58870
$528$	60.4049	2.62879
$529$	−6.45238	−0.280538
$530$	29.7027	1.29020
$531$	2.40583	0.104404
$532$	109.917	4.76549
$533$	38.2349	1.65614
$534$	−11.2636	−0.487425
$535$	15.7593	0.681336
$536$	46.8790	2.02486
$537$	6.42839	0.277406
$538$	−38.0232	−1.63930
$539$	7.15293	0.308099
$540$	−8.65520	−0.372460
$541$	−4.70582	−0.202319	−0.101159	−	0.994870i	$-0.532255\pi$
$541$	−4.70582	−0.202319	−0.101159	+	0.994870i	$0.532255\pi$
$542$	−70.8217	−3.04205
$543$	−22.6389	−0.971529
$544$	−208.010	−8.91836
$545$	−18.5246	−0.793506
$546$	28.2040	1.20702
$547$	−2.20491	−0.0942749	−0.0471375	−	0.998888i	$-0.515010\pi$
$547$	−2.20491	−0.0942749	−0.0471375	+	0.998888i	$0.515010\pi$
$548$	−84.7669	−3.62106
$549$	−8.00620	−0.341696
$550$	26.2799	1.12058
$551$	19.4619	0.829106
$552$	−42.1816	−1.79537
$553$	−43.7652	−1.86108
$554$	−13.2361	−0.562347
$555$	6.67017	0.283133
$556$	−38.3778	−1.62758
$557$	19.7969	0.838822	0.419411	−	0.907796i	$-0.362237\pi$
$557$	19.7969	0.838822	0.419411	+	0.907796i	$0.362237\pi$
$558$	13.4316	0.568606
$559$	21.0248	0.889253
$560$	78.9654	3.33690
$561$	26.2524	1.10838
$562$	−22.6920	−0.957206
$563$	4.01634	0.169269	0.0846343	−	0.996412i	$-0.473028\pi$
$563$	4.01634	0.169269	0.0846343	+	0.996412i	$0.473028\pi$
$564$	−8.11319	−0.341627
$565$	16.2648	0.684264
$566$	−41.2980	−1.73588
$567$	−3.00947	−0.126386
$568$	−158.213	−6.63845
$569$	12.5039	0.524190	0.262095	−	0.965042i	$-0.415587\pi$
$569$	12.5039	0.524190	0.262095	+	0.965042i	$0.415587\pi$
$570$	−26.7712	−1.12132
$571$	26.0228	1.08902	0.544510	−	0.838755i	$-0.316716\pi$
$571$	26.0228	1.08902	0.544510	+	0.838755i	$0.316716\pi$
$572$	67.1648	2.80830
$573$	−0.884621	−0.0369556
$574$	94.9056	3.96128
$575$	−11.0570	−0.461111
$576$	41.8665	1.74444
$577$	−18.8752	−0.785787	−0.392893	−	0.919584i	$-0.628526\pi$
$577$	−18.8752	−0.785787	−0.392893	+	0.919584i	$0.628526\pi$
$578$	−111.182	−4.62454
$579$	−12.0992	−0.502824
$580$	26.4254	1.09725
$581$	51.8117	2.14951
$582$	27.3184	1.13238
$583$	24.5946	1.01861
$584$	−78.0897	−3.23137
$585$	−5.09195	−0.210526
$586$	−46.4530	−1.91896
$587$	22.0838	0.911497	0.455749	−	0.890109i	$-0.349372\pi$
$587$	22.0838	0.911497	0.455749	+	0.890109i	$0.349372\pi$
$588$	11.7854	0.486024
$589$	30.7956	1.26891
$590$	10.1040	0.415974
$591$	10.3552	0.425956
$592$	−76.6999	−3.15235
$593$	13.9464	0.572709	0.286354	−	0.958124i	$-0.407557\pi$
$593$	13.9464	0.572709	0.286354	+	0.958124i	$0.407557\pi$
$594$	−9.66833	−0.396697
$595$	34.3190	1.40694
$596$	−15.0006	−0.614448
$597$	−4.65662	−0.190583
$598$	−38.1232	−1.55897
$599$	−31.7142	−1.29581	−0.647903	−	0.761723i	$-0.724354\pi$
$599$	−31.7142	−1.29581	−0.647903	+	0.761723i	$0.724354\pi$
$600$	28.1856	1.15067
$601$	−19.9500	−0.813777	−0.406888	−	0.913478i	$-0.633386\pi$
$601$	−19.9500	−0.813777	−0.406888	+	0.913478i	$0.633386\pi$
$602$	52.1871	2.12699
$603$	−4.52087	−0.184104
$604$	124.387	5.06124
$605$	−1.65133	−0.0671363
$606$	−21.3796	−0.868488
$607$	−3.87655	−0.157344	−0.0786721	−	0.996901i	$-0.525068\pi$
$607$	−3.87655	−0.157344	−0.0786721	+	0.996901i	$0.525068\pi$
$608$	175.641	7.12320
$609$	9.18827	0.372328
$610$	−33.6243	−1.36141
$611$	−4.77308	−0.193098
$612$	43.2545	1.74846
$613$	−23.8796	−0.964487	−0.482244	−	0.876037i	$-0.660178\pi$
$613$	−23.8796	−0.964487	−0.482244	+	0.876037i	$0.660178\pi$
$614$	66.3223	2.67655
$615$	−17.1343	−0.690920
$616$	108.522	4.37246
$617$	−10.3404	−0.416287	−0.208144	−	0.978098i	$-0.566742\pi$
$617$	−10.3404	−0.416287	−0.208144	+	0.978098i	$0.566742\pi$
$618$	−0.219013	−0.00880999
$619$	−1.76798	−0.0710613	−0.0355307	−	0.999369i	$-0.511312\pi$
$619$	−1.76798	−0.0710613	−0.0355307	+	0.999369i	$0.511312\pi$
$620$	41.8143	1.67930
$621$	4.06788	0.163238
$622$	−35.8310	−1.43669
$623$	−12.1923	−0.488476
$624$	58.5520	2.34396
$625$	−4.02105	−0.160842
$626$	9.56833	0.382427
$627$	−22.1673	−0.885275
$628$	6.14433	0.245186
$629$	−33.3343	−1.32913
$630$	−12.6391	−0.503554
$631$	17.1224	0.681633	0.340816	−	0.940130i	$-0.389297\pi$
$631$	17.1224	0.681633	0.340816	+	0.940130i	$0.389297\pi$
$632$	−150.798	−5.99841
$633$	−8.70631	−0.346045
$634$	−0.295531	−0.0117370
$635$	−25.6719	−1.01876
$636$	40.5231	1.60685
$637$	6.93352	0.274716
$638$	29.5186	1.16865
$639$	15.2576	0.603580
$640$	92.5846	3.65973
$641$	43.9825	1.73720	0.868601	−	0.495511i	$-0.165019\pi$
$641$	43.9825	1.73720	0.868601	+	0.495511i	$0.165019\pi$
$642$	29.0051	1.14474
$643$	−28.7556	−1.13401	−0.567005	−	0.823714i	$-0.691898\pi$
$643$	−28.7556	−1.13401	−0.567005	+	0.823714i	$0.691898\pi$
$644$	−70.1439	−2.76406
$645$	−9.42185	−0.370985
$646$	133.790	5.26388
$647$	1.09639	0.0431037	0.0215518	−	0.999768i	$-0.493139\pi$
$647$	1.09639	0.0431037	0.0215518	+	0.999768i	$0.493139\pi$
$648$	−10.3695	−0.407350
$649$	8.36635	0.328408
$650$	25.4737	0.999162
$651$	14.5391	0.569832
$652$	54.3462	2.12836
$653$	−44.5291	−1.74256	−0.871279	−	0.490788i	$-0.836709\pi$
$653$	−44.5291	−1.74256	−0.871279	+	0.490788i	$0.836709\pi$
$654$	−34.0946	−1.33320
$655$	−1.88071	−0.0734854
$656$	197.026	7.69256
$657$	7.53074	0.293802
$658$	−11.8476	−0.461868
$659$	14.8075	0.576819	0.288409	−	0.957507i	$-0.406874\pi$
$659$	14.8075	0.576819	0.288409	+	0.957507i	$0.406874\pi$
$660$	−30.0987	−1.17159
$661$	14.4440	0.561805	0.280902	−	0.959736i	$-0.409366\pi$
$661$	14.4440	0.561805	0.280902	+	0.959736i	$0.409366\pi$
$662$	19.3857	0.753447
$663$	25.4471	0.988285
$664$	178.523	6.92802
$665$	−28.9785	−1.12374
$666$	12.2765	0.475704
$667$	−12.4197	−0.480894
$668$	−88.1111	−3.40912
$669$	−1.44816	−0.0559891
$670$	−18.9867	−0.733519
$671$	−27.8418	−1.07482
$672$	82.9230	3.19883
$673$	4.76146	0.183541	0.0917704	−	0.995780i	$-0.470747\pi$
$673$	4.76146	0.183541	0.0917704	+	0.995780i	$0.470747\pi$
$674$	29.3999	1.13244
$675$	−2.71814	−0.104621
$676$	−9.38158	−0.360830
$677$	48.3344	1.85764	0.928822	−	0.370527i	$-0.120823\pi$
$677$	48.3344	1.85764	0.928822	+	0.370527i	$0.120823\pi$
$678$	29.9354	1.14966
$679$	29.5708	1.13482
$680$	118.250	4.53467
$681$	−13.1542	−0.504069
$682$	46.7088	1.78857
$683$	−15.8783	−0.607564	−0.303782	−	0.952741i	$-0.598250\pi$
$683$	−15.8783	−0.607564	−0.303782	+	0.952741i	$0.598250\pi$
$684$	−36.5236	−1.39652
$685$	22.3480	0.853874
$686$	−41.3590	−1.57910
$687$	−17.5412	−0.669237
$688$	108.341	4.13047
$689$	23.8402	0.908240
$690$	17.0842	0.650383
$691$	−6.91889	−0.263207	−0.131604	−	0.991302i	$-0.542013\pi$
$691$	−6.91889	−0.263207	−0.131604	+	0.991302i	$0.542013\pi$
$692$	−114.384	−4.34823
$693$	−10.4655	−0.397552
$694$	−8.02300	−0.304549
$695$	10.1180	0.383796
$696$	31.6592	1.20004
$697$	85.6288	3.24342
$698$	19.6138	0.742391
$699$	−22.9591	−0.868393
$700$	46.8698	1.77151
$701$	2.79296	0.105489	0.0527443	−	0.998608i	$-0.483203\pi$
$701$	2.79296	0.105489	0.0527443	+	0.998608i	$0.483203\pi$
$702$	−9.37176	−0.353714
$703$	28.1471	1.06159
$704$	145.592	5.48720
$705$	2.13897	0.0805582
$706$	85.0648	3.20146
$707$	−23.1424	−0.870359
$708$	13.7847	0.518061
$709$	8.81439	0.331031	0.165516	−	0.986207i	$-0.447071\pi$
$709$	8.81439	0.331031	0.165516	+	0.986207i	$0.447071\pi$
$710$	64.0784	2.40482
$711$	14.5425	0.545386
$712$	−42.0100	−1.57439
$713$	−19.6524	−0.735987
$714$	63.1641	2.36386
$715$	−17.7074	−0.662219
$716$	36.8328	1.37651
$717$	17.2723	0.645045
$718$	−104.018	−3.88193
$719$	30.9326	1.15359	0.576795	−	0.816889i	$-0.304303\pi$
$719$	30.9326	1.15359	0.576795	+	0.816889i	$0.304303\pi$
$720$	−26.2390	−0.977869
$721$	−0.237071	−0.00882897
$722$	−60.1460	−2.23840
$723$	18.7096	0.695817
$724$	−129.714	−4.82080
$725$	8.29880	0.308210
$726$	−3.03929	−0.112799
$727$	−14.8355	−0.550220	−0.275110	−	0.961413i	$-0.588714\pi$
$727$	−14.8355	−0.550220	−0.275110	+	0.961413i	$0.588714\pi$
$728$	105.193	3.89870
$729$	1.00000	0.0370370
$730$	31.6274	1.17058
$731$	47.0859	1.74154
$732$	−45.8732	−1.69552
$733$	28.7870	1.06327	0.531636	−	0.846973i	$-0.321577\pi$
$733$	28.7870	1.06327	0.531636	+	0.846973i	$0.321577\pi$
$734$	−63.8601	−2.35712
$735$	−3.10712	−0.114608
$736$	−112.086	−4.13156
$737$	−15.7214	−0.579107
$738$	−31.5357	−1.16084
$739$	8.77568	0.322818	0.161409	−	0.986888i	$-0.448396\pi$
$739$	8.77568	0.322818	0.161409	+	0.986888i	$0.448396\pi$
$740$	38.2181	1.40493
$741$	−21.4873	−0.789355
$742$	59.1755	2.17240
$743$	44.2858	1.62469	0.812344	−	0.583178i	$-0.198191\pi$
$743$	44.2858	1.62469	0.812344	+	0.583178i	$0.198191\pi$
$744$	50.0960	1.83661
$745$	3.95477	0.144892
$746$	−91.8616	−3.36329
$747$	−17.2162	−0.629908
$748$	150.419	5.49985
$749$	31.3966	1.14721
$750$	−32.4145	−1.18361
$751$	−19.5485	−0.713335	−0.356667	−	0.934231i	$-0.616087\pi$
$751$	−19.5485	−0.713335	−0.356667	+	0.934231i	$0.616087\pi$
$752$	−24.5958	−0.896918
$753$	16.2558	0.592394
$754$	28.6131	1.04203
$755$	−32.7935	−1.19348
$756$	−17.2434	−0.627135
$757$	−30.5538	−1.11050	−0.555248	−	0.831685i	$-0.687377\pi$
$757$	−30.5538	−1.11050	−0.555248	+	0.831685i	$0.687377\pi$
$758$	17.4197	0.632711
$759$	14.1461	0.513472
$760$	−99.8486	−3.62189
$761$	30.3122	1.09882	0.549408	−	0.835554i	$-0.314853\pi$
$761$	30.3122	1.09882	0.549408	+	0.835554i	$0.314853\pi$
$762$	−47.2492	−1.71166
$763$	−36.9057	−1.33608
$764$	−5.06862	−0.183376
$765$	−11.4037	−0.412300
$766$	−64.7429	−2.33926
$767$	8.10971	0.292825
$768$	86.6693	3.12741
$769$	−11.2152	−0.404431	−0.202215	−	0.979341i	$-0.564814\pi$
$769$	−11.2152	−0.404431	−0.202215	+	0.979341i	$0.564814\pi$
$770$	−43.9528	−1.58395
$771$	5.31616	0.191457
$772$	−69.3246	−2.49505
$773$	−25.6788	−0.923603	−0.461801	−	0.886983i	$-0.652797\pi$
$773$	−25.6788	−0.923603	−0.461801	+	0.886983i	$0.652797\pi$
$774$	−17.3410	−0.623308
$775$	13.1316	0.471702
$776$	101.889	3.65762
$777$	13.2887	0.476729
$778$	15.2413	0.546429
$779$	−72.3040	−2.59056
$780$	−29.1754	−1.04465
$781$	53.0586	1.89858
$782$	−85.3785	−3.05313
$783$	−3.05312	−0.109110
$784$	35.7286	1.27602
$785$	−1.61990	−0.0578166
$786$	−3.46145	−0.123466
$787$	−5.25535	−0.187333	−0.0936666	−	0.995604i	$-0.529859\pi$
$787$	−5.25535	−0.187333	−0.0936666	+	0.995604i	$0.529859\pi$
$788$	59.3323	2.11363
$789$	13.4500	0.478832
$790$	61.0752	2.17296
$791$	32.4036	1.15214
$792$	−36.0600	−1.28134
$793$	−26.9877	−0.958362
$794$	30.1575	1.07025
$795$	−10.6835	−0.378906
$796$	−26.6811	−0.945685
$797$	34.6139	1.22609	0.613044	−	0.790048i	$-0.289945\pi$
$797$	34.6139	1.22609	0.613044	+	0.790048i	$0.289945\pi$
$798$	−53.3351	−1.88804
$799$	−10.6895	−0.378168
$800$	74.8957	2.64796
$801$	4.05133	0.143147
$802$	−75.3804	−2.66177
$803$	26.1883	0.924166
$804$	−25.9033	−0.913538
$805$	18.4928	0.651785
$806$	45.2760	1.59478
$807$	13.6762	0.481427
$808$	−79.7396	−2.80523
$809$	−5.45926	−0.191937	−0.0959687	−	0.995384i	$-0.530595\pi$
$809$	−5.45926	−0.191937	−0.0959687	+	0.995384i	$0.530595\pi$
$810$	4.19978	0.147565
$811$	−32.7001	−1.14826	−0.574128	−	0.818766i	$-0.694659\pi$
$811$	−32.7001	−1.14826	−0.574128	+	0.818766i	$0.694659\pi$
$812$	52.6461	1.84752
$813$	25.4733	0.893387
$814$	42.6918	1.49635
$815$	−14.3279	−0.501883
$816$	131.130	4.59046
$817$	−39.7588	−1.39098
$818$	33.6594	1.17687
$819$	−10.1445	−0.354477
$820$	−98.1743	−3.42839
$821$	−16.1622	−0.564064	−0.282032	−	0.959405i	$-0.591008\pi$
$821$	−16.1622	−0.564064	−0.282032	+	0.959405i	$0.591008\pi$
$822$	41.1316	1.43463
$823$	−54.5567	−1.90173	−0.950864	−	0.309608i	$-0.899802\pi$
$823$	−54.5567	−1.90173	−0.950864	+	0.309608i	$0.899802\pi$
$824$	−0.816853	−0.0284564
$825$	−9.45239	−0.329090
$826$	20.1297	0.700401
$827$	25.2374	0.877592	0.438796	−	0.898587i	$-0.355405\pi$
$827$	25.2374	0.877592	0.438796	+	0.898587i	$0.355405\pi$
$828$	23.3077	0.809999
$829$	46.2248	1.60545	0.802727	−	0.596347i	$-0.203382\pi$
$829$	46.2248	1.60545	0.802727	+	0.596347i	$0.203382\pi$
$830$	−72.3043	−2.50972
$831$	4.76078	0.165150
$832$	141.126	4.89266
$833$	15.5279	0.538010
$834$	18.6221	0.644831
$835$	23.2297	0.803897
$836$	−127.012	−4.39279
$837$	−4.83112	−0.166988
$838$	93.5771	3.23257
$839$	0.848253	0.0292849	0.0146425	−	0.999893i	$-0.495339\pi$
$839$	0.848253	0.0292849	0.0146425	+	0.999893i	$0.495339\pi$
$840$	−47.1401	−1.62649
$841$	−19.6784	−0.678567
$842$	52.4484	1.80749
$843$	8.16192	0.281111
$844$	−49.8846	−1.71710
$845$	2.47337	0.0850864
$846$	3.93678	0.135349
$847$	−3.28988	−0.113042
$848$	122.849	4.21866
$849$	14.8541	0.509792
$850$	57.0495	1.95678
$851$	−17.9622	−0.615737
$852$	87.4213	2.99501
$853$	20.2811	0.694412	0.347206	−	0.937789i	$-0.387131\pi$
$853$	20.2811	0.694412	0.347206	+	0.937789i	$0.387131\pi$
$854$	−66.9881	−2.29229
$855$	9.62912	0.329309
$856$	108.181	3.69753
$857$	−23.4347	−0.800515	−0.400258	−	0.916403i	$-0.631079\pi$
$857$	−23.4347	−0.800515	−0.400258	+	0.916403i	$0.631079\pi$
$858$	−32.5905	−1.11262
$859$	29.9562	1.02209	0.511046	−	0.859553i	$-0.329258\pi$
$859$	29.9562	1.02209	0.511046	+	0.859553i	$0.329258\pi$
$860$	−53.9844	−1.84085
$861$	−34.1358	−1.16335
$862$	−62.9611	−2.14446
$863$	−35.0342	−1.19258	−0.596289	−	0.802770i	$-0.703359\pi$
$863$	−35.0342	−1.19258	−0.596289	+	0.802770i	$0.703359\pi$
$864$	−27.5540	−0.937408
$865$	30.1563	1.02534
$866$	−41.2837	−1.40288
$867$	39.9900	1.35813
$868$	83.3047	2.82755
$869$	50.5719	1.71553
$870$	−12.8224	−0.434721
$871$	−15.2392	−0.516360
$872$	−127.163	−4.30627
$873$	−9.82592	−0.332557
$874$	72.0926	2.43857
$875$	−35.0871	−1.18616
$876$	43.1489	1.45787
$877$	31.9235	1.07798	0.538989	−	0.842313i	$-0.318806\pi$
$877$	31.9235	1.07798	0.538989	+	0.842313i	$0.318806\pi$
$878$	−101.863	−3.43771
$879$	16.7083	0.563557
$880$	−91.2467	−3.07593
$881$	52.6511	1.77386	0.886930	−	0.461903i	$-0.152833\pi$
$881$	52.6511	1.77386	0.886930	+	0.461903i	$0.152833\pi$
$882$	−5.71867	−0.192558
$883$	30.1036	1.01307	0.506534	−	0.862220i	$-0.330927\pi$
$883$	30.1036	1.01307	0.506534	+	0.862220i	$0.330927\pi$
$884$	145.805	4.90394
$885$	−3.63421	−0.122163
$886$	−96.6410	−3.24672
$887$	3.66393	0.123023	0.0615114	−	0.998106i	$-0.480408\pi$
$887$	3.66393	0.123023	0.0615114	+	0.998106i	$0.480408\pi$
$888$	45.7876	1.53653
$889$	−51.1450	−1.71535
$890$	17.0147	0.570333
$891$	3.47752	0.116501
$892$	−8.29753	−0.277822
$893$	9.02612	0.302048
$894$	7.27877	0.243438
$895$	−9.71063	−0.324590
$896$	184.452	6.16211
$897$	13.7122	0.457837
$898$	−109.193	−3.64382
$899$	14.7500	0.491939
$900$	−15.5741	−0.519137
$901$	53.3912	1.77872
$902$	−109.666	−3.65148
$903$	−18.7707	−0.624651
$904$	111.650	3.71343
$905$	34.1980	1.13678
$906$	−60.3565	−2.00521
$907$	−28.9938	−0.962724	−0.481362	−	0.876522i	$-0.659858\pi$
$907$	−28.9938	−0.962724	−0.481362	+	0.876522i	$0.659858\pi$
$908$	−75.3695	−2.50122
$909$	7.68986	0.255057
$910$	−42.6045	−1.41233
$911$	47.3921	1.57017	0.785085	−	0.619388i	$-0.212619\pi$
$911$	47.3921	1.57017	0.785085	+	0.619388i	$0.212619\pi$
$912$	−110.725	−3.66646
$913$	−59.8698	−1.98140
$914$	13.8354	0.457635
$915$	12.0940	0.399817
$916$	−100.506	−3.32080
$917$	−3.74685	−0.123732
$918$	−20.9885	−0.692723
$919$	49.6114	1.63653	0.818265	−	0.574841i	$-0.194936\pi$
$919$	49.6114	1.63653	0.818265	+	0.574841i	$0.194936\pi$
$920$	63.7189	2.10075
$921$	−23.8549	−0.786047
$922$	1.80763	0.0595312
$923$	51.4310	1.69287
$924$	−59.9642	−1.97268
$925$	12.0023	0.394632
$926$	−88.1804	−2.89779
$927$	0.0787749	0.00258731
$928$	84.1258	2.76157
$929$	−33.2941	−1.09234	−0.546171	−	0.837673i	$-0.683915\pi$
$929$	−33.2941	−1.09234	−0.546171	+	0.837673i	$0.683915\pi$
$930$	−20.2896	−0.665322
$931$	−13.1116	−0.429715
$932$	−131.549	−4.30903
$933$	12.8878	0.421926
$934$	−60.1699	−1.96882
$935$	−39.6565	−1.29691
$936$	−34.9539	−1.14250
$937$	−26.3093	−0.859486	−0.429743	−	0.902951i	$-0.641396\pi$
$937$	−26.3093	−0.859486	−0.429743	+	0.902951i	$0.641396\pi$
$938$	−37.8263	−1.23507
$939$	−3.44156	−0.112311
$940$	12.2556	0.399735
$941$	30.9180	1.00790	0.503949	−	0.863733i	$-0.331880\pi$
$941$	30.9180	1.00790	0.503949	+	0.863733i	$0.331880\pi$
$942$	−2.98143	−0.0971401
$943$	46.1411	1.50256
$944$	41.7896	1.36013
$945$	4.54605	0.147883
$946$	−60.3036	−1.96064
$947$	−57.8322	−1.87929	−0.939646	−	0.342147i	$-0.888846\pi$
$947$	−57.8322	−1.87929	−0.939646	+	0.342147i	$0.888846\pi$
$948$	83.3241	2.70624
$949$	25.3850	0.824032
$950$	−48.1720	−1.56291
$951$	0.106297	0.00344692
$952$	235.584	7.63531
$953$	42.2308	1.36799	0.683996	−	0.729486i	$-0.260241\pi$
$953$	42.2308	1.36799	0.683996	+	0.729486i	$0.260241\pi$
$954$	−19.6631	−0.636616
$955$	1.33629	0.0432415
$956$	98.9650	3.20076
$957$	−10.6173	−0.343209
$958$	−43.4676	−1.40437
$959$	44.5229	1.43772
$960$	−63.2429	−2.04115
$961$	−7.66032	−0.247107
$962$	41.3822	1.33422
$963$	−10.4326	−0.336186
$964$	107.200	3.45269
$965$	18.2768	0.588351
$966$	34.0360	1.09509
$967$	15.9593	0.513218	0.256609	−	0.966515i	$-0.417395\pi$
$967$	15.9593	0.513218	0.256609	+	0.966515i	$0.417395\pi$
$968$	−11.3356	−0.364341
$969$	−48.1217	−1.54589
$970$	−41.2667	−1.32499
$971$	−36.2544	−1.16346	−0.581729	−	0.813383i	$-0.697624\pi$
$971$	−36.2544	−1.16346	−0.581729	+	0.813383i	$0.697624\pi$
$972$	5.72970	0.183780
$973$	20.1575	0.646221
$974$	−77.6050	−2.48663
$975$	−9.16244	−0.293433
$976$	−139.069	−4.45148
$977$	−21.9319	−0.701664	−0.350832	−	0.936439i	$-0.614101\pi$
$977$	−21.9319	−0.701664	−0.350832	+	0.936439i	$0.614101\pi$
$978$	−26.3705	−0.843235
$979$	14.0886	0.450273
$980$	−17.8029	−0.568693
$981$	12.2632	0.391534
$982$	−6.79736	−0.216913
$983$	−22.8022	−0.727277	−0.363638	−	0.931540i	$-0.618466\pi$
$983$	−22.8022	−0.727277	−0.363638	+	0.931540i	$0.618466\pi$
$984$	−117.619	−3.74955
$985$	−15.6424	−0.498409
$986$	64.0803	2.04073
$987$	4.26137	0.135641
$988$	−123.116	−3.91683
$989$	25.3723	0.806791
$990$	14.6048	0.464172
$991$	−38.5609	−1.22493	−0.612463	−	0.790499i	$-0.709821\pi$
$991$	−38.5609	−1.22493	−0.612463	+	0.790499i	$0.709821\pi$
$992$	133.117	4.22646
$993$	−6.97269	−0.221272
$994$	127.661	4.04915
$995$	7.03422	0.223000
$996$	−98.6438	−3.12565
$997$	−19.8696	−0.629278	−0.314639	−	0.949211i	$-0.601883\pi$
$997$	−19.8696	−0.629278	−0.314639	+	0.949211i	$0.601883\pi$
$998$	−8.54956	−0.270632
$999$	−4.41563	−0.139704

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6009.2.a.c.1.1	✓	92

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6009.2.a.c.1.1	✓	92	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:	\( N \)	\(=\)	\( 6009 = 3 \cdot 2003 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6009.a (trivial)

\(f(q)\)	\(=\)	\(q-2.78023 q^{2} +1.00000 q^{3} +5.72970 q^{4} -1.51058 q^{5} -2.78023 q^{6} -3.00947 q^{7} -10.3695 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-2.78023 q^{2} +1.00000 q^{3} +5.72970 q^{4} -1.51058 q^{5} -2.78023 q^{6} -3.00947 q^{7} -10.3695 q^{8} +1.00000 q^{9} +4.19978 q^{10} +3.47752 q^{11} +5.72970 q^{12} +3.37085 q^{13} +8.36703 q^{14} -1.51058 q^{15} +17.3701 q^{16} +7.54917 q^{17} -2.78023 q^{18} -6.37443 q^{19} -8.65520 q^{20} -3.00947 q^{21} -9.66833 q^{22} +4.06788 q^{23} -10.3695 q^{24} -2.71814 q^{25} -9.37176 q^{26} +1.00000 q^{27} -17.2434 q^{28} -3.05312 q^{29} +4.19978 q^{30} -4.83112 q^{31} -27.5540 q^{32} +3.47752 q^{33} -20.9885 q^{34} +4.54605 q^{35} +5.72970 q^{36} -4.41563 q^{37} +17.7224 q^{38} +3.37085 q^{39} +15.6639 q^{40} +11.3428 q^{41} +8.36703 q^{42} +6.23723 q^{43} +19.9252 q^{44} -1.51058 q^{45} -11.3096 q^{46} -1.41599 q^{47} +17.3701 q^{48} +2.05690 q^{49} +7.55706 q^{50} +7.54917 q^{51} +19.3140 q^{52} +7.07246 q^{53} -2.78023 q^{54} -5.25309 q^{55} +31.2065 q^{56} -6.37443 q^{57} +8.48839 q^{58} +2.40583 q^{59} -8.65520 q^{60} -8.00620 q^{61} +13.4316 q^{62} -3.00947 q^{63} +41.8665 q^{64} -5.09195 q^{65} -9.66833 q^{66} -4.52087 q^{67} +43.2545 q^{68} +4.06788 q^{69} -12.6391 q^{70} +15.2576 q^{71} -10.3695 q^{72} +7.53074 q^{73} +12.2765 q^{74} -2.71814 q^{75} -36.5236 q^{76} -10.4655 q^{77} -9.37176 q^{78} +14.5425 q^{79} -26.2390 q^{80} +1.00000 q^{81} -31.5357 q^{82} -17.2162 q^{83} -17.2434 q^{84} -11.4037 q^{85} -17.3410 q^{86} -3.05312 q^{87} -36.0600 q^{88} +4.05133 q^{89} +4.19978 q^{90} -10.1445 q^{91} +23.3077 q^{92} -4.83112 q^{93} +3.93678 q^{94} +9.62912 q^{95} -27.5540 q^{96} -9.82592 q^{97} -5.71867 q^{98} +3.47752 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 92 q + 17 q^{2} + 92 q^{3} + 107 q^{4} + 34 q^{5} + 17 q^{6} + 22 q^{7} + 51 q^{8} + 92 q^{9}+O(q^{10}) \) 92 * q + 17 * q^2 + 92 * q^3 + 107 * q^4 + 34 * q^5 + 17 * q^6 + 22 * q^7 + 51 * q^8 + 92 * q^9	\( 92 q + 17 q^{2} + 92 q^{3} + 107 q^{4} + 34 q^{5} + 17 q^{6} + 22 q^{7} + 51 q^{8} + 92 q^{9} + 13 q^{10} + 40 q^{11} + 107 q^{12} + 6 q^{13} + 37 q^{14} + 34 q^{15} + 133 q^{16} + 77 q^{17} + 17 q^{18} + 34 q^{19} + 55 q^{20} + 22 q^{21} + 8 q^{22} + 83 q^{23} + 51 q^{24} + 110 q^{25} + 22 q^{26} + 92 q^{27} + 32 q^{28} + 97 q^{29} + 13 q^{30} + 44 q^{31} + 104 q^{32} + 40 q^{33} + 20 q^{34} + 80 q^{35} + 107 q^{36} + 12 q^{37} + 54 q^{38} + 6 q^{39} + 23 q^{40} + 67 q^{41} + 37 q^{42} + 30 q^{43} + 87 q^{44} + 34 q^{45} + 33 q^{46} + 69 q^{47} + 133 q^{48} + 112 q^{49} + 58 q^{50} + 77 q^{51} - 3 q^{52} + 113 q^{53} + 17 q^{54} + 42 q^{55} + 92 q^{56} + 34 q^{57} - 30 q^{58} + 72 q^{59} + 55 q^{60} + 19 q^{61} + 60 q^{62} + 22 q^{63} + 147 q^{64} + 74 q^{65} + 8 q^{66} + 26 q^{67} + 171 q^{68} + 83 q^{69} - 35 q^{70} + 134 q^{71} + 51 q^{72} - 17 q^{73} + 95 q^{74} + 110 q^{75} + 27 q^{76} + 108 q^{77} + 22 q^{78} + 159 q^{79} + 79 q^{80} + 92 q^{81} - 64 q^{82} + 73 q^{83} + 32 q^{84} - 4 q^{85} + 22 q^{86} + 97 q^{87} - 16 q^{88} + 50 q^{89} + 13 q^{90} + 17 q^{91} + 154 q^{92} + 44 q^{93} + 8 q^{94} + 155 q^{95} + 104 q^{96} - 20 q^{97} + 63 q^{98} + 40 q^{99}+O(q^{100}) \) 92 * q + 17 * q^2 + 92 * q^3 + 107 * q^4 + 34 * q^5 + 17 * q^6 + 22 * q^7 + 51 * q^8 + 92 * q^9 + 13 * q^10 + 40 * q^11 + 107 * q^12 + 6 * q^13 + 37 * q^14 + 34 * q^15 + 133 * q^16 + 77 * q^17 + 17 * q^18 + 34 * q^19 + 55 * q^20 + 22 * q^21 + 8 * q^22 + 83 * q^23 + 51 * q^24 + 110 * q^25 + 22 * q^26 + 92 * q^27 + 32 * q^28 + 97 * q^29 + 13 * q^30 + 44 * q^31 + 104 * q^32 + 40 * q^33 + 20 * q^34 + 80 * q^35 + 107 * q^36 + 12 * q^37 + 54 * q^38 + 6 * q^39 + 23 * q^40 + 67 * q^41 + 37 * q^42 + 30 * q^43 + 87 * q^44 + 34 * q^45 + 33 * q^46 + 69 * q^47 + 133 * q^48 + 112 * q^49 + 58 * q^50 + 77 * q^51 - 3 * q^52 + 113 * q^53 + 17 * q^54 + 42 * q^55 + 92 * q^56 + 34 * q^57 - 30 * q^58 + 72 * q^59 + 55 * q^60 + 19 * q^61 + 60 * q^62 + 22 * q^63 + 147 * q^64 + 74 * q^65 + 8 * q^66 + 26 * q^67 + 171 * q^68 + 83 * q^69 - 35 * q^70 + 134 * q^71 + 51 * q^72 - 17 * q^73 + 95 * q^74 + 110 * q^75 + 27 * q^76 + 108 * q^77 + 22 * q^78 + 159 * q^79 + 79 * q^80 + 92 * q^81 - 64 * q^82 + 73 * q^83 + 32 * q^84 - 4 * q^85 + 22 * q^86 + 97 * q^87 - 16 * q^88 + 50 * q^89 + 13 * q^90 + 17 * q^91 + 154 * q^92 + 44 * q^93 + 8 * q^94 + 155 * q^95 + 104 * q^96 - 20 * q^97 + 63 * q^98 + 40 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more