LMFDB - Embedded newform 8619.2.a.bq.1.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8619,2,Mod(1,8619)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8619, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("8619.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8619 = 3 \cdot 13^{2} \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8619.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:	$68.8230615021$
Analytic rank:	$0$
Dimension:	$21$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
Character	$\chi$	$=$	8619.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.0929476 q^{2} +1.00000 q^{3} -1.99136 q^{4} +3.17289 q^{5} -0.0929476 q^{6} -0.513856 q^{7} +0.370988 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-0.0929476 q^{2} +1.00000 q^{3} -1.99136 q^{4} +3.17289 q^{5} -0.0929476 q^{6} -0.513856 q^{7} +0.370988 q^{8} +1.00000 q^{9} -0.294913 q^{10} +0.710085 q^{11} -1.99136 q^{12} +0.0477617 q^{14} +3.17289 q^{15} +3.94824 q^{16} -1.00000 q^{17} -0.0929476 q^{18} +6.95069 q^{19} -6.31838 q^{20} -0.513856 q^{21} -0.0660007 q^{22} +4.94547 q^{23} +0.370988 q^{24} +5.06726 q^{25} +1.00000 q^{27} +1.02327 q^{28} +1.24548 q^{29} -0.294913 q^{30} -10.7371 q^{31} -1.10895 q^{32} +0.710085 q^{33} +0.0929476 q^{34} -1.63041 q^{35} -1.99136 q^{36} +0.627020 q^{37} -0.646050 q^{38} +1.17710 q^{40} +12.4034 q^{41} +0.0477617 q^{42} +5.88517 q^{43} -1.41404 q^{44} +3.17289 q^{45} -0.459669 q^{46} -7.60138 q^{47} +3.94824 q^{48} -6.73595 q^{49} -0.470990 q^{50} -1.00000 q^{51} +1.78204 q^{53} -0.0929476 q^{54} +2.25302 q^{55} -0.190634 q^{56} +6.95069 q^{57} -0.115764 q^{58} +7.25662 q^{59} -6.31838 q^{60} +3.22157 q^{61} +0.997992 q^{62} -0.513856 q^{63} -7.79340 q^{64} -0.0660007 q^{66} +14.5650 q^{67} +1.99136 q^{68} +4.94547 q^{69} +0.151543 q^{70} -10.3399 q^{71} +0.370988 q^{72} -13.4443 q^{73} -0.0582800 q^{74} +5.06726 q^{75} -13.8413 q^{76} -0.364882 q^{77} +8.66207 q^{79} +12.5273 q^{80} +1.00000 q^{81} -1.15286 q^{82} +8.16003 q^{83} +1.02327 q^{84} -3.17289 q^{85} -0.547013 q^{86} +1.24548 q^{87} +0.263433 q^{88} -13.5553 q^{89} -0.294913 q^{90} -9.84821 q^{92} -10.7371 q^{93} +0.706531 q^{94} +22.0538 q^{95} -1.10895 q^{96} +4.22630 q^{97} +0.626091 q^{98} +0.710085 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 21 q + 9 q^{2} + 21 q^{3} + 29 q^{4} + 26 q^{5} + 9 q^{6} - 5 q^{7} + 24 q^{8} + 21 q^{9}+O(q^{10}) $ 21 * q + 9 * q^2 + 21 * q^3 + 29 * q^4 + 26 * q^5 + 9 * q^6 - 5 * q^7 + 24 * q^8 + 21 * q^9	\( 21 q + 9 q^{2} + 21 q^{3} + 29 q^{4} + 26 q^{5} + 9 q^{6} - 5 q^{7} + 24 q^{8} + 21 q^{9} + 12 q^{10} + 37 q^{11} + 29 q^{12} - 6 q^{14} + 26 q^{15} + 33 q^{16} - 21 q^{17} + 9 q^{18} - 6 q^{19} + 53 q^{20} - 5 q^{21} + 17 q^{22} - 8 q^{23} + 24 q^{24} + 57 q^{25} + 21 q^{27} + 5 q^{28} - 6 q^{29} + 12 q^{30} + 9 q^{31} + 37 q^{32} + 37 q^{33} - 9 q^{34} - 8 q^{35} + 29 q^{36} + 9 q^{37} + 11 q^{38} - 8 q^{40} + 50 q^{41} - 6 q^{42} + 18 q^{43} + 67 q^{44} + 26 q^{45} + 23 q^{46} + 71 q^{47} + 33 q^{48} + 46 q^{49} + 2 q^{50} - 21 q^{51} - 14 q^{53} + 9 q^{54} + 29 q^{55} - 17 q^{56} - 6 q^{57} + 37 q^{58} + 59 q^{59} + 53 q^{60} + 44 q^{61} + 2 q^{62} - 5 q^{63} + 44 q^{64} + 17 q^{66} + 8 q^{67} - 29 q^{68} - 8 q^{69} + 15 q^{70} + 60 q^{71} + 24 q^{72} + 13 q^{73} - 14 q^{74} + 57 q^{75} + 15 q^{76} - 28 q^{77} + 51 q^{79} + 131 q^{80} + 21 q^{81} + 18 q^{82} + 30 q^{83} + 5 q^{84} - 26 q^{85} - 15 q^{86} - 6 q^{87} + 71 q^{88} + 62 q^{89} + 12 q^{90} + 20 q^{92} + 9 q^{93} - 72 q^{94} - 16 q^{95} + 37 q^{96} - 6 q^{97} + 58 q^{98} + 37 q^{99}+O(q^{100}) \) 21 * q + 9 * q^2 + 21 * q^3 + 29 * q^4 + 26 * q^5 + 9 * q^6 - 5 * q^7 + 24 * q^8 + 21 * q^9 + 12 * q^10 + 37 * q^11 + 29 * q^12 - 6 * q^14 + 26 * q^15 + 33 * q^16 - 21 * q^17 + 9 * q^18 - 6 * q^19 + 53 * q^20 - 5 * q^21 + 17 * q^22 - 8 * q^23 + 24 * q^24 + 57 * q^25 + 21 * q^27 + 5 * q^28 - 6 * q^29 + 12 * q^30 + 9 * q^31 + 37 * q^32 + 37 * q^33 - 9 * q^34 - 8 * q^35 + 29 * q^36 + 9 * q^37 + 11 * q^38 - 8 * q^40 + 50 * q^41 - 6 * q^42 + 18 * q^43 + 67 * q^44 + 26 * q^45 + 23 * q^46 + 71 * q^47 + 33 * q^48 + 46 * q^49 + 2 * q^50 - 21 * q^51 - 14 * q^53 + 9 * q^54 + 29 * q^55 - 17 * q^56 - 6 * q^57 + 37 * q^58 + 59 * q^59 + 53 * q^60 + 44 * q^61 + 2 * q^62 - 5 * q^63 + 44 * q^64 + 17 * q^66 + 8 * q^67 - 29 * q^68 - 8 * q^69 + 15 * q^70 + 60 * q^71 + 24 * q^72 + 13 * q^73 - 14 * q^74 + 57 * q^75 + 15 * q^76 - 28 * q^77 + 51 * q^79 + 131 * q^80 + 21 * q^81 + 18 * q^82 + 30 * q^83 + 5 * q^84 - 26 * q^85 - 15 * q^86 - 6 * q^87 + 71 * q^88 + 62 * q^89 + 12 * q^90 + 20 * q^92 + 9 * q^93 - 72 * q^94 - 16 * q^95 + 37 * q^96 - 6 * q^97 + 58 * q^98 + 37 * 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 $

$2$	−0.0929476	−0.0657239	−0.0328620	−	0.999460i	$-0.510462\pi$
$2$	−0.0929476	−0.0657239	−0.0328620	+	0.999460i	$0.510462\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	−1.99136	−0.995680
$5$	3.17289	1.41896	0.709481	−	0.704725i	$-0.248930\pi$
$5$	3.17289	1.41896	0.709481	+	0.704725i	$0.248930\pi$
$6$	−0.0929476	−0.0379457
$7$	−0.513856	−0.194219	−0.0971097	−	0.995274i	$-0.530960\pi$
$7$	−0.513856	−0.194219	−0.0971097	+	0.995274i	$0.530960\pi$
$8$	0.370988	0.131164
$9$	1.00000	0.333333
$10$	−0.294913	−0.0932597
$11$	0.710085	0.214099	0.107049	−	0.994254i	$-0.465860\pi$
$11$	0.710085	0.214099	0.107049	+	0.994254i	$0.465860\pi$
$12$	−1.99136	−0.574856
$13$	0	0
$13$	0	0
$14$	0.0477617	0.0127649
$15$	3.17289	0.819238
$16$	3.94824	0.987060
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	−0.0929476	−0.0219080
$19$	6.95069	1.59460	0.797299	−	0.603585i	$-0.206262\pi$
$19$	6.95069	1.59460	0.797299	+	0.603585i	$0.206262\pi$
$20$	−6.31838	−1.41283
$21$	−0.513856	−0.112133
$22$	−0.0660007	−0.0140714
$23$	4.94547	1.03120	0.515601	−	0.856829i	$-0.327569\pi$
$23$	4.94547	1.03120	0.515601	+	0.856829i	$0.327569\pi$
$24$	0.370988	0.0757275
$25$	5.06726	1.01345
$26$	0	0
$27$	1.00000	0.192450
$28$	1.02327	0.193381
$29$	1.24548	0.231279	0.115640	−	0.993291i	$-0.463108\pi$
$29$	1.24548	0.231279	0.115640	+	0.993291i	$0.463108\pi$
$30$	−0.294913	−0.0538435
$31$	−10.7371	−1.92845	−0.964224	−	0.265089i	$-0.914599\pi$
$31$	−10.7371	−1.92845	−0.964224	+	0.265089i	$0.914599\pi$
$32$	−1.10895	−0.196037
$33$	0.710085	0.123610
$34$	0.0929476	0.0159404
$35$	−1.63041	−0.275590
$36$	−1.99136	−0.331893
$37$	0.627020	0.103081	0.0515407	−	0.998671i	$-0.483587\pi$
$37$	0.627020	0.103081	0.0515407	+	0.998671i	$0.483587\pi$
$38$	−0.646050	−0.104803
$39$	0	0
$40$	1.17710	0.186117
$41$	12.4034	1.93708	0.968539	−	0.248861i	$-0.0800561\pi$
$41$	12.4034	1.93708	0.968539	+	0.248861i	$0.0800561\pi$
$42$	0.0477617	0.00736980
$43$	5.88517	0.897480	0.448740	−	0.893662i	$-0.351873\pi$
$43$	5.88517	0.897480	0.448740	+	0.893662i	$0.351873\pi$
$44$	−1.41404	−0.213174
$45$	3.17289	0.472987
$46$	−0.459669	−0.0677746
$47$	−7.60138	−1.10878	−0.554388	−	0.832258i	$-0.687048\pi$
$47$	−7.60138	−1.10878	−0.554388	+	0.832258i	$0.687048\pi$
$48$	3.94824	0.569879
$49$	−6.73595	−0.962279
$50$	−0.470990	−0.0666080
$51$	−1.00000	−0.140028
$52$	0	0
$53$	1.78204	0.244782	0.122391	−	0.992482i	$-0.460944\pi$
$53$	1.78204	0.244782	0.122391	+	0.992482i	$0.460944\pi$
$54$	−0.0929476	−0.0126486
$55$	2.25302	0.303798
$56$	−0.190634	−0.0254746
$57$	6.95069	0.920641
$58$	−0.115764	−0.0152006
$59$	7.25662	0.944731	0.472365	−	0.881403i	$-0.343400\pi$
$59$	7.25662	0.944731	0.472365	+	0.881403i	$0.343400\pi$
$60$	−6.31838	−0.815699
$61$	3.22157	0.412480	0.206240	−	0.978501i	$-0.433877\pi$
$61$	3.22157	0.412480	0.206240	+	0.978501i	$0.433877\pi$
$62$	0.997992	0.126745
$63$	−0.513856	−0.0647398
$64$	−7.79340	−0.974175
$65$	0	0
$66$	−0.0660007	−0.00812413
$67$	14.5650	1.77940	0.889699	−	0.456548i	$-0.150914\pi$
$67$	14.5650	1.77940	0.889699	+	0.456548i	$0.150914\pi$
$68$	1.99136	0.241488
$69$	4.94547	0.595364
$70$	0.151543	0.0181129
$71$	−10.3399	−1.22712	−0.613562	−	0.789647i	$-0.710264\pi$
$71$	−10.3399	−1.22712	−0.613562	+	0.789647i	$0.710264\pi$
$72$	0.370988	0.0437213
$73$	−13.4443	−1.57354	−0.786770	−	0.617246i	$-0.788248\pi$
$73$	−13.4443	−1.57354	−0.786770	+	0.617246i	$0.788248\pi$
$74$	−0.0582800	−0.00677491
$75$	5.06726	0.585117
$76$	−13.8413	−1.58771
$77$	−0.364882	−0.0415821
$78$	0	0
$79$	8.66207	0.974559	0.487279	−	0.873246i	$-0.337989\pi$
$79$	8.66207	0.974559	0.487279	+	0.873246i	$0.337989\pi$
$80$	12.5273	1.40060
$81$	1.00000	0.111111
$82$	−1.15286	−0.127312
$83$	8.16003	0.895680	0.447840	−	0.894114i	$-0.352193\pi$
$83$	8.16003	0.895680	0.447840	+	0.894114i	$0.352193\pi$
$84$	1.02327	0.111648
$85$	−3.17289	−0.344149
$86$	−0.547013	−0.0589859
$87$	1.24548	0.133529
$88$	0.263433	0.0280820
$89$	−13.5553	−1.43686	−0.718430	−	0.695600i	$-0.755139\pi$
$89$	−13.5553	−1.43686	−0.718430	+	0.695600i	$0.755139\pi$
$90$	−0.294913	−0.0310866
$91$	0	0
$92$	−9.84821	−1.02675
$93$	−10.7371	−1.11339
$94$	0.706531	0.0728731
$95$	22.0538	2.26267
$96$	−1.10895	−0.113182
$97$	4.22630	0.429115	0.214558	−	0.976711i	$-0.431169\pi$
$97$	4.22630	0.429115	0.214558	+	0.976711i	$0.431169\pi$
$98$	0.626091	0.0632447
$99$	0.710085	0.0713662
$100$	−10.0907	−1.00907
$101$	3.01091	0.299597	0.149799	−	0.988717i	$-0.452137\pi$
$101$	3.01091	0.299597	0.149799	+	0.988717i	$0.452137\pi$
$102$	0.0929476	0.00920319
$103$	6.96909	0.686685	0.343342	−	0.939210i	$-0.388441\pi$
$103$	6.96909	0.686685	0.343342	+	0.939210i	$0.388441\pi$
$104$	0	0
$105$	−1.63041	−0.159112
$106$	−0.165636	−0.0160880
$107$	2.53170	0.244749	0.122375	−	0.992484i	$-0.460949\pi$
$107$	2.53170	0.244749	0.122375	+	0.992484i	$0.460949\pi$
$108$	−1.99136	−0.191619
$109$	−10.4929	−1.00504	−0.502519	−	0.864566i	$-0.667593\pi$
$109$	−10.4929	−1.00504	−0.502519	+	0.864566i	$0.667593\pi$
$110$	−0.209413	−0.0199668
$111$	0.627020	0.0595141
$112$	−2.02883	−0.191706
$113$	1.70521	0.160412	0.0802061	−	0.996778i	$-0.474442\pi$
$113$	1.70521	0.160412	0.0802061	+	0.996778i	$0.474442\pi$
$114$	−0.646050	−0.0605081
$115$	15.6914	1.46323
$116$	−2.48019	−0.230280
$117$	0	0
$118$	−0.674485	−0.0620914
$119$	0.513856	0.0471051
$120$	1.17710	0.107454
$121$	−10.4958	−0.954162
$122$	−0.299437	−0.0271098
$123$	12.4034	1.11837
$124$	21.3815	1.92012
$125$	0.213406	0.0190876
$126$	0.0477617	0.00425495
$127$	14.1095	1.25202	0.626008	−	0.779817i	$-0.284688\pi$
$127$	14.1095	1.25202	0.626008	+	0.779817i	$0.284688\pi$
$128$	2.94229	0.260064
$129$	5.88517	0.518161
$130$	0	0
$131$	−18.6923	−1.63315	−0.816576	−	0.577238i	$-0.804130\pi$
$131$	−18.6923	−1.63315	−0.816576	+	0.577238i	$0.804130\pi$
$132$	−1.41404	−0.123076
$133$	−3.57166	−0.309702
$134$	−1.35378	−0.116949
$135$	3.17289	0.273079
$136$	−0.370988	−0.0318119
$137$	−3.84179	−0.328226	−0.164113	−	0.986442i	$-0.552476\pi$
$137$	−3.84179	−0.328226	−0.164113	+	0.986442i	$0.552476\pi$
$138$	−0.459669	−0.0391297
$139$	−5.43821	−0.461263	−0.230631	−	0.973041i	$-0.574079\pi$
$139$	−5.43821	−0.461263	−0.230631	+	0.973041i	$0.574079\pi$
$140$	3.24674	0.274400
$141$	−7.60138	−0.640152
$142$	0.961072	0.0806514
$143$	0	0
$144$	3.94824	0.329020
$145$	3.95177	0.328176
$146$	1.24962	0.103419
$147$	−6.73595	−0.555572
$148$	−1.24862	−0.102636
$149$	10.1984	0.835485	0.417743	−	0.908565i	$-0.362821\pi$
$149$	10.1984	0.835485	0.417743	+	0.908565i	$0.362821\pi$
$150$	−0.470990	−0.0384562
$151$	10.1235	0.823842	0.411921	−	0.911220i	$-0.364858\pi$
$151$	10.1235	0.823842	0.411921	+	0.911220i	$0.364858\pi$
$152$	2.57862	0.209154
$153$	−1.00000	−0.0808452
$154$	0.0339149	0.00273294
$155$	−34.0678	−2.73639
$156$	0	0
$157$	−3.68481	−0.294080	−0.147040	−	0.989131i	$-0.546975\pi$
$157$	−3.68481	−0.294080	−0.147040	+	0.989131i	$0.546975\pi$
$158$	−0.805119	−0.0640518
$159$	1.78204	0.141325
$160$	−3.51860	−0.278169
$161$	−2.54126	−0.200279
$162$	−0.0929476	−0.00730266
$163$	5.63774	0.441582	0.220791	−	0.975321i	$-0.429136\pi$
$163$	5.63774	0.441582	0.220791	+	0.975321i	$0.429136\pi$
$164$	−24.6996	−1.92871
$165$	2.25302	0.175398
$166$	−0.758456	−0.0588676
$167$	11.7187	0.906822	0.453411	−	0.891302i	$-0.350207\pi$
$167$	11.7187	0.906822	0.453411	+	0.891302i	$0.350207\pi$
$168$	−0.190634	−0.0147078
$169$	0	0
$170$	0.294913	0.0226188
$171$	6.95069	0.531532
$172$	−11.7195	−0.893604
$173$	−5.94056	−0.451653	−0.225826	−	0.974168i	$-0.572508\pi$
$173$	−5.94056	−0.451653	−0.225826	+	0.974168i	$0.572508\pi$
$174$	−0.115764	−0.00877606
$175$	−2.60384	−0.196832
$176$	2.80358	0.211328
$177$	7.25662	0.545441
$178$	1.25993	0.0944360
$179$	−1.17793	−0.0880424	−0.0440212	−	0.999031i	$-0.514017\pi$
$179$	−1.17793	−0.0880424	−0.0440212	+	0.999031i	$0.514017\pi$
$180$	−6.31838	−0.470944
$181$	8.71950	0.648115	0.324058	−	0.946037i	$-0.394953\pi$
$181$	8.71950	0.648115	0.324058	+	0.946037i	$0.394953\pi$
$182$	0	0
$183$	3.22157	0.238145
$184$	1.83471	0.135256
$185$	1.98947	0.146269
$186$	0.997992	0.0731763
$187$	−0.710085	−0.0519265
$188$	15.1371	1.10399
$189$	−0.513856	−0.0373776
$190$	−2.04985	−0.148712
$191$	−23.1293	−1.67358	−0.836789	−	0.547525i	$-0.815570\pi$
$191$	−23.1293	−1.67358	−0.836789	+	0.547525i	$0.815570\pi$
$192$	−7.79340	−0.562440
$193$	12.9444	0.931755	0.465878	−	0.884849i	$-0.345739\pi$
$193$	12.9444	0.931755	0.465878	+	0.884849i	$0.345739\pi$
$194$	−0.392824	−0.0282031
$195$	0	0
$196$	13.4137	0.958122
$197$	23.2525	1.65667	0.828336	−	0.560232i	$-0.189288\pi$
$197$	23.2525	1.65667	0.828336	+	0.560232i	$0.189288\pi$
$198$	−0.0660007	−0.00469047
$199$	−14.7511	−1.04568	−0.522840	−	0.852431i	$-0.675128\pi$
$199$	−14.7511	−1.04568	−0.522840	+	0.852431i	$0.675128\pi$
$200$	1.87989	0.132928
$201$	14.5650	1.02734
$202$	−0.279857	−0.0196907
$203$	−0.639996	−0.0449189
$204$	1.99136	0.139423
$205$	39.3545	2.74864
$206$	−0.647760	−0.0451316
$207$	4.94547	0.343734
$208$	0	0
$209$	4.93558	0.341401
$210$	0.151543	0.0104575
$211$	23.0352	1.58581	0.792903	−	0.609347i	$-0.208568\pi$
$211$	23.0352	1.58581	0.792903	+	0.609347i	$0.208568\pi$
$212$	−3.54868	−0.243724
$213$	−10.3399	−0.708480
$214$	−0.235316	−0.0160859
$215$	18.6730	1.27349
$216$	0.370988	0.0252425
$217$	5.51735	0.374542
$218$	0.975290	0.0660550
$219$	−13.4443	−0.908484
$220$	−4.48658	−0.302485
$221$	0	0
$222$	−0.0582800	−0.00391150
$223$	6.97643	0.467177	0.233588	−	0.972336i	$-0.424953\pi$
$223$	6.97643	0.467177	0.233588	+	0.972336i	$0.424953\pi$
$224$	0.569844	0.0380743
$225$	5.06726	0.337817
$226$	−0.158495	−0.0105429
$227$	5.98914	0.397513	0.198757	−	0.980049i	$-0.436310\pi$
$227$	5.98914	0.397513	0.198757	+	0.980049i	$0.436310\pi$
$228$	−13.8413	−0.916664
$229$	29.1165	1.92408	0.962038	−	0.272917i	$-0.0879885\pi$
$229$	29.1165	1.92408	0.962038	+	0.272917i	$0.0879885\pi$
$230$	−1.45848	−0.0961695
$231$	−0.364882	−0.0240075
$232$	0.462056	0.0303355
$233$	17.8894	1.17197	0.585987	−	0.810321i	$-0.300707\pi$
$233$	17.8894	1.17197	0.585987	+	0.810321i	$0.300707\pi$
$234$	0	0
$235$	−24.1184	−1.57331
$236$	−14.4505	−0.940650
$237$	8.66207	0.562662
$238$	−0.0477617	−0.00309593
$239$	11.1361	0.720333	0.360167	−	0.932888i	$-0.382720\pi$
$239$	11.1361	0.720333	0.360167	+	0.932888i	$0.382720\pi$
$240$	12.5273	0.808637
$241$	7.27606	0.468692	0.234346	−	0.972153i	$-0.424705\pi$
$241$	7.27606	0.468692	0.234346	+	0.972153i	$0.424705\pi$
$242$	0.975558	0.0627112
$243$	1.00000	0.0641500
$244$	−6.41530	−0.410698
$245$	−21.3725	−1.36544
$246$	−1.15286	−0.0735038
$247$	0	0
$248$	−3.98335	−0.252943
$249$	8.16003	0.517121
$250$	−0.0198356	−0.00125451
$251$	−11.1151	−0.701578	−0.350789	−	0.936455i	$-0.614087\pi$
$251$	−11.1151	−0.701578	−0.350789	+	0.936455i	$0.614087\pi$
$252$	1.02327	0.0644602
$253$	3.51170	0.220779
$254$	−1.31144	−0.0822874
$255$	−3.17289	−0.198694
$256$	15.3133	0.957083
$257$	−19.5951	−1.22231	−0.611153	−	0.791512i	$-0.709294\pi$
$257$	−19.5951	−1.22231	−0.611153	+	0.791512i	$0.709294\pi$
$258$	−0.547013	−0.0340555
$259$	−0.322198	−0.0200204
$260$	0	0
$261$	1.24548	0.0770931
$262$	1.73740	0.107337
$263$	−17.8319	−1.09956	−0.549782	−	0.835308i	$-0.685289\pi$
$263$	−17.8319	−1.09956	−0.549782	+	0.835308i	$0.685289\pi$
$264$	0.263433	0.0162132
$265$	5.65422	0.347336
$266$	0.331977	0.0203548
$267$	−13.5553	−0.829571
$268$	−29.0042	−1.77171
$269$	16.6026	1.01228	0.506139	−	0.862452i	$-0.331072\pi$
$269$	16.6026	1.01228	0.506139	+	0.862452i	$0.331072\pi$
$270$	−0.294913	−0.0179478
$271$	−7.22773	−0.439053	−0.219527	−	0.975607i	$-0.570451\pi$
$271$	−7.22773	−0.439053	−0.219527	+	0.975607i	$0.570451\pi$
$272$	−3.94824	−0.239397
$273$	0	0
$274$	0.357085	0.0215723
$275$	3.59818	0.216979
$276$	−9.84821	−0.592792
$277$	−25.4209	−1.52740	−0.763698	−	0.645574i	$-0.776618\pi$
$277$	−25.4209	−1.52740	−0.763698	+	0.645574i	$0.776618\pi$
$278$	0.505469	0.0303160
$279$	−10.7371	−0.642816
$280$	−0.604863	−0.0361475
$281$	15.4198	0.919870	0.459935	−	0.887953i	$-0.347873\pi$
$281$	15.4198	0.919870	0.459935	+	0.887953i	$0.347873\pi$
$282$	0.706531	0.0420733
$283$	11.1880	0.665056	0.332528	−	0.943093i	$-0.392098\pi$
$283$	11.1880	0.665056	0.332528	+	0.943093i	$0.392098\pi$
$284$	20.5905	1.22182
$285$	22.0538	1.30635
$286$	0	0
$287$	−6.37354	−0.376218
$288$	−1.10895	−0.0653458
$289$	1.00000	0.0588235
$290$	−0.367307	−0.0215690
$291$	4.22630	0.247750
$292$	26.7725	1.56674
$293$	15.1237	0.883536	0.441768	−	0.897129i	$-0.354351\pi$
$293$	15.1237	0.883536	0.441768	+	0.897129i	$0.354351\pi$
$294$	0.626091	0.0365144
$295$	23.0245	1.34054
$296$	0.232617	0.0135206
$297$	0.710085	0.0412033
$298$	−0.947917	−0.0549114
$299$	0	0
$300$	−10.0907	−0.582589
$301$	−3.02413	−0.174308
$302$	−0.940960	−0.0541461
$303$	3.01091	0.172972
$304$	27.4430	1.57396
$305$	10.2217	0.585293
$306$	0.0929476	0.00531346
$307$	15.3277	0.874796	0.437398	−	0.899268i	$-0.355900\pi$
$307$	15.3277	0.874796	0.437398	+	0.899268i	$0.355900\pi$
$308$	0.726611	0.0414025
$309$	6.96909	0.396458
$310$	3.16652	0.179846
$311$	−20.4430	−1.15921	−0.579607	−	0.814896i	$-0.696794\pi$
$311$	−20.4430	−1.15921	−0.579607	+	0.814896i	$0.696794\pi$
$312$	0	0
$313$	24.3682	1.37737	0.688687	−	0.725059i	$-0.258187\pi$
$313$	24.3682	1.37737	0.688687	+	0.725059i	$0.258187\pi$
$314$	0.342494	0.0193281
$315$	−1.63041	−0.0918633
$316$	−17.2493	−0.970349
$317$	13.9897	0.785740	0.392870	−	0.919594i	$-0.371482\pi$
$317$	13.9897	0.785740	0.392870	+	0.919594i	$0.371482\pi$
$318$	−0.165636	−0.00928842
$319$	0.884394	0.0495166
$320$	−24.7276	−1.38232
$321$	2.53170	0.141306
$322$	0.236204	0.0131631
$323$	−6.95069	−0.386747
$324$	−1.99136	−0.110631
$325$	0	0
$326$	−0.524014	−0.0290225
$327$	−10.4929	−0.580259
$328$	4.60149	0.254075
$329$	3.90602	0.215346
$330$	−0.209413	−0.0115278
$331$	−27.7717	−1.52647	−0.763235	−	0.646121i	$-0.776390\pi$
$331$	−27.7717	−1.52647	−0.763235	+	0.646121i	$0.776390\pi$
$332$	−16.2496	−0.891811
$333$	0.627020	0.0343605
$334$	−1.08923	−0.0595999
$335$	46.2132	2.52490
$336$	−2.02883	−0.110682
$337$	−3.77643	−0.205715	−0.102858	−	0.994696i	$-0.532799\pi$
$337$	−3.77643	−0.205715	−0.102858	+	0.994696i	$0.532799\pi$
$338$	0	0
$339$	1.70521	0.0926140
$340$	6.31838	0.342662
$341$	−7.62428	−0.412878
$342$	−0.646050	−0.0349344
$343$	7.05831	0.381113
$344$	2.18333	0.117717
$345$	15.6914	0.844799
$346$	0.552161	0.0296844
$347$	−28.3039	−1.51943	−0.759717	−	0.650254i	$-0.774663\pi$
$347$	−28.3039	−1.51943	−0.759717	+	0.650254i	$0.774663\pi$
$348$	−2.48019	−0.132952
$349$	−13.8054	−0.738984	−0.369492	−	0.929234i	$-0.620468\pi$
$349$	−13.8054	−0.738984	−0.369492	+	0.929234i	$0.620468\pi$
$350$	0.242021	0.0129366
$351$	0	0
$352$	−0.787452	−0.0419713
$353$	−25.0835	−1.33506	−0.667530	−	0.744583i	$-0.732648\pi$
$353$	−25.0835	−1.33506	−0.667530	+	0.744583i	$0.732648\pi$
$354$	−0.674485	−0.0358485
$355$	−32.8075	−1.74124
$356$	26.9935	1.43065
$357$	0.513856	0.0271962
$358$	0.109486	0.00578649
$359$	−5.58828	−0.294938	−0.147469	−	0.989067i	$-0.547113\pi$
$359$	−5.58828	−0.294938	−0.147469	+	0.989067i	$0.547113\pi$
$360$	1.17710	0.0620389
$361$	29.3121	1.54274
$362$	−0.810457	−0.0425967
$363$	−10.4958	−0.550886
$364$	0	0
$365$	−42.6574	−2.23279
$366$	−0.299437	−0.0156518
$367$	9.20267	0.480375	0.240188	−	0.970726i	$-0.422791\pi$
$367$	9.20267	0.480375	0.240188	+	0.970726i	$0.422791\pi$
$368$	19.5259	1.01786
$369$	12.4034	0.645693
$370$	−0.184916	−0.00961334
$371$	−0.915712	−0.0475414
$372$	21.3815	1.10858
$373$	3.11861	0.161475	0.0807377	−	0.996735i	$-0.474272\pi$
$373$	3.11861	0.161475	0.0807377	+	0.996735i	$0.474272\pi$
$374$	0.0660007	0.00341282
$375$	0.213406	0.0110202
$376$	−2.82002	−0.145431
$377$	0	0
$378$	0.0477617	0.00245660
$379$	−4.80123	−0.246622	−0.123311	−	0.992368i	$-0.539351\pi$
$379$	−4.80123	−0.246622	−0.123311	+	0.992368i	$0.539351\pi$
$380$	−43.9171	−2.25290
$381$	14.1095	0.722852
$382$	2.14982	0.109994
$383$	−16.6011	−0.848275	−0.424137	−	0.905598i	$-0.639423\pi$
$383$	−16.6011	−0.848275	−0.424137	+	0.905598i	$0.639423\pi$
$384$	2.94229	0.150148
$385$	−1.15773	−0.0590034
$386$	−1.20315	−0.0612386
$387$	5.88517	0.299160
$388$	−8.41608	−0.427262
$389$	28.4377	1.44185	0.720925	−	0.693013i	$-0.243717\pi$
$389$	28.4377	1.44185	0.720925	+	0.693013i	$0.243717\pi$
$390$	0	0
$391$	−4.94547	−0.250103
$392$	−2.49895	−0.126216
$393$	−18.6923	−0.942900
$394$	−2.16126	−0.108883
$395$	27.4838	1.38286
$396$	−1.41404	−0.0710579
$397$	10.4611	0.525028	0.262514	−	0.964928i	$-0.415448\pi$
$397$	10.4611	0.525028	0.262514	+	0.964928i	$0.415448\pi$
$398$	1.37108	0.0687262
$399$	−3.57166	−0.178806
$400$	20.0068	1.00034
$401$	1.88450	0.0941075	0.0470538	−	0.998892i	$-0.485017\pi$
$401$	1.88450	0.0941075	0.0470538	+	0.998892i	$0.485017\pi$
$402$	−1.35378	−0.0675205
$403$	0	0
$404$	−5.99581	−0.298303
$405$	3.17289	0.157662
$406$	0.0594862	0.00295225
$407$	0.445237	0.0220696
$408$	−0.370988	−0.0183666
$409$	2.08512	0.103103	0.0515513	−	0.998670i	$-0.483583\pi$
$409$	2.08512	0.103103	0.0515513	+	0.998670i	$0.483583\pi$
$410$	−3.65791	−0.180651
$411$	−3.84179	−0.189501
$412$	−13.8780	−0.683719
$413$	−3.72886	−0.183485
$414$	−0.459669	−0.0225915
$415$	25.8909	1.27094
$416$	0	0
$417$	−5.43821	−0.266310
$418$	−0.458750	−0.0224382
$419$	−25.6130	−1.25128	−0.625638	−	0.780114i	$-0.715161\pi$
$419$	−25.6130	−1.25128	−0.625638	+	0.780114i	$0.715161\pi$
$420$	3.24674	0.158425
$421$	−16.6900	−0.813419	−0.406710	−	0.913558i	$-0.633324\pi$
$421$	−16.6900	−0.813419	−0.406710	+	0.913558i	$0.633324\pi$
$422$	−2.14107	−0.104225
$423$	−7.60138	−0.369592
$424$	0.661114	0.0321065
$425$	−5.06726	−0.245798
$426$	0.961072	0.0465641
$427$	−1.65542	−0.0801116
$428$	−5.04154	−0.243692
$429$	0	0
$430$	−1.73561	−0.0836987
$431$	−15.6532	−0.753986	−0.376993	−	0.926216i	$-0.623042\pi$
$431$	−15.6532	−0.753986	−0.376993	+	0.926216i	$0.623042\pi$
$432$	3.94824	0.189960
$433$	−11.0119	−0.529201	−0.264600	−	0.964358i	$-0.585240\pi$
$433$	−11.0119	−0.529201	−0.264600	+	0.964358i	$0.585240\pi$
$434$	−0.512825	−0.0246164
$435$	3.95177	0.189473
$436$	20.8951	1.00070
$437$	34.3744	1.64435
$438$	1.24962	0.0597091
$439$	31.6240	1.50933	0.754666	−	0.656110i	$-0.227799\pi$
$439$	31.6240	1.50933	0.754666	+	0.656110i	$0.227799\pi$
$440$	0.835844	0.0398473
$441$	−6.73595	−0.320760
$442$	0	0
$443$	−32.6966	−1.55346	−0.776730	−	0.629833i	$-0.783123\pi$
$443$	−32.6966	−1.55346	−0.776730	+	0.629833i	$0.783123\pi$
$444$	−1.24862	−0.0592570
$445$	−43.0095	−2.03885
$446$	−0.648443	−0.0307047
$447$	10.1984	0.482368
$448$	4.00469	0.189204
$449$	16.9612	0.800449	0.400224	−	0.916417i	$-0.368932\pi$
$449$	16.9612	0.800449	0.400224	+	0.916417i	$0.368932\pi$
$450$	−0.470990	−0.0222027
$451$	8.80743	0.414726
$452$	−3.39568	−0.159719
$453$	10.1235	0.475646
$454$	−0.556677	−0.0261261
$455$	0	0
$456$	2.57862	0.120755
$457$	12.3211	0.576358	0.288179	−	0.957577i	$-0.406950\pi$
$457$	12.3211	0.576358	0.288179	+	0.957577i	$0.406950\pi$
$458$	−2.70631	−0.126458
$459$	−1.00000	−0.0466760
$460$	−31.2473	−1.45691
$461$	1.57749	0.0734712	0.0367356	−	0.999325i	$-0.488304\pi$
$461$	1.57749	0.0734712	0.0367356	+	0.999325i	$0.488304\pi$
$462$	0.0339149	0.00157786
$463$	−7.79713	−0.362363	−0.181182	−	0.983450i	$-0.557992\pi$
$463$	−7.79713	−0.362363	−0.181182	+	0.983450i	$0.557992\pi$
$464$	4.91744	0.228286
$465$	−34.0678	−1.57986
$466$	−1.66278	−0.0770267
$467$	−25.8165	−1.19464	−0.597322	−	0.802002i	$-0.703768\pi$
$467$	−25.8165	−1.19464	−0.597322	+	0.802002i	$0.703768\pi$
$468$	0	0
$469$	−7.48432	−0.345594
$470$	2.24175	0.103404
$471$	−3.68481	−0.169787
$472$	2.69211	0.123915
$473$	4.17897	0.192149
$474$	−0.805119	−0.0369803
$475$	35.2209	1.61605
$476$	−1.02327	−0.0469017
$477$	1.78204	0.0815939
$478$	−1.03507	−0.0473431
$479$	−4.11443	−0.187993	−0.0939965	−	0.995573i	$-0.529964\pi$
$479$	−4.11443	−0.187993	−0.0939965	+	0.995573i	$0.529964\pi$
$480$	−3.51860	−0.160601
$481$	0	0
$482$	−0.676292	−0.0308043
$483$	−2.54126	−0.115631
$484$	20.9009	0.950040
$485$	13.4096	0.608898
$486$	−0.0929476	−0.00421619
$487$	33.9931	1.54037	0.770187	−	0.637818i	$-0.220163\pi$
$487$	33.9931	1.54037	0.770187	+	0.637818i	$0.220163\pi$
$488$	1.19516	0.0541024
$489$	5.63774	0.254947
$490$	1.98652	0.0897418
$491$	−4.01426	−0.181161	−0.0905805	−	0.995889i	$-0.528872\pi$
$491$	−4.01426	−0.181161	−0.0905805	+	0.995889i	$0.528872\pi$
$492$	−24.6996	−1.11354
$493$	−1.24548	−0.0560935
$494$	0	0
$495$	2.25302	0.101266
$496$	−42.3928	−1.90349
$497$	5.31324	0.238331
$498$	−0.758456	−0.0339872
$499$	−10.8779	−0.486960	−0.243480	−	0.969906i	$-0.578289\pi$
$499$	−10.8779	−0.486960	−0.243480	+	0.969906i	$0.578289\pi$
$500$	−0.424969	−0.0190052
$501$	11.7187	0.523554
$502$	1.03312	0.0461104
$503$	33.8098	1.50751	0.753753	−	0.657158i	$-0.228242\pi$
$503$	33.8098	1.50751	0.753753	+	0.657158i	$0.228242\pi$
$504$	−0.190634	−0.00849153
$505$	9.55331	0.425117
$506$	−0.326404	−0.0145104
$507$	0	0
$508$	−28.0971	−1.24661
$509$	29.3839	1.30242	0.651210	−	0.758898i	$-0.274262\pi$
$509$	29.3839	1.30242	0.651210	+	0.758898i	$0.274262\pi$
$510$	0.294913	0.0130590
$511$	6.90846	0.305612
$512$	−7.30791	−0.322967
$513$	6.95069	0.306880
$514$	1.82132	0.0803348
$515$	22.1122	0.974379
$516$	−11.7195	−0.515922
$517$	−5.39763	−0.237387
$518$	0.0299476	0.00131582
$519$	−5.94056	−0.260762
$520$	0	0
$521$	21.8780	0.958494	0.479247	−	0.877680i	$-0.340910\pi$
$521$	21.8780	0.958494	0.479247	+	0.877680i	$0.340910\pi$
$522$	−0.115764	−0.00506686
$523$	−29.7241	−1.29975	−0.649873	−	0.760043i	$-0.725178\pi$
$523$	−29.7241	−1.29975	−0.649873	+	0.760043i	$0.725178\pi$
$524$	37.2231	1.62610
$525$	−2.60384	−0.113641
$526$	1.65744	0.0722677
$527$	10.7371	0.467717
$528$	2.80358	0.122010
$529$	1.45764	0.0633756
$530$	−0.525546	−0.0228283
$531$	7.25662	0.314910
$532$	7.11246	0.308364
$533$	0	0
$534$	1.25993	0.0545226
$535$	8.03283	0.347290
$536$	5.40343	0.233393
$537$	−1.17793	−0.0508313
$538$	−1.54317	−0.0665309
$539$	−4.78310	−0.206023
$540$	−6.31838	−0.271900
$541$	32.2930	1.38838	0.694192	−	0.719790i	$-0.255762\pi$
$541$	32.2930	1.38838	0.694192	+	0.719790i	$0.255762\pi$
$542$	0.671800	0.0288563
$543$	8.71950	0.374190
$544$	1.10895	0.0475460
$545$	−33.2929	−1.42611
$546$	0	0
$547$	20.7225	0.886031	0.443015	−	0.896514i	$-0.353909\pi$
$547$	20.7225	0.886031	0.443015	+	0.896514i	$0.353909\pi$
$548$	7.65039	0.326808
$549$	3.22157	0.137493
$550$	−0.334443	−0.0142607
$551$	8.65692	0.368797
$552$	1.83471	0.0780903
$553$	−4.45106	−0.189278
$554$	2.36282	0.100386
$555$	1.98947	0.0844482
$556$	10.8294	0.459271
$557$	−21.9183	−0.928710	−0.464355	−	0.885649i	$-0.653714\pi$
$557$	−21.9183	−0.928710	−0.464355	+	0.885649i	$0.653714\pi$
$558$	0.997992	0.0422484
$559$	0	0
$560$	−6.43726	−0.272024
$561$	−0.710085	−0.0299798
$562$	−1.43324	−0.0604574
$563$	−31.1228	−1.31167	−0.655835	−	0.754904i	$-0.727683\pi$
$563$	−31.1228	−1.31167	−0.655835	+	0.754904i	$0.727683\pi$
$564$	15.1371	0.637387
$565$	5.41044	0.227619
$566$	−1.03990	−0.0437101
$567$	−0.513856	−0.0215799
$568$	−3.83599	−0.160954
$569$	25.3193	1.06144	0.530719	−	0.847548i	$-0.321922\pi$
$569$	25.3193	1.06144	0.530719	+	0.847548i	$0.321922\pi$
$570$	−2.04985	−0.0858587
$571$	−42.8079	−1.79145	−0.895727	−	0.444604i	$-0.853345\pi$
$571$	−42.8079	−1.79145	−0.895727	+	0.444604i	$0.853345\pi$
$572$	0	0
$573$	−23.1293	−0.966241
$574$	0.592406	0.0247265
$575$	25.0600	1.04507
$576$	−7.79340	−0.324725
$577$	10.9462	0.455695	0.227847	−	0.973697i	$-0.426831\pi$
$577$	10.9462	0.455695	0.227847	+	0.973697i	$0.426831\pi$
$578$	−0.0929476	−0.00386611
$579$	12.9444	0.537949
$580$	−7.86939	−0.326759
$581$	−4.19309	−0.173959
$582$	−0.392824	−0.0162831
$583$	1.26540	0.0524074
$584$	−4.98768	−0.206392
$585$	0	0
$586$	−1.40571	−0.0580694
$587$	46.0068	1.89890	0.949452	−	0.313912i	$-0.101640\pi$
$587$	46.0068	1.89890	0.949452	+	0.313912i	$0.101640\pi$
$588$	13.4137	0.553172
$589$	−74.6305	−3.07510
$590$	−2.14007	−0.0881053
$591$	23.2525	0.956480
$592$	2.47562	0.101748
$593$	22.8488	0.938287	0.469144	−	0.883122i	$-0.344563\pi$
$593$	22.8488	0.938287	0.469144	+	0.883122i	$0.344563\pi$
$594$	−0.0660007	−0.00270804
$595$	1.63041	0.0668404
$596$	−20.3087	−0.831876
$597$	−14.7511	−0.603724
$598$	0	0
$599$	28.4170	1.16109	0.580543	−	0.814229i	$-0.302840\pi$
$599$	28.4170	1.16109	0.580543	+	0.814229i	$0.302840\pi$
$600$	1.87989	0.0767462
$601$	−18.1852	−0.741788	−0.370894	−	0.928675i	$-0.620949\pi$
$601$	−18.1852	−0.741788	−0.370894	+	0.928675i	$0.620949\pi$
$602$	0.281086	0.0114562
$603$	14.5650	0.593132
$604$	−20.1596	−0.820284
$605$	−33.3020	−1.35392
$606$	−0.279857	−0.0113684
$607$	33.4550	1.35790	0.678949	−	0.734185i	$-0.262436\pi$
$607$	33.4550	1.35790	0.678949	+	0.734185i	$0.262436\pi$
$608$	−7.70800	−0.312601
$609$	−0.639996	−0.0259340
$610$	−0.950083	−0.0384677
$611$	0	0
$612$	1.99136	0.0804960
$613$	−17.9017	−0.723042	−0.361521	−	0.932364i	$-0.617742\pi$
$613$	−17.9017	−0.723042	−0.361521	+	0.932364i	$0.617742\pi$
$614$	−1.42467	−0.0574950
$615$	39.3545	1.58693
$616$	−0.135367	−0.00545407
$617$	−10.8918	−0.438487	−0.219244	−	0.975670i	$-0.570359\pi$
$617$	−10.8918	−0.438487	−0.219244	+	0.975670i	$0.570359\pi$
$618$	−0.647760	−0.0260567
$619$	−7.88332	−0.316858	−0.158429	−	0.987370i	$-0.550643\pi$
$619$	−7.88332	−0.316858	−0.158429	+	0.987370i	$0.550643\pi$
$620$	67.8413	2.72457
$621$	4.94547	0.198455
$622$	1.90013	0.0761881
$623$	6.96548	0.279066
$624$	0	0
$625$	−24.6592	−0.986367
$626$	−2.26497	−0.0905264
$627$	4.93558	0.197108
$628$	7.33778	0.292809
$629$	−0.627020	−0.0250009
$630$	0.151543	0.00603762
$631$	30.4418	1.21187	0.605934	−	0.795515i	$-0.292799\pi$
$631$	30.4418	1.21187	0.605934	+	0.795515i	$0.292799\pi$
$632$	3.21352	0.127827
$633$	23.0352	0.915566
$634$	−1.30031	−0.0516419
$635$	44.7680	1.77656
$636$	−3.54868	−0.140714
$637$	0	0
$638$	−0.0822024	−0.00325442
$639$	−10.3399	−0.409041
$640$	9.33557	0.369021
$641$	−8.97288	−0.354407	−0.177204	−	0.984174i	$-0.556705\pi$
$641$	−8.97288	−0.354407	−0.177204	+	0.984174i	$0.556705\pi$
$642$	−0.235316	−0.00928718
$643$	−36.6642	−1.44590	−0.722948	−	0.690903i	$-0.757213\pi$
$643$	−36.6642	−1.44590	−0.722948	+	0.690903i	$0.757213\pi$
$644$	5.06057	0.199414
$645$	18.6730	0.735250
$646$	0.646050	0.0254185
$647$	−12.7489	−0.501212	−0.250606	−	0.968089i	$-0.580630\pi$
$647$	−12.7489	−0.501212	−0.250606	+	0.968089i	$0.580630\pi$
$648$	0.370988	0.0145738
$649$	5.15281	0.202266
$650$	0	0
$651$	5.51735	0.216242
$652$	−11.2268	−0.439674
$653$	29.2046	1.14286	0.571432	−	0.820649i	$-0.306388\pi$
$653$	29.2046	1.14286	0.571432	+	0.820649i	$0.306388\pi$
$654$	0.975290	0.0381369
$655$	−59.3086	−2.31738
$656$	48.9714	1.91201
$657$	−13.4443	−0.524513
$658$	−0.363055	−0.0141534
$659$	14.0648	0.547886	0.273943	−	0.961746i	$-0.411672\pi$
$659$	14.0648	0.547886	0.273943	+	0.961746i	$0.411672\pi$
$660$	−4.48658	−0.174640
$661$	9.35689	0.363941	0.181971	−	0.983304i	$-0.441752\pi$
$661$	9.35689	0.363941	0.181971	+	0.983304i	$0.441752\pi$
$662$	2.58131	0.100326
$663$	0	0
$664$	3.02727	0.117481
$665$	−11.3325	−0.439455
$666$	−0.0582800	−0.00225830
$667$	6.15946	0.238495
$668$	−23.3362	−0.902905
$669$	6.97643	0.269725
$670$	−4.29541	−0.165946
$671$	2.28759	0.0883113
$672$	0.569844	0.0219822
$673$	−35.1658	−1.35554	−0.677771	−	0.735273i	$-0.737054\pi$
$673$	−35.1658	−1.35554	−0.677771	+	0.735273i	$0.737054\pi$
$674$	0.351010	0.0135204
$675$	5.06726	0.195039
$676$	0	0
$677$	20.3276	0.781254	0.390627	−	0.920549i	$-0.372258\pi$
$677$	20.3276	0.781254	0.390627	+	0.920549i	$0.372258\pi$
$678$	−0.158495	−0.00608696
$679$	−2.17171	−0.0833426
$680$	−1.17710	−0.0451399
$681$	5.98914	0.229504
$682$	0.708659	0.0271360
$683$	17.9018	0.684992	0.342496	−	0.939519i	$-0.388728\pi$
$683$	17.9018	0.684992	0.342496	+	0.939519i	$0.388728\pi$
$684$	−13.8413	−0.529236
$685$	−12.1896	−0.465740
$686$	−0.656053	−0.0250482
$687$	29.1165	1.11087
$688$	23.2361	0.885867
$689$	0	0
$690$	−1.45848	−0.0555235
$691$	25.5137	0.970585	0.485293	−	0.874352i	$-0.338713\pi$
$691$	25.5137	0.970585	0.485293	+	0.874352i	$0.338713\pi$
$692$	11.8298	0.449702
$693$	−0.364882	−0.0138607
$694$	2.63078	0.0998632
$695$	−17.2549	−0.654514
$696$	0.462056	0.0175142
$697$	−12.4034	−0.469811
$698$	1.28317	0.0485689
$699$	17.8894	0.676639
$700$	5.18519	0.195982
$701$	−0.249197	−0.00941203	−0.00470602	−	0.999989i	$-0.501498\pi$
$701$	−0.249197	−0.00941203	−0.00470602	+	0.999989i	$0.501498\pi$
$702$	0	0
$703$	4.35822	0.164373
$704$	−5.53398	−0.208570
$705$	−24.1184	−0.908351
$706$	2.33145	0.0877454
$707$	−1.54718	−0.0581876
$708$	−14.4505	−0.543085
$709$	16.0650	0.603333	0.301666	−	0.953414i	$-0.402457\pi$
$709$	16.0650	0.603333	0.301666	+	0.953414i	$0.402457\pi$
$710$	3.04938	0.114441
$711$	8.66207	0.324853
$712$	−5.02885	−0.188464
$713$	−53.1002	−1.98862
$714$	−0.0477617	−0.00178744
$715$	0	0
$716$	2.34568	0.0876621
$717$	11.1361	0.415885
$718$	0.519417	0.0193845
$719$	16.9451	0.631946	0.315973	−	0.948768i	$-0.397669\pi$
$719$	16.9451	0.631946	0.315973	+	0.948768i	$0.397669\pi$
$720$	12.5273	0.466867
$721$	−3.58111	−0.133368
$722$	−2.72449	−0.101395
$723$	7.27606	0.270599
$724$	−17.3637	−0.645316
$725$	6.31115	0.234390
$726$	0.975558	0.0362064
$727$	−28.4184	−1.05398	−0.526990	−	0.849872i	$-0.676679\pi$
$727$	−28.4184	−1.05398	−0.526990	+	0.849872i	$0.676679\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	3.96491	0.146748
$731$	−5.88517	−0.217671
$732$	−6.41530	−0.237116
$733$	5.39218	0.199165	0.0995823	−	0.995029i	$-0.468249\pi$
$733$	5.39218	0.199165	0.0995823	+	0.995029i	$0.468249\pi$
$734$	−0.855367	−0.0315722
$735$	−21.3725	−0.788335
$736$	−5.48430	−0.202154
$737$	10.3424	0.380967
$738$	−1.15286	−0.0424375
$739$	−27.7119	−1.01940	−0.509699	−	0.860353i	$-0.670243\pi$
$739$	−27.7119	−1.01940	−0.509699	+	0.860353i	$0.670243\pi$
$740$	−3.96175	−0.145637
$741$	0	0
$742$	0.0851132	0.00312461
$743$	34.0737	1.25004	0.625022	−	0.780607i	$-0.285090\pi$
$743$	34.0737	1.25004	0.625022	+	0.780607i	$0.285090\pi$
$744$	−3.98335	−0.146037
$745$	32.3584	1.18552
$746$	−0.289867	−0.0106128
$747$	8.16003	0.298560
$748$	1.41404	0.0517022
$749$	−1.30093	−0.0475350
$750$	−0.0198356	−0.000724294 0
$751$	−13.3233	−0.486176	−0.243088	−	0.970004i	$-0.578160\pi$
$751$	−13.3233	−0.486176	−0.243088	+	0.970004i	$0.578160\pi$
$752$	−30.0121	−1.09443
$753$	−11.1151	−0.405056
$754$	0	0
$755$	32.1209	1.16900
$756$	1.02327	0.0372161
$757$	23.0125	0.836404	0.418202	−	0.908354i	$-0.362660\pi$
$757$	23.0125	0.836404	0.418202	+	0.908354i	$0.362660\pi$
$758$	0.446263	0.0162090
$759$	3.51170	0.127467
$760$	8.18169	0.296781
$761$	−20.2044	−0.732409	−0.366205	−	0.930534i	$-0.619343\pi$
$761$	−20.2044	−0.732409	−0.366205	+	0.930534i	$0.619343\pi$
$762$	−1.31144	−0.0475086
$763$	5.39184	0.195198
$764$	46.0588	1.66635
$765$	−3.17289	−0.114716
$766$	1.54303	0.0557519
$767$	0	0
$768$	15.3133	0.552572
$769$	9.05451	0.326514	0.163257	−	0.986584i	$-0.447800\pi$
$769$	9.05451	0.326514	0.163257	+	0.986584i	$0.447800\pi$
$770$	0.107608	0.00387794
$771$	−19.5951	−0.705699
$772$	−25.7769	−0.927730
$773$	45.1643	1.62445	0.812224	−	0.583346i	$-0.198257\pi$
$773$	45.1643	1.62445	0.812224	+	0.583346i	$0.198257\pi$
$774$	−0.547013	−0.0196620
$775$	−54.4079	−1.95439
$776$	1.56790	0.0562845
$777$	−0.322198	−0.0115588
$778$	−2.64322	−0.0947641
$779$	86.2119	3.08886
$780$	0	0
$781$	−7.34223	−0.262726
$782$	0.459669	0.0164377
$783$	1.24548	0.0445097
$784$	−26.5951	−0.949827
$785$	−11.6915	−0.417288
$786$	1.73740	0.0619711
$787$	−22.0025	−0.784306	−0.392153	−	0.919900i	$-0.628270\pi$
$787$	−22.0025	−0.784306	−0.392153	+	0.919900i	$0.628270\pi$
$788$	−46.3041	−1.64952
$789$	−17.8319	−0.634834
$790$	−2.55456	−0.0908871
$791$	−0.876231	−0.0311552
$792$	0.263433	0.00936067
$793$	0	0
$794$	−0.972336	−0.0345069
$795$	5.65422	0.200534
$796$	29.3748	1.04116
$797$	31.7813	1.12575	0.562876	−	0.826542i	$-0.309695\pi$
$797$	31.7813	1.12575	0.562876	+	0.826542i	$0.309695\pi$
$798$	0.331977	0.0117519
$799$	7.60138	0.268918
$800$	−5.61936	−0.198674
$801$	−13.5553	−0.478953
$802$	−0.175160	−0.00618511
$803$	−9.54662	−0.336893
$804$	−29.0042	−1.02290
$805$	−8.06315	−0.284189
$806$	0	0
$807$	16.6026	0.584439
$808$	1.11701	0.0392963
$809$	−14.7642	−0.519081	−0.259540	−	0.965732i	$-0.583571\pi$
$809$	−14.7642	−0.519081	−0.259540	+	0.965732i	$0.583571\pi$
$810$	−0.294913	−0.0103622
$811$	0.413947	0.0145357	0.00726783	−	0.999974i	$-0.497687\pi$
$811$	0.413947	0.0145357	0.00726783	+	0.999974i	$0.497687\pi$
$812$	1.27446	0.0447249
$813$	−7.22773	−0.253487
$814$	−0.0413838	−0.00145050
$815$	17.8879	0.626587
$816$	−3.94824	−0.138216
$817$	40.9060	1.43112
$818$	−0.193807	−0.00677631
$819$	0	0
$820$	−78.3691	−2.73677
$821$	−4.53178	−0.158160	−0.0790801	−	0.996868i	$-0.525198\pi$
$821$	−4.53178	−0.158160	−0.0790801	+	0.996868i	$0.525198\pi$
$822$	0.357085	0.0124548
$823$	−0.721512	−0.0251503	−0.0125752	−	0.999921i	$-0.504003\pi$
$823$	−0.721512	−0.0251503	−0.0125752	+	0.999921i	$0.504003\pi$
$824$	2.58545	0.0900683
$825$	3.59818	0.125273
$826$	0.346589	0.0120594
$827$	11.2099	0.389806	0.194903	−	0.980823i	$-0.437561\pi$
$827$	11.2099	0.389806	0.194903	+	0.980823i	$0.437561\pi$
$828$	−9.84821	−0.342249
$829$	10.3215	0.358482	0.179241	−	0.983805i	$-0.442636\pi$
$829$	10.3215	0.358482	0.179241	+	0.983805i	$0.442636\pi$
$830$	−2.40650	−0.0835309
$831$	−25.4209	−0.881842
$832$	0	0
$833$	6.73595	0.233387
$834$	0.505469	0.0175030
$835$	37.1823	1.28675
$836$	−9.82852	−0.339926
$837$	−10.7371	−0.371130
$838$	2.38066	0.0822387
$839$	45.6743	1.57685	0.788427	−	0.615129i	$-0.210896\pi$
$839$	45.6743	1.57685	0.788427	+	0.615129i	$0.210896\pi$
$840$	−0.604863	−0.0208697
$841$	−27.4488	−0.946510
$842$	1.55129	0.0534611
$843$	15.4198	0.531087
$844$	−45.8713	−1.57896
$845$	0	0
$846$	0.706531	0.0242910
$847$	5.39332	0.185317
$848$	7.03591	0.241614
$849$	11.1880	0.383970
$850$	0.470990	0.0161548
$851$	3.10091	0.106298
$852$	20.5905	0.705420
$853$	−33.4854	−1.14652	−0.573260	−	0.819374i	$-0.694321\pi$
$853$	−33.4854	−1.14652	−0.573260	+	0.819374i	$0.694321\pi$
$854$	0.153868	0.00526525
$855$	22.0538	0.754224
$856$	0.939231	0.0321022
$857$	−35.4841	−1.21211	−0.606056	−	0.795422i	$-0.707249\pi$
$857$	−35.4841	−1.21211	−0.606056	+	0.795422i	$0.707249\pi$
$858$	0	0
$859$	−40.0729	−1.36727	−0.683634	−	0.729825i	$-0.739602\pi$
$859$	−40.0729	−1.36727	−0.683634	+	0.729825i	$0.739602\pi$
$860$	−37.1847	−1.26799
$861$	−6.37354	−0.217210
$862$	1.45492	0.0495549
$863$	27.3341	0.930463	0.465232	−	0.885189i	$-0.345971\pi$
$863$	27.3341	0.930463	0.465232	+	0.885189i	$0.345971\pi$
$864$	−1.10895	−0.0377274
$865$	−18.8488	−0.640878
$866$	1.02353	0.0347811
$867$	1.00000	0.0339618
$868$	−10.9870	−0.372924
$869$	6.15080	0.208652
$870$	−0.367307	−0.0124529
$871$	0	0
$872$	−3.89274	−0.131825
$873$	4.22630	0.143038
$874$	−3.19502	−0.108073
$875$	−0.109660	−0.00370719
$876$	26.7725	0.904560
$877$	−53.6867	−1.81287	−0.906435	−	0.422345i	$-0.861207\pi$
$877$	−53.6867	−1.81287	−0.906435	+	0.422345i	$0.861207\pi$
$878$	−2.93938	−0.0991992
$879$	15.1237	0.510110
$880$	8.89548	0.299867
$881$	0.887871	0.0299131	0.0149566	−	0.999888i	$-0.495239\pi$
$881$	0.887871	0.0299131	0.0149566	+	0.999888i	$0.495239\pi$
$882$	0.626091	0.0210816
$883$	16.3629	0.550656	0.275328	−	0.961350i	$-0.411214\pi$
$883$	16.3629	0.550656	0.275328	+	0.961350i	$0.411214\pi$
$884$	0	0
$885$	23.0245	0.773959
$886$	3.03907	0.102099
$887$	−19.3307	−0.649063	−0.324531	−	0.945875i	$-0.605207\pi$
$887$	−19.3307	−0.649063	−0.324531	+	0.945875i	$0.605207\pi$
$888$	0.232617	0.00780610
$889$	−7.25026	−0.243166
$890$	3.99764	0.134001
$891$	0.710085	0.0237887
$892$	−13.8926	−0.465159
$893$	−52.8348	−1.76805
$894$	−0.947917	−0.0317031
$895$	−3.73744	−0.124929
$896$	−1.51191	−0.0505095
$897$	0	0
$898$	−1.57650	−0.0526086
$899$	−13.3729	−0.446010
$900$	−10.0907	−0.336358
$901$	−1.78204	−0.0593683
$902$	−0.818630	−0.0272574
$903$	−3.02413	−0.100637
$904$	0.632610	0.0210403
$905$	27.6661	0.919651
$906$	−0.940960	−0.0312613
$907$	−52.2898	−1.73625	−0.868127	−	0.496342i	$-0.834676\pi$
$907$	−52.2898	−1.73625	−0.868127	+	0.496342i	$0.834676\pi$
$908$	−11.9265	−0.395796
$909$	3.01091	0.0998657
$910$	0	0
$911$	19.4555	0.644588	0.322294	−	0.946640i	$-0.395546\pi$
$911$	19.4555	0.644588	0.322294	+	0.946640i	$0.395546\pi$
$912$	27.4430	0.908728
$913$	5.79432	0.191764
$914$	−1.14522	−0.0378805
$915$	10.2217	0.337919
$916$	−57.9815	−1.91576
$917$	9.60515	0.317190
$918$	0.0929476	0.00306773
$919$	16.9897	0.560437	0.280219	−	0.959936i	$-0.409593\pi$
$919$	16.9897	0.560437	0.280219	+	0.959936i	$0.409593\pi$
$920$	5.82133	0.191924
$921$	15.3277	0.505064
$922$	−0.146624	−0.00482881
$923$	0	0
$924$	0.726611	0.0239037
$925$	3.17727	0.104468
$926$	0.724725	0.0238159
$927$	6.96909	0.228895
$928$	−1.38118	−0.0453394
$929$	52.0044	1.70621	0.853104	−	0.521741i	$-0.174717\pi$
$929$	52.0044	1.70621	0.853104	+	0.521741i	$0.174717\pi$
$930$	3.16652	0.103834
$931$	−46.8195	−1.53445
$932$	−35.6242	−1.16691
$933$	−20.4430	−0.669273
$934$	2.39958	0.0785166
$935$	−2.25302	−0.0736818
$936$	0	0
$937$	57.9027	1.89160	0.945800	−	0.324751i	$-0.105280\pi$
$937$	57.9027	1.89160	0.945800	+	0.324751i	$0.105280\pi$
$938$	0.695650	0.0227138
$939$	24.3682	0.795228
$940$	48.0284	1.56651
$941$	19.9637	0.650799	0.325400	−	0.945577i	$-0.394501\pi$
$941$	19.9637	0.650799	0.325400	+	0.945577i	$0.394501\pi$
$942$	0.342494	0.0111591
$943$	61.3404	1.99752
$944$	28.6509	0.932506
$945$	−1.63041	−0.0530373
$946$	−0.388426	−0.0126288
$947$	−1.44437	−0.0469356	−0.0234678	−	0.999725i	$-0.507471\pi$
$947$	−1.44437	−0.0469356	−0.0234678	+	0.999725i	$0.507471\pi$
$948$	−17.2493	−0.560231
$949$	0	0
$950$	−3.27370	−0.106213
$951$	13.9897	0.453647
$952$	0.190634	0.00617850
$953$	−29.2363	−0.947057	−0.473528	−	0.880779i	$-0.657020\pi$
$953$	−29.2363	−0.947057	−0.473528	+	0.880779i	$0.657020\pi$
$954$	−0.165636	−0.00536267
$955$	−73.3869	−2.37474
$956$	−22.1760	−0.717222
$957$	0.884394	0.0285884
$958$	0.382426	0.0123556
$959$	1.97413	0.0637479
$960$	−24.7276	−0.798081
$961$	84.2862	2.71891
$962$	0	0
$963$	2.53170	0.0815830
$964$	−14.4893	−0.466667
$965$	41.0711	1.32212
$966$	0.236204	0.00759974
$967$	−18.3197	−0.589122	−0.294561	−	0.955633i	$-0.595173\pi$
$967$	−18.3197	−0.589122	−0.294561	+	0.955633i	$0.595173\pi$
$968$	−3.89380	−0.125152
$969$	−6.95069	−0.223288
$970$	−1.24639	−0.0400192
$971$	13.8769	0.445331	0.222665	−	0.974895i	$-0.428524\pi$
$971$	13.8769	0.445331	0.222665	+	0.974895i	$0.428524\pi$
$972$	−1.99136	−0.0638729
$973$	2.79446	0.0895863
$974$	−3.15958	−0.101239
$975$	0	0
$976$	12.7195	0.407142
$977$	3.63021	0.116141	0.0580704	−	0.998312i	$-0.481505\pi$
$977$	3.63021	0.116141	0.0580704	+	0.998312i	$0.481505\pi$
$978$	−0.524014	−0.0167561
$979$	−9.62541	−0.307630
$980$	42.5603	1.35954
$981$	−10.4929	−0.335012
$982$	0.373116	0.0119066
$983$	24.5856	0.784160	0.392080	−	0.919931i	$-0.371756\pi$
$983$	24.5856	0.784160	0.392080	+	0.919931i	$0.371756\pi$
$984$	4.60149	0.146690
$985$	73.7777	2.35075
$986$	0.115764	0.00368668
$987$	3.90602	0.124330
$988$	0	0
$989$	29.1049	0.925483
$990$	−0.209413	−0.00665559
$991$	−21.1201	−0.670903	−0.335451	−	0.942058i	$-0.608889\pi$
$991$	−21.1201	−0.670903	−0.335451	+	0.942058i	$0.608889\pi$
$992$	11.9070	0.378048
$993$	−27.7717	−0.881308
$994$	−0.493853	−0.0156641
$995$	−46.8038	−1.48378
$996$	−16.2496	−0.514887
$997$	33.6508	1.06573	0.532866	−	0.846200i	$-0.321115\pi$
$997$	33.6508	1.06573	0.532866	+	0.846200i	$0.321115\pi$
$998$	1.01107	0.0320049
$999$	0.627020	0.0198380

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8619.2.a.bq.1.8	yes	21
13.12	even	2		8619.2.a.bp.1.14	✓	21

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8619.2.a.bp.1.14	✓	21	13.12	even	2
8619.2.a.bq.1.8	yes	21	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:	\( N \)	\(=\)	\( 8619 = 3 \cdot 13^{2} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8619.a (trivial)

\(f(q)\)	\(=\)	\(q-0.0929476 q^{2} +1.00000 q^{3} -1.99136 q^{4} +3.17289 q^{5} -0.0929476 q^{6} -0.513856 q^{7} +0.370988 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-0.0929476 q^{2} +1.00000 q^{3} -1.99136 q^{4} +3.17289 q^{5} -0.0929476 q^{6} -0.513856 q^{7} +0.370988 q^{8} +1.00000 q^{9} -0.294913 q^{10} +0.710085 q^{11} -1.99136 q^{12} +0.0477617 q^{14} +3.17289 q^{15} +3.94824 q^{16} -1.00000 q^{17} -0.0929476 q^{18} +6.95069 q^{19} -6.31838 q^{20} -0.513856 q^{21} -0.0660007 q^{22} +4.94547 q^{23} +0.370988 q^{24} +5.06726 q^{25} +1.00000 q^{27} +1.02327 q^{28} +1.24548 q^{29} -0.294913 q^{30} -10.7371 q^{31} -1.10895 q^{32} +0.710085 q^{33} +0.0929476 q^{34} -1.63041 q^{35} -1.99136 q^{36} +0.627020 q^{37} -0.646050 q^{38} +1.17710 q^{40} +12.4034 q^{41} +0.0477617 q^{42} +5.88517 q^{43} -1.41404 q^{44} +3.17289 q^{45} -0.459669 q^{46} -7.60138 q^{47} +3.94824 q^{48} -6.73595 q^{49} -0.470990 q^{50} -1.00000 q^{51} +1.78204 q^{53} -0.0929476 q^{54} +2.25302 q^{55} -0.190634 q^{56} +6.95069 q^{57} -0.115764 q^{58} +7.25662 q^{59} -6.31838 q^{60} +3.22157 q^{61} +0.997992 q^{62} -0.513856 q^{63} -7.79340 q^{64} -0.0660007 q^{66} +14.5650 q^{67} +1.99136 q^{68} +4.94547 q^{69} +0.151543 q^{70} -10.3399 q^{71} +0.370988 q^{72} -13.4443 q^{73} -0.0582800 q^{74} +5.06726 q^{75} -13.8413 q^{76} -0.364882 q^{77} +8.66207 q^{79} +12.5273 q^{80} +1.00000 q^{81} -1.15286 q^{82} +8.16003 q^{83} +1.02327 q^{84} -3.17289 q^{85} -0.547013 q^{86} +1.24548 q^{87} +0.263433 q^{88} -13.5553 q^{89} -0.294913 q^{90} -9.84821 q^{92} -10.7371 q^{93} +0.706531 q^{94} +22.0538 q^{95} -1.10895 q^{96} +4.22630 q^{97} +0.626091 q^{98} +0.710085 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 21 q + 9 q^{2} + 21 q^{3} + 29 q^{4} + 26 q^{5} + 9 q^{6} - 5 q^{7} + 24 q^{8} + 21 q^{9}+O(q^{10}) \) 21 * q + 9 * q^2 + 21 * q^3 + 29 * q^4 + 26 * q^5 + 9 * q^6 - 5 * q^7 + 24 * q^8 + 21 * q^9	\( 21 q + 9 q^{2} + 21 q^{3} + 29 q^{4} + 26 q^{5} + 9 q^{6} - 5 q^{7} + 24 q^{8} + 21 q^{9} + 12 q^{10} + 37 q^{11} + 29 q^{12} - 6 q^{14} + 26 q^{15} + 33 q^{16} - 21 q^{17} + 9 q^{18} - 6 q^{19} + 53 q^{20} - 5 q^{21} + 17 q^{22} - 8 q^{23} + 24 q^{24} + 57 q^{25} + 21 q^{27} + 5 q^{28} - 6 q^{29} + 12 q^{30} + 9 q^{31} + 37 q^{32} + 37 q^{33} - 9 q^{34} - 8 q^{35} + 29 q^{36} + 9 q^{37} + 11 q^{38} - 8 q^{40} + 50 q^{41} - 6 q^{42} + 18 q^{43} + 67 q^{44} + 26 q^{45} + 23 q^{46} + 71 q^{47} + 33 q^{48} + 46 q^{49} + 2 q^{50} - 21 q^{51} - 14 q^{53} + 9 q^{54} + 29 q^{55} - 17 q^{56} - 6 q^{57} + 37 q^{58} + 59 q^{59} + 53 q^{60} + 44 q^{61} + 2 q^{62} - 5 q^{63} + 44 q^{64} + 17 q^{66} + 8 q^{67} - 29 q^{68} - 8 q^{69} + 15 q^{70} + 60 q^{71} + 24 q^{72} + 13 q^{73} - 14 q^{74} + 57 q^{75} + 15 q^{76} - 28 q^{77} + 51 q^{79} + 131 q^{80} + 21 q^{81} + 18 q^{82} + 30 q^{83} + 5 q^{84} - 26 q^{85} - 15 q^{86} - 6 q^{87} + 71 q^{88} + 62 q^{89} + 12 q^{90} + 20 q^{92} + 9 q^{93} - 72 q^{94} - 16 q^{95} + 37 q^{96} - 6 q^{97} + 58 q^{98} + 37 q^{99}+O(q^{100}) \) 21 * q + 9 * q^2 + 21 * q^3 + 29 * q^4 + 26 * q^5 + 9 * q^6 - 5 * q^7 + 24 * q^8 + 21 * q^9 + 12 * q^10 + 37 * q^11 + 29 * q^12 - 6 * q^14 + 26 * q^15 + 33 * q^16 - 21 * q^17 + 9 * q^18 - 6 * q^19 + 53 * q^20 - 5 * q^21 + 17 * q^22 - 8 * q^23 + 24 * q^24 + 57 * q^25 + 21 * q^27 + 5 * q^28 - 6 * q^29 + 12 * q^30 + 9 * q^31 + 37 * q^32 + 37 * q^33 - 9 * q^34 - 8 * q^35 + 29 * q^36 + 9 * q^37 + 11 * q^38 - 8 * q^40 + 50 * q^41 - 6 * q^42 + 18 * q^43 + 67 * q^44 + 26 * q^45 + 23 * q^46 + 71 * q^47 + 33 * q^48 + 46 * q^49 + 2 * q^50 - 21 * q^51 - 14 * q^53 + 9 * q^54 + 29 * q^55 - 17 * q^56 - 6 * q^57 + 37 * q^58 + 59 * q^59 + 53 * q^60 + 44 * q^61 + 2 * q^62 - 5 * q^63 + 44 * q^64 + 17 * q^66 + 8 * q^67 - 29 * q^68 - 8 * q^69 + 15 * q^70 + 60 * q^71 + 24 * q^72 + 13 * q^73 - 14 * q^74 + 57 * q^75 + 15 * q^76 - 28 * q^77 + 51 * q^79 + 131 * q^80 + 21 * q^81 + 18 * q^82 + 30 * q^83 + 5 * q^84 - 26 * q^85 - 15 * q^86 - 6 * q^87 + 71 * q^88 + 62 * q^89 + 12 * q^90 + 20 * q^92 + 9 * q^93 - 72 * q^94 - 16 * q^95 + 37 * q^96 - 6 * q^97 + 58 * q^98 + 37 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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