LMFDB - Embedded newform 8470.2.a.cm.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8470,2,Mod(1,8470)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8470, 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("8470.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8470 = 2 \cdot 5 \cdot 7 \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8470.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:	$67.6332905120$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.733.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 7x + 8 $ x^3 - x^2 - 7*x + 8
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$1.17819$ of defining polynomial
Character	$\chi$	$=$	8470.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} +3.43366 q^{3} +1.00000 q^{4} +1.00000 q^{5} +3.43366 q^{6} -1.00000 q^{7} +1.00000 q^{8} +8.79005 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} +3.43366 q^{3} +1.00000 q^{4} +1.00000 q^{5} +3.43366 q^{6} -1.00000 q^{7} +1.00000 q^{8} +8.79005 q^{9} +1.00000 q^{10} +3.43366 q^{12} -5.79005 q^{13} -1.00000 q^{14} +3.43366 q^{15} +1.00000 q^{16} +2.43366 q^{17} +8.79005 q^{18} +8.22372 q^{19} +1.00000 q^{20} -3.43366 q^{21} +0.566335 q^{23} +3.43366 q^{24} +1.00000 q^{25} -5.79005 q^{26} +19.8811 q^{27} -1.00000 q^{28} +2.00000 q^{29} +3.43366 q^{30} -9.22372 q^{31} +1.00000 q^{32} +2.43366 q^{34} -1.00000 q^{35} +8.79005 q^{36} +8.86733 q^{37} +8.22372 q^{38} -19.8811 q^{39} +1.00000 q^{40} -6.35639 q^{41} -3.43366 q^{42} +0.433665 q^{43} +8.79005 q^{45} +0.566335 q^{46} -5.22372 q^{47} +3.43366 q^{48} +1.00000 q^{49} +1.00000 q^{50} +8.35639 q^{51} -5.79005 q^{52} -5.30099 q^{53} +19.8811 q^{54} -1.00000 q^{56} +28.2375 q^{57} +2.00000 q^{58} +7.51094 q^{59} +3.43366 q^{60} +0.0772765 q^{61} -9.22372 q^{62} -8.79005 q^{63} +1.00000 q^{64} -5.79005 q^{65} +3.56634 q^{67} +2.43366 q^{68} +1.94461 q^{69} -1.00000 q^{70} +15.6574 q^{71} +8.79005 q^{72} -0.433665 q^{73} +8.86733 q^{74} +3.43366 q^{75} +8.22372 q^{76} -19.8811 q^{78} -10.2237 q^{79} +1.00000 q^{80} +41.8949 q^{81} -6.35639 q^{82} -1.43366 q^{83} -3.43366 q^{84} +2.43366 q^{85} +0.433665 q^{86} +6.86733 q^{87} +9.22372 q^{89} +8.79005 q^{90} +5.79005 q^{91} +0.566335 q^{92} -31.6712 q^{93} -5.22372 q^{94} +8.22372 q^{95} +3.43366 q^{96} -4.94461 q^{97} +1.00000 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{2} + 2 q^{3} + 3 q^{4} + 3 q^{5} + 2 q^{6} - 3 q^{7} + 3 q^{8} + 13 q^{9}+O(q^{10}) $ 3 * q + 3 * q^2 + 2 * q^3 + 3 * q^4 + 3 * q^5 + 2 * q^6 - 3 * q^7 + 3 * q^8 + 13 * q^9	$ 3 q + 3 q^{2} + 2 q^{3} + 3 q^{4} + 3 q^{5} + 2 q^{6} - 3 q^{7} + 3 q^{8} + 13 q^{9} + 3 q^{10} + 2 q^{12} - 4 q^{13} - 3 q^{14} + 2 q^{15} + 3 q^{16} - q^{17} + 13 q^{18} + 3 q^{19} + 3 q^{20} - 2 q^{21} + 10 q^{23} + 2 q^{24} + 3 q^{25} - 4 q^{26} + 8 q^{27} - 3 q^{28} + 6 q^{29} + 2 q^{30} - 6 q^{31} + 3 q^{32} - q^{34} - 3 q^{35} + 13 q^{36} + 10 q^{37} + 3 q^{38} - 8 q^{39} + 3 q^{40} - 14 q^{41} - 2 q^{42} - 7 q^{43} + 13 q^{45} + 10 q^{46} + 6 q^{47} + 2 q^{48} + 3 q^{49} + 3 q^{50} + 20 q^{51} - 4 q^{52} + 9 q^{53} + 8 q^{54} - 3 q^{56} + 28 q^{57} + 6 q^{58} + 11 q^{59} + 2 q^{60} - 3 q^{61} - 6 q^{62} - 13 q^{63} + 3 q^{64} - 4 q^{65} + 19 q^{67} - q^{68} - 14 q^{69} - 3 q^{70} + 17 q^{71} + 13 q^{72} + 7 q^{73} + 10 q^{74} + 2 q^{75} + 3 q^{76} - 8 q^{78} - 9 q^{79} + 3 q^{80} + 39 q^{81} - 14 q^{82} + 4 q^{83} - 2 q^{84} - q^{85} - 7 q^{86} + 4 q^{87} + 6 q^{89} + 13 q^{90} + 4 q^{91} + 10 q^{92} - 30 q^{93} + 6 q^{94} + 3 q^{95} + 2 q^{96} + 5 q^{97} + 3 q^{98}+O(q^{100}) $ 3 * q + 3 * q^2 + 2 * q^3 + 3 * q^4 + 3 * q^5 + 2 * q^6 - 3 * q^7 + 3 * q^8 + 13 * q^9 + 3 * q^10 + 2 * q^12 - 4 * q^13 - 3 * q^14 + 2 * q^15 + 3 * q^16 - q^17 + 13 * q^18 + 3 * q^19 + 3 * q^20 - 2 * q^21 + 10 * q^23 + 2 * q^24 + 3 * q^25 - 4 * q^26 + 8 * q^27 - 3 * q^28 + 6 * q^29 + 2 * q^30 - 6 * q^31 + 3 * q^32 - q^34 - 3 * q^35 + 13 * q^36 + 10 * q^37 + 3 * q^38 - 8 * q^39 + 3 * q^40 - 14 * q^41 - 2 * q^42 - 7 * q^43 + 13 * q^45 + 10 * q^46 + 6 * q^47 + 2 * q^48 + 3 * q^49 + 3 * q^50 + 20 * q^51 - 4 * q^52 + 9 * q^53 + 8 * q^54 - 3 * q^56 + 28 * q^57 + 6 * q^58 + 11 * q^59 + 2 * q^60 - 3 * q^61 - 6 * q^62 - 13 * q^63 + 3 * q^64 - 4 * q^65 + 19 * q^67 - q^68 - 14 * q^69 - 3 * q^70 + 17 * q^71 + 13 * q^72 + 7 * q^73 + 10 * q^74 + 2 * q^75 + 3 * q^76 - 8 * q^78 - 9 * q^79 + 3 * q^80 + 39 * q^81 - 14 * q^82 + 4 * q^83 - 2 * q^84 - q^85 - 7 * q^86 + 4 * q^87 + 6 * q^89 + 13 * q^90 + 4 * q^91 + 10 * q^92 - 30 * q^93 + 6 * q^94 + 3 * q^95 + 2 * q^96 + 5 * q^97 + 3 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	3.43366	1.98243	0.991214	−	0.132271i	$-0.0422269\pi$
$3$	3.43366	1.98243	0.991214	+	0.132271i	$0.0422269\pi$
$4$	1.00000	0.500000
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	3.43366	1.40179
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	1.00000	0.353553
$9$	8.79005	2.93002
$10$	1.00000	0.316228
$11$	0	0
$11$	0	0
$12$	3.43366	0.991214
$13$	−5.79005	−1.60587	−0.802936	−	0.596065i	$-0.796730\pi$
$13$	−5.79005	−1.60587	−0.802936	+	0.596065i	$0.796730\pi$
$14$	−1.00000	−0.267261
$15$	3.43366	0.886568
$16$	1.00000	0.250000
$17$	2.43366	0.590250	0.295125	−	0.955459i	$-0.404639\pi$
$17$	2.43366	0.590250	0.295125	+	0.955459i	$0.404639\pi$
$18$	8.79005	2.07184
$19$	8.22372	1.88665	0.943325	−	0.331870i	$-0.107680\pi$
$19$	8.22372	1.88665	0.943325	+	0.331870i	$0.107680\pi$
$20$	1.00000	0.223607
$21$	−3.43366	−0.749287
$22$	0	0
$23$	0.566335	0.118089	0.0590445	−	0.998255i	$-0.481195\pi$
$23$	0.566335	0.118089	0.0590445	+	0.998255i	$0.481195\pi$
$24$	3.43366	0.700894
$25$	1.00000	0.200000
$26$	−5.79005	−1.13552
$27$	19.8811	3.82612
$28$	−1.00000	−0.188982
$29$	2.00000	0.371391	0.185695	−	0.982607i	$-0.440546\pi$
$29$	2.00000	0.371391	0.185695	+	0.982607i	$0.440546\pi$
$30$	3.43366	0.626899
$31$	−9.22372	−1.65663	−0.828314	−	0.560264i	$-0.810700\pi$
$31$	−9.22372	−1.65663	−0.828314	+	0.560264i	$0.810700\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	2.43366	0.417370
$35$	−1.00000	−0.169031
$36$	8.79005	1.46501
$37$	8.86733	1.45778	0.728890	−	0.684631i	$-0.240037\pi$
$37$	8.86733	1.45778	0.728890	+	0.684631i	$0.240037\pi$
$38$	8.22372	1.33406
$39$	−19.8811	−3.18352
$40$	1.00000	0.158114
$41$	−6.35639	−0.992701	−0.496351	−	0.868122i	$-0.665327\pi$
$41$	−6.35639	−0.992701	−0.496351	+	0.868122i	$0.665327\pi$
$42$	−3.43366	−0.529826
$43$	0.433665	0.0661332	0.0330666	−	0.999453i	$-0.489473\pi$
$43$	0.433665	0.0661332	0.0330666	+	0.999453i	$0.489473\pi$
$44$	0	0
$45$	8.79005	1.31034
$46$	0.566335	0.0835016
$47$	−5.22372	−0.761957	−0.380979	−	0.924584i	$-0.624413\pi$
$47$	−5.22372	−0.761957	−0.380979	+	0.924584i	$0.624413\pi$
$48$	3.43366	0.495607
$49$	1.00000	0.142857
$50$	1.00000	0.141421
$51$	8.35639	1.17013
$52$	−5.79005	−0.802936
$53$	−5.30099	−0.728148	−0.364074	−	0.931370i	$-0.618614\pi$
$53$	−5.30099	−0.728148	−0.364074	+	0.931370i	$0.618614\pi$
$54$	19.8811	2.70547
$55$	0	0
$56$	−1.00000	−0.133631
$57$	28.2375	3.74015
$58$	2.00000	0.262613
$59$	7.51094	0.977841	0.488921	−	0.872328i	$-0.337391\pi$
$59$	7.51094	0.977841	0.488921	+	0.872328i	$0.337391\pi$
$60$	3.43366	0.443284
$61$	0.0772765	0.00989424	0.00494712	−	0.999988i	$-0.498425\pi$
$61$	0.0772765	0.00989424	0.00494712	+	0.999988i	$0.498425\pi$
$62$	−9.22372	−1.17141
$63$	−8.79005	−1.10744
$64$	1.00000	0.125000
$65$	−5.79005	−0.718168
$66$	0	0
$67$	3.56634	0.435697	0.217849	−	0.975983i	$-0.430096\pi$
$67$	3.56634	0.435697	0.217849	+	0.975983i	$0.430096\pi$
$68$	2.43366	0.295125
$69$	1.94461	0.234103
$70$	−1.00000	−0.119523
$71$	15.6574	1.85819	0.929095	−	0.369842i	$-0.120588\pi$
$71$	15.6574	1.85819	0.929095	+	0.369842i	$0.120588\pi$
$72$	8.79005	1.03592
$73$	−0.433665	−0.0507566	−0.0253783	−	0.999678i	$-0.508079\pi$
$73$	−0.433665	−0.0507566	−0.0253783	+	0.999678i	$0.508079\pi$
$74$	8.86733	1.03081
$75$	3.43366	0.396485
$76$	8.22372	0.943325
$77$	0	0
$78$	−19.8811	−2.25109
$79$	−10.2237	−1.15026	−0.575129	−	0.818063i	$-0.695048\pi$
$79$	−10.2237	−1.15026	−0.575129	+	0.818063i	$0.695048\pi$
$80$	1.00000	0.111803
$81$	41.8949	4.65499
$82$	−6.35639	−0.701946
$83$	−1.43366	−0.157365	−0.0786826	−	0.996900i	$-0.525071\pi$
$83$	−1.43366	−0.157365	−0.0786826	+	0.996900i	$0.525071\pi$
$84$	−3.43366	−0.374644
$85$	2.43366	0.263968
$86$	0.433665	0.0467633
$87$	6.86733	0.736255
$88$	0	0
$89$	9.22372	0.977712	0.488856	−	0.872364i	$-0.337414\pi$
$89$	9.22372	0.977712	0.488856	+	0.872364i	$0.337414\pi$
$90$	8.79005	0.926553
$91$	5.79005	0.606962
$92$	0.566335	0.0590445
$93$	−31.6712	−3.28415
$94$	−5.22372	−0.538785
$95$	8.22372	0.843736
$96$	3.43366	0.350447
$97$	−4.94461	−0.502049	−0.251024	−	0.967981i	$-0.580767\pi$
$97$	−4.94461	−0.502049	−0.251024	+	0.967981i	$0.580767\pi$
$98$	1.00000	0.101015
$99$	0	0
$100$	1.00000	0.100000
$101$	−12.6574	−1.25946	−0.629728	−	0.776815i	$-0.716834\pi$
$101$	−12.6574	−1.25946	−0.629728	+	0.776815i	$0.716834\pi$
$102$	8.35639	0.827406
$103$	−4.01377	−0.395489	−0.197744	−	0.980254i	$-0.563362\pi$
$103$	−4.01377	−0.395489	−0.197744	+	0.980254i	$0.563362\pi$
$104$	−5.79005	−0.567761
$105$	−3.43366	−0.335091
$106$	−5.30099	−0.514878
$107$	−10.7901	−1.04311	−0.521557	−	0.853217i	$-0.674649\pi$
$107$	−10.7901	−1.04311	−0.521557	+	0.853217i	$0.674649\pi$
$108$	19.8811	1.91306
$109$	−17.9365	−1.71800	−0.859002	−	0.511972i	$-0.828915\pi$
$109$	−17.9365	−1.71800	−0.859002	+	0.511972i	$0.828915\pi$
$110$	0	0
$111$	30.4474	2.88994
$112$	−1.00000	−0.0944911
$113$	−8.50283	−0.799879	−0.399939	−	0.916542i	$-0.630969\pi$
$113$	−8.50283	−0.799879	−0.399939	+	0.916542i	$0.630969\pi$
$114$	28.2375	2.64468
$115$	0.566335	0.0528110
$116$	2.00000	0.185695
$117$	−50.8949	−4.70523
$118$	7.51094	0.691438
$119$	−2.43366	−0.223094
$120$	3.43366	0.313449
$121$	0	0
$122$	0.0772765	0.00699628
$123$	−21.8257	−1.96796
$124$	−9.22372	−0.828314
$125$	1.00000	0.0894427
$126$	−8.79005	−0.783080
$127$	9.27911	0.823388	0.411694	−	0.911322i	$-0.364937\pi$
$127$	9.27911	0.823388	0.411694	+	0.911322i	$0.364937\pi$
$128$	1.00000	0.0883883
$129$	1.48906	0.131104
$130$	−5.79005	−0.507821
$131$	−12.3010	−1.07474	−0.537371	−	0.843346i	$-0.680583\pi$
$131$	−12.3010	−1.07474	−0.537371	+	0.843346i	$0.680583\pi$
$132$	0	0
$133$	−8.22372	−0.713087
$134$	3.56634	0.308084
$135$	19.8811	1.71109
$136$	2.43366	0.208685
$137$	15.3702	1.31316	0.656581	−	0.754256i	$-0.272002\pi$
$137$	15.3702	1.31316	0.656581	+	0.754256i	$0.272002\pi$
$138$	1.94461	0.165536
$139$	9.09105	0.771093	0.385546	−	0.922689i	$-0.374013\pi$
$139$	9.09105	0.771093	0.385546	+	0.922689i	$0.374013\pi$
$140$	−1.00000	−0.0845154
$141$	−17.9365	−1.51053
$142$	15.6574	1.31394
$143$	0	0
$144$	8.79005	0.732504
$145$	2.00000	0.166091
$146$	−0.433665	−0.0358903
$147$	3.43366	0.283204
$148$	8.86733	0.728890
$149$	6.44743	0.528194	0.264097	−	0.964496i	$-0.414926\pi$
$149$	6.44743	0.528194	0.264097	+	0.964496i	$0.414926\pi$
$150$	3.43366	0.280358
$151$	10.5663	0.859876	0.429938	−	0.902858i	$-0.358535\pi$
$151$	10.5663	0.859876	0.429938	+	0.902858i	$0.358535\pi$
$152$	8.22372	0.667032
$153$	21.3920	1.72944
$154$	0	0
$155$	−9.22372	−0.740867
$156$	−19.8811	−1.59176
$157$	12.5028	0.997834	0.498917	−	0.866650i	$-0.333731\pi$
$157$	12.5028	0.997834	0.498917	+	0.866650i	$0.333731\pi$
$158$	−10.2237	−0.813355
$159$	−18.2018	−1.44350
$160$	1.00000	0.0790569
$161$	−0.566335	−0.0446335
$162$	41.8949	3.29157
$163$	−14.5247	−1.13766	−0.568832	−	0.822454i	$-0.692604\pi$
$163$	−14.5247	−1.13766	−0.568832	+	0.822454i	$0.692604\pi$
$164$	−6.35639	−0.496351
$165$	0	0
$166$	−1.43366	−0.111274
$167$	−9.30099	−0.719733	−0.359866	−	0.933004i	$-0.617178\pi$
$167$	−9.30099	−0.719733	−0.359866	+	0.933004i	$0.617178\pi$
$168$	−3.43366	−0.264913
$169$	20.5247	1.57882
$170$	2.43366	0.186654
$171$	72.2869	5.52792
$172$	0.433665	0.0330666
$173$	7.58011	0.576305	0.288152	−	0.957585i	$-0.406959\pi$
$173$	7.58011	0.576305	0.288152	+	0.957585i	$0.406959\pi$
$174$	6.86733	0.520611
$175$	−1.00000	−0.0755929
$176$	0	0
$177$	25.7901	1.93850
$178$	9.22372	0.691347
$179$	−20.0910	−1.50168	−0.750838	−	0.660487i	$-0.770350\pi$
$179$	−20.0910	−1.50168	−0.750838	+	0.660487i	$0.770350\pi$
$180$	8.79005	0.655172
$181$	−15.7128	−1.16792	−0.583961	−	0.811782i	$-0.698498\pi$
$181$	−15.7128	−1.16792	−0.583961	+	0.811782i	$0.698498\pi$
$182$	5.79005	0.429187
$183$	0.265341	0.0196146
$184$	0.566335	0.0417508
$185$	8.86733	0.651939
$186$	−31.6712	−2.32224
$187$	0	0
$188$	−5.22372	−0.380979
$189$	−19.8811	−1.44614
$190$	8.22372	0.596611
$191$	−3.99189	−0.288843	−0.144421	−	0.989516i	$-0.546132\pi$
$191$	−3.99189	−0.288843	−0.144421	+	0.989516i	$0.546132\pi$
$192$	3.43366	0.247803
$193$	4.50283	0.324121	0.162060	−	0.986781i	$-0.448186\pi$
$193$	4.50283	0.324121	0.162060	+	0.986781i	$0.448186\pi$
$194$	−4.94461	−0.355002
$195$	−19.8811	−1.42372
$196$	1.00000	0.0714286
$197$	10.5247	0.749855	0.374927	−	0.927054i	$-0.377668\pi$
$197$	10.5247	0.749855	0.374927	+	0.927054i	$0.377668\pi$
$198$	0	0
$199$	−11.4256	−0.809936	−0.404968	−	0.914331i	$-0.632717\pi$
$199$	−11.4256	−0.809936	−0.404968	+	0.914331i	$0.632717\pi$
$200$	1.00000	0.0707107
$201$	12.2456	0.863738
$202$	−12.6574	−0.890570
$203$	−2.00000	−0.140372
$204$	8.35639	0.585064
$205$	−6.35639	−0.443949
$206$	−4.01377	−0.279653
$207$	4.97812	0.346003
$208$	−5.79005	−0.401468
$209$	0	0
$210$	−3.43366	−0.236945
$211$	1.84545	0.127046	0.0635229	−	0.997980i	$-0.479766\pi$
$211$	1.84545	0.127046	0.0635229	+	0.997980i	$0.479766\pi$
$212$	−5.30099	−0.364074
$213$	53.7622	3.68373
$214$	−10.7901	−0.737593
$215$	0.433665	0.0295757
$216$	19.8811	1.35274
$217$	9.22372	0.626147
$218$	−17.9365	−1.21481
$219$	−1.48906	−0.100621
$220$	0	0
$221$	−14.0910	−0.947866
$222$	30.4474	2.04350
$223$	−13.1464	−0.880351	−0.440175	−	0.897912i	$-0.645084\pi$
$223$	−13.1464	−0.880351	−0.440175	+	0.897912i	$0.645084\pi$
$224$	−1.00000	−0.0668153
$225$	8.79005	0.586004
$226$	−8.50283	−0.565600
$227$	−7.28722	−0.483670	−0.241835	−	0.970317i	$-0.577749\pi$
$227$	−7.28722	−0.483670	−0.241835	+	0.970317i	$0.577749\pi$
$228$	28.2375	1.87007
$229$	−2.52471	−0.166838	−0.0834188	−	0.996515i	$-0.526584\pi$
$229$	−2.52471	−0.166838	−0.0834188	+	0.996515i	$0.526584\pi$
$230$	0.566335	0.0373430
$231$	0	0
$232$	2.00000	0.131306
$233$	21.5247	1.41013	0.705065	−	0.709142i	$-0.250918\pi$
$233$	21.5247	1.41013	0.705065	+	0.709142i	$0.250918\pi$
$234$	−50.8949	−3.32710
$235$	−5.22372	−0.340758
$236$	7.51094	0.488921
$237$	−35.1048	−2.28030
$238$	−2.43366	−0.157751
$239$	15.2791	0.988324	0.494162	−	0.869370i	$-0.335475\pi$
$239$	15.2791	0.988324	0.494162	+	0.869370i	$0.335475\pi$
$240$	3.43366	0.221642
$241$	−6.15455	−0.396450	−0.198225	−	0.980157i	$-0.563518\pi$
$241$	−6.15455	−0.396450	−0.198225	+	0.980157i	$0.563518\pi$
$242$	0	0
$243$	84.2096	5.40205
$244$	0.0772765	0.00494712
$245$	1.00000	0.0638877
$246$	−21.8257	−1.39156
$247$	−47.6158	−3.02972
$248$	−9.22372	−0.585707
$249$	−4.92272	−0.311965
$250$	1.00000	0.0632456
$251$	−2.79005	−0.176107	−0.0880533	−	0.996116i	$-0.528065\pi$
$251$	−2.79005	−0.176107	−0.0880533	+	0.996116i	$0.528065\pi$
$252$	−8.79005	−0.553721
$253$	0	0
$254$	9.27911	0.582223
$255$	8.35639	0.523297
$256$	1.00000	0.0625000
$257$	3.72089	0.232103	0.116051	−	0.993243i	$-0.462976\pi$
$257$	3.72089	0.232103	0.116051	+	0.993243i	$0.462976\pi$
$258$	1.48906	0.0927048
$259$	−8.86733	−0.550989
$260$	−5.79005	−0.359084
$261$	17.5801	1.08818
$262$	−12.3010	−0.759958
$263$	21.8811	1.34925	0.674623	−	0.738162i	$-0.264306\pi$
$263$	21.8811	1.34925	0.674623	+	0.738162i	$0.264306\pi$
$264$	0	0
$265$	−5.30099	−0.325637
$266$	−8.22372	−0.504228
$267$	31.6712	1.93824
$268$	3.56634	0.217849
$269$	−4.02188	−0.245218	−0.122609	−	0.992455i	$-0.539126\pi$
$269$	−4.02188	−0.245218	−0.122609	+	0.992455i	$0.539126\pi$
$270$	19.8811	1.20993
$271$	−18.6020	−1.12999	−0.564995	−	0.825094i	$-0.691122\pi$
$271$	−18.6020	−1.12999	−0.564995	+	0.825094i	$0.691122\pi$
$272$	2.43366	0.147563
$273$	19.8811	1.20326
$274$	15.3702	0.928545
$275$	0	0
$276$	1.94461	0.117052
$277$	29.6712	1.78277	0.891383	−	0.453250i	$-0.149736\pi$
$277$	29.6712	1.78277	0.891383	+	0.453250i	$0.149736\pi$
$278$	9.09105	0.545245
$279$	−81.0770	−4.85395
$280$	−1.00000	−0.0597614
$281$	−18.3148	−1.09257	−0.546284	−	0.837600i	$-0.683958\pi$
$281$	−18.3148	−1.09257	−0.546284	+	0.837600i	$0.683958\pi$
$282$	−17.9365	−1.06810
$283$	17.0138	1.01136	0.505682	−	0.862720i	$-0.331241\pi$
$283$	17.0138	1.01136	0.505682	+	0.862720i	$0.331241\pi$
$284$	15.6574	0.929095
$285$	28.2375	1.67264
$286$	0	0
$287$	6.35639	0.375206
$288$	8.79005	0.517959
$289$	−11.0773	−0.651604
$290$	2.00000	0.117444
$291$	−16.9781	−0.995275
$292$	−0.433665	−0.0253783
$293$	−22.9503	−1.34077	−0.670384	−	0.742014i	$-0.733871\pi$
$293$	−22.9503	−1.34077	−0.670384	+	0.742014i	$0.733871\pi$
$294$	3.43366	0.200255
$295$	7.51094	0.437304
$296$	8.86733	0.515403
$297$	0	0
$298$	6.44743	0.373490
$299$	−3.27911	−0.189636
$300$	3.43366	0.198243
$301$	−0.433665	−0.0249960
$302$	10.5663	0.608024
$303$	−43.4612	−2.49678
$304$	8.22372	0.471663
$305$	0.0772765	0.00442484
$306$	21.3920	1.22290
$307$	7.13267	0.407083	0.203542	−	0.979066i	$-0.434755\pi$
$307$	7.13267	0.407083	0.203542	+	0.979066i	$0.434755\pi$
$308$	0	0
$309$	−13.7819	−0.784027
$310$	−9.22372	−0.523872
$311$	−18.2456	−1.03461	−0.517307	−	0.855800i	$-0.673065\pi$
$311$	−18.2456	−1.03461	−0.517307	+	0.855800i	$0.673065\pi$
$312$	−19.8811	−1.12555
$313$	−27.0219	−1.52737	−0.763684	−	0.645591i	$-0.776611\pi$
$313$	−27.0219	−1.52737	−0.763684	+	0.645591i	$0.776611\pi$
$314$	12.5028	0.705575
$315$	−8.79005	−0.495263
$316$	−10.2237	−0.575129
$317$	8.01377	0.450098	0.225049	−	0.974347i	$-0.427746\pi$
$317$	8.01377	0.450098	0.225049	+	0.974347i	$0.427746\pi$
$318$	−18.2018	−1.02071
$319$	0	0
$320$	1.00000	0.0559017
$321$	−37.0494	−2.06790
$322$	−0.566335	−0.0315606
$323$	20.0138	1.11360
$324$	41.8949	2.32749
$325$	−5.79005	−0.321174
$326$	−14.5247	−0.804449
$327$	−61.5879	−3.40582
$328$	−6.35639	−0.350973
$329$	5.22372	0.287993
$330$	0	0
$331$	−12.6493	−0.695267	−0.347633	−	0.937631i	$-0.613015\pi$
$331$	−12.6493	−0.695267	−0.347633	+	0.937631i	$0.613015\pi$
$332$	−1.43366	−0.0786826
$333$	77.9443	4.27132
$334$	−9.30099	−0.508928
$335$	3.56634	0.194850
$336$	−3.43366	−0.187322
$337$	−14.8119	−0.806858	−0.403429	−	0.915011i	$-0.632182\pi$
$337$	−14.8119	−0.806858	−0.403429	+	0.915011i	$0.632182\pi$
$338$	20.5247	1.11640
$339$	−29.1959	−1.58570
$340$	2.43366	0.131984
$341$	0	0
$342$	72.2869	3.90883
$343$	−1.00000	−0.0539949
$344$	0.433665	0.0233816
$345$	1.94461	0.104694
$346$	7.58011	0.407509
$347$	−26.6793	−1.43222	−0.716109	−	0.697988i	$-0.754079\pi$
$347$	−26.6793	−1.43222	−0.716109	+	0.697988i	$0.754079\pi$
$348$	6.86733	0.368127
$349$	0.475289	0.0254416	0.0127208	−	0.999919i	$-0.495951\pi$
$349$	0.475289	0.0254416	0.0127208	+	0.999919i	$0.495951\pi$
$350$	−1.00000	−0.0534522
$351$	−115.113	−6.14426
$352$	0	0
$353$	−10.0138	−0.532979	−0.266490	−	0.963838i	$-0.585864\pi$
$353$	−10.0138	−0.532979	−0.266490	+	0.963838i	$0.585864\pi$
$354$	25.7901	1.37073
$355$	15.6574	0.831008
$356$	9.22372	0.488856
$357$	−8.35639	−0.442267
$358$	−20.0910	−1.06184
$359$	7.47529	0.394531	0.197265	−	0.980350i	$-0.436794\pi$
$359$	7.47529	0.394531	0.197265	+	0.980350i	$0.436794\pi$
$360$	8.79005	0.463276
$361$	48.6295	2.55945
$362$	−15.7128	−0.825845
$363$	0	0
$364$	5.79005	0.303481
$365$	−0.433665	−0.0226990
$366$	0.265341	0.0138696
$367$	30.1683	1.57477	0.787387	−	0.616459i	$-0.211433\pi$
$367$	30.1683	1.57477	0.787387	+	0.616459i	$0.211433\pi$
$368$	0.566335	0.0295223
$369$	−55.8730	−2.90863
$370$	8.86733	0.460990
$371$	5.30099	0.275214
$372$	−31.6712	−1.64207
$373$	13.2099	0.683985	0.341993	−	0.939703i	$-0.388898\pi$
$373$	13.2099	0.683985	0.341993	+	0.939703i	$0.388898\pi$
$374$	0	0
$375$	3.43366	0.177314
$376$	−5.22372	−0.269393
$377$	−11.5801	−0.596406
$378$	−19.8811	−1.02257
$379$	19.1602	0.984194	0.492097	−	0.870540i	$-0.336231\pi$
$379$	19.1602	0.984194	0.492097	+	0.870540i	$0.336231\pi$
$380$	8.22372	0.421868
$381$	31.8614	1.63231
$382$	−3.99189	−0.204243
$383$	6.58822	0.336642	0.168321	−	0.985732i	$-0.446165\pi$
$383$	6.58822	0.336642	0.168321	+	0.985732i	$0.446165\pi$
$384$	3.43366	0.175223
$385$	0	0
$386$	4.50283	0.229188
$387$	3.81193	0.193772
$388$	−4.94461	−0.251024
$389$	−25.1129	−1.27328	−0.636638	−	0.771163i	$-0.719675\pi$
$389$	−25.1129	−1.27328	−0.636638	+	0.771163i	$0.719675\pi$
$390$	−19.8811	−1.00672
$391$	1.37827	0.0697021
$392$	1.00000	0.0505076
$393$	−42.2375	−2.13060
$394$	10.5247	0.530227
$395$	−10.2237	−0.514411
$396$	0	0
$397$	−11.5801	−0.581189	−0.290594	−	0.956846i	$-0.593853\pi$
$397$	−11.5801	−0.581189	−0.290594	+	0.956846i	$0.593853\pi$
$398$	−11.4256	−0.572711
$399$	−28.2375	−1.41364
$400$	1.00000	0.0500000
$401$	−31.7347	−1.58475	−0.792377	−	0.610032i	$-0.791156\pi$
$401$	−31.7347	−1.58475	−0.792377	+	0.610032i	$0.791156\pi$
$402$	12.2456	0.610755
$403$	53.4058	2.66033
$404$	−12.6574	−0.629728
$405$	41.8949	2.08177
$406$	−2.00000	−0.0992583
$407$	0	0
$408$	8.35639	0.413703
$409$	2.15455	0.106536	0.0532679	−	0.998580i	$-0.483036\pi$
$409$	2.15455	0.106536	0.0532679	+	0.998580i	$0.483036\pi$
$410$	−6.35639	−0.313920
$411$	52.7760	2.60325
$412$	−4.01377	−0.197744
$413$	−7.51094	−0.369589
$414$	4.97812	0.244661
$415$	−1.43366	−0.0703758
$416$	−5.79005	−0.283881
$417$	31.2156	1.52863
$418$	0	0
$419$	−13.5385	−0.661398	−0.330699	−	0.943736i	$-0.607285\pi$
$419$	−13.5385	−0.661398	−0.330699	+	0.943736i	$0.607285\pi$
$420$	−3.43366	−0.167546
$421$	−1.42555	−0.0694772	−0.0347386	−	0.999396i	$-0.511060\pi$
$421$	−1.42555	−0.0694772	−0.0347386	+	0.999396i	$0.511060\pi$
$422$	1.84545	0.0898350
$423$	−45.9168	−2.23255
$424$	−5.30099	−0.257439
$425$	2.43366	0.118050
$426$	53.7622	2.60479
$427$	−0.0772765	−0.00373967
$428$	−10.7901	−0.521557
$429$	0	0
$430$	0.433665	0.0209132
$431$	−30.1129	−1.45049	−0.725244	−	0.688492i	$-0.758273\pi$
$431$	−30.1129	−1.45049	−0.725244	+	0.688492i	$0.758273\pi$
$432$	19.8811	0.956530
$433$	29.2513	1.40573	0.702863	−	0.711325i	$-0.251905\pi$
$433$	29.2513	1.40573	0.702863	+	0.711325i	$0.251905\pi$
$434$	9.22372	0.442753
$435$	6.86733	0.329263
$436$	−17.9365	−0.859002
$437$	4.65738	0.222793
$438$	−1.48906	−0.0711500
$439$	33.0692	1.57831	0.789153	−	0.614197i	$-0.210520\pi$
$439$	33.0692	1.57831	0.789153	+	0.614197i	$0.210520\pi$
$440$	0	0
$441$	8.79005	0.418574
$442$	−14.0910	−0.670243
$443$	−28.1821	−1.33897	−0.669486	−	0.742825i	$-0.733486\pi$
$443$	−28.1821	−1.33897	−0.669486	+	0.742825i	$0.733486\pi$
$444$	30.4474	1.44497
$445$	9.22372	0.437246
$446$	−13.1464	−0.622502
$447$	22.1383	1.04711
$448$	−1.00000	−0.0472456
$449$	−0.734659	−0.0346707	−0.0173353	−	0.999850i	$-0.505518\pi$
$449$	−0.734659	−0.0346707	−0.0173353	+	0.999850i	$0.505518\pi$
$450$	8.79005	0.414367
$451$	0	0
$452$	−8.50283	−0.399939
$453$	36.2813	1.70464
$454$	−7.28722	−0.342006
$455$	5.79005	0.271442
$456$	28.2375	1.32234
$457$	5.21561	0.243976	0.121988	−	0.992532i	$-0.461073\pi$
$457$	5.21561	0.243976	0.121988	+	0.992532i	$0.461073\pi$
$458$	−2.52471	−0.117972
$459$	48.3839	2.25837
$460$	0.566335	0.0264055
$461$	−24.9446	−1.16179	−0.580893	−	0.813980i	$-0.697297\pi$
$461$	−24.9446	−1.16179	−0.580893	+	0.813980i	$0.697297\pi$
$462$	0	0
$463$	−15.6158	−0.725726	−0.362863	−	0.931843i	$-0.618201\pi$
$463$	−15.6158	−0.725726	−0.362863	+	0.931843i	$0.618201\pi$
$464$	2.00000	0.0928477
$465$	−31.6712	−1.46871
$466$	21.5247	0.997113
$467$	−36.9030	−1.70767	−0.853833	−	0.520547i	$-0.825728\pi$
$467$	−36.9030	−1.70767	−0.853833	+	0.520547i	$0.825728\pi$
$468$	−50.8949	−2.35262
$469$	−3.56634	−0.164678
$470$	−5.22372	−0.240952
$471$	42.9305	1.97813
$472$	7.51094	0.345719
$473$	0	0
$474$	−35.1048	−1.61242
$475$	8.22372	0.377330
$476$	−2.43366	−0.111547
$477$	−46.5960	−2.13349
$478$	15.2791	0.698850
$479$	16.0910	0.735219	0.367609	−	0.929980i	$-0.380176\pi$
$479$	16.0910	0.735219	0.367609	+	0.929980i	$0.380176\pi$
$480$	3.43366	0.156725
$481$	−51.3423	−2.34101
$482$	−6.15455	−0.280332
$483$	−1.94461	−0.0884826
$484$	0	0
$485$	−4.94461	−0.224523
$486$	84.2096	3.81983
$487$	−38.6376	−1.75084	−0.875419	−	0.483364i	$-0.839415\pi$
$487$	−38.6376	−1.75084	−0.875419	+	0.483364i	$0.839415\pi$
$488$	0.0772765	0.00349814
$489$	−49.8730	−2.25533
$490$	1.00000	0.0451754
$491$	39.5604	1.78533	0.892667	−	0.450717i	$-0.148831\pi$
$491$	39.5604	1.78533	0.892667	+	0.450717i	$0.148831\pi$
$492$	−21.8257	−0.983979
$493$	4.86733	0.219213
$494$	−47.6158	−2.14233
$495$	0	0
$496$	−9.22372	−0.414157
$497$	−15.6574	−0.702330
$498$	−4.92272	−0.220593
$499$	−42.4474	−1.90021	−0.950104	−	0.311933i	$-0.899024\pi$
$499$	−42.4474	−1.90021	−0.950104	+	0.311933i	$0.899024\pi$
$500$	1.00000	0.0447214
$501$	−31.9365	−1.42682
$502$	−2.79005	−0.124526
$503$	−18.4337	−0.821916	−0.410958	−	0.911654i	$-0.634806\pi$
$503$	−18.4337	−0.821916	−0.410958	+	0.911654i	$0.634806\pi$
$504$	−8.79005	−0.391540
$505$	−12.6574	−0.563246
$506$	0	0
$507$	70.4750	3.12990
$508$	9.27911	0.411694
$509$	27.2929	1.20974	0.604868	−	0.796326i	$-0.293226\pi$
$509$	27.2929	1.20974	0.604868	+	0.796326i	$0.293226\pi$
$510$	8.35639	0.370027
$511$	0.433665	0.0191842
$512$	1.00000	0.0441942
$513$	163.497	7.21855
$514$	3.72089	0.164121
$515$	−4.01377	−0.176868
$516$	1.48906	0.0655522
$517$	0	0
$518$	−8.86733	−0.389608
$519$	26.0275	1.14248
$520$	−5.79005	−0.253911
$521$	7.55257	0.330884	0.165442	−	0.986220i	$-0.447095\pi$
$521$	7.55257	0.330884	0.165442	+	0.986220i	$0.447095\pi$
$522$	17.5801	0.769460
$523$	33.9086	1.48272	0.741361	−	0.671107i	$-0.234181\pi$
$523$	33.9086	1.48272	0.741361	+	0.671107i	$0.234181\pi$
$524$	−12.3010	−0.537371
$525$	−3.43366	−0.149857
$526$	21.8811	0.954061
$527$	−22.4474	−0.977826
$528$	0	0
$529$	−22.6793	−0.986055
$530$	−5.30099	−0.230260
$531$	66.0216	2.86509
$532$	−8.22372	−0.356543
$533$	36.8038	1.59415
$534$	31.6712	1.37054
$535$	−10.7901	−0.466495
$536$	3.56634	0.154042
$537$	−68.9859	−2.97696
$538$	−4.02188	−0.173396
$539$	0	0
$540$	19.8811	0.855546
$541$	−11.0219	−0.473868	−0.236934	−	0.971526i	$-0.576142\pi$
$541$	−11.0219	−0.473868	−0.236934	+	0.971526i	$0.576142\pi$
$542$	−18.6020	−0.799024
$543$	−53.9524	−2.31532
$544$	2.43366	0.104343
$545$	−17.9365	−0.768315
$546$	19.8811	0.850833
$547$	27.5663	1.17865	0.589326	−	0.807896i	$-0.299393\pi$
$547$	27.5663	1.17865	0.589326	+	0.807896i	$0.299393\pi$
$548$	15.3702	0.656581
$549$	0.679264	0.0289903
$550$	0	0
$551$	16.4474	0.700684
$552$	1.94461	0.0827679
$553$	10.2237	0.434757
$554$	29.6712	1.26061
$555$	30.4474	1.29242
$556$	9.09105	0.385546
$557$	−16.5385	−0.700758	−0.350379	−	0.936608i	$-0.613947\pi$
$557$	−16.5385	−0.700758	−0.350379	+	0.936608i	$0.613947\pi$
$558$	−81.0770	−3.43226
$559$	−2.51094	−0.106201
$560$	−1.00000	−0.0422577
$561$	0	0
$562$	−18.3148	−0.772562
$563$	22.5939	0.952218	0.476109	−	0.879386i	$-0.342047\pi$
$563$	22.5939	0.952218	0.476109	+	0.879386i	$0.342047\pi$
$564$	−17.9365	−0.755263
$565$	−8.50283	−0.357717
$566$	17.0138	0.715142
$567$	−41.8949	−1.75942
$568$	15.6574	0.656969
$569$	−15.1383	−0.634632	−0.317316	−	0.948320i	$-0.602782\pi$
$569$	−15.1383	−0.634632	−0.317316	+	0.948320i	$0.602782\pi$
$570$	28.2375	1.18274
$571$	10.2929	0.430744	0.215372	−	0.976532i	$-0.430904\pi$
$571$	10.2929	0.430744	0.215372	+	0.976532i	$0.430904\pi$
$572$	0	0
$573$	−13.7068	−0.572610
$574$	6.35639	0.265311
$575$	0.566335	0.0236178
$576$	8.79005	0.366252
$577$	−7.00811	−0.291752	−0.145876	−	0.989303i	$-0.546600\pi$
$577$	−7.00811	−0.291752	−0.145876	+	0.989303i	$0.546600\pi$
$578$	−11.0773	−0.460754
$579$	15.4612	0.642546
$580$	2.00000	0.0830455
$581$	1.43366	0.0594784
$582$	−16.9781	−0.703766
$583$	0	0
$584$	−0.433665	−0.0179452
$585$	−50.8949	−2.10424
$586$	−22.9503	−0.948067
$587$	26.1740	1.08032	0.540158	−	0.841564i	$-0.318365\pi$
$587$	26.1740	1.08032	0.540158	+	0.841564i	$0.318365\pi$
$588$	3.43366	0.141602
$589$	−75.8532	−3.12548
$590$	7.51094	0.309221
$591$	36.1383	1.48653
$592$	8.86733	0.364445
$593$	−32.2156	−1.32294	−0.661468	−	0.749973i	$-0.730066\pi$
$593$	−32.2156	−1.32294	−0.661468	+	0.749973i	$0.730066\pi$
$594$	0	0
$595$	−2.43366	−0.0997705
$596$	6.44743	0.264097
$597$	−39.2315	−1.60564
$598$	−3.27911	−0.134093
$599$	10.4891	0.428571	0.214286	−	0.976771i	$-0.431258\pi$
$599$	10.4891	0.428571	0.214286	+	0.976771i	$0.431258\pi$
$600$	3.43366	0.140179
$601$	−19.0692	−0.777847	−0.388924	−	0.921270i	$-0.627153\pi$
$601$	−19.0692	−0.777847	−0.388924	+	0.921270i	$0.627153\pi$
$602$	−0.433665	−0.0176749
$603$	31.3483	1.27660
$604$	10.5663	0.429938
$605$	0	0
$606$	−43.4612	−1.76549
$607$	−37.3758	−1.51704	−0.758519	−	0.651651i	$-0.774077\pi$
$607$	−37.3758	−1.51704	−0.758519	+	0.651651i	$0.774077\pi$
$608$	8.22372	0.333516
$609$	−6.86733	−0.278278
$610$	0.0772765	0.00312883
$611$	30.2456	1.22361
$612$	21.3920	0.864722
$613$	23.3285	0.942231	0.471115	−	0.882072i	$-0.343852\pi$
$613$	23.3285	0.942231	0.471115	+	0.882072i	$0.343852\pi$
$614$	7.13267	0.287851
$615$	−21.8257	−0.880097
$616$	0	0
$617$	18.8949	0.760679	0.380339	−	0.924847i	$-0.375807\pi$
$617$	18.8949	0.760679	0.380339	+	0.924847i	$0.375807\pi$
$618$	−13.7819	−0.554391
$619$	−32.1902	−1.29383	−0.646917	−	0.762561i	$-0.723942\pi$
$619$	−32.1902	−1.29383	−0.646917	+	0.762561i	$0.723942\pi$
$620$	−9.22372	−0.370433
$621$	11.2594	0.451823
$622$	−18.2456	−0.731582
$623$	−9.22372	−0.369540
$624$	−19.8811	−0.795881
$625$	1.00000	0.0400000
$626$	−27.0219	−1.08001
$627$	0	0
$628$	12.5028	0.498917
$629$	21.5801	0.860455
$630$	−8.79005	−0.350204
$631$	−11.8119	−0.470226	−0.235113	−	0.971968i	$-0.575546\pi$
$631$	−11.8119	−0.470226	−0.235113	+	0.971968i	$0.575546\pi$
$632$	−10.2237	−0.406678
$633$	6.33665	0.251859
$634$	8.01377	0.318267
$635$	9.27911	0.368230
$636$	−18.2018	−0.721750
$637$	−5.79005	−0.229410
$638$	0	0
$639$	137.629	5.44453
$640$	1.00000	0.0395285
$641$	−25.7403	−1.01668	−0.508341	−	0.861156i	$-0.669741\pi$
$641$	−25.7403	−1.01668	−0.508341	+	0.861156i	$0.669741\pi$
$642$	−37.0494	−1.46222
$643$	26.0438	1.02707	0.513533	−	0.858070i	$-0.328336\pi$
$643$	26.0438	1.02707	0.513533	+	0.858070i	$0.328336\pi$
$644$	−0.566335	−0.0223167
$645$	1.48906	0.0586316
$646$	20.0138	0.787431
$647$	3.11890	0.122617	0.0613083	−	0.998119i	$-0.480473\pi$
$647$	3.11890	0.122617	0.0613083	+	0.998119i	$0.480473\pi$
$648$	41.8949	1.64579
$649$	0	0
$650$	−5.79005	−0.227105
$651$	31.6712	1.24129
$652$	−14.5247	−0.568832
$653$	−8.27911	−0.323987	−0.161993	−	0.986792i	$-0.551792\pi$
$653$	−8.27911	−0.323987	−0.161993	+	0.986792i	$0.551792\pi$
$654$	−61.5879	−2.40828
$655$	−12.3010	−0.480640
$656$	−6.35639	−0.248175
$657$	−3.81193	−0.148718
$658$	5.22372	0.203642
$659$	−13.9527	−0.543521	−0.271760	−	0.962365i	$-0.587606\pi$
$659$	−13.9527	−0.543521	−0.271760	+	0.962365i	$0.587606\pi$
$660$	0	0
$661$	−6.51905	−0.253562	−0.126781	−	0.991931i	$-0.540465\pi$
$661$	−6.51905	−0.253562	−0.126781	+	0.991931i	$0.540465\pi$
$662$	−12.6493	−0.491628
$663$	−48.3839	−1.87908
$664$	−1.43366	−0.0556370
$665$	−8.22372	−0.318902
$666$	77.9443	3.02028
$667$	1.13267	0.0438572
$668$	−9.30099	−0.359866
$669$	−45.1405	−1.74523
$670$	3.56634	0.137780
$671$	0	0
$672$	−3.43366	−0.132456
$673$	10.6574	0.410812	0.205406	−	0.978677i	$-0.434149\pi$
$673$	10.6574	0.410812	0.205406	+	0.978677i	$0.434149\pi$
$674$	−14.8119	−0.570534
$675$	19.8811	0.765224
$676$	20.5247	0.789412
$677$	−37.4139	−1.43793	−0.718967	−	0.695044i	$-0.755385\pi$
$677$	−37.4139	−1.43793	−0.718967	+	0.695044i	$0.755385\pi$
$678$	−29.1959	−1.12126
$679$	4.94461	0.189757
$680$	2.43366	0.0933268
$681$	−25.0219	−0.958841
$682$	0	0
$683$	−19.2099	−0.735048	−0.367524	−	0.930014i	$-0.619794\pi$
$683$	−19.2099	−0.735048	−0.367524	+	0.930014i	$0.619794\pi$
$684$	72.2869	2.76396
$685$	15.3702	0.587264
$686$	−1.00000	−0.0381802
$687$	−8.66901	−0.330743
$688$	0.433665	0.0165333
$689$	30.6930	1.16931
$690$	1.94461	0.0740299
$691$	14.1048	0.536573	0.268286	−	0.963339i	$-0.413543\pi$
$691$	14.1048	0.536573	0.268286	+	0.963339i	$0.413543\pi$
$692$	7.58011	0.288152
$693$	0	0
$694$	−26.6793	−1.01273
$695$	9.09105	0.344843
$696$	6.86733	0.260305
$697$	−15.4693	−0.585942
$698$	0.475289	0.0179899
$699$	73.9086	2.79548
$700$	−1.00000	−0.0377964
$701$	−2.77628	−0.104859	−0.0524294	−	0.998625i	$-0.516696\pi$
$701$	−2.77628	−0.104859	−0.0524294	+	0.998625i	$0.516696\pi$
$702$	−115.113	−4.34465
$703$	72.9224	2.75032
$704$	0	0
$705$	−17.9365	−0.675527
$706$	−10.0138	−0.376873
$707$	12.6574	0.476030
$708$	25.7901	0.969250
$709$	−1.17644	−0.0441819	−0.0220910	−	0.999756i	$-0.507032\pi$
$709$	−1.17644	−0.0441819	−0.0220910	+	0.999756i	$0.507032\pi$
$710$	15.6574	0.587611
$711$	−89.8670	−3.37028
$712$	9.22372	0.345673
$713$	−5.22372	−0.195630
$714$	−8.35639	−0.312730
$715$	0	0
$716$	−20.0910	−0.750838
$717$	52.4633	1.95928
$718$	7.47529	0.278975
$719$	8.74032	0.325959	0.162979	−	0.986629i	$-0.447890\pi$
$719$	8.74032	0.325959	0.162979	+	0.986629i	$0.447890\pi$
$720$	8.79005	0.327586
$721$	4.01377	0.149481
$722$	48.6295	1.80980
$723$	−21.1327	−0.785932
$724$	−15.7128	−0.583961
$725$	2.00000	0.0742781
$726$	0	0
$727$	29.0057	1.07576	0.537880	−	0.843021i	$-0.319225\pi$
$727$	29.0057	1.07576	0.537880	+	0.843021i	$0.319225\pi$
$728$	5.79005	0.214594
$729$	163.463	6.05419
$730$	−0.433665	−0.0160506
$731$	1.05539	0.0390352
$732$	0.265341	0.00980730
$733$	−14.7682	−0.545475	−0.272737	−	0.962089i	$-0.587929\pi$
$733$	−14.7682	−0.545475	−0.272737	+	0.962089i	$0.587929\pi$
$734$	30.1683	1.11353
$735$	3.43366	0.126653
$736$	0.566335	0.0208754
$737$	0	0
$738$	−55.8730	−2.05671
$739$	16.0910	0.591919	0.295959	−	0.955201i	$-0.404361\pi$
$739$	16.0910	0.591919	0.295959	+	0.955201i	$0.404361\pi$
$740$	8.86733	0.325970
$741$	−163.497	−6.00620
$742$	5.30099	0.194606
$743$	−9.88921	−0.362800	−0.181400	−	0.983409i	$-0.558063\pi$
$743$	−9.88921	−0.362800	−0.181400	+	0.983409i	$0.558063\pi$
$744$	−31.6712	−1.16112
$745$	6.44743	0.236216
$746$	13.2099	0.483651
$747$	−12.6020	−0.461083
$748$	0	0
$749$	10.7901	0.394260
$750$	3.43366	0.125380
$751$	−32.4831	−1.18532	−0.592662	−	0.805451i	$-0.701923\pi$
$751$	−32.4831	−1.18532	−0.592662	+	0.805451i	$0.701923\pi$
$752$	−5.22372	−0.190489
$753$	−9.58011	−0.349118
$754$	−11.5801	−0.421723
$755$	10.5663	0.384548
$756$	−19.8811	−0.723069
$757$	32.7703	1.19106	0.595529	−	0.803334i	$-0.296943\pi$
$757$	32.7703	1.19106	0.595529	+	0.803334i	$0.296943\pi$
$758$	19.1602	0.695930
$759$	0	0
$760$	8.22372	0.298306
$761$	9.40581	0.340960	0.170480	−	0.985361i	$-0.445468\pi$
$761$	9.40581	0.340960	0.170480	+	0.985361i	$0.445468\pi$
$762$	31.8614	1.15422
$763$	17.9365	0.649345
$764$	−3.99189	−0.144421
$765$	21.3920	0.773431
$766$	6.58822	0.238042
$767$	−43.4887	−1.57029
$768$	3.43366	0.123902
$769$	1.70712	0.0615603	0.0307801	−	0.999526i	$-0.490201\pi$
$769$	1.70712	0.0615603	0.0307801	+	0.999526i	$0.490201\pi$
$770$	0	0
$771$	12.7763	0.460127
$772$	4.50283	0.162060
$773$	−0.0991587	−0.00356649	−0.00178324	−	0.999998i	$-0.500568\pi$
$773$	−0.0991587	−0.00356649	−0.00178324	+	0.999998i	$0.500568\pi$
$774$	3.81193	0.137017
$775$	−9.22372	−0.331326
$776$	−4.94461	−0.177501
$777$	−30.4474	−1.09230
$778$	−25.1129	−0.900342
$779$	−52.2731	−1.87288
$780$	−19.8811	−0.711858
$781$	0	0
$782$	1.37827	0.0492868
$783$	39.7622	1.42099
$784$	1.00000	0.0357143
$785$	12.5028	0.446245
$786$	−42.2375	−1.50656
$787$	−19.0057	−0.677479	−0.338739	−	0.940880i	$-0.610000\pi$
$787$	−19.0057	−0.677479	−0.338739	+	0.940880i	$0.610000\pi$
$788$	10.5247	0.374927
$789$	75.1324	2.67478
$790$	−10.2237	−0.363744
$791$	8.50283	0.302326
$792$	0	0
$793$	−0.447435	−0.0158889
$794$	−11.5801	−0.410963
$795$	−18.2018	−0.645553
$796$	−11.4256	−0.404968
$797$	−3.52471	−0.124852	−0.0624258	−	0.998050i	$-0.519884\pi$
$797$	−3.52471	−0.124852	−0.0624258	+	0.998050i	$0.519884\pi$
$798$	−28.2375	−0.999596
$799$	−12.7128	−0.449746
$800$	1.00000	0.0353553
$801$	81.0770	2.86471
$802$	−31.7347	−1.12059
$803$	0	0
$804$	12.2456	0.431869
$805$	−0.566335	−0.0199607
$806$	53.4058	1.88114
$807$	−13.8098	−0.486128
$808$	−12.6574	−0.445285
$809$	25.3483	0.891198	0.445599	−	0.895233i	$-0.352991\pi$
$809$	25.3483	0.891198	0.445599	+	0.895233i	$0.352991\pi$
$810$	41.8949	1.47204
$811$	−10.7565	−0.377713	−0.188857	−	0.982005i	$-0.560478\pi$
$811$	−10.7565	−0.377713	−0.188857	+	0.982005i	$0.560478\pi$
$812$	−2.00000	−0.0701862
$813$	−63.8730	−2.24012
$814$	0	0
$815$	−14.5247	−0.508778
$816$	8.35639	0.292532
$817$	3.56634	0.124770
$818$	2.15455	0.0753322
$819$	50.8949	1.77841
$820$	−6.35639	−0.221975
$821$	51.0494	1.78164	0.890819	−	0.454359i	$-0.150132\pi$
$821$	51.0494	1.78164	0.890819	+	0.454359i	$0.150132\pi$
$822$	52.7760	1.84077
$823$	−13.7622	−0.479720	−0.239860	−	0.970807i	$-0.577102\pi$
$823$	−13.7622	−0.479720	−0.239860	+	0.970807i	$0.577102\pi$
$824$	−4.01377	−0.139826
$825$	0	0
$826$	−7.51094	−0.261339
$827$	−18.5445	−0.644854	−0.322427	−	0.946594i	$-0.604499\pi$
$827$	−18.5445	−0.644854	−0.322427	+	0.946594i	$0.604499\pi$
$828$	4.97812	0.173002
$829$	8.54659	0.296835	0.148418	−	0.988925i	$-0.452582\pi$
$829$	8.54659	0.296835	0.148418	+	0.988925i	$0.452582\pi$
$830$	−1.43366	−0.0497632
$831$	101.881	3.53421
$832$	−5.79005	−0.200734
$833$	2.43366	0.0843215
$834$	31.2156	1.08091
$835$	−9.30099	−0.321874
$836$	0	0
$837$	−183.378	−6.33846
$838$	−13.5385	−0.467679
$839$	4.04728	0.139728	0.0698639	−	0.997557i	$-0.477744\pi$
$839$	4.04728	0.139728	0.0698639	+	0.997557i	$0.477744\pi$
$840$	−3.43366	−0.118473
$841$	−25.0000	−0.862069
$842$	−1.42555	−0.0491278
$843$	−62.8868	−2.16593
$844$	1.84545	0.0635229
$845$	20.5247	0.706072
$846$	−45.9168	−1.57865
$847$	0	0
$848$	−5.30099	−0.182037
$849$	58.4196	2.00495
$850$	2.43366	0.0834740
$851$	5.02188	0.172148
$852$	53.7622	1.84186
$853$	42.9778	1.47153	0.735766	−	0.677236i	$-0.236822\pi$
$853$	42.9778	1.47153	0.735766	+	0.677236i	$0.236822\pi$
$854$	−0.0772765	−0.00264435
$855$	72.2869	2.47216
$856$	−10.7901	−0.368796
$857$	15.6239	0.533701	0.266851	−	0.963738i	$-0.414017\pi$
$857$	15.6239	0.533701	0.266851	+	0.963738i	$0.414017\pi$
$858$	0	0
$859$	−22.6355	−0.772313	−0.386157	−	0.922433i	$-0.626198\pi$
$859$	−22.6355	−0.772313	−0.386157	+	0.922433i	$0.626198\pi$
$860$	0.433665	0.0147878
$861$	21.8257	0.743818
$862$	−30.1129	−1.02565
$863$	31.6239	1.07649	0.538245	−	0.842789i	$-0.319088\pi$
$863$	31.6239	1.07649	0.538245	+	0.842789i	$0.319088\pi$
$864$	19.8811	0.676369
$865$	7.58011	0.257731
$866$	29.2513	0.993998
$867$	−38.0357	−1.29176
$868$	9.22372	0.313073
$869$	0	0
$870$	6.86733	0.232824
$871$	−20.6493	−0.699674
$872$	−17.9365	−0.607406
$873$	−43.4633	−1.47101
$874$	4.65738	0.157538
$875$	−1.00000	−0.0338062
$876$	−1.48906	−0.0503106
$877$	42.8811	1.44799	0.723996	−	0.689804i	$-0.242303\pi$
$877$	42.8811	1.44799	0.723996	+	0.689804i	$0.242303\pi$
$878$	33.0692	1.11603
$879$	−78.8035	−2.65798
$880$	0	0
$881$	−29.9168	−1.00792	−0.503960	−	0.863727i	$-0.668124\pi$
$881$	−29.9168	−1.00792	−0.503960	+	0.863727i	$0.668124\pi$
$882$	8.79005	0.295976
$883$	19.0992	0.642738	0.321369	−	0.946954i	$-0.395857\pi$
$883$	19.0992	0.642738	0.321369	+	0.946954i	$0.395857\pi$
$884$	−14.0910	−0.473933
$885$	25.7901	0.866923
$886$	−28.1821	−0.946796
$887$	12.3839	0.415812	0.207906	−	0.978149i	$-0.433335\pi$
$887$	12.3839	0.415812	0.207906	+	0.978149i	$0.433335\pi$
$888$	30.4474	1.02175
$889$	−9.27911	−0.311211
$890$	9.22372	0.309180
$891$	0	0
$892$	−13.1464	−0.440175
$893$	−42.9584	−1.43755
$894$	22.1383	0.740416
$895$	−20.0910	−0.671570
$896$	−1.00000	−0.0334077
$897$	−11.2594	−0.375939
$898$	−0.734659	−0.0245159
$899$	−18.4474	−0.615256
$900$	8.79005	0.293002
$901$	−12.9008	−0.429789
$902$	0	0
$903$	−1.48906	−0.0495528
$904$	−8.50283	−0.282800
$905$	−15.7128	−0.522310
$906$	36.2813	1.20536
$907$	−45.2375	−1.50209	−0.751043	−	0.660253i	$-0.770449\pi$
$907$	−45.2375	−1.50209	−0.751043	+	0.660253i	$0.770449\pi$
$908$	−7.28722	−0.241835
$909$	−111.259	−3.69023
$910$	5.79005	0.191938
$911$	−8.78194	−0.290959	−0.145479	−	0.989361i	$-0.546472\pi$
$911$	−8.78194	−0.290959	−0.145479	+	0.989361i	$0.546472\pi$
$912$	28.2375	0.935037
$913$	0	0
$914$	5.21561	0.172517
$915$	0.265341	0.00877192
$916$	−2.52471	−0.0834188
$917$	12.3010	0.406215
$918$	48.3839	1.59691
$919$	−20.5858	−0.679062	−0.339531	−	0.940595i	$-0.610268\pi$
$919$	−20.5858	−0.679062	−0.339531	+	0.940595i	$0.610268\pi$
$920$	0.566335	0.0186715
$921$	24.4912	0.807012
$922$	−24.9446	−0.821507
$923$	−90.6571	−2.98401
$924$	0	0
$925$	8.86733	0.291556
$926$	−15.6158	−0.513166
$927$	−35.2813	−1.15879
$928$	2.00000	0.0656532
$929$	−56.8949	−1.86666	−0.933330	−	0.359019i	$-0.883111\pi$
$929$	−56.8949	−1.86666	−0.933330	+	0.359019i	$0.883111\pi$
$930$	−31.6712	−1.03854
$931$	8.22372	0.269521
$932$	21.5247	0.705065
$933$	−62.6493	−2.05105
$934$	−36.9030	−1.20750
$935$	0	0
$936$	−50.8949	−1.66355
$937$	9.69089	0.316588	0.158294	−	0.987392i	$-0.449401\pi$
$937$	9.69089	0.316588	0.158294	+	0.987392i	$0.449401\pi$
$938$	−3.56634	−0.116445
$939$	−92.7841	−3.02789
$940$	−5.22372	−0.170379
$941$	21.2537	0.692851	0.346426	−	0.938077i	$-0.387395\pi$
$941$	21.2537	0.692851	0.346426	+	0.938077i	$0.387395\pi$
$942$	42.9305	1.39875
$943$	−3.59985	−0.117227
$944$	7.51094	0.244460
$945$	−19.8811	−0.646732
$946$	0	0
$947$	−9.20749	−0.299203	−0.149602	−	0.988746i	$-0.547799\pi$
$947$	−9.20749	−0.299203	−0.149602	+	0.988746i	$0.547799\pi$
$948$	−35.1048	−1.14015
$949$	2.51094	0.0815086
$950$	8.22372	0.266813
$951$	27.5166	0.892287
$952$	−2.43366	−0.0788755
$953$	15.7625	0.510598	0.255299	−	0.966862i	$-0.417826\pi$
$953$	15.7625	0.510598	0.255299	+	0.966862i	$0.417826\pi$
$954$	−46.5960	−1.50860
$955$	−3.99189	−0.129174
$956$	15.2791	0.494162
$957$	0	0
$958$	16.0910	0.519878
$959$	−15.3702	−0.496328
$960$	3.43366	0.110821
$961$	54.0770	1.74442
$962$	−51.3423	−1.65534
$963$	−94.8451	−3.05634
$964$	−6.15455	−0.198225
$965$	4.50283	0.144951
$966$	−1.94461	−0.0625667
$967$	−0.558223	−0.0179513	−0.00897563	−	0.999960i	$-0.502857\pi$
$967$	−0.558223	−0.0179513	−0.00897563	+	0.999960i	$0.502857\pi$
$968$	0	0
$969$	68.7206	2.20762
$970$	−4.94461	−0.158762
$971$	8.67115	0.278271	0.139135	−	0.990273i	$-0.455568\pi$
$971$	8.67115	0.278271	0.139135	+	0.990273i	$0.455568\pi$
$972$	84.2096	2.70103
$973$	−9.09105	−0.291446
$974$	−38.6376	−1.23803
$975$	−19.8811	−0.636705
$976$	0.0772765	0.00247356
$977$	15.1210	0.483765	0.241882	−	0.970306i	$-0.422235\pi$
$977$	15.1210	0.483765	0.241882	+	0.970306i	$0.422235\pi$
$978$	−49.8730	−1.59476
$979$	0	0
$980$	1.00000	0.0319438
$981$	−157.663	−5.03378
$982$	39.5604	1.26242
$983$	−14.8511	−0.473677	−0.236838	−	0.971549i	$-0.576111\pi$
$983$	−14.8511	−0.473677	−0.236838	+	0.971549i	$0.576111\pi$
$984$	−21.8257	−0.695778
$985$	10.5247	0.335345
$986$	4.86733	0.155007
$987$	17.9365	0.570925
$988$	−47.6158	−1.51486
$989$	0.245600	0.00780961
$990$	0	0
$991$	−40.8257	−1.29687	−0.648436	−	0.761269i	$-0.724576\pi$
$991$	−40.8257	−1.29687	−0.648436	+	0.761269i	$0.724576\pi$
$992$	−9.22372	−0.292853
$993$	−43.4334	−1.37832
$994$	−15.6574	−0.496622
$995$	−11.4256	−0.362214
$996$	−4.92272	−0.155982
$997$	−0.629842	−0.0199473	−0.00997364	−	0.999950i	$-0.503175\pi$
$997$	−0.629842	−0.0199473	−0.00997364	+	0.999950i	$0.503175\pi$
$998$	−42.4474	−1.34365
$999$	176.292	5.57764

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8470.2.a.cm.1.3	yes	3
11.10	odd	2		8470.2.a.cj.1.3	✓	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8470.2.a.cj.1.3	✓	3	11.10	odd	2
8470.2.a.cm.1.3	yes	3	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:	\( N \)	\(=\)	\( 8470 = 2 \cdot 5 \cdot 7 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8470.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +3.43366 q^{3} +1.00000 q^{4} +1.00000 q^{5} +3.43366 q^{6} -1.00000 q^{7} +1.00000 q^{8} +8.79005 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} +3.43366 q^{3} +1.00000 q^{4} +1.00000 q^{5} +3.43366 q^{6} -1.00000 q^{7} +1.00000 q^{8} +8.79005 q^{9} +1.00000 q^{10} +3.43366 q^{12} -5.79005 q^{13} -1.00000 q^{14} +3.43366 q^{15} +1.00000 q^{16} +2.43366 q^{17} +8.79005 q^{18} +8.22372 q^{19} +1.00000 q^{20} -3.43366 q^{21} +0.566335 q^{23} +3.43366 q^{24} +1.00000 q^{25} -5.79005 q^{26} +19.8811 q^{27} -1.00000 q^{28} +2.00000 q^{29} +3.43366 q^{30} -9.22372 q^{31} +1.00000 q^{32} +2.43366 q^{34} -1.00000 q^{35} +8.79005 q^{36} +8.86733 q^{37} +8.22372 q^{38} -19.8811 q^{39} +1.00000 q^{40} -6.35639 q^{41} -3.43366 q^{42} +0.433665 q^{43} +8.79005 q^{45} +0.566335 q^{46} -5.22372 q^{47} +3.43366 q^{48} +1.00000 q^{49} +1.00000 q^{50} +8.35639 q^{51} -5.79005 q^{52} -5.30099 q^{53} +19.8811 q^{54} -1.00000 q^{56} +28.2375 q^{57} +2.00000 q^{58} +7.51094 q^{59} +3.43366 q^{60} +0.0772765 q^{61} -9.22372 q^{62} -8.79005 q^{63} +1.00000 q^{64} -5.79005 q^{65} +3.56634 q^{67} +2.43366 q^{68} +1.94461 q^{69} -1.00000 q^{70} +15.6574 q^{71} +8.79005 q^{72} -0.433665 q^{73} +8.86733 q^{74} +3.43366 q^{75} +8.22372 q^{76} -19.8811 q^{78} -10.2237 q^{79} +1.00000 q^{80} +41.8949 q^{81} -6.35639 q^{82} -1.43366 q^{83} -3.43366 q^{84} +2.43366 q^{85} +0.433665 q^{86} +6.86733 q^{87} +9.22372 q^{89} +8.79005 q^{90} +5.79005 q^{91} +0.566335 q^{92} -31.6712 q^{93} -5.22372 q^{94} +8.22372 q^{95} +3.43366 q^{96} -4.94461 q^{97} +1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{2} + 2 q^{3} + 3 q^{4} + 3 q^{5} + 2 q^{6} - 3 q^{7} + 3 q^{8} + 13 q^{9}+O(q^{10}) \) 3 * q + 3 * q^2 + 2 * q^3 + 3 * q^4 + 3 * q^5 + 2 * q^6 - 3 * q^7 + 3 * q^8 + 13 * q^9	\( 3 q + 3 q^{2} + 2 q^{3} + 3 q^{4} + 3 q^{5} + 2 q^{6} - 3 q^{7} + 3 q^{8} + 13 q^{9} + 3 q^{10} + 2 q^{12} - 4 q^{13} - 3 q^{14} + 2 q^{15} + 3 q^{16} - q^{17} + 13 q^{18} + 3 q^{19} + 3 q^{20} - 2 q^{21} + 10 q^{23} + 2 q^{24} + 3 q^{25} - 4 q^{26} + 8 q^{27} - 3 q^{28} + 6 q^{29} + 2 q^{30} - 6 q^{31} + 3 q^{32} - q^{34} - 3 q^{35} + 13 q^{36} + 10 q^{37} + 3 q^{38} - 8 q^{39} + 3 q^{40} - 14 q^{41} - 2 q^{42} - 7 q^{43} + 13 q^{45} + 10 q^{46} + 6 q^{47} + 2 q^{48} + 3 q^{49} + 3 q^{50} + 20 q^{51} - 4 q^{52} + 9 q^{53} + 8 q^{54} - 3 q^{56} + 28 q^{57} + 6 q^{58} + 11 q^{59} + 2 q^{60} - 3 q^{61} - 6 q^{62} - 13 q^{63} + 3 q^{64} - 4 q^{65} + 19 q^{67} - q^{68} - 14 q^{69} - 3 q^{70} + 17 q^{71} + 13 q^{72} + 7 q^{73} + 10 q^{74} + 2 q^{75} + 3 q^{76} - 8 q^{78} - 9 q^{79} + 3 q^{80} + 39 q^{81} - 14 q^{82} + 4 q^{83} - 2 q^{84} - q^{85} - 7 q^{86} + 4 q^{87} + 6 q^{89} + 13 q^{90} + 4 q^{91} + 10 q^{92} - 30 q^{93} + 6 q^{94} + 3 q^{95} + 2 q^{96} + 5 q^{97} + 3 q^{98}+O(q^{100}) \) 3 * q + 3 * q^2 + 2 * q^3 + 3 * q^4 + 3 * q^5 + 2 * q^6 - 3 * q^7 + 3 * q^8 + 13 * q^9 + 3 * q^10 + 2 * q^12 - 4 * q^13 - 3 * q^14 + 2 * q^15 + 3 * q^16 - q^17 + 13 * q^18 + 3 * q^19 + 3 * q^20 - 2 * q^21 + 10 * q^23 + 2 * q^24 + 3 * q^25 - 4 * q^26 + 8 * q^27 - 3 * q^28 + 6 * q^29 + 2 * q^30 - 6 * q^31 + 3 * q^32 - q^34 - 3 * q^35 + 13 * q^36 + 10 * q^37 + 3 * q^38 - 8 * q^39 + 3 * q^40 - 14 * q^41 - 2 * q^42 - 7 * q^43 + 13 * q^45 + 10 * q^46 + 6 * q^47 + 2 * q^48 + 3 * q^49 + 3 * q^50 + 20 * q^51 - 4 * q^52 + 9 * q^53 + 8 * q^54 - 3 * q^56 + 28 * q^57 + 6 * q^58 + 11 * q^59 + 2 * q^60 - 3 * q^61 - 6 * q^62 - 13 * q^63 + 3 * q^64 - 4 * q^65 + 19 * q^67 - q^68 - 14 * q^69 - 3 * q^70 + 17 * q^71 + 13 * q^72 + 7 * q^73 + 10 * q^74 + 2 * q^75 + 3 * q^76 - 8 * q^78 - 9 * q^79 + 3 * q^80 + 39 * q^81 - 14 * q^82 + 4 * q^83 - 2 * q^84 - q^85 - 7 * q^86 + 4 * q^87 + 6 * q^89 + 13 * q^90 + 4 * q^91 + 10 * q^92 - 30 * q^93 + 6 * q^94 + 3 * q^95 + 2 * q^96 + 5 * q^97 + 3 * q^98

Introduction

L-functions

Modular forms

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Database

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