LMFDB - Embedded newform 8470.2.a.bx.1.1

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:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{10})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - x - 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 770)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$1.61803$ 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)

$f(q)$	$=$	$q+1.00000 q^{2} -1.61803 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.61803 q^{6} -1.00000 q^{7} +1.00000 q^{8} -0.381966 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} -1.61803 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.61803 q^{6} -1.00000 q^{7} +1.00000 q^{8} -0.381966 q^{9} -1.00000 q^{10} -1.61803 q^{12} +0.763932 q^{13} -1.00000 q^{14} +1.61803 q^{15} +1.00000 q^{16} -0.618034 q^{17} -0.381966 q^{18} +2.61803 q^{19} -1.00000 q^{20} +1.61803 q^{21} +0.472136 q^{23} -1.61803 q^{24} +1.00000 q^{25} +0.763932 q^{26} +5.47214 q^{27} -1.00000 q^{28} -4.00000 q^{29} +1.61803 q^{30} +4.47214 q^{31} +1.00000 q^{32} -0.618034 q^{34} +1.00000 q^{35} -0.381966 q^{36} -3.70820 q^{37} +2.61803 q^{38} -1.23607 q^{39} -1.00000 q^{40} -6.38197 q^{41} +1.61803 q^{42} +2.38197 q^{43} +0.381966 q^{45} +0.472136 q^{46} -11.2361 q^{47} -1.61803 q^{48} +1.00000 q^{49} +1.00000 q^{50} +1.00000 q^{51} +0.763932 q^{52} +2.47214 q^{53} +5.47214 q^{54} -1.00000 q^{56} -4.23607 q^{57} -4.00000 q^{58} -1.38197 q^{59} +1.61803 q^{60} +8.00000 q^{61} +4.47214 q^{62} +0.381966 q^{63} +1.00000 q^{64} -0.763932 q^{65} +5.09017 q^{67} -0.618034 q^{68} -0.763932 q^{69} +1.00000 q^{70} -6.47214 q^{71} -0.381966 q^{72} +11.3262 q^{73} -3.70820 q^{74} -1.61803 q^{75} +2.61803 q^{76} -1.23607 q^{78} +11.7082 q^{79} -1.00000 q^{80} -7.70820 q^{81} -6.38197 q^{82} +0.0901699 q^{83} +1.61803 q^{84} +0.618034 q^{85} +2.38197 q^{86} +6.47214 q^{87} +3.14590 q^{89} +0.381966 q^{90} -0.763932 q^{91} +0.472136 q^{92} -7.23607 q^{93} -11.2361 q^{94} -2.61803 q^{95} -1.61803 q^{96} +13.7984 q^{97} +1.00000 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{2} - q^{3} + 2 q^{4} - 2 q^{5} - q^{6} - 2 q^{7} + 2 q^{8} - 3 q^{9}+O(q^{10}) $ 2 * q + 2 * q^2 - q^3 + 2 * q^4 - 2 * q^5 - q^6 - 2 * q^7 + 2 * q^8 - 3 * q^9	$ 2 q + 2 q^{2} - q^{3} + 2 q^{4} - 2 q^{5} - q^{6} - 2 q^{7} + 2 q^{8} - 3 q^{9} - 2 q^{10} - q^{12} + 6 q^{13} - 2 q^{14} + q^{15} + 2 q^{16} + q^{17} - 3 q^{18} + 3 q^{19} - 2 q^{20} + q^{21} - 8 q^{23} - q^{24} + 2 q^{25} + 6 q^{26} + 2 q^{27} - 2 q^{28} - 8 q^{29} + q^{30} + 2 q^{32} + q^{34} + 2 q^{35} - 3 q^{36} + 6 q^{37} + 3 q^{38} + 2 q^{39} - 2 q^{40} - 15 q^{41} + q^{42} + 7 q^{43} + 3 q^{45} - 8 q^{46} - 18 q^{47} - q^{48} + 2 q^{49} + 2 q^{50} + 2 q^{51} + 6 q^{52} - 4 q^{53} + 2 q^{54} - 2 q^{56} - 4 q^{57} - 8 q^{58} - 5 q^{59} + q^{60} + 16 q^{61} + 3 q^{63} + 2 q^{64} - 6 q^{65} - q^{67} + q^{68} - 6 q^{69} + 2 q^{70} - 4 q^{71} - 3 q^{72} + 7 q^{73} + 6 q^{74} - q^{75} + 3 q^{76} + 2 q^{78} + 10 q^{79} - 2 q^{80} - 2 q^{81} - 15 q^{82} - 11 q^{83} + q^{84} - q^{85} + 7 q^{86} + 4 q^{87} + 13 q^{89} + 3 q^{90} - 6 q^{91} - 8 q^{92} - 10 q^{93} - 18 q^{94} - 3 q^{95} - q^{96} + 3 q^{97} + 2 q^{98}+O(q^{100}) $ 2 * q + 2 * q^2 - q^3 + 2 * q^4 - 2 * q^5 - q^6 - 2 * q^7 + 2 * q^8 - 3 * q^9 - 2 * q^10 - q^12 + 6 * q^13 - 2 * q^14 + q^15 + 2 * q^16 + q^17 - 3 * q^18 + 3 * q^19 - 2 * q^20 + q^21 - 8 * q^23 - q^24 + 2 * q^25 + 6 * q^26 + 2 * q^27 - 2 * q^28 - 8 * q^29 + q^30 + 2 * q^32 + q^34 + 2 * q^35 - 3 * q^36 + 6 * q^37 + 3 * q^38 + 2 * q^39 - 2 * q^40 - 15 * q^41 + q^42 + 7 * q^43 + 3 * q^45 - 8 * q^46 - 18 * q^47 - q^48 + 2 * q^49 + 2 * q^50 + 2 * q^51 + 6 * q^52 - 4 * q^53 + 2 * q^54 - 2 * q^56 - 4 * q^57 - 8 * q^58 - 5 * q^59 + q^60 + 16 * q^61 + 3 * q^63 + 2 * q^64 - 6 * q^65 - q^67 + q^68 - 6 * q^69 + 2 * q^70 - 4 * q^71 - 3 * q^72 + 7 * q^73 + 6 * q^74 - q^75 + 3 * q^76 + 2 * q^78 + 10 * q^79 - 2 * q^80 - 2 * q^81 - 15 * q^82 - 11 * q^83 + q^84 - q^85 + 7 * q^86 + 4 * q^87 + 13 * q^89 + 3 * q^90 - 6 * q^91 - 8 * q^92 - 10 * q^93 - 18 * q^94 - 3 * q^95 - q^96 + 3 * q^97 + 2 * 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
$3$	−1.61803	−0.934172	−0.467086	−	0.884212i	$-0.654696\pi$
$3$	−1.61803	−0.934172	−0.467086	+	0.884212i	$0.654696\pi$
$4$	1.00000	0.500000
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	−1.61803	−0.660560
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	1.00000	0.353553
$9$	−0.381966	−0.127322
$10$	−1.00000	−0.316228
$11$	0	0
$11$	0	0
$12$	−1.61803	−0.467086
$13$	0.763932	0.211877	0.105938	−	0.994373i	$-0.466215\pi$
$13$	0.763932	0.211877	0.105938	+	0.994373i	$0.466215\pi$
$14$	−1.00000	−0.267261
$15$	1.61803	0.417775
$16$	1.00000	0.250000
$17$	−0.618034	−0.149895	−0.0749476	−	0.997187i	$-0.523879\pi$
$17$	−0.618034	−0.149895	−0.0749476	+	0.997187i	$0.523879\pi$
$18$	−0.381966	−0.0900303
$19$	2.61803	0.600618	0.300309	−	0.953842i	$-0.402910\pi$
$19$	2.61803	0.600618	0.300309	+	0.953842i	$0.402910\pi$
$20$	−1.00000	−0.223607
$21$	1.61803	0.353084
$22$	0	0
$23$	0.472136	0.0984472	0.0492236	−	0.998788i	$-0.484325\pi$
$23$	0.472136	0.0984472	0.0492236	+	0.998788i	$0.484325\pi$
$24$	−1.61803	−0.330280
$25$	1.00000	0.200000
$26$	0.763932	0.149819
$27$	5.47214	1.05311
$28$	−1.00000	−0.188982
$29$	−4.00000	−0.742781	−0.371391	−	0.928477i	$-0.621119\pi$
$29$	−4.00000	−0.742781	−0.371391	+	0.928477i	$0.621119\pi$
$30$	1.61803	0.295411
$31$	4.47214	0.803219	0.401610	−	0.915811i	$-0.368451\pi$
$31$	4.47214	0.803219	0.401610	+	0.915811i	$0.368451\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	−0.618034	−0.105992
$35$	1.00000	0.169031
$36$	−0.381966	−0.0636610
$37$	−3.70820	−0.609625	−0.304812	−	0.952412i	$-0.598594\pi$
$37$	−3.70820	−0.609625	−0.304812	+	0.952412i	$0.598594\pi$
$38$	2.61803	0.424701
$39$	−1.23607	−0.197929
$40$	−1.00000	−0.158114
$41$	−6.38197	−0.996696	−0.498348	−	0.866977i	$-0.666060\pi$
$41$	−6.38197	−0.996696	−0.498348	+	0.866977i	$0.666060\pi$
$42$	1.61803	0.249668
$43$	2.38197	0.363246	0.181623	−	0.983368i	$-0.441865\pi$
$43$	2.38197	0.363246	0.181623	+	0.983368i	$0.441865\pi$
$44$	0	0
$45$	0.381966	0.0569401
$46$	0.472136	0.0696126
$47$	−11.2361	−1.63895	−0.819474	−	0.573116i	$-0.805735\pi$
$47$	−11.2361	−1.63895	−0.819474	+	0.573116i	$0.805735\pi$
$48$	−1.61803	−0.233543
$49$	1.00000	0.142857
$50$	1.00000	0.141421
$51$	1.00000	0.140028
$52$	0.763932	0.105938
$53$	2.47214	0.339574	0.169787	−	0.985481i	$-0.445692\pi$
$53$	2.47214	0.339574	0.169787	+	0.985481i	$0.445692\pi$
$54$	5.47214	0.744663
$55$	0	0
$56$	−1.00000	−0.133631
$57$	−4.23607	−0.561081
$58$	−4.00000	−0.525226
$59$	−1.38197	−0.179917	−0.0899583	−	0.995946i	$-0.528673\pi$
$59$	−1.38197	−0.179917	−0.0899583	+	0.995946i	$0.528673\pi$
$60$	1.61803	0.208887
$61$	8.00000	1.02430	0.512148	−	0.858898i	$-0.328850\pi$
$61$	8.00000	1.02430	0.512148	+	0.858898i	$0.328850\pi$
$62$	4.47214	0.567962
$63$	0.381966	0.0481232
$64$	1.00000	0.125000
$65$	−0.763932	−0.0947541
$66$	0	0
$67$	5.09017	0.621863	0.310932	−	0.950432i	$-0.399359\pi$
$67$	5.09017	0.621863	0.310932	+	0.950432i	$0.399359\pi$
$68$	−0.618034	−0.0749476
$69$	−0.763932	−0.0919666
$70$	1.00000	0.119523
$71$	−6.47214	−0.768101	−0.384051	−	0.923312i	$-0.625471\pi$
$71$	−6.47214	−0.768101	−0.384051	+	0.923312i	$0.625471\pi$
$72$	−0.381966	−0.0450151
$73$	11.3262	1.32564	0.662818	−	0.748781i	$-0.269360\pi$
$73$	11.3262	1.32564	0.662818	+	0.748781i	$0.269360\pi$
$74$	−3.70820	−0.431070
$75$	−1.61803	−0.186834
$76$	2.61803	0.300309
$77$	0	0
$78$	−1.23607	−0.139957
$79$	11.7082	1.31728	0.658638	−	0.752460i	$-0.271133\pi$
$79$	11.7082	1.31728	0.658638	+	0.752460i	$0.271133\pi$
$80$	−1.00000	−0.111803
$81$	−7.70820	−0.856467
$82$	−6.38197	−0.704770
$83$	0.0901699	0.00989744	0.00494872	−	0.999988i	$-0.498425\pi$
$83$	0.0901699	0.00989744	0.00494872	+	0.999988i	$0.498425\pi$
$84$	1.61803	0.176542
$85$	0.618034	0.0670352
$86$	2.38197	0.256854
$87$	6.47214	0.693886
$88$	0	0
$89$	3.14590	0.333465	0.166732	−	0.986002i	$-0.446678\pi$
$89$	3.14590	0.333465	0.166732	+	0.986002i	$0.446678\pi$
$90$	0.381966	0.0402628
$91$	−0.763932	−0.0800818
$92$	0.472136	0.0492236
$93$	−7.23607	−0.750345
$94$	−11.2361	−1.15891
$95$	−2.61803	−0.268605
$96$	−1.61803	−0.165140
$97$	13.7984	1.40101	0.700506	−	0.713646i	$-0.252958\pi$
$97$	13.7984	1.40101	0.700506	+	0.713646i	$0.252958\pi$
$98$	1.00000	0.101015
$99$	0	0
$100$	1.00000	0.100000
$101$	−12.7639	−1.27006	−0.635029	−	0.772488i	$-0.719012\pi$
$101$	−12.7639	−1.27006	−0.635029	+	0.772488i	$0.719012\pi$
$102$	1.00000	0.0990148
$103$	13.7082	1.35071	0.675355	−	0.737493i	$-0.263991\pi$
$103$	13.7082	1.35071	0.675355	+	0.737493i	$0.263991\pi$
$104$	0.763932	0.0749097
$105$	−1.61803	−0.157904
$106$	2.47214	0.240115
$107$	−8.09017	−0.782106	−0.391053	−	0.920368i	$-0.627889\pi$
$107$	−8.09017	−0.782106	−0.391053	+	0.920368i	$0.627889\pi$
$108$	5.47214	0.526557
$109$	−10.1803	−0.975100	−0.487550	−	0.873095i	$-0.662109\pi$
$109$	−10.1803	−0.975100	−0.487550	+	0.873095i	$0.662109\pi$
$110$	0	0
$111$	6.00000	0.569495
$112$	−1.00000	−0.0944911
$113$	−10.7984	−1.01583	−0.507913	−	0.861409i	$-0.669583\pi$
$113$	−10.7984	−1.01583	−0.507913	+	0.861409i	$0.669583\pi$
$114$	−4.23607	−0.396744
$115$	−0.472136	−0.0440269
$116$	−4.00000	−0.371391
$117$	−0.291796	−0.0269766
$118$	−1.38197	−0.127220
$119$	0.618034	0.0566551
$120$	1.61803	0.147706
$121$	0	0
$122$	8.00000	0.724286
$123$	10.3262	0.931086
$124$	4.47214	0.401610
$125$	−1.00000	−0.0894427
$126$	0.381966	0.0340282
$127$	8.18034	0.725888	0.362944	−	0.931811i	$-0.381772\pi$
$127$	8.18034	0.725888	0.362944	+	0.931811i	$0.381772\pi$
$128$	1.00000	0.0883883
$129$	−3.85410	−0.339335
$130$	−0.763932	−0.0670013
$131$	−8.14590	−0.711710	−0.355855	−	0.934541i	$-0.615810\pi$
$131$	−8.14590	−0.711710	−0.355855	+	0.934541i	$0.615810\pi$
$132$	0	0
$133$	−2.61803	−0.227012
$134$	5.09017	0.439724
$135$	−5.47214	−0.470966
$136$	−0.618034	−0.0529960
$137$	−12.8541	−1.09820	−0.549100	−	0.835757i	$-0.685029\pi$
$137$	−12.8541	−1.09820	−0.549100	+	0.835757i	$0.685029\pi$
$138$	−0.763932	−0.0650302
$139$	1.52786	0.129592	0.0647959	−	0.997899i	$-0.479360\pi$
$139$	1.52786	0.129592	0.0647959	+	0.997899i	$0.479360\pi$
$140$	1.00000	0.0845154
$141$	18.1803	1.53106
$142$	−6.47214	−0.543130
$143$	0	0
$144$	−0.381966	−0.0318305
$145$	4.00000	0.332182
$146$	11.3262	0.937366
$147$	−1.61803	−0.133453
$148$	−3.70820	−0.304812
$149$	7.70820	0.631481	0.315740	−	0.948846i	$-0.397747\pi$
$149$	7.70820	0.631481	0.315740	+	0.948846i	$0.397747\pi$
$150$	−1.61803	−0.132112
$151$	−20.1803	−1.64225	−0.821126	−	0.570746i	$-0.806654\pi$
$151$	−20.1803	−1.64225	−0.821126	+	0.570746i	$0.806654\pi$
$152$	2.61803	0.212351
$153$	0.236068	0.0190850
$154$	0	0
$155$	−4.47214	−0.359211
$156$	−1.23607	−0.0989646
$157$	−6.18034	−0.493245	−0.246622	−	0.969112i	$-0.579321\pi$
$157$	−6.18034	−0.493245	−0.246622	+	0.969112i	$0.579321\pi$
$158$	11.7082	0.931455
$159$	−4.00000	−0.317221
$160$	−1.00000	−0.0790569
$161$	−0.472136	−0.0372095
$162$	−7.70820	−0.605614
$163$	−22.6180	−1.77158	−0.885791	−	0.464085i	$-0.846383\pi$
$163$	−22.6180	−1.77158	−0.885791	+	0.464085i	$0.846383\pi$
$164$	−6.38197	−0.498348
$165$	0	0
$166$	0.0901699	0.00699854
$167$	−10.0000	−0.773823	−0.386912	−	0.922117i	$-0.626458\pi$
$167$	−10.0000	−0.773823	−0.386912	+	0.922117i	$0.626458\pi$
$168$	1.61803	0.124834
$169$	−12.4164	−0.955108
$170$	0.618034	0.0474010
$171$	−1.00000	−0.0764719
$172$	2.38197	0.181623
$173$	−4.76393	−0.362195	−0.181098	−	0.983465i	$-0.557965\pi$
$173$	−4.76393	−0.362195	−0.181098	+	0.983465i	$0.557965\pi$
$174$	6.47214	0.490651
$175$	−1.00000	−0.0755929
$176$	0	0
$177$	2.23607	0.168073
$178$	3.14590	0.235795
$179$	−8.14590	−0.608853	−0.304427	−	0.952536i	$-0.598465\pi$
$179$	−8.14590	−0.608853	−0.304427	+	0.952536i	$0.598465\pi$
$180$	0.381966	0.0284701
$181$	−23.1246	−1.71884	−0.859419	−	0.511271i	$-0.829175\pi$
$181$	−23.1246	−1.71884	−0.859419	+	0.511271i	$0.829175\pi$
$182$	−0.763932	−0.0566264
$183$	−12.9443	−0.956868
$184$	0.472136	0.0348063
$185$	3.70820	0.272633
$186$	−7.23607	−0.530574
$187$	0	0
$188$	−11.2361	−0.819474
$189$	−5.47214	−0.398039
$190$	−2.61803	−0.189932
$191$	−2.00000	−0.144715	−0.0723575	−	0.997379i	$-0.523052\pi$
$191$	−2.00000	−0.144715	−0.0723575	+	0.997379i	$0.523052\pi$
$192$	−1.61803	−0.116772
$193$	3.52786	0.253941	0.126971	−	0.991906i	$-0.459475\pi$
$193$	3.52786	0.253941	0.126971	+	0.991906i	$0.459475\pi$
$194$	13.7984	0.990666
$195$	1.23607	0.0885167
$196$	1.00000	0.0714286
$197$	5.41641	0.385903	0.192952	−	0.981208i	$-0.438194\pi$
$197$	5.41641	0.385903	0.192952	+	0.981208i	$0.438194\pi$
$198$	0	0
$199$	3.23607	0.229399	0.114699	−	0.993400i	$-0.463410\pi$
$199$	3.23607	0.229399	0.114699	+	0.993400i	$0.463410\pi$
$200$	1.00000	0.0707107
$201$	−8.23607	−0.580927
$202$	−12.7639	−0.898067
$203$	4.00000	0.280745
$204$	1.00000	0.0700140
$205$	6.38197	0.445736
$206$	13.7082	0.955096
$207$	−0.180340	−0.0125345
$208$	0.763932	0.0529692
$209$	0	0
$210$	−1.61803	−0.111655
$211$	−28.0902	−1.93381	−0.966904	−	0.255142i	$-0.917878\pi$
$211$	−28.0902	−1.93381	−0.966904	+	0.255142i	$0.917878\pi$
$212$	2.47214	0.169787
$213$	10.4721	0.717539
$214$	−8.09017	−0.553033
$215$	−2.38197	−0.162449
$216$	5.47214	0.372332
$217$	−4.47214	−0.303588
$218$	−10.1803	−0.689500
$219$	−18.3262	−1.23837
$220$	0	0
$221$	−0.472136	−0.0317593
$222$	6.00000	0.402694
$223$	−22.9443	−1.53646	−0.768231	−	0.640173i	$-0.778863\pi$
$223$	−22.9443	−1.53646	−0.768231	+	0.640173i	$0.778863\pi$
$224$	−1.00000	−0.0668153
$225$	−0.381966	−0.0254644
$226$	−10.7984	−0.718297
$227$	11.5623	0.767417	0.383709	−	0.923454i	$-0.374647\pi$
$227$	11.5623	0.767417	0.383709	+	0.923454i	$0.374647\pi$
$228$	−4.23607	−0.280540
$229$	7.70820	0.509372	0.254686	−	0.967024i	$-0.418028\pi$
$229$	7.70820	0.509372	0.254686	+	0.967024i	$0.418028\pi$
$230$	−0.472136	−0.0311317
$231$	0	0
$232$	−4.00000	−0.262613
$233$	−19.3262	−1.26610	−0.633052	−	0.774109i	$-0.718198\pi$
$233$	−19.3262	−1.26610	−0.633052	+	0.774109i	$0.718198\pi$
$234$	−0.291796	−0.0190753
$235$	11.2361	0.732960
$236$	−1.38197	−0.0899583
$237$	−18.9443	−1.23056
$238$	0.618034	0.0400612
$239$	−4.29180	−0.277613	−0.138807	−	0.990320i	$-0.544327\pi$
$239$	−4.29180	−0.277613	−0.138807	+	0.990320i	$0.544327\pi$
$240$	1.61803	0.104444
$241$	13.8541	0.892421	0.446211	−	0.894928i	$-0.352773\pi$
$241$	13.8541	0.892421	0.446211	+	0.894928i	$0.352773\pi$
$242$	0	0
$243$	−3.94427	−0.253025
$244$	8.00000	0.512148
$245$	−1.00000	−0.0638877
$246$	10.3262	0.658377
$247$	2.00000	0.127257
$248$	4.47214	0.283981
$249$	−0.145898	−0.00924591
$250$	−1.00000	−0.0632456
$251$	17.8885	1.12911	0.564557	−	0.825394i	$-0.309047\pi$
$251$	17.8885	1.12911	0.564557	+	0.825394i	$0.309047\pi$
$252$	0.381966	0.0240616
$253$	0	0
$254$	8.18034	0.513280
$255$	−1.00000	−0.0626224
$256$	1.00000	0.0625000
$257$	−5.43769	−0.339194	−0.169597	−	0.985513i	$-0.554247\pi$
$257$	−5.43769	−0.339194	−0.169597	+	0.985513i	$0.554247\pi$
$258$	−3.85410	−0.239946
$259$	3.70820	0.230417
$260$	−0.763932	−0.0473771
$261$	1.52786	0.0945724
$262$	−8.14590	−0.503255
$263$	12.0000	0.739952	0.369976	−	0.929041i	$-0.379366\pi$
$263$	12.0000	0.739952	0.369976	+	0.929041i	$0.379366\pi$
$264$	0	0
$265$	−2.47214	−0.151862
$266$	−2.61803	−0.160522
$267$	−5.09017	−0.311513
$268$	5.09017	0.310932
$269$	14.0000	0.853595	0.426798	−	0.904347i	$-0.359642\pi$
$269$	14.0000	0.853595	0.426798	+	0.904347i	$0.359642\pi$
$270$	−5.47214	−0.333024
$271$	−6.29180	−0.382199	−0.191100	−	0.981571i	$-0.561205\pi$
$271$	−6.29180	−0.382199	−0.191100	+	0.981571i	$0.561205\pi$
$272$	−0.618034	−0.0374738
$273$	1.23607	0.0748102
$274$	−12.8541	−0.776545
$275$	0	0
$276$	−0.763932	−0.0459833
$277$	17.1246	1.02892	0.514459	−	0.857515i	$-0.327993\pi$
$277$	17.1246	1.02892	0.514459	+	0.857515i	$0.327993\pi$
$278$	1.52786	0.0916352
$279$	−1.70820	−0.102267
$280$	1.00000	0.0597614
$281$	−17.7984	−1.06176	−0.530881	−	0.847446i	$-0.678139\pi$
$281$	−17.7984	−1.06176	−0.530881	+	0.847446i	$0.678139\pi$
$282$	18.1803	1.08262
$283$	−9.88854	−0.587813	−0.293906	−	0.955834i	$-0.594955\pi$
$283$	−9.88854	−0.587813	−0.293906	+	0.955834i	$0.594955\pi$
$284$	−6.47214	−0.384051
$285$	4.23607	0.250923
$286$	0	0
$287$	6.38197	0.376716
$288$	−0.381966	−0.0225076
$289$	−16.6180	−0.977531
$290$	4.00000	0.234888
$291$	−22.3262	−1.30879
$292$	11.3262	0.662818
$293$	−15.7082	−0.917683	−0.458842	−	0.888518i	$-0.651735\pi$
$293$	−15.7082	−0.917683	−0.458842	+	0.888518i	$0.651735\pi$
$294$	−1.61803	−0.0943657
$295$	1.38197	0.0804612
$296$	−3.70820	−0.215535
$297$	0	0
$298$	7.70820	0.446524
$299$	0.360680	0.0208586
$300$	−1.61803	−0.0934172
$301$	−2.38197	−0.137294
$302$	−20.1803	−1.16125
$303$	20.6525	1.18645
$304$	2.61803	0.150155
$305$	−8.00000	−0.458079
$306$	0.236068	0.0134951
$307$	3.50658	0.200131	0.100065	−	0.994981i	$-0.468095\pi$
$307$	3.50658	0.200131	0.100065	+	0.994981i	$0.468095\pi$
$308$	0	0
$309$	−22.1803	−1.26180
$310$	−4.47214	−0.254000
$311$	−1.23607	−0.0700910	−0.0350455	−	0.999386i	$-0.511158\pi$
$311$	−1.23607	−0.0700910	−0.0350455	+	0.999386i	$0.511158\pi$
$312$	−1.23607	−0.0699786
$313$	−12.3262	−0.696720	−0.348360	−	0.937361i	$-0.613261\pi$
$313$	−12.3262	−0.696720	−0.348360	+	0.937361i	$0.613261\pi$
$314$	−6.18034	−0.348777
$315$	−0.381966	−0.0215213
$316$	11.7082	0.658638
$317$	17.8885	1.00472	0.502360	−	0.864658i	$-0.332465\pi$
$317$	17.8885	1.00472	0.502360	+	0.864658i	$0.332465\pi$
$318$	−4.00000	−0.224309
$319$	0	0
$320$	−1.00000	−0.0559017
$321$	13.0902	0.730622
$322$	−0.472136	−0.0263111
$323$	−1.61803	−0.0900298
$324$	−7.70820	−0.428234
$325$	0.763932	0.0423753
$326$	−22.6180	−1.25270
$327$	16.4721	0.910911
$328$	−6.38197	−0.352385
$329$	11.2361	0.619464
$330$	0	0
$331$	−3.32624	−0.182827	−0.0914133	−	0.995813i	$-0.529138\pi$
$331$	−3.32624	−0.182827	−0.0914133	+	0.995813i	$0.529138\pi$
$332$	0.0901699	0.00494872
$333$	1.41641	0.0776187
$334$	−10.0000	−0.547176
$335$	−5.09017	−0.278106
$336$	1.61803	0.0882710
$337$	1.27051	0.0692091	0.0346045	−	0.999401i	$-0.488983\pi$
$337$	1.27051	0.0692091	0.0346045	+	0.999401i	$0.488983\pi$
$338$	−12.4164	−0.675364
$339$	17.4721	0.948956
$340$	0.618034	0.0335176
$341$	0	0
$342$	−1.00000	−0.0540738
$343$	−1.00000	−0.0539949
$344$	2.38197	0.128427
$345$	0.763932	0.0411287
$346$	−4.76393	−0.256111
$347$	−2.03444	−0.109215	−0.0546073	−	0.998508i	$-0.517391\pi$
$347$	−2.03444	−0.109215	−0.0546073	+	0.998508i	$0.517391\pi$
$348$	6.47214	0.346943
$349$	−1.52786	−0.0817847	−0.0408923	−	0.999164i	$-0.513020\pi$
$349$	−1.52786	−0.0817847	−0.0408923	+	0.999164i	$0.513020\pi$
$350$	−1.00000	−0.0534522
$351$	4.18034	0.223130
$352$	0	0
$353$	−12.0344	−0.640529	−0.320264	−	0.947328i	$-0.603772\pi$
$353$	−12.0344	−0.640529	−0.320264	+	0.947328i	$0.603772\pi$
$354$	2.23607	0.118846
$355$	6.47214	0.343505
$356$	3.14590	0.166732
$357$	−1.00000	−0.0529256
$358$	−8.14590	−0.430524
$359$	−12.2918	−0.648736	−0.324368	−	0.945931i	$-0.605152\pi$
$359$	−12.2918	−0.648736	−0.324368	+	0.945931i	$0.605152\pi$
$360$	0.381966	0.0201314
$361$	−12.1459	−0.639258
$362$	−23.1246	−1.21540
$363$	0	0
$364$	−0.763932	−0.0400409
$365$	−11.3262	−0.592842
$366$	−12.9443	−0.676608
$367$	28.9443	1.51088	0.755439	−	0.655219i	$-0.227423\pi$
$367$	28.9443	1.51088	0.755439	+	0.655219i	$0.227423\pi$
$368$	0.472136	0.0246118
$369$	2.43769	0.126901
$370$	3.70820	0.192780
$371$	−2.47214	−0.128347
$372$	−7.23607	−0.375173
$373$	−12.9443	−0.670229	−0.335114	−	0.942177i	$-0.608775\pi$
$373$	−12.9443	−0.670229	−0.335114	+	0.942177i	$0.608775\pi$
$374$	0	0
$375$	1.61803	0.0835549
$376$	−11.2361	−0.579456
$377$	−3.05573	−0.157378
$378$	−5.47214	−0.281456
$379$	−16.8541	−0.865737	−0.432869	−	0.901457i	$-0.642499\pi$
$379$	−16.8541	−0.865737	−0.432869	+	0.901457i	$0.642499\pi$
$380$	−2.61803	−0.134302
$381$	−13.2361	−0.678104
$382$	−2.00000	−0.102329
$383$	4.29180	0.219301	0.109650	−	0.993970i	$-0.465027\pi$
$383$	4.29180	0.219301	0.109650	+	0.993970i	$0.465027\pi$
$384$	−1.61803	−0.0825700
$385$	0	0
$386$	3.52786	0.179564
$387$	−0.909830	−0.0462493
$388$	13.7984	0.700506
$389$	8.00000	0.405616	0.202808	−	0.979219i	$-0.434993\pi$
$389$	8.00000	0.405616	0.202808	+	0.979219i	$0.434993\pi$
$390$	1.23607	0.0625907
$391$	−0.291796	−0.0147568
$392$	1.00000	0.0505076
$393$	13.1803	0.664860
$394$	5.41641	0.272875
$395$	−11.7082	−0.589104
$396$	0	0
$397$	32.5410	1.63319	0.816593	−	0.577213i	$-0.195860\pi$
$397$	32.5410	1.63319	0.816593	+	0.577213i	$0.195860\pi$
$398$	3.23607	0.162209
$399$	4.23607	0.212069
$400$	1.00000	0.0500000
$401$	27.5066	1.37361	0.686806	−	0.726840i	$-0.259012\pi$
$401$	27.5066	1.37361	0.686806	+	0.726840i	$0.259012\pi$
$402$	−8.23607	−0.410778
$403$	3.41641	0.170183
$404$	−12.7639	−0.635029
$405$	7.70820	0.383024
$406$	4.00000	0.198517
$407$	0	0
$408$	1.00000	0.0495074
$409$	−31.3050	−1.54793	−0.773965	−	0.633228i	$-0.781729\pi$
$409$	−31.3050	−1.54793	−0.773965	+	0.633228i	$0.781729\pi$
$410$	6.38197	0.315183
$411$	20.7984	1.02591
$412$	13.7082	0.675355
$413$	1.38197	0.0680021
$414$	−0.180340	−0.00886322
$415$	−0.0901699	−0.00442627
$416$	0.763932	0.0374548
$417$	−2.47214	−0.121061
$418$	0	0
$419$	−14.7426	−0.720225	−0.360113	−	0.932909i	$-0.617262\pi$
$419$	−14.7426	−0.720225	−0.360113	+	0.932909i	$0.617262\pi$
$420$	−1.61803	−0.0789520
$421$	8.47214	0.412907	0.206453	−	0.978456i	$-0.433808\pi$
$421$	8.47214	0.412907	0.206453	+	0.978456i	$0.433808\pi$
$422$	−28.0902	−1.36741
$423$	4.29180	0.208674
$424$	2.47214	0.120058
$425$	−0.618034	−0.0299791
$426$	10.4721	0.507377
$427$	−8.00000	−0.387147
$428$	−8.09017	−0.391053
$429$	0	0
$430$	−2.38197	−0.114869
$431$	−19.7082	−0.949311	−0.474655	−	0.880172i	$-0.657427\pi$
$431$	−19.7082	−0.949311	−0.474655	+	0.880172i	$0.657427\pi$
$432$	5.47214	0.263278
$433$	1.20163	0.0577465	0.0288732	−	0.999583i	$-0.490808\pi$
$433$	1.20163	0.0577465	0.0288732	+	0.999583i	$0.490808\pi$
$434$	−4.47214	−0.214669
$435$	−6.47214	−0.310315
$436$	−10.1803	−0.487550
$437$	1.23607	0.0591292
$438$	−18.3262	−0.875662
$439$	−24.3607	−1.16267	−0.581336	−	0.813664i	$-0.697470\pi$
$439$	−24.3607	−1.16267	−0.581336	+	0.813664i	$0.697470\pi$
$440$	0	0
$441$	−0.381966	−0.0181889
$442$	−0.472136	−0.0224572
$443$	−26.3820	−1.25345	−0.626723	−	0.779243i	$-0.715604\pi$
$443$	−26.3820	−1.25345	−0.626723	+	0.779243i	$0.715604\pi$
$444$	6.00000	0.284747
$445$	−3.14590	−0.149130
$446$	−22.9443	−1.08644
$447$	−12.4721	−0.589912
$448$	−1.00000	−0.0472456
$449$	27.7426	1.30926	0.654628	−	0.755951i	$-0.272825\pi$
$449$	27.7426	1.30926	0.654628	+	0.755951i	$0.272825\pi$
$450$	−0.381966	−0.0180061
$451$	0	0
$452$	−10.7984	−0.507913
$453$	32.6525	1.53415
$454$	11.5623	0.542646
$455$	0.763932	0.0358137
$456$	−4.23607	−0.198372
$457$	5.43769	0.254365	0.127182	−	0.991879i	$-0.459407\pi$
$457$	5.43769	0.254365	0.127182	+	0.991879i	$0.459407\pi$
$458$	7.70820	0.360181
$459$	−3.38197	−0.157857
$460$	−0.472136	−0.0220135
$461$	−39.1246	−1.82221	−0.911107	−	0.412169i	$-0.864771\pi$
$461$	−39.1246	−1.82221	−0.911107	+	0.412169i	$0.864771\pi$
$462$	0	0
$463$	−30.6525	−1.42454	−0.712271	−	0.701905i	$-0.752333\pi$
$463$	−30.6525	−1.42454	−0.712271	+	0.701905i	$0.752333\pi$
$464$	−4.00000	−0.185695
$465$	7.23607	0.335565
$466$	−19.3262	−0.895271
$467$	−0.944272	−0.0436957	−0.0218478	−	0.999761i	$-0.506955\pi$
$467$	−0.944272	−0.0436957	−0.0218478	+	0.999761i	$0.506955\pi$
$468$	−0.291796	−0.0134883
$469$	−5.09017	−0.235042
$470$	11.2361	0.518281
$471$	10.0000	0.460776
$472$	−1.38197	−0.0636101
$473$	0	0
$474$	−18.9443	−0.870139
$475$	2.61803	0.120124
$476$	0.618034	0.0283275
$477$	−0.944272	−0.0432352
$478$	−4.29180	−0.196302
$479$	−23.1246	−1.05659	−0.528295	−	0.849061i	$-0.677169\pi$
$479$	−23.1246	−1.05659	−0.528295	+	0.849061i	$0.677169\pi$
$480$	1.61803	0.0738528
$481$	−2.83282	−0.129165
$482$	13.8541	0.631037
$483$	0.763932	0.0347601
$484$	0	0
$485$	−13.7984	−0.626552
$486$	−3.94427	−0.178916
$487$	−33.7771	−1.53059	−0.765293	−	0.643682i	$-0.777406\pi$
$487$	−33.7771	−1.53059	−0.765293	+	0.643682i	$0.777406\pi$
$488$	8.00000	0.362143
$489$	36.5967	1.65496
$490$	−1.00000	−0.0451754
$491$	−26.9230	−1.21502	−0.607509	−	0.794313i	$-0.707831\pi$
$491$	−26.9230	−1.21502	−0.607509	+	0.794313i	$0.707831\pi$
$492$	10.3262	0.465543
$493$	2.47214	0.111339
$494$	2.00000	0.0899843
$495$	0	0
$496$	4.47214	0.200805
$497$	6.47214	0.290315
$498$	−0.145898	−0.00653785
$499$	−1.61803	−0.0724331	−0.0362166	−	0.999344i	$-0.511531\pi$
$499$	−1.61803	−0.0724331	−0.0362166	+	0.999344i	$0.511531\pi$
$500$	−1.00000	−0.0447214
$501$	16.1803	0.722884
$502$	17.8885	0.798405
$503$	12.2918	0.548064	0.274032	−	0.961721i	$-0.411643\pi$
$503$	12.2918	0.548064	0.274032	+	0.961721i	$0.411643\pi$
$504$	0.381966	0.0170141
$505$	12.7639	0.567988
$506$	0	0
$507$	20.0902	0.892236
$508$	8.18034	0.362944
$509$	5.81966	0.257952	0.128976	−	0.991648i	$-0.458831\pi$
$509$	5.81966	0.257952	0.128976	+	0.991648i	$0.458831\pi$
$510$	−1.00000	−0.0442807
$511$	−11.3262	−0.501043
$512$	1.00000	0.0441942
$513$	14.3262	0.632519
$514$	−5.43769	−0.239846
$515$	−13.7082	−0.604056
$516$	−3.85410	−0.169667
$517$	0	0
$518$	3.70820	0.162929
$519$	7.70820	0.338353
$520$	−0.763932	−0.0335006
$521$	37.5623	1.64563	0.822817	−	0.568306i	$-0.192401\pi$
$521$	37.5623	1.64563	0.822817	+	0.568306i	$0.192401\pi$
$522$	1.52786	0.0668728
$523$	35.9230	1.57080	0.785401	−	0.618987i	$-0.212457\pi$
$523$	35.9230	1.57080	0.785401	+	0.618987i	$0.212457\pi$
$524$	−8.14590	−0.355855
$525$	1.61803	0.0706168
$526$	12.0000	0.523225
$527$	−2.76393	−0.120399
$528$	0	0
$529$	−22.7771	−0.990308
$530$	−2.47214	−0.107383
$531$	0.527864	0.0229073
$532$	−2.61803	−0.113506
$533$	−4.87539	−0.211177
$534$	−5.09017	−0.220273
$535$	8.09017	0.349769
$536$	5.09017	0.219862
$537$	13.1803	0.568774
$538$	14.0000	0.603583
$539$	0	0
$540$	−5.47214	−0.235483
$541$	−17.8197	−0.766127	−0.383064	−	0.923722i	$-0.625131\pi$
$541$	−17.8197	−0.766127	−0.383064	+	0.923722i	$0.625131\pi$
$542$	−6.29180	−0.270256
$543$	37.4164	1.60569
$544$	−0.618034	−0.0264980
$545$	10.1803	0.436078
$546$	1.23607	0.0528988
$547$	−26.9098	−1.15058	−0.575291	−	0.817949i	$-0.695111\pi$
$547$	−26.9098	−1.15058	−0.575291	+	0.817949i	$0.695111\pi$
$548$	−12.8541	−0.549100
$549$	−3.05573	−0.130415
$550$	0	0
$551$	−10.4721	−0.446128
$552$	−0.763932	−0.0325151
$553$	−11.7082	−0.497883
$554$	17.1246	0.727555
$555$	−6.00000	−0.254686
$556$	1.52786	0.0647959
$557$	−4.36068	−0.184768	−0.0923840	−	0.995723i	$-0.529449\pi$
$557$	−4.36068	−0.184768	−0.0923840	+	0.995723i	$0.529449\pi$
$558$	−1.70820	−0.0723140
$559$	1.81966	0.0769634
$560$	1.00000	0.0422577
$561$	0	0
$562$	−17.7984	−0.750779
$563$	−43.6869	−1.84118	−0.920592	−	0.390526i	$-0.872293\pi$
$563$	−43.6869	−1.84118	−0.920592	+	0.390526i	$0.872293\pi$
$564$	18.1803	0.765530
$565$	10.7984	0.454291
$566$	−9.88854	−0.415646
$567$	7.70820	0.323714
$568$	−6.47214	−0.271565
$569$	19.4508	0.815422	0.407711	−	0.913111i	$-0.366327\pi$
$569$	19.4508	0.815422	0.407711	+	0.913111i	$0.366327\pi$
$570$	4.23607	0.177429
$571$	−26.8328	−1.12292	−0.561459	−	0.827504i	$-0.689760\pi$
$571$	−26.8328	−1.12292	−0.561459	+	0.827504i	$0.689760\pi$
$572$	0	0
$573$	3.23607	0.135189
$574$	6.38197	0.266378
$575$	0.472136	0.0196894
$576$	−0.381966	−0.0159153
$577$	24.3262	1.01271	0.506357	−	0.862324i	$-0.330992\pi$
$577$	24.3262	1.01271	0.506357	+	0.862324i	$0.330992\pi$
$578$	−16.6180	−0.691219
$579$	−5.70820	−0.237225
$580$	4.00000	0.166091
$581$	−0.0901699	−0.00374088
$582$	−22.3262	−0.925452
$583$	0	0
$584$	11.3262	0.468683
$585$	0.291796	0.0120643
$586$	−15.7082	−0.648900
$587$	−10.6180	−0.438253	−0.219127	−	0.975696i	$-0.570321\pi$
$587$	−10.6180	−0.438253	−0.219127	+	0.975696i	$0.570321\pi$
$588$	−1.61803	−0.0667266
$589$	11.7082	0.482428
$590$	1.38197	0.0568946
$591$	−8.76393	−0.360500
$592$	−3.70820	−0.152406
$593$	12.6180	0.518161	0.259080	−	0.965856i	$-0.416581\pi$
$593$	12.6180	0.518161	0.259080	+	0.965856i	$0.416581\pi$
$594$	0	0
$595$	−0.618034	−0.0253369
$596$	7.70820	0.315740
$597$	−5.23607	−0.214298
$598$	0.360680	0.0147493
$599$	24.0000	0.980613	0.490307	−	0.871550i	$-0.336885\pi$
$599$	24.0000	0.980613	0.490307	+	0.871550i	$0.336885\pi$
$600$	−1.61803	−0.0660560
$601$	−6.56231	−0.267682	−0.133841	−	0.991003i	$-0.542731\pi$
$601$	−6.56231	−0.267682	−0.133841	+	0.991003i	$0.542731\pi$
$602$	−2.38197	−0.0970817
$603$	−1.94427	−0.0791769
$604$	−20.1803	−0.821126
$605$	0	0
$606$	20.6525	0.838949
$607$	34.0000	1.38002	0.690009	−	0.723801i	$-0.257607\pi$
$607$	34.0000	1.38002	0.690009	+	0.723801i	$0.257607\pi$
$608$	2.61803	0.106175
$609$	−6.47214	−0.262264
$610$	−8.00000	−0.323911
$611$	−8.58359	−0.347255
$612$	0.236068	0.00954248
$613$	−48.0689	−1.94148	−0.970742	−	0.240125i	$-0.922812\pi$
$613$	−48.0689	−1.94148	−0.970742	+	0.240125i	$0.922812\pi$
$614$	3.50658	0.141514
$615$	−10.3262	−0.416394
$616$	0	0
$617$	13.2016	0.531477	0.265739	−	0.964045i	$-0.414384\pi$
$617$	13.2016	0.531477	0.265739	+	0.964045i	$0.414384\pi$
$618$	−22.1803	−0.892224
$619$	20.4377	0.821460	0.410730	−	0.911757i	$-0.365274\pi$
$619$	20.4377	0.821460	0.410730	+	0.911757i	$0.365274\pi$
$620$	−4.47214	−0.179605
$621$	2.58359	0.103676
$622$	−1.23607	−0.0495618
$623$	−3.14590	−0.126038
$624$	−1.23607	−0.0494823
$625$	1.00000	0.0400000
$626$	−12.3262	−0.492656
$627$	0	0
$628$	−6.18034	−0.246622
$629$	2.29180	0.0913799
$630$	−0.381966	−0.0152179
$631$	−20.7639	−0.826599	−0.413300	−	0.910595i	$-0.635624\pi$
$631$	−20.7639	−0.826599	−0.413300	+	0.910595i	$0.635624\pi$
$632$	11.7082	0.465727
$633$	45.4508	1.80651
$634$	17.8885	0.710445
$635$	−8.18034	−0.324627
$636$	−4.00000	−0.158610
$637$	0.763932	0.0302681
$638$	0	0
$639$	2.47214	0.0977962
$640$	−1.00000	−0.0395285
$641$	24.6738	0.974555	0.487278	−	0.873247i	$-0.337990\pi$
$641$	24.6738	0.974555	0.487278	+	0.873247i	$0.337990\pi$
$642$	13.0902	0.516628
$643$	6.56231	0.258792	0.129396	−	0.991593i	$-0.458696\pi$
$643$	6.56231	0.258792	0.129396	+	0.991593i	$0.458696\pi$
$644$	−0.472136	−0.0186048
$645$	3.85410	0.151755
$646$	−1.61803	−0.0636607
$647$	−7.88854	−0.310131	−0.155065	−	0.987904i	$-0.549559\pi$
$647$	−7.88854	−0.310131	−0.155065	+	0.987904i	$0.549559\pi$
$648$	−7.70820	−0.302807
$649$	0	0
$650$	0.763932	0.0299639
$651$	7.23607	0.283604
$652$	−22.6180	−0.885791
$653$	−6.00000	−0.234798	−0.117399	−	0.993085i	$-0.537456\pi$
$653$	−6.00000	−0.234798	−0.117399	+	0.993085i	$0.537456\pi$
$654$	16.4721	0.644111
$655$	8.14590	0.318287
$656$	−6.38197	−0.249174
$657$	−4.32624	−0.168783
$658$	11.2361	0.438028
$659$	−21.0902	−0.821556	−0.410778	−	0.911735i	$-0.634743\pi$
$659$	−21.0902	−0.821556	−0.410778	+	0.911735i	$0.634743\pi$
$660$	0	0
$661$	37.7771	1.46936	0.734679	−	0.678415i	$-0.237333\pi$
$661$	37.7771	1.46936	0.734679	+	0.678415i	$0.237333\pi$
$662$	−3.32624	−0.129278
$663$	0.763932	0.0296687
$664$	0.0901699	0.00349927
$665$	2.61803	0.101523
$666$	1.41641	0.0548847
$667$	−1.88854	−0.0731247
$668$	−10.0000	−0.386912
$669$	37.1246	1.43532
$670$	−5.09017	−0.196650
$671$	0	0
$672$	1.61803	0.0624170
$673$	−13.8541	−0.534036	−0.267018	−	0.963691i	$-0.586038\pi$
$673$	−13.8541	−0.534036	−0.267018	+	0.963691i	$0.586038\pi$
$674$	1.27051	0.0489382
$675$	5.47214	0.210623
$676$	−12.4164	−0.477554
$677$	10.9443	0.420623	0.210311	−	0.977634i	$-0.432552\pi$
$677$	10.9443	0.420623	0.210311	+	0.977634i	$0.432552\pi$
$678$	17.4721	0.671013
$679$	−13.7984	−0.529533
$680$	0.618034	0.0237005
$681$	−18.7082	−0.716900
$682$	0	0
$683$	−45.3050	−1.73355	−0.866773	−	0.498703i	$-0.833810\pi$
$683$	−45.3050	−1.73355	−0.866773	+	0.498703i	$0.833810\pi$
$684$	−1.00000	−0.0382360
$685$	12.8541	0.491130
$686$	−1.00000	−0.0381802
$687$	−12.4721	−0.475842
$688$	2.38197	0.0908116
$689$	1.88854	0.0719478
$690$	0.763932	0.0290824
$691$	−3.20163	−0.121796	−0.0608978	−	0.998144i	$-0.519396\pi$
$691$	−3.20163	−0.121796	−0.0608978	+	0.998144i	$0.519396\pi$
$692$	−4.76393	−0.181098
$693$	0	0
$694$	−2.03444	−0.0772264
$695$	−1.52786	−0.0579552
$696$	6.47214	0.245326
$697$	3.94427	0.149400
$698$	−1.52786	−0.0578305
$699$	31.2705	1.18276
$700$	−1.00000	−0.0377964
$701$	4.18034	0.157889	0.0789446	−	0.996879i	$-0.474845\pi$
$701$	4.18034	0.157889	0.0789446	+	0.996879i	$0.474845\pi$
$702$	4.18034	0.157777
$703$	−9.70820	−0.366152
$704$	0	0
$705$	−18.1803	−0.684711
$706$	−12.0344	−0.452922
$707$	12.7639	0.480037
$708$	2.23607	0.0840366
$709$	32.7639	1.23048	0.615238	−	0.788342i	$-0.289060\pi$
$709$	32.7639	1.23048	0.615238	+	0.788342i	$0.289060\pi$
$710$	6.47214	0.242895
$711$	−4.47214	−0.167718
$712$	3.14590	0.117898
$713$	2.11146	0.0790747
$714$	−1.00000	−0.0374241
$715$	0	0
$716$	−8.14590	−0.304427
$717$	6.94427	0.259339
$718$	−12.2918	−0.458726
$719$	5.63932	0.210311	0.105156	−	0.994456i	$-0.466466\pi$
$719$	5.63932	0.210311	0.105156	+	0.994456i	$0.466466\pi$
$720$	0.381966	0.0142350
$721$	−13.7082	−0.510520
$722$	−12.1459	−0.452024
$723$	−22.4164	−0.833675
$724$	−23.1246	−0.859419
$725$	−4.00000	−0.148556
$726$	0	0
$727$	−25.3050	−0.938509	−0.469254	−	0.883063i	$-0.655477\pi$
$727$	−25.3050	−0.938509	−0.469254	+	0.883063i	$0.655477\pi$
$728$	−0.763932	−0.0283132
$729$	29.5066	1.09284
$730$	−11.3262	−0.419203
$731$	−1.47214	−0.0544489
$732$	−12.9443	−0.478434
$733$	19.2361	0.710500	0.355250	−	0.934771i	$-0.384396\pi$
$733$	19.2361	0.710500	0.355250	+	0.934771i	$0.384396\pi$
$734$	28.9443	1.06835
$735$	1.61803	0.0596821
$736$	0.472136	0.0174032
$737$	0	0
$738$	2.43769	0.0897328
$739$	26.9787	0.992428	0.496214	−	0.868200i	$-0.334723\pi$
$739$	26.9787	0.992428	0.496214	+	0.868200i	$0.334723\pi$
$740$	3.70820	0.136316
$741$	−3.23607	−0.118880
$742$	−2.47214	−0.0907550
$743$	22.0000	0.807102	0.403551	−	0.914957i	$-0.367776\pi$
$743$	22.0000	0.807102	0.403551	+	0.914957i	$0.367776\pi$
$744$	−7.23607	−0.265287
$745$	−7.70820	−0.282407
$746$	−12.9443	−0.473923
$747$	−0.0344419	−0.00126016
$748$	0	0
$749$	8.09017	0.295608
$750$	1.61803	0.0590822
$751$	−49.4164	−1.80323	−0.901615	−	0.432539i	$-0.857618\pi$
$751$	−49.4164	−1.80323	−0.901615	+	0.432539i	$0.857618\pi$
$752$	−11.2361	−0.409737
$753$	−28.9443	−1.05479
$754$	−3.05573	−0.111283
$755$	20.1803	0.734438
$756$	−5.47214	−0.199020
$757$	26.0689	0.947490	0.473745	−	0.880662i	$-0.342902\pi$
$757$	26.0689	0.947490	0.473745	+	0.880662i	$0.342902\pi$
$758$	−16.8541	−0.612169
$759$	0	0
$760$	−2.61803	−0.0949661
$761$	−6.72949	−0.243944	−0.121972	−	0.992534i	$-0.538922\pi$
$761$	−6.72949	−0.243944	−0.121972	+	0.992534i	$0.538922\pi$
$762$	−13.2361	−0.479492
$763$	10.1803	0.368553
$764$	−2.00000	−0.0723575
$765$	−0.236068	−0.00853506
$766$	4.29180	0.155069
$767$	−1.05573	−0.0381201
$768$	−1.61803	−0.0583858
$769$	−25.4164	−0.916539	−0.458270	−	0.888813i	$-0.651531\pi$
$769$	−25.4164	−0.916539	−0.458270	+	0.888813i	$0.651531\pi$
$770$	0	0
$771$	8.79837	0.316866
$772$	3.52786	0.126971
$773$	36.3607	1.30780	0.653901	−	0.756580i	$-0.273131\pi$
$773$	36.3607	1.30780	0.653901	+	0.756580i	$0.273131\pi$
$774$	−0.909830	−0.0327032
$775$	4.47214	0.160644
$776$	13.7984	0.495333
$777$	−6.00000	−0.215249
$778$	8.00000	0.286814
$779$	−16.7082	−0.598634
$780$	1.23607	0.0442583
$781$	0	0
$782$	−0.291796	−0.0104346
$783$	−21.8885	−0.782233
$784$	1.00000	0.0357143
$785$	6.18034	0.220586
$786$	13.1803	0.470127
$787$	−14.0344	−0.500274	−0.250137	−	0.968210i	$-0.580476\pi$
$787$	−14.0344	−0.500274	−0.250137	+	0.968210i	$0.580476\pi$
$788$	5.41641	0.192952
$789$	−19.4164	−0.691242
$790$	−11.7082	−0.416559
$791$	10.7984	0.383946
$792$	0	0
$793$	6.11146	0.217024
$794$	32.5410	1.15484
$795$	4.00000	0.141865
$796$	3.23607	0.114699
$797$	34.4721	1.22107	0.610533	−	0.791991i	$-0.290955\pi$
$797$	34.4721	1.22107	0.610533	+	0.791991i	$0.290955\pi$
$798$	4.23607	0.149955
$799$	6.94427	0.245671
$800$	1.00000	0.0353553
$801$	−1.20163	−0.0424574
$802$	27.5066	0.971291
$803$	0	0
$804$	−8.23607	−0.290464
$805$	0.472136	0.0166406
$806$	3.41641	0.120338
$807$	−22.6525	−0.797405
$808$	−12.7639	−0.449034
$809$	−26.4377	−0.929500	−0.464750	−	0.885442i	$-0.653856\pi$
$809$	−26.4377	−0.929500	−0.464750	+	0.885442i	$0.653856\pi$
$810$	7.70820	0.270839
$811$	−49.7984	−1.74866	−0.874329	−	0.485334i	$-0.838698\pi$
$811$	−49.7984	−1.74866	−0.874329	+	0.485334i	$0.838698\pi$
$812$	4.00000	0.140372
$813$	10.1803	0.357040
$814$	0	0
$815$	22.6180	0.792275
$816$	1.00000	0.0350070
$817$	6.23607	0.218172
$818$	−31.3050	−1.09455
$819$	0.291796	0.0101962
$820$	6.38197	0.222868
$821$	−49.4853	−1.72705	−0.863524	−	0.504307i	$-0.831748\pi$
$821$	−49.4853	−1.72705	−0.863524	+	0.504307i	$0.831748\pi$
$822$	20.7984	0.725427
$823$	51.4853	1.79466	0.897332	−	0.441356i	$-0.145502\pi$
$823$	51.4853	1.79466	0.897332	+	0.441356i	$0.145502\pi$
$824$	13.7082	0.477548
$825$	0	0
$826$	1.38197	0.0480847
$827$	−25.7426	−0.895160	−0.447580	−	0.894244i	$-0.647714\pi$
$827$	−25.7426	−0.895160	−0.447580	+	0.894244i	$0.647714\pi$
$828$	−0.180340	−0.00626724
$829$	2.94427	0.102259	0.0511294	−	0.998692i	$-0.483718\pi$
$829$	2.94427	0.102259	0.0511294	+	0.998692i	$0.483718\pi$
$830$	−0.0901699	−0.00312984
$831$	−27.7082	−0.961187
$832$	0.763932	0.0264846
$833$	−0.618034	−0.0214136
$834$	−2.47214	−0.0856031
$835$	10.0000	0.346064
$836$	0	0
$837$	24.4721	0.845881
$838$	−14.7426	−0.509276
$839$	−10.6525	−0.367764	−0.183882	−	0.982948i	$-0.558866\pi$
$839$	−10.6525	−0.367764	−0.183882	+	0.982948i	$0.558866\pi$
$840$	−1.61803	−0.0558275
$841$	−13.0000	−0.448276
$842$	8.47214	0.291969
$843$	28.7984	0.991869
$844$	−28.0902	−0.966904
$845$	12.4164	0.427137
$846$	4.29180	0.147555
$847$	0	0
$848$	2.47214	0.0848935
$849$	16.0000	0.549119
$850$	−0.618034	−0.0211984
$851$	−1.75078	−0.0600158
$852$	10.4721	0.358769
$853$	36.2918	1.24261	0.621304	−	0.783570i	$-0.286603\pi$
$853$	36.2918	1.24261	0.621304	+	0.783570i	$0.286603\pi$
$854$	−8.00000	−0.273754
$855$	1.00000	0.0341993
$856$	−8.09017	−0.276516
$857$	−47.6869	−1.62895	−0.814477	−	0.580196i	$-0.802976\pi$
$857$	−47.6869	−1.62895	−0.814477	+	0.580196i	$0.802976\pi$
$858$	0	0
$859$	43.8541	1.49628	0.748141	−	0.663539i	$-0.230947\pi$
$859$	43.8541	1.49628	0.748141	+	0.663539i	$0.230947\pi$
$860$	−2.38197	−0.0812244
$861$	−10.3262	−0.351917
$862$	−19.7082	−0.671264
$863$	40.2492	1.37010	0.685050	−	0.728496i	$-0.259780\pi$
$863$	40.2492	1.37010	0.685050	+	0.728496i	$0.259780\pi$
$864$	5.47214	0.186166
$865$	4.76393	0.161979
$866$	1.20163	0.0408329
$867$	26.8885	0.913183
$868$	−4.47214	−0.151794
$869$	0	0
$870$	−6.47214	−0.219426
$871$	3.88854	0.131758
$872$	−10.1803	−0.344750
$873$	−5.27051	−0.178380
$874$	1.23607	0.0418106
$875$	1.00000	0.0338062
$876$	−18.3262	−0.619186
$877$	21.1246	0.713327	0.356664	−	0.934233i	$-0.383914\pi$
$877$	21.1246	0.713327	0.356664	+	0.934233i	$0.383914\pi$
$878$	−24.3607	−0.822133
$879$	25.4164	0.857274
$880$	0	0
$881$	34.3394	1.15692	0.578462	−	0.815709i	$-0.303653\pi$
$881$	34.3394	1.15692	0.578462	+	0.815709i	$0.303653\pi$
$882$	−0.381966	−0.0128615
$883$	34.3820	1.15705	0.578523	−	0.815666i	$-0.303629\pi$
$883$	34.3820	1.15705	0.578523	+	0.815666i	$0.303629\pi$
$884$	−0.472136	−0.0158797
$885$	−2.23607	−0.0751646
$886$	−26.3820	−0.886319
$887$	22.0000	0.738688	0.369344	−	0.929293i	$-0.379582\pi$
$887$	22.0000	0.738688	0.369344	+	0.929293i	$0.379582\pi$
$888$	6.00000	0.201347
$889$	−8.18034	−0.274360
$890$	−3.14590	−0.105451
$891$	0	0
$892$	−22.9443	−0.768231
$893$	−29.4164	−0.984383
$894$	−12.4721	−0.417131
$895$	8.14590	0.272287
$896$	−1.00000	−0.0334077
$897$	−0.583592	−0.0194856
$898$	27.7426	0.925784
$899$	−17.8885	−0.596616
$900$	−0.381966	−0.0127322
$901$	−1.52786	−0.0509005
$902$	0	0
$903$	3.85410	0.128256
$904$	−10.7984	−0.359149
$905$	23.1246	0.768688
$906$	32.6525	1.08481
$907$	20.7984	0.690599	0.345299	−	0.938493i	$-0.387777\pi$
$907$	20.7984	0.690599	0.345299	+	0.938493i	$0.387777\pi$
$908$	11.5623	0.383709
$909$	4.87539	0.161706
$910$	0.763932	0.0253241
$911$	−26.0689	−0.863701	−0.431850	−	0.901945i	$-0.642139\pi$
$911$	−26.0689	−0.863701	−0.431850	+	0.901945i	$0.642139\pi$
$912$	−4.23607	−0.140270
$913$	0	0
$914$	5.43769	0.179863
$915$	12.9443	0.427924
$916$	7.70820	0.254686
$917$	8.14590	0.269001
$918$	−3.38197	−0.111622
$919$	21.2361	0.700513	0.350257	−	0.936654i	$-0.386094\pi$
$919$	21.2361	0.700513	0.350257	+	0.936654i	$0.386094\pi$
$920$	−0.472136	−0.0155659
$921$	−5.67376	−0.186957
$922$	−39.1246	−1.28850
$923$	−4.94427	−0.162743
$924$	0	0
$925$	−3.70820	−0.121925
$926$	−30.6525	−1.00730
$927$	−5.23607	−0.171975
$928$	−4.00000	−0.131306
$929$	19.6738	0.645475	0.322738	−	0.946488i	$-0.395397\pi$
$929$	19.6738	0.645475	0.322738	+	0.946488i	$0.395397\pi$
$930$	7.23607	0.237280
$931$	2.61803	0.0858026
$932$	−19.3262	−0.633052
$933$	2.00000	0.0654771
$934$	−0.944272	−0.0308975
$935$	0	0
$936$	−0.291796	−0.00953765
$937$	28.3951	0.927628	0.463814	−	0.885933i	$-0.346481\pi$
$937$	28.3951	0.927628	0.463814	+	0.885933i	$0.346481\pi$
$938$	−5.09017	−0.166200
$939$	19.9443	0.650857
$940$	11.2361	0.366480
$941$	−53.7771	−1.75308	−0.876541	−	0.481326i	$-0.840155\pi$
$941$	−53.7771	−1.75308	−0.876541	+	0.481326i	$0.840155\pi$
$942$	10.0000	0.325818
$943$	−3.01316	−0.0981218
$944$	−1.38197	−0.0449792
$945$	5.47214	0.178009
$946$	0	0
$947$	51.3394	1.66831	0.834153	−	0.551533i	$-0.185957\pi$
$947$	51.3394	1.66831	0.834153	+	0.551533i	$0.185957\pi$
$948$	−18.9443	−0.615281
$949$	8.65248	0.280871
$950$	2.61803	0.0849402
$951$	−28.9443	−0.938582
$952$	0.618034	0.0200306
$953$	−27.1459	−0.879342	−0.439671	−	0.898159i	$-0.644905\pi$
$953$	−27.1459	−0.879342	−0.439671	+	0.898159i	$0.644905\pi$
$954$	−0.944272	−0.0305719
$955$	2.00000	0.0647185
$956$	−4.29180	−0.138807
$957$	0	0
$958$	−23.1246	−0.747122
$959$	12.8541	0.415081
$960$	1.61803	0.0522218
$961$	−11.0000	−0.354839
$962$	−2.83282	−0.0913336
$963$	3.09017	0.0995793
$964$	13.8541	0.446211
$965$	−3.52786	−0.113566
$966$	0.763932	0.0245791
$967$	26.0000	0.836104	0.418052	−	0.908423i	$-0.362713\pi$
$967$	26.0000	0.836104	0.418052	+	0.908423i	$0.362713\pi$
$968$	0	0
$969$	2.61803	0.0841034
$970$	−13.7984	−0.443039
$971$	−21.3050	−0.683708	−0.341854	−	0.939753i	$-0.611055\pi$
$971$	−21.3050	−0.683708	−0.341854	+	0.939753i	$0.611055\pi$
$972$	−3.94427	−0.126513
$973$	−1.52786	−0.0489811
$974$	−33.7771	−1.08229
$975$	−1.23607	−0.0395859
$976$	8.00000	0.256074
$977$	31.5279	1.00867	0.504333	−	0.863509i	$-0.331738\pi$
$977$	31.5279	1.00867	0.504333	+	0.863509i	$0.331738\pi$
$978$	36.5967	1.17023
$979$	0	0
$980$	−1.00000	−0.0319438
$981$	3.88854	0.124152
$982$	−26.9230	−0.859147
$983$	31.2361	0.996276	0.498138	−	0.867098i	$-0.334017\pi$
$983$	31.2361	0.996276	0.498138	+	0.867098i	$0.334017\pi$
$984$	10.3262	0.329188
$985$	−5.41641	−0.172581
$986$	2.47214	0.0787288
$987$	−18.1803	−0.578687
$988$	2.00000	0.0636285
$989$	1.12461	0.0357606
$990$	0	0
$991$	−61.2361	−1.94523	−0.972614	−	0.232427i	$-0.925333\pi$
$991$	−61.2361	−1.94523	−0.972614	+	0.232427i	$0.925333\pi$
$992$	4.47214	0.141990
$993$	5.38197	0.170792
$994$	6.47214	0.205284
$995$	−3.23607	−0.102590
$996$	−0.145898	−0.00462296
$997$	−12.8754	−0.407768	−0.203884	−	0.978995i	$-0.565356\pi$
$997$	−12.8754	−0.407768	−0.203884	+	0.978995i	$0.565356\pi$
$998$	−1.61803	−0.0512180
$999$	−20.2918	−0.642004

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8470.2.a.bx.1.1		2
11.2	odd	10		770.2.n.c.631.1	yes	4
11.6	odd	10		770.2.n.c.421.1	✓	4
11.10	odd	2		8470.2.a.bk.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
770.2.n.c.421.1	✓	4	11.6	odd	10
770.2.n.c.631.1	yes	4	11.2	odd	10
8470.2.a.bk.1.1		2	11.10	odd	2
8470.2.a.bx.1.1		2	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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -1.61803 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.61803 q^{6} -1.00000 q^{7} +1.00000 q^{8} -0.381966 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} -1.61803 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.61803 q^{6} -1.00000 q^{7} +1.00000 q^{8} -0.381966 q^{9} -1.00000 q^{10} -1.61803 q^{12} +0.763932 q^{13} -1.00000 q^{14} +1.61803 q^{15} +1.00000 q^{16} -0.618034 q^{17} -0.381966 q^{18} +2.61803 q^{19} -1.00000 q^{20} +1.61803 q^{21} +0.472136 q^{23} -1.61803 q^{24} +1.00000 q^{25} +0.763932 q^{26} +5.47214 q^{27} -1.00000 q^{28} -4.00000 q^{29} +1.61803 q^{30} +4.47214 q^{31} +1.00000 q^{32} -0.618034 q^{34} +1.00000 q^{35} -0.381966 q^{36} -3.70820 q^{37} +2.61803 q^{38} -1.23607 q^{39} -1.00000 q^{40} -6.38197 q^{41} +1.61803 q^{42} +2.38197 q^{43} +0.381966 q^{45} +0.472136 q^{46} -11.2361 q^{47} -1.61803 q^{48} +1.00000 q^{49} +1.00000 q^{50} +1.00000 q^{51} +0.763932 q^{52} +2.47214 q^{53} +5.47214 q^{54} -1.00000 q^{56} -4.23607 q^{57} -4.00000 q^{58} -1.38197 q^{59} +1.61803 q^{60} +8.00000 q^{61} +4.47214 q^{62} +0.381966 q^{63} +1.00000 q^{64} -0.763932 q^{65} +5.09017 q^{67} -0.618034 q^{68} -0.763932 q^{69} +1.00000 q^{70} -6.47214 q^{71} -0.381966 q^{72} +11.3262 q^{73} -3.70820 q^{74} -1.61803 q^{75} +2.61803 q^{76} -1.23607 q^{78} +11.7082 q^{79} -1.00000 q^{80} -7.70820 q^{81} -6.38197 q^{82} +0.0901699 q^{83} +1.61803 q^{84} +0.618034 q^{85} +2.38197 q^{86} +6.47214 q^{87} +3.14590 q^{89} +0.381966 q^{90} -0.763932 q^{91} +0.472136 q^{92} -7.23607 q^{93} -11.2361 q^{94} -2.61803 q^{95} -1.61803 q^{96} +13.7984 q^{97} +1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} - q^{3} + 2 q^{4} - 2 q^{5} - q^{6} - 2 q^{7} + 2 q^{8} - 3 q^{9}+O(q^{10}) \) 2 * q + 2 * q^2 - q^3 + 2 * q^4 - 2 * q^5 - q^6 - 2 * q^7 + 2 * q^8 - 3 * q^9	\( 2 q + 2 q^{2} - q^{3} + 2 q^{4} - 2 q^{5} - q^{6} - 2 q^{7} + 2 q^{8} - 3 q^{9} - 2 q^{10} - q^{12} + 6 q^{13} - 2 q^{14} + q^{15} + 2 q^{16} + q^{17} - 3 q^{18} + 3 q^{19} - 2 q^{20} + q^{21} - 8 q^{23} - q^{24} + 2 q^{25} + 6 q^{26} + 2 q^{27} - 2 q^{28} - 8 q^{29} + q^{30} + 2 q^{32} + q^{34} + 2 q^{35} - 3 q^{36} + 6 q^{37} + 3 q^{38} + 2 q^{39} - 2 q^{40} - 15 q^{41} + q^{42} + 7 q^{43} + 3 q^{45} - 8 q^{46} - 18 q^{47} - q^{48} + 2 q^{49} + 2 q^{50} + 2 q^{51} + 6 q^{52} - 4 q^{53} + 2 q^{54} - 2 q^{56} - 4 q^{57} - 8 q^{58} - 5 q^{59} + q^{60} + 16 q^{61} + 3 q^{63} + 2 q^{64} - 6 q^{65} - q^{67} + q^{68} - 6 q^{69} + 2 q^{70} - 4 q^{71} - 3 q^{72} + 7 q^{73} + 6 q^{74} - q^{75} + 3 q^{76} + 2 q^{78} + 10 q^{79} - 2 q^{80} - 2 q^{81} - 15 q^{82} - 11 q^{83} + q^{84} - q^{85} + 7 q^{86} + 4 q^{87} + 13 q^{89} + 3 q^{90} - 6 q^{91} - 8 q^{92} - 10 q^{93} - 18 q^{94} - 3 q^{95} - q^{96} + 3 q^{97} + 2 q^{98}+O(q^{100}) \) 2 * q + 2 * q^2 - q^3 + 2 * q^4 - 2 * q^5 - q^6 - 2 * q^7 + 2 * q^8 - 3 * q^9 - 2 * q^10 - q^12 + 6 * q^13 - 2 * q^14 + q^15 + 2 * q^16 + q^17 - 3 * q^18 + 3 * q^19 - 2 * q^20 + q^21 - 8 * q^23 - q^24 + 2 * q^25 + 6 * q^26 + 2 * q^27 - 2 * q^28 - 8 * q^29 + q^30 + 2 * q^32 + q^34 + 2 * q^35 - 3 * q^36 + 6 * q^37 + 3 * q^38 + 2 * q^39 - 2 * q^40 - 15 * q^41 + q^42 + 7 * q^43 + 3 * q^45 - 8 * q^46 - 18 * q^47 - q^48 + 2 * q^49 + 2 * q^50 + 2 * q^51 + 6 * q^52 - 4 * q^53 + 2 * q^54 - 2 * q^56 - 4 * q^57 - 8 * q^58 - 5 * q^59 + q^60 + 16 * q^61 + 3 * q^63 + 2 * q^64 - 6 * q^65 - q^67 + q^68 - 6 * q^69 + 2 * q^70 - 4 * q^71 - 3 * q^72 + 7 * q^73 + 6 * q^74 - q^75 + 3 * q^76 + 2 * q^78 + 10 * q^79 - 2 * q^80 - 2 * q^81 - 15 * q^82 - 11 * q^83 + q^84 - q^85 + 7 * q^86 + 4 * q^87 + 13 * q^89 + 3 * q^90 - 6 * q^91 - 8 * q^92 - 10 * q^93 - 18 * q^94 - 3 * q^95 - q^96 + 3 * q^97 + 2 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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