LMFDB - Embedded newform 8470.2.a.bx.1.2

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			1.2
Root			$-0.618034$ 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)

$f(q)$	$=$	$q+1.00000 q^{2} +0.618034 q^{3} +1.00000 q^{4} -1.00000 q^{5} +0.618034 q^{6} -1.00000 q^{7} +1.00000 q^{8} -2.61803 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} +0.618034 q^{3} +1.00000 q^{4} -1.00000 q^{5} +0.618034 q^{6} -1.00000 q^{7} +1.00000 q^{8} -2.61803 q^{9} -1.00000 q^{10} +0.618034 q^{12} +5.23607 q^{13} -1.00000 q^{14} -0.618034 q^{15} +1.00000 q^{16} +1.61803 q^{17} -2.61803 q^{18} +0.381966 q^{19} -1.00000 q^{20} -0.618034 q^{21} -8.47214 q^{23} +0.618034 q^{24} +1.00000 q^{25} +5.23607 q^{26} -3.47214 q^{27} -1.00000 q^{28} -4.00000 q^{29} -0.618034 q^{30} -4.47214 q^{31} +1.00000 q^{32} +1.61803 q^{34} +1.00000 q^{35} -2.61803 q^{36} +9.70820 q^{37} +0.381966 q^{38} +3.23607 q^{39} -1.00000 q^{40} -8.61803 q^{41} -0.618034 q^{42} +4.61803 q^{43} +2.61803 q^{45} -8.47214 q^{46} -6.76393 q^{47} +0.618034 q^{48} +1.00000 q^{49} +1.00000 q^{50} +1.00000 q^{51} +5.23607 q^{52} -6.47214 q^{53} -3.47214 q^{54} -1.00000 q^{56} +0.236068 q^{57} -4.00000 q^{58} -3.61803 q^{59} -0.618034 q^{60} +8.00000 q^{61} -4.47214 q^{62} +2.61803 q^{63} +1.00000 q^{64} -5.23607 q^{65} -6.09017 q^{67} +1.61803 q^{68} -5.23607 q^{69} +1.00000 q^{70} +2.47214 q^{71} -2.61803 q^{72} -4.32624 q^{73} +9.70820 q^{74} +0.618034 q^{75} +0.381966 q^{76} +3.23607 q^{78} -1.70820 q^{79} -1.00000 q^{80} +5.70820 q^{81} -8.61803 q^{82} -11.0902 q^{83} -0.618034 q^{84} -1.61803 q^{85} +4.61803 q^{86} -2.47214 q^{87} +9.85410 q^{89} +2.61803 q^{90} -5.23607 q^{91} -8.47214 q^{92} -2.76393 q^{93} -6.76393 q^{94} -0.381966 q^{95} +0.618034 q^{96} -10.7984 q^{97} +1.00000 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	0.618034	0.356822	0.178411	−	0.983956i	$-0.442904\pi$
$3$	0.618034	0.356822	0.178411	+	0.983956i	$0.442904\pi$
$4$	1.00000	0.500000
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0.618034	0.252311
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	1.00000	0.353553
$9$	−2.61803	−0.872678
$10$	−1.00000	−0.316228
$11$	0	0
$11$	0	0
$12$	0.618034	0.178411
$13$	5.23607	1.45222	0.726112	−	0.687576i	$-0.241325\pi$
$13$	5.23607	1.45222	0.726112	+	0.687576i	$0.241325\pi$
$14$	−1.00000	−0.267261
$15$	−0.618034	−0.159576
$16$	1.00000	0.250000
$17$	1.61803	0.392431	0.196215	−	0.980561i	$-0.437135\pi$
$17$	1.61803	0.392431	0.196215	+	0.980561i	$0.437135\pi$
$18$	−2.61803	−0.617077
$19$	0.381966	0.0876290	0.0438145	−	0.999040i	$-0.486049\pi$
$19$	0.381966	0.0876290	0.0438145	+	0.999040i	$0.486049\pi$
$20$	−1.00000	−0.223607
$21$	−0.618034	−0.134866
$22$	0	0
$23$	−8.47214	−1.76656	−0.883281	−	0.468844i	$-0.844671\pi$
$23$	−8.47214	−1.76656	−0.883281	+	0.468844i	$0.844671\pi$
$24$	0.618034	0.126156
$25$	1.00000	0.200000
$26$	5.23607	1.02688
$27$	−3.47214	−0.668213
$28$	−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	$0.621119\pi$
$30$	−0.618034	−0.112837
$31$	−4.47214	−0.803219	−0.401610	−	0.915811i	$-0.631549\pi$
$31$	−4.47214	−0.803219	−0.401610	+	0.915811i	$0.631549\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	1.61803	0.277491
$35$	1.00000	0.169031
$36$	−2.61803	−0.436339
$37$	9.70820	1.59602	0.798009	−	0.602645i	$-0.205886\pi$
$37$	9.70820	1.59602	0.798009	+	0.602645i	$0.205886\pi$
$38$	0.381966	0.0619631
$39$	3.23607	0.518186
$40$	−1.00000	−0.158114
$41$	−8.61803	−1.34591	−0.672955	−	0.739683i	$-0.734975\pi$
$41$	−8.61803	−1.34591	−0.672955	+	0.739683i	$0.734975\pi$
$42$	−0.618034	−0.0953647
$43$	4.61803	0.704244	0.352122	−	0.935954i	$-0.385460\pi$
$43$	4.61803	0.704244	0.352122	+	0.935954i	$0.385460\pi$
$44$	0	0
$45$	2.61803	0.390273
$46$	−8.47214	−1.24915
$47$	−6.76393	−0.986621	−0.493310	−	0.869853i	$-0.664213\pi$
$47$	−6.76393	−0.986621	−0.493310	+	0.869853i	$0.664213\pi$
$48$	0.618034	0.0892055
$49$	1.00000	0.142857
$50$	1.00000	0.141421
$51$	1.00000	0.140028
$52$	5.23607	0.726112
$53$	−6.47214	−0.889016	−0.444508	−	0.895775i	$-0.646622\pi$
$53$	−6.47214	−0.889016	−0.444508	+	0.895775i	$0.646622\pi$
$54$	−3.47214	−0.472498
$55$	0	0
$56$	−1.00000	−0.133631
$57$	0.236068	0.0312680
$58$	−4.00000	−0.525226
$59$	−3.61803	−0.471028	−0.235514	−	0.971871i	$-0.575677\pi$
$59$	−3.61803	−0.471028	−0.235514	+	0.971871i	$0.575677\pi$
$60$	−0.618034	−0.0797878
$61$	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
$63$	2.61803	0.329841
$64$	1.00000	0.125000
$65$	−5.23607	−0.649454
$66$	0	0
$67$	−6.09017	−0.744033	−0.372016	−	0.928226i	$-0.621333\pi$
$67$	−6.09017	−0.744033	−0.372016	+	0.928226i	$0.621333\pi$
$68$	1.61803	0.196215
$69$	−5.23607	−0.630349
$70$	1.00000	0.119523
$71$	2.47214	0.293389	0.146694	−	0.989182i	$-0.453137\pi$
$71$	2.47214	0.293389	0.146694	+	0.989182i	$0.453137\pi$
$72$	−2.61803	−0.308538
$73$	−4.32624	−0.506348	−0.253174	−	0.967421i	$-0.581474\pi$
$73$	−4.32624	−0.506348	−0.253174	+	0.967421i	$0.581474\pi$
$74$	9.70820	1.12856
$75$	0.618034	0.0713644
$76$	0.381966	0.0438145
$77$	0	0
$78$	3.23607	0.366413
$79$	−1.70820	−0.192188	−0.0960940	−	0.995372i	$-0.530635\pi$
$79$	−1.70820	−0.192188	−0.0960940	+	0.995372i	$0.530635\pi$
$80$	−1.00000	−0.111803
$81$	5.70820	0.634245
$82$	−8.61803	−0.951703
$83$	−11.0902	−1.21730	−0.608652	−	0.793437i	$-0.708289\pi$
$83$	−11.0902	−1.21730	−0.608652	+	0.793437i	$0.708289\pi$
$84$	−0.618034	−0.0674330
$85$	−1.61803	−0.175500
$86$	4.61803	0.497975
$87$	−2.47214	−0.265041
$88$	0	0
$89$	9.85410	1.04453	0.522266	−	0.852782i	$-0.325087\pi$
$89$	9.85410	1.04453	0.522266	+	0.852782i	$0.325087\pi$
$90$	2.61803	0.275965
$91$	−5.23607	−0.548889
$92$	−8.47214	−0.883281
$93$	−2.76393	−0.286606
$94$	−6.76393	−0.697646
$95$	−0.381966	−0.0391889
$96$	0.618034	0.0630778
$97$	−10.7984	−1.09641	−0.548204	−	0.836344i	$-0.684688\pi$
$97$	−10.7984	−1.09641	−0.548204	+	0.836344i	$0.684688\pi$
$98$	1.00000	0.101015
$99$	0	0
$100$	1.00000	0.100000
$101$	−17.2361	−1.71505	−0.857526	−	0.514440i	$-0.828000\pi$
$101$	−17.2361	−1.71505	−0.857526	+	0.514440i	$0.828000\pi$
$102$	1.00000	0.0990148
$103$	0.291796	0.0287515	0.0143758	−	0.999897i	$-0.495424\pi$
$103$	0.291796	0.0287515	0.0143758	+	0.999897i	$0.495424\pi$
$104$	5.23607	0.513439
$105$	0.618034	0.0603139
$106$	−6.47214	−0.628629
$107$	3.09017	0.298738	0.149369	−	0.988782i	$-0.452276\pi$
$107$	3.09017	0.298738	0.149369	+	0.988782i	$0.452276\pi$
$108$	−3.47214	−0.334106
$109$	12.1803	1.16666	0.583332	−	0.812233i	$-0.301748\pi$
$109$	12.1803	1.16666	0.583332	+	0.812233i	$0.301748\pi$
$110$	0	0
$111$	6.00000	0.569495
$112$	−1.00000	−0.0944911
$113$	13.7984	1.29804	0.649021	−	0.760771i	$-0.275179\pi$
$113$	13.7984	1.29804	0.649021	+	0.760771i	$0.275179\pi$
$114$	0.236068	0.0221098
$115$	8.47214	0.790031
$116$	−4.00000	−0.371391
$117$	−13.7082	−1.26732
$118$	−3.61803	−0.333067
$119$	−1.61803	−0.148325
$120$	−0.618034	−0.0564185
$121$	0	0
$122$	8.00000	0.724286
$123$	−5.32624	−0.480251
$124$	−4.47214	−0.401610
$125$	−1.00000	−0.0894427
$126$	2.61803	0.233233
$127$	−14.1803	−1.25830	−0.629151	−	0.777283i	$-0.716597\pi$
$127$	−14.1803	−1.25830	−0.629151	+	0.777283i	$0.716597\pi$
$128$	1.00000	0.0883883
$129$	2.85410	0.251290
$130$	−5.23607	−0.459234
$131$	−14.8541	−1.29781	−0.648904	−	0.760870i	$-0.724773\pi$
$131$	−14.8541	−1.29781	−0.648904	+	0.760870i	$0.724773\pi$
$132$	0	0
$133$	−0.381966	−0.0331207
$134$	−6.09017	−0.526111
$135$	3.47214	0.298834
$136$	1.61803	0.138745
$137$	−6.14590	−0.525080	−0.262540	−	0.964921i	$-0.584560\pi$
$137$	−6.14590	−0.525080	−0.262540	+	0.964921i	$0.584560\pi$
$138$	−5.23607	−0.445724
$139$	10.4721	0.888235	0.444117	−	0.895969i	$-0.353517\pi$
$139$	10.4721	0.888235	0.444117	+	0.895969i	$0.353517\pi$
$140$	1.00000	0.0845154
$141$	−4.18034	−0.352048
$142$	2.47214	0.207457
$143$	0	0
$144$	−2.61803	−0.218169
$145$	4.00000	0.332182
$146$	−4.32624	−0.358042
$147$	0.618034	0.0509746
$148$	9.70820	0.798009
$149$	−5.70820	−0.467634	−0.233817	−	0.972281i	$-0.575122\pi$
$149$	−5.70820	−0.467634	−0.233817	+	0.972281i	$0.575122\pi$
$150$	0.618034	0.0504623
$151$	2.18034	0.177434	0.0887168	−	0.996057i	$-0.471723\pi$
$151$	2.18034	0.177434	0.0887168	+	0.996057i	$0.471723\pi$
$152$	0.381966	0.0309815
$153$	−4.23607	−0.342466
$154$	0	0
$155$	4.47214	0.359211
$156$	3.23607	0.259093
$157$	16.1803	1.29133	0.645666	−	0.763620i	$-0.276580\pi$
$157$	16.1803	1.29133	0.645666	+	0.763620i	$0.276580\pi$
$158$	−1.70820	−0.135897
$159$	−4.00000	−0.317221
$160$	−1.00000	−0.0790569
$161$	8.47214	0.667698
$162$	5.70820	0.448479
$163$	−20.3820	−1.59644	−0.798219	−	0.602367i	$-0.794224\pi$
$163$	−20.3820	−1.59644	−0.798219	+	0.602367i	$0.794224\pi$
$164$	−8.61803	−0.672955
$165$	0	0
$166$	−11.0902	−0.860764
$167$	−10.0000	−0.773823	−0.386912	−	0.922117i	$-0.626458\pi$
$167$	−10.0000	−0.773823	−0.386912	+	0.922117i	$0.626458\pi$
$168$	−0.618034	−0.0476824
$169$	14.4164	1.10895
$170$	−1.61803	−0.124098
$171$	−1.00000	−0.0764719
$172$	4.61803	0.352122
$173$	−9.23607	−0.702205	−0.351103	−	0.936337i	$-0.614193\pi$
$173$	−9.23607	−0.702205	−0.351103	+	0.936337i	$0.614193\pi$
$174$	−2.47214	−0.187412
$175$	−1.00000	−0.0755929
$176$	0	0
$177$	−2.23607	−0.168073
$178$	9.85410	0.738596
$179$	−14.8541	−1.11025	−0.555124	−	0.831768i	$-0.687329\pi$
$179$	−14.8541	−1.11025	−0.555124	+	0.831768i	$0.687329\pi$
$180$	2.61803	0.195137
$181$	17.1246	1.27286	0.636431	−	0.771333i	$-0.280410\pi$
$181$	17.1246	1.27286	0.636431	+	0.771333i	$0.280410\pi$
$182$	−5.23607	−0.388123
$183$	4.94427	0.365491
$184$	−8.47214	−0.624574
$185$	−9.70820	−0.713761
$186$	−2.76393	−0.202661
$187$	0	0
$188$	−6.76393	−0.493310
$189$	3.47214	0.252561
$190$	−0.381966	−0.0277107
$191$	−2.00000	−0.144715	−0.0723575	−	0.997379i	$-0.523052\pi$
$191$	−2.00000	−0.144715	−0.0723575	+	0.997379i	$0.523052\pi$
$192$	0.618034	0.0446028
$193$	12.4721	0.897764	0.448882	−	0.893591i	$-0.351822\pi$
$193$	12.4721	0.897764	0.448882	+	0.893591i	$0.351822\pi$
$194$	−10.7984	−0.775278
$195$	−3.23607	−0.231740
$196$	1.00000	0.0714286
$197$	−21.4164	−1.52586	−0.762928	−	0.646484i	$-0.776239\pi$
$197$	−21.4164	−1.52586	−0.762928	+	0.646484i	$0.776239\pi$
$198$	0	0
$199$	−1.23607	−0.0876225	−0.0438113	−	0.999040i	$-0.513950\pi$
$199$	−1.23607	−0.0876225	−0.0438113	+	0.999040i	$0.513950\pi$
$200$	1.00000	0.0707107
$201$	−3.76393	−0.265487
$202$	−17.2361	−1.21273
$203$	4.00000	0.280745
$204$	1.00000	0.0700140
$205$	8.61803	0.601910
$206$	0.291796	0.0203304
$207$	22.1803	1.54164
$208$	5.23607	0.363056
$209$	0	0
$210$	0.618034	0.0426484
$211$	−16.9098	−1.16412	−0.582060	−	0.813146i	$-0.697753\pi$
$211$	−16.9098	−1.16412	−0.582060	+	0.813146i	$0.697753\pi$
$212$	−6.47214	−0.444508
$213$	1.52786	0.104688
$214$	3.09017	0.211240
$215$	−4.61803	−0.314947
$216$	−3.47214	−0.236249
$217$	4.47214	0.303588
$218$	12.1803	0.824957
$219$	−2.67376	−0.180676
$220$	0	0
$221$	8.47214	0.569898
$222$	6.00000	0.402694
$223$	−5.05573	−0.338557	−0.169278	−	0.985568i	$-0.554144\pi$
$223$	−5.05573	−0.338557	−0.169278	+	0.985568i	$0.554144\pi$
$224$	−1.00000	−0.0668153
$225$	−2.61803	−0.174536
$226$	13.7984	0.917854
$227$	−8.56231	−0.568300	−0.284150	−	0.958780i	$-0.591711\pi$
$227$	−8.56231	−0.568300	−0.284150	+	0.958780i	$0.591711\pi$
$228$	0.236068	0.0156340
$229$	−5.70820	−0.377209	−0.188604	−	0.982053i	$-0.560396\pi$
$229$	−5.70820	−0.377209	−0.188604	+	0.982053i	$0.560396\pi$
$230$	8.47214	0.558636
$231$	0	0
$232$	−4.00000	−0.262613
$233$	−3.67376	−0.240676	−0.120338	−	0.992733i	$-0.538398\pi$
$233$	−3.67376	−0.240676	−0.120338	+	0.992733i	$0.538398\pi$
$234$	−13.7082	−0.896133
$235$	6.76393	0.441230
$236$	−3.61803	−0.235514
$237$	−1.05573	−0.0685769
$238$	−1.61803	−0.104882
$239$	−17.7082	−1.14545	−0.572724	−	0.819748i	$-0.694113\pi$
$239$	−17.7082	−1.14545	−0.572724	+	0.819748i	$0.694113\pi$
$240$	−0.618034	−0.0398939
$241$	7.14590	0.460308	0.230154	−	0.973154i	$-0.426077\pi$
$241$	7.14590	0.460308	0.230154	+	0.973154i	$0.426077\pi$
$242$	0	0
$243$	13.9443	0.894525
$244$	8.00000	0.512148
$245$	−1.00000	−0.0638877
$246$	−5.32624	−0.339589
$247$	2.00000	0.127257
$248$	−4.47214	−0.283981
$249$	−6.85410	−0.434361
$250$	−1.00000	−0.0632456
$251$	−17.8885	−1.12911	−0.564557	−	0.825394i	$-0.690953\pi$
$251$	−17.8885	−1.12911	−0.564557	+	0.825394i	$0.690953\pi$
$252$	2.61803	0.164921
$253$	0	0
$254$	−14.1803	−0.889754
$255$	−1.00000	−0.0626224
$256$	1.00000	0.0625000
$257$	−25.5623	−1.59453	−0.797266	−	0.603628i	$-0.793721\pi$
$257$	−25.5623	−1.59453	−0.797266	+	0.603628i	$0.793721\pi$
$258$	2.85410	0.177689
$259$	−9.70820	−0.603238
$260$	−5.23607	−0.324727
$261$	10.4721	0.648209
$262$	−14.8541	−0.917689
$263$	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
$265$	6.47214	0.397580
$266$	−0.381966	−0.0234198
$267$	6.09017	0.372712
$268$	−6.09017	−0.372016
$269$	14.0000	0.853595	0.426798	−	0.904347i	$-0.359642\pi$
$269$	14.0000	0.853595	0.426798	+	0.904347i	$0.359642\pi$
$270$	3.47214	0.211307
$271$	−19.7082	−1.19719	−0.598594	−	0.801053i	$-0.704274\pi$
$271$	−19.7082	−1.19719	−0.598594	+	0.801053i	$0.704274\pi$
$272$	1.61803	0.0981077
$273$	−3.23607	−0.195856
$274$	−6.14590	−0.371287
$275$	0	0
$276$	−5.23607	−0.315174
$277$	−23.1246	−1.38942	−0.694712	−	0.719288i	$-0.744468\pi$
$277$	−23.1246	−1.38942	−0.694712	+	0.719288i	$0.744468\pi$
$278$	10.4721	0.628077
$279$	11.7082	0.700952
$280$	1.00000	0.0597614
$281$	6.79837	0.405557	0.202778	−	0.979225i	$-0.435003\pi$
$281$	6.79837	0.405557	0.202778	+	0.979225i	$0.435003\pi$
$282$	−4.18034	−0.248936
$283$	25.8885	1.53891	0.769457	−	0.638699i	$-0.220527\pi$
$283$	25.8885	1.53891	0.769457	+	0.638699i	$0.220527\pi$
$284$	2.47214	0.146694
$285$	−0.236068	−0.0139835
$286$	0	0
$287$	8.61803	0.508706
$288$	−2.61803	−0.154269
$289$	−14.3820	−0.845998
$290$	4.00000	0.234888
$291$	−6.67376	−0.391223
$292$	−4.32624	−0.253174
$293$	−2.29180	−0.133888	−0.0669441	−	0.997757i	$-0.521325\pi$
$293$	−2.29180	−0.133888	−0.0669441	+	0.997757i	$0.521325\pi$
$294$	0.618034	0.0360445
$295$	3.61803	0.210650
$296$	9.70820	0.564278
$297$	0	0
$298$	−5.70820	−0.330667
$299$	−44.3607	−2.56544
$300$	0.618034	0.0356822
$301$	−4.61803	−0.266179
$302$	2.18034	0.125464
$303$	−10.6525	−0.611969
$304$	0.381966	0.0219073
$305$	−8.00000	−0.458079
$306$	−4.23607	−0.242160
$307$	−34.5066	−1.96939	−0.984697	−	0.174274i	$-0.944242\pi$
$307$	−34.5066	−1.96939	−0.984697	+	0.174274i	$0.944242\pi$
$308$	0	0
$309$	0.180340	0.0102592
$310$	4.47214	0.254000
$311$	3.23607	0.183501	0.0917503	−	0.995782i	$-0.470754\pi$
$311$	3.23607	0.183501	0.0917503	+	0.995782i	$0.470754\pi$
$312$	3.23607	0.183206
$313$	3.32624	0.188010	0.0940050	−	0.995572i	$-0.470033\pi$
$313$	3.32624	0.188010	0.0940050	+	0.995572i	$0.470033\pi$
$314$	16.1803	0.913109
$315$	−2.61803	−0.147510
$316$	−1.70820	−0.0960940
$317$	−17.8885	−1.00472	−0.502360	−	0.864658i	$-0.667535\pi$
$317$	−17.8885	−1.00472	−0.502360	+	0.864658i	$0.667535\pi$
$318$	−4.00000	−0.224309
$319$	0	0
$320$	−1.00000	−0.0559017
$321$	1.90983	0.106596
$322$	8.47214	0.472134
$323$	0.618034	0.0343883
$324$	5.70820	0.317122
$325$	5.23607	0.290445
$326$	−20.3820	−1.12885
$327$	7.52786	0.416292
$328$	−8.61803	−0.475851
$329$	6.76393	0.372908
$330$	0	0
$331$	12.3262	0.677511	0.338756	−	0.940874i	$-0.389994\pi$
$331$	12.3262	0.677511	0.338756	+	0.940874i	$0.389994\pi$
$332$	−11.0902	−0.608652
$333$	−25.4164	−1.39281
$334$	−10.0000	−0.547176
$335$	6.09017	0.332742
$336$	−0.618034	−0.0337165
$337$	−32.2705	−1.75789	−0.878943	−	0.476926i	$-0.841751\pi$
$337$	−32.2705	−1.75789	−0.878943	+	0.476926i	$0.841751\pi$
$338$	14.4164	0.784149
$339$	8.52786	0.463170
$340$	−1.61803	−0.0877502
$341$	0	0
$342$	−1.00000	−0.0540738
$343$	−1.00000	−0.0539949
$344$	4.61803	0.248988
$345$	5.23607	0.281900
$346$	−9.23607	−0.496534
$347$	27.0344	1.45128	0.725642	−	0.688072i	$-0.241543\pi$
$347$	27.0344	1.45128	0.725642	+	0.688072i	$0.241543\pi$
$348$	−2.47214	−0.132520
$349$	−10.4721	−0.560561	−0.280280	−	0.959918i	$-0.590427\pi$
$349$	−10.4721	−0.560561	−0.280280	+	0.959918i	$0.590427\pi$
$350$	−1.00000	−0.0534522
$351$	−18.1803	−0.970395
$352$	0	0
$353$	17.0344	0.906652	0.453326	−	0.891345i	$-0.350237\pi$
$353$	17.0344	0.906652	0.453326	+	0.891345i	$0.350237\pi$
$354$	−2.23607	−0.118846
$355$	−2.47214	−0.131207
$356$	9.85410	0.522266
$357$	−1.00000	−0.0529256
$358$	−14.8541	−0.785064
$359$	−25.7082	−1.35683	−0.678414	−	0.734680i	$-0.737332\pi$
$359$	−25.7082	−1.35683	−0.678414	+	0.734680i	$0.737332\pi$
$360$	2.61803	0.137983
$361$	−18.8541	−0.992321
$362$	17.1246	0.900050
$363$	0	0
$364$	−5.23607	−0.274445
$365$	4.32624	0.226446
$366$	4.94427	0.258441
$367$	11.0557	0.577104	0.288552	−	0.957464i	$-0.406826\pi$
$367$	11.0557	0.577104	0.288552	+	0.957464i	$0.406826\pi$
$368$	−8.47214	−0.441641
$369$	22.5623	1.17455
$370$	−9.70820	−0.504705
$371$	6.47214	0.336017
$372$	−2.76393	−0.143303
$373$	4.94427	0.256005	0.128002	−	0.991774i	$-0.459143\pi$
$373$	4.94427	0.256005	0.128002	+	0.991774i	$0.459143\pi$
$374$	0	0
$375$	−0.618034	−0.0319151
$376$	−6.76393	−0.348823
$377$	−20.9443	−1.07868
$378$	3.47214	0.178587
$379$	−10.1459	−0.521160	−0.260580	−	0.965452i	$-0.583914\pi$
$379$	−10.1459	−0.521160	−0.260580	+	0.965452i	$0.583914\pi$
$380$	−0.381966	−0.0195944
$381$	−8.76393	−0.448990
$382$	−2.00000	−0.102329
$383$	17.7082	0.904847	0.452423	−	0.891803i	$-0.350560\pi$
$383$	17.7082	0.904847	0.452423	+	0.891803i	$0.350560\pi$
$384$	0.618034	0.0315389
$385$	0	0
$386$	12.4721	0.634815
$387$	−12.0902	−0.614578
$388$	−10.7984	−0.548204
$389$	8.00000	0.405616	0.202808	−	0.979219i	$-0.434993\pi$
$389$	8.00000	0.405616	0.202808	+	0.979219i	$0.434993\pi$
$390$	−3.23607	−0.163865
$391$	−13.7082	−0.693254
$392$	1.00000	0.0505076
$393$	−9.18034	−0.463087
$394$	−21.4164	−1.07894
$395$	1.70820	0.0859491
$396$	0	0
$397$	−34.5410	−1.73356	−0.866782	−	0.498687i	$-0.833816\pi$
$397$	−34.5410	−1.73356	−0.866782	+	0.498687i	$0.833816\pi$
$398$	−1.23607	−0.0619585
$399$	−0.236068	−0.0118182
$400$	1.00000	0.0500000
$401$	−10.5066	−0.524673	−0.262337	−	0.964976i	$-0.584493\pi$
$401$	−10.5066	−0.524673	−0.262337	+	0.964976i	$0.584493\pi$
$402$	−3.76393	−0.187728
$403$	−23.4164	−1.16645
$404$	−17.2361	−0.857526
$405$	−5.70820	−0.283643
$406$	4.00000	0.198517
$407$	0	0
$408$	1.00000	0.0495074
$409$	31.3050	1.54793	0.773965	−	0.633228i	$-0.218271\pi$
$409$	31.3050	1.54793	0.773965	+	0.633228i	$0.218271\pi$
$410$	8.61803	0.425614
$411$	−3.79837	−0.187360
$412$	0.291796	0.0143758
$413$	3.61803	0.178032
$414$	22.1803	1.09010
$415$	11.0902	0.544395
$416$	5.23607	0.256719
$417$	6.47214	0.316942
$418$	0	0
$419$	27.7426	1.35532	0.677658	−	0.735377i	$-0.262995\pi$
$419$	27.7426	1.35532	0.677658	+	0.735377i	$0.262995\pi$
$420$	0.618034	0.0301570
$421$	−0.472136	−0.0230105	−0.0115052	−	0.999934i	$-0.503662\pi$
$421$	−0.472136	−0.0230105	−0.0115052	+	0.999934i	$0.503662\pi$
$422$	−16.9098	−0.823158
$423$	17.7082	0.861002
$424$	−6.47214	−0.314315
$425$	1.61803	0.0784862
$426$	1.52786	0.0740253
$427$	−8.00000	−0.387147
$428$	3.09017	0.149369
$429$	0	0
$430$	−4.61803	−0.222701
$431$	−6.29180	−0.303065	−0.151533	−	0.988452i	$-0.548421\pi$
$431$	−6.29180	−0.303065	−0.151533	+	0.988452i	$0.548421\pi$
$432$	−3.47214	−0.167053
$433$	25.7984	1.23979	0.619895	−	0.784684i	$-0.287175\pi$
$433$	25.7984	1.23979	0.619895	+	0.784684i	$0.287175\pi$
$434$	4.47214	0.214669
$435$	2.47214	0.118530
$436$	12.1803	0.583332
$437$	−3.23607	−0.154802
$438$	−2.67376	−0.127757
$439$	20.3607	0.971762	0.485881	−	0.874025i	$-0.338499\pi$
$439$	20.3607	0.971762	0.485881	+	0.874025i	$0.338499\pi$
$440$	0	0
$441$	−2.61803	−0.124668
$442$	8.47214	0.402978
$443$	−28.6180	−1.35968	−0.679842	−	0.733359i	$-0.737952\pi$
$443$	−28.6180	−1.35968	−0.679842	+	0.733359i	$0.737952\pi$
$444$	6.00000	0.284747
$445$	−9.85410	−0.467129
$446$	−5.05573	−0.239396
$447$	−3.52786	−0.166862
$448$	−1.00000	−0.0472456
$449$	−14.7426	−0.695748	−0.347874	−	0.937541i	$-0.613096\pi$
$449$	−14.7426	−0.695748	−0.347874	+	0.937541i	$0.613096\pi$
$450$	−2.61803	−0.123415
$451$	0	0
$452$	13.7984	0.649021
$453$	1.34752	0.0633122
$454$	−8.56231	−0.401849
$455$	5.23607	0.245471
$456$	0.236068	0.0110549
$457$	25.5623	1.19575	0.597877	−	0.801588i	$-0.296011\pi$
$457$	25.5623	1.19575	0.597877	+	0.801588i	$0.296011\pi$
$458$	−5.70820	−0.266727
$459$	−5.61803	−0.262227
$460$	8.47214	0.395015
$461$	1.12461	0.0523784	0.0261892	−	0.999657i	$-0.491663\pi$
$461$	1.12461	0.0523784	0.0261892	+	0.999657i	$0.491663\pi$
$462$	0	0
$463$	0.652476	0.0303231	0.0151616	−	0.999885i	$-0.495174\pi$
$463$	0.652476	0.0303231	0.0151616	+	0.999885i	$0.495174\pi$
$464$	−4.00000	−0.185695
$465$	2.76393	0.128174
$466$	−3.67376	−0.170184
$467$	16.9443	0.784087	0.392044	−	0.919947i	$-0.371768\pi$
$467$	16.9443	0.784087	0.392044	+	0.919947i	$0.371768\pi$
$468$	−13.7082	−0.633662
$469$	6.09017	0.281218
$470$	6.76393	0.311997
$471$	10.0000	0.460776
$472$	−3.61803	−0.166534
$473$	0	0
$474$	−1.05573	−0.0484912
$475$	0.381966	0.0175258
$476$	−1.61803	−0.0741625
$477$	16.9443	0.775825
$478$	−17.7082	−0.809954
$479$	17.1246	0.782443	0.391222	−	0.920296i	$-0.372053\pi$
$479$	17.1246	0.782443	0.391222	+	0.920296i	$0.372053\pi$
$480$	−0.618034	−0.0282093
$481$	50.8328	2.31778
$482$	7.14590	0.325487
$483$	5.23607	0.238249
$484$	0	0
$485$	10.7984	0.490329
$486$	13.9443	0.632525
$487$	37.7771	1.71184	0.855922	−	0.517106i	$-0.172991\pi$
$487$	37.7771	1.71184	0.855922	+	0.517106i	$0.172991\pi$
$488$	8.00000	0.362143
$489$	−12.5967	−0.569645
$490$	−1.00000	−0.0451754
$491$	37.9230	1.71144	0.855720	−	0.517439i	$-0.173115\pi$
$491$	37.9230	1.71144	0.855720	+	0.517439i	$0.173115\pi$
$492$	−5.32624	−0.240125
$493$	−6.47214	−0.291490
$494$	2.00000	0.0899843
$495$	0	0
$496$	−4.47214	−0.200805
$497$	−2.47214	−0.110890
$498$	−6.85410	−0.307140
$499$	0.618034	0.0276670	0.0138335	−	0.999904i	$-0.495597\pi$
$499$	0.618034	0.0276670	0.0138335	+	0.999904i	$0.495597\pi$
$500$	−1.00000	−0.0447214
$501$	−6.18034	−0.276117
$502$	−17.8885	−0.798405
$503$	25.7082	1.14627	0.573136	−	0.819460i	$-0.305727\pi$
$503$	25.7082	1.14627	0.573136	+	0.819460i	$0.305727\pi$
$504$	2.61803	0.116617
$505$	17.2361	0.766995
$506$	0	0
$507$	8.90983	0.395699
$508$	−14.1803	−0.629151
$509$	28.1803	1.24907	0.624536	−	0.780996i	$-0.285288\pi$
$509$	28.1803	1.24907	0.624536	+	0.780996i	$0.285288\pi$
$510$	−1.00000	−0.0442807
$511$	4.32624	0.191381
$512$	1.00000	0.0441942
$513$	−1.32624	−0.0585548
$514$	−25.5623	−1.12750
$515$	−0.291796	−0.0128581
$516$	2.85410	0.125645
$517$	0	0
$518$	−9.70820	−0.426554
$519$	−5.70820	−0.250562
$520$	−5.23607	−0.229617
$521$	17.4377	0.763959	0.381980	−	0.924171i	$-0.375242\pi$
$521$	17.4377	0.763959	0.381980	+	0.924171i	$0.375242\pi$
$522$	10.4721	0.458353
$523$	−28.9230	−1.26471	−0.632357	−	0.774677i	$-0.717912\pi$
$523$	−28.9230	−1.26471	−0.632357	+	0.774677i	$0.717912\pi$
$524$	−14.8541	−0.648904
$525$	−0.618034	−0.0269732
$526$	12.0000	0.523225
$527$	−7.23607	−0.315208
$528$	0	0
$529$	48.7771	2.12074
$530$	6.47214	0.281132
$531$	9.47214	0.411056
$532$	−0.381966	−0.0165603
$533$	−45.1246	−1.95456
$534$	6.09017	0.263547
$535$	−3.09017	−0.133600
$536$	−6.09017	−0.263055
$537$	−9.18034	−0.396161
$538$	14.0000	0.603583
$539$	0	0
$540$	3.47214	0.149417
$541$	−40.1803	−1.72749	−0.863744	−	0.503931i	$-0.831887\pi$
$541$	−40.1803	−1.72749	−0.863744	+	0.503931i	$0.831887\pi$
$542$	−19.7082	−0.846540
$543$	10.5836	0.454185
$544$	1.61803	0.0693726
$545$	−12.1803	−0.521748
$546$	−3.23607	−0.138491
$547$	−38.0902	−1.62862	−0.814309	−	0.580432i	$-0.802884\pi$
$547$	−38.0902	−1.62862	−0.814309	+	0.580432i	$0.802884\pi$
$548$	−6.14590	−0.262540
$549$	−20.9443	−0.893880
$550$	0	0
$551$	−1.52786	−0.0650892
$552$	−5.23607	−0.222862
$553$	1.70820	0.0726402
$554$	−23.1246	−0.982471
$555$	−6.00000	−0.254686
$556$	10.4721	0.444117
$557$	40.3607	1.71014	0.855068	−	0.518515i	$-0.173515\pi$
$557$	40.3607	1.71014	0.855068	+	0.518515i	$0.173515\pi$
$558$	11.7082	0.495648
$559$	24.1803	1.02272
$560$	1.00000	0.0422577
$561$	0	0
$562$	6.79837	0.286772
$563$	16.6869	0.703270	0.351635	−	0.936137i	$-0.385626\pi$
$563$	16.6869	0.703270	0.351635	+	0.936137i	$0.385626\pi$
$564$	−4.18034	−0.176024
$565$	−13.7984	−0.580502
$566$	25.8885	1.08818
$567$	−5.70820	−0.239722
$568$	2.47214	0.103729
$569$	−36.4508	−1.52810	−0.764050	−	0.645158i	$-0.776792\pi$
$569$	−36.4508	−1.52810	−0.764050	+	0.645158i	$0.776792\pi$
$570$	−0.236068	−0.00988780
$571$	26.8328	1.12292	0.561459	−	0.827504i	$-0.310240\pi$
$571$	26.8328	1.12292	0.561459	+	0.827504i	$0.310240\pi$
$572$	0	0
$573$	−1.23607	−0.0516375
$574$	8.61803	0.359710
$575$	−8.47214	−0.353312
$576$	−2.61803	−0.109085
$577$	8.67376	0.361093	0.180547	−	0.983566i	$-0.442213\pi$
$577$	8.67376	0.361093	0.180547	+	0.983566i	$0.442213\pi$
$578$	−14.3820	−0.598211
$579$	7.70820	0.320342
$580$	4.00000	0.166091
$581$	11.0902	0.460098
$582$	−6.67376	−0.276636
$583$	0	0
$584$	−4.32624	−0.179021
$585$	13.7082	0.566764
$586$	−2.29180	−0.0946732
$587$	−8.38197	−0.345961	−0.172980	−	0.984925i	$-0.555340\pi$
$587$	−8.38197	−0.345961	−0.172980	+	0.984925i	$0.555340\pi$
$588$	0.618034	0.0254873
$589$	−1.70820	−0.0703853
$590$	3.61803	0.148952
$591$	−13.2361	−0.544459
$592$	9.70820	0.399005
$593$	10.3820	0.426336	0.213168	−	0.977016i	$-0.431622\pi$
$593$	10.3820	0.426336	0.213168	+	0.977016i	$0.431622\pi$
$594$	0	0
$595$	1.61803	0.0663329
$596$	−5.70820	−0.233817
$597$	−0.763932	−0.0312657
$598$	−44.3607	−1.81404
$599$	24.0000	0.980613	0.490307	−	0.871550i	$-0.336885\pi$
$599$	24.0000	0.980613	0.490307	+	0.871550i	$0.336885\pi$
$600$	0.618034	0.0252311
$601$	13.5623	0.553218	0.276609	−	0.960983i	$-0.410789\pi$
$601$	13.5623	0.553218	0.276609	+	0.960983i	$0.410789\pi$
$602$	−4.61803	−0.188217
$603$	15.9443	0.649301
$604$	2.18034	0.0887168
$605$	0	0
$606$	−10.6525	−0.432727
$607$	34.0000	1.38002	0.690009	−	0.723801i	$-0.257607\pi$
$607$	34.0000	1.38002	0.690009	+	0.723801i	$0.257607\pi$
$608$	0.381966	0.0154908
$609$	2.47214	0.100176
$610$	−8.00000	−0.323911
$611$	−35.4164	−1.43279
$612$	−4.23607	−0.171233
$613$	10.0689	0.406678	0.203339	−	0.979108i	$-0.434821\pi$
$613$	10.0689	0.406678	0.203339	+	0.979108i	$0.434821\pi$
$614$	−34.5066	−1.39257
$615$	5.32624	0.214775
$616$	0	0
$617$	37.7984	1.52171	0.760853	−	0.648925i	$-0.224781\pi$
$617$	37.7984	1.52171	0.760853	+	0.648925i	$0.224781\pi$
$618$	0.180340	0.00725433
$619$	40.5623	1.63034	0.815168	−	0.579224i	$-0.196644\pi$
$619$	40.5623	1.63034	0.815168	+	0.579224i	$0.196644\pi$
$620$	4.47214	0.179605
$621$	29.4164	1.18044
$622$	3.23607	0.129755
$623$	−9.85410	−0.394796
$624$	3.23607	0.129546
$625$	1.00000	0.0400000
$626$	3.32624	0.132943
$627$	0	0
$628$	16.1803	0.645666
$629$	15.7082	0.626327
$630$	−2.61803	−0.104305
$631$	−25.2361	−1.00463	−0.502316	−	0.864684i	$-0.667519\pi$
$631$	−25.2361	−1.00463	−0.502316	+	0.864684i	$0.667519\pi$
$632$	−1.70820	−0.0679487
$633$	−10.4508	−0.415384
$634$	−17.8885	−0.710445
$635$	14.1803	0.562730
$636$	−4.00000	−0.158610
$637$	5.23607	0.207461
$638$	0	0
$639$	−6.47214	−0.256034
$640$	−1.00000	−0.0395285
$641$	40.3262	1.59279	0.796395	−	0.604776i	$-0.206738\pi$
$641$	40.3262	1.59279	0.796395	+	0.604776i	$0.206738\pi$
$642$	1.90983	0.0753750
$643$	−13.5623	−0.534845	−0.267423	−	0.963579i	$-0.586172\pi$
$643$	−13.5623	−0.534845	−0.267423	+	0.963579i	$0.586172\pi$
$644$	8.47214	0.333849
$645$	−2.85410	−0.112380
$646$	0.618034	0.0243162
$647$	27.8885	1.09641	0.548206	−	0.836343i	$-0.315311\pi$
$647$	27.8885	1.09641	0.548206	+	0.836343i	$0.315311\pi$
$648$	5.70820	0.224239
$649$	0	0
$650$	5.23607	0.205375
$651$	2.76393	0.108327
$652$	−20.3820	−0.798219
$653$	−6.00000	−0.234798	−0.117399	−	0.993085i	$-0.537456\pi$
$653$	−6.00000	−0.234798	−0.117399	+	0.993085i	$0.537456\pi$
$654$	7.52786	0.294363
$655$	14.8541	0.580398
$656$	−8.61803	−0.336478
$657$	11.3262	0.441879
$658$	6.76393	0.263686
$659$	−9.90983	−0.386032	−0.193016	−	0.981196i	$-0.561827\pi$
$659$	−9.90983	−0.386032	−0.193016	+	0.981196i	$0.561827\pi$
$660$	0	0
$661$	−33.7771	−1.31378	−0.656888	−	0.753988i	$-0.728128\pi$
$661$	−33.7771	−1.31378	−0.656888	+	0.753988i	$0.728128\pi$
$662$	12.3262	0.479073
$663$	5.23607	0.203352
$664$	−11.0902	−0.430382
$665$	0.381966	0.0148120
$666$	−25.4164	−0.984866
$667$	33.8885	1.31217
$668$	−10.0000	−0.386912
$669$	−3.12461	−0.120804
$670$	6.09017	0.235284
$671$	0	0
$672$	−0.618034	−0.0238412
$673$	−7.14590	−0.275454	−0.137727	−	0.990470i	$-0.543980\pi$
$673$	−7.14590	−0.275454	−0.137727	+	0.990470i	$0.543980\pi$
$674$	−32.2705	−1.24301
$675$	−3.47214	−0.133643
$676$	14.4164	0.554477
$677$	−6.94427	−0.266890	−0.133445	−	0.991056i	$-0.542604\pi$
$677$	−6.94427	−0.266890	−0.133445	+	0.991056i	$0.542604\pi$
$678$	8.52786	0.327511
$679$	10.7984	0.414404
$680$	−1.61803	−0.0620488
$681$	−5.29180	−0.202782
$682$	0	0
$683$	17.3050	0.662156	0.331078	−	0.943603i	$-0.392588\pi$
$683$	17.3050	0.662156	0.331078	+	0.943603i	$0.392588\pi$
$684$	−1.00000	−0.0382360
$685$	6.14590	0.234823
$686$	−1.00000	−0.0381802
$687$	−3.52786	−0.134596
$688$	4.61803	0.176061
$689$	−33.8885	−1.29105
$690$	5.23607	0.199334
$691$	−27.7984	−1.05750	−0.528750	−	0.848778i	$-0.677339\pi$
$691$	−27.7984	−1.05750	−0.528750	+	0.848778i	$0.677339\pi$
$692$	−9.23607	−0.351103
$693$	0	0
$694$	27.0344	1.02621
$695$	−10.4721	−0.397231
$696$	−2.47214	−0.0937061
$697$	−13.9443	−0.528177
$698$	−10.4721	−0.396376
$699$	−2.27051	−0.0858786
$700$	−1.00000	−0.0377964
$701$	−18.1803	−0.686662	−0.343331	−	0.939214i	$-0.611555\pi$
$701$	−18.1803	−0.686662	−0.343331	+	0.939214i	$0.611555\pi$
$702$	−18.1803	−0.686173
$703$	3.70820	0.139858
$704$	0	0
$705$	4.18034	0.157441
$706$	17.0344	0.641100
$707$	17.2361	0.648229
$708$	−2.23607	−0.0840366
$709$	37.2361	1.39843	0.699215	−	0.714912i	$-0.253533\pi$
$709$	37.2361	1.39843	0.699215	+	0.714912i	$0.253533\pi$
$710$	−2.47214	−0.0927776
$711$	4.47214	0.167718
$712$	9.85410	0.369298
$713$	37.8885	1.41894
$714$	−1.00000	−0.0374241
$715$	0	0
$716$	−14.8541	−0.555124
$717$	−10.9443	−0.408721
$718$	−25.7082	−0.959422
$719$	50.3607	1.87814	0.939068	−	0.343731i	$-0.111691\pi$
$719$	50.3607	1.87814	0.939068	+	0.343731i	$0.111691\pi$
$720$	2.61803	0.0975684
$721$	−0.291796	−0.0108671
$722$	−18.8541	−0.701677
$723$	4.41641	0.164248
$724$	17.1246	0.636431
$725$	−4.00000	−0.148556
$726$	0	0
$727$	37.3050	1.38356	0.691782	−	0.722106i	$-0.256826\pi$
$727$	37.3050	1.38356	0.691782	+	0.722106i	$0.256826\pi$
$728$	−5.23607	−0.194062
$729$	−8.50658	−0.315058
$730$	4.32624	0.160121
$731$	7.47214	0.276367
$732$	4.94427	0.182746
$733$	14.7639	0.545318	0.272659	−	0.962111i	$-0.412097\pi$
$733$	14.7639	0.545318	0.272659	+	0.962111i	$0.412097\pi$
$734$	11.0557	0.408074
$735$	−0.618034	−0.0227965
$736$	−8.47214	−0.312287
$737$	0	0
$738$	22.5623	0.830530
$739$	−19.9787	−0.734929	−0.367464	−	0.930038i	$-0.619774\pi$
$739$	−19.9787	−0.734929	−0.367464	+	0.930038i	$0.619774\pi$
$740$	−9.70820	−0.356881
$741$	1.23607	0.0454081
$742$	6.47214	0.237600
$743$	22.0000	0.807102	0.403551	−	0.914957i	$-0.367776\pi$
$743$	22.0000	0.807102	0.403551	+	0.914957i	$0.367776\pi$
$744$	−2.76393	−0.101331
$745$	5.70820	0.209132
$746$	4.94427	0.181023
$747$	29.0344	1.06231
$748$	0	0
$749$	−3.09017	−0.112912
$750$	−0.618034	−0.0225674
$751$	−22.5836	−0.824087	−0.412043	−	0.911164i	$-0.635185\pi$
$751$	−22.5836	−0.824087	−0.412043	+	0.911164i	$0.635185\pi$
$752$	−6.76393	−0.246655
$753$	−11.0557	−0.402893
$754$	−20.9443	−0.762745
$755$	−2.18034	−0.0793507
$756$	3.47214	0.126280
$757$	−32.0689	−1.16556	−0.582782	−	0.812629i	$-0.698036\pi$
$757$	−32.0689	−1.16556	−0.582782	+	0.812629i	$0.698036\pi$
$758$	−10.1459	−0.368516
$759$	0	0
$760$	−0.381966	−0.0138554
$761$	−40.2705	−1.45980	−0.729902	−	0.683551i	$-0.760435\pi$
$761$	−40.2705	−1.45980	−0.729902	+	0.683551i	$0.760435\pi$
$762$	−8.76393	−0.317484
$763$	−12.1803	−0.440958
$764$	−2.00000	−0.0723575
$765$	4.23607	0.153155
$766$	17.7082	0.639823
$767$	−18.9443	−0.684038
$768$	0.618034	0.0223014
$769$	1.41641	0.0510770	0.0255385	−	0.999674i	$-0.491870\pi$
$769$	1.41641	0.0510770	0.0255385	+	0.999674i	$0.491870\pi$
$770$	0	0
$771$	−15.7984	−0.568965
$772$	12.4721	0.448882
$773$	−8.36068	−0.300713	−0.150356	−	0.988632i	$-0.548042\pi$
$773$	−8.36068	−0.300713	−0.150356	+	0.988632i	$0.548042\pi$
$774$	−12.0902	−0.434572
$775$	−4.47214	−0.160644
$776$	−10.7984	−0.387639
$777$	−6.00000	−0.215249
$778$	8.00000	0.286814
$779$	−3.29180	−0.117941
$780$	−3.23607	−0.115870
$781$	0	0
$782$	−13.7082	−0.490204
$783$	13.8885	0.496336
$784$	1.00000	0.0357143
$785$	−16.1803	−0.577501
$786$	−9.18034	−0.327452
$787$	15.0344	0.535920	0.267960	−	0.963430i	$-0.413650\pi$
$787$	15.0344	0.535920	0.267960	+	0.963430i	$0.413650\pi$
$788$	−21.4164	−0.762928
$789$	7.41641	0.264031
$790$	1.70820	0.0607752
$791$	−13.7984	−0.490614
$792$	0	0
$793$	41.8885	1.48751
$794$	−34.5410	−1.22581
$795$	4.00000	0.141865
$796$	−1.23607	−0.0438113
$797$	25.5279	0.904243	0.452122	−	0.891956i	$-0.350667\pi$
$797$	25.5279	0.904243	0.452122	+	0.891956i	$0.350667\pi$
$798$	−0.236068	−0.00835672
$799$	−10.9443	−0.387181
$800$	1.00000	0.0353553
$801$	−25.7984	−0.911541
$802$	−10.5066	−0.371000
$803$	0	0
$804$	−3.76393	−0.132744
$805$	−8.47214	−0.298604
$806$	−23.4164	−0.824808
$807$	8.65248	0.304582
$808$	−17.2361	−0.606363
$809$	−46.5623	−1.63704	−0.818522	−	0.574476i	$-0.805206\pi$
$809$	−46.5623	−1.63704	−0.818522	+	0.574476i	$0.805206\pi$
$810$	−5.70820	−0.200566
$811$	−25.2016	−0.884949	−0.442474	−	0.896781i	$-0.645899\pi$
$811$	−25.2016	−0.884949	−0.442474	+	0.896781i	$0.645899\pi$
$812$	4.00000	0.140372
$813$	−12.1803	−0.427183
$814$	0	0
$815$	20.3820	0.713949
$816$	1.00000	0.0350070
$817$	1.76393	0.0617122
$818$	31.3050	1.09455
$819$	13.7082	0.479003
$820$	8.61803	0.300955
$821$	35.4853	1.23845	0.619223	−	0.785215i	$-0.287448\pi$
$821$	35.4853	1.23845	0.619223	+	0.785215i	$0.287448\pi$
$822$	−3.79837	−0.132484
$823$	−33.4853	−1.16722	−0.583612	−	0.812033i	$-0.698361\pi$
$823$	−33.4853	−1.16722	−0.583612	+	0.812033i	$0.698361\pi$
$824$	0.291796	0.0101652
$825$	0	0
$826$	3.61803	0.125888
$827$	16.7426	0.582199	0.291099	−	0.956693i	$-0.405979\pi$
$827$	16.7426	0.582199	0.291099	+	0.956693i	$0.405979\pi$
$828$	22.1803	0.770820
$829$	−14.9443	−0.519036	−0.259518	−	0.965738i	$-0.583564\pi$
$829$	−14.9443	−0.519036	−0.259518	+	0.965738i	$0.583564\pi$
$830$	11.0902	0.384945
$831$	−14.2918	−0.495777
$832$	5.23607	0.181528
$833$	1.61803	0.0560616
$834$	6.47214	0.224112
$835$	10.0000	0.346064
$836$	0	0
$837$	15.5279	0.536721
$838$	27.7426	0.958354
$839$	20.6525	0.713003	0.356501	−	0.934295i	$-0.383970\pi$
$839$	20.6525	0.713003	0.356501	+	0.934295i	$0.383970\pi$
$840$	0.618034	0.0213242
$841$	−13.0000	−0.448276
$842$	−0.472136	−0.0162709
$843$	4.20163	0.144712
$844$	−16.9098	−0.582060
$845$	−14.4164	−0.495940
$846$	17.7082	0.608821
$847$	0	0
$848$	−6.47214	−0.222254
$849$	16.0000	0.549119
$850$	1.61803	0.0554981
$851$	−82.2492	−2.81947
$852$	1.52786	0.0523438
$853$	49.7082	1.70198	0.850988	−	0.525185i	$-0.176004\pi$
$853$	49.7082	1.70198	0.850988	+	0.525185i	$0.176004\pi$
$854$	−8.00000	−0.273754
$855$	1.00000	0.0341993
$856$	3.09017	0.105620
$857$	12.6869	0.433377	0.216688	−	0.976241i	$-0.430474\pi$
$857$	12.6869	0.433377	0.216688	+	0.976241i	$0.430474\pi$
$858$	0	0
$859$	37.1459	1.26740	0.633701	−	0.773578i	$-0.281535\pi$
$859$	37.1459	1.26740	0.633701	+	0.773578i	$0.281535\pi$
$860$	−4.61803	−0.157474
$861$	5.32624	0.181518
$862$	−6.29180	−0.214299
$863$	−40.2492	−1.37010	−0.685050	−	0.728496i	$-0.740220\pi$
$863$	−40.2492	−1.37010	−0.685050	+	0.728496i	$0.740220\pi$
$864$	−3.47214	−0.118124
$865$	9.23607	0.314036
$866$	25.7984	0.876664
$867$	−8.88854	−0.301871
$868$	4.47214	0.151794
$869$	0	0
$870$	2.47214	0.0838133
$871$	−31.8885	−1.08050
$872$	12.1803	0.412478
$873$	28.2705	0.956812
$874$	−3.23607	−0.109462
$875$	1.00000	0.0338062
$876$	−2.67376	−0.0903380
$877$	−19.1246	−0.645792	−0.322896	−	0.946434i	$-0.604656\pi$
$877$	−19.1246	−0.645792	−0.322896	+	0.946434i	$0.604656\pi$
$878$	20.3607	0.687140
$879$	−1.41641	−0.0477743
$880$	0	0
$881$	−57.3394	−1.93181	−0.965907	−	0.258891i	$-0.916643\pi$
$881$	−57.3394	−1.93181	−0.965907	+	0.258891i	$0.916643\pi$
$882$	−2.61803	−0.0881538
$883$	36.6180	1.23229	0.616147	−	0.787631i	$-0.288693\pi$
$883$	36.6180	1.23229	0.616147	+	0.787631i	$0.288693\pi$
$884$	8.47214	0.284949
$885$	2.23607	0.0751646
$886$	−28.6180	−0.961442
$887$	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
$889$	14.1803	0.475593
$890$	−9.85410	−0.330310
$891$	0	0
$892$	−5.05573	−0.169278
$893$	−2.58359	−0.0864566
$894$	−3.52786	−0.117989
$895$	14.8541	0.496518
$896$	−1.00000	−0.0334077
$897$	−27.4164	−0.915407
$898$	−14.7426	−0.491968
$899$	17.8885	0.596616
$900$	−2.61803	−0.0872678
$901$	−10.4721	−0.348877
$902$	0	0
$903$	−2.85410	−0.0949786
$904$	13.7984	0.458927
$905$	−17.1246	−0.569241
$906$	1.34752	0.0447685
$907$	−3.79837	−0.126123	−0.0630615	−	0.998010i	$-0.520086\pi$
$907$	−3.79837	−0.126123	−0.0630615	+	0.998010i	$0.520086\pi$
$908$	−8.56231	−0.284150
$909$	45.1246	1.49669
$910$	5.23607	0.173574
$911$	32.0689	1.06249	0.531245	−	0.847218i	$-0.321724\pi$
$911$	32.0689	1.06249	0.531245	+	0.847218i	$0.321724\pi$
$912$	0.236068	0.00781699
$913$	0	0
$914$	25.5623	0.845526
$915$	−4.94427	−0.163453
$916$	−5.70820	−0.188604
$917$	14.8541	0.490526
$918$	−5.61803	−0.185423
$919$	16.7639	0.552991	0.276496	−	0.961015i	$-0.410827\pi$
$919$	16.7639	0.552991	0.276496	+	0.961015i	$0.410827\pi$
$920$	8.47214	0.279318
$921$	−21.3262	−0.702723
$922$	1.12461	0.0370371
$923$	12.9443	0.426066
$924$	0	0
$925$	9.70820	0.319204
$926$	0.652476	0.0214417
$927$	−0.763932	−0.0250908
$928$	−4.00000	−0.131306
$929$	35.3262	1.15902	0.579508	−	0.814966i	$-0.303245\pi$
$929$	35.3262	1.15902	0.579508	+	0.814966i	$0.303245\pi$
$930$	2.76393	0.0906329
$931$	0.381966	0.0125184
$932$	−3.67376	−0.120338
$933$	2.00000	0.0654771
$934$	16.9443	0.554434
$935$	0	0
$936$	−13.7082	−0.448067
$937$	−45.3951	−1.48299	−0.741497	−	0.670956i	$-0.765884\pi$
$937$	−45.3951	−1.48299	−0.741497	+	0.670956i	$0.765884\pi$
$938$	6.09017	0.198851
$939$	2.05573	0.0670862
$940$	6.76393	0.220615
$941$	17.7771	0.579516	0.289758	−	0.957100i	$-0.406425\pi$
$941$	17.7771	0.579516	0.289758	+	0.957100i	$0.406425\pi$
$942$	10.0000	0.325818
$943$	73.0132	2.37764
$944$	−3.61803	−0.117757
$945$	−3.47214	−0.112949
$946$	0	0
$947$	−40.3394	−1.31085	−0.655427	−	0.755258i	$-0.727511\pi$
$947$	−40.3394	−1.31085	−0.655427	+	0.755258i	$0.727511\pi$
$948$	−1.05573	−0.0342885
$949$	−22.6525	−0.735330
$950$	0.381966	0.0123926
$951$	−11.0557	−0.358507
$952$	−1.61803	−0.0524408
$953$	−33.8541	−1.09664	−0.548321	−	0.836268i	$-0.684733\pi$
$953$	−33.8541	−1.09664	−0.548321	+	0.836268i	$0.684733\pi$
$954$	16.9443	0.548591
$955$	2.00000	0.0647185
$956$	−17.7082	−0.572724
$957$	0	0
$958$	17.1246	0.553271
$959$	6.14590	0.198461
$960$	−0.618034	−0.0199470
$961$	−11.0000	−0.354839
$962$	50.8328	1.63892
$963$	−8.09017	−0.260702
$964$	7.14590	0.230154
$965$	−12.4721	−0.401492
$966$	5.23607	0.168468
$967$	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
$969$	0.381966	0.0122705
$970$	10.7984	0.346715
$971$	41.3050	1.32554	0.662769	−	0.748823i	$-0.269381\pi$
$971$	41.3050	1.32554	0.662769	+	0.748823i	$0.269381\pi$
$972$	13.9443	0.447263
$973$	−10.4721	−0.335721
$974$	37.7771	1.21046
$975$	3.23607	0.103637
$976$	8.00000	0.256074
$977$	40.4721	1.29482	0.647409	−	0.762143i	$-0.275853\pi$
$977$	40.4721	1.29482	0.647409	+	0.762143i	$0.275853\pi$
$978$	−12.5967	−0.402800
$979$	0	0
$980$	−1.00000	−0.0319438
$981$	−31.8885	−1.01812
$982$	37.9230	1.21017
$983$	26.7639	0.853637	0.426818	−	0.904337i	$-0.359634\pi$
$983$	26.7639	0.853637	0.426818	+	0.904337i	$0.359634\pi$
$984$	−5.32624	−0.169794
$985$	21.4164	0.682383
$986$	−6.47214	−0.206115
$987$	4.18034	0.133062
$988$	2.00000	0.0636285
$989$	−39.1246	−1.24409
$990$	0	0
$991$	−56.7639	−1.80317	−0.901583	−	0.432606i	$-0.857594\pi$
$991$	−56.7639	−1.80317	−0.901583	+	0.432606i	$0.857594\pi$
$992$	−4.47214	−0.141990
$993$	7.61803	0.241751
$994$	−2.47214	−0.0784114
$995$	1.23607	0.0391860
$996$	−6.85410	−0.217181
$997$	−53.1246	−1.68247	−0.841237	−	0.540667i	$-0.818172\pi$
$997$	−53.1246	−1.68247	−0.841237	+	0.540667i	$0.818172\pi$
$998$	0.618034	0.0195635
$999$	−33.7082	−1.06648

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8470.2.a.bx.1.2		2
11.7	odd	10		770.2.n.c.71.1	✓	4
11.8	odd	10		770.2.n.c.141.1	yes	4
11.10	odd	2		8470.2.a.bk.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
770.2.n.c.71.1	✓	4	11.7	odd	10
770.2.n.c.141.1	yes	4	11.8	odd	10
8470.2.a.bk.1.2		2	11.10	odd	2
8470.2.a.bx.1.2		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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +0.618034 q^{3} +1.00000 q^{4} -1.00000 q^{5} +0.618034 q^{6} -1.00000 q^{7} +1.00000 q^{8} -2.61803 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} +0.618034 q^{3} +1.00000 q^{4} -1.00000 q^{5} +0.618034 q^{6} -1.00000 q^{7} +1.00000 q^{8} -2.61803 q^{9} -1.00000 q^{10} +0.618034 q^{12} +5.23607 q^{13} -1.00000 q^{14} -0.618034 q^{15} +1.00000 q^{16} +1.61803 q^{17} -2.61803 q^{18} +0.381966 q^{19} -1.00000 q^{20} -0.618034 q^{21} -8.47214 q^{23} +0.618034 q^{24} +1.00000 q^{25} +5.23607 q^{26} -3.47214 q^{27} -1.00000 q^{28} -4.00000 q^{29} -0.618034 q^{30} -4.47214 q^{31} +1.00000 q^{32} +1.61803 q^{34} +1.00000 q^{35} -2.61803 q^{36} +9.70820 q^{37} +0.381966 q^{38} +3.23607 q^{39} -1.00000 q^{40} -8.61803 q^{41} -0.618034 q^{42} +4.61803 q^{43} +2.61803 q^{45} -8.47214 q^{46} -6.76393 q^{47} +0.618034 q^{48} +1.00000 q^{49} +1.00000 q^{50} +1.00000 q^{51} +5.23607 q^{52} -6.47214 q^{53} -3.47214 q^{54} -1.00000 q^{56} +0.236068 q^{57} -4.00000 q^{58} -3.61803 q^{59} -0.618034 q^{60} +8.00000 q^{61} -4.47214 q^{62} +2.61803 q^{63} +1.00000 q^{64} -5.23607 q^{65} -6.09017 q^{67} +1.61803 q^{68} -5.23607 q^{69} +1.00000 q^{70} +2.47214 q^{71} -2.61803 q^{72} -4.32624 q^{73} +9.70820 q^{74} +0.618034 q^{75} +0.381966 q^{76} +3.23607 q^{78} -1.70820 q^{79} -1.00000 q^{80} +5.70820 q^{81} -8.61803 q^{82} -11.0902 q^{83} -0.618034 q^{84} -1.61803 q^{85} +4.61803 q^{86} -2.47214 q^{87} +9.85410 q^{89} +2.61803 q^{90} -5.23607 q^{91} -8.47214 q^{92} -2.76393 q^{93} -6.76393 q^{94} -0.381966 q^{95} +0.618034 q^{96} -10.7984 q^{97} +1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} - q^{3} + 2 q^{4} - 2 q^{5} - q^{6} - 2 q^{7} + 2 q^{8} - 3 q^{9}+O(q^{10}) \) 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

Related objects

Downloads

Learn more