LMFDB - Embedded newform 8470.2.a.dc.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:	$0$
Dimension:	$6$
Coefficient field:	6.6.19898000.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - x^{5} - 10x^{4} + 7x^{3} + 24x^{2} - 15x - 5 $ x^6 - x^5 - 10x^4 + 7x^3 + 24x^2 - 15x - 5
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
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			$2.86564$ 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)

$f(q)$	$=$	$q+1.00000 q^{2} -2.86564 q^{3} +1.00000 q^{4} +1.00000 q^{5} -2.86564 q^{6} -1.00000 q^{7} +1.00000 q^{8} +5.21187 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} -2.86564 q^{3} +1.00000 q^{4} +1.00000 q^{5} -2.86564 q^{6} -1.00000 q^{7} +1.00000 q^{8} +5.21187 q^{9} +1.00000 q^{10} -2.86564 q^{12} +5.63670 q^{13} -1.00000 q^{14} -2.86564 q^{15} +1.00000 q^{16} +2.32986 q^{17} +5.21187 q^{18} +7.81494 q^{19} +1.00000 q^{20} +2.86564 q^{21} +7.98434 q^{23} -2.86564 q^{24} +1.00000 q^{25} +5.63670 q^{26} -6.33840 q^{27} -1.00000 q^{28} -2.99446 q^{29} -2.86564 q^{30} +5.08533 q^{31} +1.00000 q^{32} +2.32986 q^{34} -1.00000 q^{35} +5.21187 q^{36} -1.38681 q^{37} +7.81494 q^{38} -16.1527 q^{39} +1.00000 q^{40} -6.90154 q^{41} +2.86564 q^{42} +11.5492 q^{43} +5.21187 q^{45} +7.98434 q^{46} -5.14520 q^{47} -2.86564 q^{48} +1.00000 q^{49} +1.00000 q^{50} -6.67652 q^{51} +5.63670 q^{52} +0.582695 q^{53} -6.33840 q^{54} -1.00000 q^{56} -22.3948 q^{57} -2.99446 q^{58} +13.8292 q^{59} -2.86564 q^{60} -7.82443 q^{61} +5.08533 q^{62} -5.21187 q^{63} +1.00000 q^{64} +5.63670 q^{65} -0.0485769 q^{67} +2.32986 q^{68} -22.8802 q^{69} -1.00000 q^{70} +4.45361 q^{71} +5.21187 q^{72} +0.336814 q^{73} -1.38681 q^{74} -2.86564 q^{75} +7.81494 q^{76} -16.1527 q^{78} -1.05291 q^{79} +1.00000 q^{80} +2.52795 q^{81} -6.90154 q^{82} -10.8315 q^{83} +2.86564 q^{84} +2.32986 q^{85} +11.5492 q^{86} +8.58103 q^{87} -18.5268 q^{89} +5.21187 q^{90} -5.63670 q^{91} +7.98434 q^{92} -14.5727 q^{93} -5.14520 q^{94} +7.81494 q^{95} -2.86564 q^{96} -5.33180 q^{97} +1.00000 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 6 q^{2} - q^{3} + 6 q^{4} + 6 q^{5} - q^{6} - 6 q^{7} + 6 q^{8} + 3 q^{9}+O(q^{10}) $ 6 * q + 6 * q^2 - q^3 + 6 * q^4 + 6 * q^5 - q^6 - 6 * q^7 + 6 * q^8 + 3 * q^9	$ 6 q + 6 q^{2} - q^{3} + 6 q^{4} + 6 q^{5} - q^{6} - 6 q^{7} + 6 q^{8} + 3 q^{9} + 6 q^{10} - q^{12} + 9 q^{13} - 6 q^{14} - q^{15} + 6 q^{16} + 9 q^{17} + 3 q^{18} + 12 q^{19} + 6 q^{20} + q^{21} + 4 q^{23} - q^{24} + 6 q^{25} + 9 q^{26} - 4 q^{27} - 6 q^{28} + 15 q^{29} - q^{30} + 8 q^{31} + 6 q^{32} + 9 q^{34} - 6 q^{35} + 3 q^{36} - 4 q^{37} + 12 q^{38} - 19 q^{39} + 6 q^{40} + 4 q^{41} + q^{42} + 30 q^{43} + 3 q^{45} + 4 q^{46} - 7 q^{47} - q^{48} + 6 q^{49} + 6 q^{50} + 16 q^{51} + 9 q^{52} - 6 q^{53} - 4 q^{54} - 6 q^{56} - 14 q^{57} + 15 q^{58} + 4 q^{59} - q^{60} - 14 q^{61} + 8 q^{62} - 3 q^{63} + 6 q^{64} + 9 q^{65} + 18 q^{67} + 9 q^{68} - 10 q^{69} - 6 q^{70} + 23 q^{71} + 3 q^{72} + 23 q^{73} - 4 q^{74} - q^{75} + 12 q^{76} - 19 q^{78} + 21 q^{79} + 6 q^{80} - 18 q^{81} + 4 q^{82} + 25 q^{83} + q^{84} + 9 q^{85} + 30 q^{86} + 14 q^{87} - 18 q^{89} + 3 q^{90} - 9 q^{91} + 4 q^{92} - 24 q^{93} - 7 q^{94} + 12 q^{95} - q^{96} + 7 q^{97} + 6 q^{98}+O(q^{100}) $ 6 * q + 6 * q^2 - q^3 + 6 * q^4 + 6 * q^5 - q^6 - 6 * q^7 + 6 * q^8 + 3 * q^9 + 6 * q^10 - q^12 + 9 * q^13 - 6 * q^14 - q^15 + 6 * q^16 + 9 * q^17 + 3 * q^18 + 12 * q^19 + 6 * q^20 + q^21 + 4 * q^23 - q^24 + 6 * q^25 + 9 * q^26 - 4 * q^27 - 6 * q^28 + 15 * q^29 - q^30 + 8 * q^31 + 6 * q^32 + 9 * q^34 - 6 * q^35 + 3 * q^36 - 4 * q^37 + 12 * q^38 - 19 * q^39 + 6 * q^40 + 4 * q^41 + q^42 + 30 * q^43 + 3 * q^45 + 4 * q^46 - 7 * q^47 - q^48 + 6 * q^49 + 6 * q^50 + 16 * q^51 + 9 * q^52 - 6 * q^53 - 4 * q^54 - 6 * q^56 - 14 * q^57 + 15 * q^58 + 4 * q^59 - q^60 - 14 * q^61 + 8 * q^62 - 3 * q^63 + 6 * q^64 + 9 * q^65 + 18 * q^67 + 9 * q^68 - 10 * q^69 - 6 * q^70 + 23 * q^71 + 3 * q^72 + 23 * q^73 - 4 * q^74 - q^75 + 12 * q^76 - 19 * q^78 + 21 * q^79 + 6 * q^80 - 18 * q^81 + 4 * q^82 + 25 * q^83 + q^84 + 9 * q^85 + 30 * q^86 + 14 * q^87 - 18 * q^89 + 3 * q^90 - 9 * q^91 + 4 * q^92 - 24 * q^93 - 7 * q^94 + 12 * q^95 - q^96 + 7 * q^97 + 6 * 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
$3$	−2.86564	−1.65448	−0.827238	−	0.561852i	$-0.810089\pi$
$3$	−2.86564	−1.65448	−0.827238	+	0.561852i	$0.810089\pi$
$4$	1.00000	0.500000
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	−2.86564	−1.16989
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	1.00000	0.353553
$9$	5.21187	1.73729
$10$	1.00000	0.316228
$11$	0	0
$11$	0	0
$12$	−2.86564	−0.827238
$13$	5.63670	1.56334	0.781669	−	0.623693i	$-0.214369\pi$
$13$	5.63670	1.56334	0.781669	+	0.623693i	$0.214369\pi$
$14$	−1.00000	−0.267261
$15$	−2.86564	−0.739904
$16$	1.00000	0.250000
$17$	2.32986	0.565074	0.282537	−	0.959256i	$-0.408824\pi$
$17$	2.32986	0.565074	0.282537	+	0.959256i	$0.408824\pi$
$18$	5.21187	1.22845
$19$	7.81494	1.79287	0.896435	−	0.443175i	$-0.146148\pi$
$19$	7.81494	1.79287	0.896435	+	0.443175i	$0.146148\pi$
$20$	1.00000	0.223607
$21$	2.86564	0.625333
$22$	0	0
$23$	7.98434	1.66485	0.832425	−	0.554138i	$-0.186952\pi$
$23$	7.98434	1.66485	0.832425	+	0.554138i	$0.186952\pi$
$24$	−2.86564	−0.584945
$25$	1.00000	0.200000
$26$	5.63670	1.10545
$27$	−6.33840	−1.21983
$28$	−1.00000	−0.188982
$29$	−2.99446	−0.556057	−0.278029	−	0.960573i	$-0.589681\pi$
$29$	−2.99446	−0.556057	−0.278029	+	0.960573i	$0.589681\pi$
$30$	−2.86564	−0.523191
$31$	5.08533	0.913352	0.456676	−	0.889633i	$-0.349040\pi$
$31$	5.08533	0.913352	0.456676	+	0.889633i	$0.349040\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	2.32986	0.399567
$35$	−1.00000	−0.169031
$36$	5.21187	0.868644
$37$	−1.38681	−0.227989	−0.113995	−	0.993481i	$-0.536365\pi$
$37$	−1.38681	−0.227989	−0.113995	+	0.993481i	$0.536365\pi$
$38$	7.81494	1.26775
$39$	−16.1527	−2.58650
$40$	1.00000	0.158114
$41$	−6.90154	−1.07784	−0.538920	−	0.842357i	$-0.681167\pi$
$41$	−6.90154	−1.07784	−0.538920	+	0.842357i	$0.681167\pi$
$42$	2.86564	0.442177
$43$	11.5492	1.76124	0.880622	−	0.473820i	$-0.157125\pi$
$43$	11.5492	1.76124	0.880622	+	0.473820i	$0.157125\pi$
$44$	0	0
$45$	5.21187	0.776939
$46$	7.98434	1.17723
$47$	−5.14520	−0.750504	−0.375252	−	0.926923i	$-0.622444\pi$
$47$	−5.14520	−0.750504	−0.375252	+	0.926923i	$0.622444\pi$
$48$	−2.86564	−0.413619
$49$	1.00000	0.142857
$50$	1.00000	0.141421
$51$	−6.67652	−0.934900
$52$	5.63670	0.781669
$53$	0.582695	0.0800393	0.0400196	−	0.999199i	$-0.487258\pi$
$53$	0.582695	0.0800393	0.0400196	+	0.999199i	$0.487258\pi$
$54$	−6.33840	−0.862547
$55$	0	0
$56$	−1.00000	−0.133631
$57$	−22.3948	−2.96626
$58$	−2.99446	−0.393192
$59$	13.8292	1.80041	0.900204	−	0.435468i	$-0.143417\pi$
$59$	13.8292	1.80041	0.900204	+	0.435468i	$0.143417\pi$
$60$	−2.86564	−0.369952
$61$	−7.82443	−1.00182	−0.500908	−	0.865501i	$-0.667000\pi$
$61$	−7.82443	−1.00182	−0.500908	+	0.865501i	$0.667000\pi$
$62$	5.08533	0.645838
$63$	−5.21187	−0.656633
$64$	1.00000	0.125000
$65$	5.63670	0.699146
$66$	0	0
$67$	−0.0485769	−0.00593462	−0.00296731	−	0.999996i	$-0.500945\pi$
$67$	−0.0485769	−0.00593462	−0.00296731	+	0.999996i	$0.500945\pi$
$68$	2.32986	0.282537
$69$	−22.8802	−2.75445
$70$	−1.00000	−0.119523
$71$	4.45361	0.528546	0.264273	−	0.964448i	$-0.414868\pi$
$71$	4.45361	0.528546	0.264273	+	0.964448i	$0.414868\pi$
$72$	5.21187	0.614224
$73$	0.336814	0.0394210	0.0197105	−	0.999806i	$-0.493726\pi$
$73$	0.336814	0.0394210	0.0197105	+	0.999806i	$0.493726\pi$
$74$	−1.38681	−0.161213
$75$	−2.86564	−0.330895
$76$	7.81494	0.896435
$77$	0	0
$78$	−16.1527	−1.82893
$79$	−1.05291	−0.118462	−0.0592308	−	0.998244i	$-0.518865\pi$
$79$	−1.05291	−0.118462	−0.0592308	+	0.998244i	$0.518865\pi$
$80$	1.00000	0.111803
$81$	2.52795	0.280883
$82$	−6.90154	−0.762148
$83$	−10.8315	−1.18891	−0.594455	−	0.804129i	$-0.702632\pi$
$83$	−10.8315	−1.18891	−0.594455	+	0.804129i	$0.702632\pi$
$84$	2.86564	0.312666
$85$	2.32986	0.252709
$86$	11.5492	1.24539
$87$	8.58103	0.919983
$88$	0	0
$89$	−18.5268	−1.96384	−0.981921	−	0.189291i	$-0.939381\pi$
$89$	−18.5268	−1.96384	−0.981921	+	0.189291i	$0.939381\pi$
$90$	5.21187	0.549379
$91$	−5.63670	−0.590886
$92$	7.98434	0.832425
$93$	−14.5727	−1.51112
$94$	−5.14520	−0.530686
$95$	7.81494	0.801796
$96$	−2.86564	−0.292473
$97$	−5.33180	−0.541362	−0.270681	−	0.962669i	$-0.587249\pi$
$97$	−5.33180	−0.541362	−0.270681	+	0.962669i	$0.587249\pi$
$98$	1.00000	0.101015
$99$	0	0
$100$	1.00000	0.100000
$101$	18.2701	1.81795	0.908973	−	0.416854i	$-0.136867\pi$
$101$	18.2701	1.81795	0.908973	+	0.416854i	$0.136867\pi$
$102$	−6.67652	−0.661074
$103$	−11.2347	−1.10699	−0.553495	−	0.832852i	$-0.686706\pi$
$103$	−11.2347	−1.10699	−0.553495	+	0.832852i	$0.686706\pi$
$104$	5.63670	0.552723
$105$	2.86564	0.279657
$106$	0.582695	0.0565963
$107$	−4.89157	−0.472886	−0.236443	−	0.971645i	$-0.575982\pi$
$107$	−4.89157	−0.472886	−0.236443	+	0.971645i	$0.575982\pi$
$108$	−6.33840	−0.609913
$109$	−6.26442	−0.600023	−0.300011	−	0.953936i	$-0.596991\pi$
$109$	−6.26442	−0.600023	−0.300011	+	0.953936i	$0.596991\pi$
$110$	0	0
$111$	3.97408	0.377203
$112$	−1.00000	−0.0944911
$113$	10.1651	0.956251	0.478126	−	0.878291i	$-0.341316\pi$
$113$	10.1651	0.956251	0.478126	+	0.878291i	$0.341316\pi$
$114$	−22.3948	−2.09746
$115$	7.98434	0.744544
$116$	−2.99446	−0.278029
$117$	29.3777	2.71597
$118$	13.8292	1.27308
$119$	−2.32986	−0.213578
$120$	−2.86564	−0.261596
$121$	0	0
$122$	−7.82443	−0.708391
$123$	19.7773	1.78326
$124$	5.08533	0.456676
$125$	1.00000	0.0894427
$126$	−5.21187	−0.464310
$127$	−6.58727	−0.584526	−0.292263	−	0.956338i	$-0.594408\pi$
$127$	−6.58727	−0.584526	−0.292263	+	0.956338i	$0.594408\pi$
$128$	1.00000	0.0883883
$129$	−33.0959	−2.91393
$130$	5.63670	0.494371
$131$	−1.20821	−0.105562	−0.0527808	−	0.998606i	$-0.516808\pi$
$131$	−1.20821	−0.105562	−0.0527808	+	0.998606i	$0.516808\pi$
$132$	0	0
$133$	−7.81494	−0.677641
$134$	−0.0485769	−0.00419641
$135$	−6.33840	−0.545523
$136$	2.32986	0.199784
$137$	14.0829	1.20318	0.601590	−	0.798805i	$-0.294534\pi$
$137$	14.0829	1.20318	0.601590	+	0.798805i	$0.294534\pi$
$138$	−22.8802	−1.94769
$139$	−2.74290	−0.232650	−0.116325	−	0.993211i	$-0.537111\pi$
$139$	−2.74290	−0.232650	−0.116325	+	0.993211i	$0.537111\pi$
$140$	−1.00000	−0.0845154
$141$	14.7443	1.24169
$142$	4.45361	0.373739
$143$	0	0
$144$	5.21187	0.434322
$145$	−2.99446	−0.248676
$146$	0.336814	0.0278749
$147$	−2.86564	−0.236354
$148$	−1.38681	−0.113995
$149$	17.1642	1.40615	0.703075	−	0.711116i	$-0.251810\pi$
$149$	17.1642	1.40615	0.703075	+	0.711116i	$0.251810\pi$
$150$	−2.86564	−0.233978
$151$	−5.94304	−0.483637	−0.241819	−	0.970321i	$-0.577744\pi$
$151$	−5.94304	−0.483637	−0.241819	+	0.970321i	$0.577744\pi$
$152$	7.81494	0.633875
$153$	12.1429	0.981696
$154$	0	0
$155$	5.08533	0.408464
$156$	−16.1527	−1.29325
$157$	−16.7947	−1.34037	−0.670183	−	0.742196i	$-0.733784\pi$
$157$	−16.7947	−1.34037	−0.670183	+	0.742196i	$0.733784\pi$
$158$	−1.05291	−0.0837650
$159$	−1.66979	−0.132423
$160$	1.00000	0.0790569
$161$	−7.98434	−0.629254
$162$	2.52795	0.198614
$163$	2.43041	0.190364	0.0951821	−	0.995460i	$-0.469657\pi$
$163$	2.43041	0.190364	0.0951821	+	0.995460i	$0.469657\pi$
$164$	−6.90154	−0.538920
$165$	0	0
$166$	−10.8315	−0.840686
$167$	11.0306	0.853572	0.426786	−	0.904353i	$-0.359646\pi$
$167$	11.0306	0.853572	0.426786	+	0.904353i	$0.359646\pi$
$168$	2.86564	0.221089
$169$	18.7723	1.44403
$170$	2.32986	0.178692
$171$	40.7304	3.11473
$172$	11.5492	0.880622
$173$	−14.6947	−1.11721	−0.558607	−	0.829433i	$-0.688664\pi$
$173$	−14.6947	−1.11721	−0.558607	+	0.829433i	$0.688664\pi$
$174$	8.58103	0.650526
$175$	−1.00000	−0.0755929
$176$	0	0
$177$	−39.6294	−2.97873
$178$	−18.5268	−1.38865
$179$	4.51392	0.337387	0.168693	−	0.985669i	$-0.446045\pi$
$179$	4.51392	0.337387	0.168693	+	0.985669i	$0.446045\pi$
$180$	5.21187	0.388470
$181$	−10.1770	−0.756451	−0.378225	−	0.925714i	$-0.623466\pi$
$181$	−10.1770	−0.756451	−0.378225	+	0.925714i	$0.623466\pi$
$182$	−5.63670	−0.417820
$183$	22.4220	1.65748
$184$	7.98434	0.588613
$185$	−1.38681	−0.101960
$186$	−14.5727	−1.06852
$187$	0	0
$188$	−5.14520	−0.375252
$189$	6.33840	0.461051
$190$	7.81494	0.566955
$191$	11.9914	0.867669	0.433834	−	0.900993i	$-0.357160\pi$
$191$	11.9914	0.867669	0.433834	+	0.900993i	$0.357160\pi$
$192$	−2.86564	−0.206809
$193$	17.8748	1.28666	0.643330	−	0.765589i	$-0.277552\pi$
$193$	17.8748	1.28666	0.643330	+	0.765589i	$0.277552\pi$
$194$	−5.33180	−0.382801
$195$	−16.1527	−1.15672
$196$	1.00000	0.0714286
$197$	11.7871	0.839795	0.419897	−	0.907572i	$-0.362066\pi$
$197$	11.7871	0.839795	0.419897	+	0.907572i	$0.362066\pi$
$198$	0	0
$199$	−13.9342	−0.987767	−0.493884	−	0.869528i	$-0.664423\pi$
$199$	−13.9342	−0.987767	−0.493884	+	0.869528i	$0.664423\pi$
$200$	1.00000	0.0707107
$201$	0.139204	0.00981868
$202$	18.2701	1.28548
$203$	2.99446	0.210170
$204$	−6.67652	−0.467450
$205$	−6.90154	−0.482025
$206$	−11.2347	−0.782760
$207$	41.6133	2.89233
$208$	5.63670	0.390835
$209$	0	0
$210$	2.86564	0.197748
$211$	−26.4497	−1.82087	−0.910435	−	0.413653i	$-0.864253\pi$
$211$	−26.4497	−1.82087	−0.910435	+	0.413653i	$0.864253\pi$
$212$	0.582695	0.0400196
$213$	−12.7624	−0.874467
$214$	−4.89157	−0.334381
$215$	11.5492	0.787652
$216$	−6.33840	−0.431274
$217$	−5.08533	−0.345215
$218$	−6.26442	−0.424280
$219$	−0.965185	−0.0652211
$220$	0	0
$221$	13.1327	0.883401
$222$	3.97408	0.266723
$223$	24.5893	1.64662	0.823311	−	0.567591i	$-0.192124\pi$
$223$	24.5893	1.64662	0.823311	+	0.567591i	$0.192124\pi$
$224$	−1.00000	−0.0668153
$225$	5.21187	0.347458
$226$	10.1651	0.676172
$227$	5.59242	0.371182	0.185591	−	0.982627i	$-0.440580\pi$
$227$	5.59242	0.371182	0.185591	+	0.982627i	$0.440580\pi$
$228$	−22.3948	−1.48313
$229$	−8.10340	−0.535488	−0.267744	−	0.963490i	$-0.586278\pi$
$229$	−8.10340	−0.535488	−0.267744	+	0.963490i	$0.586278\pi$
$230$	7.98434	0.526472
$231$	0	0
$232$	−2.99446	−0.196596
$233$	−11.9807	−0.784882	−0.392441	−	0.919777i	$-0.628369\pi$
$233$	−11.9807	−0.784882	−0.392441	+	0.919777i	$0.628369\pi$
$234$	29.3777	1.92048
$235$	−5.14520	−0.335636
$236$	13.8292	0.900204
$237$	3.01726	0.195992
$238$	−2.32986	−0.151022
$239$	6.46612	0.418258	0.209129	−	0.977888i	$-0.432937\pi$
$239$	6.46612	0.418258	0.209129	+	0.977888i	$0.432937\pi$
$240$	−2.86564	−0.184976
$241$	−20.3441	−1.31048	−0.655238	−	0.755423i	$-0.727432\pi$
$241$	−20.3441	−1.31048	−0.655238	+	0.755423i	$0.727432\pi$
$242$	0	0
$243$	11.7710	0.755112
$244$	−7.82443	−0.500908
$245$	1.00000	0.0638877
$246$	19.7773	1.26096
$247$	44.0504	2.80286
$248$	5.08533	0.322919
$249$	31.0391	1.96702
$250$	1.00000	0.0632456
$251$	12.5842	0.794311	0.397155	−	0.917751i	$-0.369997\pi$
$251$	12.5842	0.794311	0.397155	+	0.917751i	$0.369997\pi$
$252$	−5.21187	−0.328317
$253$	0	0
$254$	−6.58727	−0.413322
$255$	−6.67652	−0.418100
$256$	1.00000	0.0625000
$257$	3.45995	0.215826	0.107913	−	0.994160i	$-0.465583\pi$
$257$	3.45995	0.215826	0.107913	+	0.994160i	$0.465583\pi$
$258$	−33.0959	−2.06046
$259$	1.38681	0.0861719
$260$	5.63670	0.349573
$261$	−15.6067	−0.966032
$262$	−1.20821	−0.0746434
$263$	8.01789	0.494404	0.247202	−	0.968964i	$-0.420489\pi$
$263$	8.01789	0.494404	0.247202	+	0.968964i	$0.420489\pi$
$264$	0	0
$265$	0.582695	0.0357946
$266$	−7.81494	−0.479165
$267$	53.0912	3.24913
$268$	−0.0485769	−0.00296731
$269$	−18.5650	−1.13193	−0.565964	−	0.824430i	$-0.691496\pi$
$269$	−18.5650	−1.13193	−0.565964	+	0.824430i	$0.691496\pi$
$270$	−6.33840	−0.385743
$271$	−25.2051	−1.53110	−0.765552	−	0.643375i	$-0.777534\pi$
$271$	−25.2051	−1.53110	−0.765552	+	0.643375i	$0.777534\pi$
$272$	2.32986	0.141268
$273$	16.1527	0.977607
$274$	14.0829	0.850777
$275$	0	0
$276$	−22.8802	−1.37723
$277$	12.2281	0.734714	0.367357	−	0.930080i	$-0.380263\pi$
$277$	12.2281	0.734714	0.367357	+	0.930080i	$0.380263\pi$
$278$	−2.74290	−0.164508
$279$	26.5041	1.58676
$280$	−1.00000	−0.0597614
$281$	−26.5934	−1.58643	−0.793216	−	0.608941i	$-0.791595\pi$
$281$	−26.5934	−1.58643	−0.793216	+	0.608941i	$0.791595\pi$
$282$	14.7443	0.878008
$283$	13.7220	0.815689	0.407845	−	0.913051i	$-0.366281\pi$
$283$	13.7220	0.815689	0.407845	+	0.913051i	$0.366281\pi$
$284$	4.45361	0.264273
$285$	−22.3948	−1.32655
$286$	0	0
$287$	6.90154	0.407385
$288$	5.21187	0.307112
$289$	−11.5718	−0.680692
$290$	−2.99446	−0.175841
$291$	15.2790	0.895670
$292$	0.336814	0.0197105
$293$	−0.714463	−0.0417394	−0.0208697	−	0.999782i	$-0.506644\pi$
$293$	−0.714463	−0.0417394	−0.0208697	+	0.999782i	$0.506644\pi$
$294$	−2.86564	−0.167127
$295$	13.8292	0.805167
$296$	−1.38681	−0.0806064
$297$	0	0
$298$	17.1642	0.994298
$299$	45.0053	2.60272
$300$	−2.86564	−0.165448
$301$	−11.5492	−0.665688
$302$	−5.94304	−0.341983
$303$	−52.3556	−3.00775
$304$	7.81494	0.448218
$305$	−7.82443	−0.448026
$306$	12.1429	0.694164
$307$	19.3442	1.10403	0.552015	−	0.833834i	$-0.313859\pi$
$307$	19.3442	1.10403	0.552015	+	0.833834i	$0.313859\pi$
$308$	0	0
$309$	32.1946	1.83149
$310$	5.08533	0.288827
$311$	1.82729	0.103616	0.0518080	−	0.998657i	$-0.483502\pi$
$311$	1.82729	0.103616	0.0518080	+	0.998657i	$0.483502\pi$
$312$	−16.1527	−0.914467
$313$	−9.83108	−0.555686	−0.277843	−	0.960627i	$-0.589619\pi$
$313$	−9.83108	−0.555686	−0.277843	+	0.960627i	$0.589619\pi$
$314$	−16.7947	−0.947782
$315$	−5.21187	−0.293655
$316$	−1.05291	−0.0592308
$317$	−13.8545	−0.778146	−0.389073	−	0.921207i	$-0.627205\pi$
$317$	−13.8545	−0.778146	−0.389073	+	0.921207i	$0.627205\pi$
$318$	−1.66979	−0.0936372
$319$	0	0
$320$	1.00000	0.0559017
$321$	14.0175	0.782378
$322$	−7.98434	−0.444950
$323$	18.2077	1.01310
$324$	2.52795	0.140442
$325$	5.63670	0.312668
$326$	2.43041	0.134608
$327$	17.9516	0.992723
$328$	−6.90154	−0.381074
$329$	5.14520	0.283664
$330$	0	0
$331$	8.02084	0.440865	0.220433	−	0.975402i	$-0.429253\pi$
$331$	8.02084	0.440865	0.220433	+	0.975402i	$0.429253\pi$
$332$	−10.8315	−0.594455
$333$	−7.22784	−0.396083
$334$	11.0306	0.603567
$335$	−0.0485769	−0.00265404
$336$	2.86564	0.156333
$337$	11.6060	0.632218	0.316109	−	0.948723i	$-0.397624\pi$
$337$	11.6060	0.632218	0.316109	+	0.948723i	$0.397624\pi$
$338$	18.7723	1.02108
$339$	−29.1294	−1.58209
$340$	2.32986	0.126354
$341$	0	0
$342$	40.7304	2.20245
$343$	−1.00000	−0.0539949
$344$	11.5492	0.622694
$345$	−22.8802	−1.23183
$346$	−14.6947	−0.789989
$347$	26.4301	1.41884	0.709420	−	0.704786i	$-0.248957\pi$
$347$	26.4301	1.41884	0.709420	+	0.704786i	$0.248957\pi$
$348$	8.58103	0.459991
$349$	−27.3777	−1.46549	−0.732747	−	0.680501i	$-0.761762\pi$
$349$	−27.3777	−1.46549	−0.732747	+	0.680501i	$0.761762\pi$
$350$	−1.00000	−0.0534522
$351$	−35.7276	−1.90700
$352$	0	0
$353$	21.3772	1.13779	0.568897	−	0.822409i	$-0.307370\pi$
$353$	21.3772	1.13779	0.568897	+	0.822409i	$0.307370\pi$
$354$	−39.6294	−2.10628
$355$	4.45361	0.236373
$356$	−18.5268	−0.981921
$357$	6.67652	0.353359
$358$	4.51392	0.238568
$359$	10.8904	0.574774	0.287387	−	0.957815i	$-0.407213\pi$
$359$	10.8904	0.574774	0.287387	+	0.957815i	$0.407213\pi$
$360$	5.21187	0.274689
$361$	42.0733	2.21439
$362$	−10.1770	−0.534891
$363$	0	0
$364$	−5.63670	−0.295443
$365$	0.336814	0.0176296
$366$	22.4220	1.17201
$367$	11.3108	0.590416	0.295208	−	0.955433i	$-0.404611\pi$
$367$	11.3108	0.590416	0.295208	+	0.955433i	$0.404611\pi$
$368$	7.98434	0.416213
$369$	−35.9699	−1.87252
$370$	−1.38681	−0.0720966
$371$	−0.582695	−0.0302520
$372$	−14.5727	−0.755559
$373$	−11.6658	−0.604034	−0.302017	−	0.953303i	$-0.597660\pi$
$373$	−11.6658	−0.604034	−0.302017	+	0.953303i	$0.597660\pi$
$374$	0	0
$375$	−2.86564	−0.147981
$376$	−5.14520	−0.265343
$377$	−16.8789	−0.869305
$378$	6.33840	0.326012
$379$	0.617088	0.0316977	0.0158488	−	0.999874i	$-0.494955\pi$
$379$	0.617088	0.0316977	0.0158488	+	0.999874i	$0.494955\pi$
$380$	7.81494	0.400898
$381$	18.8767	0.967084
$382$	11.9914	0.613535
$383$	−26.4471	−1.35138	−0.675692	−	0.737184i	$-0.736155\pi$
$383$	−26.4471	−1.35138	−0.675692	+	0.737184i	$0.736155\pi$
$384$	−2.86564	−0.146236
$385$	0	0
$386$	17.8748	0.909806
$387$	60.1931	3.05979
$388$	−5.33180	−0.270681
$389$	1.60446	0.0813492	0.0406746	−	0.999172i	$-0.487049\pi$
$389$	1.60446	0.0813492	0.0406746	+	0.999172i	$0.487049\pi$
$390$	−16.1527	−0.817924
$391$	18.6024	0.940763
$392$	1.00000	0.0505076
$393$	3.46228	0.174649
$394$	11.7871	0.593825
$395$	−1.05291	−0.0529776
$396$	0	0
$397$	5.28819	0.265407	0.132703	−	0.991156i	$-0.457634\pi$
$397$	5.28819	0.265407	0.132703	+	0.991156i	$0.457634\pi$
$398$	−13.9342	−0.698457
$399$	22.3948	1.12114
$400$	1.00000	0.0500000
$401$	−13.6059	−0.679446	−0.339723	−	0.940526i	$-0.610333\pi$
$401$	−13.6059	−0.679446	−0.339723	+	0.940526i	$0.610333\pi$
$402$	0.139204	0.00694285
$403$	28.6645	1.42788
$404$	18.2701	0.908973
$405$	2.52795	0.125615
$406$	2.99446	0.148613
$407$	0	0
$408$	−6.67652	−0.330537
$409$	3.40230	0.168233	0.0841164	−	0.996456i	$-0.473193\pi$
$409$	3.40230	0.168233	0.0841164	+	0.996456i	$0.473193\pi$
$410$	−6.90154	−0.340843
$411$	−40.3564	−1.99063
$412$	−11.2347	−0.553495
$413$	−13.8292	−0.680490
$414$	41.6133	2.04518
$415$	−10.8315	−0.531697
$416$	5.63670	0.276362
$417$	7.86016	0.384914
$418$	0	0
$419$	−39.5263	−1.93099	−0.965493	−	0.260428i	$-0.916136\pi$
$419$	−39.5263	−1.93099	−0.965493	+	0.260428i	$0.916136\pi$
$420$	2.86564	0.139829
$421$	−15.6617	−0.763304	−0.381652	−	0.924306i	$-0.624645\pi$
$421$	−15.6617	−0.763304	−0.381652	+	0.924306i	$0.624645\pi$
$422$	−26.4497	−1.28755
$423$	−26.8161	−1.30384
$424$	0.582695	0.0282982
$425$	2.32986	0.113015
$426$	−12.7624	−0.618341
$427$	7.82443	0.378651
$428$	−4.89157	−0.236443
$429$	0	0
$430$	11.5492	0.556954
$431$	−5.63150	−0.271260	−0.135630	−	0.990760i	$-0.543306\pi$
$431$	−5.63150	−0.271260	−0.135630	+	0.990760i	$0.543306\pi$
$432$	−6.33840	−0.304956
$433$	−34.5223	−1.65903	−0.829517	−	0.558481i	$-0.811384\pi$
$433$	−34.5223	−1.65903	−0.829517	+	0.558481i	$0.811384\pi$
$434$	−5.08533	−0.244104
$435$	8.58103	0.411429
$436$	−6.26442	−0.300011
$437$	62.3972	2.98486
$438$	−0.965185	−0.0461183
$439$	−30.0533	−1.43437	−0.717184	−	0.696884i	$-0.754569\pi$
$439$	−30.0533	−1.43437	−0.717184	+	0.696884i	$0.754569\pi$
$440$	0	0
$441$	5.21187	0.248184
$442$	13.1327	0.624659
$443$	−32.0651	−1.52346	−0.761730	−	0.647894i	$-0.775650\pi$
$443$	−32.0651	−1.52346	−0.761730	+	0.647894i	$0.775650\pi$
$444$	3.97408	0.188601
$445$	−18.5268	−0.878257
$446$	24.5893	1.16434
$447$	−49.1865	−2.32644
$448$	−1.00000	−0.0472456
$449$	−9.27706	−0.437811	−0.218906	−	0.975746i	$-0.570249\pi$
$449$	−9.27706	−0.437811	−0.218906	+	0.975746i	$0.570249\pi$
$450$	5.21187	0.245690
$451$	0	0
$452$	10.1651	0.478126
$453$	17.0306	0.800166
$454$	5.59242	0.262465
$455$	−5.63670	−0.264252
$456$	−22.3948	−1.04873
$457$	20.0782	0.939217	0.469608	−	0.882875i	$-0.344395\pi$
$457$	20.0782	0.939217	0.469608	+	0.882875i	$0.344395\pi$
$458$	−8.10340	−0.378647
$459$	−14.7676	−0.689291
$460$	7.98434	0.372272
$461$	−14.6490	−0.682272	−0.341136	−	0.940014i	$-0.610812\pi$
$461$	−14.6490	−0.682272	−0.341136	+	0.940014i	$0.610812\pi$
$462$	0	0
$463$	11.0973	0.515737	0.257869	−	0.966180i	$-0.416980\pi$
$463$	11.0973	0.515737	0.257869	+	0.966180i	$0.416980\pi$
$464$	−2.99446	−0.139014
$465$	−14.5727	−0.675793
$466$	−11.9807	−0.554995
$467$	24.6923	1.14262	0.571311	−	0.820734i	$-0.306435\pi$
$467$	24.6923	1.14262	0.571311	+	0.820734i	$0.306435\pi$
$468$	29.3777	1.35798
$469$	0.0485769	0.00224307
$470$	−5.14520	−0.237330
$471$	48.1276	2.21760
$472$	13.8292	0.636540
$473$	0	0
$474$	3.01726	0.138587
$475$	7.81494	0.358574
$476$	−2.32986	−0.106789
$477$	3.03693	0.139051
$478$	6.46612	0.295753
$479$	31.2114	1.42608	0.713042	−	0.701121i	$-0.247317\pi$
$479$	31.2114	1.42608	0.713042	+	0.701121i	$0.247317\pi$
$480$	−2.86564	−0.130798
$481$	−7.81700	−0.356424
$482$	−20.3441	−0.926646
$483$	22.8802	1.04109
$484$	0	0
$485$	−5.33180	−0.242104
$486$	11.7710	0.533945
$487$	3.02777	0.137201	0.0686007	−	0.997644i	$-0.478147\pi$
$487$	3.02777	0.137201	0.0686007	+	0.997644i	$0.478147\pi$
$488$	−7.82443	−0.354195
$489$	−6.96466	−0.314953
$490$	1.00000	0.0451754
$491$	14.8069	0.668228	0.334114	−	0.942533i	$-0.391563\pi$
$491$	14.8069	0.668228	0.334114	+	0.942533i	$0.391563\pi$
$492$	19.7773	0.891630
$493$	−6.97666	−0.314213
$494$	44.0504	1.98192
$495$	0	0
$496$	5.08533	0.228338
$497$	−4.45361	−0.199772
$498$	31.0391	1.39089
$499$	−6.99445	−0.313115	−0.156557	−	0.987669i	$-0.550040\pi$
$499$	−6.99445	−0.313115	−0.156557	+	0.987669i	$0.550040\pi$
$500$	1.00000	0.0447214
$501$	−31.6096	−1.41221
$502$	12.5842	0.561663
$503$	1.93800	0.0864112	0.0432056	−	0.999066i	$-0.486243\pi$
$503$	1.93800	0.0864112	0.0432056	+	0.999066i	$0.486243\pi$
$504$	−5.21187	−0.232155
$505$	18.2701	0.813011
$506$	0	0
$507$	−53.7947	−2.38910
$508$	−6.58727	−0.292263
$509$	12.9518	0.574081	0.287040	−	0.957919i	$-0.407329\pi$
$509$	12.9518	0.574081	0.287040	+	0.957919i	$0.407329\pi$
$510$	−6.67652	−0.295641
$511$	−0.336814	−0.0148998
$512$	1.00000	0.0441942
$513$	−49.5342	−2.18699
$514$	3.45995	0.152612
$515$	−11.2347	−0.495061
$516$	−33.0959	−1.45697
$517$	0	0
$518$	1.38681	0.0609327
$519$	42.1095	1.84840
$520$	5.63670	0.247185
$521$	−5.90459	−0.258685	−0.129342	−	0.991600i	$-0.541287\pi$
$521$	−5.90459	−0.258685	−0.129342	+	0.991600i	$0.541287\pi$
$522$	−15.6067	−0.683088
$523$	18.1944	0.795587	0.397794	−	0.917475i	$-0.369776\pi$
$523$	18.1944	0.795587	0.397794	+	0.917475i	$0.369776\pi$
$524$	−1.20821	−0.0527808
$525$	2.86564	0.125067
$526$	8.01789	0.349597
$527$	11.8481	0.516111
$528$	0	0
$529$	40.7497	1.77173
$530$	0.582695	0.0253106
$531$	72.0759	3.12783
$532$	−7.81494	−0.338821
$533$	−38.9019	−1.68503
$534$	53.0912	2.29748
$535$	−4.89157	−0.211481
$536$	−0.0485769	−0.00209820
$537$	−12.9353	−0.558198
$538$	−18.5650	−0.800393
$539$	0	0
$540$	−6.33840	−0.272761
$541$	15.1525	0.651457	0.325729	−	0.945463i	$-0.394390\pi$
$541$	15.1525	0.651457	0.325729	+	0.945463i	$0.394390\pi$
$542$	−25.2051	−1.08265
$543$	29.1636	1.25153
$544$	2.32986	0.0998918
$545$	−6.26442	−0.268338
$546$	16.1527	0.691272
$547$	10.3369	0.441975	0.220988	−	0.975277i	$-0.429072\pi$
$547$	10.3369	0.441975	0.220988	+	0.975277i	$0.429072\pi$
$548$	14.0829	0.601590
$549$	−40.7799	−1.74044
$550$	0	0
$551$	−23.4015	−0.996938
$552$	−22.8802	−0.973846
$553$	1.05291	0.0447743
$554$	12.2281	0.519521
$555$	3.97408	0.168690
$556$	−2.74290	−0.116325
$557$	6.41056	0.271624	0.135812	−	0.990735i	$-0.456636\pi$
$557$	6.41056	0.271624	0.135812	+	0.990735i	$0.456636\pi$
$558$	26.5041	1.12201
$559$	65.0996	2.75342
$560$	−1.00000	−0.0422577
$561$	0	0
$562$	−26.5934	−1.12178
$563$	36.3501	1.53197	0.765986	−	0.642857i	$-0.222251\pi$
$563$	36.3501	1.53197	0.765986	+	0.642857i	$0.222251\pi$
$564$	14.7443	0.620845
$565$	10.1651	0.427649
$566$	13.7220	0.576779
$567$	−2.52795	−0.106164
$568$	4.45361	0.186869
$569$	45.1901	1.89447	0.947234	−	0.320544i	$-0.103866\pi$
$569$	45.1901	1.89447	0.947234	+	0.320544i	$0.103866\pi$
$570$	−22.3948	−0.938014
$571$	−24.7479	−1.03567	−0.517834	−	0.855481i	$-0.673262\pi$
$571$	−24.7479	−1.03567	−0.517834	+	0.855481i	$0.673262\pi$
$572$	0	0
$573$	−34.3630	−1.43554
$574$	6.90154	0.288065
$575$	7.98434	0.332970
$576$	5.21187	0.217161
$577$	−2.62695	−0.109361	−0.0546807	−	0.998504i	$-0.517414\pi$
$577$	−2.62695	−0.109361	−0.0546807	+	0.998504i	$0.517414\pi$
$578$	−11.5718	−0.481322
$579$	−51.2228	−2.12875
$580$	−2.99446	−0.124338
$581$	10.8315	0.449366
$582$	15.2790	0.633334
$583$	0	0
$584$	0.336814	0.0139374
$585$	29.3777	1.21462
$586$	−0.714463	−0.0295142
$587$	12.8971	0.532319	0.266159	−	0.963929i	$-0.414245\pi$
$587$	12.8971	0.532319	0.266159	+	0.963929i	$0.414245\pi$
$588$	−2.86564	−0.118177
$589$	39.7416	1.63752
$590$	13.8292	0.569339
$591$	−33.7775	−1.38942
$592$	−1.38681	−0.0569973
$593$	33.4157	1.37222	0.686110	−	0.727498i	$-0.259317\pi$
$593$	33.4157	1.37222	0.686110	+	0.727498i	$0.259317\pi$
$594$	0	0
$595$	−2.32986	−0.0955149
$596$	17.1642	0.703075
$597$	39.9303	1.63424
$598$	45.0053	1.84040
$599$	34.4823	1.40891	0.704455	−	0.709749i	$-0.251192\pi$
$599$	34.4823	1.40891	0.704455	+	0.709749i	$0.251192\pi$
$600$	−2.86564	−0.116989
$601$	32.7460	1.33574	0.667869	−	0.744279i	$-0.267207\pi$
$601$	32.7460	1.33574	0.667869	+	0.744279i	$0.267207\pi$
$602$	−11.5492	−0.470712
$603$	−0.253176	−0.0103101
$604$	−5.94304	−0.241819
$605$	0	0
$606$	−52.3556	−2.12680
$607$	5.79559	0.235236	0.117618	−	0.993059i	$-0.462474\pi$
$607$	5.79559	0.235236	0.117618	+	0.993059i	$0.462474\pi$
$608$	7.81494	0.316938
$609$	−8.58103	−0.347721
$610$	−7.82443	−0.316802
$611$	−29.0019	−1.17329
$612$	12.1429	0.490848
$613$	39.3606	1.58976	0.794879	−	0.606767i	$-0.207534\pi$
$613$	39.3606	1.58976	0.794879	+	0.606767i	$0.207534\pi$
$614$	19.3442	0.780667
$615$	19.7773	0.797498
$616$	0	0
$617$	−16.3175	−0.656918	−0.328459	−	0.944518i	$-0.606529\pi$
$617$	−16.3175	−0.656918	−0.328459	+	0.944518i	$0.606529\pi$
$618$	32.1946	1.29506
$619$	−44.7221	−1.79753	−0.898767	−	0.438427i	$-0.855536\pi$
$619$	−44.7221	−1.79753	−0.898767	+	0.438427i	$0.855536\pi$
$620$	5.08533	0.204232
$621$	−50.6079	−2.03083
$622$	1.82729	0.0732675
$623$	18.5268	0.742263
$624$	−16.1527	−0.646626
$625$	1.00000	0.0400000
$626$	−9.83108	−0.392929
$627$	0	0
$628$	−16.7947	−0.670183
$629$	−3.23106	−0.128831
$630$	−5.21187	−0.207646
$631$	−30.1105	−1.19868	−0.599340	−	0.800495i	$-0.704570\pi$
$631$	−30.1105	−1.19868	−0.599340	+	0.800495i	$0.704570\pi$
$632$	−1.05291	−0.0418825
$633$	75.7951	3.01258
$634$	−13.8545	−0.550232
$635$	−6.58727	−0.261408
$636$	−1.66979	−0.0662115
$637$	5.63670	0.223334
$638$	0	0
$639$	23.2116	0.918237
$640$	1.00000	0.0395285
$641$	−0.937781	−0.0370401	−0.0185201	−	0.999828i	$-0.505895\pi$
$641$	−0.937781	−0.0370401	−0.0185201	+	0.999828i	$0.505895\pi$
$642$	14.0175	0.553225
$643$	44.9904	1.77425	0.887125	−	0.461529i	$-0.152699\pi$
$643$	44.9904	1.77425	0.887125	+	0.461529i	$0.152699\pi$
$644$	−7.98434	−0.314627
$645$	−33.0959	−1.30315
$646$	18.2077	0.716373
$647$	−32.9160	−1.29406	−0.647032	−	0.762463i	$-0.723990\pi$
$647$	−32.9160	−1.29406	−0.647032	+	0.762463i	$0.723990\pi$
$648$	2.52795	0.0993072
$649$	0	0
$650$	5.63670	0.221089
$651$	14.5727	0.571149
$652$	2.43041	0.0951821
$653$	−3.32100	−0.129961	−0.0649804	−	0.997887i	$-0.520698\pi$
$653$	−3.32100	−0.129961	−0.0649804	+	0.997887i	$0.520698\pi$
$654$	17.9516	0.701961
$655$	−1.20821	−0.0472086
$656$	−6.90154	−0.269460
$657$	1.75543	0.0684857
$658$	5.14520	0.200581
$659$	20.5642	0.801069	0.400535	−	0.916282i	$-0.368824\pi$
$659$	20.5642	0.801069	0.400535	+	0.916282i	$0.368824\pi$
$660$	0	0
$661$	−43.9387	−1.70902	−0.854508	−	0.519438i	$-0.826141\pi$
$661$	−43.9387	−1.70902	−0.854508	+	0.519438i	$0.826141\pi$
$662$	8.02084	0.311739
$663$	−37.6335	−1.46157
$664$	−10.8315	−0.420343
$665$	−7.81494	−0.303050
$666$	−7.22784	−0.280073
$667$	−23.9088	−0.925752
$668$	11.0306	0.426786
$669$	−70.4640	−2.72430
$670$	−0.0485769	−0.00187669
$671$	0	0
$672$	2.86564	0.110544
$673$	0.623385	0.0240297	0.0120149	−	0.999928i	$-0.496175\pi$
$673$	0.623385	0.0240297	0.0120149	+	0.999928i	$0.496175\pi$
$674$	11.6060	0.447045
$675$	−6.33840	−0.243965
$676$	18.7723	0.722013
$677$	30.6611	1.17840	0.589200	−	0.807987i	$-0.299443\pi$
$677$	30.6611	1.17840	0.589200	+	0.807987i	$0.299443\pi$
$678$	−29.1294	−1.11871
$679$	5.33180	0.204616
$680$	2.32986	0.0893460
$681$	−16.0258	−0.614112
$682$	0	0
$683$	−4.08570	−0.156335	−0.0781676	−	0.996940i	$-0.524907\pi$
$683$	−4.08570	−0.156335	−0.0781676	+	0.996940i	$0.524907\pi$
$684$	40.7304	1.55737
$685$	14.0829	0.538079
$686$	−1.00000	−0.0381802
$687$	23.2214	0.885951
$688$	11.5492	0.440311
$689$	3.28447	0.125128
$690$	−22.8802	−0.871035
$691$	−9.75385	−0.371054	−0.185527	−	0.982639i	$-0.559399\pi$
$691$	−9.75385	−0.371054	−0.185527	+	0.982639i	$0.559399\pi$
$692$	−14.6947	−0.558607
$693$	0	0
$694$	26.4301	1.00327
$695$	−2.74290	−0.104044
$696$	8.58103	0.325263
$697$	−16.0796	−0.609059
$698$	−27.3777	−1.03626
$699$	34.3323	1.29857
$700$	−1.00000	−0.0377964
$701$	8.34928	0.315348	0.157674	−	0.987491i	$-0.449600\pi$
$701$	8.34928	0.315348	0.157674	+	0.987491i	$0.449600\pi$
$702$	−35.7276	−1.34845
$703$	−10.8378	−0.408755
$704$	0	0
$705$	14.7443	0.555301
$706$	21.3772	0.804541
$707$	−18.2701	−0.687119
$708$	−39.6294	−1.48937
$709$	41.6502	1.56421	0.782103	−	0.623149i	$-0.214147\pi$
$709$	41.6502	1.56421	0.782103	+	0.623149i	$0.214147\pi$
$710$	4.45361	0.167141
$711$	−5.48763	−0.205802
$712$	−18.5268	−0.694323
$713$	40.6030	1.52059
$714$	6.67652	0.249863
$715$	0	0
$716$	4.51392	0.168693
$717$	−18.5295	−0.691998
$718$	10.8904	0.406427
$719$	27.1930	1.01413	0.507064	−	0.861909i	$-0.330731\pi$
$719$	27.1930	1.01413	0.507064	+	0.861909i	$0.330731\pi$
$720$	5.21187	0.194235
$721$	11.2347	0.418403
$722$	42.0733	1.56581
$723$	58.2986	2.16815
$724$	−10.1770	−0.378225
$725$	−2.99446	−0.111211
$726$	0	0
$727$	26.4443	0.980766	0.490383	−	0.871507i	$-0.336857\pi$
$727$	26.4443	0.980766	0.490383	+	0.871507i	$0.336857\pi$
$728$	−5.63670	−0.208910
$729$	−41.3153	−1.53020
$730$	0.336814	0.0124660
$731$	26.9081	0.995232
$732$	22.4220	0.828740
$733$	−25.1965	−0.930653	−0.465326	−	0.885139i	$-0.654063\pi$
$733$	−25.1965	−0.930653	−0.465326	+	0.885139i	$0.654063\pi$
$734$	11.3108	0.417487
$735$	−2.86564	−0.105701
$736$	7.98434	0.294307
$737$	0	0
$738$	−35.9699	−1.32407
$739$	−5.90682	−0.217286	−0.108643	−	0.994081i	$-0.534651\pi$
$739$	−5.90682	−0.217286	−0.108643	+	0.994081i	$0.534651\pi$
$740$	−1.38681	−0.0509800
$741$	−126.233	−4.63727
$742$	−0.582695	−0.0213914
$743$	−19.5944	−0.718847	−0.359424	−	0.933175i	$-0.617027\pi$
$743$	−19.5944	−0.718847	−0.359424	+	0.933175i	$0.617027\pi$
$744$	−14.5727	−0.534261
$745$	17.1642	0.628849
$746$	−11.6658	−0.427116
$747$	−56.4523	−2.06548
$748$	0	0
$749$	4.89157	0.178734
$750$	−2.86564	−0.104638
$751$	0.465285	0.0169785	0.00848925	−	0.999964i	$-0.497298\pi$
$751$	0.465285	0.0169785	0.00848925	+	0.999964i	$0.497298\pi$
$752$	−5.14520	−0.187626
$753$	−36.0619	−1.31417
$754$	−16.8789	−0.614692
$755$	−5.94304	−0.216289
$756$	6.33840	0.230525
$757$	−6.85681	−0.249215	−0.124607	−	0.992206i	$-0.539767\pi$
$757$	−6.85681	−0.249215	−0.124607	+	0.992206i	$0.539767\pi$
$758$	0.617088	0.0224136
$759$	0	0
$760$	7.81494	0.283478
$761$	−41.5119	−1.50481	−0.752403	−	0.658703i	$-0.771105\pi$
$761$	−41.5119	−1.50481	−0.752403	+	0.658703i	$0.771105\pi$
$762$	18.8767	0.683831
$763$	6.26442	0.226787
$764$	11.9914	0.433834
$765$	12.1429	0.439028
$766$	−26.4471	−0.955573
$767$	77.9510	2.81465
$768$	−2.86564	−0.103405
$769$	17.8632	0.644164	0.322082	−	0.946712i	$-0.395617\pi$
$769$	17.8632	0.644164	0.322082	+	0.946712i	$0.395617\pi$
$770$	0	0
$771$	−9.91495	−0.357078
$772$	17.8748	0.643330
$773$	9.91463	0.356604	0.178302	−	0.983976i	$-0.442940\pi$
$773$	9.91463	0.356604	0.178302	+	0.983976i	$0.442940\pi$
$774$	60.1931	2.16360
$775$	5.08533	0.182670
$776$	−5.33180	−0.191400
$777$	−3.97408	−0.142569
$778$	1.60446	0.0575226
$779$	−53.9352	−1.93243
$780$	−16.1527	−0.578360
$781$	0	0
$782$	18.6024	0.665220
$783$	18.9801	0.678293
$784$	1.00000	0.0357143
$785$	−16.7947	−0.599430
$786$	3.46228	0.123496
$787$	0.812618	0.0289667	0.0144834	−	0.999895i	$-0.495390\pi$
$787$	0.812618	0.0289667	0.0144834	+	0.999895i	$0.495390\pi$
$788$	11.7871	0.419897
$789$	−22.9763	−0.817979
$790$	−1.05291	−0.0374609
$791$	−10.1651	−0.361429
$792$	0	0
$793$	−44.1039	−1.56618
$794$	5.28819	0.187671
$795$	−1.66979	−0.0592214
$796$	−13.9342	−0.493884
$797$	32.3331	1.14530	0.572649	−	0.819800i	$-0.305916\pi$
$797$	32.3331	1.14530	0.572649	+	0.819800i	$0.305916\pi$
$798$	22.3948	0.792766
$799$	−11.9876	−0.424090
$800$	1.00000	0.0353553
$801$	−96.5595	−3.41176
$802$	−13.6059	−0.480441
$803$	0	0
$804$	0.139204	0.00490934
$805$	−7.98434	−0.281411
$806$	28.6645	1.00966
$807$	53.2005	1.87275
$808$	18.2701	0.642741
$809$	24.5501	0.863135	0.431568	−	0.902081i	$-0.357961\pi$
$809$	24.5501	0.863135	0.431568	+	0.902081i	$0.357961\pi$
$810$	2.52795	0.0888230
$811$	−41.3394	−1.45162	−0.725812	−	0.687893i	$-0.758536\pi$
$811$	−41.3394	−1.45162	−0.725812	+	0.687893i	$0.758536\pi$
$812$	2.99446	0.105085
$813$	72.2287	2.53317
$814$	0	0
$815$	2.43041	0.0851335
$816$	−6.67652	−0.233725
$817$	90.2567	3.15768
$818$	3.40230	0.118959
$819$	−29.3777	−1.02654
$820$	−6.90154	−0.241012
$821$	−16.0197	−0.559090	−0.279545	−	0.960133i	$-0.590184\pi$
$821$	−16.0197	−0.559090	−0.279545	+	0.960133i	$0.590184\pi$
$822$	−40.3564	−1.40759
$823$	19.3583	0.674787	0.337394	−	0.941364i	$-0.390455\pi$
$823$	19.3583	0.674787	0.337394	+	0.941364i	$0.390455\pi$
$824$	−11.2347	−0.391380
$825$	0	0
$826$	−13.8292	−0.481179
$827$	−43.1757	−1.50136	−0.750682	−	0.660663i	$-0.770275\pi$
$827$	−43.1757	−1.50136	−0.750682	+	0.660663i	$0.770275\pi$
$828$	41.6133	1.44616
$829$	−38.5839	−1.34007	−0.670037	−	0.742327i	$-0.733722\pi$
$829$	−38.5839	−1.34007	−0.670037	+	0.742327i	$0.733722\pi$
$830$	−10.8315	−0.375966
$831$	−35.0412	−1.21557
$832$	5.63670	0.195417
$833$	2.32986	0.0807248
$834$	7.86016	0.272175
$835$	11.0306	0.381729
$836$	0	0
$837$	−32.2329	−1.11413
$838$	−39.5263	−1.36541
$839$	9.31984	0.321757	0.160878	−	0.986974i	$-0.448567\pi$
$839$	9.31984	0.321757	0.160878	+	0.986974i	$0.448567\pi$
$840$	2.86564	0.0988738
$841$	−20.0332	−0.690800
$842$	−15.6617	−0.539738
$843$	76.2071	2.62471
$844$	−26.4497	−0.910435
$845$	18.7723	0.645788
$846$	−26.8161	−0.921956
$847$	0	0
$848$	0.582695	0.0200098
$849$	−39.3223	−1.34954
$850$	2.32986	0.0799135
$851$	−11.0727	−0.379568
$852$	−12.7624	−0.437233
$853$	−9.65688	−0.330645	−0.165323	−	0.986240i	$-0.552867\pi$
$853$	−9.65688	−0.330645	−0.165323	+	0.986240i	$0.552867\pi$
$854$	7.82443	0.267747
$855$	40.7304	1.39295
$856$	−4.89157	−0.167190
$857$	−7.88344	−0.269293	−0.134647	−	0.990894i	$-0.542990\pi$
$857$	−7.88344	−0.269293	−0.134647	+	0.990894i	$0.542990\pi$
$858$	0	0
$859$	9.00176	0.307136	0.153568	−	0.988138i	$-0.450924\pi$
$859$	9.00176	0.307136	0.153568	+	0.988138i	$0.450924\pi$
$860$	11.5492	0.393826
$861$	−19.7773	−0.674009
$862$	−5.63150	−0.191810
$863$	−14.5890	−0.496616	−0.248308	−	0.968681i	$-0.579874\pi$
$863$	−14.5890	−0.496616	−0.248308	+	0.968681i	$0.579874\pi$
$864$	−6.33840	−0.215637
$865$	−14.6947	−0.499633
$866$	−34.5223	−1.17311
$867$	33.1604	1.12619
$868$	−5.08533	−0.172607
$869$	0	0
$870$	8.58103	0.290924
$871$	−0.273813	−0.00927781
$872$	−6.26442	−0.212140
$873$	−27.7886	−0.940502
$874$	62.3972	2.11062
$875$	−1.00000	−0.0338062
$876$	−0.965185	−0.0326106
$877$	−17.0812	−0.576791	−0.288396	−	0.957511i	$-0.593122\pi$
$877$	−17.0812	−0.576791	−0.288396	+	0.957511i	$0.593122\pi$
$878$	−30.0533	−1.01425
$879$	2.04739	0.0690568
$880$	0	0
$881$	−30.9541	−1.04287	−0.521435	−	0.853291i	$-0.674603\pi$
$881$	−30.9541	−1.04287	−0.521435	+	0.853291i	$0.674603\pi$
$882$	5.21187	0.175493
$883$	−13.5966	−0.457561	−0.228781	−	0.973478i	$-0.573474\pi$
$883$	−13.5966	−0.457561	−0.228781	+	0.973478i	$0.573474\pi$
$884$	13.1327	0.441700
$885$	−39.6294	−1.33213
$886$	−32.0651	−1.07725
$887$	−17.1901	−0.577188	−0.288594	−	0.957452i	$-0.593188\pi$
$887$	−17.1901	−0.577188	−0.288594	+	0.957452i	$0.593188\pi$
$888$	3.97408	0.133361
$889$	6.58727	0.220930
$890$	−18.5268	−0.621021
$891$	0	0
$892$	24.5893	0.823311
$893$	−40.2094	−1.34556
$894$	−49.1865	−1.64504
$895$	4.51392	0.150884
$896$	−1.00000	−0.0334077
$897$	−128.969	−4.30614
$898$	−9.27706	−0.309579
$899$	−15.2278	−0.507876
$900$	5.21187	0.173729
$901$	1.35760	0.0452281
$902$	0	0
$903$	33.0959	1.10136
$904$	10.1651	0.338086
$905$	−10.1770	−0.338295
$906$	17.0306	0.565803
$907$	−23.5921	−0.783363	−0.391682	−	0.920101i	$-0.628107\pi$
$907$	−23.5921	−0.783363	−0.391682	+	0.920101i	$0.628107\pi$
$908$	5.59242	0.185591
$909$	95.2215	3.15830
$910$	−5.63670	−0.186855
$911$	50.0697	1.65888	0.829441	−	0.558594i	$-0.188659\pi$
$911$	50.0697	1.65888	0.829441	+	0.558594i	$0.188659\pi$
$912$	−22.3948	−0.741565
$913$	0	0
$914$	20.0782	0.664127
$915$	22.4220	0.741247
$916$	−8.10340	−0.267744
$917$	1.20821	0.0398985
$918$	−14.7676	−0.487403
$919$	13.1628	0.434200	0.217100	−	0.976149i	$-0.430340\pi$
$919$	13.1628	0.434200	0.217100	+	0.976149i	$0.430340\pi$
$920$	7.98434	0.263236
$921$	−55.4333	−1.82659
$922$	−14.6490	−0.482439
$923$	25.1036	0.826296
$924$	0	0
$925$	−1.38681	−0.0455979
$926$	11.0973	0.364681
$927$	−58.5539	−1.92316
$928$	−2.99446	−0.0982979
$929$	−9.87262	−0.323910	−0.161955	−	0.986798i	$-0.551780\pi$
$929$	−9.87262	−0.323910	−0.161955	+	0.986798i	$0.551780\pi$
$930$	−14.5727	−0.477858
$931$	7.81494	0.256124
$932$	−11.9807	−0.392441
$933$	−5.23634	−0.171430
$934$	24.6923	0.807956
$935$	0	0
$936$	29.3777	0.960240
$937$	−4.81983	−0.157457	−0.0787285	−	0.996896i	$-0.525086\pi$
$937$	−4.81983	−0.157457	−0.0787285	+	0.996896i	$0.525086\pi$
$938$	0.0485769	0.00158609
$939$	28.1723	0.919368
$940$	−5.14520	−0.167818
$941$	57.3904	1.87087	0.935437	−	0.353493i	$-0.115006\pi$
$941$	57.3904	1.87087	0.935437	+	0.353493i	$0.115006\pi$
$942$	48.1276	1.56808
$943$	−55.1043	−1.79444
$944$	13.8292	0.450102
$945$	6.33840	0.206188
$946$	0	0
$947$	−4.22221	−0.137203	−0.0686017	−	0.997644i	$-0.521854\pi$
$947$	−4.22221	−0.137203	−0.0686017	+	0.997644i	$0.521854\pi$
$948$	3.01726	0.0979959
$949$	1.89852	0.0616284
$950$	7.81494	0.253550
$951$	39.7019	1.28742
$952$	−2.32986	−0.0755111
$953$	5.10500	0.165367	0.0826836	−	0.996576i	$-0.473651\pi$
$953$	5.10500	0.165367	0.0826836	+	0.996576i	$0.473651\pi$
$954$	3.03693	0.0983241
$955$	11.9914	0.388033
$956$	6.46612	0.209129
$957$	0	0
$958$	31.2114	1.00839
$959$	−14.0829	−0.454759
$960$	−2.86564	−0.0924880
$961$	−5.13941	−0.165787
$962$	−7.81700	−0.252030
$963$	−25.4942	−0.821539
$964$	−20.3441	−0.655238
$965$	17.8748	0.575412
$966$	22.8802	0.736159
$967$	−61.6521	−1.98260	−0.991298	−	0.131633i	$-0.957978\pi$
$967$	−61.6521	−1.98260	−0.991298	+	0.131633i	$0.957978\pi$
$968$	0	0
$969$	−52.1766	−1.67616
$970$	−5.33180	−0.171194
$971$	−26.0709	−0.836656	−0.418328	−	0.908296i	$-0.637384\pi$
$971$	−26.0709	−0.836656	−0.418328	+	0.908296i	$0.637384\pi$
$972$	11.7710	0.377556
$973$	2.74290	0.0879334
$974$	3.02777	0.0970160
$975$	−16.1527	−0.517301
$976$	−7.82443	−0.250454
$977$	19.0437	0.609261	0.304631	−	0.952471i	$-0.401467\pi$
$977$	19.0437	0.609261	0.304631	+	0.952471i	$0.401467\pi$
$978$	−6.96466	−0.222705
$979$	0	0
$980$	1.00000	0.0319438
$981$	−32.6493	−1.04241
$982$	14.8069	0.472509
$983$	−19.3347	−0.616681	−0.308340	−	0.951276i	$-0.599774\pi$
$983$	−19.3347	−0.616681	−0.308340	+	0.951276i	$0.599774\pi$
$984$	19.7773	0.630478
$985$	11.7871	0.375568
$986$	−6.97666	−0.222182
$987$	−14.7443	−0.469315
$988$	44.0504	1.40143
$989$	92.2131	2.93221
$990$	0	0
$991$	−28.4567	−0.903958	−0.451979	−	0.892029i	$-0.649282\pi$
$991$	−28.4567	−0.903958	−0.451979	+	0.892029i	$0.649282\pi$
$992$	5.08533	0.161459
$993$	−22.9848	−0.729401
$994$	−4.45361	−0.141260
$995$	−13.9342	−0.441743
$996$	31.0391	0.983511
$997$	−14.1144	−0.447008	−0.223504	−	0.974703i	$-0.571750\pi$
$997$	−14.1144	−0.447008	−0.223504	+	0.974703i	$0.571750\pi$
$998$	−6.99445	−0.221406
$999$	8.79013	0.278107

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8470.2.a.dc.1.1		6
11.7	odd	10		770.2.n.j.71.3	✓	12
11.8	odd	10		770.2.n.j.141.3	yes	12
11.10	odd	2		8470.2.a.cw.1.1		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
770.2.n.j.71.3	✓	12	11.7	odd	10
770.2.n.j.141.3	yes	12	11.8	odd	10
8470.2.a.cw.1.1		6	11.10	odd	2
8470.2.a.dc.1.1		6	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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -2.86564 q^{3} +1.00000 q^{4} +1.00000 q^{5} -2.86564 q^{6} -1.00000 q^{7} +1.00000 q^{8} +5.21187 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} -2.86564 q^{3} +1.00000 q^{4} +1.00000 q^{5} -2.86564 q^{6} -1.00000 q^{7} +1.00000 q^{8} +5.21187 q^{9} +1.00000 q^{10} -2.86564 q^{12} +5.63670 q^{13} -1.00000 q^{14} -2.86564 q^{15} +1.00000 q^{16} +2.32986 q^{17} +5.21187 q^{18} +7.81494 q^{19} +1.00000 q^{20} +2.86564 q^{21} +7.98434 q^{23} -2.86564 q^{24} +1.00000 q^{25} +5.63670 q^{26} -6.33840 q^{27} -1.00000 q^{28} -2.99446 q^{29} -2.86564 q^{30} +5.08533 q^{31} +1.00000 q^{32} +2.32986 q^{34} -1.00000 q^{35} +5.21187 q^{36} -1.38681 q^{37} +7.81494 q^{38} -16.1527 q^{39} +1.00000 q^{40} -6.90154 q^{41} +2.86564 q^{42} +11.5492 q^{43} +5.21187 q^{45} +7.98434 q^{46} -5.14520 q^{47} -2.86564 q^{48} +1.00000 q^{49} +1.00000 q^{50} -6.67652 q^{51} +5.63670 q^{52} +0.582695 q^{53} -6.33840 q^{54} -1.00000 q^{56} -22.3948 q^{57} -2.99446 q^{58} +13.8292 q^{59} -2.86564 q^{60} -7.82443 q^{61} +5.08533 q^{62} -5.21187 q^{63} +1.00000 q^{64} +5.63670 q^{65} -0.0485769 q^{67} +2.32986 q^{68} -22.8802 q^{69} -1.00000 q^{70} +4.45361 q^{71} +5.21187 q^{72} +0.336814 q^{73} -1.38681 q^{74} -2.86564 q^{75} +7.81494 q^{76} -16.1527 q^{78} -1.05291 q^{79} +1.00000 q^{80} +2.52795 q^{81} -6.90154 q^{82} -10.8315 q^{83} +2.86564 q^{84} +2.32986 q^{85} +11.5492 q^{86} +8.58103 q^{87} -18.5268 q^{89} +5.21187 q^{90} -5.63670 q^{91} +7.98434 q^{92} -14.5727 q^{93} -5.14520 q^{94} +7.81494 q^{95} -2.86564 q^{96} -5.33180 q^{97} +1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 6 q^{2} - q^{3} + 6 q^{4} + 6 q^{5} - q^{6} - 6 q^{7} + 6 q^{8} + 3 q^{9}+O(q^{10}) \) 6 * q + 6 * q^2 - q^3 + 6 * q^4 + 6 * q^5 - q^6 - 6 * q^7 + 6 * q^8 + 3 * q^9	\( 6 q + 6 q^{2} - q^{3} + 6 q^{4} + 6 q^{5} - q^{6} - 6 q^{7} + 6 q^{8} + 3 q^{9} + 6 q^{10} - q^{12} + 9 q^{13} - 6 q^{14} - q^{15} + 6 q^{16} + 9 q^{17} + 3 q^{18} + 12 q^{19} + 6 q^{20} + q^{21} + 4 q^{23} - q^{24} + 6 q^{25} + 9 q^{26} - 4 q^{27} - 6 q^{28} + 15 q^{29} - q^{30} + 8 q^{31} + 6 q^{32} + 9 q^{34} - 6 q^{35} + 3 q^{36} - 4 q^{37} + 12 q^{38} - 19 q^{39} + 6 q^{40} + 4 q^{41} + q^{42} + 30 q^{43} + 3 q^{45} + 4 q^{46} - 7 q^{47} - q^{48} + 6 q^{49} + 6 q^{50} + 16 q^{51} + 9 q^{52} - 6 q^{53} - 4 q^{54} - 6 q^{56} - 14 q^{57} + 15 q^{58} + 4 q^{59} - q^{60} - 14 q^{61} + 8 q^{62} - 3 q^{63} + 6 q^{64} + 9 q^{65} + 18 q^{67} + 9 q^{68} - 10 q^{69} - 6 q^{70} + 23 q^{71} + 3 q^{72} + 23 q^{73} - 4 q^{74} - q^{75} + 12 q^{76} - 19 q^{78} + 21 q^{79} + 6 q^{80} - 18 q^{81} + 4 q^{82} + 25 q^{83} + q^{84} + 9 q^{85} + 30 q^{86} + 14 q^{87} - 18 q^{89} + 3 q^{90} - 9 q^{91} + 4 q^{92} - 24 q^{93} - 7 q^{94} + 12 q^{95} - q^{96} + 7 q^{97} + 6 q^{98}+O(q^{100}) \) 6 * q + 6 * q^2 - q^3 + 6 * q^4 + 6 * q^5 - q^6 - 6 * q^7 + 6 * q^8 + 3 * q^9 + 6 * q^10 - q^12 + 9 * q^13 - 6 * q^14 - q^15 + 6 * q^16 + 9 * q^17 + 3 * q^18 + 12 * q^19 + 6 * q^20 + q^21 + 4 * q^23 - q^24 + 6 * q^25 + 9 * q^26 - 4 * q^27 - 6 * q^28 + 15 * q^29 - q^30 + 8 * q^31 + 6 * q^32 + 9 * q^34 - 6 * q^35 + 3 * q^36 - 4 * q^37 + 12 * q^38 - 19 * q^39 + 6 * q^40 + 4 * q^41 + q^42 + 30 * q^43 + 3 * q^45 + 4 * q^46 - 7 * q^47 - q^48 + 6 * q^49 + 6 * q^50 + 16 * q^51 + 9 * q^52 - 6 * q^53 - 4 * q^54 - 6 * q^56 - 14 * q^57 + 15 * q^58 + 4 * q^59 - q^60 - 14 * q^61 + 8 * q^62 - 3 * q^63 + 6 * q^64 + 9 * q^65 + 18 * q^67 + 9 * q^68 - 10 * q^69 - 6 * q^70 + 23 * q^71 + 3 * q^72 + 23 * q^73 - 4 * q^74 - q^75 + 12 * q^76 - 19 * q^78 + 21 * q^79 + 6 * q^80 - 18 * q^81 + 4 * q^82 + 25 * q^83 + q^84 + 9 * q^85 + 30 * q^86 + 14 * q^87 - 18 * q^89 + 3 * q^90 - 9 * q^91 + 4 * q^92 - 24 * q^93 - 7 * q^94 + 12 * q^95 - q^96 + 7 * q^97 + 6 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more