LMFDB - Embedded newform 8670.2.a.cj.1.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8670,2,Mod(1,8670)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8670, 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("8670.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8670 = 2 \cdot 3 \cdot 5 \cdot 17^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8670.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:	$69.2302985525$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.8.20417871872.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 12x^{6} - 8x^{5} + 38x^{4} + 40x^{3} - 20x^{2} - 24x + 2 $ x^8 - 12x^6 - 8x^5 + 38x^4 + 40x^3 - 20x^2 - 24x + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 510)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$-1.55788$ of defining polynomial
Character	$\chi$	$=$	8670.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{6} +1.06098 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{6} +1.06098 q^{7} -1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{10} -3.48514 q^{11} -1.00000 q^{12} -3.47117 q^{13} -1.06098 q^{14} +1.00000 q^{15} +1.00000 q^{16} -1.00000 q^{18} -7.34218 q^{19} -1.00000 q^{20} -1.06098 q^{21} +3.48514 q^{22} -8.67926 q^{23} +1.00000 q^{24} +1.00000 q^{25} +3.47117 q^{26} -1.00000 q^{27} +1.06098 q^{28} -2.62382 q^{29} -1.00000 q^{30} -10.5015 q^{31} -1.00000 q^{32} +3.48514 q^{33} -1.06098 q^{35} +1.00000 q^{36} -5.53073 q^{37} +7.34218 q^{38} +3.47117 q^{39} +1.00000 q^{40} -2.82724 q^{41} +1.06098 q^{42} -7.02634 q^{43} -3.48514 q^{44} -1.00000 q^{45} +8.67926 q^{46} +4.88874 q^{47} -1.00000 q^{48} -5.87433 q^{49} -1.00000 q^{50} -3.47117 q^{52} +9.86766 q^{53} +1.00000 q^{54} +3.48514 q^{55} -1.06098 q^{56} +7.34218 q^{57} +2.62382 q^{58} -6.17161 q^{59} +1.00000 q^{60} +9.82606 q^{61} +10.5015 q^{62} +1.06098 q^{63} +1.00000 q^{64} +3.47117 q^{65} -3.48514 q^{66} -12.6016 q^{67} +8.67926 q^{69} +1.06098 q^{70} +12.3500 q^{71} -1.00000 q^{72} -9.60208 q^{73} +5.53073 q^{74} -1.00000 q^{75} -7.34218 q^{76} -3.69766 q^{77} -3.47117 q^{78} -4.35423 q^{79} -1.00000 q^{80} +1.00000 q^{81} +2.82724 q^{82} -9.38266 q^{83} -1.06098 q^{84} +7.02634 q^{86} +2.62382 q^{87} +3.48514 q^{88} +5.59982 q^{89} +1.00000 q^{90} -3.68283 q^{91} -8.67926 q^{92} +10.5015 q^{93} -4.88874 q^{94} +7.34218 q^{95} +1.00000 q^{96} -2.93760 q^{97} +5.87433 q^{98} -3.48514 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 8 q^{2} - 8 q^{3} + 8 q^{4} - 8 q^{5} + 8 q^{6} - 8 q^{7} - 8 q^{8} + 8 q^{9}+O(q^{10}) $ 8 * q - 8 * q^2 - 8 * q^3 + 8 * q^4 - 8 * q^5 + 8 * q^6 - 8 * q^7 - 8 * q^8 + 8 * q^9	$ 8 q - 8 q^{2} - 8 q^{3} + 8 q^{4} - 8 q^{5} + 8 q^{6} - 8 q^{7} - 8 q^{8} + 8 q^{9} + 8 q^{10} + 8 q^{11} - 8 q^{12} + 8 q^{14} + 8 q^{15} + 8 q^{16} - 8 q^{18} - 8 q^{20} + 8 q^{21} - 8 q^{22} + 8 q^{24} + 8 q^{25} - 8 q^{27} - 8 q^{28} + 8 q^{29} - 8 q^{30} - 8 q^{31} - 8 q^{32} - 8 q^{33} + 8 q^{35} + 8 q^{36} - 32 q^{37} + 8 q^{40} + 8 q^{41} - 8 q^{42} - 8 q^{43} + 8 q^{44} - 8 q^{45} - 8 q^{48} + 40 q^{49} - 8 q^{50} + 16 q^{53} + 8 q^{54} - 8 q^{55} + 8 q^{56} - 8 q^{58} + 32 q^{59} + 8 q^{60} - 8 q^{61} + 8 q^{62} - 8 q^{63} + 8 q^{64} + 8 q^{66} - 8 q^{67} - 8 q^{70} + 24 q^{71} - 8 q^{72} - 40 q^{73} + 32 q^{74} - 8 q^{75} + 16 q^{77} - 24 q^{79} - 8 q^{80} + 8 q^{81} - 8 q^{82} - 16 q^{83} + 8 q^{84} + 8 q^{86} - 8 q^{87} - 8 q^{88} + 48 q^{89} + 8 q^{90} - 24 q^{91} + 8 q^{93} + 8 q^{96} - 24 q^{97} - 40 q^{98} + 8 q^{99}+O(q^{100}) $ 8 * q - 8 * q^2 - 8 * q^3 + 8 * q^4 - 8 * q^5 + 8 * q^6 - 8 * q^7 - 8 * q^8 + 8 * q^9 + 8 * q^10 + 8 * q^11 - 8 * q^12 + 8 * q^14 + 8 * q^15 + 8 * q^16 - 8 * q^18 - 8 * q^20 + 8 * q^21 - 8 * q^22 + 8 * q^24 + 8 * q^25 - 8 * q^27 - 8 * q^28 + 8 * q^29 - 8 * q^30 - 8 * q^31 - 8 * q^32 - 8 * q^33 + 8 * q^35 + 8 * q^36 - 32 * q^37 + 8 * q^40 + 8 * q^41 - 8 * q^42 - 8 * q^43 + 8 * q^44 - 8 * q^45 - 8 * q^48 + 40 * q^49 - 8 * q^50 + 16 * q^53 + 8 * q^54 - 8 * q^55 + 8 * q^56 - 8 * q^58 + 32 * q^59 + 8 * q^60 - 8 * q^61 + 8 * q^62 - 8 * q^63 + 8 * q^64 + 8 * q^66 - 8 * q^67 - 8 * q^70 + 24 * q^71 - 8 * q^72 - 40 * q^73 + 32 * q^74 - 8 * q^75 + 16 * q^77 - 24 * q^79 - 8 * q^80 + 8 * q^81 - 8 * q^82 - 16 * q^83 + 8 * q^84 + 8 * q^86 - 8 * q^87 - 8 * q^88 + 48 * q^89 + 8 * q^90 - 24 * q^91 + 8 * q^93 + 8 * q^96 - 24 * q^97 - 40 * q^98 + 8 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	1.00000	0.500000
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	1.00000	0.408248
$7$	1.06098	0.401012	0.200506	−	0.979692i	$-0.435741\pi$
$7$	1.06098	0.401012	0.200506	+	0.979692i	$0.435741\pi$
$8$	−1.00000	−0.353553
$9$	1.00000	0.333333
$10$	1.00000	0.316228
$11$	−3.48514	−1.05081	−0.525405	−	0.850852i	$-0.676086\pi$
$11$	−3.48514	−1.05081	−0.525405	+	0.850852i	$0.676086\pi$
$12$	−1.00000	−0.288675
$13$	−3.47117	−0.962728	−0.481364	−	0.876521i	$-0.659858\pi$
$13$	−3.47117	−0.962728	−0.481364	+	0.876521i	$0.659858\pi$
$14$	−1.06098	−0.283558
$15$	1.00000	0.258199
$16$	1.00000	0.250000
$17$	0	0
$17$	0	0
$18$	−1.00000	−0.235702
$19$	−7.34218	−1.68441	−0.842206	−	0.539156i	$-0.818743\pi$
$19$	−7.34218	−1.68441	−0.842206	+	0.539156i	$0.818743\pi$
$20$	−1.00000	−0.223607
$21$	−1.06098	−0.231524
$22$	3.48514	0.743035
$23$	−8.67926	−1.80975	−0.904876	−	0.425676i	$-0.860036\pi$
$23$	−8.67926	−1.80975	−0.904876	+	0.425676i	$0.860036\pi$
$24$	1.00000	0.204124
$25$	1.00000	0.200000
$26$	3.47117	0.680752
$27$	−1.00000	−0.192450
$28$	1.06098	0.200506
$29$	−2.62382	−0.487232	−0.243616	−	0.969872i	$-0.578334\pi$
$29$	−2.62382	−0.487232	−0.243616	+	0.969872i	$0.578334\pi$
$30$	−1.00000	−0.182574
$31$	−10.5015	−1.88613	−0.943065	−	0.332608i	$-0.892071\pi$
$31$	−10.5015	−1.88613	−0.943065	+	0.332608i	$0.892071\pi$
$32$	−1.00000	−0.176777
$33$	3.48514	0.606685
$34$	0	0
$35$	−1.06098	−0.179338
$36$	1.00000	0.166667
$37$	−5.53073	−0.909247	−0.454624	−	0.890684i	$-0.650226\pi$
$37$	−5.53073	−0.909247	−0.454624	+	0.890684i	$0.650226\pi$
$38$	7.34218	1.19106
$39$	3.47117	0.555831
$40$	1.00000	0.158114
$41$	−2.82724	−0.441541	−0.220771	−	0.975326i	$-0.570857\pi$
$41$	−2.82724	−0.441541	−0.220771	+	0.975326i	$0.570857\pi$
$42$	1.06098	0.163712
$43$	−7.02634	−1.07151	−0.535753	−	0.844374i	$-0.679972\pi$
$43$	−7.02634	−1.07151	−0.535753	+	0.844374i	$0.679972\pi$
$44$	−3.48514	−0.525405
$45$	−1.00000	−0.149071
$46$	8.67926	1.27969
$47$	4.88874	0.713096	0.356548	−	0.934277i	$-0.383954\pi$
$47$	4.88874	0.713096	0.356548	+	0.934277i	$0.383954\pi$
$48$	−1.00000	−0.144338
$49$	−5.87433	−0.839190
$50$	−1.00000	−0.141421
$51$	0	0
$52$	−3.47117	−0.481364
$53$	9.86766	1.35543	0.677714	−	0.735326i	$-0.262971\pi$
$53$	9.86766	1.35543	0.677714	+	0.735326i	$0.262971\pi$
$54$	1.00000	0.136083
$55$	3.48514	0.469936
$56$	−1.06098	−0.141779
$57$	7.34218	0.972496
$58$	2.62382	0.344525
$59$	−6.17161	−0.803475	−0.401738	−	0.915755i	$-0.631594\pi$
$59$	−6.17161	−0.803475	−0.401738	+	0.915755i	$0.631594\pi$
$60$	1.00000	0.129099
$61$	9.82606	1.25810	0.629049	−	0.777365i	$-0.283444\pi$
$61$	9.82606	1.25810	0.629049	+	0.777365i	$0.283444\pi$
$62$	10.5015	1.33370
$63$	1.06098	0.133671
$64$	1.00000	0.125000
$65$	3.47117	0.430545
$66$	−3.48514	−0.428991
$67$	−12.6016	−1.53953	−0.769763	−	0.638330i	$-0.779625\pi$
$67$	−12.6016	−1.53953	−0.769763	+	0.638330i	$0.779625\pi$
$68$	0	0
$69$	8.67926	1.04486
$70$	1.06098	0.126811
$71$	12.3500	1.46568	0.732840	−	0.680401i	$-0.238195\pi$
$71$	12.3500	1.46568	0.732840	+	0.680401i	$0.238195\pi$
$72$	−1.00000	−0.117851
$73$	−9.60208	−1.12384	−0.561919	−	0.827192i	$-0.689937\pi$
$73$	−9.60208	−1.12384	−0.561919	+	0.827192i	$0.689937\pi$
$74$	5.53073	0.642935
$75$	−1.00000	−0.115470
$76$	−7.34218	−0.842206
$77$	−3.69766	−0.421387
$78$	−3.47117	−0.393032
$79$	−4.35423	−0.489889	−0.244944	−	0.969537i	$-0.578770\pi$
$79$	−4.35423	−0.489889	−0.244944	+	0.969537i	$0.578770\pi$
$80$	−1.00000	−0.111803
$81$	1.00000	0.111111
$82$	2.82724	0.312217
$83$	−9.38266	−1.02988	−0.514940	−	0.857226i	$-0.672186\pi$
$83$	−9.38266	−1.02988	−0.514940	+	0.857226i	$0.672186\pi$
$84$	−1.06098	−0.115762
$85$	0	0
$86$	7.02634	0.757670
$87$	2.62382	0.281303
$88$	3.48514	0.371517
$89$	5.59982	0.593580	0.296790	−	0.954943i	$-0.404084\pi$
$89$	5.59982	0.593580	0.296790	+	0.954943i	$0.404084\pi$
$90$	1.00000	0.105409
$91$	−3.68283	−0.386065
$92$	−8.67926	−0.904876
$93$	10.5015	1.08896
$94$	−4.88874	−0.504235
$95$	7.34218	0.753292
$96$	1.00000	0.102062
$97$	−2.93760	−0.298268	−0.149134	−	0.988817i	$-0.547649\pi$
$97$	−2.93760	−0.298268	−0.149134	+	0.988817i	$0.547649\pi$
$98$	5.87433	0.593397
$99$	−3.48514	−0.350270
$100$	1.00000	0.100000
$101$	18.4185	1.83270	0.916352	−	0.400373i	$-0.131120\pi$
$101$	18.4185	1.83270	0.916352	+	0.400373i	$0.131120\pi$
$102$	0	0
$103$	19.0667	1.87870	0.939349	−	0.342962i	$-0.111430\pi$
$103$	19.0667	1.87870	0.939349	+	0.342962i	$0.111430\pi$
$104$	3.47117	0.340376
$105$	1.06098	0.103541
$106$	−9.86766	−0.958432
$107$	−2.20671	−0.213331	−0.106665	−	0.994295i	$-0.534017\pi$
$107$	−2.20671	−0.213331	−0.106665	+	0.994295i	$0.534017\pi$
$108$	−1.00000	−0.0962250
$109$	3.48673	0.333968	0.166984	−	0.985960i	$-0.446597\pi$
$109$	3.48673	0.333968	0.166984	+	0.985960i	$0.446597\pi$
$110$	−3.48514	−0.332295
$111$	5.53073	0.524954
$112$	1.06098	0.100253
$113$	6.30915	0.593515	0.296758	−	0.954953i	$-0.404095\pi$
$113$	6.30915	0.593515	0.296758	+	0.954953i	$0.404095\pi$
$114$	−7.34218	−0.687658
$115$	8.67926	0.809345
$116$	−2.62382	−0.243616
$117$	−3.47117	−0.320909
$118$	6.17161	0.568143
$119$	0	0
$120$	−1.00000	−0.0912871
$121$	1.14622	0.104201
$122$	−9.82606	−0.889610
$123$	2.82724	0.254924
$124$	−10.5015	−0.943065
$125$	−1.00000	−0.0894427
$126$	−1.06098	−0.0945194
$127$	−5.84808	−0.518933	−0.259467	−	0.965752i	$-0.583547\pi$
$127$	−5.84808	−0.518933	−0.259467	+	0.965752i	$0.583547\pi$
$128$	−1.00000	−0.0883883
$129$	7.02634	0.618635
$130$	−3.47117	−0.304441
$131$	6.31633	0.551860	0.275930	−	0.961178i	$-0.411014\pi$
$131$	6.31633	0.551860	0.275930	+	0.961178i	$0.411014\pi$
$132$	3.48514	0.303343
$133$	−7.78989	−0.675469
$134$	12.6016	1.08861
$135$	1.00000	0.0860663
$136$	0	0
$137$	8.53613	0.729291	0.364645	−	0.931146i	$-0.381190\pi$
$137$	8.53613	0.729291	0.364645	+	0.931146i	$0.381190\pi$
$138$	−8.67926	−0.738828
$139$	−16.0849	−1.36430	−0.682152	−	0.731210i	$-0.738956\pi$
$139$	−16.0849	−1.36430	−0.682152	+	0.731210i	$0.738956\pi$
$140$	−1.06098	−0.0896690
$141$	−4.88874	−0.411706
$142$	−12.3500	−1.03639
$143$	12.0975	1.01164
$144$	1.00000	0.0833333
$145$	2.62382	0.217897
$146$	9.60208	0.794674
$147$	5.87433	0.484506
$148$	−5.53073	−0.454624
$149$	−19.2845	−1.57985	−0.789924	−	0.613205i	$-0.789880\pi$
$149$	−19.2845	−1.57985	−0.789924	+	0.613205i	$0.789880\pi$
$150$	1.00000	0.0816497
$151$	4.89015	0.397954	0.198977	−	0.980004i	$-0.436238\pi$
$151$	4.89015	0.397954	0.198977	+	0.980004i	$0.436238\pi$
$152$	7.34218	0.595530
$153$	0	0
$154$	3.69766	0.297966
$155$	10.5015	0.843503
$156$	3.47117	0.277916
$157$	12.9056	1.02998	0.514990	−	0.857196i	$-0.327796\pi$
$157$	12.9056	1.02998	0.514990	+	0.857196i	$0.327796\pi$
$158$	4.35423	0.346404
$159$	−9.86766	−0.782556
$160$	1.00000	0.0790569
$161$	−9.20850	−0.725732
$162$	−1.00000	−0.0785674
$163$	−23.3888	−1.83196	−0.915978	−	0.401229i	$-0.868583\pi$
$163$	−23.3888	−1.83196	−0.915978	+	0.401229i	$0.868583\pi$
$164$	−2.82724	−0.220771
$165$	−3.48514	−0.271318
$166$	9.38266	0.728235
$167$	−23.6763	−1.83213	−0.916063	−	0.401035i	$-0.868651\pi$
$167$	−23.6763	−1.83213	−0.916063	+	0.401035i	$0.868651\pi$
$168$	1.06098	0.0818562
$169$	−0.951010	−0.0731546
$170$	0	0
$171$	−7.34218	−0.561471
$172$	−7.02634	−0.535753
$173$	−6.09008	−0.463020	−0.231510	−	0.972832i	$-0.574367\pi$
$173$	−6.09008	−0.463020	−0.231510	+	0.972832i	$0.574367\pi$
$174$	−2.62382	−0.198912
$175$	1.06098	0.0802024
$176$	−3.48514	−0.262702
$177$	6.17161	0.463887
$178$	−5.59982	−0.419724
$179$	5.18370	0.387448	0.193724	−	0.981056i	$-0.437943\pi$
$179$	5.18370	0.387448	0.193724	+	0.981056i	$0.437943\pi$
$180$	−1.00000	−0.0745356
$181$	19.8446	1.47504	0.737519	−	0.675326i	$-0.235997\pi$
$181$	19.8446	1.47504	0.737519	+	0.675326i	$0.235997\pi$
$182$	3.68283	0.272989
$183$	−9.82606	−0.726364
$184$	8.67926	0.639844
$185$	5.53073	0.406628
$186$	−10.5015	−0.770009
$187$	0	0
$188$	4.88874	0.356548
$189$	−1.06098	−0.0771747
$190$	−7.34218	−0.532658
$191$	18.3624	1.32866	0.664329	−	0.747440i	$-0.268717\pi$
$191$	18.3624	1.32866	0.664329	+	0.747440i	$0.268717\pi$
$192$	−1.00000	−0.0721688
$193$	−14.4909	−1.04308	−0.521539	−	0.853228i	$-0.674642\pi$
$193$	−14.4909	−1.04308	−0.521539	+	0.853228i	$0.674642\pi$
$194$	2.93760	0.210907
$195$	−3.47117	−0.248575
$196$	−5.87433	−0.419595
$197$	16.1998	1.15419	0.577094	−	0.816678i	$-0.304187\pi$
$197$	16.1998	1.15419	0.577094	+	0.816678i	$0.304187\pi$
$198$	3.48514	0.247678
$199$	9.02900	0.640049	0.320025	−	0.947409i	$-0.396309\pi$
$199$	9.02900	0.640049	0.320025	+	0.947409i	$0.396309\pi$
$200$	−1.00000	−0.0707107
$201$	12.6016	0.888845
$202$	−18.4185	−1.29592
$203$	−2.78382	−0.195386
$204$	0	0
$205$	2.82724	0.197463
$206$	−19.0667	−1.32844
$207$	−8.67926	−0.603250
$208$	−3.47117	−0.240682
$209$	25.5885	1.77000
$210$	−1.06098	−0.0732144
$211$	8.64852	0.595388	0.297694	−	0.954661i	$-0.403782\pi$
$211$	8.64852	0.595388	0.297694	+	0.954661i	$0.403782\pi$
$212$	9.86766	0.677714
$213$	−12.3500	−0.846210
$214$	2.20671	0.150847
$215$	7.02634	0.479192
$216$	1.00000	0.0680414
$217$	−11.1419	−0.756360
$218$	−3.48673	−0.236151
$219$	9.60208	0.648848
$220$	3.48514	0.234968
$221$	0	0
$222$	−5.53073	−0.371199
$223$	15.5060	1.03836	0.519180	−	0.854665i	$-0.326237\pi$
$223$	15.5060	1.03836	0.519180	+	0.854665i	$0.326237\pi$
$224$	−1.06098	−0.0708895
$225$	1.00000	0.0666667
$226$	−6.30915	−0.419679
$227$	−9.48917	−0.629818	−0.314909	−	0.949122i	$-0.601974\pi$
$227$	−9.48917	−0.629818	−0.314909	+	0.949122i	$0.601974\pi$
$228$	7.34218	0.486248
$229$	−8.83945	−0.584127	−0.292064	−	0.956399i	$-0.594342\pi$
$229$	−8.83945	−0.584127	−0.292064	+	0.956399i	$0.594342\pi$
$230$	−8.67926	−0.572294
$231$	3.69766	0.243288
$232$	2.62382	0.172262
$233$	−5.94487	−0.389461	−0.194731	−	0.980857i	$-0.562383\pi$
$233$	−5.94487	−0.389461	−0.194731	+	0.980857i	$0.562383\pi$
$234$	3.47117	0.226917
$235$	−4.88874	−0.318906
$236$	−6.17161	−0.401738
$237$	4.35423	0.282838
$238$	0	0
$239$	−0.783103	−0.0506547	−0.0253274	−	0.999679i	$-0.508063\pi$
$239$	−0.783103	−0.0506547	−0.0253274	+	0.999679i	$0.508063\pi$
$240$	1.00000	0.0645497
$241$	−16.2870	−1.04914	−0.524570	−	0.851367i	$-0.675774\pi$
$241$	−16.2870	−1.04914	−0.524570	+	0.851367i	$0.675774\pi$
$242$	−1.14622	−0.0736815
$243$	−1.00000	−0.0641500
$244$	9.82606	0.629049
$245$	5.87433	0.375297
$246$	−2.82724	−0.180258
$247$	25.4859	1.62163
$248$	10.5015	0.666848
$249$	9.38266	0.594602
$250$	1.00000	0.0632456
$251$	−1.20918	−0.0763230	−0.0381615	−	0.999272i	$-0.512150\pi$
$251$	−1.20918	−0.0763230	−0.0381615	+	0.999272i	$0.512150\pi$
$252$	1.06098	0.0668353
$253$	30.2485	1.90170
$254$	5.84808	0.366941
$255$	0	0
$256$	1.00000	0.0625000
$257$	6.77280	0.422476	0.211238	−	0.977435i	$-0.432250\pi$
$257$	6.77280	0.422476	0.211238	+	0.977435i	$0.432250\pi$
$258$	−7.02634	−0.437441
$259$	−5.86798	−0.364619
$260$	3.47117	0.215273
$261$	−2.62382	−0.162411
$262$	−6.31633	−0.390224
$263$	−3.14118	−0.193693	−0.0968466	−	0.995299i	$-0.530876\pi$
$263$	−3.14118	−0.193693	−0.0968466	+	0.995299i	$0.530876\pi$
$264$	−3.48514	−0.214496
$265$	−9.86766	−0.606166
$266$	7.78989	0.477629
$267$	−5.59982	−0.342703
$268$	−12.6016	−0.769763
$269$	−6.92637	−0.422308	−0.211154	−	0.977453i	$-0.567722\pi$
$269$	−6.92637	−0.422308	−0.211154	+	0.977453i	$0.567722\pi$
$270$	−1.00000	−0.0608581
$271$	20.3367	1.23537	0.617684	−	0.786426i	$-0.288071\pi$
$271$	20.3367	1.23537	0.617684	+	0.786426i	$0.288071\pi$
$272$	0	0
$273$	3.68283	0.222895
$274$	−8.53613	−0.515686
$275$	−3.48514	−0.210162
$276$	8.67926	0.522430
$277$	−3.34799	−0.201161	−0.100581	−	0.994929i	$-0.532070\pi$
$277$	−3.34799	−0.201161	−0.100581	+	0.994929i	$0.532070\pi$
$278$	16.0849	0.964709
$279$	−10.5015	−0.628710
$280$	1.06098	0.0634055
$281$	11.4256	0.681592	0.340796	−	0.940137i	$-0.389303\pi$
$281$	11.4256	0.681592	0.340796	+	0.940137i	$0.389303\pi$
$282$	4.88874	0.291120
$283$	−10.5281	−0.625832	−0.312916	−	0.949781i	$-0.601306\pi$
$283$	−10.5281	−0.625832	−0.312916	+	0.949781i	$0.601306\pi$
$284$	12.3500	0.732840
$285$	−7.34218	−0.434913
$286$	−12.0975	−0.715340
$287$	−2.99964	−0.177063
$288$	−1.00000	−0.0589256
$289$	0	0
$290$	−2.62382	−0.154076
$291$	2.93760	0.172205
$292$	−9.60208	−0.561919
$293$	−17.7898	−1.03929	−0.519646	−	0.854381i	$-0.673936\pi$
$293$	−17.7898	−1.03929	−0.519646	+	0.854381i	$0.673936\pi$
$294$	−5.87433	−0.342598
$295$	6.17161	0.359325
$296$	5.53073	0.321467
$297$	3.48514	0.202228
$298$	19.2845	1.11712
$299$	30.1272	1.74230
$300$	−1.00000	−0.0577350
$301$	−7.45479	−0.429687
$302$	−4.89015	−0.281396
$303$	−18.4185	−1.05811
$304$	−7.34218	−0.421103
$305$	−9.82606	−0.562639
$306$	0	0
$307$	−26.8792	−1.53408	−0.767039	−	0.641600i	$-0.778271\pi$
$307$	−26.8792	−1.53408	−0.767039	+	0.641600i	$0.778271\pi$
$308$	−3.69766	−0.210694
$309$	−19.0667	−1.08467
$310$	−10.5015	−0.596447
$311$	−16.9305	−0.960042	−0.480021	−	0.877257i	$-0.659371\pi$
$311$	−16.9305	−0.960042	−0.480021	+	0.877257i	$0.659371\pi$
$312$	−3.47117	−0.196516
$313$	−0.360312	−0.0203660	−0.0101830	−	0.999948i	$-0.503241\pi$
$313$	−0.360312	−0.0203660	−0.0101830	+	0.999948i	$0.503241\pi$
$314$	−12.9056	−0.728306
$315$	−1.06098	−0.0597793
$316$	−4.35423	−0.244944
$317$	−12.5816	−0.706654	−0.353327	−	0.935500i	$-0.614950\pi$
$317$	−12.5816	−0.706654	−0.353327	+	0.935500i	$0.614950\pi$
$318$	9.86766	0.553351
$319$	9.14440	0.511988
$320$	−1.00000	−0.0559017
$321$	2.20671	0.123166
$322$	9.20850	0.513170
$323$	0	0
$324$	1.00000	0.0555556
$325$	−3.47117	−0.192546
$326$	23.3888	1.29539
$327$	−3.48673	−0.192817
$328$	2.82724	0.156108
$329$	5.18684	0.285960
$330$	3.48514	0.191851
$331$	−10.6216	−0.583815	−0.291908	−	0.956446i	$-0.594290\pi$
$331$	−10.6216	−0.583815	−0.291908	+	0.956446i	$0.594290\pi$
$332$	−9.38266	−0.514940
$333$	−5.53073	−0.303082
$334$	23.6763	1.29551
$335$	12.6016	0.688497
$336$	−1.06098	−0.0578811
$337$	−14.4546	−0.787393	−0.393697	−	0.919240i	$-0.628804\pi$
$337$	−14.4546	−0.787393	−0.393697	+	0.919240i	$0.628804\pi$
$338$	0.951010	0.0517281
$339$	−6.30915	−0.342666
$340$	0	0
$341$	36.5993	1.98196
$342$	7.34218	0.397020
$343$	−13.6594	−0.737537
$344$	7.02634	0.378835
$345$	−8.67926	−0.467276
$346$	6.09008	0.327405
$347$	−12.9506	−0.695226	−0.347613	−	0.937638i	$-0.613008\pi$
$347$	−12.9506	−0.695226	−0.347613	+	0.937638i	$0.613008\pi$
$348$	2.62382	0.140652
$349$	−5.89508	−0.315556	−0.157778	−	0.987475i	$-0.550433\pi$
$349$	−5.89508	−0.315556	−0.157778	+	0.987475i	$0.550433\pi$
$350$	−1.06098	−0.0567116
$351$	3.47117	0.185277
$352$	3.48514	0.185759
$353$	−23.5772	−1.25489	−0.627443	−	0.778663i	$-0.715898\pi$
$353$	−23.5772	−1.25489	−0.627443	+	0.778663i	$0.715898\pi$
$354$	−6.17161	−0.328017
$355$	−12.3500	−0.655472
$356$	5.59982	0.296790
$357$	0	0
$358$	−5.18370	−0.273967
$359$	26.6777	1.40799	0.703997	−	0.710203i	$-0.251397\pi$
$359$	26.6777	1.40799	0.703997	+	0.710203i	$0.251397\pi$
$360$	1.00000	0.0527046
$361$	34.9076	1.83724
$362$	−19.8446	−1.04301
$363$	−1.14622	−0.0601607
$364$	−3.68283	−0.193033
$365$	9.60208	0.502596
$366$	9.82606	0.513617
$367$	−30.6769	−1.60132	−0.800662	−	0.599117i	$-0.795518\pi$
$367$	−30.6769	−1.60132	−0.800662	+	0.599117i	$0.795518\pi$
$368$	−8.67926	−0.452438
$369$	−2.82724	−0.147180
$370$	−5.53073	−0.287529
$371$	10.4694	0.543542
$372$	10.5015	0.544479
$373$	−1.48782	−0.0770362	−0.0385181	−	0.999258i	$-0.512264\pi$
$373$	−1.48782	−0.0770362	−0.0385181	+	0.999258i	$0.512264\pi$
$374$	0	0
$375$	1.00000	0.0516398
$376$	−4.88874	−0.252117
$377$	9.10772	0.469072
$378$	1.06098	0.0545708
$379$	21.1913	1.08852	0.544262	−	0.838915i	$-0.316810\pi$
$379$	21.1913	1.08852	0.544262	+	0.838915i	$0.316810\pi$
$380$	7.34218	0.376646
$381$	5.84808	0.299606
$382$	−18.3624	−0.939503
$383$	−25.5964	−1.30791	−0.653956	−	0.756532i	$-0.726892\pi$
$383$	−25.5964	−1.30791	−0.653956	+	0.756532i	$0.726892\pi$
$384$	1.00000	0.0510310
$385$	3.69766	0.188450
$386$	14.4909	0.737567
$387$	−7.02634	−0.357169
$388$	−2.93760	−0.149134
$389$	−22.3422	−1.13279	−0.566397	−	0.824133i	$-0.691663\pi$
$389$	−22.3422	−1.13279	−0.566397	+	0.824133i	$0.691663\pi$
$390$	3.47117	0.175769
$391$	0	0
$392$	5.87433	0.296698
$393$	−6.31633	−0.318617
$394$	−16.1998	−0.816134
$395$	4.35423	0.219085
$396$	−3.48514	−0.175135
$397$	−11.3264	−0.568457	−0.284228	−	0.958757i	$-0.591737\pi$
$397$	−11.3264	−0.568457	−0.284228	+	0.958757i	$0.591737\pi$
$398$	−9.02900	−0.452583
$399$	7.78989	0.389982
$400$	1.00000	0.0500000
$401$	−16.2608	−0.812024	−0.406012	−	0.913868i	$-0.633081\pi$
$401$	−16.2608	−0.812024	−0.406012	+	0.913868i	$0.633081\pi$
$402$	−12.6016	−0.628509
$403$	36.4525	1.81583
$404$	18.4185	0.916352
$405$	−1.00000	−0.0496904
$406$	2.78382	0.138159
$407$	19.2754	0.955446
$408$	0	0
$409$	3.89097	0.192396	0.0961979	−	0.995362i	$-0.469332\pi$
$409$	3.89097	0.192396	0.0961979	+	0.995362i	$0.469332\pi$
$410$	−2.82724	−0.139628
$411$	−8.53613	−0.421056
$412$	19.0667	0.939349
$413$	−6.54794	−0.322203
$414$	8.67926	0.426562
$415$	9.38266	0.460577
$416$	3.47117	0.170188
$417$	16.0849	0.787682
$418$	−25.5885	−1.25158
$419$	24.6933	1.20635	0.603173	−	0.797610i	$-0.293903\pi$
$419$	24.6933	1.20635	0.603173	+	0.797610i	$0.293903\pi$
$420$	1.06098	0.0517704
$421$	12.0894	0.589199	0.294599	−	0.955621i	$-0.404814\pi$
$421$	12.0894	0.589199	0.294599	+	0.955621i	$0.404814\pi$
$422$	−8.64852	−0.421003
$423$	4.88874	0.237699
$424$	−9.86766	−0.479216
$425$	0	0
$426$	12.3500	0.598361
$427$	10.4252	0.504512
$428$	−2.20671	−0.106665
$429$	−12.0975	−0.584073
$430$	−7.02634	−0.338840
$431$	3.55059	0.171026	0.0855130	−	0.996337i	$-0.472747\pi$
$431$	3.55059	0.171026	0.0855130	+	0.996337i	$0.472747\pi$
$432$	−1.00000	−0.0481125
$433$	18.3807	0.883321	0.441661	−	0.897182i	$-0.354390\pi$
$433$	18.3807	0.883321	0.441661	+	0.897182i	$0.354390\pi$
$434$	11.1419	0.534828
$435$	−2.62382	−0.125803
$436$	3.48673	0.166984
$437$	63.7247	3.04837
$438$	−9.60208	−0.458805
$439$	−5.19771	−0.248073	−0.124037	−	0.992278i	$-0.539584\pi$
$439$	−5.19771	−0.248073	−0.124037	+	0.992278i	$0.539584\pi$
$440$	−3.48514	−0.166148
$441$	−5.87433	−0.279730
$442$	0	0
$443$	16.2504	0.772079	0.386040	−	0.922482i	$-0.373843\pi$
$443$	16.2504	0.772079	0.386040	+	0.922482i	$0.373843\pi$
$444$	5.53073	0.262477
$445$	−5.59982	−0.265457
$446$	−15.5060	−0.734231
$447$	19.2845	0.912125
$448$	1.06098	0.0501265
$449$	10.2565	0.484034	0.242017	−	0.970272i	$-0.422191\pi$
$449$	10.2565	0.484034	0.242017	+	0.970272i	$0.422191\pi$
$450$	−1.00000	−0.0471405
$451$	9.85335	0.463976
$452$	6.30915	0.296758
$453$	−4.89015	−0.229759
$454$	9.48917	0.445349
$455$	3.68283	0.172654
$456$	−7.34218	−0.343829
$457$	14.7915	0.691916	0.345958	−	0.938250i	$-0.387554\pi$
$457$	14.7915	0.691916	0.345958	+	0.938250i	$0.387554\pi$
$458$	8.83945	0.413040
$459$	0	0
$460$	8.67926	0.404673
$461$	6.26556	0.291816	0.145908	−	0.989298i	$-0.453390\pi$
$461$	6.26556	0.291816	0.145908	+	0.989298i	$0.453390\pi$
$462$	−3.69766	−0.172031
$463$	19.1392	0.889473	0.444736	−	0.895662i	$-0.353297\pi$
$463$	19.1392	0.889473	0.444736	+	0.895662i	$0.353297\pi$
$464$	−2.62382	−0.121808
$465$	−10.5015	−0.486997
$466$	5.94487	0.275391
$467$	12.6796	0.586743	0.293372	−	0.955998i	$-0.405223\pi$
$467$	12.6796	0.586743	0.293372	+	0.955998i	$0.405223\pi$
$468$	−3.47117	−0.160455
$469$	−13.3700	−0.617368
$470$	4.88874	0.225501
$471$	−12.9056	−0.594659
$472$	6.17161	0.284071
$473$	24.4878	1.12595
$474$	−4.35423	−0.199996
$475$	−7.34218	−0.336882
$476$	0	0
$477$	9.86766	0.451809
$478$	0.783103	0.0358183
$479$	−3.99033	−0.182323	−0.0911613	−	0.995836i	$-0.529058\pi$
$479$	−3.99033	−0.182323	−0.0911613	+	0.995836i	$0.529058\pi$
$480$	−1.00000	−0.0456435
$481$	19.1981	0.875358
$482$	16.2870	0.741854
$483$	9.20850	0.419001
$484$	1.14622	0.0521007
$485$	2.93760	0.133390
$486$	1.00000	0.0453609
$487$	1.92919	0.0874198	0.0437099	−	0.999044i	$-0.486082\pi$
$487$	1.92919	0.0874198	0.0437099	+	0.999044i	$0.486082\pi$
$488$	−9.82606	−0.444805
$489$	23.3888	1.05768
$490$	−5.87433	−0.265375
$491$	−37.9150	−1.71108	−0.855539	−	0.517738i	$-0.826774\pi$
$491$	−37.9150	−1.71108	−0.855539	+	0.517738i	$0.826774\pi$
$492$	2.82724	0.127462
$493$	0	0
$494$	−25.4859	−1.14667
$495$	3.48514	0.156645
$496$	−10.5015	−0.471533
$497$	13.1031	0.587755
$498$	−9.38266	−0.420447
$499$	−30.1433	−1.34940	−0.674700	−	0.738092i	$-0.735727\pi$
$499$	−30.1433	−1.34940	−0.674700	+	0.738092i	$0.735727\pi$
$500$	−1.00000	−0.0447214
$501$	23.6763	1.05778
$502$	1.20918	0.0539685
$503$	6.66305	0.297091	0.148545	−	0.988906i	$-0.452541\pi$
$503$	6.66305	0.297091	0.148545	+	0.988906i	$0.452541\pi$
$504$	−1.06098	−0.0472597
$505$	−18.4185	−0.819610
$506$	−30.2485	−1.34471
$507$	0.951010	0.0422359
$508$	−5.84808	−0.259467
$509$	−27.0499	−1.19897	−0.599483	−	0.800387i	$-0.704627\pi$
$509$	−27.0499	−1.19897	−0.599483	+	0.800387i	$0.704627\pi$
$510$	0	0
$511$	−10.1876	−0.450672
$512$	−1.00000	−0.0441942
$513$	7.34218	0.324165
$514$	−6.77280	−0.298736
$515$	−19.0667	−0.840179
$516$	7.02634	0.309317
$517$	−17.0379	−0.749328
$518$	5.86798	0.257824
$519$	6.09008	0.267325
$520$	−3.47117	−0.152221
$521$	−4.89694	−0.214539	−0.107269	−	0.994230i	$-0.534211\pi$
$521$	−4.89694	−0.214539	−0.107269	+	0.994230i	$0.534211\pi$
$522$	2.62382	0.114842
$523$	−10.2362	−0.447599	−0.223799	−	0.974635i	$-0.571846\pi$
$523$	−10.2362	−0.447599	−0.223799	+	0.974635i	$0.571846\pi$
$524$	6.31633	0.275930
$525$	−1.06098	−0.0463048
$526$	3.14118	0.136962
$527$	0	0
$528$	3.48514	0.151671
$529$	52.3296	2.27520
$530$	9.86766	0.428624
$531$	−6.17161	−0.267825
$532$	−7.78989	−0.337735
$533$	9.81383	0.425084
$534$	5.59982	0.242328
$535$	2.20671	0.0954043
$536$	12.6016	0.544304
$537$	−5.18370	−0.223693
$538$	6.92637	0.298617
$539$	20.4729	0.881829
$540$	1.00000	0.0430331
$541$	−36.0335	−1.54920	−0.774600	−	0.632452i	$-0.782049\pi$
$541$	−36.0335	−1.54920	−0.774600	+	0.632452i	$0.782049\pi$
$542$	−20.3367	−0.873537
$543$	−19.8446	−0.851614
$544$	0	0
$545$	−3.48673	−0.149355
$546$	−3.68283	−0.157610
$547$	−6.02142	−0.257457	−0.128729	−	0.991680i	$-0.541090\pi$
$547$	−6.02142	−0.257457	−0.128729	+	0.991680i	$0.541090\pi$
$548$	8.53613	0.364645
$549$	9.82606	0.419366
$550$	3.48514	0.148607
$551$	19.2646	0.820699
$552$	−8.67926	−0.369414
$553$	−4.61974	−0.196451
$554$	3.34799	0.142242
$555$	−5.53073	−0.234767
$556$	−16.0849	−0.682152
$557$	−15.8547	−0.671786	−0.335893	−	0.941900i	$-0.609038\pi$
$557$	−15.8547	−0.671786	−0.335893	+	0.941900i	$0.609038\pi$
$558$	10.5015	0.444565
$559$	24.3896	1.03157
$560$	−1.06098	−0.0448345
$561$	0	0
$562$	−11.4256	−0.481958
$563$	20.6547	0.870492	0.435246	−	0.900312i	$-0.356661\pi$
$563$	20.6547	0.870492	0.435246	+	0.900312i	$0.356661\pi$
$564$	−4.88874	−0.205853
$565$	−6.30915	−0.265428
$566$	10.5281	0.442530
$567$	1.06098	0.0445569
$568$	−12.3500	−0.518196
$569$	−20.4654	−0.857953	−0.428976	−	0.903316i	$-0.641126\pi$
$569$	−20.4654	−0.857953	−0.428976	+	0.903316i	$0.641126\pi$
$570$	7.34218	0.307530
$571$	29.5017	1.23461	0.617304	−	0.786724i	$-0.288225\pi$
$571$	29.5017	1.23461	0.617304	+	0.786724i	$0.288225\pi$
$572$	12.0975	0.505822
$573$	−18.3624	−0.767101
$574$	2.99964	0.125203
$575$	−8.67926	−0.361950
$576$	1.00000	0.0416667
$577$	−1.07828	−0.0448892	−0.0224446	−	0.999748i	$-0.507145\pi$
$577$	−1.07828	−0.0448892	−0.0224446	+	0.999748i	$0.507145\pi$
$578$	0	0
$579$	14.4909	0.602221
$580$	2.62382	0.108948
$581$	−9.95479	−0.412994
$582$	−2.93760	−0.121767
$583$	−34.3902	−1.42430
$584$	9.60208	0.397337
$585$	3.47117	0.143515
$586$	17.7898	0.734891
$587$	0.707987	0.0292218	0.0146109	−	0.999893i	$-0.495349\pi$
$587$	0.707987	0.0292218	0.0146109	+	0.999893i	$0.495349\pi$
$588$	5.87433	0.242253
$589$	77.1041	3.17702
$590$	−6.17161	−0.254081
$591$	−16.1998	−0.666371
$592$	−5.53073	−0.227312
$593$	1.95353	0.0802220	0.0401110	−	0.999195i	$-0.487229\pi$
$593$	1.95353	0.0802220	0.0401110	+	0.999195i	$0.487229\pi$
$594$	−3.48514	−0.142997
$595$	0	0
$596$	−19.2845	−0.789924
$597$	−9.02900	−0.369532
$598$	−30.1272	−1.23199
$599$	24.0578	0.982977	0.491489	−	0.870884i	$-0.336453\pi$
$599$	24.0578	0.982977	0.491489	+	0.870884i	$0.336453\pi$
$600$	1.00000	0.0408248
$601$	−18.7942	−0.766633	−0.383317	−	0.923617i	$-0.625218\pi$
$601$	−18.7942	−0.766633	−0.383317	+	0.923617i	$0.625218\pi$
$602$	7.45479	0.303834
$603$	−12.6016	−0.513175
$604$	4.89015	0.198977
$605$	−1.14622	−0.0466003
$606$	18.4185	0.748198
$607$	32.8223	1.33222	0.666108	−	0.745855i	$-0.267959\pi$
$607$	32.8223	1.33222	0.666108	+	0.745855i	$0.267959\pi$
$608$	7.34218	0.297765
$609$	2.78382	0.112806
$610$	9.82606	0.397846
$611$	−16.9696	−0.686517
$612$	0	0
$613$	−18.8644	−0.761926	−0.380963	−	0.924590i	$-0.624407\pi$
$613$	−18.8644	−0.761926	−0.380963	+	0.924590i	$0.624407\pi$
$614$	26.8792	1.08476
$615$	−2.82724	−0.114005
$616$	3.69766	0.148983
$617$	44.1337	1.77675	0.888377	−	0.459114i	$-0.151833\pi$
$617$	44.1337	1.77675	0.888377	+	0.459114i	$0.151833\pi$
$618$	19.0667	0.766975
$619$	28.0732	1.12836	0.564179	−	0.825652i	$-0.309193\pi$
$619$	28.0732	1.12836	0.564179	+	0.825652i	$0.309193\pi$
$620$	10.5015	0.421752
$621$	8.67926	0.348287
$622$	16.9305	0.678852
$623$	5.94128	0.238032
$624$	3.47117	0.138958
$625$	1.00000	0.0400000
$626$	0.360312	0.0144009
$627$	−25.5885	−1.02191
$628$	12.9056	0.514990
$629$	0	0
$630$	1.06098	0.0422704
$631$	−28.1129	−1.11916	−0.559579	−	0.828777i	$-0.689037\pi$
$631$	−28.1129	−1.11916	−0.559579	+	0.828777i	$0.689037\pi$
$632$	4.35423	0.173202
$633$	−8.64852	−0.343748
$634$	12.5816	0.499680
$635$	5.84808	0.232074
$636$	−9.86766	−0.391278
$637$	20.3908	0.807911
$638$	−9.14440	−0.362030
$639$	12.3500	0.488560
$640$	1.00000	0.0395285
$641$	13.1772	0.520469	0.260235	−	0.965545i	$-0.416200\pi$
$641$	13.1772	0.520469	0.260235	+	0.965545i	$0.416200\pi$
$642$	−2.20671	−0.0870918
$643$	32.9676	1.30011	0.650057	−	0.759886i	$-0.274745\pi$
$643$	32.9676	1.30011	0.650057	+	0.759886i	$0.274745\pi$
$644$	−9.20850	−0.362866
$645$	−7.02634	−0.276662
$646$	0	0
$647$	36.0021	1.41539	0.707695	−	0.706518i	$-0.249735\pi$
$647$	36.0021	1.41539	0.707695	+	0.706518i	$0.249735\pi$
$648$	−1.00000	−0.0392837
$649$	21.5089	0.844300
$650$	3.47117	0.136150
$651$	11.1419	0.436685
$652$	−23.3888	−0.915978
$653$	−2.78887	−0.109137	−0.0545685	−	0.998510i	$-0.517378\pi$
$653$	−2.78887	−0.109137	−0.0545685	+	0.998510i	$0.517378\pi$
$654$	3.48673	0.136342
$655$	−6.31633	−0.246799
$656$	−2.82724	−0.110385
$657$	−9.60208	−0.374613
$658$	−5.18684	−0.202204
$659$	17.5168	0.682356	0.341178	−	0.939999i	$-0.389174\pi$
$659$	17.5168	0.682356	0.341178	+	0.939999i	$0.389174\pi$
$660$	−3.48514	−0.135659
$661$	32.2459	1.25422	0.627109	−	0.778931i	$-0.284238\pi$
$661$	32.2459	1.25422	0.627109	+	0.778931i	$0.284238\pi$
$662$	10.6216	0.412820
$663$	0	0
$664$	9.38266	0.364118
$665$	7.78989	0.302079
$666$	5.53073	0.214312
$667$	22.7729	0.881768
$668$	−23.6763	−0.916063
$669$	−15.5060	−0.599497
$670$	−12.6016	−0.486841
$671$	−34.2452	−1.32202
$672$	1.06098	0.0409281
$673$	−18.3695	−0.708091	−0.354046	−	0.935228i	$-0.615194\pi$
$673$	−18.3695	−0.708091	−0.354046	+	0.935228i	$0.615194\pi$
$674$	14.4546	0.556771
$675$	−1.00000	−0.0384900
$676$	−0.951010	−0.0365773
$677$	12.0728	0.463995	0.231997	−	0.972716i	$-0.425474\pi$
$677$	12.0728	0.463995	0.231997	+	0.972716i	$0.425474\pi$
$678$	6.30915	0.242302
$679$	−3.11673	−0.119609
$680$	0	0
$681$	9.48917	0.363626
$682$	−36.5993	−1.40146
$683$	−20.3762	−0.779673	−0.389836	−	0.920884i	$-0.627468\pi$
$683$	−20.3762	−0.779673	−0.389836	+	0.920884i	$0.627468\pi$
$684$	−7.34218	−0.280735
$685$	−8.53613	−0.326149
$686$	13.6594	0.521517
$687$	8.83945	0.337246
$688$	−7.02634	−0.267877
$689$	−34.2523	−1.30491
$690$	8.67926	0.330414
$691$	−17.1585	−0.652740	−0.326370	−	0.945242i	$-0.605826\pi$
$691$	−17.1585	−0.652740	−0.326370	+	0.945242i	$0.605826\pi$
$692$	−6.09008	−0.231510
$693$	−3.69766	−0.140462
$694$	12.9506	0.491599
$695$	16.0849	0.610136
$696$	−2.62382	−0.0994558
$697$	0	0
$698$	5.89508	0.223132
$699$	5.94487	0.224856
$700$	1.06098	0.0401012
$701$	38.5633	1.45651	0.728257	−	0.685304i	$-0.240331\pi$
$701$	38.5633	1.45651	0.728257	+	0.685304i	$0.240331\pi$
$702$	−3.47117	−0.131011
$703$	40.6077	1.53155
$704$	−3.48514	−0.131351
$705$	4.88874	0.184121
$706$	23.5772	0.887338
$707$	19.5416	0.734936
$708$	6.17161	0.231943
$709$	−18.7230	−0.703156	−0.351578	−	0.936159i	$-0.614355\pi$
$709$	−18.7230	−0.703156	−0.351578	+	0.936159i	$0.614355\pi$
$710$	12.3500	0.463488
$711$	−4.35423	−0.163296
$712$	−5.59982	−0.209862
$713$	91.1455	3.41343
$714$	0	0
$715$	−12.0975	−0.452421
$716$	5.18370	0.193724
$717$	0.783103	0.0292455
$718$	−26.6777	−0.995602
$719$	−2.27848	−0.0849729	−0.0424865	−	0.999097i	$-0.513528\pi$
$719$	−2.27848	−0.0849729	−0.0424865	+	0.999097i	$0.513528\pi$
$720$	−1.00000	−0.0372678
$721$	20.2293	0.753380
$722$	−34.9076	−1.29913
$723$	16.2870	0.605721
$724$	19.8446	0.737519
$725$	−2.62382	−0.0974463
$726$	1.14622	0.0425401
$727$	−39.7865	−1.47560	−0.737800	−	0.675020i	$-0.764135\pi$
$727$	−39.7865	−1.47560	−0.737800	+	0.675020i	$0.764135\pi$
$728$	3.68283	0.136495
$729$	1.00000	0.0370370
$730$	−9.60208	−0.355389
$731$	0	0
$732$	−9.82606	−0.363182
$733$	−21.0684	−0.778179	−0.389089	−	0.921200i	$-0.627210\pi$
$733$	−21.0684	−0.778179	−0.389089	+	0.921200i	$0.627210\pi$
$734$	30.6769	1.13231
$735$	−5.87433	−0.216678
$736$	8.67926	0.319922
$737$	43.9182	1.61775
$738$	2.82724	0.104072
$739$	−1.62238	−0.0596803	−0.0298402	−	0.999555i	$-0.509500\pi$
$739$	−1.62238	−0.0596803	−0.0298402	+	0.999555i	$0.509500\pi$
$740$	5.53073	0.203314
$741$	−25.4859	−0.936249
$742$	−10.4694	−0.384342
$743$	−28.7018	−1.05297	−0.526483	−	0.850186i	$-0.676490\pi$
$743$	−28.7018	−1.05297	−0.526483	+	0.850186i	$0.676490\pi$
$744$	−10.5015	−0.385005
$745$	19.2845	0.706529
$746$	1.48782	0.0544728
$747$	−9.38266	−0.343293
$748$	0	0
$749$	−2.34127	−0.0855481
$750$	−1.00000	−0.0365148
$751$	18.5232	0.675921	0.337961	−	0.941160i	$-0.390263\pi$
$751$	18.5232	0.675921	0.337961	+	0.941160i	$0.390263\pi$
$752$	4.88874	0.178274
$753$	1.20918	0.0440651
$754$	−9.10772	−0.331684
$755$	−4.89015	−0.177971
$756$	−1.06098	−0.0385874
$757$	−36.5609	−1.32883	−0.664415	−	0.747364i	$-0.731319\pi$
$757$	−36.5609	−1.32883	−0.664415	+	0.747364i	$0.731319\pi$
$758$	−21.1913	−0.769703
$759$	−30.2485	−1.09795
$760$	−7.34218	−0.266329
$761$	34.9500	1.26694	0.633469	−	0.773768i	$-0.281631\pi$
$761$	34.9500	1.26694	0.633469	+	0.773768i	$0.281631\pi$
$762$	−5.84808	−0.211854
$763$	3.69934	0.133925
$764$	18.3624	0.664329
$765$	0	0
$766$	25.5964	0.924834
$767$	21.4227	0.773528
$768$	−1.00000	−0.0360844
$769$	7.41622	0.267436	0.133718	−	0.991019i	$-0.457308\pi$
$769$	7.41622	0.267436	0.133718	+	0.991019i	$0.457308\pi$
$770$	−3.69766	−0.133254
$771$	−6.77280	−0.243917
$772$	−14.4909	−0.521539
$773$	−27.7165	−0.996894	−0.498447	−	0.866920i	$-0.666096\pi$
$773$	−27.7165	−0.996894	−0.498447	+	0.866920i	$0.666096\pi$
$774$	7.02634	0.252557
$775$	−10.5015	−0.377226
$776$	2.93760	0.105454
$777$	5.86798	0.210513
$778$	22.3422	0.801006
$779$	20.7581	0.743738
$780$	−3.47117	−0.124288
$781$	−43.0416	−1.54015
$782$	0	0
$783$	2.62382	0.0937678
$784$	−5.87433	−0.209797
$785$	−12.9056	−0.460621
$786$	6.31633	0.225296
$787$	8.02886	0.286198	0.143099	−	0.989708i	$-0.454293\pi$
$787$	8.02886	0.286198	0.143099	+	0.989708i	$0.454293\pi$
$788$	16.1998	0.577094
$789$	3.14118	0.111829
$790$	−4.35423	−0.154916
$791$	6.69387	0.238007
$792$	3.48514	0.123839
$793$	−34.1079	−1.21121
$794$	11.3264	0.401959
$795$	9.86766	0.349970
$796$	9.02900	0.320025
$797$	2.75721	0.0976655	0.0488327	−	0.998807i	$-0.484450\pi$
$797$	2.75721	0.0976655	0.0488327	+	0.998807i	$0.484450\pi$
$798$	−7.78989	−0.275759
$799$	0	0
$800$	−1.00000	−0.0353553
$801$	5.59982	0.197860
$802$	16.2608	0.574188
$803$	33.4646	1.18094
$804$	12.6016	0.444423
$805$	9.20850	0.324557
$806$	−36.4525	−1.28399
$807$	6.92637	0.243820
$808$	−18.4185	−0.647959
$809$	−28.4175	−0.999106	−0.499553	−	0.866283i	$-0.666502\pi$
$809$	−28.4175	−0.999106	−0.499553	+	0.866283i	$0.666502\pi$
$810$	1.00000	0.0351364
$811$	31.9887	1.12327	0.561637	−	0.827384i	$-0.310172\pi$
$811$	31.9887	1.12327	0.561637	+	0.827384i	$0.310172\pi$
$812$	−2.78382	−0.0976928
$813$	−20.3367	−0.713240
$814$	−19.2754	−0.675602
$815$	23.3888	0.819276
$816$	0	0
$817$	51.5887	1.80486
$818$	−3.89097	−0.136044
$819$	−3.68283	−0.128688
$820$	2.82724	0.0987316
$821$	28.0998	0.980690	0.490345	−	0.871529i	$-0.336871\pi$
$821$	28.0998	0.980690	0.490345	+	0.871529i	$0.336871\pi$
$822$	8.53613	0.297732
$823$	11.2257	0.391302	0.195651	−	0.980674i	$-0.437318\pi$
$823$	11.2257	0.391302	0.195651	+	0.980674i	$0.437318\pi$
$824$	−19.0667	−0.664220
$825$	3.48514	0.121337
$826$	6.54794	0.227832
$827$	−26.0919	−0.907304	−0.453652	−	0.891179i	$-0.649879\pi$
$827$	−26.0919	−0.907304	−0.453652	+	0.891179i	$0.649879\pi$
$828$	−8.67926	−0.301625
$829$	2.95948	0.102787	0.0513935	−	0.998678i	$-0.483634\pi$
$829$	2.95948	0.102787	0.0513935	+	0.998678i	$0.483634\pi$
$830$	−9.38266	−0.325677
$831$	3.34799	0.116140
$832$	−3.47117	−0.120341
$833$	0	0
$834$	−16.0849	−0.556975
$835$	23.6763	0.819351
$836$	25.5885	0.884999
$837$	10.5015	0.362986
$838$	−24.6933	−0.853016
$839$	54.3914	1.87780	0.938900	−	0.344191i	$-0.111847\pi$
$839$	54.3914	1.87780	0.938900	+	0.344191i	$0.111847\pi$
$840$	−1.06098	−0.0366072
$841$	−22.1156	−0.762605
$842$	−12.0894	−0.416627
$843$	−11.4256	−0.393517
$844$	8.64852	0.297694
$845$	0.951010	0.0327158
$846$	−4.88874	−0.168078
$847$	1.21611	0.0417860
$848$	9.86766	0.338857
$849$	10.5281	0.361324
$850$	0	0
$851$	48.0027	1.64551
$852$	−12.3500	−0.423105
$853$	−15.1652	−0.519247	−0.259624	−	0.965710i	$-0.583598\pi$
$853$	−15.1652	−0.519247	−0.259624	+	0.965710i	$0.583598\pi$
$854$	−10.4252	−0.356744
$855$	7.34218	0.251097
$856$	2.20671	0.0754237
$857$	−29.9823	−1.02418	−0.512088	−	0.858933i	$-0.671128\pi$
$857$	−29.9823	−1.02418	−0.512088	+	0.858933i	$0.671128\pi$
$858$	12.0975	0.413002
$859$	15.6309	0.533320	0.266660	−	0.963791i	$-0.414080\pi$
$859$	15.6309	0.533320	0.266660	+	0.963791i	$0.414080\pi$
$860$	7.02634	0.239596
$861$	2.99964	0.102228
$862$	−3.55059	−0.120934
$863$	−41.4014	−1.40932	−0.704660	−	0.709545i	$-0.748901\pi$
$863$	−41.4014	−1.40932	−0.704660	+	0.709545i	$0.748901\pi$
$864$	1.00000	0.0340207
$865$	6.09008	0.207069
$866$	−18.3807	−0.624602
$867$	0	0
$868$	−11.1419	−0.378180
$869$	15.1751	0.514780
$870$	2.62382	0.0889559
$871$	43.7421	1.48214
$872$	−3.48673	−0.118076
$873$	−2.93760	−0.0994227
$874$	−63.7247	−2.15552
$875$	−1.06098	−0.0358676
$876$	9.60208	0.324424
$877$	−24.3399	−0.821902	−0.410951	−	0.911658i	$-0.634803\pi$
$877$	−24.3399	−0.821902	−0.410951	+	0.911658i	$0.634803\pi$
$878$	5.19771	0.175414
$879$	17.7898	0.600036
$880$	3.48514	0.117484
$881$	−38.6027	−1.30056	−0.650278	−	0.759696i	$-0.725348\pi$
$881$	−38.6027	−1.30056	−0.650278	+	0.759696i	$0.725348\pi$
$882$	5.87433	0.197799
$883$	−0.321515	−0.0108199	−0.00540993	−	0.999985i	$-0.501722\pi$
$883$	−0.321515	−0.0108199	−0.00540993	+	0.999985i	$0.501722\pi$
$884$	0	0
$885$	−6.17161	−0.207456
$886$	−16.2504	−0.545942
$887$	16.9896	0.570454	0.285227	−	0.958460i	$-0.407931\pi$
$887$	16.9896	0.570454	0.285227	+	0.958460i	$0.407931\pi$
$888$	−5.53073	−0.185599
$889$	−6.20468	−0.208098
$890$	5.59982	0.187706
$891$	−3.48514	−0.116757
$892$	15.5060	0.519180
$893$	−35.8940	−1.20115
$894$	−19.2845	−0.644970
$895$	−5.18370	−0.173272
$896$	−1.06098	−0.0354448
$897$	−30.1272	−1.00592
$898$	−10.2565	−0.342263
$899$	27.5542	0.918982
$900$	1.00000	0.0333333
$901$	0	0
$902$	−9.85335	−0.328081
$903$	7.45479	0.248080
$904$	−6.30915	−0.209839
$905$	−19.8446	−0.659657
$906$	4.89015	0.162464
$907$	11.3586	0.377155	0.188578	−	0.982058i	$-0.439612\pi$
$907$	11.3586	0.377155	0.188578	+	0.982058i	$0.439612\pi$
$908$	−9.48917	−0.314909
$909$	18.4185	0.610901
$910$	−3.68283	−0.122085
$911$	18.6520	0.617969	0.308984	−	0.951067i	$-0.400011\pi$
$911$	18.6520	0.617969	0.308984	+	0.951067i	$0.400011\pi$
$912$	7.34218	0.243124
$913$	32.6999	1.08221
$914$	−14.7915	−0.489258
$915$	9.82606	0.324840
$916$	−8.83945	−0.292064
$917$	6.70148	0.221303
$918$	0	0
$919$	−18.8928	−0.623217	−0.311608	−	0.950211i	$-0.600868\pi$
$919$	−18.8928	−0.623217	−0.311608	+	0.950211i	$0.600868\pi$
$920$	−8.67926	−0.286147
$921$	26.8792	0.885701
$922$	−6.26556	−0.206345
$923$	−42.8690	−1.41105
$924$	3.69766	0.121644
$925$	−5.53073	−0.181849
$926$	−19.1392	−0.628952
$927$	19.0667	0.626233
$928$	2.62382	0.0861312
$929$	−40.1377	−1.31688	−0.658438	−	0.752635i	$-0.728782\pi$
$929$	−40.1377	−1.31688	−0.658438	+	0.752635i	$0.728782\pi$
$930$	10.5015	0.344359
$931$	43.1304	1.41354
$932$	−5.94487	−0.194731
$933$	16.9305	0.554280
$934$	−12.6796	−0.414890
$935$	0	0
$936$	3.47117	0.113459
$937$	55.3629	1.80863	0.904314	−	0.426867i	$-0.140383\pi$
$937$	55.3629	1.80863	0.904314	+	0.426867i	$0.140383\pi$
$938$	13.3700	0.436545
$939$	0.360312	0.0117583
$940$	−4.88874	−0.159453
$941$	−39.9932	−1.30374	−0.651870	−	0.758331i	$-0.726015\pi$
$941$	−39.9932	−1.30374	−0.651870	+	0.758331i	$0.726015\pi$
$942$	12.9056	0.420488
$943$	24.5384	0.799080
$944$	−6.17161	−0.200869
$945$	1.06098	0.0345136
$946$	−24.4878	−0.796167
$947$	−37.5092	−1.21889	−0.609443	−	0.792830i	$-0.708607\pi$
$947$	−37.5092	−1.21889	−0.609443	+	0.792830i	$0.708607\pi$
$948$	4.35423	0.141419
$949$	33.3304	1.08195
$950$	7.34218	0.238212
$951$	12.5816	0.407987
$952$	0	0
$953$	21.6730	0.702058	0.351029	−	0.936365i	$-0.385832\pi$
$953$	21.6730	0.702058	0.351029	+	0.936365i	$0.385832\pi$
$954$	−9.86766	−0.319477
$955$	−18.3624	−0.594194
$956$	−0.783103	−0.0253274
$957$	−9.14440	−0.295596
$958$	3.99033	0.128922
$959$	9.05664	0.292454
$960$	1.00000	0.0322749
$961$	79.2821	2.55749
$962$	−19.1981	−0.618971
$963$	−2.20671	−0.0711102
$964$	−16.2870	−0.524570
$965$	14.4909	0.466478
$966$	−9.20850	−0.296279
$967$	−43.8318	−1.40954	−0.704768	−	0.709438i	$-0.748949\pi$
$967$	−43.8318	−1.40954	−0.704768	+	0.709438i	$0.748949\pi$
$968$	−1.14622	−0.0368408
$969$	0	0
$970$	−2.93760	−0.0943207
$971$	13.5720	0.435545	0.217773	−	0.976000i	$-0.430121\pi$
$971$	13.5720	0.435545	0.217773	+	0.976000i	$0.430121\pi$
$972$	−1.00000	−0.0320750
$973$	−17.0657	−0.547102
$974$	−1.92919	−0.0618151
$975$	3.47117	0.111166
$976$	9.82606	0.314525
$977$	0.273705	0.00875661	0.00437831	−	0.999990i	$-0.498606\pi$
$977$	0.273705	0.00875661	0.00437831	+	0.999990i	$0.498606\pi$
$978$	−23.3888	−0.747893
$979$	−19.5162	−0.623740
$980$	5.87433	0.187648
$981$	3.48673	0.111323
$982$	37.9150	1.20992
$983$	−20.1424	−0.642444	−0.321222	−	0.947004i	$-0.604094\pi$
$983$	−20.1424	−0.642444	−0.321222	+	0.947004i	$0.604094\pi$
$984$	−2.82724	−0.0901292
$985$	−16.1998	−0.516168
$986$	0	0
$987$	−5.18684	−0.165099
$988$	25.4859	0.810815
$989$	60.9835	1.93916
$990$	−3.48514	−0.110765
$991$	−18.0600	−0.573694	−0.286847	−	0.957976i	$-0.592607\pi$
$991$	−18.0600	−0.573694	−0.286847	+	0.957976i	$0.592607\pi$
$992$	10.5015	0.333424
$993$	10.6216	0.337066
$994$	−13.1031	−0.415605
$995$	−9.02900	−0.286239
$996$	9.38266	0.297301
$997$	28.7727	0.911241	0.455620	−	0.890174i	$-0.349417\pi$
$997$	28.7727	0.911241	0.455620	+	0.890174i	$0.349417\pi$
$998$	30.1433	0.954169
$999$	5.53073	0.174985

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8670.2.a.cj.1.6		8
17.5	odd	16		510.2.u.c.331.3	yes	16
17.7	odd	16		510.2.u.c.151.3	✓	16
17.16	even	2		8670.2.a.ck.1.3		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
510.2.u.c.151.3	✓	16	17.7	odd	16
510.2.u.c.331.3	yes	16	17.5	odd	16
8670.2.a.cj.1.6		8	1.1	even	1	trivial
8670.2.a.ck.1.3		8	17.16	even	2

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:	\( N \)	\(=\)	\( 8670 = 2 \cdot 3 \cdot 5 \cdot 17^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8670.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{6} +1.06098 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{6} +1.06098 q^{7} -1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{10} -3.48514 q^{11} -1.00000 q^{12} -3.47117 q^{13} -1.06098 q^{14} +1.00000 q^{15} +1.00000 q^{16} -1.00000 q^{18} -7.34218 q^{19} -1.00000 q^{20} -1.06098 q^{21} +3.48514 q^{22} -8.67926 q^{23} +1.00000 q^{24} +1.00000 q^{25} +3.47117 q^{26} -1.00000 q^{27} +1.06098 q^{28} -2.62382 q^{29} -1.00000 q^{30} -10.5015 q^{31} -1.00000 q^{32} +3.48514 q^{33} -1.06098 q^{35} +1.00000 q^{36} -5.53073 q^{37} +7.34218 q^{38} +3.47117 q^{39} +1.00000 q^{40} -2.82724 q^{41} +1.06098 q^{42} -7.02634 q^{43} -3.48514 q^{44} -1.00000 q^{45} +8.67926 q^{46} +4.88874 q^{47} -1.00000 q^{48} -5.87433 q^{49} -1.00000 q^{50} -3.47117 q^{52} +9.86766 q^{53} +1.00000 q^{54} +3.48514 q^{55} -1.06098 q^{56} +7.34218 q^{57} +2.62382 q^{58} -6.17161 q^{59} +1.00000 q^{60} +9.82606 q^{61} +10.5015 q^{62} +1.06098 q^{63} +1.00000 q^{64} +3.47117 q^{65} -3.48514 q^{66} -12.6016 q^{67} +8.67926 q^{69} +1.06098 q^{70} +12.3500 q^{71} -1.00000 q^{72} -9.60208 q^{73} +5.53073 q^{74} -1.00000 q^{75} -7.34218 q^{76} -3.69766 q^{77} -3.47117 q^{78} -4.35423 q^{79} -1.00000 q^{80} +1.00000 q^{81} +2.82724 q^{82} -9.38266 q^{83} -1.06098 q^{84} +7.02634 q^{86} +2.62382 q^{87} +3.48514 q^{88} +5.59982 q^{89} +1.00000 q^{90} -3.68283 q^{91} -8.67926 q^{92} +10.5015 q^{93} -4.88874 q^{94} +7.34218 q^{95} +1.00000 q^{96} -2.93760 q^{97} +5.87433 q^{98} -3.48514 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{2} - 8 q^{3} + 8 q^{4} - 8 q^{5} + 8 q^{6} - 8 q^{7} - 8 q^{8} + 8 q^{9}+O(q^{10}) \) 8 * q - 8 * q^2 - 8 * q^3 + 8 * q^4 - 8 * q^5 + 8 * q^6 - 8 * q^7 - 8 * q^8 + 8 * q^9	\( 8 q - 8 q^{2} - 8 q^{3} + 8 q^{4} - 8 q^{5} + 8 q^{6} - 8 q^{7} - 8 q^{8} + 8 q^{9} + 8 q^{10} + 8 q^{11} - 8 q^{12} + 8 q^{14} + 8 q^{15} + 8 q^{16} - 8 q^{18} - 8 q^{20} + 8 q^{21} - 8 q^{22} + 8 q^{24} + 8 q^{25} - 8 q^{27} - 8 q^{28} + 8 q^{29} - 8 q^{30} - 8 q^{31} - 8 q^{32} - 8 q^{33} + 8 q^{35} + 8 q^{36} - 32 q^{37} + 8 q^{40} + 8 q^{41} - 8 q^{42} - 8 q^{43} + 8 q^{44} - 8 q^{45} - 8 q^{48} + 40 q^{49} - 8 q^{50} + 16 q^{53} + 8 q^{54} - 8 q^{55} + 8 q^{56} - 8 q^{58} + 32 q^{59} + 8 q^{60} - 8 q^{61} + 8 q^{62} - 8 q^{63} + 8 q^{64} + 8 q^{66} - 8 q^{67} - 8 q^{70} + 24 q^{71} - 8 q^{72} - 40 q^{73} + 32 q^{74} - 8 q^{75} + 16 q^{77} - 24 q^{79} - 8 q^{80} + 8 q^{81} - 8 q^{82} - 16 q^{83} + 8 q^{84} + 8 q^{86} - 8 q^{87} - 8 q^{88} + 48 q^{89} + 8 q^{90} - 24 q^{91} + 8 q^{93} + 8 q^{96} - 24 q^{97} - 40 q^{98} + 8 q^{99}+O(q^{100}) \) 8 * q - 8 * q^2 - 8 * q^3 + 8 * q^4 - 8 * q^5 + 8 * q^6 - 8 * q^7 - 8 * q^8 + 8 * q^9 + 8 * q^10 + 8 * q^11 - 8 * q^12 + 8 * q^14 + 8 * q^15 + 8 * q^16 - 8 * q^18 - 8 * q^20 + 8 * q^21 - 8 * q^22 + 8 * q^24 + 8 * q^25 - 8 * q^27 - 8 * q^28 + 8 * q^29 - 8 * q^30 - 8 * q^31 - 8 * q^32 - 8 * q^33 + 8 * q^35 + 8 * q^36 - 32 * q^37 + 8 * q^40 + 8 * q^41 - 8 * q^42 - 8 * q^43 + 8 * q^44 - 8 * q^45 - 8 * q^48 + 40 * q^49 - 8 * q^50 + 16 * q^53 + 8 * q^54 - 8 * q^55 + 8 * q^56 - 8 * q^58 + 32 * q^59 + 8 * q^60 - 8 * q^61 + 8 * q^62 - 8 * q^63 + 8 * q^64 + 8 * q^66 - 8 * q^67 - 8 * q^70 + 24 * q^71 - 8 * q^72 - 40 * q^73 + 32 * q^74 - 8 * q^75 + 16 * q^77 - 24 * q^79 - 8 * q^80 + 8 * q^81 - 8 * q^82 - 16 * q^83 + 8 * q^84 + 8 * q^86 - 8 * q^87 - 8 * q^88 + 48 * q^89 + 8 * q^90 - 24 * q^91 + 8 * q^93 + 8 * q^96 - 24 * q^97 - 40 * q^98 + 8 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more