LMFDB - Embedded newform 8670.2.a.bi.1.1

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:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 510)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.41421$ 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)

$f(q)$	$=$	$q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{6} -2.82843 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{6} -2.82843 q^{7} +1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{10} +0.828427 q^{11} -1.00000 q^{12} -2.58579 q^{13} -2.82843 q^{14} -1.00000 q^{15} +1.00000 q^{16} +1.00000 q^{18} -6.82843 q^{19} +1.00000 q^{20} +2.82843 q^{21} +0.828427 q^{22} +2.24264 q^{23} -1.00000 q^{24} +1.00000 q^{25} -2.58579 q^{26} -1.00000 q^{27} -2.82843 q^{28} +8.00000 q^{29} -1.00000 q^{30} +4.58579 q^{31} +1.00000 q^{32} -0.828427 q^{33} -2.82843 q^{35} +1.00000 q^{36} +7.65685 q^{37} -6.82843 q^{38} +2.58579 q^{39} +1.00000 q^{40} -0.828427 q^{41} +2.82843 q^{42} -5.07107 q^{43} +0.828427 q^{44} +1.00000 q^{45} +2.24264 q^{46} -8.00000 q^{47} -1.00000 q^{48} +1.00000 q^{49} +1.00000 q^{50} -2.58579 q^{52} -0.343146 q^{53} -1.00000 q^{54} +0.828427 q^{55} -2.82843 q^{56} +6.82843 q^{57} +8.00000 q^{58} +5.07107 q^{59} -1.00000 q^{60} +12.8284 q^{61} +4.58579 q^{62} -2.82843 q^{63} +1.00000 q^{64} -2.58579 q^{65} -0.828427 q^{66} -7.41421 q^{67} -2.24264 q^{69} -2.82843 q^{70} +5.65685 q^{71} +1.00000 q^{72} -10.0000 q^{73} +7.65685 q^{74} -1.00000 q^{75} -6.82843 q^{76} -2.34315 q^{77} +2.58579 q^{78} -17.0711 q^{79} +1.00000 q^{80} +1.00000 q^{81} -0.828427 q^{82} +2.82843 q^{84} -5.07107 q^{86} -8.00000 q^{87} +0.828427 q^{88} -12.1421 q^{89} +1.00000 q^{90} +7.31371 q^{91} +2.24264 q^{92} -4.58579 q^{93} -8.00000 q^{94} -6.82843 q^{95} -1.00000 q^{96} -15.6569 q^{97} +1.00000 q^{98} +0.828427 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{2} - 2 q^{3} + 2 q^{4} + 2 q^{5} - 2 q^{6} + 2 q^{8} + 2 q^{9}+O(q^{10}) $ 2 * q + 2 * q^2 - 2 * q^3 + 2 * q^4 + 2 * q^5 - 2 * q^6 + 2 * q^8 + 2 * q^9	$ 2 q + 2 q^{2} - 2 q^{3} + 2 q^{4} + 2 q^{5} - 2 q^{6} + 2 q^{8} + 2 q^{9} + 2 q^{10} - 4 q^{11} - 2 q^{12} - 8 q^{13} - 2 q^{15} + 2 q^{16} + 2 q^{18} - 8 q^{19} + 2 q^{20} - 4 q^{22} - 4 q^{23} - 2 q^{24} + 2 q^{25} - 8 q^{26} - 2 q^{27} + 16 q^{29} - 2 q^{30} + 12 q^{31} + 2 q^{32} + 4 q^{33} + 2 q^{36} + 4 q^{37} - 8 q^{38} + 8 q^{39} + 2 q^{40} + 4 q^{41} + 4 q^{43} - 4 q^{44} + 2 q^{45} - 4 q^{46} - 16 q^{47} - 2 q^{48} + 2 q^{49} + 2 q^{50} - 8 q^{52} - 12 q^{53} - 2 q^{54} - 4 q^{55} + 8 q^{57} + 16 q^{58} - 4 q^{59} - 2 q^{60} + 20 q^{61} + 12 q^{62} + 2 q^{64} - 8 q^{65} + 4 q^{66} - 12 q^{67} + 4 q^{69} + 2 q^{72} - 20 q^{73} + 4 q^{74} - 2 q^{75} - 8 q^{76} - 16 q^{77} + 8 q^{78} - 20 q^{79} + 2 q^{80} + 2 q^{81} + 4 q^{82} + 4 q^{86} - 16 q^{87} - 4 q^{88} + 4 q^{89} + 2 q^{90} - 8 q^{91} - 4 q^{92} - 12 q^{93} - 16 q^{94} - 8 q^{95} - 2 q^{96} - 20 q^{97} + 2 q^{98} - 4 q^{99}+O(q^{100}) $ 2 * q + 2 * q^2 - 2 * q^3 + 2 * q^4 + 2 * q^5 - 2 * q^6 + 2 * q^8 + 2 * q^9 + 2 * q^10 - 4 * q^11 - 2 * q^12 - 8 * q^13 - 2 * q^15 + 2 * q^16 + 2 * q^18 - 8 * q^19 + 2 * q^20 - 4 * q^22 - 4 * q^23 - 2 * q^24 + 2 * q^25 - 8 * q^26 - 2 * q^27 + 16 * q^29 - 2 * q^30 + 12 * q^31 + 2 * q^32 + 4 * q^33 + 2 * q^36 + 4 * q^37 - 8 * q^38 + 8 * q^39 + 2 * q^40 + 4 * q^41 + 4 * q^43 - 4 * q^44 + 2 * q^45 - 4 * q^46 - 16 * q^47 - 2 * q^48 + 2 * q^49 + 2 * q^50 - 8 * q^52 - 12 * q^53 - 2 * q^54 - 4 * q^55 + 8 * q^57 + 16 * q^58 - 4 * q^59 - 2 * q^60 + 20 * q^61 + 12 * q^62 + 2 * q^64 - 8 * q^65 + 4 * q^66 - 12 * q^67 + 4 * q^69 + 2 * q^72 - 20 * q^73 + 4 * q^74 - 2 * q^75 - 8 * q^76 - 16 * q^77 + 8 * q^78 - 20 * q^79 + 2 * q^80 + 2 * q^81 + 4 * q^82 + 4 * q^86 - 16 * q^87 - 4 * q^88 + 4 * q^89 + 2 * q^90 - 8 * q^91 - 4 * q^92 - 12 * q^93 - 16 * q^94 - 8 * q^95 - 2 * q^96 - 20 * q^97 + 2 * q^98 - 4 * 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
$7$	−2.82843	−1.06904	−0.534522	−	0.845154i	$-0.679509\pi$
$7$	−2.82843	−1.06904	−0.534522	+	0.845154i	$0.679509\pi$
$8$	1.00000	0.353553
$9$	1.00000	0.333333
$10$	1.00000	0.316228
$11$	0.828427	0.249780	0.124890	−	0.992171i	$-0.460142\pi$
$11$	0.828427	0.249780	0.124890	+	0.992171i	$0.460142\pi$
$12$	−1.00000	−0.288675
$13$	−2.58579	−0.717168	−0.358584	−	0.933497i	$-0.616740\pi$
$13$	−2.58579	−0.717168	−0.358584	+	0.933497i	$0.616740\pi$
$14$	−2.82843	−0.755929
$15$	−1.00000	−0.258199
$16$	1.00000	0.250000
$17$	0	0
$17$	0	0
$18$	1.00000	0.235702
$19$	−6.82843	−1.56655	−0.783274	−	0.621676i	$-0.786452\pi$
$19$	−6.82843	−1.56655	−0.783274	+	0.621676i	$0.786452\pi$
$20$	1.00000	0.223607
$21$	2.82843	0.617213
$22$	0.828427	0.176621
$23$	2.24264	0.467623	0.233811	−	0.972282i	$-0.424880\pi$
$23$	2.24264	0.467623	0.233811	+	0.972282i	$0.424880\pi$
$24$	−1.00000	−0.204124
$25$	1.00000	0.200000
$26$	−2.58579	−0.507114
$27$	−1.00000	−0.192450
$28$	−2.82843	−0.534522
$29$	8.00000	1.48556	0.742781	−	0.669534i	$-0.233506\pi$
$29$	8.00000	1.48556	0.742781	+	0.669534i	$0.233506\pi$
$30$	−1.00000	−0.182574
$31$	4.58579	0.823632	0.411816	−	0.911267i	$-0.364895\pi$
$31$	4.58579	0.823632	0.411816	+	0.911267i	$0.364895\pi$
$32$	1.00000	0.176777
$33$	−0.828427	−0.144211
$34$	0	0
$35$	−2.82843	−0.478091
$36$	1.00000	0.166667
$37$	7.65685	1.25878	0.629390	−	0.777090i	$-0.283305\pi$
$37$	7.65685	1.25878	0.629390	+	0.777090i	$0.283305\pi$
$38$	−6.82843	−1.10772
$39$	2.58579	0.414057
$40$	1.00000	0.158114
$41$	−0.828427	−0.129379	−0.0646893	−	0.997905i	$-0.520606\pi$
$41$	−0.828427	−0.129379	−0.0646893	+	0.997905i	$0.520606\pi$
$42$	2.82843	0.436436
$43$	−5.07107	−0.773331	−0.386665	−	0.922220i	$-0.626373\pi$
$43$	−5.07107	−0.773331	−0.386665	+	0.922220i	$0.626373\pi$
$44$	0.828427	0.124890
$45$	1.00000	0.149071
$46$	2.24264	0.330659
$47$	−8.00000	−1.16692	−0.583460	−	0.812142i	$-0.698301\pi$
$47$	−8.00000	−1.16692	−0.583460	+	0.812142i	$0.698301\pi$
$48$	−1.00000	−0.144338
$49$	1.00000	0.142857
$50$	1.00000	0.141421
$51$	0	0
$52$	−2.58579	−0.358584
$53$	−0.343146	−0.0471347	−0.0235673	−	0.999722i	$-0.507502\pi$
$53$	−0.343146	−0.0471347	−0.0235673	+	0.999722i	$0.507502\pi$
$54$	−1.00000	−0.136083
$55$	0.828427	0.111705
$56$	−2.82843	−0.377964
$57$	6.82843	0.904447
$58$	8.00000	1.05045
$59$	5.07107	0.660197	0.330098	−	0.943947i	$-0.392918\pi$
$59$	5.07107	0.660197	0.330098	+	0.943947i	$0.392918\pi$
$60$	−1.00000	−0.129099
$61$	12.8284	1.64251	0.821256	−	0.570560i	$-0.193274\pi$
$61$	12.8284	1.64251	0.821256	+	0.570560i	$0.193274\pi$
$62$	4.58579	0.582395
$63$	−2.82843	−0.356348
$64$	1.00000	0.125000
$65$	−2.58579	−0.320727
$66$	−0.828427	−0.101972
$67$	−7.41421	−0.905790	−0.452895	−	0.891564i	$-0.649609\pi$
$67$	−7.41421	−0.905790	−0.452895	+	0.891564i	$0.649609\pi$
$68$	0	0
$69$	−2.24264	−0.269982
$70$	−2.82843	−0.338062
$71$	5.65685	0.671345	0.335673	−	0.941979i	$-0.391036\pi$
$71$	5.65685	0.671345	0.335673	+	0.941979i	$0.391036\pi$
$72$	1.00000	0.117851
$73$	−10.0000	−1.17041	−0.585206	−	0.810885i	$-0.698986\pi$
$73$	−10.0000	−1.17041	−0.585206	+	0.810885i	$0.698986\pi$
$74$	7.65685	0.890091
$75$	−1.00000	−0.115470
$76$	−6.82843	−0.783274
$77$	−2.34315	−0.267026
$78$	2.58579	0.292783
$79$	−17.0711	−1.92065	−0.960323	−	0.278892i	$-0.910033\pi$
$79$	−17.0711	−1.92065	−0.960323	+	0.278892i	$0.910033\pi$
$80$	1.00000	0.111803
$81$	1.00000	0.111111
$82$	−0.828427	−0.0914845
$83$	0	0		−	1.00000i	$-0.5\pi$
$83$	0	0			1.00000i	$0.5\pi$
$84$	2.82843	0.308607
$85$	0	0
$86$	−5.07107	−0.546827
$87$	−8.00000	−0.857690
$88$	0.828427	0.0883106
$89$	−12.1421	−1.28706	−0.643532	−	0.765419i	$-0.722532\pi$
$89$	−12.1421	−1.28706	−0.643532	+	0.765419i	$0.722532\pi$
$90$	1.00000	0.105409
$91$	7.31371	0.766685
$92$	2.24264	0.233811
$93$	−4.58579	−0.475524
$94$	−8.00000	−0.825137
$95$	−6.82843	−0.700582
$96$	−1.00000	−0.102062
$97$	−15.6569	−1.58971	−0.794856	−	0.606798i	$-0.792454\pi$
$97$	−15.6569	−1.58971	−0.794856	+	0.606798i	$0.792454\pi$
$98$	1.00000	0.101015
$99$	0.828427	0.0832601
$100$	1.00000	0.100000
$101$	8.72792	0.868461	0.434230	−	0.900802i	$-0.357020\pi$
$101$	8.72792	0.868461	0.434230	+	0.900802i	$0.357020\pi$
$102$	0	0
$103$	−6.82843	−0.672825	−0.336412	−	0.941715i	$-0.609214\pi$
$103$	−6.82843	−0.672825	−0.336412	+	0.941715i	$0.609214\pi$
$104$	−2.58579	−0.253557
$105$	2.82843	0.276026
$106$	−0.343146	−0.0333293
$107$	−8.00000	−0.773389	−0.386695	−	0.922208i	$-0.626383\pi$
$107$	−8.00000	−0.773389	−0.386695	+	0.922208i	$0.626383\pi$
$108$	−1.00000	−0.0962250
$109$	−15.6569	−1.49965	−0.749827	−	0.661634i	$-0.769863\pi$
$109$	−15.6569	−1.49965	−0.749827	+	0.661634i	$0.769863\pi$
$110$	0.828427	0.0789874
$111$	−7.65685	−0.726756
$112$	−2.82843	−0.267261
$113$	−1.89949	−0.178689	−0.0893447	−	0.996001i	$-0.528477\pi$
$113$	−1.89949	−0.178689	−0.0893447	+	0.996001i	$0.528477\pi$
$114$	6.82843	0.639541
$115$	2.24264	0.209127
$116$	8.00000	0.742781
$117$	−2.58579	−0.239056
$118$	5.07107	0.466830
$119$	0	0
$120$	−1.00000	−0.0912871
$121$	−10.3137	−0.937610
$122$	12.8284	1.16143
$123$	0.828427	0.0746968
$124$	4.58579	0.411816
$125$	1.00000	0.0894427
$126$	−2.82843	−0.251976
$127$	−8.00000	−0.709885	−0.354943	−	0.934888i	$-0.615500\pi$
$127$	−8.00000	−0.709885	−0.354943	+	0.934888i	$0.615500\pi$
$128$	1.00000	0.0883883
$129$	5.07107	0.446483
$130$	−2.58579	−0.226788
$131$	−4.00000	−0.349482	−0.174741	−	0.984614i	$-0.555909\pi$
$131$	−4.00000	−0.349482	−0.174741	+	0.984614i	$0.555909\pi$
$132$	−0.828427	−0.0721053
$133$	19.3137	1.67471
$134$	−7.41421	−0.640490
$135$	−1.00000	−0.0860663
$136$	0	0
$137$	12.9706	1.10815	0.554075	−	0.832467i	$-0.313072\pi$
$137$	12.9706	1.10815	0.554075	+	0.832467i	$0.313072\pi$
$138$	−2.24264	−0.190906
$139$	−6.34315	−0.538019	−0.269009	−	0.963138i	$-0.586696\pi$
$139$	−6.34315	−0.538019	−0.269009	+	0.963138i	$0.586696\pi$
$140$	−2.82843	−0.239046
$141$	8.00000	0.673722
$142$	5.65685	0.474713
$143$	−2.14214	−0.179134
$144$	1.00000	0.0833333
$145$	8.00000	0.664364
$146$	−10.0000	−0.827606
$147$	−1.00000	−0.0824786
$148$	7.65685	0.629390
$149$	18.3848	1.50614	0.753070	−	0.657941i	$-0.228572\pi$
$149$	18.3848	1.50614	0.753070	+	0.657941i	$0.228572\pi$
$150$	−1.00000	−0.0816497
$151$	−11.1716	−0.909130	−0.454565	−	0.890714i	$-0.650205\pi$
$151$	−11.1716	−0.909130	−0.454565	+	0.890714i	$0.650205\pi$
$152$	−6.82843	−0.553859
$153$	0	0
$154$	−2.34315	−0.188816
$155$	4.58579	0.368339
$156$	2.58579	0.207029
$157$	−14.5858	−1.16407	−0.582036	−	0.813163i	$-0.697744\pi$
$157$	−14.5858	−1.16407	−0.582036	+	0.813163i	$0.697744\pi$
$158$	−17.0711	−1.35810
$159$	0.343146	0.0272132
$160$	1.00000	0.0790569
$161$	−6.34315	−0.499910
$162$	1.00000	0.0785674
$163$	−0.828427	−0.0648874	−0.0324437	−	0.999474i	$-0.510329\pi$
$163$	−0.828427	−0.0648874	−0.0324437	+	0.999474i	$0.510329\pi$
$164$	−0.828427	−0.0646893
$165$	−0.828427	−0.0644930
$166$	0	0
$167$	−3.41421	−0.264200	−0.132100	−	0.991236i	$-0.542172\pi$
$167$	−3.41421	−0.264200	−0.132100	+	0.991236i	$0.542172\pi$
$168$	2.82843	0.218218
$169$	−6.31371	−0.485670
$170$	0	0
$171$	−6.82843	−0.522183
$172$	−5.07107	−0.386665
$173$	7.65685	0.582140	0.291070	−	0.956702i	$-0.405989\pi$
$173$	7.65685	0.582140	0.291070	+	0.956702i	$0.405989\pi$
$174$	−8.00000	−0.606478
$175$	−2.82843	−0.213809
$176$	0.828427	0.0624450
$177$	−5.07107	−0.381165
$178$	−12.1421	−0.910092
$179$	19.8995	1.48736	0.743679	−	0.668537i	$-0.233079\pi$
$179$	19.8995	1.48736	0.743679	+	0.668537i	$0.233079\pi$
$180$	1.00000	0.0745356
$181$	−21.7990	−1.62031	−0.810153	−	0.586218i	$-0.800616\pi$
$181$	−21.7990	−1.62031	−0.810153	+	0.586218i	$0.800616\pi$
$182$	7.31371	0.542128
$183$	−12.8284	−0.948305
$184$	2.24264	0.165330
$185$	7.65685	0.562943
$186$	−4.58579	−0.336246
$187$	0	0
$188$	−8.00000	−0.583460
$189$	2.82843	0.205738
$190$	−6.82843	−0.495386
$191$	−12.0000	−0.868290	−0.434145	−	0.900843i	$-0.642949\pi$
$191$	−12.0000	−0.868290	−0.434145	+	0.900843i	$0.642949\pi$
$192$	−1.00000	−0.0721688
$193$	−11.6569	−0.839079	−0.419539	−	0.907737i	$-0.637808\pi$
$193$	−11.6569	−0.839079	−0.419539	+	0.907737i	$0.637808\pi$
$194$	−15.6569	−1.12410
$195$	2.58579	0.185172
$196$	1.00000	0.0714286
$197$	25.7990	1.83810	0.919051	−	0.394139i	$-0.128957\pi$
$197$	25.7990	1.83810	0.919051	+	0.394139i	$0.128957\pi$
$198$	0.828427	0.0588738
$199$	9.07107	0.643031	0.321515	−	0.946904i	$-0.395808\pi$
$199$	9.07107	0.643031	0.321515	+	0.946904i	$0.395808\pi$
$200$	1.00000	0.0707107
$201$	7.41421	0.522958
$202$	8.72792	0.614094
$203$	−22.6274	−1.58813
$204$	0	0
$205$	−0.828427	−0.0578599
$206$	−6.82843	−0.475759
$207$	2.24264	0.155874
$208$	−2.58579	−0.179292
$209$	−5.65685	−0.391293
$210$	2.82843	0.195180
$211$	−2.82843	−0.194717	−0.0973585	−	0.995249i	$-0.531039\pi$
$211$	−2.82843	−0.194717	−0.0973585	+	0.995249i	$0.531039\pi$
$212$	−0.343146	−0.0235673
$213$	−5.65685	−0.387601
$214$	−8.00000	−0.546869
$215$	−5.07107	−0.345844
$216$	−1.00000	−0.0680414
$217$	−12.9706	−0.880499
$218$	−15.6569	−1.06042
$219$	10.0000	0.675737
$220$	0.828427	0.0558525
$221$	0	0
$222$	−7.65685	−0.513894
$223$	−2.14214	−0.143448	−0.0717240	−	0.997425i	$-0.522850\pi$
$223$	−2.14214	−0.143448	−0.0717240	+	0.997425i	$0.522850\pi$
$224$	−2.82843	−0.188982
$225$	1.00000	0.0666667
$226$	−1.89949	−0.126353
$227$	−8.48528	−0.563188	−0.281594	−	0.959534i	$-0.590863\pi$
$227$	−8.48528	−0.563188	−0.281594	+	0.959534i	$0.590863\pi$
$228$	6.82843	0.452224
$229$	1.31371	0.0868123	0.0434062	−	0.999058i	$-0.486179\pi$
$229$	1.31371	0.0868123	0.0434062	+	0.999058i	$0.486179\pi$
$230$	2.24264	0.147875
$231$	2.34315	0.154168
$232$	8.00000	0.525226
$233$	−2.58579	−0.169401	−0.0847003	−	0.996406i	$-0.526993\pi$
$233$	−2.58579	−0.169401	−0.0847003	+	0.996406i	$0.526993\pi$
$234$	−2.58579	−0.169038
$235$	−8.00000	−0.521862
$236$	5.07107	0.330098
$237$	17.0711	1.10889
$238$	0	0
$239$	−2.82843	−0.182956	−0.0914779	−	0.995807i	$-0.529159\pi$
$239$	−2.82843	−0.182956	−0.0914779	+	0.995807i	$0.529159\pi$
$240$	−1.00000	−0.0645497
$241$	−0.727922	−0.0468896	−0.0234448	−	0.999725i	$-0.507463\pi$
$241$	−0.727922	−0.0468896	−0.0234448	+	0.999725i	$0.507463\pi$
$242$	−10.3137	−0.662990
$243$	−1.00000	−0.0641500
$244$	12.8284	0.821256
$245$	1.00000	0.0638877
$246$	0.828427	0.0528186
$247$	17.6569	1.12348
$248$	4.58579	0.291198
$249$	0	0
$250$	1.00000	0.0632456
$251$	−24.3848	−1.53915	−0.769577	−	0.638554i	$-0.779533\pi$
$251$	−24.3848	−1.53915	−0.769577	+	0.638554i	$0.779533\pi$
$252$	−2.82843	−0.178174
$253$	1.85786	0.116803
$254$	−8.00000	−0.501965
$255$	0	0
$256$	1.00000	0.0625000
$257$	8.97056	0.559568	0.279784	−	0.960063i	$-0.409737\pi$
$257$	8.97056	0.559568	0.279784	+	0.960063i	$0.409737\pi$
$258$	5.07107	0.315711
$259$	−21.6569	−1.34569
$260$	−2.58579	−0.160364
$261$	8.00000	0.495188
$262$	−4.00000	−0.247121
$263$	−6.48528	−0.399900	−0.199950	−	0.979806i	$-0.564078\pi$
$263$	−6.48528	−0.399900	−0.199950	+	0.979806i	$0.564078\pi$
$264$	−0.828427	−0.0509862
$265$	−0.343146	−0.0210793
$266$	19.3137	1.18420
$267$	12.1421	0.743087
$268$	−7.41421	−0.452895
$269$	−9.31371	−0.567867	−0.283933	−	0.958844i	$-0.591639\pi$
$269$	−9.31371	−0.567867	−0.283933	+	0.958844i	$0.591639\pi$
$270$	−1.00000	−0.0608581
$271$	18.4853	1.12290	0.561450	−	0.827510i	$-0.310244\pi$
$271$	18.4853	1.12290	0.561450	+	0.827510i	$0.310244\pi$
$272$	0	0
$273$	−7.31371	−0.442646
$274$	12.9706	0.783580
$275$	0.828427	0.0499560
$276$	−2.24264	−0.134991
$277$	16.0000	0.961347	0.480673	−	0.876900i	$-0.340392\pi$
$277$	16.0000	0.961347	0.480673	+	0.876900i	$0.340392\pi$
$278$	−6.34315	−0.380437
$279$	4.58579	0.274544
$280$	−2.82843	−0.169031
$281$	5.51472	0.328981	0.164490	−	0.986379i	$-0.447402\pi$
$281$	5.51472	0.328981	0.164490	+	0.986379i	$0.447402\pi$
$282$	8.00000	0.476393
$283$	16.8284	1.00035	0.500173	−	0.865925i	$-0.333270\pi$
$283$	16.8284	1.00035	0.500173	+	0.865925i	$0.333270\pi$
$284$	5.65685	0.335673
$285$	6.82843	0.404481
$286$	−2.14214	−0.126667
$287$	2.34315	0.138312
$288$	1.00000	0.0589256
$289$	0	0
$290$	8.00000	0.469776
$291$	15.6569	0.917821
$292$	−10.0000	−0.585206
$293$	−22.0000	−1.28525	−0.642627	−	0.766179i	$-0.722155\pi$
$293$	−22.0000	−1.28525	−0.642627	+	0.766179i	$0.722155\pi$
$294$	−1.00000	−0.0583212
$295$	5.07107	0.295249
$296$	7.65685	0.445046
$297$	−0.828427	−0.0480702
$298$	18.3848	1.06500
$299$	−5.79899	−0.335364
$300$	−1.00000	−0.0577350
$301$	14.3431	0.826725
$302$	−11.1716	−0.642852
$303$	−8.72792	−0.501406
$304$	−6.82843	−0.391637
$305$	12.8284	0.734554
$306$	0	0
$307$	−8.58579	−0.490017	−0.245008	−	0.969521i	$-0.578791\pi$
$307$	−8.58579	−0.490017	−0.245008	+	0.969521i	$0.578791\pi$
$308$	−2.34315	−0.133513
$309$	6.82843	0.388456
$310$	4.58579	0.260455
$311$	−24.0000	−1.36092	−0.680458	−	0.732787i	$-0.738219\pi$
$311$	−24.0000	−1.36092	−0.680458	+	0.732787i	$0.738219\pi$
$312$	2.58579	0.146391
$313$	−9.31371	−0.526442	−0.263221	−	0.964736i	$-0.584785\pi$
$313$	−9.31371	−0.526442	−0.263221	+	0.964736i	$0.584785\pi$
$314$	−14.5858	−0.823124
$315$	−2.82843	−0.159364
$316$	−17.0711	−0.960323
$317$	−29.7990	−1.67368	−0.836839	−	0.547449i	$-0.815599\pi$
$317$	−29.7990	−1.67368	−0.836839	+	0.547449i	$0.815599\pi$
$318$	0.343146	0.0192427
$319$	6.62742	0.371064
$320$	1.00000	0.0559017
$321$	8.00000	0.446516
$322$	−6.34315	−0.353490
$323$	0	0
$324$	1.00000	0.0555556
$325$	−2.58579	−0.143434
$326$	−0.828427	−0.0458823
$327$	15.6569	0.865826
$328$	−0.828427	−0.0457422
$329$	22.6274	1.24749
$330$	−0.828427	−0.0456034
$331$	−7.51472	−0.413046	−0.206523	−	0.978442i	$-0.566215\pi$
$331$	−7.51472	−0.413046	−0.206523	+	0.978442i	$0.566215\pi$
$332$	0	0
$333$	7.65685	0.419593
$334$	−3.41421	−0.186817
$335$	−7.41421	−0.405082
$336$	2.82843	0.154303
$337$	0.343146	0.0186923	0.00934617	−	0.999956i	$-0.497025\pi$
$337$	0.343146	0.0186923	0.00934617	+	0.999956i	$0.497025\pi$
$338$	−6.31371	−0.343420
$339$	1.89949	0.103166
$340$	0	0
$341$	3.79899	0.205727
$342$	−6.82843	−0.369239
$343$	16.9706	0.916324
$344$	−5.07107	−0.273414
$345$	−2.24264	−0.120740
$346$	7.65685	0.411635
$347$	−28.4853	−1.52917	−0.764585	−	0.644523i	$-0.777056\pi$
$347$	−28.4853	−1.52917	−0.764585	+	0.644523i	$0.777056\pi$
$348$	−8.00000	−0.428845
$349$	−22.9706	−1.22959	−0.614793	−	0.788688i	$-0.710760\pi$
$349$	−22.9706	−1.22959	−0.614793	+	0.788688i	$0.710760\pi$
$350$	−2.82843	−0.151186
$351$	2.58579	0.138019
$352$	0.828427	0.0441553
$353$	−5.31371	−0.282820	−0.141410	−	0.989951i	$-0.545164\pi$
$353$	−5.31371	−0.282820	−0.141410	+	0.989951i	$0.545164\pi$
$354$	−5.07107	−0.269524
$355$	5.65685	0.300235
$356$	−12.1421	−0.643532
$357$	0	0
$358$	19.8995	1.05172
$359$	31.7990	1.67829	0.839143	−	0.543910i	$-0.183057\pi$
$359$	31.7990	1.67829	0.839143	+	0.543910i	$0.183057\pi$
$360$	1.00000	0.0527046
$361$	27.6274	1.45407
$362$	−21.7990	−1.14573
$363$	10.3137	0.541329
$364$	7.31371	0.383342
$365$	−10.0000	−0.523424
$366$	−12.8284	−0.670553
$367$	−18.8284	−0.982836	−0.491418	−	0.870924i	$-0.663521\pi$
$367$	−18.8284	−0.982836	−0.491418	+	0.870924i	$0.663521\pi$
$368$	2.24264	0.116906
$369$	−0.828427	−0.0431262
$370$	7.65685	0.398061
$371$	0.970563	0.0503891
$372$	−4.58579	−0.237762
$373$	−18.8701	−0.977055	−0.488527	−	0.872549i	$-0.662466\pi$
$373$	−18.8701	−0.977055	−0.488527	+	0.872549i	$0.662466\pi$
$374$	0	0
$375$	−1.00000	−0.0516398
$376$	−8.00000	−0.412568
$377$	−20.6863	−1.06540
$378$	2.82843	0.145479
$379$	26.8284	1.37808	0.689042	−	0.724722i	$-0.258032\pi$
$379$	26.8284	1.37808	0.689042	+	0.724722i	$0.258032\pi$
$380$	−6.82843	−0.350291
$381$	8.00000	0.409852
$382$	−12.0000	−0.613973
$383$	31.4558	1.60732	0.803659	−	0.595090i	$-0.202883\pi$
$383$	31.4558	1.60732	0.803659	+	0.595090i	$0.202883\pi$
$384$	−1.00000	−0.0510310
$385$	−2.34315	−0.119418
$386$	−11.6569	−0.593318
$387$	−5.07107	−0.257777
$388$	−15.6569	−0.794856
$389$	17.8995	0.907540	0.453770	−	0.891119i	$-0.350079\pi$
$389$	17.8995	0.907540	0.453770	+	0.891119i	$0.350079\pi$
$390$	2.58579	0.130936
$391$	0	0
$392$	1.00000	0.0505076
$393$	4.00000	0.201773
$394$	25.7990	1.29973
$395$	−17.0711	−0.858939
$396$	0.828427	0.0416300
$397$	32.2843	1.62030	0.810151	−	0.586222i	$-0.199385\pi$
$397$	32.2843	1.62030	0.810151	+	0.586222i	$0.199385\pi$
$398$	9.07107	0.454692
$399$	−19.3137	−0.966895
$400$	1.00000	0.0500000
$401$	4.34315	0.216886	0.108443	−	0.994103i	$-0.465413\pi$
$401$	4.34315	0.216886	0.108443	+	0.994103i	$0.465413\pi$
$402$	7.41421	0.369787
$403$	−11.8579	−0.590682
$404$	8.72792	0.434230
$405$	1.00000	0.0496904
$406$	−22.6274	−1.12298
$407$	6.34315	0.314418
$408$	0	0
$409$	−35.6569	−1.76312	−0.881559	−	0.472074i	$-0.843506\pi$
$409$	−35.6569	−1.76312	−0.881559	+	0.472074i	$0.843506\pi$
$410$	−0.828427	−0.0409131
$411$	−12.9706	−0.639791
$412$	−6.82843	−0.336412
$413$	−14.3431	−0.705780
$414$	2.24264	0.110220
$415$	0	0
$416$	−2.58579	−0.126779
$417$	6.34315	0.310625
$418$	−5.65685	−0.276686
$419$	−11.1716	−0.545767	−0.272884	−	0.962047i	$-0.587977\pi$
$419$	−11.1716	−0.545767	−0.272884	+	0.962047i	$0.587977\pi$
$420$	2.82843	0.138013
$421$	−26.4853	−1.29081	−0.645407	−	0.763839i	$-0.723312\pi$
$421$	−26.4853	−1.29081	−0.645407	+	0.763839i	$0.723312\pi$
$422$	−2.82843	−0.137686
$423$	−8.00000	−0.388973
$424$	−0.343146	−0.0166646
$425$	0	0
$426$	−5.65685	−0.274075
$427$	−36.2843	−1.75592
$428$	−8.00000	−0.386695
$429$	2.14214	0.103423
$430$	−5.07107	−0.244549
$431$	38.6274	1.86062	0.930309	−	0.366778i	$-0.119539\pi$
$431$	38.6274	1.86062	0.930309	+	0.366778i	$0.119539\pi$
$432$	−1.00000	−0.0481125
$433$	3.17157	0.152416	0.0762080	−	0.997092i	$-0.475719\pi$
$433$	3.17157	0.152416	0.0762080	+	0.997092i	$0.475719\pi$
$434$	−12.9706	−0.622607
$435$	−8.00000	−0.383571
$436$	−15.6569	−0.749827
$437$	−15.3137	−0.732554
$438$	10.0000	0.477818
$439$	17.0711	0.814758	0.407379	−	0.913259i	$-0.366443\pi$
$439$	17.0711	0.814758	0.407379	+	0.913259i	$0.366443\pi$
$440$	0.828427	0.0394937
$441$	1.00000	0.0476190
$442$	0	0
$443$	−2.82843	−0.134383	−0.0671913	−	0.997740i	$-0.521404\pi$
$443$	−2.82843	−0.134383	−0.0671913	+	0.997740i	$0.521404\pi$
$444$	−7.65685	−0.363378
$445$	−12.1421	−0.575592
$446$	−2.14214	−0.101433
$447$	−18.3848	−0.869570
$448$	−2.82843	−0.133631
$449$	−34.2843	−1.61797	−0.808987	−	0.587826i	$-0.799984\pi$
$449$	−34.2843	−1.61797	−0.808987	+	0.587826i	$0.799984\pi$
$450$	1.00000	0.0471405
$451$	−0.686292	−0.0323162
$452$	−1.89949	−0.0893447
$453$	11.1716	0.524886
$454$	−8.48528	−0.398234
$455$	7.31371	0.342872
$456$	6.82843	0.319770
$457$	20.8284	0.974313	0.487156	−	0.873315i	$-0.338034\pi$
$457$	20.8284	0.974313	0.487156	+	0.873315i	$0.338034\pi$
$458$	1.31371	0.0613856
$459$	0	0
$460$	2.24264	0.104564
$461$	37.2132	1.73319	0.866596	−	0.499011i	$-0.166303\pi$
$461$	37.2132	1.73319	0.866596	+	0.499011i	$0.166303\pi$
$462$	2.34315	0.109013
$463$	−9.45584	−0.439450	−0.219725	−	0.975562i	$-0.570516\pi$
$463$	−9.45584	−0.439450	−0.219725	+	0.975562i	$0.570516\pi$
$464$	8.00000	0.371391
$465$	−4.58579	−0.212661
$466$	−2.58579	−0.119784
$467$	−15.3137	−0.708634	−0.354317	−	0.935125i	$-0.615287\pi$
$467$	−15.3137	−0.708634	−0.354317	+	0.935125i	$0.615287\pi$
$468$	−2.58579	−0.119528
$469$	20.9706	0.968331
$470$	−8.00000	−0.369012
$471$	14.5858	0.672078
$472$	5.07107	0.233415
$473$	−4.20101	−0.193163
$474$	17.0711	0.784100
$475$	−6.82843	−0.313310
$476$	0	0
$477$	−0.343146	−0.0157116
$478$	−2.82843	−0.129369
$479$	−2.34315	−0.107061	−0.0535305	−	0.998566i	$-0.517047\pi$
$479$	−2.34315	−0.107061	−0.0535305	+	0.998566i	$0.517047\pi$
$480$	−1.00000	−0.0456435
$481$	−19.7990	−0.902756
$482$	−0.727922	−0.0331559
$483$	6.34315	0.288623
$484$	−10.3137	−0.468805
$485$	−15.6569	−0.710941
$486$	−1.00000	−0.0453609
$487$	−20.0000	−0.906287	−0.453143	−	0.891438i	$-0.649697\pi$
$487$	−20.0000	−0.906287	−0.453143	+	0.891438i	$0.649697\pi$
$488$	12.8284	0.580716
$489$	0.828427	0.0374628
$490$	1.00000	0.0451754
$491$	−14.0416	−0.633690	−0.316845	−	0.948477i	$-0.602623\pi$
$491$	−14.0416	−0.633690	−0.316845	+	0.948477i	$0.602623\pi$
$492$	0.828427	0.0373484
$493$	0	0
$494$	17.6569	0.794419
$495$	0.828427	0.0372350
$496$	4.58579	0.205908
$497$	−16.0000	−0.717698
$498$	0	0
$499$	−31.7990	−1.42352	−0.711759	−	0.702424i	$-0.752101\pi$
$499$	−31.7990	−1.42352	−0.711759	+	0.702424i	$0.752101\pi$
$500$	1.00000	0.0447214
$501$	3.41421	0.152536
$502$	−24.3848	−1.08835
$503$	6.72792	0.299983	0.149992	−	0.988687i	$-0.452075\pi$
$503$	6.72792	0.299983	0.149992	+	0.988687i	$0.452075\pi$
$504$	−2.82843	−0.125988
$505$	8.72792	0.388387
$506$	1.85786	0.0825921
$507$	6.31371	0.280402
$508$	−8.00000	−0.354943
$509$	−21.2132	−0.940259	−0.470129	−	0.882598i	$-0.655793\pi$
$509$	−21.2132	−0.940259	−0.470129	+	0.882598i	$0.655793\pi$
$510$	0	0
$511$	28.2843	1.25122
$512$	1.00000	0.0441942
$513$	6.82843	0.301482
$514$	8.97056	0.395675
$515$	−6.82843	−0.300896
$516$	5.07107	0.223241
$517$	−6.62742	−0.291473
$518$	−21.6569	−0.951548
$519$	−7.65685	−0.336099
$520$	−2.58579	−0.113394
$521$	25.3137	1.10901	0.554507	−	0.832179i	$-0.312907\pi$
$521$	25.3137	1.10901	0.554507	+	0.832179i	$0.312907\pi$
$522$	8.00000	0.350150
$523$	−16.5858	−0.725246	−0.362623	−	0.931936i	$-0.618119\pi$
$523$	−16.5858	−0.725246	−0.362623	+	0.931936i	$0.618119\pi$
$524$	−4.00000	−0.174741
$525$	2.82843	0.123443
$526$	−6.48528	−0.282772
$527$	0	0
$528$	−0.828427	−0.0360527
$529$	−17.9706	−0.781329
$530$	−0.343146	−0.0149053
$531$	5.07107	0.220066
$532$	19.3137	0.837355
$533$	2.14214	0.0927862
$534$	12.1421	0.525442
$535$	−8.00000	−0.345870
$536$	−7.41421	−0.320245
$537$	−19.8995	−0.858727
$538$	−9.31371	−0.401542
$539$	0.828427	0.0356829
$540$	−1.00000	−0.0430331
$541$	37.1127	1.59560	0.797800	−	0.602922i	$-0.205997\pi$
$541$	37.1127	1.59560	0.797800	+	0.602922i	$0.205997\pi$
$542$	18.4853	0.794011
$543$	21.7990	0.935484
$544$	0	0
$545$	−15.6569	−0.670666
$546$	−7.31371	−0.312998
$547$	19.1716	0.819717	0.409859	−	0.912149i	$-0.365578\pi$
$547$	19.1716	0.819717	0.409859	+	0.912149i	$0.365578\pi$
$548$	12.9706	0.554075
$549$	12.8284	0.547504
$550$	0.828427	0.0353243
$551$	−54.6274	−2.32721
$552$	−2.24264	−0.0954531
$553$	48.2843	2.05326
$554$	16.0000	0.679775
$555$	−7.65685	−0.325015
$556$	−6.34315	−0.269009
$557$	−26.9706	−1.14278	−0.571390	−	0.820679i	$-0.693596\pi$
$557$	−26.9706	−1.14278	−0.571390	+	0.820679i	$0.693596\pi$
$558$	4.58579	0.194132
$559$	13.1127	0.554608
$560$	−2.82843	−0.119523
$561$	0	0
$562$	5.51472	0.232624
$563$	37.9411	1.59903	0.799514	−	0.600648i	$-0.205091\pi$
$563$	37.9411	1.59903	0.799514	+	0.600648i	$0.205091\pi$
$564$	8.00000	0.336861
$565$	−1.89949	−0.0799124
$566$	16.8284	0.707352
$567$	−2.82843	−0.118783
$568$	5.65685	0.237356
$569$	10.0000	0.419222	0.209611	−	0.977785i	$-0.432780\pi$
$569$	10.0000	0.419222	0.209611	+	0.977785i	$0.432780\pi$
$570$	6.82843	0.286011
$571$	6.82843	0.285761	0.142880	−	0.989740i	$-0.454364\pi$
$571$	6.82843	0.285761	0.142880	+	0.989740i	$0.454364\pi$
$572$	−2.14214	−0.0895672
$573$	12.0000	0.501307
$574$	2.34315	0.0978010
$575$	2.24264	0.0935246
$576$	1.00000	0.0416667
$577$	−19.1716	−0.798123	−0.399062	−	0.916924i	$-0.630664\pi$
$577$	−19.1716	−0.798123	−0.399062	+	0.916924i	$0.630664\pi$
$578$	0	0
$579$	11.6569	0.484442
$580$	8.00000	0.332182
$581$	0	0
$582$	15.6569	0.648997
$583$	−0.284271	−0.0117733
$584$	−10.0000	−0.413803
$585$	−2.58579	−0.106909
$586$	−22.0000	−0.908812
$587$	−32.4853	−1.34081	−0.670406	−	0.741995i	$-0.733880\pi$
$587$	−32.4853	−1.34081	−0.670406	+	0.741995i	$0.733880\pi$
$588$	−1.00000	−0.0412393
$589$	−31.3137	−1.29026
$590$	5.07107	0.208773
$591$	−25.7990	−1.06123
$592$	7.65685	0.314695
$593$	−34.2843	−1.40789	−0.703943	−	0.710256i	$-0.748579\pi$
$593$	−34.2843	−1.40789	−0.703943	+	0.710256i	$0.748579\pi$
$594$	−0.828427	−0.0339908
$595$	0	0
$596$	18.3848	0.753070
$597$	−9.07107	−0.371254
$598$	−5.79899	−0.237138
$599$	38.8284	1.58649	0.793243	−	0.608905i	$-0.208391\pi$
$599$	38.8284	1.58649	0.793243	+	0.608905i	$0.208391\pi$
$600$	−1.00000	−0.0408248
$601$	43.5563	1.77670	0.888350	−	0.459166i	$-0.151852\pi$
$601$	43.5563	1.77670	0.888350	+	0.459166i	$0.151852\pi$
$602$	14.3431	0.584583
$603$	−7.41421	−0.301930
$604$	−11.1716	−0.454565
$605$	−10.3137	−0.419312
$606$	−8.72792	−0.354548
$607$	7.31371	0.296854	0.148427	−	0.988923i	$-0.452579\pi$
$607$	7.31371	0.296854	0.148427	+	0.988923i	$0.452579\pi$
$608$	−6.82843	−0.276929
$609$	22.6274	0.916909
$610$	12.8284	0.519408
$611$	20.6863	0.836878
$612$	0	0
$613$	−17.2132	−0.695235	−0.347617	−	0.937636i	$-0.613009\pi$
$613$	−17.2132	−0.695235	−0.347617	+	0.937636i	$0.613009\pi$
$614$	−8.58579	−0.346494
$615$	0.828427	0.0334054
$616$	−2.34315	−0.0944080
$617$	8.72792	0.351373	0.175686	−	0.984446i	$-0.443786\pi$
$617$	8.72792	0.351373	0.175686	+	0.984446i	$0.443786\pi$
$618$	6.82843	0.274680
$619$	−20.2843	−0.815294	−0.407647	−	0.913140i	$-0.633651\pi$
$619$	−20.2843	−0.815294	−0.407647	+	0.913140i	$0.633651\pi$
$620$	4.58579	0.184170
$621$	−2.24264	−0.0899941
$622$	−24.0000	−0.962312
$623$	34.3431	1.37593
$624$	2.58579	0.103514
$625$	1.00000	0.0400000
$626$	−9.31371	−0.372251
$627$	5.65685	0.225913
$628$	−14.5858	−0.582036
$629$	0	0
$630$	−2.82843	−0.112687
$631$	3.17157	0.126258	0.0631292	−	0.998005i	$-0.479892\pi$
$631$	3.17157	0.126258	0.0631292	+	0.998005i	$0.479892\pi$
$632$	−17.0711	−0.679051
$633$	2.82843	0.112420
$634$	−29.7990	−1.18347
$635$	−8.00000	−0.317470
$636$	0.343146	0.0136066
$637$	−2.58579	−0.102453
$638$	6.62742	0.262382
$639$	5.65685	0.223782
$640$	1.00000	0.0395285
$641$	45.5980	1.80101	0.900506	−	0.434844i	$-0.143196\pi$
$641$	45.5980	1.80101	0.900506	+	0.434844i	$0.143196\pi$
$642$	8.00000	0.315735
$643$	−3.85786	−0.152139	−0.0760697	−	0.997103i	$-0.524237\pi$
$643$	−3.85786	−0.152139	−0.0760697	+	0.997103i	$0.524237\pi$
$644$	−6.34315	−0.249955
$645$	5.07107	0.199673
$646$	0	0
$647$	−48.4264	−1.90384	−0.951919	−	0.306349i	$-0.900893\pi$
$647$	−48.4264	−1.90384	−0.951919	+	0.306349i	$0.900893\pi$
$648$	1.00000	0.0392837
$649$	4.20101	0.164904
$650$	−2.58579	−0.101423
$651$	12.9706	0.508356
$652$	−0.828427	−0.0324437
$653$	−14.6863	−0.574719	−0.287359	−	0.957823i	$-0.592777\pi$
$653$	−14.6863	−0.574719	−0.287359	+	0.957823i	$0.592777\pi$
$654$	15.6569	0.612231
$655$	−4.00000	−0.156293
$656$	−0.828427	−0.0323446
$657$	−10.0000	−0.390137
$658$	22.6274	0.882109
$659$	−38.2426	−1.48972	−0.744861	−	0.667220i	$-0.767484\pi$
$659$	−38.2426	−1.48972	−0.744861	+	0.667220i	$0.767484\pi$
$660$	−0.828427	−0.0322465
$661$	3.65685	0.142235	0.0711176	−	0.997468i	$-0.477343\pi$
$661$	3.65685	0.142235	0.0711176	+	0.997468i	$0.477343\pi$
$662$	−7.51472	−0.292068
$663$	0	0
$664$	0	0
$665$	19.3137	0.748953
$666$	7.65685	0.296697
$667$	17.9411	0.694683
$668$	−3.41421	−0.132100
$669$	2.14214	0.0828197
$670$	−7.41421	−0.286436
$671$	10.6274	0.410267
$672$	2.82843	0.109109
$673$	−27.9411	−1.07705	−0.538526	−	0.842609i	$-0.681018\pi$
$673$	−27.9411	−1.07705	−0.538526	+	0.842609i	$0.681018\pi$
$674$	0.343146	0.0132175
$675$	−1.00000	−0.0384900
$676$	−6.31371	−0.242835
$677$	−3.17157	−0.121893	−0.0609467	−	0.998141i	$-0.519412\pi$
$677$	−3.17157	−0.121893	−0.0609467	+	0.998141i	$0.519412\pi$
$678$	1.89949	0.0729497
$679$	44.2843	1.69947
$680$	0	0
$681$	8.48528	0.325157
$682$	3.79899	0.145471
$683$	−48.2843	−1.84755	−0.923773	−	0.382940i	$-0.874912\pi$
$683$	−48.2843	−1.84755	−0.923773	+	0.382940i	$0.874912\pi$
$684$	−6.82843	−0.261091
$685$	12.9706	0.495580
$686$	16.9706	0.647939
$687$	−1.31371	−0.0501211
$688$	−5.07107	−0.193333
$689$	0.887302	0.0338035
$690$	−2.24264	−0.0853759
$691$	8.48528	0.322795	0.161398	−	0.986889i	$-0.448400\pi$
$691$	8.48528	0.322795	0.161398	+	0.986889i	$0.448400\pi$
$692$	7.65685	0.291070
$693$	−2.34315	−0.0890087
$694$	−28.4853	−1.08129
$695$	−6.34315	−0.240609
$696$	−8.00000	−0.303239
$697$	0	0
$698$	−22.9706	−0.869449
$699$	2.58579	0.0978034
$700$	−2.82843	−0.106904
$701$	20.7279	0.782883	0.391441	−	0.920203i	$-0.371977\pi$
$701$	20.7279	0.782883	0.391441	+	0.920203i	$0.371977\pi$
$702$	2.58579	0.0975942
$703$	−52.2843	−1.97194
$704$	0.828427	0.0312225
$705$	8.00000	0.301297
$706$	−5.31371	−0.199984
$707$	−24.6863	−0.928424
$708$	−5.07107	−0.190582
$709$	20.6274	0.774679	0.387339	−	0.921937i	$-0.373394\pi$
$709$	20.6274	0.774679	0.387339	+	0.921937i	$0.373394\pi$
$710$	5.65685	0.212298
$711$	−17.0711	−0.640215
$712$	−12.1421	−0.455046
$713$	10.2843	0.385149
$714$	0	0
$715$	−2.14214	−0.0801113
$716$	19.8995	0.743679
$717$	2.82843	0.105630
$718$	31.7990	1.18673
$719$	−33.6569	−1.25519	−0.627594	−	0.778540i	$-0.715960\pi$
$719$	−33.6569	−1.25519	−0.627594	+	0.778540i	$0.715960\pi$
$720$	1.00000	0.0372678
$721$	19.3137	0.719280
$722$	27.6274	1.02819
$723$	0.727922	0.0270717
$724$	−21.7990	−0.810153
$725$	8.00000	0.297113
$726$	10.3137	0.382778
$727$	−30.3431	−1.12536	−0.562682	−	0.826673i	$-0.690231\pi$
$727$	−30.3431	−1.12536	−0.562682	+	0.826673i	$0.690231\pi$
$728$	7.31371	0.271064
$729$	1.00000	0.0370370
$730$	−10.0000	−0.370117
$731$	0	0
$732$	−12.8284	−0.474152
$733$	11.7574	0.434268	0.217134	−	0.976142i	$-0.430329\pi$
$733$	11.7574	0.434268	0.217134	+	0.976142i	$0.430329\pi$
$734$	−18.8284	−0.694970
$735$	−1.00000	−0.0368856
$736$	2.24264	0.0826648
$737$	−6.14214	−0.226248
$738$	−0.828427	−0.0304948
$739$	−31.5980	−1.16235	−0.581175	−	0.813778i	$-0.697407\pi$
$739$	−31.5980	−1.16235	−0.581175	+	0.813778i	$0.697407\pi$
$740$	7.65685	0.281472
$741$	−17.6569	−0.648641
$742$	0.970563	0.0356305
$743$	0.870058	0.0319193	0.0159597	−	0.999873i	$-0.494920\pi$
$743$	0.870058	0.0319193	0.0159597	+	0.999873i	$0.494920\pi$
$744$	−4.58579	−0.168123
$745$	18.3848	0.673566
$746$	−18.8701	−0.690882
$747$	0	0
$748$	0	0
$749$	22.6274	0.826788
$750$	−1.00000	−0.0365148
$751$	1.07107	0.0390838	0.0195419	−	0.999809i	$-0.493779\pi$
$751$	1.07107	0.0390838	0.0195419	+	0.999809i	$0.493779\pi$
$752$	−8.00000	−0.291730
$753$	24.3848	0.888631
$754$	−20.6863	−0.753350
$755$	−11.1716	−0.406575
$756$	2.82843	0.102869
$757$	28.7279	1.04413	0.522067	−	0.852904i	$-0.325161\pi$
$757$	28.7279	1.04413	0.522067	+	0.852904i	$0.325161\pi$
$758$	26.8284	0.974452
$759$	−1.85786	−0.0674362
$760$	−6.82843	−0.247693
$761$	21.3137	0.772621	0.386311	−	0.922369i	$-0.373749\pi$
$761$	21.3137	0.772621	0.386311	+	0.922369i	$0.373749\pi$
$762$	8.00000	0.289809
$763$	44.2843	1.60320
$764$	−12.0000	−0.434145
$765$	0	0
$766$	31.4558	1.13655
$767$	−13.1127	−0.473472
$768$	−1.00000	−0.0360844
$769$	10.0000	0.360609	0.180305	−	0.983611i	$-0.442292\pi$
$769$	10.0000	0.360609	0.180305	+	0.983611i	$0.442292\pi$
$770$	−2.34315	−0.0844411
$771$	−8.97056	−0.323067
$772$	−11.6569	−0.419539
$773$	−5.79899	−0.208575	−0.104288	−	0.994547i	$-0.533256\pi$
$773$	−5.79899	−0.208575	−0.104288	+	0.994547i	$0.533256\pi$
$774$	−5.07107	−0.182276
$775$	4.58579	0.164726
$776$	−15.6569	−0.562048
$777$	21.6569	0.776935
$778$	17.8995	0.641728
$779$	5.65685	0.202678
$780$	2.58579	0.0925860
$781$	4.68629	0.167689
$782$	0	0
$783$	−8.00000	−0.285897
$784$	1.00000	0.0357143
$785$	−14.5858	−0.520589
$786$	4.00000	0.142675
$787$	35.5980	1.26893	0.634465	−	0.772951i	$-0.281220\pi$
$787$	35.5980	1.26893	0.634465	+	0.772951i	$0.281220\pi$
$788$	25.7990	0.919051
$789$	6.48528	0.230882
$790$	−17.0711	−0.607361
$791$	5.37258	0.191027
$792$	0.828427	0.0294369
$793$	−33.1716	−1.17796
$794$	32.2843	1.14573
$795$	0.343146	0.0121701
$796$	9.07107	0.321515
$797$	−29.7990	−1.05553	−0.527767	−	0.849389i	$-0.676971\pi$
$797$	−29.7990	−1.05553	−0.527767	+	0.849389i	$0.676971\pi$
$798$	−19.3137	−0.683698
$799$	0	0
$800$	1.00000	0.0353553
$801$	−12.1421	−0.429021
$802$	4.34315	0.153362
$803$	−8.28427	−0.292346
$804$	7.41421	0.261479
$805$	−6.34315	−0.223567
$806$	−11.8579	−0.417675
$807$	9.31371	0.327858
$808$	8.72792	0.307047
$809$	1.79899	0.0632491	0.0316246	−	0.999500i	$-0.489932\pi$
$809$	1.79899	0.0632491	0.0316246	+	0.999500i	$0.489932\pi$
$810$	1.00000	0.0351364
$811$	−43.1127	−1.51389	−0.756946	−	0.653478i	$-0.773309\pi$
$811$	−43.1127	−1.51389	−0.756946	+	0.653478i	$0.773309\pi$
$812$	−22.6274	−0.794067
$813$	−18.4853	−0.648307
$814$	6.34315	0.222327
$815$	−0.828427	−0.0290185
$816$	0	0
$817$	34.6274	1.21146
$818$	−35.6569	−1.24671
$819$	7.31371	0.255562
$820$	−0.828427	−0.0289299
$821$	46.6274	1.62731	0.813654	−	0.581349i	$-0.197475\pi$
$821$	46.6274	1.62731	0.813654	+	0.581349i	$0.197475\pi$
$822$	−12.9706	−0.452400
$823$	45.1716	1.57458	0.787291	−	0.616582i	$-0.211483\pi$
$823$	45.1716	1.57458	0.787291	+	0.616582i	$0.211483\pi$
$824$	−6.82843	−0.237880
$825$	−0.828427	−0.0288421
$826$	−14.3431	−0.499062
$827$	33.4558	1.16337	0.581687	−	0.813413i	$-0.302393\pi$
$827$	33.4558	1.16337	0.581687	+	0.813413i	$0.302393\pi$
$828$	2.24264	0.0779372
$829$	29.7990	1.03496	0.517481	−	0.855695i	$-0.326870\pi$
$829$	29.7990	1.03496	0.517481	+	0.855695i	$0.326870\pi$
$830$	0	0
$831$	−16.0000	−0.555034
$832$	−2.58579	−0.0896460
$833$	0	0
$834$	6.34315	0.219645
$835$	−3.41421	−0.118154
$836$	−5.65685	−0.195646
$837$	−4.58579	−0.158508
$838$	−11.1716	−0.385916
$839$	−19.1127	−0.659844	−0.329922	−	0.944008i	$-0.607022\pi$
$839$	−19.1127	−0.659844	−0.329922	+	0.944008i	$0.607022\pi$
$840$	2.82843	0.0975900
$841$	35.0000	1.20690
$842$	−26.4853	−0.912743
$843$	−5.51472	−0.189937
$844$	−2.82843	−0.0973585
$845$	−6.31371	−0.217198
$846$	−8.00000	−0.275046
$847$	29.1716	1.00235
$848$	−0.343146	−0.0117837
$849$	−16.8284	−0.577550
$850$	0	0
$851$	17.1716	0.588634
$852$	−5.65685	−0.193801
$853$	53.9411	1.84691	0.923454	−	0.383708i	$-0.125353\pi$
$853$	53.9411	1.84691	0.923454	+	0.383708i	$0.125353\pi$
$854$	−36.2843	−1.24162
$855$	−6.82843	−0.233527
$856$	−8.00000	−0.273434
$857$	−29.2132	−0.997904	−0.498952	−	0.866630i	$-0.666282\pi$
$857$	−29.2132	−0.997904	−0.498952	+	0.866630i	$0.666282\pi$
$858$	2.14214	0.0731313
$859$	−12.4853	−0.425992	−0.212996	−	0.977053i	$-0.568322\pi$
$859$	−12.4853	−0.425992	−0.212996	+	0.977053i	$0.568322\pi$
$860$	−5.07107	−0.172922
$861$	−2.34315	−0.0798542
$862$	38.6274	1.31566
$863$	−8.82843	−0.300523	−0.150262	−	0.988646i	$-0.548012\pi$
$863$	−8.82843	−0.300523	−0.150262	+	0.988646i	$0.548012\pi$
$864$	−1.00000	−0.0340207
$865$	7.65685	0.260341
$866$	3.17157	0.107774
$867$	0	0
$868$	−12.9706	−0.440250
$869$	−14.1421	−0.479739
$870$	−8.00000	−0.271225
$871$	19.1716	0.649604
$872$	−15.6569	−0.530208
$873$	−15.6569	−0.529904
$874$	−15.3137	−0.517994
$875$	−2.82843	−0.0956183
$876$	10.0000	0.337869
$877$	39.9411	1.34872	0.674358	−	0.738405i	$-0.264420\pi$
$877$	39.9411	1.34872	0.674358	+	0.738405i	$0.264420\pi$
$878$	17.0711	0.576121
$879$	22.0000	0.742042
$880$	0.828427	0.0279263
$881$	14.4853	0.488022	0.244011	−	0.969773i	$-0.421537\pi$
$881$	14.4853	0.488022	0.244011	+	0.969773i	$0.421537\pi$
$882$	1.00000	0.0336718
$883$	25.5563	0.860040	0.430020	−	0.902819i	$-0.358507\pi$
$883$	25.5563	0.860040	0.430020	+	0.902819i	$0.358507\pi$
$884$	0	0
$885$	−5.07107	−0.170462
$886$	−2.82843	−0.0950229
$887$	47.8995	1.60831	0.804154	−	0.594421i	$-0.202619\pi$
$887$	47.8995	1.60831	0.804154	+	0.594421i	$0.202619\pi$
$888$	−7.65685	−0.256947
$889$	22.6274	0.758899
$890$	−12.1421	−0.407005
$891$	0.828427	0.0277534
$892$	−2.14214	−0.0717240
$893$	54.6274	1.82804
$894$	−18.3848	−0.614879
$895$	19.8995	0.665167
$896$	−2.82843	−0.0944911
$897$	5.79899	0.193623
$898$	−34.2843	−1.14408
$899$	36.6863	1.22356
$900$	1.00000	0.0333333
$901$	0	0
$902$	−0.686292	−0.0228510
$903$	−14.3431	−0.477310
$904$	−1.89949	−0.0631763
$905$	−21.7990	−0.724623
$906$	11.1716	0.371151
$907$	15.3137	0.508483	0.254242	−	0.967141i	$-0.418174\pi$
$907$	15.3137	0.508483	0.254242	+	0.967141i	$0.418174\pi$
$908$	−8.48528	−0.281594
$909$	8.72792	0.289487
$910$	7.31371	0.242447
$911$	38.6274	1.27978	0.639892	−	0.768465i	$-0.278979\pi$
$911$	38.6274	1.27978	0.639892	+	0.768465i	$0.278979\pi$
$912$	6.82843	0.226112
$913$	0	0
$914$	20.8284	0.688943
$915$	−12.8284	−0.424095
$916$	1.31371	0.0434062
$917$	11.3137	0.373612
$918$	0	0
$919$	46.0833	1.52015	0.760073	−	0.649837i	$-0.225163\pi$
$919$	46.0833	1.52015	0.760073	+	0.649837i	$0.225163\pi$
$920$	2.24264	0.0739377
$921$	8.58579	0.282911
$922$	37.2132	1.22555
$923$	−14.6274	−0.481467
$924$	2.34315	0.0770838
$925$	7.65685	0.251756
$926$	−9.45584	−0.310738
$927$	−6.82843	−0.224275
$928$	8.00000	0.262613
$929$	17.5147	0.574639	0.287320	−	0.957835i	$-0.407236\pi$
$929$	17.5147	0.574639	0.287320	+	0.957835i	$0.407236\pi$
$930$	−4.58579	−0.150374
$931$	−6.82843	−0.223793
$932$	−2.58579	−0.0847003
$933$	24.0000	0.785725
$934$	−15.3137	−0.501080
$935$	0	0
$936$	−2.58579	−0.0845191
$937$	11.8579	0.387380	0.193690	−	0.981063i	$-0.437954\pi$
$937$	11.8579	0.387380	0.193690	+	0.981063i	$0.437954\pi$
$938$	20.9706	0.684713
$939$	9.31371	0.303941
$940$	−8.00000	−0.260931
$941$	1.37258	0.0447449	0.0223725	−	0.999750i	$-0.492878\pi$
$941$	1.37258	0.0447449	0.0223725	+	0.999750i	$0.492878\pi$
$942$	14.5858	0.475231
$943$	−1.85786	−0.0605004
$944$	5.07107	0.165049
$945$	2.82843	0.0920087
$946$	−4.20101	−0.136587
$947$	−49.9411	−1.62287	−0.811434	−	0.584444i	$-0.801313\pi$
$947$	−49.9411	−1.62287	−0.811434	+	0.584444i	$0.801313\pi$
$948$	17.0711	0.554443
$949$	25.8579	0.839382
$950$	−6.82843	−0.221543
$951$	29.7990	0.966298
$952$	0	0
$953$	−28.6274	−0.927333	−0.463666	−	0.886010i	$-0.653466\pi$
$953$	−28.6274	−0.927333	−0.463666	+	0.886010i	$0.653466\pi$
$954$	−0.343146	−0.0111098
$955$	−12.0000	−0.388311
$956$	−2.82843	−0.0914779
$957$	−6.62742	−0.214234
$958$	−2.34315	−0.0757036
$959$	−36.6863	−1.18466
$960$	−1.00000	−0.0322749
$961$	−9.97056	−0.321631
$962$	−19.7990	−0.638345
$963$	−8.00000	−0.257796
$964$	−0.727922	−0.0234448
$965$	−11.6569	−0.375247
$966$	6.34315	0.204087
$967$	29.6569	0.953700	0.476850	−	0.878985i	$-0.341778\pi$
$967$	29.6569	0.953700	0.476850	+	0.878985i	$0.341778\pi$
$968$	−10.3137	−0.331495
$969$	0	0
$970$	−15.6569	−0.502711
$971$	27.6985	0.888887	0.444443	−	0.895807i	$-0.353402\pi$
$971$	27.6985	0.888887	0.444443	+	0.895807i	$0.353402\pi$
$972$	−1.00000	−0.0320750
$973$	17.9411	0.575166
$974$	−20.0000	−0.640841
$975$	2.58579	0.0828114
$976$	12.8284	0.410628
$977$	38.0000	1.21573	0.607864	−	0.794041i	$-0.292027\pi$
$977$	38.0000	1.21573	0.607864	+	0.794041i	$0.292027\pi$
$978$	0.828427	0.0264902
$979$	−10.0589	−0.321483
$980$	1.00000	0.0319438
$981$	−15.6569	−0.499885
$982$	−14.0416	−0.448086
$983$	−22.9289	−0.731319	−0.365660	−	0.930749i	$-0.619157\pi$
$983$	−22.9289	−0.731319	−0.365660	+	0.930749i	$0.619157\pi$
$984$	0.828427	0.0264093
$985$	25.7990	0.822024
$986$	0	0
$987$	−22.6274	−0.720239
$988$	17.6569	0.561739
$989$	−11.3726	−0.361627
$990$	0.828427	0.0263291
$991$	−51.0122	−1.62046	−0.810228	−	0.586115i	$-0.800657\pi$
$991$	−51.0122	−1.62046	−0.810228	+	0.586115i	$0.800657\pi$
$992$	4.58579	0.145599
$993$	7.51472	0.238472
$994$	−16.0000	−0.507489
$995$	9.07107	0.287572
$996$	0	0
$997$	−36.6274	−1.16000	−0.580001	−	0.814616i	$-0.696948\pi$
$997$	−36.6274	−1.16000	−0.580001	+	0.814616i	$0.696948\pi$
$998$	−31.7990	−1.00658
$999$	−7.65685	−0.242252

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8670.2.a.bi.1.1		2
17.8	even	8		510.2.p.a.421.2	yes	4
17.15	even	8		510.2.p.a.361.2	✓	4
17.16	even	2		8670.2.a.bj.1.2		2
51.8	odd	8		1530.2.q.a.1441.1		4
51.32	odd	8		1530.2.q.a.361.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
510.2.p.a.361.2	✓	4	17.15	even	8
510.2.p.a.421.2	yes	4	17.8	even	8
1530.2.q.a.361.1		4	51.32	odd	8
1530.2.q.a.1441.1		4	51.8	odd	8
8670.2.a.bi.1.1		2	1.1	even	1	trivial
8670.2.a.bj.1.2		2	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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{6} -2.82843 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{6} -2.82843 q^{7} +1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{10} +0.828427 q^{11} -1.00000 q^{12} -2.58579 q^{13} -2.82843 q^{14} -1.00000 q^{15} +1.00000 q^{16} +1.00000 q^{18} -6.82843 q^{19} +1.00000 q^{20} +2.82843 q^{21} +0.828427 q^{22} +2.24264 q^{23} -1.00000 q^{24} +1.00000 q^{25} -2.58579 q^{26} -1.00000 q^{27} -2.82843 q^{28} +8.00000 q^{29} -1.00000 q^{30} +4.58579 q^{31} +1.00000 q^{32} -0.828427 q^{33} -2.82843 q^{35} +1.00000 q^{36} +7.65685 q^{37} -6.82843 q^{38} +2.58579 q^{39} +1.00000 q^{40} -0.828427 q^{41} +2.82843 q^{42} -5.07107 q^{43} +0.828427 q^{44} +1.00000 q^{45} +2.24264 q^{46} -8.00000 q^{47} -1.00000 q^{48} +1.00000 q^{49} +1.00000 q^{50} -2.58579 q^{52} -0.343146 q^{53} -1.00000 q^{54} +0.828427 q^{55} -2.82843 q^{56} +6.82843 q^{57} +8.00000 q^{58} +5.07107 q^{59} -1.00000 q^{60} +12.8284 q^{61} +4.58579 q^{62} -2.82843 q^{63} +1.00000 q^{64} -2.58579 q^{65} -0.828427 q^{66} -7.41421 q^{67} -2.24264 q^{69} -2.82843 q^{70} +5.65685 q^{71} +1.00000 q^{72} -10.0000 q^{73} +7.65685 q^{74} -1.00000 q^{75} -6.82843 q^{76} -2.34315 q^{77} +2.58579 q^{78} -17.0711 q^{79} +1.00000 q^{80} +1.00000 q^{81} -0.828427 q^{82} +2.82843 q^{84} -5.07107 q^{86} -8.00000 q^{87} +0.828427 q^{88} -12.1421 q^{89} +1.00000 q^{90} +7.31371 q^{91} +2.24264 q^{92} -4.58579 q^{93} -8.00000 q^{94} -6.82843 q^{95} -1.00000 q^{96} -15.6569 q^{97} +1.00000 q^{98} +0.828427 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} - 2 q^{3} + 2 q^{4} + 2 q^{5} - 2 q^{6} + 2 q^{8} + 2 q^{9}+O(q^{10}) \) 2 * q + 2 * q^2 - 2 * q^3 + 2 * q^4 + 2 * q^5 - 2 * q^6 + 2 * q^8 + 2 * q^9	\( 2 q + 2 q^{2} - 2 q^{3} + 2 q^{4} + 2 q^{5} - 2 q^{6} + 2 q^{8} + 2 q^{9} + 2 q^{10} - 4 q^{11} - 2 q^{12} - 8 q^{13} - 2 q^{15} + 2 q^{16} + 2 q^{18} - 8 q^{19} + 2 q^{20} - 4 q^{22} - 4 q^{23} - 2 q^{24} + 2 q^{25} - 8 q^{26} - 2 q^{27} + 16 q^{29} - 2 q^{30} + 12 q^{31} + 2 q^{32} + 4 q^{33} + 2 q^{36} + 4 q^{37} - 8 q^{38} + 8 q^{39} + 2 q^{40} + 4 q^{41} + 4 q^{43} - 4 q^{44} + 2 q^{45} - 4 q^{46} - 16 q^{47} - 2 q^{48} + 2 q^{49} + 2 q^{50} - 8 q^{52} - 12 q^{53} - 2 q^{54} - 4 q^{55} + 8 q^{57} + 16 q^{58} - 4 q^{59} - 2 q^{60} + 20 q^{61} + 12 q^{62} + 2 q^{64} - 8 q^{65} + 4 q^{66} - 12 q^{67} + 4 q^{69} + 2 q^{72} - 20 q^{73} + 4 q^{74} - 2 q^{75} - 8 q^{76} - 16 q^{77} + 8 q^{78} - 20 q^{79} + 2 q^{80} + 2 q^{81} + 4 q^{82} + 4 q^{86} - 16 q^{87} - 4 q^{88} + 4 q^{89} + 2 q^{90} - 8 q^{91} - 4 q^{92} - 12 q^{93} - 16 q^{94} - 8 q^{95} - 2 q^{96} - 20 q^{97} + 2 q^{98} - 4 q^{99}+O(q^{100}) \) 2 * q + 2 * q^2 - 2 * q^3 + 2 * q^4 + 2 * q^5 - 2 * q^6 + 2 * q^8 + 2 * q^9 + 2 * q^10 - 4 * q^11 - 2 * q^12 - 8 * q^13 - 2 * q^15 + 2 * q^16 + 2 * q^18 - 8 * q^19 + 2 * q^20 - 4 * q^22 - 4 * q^23 - 2 * q^24 + 2 * q^25 - 8 * q^26 - 2 * q^27 + 16 * q^29 - 2 * q^30 + 12 * q^31 + 2 * q^32 + 4 * q^33 + 2 * q^36 + 4 * q^37 - 8 * q^38 + 8 * q^39 + 2 * q^40 + 4 * q^41 + 4 * q^43 - 4 * q^44 + 2 * q^45 - 4 * q^46 - 16 * q^47 - 2 * q^48 + 2 * q^49 + 2 * q^50 - 8 * q^52 - 12 * q^53 - 2 * q^54 - 4 * q^55 + 8 * q^57 + 16 * q^58 - 4 * q^59 - 2 * q^60 + 20 * q^61 + 12 * q^62 + 2 * q^64 - 8 * q^65 + 4 * q^66 - 12 * q^67 + 4 * q^69 + 2 * q^72 - 20 * q^73 + 4 * q^74 - 2 * q^75 - 8 * q^76 - 16 * q^77 + 8 * q^78 - 20 * q^79 + 2 * q^80 + 2 * q^81 + 4 * q^82 + 4 * q^86 - 16 * q^87 - 4 * q^88 + 4 * q^89 + 2 * q^90 - 8 * q^91 - 4 * q^92 - 12 * q^93 - 16 * q^94 - 8 * q^95 - 2 * q^96 - 20 * q^97 + 2 * q^98 - 4 * q^99

Introduction

L-functions

Modular forms

Varieties

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Database

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