LMFDB - Embedded newform 8670.2.a.ca.1.4

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:	$4$
Coefficient field:	4.4.4352.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 6x^{2} - 4x + 2 $ x^4 - 6x^2 - 4x + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 510)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$2.68554$ 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)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{6} +3.79793 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{6} +3.79793 q^{7} +1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{10} -0.224777 q^{11} +1.00000 q^{12} -0.158942 q^{13} +3.79793 q^{14} +1.00000 q^{15} +1.00000 q^{16} +1.00000 q^{18} +4.76744 q^{19} +1.00000 q^{20} +3.79793 q^{21} -0.224777 q^{22} +0.585786 q^{23} +1.00000 q^{24} +1.00000 q^{25} -0.158942 q^{26} +1.00000 q^{27} +3.79793 q^{28} -2.22478 q^{29} +1.00000 q^{30} +8.78530 q^{31} +1.00000 q^{32} -0.224777 q^{33} +3.79793 q^{35} +1.00000 q^{36} +2.76744 q^{37} +4.76744 q^{38} -0.158942 q^{39} +1.00000 q^{40} +0.712611 q^{41} +3.79793 q^{42} -3.21215 q^{43} -0.224777 q^{44} +1.00000 q^{45} +0.585786 q^{46} -3.91375 q^{47} +1.00000 q^{48} +7.42429 q^{49} +1.00000 q^{50} -0.158942 q^{52} -9.19173 q^{53} +1.00000 q^{54} -0.224777 q^{55} +3.79793 q^{56} +4.76744 q^{57} -2.22478 q^{58} +1.39150 q^{59} +1.00000 q^{60} -5.88163 q^{61} +8.78530 q^{62} +3.79793 q^{63} +1.00000 q^{64} -0.158942 q^{65} -0.224777 q^{66} +0.383719 q^{67} +0.585786 q^{69} +3.79793 q^{70} -13.4548 q^{71} +1.00000 q^{72} +5.79793 q^{73} +2.76744 q^{74} +1.00000 q^{75} +4.76744 q^{76} -0.853690 q^{77} -0.158942 q^{78} -14.1885 q^{79} +1.00000 q^{80} +1.00000 q^{81} +0.712611 q^{82} -5.53488 q^{83} +3.79793 q^{84} -3.21215 q^{86} -2.22478 q^{87} -0.224777 q^{88} -0.424292 q^{89} +1.00000 q^{90} -0.603650 q^{91} +0.585786 q^{92} +8.78530 q^{93} -3.91375 q^{94} +4.76744 q^{95} +1.00000 q^{96} -1.79793 q^{97} +7.42429 q^{98} -0.224777 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{2} + 4 q^{3} + 4 q^{4} + 4 q^{5} + 4 q^{6} + 4 q^{7} + 4 q^{8} + 4 q^{9}+O(q^{10}) $ 4 * q + 4 * q^2 + 4 * q^3 + 4 * q^4 + 4 * q^5 + 4 * q^6 + 4 * q^7 + 4 * q^8 + 4 * q^9	$ 4 q + 4 q^{2} + 4 q^{3} + 4 q^{4} + 4 q^{5} + 4 q^{6} + 4 q^{7} + 4 q^{8} + 4 q^{9} + 4 q^{10} + 4 q^{12} + 4 q^{13} + 4 q^{14} + 4 q^{15} + 4 q^{16} + 4 q^{18} + 8 q^{19} + 4 q^{20} + 4 q^{21} + 8 q^{23} + 4 q^{24} + 4 q^{25} + 4 q^{26} + 4 q^{27} + 4 q^{28} - 8 q^{29} + 4 q^{30} + 8 q^{31} + 4 q^{32} + 4 q^{35} + 4 q^{36} + 8 q^{38} + 4 q^{39} + 4 q^{40} + 12 q^{41} + 4 q^{42} + 4 q^{43} + 4 q^{45} + 8 q^{46} + 16 q^{47} + 4 q^{48} - 4 q^{49} + 4 q^{50} + 4 q^{52} + 8 q^{53} + 4 q^{54} + 4 q^{56} + 8 q^{57} - 8 q^{58} + 12 q^{59} + 4 q^{60} + 8 q^{62} + 4 q^{63} + 4 q^{64} + 4 q^{65} - 4 q^{67} + 8 q^{69} + 4 q^{70} - 20 q^{71} + 4 q^{72} + 12 q^{73} + 4 q^{75} + 8 q^{76} - 24 q^{77} + 4 q^{78} + 4 q^{80} + 4 q^{81} + 12 q^{82} + 4 q^{84} + 4 q^{86} - 8 q^{87} + 32 q^{89} + 4 q^{90} + 8 q^{91} + 8 q^{92} + 8 q^{93} + 16 q^{94} + 8 q^{95} + 4 q^{96} + 4 q^{97} - 4 q^{98}+O(q^{100}) $ 4 * q + 4 * q^2 + 4 * q^3 + 4 * q^4 + 4 * q^5 + 4 * q^6 + 4 * q^7 + 4 * q^8 + 4 * q^9 + 4 * q^10 + 4 * q^12 + 4 * q^13 + 4 * q^14 + 4 * q^15 + 4 * q^16 + 4 * q^18 + 8 * q^19 + 4 * q^20 + 4 * q^21 + 8 * q^23 + 4 * q^24 + 4 * q^25 + 4 * q^26 + 4 * q^27 + 4 * q^28 - 8 * q^29 + 4 * q^30 + 8 * q^31 + 4 * q^32 + 4 * q^35 + 4 * q^36 + 8 * q^38 + 4 * q^39 + 4 * q^40 + 12 * q^41 + 4 * q^42 + 4 * q^43 + 4 * q^45 + 8 * q^46 + 16 * q^47 + 4 * q^48 - 4 * q^49 + 4 * q^50 + 4 * q^52 + 8 * q^53 + 4 * q^54 + 4 * q^56 + 8 * q^57 - 8 * q^58 + 12 * q^59 + 4 * q^60 + 8 * q^62 + 4 * q^63 + 4 * q^64 + 4 * q^65 - 4 * q^67 + 8 * q^69 + 4 * q^70 - 20 * q^71 + 4 * q^72 + 12 * q^73 + 4 * q^75 + 8 * q^76 - 24 * q^77 + 4 * q^78 + 4 * q^80 + 4 * q^81 + 12 * q^82 + 4 * q^84 + 4 * q^86 - 8 * q^87 + 32 * q^89 + 4 * q^90 + 8 * q^91 + 8 * q^92 + 8 * q^93 + 16 * q^94 + 8 * q^95 + 4 * q^96 + 4 * q^97 - 4 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	1.00000	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
$7$	3.79793	1.43548	0.717742	−	0.696309i	$-0.245176\pi$
$7$	3.79793	1.43548	0.717742	+	0.696309i	$0.245176\pi$
$8$	1.00000	0.353553
$9$	1.00000	0.333333
$10$	1.00000	0.316228
$11$	−0.224777	−0.0677729	−0.0338865	−	0.999426i	$-0.510788\pi$
$11$	−0.224777	−0.0677729	−0.0338865	+	0.999426i	$0.510788\pi$
$12$	1.00000	0.288675
$13$	−0.158942	−0.0440825	−0.0220412	−	0.999757i	$-0.507017\pi$
$13$	−0.158942	−0.0440825	−0.0220412	+	0.999757i	$0.507017\pi$
$14$	3.79793	1.01504
$15$	1.00000	0.258199
$16$	1.00000	0.250000
$17$	0	0
$17$	0	0
$18$	1.00000	0.235702
$19$	4.76744	1.09373	0.546863	−	0.837222i	$-0.315822\pi$
$19$	4.76744	1.09373	0.546863	+	0.837222i	$0.315822\pi$
$20$	1.00000	0.223607
$21$	3.79793	0.828777
$22$	−0.224777	−0.0479227
$23$	0.585786	0.122145	0.0610725	−	0.998133i	$-0.480548\pi$
$23$	0.585786	0.122145	0.0610725	+	0.998133i	$0.480548\pi$
$24$	1.00000	0.204124
$25$	1.00000	0.200000
$26$	−0.158942	−0.0311710
$27$	1.00000	0.192450
$28$	3.79793	0.717742
$29$	−2.22478	−0.413131	−0.206565	−	0.978433i	$-0.566229\pi$
$29$	−2.22478	−0.413131	−0.206565	+	0.978433i	$0.566229\pi$
$30$	1.00000	0.182574
$31$	8.78530	1.57789	0.788943	−	0.614466i	$-0.210628\pi$
$31$	8.78530	1.57789	0.788943	+	0.614466i	$0.210628\pi$
$32$	1.00000	0.176777
$33$	−0.224777	−0.0391287
$34$	0	0
$35$	3.79793	0.641968
$36$	1.00000	0.166667
$37$	2.76744	0.454964	0.227482	−	0.973782i	$-0.426951\pi$
$37$	2.76744	0.454964	0.227482	+	0.973782i	$0.426951\pi$
$38$	4.76744	0.773381
$39$	−0.158942	−0.0254510
$40$	1.00000	0.158114
$41$	0.712611	0.111291	0.0556456	−	0.998451i	$-0.482278\pi$
$41$	0.712611	0.111291	0.0556456	+	0.998451i	$0.482278\pi$
$42$	3.79793	0.586034
$43$	−3.21215	−0.489848	−0.244924	−	0.969542i	$-0.578763\pi$
$43$	−3.21215	−0.489848	−0.244924	+	0.969542i	$0.578763\pi$
$44$	−0.224777	−0.0338865
$45$	1.00000	0.149071
$46$	0.585786	0.0863695
$47$	−3.91375	−0.570879	−0.285439	−	0.958397i	$-0.592140\pi$
$47$	−3.91375	−0.570879	−0.285439	+	0.958397i	$0.592140\pi$
$48$	1.00000	0.144338
$49$	7.42429	1.06061
$50$	1.00000	0.141421
$51$	0	0
$52$	−0.158942	−0.0220412
$53$	−9.19173	−1.26258	−0.631291	−	0.775546i	$-0.717474\pi$
$53$	−9.19173	−1.26258	−0.631291	+	0.775546i	$0.717474\pi$
$54$	1.00000	0.136083
$55$	−0.224777	−0.0303090
$56$	3.79793	0.507520
$57$	4.76744	0.631463
$58$	−2.22478	−0.292128
$59$	1.39150	0.181158	0.0905792	−	0.995889i	$-0.471128\pi$
$59$	1.39150	0.181158	0.0905792	+	0.995889i	$0.471128\pi$
$60$	1.00000	0.129099
$61$	−5.88163	−0.753066	−0.376533	−	0.926403i	$-0.622884\pi$
$61$	−5.88163	−0.753066	−0.376533	+	0.926403i	$0.622884\pi$
$62$	8.78530	1.11573
$63$	3.79793	0.478495
$64$	1.00000	0.125000
$65$	−0.158942	−0.0197143
$66$	−0.224777	−0.0276682
$67$	0.383719	0.0468787	0.0234394	−	0.999725i	$-0.492538\pi$
$67$	0.383719	0.0468787	0.0234394	+	0.999725i	$0.492538\pi$
$68$	0	0
$69$	0.585786	0.0705204
$70$	3.79793	0.453940
$71$	−13.4548	−1.59679	−0.798395	−	0.602134i	$-0.794317\pi$
$71$	−13.4548	−1.59679	−0.798395	+	0.602134i	$0.794317\pi$
$72$	1.00000	0.117851
$73$	5.79793	0.678597	0.339298	−	0.940679i	$-0.389810\pi$
$73$	5.79793	0.678597	0.339298	+	0.940679i	$0.389810\pi$
$74$	2.76744	0.321708
$75$	1.00000	0.115470
$76$	4.76744	0.546863
$77$	−0.853690	−0.0972870
$78$	−0.158942	−0.0179966
$79$	−14.1885	−1.59633	−0.798166	−	0.602438i	$-0.794196\pi$
$79$	−14.1885	−1.59633	−0.798166	+	0.602438i	$0.794196\pi$
$80$	1.00000	0.111803
$81$	1.00000	0.111111
$82$	0.712611	0.0786947
$83$	−5.53488	−0.607532	−0.303766	−	0.952747i	$-0.598244\pi$
$83$	−5.53488	−0.607532	−0.303766	+	0.952747i	$0.598244\pi$
$84$	3.79793	0.414388
$85$	0	0
$86$	−3.21215	−0.346375
$87$	−2.22478	−0.238521
$88$	−0.224777	−0.0239614
$89$	−0.424292	−0.0449749	−0.0224875	−	0.999747i	$-0.507159\pi$
$89$	−0.424292	−0.0449749	−0.0224875	+	0.999747i	$0.507159\pi$
$90$	1.00000	0.105409
$91$	−0.603650	−0.0632797
$92$	0.585786	0.0610725
$93$	8.78530	0.910993
$94$	−3.91375	−0.403672
$95$	4.76744	0.489129
$96$	1.00000	0.102062
$97$	−1.79793	−0.182552	−0.0912762	−	0.995826i	$-0.529095\pi$
$97$	−1.79793	−0.182552	−0.0912762	+	0.995826i	$0.529095\pi$
$98$	7.42429	0.749967
$99$	−0.224777	−0.0225910
$100$	1.00000	0.100000
$101$	−3.59102	−0.357320	−0.178660	−	0.983911i	$-0.557176\pi$
$101$	−3.59102	−0.357320	−0.178660	+	0.983911i	$0.557176\pi$
$102$	0	0
$103$	17.5706	1.73128	0.865641	−	0.500664i	$-0.166911\pi$
$103$	17.5706	1.73128	0.865641	+	0.500664i	$0.166911\pi$
$104$	−0.158942	−0.0155855
$105$	3.79793	0.370640
$106$	−9.19173	−0.892780
$107$	7.79538	0.753608	0.376804	−	0.926293i	$-0.377023\pi$
$107$	7.79538	0.753608	0.376804	+	0.926293i	$0.377023\pi$
$108$	1.00000	0.0962250
$109$	14.8032	1.41789	0.708943	−	0.705266i	$-0.249172\pi$
$109$	14.8032	1.41789	0.708943	+	0.705266i	$0.249172\pi$
$110$	−0.224777	−0.0214317
$111$	2.76744	0.262674
$112$	3.79793	0.358871
$113$	12.7600	1.20036	0.600182	−	0.799864i	$-0.295095\pi$
$113$	12.7600	1.20036	0.600182	+	0.799864i	$0.295095\pi$
$114$	4.76744	0.446511
$115$	0.585786	0.0546249
$116$	−2.22478	−0.206565
$117$	−0.158942	−0.0146942
$118$	1.39150	0.128098
$119$	0	0
$120$	1.00000	0.0912871
$121$	−10.9495	−0.995407
$122$	−5.88163	−0.532498
$123$	0.712611	0.0642540
$124$	8.78530	0.788943
$125$	1.00000	0.0894427
$126$	3.79793	0.338347
$127$	11.8816	1.05432	0.527162	−	0.849765i	$-0.323256\pi$
$127$	11.8816	1.05432	0.527162	+	0.849765i	$0.323256\pi$
$128$	1.00000	0.0883883
$129$	−3.21215	−0.282814
$130$	−0.158942	−0.0139401
$131$	−11.4642	−1.00163	−0.500816	−	0.865554i	$-0.666967\pi$
$131$	−11.4642	−1.00163	−0.500816	+	0.865554i	$0.666967\pi$
$132$	−0.224777	−0.0195644
$133$	18.1064	1.57002
$134$	0.383719	0.0331483
$135$	1.00000	0.0860663
$136$	0	0
$137$	1.80734	0.154411	0.0772057	−	0.997015i	$-0.475400\pi$
$137$	1.80734	0.154411	0.0772057	+	0.997015i	$0.475400\pi$
$138$	0.585786	0.0498655
$139$	−10.7101	−0.908415	−0.454208	−	0.890896i	$-0.650078\pi$
$139$	−10.7101	−0.908415	−0.454208	+	0.890896i	$0.650078\pi$
$140$	3.79793	0.320984
$141$	−3.91375	−0.329597
$142$	−13.4548	−1.12910
$143$	0.0357265	0.00298760
$144$	1.00000	0.0833333
$145$	−2.22478	−0.184758
$146$	5.79793	0.479840
$147$	7.42429	0.612345
$148$	2.76744	0.227482
$149$	18.9332	1.55107	0.775535	−	0.631304i	$-0.217480\pi$
$149$	18.9332	1.55107	0.775535	+	0.631304i	$0.217480\pi$
$150$	1.00000	0.0816497
$151$	2.16740	0.176380	0.0881902	−	0.996104i	$-0.471892\pi$
$151$	2.16740	0.176380	0.0881902	+	0.996104i	$0.471892\pi$
$152$	4.76744	0.386690
$153$	0	0
$154$	−0.853690	−0.0687923
$155$	8.78530	0.705652
$156$	−0.158942	−0.0127255
$157$	16.5222	1.31862	0.659309	−	0.751872i	$-0.270849\pi$
$157$	16.5222	1.31862	0.659309	+	0.751872i	$0.270849\pi$
$158$	−14.1885	−1.12878
$159$	−9.19173	−0.728952
$160$	1.00000	0.0790569
$161$	2.22478	0.175337
$162$	1.00000	0.0785674
$163$	−20.4921	−1.60507	−0.802534	−	0.596606i	$-0.796515\pi$
$163$	−20.4921	−1.60507	−0.802534	+	0.596606i	$0.796515\pi$
$164$	0.712611	0.0556456
$165$	−0.224777	−0.0174989
$166$	−5.53488	−0.429590
$167$	−16.0312	−1.24053	−0.620264	−	0.784393i	$-0.712975\pi$
$167$	−16.0312	−1.24053	−0.620264	+	0.784393i	$0.712975\pi$
$168$	3.79793	0.293017
$169$	−12.9747	−0.998057
$170$	0	0
$171$	4.76744	0.364575
$172$	−3.21215	−0.244924
$173$	−24.4921	−1.86210	−0.931051	−	0.364888i	$-0.881107\pi$
$173$	−24.4921	−1.86210	−0.931051	+	0.364888i	$0.881107\pi$
$174$	−2.22478	−0.168660
$175$	3.79793	0.287097
$176$	−0.224777	−0.0169432
$177$	1.39150	0.104592
$178$	−0.424292	−0.0318021
$179$	10.3263	0.771827	0.385913	−	0.922535i	$-0.373886\pi$
$179$	10.3263	0.771827	0.385913	+	0.922535i	$0.373886\pi$
$180$	1.00000	0.0745356
$181$	16.8233	1.25047	0.625234	−	0.780437i	$-0.285004\pi$
$181$	16.8233	1.25047	0.625234	+	0.780437i	$0.285004\pi$
$182$	−0.603650	−0.0447455
$183$	−5.88163	−0.434783
$184$	0.585786	0.0431847
$185$	2.76744	0.203466
$186$	8.78530	0.644170
$187$	0	0
$188$	−3.91375	−0.285439
$189$	3.79793	0.276259
$190$	4.76744	0.345866
$191$	16.7996	1.21557	0.607787	−	0.794100i	$-0.292058\pi$
$191$	16.7996	1.21557	0.607787	+	0.794100i	$0.292058\pi$
$192$	1.00000	0.0721688
$193$	−11.6222	−0.836583	−0.418292	−	0.908313i	$-0.637371\pi$
$193$	−11.6222	−0.836583	−0.418292	+	0.908313i	$0.637371\pi$
$194$	−1.79793	−0.129084
$195$	−0.158942	−0.0113820
$196$	7.42429	0.530307
$197$	15.3711	1.09514	0.547572	−	0.836758i	$-0.315552\pi$
$197$	15.3711	1.09514	0.547572	+	0.836758i	$0.315552\pi$
$198$	−0.224777	−0.0159742
$199$	−10.0431	−0.711938	−0.355969	−	0.934498i	$-0.615849\pi$
$199$	−10.0431	−0.711938	−0.355969	+	0.934498i	$0.615849\pi$
$200$	1.00000	0.0707107
$201$	0.383719	0.0270655
$202$	−3.59102	−0.252663
$203$	−8.44955	−0.593042
$204$	0	0
$205$	0.712611	0.0497709
$206$	17.5706	1.22420
$207$	0.585786	0.0407150
$208$	−0.158942	−0.0110206
$209$	−1.07161	−0.0741250
$210$	3.79793	0.262082
$211$	13.5064	0.929817	0.464908	−	0.885359i	$-0.346087\pi$
$211$	13.5064	0.929817	0.464908	+	0.885359i	$0.346087\pi$
$212$	−9.19173	−0.631291
$213$	−13.4548	−0.921907
$214$	7.79538	0.532881
$215$	−3.21215	−0.219067
$216$	1.00000	0.0680414
$217$	33.3660	2.26503
$218$	14.8032	1.00260
$219$	5.79793	0.391788
$220$	−0.224777	−0.0151545
$221$	0	0
$222$	2.76744	0.185738
$223$	−9.50637	−0.636594	−0.318297	−	0.947991i	$-0.603111\pi$
$223$	−9.50637	−0.636594	−0.318297	+	0.947991i	$0.603111\pi$
$224$	3.79793	0.253760
$225$	1.00000	0.0666667
$226$	12.7600	0.848785
$227$	−24.2807	−1.61156	−0.805782	−	0.592212i	$-0.798255\pi$
$227$	−24.2807	−1.61156	−0.805782	+	0.592212i	$0.798255\pi$
$228$	4.76744	0.315731
$229$	16.1064	1.06434	0.532171	−	0.846637i	$-0.321376\pi$
$229$	16.1064	1.06434	0.532171	+	0.846637i	$0.321376\pi$
$230$	0.585786	0.0386256
$231$	−0.853690	−0.0561687
$232$	−2.22478	−0.146064
$233$	−2.14953	−0.140821	−0.0704104	−	0.997518i	$-0.522431\pi$
$233$	−2.14953	−0.140821	−0.0704104	+	0.997518i	$0.522431\pi$
$234$	−0.158942	−0.0103903
$235$	−3.91375	−0.255305
$236$	1.39150	0.0905792
$237$	−14.1885	−0.921643
$238$	0	0
$239$	−23.6229	−1.52804	−0.764018	−	0.645194i	$-0.776776\pi$
$239$	−23.6229	−1.52804	−0.764018	+	0.645194i	$0.776776\pi$
$240$	1.00000	0.0645497
$241$	−26.3848	−1.69959	−0.849796	−	0.527111i	$-0.823275\pi$
$241$	−26.3848	−1.69959	−0.849796	+	0.527111i	$0.823275\pi$
$242$	−10.9495	−0.703859
$243$	1.00000	0.0641500
$244$	−5.88163	−0.376533
$245$	7.42429	0.474321
$246$	0.712611	0.0454344
$247$	−0.757744	−0.0482141
$248$	8.78530	0.557867
$249$	−5.53488	−0.350759
$250$	1.00000	0.0632456
$251$	20.6695	1.30465	0.652323	−	0.757941i	$-0.273795\pi$
$251$	20.6695	1.30465	0.652323	+	0.757941i	$0.273795\pi$
$252$	3.79793	0.239247
$253$	−0.131672	−0.00827812
$254$	11.8816	0.745520
$255$	0	0
$256$	1.00000	0.0625000
$257$	12.5664	0.783872	0.391936	−	0.919992i	$-0.371805\pi$
$257$	12.5664	0.783872	0.391936	+	0.919992i	$0.371805\pi$
$258$	−3.21215	−0.199979
$259$	10.5105	0.653093
$260$	−0.158942	−0.00985714
$261$	−2.22478	−0.137710
$262$	−11.4642	−0.708260
$263$	13.4000	0.826277	0.413138	−	0.910668i	$-0.364433\pi$
$263$	13.4000	0.826277	0.413138	+	0.910668i	$0.364433\pi$
$264$	−0.224777	−0.0138341
$265$	−9.19173	−0.564644
$266$	18.1064	1.11018
$267$	−0.424292	−0.0259663
$268$	0.383719	0.0234394
$269$	−23.8431	−1.45374	−0.726869	−	0.686776i	$-0.759025\pi$
$269$	−23.8431	−1.45374	−0.726869	+	0.686776i	$0.759025\pi$
$270$	1.00000	0.0608581
$271$	−0.510544	−0.0310133	−0.0155067	−	0.999880i	$-0.504936\pi$
$271$	−0.510544	−0.0310133	−0.0155067	+	0.999880i	$0.504936\pi$
$272$	0	0
$273$	−0.603650	−0.0365345
$274$	1.80734	0.109185
$275$	−0.224777	−0.0135546
$276$	0.585786	0.0352602
$277$	−29.5619	−1.77620	−0.888101	−	0.459649i	$-0.847975\pi$
$277$	−29.5619	−1.77620	−0.888101	+	0.459649i	$0.847975\pi$
$278$	−10.7101	−0.642347
$279$	8.78530	0.525962
$280$	3.79793	0.226970
$281$	−4.66103	−0.278054	−0.139027	−	0.990289i	$-0.544397\pi$
$281$	−4.66103	−0.278054	−0.139027	+	0.990289i	$0.544397\pi$
$282$	−3.91375	−0.233060
$283$	1.34583	0.0800010	0.0400005	−	0.999200i	$-0.487264\pi$
$283$	1.34583	0.0800010	0.0400005	+	0.999200i	$0.487264\pi$
$284$	−13.4548	−0.798395
$285$	4.76744	0.282399
$286$	0.0357265	0.00211255
$287$	2.70645	0.159757
$288$	1.00000	0.0589256
$289$	0	0
$290$	−2.22478	−0.130643
$291$	−1.79793	−0.105397
$292$	5.79793	0.339298
$293$	−5.76326	−0.336693	−0.168347	−	0.985728i	$-0.553843\pi$
$293$	−5.76326	−0.336693	−0.168347	+	0.985728i	$0.553843\pi$
$294$	7.42429	0.432994
$295$	1.39150	0.0810165
$296$	2.76744	0.160854
$297$	−0.224777	−0.0130429
$298$	18.9332	1.09677
$299$	−0.0931059	−0.00538445
$300$	1.00000	0.0577350
$301$	−12.1995	−0.703168
$302$	2.16740	0.124720
$303$	−3.59102	−0.206299
$304$	4.76744	0.273431
$305$	−5.88163	−0.336781
$306$	0	0
$307$	−14.2722	−0.814558	−0.407279	−	0.913304i	$-0.633522\pi$
$307$	−14.2722	−0.814558	−0.407279	+	0.913304i	$0.633522\pi$
$308$	−0.853690	−0.0486435
$309$	17.5706	0.999557
$310$	8.78530	0.498972
$311$	−25.9686	−1.47254	−0.736271	−	0.676686i	$-0.763415\pi$
$311$	−25.9686	−1.47254	−0.736271	+	0.676686i	$0.763415\pi$
$312$	−0.158942	−0.00899830
$313$	23.4497	1.32545	0.662727	−	0.748861i	$-0.269399\pi$
$313$	23.4497	1.32545	0.662727	+	0.748861i	$0.269399\pi$
$314$	16.5222	0.932404
$315$	3.79793	0.213989
$316$	−14.1885	−0.798166
$317$	26.0490	1.46306	0.731530	−	0.681810i	$-0.238807\pi$
$317$	26.0490	1.46306	0.731530	+	0.681810i	$0.238807\pi$
$318$	−9.19173	−0.515447
$319$	0.500080	0.0279991
$320$	1.00000	0.0559017
$321$	7.79538	0.435096
$322$	2.22478	0.123982
$323$	0	0
$324$	1.00000	0.0555556
$325$	−0.158942	−0.00881650
$326$	−20.4921	−1.13495
$327$	14.8032	0.818617
$328$	0.712611	0.0393474
$329$	−14.8642	−0.819487
$330$	−0.224777	−0.0123736
$331$	−19.6770	−1.08155	−0.540773	−	0.841168i	$-0.681868\pi$
$331$	−19.6770	−1.08155	−0.540773	+	0.841168i	$0.681868\pi$
$332$	−5.53488	−0.303766
$333$	2.76744	0.151655
$334$	−16.0312	−0.877186
$335$	0.383719	0.0209648
$336$	3.79793	0.207194
$337$	2.71586	0.147942	0.0739711	−	0.997260i	$-0.476433\pi$
$337$	2.71586	0.147942	0.0739711	+	0.997260i	$0.476433\pi$
$338$	−12.9747	−0.705733
$339$	12.7600	0.693030
$340$	0	0
$341$	−1.97474	−0.106938
$342$	4.76744	0.257794
$343$	1.61143	0.0870093
$344$	−3.21215	−0.173187
$345$	0.585786	0.0315377
$346$	−24.4921	−1.31671
$347$	6.62381	0.355585	0.177792	−	0.984068i	$-0.443104\pi$
$347$	6.62381	0.355585	0.177792	+	0.984068i	$0.443104\pi$
$348$	−2.22478	−0.119261
$349$	31.4792	1.68505	0.842523	−	0.538661i	$-0.181070\pi$
$349$	31.4792	1.68505	0.842523	+	0.538661i	$0.181070\pi$
$350$	3.79793	0.203008
$351$	−0.158942	−0.00848368
$352$	−0.224777	−0.0119807
$353$	−27.1256	−1.44375	−0.721876	−	0.692022i	$-0.756720\pi$
$353$	−27.1256	−1.44375	−0.721876	+	0.692022i	$0.756720\pi$
$354$	1.39150	0.0739576
$355$	−13.4548	−0.714106
$356$	−0.424292	−0.0224875
$357$	0	0
$358$	10.3263	0.545764
$359$	27.8444	1.46957	0.734786	−	0.678299i	$-0.237283\pi$
$359$	27.8444	1.46957	0.734786	+	0.678299i	$0.237283\pi$
$360$	1.00000	0.0527046
$361$	3.72847	0.196235
$362$	16.8233	0.884214
$363$	−10.9495	−0.574698
$364$	−0.603650	−0.0316398
$365$	5.79793	0.303478
$366$	−5.88163	−0.307438
$367$	15.7474	0.822008	0.411004	−	0.911634i	$-0.365178\pi$
$367$	15.7474	0.822008	0.411004	+	0.911634i	$0.365178\pi$
$368$	0.585786	0.0305362
$369$	0.712611	0.0370971
$370$	2.76744	0.143872
$371$	−34.9096	−1.81241
$372$	8.78530	0.455497
$373$	1.94747	0.100836	0.0504180	−	0.998728i	$-0.483945\pi$
$373$	1.94747	0.100836	0.0504180	+	0.998728i	$0.483945\pi$
$374$	0	0
$375$	1.00000	0.0516398
$376$	−3.91375	−0.201836
$377$	0.353610	0.0182118
$378$	3.79793	0.195345
$379$	30.3871	1.56088	0.780439	−	0.625231i	$-0.214995\pi$
$379$	30.3871	1.56088	0.780439	+	0.625231i	$0.214995\pi$
$380$	4.76744	0.244564
$381$	11.8816	0.608714
$382$	16.7996	0.859540
$383$	−33.4403	−1.70872	−0.854359	−	0.519683i	$-0.826050\pi$
$383$	−33.4403	−1.70872	−0.854359	+	0.519683i	$0.826050\pi$
$384$	1.00000	0.0510310
$385$	−0.853690	−0.0435080
$386$	−11.6222	−0.591554
$387$	−3.21215	−0.163283
$388$	−1.79793	−0.0912762
$389$	−23.0818	−1.17029	−0.585147	−	0.810927i	$-0.698963\pi$
$389$	−23.0818	−1.17029	−0.585147	+	0.810927i	$0.698963\pi$
$390$	−0.158942	−0.00804832
$391$	0	0
$392$	7.42429	0.374983
$393$	−11.4642	−0.578292
$394$	15.3711	0.774384
$395$	−14.1885	−0.713901
$396$	−0.224777	−0.0112955
$397$	−15.9679	−0.801405	−0.400703	−	0.916208i	$-0.631234\pi$
$397$	−15.9679	−0.801405	−0.400703	+	0.916208i	$0.631234\pi$
$398$	−10.0431	−0.503416
$399$	18.1064	0.906454
$400$	1.00000	0.0500000
$401$	20.0043	0.998967	0.499484	−	0.866323i	$-0.333523\pi$
$401$	20.0043	0.998967	0.499484	+	0.866323i	$0.333523\pi$
$402$	0.383719	0.0191382
$403$	−1.39635	−0.0695572
$404$	−3.59102	−0.178660
$405$	1.00000	0.0496904
$406$	−8.44955	−0.419344
$407$	−0.622058	−0.0308343
$408$	0	0
$409$	8.48853	0.419731	0.209865	−	0.977730i	$-0.432697\pi$
$409$	8.48853	0.419731	0.209865	+	0.977730i	$0.432697\pi$
$410$	0.712611	0.0351934
$411$	1.80734	0.0891495
$412$	17.5706	0.865641
$413$	5.28484	0.260050
$414$	0.585786	0.0287898
$415$	−5.53488	−0.271696
$416$	−0.158942	−0.00779276
$417$	−10.7101	−0.524474
$418$	−1.07161	−0.0524143
$419$	31.0124	1.51505	0.757527	−	0.652804i	$-0.226408\pi$
$419$	31.0124	1.51505	0.757527	+	0.652804i	$0.226408\pi$
$420$	3.79793	0.185320
$421$	30.0306	1.46360	0.731801	−	0.681518i	$-0.238680\pi$
$421$	30.0306	1.46360	0.731801	+	0.681518i	$0.238680\pi$
$422$	13.5064	0.657480
$423$	−3.91375	−0.190293
$424$	−9.19173	−0.446390
$425$	0	0
$426$	−13.4548	−0.651887
$427$	−22.3380	−1.08101
$428$	7.79538	0.376804
$429$	0.0357265	0.00172489
$430$	−3.21215	−0.154903
$431$	−13.0558	−0.628874	−0.314437	−	0.949278i	$-0.601816\pi$
$431$	−13.0558	−0.628874	−0.314437	+	0.949278i	$0.601816\pi$
$432$	1.00000	0.0481125
$433$	21.3004	1.02363	0.511816	−	0.859095i	$-0.328973\pi$
$433$	21.3004	1.02363	0.511816	+	0.859095i	$0.328973\pi$
$434$	33.3660	1.60162
$435$	−2.22478	−0.106670
$436$	14.8032	0.708943
$437$	2.79270	0.133593
$438$	5.79793	0.277036
$439$	−35.7857	−1.70796	−0.853980	−	0.520307i	$-0.825818\pi$
$439$	−35.7857	−1.70796	−0.853980	+	0.520307i	$0.825818\pi$
$440$	−0.224777	−0.0107158
$441$	7.42429	0.353538
$442$	0	0
$443$	23.7990	1.13072	0.565362	−	0.824843i	$-0.308736\pi$
$443$	23.7990	1.13072	0.565362	+	0.824843i	$0.308736\pi$
$444$	2.76744	0.131337
$445$	−0.424292	−0.0201134
$446$	−9.50637	−0.450140
$447$	18.9332	0.895511
$448$	3.79793	0.179435
$449$	−8.24104	−0.388919	−0.194459	−	0.980911i	$-0.562295\pi$
$449$	−8.24104	−0.388919	−0.194459	+	0.980911i	$0.562295\pi$
$450$	1.00000	0.0471405
$451$	−0.160179	−0.00754253
$452$	12.7600	0.600182
$453$	2.16740	0.101833
$454$	−24.2807	−1.13955
$455$	−0.603650	−0.0282995
$456$	4.76744	0.223256
$457$	−33.8408	−1.58301	−0.791503	−	0.611165i	$-0.790701\pi$
$457$	−33.8408	−1.58301	−0.791503	+	0.611165i	$0.790701\pi$
$458$	16.1064	0.752603
$459$	0	0
$460$	0.585786	0.0273124
$461$	3.71300	0.172931	0.0864657	−	0.996255i	$-0.472443\pi$
$461$	3.71300	0.172931	0.0864657	+	0.996255i	$0.472443\pi$
$462$	−0.853690	−0.0397172
$463$	14.7779	0.686787	0.343394	−	0.939192i	$-0.388423\pi$
$463$	14.7779	0.686787	0.343394	+	0.939192i	$0.388423\pi$
$464$	−2.22478	−0.103283
$465$	8.78530	0.407409
$466$	−2.14953	−0.0995753
$467$	7.19684	0.333030	0.166515	−	0.986039i	$-0.446749\pi$
$467$	7.19684	0.333030	0.166515	+	0.986039i	$0.446749\pi$
$468$	−0.158942	−0.00734708
$469$	1.45734	0.0672937
$470$	−3.91375	−0.180528
$471$	16.5222	0.761305
$472$	1.39150	0.0640491
$473$	0.722018	0.0331984
$474$	−14.1885	−0.651700
$475$	4.76744	0.218745
$476$	0	0
$477$	−9.19173	−0.420860
$478$	−23.6229	−1.08049
$479$	28.6465	1.30889	0.654446	−	0.756108i	$-0.272902\pi$
$479$	28.6465	1.30889	0.654446	+	0.756108i	$0.272902\pi$
$480$	1.00000	0.0456435
$481$	−0.439861	−0.0200559
$482$	−26.3848	−1.20179
$483$	2.22478	0.101231
$484$	−10.9495	−0.497703
$485$	−1.79793	−0.0816399
$486$	1.00000	0.0453609
$487$	−36.5603	−1.65670	−0.828352	−	0.560208i	$-0.810721\pi$
$487$	−36.5603	−1.65670	−0.828352	+	0.560208i	$0.810721\pi$
$488$	−5.88163	−0.266249
$489$	−20.4921	−0.926686
$490$	7.42429	0.335395
$491$	−4.53781	−0.204789	−0.102394	−	0.994744i	$-0.532650\pi$
$491$	−4.53781	−0.204789	−0.102394	+	0.994744i	$0.532650\pi$
$492$	0.712611	0.0321270
$493$	0	0
$494$	−0.757744	−0.0340925
$495$	−0.224777	−0.0101030
$496$	8.78530	0.394472
$497$	−51.1004	−2.29217
$498$	−5.53488	−0.248024
$499$	41.7257	1.86790	0.933949	−	0.357407i	$-0.116339\pi$
$499$	41.7257	1.86790	0.933949	+	0.357407i	$0.116339\pi$
$500$	1.00000	0.0447214
$501$	−16.0312	−0.716220
$502$	20.6695	0.922524
$503$	−41.0757	−1.83147	−0.915737	−	0.401779i	$-0.868392\pi$
$503$	−41.0757	−1.83147	−0.915737	+	0.401779i	$0.868392\pi$
$504$	3.79793	0.169173
$505$	−3.59102	−0.159798
$506$	−0.131672	−0.00585352
$507$	−12.9747	−0.576228
$508$	11.8816	0.527162
$509$	−19.6364	−0.870370	−0.435185	−	0.900341i	$-0.643317\pi$
$509$	−19.6364	−0.870370	−0.435185	+	0.900341i	$0.643317\pi$
$510$	0	0
$511$	22.0202	0.974114
$512$	1.00000	0.0441942
$513$	4.76744	0.210488
$514$	12.5664	0.554281
$515$	17.5706	0.774253
$516$	−3.21215	−0.141407
$517$	0.879722	0.0386901
$518$	10.5105	0.461807
$519$	−24.4921	−1.07509
$520$	−0.158942	−0.00697005
$521$	−17.1856	−0.752913	−0.376457	−	0.926434i	$-0.622858\pi$
$521$	−17.1856	−0.752913	−0.376457	+	0.926434i	$0.622858\pi$
$522$	−2.22478	−0.0973759
$523$	−16.4240	−0.718173	−0.359086	−	0.933304i	$-0.616912\pi$
$523$	−16.4240	−0.718173	−0.359086	+	0.933304i	$0.616912\pi$
$524$	−11.4642	−0.500816
$525$	3.79793	0.165755
$526$	13.4000	0.584266
$527$	0	0
$528$	−0.224777	−0.00978218
$529$	−22.6569	−0.985081
$530$	−9.19173	−0.399263
$531$	1.39150	0.0603861
$532$	18.1064	0.785012
$533$	−0.113264	−0.00490599
$534$	−0.424292	−0.0183609
$535$	7.79538	0.337024
$536$	0.383719	0.0165741
$537$	10.3263	0.445614
$538$	−23.8431	−1.02795
$539$	−1.66881	−0.0718809
$540$	1.00000	0.0430331
$541$	21.0734	0.906015	0.453007	−	0.891507i	$-0.350351\pi$
$541$	21.0734	0.906015	0.453007	+	0.891507i	$0.350351\pi$
$542$	−0.510544	−0.0219297
$543$	16.8233	0.721958
$544$	0	0
$545$	14.8032	0.634098
$546$	−0.603650	−0.0258338
$547$	−2.09635	−0.0896335	−0.0448167	−	0.998995i	$-0.514270\pi$
$547$	−2.09635	−0.0896335	−0.0448167	+	0.998995i	$0.514270\pi$
$548$	1.80734	0.0772057
$549$	−5.88163	−0.251022
$550$	−0.224777	−0.00958454
$551$	−10.6065	−0.451852
$552$	0.585786	0.0249327
$553$	−53.8870	−2.29151
$554$	−29.5619	−1.25596
$555$	2.76744	0.117471
$556$	−10.7101	−0.454208
$557$	8.90957	0.377511	0.188755	−	0.982024i	$-0.439555\pi$
$557$	8.90957	0.377511	0.188755	+	0.982024i	$0.439555\pi$
$558$	8.78530	0.371911
$559$	0.510544	0.0215937
$560$	3.79793	0.160492
$561$	0	0
$562$	−4.66103	−0.196614
$563$	15.8087	0.666257	0.333128	−	0.942882i	$-0.391896\pi$
$563$	15.8087	0.666257	0.333128	+	0.942882i	$0.391896\pi$
$564$	−3.91375	−0.164799
$565$	12.7600	0.536819
$566$	1.34583	0.0565693
$567$	3.79793	0.159498
$568$	−13.4548	−0.564550
$569$	26.7945	1.12328	0.561641	−	0.827381i	$-0.310170\pi$
$569$	26.7945	1.12328	0.561641	+	0.827381i	$0.310170\pi$
$570$	4.76744	0.199686
$571$	−9.89720	−0.414185	−0.207092	−	0.978321i	$-0.566400\pi$
$571$	−9.89720	−0.414185	−0.207092	+	0.978321i	$0.566400\pi$
$572$	0.0357265	0.00149380
$573$	16.7996	0.701811
$574$	2.70645	0.112965
$575$	0.585786	0.0244290
$576$	1.00000	0.0416667
$577$	7.92060	0.329739	0.164870	−	0.986315i	$-0.447280\pi$
$577$	7.92060	0.329739	0.164870	+	0.986315i	$0.447280\pi$
$578$	0	0
$579$	−11.6222	−0.483002
$580$	−2.22478	−0.0923789
$581$	−21.0211	−0.872102
$582$	−1.79793	−0.0745267
$583$	2.06609	0.0855689
$584$	5.79793	0.239920
$585$	−0.158942	−0.00657143
$586$	−5.76326	−0.238078
$587$	−18.3789	−0.758577	−0.379289	−	0.925278i	$-0.623831\pi$
$587$	−18.3789	−0.758577	−0.379289	+	0.925278i	$0.623831\pi$
$588$	7.42429	0.306173
$589$	41.8834	1.72577
$590$	1.39150	0.0572873
$591$	15.3711	0.632282
$592$	2.76744	0.113741
$593$	6.54040	0.268582	0.134291	−	0.990942i	$-0.457124\pi$
$593$	6.54040	0.268582	0.134291	+	0.990942i	$0.457124\pi$
$594$	−0.224777	−0.00922273
$595$	0	0
$596$	18.9332	0.775535
$597$	−10.0431	−0.411038
$598$	−0.0931059	−0.00380738
$599$	42.7922	1.74844	0.874221	−	0.485529i	$-0.161373\pi$
$599$	42.7922	1.74844	0.874221	+	0.485529i	$0.161373\pi$
$600$	1.00000	0.0408248
$601$	−22.9037	−0.934260	−0.467130	−	0.884189i	$-0.654712\pi$
$601$	−22.9037	−0.934260	−0.467130	+	0.884189i	$0.654712\pi$
$602$	−12.1995	−0.497215
$603$	0.383719	0.0156262
$604$	2.16740	0.0881902
$605$	−10.9495	−0.445159
$606$	−3.59102	−0.145875
$607$	30.3319	1.23113	0.615566	−	0.788085i	$-0.288927\pi$
$607$	30.3319	1.23113	0.615566	+	0.788085i	$0.288927\pi$
$608$	4.76744	0.193345
$609$	−8.44955	−0.342393
$610$	−5.88163	−0.238140
$611$	0.622058	0.0251658
$612$	0	0
$613$	33.6653	1.35973	0.679865	−	0.733338i	$-0.262039\pi$
$613$	33.6653	1.35973	0.679865	+	0.733338i	$0.262039\pi$
$614$	−14.2722	−0.575979
$615$	0.712611	0.0287353
$616$	−0.853690	−0.0343961
$617$	−10.4000	−0.418687	−0.209344	−	0.977842i	$-0.567133\pi$
$617$	−10.4000	−0.418687	−0.209344	+	0.977842i	$0.567133\pi$
$618$	17.5706	0.706793
$619$	24.0000	0.964641	0.482321	−	0.875995i	$-0.339794\pi$
$619$	24.0000	0.964641	0.482321	+	0.875995i	$0.339794\pi$
$620$	8.78530	0.352826
$621$	0.585786	0.0235068
$622$	−25.9686	−1.04125
$623$	−1.61143	−0.0645607
$624$	−0.158942	−0.00636276
$625$	1.00000	0.0400000
$626$	23.4497	0.937238
$627$	−1.07161	−0.0427961
$628$	16.5222	0.659309
$629$	0	0
$630$	3.79793	0.151313
$631$	12.5528	0.499717	0.249859	−	0.968282i	$-0.419616\pi$
$631$	12.5528	0.499717	0.249859	+	0.968282i	$0.419616\pi$
$632$	−14.1885	−0.564388
$633$	13.5064	0.536830
$634$	26.0490	1.03454
$635$	11.8816	0.471508
$636$	−9.19173	−0.364476
$637$	−1.18003	−0.0467545
$638$	0.500080	0.0197983
$639$	−13.4548	−0.532263
$640$	1.00000	0.0395285
$641$	−0.163096	−0.00644190	−0.00322095	−	0.999995i	$-0.501025\pi$
$641$	−0.163096	−0.00644190	−0.00322095	+	0.999995i	$0.501025\pi$
$642$	7.79538	0.307659
$643$	15.3996	0.607301	0.303650	−	0.952784i	$-0.401795\pi$
$643$	15.3996	0.607301	0.303650	+	0.952784i	$0.401795\pi$
$644$	2.22478	0.0876685
$645$	−3.21215	−0.126478
$646$	0	0
$647$	22.1908	0.872410	0.436205	−	0.899847i	$-0.356322\pi$
$647$	22.1908	0.872410	0.436205	+	0.899847i	$0.356322\pi$
$648$	1.00000	0.0392837
$649$	−0.312779	−0.0122776
$650$	−0.158942	−0.00623420
$651$	33.3660	1.30772
$652$	−20.4921	−0.802534
$653$	−19.7716	−0.773723	−0.386862	−	0.922138i	$-0.626441\pi$
$653$	−19.7716	−0.773723	−0.386862	+	0.922138i	$0.626441\pi$
$654$	14.8032	0.578850
$655$	−11.4642	−0.447943
$656$	0.712611	0.0278228
$657$	5.79793	0.226199
$658$	−14.8642	−0.579465
$659$	8.57601	0.334074	0.167037	−	0.985951i	$-0.446580\pi$
$659$	8.57601	0.334074	0.167037	+	0.985951i	$0.446580\pi$
$660$	−0.224777	−0.00874945
$661$	−3.89359	−0.151443	−0.0757216	−	0.997129i	$-0.524126\pi$
$661$	−3.89359	−0.151443	−0.0757216	+	0.997129i	$0.524126\pi$
$662$	−19.6770	−0.764769
$663$	0	0
$664$	−5.53488	−0.214795
$665$	18.1064	0.702136
$666$	2.76744	0.107236
$667$	−1.30324	−0.0504618
$668$	−16.0312	−0.620264
$669$	−9.50637	−0.367537
$670$	0.383719	0.0148244
$671$	1.32206	0.0510375
$672$	3.79793	0.146508
$673$	27.2057	1.04870	0.524352	−	0.851502i	$-0.324308\pi$
$673$	27.2057	1.04870	0.524352	+	0.851502i	$0.324308\pi$
$674$	2.71586	0.104611
$675$	1.00000	0.0384900
$676$	−12.9747	−0.499028
$677$	−5.20044	−0.199869	−0.0999347	−	0.994994i	$-0.531863\pi$
$677$	−5.20044	−0.199869	−0.0999347	+	0.994994i	$0.531863\pi$
$678$	12.7600	0.490046
$679$	−6.82843	−0.262051
$680$	0	0
$681$	−24.2807	−0.930437
$682$	−1.97474	−0.0756166
$683$	19.0192	0.727751	0.363875	−	0.931448i	$-0.381453\pi$
$683$	19.0192	0.727751	0.363875	+	0.931448i	$0.381453\pi$
$684$	4.76744	0.182288
$685$	1.80734	0.0690549
$686$	1.61143	0.0615248
$687$	16.1064	0.614498
$688$	−3.21215	−0.122462
$689$	1.46095	0.0556577
$690$	0.585786	0.0223005
$691$	−24.4853	−0.931464	−0.465732	−	0.884926i	$-0.654209\pi$
$691$	−24.4853	−0.931464	−0.465732	+	0.884926i	$0.654209\pi$
$692$	−24.4921	−0.931051
$693$	−0.853690	−0.0324290
$694$	6.62381	0.251436
$695$	−10.7101	−0.406256
$696$	−2.22478	−0.0843300
$697$	0	0
$698$	31.4792	1.19151
$699$	−2.14953	−0.0813029
$700$	3.79793	0.143548
$701$	−1.99191	−0.0752333	−0.0376167	−	0.999292i	$-0.511977\pi$
$701$	−1.99191	−0.0752333	−0.0376167	+	0.999292i	$0.511977\pi$
$702$	−0.158942	−0.00599887
$703$	13.1936	0.497606
$704$	−0.224777	−0.00847162
$705$	−3.91375	−0.147400
$706$	−27.1256	−1.02089
$707$	−13.6384	−0.512927
$708$	1.39150	0.0522959
$709$	−10.6036	−0.398228	−0.199114	−	0.979976i	$-0.563806\pi$
$709$	−10.6036	−0.398228	−0.199114	+	0.979976i	$0.563806\pi$
$710$	−13.4548	−0.504949
$711$	−14.1885	−0.532111
$712$	−0.424292	−0.0159010
$713$	5.14631	0.192731
$714$	0	0
$715$	0.0357265	0.00133610
$716$	10.3263	0.385913
$717$	−23.6229	−0.882213
$718$	27.8444	1.03914
$719$	−48.2107	−1.79795	−0.898977	−	0.437995i	$-0.855689\pi$
$719$	−48.2107	−1.79795	−0.898977	+	0.437995i	$0.855689\pi$
$720$	1.00000	0.0372678
$721$	66.7320	2.48523
$722$	3.72847	0.138759
$723$	−26.3848	−0.981260
$724$	16.8233	0.625234
$725$	−2.22478	−0.0826262
$726$	−10.9495	−0.406373
$727$	7.79079	0.288944	0.144472	−	0.989509i	$-0.453852\pi$
$727$	7.79079	0.288944	0.144472	+	0.989509i	$0.453852\pi$
$728$	−0.603650	−0.0223727
$729$	1.00000	0.0370370
$730$	5.79793	0.214591
$731$	0	0
$732$	−5.88163	−0.217391
$733$	−8.21669	−0.303490	−0.151745	−	0.988420i	$-0.548489\pi$
$733$	−8.21669	−0.303490	−0.151745	+	0.988420i	$0.548489\pi$
$734$	15.7474	0.581247
$735$	7.42429	0.273849
$736$	0.585786	0.0215924
$737$	−0.0862514	−0.00317711
$738$	0.712611	0.0262316
$739$	−32.8940	−1.21003	−0.605013	−	0.796216i	$-0.706832\pi$
$739$	−32.8940	−1.21003	−0.605013	+	0.796216i	$0.706832\pi$
$740$	2.76744	0.101733
$741$	−0.757744	−0.0278364
$742$	−34.9096	−1.28157
$743$	−14.9018	−0.546694	−0.273347	−	0.961915i	$-0.588131\pi$
$743$	−14.9018	−0.546694	−0.273347	+	0.961915i	$0.588131\pi$
$744$	8.78530	0.322085
$745$	18.9332	0.693660
$746$	1.94747	0.0713019
$747$	−5.53488	−0.202511
$748$	0	0
$749$	29.6063	1.08179
$750$	1.00000	0.0365148
$751$	2.90140	0.105874	0.0529369	−	0.998598i	$-0.483142\pi$
$751$	2.90140	0.105874	0.0529369	+	0.998598i	$0.483142\pi$
$752$	−3.91375	−0.142720
$753$	20.6695	0.753238
$754$	0.353610	0.0128777
$755$	2.16740	0.0788797
$756$	3.79793	0.138129
$757$	−35.9099	−1.30517	−0.652583	−	0.757717i	$-0.726315\pi$
$757$	−35.9099	−1.30517	−0.652583	+	0.757717i	$0.726315\pi$
$758$	30.3871	1.10371
$759$	−0.131672	−0.00477938
$760$	4.76744	0.172933
$761$	18.8385	0.682895	0.341447	−	0.939901i	$-0.389083\pi$
$761$	18.8385	0.682895	0.341447	+	0.939901i	$0.389083\pi$
$762$	11.8816	0.430426
$763$	56.2214	2.03535
$764$	16.7996	0.607787
$765$	0	0
$766$	−33.4403	−1.20825
$767$	−0.221168	−0.00798591
$768$	1.00000	0.0360844
$769$	−8.30231	−0.299389	−0.149695	−	0.988732i	$-0.547829\pi$
$769$	−8.30231	−0.299389	−0.149695	+	0.988732i	$0.547829\pi$
$770$	−0.853690	−0.0307648
$771$	12.5664	0.452569
$772$	−11.6222	−0.418292
$773$	7.83395	0.281767	0.140884	−	0.990026i	$-0.455006\pi$
$773$	7.83395	0.281767	0.140884	+	0.990026i	$0.455006\pi$
$774$	−3.21215	−0.115458
$775$	8.78530	0.315577
$776$	−1.79793	−0.0645420
$777$	10.5105	0.377064
$778$	−23.0818	−0.827523
$779$	3.39733	0.121722
$780$	−0.158942	−0.00569102
$781$	3.02433	0.108219
$782$	0	0
$783$	−2.22478	−0.0795071
$784$	7.42429	0.265153
$785$	16.5222	0.589704
$786$	−11.4642	−0.408914
$787$	18.5243	0.660318	0.330159	−	0.943925i	$-0.392898\pi$
$787$	18.5243	0.660318	0.330159	+	0.943925i	$0.392898\pi$
$788$	15.3711	0.547572
$789$	13.4000	0.477051
$790$	−14.1885	−0.504804
$791$	48.4618	1.72310
$792$	−0.224777	−0.00798712
$793$	0.934836	0.0331970
$794$	−15.9679	−0.566679
$795$	−9.19173	−0.325997
$796$	−10.0431	−0.355969
$797$	−30.0811	−1.06553	−0.532764	−	0.846264i	$-0.678847\pi$
$797$	−30.0811	−1.06553	−0.532764	+	0.846264i	$0.678847\pi$
$798$	18.1064	0.640960
$799$	0	0
$800$	1.00000	0.0353553
$801$	−0.424292	−0.0149916
$802$	20.0043	0.706376
$803$	−1.30324	−0.0459905
$804$	0.383719	0.0135327
$805$	2.22478	0.0784131
$806$	−1.39635	−0.0491843
$807$	−23.8431	−0.839316
$808$	−3.59102	−0.126332
$809$	36.7983	1.29376	0.646880	−	0.762591i	$-0.276073\pi$
$809$	36.7983	1.29376	0.646880	+	0.762591i	$0.276073\pi$
$810$	1.00000	0.0351364
$811$	−35.5229	−1.24738	−0.623689	−	0.781672i	$-0.714367\pi$
$811$	−35.5229	−1.24738	−0.623689	+	0.781672i	$0.714367\pi$
$812$	−8.44955	−0.296521
$813$	−0.510544	−0.0179056
$814$	−0.622058	−0.0218031
$815$	−20.4921	−0.717808
$816$	0	0
$817$	−15.3137	−0.535759
$818$	8.48853	0.296794
$819$	−0.603650	−0.0210932
$820$	0.712611	0.0248855
$821$	−30.3417	−1.05893	−0.529465	−	0.848331i	$-0.677607\pi$
$821$	−30.3417	−1.05893	−0.529465	+	0.848331i	$0.677607\pi$
$822$	1.80734	0.0630382
$823$	−7.41261	−0.258387	−0.129194	−	0.991619i	$-0.541239\pi$
$823$	−7.41261	−0.258387	−0.129194	+	0.991619i	$0.541239\pi$
$824$	17.5706	0.612101
$825$	−0.224777	−0.00782575
$826$	5.28484	0.183883
$827$	3.06516	0.106586	0.0532931	−	0.998579i	$-0.483028\pi$
$827$	3.06516	0.106586	0.0532931	+	0.998579i	$0.483028\pi$
$828$	0.585786	0.0203575
$829$	−37.3183	−1.29612	−0.648059	−	0.761590i	$-0.724419\pi$
$829$	−37.3183	−1.29612	−0.648059	+	0.761590i	$0.724419\pi$
$830$	−5.53488	−0.192118
$831$	−29.5619	−1.02549
$832$	−0.158942	−0.00551031
$833$	0	0
$834$	−10.7101	−0.370859
$835$	−16.0312	−0.554781
$836$	−1.07161	−0.0370625
$837$	8.78530	0.303664
$838$	31.0124	1.07130
$839$	24.8736	0.858731	0.429365	−	0.903131i	$-0.358737\pi$
$839$	24.8736	0.858731	0.429365	+	0.903131i	$0.358737\pi$
$840$	3.79793	0.131041
$841$	−24.0504	−0.829323
$842$	30.0306	1.03492
$843$	−4.66103	−0.160534
$844$	13.5064	0.464908
$845$	−12.9747	−0.446345
$846$	−3.91375	−0.134557
$847$	−41.5854	−1.42889
$848$	−9.19173	−0.315645
$849$	1.34583	0.0461886
$850$	0	0
$851$	1.62113	0.0555715
$852$	−13.4548	−0.460953
$853$	35.9123	1.22961	0.614806	−	0.788678i	$-0.289234\pi$
$853$	35.9123	1.22961	0.614806	+	0.788678i	$0.289234\pi$
$854$	−22.3380	−0.764392
$855$	4.76744	0.163043
$856$	7.79538	0.266441
$857$	30.7394	1.05004	0.525018	−	0.851091i	$-0.324058\pi$
$857$	30.7394	1.05004	0.525018	+	0.851091i	$0.324058\pi$
$858$	0.0357265	0.00121968
$859$	0.838891	0.0286226	0.0143113	−	0.999898i	$-0.495444\pi$
$859$	0.838891	0.0286226	0.0143113	+	0.999898i	$0.495444\pi$
$860$	−3.21215	−0.109533
$861$	2.70645	0.0922355
$862$	−13.0558	−0.444681
$863$	44.7753	1.52417	0.762084	−	0.647478i	$-0.224176\pi$
$863$	44.7753	1.52417	0.762084	+	0.647478i	$0.224176\pi$
$864$	1.00000	0.0340207
$865$	−24.4921	−0.832758
$866$	21.3004	0.723817
$867$	0	0
$868$	33.3660	1.13252
$869$	3.18926	0.108188
$870$	−2.22478	−0.0754270
$871$	−0.0609889	−0.00206653
$872$	14.8032	0.501298
$873$	−1.79793	−0.0608508
$874$	2.79270	0.0944645
$875$	3.79793	0.128394
$876$	5.79793	0.195894
$877$	−5.10548	−0.172400	−0.0861999	−	0.996278i	$-0.527472\pi$
$877$	−5.10548	−0.172400	−0.0861999	+	0.996278i	$0.527472\pi$
$878$	−35.7857	−1.20771
$879$	−5.76326	−0.194390
$880$	−0.224777	−0.00757725
$881$	−54.0245	−1.82013	−0.910065	−	0.414465i	$-0.863969\pi$
$881$	−54.0245	−1.82013	−0.910065	+	0.414465i	$0.863969\pi$
$882$	7.42429	0.249989
$883$	−50.6492	−1.70448	−0.852241	−	0.523149i	$-0.824757\pi$
$883$	−50.6492	−1.70448	−0.852241	+	0.523149i	$0.824757\pi$
$884$	0	0
$885$	1.39150	0.0467749
$886$	23.7990	0.799543
$887$	22.2830	0.748188	0.374094	−	0.927391i	$-0.377954\pi$
$887$	22.2830	0.748188	0.374094	+	0.927391i	$0.377954\pi$
$888$	2.76744	0.0928691
$889$	45.1256	1.51347
$890$	−0.424292	−0.0142223
$891$	−0.224777	−0.00753033
$892$	−9.50637	−0.318297
$893$	−18.6586	−0.624385
$894$	18.9332	0.633222
$895$	10.3263	0.345171
$896$	3.79793	0.126880
$897$	−0.0931059	−0.00310871
$898$	−8.24104	−0.275007
$899$	−19.5453	−0.651874
$900$	1.00000	0.0333333
$901$	0	0
$902$	−0.160179	−0.00533337
$903$	−12.1995	−0.405974
$904$	12.7600	0.424393
$905$	16.8233	0.559226
$906$	2.16740	0.0720070
$907$	−4.79595	−0.159247	−0.0796234	−	0.996825i	$-0.525372\pi$
$907$	−4.79595	−0.159247	−0.0796234	+	0.996825i	$0.525372\pi$
$908$	−24.2807	−0.805782
$909$	−3.59102	−0.119107
$910$	−0.603650	−0.0200108
$911$	−49.3277	−1.63430	−0.817150	−	0.576425i	$-0.804447\pi$
$911$	−49.3277	−1.63430	−0.817150	+	0.576425i	$0.804447\pi$
$912$	4.76744	0.157866
$913$	1.24412	0.0411742
$914$	−33.8408	−1.11935
$915$	−5.88163	−0.194441
$916$	16.1064	0.532171
$917$	−43.5402	−1.43783
$918$	0	0
$919$	−33.9041	−1.11839	−0.559196	−	0.829036i	$-0.688890\pi$
$919$	−33.9041	−1.11839	−0.559196	+	0.829036i	$0.688890\pi$
$920$	0.585786	0.0193128
$921$	−14.2722	−0.470285
$922$	3.71300	0.122281
$923$	2.13853	0.0703904
$924$	−0.853690	−0.0280843
$925$	2.76744	0.0909928
$926$	14.7779	0.485632
$927$	17.5706	0.577094
$928$	−2.22478	−0.0730319
$929$	−14.4740	−0.474877	−0.237439	−	0.971403i	$-0.576308\pi$
$929$	−14.4740	−0.474877	−0.237439	+	0.971403i	$0.576308\pi$
$930$	8.78530	0.288081
$931$	35.3949	1.16002
$932$	−2.14953	−0.0704104
$933$	−25.9686	−0.850173
$934$	7.19684	0.235488
$935$	0	0
$936$	−0.158942	−0.00519517
$937$	−6.79858	−0.222100	−0.111050	−	0.993815i	$-0.535421\pi$
$937$	−6.79858	−0.222100	−0.111050	+	0.993815i	$0.535421\pi$
$938$	1.45734	0.0475838
$939$	23.4497	0.765251
$940$	−3.91375	−0.127652
$941$	4.89761	0.159658	0.0798288	−	0.996809i	$-0.474563\pi$
$941$	4.89761	0.159658	0.0798288	+	0.996809i	$0.474563\pi$
$942$	16.5222	0.538324
$943$	0.417438	0.0135937
$944$	1.39150	0.0452896
$945$	3.79793	0.123547
$946$	0.722018	0.0234748
$947$	3.64680	0.118505	0.0592525	−	0.998243i	$-0.481128\pi$
$947$	3.64680	0.118505	0.0592525	+	0.998243i	$0.481128\pi$
$948$	−14.1885	−0.460821
$949$	−0.921533	−0.0299142
$950$	4.76744	0.154676
$951$	26.0490	0.844698
$952$	0	0
$953$	−5.86111	−0.189860	−0.0949300	−	0.995484i	$-0.530263\pi$
$953$	−5.86111	−0.189860	−0.0949300	+	0.995484i	$0.530263\pi$
$954$	−9.19173	−0.297593
$955$	16.7996	0.543621
$956$	−23.6229	−0.764018
$957$	0.500080	0.0161653
$958$	28.6465	0.925527
$959$	6.86415	0.221655
$960$	1.00000	0.0322749
$961$	46.1815	1.48973
$962$	−0.439861	−0.0141817
$963$	7.79538	0.251203
$964$	−26.3848	−0.849796
$965$	−11.6222	−0.374131
$966$	2.22478	0.0715810
$967$	−18.4665	−0.593843	−0.296921	−	0.954902i	$-0.595960\pi$
$967$	−18.4665	−0.593843	−0.296921	+	0.954902i	$0.595960\pi$
$968$	−10.9495	−0.351929
$969$	0	0
$970$	−1.79793	−0.0577281
$971$	23.4979	0.754084	0.377042	−	0.926196i	$-0.376941\pi$
$971$	23.4979	0.754084	0.377042	+	0.926196i	$0.376941\pi$
$972$	1.00000	0.0320750
$973$	−40.6761	−1.30402
$974$	−36.5603	−1.17147
$975$	−0.158942	−0.00509021
$976$	−5.88163	−0.188266
$977$	−43.8697	−1.40352	−0.701758	−	0.712416i	$-0.747601\pi$
$977$	−43.8697	−1.40352	−0.701758	+	0.712416i	$0.747601\pi$
$978$	−20.4921	−0.655266
$979$	0.0953714	0.00304808
$980$	7.42429	0.237160
$981$	14.8032	0.472629
$982$	−4.53781	−0.144807
$983$	−4.20826	−0.134223	−0.0671113	−	0.997745i	$-0.521378\pi$
$983$	−4.20826	−0.134223	−0.0671113	+	0.997745i	$0.521378\pi$
$984$	0.712611	0.0227172
$985$	15.3711	0.489764
$986$	0	0
$987$	−14.8642	−0.473131
$988$	−0.757744	−0.0241071
$989$	−1.88163	−0.0598324
$990$	−0.224777	−0.00714390
$991$	−37.1247	−1.17930	−0.589652	−	0.807657i	$-0.700735\pi$
$991$	−37.1247	−1.17930	−0.589652	+	0.807657i	$0.700735\pi$
$992$	8.78530	0.278934
$993$	−19.6770	−0.624431
$994$	−51.1004	−1.62081
$995$	−10.0431	−0.318388
$996$	−5.53488	−0.175379
$997$	−14.2031	−0.449817	−0.224909	−	0.974380i	$-0.572208\pi$
$997$	−14.2031	−0.449817	−0.224909	+	0.974380i	$0.572208\pi$
$998$	41.7257	1.32080
$999$	2.76744	0.0875579

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8670.2.a.ca.1.4		4
17.8	even	8		510.2.p.c.421.2	yes	8
17.15	even	8		510.2.p.c.361.2	✓	8
17.16	even	2		8670.2.a.bx.1.1		4
51.8	odd	8		1530.2.q.j.1441.2		8
51.32	odd	8		1530.2.q.j.361.2		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
510.2.p.c.361.2	✓	8	17.15	even	8
510.2.p.c.421.2	yes	8	17.8	even	8
1530.2.q.j.361.2		8	51.32	odd	8
1530.2.q.j.1441.2		8	51.8	odd	8
8670.2.a.bx.1.1		4	17.16	even	2
8670.2.a.ca.1.4		4	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

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

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{6} +3.79793 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{6} +3.79793 q^{7} +1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{10} -0.224777 q^{11} +1.00000 q^{12} -0.158942 q^{13} +3.79793 q^{14} +1.00000 q^{15} +1.00000 q^{16} +1.00000 q^{18} +4.76744 q^{19} +1.00000 q^{20} +3.79793 q^{21} -0.224777 q^{22} +0.585786 q^{23} +1.00000 q^{24} +1.00000 q^{25} -0.158942 q^{26} +1.00000 q^{27} +3.79793 q^{28} -2.22478 q^{29} +1.00000 q^{30} +8.78530 q^{31} +1.00000 q^{32} -0.224777 q^{33} +3.79793 q^{35} +1.00000 q^{36} +2.76744 q^{37} +4.76744 q^{38} -0.158942 q^{39} +1.00000 q^{40} +0.712611 q^{41} +3.79793 q^{42} -3.21215 q^{43} -0.224777 q^{44} +1.00000 q^{45} +0.585786 q^{46} -3.91375 q^{47} +1.00000 q^{48} +7.42429 q^{49} +1.00000 q^{50} -0.158942 q^{52} -9.19173 q^{53} +1.00000 q^{54} -0.224777 q^{55} +3.79793 q^{56} +4.76744 q^{57} -2.22478 q^{58} +1.39150 q^{59} +1.00000 q^{60} -5.88163 q^{61} +8.78530 q^{62} +3.79793 q^{63} +1.00000 q^{64} -0.158942 q^{65} -0.224777 q^{66} +0.383719 q^{67} +0.585786 q^{69} +3.79793 q^{70} -13.4548 q^{71} +1.00000 q^{72} +5.79793 q^{73} +2.76744 q^{74} +1.00000 q^{75} +4.76744 q^{76} -0.853690 q^{77} -0.158942 q^{78} -14.1885 q^{79} +1.00000 q^{80} +1.00000 q^{81} +0.712611 q^{82} -5.53488 q^{83} +3.79793 q^{84} -3.21215 q^{86} -2.22478 q^{87} -0.224777 q^{88} -0.424292 q^{89} +1.00000 q^{90} -0.603650 q^{91} +0.585786 q^{92} +8.78530 q^{93} -3.91375 q^{94} +4.76744 q^{95} +1.00000 q^{96} -1.79793 q^{97} +7.42429 q^{98} -0.224777 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{2} + 4 q^{3} + 4 q^{4} + 4 q^{5} + 4 q^{6} + 4 q^{7} + 4 q^{8} + 4 q^{9}+O(q^{10}) \) 4 * q + 4 * q^2 + 4 * q^3 + 4 * q^4 + 4 * q^5 + 4 * q^6 + 4 * q^7 + 4 * q^8 + 4 * q^9	\( 4 q + 4 q^{2} + 4 q^{3} + 4 q^{4} + 4 q^{5} + 4 q^{6} + 4 q^{7} + 4 q^{8} + 4 q^{9} + 4 q^{10} + 4 q^{12} + 4 q^{13} + 4 q^{14} + 4 q^{15} + 4 q^{16} + 4 q^{18} + 8 q^{19} + 4 q^{20} + 4 q^{21} + 8 q^{23} + 4 q^{24} + 4 q^{25} + 4 q^{26} + 4 q^{27} + 4 q^{28} - 8 q^{29} + 4 q^{30} + 8 q^{31} + 4 q^{32} + 4 q^{35} + 4 q^{36} + 8 q^{38} + 4 q^{39} + 4 q^{40} + 12 q^{41} + 4 q^{42} + 4 q^{43} + 4 q^{45} + 8 q^{46} + 16 q^{47} + 4 q^{48} - 4 q^{49} + 4 q^{50} + 4 q^{52} + 8 q^{53} + 4 q^{54} + 4 q^{56} + 8 q^{57} - 8 q^{58} + 12 q^{59} + 4 q^{60} + 8 q^{62} + 4 q^{63} + 4 q^{64} + 4 q^{65} - 4 q^{67} + 8 q^{69} + 4 q^{70} - 20 q^{71} + 4 q^{72} + 12 q^{73} + 4 q^{75} + 8 q^{76} - 24 q^{77} + 4 q^{78} + 4 q^{80} + 4 q^{81} + 12 q^{82} + 4 q^{84} + 4 q^{86} - 8 q^{87} + 32 q^{89} + 4 q^{90} + 8 q^{91} + 8 q^{92} + 8 q^{93} + 16 q^{94} + 8 q^{95} + 4 q^{96} + 4 q^{97} - 4 q^{98}+O(q^{100}) \) 4 * q + 4 * q^2 + 4 * q^3 + 4 * q^4 + 4 * q^5 + 4 * q^6 + 4 * q^7 + 4 * q^8 + 4 * q^9 + 4 * q^10 + 4 * q^12 + 4 * q^13 + 4 * q^14 + 4 * q^15 + 4 * q^16 + 4 * q^18 + 8 * q^19 + 4 * q^20 + 4 * q^21 + 8 * q^23 + 4 * q^24 + 4 * q^25 + 4 * q^26 + 4 * q^27 + 4 * q^28 - 8 * q^29 + 4 * q^30 + 8 * q^31 + 4 * q^32 + 4 * q^35 + 4 * q^36 + 8 * q^38 + 4 * q^39 + 4 * q^40 + 12 * q^41 + 4 * q^42 + 4 * q^43 + 4 * q^45 + 8 * q^46 + 16 * q^47 + 4 * q^48 - 4 * q^49 + 4 * q^50 + 4 * q^52 + 8 * q^53 + 4 * q^54 + 4 * q^56 + 8 * q^57 - 8 * q^58 + 12 * q^59 + 4 * q^60 + 8 * q^62 + 4 * q^63 + 4 * q^64 + 4 * q^65 - 4 * q^67 + 8 * q^69 + 4 * q^70 - 20 * q^71 + 4 * q^72 + 12 * q^73 + 4 * q^75 + 8 * q^76 - 24 * q^77 + 4 * q^78 + 4 * q^80 + 4 * q^81 + 12 * q^82 + 4 * q^84 + 4 * q^86 - 8 * q^87 + 32 * q^89 + 4 * q^90 + 8 * q^91 + 8 * q^92 + 8 * q^93 + 16 * q^94 + 8 * q^95 + 4 * q^96 + 4 * q^97 - 4 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more