LMFDB - Embedded newform 8670.2.a.cf.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:	$6$
Coefficient field:	6.6.30652992.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 15x^{4} + 63x^{2} - 73 $ x^6 - 15x^4 + 63x^2 - 73
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$1.39134$ 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)

$f(q)$	$=$	$q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{6} -4.52299 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{6} -4.52299 q^{7} +1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} +0.738635 q^{11} -1.00000 q^{12} +2.22352 q^{13} -4.52299 q^{14} +1.00000 q^{15} +1.00000 q^{16} +1.00000 q^{18} -3.49425 q^{19} -1.00000 q^{20} +4.52299 q^{21} +0.738635 q^{22} +4.95837 q^{23} -1.00000 q^{24} +1.00000 q^{25} +2.22352 q^{26} -1.00000 q^{27} -4.52299 q^{28} +2.39922 q^{29} +1.00000 q^{30} -3.26272 q^{31} +1.00000 q^{32} -0.738635 q^{33} +4.52299 q^{35} +1.00000 q^{36} -9.10743 q^{37} -3.49425 q^{38} -2.22352 q^{39} -1.00000 q^{40} +7.68524 q^{41} +4.52299 q^{42} +11.8547 q^{43} +0.738635 q^{44} -1.00000 q^{45} +4.95837 q^{46} +0.452476 q^{47} -1.00000 q^{48} +13.4575 q^{49} +1.00000 q^{50} +2.22352 q^{52} +0.507697 q^{53} -1.00000 q^{54} -0.738635 q^{55} -4.52299 q^{56} +3.49425 q^{57} +2.39922 q^{58} -6.32554 q^{59} +1.00000 q^{60} -6.20519 q^{61} -3.26272 q^{62} -4.52299 q^{63} +1.00000 q^{64} -2.22352 q^{65} -0.738635 q^{66} -5.47034 q^{67} -4.95837 q^{69} +4.52299 q^{70} -2.39314 q^{71} +1.00000 q^{72} +9.24392 q^{73} -9.10743 q^{74} -1.00000 q^{75} -3.49425 q^{76} -3.34084 q^{77} -2.22352 q^{78} -8.18405 q^{79} -1.00000 q^{80} +1.00000 q^{81} +7.68524 q^{82} +5.22184 q^{83} +4.52299 q^{84} +11.8547 q^{86} -2.39922 q^{87} +0.738635 q^{88} +15.2388 q^{89} -1.00000 q^{90} -10.0570 q^{91} +4.95837 q^{92} +3.26272 q^{93} +0.452476 q^{94} +3.49425 q^{95} -1.00000 q^{96} -8.53630 q^{97} +13.4575 q^{98} +0.738635 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 6 q^{2} - 6 q^{3} + 6 q^{4} - 6 q^{5} - 6 q^{6} - 6 q^{7} + 6 q^{8} + 6 q^{9}+O(q^{10}) $ 6 * q + 6 * q^2 - 6 * q^3 + 6 * q^4 - 6 * q^5 - 6 * q^6 - 6 * q^7 + 6 * q^8 + 6 * q^9	\( 6 q + 6 q^{2} - 6 q^{3} + 6 q^{4} - 6 q^{5} - 6 q^{6} - 6 q^{7} + 6 q^{8} + 6 q^{9} - 6 q^{10} - 6 q^{11} - 6 q^{12} + 6 q^{13} - 6 q^{14} + 6 q^{15} + 6 q^{16} + 6 q^{18} + 6 q^{19} - 6 q^{20} + 6 q^{21} - 6 q^{22} - 6 q^{23} - 6 q^{24} + 6 q^{25} + 6 q^{26} - 6 q^{27} - 6 q^{28} - 12 q^{29} + 6 q^{30} - 6 q^{31} + 6 q^{32} + 6 q^{33} + 6 q^{35} + 6 q^{36} - 6 q^{37} + 6 q^{38} - 6 q^{39} - 6 q^{40} - 12 q^{41} + 6 q^{42} + 18 q^{43} - 6 q^{44} - 6 q^{45} - 6 q^{46} - 6 q^{47} - 6 q^{48} + 18 q^{49} + 6 q^{50} + 6 q^{52} - 6 q^{53} - 6 q^{54} + 6 q^{55} - 6 q^{56} - 6 q^{57} - 12 q^{58} - 30 q^{59} + 6 q^{60} + 24 q^{61} - 6 q^{62} - 6 q^{63} + 6 q^{64} - 6 q^{65} + 6 q^{66} + 6 q^{69} + 6 q^{70} - 24 q^{71} + 6 q^{72} - 6 q^{73} - 6 q^{74} - 6 q^{75} + 6 q^{76} - 6 q^{78} - 6 q^{79} - 6 q^{80} + 6 q^{81} - 12 q^{82} + 18 q^{83} + 6 q^{84} + 18 q^{86} + 12 q^{87} - 6 q^{88} - 12 q^{89} - 6 q^{90} - 6 q^{91} - 6 q^{92} + 6 q^{93} - 6 q^{94} - 6 q^{95} - 6 q^{96} - 18 q^{97} + 18 q^{98} - 6 q^{99}+O(q^{100}) \) 6 * q + 6 * q^2 - 6 * q^3 + 6 * q^4 - 6 * q^5 - 6 * q^6 - 6 * q^7 + 6 * q^8 + 6 * q^9 - 6 * q^10 - 6 * q^11 - 6 * q^12 + 6 * q^13 - 6 * q^14 + 6 * q^15 + 6 * q^16 + 6 * q^18 + 6 * q^19 - 6 * q^20 + 6 * q^21 - 6 * q^22 - 6 * q^23 - 6 * q^24 + 6 * q^25 + 6 * q^26 - 6 * q^27 - 6 * q^28 - 12 * q^29 + 6 * q^30 - 6 * q^31 + 6 * q^32 + 6 * q^33 + 6 * q^35 + 6 * q^36 - 6 * q^37 + 6 * q^38 - 6 * q^39 - 6 * q^40 - 12 * q^41 + 6 * q^42 + 18 * q^43 - 6 * q^44 - 6 * q^45 - 6 * q^46 - 6 * q^47 - 6 * q^48 + 18 * q^49 + 6 * q^50 + 6 * q^52 - 6 * q^53 - 6 * q^54 + 6 * q^55 - 6 * q^56 - 6 * q^57 - 12 * q^58 - 30 * q^59 + 6 * q^60 + 24 * q^61 - 6 * q^62 - 6 * q^63 + 6 * q^64 - 6 * q^65 + 6 * q^66 + 6 * q^69 + 6 * q^70 - 24 * q^71 + 6 * q^72 - 6 * q^73 - 6 * q^74 - 6 * q^75 + 6 * q^76 - 6 * q^78 - 6 * q^79 - 6 * q^80 + 6 * q^81 - 12 * q^82 + 18 * q^83 + 6 * q^84 + 18 * q^86 + 12 * q^87 - 6 * q^88 - 12 * q^89 - 6 * q^90 - 6 * q^91 - 6 * q^92 + 6 * q^93 - 6 * q^94 - 6 * q^95 - 6 * q^96 - 18 * q^97 + 18 * q^98 - 6 * 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
$7$	−4.52299	−1.70953	−0.854765	−	0.519015i	$-0.826299\pi$
$7$	−4.52299	−1.70953	−0.854765	+	0.519015i	$0.826299\pi$
$8$	1.00000	0.353553
$9$	1.00000	0.333333
$10$	−1.00000	−0.316228
$11$	0.738635	0.222707	0.111353	−	0.993781i	$-0.464481\pi$
$11$	0.738635	0.222707	0.111353	+	0.993781i	$0.464481\pi$
$12$	−1.00000	−0.288675
$13$	2.22352	0.616694	0.308347	−	0.951274i	$-0.400224\pi$
$13$	2.22352	0.616694	0.308347	+	0.951274i	$0.400224\pi$
$14$	−4.52299	−1.20882
$15$	1.00000	0.258199
$16$	1.00000	0.250000
$17$	0	0
$17$	0	0
$18$	1.00000	0.235702
$19$	−3.49425	−0.801635	−0.400817	−	0.916158i	$-0.631274\pi$
$19$	−3.49425	−0.801635	−0.400817	+	0.916158i	$0.631274\pi$
$20$	−1.00000	−0.223607
$21$	4.52299	0.986998
$22$	0.738635	0.157477
$23$	4.95837	1.03389	0.516946	−	0.856018i	$-0.327069\pi$
$23$	4.95837	1.03389	0.516946	+	0.856018i	$0.327069\pi$
$24$	−1.00000	−0.204124
$25$	1.00000	0.200000
$26$	2.22352	0.436069
$27$	−1.00000	−0.192450
$28$	−4.52299	−0.854765
$29$	2.39922	0.445524	0.222762	−	0.974873i	$-0.428493\pi$
$29$	2.39922	0.445524	0.222762	+	0.974873i	$0.428493\pi$
$30$	1.00000	0.182574
$31$	−3.26272	−0.586003	−0.293001	−	0.956112i	$-0.594654\pi$
$31$	−3.26272	−0.586003	−0.293001	+	0.956112i	$0.594654\pi$
$32$	1.00000	0.176777
$33$	−0.738635	−0.128580
$34$	0	0
$35$	4.52299	0.764525
$36$	1.00000	0.166667
$37$	−9.10743	−1.49725	−0.748626	−	0.662993i	$-0.769286\pi$
$37$	−9.10743	−1.49725	−0.748626	+	0.662993i	$0.769286\pi$
$38$	−3.49425	−0.566842
$39$	−2.22352	−0.356049
$40$	−1.00000	−0.158114
$41$	7.68524	1.20023	0.600116	−	0.799913i	$-0.295121\pi$
$41$	7.68524	1.20023	0.600116	+	0.799913i	$0.295121\pi$
$42$	4.52299	0.697913
$43$	11.8547	1.80783	0.903915	−	0.427712i	$-0.140680\pi$
$43$	11.8547	1.80783	0.903915	+	0.427712i	$0.140680\pi$
$44$	0.738635	0.111353
$45$	−1.00000	−0.149071
$46$	4.95837	0.731072
$47$	0.452476	0.0660005	0.0330002	−	0.999455i	$-0.489494\pi$
$47$	0.452476	0.0660005	0.0330002	+	0.999455i	$0.489494\pi$
$48$	−1.00000	−0.144338
$49$	13.4575	1.92249
$50$	1.00000	0.141421
$51$	0	0
$52$	2.22352	0.308347
$53$	0.507697	0.0697375	0.0348687	−	0.999392i	$-0.488899\pi$
$53$	0.507697	0.0697375	0.0348687	+	0.999392i	$0.488899\pi$
$54$	−1.00000	−0.136083
$55$	−0.738635	−0.0995975
$56$	−4.52299	−0.604410
$57$	3.49425	0.462824
$58$	2.39922	0.315033
$59$	−6.32554	−0.823516	−0.411758	−	0.911293i	$-0.635085\pi$
$59$	−6.32554	−0.823516	−0.411758	+	0.911293i	$0.635085\pi$
$60$	1.00000	0.129099
$61$	−6.20519	−0.794493	−0.397247	−	0.917712i	$-0.630034\pi$
$61$	−6.20519	−0.794493	−0.397247	+	0.917712i	$0.630034\pi$
$62$	−3.26272	−0.414366
$63$	−4.52299	−0.569843
$64$	1.00000	0.125000
$65$	−2.22352	−0.275794
$66$	−0.738635	−0.0909196
$67$	−5.47034	−0.668308	−0.334154	−	0.942518i	$-0.608451\pi$
$67$	−5.47034	−0.668308	−0.334154	+	0.942518i	$0.608451\pi$
$68$	0	0
$69$	−4.95837	−0.596918
$70$	4.52299	0.540601
$71$	−2.39314	−0.284013	−0.142007	−	0.989866i	$-0.545355\pi$
$71$	−2.39314	−0.284013	−0.142007	+	0.989866i	$0.545355\pi$
$72$	1.00000	0.117851
$73$	9.24392	1.08192	0.540959	−	0.841049i	$-0.318061\pi$
$73$	9.24392	1.08192	0.540959	+	0.841049i	$0.318061\pi$
$74$	−9.10743	−1.05872
$75$	−1.00000	−0.115470
$76$	−3.49425	−0.400817
$77$	−3.34084	−0.380724
$78$	−2.22352	−0.251764
$79$	−8.18405	−0.920777	−0.460389	−	0.887717i	$-0.652290\pi$
$79$	−8.18405	−0.920777	−0.460389	+	0.887717i	$0.652290\pi$
$80$	−1.00000	−0.111803
$81$	1.00000	0.111111
$82$	7.68524	0.848692
$83$	5.22184	0.573172	0.286586	−	0.958055i	$-0.407480\pi$
$83$	5.22184	0.573172	0.286586	+	0.958055i	$0.407480\pi$
$84$	4.52299	0.493499
$85$	0	0
$86$	11.8547	1.27833
$87$	−2.39922	−0.257223
$88$	0.738635	0.0787387
$89$	15.2388	1.61531	0.807656	−	0.589654i	$-0.200736\pi$
$89$	15.2388	1.61531	0.807656	+	0.589654i	$0.200736\pi$
$90$	−1.00000	−0.105409
$91$	−10.0570	−1.05426
$92$	4.95837	0.516946
$93$	3.26272	0.338329
$94$	0.452476	0.0466694
$95$	3.49425	0.358502
$96$	−1.00000	−0.102062
$97$	−8.53630	−0.866730	−0.433365	−	0.901218i	$-0.642674\pi$
$97$	−8.53630	−0.866730	−0.433365	+	0.901218i	$0.642674\pi$
$98$	13.4575	1.35941
$99$	0.738635	0.0742356
$100$	1.00000	0.100000
$101$	−17.2781	−1.71923	−0.859616	−	0.510940i	$-0.829297\pi$
$101$	−17.2781	−1.71923	−0.859616	+	0.510940i	$0.829297\pi$
$102$	0	0
$103$	1.91428	0.188620	0.0943099	−	0.995543i	$-0.469936\pi$
$103$	1.91428	0.188620	0.0943099	+	0.995543i	$0.469936\pi$
$104$	2.22352	0.218034
$105$	−4.52299	−0.441399
$106$	0.507697	0.0493119
$107$	4.55688	0.440530	0.220265	−	0.975440i	$-0.429308\pi$
$107$	4.55688	0.440530	0.220265	+	0.975440i	$0.429308\pi$
$108$	−1.00000	−0.0962250
$109$	−16.0420	−1.53655	−0.768274	−	0.640121i	$-0.778884\pi$
$109$	−16.0420	−1.53655	−0.768274	+	0.640121i	$0.778884\pi$
$110$	−0.738635	−0.0704260
$111$	9.10743	0.864439
$112$	−4.52299	−0.427383
$113$	8.36190	0.786621	0.393310	−	0.919406i	$-0.371330\pi$
$113$	8.36190	0.786621	0.393310	+	0.919406i	$0.371330\pi$
$114$	3.49425	0.327266
$115$	−4.95837	−0.462371
$116$	2.39922	0.222762
$117$	2.22352	0.205565
$118$	−6.32554	−0.582313
$119$	0	0
$120$	1.00000	0.0912871
$121$	−10.4544	−0.950402
$122$	−6.20519	−0.561791
$123$	−7.68524	−0.692955
$124$	−3.26272	−0.293001
$125$	−1.00000	−0.0894427
$126$	−4.52299	−0.402940
$127$	−8.98613	−0.797390	−0.398695	−	0.917084i	$-0.630537\pi$
$127$	−8.98613	−0.797390	−0.398695	+	0.917084i	$0.630537\pi$
$128$	1.00000	0.0883883
$129$	−11.8547	−1.04375
$130$	−2.22352	−0.195016
$131$	−18.6648	−1.63075	−0.815375	−	0.578933i	$-0.803469\pi$
$131$	−18.6648	−1.63075	−0.815375	+	0.578933i	$0.803469\pi$
$132$	−0.738635	−0.0642899
$133$	15.8044	1.37042
$134$	−5.47034	−0.472565
$135$	1.00000	0.0860663
$136$	0	0
$137$	4.42434	0.377997	0.188998	−	0.981977i	$-0.439476\pi$
$137$	4.42434	0.377997	0.188998	+	0.981977i	$0.439476\pi$
$138$	−4.95837	−0.422085
$139$	−8.44607	−0.716386	−0.358193	−	0.933648i	$-0.616607\pi$
$139$	−8.44607	−0.716386	−0.358193	+	0.933648i	$0.616607\pi$
$140$	4.52299	0.382263
$141$	−0.452476	−0.0381054
$142$	−2.39314	−0.200828
$143$	1.64237	0.137342
$144$	1.00000	0.0833333
$145$	−2.39922	−0.199244
$146$	9.24392	0.765032
$147$	−13.4575	−1.10995
$148$	−9.10743	−0.748626
$149$	18.9489	1.55235	0.776177	−	0.630515i	$-0.217156\pi$
$149$	18.9489	1.55235	0.776177	+	0.630515i	$0.217156\pi$
$150$	−1.00000	−0.0816497
$151$	16.7797	1.36551	0.682754	−	0.730648i	$-0.260782\pi$
$151$	16.7797	1.36551	0.682754	+	0.730648i	$0.260782\pi$
$152$	−3.49425	−0.283421
$153$	0	0
$154$	−3.34084	−0.269212
$155$	3.26272	0.262068
$156$	−2.22352	−0.178024
$157$	−16.0275	−1.27913	−0.639566	−	0.768736i	$-0.720886\pi$
$157$	−16.0275	−1.27913	−0.639566	+	0.768736i	$0.720886\pi$
$158$	−8.18405	−0.651088
$159$	−0.507697	−0.0402630
$160$	−1.00000	−0.0790569
$161$	−22.4267	−1.76747
$162$	1.00000	0.0785674
$163$	6.10920	0.478510	0.239255	−	0.970957i	$-0.423097\pi$
$163$	6.10920	0.478510	0.239255	+	0.970957i	$0.423097\pi$
$164$	7.68524	0.600116
$165$	0.738635	0.0575026
$166$	5.22184	0.405293
$167$	−9.85481	−0.762588	−0.381294	−	0.924454i	$-0.624521\pi$
$167$	−9.85481	−0.762588	−0.381294	+	0.924454i	$0.624521\pi$
$168$	4.52299	0.348956
$169$	−8.05595	−0.619688
$170$	0	0
$171$	−3.49425	−0.267212
$172$	11.8547	0.903915
$173$	−9.99019	−0.759540	−0.379770	−	0.925081i	$-0.623997\pi$
$173$	−9.99019	−0.759540	−0.379770	+	0.925081i	$0.623997\pi$
$174$	−2.39922	−0.181884
$175$	−4.52299	−0.341906
$176$	0.738635	0.0556767
$177$	6.32554	0.475457
$178$	15.2388	1.14220
$179$	−21.2408	−1.58761	−0.793806	−	0.608171i	$-0.791904\pi$
$179$	−21.2408	−1.58761	−0.793806	+	0.608171i	$0.791904\pi$
$180$	−1.00000	−0.0745356
$181$	25.4999	1.89539	0.947696	−	0.319175i	$-0.103406\pi$
$181$	25.4999	1.89539	0.947696	+	0.319175i	$0.103406\pi$
$182$	−10.0570	−0.745473
$183$	6.20519	0.458701
$184$	4.95837	0.365536
$185$	9.10743	0.669591
$186$	3.26272	0.239235
$187$	0	0
$188$	0.452476	0.0330002
$189$	4.52299	0.328999
$190$	3.49425	0.253499
$191$	−14.4028	−1.04215	−0.521074	−	0.853511i	$-0.674469\pi$
$191$	−14.4028	−1.04215	−0.521074	+	0.853511i	$0.674469\pi$
$192$	−1.00000	−0.0721688
$193$	5.14964	0.370679	0.185340	−	0.982675i	$-0.440662\pi$
$193$	5.14964	0.370679	0.185340	+	0.982675i	$0.440662\pi$
$194$	−8.53630	−0.612871
$195$	2.22352	0.159230
$196$	13.4575	0.961247
$197$	21.5734	1.53704	0.768519	−	0.639826i	$-0.220994\pi$
$197$	21.5734	1.53704	0.768519	+	0.639826i	$0.220994\pi$
$198$	0.738635	0.0524925
$199$	7.72061	0.547299	0.273650	−	0.961829i	$-0.411769\pi$
$199$	7.72061	0.547299	0.273650	+	0.961829i	$0.411769\pi$
$200$	1.00000	0.0707107
$201$	5.47034	0.385848
$202$	−17.2781	−1.21568
$203$	−10.8516	−0.761636
$204$	0	0
$205$	−7.68524	−0.536760
$206$	1.91428	0.133374
$207$	4.95837	0.344631
$208$	2.22352	0.154174
$209$	−2.58097	−0.178530
$210$	−4.52299	−0.312116
$211$	5.15684	0.355012	0.177506	−	0.984120i	$-0.443197\pi$
$211$	5.15684	0.355012	0.177506	+	0.984120i	$0.443197\pi$
$212$	0.507697	0.0348687
$213$	2.39314	0.163975
$214$	4.55688	0.311502
$215$	−11.8547	−0.808486
$216$	−1.00000	−0.0680414
$217$	14.7573	1.00179
$218$	−16.0420	−1.08650
$219$	−9.24392	−0.624646
$220$	−0.738635	−0.0497987
$221$	0	0
$222$	9.10743	0.611250
$223$	−25.5891	−1.71357	−0.856785	−	0.515673i	$-0.827542\pi$
$223$	−25.5891	−1.71357	−0.856785	+	0.515673i	$0.827542\pi$
$224$	−4.52299	−0.302205
$225$	1.00000	0.0666667
$226$	8.36190	0.556225
$227$	−5.91894	−0.392854	−0.196427	−	0.980518i	$-0.562934\pi$
$227$	−5.91894	−0.392854	−0.196427	+	0.980518i	$0.562934\pi$
$228$	3.49425	0.231412
$229$	−28.6856	−1.89560	−0.947798	−	0.318872i	$-0.896696\pi$
$229$	−28.6856	−1.89560	−0.947798	+	0.318872i	$0.896696\pi$
$230$	−4.95837	−0.326945
$231$	3.34084	0.219811
$232$	2.39922	0.157516
$233$	0.753654	0.0493735	0.0246868	−	0.999695i	$-0.492141\pi$
$233$	0.753654	0.0493735	0.0246868	+	0.999695i	$0.492141\pi$
$234$	2.22352	0.145356
$235$	−0.452476	−0.0295163
$236$	−6.32554	−0.411758
$237$	8.18405	0.531611
$238$	0	0
$239$	−21.6008	−1.39724	−0.698620	−	0.715493i	$-0.746202\pi$
$239$	−21.6008	−1.39724	−0.698620	+	0.715493i	$0.746202\pi$
$240$	1.00000	0.0645497
$241$	−15.8438	−1.02059	−0.510293	−	0.860001i	$-0.670463\pi$
$241$	−15.8438	−1.02059	−0.510293	+	0.860001i	$0.670463\pi$
$242$	−10.4544	−0.672035
$243$	−1.00000	−0.0641500
$244$	−6.20519	−0.397247
$245$	−13.4575	−0.859765
$246$	−7.68524	−0.489993
$247$	−7.76953	−0.494364
$248$	−3.26272	−0.207183
$249$	−5.22184	−0.330921
$250$	−1.00000	−0.0632456
$251$	−2.13872	−0.134995	−0.0674975	−	0.997719i	$-0.521501\pi$
$251$	−2.13872	−0.134995	−0.0674975	+	0.997719i	$0.521501\pi$
$252$	−4.52299	−0.284922
$253$	3.66243	0.230255
$254$	−8.98613	−0.563840
$255$	0	0
$256$	1.00000	0.0625000
$257$	17.3717	1.08361	0.541807	−	0.840503i	$-0.317740\pi$
$257$	17.3717	1.08361	0.541807	+	0.840503i	$0.317740\pi$
$258$	−11.8547	−0.738044
$259$	41.1928	2.55960
$260$	−2.22352	−0.137897
$261$	2.39922	0.148508
$262$	−18.6648	−1.15311
$263$	−27.8246	−1.71574	−0.857869	−	0.513869i	$-0.828212\pi$
$263$	−27.8246	−1.71574	−0.857869	+	0.513869i	$0.828212\pi$
$264$	−0.738635	−0.0454598
$265$	−0.507697	−0.0311876
$266$	15.8044	0.969033
$267$	−15.2388	−0.932601
$268$	−5.47034	−0.334154
$269$	−0.674940	−0.0411518	−0.0205759	−	0.999788i	$-0.506550\pi$
$269$	−0.674940	−0.0411518	−0.0205759	+	0.999788i	$0.506550\pi$
$270$	1.00000	0.0608581
$271$	6.90710	0.419576	0.209788	−	0.977747i	$-0.432723\pi$
$271$	6.90710	0.419576	0.209788	+	0.977747i	$0.432723\pi$
$272$	0	0
$273$	10.0570	0.608676
$274$	4.42434	0.267284
$275$	0.738635	0.0445413
$276$	−4.95837	−0.298459
$277$	−30.9896	−1.86198	−0.930992	−	0.365039i	$-0.881056\pi$
$277$	−30.9896	−1.86198	−0.930992	+	0.365039i	$0.881056\pi$
$278$	−8.44607	−0.506561
$279$	−3.26272	−0.195334
$280$	4.52299	0.270300
$281$	−23.8519	−1.42289	−0.711443	−	0.702744i	$-0.751958\pi$
$281$	−23.8519	−1.42289	−0.711443	+	0.702744i	$0.751958\pi$
$282$	−0.452476	−0.0269446
$283$	−11.2449	−0.668440	−0.334220	−	0.942495i	$-0.608473\pi$
$283$	−11.2449	−0.668440	−0.334220	+	0.942495i	$0.608473\pi$
$284$	−2.39314	−0.142007
$285$	−3.49425	−0.206981
$286$	1.64237	0.0971154
$287$	−34.7603	−2.05183
$288$	1.00000	0.0589256
$289$	0	0
$290$	−2.39922	−0.140887
$291$	8.53630	0.500407
$292$	9.24392	0.540959
$293$	−16.8648	−0.985252	−0.492626	−	0.870241i	$-0.663963\pi$
$293$	−16.8648	−0.985252	−0.492626	+	0.870241i	$0.663963\pi$
$294$	−13.4575	−0.784855
$295$	6.32554	0.368287
$296$	−9.10743	−0.529358
$297$	−0.738635	−0.0428599
$298$	18.9489	1.09768
$299$	11.0251	0.637595
$300$	−1.00000	−0.0577350
$301$	−53.6189	−3.09054
$302$	16.7797	0.965561
$303$	17.2781	0.992599
$304$	−3.49425	−0.200409
$305$	6.20519	0.355308
$306$	0	0
$307$	−20.4619	−1.16782	−0.583911	−	0.811818i	$-0.698478\pi$
$307$	−20.4619	−1.16782	−0.583911	+	0.811818i	$0.698478\pi$
$308$	−3.34084	−0.190362
$309$	−1.91428	−0.108900
$310$	3.26272	0.185310
$311$	−18.4598	−1.04676	−0.523379	−	0.852100i	$-0.675329\pi$
$311$	−18.4598	−1.04676	−0.523379	+	0.852100i	$0.675329\pi$
$312$	−2.22352	−0.125882
$313$	7.43984	0.420525	0.210262	−	0.977645i	$-0.432568\pi$
$313$	7.43984	0.420525	0.210262	+	0.977645i	$0.432568\pi$
$314$	−16.0275	−0.904483
$315$	4.52299	0.254842
$316$	−8.18405	−0.460389
$317$	11.6483	0.654234	0.327117	−	0.944984i	$-0.393923\pi$
$317$	11.6483	0.654234	0.327117	+	0.944984i	$0.393923\pi$
$318$	−0.507697	−0.0284702
$319$	1.77215	0.0992211
$320$	−1.00000	−0.0559017
$321$	−4.55688	−0.254340
$322$	−22.4267	−1.24979
$323$	0	0
$324$	1.00000	0.0555556
$325$	2.22352	0.123339
$326$	6.10920	0.338358
$327$	16.0420	0.887126
$328$	7.68524	0.424346
$329$	−2.04655	−0.112830
$330$	0.738635	0.0406605
$331$	4.63912	0.254989	0.127495	−	0.991839i	$-0.459306\pi$
$331$	4.63912	0.254989	0.127495	+	0.991839i	$0.459306\pi$
$332$	5.22184	0.286586
$333$	−9.10743	−0.499084
$334$	−9.85481	−0.539231
$335$	5.47034	0.298877
$336$	4.52299	0.246749
$337$	16.8472	0.917723	0.458862	−	0.888508i	$-0.348257\pi$
$337$	16.8472	0.917723	0.458862	+	0.888508i	$0.348257\pi$
$338$	−8.05595	−0.438186
$339$	−8.36190	−0.454156
$340$	0	0
$341$	−2.40996	−0.130507
$342$	−3.49425	−0.188947
$343$	−29.2070	−1.57703
$344$	11.8547	0.639164
$345$	4.95837	0.266950
$346$	−9.99019	−0.537076
$347$	6.97037	0.374189	0.187095	−	0.982342i	$-0.440093\pi$
$347$	6.97037	0.374189	0.187095	+	0.982342i	$0.440093\pi$
$348$	−2.39922	−0.128612
$349$	−2.62562	−0.140546	−0.0702732	−	0.997528i	$-0.522387\pi$
$349$	−2.62562	−0.140546	−0.0702732	+	0.997528i	$0.522387\pi$
$350$	−4.52299	−0.241764
$351$	−2.22352	−0.118683
$352$	0.738635	0.0393694
$353$	24.5423	1.30626	0.653128	−	0.757248i	$-0.273456\pi$
$353$	24.5423	1.30626	0.653128	+	0.757248i	$0.273456\pi$
$354$	6.32554	0.336199
$355$	2.39314	0.127015
$356$	15.2388	0.807656
$357$	0	0
$358$	−21.2408	−1.12261
$359$	29.0611	1.53379	0.766893	−	0.641775i	$-0.221802\pi$
$359$	29.0611	1.53379	0.766893	+	0.641775i	$0.221802\pi$
$360$	−1.00000	−0.0527046
$361$	−6.79025	−0.357381
$362$	25.4999	1.34024
$363$	10.4544	0.548715
$364$	−10.0570	−0.527129
$365$	−9.24392	−0.483849
$366$	6.20519	0.324350
$367$	−18.9472	−0.989034	−0.494517	−	0.869168i	$-0.664655\pi$
$367$	−18.9472	−0.989034	−0.494517	+	0.869168i	$0.664655\pi$
$368$	4.95837	0.258473
$369$	7.68524	0.400077
$370$	9.10743	0.473473
$371$	−2.29631	−0.119218
$372$	3.26272	0.169164
$373$	−20.9434	−1.08441	−0.542203	−	0.840247i	$-0.682410\pi$
$373$	−20.9434	−1.08441	−0.542203	+	0.840247i	$0.682410\pi$
$374$	0	0
$375$	1.00000	0.0516398
$376$	0.452476	0.0233347
$377$	5.33472	0.274752
$378$	4.52299	0.232638
$379$	2.95276	0.151673	0.0758367	−	0.997120i	$-0.475837\pi$
$379$	2.95276	0.151673	0.0758367	+	0.997120i	$0.475837\pi$
$380$	3.49425	0.179251
$381$	8.98613	0.460373
$382$	−14.4028	−0.736910
$383$	19.2691	0.984604	0.492302	−	0.870424i	$-0.336156\pi$
$383$	19.2691	0.984604	0.492302	+	0.870424i	$0.336156\pi$
$384$	−1.00000	−0.0510310
$385$	3.34084	0.170265
$386$	5.14964	0.262110
$387$	11.8547	0.602610
$388$	−8.53630	−0.433365
$389$	−25.7768	−1.30693	−0.653467	−	0.756955i	$-0.726686\pi$
$389$	−25.7768	−1.30693	−0.653467	+	0.756955i	$0.726686\pi$
$390$	2.22352	0.112592
$391$	0	0
$392$	13.4575	0.679704
$393$	18.6648	0.941514
$394$	21.5734	1.08685
$395$	8.18405	0.411784
$396$	0.738635	0.0371178
$397$	30.5177	1.53164	0.765819	−	0.643056i	$-0.222334\pi$
$397$	30.5177	1.53164	0.765819	+	0.643056i	$0.222334\pi$
$398$	7.72061	0.386999
$399$	−15.8044	−0.791212
$400$	1.00000	0.0500000
$401$	27.6364	1.38009	0.690047	−	0.723765i	$-0.257590\pi$
$401$	27.6364	1.38009	0.690047	+	0.723765i	$0.257590\pi$
$402$	5.47034	0.272836
$403$	−7.25474	−0.361384
$404$	−17.2781	−0.859616
$405$	−1.00000	−0.0496904
$406$	−10.8516	−0.538558
$407$	−6.72706	−0.333448
$408$	0	0
$409$	7.94619	0.392914	0.196457	−	0.980512i	$-0.437056\pi$
$409$	7.94619	0.392914	0.196457	+	0.980512i	$0.437056\pi$
$410$	−7.68524	−0.379547
$411$	−4.42434	−0.218237
$412$	1.91428	0.0943099
$413$	28.6104	1.40782
$414$	4.95837	0.243691
$415$	−5.22184	−0.256330
$416$	2.22352	0.109017
$417$	8.44607	0.413606
$418$	−2.58097	−0.126239
$419$	−34.2779	−1.67459	−0.837293	−	0.546755i	$-0.815863\pi$
$419$	−34.2779	−1.67459	−0.837293	+	0.546755i	$0.815863\pi$
$420$	−4.52299	−0.220699
$421$	23.6019	1.15029	0.575144	−	0.818052i	$-0.304946\pi$
$421$	23.6019	1.15029	0.575144	+	0.818052i	$0.304946\pi$
$422$	5.15684	0.251031
$423$	0.452476	0.0220002
$424$	0.507697	0.0246559
$425$	0	0
$426$	2.39314	0.115948
$427$	28.0660	1.35821
$428$	4.55688	0.220265
$429$	−1.64237	−0.0792944
$430$	−11.8547	−0.571686
$431$	0.190639	0.00918278	0.00459139	−	0.999989i	$-0.498539\pi$
$431$	0.190639	0.00918278	0.00459139	+	0.999989i	$0.498539\pi$
$432$	−1.00000	−0.0481125
$433$	5.65799	0.271906	0.135953	−	0.990715i	$-0.456590\pi$
$433$	5.65799	0.271906	0.135953	+	0.990715i	$0.456590\pi$
$434$	14.7573	0.708372
$435$	2.39922	0.115034
$436$	−16.0420	−0.768274
$437$	−17.3258	−0.828804
$438$	−9.24392	−0.441692
$439$	−26.3307	−1.25670	−0.628348	−	0.777933i	$-0.716269\pi$
$439$	−26.3307	−1.25670	−0.628348	+	0.777933i	$0.716269\pi$
$440$	−0.738635	−0.0352130
$441$	13.4575	0.640831
$442$	0	0
$443$	−5.66985	−0.269383	−0.134691	−	0.990888i	$-0.543004\pi$
$443$	−5.66985	−0.269383	−0.134691	+	0.990888i	$0.543004\pi$
$444$	9.10743	0.432219
$445$	−15.2388	−0.722389
$446$	−25.5891	−1.21168
$447$	−18.9489	−0.896252
$448$	−4.52299	−0.213691
$449$	−25.0536	−1.18235	−0.591175	−	0.806543i	$-0.701336\pi$
$449$	−25.0536	−1.18235	−0.591175	+	0.806543i	$0.701336\pi$
$450$	1.00000	0.0471405
$451$	5.67658	0.267300
$452$	8.36190	0.393310
$453$	−16.7797	−0.788377
$454$	−5.91894	−0.277790
$455$	10.0570	0.471478
$456$	3.49425	0.163633
$457$	26.9234	1.25942	0.629712	−	0.776829i	$-0.283173\pi$
$457$	26.9234	1.25942	0.629712	+	0.776829i	$0.283173\pi$
$458$	−28.6856	−1.34039
$459$	0	0
$460$	−4.95837	−0.231185
$461$	−20.9794	−0.977109	−0.488555	−	0.872533i	$-0.662476\pi$
$461$	−20.9794	−0.977109	−0.488555	+	0.872533i	$0.662476\pi$
$462$	3.34084	0.155430
$463$	−16.6982	−0.776032	−0.388016	−	0.921653i	$-0.626840\pi$
$463$	−16.6982	−0.776032	−0.388016	+	0.921653i	$0.626840\pi$
$464$	2.39922	0.111381
$465$	−3.26272	−0.151305
$466$	0.753654	0.0349124
$467$	10.4379	0.483008	0.241504	−	0.970400i	$-0.422359\pi$
$467$	10.4379	0.483008	0.241504	+	0.970400i	$0.422359\pi$
$468$	2.22352	0.102782
$469$	24.7423	1.14249
$470$	−0.452476	−0.0208712
$471$	16.0275	0.738507
$472$	−6.32554	−0.291157
$473$	8.75632	0.402616
$474$	8.18405	0.375906
$475$	−3.49425	−0.160327
$476$	0	0
$477$	0.507697	0.0232458
$478$	−21.6008	−0.987998
$479$	−28.5898	−1.30630	−0.653151	−	0.757228i	$-0.726553\pi$
$479$	−28.5898	−1.30630	−0.653151	+	0.757228i	$0.726553\pi$
$480$	1.00000	0.0456435
$481$	−20.2506	−0.923346
$482$	−15.8438	−0.721664
$483$	22.4267	1.02045
$484$	−10.4544	−0.475201
$485$	8.53630	0.387614
$486$	−1.00000	−0.0453609
$487$	−2.27544	−0.103110	−0.0515549	−	0.998670i	$-0.516418\pi$
$487$	−2.27544	−0.103110	−0.0515549	+	0.998670i	$0.516418\pi$
$488$	−6.20519	−0.280896
$489$	−6.10920	−0.276268
$490$	−13.4575	−0.607946
$491$	−21.5853	−0.974131	−0.487066	−	0.873365i	$-0.661933\pi$
$491$	−21.5853	−0.974131	−0.487066	+	0.873365i	$0.661933\pi$
$492$	−7.68524	−0.346477
$493$	0	0
$494$	−7.76953	−0.349568
$495$	−0.738635	−0.0331992
$496$	−3.26272	−0.146501
$497$	10.8241	0.485529
$498$	−5.22184	−0.233996
$499$	−16.1056	−0.720985	−0.360493	−	0.932762i	$-0.617391\pi$
$499$	−16.1056	−0.720985	−0.360493	+	0.932762i	$0.617391\pi$
$500$	−1.00000	−0.0447214
$501$	9.85481	0.440280
$502$	−2.13872	−0.0954559
$503$	19.4219	0.865978	0.432989	−	0.901399i	$-0.357459\pi$
$503$	19.4219	0.865978	0.432989	+	0.901399i	$0.357459\pi$
$504$	−4.52299	−0.201470
$505$	17.2781	0.768864
$506$	3.66243	0.162815
$507$	8.05595	0.357777
$508$	−8.98613	−0.398695
$509$	3.08745	0.136849	0.0684243	−	0.997656i	$-0.478203\pi$
$509$	3.08745	0.136849	0.0684243	+	0.997656i	$0.478203\pi$
$510$	0	0
$511$	−41.8102	−1.84957
$512$	1.00000	0.0441942
$513$	3.49425	0.154275
$514$	17.3717	0.766231
$515$	−1.91428	−0.0843533
$516$	−11.8547	−0.521876
$517$	0.334215	0.0146987
$518$	41.1928	1.80991
$519$	9.99019	0.438521
$520$	−2.22352	−0.0975079
$521$	−22.6471	−0.992186	−0.496093	−	0.868269i	$-0.665233\pi$
$521$	−22.6471	−0.992186	−0.496093	+	0.868269i	$0.665233\pi$
$522$	2.39922	0.105011
$523$	−32.2345	−1.40952	−0.704759	−	0.709447i	$-0.748945\pi$
$523$	−32.2345	−1.40952	−0.704759	+	0.709447i	$0.748945\pi$
$524$	−18.6648	−0.815375
$525$	4.52299	0.197400
$526$	−27.8246	−1.21321
$527$	0	0
$528$	−0.738635	−0.0321449
$529$	1.58546	0.0689329
$530$	−0.507697	−0.0220529
$531$	−6.32554	−0.274505
$532$	15.8044	0.685210
$533$	17.0883	0.740176
$534$	−15.2388	−0.659448
$535$	−4.55688	−0.197011
$536$	−5.47034	−0.236283
$537$	21.2408	0.916609
$538$	−0.674940	−0.0290987
$539$	9.94015	0.428152
$540$	1.00000	0.0430331
$541$	10.8967	0.468487	0.234244	−	0.972178i	$-0.424739\pi$
$541$	10.8967	0.468487	0.234244	+	0.972178i	$0.424739\pi$
$542$	6.90710	0.296685
$543$	−25.4999	−1.09430
$544$	0	0
$545$	16.0420	0.687165
$546$	10.0570	0.430399
$547$	27.0284	1.15565	0.577826	−	0.816160i	$-0.303901\pi$
$547$	27.0284	1.15565	0.577826	+	0.816160i	$0.303901\pi$
$548$	4.42434	0.188998
$549$	−6.20519	−0.264831
$550$	0.738635	0.0314955
$551$	−8.38346	−0.357147
$552$	−4.95837	−0.211042
$553$	37.0164	1.57410
$554$	−30.9896	−1.31662
$555$	−9.10743	−0.386589
$556$	−8.44607	−0.358193
$557$	24.6707	1.04533	0.522665	−	0.852538i	$-0.324938\pi$
$557$	24.6707	1.04533	0.522665	+	0.852538i	$0.324938\pi$
$558$	−3.26272	−0.138122
$559$	26.3593	1.11488
$560$	4.52299	0.191131
$561$	0	0
$562$	−23.8519	−1.00613
$563$	−37.7844	−1.59242	−0.796212	−	0.605018i	$-0.793166\pi$
$563$	−37.7844	−1.59242	−0.796212	+	0.605018i	$0.793166\pi$
$564$	−0.452476	−0.0190527
$565$	−8.36190	−0.351788
$566$	−11.2449	−0.472658
$567$	−4.52299	−0.189948
$568$	−2.39314	−0.100414
$569$	31.1562	1.30614	0.653068	−	0.757299i	$-0.273482\pi$
$569$	31.1562	1.30614	0.653068	+	0.757299i	$0.273482\pi$
$570$	−3.49425	−0.146358
$571$	8.02223	0.335720	0.167860	−	0.985811i	$-0.446314\pi$
$571$	8.02223	0.335720	0.167860	+	0.985811i	$0.446314\pi$
$572$	1.64237	0.0686710
$573$	14.4028	0.601685
$574$	−34.7603	−1.45087
$575$	4.95837	0.206778
$576$	1.00000	0.0416667
$577$	−38.3002	−1.59446	−0.797229	−	0.603677i	$-0.793702\pi$
$577$	−38.3002	−1.59446	−0.797229	+	0.603677i	$0.793702\pi$
$578$	0	0
$579$	−5.14964	−0.214012
$580$	−2.39922	−0.0996221
$581$	−23.6183	−0.979854
$582$	8.53630	0.353841
$583$	0.375002	0.0155310
$584$	9.24392	0.382516
$585$	−2.22352	−0.0919313
$586$	−16.8648	−0.696678
$587$	12.6087	0.520417	0.260209	−	0.965552i	$-0.416209\pi$
$587$	12.6087	0.520417	0.260209	+	0.965552i	$0.416209\pi$
$588$	−13.4575	−0.554976
$589$	11.4008	0.469760
$590$	6.32554	0.260418
$591$	−21.5734	−0.887410
$592$	−9.10743	−0.374313
$593$	−19.8604	−0.815569	−0.407784	−	0.913078i	$-0.633699\pi$
$593$	−19.8604	−0.815569	−0.407784	+	0.913078i	$0.633699\pi$
$594$	−0.738635	−0.0303065
$595$	0	0
$596$	18.9489	0.776177
$597$	−7.72061	−0.315983
$598$	11.0251	0.450848
$599$	−27.7802	−1.13507	−0.567534	−	0.823350i	$-0.692102\pi$
$599$	−27.7802	−1.13507	−0.567534	+	0.823350i	$0.692102\pi$
$600$	−1.00000	−0.0408248
$601$	−5.10439	−0.208212	−0.104106	−	0.994566i	$-0.533198\pi$
$601$	−5.10439	−0.208212	−0.104106	+	0.994566i	$0.533198\pi$
$602$	−53.6189	−2.18534
$603$	−5.47034	−0.222769
$604$	16.7797	0.682754
$605$	10.4544	0.425033
$606$	17.2781	0.701874
$607$	36.4466	1.47932	0.739662	−	0.672979i	$-0.234985\pi$
$607$	36.4466	1.47932	0.739662	+	0.672979i	$0.234985\pi$
$608$	−3.49425	−0.141710
$609$	10.8516	0.439731
$610$	6.20519	0.251241
$611$	1.00609	0.0407021
$612$	0	0
$613$	44.5224	1.79824	0.899122	−	0.437698i	$-0.144206\pi$
$613$	44.5224	1.79824	0.899122	+	0.437698i	$0.144206\pi$
$614$	−20.4619	−0.825775
$615$	7.68524	0.309899
$616$	−3.34084	−0.134606
$617$	−0.221908	−0.00893367	−0.00446684	−	0.999990i	$-0.501422\pi$
$617$	−0.221908	−0.00893367	−0.00446684	+	0.999990i	$0.501422\pi$
$618$	−1.91428	−0.0770037
$619$	36.1359	1.45243	0.726213	−	0.687470i	$-0.241279\pi$
$619$	36.1359	1.45243	0.726213	+	0.687470i	$0.241279\pi$
$620$	3.26272	0.131034
$621$	−4.95837	−0.198973
$622$	−18.4598	−0.740170
$623$	−68.9251	−2.76142
$624$	−2.22352	−0.0890121
$625$	1.00000	0.0400000
$626$	7.43984	0.297356
$627$	2.58097	0.103074
$628$	−16.0275	−0.639566
$629$	0	0
$630$	4.52299	0.180200
$631$	10.0206	0.398914	0.199457	−	0.979907i	$-0.436082\pi$
$631$	10.0206	0.398914	0.199457	+	0.979907i	$0.436082\pi$
$632$	−8.18405	−0.325544
$633$	−5.15684	−0.204966
$634$	11.6483	0.462613
$635$	8.98613	0.356604
$636$	−0.507697	−0.0201315
$637$	29.9230	1.18559
$638$	1.77215	0.0701599
$639$	−2.39314	−0.0946711
$640$	−1.00000	−0.0395285
$641$	−9.14405	−0.361168	−0.180584	−	0.983560i	$-0.557799\pi$
$641$	−9.14405	−0.361168	−0.180584	+	0.983560i	$0.557799\pi$
$642$	−4.55688	−0.179846
$643$	10.1515	0.400336	0.200168	−	0.979762i	$-0.435851\pi$
$643$	10.1515	0.400336	0.200168	+	0.979762i	$0.435851\pi$
$644$	−22.4267	−0.883735
$645$	11.8547	0.466780
$646$	0	0
$647$	−21.1354	−0.830919	−0.415460	−	0.909612i	$-0.636379\pi$
$647$	−21.1354	−0.830919	−0.415460	+	0.909612i	$0.636379\pi$
$648$	1.00000	0.0392837
$649$	−4.67227	−0.183402
$650$	2.22352	0.0872137
$651$	−14.7573	−0.578383
$652$	6.10920	0.239255
$653$	33.6283	1.31598	0.657989	−	0.753028i	$-0.271407\pi$
$653$	33.6283	1.31598	0.657989	+	0.753028i	$0.271407\pi$
$654$	16.0420	0.627293
$655$	18.6648	0.729294
$656$	7.68524	0.300058
$657$	9.24392	0.360640
$658$	−2.04655	−0.0797827
$659$	38.9551	1.51748	0.758738	−	0.651395i	$-0.225816\pi$
$659$	38.9551	1.51748	0.758738	+	0.651395i	$0.225816\pi$
$660$	0.738635	0.0287513
$661$	−0.813681	−0.0316485	−0.0158243	−	0.999875i	$-0.505037\pi$
$661$	−0.813681	−0.0316485	−0.0158243	+	0.999875i	$0.505037\pi$
$662$	4.63912	0.180305
$663$	0	0
$664$	5.22184	0.202647
$665$	−15.8044	−0.612870
$666$	−9.10743	−0.352906
$667$	11.8962	0.460623
$668$	−9.85481	−0.381294
$669$	25.5891	0.989330
$670$	5.47034	0.211338
$671$	−4.58337	−0.176939
$672$	4.52299	0.174478
$673$	−9.00543	−0.347134	−0.173567	−	0.984822i	$-0.555529\pi$
$673$	−9.00543	−0.347134	−0.173567	+	0.984822i	$0.555529\pi$
$674$	16.8472	0.648928
$675$	−1.00000	−0.0384900
$676$	−8.05595	−0.309844
$677$	31.2670	1.20169	0.600844	−	0.799366i	$-0.294831\pi$
$677$	31.2670	1.20169	0.600844	+	0.799366i	$0.294831\pi$
$678$	−8.36190	−0.321137
$679$	38.6096	1.48170
$680$	0	0
$681$	5.91894	0.226814
$682$	−2.40996	−0.0922822
$683$	−44.8872	−1.71756	−0.858780	−	0.512345i	$-0.828777\pi$
$683$	−44.8872	−1.71756	−0.858780	+	0.512345i	$0.828777\pi$
$684$	−3.49425	−0.133606
$685$	−4.42434	−0.169045
$686$	−29.2070	−1.11513
$687$	28.6856	1.09442
$688$	11.8547	0.451958
$689$	1.12887	0.0430067
$690$	4.95837	0.188762
$691$	37.1393	1.41285	0.706424	−	0.707789i	$-0.250307\pi$
$691$	37.1393	1.41285	0.706424	+	0.707789i	$0.250307\pi$
$692$	−9.99019	−0.379770
$693$	−3.34084	−0.126908
$694$	6.97037	0.264592
$695$	8.44607	0.320378
$696$	−2.39922	−0.0909421
$697$	0	0
$698$	−2.62562	−0.0993813
$699$	−0.753654	−0.0285058
$700$	−4.52299	−0.170953
$701$	5.20407	0.196555	0.0982774	−	0.995159i	$-0.468667\pi$
$701$	5.20407	0.196555	0.0982774	+	0.995159i	$0.468667\pi$
$702$	−2.22352	−0.0839214
$703$	31.8236	1.20025
$704$	0.738635	0.0278383
$705$	0.452476	0.0170412
$706$	24.5423	0.923662
$707$	78.1486	2.93908
$708$	6.32554	0.237728
$709$	−45.5584	−1.71098	−0.855491	−	0.517818i	$-0.826745\pi$
$709$	−45.5584	−1.71098	−0.855491	+	0.517818i	$0.826745\pi$
$710$	2.39314	0.0898129
$711$	−8.18405	−0.306926
$712$	15.2388	0.571099
$713$	−16.1778	−0.605863
$714$	0	0
$715$	−1.64237	−0.0614212
$716$	−21.2408	−0.793806
$717$	21.6008	0.806697
$718$	29.0611	1.08455
$719$	−41.1142	−1.53330	−0.766650	−	0.642065i	$-0.778078\pi$
$719$	−41.1142	−1.53330	−0.766650	+	0.642065i	$0.778078\pi$
$720$	−1.00000	−0.0372678
$721$	−8.65828	−0.322451
$722$	−6.79025	−0.252707
$723$	15.8438	0.589236
$724$	25.4999	0.947696
$725$	2.39922	0.0891047
$726$	10.4544	0.388000
$727$	−3.12419	−0.115870	−0.0579349	−	0.998320i	$-0.518452\pi$
$727$	−3.12419	−0.115870	−0.0579349	+	0.998320i	$0.518452\pi$
$728$	−10.0570	−0.372736
$729$	1.00000	0.0370370
$730$	−9.24392	−0.342133
$731$	0	0
$732$	6.20519	0.229350
$733$	18.2968	0.675807	0.337904	−	0.941181i	$-0.390282\pi$
$733$	18.2968	0.675807	0.337904	+	0.941181i	$0.390282\pi$
$734$	−18.9472	−0.699352
$735$	13.4575	0.496386
$736$	4.95837	0.182768
$737$	−4.04058	−0.148837
$738$	7.68524	0.282897
$739$	−53.7054	−1.97559	−0.987793	−	0.155771i	$-0.950214\pi$
$739$	−53.7054	−1.97559	−0.987793	+	0.155771i	$0.950214\pi$
$740$	9.10743	0.334796
$741$	7.76953	0.285421
$742$	−2.29631	−0.0843001
$743$	46.8227	1.71776	0.858880	−	0.512177i	$-0.171161\pi$
$743$	46.8227	1.71776	0.858880	+	0.512177i	$0.171161\pi$
$744$	3.26272	0.119617
$745$	−18.9489	−0.694234
$746$	−20.9434	−0.766791
$747$	5.22184	0.191057
$748$	0	0
$749$	−20.6107	−0.753100
$750$	1.00000	0.0365148
$751$	13.2255	0.482605	0.241302	−	0.970450i	$-0.422425\pi$
$751$	13.2255	0.482605	0.241302	+	0.970450i	$0.422425\pi$
$752$	0.452476	0.0165001
$753$	2.13872	0.0779394
$754$	5.33472	0.194279
$755$	−16.7797	−0.610674
$756$	4.52299	0.164500
$757$	13.4426	0.488581	0.244291	−	0.969702i	$-0.421445\pi$
$757$	13.4426	0.488581	0.244291	+	0.969702i	$0.421445\pi$
$758$	2.95276	0.107249
$759$	−3.66243	−0.132938
$760$	3.49425	0.126750
$761$	−2.75636	−0.0999178	−0.0499589	−	0.998751i	$-0.515909\pi$
$761$	−2.75636	−0.0999178	−0.0499589	+	0.998751i	$0.515909\pi$
$762$	8.98613	0.325533
$763$	72.5580	2.62678
$764$	−14.4028	−0.521074
$765$	0	0
$766$	19.2691	0.696220
$767$	−14.0650	−0.507857
$768$	−1.00000	−0.0360844
$769$	41.0278	1.47950	0.739749	−	0.672883i	$-0.234944\pi$
$769$	41.0278	1.47950	0.739749	+	0.672883i	$0.234944\pi$
$770$	3.34084	0.120395
$771$	−17.3717	−0.625625
$772$	5.14964	0.185340
$773$	51.7055	1.85972	0.929858	−	0.367918i	$-0.119929\pi$
$773$	51.7055	1.85972	0.929858	+	0.367918i	$0.119929\pi$
$774$	11.8547	0.426110
$775$	−3.26272	−0.117201
$776$	−8.53630	−0.306435
$777$	−41.1928	−1.47778
$778$	−25.7768	−0.924142
$779$	−26.8541	−0.962148
$780$	2.22352	0.0796149
$781$	−1.76766	−0.0632517
$782$	0	0
$783$	−2.39922	−0.0857411
$784$	13.4575	0.480624
$785$	16.0275	0.572045
$786$	18.6648	0.665751
$787$	−5.79572	−0.206595	−0.103297	−	0.994651i	$-0.532939\pi$
$787$	−5.79572	−0.206595	−0.103297	+	0.994651i	$0.532939\pi$
$788$	21.5734	0.768519
$789$	27.8246	0.990582
$790$	8.18405	0.291175
$791$	−37.8208	−1.34475
$792$	0.738635	0.0262462
$793$	−13.7974	−0.489959
$794$	30.5177	1.08303
$795$	0.507697	0.0180061
$796$	7.72061	0.273650
$797$	−3.09738	−0.109715	−0.0548574	−	0.998494i	$-0.517470\pi$
$797$	−3.09738	−0.109715	−0.0548574	+	0.998494i	$0.517470\pi$
$798$	−15.8044	−0.559471
$799$	0	0
$800$	1.00000	0.0353553
$801$	15.2388	0.538437
$802$	27.6364	0.975874
$803$	6.82788	0.240951
$804$	5.47034	0.192924
$805$	22.4267	0.790437
$806$	−7.25474	−0.255537
$807$	0.674940	0.0237590
$808$	−17.2781	−0.607840
$809$	−19.8843	−0.699094	−0.349547	−	0.936919i	$-0.613665\pi$
$809$	−19.8843	−0.699094	−0.349547	+	0.936919i	$0.613665\pi$
$810$	−1.00000	−0.0351364
$811$	−20.0476	−0.703965	−0.351983	−	0.936007i	$-0.614492\pi$
$811$	−20.0476	−0.703965	−0.351983	+	0.936007i	$0.614492\pi$
$812$	−10.8516	−0.380818
$813$	−6.90710	−0.242243
$814$	−6.72706	−0.235783
$815$	−6.10920	−0.213996
$816$	0	0
$817$	−41.4234	−1.44922
$818$	7.94619	0.277832
$819$	−10.0570	−0.351419
$820$	−7.68524	−0.268380
$821$	−7.45825	−0.260295	−0.130147	−	0.991495i	$-0.541545\pi$
$821$	−7.45825	−0.260295	−0.130147	+	0.991495i	$0.541545\pi$
$822$	−4.42434	−0.154317
$823$	−22.7139	−0.791756	−0.395878	−	0.918303i	$-0.629560\pi$
$823$	−22.7139	−0.791756	−0.395878	+	0.918303i	$0.629560\pi$
$824$	1.91428	0.0666871
$825$	−0.738635	−0.0257160
$826$	28.6104	0.995483
$827$	−49.3035	−1.71445	−0.857226	−	0.514941i	$-0.827814\pi$
$827$	−49.3035	−1.71445	−0.857226	+	0.514941i	$0.827814\pi$
$828$	4.95837	0.172315
$829$	−4.62363	−0.160585	−0.0802927	−	0.996771i	$-0.525586\pi$
$829$	−4.62363	−0.160585	−0.0802927	+	0.996771i	$0.525586\pi$
$830$	−5.22184	−0.181253
$831$	30.9896	1.07502
$832$	2.22352	0.0770868
$833$	0	0
$834$	8.44607	0.292463
$835$	9.85481	0.341040
$836$	−2.58097	−0.0892648
$837$	3.26272	0.112776
$838$	−34.2779	−1.18411
$839$	56.6787	1.95677	0.978383	−	0.206799i	$-0.0663047\pi$
$839$	56.6787	1.95677	0.978383	+	0.206799i	$0.0663047\pi$
$840$	−4.52299	−0.156058
$841$	−23.2438	−0.801509
$842$	23.6019	0.813377
$843$	23.8519	0.821503
$844$	5.15684	0.177506
$845$	8.05595	0.277133
$846$	0.452476	0.0155565
$847$	47.2853	1.62474
$848$	0.507697	0.0174344
$849$	11.2449	0.385924
$850$	0	0
$851$	−45.1580	−1.54800
$852$	2.39314	0.0819876
$853$	−25.6128	−0.876966	−0.438483	−	0.898739i	$-0.644484\pi$
$853$	−25.6128	−0.876966	−0.438483	+	0.898739i	$0.644484\pi$
$854$	28.0660	0.960400
$855$	3.49425	0.119501
$856$	4.55688	0.155751
$857$	8.36989	0.285910	0.142955	−	0.989729i	$-0.454340\pi$
$857$	8.36989	0.285910	0.142955	+	0.989729i	$0.454340\pi$
$858$	−1.64237	−0.0560696
$859$	11.3010	0.385586	0.192793	−	0.981239i	$-0.438245\pi$
$859$	11.3010	0.385586	0.192793	+	0.981239i	$0.438245\pi$
$860$	−11.8547	−0.404243
$861$	34.7603	1.18463
$862$	0.190639	0.00649321
$863$	−25.9987	−0.885008	−0.442504	−	0.896767i	$-0.645910\pi$
$863$	−25.9987	−0.885008	−0.442504	+	0.896767i	$0.645910\pi$
$864$	−1.00000	−0.0340207
$865$	9.99019	0.339677
$866$	5.65799	0.192266
$867$	0	0
$868$	14.7573	0.500895
$869$	−6.04502	−0.205063
$870$	2.39922	0.0813411
$871$	−12.1634	−0.412142
$872$	−16.0420	−0.543252
$873$	−8.53630	−0.288910
$874$	−17.3258	−0.586053
$875$	4.52299	0.152905
$876$	−9.24392	−0.312323
$877$	−12.5818	−0.424859	−0.212429	−	0.977176i	$-0.568138\pi$
$877$	−12.5818	−0.424859	−0.212429	+	0.977176i	$0.568138\pi$
$878$	−26.3307	−0.888618
$879$	16.8648	0.568835
$880$	−0.738635	−0.0248994
$881$	−36.2939	−1.22277	−0.611387	−	0.791332i	$-0.709388\pi$
$881$	−36.2939	−1.22277	−0.611387	+	0.791332i	$0.709388\pi$
$882$	13.4575	0.453136
$883$	25.1862	0.847583	0.423791	−	0.905760i	$-0.360699\pi$
$883$	25.1862	0.847583	0.423791	+	0.905760i	$0.360699\pi$
$884$	0	0
$885$	−6.32554	−0.212631
$886$	−5.66985	−0.190482
$887$	15.8117	0.530904	0.265452	−	0.964124i	$-0.414479\pi$
$887$	15.8117	0.530904	0.265452	+	0.964124i	$0.414479\pi$
$888$	9.10743	0.305625
$889$	40.6442	1.36316
$890$	−15.2388	−0.510806
$891$	0.738635	0.0247452
$892$	−25.5891	−0.856785
$893$	−1.58106	−0.0529083
$894$	−18.9489	−0.633746
$895$	21.2408	0.710002
$896$	−4.52299	−0.151103
$897$	−11.0251	−0.368116
$898$	−25.0536	−0.836048
$899$	−7.82799	−0.261078
$900$	1.00000	0.0333333
$901$	0	0
$902$	5.67658	0.189010
$903$	53.6189	1.78432
$904$	8.36190	0.278113
$905$	−25.4999	−0.847645
$906$	−16.7797	−0.557467
$907$	7.74749	0.257251	0.128626	−	0.991693i	$-0.458943\pi$
$907$	7.74749	0.257251	0.128626	+	0.991693i	$0.458943\pi$
$908$	−5.91894	−0.196427
$909$	−17.2781	−0.573077
$910$	10.0570	0.333385
$911$	−10.0941	−0.334433	−0.167216	−	0.985920i	$-0.553478\pi$
$911$	−10.0941	−0.334433	−0.167216	+	0.985920i	$0.553478\pi$
$912$	3.49425	0.115706
$913$	3.85703	0.127649
$914$	26.9234	0.890547
$915$	−6.20519	−0.205137
$916$	−28.6856	−0.947798
$917$	84.4207	2.78782
$918$	0	0
$919$	45.4392	1.49890	0.749451	−	0.662060i	$-0.230318\pi$
$919$	45.4392	1.49890	0.749451	+	0.662060i	$0.230318\pi$
$920$	−4.95837	−0.163473
$921$	20.4619	0.674243
$922$	−20.9794	−0.690920
$923$	−5.32120	−0.175149
$924$	3.34084	0.109906
$925$	−9.10743	−0.299450
$926$	−16.6982	−0.548738
$927$	1.91428	0.0628732
$928$	2.39922	0.0787582
$929$	0.431256	0.0141491	0.00707453	−	0.999975i	$-0.497748\pi$
$929$	0.431256	0.0141491	0.00707453	+	0.999975i	$0.497748\pi$
$930$	−3.26272	−0.106989
$931$	−47.0237	−1.54114
$932$	0.753654	0.0246868
$933$	18.4598	0.604346
$934$	10.4379	0.341539
$935$	0	0
$936$	2.22352	0.0726781
$937$	24.5285	0.801310	0.400655	−	0.916229i	$-0.368783\pi$
$937$	24.5285	0.801310	0.400655	+	0.916229i	$0.368783\pi$
$938$	24.7423	0.807865
$939$	−7.43984	−0.242790
$940$	−0.452476	−0.0147582
$941$	6.76092	0.220400	0.110200	−	0.993909i	$-0.464851\pi$
$941$	6.76092	0.220400	0.110200	+	0.993909i	$0.464851\pi$
$942$	16.0275	0.522204
$943$	38.1063	1.24091
$944$	−6.32554	−0.205879
$945$	−4.52299	−0.147133
$946$	8.75632	0.284692
$947$	7.29976	0.237210	0.118605	−	0.992942i	$-0.462158\pi$
$947$	7.29976	0.237210	0.118605	+	0.992942i	$0.462158\pi$
$948$	8.18405	0.265806
$949$	20.5541	0.667213
$950$	−3.49425	−0.113368
$951$	−11.6483	−0.377722
$952$	0	0
$953$	−0.778337	−0.0252128	−0.0126064	−	0.999921i	$-0.504013\pi$
$953$	−0.778337	−0.0252128	−0.0126064	+	0.999921i	$0.504013\pi$
$954$	0.507697	0.0164373
$955$	14.4028	0.466063
$956$	−21.6008	−0.698620
$957$	−1.77215	−0.0572853
$958$	−28.5898	−0.923694
$959$	−20.0113	−0.646197
$960$	1.00000	0.0322749
$961$	−20.3546	−0.656601
$962$	−20.2506	−0.652904
$963$	4.55688	0.146843
$964$	−15.8438	−0.510293
$965$	−5.14964	−0.165773
$966$	22.4267	0.721567
$967$	−56.0313	−1.80184	−0.900922	−	0.433981i	$-0.857109\pi$
$967$	−56.0313	−1.80184	−0.900922	+	0.433981i	$0.857109\pi$
$968$	−10.4544	−0.336018
$969$	0	0
$970$	8.53630	0.274084
$971$	−32.1298	−1.03109	−0.515547	−	0.856861i	$-0.672411\pi$
$971$	−32.1298	−1.03109	−0.515547	+	0.856861i	$0.672411\pi$
$972$	−1.00000	−0.0320750
$973$	38.2015	1.22468
$974$	−2.27544	−0.0729097
$975$	−2.22352	−0.0712097
$976$	−6.20519	−0.198623
$977$	−17.6026	−0.563156	−0.281578	−	0.959538i	$-0.590858\pi$
$977$	−17.6026	−0.563156	−0.281578	+	0.959538i	$0.590858\pi$
$978$	−6.10920	−0.195351
$979$	11.2559	0.359741
$980$	−13.4575	−0.429883
$981$	−16.0420	−0.512183
$982$	−21.5853	−0.688815
$983$	−43.5076	−1.38768	−0.693839	−	0.720130i	$-0.744082\pi$
$983$	−43.5076	−1.38768	−0.693839	+	0.720130i	$0.744082\pi$
$984$	−7.68524	−0.244996
$985$	−21.5734	−0.687385
$986$	0	0
$987$	2.04655	0.0651423
$988$	−7.76953	−0.247182
$989$	58.7802	1.86910
$990$	−0.738635	−0.0234753
$991$	34.8086	1.10573	0.552865	−	0.833271i	$-0.313534\pi$
$991$	34.8086	1.10573	0.552865	+	0.833271i	$0.313534\pi$
$992$	−3.26272	−0.103592
$993$	−4.63912	−0.147218
$994$	10.8241	0.343321
$995$	−7.72061	−0.244760
$996$	−5.22184	−0.165460
$997$	24.0259	0.760908	0.380454	−	0.924800i	$-0.375768\pi$
$997$	24.0259	0.760908	0.380454	+	0.924800i	$0.375768\pi$
$998$	−16.1056	−0.509813
$999$	9.10743	0.288146

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8670.2.a.cf.1.1	✓	6
17.16	even	2		8670.2.a.ci.1.6	yes	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8670.2.a.cf.1.1	✓	6	1.1	even	1	trivial
8670.2.a.ci.1.6	yes	6	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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{6} -4.52299 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{6} -4.52299 q^{7} +1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} +0.738635 q^{11} -1.00000 q^{12} +2.22352 q^{13} -4.52299 q^{14} +1.00000 q^{15} +1.00000 q^{16} +1.00000 q^{18} -3.49425 q^{19} -1.00000 q^{20} +4.52299 q^{21} +0.738635 q^{22} +4.95837 q^{23} -1.00000 q^{24} +1.00000 q^{25} +2.22352 q^{26} -1.00000 q^{27} -4.52299 q^{28} +2.39922 q^{29} +1.00000 q^{30} -3.26272 q^{31} +1.00000 q^{32} -0.738635 q^{33} +4.52299 q^{35} +1.00000 q^{36} -9.10743 q^{37} -3.49425 q^{38} -2.22352 q^{39} -1.00000 q^{40} +7.68524 q^{41} +4.52299 q^{42} +11.8547 q^{43} +0.738635 q^{44} -1.00000 q^{45} +4.95837 q^{46} +0.452476 q^{47} -1.00000 q^{48} +13.4575 q^{49} +1.00000 q^{50} +2.22352 q^{52} +0.507697 q^{53} -1.00000 q^{54} -0.738635 q^{55} -4.52299 q^{56} +3.49425 q^{57} +2.39922 q^{58} -6.32554 q^{59} +1.00000 q^{60} -6.20519 q^{61} -3.26272 q^{62} -4.52299 q^{63} +1.00000 q^{64} -2.22352 q^{65} -0.738635 q^{66} -5.47034 q^{67} -4.95837 q^{69} +4.52299 q^{70} -2.39314 q^{71} +1.00000 q^{72} +9.24392 q^{73} -9.10743 q^{74} -1.00000 q^{75} -3.49425 q^{76} -3.34084 q^{77} -2.22352 q^{78} -8.18405 q^{79} -1.00000 q^{80} +1.00000 q^{81} +7.68524 q^{82} +5.22184 q^{83} +4.52299 q^{84} +11.8547 q^{86} -2.39922 q^{87} +0.738635 q^{88} +15.2388 q^{89} -1.00000 q^{90} -10.0570 q^{91} +4.95837 q^{92} +3.26272 q^{93} +0.452476 q^{94} +3.49425 q^{95} -1.00000 q^{96} -8.53630 q^{97} +13.4575 q^{98} +0.738635 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 6 q^{2} - 6 q^{3} + 6 q^{4} - 6 q^{5} - 6 q^{6} - 6 q^{7} + 6 q^{8} + 6 q^{9}+O(q^{10}) \) 6 * q + 6 * q^2 - 6 * q^3 + 6 * q^4 - 6 * q^5 - 6 * q^6 - 6 * q^7 + 6 * q^8 + 6 * q^9	\( 6 q + 6 q^{2} - 6 q^{3} + 6 q^{4} - 6 q^{5} - 6 q^{6} - 6 q^{7} + 6 q^{8} + 6 q^{9} - 6 q^{10} - 6 q^{11} - 6 q^{12} + 6 q^{13} - 6 q^{14} + 6 q^{15} + 6 q^{16} + 6 q^{18} + 6 q^{19} - 6 q^{20} + 6 q^{21} - 6 q^{22} - 6 q^{23} - 6 q^{24} + 6 q^{25} + 6 q^{26} - 6 q^{27} - 6 q^{28} - 12 q^{29} + 6 q^{30} - 6 q^{31} + 6 q^{32} + 6 q^{33} + 6 q^{35} + 6 q^{36} - 6 q^{37} + 6 q^{38} - 6 q^{39} - 6 q^{40} - 12 q^{41} + 6 q^{42} + 18 q^{43} - 6 q^{44} - 6 q^{45} - 6 q^{46} - 6 q^{47} - 6 q^{48} + 18 q^{49} + 6 q^{50} + 6 q^{52} - 6 q^{53} - 6 q^{54} + 6 q^{55} - 6 q^{56} - 6 q^{57} - 12 q^{58} - 30 q^{59} + 6 q^{60} + 24 q^{61} - 6 q^{62} - 6 q^{63} + 6 q^{64} - 6 q^{65} + 6 q^{66} + 6 q^{69} + 6 q^{70} - 24 q^{71} + 6 q^{72} - 6 q^{73} - 6 q^{74} - 6 q^{75} + 6 q^{76} - 6 q^{78} - 6 q^{79} - 6 q^{80} + 6 q^{81} - 12 q^{82} + 18 q^{83} + 6 q^{84} + 18 q^{86} + 12 q^{87} - 6 q^{88} - 12 q^{89} - 6 q^{90} - 6 q^{91} - 6 q^{92} + 6 q^{93} - 6 q^{94} - 6 q^{95} - 6 q^{96} - 18 q^{97} + 18 q^{98} - 6 q^{99}+O(q^{100}) \) 6 * q + 6 * q^2 - 6 * q^3 + 6 * q^4 - 6 * q^5 - 6 * q^6 - 6 * q^7 + 6 * q^8 + 6 * q^9 - 6 * q^10 - 6 * q^11 - 6 * q^12 + 6 * q^13 - 6 * q^14 + 6 * q^15 + 6 * q^16 + 6 * q^18 + 6 * q^19 - 6 * q^20 + 6 * q^21 - 6 * q^22 - 6 * q^23 - 6 * q^24 + 6 * q^25 + 6 * q^26 - 6 * q^27 - 6 * q^28 - 12 * q^29 + 6 * q^30 - 6 * q^31 + 6 * q^32 + 6 * q^33 + 6 * q^35 + 6 * q^36 - 6 * q^37 + 6 * q^38 - 6 * q^39 - 6 * q^40 - 12 * q^41 + 6 * q^42 + 18 * q^43 - 6 * q^44 - 6 * q^45 - 6 * q^46 - 6 * q^47 - 6 * q^48 + 18 * q^49 + 6 * q^50 + 6 * q^52 - 6 * q^53 - 6 * q^54 + 6 * q^55 - 6 * q^56 - 6 * q^57 - 12 * q^58 - 30 * q^59 + 6 * q^60 + 24 * q^61 - 6 * q^62 - 6 * q^63 + 6 * q^64 - 6 * q^65 + 6 * q^66 + 6 * q^69 + 6 * q^70 - 24 * q^71 + 6 * q^72 - 6 * q^73 - 6 * q^74 - 6 * q^75 + 6 * q^76 - 6 * q^78 - 6 * q^79 - 6 * q^80 + 6 * q^81 - 12 * q^82 + 18 * q^83 + 6 * q^84 + 18 * q^86 + 12 * q^87 - 6 * q^88 - 12 * q^89 - 6 * q^90 - 6 * q^91 - 6 * q^92 + 6 * q^93 - 6 * q^94 - 6 * q^95 - 6 * q^96 - 18 * q^97 + 18 * q^98 - 6 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more