LMFDB - Embedded newform 8670.2.a.cg.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:	6.6.45769536.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 15x^{4} + 72x^{2} - 109 $ x^6 - 15x^4 + 72x^2 - 109
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			$-2.55580$ 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)

$f(q)$	$=$	$q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{6} -4.91571 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{6} -4.91571 q^{7} +1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{10} -3.76700 q^{11} -1.00000 q^{12} +1.24753 q^{13} -4.91571 q^{14} -1.00000 q^{15} +1.00000 q^{16} +1.00000 q^{18} +5.93941 q^{19} +1.00000 q^{20} +4.91571 q^{21} -3.76700 q^{22} +0.0278577 q^{23} -1.00000 q^{24} +1.00000 q^{25} +1.24753 q^{26} -1.00000 q^{27} -4.91571 q^{28} +8.43439 q^{29} -1.00000 q^{30} -9.60736 q^{31} +1.00000 q^{32} +3.76700 q^{33} -4.91571 q^{35} +1.00000 q^{36} +7.21480 q^{37} +5.93941 q^{38} -1.24753 q^{39} +1.00000 q^{40} +5.63839 q^{41} +4.91571 q^{42} +1.61253 q^{43} -3.76700 q^{44} +1.00000 q^{45} +0.0278577 q^{46} -8.30755 q^{47} -1.00000 q^{48} +17.1642 q^{49} +1.00000 q^{50} +1.24753 q^{52} -1.77218 q^{53} -1.00000 q^{54} -3.76700 q^{55} -4.91571 q^{56} -5.93941 q^{57} +8.43439 q^{58} -3.05594 q^{59} -1.00000 q^{60} -13.8481 q^{61} -9.60736 q^{62} -4.91571 q^{63} +1.00000 q^{64} +1.24753 q^{65} +3.76700 q^{66} +8.38964 q^{67} -0.0278577 q^{69} -4.91571 q^{70} -3.51662 q^{71} +1.00000 q^{72} +8.51582 q^{73} +7.21480 q^{74} -1.00000 q^{75} +5.93941 q^{76} +18.5175 q^{77} -1.24753 q^{78} -9.43518 q^{79} +1.00000 q^{80} +1.00000 q^{81} +5.63839 q^{82} -16.4475 q^{83} +4.91571 q^{84} +1.61253 q^{86} -8.43439 q^{87} -3.76700 q^{88} +1.69173 q^{89} +1.00000 q^{90} -6.13248 q^{91} +0.0278577 q^{92} +9.60736 q^{93} -8.30755 q^{94} +5.93941 q^{95} -1.00000 q^{96} +2.10492 q^{97} +17.1642 q^{98} -3.76700 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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	\( 6 q + 6 q^{2} - 6 q^{3} + 6 q^{4} + 6 q^{5} - 6 q^{6} - 6 q^{7} + 6 q^{8} + 6 q^{9} + 6 q^{10} - 6 q^{11} - 6 q^{12} - 6 q^{13} - 6 q^{14} - 6 q^{15} + 6 q^{16} + 6 q^{18} + 6 q^{19} + 6 q^{20} + 6 q^{21} - 6 q^{22} - 6 q^{23} - 6 q^{24} + 6 q^{25} - 6 q^{26} - 6 q^{27} - 6 q^{28} - 12 q^{29} - 6 q^{30} - 30 q^{31} + 6 q^{32} + 6 q^{33} - 6 q^{35} + 6 q^{36} - 6 q^{37} + 6 q^{38} + 6 q^{39} + 6 q^{40} - 12 q^{41} + 6 q^{42} + 6 q^{43} - 6 q^{44} + 6 q^{45} - 6 q^{46} - 18 q^{47} - 6 q^{48} + 18 q^{49} + 6 q^{50} - 6 q^{52} - 18 q^{53} - 6 q^{54} - 6 q^{55} - 6 q^{56} - 6 q^{57} - 12 q^{58} + 6 q^{59} - 6 q^{60} - 24 q^{61} - 30 q^{62} - 6 q^{63} + 6 q^{64} - 6 q^{65} + 6 q^{66} + 6 q^{69} - 6 q^{70} - 24 q^{71} + 6 q^{72} + 18 q^{73} - 6 q^{74} - 6 q^{75} + 6 q^{76} - 12 q^{77} + 6 q^{78} - 30 q^{79} + 6 q^{80} + 6 q^{81} - 12 q^{82} - 18 q^{83} + 6 q^{84} + 6 q^{86} + 12 q^{87} - 6 q^{88} + 12 q^{89} + 6 q^{90} - 18 q^{91} - 6 q^{92} + 30 q^{93} - 18 q^{94} + 6 q^{95} - 6 q^{96} - 18 q^{97} + 18 q^{98} - 6 q^{99}+O(q^{100}) \) 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 - 30 * 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 + 6 * q^43 - 6 * q^44 + 6 * q^45 - 6 * q^46 - 18 * q^47 - 6 * q^48 + 18 * q^49 + 6 * q^50 - 6 * q^52 - 18 * q^53 - 6 * q^54 - 6 * q^55 - 6 * q^56 - 6 * q^57 - 12 * q^58 + 6 * q^59 - 6 * q^60 - 24 * q^61 - 30 * 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 + 18 * q^73 - 6 * q^74 - 6 * q^75 + 6 * q^76 - 12 * q^77 + 6 * q^78 - 30 * q^79 + 6 * q^80 + 6 * q^81 - 12 * q^82 - 18 * q^83 + 6 * q^84 + 6 * q^86 + 12 * q^87 - 6 * q^88 + 12 * q^89 + 6 * q^90 - 18 * q^91 - 6 * q^92 + 30 * q^93 - 18 * 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
$7$	−4.91571	−1.85796	−0.928981	−	0.370127i	$-0.879314\pi$
$7$	−4.91571	−1.85796	−0.928981	+	0.370127i	$0.879314\pi$
$8$	1.00000	0.353553
$9$	1.00000	0.333333
$10$	1.00000	0.316228
$11$	−3.76700	−1.13579	−0.567897	−	0.823100i	$-0.692243\pi$
$11$	−3.76700	−1.13579	−0.567897	+	0.823100i	$0.692243\pi$
$12$	−1.00000	−0.288675
$13$	1.24753	0.346002	0.173001	−	0.984922i	$-0.444654\pi$
$13$	1.24753	0.346002	0.173001	+	0.984922i	$0.444654\pi$
$14$	−4.91571	−1.31378
$15$	−1.00000	−0.258199
$16$	1.00000	0.250000
$17$	0	0
$17$	0	0
$18$	1.00000	0.235702
$19$	5.93941	1.36259	0.681297	−	0.732007i	$-0.261416\pi$
$19$	5.93941	1.36259	0.681297	+	0.732007i	$0.261416\pi$
$20$	1.00000	0.223607
$21$	4.91571	1.07269
$22$	−3.76700	−0.803128
$23$	0.0278577	0.00580872	0.00290436	−	0.999996i	$-0.499076\pi$
$23$	0.0278577	0.00580872	0.00290436	+	0.999996i	$0.499076\pi$
$24$	−1.00000	−0.204124
$25$	1.00000	0.200000
$26$	1.24753	0.244660
$27$	−1.00000	−0.192450
$28$	−4.91571	−0.928981
$29$	8.43439	1.56623	0.783113	−	0.621879i	$-0.213630\pi$
$29$	8.43439	1.56623	0.783113	+	0.621879i	$0.213630\pi$
$30$	−1.00000	−0.182574
$31$	−9.60736	−1.72553	−0.862766	−	0.505603i	$-0.831270\pi$
$31$	−9.60736	−1.72553	−0.862766	+	0.505603i	$0.831270\pi$
$32$	1.00000	0.176777
$33$	3.76700	0.655751
$34$	0	0
$35$	−4.91571	−0.830906
$36$	1.00000	0.166667
$37$	7.21480	1.18611	0.593053	−	0.805164i	$-0.297923\pi$
$37$	7.21480	1.18611	0.593053	+	0.805164i	$0.297923\pi$
$38$	5.93941	0.963500
$39$	−1.24753	−0.199764
$40$	1.00000	0.158114
$41$	5.63839	0.880568	0.440284	−	0.897859i	$-0.354878\pi$
$41$	5.63839	0.880568	0.440284	+	0.897859i	$0.354878\pi$
$42$	4.91571	0.758510
$43$	1.61253	0.245909	0.122954	−	0.992412i	$-0.460763\pi$
$43$	1.61253	0.245909	0.122954	+	0.992412i	$0.460763\pi$
$44$	−3.76700	−0.567897
$45$	1.00000	0.149071
$46$	0.0278577	0.00410739
$47$	−8.30755	−1.21178	−0.605891	−	0.795548i	$-0.707183\pi$
$47$	−8.30755	−1.21178	−0.605891	+	0.795548i	$0.707183\pi$
$48$	−1.00000	−0.144338
$49$	17.1642	2.45202
$50$	1.00000	0.141421
$51$	0	0
$52$	1.24753	0.173001
$53$	−1.77218	−0.243427	−0.121714	−	0.992565i	$-0.538839\pi$
$53$	−1.77218	−0.243427	−0.121714	+	0.992565i	$0.538839\pi$
$54$	−1.00000	−0.136083
$55$	−3.76700	−0.507943
$56$	−4.91571	−0.656889
$57$	−5.93941	−0.786694
$58$	8.43439	1.10749
$59$	−3.05594	−0.397850	−0.198925	−	0.980015i	$-0.563745\pi$
$59$	−3.05594	−0.397850	−0.198925	+	0.980015i	$0.563745\pi$
$60$	−1.00000	−0.129099
$61$	−13.8481	−1.77307	−0.886534	−	0.462664i	$-0.846894\pi$
$61$	−13.8481	−1.77307	−0.886534	+	0.462664i	$0.846894\pi$
$62$	−9.60736	−1.22014
$63$	−4.91571	−0.619321
$64$	1.00000	0.125000
$65$	1.24753	0.154737
$66$	3.76700	0.463686
$67$	8.38964	1.02496	0.512479	−	0.858700i	$-0.328727\pi$
$67$	8.38964	1.02496	0.512479	+	0.858700i	$0.328727\pi$
$68$	0	0
$69$	−0.0278577	−0.00335367
$70$	−4.91571	−0.587539
$71$	−3.51662	−0.417346	−0.208673	−	0.977986i	$-0.566914\pi$
$71$	−3.51662	−0.417346	−0.208673	+	0.977986i	$0.566914\pi$
$72$	1.00000	0.117851
$73$	8.51582	0.996702	0.498351	−	0.866975i	$-0.333939\pi$
$73$	8.51582	0.996702	0.498351	+	0.866975i	$0.333939\pi$
$74$	7.21480	0.838703
$75$	−1.00000	−0.115470
$76$	5.93941	0.681297
$77$	18.5175	2.11026
$78$	−1.24753	−0.141255
$79$	−9.43518	−1.06154	−0.530770	−	0.847516i	$-0.678097\pi$
$79$	−9.43518	−1.06154	−0.530770	+	0.847516i	$0.678097\pi$
$80$	1.00000	0.111803
$81$	1.00000	0.111111
$82$	5.63839	0.622655
$83$	−16.4475	−1.80535	−0.902673	−	0.430328i	$-0.858398\pi$
$83$	−16.4475	−1.80535	−0.902673	+	0.430328i	$0.858398\pi$
$84$	4.91571	0.536347
$85$	0	0
$86$	1.61253	0.173884
$87$	−8.43439	−0.904261
$88$	−3.76700	−0.401564
$89$	1.69173	0.179323	0.0896617	−	0.995972i	$-0.471421\pi$
$89$	1.69173	0.179323	0.0896617	+	0.995972i	$0.471421\pi$
$90$	1.00000	0.105409
$91$	−6.13248	−0.642859
$92$	0.0278577	0.00290436
$93$	9.60736	0.996237
$94$	−8.30755	−0.856859
$95$	5.93941	0.609371
$96$	−1.00000	−0.102062
$97$	2.10492	0.213723	0.106861	−	0.994274i	$-0.465920\pi$
$97$	2.10492	0.213723	0.106861	+	0.994274i	$0.465920\pi$
$98$	17.1642	1.73384
$99$	−3.76700	−0.378598
$100$	1.00000	0.100000
$101$	−5.81786	−0.578898	−0.289449	−	0.957193i	$-0.593472\pi$
$101$	−5.81786	−0.578898	−0.289449	+	0.957193i	$0.593472\pi$
$102$	0	0
$103$	−17.1710	−1.69191	−0.845953	−	0.533257i	$-0.820968\pi$
$103$	−17.1710	−1.69191	−0.845953	+	0.533257i	$0.820968\pi$
$104$	1.24753	0.122330
$105$	4.91571	0.479724
$106$	−1.77218	−0.172129
$107$	3.63525	0.351433	0.175716	−	0.984441i	$-0.443776\pi$
$107$	3.63525	0.351433	0.175716	+	0.984441i	$0.443776\pi$
$108$	−1.00000	−0.0962250
$109$	6.73721	0.645308	0.322654	−	0.946517i	$-0.395425\pi$
$109$	6.73721	0.645308	0.322654	+	0.946517i	$0.395425\pi$
$110$	−3.76700	−0.359170
$111$	−7.21480	−0.684798
$112$	−4.91571	−0.464490
$113$	1.64905	0.155130	0.0775649	−	0.996987i	$-0.475285\pi$
$113$	1.64905	0.155130	0.0775649	+	0.996987i	$0.475285\pi$
$114$	−5.93941	−0.556277
$115$	0.0278577	0.00259774
$116$	8.43439	0.783113
$117$	1.24753	0.115334
$118$	−3.05594	−0.281323
$119$	0	0
$120$	−1.00000	−0.0912871
$121$	3.19032	0.290029
$122$	−13.8481	−1.25375
$123$	−5.63839	−0.508396
$124$	−9.60736	−0.862766
$125$	1.00000	0.0894427
$126$	−4.91571	−0.437926
$127$	−18.2045	−1.61539	−0.807694	−	0.589601i	$-0.799285\pi$
$127$	−18.2045	−1.61539	−0.807694	+	0.589601i	$0.799285\pi$
$128$	1.00000	0.0883883
$129$	−1.61253	−0.141975
$130$	1.24753	0.109415
$131$	1.34611	0.117610	0.0588050	−	0.998269i	$-0.481271\pi$
$131$	1.34611	0.117610	0.0588050	+	0.998269i	$0.481271\pi$
$132$	3.76700	0.327876
$133$	−29.1964	−2.53165
$134$	8.38964	0.724755
$135$	−1.00000	−0.0860663
$136$	0	0
$137$	−13.9372	−1.19074	−0.595368	−	0.803453i	$-0.702994\pi$
$137$	−13.9372	−1.19074	−0.595368	+	0.803453i	$0.702994\pi$
$138$	−0.0278577	−0.00237140
$139$	22.4721	1.90606	0.953031	−	0.302873i	$-0.0979459\pi$
$139$	22.4721	1.90606	0.953031	+	0.302873i	$0.0979459\pi$
$140$	−4.91571	−0.415453
$141$	8.30755	0.699622
$142$	−3.51662	−0.295108
$143$	−4.69944	−0.392987
$144$	1.00000	0.0833333
$145$	8.43439	0.700438
$146$	8.51582	0.704775
$147$	−17.1642	−1.41568
$148$	7.21480	0.593053
$149$	−8.45265	−0.692468	−0.346234	−	0.938148i	$-0.612540\pi$
$149$	−8.45265	−0.692468	−0.346234	+	0.938148i	$0.612540\pi$
$150$	−1.00000	−0.0816497
$151$	−12.1022	−0.984861	−0.492430	−	0.870352i	$-0.663891\pi$
$151$	−12.1022	−0.984861	−0.492430	+	0.870352i	$0.663891\pi$
$152$	5.93941	0.481750
$153$	0	0
$154$	18.5175	1.49218
$155$	−9.60736	−0.771682
$156$	−1.24753	−0.0998822
$157$	−16.5686	−1.32232	−0.661158	−	0.750247i	$-0.729935\pi$
$157$	−16.5686	−1.32232	−0.661158	+	0.750247i	$0.729935\pi$
$158$	−9.43518	−0.750623
$159$	1.77218	0.140543
$160$	1.00000	0.0790569
$161$	−0.136940	−0.0107924
$162$	1.00000	0.0785674
$163$	−6.56966	−0.514575	−0.257288	−	0.966335i	$-0.582829\pi$
$163$	−6.56966	−0.514575	−0.257288	+	0.966335i	$0.582829\pi$
$164$	5.63839	0.440284
$165$	3.76700	0.293261
$166$	−16.4475	−1.27657
$167$	−14.6940	−1.13705	−0.568526	−	0.822665i	$-0.692486\pi$
$167$	−14.6940	−1.13705	−0.568526	+	0.822665i	$0.692486\pi$
$168$	4.91571	0.379255
$169$	−11.4437	−0.880283
$170$	0	0
$171$	5.93941	0.454198
$172$	1.61253	0.122954
$173$	−24.0405	−1.82777	−0.913884	−	0.405975i	$-0.866932\pi$
$173$	−24.0405	−1.82777	−0.913884	+	0.405975i	$0.866932\pi$
$174$	−8.43439	−0.639409
$175$	−4.91571	−0.371592
$176$	−3.76700	−0.283949
$177$	3.05594	0.229699
$178$	1.69173	0.126801
$179$	17.8062	1.33090	0.665449	−	0.746444i	$-0.268240\pi$
$179$	17.8062	1.33090	0.665449	+	0.746444i	$0.268240\pi$
$180$	1.00000	0.0745356
$181$	−14.7900	−1.09934	−0.549668	−	0.835383i	$-0.685246\pi$
$181$	−14.7900	−1.09934	−0.549668	+	0.835383i	$0.685246\pi$
$182$	−6.13248	−0.454570
$183$	13.8481	1.02368
$184$	0.0278577	0.00205369
$185$	7.21480	0.530442
$186$	9.60736	0.704446
$187$	0	0
$188$	−8.30755	−0.605891
$189$	4.91571	0.357565
$190$	5.93941	0.430890
$191$	6.42130	0.464629	0.232314	−	0.972641i	$-0.425370\pi$
$191$	6.42130	0.464629	0.232314	+	0.972641i	$0.425370\pi$
$192$	−1.00000	−0.0721688
$193$	5.16372	0.371692	0.185846	−	0.982579i	$-0.440497\pi$
$193$	5.16372	0.371692	0.185846	+	0.982579i	$0.440497\pi$
$194$	2.10492	0.151125
$195$	−1.24753	−0.0893374
$196$	17.1642	1.22601
$197$	1.09748	0.0781924	0.0390962	−	0.999235i	$-0.487552\pi$
$197$	1.09748	0.0781924	0.0390962	+	0.999235i	$0.487552\pi$
$198$	−3.76700	−0.267709
$199$	7.58501	0.537687	0.268844	−	0.963184i	$-0.413359\pi$
$199$	7.58501	0.537687	0.268844	+	0.963184i	$0.413359\pi$
$200$	1.00000	0.0707107
$201$	−8.38964	−0.591760
$202$	−5.81786	−0.409343
$203$	−41.4610	−2.90999
$204$	0	0
$205$	5.63839	0.393802
$206$	−17.1710	−1.19636
$207$	0.0278577	0.00193624
$208$	1.24753	0.0865005
$209$	−22.3738	−1.54763
$210$	4.91571	0.339216
$211$	0.740404	0.0509715	0.0254858	−	0.999675i	$-0.491887\pi$
$211$	0.740404	0.0509715	0.0254858	+	0.999675i	$0.491887\pi$
$212$	−1.77218	−0.121714
$213$	3.51662	0.240955
$214$	3.63525	0.248501
$215$	1.61253	0.109974
$216$	−1.00000	−0.0680414
$217$	47.2269	3.20597
$218$	6.73721	0.456302
$219$	−8.51582	−0.575446
$220$	−3.76700	−0.253971
$221$	0	0
$222$	−7.21480	−0.484226
$223$	−10.8974	−0.729742	−0.364871	−	0.931058i	$-0.618887\pi$
$223$	−10.8974	−0.729742	−0.364871	+	0.931058i	$0.618887\pi$
$224$	−4.91571	−0.328444
$225$	1.00000	0.0666667
$226$	1.64905	0.109693
$227$	20.0420	1.33023	0.665116	−	0.746740i	$-0.268382\pi$
$227$	20.0420	1.33023	0.665116	+	0.746740i	$0.268382\pi$
$228$	−5.93941	−0.393347
$229$	1.13944	0.0752963	0.0376481	−	0.999291i	$-0.488013\pi$
$229$	1.13944	0.0752963	0.0376481	+	0.999291i	$0.488013\pi$
$230$	0.0278577	0.00183688
$231$	−18.5175	−1.21836
$232$	8.43439	0.553745
$233$	−24.6249	−1.61323	−0.806615	−	0.591077i	$-0.798703\pi$
$233$	−24.6249	−1.61323	−0.806615	+	0.591077i	$0.798703\pi$
$234$	1.24753	0.0815535
$235$	−8.30755	−0.541925
$236$	−3.05594	−0.198925
$237$	9.43518	0.612881
$238$	0	0
$239$	5.39874	0.349216	0.174608	−	0.984638i	$-0.444134\pi$
$239$	5.39874	0.349216	0.174608	+	0.984638i	$0.444134\pi$
$240$	−1.00000	−0.0645497
$241$	−16.5400	−1.06543	−0.532717	−	0.846293i	$-0.678829\pi$
$241$	−16.5400	−1.06543	−0.532717	+	0.846293i	$0.678829\pi$
$242$	3.19032	0.205081
$243$	−1.00000	−0.0641500
$244$	−13.8481	−0.886534
$245$	17.1642	1.09658
$246$	−5.63839	−0.359490
$247$	7.40959	0.471461
$248$	−9.60736	−0.610068
$249$	16.4475	1.04232
$250$	1.00000	0.0632456
$251$	25.8885	1.63407	0.817034	−	0.576589i	$-0.195617\pi$
$251$	25.8885	1.63407	0.817034	+	0.576589i	$0.195617\pi$
$252$	−4.91571	−0.309660
$253$	−0.104940	−0.00659751
$254$	−18.2045	−1.14225
$255$	0	0
$256$	1.00000	0.0625000
$257$	−21.7890	−1.35916	−0.679580	−	0.733601i	$-0.737838\pi$
$257$	−21.7890	−1.35916	−0.679580	+	0.733601i	$0.737838\pi$
$258$	−1.61253	−0.100392
$259$	−35.4658	−2.20374
$260$	1.24753	0.0773684
$261$	8.43439	0.522076
$262$	1.34611	0.0831628
$263$	28.9528	1.78531	0.892653	−	0.450744i	$-0.148841\pi$
$263$	28.9528	1.78531	0.892653	+	0.450744i	$0.148841\pi$
$264$	3.76700	0.231843
$265$	−1.77218	−0.108864
$266$	−29.1964	−1.79015
$267$	−1.69173	−0.103532
$268$	8.38964	0.512479
$269$	−24.3155	−1.48254	−0.741270	−	0.671207i	$-0.765776\pi$
$269$	−24.3155	−1.48254	−0.741270	+	0.671207i	$0.765776\pi$
$270$	−1.00000	−0.0608581
$271$	0.266704	0.0162011	0.00810057	−	0.999967i	$-0.497421\pi$
$271$	0.266704	0.0162011	0.00810057	+	0.999967i	$0.497421\pi$
$272$	0	0
$273$	6.13248	0.371155
$274$	−13.9372	−0.841977
$275$	−3.76700	−0.227159
$276$	−0.0278577	−0.00167683
$277$	−9.42385	−0.566224	−0.283112	−	0.959087i	$-0.591367\pi$
$277$	−9.42385	−0.566224	−0.283112	+	0.959087i	$0.591367\pi$
$278$	22.4721	1.34779
$279$	−9.60736	−0.575178
$280$	−4.91571	−0.293770
$281$	−5.89389	−0.351600	−0.175800	−	0.984426i	$-0.556251\pi$
$281$	−5.89389	−0.351600	−0.175800	+	0.984426i	$0.556251\pi$
$282$	8.30755	0.494708
$283$	18.3030	1.08800	0.543999	−	0.839086i	$-0.316910\pi$
$283$	18.3030	1.08800	0.543999	+	0.839086i	$0.316910\pi$
$284$	−3.51662	−0.208673
$285$	−5.93941	−0.351820
$286$	−4.69944	−0.277884
$287$	−27.7166	−1.63606
$288$	1.00000	0.0589256
$289$	0	0
$290$	8.43439	0.495284
$291$	−2.10492	−0.123393
$292$	8.51582	0.498351
$293$	2.93502	0.171465	0.0857327	−	0.996318i	$-0.472677\pi$
$293$	2.93502	0.171465	0.0857327	+	0.996318i	$0.472677\pi$
$294$	−17.1642	−1.00103
$295$	−3.05594	−0.177924
$296$	7.21480	0.419352
$297$	3.76700	0.218584
$298$	−8.45265	−0.489649
$299$	0.0347532	0.00200983
$300$	−1.00000	−0.0577350
$301$	−7.92673	−0.456889
$302$	−12.1022	−0.696402
$303$	5.81786	0.334227
$304$	5.93941	0.340649
$305$	−13.8481	−0.792940
$306$	0	0
$307$	−3.41850	−0.195104	−0.0975521	−	0.995230i	$-0.531101\pi$
$307$	−3.41850	−0.195104	−0.0975521	+	0.995230i	$0.531101\pi$
$308$	18.5175	1.05513
$309$	17.1710	0.976823
$310$	−9.60736	−0.545661
$311$	−14.3969	−0.816374	−0.408187	−	0.912898i	$-0.633839\pi$
$311$	−14.3969	−0.816374	−0.408187	+	0.912898i	$0.633839\pi$
$312$	−1.24753	−0.0706274
$313$	−14.1099	−0.797540	−0.398770	−	0.917051i	$-0.630563\pi$
$313$	−14.1099	−0.797540	−0.398770	+	0.917051i	$0.630563\pi$
$314$	−16.5686	−0.935018
$315$	−4.91571	−0.276969
$316$	−9.43518	−0.530770
$317$	−10.2965	−0.578311	−0.289155	−	0.957282i	$-0.593374\pi$
$317$	−10.2965	−0.578311	−0.289155	+	0.957282i	$0.593374\pi$
$318$	1.77218	0.0993787
$319$	−31.7724	−1.77891
$320$	1.00000	0.0559017
$321$	−3.63525	−0.202900
$322$	−0.136940	−0.00763137
$323$	0	0
$324$	1.00000	0.0555556
$325$	1.24753	0.0692004
$326$	−6.56966	−0.363860
$327$	−6.73721	−0.372569
$328$	5.63839	0.311328
$329$	40.8375	2.25144
$330$	3.76700	0.207367
$331$	21.7559	1.19581	0.597906	−	0.801566i	$-0.296000\pi$
$331$	21.7559	1.19581	0.597906	+	0.801566i	$0.296000\pi$
$332$	−16.4475	−0.902673
$333$	7.21480	0.395368
$334$	−14.6940	−0.804017
$335$	8.38964	0.458375
$336$	4.91571	0.268174
$337$	−28.5839	−1.55706	−0.778532	−	0.627605i	$-0.784035\pi$
$337$	−28.5839	−1.55706	−0.778532	+	0.627605i	$0.784035\pi$
$338$	−11.4437	−0.622454
$339$	−1.64905	−0.0895643
$340$	0	0
$341$	36.1910	1.95985
$342$	5.93941	0.321167
$343$	−49.9640	−2.69780
$344$	1.61253	0.0869419
$345$	−0.0278577	−0.00149981
$346$	−24.0405	−1.29243
$347$	3.23440	0.173632	0.0868158	−	0.996224i	$-0.472331\pi$
$347$	3.23440	0.173632	0.0868158	+	0.996224i	$0.472331\pi$
$348$	−8.43439	−0.452131
$349$	22.5151	1.20520	0.602602	−	0.798042i	$-0.294130\pi$
$349$	22.5151	1.20520	0.602602	+	0.798042i	$0.294130\pi$
$350$	−4.91571	−0.262756
$351$	−1.24753	−0.0665881
$352$	−3.76700	−0.200782
$353$	−11.9822	−0.637747	−0.318873	−	0.947797i	$-0.603304\pi$
$353$	−11.9822	−0.637747	−0.318873	+	0.947797i	$0.603304\pi$
$354$	3.05594	0.162422
$355$	−3.51662	−0.186643
$356$	1.69173	0.0896617
$357$	0	0
$358$	17.8062	0.941086
$359$	−21.1732	−1.11748	−0.558739	−	0.829344i	$-0.688715\pi$
$359$	−21.1732	−1.11748	−0.558739	+	0.829344i	$0.688715\pi$
$360$	1.00000	0.0527046
$361$	16.2766	0.856664
$362$	−14.7900	−0.777347
$363$	−3.19032	−0.167448
$364$	−6.13248	−0.321429
$365$	8.51582	0.445739
$366$	13.8481	0.723852
$367$	8.50618	0.444019	0.222009	−	0.975045i	$-0.428738\pi$
$367$	8.50618	0.444019	0.222009	+	0.975045i	$0.428738\pi$
$368$	0.0278577	0.00145218
$369$	5.63839	0.293523
$370$	7.21480	0.375079
$371$	8.71149	0.452278
$372$	9.60736	0.498118
$373$	−16.4582	−0.852175	−0.426087	−	0.904682i	$-0.640108\pi$
$373$	−16.4582	−0.852175	−0.426087	+	0.904682i	$0.640108\pi$
$374$	0	0
$375$	−1.00000	−0.0516398
$376$	−8.30755	−0.428429
$377$	10.5221	0.541918
$378$	4.91571	0.252837
$379$	18.6157	0.956225	0.478112	−	0.878299i	$-0.341321\pi$
$379$	18.6157	0.956225	0.478112	+	0.878299i	$0.341321\pi$
$380$	5.93941	0.304685
$381$	18.2045	0.932645
$382$	6.42130	0.328542
$383$	−1.01256	−0.0517392	−0.0258696	−	0.999665i	$-0.508235\pi$
$383$	−1.01256	−0.0517392	−0.0258696	+	0.999665i	$0.508235\pi$
$384$	−1.00000	−0.0510310
$385$	18.5175	0.943738
$386$	5.16372	0.262826
$387$	1.61253	0.0819696
$388$	2.10492	0.106861
$389$	26.5768	1.34750	0.673748	−	0.738962i	$-0.264683\pi$
$389$	26.5768	1.34750	0.673748	+	0.738962i	$0.264683\pi$
$390$	−1.24753	−0.0631711
$391$	0	0
$392$	17.1642	0.866921
$393$	−1.34611	−0.0679022
$394$	1.09748	0.0552904
$395$	−9.43518	−0.474735
$396$	−3.76700	−0.189299
$397$	5.18223	0.260088	0.130044	−	0.991508i	$-0.458488\pi$
$397$	5.18223	0.260088	0.130044	+	0.991508i	$0.458488\pi$
$398$	7.58501	0.380202
$399$	29.1964	1.46165
$400$	1.00000	0.0500000
$401$	3.45321	0.172445	0.0862225	−	0.996276i	$-0.472520\pi$
$401$	3.45321	0.172445	0.0862225	+	0.996276i	$0.472520\pi$
$402$	−8.38964	−0.418437
$403$	−11.9855	−0.597038
$404$	−5.81786	−0.289449
$405$	1.00000	0.0496904
$406$	−41.4610	−2.05767
$407$	−27.1782	−1.34717
$408$	0	0
$409$	−17.7621	−0.878282	−0.439141	−	0.898418i	$-0.644717\pi$
$409$	−17.7621	−0.878282	−0.439141	+	0.898418i	$0.644717\pi$
$410$	5.63839	0.278460
$411$	13.9372	0.687472
$412$	−17.1710	−0.845953
$413$	15.0221	0.739190
$414$	0.0278577	0.00136913
$415$	−16.4475	−0.807375
$416$	1.24753	0.0611651
$417$	−22.4721	−1.10047
$418$	−22.3738	−1.09434
$419$	14.9169	0.728740	0.364370	−	0.931254i	$-0.381284\pi$
$419$	14.9169	0.728740	0.364370	+	0.931254i	$0.381284\pi$
$420$	4.91571	0.239862
$421$	−15.5528	−0.757996	−0.378998	−	0.925397i	$-0.623731\pi$
$421$	−15.5528	−0.757996	−0.378998	+	0.925397i	$0.623731\pi$
$422$	0.740404	0.0360423
$423$	−8.30755	−0.403927
$424$	−1.77218	−0.0860645
$425$	0	0
$426$	3.51662	0.170381
$427$	68.0732	3.29429
$428$	3.63525	0.175716
$429$	4.69944	0.226891
$430$	1.61253	0.0777632
$431$	−3.66010	−0.176301	−0.0881504	−	0.996107i	$-0.528096\pi$
$431$	−3.66010	−0.176301	−0.0881504	+	0.996107i	$0.528096\pi$
$432$	−1.00000	−0.0481125
$433$	12.0692	0.580008	0.290004	−	0.957025i	$-0.406343\pi$
$433$	12.0692	0.580008	0.290004	+	0.957025i	$0.406343\pi$
$434$	47.2269	2.26697
$435$	−8.43439	−0.404398
$436$	6.73721	0.322654
$437$	0.165458	0.00791493
$438$	−8.51582	−0.406902
$439$	−12.2762	−0.585910	−0.292955	−	0.956126i	$-0.594639\pi$
$439$	−12.2762	−0.585910	−0.292955	+	0.956126i	$0.594639\pi$
$440$	−3.76700	−0.179585
$441$	17.1642	0.817341
$442$	0	0
$443$	36.2172	1.72073	0.860367	−	0.509676i	$-0.170235\pi$
$443$	36.2172	1.72073	0.860367	+	0.509676i	$0.170235\pi$
$444$	−7.21480	−0.342399
$445$	1.69173	0.0801959
$446$	−10.8974	−0.516006
$447$	8.45265	0.399797
$448$	−4.91571	−0.232245
$449$	35.5862	1.67941	0.839707	−	0.543039i	$-0.182727\pi$
$449$	35.5862	1.67941	0.839707	+	0.543039i	$0.182727\pi$
$450$	1.00000	0.0471405
$451$	−21.2398	−1.00014
$452$	1.64905	0.0775649
$453$	12.1022	0.568610
$454$	20.0420	0.940616
$455$	−6.13248	−0.287495
$456$	−5.93941	−0.278138
$457$	−21.0772	−0.985948	−0.492974	−	0.870044i	$-0.664090\pi$
$457$	−21.0772	−0.985948	−0.492974	+	0.870044i	$0.664090\pi$
$458$	1.13944	0.0532425
$459$	0	0
$460$	0.0278577	0.00129887
$461$	17.0582	0.794480	0.397240	−	0.917715i	$-0.369968\pi$
$461$	17.0582	0.794480	0.397240	+	0.917715i	$0.369968\pi$
$462$	−18.5175	−0.861511
$463$	19.0757	0.886523	0.443262	−	0.896392i	$-0.353821\pi$
$463$	19.0757	0.886523	0.443262	+	0.896392i	$0.353821\pi$
$464$	8.43439	0.391557
$465$	9.60736	0.445531
$466$	−24.6249	−1.14073
$467$	16.4195	0.759803	0.379901	−	0.925027i	$-0.375958\pi$
$467$	16.4195	0.759803	0.379901	+	0.925027i	$0.375958\pi$
$468$	1.24753	0.0576670
$469$	−41.2410	−1.90433
$470$	−8.30755	−0.383199
$471$	16.5686	0.763439
$472$	−3.05594	−0.140661
$473$	−6.07441	−0.279302
$474$	9.43518	0.433372
$475$	5.93941	0.272519
$476$	0	0
$477$	−1.77218	−0.0811423
$478$	5.39874	0.246933
$479$	42.4013	1.93737	0.968683	−	0.248300i	$-0.0798719\pi$
$479$	42.4013	1.93737	0.968683	+	0.248300i	$0.0798719\pi$
$480$	−1.00000	−0.0456435
$481$	9.00067	0.410395
$482$	−16.5400	−0.753376
$483$	0.136940	0.00623099
$484$	3.19032	0.145014
$485$	2.10492	0.0955797
$486$	−1.00000	−0.0453609
$487$	−14.8705	−0.673848	−0.336924	−	0.941532i	$-0.609387\pi$
$487$	−14.8705	−0.673848	−0.336924	+	0.941532i	$0.609387\pi$
$488$	−13.8481	−0.626874
$489$	6.56966	0.297090
$490$	17.1642	0.775398
$491$	−22.9024	−1.03357	−0.516785	−	0.856115i	$-0.672871\pi$
$491$	−22.9024	−1.03357	−0.516785	+	0.856115i	$0.672871\pi$
$492$	−5.63839	−0.254198
$493$	0	0
$494$	7.40959	0.333373
$495$	−3.76700	−0.169314
$496$	−9.60736	−0.431383
$497$	17.2866	0.775412
$498$	16.4475	0.737029
$499$	−8.27745	−0.370550	−0.185275	−	0.982687i	$-0.559318\pi$
$499$	−8.27745	−0.370550	−0.185275	+	0.982687i	$0.559318\pi$
$500$	1.00000	0.0447214
$501$	14.6940	0.656477
$502$	25.8885	1.15546
$503$	−4.19149	−0.186889	−0.0934446	−	0.995624i	$-0.529788\pi$
$503$	−4.19149	−0.186889	−0.0934446	+	0.995624i	$0.529788\pi$
$504$	−4.91571	−0.218963
$505$	−5.81786	−0.258891
$506$	−0.104940	−0.00466515
$507$	11.4437	0.508231
$508$	−18.2045	−0.807694
$509$	32.8386	1.45554	0.727772	−	0.685819i	$-0.240556\pi$
$509$	32.8386	1.45554	0.727772	+	0.685819i	$0.240556\pi$
$510$	0	0
$511$	−41.8613	−1.85183
$512$	1.00000	0.0441942
$513$	−5.93941	−0.262231
$514$	−21.7890	−0.961072
$515$	−17.1710	−0.756644
$516$	−1.61253	−0.0709877
$517$	31.2946	1.37633
$518$	−35.4658	−1.55828
$519$	24.0405	1.05526
$520$	1.24753	0.0547077
$521$	−20.1397	−0.882335	−0.441167	−	0.897425i	$-0.645436\pi$
$521$	−20.1397	−0.882335	−0.441167	+	0.897425i	$0.645436\pi$
$522$	8.43439	0.369163
$523$	18.4845	0.808272	0.404136	−	0.914699i	$-0.367572\pi$
$523$	18.4845	0.808272	0.404136	+	0.914699i	$0.367572\pi$
$524$	1.34611	0.0588050
$525$	4.91571	0.214539
$526$	28.9528	1.26240
$527$	0	0
$528$	3.76700	0.163938
$529$	−22.9992	−0.999966
$530$	−1.77218	−0.0769784
$531$	−3.05594	−0.132617
$532$	−29.1964	−1.26582
$533$	7.03405	0.304678
$534$	−1.69173	−0.0732085
$535$	3.63525	0.157166
$536$	8.38964	0.362377
$537$	−17.8062	−0.768394
$538$	−24.3155	−1.04831
$539$	−64.6574	−2.78499
$540$	−1.00000	−0.0430331
$541$	3.28056	0.141042	0.0705212	−	0.997510i	$-0.477534\pi$
$541$	3.28056	0.141042	0.0705212	+	0.997510i	$0.477534\pi$
$542$	0.266704	0.0114559
$543$	14.7900	0.634702
$544$	0	0
$545$	6.73721	0.288590
$546$	6.13248	0.262446
$547$	11.6855	0.499638	0.249819	−	0.968293i	$-0.419629\pi$
$547$	11.6855	0.499638	0.249819	+	0.968293i	$0.419629\pi$
$548$	−13.9372	−0.595368
$549$	−13.8481	−0.591022
$550$	−3.76700	−0.160626
$551$	50.0953	2.13413
$552$	−0.0278577	−0.00118570
$553$	46.3806	1.97230
$554$	−9.42385	−0.400381
$555$	−7.21480	−0.306251
$556$	22.4721	0.953031
$557$	−21.0241	−0.890819	−0.445410	−	0.895327i	$-0.646942\pi$
$557$	−21.0241	−0.890819	−0.445410	+	0.895327i	$0.646942\pi$
$558$	−9.60736	−0.406712
$559$	2.01168	0.0850849
$560$	−4.91571	−0.207726
$561$	0	0
$562$	−5.89389	−0.248619
$563$	−16.1787	−0.681850	−0.340925	−	0.940090i	$-0.610740\pi$
$563$	−16.1787	−0.681850	−0.340925	+	0.940090i	$0.610740\pi$
$564$	8.30755	0.349811
$565$	1.64905	0.0693762
$566$	18.3030	0.769331
$567$	−4.91571	−0.206440
$568$	−3.51662	−0.147554
$569$	−24.4638	−1.02558	−0.512788	−	0.858515i	$-0.671387\pi$
$569$	−24.4638	−1.02558	−0.512788	+	0.858515i	$0.671387\pi$
$570$	−5.93941	−0.248775
$571$	−4.60723	−0.192807	−0.0964034	−	0.995342i	$-0.530734\pi$
$571$	−4.60723	−0.192807	−0.0964034	+	0.995342i	$0.530734\pi$
$572$	−4.69944	−0.196494
$573$	−6.42130	−0.268254
$574$	−27.7166	−1.15687
$575$	0.0278577	0.00116174
$576$	1.00000	0.0416667
$577$	13.0233	0.542166	0.271083	−	0.962556i	$-0.412618\pi$
$577$	13.0233	0.542166	0.271083	+	0.962556i	$0.412618\pi$
$578$	0	0
$579$	−5.16372	−0.214597
$580$	8.43439	0.350219
$581$	80.8510	3.35426
$582$	−2.10492	−0.0872519
$583$	6.67579	0.276483
$584$	8.51582	0.352387
$585$	1.24753	0.0515790
$586$	2.93502	0.121244
$587$	−43.6762	−1.80271	−0.901354	−	0.433082i	$-0.857426\pi$
$587$	−43.6762	−1.80271	−0.901354	+	0.433082i	$0.857426\pi$
$588$	−17.1642	−0.707838
$589$	−57.0621	−2.35120
$590$	−3.05594	−0.125811
$591$	−1.09748	−0.0451444
$592$	7.21480	0.296526
$593$	−25.9667	−1.06632	−0.533162	−	0.846013i	$-0.678996\pi$
$593$	−25.9667	−1.06632	−0.533162	+	0.846013i	$0.678996\pi$
$594$	3.76700	0.154562
$595$	0	0
$596$	−8.45265	−0.346234
$597$	−7.58501	−0.310434
$598$	0.0347532	0.00142116
$599$	−30.8344	−1.25986	−0.629929	−	0.776652i	$-0.716916\pi$
$599$	−30.8344	−1.25986	−0.629929	+	0.776652i	$0.716916\pi$
$600$	−1.00000	−0.0408248
$601$	−23.7535	−0.968925	−0.484462	−	0.874812i	$-0.660985\pi$
$601$	−23.7535	−0.968925	−0.484462	+	0.874812i	$0.660985\pi$
$602$	−7.92673	−0.323069
$603$	8.38964	0.341653
$604$	−12.1022	−0.492430
$605$	3.19032	0.129705
$606$	5.81786	0.236334
$607$	−25.0819	−1.01804	−0.509022	−	0.860754i	$-0.669993\pi$
$607$	−25.0819	−1.01804	−0.509022	+	0.860754i	$0.669993\pi$
$608$	5.93941	0.240875
$609$	41.4610	1.68008
$610$	−13.8481	−0.560693
$611$	−10.3639	−0.419279
$612$	0	0
$613$	−13.9630	−0.563962	−0.281981	−	0.959420i	$-0.590991\pi$
$613$	−13.9630	−0.563962	−0.281981	+	0.959420i	$0.590991\pi$
$614$	−3.41850	−0.137959
$615$	−5.63839	−0.227362
$616$	18.5175	0.746091
$617$	−10.8457	−0.436631	−0.218315	−	0.975878i	$-0.570056\pi$
$617$	−10.8457	−0.436631	−0.218315	+	0.975878i	$0.570056\pi$
$618$	17.1710	0.690718
$619$	−37.2414	−1.49686	−0.748430	−	0.663214i	$-0.769192\pi$
$619$	−37.2414	−1.49686	−0.748430	+	0.663214i	$0.769192\pi$
$620$	−9.60736	−0.385841
$621$	−0.0278577	−0.00111789
$622$	−14.3969	−0.577264
$623$	−8.31606	−0.333176
$624$	−1.24753	−0.0499411
$625$	1.00000	0.0400000
$626$	−14.1099	−0.563946
$627$	22.3738	0.893523
$628$	−16.5686	−0.661158
$629$	0	0
$630$	−4.91571	−0.195846
$631$	11.8360	0.471182	0.235591	−	0.971852i	$-0.424297\pi$
$631$	11.8360	0.471182	0.235591	+	0.971852i	$0.424297\pi$
$632$	−9.43518	−0.375311
$633$	−0.740404	−0.0294284
$634$	−10.2965	−0.408928
$635$	−18.2045	−0.722424
$636$	1.77218	0.0702713
$637$	21.4128	0.848405
$638$	−31.7724	−1.25788
$639$	−3.51662	−0.139115
$640$	1.00000	0.0395285
$641$	−24.5703	−0.970470	−0.485235	−	0.874384i	$-0.661266\pi$
$641$	−24.5703	−0.970470	−0.485235	+	0.874384i	$0.661266\pi$
$642$	−3.63525	−0.143472
$643$	24.8137	0.978555	0.489278	−	0.872128i	$-0.337260\pi$
$643$	24.8137	0.978555	0.489278	+	0.872128i	$0.337260\pi$
$644$	−0.136940	−0.00539619
$645$	−1.61253	−0.0634934
$646$	0	0
$647$	−49.1614	−1.93273	−0.966366	−	0.257169i	$-0.917210\pi$
$647$	−49.1614	−1.93273	−0.966366	+	0.257169i	$0.917210\pi$
$648$	1.00000	0.0392837
$649$	11.5118	0.451876
$650$	1.24753	0.0489321
$651$	−47.2269	−1.85097
$652$	−6.56966	−0.257288
$653$	−38.9285	−1.52339	−0.761695	−	0.647936i	$-0.775632\pi$
$653$	−38.9285	−1.52339	−0.761695	+	0.647936i	$0.775632\pi$
$654$	−6.73721	−0.263446
$655$	1.34611	0.0525968
$656$	5.63839	0.220142
$657$	8.51582	0.332234
$658$	40.8375	1.59201
$659$	13.4917	0.525561	0.262781	−	0.964856i	$-0.415360\pi$
$659$	13.4917	0.525561	0.262781	+	0.964856i	$0.415360\pi$
$660$	3.76700	0.146630
$661$	38.4848	1.49689	0.748443	−	0.663199i	$-0.230802\pi$
$661$	38.4848	1.49689	0.748443	+	0.663199i	$0.230802\pi$
$662$	21.7559	0.845567
$663$	0	0
$664$	−16.4475	−0.638286
$665$	−29.1964	−1.13219
$666$	7.21480	0.279568
$667$	0.234962	0.00909778
$668$	−14.6940	−0.568526
$669$	10.8974	0.421317
$670$	8.38964	0.324120
$671$	52.1658	2.01384
$672$	4.91571	0.189627
$673$	30.4147	1.17240	0.586201	−	0.810166i	$-0.300623\pi$
$673$	30.4147	1.17240	0.586201	+	0.810166i	$0.300623\pi$
$674$	−28.5839	−1.10101
$675$	−1.00000	−0.0384900
$676$	−11.4437	−0.440141
$677$	−31.2700	−1.20180	−0.600901	−	0.799323i	$-0.705191\pi$
$677$	−31.2700	−1.20180	−0.600901	+	0.799323i	$0.705191\pi$
$678$	−1.64905	−0.0633315
$679$	−10.3472	−0.397089
$680$	0	0
$681$	−20.0420	−0.768010
$682$	36.1910	1.38582
$683$	17.6835	0.676640	0.338320	−	0.941031i	$-0.390141\pi$
$683$	17.6835	0.676640	0.338320	+	0.941031i	$0.390141\pi$
$684$	5.93941	0.227099
$685$	−13.9372	−0.532513
$686$	−49.9640	−1.90763
$687$	−1.13944	−0.0434723
$688$	1.61253	0.0614772
$689$	−2.21084	−0.0842263
$690$	−0.0278577	−0.00106052
$691$	28.1548	1.07106	0.535529	−	0.844517i	$-0.320112\pi$
$691$	28.1548	1.07106	0.535529	+	0.844517i	$0.320112\pi$
$692$	−24.0405	−0.913884
$693$	18.5175	0.703421
$694$	3.23440	0.122776
$695$	22.4721	0.852417
$696$	−8.43439	−0.319705
$697$	0	0
$698$	22.5151	0.852209
$699$	24.6249	0.931399
$700$	−4.91571	−0.185796
$701$	4.53409	0.171250	0.0856250	−	0.996327i	$-0.472711\pi$
$701$	4.53409	0.171250	0.0856250	+	0.996327i	$0.472711\pi$
$702$	−1.24753	−0.0470849
$703$	42.8517	1.61618
$704$	−3.76700	−0.141974
$705$	8.30755	0.312881
$706$	−11.9822	−0.450955
$707$	28.5989	1.07557
$708$	3.05594	0.114849
$709$	9.59293	0.360270	0.180135	−	0.983642i	$-0.442347\pi$
$709$	9.59293	0.360270	0.180135	+	0.983642i	$0.442347\pi$
$710$	−3.51662	−0.131976
$711$	−9.43518	−0.353847
$712$	1.69173	0.0634004
$713$	−0.267639	−0.0100231
$714$	0	0
$715$	−4.69944	−0.175749
$716$	17.8062	0.665449
$717$	−5.39874	−0.201620
$718$	−21.1732	−0.790176
$719$	24.7840	0.924286	0.462143	−	0.886805i	$-0.347081\pi$
$719$	24.7840	0.924286	0.462143	+	0.886805i	$0.347081\pi$
$720$	1.00000	0.0372678
$721$	84.4075	3.14350
$722$	16.2766	0.605753
$723$	16.5400	0.615129
$724$	−14.7900	−0.549668
$725$	8.43439	0.313245
$726$	−3.19032	−0.118404
$727$	27.5496	1.02176	0.510879	−	0.859653i	$-0.329320\pi$
$727$	27.5496	1.02176	0.510879	+	0.859653i	$0.329320\pi$
$728$	−6.13248	−0.227285
$729$	1.00000	0.0370370
$730$	8.51582	0.315185
$731$	0	0
$732$	13.8481	0.511840
$733$	−10.9034	−0.402727	−0.201364	−	0.979517i	$-0.564537\pi$
$733$	−10.9034	−0.402727	−0.201364	+	0.979517i	$0.564537\pi$
$734$	8.50618	0.313969
$735$	−17.1642	−0.633110
$736$	0.0278577	0.00102685
$737$	−31.6038	−1.16414
$738$	5.63839	0.207552
$739$	1.16462	0.0428414	0.0214207	−	0.999771i	$-0.493181\pi$
$739$	1.16462	0.0428414	0.0214207	+	0.999771i	$0.493181\pi$
$740$	7.21480	0.265221
$741$	−7.40959	−0.272198
$742$	8.71149	0.319809
$743$	11.4977	0.421811	0.210905	−	0.977506i	$-0.432359\pi$
$743$	11.4977	0.421811	0.210905	+	0.977506i	$0.432359\pi$
$744$	9.60736	0.352223
$745$	−8.45265	−0.309681
$746$	−16.4582	−0.602579
$747$	−16.4475	−0.601782
$748$	0	0
$749$	−17.8698	−0.652949
$750$	−1.00000	−0.0365148
$751$	23.4085	0.854189	0.427095	−	0.904207i	$-0.359537\pi$
$751$	23.4085	0.854189	0.427095	+	0.904207i	$0.359537\pi$
$752$	−8.30755	−0.302945
$753$	−25.8885	−0.943430
$754$	10.5221	0.383194
$755$	−12.1022	−0.440443
$756$	4.91571	0.178782
$757$	35.4981	1.29020	0.645099	−	0.764099i	$-0.276816\pi$
$757$	35.4981	1.29020	0.645099	+	0.764099i	$0.276816\pi$
$758$	18.6157	0.676153
$759$	0.104940	0.00380908
$760$	5.93941	0.215445
$761$	−43.8776	−1.59056	−0.795281	−	0.606241i	$-0.792677\pi$
$761$	−43.8776	−1.59056	−0.795281	+	0.606241i	$0.792677\pi$
$762$	18.2045	0.659480
$763$	−33.1181	−1.19896
$764$	6.42130	0.232314
$765$	0	0
$766$	−1.01256	−0.0365851
$767$	−3.81238	−0.137657
$768$	−1.00000	−0.0360844
$769$	−28.7693	−1.03745	−0.518723	−	0.854942i	$-0.673593\pi$
$769$	−28.7693	−1.03745	−0.518723	+	0.854942i	$0.673593\pi$
$770$	18.5175	0.667324
$771$	21.7890	0.784712
$772$	5.16372	0.185846
$773$	7.75536	0.278941	0.139471	−	0.990226i	$-0.455460\pi$
$773$	7.75536	0.278941	0.139471	+	0.990226i	$0.455460\pi$
$774$	1.61253	0.0579612
$775$	−9.60736	−0.345107
$776$	2.10492	0.0755624
$777$	35.4658	1.27233
$778$	26.5768	0.952823
$779$	33.4887	1.19986
$780$	−1.24753	−0.0446687
$781$	13.2471	0.474019
$782$	0	0
$783$	−8.43439	−0.301420
$784$	17.1642	0.613006
$785$	−16.5686	−0.591358
$786$	−1.34611	−0.0480141
$787$	−18.1841	−0.648195	−0.324097	−	0.946024i	$-0.605061\pi$
$787$	−18.1841	−0.648195	−0.324097	+	0.946024i	$0.605061\pi$
$788$	1.09748	0.0390962
$789$	−28.9528	−1.03075
$790$	−9.43518	−0.335689
$791$	−8.10626	−0.288225
$792$	−3.76700	−0.133855
$793$	−17.2759	−0.613485
$794$	5.18223	0.183910
$795$	1.77218	0.0628526
$796$	7.58501	0.268844
$797$	30.2919	1.07299	0.536497	−	0.843902i	$-0.319747\pi$
$797$	30.2919	1.07299	0.536497	+	0.843902i	$0.319747\pi$
$798$	29.1964	1.03354
$799$	0	0
$800$	1.00000	0.0353553
$801$	1.69173	0.0597745
$802$	3.45321	0.121937
$803$	−32.0791	−1.13205
$804$	−8.38964	−0.295880
$805$	−0.136940	−0.00482650
$806$	−11.9855	−0.422170
$807$	24.3155	0.855945
$808$	−5.81786	−0.204671
$809$	20.5577	0.722770	0.361385	−	0.932417i	$-0.382304\pi$
$809$	20.5577	0.722770	0.361385	+	0.932417i	$0.382304\pi$
$810$	1.00000	0.0351364
$811$	41.5246	1.45813	0.729063	−	0.684447i	$-0.239956\pi$
$811$	41.5246	1.45813	0.729063	+	0.684447i	$0.239956\pi$
$812$	−41.4610	−1.45499
$813$	−0.266704	−0.00935373
$814$	−27.1782	−0.952594
$815$	−6.56966	−0.230125
$816$	0	0
$817$	9.57749	0.335074
$818$	−17.7621	−0.621039
$819$	−6.13248	−0.214286
$820$	5.63839	0.196901
$821$	31.6216	1.10360	0.551800	−	0.833976i	$-0.313941\pi$
$821$	31.6216	1.10360	0.551800	+	0.833976i	$0.313941\pi$
$822$	13.9372	0.486116
$823$	13.3567	0.465586	0.232793	−	0.972526i	$-0.425214\pi$
$823$	13.3567	0.465586	0.232793	+	0.972526i	$0.425214\pi$
$824$	−17.1710	−0.598179
$825$	3.76700	0.131150
$826$	15.0221	0.522687
$827$	−45.0725	−1.56732	−0.783662	−	0.621187i	$-0.786651\pi$
$827$	−45.0725	−1.56732	−0.783662	+	0.621187i	$0.786651\pi$
$828$	0.0278577	0.000968121 0
$829$	−12.0668	−0.419097	−0.209549	−	0.977798i	$-0.567199\pi$
$829$	−12.0668	−0.419097	−0.209549	+	0.977798i	$0.567199\pi$
$830$	−16.4475	−0.570900
$831$	9.42385	0.326910
$832$	1.24753	0.0432503
$833$	0	0
$834$	−22.4721	−0.778146
$835$	−14.6940	−0.508505
$836$	−22.3738	−0.773814
$837$	9.60736	0.332079
$838$	14.9169	0.515297
$839$	−53.8764	−1.86002	−0.930011	−	0.367533i	$-0.880203\pi$
$839$	−53.8764	−1.86002	−0.930011	+	0.367533i	$0.880203\pi$
$840$	4.91571	0.169608
$841$	42.1389	1.45307
$842$	−15.5528	−0.535984
$843$	5.89389	0.202996
$844$	0.740404	0.0254858
$845$	−11.4437	−0.393674
$846$	−8.30755	−0.285620
$847$	−15.6827	−0.538862
$848$	−1.77218	−0.0608568
$849$	−18.3030	−0.628156
$850$	0	0
$851$	0.200987	0.00688976
$852$	3.51662	0.120477
$853$	−46.1484	−1.58009	−0.790045	−	0.613049i	$-0.789943\pi$
$853$	−46.1484	−1.58009	−0.790045	+	0.613049i	$0.789943\pi$
$854$	68.0732	2.32942
$855$	5.93941	0.203124
$856$	3.63525	0.124250
$857$	11.3849	0.388902	0.194451	−	0.980912i	$-0.437707\pi$
$857$	11.3849	0.388902	0.194451	+	0.980912i	$0.437707\pi$
$858$	4.69944	0.160436
$859$	−47.6055	−1.62428	−0.812139	−	0.583464i	$-0.801697\pi$
$859$	−47.6055	−1.62428	−0.812139	+	0.583464i	$0.801697\pi$
$860$	1.61253	0.0549869
$861$	27.7166	0.944581
$862$	−3.66010	−0.124663
$863$	−42.6254	−1.45099	−0.725493	−	0.688230i	$-0.758388\pi$
$863$	−42.6254	−1.45099	−0.725493	+	0.688230i	$0.758388\pi$
$864$	−1.00000	−0.0340207
$865$	−24.0405	−0.817403
$866$	12.0692	0.410128
$867$	0	0
$868$	47.2269	1.60299
$869$	35.5424	1.20569
$870$	−8.43439	−0.285953
$871$	10.4663	0.354638
$872$	6.73721	0.228151
$873$	2.10492	0.0712409
$874$	0.165458	0.00559670
$875$	−4.91571	−0.166181
$876$	−8.51582	−0.287723
$877$	11.0992	0.374794	0.187397	−	0.982284i	$-0.439995\pi$
$877$	11.0992	0.374794	0.187397	+	0.982284i	$0.439995\pi$
$878$	−12.2762	−0.414301
$879$	−2.93502	−0.0989956
$880$	−3.76700	−0.126986
$881$	−18.3086	−0.616834	−0.308417	−	0.951251i	$-0.599799\pi$
$881$	−18.3086	−0.616834	−0.308417	+	0.951251i	$0.599799\pi$
$882$	17.1642	0.577947
$883$	−3.45425	−0.116245	−0.0581224	−	0.998309i	$-0.518511\pi$
$883$	−3.45425	−0.116245	−0.0581224	+	0.998309i	$0.518511\pi$
$884$	0	0
$885$	3.05594	0.102724
$886$	36.2172	1.21674
$887$	−50.5462	−1.69717	−0.848587	−	0.529056i	$-0.822546\pi$
$887$	−50.5462	−1.69717	−0.848587	+	0.529056i	$0.822546\pi$
$888$	−7.21480	−0.242113
$889$	89.4880	3.00133
$890$	1.69173	0.0567070
$891$	−3.76700	−0.126199
$892$	−10.8974	−0.364871
$893$	−49.3420	−1.65117
$894$	8.45265	0.282699
$895$	17.8062	0.595195
$896$	−4.91571	−0.164222
$897$	−0.0347532	−0.00116038
$898$	35.5862	1.18753
$899$	−81.0322	−2.70258
$900$	1.00000	0.0333333
$901$	0	0
$902$	−21.2398	−0.707209
$903$	7.92673	0.263785
$904$	1.64905	0.0548467
$905$	−14.7900	−0.491638
$906$	12.1022	0.402068
$907$	−38.1032	−1.26520	−0.632598	−	0.774480i	$-0.718012\pi$
$907$	−38.1032	−1.26520	−0.632598	+	0.774480i	$0.718012\pi$
$908$	20.0420	0.665116
$909$	−5.81786	−0.192966
$910$	−6.13248	−0.203290
$911$	−29.7489	−0.985625	−0.492812	−	0.870136i	$-0.664031\pi$
$911$	−29.7489	−0.985625	−0.492812	+	0.870136i	$0.664031\pi$
$912$	−5.93941	−0.196674
$913$	61.9577	2.05050
$914$	−21.0772	−0.697170
$915$	13.8481	0.457804
$916$	1.13944	0.0376481
$917$	−6.61707	−0.218515
$918$	0	0
$919$	−18.7875	−0.619742	−0.309871	−	0.950779i	$-0.600286\pi$
$919$	−18.7875	−0.619742	−0.309871	+	0.950779i	$0.600286\pi$
$920$	0.0278577	0.000918440 0
$921$	3.41850	0.112643
$922$	17.0582	0.561782
$923$	−4.38708	−0.144402
$924$	−18.5175	−0.609180
$925$	7.21480	0.237221
$926$	19.0757	0.626867
$927$	−17.1710	−0.563969
$928$	8.43439	0.276872
$929$	−1.43063	−0.0469374	−0.0234687	−	0.999725i	$-0.507471\pi$
$929$	−1.43063	−0.0469374	−0.0234687	+	0.999725i	$0.507471\pi$
$930$	9.60736	0.315038
$931$	101.945	3.34111
$932$	−24.6249	−0.806615
$933$	14.3969	0.471334
$934$	16.4195	0.537262
$935$	0	0
$936$	1.24753	0.0407767
$937$	24.9927	0.816476	0.408238	−	0.912875i	$-0.366143\pi$
$937$	24.9927	0.816476	0.408238	+	0.912875i	$0.366143\pi$
$938$	−41.2410	−1.34657
$939$	14.1099	0.460460
$940$	−8.30755	−0.270963
$941$	48.7025	1.58766	0.793828	−	0.608142i	$-0.208085\pi$
$941$	48.7025	1.58766	0.793828	+	0.608142i	$0.208085\pi$
$942$	16.5686	0.539833
$943$	0.157072	0.00511497
$944$	−3.05594	−0.0994625
$945$	4.91571	0.159908
$946$	−6.07441	−0.197496
$947$	−52.3701	−1.70180	−0.850900	−	0.525328i	$-0.823943\pi$
$947$	−52.3701	−1.70180	−0.850900	+	0.525328i	$0.823943\pi$
$948$	9.43518	0.306440
$949$	10.6237	0.344861
$950$	5.93941	0.192700
$951$	10.2965	0.333888
$952$	0	0
$953$	49.6993	1.60992	0.804960	−	0.593329i	$-0.202187\pi$
$953$	49.6993	1.60992	0.804960	+	0.593329i	$0.202187\pi$
$954$	−1.77218	−0.0573763
$955$	6.42130	0.207788
$956$	5.39874	0.174608
$957$	31.7724	1.02705
$958$	42.4013	1.36992
$959$	68.5112	2.21234
$960$	−1.00000	−0.0322749
$961$	61.3013	1.97746
$962$	9.00067	0.290193
$963$	3.63525	0.117144
$964$	−16.5400	−0.532717
$965$	5.16372	0.166226
$966$	0.136940	0.00440597
$967$	34.7385	1.11712	0.558558	−	0.829466i	$-0.311355\pi$
$967$	34.7385	1.11712	0.558558	+	0.829466i	$0.311355\pi$
$968$	3.19032	0.102541
$969$	0	0
$970$	2.10492	0.0675850
$971$	12.9745	0.416372	0.208186	−	0.978089i	$-0.433244\pi$
$971$	12.9745	0.416372	0.208186	+	0.978089i	$0.433244\pi$
$972$	−1.00000	−0.0320750
$973$	−110.466	−3.54139
$974$	−14.8705	−0.476482
$975$	−1.24753	−0.0399529
$976$	−13.8481	−0.443267
$977$	−9.92899	−0.317656	−0.158828	−	0.987306i	$-0.550772\pi$
$977$	−9.92899	−0.317656	−0.158828	+	0.987306i	$0.550772\pi$
$978$	6.56966	0.210074
$979$	−6.37277	−0.203674
$980$	17.1642	0.548289
$981$	6.73721	0.215103
$982$	−22.9024	−0.730844
$983$	−22.9318	−0.731412	−0.365706	−	0.930730i	$-0.619172\pi$
$983$	−22.9318	−0.731412	−0.365706	+	0.930730i	$0.619172\pi$
$984$	−5.63839	−0.179745
$985$	1.09748	0.0349687
$986$	0	0
$987$	−40.8375	−1.29987
$988$	7.40959	0.235730
$989$	0.0449213	0.00142842
$990$	−3.76700	−0.119723
$991$	54.2442	1.72312	0.861562	−	0.507652i	$-0.169487\pi$
$991$	54.2442	1.72312	0.861562	+	0.507652i	$0.169487\pi$
$992$	−9.60736	−0.305034
$993$	−21.7559	−0.690402
$994$	17.2866	0.548299
$995$	7.58501	0.240461
$996$	16.4475	0.521158
$997$	46.3593	1.46821	0.734106	−	0.679034i	$-0.237601\pi$
$997$	46.3593	1.46821	0.734106	+	0.679034i	$0.237601\pi$
$998$	−8.27745	−0.262018
$999$	−7.21480	−0.228266

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8670.2.a.cg.1.1	✓	6	1.1	even	1	trivial
8670.2.a.ch.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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{6} -4.91571 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{6} -4.91571 q^{7} +1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{10} -3.76700 q^{11} -1.00000 q^{12} +1.24753 q^{13} -4.91571 q^{14} -1.00000 q^{15} +1.00000 q^{16} +1.00000 q^{18} +5.93941 q^{19} +1.00000 q^{20} +4.91571 q^{21} -3.76700 q^{22} +0.0278577 q^{23} -1.00000 q^{24} +1.00000 q^{25} +1.24753 q^{26} -1.00000 q^{27} -4.91571 q^{28} +8.43439 q^{29} -1.00000 q^{30} -9.60736 q^{31} +1.00000 q^{32} +3.76700 q^{33} -4.91571 q^{35} +1.00000 q^{36} +7.21480 q^{37} +5.93941 q^{38} -1.24753 q^{39} +1.00000 q^{40} +5.63839 q^{41} +4.91571 q^{42} +1.61253 q^{43} -3.76700 q^{44} +1.00000 q^{45} +0.0278577 q^{46} -8.30755 q^{47} -1.00000 q^{48} +17.1642 q^{49} +1.00000 q^{50} +1.24753 q^{52} -1.77218 q^{53} -1.00000 q^{54} -3.76700 q^{55} -4.91571 q^{56} -5.93941 q^{57} +8.43439 q^{58} -3.05594 q^{59} -1.00000 q^{60} -13.8481 q^{61} -9.60736 q^{62} -4.91571 q^{63} +1.00000 q^{64} +1.24753 q^{65} +3.76700 q^{66} +8.38964 q^{67} -0.0278577 q^{69} -4.91571 q^{70} -3.51662 q^{71} +1.00000 q^{72} +8.51582 q^{73} +7.21480 q^{74} -1.00000 q^{75} +5.93941 q^{76} +18.5175 q^{77} -1.24753 q^{78} -9.43518 q^{79} +1.00000 q^{80} +1.00000 q^{81} +5.63839 q^{82} -16.4475 q^{83} +4.91571 q^{84} +1.61253 q^{86} -8.43439 q^{87} -3.76700 q^{88} +1.69173 q^{89} +1.00000 q^{90} -6.13248 q^{91} +0.0278577 q^{92} +9.60736 q^{93} -8.30755 q^{94} +5.93941 q^{95} -1.00000 q^{96} +2.10492 q^{97} +17.1642 q^{98} -3.76700 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 6 q^{2} - 6 q^{3} + 6 q^{4} + 6 q^{5} - 6 q^{6} - 6 q^{7} + 6 q^{8} + 6 q^{9}+O(q^{10}) \) 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	\( 6 q + 6 q^{2} - 6 q^{3} + 6 q^{4} + 6 q^{5} - 6 q^{6} - 6 q^{7} + 6 q^{8} + 6 q^{9} + 6 q^{10} - 6 q^{11} - 6 q^{12} - 6 q^{13} - 6 q^{14} - 6 q^{15} + 6 q^{16} + 6 q^{18} + 6 q^{19} + 6 q^{20} + 6 q^{21} - 6 q^{22} - 6 q^{23} - 6 q^{24} + 6 q^{25} - 6 q^{26} - 6 q^{27} - 6 q^{28} - 12 q^{29} - 6 q^{30} - 30 q^{31} + 6 q^{32} + 6 q^{33} - 6 q^{35} + 6 q^{36} - 6 q^{37} + 6 q^{38} + 6 q^{39} + 6 q^{40} - 12 q^{41} + 6 q^{42} + 6 q^{43} - 6 q^{44} + 6 q^{45} - 6 q^{46} - 18 q^{47} - 6 q^{48} + 18 q^{49} + 6 q^{50} - 6 q^{52} - 18 q^{53} - 6 q^{54} - 6 q^{55} - 6 q^{56} - 6 q^{57} - 12 q^{58} + 6 q^{59} - 6 q^{60} - 24 q^{61} - 30 q^{62} - 6 q^{63} + 6 q^{64} - 6 q^{65} + 6 q^{66} + 6 q^{69} - 6 q^{70} - 24 q^{71} + 6 q^{72} + 18 q^{73} - 6 q^{74} - 6 q^{75} + 6 q^{76} - 12 q^{77} + 6 q^{78} - 30 q^{79} + 6 q^{80} + 6 q^{81} - 12 q^{82} - 18 q^{83} + 6 q^{84} + 6 q^{86} + 12 q^{87} - 6 q^{88} + 12 q^{89} + 6 q^{90} - 18 q^{91} - 6 q^{92} + 30 q^{93} - 18 q^{94} + 6 q^{95} - 6 q^{96} - 18 q^{97} + 18 q^{98} - 6 q^{99}+O(q^{100}) \) 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 - 30 * 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 + 6 * q^43 - 6 * q^44 + 6 * q^45 - 6 * q^46 - 18 * q^47 - 6 * q^48 + 18 * q^49 + 6 * q^50 - 6 * q^52 - 18 * q^53 - 6 * q^54 - 6 * q^55 - 6 * q^56 - 6 * q^57 - 12 * q^58 + 6 * q^59 - 6 * q^60 - 24 * q^61 - 30 * 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 + 18 * q^73 - 6 * q^74 - 6 * q^75 + 6 * q^76 - 12 * q^77 + 6 * q^78 - 30 * q^79 + 6 * q^80 + 6 * q^81 - 12 * q^82 - 18 * q^83 + 6 * q^84 + 6 * q^86 + 12 * q^87 - 6 * q^88 + 12 * q^89 + 6 * q^90 - 18 * q^91 - 6 * q^92 + 30 * q^93 - 18 * 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

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