LMFDB - Embedded newform 7098.2.a.br.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7098,2,Mod(1,7098)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7098, 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("7098.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7098 = 2 \cdot 3 \cdot 7 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7098.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:	$56.6778153547$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{129}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 32 $ x^2 - x - 32
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 546)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-5.17891$ of defining polynomial
Character	$\chi$	$=$	7098.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -2.00000 q^{5} -1.00000 q^{6} -1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -2.00000 q^{5} -1.00000 q^{6} -1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} -2.00000 q^{10} +6.17891 q^{11} -1.00000 q^{12} -1.00000 q^{14} +2.00000 q^{15} +1.00000 q^{16} +5.00000 q^{17} +1.00000 q^{18} -8.17891 q^{19} -2.00000 q^{20} +1.00000 q^{21} +6.17891 q^{22} +3.17891 q^{23} -1.00000 q^{24} -1.00000 q^{25} -1.00000 q^{27} -1.00000 q^{28} +8.17891 q^{29} +2.00000 q^{30} -7.17891 q^{31} +1.00000 q^{32} -6.17891 q^{33} +5.00000 q^{34} +2.00000 q^{35} +1.00000 q^{36} +2.00000 q^{37} -8.17891 q^{38} -2.00000 q^{40} +4.17891 q^{41} +1.00000 q^{42} +0.821092 q^{43} +6.17891 q^{44} -2.00000 q^{45} +3.17891 q^{46} +4.17891 q^{47} -1.00000 q^{48} +1.00000 q^{49} -1.00000 q^{50} -5.00000 q^{51} -3.00000 q^{53} -1.00000 q^{54} -12.3578 q^{55} -1.00000 q^{56} +8.17891 q^{57} +8.17891 q^{58} -0.821092 q^{59} +2.00000 q^{60} -3.00000 q^{61} -7.17891 q^{62} -1.00000 q^{63} +1.00000 q^{64} -6.17891 q^{66} -15.1789 q^{67} +5.00000 q^{68} -3.17891 q^{69} +2.00000 q^{70} +9.17891 q^{71} +1.00000 q^{72} +4.00000 q^{73} +2.00000 q^{74} +1.00000 q^{75} -8.17891 q^{76} -6.17891 q^{77} +1.82109 q^{79} -2.00000 q^{80} +1.00000 q^{81} +4.17891 q^{82} -5.17891 q^{83} +1.00000 q^{84} -10.0000 q^{85} +0.821092 q^{86} -8.17891 q^{87} +6.17891 q^{88} +9.00000 q^{89} -2.00000 q^{90} +3.17891 q^{92} +7.17891 q^{93} +4.17891 q^{94} +16.3578 q^{95} -1.00000 q^{96} +12.3578 q^{97} +1.00000 q^{98} +6.17891 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{2} - 2 q^{3} + 2 q^{4} - 4 q^{5} - 2 q^{6} - 2 q^{7} + 2 q^{8} + 2 q^{9}+O(q^{10}) $ 2 * q + 2 * q^2 - 2 * q^3 + 2 * q^4 - 4 * q^5 - 2 * q^6 - 2 * q^7 + 2 * q^8 + 2 * q^9	$ 2 q + 2 q^{2} - 2 q^{3} + 2 q^{4} - 4 q^{5} - 2 q^{6} - 2 q^{7} + 2 q^{8} + 2 q^{9} - 4 q^{10} + q^{11} - 2 q^{12} - 2 q^{14} + 4 q^{15} + 2 q^{16} + 10 q^{17} + 2 q^{18} - 5 q^{19} - 4 q^{20} + 2 q^{21} + q^{22} - 5 q^{23} - 2 q^{24} - 2 q^{25} - 2 q^{27} - 2 q^{28} + 5 q^{29} + 4 q^{30} - 3 q^{31} + 2 q^{32} - q^{33} + 10 q^{34} + 4 q^{35} + 2 q^{36} + 4 q^{37} - 5 q^{38} - 4 q^{40} - 3 q^{41} + 2 q^{42} + 13 q^{43} + q^{44} - 4 q^{45} - 5 q^{46} - 3 q^{47} - 2 q^{48} + 2 q^{49} - 2 q^{50} - 10 q^{51} - 6 q^{53} - 2 q^{54} - 2 q^{55} - 2 q^{56} + 5 q^{57} + 5 q^{58} - 13 q^{59} + 4 q^{60} - 6 q^{61} - 3 q^{62} - 2 q^{63} + 2 q^{64} - q^{66} - 19 q^{67} + 10 q^{68} + 5 q^{69} + 4 q^{70} + 7 q^{71} + 2 q^{72} + 8 q^{73} + 4 q^{74} + 2 q^{75} - 5 q^{76} - q^{77} + 15 q^{79} - 4 q^{80} + 2 q^{81} - 3 q^{82} + q^{83} + 2 q^{84} - 20 q^{85} + 13 q^{86} - 5 q^{87} + q^{88} + 18 q^{89} - 4 q^{90} - 5 q^{92} + 3 q^{93} - 3 q^{94} + 10 q^{95} - 2 q^{96} + 2 q^{97} + 2 q^{98} + q^{99}+O(q^{100}) $ 2 * q + 2 * q^2 - 2 * q^3 + 2 * q^4 - 4 * q^5 - 2 * q^6 - 2 * q^7 + 2 * q^8 + 2 * q^9 - 4 * q^10 + q^11 - 2 * q^12 - 2 * q^14 + 4 * q^15 + 2 * q^16 + 10 * q^17 + 2 * q^18 - 5 * q^19 - 4 * q^20 + 2 * q^21 + q^22 - 5 * q^23 - 2 * q^24 - 2 * q^25 - 2 * q^27 - 2 * q^28 + 5 * q^29 + 4 * q^30 - 3 * q^31 + 2 * q^32 - q^33 + 10 * q^34 + 4 * q^35 + 2 * q^36 + 4 * q^37 - 5 * q^38 - 4 * q^40 - 3 * q^41 + 2 * q^42 + 13 * q^43 + q^44 - 4 * q^45 - 5 * q^46 - 3 * q^47 - 2 * q^48 + 2 * q^49 - 2 * q^50 - 10 * q^51 - 6 * q^53 - 2 * q^54 - 2 * q^55 - 2 * q^56 + 5 * q^57 + 5 * q^58 - 13 * q^59 + 4 * q^60 - 6 * q^61 - 3 * q^62 - 2 * q^63 + 2 * q^64 - q^66 - 19 * q^67 + 10 * q^68 + 5 * q^69 + 4 * q^70 + 7 * q^71 + 2 * q^72 + 8 * q^73 + 4 * q^74 + 2 * q^75 - 5 * q^76 - q^77 + 15 * q^79 - 4 * q^80 + 2 * q^81 - 3 * q^82 + q^83 + 2 * q^84 - 20 * q^85 + 13 * q^86 - 5 * q^87 + q^88 + 18 * q^89 - 4 * q^90 - 5 * q^92 + 3 * q^93 - 3 * q^94 + 10 * q^95 - 2 * q^96 + 2 * q^97 + 2 * q^98 + 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
$5$	−2.00000	−0.894427	−0.447214	−	0.894427i	$-0.647584\pi$
$5$	−2.00000	−0.894427	−0.447214	+	0.894427i	$0.647584\pi$
$6$	−1.00000	−0.408248
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	1.00000	0.353553
$9$	1.00000	0.333333
$10$	−2.00000	−0.632456
$11$	6.17891	1.86301	0.931505	−	0.363727i	$-0.118496\pi$
$11$	6.17891	1.86301	0.931505	+	0.363727i	$0.118496\pi$
$12$	−1.00000	−0.288675
$13$	0	0
$13$	0	0
$14$	−1.00000	−0.267261
$15$	2.00000	0.516398
$16$	1.00000	0.250000
$17$	5.00000	1.21268	0.606339	−	0.795206i	$-0.292637\pi$
$17$	5.00000	1.21268	0.606339	+	0.795206i	$0.292637\pi$
$18$	1.00000	0.235702
$19$	−8.17891	−1.87637	−0.938185	−	0.346134i	$-0.887494\pi$
$19$	−8.17891	−1.87637	−0.938185	+	0.346134i	$0.887494\pi$
$20$	−2.00000	−0.447214
$21$	1.00000	0.218218
$22$	6.17891	1.31735
$23$	3.17891	0.662848	0.331424	−	0.943482i	$-0.392471\pi$
$23$	3.17891	0.662848	0.331424	+	0.943482i	$0.392471\pi$
$24$	−1.00000	−0.204124
$25$	−1.00000	−0.200000
$26$	0	0
$27$	−1.00000	−0.192450
$28$	−1.00000	−0.188982
$29$	8.17891	1.51879	0.759393	−	0.650633i	$-0.225496\pi$
$29$	8.17891	1.51879	0.759393	+	0.650633i	$0.225496\pi$
$30$	2.00000	0.365148
$31$	−7.17891	−1.28937	−0.644685	−	0.764448i	$-0.723011\pi$
$31$	−7.17891	−1.28937	−0.644685	+	0.764448i	$0.723011\pi$
$32$	1.00000	0.176777
$33$	−6.17891	−1.07561
$34$	5.00000	0.857493
$35$	2.00000	0.338062
$36$	1.00000	0.166667
$37$	2.00000	0.328798	0.164399	−	0.986394i	$-0.447432\pi$
$37$	2.00000	0.328798	0.164399	+	0.986394i	$0.447432\pi$
$38$	−8.17891	−1.32679
$39$	0	0
$40$	−2.00000	−0.316228
$41$	4.17891	0.652636	0.326318	−	0.945260i	$-0.394192\pi$
$41$	4.17891	0.652636	0.326318	+	0.945260i	$0.394192\pi$
$42$	1.00000	0.154303
$43$	0.821092	0.125215	0.0626077	−	0.998038i	$-0.480058\pi$
$43$	0.821092	0.125215	0.0626077	+	0.998038i	$0.480058\pi$
$44$	6.17891	0.931505
$45$	−2.00000	−0.298142
$46$	3.17891	0.468704
$47$	4.17891	0.609556	0.304778	−	0.952423i	$-0.401418\pi$
$47$	4.17891	0.609556	0.304778	+	0.952423i	$0.401418\pi$
$48$	−1.00000	−0.144338
$49$	1.00000	0.142857
$50$	−1.00000	−0.141421
$51$	−5.00000	−0.700140
$52$	0	0
$53$	−3.00000	−0.412082	−0.206041	−	0.978543i	$-0.566058\pi$
$53$	−3.00000	−0.412082	−0.206041	+	0.978543i	$0.566058\pi$
$54$	−1.00000	−0.136083
$55$	−12.3578	−1.66633
$56$	−1.00000	−0.133631
$57$	8.17891	1.08332
$58$	8.17891	1.07394
$59$	−0.821092	−0.106897	−0.0534485	−	0.998571i	$-0.517021\pi$
$59$	−0.821092	−0.106897	−0.0534485	+	0.998571i	$0.517021\pi$
$60$	2.00000	0.258199
$61$	−3.00000	−0.384111	−0.192055	−	0.981384i	$-0.561515\pi$
$61$	−3.00000	−0.384111	−0.192055	+	0.981384i	$0.561515\pi$
$62$	−7.17891	−0.911722
$63$	−1.00000	−0.125988
$64$	1.00000	0.125000
$65$	0	0
$66$	−6.17891	−0.760571
$67$	−15.1789	−1.85440	−0.927199	−	0.374568i	$-0.877791\pi$
$67$	−15.1789	−1.85440	−0.927199	+	0.374568i	$0.877791\pi$
$68$	5.00000	0.606339
$69$	−3.17891	−0.382696
$70$	2.00000	0.239046
$71$	9.17891	1.08934	0.544668	−	0.838652i	$-0.316656\pi$
$71$	9.17891	1.08934	0.544668	+	0.838652i	$0.316656\pi$
$72$	1.00000	0.117851
$73$	4.00000	0.468165	0.234082	−	0.972217i	$-0.424791\pi$
$73$	4.00000	0.468165	0.234082	+	0.972217i	$0.424791\pi$
$74$	2.00000	0.232495
$75$	1.00000	0.115470
$76$	−8.17891	−0.938185
$77$	−6.17891	−0.704152
$78$	0	0
$79$	1.82109	0.204889	0.102444	−	0.994739i	$-0.467334\pi$
$79$	1.82109	0.204889	0.102444	+	0.994739i	$0.467334\pi$
$80$	−2.00000	−0.223607
$81$	1.00000	0.111111
$82$	4.17891	0.461483
$83$	−5.17891	−0.568459	−0.284230	−	0.958756i	$-0.591738\pi$
$83$	−5.17891	−0.568459	−0.284230	+	0.958756i	$0.591738\pi$
$84$	1.00000	0.109109
$85$	−10.0000	−1.08465
$86$	0.821092	0.0885406
$87$	−8.17891	−0.876871
$88$	6.17891	0.658674
$89$	9.00000	0.953998	0.476999	−	0.878904i	$-0.341725\pi$
$89$	9.00000	0.953998	0.476999	+	0.878904i	$0.341725\pi$
$90$	−2.00000	−0.210819
$91$	0	0
$92$	3.17891	0.331424
$93$	7.17891	0.744418
$94$	4.17891	0.431021
$95$	16.3578	1.67828
$96$	−1.00000	−0.102062
$97$	12.3578	1.25475	0.627373	−	0.778719i	$-0.284130\pi$
$97$	12.3578	1.25475	0.627373	+	0.778719i	$0.284130\pi$
$98$	1.00000	0.101015
$99$	6.17891	0.621004
$100$	−1.00000	−0.100000
$101$	−16.3578	−1.62766	−0.813832	−	0.581101i	$-0.802622\pi$
$101$	−16.3578	−1.62766	−0.813832	+	0.581101i	$0.802622\pi$
$102$	−5.00000	−0.495074
$103$	3.17891	0.313227	0.156614	−	0.987660i	$-0.449942\pi$
$103$	3.17891	0.313227	0.156614	+	0.987660i	$0.449942\pi$
$104$	0	0
$105$	−2.00000	−0.195180
$106$	−3.00000	−0.291386
$107$	7.82109	0.756093	0.378047	−	0.925787i	$-0.376596\pi$
$107$	7.82109	0.756093	0.378047	+	0.925787i	$0.376596\pi$
$108$	−1.00000	−0.0962250
$109$	4.00000	0.383131	0.191565	−	0.981480i	$-0.438644\pi$
$109$	4.00000	0.383131	0.191565	+	0.981480i	$0.438644\pi$
$110$	−12.3578	−1.17827
$111$	−2.00000	−0.189832
$112$	−1.00000	−0.0944911
$113$	−4.35782	−0.409949	−0.204974	−	0.978767i	$-0.565711\pi$
$113$	−4.35782	−0.409949	−0.204974	+	0.978767i	$0.565711\pi$
$114$	8.17891	0.766025
$115$	−6.35782	−0.592869
$116$	8.17891	0.759393
$117$	0	0
$118$	−0.821092	−0.0755876
$119$	−5.00000	−0.458349
$120$	2.00000	0.182574
$121$	27.1789	2.47081
$122$	−3.00000	−0.271607
$123$	−4.17891	−0.376799
$124$	−7.17891	−0.644685
$125$	12.0000	1.07331
$126$	−1.00000	−0.0890871
$127$	12.0000	1.06483	0.532414	−	0.846484i	$-0.321285\pi$
$127$	12.0000	1.06483	0.532414	+	0.846484i	$0.321285\pi$
$128$	1.00000	0.0883883
$129$	−0.821092	−0.0722931
$130$	0	0
$131$	19.5367	1.70693	0.853466	−	0.521149i	$-0.174496\pi$
$131$	19.5367	1.70693	0.853466	+	0.521149i	$0.174496\pi$
$132$	−6.17891	−0.537805
$133$	8.17891	0.709201
$134$	−15.1789	−1.31126
$135$	2.00000	0.172133
$136$	5.00000	0.428746
$137$	−16.3578	−1.39754	−0.698771	−	0.715345i	$-0.746269\pi$
$137$	−16.3578	−1.39754	−0.698771	+	0.715345i	$0.746269\pi$
$138$	−3.17891	−0.270607
$139$	−18.1789	−1.54191	−0.770957	−	0.636887i	$-0.780222\pi$
$139$	−18.1789	−1.54191	−0.770957	+	0.636887i	$0.780222\pi$
$140$	2.00000	0.169031
$141$	−4.17891	−0.351928
$142$	9.17891	0.770277
$143$	0	0
$144$	1.00000	0.0833333
$145$	−16.3578	−1.35844
$146$	4.00000	0.331042
$147$	−1.00000	−0.0824786
$148$	2.00000	0.164399
$149$	4.82109	0.394959	0.197480	−	0.980307i	$-0.436724\pi$
$149$	4.82109	0.394959	0.197480	+	0.980307i	$0.436724\pi$
$150$	1.00000	0.0816497
$151$	−14.5367	−1.18298	−0.591491	−	0.806312i	$-0.701460\pi$
$151$	−14.5367	−1.18298	−0.591491	+	0.806312i	$0.701460\pi$
$152$	−8.17891	−0.663397
$153$	5.00000	0.404226
$154$	−6.17891	−0.497911
$155$	14.3578	1.15325
$156$	0	0
$157$	2.00000	0.159617	0.0798087	−	0.996810i	$-0.474569\pi$
$157$	2.00000	0.159617	0.0798087	+	0.996810i	$0.474569\pi$
$158$	1.82109	0.144878
$159$	3.00000	0.237915
$160$	−2.00000	−0.158114
$161$	−3.17891	−0.250533
$162$	1.00000	0.0785674
$163$	−8.82109	−0.690921	−0.345461	−	0.938433i	$-0.612277\pi$
$163$	−8.82109	−0.690921	−0.345461	+	0.938433i	$0.612277\pi$
$164$	4.17891	0.326318
$165$	12.3578	0.962055
$166$	−5.17891	−0.401961
$167$	−8.00000	−0.619059	−0.309529	−	0.950890i	$-0.600171\pi$
$167$	−8.00000	−0.619059	−0.309529	+	0.950890i	$0.600171\pi$
$168$	1.00000	0.0771517
$169$	0	0
$170$	−10.0000	−0.766965
$171$	−8.17891	−0.625457
$172$	0.821092	0.0626077
$173$	4.00000	0.304114	0.152057	−	0.988372i	$-0.451410\pi$
$173$	4.00000	0.304114	0.152057	+	0.988372i	$0.451410\pi$
$174$	−8.17891	−0.620041
$175$	1.00000	0.0755929
$176$	6.17891	0.465753
$177$	0.821092	0.0617170
$178$	9.00000	0.674579
$179$	−14.3578	−1.07315	−0.536577	−	0.843851i	$-0.680283\pi$
$179$	−14.3578	−1.07315	−0.536577	+	0.843851i	$0.680283\pi$
$180$	−2.00000	−0.149071
$181$	7.82109	0.581337	0.290669	−	0.956824i	$-0.406122\pi$
$181$	7.82109	0.581337	0.290669	+	0.956824i	$0.406122\pi$
$182$	0	0
$183$	3.00000	0.221766
$184$	3.17891	0.234352
$185$	−4.00000	−0.294086
$186$	7.17891	0.526383
$187$	30.8945	2.25923
$188$	4.17891	0.304778
$189$	1.00000	0.0727393
$190$	16.3578	1.18672
$191$	27.5367	1.99249	0.996244	−	0.0865933i	$-0.0275981\pi$
$191$	27.5367	1.99249	0.996244	+	0.0865933i	$0.0275981\pi$
$192$	−1.00000	−0.0721688
$193$	20.5367	1.47827	0.739133	−	0.673560i	$-0.235236\pi$
$193$	20.5367	1.47827	0.739133	+	0.673560i	$0.235236\pi$
$194$	12.3578	0.887240
$195$	0	0
$196$	1.00000	0.0714286
$197$	17.0000	1.21120	0.605600	−	0.795769i	$-0.292933\pi$
$197$	17.0000	1.21120	0.605600	+	0.795769i	$0.292933\pi$
$198$	6.17891	0.439116
$199$	12.8211	0.908863	0.454432	−	0.890782i	$-0.349842\pi$
$199$	12.8211	0.908863	0.454432	+	0.890782i	$0.349842\pi$
$200$	−1.00000	−0.0707107
$201$	15.1789	1.07064
$202$	−16.3578	−1.15093
$203$	−8.17891	−0.574047
$204$	−5.00000	−0.350070
$205$	−8.35782	−0.583735
$206$	3.17891	0.221485
$207$	3.17891	0.220949
$208$	0	0
$209$	−50.5367	−3.49570
$210$	−2.00000	−0.138013
$211$	16.3578	1.12612	0.563059	−	0.826417i	$-0.309624\pi$
$211$	16.3578	1.12612	0.563059	+	0.826417i	$0.309624\pi$
$212$	−3.00000	−0.206041
$213$	−9.17891	−0.628928
$214$	7.82109	0.534639
$215$	−1.64218	−0.111996
$216$	−1.00000	−0.0680414
$217$	7.17891	0.487336
$218$	4.00000	0.270914
$219$	−4.00000	−0.270295
$220$	−12.3578	−0.833164
$221$	0	0
$222$	−2.00000	−0.134231
$223$	9.17891	0.614665	0.307333	−	0.951602i	$-0.400564\pi$
$223$	9.17891	0.614665	0.307333	+	0.951602i	$0.400564\pi$
$224$	−1.00000	−0.0668153
$225$	−1.00000	−0.0666667
$226$	−4.35782	−0.289878
$227$	8.35782	0.554728	0.277364	−	0.960765i	$-0.410539\pi$
$227$	8.35782	0.554728	0.277364	+	0.960765i	$0.410539\pi$
$228$	8.17891	0.541661
$229$	1.00000	0.0660819	0.0330409	−	0.999454i	$-0.489481\pi$
$229$	1.00000	0.0660819	0.0330409	+	0.999454i	$0.489481\pi$
$230$	−6.35782	−0.419222
$231$	6.17891	0.406542
$232$	8.17891	0.536972
$233$	−0.357817	−0.0234414	−0.0117207	−	0.999931i	$-0.503731\pi$
$233$	−0.357817	−0.0234414	−0.0117207	+	0.999931i	$0.503731\pi$
$234$	0	0
$235$	−8.35782	−0.545204
$236$	−0.821092	−0.0534485
$237$	−1.82109	−0.118293
$238$	−5.00000	−0.324102
$239$	−1.17891	−0.0762572	−0.0381286	−	0.999273i	$-0.512140\pi$
$239$	−1.17891	−0.0762572	−0.0381286	+	0.999273i	$0.512140\pi$
$240$	2.00000	0.129099
$241$	−18.3578	−1.18253	−0.591265	−	0.806477i	$-0.701371\pi$
$241$	−18.3578	−1.18253	−0.591265	+	0.806477i	$0.701371\pi$
$242$	27.1789	1.74713
$243$	−1.00000	−0.0641500
$244$	−3.00000	−0.192055
$245$	−2.00000	−0.127775
$246$	−4.17891	−0.266437
$247$	0	0
$248$	−7.17891	−0.455861
$249$	5.17891	0.328200
$250$	12.0000	0.758947
$251$	10.8211	0.683021	0.341511	−	0.939878i	$-0.389061\pi$
$251$	10.8211	0.683021	0.341511	+	0.939878i	$0.389061\pi$
$252$	−1.00000	−0.0629941
$253$	19.6422	1.23489
$254$	12.0000	0.752947
$255$	10.0000	0.626224
$256$	1.00000	0.0625000
$257$	25.7156	1.60410	0.802049	−	0.597259i	$-0.203743\pi$
$257$	25.7156	1.60410	0.802049	+	0.597259i	$0.203743\pi$
$258$	−0.821092	−0.0511189
$259$	−2.00000	−0.124274
$260$	0	0
$261$	8.17891	0.506262
$262$	19.5367	1.20698
$263$	−12.0000	−0.739952	−0.369976	−	0.929041i	$-0.620634\pi$
$263$	−12.0000	−0.739952	−0.369976	+	0.929041i	$0.620634\pi$
$264$	−6.17891	−0.380286
$265$	6.00000	0.368577
$266$	8.17891	0.501481
$267$	−9.00000	−0.550791
$268$	−15.1789	−0.927199
$269$	18.3578	1.11930	0.559648	−	0.828730i	$-0.310936\pi$
$269$	18.3578	1.11930	0.559648	+	0.828730i	$0.310936\pi$
$270$	2.00000	0.121716
$271$	31.1789	1.89398	0.946992	−	0.321257i	$-0.104105\pi$
$271$	31.1789	1.89398	0.946992	+	0.321257i	$0.104105\pi$
$272$	5.00000	0.303170
$273$	0	0
$274$	−16.3578	−0.988212
$275$	−6.17891	−0.372602
$276$	−3.17891	−0.191348
$277$	28.3578	1.70386	0.851928	−	0.523659i	$-0.175433\pi$
$277$	28.3578	1.70386	0.851928	+	0.523659i	$0.175433\pi$
$278$	−18.1789	−1.09030
$279$	−7.17891	−0.429790
$280$	2.00000	0.119523
$281$	5.64218	0.336584	0.168292	−	0.985737i	$-0.446175\pi$
$281$	5.64218	0.336584	0.168292	+	0.985737i	$0.446175\pi$
$282$	−4.17891	−0.248850
$283$	30.3578	1.80458	0.902292	−	0.431125i	$-0.141883\pi$
$283$	30.3578	1.80458	0.902292	+	0.431125i	$0.141883\pi$
$284$	9.17891	0.544668
$285$	−16.3578	−0.968953
$286$	0	0
$287$	−4.17891	−0.246673
$288$	1.00000	0.0589256
$289$	8.00000	0.470588
$290$	−16.3578	−0.960564
$291$	−12.3578	−0.724428
$292$	4.00000	0.234082
$293$	12.0000	0.701047	0.350524	−	0.936554i	$-0.386004\pi$
$293$	12.0000	0.701047	0.350524	+	0.936554i	$0.386004\pi$
$294$	−1.00000	−0.0583212
$295$	1.64218	0.0956116
$296$	2.00000	0.116248
$297$	−6.17891	−0.358537
$298$	4.82109	0.279278
$299$	0	0
$300$	1.00000	0.0577350
$301$	−0.821092	−0.0473269
$302$	−14.5367	−0.836495
$303$	16.3578	0.939732
$304$	−8.17891	−0.469093
$305$	6.00000	0.343559
$306$	5.00000	0.285831
$307$	24.8945	1.42081	0.710403	−	0.703795i	$-0.248513\pi$
$307$	24.8945	1.42081	0.710403	+	0.703795i	$0.248513\pi$
$308$	−6.17891	−0.352076
$309$	−3.17891	−0.180842
$310$	14.3578	0.815469
$311$	−8.53673	−0.484073	−0.242037	−	0.970267i	$-0.577815\pi$
$311$	−8.53673	−0.484073	−0.242037	+	0.970267i	$0.577815\pi$
$312$	0	0
$313$	4.00000	0.226093	0.113047	−	0.993590i	$-0.463939\pi$
$313$	4.00000	0.226093	0.113047	+	0.993590i	$0.463939\pi$
$314$	2.00000	0.112867
$315$	2.00000	0.112687
$316$	1.82109	0.102444
$317$	7.17891	0.403208	0.201604	−	0.979467i	$-0.435385\pi$
$317$	7.17891	0.403208	0.201604	+	0.979467i	$0.435385\pi$
$318$	3.00000	0.168232
$319$	50.5367	2.82951
$320$	−2.00000	−0.111803
$321$	−7.82109	−0.436531
$322$	−3.17891	−0.177154
$323$	−40.8945	−2.27543
$324$	1.00000	0.0555556
$325$	0	0
$326$	−8.82109	−0.488555
$327$	−4.00000	−0.221201
$328$	4.17891	0.230742
$329$	−4.17891	−0.230391
$330$	12.3578	0.680275
$331$	−24.3578	−1.33883	−0.669413	−	0.742890i	$-0.733454\pi$
$331$	−24.3578	−1.33883	−0.669413	+	0.742890i	$0.733454\pi$
$332$	−5.17891	−0.284230
$333$	2.00000	0.109599
$334$	−8.00000	−0.437741
$335$	30.3578	1.65862
$336$	1.00000	0.0545545
$337$	−0.536725	−0.0292373	−0.0146186	−	0.999893i	$-0.504653\pi$
$337$	−0.536725	−0.0292373	−0.0146186	+	0.999893i	$0.504653\pi$
$338$	0	0
$339$	4.35782	0.236684
$340$	−10.0000	−0.542326
$341$	−44.3578	−2.40211
$342$	−8.17891	−0.442265
$343$	−1.00000	−0.0539949
$344$	0.821092	0.0442703
$345$	6.35782	0.342293
$346$	4.00000	0.215041
$347$	−7.82109	−0.419858	−0.209929	−	0.977717i	$-0.567323\pi$
$347$	−7.82109	−0.419858	−0.209929	+	0.977717i	$0.567323\pi$
$348$	−8.17891	−0.438436
$349$	−3.17891	−0.170163	−0.0850815	−	0.996374i	$-0.527115\pi$
$349$	−3.17891	−0.170163	−0.0850815	+	0.996374i	$0.527115\pi$
$350$	1.00000	0.0534522
$351$	0	0
$352$	6.17891	0.329337
$353$	−7.17891	−0.382095	−0.191047	−	0.981581i	$-0.561188\pi$
$353$	−7.17891	−0.382095	−0.191047	+	0.981581i	$0.561188\pi$
$354$	0.821092	0.0436405
$355$	−18.3578	−0.974332
$356$	9.00000	0.476999
$357$	5.00000	0.264628
$358$	−14.3578	−0.758834
$359$	12.3578	0.652221	0.326110	−	0.945332i	$-0.394262\pi$
$359$	12.3578	0.652221	0.326110	+	0.945332i	$0.394262\pi$
$360$	−2.00000	−0.105409
$361$	47.8945	2.52077
$362$	7.82109	0.411067
$363$	−27.1789	−1.42652
$364$	0	0
$365$	−8.00000	−0.418739
$366$	3.00000	0.156813
$367$	3.17891	0.165938	0.0829688	−	0.996552i	$-0.473560\pi$
$367$	3.17891	0.165938	0.0829688	+	0.996552i	$0.473560\pi$
$368$	3.17891	0.165712
$369$	4.17891	0.217545
$370$	−4.00000	−0.207950
$371$	3.00000	0.155752
$372$	7.17891	0.372209
$373$	−20.0000	−1.03556	−0.517780	−	0.855514i	$-0.673242\pi$
$373$	−20.0000	−1.03556	−0.517780	+	0.855514i	$0.673242\pi$
$374$	30.8945	1.59752
$375$	−12.0000	−0.619677
$376$	4.17891	0.215511
$377$	0	0
$378$	1.00000	0.0514344
$379$	20.7156	1.06409	0.532045	−	0.846716i	$-0.321424\pi$
$379$	20.7156	1.06409	0.532045	+	0.846716i	$0.321424\pi$
$380$	16.3578	0.839138
$381$	−12.0000	−0.614779
$382$	27.5367	1.40890
$383$	9.82109	0.501834	0.250917	−	0.968009i	$-0.419268\pi$
$383$	9.82109	0.501834	0.250917	+	0.968009i	$0.419268\pi$
$384$	−1.00000	−0.0510310
$385$	12.3578	0.629813
$386$	20.5367	1.04529
$387$	0.821092	0.0417384
$388$	12.3578	0.627373
$389$	−0.821092	−0.0416310	−0.0208155	−	0.999783i	$-0.506626\pi$
$389$	−0.821092	−0.0416310	−0.0208155	+	0.999783i	$0.506626\pi$
$390$	0	0
$391$	15.8945	0.803822
$392$	1.00000	0.0505076
$393$	−19.5367	−0.985497
$394$	17.0000	0.856448
$395$	−3.64218	−0.183258
$396$	6.17891	0.310502
$397$	3.00000	0.150566	0.0752828	−	0.997162i	$-0.476014\pi$
$397$	3.00000	0.150566	0.0752828	+	0.997162i	$0.476014\pi$
$398$	12.8211	0.642663
$399$	−8.17891	−0.409458
$400$	−1.00000	−0.0500000
$401$	32.3578	1.61587	0.807936	−	0.589270i	$-0.200585\pi$
$401$	32.3578	1.61587	0.807936	+	0.589270i	$0.200585\pi$
$402$	15.1789	0.757055
$403$	0	0
$404$	−16.3578	−0.813832
$405$	−2.00000	−0.0993808
$406$	−8.17891	−0.405912
$407$	12.3578	0.612554
$408$	−5.00000	−0.247537
$409$	−13.6422	−0.674563	−0.337281	−	0.941404i	$-0.609507\pi$
$409$	−13.6422	−0.674563	−0.337281	+	0.941404i	$0.609507\pi$
$410$	−8.35782	−0.412763
$411$	16.3578	0.806872
$412$	3.17891	0.156614
$413$	0.821092	0.0404033
$414$	3.17891	0.156235
$415$	10.3578	0.508445
$416$	0	0
$417$	18.1789	0.890225
$418$	−50.5367	−2.47183
$419$	29.8945	1.46044	0.730222	−	0.683210i	$-0.239417\pi$
$419$	29.8945	1.46044	0.730222	+	0.683210i	$0.239417\pi$
$420$	−2.00000	−0.0975900
$421$	−34.7156	−1.69194	−0.845968	−	0.533233i	$-0.820977\pi$
$421$	−34.7156	−1.69194	−0.845968	+	0.533233i	$0.820977\pi$
$422$	16.3578	0.796286
$423$	4.17891	0.203185
$424$	−3.00000	−0.145693
$425$	−5.00000	−0.242536
$426$	−9.17891	−0.444720
$427$	3.00000	0.145180
$428$	7.82109	0.378047
$429$	0	0
$430$	−1.64218	−0.0791931
$431$	−1.17891	−0.0567860	−0.0283930	−	0.999597i	$-0.509039\pi$
$431$	−1.17891	−0.0567860	−0.0283930	+	0.999597i	$0.509039\pi$
$432$	−1.00000	−0.0481125
$433$	11.6422	0.559488	0.279744	−	0.960075i	$-0.409750\pi$
$433$	11.6422	0.559488	0.279744	+	0.960075i	$0.409750\pi$
$434$	7.17891	0.344599
$435$	16.3578	0.784297
$436$	4.00000	0.191565
$437$	−26.0000	−1.24375
$438$	−4.00000	−0.191127
$439$	−4.71563	−0.225065	−0.112532	−	0.993648i	$-0.535896\pi$
$439$	−4.71563	−0.225065	−0.112532	+	0.993648i	$0.535896\pi$
$440$	−12.3578	−0.589136
$441$	1.00000	0.0476190
$442$	0	0
$443$	−8.17891	−0.388592	−0.194296	−	0.980943i	$-0.562242\pi$
$443$	−8.17891	−0.388592	−0.194296	+	0.980943i	$0.562242\pi$
$444$	−2.00000	−0.0949158
$445$	−18.0000	−0.853282
$446$	9.17891	0.434634
$447$	−4.82109	−0.228030
$448$	−1.00000	−0.0472456
$449$	−14.0000	−0.660701	−0.330350	−	0.943858i	$-0.607167\pi$
$449$	−14.0000	−0.660701	−0.330350	+	0.943858i	$0.607167\pi$
$450$	−1.00000	−0.0471405
$451$	25.8211	1.21587
$452$	−4.35782	−0.204974
$453$	14.5367	0.682995
$454$	8.35782	0.392252
$455$	0	0
$456$	8.17891	0.383012
$457$	−31.8945	−1.49196	−0.745982	−	0.665966i	$-0.768019\pi$
$457$	−31.8945	−1.49196	−0.745982	+	0.665966i	$0.768019\pi$
$458$	1.00000	0.0467269
$459$	−5.00000	−0.233380
$460$	−6.35782	−0.296435
$461$	−24.7156	−1.15112	−0.575561	−	0.817759i	$-0.695216\pi$
$461$	−24.7156	−1.15112	−0.575561	+	0.817759i	$0.695216\pi$
$462$	6.17891	0.287469
$463$	−28.8945	−1.34284	−0.671422	−	0.741076i	$-0.734316\pi$
$463$	−28.8945	−1.34284	−0.671422	+	0.741076i	$0.734316\pi$
$464$	8.17891	0.379696
$465$	−14.3578	−0.665828
$466$	−0.357817	−0.0165755
$467$	33.8945	1.56845	0.784226	−	0.620475i	$-0.213060\pi$
$467$	33.8945	1.56845	0.784226	+	0.620475i	$0.213060\pi$
$468$	0	0
$469$	15.1789	0.700897
$470$	−8.35782	−0.385517
$471$	−2.00000	−0.0921551
$472$	−0.821092	−0.0377938
$473$	5.07345	0.233277
$474$	−1.82109	−0.0836455
$475$	8.17891	0.375274
$476$	−5.00000	−0.229175
$477$	−3.00000	−0.137361
$478$	−1.17891	−0.0539220
$479$	−29.8211	−1.36256	−0.681280	−	0.732023i	$-0.738576\pi$
$479$	−29.8211	−1.36256	−0.681280	+	0.732023i	$0.738576\pi$
$480$	2.00000	0.0912871
$481$	0	0
$482$	−18.3578	−0.836176
$483$	3.17891	0.144645
$484$	27.1789	1.23540
$485$	−24.7156	−1.12228
$486$	−1.00000	−0.0453609
$487$	−38.1789	−1.73005	−0.865026	−	0.501727i	$-0.832698\pi$
$487$	−38.1789	−1.73005	−0.865026	+	0.501727i	$0.832698\pi$
$488$	−3.00000	−0.135804
$489$	8.82109	0.398904
$490$	−2.00000	−0.0903508
$491$	0	0		−	1.00000i	$-0.5\pi$
$491$	0	0			1.00000i	$0.5\pi$
$492$	−4.17891	−0.188400
$493$	40.8945	1.84180
$494$	0	0
$495$	−12.3578	−0.555443
$496$	−7.17891	−0.322343
$497$	−9.17891	−0.411730
$498$	5.17891	0.232072
$499$	−3.17891	−0.142307	−0.0711537	−	0.997465i	$-0.522668\pi$
$499$	−3.17891	−0.142307	−0.0711537	+	0.997465i	$0.522668\pi$
$500$	12.0000	0.536656
$501$	8.00000	0.357414
$502$	10.8211	0.482969
$503$	−24.0000	−1.07011	−0.535054	−	0.844818i	$-0.679709\pi$
$503$	−24.0000	−1.07011	−0.535054	+	0.844818i	$0.679709\pi$
$504$	−1.00000	−0.0445435
$505$	32.7156	1.45583
$506$	19.6422	0.873202
$507$	0	0
$508$	12.0000	0.532414
$509$	28.7156	1.27280	0.636399	−	0.771360i	$-0.280423\pi$
$509$	28.7156	1.27280	0.636399	+	0.771360i	$0.280423\pi$
$510$	10.0000	0.442807
$511$	−4.00000	−0.176950
$512$	1.00000	0.0441942
$513$	8.17891	0.361108
$514$	25.7156	1.13427
$515$	−6.35782	−0.280159
$516$	−0.821092	−0.0361465
$517$	25.8211	1.13561
$518$	−2.00000	−0.0878750
$519$	−4.00000	−0.175581
$520$	0	0
$521$	−4.17891	−0.183081	−0.0915406	−	0.995801i	$-0.529179\pi$
$521$	−4.17891	−0.183081	−0.0915406	+	0.995801i	$0.529179\pi$
$522$	8.17891	0.357981
$523$	−22.5367	−0.985462	−0.492731	−	0.870182i	$-0.664001\pi$
$523$	−22.5367	−0.985462	−0.492731	+	0.870182i	$0.664001\pi$
$524$	19.5367	0.853466
$525$	−1.00000	−0.0436436
$526$	−12.0000	−0.523225
$527$	−35.8945	−1.56359
$528$	−6.17891	−0.268902
$529$	−12.8945	−0.560632
$530$	6.00000	0.260623
$531$	−0.821092	−0.0356323
$532$	8.17891	0.354601
$533$	0	0
$534$	−9.00000	−0.389468
$535$	−15.6422	−0.676271
$536$	−15.1789	−0.655629
$537$	14.3578	0.619586
$538$	18.3578	0.791462
$539$	6.17891	0.266144
$540$	2.00000	0.0860663
$541$	2.35782	0.101370	0.0506852	−	0.998715i	$-0.483859\pi$
$541$	2.35782	0.101370	0.0506852	+	0.998715i	$0.483859\pi$
$542$	31.1789	1.33925
$543$	−7.82109	−0.335635
$544$	5.00000	0.214373
$545$	−8.00000	−0.342682
$546$	0	0
$547$	36.0000	1.53925	0.769624	−	0.638497i	$-0.220443\pi$
$547$	36.0000	1.53925	0.769624	+	0.638497i	$0.220443\pi$
$548$	−16.3578	−0.698771
$549$	−3.00000	−0.128037
$550$	−6.17891	−0.263470
$551$	−66.8945	−2.84980
$552$	−3.17891	−0.135303
$553$	−1.82109	−0.0774407
$554$	28.3578	1.20481
$555$	4.00000	0.169791
$556$	−18.1789	−0.770957
$557$	−21.7156	−0.920121	−0.460060	−	0.887888i	$-0.652172\pi$
$557$	−21.7156	−0.920121	−0.460060	+	0.887888i	$0.652172\pi$
$558$	−7.17891	−0.303907
$559$	0	0
$560$	2.00000	0.0845154
$561$	−30.8945	−1.30437
$562$	5.64218	0.238001
$563$	4.35782	0.183660	0.0918300	−	0.995775i	$-0.470728\pi$
$563$	4.35782	0.183660	0.0918300	+	0.995775i	$0.470728\pi$
$564$	−4.17891	−0.175964
$565$	8.71563	0.366669
$566$	30.3578	1.27603
$567$	−1.00000	−0.0419961
$568$	9.17891	0.385138
$569$	−26.3578	−1.10498	−0.552489	−	0.833520i	$-0.686322\pi$
$569$	−26.3578	−1.10498	−0.552489	+	0.833520i	$0.686322\pi$
$570$	−16.3578	−0.685154
$571$	−22.8211	−0.955033	−0.477516	−	0.878623i	$-0.658463\pi$
$571$	−22.8211	−0.955033	−0.477516	+	0.878623i	$0.658463\pi$
$572$	0	0
$573$	−27.5367	−1.15036
$574$	−4.17891	−0.174424
$575$	−3.17891	−0.132570
$576$	1.00000	0.0416667
$577$	−5.64218	−0.234887	−0.117444	−	0.993080i	$-0.537470\pi$
$577$	−5.64218	−0.234887	−0.117444	+	0.993080i	$0.537470\pi$
$578$	8.00000	0.332756
$579$	−20.5367	−0.853477
$580$	−16.3578	−0.679221
$581$	5.17891	0.214857
$582$	−12.3578	−0.512248
$583$	−18.5367	−0.767713
$584$	4.00000	0.165521
$585$	0	0
$586$	12.0000	0.495715
$587$	−11.1789	−0.461403	−0.230701	−	0.973025i	$-0.574102\pi$
$587$	−11.1789	−0.461403	−0.230701	+	0.973025i	$0.574102\pi$
$588$	−1.00000	−0.0412393
$589$	58.7156	2.41934
$590$	1.64218	0.0676076
$591$	−17.0000	−0.699287
$592$	2.00000	0.0821995
$593$	−33.0000	−1.35515	−0.677574	−	0.735455i	$-0.736969\pi$
$593$	−33.0000	−1.35515	−0.677574	+	0.735455i	$0.736969\pi$
$594$	−6.17891	−0.253524
$595$	10.0000	0.409960
$596$	4.82109	0.197480
$597$	−12.8211	−0.524732
$598$	0	0
$599$	28.8211	1.17760	0.588799	−	0.808280i	$-0.299601\pi$
$599$	28.8211	1.17760	0.588799	+	0.808280i	$0.299601\pi$
$600$	1.00000	0.0408248
$601$	−22.0000	−0.897399	−0.448699	−	0.893683i	$-0.648113\pi$
$601$	−22.0000	−0.897399	−0.448699	+	0.893683i	$0.648113\pi$
$602$	−0.821092	−0.0334652
$603$	−15.1789	−0.618133
$604$	−14.5367	−0.591491
$605$	−54.3578	−2.20996
$606$	16.3578	0.664491
$607$	33.5367	1.36121	0.680607	−	0.732649i	$-0.261716\pi$
$607$	33.5367	1.36121	0.680607	+	0.732649i	$0.261716\pi$
$608$	−8.17891	−0.331699
$609$	8.17891	0.331426
$610$	6.00000	0.242933
$611$	0	0
$612$	5.00000	0.202113
$613$	38.3578	1.54926	0.774629	−	0.632416i	$-0.217937\pi$
$613$	38.3578	1.54926	0.774629	+	0.632416i	$0.217937\pi$
$614$	24.8945	1.00466
$615$	8.35782	0.337020
$616$	−6.17891	−0.248955
$617$	32.0000	1.28827	0.644136	−	0.764911i	$-0.277217\pi$
$617$	32.0000	1.28827	0.644136	+	0.764911i	$0.277217\pi$
$618$	−3.17891	−0.127874
$619$	−8.53673	−0.343120	−0.171560	−	0.985174i	$-0.554881\pi$
$619$	−8.53673	−0.343120	−0.171560	+	0.985174i	$0.554881\pi$
$620$	14.3578	0.576624
$621$	−3.17891	−0.127565
$622$	−8.53673	−0.342291
$623$	−9.00000	−0.360577
$624$	0	0
$625$	−19.0000	−0.760000
$626$	4.00000	0.159872
$627$	50.5367	2.01824
$628$	2.00000	0.0798087
$629$	10.0000	0.398726
$630$	2.00000	0.0796819
$631$	−30.8945	−1.22989	−0.614946	−	0.788569i	$-0.710822\pi$
$631$	−30.8945	−1.22989	−0.614946	+	0.788569i	$0.710822\pi$
$632$	1.82109	0.0724391
$633$	−16.3578	−0.650165
$634$	7.17891	0.285111
$635$	−24.0000	−0.952411
$636$	3.00000	0.118958
$637$	0	0
$638$	50.5367	2.00077
$639$	9.17891	0.363112
$640$	−2.00000	−0.0790569
$641$	−41.0735	−1.62230	−0.811152	−	0.584836i	$-0.801159\pi$
$641$	−41.0735	−1.62230	−0.811152	+	0.584836i	$0.801159\pi$
$642$	−7.82109	−0.308674
$643$	−20.1789	−0.795778	−0.397889	−	0.917433i	$-0.630257\pi$
$643$	−20.1789	−0.795778	−0.397889	+	0.917433i	$0.630257\pi$
$644$	−3.17891	−0.125267
$645$	1.64218	0.0646609
$646$	−40.8945	−1.60897
$647$	−8.89454	−0.349681	−0.174840	−	0.984597i	$-0.555941\pi$
$647$	−8.89454	−0.349681	−0.174840	+	0.984597i	$0.555941\pi$
$648$	1.00000	0.0392837
$649$	−5.07345	−0.199150
$650$	0	0
$651$	−7.17891	−0.281364
$652$	−8.82109	−0.345461
$653$	21.0000	0.821794	0.410897	−	0.911682i	$-0.365216\pi$
$653$	21.0000	0.821794	0.410897	+	0.911682i	$0.365216\pi$
$654$	−4.00000	−0.156412
$655$	−39.0735	−1.52673
$656$	4.17891	0.163159
$657$	4.00000	0.156055
$658$	−4.17891	−0.162911
$659$	−16.5367	−0.644179	−0.322090	−	0.946709i	$-0.604385\pi$
$659$	−16.5367	−0.644179	−0.322090	+	0.946709i	$0.604385\pi$
$660$	12.3578	0.481027
$661$	−17.5367	−0.682100	−0.341050	−	0.940045i	$-0.610782\pi$
$661$	−17.5367	−0.682100	−0.341050	+	0.940045i	$0.610782\pi$
$662$	−24.3578	−0.946693
$663$	0	0
$664$	−5.17891	−0.200981
$665$	−16.3578	−0.634329
$666$	2.00000	0.0774984
$667$	26.0000	1.00672
$668$	−8.00000	−0.309529
$669$	−9.17891	−0.354877
$670$	30.3578	1.17282
$671$	−18.5367	−0.715602
$672$	1.00000	0.0385758
$673$	−1.00000	−0.0385472	−0.0192736	−	0.999814i	$-0.506135\pi$
$673$	−1.00000	−0.0385472	−0.0192736	+	0.999814i	$0.506135\pi$
$674$	−0.536725	−0.0206739
$675$	1.00000	0.0384900
$676$	0	0
$677$	12.0000	0.461197	0.230599	−	0.973049i	$-0.425932\pi$
$677$	12.0000	0.461197	0.230599	+	0.973049i	$0.425932\pi$
$678$	4.35782	0.167361
$679$	−12.3578	−0.474249
$680$	−10.0000	−0.383482
$681$	−8.35782	−0.320272
$682$	−44.3578	−1.69855
$683$	−14.3578	−0.549387	−0.274693	−	0.961532i	$-0.588576\pi$
$683$	−14.3578	−0.549387	−0.274693	+	0.961532i	$0.588576\pi$
$684$	−8.17891	−0.312728
$685$	32.7156	1.25000
$686$	−1.00000	−0.0381802
$687$	−1.00000	−0.0381524
$688$	0.821092	0.0313038
$689$	0	0
$690$	6.35782	0.242038
$691$	−40.0000	−1.52167	−0.760836	−	0.648944i	$-0.775211\pi$
$691$	−40.0000	−1.52167	−0.760836	+	0.648944i	$0.775211\pi$
$692$	4.00000	0.152057
$693$	−6.17891	−0.234717
$694$	−7.82109	−0.296885
$695$	36.3578	1.37913
$696$	−8.17891	−0.310021
$697$	20.8945	0.791437
$698$	−3.17891	−0.120323
$699$	0.357817	0.0135339
$700$	1.00000	0.0377964
$701$	23.7156	0.895727	0.447864	−	0.894102i	$-0.352185\pi$
$701$	23.7156	0.895727	0.447864	+	0.894102i	$0.352185\pi$
$702$	0	0
$703$	−16.3578	−0.616947
$704$	6.17891	0.232876
$705$	8.35782	0.314774
$706$	−7.17891	−0.270182
$707$	16.3578	0.615199
$708$	0.821092	0.0308585
$709$	−16.0000	−0.600893	−0.300446	−	0.953799i	$-0.597136\pi$
$709$	−16.0000	−0.600893	−0.300446	+	0.953799i	$0.597136\pi$
$710$	−18.3578	−0.688957
$711$	1.82109	0.0682963
$712$	9.00000	0.337289
$713$	−22.8211	−0.854657
$714$	5.00000	0.187120
$715$	0	0
$716$	−14.3578	−0.536577
$717$	1.17891	0.0440271
$718$	12.3578	0.461190
$719$	−25.8211	−0.962964	−0.481482	−	0.876456i	$-0.659901\pi$
$719$	−25.8211	−0.962964	−0.481482	+	0.876456i	$0.659901\pi$
$720$	−2.00000	−0.0745356
$721$	−3.17891	−0.118389
$722$	47.8945	1.78245
$723$	18.3578	0.682735
$724$	7.82109	0.290669
$725$	−8.17891	−0.303757
$726$	−27.1789	−1.00870
$727$	0.463275	0.0171819	0.00859096	−	0.999963i	$-0.497265\pi$
$727$	0.463275	0.0171819	0.00859096	+	0.999963i	$0.497265\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	−8.00000	−0.296093
$731$	4.10546	0.151846
$732$	3.00000	0.110883
$733$	−25.7156	−0.949829	−0.474914	−	0.880032i	$-0.657521\pi$
$733$	−25.7156	−0.949829	−0.474914	+	0.880032i	$0.657521\pi$
$734$	3.17891	0.117336
$735$	2.00000	0.0737711
$736$	3.17891	0.117176
$737$	−93.7891	−3.45477
$738$	4.17891	0.153828
$739$	−4.82109	−0.177347	−0.0886734	−	0.996061i	$-0.528263\pi$
$739$	−4.82109	−0.177347	−0.0886734	+	0.996061i	$0.528263\pi$
$740$	−4.00000	−0.147043
$741$	0	0
$742$	3.00000	0.110133
$743$	43.1789	1.58408	0.792040	−	0.610469i	$-0.209019\pi$
$743$	43.1789	1.58408	0.792040	+	0.610469i	$0.209019\pi$
$744$	7.17891	0.263192
$745$	−9.64218	−0.353262
$746$	−20.0000	−0.732252
$747$	−5.17891	−0.189486
$748$	30.8945	1.12962
$749$	−7.82109	−0.285776
$750$	−12.0000	−0.438178
$751$	20.8945	0.762453	0.381226	−	0.924482i	$-0.375502\pi$
$751$	20.8945	0.762453	0.381226	+	0.924482i	$0.375502\pi$
$752$	4.17891	0.152389
$753$	−10.8211	−0.394343
$754$	0	0
$755$	29.0735	1.05809
$756$	1.00000	0.0363696
$757$	2.35782	0.0856963	0.0428482	−	0.999082i	$-0.486357\pi$
$757$	2.35782	0.0856963	0.0428482	+	0.999082i	$0.486357\pi$
$758$	20.7156	0.752426
$759$	−19.6422	−0.712966
$760$	16.3578	0.593360
$761$	47.4313	1.71938	0.859691	−	0.510814i	$-0.170656\pi$
$761$	47.4313	1.71938	0.859691	+	0.510814i	$0.170656\pi$
$762$	−12.0000	−0.434714
$763$	−4.00000	−0.144810
$764$	27.5367	0.996244
$765$	−10.0000	−0.361551
$766$	9.82109	0.354850
$767$	0	0
$768$	−1.00000	−0.0360844
$769$	−4.35782	−0.157147	−0.0785734	−	0.996908i	$-0.525037\pi$
$769$	−4.35782	−0.157147	−0.0785734	+	0.996908i	$0.525037\pi$
$770$	12.3578	0.445345
$771$	−25.7156	−0.926126
$772$	20.5367	0.739133
$773$	−30.7156	−1.10476	−0.552382	−	0.833591i	$-0.686281\pi$
$773$	−30.7156	−1.10476	−0.552382	+	0.833591i	$0.686281\pi$
$774$	0.821092	0.0295135
$775$	7.17891	0.257874
$776$	12.3578	0.443620
$777$	2.00000	0.0717496
$778$	−0.821092	−0.0294376
$779$	−34.1789	−1.22459
$780$	0	0
$781$	56.7156	2.02944
$782$	15.8945	0.568388
$783$	−8.17891	−0.292290
$784$	1.00000	0.0357143
$785$	−4.00000	−0.142766
$786$	−19.5367	−0.696852
$787$	−22.1789	−0.790593	−0.395296	−	0.918554i	$-0.629358\pi$
$787$	−22.1789	−0.790593	−0.395296	+	0.918554i	$0.629358\pi$
$788$	17.0000	0.605600
$789$	12.0000	0.427211
$790$	−3.64218	−0.129583
$791$	4.35782	0.154946
$792$	6.17891	0.219558
$793$	0	0
$794$	3.00000	0.106466
$795$	−6.00000	−0.212798
$796$	12.8211	0.454432
$797$	0	0		−	1.00000i	$-0.5\pi$
$797$	0	0			1.00000i	$0.5\pi$
$798$	−8.17891	−0.289530
$799$	20.8945	0.739196
$800$	−1.00000	−0.0353553
$801$	9.00000	0.317999
$802$	32.3578	1.14259
$803$	24.7156	0.872196
$804$	15.1789	0.535319
$805$	6.35782	0.224084
$806$	0	0
$807$	−18.3578	−0.646226
$808$	−16.3578	−0.575466
$809$	−20.0000	−0.703163	−0.351581	−	0.936157i	$-0.614356\pi$
$809$	−20.0000	−0.703163	−0.351581	+	0.936157i	$0.614356\pi$
$810$	−2.00000	−0.0702728
$811$	32.7156	1.14880	0.574401	−	0.818574i	$-0.305235\pi$
$811$	32.7156	1.14880	0.574401	+	0.818574i	$0.305235\pi$
$812$	−8.17891	−0.287023
$813$	−31.1789	−1.09349
$814$	12.3578	0.433141
$815$	17.6422	0.617979
$816$	−5.00000	−0.175035
$817$	−6.71563	−0.234950
$818$	−13.6422	−0.476988
$819$	0	0
$820$	−8.35782	−0.291868
$821$	45.7156	1.59549	0.797743	−	0.602997i	$-0.206027\pi$
$821$	45.7156	1.59549	0.797743	+	0.602997i	$0.206027\pi$
$822$	16.3578	0.570544
$823$	−12.7156	−0.443239	−0.221620	−	0.975133i	$-0.571134\pi$
$823$	−12.7156	−0.443239	−0.221620	+	0.975133i	$0.571134\pi$
$824$	3.17891	0.110743
$825$	6.17891	0.215122
$826$	0.821092	0.0285694
$827$	4.71563	0.163979	0.0819893	−	0.996633i	$-0.473873\pi$
$827$	4.71563	0.163979	0.0819893	+	0.996633i	$0.473873\pi$
$828$	3.17891	0.110475
$829$	8.17891	0.284065	0.142033	−	0.989862i	$-0.454636\pi$
$829$	8.17891	0.284065	0.142033	+	0.989862i	$0.454636\pi$
$830$	10.3578	0.359525
$831$	−28.3578	−0.983722
$832$	0	0
$833$	5.00000	0.173240
$834$	18.1789	0.629484
$835$	16.0000	0.553703
$836$	−50.5367	−1.74785
$837$	7.17891	0.248139
$838$	29.8945	1.03269
$839$	34.3578	1.18616	0.593082	−	0.805142i	$-0.297911\pi$
$839$	34.3578	1.18616	0.593082	+	0.805142i	$0.297911\pi$
$840$	−2.00000	−0.0690066
$841$	37.8945	1.30671
$842$	−34.7156	−1.19638
$843$	−5.64218	−0.194327
$844$	16.3578	0.563059
$845$	0	0
$846$	4.17891	0.143674
$847$	−27.1789	−0.933878
$848$	−3.00000	−0.103020
$849$	−30.3578	−1.04188
$850$	−5.00000	−0.171499
$851$	6.35782	0.217943
$852$	−9.17891	−0.314464
$853$	−17.7156	−0.606572	−0.303286	−	0.952900i	$-0.598084\pi$
$853$	−17.7156	−0.606572	−0.303286	+	0.952900i	$0.598084\pi$
$854$	3.00000	0.102658
$855$	16.3578	0.559426
$856$	7.82109	0.267319
$857$	34.7156	1.18586	0.592932	−	0.805253i	$-0.297970\pi$
$857$	34.7156	1.18586	0.592932	+	0.805253i	$0.297970\pi$
$858$	0	0
$859$	−36.8945	−1.25883	−0.629413	−	0.777071i	$-0.716705\pi$
$859$	−36.8945	−1.25883	−0.629413	+	0.777071i	$0.716705\pi$
$860$	−1.64218	−0.0559980
$861$	4.17891	0.142417
$862$	−1.17891	−0.0401538
$863$	36.3578	1.23763	0.618817	−	0.785535i	$-0.287612\pi$
$863$	36.3578	1.23763	0.618817	+	0.785535i	$0.287612\pi$
$864$	−1.00000	−0.0340207
$865$	−8.00000	−0.272008
$866$	11.6422	0.395617
$867$	−8.00000	−0.271694
$868$	7.17891	0.243668
$869$	11.2524	0.381710
$870$	16.3578	0.554582
$871$	0	0
$872$	4.00000	0.135457
$873$	12.3578	0.418249
$874$	−26.0000	−0.879463
$875$	−12.0000	−0.405674
$876$	−4.00000	−0.135147
$877$	−17.6422	−0.595734	−0.297867	−	0.954607i	$-0.596275\pi$
$877$	−17.6422	−0.595734	−0.297867	+	0.954607i	$0.596275\pi$
$878$	−4.71563	−0.159145
$879$	−12.0000	−0.404750
$880$	−12.3578	−0.416582
$881$	45.5367	1.53417	0.767086	−	0.641545i	$-0.221706\pi$
$881$	45.5367	1.53417	0.767086	+	0.641545i	$0.221706\pi$
$882$	1.00000	0.0336718
$883$	−37.5367	−1.26321	−0.631606	−	0.775290i	$-0.717604\pi$
$883$	−37.5367	−1.26321	−0.631606	+	0.775290i	$0.717604\pi$
$884$	0	0
$885$	−1.64218	−0.0552014
$886$	−8.17891	−0.274776
$887$	−11.4633	−0.384899	−0.192450	−	0.981307i	$-0.561643\pi$
$887$	−11.4633	−0.384899	−0.192450	+	0.981307i	$0.561643\pi$
$888$	−2.00000	−0.0671156
$889$	−12.0000	−0.402467
$890$	−18.0000	−0.603361
$891$	6.17891	0.207001
$892$	9.17891	0.307333
$893$	−34.1789	−1.14375
$894$	−4.82109	−0.161241
$895$	28.7156	0.959858
$896$	−1.00000	−0.0334077
$897$	0	0
$898$	−14.0000	−0.467186
$899$	−58.7156	−1.95828
$900$	−1.00000	−0.0333333
$901$	−15.0000	−0.499722
$902$	25.8211	0.859748
$903$	0.821092	0.0273242
$904$	−4.35782	−0.144939
$905$	−15.6422	−0.519964
$906$	14.5367	0.482950
$907$	−52.6102	−1.74689	−0.873446	−	0.486921i	$-0.838120\pi$
$907$	−52.6102	−1.74689	−0.873446	+	0.486921i	$0.838120\pi$
$908$	8.35782	0.277364
$909$	−16.3578	−0.542555
$910$	0	0
$911$	−16.7156	−0.553814	−0.276907	−	0.960897i	$-0.589309\pi$
$911$	−16.7156	−0.553814	−0.276907	+	0.960897i	$0.589309\pi$
$912$	8.17891	0.270831
$913$	−32.0000	−1.05905
$914$	−31.8945	−1.05498
$915$	−6.00000	−0.198354
$916$	1.00000	0.0330409
$917$	−19.5367	−0.645159
$918$	−5.00000	−0.165025
$919$	21.2524	0.701051	0.350525	−	0.936553i	$-0.386003\pi$
$919$	21.2524	0.701051	0.350525	+	0.936553i	$0.386003\pi$
$920$	−6.35782	−0.209611
$921$	−24.8945	−0.820303
$922$	−24.7156	−0.813966
$923$	0	0
$924$	6.17891	0.203271
$925$	−2.00000	−0.0657596
$926$	−28.8945	−0.949534
$927$	3.17891	0.104409
$928$	8.17891	0.268486
$929$	39.0000	1.27955	0.639774	−	0.768563i	$-0.279028\pi$
$929$	39.0000	1.27955	0.639774	+	0.768563i	$0.279028\pi$
$930$	−14.3578	−0.470811
$931$	−8.17891	−0.268053
$932$	−0.357817	−0.0117207
$933$	8.53673	0.279480
$934$	33.8945	1.10906
$935$	−61.7891	−2.02072
$936$	0	0
$937$	−47.0735	−1.53782	−0.768911	−	0.639355i	$-0.779201\pi$
$937$	−47.0735	−1.53782	−0.768911	+	0.639355i	$0.779201\pi$
$938$	15.1789	0.495609
$939$	−4.00000	−0.130535
$940$	−8.35782	−0.272602
$941$	−20.0000	−0.651981	−0.325991	−	0.945373i	$-0.605698\pi$
$941$	−20.0000	−0.651981	−0.325991	+	0.945373i	$0.605698\pi$
$942$	−2.00000	−0.0651635
$943$	13.2844	0.432598
$944$	−0.821092	−0.0267243
$945$	−2.00000	−0.0650600
$946$	5.07345	0.164952
$947$	39.2524	1.27553	0.637765	−	0.770231i	$-0.279859\pi$
$947$	39.2524	1.27553	0.637765	+	0.770231i	$0.279859\pi$
$948$	−1.82109	−0.0591463
$949$	0	0
$950$	8.17891	0.265359
$951$	−7.17891	−0.232792
$952$	−5.00000	−0.162051
$953$	42.7156	1.38370	0.691848	−	0.722044i	$-0.256797\pi$
$953$	42.7156	1.38370	0.691848	+	0.722044i	$0.256797\pi$
$954$	−3.00000	−0.0971286
$955$	−55.0735	−1.78213
$956$	−1.17891	−0.0381286
$957$	−50.5367	−1.63362
$958$	−29.8211	−0.963476
$959$	16.3578	0.528221
$960$	2.00000	0.0645497
$961$	20.5367	0.662475
$962$	0	0
$963$	7.82109	0.252031
$964$	−18.3578	−0.591265
$965$	−41.0735	−1.32220
$966$	3.17891	0.102280
$967$	34.3578	1.10487	0.552436	−	0.833555i	$-0.313698\pi$
$967$	34.3578	1.10487	0.552436	+	0.833555i	$0.313698\pi$
$968$	27.1789	0.873563
$969$	40.8945	1.31372
$970$	−24.7156	−0.793571
$971$	−16.8211	−0.539815	−0.269907	−	0.962886i	$-0.586993\pi$
$971$	−16.8211	−0.539815	−0.269907	+	0.962886i	$0.586993\pi$
$972$	−1.00000	−0.0320750
$973$	18.1789	0.582789
$974$	−38.1789	−1.22333
$975$	0	0
$976$	−3.00000	−0.0960277
$977$	12.0000	0.383914	0.191957	−	0.981403i	$-0.438517\pi$
$977$	12.0000	0.383914	0.191957	+	0.981403i	$0.438517\pi$
$978$	8.82109	0.282067
$979$	55.6102	1.77731
$980$	−2.00000	−0.0638877
$981$	4.00000	0.127710
$982$	0	0
$983$	55.0735	1.75657	0.878285	−	0.478137i	$-0.158688\pi$
$983$	55.0735	1.75657	0.878285	+	0.478137i	$0.158688\pi$
$984$	−4.17891	−0.133219
$985$	−34.0000	−1.08333
$986$	40.8945	1.30235
$987$	4.17891	0.133016
$988$	0	0
$989$	2.61018	0.0829987
$990$	−12.3578	−0.392757
$991$	−15.4633	−0.491207	−0.245604	−	0.969370i	$-0.578986\pi$
$991$	−15.4633	−0.491207	−0.245604	+	0.969370i	$0.578986\pi$
$992$	−7.17891	−0.227931
$993$	24.3578	0.772972
$994$	−9.17891	−0.291137
$995$	−25.6422	−0.812912
$996$	5.17891	0.164100
$997$	57.7156	1.82787	0.913936	−	0.405858i	$-0.133027\pi$
$997$	57.7156	1.82787	0.913936	+	0.405858i	$0.133027\pi$
$998$	−3.17891	−0.100627
$999$	−2.00000	−0.0632772

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7098.2.a.br.1.2		2
13.3	even	3		546.2.l.k.295.1	yes	4
13.9	even	3		546.2.l.k.211.1	✓	4
13.12	even	2		7098.2.a.bk.1.1		2
39.29	odd	6		1638.2.r.ba.1387.2		4
39.35	odd	6		1638.2.r.ba.757.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
546.2.l.k.211.1	✓	4	13.9	even	3
546.2.l.k.295.1	yes	4	13.3	even	3
1638.2.r.ba.757.2		4	39.35	odd	6
1638.2.r.ba.1387.2		4	39.29	odd	6
7098.2.a.bk.1.1		2	13.12	even	2
7098.2.a.br.1.2		2	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 7098 = 2 \cdot 3 \cdot 7 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7098.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -2.00000 q^{5} -1.00000 q^{6} -1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -2.00000 q^{5} -1.00000 q^{6} -1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} -2.00000 q^{10} +6.17891 q^{11} -1.00000 q^{12} -1.00000 q^{14} +2.00000 q^{15} +1.00000 q^{16} +5.00000 q^{17} +1.00000 q^{18} -8.17891 q^{19} -2.00000 q^{20} +1.00000 q^{21} +6.17891 q^{22} +3.17891 q^{23} -1.00000 q^{24} -1.00000 q^{25} -1.00000 q^{27} -1.00000 q^{28} +8.17891 q^{29} +2.00000 q^{30} -7.17891 q^{31} +1.00000 q^{32} -6.17891 q^{33} +5.00000 q^{34} +2.00000 q^{35} +1.00000 q^{36} +2.00000 q^{37} -8.17891 q^{38} -2.00000 q^{40} +4.17891 q^{41} +1.00000 q^{42} +0.821092 q^{43} +6.17891 q^{44} -2.00000 q^{45} +3.17891 q^{46} +4.17891 q^{47} -1.00000 q^{48} +1.00000 q^{49} -1.00000 q^{50} -5.00000 q^{51} -3.00000 q^{53} -1.00000 q^{54} -12.3578 q^{55} -1.00000 q^{56} +8.17891 q^{57} +8.17891 q^{58} -0.821092 q^{59} +2.00000 q^{60} -3.00000 q^{61} -7.17891 q^{62} -1.00000 q^{63} +1.00000 q^{64} -6.17891 q^{66} -15.1789 q^{67} +5.00000 q^{68} -3.17891 q^{69} +2.00000 q^{70} +9.17891 q^{71} +1.00000 q^{72} +4.00000 q^{73} +2.00000 q^{74} +1.00000 q^{75} -8.17891 q^{76} -6.17891 q^{77} +1.82109 q^{79} -2.00000 q^{80} +1.00000 q^{81} +4.17891 q^{82} -5.17891 q^{83} +1.00000 q^{84} -10.0000 q^{85} +0.821092 q^{86} -8.17891 q^{87} +6.17891 q^{88} +9.00000 q^{89} -2.00000 q^{90} +3.17891 q^{92} +7.17891 q^{93} +4.17891 q^{94} +16.3578 q^{95} -1.00000 q^{96} +12.3578 q^{97} +1.00000 q^{98} +6.17891 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} - 2 q^{3} + 2 q^{4} - 4 q^{5} - 2 q^{6} - 2 q^{7} + 2 q^{8} + 2 q^{9}+O(q^{10}) \) 2 * q + 2 * q^2 - 2 * q^3 + 2 * q^4 - 4 * q^5 - 2 * q^6 - 2 * q^7 + 2 * q^8 + 2 * q^9	\( 2 q + 2 q^{2} - 2 q^{3} + 2 q^{4} - 4 q^{5} - 2 q^{6} - 2 q^{7} + 2 q^{8} + 2 q^{9} - 4 q^{10} + q^{11} - 2 q^{12} - 2 q^{14} + 4 q^{15} + 2 q^{16} + 10 q^{17} + 2 q^{18} - 5 q^{19} - 4 q^{20} + 2 q^{21} + q^{22} - 5 q^{23} - 2 q^{24} - 2 q^{25} - 2 q^{27} - 2 q^{28} + 5 q^{29} + 4 q^{30} - 3 q^{31} + 2 q^{32} - q^{33} + 10 q^{34} + 4 q^{35} + 2 q^{36} + 4 q^{37} - 5 q^{38} - 4 q^{40} - 3 q^{41} + 2 q^{42} + 13 q^{43} + q^{44} - 4 q^{45} - 5 q^{46} - 3 q^{47} - 2 q^{48} + 2 q^{49} - 2 q^{50} - 10 q^{51} - 6 q^{53} - 2 q^{54} - 2 q^{55} - 2 q^{56} + 5 q^{57} + 5 q^{58} - 13 q^{59} + 4 q^{60} - 6 q^{61} - 3 q^{62} - 2 q^{63} + 2 q^{64} - q^{66} - 19 q^{67} + 10 q^{68} + 5 q^{69} + 4 q^{70} + 7 q^{71} + 2 q^{72} + 8 q^{73} + 4 q^{74} + 2 q^{75} - 5 q^{76} - q^{77} + 15 q^{79} - 4 q^{80} + 2 q^{81} - 3 q^{82} + q^{83} + 2 q^{84} - 20 q^{85} + 13 q^{86} - 5 q^{87} + q^{88} + 18 q^{89} - 4 q^{90} - 5 q^{92} + 3 q^{93} - 3 q^{94} + 10 q^{95} - 2 q^{96} + 2 q^{97} + 2 q^{98} + q^{99}+O(q^{100}) \) 2 * q + 2 * q^2 - 2 * q^3 + 2 * q^4 - 4 * q^5 - 2 * q^6 - 2 * q^7 + 2 * q^8 + 2 * q^9 - 4 * q^10 + q^11 - 2 * q^12 - 2 * q^14 + 4 * q^15 + 2 * q^16 + 10 * q^17 + 2 * q^18 - 5 * q^19 - 4 * q^20 + 2 * q^21 + q^22 - 5 * q^23 - 2 * q^24 - 2 * q^25 - 2 * q^27 - 2 * q^28 + 5 * q^29 + 4 * q^30 - 3 * q^31 + 2 * q^32 - q^33 + 10 * q^34 + 4 * q^35 + 2 * q^36 + 4 * q^37 - 5 * q^38 - 4 * q^40 - 3 * q^41 + 2 * q^42 + 13 * q^43 + q^44 - 4 * q^45 - 5 * q^46 - 3 * q^47 - 2 * q^48 + 2 * q^49 - 2 * q^50 - 10 * q^51 - 6 * q^53 - 2 * q^54 - 2 * q^55 - 2 * q^56 + 5 * q^57 + 5 * q^58 - 13 * q^59 + 4 * q^60 - 6 * q^61 - 3 * q^62 - 2 * q^63 + 2 * q^64 - q^66 - 19 * q^67 + 10 * q^68 + 5 * q^69 + 4 * q^70 + 7 * q^71 + 2 * q^72 + 8 * q^73 + 4 * q^74 + 2 * q^75 - 5 * q^76 - q^77 + 15 * q^79 - 4 * q^80 + 2 * q^81 - 3 * q^82 + q^83 + 2 * q^84 - 20 * q^85 + 13 * q^86 - 5 * q^87 + q^88 + 18 * q^89 - 4 * q^90 - 5 * q^92 + 3 * q^93 - 3 * q^94 + 10 * q^95 - 2 * q^96 + 2 * q^97 + 2 * q^98 + q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more