LMFDB - Embedded newform 5577.2.a.bi.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5577,2,Mod(1,5577)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5577, 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("5577.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5577 = 3 \cdot 11 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5577.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:	$44.5325692073$
Analytic rank:	$0$
Dimension:	$18$
Coefficient field:	$\mathbb{Q}[x]/(x^{18} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{18} - x^{17} - 29 x^{16} + 26 x^{15} + 347 x^{14} - 277 x^{13} - 2215 x^{12} + 1560 x^{11} + \cdots - 71 $ x^18 - x^17 - 29x^16 + 26x^15 + 347x^14 - 277x^13 - 2215x^12 + 1560x^11 + 8156x^10 - 4977x^9 - 17520x^8 + 8853x^7 + 21169x^6 - 7818x^5 - 13097x^4 + 2237x^3 + 3379x^2 + 285x - 71
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$1.65040$ of defining polynomial
Character	$\chi$	$=$	5577.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.65040 q^{2} +1.00000 q^{3} +0.723807 q^{4} +4.43233 q^{5} -1.65040 q^{6} -3.18712 q^{7} +2.10622 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-1.65040 q^{2} +1.00000 q^{3} +0.723807 q^{4} +4.43233 q^{5} -1.65040 q^{6} -3.18712 q^{7} +2.10622 q^{8} +1.00000 q^{9} -7.31510 q^{10} +1.00000 q^{11} +0.723807 q^{12} +5.26002 q^{14} +4.43233 q^{15} -4.92372 q^{16} -1.25246 q^{17} -1.65040 q^{18} +5.97365 q^{19} +3.20815 q^{20} -3.18712 q^{21} -1.65040 q^{22} +6.94178 q^{23} +2.10622 q^{24} +14.6456 q^{25} +1.00000 q^{27} -2.30686 q^{28} +5.43785 q^{29} -7.31510 q^{30} +7.54645 q^{31} +3.91363 q^{32} +1.00000 q^{33} +2.06706 q^{34} -14.1264 q^{35} +0.723807 q^{36} +2.52374 q^{37} -9.85889 q^{38} +9.33549 q^{40} -10.7235 q^{41} +5.26002 q^{42} +0.983186 q^{43} +0.723807 q^{44} +4.43233 q^{45} -11.4567 q^{46} -11.7924 q^{47} -4.92372 q^{48} +3.15776 q^{49} -24.1710 q^{50} -1.25246 q^{51} +6.73492 q^{53} -1.65040 q^{54} +4.43233 q^{55} -6.71280 q^{56} +5.97365 q^{57} -8.97460 q^{58} -2.73913 q^{59} +3.20815 q^{60} -2.73920 q^{61} -12.4546 q^{62} -3.18712 q^{63} +3.38839 q^{64} -1.65040 q^{66} -9.44951 q^{67} -0.906541 q^{68} +6.94178 q^{69} +23.3141 q^{70} +6.30386 q^{71} +2.10622 q^{72} +1.95737 q^{73} -4.16518 q^{74} +14.6456 q^{75} +4.32377 q^{76} -3.18712 q^{77} -9.68221 q^{79} -21.8236 q^{80} +1.00000 q^{81} +17.6980 q^{82} -1.47921 q^{83} -2.30686 q^{84} -5.55133 q^{85} -1.62265 q^{86} +5.43785 q^{87} +2.10622 q^{88} -2.64258 q^{89} -7.31510 q^{90} +5.02450 q^{92} +7.54645 q^{93} +19.4621 q^{94} +26.4772 q^{95} +3.91363 q^{96} +2.66049 q^{97} -5.21155 q^{98} +1.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 18 q - q^{2} + 18 q^{3} + 23 q^{4} + 6 q^{5} - q^{6} - q^{7} - 6 q^{8} + 18 q^{9}+O(q^{10}) $ 18 * q - q^2 + 18 * q^3 + 23 * q^4 + 6 * q^5 - q^6 - q^7 - 6 * q^8 + 18 * q^9	$ 18 q - q^{2} + 18 q^{3} + 23 q^{4} + 6 q^{5} - q^{6} - q^{7} - 6 q^{8} + 18 q^{9} - 7 q^{10} + 18 q^{11} + 23 q^{12} + 7 q^{14} + 6 q^{15} + 25 q^{16} + 19 q^{17} - q^{18} + 2 q^{20} - q^{21} - q^{22} + 38 q^{23} - 6 q^{24} + 36 q^{25} + 18 q^{27} - 33 q^{28} + 47 q^{29} - 7 q^{30} + 22 q^{31} - 33 q^{32} + 18 q^{33} - 4 q^{34} + 20 q^{35} + 23 q^{36} + 41 q^{37} + 14 q^{38} - 28 q^{40} - 8 q^{41} + 7 q^{42} + 28 q^{43} + 23 q^{44} + 6 q^{45} - 6 q^{46} - 11 q^{47} + 25 q^{48} + 9 q^{49} + q^{50} + 19 q^{51} + 49 q^{53} - q^{54} + 6 q^{55} + 35 q^{56} + 27 q^{58} + 2 q^{59} + 2 q^{60} - 13 q^{61} + 46 q^{62} - q^{63} + 40 q^{64} - q^{66} - 4 q^{67} + 30 q^{68} + 38 q^{69} + 83 q^{70} - 2 q^{71} - 6 q^{72} - 39 q^{73} + 42 q^{74} + 36 q^{75} - 20 q^{76} - q^{77} + 18 q^{79} - 18 q^{80} + 18 q^{81} - 12 q^{82} - 5 q^{83} - 33 q^{84} - 2 q^{85} - 57 q^{86} + 47 q^{87} - 6 q^{88} + 9 q^{89} - 7 q^{90} + 86 q^{92} + 22 q^{93} - 27 q^{94} + 74 q^{95} - 33 q^{96} + 17 q^{97} - 4 q^{98} + 18 q^{99}+O(q^{100}) $ 18 * q - q^2 + 18 * q^3 + 23 * q^4 + 6 * q^5 - q^6 - q^7 - 6 * q^8 + 18 * q^9 - 7 * q^10 + 18 * q^11 + 23 * q^12 + 7 * q^14 + 6 * q^15 + 25 * q^16 + 19 * q^17 - q^18 + 2 * q^20 - q^21 - q^22 + 38 * q^23 - 6 * q^24 + 36 * q^25 + 18 * q^27 - 33 * q^28 + 47 * q^29 - 7 * q^30 + 22 * q^31 - 33 * q^32 + 18 * q^33 - 4 * q^34 + 20 * q^35 + 23 * q^36 + 41 * q^37 + 14 * q^38 - 28 * q^40 - 8 * q^41 + 7 * q^42 + 28 * q^43 + 23 * q^44 + 6 * q^45 - 6 * q^46 - 11 * q^47 + 25 * q^48 + 9 * q^49 + q^50 + 19 * q^51 + 49 * q^53 - q^54 + 6 * q^55 + 35 * q^56 + 27 * q^58 + 2 * q^59 + 2 * q^60 - 13 * q^61 + 46 * q^62 - q^63 + 40 * q^64 - q^66 - 4 * q^67 + 30 * q^68 + 38 * q^69 + 83 * q^70 - 2 * q^71 - 6 * q^72 - 39 * q^73 + 42 * q^74 + 36 * q^75 - 20 * q^76 - q^77 + 18 * q^79 - 18 * q^80 + 18 * q^81 - 12 * q^82 - 5 * q^83 - 33 * q^84 - 2 * q^85 - 57 * q^86 + 47 * q^87 - 6 * q^88 + 9 * q^89 - 7 * q^90 + 86 * q^92 + 22 * q^93 - 27 * q^94 + 74 * q^95 - 33 * q^96 + 17 * q^97 - 4 * q^98 + 18 * 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 $

$2$	−1.65040	−1.16701	−0.583503	−	0.812111i	$-0.698318\pi$
$2$	−1.65040	−1.16701	−0.583503	+	0.812111i	$0.698318\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0.723807	0.361903
$5$	4.43233	1.98220	0.991100	−	0.133122i	$-0.0425002\pi$
$5$	4.43233	1.98220	0.991100	+	0.133122i	$0.0425002\pi$
$6$	−1.65040	−0.673771
$7$	−3.18712	−1.20462	−0.602310	−	0.798262i	$-0.705753\pi$
$7$	−3.18712	−1.20462	−0.602310	+	0.798262i	$0.705753\pi$
$8$	2.10622	0.744663
$9$	1.00000	0.333333
$10$	−7.31510	−2.31324
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0.723807	0.208945
$13$	0	0
$13$	0	0
$14$	5.26002	1.40580
$15$	4.43233	1.14442
$16$	−4.92372	−1.23093
$17$	−1.25246	−0.303767	−0.151884	−	0.988398i	$-0.548534\pi$
$17$	−1.25246	−0.303767	−0.151884	+	0.988398i	$0.548534\pi$
$18$	−1.65040	−0.389002
$19$	5.97365	1.37045	0.685225	−	0.728331i	$-0.259704\pi$
$19$	5.97365	1.37045	0.685225	+	0.728331i	$0.259704\pi$
$20$	3.20815	0.717365
$21$	−3.18712	−0.695487
$22$	−1.65040	−0.351866
$23$	6.94178	1.44746	0.723730	−	0.690083i	$-0.242426\pi$
$23$	6.94178	1.44746	0.723730	+	0.690083i	$0.242426\pi$
$24$	2.10622	0.429931
$25$	14.6456	2.92911
$26$	0	0
$27$	1.00000	0.192450
$28$	−2.30686	−0.435956
$29$	5.43785	1.00978	0.504891	−	0.863183i	$-0.331533\pi$
$29$	5.43785	1.00978	0.504891	+	0.863183i	$0.331533\pi$
$30$	−7.31510	−1.33555
$31$	7.54645	1.35538	0.677691	−	0.735347i	$-0.262981\pi$
$31$	7.54645	1.35538	0.677691	+	0.735347i	$0.262981\pi$
$32$	3.91363	0.691839
$33$	1.00000	0.174078
$34$	2.06706	0.354498
$35$	−14.1264	−2.38780
$36$	0.723807	0.120634
$37$	2.52374	0.414901	0.207450	−	0.978246i	$-0.433483\pi$
$37$	2.52374	0.414901	0.207450	+	0.978246i	$0.433483\pi$
$38$	−9.85889	−1.59932
$39$	0	0
$40$	9.33549	1.47607
$41$	−10.7235	−1.67473	−0.837366	−	0.546643i	$-0.815906\pi$
$41$	−10.7235	−1.67473	−0.837366	+	0.546643i	$0.815906\pi$
$42$	5.26002	0.811638
$43$	0.983186	0.149934	0.0749672	−	0.997186i	$-0.476115\pi$
$43$	0.983186	0.149934	0.0749672	+	0.997186i	$0.476115\pi$
$44$	0.723807	0.109118
$45$	4.43233	0.660733
$46$	−11.4567	−1.68919
$47$	−11.7924	−1.72009	−0.860046	−	0.510217i	$-0.829565\pi$
$47$	−11.7924	−1.72009	−0.860046	+	0.510217i	$0.829565\pi$
$48$	−4.92372	−0.710677
$49$	3.15776	0.451108
$50$	−24.1710	−3.41829
$51$	−1.25246	−0.175380
$52$	0	0
$53$	6.73492	0.925112	0.462556	−	0.886590i	$-0.346932\pi$
$53$	6.73492	0.925112	0.462556	+	0.886590i	$0.346932\pi$
$54$	−1.65040	−0.224590
$55$	4.43233	0.597656
$56$	−6.71280	−0.897035
$57$	5.97365	0.791230
$58$	−8.97460	−1.17842
$59$	−2.73913	−0.356604	−0.178302	−	0.983976i	$-0.557060\pi$
$59$	−2.73913	−0.356604	−0.178302	+	0.983976i	$0.557060\pi$
$60$	3.20815	0.414171
$61$	−2.73920	−0.350719	−0.175359	−	0.984505i	$-0.556109\pi$
$61$	−2.73920	−0.350719	−0.175359	+	0.984505i	$0.556109\pi$
$62$	−12.4546	−1.58174
$63$	−3.18712	−0.401540
$64$	3.38839	0.423549
$65$	0	0
$66$	−1.65040	−0.203150
$67$	−9.44951	−1.15444	−0.577221	−	0.816588i	$-0.695863\pi$
$67$	−9.44951	−1.15444	−0.577221	+	0.816588i	$0.695863\pi$
$68$	−0.906541	−0.109934
$69$	6.94178	0.835692
$70$	23.3141	2.78657
$71$	6.30386	0.748131	0.374065	−	0.927402i	$-0.377964\pi$
$71$	6.30386	0.748131	0.374065	+	0.927402i	$0.377964\pi$
$72$	2.10622	0.248221
$73$	1.95737	0.229093	0.114546	−	0.993418i	$-0.463459\pi$
$73$	1.95737	0.229093	0.114546	+	0.993418i	$0.463459\pi$
$74$	−4.16518	−0.484192
$75$	14.6456	1.69112
$76$	4.32377	0.495970
$77$	−3.18712	−0.363206
$78$	0	0
$79$	−9.68221	−1.08933	−0.544667	−	0.838653i	$-0.683344\pi$
$79$	−9.68221	−1.08933	−0.544667	+	0.838653i	$0.683344\pi$
$80$	−21.8236	−2.43995
$81$	1.00000	0.111111
$82$	17.6980	1.95442
$83$	−1.47921	−0.162365	−0.0811823	−	0.996699i	$-0.525870\pi$
$83$	−1.47921	−0.162365	−0.0811823	+	0.996699i	$0.525870\pi$
$84$	−2.30686	−0.251699
$85$	−5.55133	−0.602127
$86$	−1.62265	−0.174974
$87$	5.43785	0.582998
$88$	2.10622	0.224524
$89$	−2.64258	−0.280113	−0.140056	−	0.990144i	$-0.544728\pi$
$89$	−2.64258	−0.280113	−0.140056	+	0.990144i	$0.544728\pi$
$90$	−7.31510	−0.771080
$91$	0	0
$92$	5.02450	0.523841
$93$	7.54645	0.782530
$94$	19.4621	2.00736
$95$	26.4772	2.71651
$96$	3.91363	0.399434
$97$	2.66049	0.270132	0.135066	−	0.990837i	$-0.456875\pi$
$97$	2.66049	0.270132	0.135066	+	0.990837i	$0.456875\pi$
$98$	−5.21155	−0.526446
$99$	1.00000	0.100504
$100$	10.6006	1.06006
$101$	−1.83603	−0.182691	−0.0913457	−	0.995819i	$-0.529117\pi$
$101$	−1.83603	−0.182691	−0.0913457	+	0.995819i	$0.529117\pi$
$102$	2.06706	0.204669
$103$	15.4222	1.51960	0.759798	−	0.650160i	$-0.225298\pi$
$103$	15.4222	1.51960	0.759798	+	0.650160i	$0.225298\pi$
$104$	0	0
$105$	−14.1264	−1.37859
$106$	−11.1153	−1.07961
$107$	9.53132	0.921428	0.460714	−	0.887549i	$-0.347593\pi$
$107$	9.53132	0.921428	0.460714	+	0.887549i	$0.347593\pi$
$108$	0.723807	0.0696483
$109$	−13.1685	−1.26132	−0.630658	−	0.776061i	$-0.717215\pi$
$109$	−13.1685	−1.26132	−0.630658	+	0.776061i	$0.717215\pi$
$110$	−7.31510	−0.697468
$111$	2.52374	0.239543
$112$	15.6925	1.48280
$113$	−10.3278	−0.971562	−0.485781	−	0.874080i	$-0.661465\pi$
$113$	−10.3278	−0.971562	−0.485781	+	0.874080i	$0.661465\pi$
$114$	−9.85889	−0.923370
$115$	30.7683	2.86915
$116$	3.93595	0.365444
$117$	0	0
$118$	4.52064	0.416159
$119$	3.99176	0.365924
$120$	9.33549	0.852209
$121$	1.00000	0.0909091
$122$	4.52076	0.409291
$123$	−10.7235	−0.966907
$124$	5.46217	0.490517
$125$	42.7524	3.82389
$126$	5.26002	0.468599
$127$	14.3158	1.27032	0.635162	−	0.772379i	$-0.280933\pi$
$127$	14.3158	1.27032	0.635162	+	0.772379i	$0.280933\pi$
$128$	−13.4195	−1.18612
$129$	0.983186	0.0865647
$130$	0	0
$131$	2.94078	0.256937	0.128469	−	0.991714i	$-0.458994\pi$
$131$	2.94078	0.256937	0.128469	+	0.991714i	$0.458994\pi$
$132$	0.723807	0.0629993
$133$	−19.0388	−1.65087
$134$	15.5954	1.34724
$135$	4.43233	0.381474
$136$	−2.63797	−0.226204
$137$	−14.8378	−1.26768	−0.633838	−	0.773466i	$-0.718522\pi$
$137$	−14.8378	−1.26768	−0.633838	+	0.773466i	$0.718522\pi$
$138$	−11.4567	−0.975257
$139$	7.48874	0.635186	0.317593	−	0.948227i	$-0.397125\pi$
$139$	7.48874	0.635186	0.317593	+	0.948227i	$0.397125\pi$
$140$	−10.2248	−0.864151
$141$	−11.7924	−0.993095
$142$	−10.4039	−0.873073
$143$	0	0
$144$	−4.92372	−0.410310
$145$	24.1023	2.00159
$146$	−3.23043	−0.267352
$147$	3.15776	0.260447
$148$	1.82670	0.150154
$149$	−10.4158	−0.853297	−0.426649	−	0.904417i	$-0.640306\pi$
$149$	−10.4158	−0.853297	−0.426649	+	0.904417i	$0.640306\pi$
$150$	−24.1710	−1.97355
$151$	−23.0841	−1.87856	−0.939279	−	0.343155i	$-0.888504\pi$
$151$	−23.0841	−1.87856	−0.939279	+	0.343155i	$0.888504\pi$
$152$	12.5819	1.02052
$153$	−1.25246	−0.101256
$154$	5.26002	0.423864
$155$	33.4484	2.68664
$156$	0	0
$157$	−0.240756	−0.0192144	−0.00960721	−	0.999954i	$-0.503058\pi$
$157$	−0.240756	−0.0192144	−0.00960721	+	0.999954i	$0.503058\pi$
$158$	15.9795	1.27126
$159$	6.73492	0.534114
$160$	17.3465	1.37136
$161$	−22.1243	−1.74364
$162$	−1.65040	−0.129667
$163$	14.6327	1.14612	0.573061	−	0.819513i	$-0.305756\pi$
$163$	14.6327	1.14612	0.573061	+	0.819513i	$0.305756\pi$
$164$	−7.76175	−0.606091
$165$	4.43233	0.345057
$166$	2.44129	0.189480
$167$	−2.39357	−0.185220	−0.0926102	−	0.995702i	$-0.529521\pi$
$167$	−2.39357	−0.185220	−0.0926102	+	0.995702i	$0.529521\pi$
$168$	−6.71280	−0.517903
$169$	0	0
$170$	9.16190	0.702686
$171$	5.97365	0.456817
$172$	0.711636	0.0542618
$173$	3.53157	0.268501	0.134250	−	0.990947i	$-0.457137\pi$
$173$	3.53157	0.268501	0.134250	+	0.990947i	$0.457137\pi$
$174$	−8.97460	−0.680363
$175$	−46.6772	−3.52847
$176$	−4.92372	−0.371139
$177$	−2.73913	−0.205885
$178$	4.36130	0.326893
$179$	10.9825	0.820874	0.410437	−	0.911889i	$-0.365376\pi$
$179$	10.9825	0.820874	0.410437	+	0.911889i	$0.365376\pi$
$180$	3.20815	0.239122
$181$	−22.3649	−1.66237	−0.831184	−	0.555998i	$-0.812336\pi$
$181$	−22.3649	−1.66237	−0.831184	+	0.555998i	$0.812336\pi$
$182$	0	0
$183$	−2.73920	−0.202487
$184$	14.6209	1.07787
$185$	11.1861	0.822416
$186$	−12.4546	−0.913218
$187$	−1.25246	−0.0915892
$188$	−8.53538	−0.622507
$189$	−3.18712	−0.231829
$190$	−43.6979	−3.17018
$191$	−4.49076	−0.324940	−0.162470	−	0.986713i	$-0.551946\pi$
$191$	−4.49076	−0.324940	−0.162470	+	0.986713i	$0.551946\pi$
$192$	3.38839	0.244536
$193$	3.84635	0.276866	0.138433	−	0.990372i	$-0.455793\pi$
$193$	3.84635	0.276866	0.138433	+	0.990372i	$0.455793\pi$
$194$	−4.39086	−0.315245
$195$	0	0
$196$	2.28560	0.163257
$197$	8.84721	0.630338	0.315169	−	0.949036i	$-0.397939\pi$
$197$	8.84721	0.630338	0.315169	+	0.949036i	$0.397939\pi$
$198$	−1.65040	−0.117289
$199$	10.3152	0.731226	0.365613	−	0.930767i	$-0.380859\pi$
$199$	10.3152	0.731226	0.365613	+	0.930767i	$0.380859\pi$
$200$	30.8469	2.18120
$201$	−9.44951	−0.666517
$202$	3.03017	0.213202
$203$	−17.3311	−1.21640
$204$	−0.906541	−0.0634706
$205$	−47.5302	−3.31965
$206$	−25.4527	−1.77338
$207$	6.94178	0.482487
$208$	0	0
$209$	5.97365	0.413206
$210$	23.3141	1.60883
$211$	12.2727	0.844889	0.422445	−	0.906389i	$-0.361172\pi$
$211$	12.2727	0.844889	0.422445	+	0.906389i	$0.361172\pi$
$212$	4.87478	0.334801
$213$	6.30386	0.431933
$214$	−15.7305	−1.07531
$215$	4.35781	0.297200
$216$	2.10622	0.143310
$217$	−24.0515	−1.63272
$218$	21.7333	1.47196
$219$	1.95737	0.132267
$220$	3.20815	0.216294
$221$	0	0
$222$	−4.16518	−0.279548
$223$	10.0418	0.672448	0.336224	−	0.941782i	$-0.390850\pi$
$223$	10.0418	0.672448	0.336224	+	0.941782i	$0.390850\pi$
$224$	−12.4732	−0.833403
$225$	14.6456	0.976371
$226$	17.0450	1.13382
$227$	29.2155	1.93910	0.969552	−	0.244886i	$-0.0787507\pi$
$227$	29.2155	1.93910	0.969552	+	0.244886i	$0.0787507\pi$
$228$	4.32377	0.286349
$229$	8.53086	0.563735	0.281867	−	0.959453i	$-0.409046\pi$
$229$	8.53086	0.563735	0.281867	+	0.959453i	$0.409046\pi$
$230$	−50.7798	−3.34832
$231$	−3.18712	−0.209697
$232$	11.4533	0.751948
$233$	21.2941	1.39502	0.697511	−	0.716574i	$-0.254291\pi$
$233$	21.2941	1.39502	0.697511	+	0.716574i	$0.254291\pi$
$234$	0	0
$235$	−52.2676	−3.40956
$236$	−1.98260	−0.129056
$237$	−9.68221	−0.628927
$238$	−6.58798	−0.427035
$239$	17.4852	1.13103	0.565513	−	0.824739i	$-0.308678\pi$
$239$	17.4852	1.13103	0.565513	+	0.824739i	$0.308678\pi$
$240$	−21.8236	−1.40870
$241$	14.4277	0.929369	0.464684	−	0.885476i	$-0.346168\pi$
$241$	14.4277	0.929369	0.464684	+	0.885476i	$0.346168\pi$
$242$	−1.65040	−0.106091
$243$	1.00000	0.0641500
$244$	−1.98265	−0.126926
$245$	13.9962	0.894186
$246$	17.6980	1.12839
$247$	0	0
$248$	15.8945	1.00930
$249$	−1.47921	−0.0937412
$250$	−70.5583	−4.46250
$251$	8.52770	0.538263	0.269132	−	0.963103i	$-0.413263\pi$
$251$	8.52770	0.538263	0.269132	+	0.963103i	$0.413263\pi$
$252$	−2.30686	−0.145319
$253$	6.94178	0.436426
$254$	−23.6268	−1.48248
$255$	−5.55133	−0.347638
$256$	15.3706	0.960664
$257$	13.9339	0.869171	0.434586	−	0.900631i	$-0.356895\pi$
$257$	13.9339	0.869171	0.434586	+	0.900631i	$0.356895\pi$
$258$	−1.62265	−0.101021
$259$	−8.04348	−0.499798
$260$	0	0
$261$	5.43785	0.336594
$262$	−4.85346	−0.299848
$263$	−11.8035	−0.727833	−0.363916	−	0.931432i	$-0.618561\pi$
$263$	−11.8035	−0.727833	−0.363916	+	0.931432i	$0.618561\pi$
$264$	2.10622	0.129629
$265$	29.8514	1.83376
$266$	31.4215	1.92658
$267$	−2.64258	−0.161723
$268$	−6.83962	−0.417796
$269$	19.1324	1.16652	0.583260	−	0.812285i	$-0.301777\pi$
$269$	19.1324	1.16652	0.583260	+	0.812285i	$0.301777\pi$
$270$	−7.31510	−0.445183
$271$	0.254268	0.0154457	0.00772284	−	0.999970i	$-0.497542\pi$
$271$	0.254268	0.0154457	0.00772284	+	0.999970i	$0.497542\pi$
$272$	6.16678	0.373916
$273$	0	0
$274$	24.4882	1.47939
$275$	14.6456	0.883161
$276$	5.02450	0.302440
$277$	−23.0955	−1.38768	−0.693838	−	0.720132i	$-0.744081\pi$
$277$	−23.0955	−1.38768	−0.693838	+	0.720132i	$0.744081\pi$
$278$	−12.3594	−0.741266
$279$	7.54645	0.451794
$280$	−29.7533	−1.77810
$281$	−17.1299	−1.02189	−0.510943	−	0.859615i	$-0.670704\pi$
$281$	−17.1299	−1.02189	−0.510943	+	0.859615i	$0.670704\pi$
$282$	19.4621	1.15895
$283$	−17.1399	−1.01886	−0.509431	−	0.860512i	$-0.670144\pi$
$283$	−17.1399	−1.01886	−0.509431	+	0.860512i	$0.670144\pi$
$284$	4.56278	0.270751
$285$	26.4772	1.56838
$286$	0	0
$287$	34.1772	2.01741
$288$	3.91363	0.230613
$289$	−15.4313	−0.907726
$290$	−39.7784	−2.33587
$291$	2.66049	0.155961
$292$	1.41676	0.0829094
$293$	−21.7632	−1.27142	−0.635711	−	0.771927i	$-0.719293\pi$
$293$	−21.7632	−1.27142	−0.635711	+	0.771927i	$0.719293\pi$
$294$	−5.21155	−0.303944
$295$	−12.1407	−0.706860
$296$	5.31557	0.308961
$297$	1.00000	0.0580259
$298$	17.1902	0.995803
$299$	0	0
$300$	10.6006	0.612024
$301$	−3.13353	−0.180614
$302$	38.0979	2.19229
$303$	−1.83603	−0.105477
$304$	−29.4126	−1.68693
$305$	−12.1410	−0.695194
$306$	2.06706	0.118166
$307$	−4.44106	−0.253465	−0.126732	−	0.991937i	$-0.540449\pi$
$307$	−4.44106	−0.253465	−0.126732	+	0.991937i	$0.540449\pi$
$308$	−2.30686	−0.131446
$309$	15.4222	0.877339
$310$	−55.2031	−3.13532
$311$	−28.0730	−1.59188	−0.795938	−	0.605378i	$-0.793022\pi$
$311$	−28.0730	−1.59188	−0.795938	+	0.605378i	$0.793022\pi$
$312$	0	0
$313$	−2.10380	−0.118914	−0.0594569	−	0.998231i	$-0.518937\pi$
$313$	−2.10380	−0.118914	−0.0594569	+	0.998231i	$0.518937\pi$
$314$	0.397343	0.0224233
$315$	−14.1264	−0.795932
$316$	−7.00805	−0.394233
$317$	−3.12618	−0.175584	−0.0877920	−	0.996139i	$-0.527981\pi$
$317$	−3.12618	−0.175584	−0.0877920	+	0.996139i	$0.527981\pi$
$318$	−11.1153	−0.623314
$319$	5.43785	0.304461
$320$	15.0185	0.839558
$321$	9.53132	0.531986
$322$	36.5138	2.03484
$323$	−7.48178	−0.416298
$324$	0.723807	0.0402115
$325$	0	0
$326$	−24.1498	−1.33753
$327$	−13.1685	−0.728221
$328$	−22.5861	−1.24711
$329$	37.5837	2.07206
$330$	−7.31510	−0.402683
$331$	−17.4451	−0.958867	−0.479434	−	0.877578i	$-0.659158\pi$
$331$	−17.4451	−0.958867	−0.479434	+	0.877578i	$0.659158\pi$
$332$	−1.07066	−0.0587603
$333$	2.52374	0.138300
$334$	3.95035	0.216153
$335$	−41.8834	−2.28833
$336$	15.6925	0.856096
$337$	5.71974	0.311574	0.155787	−	0.987791i	$-0.450209\pi$
$337$	5.71974	0.311574	0.155787	+	0.987791i	$0.450209\pi$
$338$	0	0
$339$	−10.3278	−0.560932
$340$	−4.01809	−0.217912
$341$	7.54645	0.408663
$342$	−9.85889	−0.533108
$343$	12.2457	0.661206
$344$	2.07081	0.111651
$345$	30.7683	1.65651
$346$	−5.82850	−0.313342
$347$	31.9655	1.71600	0.857999	−	0.513652i	$-0.171708\pi$
$347$	31.9655	1.71600	0.857999	+	0.513652i	$0.171708\pi$
$348$	3.93595	0.210989
$349$	−29.0972	−1.55754	−0.778770	−	0.627310i	$-0.784156\pi$
$349$	−29.0972	−1.55754	−0.778770	+	0.627310i	$0.784156\pi$
$350$	77.0359	4.11774
$351$	0	0
$352$	3.91363	0.208597
$353$	−21.6871	−1.15429	−0.577145	−	0.816642i	$-0.695833\pi$
$353$	−21.6871	−1.15429	−0.577145	+	0.816642i	$0.695833\pi$
$354$	4.52064	0.240269
$355$	27.9408	1.48294
$356$	−1.91272	−0.101374
$357$	3.99176	0.211266
$358$	−18.1256	−0.957965
$359$	9.84088	0.519382	0.259691	−	0.965692i	$-0.416379\pi$
$359$	9.84088	0.519382	0.259691	+	0.965692i	$0.416379\pi$
$360$	9.33549	0.492023
$361$	16.6845	0.878133
$362$	36.9109	1.93999
$363$	1.00000	0.0524864
$364$	0	0
$365$	8.67570	0.454107
$366$	4.52076	0.236304
$367$	10.0567	0.524958	0.262479	−	0.964938i	$-0.415460\pi$
$367$	10.0567	0.524958	0.262479	+	0.964938i	$0.415460\pi$
$368$	−34.1793	−1.78172
$369$	−10.7235	−0.558244
$370$	−18.4614	−0.959765
$371$	−21.4650	−1.11441
$372$	5.46217	0.283200
$373$	−5.89395	−0.305177	−0.152588	−	0.988290i	$-0.548761\pi$
$373$	−5.89395	−0.305177	−0.152588	+	0.988290i	$0.548761\pi$
$374$	2.06706	0.106885
$375$	42.7524	2.20772
$376$	−24.8373	−1.28089
$377$	0	0
$378$	5.26002	0.270546
$379$	−2.83571	−0.145661	−0.0728303	−	0.997344i	$-0.523203\pi$
$379$	−2.83571	−0.145661	−0.0728303	+	0.997344i	$0.523203\pi$
$380$	19.1644	0.983112
$381$	14.3158	0.733422
$382$	7.41153	0.379207
$383$	6.80793	0.347869	0.173935	−	0.984757i	$-0.444352\pi$
$383$	6.80793	0.347869	0.173935	+	0.984757i	$0.444352\pi$
$384$	−13.4195	−0.684808
$385$	−14.1264	−0.719948
$386$	−6.34799	−0.323104
$387$	0.983186	0.0499781
$388$	1.92568	0.0977616
$389$	−26.6217	−1.34978	−0.674888	−	0.737920i	$-0.735808\pi$
$389$	−26.6217	−1.34978	−0.674888	+	0.737920i	$0.735808\pi$
$390$	0	0
$391$	−8.69432	−0.439691
$392$	6.65094	0.335923
$393$	2.94078	0.148343
$394$	−14.6014	−0.735608
$395$	−42.9148	−2.15928
$396$	0.723807	0.0363727
$397$	11.9046	0.597473	0.298736	−	0.954336i	$-0.403435\pi$
$397$	11.9046	0.597473	0.298736	+	0.954336i	$0.403435\pi$
$398$	−17.0242	−0.853345
$399$	−19.0388	−0.953131
$400$	−72.1107	−3.60553
$401$	−32.2967	−1.61282	−0.806410	−	0.591357i	$-0.798593\pi$
$401$	−32.2967	−1.61282	−0.806410	+	0.591357i	$0.798593\pi$
$402$	15.5954	0.777829
$403$	0	0
$404$	−1.32893	−0.0661167
$405$	4.43233	0.220244
$406$	28.6032	1.41955
$407$	2.52374	0.125097
$408$	−2.63797	−0.130599
$409$	16.7711	0.829277	0.414639	−	0.909986i	$-0.363908\pi$
$409$	16.7711	0.829277	0.414639	+	0.909986i	$0.363908\pi$
$410$	78.4436	3.87405
$411$	−14.8378	−0.731893
$412$	11.1627	0.549947
$413$	8.72993	0.429572
$414$	−11.4567	−0.563065
$415$	−6.55636	−0.321839
$416$	0	0
$417$	7.48874	0.366725
$418$	−9.85889	−0.482214
$419$	0.367411	0.0179492	0.00897461	−	0.999960i	$-0.497143\pi$
$419$	0.367411	0.0179492	0.00897461	+	0.999960i	$0.497143\pi$
$420$	−10.2248	−0.498918
$421$	12.5507	0.611685	0.305843	−	0.952082i	$-0.401062\pi$
$421$	12.5507	0.611685	0.305843	+	0.952082i	$0.401062\pi$
$422$	−20.2549	−0.985991
$423$	−11.7924	−0.573364
$424$	14.1852	0.688896
$425$	−18.3430	−0.889768
$426$	−10.4039	−0.504069
$427$	8.73017	0.422482
$428$	6.89883	0.333468
$429$	0	0
$430$	−7.19210	−0.346834
$431$	−2.40315	−0.115756	−0.0578779	−	0.998324i	$-0.518433\pi$
$431$	−2.40315	−0.115756	−0.0578779	+	0.998324i	$0.518433\pi$
$432$	−4.92372	−0.236892
$433$	21.0141	1.00987	0.504937	−	0.863156i	$-0.331516\pi$
$433$	21.0141	1.00987	0.504937	+	0.863156i	$0.331516\pi$
$434$	39.6944	1.90539
$435$	24.1023	1.15562
$436$	−9.53147	−0.456474
$437$	41.4678	1.98367
$438$	−3.23043	−0.154356
$439$	20.6690	0.986479	0.493240	−	0.869893i	$-0.335813\pi$
$439$	20.6690	0.986479	0.493240	+	0.869893i	$0.335813\pi$
$440$	9.33549	0.445052
$441$	3.15776	0.150369
$442$	0	0
$443$	13.5345	0.643042	0.321521	−	0.946903i	$-0.395806\pi$
$443$	13.5345	0.643042	0.321521	+	0.946903i	$0.395806\pi$
$444$	1.82670	0.0866915
$445$	−11.7128	−0.555239
$446$	−16.5729	−0.784751
$447$	−10.4158	−0.492651
$448$	−10.7992	−0.510215
$449$	36.4031	1.71797	0.858983	−	0.512004i	$-0.171097\pi$
$449$	36.4031	1.71797	0.858983	+	0.512004i	$0.171097\pi$
$450$	−24.1710	−1.13943
$451$	−10.7235	−0.504951
$452$	−7.47537	−0.351612
$453$	−23.0841	−1.08459
$454$	−48.2172	−2.26295
$455$	0	0
$456$	12.5819	0.589199
$457$	32.0495	1.49921	0.749607	−	0.661883i	$-0.230243\pi$
$457$	32.0495	1.49921	0.749607	+	0.661883i	$0.230243\pi$
$458$	−14.0793	−0.657882
$459$	−1.25246	−0.0584600
$460$	22.2703	1.03836
$461$	−1.30249	−0.0606632	−0.0303316	−	0.999540i	$-0.509656\pi$
$461$	−1.30249	−0.0606632	−0.0303316	+	0.999540i	$0.509656\pi$
$462$	5.26002	0.244718
$463$	0.658424	0.0305995	0.0152998	−	0.999883i	$-0.495130\pi$
$463$	0.658424	0.0305995	0.0152998	+	0.999883i	$0.495130\pi$
$464$	−26.7744	−1.24297
$465$	33.4484	1.55113
$466$	−35.1437	−1.62800
$467$	−11.5932	−0.536468	−0.268234	−	0.963354i	$-0.586440\pi$
$467$	−11.5932	−0.536468	−0.268234	+	0.963354i	$0.586440\pi$
$468$	0	0
$469$	30.1168	1.39066
$470$	86.2623	3.97898
$471$	−0.240756	−0.0110934
$472$	−5.76921	−0.265550
$473$	0.983186	0.0452069
$474$	15.9795	0.733962
$475$	87.4876	4.01420
$476$	2.88926	0.132429
$477$	6.73492	0.308371
$478$	−28.8576	−1.31991
$479$	−18.4567	−0.843308	−0.421654	−	0.906757i	$-0.638550\pi$
$479$	−18.4567	−0.843308	−0.421654	+	0.906757i	$0.638550\pi$
$480$	17.3465	0.791757
$481$	0	0
$482$	−23.8114	−1.08458
$483$	−22.1243	−1.00669
$484$	0.723807	0.0329003
$485$	11.7922	0.535455
$486$	−1.65040	−0.0748635
$487$	−28.2014	−1.27793	−0.638964	−	0.769236i	$-0.720637\pi$
$487$	−28.2014	−1.27793	−0.638964	+	0.769236i	$0.720637\pi$
$488$	−5.76937	−0.261167
$489$	14.6327	0.661714
$490$	−23.0993	−1.04352
$491$	−25.2708	−1.14046	−0.570229	−	0.821486i	$-0.693145\pi$
$491$	−25.2708	−1.14046	−0.570229	+	0.821486i	$0.693145\pi$
$492$	−7.76175	−0.349927
$493$	−6.81071	−0.306739
$494$	0	0
$495$	4.43233	0.199219
$496$	−37.1566	−1.66838
$497$	−20.0912	−0.901213
$498$	2.44129	0.109397
$499$	9.26437	0.414730	0.207365	−	0.978264i	$-0.433511\pi$
$499$	9.26437	0.414730	0.207365	+	0.978264i	$0.433511\pi$
$500$	30.9445	1.38388
$501$	−2.39357	−0.106937
$502$	−14.0741	−0.628157
$503$	22.1964	0.989689	0.494844	−	0.868982i	$-0.335225\pi$
$503$	22.1964	0.989689	0.494844	+	0.868982i	$0.335225\pi$
$504$	−6.71280	−0.299012
$505$	−8.13788	−0.362131
$506$	−11.4567	−0.509311
$507$	0	0
$508$	10.3619	0.459734
$509$	−1.38567	−0.0614187	−0.0307093	−	0.999528i	$-0.509777\pi$
$509$	−1.38567	−0.0614187	−0.0307093	+	0.999528i	$0.509777\pi$
$510$	9.16190	0.405696
$511$	−6.23837	−0.275969
$512$	1.47128	0.0650219
$513$	5.97365	0.263743
$514$	−22.9964	−1.01433
$515$	68.3563	3.01214
$516$	0.711636	0.0313280
$517$	−11.7924	−0.518627
$518$	13.2749	0.583267
$519$	3.53157	0.155019
$520$	0	0
$521$	21.7583	0.953246	0.476623	−	0.879108i	$-0.341861\pi$
$521$	21.7583	0.953246	0.476623	+	0.879108i	$0.341861\pi$
$522$	−8.97460	−0.392808
$523$	6.99094	0.305693	0.152846	−	0.988250i	$-0.451156\pi$
$523$	6.99094	0.305693	0.152846	+	0.988250i	$0.451156\pi$
$524$	2.12856	0.0929865
$525$	−46.6772	−2.03716
$526$	19.4804	0.849385
$527$	−9.45165	−0.411721
$528$	−4.92372	−0.214277
$529$	25.1882	1.09514
$530$	−49.2666	−2.14001
$531$	−2.73913	−0.118868
$532$	−13.7804	−0.597456
$533$	0	0
$534$	4.36130	0.188732
$535$	42.2460	1.82645
$536$	−19.9028	−0.859669
$537$	10.9825	0.473932
$538$	−31.5760	−1.36134
$539$	3.15776	0.136014
$540$	3.20815	0.138057
$541$	0.121298	0.00521501	0.00260750	−	0.999997i	$-0.499170\pi$
$541$	0.121298	0.00521501	0.00260750	+	0.999997i	$0.499170\pi$
$542$	−0.419643	−0.0180252
$543$	−22.3649	−0.959768
$544$	−4.90168	−0.210158
$545$	−58.3673	−2.50018
$546$	0	0
$547$	−38.9915	−1.66716	−0.833578	−	0.552402i	$-0.813711\pi$
$547$	−38.9915	−1.66716	−0.833578	+	0.552402i	$0.813711\pi$
$548$	−10.7397	−0.458776
$549$	−2.73920	−0.116906
$550$	−24.1710	−1.03065
$551$	32.4838	1.38386
$552$	14.6209	0.622308
$553$	30.8584	1.31223
$554$	38.1167	1.61943
$555$	11.1861	0.474822
$556$	5.42040	0.229876
$557$	−8.73146	−0.369964	−0.184982	−	0.982742i	$-0.559223\pi$
$557$	−8.73146	−0.369964	−0.184982	+	0.982742i	$0.559223\pi$
$558$	−12.4546	−0.527247
$559$	0	0
$560$	69.5544	2.93921
$561$	−1.25246	−0.0528790
$562$	28.2712	1.19255
$563$	−33.1165	−1.39569	−0.697846	−	0.716247i	$-0.745858\pi$
$563$	−33.1165	−1.39569	−0.697846	+	0.716247i	$0.745858\pi$
$564$	−8.53538	−0.359405
$565$	−45.7765	−1.92583
$566$	28.2876	1.18902
$567$	−3.18712	−0.133847
$568$	13.2773	0.557105
$569$	5.38670	0.225822	0.112911	−	0.993605i	$-0.463982\pi$
$569$	5.38670	0.225822	0.112911	+	0.993605i	$0.463982\pi$
$570$	−43.6979	−1.83030
$571$	10.8260	0.453053	0.226526	−	0.974005i	$-0.427263\pi$
$571$	10.8260	0.453053	0.226526	+	0.974005i	$0.427263\pi$
$572$	0	0
$573$	−4.49076	−0.187604
$574$	−56.4059	−2.35434
$575$	101.666	4.23978
$576$	3.38839	0.141183
$577$	−10.8717	−0.452596	−0.226298	−	0.974058i	$-0.572662\pi$
$577$	−10.8717	−0.452596	−0.226298	+	0.974058i	$0.572662\pi$
$578$	25.4678	1.05932
$579$	3.84635	0.159849
$580$	17.4454	0.724383
$581$	4.71443	0.195588
$582$	−4.39086	−0.182007
$583$	6.73492	0.278932
$584$	4.12266	0.170597
$585$	0	0
$586$	35.9180	1.48376
$587$	38.6108	1.59364	0.796820	−	0.604217i	$-0.206514\pi$
$587$	38.6108	1.59364	0.796820	+	0.604217i	$0.206514\pi$
$588$	2.28560	0.0942567
$589$	45.0799	1.85748
$590$	20.0370	0.824910
$591$	8.84721	0.363926
$592$	−12.4262	−0.510714
$593$	35.6179	1.46265	0.731326	−	0.682028i	$-0.238902\pi$
$593$	35.6179	1.46265	0.731326	+	0.682028i	$0.238902\pi$
$594$	−1.65040	−0.0677166
$595$	17.6928	0.725334
$596$	−7.53904	−0.308811
$597$	10.3152	0.422174
$598$	0	0
$599$	43.1866	1.76456	0.882278	−	0.470729i	$-0.156009\pi$
$599$	43.1866	1.76456	0.882278	+	0.470729i	$0.156009\pi$
$600$	30.8469	1.25932
$601$	−21.7867	−0.888697	−0.444348	−	0.895854i	$-0.646565\pi$
$601$	−21.7867	−0.888697	−0.444348	+	0.895854i	$0.646565\pi$
$602$	5.17157	0.210778
$603$	−9.44951	−0.384814
$604$	−16.7084	−0.679856
$605$	4.43233	0.180200
$606$	3.03017	0.123092
$607$	6.73646	0.273424	0.136712	−	0.990611i	$-0.456346\pi$
$607$	6.73646	0.273424	0.136712	+	0.990611i	$0.456346\pi$
$608$	23.3787	0.948131
$609$	−17.3311	−0.702291
$610$	20.0375	0.811296
$611$	0	0
$612$	−0.906541	−0.0366448
$613$	−11.3947	−0.460226	−0.230113	−	0.973164i	$-0.573910\pi$
$613$	−11.3947	−0.460226	−0.230113	+	0.973164i	$0.573910\pi$
$614$	7.32951	0.295795
$615$	−47.5302	−1.91660
$616$	−6.71280	−0.270466
$617$	13.5499	0.545497	0.272749	−	0.962085i	$-0.412067\pi$
$617$	13.5499	0.545497	0.272749	+	0.962085i	$0.412067\pi$
$618$	−25.4527	−1.02386
$619$	33.6655	1.35313	0.676565	−	0.736383i	$-0.263468\pi$
$619$	33.6655	1.35313	0.676565	+	0.736383i	$0.263468\pi$
$620$	24.2102	0.972303
$621$	6.94178	0.278564
$622$	46.3316	1.85773
$623$	8.42222	0.337429
$624$	0	0
$625$	116.265	4.65060
$626$	3.47210	0.138773
$627$	5.97365	0.238565
$628$	−0.174261	−0.00695376
$629$	−3.16090	−0.126033
$630$	23.3141	0.928857
$631$	−14.6760	−0.584242	−0.292121	−	0.956381i	$-0.594361\pi$
$631$	−14.6760	−0.584242	−0.292121	+	0.956381i	$0.594361\pi$
$632$	−20.3929	−0.811186
$633$	12.2727	0.487797
$634$	5.15944	0.204908
$635$	63.4525	2.51803
$636$	4.87478	0.193298
$637$	0	0
$638$	−8.97460	−0.355308
$639$	6.30386	0.249377
$640$	−59.4795	−2.35113
$641$	24.8052	0.979746	0.489873	−	0.871794i	$-0.337043\pi$
$641$	24.8052	0.979746	0.489873	+	0.871794i	$0.337043\pi$
$642$	−15.7305	−0.620831
$643$	28.5738	1.12684	0.563421	−	0.826170i	$-0.309485\pi$
$643$	28.5738	1.12684	0.563421	+	0.826170i	$0.309485\pi$
$644$	−16.0137	−0.631029
$645$	4.35781	0.171588
$646$	12.3479	0.485822
$647$	−42.4198	−1.66769	−0.833847	−	0.551996i	$-0.813866\pi$
$647$	−42.4198	−1.66769	−0.833847	+	0.551996i	$0.813866\pi$
$648$	2.10622	0.0827403
$649$	−2.73913	−0.107520
$650$	0	0
$651$	−24.0515	−0.942651
$652$	10.5912	0.414785
$653$	23.6662	0.926131	0.463066	−	0.886324i	$-0.346749\pi$
$653$	23.6662	0.926131	0.463066	+	0.886324i	$0.346749\pi$
$654$	21.7333	0.849838
$655$	13.0345	0.509301
$656$	52.7996	2.06148
$657$	1.95737	0.0763642
$658$	−62.0280	−2.41810
$659$	−5.14661	−0.200483	−0.100242	−	0.994963i	$-0.531962\pi$
$659$	−5.14661	−0.200483	−0.100242	+	0.994963i	$0.531962\pi$
$660$	3.20815	0.124877
$661$	4.22464	0.164319	0.0821596	−	0.996619i	$-0.473818\pi$
$661$	4.22464	0.164319	0.0821596	+	0.996619i	$0.473818\pi$
$662$	28.7913	1.11900
$663$	0	0
$664$	−3.11555	−0.120907
$665$	−84.3862	−3.27235
$666$	−4.16518	−0.161397
$667$	37.7483	1.46162
$668$	−1.73249	−0.0670319
$669$	10.0418	0.388238
$670$	69.1241	2.67050
$671$	−2.73920	−0.105746
$672$	−12.4732	−0.481166
$673$	19.4722	0.750598	0.375299	−	0.926904i	$-0.377540\pi$
$673$	19.4722	0.750598	0.375299	+	0.926904i	$0.377540\pi$
$674$	−9.43984	−0.363609
$675$	14.6456	0.563708
$676$	0	0
$677$	−2.58030	−0.0991691	−0.0495845	−	0.998770i	$-0.515790\pi$
$677$	−2.58030	−0.0991691	−0.0495845	+	0.998770i	$0.515790\pi$
$678$	17.0450	0.654611
$679$	−8.47931	−0.325406
$680$	−11.6924	−0.448381
$681$	29.2155	1.11954
$682$	−12.4546	−0.476912
$683$	−13.4752	−0.515615	−0.257808	−	0.966196i	$-0.583000\pi$
$683$	−13.4752	−0.515615	−0.257808	+	0.966196i	$0.583000\pi$
$684$	4.32377	0.165323
$685$	−65.7659	−2.51279
$686$	−20.2103	−0.771632
$687$	8.53086	0.325473
$688$	−4.84093	−0.184559
$689$	0	0
$690$	−50.7798	−1.93315
$691$	5.19075	0.197465	0.0987327	−	0.995114i	$-0.468521\pi$
$691$	5.19075	0.197465	0.0987327	+	0.995114i	$0.468521\pi$
$692$	2.55618	0.0971713
$693$	−3.18712	−0.121069
$694$	−52.7557	−2.00258
$695$	33.1926	1.25907
$696$	11.4533	0.434137
$697$	13.4308	0.508728
$698$	48.0220	1.81766
$699$	21.2941	0.805416
$700$	−33.7853	−1.27696
$701$	−28.2227	−1.06596	−0.532978	−	0.846129i	$-0.678927\pi$
$701$	−28.2227	−1.06596	−0.532978	+	0.846129i	$0.678927\pi$
$702$	0	0
$703$	15.0760	0.568601
$704$	3.38839	0.127705
$705$	−52.2676	−1.96851
$706$	35.7924	1.34706
$707$	5.85164	0.220074
$708$	−1.98260	−0.0745106
$709$	−29.3152	−1.10095	−0.550477	−	0.834850i	$-0.685554\pi$
$709$	−29.3152	−1.10095	−0.550477	+	0.834850i	$0.685554\pi$
$710$	−46.1134	−1.73060
$711$	−9.68221	−0.363111
$712$	−5.56586	−0.208590
$713$	52.3858	1.96186
$714$	−6.58798	−0.246549
$715$	0	0
$716$	7.94924	0.297077
$717$	17.4852	0.652998
$718$	−16.2414	−0.606122
$719$	19.7683	0.737233	0.368616	−	0.929582i	$-0.379832\pi$
$719$	19.7683	0.737233	0.368616	+	0.929582i	$0.379832\pi$
$720$	−21.8236	−0.813316
$721$	−49.1525	−1.83053
$722$	−27.5361	−1.02479
$723$	14.4277	0.536571
$724$	−16.1878	−0.601616
$725$	79.6404	2.95777
$726$	−1.65040	−0.0612519
$727$	−5.14200	−0.190706	−0.0953531	−	0.995444i	$-0.530398\pi$
$727$	−5.14200	−0.190706	−0.0953531	+	0.995444i	$0.530398\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	−14.3183	−0.529946
$731$	−1.23140	−0.0455451
$732$	−1.98265	−0.0732809
$733$	−5.00582	−0.184894	−0.0924470	−	0.995718i	$-0.529469\pi$
$733$	−5.00582	−0.184894	−0.0924470	+	0.995718i	$0.529469\pi$
$734$	−16.5976	−0.612629
$735$	13.9962	0.516258
$736$	27.1676	1.00141
$737$	−9.44951	−0.348077
$738$	17.6980	0.651474
$739$	−19.2823	−0.709309	−0.354655	−	0.934997i	$-0.615402\pi$
$739$	−19.2823	−0.709309	−0.354655	+	0.934997i	$0.615402\pi$
$740$	8.09655	0.297635
$741$	0	0
$742$	35.4258	1.30052
$743$	18.5891	0.681969	0.340985	−	0.940069i	$-0.389240\pi$
$743$	18.5891	0.681969	0.340985	+	0.940069i	$0.389240\pi$
$744$	15.8945	0.582721
$745$	−46.1664	−1.69141
$746$	9.72734	0.356143
$747$	−1.47921	−0.0541215
$748$	−0.906541	−0.0331464
$749$	−30.3775	−1.10997
$750$	−70.5583	−2.57643
$751$	−32.1508	−1.17320	−0.586599	−	0.809878i	$-0.699533\pi$
$751$	−32.1508	−1.17320	−0.586599	+	0.809878i	$0.699533\pi$
$752$	58.0622	2.11731
$753$	8.52770	0.310766
$754$	0	0
$755$	−102.316	−3.72368
$756$	−2.30686	−0.0838997
$757$	−33.9315	−1.23326	−0.616630	−	0.787253i	$-0.711502\pi$
$757$	−33.9315	−1.23326	−0.616630	+	0.787253i	$0.711502\pi$
$758$	4.68004	0.169987
$759$	6.94178	0.251970
$760$	55.7670	2.02288
$761$	−25.2771	−0.916293	−0.458146	−	0.888877i	$-0.651486\pi$
$761$	−25.2771	−0.916293	−0.458146	+	0.888877i	$0.651486\pi$
$762$	−23.6268	−0.855908
$763$	41.9697	1.51941
$764$	−3.25044	−0.117597
$765$	−5.55133	−0.200709
$766$	−11.2358	−0.405965
$767$	0	0
$768$	15.3706	0.554640
$769$	−43.1209	−1.55498	−0.777489	−	0.628897i	$-0.783507\pi$
$769$	−43.1209	−1.55498	−0.777489	+	0.628897i	$0.783507\pi$
$770$	23.3141	0.840183
$771$	13.9339	0.501816
$772$	2.78401	0.100199
$773$	−24.5523	−0.883086	−0.441543	−	0.897240i	$-0.645569\pi$
$773$	−24.5523	−0.883086	−0.441543	+	0.897240i	$0.645569\pi$
$774$	−1.62265	−0.0583248
$775$	110.522	3.97007
$776$	5.60359	0.201157
$777$	−8.04348	−0.288558
$778$	43.9364	1.57520
$779$	−64.0586	−2.29514
$780$	0	0
$781$	6.30386	0.225570
$782$	14.3491	0.513122
$783$	5.43785	0.194333
$784$	−15.5479	−0.555282
$785$	−1.06711	−0.0380868
$786$	−4.85346	−0.173117
$787$	32.1946	1.14761	0.573807	−	0.818991i	$-0.305466\pi$
$787$	32.1946	1.14761	0.573807	+	0.818991i	$0.305466\pi$
$788$	6.40367	0.228121
$789$	−11.8035	−0.420214
$790$	70.8264	2.51989
$791$	32.9161	1.17036
$792$	2.10622	0.0748414
$793$	0	0
$794$	−19.6472	−0.697255
$795$	29.8514	1.05872
$796$	7.46622	0.264633
$797$	13.5444	0.479768	0.239884	−	0.970802i	$-0.422891\pi$
$797$	13.5444	0.479768	0.239884	+	0.970802i	$0.422891\pi$
$798$	31.4215	1.11231
$799$	14.7695	0.522507
$800$	57.3174	2.02648
$801$	−2.64258	−0.0933709
$802$	53.3023	1.88217
$803$	1.95737	0.0690740
$804$	−6.83962	−0.241215
$805$	−98.0622	−3.45624
$806$	0	0
$807$	19.1324	0.673491
$808$	−3.86708	−0.136044
$809$	21.4014	0.752431	0.376216	−	0.926532i	$-0.377225\pi$
$809$	21.4014	0.752431	0.376216	+	0.926532i	$0.377225\pi$
$810$	−7.31510	−0.257027
$811$	−13.5855	−0.477053	−0.238526	−	0.971136i	$-0.576664\pi$
$811$	−13.5855	−0.477053	−0.238526	+	0.971136i	$0.576664\pi$
$812$	−12.5444	−0.440221
$813$	0.254268	0.00891757
$814$	−4.16518	−0.145989
$815$	64.8570	2.27184
$816$	6.16678	0.215880
$817$	5.87321	0.205478
$818$	−27.6789	−0.967772
$819$	0	0
$820$	−34.4027	−1.20139
$821$	49.1005	1.71362	0.856809	−	0.515634i	$-0.172443\pi$
$821$	49.1005	1.71362	0.856809	+	0.515634i	$0.172443\pi$
$822$	24.4882	0.854124
$823$	−37.7611	−1.31627	−0.658135	−	0.752900i	$-0.728654\pi$
$823$	−37.7611	−1.31627	−0.658135	+	0.752900i	$0.728654\pi$
$824$	32.4826	1.13159
$825$	14.6456	0.509893
$826$	−14.4078	−0.501313
$827$	−10.6306	−0.369664	−0.184832	−	0.982770i	$-0.559174\pi$
$827$	−10.6306	−0.369664	−0.184832	+	0.982770i	$0.559174\pi$
$828$	5.02450	0.174614
$829$	−10.5177	−0.365294	−0.182647	−	0.983179i	$-0.558467\pi$
$829$	−10.5177	−0.365294	−0.182647	+	0.983179i	$0.558467\pi$
$830$	10.8206	0.375588
$831$	−23.0955	−0.801175
$832$	0	0
$833$	−3.95497	−0.137032
$834$	−12.3594	−0.427970
$835$	−10.6091	−0.367144
$836$	4.32377	0.149541
$837$	7.54645	0.260843
$838$	−0.606374	−0.0209469
$839$	−40.2298	−1.38889	−0.694444	−	0.719546i	$-0.744350\pi$
$839$	−40.2298	−1.38889	−0.694444	+	0.719546i	$0.744350\pi$
$840$	−29.7533	−1.02659
$841$	0.570188	0.0196616
$842$	−20.7137	−0.713841
$843$	−17.1299	−0.589986
$844$	8.88308	0.305768
$845$	0	0
$846$	19.4621	0.669119
$847$	−3.18712	−0.109511
$848$	−33.1608	−1.13875
$849$	−17.1399	−0.588240
$850$	30.2733	1.03837
$851$	17.5193	0.600553
$852$	4.56278	0.156318
$853$	−9.34653	−0.320019	−0.160009	−	0.987115i	$-0.551152\pi$
$853$	−9.34653	−0.320019	−0.160009	+	0.987115i	$0.551152\pi$
$854$	−14.4082	−0.493040
$855$	26.4772	0.905502
$856$	20.0751	0.686153
$857$	−16.2386	−0.554698	−0.277349	−	0.960769i	$-0.589456\pi$
$857$	−16.2386	−0.554698	−0.277349	+	0.960769i	$0.589456\pi$
$858$	0	0
$859$	−4.38925	−0.149759	−0.0748796	−	0.997193i	$-0.523857\pi$
$859$	−4.38925	−0.149759	−0.0748796	+	0.997193i	$0.523857\pi$
$860$	3.15421	0.107558
$861$	34.1772	1.16475
$862$	3.96615	0.135088
$863$	−28.3198	−0.964018	−0.482009	−	0.876166i	$-0.660093\pi$
$863$	−28.3198	−0.964018	−0.482009	+	0.876166i	$0.660093\pi$
$864$	3.91363	0.133145
$865$	15.6531	0.532222
$866$	−34.6816	−1.17853
$867$	−15.4313	−0.524076
$868$	−17.4086	−0.590887
$869$	−9.68221	−0.328446
$870$	−39.7784	−1.34861
$871$	0	0
$872$	−27.7359	−0.939255
$873$	2.66049	0.0900439
$874$	−68.4382	−2.31496
$875$	−136.257	−4.60633
$876$	1.41676	0.0478677
$877$	16.3947	0.553610	0.276805	−	0.960926i	$-0.410724\pi$
$877$	16.3947	0.553610	0.276805	+	0.960926i	$0.410724\pi$
$878$	−34.1121	−1.15123
$879$	−21.7632	−0.734056
$880$	−21.8236	−0.735672
$881$	−40.0554	−1.34950	−0.674750	−	0.738047i	$-0.735748\pi$
$881$	−40.0554	−1.34950	−0.674750	+	0.738047i	$0.735748\pi$
$882$	−5.21155	−0.175482
$883$	−19.0596	−0.641405	−0.320703	−	0.947180i	$-0.603919\pi$
$883$	−19.0596	−0.641405	−0.320703	+	0.947180i	$0.603919\pi$
$884$	0	0
$885$	−12.1407	−0.408106
$886$	−22.3372	−0.750434
$887$	−21.8486	−0.733604	−0.366802	−	0.930299i	$-0.619547\pi$
$887$	−21.8486	−0.733604	−0.366802	+	0.930299i	$0.619547\pi$
$888$	5.31557	0.178379
$889$	−45.6263	−1.53026
$890$	19.3307	0.647968
$891$	1.00000	0.0335013
$892$	7.26831	0.243361
$893$	−70.4434	−2.35730
$894$	17.1902	0.574927
$895$	48.6783	1.62714
$896$	42.7694	1.42883
$897$	0	0
$898$	−60.0795	−2.00488
$899$	41.0364	1.36864
$900$	10.6006	0.353352
$901$	−8.43524	−0.281019
$902$	17.6980	0.589281
$903$	−3.13353	−0.104277
$904$	−21.7528	−0.723486
$905$	−99.1285	−3.29514
$906$	38.0979	1.26572
$907$	30.6376	1.01731	0.508653	−	0.860972i	$-0.330144\pi$
$907$	30.6376	1.01731	0.508653	+	0.860972i	$0.330144\pi$
$908$	21.1464	0.701768
$909$	−1.83603	−0.0608972
$910$	0	0
$911$	24.5945	0.814853	0.407426	−	0.913238i	$-0.366426\pi$
$911$	24.5945	0.814853	0.407426	+	0.913238i	$0.366426\pi$
$912$	−29.4126	−0.973948
$913$	−1.47921	−0.0489548
$914$	−52.8944	−1.74959
$915$	−12.1410	−0.401371
$916$	6.17469	0.204018
$917$	−9.37264	−0.309512
$918$	2.06706	0.0682232
$919$	36.9753	1.21970	0.609852	−	0.792515i	$-0.291229\pi$
$919$	36.9753	1.21970	0.609852	+	0.792515i	$0.291229\pi$
$920$	64.8048	2.13655
$921$	−4.44106	−0.146338
$922$	2.14963	0.0707943
$923$	0	0
$924$	−2.30686	−0.0758902
$925$	36.9617	1.21529
$926$	−1.08666	−0.0357099
$927$	15.4222	0.506532
$928$	21.2817	0.698608
$929$	−38.9335	−1.27737	−0.638684	−	0.769469i	$-0.720521\pi$
$929$	−38.9335	−1.27737	−0.638684	+	0.769469i	$0.720521\pi$
$930$	−55.2031	−1.81018
$931$	18.8633	0.618221
$932$	15.4128	0.504863
$933$	−28.0730	−0.919070
$934$	19.1333	0.626061
$935$	−5.55133	−0.181548
$936$	0	0
$937$	−53.2612	−1.73997	−0.869983	−	0.493081i	$-0.835870\pi$
$937$	−53.2612	−1.73997	−0.869983	+	0.493081i	$0.835870\pi$
$938$	−49.7046	−1.62291
$939$	−2.10380	−0.0686549
$940$	−37.8317	−1.23393
$941$	−15.3925	−0.501781	−0.250890	−	0.968016i	$-0.580723\pi$
$941$	−15.3925	−0.501781	−0.250890	+	0.968016i	$0.580723\pi$
$942$	0.397343	0.0129461
$943$	−74.4402	−2.42411
$944$	13.4867	0.438954
$945$	−14.1264	−0.459532
$946$	−1.62265	−0.0527568
$947$	16.1174	0.523744	0.261872	−	0.965103i	$-0.415660\pi$
$947$	16.1174	0.523744	0.261872	+	0.965103i	$0.415660\pi$
$948$	−7.00805	−0.227611
$949$	0	0
$950$	−144.389	−4.68460
$951$	−3.12618	−0.101373
$952$	8.40753	0.272490
$953$	−42.9959	−1.39277	−0.696386	−	0.717667i	$-0.745210\pi$
$953$	−42.9959	−1.39277	−0.696386	+	0.717667i	$0.745210\pi$
$954$	−11.1153	−0.359870
$955$	−19.9045	−0.644096
$956$	12.6559	0.409322
$957$	5.43785	0.175781
$958$	30.4609	0.984146
$959$	47.2898	1.52707
$960$	15.0185	0.484719
$961$	25.9489	0.837062
$962$	0	0
$963$	9.53132	0.307143
$964$	10.4429	0.336342
$965$	17.0483	0.548804
$966$	36.5138	1.17481
$967$	−13.9196	−0.447623	−0.223812	−	0.974632i	$-0.571850\pi$
$967$	−13.9196	−0.447623	−0.223812	+	0.974632i	$0.571850\pi$
$968$	2.10622	0.0676966
$969$	−7.48178	−0.240349
$970$	−19.4618	−0.624879
$971$	9.78982	0.314170	0.157085	−	0.987585i	$-0.449790\pi$
$971$	9.78982	0.314170	0.157085	+	0.987585i	$0.449790\pi$
$972$	0.723807	0.0232161
$973$	−23.8675	−0.765158
$974$	46.5435	1.49135
$975$	0	0
$976$	13.4870	0.431710
$977$	−52.1999	−1.67002	−0.835011	−	0.550234i	$-0.814539\pi$
$977$	−52.1999	−1.67002	−0.835011	+	0.550234i	$0.814539\pi$
$978$	−24.1498	−0.772224
$979$	−2.64258	−0.0844572
$980$	10.1306	0.323609
$981$	−13.1685	−0.420439
$982$	41.7069	1.33092
$983$	−5.05229	−0.161143	−0.0805715	−	0.996749i	$-0.525675\pi$
$983$	−5.05229	−0.161143	−0.0805715	+	0.996749i	$0.525675\pi$
$984$	−22.5861	−0.720020
$985$	39.2138	1.24946
$986$	11.2404	0.357966
$987$	37.5837	1.19630
$988$	0	0
$989$	6.82505	0.217024
$990$	−7.31510	−0.232489
$991$	27.1925	0.863798	0.431899	−	0.901922i	$-0.357844\pi$
$991$	27.1925	0.863798	0.431899	+	0.901922i	$0.357844\pi$
$992$	29.5340	0.937707
$993$	−17.4451	−0.553602
$994$	33.1584	1.05172
$995$	45.7205	1.44944
$996$	−1.07066	−0.0339253
$997$	30.8689	0.977628	0.488814	−	0.872388i	$-0.337430\pi$
$997$	30.8689	0.977628	0.488814	+	0.872388i	$0.337430\pi$
$998$	−15.2899	−0.483993
$999$	2.52374	0.0798477

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5577.2.a.bi.1.5	✓	18
13.12	even	2		5577.2.a.bk.1.14	yes	18

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5577.2.a.bi.1.5	✓	18	1.1	even	1	trivial
5577.2.a.bk.1.14	yes	18	13.12	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:	\( N \)	\(=\)	\( 5577 = 3 \cdot 11 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5577.a (trivial)

\(f(q)\)	\(=\)	\(q-1.65040 q^{2} +1.00000 q^{3} +0.723807 q^{4} +4.43233 q^{5} -1.65040 q^{6} -3.18712 q^{7} +2.10622 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-1.65040 q^{2} +1.00000 q^{3} +0.723807 q^{4} +4.43233 q^{5} -1.65040 q^{6} -3.18712 q^{7} +2.10622 q^{8} +1.00000 q^{9} -7.31510 q^{10} +1.00000 q^{11} +0.723807 q^{12} +5.26002 q^{14} +4.43233 q^{15} -4.92372 q^{16} -1.25246 q^{17} -1.65040 q^{18} +5.97365 q^{19} +3.20815 q^{20} -3.18712 q^{21} -1.65040 q^{22} +6.94178 q^{23} +2.10622 q^{24} +14.6456 q^{25} +1.00000 q^{27} -2.30686 q^{28} +5.43785 q^{29} -7.31510 q^{30} +7.54645 q^{31} +3.91363 q^{32} +1.00000 q^{33} +2.06706 q^{34} -14.1264 q^{35} +0.723807 q^{36} +2.52374 q^{37} -9.85889 q^{38} +9.33549 q^{40} -10.7235 q^{41} +5.26002 q^{42} +0.983186 q^{43} +0.723807 q^{44} +4.43233 q^{45} -11.4567 q^{46} -11.7924 q^{47} -4.92372 q^{48} +3.15776 q^{49} -24.1710 q^{50} -1.25246 q^{51} +6.73492 q^{53} -1.65040 q^{54} +4.43233 q^{55} -6.71280 q^{56} +5.97365 q^{57} -8.97460 q^{58} -2.73913 q^{59} +3.20815 q^{60} -2.73920 q^{61} -12.4546 q^{62} -3.18712 q^{63} +3.38839 q^{64} -1.65040 q^{66} -9.44951 q^{67} -0.906541 q^{68} +6.94178 q^{69} +23.3141 q^{70} +6.30386 q^{71} +2.10622 q^{72} +1.95737 q^{73} -4.16518 q^{74} +14.6456 q^{75} +4.32377 q^{76} -3.18712 q^{77} -9.68221 q^{79} -21.8236 q^{80} +1.00000 q^{81} +17.6980 q^{82} -1.47921 q^{83} -2.30686 q^{84} -5.55133 q^{85} -1.62265 q^{86} +5.43785 q^{87} +2.10622 q^{88} -2.64258 q^{89} -7.31510 q^{90} +5.02450 q^{92} +7.54645 q^{93} +19.4621 q^{94} +26.4772 q^{95} +3.91363 q^{96} +2.66049 q^{97} -5.21155 q^{98} +1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 18 q - q^{2} + 18 q^{3} + 23 q^{4} + 6 q^{5} - q^{6} - q^{7} - 6 q^{8} + 18 q^{9}+O(q^{10}) \) 18 * q - q^2 + 18 * q^3 + 23 * q^4 + 6 * q^5 - q^6 - q^7 - 6 * q^8 + 18 * q^9	\( 18 q - q^{2} + 18 q^{3} + 23 q^{4} + 6 q^{5} - q^{6} - q^{7} - 6 q^{8} + 18 q^{9} - 7 q^{10} + 18 q^{11} + 23 q^{12} + 7 q^{14} + 6 q^{15} + 25 q^{16} + 19 q^{17} - q^{18} + 2 q^{20} - q^{21} - q^{22} + 38 q^{23} - 6 q^{24} + 36 q^{25} + 18 q^{27} - 33 q^{28} + 47 q^{29} - 7 q^{30} + 22 q^{31} - 33 q^{32} + 18 q^{33} - 4 q^{34} + 20 q^{35} + 23 q^{36} + 41 q^{37} + 14 q^{38} - 28 q^{40} - 8 q^{41} + 7 q^{42} + 28 q^{43} + 23 q^{44} + 6 q^{45} - 6 q^{46} - 11 q^{47} + 25 q^{48} + 9 q^{49} + q^{50} + 19 q^{51} + 49 q^{53} - q^{54} + 6 q^{55} + 35 q^{56} + 27 q^{58} + 2 q^{59} + 2 q^{60} - 13 q^{61} + 46 q^{62} - q^{63} + 40 q^{64} - q^{66} - 4 q^{67} + 30 q^{68} + 38 q^{69} + 83 q^{70} - 2 q^{71} - 6 q^{72} - 39 q^{73} + 42 q^{74} + 36 q^{75} - 20 q^{76} - q^{77} + 18 q^{79} - 18 q^{80} + 18 q^{81} - 12 q^{82} - 5 q^{83} - 33 q^{84} - 2 q^{85} - 57 q^{86} + 47 q^{87} - 6 q^{88} + 9 q^{89} - 7 q^{90} + 86 q^{92} + 22 q^{93} - 27 q^{94} + 74 q^{95} - 33 q^{96} + 17 q^{97} - 4 q^{98} + 18 q^{99}+O(q^{100}) \) 18 * q - q^2 + 18 * q^3 + 23 * q^4 + 6 * q^5 - q^6 - q^7 - 6 * q^8 + 18 * q^9 - 7 * q^10 + 18 * q^11 + 23 * q^12 + 7 * q^14 + 6 * q^15 + 25 * q^16 + 19 * q^17 - q^18 + 2 * q^20 - q^21 - q^22 + 38 * q^23 - 6 * q^24 + 36 * q^25 + 18 * q^27 - 33 * q^28 + 47 * q^29 - 7 * q^30 + 22 * q^31 - 33 * q^32 + 18 * q^33 - 4 * q^34 + 20 * q^35 + 23 * q^36 + 41 * q^37 + 14 * q^38 - 28 * q^40 - 8 * q^41 + 7 * q^42 + 28 * q^43 + 23 * q^44 + 6 * q^45 - 6 * q^46 - 11 * q^47 + 25 * q^48 + 9 * q^49 + q^50 + 19 * q^51 + 49 * q^53 - q^54 + 6 * q^55 + 35 * q^56 + 27 * q^58 + 2 * q^59 + 2 * q^60 - 13 * q^61 + 46 * q^62 - q^63 + 40 * q^64 - q^66 - 4 * q^67 + 30 * q^68 + 38 * q^69 + 83 * q^70 - 2 * q^71 - 6 * q^72 - 39 * q^73 + 42 * q^74 + 36 * q^75 - 20 * q^76 - q^77 + 18 * q^79 - 18 * q^80 + 18 * q^81 - 12 * q^82 - 5 * q^83 - 33 * q^84 - 2 * q^85 - 57 * q^86 + 47 * q^87 - 6 * q^88 + 9 * q^89 - 7 * q^90 + 86 * q^92 + 22 * q^93 - 27 * q^94 + 74 * q^95 - 33 * q^96 + 17 * q^97 - 4 * q^98 + 18 * q^99

Introduction

L-functions

Modular forms

Varieties

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Representations

Groups

Database

Properties

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