LMFDB - Embedded newform 5577.2.a.bh.1.8

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} + 28 x^{15} + 341 x^{14} - 315 x^{13} - 2097 x^{12} + 1830 x^{11} + \cdots - 169 $ x^18 - x^17 - 29x^16 + 28x^15 + 341x^14 - 315x^13 - 2097x^12 + 1830x^11 + 7272x^10 - 5901x^9 - 14318x^8 + 10617x^7 + 15315x^6 - 10102x^5 - 8079x^4 + 4329x^3 + 1819x^2 - 533x - 169
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			1.8
Root			$0.840990$ 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)

$f(q)$	$=$	$q-0.840990 q^{2} -1.00000 q^{3} -1.29274 q^{4} -2.01174 q^{5} +0.840990 q^{6} -4.87530 q^{7} +2.76916 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-0.840990 q^{2} -1.00000 q^{3} -1.29274 q^{4} -2.01174 q^{5} +0.840990 q^{6} -4.87530 q^{7} +2.76916 q^{8} +1.00000 q^{9} +1.69185 q^{10} +1.00000 q^{11} +1.29274 q^{12} +4.10008 q^{14} +2.01174 q^{15} +0.256638 q^{16} +2.60078 q^{17} -0.840990 q^{18} -4.98749 q^{19} +2.60065 q^{20} +4.87530 q^{21} -0.840990 q^{22} +4.29377 q^{23} -2.76916 q^{24} -0.952912 q^{25} -1.00000 q^{27} +6.30248 q^{28} -3.22217 q^{29} -1.69185 q^{30} -1.26334 q^{31} -5.75415 q^{32} -1.00000 q^{33} -2.18723 q^{34} +9.80783 q^{35} -1.29274 q^{36} -7.08844 q^{37} +4.19442 q^{38} -5.57082 q^{40} -5.87044 q^{41} -4.10008 q^{42} -8.59702 q^{43} -1.29274 q^{44} -2.01174 q^{45} -3.61102 q^{46} +7.80609 q^{47} -0.256638 q^{48} +16.7686 q^{49} +0.801390 q^{50} -2.60078 q^{51} -3.45831 q^{53} +0.840990 q^{54} -2.01174 q^{55} -13.5005 q^{56} +4.98749 q^{57} +2.70982 q^{58} -4.95718 q^{59} -2.60065 q^{60} -9.00424 q^{61} +1.06245 q^{62} -4.87530 q^{63} +4.32590 q^{64} +0.840990 q^{66} -6.12478 q^{67} -3.36212 q^{68} -4.29377 q^{69} -8.24829 q^{70} +2.00789 q^{71} +2.76916 q^{72} -11.1299 q^{73} +5.96130 q^{74} +0.952912 q^{75} +6.44750 q^{76} -4.87530 q^{77} -6.28838 q^{79} -0.516289 q^{80} +1.00000 q^{81} +4.93698 q^{82} -9.11002 q^{83} -6.30248 q^{84} -5.23209 q^{85} +7.23000 q^{86} +3.22217 q^{87} +2.76916 q^{88} +1.64327 q^{89} +1.69185 q^{90} -5.55071 q^{92} +1.26334 q^{93} -6.56484 q^{94} +10.0335 q^{95} +5.75415 q^{96} -6.83038 q^{97} -14.1022 q^{98} +1.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 18 q - q^{2} - 18 q^{3} + 23 q^{4} + 6 q^{5} + q^{6} - q^{7} + 18 q^{9}+O(q^{10}) $ 18 * q - q^2 - 18 * q^3 + 23 * q^4 + 6 * q^5 + q^6 - q^7 + 18 * q^9	$ 18 q - q^{2} - 18 q^{3} + 23 q^{4} + 6 q^{5} + q^{6} - q^{7} + 18 q^{9} + 11 q^{10} + 18 q^{11} - 23 q^{12} - 13 q^{14} - 6 q^{15} + 41 q^{16} - q^{17} - q^{18} + 12 q^{19} + 8 q^{20} + q^{21} - q^{22} + 44 q^{23} + 48 q^{25} - 18 q^{27} - 5 q^{28} - 17 q^{29} - 11 q^{30} - 18 q^{31} - q^{32} - 18 q^{33} + 8 q^{34} + 8 q^{35} + 23 q^{36} + 39 q^{37} + 4 q^{38} + 8 q^{40} + 6 q^{41} + 13 q^{42} + 36 q^{43} + 23 q^{44} + 6 q^{45} - 36 q^{46} + 19 q^{47} - 41 q^{48} + 41 q^{49} - 19 q^{50} + q^{51} + 7 q^{53} + q^{54} + 6 q^{55} - 73 q^{56} - 12 q^{57} + 69 q^{58} - 16 q^{59} - 8 q^{60} + 21 q^{61} + 8 q^{62} - q^{63} + 96 q^{64} + q^{66} + 40 q^{68} - 44 q^{69} - 107 q^{70} + 12 q^{71} - 7 q^{73} - 14 q^{74} - 48 q^{75} + 30 q^{76} - q^{77} + 62 q^{79} + 58 q^{80} + 18 q^{81} + 44 q^{82} - 11 q^{83} + 5 q^{84} - 46 q^{85} + 19 q^{86} + 17 q^{87} + 17 q^{89} + 11 q^{90} + 100 q^{92} + 18 q^{93} + 31 q^{94} + 42 q^{95} + q^{96} + 35 q^{97} - 94 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 + 18 * q^9 + 11 * q^10 + 18 * q^11 - 23 * q^12 - 13 * q^14 - 6 * q^15 + 41 * q^16 - q^17 - q^18 + 12 * q^19 + 8 * q^20 + q^21 - q^22 + 44 * q^23 + 48 * q^25 - 18 * q^27 - 5 * q^28 - 17 * q^29 - 11 * q^30 - 18 * q^31 - q^32 - 18 * q^33 + 8 * q^34 + 8 * q^35 + 23 * q^36 + 39 * q^37 + 4 * q^38 + 8 * q^40 + 6 * q^41 + 13 * q^42 + 36 * q^43 + 23 * q^44 + 6 * q^45 - 36 * q^46 + 19 * q^47 - 41 * q^48 + 41 * q^49 - 19 * q^50 + q^51 + 7 * q^53 + q^54 + 6 * q^55 - 73 * q^56 - 12 * q^57 + 69 * q^58 - 16 * q^59 - 8 * q^60 + 21 * q^61 + 8 * q^62 - q^63 + 96 * q^64 + q^66 + 40 * q^68 - 44 * q^69 - 107 * q^70 + 12 * q^71 - 7 * q^73 - 14 * q^74 - 48 * q^75 + 30 * q^76 - q^77 + 62 * q^79 + 58 * q^80 + 18 * q^81 + 44 * q^82 - 11 * q^83 + 5 * q^84 - 46 * q^85 + 19 * q^86 + 17 * q^87 + 17 * q^89 + 11 * q^90 + 100 * q^92 + 18 * q^93 + 31 * q^94 + 42 * q^95 + q^96 + 35 * q^97 - 94 * 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 $

$2$	−0.840990	−0.594670	−0.297335	−	0.954773i	$-0.596098\pi$
$2$	−0.840990	−0.594670	−0.297335	+	0.954773i	$0.596098\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	−1.29274	−0.646368
$5$	−2.01174	−0.899676	−0.449838	−	0.893110i	$-0.648518\pi$
$5$	−2.01174	−0.899676	−0.449838	+	0.893110i	$0.648518\pi$
$6$	0.840990	0.343333
$7$	−4.87530	−1.84269	−0.921346	−	0.388744i	$-0.872909\pi$
$7$	−4.87530	−1.84269	−0.921346	+	0.388744i	$0.872909\pi$
$8$	2.76916	0.979045
$9$	1.00000	0.333333
$10$	1.69185	0.535010
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	1.29274	0.373181
$13$	0	0
$13$	0	0
$14$	4.10008	1.09579
$15$	2.01174	0.519428
$16$	0.256638	0.0641596
$17$	2.60078	0.630782	0.315391	−	0.948962i	$-0.397864\pi$
$17$	2.60078	0.630782	0.315391	+	0.948962i	$0.397864\pi$
$18$	−0.840990	−0.198223
$19$	−4.98749	−1.14421	−0.572104	−	0.820181i	$-0.693873\pi$
$19$	−4.98749	−1.14421	−0.572104	+	0.820181i	$0.693873\pi$
$20$	2.60065	0.581522
$21$	4.87530	1.06388
$22$	−0.840990	−0.179300
$23$	4.29377	0.895313	0.447657	−	0.894206i	$-0.352259\pi$
$23$	4.29377	0.895313	0.447657	+	0.894206i	$0.352259\pi$
$24$	−2.76916	−0.565252
$25$	−0.952912	−0.190582
$26$	0	0
$27$	−1.00000	−0.192450
$28$	6.30248	1.19106
$29$	−3.22217	−0.598343	−0.299171	−	0.954199i	$-0.596710\pi$
$29$	−3.22217	−0.598343	−0.299171	+	0.954199i	$0.596710\pi$
$30$	−1.69185	−0.308888
$31$	−1.26334	−0.226902	−0.113451	−	0.993544i	$-0.536191\pi$
$31$	−1.26334	−0.226902	−0.113451	+	0.993544i	$0.536191\pi$
$32$	−5.75415	−1.01720
$33$	−1.00000	−0.174078
$34$	−2.18723	−0.375107
$35$	9.80783	1.65783
$36$	−1.29274	−0.215456
$37$	−7.08844	−1.16533	−0.582666	−	0.812712i	$-0.697990\pi$
$37$	−7.08844	−1.16533	−0.582666	+	0.812712i	$0.697990\pi$
$38$	4.19442	0.680426
$39$	0	0
$40$	−5.57082	−0.880824
$41$	−5.87044	−0.916809	−0.458405	−	0.888744i	$-0.651579\pi$
$41$	−5.87044	−0.916809	−0.458405	+	0.888744i	$0.651579\pi$
$42$	−4.10008	−0.632656
$43$	−8.59702	−1.31103	−0.655516	−	0.755181i	$-0.727549\pi$
$43$	−8.59702	−1.31103	−0.655516	+	0.755181i	$0.727549\pi$
$44$	−1.29274	−0.194887
$45$	−2.01174	−0.299892
$46$	−3.61102	−0.532416
$47$	7.80609	1.13863	0.569317	−	0.822118i	$-0.307208\pi$
$47$	7.80609	1.13863	0.569317	+	0.822118i	$0.307208\pi$
$48$	−0.256638	−0.0370426
$49$	16.7686	2.39551
$50$	0.801390	0.113334
$51$	−2.60078	−0.364182
$52$	0	0
$53$	−3.45831	−0.475035	−0.237518	−	0.971383i	$-0.576334\pi$
$53$	−3.45831	−0.475035	−0.237518	+	0.971383i	$0.576334\pi$
$54$	0.840990	0.114444
$55$	−2.01174	−0.271263
$56$	−13.5005	−1.80408
$57$	4.98749	0.660609
$58$	2.70982	0.355816
$59$	−4.95718	−0.645370	−0.322685	−	0.946506i	$-0.604585\pi$
$59$	−4.95718	−0.645370	−0.322685	+	0.946506i	$0.604585\pi$
$60$	−2.60065	−0.335742
$61$	−9.00424	−1.15288	−0.576438	−	0.817141i	$-0.695558\pi$
$61$	−9.00424	−1.15288	−0.576438	+	0.817141i	$0.695558\pi$
$62$	1.06245	0.134932
$63$	−4.87530	−0.614231
$64$	4.32590	0.540738
$65$	0	0
$66$	0.840990	0.103519
$67$	−6.12478	−0.748261	−0.374131	−	0.927376i	$-0.622059\pi$
$67$	−6.12478	−0.748261	−0.374131	+	0.927376i	$0.622059\pi$
$68$	−3.36212	−0.407717
$69$	−4.29377	−0.516909
$70$	−8.24829	−0.985859
$71$	2.00789	0.238292	0.119146	−	0.992877i	$-0.461984\pi$
$71$	2.00789	0.238292	0.119146	+	0.992877i	$0.461984\pi$
$72$	2.76916	0.326348
$73$	−11.1299	−1.30266	−0.651331	−	0.758794i	$-0.725789\pi$
$73$	−11.1299	−1.30266	−0.651331	+	0.758794i	$0.725789\pi$
$74$	5.96130	0.692987
$75$	0.952912	0.110033
$76$	6.44750	0.739579
$77$	−4.87530	−0.555593
$78$	0	0
$79$	−6.28838	−0.707498	−0.353749	−	0.935340i	$-0.615093\pi$
$79$	−6.28838	−0.707498	−0.353749	+	0.935340i	$0.615093\pi$
$80$	−0.516289	−0.0577229
$81$	1.00000	0.111111
$82$	4.93698	0.545199
$83$	−9.11002	−0.999955	−0.499978	−	0.866038i	$-0.666658\pi$
$83$	−9.11002	−0.999955	−0.499978	+	0.866038i	$0.666658\pi$
$84$	−6.30248	−0.687657
$85$	−5.23209	−0.567500
$86$	7.23000	0.779631
$87$	3.22217	0.345453
$88$	2.76916	0.295193
$89$	1.64327	0.174186	0.0870932	−	0.996200i	$-0.472242\pi$
$89$	1.64327	0.174186	0.0870932	+	0.996200i	$0.472242\pi$
$90$	1.69185	0.178337
$91$	0	0
$92$	−5.55071	−0.578702
$93$	1.26334	0.131002
$94$	−6.56484	−0.677112
$95$	10.0335	1.02942
$96$	5.75415	0.587280
$97$	−6.83038	−0.693520	−0.346760	−	0.937954i	$-0.612718\pi$
$97$	−6.83038	−0.693520	−0.346760	+	0.937954i	$0.612718\pi$
$98$	−14.1022	−1.42454
$99$	1.00000	0.100504
$100$	1.23186	0.123186
$101$	−4.94474	−0.492020	−0.246010	−	0.969267i	$-0.579120\pi$
$101$	−4.94474	−0.492020	−0.246010	+	0.969267i	$0.579120\pi$
$102$	2.18723	0.216568
$103$	−12.8013	−1.26135	−0.630674	−	0.776048i	$-0.717221\pi$
$103$	−12.8013	−1.26135	−0.630674	+	0.776048i	$0.717221\pi$
$104$	0	0
$105$	−9.80783	−0.957147
$106$	2.90840	0.282489
$107$	16.2194	1.56799	0.783996	−	0.620765i	$-0.213178\pi$
$107$	16.2194	1.56799	0.783996	+	0.620765i	$0.213178\pi$
$108$	1.29274	0.124394
$109$	−15.8979	−1.52274	−0.761369	−	0.648319i	$-0.775472\pi$
$109$	−15.8979	−1.52274	−0.761369	+	0.648319i	$0.775472\pi$
$110$	1.69185	0.161312
$111$	7.08844	0.672805
$112$	−1.25119	−0.118226
$113$	−16.7083	−1.57178	−0.785890	−	0.618366i	$-0.787795\pi$
$113$	−16.7083	−1.57178	−0.785890	+	0.618366i	$0.787795\pi$
$114$	−4.19442	−0.392844
$115$	−8.63794	−0.805492
$116$	4.16542	0.386750
$117$	0	0
$118$	4.16894	0.383782
$119$	−12.6796	−1.16234
$120$	5.57082	0.508544
$121$	1.00000	0.0909091
$122$	7.57248	0.685580
$123$	5.87044	0.529320
$124$	1.63316	0.146662
$125$	11.9757	1.07114
$126$	4.10008	0.365264
$127$	9.85133	0.874164	0.437082	−	0.899422i	$-0.356012\pi$
$127$	9.85133	0.874164	0.437082	+	0.899422i	$0.356012\pi$
$128$	7.87025	0.695639
$129$	8.59702	0.756925
$130$	0	0
$131$	−20.4261	−1.78464	−0.892320	−	0.451404i	$-0.850923\pi$
$131$	−20.4261	−1.78464	−0.892320	+	0.451404i	$0.850923\pi$
$132$	1.29274	0.112518
$133$	24.3155	2.10842
$134$	5.15088	0.444968
$135$	2.01174	0.173143
$136$	7.20197	0.617564
$137$	20.3657	1.73996	0.869982	−	0.493084i	$-0.164130\pi$
$137$	20.3657	1.73996	0.869982	+	0.493084i	$0.164130\pi$
$138$	3.61102	0.307390
$139$	−6.14415	−0.521140	−0.260570	−	0.965455i	$-0.583911\pi$
$139$	−6.14415	−0.521140	−0.260570	+	0.965455i	$0.583911\pi$
$140$	−12.6789	−1.07157
$141$	−7.80609	−0.657391
$142$	−1.68861	−0.141705
$143$	0	0
$144$	0.256638	0.0213865
$145$	6.48217	0.538315
$146$	9.36017	0.774653
$147$	−16.7686	−1.38305
$148$	9.16348	0.753233
$149$	−14.1553	−1.15965	−0.579824	−	0.814742i	$-0.696879\pi$
$149$	−14.1553	−1.15965	−0.579824	+	0.814742i	$0.696879\pi$
$150$	−0.801390	−0.0654332
$151$	14.8059	1.20489	0.602444	−	0.798161i	$-0.294194\pi$
$151$	14.8059	1.20489	0.602444	+	0.798161i	$0.294194\pi$
$152$	−13.8111	−1.12023
$153$	2.60078	0.210261
$154$	4.10008	0.330394
$155$	2.54150	0.204139
$156$	0	0
$157$	0.997245	0.0795888	0.0397944	−	0.999208i	$-0.487330\pi$
$157$	0.997245	0.0795888	0.0397944	+	0.999208i	$0.487330\pi$
$158$	5.28846	0.420727
$159$	3.45831	0.274262
$160$	11.5758	0.915150
$161$	−20.9334	−1.64979
$162$	−0.840990	−0.0660744
$163$	−14.6295	−1.14587	−0.572937	−	0.819600i	$-0.694196\pi$
$163$	−14.6295	−1.14587	−0.572937	+	0.819600i	$0.694196\pi$
$164$	7.58893	0.592596
$165$	2.01174	0.156614
$166$	7.66144	0.594643
$167$	−14.2158	−1.10005	−0.550026	−	0.835148i	$-0.685382\pi$
$167$	−14.2158	−1.10005	−0.550026	+	0.835148i	$0.685382\pi$
$168$	13.5005	1.04159
$169$	0	0
$170$	4.40013	0.337475
$171$	−4.98749	−0.381403
$172$	11.1137	0.847410
$173$	−7.24312	−0.550684	−0.275342	−	0.961346i	$-0.588791\pi$
$173$	−7.24312	−0.550684	−0.275342	+	0.961346i	$0.588791\pi$
$174$	−2.70982	−0.205431
$175$	4.64574	0.351185
$176$	0.256638	0.0193449
$177$	4.95718	0.372604
$178$	−1.38197	−0.103583
$179$	−9.16427	−0.684970	−0.342485	−	0.939523i	$-0.611269\pi$
$179$	−9.16427	−0.684970	−0.342485	+	0.939523i	$0.611269\pi$
$180$	2.60065	0.193841
$181$	9.65167	0.717403	0.358701	−	0.933452i	$-0.383220\pi$
$181$	9.65167	0.717403	0.358701	+	0.933452i	$0.383220\pi$
$182$	0	0
$183$	9.00424	0.665613
$184$	11.8901	0.876552
$185$	14.2601	1.04842
$186$	−1.06245	−0.0779029
$187$	2.60078	0.190188
$188$	−10.0912	−0.735977
$189$	4.87530	0.354626
$190$	−8.43808	−0.612163
$191$	6.08075	0.439988	0.219994	−	0.975501i	$-0.429396\pi$
$191$	6.08075	0.439988	0.219994	+	0.975501i	$0.429396\pi$
$192$	−4.32590	−0.312195
$193$	26.3118	1.89396	0.946982	−	0.321287i	$-0.104115\pi$
$193$	26.3118	1.89396	0.946982	+	0.321287i	$0.104115\pi$
$194$	5.74428	0.412416
$195$	0	0
$196$	−21.6774	−1.54838
$197$	−8.05101	−0.573611	−0.286806	−	0.957989i	$-0.592593\pi$
$197$	−8.05101	−0.573611	−0.286806	+	0.957989i	$0.592593\pi$
$198$	−0.840990	−0.0597665
$199$	25.2030	1.78659	0.893297	−	0.449466i	$-0.148386\pi$
$199$	25.2030	1.78659	0.893297	+	0.449466i	$0.148386\pi$
$200$	−2.63876	−0.186589
$201$	6.12478	0.432009
$202$	4.15848	0.292589
$203$	15.7091	1.10256
$204$	3.36212	0.235396
$205$	11.8098	0.824832
$206$	10.7657	0.750085
$207$	4.29377	0.298438
$208$	0	0
$209$	−4.98749	−0.344992
$210$	8.24829	0.569186
$211$	12.6920	0.873754	0.436877	−	0.899521i	$-0.356085\pi$
$211$	12.6920	0.873754	0.436877	+	0.899521i	$0.356085\pi$
$212$	4.47068	0.307048
$213$	−2.00789	−0.137578
$214$	−13.6404	−0.932438
$215$	17.2949	1.17951
$216$	−2.76916	−0.188417
$217$	6.15916	0.418111
$218$	13.3699	0.905526
$219$	11.1299	0.752092
$220$	2.60065	0.175335
$221$	0	0
$222$	−5.96130	−0.400097
$223$	−19.9794	−1.33792	−0.668960	−	0.743299i	$-0.733260\pi$
$223$	−19.9794	−1.33792	−0.668960	+	0.743299i	$0.733260\pi$
$224$	28.0532	1.87438
$225$	−0.952912	−0.0635275
$226$	14.0515	0.934690
$227$	−6.75814	−0.448553	−0.224277	−	0.974526i	$-0.572002\pi$
$227$	−6.75814	−0.448553	−0.224277	+	0.974526i	$0.572002\pi$
$228$	−6.44750	−0.426996
$229$	−20.2939	−1.34106	−0.670528	−	0.741884i	$-0.733932\pi$
$229$	−20.2939	−1.34106	−0.670528	+	0.741884i	$0.733932\pi$
$230$	7.26442	0.479002
$231$	4.87530	0.320772
$232$	−8.92271	−0.585804
$233$	−4.50072	−0.294852	−0.147426	−	0.989073i	$-0.547099\pi$
$233$	−4.50072	−0.294852	−0.147426	+	0.989073i	$0.547099\pi$
$234$	0	0
$235$	−15.7038	−1.02440
$236$	6.40832	0.417146
$237$	6.28838	0.408474
$238$	10.6634	0.691207
$239$	−7.43587	−0.480987	−0.240493	−	0.970651i	$-0.577309\pi$
$239$	−7.43587	−0.480987	−0.240493	+	0.970651i	$0.577309\pi$
$240$	0.516289	0.0333263
$241$	−27.3281	−1.76035	−0.880177	−	0.474645i	$-0.842576\pi$
$241$	−27.3281	−1.76035	−0.880177	+	0.474645i	$0.842576\pi$
$242$	−0.840990	−0.0540609
$243$	−1.00000	−0.0641500
$244$	11.6401	0.745182
$245$	−33.7340	−2.15519
$246$	−4.93698	−0.314771
$247$	0	0
$248$	−3.49838	−0.222147
$249$	9.11002	0.577324
$250$	−10.0714	−0.636974
$251$	22.0252	1.39022	0.695108	−	0.718905i	$-0.255356\pi$
$251$	22.0252	1.39022	0.695108	+	0.718905i	$0.255356\pi$
$252$	6.30248	0.397019
$253$	4.29377	0.269947
$254$	−8.28487	−0.519839
$255$	5.23209	0.327646
$256$	−15.2706	−0.954413
$257$	16.8275	1.04967	0.524837	−	0.851203i	$-0.324126\pi$
$257$	16.8275	1.04967	0.524837	+	0.851203i	$0.324126\pi$
$258$	−7.23000	−0.450120
$259$	34.5583	2.14735
$260$	0	0
$261$	−3.22217	−0.199448
$262$	17.1782	1.06127
$263$	1.31458	0.0810604	0.0405302	−	0.999178i	$-0.487095\pi$
$263$	1.31458	0.0810604	0.0405302	+	0.999178i	$0.487095\pi$
$264$	−2.76916	−0.170430
$265$	6.95721	0.427378
$266$	−20.4491	−1.25381
$267$	−1.64327	−0.100567
$268$	7.91772	0.483652
$269$	−10.1617	−0.619571	−0.309786	−	0.950806i	$-0.600257\pi$
$269$	−10.1617	−0.619571	−0.309786	+	0.950806i	$0.600257\pi$
$270$	−1.69185	−0.102963
$271$	25.0439	1.52131	0.760655	−	0.649156i	$-0.224878\pi$
$271$	25.0439	1.52131	0.760655	+	0.649156i	$0.224878\pi$
$272$	0.667461	0.0404707
$273$	0	0
$274$	−17.1274	−1.03470
$275$	−0.952912	−0.0574628
$276$	5.55071	0.334114
$277$	−6.09075	−0.365958	−0.182979	−	0.983117i	$-0.558574\pi$
$277$	−6.09075	−0.365958	−0.182979	+	0.983117i	$0.558574\pi$
$278$	5.16717	0.309906
$279$	−1.26334	−0.0756341
$280$	27.1594	1.62309
$281$	−4.81935	−0.287498	−0.143749	−	0.989614i	$-0.545916\pi$
$281$	−4.81935	−0.287498	−0.143749	+	0.989614i	$0.545916\pi$
$282$	6.56484	0.390931
$283$	−20.9873	−1.24756	−0.623782	−	0.781598i	$-0.714405\pi$
$283$	−20.9873	−1.24756	−0.623782	+	0.781598i	$0.714405\pi$
$284$	−2.59567	−0.154025
$285$	−10.0335	−0.594334
$286$	0	0
$287$	28.6202	1.68940
$288$	−5.75415	−0.339066
$289$	−10.2359	−0.602114
$290$	−5.45144	−0.320119
$291$	6.83038	0.400404
$292$	14.3881	0.841998
$293$	−26.5099	−1.54872	−0.774362	−	0.632743i	$-0.781929\pi$
$293$	−26.5099	−1.54872	−0.774362	+	0.632743i	$0.781929\pi$
$294$	14.1022	0.822458
$295$	9.97254	0.580624
$296$	−19.6290	−1.14091
$297$	−1.00000	−0.0580259
$298$	11.9045	0.689607
$299$	0	0
$300$	−1.23186	−0.0711217
$301$	41.9131	2.41583
$302$	−12.4516	−0.716510
$303$	4.94474	0.284068
$304$	−1.27998	−0.0734119
$305$	18.1142	1.03721
$306$	−2.18723	−0.125036
$307$	21.5192	1.22817	0.614084	−	0.789241i	$-0.289526\pi$
$307$	21.5192	1.22817	0.614084	+	0.789241i	$0.289526\pi$
$308$	6.30248	0.359117
$309$	12.8013	0.728240
$310$	−2.13738	−0.121395
$311$	32.4339	1.83916	0.919578	−	0.392908i	$-0.128531\pi$
$311$	32.4339	1.83916	0.919578	+	0.392908i	$0.128531\pi$
$312$	0	0
$313$	25.6765	1.45132	0.725661	−	0.688052i	$-0.241534\pi$
$313$	25.6765	1.45132	0.725661	+	0.688052i	$0.241534\pi$
$314$	−0.838673	−0.0473290
$315$	9.80783	0.552609
$316$	8.12921	0.457304
$317$	23.5425	1.32228	0.661140	−	0.750263i	$-0.270073\pi$
$317$	23.5425	1.32228	0.661140	+	0.750263i	$0.270073\pi$
$318$	−2.90840	−0.163095
$319$	−3.22217	−0.180407
$320$	−8.70258	−0.486489
$321$	−16.2194	−0.905281
$322$	17.6048	0.981078
$323$	−12.9714	−0.721746
$324$	−1.29274	−0.0718187
$325$	0	0
$326$	12.3033	0.681416
$327$	15.8979	0.879153
$328$	−16.2562	−0.897598
$329$	−38.0571	−2.09815
$330$	−1.69185	−0.0931333
$331$	16.0387	0.881564	0.440782	−	0.897614i	$-0.354701\pi$
$331$	16.0387	0.881564	0.440782	+	0.897614i	$0.354701\pi$
$332$	11.7769	0.646339
$333$	−7.08844	−0.388444
$334$	11.9553	0.654167
$335$	12.3215	0.673193
$336$	1.25119	0.0682580
$337$	12.8213	0.698423	0.349212	−	0.937044i	$-0.386449\pi$
$337$	12.8213	0.698423	0.349212	+	0.937044i	$0.386449\pi$
$338$	0	0
$339$	16.7083	0.907468
$340$	6.76371	0.366814
$341$	−1.26334	−0.0684136
$342$	4.19442	0.226809
$343$	−47.6249	−2.57150
$344$	−23.8065	−1.28356
$345$	8.63794	0.465051
$346$	6.09139	0.327475
$347$	16.4456	0.882846	0.441423	−	0.897299i	$-0.354474\pi$
$347$	16.4456	0.882846	0.441423	+	0.897299i	$0.354474\pi$
$348$	−4.16542	−0.223290
$349$	−0.203678	−0.0109026	−0.00545132	−	0.999985i	$-0.501735\pi$
$349$	−0.203678	−0.0109026	−0.00545132	+	0.999985i	$0.501735\pi$
$350$	−3.90702	−0.208839
$351$	0	0
$352$	−5.75415	−0.306697
$353$	−19.4998	−1.03787	−0.518935	−	0.854814i	$-0.673671\pi$
$353$	−19.4998	−1.03787	−0.518935	+	0.854814i	$0.673671\pi$
$354$	−4.16894	−0.221576
$355$	−4.03934	−0.214386
$356$	−2.12431	−0.112588
$357$	12.6796	0.671076
$358$	7.70706	0.407331
$359$	30.7593	1.62341	0.811706	−	0.584067i	$-0.198539\pi$
$359$	30.7593	1.62341	0.811706	+	0.584067i	$0.198539\pi$
$360$	−5.57082	−0.293608
$361$	5.87501	0.309211
$362$	−8.11695	−0.426618
$363$	−1.00000	−0.0524864
$364$	0	0
$365$	22.3905	1.17197
$366$	−7.57248	−0.395820
$367$	−3.81651	−0.199220	−0.0996101	−	0.995027i	$-0.531760\pi$
$367$	−3.81651	−0.199220	−0.0996101	+	0.995027i	$0.531760\pi$
$368$	1.10195	0.0574430
$369$	−5.87044	−0.305603
$370$	−11.9926	−0.623464
$371$	16.8603	0.875344
$372$	−1.63316	−0.0846755
$373$	17.6333	0.913020	0.456510	−	0.889718i	$-0.349099\pi$
$373$	17.6333	0.913020	0.456510	+	0.889718i	$0.349099\pi$
$374$	−2.18723	−0.113099
$375$	−11.9757	−0.618422
$376$	21.6163	1.11477
$377$	0	0
$378$	−4.10008	−0.210885
$379$	−16.7084	−0.858254	−0.429127	−	0.903244i	$-0.641179\pi$
$379$	−16.7084	−0.858254	−0.429127	+	0.903244i	$0.641179\pi$
$380$	−12.9707	−0.665382
$381$	−9.85133	−0.504699
$382$	−5.11385	−0.261647
$383$	−5.72163	−0.292361	−0.146181	−	0.989258i	$-0.546698\pi$
$383$	−5.72163	−0.292361	−0.146181	+	0.989258i	$0.546698\pi$
$384$	−7.87025	−0.401627
$385$	9.80783	0.499853
$386$	−22.1279	−1.12628
$387$	−8.59702	−0.437011
$388$	8.82988	0.448269
$389$	−31.4980	−1.59701	−0.798507	−	0.601986i	$-0.794376\pi$
$389$	−31.4980	−1.59701	−0.798507	+	0.601986i	$0.794376\pi$
$390$	0	0
$391$	11.1672	0.564748
$392$	46.4349	2.34532
$393$	20.4261	1.03036
$394$	6.77082	0.341109
$395$	12.6506	0.636519
$396$	−1.29274	−0.0649624
$397$	−5.19179	−0.260569	−0.130284	−	0.991477i	$-0.541589\pi$
$397$	−5.19179	−0.260569	−0.130284	+	0.991477i	$0.541589\pi$
$398$	−21.1955	−1.06243
$399$	−24.3155	−1.21730
$400$	−0.244554	−0.0122277
$401$	−27.7006	−1.38330	−0.691651	−	0.722231i	$-0.743117\pi$
$401$	−27.7006	−1.38330	−0.691651	+	0.722231i	$0.743117\pi$
$402$	−5.15088	−0.256902
$403$	0	0
$404$	6.39224	0.318026
$405$	−2.01174	−0.0999640
$406$	−13.2112	−0.655660
$407$	−7.08844	−0.351361
$408$	−7.20197	−0.356551
$409$	−16.3362	−0.807774	−0.403887	−	0.914809i	$-0.632341\pi$
$409$	−16.3362	−0.807774	−0.403887	+	0.914809i	$0.632341\pi$
$410$	−9.93191	−0.490502
$411$	−20.3657	−1.00457
$412$	16.5487	0.815295
$413$	24.1678	1.18922
$414$	−3.61102	−0.177472
$415$	18.3270	0.899636
$416$	0	0
$417$	6.14415	0.300880
$418$	4.19442	0.205156
$419$	−23.8451	−1.16491	−0.582454	−	0.812864i	$-0.697907\pi$
$419$	−23.8451	−1.16491	−0.582454	+	0.812864i	$0.697907\pi$
$420$	12.6789	0.618669
$421$	−5.12520	−0.249787	−0.124893	−	0.992170i	$-0.539859\pi$
$421$	−5.12520	−0.249787	−0.124893	+	0.992170i	$0.539859\pi$
$422$	−10.6738	−0.519595
$423$	7.80609	0.379545
$424$	−9.57660	−0.465081
$425$	−2.47832	−0.120216
$426$	1.68861	0.0818135
$427$	43.8984	2.12439
$428$	−20.9675	−1.01350
$429$	0	0
$430$	−14.5449	−0.701416
$431$	3.60354	0.173576	0.0867882	−	0.996227i	$-0.472340\pi$
$431$	3.60354	0.173576	0.0867882	+	0.996227i	$0.472340\pi$
$432$	−0.256638	−0.0123475
$433$	12.5911	0.605090	0.302545	−	0.953135i	$-0.402164\pi$
$433$	12.5911	0.605090	0.302545	+	0.953135i	$0.402164\pi$
$434$	−5.17979	−0.248638
$435$	−6.48217	−0.310796
$436$	20.5517	0.984249
$437$	−21.4151	−1.02442
$438$	−9.36017	−0.447246
$439$	3.40984	0.162743	0.0813714	−	0.996684i	$-0.474070\pi$
$439$	3.40984	0.162743	0.0813714	+	0.996684i	$0.474070\pi$
$440$	−5.57082	−0.265578
$441$	16.7686	0.798505
$442$	0	0
$443$	−21.3065	−1.01230	−0.506152	−	0.862444i	$-0.668932\pi$
$443$	−21.3065	−1.01230	−0.506152	+	0.862444i	$0.668932\pi$
$444$	−9.16348	−0.434879
$445$	−3.30583	−0.156711
$446$	16.8025	0.795620
$447$	14.1553	0.669523
$448$	−21.0901	−0.996413
$449$	−39.8010	−1.87833	−0.939163	−	0.343471i	$-0.888397\pi$
$449$	−39.8010	−1.87833	−0.939163	+	0.343471i	$0.888397\pi$
$450$	0.801390	0.0377779
$451$	−5.87044	−0.276428
$452$	21.5994	1.01595
$453$	−14.8059	−0.695642
$454$	5.68352	0.266741
$455$	0	0
$456$	13.8111	0.646766
$457$	−12.7070	−0.594408	−0.297204	−	0.954814i	$-0.596054\pi$
$457$	−12.7070	−0.594408	−0.297204	+	0.954814i	$0.596054\pi$
$458$	17.0669	0.797485
$459$	−2.60078	−0.121394
$460$	11.1666	0.520644
$461$	6.09434	0.283842	0.141921	−	0.989878i	$-0.454672\pi$
$461$	6.09434	0.283842	0.141921	+	0.989878i	$0.454672\pi$
$462$	−4.10008	−0.190753
$463$	−3.42485	−0.159167	−0.0795833	−	0.996828i	$-0.525359\pi$
$463$	−3.42485	−0.159167	−0.0795833	+	0.996828i	$0.525359\pi$
$464$	−0.826934	−0.0383894
$465$	−2.54150	−0.117859
$466$	3.78506	0.175340
$467$	1.84361	0.0853121	0.0426560	−	0.999090i	$-0.486418\pi$
$467$	1.84361	0.0853121	0.0426560	+	0.999090i	$0.486418\pi$
$468$	0	0
$469$	29.8602	1.37881
$470$	13.2067	0.609181
$471$	−0.997245	−0.0459506
$472$	−13.7272	−0.631846
$473$	−8.59702	−0.395291
$474$	−5.28846	−0.242907
$475$	4.75264	0.218066
$476$	16.3914	0.751298
$477$	−3.45831	−0.158345
$478$	6.25349	0.286028
$479$	−14.0827	−0.643456	−0.321728	−	0.946832i	$-0.604264\pi$
$479$	−14.0827	−0.643456	−0.321728	+	0.946832i	$0.604264\pi$
$480$	−11.5758	−0.528362
$481$	0	0
$482$	22.9826	1.04683
$483$	20.9334	0.952505
$484$	−1.29274	−0.0587607
$485$	13.7409	0.623944
$486$	0.840990	0.0381481
$487$	−6.25292	−0.283347	−0.141673	−	0.989913i	$-0.545248\pi$
$487$	−6.25292	−0.283347	−0.141673	+	0.989913i	$0.545248\pi$
$488$	−24.9342	−1.12872
$489$	14.6295	0.661570
$490$	28.3700	1.28162
$491$	−8.20835	−0.370438	−0.185219	−	0.982697i	$-0.559299\pi$
$491$	−8.20835	−0.370438	−0.185219	+	0.982697i	$0.559299\pi$
$492$	−7.58893	−0.342136
$493$	−8.38017	−0.377424
$494$	0	0
$495$	−2.01174	−0.0904209
$496$	−0.324221	−0.0145580
$497$	−9.78906	−0.439099
$498$	−7.66144	−0.343317
$499$	8.87854	0.397458	0.198729	−	0.980054i	$-0.436319\pi$
$499$	8.87854	0.397458	0.198729	+	0.980054i	$0.436319\pi$
$500$	−15.4814	−0.692350
$501$	14.2158	0.635115
$502$	−18.5230	−0.826720
$503$	4.69718	0.209437	0.104719	−	0.994502i	$-0.466606\pi$
$503$	4.69718	0.209437	0.104719	+	0.994502i	$0.466606\pi$
$504$	−13.5005	−0.601360
$505$	9.94752	0.442659
$506$	−3.61102	−0.160529
$507$	0	0
$508$	−12.7352	−0.565032
$509$	9.43747	0.418309	0.209154	−	0.977883i	$-0.432929\pi$
$509$	9.43747	0.418309	0.209154	+	0.977883i	$0.432929\pi$
$510$	−4.40013	−0.194841
$511$	54.2619	2.40040
$512$	−2.89808	−0.128078
$513$	4.98749	0.220203
$514$	−14.1518	−0.624209
$515$	25.7528	1.13480
$516$	−11.1137	−0.489252
$517$	7.80609	0.343311
$518$	−29.0632	−1.27696
$519$	7.24312	0.317938
$520$	0	0
$521$	−32.7141	−1.43323	−0.716615	−	0.697469i	$-0.754310\pi$
$521$	−32.7141	−1.43323	−0.716615	+	0.697469i	$0.754310\pi$
$522$	2.70982	0.118605
$523$	−26.3272	−1.15121	−0.575603	−	0.817729i	$-0.695233\pi$
$523$	−26.3272	−1.15121	−0.575603	+	0.817729i	$0.695233\pi$
$524$	26.4056	1.15353
$525$	−4.64574	−0.202757
$526$	−1.10555	−0.0482041
$527$	−3.28567	−0.143126
$528$	−0.256638	−0.0111688
$529$	−4.56353	−0.198414
$530$	−5.85094	−0.254149
$531$	−4.95718	−0.215123
$532$	−31.4335	−1.36282
$533$	0	0
$534$	1.38197	0.0598039
$535$	−32.6293	−1.41069
$536$	−16.9605	−0.732581
$537$	9.16427	0.395468
$538$	8.54591	0.368440
$539$	16.7686	0.722275
$540$	−2.60065	−0.111914
$541$	34.8293	1.49743	0.748715	−	0.662892i	$-0.230671\pi$
$541$	34.8293	1.49743	0.748715	+	0.662892i	$0.230671\pi$
$542$	−21.0617	−0.904677
$543$	−9.65167	−0.414193
$544$	−14.9653	−0.641631
$545$	31.9823	1.36997
$546$	0	0
$547$	−44.9204	−1.92066	−0.960328	−	0.278873i	$-0.910039\pi$
$547$	−44.9204	−1.92066	−0.960328	+	0.278873i	$0.910039\pi$
$548$	−26.3275	−1.12466
$549$	−9.00424	−0.384292
$550$	0.801390	0.0341714
$551$	16.0705	0.684628
$552$	−11.8901	−0.506078
$553$	30.6578	1.30370
$554$	5.12226	0.217624
$555$	−14.2601	−0.605306
$556$	7.94277	0.336848
$557$	−24.3731	−1.03272	−0.516360	−	0.856372i	$-0.672713\pi$
$557$	−24.3731	−1.03272	−0.516360	+	0.856372i	$0.672713\pi$
$558$	1.06245	0.0449773
$559$	0	0
$560$	2.51707	0.106366
$561$	−2.60078	−0.109805
$562$	4.05302	0.170966
$563$	34.1717	1.44017	0.720083	−	0.693888i	$-0.244104\pi$
$563$	34.1717	1.44017	0.720083	+	0.693888i	$0.244104\pi$
$564$	10.0912	0.424917
$565$	33.6126	1.41409
$566$	17.6501	0.741889
$567$	−4.87530	−0.204744
$568$	5.56016	0.233299
$569$	−31.3053	−1.31239	−0.656194	−	0.754592i	$-0.727835\pi$
$569$	−31.3053	−1.31239	−0.656194	+	0.754592i	$0.727835\pi$
$570$	8.43808	0.353432
$571$	27.8943	1.16734	0.583671	−	0.811990i	$-0.301616\pi$
$571$	27.8943	1.16734	0.583671	+	0.811990i	$0.301616\pi$
$572$	0	0
$573$	−6.08075	−0.254027
$574$	−24.0693	−1.00463
$575$	−4.09159	−0.170631
$576$	4.32590	0.180246
$577$	33.8441	1.40895	0.704474	−	0.709730i	$-0.251183\pi$
$577$	33.8441	1.40895	0.704474	+	0.709730i	$0.251183\pi$
$578$	8.60832	0.358059
$579$	−26.3118	−1.09348
$580$	−8.37973	−0.347949
$581$	44.4141	1.84261
$582$	−5.74428	−0.238108
$583$	−3.45831	−0.143229
$584$	−30.8206	−1.27536
$585$	0	0
$586$	22.2946	0.920979
$587$	30.6412	1.26470	0.632349	−	0.774683i	$-0.282091\pi$
$587$	30.6412	1.26470	0.632349	+	0.774683i	$0.282091\pi$
$588$	21.6774	0.893960
$589$	6.30088	0.259623
$590$	−8.38681	−0.345279
$591$	8.05101	0.331175
$592$	−1.81917	−0.0747672
$593$	42.2576	1.73531	0.867656	−	0.497165i	$-0.165625\pi$
$593$	42.2576	1.73531	0.867656	+	0.497165i	$0.165625\pi$
$594$	0.840990	0.0345062
$595$	25.5080	1.04573
$596$	18.2991	0.749559
$597$	−25.2030	−1.03149
$598$	0	0
$599$	18.4386	0.753380	0.376690	−	0.926339i	$-0.377062\pi$
$599$	18.4386	0.753380	0.376690	+	0.926339i	$0.377062\pi$
$600$	2.63876	0.107727
$601$	11.7070	0.477539	0.238770	−	0.971076i	$-0.423256\pi$
$601$	11.7070	0.477539	0.238770	+	0.971076i	$0.423256\pi$
$602$	−35.2485	−1.43662
$603$	−6.12478	−0.249420
$604$	−19.1401	−0.778800
$605$	−2.01174	−0.0817888
$606$	−4.15848	−0.168927
$607$	32.2672	1.30969	0.654843	−	0.755765i	$-0.272735\pi$
$607$	32.2672	1.30969	0.654843	+	0.755765i	$0.272735\pi$
$608$	28.6987	1.16389
$609$	−15.7091	−0.636564
$610$	−15.2338	−0.616800
$611$	0	0
$612$	−3.36212	−0.135906
$613$	32.4201	1.30944	0.654718	−	0.755873i	$-0.272787\pi$
$613$	32.4201	1.30944	0.654718	+	0.755873i	$0.272787\pi$
$614$	−18.0975	−0.730354
$615$	−11.8098	−0.476217
$616$	−13.5005	−0.543950
$617$	43.5238	1.75220	0.876100	−	0.482129i	$-0.160136\pi$
$617$	43.5238	1.75220	0.876100	+	0.482129i	$0.160136\pi$
$618$	−10.7657	−0.433062
$619$	23.8297	0.957795	0.478897	−	0.877871i	$-0.341037\pi$
$619$	23.8297	0.957795	0.478897	+	0.877871i	$0.341037\pi$
$620$	−3.28549	−0.131949
$621$	−4.29377	−0.172303
$622$	−27.2765	−1.09369
$623$	−8.01144	−0.320972
$624$	0	0
$625$	−19.3274	−0.773096
$626$	−21.5937	−0.863058
$627$	4.98749	0.199181
$628$	−1.28917	−0.0514437
$629$	−18.4355	−0.735071
$630$	−8.24829	−0.328620
$631$	−33.5118	−1.33408	−0.667042	−	0.745020i	$-0.732440\pi$
$631$	−33.5118	−1.33408	−0.667042	+	0.745020i	$0.732440\pi$
$632$	−17.4135	−0.692672
$633$	−12.6920	−0.504462
$634$	−19.7990	−0.786320
$635$	−19.8183	−0.786465
$636$	−4.47068	−0.177274
$637$	0	0
$638$	2.70982	0.107283
$639$	2.00789	0.0794308
$640$	−15.8329	−0.625850
$641$	−5.23256	−0.206673	−0.103337	−	0.994646i	$-0.532952\pi$
$641$	−5.23256	−0.206673	−0.103337	+	0.994646i	$0.532952\pi$
$642$	13.6404	0.538343
$643$	−19.9991	−0.788688	−0.394344	−	0.918963i	$-0.629028\pi$
$643$	−19.9991	−0.788688	−0.394344	+	0.918963i	$0.629028\pi$
$644$	27.0614	1.06637
$645$	−17.2949	−0.680988
$646$	10.9088	0.429200
$647$	11.6568	0.458274	0.229137	−	0.973394i	$-0.426410\pi$
$647$	11.6568	0.458274	0.229137	+	0.973394i	$0.426410\pi$
$648$	2.76916	0.108783
$649$	−4.95718	−0.194586
$650$	0	0
$651$	−6.15916	−0.241396
$652$	18.9121	0.740656
$653$	25.4449	0.995737	0.497869	−	0.867252i	$-0.334116\pi$
$653$	25.4449	0.995737	0.497869	+	0.867252i	$0.334116\pi$
$654$	−13.3699	−0.522806
$655$	41.0920	1.60560
$656$	−1.50658	−0.0588221
$657$	−11.1299	−0.434220
$658$	32.0056	1.24771
$659$	−34.2395	−1.33378	−0.666891	−	0.745155i	$-0.732375\pi$
$659$	−34.2395	−1.33378	−0.666891	+	0.745155i	$0.732375\pi$
$660$	−2.60065	−0.101230
$661$	29.3952	1.14334	0.571670	−	0.820483i	$-0.306296\pi$
$661$	29.3952	1.14334	0.571670	+	0.820483i	$0.306296\pi$
$662$	−13.4883	−0.524240
$663$	0	0
$664$	−25.2271	−0.979001
$665$	−48.9164	−1.89690
$666$	5.96130	0.230996
$667$	−13.8353	−0.535704
$668$	18.3773	0.711038
$669$	19.9794	0.772448
$670$	−10.3622	−0.400327
$671$	−9.00424	−0.347605
$672$	−28.0532	−1.08218
$673$	−29.9716	−1.15532	−0.577660	−	0.816277i	$-0.696034\pi$
$673$	−29.9716	−1.15532	−0.577660	+	0.816277i	$0.696034\pi$
$674$	−10.7826	−0.415331
$675$	0.952912	0.0366776
$676$	0	0
$677$	−28.4543	−1.09359	−0.546794	−	0.837267i	$-0.684152\pi$
$677$	−28.4543	−1.09359	−0.546794	+	0.837267i	$0.684152\pi$
$678$	−14.0515	−0.539644
$679$	33.3002	1.27794
$680$	−14.4885	−0.555608
$681$	6.75814	0.258972
$682$	1.06245	0.0406835
$683$	37.7820	1.44569	0.722843	−	0.691012i	$-0.242835\pi$
$683$	37.7820	1.44569	0.722843	+	0.691012i	$0.242835\pi$
$684$	6.44750	0.246526
$685$	−40.9705	−1.56540
$686$	40.0521	1.52919
$687$	20.2939	0.774259
$688$	−2.20632	−0.0841153
$689$	0	0
$690$	−7.26442	−0.276552
$691$	−48.2394	−1.83511	−0.917557	−	0.397604i	$-0.869842\pi$
$691$	−48.2394	−1.83511	−0.917557	+	0.397604i	$0.869842\pi$
$692$	9.36344	0.355945
$693$	−4.87530	−0.185198
$694$	−13.8306	−0.525002
$695$	12.3604	0.468858
$696$	8.92271	0.338214
$697$	−15.2677	−0.578307
$698$	0.171291	0.00648347
$699$	4.50072	0.170233
$700$	−6.00571	−0.226995
$701$	−33.0150	−1.24696	−0.623480	−	0.781839i	$-0.714282\pi$
$701$	−33.0150	−1.24696	−0.623480	+	0.781839i	$0.714282\pi$
$702$	0	0
$703$	35.3535	1.33338
$704$	4.32590	0.163039
$705$	15.7038	0.591439
$706$	16.3991	0.617190
$707$	24.1071	0.906641
$708$	−6.40832	−0.240840
$709$	−5.18264	−0.194638	−0.0973190	−	0.995253i	$-0.531027\pi$
$709$	−5.18264	−0.194638	−0.0973190	+	0.995253i	$0.531027\pi$
$710$	3.39705	0.127489
$711$	−6.28838	−0.235833
$712$	4.55047	0.170536
$713$	−5.42448	−0.203149
$714$	−10.6634	−0.399068
$715$	0	0
$716$	11.8470	0.442743
$717$	7.43587	0.277698
$718$	−25.8682	−0.965394
$719$	11.2315	0.418863	0.209432	−	0.977823i	$-0.432839\pi$
$719$	11.2315	0.418863	0.209432	+	0.977823i	$0.432839\pi$
$720$	−0.516289	−0.0192410
$721$	62.4102	2.32428
$722$	−4.94083	−0.183879
$723$	27.3281	1.01634
$724$	−12.4771	−0.463706
$725$	3.07045	0.114034
$726$	0.840990	0.0312121
$727$	3.90726	0.144912	0.0724561	−	0.997372i	$-0.476916\pi$
$727$	3.90726	0.144912	0.0724561	+	0.997372i	$0.476916\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	−18.8302	−0.696937
$731$	−22.3590	−0.826976
$732$	−11.6401	−0.430231
$733$	−6.57061	−0.242691	−0.121346	−	0.992610i	$-0.538721\pi$
$733$	−6.57061	−0.242691	−0.121346	+	0.992610i	$0.538721\pi$
$734$	3.20965	0.118470
$735$	33.7340	1.24430
$736$	−24.7070	−0.910712
$737$	−6.12478	−0.225609
$738$	4.93698	0.181733
$739$	21.7981	0.801855	0.400928	−	0.916110i	$-0.368688\pi$
$739$	21.7981	0.801855	0.400928	+	0.916110i	$0.368688\pi$
$740$	−18.4345	−0.677666
$741$	0	0
$742$	−14.1793	−0.520540
$743$	17.5046	0.642181	0.321091	−	0.947048i	$-0.395951\pi$
$743$	17.5046	0.642181	0.321091	+	0.947048i	$0.395951\pi$
$744$	3.49838	0.128257
$745$	28.4767	1.04331
$746$	−14.8295	−0.542945
$747$	−9.11002	−0.333318
$748$	−3.36212	−0.122931
$749$	−79.0747	−2.88933
$750$	10.0714	0.367757
$751$	−22.9573	−0.837723	−0.418861	−	0.908050i	$-0.637571\pi$
$751$	−22.9573	−0.837723	−0.418861	+	0.908050i	$0.637571\pi$
$752$	2.00334	0.0730544
$753$	−22.0252	−0.802642
$754$	0	0
$755$	−29.7856	−1.08401
$756$	−6.30248	−0.229219
$757$	−2.17446	−0.0790320	−0.0395160	−	0.999219i	$-0.512582\pi$
$757$	−2.17446	−0.0790320	−0.0395160	+	0.999219i	$0.512582\pi$
$758$	14.0516	0.510378
$759$	−4.29377	−0.155854
$760$	27.7844	1.00785
$761$	8.24380	0.298838	0.149419	−	0.988774i	$-0.452260\pi$
$761$	8.24380	0.298838	0.149419	+	0.988774i	$0.452260\pi$
$762$	8.28487	0.300129
$763$	77.5069	2.80594
$764$	−7.86080	−0.284394
$765$	−5.23209	−0.189167
$766$	4.81183	0.173859
$767$	0	0
$768$	15.2706	0.551030
$769$	−50.1567	−1.80870	−0.904349	−	0.426794i	$-0.859643\pi$
$769$	−50.1567	−1.80870	−0.904349	+	0.426794i	$0.859643\pi$
$770$	−8.24829	−0.297248
$771$	−16.8275	−0.606029
$772$	−34.0142	−1.22420
$773$	47.6685	1.71452	0.857258	−	0.514888i	$-0.172166\pi$
$773$	47.6685	1.71452	0.857258	+	0.514888i	$0.172166\pi$
$774$	7.23000	0.259877
$775$	1.20385	0.0432436
$776$	−18.9144	−0.678988
$777$	−34.5583	−1.23977
$778$	26.4895	0.949695
$779$	29.2788	1.04902
$780$	0	0
$781$	2.00789	0.0718478
$782$	−9.39147	−0.335838
$783$	3.22217	0.115151
$784$	4.30347	0.153695
$785$	−2.00619	−0.0716042
$786$	−17.1782	−0.612725
$787$	10.8580	0.387045	0.193523	−	0.981096i	$-0.438009\pi$
$787$	10.8580	0.387045	0.193523	+	0.981096i	$0.438009\pi$
$788$	10.4078	0.370764
$789$	−1.31458	−0.0468002
$790$	−10.6390	−0.378519
$791$	81.4578	2.89631
$792$	2.76916	0.0983977
$793$	0	0
$794$	4.36624	0.154952
$795$	−6.95721	−0.246747
$796$	−32.5809	−1.15480
$797$	−32.0777	−1.13625	−0.568125	−	0.822942i	$-0.692331\pi$
$797$	−32.0777	−1.13625	−0.568125	+	0.822942i	$0.692331\pi$
$798$	20.4491	0.723890
$799$	20.3019	0.718231
$800$	5.48320	0.193860
$801$	1.64327	0.0580621
$802$	23.2959	0.822608
$803$	−11.1299	−0.392767
$804$	−7.91772	−0.279237
$805$	42.1126	1.48427
$806$	0	0
$807$	10.1617	0.357710
$808$	−13.6928	−0.481710
$809$	40.5016	1.42396	0.711980	−	0.702200i	$-0.247799\pi$
$809$	40.5016	1.42396	0.711980	+	0.702200i	$0.247799\pi$
$810$	1.69185	0.0594456
$811$	11.3049	0.396969	0.198484	−	0.980104i	$-0.436398\pi$
$811$	11.3049	0.396969	0.198484	+	0.980104i	$0.436398\pi$
$812$	−20.3077	−0.712660
$813$	−25.0439	−0.878329
$814$	5.96130	0.208944
$815$	29.4308	1.03091
$816$	−0.667461	−0.0233658
$817$	42.8775	1.50009
$818$	13.7386	0.480358
$819$	0	0
$820$	−15.2669	−0.533145
$821$	21.5608	0.752478	0.376239	−	0.926523i	$-0.377217\pi$
$821$	21.5608	0.752478	0.376239	+	0.926523i	$0.377217\pi$
$822$	17.1274	0.597386
$823$	−8.38221	−0.292186	−0.146093	−	0.989271i	$-0.546670\pi$
$823$	−8.38221	−0.292186	−0.146093	+	0.989271i	$0.546670\pi$
$824$	−35.4488	−1.23492
$825$	0.952912	0.0331761
$826$	−20.3248	−0.707192
$827$	22.4719	0.781425	0.390712	−	0.920513i	$-0.372229\pi$
$827$	22.4719	0.781425	0.390712	+	0.920513i	$0.372229\pi$
$828$	−5.55071	−0.192901
$829$	−35.3700	−1.22845	−0.614225	−	0.789131i	$-0.710531\pi$
$829$	−35.3700	−1.22845	−0.614225	+	0.789131i	$0.710531\pi$
$830$	−15.4128	−0.534986
$831$	6.09075	0.211286
$832$	0	0
$833$	43.6115	1.51105
$834$	−5.16717	−0.178924
$835$	28.5984	0.989690
$836$	6.44750	0.222992
$837$	1.26334	0.0436673
$838$	20.0535	0.692735
$839$	−16.8236	−0.580816	−0.290408	−	0.956903i	$-0.593791\pi$
$839$	−16.8236	−0.580816	−0.290408	+	0.956903i	$0.593791\pi$
$840$	−27.1594	−0.937090
$841$	−18.6176	−0.641986
$842$	4.31024	0.148541
$843$	4.81935	0.165987
$844$	−16.4074	−0.564766
$845$	0	0
$846$	−6.56484	−0.225704
$847$	−4.87530	−0.167517
$848$	−0.887535	−0.0304781
$849$	20.9873	0.720282
$850$	2.08424	0.0714888
$851$	−30.4361	−1.04334
$852$	2.59567	0.0889261
$853$	40.2429	1.37789	0.688946	−	0.724813i	$-0.258074\pi$
$853$	40.2429	1.37789	0.688946	+	0.724813i	$0.258074\pi$
$854$	−36.9181	−1.26331
$855$	10.0335	0.343139
$856$	44.9142	1.53514
$857$	−49.0038	−1.67394	−0.836970	−	0.547249i	$-0.815675\pi$
$857$	−49.0038	−1.67394	−0.836970	+	0.547249i	$0.815675\pi$
$858$	0	0
$859$	50.9511	1.73843	0.869215	−	0.494434i	$-0.164625\pi$
$859$	50.9511	1.73843	0.869215	+	0.494434i	$0.164625\pi$
$860$	−22.3578	−0.762394
$861$	−28.6202	−0.975374
$862$	−3.03054	−0.103221
$863$	−3.09114	−0.105223	−0.0526117	−	0.998615i	$-0.516755\pi$
$863$	−3.09114	−0.105223	−0.0526117	+	0.998615i	$0.516755\pi$
$864$	5.75415	0.195760
$865$	14.5713	0.495438
$866$	−10.5890	−0.359828
$867$	10.2359	0.347631
$868$	−7.96216	−0.270253
$869$	−6.28838	−0.213319
$870$	5.45144	0.184821
$871$	0	0
$872$	−44.0237	−1.49083
$873$	−6.83038	−0.231173
$874$	18.0099	0.609194
$875$	−58.3852	−1.97378
$876$	−14.3881	−0.486128
$877$	31.5448	1.06519	0.532596	−	0.846370i	$-0.321217\pi$
$877$	31.5448	1.06519	0.532596	+	0.846370i	$0.321217\pi$
$878$	−2.86764	−0.0967782
$879$	26.5099	0.894156
$880$	−0.516289	−0.0174041
$881$	−23.3775	−0.787608	−0.393804	−	0.919195i	$-0.628841\pi$
$881$	−23.3775	−0.787608	−0.393804	+	0.919195i	$0.628841\pi$
$882$	−14.1022	−0.474846
$883$	2.46440	0.0829335	0.0414668	−	0.999140i	$-0.486797\pi$
$883$	2.46440	0.0829335	0.0414668	+	0.999140i	$0.486797\pi$
$884$	0	0
$885$	−9.97254	−0.335223
$886$	17.9186	0.601986
$887$	0.991774	0.0333005	0.0166503	−	0.999861i	$-0.494700\pi$
$887$	0.991774	0.0333005	0.0166503	+	0.999861i	$0.494700\pi$
$888$	19.6290	0.658706
$889$	−48.0282	−1.61081
$890$	2.78017	0.0931915
$891$	1.00000	0.0335013
$892$	25.8281	0.864788
$893$	−38.9328	−1.30283
$894$	−11.9045	−0.398145
$895$	18.4361	0.616251
$896$	−38.3699	−1.28185
$897$	0	0
$898$	33.4723	1.11698
$899$	4.07069	0.135765
$900$	1.23186	0.0410621
$901$	−8.99431	−0.299644
$902$	4.93698	0.164384
$903$	−41.9131	−1.39478
$904$	−46.2678	−1.53884
$905$	−19.4166	−0.645430
$906$	12.4516	0.413677
$907$	16.7963	0.557712	0.278856	−	0.960333i	$-0.410045\pi$
$907$	16.7963	0.557712	0.278856	+	0.960333i	$0.410045\pi$
$908$	8.73648	0.289930
$909$	−4.94474	−0.164007
$910$	0	0
$911$	16.2485	0.538335	0.269168	−	0.963093i	$-0.413251\pi$
$911$	16.2485	0.538335	0.269168	+	0.963093i	$0.413251\pi$
$912$	1.27998	0.0423844
$913$	−9.11002	−0.301498
$914$	10.6864	0.353476
$915$	−18.1142	−0.598836
$916$	26.2346	0.866816
$917$	99.5836	3.28854
$918$	2.18723	0.0721894
$919$	31.2597	1.03116	0.515581	−	0.856841i	$-0.327576\pi$
$919$	31.2597	1.03116	0.515581	+	0.856841i	$0.327576\pi$
$920$	−23.9198	−0.788613
$921$	−21.5192	−0.709083
$922$	−5.12528	−0.168792
$923$	0	0
$924$	−6.30248	−0.207336
$925$	6.75466	0.222092
$926$	2.88027	0.0946515
$927$	−12.8013	−0.420449
$928$	18.5409	0.608633
$929$	−26.5150	−0.869930	−0.434965	−	0.900447i	$-0.643239\pi$
$929$	−26.5150	−0.869930	−0.434965	+	0.900447i	$0.643239\pi$
$930$	2.13738	0.0700874
$931$	−83.6331	−2.74097
$932$	5.81825	0.190583
$933$	−32.4339	−1.06184
$934$	−1.55046	−0.0507325
$935$	−5.23209	−0.171108
$936$	0	0
$937$	54.9663	1.79567	0.897836	−	0.440331i	$-0.145139\pi$
$937$	54.9663	1.79567	0.897836	+	0.440331i	$0.145139\pi$
$938$	−25.1121	−0.819939
$939$	−25.6765	−0.837922
$940$	20.3009	0.662141
$941$	28.9898	0.945040	0.472520	−	0.881320i	$-0.343344\pi$
$941$	28.9898	0.945040	0.472520	+	0.881320i	$0.343344\pi$
$942$	0.838673	0.0273254
$943$	−25.2063	−0.820831
$944$	−1.27220	−0.0414067
$945$	−9.80783	−0.319049
$946$	7.23000	0.235068
$947$	6.34370	0.206143	0.103071	−	0.994674i	$-0.467133\pi$
$947$	6.34370	0.206143	0.103071	+	0.994674i	$0.467133\pi$
$948$	−8.12921	−0.264025
$949$	0	0
$950$	−3.99692	−0.129677
$951$	−23.5425	−0.763419
$952$	−35.1118	−1.13798
$953$	−36.1877	−1.17224	−0.586118	−	0.810226i	$-0.699344\pi$
$953$	−36.1877	−1.17224	−0.586118	+	0.810226i	$0.699344\pi$
$954$	2.90840	0.0941630
$955$	−12.2329	−0.395846
$956$	9.61262	0.310894
$957$	3.22217	0.104158
$958$	11.8434	0.382644
$959$	−99.2892	−3.20622
$960$	8.70258	0.280874
$961$	−29.4040	−0.948515
$962$	0	0
$963$	16.2194	0.522664
$964$	35.3280	1.13784
$965$	−52.9324	−1.70395
$966$	−17.6048	−0.566426
$967$	26.6405	0.856701	0.428350	−	0.903613i	$-0.359095\pi$
$967$	26.6405	0.856701	0.428350	+	0.903613i	$0.359095\pi$
$968$	2.76916	0.0890041
$969$	12.9714	0.416700
$970$	−11.5560	−0.371041
$971$	−8.68423	−0.278690	−0.139345	−	0.990244i	$-0.544500\pi$
$971$	−8.68423	−0.278690	−0.139345	+	0.990244i	$0.544500\pi$
$972$	1.29274	0.0414645
$973$	29.9546	0.960301
$974$	5.25864	0.168498
$975$	0	0
$976$	−2.31084	−0.0739680
$977$	41.5142	1.32816	0.664078	−	0.747663i	$-0.268824\pi$
$977$	41.5142	1.32816	0.664078	+	0.747663i	$0.268824\pi$
$978$	−12.3033	−0.393416
$979$	1.64327	0.0525191
$980$	43.6092	1.39304
$981$	−15.8979	−0.507579
$982$	6.90314	0.220288
$983$	57.8645	1.84559	0.922796	−	0.385290i	$-0.125899\pi$
$983$	57.8645	1.84559	0.922796	+	0.385290i	$0.125899\pi$
$984$	16.2562	0.518228
$985$	16.1965	0.516064
$986$	7.04764	0.224443
$987$	38.0571	1.21137
$988$	0	0
$989$	−36.9136	−1.17378
$990$	1.69185	0.0537706
$991$	7.45617	0.236853	0.118426	−	0.992963i	$-0.462215\pi$
$991$	7.45617	0.236853	0.118426	+	0.992963i	$0.462215\pi$
$992$	7.26943	0.230805
$993$	−16.0387	−0.508971
$994$	8.23250	0.261119
$995$	−50.7019	−1.60736
$996$	−11.7769	−0.373164
$997$	−8.26801	−0.261850	−0.130925	−	0.991392i	$-0.541795\pi$
$997$	−8.26801	−0.261850	−0.130925	+	0.991392i	$0.541795\pi$
$998$	−7.46676	−0.236356
$999$	7.08844	0.224268

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5577.2.a.bh.1.8	✓	18
13.12	even	2		5577.2.a.bj.1.11	yes	18

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5577.2.a.bh.1.8	✓	18	1.1	even	1	trivial
5577.2.a.bj.1.11	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)

\(f(q)\)	\(=\)	\(q-0.840990 q^{2} -1.00000 q^{3} -1.29274 q^{4} -2.01174 q^{5} +0.840990 q^{6} -4.87530 q^{7} +2.76916 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-0.840990 q^{2} -1.00000 q^{3} -1.29274 q^{4} -2.01174 q^{5} +0.840990 q^{6} -4.87530 q^{7} +2.76916 q^{8} +1.00000 q^{9} +1.69185 q^{10} +1.00000 q^{11} +1.29274 q^{12} +4.10008 q^{14} +2.01174 q^{15} +0.256638 q^{16} +2.60078 q^{17} -0.840990 q^{18} -4.98749 q^{19} +2.60065 q^{20} +4.87530 q^{21} -0.840990 q^{22} +4.29377 q^{23} -2.76916 q^{24} -0.952912 q^{25} -1.00000 q^{27} +6.30248 q^{28} -3.22217 q^{29} -1.69185 q^{30} -1.26334 q^{31} -5.75415 q^{32} -1.00000 q^{33} -2.18723 q^{34} +9.80783 q^{35} -1.29274 q^{36} -7.08844 q^{37} +4.19442 q^{38} -5.57082 q^{40} -5.87044 q^{41} -4.10008 q^{42} -8.59702 q^{43} -1.29274 q^{44} -2.01174 q^{45} -3.61102 q^{46} +7.80609 q^{47} -0.256638 q^{48} +16.7686 q^{49} +0.801390 q^{50} -2.60078 q^{51} -3.45831 q^{53} +0.840990 q^{54} -2.01174 q^{55} -13.5005 q^{56} +4.98749 q^{57} +2.70982 q^{58} -4.95718 q^{59} -2.60065 q^{60} -9.00424 q^{61} +1.06245 q^{62} -4.87530 q^{63} +4.32590 q^{64} +0.840990 q^{66} -6.12478 q^{67} -3.36212 q^{68} -4.29377 q^{69} -8.24829 q^{70} +2.00789 q^{71} +2.76916 q^{72} -11.1299 q^{73} +5.96130 q^{74} +0.952912 q^{75} +6.44750 q^{76} -4.87530 q^{77} -6.28838 q^{79} -0.516289 q^{80} +1.00000 q^{81} +4.93698 q^{82} -9.11002 q^{83} -6.30248 q^{84} -5.23209 q^{85} +7.23000 q^{86} +3.22217 q^{87} +2.76916 q^{88} +1.64327 q^{89} +1.69185 q^{90} -5.55071 q^{92} +1.26334 q^{93} -6.56484 q^{94} +10.0335 q^{95} +5.75415 q^{96} -6.83038 q^{97} -14.1022 q^{98} +1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 18 q - q^{2} - 18 q^{3} + 23 q^{4} + 6 q^{5} + q^{6} - q^{7} + 18 q^{9}+O(q^{10}) \) 18 * q - q^2 - 18 * q^3 + 23 * q^4 + 6 * q^5 + q^6 - q^7 + 18 * q^9	\( 18 q - q^{2} - 18 q^{3} + 23 q^{4} + 6 q^{5} + q^{6} - q^{7} + 18 q^{9} + 11 q^{10} + 18 q^{11} - 23 q^{12} - 13 q^{14} - 6 q^{15} + 41 q^{16} - q^{17} - q^{18} + 12 q^{19} + 8 q^{20} + q^{21} - q^{22} + 44 q^{23} + 48 q^{25} - 18 q^{27} - 5 q^{28} - 17 q^{29} - 11 q^{30} - 18 q^{31} - q^{32} - 18 q^{33} + 8 q^{34} + 8 q^{35} + 23 q^{36} + 39 q^{37} + 4 q^{38} + 8 q^{40} + 6 q^{41} + 13 q^{42} + 36 q^{43} + 23 q^{44} + 6 q^{45} - 36 q^{46} + 19 q^{47} - 41 q^{48} + 41 q^{49} - 19 q^{50} + q^{51} + 7 q^{53} + q^{54} + 6 q^{55} - 73 q^{56} - 12 q^{57} + 69 q^{58} - 16 q^{59} - 8 q^{60} + 21 q^{61} + 8 q^{62} - q^{63} + 96 q^{64} + q^{66} + 40 q^{68} - 44 q^{69} - 107 q^{70} + 12 q^{71} - 7 q^{73} - 14 q^{74} - 48 q^{75} + 30 q^{76} - q^{77} + 62 q^{79} + 58 q^{80} + 18 q^{81} + 44 q^{82} - 11 q^{83} + 5 q^{84} - 46 q^{85} + 19 q^{86} + 17 q^{87} + 17 q^{89} + 11 q^{90} + 100 q^{92} + 18 q^{93} + 31 q^{94} + 42 q^{95} + q^{96} + 35 q^{97} - 94 q^{98} + 18 q^{99}+O(q^{100}) \) 18 * q - q^2 - 18 * q^3 + 23 * q^4 + 6 * q^5 + q^6 - q^7 + 18 * q^9 + 11 * q^10 + 18 * q^11 - 23 * q^12 - 13 * q^14 - 6 * q^15 + 41 * q^16 - q^17 - q^18 + 12 * q^19 + 8 * q^20 + q^21 - q^22 + 44 * q^23 + 48 * q^25 - 18 * q^27 - 5 * q^28 - 17 * q^29 - 11 * q^30 - 18 * q^31 - q^32 - 18 * q^33 + 8 * q^34 + 8 * q^35 + 23 * q^36 + 39 * q^37 + 4 * q^38 + 8 * q^40 + 6 * q^41 + 13 * q^42 + 36 * q^43 + 23 * q^44 + 6 * q^45 - 36 * q^46 + 19 * q^47 - 41 * q^48 + 41 * q^49 - 19 * q^50 + q^51 + 7 * q^53 + q^54 + 6 * q^55 - 73 * q^56 - 12 * q^57 + 69 * q^58 - 16 * q^59 - 8 * q^60 + 21 * q^61 + 8 * q^62 - q^63 + 96 * q^64 + q^66 + 40 * q^68 - 44 * q^69 - 107 * q^70 + 12 * q^71 - 7 * q^73 - 14 * q^74 - 48 * q^75 + 30 * q^76 - q^77 + 62 * q^79 + 58 * q^80 + 18 * q^81 + 44 * q^82 - 11 * q^83 + 5 * q^84 - 46 * q^85 + 19 * q^86 + 17 * q^87 + 17 * q^89 + 11 * q^90 + 100 * q^92 + 18 * q^93 + 31 * q^94 + 42 * q^95 + q^96 + 35 * q^97 - 94 * q^98 + 18 * q^99

Introduction

L-functions

Modular forms

Varieties

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Database

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