LMFDB - Embedded newform 6422.2.a.bj.1.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6422,2,Mod(1,6422)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6422, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("6422.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6422 = 2 \cdot 13^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6422.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:	$51.2799281781$
Analytic rank:	$0$
Dimension:	$9$
Coefficient field:	$\mathbb{Q}[x]/(x^{9} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{9} - x^{8} - 12x^{7} + 14x^{6} + 35x^{5} - 35x^{4} - 28x^{3} + 10x^{2} + 4x - 1 $ x^9 - x^8 - 12x^7 + 14x^6 + 35x^5 - 35x^4 - 28x^3 + 10x^2 + 4*x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$-2.97314$ of defining polynomial
Character	$\chi$	$=$	6422.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +1.04618 q^{3} +1.00000 q^{4} -0.929245 q^{5} -1.04618 q^{6} +2.89421 q^{7} -1.00000 q^{8} -1.90551 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} +1.04618 q^{3} +1.00000 q^{4} -0.929245 q^{5} -1.04618 q^{6} +2.89421 q^{7} -1.00000 q^{8} -1.90551 q^{9} +0.929245 q^{10} -3.97504 q^{11} +1.04618 q^{12} -2.89421 q^{14} -0.972156 q^{15} +1.00000 q^{16} -4.54056 q^{17} +1.90551 q^{18} +1.00000 q^{19} -0.929245 q^{20} +3.02786 q^{21} +3.97504 q^{22} -4.09689 q^{23} -1.04618 q^{24} -4.13650 q^{25} -5.13204 q^{27} +2.89421 q^{28} +1.64963 q^{29} +0.972156 q^{30} +7.79245 q^{31} -1.00000 q^{32} -4.15860 q^{33} +4.54056 q^{34} -2.68943 q^{35} -1.90551 q^{36} -2.85188 q^{37} -1.00000 q^{38} +0.929245 q^{40} +0.453243 q^{41} -3.02786 q^{42} +4.16877 q^{43} -3.97504 q^{44} +1.77069 q^{45} +4.09689 q^{46} +9.08685 q^{47} +1.04618 q^{48} +1.37645 q^{49} +4.13650 q^{50} -4.75023 q^{51} -3.63605 q^{53} +5.13204 q^{54} +3.69378 q^{55} -2.89421 q^{56} +1.04618 q^{57} -1.64963 q^{58} -9.02326 q^{59} -0.972156 q^{60} +4.87261 q^{61} -7.79245 q^{62} -5.51495 q^{63} +1.00000 q^{64} +4.15860 q^{66} +14.2461 q^{67} -4.54056 q^{68} -4.28608 q^{69} +2.68943 q^{70} +10.8788 q^{71} +1.90551 q^{72} +11.6900 q^{73} +2.85188 q^{74} -4.32752 q^{75} +1.00000 q^{76} -11.5046 q^{77} +9.60591 q^{79} -0.929245 q^{80} +0.347501 q^{81} -0.453243 q^{82} +4.33877 q^{83} +3.02786 q^{84} +4.21929 q^{85} -4.16877 q^{86} +1.72581 q^{87} +3.97504 q^{88} -6.23045 q^{89} -1.77069 q^{90} -4.09689 q^{92} +8.15229 q^{93} -9.08685 q^{94} -0.929245 q^{95} -1.04618 q^{96} +11.5845 q^{97} -1.37645 q^{98} +7.57447 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q - 9 q^{2} + q^{3} + 9 q^{4} - q^{5} - q^{6} + 13 q^{7} - 9 q^{8} - 2 q^{9}+O(q^{10}) $ 9 * q - 9 * q^2 + q^3 + 9 * q^4 - q^5 - q^6 + 13 * q^7 - 9 * q^8 - 2 * q^9	$ 9 q - 9 q^{2} + q^{3} + 9 q^{4} - q^{5} - q^{6} + 13 q^{7} - 9 q^{8} - 2 q^{9} + q^{10} + 3 q^{11} + q^{12} - 13 q^{14} + 3 q^{15} + 9 q^{16} - 8 q^{17} + 2 q^{18} + 9 q^{19} - q^{20} + 24 q^{21} - 3 q^{22} - 10 q^{23} - q^{24} + 8 q^{25} + 10 q^{27} + 13 q^{28} - 20 q^{29} - 3 q^{30} + q^{31} - 9 q^{32} - 2 q^{33} + 8 q^{34} + 4 q^{35} - 2 q^{36} + 15 q^{37} - 9 q^{38} + q^{40} + 19 q^{41} - 24 q^{42} - 16 q^{43} + 3 q^{44} + 15 q^{45} + 10 q^{46} + 18 q^{47} + q^{48} + 18 q^{49} - 8 q^{50} - 11 q^{51} + 17 q^{53} - 10 q^{54} - 26 q^{55} - 13 q^{56} + q^{57} + 20 q^{58} + 24 q^{59} + 3 q^{60} + 6 q^{61} - q^{62} + q^{63} + 9 q^{64} + 2 q^{66} + 29 q^{67} - 8 q^{68} - 12 q^{69} - 4 q^{70} + 23 q^{71} + 2 q^{72} + 38 q^{73} - 15 q^{74} + 11 q^{75} + 9 q^{76} - 40 q^{77} - 20 q^{79} - q^{80} - 31 q^{81} - 19 q^{82} + 20 q^{83} + 24 q^{84} + 39 q^{85} + 16 q^{86} - 10 q^{87} - 3 q^{88} - 7 q^{89} - 15 q^{90} - 10 q^{92} + 11 q^{93} - 18 q^{94} - q^{95} - q^{96} + 28 q^{97} - 18 q^{98} + 25 q^{99}+O(q^{100}) $ 9 * q - 9 * q^2 + q^3 + 9 * q^4 - q^5 - q^6 + 13 * q^7 - 9 * q^8 - 2 * q^9 + q^10 + 3 * q^11 + q^12 - 13 * q^14 + 3 * q^15 + 9 * q^16 - 8 * q^17 + 2 * q^18 + 9 * q^19 - q^20 + 24 * q^21 - 3 * q^22 - 10 * q^23 - q^24 + 8 * q^25 + 10 * q^27 + 13 * q^28 - 20 * q^29 - 3 * q^30 + q^31 - 9 * q^32 - 2 * q^33 + 8 * q^34 + 4 * q^35 - 2 * q^36 + 15 * q^37 - 9 * q^38 + q^40 + 19 * q^41 - 24 * q^42 - 16 * q^43 + 3 * q^44 + 15 * q^45 + 10 * q^46 + 18 * q^47 + q^48 + 18 * q^49 - 8 * q^50 - 11 * q^51 + 17 * q^53 - 10 * q^54 - 26 * q^55 - 13 * q^56 + q^57 + 20 * q^58 + 24 * q^59 + 3 * q^60 + 6 * q^61 - q^62 + q^63 + 9 * q^64 + 2 * q^66 + 29 * q^67 - 8 * q^68 - 12 * q^69 - 4 * q^70 + 23 * q^71 + 2 * q^72 + 38 * q^73 - 15 * q^74 + 11 * q^75 + 9 * q^76 - 40 * q^77 - 20 * q^79 - q^80 - 31 * q^81 - 19 * q^82 + 20 * q^83 + 24 * q^84 + 39 * q^85 + 16 * q^86 - 10 * q^87 - 3 * q^88 - 7 * q^89 - 15 * q^90 - 10 * q^92 + 11 * q^93 - 18 * q^94 - q^95 - q^96 + 28 * q^97 - 18 * q^98 + 25 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	1.04618	0.604011	0.302006	−	0.953306i	$-0.402344\pi$
$3$	1.04618	0.604011	0.302006	+	0.953306i	$0.402344\pi$
$4$	1.00000	0.500000
$5$	−0.929245	−0.415571	−0.207785	−	0.978174i	$-0.566626\pi$
$5$	−0.929245	−0.415571	−0.207785	+	0.978174i	$0.566626\pi$
$6$	−1.04618	−0.427101
$7$	2.89421	1.09391	0.546954	−	0.837162i	$-0.315787\pi$
$7$	2.89421	1.09391	0.546954	+	0.837162i	$0.315787\pi$
$8$	−1.00000	−0.353553
$9$	−1.90551	−0.635170
$10$	0.929245	0.293853
$11$	−3.97504	−1.19852	−0.599259	−	0.800555i	$-0.704538\pi$
$11$	−3.97504	−1.19852	−0.599259	+	0.800555i	$0.704538\pi$
$12$	1.04618	0.302006
$13$	0	0
$13$	0	0
$14$	−2.89421	−0.773510
$15$	−0.972156	−0.251010
$16$	1.00000	0.250000
$17$	−4.54056	−1.10125	−0.550623	−	0.834754i	$-0.685610\pi$
$17$	−4.54056	−1.10125	−0.550623	+	0.834754i	$0.685610\pi$
$18$	1.90551	0.449133
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	−0.929245	−0.207785
$21$	3.02786	0.660733
$22$	3.97504	0.847480
$23$	−4.09689	−0.854262	−0.427131	−	0.904190i	$-0.640476\pi$
$23$	−4.09689	−0.854262	−0.427131	+	0.904190i	$0.640476\pi$
$24$	−1.04618	−0.213550
$25$	−4.13650	−0.827301
$26$	0	0
$27$	−5.13204	−0.987662
$28$	2.89421	0.546954
$29$	1.64963	0.306329	0.153164	−	0.988201i	$-0.451054\pi$
$29$	1.64963	0.306329	0.153164	+	0.988201i	$0.451054\pi$
$30$	0.972156	0.177491
$31$	7.79245	1.39957	0.699783	−	0.714356i	$-0.253280\pi$
$31$	7.79245	1.39957	0.699783	+	0.714356i	$0.253280\pi$
$32$	−1.00000	−0.176777
$33$	−4.15860	−0.723919
$34$	4.54056	0.778699
$35$	−2.68943	−0.454597
$36$	−1.90551	−0.317585
$37$	−2.85188	−0.468847	−0.234423	−	0.972135i	$-0.575320\pi$
$37$	−2.85188	−0.468847	−0.234423	+	0.972135i	$0.575320\pi$
$38$	−1.00000	−0.162221
$39$	0	0
$40$	0.929245	0.146926
$41$	0.453243	0.0707847	0.0353923	−	0.999373i	$-0.488732\pi$
$41$	0.453243	0.0707847	0.0353923	+	0.999373i	$0.488732\pi$
$42$	−3.02786	−0.467209
$43$	4.16877	0.635732	0.317866	−	0.948136i	$-0.397034\pi$
$43$	4.16877	0.635732	0.317866	+	0.948136i	$0.397034\pi$
$44$	−3.97504	−0.599259
$45$	1.77069	0.263958
$46$	4.09689	0.604054
$47$	9.08685	1.32545	0.662727	−	0.748862i	$-0.269399\pi$
$47$	9.08685	1.32545	0.662727	+	0.748862i	$0.269399\pi$
$48$	1.04618	0.151003
$49$	1.37645	0.196636
$50$	4.13650	0.584990
$51$	−4.75023	−0.665166
$52$	0	0
$53$	−3.63605	−0.499450	−0.249725	−	0.968317i	$-0.580340\pi$
$53$	−3.63605	−0.499450	−0.249725	+	0.968317i	$0.580340\pi$
$54$	5.13204	0.698382
$55$	3.69378	0.498069
$56$	−2.89421	−0.386755
$57$	1.04618	0.138570
$58$	−1.64963	−0.216607
$59$	−9.02326	−1.17473	−0.587364	−	0.809323i	$-0.699834\pi$
$59$	−9.02326	−1.17473	−0.587364	+	0.809323i	$0.699834\pi$
$60$	−0.972156	−0.125505
$61$	4.87261	0.623874	0.311937	−	0.950103i	$-0.399022\pi$
$61$	4.87261	0.623874	0.311937	+	0.950103i	$0.399022\pi$
$62$	−7.79245	−0.989642
$63$	−5.51495	−0.694818
$64$	1.00000	0.125000
$65$	0	0
$66$	4.15860	0.511888
$67$	14.2461	1.74043	0.870217	−	0.492669i	$-0.163979\pi$
$67$	14.2461	1.74043	0.870217	+	0.492669i	$0.163979\pi$
$68$	−4.54056	−0.550623
$69$	−4.28608	−0.515984
$70$	2.68943	0.321448
$71$	10.8788	1.29107	0.645535	−	0.763731i	$-0.276634\pi$
$71$	10.8788	1.29107	0.645535	+	0.763731i	$0.276634\pi$
$72$	1.90551	0.224567
$73$	11.6900	1.36821	0.684105	−	0.729384i	$-0.260193\pi$
$73$	11.6900	1.36821	0.684105	+	0.729384i	$0.260193\pi$
$74$	2.85188	0.331525
$75$	−4.32752	−0.499699
$76$	1.00000	0.114708
$77$	−11.5046	−1.31107
$78$	0	0
$79$	9.60591	1.08075	0.540375	−	0.841425i	$-0.318283\pi$
$79$	9.60591	1.08075	0.540375	+	0.841425i	$0.318283\pi$
$80$	−0.929245	−0.103893
$81$	0.347501	0.0386112
$82$	−0.453243	−0.0500523
$83$	4.33877	0.476242	0.238121	−	0.971235i	$-0.423469\pi$
$83$	4.33877	0.476242	0.238121	+	0.971235i	$0.423469\pi$
$84$	3.02786	0.330367
$85$	4.21929	0.457646
$86$	−4.16877	−0.449530
$87$	1.72581	0.185026
$88$	3.97504	0.423740
$89$	−6.23045	−0.660427	−0.330213	−	0.943906i	$-0.607121\pi$
$89$	−6.23045	−0.660427	−0.330213	+	0.943906i	$0.607121\pi$
$90$	−1.77069	−0.186647
$91$	0	0
$92$	−4.09689	−0.427131
$93$	8.15229	0.845354
$94$	−9.08685	−0.937237
$95$	−0.929245	−0.0953385
$96$	−1.04618	−0.106775
$97$	11.5845	1.17623	0.588116	−	0.808777i	$-0.299870\pi$
$97$	11.5845	1.17623	0.588116	+	0.808777i	$0.299870\pi$
$98$	−1.37645	−0.139043
$99$	7.57447	0.761263
$100$	−4.13650	−0.413650
$101$	9.68495	0.963689	0.481844	−	0.876257i	$-0.339967\pi$
$101$	9.68495	0.963689	0.481844	+	0.876257i	$0.339967\pi$
$102$	4.75023	0.470343
$103$	−0.0784942	−0.00773427	−0.00386713	−	0.999993i	$-0.501231\pi$
$103$	−0.0784942	−0.00773427	−0.00386713	+	0.999993i	$0.501231\pi$
$104$	0	0
$105$	−2.81362	−0.274582
$106$	3.63605	0.353165
$107$	1.97833	0.191253	0.0956264	−	0.995417i	$-0.469515\pi$
$107$	1.97833	0.191253	0.0956264	+	0.995417i	$0.469515\pi$
$108$	−5.13204	−0.493831
$109$	−17.0468	−1.63279	−0.816395	−	0.577494i	$-0.804031\pi$
$109$	−17.0468	−1.63279	−0.816395	+	0.577494i	$0.804031\pi$
$110$	−3.69378	−0.352188
$111$	−2.98358	−0.283189
$112$	2.89421	0.273477
$113$	7.23739	0.680836	0.340418	−	0.940274i	$-0.389431\pi$
$113$	7.23739	0.680836	0.340418	+	0.940274i	$0.389431\pi$
$114$	−1.04618	−0.0979836
$115$	3.80702	0.355006
$116$	1.64963	0.153164
$117$	0	0
$118$	9.02326	0.830659
$119$	−13.1413	−1.20466
$120$	0.972156	0.0887453
$121$	4.80091	0.436446
$122$	−4.87261	−0.441145
$123$	0.474173	0.0427547
$124$	7.79245	0.699783
$125$	8.49005	0.759373
$126$	5.51495	0.491311
$127$	−10.6484	−0.944893	−0.472447	−	0.881359i	$-0.656629\pi$
$127$	−10.6484	−0.944893	−0.472447	+	0.881359i	$0.656629\pi$
$128$	−1.00000	−0.0883883
$129$	4.36128	0.383989
$130$	0	0
$131$	17.6085	1.53846	0.769232	−	0.638970i	$-0.220639\pi$
$131$	17.6085	1.53846	0.769232	+	0.638970i	$0.220639\pi$
$132$	−4.15860	−0.361959
$133$	2.89421	0.250960
$134$	−14.2461	−1.23067
$135$	4.76892	0.410443
$136$	4.54056	0.389349
$137$	10.1912	0.870689	0.435345	−	0.900264i	$-0.356626\pi$
$137$	10.1912	0.870689	0.435345	+	0.900264i	$0.356626\pi$
$138$	4.28608	0.364856
$139$	18.1509	1.53954	0.769771	−	0.638320i	$-0.220370\pi$
$139$	18.1509	1.53954	0.769771	+	0.638320i	$0.220370\pi$
$140$	−2.68943	−0.227298
$141$	9.50647	0.800589
$142$	−10.8788	−0.912924
$143$	0	0
$144$	−1.90551	−0.158793
$145$	−1.53291	−0.127301
$146$	−11.6900	−0.967470
$147$	1.44002	0.118771
$148$	−2.85188	−0.234423
$149$	2.69811	0.221038	0.110519	−	0.993874i	$-0.464749\pi$
$149$	2.69811	0.221038	0.110519	+	0.993874i	$0.464749\pi$
$150$	4.32752	0.353341
$151$	−5.24220	−0.426604	−0.213302	−	0.976986i	$-0.568422\pi$
$151$	−5.24220	−0.426604	−0.213302	+	0.976986i	$0.568422\pi$
$152$	−1.00000	−0.0811107
$153$	8.65208	0.699479
$154$	11.5046	0.927066
$155$	−7.24109	−0.581619
$156$	0	0
$157$	2.20220	0.175755	0.0878774	−	0.996131i	$-0.471992\pi$
$157$	2.20220	0.175755	0.0878774	+	0.996131i	$0.471992\pi$
$158$	−9.60591	−0.764205
$159$	−3.80396	−0.301674
$160$	0.929245	0.0734632
$161$	−11.8573	−0.934484
$162$	−0.347501	−0.0273022
$163$	21.2249	1.66246	0.831232	−	0.555926i	$-0.187636\pi$
$163$	21.2249	1.66246	0.831232	+	0.555926i	$0.187636\pi$
$164$	0.453243	0.0353923
$165$	3.86435	0.300840
$166$	−4.33877	−0.336754
$167$	16.6096	1.28529	0.642645	−	0.766164i	$-0.277837\pi$
$167$	16.6096	1.28529	0.642645	+	0.766164i	$0.277837\pi$
$168$	−3.02786	−0.233605
$169$	0	0
$170$	−4.21929	−0.323605
$171$	−1.90551	−0.145718
$172$	4.16877	0.317866
$173$	−5.22082	−0.396932	−0.198466	−	0.980108i	$-0.563596\pi$
$173$	−5.22082	−0.396932	−0.198466	+	0.980108i	$0.563596\pi$
$174$	−1.72581	−0.130833
$175$	−11.9719	−0.904992
$176$	−3.97504	−0.299630
$177$	−9.43994	−0.709550
$178$	6.23045	0.466992
$179$	−3.64551	−0.272478	−0.136239	−	0.990676i	$-0.543502\pi$
$179$	−3.64551	−0.272478	−0.136239	+	0.990676i	$0.543502\pi$
$180$	1.77069	0.131979
$181$	−14.4527	−1.07426	−0.537131	−	0.843499i	$-0.680492\pi$
$181$	−14.4527	−1.07426	−0.537131	+	0.843499i	$0.680492\pi$
$182$	0	0
$183$	5.09762	0.376827
$184$	4.09689	0.302027
$185$	2.65010	0.194839
$186$	−8.15229	−0.597755
$187$	18.0489	1.31986
$188$	9.08685	0.662727
$189$	−14.8532	−1.08041
$190$	0.929245	0.0674145
$191$	−7.34389	−0.531386	−0.265693	−	0.964058i	$-0.585601\pi$
$191$	−7.34389	−0.531386	−0.265693	+	0.964058i	$0.585601\pi$
$192$	1.04618	0.0755014
$193$	−9.57969	−0.689561	−0.344781	−	0.938683i	$-0.612047\pi$
$193$	−9.57969	−0.689561	−0.344781	+	0.938683i	$0.612047\pi$
$194$	−11.5845	−0.831722
$195$	0	0
$196$	1.37645	0.0983182
$197$	−1.25654	−0.0895246	−0.0447623	−	0.998998i	$-0.514253\pi$
$197$	−1.25654	−0.0895246	−0.0447623	+	0.998998i	$0.514253\pi$
$198$	−7.57447	−0.538294
$199$	5.97439	0.423513	0.211757	−	0.977322i	$-0.432082\pi$
$199$	5.97439	0.423513	0.211757	+	0.977322i	$0.432082\pi$
$200$	4.13650	0.292495
$201$	14.9039	1.05124
$202$	−9.68495	−0.681431
$203$	4.77438	0.335096
$204$	−4.75023	−0.332583
$205$	−0.421174	−0.0294160
$206$	0.0784942	0.00546895
$207$	7.80668	0.542601
$208$	0	0
$209$	−3.97504	−0.274959
$210$	2.81362	0.194158
$211$	−2.93090	−0.201772	−0.100886	−	0.994898i	$-0.532168\pi$
$211$	−2.93090	−0.201772	−0.100886	+	0.994898i	$0.532168\pi$
$212$	−3.63605	−0.249725
$213$	11.3811	0.779821
$214$	−1.97833	−0.135236
$215$	−3.87381	−0.264192
$216$	5.13204	0.349191
$217$	22.5530	1.53100
$218$	17.0468	1.15456
$219$	12.2298	0.826414
$220$	3.69378	0.249035
$221$	0	0
$222$	2.98358	0.200245
$223$	−21.7936	−1.45941	−0.729705	−	0.683762i	$-0.760343\pi$
$223$	−21.7936	−1.45941	−0.729705	+	0.683762i	$0.760343\pi$
$224$	−2.89421	−0.193378
$225$	7.88215	0.525477
$226$	−7.23739	−0.481424
$227$	18.0721	1.19949	0.599743	−	0.800192i	$-0.295269\pi$
$227$	18.0721	1.19949	0.599743	+	0.800192i	$0.295269\pi$
$228$	1.04618	0.0692849
$229$	20.7571	1.37167	0.685833	−	0.727759i	$-0.259438\pi$
$229$	20.7571	1.37167	0.685833	+	0.727759i	$0.259438\pi$
$230$	−3.80702	−0.251027
$231$	−12.0359	−0.791901
$232$	−1.64963	−0.108304
$233$	−26.8846	−1.76127	−0.880636	−	0.473794i	$-0.842884\pi$
$233$	−26.8846	−1.76127	−0.880636	+	0.473794i	$0.842884\pi$
$234$	0	0
$235$	−8.44390	−0.550820
$236$	−9.02326	−0.587364
$237$	10.0495	0.652785
$238$	13.1413	0.851826
$239$	−12.0208	−0.777559	−0.388779	−	0.921331i	$-0.627103\pi$
$239$	−12.0208	−0.777559	−0.388779	+	0.921331i	$0.627103\pi$
$240$	−0.972156	−0.0627524
$241$	16.9086	1.08918	0.544588	−	0.838704i	$-0.316686\pi$
$241$	16.9086	1.08918	0.544588	+	0.838704i	$0.316686\pi$
$242$	−4.80091	−0.308614
$243$	15.7597	1.01098
$244$	4.87261	0.311937
$245$	−1.27906	−0.0817163
$246$	−0.474173	−0.0302322
$247$	0	0
$248$	−7.79245	−0.494821
$249$	4.53913	0.287656
$250$	−8.49005	−0.536958
$251$	24.5896	1.55208	0.776041	−	0.630682i	$-0.217225\pi$
$251$	24.5896	1.55208	0.776041	+	0.630682i	$0.217225\pi$
$252$	−5.51495	−0.347409
$253$	16.2853	1.02385
$254$	10.6484	0.668140
$255$	4.41413	0.276423
$256$	1.00000	0.0625000
$257$	−20.6988	−1.29115	−0.645577	−	0.763695i	$-0.723383\pi$
$257$	−20.6988	−1.29115	−0.645577	+	0.763695i	$0.723383\pi$
$258$	−4.36128	−0.271521
$259$	−8.25395	−0.512876
$260$	0	0
$261$	−3.14339	−0.194571
$262$	−17.6085	−1.08786
$263$	−24.3236	−1.49986	−0.749930	−	0.661517i	$-0.769913\pi$
$263$	−24.3236	−1.49986	−0.749930	+	0.661517i	$0.769913\pi$
$264$	4.15860	0.255944
$265$	3.37878	0.207557
$266$	−2.89421	−0.177455
$267$	−6.51817	−0.398905
$268$	14.2461	0.870217
$269$	−0.385169	−0.0234842	−0.0117421	−	0.999931i	$-0.503738\pi$
$269$	−0.385169	−0.0234842	−0.0117421	+	0.999931i	$0.503738\pi$
$270$	−4.76892	−0.290227
$271$	−21.8256	−1.32581	−0.662906	−	0.748702i	$-0.730677\pi$
$271$	−21.8256	−1.32581	−0.662906	+	0.748702i	$0.730677\pi$
$272$	−4.54056	−0.275312
$273$	0	0
$274$	−10.1912	−0.615670
$275$	16.4428	0.991535
$276$	−4.28608	−0.257992
$277$	−12.9100	−0.775687	−0.387843	−	0.921725i	$-0.626780\pi$
$277$	−12.9100	−0.775687	−0.387843	+	0.921725i	$0.626780\pi$
$278$	−18.1509	−1.08862
$279$	−14.8486	−0.888962
$280$	2.68943	0.160724
$281$	6.07209	0.362230	0.181115	−	0.983462i	$-0.442029\pi$
$281$	6.07209	0.362230	0.181115	+	0.983462i	$0.442029\pi$
$282$	−9.50647	−0.566102
$283$	12.3131	0.731940	0.365970	−	0.930627i	$-0.380737\pi$
$283$	12.3131	0.731940	0.365970	+	0.930627i	$0.380737\pi$
$284$	10.8788	0.645535
$285$	−0.972156	−0.0575855
$286$	0	0
$287$	1.31178	0.0774320
$288$	1.90551	0.112283
$289$	3.61665	0.212744
$290$	1.53291	0.0900156
$291$	12.1195	0.710458
$292$	11.6900	0.684105
$293$	7.44576	0.434986	0.217493	−	0.976062i	$-0.430212\pi$
$293$	7.44576	0.434986	0.217493	+	0.976062i	$0.430212\pi$
$294$	−1.44002	−0.0839835
$295$	8.38482	0.488183
$296$	2.85188	0.165762
$297$	20.4000	1.18373
$298$	−2.69811	−0.156297
$299$	0	0
$300$	−4.32752	−0.249850
$301$	12.0653	0.695433
$302$	5.24220	0.301655
$303$	10.1322	0.582079
$304$	1.00000	0.0573539
$305$	−4.52785	−0.259264
$306$	−8.65208	−0.494606
$307$	−18.8708	−1.07701	−0.538506	−	0.842622i	$-0.681011\pi$
$307$	−18.8708	−1.07701	−0.538506	+	0.842622i	$0.681011\pi$
$308$	−11.5046	−0.655535
$309$	−0.0821190	−0.00467159
$310$	7.24109	0.411266
$311$	5.01837	0.284566	0.142283	−	0.989826i	$-0.454556\pi$
$311$	5.01837	0.284566	0.142283	+	0.989826i	$0.454556\pi$
$312$	0	0
$313$	−22.5284	−1.27338	−0.636689	−	0.771121i	$-0.719696\pi$
$313$	−22.5284	−1.27338	−0.636689	+	0.771121i	$0.719696\pi$
$314$	−2.20220	−0.124277
$315$	5.12474	0.288746
$316$	9.60591	0.540375
$317$	−17.2007	−0.966086	−0.483043	−	0.875597i	$-0.660468\pi$
$317$	−17.2007	−0.966086	−0.483043	+	0.875597i	$0.660468\pi$
$318$	3.80396	0.213316
$319$	−6.55734	−0.367141
$320$	−0.929245	−0.0519464
$321$	2.06969	0.115519
$322$	11.8573	0.660780
$323$	−4.54056	−0.252643
$324$	0.347501	0.0193056
$325$	0	0
$326$	−21.2249	−1.17554
$327$	−17.8340	−0.986224
$328$	−0.453243	−0.0250262
$329$	26.2993	1.44992
$330$	−3.86435	−0.212726
$331$	7.57755	0.416500	0.208250	−	0.978076i	$-0.433223\pi$
$331$	7.57755	0.416500	0.208250	+	0.978076i	$0.433223\pi$
$332$	4.33877	0.238121
$333$	5.43429	0.297797
$334$	−16.6096	−0.908837
$335$	−13.2381	−0.723273
$336$	3.02786	0.165183
$337$	12.4953	0.680665	0.340333	−	0.940305i	$-0.389460\pi$
$337$	12.4953	0.680665	0.340333	+	0.940305i	$0.389460\pi$
$338$	0	0
$339$	7.57160	0.411233
$340$	4.21929	0.228823
$341$	−30.9753	−1.67740
$342$	1.90551	0.103038
$343$	−16.2757	−0.878807
$344$	−4.16877	−0.224765
$345$	3.98282	0.214428
$346$	5.22082	0.280673
$347$	−9.64219	−0.517620	−0.258810	−	0.965928i	$-0.583330\pi$
$347$	−9.64219	−0.517620	−0.258810	+	0.965928i	$0.583330\pi$
$348$	1.72581	0.0925130
$349$	−30.2434	−1.61889	−0.809446	−	0.587195i	$-0.800232\pi$
$349$	−30.2434	−1.61889	−0.809446	+	0.587195i	$0.800232\pi$
$350$	11.9719	0.639926
$351$	0	0
$352$	3.97504	0.211870
$353$	−11.1133	−0.591500	−0.295750	−	0.955265i	$-0.595569\pi$
$353$	−11.1133	−0.591500	−0.295750	+	0.955265i	$0.595569\pi$
$354$	9.43994	0.501727
$355$	−10.1090	−0.536531
$356$	−6.23045	−0.330213
$357$	−13.7482	−0.727631
$358$	3.64551	0.192671
$359$	29.5541	1.55981	0.779904	−	0.625899i	$-0.215268\pi$
$359$	29.5541	1.55981	0.779904	+	0.625899i	$0.215268\pi$
$360$	−1.77069	−0.0933233
$361$	1.00000	0.0526316
$362$	14.4527	0.759618
$363$	5.02261	0.263618
$364$	0	0
$365$	−10.8629	−0.568588
$366$	−5.09762	−0.266457
$367$	30.4669	1.59036	0.795179	−	0.606375i	$-0.207377\pi$
$367$	30.4669	1.59036	0.795179	+	0.606375i	$0.207377\pi$
$368$	−4.09689	−0.213565
$369$	−0.863659	−0.0449603
$370$	−2.65010	−0.137772
$371$	−10.5235	−0.546353
$372$	8.15229	0.422677
$373$	28.9669	1.49985	0.749924	−	0.661524i	$-0.230090\pi$
$373$	28.9669	1.49985	0.749924	+	0.661524i	$0.230090\pi$
$374$	−18.0489	−0.933285
$375$	8.88211	0.458670
$376$	−9.08685	−0.468618
$377$	0	0
$378$	14.8532	0.763966
$379$	32.9190	1.69094	0.845468	−	0.534026i	$-0.179321\pi$
$379$	32.9190	1.69094	0.845468	+	0.534026i	$0.179321\pi$
$380$	−0.929245	−0.0476692
$381$	−11.1401	−0.570726
$382$	7.34389	0.375746
$383$	17.1752	0.877610	0.438805	−	0.898582i	$-0.355402\pi$
$383$	17.1752	0.877610	0.438805	+	0.898582i	$0.355402\pi$
$384$	−1.04618	−0.0533876
$385$	10.6906	0.544842
$386$	9.57969	0.487593
$387$	−7.94364	−0.403798
$388$	11.5845	0.588116
$389$	1.10030	0.0557876	0.0278938	−	0.999611i	$-0.491120\pi$
$389$	1.10030	0.0557876	0.0278938	+	0.999611i	$0.491120\pi$
$390$	0	0
$391$	18.6022	0.940753
$392$	−1.37645	−0.0695214
$393$	18.4217	0.929250
$394$	1.25654	0.0633035
$395$	−8.92624	−0.449128
$396$	7.57447	0.380631
$397$	−23.3055	−1.16967	−0.584836	−	0.811152i	$-0.698841\pi$
$397$	−23.3055	−1.16967	−0.584836	+	0.811152i	$0.698841\pi$
$398$	−5.97439	−0.299469
$399$	3.02786	0.151583
$400$	−4.13650	−0.206825
$401$	−27.1581	−1.35621	−0.678106	−	0.734964i	$-0.737199\pi$
$401$	−27.1581	−1.35621	−0.678106	+	0.734964i	$0.737199\pi$
$402$	−14.9039	−0.743340
$403$	0	0
$404$	9.68495	0.481844
$405$	−0.322913	−0.0160457
$406$	−4.77438	−0.236948
$407$	11.3363	0.561921
$408$	4.75023	0.235172
$409$	4.47035	0.221044	0.110522	−	0.993874i	$-0.464748\pi$
$409$	4.47035	0.221044	0.110522	+	0.993874i	$0.464748\pi$
$410$	0.421174	0.0208003
$411$	10.6618	0.525906
$412$	−0.0784942	−0.00386713
$413$	−26.1152	−1.28505
$414$	−7.80668	−0.383677
$415$	−4.03178	−0.197912
$416$	0	0
$417$	18.9891	0.929901
$418$	3.97504	0.194425
$419$	7.62391	0.372453	0.186226	−	0.982507i	$-0.440374\pi$
$419$	7.62391	0.372453	0.186226	+	0.982507i	$0.440374\pi$
$420$	−2.81362	−0.137291
$421$	31.6349	1.54179	0.770896	−	0.636962i	$-0.219809\pi$
$421$	31.6349	1.54179	0.770896	+	0.636962i	$0.219809\pi$
$422$	2.93090	0.142674
$423$	−17.3151	−0.841888
$424$	3.63605	0.176582
$425$	18.7820	0.911062
$426$	−11.3811	−0.551417
$427$	14.1024	0.682461
$428$	1.97833	0.0956264
$429$	0	0
$430$	3.87381	0.186812
$431$	5.99194	0.288621	0.144311	−	0.989532i	$-0.453904\pi$
$431$	5.99194	0.288621	0.144311	+	0.989532i	$0.453904\pi$
$432$	−5.13204	−0.246915
$433$	22.3035	1.07184	0.535919	−	0.844270i	$-0.319965\pi$
$433$	22.3035	1.07184	0.535919	+	0.844270i	$0.319965\pi$
$434$	−22.5530	−1.08258
$435$	−1.60370	−0.0768914
$436$	−17.0468	−0.816395
$437$	−4.09689	−0.195981
$438$	−12.2298	−0.584363
$439$	−8.13560	−0.388291	−0.194146	−	0.980973i	$-0.562193\pi$
$439$	−8.13560	−0.388291	−0.194146	+	0.980973i	$0.562193\pi$
$440$	−3.69378	−0.176094
$441$	−2.62285	−0.124898
$442$	0	0
$443$	9.37320	0.445334	0.222667	−	0.974895i	$-0.428524\pi$
$443$	9.37320	0.445334	0.222667	+	0.974895i	$0.428524\pi$
$444$	−2.98358	−0.141594
$445$	5.78961	0.274454
$446$	21.7936	1.03196
$447$	2.82271	0.133509
$448$	2.89421	0.136739
$449$	−23.5049	−1.10926	−0.554632	−	0.832096i	$-0.687141\pi$
$449$	−23.5049	−1.10926	−0.554632	+	0.832096i	$0.687141\pi$
$450$	−7.88215	−0.371568
$451$	−1.80166	−0.0848367
$452$	7.23739	0.340418
$453$	−5.48428	−0.257674
$454$	−18.0721	−0.848165
$455$	0	0
$456$	−1.04618	−0.0489918
$457$	−7.02736	−0.328726	−0.164363	−	0.986400i	$-0.552557\pi$
$457$	−7.02736	−0.328726	−0.164363	+	0.986400i	$0.552557\pi$
$458$	−20.7571	−0.969915
$459$	23.3023	1.08766
$460$	3.80702	0.177503
$461$	−2.09583	−0.0976127	−0.0488064	−	0.998808i	$-0.515542\pi$
$461$	−2.09583	−0.0976127	−0.0488064	+	0.998808i	$0.515542\pi$
$462$	12.0359	0.559959
$463$	22.3671	1.03949	0.519743	−	0.854323i	$-0.326028\pi$
$463$	22.3671	1.03949	0.519743	+	0.854323i	$0.326028\pi$
$464$	1.64963	0.0765822
$465$	−7.57548	−0.351304
$466$	26.8846	1.24541
$467$	36.8013	1.70296	0.851480	−	0.524387i	$-0.175705\pi$
$467$	36.8013	1.70296	0.851480	+	0.524387i	$0.175705\pi$
$468$	0	0
$469$	41.2311	1.90388
$470$	8.44390	0.389488
$471$	2.30390	0.106158
$472$	9.02326	0.415329
$473$	−16.5710	−0.761936
$474$	−10.0495	−0.461589
$475$	−4.13650	−0.189796
$476$	−13.1413	−0.602332
$477$	6.92854	0.317236
$478$	12.0208	0.549817
$479$	18.0660	0.825459	0.412729	−	0.910854i	$-0.364576\pi$
$479$	18.0660	0.825459	0.412729	+	0.910854i	$0.364576\pi$
$480$	0.972156	0.0443726
$481$	0	0
$482$	−16.9086	−0.770164
$483$	−12.4048	−0.564439
$484$	4.80091	0.218223
$485$	−10.7649	−0.488808
$486$	−15.7597	−0.714873
$487$	−19.7205	−0.893622	−0.446811	−	0.894628i	$-0.647440\pi$
$487$	−19.7205	−0.893622	−0.446811	+	0.894628i	$0.647440\pi$
$488$	−4.87261	−0.220573
$489$	22.2050	1.00415
$490$	1.27906	0.0577822
$491$	−26.6062	−1.20072	−0.600360	−	0.799730i	$-0.704976\pi$
$491$	−26.6062	−1.20072	−0.600360	+	0.799730i	$0.704976\pi$
$492$	0.474173	0.0213774
$493$	−7.49024	−0.337343
$494$	0	0
$495$	−7.03854	−0.316359
$496$	7.79245	0.349891
$497$	31.4854	1.41231
$498$	−4.53913	−0.203403
$499$	5.10306	0.228444	0.114222	−	0.993455i	$-0.463562\pi$
$499$	5.10306	0.228444	0.114222	+	0.993455i	$0.463562\pi$
$500$	8.49005	0.379686
$501$	17.3766	0.776330
$502$	−24.5896	−1.09749
$503$	27.0694	1.20697	0.603483	−	0.797376i	$-0.293779\pi$
$503$	27.0694	1.20697	0.603483	+	0.797376i	$0.293779\pi$
$504$	5.51495	0.245655
$505$	−8.99969	−0.400481
$506$	−16.2853	−0.723970
$507$	0	0
$508$	−10.6484	−0.472447
$509$	−9.16648	−0.406297	−0.203148	−	0.979148i	$-0.565117\pi$
$509$	−9.16648	−0.406297	−0.203148	+	0.979148i	$0.565117\pi$
$510$	−4.41413	−0.195461
$511$	33.8333	1.49670
$512$	−1.00000	−0.0441942
$513$	−5.13204	−0.226585
$514$	20.6988	0.912984
$515$	0.0729404	0.00321414
$516$	4.36128	0.191995
$517$	−36.1205	−1.58858
$518$	8.25395	0.362658
$519$	−5.46191	−0.239751
$520$	0	0
$521$	−30.1666	−1.32162	−0.660811	−	0.750553i	$-0.729787\pi$
$521$	−30.1666	−1.32162	−0.660811	+	0.750553i	$0.729787\pi$
$522$	3.14339	0.137582
$523$	−9.39066	−0.410625	−0.205312	−	0.978697i	$-0.565821\pi$
$523$	−9.39066	−0.410625	−0.205312	+	0.978697i	$0.565821\pi$
$524$	17.6085	0.769232
$525$	−12.5248	−0.546625
$526$	24.3236	1.06056
$527$	−35.3821	−1.54127
$528$	−4.15860	−0.180980
$529$	−6.21545	−0.270237
$530$	−3.37878	−0.146765
$531$	17.1939	0.746153
$532$	2.89421	0.125480
$533$	0	0
$534$	6.51817	0.282069
$535$	−1.83836	−0.0794791
$536$	−14.2461	−0.615336
$537$	−3.81385	−0.164580
$538$	0.385169	0.0166058
$539$	−5.47145	−0.235672
$540$	4.76892	0.205222
$541$	−13.6802	−0.588160	−0.294080	−	0.955781i	$-0.595013\pi$
$541$	−13.6802	−0.588160	−0.294080	+	0.955781i	$0.595013\pi$
$542$	21.8256	0.937491
$543$	−15.1201	−0.648866
$544$	4.54056	0.194675
$545$	15.8407	0.678540
$546$	0	0
$547$	−6.13964	−0.262512	−0.131256	−	0.991349i	$-0.541901\pi$
$547$	−6.13964	−0.262512	−0.131256	+	0.991349i	$0.541901\pi$
$548$	10.1912	0.435345
$549$	−9.28481	−0.396266
$550$	−16.4428	−0.701121
$551$	1.64963	0.0702766
$552$	4.28608	0.182428
$553$	27.8015	1.18224
$554$	12.9100	0.548494
$555$	2.77248	0.117685
$556$	18.1509	0.769771
$557$	−12.9201	−0.547444	−0.273722	−	0.961809i	$-0.588255\pi$
$557$	−12.9201	−0.547444	−0.273722	+	0.961809i	$0.588255\pi$
$558$	14.8486	0.628591
$559$	0	0
$560$	−2.68943	−0.113649
$561$	18.8823	0.797213
$562$	−6.07209	−0.256136
$563$	−39.3705	−1.65927	−0.829634	−	0.558307i	$-0.811451\pi$
$563$	−39.3705	−1.65927	−0.829634	+	0.558307i	$0.811451\pi$
$564$	9.50647	0.400294
$565$	−6.72530	−0.282936
$566$	−12.3131	−0.517560
$567$	1.00574	0.0422371
$568$	−10.8788	−0.456462
$569$	27.7960	1.16527	0.582634	−	0.812734i	$-0.302022\pi$
$569$	27.7960	1.16527	0.582634	+	0.812734i	$0.302022\pi$
$570$	0.972156	0.0407191
$571$	−4.79328	−0.200592	−0.100296	−	0.994958i	$-0.531979\pi$
$571$	−4.79328	−0.200592	−0.100296	+	0.994958i	$0.531979\pi$
$572$	0	0
$573$	−7.68303	−0.320963
$574$	−1.31178	−0.0547527
$575$	16.9468	0.706731
$576$	−1.90551	−0.0793963
$577$	−19.1405	−0.796829	−0.398415	−	0.917205i	$-0.630439\pi$
$577$	−19.1405	−0.796829	−0.398415	+	0.917205i	$0.630439\pi$
$578$	−3.61665	−0.150433
$579$	−10.0221	−0.416503
$580$	−1.53291	−0.0636506
$581$	12.5573	0.520965
$582$	−12.1195	−0.502369
$583$	14.4534	0.598600
$584$	−11.6900	−0.483735
$585$	0	0
$586$	−7.44576	−0.307582
$587$	15.8721	0.655110	0.327555	−	0.944832i	$-0.393775\pi$
$587$	15.8721	0.655110	0.327555	+	0.944832i	$0.393775\pi$
$588$	1.44002	0.0593853
$589$	7.79245	0.321082
$590$	−8.38482	−0.345198
$591$	−1.31456	−0.0540739
$592$	−2.85188	−0.117212
$593$	−10.8277	−0.444642	−0.222321	−	0.974973i	$-0.571363\pi$
$593$	−10.8277	−0.444642	−0.222321	+	0.974973i	$0.571363\pi$
$594$	−20.4000	−0.837024
$595$	12.2115	0.500623
$596$	2.69811	0.110519
$597$	6.25028	0.255807
$598$	0	0
$599$	−0.831670	−0.0339811	−0.0169906	−	0.999856i	$-0.505409\pi$
$599$	−0.831670	−0.0339811	−0.0169906	+	0.999856i	$0.505409\pi$
$600$	4.32752	0.176670
$601$	35.6665	1.45487	0.727433	−	0.686179i	$-0.240713\pi$
$601$	35.6665	1.45487	0.727433	+	0.686179i	$0.240713\pi$
$602$	−12.0653	−0.491745
$603$	−27.1460	−1.10547
$604$	−5.24220	−0.213302
$605$	−4.46122	−0.181374
$606$	−10.1322	−0.411592
$607$	16.8104	0.682311	0.341156	−	0.940007i	$-0.389182\pi$
$607$	16.8104	0.682311	0.341156	+	0.940007i	$0.389182\pi$
$608$	−1.00000	−0.0405554
$609$	4.99485	0.202402
$610$	4.52785	0.183327
$611$	0	0
$612$	8.65208	0.349739
$613$	−27.5552	−1.11294	−0.556471	−	0.830867i	$-0.687845\pi$
$613$	−27.5552	−1.11294	−0.556471	+	0.830867i	$0.687845\pi$
$614$	18.8708	0.761563
$615$	−0.440623	−0.0177676
$616$	11.5046	0.463533
$617$	40.6943	1.63829	0.819145	−	0.573586i	$-0.194448\pi$
$617$	40.6943	1.63829	0.819145	+	0.573586i	$0.194448\pi$
$618$	0.0821190	0.00330331
$619$	1.84653	0.0742183	0.0371092	−	0.999311i	$-0.488185\pi$
$619$	1.84653	0.0742183	0.0371092	+	0.999311i	$0.488185\pi$
$620$	−7.24109	−0.290809
$621$	21.0254	0.843721
$622$	−5.01837	−0.201218
$623$	−18.0322	−0.722446
$624$	0	0
$625$	12.7932	0.511728
$626$	22.5284	0.900414
$627$	−4.15860	−0.166078
$628$	2.20220	0.0878774
$629$	12.9491	0.516316
$630$	−5.12474	−0.204174
$631$	4.53563	0.180560	0.0902802	−	0.995916i	$-0.471224\pi$
$631$	4.53563	0.180560	0.0902802	+	0.995916i	$0.471224\pi$
$632$	−9.60591	−0.382103
$633$	−3.06625	−0.121872
$634$	17.2007	0.683126
$635$	9.89497	0.392670
$636$	−3.80396	−0.150837
$637$	0	0
$638$	6.55734	0.259608
$639$	−20.7296	−0.820049
$640$	0.929245	0.0367316
$641$	−7.09568	−0.280263	−0.140131	−	0.990133i	$-0.544752\pi$
$641$	−7.09568	−0.280263	−0.140131	+	0.990133i	$0.544752\pi$
$642$	−2.06969	−0.0816842
$643$	−31.5892	−1.24576	−0.622879	−	0.782318i	$-0.714037\pi$
$643$	−31.5892	−1.24576	−0.622879	+	0.782318i	$0.714037\pi$
$644$	−11.8573	−0.467242
$645$	−4.05270	−0.159575
$646$	4.54056	0.178646
$647$	3.99801	0.157178	0.0785891	−	0.996907i	$-0.474959\pi$
$647$	3.99801	0.157178	0.0785891	+	0.996907i	$0.474959\pi$
$648$	−0.347501	−0.0136511
$649$	35.8678	1.40793
$650$	0	0
$651$	23.5945	0.924740
$652$	21.2249	0.831232
$653$	21.4463	0.839258	0.419629	−	0.907696i	$-0.362160\pi$
$653$	21.4463	0.839258	0.419629	+	0.907696i	$0.362160\pi$
$654$	17.8340	0.697366
$655$	−16.3626	−0.639341
$656$	0.453243	0.0176962
$657$	−22.2754	−0.869045
$658$	−26.2993	−1.02525
$659$	−21.7818	−0.848497	−0.424248	−	0.905546i	$-0.639462\pi$
$659$	−21.7818	−0.848497	−0.424248	+	0.905546i	$0.639462\pi$
$660$	3.86435	0.150420
$661$	10.0216	0.389795	0.194898	−	0.980824i	$-0.437563\pi$
$661$	10.0216	0.389795	0.194898	+	0.980824i	$0.437563\pi$
$662$	−7.57755	−0.294510
$663$	0	0
$664$	−4.33877	−0.168377
$665$	−2.68943	−0.104292
$666$	−5.43429	−0.210575
$667$	−6.75836	−0.261685
$668$	16.6096	0.642645
$669$	−22.8000	−0.881501
$670$	13.2381	0.511431
$671$	−19.3688	−0.747724
$672$	−3.02786	−0.116802
$673$	−2.72873	−0.105185	−0.0525925	−	0.998616i	$-0.516748\pi$
$673$	−2.72873	−0.105185	−0.0525925	+	0.998616i	$0.516748\pi$
$674$	−12.4953	−0.481303
$675$	21.2287	0.817093
$676$	0	0
$677$	25.2469	0.970315	0.485158	−	0.874427i	$-0.338762\pi$
$677$	25.2469	0.970315	0.485158	+	0.874427i	$0.338762\pi$
$678$	−7.57160	−0.290786
$679$	33.5281	1.28669
$680$	−4.21929	−0.161802
$681$	18.9066	0.724504
$682$	30.9753	1.18610
$683$	42.1086	1.61124	0.805620	−	0.592432i	$-0.201832\pi$
$683$	42.1086	1.61124	0.805620	+	0.592432i	$0.201832\pi$
$684$	−1.90551	−0.0728590
$685$	−9.47008	−0.361833
$686$	16.2757	0.621410
$687$	21.7156	0.828502
$688$	4.16877	0.158933
$689$	0	0
$690$	−3.98282	−0.151623
$691$	39.6111	1.50688	0.753438	−	0.657519i	$-0.228394\pi$
$691$	39.6111	1.50688	0.753438	+	0.657519i	$0.228394\pi$
$692$	−5.22082	−0.198466
$693$	21.9221	0.832752
$694$	9.64219	0.366013
$695$	−16.8667	−0.639789
$696$	−1.72581	−0.0654166
$697$	−2.05798	−0.0779514
$698$	30.2434	1.14473
$699$	−28.1261	−1.06383
$700$	−11.9719	−0.452496
$701$	36.4844	1.37800	0.688998	−	0.724764i	$-0.258051\pi$
$701$	36.4844	1.37800	0.688998	+	0.724764i	$0.258051\pi$
$702$	0	0
$703$	−2.85188	−0.107561
$704$	−3.97504	−0.149815
$705$	−8.83383	−0.332701
$706$	11.1133	0.418253
$707$	28.0303	1.05419
$708$	−9.43994	−0.354775
$709$	−38.2462	−1.43637	−0.718183	−	0.695854i	$-0.755026\pi$
$709$	−38.2462	−1.43637	−0.718183	+	0.695854i	$0.755026\pi$
$710$	10.1090	0.379385
$711$	−18.3042	−0.686460
$712$	6.23045	0.233496
$713$	−31.9248	−1.19559
$714$	13.7482	0.514512
$715$	0	0
$716$	−3.64551	−0.136239
$717$	−12.5759	−0.469654
$718$	−29.5541	−1.10295
$719$	36.6382	1.36637	0.683187	−	0.730244i	$-0.260593\pi$
$719$	36.6382	1.36637	0.683187	+	0.730244i	$0.260593\pi$
$720$	1.77069	0.0659895
$721$	−0.227179	−0.00846058
$722$	−1.00000	−0.0372161
$723$	17.6894	0.657875
$724$	−14.4527	−0.537131
$725$	−6.82370	−0.253426
$726$	−5.02261	−0.186406
$727$	−25.0966	−0.930781	−0.465391	−	0.885105i	$-0.654086\pi$
$727$	−25.0966	−0.930781	−0.465391	+	0.885105i	$0.654086\pi$
$728$	0	0
$729$	15.4449	0.572034
$730$	10.8629	0.402052
$731$	−18.9285	−0.700097
$732$	5.09762	0.188413
$733$	10.8948	0.402410	0.201205	−	0.979549i	$-0.435514\pi$
$733$	10.8948	0.402410	0.201205	+	0.979549i	$0.435514\pi$
$734$	−30.4669	−1.12455
$735$	−1.33813	−0.0493576
$736$	4.09689	0.151014
$737$	−56.6286	−2.08594
$738$	0.863659	0.0317917
$739$	−7.22806	−0.265888	−0.132944	−	0.991124i	$-0.542443\pi$
$739$	−7.22806	−0.265888	−0.132944	+	0.991124i	$0.542443\pi$
$740$	2.65010	0.0974195
$741$	0	0
$742$	10.5235	0.386330
$743$	53.9055	1.97760	0.988801	−	0.149239i	$-0.0476824\pi$
$743$	53.9055	1.97760	0.988801	+	0.149239i	$0.0476824\pi$
$744$	−8.15229	−0.298878
$745$	−2.50720	−0.0918569
$746$	−28.9669	−1.06055
$747$	−8.26757	−0.302495
$748$	18.0489	0.659932
$749$	5.72572	0.209213
$750$	−8.88211	−0.324329
$751$	−15.8038	−0.576689	−0.288344	−	0.957527i	$-0.593105\pi$
$751$	−15.8038	−0.576689	−0.288344	+	0.957527i	$0.593105\pi$
$752$	9.08685	0.331363
$753$	25.7251	0.937475
$754$	0	0
$755$	4.87129	0.177284
$756$	−14.8532	−0.540206
$757$	−19.9728	−0.725925	−0.362963	−	0.931804i	$-0.618235\pi$
$757$	−19.9728	−0.725925	−0.362963	+	0.931804i	$0.618235\pi$
$758$	−32.9190	−1.19567
$759$	17.0373	0.618416
$760$	0.929245	0.0337072
$761$	−13.8528	−0.502164	−0.251082	−	0.967966i	$-0.580786\pi$
$761$	−13.8528	−0.502164	−0.251082	+	0.967966i	$0.580786\pi$
$762$	11.1401	0.403564
$763$	−49.3371	−1.78612
$764$	−7.34389	−0.265693
$765$	−8.03990	−0.290683
$766$	−17.1752	−0.620564
$767$	0	0
$768$	1.04618	0.0377507
$769$	−18.3743	−0.662594	−0.331297	−	0.943526i	$-0.607486\pi$
$769$	−18.3743	−0.662594	−0.331297	+	0.943526i	$0.607486\pi$
$770$	−10.6906	−0.385262
$771$	−21.6546	−0.779872
$772$	−9.57969	−0.344781
$773$	26.3325	0.947115	0.473558	−	0.880763i	$-0.342970\pi$
$773$	26.3325	0.947115	0.473558	+	0.880763i	$0.342970\pi$
$774$	7.94364	0.285528
$775$	−32.2335	−1.15786
$776$	−11.5845	−0.415861
$777$	−8.63511	−0.309783
$778$	−1.10030	−0.0394478
$779$	0.453243	0.0162391
$780$	0	0
$781$	−43.2434	−1.54737
$782$	−18.6022	−0.665213
$783$	−8.46597	−0.302549
$784$	1.37645	0.0491591
$785$	−2.04638	−0.0730386
$786$	−18.4217	−0.657079
$787$	54.9301	1.95805	0.979023	−	0.203748i	$-0.0653122\pi$
$787$	54.9301	1.95805	0.979023	+	0.203748i	$0.0653122\pi$
$788$	−1.25654	−0.0447623
$789$	−25.4469	−0.905932
$790$	8.92624	0.317581
$791$	20.9465	0.744773
$792$	−7.57447	−0.269147
$793$	0	0
$794$	23.3055	0.827083
$795$	3.53481	0.125367
$796$	5.97439	0.211757
$797$	−25.0272	−0.886510	−0.443255	−	0.896396i	$-0.646176\pi$
$797$	−25.0272	−0.886510	−0.443255	+	0.896396i	$0.646176\pi$
$798$	−3.02786	−0.107185
$799$	−41.2593	−1.45965
$800$	4.13650	0.146248
$801$	11.8722	0.419483
$802$	27.1581	0.958986
$803$	−46.4681	−1.63982
$804$	14.9039	0.525621
$805$	11.0183	0.388344
$806$	0	0
$807$	−0.402955	−0.0141847
$808$	−9.68495	−0.340716
$809$	−16.4556	−0.578549	−0.289275	−	0.957246i	$-0.593414\pi$
$809$	−16.4556	−0.578549	−0.289275	+	0.957246i	$0.593414\pi$
$810$	0.322913	0.0113460
$811$	18.3589	0.644667	0.322334	−	0.946626i	$-0.395533\pi$
$811$	18.3589	0.644667	0.322334	+	0.946626i	$0.395533\pi$
$812$	4.77438	0.167548
$813$	−22.8335	−0.800806
$814$	−11.3363	−0.397338
$815$	−19.7231	−0.690871
$816$	−4.75023	−0.166291
$817$	4.16877	0.145847
$818$	−4.47035	−0.156302
$819$	0	0
$820$	−0.421174	−0.0147080
$821$	56.5259	1.97277	0.986384	−	0.164460i	$-0.0525882\pi$
$821$	56.5259	1.97277	0.986384	+	0.164460i	$0.0525882\pi$
$822$	−10.6618	−0.371872
$823$	−22.1382	−0.771690	−0.385845	−	0.922564i	$-0.626090\pi$
$823$	−22.1382	−0.771690	−0.385845	+	0.922564i	$0.626090\pi$
$824$	0.0784942	0.00273448
$825$	17.2021	0.598899
$826$	26.1152	0.908665
$827$	−14.6709	−0.510157	−0.255078	−	0.966920i	$-0.582101\pi$
$827$	−14.6709	−0.510157	−0.255078	+	0.966920i	$0.582101\pi$
$828$	7.80668	0.271301
$829$	32.7680	1.13808	0.569040	−	0.822310i	$-0.307315\pi$
$829$	32.7680	1.13808	0.569040	+	0.822310i	$0.307315\pi$
$830$	4.03178	0.139945
$831$	−13.5062	−0.468524
$832$	0	0
$833$	−6.24987	−0.216545
$834$	−18.9891	−0.657539
$835$	−15.4344	−0.534129
$836$	−3.97504	−0.137479
$837$	−39.9912	−1.38230
$838$	−7.62391	−0.263364
$839$	13.6785	0.472234	0.236117	−	0.971725i	$-0.424125\pi$
$839$	13.6785	0.472234	0.236117	+	0.971725i	$0.424125\pi$
$840$	2.81362	0.0970792
$841$	−26.2787	−0.906163
$842$	−31.6349	−1.09021
$843$	6.35249	0.218791
$844$	−2.93090	−0.100886
$845$	0	0
$846$	17.3151	0.595305
$847$	13.8948	0.477432
$848$	−3.63605	−0.124863
$849$	12.8817	0.442100
$850$	−18.7820	−0.644218
$851$	11.6839	0.400518
$852$	11.3811	0.389911
$853$	41.0799	1.40655	0.703274	−	0.710919i	$-0.251721\pi$
$853$	41.0799	1.40655	0.703274	+	0.710919i	$0.251721\pi$
$854$	−14.1024	−0.482573
$855$	1.77069	0.0605562
$856$	−1.97833	−0.0676181
$857$	11.4735	0.391929	0.195964	−	0.980611i	$-0.437216\pi$
$857$	11.4735	0.391929	0.195964	+	0.980611i	$0.437216\pi$
$858$	0	0
$859$	−10.2630	−0.350170	−0.175085	−	0.984553i	$-0.556020\pi$
$859$	−10.2630	−0.350170	−0.175085	+	0.984553i	$0.556020\pi$
$860$	−3.87381	−0.132096
$861$	1.37236	0.0467698
$862$	−5.99194	−0.204086
$863$	53.9995	1.83816	0.919082	−	0.394067i	$-0.128932\pi$
$863$	53.9995	1.83816	0.919082	+	0.394067i	$0.128932\pi$
$864$	5.13204	0.174596
$865$	4.85142	0.164953
$866$	−22.3035	−0.757904
$867$	3.78366	0.128500
$868$	22.5530	0.765498
$869$	−38.1838	−1.29530
$870$	1.60370	0.0543705
$871$	0	0
$872$	17.0468	0.577278
$873$	−22.0745	−0.747107
$874$	4.09689	0.138580
$875$	24.5720	0.830685
$876$	12.2298	0.413207
$877$	−46.9344	−1.58486	−0.792431	−	0.609962i	$-0.791185\pi$
$877$	−46.9344	−1.58486	−0.792431	+	0.609962i	$0.791185\pi$
$878$	8.13560	0.274563
$879$	7.78960	0.262737
$880$	3.69378	0.124517
$881$	36.7205	1.23714	0.618572	−	0.785728i	$-0.287712\pi$
$881$	36.7205	1.23714	0.618572	+	0.785728i	$0.287712\pi$
$882$	2.62285	0.0883159
$883$	−40.7395	−1.37099	−0.685497	−	0.728076i	$-0.740415\pi$
$883$	−40.7395	−1.37099	−0.685497	+	0.728076i	$0.740415\pi$
$884$	0	0
$885$	8.77202	0.294868
$886$	−9.37320	−0.314899
$887$	38.9664	1.30836	0.654182	−	0.756337i	$-0.273013\pi$
$887$	38.9664	1.30836	0.654182	+	0.756337i	$0.273013\pi$
$888$	2.98358	0.100122
$889$	−30.8187	−1.03363
$890$	−5.78961	−0.194068
$891$	−1.38133	−0.0462762
$892$	−21.7936	−0.729705
$893$	9.08685	0.304080
$894$	−2.82271	−0.0944054
$895$	3.38757	0.113234
$896$	−2.89421	−0.0966888
$897$	0	0
$898$	23.5049	0.784368
$899$	12.8547	0.428727
$900$	7.88215	0.262738
$901$	16.5097	0.550018
$902$	1.80166	0.0599886
$903$	12.6225	0.420049
$904$	−7.23739	−0.240712
$905$	13.4301	0.446432
$906$	5.48428	0.182203
$907$	57.5459	1.91078	0.955391	−	0.295345i	$-0.0954347\pi$
$907$	57.5459	1.91078	0.955391	+	0.295345i	$0.0954347\pi$
$908$	18.0721	0.599743
$909$	−18.4548	−0.612106
$910$	0	0
$911$	14.3410	0.475140	0.237570	−	0.971370i	$-0.423649\pi$
$911$	14.3410	0.475140	0.237570	+	0.971370i	$0.423649\pi$
$912$	1.04618	0.0346424
$913$	−17.2468	−0.570785
$914$	7.02736	0.232445
$915$	−4.73694	−0.156598
$916$	20.7571	0.685833
$917$	50.9628	1.68294
$918$	−23.3023	−0.769091
$919$	−9.88857	−0.326194	−0.163097	−	0.986610i	$-0.552148\pi$
$919$	−9.88857	−0.326194	−0.163097	+	0.986610i	$0.552148\pi$
$920$	−3.80702	−0.125514
$921$	−19.7422	−0.650528
$922$	2.09583	0.0690226
$923$	0	0
$924$	−12.0359	−0.395951
$925$	11.7968	0.387877
$926$	−22.3671	−0.735028
$927$	0.149572	0.00491258
$928$	−1.64963	−0.0541518
$929$	29.4749	0.967041	0.483521	−	0.875333i	$-0.339358\pi$
$929$	29.4749	0.967041	0.483521	+	0.875333i	$0.339358\pi$
$930$	7.57548	0.248410
$931$	1.37645	0.0451115
$932$	−26.8846	−0.880636
$933$	5.25011	0.171881
$934$	−36.8013	−1.20417
$935$	−16.7718	−0.548497
$936$	0	0
$937$	−11.6698	−0.381237	−0.190619	−	0.981664i	$-0.561049\pi$
$937$	−11.6698	−0.381237	−0.190619	+	0.981664i	$0.561049\pi$
$938$	−41.2311	−1.34624
$939$	−23.5687	−0.769135
$940$	−8.44390	−0.275410
$941$	−14.3074	−0.466407	−0.233203	−	0.972428i	$-0.574921\pi$
$941$	−14.3074	−0.466407	−0.233203	+	0.972428i	$0.574921\pi$
$942$	−2.30390	−0.0750650
$943$	−1.85689	−0.0604686
$944$	−9.02326	−0.293682
$945$	13.8023	0.448988
$946$	16.5710	0.538770
$947$	−26.7816	−0.870285	−0.435142	−	0.900362i	$-0.643302\pi$
$947$	−26.7816	−0.870285	−0.435142	+	0.900362i	$0.643302\pi$
$948$	10.0495	0.326392
$949$	0	0
$950$	4.13650	0.134206
$951$	−17.9950	−0.583527
$952$	13.1413	0.425913
$953$	37.3685	1.21048	0.605242	−	0.796042i	$-0.293076\pi$
$953$	37.3685	1.21048	0.605242	+	0.796042i	$0.293076\pi$
$954$	−6.92854	−0.224320
$955$	6.82427	0.220828
$956$	−12.0208	−0.388779
$957$	−6.86015	−0.221757
$958$	−18.0660	−0.583687
$959$	29.4954	0.952455
$960$	−0.972156	−0.0313762
$961$	29.7223	0.958783
$962$	0	0
$963$	−3.76974	−0.121478
$964$	16.9086	0.544588
$965$	8.90188	0.286562
$966$	12.4048	0.399119
$967$	23.2450	0.747508	0.373754	−	0.927528i	$-0.378070\pi$
$967$	23.2450	0.747508	0.373754	+	0.927528i	$0.378070\pi$
$968$	−4.80091	−0.154307
$969$	−4.75023	−0.152599
$970$	10.7649	0.345639
$971$	−3.68714	−0.118326	−0.0591630	−	0.998248i	$-0.518843\pi$
$971$	−3.68714	−0.118326	−0.0591630	+	0.998248i	$0.518843\pi$
$972$	15.7597	0.505492
$973$	52.5326	1.68412
$974$	19.7205	0.631886
$975$	0	0
$976$	4.87261	0.155968
$977$	−29.2065	−0.934399	−0.467200	−	0.884152i	$-0.654737\pi$
$977$	−29.2065	−0.934399	−0.467200	+	0.884152i	$0.654737\pi$
$978$	−22.2050	−0.710039
$979$	24.7663	0.791533
$980$	−1.27906	−0.0408582
$981$	32.4829	1.03710
$982$	26.6062	0.849037
$983$	10.6377	0.339291	0.169646	−	0.985505i	$-0.445738\pi$
$983$	10.6377	0.339291	0.169646	+	0.985505i	$0.445738\pi$
$984$	−0.474173	−0.0151161
$985$	1.16763	0.0372038
$986$	7.49024	0.238538
$987$	27.5137	0.875771
$988$	0	0
$989$	−17.0790	−0.543081
$990$	7.03854	0.223699
$991$	35.2488	1.11972	0.559858	−	0.828589i	$-0.310856\pi$
$991$	35.2488	1.11972	0.559858	+	0.828589i	$0.310856\pi$
$992$	−7.79245	−0.247411
$993$	7.92747	0.251571
$994$	−31.4854	−0.998656
$995$	−5.55167	−0.176000
$996$	4.53913	0.143828
$997$	34.8962	1.10517	0.552586	−	0.833456i	$-0.313641\pi$
$997$	34.8962	1.10517	0.552586	+	0.833456i	$0.313641\pi$
$998$	−5.10306	−0.161534
$999$	14.6360	0.463062

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6422.2.a.bj.1.6	✓	9
13.12	even	2		6422.2.a.bl.1.6	yes	9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6422.2.a.bj.1.6	✓	9	1.1	even	1	trivial
6422.2.a.bl.1.6	yes	9	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:	\( N \)	\(=\)	\( 6422 = 2 \cdot 13^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6422.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.04618 q^{3} +1.00000 q^{4} -0.929245 q^{5} -1.04618 q^{6} +2.89421 q^{7} -1.00000 q^{8} -1.90551 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} +1.04618 q^{3} +1.00000 q^{4} -0.929245 q^{5} -1.04618 q^{6} +2.89421 q^{7} -1.00000 q^{8} -1.90551 q^{9} +0.929245 q^{10} -3.97504 q^{11} +1.04618 q^{12} -2.89421 q^{14} -0.972156 q^{15} +1.00000 q^{16} -4.54056 q^{17} +1.90551 q^{18} +1.00000 q^{19} -0.929245 q^{20} +3.02786 q^{21} +3.97504 q^{22} -4.09689 q^{23} -1.04618 q^{24} -4.13650 q^{25} -5.13204 q^{27} +2.89421 q^{28} +1.64963 q^{29} +0.972156 q^{30} +7.79245 q^{31} -1.00000 q^{32} -4.15860 q^{33} +4.54056 q^{34} -2.68943 q^{35} -1.90551 q^{36} -2.85188 q^{37} -1.00000 q^{38} +0.929245 q^{40} +0.453243 q^{41} -3.02786 q^{42} +4.16877 q^{43} -3.97504 q^{44} +1.77069 q^{45} +4.09689 q^{46} +9.08685 q^{47} +1.04618 q^{48} +1.37645 q^{49} +4.13650 q^{50} -4.75023 q^{51} -3.63605 q^{53} +5.13204 q^{54} +3.69378 q^{55} -2.89421 q^{56} +1.04618 q^{57} -1.64963 q^{58} -9.02326 q^{59} -0.972156 q^{60} +4.87261 q^{61} -7.79245 q^{62} -5.51495 q^{63} +1.00000 q^{64} +4.15860 q^{66} +14.2461 q^{67} -4.54056 q^{68} -4.28608 q^{69} +2.68943 q^{70} +10.8788 q^{71} +1.90551 q^{72} +11.6900 q^{73} +2.85188 q^{74} -4.32752 q^{75} +1.00000 q^{76} -11.5046 q^{77} +9.60591 q^{79} -0.929245 q^{80} +0.347501 q^{81} -0.453243 q^{82} +4.33877 q^{83} +3.02786 q^{84} +4.21929 q^{85} -4.16877 q^{86} +1.72581 q^{87} +3.97504 q^{88} -6.23045 q^{89} -1.77069 q^{90} -4.09689 q^{92} +8.15229 q^{93} -9.08685 q^{94} -0.929245 q^{95} -1.04618 q^{96} +11.5845 q^{97} -1.37645 q^{98} +7.57447 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q - 9 q^{2} + q^{3} + 9 q^{4} - q^{5} - q^{6} + 13 q^{7} - 9 q^{8} - 2 q^{9}+O(q^{10}) \) 9 * q - 9 * q^2 + q^3 + 9 * q^4 - q^5 - q^6 + 13 * q^7 - 9 * q^8 - 2 * q^9	\( 9 q - 9 q^{2} + q^{3} + 9 q^{4} - q^{5} - q^{6} + 13 q^{7} - 9 q^{8} - 2 q^{9} + q^{10} + 3 q^{11} + q^{12} - 13 q^{14} + 3 q^{15} + 9 q^{16} - 8 q^{17} + 2 q^{18} + 9 q^{19} - q^{20} + 24 q^{21} - 3 q^{22} - 10 q^{23} - q^{24} + 8 q^{25} + 10 q^{27} + 13 q^{28} - 20 q^{29} - 3 q^{30} + q^{31} - 9 q^{32} - 2 q^{33} + 8 q^{34} + 4 q^{35} - 2 q^{36} + 15 q^{37} - 9 q^{38} + q^{40} + 19 q^{41} - 24 q^{42} - 16 q^{43} + 3 q^{44} + 15 q^{45} + 10 q^{46} + 18 q^{47} + q^{48} + 18 q^{49} - 8 q^{50} - 11 q^{51} + 17 q^{53} - 10 q^{54} - 26 q^{55} - 13 q^{56} + q^{57} + 20 q^{58} + 24 q^{59} + 3 q^{60} + 6 q^{61} - q^{62} + q^{63} + 9 q^{64} + 2 q^{66} + 29 q^{67} - 8 q^{68} - 12 q^{69} - 4 q^{70} + 23 q^{71} + 2 q^{72} + 38 q^{73} - 15 q^{74} + 11 q^{75} + 9 q^{76} - 40 q^{77} - 20 q^{79} - q^{80} - 31 q^{81} - 19 q^{82} + 20 q^{83} + 24 q^{84} + 39 q^{85} + 16 q^{86} - 10 q^{87} - 3 q^{88} - 7 q^{89} - 15 q^{90} - 10 q^{92} + 11 q^{93} - 18 q^{94} - q^{95} - q^{96} + 28 q^{97} - 18 q^{98} + 25 q^{99}+O(q^{100}) \) 9 * q - 9 * q^2 + q^3 + 9 * q^4 - q^5 - q^6 + 13 * q^7 - 9 * q^8 - 2 * q^9 + q^10 + 3 * q^11 + q^12 - 13 * q^14 + 3 * q^15 + 9 * q^16 - 8 * q^17 + 2 * q^18 + 9 * q^19 - q^20 + 24 * q^21 - 3 * q^22 - 10 * q^23 - q^24 + 8 * q^25 + 10 * q^27 + 13 * q^28 - 20 * q^29 - 3 * q^30 + q^31 - 9 * q^32 - 2 * q^33 + 8 * q^34 + 4 * q^35 - 2 * q^36 + 15 * q^37 - 9 * q^38 + q^40 + 19 * q^41 - 24 * q^42 - 16 * q^43 + 3 * q^44 + 15 * q^45 + 10 * q^46 + 18 * q^47 + q^48 + 18 * q^49 - 8 * q^50 - 11 * q^51 + 17 * q^53 - 10 * q^54 - 26 * q^55 - 13 * q^56 + q^57 + 20 * q^58 + 24 * q^59 + 3 * q^60 + 6 * q^61 - q^62 + q^63 + 9 * q^64 + 2 * q^66 + 29 * q^67 - 8 * q^68 - 12 * q^69 - 4 * q^70 + 23 * q^71 + 2 * q^72 + 38 * q^73 - 15 * q^74 + 11 * q^75 + 9 * q^76 - 40 * q^77 - 20 * q^79 - q^80 - 31 * q^81 - 19 * q^82 + 20 * q^83 + 24 * q^84 + 39 * q^85 + 16 * q^86 - 10 * q^87 - 3 * q^88 - 7 * q^89 - 15 * q^90 - 10 * q^92 + 11 * q^93 - 18 * q^94 - q^95 - q^96 + 28 * q^97 - 18 * q^98 + 25 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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