LMFDB - Embedded newform 8030.2.a.k.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8030,2,Mod(1,8030)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8030, 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("8030.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8030 = 2 \cdot 5 \cdot 11 \cdot 73 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8030.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:	$64.1198728231$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{13}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 3 $ x^2 - x - 3
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.30278$ of defining polynomial
Character	$\chi$	$=$	8030.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +1.30278 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.30278 q^{6} -4.60555 q^{7} -1.00000 q^{8} -1.30278 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} +1.30278 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.30278 q^{6} -4.60555 q^{7} -1.00000 q^{8} -1.30278 q^{9} +1.00000 q^{10} +1.00000 q^{11} +1.30278 q^{12} +3.30278 q^{13} +4.60555 q^{14} -1.30278 q^{15} +1.00000 q^{16} +1.30278 q^{18} -2.30278 q^{19} -1.00000 q^{20} -6.00000 q^{21} -1.00000 q^{22} +2.00000 q^{23} -1.30278 q^{24} +1.00000 q^{25} -3.30278 q^{26} -5.60555 q^{27} -4.60555 q^{28} +4.60555 q^{29} +1.30278 q^{30} +0.605551 q^{31} -1.00000 q^{32} +1.30278 q^{33} +4.60555 q^{35} -1.30278 q^{36} +2.00000 q^{37} +2.30278 q^{38} +4.30278 q^{39} +1.00000 q^{40} +6.00000 q^{41} +6.00000 q^{42} +4.30278 q^{43} +1.00000 q^{44} +1.30278 q^{45} -2.00000 q^{46} +7.51388 q^{47} +1.30278 q^{48} +14.2111 q^{49} -1.00000 q^{50} +3.30278 q^{52} -5.21110 q^{53} +5.60555 q^{54} -1.00000 q^{55} +4.60555 q^{56} -3.00000 q^{57} -4.60555 q^{58} -7.30278 q^{59} -1.30278 q^{60} -4.09167 q^{61} -0.605551 q^{62} +6.00000 q^{63} +1.00000 q^{64} -3.30278 q^{65} -1.30278 q^{66} +3.90833 q^{67} +2.60555 q^{69} -4.60555 q^{70} -10.1194 q^{71} +1.30278 q^{72} +1.00000 q^{73} -2.00000 q^{74} +1.30278 q^{75} -2.30278 q^{76} -4.60555 q^{77} -4.30278 q^{78} -8.60555 q^{79} -1.00000 q^{80} -3.39445 q^{81} -6.00000 q^{82} +14.5139 q^{83} -6.00000 q^{84} -4.30278 q^{86} +6.00000 q^{87} -1.00000 q^{88} -2.09167 q^{89} -1.30278 q^{90} -15.2111 q^{91} +2.00000 q^{92} +0.788897 q^{93} -7.51388 q^{94} +2.30278 q^{95} -1.30278 q^{96} -13.2111 q^{97} -14.2111 q^{98} -1.30278 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} - q^{3} + 2 q^{4} - 2 q^{5} + q^{6} - 2 q^{7} - 2 q^{8} + q^{9}+O(q^{10}) $ 2 * q - 2 * q^2 - q^3 + 2 * q^4 - 2 * q^5 + q^6 - 2 * q^7 - 2 * q^8 + q^9	$ 2 q - 2 q^{2} - q^{3} + 2 q^{4} - 2 q^{5} + q^{6} - 2 q^{7} - 2 q^{8} + q^{9} + 2 q^{10} + 2 q^{11} - q^{12} + 3 q^{13} + 2 q^{14} + q^{15} + 2 q^{16} - q^{18} - q^{19} - 2 q^{20} - 12 q^{21} - 2 q^{22} + 4 q^{23} + q^{24} + 2 q^{25} - 3 q^{26} - 4 q^{27} - 2 q^{28} + 2 q^{29} - q^{30} - 6 q^{31} - 2 q^{32} - q^{33} + 2 q^{35} + q^{36} + 4 q^{37} + q^{38} + 5 q^{39} + 2 q^{40} + 12 q^{41} + 12 q^{42} + 5 q^{43} + 2 q^{44} - q^{45} - 4 q^{46} - 3 q^{47} - q^{48} + 14 q^{49} - 2 q^{50} + 3 q^{52} + 4 q^{53} + 4 q^{54} - 2 q^{55} + 2 q^{56} - 6 q^{57} - 2 q^{58} - 11 q^{59} + q^{60} - 19 q^{61} + 6 q^{62} + 12 q^{63} + 2 q^{64} - 3 q^{65} + q^{66} - 3 q^{67} - 2 q^{69} - 2 q^{70} + 5 q^{71} - q^{72} + 2 q^{73} - 4 q^{74} - q^{75} - q^{76} - 2 q^{77} - 5 q^{78} - 10 q^{79} - 2 q^{80} - 14 q^{81} - 12 q^{82} + 11 q^{83} - 12 q^{84} - 5 q^{86} + 12 q^{87} - 2 q^{88} - 15 q^{89} + q^{90} - 16 q^{91} + 4 q^{92} + 16 q^{93} + 3 q^{94} + q^{95} + q^{96} - 12 q^{97} - 14 q^{98} + q^{99}+O(q^{100}) $ 2 * q - 2 * q^2 - q^3 + 2 * q^4 - 2 * q^5 + q^6 - 2 * q^7 - 2 * q^8 + q^9 + 2 * q^10 + 2 * q^11 - q^12 + 3 * q^13 + 2 * q^14 + q^15 + 2 * q^16 - q^18 - q^19 - 2 * q^20 - 12 * q^21 - 2 * q^22 + 4 * q^23 + q^24 + 2 * q^25 - 3 * q^26 - 4 * q^27 - 2 * q^28 + 2 * q^29 - q^30 - 6 * q^31 - 2 * q^32 - q^33 + 2 * q^35 + q^36 + 4 * q^37 + q^38 + 5 * q^39 + 2 * q^40 + 12 * q^41 + 12 * q^42 + 5 * q^43 + 2 * q^44 - q^45 - 4 * q^46 - 3 * q^47 - q^48 + 14 * q^49 - 2 * q^50 + 3 * q^52 + 4 * q^53 + 4 * q^54 - 2 * q^55 + 2 * q^56 - 6 * q^57 - 2 * q^58 - 11 * q^59 + q^60 - 19 * q^61 + 6 * q^62 + 12 * q^63 + 2 * q^64 - 3 * q^65 + q^66 - 3 * q^67 - 2 * q^69 - 2 * q^70 + 5 * q^71 - q^72 + 2 * q^73 - 4 * q^74 - q^75 - q^76 - 2 * q^77 - 5 * q^78 - 10 * q^79 - 2 * q^80 - 14 * q^81 - 12 * q^82 + 11 * q^83 - 12 * q^84 - 5 * q^86 + 12 * q^87 - 2 * q^88 - 15 * q^89 + q^90 - 16 * q^91 + 4 * q^92 + 16 * q^93 + 3 * q^94 + q^95 + q^96 - 12 * q^97 - 14 * q^98 + q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	1.30278	0.752158	0.376079	−	0.926588i	$-0.377272\pi$
$3$	1.30278	0.752158	0.376079	+	0.926588i	$0.377272\pi$
$4$	1.00000	0.500000
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	−1.30278	−0.531856
$7$	−4.60555	−1.74073	−0.870367	−	0.492403i	$-0.836119\pi$
$7$	−4.60555	−1.74073	−0.870367	+	0.492403i	$0.836119\pi$
$8$	−1.00000	−0.353553
$9$	−1.30278	−0.434259
$10$	1.00000	0.316228
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	1.30278	0.376079
$13$	3.30278	0.916025	0.458013	−	0.888946i	$-0.348561\pi$
$13$	3.30278	0.916025	0.458013	+	0.888946i	$0.348561\pi$
$14$	4.60555	1.23089
$15$	−1.30278	−0.336375
$16$	1.00000	0.250000
$17$	0	0		−	1.00000i	$-0.5\pi$
$17$	0	0			1.00000i	$0.5\pi$
$18$	1.30278	0.307067
$19$	−2.30278	−0.528293	−0.264146	−	0.964483i	$-0.585090\pi$
$19$	−2.30278	−0.528293	−0.264146	+	0.964483i	$0.585090\pi$
$20$	−1.00000	−0.223607
$21$	−6.00000	−1.30931
$22$	−1.00000	−0.213201
$23$	2.00000	0.417029	0.208514	−	0.978019i	$-0.433137\pi$
$23$	2.00000	0.417029	0.208514	+	0.978019i	$0.433137\pi$
$24$	−1.30278	−0.265928
$25$	1.00000	0.200000
$26$	−3.30278	−0.647728
$27$	−5.60555	−1.07879
$28$	−4.60555	−0.870367
$29$	4.60555	0.855229	0.427615	−	0.903961i	$-0.359354\pi$
$29$	4.60555	0.855229	0.427615	+	0.903961i	$0.359354\pi$
$30$	1.30278	0.237853
$31$	0.605551	0.108760	0.0543801	−	0.998520i	$-0.482682\pi$
$31$	0.605551	0.108760	0.0543801	+	0.998520i	$0.482682\pi$
$32$	−1.00000	−0.176777
$33$	1.30278	0.226784
$34$	0	0
$35$	4.60555	0.778480
$36$	−1.30278	−0.217129
$37$	2.00000	0.328798	0.164399	−	0.986394i	$-0.447432\pi$
$37$	2.00000	0.328798	0.164399	+	0.986394i	$0.447432\pi$
$38$	2.30278	0.373560
$39$	4.30278	0.688996
$40$	1.00000	0.158114
$41$	6.00000	0.937043	0.468521	−	0.883452i	$-0.344787\pi$
$41$	6.00000	0.937043	0.468521	+	0.883452i	$0.344787\pi$
$42$	6.00000	0.925820
$43$	4.30278	0.656167	0.328084	−	0.944649i	$-0.393597\pi$
$43$	4.30278	0.656167	0.328084	+	0.944649i	$0.393597\pi$
$44$	1.00000	0.150756
$45$	1.30278	0.194206
$46$	−2.00000	−0.294884
$47$	7.51388	1.09601	0.548006	−	0.836474i	$-0.315387\pi$
$47$	7.51388	1.09601	0.548006	+	0.836474i	$0.315387\pi$
$48$	1.30278	0.188039
$49$	14.2111	2.03016
$50$	−1.00000	−0.141421
$51$	0	0
$52$	3.30278	0.458013
$53$	−5.21110	−0.715800	−0.357900	−	0.933760i	$-0.616507\pi$
$53$	−5.21110	−0.715800	−0.357900	+	0.933760i	$0.616507\pi$
$54$	5.60555	0.762819
$55$	−1.00000	−0.134840
$56$	4.60555	0.615443
$57$	−3.00000	−0.397360
$58$	−4.60555	−0.604739
$59$	−7.30278	−0.950740	−0.475370	−	0.879786i	$-0.657686\pi$
$59$	−7.30278	−0.950740	−0.475370	+	0.879786i	$0.657686\pi$
$60$	−1.30278	−0.168188
$61$	−4.09167	−0.523885	−0.261943	−	0.965083i	$-0.584363\pi$
$61$	−4.09167	−0.523885	−0.261943	+	0.965083i	$0.584363\pi$
$62$	−0.605551	−0.0769051
$63$	6.00000	0.755929
$64$	1.00000	0.125000
$65$	−3.30278	−0.409659
$66$	−1.30278	−0.160361
$67$	3.90833	0.477478	0.238739	−	0.971084i	$-0.423266\pi$
$67$	3.90833	0.477478	0.238739	+	0.971084i	$0.423266\pi$
$68$	0	0
$69$	2.60555	0.313672
$70$	−4.60555	−0.550469
$71$	−10.1194	−1.20096	−0.600478	−	0.799642i	$-0.705023\pi$
$71$	−10.1194	−1.20096	−0.600478	+	0.799642i	$0.705023\pi$
$72$	1.30278	0.153534
$73$	1.00000	0.117041
$73$	1.00000	0.117041
$74$	−2.00000	−0.232495
$75$	1.30278	0.150432
$76$	−2.30278	−0.264146
$77$	−4.60555	−0.524851
$78$	−4.30278	−0.487193
$79$	−8.60555	−0.968200	−0.484100	−	0.875013i	$-0.660853\pi$
$79$	−8.60555	−0.968200	−0.484100	+	0.875013i	$0.660853\pi$
$80$	−1.00000	−0.111803
$81$	−3.39445	−0.377161
$82$	−6.00000	−0.662589
$83$	14.5139	1.59311	0.796553	−	0.604569i	$-0.206655\pi$
$83$	14.5139	1.59311	0.796553	+	0.604569i	$0.206655\pi$
$84$	−6.00000	−0.654654
$85$	0	0
$86$	−4.30278	−0.463980
$87$	6.00000	0.643268
$88$	−1.00000	−0.106600
$89$	−2.09167	−0.221717	−0.110858	−	0.993836i	$-0.535360\pi$
$89$	−2.09167	−0.221717	−0.110858	+	0.993836i	$0.535360\pi$
$90$	−1.30278	−0.137325
$91$	−15.2111	−1.59456
$92$	2.00000	0.208514
$93$	0.788897	0.0818049
$94$	−7.51388	−0.774997
$95$	2.30278	0.236260
$96$	−1.30278	−0.132964
$97$	−13.2111	−1.34138	−0.670692	−	0.741736i	$-0.734003\pi$
$97$	−13.2111	−1.34138	−0.670692	+	0.741736i	$0.734003\pi$
$98$	−14.2111	−1.43554
$99$	−1.30278	−0.130934
$100$	1.00000	0.100000
$101$	−18.4222	−1.83308	−0.916539	−	0.399945i	$-0.869029\pi$
$101$	−18.4222	−1.83308	−0.916539	+	0.399945i	$0.869029\pi$
$102$	0	0
$103$	9.21110	0.907597	0.453798	−	0.891104i	$-0.350069\pi$
$103$	9.21110	0.907597	0.453798	+	0.891104i	$0.350069\pi$
$104$	−3.30278	−0.323864
$105$	6.00000	0.585540
$106$	5.21110	0.506147
$107$	−13.5139	−1.30644	−0.653218	−	0.757170i	$-0.726581\pi$
$107$	−13.5139	−1.30644	−0.653218	+	0.757170i	$0.726581\pi$
$108$	−5.60555	−0.539394
$109$	15.2111	1.45696	0.728480	−	0.685067i	$-0.240227\pi$
$109$	15.2111	1.45696	0.728480	+	0.685067i	$0.240227\pi$
$110$	1.00000	0.0953463
$111$	2.60555	0.247308
$112$	−4.60555	−0.435184
$113$	−18.9083	−1.77875	−0.889373	−	0.457183i	$-0.848859\pi$
$113$	−18.9083	−1.77875	−0.889373	+	0.457183i	$0.848859\pi$
$114$	3.00000	0.280976
$115$	−2.00000	−0.186501
$116$	4.60555	0.427615
$117$	−4.30278	−0.397792
$118$	7.30278	0.672275
$119$	0	0
$120$	1.30278	0.118927
$121$	1.00000	0.0909091
$122$	4.09167	0.370443
$123$	7.81665	0.704804
$124$	0.605551	0.0543801
$125$	−1.00000	−0.0894427
$126$	−6.00000	−0.534522
$127$	21.7250	1.92778	0.963890	−	0.266300i	$-0.0858012\pi$
$127$	21.7250	1.92778	0.963890	+	0.266300i	$0.0858012\pi$
$128$	−1.00000	−0.0883883
$129$	5.60555	0.493541
$130$	3.30278	0.289673
$131$	21.2111	1.85322	0.926611	−	0.376021i	$-0.122708\pi$
$131$	21.2111	1.85322	0.926611	+	0.376021i	$0.122708\pi$
$132$	1.30278	0.113392
$133$	10.6056	0.919618
$134$	−3.90833	−0.337628
$135$	5.60555	0.482449
$136$	0	0
$137$	−1.81665	−0.155207	−0.0776036	−	0.996984i	$-0.524727\pi$
$137$	−1.81665	−0.155207	−0.0776036	+	0.996984i	$0.524727\pi$
$138$	−2.60555	−0.221799
$139$	9.81665	0.832638	0.416319	−	0.909219i	$-0.363320\pi$
$139$	9.81665	0.832638	0.416319	+	0.909219i	$0.363320\pi$
$140$	4.60555	0.389240
$141$	9.78890	0.824374
$142$	10.1194	0.849204
$143$	3.30278	0.276192
$144$	−1.30278	−0.108565
$145$	−4.60555	−0.382470
$146$	−1.00000	−0.0827606
$147$	18.5139	1.52700
$148$	2.00000	0.164399
$149$	2.90833	0.238259	0.119130	−	0.992879i	$-0.461990\pi$
$149$	2.90833	0.238259	0.119130	+	0.992879i	$0.461990\pi$
$150$	−1.30278	−0.106371
$151$	−17.3028	−1.40808	−0.704040	−	0.710160i	$-0.748622\pi$
$151$	−17.3028	−1.40808	−0.704040	+	0.710160i	$0.748622\pi$
$152$	2.30278	0.186780
$153$	0	0
$154$	4.60555	0.371126
$155$	−0.605551	−0.0486390
$156$	4.30278	0.344498
$157$	3.39445	0.270907	0.135453	−	0.990784i	$-0.456751\pi$
$157$	3.39445	0.270907	0.135453	+	0.990784i	$0.456751\pi$
$158$	8.60555	0.684621
$159$	−6.78890	−0.538395
$160$	1.00000	0.0790569
$161$	−9.21110	−0.725937
$162$	3.39445	0.266693
$163$	−5.81665	−0.455595	−0.227798	−	0.973708i	$-0.573153\pi$
$163$	−5.81665	−0.455595	−0.227798	+	0.973708i	$0.573153\pi$
$164$	6.00000	0.468521
$165$	−1.30278	−0.101421
$166$	−14.5139	−1.12650
$167$	−6.00000	−0.464294	−0.232147	−	0.972681i	$-0.574575\pi$
$167$	−6.00000	−0.464294	−0.232147	+	0.972681i	$0.574575\pi$
$168$	6.00000	0.462910
$169$	−2.09167	−0.160898
$170$	0	0
$171$	3.00000	0.229416
$172$	4.30278	0.328084
$173$	−25.6333	−1.94886	−0.974432	−	0.224682i	$-0.927866\pi$
$173$	−25.6333	−1.94886	−0.974432	+	0.224682i	$0.927866\pi$
$174$	−6.00000	−0.454859
$175$	−4.60555	−0.348147
$176$	1.00000	0.0753778
$177$	−9.51388	−0.715107
$178$	2.09167	0.156778
$179$	−2.30278	−0.172118	−0.0860588	−	0.996290i	$-0.527427\pi$
$179$	−2.30278	−0.172118	−0.0860588	+	0.996290i	$0.527427\pi$
$180$	1.30278	0.0971032
$181$	−24.6056	−1.82892	−0.914458	−	0.404681i	$-0.867383\pi$
$181$	−24.6056	−1.82892	−0.914458	+	0.404681i	$0.867383\pi$
$182$	15.2111	1.12752
$183$	−5.33053	−0.394044
$184$	−2.00000	−0.147442
$185$	−2.00000	−0.147043
$186$	−0.788897	−0.0578448
$187$	0	0
$188$	7.51388	0.548006
$189$	25.8167	1.87789
$190$	−2.30278	−0.167061
$191$	2.78890	0.201798	0.100899	−	0.994897i	$-0.467828\pi$
$191$	2.78890	0.201798	0.100899	+	0.994897i	$0.467828\pi$
$192$	1.30278	0.0940197
$193$	−17.0278	−1.22568	−0.612842	−	0.790205i	$-0.709974\pi$
$193$	−17.0278	−1.22568	−0.612842	+	0.790205i	$0.709974\pi$
$194$	13.2111	0.948502
$195$	−4.30278	−0.308128
$196$	14.2111	1.01508
$197$	8.90833	0.634692	0.317346	−	0.948310i	$-0.397208\pi$
$197$	8.90833	0.634692	0.317346	+	0.948310i	$0.397208\pi$
$198$	1.30278	0.0925842
$199$	−20.4222	−1.44769	−0.723846	−	0.689962i	$-0.757627\pi$
$199$	−20.4222	−1.44769	−0.723846	+	0.689962i	$0.757627\pi$
$200$	−1.00000	−0.0707107
$201$	5.09167	0.359139
$202$	18.4222	1.29618
$203$	−21.2111	−1.48873
$204$	0	0
$205$	−6.00000	−0.419058
$206$	−9.21110	−0.641768
$207$	−2.60555	−0.181098
$208$	3.30278	0.229006
$209$	−2.30278	−0.159286
$210$	−6.00000	−0.414039
$211$	−16.7250	−1.15140	−0.575698	−	0.817663i	$-0.695269\pi$
$211$	−16.7250	−1.15140	−0.575698	+	0.817663i	$0.695269\pi$
$212$	−5.21110	−0.357900
$213$	−13.1833	−0.903308
$214$	13.5139	0.923790
$215$	−4.30278	−0.293447
$216$	5.60555	0.381409
$217$	−2.78890	−0.189323
$218$	−15.2111	−1.03023
$219$	1.30278	0.0880334
$220$	−1.00000	−0.0674200
$221$	0	0
$222$	−2.60555	−0.174873
$223$	15.8167	1.05916	0.529581	−	0.848260i	$-0.322349\pi$
$223$	15.8167	1.05916	0.529581	+	0.848260i	$0.322349\pi$
$224$	4.60555	0.307721
$225$	−1.30278	−0.0868517
$226$	18.9083	1.25776
$227$	21.3944	1.42000	0.710000	−	0.704202i	$-0.248695\pi$
$227$	21.3944	1.42000	0.710000	+	0.704202i	$0.248695\pi$
$228$	−3.00000	−0.198680
$229$	12.1194	0.800874	0.400437	−	0.916324i	$-0.368858\pi$
$229$	12.1194	0.800874	0.400437	+	0.916324i	$0.368858\pi$
$230$	2.00000	0.131876
$231$	−6.00000	−0.394771
$232$	−4.60555	−0.302369
$233$	25.2111	1.65163	0.825817	−	0.563938i	$-0.190714\pi$
$233$	25.2111	1.65163	0.825817	+	0.563938i	$0.190714\pi$
$234$	4.30278	0.281281
$235$	−7.51388	−0.490151
$236$	−7.30278	−0.475370
$237$	−11.2111	−0.728239
$238$	0	0
$239$	−1.48612	−0.0961292	−0.0480646	−	0.998844i	$-0.515305\pi$
$239$	−1.48612	−0.0961292	−0.0480646	+	0.998844i	$0.515305\pi$
$240$	−1.30278	−0.0840938
$241$	−22.0000	−1.41714	−0.708572	−	0.705638i	$-0.750660\pi$
$241$	−22.0000	−1.41714	−0.708572	+	0.705638i	$0.750660\pi$
$242$	−1.00000	−0.0642824
$243$	12.3944	0.795104
$244$	−4.09167	−0.261943
$245$	−14.2111	−0.907914
$246$	−7.81665	−0.498372
$247$	−7.60555	−0.483930
$248$	−0.605551	−0.0384525
$249$	18.9083	1.19827
$250$	1.00000	0.0632456
$251$	−22.4222	−1.41528	−0.707639	−	0.706575i	$-0.750239\pi$
$251$	−22.4222	−1.41528	−0.707639	+	0.706575i	$0.750239\pi$
$252$	6.00000	0.377964
$253$	2.00000	0.125739
$254$	−21.7250	−1.36315
$255$	0	0
$256$	1.00000	0.0625000
$257$	−20.6056	−1.28534	−0.642669	−	0.766144i	$-0.722173\pi$
$257$	−20.6056	−1.28534	−0.642669	+	0.766144i	$0.722173\pi$
$258$	−5.60555	−0.348986
$259$	−9.21110	−0.572350
$260$	−3.30278	−0.204829
$261$	−6.00000	−0.371391
$262$	−21.2111	−1.31043
$263$	−8.78890	−0.541947	−0.270973	−	0.962587i	$-0.587346\pi$
$263$	−8.78890	−0.541947	−0.270973	+	0.962587i	$0.587346\pi$
$264$	−1.30278	−0.0801803
$265$	5.21110	0.320115
$266$	−10.6056	−0.650268
$267$	−2.72498	−0.166766
$268$	3.90833	0.238739
$269$	0.605551	0.0369211	0.0184606	−	0.999830i	$-0.494123\pi$
$269$	0.605551	0.0369211	0.0184606	+	0.999830i	$0.494123\pi$
$270$	−5.60555	−0.341143
$271$	−11.9083	−0.723379	−0.361690	−	0.932299i	$-0.617800\pi$
$271$	−11.9083	−0.723379	−0.361690	+	0.932299i	$0.617800\pi$
$272$	0	0
$273$	−19.8167	−1.19936
$274$	1.81665	0.109748
$275$	1.00000	0.0603023
$276$	2.60555	0.156836
$277$	19.0917	1.14711	0.573554	−	0.819168i	$-0.305564\pi$
$277$	19.0917	1.14711	0.573554	+	0.819168i	$0.305564\pi$
$278$	−9.81665	−0.588764
$279$	−0.788897	−0.0472301
$280$	−4.60555	−0.275234
$281$	−9.90833	−0.591081	−0.295541	−	0.955330i	$-0.595500\pi$
$281$	−9.90833	−0.591081	−0.295541	+	0.955330i	$0.595500\pi$
$282$	−9.78890	−0.582920
$283$	−31.0278	−1.84441	−0.922204	−	0.386703i	$-0.873614\pi$
$283$	−31.0278	−1.84441	−0.922204	+	0.386703i	$0.873614\pi$
$284$	−10.1194	−0.600478
$285$	3.00000	0.177705
$286$	−3.30278	−0.195297
$287$	−27.6333	−1.63114
$288$	1.30278	0.0767668
$289$	−17.0000	−1.00000
$290$	4.60555	0.270447
$291$	−17.2111	−1.00893
$292$	1.00000	0.0585206
$293$	0	0		−	1.00000i	$-0.5\pi$
$293$	0	0			1.00000i	$0.5\pi$
$294$	−18.5139	−1.07975
$295$	7.30278	0.425184
$296$	−2.00000	−0.116248
$297$	−5.60555	−0.325267
$298$	−2.90833	−0.168475
$299$	6.60555	0.382009
$300$	1.30278	0.0752158
$301$	−19.8167	−1.14221
$302$	17.3028	0.995663
$303$	−24.0000	−1.37876
$304$	−2.30278	−0.132073
$305$	4.09167	0.234289
$306$	0	0
$307$	−4.30278	−0.245572	−0.122786	−	0.992433i	$-0.539183\pi$
$307$	−4.30278	−0.245572	−0.122786	+	0.992433i	$0.539183\pi$
$308$	−4.60555	−0.262426
$309$	12.0000	0.682656
$310$	0.605551	0.0343930
$311$	22.6972	1.28704	0.643521	−	0.765429i	$-0.277473\pi$
$311$	22.6972	1.28704	0.643521	+	0.765429i	$0.277473\pi$
$312$	−4.30278	−0.243597
$313$	−20.1194	−1.13722	−0.568609	−	0.822608i	$-0.692518\pi$
$313$	−20.1194	−1.13722	−0.568609	+	0.822608i	$0.692518\pi$
$314$	−3.39445	−0.191560
$315$	−6.00000	−0.338062
$316$	−8.60555	−0.484100
$317$	−28.5416	−1.60306	−0.801529	−	0.597956i	$-0.795980\pi$
$317$	−28.5416	−1.60306	−0.801529	+	0.597956i	$0.795980\pi$
$318$	6.78890	0.380702
$319$	4.60555	0.257861
$320$	−1.00000	−0.0559017
$321$	−17.6056	−0.982646
$322$	9.21110	0.513315
$323$	0	0
$324$	−3.39445	−0.188580
$325$	3.30278	0.183205
$326$	5.81665	0.322155
$327$	19.8167	1.09586
$328$	−6.00000	−0.331295
$329$	−34.6056	−1.90787
$330$	1.30278	0.0717154
$331$	−3.48612	−0.191615	−0.0958073	−	0.995400i	$-0.530543\pi$
$331$	−3.48612	−0.191615	−0.0958073	+	0.995400i	$0.530543\pi$
$332$	14.5139	0.796553
$333$	−2.60555	−0.142783
$334$	6.00000	0.328305
$335$	−3.90833	−0.213535
$336$	−6.00000	−0.327327
$337$	5.39445	0.293854	0.146927	−	0.989147i	$-0.453062\pi$
$337$	5.39445	0.293854	0.146927	+	0.989147i	$0.453062\pi$
$338$	2.09167	0.113772
$339$	−24.6333	−1.33790
$340$	0	0
$341$	0.605551	0.0327924
$342$	−3.00000	−0.162221
$343$	−33.2111	−1.79323
$344$	−4.30278	−0.231990
$345$	−2.60555	−0.140278
$346$	25.6333	1.37806
$347$	1.81665	0.0975231	0.0487615	−	0.998810i	$-0.484473\pi$
$347$	1.81665	0.0975231	0.0487615	+	0.998810i	$0.484473\pi$
$348$	6.00000	0.321634
$349$	0.486122	0.0260215	0.0130108	−	0.999915i	$-0.495858\pi$
$349$	0.486122	0.0260215	0.0130108	+	0.999915i	$0.495858\pi$
$350$	4.60555	0.246177
$351$	−18.5139	−0.988198
$352$	−1.00000	−0.0533002
$353$	22.4222	1.19341	0.596707	−	0.802459i	$-0.296476\pi$
$353$	22.4222	1.19341	0.596707	+	0.802459i	$0.296476\pi$
$354$	9.51388	0.505657
$355$	10.1194	0.537084
$356$	−2.09167	−0.110858
$357$	0	0
$358$	2.30278	0.121705
$359$	13.2111	0.697255	0.348628	−	0.937261i	$-0.386648\pi$
$359$	13.2111	0.697255	0.348628	+	0.937261i	$0.386648\pi$
$360$	−1.30278	−0.0686623
$361$	−13.6972	−0.720907
$362$	24.6056	1.29324
$363$	1.30278	0.0683780
$364$	−15.2111	−0.797278
$365$	−1.00000	−0.0523424
$366$	5.33053	0.278631
$367$	−29.6333	−1.54685	−0.773423	−	0.633890i	$-0.781457\pi$
$367$	−29.6333	−1.54685	−0.773423	+	0.633890i	$0.781457\pi$
$368$	2.00000	0.104257
$369$	−7.81665	−0.406919
$370$	2.00000	0.103975
$371$	24.0000	1.24602
$372$	0.788897	0.0409024
$373$	13.6333	0.705906	0.352953	−	0.935641i	$-0.385178\pi$
$373$	13.6333	0.705906	0.352953	+	0.935641i	$0.385178\pi$
$374$	0	0
$375$	−1.30278	−0.0672750
$376$	−7.51388	−0.387499
$377$	15.2111	0.783412
$378$	−25.8167	−1.32787
$379$	−4.72498	−0.242706	−0.121353	−	0.992609i	$-0.538723\pi$
$379$	−4.72498	−0.242706	−0.121353	+	0.992609i	$0.538723\pi$
$380$	2.30278	0.118130
$381$	28.3028	1.45000
$382$	−2.78890	−0.142692
$383$	29.6333	1.51419	0.757096	−	0.653304i	$-0.226618\pi$
$383$	29.6333	1.51419	0.757096	+	0.653304i	$0.226618\pi$
$384$	−1.30278	−0.0664820
$385$	4.60555	0.234721
$386$	17.0278	0.866690
$387$	−5.60555	−0.284946
$388$	−13.2111	−0.670692
$389$	−18.0000	−0.912636	−0.456318	−	0.889817i	$-0.650832\pi$
$389$	−18.0000	−0.912636	−0.456318	+	0.889817i	$0.650832\pi$
$390$	4.30278	0.217880
$391$	0	0
$392$	−14.2111	−0.717769
$393$	27.6333	1.39392
$394$	−8.90833	−0.448795
$395$	8.60555	0.432992
$396$	−1.30278	−0.0654669
$397$	−20.3028	−1.01897	−0.509484	−	0.860480i	$-0.670164\pi$
$397$	−20.3028	−1.01897	−0.509484	+	0.860480i	$0.670164\pi$
$398$	20.4222	1.02367
$399$	13.8167	0.691698
$400$	1.00000	0.0500000
$401$	−30.9361	−1.54487	−0.772437	−	0.635091i	$-0.780962\pi$
$401$	−30.9361	−1.54487	−0.772437	+	0.635091i	$0.780962\pi$
$402$	−5.09167	−0.253950
$403$	2.00000	0.0996271
$404$	−18.4222	−0.916539
$405$	3.39445	0.168672
$406$	21.2111	1.05269
$407$	2.00000	0.0991363
$408$	0	0
$409$	3.33053	0.164684	0.0823421	−	0.996604i	$-0.473760\pi$
$409$	3.33053	0.164684	0.0823421	+	0.996604i	$0.473760\pi$
$410$	6.00000	0.296319
$411$	−2.36669	−0.116740
$412$	9.21110	0.453798
$413$	33.6333	1.65499
$414$	2.60555	0.128056
$415$	−14.5139	−0.712458
$416$	−3.30278	−0.161932
$417$	12.7889	0.626275
$418$	2.30278	0.112632
$419$	19.6333	0.959150	0.479575	−	0.877501i	$-0.340791\pi$
$419$	19.6333	0.959150	0.479575	+	0.877501i	$0.340791\pi$
$420$	6.00000	0.292770
$421$	10.8806	0.530287	0.265143	−	0.964209i	$-0.414581\pi$
$421$	10.8806	0.530287	0.265143	+	0.964209i	$0.414581\pi$
$422$	16.7250	0.814159
$423$	−9.78890	−0.475952
$424$	5.21110	0.253074
$425$	0	0
$426$	13.1833	0.638735
$427$	18.8444	0.911945
$428$	−13.5139	−0.653218
$429$	4.30278	0.207740
$430$	4.30278	0.207498
$431$	−20.9361	−1.00846	−0.504228	−	0.863571i	$-0.668223\pi$
$431$	−20.9361	−1.00846	−0.504228	+	0.863571i	$0.668223\pi$
$432$	−5.60555	−0.269697
$433$	−16.1194	−0.774650	−0.387325	−	0.921943i	$-0.626601\pi$
$433$	−16.1194	−0.774650	−0.387325	+	0.921943i	$0.626601\pi$
$434$	2.78890	0.133871
$435$	−6.00000	−0.287678
$436$	15.2111	0.728480
$437$	−4.60555	−0.220313
$438$	−1.30278	−0.0622490
$439$	12.2389	0.584129	0.292064	−	0.956399i	$-0.405658\pi$
$439$	12.2389	0.584129	0.292064	+	0.956399i	$0.405658\pi$
$440$	1.00000	0.0476731
$441$	−18.5139	−0.881613
$442$	0	0
$443$	35.6333	1.69299	0.846495	−	0.532397i	$-0.178709\pi$
$443$	35.6333	1.69299	0.846495	+	0.532397i	$0.178709\pi$
$444$	2.60555	0.123654
$445$	2.09167	0.0991548
$446$	−15.8167	−0.748940
$447$	3.78890	0.179209
$448$	−4.60555	−0.217592
$449$	−18.0000	−0.849473	−0.424736	−	0.905317i	$-0.639633\pi$
$449$	−18.0000	−0.849473	−0.424736	+	0.905317i	$0.639633\pi$
$450$	1.30278	0.0614134
$451$	6.00000	0.282529
$452$	−18.9083	−0.889373
$453$	−22.5416	−1.05910
$454$	−21.3944	−1.00409
$455$	15.2111	0.713107
$456$	3.00000	0.140488
$457$	−5.69722	−0.266505	−0.133252	−	0.991082i	$-0.542542\pi$
$457$	−5.69722	−0.266505	−0.133252	+	0.991082i	$0.542542\pi$
$458$	−12.1194	−0.566304
$459$	0	0
$460$	−2.00000	−0.0932505
$461$	−3.27502	−0.152533	−0.0762664	−	0.997087i	$-0.524300\pi$
$461$	−3.27502	−0.152533	−0.0762664	+	0.997087i	$0.524300\pi$
$462$	6.00000	0.279145
$463$	17.8167	0.828010	0.414005	−	0.910275i	$-0.364130\pi$
$463$	17.8167	0.828010	0.414005	+	0.910275i	$0.364130\pi$
$464$	4.60555	0.213807
$465$	−0.788897	−0.0365842
$466$	−25.2111	−1.16788
$467$	0	0		−	1.00000i	$-0.5\pi$
$467$	0	0			1.00000i	$0.5\pi$
$468$	−4.30278	−0.198896
$469$	−18.0000	−0.831163
$470$	7.51388	0.346589
$471$	4.42221	0.203764
$472$	7.30278	0.336137
$473$	4.30278	0.197842
$474$	11.2111	0.514943
$475$	−2.30278	−0.105659
$476$	0	0
$477$	6.78890	0.310842
$478$	1.48612	0.0679736
$479$	35.8167	1.63650	0.818252	−	0.574859i	$-0.194943\pi$
$479$	35.8167	1.63650	0.818252	+	0.574859i	$0.194943\pi$
$480$	1.30278	0.0594633
$481$	6.60555	0.301187
$482$	22.0000	1.00207
$483$	−12.0000	−0.546019
$484$	1.00000	0.0454545
$485$	13.2111	0.599885
$486$	−12.3944	−0.562224
$487$	38.4222	1.74108	0.870538	−	0.492101i	$-0.163771\pi$
$487$	38.4222	1.74108	0.870538	+	0.492101i	$0.163771\pi$
$488$	4.09167	0.185221
$489$	−7.57779	−0.342680
$490$	14.2111	0.641992
$491$	−42.0000	−1.89543	−0.947717	−	0.319113i	$-0.896615\pi$
$491$	−42.0000	−1.89543	−0.947717	+	0.319113i	$0.896615\pi$
$492$	7.81665	0.352402
$493$	0	0
$494$	7.60555	0.342190
$495$	1.30278	0.0585554
$496$	0.605551	0.0271901
$497$	46.6056	2.09054
$498$	−18.9083	−0.847302
$499$	−20.7889	−0.930639	−0.465319	−	0.885143i	$-0.654061\pi$
$499$	−20.7889	−0.930639	−0.465319	+	0.885143i	$0.654061\pi$
$500$	−1.00000	−0.0447214
$501$	−7.81665	−0.349222
$502$	22.4222	1.00075
$503$	−1.69722	−0.0756755	−0.0378377	−	0.999284i	$-0.512047\pi$
$503$	−1.69722	−0.0756755	−0.0378377	+	0.999284i	$0.512047\pi$
$504$	−6.00000	−0.267261
$505$	18.4222	0.819777
$506$	−2.00000	−0.0889108
$507$	−2.72498	−0.121021
$508$	21.7250	0.963890
$509$	−1.63331	−0.0723951	−0.0361975	−	0.999345i	$-0.511525\pi$
$509$	−1.63331	−0.0723951	−0.0361975	+	0.999345i	$0.511525\pi$
$510$	0	0
$511$	−4.60555	−0.203738
$512$	−1.00000	−0.0441942
$513$	12.9083	0.569917
$514$	20.6056	0.908872
$515$	−9.21110	−0.405890
$516$	5.60555	0.246771
$517$	7.51388	0.330460
$518$	9.21110	0.404713
$519$	−33.3944	−1.46585
$520$	3.30278	0.144836
$521$	1.81665	0.0795890	0.0397945	−	0.999208i	$-0.487330\pi$
$521$	1.81665	0.0795890	0.0397945	+	0.999208i	$0.487330\pi$
$522$	6.00000	0.262613
$523$	−1.63331	−0.0714196	−0.0357098	−	0.999362i	$-0.511369\pi$
$523$	−1.63331	−0.0714196	−0.0357098	+	0.999362i	$0.511369\pi$
$524$	21.2111	0.926611
$525$	−6.00000	−0.261861
$526$	8.78890	0.383214
$527$	0	0
$528$	1.30278	0.0566960
$529$	−19.0000	−0.826087
$530$	−5.21110	−0.226356
$531$	9.51388	0.412867
$532$	10.6056	0.459809
$533$	19.8167	0.858355
$534$	2.72498	0.117921
$535$	13.5139	0.584256
$536$	−3.90833	−0.168814
$537$	−3.00000	−0.129460
$538$	−0.605551	−0.0261072
$539$	14.2111	0.612116
$540$	5.60555	0.241225
$541$	−43.0278	−1.84991	−0.924954	−	0.380079i	$-0.875897\pi$
$541$	−43.0278	−1.84991	−0.924954	+	0.380079i	$0.875897\pi$
$542$	11.9083	0.511506
$543$	−32.0555	−1.37563
$544$	0	0
$545$	−15.2111	−0.651572
$546$	19.8167	0.848074
$547$	−25.3944	−1.08579	−0.542894	−	0.839801i	$-0.682672\pi$
$547$	−25.3944	−1.08579	−0.542894	+	0.839801i	$0.682672\pi$
$548$	−1.81665	−0.0776036
$549$	5.33053	0.227502
$550$	−1.00000	−0.0426401
$551$	−10.6056	−0.451812
$552$	−2.60555	−0.110900
$553$	39.6333	1.68538
$554$	−19.0917	−0.811127
$555$	−2.60555	−0.110599
$556$	9.81665	0.416319
$557$	8.60555	0.364629	0.182314	−	0.983240i	$-0.441641\pi$
$557$	8.60555	0.364629	0.182314	+	0.983240i	$0.441641\pi$
$558$	0.788897	0.0333967
$559$	14.2111	0.601066
$560$	4.60555	0.194620
$561$	0	0
$562$	9.90833	0.417958
$563$	25.2111	1.06252	0.531261	−	0.847208i	$-0.321718\pi$
$563$	25.2111	1.06252	0.531261	+	0.847208i	$0.321718\pi$
$564$	9.78890	0.412187
$565$	18.9083	0.795479
$566$	31.0278	1.30419
$567$	15.6333	0.656537
$568$	10.1194	0.424602
$569$	30.8444	1.29306	0.646532	−	0.762886i	$-0.276218\pi$
$569$	30.8444	1.29306	0.646532	+	0.762886i	$0.276218\pi$
$570$	−3.00000	−0.125656
$571$	−2.78890	−0.116712	−0.0583559	−	0.998296i	$-0.518586\pi$
$571$	−2.78890	−0.116712	−0.0583559	+	0.998296i	$0.518586\pi$
$572$	3.30278	0.138096
$573$	3.63331	0.151784
$574$	27.6333	1.15339
$575$	2.00000	0.0834058
$576$	−1.30278	−0.0542823
$577$	−29.4861	−1.22752	−0.613762	−	0.789491i	$-0.710344\pi$
$577$	−29.4861	−1.22752	−0.613762	+	0.789491i	$0.710344\pi$
$578$	17.0000	0.707107
$579$	−22.1833	−0.921908
$580$	−4.60555	−0.191235
$581$	−66.8444	−2.77317
$582$	17.2111	0.713423
$583$	−5.21110	−0.215822
$584$	−1.00000	−0.0413803
$585$	4.30278	0.177898
$586$	0	0
$587$	11.6333	0.480158	0.240079	−	0.970753i	$-0.422827\pi$
$587$	11.6333	0.480158	0.240079	+	0.970753i	$0.422827\pi$
$588$	18.5139	0.763499
$589$	−1.39445	−0.0574573
$590$	−7.30278	−0.300651
$591$	11.6056	0.477389
$592$	2.00000	0.0821995
$593$	−11.8806	−0.487877	−0.243938	−	0.969791i	$-0.578439\pi$
$593$	−11.8806	−0.487877	−0.243938	+	0.969791i	$0.578439\pi$
$594$	5.60555	0.229999
$595$	0	0
$596$	2.90833	0.119130
$597$	−26.6056	−1.08889
$598$	−6.60555	−0.270121
$599$	9.39445	0.383847	0.191923	−	0.981410i	$-0.438527\pi$
$599$	9.39445	0.383847	0.191923	+	0.981410i	$0.438527\pi$
$600$	−1.30278	−0.0531856
$601$	13.7250	0.559854	0.279927	−	0.960021i	$-0.409690\pi$
$601$	13.7250	0.559854	0.279927	+	0.960021i	$0.409690\pi$
$602$	19.8167	0.807667
$603$	−5.09167	−0.207349
$604$	−17.3028	−0.704040
$605$	−1.00000	−0.0406558
$606$	24.0000	0.974933
$607$	1.33053	0.0540046	0.0270023	−	0.999635i	$-0.491404\pi$
$607$	1.33053	0.0540046	0.0270023	+	0.999635i	$0.491404\pi$
$608$	2.30278	0.0933899
$609$	−27.6333	−1.11976
$610$	−4.09167	−0.165667
$611$	24.8167	1.00397
$612$	0	0
$613$	−25.9083	−1.04643	−0.523214	−	0.852201i	$-0.675267\pi$
$613$	−25.9083	−1.04643	−0.523214	+	0.852201i	$0.675267\pi$
$614$	4.30278	0.173646
$615$	−7.81665	−0.315198
$616$	4.60555	0.185563
$617$	6.33053	0.254858	0.127429	−	0.991848i	$-0.459328\pi$
$617$	6.33053	0.254858	0.127429	+	0.991848i	$0.459328\pi$
$618$	−12.0000	−0.482711
$619$	−27.0278	−1.08634	−0.543169	−	0.839624i	$-0.682776\pi$
$619$	−27.0278	−1.08634	−0.543169	+	0.839624i	$0.682776\pi$
$620$	−0.605551	−0.0243195
$621$	−11.2111	−0.449886
$622$	−22.6972	−0.910076
$623$	9.63331	0.385950
$624$	4.30278	0.172249
$625$	1.00000	0.0400000
$626$	20.1194	0.804134
$627$	−3.00000	−0.119808
$628$	3.39445	0.135453
$629$	0	0
$630$	6.00000	0.239046
$631$	37.2111	1.48135	0.740675	−	0.671863i	$-0.234506\pi$
$631$	37.2111	1.48135	0.740675	+	0.671863i	$0.234506\pi$
$632$	8.60555	0.342310
$633$	−21.7889	−0.866031
$634$	28.5416	1.13353
$635$	−21.7250	−0.862130
$636$	−6.78890	−0.269197
$637$	46.9361	1.85968
$638$	−4.60555	−0.182336
$639$	13.1833	0.521525
$640$	1.00000	0.0395285
$641$	−34.0000	−1.34292	−0.671460	−	0.741041i	$-0.734332\pi$
$641$	−34.0000	−1.34292	−0.671460	+	0.741041i	$0.734332\pi$
$642$	17.6056	0.694836
$643$	24.0000	0.946468	0.473234	−	0.880937i	$-0.343087\pi$
$643$	24.0000	0.946468	0.473234	+	0.880937i	$0.343087\pi$
$644$	−9.21110	−0.362968
$645$	−5.60555	−0.220718
$646$	0	0
$647$	−49.5694	−1.94877	−0.974387	−	0.224878i	$-0.927802\pi$
$647$	−49.5694	−1.94877	−0.974387	+	0.224878i	$0.927802\pi$
$648$	3.39445	0.133347
$649$	−7.30278	−0.286659
$650$	−3.30278	−0.129546
$651$	−3.63331	−0.142401
$652$	−5.81665	−0.227798
$653$	25.7250	1.00670	0.503348	−	0.864084i	$-0.332101\pi$
$653$	25.7250	1.00670	0.503348	+	0.864084i	$0.332101\pi$
$654$	−19.8167	−0.774892
$655$	−21.2111	−0.828786
$656$	6.00000	0.234261
$657$	−1.30278	−0.0508261
$658$	34.6056	1.34906
$659$	−11.5139	−0.448517	−0.224259	−	0.974530i	$-0.571996\pi$
$659$	−11.5139	−0.448517	−0.224259	+	0.974530i	$0.571996\pi$
$660$	−1.30278	−0.0507105
$661$	44.8444	1.74425	0.872123	−	0.489287i	$-0.162743\pi$
$661$	44.8444	1.74425	0.872123	+	0.489287i	$0.162743\pi$
$662$	3.48612	0.135492
$663$	0	0
$664$	−14.5139	−0.563248
$665$	−10.6056	−0.411266
$666$	2.60555	0.100963
$667$	9.21110	0.356655
$668$	−6.00000	−0.232147
$669$	20.6056	0.796657
$670$	3.90833	0.150992
$671$	−4.09167	−0.157957
$672$	6.00000	0.231455
$673$	26.3305	1.01497	0.507484	−	0.861661i	$-0.330576\pi$
$673$	26.3305	1.01497	0.507484	+	0.861661i	$0.330576\pi$
$674$	−5.39445	−0.207786
$675$	−5.60555	−0.215758
$676$	−2.09167	−0.0804490
$677$	25.7527	0.989758	0.494879	−	0.868962i	$-0.335212\pi$
$677$	25.7527	0.989758	0.494879	+	0.868962i	$0.335212\pi$
$678$	24.6333	0.946036
$679$	60.8444	2.33499
$680$	0	0
$681$	27.8722	1.06806
$682$	−0.605551	−0.0231878
$683$	−38.2389	−1.46317	−0.731585	−	0.681751i	$-0.761219\pi$
$683$	−38.2389	−1.46317	−0.731585	+	0.681751i	$0.761219\pi$
$684$	3.00000	0.114708
$685$	1.81665	0.0694108
$686$	33.2111	1.26801
$687$	15.7889	0.602384
$688$	4.30278	0.164042
$689$	−17.2111	−0.655691
$690$	2.60555	0.0991916
$691$	−25.7250	−0.978624	−0.489312	−	0.872109i	$-0.662752\pi$
$691$	−25.7250	−0.978624	−0.489312	+	0.872109i	$0.662752\pi$
$692$	−25.6333	−0.974432
$693$	6.00000	0.227921
$694$	−1.81665	−0.0689592
$695$	−9.81665	−0.372367
$696$	−6.00000	−0.227429
$697$	0	0
$698$	−0.486122	−0.0184000
$699$	32.8444	1.24229
$700$	−4.60555	−0.174073
$701$	−32.8444	−1.24052	−0.620258	−	0.784398i	$-0.712972\pi$
$701$	−32.8444	−1.24052	−0.620258	+	0.784398i	$0.712972\pi$
$702$	18.5139	0.698761
$703$	−4.60555	−0.173702
$704$	1.00000	0.0376889
$705$	−9.78890	−0.368671
$706$	−22.4222	−0.843871
$707$	84.8444	3.19090
$708$	−9.51388	−0.357553
$709$	25.1194	0.943380	0.471690	−	0.881764i	$-0.343644\pi$
$709$	25.1194	0.943380	0.471690	+	0.881764i	$0.343644\pi$
$710$	−10.1194	−0.379775
$711$	11.2111	0.420449
$712$	2.09167	0.0783888
$713$	1.21110	0.0453561
$714$	0	0
$715$	−3.30278	−0.123517
$716$	−2.30278	−0.0860588
$717$	−1.93608	−0.0723044
$718$	−13.2111	−0.493034
$719$	−21.6333	−0.806786	−0.403393	−	0.915027i	$-0.632169\pi$
$719$	−21.6333	−0.806786	−0.403393	+	0.915027i	$0.632169\pi$
$720$	1.30278	0.0485516
$721$	−42.4222	−1.57989
$722$	13.6972	0.509758
$723$	−28.6611	−1.06592
$724$	−24.6056	−0.914458
$725$	4.60555	0.171046
$726$	−1.30278	−0.0483505
$727$	−40.6611	−1.50804	−0.754018	−	0.656854i	$-0.771887\pi$
$727$	−40.6611	−1.50804	−0.754018	+	0.656854i	$0.771887\pi$
$728$	15.2111	0.563761
$729$	26.3305	0.975205
$730$	1.00000	0.0370117
$731$	0	0
$732$	−5.33053	−0.197022
$733$	−24.6056	−0.908827	−0.454413	−	0.890791i	$-0.650151\pi$
$733$	−24.6056	−0.908827	−0.454413	+	0.890791i	$0.650151\pi$
$734$	29.6333	1.09379
$735$	−18.5139	−0.682895
$736$	−2.00000	−0.0737210
$737$	3.90833	0.143965
$738$	7.81665	0.287735
$739$	15.6333	0.575081	0.287540	−	0.957769i	$-0.407162\pi$
$739$	15.6333	0.575081	0.287540	+	0.957769i	$0.407162\pi$
$740$	−2.00000	−0.0735215
$741$	−9.90833	−0.363991
$742$	−24.0000	−0.881068
$743$	30.2389	1.10936	0.554678	−	0.832065i	$-0.312841\pi$
$743$	30.2389	1.10936	0.554678	+	0.832065i	$0.312841\pi$
$744$	−0.788897	−0.0289224
$745$	−2.90833	−0.106553
$746$	−13.6333	−0.499151
$747$	−18.9083	−0.691820
$748$	0	0
$749$	62.2389	2.27416
$750$	1.30278	0.0475706
$751$	−34.4222	−1.25608	−0.628042	−	0.778179i	$-0.716143\pi$
$751$	−34.4222	−1.25608	−0.628042	+	0.778179i	$0.716143\pi$
$752$	7.51388	0.274003
$753$	−29.2111	−1.06451
$754$	−15.2111	−0.553956
$755$	17.3028	0.629712
$756$	25.8167	0.938943
$757$	12.3028	0.447152	0.223576	−	0.974687i	$-0.428227\pi$
$757$	12.3028	0.447152	0.223576	+	0.974687i	$0.428227\pi$
$758$	4.72498	0.171619
$759$	2.60555	0.0945755
$760$	−2.30278	−0.0835305
$761$	8.30278	0.300975	0.150488	−	0.988612i	$-0.451916\pi$
$761$	8.30278	0.300975	0.150488	+	0.988612i	$0.451916\pi$
$762$	−28.3028	−1.02530
$763$	−70.0555	−2.53618
$764$	2.78890	0.100899
$765$	0	0
$766$	−29.6333	−1.07069
$767$	−24.1194	−0.870902
$768$	1.30278	0.0470099
$769$	23.6972	0.854544	0.427272	−	0.904123i	$-0.359475\pi$
$769$	23.6972	0.854544	0.427272	+	0.904123i	$0.359475\pi$
$770$	−4.60555	−0.165973
$771$	−26.8444	−0.966778
$772$	−17.0278	−0.612842
$773$	−18.7889	−0.675790	−0.337895	−	0.941184i	$-0.609715\pi$
$773$	−18.7889	−0.675790	−0.337895	+	0.941184i	$0.609715\pi$
$774$	5.60555	0.201487
$775$	0.605551	0.0217520
$776$	13.2111	0.474251
$777$	−12.0000	−0.430498
$778$	18.0000	0.645331
$779$	−13.8167	−0.495033
$780$	−4.30278	−0.154064
$781$	−10.1194	−0.362102
$782$	0	0
$783$	−25.8167	−0.922612
$784$	14.2111	0.507539
$785$	−3.39445	−0.121153
$786$	−27.6333	−0.985647
$787$	7.21110	0.257048	0.128524	−	0.991706i	$-0.458976\pi$
$787$	7.21110	0.257048	0.128524	+	0.991706i	$0.458976\pi$
$788$	8.90833	0.317346
$789$	−11.4500	−0.407629
$790$	−8.60555	−0.306172
$791$	87.0833	3.09632
$792$	1.30278	0.0462921
$793$	−13.5139	−0.479892
$794$	20.3028	0.720519
$795$	6.78890	0.240777
$796$	−20.4222	−0.723846
$797$	10.9361	0.387376	0.193688	−	0.981063i	$-0.437955\pi$
$797$	10.9361	0.387376	0.193688	+	0.981063i	$0.437955\pi$
$798$	−13.8167	−0.489104
$799$	0	0
$800$	−1.00000	−0.0353553
$801$	2.72498	0.0962825
$802$	30.9361	1.09239
$803$	1.00000	0.0352892
$804$	5.09167	0.179569
$805$	9.21110	0.324649
$806$	−2.00000	−0.0704470
$807$	0.788897	0.0277705
$808$	18.4222	0.648091
$809$	15.0278	0.528348	0.264174	−	0.964475i	$-0.414901\pi$
$809$	15.0278	0.528348	0.264174	+	0.964475i	$0.414901\pi$
$810$	−3.39445	−0.119269
$811$	10.7889	0.378849	0.189425	−	0.981895i	$-0.439338\pi$
$811$	10.7889	0.378849	0.189425	+	0.981895i	$0.439338\pi$
$812$	−21.2111	−0.744364
$813$	−15.5139	−0.544095
$814$	−2.00000	−0.0701000
$815$	5.81665	0.203748
$816$	0	0
$817$	−9.90833	−0.346648
$818$	−3.33053	−0.116449
$819$	19.8167	0.692450
$820$	−6.00000	−0.209529
$821$	2.66947	0.0931651	0.0465825	−	0.998914i	$-0.485167\pi$
$821$	2.66947	0.0931651	0.0465825	+	0.998914i	$0.485167\pi$
$822$	2.36669	0.0825479
$823$	−30.3028	−1.05629	−0.528144	−	0.849155i	$-0.677112\pi$
$823$	−30.3028	−1.05629	−0.528144	+	0.849155i	$0.677112\pi$
$824$	−9.21110	−0.320884
$825$	1.30278	0.0453568
$826$	−33.6333	−1.17025
$827$	−24.7889	−0.861994	−0.430997	−	0.902353i	$-0.641838\pi$
$827$	−24.7889	−0.861994	−0.430997	+	0.902353i	$0.641838\pi$
$828$	−2.60555	−0.0905492
$829$	18.8444	0.654493	0.327247	−	0.944939i	$-0.393879\pi$
$829$	18.8444	0.654493	0.327247	+	0.944939i	$0.393879\pi$
$830$	14.5139	0.503784
$831$	24.8722	0.862806
$832$	3.30278	0.114503
$833$	0	0
$834$	−12.7889	−0.442843
$835$	6.00000	0.207639
$836$	−2.30278	−0.0796432
$837$	−3.39445	−0.117329
$838$	−19.6333	−0.678221
$839$	22.1472	0.764606	0.382303	−	0.924037i	$-0.375131\pi$
$839$	22.1472	0.764606	0.382303	+	0.924037i	$0.375131\pi$
$840$	−6.00000	−0.207020
$841$	−7.78890	−0.268583
$842$	−10.8806	−0.374969
$843$	−12.9083	−0.444586
$844$	−16.7250	−0.575698
$845$	2.09167	0.0719557
$846$	9.78890	0.336549
$847$	−4.60555	−0.158249
$848$	−5.21110	−0.178950
$849$	−40.4222	−1.38729
$850$	0	0
$851$	4.00000	0.137118
$852$	−13.1833	−0.451654
$853$	−51.8167	−1.77417	−0.887084	−	0.461608i	$-0.847273\pi$
$853$	−51.8167	−1.77417	−0.887084	+	0.461608i	$0.847273\pi$
$854$	−18.8444	−0.644842
$855$	−3.00000	−0.102598
$856$	13.5139	0.461895
$857$	17.9361	0.612685	0.306342	−	0.951921i	$-0.400895\pi$
$857$	17.9361	0.612685	0.306342	+	0.951921i	$0.400895\pi$
$858$	−4.30278	−0.146894
$859$	−24.1194	−0.822944	−0.411472	−	0.911422i	$-0.634985\pi$
$859$	−24.1194	−0.822944	−0.411472	+	0.911422i	$0.634985\pi$
$860$	−4.30278	−0.146723
$861$	−36.0000	−1.22688
$862$	20.9361	0.713086
$863$	14.2750	0.485927	0.242964	−	0.970035i	$-0.421880\pi$
$863$	14.2750	0.485927	0.242964	+	0.970035i	$0.421880\pi$
$864$	5.60555	0.190705
$865$	25.6333	0.871559
$866$	16.1194	0.547761
$867$	−22.1472	−0.752158
$868$	−2.78890	−0.0946613
$869$	−8.60555	−0.291923
$870$	6.00000	0.203419
$871$	12.9083	0.437382
$872$	−15.2111	−0.515113
$873$	17.2111	0.582508
$874$	4.60555	0.155785
$875$	4.60555	0.155696
$876$	1.30278	0.0440167
$877$	0.605551	0.0204480	0.0102240	−	0.999948i	$-0.496746\pi$
$877$	0.605551	0.0204480	0.0102240	+	0.999948i	$0.496746\pi$
$878$	−12.2389	−0.413041
$879$	0	0
$880$	−1.00000	−0.0337100
$881$	33.0278	1.11273	0.556367	−	0.830937i	$-0.312195\pi$
$881$	33.0278	1.11273	0.556367	+	0.830937i	$0.312195\pi$
$882$	18.5139	0.623395
$883$	−40.4222	−1.36032	−0.680158	−	0.733066i	$-0.738089\pi$
$883$	−40.4222	−1.36032	−0.680158	+	0.733066i	$0.738089\pi$
$884$	0	0
$885$	9.51388	0.319806
$886$	−35.6333	−1.19712
$887$	51.2111	1.71950	0.859750	−	0.510715i	$-0.170619\pi$
$887$	51.2111	1.71950	0.859750	+	0.510715i	$0.170619\pi$
$888$	−2.60555	−0.0874366
$889$	−100.056	−3.35575
$890$	−2.09167	−0.0701130
$891$	−3.39445	−0.113718
$892$	15.8167	0.529581
$893$	−17.3028	−0.579015
$894$	−3.78890	−0.126720
$895$	2.30278	0.0769733
$896$	4.60555	0.153861
$897$	8.60555	0.287331
$898$	18.0000	0.600668
$899$	2.78890	0.0930149
$900$	−1.30278	−0.0434259
$901$	0	0
$902$	−6.00000	−0.199778
$903$	−25.8167	−0.859124
$904$	18.9083	0.628882
$905$	24.6056	0.817916
$906$	22.5416	0.748896
$907$	44.7889	1.48719	0.743595	−	0.668630i	$-0.233119\pi$
$907$	44.7889	1.48719	0.743595	+	0.668630i	$0.233119\pi$
$908$	21.3944	0.710000
$909$	24.0000	0.796030
$910$	−15.2111	−0.504243
$911$	−14.1194	−0.467798	−0.233899	−	0.972261i	$-0.575148\pi$
$911$	−14.1194	−0.467798	−0.233899	+	0.972261i	$0.575148\pi$
$912$	−3.00000	−0.0993399
$913$	14.5139	0.480339
$914$	5.69722	0.188447
$915$	5.33053	0.176222
$916$	12.1194	0.400437
$917$	−97.6888	−3.22597
$918$	0	0
$919$	−27.9361	−0.921526	−0.460763	−	0.887523i	$-0.652424\pi$
$919$	−27.9361	−0.921526	−0.460763	+	0.887523i	$0.652424\pi$
$920$	2.00000	0.0659380
$921$	−5.60555	−0.184709
$922$	3.27502	0.107857
$923$	−33.4222	−1.10011
$924$	−6.00000	−0.197386
$925$	2.00000	0.0657596
$926$	−17.8167	−0.585492
$927$	−12.0000	−0.394132
$928$	−4.60555	−0.151185
$929$	−5.76114	−0.189017	−0.0945085	−	0.995524i	$-0.530128\pi$
$929$	−5.76114	−0.189017	−0.0945085	+	0.995524i	$0.530128\pi$
$930$	0.788897	0.0258690
$931$	−32.7250	−1.07252
$932$	25.2111	0.825817
$933$	29.5694	0.968058
$934$	0	0
$935$	0	0
$936$	4.30278	0.140641
$937$	−13.0917	−0.427686	−0.213843	−	0.976868i	$-0.568598\pi$
$937$	−13.0917	−0.427686	−0.213843	+	0.976868i	$0.568598\pi$
$938$	18.0000	0.587721
$939$	−26.2111	−0.855367
$940$	−7.51388	−0.245076
$941$	−31.3583	−1.02225	−0.511125	−	0.859506i	$-0.670771\pi$
$941$	−31.3583	−1.02225	−0.511125	+	0.859506i	$0.670771\pi$
$942$	−4.42221	−0.144083
$943$	12.0000	0.390774
$944$	−7.30278	−0.237685
$945$	−25.8167	−0.839816
$946$	−4.30278	−0.139895
$947$	−37.1194	−1.20622	−0.603110	−	0.797658i	$-0.706072\pi$
$947$	−37.1194	−1.20622	−0.603110	+	0.797658i	$0.706072\pi$
$948$	−11.2111	−0.364120
$949$	3.30278	0.107213
$950$	2.30278	0.0747119
$951$	−37.1833	−1.20575
$952$	0	0
$953$	34.3305	1.11208	0.556038	−	0.831157i	$-0.312321\pi$
$953$	34.3305	1.11208	0.556038	+	0.831157i	$0.312321\pi$
$954$	−6.78890	−0.219799
$955$	−2.78890	−0.0902466
$956$	−1.48612	−0.0480646
$957$	6.00000	0.193952
$958$	−35.8167	−1.15718
$959$	8.36669	0.270175
$960$	−1.30278	−0.0420469
$961$	−30.6333	−0.988171
$962$	−6.60555	−0.212972
$963$	17.6056	0.567331
$964$	−22.0000	−0.708572
$965$	17.0278	0.548143
$966$	12.0000	0.386094
$967$	14.9361	0.480312	0.240156	−	0.970734i	$-0.422801\pi$
$967$	14.9361	0.480312	0.240156	+	0.970734i	$0.422801\pi$
$968$	−1.00000	−0.0321412
$969$	0	0
$970$	−13.2111	−0.424183
$971$	−1.09167	−0.0350335	−0.0175167	−	0.999847i	$-0.505576\pi$
$971$	−1.09167	−0.0350335	−0.0175167	+	0.999847i	$0.505576\pi$
$972$	12.3944	0.397552
$973$	−45.2111	−1.44940
$974$	−38.4222	−1.23113
$975$	4.30278	0.137799
$976$	−4.09167	−0.130971
$977$	1.66947	0.0534110	0.0267055	−	0.999643i	$-0.491498\pi$
$977$	1.66947	0.0534110	0.0267055	+	0.999643i	$0.491498\pi$
$978$	7.57779	0.242311
$979$	−2.09167	−0.0668502
$980$	−14.2111	−0.453957
$981$	−19.8167	−0.632697
$982$	42.0000	1.34027
$983$	55.3860	1.76654	0.883270	−	0.468864i	$-0.155337\pi$
$983$	55.3860	1.76654	0.883270	+	0.468864i	$0.155337\pi$
$984$	−7.81665	−0.249186
$985$	−8.90833	−0.283843
$986$	0	0
$987$	−45.0833	−1.43502
$988$	−7.60555	−0.241965
$989$	8.60555	0.273641
$990$	−1.30278	−0.0414049
$991$	−32.0555	−1.01828	−0.509138	−	0.860685i	$-0.670036\pi$
$991$	−32.0555	−1.01828	−0.509138	+	0.860685i	$0.670036\pi$
$992$	−0.605551	−0.0192263
$993$	−4.54163	−0.144124
$994$	−46.6056	−1.47824
$995$	20.4222	0.647427
$996$	18.9083	0.599133
$997$	−29.0278	−0.919318	−0.459659	−	0.888095i	$-0.652028\pi$
$997$	−29.0278	−0.919318	−0.459659	+	0.888095i	$0.652028\pi$
$998$	20.7889	0.658061
$999$	−11.2111	−0.354704

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8030.2.a.k.1.2	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8030.2.a.k.1.2	✓	2	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 8030 = 2 \cdot 5 \cdot 11 \cdot 73 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8030.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.30278 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.30278 q^{6} -4.60555 q^{7} -1.00000 q^{8} -1.30278 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} +1.30278 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.30278 q^{6} -4.60555 q^{7} -1.00000 q^{8} -1.30278 q^{9} +1.00000 q^{10} +1.00000 q^{11} +1.30278 q^{12} +3.30278 q^{13} +4.60555 q^{14} -1.30278 q^{15} +1.00000 q^{16} +1.30278 q^{18} -2.30278 q^{19} -1.00000 q^{20} -6.00000 q^{21} -1.00000 q^{22} +2.00000 q^{23} -1.30278 q^{24} +1.00000 q^{25} -3.30278 q^{26} -5.60555 q^{27} -4.60555 q^{28} +4.60555 q^{29} +1.30278 q^{30} +0.605551 q^{31} -1.00000 q^{32} +1.30278 q^{33} +4.60555 q^{35} -1.30278 q^{36} +2.00000 q^{37} +2.30278 q^{38} +4.30278 q^{39} +1.00000 q^{40} +6.00000 q^{41} +6.00000 q^{42} +4.30278 q^{43} +1.00000 q^{44} +1.30278 q^{45} -2.00000 q^{46} +7.51388 q^{47} +1.30278 q^{48} +14.2111 q^{49} -1.00000 q^{50} +3.30278 q^{52} -5.21110 q^{53} +5.60555 q^{54} -1.00000 q^{55} +4.60555 q^{56} -3.00000 q^{57} -4.60555 q^{58} -7.30278 q^{59} -1.30278 q^{60} -4.09167 q^{61} -0.605551 q^{62} +6.00000 q^{63} +1.00000 q^{64} -3.30278 q^{65} -1.30278 q^{66} +3.90833 q^{67} +2.60555 q^{69} -4.60555 q^{70} -10.1194 q^{71} +1.30278 q^{72} +1.00000 q^{73} -2.00000 q^{74} +1.30278 q^{75} -2.30278 q^{76} -4.60555 q^{77} -4.30278 q^{78} -8.60555 q^{79} -1.00000 q^{80} -3.39445 q^{81} -6.00000 q^{82} +14.5139 q^{83} -6.00000 q^{84} -4.30278 q^{86} +6.00000 q^{87} -1.00000 q^{88} -2.09167 q^{89} -1.30278 q^{90} -15.2111 q^{91} +2.00000 q^{92} +0.788897 q^{93} -7.51388 q^{94} +2.30278 q^{95} -1.30278 q^{96} -13.2111 q^{97} -14.2111 q^{98} -1.30278 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} - q^{3} + 2 q^{4} - 2 q^{5} + q^{6} - 2 q^{7} - 2 q^{8} + q^{9}+O(q^{10}) \) 2 * q - 2 * q^2 - q^3 + 2 * q^4 - 2 * q^5 + q^6 - 2 * q^7 - 2 * q^8 + q^9	\( 2 q - 2 q^{2} - q^{3} + 2 q^{4} - 2 q^{5} + q^{6} - 2 q^{7} - 2 q^{8} + q^{9} + 2 q^{10} + 2 q^{11} - q^{12} + 3 q^{13} + 2 q^{14} + q^{15} + 2 q^{16} - q^{18} - q^{19} - 2 q^{20} - 12 q^{21} - 2 q^{22} + 4 q^{23} + q^{24} + 2 q^{25} - 3 q^{26} - 4 q^{27} - 2 q^{28} + 2 q^{29} - q^{30} - 6 q^{31} - 2 q^{32} - q^{33} + 2 q^{35} + q^{36} + 4 q^{37} + q^{38} + 5 q^{39} + 2 q^{40} + 12 q^{41} + 12 q^{42} + 5 q^{43} + 2 q^{44} - q^{45} - 4 q^{46} - 3 q^{47} - q^{48} + 14 q^{49} - 2 q^{50} + 3 q^{52} + 4 q^{53} + 4 q^{54} - 2 q^{55} + 2 q^{56} - 6 q^{57} - 2 q^{58} - 11 q^{59} + q^{60} - 19 q^{61} + 6 q^{62} + 12 q^{63} + 2 q^{64} - 3 q^{65} + q^{66} - 3 q^{67} - 2 q^{69} - 2 q^{70} + 5 q^{71} - q^{72} + 2 q^{73} - 4 q^{74} - q^{75} - q^{76} - 2 q^{77} - 5 q^{78} - 10 q^{79} - 2 q^{80} - 14 q^{81} - 12 q^{82} + 11 q^{83} - 12 q^{84} - 5 q^{86} + 12 q^{87} - 2 q^{88} - 15 q^{89} + q^{90} - 16 q^{91} + 4 q^{92} + 16 q^{93} + 3 q^{94} + q^{95} + q^{96} - 12 q^{97} - 14 q^{98} + q^{99}+O(q^{100}) \) 2 * q - 2 * q^2 - q^3 + 2 * q^4 - 2 * q^5 + q^6 - 2 * q^7 - 2 * q^8 + q^9 + 2 * q^10 + 2 * q^11 - q^12 + 3 * q^13 + 2 * q^14 + q^15 + 2 * q^16 - q^18 - q^19 - 2 * q^20 - 12 * q^21 - 2 * q^22 + 4 * q^23 + q^24 + 2 * q^25 - 3 * q^26 - 4 * q^27 - 2 * q^28 + 2 * q^29 - q^30 - 6 * q^31 - 2 * q^32 - q^33 + 2 * q^35 + q^36 + 4 * q^37 + q^38 + 5 * q^39 + 2 * q^40 + 12 * q^41 + 12 * q^42 + 5 * q^43 + 2 * q^44 - q^45 - 4 * q^46 - 3 * q^47 - q^48 + 14 * q^49 - 2 * q^50 + 3 * q^52 + 4 * q^53 + 4 * q^54 - 2 * q^55 + 2 * q^56 - 6 * q^57 - 2 * q^58 - 11 * q^59 + q^60 - 19 * q^61 + 6 * q^62 + 12 * q^63 + 2 * q^64 - 3 * q^65 + q^66 - 3 * q^67 - 2 * q^69 - 2 * q^70 + 5 * q^71 - q^72 + 2 * q^73 - 4 * q^74 - q^75 - q^76 - 2 * q^77 - 5 * q^78 - 10 * q^79 - 2 * q^80 - 14 * q^81 - 12 * q^82 + 11 * q^83 - 12 * q^84 - 5 * q^86 + 12 * q^87 - 2 * q^88 - 15 * q^89 + q^90 - 16 * q^91 + 4 * q^92 + 16 * q^93 + 3 * q^94 + q^95 + q^96 - 12 * q^97 - 14 * q^98 + q^99

Introduction

L-functions

Modular forms

Varieties

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Representations

Groups

Database

Properties

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