LMFDB - Embedded newform 6422.2.a.bb.1.1

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:	$1$
Dimension:	$4$
Coefficient field:	4.4.10273.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 5x^{2} + x + 2 $ x^4 - 2x^3 - 5x^2 + x + 2
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 494)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$3.35017$ 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)

$f(q)$	$=$	$q+1.00000 q^{2} -3.35017 q^{3} +1.00000 q^{4} +3.12030 q^{5} -3.35017 q^{6} +1.75319 q^{7} +1.00000 q^{8} +8.22366 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} -3.35017 q^{3} +1.00000 q^{4} +3.12030 q^{5} -3.35017 q^{6} +1.75319 q^{7} +1.00000 q^{8} +8.22366 q^{9} +3.12030 q^{10} -2.52331 q^{11} -3.35017 q^{12} +1.75319 q^{14} -10.4535 q^{15} +1.00000 q^{16} +1.05674 q^{17} +8.22366 q^{18} +1.00000 q^{19} +3.12030 q^{20} -5.87349 q^{21} -2.52331 q^{22} -5.35017 q^{23} -3.35017 q^{24} +4.73625 q^{25} -17.5002 q^{27} +1.75319 q^{28} -7.08643 q^{29} -10.4535 q^{30} -2.77012 q^{31} +1.00000 q^{32} +8.45353 q^{33} +1.05674 q^{34} +5.47047 q^{35} +8.22366 q^{36} +3.05674 q^{37} +1.00000 q^{38} +3.12030 q^{40} -3.89042 q^{41} -5.87349 q^{42} -10.2973 q^{43} -2.52331 q^{44} +25.6603 q^{45} -5.35017 q^{46} -10.8545 q^{47} -3.35017 q^{48} -3.92633 q^{49} +4.73625 q^{50} -3.54025 q^{51} -6.87349 q^{53} -17.5002 q^{54} -7.87349 q^{55} +1.75319 q^{56} -3.35017 q^{57} -7.08643 q^{58} -7.32934 q^{59} -10.4535 q^{60} -7.73625 q^{61} -2.77012 q^{62} +14.4176 q^{63} +1.00000 q^{64} +8.45353 q^{66} -5.61985 q^{67} +1.05674 q^{68} +17.9240 q^{69} +5.47047 q^{70} -10.2635 q^{71} +8.22366 q^{72} -3.11347 q^{73} +3.05674 q^{74} -15.8673 q^{75} +1.00000 q^{76} -4.42384 q^{77} +15.3377 q^{79} +3.12030 q^{80} +33.9576 q^{81} -3.89042 q^{82} +8.27650 q^{83} -5.87349 q^{84} +3.29733 q^{85} -10.2973 q^{86} +23.7408 q^{87} -2.52331 q^{88} +3.85062 q^{89} +25.6603 q^{90} -5.35017 q^{92} +9.28039 q^{93} -10.8545 q^{94} +3.12030 q^{95} -3.35017 q^{96} -12.5191 q^{97} -3.92633 q^{98} -20.7509 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{2} - 2 q^{3} + 4 q^{4} + q^{5} - 2 q^{6} - q^{7} + 4 q^{8} + 2 q^{9}+O(q^{10}) $ 4 * q + 4 * q^2 - 2 * q^3 + 4 * q^4 + q^5 - 2 * q^6 - q^7 + 4 * q^8 + 2 * q^9	$ 4 q + 4 q^{2} - 2 q^{3} + 4 q^{4} + q^{5} - 2 q^{6} - q^{7} + 4 q^{8} + 2 q^{9} + q^{10} - 2 q^{11} - 2 q^{12} - q^{14} - 11 q^{15} + 4 q^{16} + q^{17} + 2 q^{18} + 4 q^{19} + q^{20} - 4 q^{21} - 2 q^{22} - 10 q^{23} - 2 q^{24} + 3 q^{25} - 23 q^{27} - q^{28} - q^{29} - 11 q^{30} - 11 q^{31} + 4 q^{32} + 3 q^{33} + q^{34} - q^{35} + 2 q^{36} + 9 q^{37} + 4 q^{38} + q^{40} - 4 q^{41} - 4 q^{42} - 15 q^{43} - 2 q^{44} + 33 q^{45} - 10 q^{46} - 25 q^{47} - 2 q^{48} - 11 q^{49} + 3 q^{50} - 14 q^{51} - 8 q^{53} - 23 q^{54} - 12 q^{55} - q^{56} - 2 q^{57} - q^{58} - 28 q^{59} - 11 q^{60} - 15 q^{61} - 11 q^{62} + 20 q^{63} + 4 q^{64} + 3 q^{66} + q^{68} + 18 q^{69} - q^{70} + q^{71} + 2 q^{72} - 6 q^{73} + 9 q^{74} - 13 q^{75} + 4 q^{76} - 11 q^{77} - 12 q^{79} + q^{80} + 40 q^{81} - 4 q^{82} + 17 q^{83} - 4 q^{84} - 13 q^{85} - 15 q^{86} + 25 q^{87} - 2 q^{88} + 15 q^{89} + 33 q^{90} - 10 q^{92} + 3 q^{93} - 25 q^{94} + q^{95} - 2 q^{96} - 2 q^{97} - 11 q^{98} - 26 q^{99}+O(q^{100}) $ 4 * q + 4 * q^2 - 2 * q^3 + 4 * q^4 + q^5 - 2 * q^6 - q^7 + 4 * q^8 + 2 * q^9 + q^10 - 2 * q^11 - 2 * q^12 - q^14 - 11 * q^15 + 4 * q^16 + q^17 + 2 * q^18 + 4 * q^19 + q^20 - 4 * q^21 - 2 * q^22 - 10 * q^23 - 2 * q^24 + 3 * q^25 - 23 * q^27 - q^28 - q^29 - 11 * q^30 - 11 * q^31 + 4 * q^32 + 3 * q^33 + q^34 - q^35 + 2 * q^36 + 9 * q^37 + 4 * q^38 + q^40 - 4 * q^41 - 4 * q^42 - 15 * q^43 - 2 * q^44 + 33 * q^45 - 10 * q^46 - 25 * q^47 - 2 * q^48 - 11 * q^49 + 3 * q^50 - 14 * q^51 - 8 * q^53 - 23 * q^54 - 12 * q^55 - q^56 - 2 * q^57 - q^58 - 28 * q^59 - 11 * q^60 - 15 * q^61 - 11 * q^62 + 20 * q^63 + 4 * q^64 + 3 * q^66 + q^68 + 18 * q^69 - q^70 + q^71 + 2 * q^72 - 6 * q^73 + 9 * q^74 - 13 * q^75 + 4 * q^76 - 11 * q^77 - 12 * q^79 + q^80 + 40 * q^81 - 4 * q^82 + 17 * q^83 - 4 * q^84 - 13 * q^85 - 15 * q^86 + 25 * q^87 - 2 * q^88 + 15 * q^89 + 33 * q^90 - 10 * q^92 + 3 * q^93 - 25 * q^94 + q^95 - 2 * q^96 - 2 * q^97 - 11 * q^98 - 26 * 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
$3$	−3.35017	−1.93422	−0.967112	−	0.254352i	$-0.918138\pi$
$3$	−3.35017	−1.93422	−0.967112	+	0.254352i	$0.918138\pi$
$4$	1.00000	0.500000
$5$	3.12030	1.39544	0.697720	−	0.716371i	$-0.254198\pi$
$5$	3.12030	1.39544	0.697720	+	0.716371i	$0.254198\pi$
$6$	−3.35017	−1.36770
$7$	1.75319	0.662643	0.331322	−	0.943518i	$-0.392506\pi$
$7$	1.75319	0.662643	0.331322	+	0.943518i	$0.392506\pi$
$8$	1.00000	0.353553
$9$	8.22366	2.74122
$10$	3.12030	0.986725
$11$	−2.52331	−0.760807	−0.380404	−	0.924821i	$-0.624215\pi$
$11$	−2.52331	−0.760807	−0.380404	+	0.924821i	$0.624215\pi$
$12$	−3.35017	−0.967112
$13$	0	0
$13$	0	0
$14$	1.75319	0.468559
$15$	−10.4535	−2.69909
$16$	1.00000	0.250000
$17$	1.05674	0.256296	0.128148	−	0.991755i	$-0.459097\pi$
$17$	1.05674	0.256296	0.128148	+	0.991755i	$0.459097\pi$
$18$	8.22366	1.93833
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	3.12030	0.697720
$21$	−5.87349	−1.28170
$22$	−2.52331	−0.537972
$23$	−5.35017	−1.11559	−0.557794	−	0.829979i	$-0.688352\pi$
$23$	−5.35017	−1.11559	−0.557794	+	0.829979i	$0.688352\pi$
$24$	−3.35017	−0.683851
$25$	4.73625	0.947251
$26$	0	0
$27$	−17.5002	−3.36791
$28$	1.75319	0.331322
$29$	−7.08643	−1.31592	−0.657958	−	0.753054i	$-0.728580\pi$
$29$	−7.08643	−1.31592	−0.657958	+	0.753054i	$0.728580\pi$
$30$	−10.4535	−1.90855
$31$	−2.77012	−0.497529	−0.248764	−	0.968564i	$-0.580025\pi$
$31$	−2.77012	−0.497529	−0.248764	+	0.968564i	$0.580025\pi$
$32$	1.00000	0.176777
$33$	8.45353	1.47157
$34$	1.05674	0.181229
$35$	5.47047	0.924678
$36$	8.22366	1.37061
$37$	3.05674	0.502524	0.251262	−	0.967919i	$-0.419154\pi$
$37$	3.05674	0.502524	0.251262	+	0.967919i	$0.419154\pi$
$38$	1.00000	0.162221
$39$	0	0
$40$	3.12030	0.493362
$41$	−3.89042	−0.607582	−0.303791	−	0.952739i	$-0.598252\pi$
$41$	−3.89042	−0.607582	−0.303791	+	0.952739i	$0.598252\pi$
$42$	−5.87349	−0.906299
$43$	−10.2973	−1.57033	−0.785164	−	0.619288i	$-0.787421\pi$
$43$	−10.2973	−1.57033	−0.785164	+	0.619288i	$0.787421\pi$
$44$	−2.52331	−0.380404
$45$	25.6603	3.82521
$46$	−5.35017	−0.788840
$47$	−10.8545	−1.58329	−0.791647	−	0.610979i	$-0.790776\pi$
$47$	−10.8545	−1.58329	−0.791647	+	0.610979i	$0.790776\pi$
$48$	−3.35017	−0.483556
$49$	−3.92633	−0.560904
$50$	4.73625	0.669807
$51$	−3.54025	−0.495734
$52$	0	0
$53$	−6.87349	−0.944146	−0.472073	−	0.881559i	$-0.656494\pi$
$53$	−6.87349	−0.944146	−0.472073	+	0.881559i	$0.656494\pi$
$54$	−17.5002	−2.38147
$55$	−7.87349	−1.06166
$56$	1.75319	0.234280
$57$	−3.35017	−0.443741
$58$	−7.08643	−0.930493
$59$	−7.32934	−0.954199	−0.477100	−	0.878849i	$-0.658312\pi$
$59$	−7.32934	−0.954199	−0.477100	+	0.878849i	$0.658312\pi$
$60$	−10.4535	−1.34955
$61$	−7.73625	−0.990526	−0.495263	−	0.868743i	$-0.664928\pi$
$61$	−7.73625	−0.990526	−0.495263	+	0.868743i	$0.664928\pi$
$62$	−2.77012	−0.351806
$63$	14.4176	1.81645
$64$	1.00000	0.125000
$65$	0	0
$66$	8.45353	1.04056
$67$	−5.61985	−0.686574	−0.343287	−	0.939231i	$-0.611540\pi$
$67$	−5.61985	−0.686574	−0.343287	+	0.939231i	$0.611540\pi$
$68$	1.05674	0.128148
$69$	17.9240	2.15780
$70$	5.47047	0.653846
$71$	−10.2635	−1.21805	−0.609024	−	0.793152i	$-0.708439\pi$
$71$	−10.2635	−1.21805	−0.609024	+	0.793152i	$0.708439\pi$
$72$	8.22366	0.969167
$73$	−3.11347	−0.364404	−0.182202	−	0.983261i	$-0.558323\pi$
$73$	−3.11347	−0.364404	−0.182202	+	0.983261i	$0.558323\pi$
$74$	3.05674	0.355338
$75$	−15.8673	−1.83219
$76$	1.00000	0.114708
$77$	−4.42384	−0.504144
$78$	0	0
$79$	15.3377	1.72563	0.862815	−	0.505519i	$-0.168699\pi$
$79$	15.3377	1.72563	0.862815	+	0.505519i	$0.168699\pi$
$80$	3.12030	0.348860
$81$	33.9576	3.77307
$82$	−3.89042	−0.429625
$83$	8.27650	0.908464	0.454232	−	0.890883i	$-0.349914\pi$
$83$	8.27650	0.908464	0.454232	+	0.890883i	$0.349914\pi$
$84$	−5.87349	−0.640850
$85$	3.29733	0.357646
$86$	−10.2973	−1.11039
$87$	23.7408	2.54528
$88$	−2.52331	−0.268986
$89$	3.85062	0.408165	0.204082	−	0.978954i	$-0.434579\pi$
$89$	3.85062	0.408165	0.204082	+	0.978954i	$0.434579\pi$
$90$	25.6603	2.70483
$91$	0	0
$92$	−5.35017	−0.557794
$93$	9.28039	0.962332
$94$	−10.8545	−1.11956
$95$	3.12030	0.320136
$96$	−3.35017	−0.341926
$97$	−12.5191	−1.27113	−0.635563	−	0.772049i	$-0.719232\pi$
$97$	−12.5191	−1.27113	−0.635563	+	0.772049i	$0.719232\pi$
$98$	−3.92633	−0.396619
$99$	−20.7509	−2.08554
$100$	4.73625	0.473625
$101$	4.63350	0.461050	0.230525	−	0.973066i	$-0.425956\pi$
$101$	4.63350	0.461050	0.230525	+	0.973066i	$0.425956\pi$
$102$	−3.54025	−0.350537
$103$	−12.1292	−1.19512	−0.597561	−	0.801824i	$-0.703863\pi$
$103$	−12.1292	−1.19512	−0.597561	+	0.801824i	$0.703863\pi$
$104$	0	0
$105$	−18.3270	−1.78853
$106$	−6.87349	−0.667612
$107$	20.2005	1.95286	0.976428	−	0.215842i	$-0.0692496\pi$
$107$	20.2005	1.95286	0.976428	+	0.215842i	$0.0692496\pi$
$108$	−17.5002	−1.68395
$109$	−0.677480	−0.0648909	−0.0324454	−	0.999474i	$-0.510330\pi$
$109$	−0.677480	−0.0648909	−0.0324454	+	0.999474i	$0.510330\pi$
$110$	−7.87349	−0.750707
$111$	−10.2406	−0.971994
$112$	1.75319	0.165661
$113$	4.86044	0.457232	0.228616	−	0.973517i	$-0.426580\pi$
$113$	4.86044	0.457232	0.228616	+	0.973517i	$0.426580\pi$
$114$	−3.35017	−0.313772
$115$	−16.6941	−1.55674
$116$	−7.08643	−0.657958
$117$	0	0
$118$	−7.32934	−0.674721
$119$	1.85266	0.169833
$120$	−10.4535	−0.954273
$121$	−4.63289	−0.421172
$122$	−7.73625	−0.700407
$123$	13.0336	1.17520
$124$	−2.77012	−0.248764
$125$	−0.822967	−0.0736084
$126$	14.4176	1.28442
$127$	−16.5171	−1.46566	−0.732828	−	0.680414i	$-0.761800\pi$
$127$	−16.5171	−1.46566	−0.732828	+	0.680414i	$0.761800\pi$
$128$	1.00000	0.0883883
$129$	34.4978	3.03737
$130$	0	0
$131$	18.6394	1.62853	0.814267	−	0.580490i	$-0.197139\pi$
$131$	18.6394	1.62853	0.814267	+	0.580490i	$0.197139\pi$
$132$	8.45353	0.735786
$133$	1.75319	0.152021
$134$	−5.61985	−0.485481
$135$	−54.6057	−4.69971
$136$	1.05674	0.0906144
$137$	8.22570	0.702769	0.351384	−	0.936231i	$-0.385711\pi$
$137$	8.22570	0.702769	0.351384	+	0.936231i	$0.385711\pi$
$138$	17.9240	1.52579
$139$	1.32313	0.112226	0.0561131	−	0.998424i	$-0.482129\pi$
$139$	1.32313	0.112226	0.0561131	+	0.998424i	$0.482129\pi$
$140$	5.47047	0.462339
$141$	36.3645	3.06244
$142$	−10.2635	−0.861290
$143$	0	0
$144$	8.22366	0.685305
$145$	−22.1118	−1.83628
$146$	−3.11347	−0.257673
$147$	13.1539	1.08491
$148$	3.05674	0.251262
$149$	21.3062	1.74547	0.872736	−	0.488193i	$-0.162344\pi$
$149$	21.3062	1.74547	0.872736	+	0.488193i	$0.162344\pi$
$150$	−15.8673	−1.29556
$151$	−15.7761	−1.28384	−0.641919	−	0.766773i	$-0.721861\pi$
$151$	−15.7761	−1.28384	−0.641919	+	0.766773i	$0.721861\pi$
$152$	1.00000	0.0811107
$153$	8.69024	0.702564
$154$	−4.42384	−0.356484
$155$	−8.64361	−0.694271
$156$	0	0
$157$	17.5337	1.39935	0.699673	−	0.714464i	$-0.253329\pi$
$157$	17.5337	1.39935	0.699673	+	0.714464i	$0.253329\pi$
$158$	15.3377	1.22020
$159$	23.0274	1.82619
$160$	3.12030	0.246681
$161$	−9.37986	−0.739237
$162$	33.9576	2.66796
$163$	16.7150	1.30922	0.654608	−	0.755968i	$-0.272834\pi$
$163$	16.7150	1.30922	0.654608	+	0.755968i	$0.272834\pi$
$164$	−3.89042	−0.303791
$165$	26.3775	2.05349
$166$	8.27650	0.642381
$167$	−16.7150	−1.29344	−0.646721	−	0.762727i	$-0.723860\pi$
$167$	−16.7150	−1.29344	−0.646721	+	0.762727i	$0.723860\pi$
$168$	−5.87349	−0.453149
$169$	0	0
$170$	3.29733	0.252894
$171$	8.22366	0.628879
$172$	−10.2973	−0.785164
$173$	12.3968	0.942511	0.471256	−	0.881997i	$-0.343801\pi$
$173$	12.3968	0.942511	0.471256	+	0.881997i	$0.343801\pi$
$174$	23.7408	1.79978
$175$	8.30355	0.627689
$176$	−2.52331	−0.190202
$177$	24.5546	1.84563
$178$	3.85062	0.288616
$179$	4.96906	0.371405	0.185703	−	0.982606i	$-0.440544\pi$
$179$	4.96906	0.371405	0.185703	+	0.982606i	$0.440544\pi$
$180$	25.6603	1.91260
$181$	−4.90257	−0.364405	−0.182203	−	0.983261i	$-0.558323\pi$
$181$	−4.90257	−0.364405	−0.182203	+	0.983261i	$0.558323\pi$
$182$	0	0
$183$	25.9178	1.91590
$184$	−5.35017	−0.394420
$185$	9.53792	0.701242
$186$	9.28039	0.680472
$187$	−2.66648	−0.194992
$188$	−10.8545	−0.791647
$189$	−30.6811	−2.23172
$190$	3.12030	0.226370
$191$	−16.1589	−1.16921	−0.584607	−	0.811317i	$-0.698751\pi$
$191$	−16.1589	−1.16921	−0.584607	+	0.811317i	$0.698751\pi$
$192$	−3.35017	−0.241778
$193$	−4.80310	−0.345735	−0.172867	−	0.984945i	$-0.555303\pi$
$193$	−4.80310	−0.345735	−0.172867	+	0.984945i	$0.555303\pi$
$194$	−12.5191	−0.898821
$195$	0	0
$196$	−3.92633	−0.280452
$197$	−9.22162	−0.657013	−0.328507	−	0.944502i	$-0.606545\pi$
$197$	−9.22162	−0.657013	−0.328507	+	0.944502i	$0.606545\pi$
$198$	−20.7509	−1.47470
$199$	−5.00186	−0.354572	−0.177286	−	0.984159i	$-0.556732\pi$
$199$	−5.00186	−0.354572	−0.177286	+	0.984159i	$0.556732\pi$
$200$	4.73625	0.334904
$201$	18.8275	1.32799
$202$	4.63350	0.326012
$203$	−12.4238	−0.871983
$204$	−3.54025	−0.247867
$205$	−12.1393	−0.847843
$206$	−12.1292	−0.845079
$207$	−43.9980	−3.05807
$208$	0	0
$209$	−2.52331	−0.174541
$210$	−18.3270	−1.26468
$211$	6.12808	0.421875	0.210937	−	0.977500i	$-0.432348\pi$
$211$	6.12808	0.421875	0.210937	+	0.977500i	$0.432348\pi$
$212$	−6.87349	−0.472073
$213$	34.3844	2.35598
$214$	20.2005	1.38088
$215$	−32.1307	−2.19130
$216$	−17.5002	−1.19074
$217$	−4.85655	−0.329684
$218$	−0.677480	−0.0458848
$219$	10.4307	0.704839
$220$	−7.87349	−0.530830
$221$	0	0
$222$	−10.2406	−0.687304
$223$	9.17110	0.614142	0.307071	−	0.951687i	$-0.400651\pi$
$223$	9.17110	0.614142	0.307071	+	0.951687i	$0.400651\pi$
$224$	1.75319	0.117140
$225$	38.9493	2.59662
$226$	4.86044	0.323312
$227$	−5.70220	−0.378468	−0.189234	−	0.981932i	$-0.560601\pi$
$227$	−5.70220	−0.378468	−0.189234	+	0.981932i	$0.560601\pi$
$228$	−3.35017	−0.221871
$229$	14.0016	0.925250	0.462625	−	0.886554i	$-0.346908\pi$
$229$	14.0016	0.925250	0.462625	+	0.886554i	$0.346908\pi$
$230$	−16.6941	−1.10078
$231$	14.8206	0.975127
$232$	−7.08643	−0.465247
$233$	−1.49069	−0.0976584	−0.0488292	−	0.998807i	$-0.515549\pi$
$233$	−1.49069	−0.0976584	−0.0488292	+	0.998807i	$0.515549\pi$
$234$	0	0
$235$	−33.8693	−2.20939
$236$	−7.32934	−0.477100
$237$	−51.3841	−3.33775
$238$	1.85266	0.120090
$239$	4.60088	0.297606	0.148803	−	0.988867i	$-0.452458\pi$
$239$	4.60088	0.297606	0.148803	+	0.988867i	$0.452458\pi$
$240$	−10.4535	−0.674773
$241$	6.03837	0.388966	0.194483	−	0.980906i	$-0.437697\pi$
$241$	6.03837	0.388966	0.194483	+	0.980906i	$0.437697\pi$
$242$	−4.63289	−0.297814
$243$	−61.2633	−3.93004
$244$	−7.73625	−0.495263
$245$	−12.2513	−0.782708
$246$	13.0336	0.830991
$247$	0	0
$248$	−2.77012	−0.175903
$249$	−27.7277	−1.75717
$250$	−0.822967	−0.0520490
$251$	30.5546	1.92859	0.964294	−	0.264835i	$-0.0853175\pi$
$251$	30.5546	1.92859	0.964294	+	0.264835i	$0.0853175\pi$
$252$	14.4176	0.908225
$253$	13.5002	0.848748
$254$	−16.5171	−1.03637
$255$	−11.0466	−0.691767
$256$	1.00000	0.0625000
$257$	−0.315342	−0.0196705	−0.00983524	−	0.999952i	$-0.503131\pi$
$257$	−0.315342	−0.0196705	−0.00983524	+	0.999952i	$0.503131\pi$
$258$	34.4978	2.14774
$259$	5.35904	0.332994
$260$	0	0
$261$	−58.2764	−3.60722
$262$	18.6394	1.15155
$263$	2.56605	0.158229	0.0791146	−	0.996866i	$-0.474791\pi$
$263$	2.56605	0.158229	0.0791146	+	0.996866i	$0.474791\pi$
$264$	8.45353	0.520279
$265$	−21.4473	−1.31750
$266$	1.75319	0.107495
$267$	−12.9002	−0.789482
$268$	−5.61985	−0.343287
$269$	−9.40691	−0.573549	−0.286775	−	0.957998i	$-0.592583\pi$
$269$	−9.40691	−0.573549	−0.286775	+	0.957998i	$0.592583\pi$
$270$	−54.6057	−3.32320
$271$	8.10151	0.492131	0.246066	−	0.969253i	$-0.420862\pi$
$271$	8.10151	0.492131	0.246066	+	0.969253i	$0.420862\pi$
$272$	1.05674	0.0640740
$273$	0	0
$274$	8.22570	0.496932
$275$	−11.9511	−0.720675
$276$	17.9240	1.07890
$277$	0.370393	0.0222548	0.0111274	−	0.999938i	$-0.496458\pi$
$277$	0.370393	0.0222548	0.0111274	+	0.999938i	$0.496458\pi$
$278$	1.32313	0.0793559
$279$	−22.7806	−1.36384
$280$	5.47047	0.326923
$281$	−7.77899	−0.464055	−0.232028	−	0.972709i	$-0.574536\pi$
$281$	−7.77899	−0.464055	−0.232028	+	0.972709i	$0.574536\pi$
$282$	36.3645	2.16547
$283$	5.88174	0.349633	0.174817	−	0.984601i	$-0.444067\pi$
$283$	5.88174	0.349633	0.174817	+	0.984601i	$0.444067\pi$
$284$	−10.2635	−0.609024
$285$	−10.4535	−0.619214
$286$	0	0
$287$	−6.82064	−0.402610
$288$	8.22366	0.484584
$289$	−15.8833	−0.934312
$290$	−22.1118	−1.29845
$291$	41.9413	2.45864
$292$	−3.11347	−0.182202
$293$	−1.57597	−0.0920694	−0.0460347	−	0.998940i	$-0.514658\pi$
$293$	−1.57597	−0.0920694	−0.0460347	+	0.998940i	$0.514658\pi$
$294$	13.1539	0.767150
$295$	−22.8697	−1.33153
$296$	3.05674	0.177669
$297$	44.1584	2.56233
$298$	21.3062	1.23423
$299$	0	0
$300$	−15.8673	−0.916097
$301$	−18.0532	−1.04057
$302$	−15.7761	−0.907810
$303$	−15.5230	−0.891775
$304$	1.00000	0.0573539
$305$	−24.1394	−1.38222
$306$	8.69024	0.496788
$307$	−10.9762	−0.626447	−0.313224	−	0.949679i	$-0.601409\pi$
$307$	−10.9762	−0.626447	−0.313224	+	0.949679i	$0.601409\pi$
$308$	−4.42384	−0.252072
$309$	40.6348	2.31163
$310$	−8.64361	−0.490924
$311$	17.4512	0.989567	0.494784	−	0.869016i	$-0.335247\pi$
$311$	17.4512	0.989567	0.494784	+	0.869016i	$0.335247\pi$
$312$	0	0
$313$	10.3048	0.582460	0.291230	−	0.956653i	$-0.405936\pi$
$313$	10.3048	0.582460	0.291230	+	0.956653i	$0.405936\pi$
$314$	17.5337	0.989486
$315$	44.9873	2.53475
$316$	15.3377	0.862815
$317$	20.5934	1.15664	0.578321	−	0.815810i	$-0.303708\pi$
$317$	20.5934	1.15664	0.578321	+	0.815810i	$0.303708\pi$
$318$	23.0274	1.29131
$319$	17.8813	1.00116
$320$	3.12030	0.174430
$321$	−67.6752	−3.77726
$322$	−9.37986	−0.522719
$323$	1.05674	0.0587984
$324$	33.9576	1.88653
$325$	0	0
$326$	16.7150	0.925756
$327$	2.26968	0.125513
$328$	−3.89042	−0.214813
$329$	−19.0300	−1.04916
$330$	26.3775	1.45204
$331$	−23.6585	−1.30039	−0.650196	−	0.759767i	$-0.725313\pi$
$331$	−23.6585	−1.30039	−0.650196	+	0.759767i	$0.725313\pi$
$332$	8.27650	0.454232
$333$	25.1376	1.37753
$334$	−16.7150	−0.914602
$335$	−17.5356	−0.958072
$336$	−5.87349	−0.320425
$337$	−23.9472	−1.30448	−0.652242	−	0.758010i	$-0.726172\pi$
$337$	−23.9472	−1.30448	−0.652242	+	0.758010i	$0.726172\pi$
$338$	0	0
$339$	−16.2833	−0.884389
$340$	3.29733	0.178823
$341$	6.98989	0.378524
$342$	8.22366	0.444685
$343$	−19.1559	−1.03432
$344$	−10.2973	−0.555195
$345$	55.9282	3.01107
$346$	12.3968	0.666456
$347$	−10.6442	−0.571412	−0.285706	−	0.958317i	$-0.592228\pi$
$347$	−10.6442	−0.571412	−0.285706	+	0.958317i	$0.592228\pi$
$348$	23.7408	1.27264
$349$	−23.2492	−1.24450	−0.622250	−	0.782819i	$-0.713781\pi$
$349$	−23.2492	−1.24450	−0.622250	+	0.782819i	$0.713781\pi$
$350$	8.30355	0.443843
$351$	0	0
$352$	−2.52331	−0.134493
$353$	12.2445	0.651708	0.325854	−	0.945420i	$-0.394348\pi$
$353$	12.2445	0.651708	0.325854	+	0.945420i	$0.394348\pi$
$354$	24.5546	1.30506
$355$	−32.0250	−1.69971
$356$	3.85062	0.204082
$357$	−6.20672	−0.328495
$358$	4.96906	0.262623
$359$	−14.1066	−0.744520	−0.372260	−	0.928128i	$-0.621417\pi$
$359$	−14.1066	−0.744520	−0.372260	+	0.928128i	$0.621417\pi$
$360$	25.6603	1.35241
$361$	1.00000	0.0526316
$362$	−4.90257	−0.257673
$363$	15.5210	0.814641
$364$	0	0
$365$	−9.71496	−0.508504
$366$	25.9178	1.35474
$367$	13.6995	0.715106	0.357553	−	0.933893i	$-0.383611\pi$
$367$	13.6995	0.715106	0.357553	+	0.933893i	$0.383611\pi$
$368$	−5.35017	−0.278897
$369$	−31.9935	−1.66551
$370$	9.53792	0.495853
$371$	−12.0505	−0.625632
$372$	9.28039	0.481166
$373$	−10.1304	−0.524533	−0.262266	−	0.964996i	$-0.584470\pi$
$373$	−10.1304	−0.524533	−0.262266	+	0.964996i	$0.584470\pi$
$374$	−2.66648	−0.137880
$375$	2.75708	0.142375
$376$	−10.8545	−0.559779
$377$	0	0
$378$	−30.6811	−1.57806
$379$	−5.51320	−0.283194	−0.141597	−	0.989924i	$-0.545224\pi$
$379$	−5.51320	−0.283194	−0.141597	+	0.989924i	$0.545224\pi$
$380$	3.12030	0.160068
$381$	55.3351	2.83490
$382$	−16.1589	−0.826759
$383$	−27.7594	−1.41844	−0.709221	−	0.704987i	$-0.750953\pi$
$383$	−27.7594	−1.41844	−0.709221	+	0.704987i	$0.750953\pi$
$384$	−3.35017	−0.170963
$385$	−13.8037	−0.703502
$386$	−4.80310	−0.244471
$387$	−84.6817	−4.30461
$388$	−12.5191	−0.635563
$389$	8.45539	0.428705	0.214353	−	0.976756i	$-0.431236\pi$
$389$	8.45539	0.428705	0.214353	+	0.976756i	$0.431236\pi$
$390$	0	0
$391$	−5.65372	−0.285921
$392$	−3.92633	−0.198310
$393$	−62.4453	−3.14995
$394$	−9.22162	−0.464579
$395$	47.8583	2.40801
$396$	−20.7509	−1.04277
$397$	−23.5763	−1.18326	−0.591630	−	0.806209i	$-0.701515\pi$
$397$	−23.5763	−1.18326	−0.591630	+	0.806209i	$0.701515\pi$
$398$	−5.00186	−0.250720
$399$	−5.87349	−0.294042
$400$	4.73625	0.236813
$401$	−32.4679	−1.62137	−0.810684	−	0.585484i	$-0.800904\pi$
$401$	−32.4679	−1.62137	−0.810684	+	0.585484i	$0.800904\pi$
$402$	18.8275	0.939029
$403$	0	0
$404$	4.63350	0.230525
$405$	105.958	5.26508
$406$	−12.4238	−0.616585
$407$	−7.71310	−0.382324
$408$	−3.54025	−0.175268
$409$	9.61967	0.475662	0.237831	−	0.971307i	$-0.423564\pi$
$409$	9.61967	0.475662	0.237831	+	0.971307i	$0.423564\pi$
$410$	−12.1393	−0.599516
$411$	−27.5575	−1.35931
$412$	−12.1292	−0.597561
$413$	−12.8497	−0.632294
$414$	−43.9980	−2.16238
$415$	25.8251	1.26771
$416$	0	0
$417$	−4.43271	−0.217071
$418$	−2.52331	−0.123419
$419$	−6.62996	−0.323895	−0.161947	−	0.986799i	$-0.551777\pi$
$419$	−6.62996	−0.323895	−0.161947	+	0.986799i	$0.551777\pi$
$420$	−18.3270	−0.894267
$421$	36.5861	1.78310	0.891549	−	0.452924i	$-0.149619\pi$
$421$	36.5861	1.78310	0.891549	+	0.452924i	$0.149619\pi$
$422$	6.12808	0.298310
$423$	−89.2638	−4.34016
$424$	−6.87349	−0.333806
$425$	5.00497	0.242777
$426$	34.3844	1.66593
$427$	−13.5631	−0.656365
$428$	20.2005	0.976428
$429$	0	0
$430$	−32.1307	−1.54948
$431$	25.4091	1.22391	0.611956	−	0.790892i	$-0.290383\pi$
$431$	25.4091	1.22391	0.611956	+	0.790892i	$0.290383\pi$
$432$	−17.5002	−0.841977
$433$	7.73733	0.371832	0.185916	−	0.982566i	$-0.440475\pi$
$433$	7.73733	0.371832	0.185916	+	0.982566i	$0.440475\pi$
$434$	−4.85655	−0.233122
$435$	74.0782	3.55178
$436$	−0.677480	−0.0324454
$437$	−5.35017	−0.255933
$438$	10.4307	0.498397
$439$	13.3006	0.634804	0.317402	−	0.948291i	$-0.397190\pi$
$439$	13.3006	0.634804	0.317402	+	0.948291i	$0.397190\pi$
$440$	−7.87349	−0.375354
$441$	−32.2888	−1.53756
$442$	0	0
$443$	36.3098	1.72513	0.862565	−	0.505946i	$-0.168856\pi$
$443$	36.3098	1.72513	0.862565	+	0.505946i	$0.168856\pi$
$444$	−10.2406	−0.485997
$445$	12.0151	0.569569
$446$	9.17110	0.434264
$447$	−71.3794	−3.37613
$448$	1.75319	0.0828304
$449$	−3.47094	−0.163804	−0.0819019	−	0.996640i	$-0.526099\pi$
$449$	−3.47094	−0.163804	−0.0819019	+	0.996640i	$0.526099\pi$
$450$	38.9493	1.83609
$451$	9.81675	0.462253
$452$	4.86044	0.228616
$453$	52.8525	2.48323
$454$	−5.70220	−0.267618
$455$	0	0
$456$	−3.35017	−0.156886
$457$	−33.0892	−1.54785	−0.773924	−	0.633278i	$-0.781709\pi$
$457$	−33.0892	−1.54785	−0.773924	+	0.633278i	$0.781709\pi$
$458$	14.0016	0.654250
$459$	−18.4930	−0.863182
$460$	−16.6941	−0.778368
$461$	24.5611	1.14392	0.571962	−	0.820280i	$-0.306183\pi$
$461$	24.5611	1.14392	0.571962	+	0.820280i	$0.306183\pi$
$462$	14.8206	0.689519
$463$	10.1113	0.469913	0.234957	−	0.972006i	$-0.424505\pi$
$463$	10.1113	0.469913	0.234957	+	0.972006i	$0.424505\pi$
$464$	−7.08643	−0.328979
$465$	28.9576	1.34288
$466$	−1.49069	−0.0690549
$467$	−27.4414	−1.26984	−0.634918	−	0.772580i	$-0.718966\pi$
$467$	−27.4414	−1.26984	−0.634918	+	0.772580i	$0.718966\pi$
$468$	0	0
$469$	−9.85266	−0.454953
$470$	−33.8693	−1.56227
$471$	−58.7411	−2.70665
$472$	−7.32934	−0.337360
$473$	25.9834	1.19472
$474$	−51.3841	−2.36015
$475$	4.73625	0.217314
$476$	1.85266	0.0849164
$477$	−56.5252	−2.58811
$478$	4.60088	0.210439
$479$	32.6665	1.49257	0.746285	−	0.665627i	$-0.231836\pi$
$479$	32.6665	1.49257	0.746285	+	0.665627i	$0.231836\pi$
$480$	−10.4535	−0.477136
$481$	0	0
$482$	6.03837	0.275040
$483$	31.4242	1.42985
$484$	−4.63289	−0.210586
$485$	−39.0634	−1.77378
$486$	−61.2633	−2.77896
$487$	24.6555	1.11725	0.558623	−	0.829422i	$-0.311330\pi$
$487$	24.6555	1.11725	0.558623	+	0.829422i	$0.311330\pi$
$488$	−7.73625	−0.350204
$489$	−55.9980	−2.53232
$490$	−12.2513	−0.553458
$491$	−22.5780	−1.01893	−0.509466	−	0.860491i	$-0.670157\pi$
$491$	−22.5780	−1.01893	−0.509466	+	0.860491i	$0.670157\pi$
$492$	13.0336	0.587599
$493$	−7.48848	−0.337264
$494$	0	0
$495$	−64.7489	−2.91024
$496$	−2.77012	−0.124382
$497$	−17.9938	−0.807131
$498$	−27.7277	−1.24251
$499$	−8.47854	−0.379552	−0.189776	−	0.981827i	$-0.560776\pi$
$499$	−8.47854	−0.379552	−0.189776	+	0.981827i	$0.560776\pi$
$500$	−0.822967	−0.0368042
$501$	55.9980	2.50181
$502$	30.5546	1.36372
$503$	29.4952	1.31512	0.657562	−	0.753400i	$-0.271588\pi$
$503$	29.4952	1.31512	0.657562	+	0.753400i	$0.271588\pi$
$504$	14.4176	0.642212
$505$	14.4579	0.643368
$506$	13.5002	0.600155
$507$	0	0
$508$	−16.5171	−0.732828
$509$	−21.0327	−0.932258	−0.466129	−	0.884717i	$-0.654352\pi$
$509$	−21.0327	−0.932258	−0.466129	+	0.884717i	$0.654352\pi$
$510$	−11.0466	−0.489153
$511$	−5.45850	−0.241470
$512$	1.00000	0.0441942
$513$	−17.5002	−0.772651
$514$	−0.315342	−0.0139091
$515$	−37.8466	−1.66772
$516$	34.4978	1.51868
$517$	27.3893	1.20458
$518$	5.35904	0.235462
$519$	−41.5314	−1.82303
$520$	0	0
$521$	−20.3903	−0.893315	−0.446658	−	0.894705i	$-0.647386\pi$
$521$	−20.3903	−0.893315	−0.446658	+	0.894705i	$0.647386\pi$
$522$	−58.2764	−2.55069
$523$	−18.1548	−0.793853	−0.396926	−	0.917850i	$-0.629923\pi$
$523$	−18.1548	−0.793853	−0.396926	+	0.917850i	$0.629923\pi$
$524$	18.6394	0.814267
$525$	−27.8183	−1.21409
$526$	2.56605	0.111885
$527$	−2.92729	−0.127515
$528$	8.45353	0.367893
$529$	5.62435	0.244537
$530$	−21.4473	−0.931612
$531$	−60.2740	−2.61567
$532$	1.75319	0.0760104
$533$	0	0
$534$	−12.9002	−0.558248
$535$	63.0316	2.72509
$536$	−5.61985	−0.242741
$537$	−16.6472	−0.718380
$538$	−9.40691	−0.405561
$539$	9.90736	0.426740
$540$	−54.6057	−2.34986
$541$	8.55425	0.367776	0.183888	−	0.982947i	$-0.441132\pi$
$541$	8.55425	0.367776	0.183888	+	0.982947i	$0.441132\pi$
$542$	8.10151	0.347989
$543$	16.4245	0.704841
$544$	1.05674	0.0453072
$545$	−2.11394	−0.0905512
$546$	0	0
$547$	−40.5774	−1.73497	−0.867483	−	0.497467i	$-0.834264\pi$
$547$	−40.5774	−1.73497	−0.867483	+	0.497467i	$0.834264\pi$
$548$	8.22570	0.351384
$549$	−63.6203	−2.71525
$550$	−11.9511	−0.509594
$551$	−7.08643	−0.301892
$552$	17.9240	0.762896
$553$	26.8900	1.14348
$554$	0.370393	0.0157365
$555$	−31.9537	−1.35636
$556$	1.32313	0.0561131
$557$	16.8406	0.713558	0.356779	−	0.934189i	$-0.383875\pi$
$557$	16.8406	0.713558	0.356779	+	0.934189i	$0.383875\pi$
$558$	−22.7806	−0.964378
$559$	0	0
$560$	5.47047	0.231170
$561$	8.93315	0.377158
$562$	−7.77899	−0.328137
$563$	9.52828	0.401569	0.200785	−	0.979635i	$-0.435651\pi$
$563$	9.52828	0.401569	0.200785	+	0.979635i	$0.435651\pi$
$564$	36.3645	1.53122
$565$	15.1660	0.638040
$566$	5.88174	0.247228
$567$	59.5341	2.50020
$568$	−10.2635	−0.430645
$569$	37.8584	1.58711	0.793553	−	0.608501i	$-0.208229\pi$
$569$	37.8584	1.58711	0.793553	+	0.608501i	$0.208229\pi$
$570$	−10.4535	−0.437850
$571$	0.364966	0.0152733	0.00763667	−	0.999971i	$-0.497569\pi$
$571$	0.364966	0.0152733	0.00763667	+	0.999971i	$0.497569\pi$
$572$	0	0
$573$	54.1349	2.26152
$574$	−6.82064	−0.284688
$575$	−25.3398	−1.05674
$576$	8.22366	0.342652
$577$	−14.8500	−0.618216	−0.309108	−	0.951027i	$-0.600030\pi$
$577$	−14.8500	−0.618216	−0.309108	+	0.951027i	$0.600030\pi$
$578$	−15.8833	−0.660659
$579$	16.0912	0.668728
$580$	−22.1118	−0.918141
$581$	14.5103	0.601987
$582$	41.9413	1.73852
$583$	17.3440	0.718313
$584$	−3.11347	−0.128836
$585$	0	0
$586$	−1.57597	−0.0651029
$587$	10.4005	0.429275	0.214638	−	0.976694i	$-0.431143\pi$
$587$	10.4005	0.429275	0.214638	+	0.976694i	$0.431143\pi$
$588$	13.1539	0.542457
$589$	−2.77012	−0.114141
$590$	−22.8697	−0.941532
$591$	30.8940	1.27081
$592$	3.05674	0.125631
$593$	−39.4792	−1.62122	−0.810608	−	0.585589i	$-0.800863\pi$
$593$	−39.4792	−1.62122	−0.810608	+	0.585589i	$0.800863\pi$
$594$	44.1584	1.81184
$595$	5.78084	0.236991
$596$	21.3062	0.872736
$597$	16.7571	0.685822
$598$	0	0
$599$	−20.9134	−0.854499	−0.427250	−	0.904134i	$-0.640517\pi$
$599$	−20.9134	−0.854499	−0.427250	+	0.904134i	$0.640517\pi$
$600$	−15.8673	−0.647779
$601$	20.8533	0.850622	0.425311	−	0.905047i	$-0.360165\pi$
$601$	20.8533	0.850622	0.425311	+	0.905047i	$0.360165\pi$
$602$	−18.0532	−0.735792
$603$	−46.2157	−1.88205
$604$	−15.7761	−0.641919
$605$	−14.4560	−0.587720
$606$	−15.5230	−0.630580
$607$	14.6749	0.595635	0.297817	−	0.954623i	$-0.403741\pi$
$607$	14.6749	0.595635	0.297817	+	0.954623i	$0.403741\pi$
$608$	1.00000	0.0405554
$609$	41.6220	1.68661
$610$	−24.1394	−0.977376
$611$	0	0
$612$	8.69024	0.351282
$613$	16.8502	0.680571	0.340286	−	0.940322i	$-0.389476\pi$
$613$	16.8502	0.680571	0.340286	+	0.940322i	$0.389476\pi$
$614$	−10.9762	−0.442965
$615$	40.6687	1.63992
$616$	−4.42384	−0.178242
$617$	30.3076	1.22014	0.610069	−	0.792349i	$-0.291142\pi$
$617$	30.3076	1.22014	0.610069	+	0.792349i	$0.291142\pi$
$618$	40.6348	1.63457
$619$	39.7126	1.59619	0.798093	−	0.602535i	$-0.205842\pi$
$619$	39.7126	1.59619	0.798093	+	0.602535i	$0.205842\pi$
$620$	−8.64361	−0.347136
$621$	93.6289	3.75720
$622$	17.4512	0.699730
$623$	6.75086	0.270468
$624$	0	0
$625$	−26.2492	−1.04997
$626$	10.3048	0.411861
$627$	8.45353	0.337602
$628$	17.5337	0.699673
$629$	3.23016	0.128795
$630$	44.9873	1.79234
$631$	−0.536708	−0.0213660	−0.0106830	−	0.999943i	$-0.503401\pi$
$631$	−0.536708	−0.0213660	−0.0106830	+	0.999943i	$0.503401\pi$
$632$	15.3377	0.610102
$633$	−20.5301	−0.816000
$634$	20.5934	0.817869
$635$	−51.5382	−2.04523
$636$	23.0274	0.913094
$637$	0	0
$638$	17.8813	0.707926
$639$	−84.4032	−3.33894
$640$	3.12030	0.123341
$641$	20.0998	0.793894	0.396947	−	0.917842i	$-0.370070\pi$
$641$	20.0998	0.793894	0.396947	+	0.917842i	$0.370070\pi$
$642$	−67.6752	−2.67093
$643$	−19.9896	−0.788313	−0.394156	−	0.919043i	$-0.628963\pi$
$643$	−19.9896	−0.788313	−0.394156	+	0.919043i	$0.628963\pi$
$644$	−9.37986	−0.369618
$645$	107.643	4.23846
$646$	1.05674	0.0415767
$647$	−2.30630	−0.0906699	−0.0453350	−	0.998972i	$-0.514436\pi$
$647$	−2.30630	−0.0906699	−0.0453350	+	0.998972i	$0.514436\pi$
$648$	33.9576	1.33398
$649$	18.4942	0.725962
$650$	0	0
$651$	16.2703	0.637683
$652$	16.7150	0.654608
$653$	7.59591	0.297251	0.148625	−	0.988894i	$-0.452515\pi$
$653$	7.59591	0.297251	0.148625	+	0.988894i	$0.452515\pi$
$654$	2.26968	0.0887514
$655$	58.1606	2.27252
$656$	−3.89042	−0.151895
$657$	−25.6041	−0.998912
$658$	−19.0300	−0.741867
$659$	−1.10683	−0.0431159	−0.0215580	−	0.999768i	$-0.506863\pi$
$659$	−1.10683	−0.0431159	−0.0215580	+	0.999768i	$0.506863\pi$
$660$	26.3775	1.02674
$661$	−29.8587	−1.16137	−0.580685	−	0.814128i	$-0.697215\pi$
$661$	−29.8587	−1.16137	−0.580685	+	0.814128i	$0.697215\pi$
$662$	−23.6585	−0.919515
$663$	0	0
$664$	8.27650	0.321191
$665$	5.47047	0.212136
$666$	25.1376	0.974060
$667$	37.9136	1.46802
$668$	−16.7150	−0.646721
$669$	−30.7248	−1.18789
$670$	−17.5356	−0.677459
$671$	19.5210	0.753599
$672$	−5.87349	−0.226575
$673$	−42.1315	−1.62405	−0.812025	−	0.583622i	$-0.801635\pi$
$673$	−42.1315	−1.62405	−0.812025	+	0.583622i	$0.801635\pi$
$674$	−23.9472	−0.922410
$675$	−82.8852	−3.19025
$676$	0	0
$677$	12.4785	0.479589	0.239795	−	0.970824i	$-0.422920\pi$
$677$	12.4785	0.479589	0.239795	+	0.970824i	$0.422920\pi$
$678$	−16.2833	−0.625357
$679$	−21.9484	−0.842302
$680$	3.29733	0.126447
$681$	19.1034	0.732042
$682$	6.98989	0.267657
$683$	22.4576	0.859317	0.429658	−	0.902992i	$-0.358634\pi$
$683$	22.4576	0.859317	0.429658	+	0.902992i	$0.358634\pi$
$684$	8.22366	0.314439
$685$	25.6666	0.980671
$686$	−19.1559	−0.731376
$687$	−46.9077	−1.78964
$688$	−10.2973	−0.392582
$689$	0	0
$690$	55.9282	2.12915
$691$	39.1064	1.48768	0.743838	−	0.668360i	$-0.233003\pi$
$691$	39.1064	1.48768	0.743838	+	0.668360i	$0.233003\pi$
$692$	12.3968	0.471256
$693$	−36.3802	−1.38197
$694$	−10.6442	−0.404049
$695$	4.12855	0.156605
$696$	23.7408	0.899891
$697$	−4.11115	−0.155721
$698$	−23.2492	−0.879994
$699$	4.99407	0.188893
$700$	8.30355	0.313845
$701$	33.7406	1.27437	0.637183	−	0.770712i	$-0.280099\pi$
$701$	33.7406	1.27437	0.637183	+	0.770712i	$0.280099\pi$
$702$	0	0
$703$	3.05674	0.115287
$704$	−2.52331	−0.0951009
$705$	113.468	4.27345
$706$	12.2445	0.460827
$707$	8.12340	0.305512
$708$	24.5546	0.922817
$709$	5.81429	0.218360	0.109180	−	0.994022i	$-0.465177\pi$
$709$	5.81429	0.218360	0.109180	+	0.994022i	$0.465177\pi$
$710$	−32.0250	−1.20188
$711$	126.132	4.73033
$712$	3.85062	0.144308
$713$	14.8206	0.555037
$714$	−6.20672	−0.232281
$715$	0	0
$716$	4.96906	0.185703
$717$	−15.4137	−0.575636
$718$	−14.1066	−0.526455
$719$	−34.8766	−1.30068	−0.650338	−	0.759645i	$-0.725373\pi$
$719$	−34.8766	−1.30068	−0.650338	+	0.759645i	$0.725373\pi$
$720$	25.6603	0.956301
$721$	−21.2647	−0.791939
$722$	1.00000	0.0372161
$723$	−20.2296	−0.752347
$724$	−4.90257	−0.182203
$725$	−33.5631	−1.24650
$726$	15.5210	0.576038
$727$	45.9146	1.70288	0.851439	−	0.524454i	$-0.175731\pi$
$727$	45.9146	1.70288	0.851439	+	0.524454i	$0.175731\pi$
$728$	0	0
$729$	103.370	3.82852
$730$	−9.71496	−0.359567
$731$	−10.8816	−0.402469
$732$	25.9178	0.957949
$733$	18.1554	0.670587	0.335293	−	0.942114i	$-0.391165\pi$
$733$	18.1554	0.670587	0.335293	+	0.942114i	$0.391165\pi$
$734$	13.6995	0.505656
$735$	41.0440	1.51393
$736$	−5.35017	−0.197210
$737$	14.1806	0.522351
$738$	−31.9935	−1.17770
$739$	−8.21569	−0.302219	−0.151110	−	0.988517i	$-0.548285\pi$
$739$	−8.21569	−0.302219	−0.151110	+	0.988517i	$0.548285\pi$
$740$	9.53792	0.350621
$741$	0	0
$742$	−12.0505	−0.442388
$743$	14.8518	0.544858	0.272429	−	0.962176i	$-0.412173\pi$
$743$	14.8518	0.544858	0.272429	+	0.962176i	$0.412173\pi$
$744$	9.28039	0.340236
$745$	66.4816	2.43570
$746$	−10.1304	−0.370901
$747$	68.0631	2.49030
$748$	−2.66648	−0.0974960
$749$	35.4153	1.29405
$750$	2.75708	0.100674
$751$	21.5149	0.785089	0.392545	−	0.919733i	$-0.371595\pi$
$751$	21.5149	0.785089	0.392545	+	0.919733i	$0.371595\pi$
$752$	−10.8545	−0.395823
$753$	−102.363	−3.73032
$754$	0	0
$755$	−49.2260	−1.79152
$756$	−30.6811	−1.11586
$757$	−38.3135	−1.39253	−0.696264	−	0.717786i	$-0.745156\pi$
$757$	−38.3135	−1.39253	−0.696264	+	0.717786i	$0.745156\pi$
$758$	−5.51320	−0.200249
$759$	−45.2279	−1.64167
$760$	3.12030	0.113185
$761$	22.3954	0.811832	0.405916	−	0.913910i	$-0.366953\pi$
$761$	22.3954	0.811832	0.405916	+	0.913910i	$0.366953\pi$
$762$	55.3351	2.00458
$763$	−1.18775	−0.0429995
$764$	−16.1589	−0.584607
$765$	27.1161	0.980385
$766$	−27.7594	−1.00299
$767$	0	0
$768$	−3.35017	−0.120889
$769$	40.8612	1.47349	0.736747	−	0.676169i	$-0.236361\pi$
$769$	40.8612	1.47349	0.736747	+	0.676169i	$0.236361\pi$
$770$	−13.8037	−0.497451
$771$	1.05645	0.0380471
$772$	−4.80310	−0.172867
$773$	−5.42413	−0.195092	−0.0975462	−	0.995231i	$-0.531099\pi$
$773$	−5.42413	−0.195092	−0.0975462	+	0.995231i	$0.531099\pi$
$774$	−84.6817	−3.04382
$775$	−13.1200	−0.471285
$776$	−12.5191	−0.449411
$777$	−17.9537	−0.644085
$778$	8.45539	0.303140
$779$	−3.89042	−0.139389
$780$	0	0
$781$	25.8979	0.926700
$782$	−5.65372	−0.202177
$783$	124.014	4.43188
$784$	−3.92633	−0.140226
$785$	54.7105	1.95270
$786$	−62.4453	−2.22735
$787$	−12.3811	−0.441339	−0.220669	−	0.975349i	$-0.570824\pi$
$787$	−12.3811	−0.441339	−0.220669	+	0.975349i	$0.570824\pi$
$788$	−9.22162	−0.328507
$789$	−8.59670	−0.306051
$790$	47.8583	1.70272
$791$	8.52128	0.302982
$792$	−20.7509	−0.737350
$793$	0	0
$794$	−23.5763	−0.836691
$795$	71.8522	2.54834
$796$	−5.00186	−0.177286
$797$	−40.7790	−1.44447	−0.722234	−	0.691649i	$-0.756884\pi$
$797$	−40.7790	−1.44447	−0.722234	+	0.691649i	$0.756884\pi$
$798$	−5.87349	−0.207919
$799$	−11.4704	−0.405792
$800$	4.73625	0.167452
$801$	31.6662	1.11887
$802$	−32.4679	−1.14648
$803$	7.85626	0.277242
$804$	18.8275	0.663994
$805$	−29.2680	−1.03156
$806$	0	0
$807$	31.5148	1.10937
$808$	4.63350	0.163006
$809$	−11.3042	−0.397433	−0.198716	−	0.980057i	$-0.563677\pi$
$809$	−11.3042	−0.397433	−0.198716	+	0.980057i	$0.563677\pi$
$810$	105.958	3.72298
$811$	9.98496	0.350619	0.175310	−	0.984513i	$-0.443907\pi$
$811$	9.98496	0.350619	0.175310	+	0.984513i	$0.443907\pi$
$812$	−12.4238	−0.435991
$813$	−27.1414	−0.951892
$814$	−7.71310	−0.270344
$815$	52.1556	1.82693
$816$	−3.54025	−0.123933
$817$	−10.2973	−0.360258
$818$	9.61967	0.336344
$819$	0	0
$820$	−12.1393	−0.423922
$821$	−9.93501	−0.346734	−0.173367	−	0.984857i	$-0.555465\pi$
$821$	−9.93501	−0.346734	−0.173367	+	0.984857i	$0.555465\pi$
$822$	−27.5575	−0.961178
$823$	−48.6265	−1.69501	−0.847507	−	0.530784i	$-0.821898\pi$
$823$	−48.6265	−1.69501	−0.847507	+	0.530784i	$0.821898\pi$
$824$	−12.1292	−0.422539
$825$	40.0381	1.39395
$826$	−12.8497	−0.447099
$827$	−3.97235	−0.138132	−0.0690660	−	0.997612i	$-0.522002\pi$
$827$	−3.97235	−0.138132	−0.0690660	+	0.997612i	$0.522002\pi$
$828$	−43.9980	−1.52904
$829$	−35.3928	−1.22924	−0.614621	−	0.788823i	$-0.710691\pi$
$829$	−35.3928	−1.22924	−0.614621	+	0.788823i	$0.710691\pi$
$830$	25.8251	0.896404
$831$	−1.24088	−0.0430457
$832$	0	0
$833$	−4.14909	−0.143758
$834$	−4.43271	−0.153492
$835$	−52.1556	−1.80492
$836$	−2.52331	−0.0872706
$837$	48.4776	1.67563
$838$	−6.62996	−0.229028
$839$	−9.82872	−0.339325	−0.169662	−	0.985502i	$-0.554268\pi$
$839$	−9.82872	−0.339325	−0.169662	+	0.985502i	$0.554268\pi$
$840$	−18.3270	−0.632342
$841$	21.2174	0.731636
$842$	36.5861	1.26084
$843$	26.0610	0.897587
$844$	6.12808	0.210937
$845$	0	0
$846$	−89.2638	−3.06895
$847$	−8.12233	−0.279087
$848$	−6.87349	−0.236036
$849$	−19.7048	−0.676269
$850$	5.00497	0.171669
$851$	−16.3541	−0.560610
$852$	34.3844	1.17799
$853$	45.5556	1.55980	0.779898	−	0.625907i	$-0.215271\pi$
$853$	45.5556	1.55980	0.779898	+	0.625907i	$0.215271\pi$
$854$	−13.5631	−0.464120
$855$	25.6603	0.877562
$856$	20.2005	0.690439
$857$	51.5808	1.76197	0.880983	−	0.473147i	$-0.156882\pi$
$857$	51.5808	1.76197	0.880983	+	0.473147i	$0.156882\pi$
$858$	0	0
$859$	−14.9638	−0.510558	−0.255279	−	0.966867i	$-0.582167\pi$
$859$	−14.9638	−0.510558	−0.255279	+	0.966867i	$0.582167\pi$
$860$	−32.1307	−1.09565
$861$	22.8503	0.778737
$862$	25.4091	0.865437
$863$	24.9784	0.850275	0.425138	−	0.905129i	$-0.360226\pi$
$863$	24.9784	0.850275	0.425138	+	0.905129i	$0.360226\pi$
$864$	−17.5002	−0.595368
$865$	38.6817	1.31522
$866$	7.73733	0.262925
$867$	53.2118	1.80717
$868$	−4.85655	−0.164842
$869$	−38.7019	−1.31287
$870$	74.0782	2.51149
$871$	0	0
$872$	−0.677480	−0.0229424
$873$	−102.953	−3.48443
$874$	−5.35017	−0.180972
$875$	−1.44282	−0.0487761
$876$	10.4307	0.352420
$877$	7.91854	0.267390	0.133695	−	0.991023i	$-0.457316\pi$
$877$	7.91854	0.267390	0.133695	+	0.991023i	$0.457316\pi$
$878$	13.3006	0.448874
$879$	5.27979	0.178083
$880$	−7.87349	−0.265415
$881$	−37.1403	−1.25129	−0.625644	−	0.780108i	$-0.715164\pi$
$881$	−37.1403	−1.25129	−0.625644	+	0.780108i	$0.715164\pi$
$882$	−32.2888	−1.08722
$883$	−14.0208	−0.471838	−0.235919	−	0.971773i	$-0.575810\pi$
$883$	−14.0208	−0.471838	−0.235919	+	0.971773i	$0.575810\pi$
$884$	0	0
$885$	76.6176	2.57547
$886$	36.3098	1.21985
$887$	20.8836	0.701203	0.350601	−	0.936525i	$-0.385977\pi$
$887$	20.8836	0.701203	0.350601	+	0.936525i	$0.385977\pi$
$888$	−10.2406	−0.343652
$889$	−28.9576	−0.971206
$890$	12.0151	0.402746
$891$	−85.6856	−2.87058
$892$	9.17110	0.307071
$893$	−10.8545	−0.363232
$894$	−71.3794	−2.38729
$895$	15.5049	0.518273
$896$	1.75319	0.0585699
$897$	0	0
$898$	−3.47094	−0.115827
$899$	19.6303	0.654707
$900$	38.9493	1.29831
$901$	−7.26346	−0.241981
$902$	9.81675	0.326862
$903$	60.4812	2.01269
$904$	4.86044	0.161656
$905$	−15.2975	−0.508505
$906$	52.8525	1.75591
$907$	−9.33402	−0.309931	−0.154965	−	0.987920i	$-0.549527\pi$
$907$	−9.33402	−0.309931	−0.154965	+	0.987920i	$0.549527\pi$
$908$	−5.70220	−0.189234
$909$	38.1043	1.26384
$910$	0	0
$911$	8.34471	0.276473	0.138236	−	0.990399i	$-0.455857\pi$
$911$	8.34471	0.276473	0.138236	+	0.990399i	$0.455857\pi$
$912$	−3.35017	−0.110935
$913$	−20.8842	−0.691166
$914$	−33.0892	−1.09449
$915$	80.8712	2.67352
$916$	14.0016	0.462625
$917$	32.6784	1.07914
$918$	−18.4930	−0.610362
$919$	−17.3064	−0.570885	−0.285442	−	0.958396i	$-0.592140\pi$
$919$	−17.3064	−0.570885	−0.285442	+	0.958396i	$0.592140\pi$
$920$	−16.6941	−0.550389
$921$	36.7723	1.21169
$922$	24.5611	0.808876
$923$	0	0
$924$	14.8206	0.487563
$925$	14.4775	0.476017
$926$	10.1113	0.332279
$927$	−99.7461	−3.27609
$928$	−7.08643	−0.232623
$929$	35.9232	1.17860	0.589302	−	0.807913i	$-0.299403\pi$
$929$	35.9232	1.17860	0.589302	+	0.807913i	$0.299403\pi$
$930$	28.9576	0.949557
$931$	−3.92633	−0.128680
$932$	−1.49069	−0.0488292
$933$	−58.4646	−1.91404
$934$	−27.4414	−0.897910
$935$	−8.32020	−0.272099
$936$	0	0
$937$	−53.1464	−1.73622	−0.868109	−	0.496374i	$-0.834665\pi$
$937$	−53.1464	−1.73622	−0.868109	+	0.496374i	$0.834665\pi$
$938$	−9.85266	−0.321701
$939$	−34.5227	−1.12661
$940$	−33.8693	−1.10469
$941$	4.94812	0.161304	0.0806520	−	0.996742i	$-0.474300\pi$
$941$	4.94812	0.161304	0.0806520	+	0.996742i	$0.474300\pi$
$942$	−58.7411	−1.91389
$943$	20.8144	0.677811
$944$	−7.32934	−0.238550
$945$	−95.7341	−3.11423
$946$	25.9834	0.844793
$947$	−12.2379	−0.397680	−0.198840	−	0.980032i	$-0.563717\pi$
$947$	−12.2379	−0.397680	−0.198840	+	0.980032i	$0.563717\pi$
$948$	−51.3841	−1.66888
$949$	0	0
$950$	4.73625	0.153664
$951$	−68.9915	−2.23720
$952$	1.85266	0.0600450
$953$	−12.5250	−0.405724	−0.202862	−	0.979207i	$-0.565024\pi$
$953$	−12.5250	−0.405724	−0.202862	+	0.979207i	$0.565024\pi$
$954$	−56.5252	−1.83007
$955$	−50.4204	−1.63157
$956$	4.60088	0.148803
$957$	−59.9054	−1.93647
$958$	32.6665	1.05541
$959$	14.4212	0.465685
$960$	−10.4535	−0.337386
$961$	−23.3264	−0.752465
$962$	0	0
$963$	166.122	5.35321
$964$	6.03837	0.194483
$965$	−14.9871	−0.482452
$966$	31.4242	1.01106
$967$	−4.94033	−0.158870	−0.0794352	−	0.996840i	$-0.525312\pi$
$967$	−4.94033	−0.158870	−0.0794352	+	0.996840i	$0.525312\pi$
$968$	−4.63289	−0.148907
$969$	−3.54025	−0.113729
$970$	−39.0634	−1.25425
$971$	32.0911	1.02985	0.514926	−	0.857234i	$-0.327819\pi$
$971$	32.0911	1.02985	0.514926	+	0.857234i	$0.327819\pi$
$972$	−61.2633	−1.96502
$973$	2.31969	0.0743659
$974$	24.6555	0.790012
$975$	0	0
$976$	−7.73625	−0.247631
$977$	−51.2010	−1.63807	−0.819033	−	0.573747i	$-0.805489\pi$
$977$	−51.2010	−1.63807	−0.819033	+	0.573747i	$0.805489\pi$
$978$	−55.9980	−1.79062
$979$	−9.71632	−0.310535
$980$	−12.2513	−0.391354
$981$	−5.57137	−0.177880
$982$	−22.5780	−0.720493
$983$	−43.6594	−1.39252	−0.696259	−	0.717791i	$-0.745153\pi$
$983$	−43.6594	−1.39252	−0.696259	+	0.717791i	$0.745153\pi$
$984$	13.0336	0.415495
$985$	−28.7742	−0.916822
$986$	−7.48848	−0.238482
$987$	63.7538	2.02931
$988$	0	0
$989$	55.0925	1.75184
$990$	−64.7489	−2.05785
$991$	16.9150	0.537321	0.268661	−	0.963235i	$-0.413419\pi$
$991$	16.9150	0.537321	0.268661	+	0.963235i	$0.413419\pi$
$992$	−2.77012	−0.0879515
$993$	79.2602	2.51525
$994$	−17.9938	−0.570728
$995$	−15.6073	−0.494784
$996$	−27.7277	−0.878586
$997$	−10.3637	−0.328221	−0.164111	−	0.986442i	$-0.552475\pi$
$997$	−10.3637	−0.328221	−0.164111	+	0.986442i	$0.552475\pi$
$998$	−8.47854	−0.268384
$999$	−53.4934	−1.69246

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6422.2.a.bb.1.1		4
13.4	even	6		494.2.g.e.419.4	yes	8
13.10	even	6		494.2.g.e.191.4	✓	8
13.12	even	2		6422.2.a.z.1.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
494.2.g.e.191.4	✓	8	13.10	even	6
494.2.g.e.419.4	yes	8	13.4	even	6
6422.2.a.z.1.1		4	13.12	even	2
6422.2.a.bb.1.1		4	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:	\( N \)	\(=\)	\( 6422 = 2 \cdot 13^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6422.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -3.35017 q^{3} +1.00000 q^{4} +3.12030 q^{5} -3.35017 q^{6} +1.75319 q^{7} +1.00000 q^{8} +8.22366 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} -3.35017 q^{3} +1.00000 q^{4} +3.12030 q^{5} -3.35017 q^{6} +1.75319 q^{7} +1.00000 q^{8} +8.22366 q^{9} +3.12030 q^{10} -2.52331 q^{11} -3.35017 q^{12} +1.75319 q^{14} -10.4535 q^{15} +1.00000 q^{16} +1.05674 q^{17} +8.22366 q^{18} +1.00000 q^{19} +3.12030 q^{20} -5.87349 q^{21} -2.52331 q^{22} -5.35017 q^{23} -3.35017 q^{24} +4.73625 q^{25} -17.5002 q^{27} +1.75319 q^{28} -7.08643 q^{29} -10.4535 q^{30} -2.77012 q^{31} +1.00000 q^{32} +8.45353 q^{33} +1.05674 q^{34} +5.47047 q^{35} +8.22366 q^{36} +3.05674 q^{37} +1.00000 q^{38} +3.12030 q^{40} -3.89042 q^{41} -5.87349 q^{42} -10.2973 q^{43} -2.52331 q^{44} +25.6603 q^{45} -5.35017 q^{46} -10.8545 q^{47} -3.35017 q^{48} -3.92633 q^{49} +4.73625 q^{50} -3.54025 q^{51} -6.87349 q^{53} -17.5002 q^{54} -7.87349 q^{55} +1.75319 q^{56} -3.35017 q^{57} -7.08643 q^{58} -7.32934 q^{59} -10.4535 q^{60} -7.73625 q^{61} -2.77012 q^{62} +14.4176 q^{63} +1.00000 q^{64} +8.45353 q^{66} -5.61985 q^{67} +1.05674 q^{68} +17.9240 q^{69} +5.47047 q^{70} -10.2635 q^{71} +8.22366 q^{72} -3.11347 q^{73} +3.05674 q^{74} -15.8673 q^{75} +1.00000 q^{76} -4.42384 q^{77} +15.3377 q^{79} +3.12030 q^{80} +33.9576 q^{81} -3.89042 q^{82} +8.27650 q^{83} -5.87349 q^{84} +3.29733 q^{85} -10.2973 q^{86} +23.7408 q^{87} -2.52331 q^{88} +3.85062 q^{89} +25.6603 q^{90} -5.35017 q^{92} +9.28039 q^{93} -10.8545 q^{94} +3.12030 q^{95} -3.35017 q^{96} -12.5191 q^{97} -3.92633 q^{98} -20.7509 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{2} - 2 q^{3} + 4 q^{4} + q^{5} - 2 q^{6} - q^{7} + 4 q^{8} + 2 q^{9}+O(q^{10}) \) 4 * q + 4 * q^2 - 2 * q^3 + 4 * q^4 + q^5 - 2 * q^6 - q^7 + 4 * q^8 + 2 * q^9	\( 4 q + 4 q^{2} - 2 q^{3} + 4 q^{4} + q^{5} - 2 q^{6} - q^{7} + 4 q^{8} + 2 q^{9} + q^{10} - 2 q^{11} - 2 q^{12} - q^{14} - 11 q^{15} + 4 q^{16} + q^{17} + 2 q^{18} + 4 q^{19} + q^{20} - 4 q^{21} - 2 q^{22} - 10 q^{23} - 2 q^{24} + 3 q^{25} - 23 q^{27} - q^{28} - q^{29} - 11 q^{30} - 11 q^{31} + 4 q^{32} + 3 q^{33} + q^{34} - q^{35} + 2 q^{36} + 9 q^{37} + 4 q^{38} + q^{40} - 4 q^{41} - 4 q^{42} - 15 q^{43} - 2 q^{44} + 33 q^{45} - 10 q^{46} - 25 q^{47} - 2 q^{48} - 11 q^{49} + 3 q^{50} - 14 q^{51} - 8 q^{53} - 23 q^{54} - 12 q^{55} - q^{56} - 2 q^{57} - q^{58} - 28 q^{59} - 11 q^{60} - 15 q^{61} - 11 q^{62} + 20 q^{63} + 4 q^{64} + 3 q^{66} + q^{68} + 18 q^{69} - q^{70} + q^{71} + 2 q^{72} - 6 q^{73} + 9 q^{74} - 13 q^{75} + 4 q^{76} - 11 q^{77} - 12 q^{79} + q^{80} + 40 q^{81} - 4 q^{82} + 17 q^{83} - 4 q^{84} - 13 q^{85} - 15 q^{86} + 25 q^{87} - 2 q^{88} + 15 q^{89} + 33 q^{90} - 10 q^{92} + 3 q^{93} - 25 q^{94} + q^{95} - 2 q^{96} - 2 q^{97} - 11 q^{98} - 26 q^{99}+O(q^{100}) \) 4 * q + 4 * q^2 - 2 * q^3 + 4 * q^4 + q^5 - 2 * q^6 - q^7 + 4 * q^8 + 2 * q^9 + q^10 - 2 * q^11 - 2 * q^12 - q^14 - 11 * q^15 + 4 * q^16 + q^17 + 2 * q^18 + 4 * q^19 + q^20 - 4 * q^21 - 2 * q^22 - 10 * q^23 - 2 * q^24 + 3 * q^25 - 23 * q^27 - q^28 - q^29 - 11 * q^30 - 11 * q^31 + 4 * q^32 + 3 * q^33 + q^34 - q^35 + 2 * q^36 + 9 * q^37 + 4 * q^38 + q^40 - 4 * q^41 - 4 * q^42 - 15 * q^43 - 2 * q^44 + 33 * q^45 - 10 * q^46 - 25 * q^47 - 2 * q^48 - 11 * q^49 + 3 * q^50 - 14 * q^51 - 8 * q^53 - 23 * q^54 - 12 * q^55 - q^56 - 2 * q^57 - q^58 - 28 * q^59 - 11 * q^60 - 15 * q^61 - 11 * q^62 + 20 * q^63 + 4 * q^64 + 3 * q^66 + q^68 + 18 * q^69 - q^70 + q^71 + 2 * q^72 - 6 * q^73 + 9 * q^74 - 13 * q^75 + 4 * q^76 - 11 * q^77 - 12 * q^79 + q^80 + 40 * q^81 - 4 * q^82 + 17 * q^83 - 4 * q^84 - 13 * q^85 - 15 * q^86 + 25 * q^87 - 2 * q^88 + 15 * q^89 + 33 * q^90 - 10 * q^92 + 3 * q^93 - 25 * q^94 + q^95 - 2 * q^96 - 2 * q^97 - 11 * q^98 - 26 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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