LMFDB - Embedded newform 6422.2.a.bd.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("6422.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6422 = 2 \cdot 13^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6422.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$51.2799281781$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.6.316645497.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} - 12x^{4} + 24x^{3} + 13x^{2} - 24x + 5 $ x^6 - 2x^5 - 12x^4 + 24x^3 + 13x^2 - 24*x + 5
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
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			1.2
Root			$-1.13360$ of defining polynomial
Character	$\chi$	$=$	6422.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} -1.13360 q^{3} +1.00000 q^{4} +2.43014 q^{5} -1.13360 q^{6} -3.63269 q^{7} +1.00000 q^{8} -1.71494 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} -1.13360 q^{3} +1.00000 q^{4} +2.43014 q^{5} -1.13360 q^{6} -3.63269 q^{7} +1.00000 q^{8} -1.71494 q^{9} +2.43014 q^{10} +2.43014 q^{11} -1.13360 q^{12} -3.63269 q^{14} -2.75481 q^{15} +1.00000 q^{16} +0.797889 q^{17} -1.71494 q^{18} -1.00000 q^{19} +2.43014 q^{20} +4.11803 q^{21} +2.43014 q^{22} +1.81120 q^{23} -1.13360 q^{24} +0.905576 q^{25} +5.34488 q^{27} -3.63269 q^{28} +0.247453 q^{29} -2.75481 q^{30} +5.82977 q^{31} +1.00000 q^{32} -2.75481 q^{33} +0.797889 q^{34} -8.82794 q^{35} -1.71494 q^{36} +3.20146 q^{37} -1.00000 q^{38} +2.43014 q^{40} -8.77777 q^{41} +4.11803 q^{42} +0.823322 q^{43} +2.43014 q^{44} -4.16755 q^{45} +1.81120 q^{46} +5.43014 q^{47} -1.13360 q^{48} +6.19643 q^{49} +0.905576 q^{50} -0.904489 q^{51} -11.0106 q^{53} +5.34488 q^{54} +5.90558 q^{55} -3.63269 q^{56} +1.13360 q^{57} +0.247453 q^{58} +5.34832 q^{59} -2.75481 q^{60} +7.97713 q^{61} +5.82977 q^{62} +6.22986 q^{63} +1.00000 q^{64} -2.75481 q^{66} -12.0699 q^{67} +0.797889 q^{68} -2.05318 q^{69} -8.82794 q^{70} +2.34301 q^{71} -1.71494 q^{72} +6.56148 q^{73} +3.20146 q^{74} -1.02656 q^{75} -1.00000 q^{76} -8.82794 q^{77} +5.06668 q^{79} +2.43014 q^{80} -0.914140 q^{81} -8.77777 q^{82} -1.88843 q^{83} +4.11803 q^{84} +1.93898 q^{85} +0.823322 q^{86} -0.280514 q^{87} +2.43014 q^{88} +15.6882 q^{89} -4.16755 q^{90} +1.81120 q^{92} -6.60865 q^{93} +5.43014 q^{94} -2.43014 q^{95} -1.13360 q^{96} +17.2242 q^{97} +6.19643 q^{98} -4.16755 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 6 q^{2} + 2 q^{3} + 6 q^{4} + 2 q^{5} + 2 q^{6} + q^{7} + 6 q^{8} + 10 q^{9}+O(q^{10}) $ 6 * q + 6 * q^2 + 2 * q^3 + 6 * q^4 + 2 * q^5 + 2 * q^6 + q^7 + 6 * q^8 + 10 * q^9	$ 6 q + 6 q^{2} + 2 q^{3} + 6 q^{4} + 2 q^{5} + 2 q^{6} + q^{7} + 6 q^{8} + 10 q^{9} + 2 q^{10} + 2 q^{11} + 2 q^{12} + q^{14} - 9 q^{15} + 6 q^{16} - 2 q^{17} + 10 q^{18} - 6 q^{19} + 2 q^{20} + q^{21} + 2 q^{22} + 8 q^{23} + 2 q^{24} + 16 q^{25} - 4 q^{27} + q^{28} + 20 q^{29} - 9 q^{30} + 3 q^{31} + 6 q^{32} - 9 q^{33} - 2 q^{34} + 7 q^{35} + 10 q^{36} - 9 q^{37} - 6 q^{38} + 2 q^{40} + 3 q^{41} + q^{42} + 13 q^{43} + 2 q^{44} + 38 q^{45} + 8 q^{46} + 20 q^{47} + 2 q^{48} - 7 q^{49} + 16 q^{50} + 2 q^{51} + 25 q^{53} - 4 q^{54} + 46 q^{55} + q^{56} - 2 q^{57} + 20 q^{58} - 9 q^{60} + 6 q^{61} + 3 q^{62} + 46 q^{63} + 6 q^{64} - 9 q^{66} - 32 q^{67} - 2 q^{68} - 29 q^{69} + 7 q^{70} + 39 q^{71} + 10 q^{72} + 7 q^{73} - 9 q^{74} - 15 q^{75} - 6 q^{76} + 7 q^{77} + 18 q^{79} + 2 q^{80} + 54 q^{81} + 3 q^{82} - 7 q^{83} + q^{84} - 2 q^{85} + 13 q^{86} + 12 q^{87} + 2 q^{88} + 9 q^{89} + 38 q^{90} + 8 q^{92} - 47 q^{93} + 20 q^{94} - 2 q^{95} + 2 q^{96} - 4 q^{97} - 7 q^{98} + 38 q^{99}+O(q^{100}) $ 6 * q + 6 * q^2 + 2 * q^3 + 6 * q^4 + 2 * q^5 + 2 * q^6 + q^7 + 6 * q^8 + 10 * q^9 + 2 * q^10 + 2 * q^11 + 2 * q^12 + q^14 - 9 * q^15 + 6 * q^16 - 2 * q^17 + 10 * q^18 - 6 * q^19 + 2 * q^20 + q^21 + 2 * q^22 + 8 * q^23 + 2 * q^24 + 16 * q^25 - 4 * q^27 + q^28 + 20 * q^29 - 9 * q^30 + 3 * q^31 + 6 * q^32 - 9 * q^33 - 2 * q^34 + 7 * q^35 + 10 * q^36 - 9 * q^37 - 6 * q^38 + 2 * q^40 + 3 * q^41 + q^42 + 13 * q^43 + 2 * q^44 + 38 * q^45 + 8 * q^46 + 20 * q^47 + 2 * q^48 - 7 * q^49 + 16 * q^50 + 2 * q^51 + 25 * q^53 - 4 * q^54 + 46 * q^55 + q^56 - 2 * q^57 + 20 * q^58 - 9 * q^60 + 6 * q^61 + 3 * q^62 + 46 * q^63 + 6 * q^64 - 9 * q^66 - 32 * q^67 - 2 * q^68 - 29 * q^69 + 7 * q^70 + 39 * q^71 + 10 * q^72 + 7 * q^73 - 9 * q^74 - 15 * q^75 - 6 * q^76 + 7 * q^77 + 18 * q^79 + 2 * q^80 + 54 * q^81 + 3 * q^82 - 7 * q^83 + q^84 - 2 * q^85 + 13 * q^86 + 12 * q^87 + 2 * q^88 + 9 * q^89 + 38 * q^90 + 8 * q^92 - 47 * q^93 + 20 * q^94 - 2 * q^95 + 2 * q^96 - 4 * q^97 - 7 * q^98 + 38 * 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
$3$	−1.13360	−0.654486	−0.327243	−	0.944940i	$-0.606120\pi$
$3$	−1.13360	−0.654486	−0.327243	+	0.944940i	$0.606120\pi$
$4$	1.00000	0.500000
$5$	2.43014	1.08679	0.543396	−	0.839477i	$-0.317138\pi$
$5$	2.43014	1.08679	0.543396	+	0.839477i	$0.317138\pi$
$6$	−1.13360	−0.462792
$7$	−3.63269	−1.37303	−0.686514	−	0.727117i	$-0.740860\pi$
$7$	−3.63269	−1.37303	−0.686514	+	0.727117i	$0.740860\pi$
$8$	1.00000	0.353553
$9$	−1.71494	−0.571648
$10$	2.43014	0.768477
$11$	2.43014	0.732714	0.366357	−	0.930474i	$-0.380605\pi$
$11$	2.43014	0.732714	0.366357	+	0.930474i	$0.380605\pi$
$12$	−1.13360	−0.327243
$13$	0	0
$13$	0	0
$14$	−3.63269	−0.970877
$15$	−2.75481	−0.711290
$16$	1.00000	0.250000
$17$	0.797889	0.193516	0.0967582	−	0.995308i	$-0.469153\pi$
$17$	0.797889	0.193516	0.0967582	+	0.995308i	$0.469153\pi$
$18$	−1.71494	−0.404216
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	2.43014	0.543396
$21$	4.11803	0.898628
$22$	2.43014	0.518107
$23$	1.81120	0.377660	0.188830	−	0.982010i	$-0.439530\pi$
$23$	1.81120	0.377660	0.188830	+	0.982010i	$0.439530\pi$
$24$	−1.13360	−0.231396
$25$	0.905576	0.181115
$26$	0	0
$27$	5.34488	1.02862
$28$	−3.63269	−0.686514
$29$	0.247453	0.0459509	0.0229755	−	0.999736i	$-0.492686\pi$
$29$	0.247453	0.0459509	0.0229755	+	0.999736i	$0.492686\pi$
$30$	−2.75481	−0.502958
$31$	5.82977	1.04706	0.523529	−	0.852008i	$-0.324615\pi$
$31$	5.82977	1.04706	0.523529	+	0.852008i	$0.324615\pi$
$32$	1.00000	0.176777
$33$	−2.75481	−0.479552
$34$	0.797889	0.136837
$35$	−8.82794	−1.49219
$36$	−1.71494	−0.285824
$37$	3.20146	0.526317	0.263159	−	0.964753i	$-0.415236\pi$
$37$	3.20146	0.526317	0.263159	+	0.964753i	$0.415236\pi$
$38$	−1.00000	−0.162221
$39$	0	0
$40$	2.43014	0.384239
$41$	−8.77777	−1.37086	−0.685429	−	0.728140i	$-0.740385\pi$
$41$	−8.77777	−1.37086	−0.685429	+	0.728140i	$0.740385\pi$
$42$	4.11803	0.635426
$43$	0.823322	0.125555	0.0627777	−	0.998028i	$-0.480004\pi$
$43$	0.823322	0.125555	0.0627777	+	0.998028i	$0.480004\pi$
$44$	2.43014	0.366357
$45$	−4.16755	−0.621262
$46$	1.81120	0.267046
$47$	5.43014	0.792067	0.396034	−	0.918236i	$-0.370386\pi$
$47$	5.43014	0.792067	0.396034	+	0.918236i	$0.370386\pi$
$48$	−1.13360	−0.163622
$49$	6.19643	0.885205
$50$	0.905576	0.128068
$51$	−0.904489	−0.126654
$52$	0	0
$53$	−11.0106	−1.51243	−0.756213	−	0.654326i	$-0.772953\pi$
$53$	−11.0106	−1.51243	−0.756213	+	0.654326i	$0.772953\pi$
$54$	5.34488	0.727345
$55$	5.90558	0.796308
$56$	−3.63269	−0.485439
$57$	1.13360	0.150149
$58$	0.247453	0.0324922
$59$	5.34832	0.696293	0.348146	−	0.937440i	$-0.386811\pi$
$59$	5.34832	0.696293	0.348146	+	0.937440i	$0.386811\pi$
$60$	−2.75481	−0.355645
$61$	7.97713	1.02137	0.510683	−	0.859769i	$-0.329393\pi$
$61$	7.97713	1.02137	0.510683	+	0.859769i	$0.329393\pi$
$62$	5.82977	0.740381
$63$	6.22986	0.784888
$64$	1.00000	0.125000
$65$	0	0
$66$	−2.75481	−0.339094
$67$	−12.0699	−1.47458	−0.737288	−	0.675579i	$-0.763894\pi$
$67$	−12.0699	−1.47458	−0.737288	+	0.675579i	$0.763894\pi$
$68$	0.797889	0.0967582
$69$	−2.05318	−0.247174
$70$	−8.82794	−1.05514
$71$	2.34301	0.278065	0.139032	−	0.990288i	$-0.455601\pi$
$71$	2.34301	0.278065	0.139032	+	0.990288i	$0.455601\pi$
$72$	−1.71494	−0.202108
$73$	6.56148	0.767963	0.383981	−	0.923341i	$-0.374553\pi$
$73$	6.56148	0.767963	0.383981	+	0.923341i	$0.374553\pi$
$74$	3.20146	0.372163
$75$	−1.02656	−0.118537
$76$	−1.00000	−0.114708
$77$	−8.82794	−1.00604
$78$	0	0
$79$	5.06668	0.570046	0.285023	−	0.958521i	$-0.407999\pi$
$79$	5.06668	0.570046	0.285023	+	0.958521i	$0.407999\pi$
$80$	2.43014	0.271698
$81$	−0.914140	−0.101571
$82$	−8.77777	−0.969343
$83$	−1.88843	−0.207282	−0.103641	−	0.994615i	$-0.533049\pi$
$83$	−1.88843	−0.207282	−0.103641	+	0.994615i	$0.533049\pi$
$84$	4.11803	0.449314
$85$	1.93898	0.210312
$86$	0.823322	0.0887811
$87$	−0.280514	−0.0300742
$88$	2.43014	0.259054
$89$	15.6882	1.66295	0.831474	−	0.555564i	$-0.187498\pi$
$89$	15.6882	1.66295	0.831474	+	0.555564i	$0.187498\pi$
$90$	−4.16755	−0.439298
$91$	0	0
$92$	1.81120	0.188830
$93$	−6.60865	−0.685285
$94$	5.43014	0.560076
$95$	−2.43014	−0.249327
$96$	−1.13360	−0.115698
$97$	17.2242	1.74885	0.874425	−	0.485162i	$-0.161239\pi$
$97$	17.2242	1.74885	0.874425	+	0.485162i	$0.161239\pi$
$98$	6.19643	0.625934
$99$	−4.16755	−0.418855
$100$	0.905576	0.0905576
$101$	−0.771578	−0.0767748	−0.0383874	−	0.999263i	$-0.512222\pi$
$101$	−0.771578	−0.0767748	−0.0383874	+	0.999263i	$0.512222\pi$
$102$	−0.904489	−0.0895578
$103$	13.5855	1.33862	0.669310	−	0.742983i	$-0.266590\pi$
$103$	13.5855	1.33862	0.669310	+	0.742983i	$0.266590\pi$
$104$	0	0
$105$	10.0074	0.976621
$106$	−11.0106	−1.06945
$107$	12.7420	1.23182	0.615908	−	0.787818i	$-0.288789\pi$
$107$	12.7420	1.23182	0.615908	+	0.787818i	$0.288789\pi$
$108$	5.34488	0.514311
$109$	−9.21911	−0.883030	−0.441515	−	0.897254i	$-0.645559\pi$
$109$	−9.21911	−0.883030	−0.441515	+	0.897254i	$0.645559\pi$
$110$	5.90558	0.563075
$111$	−3.62919	−0.344468
$112$	−3.63269	−0.343257
$113$	2.03459	0.191399	0.0956993	−	0.995410i	$-0.469491\pi$
$113$	2.03459	0.191399	0.0956993	+	0.995410i	$0.469491\pi$
$114$	1.13360	0.106172
$115$	4.40146	0.410438
$116$	0.247453	0.0229755
$117$	0	0
$118$	5.34832	0.492353
$119$	−2.89848	−0.265703
$120$	−2.75481	−0.251479
$121$	−5.09442	−0.463129
$122$	7.97713	0.722215
$123$	9.95051	0.897207
$124$	5.82977	0.523529
$125$	−9.95002	−0.889957
$126$	6.22986	0.555000
$127$	−10.6634	−0.946227	−0.473113	−	0.881002i	$-0.656870\pi$
$127$	−10.6634	−0.946227	−0.473113	+	0.881002i	$0.656870\pi$
$128$	1.00000	0.0883883
$129$	−0.933320	−0.0821743
$130$	0	0
$131$	−11.6463	−1.01754	−0.508772	−	0.860901i	$-0.669900\pi$
$131$	−11.6463	−1.01754	−0.508772	+	0.860901i	$0.669900\pi$
$132$	−2.75481	−0.239776
$133$	3.63269	0.314994
$134$	−12.0699	−1.04268
$135$	12.9888	1.11790
$136$	0.797889	0.0684184
$137$	9.68561	0.827498	0.413749	−	0.910391i	$-0.364219\pi$
$137$	9.68561	0.827498	0.413749	+	0.910391i	$0.364219\pi$
$138$	−2.05318	−0.174778
$139$	12.3751	1.04964	0.524822	−	0.851212i	$-0.324132\pi$
$139$	12.3751	1.04964	0.524822	+	0.851212i	$0.324132\pi$
$140$	−8.82794	−0.746097
$141$	−6.15562	−0.518397
$142$	2.34301	0.196621
$143$	0	0
$144$	−1.71494	−0.142912
$145$	0.601346	0.0499391
$146$	6.56148	0.543032
$147$	−7.02430	−0.579354
$148$	3.20146	0.263159
$149$	6.80183	0.557228	0.278614	−	0.960403i	$-0.410125\pi$
$149$	6.80183	0.557228	0.278614	+	0.960403i	$0.410125\pi$
$150$	−1.02656	−0.0838186
$151$	−1.93149	−0.157183	−0.0785913	−	0.996907i	$-0.525042\pi$
$151$	−1.93149	−0.157183	−0.0785913	+	0.996907i	$0.525042\pi$
$152$	−1.00000	−0.0811107
$153$	−1.36833	−0.110623
$154$	−8.82794	−0.711376
$155$	14.1671	1.13793
$156$	0	0
$157$	5.78847	0.461971	0.230985	−	0.972957i	$-0.425805\pi$
$157$	5.78847	0.461971	0.230985	+	0.972957i	$0.425805\pi$
$158$	5.06668	0.403083
$159$	12.4817	0.989862
$160$	2.43014	0.192119
$161$	−6.57951	−0.518538
$162$	−0.914140	−0.0718216
$163$	17.5069	1.37125	0.685625	−	0.727955i	$-0.259529\pi$
$163$	17.5069	1.37125	0.685625	+	0.727955i	$0.259529\pi$
$164$	−8.77777	−0.685429
$165$	−6.69458	−0.521172
$166$	−1.88843	−0.146570
$167$	12.7122	0.983701	0.491851	−	0.870680i	$-0.336321\pi$
$167$	12.7122	0.983701	0.491851	+	0.870680i	$0.336321\pi$
$168$	4.11803	0.317713
$169$	0	0
$170$	1.93898	0.148713
$171$	1.71494	0.131145
$172$	0.823322	0.0627777
$173$	19.6657	1.49516	0.747579	−	0.664173i	$-0.231216\pi$
$173$	19.6657	1.49516	0.747579	+	0.664173i	$0.231216\pi$
$174$	−0.280514	−0.0212657
$175$	−3.28968	−0.248676
$176$	2.43014	0.183179
$177$	−6.06288	−0.455714
$178$	15.6882	1.17588
$179$	26.3461	1.96920	0.984599	−	0.174826i	$-0.0559363\pi$
$179$	26.3461	1.96920	0.984599	+	0.174826i	$0.0559363\pi$
$180$	−4.16755	−0.310631
$181$	15.0101	1.11569	0.557846	−	0.829945i	$-0.311628\pi$
$181$	15.0101	1.11569	0.557846	+	0.829945i	$0.311628\pi$
$182$	0	0
$183$	−9.04290	−0.668470
$184$	1.81120	0.133523
$185$	7.78000	0.571997
$186$	−6.60865	−0.484569
$187$	1.93898	0.141792
$188$	5.43014	0.396034
$189$	−19.4163	−1.41233
$190$	−2.43014	−0.176301
$191$	0.587553	0.0425138	0.0212569	−	0.999774i	$-0.493233\pi$
$191$	0.587553	0.0425138	0.0212569	+	0.999774i	$0.493233\pi$
$192$	−1.13360	−0.0818108
$193$	−3.82268	−0.275163	−0.137581	−	0.990490i	$-0.543933\pi$
$193$	−3.82268	−0.275163	−0.137581	+	0.990490i	$0.543933\pi$
$194$	17.2242	1.23662
$195$	0	0
$196$	6.19643	0.442602
$197$	10.5501	0.751661	0.375831	−	0.926688i	$-0.377357\pi$
$197$	10.5501	0.751661	0.375831	+	0.926688i	$0.377357\pi$
$198$	−4.16755	−0.296175
$199$	−3.54304	−0.251160	−0.125580	−	0.992084i	$-0.540079\pi$
$199$	−3.54304	−0.251160	−0.125580	+	0.992084i	$0.540079\pi$
$200$	0.905576	0.0640339
$201$	13.6825	0.965090
$202$	−0.771578	−0.0542880
$203$	−0.898921	−0.0630919
$204$	−0.904489	−0.0633269
$205$	−21.3312	−1.48984
$206$	13.5855	0.946548
$207$	−3.10610	−0.215889
$208$	0	0
$209$	−2.43014	−0.168096
$210$	10.0074	0.690575
$211$	3.47824	0.239452	0.119726	−	0.992807i	$-0.461798\pi$
$211$	3.47824	0.239452	0.119726	+	0.992807i	$0.461798\pi$
$212$	−11.0106	−0.756213
$213$	−2.65605	−0.181989
$214$	12.7420	0.871025
$215$	2.00079	0.136452
$216$	5.34488	0.363673
$217$	−21.1777	−1.43764
$218$	−9.21911	−0.624397
$219$	−7.43811	−0.502621
$220$	5.90558	0.398154
$221$	0	0
$222$	−3.62919	−0.243575
$223$	−4.31423	−0.288902	−0.144451	−	0.989512i	$-0.546142\pi$
$223$	−4.31423	−0.288902	−0.144451	+	0.989512i	$0.546142\pi$
$224$	−3.63269	−0.242719
$225$	−1.55301	−0.103534
$226$	2.03459	0.135339
$227$	−3.62279	−0.240453	−0.120226	−	0.992747i	$-0.538362\pi$
$227$	−3.62279	−0.240453	−0.120226	+	0.992747i	$0.538362\pi$
$228$	1.13360	0.0750747
$229$	−4.48761	−0.296549	−0.148275	−	0.988946i	$-0.547372\pi$
$229$	−4.48761	−0.296549	−0.148275	+	0.988946i	$0.547372\pi$
$230$	4.40146	0.290224
$231$	10.0074	0.658437
$232$	0.247453	0.0162461
$233$	−14.0857	−0.922784	−0.461392	−	0.887196i	$-0.652650\pi$
$233$	−14.0857	−0.922784	−0.461392	+	0.887196i	$0.652650\pi$
$234$	0	0
$235$	13.1960	0.860812
$236$	5.34832	0.348146
$237$	−5.74361	−0.373087
$238$	−2.89848	−0.187881
$239$	13.9540	0.902609	0.451305	−	0.892370i	$-0.350959\pi$
$239$	13.9540	0.902609	0.451305	+	0.892370i	$0.350959\pi$
$240$	−2.75481	−0.177822
$241$	−16.2416	−1.04621	−0.523106	−	0.852268i	$-0.675227\pi$
$241$	−16.2416	−1.04621	−0.523106	+	0.852268i	$0.675227\pi$
$242$	−5.09442	−0.327482
$243$	−14.9984	−0.962145
$244$	7.97713	0.510683
$245$	15.0582	0.962033
$246$	9.95051	0.634421
$247$	0	0
$248$	5.82977	0.370191
$249$	2.14073	0.135663
$250$	−9.95002	−0.629295
$251$	−16.6641	−1.05183	−0.525915	−	0.850537i	$-0.676277\pi$
$251$	−16.6641	−1.05183	−0.525915	+	0.850537i	$0.676277\pi$
$252$	6.22986	0.392444
$253$	4.40146	0.276717
$254$	−10.6634	−0.669083
$255$	−2.19804	−0.137646
$256$	1.00000	0.0625000
$257$	−31.0423	−1.93637	−0.968183	−	0.250245i	$-0.919489\pi$
$257$	−31.0423	−1.93637	−0.968183	+	0.250245i	$0.919489\pi$
$258$	−0.933320	−0.0581060
$259$	−11.6299	−0.722648
$260$	0	0
$261$	−0.424368	−0.0262677
$262$	−11.6463	−0.719512
$263$	28.7011	1.76979	0.884893	−	0.465794i	$-0.154231\pi$
$263$	28.7011	1.76979	0.884893	+	0.465794i	$0.154231\pi$
$264$	−2.75481	−0.169547
$265$	−26.7574	−1.64369
$266$	3.63269	0.222734
$267$	−17.7842	−1.08838
$268$	−12.0699	−0.737288
$269$	−24.4063	−1.48808	−0.744040	−	0.668135i	$-0.767093\pi$
$269$	−24.4063	−1.48808	−0.744040	+	0.668135i	$0.767093\pi$
$270$	12.9888	0.790473
$271$	−2.85042	−0.173151	−0.0865753	−	0.996245i	$-0.527592\pi$
$271$	−2.85042	−0.173151	−0.0865753	+	0.996245i	$0.527592\pi$
$272$	0.797889	0.0483791
$273$	0	0
$274$	9.68561	0.585129
$275$	2.20067	0.132706
$276$	−2.05318	−0.123587
$277$	6.23242	0.374470	0.187235	−	0.982315i	$-0.440047\pi$
$277$	6.23242	0.374470	0.187235	+	0.982315i	$0.440047\pi$
$278$	12.3751	0.742211
$279$	−9.99772	−0.598548
$280$	−8.82794	−0.527570
$281$	1.43600	0.0856644	0.0428322	−	0.999082i	$-0.486362\pi$
$281$	1.43600	0.0856644	0.0428322	+	0.999082i	$0.486362\pi$
$282$	−6.15562	−0.366562
$283$	29.6293	1.76128	0.880638	−	0.473789i	$-0.157114\pi$
$283$	29.6293	1.76128	0.880638	+	0.473789i	$0.157114\pi$
$284$	2.34301	0.139032
$285$	2.75481	0.163181
$286$	0	0
$287$	31.8869	1.88223
$288$	−1.71494	−0.101054
$289$	−16.3634	−0.962551
$290$	0.601346	0.0353122
$291$	−19.5254	−1.14460
$292$	6.56148	0.383981
$293$	11.1435	0.651011	0.325505	−	0.945540i	$-0.394466\pi$
$293$	11.1435	0.651011	0.325505	+	0.945540i	$0.394466\pi$
$294$	−7.02430	−0.409665
$295$	12.9972	0.756725
$296$	3.20146	0.186081
$297$	12.9888	0.753686
$298$	6.80183	0.394020
$299$	0	0
$300$	−1.02656	−0.0592687
$301$	−2.99087	−0.172391
$302$	−1.93149	−0.111145
$303$	0.874663	0.0502481
$304$	−1.00000	−0.0573539
$305$	19.3855	1.11001
$306$	−1.36833	−0.0782224
$307$	−6.45650	−0.368492	−0.184246	−	0.982880i	$-0.558984\pi$
$307$	−6.45650	−0.368492	−0.184246	+	0.982880i	$0.558984\pi$
$308$	−8.82794	−0.503019
$309$	−15.4006	−0.876109
$310$	14.1671	0.804640
$311$	29.6506	1.68133	0.840665	−	0.541555i	$-0.182164\pi$
$311$	29.6506	1.68133	0.840665	+	0.541555i	$0.182164\pi$
$312$	0	0
$313$	19.7505	1.11636	0.558182	−	0.829718i	$-0.311499\pi$
$313$	19.7505	1.11636	0.558182	+	0.829718i	$0.311499\pi$
$314$	5.78847	0.326663
$315$	15.1394	0.853009
$316$	5.06668	0.285023
$317$	9.06490	0.509135	0.254568	−	0.967055i	$-0.418067\pi$
$317$	9.06490	0.509135	0.254568	+	0.967055i	$0.418067\pi$
$318$	12.4817	0.699938
$319$	0.601346	0.0336689
$320$	2.43014	0.135849
$321$	−14.4444	−0.806206
$322$	−6.57951	−0.366662
$323$	−0.797889	−0.0443957
$324$	−0.914140	−0.0507856
$325$	0	0
$326$	17.5069	0.969620
$327$	10.4508	0.577931
$328$	−8.77777	−0.484671
$329$	−19.7260	−1.08753
$330$	−6.69458	−0.368525
$331$	−33.2007	−1.82487	−0.912437	−	0.409216i	$-0.865802\pi$
$331$	−33.2007	−1.82487	−0.912437	+	0.409216i	$0.865802\pi$
$332$	−1.88843	−0.103641
$333$	−5.49033	−0.300868
$334$	12.7122	0.695582
$335$	−29.3316	−1.60256
$336$	4.11803	0.224657
$337$	−25.8790	−1.40972	−0.704860	−	0.709346i	$-0.748990\pi$
$337$	−25.8790	−1.40972	−0.704860	+	0.709346i	$0.748990\pi$
$338$	0	0
$339$	−2.30642	−0.125268
$340$	1.93898	0.105156
$341$	14.1671	0.767194
$342$	1.71494	0.0927335
$343$	2.91911	0.157617
$344$	0.823322	0.0443905
$345$	−4.98951	−0.268626
$346$	19.6657	1.05724
$347$	15.8944	0.853256	0.426628	−	0.904427i	$-0.359701\pi$
$347$	15.8944	0.853256	0.426628	+	0.904427i	$0.359701\pi$
$348$	−0.280514	−0.0150371
$349$	−36.2764	−1.94183	−0.970914	−	0.239427i	$-0.923040\pi$
$349$	−36.2764	−1.94183	−0.970914	+	0.239427i	$0.923040\pi$
$350$	−3.28968	−0.175841
$351$	0	0
$352$	2.43014	0.129527
$353$	−19.8977	−1.05905	−0.529524	−	0.848295i	$-0.677630\pi$
$353$	−19.8977	−1.05905	−0.529524	+	0.848295i	$0.677630\pi$
$354$	−6.06288	−0.322238
$355$	5.69385	0.302198
$356$	15.6882	0.831474
$357$	3.28573	0.173899
$358$	26.3461	1.39243
$359$	−5.75000	−0.303473	−0.151737	−	0.988421i	$-0.548487\pi$
$359$	−5.75000	−0.303473	−0.151737	+	0.988421i	$0.548487\pi$
$360$	−4.16755	−0.219649
$361$	1.00000	0.0526316
$362$	15.0101	0.788913
$363$	5.77506	0.303112
$364$	0	0
$365$	15.9453	0.834615
$366$	−9.04290	−0.472680
$367$	0.0134648	0.000702858 0	0.000351429	−	1.00000i	$-0.499888\pi$
$367$	0.0134648	0.000702858 0	0.000351429		1.00000i	$0.499888\pi$
$368$	1.81120	0.0944151
$369$	15.0534	0.783648
$370$	7.78000	0.404463
$371$	39.9982	2.07660
$372$	−6.60865	−0.342642
$373$	37.0227	1.91696	0.958481	−	0.285156i	$-0.0920456\pi$
$373$	37.0227	1.91696	0.958481	+	0.285156i	$0.0920456\pi$
$374$	1.93898	0.100262
$375$	11.2794	0.582465
$376$	5.43014	0.280038
$377$	0	0
$378$	−19.4163	−0.998665
$379$	−20.7672	−1.06674	−0.533369	−	0.845883i	$-0.679074\pi$
$379$	−20.7672	−1.06674	−0.533369	+	0.845883i	$0.679074\pi$
$380$	−2.43014	−0.124664
$381$	12.0881	0.619292
$382$	0.587553	0.0300618
$383$	32.8204	1.67705	0.838523	−	0.544867i	$-0.183420\pi$
$383$	32.8204	1.67705	0.838523	+	0.544867i	$0.183420\pi$
$384$	−1.13360	−0.0578490
$385$	−21.4531	−1.09335
$386$	−3.82268	−0.194569
$387$	−1.41195	−0.0717735
$388$	17.2242	0.874425
$389$	−0.591882	−0.0300096	−0.0150048	−	0.999887i	$-0.504776\pi$
$389$	−0.591882	−0.0300096	−0.0150048	+	0.999887i	$0.504776\pi$
$390$	0	0
$391$	1.44513	0.0730835
$392$	6.19643	0.312967
$393$	13.2023	0.665969
$394$	10.5501	0.531505
$395$	12.3127	0.619521
$396$	−4.16755	−0.209427
$397$	−39.0107	−1.95789	−0.978947	−	0.204117i	$-0.934568\pi$
$397$	−39.0107	−1.95789	−0.978947	+	0.204117i	$0.934568\pi$
$398$	−3.54304	−0.177597
$399$	−4.11803	−0.206159
$400$	0.905576	0.0452788
$401$	−0.807239	−0.0403116	−0.0201558	−	0.999797i	$-0.506416\pi$
$401$	−0.807239	−0.0403116	−0.0201558	+	0.999797i	$0.506416\pi$
$402$	13.6825	0.682421
$403$	0	0
$404$	−0.771578	−0.0383874
$405$	−2.22149	−0.110387
$406$	−0.898921	−0.0446127
$407$	7.78000	0.385640
$408$	−0.904489	−0.0447789
$409$	4.50732	0.222873	0.111436	−	0.993772i	$-0.464455\pi$
$409$	4.50732	0.222873	0.111436	+	0.993772i	$0.464455\pi$
$410$	−21.3312	−1.05347
$411$	−10.9796	−0.541586
$412$	13.5855	0.669310
$413$	−19.4288	−0.956029
$414$	−3.10610	−0.152656
$415$	−4.58914	−0.225272
$416$	0	0
$417$	−14.0285	−0.686978
$418$	−2.43014	−0.118862
$419$	35.6494	1.74159	0.870794	−	0.491647i	$-0.163605\pi$
$419$	35.6494	1.74159	0.870794	+	0.491647i	$0.163605\pi$
$420$	10.0074	0.488310
$421$	−16.4984	−0.804082	−0.402041	−	0.915622i	$-0.631699\pi$
$421$	−16.4984	−0.804082	−0.402041	+	0.915622i	$0.631699\pi$
$422$	3.47824	0.169318
$423$	−9.31238	−0.452783
$424$	−11.0106	−0.534723
$425$	0.722549	0.0350488
$426$	−2.65605	−0.128686
$427$	−28.9784	−1.40236
$428$	12.7420	0.615908
$429$	0	0
$430$	2.00079	0.0964865
$431$	−2.66857	−0.128540	−0.0642702	−	0.997933i	$-0.520472\pi$
$431$	−2.66857	−0.128540	−0.0642702	+	0.997933i	$0.520472\pi$
$432$	5.34488	0.257155
$433$	7.56295	0.363452	0.181726	−	0.983349i	$-0.441832\pi$
$433$	7.56295	0.363452	0.181726	+	0.983349i	$0.441832\pi$
$434$	−21.1777	−1.01656
$435$	−0.681688	−0.0326844
$436$	−9.21911	−0.441515
$437$	−1.81120	−0.0866412
$438$	−7.43811	−0.355407
$439$	11.1133	0.530409	0.265204	−	0.964192i	$-0.414561\pi$
$439$	11.1133	0.530409	0.265204	+	0.964192i	$0.414561\pi$
$440$	5.90558	0.281537
$441$	−10.6265	−0.506025
$442$	0	0
$443$	20.6644	0.981796	0.490898	−	0.871217i	$-0.336669\pi$
$443$	20.6644	0.981796	0.490898	+	0.871217i	$0.336669\pi$
$444$	−3.62919	−0.172234
$445$	38.1246	1.80728
$446$	−4.31423	−0.204285
$447$	−7.71058	−0.364698
$448$	−3.63269	−0.171628
$449$	−27.3801	−1.29215	−0.646073	−	0.763276i	$-0.723590\pi$
$449$	−27.3801	−1.29215	−0.646073	+	0.763276i	$0.723590\pi$
$450$	−1.55301	−0.0732096
$451$	−21.3312	−1.00445
$452$	2.03459	0.0956993
$453$	2.18955	0.102874
$454$	−3.62279	−0.170026
$455$	0	0
$456$	1.13360	0.0530858
$457$	15.2658	0.714102	0.357051	−	0.934085i	$-0.383782\pi$
$457$	15.2658	0.714102	0.357051	+	0.934085i	$0.383782\pi$
$458$	−4.48761	−0.209692
$459$	4.26462	0.199055
$460$	4.40146	0.205219
$461$	21.6350	1.00764	0.503821	−	0.863808i	$-0.331927\pi$
$461$	21.6350	1.00764	0.503821	+	0.863808i	$0.331927\pi$
$462$	10.0074	0.465586
$463$	1.00783	0.0468380	0.0234190	−	0.999726i	$-0.492545\pi$
$463$	1.00783	0.0468380	0.0234190	+	0.999726i	$0.492545\pi$
$464$	0.247453	0.0114877
$465$	−16.0599	−0.744761
$466$	−14.0857	−0.652507
$467$	−6.24179	−0.288836	−0.144418	−	0.989517i	$-0.546131\pi$
$467$	−6.24179	−0.288836	−0.144418	+	0.989517i	$0.546131\pi$
$468$	0	0
$469$	43.8463	2.02463
$470$	13.1960	0.608686
$471$	−6.56184	−0.302353
$472$	5.34832	0.246177
$473$	2.00079	0.0919963
$474$	−5.74361	−0.263813
$475$	−0.905576	−0.0415507
$476$	−2.89848	−0.132852
$477$	18.8826	0.864575
$478$	13.9540	0.638241
$479$	−2.77283	−0.126694	−0.0633470	−	0.997992i	$-0.520177\pi$
$479$	−2.77283	−0.126694	−0.0633470	+	0.997992i	$0.520177\pi$
$480$	−2.75481	−0.125739
$481$	0	0
$482$	−16.2416	−0.739783
$483$	7.45856	0.339376
$484$	−5.09442	−0.231565
$485$	41.8571	1.90063
$486$	−14.9984	−0.680339
$487$	−35.2602	−1.59779	−0.798897	−	0.601468i	$-0.794583\pi$
$487$	−35.2602	−1.59779	−0.798897	+	0.601468i	$0.794583\pi$
$488$	7.97713	0.361108
$489$	−19.8459	−0.897464
$490$	15.0582	0.680260
$491$	41.1833	1.85857	0.929287	−	0.369358i	$-0.120423\pi$
$491$	41.1833	1.85857	0.929287	+	0.369358i	$0.120423\pi$
$492$	9.95051	0.448604
$493$	0.197440	0.00889226
$494$	0	0
$495$	−10.1277	−0.455207
$496$	5.82977	0.261764
$497$	−8.51144	−0.381790
$498$	2.14073	0.0959282
$499$	15.0167	0.672240	0.336120	−	0.941819i	$-0.390885\pi$
$499$	15.0167	0.672240	0.336120	+	0.941819i	$0.390885\pi$
$500$	−9.95002	−0.444978
$501$	−14.4106	−0.643819
$502$	−16.6641	−0.743757
$503$	16.2937	0.726502	0.363251	−	0.931691i	$-0.381667\pi$
$503$	16.2937	0.726502	0.363251	+	0.931691i	$0.381667\pi$
$504$	6.22986	0.277500
$505$	−1.87504	−0.0834382
$506$	4.40146	0.195669
$507$	0	0
$508$	−10.6634	−0.473113
$509$	−35.0554	−1.55380	−0.776902	−	0.629621i	$-0.783210\pi$
$509$	−35.0554	−1.55380	−0.776902	+	0.629621i	$0.783210\pi$
$510$	−2.19804	−0.0973306
$511$	−23.8358	−1.05443
$512$	1.00000	0.0441942
$513$	−5.34488	−0.235982
$514$	−31.0423	−1.36922
$515$	33.0147	1.45480
$516$	−0.933320	−0.0410871
$517$	13.1960	0.580359
$518$	−11.6299	−0.510990
$519$	−22.2931	−0.978560
$520$	0	0
$521$	10.2291	0.448143	0.224072	−	0.974573i	$-0.428065\pi$
$521$	10.2291	0.448143	0.224072	+	0.974573i	$0.428065\pi$
$522$	−0.424368	−0.0185741
$523$	0.805922	0.0352405	0.0176203	−	0.999845i	$-0.494391\pi$
$523$	0.805922	0.0352405	0.0176203	+	0.999845i	$0.494391\pi$
$524$	−11.6463	−0.508772
$525$	3.72919	0.162755
$526$	28.7011	1.25143
$527$	4.65151	0.202623
$528$	−2.75481	−0.119888
$529$	−19.7196	−0.857373
$530$	−26.7574	−1.16227
$531$	−9.17207	−0.398034
$532$	3.63269	0.157497
$533$	0	0
$534$	−17.7842	−0.769598
$535$	30.9648	1.33873
$536$	−12.0699	−0.521341
$537$	−29.8660	−1.28881
$538$	−24.4063	−1.05223
$539$	15.0582	0.648602
$540$	12.9888	0.558949
$541$	37.0551	1.59312	0.796562	−	0.604557i	$-0.206650\pi$
$541$	37.0551	1.59312	0.796562	+	0.604557i	$0.206650\pi$
$542$	−2.85042	−0.122436
$543$	−17.0155	−0.730205
$544$	0.797889	0.0342092
$545$	−22.4037	−0.959670
$546$	0	0
$547$	34.1469	1.46002	0.730008	−	0.683438i	$-0.239516\pi$
$547$	34.1469	1.46002	0.730008	+	0.683438i	$0.239516\pi$
$548$	9.68561	0.413749
$549$	−13.6803	−0.583862
$550$	2.20067	0.0938371
$551$	−0.247453	−0.0105419
$552$	−2.05318	−0.0873890
$553$	−18.4057	−0.782689
$554$	6.23242	0.264790
$555$	−8.81944	−0.374364
$556$	12.3751	0.524822
$557$	15.4091	0.652904	0.326452	−	0.945214i	$-0.394147\pi$
$557$	15.4091	0.652904	0.326452	+	0.945214i	$0.394147\pi$
$558$	−9.99772	−0.423237
$559$	0	0
$560$	−8.82794	−0.373049
$561$	−2.19804	−0.0928011
$562$	1.43600	0.0605739
$563$	−11.8217	−0.498224	−0.249112	−	0.968475i	$-0.580139\pi$
$563$	−11.8217	−0.498224	−0.249112	+	0.968475i	$0.580139\pi$
$564$	−6.15562	−0.259199
$565$	4.94435	0.208010
$566$	29.6293	1.24541
$567$	3.32079	0.139460
$568$	2.34301	0.0983107
$569$	22.9689	0.962904	0.481452	−	0.876472i	$-0.340110\pi$
$569$	22.9689	0.962904	0.481452	+	0.876472i	$0.340110\pi$
$570$	2.75481	0.115386
$571$	−27.1037	−1.13425	−0.567127	−	0.823630i	$-0.691945\pi$
$571$	−27.1037	−1.13425	−0.567127	+	0.823630i	$0.691945\pi$
$572$	0	0
$573$	−0.666052	−0.0278247
$574$	31.8869	1.33093
$575$	1.64017	0.0684000
$576$	−1.71494	−0.0714560
$577$	−36.2126	−1.50755	−0.753776	−	0.657132i	$-0.771769\pi$
$577$	−36.2126	−1.50755	−0.753776	+	0.657132i	$0.771769\pi$
$578$	−16.3634	−0.680627
$579$	4.33341	0.180090
$580$	0.601346	0.0249695
$581$	6.86006	0.284603
$582$	−19.5254	−0.809353
$583$	−26.7574	−1.10818
$584$	6.56148	0.271516
$585$	0	0
$586$	11.1435	0.460334
$587$	−40.6298	−1.67697	−0.838485	−	0.544924i	$-0.816559\pi$
$587$	−40.6298	−1.67697	−0.838485	+	0.544924i	$0.816559\pi$
$588$	−7.02430	−0.289677
$589$	−5.82977	−0.240211
$590$	12.9972	0.535085
$591$	−11.9596	−0.491952
$592$	3.20146	0.131579
$593$	−10.9265	−0.448696	−0.224348	−	0.974509i	$-0.572025\pi$
$593$	−10.9265	−0.448696	−0.224348	+	0.974509i	$0.572025\pi$
$594$	12.9888	0.532937
$595$	−7.04371	−0.288764
$596$	6.80183	0.278614
$597$	4.01641	0.164381
$598$	0	0
$599$	−23.0014	−0.939811	−0.469905	−	0.882717i	$-0.655712\pi$
$599$	−23.0014	−0.939811	−0.469905	+	0.882717i	$0.655712\pi$
$600$	−1.02656	−0.0419093
$601$	34.5104	1.40771	0.703855	−	0.710344i	$-0.251460\pi$
$601$	34.5104	1.40771	0.703855	+	0.710344i	$0.251460\pi$
$602$	−2.99087	−0.121899
$603$	20.6992	0.842938
$604$	−1.93149	−0.0785913
$605$	−12.3802	−0.503325
$606$	0.874663	0.0355308
$607$	21.8306	0.886075	0.443038	−	0.896503i	$-0.353901\pi$
$607$	21.8306	0.886075	0.443038	+	0.896503i	$0.353901\pi$
$608$	−1.00000	−0.0405554
$609$	1.01902	0.0412928
$610$	19.3855	0.784897
$611$	0	0
$612$	−1.36833	−0.0553116
$613$	−35.0986	−1.41762	−0.708809	−	0.705401i	$-0.750767\pi$
$613$	−35.0986	−1.41762	−0.708809	+	0.705401i	$0.750767\pi$
$614$	−6.45650	−0.260563
$615$	24.1811	0.975077
$616$	−8.82794	−0.355688
$617$	−44.7520	−1.80165	−0.900824	−	0.434183i	$-0.857037\pi$
$617$	−44.7520	−1.80165	−0.900824	+	0.434183i	$0.857037\pi$
$618$	−15.4006	−0.619503
$619$	27.5430	1.10705	0.553524	−	0.832833i	$-0.313283\pi$
$619$	27.5430	1.10705	0.553524	+	0.832833i	$0.313283\pi$
$620$	14.1671	0.568966
$621$	9.68062	0.388470
$622$	29.6506	1.18888
$623$	−56.9904	−2.28327
$624$	0	0
$625$	−28.7078	−1.14831
$626$	19.7505	0.789389
$627$	2.75481	0.110017
$628$	5.78847	0.230985
$629$	2.55441	0.101851
$630$	15.1394	0.603169
$631$	1.94828	0.0775600	0.0387800	−	0.999248i	$-0.487653\pi$
$631$	1.94828	0.0775600	0.0387800	+	0.999248i	$0.487653\pi$
$632$	5.06668	0.201542
$633$	−3.94294	−0.156718
$634$	9.06490	0.360013
$635$	−25.9136	−1.02835
$636$	12.4817	0.494931
$637$	0	0
$638$	0.601346	0.0238075
$639$	−4.01814	−0.158955
$640$	2.43014	0.0960597
$641$	−6.29233	−0.248532	−0.124266	−	0.992249i	$-0.539658\pi$
$641$	−6.29233	−0.248532	−0.124266	+	0.992249i	$0.539658\pi$
$642$	−14.4444	−0.570074
$643$	−33.5838	−1.32442	−0.662208	−	0.749320i	$-0.730380\pi$
$643$	−33.5838	−1.32442	−0.662208	+	0.749320i	$0.730380\pi$
$644$	−6.57951	−0.259269
$645$	−2.26810	−0.0893063
$646$	−0.797889	−0.0313925
$647$	−22.3809	−0.879883	−0.439942	−	0.898026i	$-0.645001\pi$
$647$	−22.3809	−0.879883	−0.439942	+	0.898026i	$0.645001\pi$
$648$	−0.914140	−0.0359108
$649$	12.9972	0.510184
$650$	0	0
$651$	24.0072	0.940915
$652$	17.5069	0.685625
$653$	−25.5372	−0.999348	−0.499674	−	0.866214i	$-0.666547\pi$
$653$	−25.5372	−0.999348	−0.499674	+	0.866214i	$0.666547\pi$
$654$	10.4508	0.408659
$655$	−28.3022	−1.10586
$656$	−8.77777	−0.342714
$657$	−11.2526	−0.439004
$658$	−19.7260	−0.769000
$659$	26.6492	1.03810	0.519052	−	0.854743i	$-0.326285\pi$
$659$	26.6492	1.03810	0.519052	+	0.854743i	$0.326285\pi$
$660$	−6.69458	−0.260586
$661$	−9.36093	−0.364098	−0.182049	−	0.983289i	$-0.558273\pi$
$661$	−9.36093	−0.364098	−0.182049	+	0.983289i	$0.558273\pi$
$662$	−33.2007	−1.29038
$663$	0	0
$664$	−1.88843	−0.0732851
$665$	8.82794	0.342333
$666$	−5.49033	−0.212746
$667$	0.448186	0.0173538
$668$	12.7122	0.491851
$669$	4.89062	0.189082
$670$	−29.3316	−1.13318
$671$	19.3855	0.748370
$672$	4.11803	0.158856
$673$	42.7253	1.64694	0.823470	−	0.567360i	$-0.192035\pi$
$673$	42.7253	1.64694	0.823470	+	0.567360i	$0.192035\pi$
$674$	−25.8790	−0.996823
$675$	4.84019	0.186299
$676$	0	0
$677$	9.00585	0.346123	0.173062	−	0.984911i	$-0.444634\pi$
$677$	9.00585	0.346123	0.173062	+	0.984911i	$0.444634\pi$
$678$	−2.30642	−0.0885776
$679$	−62.5700	−2.40122
$680$	1.93898	0.0743565
$681$	4.10680	0.157373
$682$	14.1671	0.542488
$683$	28.7776	1.10114	0.550572	−	0.834787i	$-0.314409\pi$
$683$	28.7776	1.10114	0.550572	+	0.834787i	$0.314409\pi$
$684$	1.71494	0.0655725
$685$	23.5374	0.899317
$686$	2.91911	0.111452
$687$	5.08717	0.194088
$688$	0.823322	0.0313888
$689$	0	0
$690$	−4.98951	−0.189947
$691$	37.5254	1.42753	0.713766	−	0.700384i	$-0.246988\pi$
$691$	37.5254	1.42753	0.713766	+	0.700384i	$0.246988\pi$
$692$	19.6657	0.747579
$693$	15.1394	0.575099
$694$	15.8944	0.603343
$695$	30.0733	1.14074
$696$	−0.280514	−0.0106329
$697$	−7.00369	−0.265283
$698$	−36.2764	−1.37308
$699$	15.9676	0.603949
$700$	−3.28968	−0.124338
$701$	9.99062	0.377340	0.188670	−	0.982040i	$-0.439582\pi$
$701$	9.99062	0.377340	0.188670	+	0.982040i	$0.439582\pi$
$702$	0	0
$703$	−3.20146	−0.120746
$704$	2.43014	0.0915893
$705$	−14.9590	−0.563389
$706$	−19.8977	−0.748861
$707$	2.80290	0.105414
$708$	−6.06288	−0.227857
$709$	−12.0806	−0.453698	−0.226849	−	0.973930i	$-0.572842\pi$
$709$	−12.0806	−0.453698	−0.226849	+	0.973930i	$0.572842\pi$
$710$	5.69385	0.213686
$711$	−8.68907	−0.325866
$712$	15.6882	0.587941
$713$	10.5589	0.395432
$714$	3.28573	0.122965
$715$	0	0
$716$	26.3461	0.984599
$717$	−15.8183	−0.590745
$718$	−5.75000	−0.214588
$719$	−24.0155	−0.895627	−0.447813	−	0.894127i	$-0.647797\pi$
$719$	−24.0155	−0.895627	−0.447813	+	0.894127i	$0.647797\pi$
$720$	−4.16755	−0.155315
$721$	−49.3520	−1.83796
$722$	1.00000	0.0372161
$723$	18.4115	0.684731
$724$	15.0101	0.557846
$725$	0.224088	0.00832241
$726$	5.77506	0.214332
$727$	−41.0623	−1.52291	−0.761457	−	0.648215i	$-0.775516\pi$
$727$	−41.0623	−1.52291	−0.761457	+	0.648215i	$0.775516\pi$
$728$	0	0
$729$	19.7446	0.731282
$730$	15.9453	0.590162
$731$	0.656919	0.0242970
$732$	−9.04290	−0.334235
$733$	42.2425	1.56026	0.780131	−	0.625616i	$-0.215152\pi$
$733$	42.2425	1.56026	0.780131	+	0.625616i	$0.215152\pi$
$734$	0.0134648	0.000496996 0
$735$	−17.0700	−0.629637
$736$	1.81120	0.0667616
$737$	−29.3316	−1.08044
$738$	15.0534	0.554123
$739$	−1.74107	−0.0640462	−0.0320231	−	0.999487i	$-0.510195\pi$
$739$	−1.74107	−0.0640462	−0.0320231	+	0.999487i	$0.510195\pi$
$740$	7.78000	0.285999
$741$	0	0
$742$	39.9982	1.46838
$743$	−40.8210	−1.49758	−0.748789	−	0.662809i	$-0.769364\pi$
$743$	−40.8210	−1.49758	−0.748789	+	0.662809i	$0.769364\pi$
$744$	−6.60865	−0.242285
$745$	16.5294	0.605591
$746$	37.0227	1.35550
$747$	3.23854	0.118492
$748$	1.93898	0.0708962
$749$	−46.2877	−1.69132
$750$	11.2794	0.411865
$751$	20.8133	0.759490	0.379745	−	0.925091i	$-0.376012\pi$
$751$	20.8133	0.759490	0.379745	+	0.925091i	$0.376012\pi$
$752$	5.43014	0.198017
$753$	18.8905	0.688409
$754$	0	0
$755$	−4.69379	−0.170825
$756$	−19.4163	−0.706163
$757$	−23.1734	−0.842252	−0.421126	−	0.907002i	$-0.638365\pi$
$757$	−23.1734	−0.842252	−0.421126	+	0.907002i	$0.638365\pi$
$758$	−20.7672	−0.754297
$759$	−4.98951	−0.181108
$760$	−2.43014	−0.0881504
$761$	−44.0561	−1.59703	−0.798517	−	0.601973i	$-0.794382\pi$
$761$	−44.0561	−1.59703	−0.798517	+	0.601973i	$0.794382\pi$
$762$	12.0881	0.437906
$763$	33.4902	1.21243
$764$	0.587553	0.0212569
$765$	−3.32524	−0.120224
$766$	32.8204	1.18585
$767$	0	0
$768$	−1.13360	−0.0409054
$769$	12.3942	0.446947	0.223473	−	0.974710i	$-0.428260\pi$
$769$	12.3942	0.446947	0.223473	+	0.974710i	$0.428260\pi$
$770$	−21.4531	−0.773117
$771$	35.1897	1.26732
$772$	−3.82268	−0.137581
$773$	−24.1191	−0.867504	−0.433752	−	0.901032i	$-0.642811\pi$
$773$	−24.1191	−0.867504	−0.433752	+	0.901032i	$0.642811\pi$
$774$	−1.41195	−0.0507515
$775$	5.27930	0.189638
$776$	17.2242	0.618312
$777$	13.1837	0.472963
$778$	−0.591882	−0.0212200
$779$	8.77777	0.314496
$780$	0	0
$781$	5.69385	0.203742
$782$	1.44513	0.0516778
$783$	1.32261	0.0472661
$784$	6.19643	0.221301
$785$	14.0668	0.502066
$786$	13.2023	0.470911
$787$	43.5990	1.55414	0.777069	−	0.629416i	$-0.216706\pi$
$787$	43.5990	1.55414	0.777069	+	0.629416i	$0.216706\pi$
$788$	10.5501	0.375831
$789$	−32.5357	−1.15830
$790$	12.3127	0.438067
$791$	−7.39105	−0.262795
$792$	−4.16755	−0.148087
$793$	0	0
$794$	−39.0107	−1.38444
$795$	30.3322	1.07577
$796$	−3.54304	−0.125580
$797$	2.63165	0.0932178	0.0466089	−	0.998913i	$-0.485159\pi$
$797$	2.63165	0.0932178	0.0466089	+	0.998913i	$0.485159\pi$
$798$	−4.11803	−0.145777
$799$	4.33265	0.153278
$800$	0.905576	0.0320169
$801$	−26.9044	−0.950620
$802$	−0.807239	−0.0285046
$803$	15.9453	0.562697
$804$	13.6825	0.482545
$805$	−15.9891	−0.563543
$806$	0	0
$807$	27.6671	0.973927
$808$	−0.771578	−0.0271440
$809$	12.1738	0.428007	0.214003	−	0.976833i	$-0.431350\pi$
$809$	12.1738	0.428007	0.214003	+	0.976833i	$0.431350\pi$
$810$	−2.22149	−0.0780551
$811$	−33.1949	−1.16563	−0.582815	−	0.812605i	$-0.698049\pi$
$811$	−33.1949	−1.16563	−0.582815	+	0.812605i	$0.698049\pi$
$812$	−0.898921	−0.0315459
$813$	3.23125	0.113325
$814$	7.78000	0.272689
$815$	42.5443	1.49026
$816$	−0.904489	−0.0316635
$817$	−0.823322	−0.0288044
$818$	4.50732	0.157595
$819$	0	0
$820$	−21.3312	−0.744918
$821$	29.7028	1.03663	0.518317	−	0.855189i	$-0.326559\pi$
$821$	29.7028	1.03663	0.518317	+	0.855189i	$0.326559\pi$
$822$	−10.9796	−0.382959
$823$	−12.7914	−0.445879	−0.222939	−	0.974832i	$-0.571565\pi$
$823$	−12.7914	−0.445879	−0.222939	+	0.974832i	$0.571565\pi$
$824$	13.5855	0.473274
$825$	−2.49469	−0.0868540
$826$	−19.4288	−0.676014
$827$	18.7359	0.651512	0.325756	−	0.945454i	$-0.394381\pi$
$827$	18.7359	0.651512	0.325756	+	0.945454i	$0.394381\pi$
$828$	−3.10610	−0.107944
$829$	−16.7616	−0.582153	−0.291076	−	0.956700i	$-0.594013\pi$
$829$	−16.7616	−0.582153	−0.291076	+	0.956700i	$0.594013\pi$
$830$	−4.58914	−0.159291
$831$	−7.06509	−0.245085
$832$	0	0
$833$	4.94406	0.171302
$834$	−14.0285	−0.485767
$835$	30.8925	1.06908
$836$	−2.43014	−0.0840481
$837$	31.1594	1.07703
$838$	35.6494	1.23149
$839$	12.9940	0.448604	0.224302	−	0.974520i	$-0.427990\pi$
$839$	12.9940	0.448604	0.224302	+	0.974520i	$0.427990\pi$
$840$	10.0074	0.345288
$841$	−28.9388	−0.997889
$842$	−16.4984	−0.568572
$843$	−1.62785	−0.0560662
$844$	3.47824	0.119726
$845$	0	0
$846$	−9.31238	−0.320166
$847$	18.5065	0.635890
$848$	−11.0106	−0.378106
$849$	−33.5878	−1.15273
$850$	0.722549	0.0247832
$851$	5.79848	0.198769
$852$	−2.65605	−0.0909947
$853$	−10.3397	−0.354026	−0.177013	−	0.984209i	$-0.556643\pi$
$853$	−10.3397	−0.354026	−0.177013	+	0.984209i	$0.556643\pi$
$854$	−28.9784	−0.991621
$855$	4.16755	0.142527
$856$	12.7420	0.435512
$857$	3.34059	0.114112	0.0570561	−	0.998371i	$-0.481829\pi$
$857$	3.34059	0.114112	0.0570561	+	0.998371i	$0.481829\pi$
$858$	0	0
$859$	−54.0193	−1.84311	−0.921557	−	0.388244i	$-0.873082\pi$
$859$	−54.0193	−1.84311	−0.921557	+	0.388244i	$0.873082\pi$
$860$	2.00079	0.0682262
$861$	−36.1471	−1.23189
$862$	−2.66857	−0.0908918
$863$	−26.2611	−0.893940	−0.446970	−	0.894549i	$-0.647497\pi$
$863$	−26.2611	−0.893940	−0.446970	+	0.894549i	$0.647497\pi$
$864$	5.34488	0.181836
$865$	47.7905	1.62492
$866$	7.56295	0.256999
$867$	18.5496	0.629977
$868$	−21.1777	−0.718819
$869$	12.3127	0.417681
$870$	−0.681688	−0.0231114
$871$	0	0
$872$	−9.21911	−0.312198
$873$	−29.5385	−0.999726
$874$	−1.81120	−0.0612646
$875$	36.1453	1.22194
$876$	−7.43811	−0.251310
$877$	−10.3950	−0.351015	−0.175507	−	0.984478i	$-0.556157\pi$
$877$	−10.3950	−0.351015	−0.175507	+	0.984478i	$0.556157\pi$
$878$	11.1133	0.375056
$879$	−12.6323	−0.426077
$880$	5.90558	0.199077
$881$	55.6406	1.87458	0.937290	−	0.348551i	$-0.113326\pi$
$881$	55.6406	1.87458	0.937290	+	0.348551i	$0.113326\pi$
$882$	−10.6265	−0.357814
$883$	14.5307	0.488997	0.244498	−	0.969650i	$-0.421377\pi$
$883$	14.5307	0.488997	0.244498	+	0.969650i	$0.421377\pi$
$884$	0	0
$885$	−14.7336	−0.495266
$886$	20.6644	0.694234
$887$	19.2781	0.647295	0.323648	−	0.946178i	$-0.395091\pi$
$887$	19.2781	0.647295	0.323648	+	0.946178i	$0.395091\pi$
$888$	−3.62919	−0.121788
$889$	38.7369	1.29920
$890$	38.1246	1.27794
$891$	−2.22149	−0.0744226
$892$	−4.31423	−0.144451
$893$	−5.43014	−0.181713
$894$	−7.71058	−0.257880
$895$	64.0246	2.14011
$896$	−3.63269	−0.121360
$897$	0	0
$898$	−27.3801	−0.913685
$899$	1.44260	0.0481133
$900$	−1.55301	−0.0517670
$901$	−8.78525	−0.292679
$902$	−21.3312	−0.710251
$903$	3.39046	0.112828
$904$	2.03459	0.0676696
$905$	36.4766	1.21252
$906$	2.18955	0.0727428
$907$	−29.8204	−0.990171	−0.495086	−	0.868844i	$-0.664863\pi$
$907$	−29.8204	−0.990171	−0.495086	+	0.868844i	$0.664863\pi$
$908$	−3.62279	−0.120226
$909$	1.32321	0.0438882
$910$	0	0
$911$	−1.44844	−0.0479891	−0.0239945	−	0.999712i	$-0.507638\pi$
$911$	−1.44844	−0.0479891	−0.0239945	+	0.999712i	$0.507638\pi$
$912$	1.13360	0.0375374
$913$	−4.58914	−0.151878
$914$	15.2658	0.504947
$915$	−21.9755	−0.726488
$916$	−4.48761	−0.148275
$917$	42.3075	1.39712
$918$	4.26462	0.140753
$919$	−13.5150	−0.445818	−0.222909	−	0.974839i	$-0.571555\pi$
$919$	−13.5150	−0.445818	−0.222909	+	0.974839i	$0.571555\pi$
$920$	4.40146	0.145112
$921$	7.31911	0.241173
$922$	21.6350	0.712510
$923$	0	0
$924$	10.0074	0.329219
$925$	2.89917	0.0953241
$926$	1.00783	0.0331195
$927$	−23.2984	−0.765220
$928$	0.247453	0.00812305
$929$	−31.0582	−1.01899	−0.509494	−	0.860474i	$-0.670167\pi$
$929$	−31.0582	−1.01899	−0.509494	+	0.860474i	$0.670167\pi$
$930$	−16.0599	−0.526626
$931$	−6.19643	−0.203080
$932$	−14.0857	−0.461392
$933$	−33.6120	−1.10041
$934$	−6.24179	−0.204238
$935$	4.71199	0.154099
$936$	0	0
$937$	−35.5607	−1.16172	−0.580859	−	0.814004i	$-0.697283\pi$
$937$	−35.5607	−1.16172	−0.580859	+	0.814004i	$0.697283\pi$
$938$	43.8463	1.43163
$939$	−22.3892	−0.730645
$940$	13.1960	0.430406
$941$	−51.1511	−1.66748	−0.833740	−	0.552157i	$-0.813805\pi$
$941$	−51.1511	−1.66748	−0.833740	+	0.552157i	$0.813805\pi$
$942$	−6.56184	−0.213796
$943$	−15.8983	−0.517719
$944$	5.34832	0.174073
$945$	−47.1842	−1.53490
$946$	2.00079	0.0650512
$947$	32.8872	1.06869	0.534345	−	0.845267i	$-0.320558\pi$
$947$	32.8872	1.06869	0.534345	+	0.845267i	$0.320558\pi$
$948$	−5.74361	−0.186544
$949$	0	0
$950$	−0.905576	−0.0293808
$951$	−10.2760	−0.333222
$952$	−2.89848	−0.0939403
$953$	48.0653	1.55699	0.778494	−	0.627652i	$-0.215984\pi$
$953$	48.0653	1.55699	0.778494	+	0.627652i	$0.215984\pi$
$954$	18.8826	0.611347
$955$	1.42784	0.0462037
$956$	13.9540	0.451305
$957$	−0.681688	−0.0220358
$958$	−2.77283	−0.0895862
$959$	−35.1848	−1.13618
$960$	−2.75481	−0.0889112
$961$	2.98620	0.0963290
$962$	0	0
$963$	−21.8518	−0.704164
$964$	−16.2416	−0.523106
$965$	−9.28965	−0.299044
$966$	7.45856	0.239975
$967$	27.3359	0.879062	0.439531	−	0.898227i	$-0.355145\pi$
$967$	27.3359	0.879062	0.439531	+	0.898227i	$0.355145\pi$
$968$	−5.09442	−0.163741
$969$	0.904489	0.0290564
$970$	41.8571	1.34395
$971$	−14.1785	−0.455011	−0.227505	−	0.973777i	$-0.573057\pi$
$971$	−14.1785	−0.455011	−0.227505	+	0.973777i	$0.573057\pi$
$972$	−14.9984	−0.481072
$973$	−44.9550	−1.44119
$974$	−35.2602	−1.12981
$975$	0	0
$976$	7.97713	0.255342
$977$	−28.8515	−0.923040	−0.461520	−	0.887130i	$-0.652696\pi$
$977$	−28.8515	−0.923040	−0.461520	+	0.887130i	$0.652696\pi$
$978$	−19.8459	−0.634603
$979$	38.1246	1.21847
$980$	15.0582	0.481016
$981$	15.8102	0.504782
$982$	41.1833	1.31421
$983$	−19.2569	−0.614199	−0.307100	−	0.951677i	$-0.599358\pi$
$983$	−19.2569	−0.614199	−0.307100	+	0.951677i	$0.599358\pi$
$984$	9.95051	0.317211
$985$	25.6381	0.816899
$986$	0.197440	0.00628778
$987$	22.3615	0.711773
$988$	0	0
$989$	1.49120	0.0474173
$990$	−10.1277	−0.321880
$991$	−19.1072	−0.606960	−0.303480	−	0.952838i	$-0.598149\pi$
$991$	−19.1072	−0.606960	−0.303480	+	0.952838i	$0.598149\pi$
$992$	5.82977	0.185095
$993$	37.6364	1.19436
$994$	−8.51144	−0.269967
$995$	−8.61009	−0.272958
$996$	2.14073	0.0678315
$997$	−17.0699	−0.540609	−0.270304	−	0.962775i	$-0.587124\pi$
$997$	−17.0699	−0.540609	−0.270304	+	0.962775i	$0.587124\pi$
$998$	15.0167	0.475345
$999$	17.1114	0.541382

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6422.2.a.bd.1.2		6
13.3	even	3		494.2.g.f.191.5	✓	12
13.9	even	3		494.2.g.f.419.5	yes	12
13.12	even	2		6422.2.a.bc.1.2		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
494.2.g.f.191.5	✓	12	13.3	even	3
494.2.g.f.419.5	yes	12	13.9	even	3
6422.2.a.bc.1.2		6	13.12	even	2
6422.2.a.bd.1.2		6	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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -1.13360 q^{3} +1.00000 q^{4} +2.43014 q^{5} -1.13360 q^{6} -3.63269 q^{7} +1.00000 q^{8} -1.71494 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} -1.13360 q^{3} +1.00000 q^{4} +2.43014 q^{5} -1.13360 q^{6} -3.63269 q^{7} +1.00000 q^{8} -1.71494 q^{9} +2.43014 q^{10} +2.43014 q^{11} -1.13360 q^{12} -3.63269 q^{14} -2.75481 q^{15} +1.00000 q^{16} +0.797889 q^{17} -1.71494 q^{18} -1.00000 q^{19} +2.43014 q^{20} +4.11803 q^{21} +2.43014 q^{22} +1.81120 q^{23} -1.13360 q^{24} +0.905576 q^{25} +5.34488 q^{27} -3.63269 q^{28} +0.247453 q^{29} -2.75481 q^{30} +5.82977 q^{31} +1.00000 q^{32} -2.75481 q^{33} +0.797889 q^{34} -8.82794 q^{35} -1.71494 q^{36} +3.20146 q^{37} -1.00000 q^{38} +2.43014 q^{40} -8.77777 q^{41} +4.11803 q^{42} +0.823322 q^{43} +2.43014 q^{44} -4.16755 q^{45} +1.81120 q^{46} +5.43014 q^{47} -1.13360 q^{48} +6.19643 q^{49} +0.905576 q^{50} -0.904489 q^{51} -11.0106 q^{53} +5.34488 q^{54} +5.90558 q^{55} -3.63269 q^{56} +1.13360 q^{57} +0.247453 q^{58} +5.34832 q^{59} -2.75481 q^{60} +7.97713 q^{61} +5.82977 q^{62} +6.22986 q^{63} +1.00000 q^{64} -2.75481 q^{66} -12.0699 q^{67} +0.797889 q^{68} -2.05318 q^{69} -8.82794 q^{70} +2.34301 q^{71} -1.71494 q^{72} +6.56148 q^{73} +3.20146 q^{74} -1.02656 q^{75} -1.00000 q^{76} -8.82794 q^{77} +5.06668 q^{79} +2.43014 q^{80} -0.914140 q^{81} -8.77777 q^{82} -1.88843 q^{83} +4.11803 q^{84} +1.93898 q^{85} +0.823322 q^{86} -0.280514 q^{87} +2.43014 q^{88} +15.6882 q^{89} -4.16755 q^{90} +1.81120 q^{92} -6.60865 q^{93} +5.43014 q^{94} -2.43014 q^{95} -1.13360 q^{96} +17.2242 q^{97} +6.19643 q^{98} -4.16755 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 6 q^{2} + 2 q^{3} + 6 q^{4} + 2 q^{5} + 2 q^{6} + q^{7} + 6 q^{8} + 10 q^{9}+O(q^{10}) \) 6 * q + 6 * q^2 + 2 * q^3 + 6 * q^4 + 2 * q^5 + 2 * q^6 + q^7 + 6 * q^8 + 10 * q^9	\( 6 q + 6 q^{2} + 2 q^{3} + 6 q^{4} + 2 q^{5} + 2 q^{6} + q^{7} + 6 q^{8} + 10 q^{9} + 2 q^{10} + 2 q^{11} + 2 q^{12} + q^{14} - 9 q^{15} + 6 q^{16} - 2 q^{17} + 10 q^{18} - 6 q^{19} + 2 q^{20} + q^{21} + 2 q^{22} + 8 q^{23} + 2 q^{24} + 16 q^{25} - 4 q^{27} + q^{28} + 20 q^{29} - 9 q^{30} + 3 q^{31} + 6 q^{32} - 9 q^{33} - 2 q^{34} + 7 q^{35} + 10 q^{36} - 9 q^{37} - 6 q^{38} + 2 q^{40} + 3 q^{41} + q^{42} + 13 q^{43} + 2 q^{44} + 38 q^{45} + 8 q^{46} + 20 q^{47} + 2 q^{48} - 7 q^{49} + 16 q^{50} + 2 q^{51} + 25 q^{53} - 4 q^{54} + 46 q^{55} + q^{56} - 2 q^{57} + 20 q^{58} - 9 q^{60} + 6 q^{61} + 3 q^{62} + 46 q^{63} + 6 q^{64} - 9 q^{66} - 32 q^{67} - 2 q^{68} - 29 q^{69} + 7 q^{70} + 39 q^{71} + 10 q^{72} + 7 q^{73} - 9 q^{74} - 15 q^{75} - 6 q^{76} + 7 q^{77} + 18 q^{79} + 2 q^{80} + 54 q^{81} + 3 q^{82} - 7 q^{83} + q^{84} - 2 q^{85} + 13 q^{86} + 12 q^{87} + 2 q^{88} + 9 q^{89} + 38 q^{90} + 8 q^{92} - 47 q^{93} + 20 q^{94} - 2 q^{95} + 2 q^{96} - 4 q^{97} - 7 q^{98} + 38 q^{99}+O(q^{100}) \) 6 * q + 6 * q^2 + 2 * q^3 + 6 * q^4 + 2 * q^5 + 2 * q^6 + q^7 + 6 * q^8 + 10 * q^9 + 2 * q^10 + 2 * q^11 + 2 * q^12 + q^14 - 9 * q^15 + 6 * q^16 - 2 * q^17 + 10 * q^18 - 6 * q^19 + 2 * q^20 + q^21 + 2 * q^22 + 8 * q^23 + 2 * q^24 + 16 * q^25 - 4 * q^27 + q^28 + 20 * q^29 - 9 * q^30 + 3 * q^31 + 6 * q^32 - 9 * q^33 - 2 * q^34 + 7 * q^35 + 10 * q^36 - 9 * q^37 - 6 * q^38 + 2 * q^40 + 3 * q^41 + q^42 + 13 * q^43 + 2 * q^44 + 38 * q^45 + 8 * q^46 + 20 * q^47 + 2 * q^48 - 7 * q^49 + 16 * q^50 + 2 * q^51 + 25 * q^53 - 4 * q^54 + 46 * q^55 + q^56 - 2 * q^57 + 20 * q^58 - 9 * q^60 + 6 * q^61 + 3 * q^62 + 46 * q^63 + 6 * q^64 - 9 * q^66 - 32 * q^67 - 2 * q^68 - 29 * q^69 + 7 * q^70 + 39 * q^71 + 10 * q^72 + 7 * q^73 - 9 * q^74 - 15 * q^75 - 6 * q^76 + 7 * q^77 + 18 * q^79 + 2 * q^80 + 54 * q^81 + 3 * q^82 - 7 * q^83 + q^84 - 2 * q^85 + 13 * q^86 + 12 * q^87 + 2 * q^88 + 9 * q^89 + 38 * q^90 + 8 * q^92 - 47 * q^93 + 20 * q^94 - 2 * q^95 + 2 * q^96 - 4 * q^97 - 7 * q^98 + 38 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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