LMFDB - Embedded newform 6422.2.a.w.1.3

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:	$3$
Coefficient field:	3.3.361.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 6x + 7 $ x^3 - x^2 - 6*x + 7
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			1.3
Root			$-2.50702$ 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)

$f(q)$	$=$	$q+1.00000 q^{2} +2.50702 q^{3} +1.00000 q^{4} -2.28514 q^{5} +2.50702 q^{6} -2.44375 q^{7} +1.00000 q^{8} +3.28514 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} +2.50702 q^{3} +1.00000 q^{4} -2.28514 q^{5} +2.50702 q^{6} -2.44375 q^{7} +1.00000 q^{8} +3.28514 q^{9} -2.28514 q^{10} +1.22188 q^{11} +2.50702 q^{12} -2.44375 q^{14} -5.72889 q^{15} +1.00000 q^{16} -4.50702 q^{17} +3.28514 q^{18} +1.00000 q^{19} -2.28514 q^{20} -6.12653 q^{21} +1.22188 q^{22} -3.34841 q^{23} +2.50702 q^{24} +0.221876 q^{25} +0.714858 q^{27} -2.44375 q^{28} -4.23591 q^{29} -5.72889 q^{30} +1.29918 q^{31} +1.00000 q^{32} +3.06327 q^{33} -4.50702 q^{34} +5.58432 q^{35} +3.28514 q^{36} -0.985963 q^{37} +1.00000 q^{38} -2.28514 q^{40} -1.93673 q^{41} -6.12653 q^{42} -4.44375 q^{43} +1.22188 q^{44} -7.50702 q^{45} -3.34841 q^{46} +0.443752 q^{47} +2.50702 q^{48} -1.02807 q^{49} +0.221876 q^{50} -11.2992 q^{51} +8.72889 q^{53} +0.714858 q^{54} -2.79216 q^{55} -2.44375 q^{56} +2.50702 q^{57} -4.23591 q^{58} -4.57028 q^{59} -5.72889 q^{60} +0.443752 q^{61} +1.29918 q^{62} -8.02807 q^{63} +1.00000 q^{64} +3.06327 q^{66} -13.1406 q^{67} -4.50702 q^{68} -8.39452 q^{69} +5.58432 q^{70} +4.41168 q^{71} +3.28514 q^{72} -8.69682 q^{73} -0.985963 q^{74} +0.556248 q^{75} +1.00000 q^{76} -2.98596 q^{77} +14.0281 q^{79} -2.28514 q^{80} -8.06327 q^{81} -1.93673 q^{82} +4.18980 q^{83} -6.12653 q^{84} +10.2992 q^{85} -4.44375 q^{86} -10.6195 q^{87} +1.22188 q^{88} -1.22188 q^{89} -7.50702 q^{90} -3.34841 q^{92} +3.25707 q^{93} +0.443752 q^{94} -2.28514 q^{95} +2.50702 q^{96} -11.6656 q^{97} -1.02807 q^{98} +4.01404 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{2} - q^{3} + 3 q^{4} - q^{5} - q^{6} - 2 q^{7} + 3 q^{8} + 4 q^{9}+O(q^{10}) $ 3 * q + 3 * q^2 - q^3 + 3 * q^4 - q^5 - q^6 - 2 * q^7 + 3 * q^8 + 4 * q^9	$ 3 q + 3 q^{2} - q^{3} + 3 q^{4} - q^{5} - q^{6} - 2 q^{7} + 3 q^{8} + 4 q^{9} - q^{10} + q^{11} - q^{12} - 2 q^{14} - 6 q^{15} + 3 q^{16} - 5 q^{17} + 4 q^{18} + 3 q^{19} - q^{20} - 12 q^{21} + q^{22} - q^{23} - q^{24} - 2 q^{25} + 8 q^{27} - 2 q^{28} + 7 q^{29} - 6 q^{30} - 19 q^{31} + 3 q^{32} + 6 q^{33} - 5 q^{34} - 12 q^{35} + 4 q^{36} - 20 q^{37} + 3 q^{38} - q^{40} - 9 q^{41} - 12 q^{42} - 8 q^{43} + q^{44} - 14 q^{45} - q^{46} - 4 q^{47} - q^{48} + 31 q^{49} - 2 q^{50} - 11 q^{51} + 15 q^{53} + 8 q^{54} + 6 q^{55} - 2 q^{56} - q^{57} + 7 q^{58} - 2 q^{59} - 6 q^{60} - 4 q^{61} - 19 q^{62} + 10 q^{63} + 3 q^{64} + 6 q^{66} - 16 q^{67} - 5 q^{68} - 6 q^{69} - 12 q^{70} + q^{71} + 4 q^{72} - 8 q^{73} - 20 q^{74} + 7 q^{75} + 3 q^{76} - 26 q^{77} + 8 q^{79} - q^{80} - 21 q^{81} - 9 q^{82} + 3 q^{83} - 12 q^{84} + 8 q^{85} - 8 q^{86} - 34 q^{87} + q^{88} - q^{89} - 14 q^{90} - q^{92} + 38 q^{93} - 4 q^{94} - q^{95} - q^{96} - 27 q^{97} + 31 q^{98} - 5 q^{99}+O(q^{100}) $ 3 * q + 3 * q^2 - q^3 + 3 * q^4 - q^5 - q^6 - 2 * q^7 + 3 * q^8 + 4 * q^9 - q^10 + q^11 - q^12 - 2 * q^14 - 6 * q^15 + 3 * q^16 - 5 * q^17 + 4 * q^18 + 3 * q^19 - q^20 - 12 * q^21 + q^22 - q^23 - q^24 - 2 * q^25 + 8 * q^27 - 2 * q^28 + 7 * q^29 - 6 * q^30 - 19 * q^31 + 3 * q^32 + 6 * q^33 - 5 * q^34 - 12 * q^35 + 4 * q^36 - 20 * q^37 + 3 * q^38 - q^40 - 9 * q^41 - 12 * q^42 - 8 * q^43 + q^44 - 14 * q^45 - q^46 - 4 * q^47 - q^48 + 31 * q^49 - 2 * q^50 - 11 * q^51 + 15 * q^53 + 8 * q^54 + 6 * q^55 - 2 * q^56 - q^57 + 7 * q^58 - 2 * q^59 - 6 * q^60 - 4 * q^61 - 19 * q^62 + 10 * q^63 + 3 * q^64 + 6 * q^66 - 16 * q^67 - 5 * q^68 - 6 * q^69 - 12 * q^70 + q^71 + 4 * q^72 - 8 * q^73 - 20 * q^74 + 7 * q^75 + 3 * q^76 - 26 * q^77 + 8 * q^79 - q^80 - 21 * q^81 - 9 * q^82 + 3 * q^83 - 12 * q^84 + 8 * q^85 - 8 * q^86 - 34 * q^87 + q^88 - q^89 - 14 * q^90 - q^92 + 38 * q^93 - 4 * q^94 - q^95 - q^96 - 27 * q^97 + 31 * q^98 - 5 * 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
$3$	2.50702	1.44743	0.723714	−	0.690100i	$-0.242434\pi$
$3$	2.50702	1.44743	0.723714	+	0.690100i	$0.242434\pi$
$4$	1.00000	0.500000
$5$	−2.28514	−1.02195	−0.510973	−	0.859597i	$-0.670715\pi$
$5$	−2.28514	−1.02195	−0.510973	+	0.859597i	$0.670715\pi$
$6$	2.50702	1.02349
$7$	−2.44375	−0.923652	−0.461826	−	0.886971i	$-0.652805\pi$
$7$	−2.44375	−0.923652	−0.461826	+	0.886971i	$0.652805\pi$
$8$	1.00000	0.353553
$9$	3.28514	1.09505
$10$	−2.28514	−0.722626
$11$	1.22188	0.368410	0.184205	−	0.982888i	$-0.441029\pi$
$11$	1.22188	0.368410	0.184205	+	0.982888i	$0.441029\pi$
$12$	2.50702	0.723714
$13$	0	0
$13$	0	0
$14$	−2.44375	−0.653120
$15$	−5.72889	−1.47919
$16$	1.00000	0.250000
$17$	−4.50702	−1.09311	−0.546556	−	0.837422i	$-0.684062\pi$
$17$	−4.50702	−1.09311	−0.546556	+	0.837422i	$0.684062\pi$
$18$	3.28514	0.774316
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	−2.28514	−0.510973
$21$	−6.12653	−1.33692
$22$	1.22188	0.260505
$23$	−3.34841	−0.698191	−0.349096	−	0.937087i	$-0.613511\pi$
$23$	−3.34841	−0.698191	−0.349096	+	0.937087i	$0.613511\pi$
$24$	2.50702	0.511743
$25$	0.221876	0.0443752
$26$	0	0
$27$	0.714858	0.137574
$28$	−2.44375	−0.461826
$29$	−4.23591	−0.786589	−0.393295	−	0.919412i	$-0.628665\pi$
$29$	−4.23591	−0.786589	−0.393295	+	0.919412i	$0.628665\pi$
$30$	−5.72889	−1.04595
$31$	1.29918	0.233340	0.116670	−	0.993171i	$-0.462778\pi$
$31$	1.29918	0.233340	0.116670	+	0.993171i	$0.462778\pi$
$32$	1.00000	0.176777
$33$	3.06327	0.533246
$34$	−4.50702	−0.772947
$35$	5.58432	0.943923
$36$	3.28514	0.547524
$37$	−0.985963	−0.162091	−0.0810456	−	0.996710i	$-0.525826\pi$
$37$	−0.985963	−0.162091	−0.0810456	+	0.996710i	$0.525826\pi$
$38$	1.00000	0.162221
$39$	0	0
$40$	−2.28514	−0.361313
$41$	−1.93673	−0.302467	−0.151233	−	0.988498i	$-0.548325\pi$
$41$	−1.93673	−0.302467	−0.151233	+	0.988498i	$0.548325\pi$
$42$	−6.12653	−0.945345
$43$	−4.44375	−0.677666	−0.338833	−	0.940847i	$-0.610032\pi$
$43$	−4.44375	−0.677666	−0.338833	+	0.940847i	$0.610032\pi$
$44$	1.22188	0.184205
$45$	−7.50702	−1.11908
$46$	−3.34841	−0.493696
$47$	0.443752	0.0647279	0.0323640	−	0.999476i	$-0.489696\pi$
$47$	0.443752	0.0647279	0.0323640	+	0.999476i	$0.489696\pi$
$48$	2.50702	0.361857
$49$	−1.02807	−0.146868
$50$	0.221876	0.0313780
$51$	−11.2992	−1.58220
$52$	0	0
$53$	8.72889	1.19901	0.599503	−	0.800373i	$-0.295365\pi$
$53$	8.72889	1.19901	0.599503	+	0.800373i	$0.295365\pi$
$54$	0.714858	0.0972798
$55$	−2.79216	−0.376495
$56$	−2.44375	−0.326560
$57$	2.50702	0.332063
$58$	−4.23591	−0.556203
$59$	−4.57028	−0.595000	−0.297500	−	0.954722i	$-0.596153\pi$
$59$	−4.57028	−0.595000	−0.297500	+	0.954722i	$0.596153\pi$
$60$	−5.72889	−0.739597
$61$	0.443752	0.0568167	0.0284083	−	0.999596i	$-0.490956\pi$
$61$	0.443752	0.0568167	0.0284083	+	0.999596i	$0.490956\pi$
$62$	1.29918	0.164996
$63$	−8.02807	−1.01144
$64$	1.00000	0.125000
$65$	0	0
$66$	3.06327	0.377062
$67$	−13.1406	−1.60538	−0.802688	−	0.596399i	$-0.796598\pi$
$67$	−13.1406	−1.60538	−0.802688	+	0.596399i	$0.796598\pi$
$68$	−4.50702	−0.546556
$69$	−8.39452	−1.01058
$70$	5.58432	0.667454
$71$	4.41168	0.523570	0.261785	−	0.965126i	$-0.415689\pi$
$71$	4.41168	0.523570	0.261785	+	0.965126i	$0.415689\pi$
$72$	3.28514	0.387158
$73$	−8.69682	−1.01789	−0.508943	−	0.860800i	$-0.669964\pi$
$73$	−8.69682	−1.01789	−0.508943	+	0.860800i	$0.669964\pi$
$74$	−0.985963	−0.114616
$75$	0.556248	0.0642299
$76$	1.00000	0.114708
$77$	−2.98596	−0.340282
$78$	0	0
$79$	14.0281	1.57828	0.789141	−	0.614212i	$-0.210526\pi$
$79$	14.0281	1.57828	0.789141	+	0.614212i	$0.210526\pi$
$80$	−2.28514	−0.255487
$81$	−8.06327	−0.895918
$82$	−1.93673	−0.213876
$83$	4.18980	0.459890	0.229945	−	0.973204i	$-0.426145\pi$
$83$	4.18980	0.459890	0.229945	+	0.973204i	$0.426145\pi$
$84$	−6.12653	−0.668460
$85$	10.2992	1.11710
$86$	−4.44375	−0.479182
$87$	−10.6195	−1.13853
$88$	1.22188	0.130252
$89$	−1.22188	−0.129519	−0.0647593	−	0.997901i	$-0.520628\pi$
$89$	−1.22188	−0.129519	−0.0647593	+	0.997901i	$0.520628\pi$
$90$	−7.50702	−0.791309
$91$	0	0
$92$	−3.34841	−0.349096
$93$	3.25707	0.337742
$94$	0.443752	0.0457696
$95$	−2.28514	−0.234451
$96$	2.50702	0.255872
$97$	−11.6656	−1.18447	−0.592233	−	0.805767i	$-0.701753\pi$
$97$	−11.6656	−1.18447	−0.592233	+	0.805767i	$0.701753\pi$
$98$	−1.02807	−0.103851
$99$	4.01404	0.403426
$100$	0.221876	0.0221876
$101$	−14.1546	−1.40844	−0.704218	−	0.709984i	$-0.748702\pi$
$101$	−14.1546	−1.40844	−0.704218	+	0.709984i	$0.748702\pi$
$102$	−11.2992	−1.11879
$103$	2.69682	0.265725	0.132863	−	0.991134i	$-0.457583\pi$
$103$	2.69682	0.265725	0.132863	+	0.991134i	$0.457583\pi$
$104$	0	0
$105$	14.0000	1.36626
$106$	8.72889	0.847825
$107$	−18.7289	−1.81059	−0.905295	−	0.424783i	$-0.860350\pi$
$107$	−18.7289	−1.81059	−0.905295	+	0.424783i	$0.860350\pi$
$108$	0.714858	0.0687872
$109$	−7.01404	−0.671823	−0.335911	−	0.941894i	$-0.609044\pi$
$109$	−7.01404	−0.671823	−0.335911	+	0.941894i	$0.609044\pi$
$110$	−2.79216	−0.266222
$111$	−2.47183	−0.234615
$112$	−2.44375	−0.230913
$113$	4.57028	0.429936	0.214968	−	0.976621i	$-0.431035\pi$
$113$	4.57028	0.429936	0.214968	+	0.976621i	$0.431035\pi$
$114$	2.50702	0.234804
$115$	7.65159	0.713515
$116$	−4.23591	−0.393295
$117$	0	0
$118$	−4.57028	−0.420729
$119$	11.0140	1.00966
$120$	−5.72889	−0.522974
$121$	−9.50702	−0.864274
$122$	0.443752	0.0401754
$123$	−4.85543	−0.437799
$124$	1.29918	0.116670
$125$	10.9187	0.976598
$126$	−8.02807	−0.715198
$127$	−22.4718	−1.99405	−0.997026	−	0.0770647i	$-0.975445\pi$
$127$	−22.4718	−1.99405	−0.997026	+	0.0770647i	$0.975445\pi$
$128$	1.00000	0.0883883
$129$	−11.1406	−0.980872
$130$	0	0
$131$	−8.47183	−0.740187	−0.370093	−	0.928995i	$-0.620674\pi$
$131$	−8.47183	−0.740187	−0.370093	+	0.928995i	$0.620674\pi$
$132$	3.06327	0.266623
$133$	−2.44375	−0.211900
$134$	−13.1406	−1.13517
$135$	−1.63355	−0.140594
$136$	−4.50702	−0.386474
$137$	−0.668743	−0.0571346	−0.0285673	−	0.999592i	$-0.509094\pi$
$137$	−0.668743	−0.0571346	−0.0285673	+	0.999592i	$0.509094\pi$
$138$	−8.39452	−0.714589
$139$	15.2671	1.29494	0.647469	−	0.762091i	$-0.275827\pi$
$139$	15.2671	1.29494	0.647469	+	0.762091i	$0.275827\pi$
$140$	5.58432	0.471961
$141$	1.11250	0.0936890
$142$	4.41168	0.370220
$143$	0	0
$144$	3.28514	0.273762
$145$	9.67967	0.803852
$146$	−8.69682	−0.719754
$147$	−2.57740	−0.212581
$148$	−0.985963	−0.0810456
$149$	−1.96481	−0.160963	−0.0804817	−	0.996756i	$-0.525646\pi$
$149$	−1.96481	−0.160963	−0.0804817	+	0.996756i	$0.525646\pi$
$150$	0.556248	0.0454174
$151$	23.8483	1.94075	0.970374	−	0.241608i	$-0.0776747\pi$
$151$	23.8483	1.94075	0.970374	+	0.241608i	$0.0776747\pi$
$152$	1.00000	0.0811107
$153$	−14.8062	−1.19701
$154$	−2.98596	−0.240616
$155$	−2.96881	−0.238461
$156$	0	0
$157$	13.1406	1.04873	0.524366	−	0.851493i	$-0.324302\pi$
$157$	13.1406	1.04873	0.524366	+	0.851493i	$0.324302\pi$
$158$	14.0281	1.11601
$159$	21.8835	1.73547
$160$	−2.28514	−0.180656
$161$	8.18268	0.644886
$162$	−8.06327	−0.633510
$163$	−4.06327	−0.318260	−0.159130	−	0.987258i	$-0.550869\pi$
$163$	−4.06327	−0.318260	−0.159130	+	0.987258i	$0.550869\pi$
$164$	−1.93673	−0.151233
$165$	−7.00000	−0.544949
$166$	4.18980	0.325191
$167$	−1.93673	−0.149869	−0.0749345	−	0.997188i	$-0.523875\pi$
$167$	−1.93673	−0.149869	−0.0749345	+	0.997188i	$0.523875\pi$
$168$	−6.12653	−0.472672
$169$	0	0
$170$	10.2992	0.789911
$171$	3.28514	0.251221
$172$	−4.44375	−0.338833
$173$	−5.61951	−0.427244	−0.213622	−	0.976916i	$-0.568526\pi$
$173$	−5.61951	−0.427244	−0.213622	+	0.976916i	$0.568526\pi$
$174$	−10.6195	−0.805063
$175$	−0.542210	−0.0409873
$176$	1.22188	0.0921024
$177$	−11.4578	−0.861220
$178$	−1.22188	−0.0915835
$179$	17.3273	1.29510	0.647550	−	0.762023i	$-0.275794\pi$
$179$	17.3273	1.29510	0.647550	+	0.762023i	$0.275794\pi$
$180$	−7.50702	−0.559540
$181$	20.4086	1.51696	0.758478	−	0.651698i	$-0.225943\pi$
$181$	20.4086	1.51696	0.758478	+	0.651698i	$0.225943\pi$
$182$	0	0
$183$	1.11250	0.0822380
$184$	−3.34841	−0.246848
$185$	2.25307	0.165649
$186$	3.25707	0.238820
$187$	−5.50702	−0.402713
$188$	0.443752	0.0323640
$189$	−1.74693	−0.127071
$190$	−2.28514	−0.165782
$191$	7.99600	0.578570	0.289285	−	0.957243i	$-0.406582\pi$
$191$	7.99600	0.578570	0.289285	+	0.957243i	$0.406582\pi$
$192$	2.50702	0.180928
$193$	−7.83828	−0.564211	−0.282106	−	0.959383i	$-0.591033\pi$
$193$	−7.83828	−0.564211	−0.282106	+	0.959383i	$0.591033\pi$
$194$	−11.6656	−0.837543
$195$	0	0
$196$	−1.02807	−0.0734339
$197$	7.14057	0.508745	0.254372	−	0.967106i	$-0.418131\pi$
$197$	7.14057	0.508745	0.254372	+	0.967106i	$0.418131\pi$
$198$	4.01404	0.285265
$199$	−21.6295	−1.53328	−0.766639	−	0.642078i	$-0.778072\pi$
$199$	−21.6295	−1.53328	−0.766639	+	0.642078i	$0.778072\pi$
$200$	0.221876	0.0156890
$201$	−32.9437	−2.32367
$202$	−14.1546	−0.995915
$203$	10.3515	0.726535
$204$	−11.2992	−0.791101
$205$	4.42571	0.309105
$206$	2.69682	0.187896
$207$	−11.0000	−0.764553
$208$	0	0
$209$	1.22188	0.0845189
$210$	14.0000	0.966092
$211$	6.08131	0.418654	0.209327	−	0.977846i	$-0.432873\pi$
$211$	6.08131	0.418654	0.209327	+	0.977846i	$0.432873\pi$
$212$	8.72889	0.599503
$213$	11.0602	0.757829
$214$	−18.7289	−1.28028
$215$	10.1546	0.692538
$216$	0.714858	0.0486399
$217$	−3.17487	−0.215524
$218$	−7.01404	−0.475051
$219$	−21.8031	−1.47332
$220$	−2.79216	−0.188247
$221$	0	0
$222$	−2.47183	−0.165898
$223$	12.0913	0.809696	0.404848	−	0.914384i	$-0.367324\pi$
$223$	12.0913	0.809696	0.404848	+	0.914384i	$0.367324\pi$
$224$	−2.44375	−0.163280
$225$	0.728895	0.0485930
$226$	4.57028	0.304011
$227$	29.5139	1.95891	0.979454	−	0.201665i	$-0.0646354\pi$
$227$	29.5139	1.95891	0.979454	+	0.201665i	$0.0646354\pi$
$228$	2.50702	0.166031
$229$	−15.7922	−1.04358	−0.521788	−	0.853075i	$-0.674735\pi$
$229$	−15.7922	−1.04358	−0.521788	+	0.853075i	$0.674735\pi$
$230$	7.65159	0.504531
$231$	−7.48586	−0.492534
$232$	−4.23591	−0.278101
$233$	23.7710	1.55729	0.778645	−	0.627464i	$-0.215907\pi$
$233$	23.7710	1.55729	0.778645	+	0.627464i	$0.215907\pi$
$234$	0	0
$235$	−1.01404	−0.0661485
$236$	−4.57028	−0.297500
$237$	35.1686	2.28445
$238$	11.0140	0.713934
$239$	−25.1406	−1.62621	−0.813104	−	0.582118i	$-0.802224\pi$
$239$	−25.1406	−1.62621	−0.813104	+	0.582118i	$0.802224\pi$
$240$	−5.72889	−0.369799
$241$	−23.5522	−1.51713	−0.758567	−	0.651595i	$-0.774100\pi$
$241$	−23.5522	−1.51713	−0.758567	+	0.651595i	$0.774100\pi$
$242$	−9.50702	−0.611134
$243$	−22.3593	−1.43435
$244$	0.443752	0.0284083
$245$	2.34930	0.150091
$246$	−4.85543	−0.309571
$247$	0	0
$248$	1.29918	0.0824980
$249$	10.5039	0.665658
$250$	10.9187	0.690559
$251$	24.6264	1.55441	0.777203	−	0.629249i	$-0.216638\pi$
$251$	24.6264	1.55441	0.777203	+	0.629249i	$0.216638\pi$
$252$	−8.02807	−0.505721
$253$	−4.09134	−0.257220
$254$	−22.4718	−1.41001
$255$	25.8202	1.61693
$256$	1.00000	0.0625000
$257$	20.0000	1.24757	0.623783	−	0.781598i	$-0.285595\pi$
$257$	20.0000	1.24757	0.623783	+	0.781598i	$0.285595\pi$
$258$	−11.1406	−0.693582
$259$	2.40945	0.149716
$260$	0	0
$261$	−13.9156	−0.861353
$262$	−8.47183	−0.523391
$263$	−20.6336	−1.27232	−0.636160	−	0.771557i	$-0.719478\pi$
$263$	−20.6336	−1.27232	−0.636160	+	0.771557i	$0.719478\pi$
$264$	3.06327	0.188531
$265$	−19.9468	−1.22532
$266$	−2.44375	−0.149836
$267$	−3.06327	−0.187469
$268$	−13.1406	−0.802688
$269$	−4.02807	−0.245596	−0.122798	−	0.992432i	$-0.539187\pi$
$269$	−4.02807	−0.245596	−0.122798	+	0.992432i	$0.539187\pi$
$270$	−1.63355	−0.0994148
$271$	27.3593	1.66196	0.830981	−	0.556302i	$-0.187780\pi$
$271$	27.3593	1.66196	0.830981	+	0.556302i	$0.187780\pi$
$272$	−4.50702	−0.273278
$273$	0	0
$274$	−0.668743	−0.0404002
$275$	0.271105	0.0163483
$276$	−8.39452	−0.505291
$277$	1.55625	0.0935059	0.0467529	−	0.998906i	$-0.485113\pi$
$277$	1.55625	0.0935059	0.0467529	+	0.998906i	$0.485113\pi$
$278$	15.2671	0.915660
$279$	4.26799	0.255518
$280$	5.58432	0.333727
$281$	−16.3344	−0.974427	−0.487213	−	0.873283i	$-0.661987\pi$
$281$	−16.3344	−0.974427	−0.487213	+	0.873283i	$0.661987\pi$
$282$	1.11250	0.0662481
$283$	−25.3313	−1.50579	−0.752893	−	0.658142i	$-0.771342\pi$
$283$	−25.3313	−1.50579	−0.752893	+	0.658142i	$0.771342\pi$
$284$	4.41168	0.261785
$285$	−5.72889	−0.339350
$286$	0	0
$287$	4.73290	0.279374
$288$	3.28514	0.193579
$289$	3.31322	0.194895
$290$	9.67967	0.568410
$291$	−29.2459	−1.71443
$292$	−8.69682	−0.508943
$293$	14.3453	0.838061	0.419031	−	0.907972i	$-0.362370\pi$
$293$	14.3453	0.838061	0.419031	+	0.907972i	$0.362370\pi$
$294$	−2.57740	−0.150317
$295$	10.4438	0.608059
$296$	−0.985963	−0.0573079
$297$	0.873467	0.0506837
$298$	−1.96481	−0.113818
$299$	0	0
$300$	0.556248	0.0321150
$301$	10.8594	0.625927
$302$	23.8483	1.37232
$303$	−35.4859	−2.03861
$304$	1.00000	0.0573539
$305$	−1.01404	−0.0580636
$306$	−14.8062	−0.846414
$307$	−29.1686	−1.66474	−0.832371	−	0.554218i	$-0.813017\pi$
$307$	−29.1686	−1.66474	−0.832371	+	0.554218i	$0.813017\pi$
$308$	−2.98596	−0.170141
$309$	6.76097	0.384618
$310$	−2.96881	−0.168617
$311$	−20.5070	−1.16285	−0.581423	−	0.813601i	$-0.697504\pi$
$311$	−20.5070	−1.16285	−0.581423	+	0.813601i	$0.697504\pi$
$312$	0	0
$313$	12.0321	0.680093	0.340047	−	0.940409i	$-0.389557\pi$
$313$	12.0321	0.680093	0.340047	+	0.940409i	$0.389557\pi$
$314$	13.1406	0.741565
$315$	18.3453	1.03364
$316$	14.0281	0.789141
$317$	−21.1686	−1.18895	−0.594475	−	0.804114i	$-0.702640\pi$
$317$	−21.1686	−1.18895	−0.594475	+	0.804114i	$0.702640\pi$
$318$	21.8835	1.22717
$319$	−5.17576	−0.289787
$320$	−2.28514	−0.127743
$321$	−46.9537	−2.62070
$322$	8.18268	0.456003
$323$	−4.50702	−0.250777
$324$	−8.06327	−0.447959
$325$	0	0
$326$	−4.06327	−0.225044
$327$	−17.5843	−0.972415
$328$	−1.93673	−0.106938
$329$	−1.08442	−0.0597861
$330$	−7.00000	−0.385337
$331$	−1.97193	−0.108387	−0.0541934	−	0.998530i	$-0.517259\pi$
$331$	−1.97193	−0.108387	−0.0541934	+	0.998530i	$0.517259\pi$
$332$	4.18980	0.229945
$333$	−3.23903	−0.177498
$334$	−1.93673	−0.105973
$335$	30.0281	1.64061
$336$	−6.12653	−0.334230
$337$	−0.0984581	−0.00536335	−0.00268168	−	0.999996i	$-0.500854\pi$
$337$	−0.0984581	−0.00536335	−0.00268168	+	0.999996i	$0.500854\pi$
$338$	0	0
$339$	11.4578	0.622302
$340$	10.2992	0.558551
$341$	1.58744	0.0859645
$342$	3.28514	0.177640
$343$	19.6186	1.05931
$344$	−4.44375	−0.239591
$345$	19.1827	1.03276
$346$	−5.61951	−0.302107
$347$	10.1265	0.543621	0.271810	−	0.962351i	$-0.412378\pi$
$347$	10.1265	0.543621	0.271810	+	0.962351i	$0.412378\pi$
$348$	−10.6195	−0.569266
$349$	−12.0633	−0.645732	−0.322866	−	0.946445i	$-0.604646\pi$
$349$	−12.0633	−0.645732	−0.322866	+	0.946445i	$0.604646\pi$
$350$	−0.542210	−0.0289824
$351$	0	0
$352$	1.22188	0.0651262
$353$	10.9156	0.580978	0.290489	−	0.956878i	$-0.406182\pi$
$353$	10.9156	0.580978	0.290489	+	0.956878i	$0.406182\pi$
$354$	−11.4578	−0.608975
$355$	−10.0813	−0.535060
$356$	−1.22188	−0.0647593
$357$	27.6124	1.46140
$358$	17.3273	0.915774
$359$	17.2671	0.911323	0.455661	−	0.890153i	$-0.349403\pi$
$359$	17.2671	0.911323	0.455661	+	0.890153i	$0.349403\pi$
$360$	−7.50702	−0.395655
$361$	1.00000	0.0526316
$362$	20.4086	1.07265
$363$	−23.8343	−1.25097
$364$	0	0
$365$	19.8735	1.04022
$366$	1.11250	0.0581511
$367$	−13.7710	−0.718841	−0.359420	−	0.933176i	$-0.617026\pi$
$367$	−13.7710	−0.718841	−0.359420	+	0.933176i	$0.617026\pi$
$368$	−3.34841	−0.174548
$369$	−6.36245	−0.331216
$370$	2.25307	0.117131
$371$	−21.3313	−1.10746
$372$	3.25707	0.168871
$373$	−7.46491	−0.386518	−0.193259	−	0.981148i	$-0.561906\pi$
$373$	−7.46491	−0.386518	−0.193259	+	0.981148i	$0.561906\pi$
$374$	−5.50702	−0.284761
$375$	27.3734	1.41355
$376$	0.443752	0.0228848
$377$	0	0
$378$	−1.74693	−0.0898526
$379$	14.7249	0.756367	0.378183	−	0.925731i	$-0.376549\pi$
$379$	14.7249	0.756367	0.378183	+	0.925731i	$0.376549\pi$
$380$	−2.28514	−0.117225
$381$	−56.3373	−2.88625
$382$	7.99600	0.409111
$383$	26.7570	1.36722	0.683609	−	0.729849i	$-0.260410\pi$
$383$	26.7570	1.36722	0.683609	+	0.729849i	$0.260410\pi$
$384$	2.50702	0.127936
$385$	6.82335	0.347750
$386$	−7.83828	−0.398958
$387$	−14.5984	−0.742076
$388$	−11.6656	−0.592233
$389$	21.8374	1.10720	0.553600	−	0.832783i	$-0.313254\pi$
$389$	21.8374	1.10720	0.553600	+	0.832783i	$0.313254\pi$
$390$	0	0
$391$	15.0913	0.763202
$392$	−1.02807	−0.0519256
$393$	−21.2390	−1.07137
$394$	7.14057	0.359737
$395$	−32.0561	−1.61292
$396$	4.01404	0.201713
$397$	19.6788	0.987650	0.493825	−	0.869561i	$-0.335598\pi$
$397$	19.6788	0.987650	0.493825	+	0.869561i	$0.335598\pi$
$398$	−21.6295	−1.08419
$399$	−6.12653	−0.306710
$400$	0.221876	0.0110938
$401$	0.130535	0.00651862	0.00325931	−	0.999995i	$-0.498963\pi$
$401$	0.130535	0.00651862	0.00325931	+	0.999995i	$0.498963\pi$
$402$	−32.9437	−1.64308
$403$	0	0
$404$	−14.1546	−0.704218
$405$	18.4257	0.915581
$406$	10.3515	0.513737
$407$	−1.20472	−0.0597160
$408$	−11.2992	−0.559393
$409$	3.77501	0.186662	0.0933311	−	0.995635i	$-0.470248\pi$
$409$	3.77501	0.186662	0.0933311	+	0.995635i	$0.470248\pi$
$410$	4.42571	0.218570
$411$	−1.67655	−0.0826982
$412$	2.69682	0.132863
$413$	11.1686	0.549573
$414$	−11.0000	−0.540621
$415$	−9.57429	−0.469983
$416$	0	0
$417$	38.2749	1.87433
$418$	1.22188	0.0597639
$419$	12.4438	0.607917	0.303959	−	0.952685i	$-0.401692\pi$
$419$	12.4438	0.607917	0.303959	+	0.952685i	$0.401692\pi$
$420$	14.0000	0.683130
$421$	−21.0702	−1.02690	−0.513449	−	0.858120i	$-0.671632\pi$
$421$	−21.0702	−1.02690	−0.513449	+	0.858120i	$0.671632\pi$
$422$	6.08131	0.296033
$423$	1.45779	0.0708802
$424$	8.72889	0.423913
$425$	−1.00000	−0.0485071
$426$	11.0602	0.535866
$427$	−1.08442	−0.0524788
$428$	−18.7289	−0.905295
$429$	0	0
$430$	10.1546	0.489699
$431$	−4.09134	−0.197073	−0.0985365	−	0.995133i	$-0.531416\pi$
$431$	−4.09134	−0.197073	−0.0985365	+	0.995133i	$0.531416\pi$
$432$	0.714858	0.0343936
$433$	−29.2609	−1.40619	−0.703094	−	0.711097i	$-0.748199\pi$
$433$	−29.2609	−1.40619	−0.703094	+	0.711097i	$0.748199\pi$
$434$	−3.17487	−0.152399
$435$	24.2671	1.16352
$436$	−7.01404	−0.335911
$437$	−3.34841	−0.160176
$438$	−21.8031	−1.04179
$439$	31.3593	1.49670	0.748350	−	0.663304i	$-0.230847\pi$
$439$	31.3593	1.49670	0.748350	+	0.663304i	$0.230847\pi$
$440$	−2.79216	−0.133111
$441$	−3.37737	−0.160827
$442$	0	0
$443$	−11.3313	−0.538364	−0.269182	−	0.963089i	$-0.586753\pi$
$443$	−11.3313	−0.538364	−0.269182	+	0.963089i	$0.586753\pi$
$444$	−2.47183	−0.117308
$445$	2.79216	0.132361
$446$	12.0913	0.572542
$447$	−4.92581	−0.232983
$448$	−2.44375	−0.115456
$449$	−1.04923	−0.0495162	−0.0247581	−	0.999693i	$-0.507882\pi$
$449$	−1.04923	−0.0495162	−0.0247581	+	0.999693i	$0.507882\pi$
$450$	0.728895	0.0343604
$451$	−2.36645	−0.111432
$452$	4.57028	0.214968
$453$	59.7882	2.80909
$454$	29.5139	1.38516
$455$	0	0
$456$	2.50702	0.117402
$457$	−34.6264	−1.61976	−0.809878	−	0.586598i	$-0.800467\pi$
$457$	−34.6264	−1.61976	−0.809878	+	0.586598i	$0.800467\pi$
$458$	−15.7922	−0.737919
$459$	−3.22188	−0.150384
$460$	7.65159	0.356757
$461$	−1.72978	−0.0805640	−0.0402820	−	0.999188i	$-0.512826\pi$
$461$	−1.72978	−0.0805640	−0.0402820	+	0.999188i	$0.512826\pi$
$462$	−7.48586	−0.348274
$463$	0.0641544	0.00298151	0.00149075	−	0.999999i	$-0.499525\pi$
$463$	0.0641544	0.00298151	0.00149075	+	0.999999i	$0.499525\pi$
$464$	−4.23591	−0.196647
$465$	−7.44286	−0.345155
$466$	23.7710	1.10117
$467$	−24.5984	−1.13828	−0.569138	−	0.822242i	$-0.692723\pi$
$467$	−24.5984	−1.13828	−0.569138	+	0.822242i	$0.692723\pi$
$468$	0	0
$469$	32.1123	1.48281
$470$	−1.01404	−0.0467741
$471$	32.9437	1.51796
$472$	−4.57028	−0.210364
$473$	−5.42972	−0.249659
$474$	35.1686	1.61535
$475$	0.221876	0.0101804
$476$	11.0140	0.504828
$477$	28.6757	1.31297
$478$	−25.1406	−1.14990
$479$	−4.63444	−0.211753	−0.105876	−	0.994379i	$-0.533765\pi$
$479$	−4.63444	−0.211753	−0.105876	+	0.994379i	$0.533765\pi$
$480$	−5.72889	−0.261487
$481$	0	0
$482$	−23.5522	−1.07278
$483$	20.5141	0.933425
$484$	−9.50702	−0.432137
$485$	26.6576	1.21046
$486$	−22.3593	−1.01424
$487$	−16.0452	−0.727079	−0.363539	−	0.931579i	$-0.618432\pi$
$487$	−16.0452	−0.727079	−0.363539	+	0.931579i	$0.618432\pi$
$488$	0.443752	0.0200877
$489$	−10.1867	−0.460658
$490$	2.34930	0.106130
$491$	4.15461	0.187495	0.0937474	−	0.995596i	$-0.470115\pi$
$491$	4.15461	0.187495	0.0937474	+	0.995596i	$0.470115\pi$
$492$	−4.85543	−0.218900
$493$	19.0913	0.859831
$494$	0	0
$495$	−9.17265	−0.412280
$496$	1.29918	0.0583349
$497$	−10.7810	−0.483596
$498$	10.5039	0.470691
$499$	−8.95077	−0.400692	−0.200346	−	0.979725i	$-0.564207\pi$
$499$	−8.95077	−0.400692	−0.200346	+	0.979725i	$0.564207\pi$
$500$	10.9187	0.488299
$501$	−4.85543	−0.216925
$502$	24.6264	1.09913
$503$	40.3092	1.79730	0.898650	−	0.438667i	$-0.144549\pi$
$503$	40.3092	1.79730	0.898650	+	0.438667i	$0.144549\pi$
$504$	−8.02807	−0.357599
$505$	32.3453	1.43935
$506$	−4.09134	−0.181882
$507$	0	0
$508$	−22.4718	−0.997026
$509$	−34.1827	−1.51512	−0.757560	−	0.652765i	$-0.773609\pi$
$509$	−34.1827	−1.51512	−0.757560	+	0.652765i	$0.773609\pi$
$510$	25.8202	1.14334
$511$	21.2529	0.940172
$512$	1.00000	0.0441942
$513$	0.714858	0.0315617
$514$	20.0000	0.882162
$515$	−6.16261	−0.271557
$516$	−11.1406	−0.490436
$517$	0.542210	0.0238464
$518$	2.40945	0.105865
$519$	−14.0882	−0.618405
$520$	0	0
$521$	25.4859	1.11656	0.558278	−	0.829654i	$-0.311462\pi$
$521$	25.4859	1.11656	0.558278	+	0.829654i	$0.311462\pi$
$522$	−13.9156	−0.609068
$523$	−6.66474	−0.291429	−0.145714	−	0.989327i	$-0.546548\pi$
$523$	−6.66474	−0.291429	−0.145714	+	0.989327i	$0.546548\pi$
$524$	−8.47183	−0.370093
$525$	−1.35933	−0.0593261
$526$	−20.6336	−0.899666
$527$	−5.85543	−0.255066
$528$	3.06327	0.133312
$529$	−11.7882	−0.512529
$530$	−19.9468	−0.866432
$531$	−15.0140	−0.651554
$532$	−2.44375	−0.105950
$533$	0	0
$534$	−3.06327	−0.132560
$535$	42.7982	1.85033
$536$	−13.1406	−0.567586
$537$	43.4397	1.87456
$538$	−4.02807	−0.173663
$539$	−1.25618	−0.0541075
$540$	−1.63355	−0.0702969
$541$	29.2007	1.25544	0.627719	−	0.778440i	$-0.283989\pi$
$541$	29.2007	1.25544	0.627719	+	0.778440i	$0.283989\pi$
$542$	27.3593	1.17518
$543$	51.1646	2.19568
$544$	−4.50702	−0.193237
$545$	16.0281	0.686567
$546$	0	0
$547$	13.0100	0.556269	0.278134	−	0.960542i	$-0.410284\pi$
$547$	13.0100	0.556269	0.278134	+	0.960542i	$0.410284\pi$
$548$	−0.668743	−0.0285673
$549$	1.45779	0.0622169
$550$	0.271105	0.0115600
$551$	−4.23591	−0.180456
$552$	−8.39452	−0.357295
$553$	−34.2811	−1.45778
$554$	1.55625	0.0661186
$555$	5.64848	0.239764
$556$	15.2671	0.647469
$557$	−1.56717	−0.0664031	−0.0332016	−	0.999449i	$-0.510570\pi$
$557$	−1.56717	−0.0664031	−0.0332016	+	0.999449i	$0.510570\pi$
$558$	4.26799	0.180678
$559$	0	0
$560$	5.58432	0.235981
$561$	−13.8062	−0.582898
$562$	−16.3344	−0.689024
$563$	−22.5531	−0.950501	−0.475251	−	0.879850i	$-0.657643\pi$
$563$	−22.5531	−0.950501	−0.475251	+	0.879850i	$0.657643\pi$
$564$	1.11250	0.0468445
$565$	−10.4438	−0.439372
$566$	−25.3313	−1.06475
$567$	19.7046	0.827517
$568$	4.41168	0.185110
$569$	43.4155	1.82007	0.910036	−	0.414530i	$-0.136054\pi$
$569$	43.4155	1.82007	0.910036	+	0.414530i	$0.136054\pi$
$570$	−5.72889	−0.239957
$571$	−4.94988	−0.207146	−0.103573	−	0.994622i	$-0.533028\pi$
$571$	−4.94988	−0.207146	−0.103573	+	0.994622i	$0.533028\pi$
$572$	0	0
$573$	20.0461	0.837438
$574$	4.73290	0.197547
$575$	−0.742932	−0.0309824
$576$	3.28514	0.136881
$577$	32.3092	1.34505	0.672525	−	0.740074i	$-0.265210\pi$
$577$	32.3092	1.34505	0.672525	+	0.740074i	$0.265210\pi$
$578$	3.31322	0.137812
$579$	−19.6507	−0.816655
$580$	9.67967	0.401926
$581$	−10.2388	−0.424778
$582$	−29.2459	−1.21228
$583$	10.6656	0.441725
$584$	−8.69682	−0.359877
$585$	0	0
$586$	14.3453	0.592599
$587$	−16.6344	−0.686577	−0.343288	−	0.939230i	$-0.611541\pi$
$587$	−16.6344	−0.686577	−0.343288	+	0.939230i	$0.611541\pi$
$588$	−2.57740	−0.106290
$589$	1.29918	0.0535318
$590$	10.4438	0.429962
$591$	17.9015	0.736371
$592$	−0.985963	−0.0405228
$593$	−14.9437	−0.613662	−0.306831	−	0.951764i	$-0.599269\pi$
$593$	−14.9437	−0.613662	−0.306831	+	0.951764i	$0.599269\pi$
$594$	0.873467	0.0358388
$595$	−25.1686	−1.03181
$596$	−1.96481	−0.0804817
$597$	−54.2257	−2.21931
$598$	0	0
$599$	−7.39364	−0.302096	−0.151048	−	0.988526i	$-0.548265\pi$
$599$	−7.39364	−0.302096	−0.151048	+	0.988526i	$0.548265\pi$
$600$	0.556248	0.0227087
$601$	−33.8312	−1.38000	−0.690001	−	0.723809i	$-0.742390\pi$
$601$	−33.8312	−1.38000	−0.690001	+	0.723809i	$0.742390\pi$
$602$	10.8594	0.442597
$603$	−43.1686	−1.75796
$604$	23.8483	0.970374
$605$	21.7249	0.883242
$606$	−35.4859	−1.44151
$607$	−3.93585	−0.159751	−0.0798755	−	0.996805i	$-0.525452\pi$
$607$	−3.93585	−0.159751	−0.0798755	+	0.996805i	$0.525452\pi$
$608$	1.00000	0.0405554
$609$	25.9515	1.05161
$610$	−1.01404	−0.0410572
$611$	0	0
$612$	−14.8062	−0.598505
$613$	36.6866	1.48176	0.740879	−	0.671639i	$-0.234409\pi$
$613$	36.6866	1.48176	0.740879	+	0.671639i	$0.234409\pi$
$614$	−29.1686	−1.17715
$615$	11.0953	0.447407
$616$	−2.98596	−0.120308
$617$	6.73290	0.271056	0.135528	−	0.990773i	$-0.456727\pi$
$617$	6.73290	0.271056	0.135528	+	0.990773i	$0.456727\pi$
$618$	6.76097	0.271966
$619$	8.03208	0.322836	0.161418	−	0.986886i	$-0.448393\pi$
$619$	8.03208	0.322836	0.161418	+	0.986886i	$0.448393\pi$
$620$	−2.96881	−0.119230
$621$	−2.39364	−0.0960533
$622$	−20.5070	−0.822257
$623$	2.98596	0.119630
$624$	0	0
$625$	−26.0602	−1.04241
$626$	12.0321	0.480899
$627$	3.06327	0.122335
$628$	13.1406	0.524366
$629$	4.44375	0.177184
$630$	18.3453	0.730894
$631$	−20.8234	−0.828964	−0.414482	−	0.910057i	$-0.636037\pi$
$631$	−20.8234	−0.828964	−0.414482	+	0.910057i	$0.636037\pi$
$632$	14.0281	0.558007
$633$	15.2459	0.605972
$634$	−21.1686	−0.840714
$635$	51.3513	2.03782
$636$	21.8835	0.867737
$637$	0	0
$638$	−5.17576	−0.204910
$639$	14.4930	0.573333
$640$	−2.28514	−0.0903282
$641$	42.4077	1.67500	0.837501	−	0.546436i	$-0.184016\pi$
$641$	42.4077	1.67500	0.837501	+	0.546436i	$0.184016\pi$
$642$	−46.9537	−1.85311
$643$	18.9788	0.748453	0.374226	−	0.927337i	$-0.377908\pi$
$643$	18.9788	0.748453	0.374226	+	0.927337i	$0.377908\pi$
$644$	8.18268	0.322443
$645$	25.4578	1.00240
$646$	−4.50702	−0.177326
$647$	29.7069	1.16790	0.583948	−	0.811791i	$-0.301507\pi$
$647$	29.7069	1.16790	0.583948	+	0.811791i	$0.301507\pi$
$648$	−8.06327	−0.316755
$649$	−5.58432	−0.219204
$650$	0	0
$651$	−7.95947	−0.311956
$652$	−4.06327	−0.159130
$653$	−31.5202	−1.23348	−0.616740	−	0.787167i	$-0.711547\pi$
$653$	−31.5202	−1.23348	−0.616740	+	0.787167i	$0.711547\pi$
$654$	−17.5843	−0.687601
$655$	19.3593	0.756432
$656$	−1.93673	−0.0756167
$657$	−28.5703	−1.11463
$658$	−1.08442	−0.0422751
$659$	21.8022	0.849293	0.424646	−	0.905359i	$-0.360398\pi$
$659$	21.8022	0.849293	0.424646	+	0.905359i	$0.360398\pi$
$660$	−7.00000	−0.272475
$661$	49.6044	1.92939	0.964694	−	0.263375i	$-0.0848356\pi$
$661$	49.6044	1.92939	0.964694	+	0.263375i	$0.0848356\pi$
$662$	−1.97193	−0.0766411
$663$	0	0
$664$	4.18980	0.162596
$665$	5.58432	0.216551
$666$	−3.23903	−0.125510
$667$	14.1836	0.549190
$668$	−1.93673	−0.0749345
$669$	30.3132	1.17198
$670$	30.0281	1.16009
$671$	0.542210	0.0209318
$672$	−6.12653	−0.236336
$673$	20.8875	0.805154	0.402577	−	0.915386i	$-0.368115\pi$
$673$	20.8875	0.805154	0.402577	+	0.915386i	$0.368115\pi$
$674$	−0.0984581	−0.00379246
$675$	0.158610	0.00610490
$676$	0	0
$677$	−46.3201	−1.78023	−0.890114	−	0.455738i	$-0.849375\pi$
$677$	−46.3201	−1.78023	−0.890114	+	0.455738i	$0.849375\pi$
$678$	11.4578	0.440034
$679$	28.5079	1.09403
$680$	10.2992	0.394956
$681$	73.9920	2.83538
$682$	1.58744	0.0607861
$683$	−38.4014	−1.46939	−0.734695	−	0.678397i	$-0.762675\pi$
$683$	−38.4014	−1.46939	−0.734695	+	0.678397i	$0.762675\pi$
$684$	3.28514	0.125611
$685$	1.52817	0.0583885
$686$	19.6186	0.749043
$687$	−39.5912	−1.51050
$688$	−4.44375	−0.169416
$689$	0	0
$690$	19.1827	0.730272
$691$	24.1936	0.920368	0.460184	−	0.887824i	$-0.347783\pi$
$691$	24.1936	0.920368	0.460184	+	0.887824i	$0.347783\pi$
$692$	−5.61951	−0.213622
$693$	−9.80931	−0.372625
$694$	10.1265	0.384398
$695$	−34.8875	−1.32336
$696$	−10.6195	−0.402532
$697$	8.72889	0.330630
$698$	−12.0633	−0.456601
$699$	59.5944	2.25407
$700$	−0.542210	−0.0204936
$701$	27.5500	1.04055	0.520275	−	0.853999i	$-0.325829\pi$
$701$	27.5500	1.04055	0.520275	+	0.853999i	$0.325829\pi$
$702$	0	0
$703$	−0.985963	−0.0371863
$704$	1.22188	0.0460512
$705$	−2.54221	−0.0957452
$706$	10.9156	0.410813
$707$	34.5904	1.30090
$708$	−11.4578	−0.430610
$709$	18.0000	0.676004	0.338002	−	0.941145i	$-0.390249\pi$
$709$	18.0000	0.676004	0.338002	+	0.941145i	$0.390249\pi$
$710$	−10.0813	−0.378345
$711$	46.0842	1.72829
$712$	−1.22188	−0.0457917
$713$	−4.35018	−0.162916
$714$	27.6124	1.03337
$715$	0	0
$716$	17.3273	0.647550
$717$	−63.0279	−2.35382
$718$	17.2671	0.644403
$719$	6.85943	0.255814	0.127907	−	0.991786i	$-0.459174\pi$
$719$	6.85943	0.255814	0.127907	+	0.991786i	$0.459174\pi$
$720$	−7.50702	−0.279770
$721$	−6.59035	−0.245438
$722$	1.00000	0.0372161
$723$	−59.0459	−2.19594
$724$	20.4086	0.758478
$725$	−0.939848	−0.0349051
$726$	−23.8343	−0.884573
$727$	30.5351	1.13248	0.566242	−	0.824239i	$-0.308397\pi$
$727$	30.5351	1.13248	0.566242	+	0.824239i	$0.308397\pi$
$728$	0	0
$729$	−31.8655	−1.18020
$730$	19.8735	0.735550
$731$	20.0281	0.740765
$732$	1.11250	0.0411190
$733$	14.3985	0.531822	0.265911	−	0.963998i	$-0.414327\pi$
$733$	14.3985	0.531822	0.265911	+	0.963998i	$0.414327\pi$
$734$	−13.7710	−0.508297
$735$	5.88973	0.217246
$736$	−3.34841	−0.123424
$737$	−16.0561	−0.591436
$738$	−6.36245	−0.234205
$739$	−23.9748	−0.881929	−0.440964	−	0.897525i	$-0.645364\pi$
$739$	−23.9748	−0.881929	−0.440964	+	0.897525i	$0.645364\pi$
$740$	2.25307	0.0828243
$741$	0	0
$742$	−21.3313	−0.783095
$743$	35.3834	1.29809	0.649046	−	0.760750i	$-0.275168\pi$
$743$	35.3834	1.29809	0.649046	+	0.760750i	$0.275168\pi$
$744$	3.25707	0.119410
$745$	4.48987	0.164496
$746$	−7.46491	−0.273310
$747$	13.7641	0.503602
$748$	−5.50702	−0.201357
$749$	45.7688	1.67235
$750$	27.3734	0.999534
$751$	35.6685	1.30156	0.650782	−	0.759265i	$-0.274441\pi$
$751$	35.6685	1.30156	0.650782	+	0.759265i	$0.274441\pi$
$752$	0.443752	0.0161820
$753$	61.7389	2.24989
$754$	0	0
$755$	−54.4968	−1.98334
$756$	−1.74693	−0.0635354
$757$	−51.5420	−1.87333	−0.936663	−	0.350232i	$-0.886103\pi$
$757$	−51.5420	−1.87333	−0.936663	+	0.350232i	$0.886103\pi$
$758$	14.7249	0.534832
$759$	−10.2571	−0.372308
$760$	−2.28514	−0.0828908
$761$	−34.5623	−1.25288	−0.626441	−	0.779469i	$-0.715489\pi$
$761$	−34.5623	−1.25288	−0.626441	+	0.779469i	$0.715489\pi$
$762$	−56.3373	−2.04088
$763$	17.1406	0.620530
$764$	7.99600	0.289285
$765$	33.8343	1.22328
$766$	26.7570	0.966769
$767$	0	0
$768$	2.50702	0.0904642
$769$	−51.3794	−1.85279	−0.926394	−	0.376555i	$-0.877109\pi$
$769$	−51.3794	−1.85279	−0.926394	+	0.376555i	$0.877109\pi$
$770$	6.82335	0.245896
$771$	50.1404	1.80576
$772$	−7.83828	−0.282106
$773$	10.3515	0.372318	0.186159	−	0.982520i	$-0.440396\pi$
$773$	10.3515	0.372318	0.186159	+	0.982520i	$0.440396\pi$
$774$	−14.5984	−0.524727
$775$	0.288257	0.0103545
$776$	−11.6656	−0.418772
$777$	6.04053	0.216703
$778$	21.8374	0.782908
$779$	−1.93673	−0.0693907
$780$	0	0
$781$	5.39052	0.192888
$782$	15.0913	0.539665
$783$	−3.02807	−0.108215
$784$	−1.02807	−0.0367169
$785$	−30.0281	−1.07175
$786$	−21.2390	−0.757571
$787$	54.8795	1.95624	0.978121	−	0.208035i	$-0.0667067\pi$
$787$	54.8795	1.95624	0.978121	+	0.208035i	$0.0667067\pi$
$788$	7.14057	0.254372
$789$	−51.7287	−1.84159
$790$	−32.0561	−1.14051
$791$	−11.1686	−0.397111
$792$	4.01404	0.142633
$793$	0	0
$794$	19.6788	0.698374
$795$	−50.0069	−1.77356
$796$	−21.6295	−0.766639
$797$	−26.9889	−0.955995	−0.477998	−	0.878361i	$-0.658637\pi$
$797$	−26.9889	−0.955995	−0.477998	+	0.878361i	$0.658637\pi$
$798$	−6.12653	−0.216877
$799$	−2.00000	−0.0707549
$800$	0.221876	0.00784451
$801$	−4.01404	−0.141829
$802$	0.130535	0.00460936
$803$	−10.6264	−0.374999
$804$	−32.9437	−1.16183
$805$	−18.6986	−0.659039
$806$	0	0
$807$	−10.0985	−0.355483
$808$	−14.1546	−0.497957
$809$	−45.7779	−1.60947	−0.804733	−	0.593637i	$-0.797692\pi$
$809$	−45.7779	−1.60947	−0.804733	+	0.593637i	$0.797692\pi$
$810$	18.4257	0.647414
$811$	−5.55625	−0.195106	−0.0975531	−	0.995230i	$-0.531102\pi$
$811$	−5.55625	−0.195106	−0.0975531	+	0.995230i	$0.531102\pi$
$812$	10.3515	0.363267
$813$	68.5904	2.40557
$814$	−1.20472	−0.0422256
$815$	9.28514	0.325244
$816$	−11.2992	−0.395550
$817$	−4.44375	−0.155467
$818$	3.77501	0.131990
$819$	0	0
$820$	4.42571	0.154553
$821$	19.6467	0.685675	0.342837	−	0.939395i	$-0.388612\pi$
$821$	19.6467	0.685675	0.342837	+	0.939395i	$0.388612\pi$
$822$	−1.67655	−0.0584764
$823$	15.8202	0.551459	0.275729	−	0.961235i	$-0.411081\pi$
$823$	15.8202	0.551459	0.275729	+	0.961235i	$0.411081\pi$
$824$	2.69682	0.0939481
$825$	0.679666	0.0236629
$826$	11.1686	0.388607
$827$	−10.0624	−0.349903	−0.174952	−	0.984577i	$-0.555977\pi$
$827$	−10.0624	−0.349903	−0.174952	+	0.984577i	$0.555977\pi$
$828$	−11.0000	−0.382276
$829$	9.23503	0.320746	0.160373	−	0.987057i	$-0.448730\pi$
$829$	9.23503	0.320746	0.160373	+	0.987057i	$0.448730\pi$
$830$	−9.57429	−0.332328
$831$	3.90154	0.135343
$832$	0	0
$833$	4.63355	0.160543
$834$	38.2749	1.32535
$835$	4.42571	0.153158
$836$	1.22188	0.0422595
$837$	0.928728	0.0321016
$838$	12.4438	0.429862
$839$	−20.5732	−0.710266	−0.355133	−	0.934816i	$-0.615564\pi$
$839$	−20.5732	−0.710266	−0.355133	+	0.934816i	$0.615564\pi$
$840$	14.0000	0.483046
$841$	−11.0570	−0.381277
$842$	−21.0702	−0.726127
$843$	−40.9506	−1.41041
$844$	6.08131	0.209327
$845$	0	0
$846$	1.45779	0.0501198
$847$	23.2328	0.798288
$848$	8.72889	0.299751
$849$	−63.5059	−2.17952
$850$	−1.00000	−0.0342997
$851$	3.30141	0.113171
$852$	11.0602	0.378915
$853$	−20.7539	−0.710598	−0.355299	−	0.934753i	$-0.615621\pi$
$853$	−20.7539	−0.710598	−0.355299	+	0.934753i	$0.615621\pi$
$854$	−1.08442	−0.0371081
$855$	−7.50702	−0.256735
$856$	−18.7289	−0.640140
$857$	11.9015	0.406549	0.203274	−	0.979122i	$-0.434842\pi$
$857$	11.9015	0.406549	0.203274	+	0.979122i	$0.434842\pi$
$858$	0	0
$859$	−21.2248	−0.724181	−0.362090	−	0.932143i	$-0.617937\pi$
$859$	−21.2248	−0.724181	−0.362090	+	0.932143i	$0.617937\pi$
$860$	10.1546	0.346269
$861$	11.8655	0.404374
$862$	−4.09134	−0.139352
$863$	20.8984	0.711391	0.355695	−	0.934602i	$-0.384244\pi$
$863$	20.8984	0.711391	0.355695	+	0.934602i	$0.384244\pi$
$864$	0.714858	0.0243199
$865$	12.8414	0.436620
$866$	−29.2609	−0.994325
$867$	8.30630	0.282097
$868$	−3.17487	−0.107762
$869$	17.1406	0.581454
$870$	24.2671	0.822732
$871$	0	0
$872$	−7.01404	−0.237525
$873$	−38.3233	−1.29705
$874$	−3.34841	−0.113262
$875$	−26.6826	−0.902036
$876$	−21.8031	−0.736658
$877$	−48.6545	−1.64295	−0.821473	−	0.570247i	$-0.806847\pi$
$877$	−48.6545	−1.64295	−0.821473	+	0.570247i	$0.806847\pi$
$878$	31.3593	1.05833
$879$	35.9639	1.21303
$880$	−2.79216	−0.0941237
$881$	−27.1927	−0.916146	−0.458073	−	0.888915i	$-0.651460\pi$
$881$	−27.1927	−0.916146	−0.458073	+	0.888915i	$0.651460\pi$
$882$	−3.37737	−0.113722
$883$	11.2609	0.378959	0.189479	−	0.981885i	$-0.439320\pi$
$883$	11.2609	0.378959	0.189479	+	0.981885i	$0.439320\pi$
$884$	0	0
$885$	26.1827	0.880121
$886$	−11.3313	−0.380681
$887$	2.63266	0.0883962	0.0441981	−	0.999023i	$-0.485927\pi$
$887$	2.63266	0.0883962	0.0441981	+	0.999023i	$0.485927\pi$
$888$	−2.47183	−0.0829491
$889$	54.9156	1.84181
$890$	2.79216	0.0935935
$891$	−9.85231	−0.330065
$892$	12.0913	0.404848
$893$	0.443752	0.0148496
$894$	−4.92581	−0.164744
$895$	−39.5952	−1.32352
$896$	−2.44375	−0.0816400
$897$	0	0
$898$	−1.04923	−0.0350132
$899$	−5.50321	−0.183542
$900$	0.728895	0.0242965
$901$	−39.3413	−1.31065
$902$	−2.36645	−0.0787941
$903$	27.2248	0.905984
$904$	4.57028	0.152005
$905$	−46.6365	−1.55025
$906$	59.7882	1.98633
$907$	−17.8975	−0.594278	−0.297139	−	0.954834i	$-0.596032\pi$
$907$	−17.8975	−0.594278	−0.297139	+	0.954834i	$0.596032\pi$
$908$	29.5139	0.979454
$909$	−46.4999	−1.54230
$910$	0	0
$911$	12.6906	0.420458	0.210229	−	0.977652i	$-0.432579\pi$
$911$	12.6906	0.420458	0.210229	+	0.977652i	$0.432579\pi$
$912$	2.50702	0.0830157
$913$	5.11942	0.169428
$914$	−34.6264	−1.14534
$915$	−2.54221	−0.0840429
$916$	−15.7922	−0.521788
$917$	20.7030	0.683675
$918$	−3.22188	−0.106338
$919$	37.0693	1.22280	0.611402	−	0.791320i	$-0.290606\pi$
$919$	37.0693	1.22280	0.611402	+	0.791320i	$0.290606\pi$
$920$	7.65159	0.252265
$921$	−73.1263	−2.40959
$922$	−1.72978	−0.0569674
$923$	0	0
$924$	−7.48586	−0.246267
$925$	−0.218762	−0.00719284
$926$	0.0641544	0.00210824
$927$	8.85943	0.290982
$928$	−4.23591	−0.139051
$929$	−20.6545	−0.677652	−0.338826	−	0.940849i	$-0.610030\pi$
$929$	−20.6545	−0.677652	−0.338826	+	0.940849i	$0.610030\pi$
$930$	−7.44286	−0.244061
$931$	−1.02807	−0.0336938
$932$	23.7710	0.778645
$933$	−51.4115	−1.68314
$934$	−24.5984	−0.804883
$935$	12.5843	0.411551
$936$	0	0
$937$	−17.4819	−0.571108	−0.285554	−	0.958363i	$-0.592178\pi$
$937$	−17.4819	−0.571108	−0.285554	+	0.958363i	$0.592178\pi$
$938$	32.1123	1.04850
$939$	30.1646	0.984386
$940$	−1.01404	−0.0330742
$941$	−8.60459	−0.280502	−0.140251	−	0.990116i	$-0.544791\pi$
$941$	−8.60459	−0.280502	−0.140251	+	0.990116i	$0.544791\pi$
$942$	32.9437	1.07336
$943$	6.48498	0.211180
$944$	−4.57028	−0.148750
$945$	3.99199	0.129860
$946$	−5.42972	−0.176535
$947$	9.80709	0.318687	0.159344	−	0.987223i	$-0.449062\pi$
$947$	9.80709	0.318687	0.159344	+	0.987223i	$0.449062\pi$
$948$	35.1686	1.14222
$949$	0	0
$950$	0.221876	0.00719861
$951$	−53.0702	−1.72092
$952$	11.0140	0.356967
$953$	4.66874	0.151235	0.0756177	−	0.997137i	$-0.475907\pi$
$953$	4.66874	0.151235	0.0756177	+	0.997137i	$0.475907\pi$
$954$	28.6757	0.928409
$955$	−18.2720	−0.591268
$956$	−25.1406	−0.813104
$957$	−12.9757	−0.419446
$958$	−4.63444	−0.149732
$959$	1.63424	0.0527724
$960$	−5.72889	−0.184899
$961$	−29.3121	−0.945553
$962$	0	0
$963$	−61.5271	−1.98268
$964$	−23.5522	−0.758567
$965$	17.9116	0.576594
$966$	20.5141	0.660032
$967$	11.7750	0.378659	0.189329	−	0.981914i	$-0.439369\pi$
$967$	11.7750	0.378659	0.189329	+	0.981914i	$0.439369\pi$
$968$	−9.50702	−0.305567
$969$	−11.2992	−0.362982
$970$	26.6576	0.855925
$971$	−59.9898	−1.92516	−0.962582	−	0.270992i	$-0.912648\pi$
$971$	−59.9898	−1.92516	−0.962582	+	0.270992i	$0.912648\pi$
$972$	−22.3593	−0.717176
$973$	−37.3090	−1.19607
$974$	−16.0452	−0.514122
$975$	0	0
$976$	0.443752	0.0142042
$977$	−25.4406	−0.813918	−0.406959	−	0.913446i	$-0.633411\pi$
$977$	−25.4406	−0.813918	−0.406959	+	0.913446i	$0.633411\pi$
$978$	−10.1867	−0.325734
$979$	−1.49298	−0.0477159
$980$	2.34930	0.0750455
$981$	−23.0421	−0.735678
$982$	4.15461	0.132579
$983$	27.4569	0.875739	0.437870	−	0.899039i	$-0.355733\pi$
$983$	27.4569	0.875739	0.437870	+	0.899039i	$0.355733\pi$
$984$	−4.85543	−0.154785
$985$	−16.3172	−0.519910
$986$	19.0913	0.607992
$987$	−2.71866	−0.0865360
$988$	0	0
$989$	14.8795	0.473141
$990$	−9.17265	−0.291526
$991$	8.69682	0.276264	0.138132	−	0.990414i	$-0.455890\pi$
$991$	8.69682	0.276264	0.138132	+	0.990414i	$0.455890\pi$
$992$	1.29918	0.0412490
$993$	−4.94365	−0.156882
$994$	−10.7810	−0.341954
$995$	49.4266	1.56693
$996$	10.5039	0.332829
$997$	−48.9717	−1.55095	−0.775475	−	0.631378i	$-0.782490\pi$
$997$	−48.9717	−1.55095	−0.775475	+	0.631378i	$0.782490\pi$
$998$	−8.95077	−0.283332
$999$	−0.704823	−0.0222996

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6422.2.a.w.1.3		3
13.12	even	2		494.2.a.f.1.3	✓	3
39.38	odd	2		4446.2.a.bk.1.1		3
52.51	odd	2		3952.2.a.o.1.1		3
247.246	odd	2		9386.2.a.bc.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
494.2.a.f.1.3	✓	3	13.12	even	2
3952.2.a.o.1.1		3	52.51	odd	2
4446.2.a.bk.1.1		3	39.38	odd	2
6422.2.a.w.1.3		3	1.1	even	1	trivial
9386.2.a.bc.1.1		3	247.246	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +2.50702 q^{3} +1.00000 q^{4} -2.28514 q^{5} +2.50702 q^{6} -2.44375 q^{7} +1.00000 q^{8} +3.28514 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} +2.50702 q^{3} +1.00000 q^{4} -2.28514 q^{5} +2.50702 q^{6} -2.44375 q^{7} +1.00000 q^{8} +3.28514 q^{9} -2.28514 q^{10} +1.22188 q^{11} +2.50702 q^{12} -2.44375 q^{14} -5.72889 q^{15} +1.00000 q^{16} -4.50702 q^{17} +3.28514 q^{18} +1.00000 q^{19} -2.28514 q^{20} -6.12653 q^{21} +1.22188 q^{22} -3.34841 q^{23} +2.50702 q^{24} +0.221876 q^{25} +0.714858 q^{27} -2.44375 q^{28} -4.23591 q^{29} -5.72889 q^{30} +1.29918 q^{31} +1.00000 q^{32} +3.06327 q^{33} -4.50702 q^{34} +5.58432 q^{35} +3.28514 q^{36} -0.985963 q^{37} +1.00000 q^{38} -2.28514 q^{40} -1.93673 q^{41} -6.12653 q^{42} -4.44375 q^{43} +1.22188 q^{44} -7.50702 q^{45} -3.34841 q^{46} +0.443752 q^{47} +2.50702 q^{48} -1.02807 q^{49} +0.221876 q^{50} -11.2992 q^{51} +8.72889 q^{53} +0.714858 q^{54} -2.79216 q^{55} -2.44375 q^{56} +2.50702 q^{57} -4.23591 q^{58} -4.57028 q^{59} -5.72889 q^{60} +0.443752 q^{61} +1.29918 q^{62} -8.02807 q^{63} +1.00000 q^{64} +3.06327 q^{66} -13.1406 q^{67} -4.50702 q^{68} -8.39452 q^{69} +5.58432 q^{70} +4.41168 q^{71} +3.28514 q^{72} -8.69682 q^{73} -0.985963 q^{74} +0.556248 q^{75} +1.00000 q^{76} -2.98596 q^{77} +14.0281 q^{79} -2.28514 q^{80} -8.06327 q^{81} -1.93673 q^{82} +4.18980 q^{83} -6.12653 q^{84} +10.2992 q^{85} -4.44375 q^{86} -10.6195 q^{87} +1.22188 q^{88} -1.22188 q^{89} -7.50702 q^{90} -3.34841 q^{92} +3.25707 q^{93} +0.443752 q^{94} -2.28514 q^{95} +2.50702 q^{96} -11.6656 q^{97} -1.02807 q^{98} +4.01404 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{2} - q^{3} + 3 q^{4} - q^{5} - q^{6} - 2 q^{7} + 3 q^{8} + 4 q^{9}+O(q^{10}) \) 3 * q + 3 * q^2 - q^3 + 3 * q^4 - q^5 - q^6 - 2 * q^7 + 3 * q^8 + 4 * q^9	\( 3 q + 3 q^{2} - q^{3} + 3 q^{4} - q^{5} - q^{6} - 2 q^{7} + 3 q^{8} + 4 q^{9} - q^{10} + q^{11} - q^{12} - 2 q^{14} - 6 q^{15} + 3 q^{16} - 5 q^{17} + 4 q^{18} + 3 q^{19} - q^{20} - 12 q^{21} + q^{22} - q^{23} - q^{24} - 2 q^{25} + 8 q^{27} - 2 q^{28} + 7 q^{29} - 6 q^{30} - 19 q^{31} + 3 q^{32} + 6 q^{33} - 5 q^{34} - 12 q^{35} + 4 q^{36} - 20 q^{37} + 3 q^{38} - q^{40} - 9 q^{41} - 12 q^{42} - 8 q^{43} + q^{44} - 14 q^{45} - q^{46} - 4 q^{47} - q^{48} + 31 q^{49} - 2 q^{50} - 11 q^{51} + 15 q^{53} + 8 q^{54} + 6 q^{55} - 2 q^{56} - q^{57} + 7 q^{58} - 2 q^{59} - 6 q^{60} - 4 q^{61} - 19 q^{62} + 10 q^{63} + 3 q^{64} + 6 q^{66} - 16 q^{67} - 5 q^{68} - 6 q^{69} - 12 q^{70} + q^{71} + 4 q^{72} - 8 q^{73} - 20 q^{74} + 7 q^{75} + 3 q^{76} - 26 q^{77} + 8 q^{79} - q^{80} - 21 q^{81} - 9 q^{82} + 3 q^{83} - 12 q^{84} + 8 q^{85} - 8 q^{86} - 34 q^{87} + q^{88} - q^{89} - 14 q^{90} - q^{92} + 38 q^{93} - 4 q^{94} - q^{95} - q^{96} - 27 q^{97} + 31 q^{98} - 5 q^{99}+O(q^{100}) \) 3 * q + 3 * q^2 - q^3 + 3 * q^4 - q^5 - q^6 - 2 * q^7 + 3 * q^8 + 4 * q^9 - q^10 + q^11 - q^12 - 2 * q^14 - 6 * q^15 + 3 * q^16 - 5 * q^17 + 4 * q^18 + 3 * q^19 - q^20 - 12 * q^21 + q^22 - q^23 - q^24 - 2 * q^25 + 8 * q^27 - 2 * q^28 + 7 * q^29 - 6 * q^30 - 19 * q^31 + 3 * q^32 + 6 * q^33 - 5 * q^34 - 12 * q^35 + 4 * q^36 - 20 * q^37 + 3 * q^38 - q^40 - 9 * q^41 - 12 * q^42 - 8 * q^43 + q^44 - 14 * q^45 - q^46 - 4 * q^47 - q^48 + 31 * q^49 - 2 * q^50 - 11 * q^51 + 15 * q^53 + 8 * q^54 + 6 * q^55 - 2 * q^56 - q^57 + 7 * q^58 - 2 * q^59 - 6 * q^60 - 4 * q^61 - 19 * q^62 + 10 * q^63 + 3 * q^64 + 6 * q^66 - 16 * q^67 - 5 * q^68 - 6 * q^69 - 12 * q^70 + q^71 + 4 * q^72 - 8 * q^73 - 20 * q^74 + 7 * q^75 + 3 * q^76 - 26 * q^77 + 8 * q^79 - q^80 - 21 * q^81 - 9 * q^82 + 3 * q^83 - 12 * q^84 + 8 * q^85 - 8 * q^86 - 34 * q^87 + q^88 - q^89 - 14 * q^90 - q^92 + 38 * q^93 - 4 * q^94 - q^95 - q^96 - 27 * q^97 + 31 * q^98 - 5 * q^99

Introduction

L-functions

Modular forms

Varieties

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Representations

Groups

Database

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