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Properties

Label 8047.2.a.d
Level $8047$
Weight $2$
Character orbit 8047.a
Self dual yes
Analytic conductor $64.256$
Analytic rank $0$
Dimension $156$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [8047,2,Mod(1,8047)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8047, 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("8047.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8047 = 13 \cdot 619 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8047.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(64.2556185065\)
Analytic rank: \(0\)
Dimension: \(156\)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 156 q + 13 q^{2} + 23 q^{3} + 161 q^{4} + 39 q^{5} + 25 q^{6} + 19 q^{7} + 42 q^{8} + 169 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 156 q + 13 q^{2} + 23 q^{3} + 161 q^{4} + 39 q^{5} + 25 q^{6} + 19 q^{7} + 42 q^{8} + 169 q^{9} + 11 q^{10} + 23 q^{11} + 57 q^{12} - 156 q^{13} + 18 q^{14} + 32 q^{15} + 159 q^{16} + 119 q^{17} + 36 q^{18} + 35 q^{19} + 109 q^{20} + 33 q^{21} + 11 q^{22} + 55 q^{23} + 63 q^{24} + 189 q^{25} - 13 q^{26} + 89 q^{27} + 54 q^{28} - 55 q^{29} + 47 q^{31} + 112 q^{32} + 109 q^{33} + 51 q^{34} + 25 q^{35} + 162 q^{36} + 53 q^{37} + 37 q^{38} - 23 q^{39} + 25 q^{40} + 113 q^{41} + 26 q^{42} + 31 q^{43} + 86 q^{44} + 144 q^{45} + 37 q^{46} + 115 q^{47} + 129 q^{48} + 189 q^{49} + 72 q^{50} - 4 q^{51} - 161 q^{52} + 51 q^{53} + 108 q^{54} + 22 q^{55} + 39 q^{56} + 102 q^{57} + 31 q^{58} + 75 q^{59} + 97 q^{60} + 7 q^{61} + 77 q^{62} + 94 q^{63} + 158 q^{64} - 39 q^{65} + 48 q^{66} + 37 q^{67} + 235 q^{68} + 27 q^{69} + 38 q^{70} + 70 q^{71} + 152 q^{72} + 155 q^{73} - 18 q^{74} + 80 q^{75} + 21 q^{76} + 101 q^{77} - 25 q^{78} + 10 q^{79} + 211 q^{80} + 220 q^{81} + 45 q^{82} + 132 q^{83} + 86 q^{84} + 74 q^{85} + 35 q^{86} + 53 q^{87} + 51 q^{88} + 190 q^{89} - 27 q^{90} - 19 q^{91} + 125 q^{92} + 96 q^{93} - 19 q^{94} + 72 q^{95} + 146 q^{96} + 155 q^{97} + 135 q^{98} + 89 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1 −2.78003 2.35788 5.72856 3.19176 −6.55498 −2.60821 −10.3655 2.55961 −8.87319
1.2 −2.75185 0.388525 5.57269 0.235747 −1.06916 −0.913788 −9.83151 −2.84905 −0.648741
1.3 −2.71116 −0.430887 5.35037 1.49097 1.16820 4.38752 −9.08338 −2.81434 −4.04226
1.4 −2.66092 −2.09208 5.08047 4.25784 5.56685 −2.19372 −8.19688 1.37680 −11.3297
1.5 −2.65345 0.515937 5.04077 −0.00500829 −1.36901 0.475978 −8.06853 −2.73381 0.0132892
1.6 −2.61991 −3.07843 4.86391 −2.25474 8.06519 3.02795 −7.50318 6.47672 5.90721
1.7 −2.53163 −1.76740 4.40915 3.70059 4.47441 1.97275 −6.09908 0.123713 −9.36852
1.8 −2.51520 3.16825 4.32624 1.81160 −7.96880 1.29182 −5.85097 7.03782 −4.55654
1.9 −2.51030 2.30230 4.30163 −2.11540 −5.77947 3.11614 −5.77778 2.30059 5.31029
1.10 −2.48921 1.66409 4.19617 0.253243 −4.14227 −2.12862 −5.46674 −0.230804 −0.630376
1.11 −2.43558 −1.59767 3.93207 −2.50698 3.89125 −1.32806 −4.70573 −0.447465 6.10595
1.12 −2.42534 −2.30091 3.88226 −2.29855 5.58047 −0.451041 −4.56512 2.29416 5.57476
1.13 −2.42132 −1.26535 3.86281 0.549340 3.06382 1.75655 −4.51045 −1.39889 −1.33013
1.14 −2.40103 1.80838 3.76492 2.80011 −4.34197 −2.97647 −4.23763 0.270237 −6.72314
1.15 −2.37112 0.716095 3.62222 −3.83331 −1.69795 −1.71381 −3.84647 −2.48721 9.08923
1.16 −2.35417 1.10999 3.54214 −1.30122 −2.61310 −1.47122 −3.63046 −1.76793 3.06330
1.17 −2.22460 3.13495 2.94885 −1.36986 −6.97401 0.348323 −2.11081 6.82790 3.04739
1.18 −2.19459 1.61581 2.81621 −2.64713 −3.54605 3.12381 −1.79125 −0.389142 5.80935
1.19 −2.16081 −2.05453 2.66911 −1.91838 4.43946 −4.69995 −1.44582 1.22110 4.14525
1.20 −2.15517 0.169019 2.64475 4.16569 −0.364265 2.92802 −1.38954 −2.97143 −8.97775
See next 80 embeddings (of 156 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.156
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(13\) \(1\)
\(619\) \(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 8047.2.a.d ✓ 156
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8047.2.a.d ✓ 156 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{156} - 13 T_{2}^{155} - 152 T_{2}^{154} + 2677 T_{2}^{153} + 8986 T_{2}^{152} + \cdots - 6152614763945 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8047))\). Copy content Toggle raw display