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Properties

Label 8032.2.a.i
Level $8032$
Weight $2$
Character orbit 8032.a
Self dual yes
Analytic conductor $64.136$
Analytic rank $0$
Dimension $30$
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] = [8032,2,Mod(1,8032)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8032, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("8032.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8032 = 2^{5} \cdot 251 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8032.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: \(64.1358429035\)
Analytic rank: \(0\)
Dimension: \(30\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 30 q + 3 q^{3} + 13 q^{7} + 35 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 30 q + 3 q^{3} + 13 q^{7} + 35 q^{9} + 13 q^{11} - 7 q^{13} + 28 q^{15} - 9 q^{17} - 17 q^{19} - 6 q^{21} + 43 q^{23} + 34 q^{25} + 12 q^{27} - q^{29} + 39 q^{31} + 17 q^{35} - q^{37} + 48 q^{39} - 3 q^{41} - 19 q^{43} + 66 q^{47} + 25 q^{49} - 14 q^{51} + 3 q^{53} + 50 q^{55} - 14 q^{57} + 27 q^{59} + 15 q^{61} + 75 q^{63} - 6 q^{65} + 8 q^{67} + 18 q^{69} + 64 q^{71} - 15 q^{73} + 9 q^{75} + 71 q^{79} + 6 q^{81} + 60 q^{83} + 15 q^{85} + 64 q^{87} - 32 q^{89} - 26 q^{91} - 4 q^{93} + 72 q^{95} - 4 q^{97} + 13 q^{99}+O(q^{100}) \) 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 0 −3.18079 0 −0.628547 0 2.91752 0 7.11746 0
1.2 0 −2.91073 0 −4.30087 0 1.78682 0 5.47235 0
1.3 0 −2.87266 0 1.81575 0 4.67512 0 5.25220 0
1.4 0 −2.63245 0 −1.53692 0 −2.44148 0 3.92982 0
1.5 0 −2.41377 0 −0.771409 0 0.588687 0 2.82630 0
1.6 0 −2.05220 0 −2.64875 0 −3.00423 0 1.21151 0
1.7 0 −1.95667 0 −3.45496 0 1.56067 0 0.828557 0
1.8 0 −1.70694 0 1.80279 0 −0.510098 0 −0.0863705 0
1.9 0 −1.69973 0 2.39251 0 −2.42748 0 −0.110913 0
1.10 0 −1.53827 0 2.32384 0 4.76077 0 −0.633727 0
1.11 0 −1.12351 0 0.837743 0 −0.914625 0 −1.73772 0
1.12 0 −0.797542 0 −1.59101 0 1.45275 0 −2.36393 0
1.13 0 −0.513479 0 3.65851 0 −0.146814 0 −2.73634 0
1.14 0 −0.0957135 0 3.87548 0 3.14340 0 −2.99084 0
1.15 0 −0.0692070 0 0.775693 0 −4.19948 0 −2.99521 0
1.16 0 0.486172 0 −3.69982 0 0.0825276 0 −2.76364 0
1.17 0 0.679540 0 −1.36350 0 −2.99119 0 −2.53823 0
1.18 0 0.700584 0 −3.32459 0 −0.693647 0 −2.50918 0
1.19 0 1.07677 0 −3.37253 0 5.03712 0 −1.84058 0
1.20 0 1.18444 0 −0.0964158 0 −1.78364 0 −1.59711 0
See all 30 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.30
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(251\) \(-1\)

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 8032.2.a.i yes 30
4.b odd 2 1 8032.2.a.h ✓ 30
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8032.2.a.h ✓ 30 4.b odd 2 1
8032.2.a.i yes 30 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8032))\):

\( T_{3}^{30} - 3 T_{3}^{29} - 58 T_{3}^{28} + 173 T_{3}^{27} + 1490 T_{3}^{26} - 4418 T_{3}^{25} + \cdots - 14616 \) Copy content Toggle raw display
\( T_{7}^{30} - 13 T_{7}^{29} - 33 T_{7}^{28} + 1047 T_{7}^{27} - 1293 T_{7}^{26} - 35345 T_{7}^{25} + \cdots - 533733 \) Copy content Toggle raw display
\( T_{11}^{30} - 13 T_{11}^{29} - 101 T_{11}^{28} + 1880 T_{11}^{27} + 2741 T_{11}^{26} - 117520 T_{11}^{25} + \cdots + 6097731584 \) Copy content Toggle raw display