Properties

Label 4008.2.a.g
Level 4008
Weight 2
Character orbit 4008.a
Self dual yes
Analytic conductor 32.004
Analytic rank 0
Dimension 7
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 4008 = 2^{3} \cdot 3 \cdot 167 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 4008.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(32.0040411301\)
Analytic rank: \(0\)
Dimension: \(7\)
Coefficient field: \(\mathbb{Q}[x]/(x^{7} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{6}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{3} -\beta_{1} q^{5} + ( 1 + \beta_{2} ) q^{7} + q^{9} +O(q^{10})\) \( q - q^{3} -\beta_{1} q^{5} + ( 1 + \beta_{2} ) q^{7} + q^{9} + ( -\beta_{1} + \beta_{3} + \beta_{4} ) q^{11} + ( -\beta_{1} + \beta_{3} + \beta_{4} - \beta_{5} ) q^{13} + \beta_{1} q^{15} + ( 1 + \beta_{2} + \beta_{4} ) q^{17} + ( 1 - 2 \beta_{1} + \beta_{3} - \beta_{5} + \beta_{6} ) q^{19} + ( -1 - \beta_{2} ) q^{21} + ( 2 + \beta_{4} ) q^{23} + ( \beta_{3} + \beta_{4} + \beta_{5} - \beta_{6} ) q^{25} - q^{27} + ( -1 + 2 \beta_{2} + \beta_{3} - \beta_{4} - \beta_{5} + \beta_{6} ) q^{29} + ( 2 - \beta_{1} - \beta_{3} ) q^{31} + ( \beta_{1} - \beta_{3} - \beta_{4} ) q^{33} + ( 3 - 2 \beta_{1} - 2 \beta_{2} - \beta_{6} ) q^{35} + ( -4 - \beta_{2} - \beta_{3} + 2 \beta_{5} + \beta_{6} ) q^{37} + ( \beta_{1} - \beta_{3} - \beta_{4} + \beta_{5} ) q^{39} + ( 1 + \beta_{1} + 2 \beta_{2} - \beta_{3} - 2 \beta_{4} + \beta_{6} ) q^{41} + ( 2 \beta_{2} - 2 \beta_{3} - \beta_{4} - \beta_{5} - 2 \beta_{6} ) q^{43} -\beta_{1} q^{45} + ( 4 - \beta_{2} + \beta_{3} - \beta_{4} + \beta_{5} - \beta_{6} ) q^{47} + ( -2 + \beta_{1} + \beta_{2} + \beta_{5} ) q^{49} + ( -1 - \beta_{2} - \beta_{4} ) q^{51} + ( 1 + \beta_{1} + 2 \beta_{2} - \beta_{3} - 2 \beta_{4} - 3 \beta_{5} - \beta_{6} ) q^{53} + ( 3 - 2 \beta_{1} - 2 \beta_{2} - \beta_{3} + \beta_{4} + \beta_{5} - \beta_{6} ) q^{55} + ( -1 + 2 \beta_{1} - \beta_{3} + \beta_{5} - \beta_{6} ) q^{57} + ( 1 + \beta_{1} - \beta_{3} - \beta_{5} - \beta_{6} ) q^{59} + ( -3 + \beta_{1} + 2 \beta_{2} + 2 \beta_{3} + \beta_{5} - \beta_{6} ) q^{61} + ( 1 + \beta_{2} ) q^{63} + ( -\beta_{2} - \beta_{3} + \beta_{5} + \beta_{6} ) q^{65} + ( -\beta_{1} + \beta_{3} - \beta_{4} + \beta_{5} + 2 \beta_{6} ) q^{67} + ( -2 - \beta_{4} ) q^{69} + ( 4 - \beta_{3} + \beta_{4} - \beta_{5} ) q^{71} + ( -2 - \beta_{1} - 3 \beta_{2} - \beta_{3} + 3 \beta_{6} ) q^{73} + ( -\beta_{3} - \beta_{4} - \beta_{5} + \beta_{6} ) q^{75} + ( 1 - 2 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} + 2 \beta_{4} + 2 \beta_{5} + \beta_{6} ) q^{77} + ( -\beta_{3} + \beta_{5} + 2 \beta_{6} ) q^{79} + q^{81} + ( 1 - \beta_{1} + 2 \beta_{3} - 2 \beta_{4} + 5 \beta_{6} ) q^{83} + ( 2 - 3 \beta_{1} - 4 \beta_{2} - \beta_{3} + \beta_{4} - 2 \beta_{6} ) q^{85} + ( 1 - 2 \beta_{2} - \beta_{3} + \beta_{4} + \beta_{5} - \beta_{6} ) q^{87} + ( 3 + 2 \beta_{1} - \beta_{3} - \beta_{5} - 3 \beta_{6} ) q^{89} + ( -1 - 2 \beta_{1} - 3 \beta_{2} + \beta_{3} + \beta_{4} + \beta_{5} + 2 \beta_{6} ) q^{91} + ( -2 + \beta_{1} + \beta_{3} ) q^{93} + ( 5 + \beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} ) q^{95} + ( 1 + 2 \beta_{1} + 2 \beta_{2} + \beta_{4} - 2 \beta_{5} - 3 \beta_{6} ) q^{97} + ( -\beta_{1} + \beta_{3} + \beta_{4} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 7q - 7q^{3} - 3q^{5} + 8q^{7} + 7q^{9} + O(q^{10}) \) \( 7q - 7q^{3} - 3q^{5} + 8q^{7} + 7q^{9} + q^{11} - 2q^{13} + 3q^{15} + 11q^{17} + 2q^{19} - 8q^{21} + 17q^{23} + 4q^{25} - 7q^{27} - 7q^{29} + 10q^{31} - q^{33} + 10q^{35} - 21q^{37} + 2q^{39} + 8q^{41} - 12q^{43} - 3q^{45} + 25q^{47} - 7q^{49} - 11q^{51} - 7q^{53} + 15q^{55} - 2q^{57} + 3q^{59} - 14q^{61} + 8q^{63} + 4q^{65} + 4q^{67} - 17q^{69} + 27q^{71} - 12q^{73} - 4q^{75} + 16q^{77} + 8q^{79} + 7q^{81} + 15q^{83} - 3q^{85} + 7q^{87} + 14q^{89} - 3q^{91} - 10q^{93} + 37q^{95} + 3q^{97} + q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{7} - 2 x^{6} - 7 x^{5} + 11 x^{4} + 12 x^{3} - 14 x^{2} - 6 x + 3\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( -\nu^{5} + \nu^{4} + 7 \nu^{3} - 2 \nu^{2} - 8 \nu - 1 \)
\(\beta_{2}\)\(=\)\( \nu^{5} - \nu^{4} - 7 \nu^{3} + 3 \nu^{2} + 8 \nu - 1 \)
\(\beta_{3}\)\(=\)\( -\nu^{6} + \nu^{5} + 7 \nu^{4} - 2 \nu^{3} - 9 \nu^{2} - \nu + 1 \)
\(\beta_{4}\)\(=\)\( \nu^{6} - \nu^{5} - 7 \nu^{4} + 2 \nu^{3} + 9 \nu^{2} + 3 \nu - 1 \)
\(\beta_{5}\)\(=\)\( -\nu^{6} + \nu^{5} + 8 \nu^{4} - 3 \nu^{3} - 15 \nu^{2} + 6 \)
\(\beta_{6}\)\(=\)\( 2 \nu^{6} - 2 \nu^{5} - 15 \nu^{4} + 6 \nu^{3} + 23 \nu^{2} - 2 \nu - 5 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{4} + \beta_{3}\)\()/2\)
\(\nu^{2}\)\(=\)\(\beta_{2} + \beta_{1} + 2\)
\(\nu^{3}\)\(=\)\((\)\(2 \beta_{6} + 2 \beta_{5} + 3 \beta_{4} + 5 \beta_{3} + 2 \beta_{2} + 2 \beta_{1}\)\()/2\)
\(\nu^{4}\)\(=\)\(\beta_{6} + 2 \beta_{5} + \beta_{4} + \beta_{3} + 7 \beta_{2} + 7 \beta_{1} + 7\)
\(\nu^{5}\)\(=\)\((\)\(16 \beta_{6} + 18 \beta_{5} + 15 \beta_{4} + 29 \beta_{3} + 24 \beta_{2} + 22 \beta_{1} + 4\)\()/2\)
\(\nu^{6}\)\(=\)\(13 \beta_{6} + 21 \beta_{5} + 11 \beta_{4} + 15 \beta_{3} + 50 \beta_{2} + 49 \beta_{1} + 34\)

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.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
2.80982
1.47270
−0.674271
−2.05123
1.31154
−1.20126
0.332704
0 −1.00000 0 −3.20729 0 3.68779 0 1.00000 0
1.2 0 −1.00000 0 −3.01562 0 −1.84677 0 1.00000 0
1.3 0 −1.00000 0 −1.68509 0 −2.23045 0 1.00000 0
1.4 0 −1.00000 0 −0.597616 0 2.60994 0 1.00000 0
1.5 0 −1.00000 0 0.0621653 0 0.782308 0 1.00000 0
1.6 0 −1.00000 0 1.82640 0 2.26944 0 1.00000 0
1.7 0 −1.00000 0 3.61705 0 2.72774 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.7
Significant digits:
Format:

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 4008.2.a.g 7
4.b odd 2 1 8016.2.a.w 7
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4008.2.a.g 7 1.a even 1 1 trivial
8016.2.a.w 7 4.b odd 2 1

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(1\)
\(167\) \(1\)

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4008))\):

\( T_{5}^{7} + 3 T_{5}^{6} - 15 T_{5}^{5} - 50 T_{5}^{4} + 23 T_{5}^{3} + 133 T_{5}^{2} + 56 T_{5} - 4 \)
\( T_{7}^{7} - 8 T_{7}^{6} + 11 T_{7}^{5} + 55 T_{7}^{4} - 147 T_{7}^{3} - 37 T_{7}^{2} + 336 T_{7} - 192 \)