Properties

Label 4004.2.a.d
Level 4004
Weight 2
Character orbit 4004.a
Self dual yes
Analytic conductor 31.972
Analytic rank 1
Dimension 4
CM no
Inner twists 1

Related objects

Downloads

Learn more about

Newspace parameters

Level: \( N \) = \( 4004 = 2^{2} \cdot 7 \cdot 11 \cdot 13 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 4004.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(31.9721009693\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: 4.4.3981.1
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -1 - \beta_{3} ) q^{3} + ( \beta_{2} + \beta_{3} ) q^{5} + q^{7} + ( 1 - \beta_{1} + 2 \beta_{3} ) q^{9} +O(q^{10})\) \( q + ( -1 - \beta_{3} ) q^{3} + ( \beta_{2} + \beta_{3} ) q^{5} + q^{7} + ( 1 - \beta_{1} + 2 \beta_{3} ) q^{9} + q^{11} + q^{13} + ( -1 - \beta_{1} ) q^{15} + ( -2 - \beta_{2} - \beta_{3} ) q^{17} + ( -2 - \beta_{1} - \beta_{2} ) q^{19} + ( -1 - \beta_{3} ) q^{21} + ( 3 \beta_{1} - \beta_{2} ) q^{23} + ( -2 + 2 \beta_{1} - 2 \beta_{2} - 3 \beta_{3} ) q^{25} + ( -3 + 3 \beta_{1} + \beta_{2} ) q^{27} + ( -2 + 3 \beta_{1} - 2 \beta_{3} ) q^{29} + ( -1 + 2 \beta_{1} - 2 \beta_{3} ) q^{31} + ( -1 - \beta_{3} ) q^{33} + ( \beta_{2} + \beta_{3} ) q^{35} + ( -3 - 3 \beta_{1} + 3 \beta_{3} ) q^{37} + ( -1 - \beta_{3} ) q^{39} + ( -3 - 2 \beta_{1} - 3 \beta_{2} ) q^{41} + ( 1 - 3 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} ) q^{43} + ( 2 + \beta_{1} - 2 \beta_{2} - 2 \beta_{3} ) q^{45} + ( -3 \beta_{1} - 3 \beta_{2} ) q^{47} + q^{49} + ( 3 + \beta_{1} + 2 \beta_{3} ) q^{51} + ( \beta_{1} + \beta_{2} - 3 \beta_{3} ) q^{53} + ( \beta_{2} + \beta_{3} ) q^{55} + ( 1 + 3 \beta_{1} + \beta_{2} + \beta_{3} ) q^{57} + ( -1 - \beta_{1} + 2 \beta_{2} + 2 \beta_{3} ) q^{59} + ( -4 + 3 \beta_{1} + 4 \beta_{2} + \beta_{3} ) q^{61} + ( 1 - \beta_{1} + 2 \beta_{3} ) q^{63} + ( \beta_{2} + \beta_{3} ) q^{65} + ( 1 - 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} ) q^{67} + ( -5 - \beta_{1} - 3 \beta_{2} - \beta_{3} ) q^{69} + ( 2 - 4 \beta_{1} + 4 \beta_{3} ) q^{71} + ( -4 + 7 \beta_{1} + 2 \beta_{2} - 3 \beta_{3} ) q^{73} + ( 5 - \beta_{1} - 2 \beta_{2} + 3 \beta_{3} ) q^{75} + q^{77} + ( 3 - \beta_{2} + 5 \beta_{3} ) q^{79} + ( -1 - 2 \beta_{1} - 3 \beta_{2} - 2 \beta_{3} ) q^{81} + ( 4 - 7 \beta_{1} + \beta_{3} ) q^{83} + ( -3 - 2 \beta_{1} + \beta_{3} ) q^{85} + ( 5 - 5 \beta_{1} - 3 \beta_{2} + 4 \beta_{3} ) q^{87} + ( 4 - 5 \beta_{1} + 3 \beta_{2} + 2 \beta_{3} ) q^{89} + q^{91} + ( 5 - 4 \beta_{1} - 2 \beta_{2} + 3 \beta_{3} ) q^{93} + ( -2 - 2 \beta_{1} - 2 \beta_{2} - \beta_{3} ) q^{95} + ( -4 - 2 \beta_{1} - \beta_{2} - 2 \beta_{3} ) q^{97} + ( 1 - \beta_{1} + 2 \beta_{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 3q^{3} - q^{5} + 4q^{7} + q^{9} + O(q^{10}) \) \( 4q - 3q^{3} - q^{5} + 4q^{7} + q^{9} + 4q^{11} + 4q^{13} - 5q^{15} - 7q^{17} - 9q^{19} - 3q^{21} + 3q^{23} - 3q^{25} - 9q^{27} - 3q^{29} - 3q^{33} - q^{35} - 18q^{37} - 3q^{39} - 14q^{41} - q^{43} + 11q^{45} - 3q^{47} + 4q^{49} + 11q^{51} + 4q^{53} - q^{55} + 6q^{57} - 7q^{59} - 14q^{61} + q^{63} - q^{65} - 20q^{69} - 6q^{73} + 16q^{75} + 4q^{77} + 7q^{79} - 4q^{81} + 8q^{83} - 15q^{85} + 11q^{87} + 9q^{89} + 4q^{91} + 13q^{93} - 9q^{95} - 16q^{97} + q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - x^{3} - 4 x^{2} + 2 x + 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - \nu - 2 \)
\(\beta_{3}\)\(=\)\( \nu^{3} - \nu^{2} - 3 \nu + 1 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + \beta_{1} + 2\)
\(\nu^{3}\)\(=\)\(\beta_{3} + \beta_{2} + 4 \beta_{1} + 1\)

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
−0.318459
2.28400
0.785261
−1.75080
0 −2.82166 0 0.241539 0 1.00000 0 4.96179 0
1.2 0 −1.84617 0 1.77882 0 1.00000 0 0.408340 0
1.3 0 0.488200 0 −3.65683 0 1.00000 0 −2.76166 0
1.4 0 1.17963 0 0.636469 0 1.00000 0 −1.60847 0
\(n\): e.g. 2-40 or 990-1000
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 4004.2.a.d 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4004.2.a.d 4 1.a even 1 1 trivial

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(7\) \(-1\)
\(11\) \(-1\)
\(13\) \(-1\)

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{4} + 3 T_{3}^{3} - 2 T_{3}^{2} - 6 T_{3} + 3 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4004))\).