Properties

Label 804.2.a.f
Level 804
Weight 2
Character orbit 804.a
Self dual yes
Analytic conductor 6.420
Analytic rank 0
Dimension 5
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 804 = 2^{2} \cdot 3 \cdot 67 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 804.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(6.41997232251\)
Analytic rank: \(0\)
Dimension: \(5\)
Coefficient field: 5.5.24571284.1
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
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 \(1,\beta_1,\beta_2,\beta_3,\beta_4\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{3} + ( 1 - \beta_{1} ) q^{5} + ( 1 - \beta_{2} ) q^{7} + q^{9} +O(q^{10})\) \( q + q^{3} + ( 1 - \beta_{1} ) q^{5} + ( 1 - \beta_{2} ) q^{7} + q^{9} + ( 1 + \beta_{4} ) q^{11} + ( 1 - \beta_{4} ) q^{13} + ( 1 - \beta_{1} ) q^{15} + ( \beta_{2} - \beta_{3} ) q^{17} + ( 2 + 2 \beta_{1} + \beta_{2} + \beta_{3} ) q^{19} + ( 1 - \beta_{2} ) q^{21} + ( \beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} ) q^{23} + ( 2 + \beta_{2} ) q^{25} + q^{27} + ( 1 + \beta_{2} + \beta_{3} - \beta_{4} ) q^{29} + ( -\beta_{2} - 2 \beta_{3} + \beta_{4} ) q^{31} + ( 1 + \beta_{4} ) q^{33} + ( -2 - \beta_{1} - 2 \beta_{2} - 2 \beta_{3} + \beta_{4} ) q^{35} + ( 2 - \beta_{3} + \beta_{4} ) q^{37} + ( 1 - \beta_{4} ) q^{39} + ( 2 + \beta_{1} + 2 \beta_{3} + \beta_{4} ) q^{41} + ( 2 \beta_{1} + \beta_{2} - \beta_{4} ) q^{43} + ( 1 - \beta_{1} ) q^{45} + ( -2 - \beta_{2} + \beta_{3} - 2 \beta_{4} ) q^{47} + ( 5 - \beta_{2} + 2 \beta_{3} + \beta_{4} ) q^{49} + ( \beta_{2} - \beta_{3} ) q^{51} + ( -2 + \beta_{1} - \beta_{4} ) q^{53} + ( 3 + 2 \beta_{3} + \beta_{4} ) q^{55} + ( 2 + 2 \beta_{1} + \beta_{2} + \beta_{3} ) q^{57} + ( -3 + \beta_{1} - \beta_{2} - \beta_{3} ) q^{59} + ( 1 - 2 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} + \beta_{4} ) q^{61} + ( 1 - \beta_{2} ) q^{63} + ( -1 - 2 \beta_{1} - 2 \beta_{3} - \beta_{4} ) q^{65} + q^{67} + ( \beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} ) q^{69} + ( -5 + \beta_{4} ) q^{71} + ( 3 - 2 \beta_{1} - \beta_{3} ) q^{73} + ( 2 + \beta_{2} ) q^{75} + ( -1 - 2 \beta_{1} - 4 \beta_{2} - 2 \beta_{3} + 3 \beta_{4} ) q^{77} + ( -2 \beta_{1} + 2 \beta_{2} ) q^{79} + q^{81} + ( -4 - 3 \beta_{1} - \beta_{4} ) q^{83} + ( 1 - 2 \beta_{1} + 2 \beta_{3} - 3 \beta_{4} ) q^{85} + ( 1 + \beta_{2} + \beta_{3} - \beta_{4} ) q^{87} + ( -2 + 2 \beta_{1} + \beta_{2} - \beta_{3} ) q^{89} + ( 3 + 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} - 3 \beta_{4} ) q^{91} + ( -\beta_{2} - 2 \beta_{3} + \beta_{4} ) q^{93} + ( -5 - 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} + \beta_{4} ) q^{95} + ( 3 - 2 \beta_{1} + 2 \beta_{2} - 3 \beta_{4} ) q^{97} + ( 1 + \beta_{4} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5q + 5q^{3} + 3q^{5} + 5q^{7} + 5q^{9} + O(q^{10}) \) \( 5q + 5q^{3} + 3q^{5} + 5q^{7} + 5q^{9} + 6q^{11} + 4q^{13} + 3q^{15} + q^{17} + 13q^{19} + 5q^{21} + 10q^{25} + 5q^{27} + 3q^{29} + 3q^{31} + 6q^{33} - 9q^{35} + 12q^{37} + 4q^{39} + 11q^{41} + 3q^{43} + 3q^{45} - 13q^{47} + 24q^{49} + q^{51} - 9q^{53} + 14q^{55} + 13q^{57} - 12q^{59} + 4q^{61} + 5q^{63} - 8q^{65} + 5q^{67} - 24q^{71} + 12q^{73} + 10q^{75} - 4q^{77} - 4q^{79} + 5q^{81} - 27q^{83} - 4q^{85} + 3q^{87} - 5q^{89} + 14q^{91} + 3q^{93} - 30q^{95} + 8q^{97} + 6q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{5} - 2 x^{4} - 15 x^{3} + 10 x^{2} + 64 x + 38\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 2 \nu - 6 \)
\(\beta_{3}\)\(=\)\((\)\( \nu^{4} - 5 \nu^{3} - 8 \nu^{2} + 34 \nu + 34 \)\()/4\)
\(\beta_{4}\)\(=\)\((\)\( \nu^{4} - 3 \nu^{3} - 10 \nu^{2} + 18 \nu + 28 \)\()/2\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + 2 \beta_{1} + 6\)
\(\nu^{3}\)\(=\)\(\beta_{4} - 2 \beta_{3} + \beta_{2} + 10 \beta_{1} + 9\)
\(\nu^{4}\)\(=\)\(5 \beta_{4} - 6 \beta_{3} + 13 \beta_{2} + 32 \beta_{1} + 59\)

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
3.96721
2.95721
−0.789611
−1.65253
−2.48228
0 1.00000 0 −2.96721 0 −0.804360 0 1.00000 0
1.2 0 1.00000 0 −1.95721 0 4.16931 0 1.00000 0
1.3 0 1.00000 0 1.78961 0 4.79729 0 1.00000 0
1.4 0 1.00000 0 2.65253 0 0.964067 0 1.00000 0
1.5 0 1.00000 0 3.48228 0 −4.12631 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.5
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 804.2.a.f 5
3.b odd 2 1 2412.2.a.j 5
4.b odd 2 1 3216.2.a.ba 5
12.b even 2 1 9648.2.a.ca 5
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
804.2.a.f 5 1.a even 1 1 trivial
2412.2.a.j 5 3.b odd 2 1
3216.2.a.ba 5 4.b odd 2 1
9648.2.a.ca 5 12.b even 2 1

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(-1\)
\(67\) \(-1\)

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{5} - 3 T_{5}^{4} - 13 T_{5}^{3} + 37 T_{5}^{2} + 36 T_{5} - 96 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(804))\).