Properties

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

Related objects

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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: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

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

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
0 1.00000 0 −1.00000 0 −3.00000 0 1.00000 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 804.2.a.d 1
3.b odd 2 1 2412.2.a.d 1
4.b odd 2 1 3216.2.a.c 1
12.b even 2 1 9648.2.a.m 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
804.2.a.d 1 1.a even 1 1 trivial
2412.2.a.d 1 3.b odd 2 1
3216.2.a.c 1 4.b odd 2 1
9648.2.a.m 1 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 \( T_{5} + 1 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(804))\).