Properties

Label 804.2.a.a
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

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

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

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( 1 + T \)
$5$ \( 1 + 3 T + 5 T^{2} \)
$7$ \( 1 - 3 T + 7 T^{2} \)
$11$ \( 1 + 2 T + 11 T^{2} \)
$13$ \( 1 - 2 T + 13 T^{2} \)
$17$ \( 1 + 17 T^{2} \)
$19$ \( 1 + 4 T + 19 T^{2} \)
$23$ \( 1 + 5 T + 23 T^{2} \)
$29$ \( 1 + 4 T + 29 T^{2} \)
$31$ \( 1 - 5 T + 31 T^{2} \)
$37$ \( 1 + 11 T + 37 T^{2} \)
$41$ \( 1 + 5 T + 41 T^{2} \)
$43$ \( 1 + 5 T + 43 T^{2} \)
$47$ \( 1 + 8 T + 47 T^{2} \)
$53$ \( 1 - T + 53 T^{2} \)
$59$ \( 1 - 9 T + 59 T^{2} \)
$61$ \( 1 + 12 T + 61 T^{2} \)
$67$ \( 1 - T \)
$71$ \( 1 + 71 T^{2} \)
$73$ \( 1 + 9 T + 73 T^{2} \)
$79$ \( 1 + 79 T^{2} \)
$83$ \( 1 + T + 83 T^{2} \)
$89$ \( 1 - 14 T + 89 T^{2} \)
$97$ \( 1 + 6 T + 97 T^{2} \)
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