Properties

Label 76.2.a.a
Level $76$
Weight $2$
Character orbit 76.a
Self dual yes
Analytic conductor $0.607$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 76 = 2^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 76.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(0.606863055362\)
Analytic rank: \(0\)
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

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

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 2.00000 0 −1.00000 0 −3.00000 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(19\) \(1\)

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 76.2.a.a 1
3.b odd 2 1 684.2.a.b 1
4.b odd 2 1 304.2.a.a 1
5.b even 2 1 1900.2.a.b 1
5.c odd 4 2 1900.2.c.b 2
7.b odd 2 1 3724.2.a.a 1
8.b even 2 1 1216.2.a.c 1
8.d odd 2 1 1216.2.a.q 1
11.b odd 2 1 9196.2.a.f 1
12.b even 2 1 2736.2.a.q 1
19.b odd 2 1 1444.2.a.a 1
19.c even 3 2 1444.2.e.a 2
19.d odd 6 2 1444.2.e.c 2
20.d odd 2 1 7600.2.a.p 1
76.d even 2 1 5776.2.a.p 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
76.2.a.a 1 1.a even 1 1 trivial
304.2.a.a 1 4.b odd 2 1
684.2.a.b 1 3.b odd 2 1
1216.2.a.c 1 8.b even 2 1
1216.2.a.q 1 8.d odd 2 1
1444.2.a.a 1 19.b odd 2 1
1444.2.e.a 2 19.c even 3 2
1444.2.e.c 2 19.d odd 6 2
1900.2.a.b 1 5.b even 2 1
1900.2.c.b 2 5.c odd 4 2
2736.2.a.q 1 12.b even 2 1
3724.2.a.a 1 7.b odd 2 1
5776.2.a.p 1 76.d even 2 1
7600.2.a.p 1 20.d odd 2 1
9196.2.a.f 1 11.b odd 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(\Gamma_0(76))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \) Copy content Toggle raw display
$3$ \( T - 2 \) Copy content Toggle raw display
$5$ \( T + 1 \) Copy content Toggle raw display
$7$ \( T + 3 \) Copy content Toggle raw display
$11$ \( T - 5 \) Copy content Toggle raw display
$13$ \( T + 4 \) Copy content Toggle raw display
$17$ \( T + 3 \) Copy content Toggle raw display
$19$ \( T + 1 \) Copy content Toggle raw display
$23$ \( T - 8 \) Copy content Toggle raw display
$29$ \( T + 2 \) Copy content Toggle raw display
$31$ \( T - 4 \) Copy content Toggle raw display
$37$ \( T - 10 \) Copy content Toggle raw display
$41$ \( T - 10 \) Copy content Toggle raw display
$43$ \( T - 1 \) Copy content Toggle raw display
$47$ \( T + 1 \) Copy content Toggle raw display
$53$ \( T + 4 \) Copy content Toggle raw display
$59$ \( T - 6 \) Copy content Toggle raw display
$61$ \( T + 13 \) Copy content Toggle raw display
$67$ \( T + 12 \) Copy content Toggle raw display
$71$ \( T - 2 \) Copy content Toggle raw display
$73$ \( T - 9 \) Copy content Toggle raw display
$79$ \( T - 8 \) Copy content Toggle raw display
$83$ \( T + 12 \) Copy content Toggle raw display
$89$ \( T - 12 \) Copy content Toggle raw display
$97$ \( T + 8 \) Copy content Toggle raw display
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