Properties

Label 49.2.a.a
Level $49$
Weight $2$
Character orbit 49.a
Self dual yes
Analytic conductor $0.391$
Analytic rank $0$
Dimension $1$
CM discriminant -7
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 49 = 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 49.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(0.391266969904\)
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: $N(\mathrm{U}(1))$

$q$-expansion

\(f(q)\) \(=\) \( q + q^{2} - q^{4} - 3 q^{8} - 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{2} - q^{4} - 3 q^{8} - 3 q^{9} + 4 q^{11} - q^{16} - 3 q^{18} + 4 q^{22} + 8 q^{23} - 5 q^{25} + 2 q^{29} + 5 q^{32} + 3 q^{36} - 6 q^{37} - 12 q^{43} - 4 q^{44} + 8 q^{46} - 5 q^{50} - 10 q^{53} + 2 q^{58} + 7 q^{64} + 4 q^{67} + 16 q^{71} + 9 q^{72} - 6 q^{74} + 8 q^{79} + 9 q^{81} - 12 q^{86} - 12 q^{88} - 8 q^{92} - 12 q^{99}+O(q^{100}) \) 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
1.00000 0 −1.00000 0 0 0 −3.00000 −3.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(7\) \(-1\)

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 CM by \(\Q(\sqrt{-7}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 49.2.a.a 1
3.b odd 2 1 441.2.a.c 1
4.b odd 2 1 784.2.a.f 1
5.b even 2 1 1225.2.a.c 1
5.c odd 4 2 1225.2.b.c 2
7.b odd 2 1 CM 49.2.a.a 1
7.c even 3 2 49.2.c.a 2
7.d odd 6 2 49.2.c.a 2
8.b even 2 1 3136.2.a.n 1
8.d odd 2 1 3136.2.a.o 1
11.b odd 2 1 5929.2.a.c 1
12.b even 2 1 7056.2.a.bg 1
13.b even 2 1 8281.2.a.d 1
21.c even 2 1 441.2.a.c 1
21.g even 6 2 441.2.e.d 2
21.h odd 6 2 441.2.e.d 2
28.d even 2 1 784.2.a.f 1
28.f even 6 2 784.2.i.f 2
28.g odd 6 2 784.2.i.f 2
35.c odd 2 1 1225.2.a.c 1
35.f even 4 2 1225.2.b.c 2
56.e even 2 1 3136.2.a.o 1
56.h odd 2 1 3136.2.a.n 1
77.b even 2 1 5929.2.a.c 1
84.h odd 2 1 7056.2.a.bg 1
91.b odd 2 1 8281.2.a.d 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
49.2.a.a 1 1.a even 1 1 trivial
49.2.a.a 1 7.b odd 2 1 CM
49.2.c.a 2 7.c even 3 2
49.2.c.a 2 7.d odd 6 2
441.2.a.c 1 3.b odd 2 1
441.2.a.c 1 21.c even 2 1
441.2.e.d 2 21.g even 6 2
441.2.e.d 2 21.h odd 6 2
784.2.a.f 1 4.b odd 2 1
784.2.a.f 1 28.d even 2 1
784.2.i.f 2 28.f even 6 2
784.2.i.f 2 28.g odd 6 2
1225.2.a.c 1 5.b even 2 1
1225.2.a.c 1 35.c odd 2 1
1225.2.b.c 2 5.c odd 4 2
1225.2.b.c 2 35.f even 4 2
3136.2.a.n 1 8.b even 2 1
3136.2.a.n 1 56.h odd 2 1
3136.2.a.o 1 8.d odd 2 1
3136.2.a.o 1 56.e even 2 1
5929.2.a.c 1 11.b odd 2 1
5929.2.a.c 1 77.b even 2 1
7056.2.a.bg 1 12.b even 2 1
7056.2.a.bg 1 84.h odd 2 1
8281.2.a.d 1 13.b even 2 1
8281.2.a.d 1 91.b odd 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(\Gamma_0(49))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T - 1 \) Copy content Toggle raw display
$3$ \( T \) Copy content Toggle raw display
$5$ \( T \) Copy content Toggle raw display
$7$ \( T \) Copy content Toggle raw display
$11$ \( T - 4 \) Copy content Toggle raw display
$13$ \( T \) Copy content Toggle raw display
$17$ \( T \) Copy content Toggle raw display
$19$ \( T \) Copy content Toggle raw display
$23$ \( T - 8 \) Copy content Toggle raw display
$29$ \( T - 2 \) Copy content Toggle raw display
$31$ \( T \) Copy content Toggle raw display
$37$ \( T + 6 \) Copy content Toggle raw display
$41$ \( T \) Copy content Toggle raw display
$43$ \( T + 12 \) Copy content Toggle raw display
$47$ \( T \) Copy content Toggle raw display
$53$ \( T + 10 \) Copy content Toggle raw display
$59$ \( T \) Copy content Toggle raw display
$61$ \( T \) Copy content Toggle raw display
$67$ \( T - 4 \) Copy content Toggle raw display
$71$ \( T - 16 \) Copy content Toggle raw display
$73$ \( T \) Copy content Toggle raw display
$79$ \( T - 8 \) Copy content Toggle raw display
$83$ \( T \) Copy content Toggle raw display
$89$ \( T \) Copy content Toggle raw display
$97$ \( T \) Copy content Toggle raw display
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