Properties

Label 49.4.a.a
Level $49$
Weight $4$
Character orbit 49.a
Self dual yes
Analytic conductor $2.891$
Analytic rank $1$
Dimension $1$
CM discriminant -7
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 49 = 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 49.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(2.89109359028\)
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: $N(\mathrm{U}(1))$

$q$-expansion

\(f(q)\) \(=\) \( q - 5 q^{2} + 17 q^{4} - 45 q^{8} - 27 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 5 q^{2} + 17 q^{4} - 45 q^{8} - 27 q^{9} - 68 q^{11} + 89 q^{16} + 135 q^{18} + 340 q^{22} - 40 q^{23} - 125 q^{25} - 166 q^{29} - 85 q^{32} - 459 q^{36} + 450 q^{37} - 180 q^{43} - 1156 q^{44} + 200 q^{46} + 625 q^{50} + 590 q^{53} + 830 q^{58} - 287 q^{64} - 740 q^{67} + 688 q^{71} + 1215 q^{72} - 2250 q^{74} - 1384 q^{79} + 729 q^{81} + 900 q^{86} + 3060 q^{88} - 680 q^{92} + 1836 q^{99}+O(q^{100}) \) 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
−5.00000 0 17.0000 0 0 0 −45.0000 −27.0000 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 49.4.a.a 1
3.b odd 2 1 441.4.a.m 1
4.b odd 2 1 784.4.a.k 1
5.b even 2 1 1225.4.a.l 1
7.b odd 2 1 CM 49.4.a.a 1
7.c even 3 2 49.4.c.d 2
7.d odd 6 2 49.4.c.d 2
21.c even 2 1 441.4.a.m 1
21.g even 6 2 441.4.e.a 2
21.h odd 6 2 441.4.e.a 2
28.d even 2 1 784.4.a.k 1
35.c odd 2 1 1225.4.a.l 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
49.4.a.a 1 1.a even 1 1 trivial
49.4.a.a 1 7.b odd 2 1 CM
49.4.c.d 2 7.c even 3 2
49.4.c.d 2 7.d odd 6 2
441.4.a.m 1 3.b odd 2 1
441.4.a.m 1 21.c even 2 1
441.4.e.a 2 21.g even 6 2
441.4.e.a 2 21.h odd 6 2
784.4.a.k 1 4.b odd 2 1
784.4.a.k 1 28.d even 2 1
1225.4.a.l 1 5.b even 2 1
1225.4.a.l 1 35.c odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(49))\):

\( T_{2} + 5 \) Copy content Toggle raw display
\( T_{3} \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T + 5 \) 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
$11$ \( T + 68 \) 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
$23$ \( T + 40 \) Copy content Toggle raw display
$29$ \( T + 166 \) Copy content Toggle raw display
$31$ \( T \) Copy content Toggle raw display
$37$ \( T - 450 \) Copy content Toggle raw display
$41$ \( T \) Copy content Toggle raw display
$43$ \( T + 180 \) Copy content Toggle raw display
$47$ \( T \) Copy content Toggle raw display
$53$ \( T - 590 \) Copy content Toggle raw display
$59$ \( T \) Copy content Toggle raw display
$61$ \( T \) Copy content Toggle raw display
$67$ \( T + 740 \) Copy content Toggle raw display
$71$ \( T - 688 \) Copy content Toggle raw display
$73$ \( T \) Copy content Toggle raw display
$79$ \( T + 1384 \) 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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