Properties

Label 1225.4.a.l
Level $1225$
Weight $4$
Character orbit 1225.a
Self dual yes
Analytic conductor $72.277$
Analytic rank $1$
Dimension $1$
CM discriminant -7
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1225 = 5^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1225.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(72.2773397570\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 49)
Fricke sign: \(-1\)
Sato-Tate group: $N(\mathrm{U}(1))$

$q$-expansion

\(f(q)\) \(=\) \( q + 5q^{2} + 17q^{4} + 45q^{8} - 27q^{9} + O(q^{10}) \) \( q + 5q^{2} + 17q^{4} + 45q^{8} - 27q^{9} - 68q^{11} + 89q^{16} - 135q^{18} - 340q^{22} + 40q^{23} - 166q^{29} + 85q^{32} - 459q^{36} - 450q^{37} + 180q^{43} - 1156q^{44} + 200q^{46} - 590q^{53} - 830q^{58} - 287q^{64} + 740q^{67} + 688q^{71} - 1215q^{72} - 2250q^{74} - 1384q^{79} + 729q^{81} + 900q^{86} - 3060q^{88} + 680q^{92} + 1836q^{99} + O(q^{100}) \)

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
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
\(5\) \(1\)
\(7\) \(-1\)

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 1225.4.a.l 1
5.b even 2 1 49.4.a.a 1
7.b odd 2 1 CM 1225.4.a.l 1
15.d odd 2 1 441.4.a.m 1
20.d odd 2 1 784.4.a.k 1
35.c odd 2 1 49.4.a.a 1
35.i odd 6 2 49.4.c.d 2
35.j even 6 2 49.4.c.d 2
105.g even 2 1 441.4.a.m 1
105.o odd 6 2 441.4.e.a 2
105.p even 6 2 441.4.e.a 2
140.c even 2 1 784.4.a.k 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
49.4.a.a 1 5.b even 2 1
49.4.a.a 1 35.c odd 2 1
49.4.c.d 2 35.i odd 6 2
49.4.c.d 2 35.j even 6 2
441.4.a.m 1 15.d odd 2 1
441.4.a.m 1 105.g even 2 1
441.4.e.a 2 105.o odd 6 2
441.4.e.a 2 105.p even 6 2
784.4.a.k 1 20.d odd 2 1
784.4.a.k 1 140.c even 2 1
1225.4.a.l 1 1.a even 1 1 trivial
1225.4.a.l 1 7.b odd 2 1 CM

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1225))\):

\( T_{2} - 5 \)
\( T_{3} \)
\( T_{19} \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -5 + T \)
$3$ \( T \)
$5$ \( T \)
$7$ \( T \)
$11$ \( 68 + T \)
$13$ \( T \)
$17$ \( T \)
$19$ \( T \)
$23$ \( -40 + T \)
$29$ \( 166 + T \)
$31$ \( T \)
$37$ \( 450 + T \)
$41$ \( T \)
$43$ \( -180 + T \)
$47$ \( T \)
$53$ \( 590 + T \)
$59$ \( T \)
$61$ \( T \)
$67$ \( -740 + T \)
$71$ \( -688 + T \)
$73$ \( T \)
$79$ \( 1384 + T \)
$83$ \( T \)
$89$ \( T \)
$97$ \( T \)
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