Properties

Label 1225.2.a.c
Level $1225$
Weight $2$
Character orbit 1225.a
Self dual yes
Analytic conductor $9.782$
Analytic rank $0$
Dimension $1$
CM discriminant -7
Inner twists $2$

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Newspace parameters

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

Newform invariants

Self dual: yes
Analytic conductor: \(9.78167424761\)
Analytic rank: \(0\)
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

\(f(q)\) \(=\) \( q - q^{2} - q^{4} + 3 q^{8} - 3 q^{9} + O(q^{10}) \) \( q - q^{2} - q^{4} + 3 q^{8} - 3 q^{9} + 4 q^{11} - q^{16} + 3 q^{18} - 4 q^{22} - 8 q^{23} + 2 q^{29} - 5 q^{32} + 3 q^{36} + 6 q^{37} + 12 q^{43} - 4 q^{44} + 8 q^{46} + 10 q^{53} - 2 q^{58} + 7 q^{64} - 4 q^{67} + 16 q^{71} - 9 q^{72} - 6 q^{74} + 8 q^{79} + 9 q^{81} - 12 q^{86} + 12 q^{88} + 8 q^{92} - 12 q^{99} + O(q^{100}) \)

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

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 1225.2.a.c 1
5.b even 2 1 49.2.a.a 1
5.c odd 4 2 1225.2.b.c 2
7.b odd 2 1 CM 1225.2.a.c 1
15.d odd 2 1 441.2.a.c 1
20.d odd 2 1 784.2.a.f 1
35.c odd 2 1 49.2.a.a 1
35.f even 4 2 1225.2.b.c 2
35.i odd 6 2 49.2.c.a 2
35.j even 6 2 49.2.c.a 2
40.e odd 2 1 3136.2.a.o 1
40.f even 2 1 3136.2.a.n 1
55.d odd 2 1 5929.2.a.c 1
60.h even 2 1 7056.2.a.bg 1
65.d even 2 1 8281.2.a.d 1
105.g even 2 1 441.2.a.c 1
105.o odd 6 2 441.2.e.d 2
105.p even 6 2 441.2.e.d 2
140.c even 2 1 784.2.a.f 1
140.p odd 6 2 784.2.i.f 2
140.s even 6 2 784.2.i.f 2
280.c odd 2 1 3136.2.a.n 1
280.n even 2 1 3136.2.a.o 1
385.h even 2 1 5929.2.a.c 1
420.o odd 2 1 7056.2.a.bg 1
455.h 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 5.b even 2 1
49.2.a.a 1 35.c odd 2 1
49.2.c.a 2 35.i odd 6 2
49.2.c.a 2 35.j even 6 2
441.2.a.c 1 15.d odd 2 1
441.2.a.c 1 105.g even 2 1
441.2.e.d 2 105.o odd 6 2
441.2.e.d 2 105.p even 6 2
784.2.a.f 1 20.d odd 2 1
784.2.a.f 1 140.c even 2 1
784.2.i.f 2 140.p odd 6 2
784.2.i.f 2 140.s even 6 2
1225.2.a.c 1 1.a even 1 1 trivial
1225.2.a.c 1 7.b odd 2 1 CM
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 40.f even 2 1
3136.2.a.n 1 280.c odd 2 1
3136.2.a.o 1 40.e odd 2 1
3136.2.a.o 1 280.n even 2 1
5929.2.a.c 1 55.d odd 2 1
5929.2.a.c 1 385.h even 2 1
7056.2.a.bg 1 60.h even 2 1
7056.2.a.bg 1 420.o odd 2 1
8281.2.a.d 1 65.d even 2 1
8281.2.a.d 1 455.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1225))\):

\( T_{2} + 1 \)
\( T_{3} \)

Hecke characteristic polynomials

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