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

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

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

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