Properties

Label 1225.1.g.b
Level $1225$
Weight $1$
Character orbit 1225.g
Analytic conductor $0.611$
Analytic rank $0$
Dimension $4$
Projective image $D_{6}$
CM discriminant -7
Inner twists $8$

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Newspace parameters

Level: \( N \) \(=\) \( 1225 = 5^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1225.g (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.611354640475\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(i)\)
Coefficient field: \(\Q(i, \sqrt{6})\)
Defining polynomial: \(x^{4} + 9\)
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image \(D_{6}\)
Projective field Galois closure of 6.2.153125.1

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\beta_{1} q^{2} + 2 \beta_{2} q^{4} -\beta_{3} q^{8} + \beta_{2} q^{9} +O(q^{10})\) \( q -\beta_{1} q^{2} + 2 \beta_{2} q^{4} -\beta_{3} q^{8} + \beta_{2} q^{9} - q^{11} - q^{16} -\beta_{3} q^{18} + \beta_{1} q^{22} + \beta_{3} q^{23} + \beta_{2} q^{29} -2 q^{36} + \beta_{1} q^{37} + \beta_{3} q^{43} -2 \beta_{2} q^{44} + 3 q^{46} -\beta_{3} q^{58} + \beta_{2} q^{64} + \beta_{1} q^{67} + q^{71} + \beta_{1} q^{72} -3 \beta_{2} q^{74} -\beta_{2} q^{79} - q^{81} + 3 q^{86} + \beta_{3} q^{88} -2 \beta_{1} q^{92} -\beta_{2} q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + O(q^{10}) \) \( 4q - 4q^{11} - 4q^{16} - 8q^{36} + 12q^{46} + 4q^{71} - 4q^{81} + 12q^{86} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} + 9\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} \)\(/3\)
\(\beta_{3}\)\(=\)\( \nu^{3} \)\(/3\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(3 \beta_{2}\)
\(\nu^{3}\)\(=\)\(3 \beta_{3}\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1225\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(1177\)
\(\chi(n)\) \(1\) \(\beta_{2}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
393.1
1.22474 1.22474i
−1.22474 + 1.22474i
1.22474 + 1.22474i
−1.22474 1.22474i
−1.22474 + 1.22474i 0 2.00000i 0 0 0 1.22474 + 1.22474i 1.00000i 0
393.2 1.22474 1.22474i 0 2.00000i 0 0 0 −1.22474 1.22474i 1.00000i 0
932.1 −1.22474 1.22474i 0 2.00000i 0 0 0 1.22474 1.22474i 1.00000i 0
932.2 1.22474 + 1.22474i 0 2.00000i 0 0 0 −1.22474 + 1.22474i 1.00000i 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 CM by \(\Q(\sqrt{-7}) \)
5.b even 2 1 inner
5.c odd 4 2 inner
35.c odd 2 1 inner
35.f even 4 2 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1225.1.g.b 4
5.b even 2 1 inner 1225.1.g.b 4
5.c odd 4 2 inner 1225.1.g.b 4
7.b odd 2 1 CM 1225.1.g.b 4
7.c even 3 2 1225.1.q.b 8
7.d odd 6 2 1225.1.q.b 8
35.c odd 2 1 inner 1225.1.g.b 4
35.f even 4 2 inner 1225.1.g.b 4
35.i odd 6 2 1225.1.q.b 8
35.j even 6 2 1225.1.q.b 8
35.k even 12 4 1225.1.q.b 8
35.l odd 12 4 1225.1.q.b 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1225.1.g.b 4 1.a even 1 1 trivial
1225.1.g.b 4 5.b even 2 1 inner
1225.1.g.b 4 5.c odd 4 2 inner
1225.1.g.b 4 7.b odd 2 1 CM
1225.1.g.b 4 35.c odd 2 1 inner
1225.1.g.b 4 35.f even 4 2 inner
1225.1.q.b 8 7.c even 3 2
1225.1.q.b 8 7.d odd 6 2
1225.1.q.b 8 35.i odd 6 2
1225.1.q.b 8 35.j even 6 2
1225.1.q.b 8 35.k even 12 4
1225.1.q.b 8 35.l odd 12 4

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{4} + 9 \) acting on \(S_{1}^{\mathrm{new}}(1225, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 9 + T^{4} \)
$3$ \( T^{4} \)
$5$ \( T^{4} \)
$7$ \( T^{4} \)
$11$ \( ( 1 + T )^{4} \)
$13$ \( T^{4} \)
$17$ \( T^{4} \)
$19$ \( T^{4} \)
$23$ \( 9 + T^{4} \)
$29$ \( ( 1 + T^{2} )^{2} \)
$31$ \( T^{4} \)
$37$ \( 9 + T^{4} \)
$41$ \( T^{4} \)
$43$ \( 9 + T^{4} \)
$47$ \( T^{4} \)
$53$ \( T^{4} \)
$59$ \( T^{4} \)
$61$ \( T^{4} \)
$67$ \( 9 + T^{4} \)
$71$ \( ( -1 + T )^{4} \)
$73$ \( T^{4} \)
$79$ \( ( 1 + T^{2} )^{2} \)
$83$ \( T^{4} \)
$89$ \( T^{4} \)
$97$ \( T^{4} \)
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