Properties

Label 1225.1.g.a
Level 1225
Weight 1
Character orbit 1225.g
Analytic conductor 0.611
Analytic rank 0
Dimension 2
Projective image \(D_{2}\)
CM/RM discs -7, -35, 5
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: \(2\)
Coefficient field: \(\Q(i)\)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image \(D_{2}\)
Projective field Galois closure of \(\Q(\sqrt{5}, \sqrt{-7})\)
Artin image $OD_{16}$
Artin field Galois closure of 8.4.9191328125.1

$q$-expansion

The \(q\)-expansion and trace form are shown below.

\(f(q)\) \(=\) \( q -i q^{4} + i q^{9} +O(q^{10})\) \( q -i q^{4} + i q^{9} + 2 q^{11} - q^{16} -2 i q^{29} + q^{36} -2 i q^{44} + i q^{64} -2 q^{71} + 2 i q^{79} - q^{81} + 2 i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + O(q^{10}) \) \( 2q + 4q^{11} - 2q^{16} + 2q^{36} - 4q^{71} - 2q^{81} + O(q^{100}) \)

Character values

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

\(n\) \(101\) \(1177\)
\(\chi(n)\) \(1\) \(i\)

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
1.00000i
1.00000i
0 0 1.00000i 0 0 0 0 1.00000i 0
932.1 0 0 1.00000i 0 0 0 0 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
5.b even 2 1 RM by \(\Q(\sqrt{5}) \)
7.b odd 2 1 CM by \(\Q(\sqrt{-7}) \)
35.c odd 2 1 CM by \(\Q(\sqrt{-35}) \)
5.c odd 4 2 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.a 2
5.b even 2 1 RM 1225.1.g.a 2
5.c odd 4 2 inner 1225.1.g.a 2
7.b odd 2 1 CM 1225.1.g.a 2
7.c even 3 2 1225.1.q.a 4
7.d odd 6 2 1225.1.q.a 4
35.c odd 2 1 CM 1225.1.g.a 2
35.f even 4 2 inner 1225.1.g.a 2
35.i odd 6 2 1225.1.q.a 4
35.j even 6 2 1225.1.q.a 4
35.k even 12 4 1225.1.q.a 4
35.l odd 12 4 1225.1.q.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1225.1.g.a 2 1.a even 1 1 trivial
1225.1.g.a 2 5.b even 2 1 RM
1225.1.g.a 2 5.c odd 4 2 inner
1225.1.g.a 2 7.b odd 2 1 CM
1225.1.g.a 2 35.c odd 2 1 CM
1225.1.g.a 2 35.f even 4 2 inner
1225.1.q.a 4 7.c even 3 2
1225.1.q.a 4 7.d odd 6 2
1225.1.q.a 4 35.i odd 6 2
1225.1.q.a 4 35.j even 6 2
1225.1.q.a 4 35.k even 12 4
1225.1.q.a 4 35.l odd 12 4

Hecke kernels

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

Hecke characteristic polynomials

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