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 disc. -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 \(4\) and degree \(2\))

Newform invariants

Self dual: No
Analytic conductor: \(0.611354640475\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 1 \)
Projective image \(D_{2}\)
Projective field Galois closure of \(\Q(\sqrt{5}, \sqrt{-7})\)
Artin image size \(16\)
Artin image $OD_{16}$
Artin field Galois closure of 8.4.9191328125.1

$q$-expansion

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

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

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. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
5.b Even 1 RM by \(\Q(\sqrt{5}) \) yes
7.b Odd 1 CM by \(\Q(\sqrt{-7}) \) yes
35.c Odd 1 CM by \(\Q(\sqrt{-35}) \) yes
5.c Odd 2 yes
35.f Even 2 yes

Hecke kernels

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