Properties

Label 225.1.g.a
Level 225
Weight 1
Character orbit 225.g
Analytic conductor 0.112
Analytic rank 0
Dimension 2
Projective image \(D_{2}\)
CM/RM discs -3, -15, 5
Inner twists 8

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Newspace parameters

Level: \( N \) \(=\) \( 225 = 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 225.g (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.112289627842\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image \(D_{2}\)
Projective field Galois closure of \(\Q(\sqrt{-3}, \sqrt{5})\)
Artin image $OD_{16}$
Artin field Galois closure of 8.4.56953125.1

$q$-expansion

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

\(f(q)\) \(=\) \( q + i q^{4} +O(q^{10})\) \( q + i q^{4} - q^{16} -2 i q^{19} -2 q^{31} + i q^{49} + 2 q^{61} -i q^{64} + 2 q^{76} + 2 i q^{79} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + O(q^{10}) \) \( 2q - 2q^{16} - 4q^{31} + 4q^{61} + 4q^{76} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/225\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(127\)
\(\chi(n)\) \(1\) \(-i\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
82.1
1.00000i
1.00000i
0 0 1.00000i 0 0 0 0 0 0
118.1 0 0 1.00000i 0 0 0 0 0 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
3.b odd 2 1 CM by \(\Q(\sqrt{-3}) \)
5.b even 2 1 RM by \(\Q(\sqrt{5}) \)
15.d odd 2 1 CM by \(\Q(\sqrt{-15}) \)
5.c odd 4 2 inner
15.e even 4 2 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 225.1.g.a 2
3.b odd 2 1 CM 225.1.g.a 2
4.b odd 2 1 3600.1.bh.a 2
5.b even 2 1 RM 225.1.g.a 2
5.c odd 4 2 inner 225.1.g.a 2
9.c even 3 2 2025.1.p.a 4
9.d odd 6 2 2025.1.p.a 4
12.b even 2 1 3600.1.bh.a 2
15.d odd 2 1 CM 225.1.g.a 2
15.e even 4 2 inner 225.1.g.a 2
20.d odd 2 1 3600.1.bh.a 2
20.e even 4 2 3600.1.bh.a 2
45.h odd 6 2 2025.1.p.a 4
45.j even 6 2 2025.1.p.a 4
45.k odd 12 4 2025.1.p.a 4
45.l even 12 4 2025.1.p.a 4
60.h even 2 1 3600.1.bh.a 2
60.l odd 4 2 3600.1.bh.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
225.1.g.a 2 1.a even 1 1 trivial
225.1.g.a 2 3.b odd 2 1 CM
225.1.g.a 2 5.b even 2 1 RM
225.1.g.a 2 5.c odd 4 2 inner
225.1.g.a 2 15.d odd 2 1 CM
225.1.g.a 2 15.e even 4 2 inner
2025.1.p.a 4 9.c even 3 2
2025.1.p.a 4 9.d odd 6 2
2025.1.p.a 4 45.h odd 6 2
2025.1.p.a 4 45.j even 6 2
2025.1.p.a 4 45.k odd 12 4
2025.1.p.a 4 45.l even 12 4
3600.1.bh.a 2 4.b odd 2 1
3600.1.bh.a 2 12.b even 2 1
3600.1.bh.a 2 20.d odd 2 1
3600.1.bh.a 2 20.e even 4 2
3600.1.bh.a 2 60.h even 2 1
3600.1.bh.a 2 60.l odd 4 2

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(225, [\chi])\).

Hecke characteristic polynomials

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