Properties

Label 225.2.b.c
Level $225$
Weight $2$
Character orbit 225.b
Analytic conductor $1.797$
Analytic rank $0$
Dimension $2$
CM discriminant -3
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 225 = 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 225.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.79663404548\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 5 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 2 q^{4} + \beta q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + 2 q^{4} + \beta q^{7} - \beta q^{13} + 4 q^{16} + q^{19} + 2 \beta q^{28} - 7 q^{31} - 2 \beta q^{37} - \beta q^{43} - 18 q^{49} - 2 \beta q^{52} - 13 q^{61} + 8 q^{64} + \beta q^{67} + 2 \beta q^{73} + 2 q^{76} + 4 q^{79} + 25 q^{91} + \beta q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 4 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 4 q^{4} + 8 q^{16} + 2 q^{19} - 14 q^{31} - 36 q^{49} - 26 q^{61} + 16 q^{64} + 4 q^{76} + 8 q^{79} + 50 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
199.1
1.00000i
1.00000i
0 0 2.00000 0 0 5.00000i 0 0 0
199.2 0 0 2.00000 0 0 5.00000i 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 inner
15.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 225.2.b.c 2
3.b odd 2 1 CM 225.2.b.c 2
4.b odd 2 1 3600.2.f.k 2
5.b even 2 1 inner 225.2.b.c 2
5.c odd 4 1 225.2.a.c 1
5.c odd 4 1 225.2.a.d yes 1
12.b even 2 1 3600.2.f.k 2
15.d odd 2 1 inner 225.2.b.c 2
15.e even 4 1 225.2.a.c 1
15.e even 4 1 225.2.a.d yes 1
20.d odd 2 1 3600.2.f.k 2
20.e even 4 1 3600.2.a.b 1
20.e even 4 1 3600.2.a.br 1
60.h even 2 1 3600.2.f.k 2
60.l odd 4 1 3600.2.a.b 1
60.l odd 4 1 3600.2.a.br 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
225.2.a.c 1 5.c odd 4 1
225.2.a.c 1 15.e even 4 1
225.2.a.d yes 1 5.c odd 4 1
225.2.a.d yes 1 15.e even 4 1
225.2.b.c 2 1.a even 1 1 trivial
225.2.b.c 2 3.b odd 2 1 CM
225.2.b.c 2 5.b even 2 1 inner
225.2.b.c 2 15.d odd 2 1 inner
3600.2.a.b 1 20.e even 4 1
3600.2.a.b 1 60.l odd 4 1
3600.2.a.br 1 20.e even 4 1
3600.2.a.br 1 60.l odd 4 1
3600.2.f.k 2 4.b odd 2 1
3600.2.f.k 2 12.b even 2 1
3600.2.f.k 2 20.d odd 2 1
3600.2.f.k 2 60.h even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} \) acting on \(S_{2}^{\mathrm{new}}(225, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 25 \) Copy content Toggle raw display
$11$ \( T^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 25 \) Copy content Toggle raw display
$17$ \( T^{2} \) Copy content Toggle raw display
$19$ \( (T - 1)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} \) Copy content Toggle raw display
$29$ \( T^{2} \) Copy content Toggle raw display
$31$ \( (T + 7)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 100 \) Copy content Toggle raw display
$41$ \( T^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 25 \) Copy content Toggle raw display
$47$ \( T^{2} \) Copy content Toggle raw display
$53$ \( T^{2} \) Copy content Toggle raw display
$59$ \( T^{2} \) Copy content Toggle raw display
$61$ \( (T + 13)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 25 \) Copy content Toggle raw display
$71$ \( T^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 100 \) Copy content Toggle raw display
$79$ \( (T - 4)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} \) Copy content Toggle raw display
$89$ \( T^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 25 \) Copy content Toggle raw display
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