Properties

Label 225.2.a.f
Level $225$
Weight $2$
Character orbit 225.a
Self dual yes
Analytic conductor $1.797$
Analytic rank $0$
Dimension $2$
CM discriminant -15
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 225 = 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 225.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(1.79663404548\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{5}) \)
Defining polynomial: \( x^{2} - x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 45)
Fricke sign: \(-1\)
Sato-Tate group: $N(\mathrm{U}(1))$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{5}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta q^{2} + 3 q^{4} - \beta q^{8} +O(q^{10}) \) Copy content Toggle raw display \( q - \beta q^{2} + 3 q^{4} - \beta q^{8} - q^{16} + 2 \beta q^{17} + 4 q^{19} + 4 \beta q^{23} + 8 q^{31} + 3 \beta q^{32} - 10 q^{34} - 4 \beta q^{38} - 20 q^{46} - 4 \beta q^{47} - 7 q^{49} - 2 \beta q^{53} + 2 q^{61} - 8 \beta q^{62} - 13 q^{64} + 6 \beta q^{68} + 12 q^{76} + 16 q^{79} - 8 \beta q^{83} + 12 \beta q^{92} + 20 q^{94} + 7 \beta q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 6 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 6 q^{4} - 2 q^{16} + 8 q^{19} + 16 q^{31} - 20 q^{34} - 40 q^{46} - 14 q^{49} + 4 q^{61} - 26 q^{64} + 24 q^{76} + 32 q^{79} + 40 q^{94}+O(q^{100}) \) Copy content Toggle raw display

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
1.61803
−0.618034
−2.23607 0 3.00000 0 0 0 −2.23607 0 0
1.2 2.23607 0 3.00000 0 0 0 2.23607 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(1\)
\(5\) \(-1\)

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
15.d odd 2 1 CM by \(\Q(\sqrt{-15}) \)
3.b odd 2 1 inner
5.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 225.2.a.f 2
3.b odd 2 1 inner 225.2.a.f 2
4.b odd 2 1 3600.2.a.bs 2
5.b even 2 1 inner 225.2.a.f 2
5.c odd 4 2 45.2.b.a 2
12.b even 2 1 3600.2.a.bs 2
15.d odd 2 1 CM 225.2.a.f 2
15.e even 4 2 45.2.b.a 2
20.d odd 2 1 3600.2.a.bs 2
20.e even 4 2 720.2.f.d 2
35.f even 4 2 2205.2.d.a 2
40.i odd 4 2 2880.2.f.k 2
40.k even 4 2 2880.2.f.j 2
45.k odd 12 4 405.2.j.c 4
45.l even 12 4 405.2.j.c 4
60.h even 2 1 3600.2.a.bs 2
60.l odd 4 2 720.2.f.d 2
105.k odd 4 2 2205.2.d.a 2
120.q odd 4 2 2880.2.f.j 2
120.w even 4 2 2880.2.f.k 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
45.2.b.a 2 5.c odd 4 2
45.2.b.a 2 15.e even 4 2
225.2.a.f 2 1.a even 1 1 trivial
225.2.a.f 2 3.b odd 2 1 inner
225.2.a.f 2 5.b even 2 1 inner
225.2.a.f 2 15.d odd 2 1 CM
405.2.j.c 4 45.k odd 12 4
405.2.j.c 4 45.l even 12 4
720.2.f.d 2 20.e even 4 2
720.2.f.d 2 60.l odd 4 2
2205.2.d.a 2 35.f even 4 2
2205.2.d.a 2 105.k odd 4 2
2880.2.f.j 2 40.k even 4 2
2880.2.f.j 2 120.q odd 4 2
2880.2.f.k 2 40.i odd 4 2
2880.2.f.k 2 120.w even 4 2
3600.2.a.bs 2 4.b odd 2 1
3600.2.a.bs 2 12.b even 2 1
3600.2.a.bs 2 20.d odd 2 1
3600.2.a.bs 2 60.h even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(225))\):

\( T_{2}^{2} - 5 \) Copy content Toggle raw display
\( T_{7} \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} - 5 \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} \) Copy content Toggle raw display
$13$ \( T^{2} \) Copy content Toggle raw display
$17$ \( T^{2} - 20 \) Copy content Toggle raw display
$19$ \( (T - 4)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} - 80 \) Copy content Toggle raw display
$29$ \( T^{2} \) Copy content Toggle raw display
$31$ \( (T - 8)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} \) Copy content Toggle raw display
$41$ \( T^{2} \) Copy content Toggle raw display
$43$ \( T^{2} \) Copy content Toggle raw display
$47$ \( T^{2} - 80 \) Copy content Toggle raw display
$53$ \( T^{2} - 20 \) Copy content Toggle raw display
$59$ \( T^{2} \) Copy content Toggle raw display
$61$ \( (T - 2)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} \) Copy content Toggle raw display
$71$ \( T^{2} \) Copy content Toggle raw display
$73$ \( T^{2} \) Copy content Toggle raw display
$79$ \( (T - 16)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} - 320 \) Copy content Toggle raw display
$89$ \( T^{2} \) Copy content Toggle raw display
$97$ \( T^{2} \) Copy content Toggle raw display
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