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: \(x^{2} - x - 1\)
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.

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

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 \(S_{2}^{\mathrm{new}}(\Gamma_0(225))\):

\( T_{2}^{2} - 5 \)
\( T_{7} \)

Hecke characteristic polynomials

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