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 disc. -15
Inner twists 4

Related objects

Downloads

Learn more about

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: \(\Q(\sqrt{5}) \)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2 \)
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.

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

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:

Inner twists

Char. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
15.d Odd 1 CM by \(\Q(\sqrt{-15}) \) yes
3.b Odd 1 yes
5.b Even 1 yes

Atkin-Lehner signs

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

Hecke kernels

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

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