Properties

Label 675.2.a.e
Level $675$
Weight $2$
Character orbit 675.a
Self dual yes
Analytic conductor $5.390$
Analytic rank $1$
Dimension $1$
CM discriminant -3
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 675 = 3^{3} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 675.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(5.38990213644\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 27)
Fricke sign: \(1\)
Sato-Tate group: $N(\mathrm{U}(1))$

$q$-expansion

\(f(q)\) \(=\) \( q - 2 q^{4} + q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q - 2 q^{4} + q^{7} - 5 q^{13} + 4 q^{16} - 7 q^{19} - 2 q^{28} - 4 q^{31} - 11 q^{37} - 8 q^{43} - 6 q^{49} + 10 q^{52} - q^{61} - 8 q^{64} - 5 q^{67} + 7 q^{73} + 14 q^{76} + 17 q^{79} - 5 q^{91} + 19 q^{97}+O(q^{100}) \) 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
0
0 0 −2.00000 0 0 1.00000 0 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
3.b odd 2 1 CM by \(\Q(\sqrt{-3}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 675.2.a.e 1
3.b odd 2 1 CM 675.2.a.e 1
5.b even 2 1 27.2.a.a 1
5.c odd 4 2 675.2.b.f 2
15.d odd 2 1 27.2.a.a 1
15.e even 4 2 675.2.b.f 2
20.d odd 2 1 432.2.a.e 1
35.c odd 2 1 1323.2.a.i 1
40.e odd 2 1 1728.2.a.o 1
40.f even 2 1 1728.2.a.n 1
45.h odd 6 2 81.2.c.a 2
45.j even 6 2 81.2.c.a 2
55.d odd 2 1 3267.2.a.f 1
60.h even 2 1 432.2.a.e 1
65.d even 2 1 4563.2.a.e 1
85.c even 2 1 7803.2.a.k 1
95.d odd 2 1 9747.2.a.f 1
105.g even 2 1 1323.2.a.i 1
120.i odd 2 1 1728.2.a.n 1
120.m even 2 1 1728.2.a.o 1
135.n odd 18 6 729.2.e.f 6
135.p even 18 6 729.2.e.f 6
165.d even 2 1 3267.2.a.f 1
180.n even 6 2 1296.2.i.i 2
180.p odd 6 2 1296.2.i.i 2
195.e odd 2 1 4563.2.a.e 1
255.h odd 2 1 7803.2.a.k 1
285.b even 2 1 9747.2.a.f 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
27.2.a.a 1 5.b even 2 1
27.2.a.a 1 15.d odd 2 1
81.2.c.a 2 45.h odd 6 2
81.2.c.a 2 45.j even 6 2
432.2.a.e 1 20.d odd 2 1
432.2.a.e 1 60.h even 2 1
675.2.a.e 1 1.a even 1 1 trivial
675.2.a.e 1 3.b odd 2 1 CM
675.2.b.f 2 5.c odd 4 2
675.2.b.f 2 15.e even 4 2
729.2.e.f 6 135.n odd 18 6
729.2.e.f 6 135.p even 18 6
1296.2.i.i 2 180.n even 6 2
1296.2.i.i 2 180.p odd 6 2
1323.2.a.i 1 35.c odd 2 1
1323.2.a.i 1 105.g even 2 1
1728.2.a.n 1 40.f even 2 1
1728.2.a.n 1 120.i odd 2 1
1728.2.a.o 1 40.e odd 2 1
1728.2.a.o 1 120.m even 2 1
3267.2.a.f 1 55.d odd 2 1
3267.2.a.f 1 165.d even 2 1
4563.2.a.e 1 65.d even 2 1
4563.2.a.e 1 195.e odd 2 1
7803.2.a.k 1 85.c even 2 1
7803.2.a.k 1 255.h odd 2 1
9747.2.a.f 1 95.d odd 2 1
9747.2.a.f 1 285.b even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(675))\):

\( T_{2} \) Copy content Toggle raw display
\( T_{7} - 1 \) Copy content Toggle raw display
\( T_{11} \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \) Copy content Toggle raw display
$3$ \( T \) Copy content Toggle raw display
$5$ \( T \) Copy content Toggle raw display
$7$ \( T - 1 \) Copy content Toggle raw display
$11$ \( T \) Copy content Toggle raw display
$13$ \( T + 5 \) Copy content Toggle raw display
$17$ \( T \) Copy content Toggle raw display
$19$ \( T + 7 \) Copy content Toggle raw display
$23$ \( T \) Copy content Toggle raw display
$29$ \( T \) Copy content Toggle raw display
$31$ \( T + 4 \) Copy content Toggle raw display
$37$ \( T + 11 \) Copy content Toggle raw display
$41$ \( T \) Copy content Toggle raw display
$43$ \( T + 8 \) Copy content Toggle raw display
$47$ \( T \) Copy content Toggle raw display
$53$ \( T \) Copy content Toggle raw display
$59$ \( T \) Copy content Toggle raw display
$61$ \( T + 1 \) Copy content Toggle raw display
$67$ \( T + 5 \) Copy content Toggle raw display
$71$ \( T \) Copy content Toggle raw display
$73$ \( T - 7 \) Copy content Toggle raw display
$79$ \( T - 17 \) Copy content Toggle raw display
$83$ \( T \) Copy content Toggle raw display
$89$ \( T \) Copy content Toggle raw display
$97$ \( T - 19 \) Copy content Toggle raw display
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