Properties

Label 2700.2.a.b
Level $2700$
Weight $2$
Character orbit 2700.a
Self dual yes
Analytic conductor $21.560$
Analytic rank $1$
Dimension $1$
CM discriminant -3
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 2700 = 2^{2} \cdot 3^{3} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2700.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(21.5596085457\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 108)
Fricke sign: \(1\)
Sato-Tate group: $N(\mathrm{U}(1))$

$q$-expansion

\(f(q)\) \(=\) \( q - 5 q^{7} + O(q^{10}) \) \( q - 5 q^{7} + 7 q^{13} - q^{19} - 4 q^{31} + q^{37} - 8 q^{43} + 18 q^{49} - 13 q^{61} - 11 q^{67} - 17 q^{73} - 13 q^{79} - 35 q^{91} - 5 q^{97} + O(q^{100}) \)

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
0
0 0 0 0 0 −5.00000 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(-1\)
\(5\) \(1\)

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 2700.2.a.b 1
3.b odd 2 1 CM 2700.2.a.b 1
5.b even 2 1 108.2.a.a 1
5.c odd 4 2 2700.2.d.g 2
15.d odd 2 1 108.2.a.a 1
15.e even 4 2 2700.2.d.g 2
20.d odd 2 1 432.2.a.d 1
35.c odd 2 1 5292.2.a.j 1
40.e odd 2 1 1728.2.a.m 1
40.f even 2 1 1728.2.a.p 1
45.h odd 6 2 324.2.e.b 2
45.j even 6 2 324.2.e.b 2
60.h even 2 1 432.2.a.d 1
105.g even 2 1 5292.2.a.j 1
120.i odd 2 1 1728.2.a.p 1
120.m even 2 1 1728.2.a.m 1
180.n even 6 2 1296.2.i.j 2
180.p odd 6 2 1296.2.i.j 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
108.2.a.a 1 5.b even 2 1
108.2.a.a 1 15.d odd 2 1
324.2.e.b 2 45.h odd 6 2
324.2.e.b 2 45.j even 6 2
432.2.a.d 1 20.d odd 2 1
432.2.a.d 1 60.h even 2 1
1296.2.i.j 2 180.n even 6 2
1296.2.i.j 2 180.p odd 6 2
1728.2.a.m 1 40.e odd 2 1
1728.2.a.m 1 120.m even 2 1
1728.2.a.p 1 40.f even 2 1
1728.2.a.p 1 120.i odd 2 1
2700.2.a.b 1 1.a even 1 1 trivial
2700.2.a.b 1 3.b odd 2 1 CM
2700.2.d.g 2 5.c odd 4 2
2700.2.d.g 2 15.e even 4 2
5292.2.a.j 1 35.c odd 2 1
5292.2.a.j 1 105.g even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(2700))\):

\( T_{7} + 5 \)
\( T_{11} \)
\( T_{13} - 7 \)
\( T_{17} \)

Hecke characteristic polynomials

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