Properties

Label 270.2.a.c
Level $270$
Weight $2$
Character orbit 270.a
Self dual yes
Analytic conductor $2.156$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 270 = 2 \cdot 3^{3} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 270.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(2.15596085457\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + q^{2} + q^{4} - q^{5} + 2 q^{7} + q^{8} + O(q^{10}) \) \( q + q^{2} + q^{4} - q^{5} + 2 q^{7} + q^{8} - q^{10} + 3 q^{11} + 5 q^{13} + 2 q^{14} + q^{16} - 3 q^{17} - 4 q^{19} - q^{20} + 3 q^{22} - 9 q^{23} + q^{25} + 5 q^{26} + 2 q^{28} - 3 q^{29} + 5 q^{31} + q^{32} - 3 q^{34} - 2 q^{35} - 10 q^{37} - 4 q^{38} - q^{40} - q^{43} + 3 q^{44} - 9 q^{46} + 9 q^{47} - 3 q^{49} + q^{50} + 5 q^{52} - 12 q^{53} - 3 q^{55} + 2 q^{56} - 3 q^{58} + 12 q^{59} + 2 q^{61} + 5 q^{62} + q^{64} - 5 q^{65} - 4 q^{67} - 3 q^{68} - 2 q^{70} + 12 q^{71} - 10 q^{73} - 10 q^{74} - 4 q^{76} + 6 q^{77} - 13 q^{79} - q^{80} + 6 q^{83} + 3 q^{85} - q^{86} + 3 q^{88} - 12 q^{89} + 10 q^{91} - 9 q^{92} + 9 q^{94} + 4 q^{95} + 2 q^{97} - 3 q^{98} + O(q^{100}) \)

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
1.00000 0 1.00000 −1.00000 0 2.00000 1.00000 0 −1.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

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

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 270.2.a.c yes 1
3.b odd 2 1 270.2.a.b 1
4.b odd 2 1 2160.2.a.b 1
5.b even 2 1 1350.2.a.d 1
5.c odd 4 2 1350.2.c.j 2
8.b even 2 1 8640.2.a.bx 1
8.d odd 2 1 8640.2.a.bn 1
9.c even 3 2 810.2.e.d 2
9.d odd 6 2 810.2.e.i 2
12.b even 2 1 2160.2.a.q 1
15.d odd 2 1 1350.2.a.o 1
15.e even 4 2 1350.2.c.c 2
24.f even 2 1 8640.2.a.e 1
24.h odd 2 1 8640.2.a.y 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
270.2.a.b 1 3.b odd 2 1
270.2.a.c yes 1 1.a even 1 1 trivial
810.2.e.d 2 9.c even 3 2
810.2.e.i 2 9.d odd 6 2
1350.2.a.d 1 5.b even 2 1
1350.2.a.o 1 15.d odd 2 1
1350.2.c.c 2 15.e even 4 2
1350.2.c.j 2 5.c odd 4 2
2160.2.a.b 1 4.b odd 2 1
2160.2.a.q 1 12.b even 2 1
8640.2.a.e 1 24.f even 2 1
8640.2.a.y 1 24.h odd 2 1
8640.2.a.bn 1 8.d odd 2 1
8640.2.a.bx 1 8.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(270))\):

\( T_{11} - 3 \)
\( T_{13} - 5 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -1 + T \)
$3$ \( T \)
$5$ \( 1 + T \)
$7$ \( -2 + T \)
$11$ \( -3 + T \)
$13$ \( -5 + T \)
$17$ \( 3 + T \)
$19$ \( 4 + T \)
$23$ \( 9 + T \)
$29$ \( 3 + T \)
$31$ \( -5 + T \)
$37$ \( 10 + T \)
$41$ \( T \)
$43$ \( 1 + T \)
$47$ \( -9 + T \)
$53$ \( 12 + T \)
$59$ \( -12 + T \)
$61$ \( -2 + T \)
$67$ \( 4 + T \)
$71$ \( -12 + T \)
$73$ \( 10 + T \)
$79$ \( 13 + T \)
$83$ \( -6 + T \)
$89$ \( 12 + T \)
$97$ \( -2 + T \)
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