Properties

Label 2730.2.a.o
Level 2730
Weight 2
Character orbit 2730.a
Self dual yes
Analytic conductor 21.799
Analytic rank 0
Dimension 1
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 2730 = 2 \cdot 3 \cdot 5 \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2730.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(21.7991597518\)
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

\(f(q)\) \(=\) \( q - q^{2} + q^{3} + q^{4} - q^{5} - q^{6} + q^{7} - q^{8} + q^{9} + O(q^{10}) \) \( q - q^{2} + q^{3} + q^{4} - q^{5} - q^{6} + q^{7} - q^{8} + q^{9} + q^{10} + q^{12} + q^{13} - q^{14} - q^{15} + q^{16} - 6q^{17} - q^{18} - 4q^{19} - q^{20} + q^{21} - q^{24} + q^{25} - q^{26} + q^{27} + q^{28} + 6q^{29} + q^{30} + 8q^{31} - q^{32} + 6q^{34} - q^{35} + q^{36} + 2q^{37} + 4q^{38} + q^{39} + q^{40} + 6q^{41} - q^{42} + 8q^{43} - q^{45} + 12q^{47} + q^{48} + q^{49} - q^{50} - 6q^{51} + q^{52} - 6q^{53} - q^{54} - q^{56} - 4q^{57} - 6q^{58} - 12q^{59} - q^{60} + 14q^{61} - 8q^{62} + q^{63} + q^{64} - q^{65} - 4q^{67} - 6q^{68} + q^{70} - 12q^{71} - q^{72} + 2q^{73} - 2q^{74} + q^{75} - 4q^{76} - q^{78} + 8q^{79} - q^{80} + q^{81} - 6q^{82} + q^{84} + 6q^{85} - 8q^{86} + 6q^{87} + 6q^{89} + q^{90} + q^{91} + 8q^{93} - 12q^{94} + 4q^{95} - q^{96} + 2q^{97} - q^{98} + O(q^{100}) \)

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

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(3\) \(-1\)
\(5\) \(1\)
\(7\) \(-1\)
\(13\) \(-1\)

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 2730.2.a.o 1
3.b odd 2 1 8190.2.a.bx 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2730.2.a.o 1 1.a even 1 1 trivial
8190.2.a.bx 1 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(2730))\):

\( T_{11} \)
\( T_{17} + 6 \)
\( T_{19} + 4 \)
\( T_{23} \)
\( T_{29} - 6 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + T \)
$3$ \( 1 - T \)
$5$ \( 1 + T \)
$7$ \( 1 - T \)
$11$ \( 1 + 11 T^{2} \)
$13$ \( 1 - T \)
$17$ \( 1 + 6 T + 17 T^{2} \)
$19$ \( 1 + 4 T + 19 T^{2} \)
$23$ \( 1 + 23 T^{2} \)
$29$ \( 1 - 6 T + 29 T^{2} \)
$31$ \( 1 - 8 T + 31 T^{2} \)
$37$ \( 1 - 2 T + 37 T^{2} \)
$41$ \( 1 - 6 T + 41 T^{2} \)
$43$ \( 1 - 8 T + 43 T^{2} \)
$47$ \( 1 - 12 T + 47 T^{2} \)
$53$ \( 1 + 6 T + 53 T^{2} \)
$59$ \( 1 + 12 T + 59 T^{2} \)
$61$ \( 1 - 14 T + 61 T^{2} \)
$67$ \( 1 + 4 T + 67 T^{2} \)
$71$ \( 1 + 12 T + 71 T^{2} \)
$73$ \( 1 - 2 T + 73 T^{2} \)
$79$ \( 1 - 8 T + 79 T^{2} \)
$83$ \( 1 + 83 T^{2} \)
$89$ \( 1 - 6 T + 89 T^{2} \)
$97$ \( 1 - 2 T + 97 T^{2} \)
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