Properties

Label 1350.2.a.t
Level 1350
Weight 2
Character orbit 1350.a
Self dual yes
Analytic conductor 10.780
Analytic rank 0
Dimension 1
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 1350 = 2 \cdot 3^{3} \cdot 5^{2} \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 1350.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(10.7798042729\)
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

\(f(q)\) \(=\) \( q + q^{2} + q^{4} + 2q^{7} + q^{8} + O(q^{10}) \) \( q + q^{2} + q^{4} + 2q^{7} + q^{8} + 3q^{11} + 5q^{13} + 2q^{14} + q^{16} - 6q^{17} - 4q^{19} + 3q^{22} + 3q^{23} + 5q^{26} + 2q^{28} + 2q^{31} + q^{32} - 6q^{34} + 11q^{37} - 4q^{38} - 6q^{41} - 4q^{43} + 3q^{44} + 3q^{46} - 3q^{47} - 3q^{49} + 5q^{52} + 12q^{53} + 2q^{56} - 9q^{59} + 11q^{61} + 2q^{62} + q^{64} + 14q^{67} - 6q^{68} + 15q^{71} + 2q^{73} + 11q^{74} - 4q^{76} + 6q^{77} - 10q^{79} - 6q^{82} - 4q^{86} + 3q^{88} - 6q^{89} + 10q^{91} + 3q^{92} - 3q^{94} - 7q^{97} - 3q^{98} + O(q^{100}) \)

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

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 1350.2.a.t yes 1
3.b odd 2 1 1350.2.a.i yes 1
5.b even 2 1 1350.2.a.e 1
5.c odd 4 2 1350.2.c.i 2
15.d odd 2 1 1350.2.a.n yes 1
15.e even 4 2 1350.2.c.d 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1350.2.a.e 1 5.b even 2 1
1350.2.a.i yes 1 3.b odd 2 1
1350.2.a.n yes 1 15.d odd 2 1
1350.2.a.t yes 1 1.a even 1 1 trivial
1350.2.c.d 2 15.e even 4 2
1350.2.c.i 2 5.c odd 4 2

Atkin-Lehner signs

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

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1350))\):

\( T_{7} - 2 \)
\( T_{11} - 3 \)
\( T_{13} - 5 \)
\( T_{17} + 6 \)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 - T \)
$3$ \( \)
$5$ \( \)
$7$ \( 1 - 2 T + 7 T^{2} \)
$11$ \( 1 - 3 T + 11 T^{2} \)
$13$ \( 1 - 5 T + 13 T^{2} \)
$17$ \( 1 + 6 T + 17 T^{2} \)
$19$ \( 1 + 4 T + 19 T^{2} \)
$23$ \( 1 - 3 T + 23 T^{2} \)
$29$ \( 1 + 29 T^{2} \)
$31$ \( 1 - 2 T + 31 T^{2} \)
$37$ \( 1 - 11 T + 37 T^{2} \)
$41$ \( 1 + 6 T + 41 T^{2} \)
$43$ \( 1 + 4 T + 43 T^{2} \)
$47$ \( 1 + 3 T + 47 T^{2} \)
$53$ \( 1 - 12 T + 53 T^{2} \)
$59$ \( 1 + 9 T + 59 T^{2} \)
$61$ \( 1 - 11 T + 61 T^{2} \)
$67$ \( 1 - 14 T + 67 T^{2} \)
$71$ \( 1 - 15 T + 71 T^{2} \)
$73$ \( 1 - 2 T + 73 T^{2} \)
$79$ \( 1 + 10 T + 79 T^{2} \)
$83$ \( 1 + 83 T^{2} \)
$89$ \( 1 + 6 T + 89 T^{2} \)
$97$ \( 1 + 7 T + 97 T^{2} \)
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