Properties

Label 1350.2.a.b
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

Related objects

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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: no (minimal twist has level 270)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - q^{2} + q^{4} - 4q^{7} - q^{8} + O(q^{10}) \) \( q - q^{2} + q^{4} - 4q^{7} - q^{8} + 5q^{11} + 3q^{13} + 4q^{14} + q^{16} - q^{17} - 6q^{19} - 5q^{22} - q^{23} - 3q^{26} - 4q^{28} + 9q^{29} - 5q^{31} - q^{32} + q^{34} + 2q^{37} + 6q^{38} + 2q^{41} - q^{43} + 5q^{44} + q^{46} + 13q^{47} + 9q^{49} + 3q^{52} + 4q^{56} - 9q^{58} + 4q^{59} + 8q^{61} + 5q^{62} + q^{64} + 4q^{67} - q^{68} + 6q^{71} + 2q^{73} - 2q^{74} - 6q^{76} - 20q^{77} + 9q^{79} - 2q^{82} - 4q^{83} + q^{86} - 5q^{88} + 14q^{89} - 12q^{91} - q^{92} - 13q^{94} + 10q^{97} - 9q^{98} + O(q^{100}) \)

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 −4.00000 −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.b 1
3.b odd 2 1 1350.2.a.l 1
5.b even 2 1 1350.2.a.v 1
5.c odd 4 2 270.2.c.b yes 2
15.d odd 2 1 1350.2.a.j 1
15.e even 4 2 270.2.c.a 2
20.e even 4 2 2160.2.f.e 2
45.k odd 12 4 810.2.i.c 4
45.l even 12 4 810.2.i.d 4
60.l odd 4 2 2160.2.f.d 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
270.2.c.a 2 15.e even 4 2
270.2.c.b yes 2 5.c odd 4 2
810.2.i.c 4 45.k odd 12 4
810.2.i.d 4 45.l even 12 4
1350.2.a.b 1 1.a even 1 1 trivial
1350.2.a.j 1 15.d odd 2 1
1350.2.a.l 1 3.b odd 2 1
1350.2.a.v 1 5.b even 2 1
2160.2.f.d 2 60.l odd 4 2
2160.2.f.e 2 20.e even 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(1350))\):

\( T_{7} + 4 \)
\( T_{11} - 5 \)
\( T_{13} - 3 \)
\( T_{17} + 1 \)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 + T \)
$3$ \( \)
$5$ \( \)
$7$ \( 1 + 4 T + 7 T^{2} \)
$11$ \( 1 - 5 T + 11 T^{2} \)
$13$ \( 1 - 3 T + 13 T^{2} \)
$17$ \( 1 + T + 17 T^{2} \)
$19$ \( 1 + 6 T + 19 T^{2} \)
$23$ \( 1 + T + 23 T^{2} \)
$29$ \( 1 - 9 T + 29 T^{2} \)
$31$ \( 1 + 5 T + 31 T^{2} \)
$37$ \( 1 - 2 T + 37 T^{2} \)
$41$ \( 1 - 2 T + 41 T^{2} \)
$43$ \( 1 + T + 43 T^{2} \)
$47$ \( 1 - 13 T + 47 T^{2} \)
$53$ \( 1 + 53 T^{2} \)
$59$ \( 1 - 4 T + 59 T^{2} \)
$61$ \( 1 - 8 T + 61 T^{2} \)
$67$ \( 1 - 4 T + 67 T^{2} \)
$71$ \( 1 - 6 T + 71 T^{2} \)
$73$ \( 1 - 2 T + 73 T^{2} \)
$79$ \( 1 - 9 T + 79 T^{2} \)
$83$ \( 1 + 4 T + 83 T^{2} \)
$89$ \( 1 - 14 T + 89 T^{2} \)
$97$ \( 1 - 10 T + 97 T^{2} \)
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