Properties

Label 7350.2.a.b
Level 7350
Weight 2
Character orbit 7350.a
Self dual yes
Analytic conductor 58.690
Analytic rank 1
Dimension 1
CM no
Inner twists 1

Related objects

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Newspace parameters

Level: \( N \) \(=\) \( 7350 = 2 \cdot 3 \cdot 5^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7350.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(58.6900454856\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 210)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - q^{2} - q^{3} + q^{4} + q^{6} - q^{8} + q^{9} + O(q^{10}) \) \( q - q^{2} - q^{3} + q^{4} + q^{6} - q^{8} + q^{9} - 5q^{11} - q^{12} - q^{13} + q^{16} + 2q^{17} - q^{18} - 7q^{19} + 5q^{22} + 3q^{23} + q^{24} + q^{26} - q^{27} + 6q^{31} - q^{32} + 5q^{33} - 2q^{34} + q^{36} - 5q^{37} + 7q^{38} + q^{39} + 9q^{41} + 10q^{43} - 5q^{44} - 3q^{46} + 13q^{47} - q^{48} - 2q^{51} - q^{52} - q^{53} + q^{54} + 7q^{57} - 4q^{59} + 2q^{61} - 6q^{62} + q^{64} - 5q^{66} + 6q^{67} + 2q^{68} - 3q^{69} - 2q^{71} - q^{72} - 4q^{73} + 5q^{74} - 7q^{76} - q^{78} - 14q^{79} + q^{81} - 9q^{82} + 10q^{83} - 10q^{86} + 5q^{88} - 10q^{89} + 3q^{92} - 6q^{93} - 13q^{94} + q^{96} - 8q^{97} - 5q^{99} + O(q^{100}) \)

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 0 1.00000 0 −1.00000 1.00000 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 7350.2.a.b 1
5.b even 2 1 7350.2.a.ch 1
5.c odd 4 2 1470.2.g.a 2
7.b odd 2 1 7350.2.a.t 1
7.d odd 6 2 1050.2.i.o 2
35.c odd 2 1 7350.2.a.bn 1
35.f even 4 2 1470.2.g.f 2
35.i odd 6 2 1050.2.i.f 2
35.k even 12 4 210.2.n.a 4
35.l odd 12 4 1470.2.n.i 4
105.w odd 12 4 630.2.u.c 4
140.x odd 12 4 1680.2.di.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
210.2.n.a 4 35.k even 12 4
630.2.u.c 4 105.w odd 12 4
1050.2.i.f 2 35.i odd 6 2
1050.2.i.o 2 7.d odd 6 2
1470.2.g.a 2 5.c odd 4 2
1470.2.g.f 2 35.f even 4 2
1470.2.n.i 4 35.l odd 12 4
1680.2.di.a 4 140.x odd 12 4
7350.2.a.b 1 1.a even 1 1 trivial
7350.2.a.t 1 7.b odd 2 1
7350.2.a.bn 1 35.c odd 2 1
7350.2.a.ch 1 5.b even 2 1

Atkin-Lehner signs

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

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(7350))\):

\( T_{11} + 5 \)
\( T_{13} + 1 \)
\( T_{17} - 2 \)
\( T_{19} + 7 \)
\( T_{23} - 3 \)
\( T_{31} - 6 \)

Hecke characteristic polynomials

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