Properties

Label 7350.2.a.cm
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

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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 1050)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + q^{2} + q^{3} + q^{4} + q^{6} + q^{8} + q^{9} + O(q^{10}) \) \( q + q^{2} + q^{3} + q^{4} + q^{6} + q^{8} + q^{9} - 2q^{11} + q^{12} - 7q^{13} + q^{16} + 7q^{17} + q^{18} - 8q^{19} - 2q^{22} - 5q^{23} + q^{24} - 7q^{26} + q^{27} + 9q^{29} - q^{31} + q^{32} - 2q^{33} + 7q^{34} + q^{36} - 2q^{37} - 8q^{38} - 7q^{39} - 11q^{41} + 3q^{43} - 2q^{44} - 5q^{46} - 4q^{47} + q^{48} + 7q^{51} - 7q^{52} + 3q^{53} + q^{54} - 8q^{57} + 9q^{58} - 7q^{59} + 5q^{61} - q^{62} + q^{64} - 2q^{66} - 12q^{67} + 7q^{68} - 5q^{69} - 4q^{71} + q^{72} - 10q^{73} - 2q^{74} - 8q^{76} - 7q^{78} - 6q^{79} + q^{81} - 11q^{82} + 9q^{83} + 3q^{86} + 9q^{87} - 2q^{88} + 10q^{89} - 5q^{92} - q^{93} - 4q^{94} + q^{96} - 10q^{97} - 2q^{99} + O(q^{100}) \)

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.cm 1
5.b even 2 1 7350.2.a.i 1
7.b odd 2 1 1050.2.a.n yes 1
21.c even 2 1 3150.2.a.r 1
28.d even 2 1 8400.2.a.cb 1
35.c odd 2 1 1050.2.a.f 1
35.f even 4 2 1050.2.g.b 2
105.g even 2 1 3150.2.a.bd 1
105.k odd 4 2 3150.2.g.p 2
140.c even 2 1 8400.2.a.bb 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1050.2.a.f 1 35.c odd 2 1
1050.2.a.n yes 1 7.b odd 2 1
1050.2.g.b 2 35.f even 4 2
3150.2.a.r 1 21.c even 2 1
3150.2.a.bd 1 105.g even 2 1
3150.2.g.p 2 105.k odd 4 2
7350.2.a.i 1 5.b even 2 1
7350.2.a.cm 1 1.a even 1 1 trivial
8400.2.a.bb 1 140.c even 2 1
8400.2.a.cb 1 28.d 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(7350))\):

\( T_{11} + 2 \)
\( T_{13} + 7 \)
\( T_{17} - 7 \)
\( T_{19} + 8 \)
\( T_{23} + 5 \)
\( T_{31} + 1 \)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 - T \)
$3$ \( 1 - T \)
$5$ 1
$7$ 1
$11$ \( 1 + 2 T + 11 T^{2} \)
$13$ \( 1 + 7 T + 13 T^{2} \)
$17$ \( 1 - 7 T + 17 T^{2} \)
$19$ \( 1 + 8 T + 19 T^{2} \)
$23$ \( 1 + 5 T + 23 T^{2} \)
$29$ \( 1 - 9 T + 29 T^{2} \)
$31$ \( 1 + T + 31 T^{2} \)
$37$ \( 1 + 2 T + 37 T^{2} \)
$41$ \( 1 + 11 T + 41 T^{2} \)
$43$ \( 1 - 3 T + 43 T^{2} \)
$47$ \( 1 + 4 T + 47 T^{2} \)
$53$ \( 1 - 3 T + 53 T^{2} \)
$59$ \( 1 + 7 T + 59 T^{2} \)
$61$ \( 1 - 5 T + 61 T^{2} \)
$67$ \( 1 + 12 T + 67 T^{2} \)
$71$ \( 1 + 4 T + 71 T^{2} \)
$73$ \( 1 + 10 T + 73 T^{2} \)
$79$ \( 1 + 6 T + 79 T^{2} \)
$83$ \( 1 - 9 T + 83 T^{2} \)
$89$ \( 1 - 10 T + 89 T^{2} \)
$97$ \( 1 + 10 T + 97 T^{2} \)
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