Properties

Label 7350.2.a.bl
Level 7350
Weight 2
Character orbit 7350.a
Self dual yes
Analytic conductor 58.690
Analytic rank 0
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: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 42)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

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

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.bl 1
5.b even 2 1 294.2.a.e 1
7.b odd 2 1 7350.2.a.q 1
7.c even 3 2 1050.2.i.l 2
15.d odd 2 1 882.2.a.c 1
20.d odd 2 1 2352.2.a.t 1
35.c odd 2 1 294.2.a.f 1
35.i odd 6 2 294.2.e.b 2
35.j even 6 2 42.2.e.a 2
35.l odd 12 4 1050.2.o.a 4
40.e odd 2 1 9408.2.a.q 1
40.f even 2 1 9408.2.a.ce 1
60.h even 2 1 7056.2.a.w 1
105.g even 2 1 882.2.a.d 1
105.o odd 6 2 126.2.g.c 2
105.p even 6 2 882.2.g.i 2
140.c even 2 1 2352.2.a.f 1
140.p odd 6 2 336.2.q.b 2
140.s even 6 2 2352.2.q.u 2
280.c odd 2 1 9408.2.a.z 1
280.n even 2 1 9408.2.a.cr 1
280.bf even 6 2 1344.2.q.g 2
280.bi odd 6 2 1344.2.q.s 2
315.r even 6 2 1134.2.e.l 2
315.v odd 6 2 1134.2.h.l 2
315.bo even 6 2 1134.2.h.e 2
315.br odd 6 2 1134.2.e.e 2
420.o odd 2 1 7056.2.a.bl 1
420.ba even 6 2 1008.2.s.k 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
42.2.e.a 2 35.j even 6 2
126.2.g.c 2 105.o odd 6 2
294.2.a.e 1 5.b even 2 1
294.2.a.f 1 35.c odd 2 1
294.2.e.b 2 35.i odd 6 2
336.2.q.b 2 140.p odd 6 2
882.2.a.c 1 15.d odd 2 1
882.2.a.d 1 105.g even 2 1
882.2.g.i 2 105.p even 6 2
1008.2.s.k 2 420.ba even 6 2
1050.2.i.l 2 7.c even 3 2
1050.2.o.a 4 35.l odd 12 4
1134.2.e.e 2 315.br odd 6 2
1134.2.e.l 2 315.r even 6 2
1134.2.h.e 2 315.bo even 6 2
1134.2.h.l 2 315.v odd 6 2
1344.2.q.g 2 280.bf even 6 2
1344.2.q.s 2 280.bi odd 6 2
2352.2.a.f 1 140.c even 2 1
2352.2.a.t 1 20.d odd 2 1
2352.2.q.u 2 140.s even 6 2
7056.2.a.w 1 60.h even 2 1
7056.2.a.bl 1 420.o odd 2 1
7350.2.a.q 1 7.b odd 2 1
7350.2.a.bl 1 1.a even 1 1 trivial
9408.2.a.q 1 40.e odd 2 1
9408.2.a.z 1 280.c odd 2 1
9408.2.a.ce 1 40.f even 2 1
9408.2.a.cr 1 280.n 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(7350))\):

\( T_{11} - 5 \)
\( T_{13} \)
\( T_{17} - 4 \)
\( T_{19} - 8 \)
\( T_{23} - 4 \)
\( T_{31} - 3 \)

Hecke characteristic polynomials

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