Properties

Label 7350.2.a.bg
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

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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 30)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

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

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.bg 1
5.b even 2 1 7350.2.a.cc 1
5.c odd 4 2 1470.2.g.g 2
7.b odd 2 1 150.2.a.a 1
21.c even 2 1 450.2.a.f 1
28.d even 2 1 1200.2.a.m 1
35.c odd 2 1 150.2.a.c 1
35.f even 4 2 30.2.c.a 2
35.k even 12 4 1470.2.n.h 4
35.l odd 12 4 1470.2.n.a 4
56.e even 2 1 4800.2.a.m 1
56.h odd 2 1 4800.2.a.cg 1
84.h odd 2 1 3600.2.a.o 1
105.g even 2 1 450.2.a.b 1
105.k odd 4 2 90.2.c.a 2
140.c even 2 1 1200.2.a.g 1
140.j odd 4 2 240.2.f.a 2
280.c odd 2 1 4800.2.a.l 1
280.n even 2 1 4800.2.a.cj 1
280.s even 4 2 960.2.f.h 2
280.y odd 4 2 960.2.f.i 2
315.cb even 12 4 810.2.i.e 4
315.cf odd 12 4 810.2.i.b 4
420.o odd 2 1 3600.2.a.bg 1
420.w even 4 2 720.2.f.f 2
560.r even 4 2 3840.2.d.g 2
560.u odd 4 2 3840.2.d.j 2
560.bm odd 4 2 3840.2.d.x 2
560.bn even 4 2 3840.2.d.y 2
840.bm even 4 2 2880.2.f.c 2
840.bp odd 4 2 2880.2.f.e 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
30.2.c.a 2 35.f even 4 2
90.2.c.a 2 105.k odd 4 2
150.2.a.a 1 7.b odd 2 1
150.2.a.c 1 35.c odd 2 1
240.2.f.a 2 140.j odd 4 2
450.2.a.b 1 105.g even 2 1
450.2.a.f 1 21.c even 2 1
720.2.f.f 2 420.w even 4 2
810.2.i.b 4 315.cf odd 12 4
810.2.i.e 4 315.cb even 12 4
960.2.f.h 2 280.s even 4 2
960.2.f.i 2 280.y odd 4 2
1200.2.a.g 1 140.c even 2 1
1200.2.a.m 1 28.d even 2 1
1470.2.g.g 2 5.c odd 4 2
1470.2.n.a 4 35.l odd 12 4
1470.2.n.h 4 35.k even 12 4
2880.2.f.c 2 840.bm even 4 2
2880.2.f.e 2 840.bp odd 4 2
3600.2.a.o 1 84.h odd 2 1
3600.2.a.bg 1 420.o odd 2 1
3840.2.d.g 2 560.r even 4 2
3840.2.d.j 2 560.u odd 4 2
3840.2.d.x 2 560.bm odd 4 2
3840.2.d.y 2 560.bn even 4 2
4800.2.a.l 1 280.c odd 2 1
4800.2.a.m 1 56.e even 2 1
4800.2.a.cg 1 56.h odd 2 1
4800.2.a.cj 1 280.n even 2 1
7350.2.a.bg 1 1.a even 1 1 trivial
7350.2.a.cc 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(7350))\):

\( T_{11} - 2 \)
\( T_{13} + 6 \)
\( T_{17} + 2 \)
\( T_{19} \)
\( T_{23} + 4 \)
\( T_{31} - 8 \)

Hecke characteristic polynomials

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