Properties

Label 7350.2.a.ct
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

\(f(q)\) \(=\) \( q + q^{2} + q^{3} + q^{4} + q^{6} + q^{8} + q^{9} + O(q^{10}) \) \( q + q^{2} + q^{3} + q^{4} + q^{6} + q^{8} + q^{9} + q^{12} + 2q^{13} + q^{16} + 6q^{17} + q^{18} + 4q^{19} + q^{24} + 2q^{26} + q^{27} - 6q^{29} - 8q^{31} + q^{32} + 6q^{34} + q^{36} - 2q^{37} + 4q^{38} + 2q^{39} + 6q^{41} + 4q^{43} + q^{48} + 6q^{51} + 2q^{52} + 6q^{53} + q^{54} + 4q^{57} - 6q^{58} + 10q^{61} - 8q^{62} + q^{64} + 4q^{67} + 6q^{68} + q^{72} + 2q^{73} - 2q^{74} + 4q^{76} + 2q^{78} + 8q^{79} + q^{81} + 6q^{82} + 12q^{83} + 4q^{86} - 6q^{87} - 18q^{89} - 8q^{93} + q^{96} + 2q^{97} + O(q^{100}) \)

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:

Atkin-Lehner signs

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

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.ct 1
5.b even 2 1 1470.2.a.d 1
7.b odd 2 1 150.2.a.b 1
15.d odd 2 1 4410.2.a.z 1
21.c even 2 1 450.2.a.d 1
28.d even 2 1 1200.2.a.k 1
35.c odd 2 1 30.2.a.a 1
35.f even 4 2 150.2.c.a 2
35.i odd 6 2 1470.2.i.o 2
35.j even 6 2 1470.2.i.q 2
56.e even 2 1 4800.2.a.d 1
56.h odd 2 1 4800.2.a.cq 1
84.h odd 2 1 3600.2.a.f 1
105.g even 2 1 90.2.a.c 1
105.k odd 4 2 450.2.c.b 2
140.c even 2 1 240.2.a.b 1
140.j odd 4 2 1200.2.f.e 2
280.c odd 2 1 960.2.a.e 1
280.n even 2 1 960.2.a.p 1
280.s even 4 2 4800.2.f.p 2
280.y odd 4 2 4800.2.f.w 2
315.z even 6 2 810.2.e.b 2
315.bg odd 6 2 810.2.e.l 2
385.h even 2 1 3630.2.a.w 1
420.o odd 2 1 720.2.a.j 1
420.w even 4 2 3600.2.f.i 2
455.h odd 2 1 5070.2.a.w 1
455.u even 4 2 5070.2.b.k 2
560.be even 4 2 3840.2.k.f 2
560.bf odd 4 2 3840.2.k.y 2
595.b odd 2 1 8670.2.a.g 1
840.b odd 2 1 2880.2.a.q 1
840.u even 2 1 2880.2.a.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
30.2.a.a 1 35.c odd 2 1
90.2.a.c 1 105.g even 2 1
150.2.a.b 1 7.b odd 2 1
150.2.c.a 2 35.f even 4 2
240.2.a.b 1 140.c even 2 1
450.2.a.d 1 21.c even 2 1
450.2.c.b 2 105.k odd 4 2
720.2.a.j 1 420.o odd 2 1
810.2.e.b 2 315.z even 6 2
810.2.e.l 2 315.bg odd 6 2
960.2.a.e 1 280.c odd 2 1
960.2.a.p 1 280.n even 2 1
1200.2.a.k 1 28.d even 2 1
1200.2.f.e 2 140.j odd 4 2
1470.2.a.d 1 5.b even 2 1
1470.2.i.o 2 35.i odd 6 2
1470.2.i.q 2 35.j even 6 2
2880.2.a.a 1 840.u even 2 1
2880.2.a.q 1 840.b odd 2 1
3600.2.a.f 1 84.h odd 2 1
3600.2.f.i 2 420.w even 4 2
3630.2.a.w 1 385.h even 2 1
3840.2.k.f 2 560.be even 4 2
3840.2.k.y 2 560.bf odd 4 2
4410.2.a.z 1 15.d odd 2 1
4800.2.a.d 1 56.e even 2 1
4800.2.a.cq 1 56.h odd 2 1
4800.2.f.p 2 280.s even 4 2
4800.2.f.w 2 280.y odd 4 2
5070.2.a.w 1 455.h odd 2 1
5070.2.b.k 2 455.u even 4 2
7350.2.a.ct 1 1.a even 1 1 trivial
8670.2.a.g 1 595.b odd 2 1

Hecke kernels

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

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -1 + T \)
$3$ \( -1 + T \)
$5$ \( T \)
$7$ \( T \)
$11$ \( T \)
$13$ \( -2 + T \)
$17$ \( -6 + T \)
$19$ \( -4 + T \)
$23$ \( T \)
$29$ \( 6 + T \)
$31$ \( 8 + T \)
$37$ \( 2 + T \)
$41$ \( -6 + T \)
$43$ \( -4 + T \)
$47$ \( T \)
$53$ \( -6 + T \)
$59$ \( T \)
$61$ \( -10 + T \)
$67$ \( -4 + T \)
$71$ \( T \)
$73$ \( -2 + T \)
$79$ \( -8 + T \)
$83$ \( -12 + T \)
$89$ \( 18 + T \)
$97$ \( -2 + T \)
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