Properties

Label 7350.2.a.w
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

Downloads

Learn more about

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

\(f(q)\) \(=\) \( q - q^{2} + q^{3} + q^{4} - q^{6} - q^{8} + q^{9} + O(q^{10}) \) \( q - q^{2} + q^{3} + q^{4} - q^{6} - q^{8} + q^{9} - 4q^{11} + q^{12} - 2q^{13} + q^{16} + 2q^{17} - q^{18} - 4q^{19} + 4q^{22} + 8q^{23} - q^{24} + 2q^{26} + q^{27} - 2q^{29} - q^{32} - 4q^{33} - 2q^{34} + q^{36} - 6q^{37} + 4q^{38} - 2q^{39} + 6q^{41} + 4q^{43} - 4q^{44} - 8q^{46} + q^{48} + 2q^{51} - 2q^{52} + 10q^{53} - q^{54} - 4q^{57} + 2q^{58} - 12q^{59} - 14q^{61} + q^{64} + 4q^{66} + 12q^{67} + 2q^{68} + 8q^{69} - 8q^{71} - q^{72} + 10q^{73} + 6q^{74} - 4q^{76} + 2q^{78} + 16q^{79} + q^{81} - 6q^{82} - 12q^{83} - 4q^{86} - 2q^{87} + 4q^{88} - 10q^{89} + 8q^{92} - q^{96} + 2q^{97} - 4q^{99} + O(q^{100}) \)

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.w 1
5.b even 2 1 1470.2.a.j 1
7.b odd 2 1 1050.2.a.c 1
15.d odd 2 1 4410.2.a.t 1
21.c even 2 1 3150.2.a.bp 1
28.d even 2 1 8400.2.a.ce 1
35.c odd 2 1 210.2.a.e 1
35.f even 4 2 1050.2.g.g 2
35.i odd 6 2 1470.2.i.a 2
35.j even 6 2 1470.2.i.j 2
105.g even 2 1 630.2.a.a 1
105.k odd 4 2 3150.2.g.q 2
140.c even 2 1 1680.2.a.j 1
280.c odd 2 1 6720.2.a.j 1
280.n even 2 1 6720.2.a.bq 1
420.o odd 2 1 5040.2.a.k 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
210.2.a.e 1 35.c odd 2 1
630.2.a.a 1 105.g even 2 1
1050.2.a.c 1 7.b odd 2 1
1050.2.g.g 2 35.f even 4 2
1470.2.a.j 1 5.b even 2 1
1470.2.i.a 2 35.i odd 6 2
1470.2.i.j 2 35.j even 6 2
1680.2.a.j 1 140.c even 2 1
3150.2.a.bp 1 21.c even 2 1
3150.2.g.q 2 105.k odd 4 2
4410.2.a.t 1 15.d odd 2 1
5040.2.a.k 1 420.o odd 2 1
6720.2.a.j 1 280.c odd 2 1
6720.2.a.bq 1 280.n even 2 1
7350.2.a.w 1 1.a even 1 1 trivial
8400.2.a.ce 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(7350))\):

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + T \)
$3$ \( 1 - T \)
$5$ 1
$7$ 1
$11$ \( 1 + 4 T + 11 T^{2} \)
$13$ \( 1 + 2 T + 13 T^{2} \)
$17$ \( 1 - 2 T + 17 T^{2} \)
$19$ \( 1 + 4 T + 19 T^{2} \)
$23$ \( 1 - 8 T + 23 T^{2} \)
$29$ \( 1 + 2 T + 29 T^{2} \)
$31$ \( 1 + 31 T^{2} \)
$37$ \( 1 + 6 T + 37 T^{2} \)
$41$ \( 1 - 6 T + 41 T^{2} \)
$43$ \( 1 - 4 T + 43 T^{2} \)
$47$ \( 1 + 47 T^{2} \)
$53$ \( 1 - 10 T + 53 T^{2} \)
$59$ \( 1 + 12 T + 59 T^{2} \)
$61$ \( 1 + 14 T + 61 T^{2} \)
$67$ \( 1 - 12 T + 67 T^{2} \)
$71$ \( 1 + 8 T + 71 T^{2} \)
$73$ \( 1 - 10 T + 73 T^{2} \)
$79$ \( 1 - 16 T + 79 T^{2} \)
$83$ \( 1 + 12 T + 83 T^{2} \)
$89$ \( 1 + 10 T + 89 T^{2} \)
$97$ \( 1 - 2 T + 97 T^{2} \)
show more
show less