Properties

Label 7350.2.a.dh
Level 7350
Weight 2
Character orbit 7350.a
Self dual yes
Analytic conductor 58.690
Analytic rank 0
Dimension 2
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: \(2\)
Coefficient field: \(\Q(\sqrt{2}) \)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1470)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{2}\). We also show the integral \(q\)-expansion of the trace form.

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

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
−1.41421
1.41421
1.00000 −1.00000 1.00000 0 −1.00000 0 1.00000 1.00000 0
1.2 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.dh 2
5.b even 2 1 1470.2.a.t yes 2
7.b odd 2 1 7350.2.a.dl 2
15.d odd 2 1 4410.2.a.bz 2
35.c odd 2 1 1470.2.a.s 2
35.i odd 6 2 1470.2.i.x 4
35.j even 6 2 1470.2.i.w 4
105.g even 2 1 4410.2.a.bw 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1470.2.a.s 2 35.c odd 2 1
1470.2.a.t yes 2 5.b even 2 1
1470.2.i.w 4 35.j even 6 2
1470.2.i.x 4 35.i odd 6 2
4410.2.a.bw 2 105.g even 2 1
4410.2.a.bz 2 15.d odd 2 1
7350.2.a.dh 2 1.a even 1 1 trivial
7350.2.a.dl 2 7.b odd 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(7350))\):

\( T_{11}^{2} + 4 T_{11} - 14 \)
\( T_{13}^{2} - 32 \)
\( T_{17}^{2} - 8 T_{17} + 14 \)
\( T_{19}^{2} - 8 T_{19} + 8 \)
\( T_{23}^{2} + 12 T_{23} + 28 \)
\( T_{31}^{2} - 12 T_{31} + 18 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 - T )^{2} \)
$3$ \( ( 1 + T )^{2} \)
$5$ 1
$7$ 1
$11$ \( 1 + 4 T + 8 T^{2} + 44 T^{3} + 121 T^{4} \)
$13$ \( 1 - 6 T^{2} + 169 T^{4} \)
$17$ \( 1 - 8 T + 48 T^{2} - 136 T^{3} + 289 T^{4} \)
$19$ \( 1 - 8 T + 46 T^{2} - 152 T^{3} + 361 T^{4} \)
$23$ \( 1 + 12 T + 74 T^{2} + 276 T^{3} + 529 T^{4} \)
$29$ \( 1 - 8 T + 72 T^{2} - 232 T^{3} + 841 T^{4} \)
$31$ \( 1 - 12 T + 80 T^{2} - 372 T^{3} + 961 T^{4} \)
$37$ \( 1 + 8 T + 72 T^{2} + 296 T^{3} + 1369 T^{4} \)
$41$ \( 1 + 4 T + 14 T^{2} + 164 T^{3} + 1681 T^{4} \)
$43$ \( 1 + 4 T + 40 T^{2} + 172 T^{3} + 1849 T^{4} \)
$47$ \( 1 - 4 T + 80 T^{2} - 188 T^{3} + 2209 T^{4} \)
$53$ \( 1 + 12 T + 110 T^{2} + 636 T^{3} + 2809 T^{4} \)
$59$ \( 1 - 12 T + 146 T^{2} - 708 T^{3} + 3481 T^{4} \)
$61$ \( 1 - 12 T + 126 T^{2} - 732 T^{3} + 3721 T^{4} \)
$67$ \( 1 + 4 T + 88 T^{2} + 268 T^{3} + 4489 T^{4} \)
$71$ \( 1 - 8 T + 86 T^{2} - 568 T^{3} + 5041 T^{4} \)
$73$ \( ( 1 + 2 T + 73 T^{2} )^{2} \)
$79$ \( ( 1 - 8 T + 79 T^{2} )^{2} \)
$83$ \( 1 - 16 T + 222 T^{2} - 1328 T^{3} + 6889 T^{4} \)
$89$ \( 1 + 4 T + 174 T^{2} + 356 T^{3} + 7921 T^{4} \)
$97$ \( 1 - 4 T + 126 T^{2} - 388 T^{3} + 9409 T^{4} \)
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