Properties

Label 7350.2.a.dm
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: \(\Q(\sqrt{2}) \)
Defining polynomial: \(x^{2} - 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
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.

\(f(q)\) \(=\) \( q + q^{2} + q^{3} + q^{4} + q^{6} + q^{8} + q^{9} +O(q^{10})\) \( q + q^{2} + q^{3} + q^{4} + q^{6} + q^{8} + q^{9} + ( 2 + \beta ) q^{11} + q^{12} + 3 q^{13} + q^{16} + ( 3 + 2 \beta ) q^{17} + q^{18} -\beta q^{19} + ( 2 + \beta ) q^{22} + ( -1 + \beta ) q^{23} + q^{24} + 3 q^{26} + q^{27} + q^{29} + ( 1 + \beta ) q^{31} + q^{32} + ( 2 + \beta ) q^{33} + ( 3 + 2 \beta ) q^{34} + q^{36} -5 \beta q^{37} -\beta q^{38} + 3 q^{39} + ( 3 + 2 \beta ) q^{41} + ( -3 + \beta ) q^{43} + ( 2 + \beta ) q^{44} + ( -1 + \beta ) q^{46} + ( 4 - 4 \beta ) q^{47} + q^{48} + ( 3 + 2 \beta ) q^{51} + 3 q^{52} + ( -7 - 4 \beta ) q^{53} + q^{54} -\beta q^{57} + q^{58} + ( -3 - 3 \beta ) q^{59} + ( 3 - 2 \beta ) q^{61} + ( 1 + \beta ) q^{62} + q^{64} + ( 2 + \beta ) q^{66} + ( 4 + 8 \beta ) q^{67} + ( 3 + 2 \beta ) q^{68} + ( -1 + \beta ) q^{69} + ( 2 - 8 \beta ) q^{71} + q^{72} + ( 8 + 5 \beta ) q^{73} -5 \beta q^{74} -\beta q^{76} + 3 q^{78} + ( -6 + 5 \beta ) q^{79} + q^{81} + ( 3 + 2 \beta ) q^{82} + ( 7 - 5 \beta ) q^{83} + ( -3 + \beta ) q^{86} + q^{87} + ( 2 + \beta ) q^{88} -4 q^{89} + ( -1 + \beta ) q^{92} + ( 1 + \beta ) q^{93} + ( 4 - 4 \beta ) q^{94} + q^{96} + ( -6 + 4 \beta ) q^{97} + ( 2 + \beta ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{2} + 2q^{3} + 2q^{4} + 2q^{6} + 2q^{8} + 2q^{9} + O(q^{10}) \) \( 2q + 2q^{2} + 2q^{3} + 2q^{4} + 2q^{6} + 2q^{8} + 2q^{9} + 4q^{11} + 2q^{12} + 6q^{13} + 2q^{16} + 6q^{17} + 2q^{18} + 4q^{22} - 2q^{23} + 2q^{24} + 6q^{26} + 2q^{27} + 2q^{29} + 2q^{31} + 2q^{32} + 4q^{33} + 6q^{34} + 2q^{36} + 6q^{39} + 6q^{41} - 6q^{43} + 4q^{44} - 2q^{46} + 8q^{47} + 2q^{48} + 6q^{51} + 6q^{52} - 14q^{53} + 2q^{54} + 2q^{58} - 6q^{59} + 6q^{61} + 2q^{62} + 2q^{64} + 4q^{66} + 8q^{67} + 6q^{68} - 2q^{69} + 4q^{71} + 2q^{72} + 16q^{73} + 6q^{78} - 12q^{79} + 2q^{81} + 6q^{82} + 14q^{83} - 6q^{86} + 2q^{87} + 4q^{88} - 8q^{89} - 2q^{92} + 2q^{93} + 8q^{94} + 2q^{96} - 12q^{97} + 4q^{99} + O(q^{100}) \)

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.dm yes 2
5.b even 2 1 7350.2.a.dc 2
7.b odd 2 1 7350.2.a.dj yes 2
35.c odd 2 1 7350.2.a.dg yes 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
7350.2.a.dc 2 5.b even 2 1
7350.2.a.dg yes 2 35.c odd 2 1
7350.2.a.dj yes 2 7.b odd 2 1
7350.2.a.dm yes 2 1.a even 1 1 trivial

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 \(S_{2}^{\mathrm{new}}(\Gamma_0(7350))\):

\( T_{11}^{2} - 4 T_{11} + 2 \)
\( T_{13} - 3 \)
\( T_{17}^{2} - 6 T_{17} + 1 \)
\( T_{19}^{2} - 2 \)
\( T_{23}^{2} + 2 T_{23} - 1 \)
\( T_{31}^{2} - 2 T_{31} - 1 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 - T )^{2} \)
$3$ \( ( 1 - T )^{2} \)
$5$ 1
$7$ 1
$11$ \( 1 - 4 T + 24 T^{2} - 44 T^{3} + 121 T^{4} \)
$13$ \( ( 1 - 3 T + 13 T^{2} )^{2} \)
$17$ \( 1 - 6 T + 35 T^{2} - 102 T^{3} + 289 T^{4} \)
$19$ \( 1 + 36 T^{2} + 361 T^{4} \)
$23$ \( 1 + 2 T + 45 T^{2} + 46 T^{3} + 529 T^{4} \)
$29$ \( ( 1 - T + 29 T^{2} )^{2} \)
$31$ \( 1 - 2 T + 61 T^{2} - 62 T^{3} + 961 T^{4} \)
$37$ \( 1 + 24 T^{2} + 1369 T^{4} \)
$41$ \( 1 - 6 T + 83 T^{2} - 246 T^{3} + 1681 T^{4} \)
$43$ \( 1 + 6 T + 93 T^{2} + 258 T^{3} + 1849 T^{4} \)
$47$ \( 1 - 8 T + 78 T^{2} - 376 T^{3} + 2209 T^{4} \)
$53$ \( 1 + 14 T + 123 T^{2} + 742 T^{3} + 2809 T^{4} \)
$59$ \( 1 + 6 T + 109 T^{2} + 354 T^{3} + 3481 T^{4} \)
$61$ \( 1 - 6 T + 123 T^{2} - 366 T^{3} + 3721 T^{4} \)
$67$ \( 1 - 8 T + 22 T^{2} - 536 T^{3} + 4489 T^{4} \)
$71$ \( 1 - 4 T + 18 T^{2} - 284 T^{3} + 5041 T^{4} \)
$73$ \( 1 - 16 T + 160 T^{2} - 1168 T^{3} + 5329 T^{4} \)
$79$ \( 1 + 12 T + 144 T^{2} + 948 T^{3} + 6241 T^{4} \)
$83$ \( 1 - 14 T + 165 T^{2} - 1162 T^{3} + 6889 T^{4} \)
$89$ \( ( 1 + 4 T + 89 T^{2} )^{2} \)
$97$ \( 1 + 12 T + 198 T^{2} + 1164 T^{3} + 9409 T^{4} \)
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