Properties

Label 7350.2.a.ds
Level $7350$
Weight $2$
Character orbit 7350.a
Self dual yes
Analytic conductor $58.690$
Analytic rank $1$
Dimension $4$
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: \(1\)
Dimension: \(4\)
Coefficient field: 4.4.10304.1
Defining polynomial: \(x^{4} - 2 x^{3} - 7 x^{2} + 8 x + 8\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2 \)
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 a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. 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} + ( -\beta_{1} + \beta_{2} ) q^{11} + q^{12} + ( \beta_{1} - 2 \beta_{2} ) q^{13} + q^{16} + ( 1 - \beta_{1} + \beta_{2} - \beta_{3} ) q^{17} - q^{18} + ( -3 + \beta_{1} - 2 \beta_{2} - \beta_{3} ) q^{19} + ( \beta_{1} - \beta_{2} ) q^{22} + ( -1 - \beta_{1} + 2 \beta_{2} + \beta_{3} ) q^{23} - q^{24} + ( -\beta_{1} + 2 \beta_{2} ) q^{26} + q^{27} + ( -2 + \beta_{1} + \beta_{2} + 2 \beta_{3} ) q^{29} + ( -4 + \beta_{1} - \beta_{2} ) q^{31} - q^{32} + ( -\beta_{1} + \beta_{2} ) q^{33} + ( -1 + \beta_{1} - \beta_{2} + \beta_{3} ) q^{34} + q^{36} + ( 2 + \beta_{1} + 3 \beta_{2} ) q^{37} + ( 3 - \beta_{1} + 2 \beta_{2} + \beta_{3} ) q^{38} + ( \beta_{1} - 2 \beta_{2} ) q^{39} + ( -3 - 2 \beta_{2} - \beta_{3} ) q^{41} + ( -1 - 2 \beta_{1} - \beta_{2} + \beta_{3} ) q^{43} + ( -\beta_{1} + \beta_{2} ) q^{44} + ( 1 + \beta_{1} - 2 \beta_{2} - \beta_{3} ) q^{46} + ( 3 + 2 \beta_{1} - \beta_{2} + \beta_{3} ) q^{47} + q^{48} + ( 1 - \beta_{1} + \beta_{2} - \beta_{3} ) q^{51} + ( \beta_{1} - 2 \beta_{2} ) q^{52} + ( -5 - \beta_{1} + 2 \beta_{2} + \beta_{3} ) q^{53} - q^{54} + ( -3 + \beta_{1} - 2 \beta_{2} - \beta_{3} ) q^{57} + ( 2 - \beta_{1} - \beta_{2} - 2 \beta_{3} ) q^{58} + ( -5 + \beta_{1} + 4 \beta_{2} + \beta_{3} ) q^{59} + ( -3 + 3 \beta_{3} ) q^{61} + ( 4 - \beta_{1} + \beta_{2} ) q^{62} + q^{64} + ( \beta_{1} - \beta_{2} ) q^{66} + ( 1 - \beta_{2} - \beta_{3} ) q^{67} + ( 1 - \beta_{1} + \beta_{2} - \beta_{3} ) q^{68} + ( -1 - \beta_{1} + 2 \beta_{2} + \beta_{3} ) q^{69} + ( -5 - \beta_{1} - 2 \beta_{2} - 3 \beta_{3} ) q^{71} - q^{72} + ( -3 + 2 \beta_{1} + 2 \beta_{2} - \beta_{3} ) q^{73} + ( -2 - \beta_{1} - 3 \beta_{2} ) q^{74} + ( -3 + \beta_{1} - 2 \beta_{2} - \beta_{3} ) q^{76} + ( -\beta_{1} + 2 \beta_{2} ) q^{78} + ( -2 + 2 \beta_{1} + 4 \beta_{2} + 2 \beta_{3} ) q^{79} + q^{81} + ( 3 + 2 \beta_{2} + \beta_{3} ) q^{82} + ( 2 - 2 \beta_{2} + 2 \beta_{3} ) q^{83} + ( 1 + 2 \beta_{1} + \beta_{2} - \beta_{3} ) q^{86} + ( -2 + \beta_{1} + \beta_{2} + 2 \beta_{3} ) q^{87} + ( \beta_{1} - \beta_{2} ) q^{88} + ( -11 - 4 \beta_{2} - \beta_{3} ) q^{89} + ( -1 - \beta_{1} + 2 \beta_{2} + \beta_{3} ) q^{92} + ( -4 + \beta_{1} - \beta_{2} ) q^{93} + ( -3 - 2 \beta_{1} + \beta_{2} - \beta_{3} ) q^{94} - q^{96} + ( 5 - 2 \beta_{1} - 4 \beta_{2} - \beta_{3} ) q^{97} + ( -\beta_{1} + \beta_{2} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 4q^{2} + 4q^{3} + 4q^{4} - 4q^{6} - 4q^{8} + 4q^{9} + O(q^{10}) \) \( 4q - 4q^{2} + 4q^{3} + 4q^{4} - 4q^{6} - 4q^{8} + 4q^{9} + 4q^{12} + 4q^{16} + 4q^{17} - 4q^{18} - 12q^{19} - 4q^{23} - 4q^{24} + 4q^{27} - 8q^{29} - 16q^{31} - 4q^{32} - 4q^{34} + 4q^{36} + 8q^{37} + 12q^{38} - 12q^{41} - 4q^{43} + 4q^{46} + 12q^{47} + 4q^{48} + 4q^{51} - 20q^{53} - 4q^{54} - 12q^{57} + 8q^{58} - 20q^{59} - 12q^{61} + 16q^{62} + 4q^{64} + 4q^{67} + 4q^{68} - 4q^{69} - 20q^{71} - 4q^{72} - 12q^{73} - 8q^{74} - 12q^{76} - 8q^{79} + 4q^{81} + 12q^{82} + 8q^{83} + 4q^{86} - 8q^{87} - 44q^{89} - 4q^{92} - 16q^{93} - 12q^{94} - 4q^{96} + 20q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - 2 x^{3} - 7 x^{2} + 8 x + 8\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( \nu^{3} - \nu^{2} - 4 \nu \)\()/2\)
\(\beta_{2}\)\(=\)\((\)\( \nu^{2} - \nu - 4 \)\()/2\)
\(\beta_{3}\)\(=\)\((\)\( -\nu^{3} + \nu^{2} + 8 \nu - 2 \)\()/2\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{3} + \beta_{1} + 1\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{3} + 4 \beta_{2} + \beta_{1} + 9\)\()/2\)
\(\nu^{3}\)\(=\)\((\)\(5 \beta_{3} + 4 \beta_{2} + 9 \beta_{1} + 13\)\()/2\)

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
3.16053
−0.692297
1.69230
−2.16053
−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
1.3 −1.00000 1.00000 1.00000 0 −1.00000 0 −1.00000 1.00000 0
1.4 −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.ds 4
5.b even 2 1 7350.2.a.dt 4
5.c odd 4 2 1470.2.g.j 8
7.b odd 2 1 7350.2.a.dr 4
35.c odd 2 1 7350.2.a.du 4
35.f even 4 2 1470.2.g.k yes 8
35.k even 12 4 1470.2.n.k 16
35.l odd 12 4 1470.2.n.l 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1470.2.g.j 8 5.c odd 4 2
1470.2.g.k yes 8 35.f even 4 2
1470.2.n.k 16 35.k even 12 4
1470.2.n.l 16 35.l odd 12 4
7350.2.a.dr 4 7.b odd 2 1
7350.2.a.ds 4 1.a even 1 1 trivial
7350.2.a.dt 4 5.b even 2 1
7350.2.a.du 4 35.c odd 2 1

Hecke kernels

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

\( T_{11}^{4} - 18 T_{11}^{2} - 16 T_{11} + 32 \)
\( T_{13}^{4} - 26 T_{13}^{2} + 48 T_{13} - 16 \)
\( T_{17}^{4} - 4 T_{17}^{3} - 32 T_{17}^{2} + 8 T_{17} + 124 \)
\( T_{19}^{4} + 12 T_{19}^{3} + 12 T_{19}^{2} - 224 T_{19} - 448 \)
\( T_{23}^{4} + 4 T_{23}^{3} - 36 T_{23}^{2} + 128 \)
\( T_{31}^{4} + 16 T_{31}^{3} + 78 T_{31}^{2} + 128 T_{31} + 64 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T )^{4} \)
$3$ \( ( -1 + T )^{4} \)
$5$ \( T^{4} \)
$7$ \( T^{4} \)
$11$ \( 32 - 16 T - 18 T^{2} + T^{4} \)
$13$ \( -16 + 48 T - 26 T^{2} + T^{4} \)
$17$ \( 124 + 8 T - 32 T^{2} - 4 T^{3} + T^{4} \)
$19$ \( -448 - 224 T + 12 T^{2} + 12 T^{3} + T^{4} \)
$23$ \( 128 - 36 T^{2} + 4 T^{3} + T^{4} \)
$29$ \( 1568 - 336 T - 66 T^{2} + 8 T^{3} + T^{4} \)
$31$ \( 64 + 128 T + 78 T^{2} + 16 T^{3} + T^{4} \)
$37$ \( 200 + 120 T - 42 T^{2} - 8 T^{3} + T^{4} \)
$41$ \( -376 - 120 T + 26 T^{2} + 12 T^{3} + T^{4} \)
$43$ \( 736 - 368 T - 110 T^{2} + 4 T^{3} + T^{4} \)
$47$ \( -1024 + 512 T - 30 T^{2} - 12 T^{3} + T^{4} \)
$53$ \( 64 + 160 T + 108 T^{2} + 20 T^{3} + T^{4} \)
$59$ \( -3008 - 736 T + 52 T^{2} + 20 T^{3} + T^{4} \)
$61$ \( -648 - 1512 T - 126 T^{2} + 12 T^{3} + T^{4} \)
$67$ \( 32 + 16 T - 14 T^{2} - 4 T^{3} + T^{4} \)
$71$ \( 2944 - 1472 T - 36 T^{2} + 20 T^{3} + T^{4} \)
$73$ \( -248 - 536 T - 86 T^{2} + 12 T^{3} + T^{4} \)
$79$ \( -3584 - 1792 T - 176 T^{2} + 8 T^{3} + T^{4} \)
$83$ \( 128 - 64 T - 88 T^{2} - 8 T^{3} + T^{4} \)
$89$ \( 5048 + 3688 T + 658 T^{2} + 44 T^{3} + T^{4} \)
$97$ \( -10376 + 2184 T - 14 T^{2} - 20 T^{3} + T^{4} \)
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