Properties

Label 7350.2.a.du
Level 7350
Weight 2
Character orbit 7350.a
Self dual yes
Analytic conductor 58.690
Analytic rank 0
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: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.10304.1
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:

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.du 4
5.b even 2 1 7350.2.a.dr 4
5.c odd 4 2 1470.2.g.k yes 8
7.b odd 2 1 7350.2.a.dt 4
35.c odd 2 1 7350.2.a.ds 4
35.f even 4 2 1470.2.g.j 8
35.k even 12 4 1470.2.n.l 16
35.l odd 12 4 1470.2.n.k 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1470.2.g.j 8 35.f even 4 2
1470.2.g.k yes 8 5.c odd 4 2
1470.2.n.k 16 35.l odd 12 4
1470.2.n.l 16 35.k even 12 4
7350.2.a.dr 4 5.b even 2 1
7350.2.a.ds 4 35.c odd 2 1
7350.2.a.dt 4 7.b odd 2 1
7350.2.a.du 4 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(7350))\):

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 - T )^{4} \)
$3$ \( ( 1 - T )^{4} \)
$5$ 1
$7$ 1
$11$ \( 1 + 26 T^{2} - 16 T^{3} + 362 T^{4} - 176 T^{5} + 3146 T^{6} + 14641 T^{8} \)
$13$ \( 1 + 26 T^{2} + 48 T^{3} + 322 T^{4} + 624 T^{5} + 4394 T^{6} + 28561 T^{8} \)
$17$ \( 1 - 4 T + 36 T^{2} - 196 T^{3} + 770 T^{4} - 3332 T^{5} + 10404 T^{6} - 19652 T^{7} + 83521 T^{8} \)
$19$ \( 1 - 12 T + 88 T^{2} - 460 T^{3} + 2174 T^{4} - 8740 T^{5} + 31768 T^{6} - 82308 T^{7} + 130321 T^{8} \)
$23$ \( 1 - 4 T + 56 T^{2} - 276 T^{3} + 1646 T^{4} - 6348 T^{5} + 29624 T^{6} - 48668 T^{7} + 279841 T^{8} \)
$29$ \( 1 + 8 T + 50 T^{2} + 360 T^{3} + 2786 T^{4} + 10440 T^{5} + 42050 T^{6} + 195112 T^{7} + 707281 T^{8} \)
$31$ \( 1 - 16 T + 202 T^{2} - 1616 T^{3} + 10666 T^{4} - 50096 T^{5} + 194122 T^{6} - 476656 T^{7} + 923521 T^{8} \)
$37$ \( 1 + 8 T + 106 T^{2} + 768 T^{3} + 5306 T^{4} + 28416 T^{5} + 145114 T^{6} + 405224 T^{7} + 1874161 T^{8} \)
$41$ \( 1 - 12 T + 190 T^{2} - 1356 T^{3} + 11842 T^{4} - 55596 T^{5} + 319390 T^{6} - 827052 T^{7} + 2825761 T^{8} \)
$43$ \( 1 - 4 T + 62 T^{2} - 148 T^{3} + 2370 T^{4} - 6364 T^{5} + 114638 T^{6} - 318028 T^{7} + 3418801 T^{8} \)
$47$ \( 1 - 12 T + 158 T^{2} - 1180 T^{3} + 9410 T^{4} - 55460 T^{5} + 349022 T^{6} - 1245876 T^{7} + 4879681 T^{8} \)
$53$ \( 1 - 20 T + 320 T^{2} - 3340 T^{3} + 28366 T^{4} - 177020 T^{5} + 898880 T^{6} - 2977540 T^{7} + 7890481 T^{8} \)
$59$ \( 1 - 20 T + 288 T^{2} - 2804 T^{3} + 24014 T^{4} - 165436 T^{5} + 1002528 T^{6} - 4107580 T^{7} + 12117361 T^{8} \)
$61$ \( 1 - 12 T + 118 T^{2} - 684 T^{3} + 6306 T^{4} - 41724 T^{5} + 439078 T^{6} - 2723772 T^{7} + 13845841 T^{8} \)
$67$ \( 1 + 4 T + 254 T^{2} + 788 T^{3} + 25090 T^{4} + 52796 T^{5} + 1140206 T^{6} + 1203052 T^{7} + 20151121 T^{8} \)
$71$ \( 1 + 20 T + 248 T^{2} + 2788 T^{3} + 28078 T^{4} + 197948 T^{5} + 1250168 T^{6} + 7158220 T^{7} + 25411681 T^{8} \)
$73$ \( 1 + 12 T + 206 T^{2} + 2092 T^{3} + 19170 T^{4} + 152716 T^{5} + 1097774 T^{6} + 4668204 T^{7} + 28398241 T^{8} \)
$79$ \( 1 + 8 T + 140 T^{2} + 104 T^{3} + 6054 T^{4} + 8216 T^{5} + 873740 T^{6} + 3944312 T^{7} + 38950081 T^{8} \)
$83$ \( 1 - 8 T + 244 T^{2} - 2056 T^{3} + 26854 T^{4} - 170648 T^{5} + 1680916 T^{6} - 4574296 T^{7} + 47458321 T^{8} \)
$89$ \( 1 - 44 T + 1014 T^{2} - 15436 T^{3} + 169698 T^{4} - 1373804 T^{5} + 8031894 T^{6} - 31018636 T^{7} + 62742241 T^{8} \)
$97$ \( 1 - 20 T + 374 T^{2} - 3636 T^{3} + 43362 T^{4} - 352692 T^{5} + 3518966 T^{6} - 18253460 T^{7} + 88529281 T^{8} \)
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