Properties

Label 5376.2.a.bp
Level 5376
Weight 2
Character orbit 5376.a
Self dual yes
Analytic conductor 42.928
Analytic rank 0
Dimension 4
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 5376 = 2^{8} \cdot 3 \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 5376.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(42.9275761266\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.19664.1
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 168)
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.

\(f(q)\) \(=\) \( q + q^{3} + ( -1 - \beta_{3} ) q^{5} - q^{7} + q^{9} +O(q^{10})\) \( q + q^{3} + ( -1 - \beta_{3} ) q^{5} - q^{7} + q^{9} + ( 2 + \beta_{2} ) q^{11} + ( -1 + \beta_{1} ) q^{13} + ( -1 - \beta_{3} ) q^{15} + ( \beta_{1} - \beta_{3} ) q^{17} + ( 1 - \beta_{2} - \beta_{3} ) q^{19} - q^{21} + ( -2 + \beta_{1} - \beta_{3} ) q^{23} + ( 4 - 2 \beta_{1} + \beta_{2} + \beta_{3} ) q^{25} + q^{27} + ( 1 - 2 \beta_{1} + \beta_{2} + \beta_{3} ) q^{29} + ( 1 + \beta_{2} - \beta_{3} ) q^{31} + ( 2 + \beta_{2} ) q^{33} + ( 1 + \beta_{3} ) q^{35} + ( -1 - \beta_{2} + \beta_{3} ) q^{37} + ( -1 + \beta_{1} ) q^{39} + ( \beta_{1} - \beta_{3} ) q^{41} + ( 2 + \beta_{1} - \beta_{2} + \beta_{3} ) q^{43} + ( -1 - \beta_{3} ) q^{45} + ( -2 - 2 \beta_{2} - 2 \beta_{3} ) q^{47} + q^{49} + ( \beta_{1} - \beta_{3} ) q^{51} + ( 1 - \beta_{2} - \beta_{3} ) q^{53} + ( -1 - \beta_{2} - 3 \beta_{3} ) q^{55} + ( 1 - \beta_{2} - \beta_{3} ) q^{57} + 4 q^{59} + ( 1 + \beta_{1} + 2 \beta_{3} ) q^{61} - q^{63} + 4 \beta_{3} q^{65} + ( 2 - \beta_{1} - \beta_{2} - \beta_{3} ) q^{67} + ( -2 + \beta_{1} - \beta_{3} ) q^{69} + ( 4 - \beta_{1} + 2 \beta_{2} - \beta_{3} ) q^{71} + ( 2 + 2 \beta_{1} + 2 \beta_{3} ) q^{73} + ( 4 - 2 \beta_{1} + \beta_{2} + \beta_{3} ) q^{75} + ( -2 - \beta_{2} ) q^{77} + ( -4 - 2 \beta_{1} + 2 \beta_{3} ) q^{79} + q^{81} + ( 8 + 2 \beta_{2} ) q^{83} + ( 7 - 2 \beta_{1} + \beta_{2} + 3 \beta_{3} ) q^{85} + ( 1 - 2 \beta_{1} + \beta_{2} + \beta_{3} ) q^{87} + ( -4 - \beta_{1} - 3 \beta_{3} ) q^{89} + ( 1 - \beta_{1} ) q^{91} + ( 1 + \beta_{2} - \beta_{3} ) q^{93} + ( 6 - 2 \beta_{1} + 2 \beta_{2} ) q^{95} + ( 4 - 2 \beta_{1} - 2 \beta_{2} ) q^{97} + ( 2 + \beta_{2} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 4q^{3} - 2q^{5} - 4q^{7} + 4q^{9} + O(q^{10}) \) \( 4q + 4q^{3} - 2q^{5} - 4q^{7} + 4q^{9} + 6q^{11} - 4q^{13} - 2q^{15} + 2q^{17} + 8q^{19} - 4q^{21} - 6q^{23} + 12q^{25} + 4q^{27} + 4q^{31} + 6q^{33} + 2q^{35} - 4q^{37} - 4q^{39} + 2q^{41} + 8q^{43} - 2q^{45} + 4q^{49} + 2q^{51} + 8q^{53} + 4q^{55} + 8q^{57} + 16q^{59} - 4q^{63} - 8q^{65} + 12q^{67} - 6q^{69} + 14q^{71} + 4q^{73} + 12q^{75} - 6q^{77} - 20q^{79} + 4q^{81} + 28q^{83} + 20q^{85} - 10q^{89} + 4q^{91} + 4q^{93} + 20q^{95} + 20q^{97} + 6q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - 2 x^{3} - 5 x^{2} + 2 x + 2\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( 2 \nu - 1 \)
\(\beta_{2}\)\(=\)\( \nu^{3} - \nu^{2} - 6 \nu - 2 \)
\(\beta_{3}\)\(=\)\( \nu^{3} - 3 \nu^{2} - 2 \nu + 3 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{1} + 1\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(-\beta_{3} + \beta_{2} + 2 \beta_{1} + 7\)\()/2\)
\(\nu^{3}\)\(=\)\((\)\(-\beta_{3} + 3 \beta_{2} + 8 \beta_{1} + 17\)\()/2\)

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
−0.512486
0.771876
3.28139
−1.54078
0 1.00000 0 −4.10245 0 −1.00000 0 1.00000 0
1.2 0 1.00000 0 −1.12875 0 −1.00000 0 1.00000 0
1.3 0 1.00000 0 −0.467138 0 −1.00000 0 1.00000 0
1.4 0 1.00000 0 3.69833 0 −1.00000 0 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 5376.2.a.bp 4
4.b odd 2 1 5376.2.a.bl 4
8.b even 2 1 5376.2.a.bm 4
8.d odd 2 1 5376.2.a.bq 4
16.e even 4 2 168.2.c.b 8
16.f odd 4 2 672.2.c.b 8
48.i odd 4 2 504.2.c.f 8
48.k even 4 2 2016.2.c.e 8
112.j even 4 2 4704.2.c.c 8
112.l odd 4 2 1176.2.c.c 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
168.2.c.b 8 16.e even 4 2
504.2.c.f 8 48.i odd 4 2
672.2.c.b 8 16.f odd 4 2
1176.2.c.c 8 112.l odd 4 2
2016.2.c.e 8 48.k even 4 2
4704.2.c.c 8 112.j even 4 2
5376.2.a.bl 4 4.b odd 2 1
5376.2.a.bm 4 8.b even 2 1
5376.2.a.bp 4 1.a even 1 1 trivial
5376.2.a.bq 4 8.d odd 2 1

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(3\) \(-1\)
\(7\) \(1\)

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(5376))\):

\( T_{5}^{4} + 2 T_{5}^{3} - 14 T_{5}^{2} - 24 T_{5} - 8 \)
\( T_{11}^{4} - 6 T_{11}^{3} - 14 T_{11}^{2} + 136 T_{11} - 200 \)
\( T_{13}^{4} + 4 T_{13}^{3} - 20 T_{13}^{2} - 80 T_{13} - 32 \)
\( T_{19}^{4} - 8 T_{19}^{3} - 12 T_{19}^{2} + 96 T_{19} + 128 \)
\( T_{29}^{4} - 108 T_{29}^{2} - 32 T_{29} + 2624 \)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( ( 1 - T )^{4} \)
$5$ \( 1 + 2 T + 6 T^{2} + 6 T^{3} + 2 T^{4} + 30 T^{5} + 150 T^{6} + 250 T^{7} + 625 T^{8} \)
$7$ \( ( 1 + T )^{4} \)
$11$ \( 1 - 6 T + 30 T^{2} - 62 T^{3} + 218 T^{4} - 682 T^{5} + 3630 T^{6} - 7986 T^{7} + 14641 T^{8} \)
$13$ \( 1 + 4 T + 32 T^{2} + 76 T^{3} + 462 T^{4} + 988 T^{5} + 5408 T^{6} + 8788 T^{7} + 28561 T^{8} \)
$17$ \( 1 - 2 T + 38 T^{2} - 70 T^{3} + 706 T^{4} - 1190 T^{5} + 10982 T^{6} - 9826 T^{7} + 83521 T^{8} \)
$19$ \( 1 - 8 T + 64 T^{2} - 360 T^{3} + 1838 T^{4} - 6840 T^{5} + 23104 T^{6} - 54872 T^{7} + 130321 T^{8} \)
$23$ \( 1 + 6 T + 74 T^{2} + 334 T^{3} + 2282 T^{4} + 7682 T^{5} + 39146 T^{6} + 73002 T^{7} + 279841 T^{8} \)
$29$ \( 1 + 8 T^{2} - 32 T^{3} + 1406 T^{4} - 928 T^{5} + 6728 T^{6} + 707281 T^{8} \)
$31$ \( 1 - 4 T + 80 T^{2} - 244 T^{3} + 3294 T^{4} - 7564 T^{5} + 76880 T^{6} - 119164 T^{7} + 923521 T^{8} \)
$37$ \( 1 + 4 T + 104 T^{2} + 316 T^{3} + 5214 T^{4} + 11692 T^{5} + 142376 T^{6} + 202612 T^{7} + 1874161 T^{8} \)
$41$ \( 1 - 2 T + 134 T^{2} - 214 T^{3} + 7618 T^{4} - 8774 T^{5} + 225254 T^{6} - 137842 T^{7} + 2825761 T^{8} \)
$43$ \( 1 - 8 T + 116 T^{2} - 552 T^{3} + 5526 T^{4} - 23736 T^{5} + 214484 T^{6} - 636056 T^{7} + 3418801 T^{8} \)
$47$ \( 1 + 44 T^{2} - 128 T^{3} + 3302 T^{4} - 6016 T^{5} + 97196 T^{6} + 4879681 T^{8} \)
$53$ \( 1 - 8 T + 200 T^{2} - 1176 T^{3} + 15710 T^{4} - 62328 T^{5} + 561800 T^{6} - 1191016 T^{7} + 7890481 T^{8} \)
$59$ \( ( 1 - 4 T + 59 T^{2} )^{4} \)
$61$ \( 1 + 136 T^{2} + 544 T^{3} + 8510 T^{4} + 33184 T^{5} + 506056 T^{6} + 13845841 T^{8} \)
$67$ \( 1 - 12 T + 244 T^{2} - 1948 T^{3} + 23606 T^{4} - 130516 T^{5} + 1095316 T^{6} - 3609156 T^{7} + 20151121 T^{8} \)
$71$ \( 1 - 14 T + 194 T^{2} - 1686 T^{3} + 14330 T^{4} - 119706 T^{5} + 977954 T^{6} - 5010754 T^{7} + 25411681 T^{8} \)
$73$ \( 1 - 4 T + 92 T^{2} + 580 T^{3} + 166 T^{4} + 42340 T^{5} + 490268 T^{6} - 1556068 T^{7} + 28398241 T^{8} \)
$79$ \( 1 + 20 T + 340 T^{2} + 3972 T^{3} + 38678 T^{4} + 313788 T^{5} + 2121940 T^{6} + 9860780 T^{7} + 38950081 T^{8} \)
$83$ \( 1 - 28 T + 516 T^{2} - 6268 T^{3} + 64454 T^{4} - 520244 T^{5} + 3554724 T^{6} - 16010036 T^{7} + 47458321 T^{8} \)
$89$ \( 1 + 10 T + 198 T^{2} + 494 T^{3} + 13122 T^{4} + 43966 T^{5} + 1568358 T^{6} + 7049690 T^{7} + 62742241 T^{8} \)
$97$ \( 1 - 20 T + 300 T^{2} - 2572 T^{3} + 25574 T^{4} - 249484 T^{5} + 2822700 T^{6} - 18253460 T^{7} + 88529281 T^{8} \)
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