Properties

Label 5376.2.c.bb
Level $5376$
Weight $2$
Character orbit 5376.c
Analytic conductor $42.928$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 5376 = 2^{8} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5376.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(42.9275761266\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 168)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -i q^{3} + 2 i q^{5} + q^{7} - q^{9} +O(q^{10})\) \( q -i q^{3} + 2 i q^{5} + q^{7} - q^{9} -6 i q^{13} + 2 q^{15} -2 q^{17} + 4 i q^{19} -i q^{21} -4 q^{23} + q^{25} + i q^{27} + 10 i q^{29} + 8 q^{31} + 2 i q^{35} + 6 i q^{37} -6 q^{39} + 2 q^{41} + 4 i q^{43} -2 i q^{45} -8 q^{47} + q^{49} + 2 i q^{51} -10 i q^{53} + 4 q^{57} -12 i q^{59} + 2 i q^{61} - q^{63} + 12 q^{65} + 12 i q^{67} + 4 i q^{69} -12 q^{71} + 14 q^{73} -i q^{75} + 8 q^{79} + q^{81} + 12 i q^{83} -4 i q^{85} + 10 q^{87} + 2 q^{89} -6 i q^{91} -8 i q^{93} -8 q^{95} + 10 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{7} - 2q^{9} + O(q^{10}) \) \( 2q + 2q^{7} - 2q^{9} + 4q^{15} - 4q^{17} - 8q^{23} + 2q^{25} + 16q^{31} - 12q^{39} + 4q^{41} - 16q^{47} + 2q^{49} + 8q^{57} - 2q^{63} + 24q^{65} - 24q^{71} + 28q^{73} + 16q^{79} + 2q^{81} + 20q^{87} + 4q^{89} - 16q^{95} + 20q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/5376\mathbb{Z}\right)^\times\).

\(n\) \(1793\) \(2815\) \(4609\) \(5125\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2689.1
1.00000i
1.00000i
0 1.00000i 0 2.00000i 0 1.00000 0 −1.00000 0
2689.2 0 1.00000i 0 2.00000i 0 1.00000 0 −1.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 5376.2.c.bb 2
4.b odd 2 1 5376.2.c.d 2
8.b even 2 1 inner 5376.2.c.bb 2
8.d odd 2 1 5376.2.c.d 2
16.e even 4 1 336.2.a.e 1
16.e even 4 1 1344.2.a.b 1
16.f odd 4 1 168.2.a.a 1
16.f odd 4 1 1344.2.a.m 1
48.i odd 4 1 1008.2.a.b 1
48.i odd 4 1 4032.2.a.bc 1
48.k even 4 1 504.2.a.e 1
48.k even 4 1 4032.2.a.bh 1
80.j even 4 1 4200.2.t.j 2
80.k odd 4 1 4200.2.a.t 1
80.q even 4 1 8400.2.a.y 1
80.s even 4 1 4200.2.t.j 2
112.j even 4 1 1176.2.a.f 1
112.j even 4 1 9408.2.a.be 1
112.l odd 4 1 2352.2.a.c 1
112.l odd 4 1 9408.2.a.da 1
112.u odd 12 2 1176.2.q.f 2
112.v even 12 2 1176.2.q.d 2
112.w even 12 2 2352.2.q.d 2
112.x odd 12 2 2352.2.q.w 2
336.v odd 4 1 3528.2.a.v 1
336.y even 4 1 7056.2.a.bq 1
336.br odd 12 2 3528.2.s.g 2
336.bu even 12 2 3528.2.s.w 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
168.2.a.a 1 16.f odd 4 1
336.2.a.e 1 16.e even 4 1
504.2.a.e 1 48.k even 4 1
1008.2.a.b 1 48.i odd 4 1
1176.2.a.f 1 112.j even 4 1
1176.2.q.d 2 112.v even 12 2
1176.2.q.f 2 112.u odd 12 2
1344.2.a.b 1 16.e even 4 1
1344.2.a.m 1 16.f odd 4 1
2352.2.a.c 1 112.l odd 4 1
2352.2.q.d 2 112.w even 12 2
2352.2.q.w 2 112.x odd 12 2
3528.2.a.v 1 336.v odd 4 1
3528.2.s.g 2 336.br odd 12 2
3528.2.s.w 2 336.bu even 12 2
4032.2.a.bc 1 48.i odd 4 1
4032.2.a.bh 1 48.k even 4 1
4200.2.a.t 1 80.k odd 4 1
4200.2.t.j 2 80.j even 4 1
4200.2.t.j 2 80.s even 4 1
5376.2.c.d 2 4.b odd 2 1
5376.2.c.d 2 8.d odd 2 1
5376.2.c.bb 2 1.a even 1 1 trivial
5376.2.c.bb 2 8.b even 2 1 inner
7056.2.a.bq 1 336.y even 4 1
8400.2.a.y 1 80.q even 4 1
9408.2.a.be 1 112.j even 4 1
9408.2.a.da 1 112.l odd 4 1

Hecke kernels

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

\( T_{5}^{2} + 4 \)
\( T_{11} \)
\( T_{13}^{2} + 36 \)
\( T_{17} + 2 \)
\( T_{23} + 4 \)
\( T_{31} - 8 \)
\( T_{47} + 8 \)
\( T_{71} + 12 \)
\( T_{79} - 8 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( 1 + T^{2} \)
$5$ \( ( 1 - 4 T + 5 T^{2} )( 1 + 4 T + 5 T^{2} ) \)
$7$ \( ( 1 - T )^{2} \)
$11$ \( ( 1 - 11 T^{2} )^{2} \)
$13$ \( ( 1 - 4 T + 13 T^{2} )( 1 + 4 T + 13 T^{2} ) \)
$17$ \( ( 1 + 2 T + 17 T^{2} )^{2} \)
$19$ \( 1 - 22 T^{2} + 361 T^{4} \)
$23$ \( ( 1 + 4 T + 23 T^{2} )^{2} \)
$29$ \( ( 1 - 4 T + 29 T^{2} )( 1 + 4 T + 29 T^{2} ) \)
$31$ \( ( 1 - 8 T + 31 T^{2} )^{2} \)
$37$ \( 1 - 38 T^{2} + 1369 T^{4} \)
$41$ \( ( 1 - 2 T + 41 T^{2} )^{2} \)
$43$ \( 1 - 70 T^{2} + 1849 T^{4} \)
$47$ \( ( 1 + 8 T + 47 T^{2} )^{2} \)
$53$ \( 1 - 6 T^{2} + 2809 T^{4} \)
$59$ \( 1 + 26 T^{2} + 3481 T^{4} \)
$61$ \( 1 - 118 T^{2} + 3721 T^{4} \)
$67$ \( 1 + 10 T^{2} + 4489 T^{4} \)
$71$ \( ( 1 + 12 T + 71 T^{2} )^{2} \)
$73$ \( ( 1 - 14 T + 73 T^{2} )^{2} \)
$79$ \( ( 1 - 8 T + 79 T^{2} )^{2} \)
$83$ \( 1 - 22 T^{2} + 6889 T^{4} \)
$89$ \( ( 1 - 2 T + 89 T^{2} )^{2} \)
$97$ \( ( 1 - 10 T + 97 T^{2} )^{2} \)
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