# Properties

 Label 5376.2.c.bn Level $5376$ Weight $2$ Character orbit 5376.c Analytic conductor $42.928$ Analytic rank $0$ Dimension $4$ 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: $$4$$ Coefficient field: $$\Q(\zeta_{12})$$ Defining polynomial: $$x^{4} - x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{4}$$ Twist minimal: no (minimal twist has level 672) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{12}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \zeta_{12}^{3} q^{3} + ( 2 - 4 \zeta_{12}^{2} ) q^{5} + q^{7} - q^{9} +O(q^{10})$$ $$q + \zeta_{12}^{3} q^{3} + ( 2 - 4 \zeta_{12}^{2} ) q^{5} + q^{7} - q^{9} + ( -2 + 4 \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{11} + 2 \zeta_{12}^{3} q^{13} + ( 4 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{15} + ( 4 - 4 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{17} + ( -4 + 8 \zeta_{12}^{2} ) q^{19} + \zeta_{12}^{3} q^{21} + ( -2 + 4 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{23} -7 q^{25} -\zeta_{12}^{3} q^{27} + ( 4 - 8 \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{29} + ( -4 - 8 \zeta_{12} + 4 \zeta_{12}^{3} ) q^{31} + ( 2 - 4 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{33} + ( 2 - 4 \zeta_{12}^{2} ) q^{35} + 2 \zeta_{12}^{3} q^{37} -2 q^{39} + ( -8 - 4 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{41} + 8 \zeta_{12}^{3} q^{43} + ( -2 + 4 \zeta_{12}^{2} ) q^{45} + ( 4 + 8 \zeta_{12} - 4 \zeta_{12}^{3} ) q^{47} + q^{49} + ( 2 - 4 \zeta_{12}^{2} + 4 \zeta_{12}^{3} ) q^{51} + 2 \zeta_{12}^{3} q^{53} + ( 12 - 8 \zeta_{12} + 4 \zeta_{12}^{3} ) q^{55} + ( -8 \zeta_{12} + 4 \zeta_{12}^{3} ) q^{57} + ( 4 - 8 \zeta_{12}^{2} + 8 \zeta_{12}^{3} ) q^{59} + ( 4 - 8 \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{61} - q^{63} + ( 8 \zeta_{12} - 4 \zeta_{12}^{3} ) q^{65} + ( 4 - 8 \zeta_{12}^{2} + 4 \zeta_{12}^{3} ) q^{67} + ( -2 + 4 \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{69} + ( -6 - 4 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{71} + ( -6 - 8 \zeta_{12} + 4 \zeta_{12}^{3} ) q^{73} -7 \zeta_{12}^{3} q^{75} + ( -2 + 4 \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{77} + ( -4 - 8 \zeta_{12} + 4 \zeta_{12}^{3} ) q^{79} + q^{81} -4 \zeta_{12}^{3} q^{83} + ( 8 - 16 \zeta_{12}^{2} + 12 \zeta_{12}^{3} ) q^{85} + ( -2 + 8 \zeta_{12} - 4 \zeta_{12}^{3} ) q^{87} + ( -4 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{89} + 2 \zeta_{12}^{3} q^{91} + ( 4 - 8 \zeta_{12}^{2} - 4 \zeta_{12}^{3} ) q^{93} + 24 q^{95} + ( -2 - 8 \zeta_{12} + 4 \zeta_{12}^{3} ) q^{97} + ( 2 - 4 \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 4q^{7} - 4q^{9} + O(q^{10})$$ $$4q + 4q^{7} - 4q^{9} + 16q^{17} - 8q^{23} - 28q^{25} - 16q^{31} + 8q^{33} - 8q^{39} - 32q^{41} + 16q^{47} + 4q^{49} + 48q^{55} - 4q^{63} - 24q^{71} - 24q^{73} - 16q^{79} + 4q^{81} - 8q^{87} + 96q^{95} - 8q^{97} + O(q^{100})$$

## 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
 −0.866025 − 0.500000i 0.866025 − 0.500000i 0.866025 + 0.500000i −0.866025 + 0.500000i
0 1.00000i 0 3.46410i 0 1.00000 0 −1.00000 0
2689.2 0 1.00000i 0 3.46410i 0 1.00000 0 −1.00000 0
2689.3 0 1.00000i 0 3.46410i 0 1.00000 0 −1.00000 0
2689.4 0 1.00000i 0 3.46410i 0 1.00000 0 −1.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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.bn 4
4.b odd 2 1 5376.2.c.bh 4
8.b even 2 1 inner 5376.2.c.bn 4
8.d odd 2 1 5376.2.c.bh 4
16.e even 4 1 672.2.a.j yes 2
16.e even 4 1 1344.2.a.u 2
16.f odd 4 1 672.2.a.i 2
16.f odd 4 1 1344.2.a.v 2
48.i odd 4 1 2016.2.a.s 2
48.i odd 4 1 4032.2.a.br 2
48.k even 4 1 2016.2.a.t 2
48.k even 4 1 4032.2.a.bs 2
112.j even 4 1 4704.2.a.bn 2
112.j even 4 1 9408.2.a.do 2
112.l odd 4 1 4704.2.a.bm 2
112.l odd 4 1 9408.2.a.dx 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
672.2.a.i 2 16.f odd 4 1
672.2.a.j yes 2 16.e even 4 1
1344.2.a.u 2 16.e even 4 1
1344.2.a.v 2 16.f odd 4 1
2016.2.a.s 2 48.i odd 4 1
2016.2.a.t 2 48.k even 4 1
4032.2.a.br 2 48.i odd 4 1
4032.2.a.bs 2 48.k even 4 1
4704.2.a.bm 2 112.l odd 4 1
4704.2.a.bn 2 112.j even 4 1
5376.2.c.bh 4 4.b odd 2 1
5376.2.c.bh 4 8.d odd 2 1
5376.2.c.bn 4 1.a even 1 1 trivial
5376.2.c.bn 4 8.b even 2 1 inner
9408.2.a.do 2 112.j even 4 1
9408.2.a.dx 2 112.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 $$S_{2}^{\mathrm{new}}(5376, [\chi])$$:

 $$T_{5}^{2} + 12$$ $$T_{11}^{4} + 32 T_{11}^{2} + 64$$ $$T_{13}^{2} + 4$$ $$T_{17}^{2} - 8 T_{17} + 4$$ $$T_{23}^{2} + 4 T_{23} - 8$$ $$T_{31}^{2} + 8 T_{31} - 32$$ $$T_{47}^{2} - 8 T_{47} - 32$$ $$T_{71}^{2} + 12 T_{71} + 24$$ $$T_{79}^{2} + 8 T_{79} - 32$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ $$( 1 + T^{2} )^{2}$$
$5$ $$( 1 + 2 T^{2} + 25 T^{4} )^{2}$$
$7$ $$( 1 - T )^{4}$$
$11$ $$1 - 12 T^{2} + 86 T^{4} - 1452 T^{6} + 14641 T^{8}$$
$13$ $$( 1 - 22 T^{2} + 169 T^{4} )^{2}$$
$17$ $$( 1 - 8 T + 38 T^{2} - 136 T^{3} + 289 T^{4} )^{2}$$
$19$ $$( 1 + 10 T^{2} + 361 T^{4} )^{2}$$
$23$ $$( 1 + 4 T + 38 T^{2} + 92 T^{3} + 529 T^{4} )^{2}$$
$29$ $$1 - 12 T^{2} + 950 T^{4} - 10092 T^{6} + 707281 T^{8}$$
$31$ $$( 1 + 8 T + 30 T^{2} + 248 T^{3} + 961 T^{4} )^{2}$$
$37$ $$( 1 - 12 T + 37 T^{2} )^{2}( 1 + 12 T + 37 T^{2} )^{2}$$
$41$ $$( 1 + 16 T + 134 T^{2} + 656 T^{3} + 1681 T^{4} )^{2}$$
$43$ $$( 1 - 22 T^{2} + 1849 T^{4} )^{2}$$
$47$ $$( 1 - 8 T + 62 T^{2} - 376 T^{3} + 2209 T^{4} )^{2}$$
$53$ $$( 1 - 102 T^{2} + 2809 T^{4} )^{2}$$
$59$ $$1 - 12 T^{2} - 5290 T^{4} - 41772 T^{6} + 12117361 T^{8}$$
$61$ $$1 - 140 T^{2} + 11574 T^{4} - 520940 T^{6} + 13845841 T^{8}$$
$67$ $$1 - 140 T^{2} + 10806 T^{4} - 628460 T^{6} + 20151121 T^{8}$$
$71$ $$( 1 + 12 T + 166 T^{2} + 852 T^{3} + 5041 T^{4} )^{2}$$
$73$ $$( 1 + 12 T + 134 T^{2} + 876 T^{3} + 5329 T^{4} )^{2}$$
$79$ $$( 1 + 8 T + 126 T^{2} + 632 T^{3} + 6241 T^{4} )^{2}$$
$83$ $$( 1 - 150 T^{2} + 6889 T^{4} )^{2}$$
$89$ $$( 1 + 166 T^{2} + 7921 T^{4} )^{2}$$
$97$ $$( 1 + 4 T + 150 T^{2} + 388 T^{3} + 9409 T^{4} )^{2}$$
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