# Properties

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

# Related objects

## 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 42) 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.

 $$f(q)$$ $$=$$ $$q + i q^{3} + 2 i q^{5} - q^{7} - q^{9} +O(q^{10})$$ $$q + i q^{3} + 2 i q^{5} - q^{7} - q^{9} -4 i q^{11} + 6 i q^{13} -2 q^{15} + 2 q^{17} + 4 i q^{19} -i q^{21} + 8 q^{23} + q^{25} -i q^{27} -2 i q^{29} + 4 q^{33} -2 i q^{35} + 10 i q^{37} -6 q^{39} + 6 q^{41} -4 i q^{43} -2 i q^{45} + q^{49} + 2 i q^{51} -6 i q^{53} + 8 q^{55} -4 q^{57} + 4 i q^{59} + 6 i q^{61} + q^{63} -12 q^{65} -4 i q^{67} + 8 i q^{69} + 8 q^{71} -10 q^{73} + i q^{75} + 4 i q^{77} + q^{81} + 4 i q^{83} + 4 i q^{85} + 2 q^{87} + 6 q^{89} -6 i q^{91} -8 q^{95} -14 q^{97} + 4 i q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{7} - 2 q^{9} + O(q^{10})$$ $$2 q - 2 q^{7} - 2 q^{9} - 4 q^{15} + 4 q^{17} + 16 q^{23} + 2 q^{25} + 8 q^{33} - 12 q^{39} + 12 q^{41} + 2 q^{49} + 16 q^{55} - 8 q^{57} + 2 q^{63} - 24 q^{65} + 16 q^{71} - 20 q^{73} + 2 q^{81} + 4 q^{87} + 12 q^{89} - 16 q^{95} - 28 q^{97} + O(q^{100})$$

## 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: 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.e 2
4.b odd 2 1 5376.2.c.bc 2
8.b even 2 1 inner 5376.2.c.e 2
8.d odd 2 1 5376.2.c.bc 2
16.e even 4 1 336.2.a.d 1
16.e even 4 1 1344.2.a.i 1
16.f odd 4 1 42.2.a.a 1
16.f odd 4 1 1344.2.a.q 1
48.i odd 4 1 1008.2.a.j 1
48.i odd 4 1 4032.2.a.m 1
48.k even 4 1 126.2.a.a 1
48.k even 4 1 4032.2.a.e 1
80.j even 4 1 1050.2.g.a 2
80.k odd 4 1 1050.2.a.i 1
80.q even 4 1 8400.2.a.k 1
80.s even 4 1 1050.2.g.a 2
112.j even 4 1 294.2.a.g 1
112.j even 4 1 9408.2.a.n 1
112.l odd 4 1 2352.2.a.l 1
112.l odd 4 1 9408.2.a.bw 1
112.u odd 12 2 294.2.e.c 2
112.v even 12 2 294.2.e.a 2
112.w even 12 2 2352.2.q.i 2
112.x odd 12 2 2352.2.q.n 2
144.u even 12 2 1134.2.f.j 2
144.v odd 12 2 1134.2.f.g 2
176.i even 4 1 5082.2.a.d 1
208.o odd 4 1 7098.2.a.f 1
240.t even 4 1 3150.2.a.bo 1
240.z odd 4 1 3150.2.g.r 2
240.bd odd 4 1 3150.2.g.r 2
336.v odd 4 1 882.2.a.b 1
336.y even 4 1 7056.2.a.k 1
336.br odd 12 2 882.2.g.j 2
336.bu even 12 2 882.2.g.h 2
560.be even 4 1 7350.2.a.f 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
42.2.a.a 1 16.f odd 4 1
126.2.a.a 1 48.k even 4 1
294.2.a.g 1 112.j even 4 1
294.2.e.a 2 112.v even 12 2
294.2.e.c 2 112.u odd 12 2
336.2.a.d 1 16.e even 4 1
882.2.a.b 1 336.v odd 4 1
882.2.g.h 2 336.bu even 12 2
882.2.g.j 2 336.br odd 12 2
1008.2.a.j 1 48.i odd 4 1
1050.2.a.i 1 80.k odd 4 1
1050.2.g.a 2 80.j even 4 1
1050.2.g.a 2 80.s even 4 1
1134.2.f.g 2 144.v odd 12 2
1134.2.f.j 2 144.u even 12 2
1344.2.a.i 1 16.e even 4 1
1344.2.a.q 1 16.f odd 4 1
2352.2.a.l 1 112.l odd 4 1
2352.2.q.i 2 112.w even 12 2
2352.2.q.n 2 112.x odd 12 2
3150.2.a.bo 1 240.t even 4 1
3150.2.g.r 2 240.z odd 4 1
3150.2.g.r 2 240.bd odd 4 1
4032.2.a.e 1 48.k even 4 1
4032.2.a.m 1 48.i odd 4 1
5082.2.a.d 1 176.i even 4 1
5376.2.c.e 2 1.a even 1 1 trivial
5376.2.c.e 2 8.b even 2 1 inner
5376.2.c.bc 2 4.b odd 2 1
5376.2.c.bc 2 8.d odd 2 1
7056.2.a.k 1 336.y even 4 1
7098.2.a.f 1 208.o odd 4 1
7350.2.a.f 1 560.be even 4 1
8400.2.a.k 1 80.q even 4 1
9408.2.a.n 1 112.j even 4 1
9408.2.a.bw 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 $$S_{2}^{\mathrm{new}}(5376, [\chi])$$:

 $$T_{5}^{2} + 4$$ $$T_{11}^{2} + 16$$ $$T_{13}^{2} + 36$$ $$T_{17} - 2$$ $$T_{23} - 8$$ $$T_{31}$$ $$T_{47}$$ $$T_{71} - 8$$ $$T_{79}$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$1 + T^{2}$$
$5$ $$4 + T^{2}$$
$7$ $$( 1 + T )^{2}$$
$11$ $$16 + T^{2}$$
$13$ $$36 + T^{2}$$
$17$ $$( -2 + T )^{2}$$
$19$ $$16 + T^{2}$$
$23$ $$( -8 + T )^{2}$$
$29$ $$4 + T^{2}$$
$31$ $$T^{2}$$
$37$ $$100 + T^{2}$$
$41$ $$( -6 + T )^{2}$$
$43$ $$16 + T^{2}$$
$47$ $$T^{2}$$
$53$ $$36 + T^{2}$$
$59$ $$16 + T^{2}$$
$61$ $$36 + T^{2}$$
$67$ $$16 + T^{2}$$
$71$ $$( -8 + T )^{2}$$
$73$ $$( 10 + T )^{2}$$
$79$ $$T^{2}$$
$83$ $$16 + T^{2}$$
$89$ $$( -6 + T )^{2}$$
$97$ $$( 14 + T )^{2}$$