# Properties

 Label 5376.2.c.f 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 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.

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

## 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: 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.f 2
4.b odd 2 1 5376.2.c.bd 2
8.b even 2 1 inner 5376.2.c.f 2
8.d odd 2 1 5376.2.c.bd 2
16.e even 4 1 336.2.a.c 1
16.e even 4 1 1344.2.a.n 1
16.f odd 4 1 168.2.a.b 1
16.f odd 4 1 1344.2.a.c 1
48.i odd 4 1 1008.2.a.e 1
48.i odd 4 1 4032.2.a.bj 1
48.k even 4 1 504.2.a.b 1
48.k even 4 1 4032.2.a.be 1
80.j even 4 1 4200.2.t.m 2
80.k odd 4 1 4200.2.a.i 1
80.q even 4 1 8400.2.a.bx 1
80.s even 4 1 4200.2.t.m 2
112.j even 4 1 1176.2.a.a 1
112.j even 4 1 9408.2.a.cy 1
112.l odd 4 1 2352.2.a.q 1
112.l odd 4 1 9408.2.a.bc 1
112.u odd 12 2 1176.2.q.b 2
112.v even 12 2 1176.2.q.j 2
112.w even 12 2 2352.2.q.o 2
112.x odd 12 2 2352.2.q.j 2
336.v odd 4 1 3528.2.a.w 1
336.y even 4 1 7056.2.a.br 1
336.br odd 12 2 3528.2.s.h 2
336.bu even 12 2 3528.2.s.v 2

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

## Hecke kernels

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

 $$T_{5}^{2} + 4$$ $$T_{11}$$ $$T_{13}^{2} + 4$$ $$T_{17} - 6$$ $$T_{23} + 4$$ $$T_{31} - 8$$ $$T_{47} - 8$$ $$T_{71} - 4$$ $$T_{79} + 8$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ $$1 + T^{2}$$
$5$ $$( 1 - 4 T + 5 T^{2} )( 1 + 4 T + 5 T^{2} )$$
$7$ $$( 1 + T )^{2}$$
$11$ $$( 1 - 11 T^{2} )^{2}$$
$13$ $$1 - 22 T^{2} + 169 T^{4}$$
$17$ $$( 1 - 6 T + 17 T^{2} )^{2}$$
$19$ $$1 - 22 T^{2} + 361 T^{4}$$
$23$ $$( 1 + 4 T + 23 T^{2} )^{2}$$
$29$ $$1 - 22 T^{2} + 841 T^{4}$$
$31$ $$( 1 - 8 T + 31 T^{2} )^{2}$$
$37$ $$1 + 26 T^{2} + 1369 T^{4}$$
$41$ $$( 1 - 10 T + 41 T^{2} )^{2}$$
$43$ $$1 + 58 T^{2} + 1849 T^{4}$$
$47$ $$( 1 - 8 T + 47 T^{2} )^{2}$$
$53$ $$1 - 70 T^{2} + 2809 T^{4}$$
$59$ $$1 - 102 T^{2} + 3481 T^{4}$$
$61$ $$( 1 - 12 T + 61 T^{2} )( 1 + 12 T + 61 T^{2} )$$
$67$ $$1 + 10 T^{2} + 4489 T^{4}$$
$71$ $$( 1 - 4 T + 71 T^{2} )^{2}$$
$73$ $$( 1 + 2 T + 73 T^{2} )^{2}$$
$79$ $$( 1 + 8 T + 79 T^{2} )^{2}$$
$83$ $$1 - 150 T^{2} + 6889 T^{4}$$
$89$ $$( 1 + 6 T + 89 T^{2} )^{2}$$
$97$ $$( 1 - 10 T + 97 T^{2} )^{2}$$