# Properties

 Label 2352.2.k.e Level $2352$ Weight $2$ Character orbit 2352.k Analytic conductor $18.781$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2352 = 2^{4} \cdot 3 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2352.k (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$18.7808145554$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{12})$$ Defining polynomial: $$x^{4} - x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{2}\cdot 3$$ 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 a primitive root of unity $$\zeta_{12}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( 2 \zeta_{12} - \zeta_{12}^{3} ) q^{3} + ( -2 \zeta_{12} + \zeta_{12}^{3} ) q^{5} + 3 q^{9} +O(q^{10})$$ $$q + ( 2 \zeta_{12} - \zeta_{12}^{3} ) q^{3} + ( -2 \zeta_{12} + \zeta_{12}^{3} ) q^{5} + 3 q^{9} + 3 \zeta_{12}^{3} q^{11} + ( 2 - 4 \zeta_{12}^{2} ) q^{13} -3 q^{15} + ( 4 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{17} + ( -2 + 4 \zeta_{12}^{2} ) q^{19} + 6 \zeta_{12}^{3} q^{23} -2 q^{25} + ( 6 \zeta_{12} - 3 \zeta_{12}^{3} ) q^{27} + 3 \zeta_{12}^{3} q^{29} + ( -1 + 2 \zeta_{12}^{2} ) q^{31} + ( -3 + 6 \zeta_{12}^{2} ) q^{33} -2 q^{37} -6 \zeta_{12}^{3} q^{39} + ( 8 \zeta_{12} - 4 \zeta_{12}^{3} ) q^{41} + 8 q^{43} + ( -6 \zeta_{12} + 3 \zeta_{12}^{3} ) q^{45} + ( 8 \zeta_{12} - 4 \zeta_{12}^{3} ) q^{47} + 6 q^{51} -9 \zeta_{12}^{3} q^{53} + ( 3 - 6 \zeta_{12}^{2} ) q^{55} + 6 \zeta_{12}^{3} q^{57} + ( -2 \zeta_{12} + \zeta_{12}^{3} ) q^{59} + 6 \zeta_{12}^{3} q^{65} -2 q^{67} + ( -6 + 12 \zeta_{12}^{2} ) q^{69} + 12 \zeta_{12}^{3} q^{71} + ( -4 + 8 \zeta_{12}^{2} ) q^{73} + ( -4 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{75} + q^{79} + 9 q^{81} + ( 10 \zeta_{12} - 5 \zeta_{12}^{3} ) q^{83} -6 q^{85} + ( -3 + 6 \zeta_{12}^{2} ) q^{87} + ( 12 \zeta_{12} - 6 \zeta_{12}^{3} ) q^{89} + 3 \zeta_{12}^{3} q^{93} -6 \zeta_{12}^{3} q^{95} + ( 3 - 6 \zeta_{12}^{2} ) q^{97} + 9 \zeta_{12}^{3} q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 12q^{9} + O(q^{10})$$ $$4q + 12q^{9} - 12q^{15} - 8q^{25} - 8q^{37} + 32q^{43} + 24q^{51} - 8q^{67} + 4q^{79} + 36q^{81} - 24q^{85} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/2352\mathbb{Z}\right)^\times$$.

 $$n$$ $$785$$ $$1471$$ $$1765$$ $$2257$$ $$\chi(n)$$ $$-1$$ $$1$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
881.1
 −0.866025 − 0.500000i −0.866025 + 0.500000i 0.866025 − 0.500000i 0.866025 + 0.500000i
0 −1.73205 0 1.73205 0 0 0 3.00000 0
881.2 0 −1.73205 0 1.73205 0 0 0 3.00000 0
881.3 0 1.73205 0 −1.73205 0 0 0 3.00000 0
881.4 0 1.73205 0 −1.73205 0 0 0 3.00000 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
3.b odd 2 1 inner
7.b odd 2 1 inner
21.c even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2352.2.k.e 4
3.b odd 2 1 inner 2352.2.k.e 4
4.b odd 2 1 294.2.d.a 4
7.b odd 2 1 inner 2352.2.k.e 4
7.c even 3 1 336.2.bc.e 4
7.d odd 6 1 336.2.bc.e 4
12.b even 2 1 294.2.d.a 4
21.c even 2 1 inner 2352.2.k.e 4
21.g even 6 1 336.2.bc.e 4
21.h odd 6 1 336.2.bc.e 4
28.d even 2 1 294.2.d.a 4
28.f even 6 1 42.2.f.a 4
28.f even 6 1 294.2.f.a 4
28.g odd 6 1 42.2.f.a 4
28.g odd 6 1 294.2.f.a 4
84.h odd 2 1 294.2.d.a 4
84.j odd 6 1 42.2.f.a 4
84.j odd 6 1 294.2.f.a 4
84.n even 6 1 42.2.f.a 4
84.n even 6 1 294.2.f.a 4
140.p odd 6 1 1050.2.s.b 4
140.s even 6 1 1050.2.s.b 4
140.w even 12 1 1050.2.u.a 4
140.w even 12 1 1050.2.u.d 4
140.x odd 12 1 1050.2.u.a 4
140.x odd 12 1 1050.2.u.d 4
252.n even 6 1 1134.2.t.d 4
252.o even 6 1 1134.2.t.d 4
252.r odd 6 1 1134.2.l.c 4
252.u odd 6 1 1134.2.l.c 4
252.bb even 6 1 1134.2.l.c 4
252.bj even 6 1 1134.2.l.c 4
252.bl odd 6 1 1134.2.t.d 4
252.bn odd 6 1 1134.2.t.d 4
420.ba even 6 1 1050.2.s.b 4
420.be odd 6 1 1050.2.s.b 4
420.bp odd 12 1 1050.2.u.a 4
420.bp odd 12 1 1050.2.u.d 4
420.br even 12 1 1050.2.u.a 4
420.br even 12 1 1050.2.u.d 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
42.2.f.a 4 28.f even 6 1
42.2.f.a 4 28.g odd 6 1
42.2.f.a 4 84.j odd 6 1
42.2.f.a 4 84.n even 6 1
294.2.d.a 4 4.b odd 2 1
294.2.d.a 4 12.b even 2 1
294.2.d.a 4 28.d even 2 1
294.2.d.a 4 84.h odd 2 1
294.2.f.a 4 28.f even 6 1
294.2.f.a 4 28.g odd 6 1
294.2.f.a 4 84.j odd 6 1
294.2.f.a 4 84.n even 6 1
336.2.bc.e 4 7.c even 3 1
336.2.bc.e 4 7.d odd 6 1
336.2.bc.e 4 21.g even 6 1
336.2.bc.e 4 21.h odd 6 1
1050.2.s.b 4 140.p odd 6 1
1050.2.s.b 4 140.s even 6 1
1050.2.s.b 4 420.ba even 6 1
1050.2.s.b 4 420.be odd 6 1
1050.2.u.a 4 140.w even 12 1
1050.2.u.a 4 140.x odd 12 1
1050.2.u.a 4 420.bp odd 12 1
1050.2.u.a 4 420.br even 12 1
1050.2.u.d 4 140.w even 12 1
1050.2.u.d 4 140.x odd 12 1
1050.2.u.d 4 420.bp odd 12 1
1050.2.u.d 4 420.br even 12 1
1134.2.l.c 4 252.r odd 6 1
1134.2.l.c 4 252.u odd 6 1
1134.2.l.c 4 252.bb even 6 1
1134.2.l.c 4 252.bj even 6 1
1134.2.t.d 4 252.n even 6 1
1134.2.t.d 4 252.o even 6 1
1134.2.t.d 4 252.bl odd 6 1
1134.2.t.d 4 252.bn odd 6 1
2352.2.k.e 4 1.a even 1 1 trivial
2352.2.k.e 4 3.b odd 2 1 inner
2352.2.k.e 4 7.b odd 2 1 inner
2352.2.k.e 4 21.c even 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(2352, [\chi])$$:

 $$T_{5}^{2} - 3$$ $$T_{13}^{2} + 12$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$( -3 + T^{2} )^{2}$$
$5$ $$( -3 + T^{2} )^{2}$$
$7$ $$T^{4}$$
$11$ $$( 9 + T^{2} )^{2}$$
$13$ $$( 12 + T^{2} )^{2}$$
$17$ $$( -12 + T^{2} )^{2}$$
$19$ $$( 12 + T^{2} )^{2}$$
$23$ $$( 36 + T^{2} )^{2}$$
$29$ $$( 9 + T^{2} )^{2}$$
$31$ $$( 3 + T^{2} )^{2}$$
$37$ $$( 2 + T )^{4}$$
$41$ $$( -48 + T^{2} )^{2}$$
$43$ $$( -8 + T )^{4}$$
$47$ $$( -48 + T^{2} )^{2}$$
$53$ $$( 81 + T^{2} )^{2}$$
$59$ $$( -3 + T^{2} )^{2}$$
$61$ $$T^{4}$$
$67$ $$( 2 + T )^{4}$$
$71$ $$( 144 + T^{2} )^{2}$$
$73$ $$( 48 + T^{2} )^{2}$$
$79$ $$( -1 + T )^{4}$$
$83$ $$( -75 + T^{2} )^{2}$$
$89$ $$( -108 + T^{2} )^{2}$$
$97$ $$( 27 + T^{2} )^{2}$$