# Properties

 Label 2352.2.q.p Level $2352$ Weight $2$ Character orbit 2352.q Analytic conductor $18.781$ Analytic rank $1$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$18.7808145554$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Defining polynomial: $$x^{2} - x + 1$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 588) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( 1 - \zeta_{6} ) q^{3} -2 \zeta_{6} q^{5} -\zeta_{6} q^{9} +O(q^{10})$$ $$q + ( 1 - \zeta_{6} ) q^{3} -2 \zeta_{6} q^{5} -\zeta_{6} q^{9} + ( 2 - 2 \zeta_{6} ) q^{11} -4 q^{13} -2 q^{15} + ( -6 + 6 \zeta_{6} ) q^{17} + 8 \zeta_{6} q^{19} -6 \zeta_{6} q^{23} + ( 1 - \zeta_{6} ) q^{25} - q^{27} -10 q^{29} + ( 4 - 4 \zeta_{6} ) q^{31} -2 \zeta_{6} q^{33} -6 \zeta_{6} q^{37} + ( -4 + 4 \zeta_{6} ) q^{39} -6 q^{41} -4 q^{43} + ( -2 + 2 \zeta_{6} ) q^{45} + 8 \zeta_{6} q^{47} + 6 \zeta_{6} q^{51} + ( -2 + 2 \zeta_{6} ) q^{53} -4 q^{55} + 8 q^{57} + ( -4 + 4 \zeta_{6} ) q^{59} + 8 \zeta_{6} q^{61} + 8 \zeta_{6} q^{65} + ( -8 + 8 \zeta_{6} ) q^{67} -6 q^{69} + 10 q^{71} + ( -4 + 4 \zeta_{6} ) q^{73} -\zeta_{6} q^{75} + 4 \zeta_{6} q^{79} + ( -1 + \zeta_{6} ) q^{81} -12 q^{83} + 12 q^{85} + ( -10 + 10 \zeta_{6} ) q^{87} + 14 \zeta_{6} q^{89} -4 \zeta_{6} q^{93} + ( 16 - 16 \zeta_{6} ) q^{95} + 4 q^{97} -2 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + q^{3} - 2 q^{5} - q^{9} + O(q^{10})$$ $$2 q + q^{3} - 2 q^{5} - q^{9} + 2 q^{11} - 8 q^{13} - 4 q^{15} - 6 q^{17} + 8 q^{19} - 6 q^{23} + q^{25} - 2 q^{27} - 20 q^{29} + 4 q^{31} - 2 q^{33} - 6 q^{37} - 4 q^{39} - 12 q^{41} - 8 q^{43} - 2 q^{45} + 8 q^{47} + 6 q^{51} - 2 q^{53} - 8 q^{55} + 16 q^{57} - 4 q^{59} + 8 q^{61} + 8 q^{65} - 8 q^{67} - 12 q^{69} + 20 q^{71} - 4 q^{73} - q^{75} + 4 q^{79} - q^{81} - 24 q^{83} + 24 q^{85} - 10 q^{87} + 14 q^{89} - 4 q^{93} + 16 q^{95} + 8 q^{97} - 4 q^{99} + O(q^{100})$$

## Character values

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

 $$n$$ $$785$$ $$1471$$ $$1765$$ $$2257$$ $$\chi(n)$$ $$1$$ $$1$$ $$1$$ $$-\zeta_{6}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
961.1
 0.5 − 0.866025i 0.5 + 0.866025i
0 0.500000 + 0.866025i 0 −1.00000 + 1.73205i 0 0 0 −0.500000 + 0.866025i 0
1537.1 0 0.500000 0.866025i 0 −1.00000 1.73205i 0 0 0 −0.500000 0.866025i 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
7.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2352.2.q.p 2
4.b odd 2 1 588.2.i.a 2
7.b odd 2 1 2352.2.q.k 2
7.c even 3 1 2352.2.a.j 1
7.c even 3 1 inner 2352.2.q.p 2
7.d odd 6 1 2352.2.a.p 1
7.d odd 6 1 2352.2.q.k 2
12.b even 2 1 1764.2.k.i 2
21.g even 6 1 7056.2.a.bu 1
21.h odd 6 1 7056.2.a.n 1
28.d even 2 1 588.2.i.g 2
28.f even 6 1 588.2.a.b 1
28.f even 6 1 588.2.i.g 2
28.g odd 6 1 588.2.a.e yes 1
28.g odd 6 1 588.2.i.a 2
56.j odd 6 1 9408.2.a.bf 1
56.k odd 6 1 9408.2.a.l 1
56.m even 6 1 9408.2.a.cu 1
56.p even 6 1 9408.2.a.ca 1
84.h odd 2 1 1764.2.k.c 2
84.j odd 6 1 1764.2.a.i 1
84.j odd 6 1 1764.2.k.c 2
84.n even 6 1 1764.2.a.b 1
84.n even 6 1 1764.2.k.i 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
588.2.a.b 1 28.f even 6 1
588.2.a.e yes 1 28.g odd 6 1
588.2.i.a 2 4.b odd 2 1
588.2.i.a 2 28.g odd 6 1
588.2.i.g 2 28.d even 2 1
588.2.i.g 2 28.f even 6 1
1764.2.a.b 1 84.n even 6 1
1764.2.a.i 1 84.j odd 6 1
1764.2.k.c 2 84.h odd 2 1
1764.2.k.c 2 84.j odd 6 1
1764.2.k.i 2 12.b even 2 1
1764.2.k.i 2 84.n even 6 1
2352.2.a.j 1 7.c even 3 1
2352.2.a.p 1 7.d odd 6 1
2352.2.q.k 2 7.b odd 2 1
2352.2.q.k 2 7.d odd 6 1
2352.2.q.p 2 1.a even 1 1 trivial
2352.2.q.p 2 7.c even 3 1 inner
7056.2.a.n 1 21.h odd 6 1
7056.2.a.bu 1 21.g even 6 1
9408.2.a.l 1 56.k odd 6 1
9408.2.a.bf 1 56.j odd 6 1
9408.2.a.ca 1 56.p even 6 1
9408.2.a.cu 1 56.m even 6 1

## Hecke kernels

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

 $$T_{5}^{2} + 2 T_{5} + 4$$ $$T_{11}^{2} - 2 T_{11} + 4$$ $$T_{13} + 4$$ $$T_{17}^{2} + 6 T_{17} + 36$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$1 - T + T^{2}$$
$5$ $$4 + 2 T + T^{2}$$
$7$ $$T^{2}$$
$11$ $$4 - 2 T + T^{2}$$
$13$ $$( 4 + T )^{2}$$
$17$ $$36 + 6 T + T^{2}$$
$19$ $$64 - 8 T + T^{2}$$
$23$ $$36 + 6 T + T^{2}$$
$29$ $$( 10 + T )^{2}$$
$31$ $$16 - 4 T + T^{2}$$
$37$ $$36 + 6 T + T^{2}$$
$41$ $$( 6 + T )^{2}$$
$43$ $$( 4 + T )^{2}$$
$47$ $$64 - 8 T + T^{2}$$
$53$ $$4 + 2 T + T^{2}$$
$59$ $$16 + 4 T + T^{2}$$
$61$ $$64 - 8 T + T^{2}$$
$67$ $$64 + 8 T + T^{2}$$
$71$ $$( -10 + T )^{2}$$
$73$ $$16 + 4 T + T^{2}$$
$79$ $$16 - 4 T + T^{2}$$
$83$ $$( 12 + T )^{2}$$
$89$ $$196 - 14 T + T^{2}$$
$97$ $$( -4 + T )^{2}$$