# Properties

 Label 2352.2.b.i Level $2352$ Weight $2$ Character orbit 2352.b Analytic conductor $18.781$ Analytic rank $0$ Dimension $4$ 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.b (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$18.7808145554$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: 4.0.2048.2 Defining polynomial: $$x^{4} + 4 x^{2} + 2$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - q^{3} + \beta_{1} q^{5} + q^{9} +O(q^{10})$$ $$q - q^{3} + \beta_{1} q^{5} + q^{9} + ( 2 \beta_{1} - 2 \beta_{3} ) q^{11} + ( -2 \beta_{1} + \beta_{3} ) q^{13} -\beta_{1} q^{15} + ( -2 \beta_{1} - \beta_{3} ) q^{17} -4 \beta_{2} q^{19} + ( 2 \beta_{1} + 2 \beta_{3} ) q^{23} + ( 3 + \beta_{2} ) q^{25} - q^{27} + ( -4 + \beta_{2} ) q^{29} + ( 4 - 4 \beta_{2} ) q^{31} + ( -2 \beta_{1} + 2 \beta_{3} ) q^{33} -\beta_{2} q^{37} + ( 2 \beta_{1} - \beta_{3} ) q^{39} + ( 3 \beta_{1} + 2 \beta_{3} ) q^{41} -4 \beta_{3} q^{43} + \beta_{1} q^{45} + ( -4 - 4 \beta_{2} ) q^{47} + ( 2 \beta_{1} + \beta_{3} ) q^{51} + ( -4 + 4 \beta_{2} ) q^{53} + ( -4 + 4 \beta_{2} ) q^{55} + 4 \beta_{2} q^{57} -4 \beta_{2} q^{59} + ( 2 \beta_{1} + 3 \beta_{3} ) q^{61} + ( 4 - 3 \beta_{2} ) q^{65} + ( 4 \beta_{1} - 8 \beta_{3} ) q^{67} + ( -2 \beta_{1} - 2 \beta_{3} ) q^{69} + ( -6 \beta_{1} - 6 \beta_{3} ) q^{71} + ( -\beta_{1} + 4 \beta_{3} ) q^{73} + ( -3 - \beta_{2} ) q^{75} + ( 8 \beta_{1} + 4 \beta_{3} ) q^{79} + q^{81} + ( -12 + 4 \beta_{2} ) q^{83} + ( 4 - \beta_{2} ) q^{85} + ( 4 - \beta_{2} ) q^{87} + ( 2 \beta_{1} - 7 \beta_{3} ) q^{89} + ( -4 + 4 \beta_{2} ) q^{93} + ( 4 \beta_{1} - 4 \beta_{3} ) q^{95} -5 \beta_{1} q^{97} + ( 2 \beta_{1} - 2 \beta_{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 4q^{3} + 4q^{9} + O(q^{10})$$ $$4q - 4q^{3} + 4q^{9} + 12q^{25} - 4q^{27} - 16q^{29} + 16q^{31} - 16q^{47} - 16q^{53} - 16q^{55} + 16q^{65} - 12q^{75} + 4q^{81} - 48q^{83} + 16q^{85} + 16q^{87} - 16q^{93} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 4 x^{2} + 2$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} + 2$$ $$\beta_{3}$$ $$=$$ $$\nu^{3} + 3 \nu$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} - 2$$ $$\nu^{3}$$ $$=$$ $$\beta_{3} - 3 \beta_{1}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1567.1
 − 1.84776i − 0.765367i 0.765367i 1.84776i
0 −1.00000 0 1.84776i 0 0 0 1.00000 0
1567.2 0 −1.00000 0 0.765367i 0 0 0 1.00000 0
1567.3 0 −1.00000 0 0.765367i 0 0 0 1.00000 0
1567.4 0 −1.00000 0 1.84776i 0 0 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
28.d 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.b.i 4
3.b odd 2 1 7056.2.b.u 4
4.b odd 2 1 2352.2.b.j yes 4
7.b odd 2 1 2352.2.b.j yes 4
7.c even 3 2 2352.2.bl.s 8
7.d odd 6 2 2352.2.bl.p 8
12.b even 2 1 7056.2.b.t 4
21.c even 2 1 7056.2.b.t 4
28.d even 2 1 inner 2352.2.b.i 4
28.f even 6 2 2352.2.bl.s 8
28.g odd 6 2 2352.2.bl.p 8
84.h odd 2 1 7056.2.b.u 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2352.2.b.i 4 1.a even 1 1 trivial
2352.2.b.i 4 28.d even 2 1 inner
2352.2.b.j yes 4 4.b odd 2 1
2352.2.b.j yes 4 7.b odd 2 1
2352.2.bl.p 8 7.d odd 6 2
2352.2.bl.p 8 28.g odd 6 2
2352.2.bl.s 8 7.c even 3 2
2352.2.bl.s 8 28.f even 6 2
7056.2.b.t 4 12.b even 2 1
7056.2.b.t 4 21.c even 2 1
7056.2.b.u 4 3.b odd 2 1
7056.2.b.u 4 84.h odd 2 1

## Hecke kernels

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

 $$T_{5}^{4} + 4 T_{5}^{2} + 2$$ $$T_{11}^{4} + 32 T_{11}^{2} + 128$$ $$T_{13}^{4} + 20 T_{13}^{2} + 2$$ $$T_{19}^{2} - 32$$ $$T_{31}^{2} - 8 T_{31} - 16$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$( 1 + T )^{4}$$
$5$ $$2 + 4 T^{2} + T^{4}$$
$7$ $$T^{4}$$
$11$ $$128 + 32 T^{2} + T^{4}$$
$13$ $$2 + 20 T^{2} + T^{4}$$
$17$ $$98 + 20 T^{2} + T^{4}$$
$19$ $$( -32 + T^{2} )^{2}$$
$23$ $$128 + 32 T^{2} + T^{4}$$
$29$ $$( 14 + 8 T + T^{2} )^{2}$$
$31$ $$( -16 - 8 T + T^{2} )^{2}$$
$37$ $$( -2 + T^{2} )^{2}$$
$41$ $$578 + 52 T^{2} + T^{4}$$
$43$ $$512 + 64 T^{2} + T^{4}$$
$47$ $$( -16 + 8 T + T^{2} )^{2}$$
$53$ $$( -16 + 8 T + T^{2} )^{2}$$
$59$ $$( -32 + T^{2} )^{2}$$
$61$ $$98 + 52 T^{2} + T^{4}$$
$67$ $$25088 + 320 T^{2} + T^{4}$$
$71$ $$10368 + 288 T^{2} + T^{4}$$
$73$ $$1058 + 68 T^{2} + T^{4}$$
$79$ $$25088 + 320 T^{2} + T^{4}$$
$83$ $$( 112 + 24 T + T^{2} )^{2}$$
$89$ $$10658 + 212 T^{2} + T^{4}$$
$97$ $$1250 + 100 T^{2} + T^{4}$$