# Properties

 Label 2352.3.m.l Level $2352$ Weight $3$ Character orbit 2352.m Analytic conductor $64.087$ 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: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 2352.m (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$64.0873581775$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\sqrt{2}, \sqrt{-3})$$ Defining polynomial: $$x^{4} + 2 x^{2} + 4$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{2}$$ 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.

 $$f(q)$$ $$=$$ $$q -\beta_{2} q^{3} + ( 6 + \beta_{1} ) q^{5} -3 q^{9} +O(q^{10})$$ $$q -\beta_{2} q^{3} + ( 6 + \beta_{1} ) q^{5} -3 q^{9} + ( -6 \beta_{2} + 4 \beta_{3} ) q^{11} + ( 12 + 7 \beta_{1} ) q^{13} + ( -6 \beta_{2} + \beta_{3} ) q^{15} + ( 6 + 5 \beta_{1} ) q^{17} + 6 \beta_{3} q^{19} + ( -18 \beta_{2} + 4 \beta_{3} ) q^{23} + ( 13 + 12 \beta_{1} ) q^{25} + 3 \beta_{2} q^{27} + ( 24 + 6 \beta_{1} ) q^{29} + ( 4 \beta_{2} - 18 \beta_{3} ) q^{31} + ( -18 - 12 \beta_{1} ) q^{33} -24 \beta_{1} q^{37} + ( -12 \beta_{2} + 7 \beta_{3} ) q^{39} + ( -18 + 7 \beta_{1} ) q^{41} + ( 12 \beta_{2} - 20 \beta_{3} ) q^{43} + ( -18 - 3 \beta_{1} ) q^{45} + ( -8 \beta_{2} + 6 \beta_{3} ) q^{47} + ( -6 \beta_{2} + 5 \beta_{3} ) q^{51} + ( -22 - 36 \beta_{1} ) q^{53} + ( -44 \beta_{2} + 30 \beta_{3} ) q^{55} -18 \beta_{1} q^{57} + ( -28 \beta_{2} - 18 \beta_{3} ) q^{59} + ( 24 - 41 \beta_{1} ) q^{61} + ( 86 + 54 \beta_{1} ) q^{65} + ( 12 \beta_{2} - 4 \beta_{3} ) q^{67} + ( -54 - 12 \beta_{1} ) q^{69} + ( -18 \beta_{2} - 24 \beta_{3} ) q^{71} + ( 12 + 35 \beta_{1} ) q^{73} + ( -13 \beta_{2} + 12 \beta_{3} ) q^{75} + ( -24 \beta_{2} + 20 \beta_{3} ) q^{79} + 9 q^{81} + ( -20 \beta_{2} - 12 \beta_{3} ) q^{83} + ( 46 + 36 \beta_{1} ) q^{85} + ( -24 \beta_{2} + 6 \beta_{3} ) q^{87} + ( -30 - 19 \beta_{1} ) q^{89} + ( 12 + 54 \beta_{1} ) q^{93} + ( -12 \beta_{2} + 36 \beta_{3} ) q^{95} + ( 108 + 11 \beta_{1} ) q^{97} + ( 18 \beta_{2} - 12 \beta_{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 24q^{5} - 12q^{9} + O(q^{10})$$ $$4q + 24q^{5} - 12q^{9} + 48q^{13} + 24q^{17} + 52q^{25} + 96q^{29} - 72q^{33} - 72q^{41} - 72q^{45} - 88q^{53} + 96q^{61} + 344q^{65} - 216q^{69} + 48q^{73} + 36q^{81} + 184q^{85} - 120q^{89} + 48q^{93} + 432q^{97} + O(q^{100})$$

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

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu^{3}$$$$/2$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} + 1$$ $$\beta_{3}$$ $$=$$ $$($$$$\nu^{3} + 4 \nu$$$$)/2$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{3} - \beta_{1}$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} - 1$$ $$\nu^{3}$$ $$=$$ $$2 \beta_{1}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1471.1
 0.707107 + 1.22474i −0.707107 − 1.22474i 0.707107 − 1.22474i −0.707107 + 1.22474i
0 1.73205i 0 4.58579 0 0 0 −3.00000 0
1471.2 0 1.73205i 0 7.41421 0 0 0 −3.00000 0
1471.3 0 1.73205i 0 4.58579 0 0 0 −3.00000 0
1471.4 0 1.73205i 0 7.41421 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
4.b odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2352.3.m.l yes 4
4.b odd 2 1 inner 2352.3.m.l yes 4
7.b odd 2 1 2352.3.m.d 4
28.d even 2 1 2352.3.m.d 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2352.3.m.d 4 7.b odd 2 1
2352.3.m.d 4 28.d even 2 1
2352.3.m.l yes 4 1.a even 1 1 trivial
2352.3.m.l yes 4 4.b odd 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{2} - 12 T_{5} + 34$$ acting on $$S_{3}^{\mathrm{new}}(2352, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$( 3 + T^{2} )^{2}$$
$5$ $$( 34 - 12 T + T^{2} )^{2}$$
$7$ $$T^{4}$$
$11$ $$144 + 408 T^{2} + T^{4}$$
$13$ $$( 46 - 24 T + T^{2} )^{2}$$
$17$ $$( -14 - 12 T + T^{2} )^{2}$$
$19$ $$( 216 + T^{2} )^{2}$$
$23$ $$767376 + 2136 T^{2} + T^{4}$$
$29$ $$( 504 - 48 T + T^{2} )^{2}$$
$31$ $$3594816 + 3984 T^{2} + T^{4}$$
$37$ $$( -1152 + T^{2} )^{2}$$
$41$ $$( 226 + 36 T + T^{2} )^{2}$$
$43$ $$3873024 + 5664 T^{2} + T^{4}$$
$47$ $$576 + 816 T^{2} + T^{4}$$
$53$ $$( -2108 + 44 T + T^{2} )^{2}$$
$59$ $$166464 + 8592 T^{2} + T^{4}$$
$61$ $$( -2786 - 48 T + T^{2} )^{2}$$
$67$ $$112896 + 1056 T^{2} + T^{4}$$
$71$ $$6170256 + 8856 T^{2} + T^{4}$$
$73$ $$( -2306 - 24 T + T^{2} )^{2}$$
$79$ $$451584 + 8256 T^{2} + T^{4}$$
$83$ $$112896 + 4128 T^{2} + T^{4}$$
$89$ $$( 178 + 60 T + T^{2} )^{2}$$
$97$ $$( 11422 - 216 T + T^{2} )^{2}$$