# Properties

 Label 2352.3.m.j 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: $$\Q(\sqrt{-3}, \sqrt{37})$$ Defining polynomial: $$x^{4} - x^{3} + 10 x^{2} + 9 x + 81$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{4}$$ 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.

 $$f(q)$$ $$=$$ $$q -\beta_{1} q^{3} + ( 3 + \beta_{2} ) q^{5} -3 q^{9} +O(q^{10})$$ $$q -\beta_{1} q^{3} + ( 3 + \beta_{2} ) q^{5} -3 q^{9} + ( \beta_{1} + \beta_{3} ) q^{11} + ( -3 \beta_{1} + \beta_{3} ) q^{15} + ( -9 + \beta_{2} ) q^{17} + ( -2 \beta_{1} - 2 \beta_{3} ) q^{19} + ( 11 \beta_{1} - \beta_{3} ) q^{23} + ( 21 + 6 \beta_{2} ) q^{25} + 3 \beta_{1} q^{27} + ( 12 + 6 \beta_{2} ) q^{29} + ( 10 \beta_{1} - 2 \beta_{3} ) q^{31} + ( 3 - 3 \beta_{2} ) q^{33} + ( 16 - 6 \beta_{2} ) q^{37} + ( 33 - \beta_{2} ) q^{41} -16 \beta_{1} q^{43} + ( -9 - 3 \beta_{2} ) q^{45} + ( -12 \beta_{1} + 4 \beta_{3} ) q^{47} + ( 9 \beta_{1} + \beta_{3} ) q^{51} + ( 48 + 6 \beta_{2} ) q^{53} + ( -34 \beta_{1} + 2 \beta_{3} ) q^{55} + ( -6 + 6 \beta_{2} ) q^{57} + 4 \beta_{3} q^{59} + ( 66 + 6 \beta_{2} ) q^{61} + ( 6 \beta_{1} + 6 \beta_{3} ) q^{67} + ( 33 + 3 \beta_{2} ) q^{69} + ( -7 \beta_{1} + 5 \beta_{3} ) q^{71} + ( -6 + 6 \beta_{2} ) q^{73} + ( -21 \beta_{1} + 6 \beta_{3} ) q^{75} + ( 50 \beta_{1} - 6 \beta_{3} ) q^{79} + 9 q^{81} + ( -24 \beta_{1} - 4 \beta_{3} ) q^{83} + ( 10 - 6 \beta_{2} ) q^{85} + ( -12 \beta_{1} + 6 \beta_{3} ) q^{87} + ( 69 - 13 \beta_{2} ) q^{89} + ( 30 + 6 \beta_{2} ) q^{93} + ( 68 \beta_{1} - 4 \beta_{3} ) q^{95} + ( 150 - 6 \beta_{2} ) q^{97} + ( -3 \beta_{1} - 3 \beta_{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 12q^{5} - 12q^{9} + O(q^{10})$$ $$4q + 12q^{5} - 12q^{9} - 36q^{17} + 84q^{25} + 48q^{29} + 12q^{33} + 64q^{37} + 132q^{41} - 36q^{45} + 192q^{53} - 24q^{57} + 264q^{61} + 132q^{69} - 24q^{73} + 36q^{81} + 40q^{85} + 276q^{89} + 120q^{93} + 600q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - x^{3} + 10 x^{2} + 9 x + 81$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-\nu^{3} + 10 \nu^{2} - 10 \nu + 36$$$$)/45$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{3} + 14$$$$)/5$$ $$\beta_{3}$$ $$=$$ $$($$$$2 \nu^{3} - 2 \nu^{2} + 38 \nu + 9$$$$)/9$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{3} - \beta_{2} + \beta_{1} + 1$$$$)/4$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{3} + \beta_{2} + 19 \beta_{1} - 19$$$$)/4$$ $$\nu^{3}$$ $$=$$ $$5 \beta_{2} - 14$$

## 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
 1.77069 + 3.06693i −1.27069 − 2.20090i 1.77069 − 3.06693i −1.27069 + 2.20090i
0 1.73205i 0 −3.08276 0 0 0 −3.00000 0
1471.2 0 1.73205i 0 9.08276 0 0 0 −3.00000 0
1471.3 0 1.73205i 0 −3.08276 0 0 0 −3.00000 0
1471.4 0 1.73205i 0 9.08276 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.j yes 4
4.b odd 2 1 inner 2352.3.m.j yes 4
7.b odd 2 1 2352.3.m.e 4
28.d even 2 1 2352.3.m.e 4

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{2} - 6 T_{5} - 28$$ acting on $$S_{3}^{\mathrm{new}}(2352, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$( 3 + T^{2} )^{2}$$
$5$ $$( -28 - 6 T + T^{2} )^{2}$$
$7$ $$T^{4}$$
$11$ $$11664 + 228 T^{2} + T^{4}$$
$13$ $$T^{4}$$
$17$ $$( 44 + 18 T + T^{2} )^{2}$$
$19$ $$186624 + 912 T^{2} + T^{4}$$
$23$ $$63504 + 948 T^{2} + T^{4}$$
$29$ $$( -1188 - 24 T + T^{2} )^{2}$$
$31$ $$20736 + 1488 T^{2} + T^{4}$$
$37$ $$( -1076 - 32 T + T^{2} )^{2}$$
$41$ $$( 1052 - 66 T + T^{2} )^{2}$$
$43$ $$( 768 + T^{2} )^{2}$$
$47$ $$1806336 + 4416 T^{2} + T^{4}$$
$53$ $$( 972 - 96 T + T^{2} )^{2}$$
$59$ $$( 1776 + T^{2} )^{2}$$
$61$ $$( 3024 - 132 T + T^{2} )^{2}$$
$67$ $$15116544 + 8208 T^{2} + T^{4}$$
$71$ $$6906384 + 5844 T^{2} + T^{4}$$
$73$ $$( -1296 + 12 T + T^{2} )^{2}$$
$79$ $$12278016 + 22992 T^{2} + T^{4}$$
$83$ $$2304 + 7008 T^{2} + T^{4}$$
$89$ $$( -1492 - 138 T + T^{2} )^{2}$$
$97$ $$( 21168 - 300 T + T^{2} )^{2}$$