# Properties

 Label 2352.3.m.k 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: $$\Q(\sqrt{-3}, \sqrt{-7})$$ Defining polynomial: $$x^{4} - x^{3} - x^{2} - 2 x + 4$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{4}$$ Twist minimal: no (minimal twist has level 336) 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.

 $$f(q)$$ $$=$$ $$q -\beta_{2} q^{3} + ( 5 - \beta_{1} ) q^{5} -3 q^{9} +O(q^{10})$$ $$q -\beta_{2} q^{3} + ( 5 - \beta_{1} ) q^{5} -3 q^{9} + ( 3 \beta_{2} + 5 \beta_{3} ) q^{11} + ( -8 + 2 \beta_{1} ) q^{13} + ( -5 \beta_{2} + 3 \beta_{3} ) q^{15} + ( -5 + \beta_{1} ) q^{17} -8 \beta_{3} q^{19} + ( -7 \beta_{2} + 11 \beta_{3} ) q^{23} + ( 21 - 10 \beta_{1} ) q^{25} + 3 \beta_{2} q^{27} + ( -14 + 4 \beta_{1} ) q^{29} -12 \beta_{2} q^{31} + ( 9 + 5 \beta_{1} ) q^{33} + ( -52 - 2 \beta_{1} ) q^{37} + ( 8 \beta_{2} - 6 \beta_{3} ) q^{39} + ( -23 + 7 \beta_{1} ) q^{41} + ( -8 \beta_{2} + 8 \beta_{3} ) q^{43} + ( -15 + 3 \beta_{1} ) q^{45} + ( -2 \beta_{2} - 6 \beta_{3} ) q^{47} + ( 5 \beta_{2} - 3 \beta_{3} ) q^{51} + ( 28 + 14 \beta_{1} ) q^{53} + ( -20 \beta_{2} + 16 \beta_{3} ) q^{55} -8 \beta_{1} q^{57} + ( -42 \beta_{2} - 10 \beta_{3} ) q^{59} + ( -42 - 8 \beta_{1} ) q^{61} + ( -82 + 18 \beta_{1} ) q^{65} + ( -54 \beta_{2} + 6 \beta_{3} ) q^{67} + ( -21 + 11 \beta_{1} ) q^{69} + ( -33 \beta_{2} + 5 \beta_{3} ) q^{71} + ( 8 + 10 \beta_{1} ) q^{73} + ( -21 \beta_{2} + 30 \beta_{3} ) q^{75} + ( -34 \beta_{2} - 6 \beta_{3} ) q^{79} + 9 q^{81} + ( 24 \beta_{2} - 28 \beta_{3} ) q^{83} + ( -46 + 10 \beta_{1} ) q^{85} + ( 14 \beta_{2} - 12 \beta_{3} ) q^{87} + ( -59 - 5 \beta_{1} ) q^{89} -36 q^{93} + ( 56 \beta_{2} - 40 \beta_{3} ) q^{95} + ( 76 - 10 \beta_{1} ) q^{97} + ( -9 \beta_{2} - 15 \beta_{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 20q^{5} - 12q^{9} + O(q^{10})$$ $$4q + 20q^{5} - 12q^{9} - 32q^{13} - 20q^{17} + 84q^{25} - 56q^{29} + 36q^{33} - 208q^{37} - 92q^{41} - 60q^{45} + 112q^{53} - 168q^{61} - 328q^{65} - 84q^{69} + 32q^{73} + 36q^{81} - 184q^{85} - 236q^{89} - 144q^{93} + 304q^{97} + O(q^{100})$$

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

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$-\nu^{3} + \nu^{2} + 3 \nu + 1$$ $$\beta_{2}$$ $$=$$ $$\nu^{3} + \nu^{2} - \nu - 3$$ $$\beta_{3}$$ $$=$$ $$2 \nu^{3} - 5$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{3} - \beta_{2} + \beta_{1} + 1$$$$)/4$$ $$\nu^{2}$$ $$=$$ $$($$$$-\beta_{3} + 3 \beta_{2} + \beta_{1} + 3$$$$)/4$$ $$\nu^{3}$$ $$=$$ $$($$$$\beta_{3} + 5$$$$)/2$$

## 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
 1.39564 + 0.228425i −0.895644 − 1.09445i 1.39564 − 0.228425i −0.895644 + 1.09445i
0 1.73205i 0 0.417424 0 0 0 −3.00000 0
1471.2 0 1.73205i 0 9.58258 0 0 0 −3.00000 0
1471.3 0 1.73205i 0 0.417424 0 0 0 −3.00000 0
1471.4 0 1.73205i 0 9.58258 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.k 4
4.b odd 2 1 inner 2352.3.m.k 4
7.b odd 2 1 336.3.m.a 4
21.c even 2 1 1008.3.m.f 4
28.d even 2 1 336.3.m.a 4
56.e even 2 1 1344.3.m.c 4
56.h odd 2 1 1344.3.m.c 4
84.h odd 2 1 1008.3.m.f 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
336.3.m.a 4 7.b odd 2 1
336.3.m.a 4 28.d even 2 1
1008.3.m.f 4 21.c even 2 1
1008.3.m.f 4 84.h odd 2 1
1344.3.m.c 4 56.e even 2 1
1344.3.m.c 4 56.h odd 2 1
2352.3.m.k 4 1.a even 1 1 trivial
2352.3.m.k 4 4.b odd 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{2} - 10 T_{5} + 4$$ acting on $$S_{3}^{\mathrm{new}}(2352, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$( 3 + T^{2} )^{2}$$
$5$ $$( 4 - 10 T + T^{2} )^{2}$$
$7$ $$T^{4}$$
$11$ $$21904 + 404 T^{2} + T^{4}$$
$13$ $$( -20 + 16 T + T^{2} )^{2}$$
$17$ $$( 4 + 10 T + T^{2} )^{2}$$
$19$ $$( 448 + T^{2} )^{2}$$
$23$ $$490000 + 1988 T^{2} + T^{4}$$
$29$ $$( -140 + 28 T + T^{2} )^{2}$$
$31$ $$( 432 + T^{2} )^{2}$$
$37$ $$( 2620 + 104 T + T^{2} )^{2}$$
$41$ $$( -500 + 46 T + T^{2} )^{2}$$
$43$ $$65536 + 1280 T^{2} + T^{4}$$
$47$ $$57600 + 528 T^{2} + T^{4}$$
$53$ $$( -3332 - 56 T + T^{2} )^{2}$$
$59$ $$21086464 + 11984 T^{2} + T^{4}$$
$61$ $$( 420 + 84 T + T^{2} )^{2}$$
$67$ $$72182016 + 18000 T^{2} + T^{4}$$
$71$ $$9560464 + 6884 T^{2} + T^{4}$$
$73$ $$( -2036 - 16 T + T^{2} )^{2}$$
$79$ $$10342656 + 7440 T^{2} + T^{4}$$
$83$ $$14137600 + 14432 T^{2} + T^{4}$$
$89$ $$( 2956 + 118 T + T^{2} )^{2}$$
$97$ $$( 3676 - 152 T + T^{2} )^{2}$$