# Properties

 Label 2352.3.m.f 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.

 $$f(q)$$ $$=$$ $$q -\beta_{2} q^{3} + ( -1 + \beta_{1} ) q^{5} -3 q^{9} +O(q^{10})$$ $$q -\beta_{2} q^{3} + ( -1 + \beta_{1} ) q^{5} -3 q^{9} + ( \beta_{2} - \beta_{3} ) q^{11} + ( -8 + 2 \beta_{1} ) q^{13} + ( \beta_{2} - 3 \beta_{3} ) q^{15} + ( 13 + 3 \beta_{1} ) q^{17} + ( 4 \beta_{2} + 4 \beta_{3} ) q^{19} + ( -5 \beta_{2} - 7 \beta_{3} ) q^{23} + ( -3 - 2 \beta_{1} ) q^{25} + 3 \beta_{2} q^{27} -2 q^{29} + ( 8 \beta_{2} + 12 \beta_{3} ) q^{31} + ( 3 - \beta_{1} ) q^{33} + ( 20 + 6 \beta_{1} ) q^{37} + ( 8 \beta_{2} - 6 \beta_{3} ) q^{39} + ( -17 + 13 \beta_{1} ) q^{41} + ( 16 \beta_{2} - 16 \beta_{3} ) q^{43} + ( 3 - 3 \beta_{1} ) q^{45} + ( -14 \beta_{2} + 6 \beta_{3} ) q^{47} + ( -13 \beta_{2} - 9 \beta_{3} ) q^{51} + ( 4 - 10 \beta_{1} ) q^{53} + ( -8 \beta_{2} + 4 \beta_{3} ) q^{55} + ( 12 + 4 \beta_{1} ) q^{57} + ( 18 \beta_{2} - 22 \beta_{3} ) q^{59} + ( 66 - 12 \beta_{1} ) q^{61} + ( 50 - 10 \beta_{1} ) q^{65} + ( 14 \beta_{2} + 18 \beta_{3} ) q^{67} + ( -15 - 7 \beta_{1} ) q^{69} + ( -3 \beta_{2} + 23 \beta_{3} ) q^{71} + ( 68 + 6 \beta_{1} ) q^{73} + ( 3 \beta_{2} + 6 \beta_{3} ) q^{75} + ( -46 \beta_{2} + 6 \beta_{3} ) q^{79} + 9 q^{81} + ( 24 \beta_{2} - 28 \beta_{3} ) q^{83} + ( 50 + 10 \beta_{1} ) q^{85} + 2 \beta_{2} q^{87} + ( 43 + 17 \beta_{1} ) q^{89} + ( 24 + 12 \beta_{1} ) q^{93} + ( 24 \beta_{2} + 8 \beta_{3} ) q^{95} + ( -80 - 6 \beta_{1} ) q^{97} + ( -3 \beta_{2} + 3 \beta_{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 4q^{5} - 12q^{9} + O(q^{10})$$ $$4q - 4q^{5} - 12q^{9} - 32q^{13} + 52q^{17} - 12q^{25} - 8q^{29} + 12q^{33} + 80q^{37} - 68q^{41} + 12q^{45} + 16q^{53} + 48q^{57} + 264q^{61} + 200q^{65} - 60q^{69} + 272q^{73} + 36q^{81} + 200q^{85} + 172q^{89} + 96q^{93} - 320q^{97} + O(q^{100})$$

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
 −0.895644 − 1.09445i 1.39564 + 0.228425i −0.895644 + 1.09445i 1.39564 − 0.228425i
0 1.73205i 0 −5.58258 0 0 0 −3.00000 0
1471.2 0 1.73205i 0 3.58258 0 0 0 −3.00000 0
1471.3 0 1.73205i 0 −5.58258 0 0 0 −3.00000 0
1471.4 0 1.73205i 0 3.58258 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.f 4
4.b odd 2 1 inner 2352.3.m.f 4
7.b odd 2 1 336.3.m.c 4
21.c even 2 1 1008.3.m.b 4
28.d even 2 1 336.3.m.c 4
56.e even 2 1 1344.3.m.a 4
56.h odd 2 1 1344.3.m.a 4
84.h odd 2 1 1008.3.m.b 4

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{2} + 2 T_{5} - 20$$ acting on $$S_{3}^{\mathrm{new}}(2352, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$( 3 + T^{2} )^{2}$$
$5$ $$( -20 + 2 T + T^{2} )^{2}$$
$7$ $$T^{4}$$
$11$ $$16 + 20 T^{2} + T^{4}$$
$13$ $$( -20 + 16 T + T^{2} )^{2}$$
$17$ $$( -20 - 26 T + T^{2} )^{2}$$
$19$ $$4096 + 320 T^{2} + T^{4}$$
$23$ $$71824 + 836 T^{2} + T^{4}$$
$29$ $$( 2 + T )^{4}$$
$31$ $$665856 + 2400 T^{2} + T^{4}$$
$37$ $$( -356 - 40 T + T^{2} )^{2}$$
$41$ $$( -3260 + 34 T + T^{2} )^{2}$$
$43$ $$1048576 + 5120 T^{2} + T^{4}$$
$47$ $$112896 + 1680 T^{2} + T^{4}$$
$53$ $$( -2084 - 8 T + T^{2} )^{2}$$
$59$ $$5837056 + 8720 T^{2} + T^{4}$$
$61$ $$( 1332 - 132 T + T^{2} )^{2}$$
$67$ $$2822400 + 5712 T^{2} + T^{4}$$
$71$ $$13512976 + 7460 T^{2} + T^{4}$$
$73$ $$( 3868 - 136 T + T^{2} )^{2}$$
$79$ $$37161216 + 13200 T^{2} + T^{4}$$
$83$ $$14137600 + 14432 T^{2} + T^{4}$$
$89$ $$( -4220 - 86 T + T^{2} )^{2}$$
$97$ $$( 5644 + 160 T + T^{2} )^{2}$$