# Properties

 Label 2352.2.bl.n Level $2352$ Weight $2$ Character orbit 2352.bl Analytic conductor $18.781$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2352 = 2^{4} \cdot 3 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2352.bl (of order $$6$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$18.7808145554$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{6})$$ 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: $$3$$ Twist minimal: no (minimal twist has level 336) Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + ( 1 + \beta_{1} ) q^{3} + \beta_{3} q^{5} + \beta_{1} q^{9} +O(q^{10})$$ $$q + ( 1 + \beta_{1} ) q^{3} + \beta_{3} q^{5} + \beta_{1} q^{9} + ( \beta_{2} - \beta_{3} ) q^{11} -2 \beta_{2} q^{13} + \beta_{2} q^{15} + ( \beta_{2} - \beta_{3} ) q^{17} + 2 \beta_{1} q^{19} + 3 \beta_{3} q^{23} + ( 1 + \beta_{1} ) q^{25} - q^{27} + 6 q^{29} + ( 8 + 8 \beta_{1} ) q^{31} -\beta_{3} q^{33} + 4 \beta_{1} q^{37} + ( -2 \beta_{2} + 2 \beta_{3} ) q^{39} + 3 \beta_{2} q^{41} + 2 \beta_{2} q^{43} + ( \beta_{2} - \beta_{3} ) q^{45} + 12 \beta_{1} q^{47} -\beta_{3} q^{51} + ( 6 + 6 \beta_{1} ) q^{53} -6 q^{55} -2 q^{57} + ( 12 + 12 \beta_{1} ) q^{59} -12 \beta_{1} q^{65} + 3 \beta_{2} q^{69} -5 \beta_{2} q^{71} + ( -6 \beta_{2} + 6 \beta_{3} ) q^{73} + \beta_{1} q^{75} -4 \beta_{3} q^{79} + ( -1 - \beta_{1} ) q^{81} -6 q^{85} + ( 6 + 6 \beta_{1} ) q^{87} -5 \beta_{3} q^{89} + 8 \beta_{1} q^{93} + ( 2 \beta_{2} - 2 \beta_{3} ) q^{95} -2 \beta_{2} q^{97} -\beta_{2} q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 2q^{3} - 2q^{9} + O(q^{10})$$ $$4q + 2q^{3} - 2q^{9} - 4q^{19} + 2q^{25} - 4q^{27} + 24q^{29} + 16q^{31} - 8q^{37} - 24q^{47} + 12q^{53} - 24q^{55} - 8q^{57} + 24q^{59} + 24q^{65} - 2q^{75} - 2q^{81} - 24q^{85} + 12q^{87} - 16q^{93} + O(q^{100})$$

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

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

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/2352\mathbb{Z}\right)^\times$$.

 $$n$$ $$785$$ $$1471$$ $$1765$$ $$2257$$ $$\chi(n)$$ $$1$$ $$-1$$ $$1$$ $$-\beta_{1}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
31.1
 −0.707107 + 1.22474i 0.707107 − 1.22474i −0.707107 − 1.22474i 0.707107 + 1.22474i
0 0.500000 0.866025i 0 −2.12132 + 1.22474i 0 0 0 −0.500000 0.866025i 0
31.2 0 0.500000 0.866025i 0 2.12132 1.22474i 0 0 0 −0.500000 0.866025i 0
607.1 0 0.500000 + 0.866025i 0 −2.12132 1.22474i 0 0 0 −0.500000 + 0.866025i 0
607.2 0 0.500000 + 0.866025i 0 2.12132 + 1.22474i 0 0 0 −0.500000 + 0.866025i 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
7.c even 3 1 inner
28.d even 2 1 inner
28.f even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2352.2.bl.n 4
4.b odd 2 1 2352.2.bl.m 4
7.b odd 2 1 2352.2.bl.m 4
7.c even 3 1 336.2.b.b 2
7.c even 3 1 inner 2352.2.bl.n 4
7.d odd 6 1 336.2.b.c yes 2
7.d odd 6 1 2352.2.bl.m 4
21.g even 6 1 1008.2.b.d 2
21.h odd 6 1 1008.2.b.e 2
28.d even 2 1 inner 2352.2.bl.n 4
28.f even 6 1 336.2.b.b 2
28.f even 6 1 inner 2352.2.bl.n 4
28.g odd 6 1 336.2.b.c yes 2
28.g odd 6 1 2352.2.bl.m 4
56.j odd 6 1 1344.2.b.a 2
56.k odd 6 1 1344.2.b.a 2
56.m even 6 1 1344.2.b.d 2
56.p even 6 1 1344.2.b.d 2
84.j odd 6 1 1008.2.b.e 2
84.n even 6 1 1008.2.b.d 2
168.s odd 6 1 4032.2.b.f 2
168.v even 6 1 4032.2.b.d 2
168.ba even 6 1 4032.2.b.d 2
168.be odd 6 1 4032.2.b.f 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
336.2.b.b 2 7.c even 3 1
336.2.b.b 2 28.f even 6 1
336.2.b.c yes 2 7.d odd 6 1
336.2.b.c yes 2 28.g odd 6 1
1008.2.b.d 2 21.g even 6 1
1008.2.b.d 2 84.n even 6 1
1008.2.b.e 2 21.h odd 6 1
1008.2.b.e 2 84.j odd 6 1
1344.2.b.a 2 56.j odd 6 1
1344.2.b.a 2 56.k odd 6 1
1344.2.b.d 2 56.m even 6 1
1344.2.b.d 2 56.p even 6 1
2352.2.bl.m 4 4.b odd 2 1
2352.2.bl.m 4 7.b odd 2 1
2352.2.bl.m 4 7.d odd 6 1
2352.2.bl.m 4 28.g odd 6 1
2352.2.bl.n 4 1.a even 1 1 trivial
2352.2.bl.n 4 7.c even 3 1 inner
2352.2.bl.n 4 28.d even 2 1 inner
2352.2.bl.n 4 28.f even 6 1 inner
4032.2.b.d 2 168.v even 6 1
4032.2.b.d 2 168.ba even 6 1
4032.2.b.f 2 168.s odd 6 1
4032.2.b.f 2 168.be odd 6 1

## Hecke kernels

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

 $$T_{5}^{4} - 6 T_{5}^{2} + 36$$ $$T_{11}^{4} - 6 T_{11}^{2} + 36$$ $$T_{17}^{4} - 6 T_{17}^{2} + 36$$ $$T_{19}^{2} + 2 T_{19} + 4$$ $$T_{31}^{2} - 8 T_{31} + 64$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$( 1 - T + T^{2} )^{2}$$
$5$ $$36 - 6 T^{2} + T^{4}$$
$7$ $$T^{4}$$
$11$ $$36 - 6 T^{2} + T^{4}$$
$13$ $$( 24 + T^{2} )^{2}$$
$17$ $$36 - 6 T^{2} + T^{4}$$
$19$ $$( 4 + 2 T + T^{2} )^{2}$$
$23$ $$2916 - 54 T^{2} + T^{4}$$
$29$ $$( -6 + T )^{4}$$
$31$ $$( 64 - 8 T + T^{2} )^{2}$$
$37$ $$( 16 + 4 T + T^{2} )^{2}$$
$41$ $$( 54 + T^{2} )^{2}$$
$43$ $$( 24 + T^{2} )^{2}$$
$47$ $$( 144 + 12 T + T^{2} )^{2}$$
$53$ $$( 36 - 6 T + T^{2} )^{2}$$
$59$ $$( 144 - 12 T + T^{2} )^{2}$$
$61$ $$T^{4}$$
$67$ $$T^{4}$$
$71$ $$( 150 + T^{2} )^{2}$$
$73$ $$46656 - 216 T^{2} + T^{4}$$
$79$ $$9216 - 96 T^{2} + T^{4}$$
$83$ $$T^{4}$$
$89$ $$22500 - 150 T^{2} + T^{4}$$
$97$ $$( 24 + T^{2} )^{2}$$