# Properties

 Label 2352.3.f.e Level $2352$ Weight $3$ Character orbit 2352.f 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.f (of order $$2$$, degree $$1$$, not 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: $$2^{4}\cdot 3$$ Twist minimal: no (minimal twist has level 42) 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.

 $$f(q)$$ $$=$$ $$q + \beta_{2} q^{3} + ( -\beta_{1} + 2 \beta_{2} ) q^{5} -3 q^{9} +O(q^{10})$$ $$q + \beta_{2} q^{3} + ( -\beta_{1} + 2 \beta_{2} ) q^{5} -3 q^{9} -6 q^{11} + ( -4 \beta_{1} - \beta_{2} ) q^{13} + ( -6 + \beta_{3} ) q^{15} + ( -\beta_{1} + 8 \beta_{2} ) q^{17} + ( -\beta_{1} + 7 \beta_{2} ) q^{19} + ( 12 + 3 \beta_{3} ) q^{23} + ( -11 + 4 \beta_{3} ) q^{25} -3 \beta_{2} q^{27} -4 \beta_{3} q^{29} + ( 3 \beta_{1} + 17 \beta_{2} ) q^{31} -6 \beta_{2} q^{33} + ( -11 - 2 \beta_{3} ) q^{37} + ( 3 + 4 \beta_{3} ) q^{39} + ( -2 \beta_{1} - 26 \beta_{2} ) q^{41} + ( -7 + \beta_{3} ) q^{43} + ( 3 \beta_{1} - 6 \beta_{2} ) q^{45} + ( \beta_{1} + 22 \beta_{2} ) q^{47} + ( -24 + \beta_{3} ) q^{51} + ( -60 - 3 \beta_{3} ) q^{53} + ( 6 \beta_{1} - 12 \beta_{2} ) q^{55} + ( -21 + \beta_{3} ) q^{57} + ( -7 \beta_{1} - 4 \beta_{2} ) q^{59} + ( 4 \beta_{1} - 12 \beta_{2} ) q^{61} + ( -90 + 7 \beta_{3} ) q^{65} + ( 55 - 7 \beta_{3} ) q^{67} + ( 9 \beta_{1} + 12 \beta_{2} ) q^{69} + ( -78 - 7 \beta_{3} ) q^{71} + ( -20 \beta_{1} + 11 \beta_{2} ) q^{73} + ( 12 \beta_{1} - 11 \beta_{2} ) q^{75} + ( -5 - 11 \beta_{3} ) q^{79} + 9 q^{81} + ( -19 \beta_{1} - 10 \beta_{2} ) q^{83} + ( -72 + 10 \beta_{3} ) q^{85} -12 \beta_{1} q^{87} -12 \beta_{2} q^{89} + ( -51 - 3 \beta_{3} ) q^{93} + ( -66 + 9 \beta_{3} ) q^{95} + ( 2 \beta_{1} - 12 \beta_{2} ) q^{97} + 18 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 12q^{9} + O(q^{10})$$ $$4q - 12q^{9} - 24q^{11} - 24q^{15} + 48q^{23} - 44q^{25} - 44q^{37} + 12q^{39} - 28q^{43} - 96q^{51} - 240q^{53} - 84q^{57} - 360q^{65} + 220q^{67} - 312q^{71} - 20q^{79} + 36q^{81} - 288q^{85} - 204q^{93} - 264q^{95} + 72q^{99} + O(q^{100})$$

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

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu^{3} + 4 \nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} + 1$$ $$\beta_{3}$$ $$=$$ $$-3 \nu^{3}$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{3} + 3 \beta_{1}$$$$)/12$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} - 1$$ $$\nu^{3}$$ $$=$$ $$($$$$-\beta_{3}$$$$)/3$$

## Character values

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

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
97.1
 −0.707107 + 1.22474i 0.707107 − 1.22474i 0.707107 + 1.22474i −0.707107 − 1.22474i
0 1.73205i 0 8.36308i 0 0 0 −3.00000 0
97.2 0 1.73205i 0 1.43488i 0 0 0 −3.00000 0
97.3 0 1.73205i 0 1.43488i 0 0 0 −3.00000 0
97.4 0 1.73205i 0 8.36308i 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
7.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.f.e 4
4.b odd 2 1 294.3.c.a 4
7.b odd 2 1 inner 2352.3.f.e 4
7.c even 3 1 336.3.bh.e 4
7.d odd 6 1 336.3.bh.e 4
12.b even 2 1 882.3.c.b 4
21.g even 6 1 1008.3.cg.h 4
21.h odd 6 1 1008.3.cg.h 4
28.d even 2 1 294.3.c.a 4
28.f even 6 1 42.3.g.a 4
28.f even 6 1 294.3.g.a 4
28.g odd 6 1 42.3.g.a 4
28.g odd 6 1 294.3.g.a 4
84.h odd 2 1 882.3.c.b 4
84.j odd 6 1 126.3.n.a 4
84.j odd 6 1 882.3.n.e 4
84.n even 6 1 126.3.n.a 4
84.n even 6 1 882.3.n.e 4
140.p odd 6 1 1050.3.p.a 4
140.s even 6 1 1050.3.p.a 4
140.w even 12 2 1050.3.q.a 8
140.x odd 12 2 1050.3.q.a 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
42.3.g.a 4 28.f even 6 1
42.3.g.a 4 28.g odd 6 1
126.3.n.a 4 84.j odd 6 1
126.3.n.a 4 84.n even 6 1
294.3.c.a 4 4.b odd 2 1
294.3.c.a 4 28.d even 2 1
294.3.g.a 4 28.f even 6 1
294.3.g.a 4 28.g odd 6 1
336.3.bh.e 4 7.c even 3 1
336.3.bh.e 4 7.d odd 6 1
882.3.c.b 4 12.b even 2 1
882.3.c.b 4 84.h odd 2 1
882.3.n.e 4 84.j odd 6 1
882.3.n.e 4 84.n even 6 1
1008.3.cg.h 4 21.g even 6 1
1008.3.cg.h 4 21.h odd 6 1
1050.3.p.a 4 140.p odd 6 1
1050.3.p.a 4 140.s even 6 1
1050.3.q.a 8 140.w even 12 2
1050.3.q.a 8 140.x odd 12 2
2352.3.f.e 4 1.a even 1 1 trivial
2352.3.f.e 4 7.b odd 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{3}^{\mathrm{new}}(2352, [\chi])$$:

 $$T_{5}^{4} + 72 T_{5}^{2} + 144$$ $$T_{11} + 6$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$( 3 + T^{2} )^{2}$$
$5$ $$144 + 72 T^{2} + T^{4}$$
$7$ $$T^{4}$$
$11$ $$( 6 + T )^{4}$$
$13$ $$145161 + 774 T^{2} + T^{4}$$
$17$ $$28224 + 432 T^{2} + T^{4}$$
$19$ $$15129 + 342 T^{2} + T^{4}$$
$23$ $$( -504 - 24 T + T^{2} )^{2}$$
$29$ $$( -1152 + T^{2} )^{2}$$
$31$ $$423801 + 2166 T^{2} + T^{4}$$
$37$ $$( -167 + 22 T + T^{2} )^{2}$$
$41$ $$3732624 + 4248 T^{2} + T^{4}$$
$43$ $$( -23 + 14 T + T^{2} )^{2}$$
$47$ $$2039184 + 2952 T^{2} + T^{4}$$
$53$ $$( 2952 + 120 T + T^{2} )^{2}$$
$59$ $$1272384 + 2448 T^{2} + T^{4}$$
$61$ $$2304 + 1632 T^{2} + T^{4}$$
$67$ $$( -503 - 110 T + T^{2} )^{2}$$
$71$ $$( 2556 + 156 T + T^{2} )^{2}$$
$73$ $$85322169 + 19926 T^{2} + T^{4}$$
$79$ $$( -8687 + 10 T + T^{2} )^{2}$$
$83$ $$69956496 + 17928 T^{2} + T^{4}$$
$89$ $$( 432 + T^{2} )^{2}$$
$97$ $$112896 + 1056 T^{2} + T^{4}$$