# Properties

 Label 2352.3.f.f 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: $$\Q(\sqrt{-3}, \sqrt{65})$$ Defining polynomial: $$x^{4} - x^{3} + 17 x^{2} + 16 x + 256$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{2}\cdot 3$$ Twist minimal: no (minimal twist has level 84) 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.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{3} + ( 2 \beta_{1} + \beta_{2} ) q^{5} -3 q^{9} +O(q^{10})$$ $$q + \beta_{1} q^{3} + ( 2 \beta_{1} + \beta_{2} ) q^{5} -3 q^{9} + ( 8 - \beta_{3} ) q^{11} + ( \beta_{1} + \beta_{2} ) q^{13} + ( -4 - \beta_{3} ) q^{15} + ( -4 \beta_{1} - 2 \beta_{2} ) q^{17} + ( -13 \beta_{1} + \beta_{2} ) q^{19} + 24 q^{23} + ( -29 - 3 \beta_{3} ) q^{25} -3 \beta_{1} q^{27} + ( 2 - \beta_{3} ) q^{29} + ( -11 \beta_{1} - 6 \beta_{2} ) q^{31} + ( 6 \beta_{1} - 3 \beta_{2} ) q^{33} + ( -37 + \beta_{3} ) q^{37} + ( -1 - \beta_{3} ) q^{39} + ( -14 \beta_{1} - 4 \beta_{2} ) q^{41} + ( -19 + 3 \beta_{3} ) q^{43} + ( -6 \beta_{1} - 3 \beta_{2} ) q^{45} + ( -14 \beta_{1} + 2 \beta_{2} ) q^{47} + ( 8 + 2 \beta_{3} ) q^{51} + ( -50 + \beta_{3} ) q^{53} + ( -36 \beta_{1} + 3 \beta_{2} ) q^{55} + ( 41 - \beta_{3} ) q^{57} + ( -40 \beta_{1} + \beta_{2} ) q^{59} + ( 36 \beta_{1} + 8 \beta_{2} ) q^{61} + ( -50 - 2 \beta_{3} ) q^{65} + ( -1 - 5 \beta_{3} ) q^{67} + 24 \beta_{1} q^{69} + ( 54 - 6 \beta_{3} ) q^{71} + ( -11 \beta_{1} - 7 \beta_{2} ) q^{73} + ( -35 \beta_{1} - 9 \beta_{2} ) q^{75} + ( 29 - 2 \beta_{3} ) q^{79} + 9 q^{81} + ( 2 \beta_{1} - 17 \beta_{2} ) q^{83} + ( 108 + 6 \beta_{3} ) q^{85} -3 \beta_{2} q^{87} + ( -36 \beta_{1} - 6 \beta_{2} ) q^{89} + ( 21 + 6 \beta_{3} ) q^{93} + ( 6 + 12 \beta_{3} ) q^{95} + ( 12 \beta_{1} + 7 \beta_{2} ) q^{97} + ( -24 + 3 \beta_{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 12q^{9} + O(q^{10})$$ $$4q - 12q^{9} + 30q^{11} - 18q^{15} + 96q^{23} - 122q^{25} + 6q^{29} - 146q^{37} - 6q^{39} - 70q^{43} + 36q^{51} - 198q^{53} + 162q^{57} - 204q^{65} - 14q^{67} + 204q^{71} + 112q^{79} + 36q^{81} + 444q^{85} + 96q^{93} + 48q^{95} - 90q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - x^{3} + 17 x^{2} + 16 x + 256$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-\nu^{3} + 17 \nu^{2} - 17 \nu + 120$$$$)/136$$ $$\beta_{2}$$ $$=$$ $$($$$$9 \nu^{3} - 17 \nu^{2} + 289 \nu + 8$$$$)/136$$ $$\beta_{3}$$ $$=$$ $$($$$$-3 \nu^{3} - 65$$$$)/17$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{3} + 3 \beta_{2} + 3 \beta_{1} + 1$$$$)/6$$ $$\nu^{2}$$ $$=$$ $$($$$$-\beta_{3} + 3 \beta_{2} + 51 \beta_{1} - 49$$$$)/6$$ $$\nu^{3}$$ $$=$$ $$($$$$-17 \beta_{3} - 65$$$$)/3$$

## 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
 2.26556 − 3.92407i −1.76556 + 3.05805i −1.76556 − 3.05805i 2.26556 + 3.92407i
0 1.73205i 0 9.58020i 0 0 0 −3.00000 0
97.2 0 1.73205i 0 4.38404i 0 0 0 −3.00000 0
97.3 0 1.73205i 0 4.38404i 0 0 0 −3.00000 0
97.4 0 1.73205i 0 9.58020i 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.f 4
4.b odd 2 1 588.3.d.b 4
7.b odd 2 1 inner 2352.3.f.f 4
7.c even 3 1 336.3.bh.f 4
7.d odd 6 1 336.3.bh.f 4
12.b even 2 1 1764.3.d.f 4
21.g even 6 1 1008.3.cg.m 4
21.h odd 6 1 1008.3.cg.m 4
28.d even 2 1 588.3.d.b 4
28.f even 6 1 84.3.m.b 4
28.f even 6 1 588.3.m.d 4
28.g odd 6 1 84.3.m.b 4
28.g odd 6 1 588.3.m.d 4
84.h odd 2 1 1764.3.d.f 4
84.j odd 6 1 252.3.z.e 4
84.j odd 6 1 1764.3.z.h 4
84.n even 6 1 252.3.z.e 4
84.n even 6 1 1764.3.z.h 4
140.p odd 6 1 2100.3.bd.f 4
140.s even 6 1 2100.3.bd.f 4
140.w even 12 2 2100.3.be.d 8
140.x odd 12 2 2100.3.be.d 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
84.3.m.b 4 28.f even 6 1
84.3.m.b 4 28.g odd 6 1
252.3.z.e 4 84.j odd 6 1
252.3.z.e 4 84.n even 6 1
336.3.bh.f 4 7.c even 3 1
336.3.bh.f 4 7.d odd 6 1
588.3.d.b 4 4.b odd 2 1
588.3.d.b 4 28.d even 2 1
588.3.m.d 4 28.f even 6 1
588.3.m.d 4 28.g odd 6 1
1008.3.cg.m 4 21.g even 6 1
1008.3.cg.m 4 21.h odd 6 1
1764.3.d.f 4 12.b even 2 1
1764.3.d.f 4 84.h odd 2 1
1764.3.z.h 4 84.j odd 6 1
1764.3.z.h 4 84.n even 6 1
2100.3.bd.f 4 140.p odd 6 1
2100.3.bd.f 4 140.s even 6 1
2100.3.be.d 8 140.w even 12 2
2100.3.be.d 8 140.x odd 12 2
2352.3.f.f 4 1.a even 1 1 trivial
2352.3.f.f 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 $$S_{3}^{\mathrm{new}}(2352, [\chi])$$:

 $$T_{5}^{4} + 111 T_{5}^{2} + 1764$$ $$T_{11}^{2} - 15 T_{11} - 90$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$( 3 + T^{2} )^{2}$$
$5$ $$1764 + 111 T^{2} + T^{4}$$
$7$ $$T^{4}$$
$11$ $$( -90 - 15 T + T^{2} )^{2}$$
$13$ $$2304 + 99 T^{2} + T^{4}$$
$17$ $$28224 + 444 T^{2} + T^{4}$$
$19$ $$248004 + 1191 T^{2} + T^{4}$$
$23$ $$( -24 + T )^{4}$$
$29$ $$( -144 - 3 T + T^{2} )^{2}$$
$31$ $$2442969 + 3894 T^{2} + T^{4}$$
$37$ $$( 1186 + 73 T + T^{2} )^{2}$$
$41$ $$121104 + 2424 T^{2} + T^{4}$$
$43$ $$( -1010 + 35 T + T^{2} )^{2}$$
$47$ $$230400 + 1740 T^{2} + T^{4}$$
$53$ $$( 2304 + 99 T + T^{2} )^{2}$$
$59$ $$23736384 + 9939 T^{2} + T^{4}$$
$61$ $$2304 + 12384 T^{2} + T^{4}$$
$67$ $$( -3644 + 7 T + T^{2} )^{2}$$
$71$ $$( -2664 - 102 T + T^{2} )^{2}$$
$73$ $$4928400 + 5115 T^{2} + T^{4}$$
$79$ $$( 199 - 56 T + T^{2} )^{2}$$
$83$ $$189282564 + 28839 T^{2} + T^{4}$$
$89$ $$2286144 + 10044 T^{2} + T^{4}$$
$97$ $$4717584 + 5211 T^{2} + T^{4}$$