# Properties

 Label 2352.3.m.h Level $2352$ Weight $3$ Character orbit 2352.m Analytic conductor $64.087$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

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## 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: $$\Q(\sqrt{-3}, \sqrt{-19})$$ Defining polynomial: $$x^{4} - x^{3} - 4 x^{2} - 5 x + 25$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{2}$$ 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.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{3} + \beta_{2} q^{5} -3 q^{9} +O(q^{10})$$ $$q + \beta_{1} q^{3} + \beta_{2} q^{5} -3 q^{9} + ( -7 \beta_{1} - 5 \beta_{3} ) q^{11} + ( -7 + \beta_{2} ) q^{13} + ( -\beta_{1} - 3 \beta_{3} ) q^{15} + ( 12 - 4 \beta_{2} ) q^{17} + ( -8 \beta_{1} - 13 \beta_{3} ) q^{19} + 4 \beta_{3} q^{23} + ( -11 + \beta_{2} ) q^{25} -3 \beta_{1} q^{27} + ( -6 + 11 \beta_{2} ) q^{29} + ( 21 \beta_{1} + 12 \beta_{3} ) q^{31} + ( 16 - 5 \beta_{2} ) q^{33} + ( 13 - 7 \beta_{2} ) q^{37} + ( -8 \beta_{1} - 3 \beta_{3} ) q^{39} + ( 6 - 10 \beta_{2} ) q^{41} + ( 22 \beta_{1} - 5 \beta_{3} ) q^{43} -3 \beta_{2} q^{45} + 14 \beta_{1} q^{47} + ( 16 \beta_{1} + 12 \beta_{3} ) q^{51} + ( -82 - 5 \beta_{2} ) q^{53} + ( 27 \beta_{1} + 11 \beta_{3} ) q^{55} + ( 11 - 13 \beta_{2} ) q^{57} + ( -13 \beta_{1} - 17 \beta_{3} ) q^{59} + ( 50 - 4 \beta_{2} ) q^{61} + ( 14 - 6 \beta_{2} ) q^{65} + ( 6 \beta_{1} + 9 \beta_{3} ) q^{67} + ( 4 + 4 \beta_{2} ) q^{69} + ( 26 \beta_{1} + 40 \beta_{3} ) q^{71} + ( -95 - \beta_{2} ) q^{73} + ( -12 \beta_{1} - 3 \beta_{3} ) q^{75} + ( 19 \beta_{1} - 18 \beta_{3} ) q^{79} + 9 q^{81} + ( 47 \beta_{1} - 23 \beta_{3} ) q^{83} + ( -56 + 8 \beta_{2} ) q^{85} + ( -17 \beta_{1} - 33 \beta_{3} ) q^{87} + ( 60 + 2 \beta_{2} ) q^{89} + ( -51 + 12 \beta_{2} ) q^{93} + ( 60 \beta_{1} - 2 \beta_{3} ) q^{95} + ( -28 + 31 \beta_{2} ) q^{97} + ( 21 \beta_{1} + 15 \beta_{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 2q^{5} - 12q^{9} + O(q^{10})$$ $$4q + 2q^{5} - 12q^{9} - 26q^{13} + 40q^{17} - 42q^{25} - 2q^{29} + 54q^{33} + 38q^{37} + 4q^{41} - 6q^{45} - 338q^{53} + 18q^{57} + 192q^{61} + 44q^{65} + 24q^{69} - 382q^{73} + 36q^{81} - 208q^{85} + 244q^{89} - 180q^{93} - 50q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - x^{3} - 4 x^{2} - 5 x + 25$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$\nu^{3} + 4 \nu^{2} - 4 \nu - 15$$$$)/10$$ $$\beta_{2}$$ $$=$$ $$($$$$-\nu^{3} + \nu^{2} + 9 \nu + 5$$$$)/5$$ $$\beta_{3}$$ $$=$$ $$($$$$-3 \nu^{3} - 2 \nu^{2} + 2 \nu + 25$$$$)/10$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$-\beta_{3} + \beta_{2} - \beta_{1}$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{3} + \beta_{2} + 5 \beta_{1} + 4$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$-4 \beta_{3} - 2 \beta_{1} + 7$$

## 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
 −1.63746 + 1.52274i 2.13746 − 0.656712i −1.63746 − 1.52274i 2.13746 + 0.656712i
0 1.73205i 0 −3.27492 0 0 0 −3.00000 0
1471.2 0 1.73205i 0 4.27492 0 0 0 −3.00000 0
1471.3 0 1.73205i 0 −3.27492 0 0 0 −3.00000 0
1471.4 0 1.73205i 0 4.27492 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.h 4
4.b odd 2 1 inner 2352.3.m.h 4
7.b odd 2 1 2352.3.m.g 4
7.d odd 6 1 336.3.be.a 4
7.d odd 6 1 336.3.be.b yes 4
21.g even 6 1 1008.3.cd.f 4
21.g even 6 1 1008.3.cd.g 4
28.d even 2 1 2352.3.m.g 4
28.f even 6 1 336.3.be.a 4
28.f even 6 1 336.3.be.b yes 4
84.j odd 6 1 1008.3.cd.f 4
84.j odd 6 1 1008.3.cd.g 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
336.3.be.a 4 7.d odd 6 1
336.3.be.a 4 28.f even 6 1
336.3.be.b yes 4 7.d odd 6 1
336.3.be.b yes 4 28.f even 6 1
1008.3.cd.f 4 21.g even 6 1
1008.3.cd.f 4 84.j odd 6 1
1008.3.cd.g 4 21.g even 6 1
1008.3.cd.g 4 84.j odd 6 1
2352.3.m.g 4 7.b odd 2 1
2352.3.m.g 4 28.d even 2 1
2352.3.m.h 4 1.a even 1 1 trivial
2352.3.m.h 4 4.b odd 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{2} - T_{5} - 14$$ acting on $$S_{3}^{\mathrm{new}}(2352, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$( 3 + T^{2} )^{2}$$
$5$ $$( -14 - T + T^{2} )^{2}$$
$7$ $$T^{4}$$
$11$ $$3364 + 359 T^{2} + T^{4}$$
$13$ $$( 28 + 13 T + T^{2} )^{2}$$
$17$ $$( -128 - 20 T + T^{2} )^{2}$$
$19$ $$633616 + 1619 T^{2} + T^{4}$$
$23$ $$4096 + 176 T^{2} + T^{4}$$
$29$ $$( -1724 + T + T^{2} )^{2}$$
$31$ $$81 + 2718 T^{2} + T^{4}$$
$37$ $$( -608 - 19 T + T^{2} )^{2}$$
$41$ $$( -1424 - 2 T + T^{2} )^{2}$$
$43$ $$2829124 + 3839 T^{2} + T^{4}$$
$47$ $$( 588 + T^{2} )^{2}$$
$53$ $$( 6784 + 169 T + T^{2} )^{2}$$
$59$ $$1721344 + 2867 T^{2} + T^{4}$$
$61$ $$( 2076 - 96 T + T^{2} )^{2}$$
$67$ $$142884 + 783 T^{2} + T^{4}$$
$71$ $$56130064 + 15416 T^{2} + T^{4}$$
$73$ $$( 9106 + 191 T + T^{2} )^{2}$$
$79$ $$660969 + 7782 T^{2} + T^{4}$$
$83$ $$60124516 + 25559 T^{2} + T^{4}$$
$89$ $$( 3664 - 122 T + T^{2} )^{2}$$
$97$ $$( -13538 + 25 T + T^{2} )^{2}$$
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