# Properties

 Label 2352.2.h.n Level $2352$ Weight $2$ Character orbit 2352.h Analytic conductor $18.781$ Analytic rank $0$ Dimension $8$ 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.h (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$18.7808145554$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: 8.0.8275904784.2 Defining polynomial: $$x^{8} - 3 x^{7} + 5 x^{6} - 6 x^{5} + 4 x^{4} - 18 x^{3} + 45 x^{2} - 81 x + 81$$ 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 $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{3} q^{3} -\beta_{7} q^{5} + ( -\beta_{2} + \beta_{7} ) q^{9} +O(q^{10})$$ $$q + \beta_{3} q^{3} -\beta_{7} q^{5} + ( -\beta_{2} + \beta_{7} ) q^{9} + ( \beta_{1} + \beta_{3} + \beta_{5} ) q^{11} + 2 q^{13} + ( -\beta_{1} + \beta_{5} - \beta_{6} ) q^{15} + ( \beta_{2} - \beta_{4} - 2 \beta_{7} ) q^{17} + ( \beta_{3} - \beta_{5} ) q^{19} + ( -\beta_{3} - \beta_{5} ) q^{23} + ( -4 - 2 \beta_{2} - 2 \beta_{4} ) q^{25} + ( \beta_{1} + \beta_{3} - \beta_{5} - 2 \beta_{6} ) q^{27} + ( -\beta_{2} + \beta_{4} - \beta_{7} ) q^{29} + \beta_{6} q^{31} + ( 2 - 2 \beta_{2} - \beta_{4} - \beta_{7} ) q^{33} + ( -2 + \beta_{2} + \beta_{4} ) q^{37} + 2 \beta_{3} q^{39} + ( 2 \beta_{2} - 2 \beta_{4} - 2 \beta_{7} ) q^{41} + ( -2 \beta_{3} + 2 \beta_{5} + 4 \beta_{6} ) q^{43} + ( 5 + \beta_{2} + 2 \beta_{4} + 2 \beta_{7} ) q^{45} + ( 3 \beta_{3} + 3 \beta_{5} ) q^{47} + ( -\beta_{1} - \beta_{3} + 4 \beta_{5} + 2 \beta_{6} ) q^{51} + ( -2 \beta_{2} + 2 \beta_{4} - \beta_{7} ) q^{53} + ( 4 \beta_{3} - 4 \beta_{5} + \beta_{6} ) q^{55} + ( -3 - \beta_{2} + \beta_{7} ) q^{57} + ( \beta_{1} - 3 \beta_{3} - 3 \beta_{5} ) q^{59} + ( -8 - \beta_{2} - \beta_{4} ) q^{61} -2 \beta_{7} q^{65} + ( -3 \beta_{3} + 3 \beta_{5} - 4 \beta_{6} ) q^{67} + ( -3 + \beta_{2} - \beta_{7} ) q^{69} + 2 \beta_{1} q^{71} + ( -4 - \beta_{2} - \beta_{4} ) q^{73} + ( 2 \beta_{1} - 2 \beta_{3} + 4 \beta_{5} - 4 \beta_{6} ) q^{75} -3 \beta_{6} q^{79} + ( -2 - 2 \beta_{2} + \beta_{4} - \beta_{7} ) q^{81} + \beta_{1} q^{83} + ( -10 - 3 \beta_{2} - 3 \beta_{4} ) q^{85} + ( -2 \beta_{1} + \beta_{3} - \beta_{5} - 5 \beta_{6} ) q^{87} + ( -3 \beta_{2} + 3 \beta_{4} + 4 \beta_{7} ) q^{89} + ( -1 - \beta_{4} ) q^{93} + ( -2 \beta_{1} + \beta_{3} + \beta_{5} ) q^{95} + ( 1 + \beta_{2} + \beta_{4} ) q^{97} + ( 4 \beta_{3} + 3 \beta_{5} - 6 \beta_{6} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q + 2 q^{9} + O(q^{10})$$ $$8 q + 2 q^{9} + 16 q^{13} - 24 q^{25} + 22 q^{33} - 20 q^{37} + 34 q^{45} - 22 q^{57} - 60 q^{61} - 26 q^{69} - 28 q^{73} - 14 q^{81} - 68 q^{85} - 6 q^{93} + 4 q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 3 x^{7} + 5 x^{6} - 6 x^{5} + 4 x^{4} - 18 x^{3} + 45 x^{2} - 81 x + 81$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$4 \nu^{7} - 3 \nu^{6} + 2 \nu^{5} - 6 \nu^{4} - 20 \nu^{3} - 90 \nu^{2} + 54 \nu - 81$$$$)/54$$ $$\beta_{2}$$ $$=$$ $$($$$$8 \nu^{7} - 3 \nu^{6} + 13 \nu^{5} + 3 \nu^{4} + 5 \nu^{3} - 87 \nu^{2} + 45 \nu - 270$$$$)/54$$ $$\beta_{3}$$ $$=$$ $$($$$$4 \nu^{7} - 6 \nu^{6} + 11 \nu^{5} - 3 \nu^{4} + 7 \nu^{3} - 57 \nu^{2} + 99 \nu - 189$$$$)/18$$ $$\beta_{4}$$ $$=$$ $$($$$$-4 \nu^{7} + 6 \nu^{6} - 11 \nu^{5} + 3 \nu^{4} - 7 \nu^{3} + 57 \nu^{2} - 63 \nu + 171$$$$)/18$$ $$\beta_{5}$$ $$=$$ $$($$$$-4 \nu^{7} + 5 \nu^{6} - 11 \nu^{5} + 7 \nu^{4} - 7 \nu^{3} + 53 \nu^{2} - 75 \nu + 180$$$$)/18$$ $$\beta_{6}$$ $$=$$ $$($$$$7 \nu^{7} - 9 \nu^{6} + 17 \nu^{5} - 9 \nu^{4} + 19 \nu^{3} - 105 \nu^{2} + 144 \nu - 297$$$$)/27$$ $$\beta_{7}$$ $$=$$ $$($$$$13 \nu^{7} - 15 \nu^{6} + 38 \nu^{5} - 12 \nu^{4} + 25 \nu^{3} - 192 \nu^{2} + 225 \nu - 621$$$$)/27$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{4} + \beta_{3} + 1$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$($$$$-\beta_{7} - \beta_{6} - 2 \beta_{5} + \beta_{3} + \beta_{2} - \beta_{1}$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$2 \beta_{6} + \beta_{5} - \beta_{3} - \beta_{1}$$ $$\nu^{4}$$ $$=$$ $$($$$$\beta_{7} - \beta_{6} + 7 \beta_{5} - \beta_{4} + 4 \beta_{3} + 2 \beta_{2} - \beta_{1} + 2$$$$)/2$$ $$\nu^{5}$$ $$=$$ $$($$$$6 \beta_{7} - 8 \beta_{6} - \beta_{5} + 2 \beta_{4} - 3 \beta_{2} - 2 \beta_{1} + 23$$$$)/2$$ $$\nu^{6}$$ $$=$$ $$4 \beta_{7} + 10 \beta_{4} + 2 \beta_{2} + 7$$ $$\nu^{7}$$ $$=$$ $$($$$$-18 \beta_{7} - 24 \beta_{5} - \beta_{4} + 5 \beta_{3} + 30 \beta_{2} - 6 \beta_{1} + 29$$$$)/2$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
2255.1
 −1.37009 + 1.05965i −1.37009 − 1.05965i 0.906034 − 1.47618i 0.906034 + 1.47618i 1.73142 − 0.0465589i 1.73142 + 0.0465589i 0.232633 − 1.71636i 0.232633 + 1.71636i
0 −1.60273 0.656712i 0 0.670944i 0 0 0 2.13746 + 2.10506i 0
2255.2 0 −1.60273 + 0.656712i 0 0.670944i 0 0 0 2.13746 2.10506i 0
2255.3 0 −0.825391 1.52274i 0 3.94333i 0 0 0 −1.63746 + 2.51371i 0
2255.4 0 −0.825391 + 1.52274i 0 3.94333i 0 0 0 −1.63746 2.51371i 0
2255.5 0 0.825391 1.52274i 0 3.94333i 0 0 0 −1.63746 2.51371i 0
2255.6 0 0.825391 + 1.52274i 0 3.94333i 0 0 0 −1.63746 + 2.51371i 0
2255.7 0 1.60273 0.656712i 0 0.670944i 0 0 0 2.13746 2.10506i 0
2255.8 0 1.60273 + 0.656712i 0 0.670944i 0 0 0 2.13746 + 2.10506i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 2255.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
4.b odd 2 1 inner
12.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2352.2.h.n 8
3.b odd 2 1 inner 2352.2.h.n 8
4.b odd 2 1 inner 2352.2.h.n 8
7.b odd 2 1 2352.2.h.m 8
7.c even 3 1 336.2.bj.e 8
7.c even 3 1 336.2.bj.g yes 8
12.b even 2 1 inner 2352.2.h.n 8
21.c even 2 1 2352.2.h.m 8
21.h odd 6 1 336.2.bj.e 8
21.h odd 6 1 336.2.bj.g yes 8
28.d even 2 1 2352.2.h.m 8
28.g odd 6 1 336.2.bj.e 8
28.g odd 6 1 336.2.bj.g yes 8
84.h odd 2 1 2352.2.h.m 8
84.n even 6 1 336.2.bj.e 8
84.n even 6 1 336.2.bj.g yes 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
336.2.bj.e 8 7.c even 3 1
336.2.bj.e 8 21.h odd 6 1
336.2.bj.e 8 28.g odd 6 1
336.2.bj.e 8 84.n even 6 1
336.2.bj.g yes 8 7.c even 3 1
336.2.bj.g yes 8 21.h odd 6 1
336.2.bj.g yes 8 28.g odd 6 1
336.2.bj.g yes 8 84.n even 6 1
2352.2.h.m 8 7.b odd 2 1
2352.2.h.m 8 21.c even 2 1
2352.2.h.m 8 28.d even 2 1
2352.2.h.m 8 84.h odd 2 1
2352.2.h.n 8 1.a even 1 1 trivial
2352.2.h.n 8 3.b odd 2 1 inner
2352.2.h.n 8 4.b odd 2 1 inner
2352.2.h.n 8 12.b even 2 1 inner

## Hecke kernels

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

 $$T_{5}^{4} + 16 T_{5}^{2} + 7$$ $$T_{11}^{4} - 40 T_{11}^{2} + 343$$ $$T_{13} - 2$$ $$T_{47}^{4} - 117 T_{47}^{2} + 2268$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$81 - 9 T^{2} + 4 T^{4} - T^{6} + T^{8}$$
$5$ $$( 7 + 16 T^{2} + T^{4} )^{2}$$
$7$ $$T^{8}$$
$11$ $$( 343 - 40 T^{2} + T^{4} )^{2}$$
$13$ $$( -2 + T )^{8}$$
$17$ $$( 448 + 43 T^{2} + T^{4} )^{2}$$
$19$ $$( 16 + 11 T^{2} + T^{4} )^{2}$$
$23$ $$( 28 - 13 T^{2} + T^{4} )^{2}$$
$29$ $$( 1792 + 85 T^{2} + T^{4} )^{2}$$
$31$ $$( 3 + T^{2} )^{4}$$
$37$ $$( -8 + 5 T + T^{2} )^{4}$$
$41$ $$( 448 + 100 T^{2} + T^{4} )^{2}$$
$43$ $$( 64 + 92 T^{2} + T^{4} )^{2}$$
$47$ $$( 2268 - 117 T^{2} + T^{4} )^{2}$$
$53$ $$( 12943 + 232 T^{2} + T^{4} )^{2}$$
$59$ $$( 5887 - 160 T^{2} + T^{4} )^{2}$$
$61$ $$( 42 + 15 T + T^{2} )^{4}$$
$67$ $$( 2304 + 267 T^{2} + T^{4} )^{2}$$
$71$ $$( 1792 - 124 T^{2} + T^{4} )^{2}$$
$73$ $$( -2 + 7 T + T^{2} )^{4}$$
$79$ $$( 27 + T^{2} )^{4}$$
$83$ $$( 112 - 31 T^{2} + T^{4} )^{2}$$
$89$ $$( 10108 + 247 T^{2} + T^{4} )^{2}$$
$97$ $$( -14 - T + T^{2} )^{4}$$