# Properties

 Label 2352.2.h.h Level $2352$ Weight $2$ Character orbit 2352.h 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.h (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$18.7808145554$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{8})$$ Defining polynomial: $$x^{4} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2$$ 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 primitive root of unity $$\zeta_{8}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -\zeta_{8} + \zeta_{8}^{2} + \zeta_{8}^{3} ) q^{3} + ( \zeta_{8} + \zeta_{8}^{3} ) q^{5} + ( 1 - 2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{9} +O(q^{10})$$ $$q + ( -\zeta_{8} + \zeta_{8}^{2} + \zeta_{8}^{3} ) q^{3} + ( \zeta_{8} + \zeta_{8}^{3} ) q^{5} + ( 1 - 2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{9} + ( \zeta_{8} - \zeta_{8}^{3} ) q^{11} -2 q^{13} + ( -\zeta_{8} - 2 \zeta_{8}^{2} + \zeta_{8}^{3} ) q^{15} + ( 5 \zeta_{8} + 5 \zeta_{8}^{3} ) q^{17} -4 \zeta_{8}^{2} q^{19} + ( -5 \zeta_{8} + 5 \zeta_{8}^{3} ) q^{23} + 3 q^{25} + ( \zeta_{8} + 5 \zeta_{8}^{2} - \zeta_{8}^{3} ) q^{27} + ( 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{29} -6 \zeta_{8}^{2} q^{31} + ( -2 + \zeta_{8} + \zeta_{8}^{3} ) q^{33} -4 q^{37} + ( 2 \zeta_{8} - 2 \zeta_{8}^{2} - 2 \zeta_{8}^{3} ) q^{39} + ( -\zeta_{8} - \zeta_{8}^{3} ) q^{41} + 4 \zeta_{8}^{2} q^{43} + ( 4 + \zeta_{8} + \zeta_{8}^{3} ) q^{45} + ( -6 \zeta_{8} + 6 \zeta_{8}^{3} ) q^{47} + ( -5 \zeta_{8} - 10 \zeta_{8}^{2} + 5 \zeta_{8}^{3} ) q^{51} + ( 8 \zeta_{8} + 8 \zeta_{8}^{3} ) q^{53} + 2 \zeta_{8}^{2} q^{55} + ( 4 + 4 \zeta_{8} + 4 \zeta_{8}^{3} ) q^{57} + ( -8 \zeta_{8} + 8 \zeta_{8}^{3} ) q^{59} + 6 q^{61} + ( -2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{65} + 12 \zeta_{8}^{2} q^{67} + ( 10 - 5 \zeta_{8} - 5 \zeta_{8}^{3} ) q^{69} + ( -5 \zeta_{8} + 5 \zeta_{8}^{3} ) q^{71} + 2 q^{73} + ( -3 \zeta_{8} + 3 \zeta_{8}^{2} + 3 \zeta_{8}^{3} ) q^{75} -12 \zeta_{8}^{2} q^{79} + ( -7 - 4 \zeta_{8} - 4 \zeta_{8}^{3} ) q^{81} + ( 10 \zeta_{8} - 10 \zeta_{8}^{3} ) q^{83} -10 q^{85} + ( -2 \zeta_{8} - 4 \zeta_{8}^{2} + 2 \zeta_{8}^{3} ) q^{87} + ( -7 \zeta_{8} - 7 \zeta_{8}^{3} ) q^{89} + ( 6 + 6 \zeta_{8} + 6 \zeta_{8}^{3} ) q^{93} + ( 4 \zeta_{8} - 4 \zeta_{8}^{3} ) q^{95} -14 q^{97} + ( \zeta_{8} - 4 \zeta_{8}^{2} - \zeta_{8}^{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 4q^{9} + O(q^{10})$$ $$4q + 4q^{9} - 8q^{13} + 12q^{25} - 8q^{33} - 16q^{37} + 16q^{45} + 16q^{57} + 24q^{61} + 40q^{69} + 8q^{73} - 28q^{81} - 40q^{85} + 24q^{93} - 56q^{97} + O(q^{100})$$

## 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
 0.707107 − 0.707107i 0.707107 + 0.707107i −0.707107 + 0.707107i −0.707107 − 0.707107i
0 −1.41421 1.00000i 0 1.41421i 0 0 0 1.00000 + 2.82843i 0
2255.2 0 −1.41421 + 1.00000i 0 1.41421i 0 0 0 1.00000 2.82843i 0
2255.3 0 1.41421 1.00000i 0 1.41421i 0 0 0 1.00000 2.82843i 0
2255.4 0 1.41421 + 1.00000i 0 1.41421i 0 0 0 1.00000 + 2.82843i 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
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.h 4
3.b odd 2 1 inner 2352.2.h.h 4
4.b odd 2 1 inner 2352.2.h.h 4
7.b odd 2 1 336.2.h.a 4
12.b even 2 1 inner 2352.2.h.h 4
21.c even 2 1 336.2.h.a 4
28.d even 2 1 336.2.h.a 4
56.e even 2 1 1344.2.h.c 4
56.h odd 2 1 1344.2.h.c 4
84.h odd 2 1 336.2.h.a 4
168.e odd 2 1 1344.2.h.c 4
168.i even 2 1 1344.2.h.c 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
336.2.h.a 4 7.b odd 2 1
336.2.h.a 4 21.c even 2 1
336.2.h.a 4 28.d even 2 1
336.2.h.a 4 84.h odd 2 1
1344.2.h.c 4 56.e even 2 1
1344.2.h.c 4 56.h odd 2 1
1344.2.h.c 4 168.e odd 2 1
1344.2.h.c 4 168.i even 2 1
2352.2.h.h 4 1.a even 1 1 trivial
2352.2.h.h 4 3.b odd 2 1 inner
2352.2.h.h 4 4.b odd 2 1 inner
2352.2.h.h 4 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 $$S_{2}^{\mathrm{new}}(2352, [\chi])$$:

 $$T_{5}^{2} + 2$$ $$T_{11}^{2} - 2$$ $$T_{13} + 2$$ $$T_{47}^{2} - 72$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$9 - 2 T^{2} + T^{4}$$
$5$ $$( 2 + T^{2} )^{2}$$
$7$ $$T^{4}$$
$11$ $$( -2 + T^{2} )^{2}$$
$13$ $$( 2 + T )^{4}$$
$17$ $$( 50 + T^{2} )^{2}$$
$19$ $$( 16 + T^{2} )^{2}$$
$23$ $$( -50 + T^{2} )^{2}$$
$29$ $$( 8 + T^{2} )^{2}$$
$31$ $$( 36 + T^{2} )^{2}$$
$37$ $$( 4 + T )^{4}$$
$41$ $$( 2 + T^{2} )^{2}$$
$43$ $$( 16 + T^{2} )^{2}$$
$47$ $$( -72 + T^{2} )^{2}$$
$53$ $$( 128 + T^{2} )^{2}$$
$59$ $$( -128 + T^{2} )^{2}$$
$61$ $$( -6 + T )^{4}$$
$67$ $$( 144 + T^{2} )^{2}$$
$71$ $$( -50 + T^{2} )^{2}$$
$73$ $$( -2 + T )^{4}$$
$79$ $$( 144 + T^{2} )^{2}$$
$83$ $$( -200 + T^{2} )^{2}$$
$89$ $$( 98 + T^{2} )^{2}$$
$97$ $$( 14 + T )^{4}$$