# Properties

 Label 2352.2.h.d Level $2352$ Weight $2$ Character orbit 2352.h Analytic conductor $18.781$ Analytic rank $0$ Dimension $2$ CM discriminant -3 Inner twists $4$

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## 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: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Defining polynomial: $$x^{2} - x + 1$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 336) Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -1 + 2 \zeta_{6} ) q^{3} -3 q^{9} +O(q^{10})$$ $$q + ( -1 + 2 \zeta_{6} ) q^{3} -3 q^{9} + 5 q^{13} + ( 5 - 10 \zeta_{6} ) q^{19} + 5 q^{25} + ( 3 - 6 \zeta_{6} ) q^{27} + ( 1 - 2 \zeta_{6} ) q^{31} + 11 q^{37} + ( -5 + 10 \zeta_{6} ) q^{39} + ( -1 + 2 \zeta_{6} ) q^{43} + 15 q^{57} -14 q^{61} + ( -9 + 18 \zeta_{6} ) q^{67} + 17 q^{73} + ( -5 + 10 \zeta_{6} ) q^{75} + ( 3 - 6 \zeta_{6} ) q^{79} + 9 q^{81} + 3 q^{93} + 14 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 6 q^{9} + O(q^{10})$$ $$2 q - 6 q^{9} + 10 q^{13} + 10 q^{25} + 22 q^{37} + 30 q^{57} - 28 q^{61} + 34 q^{73} + 18 q^{81} + 6 q^{93} + 28 q^{97} + O(q^{100})$$

## 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
 0.5 − 0.866025i 0.5 + 0.866025i
0 1.73205i 0 0 0 0 0 −3.00000 0
2255.2 0 1.73205i 0 0 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
3.b odd 2 1 CM by $$\Q(\sqrt{-3})$$
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.d 2
3.b odd 2 1 CM 2352.2.h.d 2
4.b odd 2 1 inner 2352.2.h.d 2
7.b odd 2 1 2352.2.h.b 2
7.d odd 6 1 336.2.bj.a 2
7.d odd 6 1 336.2.bj.d yes 2
12.b even 2 1 inner 2352.2.h.d 2
21.c even 2 1 2352.2.h.b 2
21.g even 6 1 336.2.bj.a 2
21.g even 6 1 336.2.bj.d yes 2
28.d even 2 1 2352.2.h.b 2
28.f even 6 1 336.2.bj.a 2
28.f even 6 1 336.2.bj.d yes 2
84.h odd 2 1 2352.2.h.b 2
84.j odd 6 1 336.2.bj.a 2
84.j odd 6 1 336.2.bj.d yes 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
336.2.bj.a 2 7.d odd 6 1
336.2.bj.a 2 21.g even 6 1
336.2.bj.a 2 28.f even 6 1
336.2.bj.a 2 84.j odd 6 1
336.2.bj.d yes 2 7.d odd 6 1
336.2.bj.d yes 2 21.g even 6 1
336.2.bj.d yes 2 28.f even 6 1
336.2.bj.d yes 2 84.j odd 6 1
2352.2.h.b 2 7.b odd 2 1
2352.2.h.b 2 21.c even 2 1
2352.2.h.b 2 28.d even 2 1
2352.2.h.b 2 84.h odd 2 1
2352.2.h.d 2 1.a even 1 1 trivial
2352.2.h.d 2 3.b odd 2 1 CM
2352.2.h.d 2 4.b odd 2 1 inner
2352.2.h.d 2 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 $$S_{2}^{\mathrm{new}}(2352, [\chi])$$:

 $$T_{5}$$ $$T_{11}$$ $$T_{13} - 5$$ $$T_{47}$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$3 + T^{2}$$
$5$ $$T^{2}$$
$7$ $$T^{2}$$
$11$ $$T^{2}$$
$13$ $$( -5 + T )^{2}$$
$17$ $$T^{2}$$
$19$ $$75 + T^{2}$$
$23$ $$T^{2}$$
$29$ $$T^{2}$$
$31$ $$3 + T^{2}$$
$37$ $$( -11 + T )^{2}$$
$41$ $$T^{2}$$
$43$ $$3 + T^{2}$$
$47$ $$T^{2}$$
$53$ $$T^{2}$$
$59$ $$T^{2}$$
$61$ $$( 14 + T )^{2}$$
$67$ $$243 + T^{2}$$
$71$ $$T^{2}$$
$73$ $$( -17 + T )^{2}$$
$79$ $$27 + T^{2}$$
$83$ $$T^{2}$$
$89$ $$T^{2}$$
$97$ $$( -14 + T )^{2}$$
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