# Properties

 Label 2352.2.b.d Level $2352$ Weight $2$ Character orbit 2352.b Analytic conductor $18.781$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2352 = 2^{4} \cdot 3 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2352.b (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: $$\Z[a_1, \ldots, a_{13}]$$ 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 $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - q^{3} + ( 2 - 4 \zeta_{6} ) q^{5} + q^{9} +O(q^{10})$$ $$q - q^{3} + ( 2 - 4 \zeta_{6} ) q^{5} + q^{9} + ( -2 + 4 \zeta_{6} ) q^{11} + ( -1 + 2 \zeta_{6} ) q^{13} + ( -2 + 4 \zeta_{6} ) q^{15} + 5 q^{19} + ( -4 + 8 \zeta_{6} ) q^{23} -7 q^{25} - q^{27} -5 q^{31} + ( 2 - 4 \zeta_{6} ) q^{33} -11 q^{37} + ( 1 - 2 \zeta_{6} ) q^{39} + ( -2 + 4 \zeta_{6} ) q^{41} + ( -5 + 10 \zeta_{6} ) q^{43} + ( 2 - 4 \zeta_{6} ) q^{45} -6 q^{47} + 12 q^{53} + 12 q^{55} -5 q^{57} + 12 q^{59} + ( -8 + 16 \zeta_{6} ) q^{61} + 6 q^{65} + ( 5 - 10 \zeta_{6} ) q^{67} + ( 4 - 8 \zeta_{6} ) q^{69} + ( -2 + 4 \zeta_{6} ) q^{71} + ( 3 - 6 \zeta_{6} ) q^{73} + 7 q^{75} + ( -7 + 14 \zeta_{6} ) q^{79} + q^{81} + 18 q^{83} + ( 4 - 8 \zeta_{6} ) q^{89} + 5 q^{93} + ( 10 - 20 \zeta_{6} ) q^{95} + ( 4 - 8 \zeta_{6} ) q^{97} + ( -2 + 4 \zeta_{6} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{3} + 2q^{9} + O(q^{10})$$ $$2q - 2q^{3} + 2q^{9} + 10q^{19} - 14q^{25} - 2q^{27} - 10q^{31} - 22q^{37} - 12q^{47} + 24q^{53} + 24q^{55} - 10q^{57} + 24q^{59} + 12q^{65} + 14q^{75} + 2q^{81} + 36q^{83} + 10q^{93} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1567.1
 0.5 + 0.866025i 0.5 − 0.866025i
0 −1.00000 0 3.46410i 0 0 0 1.00000 0
1567.2 0 −1.00000 0 3.46410i 0 0 0 1.00000 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
28.d 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.b.d 2
3.b odd 2 1 7056.2.b.k 2
4.b odd 2 1 2352.2.b.e 2
7.b odd 2 1 2352.2.b.e 2
7.c even 3 1 336.2.bl.h yes 2
7.c even 3 1 2352.2.bl.g 2
7.d odd 6 1 336.2.bl.d 2
7.d odd 6 1 2352.2.bl.a 2
12.b even 2 1 7056.2.b.d 2
21.c even 2 1 7056.2.b.d 2
21.g even 6 1 1008.2.cs.b 2
21.h odd 6 1 1008.2.cs.a 2
28.d even 2 1 inner 2352.2.b.d 2
28.f even 6 1 336.2.bl.h yes 2
28.f even 6 1 2352.2.bl.g 2
28.g odd 6 1 336.2.bl.d 2
28.g odd 6 1 2352.2.bl.a 2
56.j odd 6 1 1344.2.bl.e 2
56.k odd 6 1 1344.2.bl.e 2
56.m even 6 1 1344.2.bl.a 2
56.p even 6 1 1344.2.bl.a 2
84.h odd 2 1 7056.2.b.k 2
84.j odd 6 1 1008.2.cs.a 2
84.n even 6 1 1008.2.cs.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
336.2.bl.d 2 7.d odd 6 1
336.2.bl.d 2 28.g odd 6 1
336.2.bl.h yes 2 7.c even 3 1
336.2.bl.h yes 2 28.f even 6 1
1008.2.cs.a 2 21.h odd 6 1
1008.2.cs.a 2 84.j odd 6 1
1008.2.cs.b 2 21.g even 6 1
1008.2.cs.b 2 84.n even 6 1
1344.2.bl.a 2 56.m even 6 1
1344.2.bl.a 2 56.p even 6 1
1344.2.bl.e 2 56.j odd 6 1
1344.2.bl.e 2 56.k odd 6 1
2352.2.b.d 2 1.a even 1 1 trivial
2352.2.b.d 2 28.d even 2 1 inner
2352.2.b.e 2 4.b odd 2 1
2352.2.b.e 2 7.b odd 2 1
2352.2.bl.a 2 7.d odd 6 1
2352.2.bl.a 2 28.g odd 6 1
2352.2.bl.g 2 7.c even 3 1
2352.2.bl.g 2 28.f even 6 1
7056.2.b.d 2 12.b even 2 1
7056.2.b.d 2 21.c even 2 1
7056.2.b.k 2 3.b odd 2 1
7056.2.b.k 2 84.h odd 2 1

## Hecke kernels

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

 $$T_{5}^{2} + 12$$ $$T_{11}^{2} + 12$$ $$T_{13}^{2} + 3$$ $$T_{19} - 5$$ $$T_{31} + 5$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ $$( 1 + T )^{2}$$
$5$ $$1 + 2 T^{2} + 25 T^{4}$$
$7$ 1
$11$ $$1 - 10 T^{2} + 121 T^{4}$$
$13$ $$( 1 - 7 T + 13 T^{2} )( 1 + 7 T + 13 T^{2} )$$
$17$ $$( 1 - 17 T^{2} )^{2}$$
$19$ $$( 1 - 5 T + 19 T^{2} )^{2}$$
$23$ $$1 + 2 T^{2} + 529 T^{4}$$
$29$ $$( 1 + 29 T^{2} )^{2}$$
$31$ $$( 1 + 5 T + 31 T^{2} )^{2}$$
$37$ $$( 1 + 11 T + 37 T^{2} )^{2}$$
$41$ $$1 - 70 T^{2} + 1681 T^{4}$$
$43$ $$1 - 11 T^{2} + 1849 T^{4}$$
$47$ $$( 1 + 6 T + 47 T^{2} )^{2}$$
$53$ $$( 1 - 12 T + 53 T^{2} )^{2}$$
$59$ $$( 1 - 12 T + 59 T^{2} )^{2}$$
$61$ $$1 + 70 T^{2} + 3721 T^{4}$$
$67$ $$1 - 59 T^{2} + 4489 T^{4}$$
$71$ $$1 - 130 T^{2} + 5041 T^{4}$$
$73$ $$1 - 119 T^{2} + 5329 T^{4}$$
$79$ $$( 1 - 13 T + 79 T^{2} )( 1 + 13 T + 79 T^{2} )$$
$83$ $$( 1 - 18 T + 83 T^{2} )^{2}$$
$89$ $$1 - 130 T^{2} + 7921 T^{4}$$
$97$ $$1 - 146 T^{2} + 9409 T^{4}$$