# Properties

 Label 2352.2.b.f 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: $$\Z[a_1, \ldots, a_{5}]$$ 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.

 $$f(q)$$ $$=$$ $$q + q^{3} + ( 1 - 2 \zeta_{6} ) q^{5} + q^{9} +O(q^{10})$$ $$q + q^{3} + ( 1 - 2 \zeta_{6} ) q^{5} + q^{9} + ( -3 + 6 \zeta_{6} ) q^{11} + ( 4 - 8 \zeta_{6} ) q^{13} + ( 1 - 2 \zeta_{6} ) q^{15} + ( 2 - 4 \zeta_{6} ) q^{17} -2 q^{19} + ( 4 - 8 \zeta_{6} ) q^{23} + 2 q^{25} + q^{27} -9 q^{29} - q^{31} + ( -3 + 6 \zeta_{6} ) q^{33} -2 q^{37} + ( 4 - 8 \zeta_{6} ) q^{39} + ( -2 + 4 \zeta_{6} ) q^{41} + ( 2 - 4 \zeta_{6} ) q^{43} + ( 1 - 2 \zeta_{6} ) q^{45} + ( 2 - 4 \zeta_{6} ) q^{51} + 9 q^{53} + 9 q^{55} -2 q^{57} + 3 q^{59} + ( 4 - 8 \zeta_{6} ) q^{61} -12 q^{65} + ( 4 - 8 \zeta_{6} ) q^{69} + ( 4 - 8 \zeta_{6} ) q^{71} + ( 4 - 8 \zeta_{6} ) q^{73} + 2 q^{75} + ( -1 + 2 \zeta_{6} ) q^{79} + q^{81} + 15 q^{83} -6 q^{85} -9 q^{87} + ( 6 - 12 \zeta_{6} ) q^{89} - q^{93} + ( -2 + 4 \zeta_{6} ) q^{95} + ( 5 - 10 \zeta_{6} ) q^{97} + ( -3 + 6 \zeta_{6} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 2q^{3} + 2q^{9} + O(q^{10})$$ $$2q + 2q^{3} + 2q^{9} - 4q^{19} + 4q^{25} + 2q^{27} - 18q^{29} - 2q^{31} - 4q^{37} + 18q^{53} + 18q^{55} - 4q^{57} + 6q^{59} - 24q^{65} + 4q^{75} + 2q^{81} + 30q^{83} - 12q^{85} - 18q^{87} - 2q^{93} + O(q^{100})$$

## 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 1.73205i 0 0 0 1.00000 0
1567.2 0 1.00000 0 1.73205i 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.f 2
3.b odd 2 1 7056.2.b.f 2
4.b odd 2 1 2352.2.b.b 2
7.b odd 2 1 2352.2.b.b 2
7.c even 3 1 336.2.bl.b 2
7.c even 3 1 2352.2.bl.e 2
7.d odd 6 1 336.2.bl.f yes 2
7.d odd 6 1 2352.2.bl.k 2
12.b even 2 1 7056.2.b.j 2
21.c even 2 1 7056.2.b.j 2
21.g even 6 1 1008.2.cs.l 2
21.h odd 6 1 1008.2.cs.k 2
28.d even 2 1 inner 2352.2.b.f 2
28.f even 6 1 336.2.bl.b 2
28.f even 6 1 2352.2.bl.e 2
28.g odd 6 1 336.2.bl.f yes 2
28.g odd 6 1 2352.2.bl.k 2
56.j odd 6 1 1344.2.bl.c 2
56.k odd 6 1 1344.2.bl.c 2
56.m even 6 1 1344.2.bl.g 2
56.p even 6 1 1344.2.bl.g 2
84.h odd 2 1 7056.2.b.f 2
84.j odd 6 1 1008.2.cs.k 2
84.n even 6 1 1008.2.cs.l 2

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

## Hecke kernels

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

 $$T_{5}^{2} + 3$$ $$T_{11}^{2} + 27$$ $$T_{13}^{2} + 48$$ $$T_{19} + 2$$ $$T_{31} + 1$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$( -1 + T )^{2}$$
$5$ $$3 + T^{2}$$
$7$ $$T^{2}$$
$11$ $$27 + T^{2}$$
$13$ $$48 + T^{2}$$
$17$ $$12 + T^{2}$$
$19$ $$( 2 + T )^{2}$$
$23$ $$48 + T^{2}$$
$29$ $$( 9 + T )^{2}$$
$31$ $$( 1 + T )^{2}$$
$37$ $$( 2 + T )^{2}$$
$41$ $$12 + T^{2}$$
$43$ $$12 + T^{2}$$
$47$ $$T^{2}$$
$53$ $$( -9 + T )^{2}$$
$59$ $$( -3 + T )^{2}$$
$61$ $$48 + T^{2}$$
$67$ $$T^{2}$$
$71$ $$48 + T^{2}$$
$73$ $$48 + T^{2}$$
$79$ $$3 + T^{2}$$
$83$ $$( -15 + T )^{2}$$
$89$ $$108 + T^{2}$$
$97$ $$75 + T^{2}$$