# Properties

 Label 2352.2.q.t Level $2352$ Weight $2$ Character orbit 2352.q 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.q (of order $$3$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$18.7808145554$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Defining polynomial: $$x^{2} - x + 1$$ x^2 - x + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 1176) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + ( - \zeta_{6} + 1) q^{3} - \zeta_{6} q^{9} +O(q^{10})$$ q + (-z + 1) * q^3 - z * q^9 $$q + ( - \zeta_{6} + 1) q^{3} - \zeta_{6} q^{9} + 4 q^{13} + (4 \zeta_{6} - 4) q^{17} - 4 \zeta_{6} q^{19} + 4 \zeta_{6} q^{23} + ( - 5 \zeta_{6} + 5) q^{25} - q^{27} + 2 q^{29} + ( - 8 \zeta_{6} + 8) q^{31} + 6 \zeta_{6} q^{37} + ( - 4 \zeta_{6} + 4) q^{39} + 12 q^{41} - 4 q^{43} - 8 \zeta_{6} q^{47} + 4 \zeta_{6} q^{51} + (6 \zeta_{6} - 6) q^{53} - 4 q^{57} + ( - 12 \zeta_{6} + 12) q^{59} - 4 \zeta_{6} q^{61} + (4 \zeta_{6} - 4) q^{67} + 4 q^{69} + 12 q^{71} + (8 \zeta_{6} - 8) q^{73} - 5 \zeta_{6} q^{75} - 16 \zeta_{6} q^{79} + (\zeta_{6} - 1) q^{81} + 4 q^{83} + ( - 2 \zeta_{6} + 2) q^{87} - 4 \zeta_{6} q^{89} - 8 \zeta_{6} q^{93} + 16 q^{97} +O(q^{100})$$ q + (-z + 1) * q^3 - z * q^9 + 4 * q^13 + (4*z - 4) * q^17 - 4*z * q^19 + 4*z * q^23 + (-5*z + 5) * q^25 - q^27 + 2 * q^29 + (-8*z + 8) * q^31 + 6*z * q^37 + (-4*z + 4) * q^39 + 12 * q^41 - 4 * q^43 - 8*z * q^47 + 4*z * q^51 + (6*z - 6) * q^53 - 4 * q^57 + (-12*z + 12) * q^59 - 4*z * q^61 + (4*z - 4) * q^67 + 4 * q^69 + 12 * q^71 + (8*z - 8) * q^73 - 5*z * q^75 - 16*z * q^79 + (z - 1) * q^81 + 4 * q^83 + (-2*z + 2) * q^87 - 4*z * q^89 - 8*z * q^93 + 16 * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + q^{3} - q^{9}+O(q^{10})$$ 2 * q + q^3 - q^9 $$2 q + q^{3} - q^{9} + 8 q^{13} - 4 q^{17} - 4 q^{19} + 4 q^{23} + 5 q^{25} - 2 q^{27} + 4 q^{29} + 8 q^{31} + 6 q^{37} + 4 q^{39} + 24 q^{41} - 8 q^{43} - 8 q^{47} + 4 q^{51} - 6 q^{53} - 8 q^{57} + 12 q^{59} - 4 q^{61} - 4 q^{67} + 8 q^{69} + 24 q^{71} - 8 q^{73} - 5 q^{75} - 16 q^{79} - q^{81} + 8 q^{83} + 2 q^{87} - 4 q^{89} - 8 q^{93} + 32 q^{97}+O(q^{100})$$ 2 * q + q^3 - q^9 + 8 * q^13 - 4 * q^17 - 4 * q^19 + 4 * q^23 + 5 * q^25 - 2 * q^27 + 4 * q^29 + 8 * q^31 + 6 * q^37 + 4 * q^39 + 24 * q^41 - 8 * q^43 - 8 * q^47 + 4 * q^51 - 6 * q^53 - 8 * q^57 + 12 * q^59 - 4 * q^61 - 4 * q^67 + 8 * q^69 + 24 * q^71 - 8 * q^73 - 5 * q^75 - 16 * q^79 - q^81 + 8 * q^83 + 2 * q^87 - 4 * q^89 - 8 * q^93 + 32 * q^97

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/2352\mathbb{Z}\right)^\times$$.

 $$n$$ $$785$$ $$1471$$ $$1765$$ $$2257$$ $$\chi(n)$$ $$1$$ $$1$$ $$1$$ $$-\zeta_{6}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
961.1
 0.5 − 0.866025i 0.5 + 0.866025i
0 0.500000 + 0.866025i 0 0 0 0 0 −0.500000 + 0.866025i 0
1537.1 0 0.500000 0.866025i 0 0 0 0 0 −0.500000 0.866025i 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
7.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2352.2.q.t 2
4.b odd 2 1 1176.2.q.c 2
7.b odd 2 1 2352.2.q.h 2
7.c even 3 1 2352.2.a.h 1
7.c even 3 1 inner 2352.2.q.t 2
7.d odd 6 1 2352.2.a.r 1
7.d odd 6 1 2352.2.q.h 2
12.b even 2 1 3528.2.s.n 2
21.g even 6 1 7056.2.a.ba 1
21.h odd 6 1 7056.2.a.bc 1
28.d even 2 1 1176.2.q.h 2
28.f even 6 1 1176.2.a.b 1
28.f even 6 1 1176.2.q.h 2
28.g odd 6 1 1176.2.a.h yes 1
28.g odd 6 1 1176.2.q.c 2
56.j odd 6 1 9408.2.a.v 1
56.k odd 6 1 9408.2.a.u 1
56.m even 6 1 9408.2.a.cl 1
56.p even 6 1 9408.2.a.ck 1
84.h odd 2 1 3528.2.s.m 2
84.j odd 6 1 3528.2.a.m 1
84.j odd 6 1 3528.2.s.m 2
84.n even 6 1 3528.2.a.n 1
84.n even 6 1 3528.2.s.n 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1176.2.a.b 1 28.f even 6 1
1176.2.a.h yes 1 28.g odd 6 1
1176.2.q.c 2 4.b odd 2 1
1176.2.q.c 2 28.g odd 6 1
1176.2.q.h 2 28.d even 2 1
1176.2.q.h 2 28.f even 6 1
2352.2.a.h 1 7.c even 3 1
2352.2.a.r 1 7.d odd 6 1
2352.2.q.h 2 7.b odd 2 1
2352.2.q.h 2 7.d odd 6 1
2352.2.q.t 2 1.a even 1 1 trivial
2352.2.q.t 2 7.c even 3 1 inner
3528.2.a.m 1 84.j odd 6 1
3528.2.a.n 1 84.n even 6 1
3528.2.s.m 2 84.h odd 2 1
3528.2.s.m 2 84.j odd 6 1
3528.2.s.n 2 12.b even 2 1
3528.2.s.n 2 84.n even 6 1
7056.2.a.ba 1 21.g even 6 1
7056.2.a.bc 1 21.h odd 6 1
9408.2.a.u 1 56.k odd 6 1
9408.2.a.v 1 56.j odd 6 1
9408.2.a.ck 1 56.p even 6 1
9408.2.a.cl 1 56.m even 6 1

## Hecke kernels

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

 $$T_{5}$$ T5 $$T_{11}$$ T11 $$T_{13} - 4$$ T13 - 4 $$T_{17}^{2} + 4T_{17} + 16$$ T17^2 + 4*T17 + 16

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} - T + 1$$
$5$ $$T^{2}$$
$7$ $$T^{2}$$
$11$ $$T^{2}$$
$13$ $$(T - 4)^{2}$$
$17$ $$T^{2} + 4T + 16$$
$19$ $$T^{2} + 4T + 16$$
$23$ $$T^{2} - 4T + 16$$
$29$ $$(T - 2)^{2}$$
$31$ $$T^{2} - 8T + 64$$
$37$ $$T^{2} - 6T + 36$$
$41$ $$(T - 12)^{2}$$
$43$ $$(T + 4)^{2}$$
$47$ $$T^{2} + 8T + 64$$
$53$ $$T^{2} + 6T + 36$$
$59$ $$T^{2} - 12T + 144$$
$61$ $$T^{2} + 4T + 16$$
$67$ $$T^{2} + 4T + 16$$
$71$ $$(T - 12)^{2}$$
$73$ $$T^{2} + 8T + 64$$
$79$ $$T^{2} + 16T + 256$$
$83$ $$(T - 4)^{2}$$
$89$ $$T^{2} + 4T + 16$$
$97$ $$(T - 16)^{2}$$