# Properties

 Label 2352.2.q.n 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 42) 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.

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

## 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 −1.00000 + 1.73205i 0 0 0 −0.500000 + 0.866025i 0
1537.1 0 0.500000 0.866025i 0 −1.00000 1.73205i 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.n 2
4.b odd 2 1 294.2.e.a 2
7.b odd 2 1 2352.2.q.i 2
7.c even 3 1 2352.2.a.l 1
7.c even 3 1 inner 2352.2.q.n 2
7.d odd 6 1 336.2.a.d 1
7.d odd 6 1 2352.2.q.i 2
12.b even 2 1 882.2.g.j 2
21.g even 6 1 1008.2.a.j 1
21.h odd 6 1 7056.2.a.k 1
28.d even 2 1 294.2.e.c 2
28.f even 6 1 42.2.a.a 1
28.f even 6 1 294.2.e.c 2
28.g odd 6 1 294.2.a.g 1
28.g odd 6 1 294.2.e.a 2
35.i odd 6 1 8400.2.a.k 1
56.j odd 6 1 1344.2.a.i 1
56.k odd 6 1 9408.2.a.n 1
56.m even 6 1 1344.2.a.q 1
56.p even 6 1 9408.2.a.bw 1
84.h odd 2 1 882.2.g.h 2
84.j odd 6 1 126.2.a.a 1
84.j odd 6 1 882.2.g.h 2
84.n even 6 1 882.2.a.b 1
84.n even 6 1 882.2.g.j 2
112.v even 12 2 5376.2.c.bc 2
112.x odd 12 2 5376.2.c.e 2
140.p odd 6 1 7350.2.a.f 1
140.s even 6 1 1050.2.a.i 1
140.x odd 12 2 1050.2.g.a 2
168.ba even 6 1 4032.2.a.m 1
168.be odd 6 1 4032.2.a.e 1
252.n even 6 1 1134.2.f.g 2
252.r odd 6 1 1134.2.f.j 2
252.bj even 6 1 1134.2.f.g 2
252.bn odd 6 1 1134.2.f.j 2
308.m odd 6 1 5082.2.a.d 1
364.x even 6 1 7098.2.a.f 1
420.be odd 6 1 3150.2.a.bo 1
420.br even 12 2 3150.2.g.r 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
42.2.a.a 1 28.f even 6 1
126.2.a.a 1 84.j odd 6 1
294.2.a.g 1 28.g odd 6 1
294.2.e.a 2 4.b odd 2 1
294.2.e.a 2 28.g odd 6 1
294.2.e.c 2 28.d even 2 1
294.2.e.c 2 28.f even 6 1
336.2.a.d 1 7.d odd 6 1
882.2.a.b 1 84.n even 6 1
882.2.g.h 2 84.h odd 2 1
882.2.g.h 2 84.j odd 6 1
882.2.g.j 2 12.b even 2 1
882.2.g.j 2 84.n even 6 1
1008.2.a.j 1 21.g even 6 1
1050.2.a.i 1 140.s even 6 1
1050.2.g.a 2 140.x odd 12 2
1134.2.f.g 2 252.n even 6 1
1134.2.f.g 2 252.bj even 6 1
1134.2.f.j 2 252.r odd 6 1
1134.2.f.j 2 252.bn odd 6 1
1344.2.a.i 1 56.j odd 6 1
1344.2.a.q 1 56.m even 6 1
2352.2.a.l 1 7.c even 3 1
2352.2.q.i 2 7.b odd 2 1
2352.2.q.i 2 7.d odd 6 1
2352.2.q.n 2 1.a even 1 1 trivial
2352.2.q.n 2 7.c even 3 1 inner
3150.2.a.bo 1 420.be odd 6 1
3150.2.g.r 2 420.br even 12 2
4032.2.a.e 1 168.be odd 6 1
4032.2.a.m 1 168.ba even 6 1
5082.2.a.d 1 308.m odd 6 1
5376.2.c.e 2 112.x odd 12 2
5376.2.c.bc 2 112.v even 12 2
7056.2.a.k 1 21.h odd 6 1
7098.2.a.f 1 364.x even 6 1
7350.2.a.f 1 140.p odd 6 1
8400.2.a.k 1 35.i odd 6 1
9408.2.a.n 1 56.k odd 6 1
9408.2.a.bw 1 56.p even 6 1

## Hecke kernels

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

 $$T_{5}^{2} + 2T_{5} + 4$$ T5^2 + 2*T5 + 4 $$T_{11}^{2} + 4T_{11} + 16$$ T11^2 + 4*T11 + 16 $$T_{13} + 6$$ T13 + 6 $$T_{17}^{2} - 2T_{17} + 4$$ T17^2 - 2*T17 + 4

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} - T + 1$$
$5$ $$T^{2} + 2T + 4$$
$7$ $$T^{2}$$
$11$ $$T^{2} + 4T + 16$$
$13$ $$(T + 6)^{2}$$
$17$ $$T^{2} - 2T + 4$$
$19$ $$T^{2} - 4T + 16$$
$23$ $$T^{2} - 8T + 64$$
$29$ $$(T + 2)^{2}$$
$31$ $$T^{2}$$
$37$ $$T^{2} - 10T + 100$$
$41$ $$(T - 6)^{2}$$
$43$ $$(T - 4)^{2}$$
$47$ $$T^{2}$$
$53$ $$T^{2} + 6T + 36$$
$59$ $$T^{2} + 4T + 16$$
$61$ $$T^{2} - 6T + 36$$
$67$ $$T^{2} - 4T + 16$$
$71$ $$(T + 8)^{2}$$
$73$ $$T^{2} - 10T + 100$$
$79$ $$T^{2}$$
$83$ $$(T + 4)^{2}$$
$89$ $$T^{2} + 6T + 36$$
$97$ $$(T - 14)^{2}$$