Properties

 Label 2352.2.q.j 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 168) 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 $$q + (\zeta_{6} - 1) q^{3} + 2 \zeta_{6} q^{5} - \zeta_{6} q^{9} + 2 q^{13} - 2 q^{15} + ( - 6 \zeta_{6} + 6) q^{17} + 4 \zeta_{6} q^{19} - 4 \zeta_{6} q^{23} + ( - \zeta_{6} + 1) q^{25} + q^{27} + 6 q^{29} + ( - 8 \zeta_{6} + 8) q^{31} + 10 \zeta_{6} q^{37} + (2 \zeta_{6} - 2) q^{39} + 10 q^{41} - 12 q^{43} + ( - 2 \zeta_{6} + 2) q^{45} + 8 \zeta_{6} q^{47} + 6 \zeta_{6} q^{51} + (6 \zeta_{6} - 6) q^{53} - 4 q^{57} + (4 \zeta_{6} - 4) q^{59} - 10 \zeta_{6} q^{61} + 4 \zeta_{6} q^{65} + ( - 12 \zeta_{6} + 12) q^{67} + 4 q^{69} - 4 q^{71} + ( - 2 \zeta_{6} + 2) q^{73} + \zeta_{6} q^{75} + 8 \zeta_{6} q^{79} + (\zeta_{6} - 1) q^{81} + 4 q^{83} + 12 q^{85} + (6 \zeta_{6} - 6) q^{87} + 6 \zeta_{6} q^{89} + 8 \zeta_{6} q^{93} + (8 \zeta_{6} - 8) q^{95} - 10 q^{97} +O(q^{100})$$ q + (z - 1) * q^3 + 2*z * q^5 - z * q^9 + 2 * q^13 - 2 * q^15 + (-6*z + 6) * q^17 + 4*z * q^19 - 4*z * q^23 + (-z + 1) * q^25 + q^27 + 6 * q^29 + (-8*z + 8) * q^31 + 10*z * q^37 + (2*z - 2) * q^39 + 10 * q^41 - 12 * q^43 + (-2*z + 2) * q^45 + 8*z * q^47 + 6*z * q^51 + (6*z - 6) * q^53 - 4 * q^57 + (4*z - 4) * q^59 - 10*z * q^61 + 4*z * q^65 + (-12*z + 12) * q^67 + 4 * q^69 - 4 * q^71 + (-2*z + 2) * q^73 + z * q^75 + 8*z * q^79 + (z - 1) * q^81 + 4 * q^83 + 12 * q^85 + (6*z - 6) * q^87 + 6*z * q^89 + 8*z * q^93 + (8*z - 8) * q^95 - 10 * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - q^{3} + 2 q^{5} - q^{9}+O(q^{10})$$ 2 * q - q^3 + 2 * q^5 - q^9 $$2 q - q^{3} + 2 q^{5} - q^{9} + 4 q^{13} - 4 q^{15} + 6 q^{17} + 4 q^{19} - 4 q^{23} + q^{25} + 2 q^{27} + 12 q^{29} + 8 q^{31} + 10 q^{37} - 2 q^{39} + 20 q^{41} - 24 q^{43} + 2 q^{45} + 8 q^{47} + 6 q^{51} - 6 q^{53} - 8 q^{57} - 4 q^{59} - 10 q^{61} + 4 q^{65} + 12 q^{67} + 8 q^{69} - 8 q^{71} + 2 q^{73} + q^{75} + 8 q^{79} - q^{81} + 8 q^{83} + 24 q^{85} - 6 q^{87} + 6 q^{89} + 8 q^{93} - 8 q^{95} - 20 q^{97}+O(q^{100})$$ 2 * q - q^3 + 2 * q^5 - q^9 + 4 * q^13 - 4 * q^15 + 6 * q^17 + 4 * q^19 - 4 * q^23 + q^25 + 2 * q^27 + 12 * q^29 + 8 * q^31 + 10 * q^37 - 2 * q^39 + 20 * q^41 - 24 * q^43 + 2 * q^45 + 8 * q^47 + 6 * q^51 - 6 * q^53 - 8 * q^57 - 4 * q^59 - 10 * q^61 + 4 * q^65 + 12 * q^67 + 8 * q^69 - 8 * q^71 + 2 * q^73 + q^75 + 8 * q^79 - q^81 + 8 * q^83 + 24 * q^85 - 6 * q^87 + 6 * q^89 + 8 * q^93 - 8 * q^95 - 20 * 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 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.j 2
4.b odd 2 1 1176.2.q.j 2
7.b odd 2 1 2352.2.q.o 2
7.c even 3 1 2352.2.a.q 1
7.c even 3 1 inner 2352.2.q.j 2
7.d odd 6 1 336.2.a.c 1
7.d odd 6 1 2352.2.q.o 2
12.b even 2 1 3528.2.s.h 2
21.g even 6 1 1008.2.a.e 1
21.h odd 6 1 7056.2.a.br 1
28.d even 2 1 1176.2.q.b 2
28.f even 6 1 168.2.a.b 1
28.f even 6 1 1176.2.q.b 2
28.g odd 6 1 1176.2.a.a 1
28.g odd 6 1 1176.2.q.j 2
35.i odd 6 1 8400.2.a.bx 1
56.j odd 6 1 1344.2.a.n 1
56.k odd 6 1 9408.2.a.cy 1
56.m even 6 1 1344.2.a.c 1
56.p even 6 1 9408.2.a.bc 1
84.h odd 2 1 3528.2.s.v 2
84.j odd 6 1 504.2.a.b 1
84.j odd 6 1 3528.2.s.v 2
84.n even 6 1 3528.2.a.w 1
84.n even 6 1 3528.2.s.h 2
112.v even 12 2 5376.2.c.bd 2
112.x odd 12 2 5376.2.c.f 2
140.s even 6 1 4200.2.a.i 1
140.x odd 12 2 4200.2.t.m 2
168.ba even 6 1 4032.2.a.bj 1
168.be odd 6 1 4032.2.a.be 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
168.2.a.b 1 28.f even 6 1
336.2.a.c 1 7.d odd 6 1
504.2.a.b 1 84.j odd 6 1
1008.2.a.e 1 21.g even 6 1
1176.2.a.a 1 28.g odd 6 1
1176.2.q.b 2 28.d even 2 1
1176.2.q.b 2 28.f even 6 1
1176.2.q.j 2 4.b odd 2 1
1176.2.q.j 2 28.g odd 6 1
1344.2.a.c 1 56.m even 6 1
1344.2.a.n 1 56.j odd 6 1
2352.2.a.q 1 7.c even 3 1
2352.2.q.j 2 1.a even 1 1 trivial
2352.2.q.j 2 7.c even 3 1 inner
2352.2.q.o 2 7.b odd 2 1
2352.2.q.o 2 7.d odd 6 1
3528.2.a.w 1 84.n even 6 1
3528.2.s.h 2 12.b even 2 1
3528.2.s.h 2 84.n even 6 1
3528.2.s.v 2 84.h odd 2 1
3528.2.s.v 2 84.j odd 6 1
4032.2.a.be 1 168.be odd 6 1
4032.2.a.bj 1 168.ba even 6 1
4200.2.a.i 1 140.s even 6 1
4200.2.t.m 2 140.x odd 12 2
5376.2.c.f 2 112.x odd 12 2
5376.2.c.bd 2 112.v even 12 2
7056.2.a.br 1 21.h odd 6 1
8400.2.a.bx 1 35.i odd 6 1
9408.2.a.bc 1 56.p even 6 1
9408.2.a.cy 1 56.k odd 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 $$T_{11}$$ T11 $$T_{13} - 2$$ T13 - 2 $$T_{17}^{2} - 6T_{17} + 36$$ T17^2 - 6*T17 + 36

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} + T + 1$$
$5$ $$T^{2} - 2T + 4$$
$7$ $$T^{2}$$
$11$ $$T^{2}$$
$13$ $$(T - 2)^{2}$$
$17$ $$T^{2} - 6T + 36$$
$19$ $$T^{2} - 4T + 16$$
$23$ $$T^{2} + 4T + 16$$
$29$ $$(T - 6)^{2}$$
$31$ $$T^{2} - 8T + 64$$
$37$ $$T^{2} - 10T + 100$$
$41$ $$(T - 10)^{2}$$
$43$ $$(T + 12)^{2}$$
$47$ $$T^{2} - 8T + 64$$
$53$ $$T^{2} + 6T + 36$$
$59$ $$T^{2} + 4T + 16$$
$61$ $$T^{2} + 10T + 100$$
$67$ $$T^{2} - 12T + 144$$
$71$ $$(T + 4)^{2}$$
$73$ $$T^{2} - 2T + 4$$
$79$ $$T^{2} - 8T + 64$$
$83$ $$(T - 4)^{2}$$
$89$ $$T^{2} - 6T + 36$$
$97$ $$(T + 10)^{2}$$