# Properties

 Label 2352.2.q.bg Level $2352$ Weight $2$ Character orbit 2352.q Analytic conductor $18.781$ Analytic rank $0$ Dimension $4$ 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: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\Q(\sqrt{2}, \sqrt{-3})$$ Defining polynomial: $$x^{4} + 2x^{2} + 4$$ x^4 + 2*x^2 + 4 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ 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 basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\beta_{2} + 1) q^{3} + ( - \beta_{3} - 2 \beta_{2} - \beta_1) q^{5} + \beta_{2} q^{9}+O(q^{10})$$ q + (b2 + 1) * q^3 + (-b3 - 2*b2 - b1) * q^5 + b2 * q^9 $$q + (\beta_{2} + 1) q^{3} + ( - \beta_{3} - 2 \beta_{2} - \beta_1) q^{5} + \beta_{2} q^{9} + ( - 2 \beta_{2} + 2 \beta_1 - 2) q^{11} - 3 \beta_{3} q^{13} + ( - \beta_{3} + 2) q^{15} + (6 \beta_{2} + \beta_1 + 6) q^{17} + (2 \beta_{3} + 4 \beta_{2} + 2 \beta_1) q^{19} + (2 \beta_{3} + 2 \beta_{2} + 2 \beta_1) q^{23} + ( - \beta_{2} - 4 \beta_1 - 1) q^{25} - q^{27} - 2 \beta_{3} q^{29} + 2 \beta_1 q^{31} + (2 \beta_{3} - 2 \beta_{2} + 2 \beta_1) q^{33} + ( - 4 \beta_{3} + 4 \beta_{2} - 4 \beta_1) q^{37} + 3 \beta_1 q^{39} + (3 \beta_{3} - 6) q^{41} - 8 \beta_{3} q^{43} + (2 \beta_{2} + \beta_1 + 2) q^{45} + ( - 6 \beta_{3} + 4 \beta_{2} - 6 \beta_1) q^{47} + (\beta_{3} + 6 \beta_{2} + \beta_1) q^{51} + (2 \beta_{2} + 2) q^{53} - 2 \beta_{3} q^{55} + (2 \beta_{3} - 4) q^{57} - 6 \beta_1 q^{59} + ( - 5 \beta_{3} - 4 \beta_{2} - 5 \beta_1) q^{61} + ( - 6 \beta_{3} - 6 \beta_{2} - 6 \beta_1) q^{65} + 8 \beta_1 q^{67} + (2 \beta_{3} - 2) q^{69} + ( - 6 \beta_{3} + 2) q^{71} + (12 \beta_{2} - 3 \beta_1 + 12) q^{73} + ( - 4 \beta_{3} - \beta_{2} - 4 \beta_1) q^{75} + ( - 4 \beta_{3} - 8 \beta_{2} - 4 \beta_1) q^{79} + ( - \beta_{2} - 1) q^{81} - 4 q^{83} + ( - 8 \beta_{3} + 14) q^{85} + 2 \beta_1 q^{87} + (3 \beta_{3} - 10 \beta_{2} + 3 \beta_1) q^{89} + (2 \beta_{3} + 2 \beta_1) q^{93} + (12 \beta_{2} + 8 \beta_1 + 12) q^{95} + ( - 3 \beta_{3} - 4) q^{97} + (2 \beta_{3} + 2) q^{99}+O(q^{100})$$ q + (b2 + 1) * q^3 + (-b3 - 2*b2 - b1) * q^5 + b2 * q^9 + (-2*b2 + 2*b1 - 2) * q^11 - 3*b3 * q^13 + (-b3 + 2) * q^15 + (6*b2 + b1 + 6) * q^17 + (2*b3 + 4*b2 + 2*b1) * q^19 + (2*b3 + 2*b2 + 2*b1) * q^23 + (-b2 - 4*b1 - 1) * q^25 - q^27 - 2*b3 * q^29 + 2*b1 * q^31 + (2*b3 - 2*b2 + 2*b1) * q^33 + (-4*b3 + 4*b2 - 4*b1) * q^37 + 3*b1 * q^39 + (3*b3 - 6) * q^41 - 8*b3 * q^43 + (2*b2 + b1 + 2) * q^45 + (-6*b3 + 4*b2 - 6*b1) * q^47 + (b3 + 6*b2 + b1) * q^51 + (2*b2 + 2) * q^53 - 2*b3 * q^55 + (2*b3 - 4) * q^57 - 6*b1 * q^59 + (-5*b3 - 4*b2 - 5*b1) * q^61 + (-6*b3 - 6*b2 - 6*b1) * q^65 + 8*b1 * q^67 + (2*b3 - 2) * q^69 + (-6*b3 + 2) * q^71 + (12*b2 - 3*b1 + 12) * q^73 + (-4*b3 - b2 - 4*b1) * q^75 + (-4*b3 - 8*b2 - 4*b1) * q^79 + (-b2 - 1) * q^81 - 4 * q^83 + (-8*b3 + 14) * q^85 + 2*b1 * q^87 + (3*b3 - 10*b2 + 3*b1) * q^89 + (2*b3 + 2*b1) * q^93 + (12*b2 + 8*b1 + 12) * q^95 + (-3*b3 - 4) * q^97 + (2*b3 + 2) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 2 q^{3} + 4 q^{5} - 2 q^{9}+O(q^{10})$$ 4 * q + 2 * q^3 + 4 * q^5 - 2 * q^9 $$4 q + 2 q^{3} + 4 q^{5} - 2 q^{9} - 4 q^{11} + 8 q^{15} + 12 q^{17} - 8 q^{19} - 4 q^{23} - 2 q^{25} - 4 q^{27} + 4 q^{33} - 8 q^{37} - 24 q^{41} + 4 q^{45} - 8 q^{47} - 12 q^{51} + 4 q^{53} - 16 q^{57} + 8 q^{61} + 12 q^{65} - 8 q^{69} + 8 q^{71} + 24 q^{73} + 2 q^{75} + 16 q^{79} - 2 q^{81} - 16 q^{83} + 56 q^{85} + 20 q^{89} + 24 q^{95} - 16 q^{97} + 8 q^{99}+O(q^{100})$$ 4 * q + 2 * q^3 + 4 * q^5 - 2 * q^9 - 4 * q^11 + 8 * q^15 + 12 * q^17 - 8 * q^19 - 4 * q^23 - 2 * q^25 - 4 * q^27 + 4 * q^33 - 8 * q^37 - 24 * q^41 + 4 * q^45 - 8 * q^47 - 12 * q^51 + 4 * q^53 - 16 * q^57 + 8 * q^61 + 12 * q^65 - 8 * q^69 + 8 * q^71 + 24 * q^73 + 2 * q^75 + 16 * q^79 - 2 * q^81 - 16 * q^83 + 56 * q^85 + 20 * q^89 + 24 * q^95 - 16 * q^97 + 8 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 2x^{2} + 4$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{2} ) / 2$$ (v^2) / 2 $$\beta_{3}$$ $$=$$ $$( \nu^{3} ) / 2$$ (v^3) / 2
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$2\beta_{2}$$ 2*b2 $$\nu^{3}$$ $$=$$ $$2\beta_{3}$$ 2*b3

## Character values

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

 $$n$$ $$785$$ $$1471$$ $$1765$$ $$2257$$ $$\chi(n)$$ $$1$$ $$1$$ $$1$$ $$\beta_{2}$$

## 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
 −0.707107 − 1.22474i 0.707107 + 1.22474i −0.707107 + 1.22474i 0.707107 − 1.22474i
0 0.500000 + 0.866025i 0 0.292893 0.507306i 0 0 0 −0.500000 + 0.866025i 0
961.2 0 0.500000 + 0.866025i 0 1.70711 2.95680i 0 0 0 −0.500000 + 0.866025i 0
1537.1 0 0.500000 0.866025i 0 0.292893 + 0.507306i 0 0 0 −0.500000 0.866025i 0
1537.2 0 0.500000 0.866025i 0 1.70711 + 2.95680i 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.bg 4
4.b odd 2 1 1176.2.q.m 4
7.b odd 2 1 2352.2.q.ba 4
7.c even 3 1 2352.2.a.z 2
7.c even 3 1 inner 2352.2.q.bg 4
7.d odd 6 1 2352.2.a.bg 2
7.d odd 6 1 2352.2.q.ba 4
12.b even 2 1 3528.2.s.bc 4
21.g even 6 1 7056.2.a.ce 2
21.h odd 6 1 7056.2.a.cw 2
28.d even 2 1 1176.2.q.n 4
28.f even 6 1 1176.2.a.l 2
28.f even 6 1 1176.2.q.n 4
28.g odd 6 1 1176.2.a.m yes 2
28.g odd 6 1 1176.2.q.m 4
56.j odd 6 1 9408.2.a.dh 2
56.k odd 6 1 9408.2.a.dr 2
56.m even 6 1 9408.2.a.dv 2
56.p even 6 1 9408.2.a.ed 2
84.h odd 2 1 3528.2.s.bl 4
84.j odd 6 1 3528.2.a.bc 2
84.j odd 6 1 3528.2.s.bl 4
84.n even 6 1 3528.2.a.bm 2
84.n even 6 1 3528.2.s.bc 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1176.2.a.l 2 28.f even 6 1
1176.2.a.m yes 2 28.g odd 6 1
1176.2.q.m 4 4.b odd 2 1
1176.2.q.m 4 28.g odd 6 1
1176.2.q.n 4 28.d even 2 1
1176.2.q.n 4 28.f even 6 1
2352.2.a.z 2 7.c even 3 1
2352.2.a.bg 2 7.d odd 6 1
2352.2.q.ba 4 7.b odd 2 1
2352.2.q.ba 4 7.d odd 6 1
2352.2.q.bg 4 1.a even 1 1 trivial
2352.2.q.bg 4 7.c even 3 1 inner
3528.2.a.bc 2 84.j odd 6 1
3528.2.a.bm 2 84.n even 6 1
3528.2.s.bc 4 12.b even 2 1
3528.2.s.bc 4 84.n even 6 1
3528.2.s.bl 4 84.h odd 2 1
3528.2.s.bl 4 84.j odd 6 1
7056.2.a.ce 2 21.g even 6 1
7056.2.a.cw 2 21.h odd 6 1
9408.2.a.dh 2 56.j odd 6 1
9408.2.a.dr 2 56.k odd 6 1
9408.2.a.dv 2 56.m even 6 1
9408.2.a.ed 2 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 $$S_{2}^{\mathrm{new}}(2352, [\chi])$$:

 $$T_{5}^{4} - 4T_{5}^{3} + 14T_{5}^{2} - 8T_{5} + 4$$ T5^4 - 4*T5^3 + 14*T5^2 - 8*T5 + 4 $$T_{11}^{4} + 4T_{11}^{3} + 20T_{11}^{2} - 16T_{11} + 16$$ T11^4 + 4*T11^3 + 20*T11^2 - 16*T11 + 16 $$T_{13}^{2} - 18$$ T13^2 - 18 $$T_{17}^{4} - 12T_{17}^{3} + 110T_{17}^{2} - 408T_{17} + 1156$$ T17^4 - 12*T17^3 + 110*T17^2 - 408*T17 + 1156

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$(T^{2} - T + 1)^{2}$$
$5$ $$T^{4} - 4 T^{3} + 14 T^{2} - 8 T + 4$$
$7$ $$T^{4}$$
$11$ $$T^{4} + 4 T^{3} + 20 T^{2} - 16 T + 16$$
$13$ $$(T^{2} - 18)^{2}$$
$17$ $$T^{4} - 12 T^{3} + 110 T^{2} + \cdots + 1156$$
$19$ $$T^{4} + 8 T^{3} + 56 T^{2} + 64 T + 64$$
$23$ $$T^{4} + 4 T^{3} + 20 T^{2} - 16 T + 16$$
$29$ $$(T^{2} - 8)^{2}$$
$31$ $$T^{4} + 8T^{2} + 64$$
$37$ $$T^{4} + 8 T^{3} + 80 T^{2} - 128 T + 256$$
$41$ $$(T^{2} + 12 T + 18)^{2}$$
$43$ $$(T^{2} - 128)^{2}$$
$47$ $$T^{4} + 8 T^{3} + 120 T^{2} + \cdots + 3136$$
$53$ $$(T^{2} - 2 T + 4)^{2}$$
$59$ $$T^{4} + 72T^{2} + 5184$$
$61$ $$T^{4} - 8 T^{3} + 98 T^{2} + \cdots + 1156$$
$67$ $$T^{4} + 128 T^{2} + 16384$$
$71$ $$(T^{2} - 4 T - 68)^{2}$$
$73$ $$T^{4} - 24 T^{3} + 450 T^{2} + \cdots + 15876$$
$79$ $$T^{4} - 16 T^{3} + 224 T^{2} + \cdots + 1024$$
$83$ $$(T + 4)^{4}$$
$89$ $$T^{4} - 20 T^{3} + 318 T^{2} + \cdots + 6724$$
$97$ $$(T^{2} + 8 T - 2)^{2}$$