# Properties

 Label 2352.2.h.i Level $2352$ Weight $2$ Character orbit 2352.h Analytic conductor $18.781$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2352 = 2^{4} \cdot 3 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2352.h (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$18.7808145554$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(i, \sqrt{11})$$ Defining polynomial: $$x^{4} - 5x^{2} + 9$$ x^4 - 5*x^2 + 9 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$1$$ 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 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.

 $$f(q)$$ $$=$$ $$q + (\beta_{2} - \beta_1) q^{3} + ( - 2 \beta_{3} - 1) q^{5} + ( - \beta_{3} + 2) q^{9}+O(q^{10})$$ q + (b2 - b1) * q^3 + (-2*b3 - 1) * q^5 + (-b3 + 2) * q^9 $$q + (\beta_{2} - \beta_1) q^{3} + ( - 2 \beta_{3} - 1) q^{5} + ( - \beta_{3} + 2) q^{9} + ( - \beta_{2} + 2 \beta_1) q^{11} - 4 q^{13} + (5 \beta_{2} + \beta_1) q^{15} + (2 \beta_{3} + 1) q^{17} + 7 \beta_{2} q^{19} + ( - \beta_{2} + 2 \beta_1) q^{23} - 6 q^{25} + (5 \beta_{2} - 2 \beta_1) q^{27} + (4 \beta_{3} + 2) q^{29} + 3 \beta_{2} q^{31} + (\beta_{3} - 5) q^{33} - q^{37} + ( - 4 \beta_{2} + 4 \beta_1) q^{39} + ( - 4 \beta_{3} - 2) q^{41} - 2 \beta_{2} q^{43} + ( - 5 \beta_{3} - 8) q^{45} + ( - 3 \beta_{2} + 6 \beta_1) q^{47} + ( - 5 \beta_{2} - \beta_1) q^{51} + ( - 2 \beta_{3} - 1) q^{53} - 11 \beta_{2} q^{55} + ( - 7 \beta_{3} - 7) q^{57} + (\beta_{2} - 2 \beta_1) q^{59} + 3 q^{61} + (8 \beta_{3} + 4) q^{65} + 9 \beta_{2} q^{67} + (\beta_{3} - 5) q^{69} + ( - 4 \beta_{2} + 8 \beta_1) q^{71} + 7 q^{73} + ( - 6 \beta_{2} + 6 \beta_1) q^{75} + 9 \beta_{2} q^{79} + ( - 5 \beta_{3} + 1) q^{81} + (4 \beta_{2} - 8 \beta_1) q^{83} + 11 q^{85} + ( - 10 \beta_{2} - 2 \beta_1) q^{87} + (2 \beta_{3} + 1) q^{89} + ( - 3 \beta_{3} - 3) q^{93} + ( - 7 \beta_{2} + 14 \beta_1) q^{95} + 8 q^{97} + ( - 8 \beta_{2} + 5 \beta_1) q^{99}+O(q^{100})$$ q + (b2 - b1) * q^3 + (-2*b3 - 1) * q^5 + (-b3 + 2) * q^9 + (-b2 + 2*b1) * q^11 - 4 * q^13 + (5*b2 + b1) * q^15 + (2*b3 + 1) * q^17 + 7*b2 * q^19 + (-b2 + 2*b1) * q^23 - 6 * q^25 + (5*b2 - 2*b1) * q^27 + (4*b3 + 2) * q^29 + 3*b2 * q^31 + (b3 - 5) * q^33 - q^37 + (-4*b2 + 4*b1) * q^39 + (-4*b3 - 2) * q^41 - 2*b2 * q^43 + (-5*b3 - 8) * q^45 + (-3*b2 + 6*b1) * q^47 + (-5*b2 - b1) * q^51 + (-2*b3 - 1) * q^53 - 11*b2 * q^55 + (-7*b3 - 7) * q^57 + (b2 - 2*b1) * q^59 + 3 * q^61 + (8*b3 + 4) * q^65 + 9*b2 * q^67 + (b3 - 5) * q^69 + (-4*b2 + 8*b1) * q^71 + 7 * q^73 + (-6*b2 + 6*b1) * q^75 + 9*b2 * q^79 + (-5*b3 + 1) * q^81 + (4*b2 - 8*b1) * q^83 + 11 * q^85 + (-10*b2 - 2*b1) * q^87 + (2*b3 + 1) * q^89 + (-3*b3 - 3) * q^93 + (-7*b2 + 14*b1) * q^95 + 8 * q^97 + (-8*b2 + 5*b1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 10 q^{9}+O(q^{10})$$ 4 * q + 10 * q^9 $$4 q + 10 q^{9} - 16 q^{13} - 24 q^{25} - 22 q^{33} - 4 q^{37} - 22 q^{45} - 14 q^{57} + 12 q^{61} - 22 q^{69} + 28 q^{73} + 14 q^{81} + 44 q^{85} - 6 q^{93} + 32 q^{97}+O(q^{100})$$ 4 * q + 10 * q^9 - 16 * q^13 - 24 * q^25 - 22 * q^33 - 4 * q^37 - 22 * q^45 - 14 * q^57 + 12 * q^61 - 22 * q^69 + 28 * q^73 + 14 * q^81 + 44 * q^85 - 6 * q^93 + 32 * q^97

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 5x^{2} + 9$$ :

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
2255.1
 1.65831 − 0.500000i 1.65831 + 0.500000i −1.65831 − 0.500000i −1.65831 + 0.500000i
0 −1.65831 0.500000i 0 3.31662i 0 0 0 2.50000 + 1.65831i 0
2255.2 0 −1.65831 + 0.500000i 0 3.31662i 0 0 0 2.50000 1.65831i 0
2255.3 0 1.65831 0.500000i 0 3.31662i 0 0 0 2.50000 1.65831i 0
2255.4 0 1.65831 + 0.500000i 0 3.31662i 0 0 0 2.50000 + 1.65831i 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
3.b odd 2 1 inner
4.b odd 2 1 inner
12.b 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.h.i 4
3.b odd 2 1 inner 2352.2.h.i 4
4.b odd 2 1 inner 2352.2.h.i 4
7.b odd 2 1 2352.2.h.j 4
7.c even 3 2 336.2.bj.f 8
12.b even 2 1 inner 2352.2.h.i 4
21.c even 2 1 2352.2.h.j 4
21.h odd 6 2 336.2.bj.f 8
28.d even 2 1 2352.2.h.j 4
28.g odd 6 2 336.2.bj.f 8
84.h odd 2 1 2352.2.h.j 4
84.n even 6 2 336.2.bj.f 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
336.2.bj.f 8 7.c even 3 2
336.2.bj.f 8 21.h odd 6 2
336.2.bj.f 8 28.g odd 6 2
336.2.bj.f 8 84.n even 6 2
2352.2.h.i 4 1.a even 1 1 trivial
2352.2.h.i 4 3.b odd 2 1 inner
2352.2.h.i 4 4.b odd 2 1 inner
2352.2.h.i 4 12.b even 2 1 inner
2352.2.h.j 4 7.b odd 2 1
2352.2.h.j 4 21.c even 2 1
2352.2.h.j 4 28.d even 2 1
2352.2.h.j 4 84.h odd 2 1

## Hecke kernels

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

 $$T_{5}^{2} + 11$$ T5^2 + 11 $$T_{11}^{2} - 11$$ T11^2 - 11 $$T_{13} + 4$$ T13 + 4 $$T_{47}^{2} - 99$$ T47^2 - 99

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4} - 5T^{2} + 9$$
$5$ $$(T^{2} + 11)^{2}$$
$7$ $$T^{4}$$
$11$ $$(T^{2} - 11)^{2}$$
$13$ $$(T + 4)^{4}$$
$17$ $$(T^{2} + 11)^{2}$$
$19$ $$(T^{2} + 49)^{2}$$
$23$ $$(T^{2} - 11)^{2}$$
$29$ $$(T^{2} + 44)^{2}$$
$31$ $$(T^{2} + 9)^{2}$$
$37$ $$(T + 1)^{4}$$
$41$ $$(T^{2} + 44)^{2}$$
$43$ $$(T^{2} + 4)^{2}$$
$47$ $$(T^{2} - 99)^{2}$$
$53$ $$(T^{2} + 11)^{2}$$
$59$ $$(T^{2} - 11)^{2}$$
$61$ $$(T - 3)^{4}$$
$67$ $$(T^{2} + 81)^{2}$$
$71$ $$(T^{2} - 176)^{2}$$
$73$ $$(T - 7)^{4}$$
$79$ $$(T^{2} + 81)^{2}$$
$83$ $$(T^{2} - 176)^{2}$$
$89$ $$(T^{2} + 11)^{2}$$
$97$ $$(T - 8)^{4}$$