# Properties

 Label 2352.2.h.g Level $2352$ Weight $2$ Character orbit 2352.h Analytic conductor $18.781$ Analytic rank $0$ Dimension $4$ CM discriminant -84 Inner twists $8$

# 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: $$\Q(\sqrt{-3}, \sqrt{-14})$$ Defining polynomial: $$x^{4} - 14 x^{2} + 196$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: yes Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + \beta_{2} q^{3} + \beta_{1} q^{5} -3 q^{9} +O(q^{10})$$ $$q + \beta_{2} q^{3} + \beta_{1} q^{5} -3 q^{9} -\beta_{3} q^{11} -\beta_{3} q^{15} + \beta_{1} q^{17} -4 \beta_{2} q^{19} + \beta_{3} q^{23} -9 q^{25} -3 \beta_{2} q^{27} + 2 \beta_{2} q^{31} -3 \beta_{1} q^{33} -8 q^{37} -\beta_{1} q^{41} -3 \beta_{1} q^{45} -\beta_{3} q^{51} -14 \beta_{2} q^{55} + 12 q^{57} + 3 \beta_{1} q^{69} + \beta_{3} q^{71} -9 \beta_{2} q^{75} + 9 q^{81} -14 q^{85} + 5 \beta_{1} q^{89} -6 q^{93} + 4 \beta_{3} q^{95} + 3 \beta_{3} q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 12 q^{9} + O(q^{10})$$ $$4 q - 12 q^{9} - 36 q^{25} - 32 q^{37} + 48 q^{57} + 36 q^{81} - 56 q^{85} - 24 q^{93} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 14 x^{2} + 196$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu^{3}$$$$/14$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{2} - 7$$$$)/7$$ $$\beta_{3}$$ $$=$$ $$($$$$-\nu^{3} + 28 \nu$$$$)/14$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{3} + \beta_{1}$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$7 \beta_{2} + 7$$ $$\nu^{3}$$ $$=$$ $$14 \beta_{1}$$

## 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
 3.24037 − 1.87083i −3.24037 + 1.87083i −3.24037 − 1.87083i 3.24037 + 1.87083i
0 1.73205i 0 3.74166i 0 0 0 −3.00000 0
2255.2 0 1.73205i 0 3.74166i 0 0 0 −3.00000 0
2255.3 0 1.73205i 0 3.74166i 0 0 0 −3.00000 0
2255.4 0 1.73205i 0 3.74166i 0 0 0 −3.00000 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
84.h odd 2 1 CM by $$\Q(\sqrt{-21})$$
3.b odd 2 1 inner
4.b odd 2 1 inner
7.b odd 2 1 inner
12.b even 2 1 inner
21.c even 2 1 inner
28.d 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.g 4
3.b odd 2 1 inner 2352.2.h.g 4
4.b odd 2 1 inner 2352.2.h.g 4
7.b odd 2 1 inner 2352.2.h.g 4
12.b even 2 1 inner 2352.2.h.g 4
21.c even 2 1 inner 2352.2.h.g 4
28.d even 2 1 inner 2352.2.h.g 4
84.h odd 2 1 CM 2352.2.h.g 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2352.2.h.g 4 1.a even 1 1 trivial
2352.2.h.g 4 3.b odd 2 1 inner
2352.2.h.g 4 4.b odd 2 1 inner
2352.2.h.g 4 7.b odd 2 1 inner
2352.2.h.g 4 12.b even 2 1 inner
2352.2.h.g 4 21.c even 2 1 inner
2352.2.h.g 4 28.d even 2 1 inner
2352.2.h.g 4 84.h odd 2 1 CM

## Hecke kernels

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

 $$T_{5}^{2} + 14$$ $$T_{11}^{2} - 42$$ $$T_{13}$$ $$T_{47}$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$( 3 + T^{2} )^{2}$$
$5$ $$( 14 + T^{2} )^{2}$$
$7$ $$T^{4}$$
$11$ $$( -42 + T^{2} )^{2}$$
$13$ $$T^{4}$$
$17$ $$( 14 + T^{2} )^{2}$$
$19$ $$( 48 + T^{2} )^{2}$$
$23$ $$( -42 + T^{2} )^{2}$$
$29$ $$T^{4}$$
$31$ $$( 12 + T^{2} )^{2}$$
$37$ $$( 8 + T )^{4}$$
$41$ $$( 14 + T^{2} )^{2}$$
$43$ $$T^{4}$$
$47$ $$T^{4}$$
$53$ $$T^{4}$$
$59$ $$T^{4}$$
$61$ $$T^{4}$$
$67$ $$T^{4}$$
$71$ $$( -42 + T^{2} )^{2}$$
$73$ $$T^{4}$$
$79$ $$T^{4}$$
$83$ $$T^{4}$$
$89$ $$( 350 + T^{2} )^{2}$$
$97$ $$T^{4}$$