# Properties

 Label 1232.2.e.c Level $1232$ Weight $2$ Character orbit 1232.e Analytic conductor $9.838$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1232 = 2^{4} \cdot 7 \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1232.e (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$9.83756952902$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\sqrt{-2}, \sqrt{-5})$$ Defining polynomial: $$x^{4} - 4 x^{2} + 9$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 77) 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.

 $$f(q)$$ $$=$$ $$q + \beta_{3} q^{3} + \beta_{3} q^{5} + ( \beta_{1} - 2 \beta_{2} ) q^{7} -2 q^{9} +O(q^{10})$$ $$q + \beta_{3} q^{3} + \beta_{3} q^{5} + ( \beta_{1} - 2 \beta_{2} ) q^{7} -2 q^{9} + ( 3 - \beta_{2} ) q^{11} + ( 4 \beta_{1} - 2 \beta_{2} ) q^{13} -5 q^{15} + ( 2 \beta_{1} - \beta_{2} ) q^{19} + ( 3 \beta_{1} + \beta_{2} ) q^{21} + 3 q^{23} + \beta_{3} q^{27} -\beta_{2} q^{29} -3 \beta_{3} q^{31} + ( 2 \beta_{1} - \beta_{2} + 3 \beta_{3} ) q^{33} + ( 3 \beta_{1} + \beta_{2} ) q^{35} - q^{37} + 10 \beta_{2} q^{39} + ( -6 \beta_{1} + 3 \beta_{2} ) q^{41} -3 \beta_{2} q^{43} -2 \beta_{3} q^{45} + 2 \beta_{3} q^{47} + ( -2 - 3 \beta_{3} ) q^{49} + ( 2 \beta_{1} - \beta_{2} + 3 \beta_{3} ) q^{55} + 5 \beta_{2} q^{57} -\beta_{3} q^{59} + ( -2 \beta_{1} + \beta_{2} ) q^{61} + ( -2 \beta_{1} + 4 \beta_{2} ) q^{63} + 10 \beta_{2} q^{65} -11 q^{67} + 3 \beta_{3} q^{69} -9 q^{71} + ( -2 \beta_{1} + \beta_{2} ) q^{73} + ( -3 + 3 \beta_{1} - 6 \beta_{2} - \beta_{3} ) q^{77} + 6 \beta_{2} q^{79} -11 q^{81} + ( -6 \beta_{1} + 3 \beta_{2} ) q^{83} + ( 2 \beta_{1} - \beta_{2} ) q^{87} + \beta_{3} q^{89} + ( 10 - 6 \beta_{3} ) q^{91} + 15 q^{93} + 5 \beta_{2} q^{95} -3 \beta_{3} q^{97} + ( -6 + 2 \beta_{2} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 8 q^{9} + O(q^{10})$$ $$4 q - 8 q^{9} + 12 q^{11} - 20 q^{15} + 12 q^{23} - 4 q^{37} - 8 q^{49} - 44 q^{67} - 36 q^{71} - 12 q^{77} - 44 q^{81} + 40 q^{91} + 60 q^{93} - 24 q^{99} + O(q^{100})$$

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

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

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1232\mathbb{Z}\right)^\times$$.

 $$n$$ $$309$$ $$353$$ $$463$$ $$673$$ $$\chi(n)$$ $$1$$ $$-1$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
769.1
 −1.58114 + 0.707107i 1.58114 − 0.707107i −1.58114 − 0.707107i 1.58114 + 0.707107i
0 2.23607i 0 2.23607i 0 −1.58114 2.12132i 0 −2.00000 0
769.2 0 2.23607i 0 2.23607i 0 1.58114 + 2.12132i 0 −2.00000 0
769.3 0 2.23607i 0 2.23607i 0 −1.58114 + 2.12132i 0 −2.00000 0
769.4 0 2.23607i 0 2.23607i 0 1.58114 2.12132i 0 −2.00000 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
7.b odd 2 1 inner
11.b odd 2 1 inner
77.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1232.2.e.c 4
4.b odd 2 1 77.2.b.b 4
7.b odd 2 1 inner 1232.2.e.c 4
11.b odd 2 1 inner 1232.2.e.c 4
12.b even 2 1 693.2.c.b 4
28.d even 2 1 77.2.b.b 4
28.f even 6 2 539.2.i.b 8
28.g odd 6 2 539.2.i.b 8
44.c even 2 1 77.2.b.b 4
44.g even 10 4 847.2.l.g 16
44.h odd 10 4 847.2.l.g 16
77.b even 2 1 inner 1232.2.e.c 4
84.h odd 2 1 693.2.c.b 4
132.d odd 2 1 693.2.c.b 4
308.g odd 2 1 77.2.b.b 4
308.m odd 6 2 539.2.i.b 8
308.n even 6 2 539.2.i.b 8
308.s odd 10 4 847.2.l.g 16
308.t even 10 4 847.2.l.g 16
924.n even 2 1 693.2.c.b 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
77.2.b.b 4 4.b odd 2 1
77.2.b.b 4 28.d even 2 1
77.2.b.b 4 44.c even 2 1
77.2.b.b 4 308.g odd 2 1
539.2.i.b 8 28.f even 6 2
539.2.i.b 8 28.g odd 6 2
539.2.i.b 8 308.m odd 6 2
539.2.i.b 8 308.n even 6 2
693.2.c.b 4 12.b even 2 1
693.2.c.b 4 84.h odd 2 1
693.2.c.b 4 132.d odd 2 1
693.2.c.b 4 924.n even 2 1
847.2.l.g 16 44.g even 10 4
847.2.l.g 16 44.h odd 10 4
847.2.l.g 16 308.s odd 10 4
847.2.l.g 16 308.t even 10 4
1232.2.e.c 4 1.a even 1 1 trivial
1232.2.e.c 4 7.b odd 2 1 inner
1232.2.e.c 4 11.b odd 2 1 inner
1232.2.e.c 4 77.b even 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(1232, [\chi])$$:

 $$T_{3}^{2} + 5$$ $$T_{13}^{2} - 40$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$( 5 + T^{2} )^{2}$$
$5$ $$( 5 + T^{2} )^{2}$$
$7$ $$49 + 4 T^{2} + T^{4}$$
$11$ $$( 11 - 6 T + T^{2} )^{2}$$
$13$ $$( -40 + T^{2} )^{2}$$
$17$ $$T^{4}$$
$19$ $$( -10 + T^{2} )^{2}$$
$23$ $$( -3 + T )^{4}$$
$29$ $$( 2 + T^{2} )^{2}$$
$31$ $$( 45 + T^{2} )^{2}$$
$37$ $$( 1 + T )^{4}$$
$41$ $$( -90 + T^{2} )^{2}$$
$43$ $$( 18 + T^{2} )^{2}$$
$47$ $$( 20 + T^{2} )^{2}$$
$53$ $$T^{4}$$
$59$ $$( 5 + T^{2} )^{2}$$
$61$ $$( -10 + T^{2} )^{2}$$
$67$ $$( 11 + T )^{4}$$
$71$ $$( 9 + T )^{4}$$
$73$ $$( -10 + T^{2} )^{2}$$
$79$ $$( 72 + T^{2} )^{2}$$
$83$ $$( -90 + T^{2} )^{2}$$
$89$ $$( 5 + T^{2} )^{2}$$
$97$ $$( 45 + T^{2} )^{2}$$