# Properties

 Label 1232.2.e.d 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: $$\Q(\sqrt{-2}, \sqrt{-7})$$ Defining polynomial: $$x^{4} - 2 x^{3} + 9 x^{2} - 8 x + 2$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 308) 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.

 $$f(q)$$ $$=$$ $$q -\beta_{2} q^{3} -2 \beta_{2} q^{5} -\beta_{3} q^{7} + q^{9} +O(q^{10})$$ $$q -\beta_{2} q^{3} -2 \beta_{2} q^{5} -\beta_{3} q^{7} + q^{9} + ( -2 - \beta_{3} ) q^{11} -\beta_{1} q^{13} -4 q^{15} -\beta_{1} q^{17} -2 \beta_{1} q^{19} + \beta_{1} q^{21} -4 q^{23} -3 q^{25} -4 \beta_{2} q^{27} -\beta_{2} q^{31} + ( \beta_{1} + 2 \beta_{2} ) q^{33} + 2 \beta_{1} q^{35} -4 q^{37} -2 \beta_{3} q^{39} + \beta_{1} q^{41} + 4 \beta_{3} q^{43} -2 \beta_{2} q^{45} + 7 \beta_{2} q^{47} -7 q^{49} -2 \beta_{3} q^{51} -4 q^{53} + ( 2 \beta_{1} + 4 \beta_{2} ) q^{55} -4 \beta_{3} q^{57} + \beta_{2} q^{59} -3 \beta_{1} q^{61} -\beta_{3} q^{63} -4 \beta_{3} q^{65} + 12 q^{67} + 4 \beta_{2} q^{69} -8 q^{71} + 3 \beta_{1} q^{73} + 3 \beta_{2} q^{75} + ( -7 + 2 \beta_{3} ) q^{77} + 4 \beta_{3} q^{79} -5 q^{81} -2 \beta_{1} q^{83} -4 \beta_{3} q^{85} + 12 \beta_{2} q^{89} -7 \beta_{2} q^{91} -2 q^{93} -8 \beta_{3} q^{95} -4 \beta_{2} q^{97} + ( -2 - \beta_{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 4 q^{9} + O(q^{10})$$ $$4 q + 4 q^{9} - 8 q^{11} - 16 q^{15} - 16 q^{23} - 12 q^{25} - 16 q^{37} - 28 q^{49} - 16 q^{53} + 48 q^{67} - 32 q^{71} - 28 q^{77} - 20 q^{81} - 8 q^{93} - 8 q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 2 x^{3} + 9 x^{2} - 8 x + 2$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu^{2} - \nu + 4$$ $$\beta_{2}$$ $$=$$ $$2 \nu^{3} - 3 \nu^{2} + 17 \nu - 8$$ $$\beta_{3}$$ $$=$$ $$-4 \nu^{3} + 6 \nu^{2} - 32 \nu + 15$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{3} + 2 \beta_{2} + 1$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{3} + 2 \beta_{2} + 2 \beta_{1} - 7$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$($$$$-7 \beta_{3} - 13 \beta_{2} + 3 \beta_{1} - 11$$$$)/2$$

## 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
 0.5 + 2.73709i 0.5 + 0.0913379i 0.5 − 0.0913379i 0.5 − 2.73709i
0 1.41421i 0 2.82843i 0 2.64575i 0 1.00000 0
769.2 0 1.41421i 0 2.82843i 0 2.64575i 0 1.00000 0
769.3 0 1.41421i 0 2.82843i 0 2.64575i 0 1.00000 0
769.4 0 1.41421i 0 2.82843i 0 2.64575i 0 1.00000 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.d 4
4.b odd 2 1 308.2.c.b 4
7.b odd 2 1 inner 1232.2.e.d 4
11.b odd 2 1 inner 1232.2.e.d 4
12.b even 2 1 2772.2.i.a 4
28.d even 2 1 308.2.c.b 4
28.f even 6 2 2156.2.q.a 8
28.g odd 6 2 2156.2.q.a 8
44.c even 2 1 308.2.c.b 4
77.b even 2 1 inner 1232.2.e.d 4
84.h odd 2 1 2772.2.i.a 4
132.d odd 2 1 2772.2.i.a 4
308.g odd 2 1 308.2.c.b 4
308.m odd 6 2 2156.2.q.a 8
308.n even 6 2 2156.2.q.a 8
924.n even 2 1 2772.2.i.a 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
308.2.c.b 4 4.b odd 2 1
308.2.c.b 4 28.d even 2 1
308.2.c.b 4 44.c even 2 1
308.2.c.b 4 308.g odd 2 1
1232.2.e.d 4 1.a even 1 1 trivial
1232.2.e.d 4 7.b odd 2 1 inner
1232.2.e.d 4 11.b odd 2 1 inner
1232.2.e.d 4 77.b even 2 1 inner
2156.2.q.a 8 28.f even 6 2
2156.2.q.a 8 28.g odd 6 2
2156.2.q.a 8 308.m odd 6 2
2156.2.q.a 8 308.n even 6 2
2772.2.i.a 4 12.b even 2 1
2772.2.i.a 4 84.h odd 2 1
2772.2.i.a 4 132.d odd 2 1
2772.2.i.a 4 924.n even 2 1

## Hecke kernels

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

 $$T_{3}^{2} + 2$$ $$T_{13}^{2} - 14$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$( 2 + T^{2} )^{2}$$
$5$ $$( 8 + T^{2} )^{2}$$
$7$ $$( 7 + T^{2} )^{2}$$
$11$ $$( 11 + 4 T + T^{2} )^{2}$$
$13$ $$( -14 + T^{2} )^{2}$$
$17$ $$( -14 + T^{2} )^{2}$$
$19$ $$( -56 + T^{2} )^{2}$$
$23$ $$( 4 + T )^{4}$$
$29$ $$T^{4}$$
$31$ $$( 2 + T^{2} )^{2}$$
$37$ $$( 4 + T )^{4}$$
$41$ $$( -14 + T^{2} )^{2}$$
$43$ $$( 112 + T^{2} )^{2}$$
$47$ $$( 98 + T^{2} )^{2}$$
$53$ $$( 4 + T )^{4}$$
$59$ $$( 2 + T^{2} )^{2}$$
$61$ $$( -126 + T^{2} )^{2}$$
$67$ $$( -12 + T )^{4}$$
$71$ $$( 8 + T )^{4}$$
$73$ $$( -126 + T^{2} )^{2}$$
$79$ $$( 112 + T^{2} )^{2}$$
$83$ $$( -56 + T^{2} )^{2}$$
$89$ $$( 288 + T^{2} )^{2}$$
$97$ $$( 32 + T^{2} )^{2}$$