# Properties

 Label 1232.2.e.b 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 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.

 $$f(q)$$ $$=$$ $$q + \beta_{3} q^{3} -\beta_{3} q^{5} + ( \beta_{1} - \beta_{2} ) q^{7} -2 q^{9} +O(q^{10})$$ $$q + \beta_{3} q^{3} -\beta_{3} q^{5} + ( \beta_{1} - \beta_{2} ) q^{7} -2 q^{9} + ( 1 - 2 \beta_{1} - \beta_{2} ) q^{11} + 2 \beta_{2} q^{13} + 5 q^{15} -4 \beta_{2} q^{17} + \beta_{2} q^{19} + ( 3 \beta_{1} + 4 \beta_{2} ) q^{21} + 5 q^{23} + \beta_{3} q^{27} + ( -6 \beta_{1} - 3 \beta_{2} ) q^{29} + \beta_{3} q^{31} + ( -5 \beta_{2} + \beta_{3} ) q^{33} + ( -3 \beta_{1} - 4 \beta_{2} ) q^{35} + 11 q^{37} + ( -4 \beta_{1} - 2 \beta_{2} ) q^{39} + 7 \beta_{2} q^{41} + ( 2 \beta_{1} + \beta_{2} ) q^{43} + 2 \beta_{3} q^{45} + 2 \beta_{3} q^{47} + ( 2 + 3 \beta_{3} ) q^{49} + ( 8 \beta_{1} + 4 \beta_{2} ) q^{51} + 8 q^{53} + ( 5 \beta_{2} - \beta_{3} ) q^{55} + ( -2 \beta_{1} - \beta_{2} ) q^{57} -\beta_{3} q^{59} -3 \beta_{2} q^{61} + ( -2 \beta_{1} + 2 \beta_{2} ) q^{63} + ( 4 \beta_{1} + 2 \beta_{2} ) q^{65} -9 q^{67} + 5 \beta_{3} q^{69} + q^{71} -3 \beta_{2} q^{73} + ( 5 + \beta_{1} - \beta_{2} - 3 \beta_{3} ) q^{77} + ( -4 \beta_{1} - 2 \beta_{2} ) q^{79} -11 q^{81} + \beta_{2} q^{83} + ( -8 \beta_{1} - 4 \beta_{2} ) q^{85} -15 \beta_{2} q^{87} + 3 \beta_{3} q^{89} + ( -6 - 2 \beta_{3} ) q^{91} -5 q^{93} + ( 2 \beta_{1} + \beta_{2} ) q^{95} + 7 \beta_{3} q^{97} + ( -2 + 4 \beta_{1} + 2 \beta_{2} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 8 q^{9} + O(q^{10})$$ $$4 q - 8 q^{9} + 4 q^{11} + 20 q^{15} + 20 q^{23} + 44 q^{37} + 8 q^{49} + 32 q^{53} - 36 q^{67} + 4 q^{71} + 20 q^{77} - 44 q^{81} - 24 q^{91} - 20 q^{93} - 8 q^{99} + O(q^{100})$$

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
 −0.707107 + 1.58114i 0.707107 − 1.58114i −0.707107 − 1.58114i 0.707107 + 1.58114i
0 2.23607i 0 2.23607i 0 −2.12132 + 1.58114i 0 −2.00000 0
769.2 0 2.23607i 0 2.23607i 0 2.12132 1.58114i 0 −2.00000 0
769.3 0 2.23607i 0 2.23607i 0 −2.12132 1.58114i 0 −2.00000 0
769.4 0 2.23607i 0 2.23607i 0 2.12132 + 1.58114i 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.b 4
4.b odd 2 1 308.2.c.a 4
7.b odd 2 1 inner 1232.2.e.b 4
11.b odd 2 1 inner 1232.2.e.b 4
12.b even 2 1 2772.2.i.c 4
28.d even 2 1 308.2.c.a 4
28.f even 6 2 2156.2.q.b 8
28.g odd 6 2 2156.2.q.b 8
44.c even 2 1 308.2.c.a 4
77.b even 2 1 inner 1232.2.e.b 4
84.h odd 2 1 2772.2.i.c 4
132.d odd 2 1 2772.2.i.c 4
308.g odd 2 1 308.2.c.a 4
308.m odd 6 2 2156.2.q.b 8
308.n even 6 2 2156.2.q.b 8
924.n even 2 1 2772.2.i.c 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
308.2.c.a 4 4.b odd 2 1
308.2.c.a 4 28.d even 2 1
308.2.c.a 4 44.c even 2 1
308.2.c.a 4 308.g odd 2 1
1232.2.e.b 4 1.a even 1 1 trivial
1232.2.e.b 4 7.b odd 2 1 inner
1232.2.e.b 4 11.b odd 2 1 inner
1232.2.e.b 4 77.b even 2 1 inner
2156.2.q.b 8 28.f even 6 2
2156.2.q.b 8 28.g odd 6 2
2156.2.q.b 8 308.m odd 6 2
2156.2.q.b 8 308.n even 6 2
2772.2.i.c 4 12.b even 2 1
2772.2.i.c 4 84.h odd 2 1
2772.2.i.c 4 132.d odd 2 1
2772.2.i.c 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 $$S_{2}^{\mathrm{new}}(1232, [\chi])$$:

 $$T_{3}^{2} + 5$$ $$T_{13}^{2} - 8$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$( 5 + T^{2} )^{2}$$
$5$ $$( 5 + T^{2} )^{2}$$
$7$ $$49 - 4 T^{2} + T^{4}$$
$11$ $$( 11 - 2 T + T^{2} )^{2}$$
$13$ $$( -8 + T^{2} )^{2}$$
$17$ $$( -32 + T^{2} )^{2}$$
$19$ $$( -2 + T^{2} )^{2}$$
$23$ $$( -5 + T )^{4}$$
$29$ $$( 90 + T^{2} )^{2}$$
$31$ $$( 5 + T^{2} )^{2}$$
$37$ $$( -11 + T )^{4}$$
$41$ $$( -98 + T^{2} )^{2}$$
$43$ $$( 10 + T^{2} )^{2}$$
$47$ $$( 20 + T^{2} )^{2}$$
$53$ $$( -8 + T )^{4}$$
$59$ $$( 5 + T^{2} )^{2}$$
$61$ $$( -18 + T^{2} )^{2}$$
$67$ $$( 9 + T )^{4}$$
$71$ $$( -1 + T )^{4}$$
$73$ $$( -18 + T^{2} )^{2}$$
$79$ $$( 40 + T^{2} )^{2}$$
$83$ $$( -2 + T^{2} )^{2}$$
$89$ $$( 45 + T^{2} )^{2}$$
$97$ $$( 245 + T^{2} )^{2}$$