# Properties

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

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1232 = 2^{4} \cdot 7 \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1232.q (of order $$3$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$9.83756952902$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\Q(\sqrt{2}, \sqrt{-3})$$ Defining polynomial: $$x^{4} + 2 x^{2} + 4$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 616) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + ( 1 + \beta_{1} + \beta_{2} ) q^{3} + ( -\beta_{1} - \beta_{3} ) q^{5} + ( \beta_{1} + \beta_{2} - \beta_{3} ) q^{7} + ( 2 \beta_{1} + 2 \beta_{3} ) q^{9} +O(q^{10})$$ $$q + ( 1 + \beta_{1} + \beta_{2} ) q^{3} + ( -\beta_{1} - \beta_{3} ) q^{5} + ( \beta_{1} + \beta_{2} - \beta_{3} ) q^{7} + ( 2 \beta_{1} + 2 \beta_{3} ) q^{9} + ( 1 + \beta_{2} ) q^{11} + ( -1 + 2 \beta_{3} ) q^{13} + ( 2 - \beta_{3} ) q^{15} + ( 2 + 2 \beta_{2} ) q^{17} + ( \beta_{1} + 4 \beta_{2} + \beta_{3} ) q^{19} + ( 1 + 2 \beta_{1} + 4 \beta_{2} + 2 \beta_{3} ) q^{21} + ( \beta_{1} - 4 \beta_{2} + \beta_{3} ) q^{23} + ( 3 + 3 \beta_{2} ) q^{25} + ( -1 - \beta_{3} ) q^{27} + q^{29} + ( -4 + 4 \beta_{1} - 4 \beta_{2} ) q^{31} + ( \beta_{1} + \beta_{2} + \beta_{3} ) q^{33} + ( 2 + \beta_{1} - 2 \beta_{2} ) q^{35} + ( 5 \beta_{1} - 2 \beta_{2} + 5 \beta_{3} ) q^{37} + ( -5 - 3 \beta_{1} - 5 \beta_{2} ) q^{39} + ( 2 + 3 \beta_{3} ) q^{41} + 8 q^{43} + ( 4 + 4 \beta_{2} ) q^{45} + ( 2 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} ) q^{47} + ( 5 + 2 \beta_{1} + 5 \beta_{2} + 4 \beta_{3} ) q^{49} + ( 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} ) q^{51} + ( -4 - \beta_{1} - 4 \beta_{2} ) q^{53} -\beta_{3} q^{55} + ( -6 + 5 \beta_{3} ) q^{57} + ( 3 - \beta_{1} + 3 \beta_{2} ) q^{59} + ( -2 \beta_{1} - 11 \beta_{2} - 2 \beta_{3} ) q^{61} + ( -4 - 2 \beta_{1} + 4 \beta_{2} ) q^{63} + ( \beta_{1} + 4 \beta_{2} + \beta_{3} ) q^{65} + ( 1 - 5 \beta_{1} + \beta_{2} ) q^{67} + ( 2 - 3 \beta_{3} ) q^{69} + ( 10 - 3 \beta_{3} ) q^{71} + ( 2 - \beta_{1} + 2 \beta_{2} ) q^{73} + ( 3 \beta_{1} + 3 \beta_{2} + 3 \beta_{3} ) q^{75} + ( -1 + 2 \beta_{1} + \beta_{3} ) q^{77} + ( 5 \beta_{1} - \beta_{2} + 5 \beta_{3} ) q^{79} + ( 1 + 6 \beta_{1} + \beta_{2} ) q^{81} + ( 2 + 6 \beta_{3} ) q^{83} -2 \beta_{3} q^{85} + ( 1 + \beta_{1} + \beta_{2} ) q^{87} + ( -2 \beta_{1} - 12 \beta_{2} - 2 \beta_{3} ) q^{89} + ( -8 - 3 \beta_{1} - 5 \beta_{2} - \beta_{3} ) q^{91} + 4 \beta_{2} q^{93} + ( 2 + 4 \beta_{1} + 2 \beta_{2} ) q^{95} + ( -13 - 2 \beta_{3} ) q^{97} + 2 \beta_{3} q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 2 q^{3} - 2 q^{7} + O(q^{10})$$ $$4 q + 2 q^{3} - 2 q^{7} + 2 q^{11} - 4 q^{13} + 8 q^{15} + 4 q^{17} - 8 q^{19} - 4 q^{21} + 8 q^{23} + 6 q^{25} - 4 q^{27} + 4 q^{29} - 8 q^{31} - 2 q^{33} + 12 q^{35} + 4 q^{37} - 10 q^{39} + 8 q^{41} + 32 q^{43} + 8 q^{45} + 4 q^{47} + 10 q^{49} - 4 q^{51} - 8 q^{53} - 24 q^{57} + 6 q^{59} + 22 q^{61} - 24 q^{63} - 8 q^{65} + 2 q^{67} + 8 q^{69} + 40 q^{71} + 4 q^{73} - 6 q^{75} - 4 q^{77} + 2 q^{79} + 2 q^{81} + 8 q^{83} + 2 q^{87} + 24 q^{89} - 22 q^{91} - 8 q^{93} + 4 q^{95} - 52 q^{97} + O(q^{100})$$

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

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2}$$$$/2$$ $$\beta_{3}$$ $$=$$ $$\nu^{3}$$$$/2$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$2 \beta_{2}$$ $$\nu^{3}$$ $$=$$ $$2 \beta_{3}$$

## Character values

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

 $$n$$ $$309$$ $$353$$ $$463$$ $$673$$ $$\chi(n)$$ $$1$$ $$\beta_{2}$$ $$1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
177.1
 −0.707107 − 1.22474i 0.707107 + 1.22474i −0.707107 + 1.22474i 0.707107 − 1.22474i
0 −0.207107 0.358719i 0 −0.707107 + 1.22474i 0 −2.62132 0.358719i 0 1.41421 2.44949i 0
177.2 0 1.20711 + 2.09077i 0 0.707107 1.22474i 0 1.62132 + 2.09077i 0 −1.41421 + 2.44949i 0
529.1 0 −0.207107 + 0.358719i 0 −0.707107 1.22474i 0 −2.62132 + 0.358719i 0 1.41421 + 2.44949i 0
529.2 0 1.20711 2.09077i 0 0.707107 + 1.22474i 0 1.62132 2.09077i 0 −1.41421 2.44949i 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.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1232.2.q.h 4
4.b odd 2 1 616.2.q.b 4
7.c even 3 1 inner 1232.2.q.h 4
7.c even 3 1 8624.2.a.bg 2
7.d odd 6 1 8624.2.a.cd 2
28.f even 6 1 4312.2.a.m 2
28.g odd 6 1 616.2.q.b 4
28.g odd 6 1 4312.2.a.u 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
616.2.q.b 4 4.b odd 2 1
616.2.q.b 4 28.g odd 6 1
1232.2.q.h 4 1.a even 1 1 trivial
1232.2.q.h 4 7.c even 3 1 inner
4312.2.a.m 2 28.f even 6 1
4312.2.a.u 2 28.g odd 6 1
8624.2.a.bg 2 7.c even 3 1
8624.2.a.cd 2 7.d odd 6 1

## Hecke kernels

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

 $$T_{3}^{4} - 2 T_{3}^{3} + 5 T_{3}^{2} + 2 T_{3} + 1$$ $$T_{13}^{2} + 2 T_{13} - 7$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$1 + 2 T + 5 T^{2} - 2 T^{3} + T^{4}$$
$5$ $$4 + 2 T^{2} + T^{4}$$
$7$ $$49 + 14 T - 3 T^{2} + 2 T^{3} + T^{4}$$
$11$ $$( 1 - T + T^{2} )^{2}$$
$13$ $$( -7 + 2 T + T^{2} )^{2}$$
$17$ $$( 4 - 2 T + T^{2} )^{2}$$
$19$ $$196 + 112 T + 50 T^{2} + 8 T^{3} + T^{4}$$
$23$ $$196 - 112 T + 50 T^{2} - 8 T^{3} + T^{4}$$
$29$ $$( -1 + T )^{4}$$
$31$ $$256 - 128 T + 80 T^{2} + 8 T^{3} + T^{4}$$
$37$ $$2116 + 184 T + 62 T^{2} - 4 T^{3} + T^{4}$$
$41$ $$( -14 - 4 T + T^{2} )^{2}$$
$43$ $$( -8 + T )^{4}$$
$47$ $$16 + 16 T + 20 T^{2} - 4 T^{3} + T^{4}$$
$53$ $$196 + 112 T + 50 T^{2} + 8 T^{3} + T^{4}$$
$59$ $$49 - 42 T + 29 T^{2} - 6 T^{3} + T^{4}$$
$61$ $$12769 - 2486 T + 371 T^{2} - 22 T^{3} + T^{4}$$
$67$ $$2401 + 98 T + 53 T^{2} - 2 T^{3} + T^{4}$$
$71$ $$( 82 - 20 T + T^{2} )^{2}$$
$73$ $$4 - 8 T + 14 T^{2} - 4 T^{3} + T^{4}$$
$79$ $$2401 + 98 T + 53 T^{2} - 2 T^{3} + T^{4}$$
$83$ $$( -68 - 4 T + T^{2} )^{2}$$
$89$ $$18496 - 3264 T + 440 T^{2} - 24 T^{3} + T^{4}$$
$97$ $$( 161 + 26 T + T^{2} )^{2}$$