# Properties

 Label 1232.2.q.f 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 154) 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.

 $$f(q)$$ $$=$$ $$q + ( -1 + \beta_{1} - \beta_{2} ) q^{3} + ( -\beta_{1} - 2 \beta_{2} - \beta_{3} ) q^{5} + ( \beta_{1} - \beta_{2} - \beta_{3} ) q^{7} + ( -2 \beta_{1} - 2 \beta_{3} ) q^{9} +O(q^{10})$$ $$q + ( -1 + \beta_{1} - \beta_{2} ) q^{3} + ( -\beta_{1} - 2 \beta_{2} - \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} -\beta_{3} q^{15} + ( 2 + 4 \beta_{1} + 2 \beta_{2} ) q^{17} + ( \beta_{1} + 2 \beta_{2} + \beta_{3} ) q^{19} + ( 1 - 2 \beta_{1} + 4 \beta_{2} - 2 \beta_{3} ) q^{21} + ( -3 \beta_{1} + 2 \beta_{2} - 3 \beta_{3} ) q^{23} + ( -1 - 4 \beta_{1} - \beta_{2} ) q^{25} + ( 1 - \beta_{3} ) q^{27} + ( -3 + 4 \beta_{3} ) q^{29} + ( -4 - 4 \beta_{2} ) q^{31} + ( \beta_{1} - \beta_{2} + \beta_{3} ) q^{33} + ( -3 \beta_{1} - 4 \beta_{2} - 4 \beta_{3} ) q^{35} + ( \beta_{1} - 8 \beta_{2} + \beta_{3} ) q^{37} + ( 5 - 3 \beta_{1} + 5 \beta_{2} ) q^{39} + ( -4 - \beta_{3} ) q^{41} + 4 \beta_{3} q^{43} + ( -4 - 4 \beta_{1} - 4 \beta_{2} ) q^{45} + ( -6 \beta_{1} + 2 \beta_{2} - 6 \beta_{3} ) q^{47} + ( 5 - 2 \beta_{1} + 5 \beta_{2} - 4 \beta_{3} ) q^{49} + ( -2 \beta_{1} + 6 \beta_{2} - 2 \beta_{3} ) q^{51} + ( 2 + 7 \beta_{1} + 2 \beta_{2} ) q^{53} + ( 2 - \beta_{3} ) q^{55} + \beta_{3} q^{57} + ( -7 - \beta_{1} - 7 \beta_{2} ) q^{59} + ( -2 \beta_{1} + 9 \beta_{2} - 2 \beta_{3} ) q^{61} + ( 4 - 2 \beta_{1} - 4 \beta_{2} ) q^{63} + ( -3 \beta_{1} - 2 \beta_{2} - 3 \beta_{3} ) q^{65} + ( 7 + 3 \beta_{1} + 7 \beta_{2} ) q^{67} + ( 8 + 5 \beta_{3} ) q^{69} + ( 4 + 5 \beta_{3} ) q^{71} + ( 8 - \beta_{1} + 8 \beta_{2} ) q^{73} + ( 3 \beta_{1} - 7 \beta_{2} + 3 \beta_{3} ) q^{75} + ( 1 + 2 \beta_{1} + \beta_{3} ) q^{77} + ( -3 \beta_{1} + 9 \beta_{2} - 3 \beta_{3} ) q^{79} + ( 1 - 6 \beta_{1} + \beta_{2} ) q^{81} + ( -2 + 10 \beta_{3} ) q^{83} + ( 12 - 10 \beta_{3} ) q^{85} + ( -5 + \beta_{1} - 5 \beta_{2} ) q^{87} + ( -6 \beta_{1} + 4 \beta_{2} - 6 \beta_{3} ) q^{89} + ( 8 - 3 \beta_{1} + 5 \beta_{2} - \beta_{3} ) q^{91} + ( -4 \beta_{1} + 4 \beta_{2} - 4 \beta_{3} ) q^{93} + ( 6 + 4 \beta_{1} + 6 \beta_{2} ) q^{95} + ( -1 - 2 \beta_{3} ) q^{97} -2 \beta_{3} q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 2q^{3} + 4q^{5} + 2q^{7} + O(q^{10})$$ $$4q - 2q^{3} + 4q^{5} + 2q^{7} + 2q^{11} - 4q^{13} + 4q^{17} - 4q^{19} - 4q^{21} - 4q^{23} - 2q^{25} + 4q^{27} - 12q^{29} - 8q^{31} + 2q^{33} + 8q^{35} + 16q^{37} + 10q^{39} - 16q^{41} - 8q^{45} - 4q^{47} + 10q^{49} - 12q^{51} + 4q^{53} + 8q^{55} - 14q^{59} - 18q^{61} + 24q^{63} + 4q^{65} + 14q^{67} + 32q^{69} + 16q^{71} + 16q^{73} + 14q^{75} + 4q^{77} - 18q^{79} + 2q^{81} - 8q^{83} + 48q^{85} - 10q^{87} - 8q^{89} + 22q^{91} - 8q^{93} + 12q^{95} - 4q^{97} + O(q^{100})$$

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 −1.20711 2.09077i 0 0.292893 0.507306i 0 −1.62132 2.09077i 0 −1.41421 + 2.44949i 0
177.2 0 0.207107 + 0.358719i 0 1.70711 2.95680i 0 2.62132 + 0.358719i 0 1.41421 2.44949i 0
529.1 0 −1.20711 + 2.09077i 0 0.292893 + 0.507306i 0 −1.62132 + 2.09077i 0 −1.41421 2.44949i 0
529.2 0 0.207107 0.358719i 0 1.70711 + 2.95680i 0 2.62132 0.358719i 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.f 4
4.b odd 2 1 154.2.e.e 4
7.c even 3 1 inner 1232.2.q.f 4
7.c even 3 1 8624.2.a.cc 2
7.d odd 6 1 8624.2.a.bh 2
12.b even 2 1 1386.2.k.t 4
28.d even 2 1 1078.2.e.m 4
28.f even 6 1 1078.2.a.x 2
28.f even 6 1 1078.2.e.m 4
28.g odd 6 1 154.2.e.e 4
28.g odd 6 1 1078.2.a.t 2
84.j odd 6 1 9702.2.a.ch 2
84.n even 6 1 1386.2.k.t 4
84.n even 6 1 9702.2.a.cx 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
154.2.e.e 4 4.b odd 2 1
154.2.e.e 4 28.g odd 6 1
1078.2.a.t 2 28.g odd 6 1
1078.2.a.x 2 28.f even 6 1
1078.2.e.m 4 28.d even 2 1
1078.2.e.m 4 28.f even 6 1
1232.2.q.f 4 1.a even 1 1 trivial
1232.2.q.f 4 7.c even 3 1 inner
1386.2.k.t 4 12.b even 2 1
1386.2.k.t 4 84.n even 6 1
8624.2.a.bh 2 7.d odd 6 1
8624.2.a.cc 2 7.c even 3 1
9702.2.a.ch 2 84.j odd 6 1
9702.2.a.cx 2 84.n even 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

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$1 - 2 T + 5 T^{2} + 2 T^{3} + T^{4}$$
$5$ $$4 - 8 T + 14 T^{2} - 4 T^{3} + T^{4}$$
$7$ $$49 - 14 T - 3 T^{2} - 2 T^{3} + T^{4}$$
$11$ $$( 1 - T + T^{2} )^{2}$$
$13$ $$( -7 + 2 T + T^{2} )^{2}$$
$17$ $$784 + 112 T + 44 T^{2} - 4 T^{3} + T^{4}$$
$19$ $$4 + 8 T + 14 T^{2} + 4 T^{3} + T^{4}$$
$23$ $$196 - 56 T + 30 T^{2} + 4 T^{3} + T^{4}$$
$29$ $$( -23 + 6 T + T^{2} )^{2}$$
$31$ $$( 16 + 4 T + T^{2} )^{2}$$
$37$ $$3844 - 992 T + 194 T^{2} - 16 T^{3} + T^{4}$$
$41$ $$( 14 + 8 T + T^{2} )^{2}$$
$43$ $$( -32 + T^{2} )^{2}$$
$47$ $$4624 - 272 T + 84 T^{2} + 4 T^{3} + T^{4}$$
$53$ $$8836 + 376 T + 110 T^{2} - 4 T^{3} + T^{4}$$
$59$ $$2209 + 658 T + 149 T^{2} + 14 T^{3} + T^{4}$$
$61$ $$5329 + 1314 T + 251 T^{2} + 18 T^{3} + T^{4}$$
$67$ $$961 - 434 T + 165 T^{2} - 14 T^{3} + T^{4}$$
$71$ $$( -34 - 8 T + T^{2} )^{2}$$
$73$ $$3844 - 992 T + 194 T^{2} - 16 T^{3} + T^{4}$$
$79$ $$3969 + 1134 T + 261 T^{2} + 18 T^{3} + T^{4}$$
$83$ $$( -196 + 4 T + T^{2} )^{2}$$
$89$ $$3136 - 448 T + 120 T^{2} + 8 T^{3} + T^{4}$$
$97$ $$( -7 + 2 T + T^{2} )^{2}$$