# Properties

 Label 1232.2.q.k Level $1232$ Weight $2$ Character orbit 1232.q Analytic conductor $9.838$ Analytic rank $0$ Dimension $6$ 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: $$6$$ Relative dimension: $$3$$ over $$\Q(\zeta_{3})$$ Coefficient field: 6.0.1783323.2 Defining polynomial: $$x^{6} - x^{5} + 5 x^{4} - 2 x^{3} + 19 x^{2} - 12 x + 9$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 77) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{5}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -\beta_{1} q^{3} + ( \beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} ) q^{5} + ( -1 + \beta_{1} - \beta_{3} + \beta_{4} + \beta_{5} ) q^{7} + ( -\beta_{3} + \beta_{5} ) q^{9} +O(q^{10})$$ $$q -\beta_{1} q^{3} + ( \beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} ) q^{5} + ( -1 + \beta_{1} - \beta_{3} + \beta_{4} + \beta_{5} ) q^{7} + ( -\beta_{3} + \beta_{5} ) q^{9} + ( -1 + \beta_{4} ) q^{11} + ( -4 - \beta_{2} ) q^{13} + ( 3 + 2 \beta_{2} ) q^{15} + ( 1 - \beta_{4} + \beta_{5} ) q^{17} + ( 2 \beta_{1} + 2 \beta_{2} + \beta_{3} - 3 \beta_{4} - \beta_{5} ) q^{19} + ( \beta_{1} + 2 \beta_{3} + 3 \beta_{4} - \beta_{5} ) q^{21} + ( -\beta_{3} + 4 \beta_{4} + \beta_{5} ) q^{23} + ( -2 + 3 \beta_{1} + 2 \beta_{4} ) q^{25} + ( 2 \beta_{2} + \beta_{3} ) q^{27} + ( -2 + 3 \beta_{2} - \beta_{3} ) q^{29} + ( -2 + 3 \beta_{1} + 2 \beta_{4} + \beta_{5} ) q^{31} + ( \beta_{1} + \beta_{2} ) q^{33} + ( -1 - 3 \beta_{2} - \beta_{3} - 3 \beta_{4} - \beta_{5} ) q^{35} + ( 2 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} + 2 \beta_{4} + 2 \beta_{5} ) q^{37} + ( -3 + 4 \beta_{1} + 3 \beta_{4} - \beta_{5} ) q^{39} + ( -2 - \beta_{2} + 4 \beta_{3} ) q^{41} + ( -1 - \beta_{2} + 2 \beta_{3} ) q^{43} + ( 3 - 3 \beta_{4} - \beta_{5} ) q^{45} + ( -\beta_{3} - \beta_{4} + \beta_{5} ) q^{47} + ( -3 - 3 \beta_{1} - \beta_{3} - \beta_{4} + 2 \beta_{5} ) q^{49} + ( -2 \beta_{1} - 2 \beta_{2} + \beta_{3} - \beta_{5} ) q^{51} + ( -6 + \beta_{1} + 6 \beta_{4} - 2 \beta_{5} ) q^{53} + ( -1 - \beta_{2} - \beta_{3} ) q^{55} + ( 6 - 2 \beta_{2} + \beta_{3} ) q^{57} + ( 3 - \beta_{1} - 3 \beta_{4} + 6 \beta_{5} ) q^{59} + ( -2 \beta_{3} + 8 \beta_{4} + 2 \beta_{5} ) q^{61} + ( -3 - \beta_{1} + \beta_{2} + 3 \beta_{4} + \beta_{5} ) q^{63} + ( -6 \beta_{1} - 6 \beta_{2} - 4 \beta_{3} - 7 \beta_{4} + 4 \beta_{5} ) q^{65} + ( -6 + 2 \beta_{1} + 6 \beta_{4} ) q^{67} + ( 3 \beta_{2} + \beta_{3} ) q^{69} + ( -1 + 4 \beta_{2} + 3 \beta_{3} ) q^{71} + ( 5 + 5 \beta_{1} - 5 \beta_{4} ) q^{73} + ( 2 \beta_{1} + 2 \beta_{2} + 3 \beta_{3} + 9 \beta_{4} - 3 \beta_{5} ) q^{75} + ( -\beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} ) q^{77} + ( -3 \beta_{1} - 3 \beta_{2} - \beta_{3} + \beta_{5} ) q^{79} + ( 6 - \beta_{1} - 6 \beta_{4} + 4 \beta_{5} ) q^{81} + ( 3 - 2 \beta_{2} + \beta_{3} ) q^{83} + ( 4 + \beta_{2} ) q^{85} + ( 9 + 3 \beta_{1} - 9 \beta_{4} + 4 \beta_{5} ) q^{87} + ( \beta_{1} + \beta_{2} + \beta_{3} - \beta_{5} ) q^{89} + ( 7 - 4 \beta_{1} + \beta_{2} + 3 \beta_{3} - 7 \beta_{4} - 2 \beta_{5} ) q^{91} + ( \beta_{1} + \beta_{2} + 4 \beta_{3} + 9 \beta_{4} - 4 \beta_{5} ) q^{93} + ( -6 + \beta_{1} + 6 \beta_{4} + 4 \beta_{5} ) q^{95} + ( 4 + 3 \beta_{2} ) q^{97} + \beta_{3} q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6q - q^{3} + 2q^{5} - 2q^{7} + O(q^{10})$$ $$6q - q^{3} + 2q^{5} - 2q^{7} - 3q^{11} - 22q^{13} + 14q^{15} + 3q^{17} - 11q^{19} + 10q^{21} + 12q^{23} - 3q^{25} - 4q^{27} - 18q^{29} - 3q^{31} - q^{33} - 9q^{35} + 4q^{37} - 5q^{39} - 10q^{41} - 4q^{43} + 9q^{45} - 3q^{47} - 24q^{49} + 2q^{51} - 17q^{53} - 4q^{55} + 40q^{57} + 8q^{59} + 24q^{61} - 12q^{63} - 15q^{65} - 16q^{67} - 6q^{69} - 14q^{71} + 20q^{73} + 25q^{75} - 2q^{77} + 3q^{79} + 17q^{81} + 22q^{83} + 22q^{85} + 30q^{87} - q^{89} + 15q^{91} + 26q^{93} - 17q^{95} + 18q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} - x^{5} + 5 x^{4} - 2 x^{3} + 19 x^{2} - 12 x + 9$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{5} - 5 \nu^{4} + 25 \nu^{3} - 19 \nu^{2} + 12 \nu - 60$$$$)/83$$ $$\beta_{3}$$ $$=$$ $$($$$$4 \nu^{5} - 20 \nu^{4} + 17 \nu^{3} - 76 \nu^{2} + 48 \nu - 240$$$$)/83$$ $$\beta_{4}$$ $$=$$ $$($$$$-20 \nu^{5} + 17 \nu^{4} - 85 \nu^{3} - 35 \nu^{2} - 323 \nu + 204$$$$)/249$$ $$\beta_{5}$$ $$=$$ $$($$$$-16 \nu^{5} - 3 \nu^{4} - 68 \nu^{3} - 28 \nu^{2} - 275 \nu - 36$$$$)/83$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{5} - 3 \beta_{4} - \beta_{3}$$ $$\nu^{3}$$ $$=$$ $$-\beta_{3} + 4 \beta_{2}$$ $$\nu^{4}$$ $$=$$ $$-5 \beta_{5} + 12 \beta_{4} - \beta_{1} - 12$$ $$\nu^{5}$$ $$=$$ $$-6 \beta_{5} + 3 \beta_{4} + 6 \beta_{3} - 17 \beta_{2} - 17 \beta_{1}$$

## Character values

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

 $$n$$ $$309$$ $$353$$ $$463$$ $$673$$ $$\chi(n)$$ $$1$$ $$-\beta_{4}$$ $$1$$ $$1$$

## 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
 1.09935 + 1.90412i 0.356769 + 0.617942i −0.956115 − 1.65604i 1.09935 − 1.90412i 0.356769 − 0.617942i −0.956115 + 1.65604i
0 −1.09935 1.90412i 0 0.317776 0.550404i 0 −0.317776 + 2.62660i 0 −0.917122 + 1.58850i 0
177.2 0 −0.356769 0.617942i 0 −1.10220 + 1.90907i 0 1.10220 2.40523i 0 1.24543 2.15715i 0
177.3 0 0.956115 + 1.65604i 0 1.78442 3.09071i 0 −1.78442 1.95341i 0 −0.328310 + 0.568650i 0
529.1 0 −1.09935 + 1.90412i 0 0.317776 + 0.550404i 0 −0.317776 2.62660i 0 −0.917122 1.58850i 0
529.2 0 −0.356769 + 0.617942i 0 −1.10220 1.90907i 0 1.10220 + 2.40523i 0 1.24543 + 2.15715i 0
529.3 0 0.956115 1.65604i 0 1.78442 + 3.09071i 0 −1.78442 + 1.95341i 0 −0.328310 0.568650i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 529.3 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.k 6
4.b odd 2 1 77.2.e.b 6
7.c even 3 1 inner 1232.2.q.k 6
7.c even 3 1 8624.2.a.cl 3
7.d odd 6 1 8624.2.a.ck 3
12.b even 2 1 693.2.i.g 6
28.d even 2 1 539.2.e.l 6
28.f even 6 1 539.2.a.i 3
28.f even 6 1 539.2.e.l 6
28.g odd 6 1 77.2.e.b 6
28.g odd 6 1 539.2.a.h 3
44.c even 2 1 847.2.e.d 6
44.g even 10 4 847.2.n.d 24
44.h odd 10 4 847.2.n.e 24
84.j odd 6 1 4851.2.a.bn 3
84.n even 6 1 693.2.i.g 6
84.n even 6 1 4851.2.a.bo 3
308.m odd 6 1 5929.2.a.w 3
308.n even 6 1 847.2.e.d 6
308.n even 6 1 5929.2.a.v 3
308.bb odd 30 4 847.2.n.e 24
308.bc even 30 4 847.2.n.d 24

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
77.2.e.b 6 4.b odd 2 1
77.2.e.b 6 28.g odd 6 1
539.2.a.h 3 28.g odd 6 1
539.2.a.i 3 28.f even 6 1
539.2.e.l 6 28.d even 2 1
539.2.e.l 6 28.f even 6 1
693.2.i.g 6 12.b even 2 1
693.2.i.g 6 84.n even 6 1
847.2.e.d 6 44.c even 2 1
847.2.e.d 6 308.n even 6 1
847.2.n.d 24 44.g even 10 4
847.2.n.d 24 308.bc even 30 4
847.2.n.e 24 44.h odd 10 4
847.2.n.e 24 308.bb odd 30 4
1232.2.q.k 6 1.a even 1 1 trivial
1232.2.q.k 6 7.c even 3 1 inner
4851.2.a.bn 3 84.j odd 6 1
4851.2.a.bo 3 84.n even 6 1
5929.2.a.v 3 308.n even 6 1
5929.2.a.w 3 308.m odd 6 1
8624.2.a.ck 3 7.d odd 6 1
8624.2.a.cl 3 7.c even 3 1

## Hecke kernels

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

 $$T_{3}^{6} + T_{3}^{5} + 5 T_{3}^{4} + 2 T_{3}^{3} + 19 T_{3}^{2} + 12 T_{3} + 9$$ $$T_{13}^{3} + 11 T_{13}^{2} + 36 T_{13} + 35$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{6}$$
$3$ $$9 + 12 T + 19 T^{2} + 2 T^{3} + 5 T^{4} + T^{5} + T^{6}$$
$5$ $$25 - 35 T + 59 T^{2} + 4 T^{3} + 11 T^{4} - 2 T^{5} + T^{6}$$
$7$ $$343 + 98 T + 98 T^{2} + 23 T^{3} + 14 T^{4} + 2 T^{5} + T^{6}$$
$11$ $$( 1 + T + T^{2} )^{3}$$
$13$ $$( 35 + 36 T + 11 T^{2} + T^{3} )^{2}$$
$17$ $$49 - 14 T + 25 T^{2} - 8 T^{3} + 11 T^{4} - 3 T^{5} + T^{6}$$
$19$ $$3249 - 1140 T + 1027 T^{2} + 334 T^{3} + 101 T^{4} + 11 T^{5} + T^{6}$$
$23$ $$2209 - 2021 T + 1285 T^{2} - 422 T^{3} + 101 T^{4} - 12 T^{5} + T^{6}$$
$29$ $$( -53 - 20 T + 9 T^{2} + T^{3} )^{2}$$
$31$ $$11449 + 4708 T + 2257 T^{2} + 82 T^{3} + 53 T^{4} + 3 T^{5} + T^{6}$$
$37$ $$23104 - 5472 T + 1904 T^{2} - 160 T^{3} + 52 T^{4} - 4 T^{5} + T^{6}$$
$41$ $$( -109 - 80 T + 5 T^{2} + T^{3} )^{2}$$
$43$ $$( -41 - 25 T + 2 T^{2} + T^{3} )^{2}$$
$47$ $$49 + 14 T + 25 T^{2} + 8 T^{3} + 11 T^{4} + 3 T^{5} + T^{6}$$
$53$ $$441 + 1554 T + 5119 T^{2} + 1216 T^{3} + 215 T^{4} + 17 T^{5} + T^{6}$$
$59$ $$1750329 - 207711 T + 35233 T^{2} - 1390 T^{3} + 221 T^{4} - 8 T^{5} + T^{6}$$
$61$ $$141376 - 64672 T + 20560 T^{2} - 3376 T^{3} + 404 T^{4} - 24 T^{5} + T^{6}$$
$67$ $$5184 + 4896 T + 3472 T^{2} + 944 T^{3} + 188 T^{4} + 16 T^{5} + T^{6}$$
$71$ $$( -419 - 86 T + 7 T^{2} + T^{3} )^{2}$$
$73$ $$390625 + 15625 T + 13125 T^{2} - 1750 T^{3} + 375 T^{4} - 20 T^{5} + T^{6}$$
$79$ $$19881 - 5358 T + 1867 T^{2} - 168 T^{3} + 47 T^{4} - 3 T^{5} + T^{6}$$
$83$ $$( 3 + 16 T - 11 T^{2} + T^{3} )^{2}$$
$89$ $$9 + 24 T + 67 T^{2} - 2 T^{3} + 9 T^{4} + T^{5} + T^{6}$$
$97$ $$( 47 - 12 T - 9 T^{2} + T^{3} )^{2}$$