# Properties

 Label 1232.2.q.l 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: 6.0.309123.1 Defining polynomial: $$x^{6} - 3 x^{5} + 10 x^{4} - 15 x^{3} + 19 x^{2} - 12 x + 3$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$3$$ Twist minimal: no (minimal twist has level 308) 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.

 $$f(q)$$ $$=$$ $$q + \beta_{2} q^{3} + ( -\beta_{1} - \beta_{2} - \beta_{3} + \beta_{5} ) q^{5} + ( -1 - \beta_{1} + \beta_{2} + \beta_{5} ) q^{7} + ( -\beta_{1} + 2 \beta_{2} + 2 \beta_{3} - \beta_{4} + \beta_{5} ) q^{9} +O(q^{10})$$ $$q + \beta_{2} q^{3} + ( -\beta_{1} - \beta_{2} - \beta_{3} + \beta_{5} ) q^{5} + ( -1 - \beta_{1} + \beta_{2} + \beta_{5} ) q^{7} + ( -\beta_{1} + 2 \beta_{2} + 2 \beta_{3} - \beta_{4} + \beta_{5} ) q^{9} + ( -1 + \beta_{4} ) q^{11} + ( -2 \beta_{1} - \beta_{3} ) q^{13} + ( 5 + 2 \beta_{1} - 2 \beta_{3} ) q^{15} + ( 2 - 2 \beta_{4} + \beta_{5} ) q^{17} + ( \beta_{1} + 4 \beta_{4} - \beta_{5} ) q^{19} + ( 1 + \beta_{2} + 2 \beta_{3} - 4 \beta_{4} + \beta_{5} ) q^{21} + ( \beta_{1} - \beta_{4} - \beta_{5} ) q^{23} + ( -4 - \beta_{2} + 4 \beta_{4} - 2 \beta_{5} ) q^{25} + ( -7 - \beta_{1} + 2 \beta_{3} ) q^{27} + ( 1 + 3 \beta_{1} + \beta_{3} ) q^{29} + ( 7 + \beta_{2} - 7 \beta_{4} + 3 \beta_{5} ) q^{31} + ( -\beta_{2} - \beta_{3} ) q^{33} + ( 1 + 3 \beta_{1} + 2 \beta_{2} - \beta_{3} + 4 \beta_{4} - \beta_{5} ) q^{35} + ( 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} + 4 \beta_{4} - 2 \beta_{5} ) q^{37} + ( 6 + 2 \beta_{2} - 6 \beta_{4} + 3 \beta_{5} ) q^{39} + ( 6 + 2 \beta_{1} - \beta_{3} ) q^{41} + ( -3 - 2 \beta_{1} + 3 \beta_{3} ) q^{43} + ( 6 + 6 \beta_{2} - 6 \beta_{4} + 3 \beta_{5} ) q^{45} + ( -\beta_{1} + 4 \beta_{4} + \beta_{5} ) q^{47} + ( 3 \beta_{1} + \beta_{2} + 2 \beta_{3} - \beta_{4} ) q^{49} + ( \beta_{1} + 2 \beta_{2} + 2 \beta_{3} + \beta_{4} - \beta_{5} ) q^{51} + ( -2 + \beta_{2} + 2 \beta_{4} + 4 \beta_{5} ) q^{53} + ( \beta_{1} + \beta_{3} ) q^{55} + ( -1 - \beta_{1} - 4 \beta_{3} ) q^{57} + ( 3 - 3 \beta_{2} - 3 \beta_{4} ) q^{59} + ( 2 \beta_{1} - 4 \beta_{2} - 4 \beta_{3} - 2 \beta_{4} - 2 \beta_{5} ) q^{61} + ( -8 - \beta_{2} + 3 \beta_{3} + 2 \beta_{4} + \beta_{5} ) q^{63} + ( 2 \beta_{1} + 13 \beta_{4} - 2 \beta_{5} ) q^{65} + ( 2 - 2 \beta_{2} - 2 \beta_{4} - 4 \beta_{5} ) q^{67} + ( -1 - \beta_{1} + \beta_{3} ) q^{69} + ( 4 - \beta_{1} ) q^{71} + ( -7 - \beta_{2} + 7 \beta_{4} ) q^{73} + ( -\beta_{1} - 6 \beta_{2} - 6 \beta_{3} + 2 \beta_{4} + \beta_{5} ) q^{75} + ( 1 + \beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} ) q^{77} + ( -5 \beta_{1} + 3 \beta_{2} + 3 \beta_{3} + \beta_{4} + 5 \beta_{5} ) q^{79} + ( -4 - 5 \beta_{2} + 4 \beta_{4} + 2 \beta_{5} ) q^{81} + ( -2 + \beta_{1} - 4 \beta_{3} ) q^{83} + ( -4 - 2 \beta_{1} - 3 \beta_{3} ) q^{85} + ( -7 - \beta_{2} + 7 \beta_{4} - 4 \beta_{5} ) q^{87} + ( 3 \beta_{1} + 3 \beta_{2} + 3 \beta_{3} + 3 \beta_{4} - 3 \beta_{5} ) q^{89} + ( 6 + \beta_{1} + 4 \beta_{2} + 3 \beta_{3} + \beta_{4} + 4 \beta_{5} ) q^{91} + ( 2 \beta_{1} + 9 \beta_{2} + 9 \beta_{3} - \beta_{4} - 2 \beta_{5} ) q^{93} + ( 4 - 5 \beta_{2} - 4 \beta_{4} + 4 \beta_{5} ) q^{95} + ( 10 + 6 \beta_{1} - \beta_{3} ) q^{97} + ( 1 + \beta_{1} - 2 \beta_{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q + q^{3} + 2 q^{5} - 4 q^{7} - 4 q^{9} + O(q^{10})$$ $$6 q + q^{3} + 2 q^{5} - 4 q^{7} - 4 q^{9} - 3 q^{11} + 6 q^{13} + 30 q^{15} + 5 q^{17} + 11 q^{19} - 10 q^{21} - 4 q^{23} - 11 q^{25} - 44 q^{27} - 2 q^{29} + 19 q^{31} + q^{33} + 17 q^{35} + 8 q^{37} + 17 q^{39} + 34 q^{41} - 20 q^{43} + 21 q^{45} + 13 q^{47} - 12 q^{49} - 9 q^{53} - 4 q^{55} + 4 q^{57} + 6 q^{59} - 4 q^{61} - 50 q^{63} + 37 q^{65} + 8 q^{67} - 6 q^{69} + 26 q^{71} - 22 q^{73} + 13 q^{75} + 2 q^{77} + 5 q^{79} - 19 q^{81} - 6 q^{83} - 14 q^{85} - 18 q^{87} + 3 q^{89} + 31 q^{91} - 14 q^{93} + 3 q^{95} + 50 q^{97} + 8 q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} - 3 x^{5} + 10 x^{4} - 15 x^{3} + 19 x^{2} - 12 x + 3$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu^{2} - \nu + 2$$ $$\beta_{2}$$ $$=$$ $$($$$$-\nu^{5} + \nu^{4} - 8 \nu^{3} + 5 \nu^{2} - 18 \nu + 6$$$$)/3$$ $$\beta_{3}$$ $$=$$ $$\nu^{4} - 2 \nu^{3} + 6 \nu^{2} - 5 \nu + 3$$ $$\beta_{4}$$ $$=$$ $$($$$$-2 \nu^{5} + 5 \nu^{4} - 16 \nu^{3} + 19 \nu^{2} - 21 \nu + 9$$$$)/3$$ $$\beta_{5}$$ $$=$$ $$($$$$2 \nu^{5} - 5 \nu^{4} + 19 \nu^{3} - 22 \nu^{2} + 30 \nu - 9$$$$)/3$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$-2 \beta_{5} - \beta_{4} - \beta_{3} - 2 \beta_{2} + \beta_{1} + 2$$$$)/3$$ $$\nu^{2}$$ $$=$$ $$($$$$-2 \beta_{5} - \beta_{4} - \beta_{3} - 2 \beta_{2} + 4 \beta_{1} - 4$$$$)/3$$ $$\nu^{3}$$ $$=$$ $$($$$$7 \beta_{5} + 5 \beta_{4} + 2 \beta_{3} + 4 \beta_{2} + \beta_{1} - 10$$$$)/3$$ $$\nu^{4}$$ $$=$$ $$($$$$16 \beta_{5} + 11 \beta_{4} + 8 \beta_{3} + 10 \beta_{2} - 17 \beta_{1} + 5$$$$)/3$$ $$\nu^{5}$$ $$=$$ $$($$$$-14 \beta_{5} - 16 \beta_{4} + 5 \beta_{3} - 5 \beta_{2} - 23 \beta_{1} + 47$$$$)/3$$

## Character values

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

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

## 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
 0.5 + 0.224437i 0.5 + 2.05195i 0.5 − 1.41036i 0.5 − 0.224437i 0.5 − 2.05195i 0.5 + 1.41036i
0 −0.794182 1.37556i 0 −1.64400 + 2.84748i 0 −2.64400 + 0.0963576i 0 0.238550 0.413181i 0
177.2 0 −0.296790 0.514055i 0 0.933463 1.61680i 0 −0.0665372 2.64491i 0 1.32383 2.29294i 0
177.3 0 1.59097 + 2.75564i 0 1.71053 2.96273i 0 0.710533 + 2.54856i 0 −3.56238 + 6.17023i 0
529.1 0 −0.794182 + 1.37556i 0 −1.64400 2.84748i 0 −2.64400 0.0963576i 0 0.238550 + 0.413181i 0
529.2 0 −0.296790 + 0.514055i 0 0.933463 + 1.61680i 0 −0.0665372 + 2.64491i 0 1.32383 + 2.29294i 0
529.3 0 1.59097 2.75564i 0 1.71053 + 2.96273i 0 0.710533 2.54856i 0 −3.56238 6.17023i 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.l 6
4.b odd 2 1 308.2.i.a 6
7.c even 3 1 inner 1232.2.q.l 6
7.c even 3 1 8624.2.a.ci 3
7.d odd 6 1 8624.2.a.cn 3
12.b even 2 1 2772.2.s.f 6
28.d even 2 1 2156.2.i.l 6
28.f even 6 1 2156.2.a.h 3
28.f even 6 1 2156.2.i.l 6
28.g odd 6 1 308.2.i.a 6
28.g odd 6 1 2156.2.a.i 3
84.n even 6 1 2772.2.s.f 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
308.2.i.a 6 4.b odd 2 1
308.2.i.a 6 28.g odd 6 1
1232.2.q.l 6 1.a even 1 1 trivial
1232.2.q.l 6 7.c even 3 1 inner
2156.2.a.h 3 28.f even 6 1
2156.2.a.i 3 28.g odd 6 1
2156.2.i.l 6 28.d even 2 1
2156.2.i.l 6 28.f even 6 1
2772.2.s.f 6 12.b even 2 1
2772.2.s.f 6 84.n even 6 1
8624.2.a.ci 3 7.c even 3 1
8624.2.a.cn 3 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 $$S_{2}^{\mathrm{new}}(1232, [\chi])$$:

 $$T_{3}^{6} - T_{3}^{5} + 7 T_{3}^{4} + 12 T_{3}^{3} + 33 T_{3}^{2} + 18 T_{3} + 9$$ $$T_{13}^{3} - 3 T_{13}^{2} - 24 T_{13} + 79$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{6}$$
$3$ $$9 + 18 T + 33 T^{2} + 12 T^{3} + 7 T^{4} - T^{5} + T^{6}$$
$5$ $$441 - 231 T + 163 T^{2} - 20 T^{3} + 15 T^{4} - 2 T^{5} + T^{6}$$
$7$ $$343 + 196 T + 98 T^{2} + 55 T^{3} + 14 T^{4} + 4 T^{5} + T^{6}$$
$11$ $$( 1 + T + T^{2} )^{3}$$
$13$ $$( 79 - 24 T - 3 T^{2} + T^{3} )^{2}$$
$17$ $$9 + 12 T + 31 T^{2} - 26 T^{3} + 21 T^{4} - 5 T^{5} + T^{6}$$
$19$ $$1089 - 1188 T + 933 T^{2} - 330 T^{3} + 85 T^{4} - 11 T^{5} + T^{6}$$
$23$ $$9 - 3 T + 13 T^{2} + 10 T^{3} + 15 T^{4} + 4 T^{5} + T^{6}$$
$29$ $$( -129 - 50 T + T^{2} + T^{3} )^{2}$$
$31$ $$9409 + 7760 T + 8243 T^{2} - 1714 T^{3} + 281 T^{4} - 19 T^{5} + T^{6}$$
$37$ $$64 - 224 T + 848 T^{2} + 208 T^{3} + 92 T^{4} - 8 T^{5} + T^{6}$$
$41$ $$( -33 + 76 T - 17 T^{2} + T^{3} )^{2}$$
$43$ $$( -73 - 31 T + 10 T^{2} + T^{3} )^{2}$$
$47$ $$3969 - 3276 T + 1885 T^{2} - 550 T^{3} + 117 T^{4} - 13 T^{5} + T^{6}$$
$53$ $$81 - 378 T + 1683 T^{2} - 396 T^{3} + 123 T^{4} + 9 T^{5} + T^{6}$$
$59$ $$59049 - 10935 T + 3483 T^{2} - 216 T^{3} + 81 T^{4} - 6 T^{5} + T^{6}$$
$61$ $$222784 + 47200 T + 11888 T^{2} + 544 T^{3} + 116 T^{4} + 4 T^{5} + T^{6}$$
$67$ $$5184 + 4320 T + 3024 T^{2} + 624 T^{3} + 124 T^{4} - 8 T^{5} + T^{6}$$
$71$ $$( -63 + 52 T - 13 T^{2} + T^{3} )^{2}$$
$73$ $$124609 + 54715 T + 16259 T^{2} + 2704 T^{3} + 329 T^{4} + 22 T^{5} + T^{6}$$
$79$ $$91809 + 39996 T + 15909 T^{2} + 1266 T^{3} + 157 T^{4} - 5 T^{5} + T^{6}$$
$83$ $$( -477 - 96 T + 3 T^{2} + T^{3} )^{2}$$
$89$ $$59049 + 26244 T + 10935 T^{2} + 810 T^{3} + 117 T^{4} - 3 T^{5} + T^{6}$$
$97$ $$( 1171 + 56 T - 25 T^{2} + T^{3} )^{2}$$