# Properties

 Label 1232.2.q.i 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: 6.0.64827.1 Defining polynomial: $$x^{6} - x^{5} + 3 x^{4} + 5 x^{2} - 2 x + 1$$ 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 $$1,\beta_1,\ldots,\beta_{5}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} ) q^{3} + ( 1 + \beta_{1} - 2 \beta_{4} - \beta_{5} ) q^{5} + ( 1 - \beta_{1} - \beta_{2} - \beta_{3} - 2 \beta_{4} - \beta_{5} ) q^{7} + ( 1 - 4 \beta_{1} - 3 \beta_{4} - \beta_{5} ) q^{9} +O(q^{10})$$ $$q + ( \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} ) q^{3} + ( 1 + \beta_{1} - 2 \beta_{4} - \beta_{5} ) q^{5} + ( 1 - \beta_{1} - \beta_{2} - \beta_{3} - 2 \beta_{4} - \beta_{5} ) q^{7} + ( 1 - 4 \beta_{1} - 3 \beta_{4} - \beta_{5} ) q^{9} + \beta_{5} q^{11} + ( -1 + \beta_{2} + \beta_{3} ) q^{13} + ( 1 - 4 \beta_{2} ) q^{15} + ( -2 \beta_{1} + 2 \beta_{2} - 3 \beta_{3} - 3 \beta_{4} + 2 \beta_{5} ) q^{17} + ( 2 + \beta_{4} - 2 \beta_{5} ) q^{19} + ( 1 - 3 \beta_{1} + 5 \beta_{2} - 2 \beta_{3} - \beta_{5} ) q^{21} + ( -1 - 2 \beta_{1} - \beta_{4} + \beta_{5} ) q^{23} + ( -5 \beta_{1} + 5 \beta_{2} - 3 \beta_{3} - 3 \beta_{4} - 9 \beta_{5} ) q^{25} + ( -1 + 8 \beta_{2} - 3 \beta_{3} ) q^{27} + ( \beta_{2} + 4 \beta_{3} ) q^{29} + ( -\beta_{1} + \beta_{2} + 6 \beta_{5} ) q^{31} + ( 1 + \beta_{1} + \beta_{4} - \beta_{5} ) q^{33} + ( -8 - 2 \beta_{1} + 5 \beta_{2} - 5 \beta_{3} - 5 \beta_{4} ) q^{35} + ( -6 + 2 \beta_{1} + 4 \beta_{4} + 6 \beta_{5} ) q^{37} + ( 2 \beta_{1} - 2 \beta_{2} - \beta_{3} - \beta_{4} + 2 \beta_{5} ) q^{39} + ( -3 - 3 \beta_{2} + 5 \beta_{3} ) q^{41} + ( -2 - 5 \beta_{2} + \beta_{3} ) q^{43} + ( -8 \beta_{1} + 8 \beta_{2} - 9 \beta_{3} - 9 \beta_{4} - 4 \beta_{5} ) q^{45} + ( -10 + 8 \beta_{1} + 5 \beta_{4} + 10 \beta_{5} ) q^{47} + ( -3 + 5 \beta_{1} - \beta_{2} - 4 \beta_{3} + \beta_{4} + 2 \beta_{5} ) q^{49} + ( 5 + 8 \beta_{1} + 7 \beta_{4} - 5 \beta_{5} ) q^{51} + ( 5 \beta_{1} - 5 \beta_{2} + \beta_{3} + \beta_{4} + 3 \beta_{5} ) q^{53} + ( 1 + \beta_{2} + 2 \beta_{3} ) q^{55} + ( -3 - 2 \beta_{2} + 3 \beta_{3} ) q^{57} + ( -9 \beta_{1} + 9 \beta_{2} - 3 \beta_{3} - 3 \beta_{4} - 8 \beta_{5} ) q^{59} + ( 2 - 6 \beta_{4} - 2 \beta_{5} ) q^{61} + ( -1 + 9 \beta_{1} - 4 \beta_{2} - \beta_{3} + 4 \beta_{4} - 5 \beta_{5} ) q^{63} + ( 7 - 4 \beta_{1} + 2 \beta_{4} - 7 \beta_{5} ) q^{65} + ( -6 \beta_{1} + 6 \beta_{2} - 6 \beta_{3} - 6 \beta_{4} - 4 \beta_{5} ) q^{67} + ( 2 + 7 \beta_{2} - 4 \beta_{3} ) q^{69} + ( -2 + 8 \beta_{2} - \beta_{3} ) q^{71} + ( 5 \beta_{1} - 5 \beta_{2} + 5 \beta_{3} + 5 \beta_{4} + 12 \beta_{5} ) q^{73} + ( -6 + 6 \beta_{1} - \beta_{4} + 6 \beta_{5} ) q^{75} + ( 1 + \beta_{1} - 2 \beta_{2} + \beta_{3} - \beta_{4} ) q^{77} + ( 8 - \beta_{1} - 8 \beta_{5} ) q^{79} + ( 11 \beta_{1} - 11 \beta_{2} + \beta_{3} + \beta_{4} - 5 \beta_{5} ) q^{81} + ( 8 - 8 \beta_{2} + 7 \beta_{3} ) q^{83} + ( -7 + 11 \beta_{2} - \beta_{3} ) q^{85} + ( 3 \beta_{1} - 3 \beta_{2} - 3 \beta_{3} - 3 \beta_{4} + 4 \beta_{5} ) q^{87} + ( -4 + \beta_{1} - 2 \beta_{4} + 4 \beta_{5} ) q^{89} + ( -2 - 2 \beta_{1} + 3 \beta_{2} + 2 \beta_{3} - 3 \beta_{5} ) q^{91} + ( 6 + 9 \beta_{1} + 7 \beta_{4} - 6 \beta_{5} ) q^{93} + ( 5 \beta_{1} - 5 \beta_{2} - 3 \beta_{3} - 3 \beta_{4} + 3 \beta_{5} ) q^{95} + ( -1 + 7 \beta_{2} + 3 \beta_{3} ) q^{97} + ( 1 - 4 \beta_{2} + 3 \beta_{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q - 5 q^{3} + 2 q^{5} - 4 q^{9} + O(q^{10})$$ $$6 q - 5 q^{3} + 2 q^{5} - 4 q^{9} + 3 q^{11} - 6 q^{13} - 2 q^{15} + 11 q^{17} + 7 q^{19} + 14 q^{21} - 6 q^{23} - 19 q^{25} + 16 q^{27} - 6 q^{29} + 19 q^{31} + 5 q^{33} - 35 q^{35} - 12 q^{37} + 5 q^{39} - 34 q^{41} - 24 q^{43} + 5 q^{45} - 17 q^{47} + 30 q^{51} + 3 q^{53} + 4 q^{55} - 28 q^{57} - 12 q^{59} - 14 q^{63} + 19 q^{65} + 34 q^{69} + 6 q^{71} + 26 q^{73} - 13 q^{75} + 23 q^{79} - 27 q^{81} + 18 q^{83} - 18 q^{85} + 12 q^{87} - 13 q^{89} - 21 q^{91} + 34 q^{93} + 7 q^{95} + 2 q^{97} - 8 q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} - x^{5} + 3 x^{4} + 5 x^{2} - 2 x + 1$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$-\nu^{5} + 3 \nu^{4} - 9 \nu^{3} + 5 \nu^{2} - 2 \nu + 6$$$$)/13$$ $$\beta_{3}$$ $$=$$ $$($$$$-3 \nu^{5} + 9 \nu^{4} - 14 \nu^{3} + 15 \nu^{2} - 6 \nu + 18$$$$)/13$$ $$\beta_{4}$$ $$=$$ $$($$$$-4 \nu^{5} - \nu^{4} - 10 \nu^{3} - 6 \nu^{2} - 34 \nu - 2$$$$)/13$$ $$\beta_{5}$$ $$=$$ $$($$$$-6 \nu^{5} + 5 \nu^{4} - 15 \nu^{3} - 9 \nu^{2} - 25 \nu + 10$$$$)/13$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$-\beta_{5} + \beta_{4} + \beta_{3} - \beta_{2} + \beta_{1}$$ $$\nu^{3}$$ $$=$$ $$\beta_{3} - 3 \beta_{2}$$ $$\nu^{4}$$ $$=$$ $$2 \beta_{5} - 3 \beta_{4} - 4 \beta_{1} - 2$$ $$\nu^{5}$$ $$=$$ $$\beta_{5} - 4 \beta_{4} - 4 \beta_{3} + 9 \beta_{2} - 9 \beta_{1}$$

## Character values

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

 $$n$$ $$309$$ $$353$$ $$463$$ $$673$$ $$\chi(n)$$ $$1$$ $$-1 + \beta_{5}$$ $$1$$ $$1$$

## 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
 0.900969 − 1.56052i −0.623490 + 1.07992i 0.222521 − 0.385418i 0.900969 + 1.56052i −0.623490 − 1.07992i 0.222521 + 0.385418i
0 −1.62349 2.81197i 0 0.955927 1.65571i 0 −2.20291 + 1.46533i 0 −3.77144 + 6.53232i 0
177.2 0 −0.777479 1.34663i 0 −1.92543 + 3.33494i 0 2.37047 + 1.17511i 0 0.291053 0.504118i 0
177.3 0 −0.0990311 0.171527i 0 1.96950 3.41127i 0 −0.167563 2.64044i 0 1.48039 2.56410i 0
529.1 0 −1.62349 + 2.81197i 0 0.955927 + 1.65571i 0 −2.20291 1.46533i 0 −3.77144 6.53232i 0
529.2 0 −0.777479 + 1.34663i 0 −1.92543 3.33494i 0 2.37047 1.17511i 0 0.291053 + 0.504118i 0
529.3 0 −0.0990311 + 0.171527i 0 1.96950 + 3.41127i 0 −0.167563 + 2.64044i 0 1.48039 + 2.56410i 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.i 6
4.b odd 2 1 616.2.q.c 6
7.c even 3 1 inner 1232.2.q.i 6
7.c even 3 1 8624.2.a.cq 3
7.d odd 6 1 8624.2.a.cf 3
28.f even 6 1 4312.2.a.x 3
28.g odd 6 1 616.2.q.c 6
28.g odd 6 1 4312.2.a.v 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
616.2.q.c 6 4.b odd 2 1
616.2.q.c 6 28.g odd 6 1
1232.2.q.i 6 1.a even 1 1 trivial
1232.2.q.i 6 7.c even 3 1 inner
4312.2.a.v 3 28.g odd 6 1
4312.2.a.x 3 28.f even 6 1
8624.2.a.cf 3 7.d odd 6 1
8624.2.a.cq 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 $$S_{2}^{\mathrm{new}}(1232, [\chi])$$:

 $$T_{3}^{6} + 5 T_{3}^{5} + 19 T_{3}^{4} + 28 T_{3}^{3} + 31 T_{3}^{2} + 6 T_{3} + 1$$ $$T_{13}^{3} + 3 T_{13}^{2} - 4 T_{13} + 1$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{6}$$
$3$ $$1 + 6 T + 31 T^{2} + 28 T^{3} + 19 T^{4} + 5 T^{5} + T^{6}$$
$5$ $$841 - 435 T + 283 T^{2} - 28 T^{3} + 19 T^{4} - 2 T^{5} + T^{6}$$
$7$ $$343 - 7 T^{3} + T^{6}$$
$11$ $$( 1 - T + T^{2} )^{3}$$
$13$ $$( 1 - 4 T + 3 T^{2} + T^{3} )^{2}$$
$17$ $$841 + 696 T + 895 T^{2} - 322 T^{3} + 97 T^{4} - 11 T^{5} + T^{6}$$
$19$ $$49 - 98 T + 147 T^{2} - 84 T^{3} + 35 T^{4} - 7 T^{5} + T^{6}$$
$23$ $$1 + 5 T + 19 T^{2} + 28 T^{3} + 31 T^{4} + 6 T^{5} + T^{6}$$
$29$ $$( 1 - 46 T + 3 T^{2} + T^{3} )^{2}$$
$31$ $$57121 - 28202 T + 9383 T^{2} - 1764 T^{3} + 243 T^{4} - 19 T^{5} + T^{6}$$
$37$ $$64 + 160 T + 304 T^{2} + 224 T^{3} + 124 T^{4} + 12 T^{5} + T^{6}$$
$41$ $$( -167 + 52 T + 17 T^{2} + T^{3} )^{2}$$
$43$ $$( -83 - T + 12 T^{2} + T^{3} )^{2}$$
$47$ $$851929 + 16614 T + 16015 T^{2} + 1540 T^{3} + 307 T^{4} + 17 T^{5} + T^{6}$$
$53$ $$9409 - 4462 T + 2407 T^{2} - 56 T^{3} + 55 T^{4} - 3 T^{5} + T^{6}$$
$59$ $$1042441 + 101079 T + 22053 T^{2} + 854 T^{3} + 243 T^{4} + 12 T^{5} + T^{6}$$
$61$ $$3136 + 4704 T + 7056 T^{2} + 112 T^{3} + 84 T^{4} + T^{6}$$
$67$ $$3136 + 4704 T + 7056 T^{2} + 112 T^{3} + 84 T^{4} + T^{6}$$
$71$ $$( 41 - 130 T - 3 T^{2} + T^{3} )^{2}$$
$73$ $$12769 - 18871 T + 24951 T^{2} - 4116 T^{3} + 509 T^{4} - 26 T^{5} + T^{6}$$
$79$ $$187489 - 75342 T + 20317 T^{2} - 3136 T^{3} + 355 T^{4} - 23 T^{5} + T^{6}$$
$83$ $$( 673 - 106 T - 9 T^{2} + T^{3} )^{2}$$
$89$ $$841 + 1160 T + 1223 T^{2} + 462 T^{3} + 129 T^{4} + 13 T^{5} + T^{6}$$
$97$ $$( 911 - 184 T - T^{2} + T^{3} )^{2}$$