# Properties

 Label 231.2.l.a Level 231 Weight 2 Character orbit 231.l Analytic conductor 1.845 Analytic rank 0 Dimension 8 CM no Inner twists 8

# Related objects

## Newspace parameters

 Level: $$N$$ = $$231 = 3 \cdot 7 \cdot 11$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 231.l (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$1.84454428669$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$4$$ over $$\Q(\zeta_{6})$$ Coefficient field: 8.0.12745506816.5 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( \beta_{2} + \beta_{4} ) q^{3} + 2 \beta_{4} q^{4} + ( -\beta_{2} - \beta_{7} ) q^{5} + ( -\beta_{3} - \beta_{6} ) q^{7} + ( 1 + 2 \beta_{2} - \beta_{4} + 2 \beta_{7} ) q^{9} +O(q^{10})$$ $$q + ( \beta_{2} + \beta_{4} ) q^{3} + 2 \beta_{4} q^{4} + ( -\beta_{2} - \beta_{7} ) q^{5} + ( -\beta_{3} - \beta_{6} ) q^{7} + ( 1 + 2 \beta_{2} - \beta_{4} + 2 \beta_{7} ) q^{9} + ( \beta_{1} + 2 \beta_{5} - \beta_{7} ) q^{11} + ( -2 + 2 \beta_{2} + 2 \beta_{4} + 2 \beta_{7} ) q^{12} + ( \beta_{3} + 2 \beta_{6} ) q^{13} + ( -2 - \beta_{7} ) q^{15} + ( -4 + 4 \beta_{4} ) q^{16} + ( -2 \beta_{1} + \beta_{2} - 4 \beta_{5} + 2 \beta_{7} ) q^{17} + ( \beta_{3} - \beta_{6} ) q^{19} -2 \beta_{7} q^{20} + ( -\beta_{3} + 2 \beta_{5} - \beta_{7} ) q^{21} + ( -4 \beta_{2} - 4 \beta_{7} ) q^{23} -3 \beta_{4} q^{25} + ( 5 + \beta_{7} ) q^{27} -2 \beta_{3} q^{28} + ( 4 \beta_{1} - 2 \beta_{2} + 2 \beta_{5} - \beta_{7} ) q^{29} + \beta_{4} q^{31} + ( 1 - \beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} + \beta_{5} - \beta_{6} ) q^{33} + ( 2 \beta_{1} - \beta_{2} ) q^{35} + ( 2 + 4 \beta_{7} ) q^{36} + ( 7 - 7 \beta_{4} ) q^{37} + ( -2 \beta_{1} + \beta_{2} + 2 \beta_{3} - 4 \beta_{5} + \beta_{6} + 2 \beta_{7} ) q^{39} + ( -4 \beta_{1} + 2 \beta_{2} - 2 \beta_{5} + \beta_{7} ) q^{41} + ( -\beta_{3} - 2 \beta_{6} ) q^{43} + ( -2 \beta_{1} + 2 \beta_{2} + 2 \beta_{5} ) q^{44} + ( -\beta_{2} - 4 \beta_{4} ) q^{45} + ( -4 \beta_{2} - 4 \beta_{7} ) q^{47} + ( -4 + 4 \beta_{7} ) q^{48} -7 \beta_{4} q^{49} + ( 2 \beta_{1} - \beta_{2} - 2 \beta_{3} - 2 \beta_{5} + 2 \beta_{6} + \beta_{7} ) q^{51} + ( 4 \beta_{3} + 2 \beta_{6} ) q^{52} -5 \beta_{2} q^{53} + ( -1 + \beta_{3} + 2 \beta_{6} ) q^{55} + ( 4 \beta_{1} - 2 \beta_{2} - \beta_{3} + 2 \beta_{5} - 2 \beta_{6} - \beta_{7} ) q^{57} -2 \beta_{2} q^{59} + ( 2 \beta_{2} - 4 \beta_{4} ) q^{60} + ( -4 \beta_{1} + 2 \beta_{2} - \beta_{6} ) q^{63} -8 q^{64} + ( -2 \beta_{1} + \beta_{2} + 2 \beta_{5} - \beta_{7} ) q^{65} + \beta_{4} q^{67} + ( 4 \beta_{1} - 2 \beta_{2} - 4 \beta_{5} + 2 \beta_{7} ) q^{68} + ( -8 - 4 \beta_{7} ) q^{69} + \beta_{7} q^{71} + ( 2 \beta_{3} + \beta_{6} ) q^{73} + ( 3 - 3 \beta_{2} - 3 \beta_{4} - 3 \beta_{7} ) q^{75} + ( -2 \beta_{3} - 4 \beta_{6} ) q^{76} + ( -7 \beta_{2} + \beta_{5} - 4 \beta_{7} ) q^{77} + ( -3 \beta_{3} + 3 \beta_{6} ) q^{79} + 4 \beta_{2} q^{80} + ( 4 \beta_{2} + 7 \beta_{4} ) q^{81} + ( 4 \beta_{1} - 2 \beta_{2} + 2 \beta_{5} - \beta_{7} ) q^{83} + ( -4 \beta_{1} + 2 \beta_{2} + 2 \beta_{6} ) q^{84} + ( -2 \beta_{3} - 4 \beta_{6} ) q^{85} + ( 2 \beta_{1} - \beta_{2} + 4 \beta_{3} + 4 \beta_{5} + 2 \beta_{6} - 2 \beta_{7} ) q^{87} + ( 11 \beta_{2} + 11 \beta_{7} ) q^{89} + ( 7 + 7 \beta_{4} ) q^{91} -8 \beta_{7} q^{92} + ( -1 + \beta_{2} + \beta_{4} + \beta_{7} ) q^{93} + ( -2 \beta_{1} + \beta_{2} - 4 \beta_{5} + 2 \beta_{7} ) q^{95} + 8 q^{97} + ( 2 + 2 \beta_{1} - \beta_{2} - 2 \beta_{3} + \beta_{5} - 4 \beta_{6} - \beta_{7} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q + 4q^{3} + 8q^{4} + 4q^{9} + O(q^{10})$$ $$8q + 4q^{3} + 8q^{4} + 4q^{9} - 8q^{12} - 16q^{15} - 16q^{16} - 12q^{25} + 40q^{27} + 4q^{31} + 4q^{33} + 16q^{36} + 28q^{37} - 16q^{45} - 32q^{48} - 28q^{49} - 8q^{55} - 16q^{60} - 64q^{64} + 4q^{67} - 64q^{69} + 12q^{75} + 28q^{81} + 84q^{91} - 4q^{93} + 64q^{97} + 16q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 8 x^{6} + 55 x^{4} - 72 x^{2} + 81$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$-\nu^{7} - 203 \nu$$$$)/165$$ $$\beta_{3}$$ $$=$$ $$($$$$-\nu^{6} - 148$$$$)/55$$ $$\beta_{4}$$ $$=$$ $$($$$$-8 \nu^{6} + 55 \nu^{4} - 440 \nu^{2} + 576$$$$)/495$$ $$\beta_{5}$$ $$=$$ $$($$$$-8 \nu^{7} + 55 \nu^{5} - 440 \nu^{3} + 81 \nu$$$$)/495$$ $$\beta_{6}$$ $$=$$ $$($$$$-23 \nu^{6} + 220 \nu^{4} - 1265 \nu^{2} + 1656$$$$)/495$$ $$\beta_{7}$$ $$=$$ $$($$$$8 \nu^{7} - 55 \nu^{5} + 341 \nu^{3} - 81 \nu$$$$)/297$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{6} - 4 \beta_{4} + \beta_{3} + 4$$ $$\nu^{3}$$ $$=$$ $$-3 \beta_{7} - 5 \beta_{5}$$ $$\nu^{4}$$ $$=$$ $$8 \beta_{6} - 23 \beta_{4}$$ $$\nu^{5}$$ $$=$$ $$-24 \beta_{7} - 31 \beta_{5} - 24 \beta_{2} - 31 \beta_{1}$$ $$\nu^{6}$$ $$=$$ $$-55 \beta_{3} - 148$$ $$\nu^{7}$$ $$=$$ $$-165 \beta_{2} - 203 \beta_{1}$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/231\mathbb{Z}\right)^\times$$.

 $$n$$ $$155$$ $$199$$ $$211$$ $$\chi(n)$$ $$-1$$ $$-1 + \beta_{4}$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
32.1
 −2.23256 − 1.28897i 1.00781 + 0.581861i 2.23256 + 1.28897i −1.00781 − 0.581861i −2.23256 + 1.28897i 1.00781 − 0.581861i 2.23256 − 1.28897i −1.00781 + 0.581861i
0 −0.724745 1.57313i 1.00000 1.73205i 1.22474 0.707107i 0 −1.32288 2.29129i 0 −1.94949 + 2.28024i 0
32.2 0 −0.724745 1.57313i 1.00000 1.73205i 1.22474 0.707107i 0 1.32288 + 2.29129i 0 −1.94949 + 2.28024i 0
32.3 0 1.72474 0.158919i 1.00000 1.73205i −1.22474 + 0.707107i 0 −1.32288 2.29129i 0 2.94949 0.548188i 0
32.4 0 1.72474 0.158919i 1.00000 1.73205i −1.22474 + 0.707107i 0 1.32288 + 2.29129i 0 2.94949 0.548188i 0
65.1 0 −0.724745 + 1.57313i 1.00000 + 1.73205i 1.22474 + 0.707107i 0 −1.32288 + 2.29129i 0 −1.94949 2.28024i 0
65.2 0 −0.724745 + 1.57313i 1.00000 + 1.73205i 1.22474 + 0.707107i 0 1.32288 2.29129i 0 −1.94949 2.28024i 0
65.3 0 1.72474 + 0.158919i 1.00000 + 1.73205i −1.22474 0.707107i 0 −1.32288 + 2.29129i 0 2.94949 + 0.548188i 0
65.4 0 1.72474 + 0.158919i 1.00000 + 1.73205i −1.22474 0.707107i 0 1.32288 2.29129i 0 2.94949 + 0.548188i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 65.4 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
7.c even 3 1 inner
11.b odd 2 1 inner
21.h odd 6 1 inner
33.d even 2 1 inner
77.h odd 6 1 inner
231.l even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 231.2.l.a 8
3.b odd 2 1 inner 231.2.l.a 8
7.c even 3 1 inner 231.2.l.a 8
11.b odd 2 1 inner 231.2.l.a 8
21.h odd 6 1 inner 231.2.l.a 8
33.d even 2 1 inner 231.2.l.a 8
77.h odd 6 1 inner 231.2.l.a 8
231.l even 6 1 inner 231.2.l.a 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
231.2.l.a 8 1.a even 1 1 trivial
231.2.l.a 8 3.b odd 2 1 inner
231.2.l.a 8 7.c even 3 1 inner
231.2.l.a 8 11.b odd 2 1 inner
231.2.l.a 8 21.h odd 6 1 inner
231.2.l.a 8 33.d even 2 1 inner
231.2.l.a 8 77.h odd 6 1 inner
231.2.l.a 8 231.l even 6 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}$$ acting on $$S_{2}^{\mathrm{new}}(231, [\chi])$$.

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$( 1 - 2 T^{2} + 4 T^{4} )^{4}$$
$3$ $$( 1 - 2 T + T^{2} - 6 T^{3} + 9 T^{4} )^{2}$$
$5$ $$( 1 + 8 T^{2} + 39 T^{4} + 200 T^{6} + 625 T^{8} )^{2}$$
$7$ $$( 1 + 7 T^{2} + 49 T^{4} )^{2}$$
$11$ $$1 + 20 T^{2} + 279 T^{4} + 2420 T^{6} + 14641 T^{8}$$
$13$ $$( 1 - 5 T^{2} + 169 T^{4} )^{4}$$
$17$ $$( 1 + 8 T^{2} - 225 T^{4} + 2312 T^{6} + 83521 T^{8} )^{2}$$
$19$ $$( 1 + 17 T^{2} - 72 T^{4} + 6137 T^{6} + 130321 T^{8} )^{2}$$
$23$ $$( 1 + 14 T^{2} - 333 T^{4} + 7406 T^{6} + 279841 T^{8} )^{2}$$
$29$ $$( 1 + 16 T^{2} + 841 T^{4} )^{4}$$
$31$ $$( 1 - T - 30 T^{2} - 31 T^{3} + 961 T^{4} )^{4}$$
$37$ $$( 1 - 7 T + 12 T^{2} - 259 T^{3} + 1369 T^{4} )^{4}$$
$41$ $$( 1 + 40 T^{2} + 1681 T^{4} )^{4}$$
$43$ $$( 1 - 65 T^{2} + 1849 T^{4} )^{4}$$
$47$ $$( 1 + 62 T^{2} + 1635 T^{4} + 136958 T^{6} + 4879681 T^{8} )^{2}$$
$53$ $$( 1 + 56 T^{2} + 327 T^{4} + 157304 T^{6} + 7890481 T^{8} )^{2}$$
$59$ $$( 1 + 110 T^{2} + 8619 T^{4} + 382910 T^{6} + 12117361 T^{8} )^{2}$$
$61$ $$( 1 + 61 T^{2} + 3721 T^{4} )^{4}$$
$67$ $$( 1 - T - 66 T^{2} - 67 T^{3} + 4489 T^{4} )^{4}$$
$71$ $$( 1 - 140 T^{2} + 5041 T^{4} )^{4}$$
$73$ $$( 1 + 125 T^{2} + 10296 T^{4} + 666125 T^{6} + 28398241 T^{8} )^{2}$$
$79$ $$( 1 - 31 T^{2} - 5280 T^{4} - 193471 T^{6} + 38950081 T^{8} )^{2}$$
$83$ $$( 1 + 124 T^{2} + 6889 T^{4} )^{4}$$
$89$ $$( 1 - 64 T^{2} - 3825 T^{4} - 506944 T^{6} + 62742241 T^{8} )^{2}$$
$97$ $$( 1 - 8 T + 97 T^{2} )^{8}$$