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

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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:

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

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