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Properties

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

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [231,2,Mod(67,231)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(231, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 4, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("231.67");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 231 = 3 \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 231.i (of order \(3\), degree \(2\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(1.84454428669\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 2x^{7} + 8x^{6} + 21x^{4} - 4x^{3} + 28x^{2} + 12x + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

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.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + (\beta_{4} - 1) q^{3} + (\beta_{6} + \beta_{3} + \beta_{2} + \beta_1) q^{4} + (\beta_{6} - \beta_{4} - \beta_1) q^{5} - \beta_{3} q^{6} + ( - \beta_{7} - \beta_{5} - \beta_{3}) q^{7} + ( - \beta_{7} - \beta_{3} - 2 \beta_{2} + 2) q^{8} - \beta_{4} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} + (\beta_{4} - 1) q^{3} + (\beta_{6} + \beta_{3} + \beta_{2} + \beta_1) q^{4} + (\beta_{6} - \beta_{4} - \beta_1) q^{5} - \beta_{3} q^{6} + ( - \beta_{7} - \beta_{5} - \beta_{3}) q^{7} + ( - \beta_{7} - \beta_{3} - 2 \beta_{2} + 2) q^{8} - \beta_{4} q^{9} + ( - \beta_{7} - \beta_{5} + 2 \beta_{4} - 2) q^{10} + ( - \beta_{4} + 1) q^{11} + ( - \beta_{6} - \beta_1) q^{12} - \beta_{7} q^{13} + ( - \beta_{6} - 2 \beta_{4} - \beta_{3} + \cdots + 1) q^{14}+ \cdots - q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 2 q^{2} - 4 q^{3} - 4 q^{4} - 4 q^{5} + 4 q^{6} + 2 q^{7} + 24 q^{8} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 2 q^{2} - 4 q^{3} - 4 q^{4} - 4 q^{5} + 4 q^{6} + 2 q^{7} + 24 q^{8} - 4 q^{9} - 10 q^{10} + 4 q^{11} - 4 q^{12} - 4 q^{13} - 4 q^{14} + 8 q^{15} - 12 q^{16} - 2 q^{17} - 2 q^{18} - 4 q^{21} - 4 q^{22} - 4 q^{23} - 12 q^{24} - 4 q^{25} + 4 q^{26} + 8 q^{27} - 22 q^{28} + 16 q^{29} - 10 q^{30} + 12 q^{31} - 26 q^{32} + 4 q^{33} - 32 q^{34} - 2 q^{35} + 8 q^{36} + 4 q^{37} - 8 q^{38} + 2 q^{39} + 6 q^{40} + 4 q^{41} + 20 q^{42} + 36 q^{43} + 4 q^{44} - 4 q^{45} + 14 q^{46} - 12 q^{47} + 24 q^{48} - 4 q^{49} + 4 q^{50} - 2 q^{51} + 6 q^{52} + 12 q^{53} - 2 q^{54} - 8 q^{55} + 48 q^{56} + 4 q^{58} - 12 q^{59} - 2 q^{61} - 52 q^{62} + 2 q^{63} + 112 q^{64} + 4 q^{65} + 2 q^{66} - 28 q^{67} + 48 q^{68} + 8 q^{69} - 32 q^{70} + 24 q^{71} - 12 q^{72} - 6 q^{73} + 16 q^{74} - 4 q^{75} - 36 q^{76} + 4 q^{77} - 8 q^{78} - 2 q^{79} - 16 q^{80} - 4 q^{81} + 12 q^{82} - 24 q^{83} - 4 q^{84} + 36 q^{85} - 36 q^{86} - 8 q^{87} + 12 q^{88} - 8 q^{89} + 20 q^{90} + 12 q^{91} - 32 q^{92} + 12 q^{93} - 20 q^{94} - 34 q^{95} - 26 q^{96} - 88 q^{97} + 16 q^{98} - 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 2x^{7} + 8x^{6} + 21x^{4} - 4x^{3} + 28x^{2} + 12x + 9 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 68\nu^{7} - 215\nu^{6} + 357\nu^{5} + 646\nu^{4} - 1444\nu^{3} + 1156\nu^{2} + 561\nu + 5468 ) / 4243 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 84\nu^{7} - 16\nu^{6} + 441\nu^{5} + 798\nu^{4} + 3208\nu^{3} + 1428\nu^{2} + 693\nu + 2262 ) / 4243 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -754\nu^{7} + 1760\nu^{6} - 6080\nu^{5} + 1323\nu^{4} - 13440\nu^{3} + 12640\nu^{2} - 16828\nu + 5760 ) / 12729 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -815\nu^{7} + 3388\nu^{6} - 11704\nu^{5} + 15594\nu^{4} - 25872\nu^{3} + 24332\nu^{2} - 56579\nu + 11088 ) / 12729 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 1052\nu^{7} - 2827\nu^{6} + 9766\nu^{5} - 6978\nu^{4} + 21588\nu^{3} - 20303\nu^{2} + 17165\nu - 9252 ) / 12729 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -556\nu^{7} + 510\nu^{6} - 2919\nu^{5} - 5282\nu^{4} - 8909\nu^{3} - 9452\nu^{2} - 4587\nu - 13760 ) / 4243 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{6} + 2\beta_{4} + \beta_{3} + \beta_{2} + \beta _1 - 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{7} + 5\beta_{3} + 2\beta_{2} - 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -8\beta_{6} - 2\beta_{5} - 9\beta_{4} - 10\beta_1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -8\beta_{7} - 20\beta_{6} - 8\beta_{5} - 18\beta_{4} - 33\beta_{3} - 20\beta_{2} - 33\beta _1 + 18 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -20\beta_{7} - 83\beta_{3} - 61\beta_{2} + 58 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 164\beta_{6} + 61\beta_{5} + 146\beta_{4} + 243\beta_1 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

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

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

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
67.1
1.39083 + 2.40898i
0.643668 + 1.11487i
−0.276205 − 0.478401i
−0.758290 − 1.31340i
1.39083 − 2.40898i
0.643668 − 1.11487i
−0.276205 + 0.478401i
−0.758290 + 1.31340i
−1.39083 − 2.40898i −0.500000 + 0.866025i −2.86880 + 4.96890i −0.412855 − 0.715087i 2.78165 2.63323 + 0.257073i 10.3967 −0.500000 − 0.866025i −1.14842 + 1.98912i
67.2 −0.643668 − 1.11487i −0.500000 + 0.866025i 0.171383 − 0.296844i −1.95872 − 3.39260i 1.28734 −0.234193 + 2.63537i −3.01593 −0.500000 − 0.866025i −2.52153 + 4.36742i
67.3 0.276205 + 0.478401i −0.500000 + 0.866025i 0.847422 − 1.46778i −0.795012 − 1.37700i −0.552409 0.886763 − 2.49272i 2.04107 −0.500000 − 0.866025i 0.439172 − 0.760669i
67.4 0.758290 + 1.31340i −0.500000 + 0.866025i −0.150007 + 0.259820i 1.16659 + 2.02059i −1.51658 −2.28580 + 1.33233i 2.57816 −0.500000 − 0.866025i −1.76922 + 3.06438i
100.1 −1.39083 + 2.40898i −0.500000 − 0.866025i −2.86880 − 4.96890i −0.412855 + 0.715087i 2.78165 2.63323 − 0.257073i 10.3967 −0.500000 + 0.866025i −1.14842 − 1.98912i
100.2 −0.643668 + 1.11487i −0.500000 − 0.866025i 0.171383 + 0.296844i −1.95872 + 3.39260i 1.28734 −0.234193 − 2.63537i −3.01593 −0.500000 + 0.866025i −2.52153 − 4.36742i
100.3 0.276205 − 0.478401i −0.500000 − 0.866025i 0.847422 + 1.46778i −0.795012 + 1.37700i −0.552409 0.886763 + 2.49272i 2.04107 −0.500000 + 0.866025i 0.439172 + 0.760669i
100.4 0.758290 − 1.31340i −0.500000 − 0.866025i −0.150007 − 0.259820i 1.16659 − 2.02059i −1.51658 −2.28580 − 1.33233i 2.57816 −0.500000 + 0.866025i −1.76922 − 3.06438i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 67.4
Significant digits:
Format:

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 231.2.i.e ✓ 8
3.b odd 2 1 693.2.i.i 8
7.c even 3 1 inner 231.2.i.e ✓ 8
7.c even 3 1 1617.2.a.z 4
7.d odd 6 1 1617.2.a.x 4
21.g even 6 1 4851.2.a.bu 4
21.h odd 6 1 693.2.i.i 8
21.h odd 6 1 4851.2.a.bt 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
231.2.i.e ✓ 8 1.a even 1 1 trivial
231.2.i.e ✓ 8 7.c even 3 1 inner
693.2.i.i 8 3.b odd 2 1
693.2.i.i 8 21.h odd 6 1
1617.2.a.x 4 7.d odd 6 1
1617.2.a.z 4 7.c even 3 1
4851.2.a.bt 4 21.h odd 6 1
4851.2.a.bu 4 21.g even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{8} + 2T_{2}^{7} + 8T_{2}^{6} + 21T_{2}^{4} + 4T_{2}^{3} + 28T_{2}^{2} - 12T_{2} + 9 \) acting on \(S_{2}^{\mathrm{new}}(231, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} + 2 T^{7} + \cdots + 9 \) Copy content Toggle raw display
$3$ \( (T^{2} + T + 1)^{4} \) Copy content Toggle raw display
$5$ \( T^{8} + 4 T^{7} + \cdots + 144 \) Copy content Toggle raw display
$7$ \( T^{8} - 2 T^{7} + \cdots + 2401 \) Copy content Toggle raw display
$11$ \( (T^{2} - T + 1)^{4} \) Copy content Toggle raw display
$13$ \( (T^{4} + 2 T^{3} - 8 T^{2} + \cdots - 4)^{2} \) Copy content Toggle raw display
$17$ \( T^{8} + 2 T^{7} + \cdots + 225 \) Copy content Toggle raw display
$19$ \( T^{8} + 40 T^{6} + \cdots + 7921 \) Copy content Toggle raw display
$23$ \( T^{8} + 4 T^{7} + \cdots + 216225 \) Copy content Toggle raw display
$29$ \( (T^{4} - 8 T^{3} + 12 T^{2} + \cdots + 3)^{2} \) Copy content Toggle raw display
$31$ \( T^{8} - 12 T^{7} + \cdots + 1948816 \) Copy content Toggle raw display
$37$ \( T^{8} - 4 T^{7} + \cdots + 1 \) Copy content Toggle raw display
$41$ \( (T^{4} - 2 T^{3} - 44 T^{2} + \cdots + 60)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} - 18 T^{3} + \cdots - 1385)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} + 12 T^{7} + \cdots + 81 \) Copy content Toggle raw display
$53$ \( T^{8} - 12 T^{7} + \cdots + 39087504 \) Copy content Toggle raw display
$59$ \( T^{8} + 12 T^{7} + \cdots + 62001 \) Copy content Toggle raw display
$61$ \( T^{8} + 2 T^{7} + \cdots + 400 \) Copy content Toggle raw display
$67$ \( T^{8} + 28 T^{7} + \cdots + 2131600 \) Copy content Toggle raw display
$71$ \( (T^{4} - 12 T^{3} + \cdots - 699)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} + 6 T^{7} + \cdots + 294328336 \) Copy content Toggle raw display
$79$ \( T^{8} + 2 T^{7} + \cdots + 150544 \) Copy content Toggle raw display
$83$ \( (T^{4} + 12 T^{3} + \cdots + 2592)^{2} \) Copy content Toggle raw display
$89$ \( T^{8} + 8 T^{7} + \cdots + 145154304 \) Copy content Toggle raw display
$97$ \( (T^{4} + 44 T^{3} + \cdots + 8501)^{2} \) Copy content Toggle raw display
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