↑

Properties

Label 231.2.j.g
Level $231$
Weight $2$
Character orbit 231.j
Analytic conductor $1.845$
Analytic rank $0$
Dimension $20$
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [231,2,Mod(64,231)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("231.64"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(231, base_ring=CyclotomicField(10)) chi = DirichletCharacter(H, H._module([0, 0, 6])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 231 = 3 \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 231.j (of order \(5\), degree \(4\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [20,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(1.84454428669\)
Analytic rank: \(0\)
Dimension: \(20\)
Relative dimension: \(5\) over \(\Q(\zeta_{5})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} + \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} + 12 x^{18} - 3 x^{17} + 94 x^{16} - 10 x^{15} + 662 x^{14} - 153 x^{13} + 4638 x^{12} + \cdots + 121 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

$q$-expansion

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

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{19}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{7} q^{2} - \beta_{4} q^{3} + (\beta_{19} - \beta_{10} + \cdots + \beta_{3}) q^{4} + \beta_{13} q^{5} - \beta_1 q^{6} + \beta_{9} q^{7} + ( - \beta_{19} + \beta_{18} + \cdots + \beta_1) q^{8}+ \cdots + ( - \beta_{19} - \beta_{17} + \beta_{13} + \cdots + 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q - 5 q^{3} - 14 q^{4} - 5 q^{5} + 5 q^{7} - 9 q^{8} - 5 q^{9} + 12 q^{10} - q^{11} + 36 q^{12} + 13 q^{13} - 24 q^{16} - q^{17} + 10 q^{19} - 46 q^{20} - 20 q^{21} + 26 q^{22} + 6 q^{24} - 8 q^{25}+ \cdots - q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} + 12 x^{18} - 3 x^{17} + 94 x^{16} - 10 x^{15} + 662 x^{14} - 153 x^{13} + 4638 x^{12} + \cdots + 121 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 46\!\cdots\!50 \nu^{19} + \cdots + 53\!\cdots\!27 ) / 18\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 15\!\cdots\!62 \nu^{19} + \cdots + 73\!\cdots\!95 ) / 18\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 48\!\cdots\!57 \nu^{19} + \cdots + 22\!\cdots\!27 ) / 20\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 49\!\cdots\!48 \nu^{19} + \cdots - 68\!\cdots\!52 ) / 18\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 62\!\cdots\!32 \nu^{19} + \cdots - 19\!\cdots\!06 ) / 20\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 56\!\cdots\!45 \nu^{19} + \cdots - 29\!\cdots\!56 ) / 18\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 26\!\cdots\!84 \nu^{19} + \cdots - 78\!\cdots\!74 ) / 20\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 26\!\cdots\!96 \nu^{19} + \cdots + 41\!\cdots\!85 ) / 20\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 29\!\cdots\!75 \nu^{19} + \cdots + 14\!\cdots\!88 ) / 20\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 30\!\cdots\!21 \nu^{19} + \cdots - 89\!\cdots\!91 ) / 18\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 39\!\cdots\!52 \nu^{19} + \cdots - 74\!\cdots\!47 ) / 20\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 94\!\cdots\!92 \nu^{19} + \cdots - 43\!\cdots\!83 ) / 40\!\cdots\!12 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 50\!\cdots\!16 \nu^{19} + \cdots + 45\!\cdots\!99 ) / 20\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 54\!\cdots\!50 \nu^{19} + \cdots + 33\!\cdots\!56 ) / 20\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 64\!\cdots\!20 \nu^{19} + \cdots - 66\!\cdots\!86 ) / 20\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( - 64\!\cdots\!30 \nu^{19} + \cdots + 62\!\cdots\!90 ) / 20\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( 11\!\cdots\!15 \nu^{19} + \cdots - 52\!\cdots\!28 ) / 20\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( 16\!\cdots\!50 \nu^{19} + \cdots - 14\!\cdots\!54 ) / 20\!\cdots\!60 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{19} + \beta_{18} + \beta_{10} + \beta_{6} - 4\beta_{4} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{19} - \beta_{17} - \beta_{12} + \beta_{9} + \beta_{8} + \beta_{7} + \beta_{6} + \beta_{4} - \beta_{3} - 6\beta_{2} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{17} - \beta_{16} + \beta_{15} + \beta_{11} + 7\beta_{8} + \beta_{7} - 23\beta_{6} + \beta_{5} + \beta_{2} - 1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 2 \beta_{18} + 9 \beta_{17} + 18 \beta_{16} - 9 \beta_{15} - 10 \beta_{14} - 9 \beta_{13} - 8 \beta_{12} + \cdots - 11 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - \beta_{19} - 49 \beta_{18} - 22 \beta_{17} - 10 \beta_{16} + \beta_{15} + 12 \beta_{14} + 10 \beta_{13} + \cdots - 160 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 71 \beta_{19} - 71 \beta_{18} + 69 \beta_{17} - 13 \beta_{16} + \beta_{15} + 69 \beta_{14} - \beta_{13} + \cdots + 34 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 348 \beta_{19} + 18 \beta_{18} + 113 \beta_{17} + 97 \beta_{16} - 85 \beta_{15} - 194 \beta_{14} + \cdots + 80 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 255 \beta_{19} - 635 \beta_{17} - 383 \beta_{16} + 488 \beta_{15} + 509 \beta_{14} + 509 \beta_{13} + \cdots - 400 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 232 \beta_{18} - 178 \beta_{17} - 356 \beta_{16} + 178 \beta_{15} + 980 \beta_{14} - 419 \beta_{13} + \cdots + 7048 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 2265 \beta_{19} + 4052 \beta_{18} + 2224 \beta_{17} - 2604 \beta_{16} - 834 \beta_{15} - 4828 \beta_{14} + \cdots + 6052 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 18008 \beta_{19} + 18008 \beta_{18} - 1726 \beta_{17} + 6464 \beta_{16} - 834 \beta_{15} + \cdots + 5068 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 30262 \beta_{19} - 19232 \beta_{18} - 36553 \beta_{17} - 9462 \beta_{16} + 7343 \beta_{15} + \cdots - 3011 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 25683 \beta_{19} + 67097 \beta_{17} - 35809 \beta_{16} + 15779 \beta_{15} - 15644 \beta_{14} + \cdots - 19380 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 159501 \beta_{18} + 197887 \beta_{17} + 395774 \beta_{16} - 197887 \beta_{15} - 277088 \beta_{14} + \cdots - 314427 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( - 242805 \beta_{19} - 952108 \beta_{18} - 813282 \beta_{17} - 270246 \beta_{16} + 178806 \beta_{15} + \cdots - 2796049 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( 1687959 \beta_{19} - 1687959 \beta_{18} + 1450189 \beta_{17} - 657013 \beta_{16} + 178806 \beta_{15} + \cdots + 2301504 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( 6965155 \beta_{19} + 2201571 \beta_{18} + 4359143 \beta_{17} + 3199137 \beta_{16} - 2807347 \beta_{15} + \cdots + 1663173 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( - 10583887 \beta_{19} - 16083999 \beta_{17} - 5250919 \beta_{16} + 8363279 \beta_{15} + 10667459 \beta_{14} + \cdots - 20344493 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(155\) \(199\) \(211\)
\(\chi(n)\) \(1\) \(1\) \(-\beta_{9}\)

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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
64.1
0.739775 + 2.27679i
0.648215 + 1.99500i
0.0347586 + 0.106976i
−0.557915 − 1.71709i
−0.864833 − 2.66168i
0.739775 − 2.27679i
0.648215 − 1.99500i
0.0347586 − 0.106976i
−0.557915 + 1.71709i
−0.864833 + 2.66168i
2.13576 − 1.55172i
0.705143 − 0.512316i
0.383242 − 0.278442i
−1.07866 + 0.783695i
−2.14548 + 1.55878i
2.13576 + 1.55172i
0.705143 + 0.512316i
0.383242 + 0.278442i
−1.07866 − 0.783695i
−2.14548 − 1.55878i
−0.739775 + 2.27679i −0.809017 + 0.587785i −3.01849 − 2.19306i −1.21700 − 3.74554i −0.739775 − 2.27679i 0.809017 + 0.587785i 3.35264 − 2.43583i 0.309017 − 0.951057i 9.42812
64.2 −0.648215 + 1.99500i −0.809017 + 0.587785i −1.94181 − 1.41081i 0.976330 + 3.00483i −0.648215 − 1.99500i 0.809017 + 0.587785i 0.679172 − 0.493447i 0.309017 − 0.951057i −6.62751
64.3 −0.0347586 + 0.106976i −0.809017 + 0.587785i 1.60780 + 1.16813i 0.327964 + 1.00937i −0.0347586 − 0.106976i 0.809017 + 0.587785i −0.362845 + 0.263622i 0.309017 − 0.951057i −0.119378
64.4 0.557915 − 1.71709i −0.809017 + 0.587785i −1.01908 − 0.740407i 0.858186 + 2.64122i 0.557915 + 1.71709i 0.809017 + 0.587785i 1.08138 − 0.785667i 0.309017 − 0.951057i 5.01400
64.5 0.864833 − 2.66168i −0.809017 + 0.587785i −4.71859 − 3.42825i −0.518429 − 1.59556i 0.864833 + 2.66168i 0.809017 + 0.587785i −8.67739 + 6.30449i 0.309017 − 0.951057i −4.69523
148.1 −0.739775 − 2.27679i −0.809017 − 0.587785i −3.01849 + 2.19306i −1.21700 + 3.74554i −0.739775 + 2.27679i 0.809017 − 0.587785i 3.35264 + 2.43583i 0.309017 + 0.951057i 9.42812
148.2 −0.648215 − 1.99500i −0.809017 − 0.587785i −1.94181 + 1.41081i 0.976330 − 3.00483i −0.648215 + 1.99500i 0.809017 − 0.587785i 0.679172 + 0.493447i 0.309017 + 0.951057i −6.62751
148.3 −0.0347586 − 0.106976i −0.809017 − 0.587785i 1.60780 − 1.16813i 0.327964 − 1.00937i −0.0347586 + 0.106976i 0.809017 − 0.587785i −0.362845 − 0.263622i 0.309017 + 0.951057i −0.119378
148.4 0.557915 + 1.71709i −0.809017 − 0.587785i −1.01908 + 0.740407i 0.858186 − 2.64122i 0.557915 − 1.71709i 0.809017 − 0.587785i 1.08138 + 0.785667i 0.309017 + 0.951057i 5.01400
148.5 0.864833 + 2.66168i −0.809017 − 0.587785i −4.71859 + 3.42825i −0.518429 + 1.59556i 0.864833 − 2.66168i 0.809017 − 0.587785i −8.67739 − 6.30449i 0.309017 + 0.951057i −4.69523
169.1 −2.13576 − 1.55172i 0.309017 − 0.951057i 1.53559 + 4.72606i −2.49476 + 1.81255i −2.13576 + 1.55172i −0.309017 − 0.951057i 2.42230 − 7.45506i −0.809017 − 0.587785i 8.14075
169.2 −0.705143 − 0.512316i 0.309017 − 0.951057i −0.383276 − 1.17960i 3.28814 − 2.38897i −0.705143 + 0.512316i −0.309017 − 0.951057i −0.872746 + 2.68604i −0.809017 − 0.587785i −3.54252
169.3 −0.383242 − 0.278442i 0.309017 − 0.951057i −0.548689 − 1.68869i −3.04094 + 2.20937i −0.383242 + 0.278442i −0.309017 − 0.951057i −0.552692 + 1.70101i −0.809017 − 0.587785i 1.78060
169.4 1.07866 + 0.783695i 0.309017 − 0.951057i −0.0686960 − 0.211425i 0.706442 − 0.513260i 1.07866 − 0.783695i −0.309017 − 0.951057i 0.915619 − 2.81798i −0.809017 − 0.587785i 1.16425
169.5 2.14548 + 1.55878i 0.309017 − 0.951057i 1.55524 + 4.78653i −1.38593 + 1.00694i 2.14548 − 1.55878i −0.309017 − 0.951057i −2.48543 + 7.64935i −0.809017 − 0.587785i −4.54308
190.1 −2.13576 + 1.55172i 0.309017 + 0.951057i 1.53559 − 4.72606i −2.49476 − 1.81255i −2.13576 − 1.55172i −0.309017 + 0.951057i 2.42230 + 7.45506i −0.809017 + 0.587785i 8.14075
190.2 −0.705143 + 0.512316i 0.309017 + 0.951057i −0.383276 + 1.17960i 3.28814 + 2.38897i −0.705143 − 0.512316i −0.309017 + 0.951057i −0.872746 − 2.68604i −0.809017 + 0.587785i −3.54252
190.3 −0.383242 + 0.278442i 0.309017 + 0.951057i −0.548689 + 1.68869i −3.04094 − 2.20937i −0.383242 − 0.278442i −0.309017 + 0.951057i −0.552692 − 1.70101i −0.809017 + 0.587785i 1.78060
190.4 1.07866 − 0.783695i 0.309017 + 0.951057i −0.0686960 + 0.211425i 0.706442 + 0.513260i 1.07866 + 0.783695i −0.309017 + 0.951057i 0.915619 + 2.81798i −0.809017 + 0.587785i 1.16425
190.5 2.14548 − 1.55878i 0.309017 + 0.951057i 1.55524 − 4.78653i −1.38593 − 1.00694i 2.14548 + 1.55878i −0.309017 + 0.951057i −2.48543 − 7.64935i −0.809017 + 0.587785i −4.54308
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 64.5
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.c even 5 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 231.2.j.g ✓ 20
3.b odd 2 1 693.2.m.j 20
11.c even 5 1 inner 231.2.j.g ✓ 20
11.c even 5 1 2541.2.a.bq 10
11.d odd 10 1 2541.2.a.br 10
33.f even 10 1 7623.2.a.cy 10
33.h odd 10 1 693.2.m.j 20
33.h odd 10 1 7623.2.a.cx 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
231.2.j.g ✓ 20 1.a even 1 1 trivial
231.2.j.g ✓ 20 11.c even 5 1 inner
693.2.m.j 20 3.b odd 2 1
693.2.m.j 20 33.h odd 10 1
2541.2.a.bq 10 11.c even 5 1
2541.2.a.br 10 11.d odd 10 1
7623.2.a.cx 10 33.h odd 10 1
7623.2.a.cy 10 33.f even 10 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{20} + 12 T_{2}^{18} + 3 T_{2}^{17} + 94 T_{2}^{16} + 10 T_{2}^{15} + 662 T_{2}^{14} + 153 T_{2}^{13} + \cdots + 121 \) acting on \(S_{2}^{\mathrm{new}}(231, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{20} + 12 T^{18} + \cdots + 121 \) Copy content Toggle raw display
$3$ \( (T^{4} + T^{3} + T^{2} + \cdots + 1)^{5} \) Copy content Toggle raw display
$5$ \( T^{20} + 5 T^{19} + \cdots + 18800896 \) Copy content Toggle raw display
$7$ \( (T^{4} - T^{3} + T^{2} + \cdots + 1)^{5} \) Copy content Toggle raw display
$11$ \( T^{20} + \cdots + 25937424601 \) Copy content Toggle raw display
$13$ \( T^{20} - 13 T^{19} + \cdots + 7929856 \) Copy content Toggle raw display
$17$ \( T^{20} + T^{19} + \cdots + 69488896 \) Copy content Toggle raw display
$19$ \( T^{20} + \cdots + 2595291136 \) Copy content Toggle raw display
$23$ \( (T^{10} - 106 T^{8} + \cdots - 600380)^{2} \) Copy content Toggle raw display
$29$ \( T^{20} + \cdots + 12760212756736 \) Copy content Toggle raw display
$31$ \( T^{20} + \cdots + 283449760000 \) Copy content Toggle raw display
$37$ \( T^{20} + \cdots + 2807916867856 \) Copy content Toggle raw display
$41$ \( T^{20} + \cdots + 101929216 \) Copy content Toggle raw display
$43$ \( (T^{10} - 6 T^{9} + \cdots - 21296)^{2} \) Copy content Toggle raw display
$47$ \( T^{20} + \cdots + 7259791360000 \) Copy content Toggle raw display
$53$ \( T^{20} + \cdots + 57838326016 \) Copy content Toggle raw display
$59$ \( T^{20} + \cdots + 25\!\cdots\!56 \) Copy content Toggle raw display
$61$ \( T^{20} + \cdots + 77\!\cdots\!36 \) Copy content Toggle raw display
$67$ \( (T^{10} - 38 T^{9} + \cdots + 52960256)^{2} \) Copy content Toggle raw display
$71$ \( T^{20} + \cdots + 291518744166400 \) Copy content Toggle raw display
$73$ \( T^{20} + \cdots + 57999556673536 \) Copy content Toggle raw display
$79$ \( T^{20} + \cdots + 3187281807616 \) Copy content Toggle raw display
$83$ \( T^{20} + \cdots + 607252381696 \) Copy content Toggle raw display
$89$ \( (T^{10} + 9 T^{9} + \cdots + 106384)^{2} \) Copy content Toggle raw display
$97$ \( T^{20} + \cdots + 231384088576 \) Copy content Toggle raw display
show more
show less