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Properties

Label 231.4.i.a
Level $231$
Weight $4$
Character orbit 231.i
Analytic conductor $13.629$
Analytic rank $0$
Dimension $16$
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [231,4,Mod(67,231)] 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("231.67"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(231, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 4, 0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 231 = 3 \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 231.i (of order \(3\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [16,4] 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: \(13.6294412113\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 4 x^{15} + 55 x^{14} - 108 x^{13} + 1559 x^{12} - 2354 x^{11} + 27458 x^{10} - 12372 x^{9} + \cdots + 7225344 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{4} + \beta_{3} + \beta_1 + 1) q^{2} + 3 \beta_{4} q^{3} + ( - \beta_{8} - 4 \beta_{4} + \beta_1) q^{4} + ( - \beta_{12} + 2 \beta_{4} + \cdots - 2) q^{5} + (3 \beta_{3} + 3) q^{6} + (\beta_{15} + \beta_{11} - \beta_{9} + \cdots - 1) q^{7}+ \cdots + 99 q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 4 q^{2} + 24 q^{3} - 30 q^{4} - 20 q^{5} + 24 q^{6} + 18 q^{7} - 156 q^{8} - 72 q^{9} - 94 q^{10} - 88 q^{11} + 90 q^{12} - 188 q^{13} + 52 q^{14} - 120 q^{15} + 10 q^{16} - 144 q^{17} + 36 q^{18}+ \cdots + 1584 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 4 x^{15} + 55 x^{14} - 108 x^{13} + 1559 x^{12} - 2354 x^{11} + 27458 x^{10} - 12372 x^{9} + \cdots + 7225344 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 57\!\cdots\!91 \nu^{15} + \cdots - 69\!\cdots\!64 ) / 74\!\cdots\!44 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 66\!\cdots\!29 \nu^{15} + \cdots + 67\!\cdots\!44 ) / 74\!\cdots\!44 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 78\!\cdots\!59 \nu^{15} + \cdots + 55\!\cdots\!84 ) / 62\!\cdots\!96 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 17\!\cdots\!39 \nu^{15} + \cdots - 47\!\cdots\!60 ) / 75\!\cdots\!92 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 74\!\cdots\!31 \nu^{15} + \cdots - 83\!\cdots\!96 ) / 23\!\cdots\!48 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 36\!\cdots\!19 \nu^{15} + \cdots + 67\!\cdots\!88 ) / 78\!\cdots\!16 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 75\!\cdots\!69 \nu^{15} + \cdots - 57\!\cdots\!40 ) / 62\!\cdots\!96 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 56\!\cdots\!95 \nu^{15} + \cdots - 62\!\cdots\!96 ) / 33\!\cdots\!64 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 45\!\cdots\!91 \nu^{15} + \cdots + 84\!\cdots\!52 ) / 23\!\cdots\!48 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 10\!\cdots\!11 \nu^{15} + \cdots + 20\!\cdots\!88 ) / 47\!\cdots\!96 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 37\!\cdots\!93 \nu^{15} + \cdots - 76\!\cdots\!96 ) / 15\!\cdots\!32 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 57\!\cdots\!63 \nu^{15} + \cdots + 15\!\cdots\!16 ) / 23\!\cdots\!48 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 29\!\cdots\!09 \nu^{15} + \cdots - 10\!\cdots\!80 ) / 11\!\cdots\!24 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 62\!\cdots\!09 \nu^{15} + \cdots + 50\!\cdots\!88 ) / 47\!\cdots\!96 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{8} + 11\beta_{4} + \beta_{3} - \beta_{2} + \beta _1 - 11 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{15} - \beta_{13} - \beta_{12} - \beta_{11} + \beta_{10} + \beta_{9} + \beta_{7} - \beta_{6} - \beta_{5} + 18\beta_{3} - 8 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -2\beta_{13} + \beta_{11} + \beta_{10} - \beta_{9} - 24\beta_{8} + 4\beta_{7} + \beta_{5} - 196\beta_{4} - 28\beta_1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 30 \beta_{15} - 29 \beta_{14} + 38 \beta_{12} + 29 \beta_{11} - 32 \beta_{10} - 52 \beta_{9} + \cdots + 246 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 76 \beta_{15} - 29 \beta_{14} + 76 \beta_{13} + 164 \beta_{12} + 10 \beta_{11} - 39 \beta_{10} + \cdots + 3926 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 814 \beta_{14} + 768 \beta_{13} + 117 \beta_{11} + 117 \beta_{10} + 519 \beta_{9} + 522 \beta_{8} + \cdots + 7724 \beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 2222 \beta_{15} + 1117 \beta_{14} - 5026 \beta_{12} - 1117 \beta_{11} + 548 \beta_{10} + 2392 \beta_{9} + \cdots - 82452 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 18646 \beta_{15} - 3519 \beta_{14} - 18646 \beta_{13} - 30306 \beta_{12} - 19224 \beta_{11} + \cdots - 169312 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 20320 \beta_{14} - 59346 \beta_{13} + 8257 \beta_{11} + 8257 \beta_{10} - 35963 \beta_{9} + \cdots - 449104 \beta_1 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 443158 \beta_{15} - 343717 \beta_{14} + 789370 \beta_{12} + 343717 \beta_{11} - 441544 \beta_{10} + \cdots + 4252688 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 1521242 \beta_{15} - 53173 \beta_{14} + 1521242 \beta_{13} + 3691786 \beta_{12} + 639768 \beta_{11} + \cdots + 39188488 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 10052804 \beta_{14} + 10450866 \beta_{13} + 2634207 \beta_{11} + 2634207 \beta_{10} + \cdots + 84631972 \beta_1 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 38172730 \beta_{15} + 16320669 \beta_{14} - 95491802 \beta_{12} - 16320669 \beta_{11} + \cdots - 873848312 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 246080994 \beta_{15} - 69756903 \beta_{14} - 246080994 \beta_{13} - 509054326 \beta_{12} + \cdots - 2590080056 \) Copy content Toggle raw display

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

Character values

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

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

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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
67.1
2.46352 − 4.26695i
2.32362 − 4.02463i
1.74221 − 3.01759i
0.666329 − 1.15412i
−0.583268 + 1.01025i
−0.737497 + 1.27738i
−1.65375 + 2.86438i
−2.22116 + 3.84717i
2.46352 + 4.26695i
2.32362 + 4.02463i
1.74221 + 3.01759i
0.666329 + 1.15412i
−0.583268 − 1.01025i
−0.737497 − 1.27738i
−1.65375 − 2.86438i
−2.22116 − 3.84717i
−1.96352 − 3.40092i 1.50000 − 2.59808i −3.71085 + 6.42739i −9.61634 − 16.6560i −11.7811 17.0365 + 7.26346i −2.27098 −4.50000 − 7.79423i −37.7638 + 65.4089i
67.2 −1.82362 − 3.15860i 1.50000 − 2.59808i −2.65118 + 4.59197i 1.62425 + 2.81329i −10.9417 −2.33065 + 18.3730i −9.83896 −4.50000 − 7.79423i 5.92404 − 10.2607i
67.3 −1.24221 − 2.15156i 1.50000 − 2.59808i 0.913847 − 1.58283i 2.47926 + 4.29420i −7.45324 5.11112 − 17.8010i −24.4160 −4.50000 − 7.79423i 6.15950 − 10.6686i
67.4 −0.166329 − 0.288091i 1.50000 − 2.59808i 3.94467 − 6.83237i −7.20930 − 12.4869i −0.997975 −1.79838 − 18.4327i −5.28572 −4.50000 − 7.79423i −2.39824 + 4.15387i
67.5 1.08327 + 1.87628i 1.50000 − 2.59808i 1.65306 − 2.86318i 1.06225 + 1.83988i 6.49961 7.56550 + 16.9045i 24.4951 −4.50000 − 7.79423i −2.30141 + 3.98616i
67.6 1.23750 + 2.14341i 1.50000 − 2.59808i 0.937203 − 1.62328i −4.40030 − 7.62155i 7.42498 −18.2919 − 2.89928i 24.4391 −4.50000 − 7.79423i 10.8907 − 18.8633i
67.7 2.15375 + 3.73040i 1.50000 − 2.59808i −5.27728 + 9.14051i 4.82076 + 8.34981i 12.9225 17.5449 − 5.93094i −11.0037 −4.50000 − 7.79423i −20.7654 + 35.9668i
67.8 2.72116 + 4.71320i 1.50000 − 2.59808i −10.8095 + 18.7226i 1.23942 + 2.14674i 16.3270 −15.8371 − 9.60139i −74.1188 −4.50000 − 7.79423i −6.74534 + 11.6833i
100.1 −1.96352 + 3.40092i 1.50000 + 2.59808i −3.71085 − 6.42739i −9.61634 + 16.6560i −11.7811 17.0365 − 7.26346i −2.27098 −4.50000 + 7.79423i −37.7638 − 65.4089i
100.2 −1.82362 + 3.15860i 1.50000 + 2.59808i −2.65118 − 4.59197i 1.62425 − 2.81329i −10.9417 −2.33065 − 18.3730i −9.83896 −4.50000 + 7.79423i 5.92404 + 10.2607i
100.3 −1.24221 + 2.15156i 1.50000 + 2.59808i 0.913847 + 1.58283i 2.47926 − 4.29420i −7.45324 5.11112 + 17.8010i −24.4160 −4.50000 + 7.79423i 6.15950 + 10.6686i
100.4 −0.166329 + 0.288091i 1.50000 + 2.59808i 3.94467 + 6.83237i −7.20930 + 12.4869i −0.997975 −1.79838 + 18.4327i −5.28572 −4.50000 + 7.79423i −2.39824 − 4.15387i
100.5 1.08327 − 1.87628i 1.50000 + 2.59808i 1.65306 + 2.86318i 1.06225 − 1.83988i 6.49961 7.56550 − 16.9045i 24.4951 −4.50000 + 7.79423i −2.30141 − 3.98616i
100.6 1.23750 − 2.14341i 1.50000 + 2.59808i 0.937203 + 1.62328i −4.40030 + 7.62155i 7.42498 −18.2919 + 2.89928i 24.4391 −4.50000 + 7.79423i 10.8907 + 18.8633i
100.7 2.15375 − 3.73040i 1.50000 + 2.59808i −5.27728 − 9.14051i 4.82076 − 8.34981i 12.9225 17.5449 + 5.93094i −11.0037 −4.50000 + 7.79423i −20.7654 − 35.9668i
100.8 2.72116 − 4.71320i 1.50000 + 2.59808i −10.8095 − 18.7226i 1.23942 − 2.14674i 16.3270 −15.8371 + 9.60139i −74.1188 −4.50000 + 7.79423i −6.74534 − 11.6833i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 67.8
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 231.4.i.a ✓ 16
7.c even 3 1 inner 231.4.i.a ✓ 16
7.c even 3 1 1617.4.a.u 8
7.d odd 6 1 1617.4.a.v 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
231.4.i.a ✓ 16 1.a even 1 1 trivial
231.4.i.a ✓ 16 7.c even 3 1 inner
1617.4.a.u 8 7.c even 3 1
1617.4.a.v 8 7.d odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{16} - 4 T_{2}^{15} + 55 T_{2}^{14} - 104 T_{2}^{13} + 1546 T_{2}^{12} - 2400 T_{2}^{11} + \cdots + 2214144 \) acting on \(S_{4}^{\mathrm{new}}(231, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} - 4 T^{15} + \cdots + 2214144 \) Copy content Toggle raw display
$3$ \( (T^{2} - 3 T + 9)^{8} \) Copy content Toggle raw display
$5$ \( T^{16} + \cdots + 3984079872256 \) Copy content Toggle raw display
$7$ \( T^{16} + \cdots + 19\!\cdots\!01 \) Copy content Toggle raw display
$11$ \( (T^{2} + 11 T + 121)^{8} \) Copy content Toggle raw display
$13$ \( (T^{8} + \cdots + 1416035880512)^{2} \) Copy content Toggle raw display
$17$ \( T^{16} + \cdots + 15\!\cdots\!21 \) Copy content Toggle raw display
$19$ \( T^{16} + \cdots + 31\!\cdots\!76 \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 82\!\cdots\!69 \) Copy content Toggle raw display
$29$ \( (T^{8} + \cdots - 87\!\cdots\!68)^{2} \) Copy content Toggle raw display
$31$ \( T^{16} + \cdots + 53\!\cdots\!84 \) Copy content Toggle raw display
$37$ \( T^{16} + \cdots + 98\!\cdots\!24 \) Copy content Toggle raw display
$41$ \( (T^{8} + \cdots + 83\!\cdots\!04)^{2} \) Copy content Toggle raw display
$43$ \( (T^{8} + \cdots + 86\!\cdots\!00)^{2} \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 28\!\cdots\!49 \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 39\!\cdots\!36 \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 65\!\cdots\!04 \) Copy content Toggle raw display
$61$ \( T^{16} + \cdots + 58\!\cdots\!44 \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots + 12\!\cdots\!00 \) Copy content Toggle raw display
$71$ \( (T^{8} + \cdots + 26\!\cdots\!64)^{2} \) Copy content Toggle raw display
$73$ \( T^{16} + \cdots + 10\!\cdots\!44 \) Copy content Toggle raw display
$79$ \( T^{16} + \cdots + 33\!\cdots\!64 \) Copy content Toggle raw display
$83$ \( (T^{8} + \cdots + 51\!\cdots\!24)^{2} \) Copy content Toggle raw display
$89$ \( T^{16} + \cdots + 17\!\cdots\!36 \) Copy content Toggle raw display
$97$ \( (T^{8} + \cdots - 15\!\cdots\!31)^{2} \) Copy content Toggle raw display
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