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Properties

Label 222.3.l.d
Level $222$
Weight $3$
Character orbit 222.l
Analytic conductor $6.049$
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] = [222,3,Mod(97,222)] 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("222.97"); S:= CuspForms(chi, 3); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(222, base_ring=CyclotomicField(12)) chi = DirichletCharacter(H, H._module([0, 5])) N = Newforms(chi, 3, names="a")
 
Level: \( N \) \(=\) \( 222 = 2 \cdot 3 \cdot 37 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 222.l (of order \(12\), degree \(4\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [16,8,24] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(6.04906186880\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(4\) over \(\Q(\zeta_{12})\)
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: \( x^{16} - 4 x^{15} + 8 x^{14} + 318 x^{13} + 8876 x^{12} - 14732 x^{11} + 38482 x^{10} + 1520688 x^{9} + \cdots + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 3^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( - \beta_{9} - \beta_{8}) q^{2} + ( - \beta_{9} + 1) q^{3} + ( - 2 \beta_{8} + 2 \beta_{5}) q^{4} + (\beta_{9} - \beta_{8} + \beta_{5} + \cdots + 1) q^{5} + ( - 2 \beta_{9} - 2 \beta_{8} + \cdots - 1) q^{6}+ \cdots + ( - 3 \beta_{13} + 6 \beta_{9} + \cdots + 9) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 8 q^{2} + 24 q^{3} + 6 q^{5} - 14 q^{7} + 32 q^{8} + 24 q^{9} + 24 q^{10} - 16 q^{13} - 28 q^{14} + 18 q^{15} + 32 q^{16} - 16 q^{17} - 24 q^{18} + 42 q^{19} + 12 q^{20} - 42 q^{21} + 46 q^{22} - 34 q^{23}+ \cdots + 90 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 4 x^{15} + 8 x^{14} + 318 x^{13} + 8876 x^{12} - 14732 x^{11} + 38482 x^{10} + 1520688 x^{9} + \cdots + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 24\!\cdots\!20 \nu^{15} + \cdots - 25\!\cdots\!14 ) / 17\!\cdots\!24 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 16\!\cdots\!97 \nu^{15} + \cdots + 24\!\cdots\!24 ) / 44\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 45\!\cdots\!17 \nu^{15} + \cdots + 39\!\cdots\!88 ) / 20\!\cdots\!76 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 60\!\cdots\!31 \nu^{15} + \cdots + 93\!\cdots\!24 ) / 44\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 62\!\cdots\!60 \nu^{15} + \cdots + 19\!\cdots\!60 ) / 20\!\cdots\!76 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 63\!\cdots\!84 \nu^{15} + \cdots - 11\!\cdots\!56 ) / 20\!\cdots\!76 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 69\!\cdots\!95 \nu^{15} + \cdots + 18\!\cdots\!08 ) / 20\!\cdots\!76 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 12\!\cdots\!07 \nu^{15} + \cdots + 19\!\cdots\!68 ) / 34\!\cdots\!48 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 13\!\cdots\!27 \nu^{15} + \cdots - 48\!\cdots\!20 ) / 20\!\cdots\!76 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 14\!\cdots\!07 \nu^{15} + \cdots + 24\!\cdots\!24 ) / 20\!\cdots\!76 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 20\!\cdots\!63 \nu^{15} + \cdots - 89\!\cdots\!16 ) / 20\!\cdots\!76 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 62\!\cdots\!63 \nu^{15} + \cdots + 16\!\cdots\!60 ) / 20\!\cdots\!76 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 13\!\cdots\!43 \nu^{15} + \cdots + 89\!\cdots\!24 ) / 20\!\cdots\!76 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 15\!\cdots\!71 \nu^{15} + \cdots - 25\!\cdots\!56 ) / 20\!\cdots\!76 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{14} - \beta_{12} + 3\beta_{7} - 34\beta_{5} + \beta_{4} - 3\beta_{3} - \beta_{2} + 2\beta_1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 3 \beta_{15} + 3 \beta_{14} + 3 \beta_{13} - 3 \beta_{10} + 25 \beta_{9} + 25 \beta_{8} + 3 \beta_{7} + \cdots - 30 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 72 \beta_{15} + 6 \beta_{14} + 144 \beta_{13} + 6 \beta_{12} - 6 \beta_{11} - 213 \beta_{10} + \cdots - 2502 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 23 \beta_{15} - 23 \beta_{14} + 379 \beta_{13} + 356 \beta_{12} - 117 \beta_{11} - 516 \beta_{10} + \cdots - 7739 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 1155 \beta_{15} - 5442 \beta_{14} + 5442 \beta_{12} - 639 \beta_{10} - 23783 \beta_{9} - 15693 \beta_{7} + \cdots - 12469 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 36747 \beta_{15} - 36747 \beta_{14} - 39372 \beta_{13} + 2625 \beta_{12} + 17790 \beta_{11} + \cdots + 456563 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 426512 \beta_{15} - 153981 \beta_{14} - 853024 \beta_{13} - 153981 \beta_{12} + 189666 \beta_{11} + \cdots + 14557228 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 200133 \beta_{15} + 200133 \beta_{14} - 3779577 \beta_{13} - 3579444 \beta_{12} + 2081880 \beta_{11} + \cdots + 81029620 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 17759610 \beta_{15} + 34445574 \beta_{14} - 34445574 \beta_{12} + 11792853 \beta_{10} + 378093594 \beta_{9} + \cdots + 197926602 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 338775685 \beta_{15} + 338775685 \beta_{14} + 349039913 \beta_{13} - 10264228 \beta_{12} + \cdots - 3934978331 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 2849346291 \beta_{15} + 1896496932 \beta_{14} + 5698692582 \beta_{13} + 1896496932 \beta_{12} + \cdots - 96090510199 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 16531617 \beta_{15} - 16531617 \beta_{14} + 31586524848 \beta_{13} + 31569993231 \beta_{12} + \cdots - 703538800795 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 193269681108 \beta_{15} - 240147660085 \beta_{14} + 240147660085 \beta_{12} - 140525939826 \beta_{10} + \cdots - 2192442116006 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 2915979217803 \beta_{15} - 2915979217803 \beta_{14} - 2828252494821 \beta_{13} + \cdots + 29869558046128 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/222\mathbb{Z}\right)^\times\).

\(n\) \(149\) \(187\)
\(\chi(n)\) \(1\) \(\beta_{5} - \beta_{8}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
97.1
−5.45996 + 5.45996i
0.0224565 − 0.0224565i
0.240876 − 0.240876i
6.19663 − 6.19663i
−5.45996 − 5.45996i
0.0224565 + 0.0224565i
0.240876 + 0.240876i
6.19663 + 6.19663i
6.72757 − 6.72757i
0.0303820 − 0.0303820i
−0.654811 + 0.654811i
−5.10315 + 5.10315i
6.72757 + 6.72757i
0.0303820 + 0.0303820i
−0.654811 − 0.654811i
−5.10315 − 5.10315i
−0.366025 − 1.36603i 1.50000 − 0.866025i −1.73205 + 1.00000i −2.36451 + 8.82447i −1.73205 − 1.73205i −4.27169 − 7.39879i 2.00000 + 2.00000i 1.50000 − 2.59808i 12.9199
97.2 −0.366025 − 1.36603i 1.50000 − 0.866025i −1.73205 + 1.00000i −0.357806 + 1.33535i −1.73205 − 1.73205i −0.628500 − 1.08859i 2.00000 + 2.00000i 1.50000 − 2.59808i 1.95509
97.3 −0.366025 − 1.36603i 1.50000 − 0.866025i −1.73205 + 1.00000i −0.277859 + 1.03698i −1.73205 − 1.73205i 6.12832 + 10.6146i 2.00000 + 2.00000i 1.50000 − 2.59808i 1.51825
97.4 −0.366025 − 1.36603i 1.50000 − 0.866025i −1.73205 + 1.00000i 1.90210 − 7.09873i −1.73205 − 1.73205i −4.72813 − 8.18936i 2.00000 + 2.00000i 1.50000 − 2.59808i −10.3933
103.1 −0.366025 + 1.36603i 1.50000 + 0.866025i −1.73205 − 1.00000i −2.36451 − 8.82447i −1.73205 + 1.73205i −4.27169 + 7.39879i 2.00000 − 2.00000i 1.50000 + 2.59808i 12.9199
103.2 −0.366025 + 1.36603i 1.50000 + 0.866025i −1.73205 − 1.00000i −0.357806 − 1.33535i −1.73205 + 1.73205i −0.628500 + 1.08859i 2.00000 − 2.00000i 1.50000 + 2.59808i 1.95509
103.3 −0.366025 + 1.36603i 1.50000 + 0.866025i −1.73205 − 1.00000i −0.277859 − 1.03698i −1.73205 + 1.73205i 6.12832 − 10.6146i 2.00000 − 2.00000i 1.50000 + 2.59808i 1.51825
103.4 −0.366025 + 1.36603i 1.50000 + 0.866025i −1.73205 − 1.00000i 1.90210 + 7.09873i −1.73205 + 1.73205i −4.72813 + 8.18936i 2.00000 − 2.00000i 1.50000 + 2.59808i −10.3933
193.1 1.36603 + 0.366025i 1.50000 + 0.866025i 1.73205 + 1.00000i −7.82401 + 2.09644i 1.73205 + 1.73205i −3.59433 + 6.22556i 2.00000 + 2.00000i 1.50000 + 2.59808i −11.4551
193.2 1.36603 + 0.366025i 1.50000 + 0.866025i 1.73205 + 1.00000i 1.32452 − 0.354905i 1.73205 + 1.73205i 0.749275 − 1.29778i 2.00000 + 2.00000i 1.50000 + 2.59808i 1.93924
193.3 1.36603 + 0.366025i 1.50000 + 0.866025i 1.73205 + 1.00000i 2.26051 − 0.605703i 1.73205 + 1.73205i 5.80649 − 10.0571i 2.00000 + 2.00000i 1.50000 + 2.59808i 3.30962
193.4 1.36603 + 0.366025i 1.50000 + 0.866025i 1.73205 + 1.00000i 8.33705 − 2.23391i 1.73205 + 1.73205i −6.46144 + 11.1915i 2.00000 + 2.00000i 1.50000 + 2.59808i 12.2063
199.1 1.36603 − 0.366025i 1.50000 − 0.866025i 1.73205 − 1.00000i −7.82401 − 2.09644i 1.73205 − 1.73205i −3.59433 − 6.22556i 2.00000 − 2.00000i 1.50000 − 2.59808i −11.4551
199.2 1.36603 − 0.366025i 1.50000 − 0.866025i 1.73205 − 1.00000i 1.32452 + 0.354905i 1.73205 − 1.73205i 0.749275 + 1.29778i 2.00000 − 2.00000i 1.50000 − 2.59808i 1.93924
199.3 1.36603 − 0.366025i 1.50000 − 0.866025i 1.73205 − 1.00000i 2.26051 + 0.605703i 1.73205 − 1.73205i 5.80649 + 10.0571i 2.00000 − 2.00000i 1.50000 − 2.59808i 3.30962
199.4 1.36603 − 0.366025i 1.50000 − 0.866025i 1.73205 − 1.00000i 8.33705 + 2.23391i 1.73205 − 1.73205i −6.46144 − 11.1915i 2.00000 − 2.00000i 1.50000 − 2.59808i 12.2063
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 97.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
37.g odd 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 222.3.l.d ✓ 16
37.g odd 12 1 inner 222.3.l.d ✓ 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
222.3.l.d ✓ 16 1.a even 1 1 trivial
222.3.l.d ✓ 16 37.g odd 12 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{16} - 6 T_{5}^{15} + 12 T_{5}^{14} - 234 T_{5}^{13} - 3576 T_{5}^{12} + 41850 T_{5}^{11} + \cdots + 499790736 \) acting on \(S_{3}^{\mathrm{new}}(222, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} - 2 T^{3} + 2 T^{2} + \cdots + 4)^{4} \) Copy content Toggle raw display
$3$ \( (T^{2} - 3 T + 3)^{8} \) Copy content Toggle raw display
$5$ \( T^{16} + \cdots + 499790736 \) Copy content Toggle raw display
$7$ \( T^{16} + \cdots + 4049077621824 \) Copy content Toggle raw display
$11$ \( T^{16} + \cdots + 12\!\cdots\!56 \) Copy content Toggle raw display
$13$ \( T^{16} + \cdots + 20\!\cdots\!64 \) Copy content Toggle raw display
$17$ \( T^{16} + \cdots + 44\!\cdots\!64 \) Copy content Toggle raw display
$19$ \( T^{16} + \cdots + 31\!\cdots\!76 \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 452027059421184 \) Copy content Toggle raw display
$29$ \( T^{16} + \cdots + 21\!\cdots\!04 \) Copy content Toggle raw display
$31$ \( T^{16} + \cdots + 11\!\cdots\!36 \) Copy content Toggle raw display
$37$ \( T^{16} + \cdots + 12\!\cdots\!41 \) Copy content Toggle raw display
$41$ \( T^{16} + \cdots + 42\!\cdots\!44 \) Copy content Toggle raw display
$43$ \( T^{16} + \cdots + 80\!\cdots\!36 \) Copy content Toggle raw display
$47$ \( (T^{8} + \cdots - 2331500305392)^{2} \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 12\!\cdots\!36 \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 39\!\cdots\!76 \) Copy content Toggle raw display
$61$ \( T^{16} + \cdots + 18\!\cdots\!96 \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots + 44\!\cdots\!04 \) Copy content Toggle raw display
$71$ \( T^{16} + \cdots + 95\!\cdots\!16 \) Copy content Toggle raw display
$73$ \( T^{16} + \cdots + 80\!\cdots\!16 \) Copy content Toggle raw display
$79$ \( T^{16} + \cdots + 71\!\cdots\!64 \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 97\!\cdots\!44 \) Copy content Toggle raw display
$89$ \( T^{16} + \cdots + 99\!\cdots\!04 \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots + 47\!\cdots\!01 \) Copy content Toggle raw display
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