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Properties

Label 666.2.e.d
Level $666$
Weight $2$
Character orbit 666.e
Analytic conductor $5.318$
Analytic rank $0$
Dimension $14$
Inner twists $2$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [666,2,Mod(223,666)] 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("666.223"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(666, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([4, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 666 = 2 \cdot 3^{2} \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 666.e (of order \(3\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [14,7] 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: \(5.31803677462\)
Analytic rank: \(0\)
Dimension: \(14\)
Relative dimension: \(7\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} + \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{14} + 21x^{12} + 119x^{10} + 272x^{8} + 283x^{6} + 143x^{4} + 34x^{2} + 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 3^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{13}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{10} + 1) q^{2} + \beta_{2} q^{3} + \beta_{10} q^{4} - \beta_{12} q^{5} + ( - \beta_{5} + \beta_{2}) q^{6} + ( - \beta_{13} - \beta_{10} + \cdots - \beta_1) q^{7} - q^{8} + (\beta_{13} + \beta_{10} + \cdots + \beta_1) q^{9}+ \cdots + ( - \beta_{13} + 3 \beta_{12} + \cdots + 3) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q + 7 q^{2} - q^{3} - 7 q^{4} - 3 q^{5} - 2 q^{6} + 3 q^{7} - 14 q^{8} - q^{9} - 6 q^{10} - q^{11} - q^{12} + 9 q^{13} - 3 q^{14} - q^{15} - 7 q^{16} + 18 q^{17} + q^{18} - 26 q^{19} - 3 q^{20}+ \cdots + 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{14} + 21x^{12} + 119x^{10} + 272x^{8} + 283x^{6} + 143x^{4} + 34x^{2} + 3 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 13\nu^{12} + 268\nu^{10} + 1444\nu^{8} + 2982\nu^{6} + 2537\nu^{4} + 888\nu^{2} + 103 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 34 \nu^{13} - 77 \nu^{12} + 701 \nu^{11} - 1584 \nu^{10} + 3778 \nu^{9} - 8484 \nu^{8} + 7804 \nu^{7} + \cdots - 531 ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 34 \nu^{13} - 77 \nu^{12} - 701 \nu^{11} - 1584 \nu^{10} - 3778 \nu^{9} - 8484 \nu^{8} - 7804 \nu^{7} + \cdots - 531 ) / 2 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 31 \nu^{13} + 135 \nu^{12} + 635 \nu^{11} + 2778 \nu^{10} + 3361 \nu^{9} + 14892 \nu^{8} + 6692 \nu^{7} + \cdots + 961 ) / 2 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 66 \nu^{13} + 172 \nu^{12} - 1357 \nu^{11} + 3539 \nu^{10} - 7258 \nu^{9} + 18966 \nu^{8} + \cdots + 1221 ) / 2 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 43 \nu^{13} - 164 \nu^{12} - 883 \nu^{11} - 3374 \nu^{10} - 4706 \nu^{9} - 18076 \nu^{8} + \cdots - 1157 ) / 2 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 100 \nu^{13} + 79 \nu^{12} + 2058 \nu^{11} + 1625 \nu^{10} + 11036 \nu^{9} + 8702 \nu^{8} + 22572 \nu^{7} + \cdots + 564 ) / 2 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( 135\nu^{12} + 2778\nu^{10} + 14892\nu^{8} + 30431\nu^{6} + 25351\nu^{4} + 8595\nu^{2} + 961 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 100 \nu^{13} + 249 \nu^{12} + 2058 \nu^{11} + 5123 \nu^{10} + 11036 \nu^{9} + 27450 \nu^{8} + \cdots + 1752 ) / 2 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( -200\nu^{13} - 4114\nu^{11} - 22031\nu^{9} - 44927\nu^{7} - 37282\nu^{5} - 12566\nu^{3} - 1391\nu - 1 ) / 2 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( -200\nu^{12} - 4114\nu^{10} - 22031\nu^{8} - 44927\nu^{6} - 37282\nu^{4} - 12566\nu^{2} - 1392 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 201 \nu^{13} + 200 \nu^{12} + 4135 \nu^{11} + 4114 \nu^{10} + 22150 \nu^{9} + 22031 \nu^{8} + \cdots + 1392 ) / 2 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 618 \nu^{13} + 151 \nu^{12} + 12716 \nu^{11} + 3106 \nu^{10} + 68151 \nu^{9} + 16632 \nu^{8} + \cdots + 1055 ) / 2 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( 2\beta_{12} + \beta_{11} + 2\beta_{10} - \beta_{9} - \beta_{8} + \beta_{5} + 2\beta_{4} - 2\beta_{3} + \beta_{2} + 1 ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -2\beta_{9} + 2\beta_{8} + \beta_{7} - \beta_{5} - \beta_{3} - \beta _1 - 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 8 \beta_{13} - 8 \beta_{12} - 4 \beta_{11} - 36 \beta_{10} + 5 \beta_{9} + 14 \beta_{8} + 2 \beta_{7} + \cdots - 16 ) / 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 4\beta_{11} + 28\beta_{9} - 29\beta_{8} - 9\beta_{7} + 19\beta_{5} + 12\beta_{3} - 7\beta_{2} + 11\beta _1 + 42 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 128 \beta_{13} + 76 \beta_{12} + 38 \beta_{11} + 530 \beta_{10} - 45 \beta_{9} - 189 \beta_{8} + \cdots + 242 ) / 3 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 68 \beta_{11} - 380 \beta_{9} + 399 \beta_{8} + 105 \beta_{7} - 275 \beta_{5} - 153 \beta_{3} + \cdots - 542 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 1786 \beta_{13} - 948 \beta_{12} - 474 \beta_{11} - 7322 \beta_{10} + 542 \beta_{9} + 2564 \beta_{8} + \cdots - 3381 ) / 3 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 969 \beta_{11} + 5166 \beta_{9} - 5449 \beta_{8} - 1369 \beta_{7} + 3797 \beta_{5} + 2043 \beta_{3} + \cdots + 7292 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 24426 \beta_{13} + 12646 \beta_{12} + 6323 \beta_{11} + 100006 \beta_{10} - 7145 \beta_{9} + \cdots + 46337 ) / 3 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 13338 \beta_{11} - 70325 \beta_{9} + 74278 \beta_{8} + 18434 \beta_{7} - 51891 \beta_{5} + \cdots - 99035 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 332968 \beta_{13} - 171394 \beta_{12} - 85697 \beta_{11} - 1363026 \beta_{10} + 96508 \beta_{9} + \cdots - 632153 ) / 3 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 182152 \beta_{11} + 957792 \beta_{9} - 1012013 \beta_{8} - 250357 \beta_{7} + 707435 \beta_{5} + \cdots + 1348067 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 4536346 \beta_{13} + 2331830 \beta_{12} + 1165915 \beta_{11} + 18569512 \beta_{10} - 1311783 \beta_{9} + \cdots + 8614513 ) / 3 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/666\mathbb{Z}\right)^\times\).

\(n\) \(371\) \(631\)
\(\chi(n)\) \(\beta_{10}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
223.1
− 0.871950i
0.655413i
1.42526i
− 3.69082i
− 0.650859i
− 0.472514i
1.87343i
0.871950i
− 0.655413i
− 1.42526i
3.69082i
0.650859i
0.472514i
− 1.87343i
0.500000 − 0.866025i −1.70806 − 0.287285i −0.500000 − 0.866025i 0.493206 + 0.854257i −1.10283 + 1.33558i 0.970551 − 1.68104i −1.00000 2.83493 + 0.981400i 0.986411
223.2 0.500000 − 0.866025i −0.921133 − 1.46680i −0.500000 − 0.866025i −1.82134 − 3.15466i −1.73086 + 0.0643229i 1.65326 − 2.86353i −1.00000 −1.30303 + 2.70224i −3.64269
223.3 0.500000 − 0.866025i −0.912543 + 1.47216i −0.500000 − 0.866025i −1.10763 − 1.91847i 0.818660 + 1.52637i −1.77451 + 3.07355i −1.00000 −1.33453 − 2.68682i −2.21526
223.4 0.500000 − 0.866025i −0.398701 − 1.68554i −0.500000 − 0.866025i −0.265357 − 0.459612i −1.65907 − 0.497484i −1.21257 + 2.10023i −1.00000 −2.68208 + 1.34405i −0.530714
223.5 0.500000 − 0.866025i 0.135216 + 1.72676i −0.500000 − 0.866025i 0.830588 + 1.43862i 1.56303 + 0.746282i 0.0395868 − 0.0685663i −1.00000 −2.96343 + 0.466973i 1.66118
223.6 0.500000 − 0.866025i 1.57571 − 0.719122i −0.500000 − 0.866025i 1.33280 + 2.30848i 0.165078 − 1.72417i 0.530356 − 0.918603i −1.00000 1.96573 − 2.26626i 2.66561
223.7 0.500000 − 0.866025i 1.72951 + 0.0937946i −0.500000 − 0.866025i −0.962268 − 1.66670i 0.945983 − 1.45090i 1.29333 − 2.24011i −1.00000 2.98241 + 0.324437i −1.92454
445.1 0.500000 + 0.866025i −1.70806 + 0.287285i −0.500000 + 0.866025i 0.493206 − 0.854257i −1.10283 − 1.33558i 0.970551 + 1.68104i −1.00000 2.83493 − 0.981400i 0.986411
445.2 0.500000 + 0.866025i −0.921133 + 1.46680i −0.500000 + 0.866025i −1.82134 + 3.15466i −1.73086 − 0.0643229i 1.65326 + 2.86353i −1.00000 −1.30303 − 2.70224i −3.64269
445.3 0.500000 + 0.866025i −0.912543 − 1.47216i −0.500000 + 0.866025i −1.10763 + 1.91847i 0.818660 − 1.52637i −1.77451 − 3.07355i −1.00000 −1.33453 + 2.68682i −2.21526
445.4 0.500000 + 0.866025i −0.398701 + 1.68554i −0.500000 + 0.866025i −0.265357 + 0.459612i −1.65907 + 0.497484i −1.21257 − 2.10023i −1.00000 −2.68208 − 1.34405i −0.530714
445.5 0.500000 + 0.866025i 0.135216 − 1.72676i −0.500000 + 0.866025i 0.830588 − 1.43862i 1.56303 − 0.746282i 0.0395868 + 0.0685663i −1.00000 −2.96343 − 0.466973i 1.66118
445.6 0.500000 + 0.866025i 1.57571 + 0.719122i −0.500000 + 0.866025i 1.33280 − 2.30848i 0.165078 + 1.72417i 0.530356 + 0.918603i −1.00000 1.96573 + 2.26626i 2.66561
445.7 0.500000 + 0.866025i 1.72951 − 0.0937946i −0.500000 + 0.866025i −0.962268 + 1.66670i 0.945983 + 1.45090i 1.29333 + 2.24011i −1.00000 2.98241 − 0.324437i −1.92454
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 223.7
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 666.2.e.d ✓ 14
3.b odd 2 1 1998.2.e.d 14
9.c even 3 1 inner 666.2.e.d ✓ 14
9.c even 3 1 5994.2.a.u 7
9.d odd 6 1 1998.2.e.d 14
9.d odd 6 1 5994.2.a.v 7
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
666.2.e.d ✓ 14 1.a even 1 1 trivial
666.2.e.d ✓ 14 9.c even 3 1 inner
1998.2.e.d 14 3.b odd 2 1
1998.2.e.d 14 9.d odd 6 1
5994.2.a.u 7 9.c even 3 1
5994.2.a.v 7 9.d odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{14} + 3 T_{5}^{13} + 21 T_{5}^{12} + 26 T_{5}^{11} + 198 T_{5}^{10} + 216 T_{5}^{9} + 1113 T_{5}^{8} + \cdots + 1296 \) acting on \(S_{2}^{\mathrm{new}}(666, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} - T + 1)^{7} \) Copy content Toggle raw display
$3$ \( T^{14} + T^{13} + \cdots + 2187 \) Copy content Toggle raw display
$5$ \( T^{14} + 3 T^{13} + \cdots + 1296 \) Copy content Toggle raw display
$7$ \( T^{14} - 3 T^{13} + \cdots + 144 \) Copy content Toggle raw display
$11$ \( T^{14} + T^{13} + \cdots + 7921 \) Copy content Toggle raw display
$13$ \( T^{14} - 9 T^{13} + \cdots + 2916 \) Copy content Toggle raw display
$17$ \( (T^{7} - 9 T^{6} - 16 T^{5} + \cdots + 9)^{2} \) Copy content Toggle raw display
$19$ \( (T^{7} + 13 T^{6} + \cdots + 489)^{2} \) Copy content Toggle raw display
$23$ \( T^{14} + \cdots + 119727364 \) Copy content Toggle raw display
$29$ \( T^{14} + 3 T^{13} + \cdots + 795664 \) Copy content Toggle raw display
$31$ \( T^{14} - 19 T^{13} + \cdots + 18507204 \) Copy content Toggle raw display
$37$ \( (T + 1)^{14} \) Copy content Toggle raw display
$41$ \( T^{14} + 24 T^{13} + \cdots + 205209 \) Copy content Toggle raw display
$43$ \( T^{14} - T^{13} + \cdots + 25250625 \) Copy content Toggle raw display
$47$ \( T^{14} + \cdots + 3065615424 \) Copy content Toggle raw display
$53$ \( (T^{7} - 4 T^{6} + \cdots - 6508)^{2} \) Copy content Toggle raw display
$59$ \( T^{14} + \cdots + 249221606841 \) Copy content Toggle raw display
$61$ \( T^{14} + \cdots + 67079964004 \) Copy content Toggle raw display
$67$ \( T^{14} - 15 T^{13} + \cdots + 17447329 \) Copy content Toggle raw display
$71$ \( (T^{7} + 3 T^{6} + \cdots + 1762452)^{2} \) Copy content Toggle raw display
$73$ \( (T^{7} + 7 T^{6} + \cdots + 1228992)^{2} \) Copy content Toggle raw display
$79$ \( T^{14} + \cdots + 4783520139876 \) Copy content Toggle raw display
$83$ \( T^{14} + \cdots + 229704336 \) Copy content Toggle raw display
$89$ \( (T^{7} - 13 T^{6} + \cdots + 1734)^{2} \) Copy content Toggle raw display
$97$ \( T^{14} + \cdots + 129512525447449 \) Copy content Toggle raw display
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