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Properties

Label 546.4.l.j
Level $546$
Weight $4$
Character orbit 546.l
Analytic conductor $32.215$
Analytic rank $0$
Dimension $10$
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [546,4,Mod(211,546)] 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("546.211"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(546, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 2])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 546 = 2 \cdot 3 \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 546.l (of order \(3\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [10,10,15,-20,-28] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(32.2150428631\)
Analytic rank: \(0\)
Dimension: \(10\)
Relative dimension: \(5\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - 5 x^{9} + 171 x^{8} - 654 x^{7} + 8320 x^{6} - 22692 x^{5} + 91812 x^{4} - 146557 x^{3} + \cdots + 52689 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
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 \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - 2 \beta_{6} + 2) q^{2} + ( - 3 \beta_{6} + 3) q^{3} - 4 \beta_{6} q^{4} + ( - \beta_1 - 3) q^{5} - 6 \beta_{6} q^{6} + 7 \beta_{6} q^{7} - 8 q^{8} - 9 \beta_{6} q^{9} + (6 \beta_{6} + 2 \beta_{5} - 2 \beta_1 - 6) q^{10}+ \cdots + ( - 9 \beta_{4} + 9 \beta_{3} + \cdots + 27) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q + 10 q^{2} + 15 q^{3} - 20 q^{4} - 28 q^{5} - 30 q^{6} + 35 q^{7} - 80 q^{8} - 45 q^{9} - 28 q^{10} - 13 q^{11} - 120 q^{12} + 46 q^{13} + 140 q^{14} - 42 q^{15} - 80 q^{16} - 14 q^{17} - 180 q^{18}+ \cdots + 234 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - 5 x^{9} + 171 x^{8} - 654 x^{7} + 8320 x^{6} - 22692 x^{5} + 91812 x^{4} - 146557 x^{3} + \cdots + 52689 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{8} - 4\nu^{7} + 114\nu^{6} - 328\nu^{5} + 2979\nu^{4} - 5416\nu^{3} + 14945\nu^{2} - 12291\nu - 1491 ) / 5145 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 13 \nu^{8} - 52 \nu^{7} + 2168 \nu^{6} - 6322 \nu^{5} + 111786 \nu^{4} - 213096 \nu^{3} + \cdots + 4341519 ) / 75460 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 95 \nu^{8} + 380 \nu^{7} - 17004 \nu^{6} + 49682 \nu^{5} - 827346 \nu^{4} + 1572332 \nu^{3} + \cdots - 1753773 ) / 226380 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 95 \nu^{8} - 380 \nu^{7} + 17004 \nu^{6} - 49682 \nu^{5} + 827346 \nu^{4} - 1572332 \nu^{3} + \cdots + 9224313 ) / 226380 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 67 \nu^{9} + 862 \nu^{8} + 7234 \nu^{7} + 92438 \nu^{6} + 222031 \nu^{5} + 2056718 \nu^{4} + \cdots - 19032741 ) / 11972415 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 134 \nu^{9} + 603 \nu^{8} - 23776 \nu^{7} + 80402 \nu^{6} - 1207318 \nu^{5} + 2818697 \nu^{4} + \cdots + 22623510 ) / 11972415 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 140467 \nu^{9} + 521569 \nu^{8} - 23695080 \nu^{7} + 61746274 \nu^{6} - 1072830828 \nu^{5} + \cdots - 1923136761 ) / 526786260 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 384755 \nu^{9} - 1841930 \nu^{8} + 64779060 \nu^{7} - 236883554 \nu^{6} + 3089088822 \nu^{5} + \cdots - 34851902358 ) / 526786260 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 579259 \nu^{9} + 2561289 \nu^{8} - 97094324 \nu^{7} + 320733562 \nu^{6} - 4563602924 \nu^{5} + \cdots + 22731137247 ) / 526786260 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{6} + 2\beta_{5} - \beta _1 + 1 ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{6} + 2\beta_{5} + 3\beta_{4} + 3\beta_{3} - \beta _1 - 98 ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 10 \beta_{9} - 13 \beta_{8} + 5 \beta_{7} - 52 \beta_{6} - 134 \beta_{5} - 2 \beta_{4} + 2 \beta_{3} + \cdots - 121 ) / 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 20 \beta_{9} - 26 \beta_{8} + 10 \beta_{7} - 105 \beta_{6} - 270 \beta_{5} - 247 \beta_{4} + \cdots + 6786 ) / 3 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 788 \beta_{9} + 1031 \beta_{8} - 361 \beta_{7} + 3296 \beta_{6} + 9667 \beta_{5} - 77 \beta_{4} + \cdots + 15431 ) / 3 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 2414 \beta_{9} + 3158 \beta_{8} - 1108 \beta_{7} + 10151 \beta_{6} + 29677 \beta_{5} + 18622 \beta_{4} + \cdots - 488554 ) / 3 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 56835 \beta_{9} - 74949 \beta_{8} + 25110 \beta_{7} - 251150 \beta_{6} - 691600 \beta_{5} + \cdots - 1626548 ) / 3 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 238652 \beta_{9} - 314594 \beta_{8} + 105634 \beta_{7} - 1052217 \beta_{6} - 2905524 \beta_{5} + \cdots + 34860876 ) / 3 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 4010052 \beta_{9} + 5334624 \beta_{8} - 1717704 \beta_{7} + 19163638 \beta_{6} + 48719096 \beta_{5} + \cdots + 155509399 ) / 3 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/546\mathbb{Z}\right)^\times\).

\(n\) \(157\) \(365\) \(379\)
\(\chi(n)\) \(1\) \(1\) \(-\beta_{6}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
211.1
0.500000 − 8.64371i
0.500000 − 2.84401i
0.500000 + 0.359672i
0.500000 + 1.65508i
0.500000 + 8.60693i
0.500000 + 8.64371i
0.500000 + 2.84401i
0.500000 − 0.359672i
0.500000 − 1.65508i
0.500000 − 8.60693i
1.00000 + 1.73205i 1.50000 + 2.59808i −2.00000 + 3.46410i −17.4713 −3.00000 + 5.19615i 3.50000 − 6.06218i −8.00000 −4.50000 + 7.79423i −17.4713 − 30.2612i
211.2 1.00000 + 1.73205i 1.50000 + 2.59808i −2.00000 + 3.46410i −7.42597 −3.00000 + 5.19615i 3.50000 − 6.06218i −8.00000 −4.50000 + 7.79423i −7.42597 − 12.8622i
211.3 1.00000 + 1.73205i 1.50000 + 2.59808i −2.00000 + 3.46410i −1.87703 −3.00000 + 5.19615i 3.50000 − 6.06218i −8.00000 −4.50000 + 7.79423i −1.87703 − 3.25111i
211.4 1.00000 + 1.73205i 1.50000 + 2.59808i −2.00000 + 3.46410i 0.366691 −3.00000 + 5.19615i 3.50000 − 6.06218i −8.00000 −4.50000 + 7.79423i 0.366691 + 0.635127i
211.5 1.00000 + 1.73205i 1.50000 + 2.59808i −2.00000 + 3.46410i 12.4076 −3.00000 + 5.19615i 3.50000 − 6.06218i −8.00000 −4.50000 + 7.79423i 12.4076 + 21.4907i
295.1 1.00000 − 1.73205i 1.50000 − 2.59808i −2.00000 − 3.46410i −17.4713 −3.00000 − 5.19615i 3.50000 + 6.06218i −8.00000 −4.50000 − 7.79423i −17.4713 + 30.2612i
295.2 1.00000 − 1.73205i 1.50000 − 2.59808i −2.00000 − 3.46410i −7.42597 −3.00000 − 5.19615i 3.50000 + 6.06218i −8.00000 −4.50000 − 7.79423i −7.42597 + 12.8622i
295.3 1.00000 − 1.73205i 1.50000 − 2.59808i −2.00000 − 3.46410i −1.87703 −3.00000 − 5.19615i 3.50000 + 6.06218i −8.00000 −4.50000 − 7.79423i −1.87703 + 3.25111i
295.4 1.00000 − 1.73205i 1.50000 − 2.59808i −2.00000 − 3.46410i 0.366691 −3.00000 − 5.19615i 3.50000 + 6.06218i −8.00000 −4.50000 − 7.79423i 0.366691 − 0.635127i
295.5 1.00000 − 1.73205i 1.50000 − 2.59808i −2.00000 − 3.46410i 12.4076 −3.00000 − 5.19615i 3.50000 + 6.06218i −8.00000 −4.50000 − 7.79423i 12.4076 − 21.4907i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 211.5
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 546.4.l.j ✓ 10
13.c even 3 1 inner 546.4.l.j ✓ 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
546.4.l.j ✓ 10 1.a even 1 1 trivial
546.4.l.j ✓ 10 13.c even 3 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{5} + 14T_{5}^{4} - 161T_{5}^{3} - 1889T_{5}^{2} - 2308T_{5} + 1108 \) acting on \(S_{4}^{\mathrm{new}}(546, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} - 2 T + 4)^{5} \) Copy content Toggle raw display
$3$ \( (T^{2} - 3 T + 9)^{5} \) Copy content Toggle raw display
$5$ \( (T^{5} + 14 T^{4} + \cdots + 1108)^{2} \) Copy content Toggle raw display
$7$ \( (T^{2} - 7 T + 49)^{5} \) Copy content Toggle raw display
$11$ \( T^{10} + \cdots + 353081537335296 \) Copy content Toggle raw display
$13$ \( T^{10} + \cdots + 51\!\cdots\!57 \) Copy content Toggle raw display
$17$ \( T^{10} + \cdots + 48\!\cdots\!01 \) Copy content Toggle raw display
$19$ \( T^{10} + \cdots + 22\!\cdots\!16 \) Copy content Toggle raw display
$23$ \( T^{10} + \cdots + 40\!\cdots\!56 \) Copy content Toggle raw display
$29$ \( T^{10} + \cdots + 86\!\cdots\!24 \) Copy content Toggle raw display
$31$ \( (T^{5} + 156 T^{4} + \cdots + 42086342154)^{2} \) Copy content Toggle raw display
$37$ \( T^{10} + \cdots + 24\!\cdots\!84 \) Copy content Toggle raw display
$41$ \( T^{10} + \cdots + 12\!\cdots\!56 \) Copy content Toggle raw display
$43$ \( T^{10} + \cdots + 49\!\cdots\!24 \) Copy content Toggle raw display
$47$ \( (T^{5} + 30 T^{4} + \cdots - 157948746)^{2} \) Copy content Toggle raw display
$53$ \( (T^{5} - 1149 T^{4} + \cdots + 615167770203)^{2} \) Copy content Toggle raw display
$59$ \( T^{10} + \cdots + 27\!\cdots\!16 \) Copy content Toggle raw display
$61$ \( T^{10} + \cdots + 36\!\cdots\!29 \) Copy content Toggle raw display
$67$ \( T^{10} + \cdots + 27\!\cdots\!04 \) Copy content Toggle raw display
$71$ \( T^{10} + \cdots + 10\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( (T^{5} + \cdots - 10228962399408)^{2} \) Copy content Toggle raw display
$79$ \( (T^{5} + \cdots + 21504139820592)^{2} \) Copy content Toggle raw display
$83$ \( (T^{5} + \cdots - 1437180391764)^{2} \) Copy content Toggle raw display
$89$ \( T^{10} + \cdots + 46\!\cdots\!49 \) Copy content Toggle raw display
$97$ \( T^{10} + \cdots + 32\!\cdots\!84 \) Copy content Toggle raw display
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