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Properties

Label 546.4.s.b
Level $546$
Weight $4$
Character orbit 546.s
Analytic conductor $32.215$
Analytic rank $0$
Dimension $20$
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(43,546)] 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("546.43"); 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, 1])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 546 = 2 \cdot 3 \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 546.s (of order \(6\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [20,0,30] 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: \(32.2150428631\)
Analytic rank: \(0\)
Dimension: \(20\)
Relative dimension: \(10\) over \(\Q(\zeta_{6})\)
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: \( x^{20} + 1316 x^{18} + 722042 x^{16} + 215100738 x^{14} + 38026916607 x^{12} + 4099786385782 x^{10} + \cdots + 10\!\cdots\!16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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.

\(f(q)\) \(=\) \( q - 2 \beta_{4} q^{2} + 3 \beta_{7} q^{3} + ( - 4 \beta_{7} + 4) q^{4} + (2 \beta_{7} + 2 \beta_{5} - 2 \beta_{4} + \cdots - 1) q^{5} - 6 \beta_{5} q^{6} - 7 \beta_{5} q^{7} + (8 \beta_{5} - 8 \beta_{4}) q^{8}+ \cdots + (9 \beta_{18} - 9 \beta_{17} + \cdots + 45) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q + 30 q^{3} + 40 q^{4} - 90 q^{9} - 48 q^{10} - 138 q^{11} + 240 q^{12} + 28 q^{13} + 280 q^{14} - 90 q^{15} - 160 q^{16} - 106 q^{17} + 60 q^{19} + 120 q^{20} - 64 q^{22} - 94 q^{23} - 304 q^{25} + 44 q^{26}+ \cdots + 5412 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} + 1316 x^{18} + 722042 x^{16} + 215100738 x^{14} + 38026916607 x^{12} + 4099786385782 x^{10} + \cdots + 10\!\cdots\!16 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 20\!\cdots\!19 \nu^{18} + \cdots + 33\!\cdots\!12 ) / 57\!\cdots\!84 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 45\!\cdots\!09 \nu^{18} + \cdots - 70\!\cdots\!72 ) / 78\!\cdots\!68 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 47\!\cdots\!39 \nu^{19} + \cdots + 97\!\cdots\!72 ) / 10\!\cdots\!12 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 47\!\cdots\!39 \nu^{19} + \cdots + 97\!\cdots\!72 ) / 10\!\cdots\!12 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 12\!\cdots\!89 \nu^{19} + \cdots + 82\!\cdots\!32 ) / 28\!\cdots\!12 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 34\!\cdots\!49 \nu^{19} + \cdots + 30\!\cdots\!44 ) / 61\!\cdots\!88 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 10\!\cdots\!37 \nu^{19} + \cdots - 47\!\cdots\!68 ) / 10\!\cdots\!84 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 12\!\cdots\!51 \nu^{19} + \cdots + 68\!\cdots\!32 ) / 45\!\cdots\!92 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 38\!\cdots\!71 \nu^{19} + \cdots - 94\!\cdots\!28 ) / 91\!\cdots\!84 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 53\!\cdots\!51 \nu^{19} + \cdots + 91\!\cdots\!20 ) / 11\!\cdots\!48 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 53\!\cdots\!51 \nu^{19} + \cdots - 91\!\cdots\!20 ) / 11\!\cdots\!48 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 29\!\cdots\!55 \nu^{19} + \cdots + 71\!\cdots\!08 ) / 57\!\cdots\!24 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 52\!\cdots\!13 \nu^{19} + \cdots + 20\!\cdots\!36 ) / 91\!\cdots\!84 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 52\!\cdots\!13 \nu^{19} + \cdots - 20\!\cdots\!36 ) / 91\!\cdots\!84 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( - 60\!\cdots\!29 \nu^{19} + \cdots - 86\!\cdots\!72 ) / 91\!\cdots\!84 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( - 69\!\cdots\!75 \nu^{19} + \cdots + 12\!\cdots\!88 ) / 91\!\cdots\!84 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( 13\!\cdots\!61 \nu^{19} + \cdots + 48\!\cdots\!20 ) / 91\!\cdots\!84 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( 76\!\cdots\!95 \nu^{19} + \cdots + 60\!\cdots\!28 ) / 45\!\cdots\!92 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 5 \beta_{18} + 5 \beta_{17} + 2 \beta_{16} - 2 \beta_{15} + 2 \beta_{14} + 3 \beta_{13} + 2 \beta_{12} + \cdots - 131 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 5 \beta_{19} - 3 \beta_{18} + 3 \beta_{17} + 13 \beta_{16} - 13 \beta_{15} - 13 \beta_{14} + 15 \beta_{13} + \cdots + 82 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 156 \beta_{19} - 1606 \beta_{18} - 1606 \beta_{17} - 603 \beta_{16} + 281 \beta_{15} - 593 \beta_{14} + \cdots + 28519 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 1942 \beta_{19} + 961 \beta_{18} - 961 \beta_{17} - 4395 \beta_{16} + 4870 \beta_{15} + 4870 \beta_{14} + \cdots - 24127 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 44967 \beta_{19} + 473912 \beta_{18} + 473912 \beta_{17} + 148055 \beta_{16} - 57494 \beta_{15} + \cdots - 7312590 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 795187 \beta_{19} - 365288 \beta_{18} + 365288 \beta_{17} + 1276299 \beta_{16} - 1421989 \beta_{15} + \cdots + 6324817 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 9653108 \beta_{19} - 139665058 \beta_{18} - 139665058 \beta_{17} - 35226058 \beta_{16} + \cdots + 2014880206 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 296740447 \beta_{19} + 140404198 \beta_{18} - 140404198 \beta_{17} - 365272232 \beta_{16} + \cdots - 1544934179 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 1753848035 \beta_{19} + 41492190653 \beta_{18} + 41492190653 \beta_{17} + 8576676562 \beta_{16} + \cdots - 575412444828 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 102654838510 \beta_{19} - 51393232808 \beta_{18} + 51393232808 \beta_{17} + 105673599655 \beta_{16} + \cdots + 352473989734 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 250442850559 \beta_{19} - 12412805551006 \beta_{18} - 12412805551006 \beta_{17} + \cdots + 167615022888849 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 33951155774510 \beta_{19} + 17916640023029 \beta_{18} - 17916640023029 \beta_{17} + \cdots - 74100705578667 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 12387065957038 \beta_{19} + \cdots - 49\!\cdots\!86 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 10\!\cdots\!16 \beta_{19} + \cdots + 13\!\cdots\!64 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( 10\!\cdots\!00 \beta_{19} + \cdots + 14\!\cdots\!39 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( - 34\!\cdots\!69 \beta_{19} + \cdots - 18\!\cdots\!50 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( - 62\!\cdots\!56 \beta_{19} + \cdots - 43\!\cdots\!11 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( 10\!\cdots\!30 \beta_{19} + \cdots - 65\!\cdots\!49 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(157\) \(365\) \(379\)
\(\chi(n)\) \(1\) \(1\) \(1 - \beta_{7}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
43.1
− 14.9809i
− 9.93139i
− 0.270254i
10.2135i
12.9691i
− 17.3330i
− 8.09042i
4.61550i
5.35994i
17.4480i
− 12.9691i
− 10.2135i
0.270254i
9.93139i
14.9809i
− 17.4480i
− 5.35994i
− 4.61550i
8.09042i
17.3330i
−1.73205 − 1.00000i 1.50000 − 2.59808i 2.00000 + 3.46410i − 18.7130i −5.19615 + 3.00000i −6.06218 + 3.50000i − 8.00000i −4.50000 − 7.79423i −18.7130 + 32.4118i
43.2 −1.73205 − 1.00000i 1.50000 − 2.59808i 2.00000 + 3.46410i − 13.6634i −5.19615 + 3.00000i −6.06218 + 3.50000i − 8.00000i −4.50000 − 7.79423i −13.6634 + 23.6658i
43.3 −1.73205 − 1.00000i 1.50000 − 2.59808i 2.00000 + 3.46410i − 4.00230i −5.19615 + 3.00000i −6.06218 + 3.50000i − 8.00000i −4.50000 − 7.79423i −4.00230 + 6.93220i
43.4 −1.73205 − 1.00000i 1.50000 − 2.59808i 2.00000 + 3.46410i 6.48142i −5.19615 + 3.00000i −6.06218 + 3.50000i − 8.00000i −4.50000 − 7.79423i 6.48142 − 11.2262i
43.5 −1.73205 − 1.00000i 1.50000 − 2.59808i 2.00000 + 3.46410i 9.23705i −5.19615 + 3.00000i −6.06218 + 3.50000i − 8.00000i −4.50000 − 7.79423i 9.23705 − 15.9990i
43.6 1.73205 + 1.00000i 1.50000 − 2.59808i 2.00000 + 3.46410i − 17.0651i 5.19615 − 3.00000i 6.06218 − 3.50000i 8.00000i −4.50000 − 7.79423i 17.0651 − 29.5576i
43.7 1.73205 + 1.00000i 1.50000 − 2.59808i 2.00000 + 3.46410i − 7.82247i 5.19615 − 3.00000i 6.06218 − 3.50000i 8.00000i −4.50000 − 7.79423i 7.82247 − 13.5489i
43.8 1.73205 + 1.00000i 1.50000 − 2.59808i 2.00000 + 3.46410i 4.88345i 5.19615 − 3.00000i 6.06218 − 3.50000i 8.00000i −4.50000 − 7.79423i −4.88345 + 8.45839i
43.9 1.73205 + 1.00000i 1.50000 − 2.59808i 2.00000 + 3.46410i 5.62789i 5.19615 − 3.00000i 6.06218 − 3.50000i 8.00000i −4.50000 − 7.79423i −5.62789 + 9.74779i
43.10 1.73205 + 1.00000i 1.50000 − 2.59808i 2.00000 + 3.46410i 17.7160i 5.19615 − 3.00000i 6.06218 − 3.50000i 8.00000i −4.50000 − 7.79423i −17.7160 + 30.6849i
127.1 −1.73205 + 1.00000i 1.50000 + 2.59808i 2.00000 − 3.46410i − 9.23705i −5.19615 − 3.00000i −6.06218 − 3.50000i 8.00000i −4.50000 + 7.79423i 9.23705 + 15.9990i
127.2 −1.73205 + 1.00000i 1.50000 + 2.59808i 2.00000 − 3.46410i − 6.48142i −5.19615 − 3.00000i −6.06218 − 3.50000i 8.00000i −4.50000 + 7.79423i 6.48142 + 11.2262i
127.3 −1.73205 + 1.00000i 1.50000 + 2.59808i 2.00000 − 3.46410i 4.00230i −5.19615 − 3.00000i −6.06218 − 3.50000i 8.00000i −4.50000 + 7.79423i −4.00230 − 6.93220i
127.4 −1.73205 + 1.00000i 1.50000 + 2.59808i 2.00000 − 3.46410i 13.6634i −5.19615 − 3.00000i −6.06218 − 3.50000i 8.00000i −4.50000 + 7.79423i −13.6634 − 23.6658i
127.5 −1.73205 + 1.00000i 1.50000 + 2.59808i 2.00000 − 3.46410i 18.7130i −5.19615 − 3.00000i −6.06218 − 3.50000i 8.00000i −4.50000 + 7.79423i −18.7130 − 32.4118i
127.6 1.73205 − 1.00000i 1.50000 + 2.59808i 2.00000 − 3.46410i − 17.7160i 5.19615 + 3.00000i 6.06218 + 3.50000i − 8.00000i −4.50000 + 7.79423i −17.7160 − 30.6849i
127.7 1.73205 − 1.00000i 1.50000 + 2.59808i 2.00000 − 3.46410i − 5.62789i 5.19615 + 3.00000i 6.06218 + 3.50000i − 8.00000i −4.50000 + 7.79423i −5.62789 − 9.74779i
127.8 1.73205 − 1.00000i 1.50000 + 2.59808i 2.00000 − 3.46410i − 4.88345i 5.19615 + 3.00000i 6.06218 + 3.50000i − 8.00000i −4.50000 + 7.79423i −4.88345 − 8.45839i
127.9 1.73205 − 1.00000i 1.50000 + 2.59808i 2.00000 − 3.46410i 7.82247i 5.19615 + 3.00000i 6.06218 + 3.50000i − 8.00000i −4.50000 + 7.79423i 7.82247 + 13.5489i
127.10 1.73205 − 1.00000i 1.50000 + 2.59808i 2.00000 − 3.46410i 17.0651i 5.19615 + 3.00000i 6.06218 + 3.50000i − 8.00000i −4.50000 + 7.79423i 17.0651 + 29.5576i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 43.10
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.e even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 546.4.s.b ✓ 20
13.e even 6 1 inner 546.4.s.b ✓ 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
546.4.s.b ✓ 20 1.a even 1 1 trivial
546.4.s.b ✓ 20 13.e even 6 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{20} + 1402 T_{5}^{18} + 805099 T_{5}^{16} + 245501122 T_{5}^{14} + 43374184591 T_{5}^{12} + \cdots + 15\!\cdots\!76 \) acting on \(S_{4}^{\mathrm{new}}(546, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} - 4 T^{2} + 16)^{5} \) Copy content Toggle raw display
$3$ \( (T^{2} - 3 T + 9)^{10} \) Copy content Toggle raw display
$5$ \( T^{20} + \cdots + 15\!\cdots\!76 \) Copy content Toggle raw display
$7$ \( (T^{4} - 49 T^{2} + 2401)^{5} \) Copy content Toggle raw display
$11$ \( T^{20} + \cdots + 12\!\cdots\!16 \) Copy content Toggle raw display
$13$ \( T^{20} + \cdots + 26\!\cdots\!49 \) Copy content Toggle raw display
$17$ \( T^{20} + \cdots + 83\!\cdots\!61 \) Copy content Toggle raw display
$19$ \( T^{20} + \cdots + 12\!\cdots\!84 \) Copy content Toggle raw display
$23$ \( T^{20} + \cdots + 37\!\cdots\!64 \) Copy content Toggle raw display
$29$ \( T^{20} + \cdots + 16\!\cdots\!61 \) Copy content Toggle raw display
$31$ \( T^{20} + \cdots + 10\!\cdots\!96 \) Copy content Toggle raw display
$37$ \( T^{20} + \cdots + 41\!\cdots\!64 \) Copy content Toggle raw display
$41$ \( T^{20} + \cdots + 10\!\cdots\!64 \) Copy content Toggle raw display
$43$ \( T^{20} + \cdots + 83\!\cdots\!56 \) Copy content Toggle raw display
$47$ \( T^{20} + \cdots + 17\!\cdots\!36 \) Copy content Toggle raw display
$53$ \( (T^{10} + \cdots + 55\!\cdots\!69)^{2} \) Copy content Toggle raw display
$59$ \( T^{20} + \cdots + 16\!\cdots\!04 \) Copy content Toggle raw display
$61$ \( T^{20} + \cdots + 45\!\cdots\!41 \) Copy content Toggle raw display
$67$ \( T^{20} + \cdots + 70\!\cdots\!76 \) Copy content Toggle raw display
$71$ \( T^{20} + \cdots + 81\!\cdots\!84 \) Copy content Toggle raw display
$73$ \( T^{20} + \cdots + 22\!\cdots\!04 \) Copy content Toggle raw display
$79$ \( (T^{10} + \cdots - 17\!\cdots\!48)^{2} \) Copy content Toggle raw display
$83$ \( T^{20} + \cdots + 16\!\cdots\!44 \) Copy content Toggle raw display
$89$ \( T^{20} + \cdots + 36\!\cdots\!04 \) Copy content Toggle raw display
$97$ \( T^{20} + \cdots + 32\!\cdots\!76 \) Copy content Toggle raw display
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