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Properties

Label 546.4.c.d
Level $546$
Weight $4$
Character orbit 546.c
Analytic conductor $32.215$
Analytic rank $0$
Dimension $12$
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(337,546)] 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("546.337"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(546, base_ring=CyclotomicField(2)) 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.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [12,0,36] 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: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} + 884x^{10} + 263470x^{8} + 30253092x^{6} + 990834105x^{4} + 944723920x^{2} + 120472576 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - 2 \beta_{6} q^{2} + 3 q^{3} - 4 q^{4} + ( - 3 \beta_{6} + \beta_1) q^{5} - 6 \beta_{6} q^{6} + 7 \beta_{6} q^{7} + 8 \beta_{6} q^{8} + 9 q^{9} + (2 \beta_{2} - 6) q^{10} + ( - \beta_{7} + 9 \beta_{6}) q^{11}+ \cdots + ( - 9 \beta_{7} + 81 \beta_{6}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 36 q^{3} - 48 q^{4} + 108 q^{9} - 80 q^{10} - 144 q^{12} + 20 q^{13} + 168 q^{14} + 192 q^{16} - 250 q^{17} + 212 q^{22} - 556 q^{23} - 400 q^{25} + 128 q^{26} + 324 q^{27} + 256 q^{29} - 240 q^{30}+ \cdots + 3116 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 884x^{10} + 263470x^{8} + 30253092x^{6} + 990834105x^{4} + 944723920x^{2} + 120472576 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 123215 \nu^{10} - 110023797 \nu^{8} - 33596012783 \nu^{6} - 4051690649435 \nu^{4} + \cdots - 50386184457280 ) / 73322759284824 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 90558721166 \nu^{10} + 79885559663337 \nu^{8} + \cdots + 11\!\cdots\!48 ) / 13\!\cdots\!64 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 273913871113 \nu^{10} + 241654821167328 \nu^{8} + \cdots + 13\!\cdots\!52 ) / 26\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 1681946270995 \nu^{10} + \cdots - 73\!\cdots\!88 ) / 62\!\cdots\!32 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 2295289015 \nu^{11} - 2028359285340 \nu^{9} - 604135986184114 \nu^{7} + \cdots - 13\!\cdots\!24 \nu ) / 40\!\cdots\!12 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 643807015217455 \nu^{11} + \cdots + 76\!\cdots\!56 \nu ) / 73\!\cdots\!32 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 334529010873977 \nu^{11} + 291745272157136 \nu^{10} + \cdots + 15\!\cdots\!60 ) / 36\!\cdots\!16 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 334529010873977 \nu^{11} - 43252141277632 \nu^{10} + \cdots + 14\!\cdots\!52 ) / 36\!\cdots\!16 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 49676631251349 \nu^{11} + \cdots + 77\!\cdots\!12 \nu ) / 34\!\cdots\!92 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 12\!\cdots\!35 \nu^{11} + \cdots - 86\!\cdots\!52 \nu ) / 36\!\cdots\!16 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -2\beta_{4} + 3\beta_{3} - \beta_{2} - 148 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 7\beta_{11} + \beta_{10} - 13\beta_{9} - 13\beta_{8} + 27\beta_{7} - 392\beta_{6} + 13\beta_{3} - 286\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 121\beta_{9} - 121\beta_{8} + 37\beta_{5} + 891\beta_{4} - 997\beta_{3} + 1624\beta_{2} + 43196 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 5790 \beta_{11} - 402 \beta_{10} + 5506 \beta_{9} + 5506 \beta_{8} - 12802 \beta_{7} + \cdots + 92035 \beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 61078 \beta_{9} + 61078 \beta_{8} - 15634 \beta_{5} - 341516 \beta_{4} + 348983 \beta_{3} + \cdots - 14217132 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 3064365 \beta_{11} + 122463 \beta_{10} - 2059479 \beta_{9} - 2059479 \beta_{8} + 5407933 \beta_{7} + \cdots - 31857316 \beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 27064847 \beta_{9} - 27064847 \beta_{8} + 5340151 \beta_{5} + 129912421 \beta_{4} - 128088865 \beta_{3} + \cdots + 5010367716 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 1391942036 \beta_{11} - 28452648 \beta_{10} + 766242484 \beta_{9} + 766242484 \beta_{8} + \cdots + 11582276673 \beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 11492549868 \beta_{9} + 11492549868 \beta_{8} - 1722324228 \beta_{5} - 49833437126 \beta_{4} + \cdots - 1844331516756 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 591336847907 \beta_{11} + 4058952413 \beta_{10} - 288439797585 \beta_{9} - 288439797585 \beta_{8} + \cdots - 4347194717194 \beta_1 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(157\) \(365\) \(379\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
337.1
− 15.4753i
− 14.0703i
− 0.389198i
0.909519i
7.17317i
19.8521i
− 19.8521i
− 7.17317i
− 0.909519i
0.389198i
14.0703i
15.4753i
− 2.00000i 3.00000 −4.00000 − 18.4753i − 6.00000i 7.00000i 8.00000i 9.00000 −36.9506
337.2 − 2.00000i 3.00000 −4.00000 − 17.0703i − 6.00000i 7.00000i 8.00000i 9.00000 −34.1407
337.3 − 2.00000i 3.00000 −4.00000 − 3.38920i − 6.00000i 7.00000i 8.00000i 9.00000 −6.77840
337.4 − 2.00000i 3.00000 −4.00000 − 2.09048i − 6.00000i 7.00000i 8.00000i 9.00000 −4.18096
337.5 − 2.00000i 3.00000 −4.00000 4.17317i − 6.00000i 7.00000i 8.00000i 9.00000 8.34634
337.6 − 2.00000i 3.00000 −4.00000 16.8521i − 6.00000i 7.00000i 8.00000i 9.00000 33.7043
337.7 2.00000i 3.00000 −4.00000 − 16.8521i 6.00000i − 7.00000i − 8.00000i 9.00000 33.7043
337.8 2.00000i 3.00000 −4.00000 − 4.17317i 6.00000i − 7.00000i − 8.00000i 9.00000 8.34634
337.9 2.00000i 3.00000 −4.00000 2.09048i 6.00000i − 7.00000i − 8.00000i 9.00000 −4.18096
337.10 2.00000i 3.00000 −4.00000 3.38920i 6.00000i − 7.00000i − 8.00000i 9.00000 −6.77840
337.11 2.00000i 3.00000 −4.00000 17.0703i 6.00000i − 7.00000i − 8.00000i 9.00000 −34.1407
337.12 2.00000i 3.00000 −4.00000 18.4753i 6.00000i − 7.00000i − 8.00000i 9.00000 −36.9506
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 337.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 546.4.c.d ✓ 12
13.b even 2 1 inner 546.4.c.d ✓ 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
546.4.c.d ✓ 12 1.a even 1 1 trivial
546.4.c.d ✓ 12 13.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{12} + 950T_{5}^{10} + 309985T_{5}^{8} + 37835532T_{5}^{6} + 1031752560T_{5}^{4} + 9462533584T_{5}^{2} + 24694236736 \) acting on \(S_{4}^{\mathrm{new}}(546, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 4)^{6} \) Copy content Toggle raw display
$3$ \( (T - 3)^{12} \) Copy content Toggle raw display
$5$ \( T^{12} + \cdots + 24694236736 \) Copy content Toggle raw display
$7$ \( (T^{2} + 49)^{6} \) Copy content Toggle raw display
$11$ \( T^{12} + \cdots + 60\!\cdots\!76 \) Copy content Toggle raw display
$13$ \( T^{12} + \cdots + 11\!\cdots\!29 \) Copy content Toggle raw display
$17$ \( (T^{6} + 125 T^{5} + \cdots - 4258153856)^{2} \) Copy content Toggle raw display
$19$ \( T^{12} + \cdots + 40\!\cdots\!76 \) Copy content Toggle raw display
$23$ \( (T^{6} + \cdots + 1688076040752)^{2} \) Copy content Toggle raw display
$29$ \( (T^{6} + \cdots - 4730886429696)^{2} \) Copy content Toggle raw display
$31$ \( T^{12} + \cdots + 84\!\cdots\!44 \) Copy content Toggle raw display
$37$ \( T^{12} + \cdots + 12\!\cdots\!16 \) Copy content Toggle raw display
$41$ \( T^{12} + \cdots + 16\!\cdots\!24 \) Copy content Toggle raw display
$43$ \( (T^{6} + \cdots - 42303641930752)^{2} \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 34\!\cdots\!96 \) Copy content Toggle raw display
$53$ \( (T^{6} + \cdots + 59583769414848)^{2} \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 65\!\cdots\!56 \) Copy content Toggle raw display
$61$ \( (T^{6} + \cdots - 14080656027776)^{2} \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 61\!\cdots\!44 \) Copy content Toggle raw display
$71$ \( T^{12} + \cdots + 97\!\cdots\!36 \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots + 89\!\cdots\!36 \) Copy content Toggle raw display
$79$ \( (T^{6} + \cdots - 14\!\cdots\!68)^{2} \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 33\!\cdots\!36 \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 15\!\cdots\!04 \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 41\!\cdots\!04 \) Copy content Toggle raw display
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