↑

Properties

Label 546.4.i.g
Level $546$
Weight $4$
Character orbit 546.i
Analytic conductor $32.215$
Analytic rank $0$
Dimension $12$
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [546,4,Mod(79,546)] 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("546.79"); 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, 2, 0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 546 = 2 \cdot 3 \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 546.i (of order \(3\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [12,12,18] 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: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{3})\)
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: \( x^{12} - x^{11} + 383 x^{10} + 1336 x^{9} + 131031 x^{8} + 188148 x^{7} + 5674209 x^{6} + \cdots + 348456889 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{4}\cdot 3 \)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - 2 \beta_{4} q^{2} + (3 \beta_{4} + 3) q^{3} + ( - 4 \beta_{4} - 4) q^{4} + (\beta_{4} - \beta_1) q^{5} + 6 q^{6} + (\beta_{9} + \beta_{4} - 2) q^{7} - 8 q^{8} + 9 \beta_{4} q^{9} + (2 \beta_{4} + 2 \beta_{3} - 2 \beta_1 + 2) q^{10}+ \cdots + ( - 9 \beta_{11} - 9 \beta_{9} + \cdots - 90) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 12 q^{2} + 18 q^{3} - 24 q^{4} - 7 q^{5} + 72 q^{6} - 35 q^{7} - 96 q^{8} - 54 q^{9} + 14 q^{10} + 48 q^{11} + 72 q^{12} - 156 q^{13} - 14 q^{14} - 42 q^{15} - 96 q^{16} - 218 q^{17} + 108 q^{18}+ \cdots - 864 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - x^{11} + 383 x^{10} + 1336 x^{9} + 131031 x^{8} + 188148 x^{7} + 5674209 x^{6} + \cdots + 348456889 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 48\!\cdots\!15 \nu^{11} + \cdots + 25\!\cdots\!28 ) / 10\!\cdots\!58 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 30\!\cdots\!31 \nu^{11} + \cdots - 12\!\cdots\!78 ) / 23\!\cdots\!27 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 68\!\cdots\!34 \nu^{11} + \cdots - 14\!\cdots\!67 ) / 44\!\cdots\!09 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 87\!\cdots\!78 \nu^{11} + \cdots - 77\!\cdots\!85 ) / 10\!\cdots\!58 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 64\!\cdots\!82 \nu^{11} + \cdots + 25\!\cdots\!14 ) / 53\!\cdots\!79 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 11\!\cdots\!29 \nu^{11} + \cdots + 60\!\cdots\!45 ) / 81\!\cdots\!82 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 92\!\cdots\!92 \nu^{11} + \cdots + 11\!\cdots\!56 ) / 53\!\cdots\!79 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 18\!\cdots\!14 \nu^{11} + \cdots + 14\!\cdots\!63 ) / 10\!\cdots\!58 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 34\!\cdots\!56 \nu^{11} + \cdots + 50\!\cdots\!83 ) / 15\!\cdots\!94 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 71\!\cdots\!55 \nu^{11} + \cdots + 33\!\cdots\!42 ) / 10\!\cdots\!58 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 4 \beta_{11} + \beta_{10} + \beta_{9} - \beta_{7} - 4 \beta_{6} - \beta_{5} - 130 \beta_{4} - 2 \beta_{3} + \cdots - 126 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -4\beta_{11} + 13\beta_{9} + 14\beta_{8} - 35\beta_{7} - 14\beta_{6} - 4\beta_{5} - 303\beta_{3} + 13\beta_{2} - 374 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 215 \beta_{11} - 391 \beta_{10} + 1064 \beta_{9} + 1585 \beta_{8} + 1279 \beta_{5} + 40197 \beta_{4} + \cdots + 1279 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 8780 \beta_{11} - 13923 \beta_{10} - 7497 \beta_{9} + 13923 \beta_{7} + 8358 \beta_{6} + \cdots + 265637 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 362575 \beta_{11} - 445895 \beta_{9} - 562842 \beta_{8} + 165564 \beta_{7} + 562842 \beta_{6} + \cdots + 13073268 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 2827555 \beta_{11} + 5188995 \beta_{10} - 1486567 \beta_{9} - 4401411 \beta_{8} - 4314122 \beta_{5} + \cdots - 4314122 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 160768350 \beta_{11} + 70259155 \beta_{10} + 35540357 \beta_{9} - 70259155 \beta_{7} - 200939701 \beta_{6} + \cdots - 4872063334 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 871837938 \beta_{11} + 1924411602 \beta_{9} + 2066820077 \beta_{8} - 1940321768 \beta_{7} + \cdots - 56454727882 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 14952272364 \beta_{11} - 29193806620 \beta_{10} + 44240382345 \beta_{9} + 73032297313 \beta_{8} + \cdots + 59192654709 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 819186006841 \beta_{11} - 733156772124 \beta_{10} - 395273937280 \beta_{9} + 733156772124 \beta_{7} + \cdots + 24846657119934 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(157\) \(365\) \(379\)
\(\chi(n)\) \(\beta_{4}\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
79.1
9.90206 − 17.1509i
1.85410 − 3.21140i
1.55004 − 2.68474i
−0.296086 + 0.512837i
−4.13140 + 7.15579i
−8.37871 + 14.5124i
9.90206 + 17.1509i
1.85410 + 3.21140i
1.55004 + 2.68474i
−0.296086 − 0.512837i
−4.13140 − 7.15579i
−8.37871 − 14.5124i
1.00000 − 1.73205i 1.50000 + 2.59808i −2.00000 − 3.46410i −10.4021 + 18.0169i 6.00000 −12.8917 + 13.2968i −8.00000 −4.50000 + 7.79423i 20.8041 + 36.0338i
79.2 1.00000 − 1.73205i 1.50000 + 2.59808i −2.00000 − 3.46410i −2.35410 + 4.07743i 6.00000 −6.28492 − 17.4212i −8.00000 −4.50000 + 7.79423i 4.70821 + 8.15486i
79.3 1.00000 − 1.73205i 1.50000 + 2.59808i −2.00000 − 3.46410i −2.05004 + 3.55077i 6.00000 12.2538 + 13.8868i −8.00000 −4.50000 + 7.79423i 4.10008 + 7.10154i
79.4 1.00000 − 1.73205i 1.50000 + 2.59808i −2.00000 − 3.46410i −0.203914 + 0.353189i 6.00000 16.2146 − 8.94905i −8.00000 −4.50000 + 7.79423i 0.407827 + 0.706378i
79.5 1.00000 − 1.73205i 1.50000 + 2.59808i −2.00000 − 3.46410i 3.63140 − 6.28977i 6.00000 −15.8023 − 9.65850i −8.00000 −4.50000 + 7.79423i −7.26280 − 12.5795i
79.6 1.00000 − 1.73205i 1.50000 + 2.59808i −2.00000 − 3.46410i 7.87871 − 13.6463i 6.00000 −10.9896 + 14.9074i −8.00000 −4.50000 + 7.79423i −15.7574 − 27.2927i
235.1 1.00000 + 1.73205i 1.50000 − 2.59808i −2.00000 + 3.46410i −10.4021 − 18.0169i 6.00000 −12.8917 − 13.2968i −8.00000 −4.50000 − 7.79423i 20.8041 − 36.0338i
235.2 1.00000 + 1.73205i 1.50000 − 2.59808i −2.00000 + 3.46410i −2.35410 − 4.07743i 6.00000 −6.28492 + 17.4212i −8.00000 −4.50000 − 7.79423i 4.70821 − 8.15486i
235.3 1.00000 + 1.73205i 1.50000 − 2.59808i −2.00000 + 3.46410i −2.05004 − 3.55077i 6.00000 12.2538 − 13.8868i −8.00000 −4.50000 − 7.79423i 4.10008 − 7.10154i
235.4 1.00000 + 1.73205i 1.50000 − 2.59808i −2.00000 + 3.46410i −0.203914 − 0.353189i 6.00000 16.2146 + 8.94905i −8.00000 −4.50000 − 7.79423i 0.407827 − 0.706378i
235.5 1.00000 + 1.73205i 1.50000 − 2.59808i −2.00000 + 3.46410i 3.63140 + 6.28977i 6.00000 −15.8023 + 9.65850i −8.00000 −4.50000 − 7.79423i −7.26280 + 12.5795i
235.6 1.00000 + 1.73205i 1.50000 − 2.59808i −2.00000 + 3.46410i 7.87871 + 13.6463i 6.00000 −10.9896 − 14.9074i −8.00000 −4.50000 − 7.79423i −15.7574 + 27.2927i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 79.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.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.i.g ✓ 12
7.c even 3 1 inner 546.4.i.g ✓ 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
546.4.i.g ✓ 12 1.a even 1 1 trivial
546.4.i.g ✓ 12 7.c even 3 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{12} + 7 T_{5}^{11} + 411 T_{5}^{10} - 492 T_{5}^{9} + 124611 T_{5}^{8} + 231126 T_{5}^{7} + \cdots + 351337536 \) acting on \(S_{4}^{\mathrm{new}}(546, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} - 2 T + 4)^{6} \) Copy content Toggle raw display
$3$ \( (T^{2} - 3 T + 9)^{6} \) Copy content Toggle raw display
$5$ \( T^{12} + \cdots + 351337536 \) Copy content Toggle raw display
$7$ \( T^{12} + \cdots + 16\!\cdots\!49 \) Copy content Toggle raw display
$11$ \( T^{12} + \cdots + 60\!\cdots\!25 \) Copy content Toggle raw display
$13$ \( (T + 13)^{12} \) Copy content Toggle raw display
$17$ \( T^{12} + \cdots + 23\!\cdots\!61 \) Copy content Toggle raw display
$19$ \( T^{12} + \cdots + 30\!\cdots\!29 \) Copy content Toggle raw display
$23$ \( T^{12} + \cdots + 41\!\cdots\!56 \) Copy content Toggle raw display
$29$ \( (T^{6} + \cdots - 7331045228875)^{2} \) Copy content Toggle raw display
$31$ \( T^{12} + \cdots + 10\!\cdots\!16 \) Copy content Toggle raw display
$37$ \( T^{12} + \cdots + 22\!\cdots\!00 \) Copy content Toggle raw display
$41$ \( (T^{6} + \cdots - 98379056604456)^{2} \) Copy content Toggle raw display
$43$ \( (T^{6} + \cdots - 3400314669864)^{2} \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 12\!\cdots\!76 \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots + 10\!\cdots\!61 \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 24\!\cdots\!41 \) Copy content Toggle raw display
$61$ \( T^{12} + \cdots + 21\!\cdots\!81 \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 51\!\cdots\!41 \) Copy content Toggle raw display
$71$ \( (T^{6} + \cdots - 51\!\cdots\!53)^{2} \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots + 61\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots + 25\!\cdots\!84 \) Copy content Toggle raw display
$83$ \( (T^{6} + \cdots + 31\!\cdots\!96)^{2} \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 28\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( (T^{6} + \cdots + 252477357192536)^{2} \) Copy content Toggle raw display
show more
show less