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Properties

Label 546.2.bk.b
Level $546$
Weight $2$
Character orbit 546.bk
Analytic conductor $4.360$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [546,2,Mod(25,546)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(546, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 4, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("546.25");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 546 = 2 \cdot 3 \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 546.bk (of order \(6\), degree \(2\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(4.35983195036\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 15x^{10} + 90x^{8} - 247x^{6} + 270x^{4} + 21x^{2} + 49 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
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.

\(f(q)\) \(=\) \( q - \beta_{6} q^{2} - \beta_{2} q^{3} + \beta_{2} q^{4} + \beta_{8} q^{5} + \beta_{10} q^{6} + (\beta_{11} - \beta_{10} + \beta_{6}) q^{7} - \beta_{10} q^{8} + (\beta_{2} - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{6} q^{2} - \beta_{2} q^{3} + \beta_{2} q^{4} + \beta_{8} q^{5} + \beta_{10} q^{6} + (\beta_{11} - \beta_{10} + \beta_{6}) q^{7} - \beta_{10} q^{8} + (\beta_{2} - 1) q^{9} + ( - \beta_{9} - \beta_{7}) q^{10} + (\beta_{11} + 2 \beta_{10} + \cdots + \beta_1) q^{11}+ \cdots + ( - \beta_{11} - 2 \beta_{10} + \cdots - \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 6 q^{3} + 6 q^{4} - 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 6 q^{3} + 6 q^{4} - 6 q^{9} + 6 q^{12} + 12 q^{13} - 18 q^{14} - 6 q^{16} + 18 q^{17} + 12 q^{22} - 6 q^{25} + 12 q^{27} + 12 q^{29} + 24 q^{35} - 12 q^{36} - 6 q^{38} - 6 q^{39} + 6 q^{42} + 24 q^{43} + 12 q^{48} - 18 q^{49} + 18 q^{51} + 6 q^{52} - 18 q^{53} + 48 q^{55} - 6 q^{56} + 6 q^{61} - 48 q^{62} - 12 q^{64} - 12 q^{65} - 6 q^{66} - 18 q^{68} - 6 q^{75} - 24 q^{77} + 24 q^{79} - 6 q^{81} - 12 q^{82} - 6 q^{87} + 6 q^{88} - 24 q^{91} + 6 q^{94} + 24 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 15x^{10} + 90x^{8} - 247x^{6} + 270x^{4} + 21x^{2} + 49 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 2\nu^{10} - 83\nu^{8} + 906\nu^{6} - 3874\nu^{4} + 5950\nu^{2} + 1505 ) / 2947 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -25\nu^{10} + 406\nu^{8} - 2063\nu^{6} + 2536\nu^{4} + 3931\nu^{2} + 3290 ) / 5894 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -29\nu^{10} + 151\nu^{8} + 335\nu^{6} - 3609\nu^{4} + 2977\nu^{2} + 3227 ) / 5894 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 2\nu^{11} - 83\nu^{9} + 906\nu^{7} - 3874\nu^{5} + 5950\nu^{3} - 1442\nu ) / 2947 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -40\nu^{11} + 397\nu^{9} + 404\nu^{7} - 17245\nu^{5} + 61188\nu^{3} - 56623\nu ) / 41258 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -58\nu^{10} + 723\nu^{8} - 3540\nu^{6} + 6675\nu^{4} + 902\nu^{2} - 14175 ) / 5894 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( -55\nu^{11} + 1651\nu^{9} - 15653\nu^{7} + 69487\nu^{5} - 156047\nu^{3} + 151641\nu ) / 41258 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( -89\nu^{10} + 1378\nu^{8} - 8321\nu^{6} + 23780\nu^{4} - 30699\nu^{2} + 8176 ) / 5894 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( -501\nu^{11} + 7109\nu^{9} - 40029\nu^{7} + 98967\nu^{5} - 88545\nu^{3} - 45465\nu ) / 41258 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( -332\nu^{11} + 5358\nu^{9} - 33779\nu^{7} + 94100\nu^{5} - 82129\nu^{3} - 75957\nu ) / 20629 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{7} - \beta_{4} - \beta_{3} + \beta_{2} + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{11} - \beta_{10} - \beta_{8} - 2\beta_{6} + \beta_{5} + 4\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 2\beta_{9} + 4\beta_{7} - 6\beta_{4} - 8\beta_{3} + 9\beta_{2} + 10 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 8\beta_{11} - 8\beta_{10} - 4\beta_{8} - 18\beta_{6} + 13\beta_{5} + 17\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 17\beta_{9} + 13\beta_{7} - 33\beta_{4} - 42\beta_{3} + 65\beta_{2} + 16 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 50\beta_{11} - 53\beta_{10} - 5\beta_{8} - 90\beta_{6} + 99\beta_{5} + 69\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 104\beta_{9} + 24\beta_{7} - 172\beta_{4} - 168\beta_{3} + 365\beta_{2} - 85 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 276\beta_{11} - 320\beta_{10} + 84\beta_{8} - 288\beta_{6} + 573\beta_{5} + 240\beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 489\beta_{9} - 120\beta_{7} - 836\beta_{4} - 467\beta_{3} + 1634\beta_{2} - 1083 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 1325\beta_{11} - 1792\beta_{10} + 978\beta_{8} - 98\beta_{6} + 2612\beta_{5} + 453\beta_1 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(157\) \(365\) \(379\)
\(\chi(n)\) \(-1 + \beta_{2}\) \(1\) \(-1\)

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
25.1
−0.385124 + 0.500000i
1.75780 + 0.500000i
−2.23871 + 0.500000i
2.23871 − 0.500000i
−1.75780 − 0.500000i
0.385124 − 0.500000i
−0.385124 − 0.500000i
1.75780 − 0.500000i
−2.23871 − 0.500000i
2.23871 + 0.500000i
−1.75780 + 0.500000i
0.385124 + 0.500000i
−0.866025 + 0.500000i −0.500000 + 0.866025i 0.500000 − 0.866025i −2.25312 + 1.30084i − 1.00000i 1.49160 − 2.18521i 1.00000i −0.500000 − 0.866025i 1.30084 − 2.25312i
25.2 −0.866025 + 0.500000i −0.500000 + 0.866025i 0.500000 − 0.866025i 0.294342 − 0.169938i − 1.00000i 0.420136 + 2.61218i 1.00000i −0.500000 − 0.866025i −0.169938 + 0.294342i
25.3 −0.866025 + 0.500000i −0.500000 + 0.866025i 0.500000 − 0.866025i 1.95878 − 1.13090i − 1.00000i 2.41839 + 1.07303i 1.00000i −0.500000 − 0.866025i −1.13090 + 1.95878i
25.4 0.866025 − 0.500000i −0.500000 + 0.866025i 0.500000 − 0.866025i −1.95878 + 1.13090i 1.00000i −2.41839 − 1.07303i − 1.00000i −0.500000 − 0.866025i −1.13090 + 1.95878i
25.5 0.866025 − 0.500000i −0.500000 + 0.866025i 0.500000 − 0.866025i −0.294342 + 0.169938i 1.00000i −0.420136 − 2.61218i − 1.00000i −0.500000 − 0.866025i −0.169938 + 0.294342i
25.6 0.866025 − 0.500000i −0.500000 + 0.866025i 0.500000 − 0.866025i 2.25312 − 1.30084i 1.00000i −1.49160 + 2.18521i − 1.00000i −0.500000 − 0.866025i 1.30084 − 2.25312i
415.1 −0.866025 − 0.500000i −0.500000 − 0.866025i 0.500000 + 0.866025i −2.25312 − 1.30084i 1.00000i 1.49160 + 2.18521i − 1.00000i −0.500000 + 0.866025i 1.30084 + 2.25312i
415.2 −0.866025 − 0.500000i −0.500000 − 0.866025i 0.500000 + 0.866025i 0.294342 + 0.169938i 1.00000i 0.420136 − 2.61218i − 1.00000i −0.500000 + 0.866025i −0.169938 − 0.294342i
415.3 −0.866025 − 0.500000i −0.500000 − 0.866025i 0.500000 + 0.866025i 1.95878 + 1.13090i 1.00000i 2.41839 − 1.07303i − 1.00000i −0.500000 + 0.866025i −1.13090 − 1.95878i
415.4 0.866025 + 0.500000i −0.500000 − 0.866025i 0.500000 + 0.866025i −1.95878 − 1.13090i − 1.00000i −2.41839 + 1.07303i 1.00000i −0.500000 + 0.866025i −1.13090 − 1.95878i
415.5 0.866025 + 0.500000i −0.500000 − 0.866025i 0.500000 + 0.866025i −0.294342 − 0.169938i − 1.00000i −0.420136 + 2.61218i 1.00000i −0.500000 + 0.866025i −0.169938 − 0.294342i
415.6 0.866025 + 0.500000i −0.500000 − 0.866025i 0.500000 + 0.866025i 2.25312 + 1.30084i − 1.00000i −1.49160 − 2.18521i 1.00000i −0.500000 + 0.866025i 1.30084 + 2.25312i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 25.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner
13.b even 2 1 inner
91.r even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 546.2.bk.b ✓ 12
3.b odd 2 1 1638.2.dm.c 12
7.c even 3 1 inner 546.2.bk.b ✓ 12
7.c even 3 1 3822.2.c.k 6
7.d odd 6 1 3822.2.c.j 6
13.b even 2 1 inner 546.2.bk.b ✓ 12
21.h odd 6 1 1638.2.dm.c 12
39.d odd 2 1 1638.2.dm.c 12
91.r even 6 1 inner 546.2.bk.b ✓ 12
91.r even 6 1 3822.2.c.k 6
91.s odd 6 1 3822.2.c.j 6
273.w odd 6 1 1638.2.dm.c 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
546.2.bk.b ✓ 12 1.a even 1 1 trivial
546.2.bk.b ✓ 12 7.c even 3 1 inner
546.2.bk.b ✓ 12 13.b even 2 1 inner
546.2.bk.b ✓ 12 91.r even 6 1 inner
1638.2.dm.c 12 3.b odd 2 1
1638.2.dm.c 12 21.h odd 6 1
1638.2.dm.c 12 39.d odd 2 1
1638.2.dm.c 12 273.w odd 6 1
3822.2.c.j 6 7.d odd 6 1
3822.2.c.j 6 91.s odd 6 1
3822.2.c.k 6 7.c even 3 1
3822.2.c.k 6 91.r even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{12} - 12T_{5}^{10} + 108T_{5}^{8} - 424T_{5}^{6} + 1248T_{5}^{4} - 144T_{5}^{2} + 16 \) acting on \(S_{2}^{\mathrm{new}}(546, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} - T^{2} + 1)^{3} \) Copy content Toggle raw display
$3$ \( (T^{2} + T + 1)^{6} \) Copy content Toggle raw display
$5$ \( T^{12} - 12 T^{10} + \cdots + 16 \) Copy content Toggle raw display
$7$ \( T^{12} + 9 T^{10} + \cdots + 117649 \) Copy content Toggle raw display
$11$ \( T^{12} - 15 T^{10} + \cdots + 81 \) Copy content Toggle raw display
$13$ \( (T^{6} - 6 T^{5} + \cdots + 2197)^{2} \) Copy content Toggle raw display
$17$ \( (T^{6} - 9 T^{5} + 66 T^{4} + \cdots + 49)^{2} \) Copy content Toggle raw display
$19$ \( T^{12} - 99 T^{10} + \cdots + 15752961 \) Copy content Toggle raw display
$23$ \( (T^{6} + 42 T^{4} + \cdots + 9604)^{2} \) Copy content Toggle raw display
$29$ \( (T^{3} - 3 T^{2} - 45 T - 77)^{4} \) Copy content Toggle raw display
$31$ \( (T^{4} - 16 T^{2} + 256)^{3} \) Copy content Toggle raw display
$37$ \( T^{12} - 24 T^{10} + \cdots + 16 \) Copy content Toggle raw display
$41$ \( (T^{6} + 108 T^{4} + \cdots + 676)^{2} \) Copy content Toggle raw display
$43$ \( (T^{3} - 6 T^{2} + \cdots + 104)^{4} \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 8428892481 \) Copy content Toggle raw display
$53$ \( (T^{2} + 3 T + 9)^{6} \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 1982119441 \) Copy content Toggle raw display
$61$ \( (T^{6} - 3 T^{5} + \cdots + 4489)^{2} \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 163047361 \) Copy content Toggle raw display
$71$ \( (T^{6} + 51 T^{4} + \cdots + 441)^{2} \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots + 1474777075216 \) Copy content Toggle raw display
$79$ \( (T^{6} - 12 T^{5} + \cdots + 278784)^{2} \) Copy content Toggle raw display
$83$ \( (T^{6} + 372 T^{4} + \cdots + 712336)^{2} \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 29376588816 \) Copy content Toggle raw display
$97$ \( (T^{6} + 324 T^{4} + \cdots + 324)^{2} \) Copy content Toggle raw display
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