↑

Properties

Label 546.4.i.c
Level $546$
Weight $4$
Character orbit 546.i
Analytic conductor $32.215$
Analytic rank $0$
Dimension $10$
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 = [10,-10,-15] 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: \(10\)
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: \( x^{10} + 150 x^{8} - 136 x^{7} + 20148 x^{6} - 12552 x^{5} + 357424 x^{4} - 545664 x^{3} + \cdots + 5531904 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 3 \)
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 \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - 2 \beta_{6} q^{2} + (3 \beta_{6} - 3) q^{3} + (4 \beta_{6} - 4) q^{4} + ( - \beta_{8} + \beta_{6}) q^{5} + 6 q^{6} + (\beta_{8} - \beta_{3} + \cdots + 2 \beta_1) q^{7} + 8 q^{8} - 9 \beta_{6} q^{9}+ \cdots + ( - 9 \beta_{5} + 9 \beta_{4} + \cdots - 207) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 10 q^{2} - 15 q^{3} - 20 q^{4} + 7 q^{5} + 60 q^{6} - 6 q^{7} + 80 q^{8} - 45 q^{9} + 14 q^{10} + 117 q^{11} - 60 q^{12} + 130 q^{13} - 42 q^{15} - 80 q^{16} - 3 q^{17} - 90 q^{18} + 163 q^{19} - 56 q^{20}+ \cdots - 2106 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} + 150 x^{8} - 136 x^{7} + 20148 x^{6} - 12552 x^{5} + 357424 x^{4} - 545664 x^{3} + \cdots + 5531904 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 29577007 \nu^{9} + 116407732 \nu^{8} - 388421718 \nu^{7} + 12077181080 \nu^{6} + \cdots + 15\!\cdots\!72 ) / 15\!\cdots\!84 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 1248275191 \nu^{9} + 67755667378 \nu^{8} - 226082686047 \nu^{7} + 9588540662674 \nu^{6} + \cdots + 60\!\cdots\!64 ) / 48\!\cdots\!96 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 19246210597 \nu^{9} + 316048229924 \nu^{8} - 1054569093726 \nu^{7} + \cdots + 18\!\cdots\!24 ) / 19\!\cdots\!84 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 271442725 \nu^{9} - 11156006596 \nu^{8} + 37224634254 \nu^{7} - 1727378453552 \nu^{6} + \cdots - 29\!\cdots\!36 ) / 24\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 278138093 \nu^{9} - 29577007 \nu^{8} + 41604306218 \nu^{7} - 37438358930 \nu^{6} + \cdots + 33135958091712 ) / 15\!\cdots\!84 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 172335751500 \nu^{9} - 26898048149 \nu^{8} + 26505585638212 \nu^{7} - 25623967230738 \nu^{6} + \cdots - 11\!\cdots\!00 ) / 96\!\cdots\!92 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 28985268415 \nu^{9} - 24637130565 \nu^{8} - 4362213749342 \nu^{7} + \cdots - 44\!\cdots\!64 ) / 13\!\cdots\!56 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 69661691973 \nu^{9} - 11567582600 \nu^{8} - 10420817175150 \nu^{7} + \cdots + 37\!\cdots\!92 ) / 32\!\cdots\!64 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -2\beta_{9} + 4\beta_{8} - 4\beta_{7} + 64\beta_{6} + 4\beta_{4} - 64 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -4\beta_{5} - 20\beta_{4} + 16\beta_{3} - 102\beta_{2} + 44 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 356\beta_{9} - 544\beta_{8} + 544\beta_{7} - 7696\beta_{6} + 356\beta_{5} + 544\beta_{3} + 12\beta_1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 736 \beta_{9} + 3272 \beta_{8} + 2128 \beta_{7} + 8600 \beta_{6} + 3272 \beta_{4} + 12948 \beta_{2} + \cdots - 8600 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -48968\beta_{5} - 73552\beta_{4} - 71104\beta_{3} - 6384\beta_{2} + 1006864 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 120496 \beta_{9} - 471344 \beta_{8} - 253984 \beta_{7} - 1559312 \beta_{6} + 120496 \beta_{5} + \cdots + 1703112 \beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 6548528 \beta_{9} + 9928768 \beta_{8} - 9279040 \beta_{7} + 133409920 \beta_{6} + 9928768 \beta_{4} + \cdots - 133409920 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( -18835840\beta_{5} - 66727904\beta_{4} + 29536960\beta_{3} - 225418992\beta_{2} + 264035360 \) Copy content Toggle raw display

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

Character values

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

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

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
−5.85660 + 10.1439i
1.92444 − 3.33324i
5.63705 − 9.76366i
0.519971 − 0.900617i
−2.22487 + 3.85359i
−5.85660 − 10.1439i
1.92444 + 3.33324i
5.63705 + 9.76366i
0.519971 + 0.900617i
−2.22487 − 3.85359i
−1.00000 + 1.73205i −1.50000 − 2.59808i −2.00000 − 3.46410i −6.63429 + 11.4909i 6.00000 16.7362 + 7.93089i 8.00000 −4.50000 + 7.79423i −13.2686 − 22.9818i
79.2 −1.00000 + 1.73205i −1.50000 − 2.59808i −2.00000 − 3.46410i −2.28737 + 3.96184i 6.00000 −14.5217 − 11.4943i 8.00000 −4.50000 + 7.79423i −4.57474 − 7.92368i
79.3 −1.00000 + 1.73205i −1.50000 − 2.59808i −2.00000 − 3.46410i 1.56168 − 2.70491i 6.00000 5.48807 − 17.6884i 8.00000 −4.50000 + 7.79423i 3.12336 + 5.40981i
79.4 −1.00000 + 1.73205i −1.50000 − 2.59808i −2.00000 − 3.46410i 4.88421 − 8.45969i 6.00000 −17.5911 + 5.79243i 8.00000 −4.50000 + 7.79423i 9.76841 + 16.9194i
79.5 −1.00000 + 1.73205i −1.50000 − 2.59808i −2.00000 − 3.46410i 5.97577 − 10.3503i 6.00000 6.88856 + 17.1915i 8.00000 −4.50000 + 7.79423i 11.9515 + 20.7007i
235.1 −1.00000 − 1.73205i −1.50000 + 2.59808i −2.00000 + 3.46410i −6.63429 − 11.4909i 6.00000 16.7362 − 7.93089i 8.00000 −4.50000 − 7.79423i −13.2686 + 22.9818i
235.2 −1.00000 − 1.73205i −1.50000 + 2.59808i −2.00000 + 3.46410i −2.28737 − 3.96184i 6.00000 −14.5217 + 11.4943i 8.00000 −4.50000 − 7.79423i −4.57474 + 7.92368i
235.3 −1.00000 − 1.73205i −1.50000 + 2.59808i −2.00000 + 3.46410i 1.56168 + 2.70491i 6.00000 5.48807 + 17.6884i 8.00000 −4.50000 − 7.79423i 3.12336 − 5.40981i
235.4 −1.00000 − 1.73205i −1.50000 + 2.59808i −2.00000 + 3.46410i 4.88421 + 8.45969i 6.00000 −17.5911 − 5.79243i 8.00000 −4.50000 − 7.79423i 9.76841 − 16.9194i
235.5 −1.00000 − 1.73205i −1.50000 + 2.59808i −2.00000 + 3.46410i 5.97577 + 10.3503i 6.00000 6.88856 − 17.1915i 8.00000 −4.50000 − 7.79423i 11.9515 − 20.7007i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 79.5
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.c ✓ 10
7.c even 3 1 inner 546.4.i.c ✓ 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
546.4.i.c ✓ 10 1.a even 1 1 trivial
546.4.i.c ✓ 10 7.c even 3 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{10} - 7 T_{5}^{9} + 247 T_{5}^{8} - 1456 T_{5}^{7} + 44453 T_{5}^{6} - 237720 T_{5}^{5} + \cdots + 489913956 \) 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
$5$ \( T^{10} + \cdots + 489913956 \) Copy content Toggle raw display
$7$ \( T^{10} + \cdots + 4747561509943 \) Copy content Toggle raw display
$11$ \( T^{10} + \cdots + 10\!\cdots\!49 \) Copy content Toggle raw display
$13$ \( (T - 13)^{10} \) Copy content Toggle raw display
$17$ \( T^{10} + \cdots + 420825961521 \) Copy content Toggle raw display
$19$ \( T^{10} + \cdots + 21\!\cdots\!61 \) Copy content Toggle raw display
$23$ \( T^{10} + \cdots + 11\!\cdots\!76 \) Copy content Toggle raw display
$29$ \( (T^{5} + 376 T^{4} + \cdots - 11506682067)^{2} \) Copy content Toggle raw display
$31$ \( T^{10} + \cdots + 11\!\cdots\!84 \) Copy content Toggle raw display
$37$ \( T^{10} + \cdots + 20\!\cdots\!84 \) Copy content Toggle raw display
$41$ \( (T^{5} - 50 T^{4} + \cdots + 176622972942)^{2} \) Copy content Toggle raw display
$43$ \( (T^{5} + 260 T^{4} + \cdots + 417865055396)^{2} \) Copy content Toggle raw display
$47$ \( T^{10} + \cdots + 27\!\cdots\!56 \) Copy content Toggle raw display
$53$ \( T^{10} + \cdots + 19\!\cdots\!29 \) Copy content Toggle raw display
$59$ \( T^{10} + \cdots + 50\!\cdots\!49 \) Copy content Toggle raw display
$61$ \( T^{10} + \cdots + 19\!\cdots\!29 \) Copy content Toggle raw display
$67$ \( T^{10} + \cdots + 28\!\cdots\!09 \) Copy content Toggle raw display
$71$ \( (T^{5} + \cdots - 21529685350911)^{2} \) Copy content Toggle raw display
$73$ \( T^{10} + \cdots + 17\!\cdots\!64 \) Copy content Toggle raw display
$79$ \( T^{10} + \cdots + 39\!\cdots\!56 \) Copy content Toggle raw display
$83$ \( (T^{5} + \cdots - 39186578021724)^{2} \) Copy content Toggle raw display
$89$ \( T^{10} + \cdots + 22\!\cdots\!04 \) Copy content Toggle raw display
$97$ \( (T^{5} + \cdots + 36\!\cdots\!66)^{2} \) Copy content Toggle raw display
show more
show less