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Properties

Label 546.4.l.e
Level $546$
Weight $4$
Character orbit 546.l
Analytic conductor $32.215$
Analytic rank $0$
Dimension $10$
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(211,546)] 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("546.211"); 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, 2])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 546 = 2 \cdot 3 \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 546.l (of order \(3\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [10,-10,-15,-20,-24] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == 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: \( x^{10} - 2 x^{9} + 335 x^{8} - 2684 x^{7} + 91511 x^{6} - 599219 x^{5} + 9618925 x^{4} + \cdots + 17171481600 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
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.

\(f(q)\) \(=\) \( q - 2 \beta_{3} q^{2} - 3 \beta_{3} q^{3} + (4 \beta_{3} - 4) q^{4} + ( - \beta_{2} - \beta_1 - 2) q^{5} + (6 \beta_{3} - 6) q^{6} + ( - 7 \beta_{3} + 7) q^{7} + 8 q^{8} + (9 \beta_{3} - 9) q^{9} + (4 \beta_{3} + 2 \beta_{2}) q^{10}+ \cdots + ( - 9 \beta_{6} + 9 \beta_{5} + \cdots - 36) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 10 q^{2} - 15 q^{3} - 20 q^{4} - 24 q^{5} - 30 q^{6} + 35 q^{7} + 80 q^{8} - 45 q^{9} + 24 q^{10} + 19 q^{11} + 120 q^{12} - 128 q^{13} - 140 q^{14} + 36 q^{15} - 80 q^{16} - 38 q^{17} + 180 q^{18}+ \cdots - 342 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - 2 x^{9} + 335 x^{8} - 2684 x^{7} + 91511 x^{6} - 599219 x^{5} + 9618925 x^{4} + \cdots + 17171481600 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 4712136721261 \nu^{9} + 338090823470558 \nu^{8} + \cdots + 20\!\cdots\!00 ) / 31\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 25\!\cdots\!15 \nu^{9} + \cdots + 43\!\cdots\!60 ) / 69\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 82\!\cdots\!41 \nu^{9} + \cdots - 10\!\cdots\!60 ) / 52\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 30\!\cdots\!37 \nu^{9} + \cdots + 37\!\cdots\!40 ) / 20\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 50\!\cdots\!49 \nu^{9} + \cdots + 20\!\cdots\!60 ) / 20\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 26\!\cdots\!33 \nu^{9} + \cdots + 48\!\cdots\!80 ) / 10\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 27\!\cdots\!37 \nu^{9} + \cdots + 48\!\cdots\!00 ) / 48\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 36\!\cdots\!19 \nu^{9} + \cdots - 15\!\cdots\!80 ) / 62\!\cdots\!40 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{7} + \beta_{6} - 135\beta_{3} + 5\beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -9\beta_{9} - 18\beta_{6} + 3\beta_{5} - 6\beta_{4} - 190\beta_{2} - 190\beta _1 + 681 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -90\beta_{8} + 379\beta_{7} + 355\beta_{5} - 24\beta_{4} + 25839\beta_{3} + 1964\beta _1 - 25839 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 2799 \beta_{9} + 2799 \beta_{8} - 6873 \beta_{7} + 6873 \beta_{6} - 4737 \beta_{5} + \cdots + 45931 \beta_{2} \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -39249\beta_{9} - 120421\beta_{6} - 52960\beta_{5} - 14106\beta_{4} - 638072\beta_{2} - 638072\beta _1 + 6266466 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 805977 \beta_{8} + 2152020 \beta_{7} + 1626978 \beta_{5} - 525042 \beta_{4} + 87324027 \beta_{3} + \cdots - 87324027 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 12957192 \beta_{9} + 12957192 \beta_{8} - 36586036 \beta_{7} + 36586036 \beta_{6} + \cdots + 196140353 \beta_{2} \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 234846576 \beta_{9} - 643892013 \beta_{6} - 125735901 \beta_{5} - 141654768 \beta_{4} + \cdots + 26842767963 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(157\) \(365\) \(379\)
\(\chi(n)\) \(1\) \(1\) \(-1 + \beta_{3}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
211.1
5.85512 − 10.1414i
4.84743 − 8.39600i
3.58144 − 6.20324i
−4.68481 + 8.11433i
−8.59918 + 14.8942i
5.85512 + 10.1414i
4.84743 + 8.39600i
3.58144 + 6.20324i
−4.68481 − 8.11433i
−8.59918 − 14.8942i
−1.00000 − 1.73205i −1.50000 − 2.59808i −2.00000 + 3.46410i −13.7102 −3.00000 + 5.19615i 3.50000 − 6.06218i 8.00000 −4.50000 + 7.79423i 13.7102 + 23.7468i
211.2 −1.00000 − 1.73205i −1.50000 − 2.59808i −2.00000 + 3.46410i −11.6949 −3.00000 + 5.19615i 3.50000 − 6.06218i 8.00000 −4.50000 + 7.79423i 11.6949 + 20.2561i
211.3 −1.00000 − 1.73205i −1.50000 − 2.59808i −2.00000 + 3.46410i −9.16288 −3.00000 + 5.19615i 3.50000 − 6.06218i 8.00000 −4.50000 + 7.79423i 9.16288 + 15.8706i
211.4 −1.00000 − 1.73205i −1.50000 − 2.59808i −2.00000 + 3.46410i 7.36962 −3.00000 + 5.19615i 3.50000 − 6.06218i 8.00000 −4.50000 + 7.79423i −7.36962 − 12.7646i
211.5 −1.00000 − 1.73205i −1.50000 − 2.59808i −2.00000 + 3.46410i 15.1984 −3.00000 + 5.19615i 3.50000 − 6.06218i 8.00000 −4.50000 + 7.79423i −15.1984 − 26.3243i
295.1 −1.00000 + 1.73205i −1.50000 + 2.59808i −2.00000 − 3.46410i −13.7102 −3.00000 − 5.19615i 3.50000 + 6.06218i 8.00000 −4.50000 − 7.79423i 13.7102 − 23.7468i
295.2 −1.00000 + 1.73205i −1.50000 + 2.59808i −2.00000 − 3.46410i −11.6949 −3.00000 − 5.19615i 3.50000 + 6.06218i 8.00000 −4.50000 − 7.79423i 11.6949 − 20.2561i
295.3 −1.00000 + 1.73205i −1.50000 + 2.59808i −2.00000 − 3.46410i −9.16288 −3.00000 − 5.19615i 3.50000 + 6.06218i 8.00000 −4.50000 − 7.79423i 9.16288 − 15.8706i
295.4 −1.00000 + 1.73205i −1.50000 + 2.59808i −2.00000 − 3.46410i 7.36962 −3.00000 − 5.19615i 3.50000 + 6.06218i 8.00000 −4.50000 − 7.79423i −7.36962 + 12.7646i
295.5 −1.00000 + 1.73205i −1.50000 + 2.59808i −2.00000 − 3.46410i 15.1984 −3.00000 − 5.19615i 3.50000 + 6.06218i 8.00000 −4.50000 − 7.79423i −15.1984 + 26.3243i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 211.5
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.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.l.e ✓ 10
13.c even 3 1 inner 546.4.l.e ✓ 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
546.4.l.e ✓ 10 1.a even 1 1 trivial
546.4.l.e ✓ 10 13.c even 3 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{5} + 12T_{5}^{4} - 275T_{5}^{3} - 3531T_{5}^{2} + 10876T_{5} + 164556 \) 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
$5$ \( (T^{5} + 12 T^{4} + \cdots + 164556)^{2} \) Copy content Toggle raw display
$7$ \( (T^{2} - 7 T + 49)^{5} \) Copy content Toggle raw display
$11$ \( T^{10} + \cdots + 21\!\cdots\!00 \) Copy content Toggle raw display
$13$ \( T^{10} + \cdots + 51\!\cdots\!57 \) Copy content Toggle raw display
$17$ \( T^{10} + \cdots + 22\!\cdots\!89 \) Copy content Toggle raw display
$19$ \( T^{10} + \cdots + 28\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( T^{10} + \cdots + 44\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( T^{10} + \cdots + 31\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( (T^{5} + 116 T^{4} + \cdots - 1403368902)^{2} \) Copy content Toggle raw display
$37$ \( T^{10} + \cdots + 92\!\cdots\!36 \) Copy content Toggle raw display
$41$ \( T^{10} + \cdots + 35\!\cdots\!00 \) Copy content Toggle raw display
$43$ \( T^{10} + \cdots + 19\!\cdots\!56 \) Copy content Toggle raw display
$47$ \( (T^{5} + 310 T^{4} + \cdots + 811075158306)^{2} \) Copy content Toggle raw display
$53$ \( (T^{5} - 1127 T^{4} + \cdots - 158042030175)^{2} \) Copy content Toggle raw display
$59$ \( T^{10} + \cdots + 81\!\cdots\!24 \) Copy content Toggle raw display
$61$ \( T^{10} + \cdots + 57\!\cdots\!25 \) Copy content Toggle raw display
$67$ \( T^{10} + \cdots + 75\!\cdots\!96 \) Copy content Toggle raw display
$71$ \( T^{10} + \cdots + 13\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( (T^{5} + \cdots - 41005408762608)^{2} \) Copy content Toggle raw display
$79$ \( (T^{5} + \cdots - 9388973430876)^{2} \) Copy content Toggle raw display
$83$ \( (T^{5} + \cdots + 9400585484028)^{2} \) Copy content Toggle raw display
$89$ \( T^{10} + \cdots + 55\!\cdots\!25 \) Copy content Toggle raw display
$97$ \( T^{10} + \cdots + 74\!\cdots\!04 \) Copy content Toggle raw display
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