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Properties

Label 544.4.a.m
Level $544$
Weight $4$
Character orbit 544.a
Self dual yes
Analytic conductor $32.097$
Analytic rank $0$
Dimension $8$
Inner twists $2$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [544,4,Mod(1,544)] 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("544.1"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(544, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 544 = 2^{5} \cdot 17 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 544.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,0,0,0,16,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(32.0970390431\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 4x^{7} - 34x^{6} + 116x^{5} + 383x^{4} - 964x^{3} - 1476x^{2} + 1978x + 1028 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{14} \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} + (\beta_{2} + 2) q^{5} + (\beta_{6} + \beta_1) q^{7} + (\beta_{4} - \beta_{3} + 2 \beta_{2} + 17) q^{9} + ( - \beta_{7} - \beta_1) q^{11} + (\beta_{3} + \beta_{2} - 7) q^{13} + (\beta_{7} + \beta_{6} + \cdots + 9 \beta_1) q^{15}+ \cdots + ( - 28 \beta_{7} - 50 \beta_{6} + \cdots - 209 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 16 q^{5} + 136 q^{9} - 56 q^{13} + 136 q^{17} + 312 q^{21} + 600 q^{25} - 128 q^{29} - 304 q^{33} - 112 q^{37} + 1328 q^{41} + 2592 q^{45} + 1624 q^{49} + 1216 q^{53} + 2560 q^{57} + 752 q^{61} + 1824 q^{65}+ \cdots + 4560 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 4x^{7} - 34x^{6} + 116x^{5} + 383x^{4} - 964x^{3} - 1476x^{2} + 1978x + 1028 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -8\nu^{7} + 28\nu^{6} + 286\nu^{5} - 785\nu^{4} - 2156\nu^{3} + 4033\nu^{2} - 3082\nu + 842 ) / 2601 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( 2\nu^{2} - 2\nu - 20 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -2\nu^{6} + 6\nu^{5} + 32\nu^{4} - 74\nu^{3} + 2\nu^{2} + 36\nu - 455 ) / 17 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 2\nu^{6} - 6\nu^{5} - 66\nu^{4} + 142\nu^{3} + 610\nu^{2} - 682\nu - 939 ) / 17 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -56\nu^{7} + 196\nu^{6} + 2002\nu^{5} - 5495\nu^{4} - 25496\nu^{3} + 43837\nu^{2} + 139688\nu - 77338 ) / 2601 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 80\nu^{7} - 280\nu^{6} - 2860\nu^{5} + 7850\nu^{4} + 31964\nu^{3} - 55936\nu^{2} - 88826\nu + 54004 ) / 2601 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 196\nu^{7} - 686\nu^{6} - 4406\nu^{5} + 12730\nu^{4} + 21610\nu^{3} - 45488\nu^{2} + 31292\nu - 7624 ) / 2601 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{6} + \beta_{5} + 3\beta _1 + 8 ) / 16 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{6} + \beta_{5} + 8\beta_{2} + 3\beta _1 + 168 ) / 16 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 17\beta_{6} + 13\beta_{5} + 12\beta_{2} + 79\beta _1 + 248 ) / 16 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 33\beta_{6} + 25\beta_{5} - 8\beta_{4} - 8\beta_{3} + 168\beta_{2} + 155\beta _1 + 2712 ) / 16 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 16\beta_{7} + 283\beta_{6} + 215\beta_{5} - 20\beta_{4} - 20\beta_{3} + 400\beta_{2} + 1717\beta _1 + 6368 ) / 16 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 48\beta_{7} + 767\beta_{6} + 583\beta_{5} - 188\beta_{4} - 324\beta_{3} + 3452\beta_{2} + 4765\beta _1 + 49992 ) / 16 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 740 \beta_{7} + 5101 \beta_{6} + 3889 \beta_{5} - 588 \beta_{4} - 1064 \beta_{3} + 10696 \beta_{2} + \cdots + 152972 ) / 16 \) Copy content Toggle raw display

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−3.81367
3.03774
−3.38492
1.42708
−0.427084
4.38492
−2.03774
4.81367
0 −9.96818 0 18.7155 0 −19.9877 0 72.3646 0
1.2 0 −8.34287 0 −5.61973 0 16.1101 0 42.6035 0
1.3 0 −2.38184 0 11.6852 0 −22.7367 0 −21.3268 0
1.4 0 −1.16565 0 −16.7810 0 −31.7490 0 −25.6413 0
1.5 0 1.16565 0 −16.7810 0 31.7490 0 −25.6413 0
1.6 0 2.38184 0 11.6852 0 22.7367 0 −21.3268 0
1.7 0 8.34287 0 −5.61973 0 −16.1101 0 42.6035 0
1.8 0 9.96818 0 18.7155 0 19.9877 0 72.3646 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.8
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( -1 \)
\(17\) \( -1 \)

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 544.4.a.m ✓ 8
4.b odd 2 1 inner 544.4.a.m ✓ 8
8.b even 2 1 1088.4.a.bi 8
8.d odd 2 1 1088.4.a.bi 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
544.4.a.m ✓ 8 1.a even 1 1 trivial
544.4.a.m ✓ 8 4.b odd 2 1 inner
1088.4.a.bi 8 8.b even 2 1
1088.4.a.bi 8 8.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{8} - 176T_{3}^{6} + 8112T_{3}^{4} - 49936T_{3}^{2} + 53312 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(544))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} - 176 T^{6} + \cdots + 53312 \) Copy content Toggle raw display
$5$ \( (T^{4} - 8 T^{3} + \cdots + 20624)^{2} \) Copy content Toggle raw display
$7$ \( T^{8} + \cdots + 54030307392 \) Copy content Toggle raw display
$11$ \( T^{8} + \cdots + 2514800946752 \) Copy content Toggle raw display
$13$ \( (T^{4} + 28 T^{3} + \cdots + 274576)^{2} \) Copy content Toggle raw display
$17$ \( (T - 17)^{8} \) Copy content Toggle raw display
$19$ \( T^{8} + \cdots + 14920121335808 \) Copy content Toggle raw display
$23$ \( T^{8} + \cdots + 10\!\cdots\!68 \) Copy content Toggle raw display
$29$ \( (T^{4} + 64 T^{3} + \cdots - 5044848)^{2} \) Copy content Toggle raw display
$31$ \( T^{8} + \cdots + 32\!\cdots\!32 \) Copy content Toggle raw display
$37$ \( (T^{4} + 56 T^{3} + \cdots + 1790172496)^{2} \) Copy content Toggle raw display
$41$ \( (T^{4} - 664 T^{3} + \cdots + 149710224)^{2} \) Copy content Toggle raw display
$43$ \( T^{8} + \cdots + 72\!\cdots\!48 \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 55\!\cdots\!32 \) Copy content Toggle raw display
$53$ \( (T^{4} - 608 T^{3} + \cdots + 3749859024)^{2} \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 14\!\cdots\!88 \) Copy content Toggle raw display
$61$ \( (T^{4} - 376 T^{3} + \cdots - 90804336)^{2} \) Copy content Toggle raw display
$67$ \( T^{8} + \cdots + 19727875309568 \) Copy content Toggle raw display
$71$ \( T^{8} + \cdots + 39\!\cdots\!88 \) Copy content Toggle raw display
$73$ \( (T^{4} - 1976 T^{3} + \cdots + 12350168464)^{2} \) Copy content Toggle raw display
$79$ \( T^{8} + \cdots + 25\!\cdots\!88 \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 27\!\cdots\!68 \) Copy content Toggle raw display
$89$ \( (T^{4} - 1152 T^{3} + \cdots + 578325213808)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} - 2280 T^{3} + \cdots - 269946141424)^{2} \) Copy content Toggle raw display
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