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Properties

Label 560.3.bh.d
Level $560$
Weight $3$
Character orbit 560.bh
Analytic conductor $15.259$
Analytic rank $0$
Dimension $8$
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [560,3,Mod(113,560)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(560, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([0, 0, 3, 0])) N = Newforms(chi, 3, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("560.113"); S:= CuspForms(chi, 3); N := Newforms(S);
 
Level: \( N \) \(=\) \( 560 = 2^{4} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 560.bh (of order \(4\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,0,8,0,20] 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: \(15.2588948042\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(i)\)
Coefficient field: 8.0.1218533392384.32
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 78x^{4} + 121 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 140)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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.

\(f(q)\) \(=\) \( q + (\beta_{4} - \beta_{3} + \beta_1 + 1) q^{3} + ( - \beta_{7} - \beta_{4} - \beta_{2} + \cdots + 3) q^{5} + \beta_{5} q^{7} + ( - 2 \beta_{7} + 2 \beta_{6} + \cdots + 2 \beta_{2}) q^{9} + ( - 2 \beta_{7} - 2 \beta_{6} + \cdots - 2) q^{11}+ \cdots + (26 \beta_{7} - 26 \beta_{6} + \cdots - 16 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 8 q^{3} + 20 q^{5} - 32 q^{11} - 16 q^{13} - 56 q^{15} - 16 q^{17} + 56 q^{21} + 24 q^{23} + 32 q^{25} + 8 q^{27} + 32 q^{31} - 80 q^{33} - 28 q^{35} - 208 q^{37} + 48 q^{43} - 92 q^{45} - 176 q^{47}+ \cdots - 400 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 78x^{4} + 121 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{7} - 89\nu^{3} ) / 110 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{6} - 89\nu^{2} ) / 110 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 5\nu^{6} + 11\nu^{4} + 335\nu^{2} + 429 ) / 220 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -5\nu^{6} + 11\nu^{4} - 335\nu^{2} + 429 ) / 220 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{7} - 6\nu^{6} + 11\nu^{5} + 89\nu^{3} - 424\nu^{2} + 759\nu + 110 ) / 220 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -4\nu^{7} - 3\nu^{6} - 301\nu^{3} - 212\nu^{2} + 55\nu + 55 ) / 110 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{5} - \beta_{4} - 5\beta_{3} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{7} - \beta_{5} + \beta_{4} - \beta_{3} - 8\beta_{2} - \beta _1 - 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 10\beta_{5} + 10\beta_{4} - 39 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 20\beta_{6} - 10\beta_{5} + 10\beta_{4} - 10\beta_{3} + 10\beta_{2} - 69\beta _1 - 10 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -89\beta_{5} + 89\beta_{4} + 335\beta_{3} \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -178\beta_{7} + 89\beta_{5} - 89\beta_{4} + 89\beta_{3} + 602\beta_{2} + 89\beta _1 + 89 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/560\mathbb{Z}\right)^\times\).

\(n\) \(241\) \(337\) \(351\) \(421\)
\(\chi(n)\) \(1\) \(\beta_{3}\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
113.1
−2.09065 − 2.09065i
2.09065 + 2.09065i
−0.793203 − 0.793203i
0.793203 + 0.793203i
−2.09065 + 2.09065i
2.09065 − 2.09065i
−0.793203 + 0.793203i
0.793203 − 0.793203i
0 −2.96148 − 2.96148i 0 4.64088 − 1.86071i 0 −1.87083 + 1.87083i 0 8.54074i 0
113.2 0 1.21982 + 1.21982i 0 4.10078 + 2.86071i 0 −1.87083 + 1.87083i 0 − 6.02406i 0
113.3 0 2.07763 + 2.07763i 0 3.69952 − 3.36356i 0 1.87083 − 1.87083i 0 − 0.366945i 0
113.4 0 3.66403 + 3.66403i 0 −2.44118 + 4.36356i 0 1.87083 − 1.87083i 0 17.8503i 0
337.1 0 −2.96148 + 2.96148i 0 4.64088 + 1.86071i 0 −1.87083 − 1.87083i 0 − 8.54074i 0
337.2 0 1.21982 − 1.21982i 0 4.10078 − 2.86071i 0 −1.87083 − 1.87083i 0 6.02406i 0
337.3 0 2.07763 − 2.07763i 0 3.69952 + 3.36356i 0 1.87083 + 1.87083i 0 0.366945i 0
337.4 0 3.66403 − 3.66403i 0 −2.44118 − 4.36356i 0 1.87083 + 1.87083i 0 − 17.8503i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 113.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 560.3.bh.d 8
4.b odd 2 1 140.3.l.b ✓ 8
5.c odd 4 1 inner 560.3.bh.d 8
12.b even 2 1 1260.3.y.b 8
20.d odd 2 1 700.3.l.c 8
20.e even 4 1 140.3.l.b ✓ 8
20.e even 4 1 700.3.l.c 8
28.d even 2 1 980.3.l.c 8
60.l odd 4 1 1260.3.y.b 8
140.j odd 4 1 980.3.l.c 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
140.3.l.b ✓ 8 4.b odd 2 1
140.3.l.b ✓ 8 20.e even 4 1
560.3.bh.d 8 1.a even 1 1 trivial
560.3.bh.d 8 5.c odd 4 1 inner
700.3.l.c 8 20.d odd 2 1
700.3.l.c 8 20.e even 4 1
980.3.l.c 8 28.d even 2 1
980.3.l.c 8 140.j odd 4 1
1260.3.y.b 8 12.b even 2 1
1260.3.y.b 8 60.l odd 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(560, [\chi])\):

\( T_{3}^{8} - 8T_{3}^{7} + 32T_{3}^{6} - 40T_{3}^{5} + 364T_{3}^{4} - 2512T_{3}^{3} + 9248T_{3}^{2} - 14960T_{3} + 12100 \) Copy content Toggle raw display
\( T_{11}^{4} + 16T_{11}^{3} - 56T_{11}^{2} - 960T_{11} - 2000 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} - 8 T^{7} + \cdots + 12100 \) Copy content Toggle raw display
$5$ \( T^{8} - 20 T^{7} + \cdots + 390625 \) Copy content Toggle raw display
$7$ \( (T^{4} + 49)^{2} \) Copy content Toggle raw display
$11$ \( (T^{4} + 16 T^{3} + \cdots - 2000)^{2} \) Copy content Toggle raw display
$13$ \( T^{8} + 16 T^{7} + \cdots + 363207364 \) Copy content Toggle raw display
$17$ \( T^{8} + \cdots + 143489440000 \) Copy content Toggle raw display
$19$ \( T^{8} + \cdots + 3534778116 \) Copy content Toggle raw display
$23$ \( T^{8} - 24 T^{7} + \cdots + 8456464 \) Copy content Toggle raw display
$29$ \( T^{8} + \cdots + 1240568260864 \) Copy content Toggle raw display
$31$ \( (T^{4} - 16 T^{3} + \cdots + 953408)^{2} \) Copy content Toggle raw display
$37$ \( T^{8} + \cdots + 609024160000 \) Copy content Toggle raw display
$41$ \( (T^{4} - 2952 T^{2} + \cdots + 1747376)^{2} \) Copy content Toggle raw display
$43$ \( T^{8} + \cdots + 82162489600 \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 8953787474944 \) Copy content Toggle raw display
$53$ \( T^{8} + \cdots + 1371896838400 \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 243295562500 \) Copy content Toggle raw display
$61$ \( (T^{4} - 36 T^{3} + \cdots + 14422850)^{2} \) Copy content Toggle raw display
$67$ \( T^{8} + \cdots + 119716000000 \) Copy content Toggle raw display
$71$ \( (T^{4} - 200 T^{3} + \cdots - 21038300)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} + \cdots + 613949284000000 \) Copy content Toggle raw display
$79$ \( T^{8} + \cdots + 440083638544 \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 162039915660100 \) Copy content Toggle raw display
$89$ \( T^{8} + \cdots + 78299845638400 \) Copy content Toggle raw display
$97$ \( T^{8} + \cdots + 17\!\cdots\!96 \) Copy content Toggle raw display
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