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Properties

Label 585.2.bu.b
Level $585$
Weight $2$
Character orbit 585.bu
Analytic conductor $4.671$
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] = [585,2,Mod(316,585)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(585, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 1])) N = Newforms(chi, 2, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("585.316"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 585 = 3^{2} \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 585.bu (of order \(6\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,-6,0,4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(4)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(4.67124851824\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: 8.0.56070144.2
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 4x^{7} + 16x^{6} - 34x^{5} + 63x^{4} - 74x^{3} + 70x^{2} - 38x + 13 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 195)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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.

\(f(q)\) \(=\) \( q + (\beta_{6} + \beta_{5} - \beta_{2} - 1) q^{2} + ( - \beta_{7} - \beta_{6} - \beta_{5} + \cdots + 2) q^{4} + ( - \beta_{5} + \beta_{4}) q^{5} + (\beta_{6} + \beta_{5} - 2 \beta_{4} + \cdots + 1) q^{7}+ \cdots + (\beta_{7} - 2 \beta_{6} - 5 \beta_{5} + \cdots - 1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 6 q^{2} + 4 q^{4} + 6 q^{7} + 2 q^{10} - 12 q^{11} + 6 q^{13} + 4 q^{14} + 2 q^{17} + 12 q^{19} - 4 q^{23} - 8 q^{25} + 4 q^{26} - 12 q^{28} + 6 q^{29} - 12 q^{32} + 6 q^{35} - 12 q^{37} + 16 q^{38}+ \cdots + 12 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 4x^{7} + 16x^{6} - 34x^{5} + 63x^{4} - 74x^{3} + 70x^{2} - 38x + 13 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{7} - 15\nu^{6} + 32\nu^{5} - 172\nu^{4} + 221\nu^{3} - 426\nu^{2} + 235\nu - 159 ) / 37 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -3\nu^{7} - 8\nu^{6} + 22\nu^{5} - 146\nu^{4} + 256\nu^{3} - 390\nu^{2} + 298\nu - 70 ) / 37 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -3\nu^{7} - 8\nu^{6} + 22\nu^{5} - 146\nu^{4} + 256\nu^{3} - 427\nu^{2} + 335\nu - 181 ) / 37 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 3\nu^{7} - 29\nu^{6} + 89\nu^{5} - 261\nu^{4} + 373\nu^{3} - 498\nu^{2} + 294\nu - 152 ) / 37 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -8\nu^{7} + 28\nu^{6} - 114\nu^{5} + 215\nu^{4} - 378\nu^{3} + 366\nu^{2} - 266\nu + 97 ) / 37 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 17\nu^{7} - 41\nu^{6} + 159\nu^{5} - 184\nu^{4} + 276\nu^{3} - 84\nu^{2} + 38\nu + 39 ) / 37 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{4} + \beta_{3} + \beta _1 - 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{7} + \beta_{6} - \beta_{5} + 2\beta_{3} - 2\beta _1 - 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 2\beta_{7} + 3\beta_{6} + 6\beta_{4} - 2\beta_{3} - 2\beta_{2} - 6\beta _1 + 7 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -4\beta_{7} - 3\beta_{6} + 7\beta_{5} + 6\beta_{4} - 12\beta_{3} - 5\beta_{2} + \beta _1 + 26 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -17\beta_{7} - 25\beta_{6} + 3\beta_{5} - 24\beta_{4} - 5\beta_{3} + 7\beta_{2} + 27\beta _1 - 1 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 4\beta_{7} - 16\beta_{6} - 42\beta_{5} - 54\beta_{4} + 51\beta_{3} + 42\beta_{2} + 26\beta _1 - 122 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/585\mathbb{Z}\right)^\times\).

\(n\) \(326\) \(352\) \(496\)
\(\chi(n)\) \(1\) \(1\) \(1 - \beta_{6}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
316.1
0.500000 − 1.56488i
0.500000 − 1.19293i
0.500000 + 0.564882i
0.500000 + 2.19293i
0.500000 + 1.56488i
0.500000 + 1.19293i
0.500000 − 0.564882i
0.500000 − 2.19293i
−2.10523 − 1.21545i 0 1.95466 + 3.38556i 1.00000i 0 1.12682 − 0.650571i − 4.64136i 0 1.21545 − 2.10523i
316.2 −1.78311 − 1.02948i 0 1.11966 + 1.93930i − 1.00000i 0 −2.01516 + 1.16345i − 0.492737i 0 −1.02948 + 1.78311i
316.3 −0.260797 − 0.150571i 0 −0.954656 − 1.65351i 1.00000i 0 2.97125 − 1.71545i 1.17726i 0 0.150571 − 0.260797i
316.4 1.14914 + 0.663454i 0 −0.119657 − 0.207252i − 1.00000i 0 0.917086 − 0.529480i − 2.97136i 0 0.663454 − 1.14914i
361.1 −2.10523 + 1.21545i 0 1.95466 − 3.38556i − 1.00000i 0 1.12682 + 0.650571i 4.64136i 0 1.21545 + 2.10523i
361.2 −1.78311 + 1.02948i 0 1.11966 − 1.93930i 1.00000i 0 −2.01516 − 1.16345i 0.492737i 0 −1.02948 − 1.78311i
361.3 −0.260797 + 0.150571i 0 −0.954656 + 1.65351i − 1.00000i 0 2.97125 + 1.71545i − 1.17726i 0 0.150571 + 0.260797i
361.4 1.14914 − 0.663454i 0 −0.119657 + 0.207252i 1.00000i 0 0.917086 + 0.529480i 2.97136i 0 0.663454 + 1.14914i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 316.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.e even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 585.2.bu.b 8
3.b odd 2 1 195.2.bb.c ✓ 8
13.e even 6 1 inner 585.2.bu.b 8
13.f odd 12 1 7605.2.a.cg 4
13.f odd 12 1 7605.2.a.ck 4
15.d odd 2 1 975.2.bc.i 8
15.e even 4 1 975.2.w.g 8
15.e even 4 1 975.2.w.j 8
39.h odd 6 1 195.2.bb.c ✓ 8
39.k even 12 1 2535.2.a.bi 4
39.k even 12 1 2535.2.a.bl 4
195.y odd 6 1 975.2.bc.i 8
195.bf even 12 1 975.2.w.g 8
195.bf even 12 1 975.2.w.j 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
195.2.bb.c ✓ 8 3.b odd 2 1
195.2.bb.c ✓ 8 39.h odd 6 1
585.2.bu.b 8 1.a even 1 1 trivial
585.2.bu.b 8 13.e even 6 1 inner
975.2.w.g 8 15.e even 4 1
975.2.w.g 8 195.bf even 12 1
975.2.w.j 8 15.e even 4 1
975.2.w.j 8 195.bf even 12 1
975.2.bc.i 8 15.d odd 2 1
975.2.bc.i 8 195.y odd 6 1
2535.2.a.bi 4 39.k even 12 1
2535.2.a.bl 4 39.k even 12 1
7605.2.a.cg 4 13.f odd 12 1
7605.2.a.ck 4 13.f odd 12 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{8} + 6T_{2}^{7} + 12T_{2}^{6} - 22T_{2}^{4} + 48T_{2}^{2} + 24T_{2} + 4 \) acting on \(S_{2}^{\mathrm{new}}(585, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} + 6 T^{7} + \cdots + 4 \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( (T^{2} + 1)^{4} \) Copy content Toggle raw display
$7$ \( T^{8} - 6 T^{7} + \cdots + 121 \) Copy content Toggle raw display
$11$ \( T^{8} + 12 T^{7} + \cdots + 10816 \) Copy content Toggle raw display
$13$ \( T^{8} - 6 T^{7} + \cdots + 28561 \) Copy content Toggle raw display
$17$ \( T^{8} - 2 T^{7} + \cdots + 4 \) Copy content Toggle raw display
$19$ \( T^{8} - 12 T^{7} + \cdots + 1936 \) Copy content Toggle raw display
$23$ \( T^{8} + 4 T^{7} + \cdots + 64 \) Copy content Toggle raw display
$29$ \( T^{8} - 6 T^{7} + \cdots + 8836 \) Copy content Toggle raw display
$31$ \( T^{8} + 108 T^{6} + \cdots + 9801 \) Copy content Toggle raw display
$37$ \( T^{8} + 12 T^{7} + \cdots + 256 \) Copy content Toggle raw display
$41$ \( T^{8} - 30 T^{7} + \cdots + 15507844 \) Copy content Toggle raw display
$43$ \( T^{8} - 6 T^{7} + \cdots + 316969 \) Copy content Toggle raw display
$47$ \( T^{8} + 388 T^{6} + \cdots + 47087044 \) Copy content Toggle raw display
$53$ \( (T^{4} - 16 T^{3} + \cdots - 1664)^{2} \) Copy content Toggle raw display
$59$ \( T^{8} - 30 T^{7} + \cdots + 209764 \) Copy content Toggle raw display
$61$ \( T^{8} + 178 T^{6} + \cdots + 2474329 \) Copy content Toggle raw display
$67$ \( T^{8} - 30 T^{7} + \cdots + 5470921 \) Copy content Toggle raw display
$71$ \( T^{8} + 18 T^{7} + \cdots + 8836 \) Copy content Toggle raw display
$73$ \( T^{8} + 372 T^{6} + \cdots + 177241 \) Copy content Toggle raw display
$79$ \( (T^{4} - 28 T^{3} + \cdots - 407)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + 688 T^{6} + \cdots + 667292224 \) Copy content Toggle raw display
$89$ \( T^{8} - 18 T^{7} + \cdots + 94828644 \) Copy content Toggle raw display
$97$ \( T^{8} + 6 T^{7} + \cdots + 39400729 \) Copy content Toggle raw display
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