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Properties

Label 867.2.d.f
Level $867$
Weight $2$
Character orbit 867.d
Analytic conductor $6.923$
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] = [867,2,Mod(577,867)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(867, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([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("867.577"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 867 = 3 \cdot 17^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 867.d (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(6.92302985525\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.836829184.2
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 14x^{6} + 61x^{4} + 84x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 51)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{6} q^{2} + \beta_{3} q^{3} + ( - \beta_{7} + \beta_{6} + 1) q^{4} + (2 \beta_{3} + \beta_1) q^{5} + ( - \beta_{4} + \beta_{3} + \beta_1) q^{6} + ( - \beta_{5} - \beta_{3}) q^{7} + ( - \beta_{7} + \beta_{6} + 2 \beta_{2} + 1) q^{8}+ \cdots + ( - \beta_{5} + \beta_{4} + \cdots - \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 4 q^{2} + 12 q^{4} + 12 q^{8} - 8 q^{9} + 4 q^{13} - 12 q^{15} + 12 q^{16} - 4 q^{18} - 20 q^{19} + 8 q^{21} - 4 q^{25} + 40 q^{26} - 24 q^{30} + 28 q^{32} + 12 q^{33} + 8 q^{35} - 12 q^{36} - 24 q^{38}+ \cdots + 76 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 14x^{6} + 61x^{4} + 84x^{2} + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{4} + 7\nu^{2} + 6 ) / 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{7} + 12\nu^{5} + 41\nu^{3} + 38\nu ) / 8 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{5} - 11\nu^{3} - 26\nu ) / 4 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{7} + 14\nu^{5} + 55\nu^{3} + 50\nu ) / 8 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{6} + 11\nu^{4} + 30\nu^{2} + 12 ) / 4 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( \nu^{6} + 11\nu^{4} + 26\nu^{2} - 4 ) / 4 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{7} + \beta_{6} - 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{5} - \beta_{4} + \beta_{3} - 5\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 7\beta_{7} - 7\beta_{6} + 4\beta_{2} + 22 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 11\beta_{5} + 7\beta_{4} - 11\beta_{3} + 29\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -47\beta_{7} + 51\beta_{6} - 44\beta_{2} - 134 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -91\beta_{5} - 43\beta_{4} + 99\beta_{3} - 181\beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/867\mathbb{Z}\right)^\times\).

\(n\) \(290\) \(292\)
\(\chi(n)\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
577.1
1.65222i
− 1.65222i
2.63640i
− 2.63640i
− 2.06644i
2.06644i
− 0.222191i
0.222191i
−2.06644 − 1.00000i 2.27016 − 0.347777i 2.06644i 4.33660i −0.558268 −1.00000 0.718659i
577.2 −2.06644 1.00000i 2.27016 0.347777i − 2.06644i − 4.33660i −0.558268 −1.00000 − 0.718659i
577.3 −0.222191 − 1.00000i −1.95063 0.636405i 0.222191i − 1.72844i 0.877796 −1.00000 − 0.141404i
577.4 −0.222191 1.00000i −1.95063 − 0.636405i − 0.222191i 1.72844i 0.877796 −1.00000 0.141404i
577.5 1.65222 − 1.00000i 0.729840 − 4.06644i − 1.65222i − 0.922382i −2.09859 −1.00000 − 6.71866i
577.6 1.65222 1.00000i 0.729840 4.06644i 1.65222i 0.922382i −2.09859 −1.00000 6.71866i
577.7 2.63640 − 1.00000i 4.95063 − 2.22219i − 2.63640i 2.31423i 7.77906 −1.00000 − 5.85860i
577.8 2.63640 1.00000i 4.95063 2.22219i 2.63640i − 2.31423i 7.77906 −1.00000 5.85860i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 577.8
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 867.2.d.f 8
17.b even 2 1 inner 867.2.d.f 8
17.c even 4 1 867.2.a.k 4
17.c even 4 1 867.2.a.l 4
17.d even 8 2 51.2.e.a ✓ 8
17.d even 8 2 867.2.e.g 8
17.e odd 16 4 867.2.h.i 16
17.e odd 16 4 867.2.h.k 16
51.f odd 4 1 2601.2.a.be 4
51.f odd 4 1 2601.2.a.bf 4
51.g odd 8 2 153.2.f.b 8
68.g odd 8 2 816.2.bd.e 8
204.p even 8 2 2448.2.be.x 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
51.2.e.a ✓ 8 17.d even 8 2
153.2.f.b 8 51.g odd 8 2
816.2.bd.e 8 68.g odd 8 2
867.2.a.k 4 17.c even 4 1
867.2.a.l 4 17.c even 4 1
867.2.d.f 8 1.a even 1 1 trivial
867.2.d.f 8 17.b even 2 1 inner
867.2.e.g 8 17.d even 8 2
867.2.h.i 16 17.e odd 16 4
867.2.h.k 16 17.e odd 16 4
2448.2.be.x 8 204.p even 8 2
2601.2.a.be 4 51.f odd 4 1
2601.2.a.bf 4 51.f odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{4} - 2T_{2}^{3} - 5T_{2}^{2} + 8T_{2} + 2 \) acting on \(S_{2}^{\mathrm{new}}(867, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} - 2 T^{3} - 5 T^{2} + \cdots + 2)^{2} \) Copy content Toggle raw display
$3$ \( (T^{2} + 1)^{4} \) Copy content Toggle raw display
$5$ \( T^{8} + 22 T^{6} + \cdots + 4 \) Copy content Toggle raw display
$7$ \( T^{8} + 28 T^{6} + \cdots + 256 \) Copy content Toggle raw display
$11$ \( T^{8} + 34 T^{6} + \cdots + 256 \) Copy content Toggle raw display
$13$ \( (T^{4} - 2 T^{3} - 23 T^{2} + \cdots - 64)^{2} \) Copy content Toggle raw display
$17$ \( T^{8} \) Copy content Toggle raw display
$19$ \( (T^{4} + 10 T^{3} + \cdots - 32)^{2} \) Copy content Toggle raw display
$23$ \( T^{8} + 66 T^{6} + \cdots + 33856 \) Copy content Toggle raw display
$29$ \( T^{8} + 116 T^{6} + \cdots + 256 \) Copy content Toggle raw display
$31$ \( T^{8} + 56 T^{6} + \cdots + 1024 \) Copy content Toggle raw display
$37$ \( T^{8} + 144 T^{6} + \cdots + 595984 \) Copy content Toggle raw display
$41$ \( T^{8} + 230 T^{6} + \cdots + 2775556 \) Copy content Toggle raw display
$43$ \( (T^{4} + 14 T^{3} + \cdots - 1168)^{2} \) Copy content Toggle raw display
$47$ \( (T^{4} - 4 T^{3} - 52 T^{2} + \cdots - 64)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} - 20 T^{3} + \cdots + 128)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} - 24 T^{3} + \cdots - 4672)^{2} \) Copy content Toggle raw display
$61$ \( T^{8} + 112 T^{6} + \cdots + 126736 \) Copy content Toggle raw display
$67$ \( (T^{4} - 4 T^{3} + \cdots + 2048)^{2} \) Copy content Toggle raw display
$71$ \( T^{8} + 120 T^{6} + \cdots + 1024 \) Copy content Toggle raw display
$73$ \( T^{8} + 172 T^{6} + \cdots + 107584 \) Copy content Toggle raw display
$79$ \( T^{8} + 148 T^{6} + \cdots + 256 \) Copy content Toggle raw display
$83$ \( (T^{4} - 4 T^{3} + \cdots + 5248)^{2} \) Copy content Toggle raw display
$89$ \( (T^{4} - 4 T^{3} + \cdots + 1088)^{2} \) Copy content Toggle raw display
$97$ \( T^{8} + 244 T^{6} + \cdots + 135424 \) Copy content Toggle raw display
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