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Properties

Label 867.2.h.e
Level $867$
Weight $2$
Character orbit 867.h
Analytic conductor $6.923$
Analytic rank $0$
Dimension $8$
Inner twists $8$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [867,2,Mod(688,867)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(867, base_ring=CyclotomicField(8)) chi = DirichletCharacter(H, H._module([0, 5])) 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("867.688"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 867 = 3 \cdot 17^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 867.h (of order \(8\), degree \(4\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(16)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(6.92302985525\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{8})\)
Coefficient field: \(\Q(\zeta_{16})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 51)
Sato-Tate group: $\mathrm{SU}(2)[C_{8}]$

$q$-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 primitive root of unity \(\zeta_{16}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - 2 \zeta_{16}^{6} q^{2} - \zeta_{16} q^{3} - 2 \zeta_{16}^{4} q^{4} - 3 \zeta_{16}^{5} q^{5} + 2 \zeta_{16}^{7} q^{6} - 2 \zeta_{16}^{3} q^{7} + \zeta_{16}^{2} q^{9} - 6 \zeta_{16}^{3} q^{10} - 5 \zeta_{16}^{7} q^{11} + \cdots + 5 \zeta_{16} q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 32 q^{16} + 16 q^{18} - 40 q^{33} - 48 q^{35} - 64 q^{50} - 16 q^{52} + 96 q^{67} + 8 q^{69} + 32 q^{84} + 16 q^{86}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(290\) \(292\)
\(\chi(n)\) \(1\) \(\zeta_{16}^{2}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
688.1
0.382683 − 0.923880i
−0.382683 + 0.923880i
0.382683 + 0.923880i
−0.382683 − 0.923880i
0.923880 − 0.382683i
−0.923880 + 0.382683i
0.923880 + 0.382683i
−0.923880 − 0.382683i
−1.41421 + 1.41421i −0.382683 + 0.923880i − 2.00000i −2.77164 − 1.14805i −0.765367 − 1.84776i 1.84776 − 0.765367i 0 −0.707107 − 0.707107i 5.54328 − 2.29610i
688.2 −1.41421 + 1.41421i 0.382683 − 0.923880i − 2.00000i 2.77164 + 1.14805i 0.765367 + 1.84776i −1.84776 + 0.765367i 0 −0.707107 − 0.707107i −5.54328 + 2.29610i
712.1 −1.41421 − 1.41421i −0.382683 − 0.923880i 2.00000i −2.77164 + 1.14805i −0.765367 + 1.84776i 1.84776 + 0.765367i 0 −0.707107 + 0.707107i 5.54328 + 2.29610i
712.2 −1.41421 − 1.41421i 0.382683 + 0.923880i 2.00000i 2.77164 − 1.14805i 0.765367 − 1.84776i −1.84776 − 0.765367i 0 −0.707107 + 0.707107i −5.54328 − 2.29610i
733.1 1.41421 + 1.41421i −0.923880 + 0.382683i 2.00000i 1.14805 + 2.77164i −1.84776 − 0.765367i −0.765367 + 1.84776i 0 0.707107 − 0.707107i −2.29610 + 5.54328i
733.2 1.41421 + 1.41421i 0.923880 − 0.382683i 2.00000i −1.14805 − 2.77164i 1.84776 + 0.765367i 0.765367 − 1.84776i 0 0.707107 − 0.707107i 2.29610 − 5.54328i
757.1 1.41421 − 1.41421i −0.923880 − 0.382683i − 2.00000i 1.14805 − 2.77164i −1.84776 + 0.765367i −0.765367 − 1.84776i 0 0.707107 + 0.707107i −2.29610 − 5.54328i
757.2 1.41421 − 1.41421i 0.923880 + 0.382683i − 2.00000i −1.14805 + 2.77164i 1.84776 − 0.765367i 0.765367 + 1.84776i 0 0.707107 + 0.707107i 2.29610 + 5.54328i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 688.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
17.b even 2 1 inner
17.c even 4 2 inner
17.d even 8 4 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 867.2.h.e 8
17.b even 2 1 inner 867.2.h.e 8
17.c even 4 2 inner 867.2.h.e 8
17.d even 8 4 inner 867.2.h.e 8
17.e odd 16 2 51.2.d.a ✓ 2
17.e odd 16 1 867.2.a.d 1
17.e odd 16 1 867.2.a.e 1
17.e odd 16 4 867.2.e.a 4
51.i even 16 2 153.2.d.c 2
51.i even 16 1 2601.2.a.a 1
51.i even 16 1 2601.2.a.c 1
68.i even 16 2 816.2.c.b 2
85.o even 16 1 1275.2.d.a 2
85.o even 16 1 1275.2.d.c 2
85.p odd 16 2 1275.2.g.b 2
85.r even 16 1 1275.2.d.a 2
85.r even 16 1 1275.2.d.c 2
136.q odd 16 2 3264.2.c.g 2
136.s even 16 2 3264.2.c.h 2
204.t odd 16 2 2448.2.c.f 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
51.2.d.a ✓ 2 17.e odd 16 2
153.2.d.c 2 51.i even 16 2
816.2.c.b 2 68.i even 16 2
867.2.a.d 1 17.e odd 16 1
867.2.a.e 1 17.e odd 16 1
867.2.e.a 4 17.e odd 16 4
867.2.h.e 8 1.a even 1 1 trivial
867.2.h.e 8 17.b even 2 1 inner
867.2.h.e 8 17.c even 4 2 inner
867.2.h.e 8 17.d even 8 4 inner
1275.2.d.a 2 85.o even 16 1
1275.2.d.a 2 85.r even 16 1
1275.2.d.c 2 85.o even 16 1
1275.2.d.c 2 85.r even 16 1
1275.2.g.b 2 85.p odd 16 2
2448.2.c.f 2 204.t odd 16 2
2601.2.a.a 1 51.i even 16 1
2601.2.a.c 1 51.i even 16 1
3264.2.c.g 2 136.q odd 16 2
3264.2.c.h 2 136.s even 16 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(867, [\chi])\):

\( T_{2}^{4} + 16 \) Copy content Toggle raw display
\( T_{5}^{8} + 6561 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} + 16)^{2} \) Copy content Toggle raw display
$3$ \( T^{8} + 1 \) Copy content Toggle raw display
$5$ \( T^{8} + 6561 \) Copy content Toggle raw display
$7$ \( T^{8} + 256 \) Copy content Toggle raw display
$11$ \( T^{8} + 390625 \) Copy content Toggle raw display
$13$ \( (T^{2} + 1)^{4} \) Copy content Toggle raw display
$17$ \( T^{8} \) Copy content Toggle raw display
$19$ \( (T^{4} + 625)^{2} \) Copy content Toggle raw display
$23$ \( T^{8} + 1 \) Copy content Toggle raw display
$29$ \( T^{8} + 1679616 \) Copy content Toggle raw display
$31$ \( T^{8} + 100000000 \) Copy content Toggle raw display
$37$ \( T^{8} + 256 \) Copy content Toggle raw display
$41$ \( T^{8} + 390625 \) Copy content Toggle raw display
$43$ \( (T^{4} + 1)^{2} \) Copy content Toggle raw display
$47$ \( (T^{2} + 4)^{4} \) Copy content Toggle raw display
$53$ \( (T^{4} + 1296)^{2} \) Copy content Toggle raw display
$59$ \( T^{8} \) Copy content Toggle raw display
$61$ \( T^{8} + 100000000 \) Copy content Toggle raw display
$67$ \( (T - 12)^{8} \) Copy content Toggle raw display
$71$ \( T^{8} \) Copy content Toggle raw display
$73$ \( T^{8} + 1679616 \) Copy content Toggle raw display
$79$ \( T^{8} + 65536 \) Copy content Toggle raw display
$83$ \( (T^{4} + 1296)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} + 100)^{4} \) Copy content Toggle raw display
$97$ \( T^{8} + 16777216 \) Copy content Toggle raw display
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