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Properties

Label 847.2.e.c
Level $847$
Weight $2$
Character orbit 847.e
Analytic conductor $6.763$
Analytic rank $0$
Dimension $6$
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [847,2,Mod(485,847)] 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("847.485"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(847, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([2, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 847 = 7 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 847.e (of order \(3\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [6,0,-3,0,-6,6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(6)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(6.76332905120\)
Analytic rank: \(0\)
Dimension: \(6\)
Relative dimension: \(3\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\zeta_{18})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - x^{3} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 77)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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 \(\zeta_{18}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\zeta_{18}^{5} - \zeta_{18}^{4} + \cdots + \zeta_{18}) q^{2} + ( - \zeta_{18}^{5} - \zeta_{18}^{4} + \cdots - 1) q^{3} + ( - \zeta_{18}^{5} - \zeta_{18}^{4} + \zeta_{18}) q^{4} + (\zeta_{18}^{4} + \cdots + \zeta_{18}^{2}) q^{5}+ \cdots + (8 \zeta_{18}^{4} - 8 \zeta_{18}^{3} + \cdots + 5) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 3 q^{3} - 6 q^{5} + 6 q^{6} - 6 q^{8} + 3 q^{10} - 3 q^{12} - 6 q^{13} - 12 q^{14} + 18 q^{15} + 6 q^{16} - 3 q^{17} - 12 q^{18} + 9 q^{19} + 12 q^{20} + 12 q^{21} + 6 q^{24} - 3 q^{25} - 9 q^{26}+ \cdots + 6 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/847\mathbb{Z}\right)^\times\).

\(n\) \(122\) \(365\)
\(\chi(n)\) \(-\zeta_{18}^{3}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
485.1
−0.173648 + 0.984808i
0.939693 − 0.342020i
−0.766044 − 0.642788i
−0.173648 − 0.984808i
0.939693 + 0.342020i
−0.766044 + 0.642788i
−0.766044 + 1.32683i −1.43969 − 2.49362i −0.173648 − 0.300767i −1.17365 + 2.03282i 4.41147 −2.05303 + 1.66885i −2.53209 −2.64543 + 4.58202i −1.79813 − 3.11446i
485.2 −0.173648 + 0.300767i 0.266044 + 0.460802i 0.939693 + 1.62760i −0.0603074 + 0.104455i −0.184793 2.47178 + 0.943555i −1.34730 1.35844 − 2.35289i −0.0209445 − 0.0362770i
485.3 0.939693 − 1.62760i −0.326352 − 0.565258i −0.766044 − 1.32683i −1.76604 + 3.05888i −1.22668 −0.418748 − 2.61240i 0.879385 1.28699 − 2.22913i 3.31908 + 5.74881i
606.1 −0.766044 − 1.32683i −1.43969 + 2.49362i −0.173648 + 0.300767i −1.17365 − 2.03282i 4.41147 −2.05303 − 1.66885i −2.53209 −2.64543 − 4.58202i −1.79813 + 3.11446i
606.2 −0.173648 − 0.300767i 0.266044 − 0.460802i 0.939693 − 1.62760i −0.0603074 − 0.104455i −0.184793 2.47178 − 0.943555i −1.34730 1.35844 + 2.35289i −0.0209445 + 0.0362770i
606.3 0.939693 + 1.62760i −0.326352 + 0.565258i −0.766044 + 1.32683i −1.76604 − 3.05888i −1.22668 −0.418748 + 2.61240i 0.879385 1.28699 + 2.22913i 3.31908 − 5.74881i
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 485.3
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 847.2.e.c 6
7.c even 3 1 inner 847.2.e.c 6
7.c even 3 1 5929.2.a.x 3
7.d odd 6 1 5929.2.a.u 3
11.b odd 2 1 77.2.e.a ✓ 6
11.c even 5 4 847.2.n.f 24
11.d odd 10 4 847.2.n.g 24
33.d even 2 1 693.2.i.h 6
44.c even 2 1 1232.2.q.m 6
77.b even 2 1 539.2.e.m 6
77.h odd 6 1 77.2.e.a ✓ 6
77.h odd 6 1 539.2.a.j 3
77.i even 6 1 539.2.a.g 3
77.i even 6 1 539.2.e.m 6
77.m even 15 4 847.2.n.f 24
77.o odd 30 4 847.2.n.g 24
231.k odd 6 1 4851.2.a.bk 3
231.l even 6 1 693.2.i.h 6
231.l even 6 1 4851.2.a.bj 3
308.m odd 6 1 8624.2.a.co 3
308.n even 6 1 1232.2.q.m 6
308.n even 6 1 8624.2.a.ch 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
77.2.e.a ✓ 6 11.b odd 2 1
77.2.e.a ✓ 6 77.h odd 6 1
539.2.a.g 3 77.i even 6 1
539.2.a.j 3 77.h odd 6 1
539.2.e.m 6 77.b even 2 1
539.2.e.m 6 77.i even 6 1
693.2.i.h 6 33.d even 2 1
693.2.i.h 6 231.l even 6 1
847.2.e.c 6 1.a even 1 1 trivial
847.2.e.c 6 7.c even 3 1 inner
847.2.n.f 24 11.c even 5 4
847.2.n.f 24 77.m even 15 4
847.2.n.g 24 11.d odd 10 4
847.2.n.g 24 77.o odd 30 4
1232.2.q.m 6 44.c even 2 1
1232.2.q.m 6 308.n even 6 1
4851.2.a.bj 3 231.l even 6 1
4851.2.a.bk 3 231.k odd 6 1
5929.2.a.u 3 7.d odd 6 1
5929.2.a.x 3 7.c even 3 1
8624.2.a.ch 3 308.n even 6 1
8624.2.a.co 3 308.m odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{6} + 3T_{2}^{4} + 2T_{2}^{3} + 9T_{2}^{2} + 3T_{2} + 1 \) acting on \(S_{2}^{\mathrm{new}}(847, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} + 3 T^{4} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( T^{6} + 3 T^{5} + \cdots + 1 \) Copy content Toggle raw display
$5$ \( T^{6} + 6 T^{5} + \cdots + 1 \) Copy content Toggle raw display
$7$ \( T^{6} - 17T^{3} + 343 \) Copy content Toggle raw display
$11$ \( T^{6} \) Copy content Toggle raw display
$13$ \( (T^{3} + 3 T^{2} - 6 T + 1)^{2} \) Copy content Toggle raw display
$17$ \( T^{6} + 3 T^{5} + \cdots + 16129 \) Copy content Toggle raw display
$19$ \( T^{6} - 9 T^{5} + \cdots + 81 \) Copy content Toggle raw display
$23$ \( T^{6} + 57 T^{4} + \cdots + 11449 \) Copy content Toggle raw display
$29$ \( (T^{3} - 3 T^{2} - 36 T - 51)^{2} \) Copy content Toggle raw display
$31$ \( T^{6} + 9 T^{5} + \cdots + 361 \) Copy content Toggle raw display
$37$ \( T^{6} + 36 T^{4} + \cdots + 5184 \) Copy content Toggle raw display
$41$ \( (T^{3} + 9 T^{2} + 6 T + 1)^{2} \) Copy content Toggle raw display
$43$ \( (T^{3} - 9 T - 9)^{2} \) Copy content Toggle raw display
$47$ \( T^{6} - 3 T^{5} + \cdots + 104329 \) Copy content Toggle raw display
$53$ \( T^{6} + 9 T^{5} + \cdots + 210681 \) Copy content Toggle raw display
$59$ \( T^{6} + 93 T^{4} + \cdots + 361 \) Copy content Toggle raw display
$61$ \( T^{6} - 12 T^{5} + \cdots + 576 \) Copy content Toggle raw display
$67$ \( T^{6} + 12 T^{4} + \cdots + 64 \) Copy content Toggle raw display
$71$ \( (T^{3} + 9 T^{2} + \cdots - 801)^{2} \) Copy content Toggle raw display
$73$ \( T^{6} - 6 T^{5} + \cdots + 1 \) Copy content Toggle raw display
$79$ \( T^{6} + 3 T^{5} + \cdots + 1369 \) Copy content Toggle raw display
$83$ \( (T^{3} - 15 T^{2} + \cdots + 267)^{2} \) Copy content Toggle raw display
$89$ \( T^{6} - 15 T^{5} + \cdots + 12321 \) Copy content Toggle raw display
$97$ \( (T^{3} - 45 T^{2} + \cdots - 3329)^{2} \) Copy content Toggle raw display
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