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Properties

Label 847.2.l.b
Level $847$
Weight $2$
Character orbit 847.l
Analytic conductor $6.763$
Analytic rank $0$
Dimension $8$
CM discriminant -7
Inner twists $16$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [847,2,Mod(118,847)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(847, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([5, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("847.118");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 847 = 7 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 847.l (of order \(10\), degree \(4\), not minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(6.76332905120\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{10})\)
Coefficient field: 8.0.37515625.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{7} - x^{6} + 3x^{5} - x^{4} + 6x^{3} - 4x^{2} - 8x + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 77)
Sato-Tate group: $\mathrm{U}(1)[D_{10}]$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
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.

\(f(q)\) \(=\) \( q - \beta_{4} q^{2} - 5 \beta_1 q^{4} + \beta_{5} q^{7} - 3 \beta_{7} q^{8} + ( - 3 \beta_{6} - 3 \beta_{3} + \cdots - 3) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{4} q^{2} - 5 \beta_1 q^{4} + \beta_{5} q^{7} - 3 \beta_{7} q^{8} + ( - 3 \beta_{6} - 3 \beta_{3} + \cdots - 3) q^{9}+ \cdots + (7 \beta_{7} + 7 \beta_{5} + \cdots - 7 \beta_{2}) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 10 q^{4} - 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 10 q^{4} - 6 q^{9} + 14 q^{14} - 22 q^{16} - 64 q^{23} - 10 q^{25} + 30 q^{36} + 12 q^{37} + 14 q^{49} - 20 q^{53} + 168 q^{56} - 56 q^{58} + 26 q^{64} - 32 q^{67} + 32 q^{71} - 18 q^{81} + 28 q^{86} - 80 q^{92}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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

\(\beta_{1}\)\(=\) \( ( -\nu^{7} - 7\nu^{2} ) / 4 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{7} + \nu^{6} + \nu^{5} + 5\nu^{4} + \nu^{3} - 6\nu^{2} + 4\nu + 8 ) / 8 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{7} - \nu^{6} + 3\nu^{5} - \nu^{4} + 3\nu^{3} - 4\nu^{2} - 8\nu + 16 ) / 8 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{6} + 9\nu ) / 2 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -3\nu^{7} - 13\nu^{2} ) / 4 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 3\nu^{7} - 3\nu^{6} - 3\nu^{5} + \nu^{4} - 3\nu^{3} + 18\nu^{2} - 12\nu - 24 ) / 8 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 5\nu^{7} + 5\nu^{6} - 15\nu^{5} + 5\nu^{4} + \nu^{3} + 20\nu^{2} + 40\nu - 80 ) / 8 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{6} + \beta_{4} + \beta_{3} + \beta _1 + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{5} - 3\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{7} + 5\beta_{3} ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( \beta_{6} + 3\beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -\beta_{7} - \beta_{5} + \beta_{4} + \beta_{2} - 11 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -9\beta_{6} - 5\beta_{4} - 9\beta_{3} - 9\beta _1 - 9 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -7\beta_{5} + 13\beta_1 ) / 2 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(122\) \(365\)
\(\chi(n)\) \(-1\) \(-\beta_{1}\)

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
118.1
1.18208 + 0.776336i
−0.373058 − 1.36412i
1.10362 + 0.884319i
−1.41264 + 0.0667372i
1.18208 − 0.776336i
−0.373058 + 1.36412i
1.10362 − 0.884319i
−1.41264 − 0.0667372i
−1.55513 − 2.14046i 0 −1.54508 + 4.75528i 0 0 2.51626 + 0.817582i 7.54878 − 2.45275i −2.42705 + 1.76336i 0
118.2 1.55513 + 2.14046i 0 −1.54508 + 4.75528i 0 0 −2.51626 − 0.817582i −7.54878 + 2.45275i −2.42705 + 1.76336i 0
475.1 −2.51626 − 0.817582i 0 4.04508 + 2.93893i 0 0 −1.55513 + 2.14046i −4.66540 − 6.42137i 0.927051 − 2.85317i 0
475.2 2.51626 + 0.817582i 0 4.04508 + 2.93893i 0 0 1.55513 − 2.14046i 4.66540 + 6.42137i 0.927051 − 2.85317i 0
524.1 −1.55513 + 2.14046i 0 −1.54508 − 4.75528i 0 0 2.51626 − 0.817582i 7.54878 + 2.45275i −2.42705 − 1.76336i 0
524.2 1.55513 − 2.14046i 0 −1.54508 − 4.75528i 0 0 −2.51626 + 0.817582i −7.54878 − 2.45275i −2.42705 − 1.76336i 0
699.1 −2.51626 + 0.817582i 0 4.04508 − 2.93893i 0 0 −1.55513 − 2.14046i −4.66540 + 6.42137i 0.927051 + 2.85317i 0
699.2 2.51626 − 0.817582i 0 4.04508 − 2.93893i 0 0 1.55513 + 2.14046i 4.66540 − 6.42137i 0.927051 + 2.85317i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 118.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 CM by \(\Q(\sqrt{-7}) \)
11.b odd 2 1 inner
11.c even 5 3 inner
11.d odd 10 3 inner
77.b even 2 1 inner
77.j odd 10 3 inner
77.l even 10 3 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 847.2.l.b 8
7.b odd 2 1 CM 847.2.l.b 8
11.b odd 2 1 inner 847.2.l.b 8
11.c even 5 1 77.2.b.a ✓ 2
11.c even 5 3 inner 847.2.l.b 8
11.d odd 10 1 77.2.b.a ✓ 2
11.d odd 10 3 inner 847.2.l.b 8
33.f even 10 1 693.2.c.a 2
33.h odd 10 1 693.2.c.a 2
44.g even 10 1 1232.2.e.a 2
44.h odd 10 1 1232.2.e.a 2
77.b even 2 1 inner 847.2.l.b 8
77.j odd 10 1 77.2.b.a ✓ 2
77.j odd 10 3 inner 847.2.l.b 8
77.l even 10 1 77.2.b.a ✓ 2
77.l even 10 3 inner 847.2.l.b 8
77.m even 15 2 539.2.i.a 4
77.n even 30 2 539.2.i.a 4
77.o odd 30 2 539.2.i.a 4
77.p odd 30 2 539.2.i.a 4
231.r odd 10 1 693.2.c.a 2
231.u even 10 1 693.2.c.a 2
308.s odd 10 1 1232.2.e.a 2
308.t even 10 1 1232.2.e.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
77.2.b.a ✓ 2 11.c even 5 1
77.2.b.a ✓ 2 11.d odd 10 1
77.2.b.a ✓ 2 77.j odd 10 1
77.2.b.a ✓ 2 77.l even 10 1
539.2.i.a 4 77.m even 15 2
539.2.i.a 4 77.n even 30 2
539.2.i.a 4 77.o odd 30 2
539.2.i.a 4 77.p odd 30 2
693.2.c.a 2 33.f even 10 1
693.2.c.a 2 33.h odd 10 1
693.2.c.a 2 231.r odd 10 1
693.2.c.a 2 231.u even 10 1
847.2.l.b 8 1.a even 1 1 trivial
847.2.l.b 8 7.b odd 2 1 CM
847.2.l.b 8 11.b odd 2 1 inner
847.2.l.b 8 11.c even 5 3 inner
847.2.l.b 8 11.d odd 10 3 inner
847.2.l.b 8 77.b even 2 1 inner
847.2.l.b 8 77.j odd 10 3 inner
847.2.l.b 8 77.l even 10 3 inner
1232.2.e.a 2 44.g even 10 1
1232.2.e.a 2 44.h odd 10 1
1232.2.e.a 2 308.s odd 10 1
1232.2.e.a 2 308.t even 10 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{8} - 7T_{2}^{6} + 49T_{2}^{4} - 343T_{2}^{2} + 2401 \) acting on \(S_{2}^{\mathrm{new}}(847, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} - 7 T^{6} + \cdots + 2401 \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( T^{8} - 7 T^{6} + \cdots + 2401 \) Copy content Toggle raw display
$11$ \( T^{8} \) Copy content Toggle raw display
$13$ \( T^{8} \) Copy content Toggle raw display
$17$ \( T^{8} \) Copy content Toggle raw display
$19$ \( T^{8} \) Copy content Toggle raw display
$23$ \( (T + 8)^{8} \) Copy content Toggle raw display
$29$ \( T^{8} - 112 T^{6} + \cdots + 157351936 \) Copy content Toggle raw display
$31$ \( T^{8} \) Copy content Toggle raw display
$37$ \( (T^{4} - 6 T^{3} + \cdots + 1296)^{2} \) Copy content Toggle raw display
$41$ \( T^{8} \) Copy content Toggle raw display
$43$ \( (T^{2} + 28)^{4} \) Copy content Toggle raw display
$47$ \( T^{8} \) Copy content Toggle raw display
$53$ \( (T^{4} + 10 T^{3} + \cdots + 10000)^{2} \) Copy content Toggle raw display
$59$ \( T^{8} \) Copy content Toggle raw display
$61$ \( T^{8} \) Copy content Toggle raw display
$67$ \( (T + 4)^{8} \) Copy content Toggle raw display
$71$ \( (T^{4} - 16 T^{3} + \cdots + 65536)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} \) Copy content Toggle raw display
$79$ \( T^{8} + \cdots + 4032758016 \) Copy content Toggle raw display
$83$ \( T^{8} \) Copy content Toggle raw display
$89$ \( T^{8} \) Copy content Toggle raw display
$97$ \( T^{8} \) Copy content Toggle raw display
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