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Properties

Label 77.2.l.a
Level $77$
Weight $2$
Character orbit 77.l
Analytic conductor $0.615$
Analytic rank $0$
Dimension $8$
CM discriminant -7
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [77,2,Mod(6,77)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(77, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([5, 9]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("77.6");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 77 = 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 77.l (of order \(10\), degree \(4\), 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: \(0.614848095564\)
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: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
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.

\(f(q)\) \(=\) \( q + ( - \beta_{6} + \beta_{5} + \beta_{4} + \cdots + 1) q^{2}+ \cdots + (3 \beta_{7} - 3 \beta_{5} - 3 \beta_{3} - 3) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{6} + \beta_{5} + \beta_{4} + \cdots + 1) q^{2}+ \cdots + ( - 6 \beta_{6} + 9) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 10 q^{4} - 10 q^{8} - 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 10 q^{4} - 10 q^{8} - 6 q^{9} - 4 q^{11} - 21 q^{14} + 8 q^{16} - 15 q^{18} - 14 q^{22} + 16 q^{23} - 10 q^{25} + 35 q^{28} + 30 q^{36} - 18 q^{37} + 25 q^{44} + 15 q^{46} + 14 q^{49} + 30 q^{53} - 42 q^{56} + 19 q^{58} - 34 q^{64} + 8 q^{67} - 48 q^{71} - 75 q^{72} - 14 q^{77} - 40 q^{79} - 18 q^{81} + 23 q^{86} - 8 q^{88} + 25 q^{92} + 48 q^{99}+O(q^{100}) \) 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

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

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

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

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/77\mathbb{Z}\right)^\times\).

\(n\) \(45\) \(57\)
\(\chi(n)\) \(-1\) \(-\beta_{5}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
6.1
1.10362 − 0.884319i
−1.41264 − 0.0667372i
1.10362 + 0.884319i
−1.41264 + 0.0667372i
−0.373058 − 1.36412i
1.18208 + 0.776336i
−0.373058 + 1.36412i
1.18208 − 0.776336i
−2.59471 + 0.843073i 0 4.40373 − 3.19950i 0 0 1.55513 + 2.14046i −5.52176 + 7.60006i 0.927051 + 2.85317i 0
6.2 1.47668 − 0.479802i 0 0.332338 − 0.241457i 0 0 −1.55513 − 2.14046i −1.45037 + 1.99627i 0.927051 + 2.85317i 0
13.1 −2.59471 − 0.843073i 0 4.40373 + 3.19950i 0 0 1.55513 − 2.14046i −5.52176 − 7.60006i 0.927051 − 2.85317i 0
13.2 1.47668 + 0.479802i 0 0.332338 + 0.241457i 0 0 −1.55513 + 2.14046i −1.45037 − 1.99627i 0.927051 − 2.85317i 0
41.1 0.0784543 + 0.107983i 0 0.612529 − 1.88517i 0 0 2.51626 + 0.817582i 0.505505 − 0.164249i −2.42705 + 1.76336i 0
41.2 1.03958 + 1.43086i 0 −0.348597 + 1.07287i 0 0 −2.51626 − 0.817582i 1.46663 − 0.476537i −2.42705 + 1.76336i 0
62.1 0.0784543 − 0.107983i 0 0.612529 + 1.88517i 0 0 2.51626 − 0.817582i 0.505505 + 0.164249i −2.42705 − 1.76336i 0
62.2 1.03958 − 1.43086i 0 −0.348597 − 1.07287i 0 0 −2.51626 + 0.817582i 1.46663 + 0.476537i −2.42705 − 1.76336i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 6.2
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.d odd 10 1 inner
77.l even 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 77.2.l.a ✓ 8
3.b odd 2 1 693.2.bu.a 8
7.b odd 2 1 CM 77.2.l.a ✓ 8
7.c even 3 2 539.2.s.a 16
7.d odd 6 2 539.2.s.a 16
11.b odd 2 1 847.2.l.c 8
11.c even 5 1 847.2.b.b 8
11.c even 5 1 847.2.l.a 8
11.c even 5 1 847.2.l.c 8
11.c even 5 1 847.2.l.d 8
11.d odd 10 1 inner 77.2.l.a ✓ 8
11.d odd 10 1 847.2.b.b 8
11.d odd 10 1 847.2.l.a 8
11.d odd 10 1 847.2.l.d 8
21.c even 2 1 693.2.bu.a 8
33.f even 10 1 693.2.bu.a 8
77.b even 2 1 847.2.l.c 8
77.j odd 10 1 847.2.b.b 8
77.j odd 10 1 847.2.l.a 8
77.j odd 10 1 847.2.l.c 8
77.j odd 10 1 847.2.l.d 8
77.l even 10 1 inner 77.2.l.a ✓ 8
77.l even 10 1 847.2.b.b 8
77.l even 10 1 847.2.l.a 8
77.l even 10 1 847.2.l.d 8
77.n even 30 2 539.2.s.a 16
77.o odd 30 2 539.2.s.a 16
231.r odd 10 1 693.2.bu.a 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
77.2.l.a ✓ 8 1.a even 1 1 trivial
77.2.l.a ✓ 8 7.b odd 2 1 CM
77.2.l.a ✓ 8 11.d odd 10 1 inner
77.2.l.a ✓ 8 77.l even 10 1 inner
539.2.s.a 16 7.c even 3 2
539.2.s.a 16 7.d odd 6 2
539.2.s.a 16 77.n even 30 2
539.2.s.a 16 77.o odd 30 2
693.2.bu.a 8 3.b odd 2 1
693.2.bu.a 8 21.c even 2 1
693.2.bu.a 8 33.f even 10 1
693.2.bu.a 8 231.r odd 10 1
847.2.b.b 8 11.c even 5 1
847.2.b.b 8 11.d odd 10 1
847.2.b.b 8 77.j odd 10 1
847.2.b.b 8 77.l even 10 1
847.2.l.a 8 11.c even 5 1
847.2.l.a 8 11.d odd 10 1
847.2.l.a 8 77.j odd 10 1
847.2.l.a 8 77.l even 10 1
847.2.l.c 8 11.b odd 2 1
847.2.l.c 8 11.c even 5 1
847.2.l.c 8 77.b even 2 1
847.2.l.c 8 77.j odd 10 1
847.2.l.d 8 11.c even 5 1
847.2.l.d 8 11.d odd 10 1
847.2.l.d 8 77.j odd 10 1
847.2.l.d 8 77.l even 10 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{8} - 7T_{2}^{6} + 10T_{2}^{5} + 19T_{2}^{4} - 70T_{2}^{3} + 67T_{2}^{2} - 10T_{2} + 1 \) acting on \(S_{2}^{\mathrm{new}}(77, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} - 7 T^{6} + \cdots + 1 \) 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
$11$ \( T^{8} + 4 T^{7} + \cdots + 14641 \) 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
$23$ \( (T^{4} - 8 T^{3} + \cdots - 619)^{2} \) Copy content Toggle raw display
$29$ \( T^{8} - 112 T^{6} + \cdots + 259081 \) Copy content Toggle raw display
$31$ \( T^{8} \) Copy content Toggle raw display
$37$ \( T^{8} + 18 T^{7} + \cdots + 2193361 \) Copy content Toggle raw display
$41$ \( T^{8} \) Copy content Toggle raw display
$43$ \( T^{8} + 402 T^{6} + \cdots + 16072081 \) Copy content Toggle raw display
$47$ \( T^{8} \) Copy content Toggle raw display
$53$ \( T^{8} - 30 T^{7} + \cdots + 6027025 \) Copy content Toggle raw display
$59$ \( T^{8} \) Copy content Toggle raw display
$61$ \( T^{8} \) Copy content Toggle raw display
$67$ \( (T^{4} - 4 T^{3} + \cdots + 17341)^{2} \) Copy content Toggle raw display
$71$ \( T^{8} + 48 T^{7} + \cdots + 19321 \) Copy content Toggle raw display
$73$ \( T^{8} \) Copy content Toggle raw display
$79$ \( T^{8} + 40 T^{7} + \cdots + 23338561 \) 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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