↑

Properties

Label 688.2.o.c
Level $688$
Weight $2$
Character orbit 688.o
Analytic conductor $5.494$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [688,2,Mod(351,688)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(688, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 0, 5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("688.351");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 688 = 2^{4} \cdot 43 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 688.o (of order \(6\), degree \(2\), 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: \(5.49370765906\)
Analytic rank: \(0\)
Dimension: \(6\)
Relative dimension: \(3\) over \(\Q(\zeta_{6})\)
Coefficient field: 6.0.1714608.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 3x^{5} + 12x^{4} - 19x^{3} + 30x^{2} - 21x + 7 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{4} q^{3} + (\beta_{5} + \beta_{2} - \beta_1 + 1) q^{5} + (\beta_{5} - \beta_{4} + \beta_{2} - 1) q^{7} + (2 \beta_{4} + 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{4} q^{3} + (\beta_{5} + \beta_{2} - \beta_1 + 1) q^{5} + (\beta_{5} - \beta_{4} + \beta_{2} - 1) q^{7} + (2 \beta_{4} + 2) q^{9} + ( - 2 \beta_{4} - \beta_{2} - \beta_1 - 1) q^{11} + (4 \beta_{4} - \beta_{3} + \beta_{2} + \cdots + 3) q^{13}+ \cdots + (2 \beta_{5} - 2 \beta_{4} + \cdots - 2 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 3 q^{3} + 3 q^{5} - 3 q^{7} + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 3 q^{3} + 3 q^{5} - 3 q^{7} + 6 q^{9} + 9 q^{13} - 3 q^{15} + 3 q^{17} - 3 q^{19} + 6 q^{21} + 9 q^{23} + 18 q^{25} - 30 q^{27} + 3 q^{29} + 9 q^{31} + 6 q^{33} - 9 q^{37} - 18 q^{39} + 36 q^{41} - 24 q^{43} - 6 q^{49} - 6 q^{51} - 15 q^{53} + 6 q^{55} - 3 q^{57} - 9 q^{61} + 6 q^{63} - 9 q^{67} - 9 q^{69} + 9 q^{71} + 9 q^{73} - 36 q^{75} + 30 q^{77} + 27 q^{79} - 3 q^{81} + 3 q^{83} + 9 q^{89} - 3 q^{91} - 9 q^{93} + 57 q^{95} - 36 q^{97} + 12 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - 3x^{5} + 12x^{4} - 19x^{3} + 30x^{2} - 21x + 7 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu^{2} + 3 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( -\nu^{2} + 2\nu - 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{5} + \nu^{4} + 9\nu^{3} - 4\nu^{2} + 35\nu - 28 ) / 7 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -2\nu^{5} + 5\nu^{4} - 18\nu^{3} + 22\nu^{2} - 28\nu + 7 ) / 7 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{5} - 6\nu^{4} + 23\nu^{3} - 39\nu^{2} + 63\nu - 14 ) / 7 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{2} + \beta _1 + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta _1 - 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{5} + \beta_{4} + \beta_{3} - 5\beta_{2} - 2\beta _1 - 9 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{4} + 2\beta_{3} - 3\beta_{2} - 5\beta _1 + 10 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -9\beta_{5} - 11\beta_{4} + \beta_{3} + 16\beta_{2} + \beta _1 + 58 ) / 2 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/688\mathbb{Z}\right)^\times\).

\(n\) \(431\) \(433\) \(517\)
\(\chi(n)\) \(-1\) \(1 + \beta_{4}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
351.1
0.500000 + 0.385124i
0.500000 + 2.23871i
0.500000 − 1.75780i
0.500000 − 0.385124i
0.500000 − 2.23871i
0.500000 + 1.75780i
0 −0.500000 − 0.866025i 0 −3.48605 + 2.01267i 0 −1.88437 + 3.26382i 0 1.00000 − 1.73205i 0
351.2 0 −0.500000 − 0.866025i 0 2.20393 − 1.27244i 0 −1.05787 + 1.83229i 0 1.00000 − 1.73205i 0
351.3 0 −0.500000 − 0.866025i 0 2.78212 − 1.60626i 0 1.44224 − 2.49804i 0 1.00000 − 1.73205i 0
639.1 0 −0.500000 + 0.866025i 0 −3.48605 − 2.01267i 0 −1.88437 − 3.26382i 0 1.00000 + 1.73205i 0
639.2 0 −0.500000 + 0.866025i 0 2.20393 + 1.27244i 0 −1.05787 − 1.83229i 0 1.00000 + 1.73205i 0
639.3 0 −0.500000 + 0.866025i 0 2.78212 + 1.60626i 0 1.44224 + 2.49804i 0 1.00000 + 1.73205i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 351.3
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
172.f even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 688.2.o.c ✓ 6
4.b odd 2 1 688.2.o.f yes 6
43.d odd 6 1 688.2.o.f yes 6
172.f even 6 1 inner 688.2.o.c ✓ 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
688.2.o.c ✓ 6 1.a even 1 1 trivial
688.2.o.c ✓ 6 172.f even 6 1 inner
688.2.o.f yes 6 4.b odd 2 1
688.2.o.f yes 6 43.d odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(688, [\chi])\):

\( T_{3}^{2} + T_{3} + 1 \) Copy content Toggle raw display
\( T_{5}^{6} - 3T_{5}^{5} - 12T_{5}^{4} + 45T_{5}^{3} + 168T_{5}^{2} - 855T_{5} + 1083 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \) Copy content Toggle raw display
$3$ \( (T^{2} + T + 1)^{3} \) Copy content Toggle raw display
$5$ \( T^{6} - 3 T^{5} + \cdots + 1083 \) Copy content Toggle raw display
$7$ \( T^{6} + 3 T^{5} + \cdots + 529 \) Copy content Toggle raw display
$11$ \( T^{6} + 36 T^{4} + \cdots + 192 \) Copy content Toggle raw display
$13$ \( T^{6} - 9 T^{5} + \cdots + 7921 \) Copy content Toggle raw display
$17$ \( T^{6} - 3 T^{5} + \cdots + 3969 \) Copy content Toggle raw display
$19$ \( T^{6} + 3 T^{5} + \cdots + 49 \) Copy content Toggle raw display
$23$ \( T^{6} - 9 T^{5} + \cdots + 14283 \) Copy content Toggle raw display
$29$ \( T^{6} - 3 T^{5} + \cdots + 2883 \) Copy content Toggle raw display
$31$ \( T^{6} - 9 T^{5} + \cdots + 27 \) Copy content Toggle raw display
$37$ \( T^{6} + 9 T^{5} + \cdots + 27 \) Copy content Toggle raw display
$41$ \( (T - 6)^{6} \) Copy content Toggle raw display
$43$ \( (T^{2} + 8 T + 43)^{3} \) Copy content Toggle raw display
$47$ \( T^{6} + 228 T^{4} + \cdots + 184512 \) Copy content Toggle raw display
$53$ \( T^{6} + 15 T^{5} + \cdots + 81 \) Copy content Toggle raw display
$59$ \( T^{6} + 180 T^{4} + \cdots + 84672 \) Copy content Toggle raw display
$61$ \( T^{6} + 9 T^{5} + \cdots + 121203 \) Copy content Toggle raw display
$67$ \( (T^{2} + 3 T + 3)^{3} \) Copy content Toggle raw display
$71$ \( T^{6} - 9 T^{5} + \cdots + 3272481 \) Copy content Toggle raw display
$73$ \( T^{6} - 9 T^{5} + \cdots + 64827 \) Copy content Toggle raw display
$79$ \( T^{6} - 27 T^{5} + \cdots + 27 \) Copy content Toggle raw display
$83$ \( T^{6} - 3 T^{5} + \cdots + 2152227 \) Copy content Toggle raw display
$89$ \( T^{6} - 9 T^{5} + \cdots + 1323 \) Copy content Toggle raw display
$97$ \( (T^{3} + 18 T^{2} + \cdots - 568)^{2} \) Copy content Toggle raw display
show more
show less