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Properties

Label 688.3.b.e
Level $688$
Weight $3$
Character orbit 688.b
Analytic conductor $18.747$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [688,3,Mod(257,688)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(688, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("688.257");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 688 = 2^{4} \cdot 43 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 688.b (of order \(2\), degree \(1\), 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: \(18.7466421880\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} + 20x^{4} + 121x^{2} + 214 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 43)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q - \beta_{5} q^{3} + \beta_{2} q^{5} + (\beta_{5} + \beta_{2} - \beta_1) q^{7} + ( - 5 \beta_{4} - 4 \beta_{3} - 9) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{5} q^{3} + \beta_{2} q^{5} + (\beta_{5} + \beta_{2} - \beta_1) q^{7} + ( - 5 \beta_{4} - 4 \beta_{3} - 9) q^{9} + (\beta_{4} - 3 \beta_{3} - 7) q^{11} + (5 \beta_{4} - 5 \beta_{3} + 5) q^{13} + (9 \beta_{4} - \beta_{3} - 2) q^{15} + ( - 5 \beta_{4} - 5) q^{17} + (5 \beta_{5} - 5 \beta_1) q^{19} + (10 \beta_{4} + 4 \beta_{3} + 14) q^{21} + (10 \beta_{4} - 5 \beta_{3} + 15) q^{23} + (10 \beta_{4} + 5 \beta_{3} - 9) q^{25} + (9 \beta_{5} + 5 \beta_{2} + 6 \beta_1) q^{27} + (5 \beta_{5} + 10 \beta_1) q^{29} + ( - 15 \beta_{4} + 4 \beta_{3} + 15) q^{31} + (9 \beta_{5} - \beta_{2} - 5 \beta_1) q^{33} + (14 \beta_{3} - 30) q^{35} + (5 \beta_{2} - \beta_1) q^{37} + ( - 5 \beta_{5} - 5 \beta_{2} - 15 \beta_1) q^{39} + (6 \beta_{4} + 11 \beta_{3} - 23) q^{41} + ( - 5 \beta_{5} + 15 \beta_{4} + \cdots + 5) q^{43}+ \cdots + (25 \beta_{4} + 15 \beta_{3} + 91) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 36 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 36 q^{9} - 38 q^{11} + 30 q^{13} - 28 q^{15} - 20 q^{17} + 56 q^{21} + 80 q^{23} - 84 q^{25} + 112 q^{31} - 208 q^{35} - 172 q^{41} - 10 q^{43} - 30 q^{47} - 6 q^{49} - 110 q^{53} + 420 q^{57} + 12 q^{59} + 70 q^{67} - 178 q^{79} + 382 q^{81} - 10 q^{83} + 510 q^{87} + 130 q^{95} - 380 q^{97} + 466 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} + 20x^{4} + 121x^{2} + 214 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{5} - 11\nu^{3} - 18\nu ) / 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{4} - 11\nu^{2} - 22 ) / 4 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{4} + 15\nu^{2} + 46 ) / 4 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{5} + 15\nu^{3} + 50\nu ) / 4 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{4} + \beta_{3} - 6 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{5} + \beta_{2} - 8\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -11\beta_{4} - 15\beta_{3} + 44 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -11\beta_{5} - 15\beta_{2} + 70\beta_1 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(431\) \(433\) \(517\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
257.1
1.77533i
− 2.58315i
3.18991i
− 3.18991i
2.58315i
− 1.77533i
0 − 5.61757i 0 2.98959i 0 6.83184i 0 −22.5571 0
257.2 0 − 3.59415i 0 − 7.02284i 0 − 0.845536i 0 −3.91793 0
257.3 0 − 0.724539i 0 − 7.66434i 0 − 10.1297i 0 8.47504 0
257.4 0 0.724539i 0 7.66434i 0 10.1297i 0 8.47504 0
257.5 0 3.59415i 0 7.02284i 0 0.845536i 0 −3.91793 0
257.6 0 5.61757i 0 − 2.98959i 0 − 6.83184i 0 −22.5571 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 257.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
43.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 688.3.b.e 6
4.b odd 2 1 43.3.b.b ✓ 6
12.b even 2 1 387.3.b.c 6
43.b odd 2 1 inner 688.3.b.e 6
172.d even 2 1 43.3.b.b ✓ 6
516.h odd 2 1 387.3.b.c 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
43.3.b.b ✓ 6 4.b odd 2 1
43.3.b.b ✓ 6 172.d even 2 1
387.3.b.c 6 12.b even 2 1
387.3.b.c 6 516.h odd 2 1
688.3.b.e 6 1.a even 1 1 trivial
688.3.b.e 6 43.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(688, [\chi])\):

\( T_{3}^{6} + 45T_{3}^{4} + 431T_{3}^{2} + 214 \) Copy content Toggle raw display
\( T_{11}^{3} + 19T_{11}^{2} + 48T_{11} - 244 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \) Copy content Toggle raw display
$3$ \( T^{6} + 45 T^{4} + \cdots + 214 \) Copy content Toggle raw display
$5$ \( T^{6} + 117 T^{4} + \cdots + 25894 \) Copy content Toggle raw display
$7$ \( T^{6} + 150 T^{4} + \cdots + 3424 \) Copy content Toggle raw display
$11$ \( (T^{3} + 19 T^{2} + \cdots - 244)^{2} \) Copy content Toggle raw display
$13$ \( (T^{3} - 15 T^{2} + \cdots + 3500)^{2} \) Copy content Toggle raw display
$17$ \( (T^{3} + 10 T^{2} + \cdots - 125)^{2} \) Copy content Toggle raw display
$19$ \( T^{6} + 1475 T^{4} + \cdots + 53500000 \) Copy content Toggle raw display
$23$ \( (T^{3} - 40 T^{2} + \cdots + 14125)^{2} \) Copy content Toggle raw display
$29$ \( T^{6} + 3425 T^{4} + \cdots + 163843750 \) Copy content Toggle raw display
$31$ \( (T^{3} - 56 T^{2} + \cdots + 11041)^{2} \) Copy content Toggle raw display
$37$ \( T^{6} + 2955 T^{4} + \cdots + 316432384 \) Copy content Toggle raw display
$41$ \( (T^{3} + 86 T^{2} + \cdots + 2989)^{2} \) Copy content Toggle raw display
$43$ \( T^{6} + \cdots + 6321363049 \) Copy content Toggle raw display
$47$ \( (T^{3} + 15 T^{2} + \cdots + 13500)^{2} \) Copy content Toggle raw display
$53$ \( (T^{3} + 55 T^{2} + \cdots - 140500)^{2} \) Copy content Toggle raw display
$59$ \( (T^{3} - 6 T^{2} + \cdots - 105424)^{2} \) Copy content Toggle raw display
$61$ \( T^{6} + \cdots + 150281500000 \) Copy content Toggle raw display
$67$ \( (T^{3} - 35 T^{2} + \cdots - 36500)^{2} \) Copy content Toggle raw display
$71$ \( T^{6} + \cdots + 13696000000 \) Copy content Toggle raw display
$73$ \( T^{6} + 7370 T^{4} + \cdots + 601120864 \) Copy content Toggle raw display
$79$ \( (T^{3} + 89 T^{2} + \cdots + 11236)^{2} \) Copy content Toggle raw display
$83$ \( (T^{3} + 5 T^{2} + \cdots + 59500)^{2} \) Copy content Toggle raw display
$89$ \( T^{6} + 8650 T^{4} + \cdots + 53500000 \) Copy content Toggle raw display
$97$ \( (T^{3} + 190 T^{2} + \cdots - 1046125)^{2} \) Copy content Toggle raw display
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