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Properties

Label 676.2.e.f
Level $676$
Weight $2$
Character orbit 676.e
Analytic conductor $5.398$
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] = [676,2,Mod(529,676)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(676, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("676.529");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 676 = 2^{2} \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 676.e (of order \(3\), degree \(2\), 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: \(5.39788717664\)
Analytic rank: \(0\)
Dimension: \(6\)
Relative dimension: \(3\) over \(\Q(\zeta_{3})\)
Coefficient field: 6.0.64827.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - x^{5} + 3x^{4} + 5x^{2} - 2x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( - \beta_{4} - \beta_{3} + \cdots + \beta_1) q^{3}+ \cdots + (2 \beta_{5} - \beta_{4} + 2 \beta_1 - 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{4} - \beta_{3} + \cdots + \beta_1) q^{3}+ \cdots + ( - 2 \beta_{3} + 3) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 16 q^{5} - q^{7} - 5 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 16 q^{5} - q^{7} - 5 q^{9} + 6 q^{11} + 10 q^{17} + 4 q^{19} - 28 q^{21} - 3 q^{23} + 22 q^{25} - 42 q^{27} + q^{29} - 10 q^{31} + 14 q^{33} + 12 q^{35} + 4 q^{37} + 25 q^{41} - 5 q^{43} + 11 q^{45} + 16 q^{47} - 12 q^{49} - 14 q^{51} - 6 q^{53} - 16 q^{55} + 14 q^{57} + 9 q^{59} - 8 q^{61} - 18 q^{63} - 9 q^{67} + 14 q^{69} - 9 q^{71} - 2 q^{73} - 7 q^{75} + 24 q^{77} + 18 q^{79} + q^{81} - 22 q^{83} - 15 q^{85} - 14 q^{87} + 25 q^{89} - 14 q^{93} - 27 q^{95} - 13 q^{97} + 22 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - x^{5} + 3x^{4} + 5x^{2} - 2x + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{5} + 3\nu^{4} - 9\nu^{3} + 5\nu^{2} - 2\nu + 6 ) / 13 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -3\nu^{5} + 9\nu^{4} - 14\nu^{3} + 15\nu^{2} - 6\nu + 18 ) / 13 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -4\nu^{5} - \nu^{4} - 10\nu^{3} - 6\nu^{2} - 34\nu - 2 ) / 13 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -6\nu^{5} + 5\nu^{4} - 15\nu^{3} - 9\nu^{2} - 25\nu + 10 ) / 13 \) Copy content Toggle raw display

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

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

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/676\mathbb{Z}\right)^\times\).

\(n\) \(339\) \(509\)
\(\chi(n)\) \(1\) \(-1 + \beta_{5}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
529.1
0.222521 + 0.385418i
0.900969 + 1.56052i
−0.623490 − 1.07992i
0.222521 − 0.385418i
0.900969 − 1.56052i
−0.623490 + 1.07992i
0 −0.846011 + 1.46533i 0 −1.19806 0 2.14795 + 3.72036i 0 0.0685317 + 0.118700i 0
529.2 0 −0.678448 + 1.17511i 0 −4.24698 0 −1.06853 − 1.85075i 0 0.579417 + 1.00358i 0
529.3 0 1.52446 − 2.64044i 0 −2.55496 0 −1.57942 − 2.73563i 0 −3.14795 − 5.45241i 0
653.1 0 −0.846011 − 1.46533i 0 −1.19806 0 2.14795 − 3.72036i 0 0.0685317 − 0.118700i 0
653.2 0 −0.678448 − 1.17511i 0 −4.24698 0 −1.06853 + 1.85075i 0 0.579417 − 1.00358i 0
653.3 0 1.52446 + 2.64044i 0 −2.55496 0 −1.57942 + 2.73563i 0 −3.14795 + 5.45241i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 529.3
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 676.2.e.f 6
13.b even 2 1 676.2.e.g 6
13.c even 3 1 676.2.a.g ✓ 3
13.c even 3 1 inner 676.2.e.f 6
13.d odd 4 2 676.2.h.e 12
13.e even 6 1 676.2.a.h yes 3
13.e even 6 1 676.2.e.g 6
13.f odd 12 2 676.2.d.e 6
13.f odd 12 2 676.2.h.e 12
39.h odd 6 1 6084.2.a.x 3
39.i odd 6 1 6084.2.a.bc 3
39.k even 12 2 6084.2.b.p 6
52.i odd 6 1 2704.2.a.y 3
52.j odd 6 1 2704.2.a.x 3
52.l even 12 2 2704.2.f.n 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
676.2.a.g ✓ 3 13.c even 3 1
676.2.a.h yes 3 13.e even 6 1
676.2.d.e 6 13.f odd 12 2
676.2.e.f 6 1.a even 1 1 trivial
676.2.e.f 6 13.c even 3 1 inner
676.2.e.g 6 13.b even 2 1
676.2.e.g 6 13.e even 6 1
676.2.h.e 12 13.d odd 4 2
676.2.h.e 12 13.f odd 12 2
2704.2.a.x 3 52.j odd 6 1
2704.2.a.y 3 52.i odd 6 1
2704.2.f.n 6 52.l even 12 2
6084.2.a.x 3 39.h odd 6 1
6084.2.a.bc 3 39.i odd 6 1
6084.2.b.p 6 39.k even 12 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(676, [\chi])\):

\( T_{3}^{6} + 7T_{3}^{4} + 14T_{3}^{3} + 49T_{3}^{2} + 49T_{3} + 49 \) Copy content Toggle raw display
\( T_{5}^{3} + 8T_{5}^{2} + 19T_{5} + 13 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \) Copy content Toggle raw display
$3$ \( T^{6} + 7 T^{4} + \cdots + 49 \) Copy content Toggle raw display
$5$ \( (T^{3} + 8 T^{2} + 19 T + 13)^{2} \) Copy content Toggle raw display
$7$ \( T^{6} + T^{5} + \cdots + 841 \) Copy content Toggle raw display
$11$ \( T^{6} - 6 T^{5} + \cdots + 169 \) Copy content Toggle raw display
$13$ \( T^{6} \) Copy content Toggle raw display
$17$ \( T^{6} - 10 T^{5} + \cdots + 1 \) Copy content Toggle raw display
$19$ \( T^{6} - 4 T^{5} + \cdots + 5041 \) Copy content Toggle raw display
$23$ \( T^{6} + 3 T^{5} + \cdots + 6889 \) Copy content Toggle raw display
$29$ \( T^{6} - T^{5} + \cdots + 841 \) Copy content Toggle raw display
$31$ \( (T^{3} + 5 T^{2} - 8 T - 41)^{2} \) Copy content Toggle raw display
$37$ \( T^{6} - 4 T^{5} + \cdots + 57121 \) Copy content Toggle raw display
$41$ \( T^{6} - 25 T^{5} + \cdots + 253009 \) Copy content Toggle raw display
$43$ \( T^{6} + 5 T^{5} + \cdots + 9409 \) Copy content Toggle raw display
$47$ \( (T^{3} - 8 T^{2} + \cdots + 113)^{2} \) Copy content Toggle raw display
$53$ \( (T^{3} + 3 T^{2} - 46 T + 1)^{2} \) Copy content Toggle raw display
$59$ \( T^{6} - 9 T^{5} + \cdots + 28561 \) Copy content Toggle raw display
$61$ \( T^{6} + 8 T^{5} + \cdots + 57121 \) Copy content Toggle raw display
$67$ \( T^{6} + 9 T^{5} + \cdots + 1849 \) Copy content Toggle raw display
$71$ \( T^{6} + 9 T^{5} + \cdots + 1 \) Copy content Toggle raw display
$73$ \( (T^{3} + T^{2} - 44 T - 127)^{2} \) Copy content Toggle raw display
$79$ \( (T^{3} - 9 T^{2} + \cdots + 911)^{2} \) Copy content Toggle raw display
$83$ \( (T^{3} + 11 T^{2} + \cdots + 41)^{2} \) Copy content Toggle raw display
$89$ \( T^{6} - 25 T^{5} + \cdots + 85849 \) Copy content Toggle raw display
$97$ \( T^{6} + 13 T^{5} + \cdots + 27889 \) Copy content Toggle raw display
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