Properties

Label 676.2.f.d
Level $676$
Weight $2$
Character orbit 676.f
Analytic conductor $5.398$
Analytic rank $0$
Dimension $4$
CM discriminant -4
Inner twists $4$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [676,2,Mod(99,676)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(676, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("676.99");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 676 = 2^{2} \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 676.f (of order \(4\), 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: \(5.39788717664\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(i)\)
Coefficient field: \(\Q(\zeta_{12})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 52)
Sato-Tate group: $\mathrm{U}(1)[D_{4}]$

$q$-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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{2} - 1) q^{2} - 2 \beta_{2} q^{4} + (\beta_{3} + \beta_{2} - 1) q^{5} + (2 \beta_{2} + 2) q^{8} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{2} - 1) q^{2} - 2 \beta_{2} q^{4} + (\beta_{3} + \beta_{2} - 1) q^{5} + (2 \beta_{2} + 2) q^{8} + 3 q^{9} + ( - \beta_{3} - 3 \beta_{2} + \beta_1 - 1) q^{10} - 4 q^{16} + (4 \beta_{3} - \beta_{2} - 4 \beta_1 + 4) q^{17} + (3 \beta_{2} - 3) q^{18} + (4 \beta_{2} - 2 \beta_1 + 4) q^{20} + ( - 3 \beta_{3} - \beta_{2} + 3 \beta_1 - 3) q^{25} + (5 \beta_{3} - 5 \beta_{2} + 5 \beta_1 - 2) q^{29} + ( - 4 \beta_{2} + 4) q^{32} + ( - 3 \beta_{2} + 8 \beta_1 - 3) q^{34} - 6 \beta_{2} q^{36} + ( - \beta_{2} + 7 \beta_1 - 1) q^{37} + (2 \beta_{3} - 2 \beta_{2} + 2 \beta_1 - 6) q^{40} + (9 \beta_{3} - 4 \beta_{2} + 4) q^{41} + (3 \beta_{3} + 3 \beta_{2} - 3) q^{45} + 7 \beta_{2} q^{49} + (4 \beta_{2} - 6 \beta_1 + 4) q^{50} + (2 \beta_{3} - 2 \beta_{2} + 2 \beta_1 + 7) q^{53} + ( - 10 \beta_{3} + 3 \beta_{2} - 3) q^{58} + (6 \beta_{3} - 6 \beta_{2} + 6 \beta_1 - 5) q^{61} + 8 \beta_{2} q^{64} + ( - 8 \beta_{3} + 8 \beta_{2} - 8 \beta_1 - 2) q^{68} + (6 \beta_{2} + 6) q^{72} + ( - 3 \beta_{2} - 5 \beta_1 - 3) q^{73} + ( - 7 \beta_{3} + 7 \beta_{2} - 7 \beta_1 - 5) q^{74} + ( - 4 \beta_{3} - 4 \beta_{2} + 4) q^{80} + 9 q^{81} + ( - 9 \beta_{3} - \beta_{2} + 9 \beta_1 - 9) q^{82} + ( - 10 \beta_{2} + 11 \beta_1 - 10) q^{85} + ( - 3 \beta_{2} - 3) q^{89} + ( - 3 \beta_{3} - 9 \beta_{2} + 3 \beta_1 - 3) q^{90} + (5 \beta_{2} - 5) q^{97} + ( - 7 \beta_{2} - 7) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{2} - 6 q^{5} + 8 q^{8} + 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{2} - 6 q^{5} + 8 q^{8} + 12 q^{9} - 16 q^{16} - 12 q^{18} + 12 q^{20} - 8 q^{29} + 16 q^{32} + 4 q^{34} + 10 q^{37} - 24 q^{40} - 2 q^{41} - 18 q^{45} + 4 q^{50} + 28 q^{53} + 8 q^{58} - 20 q^{61} - 8 q^{68} + 24 q^{72} - 22 q^{73} - 20 q^{74} + 24 q^{80} + 36 q^{81} - 18 q^{85} - 12 q^{89} - 20 q^{97} - 28 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( \zeta_{12}^{2} + \zeta_{12} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \zeta_{12}^{3} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -\zeta_{12}^{2} + \zeta_{12} \) Copy content Toggle raw display
\(\zeta_{12}\)\(=\) \( ( \beta_{3} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\zeta_{12}^{2}\)\(=\) \( ( -\beta_{3} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\zeta_{12}^{3}\)\(=\) \( \beta_{2} \) Copy content Toggle raw display

Character values

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

\(n\) \(339\) \(509\)
\(\chi(n)\) \(-1\) \(-\beta_{2}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
99.1
−0.866025 0.500000i
0.866025 0.500000i
−0.866025 + 0.500000i
0.866025 + 0.500000i
−1.00000 1.00000i 0 2.00000i −2.36603 2.36603i 0 0 2.00000 2.00000i 3.00000 4.73205i
99.2 −1.00000 1.00000i 0 2.00000i −0.633975 0.633975i 0 0 2.00000 2.00000i 3.00000 1.26795i
239.1 −1.00000 + 1.00000i 0 2.00000i −2.36603 + 2.36603i 0 0 2.00000 + 2.00000i 3.00000 4.73205i
239.2 −1.00000 + 1.00000i 0 2.00000i −0.633975 + 0.633975i 0 0 2.00000 + 2.00000i 3.00000 1.26795i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 CM by \(\Q(\sqrt{-1}) \)
13.d odd 4 1 inner
52.f even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 676.2.f.d 4
4.b odd 2 1 CM 676.2.f.d 4
13.b even 2 1 676.2.f.e 4
13.c even 3 1 676.2.l.d 4
13.c even 3 1 676.2.l.e 4
13.d odd 4 1 inner 676.2.f.d 4
13.d odd 4 1 676.2.f.e 4
13.e even 6 1 52.2.l.a 4
13.e even 6 1 676.2.l.c 4
13.f odd 12 1 52.2.l.a 4
13.f odd 12 1 676.2.l.c 4
13.f odd 12 1 676.2.l.d 4
13.f odd 12 1 676.2.l.e 4
39.h odd 6 1 468.2.cb.d 4
39.k even 12 1 468.2.cb.d 4
52.b odd 2 1 676.2.f.e 4
52.f even 4 1 inner 676.2.f.d 4
52.f even 4 1 676.2.f.e 4
52.i odd 6 1 52.2.l.a 4
52.i odd 6 1 676.2.l.c 4
52.j odd 6 1 676.2.l.d 4
52.j odd 6 1 676.2.l.e 4
52.l even 12 1 52.2.l.a 4
52.l even 12 1 676.2.l.c 4
52.l even 12 1 676.2.l.d 4
52.l even 12 1 676.2.l.e 4
104.p odd 6 1 832.2.bu.d 4
104.s even 6 1 832.2.bu.d 4
104.u even 12 1 832.2.bu.d 4
104.x odd 12 1 832.2.bu.d 4
156.r even 6 1 468.2.cb.d 4
156.v odd 12 1 468.2.cb.d 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
52.2.l.a 4 13.e even 6 1
52.2.l.a 4 13.f odd 12 1
52.2.l.a 4 52.i odd 6 1
52.2.l.a 4 52.l even 12 1
468.2.cb.d 4 39.h odd 6 1
468.2.cb.d 4 39.k even 12 1
468.2.cb.d 4 156.r even 6 1
468.2.cb.d 4 156.v odd 12 1
676.2.f.d 4 1.a even 1 1 trivial
676.2.f.d 4 4.b odd 2 1 CM
676.2.f.d 4 13.d odd 4 1 inner
676.2.f.d 4 52.f even 4 1 inner
676.2.f.e 4 13.b even 2 1
676.2.f.e 4 13.d odd 4 1
676.2.f.e 4 52.b odd 2 1
676.2.f.e 4 52.f even 4 1
676.2.l.c 4 13.e even 6 1
676.2.l.c 4 13.f odd 12 1
676.2.l.c 4 52.i odd 6 1
676.2.l.c 4 52.l even 12 1
676.2.l.d 4 13.c even 3 1
676.2.l.d 4 13.f odd 12 1
676.2.l.d 4 52.j odd 6 1
676.2.l.d 4 52.l even 12 1
676.2.l.e 4 13.c even 3 1
676.2.l.e 4 13.f odd 12 1
676.2.l.e 4 52.j odd 6 1
676.2.l.e 4 52.l even 12 1
832.2.bu.d 4 104.p odd 6 1
832.2.bu.d 4 104.s even 6 1
832.2.bu.d 4 104.u even 12 1
832.2.bu.d 4 104.x odd 12 1

Hecke kernels

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

\( T_{3} \) Copy content Toggle raw display
\( T_{5}^{4} + 6T_{5}^{3} + 18T_{5}^{2} + 18T_{5} + 9 \) Copy content Toggle raw display
\( T_{7} \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 2 T + 2)^{2} \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} + 6 T^{3} + 18 T^{2} + 18 T + 9 \) Copy content Toggle raw display
$7$ \( T^{4} \) Copy content Toggle raw display
$11$ \( T^{4} \) Copy content Toggle raw display
$13$ \( T^{4} \) Copy content Toggle raw display
$17$ \( T^{4} + 98T^{2} + 2209 \) Copy content Toggle raw display
$19$ \( T^{4} \) Copy content Toggle raw display
$23$ \( T^{4} \) Copy content Toggle raw display
$29$ \( (T^{2} + 4 T - 71)^{2} \) Copy content Toggle raw display
$31$ \( T^{4} \) Copy content Toggle raw display
$37$ \( T^{4} - 10 T^{3} + 50 T^{2} + \cdots + 3721 \) Copy content Toggle raw display
$41$ \( T^{4} + 2 T^{3} + 2 T^{2} + \cdots + 14641 \) Copy content Toggle raw display
$43$ \( T^{4} \) Copy content Toggle raw display
$47$ \( T^{4} \) Copy content Toggle raw display
$53$ \( (T^{2} - 14 T + 37)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} \) Copy content Toggle raw display
$61$ \( (T^{2} + 10 T - 83)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} \) Copy content Toggle raw display
$71$ \( T^{4} \) Copy content Toggle raw display
$73$ \( T^{4} + 22 T^{3} + 242 T^{2} + \cdots + 529 \) Copy content Toggle raw display
$79$ \( T^{4} \) Copy content Toggle raw display
$83$ \( T^{4} \) Copy content Toggle raw display
$89$ \( (T^{2} + 6 T + 18)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} + 10 T + 50)^{2} \) Copy content Toggle raw display
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