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Properties

Label 676.2.f.g
Level $676$
Weight $2$
Character orbit 676.f
Analytic conductor $5.398$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

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: \(8\)
Relative dimension: \(4\) over \(\Q(i)\)
Coefficient field: 8.0.18939904.2
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 4x^{7} + 14x^{6} - 28x^{5} + 43x^{4} - 44x^{3} + 30x^{2} - 12x + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 52)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{7} + \beta_{6} - \beta_{3}) q^{2} + ( - \beta_{6} + \beta_{5} + 2 \beta_1 - 1) q^{3} + ( - \beta_{7} + \beta_{6} + \cdots - \beta_{2}) q^{4}+ \cdots + (2 \beta_{6} + 2 \beta_{5} - 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{7} + \beta_{6} - \beta_{3}) q^{2} + ( - \beta_{6} + \beta_{5} + 2 \beta_1 - 1) q^{3} + ( - \beta_{7} + \beta_{6} + \cdots - \beta_{2}) q^{4}+ \cdots + ( - 2 \beta_{7} - 4 \beta_{6} + \cdots - 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 4 q^{2} - 8 q^{6} + 4 q^{8} - 16 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 4 q^{2} - 8 q^{6} + 4 q^{8} - 16 q^{9} - 20 q^{14} - 4 q^{16} + 8 q^{20} - 16 q^{21} + 24 q^{22} - 32 q^{24} - 12 q^{28} - 16 q^{29} + 4 q^{32} - 8 q^{33} - 4 q^{34} + 32 q^{37} + 4 q^{40} + 4 q^{42} + 28 q^{44} - 16 q^{45} + 20 q^{46} - 44 q^{48} + 16 q^{50} + 32 q^{53} + 32 q^{54} - 48 q^{57} - 12 q^{58} + 12 q^{60} - 16 q^{61} - 32 q^{66} + 44 q^{68} - 8 q^{70} - 16 q^{72} + 32 q^{73} + 44 q^{74} - 32 q^{76} - 16 q^{80} - 24 q^{81} + 48 q^{84} - 16 q^{85} - 32 q^{86} - 16 q^{89} + 24 q^{92} - 32 q^{93} + 20 q^{94} + 24 q^{96} + 8 q^{97} + 16 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 4x^{7} + 14x^{6} - 28x^{5} + 43x^{4} - 44x^{3} + 30x^{2} - 12x + 2 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( 2\nu^{7} - 7\nu^{6} + 25\nu^{5} - 45\nu^{4} + 68\nu^{3} - 60\nu^{2} + 36\nu - 8 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -2\nu^{7} + 7\nu^{6} - 25\nu^{5} + 45\nu^{4} - 68\nu^{3} + 61\nu^{2} - 37\nu + 10 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( -4\nu^{7} + 14\nu^{6} - 49\nu^{5} + 87\nu^{4} - 127\nu^{3} + 109\nu^{2} - 61\nu + 15 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( 5\nu^{7} - 17\nu^{6} + 60\nu^{5} - 105\nu^{4} + 155\nu^{3} - 133\nu^{2} + 77\nu - 19 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( -5\nu^{7} + 18\nu^{6} - 63\nu^{5} + 115\nu^{4} - 170\nu^{3} + 152\nu^{2} - 89\nu + 23 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( -8\nu^{7} + 28\nu^{6} - 98\nu^{5} + 175\nu^{4} - 256\nu^{3} + 223\nu^{2} - 126\nu + 31 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} + \beta_{2} + \beta _1 - 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{7} - \beta_{6} + \beta_{5} + 2\beta_{3} + \beta_{2} - 2\beta _1 - 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 3\beta_{7} - 2\beta_{6} + 2\beta_{5} - 2\beta_{4} - \beta_{3} - 3\beta_{2} - 5\beta _1 + 7 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 3\beta_{6} - 3\beta_{5} - 5\beta_{4} - 10\beta_{3} - 5\beta_{2} + 3\beta _1 + 9 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -15\beta_{7} + 15\beta_{6} - 13\beta_{5} + 5\beta_{4} - 9\beta_{3} + 11\beta_{2} + 22\beta _1 - 24 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -19\beta_{7} + 4\beta_{6} + 3\beta_{5} + 35\beta_{4} + 33\beta_{3} + 30\beta_{2} + 7\beta _1 - 61 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(339\) \(509\)
\(\chi(n)\) \(-1\) \(\beta_{7}\)

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
0.500000 + 0.691860i
0.500000 − 0.0297061i
0.500000 − 2.10607i
0.500000 + 1.44392i
0.500000 − 0.691860i
0.500000 + 0.0297061i
0.500000 + 2.10607i
0.500000 − 1.44392i
−0.842772 + 1.13567i 2.79793i −0.579471 − 1.91421i 0.707107 + 0.707107i −3.17751 − 2.35802i 1.97844 + 1.97844i 2.66227 + 0.955161i −4.82843 −1.39897 + 0.207107i
99.2 0.332548 + 1.37456i − 1.47363i −1.77882 + 0.914214i −0.707107 − 0.707107i 2.02559 − 0.490051i 1.04201 + 1.04201i −1.84818 − 2.14108i 0.828427 0.736813 − 1.20711i
99.3 1.13567 − 0.842772i − 2.79793i 0.579471 − 1.91421i 0.707107 + 0.707107i −2.35802 − 3.17751i −1.97844 − 1.97844i −0.955161 − 2.66227i −4.82843 1.39897 + 0.207107i
99.4 1.37456 + 0.332548i 1.47363i 1.77882 + 0.914214i −0.707107 − 0.707107i −0.490051 + 2.02559i −1.04201 − 1.04201i 2.14108 + 1.84818i 0.828427 −0.736813 − 1.20711i
239.1 −0.842772 − 1.13567i − 2.79793i −0.579471 + 1.91421i 0.707107 − 0.707107i −3.17751 + 2.35802i 1.97844 − 1.97844i 2.66227 − 0.955161i −4.82843 −1.39897 − 0.207107i
239.2 0.332548 − 1.37456i 1.47363i −1.77882 − 0.914214i −0.707107 + 0.707107i 2.02559 + 0.490051i 1.04201 − 1.04201i −1.84818 + 2.14108i 0.828427 0.736813 + 1.20711i
239.3 1.13567 + 0.842772i 2.79793i 0.579471 + 1.91421i 0.707107 − 0.707107i −2.35802 + 3.17751i −1.97844 + 1.97844i −0.955161 + 2.66227i −4.82843 1.39897 − 0.207107i
239.4 1.37456 − 0.332548i − 1.47363i 1.77882 − 0.914214i −0.707107 + 0.707107i −0.490051 − 2.02559i −1.04201 + 1.04201i 2.14108 − 1.84818i 0.828427 −0.736813 + 1.20711i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 99.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
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.g 8
4.b odd 2 1 inner 676.2.f.g 8
13.b even 2 1 52.2.f.b ✓ 8
13.c even 3 2 676.2.l.h 16
13.d odd 4 1 52.2.f.b ✓ 8
13.d odd 4 1 inner 676.2.f.g 8
13.e even 6 2 676.2.l.l 16
13.f odd 12 2 676.2.l.h 16
13.f odd 12 2 676.2.l.l 16
39.d odd 2 1 468.2.n.i 8
39.f even 4 1 468.2.n.i 8
52.b odd 2 1 52.2.f.b ✓ 8
52.f even 4 1 52.2.f.b ✓ 8
52.f even 4 1 inner 676.2.f.g 8
52.i odd 6 2 676.2.l.l 16
52.j odd 6 2 676.2.l.h 16
52.l even 12 2 676.2.l.h 16
52.l even 12 2 676.2.l.l 16
104.e even 2 1 832.2.k.h 8
104.h odd 2 1 832.2.k.h 8
104.j odd 4 1 832.2.k.h 8
104.m even 4 1 832.2.k.h 8
156.h even 2 1 468.2.n.i 8
156.l odd 4 1 468.2.n.i 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
52.2.f.b ✓ 8 13.b even 2 1
52.2.f.b ✓ 8 13.d odd 4 1
52.2.f.b ✓ 8 52.b odd 2 1
52.2.f.b ✓ 8 52.f even 4 1
468.2.n.i 8 39.d odd 2 1
468.2.n.i 8 39.f even 4 1
468.2.n.i 8 156.h even 2 1
468.2.n.i 8 156.l odd 4 1
676.2.f.g 8 1.a even 1 1 trivial
676.2.f.g 8 4.b odd 2 1 inner
676.2.f.g 8 13.d odd 4 1 inner
676.2.f.g 8 52.f even 4 1 inner
676.2.l.h 16 13.c even 3 2
676.2.l.h 16 13.f odd 12 2
676.2.l.h 16 52.j odd 6 2
676.2.l.h 16 52.l even 12 2
676.2.l.l 16 13.e even 6 2
676.2.l.l 16 13.f odd 12 2
676.2.l.l 16 52.i odd 6 2
676.2.l.l 16 52.l even 12 2
832.2.k.h 8 104.e even 2 1
832.2.k.h 8 104.h odd 2 1
832.2.k.h 8 104.j odd 4 1
832.2.k.h 8 104.m even 4 1

Hecke kernels

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

\( T_{3}^{4} + 10T_{3}^{2} + 17 \) Copy content Toggle raw display
\( T_{5}^{4} + 1 \) Copy content Toggle raw display
\( T_{7}^{8} + 66T_{7}^{4} + 289 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} - 4 T^{7} + \cdots + 16 \) Copy content Toggle raw display
$3$ \( (T^{4} + 10 T^{2} + 17)^{2} \) Copy content Toggle raw display
$5$ \( (T^{4} + 1)^{2} \) Copy content Toggle raw display
$7$ \( T^{8} + 66T^{4} + 289 \) Copy content Toggle raw display
$11$ \( T^{8} + 648T^{4} + 4624 \) Copy content Toggle raw display
$13$ \( T^{8} \) Copy content Toggle raw display
$17$ \( (T^{4} + 34 T^{2} + 1)^{2} \) Copy content Toggle raw display
$19$ \( T^{8} + 2592 T^{4} + 73984 \) Copy content Toggle raw display
$23$ \( (T^{4} - 20 T^{2} + 68)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} + 4 T + 2)^{4} \) Copy content Toggle raw display
$31$ \( T^{8} + 1056 T^{4} + 73984 \) Copy content Toggle raw display
$37$ \( (T^{4} - 16 T^{3} + \cdots + 529)^{2} \) Copy content Toggle raw display
$41$ \( (T^{4} + 1296)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} - 58 T^{2} + 833)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} + 66T^{4} + 289 \) Copy content Toggle raw display
$53$ \( (T^{2} - 8 T - 2)^{4} \) Copy content Toggle raw display
$59$ \( T^{8} + 10368 T^{4} + 1183744 \) Copy content Toggle raw display
$61$ \( (T^{2} + 4 T - 4)^{4} \) Copy content Toggle raw display
$67$ \( T^{8} + 264T^{4} + 4624 \) Copy content Toggle raw display
$71$ \( T^{8} + 29538 T^{4} + 80874049 \) Copy content Toggle raw display
$73$ \( (T^{4} - 16 T^{3} + \cdots + 1024)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} + 92 T^{2} + 68)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + 8328 T^{4} + 4624 \) Copy content Toggle raw display
$89$ \( (T^{4} + 8 T^{3} + \cdots + 784)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} - 4 T^{3} + 8 T^{2} + \cdots + 4)^{2} \) Copy content Toggle raw display
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