Properties

Label 819.2.g.b
Level $819$
Weight $2$
Character orbit 819.g
Analytic conductor $6.540$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [819,2,Mod(818,819)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(819, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("819.818");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 819 = 3^{2} \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 819.g (of order \(2\), degree \(1\), 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: \(6.53974792554\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.40960000.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 7x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: yes
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{3} q^{2} + 3 q^{4} - 2 \beta_1 q^{5} + (2 \beta_{7} + \beta_{2}) q^{7} - \beta_{3} q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{3} q^{2} + 3 q^{4} - 2 \beta_1 q^{5} + (2 \beta_{7} + \beta_{2}) q^{7} - \beta_{3} q^{8} + 2 \beta_{6} q^{10} - 2 \beta_{3} q^{11} + ( - 2 \beta_{7} + \beta_{6}) q^{13} + (3 \beta_{5} - 4 \beta_{4}) q^{14} - q^{16} + (2 \beta_{5} - \beta_{4}) q^{17} - 3 \beta_{7} q^{19} - 6 \beta_1 q^{20} + 10 q^{22} + q^{25} + ( - 4 \beta_{5} + 2 \beta_{4} - 5 \beta_1) q^{26} + (6 \beta_{7} + 3 \beta_{2}) q^{28} - 3 \beta_{4} q^{29} + \beta_{7} q^{31} + 3 \beta_{3} q^{32} + 5 \beta_{7} q^{34} + (2 \beta_{5} + 2 \beta_{4}) q^{35} + ( - 2 \beta_{7} - 4 \beta_{2}) q^{37} + ( - 6 \beta_{5} + 3 \beta_{4}) q^{38} + 2 \beta_{6} q^{40} + 10 \beta_1 q^{41} - 4 q^{43} - 6 \beta_{3} q^{44} - 8 \beta_1 q^{47} + ( - 3 \beta_{6} + 2) q^{49} - \beta_{3} q^{50} + ( - 6 \beta_{7} + 3 \beta_{6}) q^{52} - \beta_{4} q^{53} + 4 \beta_{6} q^{55} + (3 \beta_{5} - 4 \beta_{4}) q^{56} + (3 \beta_{7} + 6 \beta_{2}) q^{58} + 6 \beta_1 q^{59} - 6 \beta_{6} q^{61} + (2 \beta_{5} - \beta_{4}) q^{62} - 13 q^{64} + ( - 4 \beta_{4} + 2 \beta_{3}) q^{65} + ( - 5 \beta_{7} - 10 \beta_{2}) q^{67} + (6 \beta_{5} - 3 \beta_{4}) q^{68} + (2 \beta_{7} - 6 \beta_{2}) q^{70} + 4 \beta_{3} q^{71} - 10 \beta_{7} q^{73} + 10 \beta_{4} q^{74} - 9 \beta_{7} q^{76} + (6 \beta_{5} - 8 \beta_{4}) q^{77} - 4 q^{79} + 2 \beta_1 q^{80} - 10 \beta_{6} q^{82} + 6 \beta_1 q^{83} + ( - 2 \beta_{7} - 4 \beta_{2}) q^{85} + 4 \beta_{3} q^{86} + 10 q^{88} - 6 \beta_1 q^{89} + (\beta_{7} + 2 \beta_{6} - 3 \beta_{2} - 6) q^{91} + 8 \beta_{6} q^{94} - 6 \beta_{4} q^{95} - 10 \beta_{7} q^{97} + ( - 2 \beta_{3} + 15 \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 24 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 24 q^{4} - 8 q^{16} + 80 q^{22} + 8 q^{25} - 32 q^{43} + 16 q^{49} - 104 q^{64} - 32 q^{79} + 80 q^{88} - 48 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 7x^{4} + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{6} + 8\nu^{2} ) / 3 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{7} + \nu^{5} + 8\nu^{3} + 8\nu ) / 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 2\nu^{4} + 7 ) / 3 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 2\nu^{7} + \nu^{5} + 13\nu^{3} + 5\nu ) / 3 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -\nu^{7} + \nu^{5} - 8\nu^{3} + 8\nu ) / 3 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( \nu^{6} + 6\nu^{2} \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 2\nu^{7} - \nu^{5} + 13\nu^{3} - 5\nu ) / 3 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{7} + \beta_{5} - \beta_{4} + \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{6} + 3\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -\beta_{7} - 2\beta_{5} - \beta_{4} + 2\beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 3\beta_{3} - 7 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -8\beta_{7} - 5\beta_{5} + 8\beta_{4} - 5\beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 4\beta_{6} - 9\beta_1 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 8\beta_{7} + 13\beta_{5} + 8\beta_{4} - 13\beta_{2} ) / 2 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/819\mathbb{Z}\right)^\times\).

\(n\) \(92\) \(379\) \(703\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
818.1
0.437016 + 0.437016i
−0.437016 0.437016i
0.437016 0.437016i
−0.437016 + 0.437016i
−1.14412 1.14412i
1.14412 + 1.14412i
−1.14412 + 1.14412i
1.14412 1.14412i
−2.23607 0 3.00000 2.00000i 0 −2.12132 + 1.58114i −2.23607 0 4.47214i
818.2 −2.23607 0 3.00000 2.00000i 0 2.12132 1.58114i −2.23607 0 4.47214i
818.3 −2.23607 0 3.00000 2.00000i 0 −2.12132 1.58114i −2.23607 0 4.47214i
818.4 −2.23607 0 3.00000 2.00000i 0 2.12132 + 1.58114i −2.23607 0 4.47214i
818.5 2.23607 0 3.00000 2.00000i 0 −2.12132 1.58114i 2.23607 0 4.47214i
818.6 2.23607 0 3.00000 2.00000i 0 2.12132 + 1.58114i 2.23607 0 4.47214i
818.7 2.23607 0 3.00000 2.00000i 0 −2.12132 + 1.58114i 2.23607 0 4.47214i
818.8 2.23607 0 3.00000 2.00000i 0 2.12132 1.58114i 2.23607 0 4.47214i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 818.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
7.b odd 2 1 inner
13.b even 2 1 inner
21.c even 2 1 inner
39.d odd 2 1 inner
91.b odd 2 1 inner
273.g even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 819.2.g.b 8
3.b odd 2 1 inner 819.2.g.b 8
7.b odd 2 1 inner 819.2.g.b 8
13.b even 2 1 inner 819.2.g.b 8
21.c even 2 1 inner 819.2.g.b 8
39.d odd 2 1 inner 819.2.g.b 8
91.b odd 2 1 inner 819.2.g.b 8
273.g even 2 1 inner 819.2.g.b 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
819.2.g.b 8 1.a even 1 1 trivial
819.2.g.b 8 3.b odd 2 1 inner
819.2.g.b 8 7.b odd 2 1 inner
819.2.g.b 8 13.b even 2 1 inner
819.2.g.b 8 21.c even 2 1 inner
819.2.g.b 8 39.d odd 2 1 inner
819.2.g.b 8 91.b odd 2 1 inner
819.2.g.b 8 273.g even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} - 5 \) acting on \(S_{2}^{\mathrm{new}}(819, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} - 5)^{4} \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( (T^{2} + 4)^{4} \) Copy content Toggle raw display
$7$ \( (T^{4} - 4 T^{2} + 49)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} - 20)^{4} \) Copy content Toggle raw display
$13$ \( (T^{4} - 6 T^{2} + 169)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} - 10)^{4} \) Copy content Toggle raw display
$19$ \( (T^{2} - 18)^{4} \) Copy content Toggle raw display
$23$ \( T^{8} \) Copy content Toggle raw display
$29$ \( (T^{2} + 18)^{4} \) Copy content Toggle raw display
$31$ \( (T^{2} - 2)^{4} \) Copy content Toggle raw display
$37$ \( (T^{2} + 40)^{4} \) Copy content Toggle raw display
$41$ \( (T^{2} + 100)^{4} \) Copy content Toggle raw display
$43$ \( (T + 4)^{8} \) Copy content Toggle raw display
$47$ \( (T^{2} + 64)^{4} \) Copy content Toggle raw display
$53$ \( (T^{2} + 2)^{4} \) Copy content Toggle raw display
$59$ \( (T^{2} + 36)^{4} \) Copy content Toggle raw display
$61$ \( (T^{2} + 180)^{4} \) Copy content Toggle raw display
$67$ \( (T^{2} + 250)^{4} \) Copy content Toggle raw display
$71$ \( (T^{2} - 80)^{4} \) Copy content Toggle raw display
$73$ \( (T^{2} - 200)^{4} \) Copy content Toggle raw display
$79$ \( (T + 4)^{8} \) Copy content Toggle raw display
$83$ \( (T^{2} + 36)^{4} \) Copy content Toggle raw display
$89$ \( (T^{2} + 36)^{4} \) Copy content Toggle raw display
$97$ \( (T^{2} - 200)^{4} \) Copy content Toggle raw display
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