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Properties

Label 819.2.o.c
Level $819$
Weight $2$
Character orbit 819.o
Analytic conductor $6.540$
Analytic rank $0$
Dimension $4$
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] = [819,2,Mod(568,819)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(819, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 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("819.568");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 819 = 3^{2} \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 819.o (of order \(3\), 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: \(6.53974792554\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{5})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} + 2x^{2} + x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 91)
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{3} + \beta_1 + 1) q^{2} + (3 \beta_{2} + 3 \beta_1) q^{4} + (\beta_{2} - 1) q^{5} + \beta_{3} q^{7} + (4 \beta_{2} - 1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{3} + \beta_1 + 1) q^{2} + (3 \beta_{2} + 3 \beta_1) q^{4} + (\beta_{2} - 1) q^{5} + \beta_{3} q^{7} + (4 \beta_{2} - 1) q^{8} + ( - 2 \beta_{3} - 3 \beta_1 - 2) q^{10} + ( - 3 \beta_{3} + 3 \beta_1 - 3) q^{11} + ( - 3 \beta_{3} - 4) q^{13} + (\beta_{2} - 1) q^{14} + ( - 5 \beta_{3} - 3 \beta_1 - 5) q^{16} + ( - 5 \beta_{3} + 4 \beta_{2} + 4 \beta_1) q^{17} + (3 \beta_{3} - 3 \beta_{2} - 3 \beta_1) q^{19} + ( - 3 \beta_{3} - 6 \beta_{2} - 6 \beta_1) q^{20} + (3 \beta_{2} + 3 \beta_1) q^{22} + (2 \beta_{3} - 4 \beta_1 + 2) q^{23} + ( - 3 \beta_{2} - 3) q^{25} + ( - 4 \beta_{3} - 3 \beta_{2} + \cdots - 1) q^{26}+ \cdots + ( - \beta_{3} - \beta_{2} - \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 3 q^{2} - 3 q^{4} - 6 q^{5} - 2 q^{7} - 12 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 3 q^{2} - 3 q^{4} - 6 q^{5} - 2 q^{7} - 12 q^{8} - 7 q^{10} - 3 q^{11} - 10 q^{13} - 6 q^{14} - 13 q^{16} + 6 q^{17} - 3 q^{19} + 12 q^{20} - 3 q^{22} - 6 q^{25} + 6 q^{26} - 3 q^{28} + 3 q^{29} + 8 q^{31} + 15 q^{32} - 2 q^{34} + 3 q^{35} - 8 q^{37} + 6 q^{38} + 38 q^{40} + 6 q^{41} - 5 q^{43} - 36 q^{44} + 10 q^{46} - 2 q^{49} + 3 q^{50} + 21 q^{52} - 24 q^{53} - 3 q^{55} + 6 q^{56} - 17 q^{58} + 12 q^{61} - 9 q^{62} + 8 q^{64} + 15 q^{65} + 12 q^{67} - 21 q^{68} + 14 q^{70} - 6 q^{71} - 8 q^{73} + 12 q^{74} + 18 q^{76} + 6 q^{77} + 16 q^{79} + 27 q^{80} - 4 q^{82} + q^{85} - 60 q^{86} - 21 q^{88} + 21 q^{89} + 14 q^{91} + 60 q^{92} + 5 q^{94} - 3 q^{95} - 31 q^{97} + 3 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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

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

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

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

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

Character values

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

\(n\) \(92\) \(379\) \(703\)
\(\chi(n)\) \(1\) \(\beta_{3}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
568.1
−0.309017 + 0.535233i
0.809017 − 1.40126i
−0.309017 − 0.535233i
0.809017 + 1.40126i
0.190983 − 0.330792i 0 0.927051 + 1.60570i −0.381966 0 −0.500000 − 0.866025i 1.47214 0 −0.0729490 + 0.126351i
568.2 1.30902 − 2.26728i 0 −2.42705 − 4.20378i −2.61803 0 −0.500000 − 0.866025i −7.47214 0 −3.42705 + 5.93583i
757.1 0.190983 + 0.330792i 0 0.927051 − 1.60570i −0.381966 0 −0.500000 + 0.866025i 1.47214 0 −0.0729490 − 0.126351i
757.2 1.30902 + 2.26728i 0 −2.42705 + 4.20378i −2.61803 0 −0.500000 + 0.866025i −7.47214 0 −3.42705 − 5.93583i
\(n\): e.g. 2-40 or 990-1000
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 819.2.o.c 4
3.b odd 2 1 91.2.f.a ✓ 4
12.b even 2 1 1456.2.s.h 4
13.c even 3 1 inner 819.2.o.c 4
21.c even 2 1 637.2.f.c 4
21.g even 6 1 637.2.g.c 4
21.g even 6 1 637.2.h.f 4
21.h odd 6 1 637.2.g.b 4
21.h odd 6 1 637.2.h.g 4
39.h odd 6 1 1183.2.a.c 2
39.i odd 6 1 91.2.f.a ✓ 4
39.i odd 6 1 1183.2.a.g 2
39.k even 12 2 1183.2.c.c 4
156.p even 6 1 1456.2.s.h 4
273.r even 6 1 637.2.g.c 4
273.s odd 6 1 637.2.g.b 4
273.u even 6 1 8281.2.a.n 2
273.bf even 6 1 637.2.h.f 4
273.bm odd 6 1 637.2.h.g 4
273.bn even 6 1 637.2.f.c 4
273.bn even 6 1 8281.2.a.bb 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
91.2.f.a ✓ 4 3.b odd 2 1
91.2.f.a ✓ 4 39.i odd 6 1
637.2.f.c 4 21.c even 2 1
637.2.f.c 4 273.bn even 6 1
637.2.g.b 4 21.h odd 6 1
637.2.g.b 4 273.s odd 6 1
637.2.g.c 4 21.g even 6 1
637.2.g.c 4 273.r even 6 1
637.2.h.f 4 21.g even 6 1
637.2.h.f 4 273.bf even 6 1
637.2.h.g 4 21.h odd 6 1
637.2.h.g 4 273.bm odd 6 1
819.2.o.c 4 1.a even 1 1 trivial
819.2.o.c 4 13.c even 3 1 inner
1183.2.a.c 2 39.h odd 6 1
1183.2.a.g 2 39.i odd 6 1
1183.2.c.c 4 39.k even 12 2
1456.2.s.h 4 12.b even 2 1
1456.2.s.h 4 156.p even 6 1
8281.2.a.n 2 273.u even 6 1
8281.2.a.bb 2 273.bn even 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(819, [\chi])\):

\( T_{2}^{4} - 3T_{2}^{3} + 8T_{2}^{2} - 3T_{2} + 1 \) Copy content Toggle raw display
\( T_{11}^{4} + 3T_{11}^{3} + 18T_{11}^{2} - 27T_{11} + 81 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} - 3 T^{3} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( (T^{2} + 3 T + 1)^{2} \) Copy content Toggle raw display
$7$ \( (T^{2} + T + 1)^{2} \) Copy content Toggle raw display
$11$ \( T^{4} + 3 T^{3} + \cdots + 81 \) Copy content Toggle raw display
$13$ \( (T^{2} + 5 T + 13)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} - 6 T^{3} + \cdots + 121 \) Copy content Toggle raw display
$19$ \( T^{4} + 3 T^{3} + \cdots + 81 \) Copy content Toggle raw display
$23$ \( T^{4} + 20T^{2} + 400 \) Copy content Toggle raw display
$29$ \( T^{4} - 3 T^{3} + \cdots + 841 \) Copy content Toggle raw display
$31$ \( (T^{2} - 4 T - 41)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} + 4 T + 16)^{2} \) Copy content Toggle raw display
$41$ \( T^{4} - 6 T^{3} + \cdots + 16 \) Copy content Toggle raw display
$43$ \( T^{4} + 5 T^{3} + \cdots + 9025 \) Copy content Toggle raw display
$47$ \( (T^{2} - 5)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} + 12 T + 31)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} + 5T^{2} + 25 \) Copy content Toggle raw display
$61$ \( (T^{2} - 6 T + 36)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} - 12 T^{3} + \cdots + 81 \) Copy content Toggle raw display
$71$ \( T^{4} + 6 T^{3} + \cdots + 13456 \) Copy content Toggle raw display
$73$ \( (T + 2)^{4} \) Copy content Toggle raw display
$79$ \( (T - 4)^{4} \) Copy content Toggle raw display
$83$ \( (T^{2} - 45)^{2} \) Copy content Toggle raw display
$89$ \( T^{4} - 21 T^{3} + \cdots + 6241 \) Copy content Toggle raw display
$97$ \( T^{4} + 31 T^{3} + \cdots + 52441 \) Copy content Toggle raw display
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