Properties

Label 819.2.a.i
Level $819$
Weight $2$
Character orbit 819.a
Self dual yes
Analytic conductor $6.540$
Analytic rank $1$
Dimension $3$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [819,2,Mod(1,819)]
 
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([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("819.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 819 = 3^{2} \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 819.a (trivial)

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: yes
Analytic conductor: \(6.53974792554\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.316.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 4x + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 91)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

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

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + (\beta_{2} + 1) q^{4} + (\beta_1 - 1) q^{5} - q^{7} + ( - \beta_{2} - 1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} + (\beta_{2} + 1) q^{4} + (\beta_1 - 1) q^{5} - q^{7} + ( - \beta_{2} - 1) q^{8} + ( - \beta_{2} + \beta_1 - 3) q^{10} + ( - \beta_{2} + \beta_1 - 1) q^{11} + q^{13} + \beta_1 q^{14} + ( - \beta_{2} + 2 \beta_1 - 1) q^{16} + ( - \beta_{2} - \beta_1 - 1) q^{17} + ( - \beta_1 - 1) q^{19} + 2 \beta_1 q^{20} + (2 \beta_1 - 2) q^{22} + (\beta_{2} + 2 \beta_1 - 4) q^{23} + (\beta_{2} - 2 \beta_1 - 1) q^{25} - \beta_1 q^{26} + ( - \beta_{2} - 1) q^{28} + ( - \beta_{2} - 8) q^{29} + (2 \beta_{2} - \beta_1 - 1) q^{31} + (\beta_{2} + 2 \beta_1 - 3) q^{32} + (2 \beta_{2} + 2 \beta_1 + 4) q^{34} + ( - \beta_1 + 1) q^{35} + (\beta_{2} + 3 \beta_1 - 1) q^{37} + (\beta_{2} + \beta_1 + 3) q^{38} - 2 \beta_1 q^{40} + (2 \beta_{2} - 2 \beta_1) q^{41} + ( - 3 \beta_{2} - 2 \beta_1 + 4) q^{43} - 4 q^{44} + ( - 3 \beta_{2} + 3 \beta_1 - 7) q^{46} + (4 \beta_{2} - \beta_1 + 3) q^{47} + q^{49} + (\beta_{2} + 5) q^{50} + (\beta_{2} + 1) q^{52} + (3 \beta_{2} - 2 \beta_1 - 2) q^{53} + (\beta_{2} - 3 \beta_1 + 3) q^{55} + (\beta_{2} + 1) q^{56} + (\beta_{2} + 9 \beta_1 + 1) q^{58} + ( - 4 \beta_{2} - 2 \beta_1 + 2) q^{59} - 2 q^{61} + ( - \beta_{2} - \beta_1 + 1) q^{62} + ( - \beta_{2} - 2 \beta_1 - 5) q^{64} + (\beta_1 - 1) q^{65} + (4 \beta_{2} - 6 \beta_1 - 2) q^{67} + ( - 2 \beta_{2} - 4 \beta_1 - 6) q^{68} + (\beta_{2} - \beta_1 + 3) q^{70} + (\beta_{2} - 3 \beta_1 + 3) q^{71} + ( - 4 \beta_{2} - \beta_1 - 3) q^{73} + ( - 4 \beta_{2} - 10) q^{74} + ( - 2 \beta_{2} - 2 \beta_1 - 2) q^{76} + (\beta_{2} - \beta_1 + 1) q^{77} + ( - \beta_{2} + 4 \beta_1 - 6) q^{79} + (2 \beta_{2} - 4 \beta_1 + 6) q^{80} + ( - 2 \beta_1 + 4) q^{82} + ( - 4 \beta_{2} + 9 \beta_1 + 1) q^{83} + ( - \beta_{2} - \beta_1 - 3) q^{85} + (5 \beta_{2} - \beta_1 + 9) q^{86} + 4 q^{88} + (2 \beta_{2} - 5 \beta_1 + 1) q^{89} - q^{91} + ( - 2 \beta_{2} + 6 \beta_1 + 2) q^{92} + ( - 3 \beta_{2} - 7 \beta_1 - 1) q^{94} + ( - \beta_{2} - 2) q^{95} + ( - \beta_1 - 3) q^{97} - \beta_1 q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - q^{2} + 3 q^{4} - 2 q^{5} - 3 q^{7} - 3 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - q^{2} + 3 q^{4} - 2 q^{5} - 3 q^{7} - 3 q^{8} - 8 q^{10} - 2 q^{11} + 3 q^{13} + q^{14} - q^{16} - 4 q^{17} - 4 q^{19} + 2 q^{20} - 4 q^{22} - 10 q^{23} - 5 q^{25} - q^{26} - 3 q^{28} - 24 q^{29} - 4 q^{31} - 7 q^{32} + 14 q^{34} + 2 q^{35} + 10 q^{38} - 2 q^{40} - 2 q^{41} + 10 q^{43} - 12 q^{44} - 18 q^{46} + 8 q^{47} + 3 q^{49} + 15 q^{50} + 3 q^{52} - 8 q^{53} + 6 q^{55} + 3 q^{56} + 12 q^{58} + 4 q^{59} - 6 q^{61} + 2 q^{62} - 17 q^{64} - 2 q^{65} - 12 q^{67} - 22 q^{68} + 8 q^{70} + 6 q^{71} - 10 q^{73} - 30 q^{74} - 8 q^{76} + 2 q^{77} - 14 q^{79} + 14 q^{80} + 10 q^{82} + 12 q^{83} - 10 q^{85} + 26 q^{86} + 12 q^{88} - 2 q^{89} - 3 q^{91} + 12 q^{92} - 10 q^{94} - 6 q^{95} - 10 q^{97} - q^{98}+O(q^{100}) \) Copy content Toggle raw display

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

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
2.34292
0.470683
−1.81361
−2.34292 0 3.48929 1.34292 0 −1.00000 −3.48929 0 −3.14637
1.2 −0.470683 0 −1.77846 −0.529317 0 −1.00000 1.77846 0 0.249141
1.3 1.81361 0 1.28917 −2.81361 0 −1.00000 −1.28917 0 −5.10278
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(7\) \(1\)
\(13\) \(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 819.2.a.i 3
3.b odd 2 1 91.2.a.d 3
7.b odd 2 1 5733.2.a.x 3
12.b even 2 1 1456.2.a.t 3
15.d odd 2 1 2275.2.a.m 3
21.c even 2 1 637.2.a.j 3
21.g even 6 2 637.2.e.i 6
21.h odd 6 2 637.2.e.j 6
24.f even 2 1 5824.2.a.bs 3
24.h odd 2 1 5824.2.a.by 3
39.d odd 2 1 1183.2.a.i 3
39.f even 4 2 1183.2.c.f 6
273.g even 2 1 8281.2.a.bg 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
91.2.a.d 3 3.b odd 2 1
637.2.a.j 3 21.c even 2 1
637.2.e.i 6 21.g even 6 2
637.2.e.j 6 21.h odd 6 2
819.2.a.i 3 1.a even 1 1 trivial
1183.2.a.i 3 39.d odd 2 1
1183.2.c.f 6 39.f even 4 2
1456.2.a.t 3 12.b even 2 1
2275.2.a.m 3 15.d odd 2 1
5733.2.a.x 3 7.b odd 2 1
5824.2.a.bs 3 24.f even 2 1
5824.2.a.by 3 24.h odd 2 1
8281.2.a.bg 3 273.g even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(819))\):

\( T_{2}^{3} + T_{2}^{2} - 4T_{2} - 2 \) Copy content Toggle raw display
\( T_{5}^{3} + 2T_{5}^{2} - 3T_{5} - 2 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} + T^{2} - 4T - 2 \) Copy content Toggle raw display
$3$ \( T^{3} \) Copy content Toggle raw display
$5$ \( T^{3} + 2 T^{2} - 3 T - 2 \) Copy content Toggle raw display
$7$ \( (T + 1)^{3} \) Copy content Toggle raw display
$11$ \( T^{3} + 2 T^{2} - 6 T - 8 \) Copy content Toggle raw display
$13$ \( (T - 1)^{3} \) Copy content Toggle raw display
$17$ \( T^{3} + 4 T^{2} - 10 T + 4 \) Copy content Toggle raw display
$19$ \( T^{3} + 4T^{2} + T - 4 \) Copy content Toggle raw display
$23$ \( T^{3} + 10 T^{2} + T - 136 \) Copy content Toggle raw display
$29$ \( T^{3} + 24 T^{2} + 185 T + 454 \) Copy content Toggle raw display
$31$ \( T^{3} + 4 T^{2} - 19 T + 16 \) Copy content Toggle raw display
$37$ \( T^{3} - 58T - 124 \) Copy content Toggle raw display
$41$ \( T^{3} + 2 T^{2} - 28 T + 8 \) Copy content Toggle raw display
$43$ \( T^{3} - 10 T^{2} - 71 T + 628 \) Copy content Toggle raw display
$47$ \( T^{3} - 8 T^{2} - 79 T + 544 \) Copy content Toggle raw display
$53$ \( T^{3} + 8 T^{2} - 35 T + 22 \) Copy content Toggle raw display
$59$ \( T^{3} - 4 T^{2} - 156 T + 688 \) Copy content Toggle raw display
$61$ \( (T + 2)^{3} \) Copy content Toggle raw display
$67$ \( T^{3} + 12 T^{2} - 124 T - 976 \) Copy content Toggle raw display
$71$ \( T^{3} - 6 T^{2} - 22 T - 16 \) Copy content Toggle raw display
$73$ \( T^{3} + 10 T^{2} - 99 T - 274 \) Copy content Toggle raw display
$79$ \( T^{3} + 14 T^{2} + 5 T - 16 \) Copy content Toggle raw display
$83$ \( T^{3} - 12 T^{2} - 271 T + 3268 \) Copy content Toggle raw display
$89$ \( T^{3} + 2 T^{2} - 95 T - 422 \) Copy content Toggle raw display
$97$ \( T^{3} + 10 T^{2} + 29 T + 22 \) Copy content Toggle raw display
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