Properties

Label 819.2.o.e
Level $819$
Weight $2$
Character orbit 819.o
Analytic conductor $6.540$
Analytic rank $0$
Dimension $6$
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: \(6\)
Relative dimension: \(3\) over \(\Q(\zeta_{3})\)
Coefficient field: 6.0.64827.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - x^{5} + 3x^{4} + 5x^{2} - 2x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 273)
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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{5} + \beta_{4} - 1) q^{2} + ( - \beta_{5} - 2 \beta_{4} - 2 \beta_{3} + \beta_{2} - \beta_1) q^{4} + ( - \beta_{3} - \beta_{2}) q^{5} - \beta_{5} q^{7} + (\beta_{3} - 2 \beta_{2} + 2) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{5} + \beta_{4} - 1) q^{2} + ( - \beta_{5} - 2 \beta_{4} - 2 \beta_{3} + \beta_{2} - \beta_1) q^{4} + ( - \beta_{3} - \beta_{2}) q^{5} - \beta_{5} q^{7} + (\beta_{3} - 2 \beta_{2} + 2) q^{8} + ( - 3 \beta_{5} - \beta_{4} - \beta_1 + 3) q^{10} + (2 \beta_{5} + 3 \beta_{4} + \beta_1 - 2) q^{11} + ( - 2 \beta_{5} - \beta_{4} - 2 \beta_{3} + 3 \beta_{2} - 3 \beta_1 - 1) q^{13} + (\beta_{3} + 1) q^{14} + ( - \beta_{4} - \beta_1) q^{16} + ( - \beta_{5} - \beta_{4} - \beta_{3}) q^{17} + (5 \beta_{5} + 2 \beta_{4} + 2 \beta_{3}) q^{19} + (4 \beta_{5} + 2 \beta_{4} + 2 \beta_{3} - 3 \beta_{2} + 3 \beta_1) q^{20} + ( - 7 \beta_{5} - 5 \beta_{4} - 5 \beta_{3} + 3 \beta_{2} - 3 \beta_1) q^{22} + ( - 4 \beta_{5} - 4 \beta_{4} - 5 \beta_1 + 4) q^{23} + ( - \beta_{3} - 2 \beta_{2}) q^{25} + ( - 3 \beta_{5} - 2 \beta_{4} + 3 \beta_{3} - \beta_{2} - \beta_1 + 4) q^{26} + (\beta_{5} + 2 \beta_{4} + \beta_1 - 1) q^{28} + ( - 4 \beta_{5} - 5 \beta_{4} - 6 \beta_1 + 4) q^{29} + (5 \beta_{3} - 3 \beta_{2} - 1) q^{31} + (5 \beta_{5} + 3 \beta_{4} + 3 \beta_{3} - 5 \beta_{2} + 5 \beta_1) q^{32} + (2 \beta_{3} - \beta_{2} + 3) q^{34} + (\beta_{4} + \beta_{3} + \beta_{2} - \beta_1) q^{35} + ( - 2 \beta_{5} - 5 \beta_{4} + 3 \beta_1 + 2) q^{37} + ( - 7 \beta_{3} + 2 \beta_{2} - 9) q^{38} + ( - 4 \beta_{3} + 1) q^{40} + ( - 2 \beta_{5} - 4 \beta_{4} + 2) q^{41} + (6 \beta_{5} + 2 \beta_{4} + 2 \beta_{3} - 5 \beta_{2} + 5 \beta_1) q^{43} + (6 \beta_{3} - 3 \beta_{2} + 10) q^{44} + (7 \beta_{5} + 8 \beta_{4} + 8 \beta_{3} - 4 \beta_{2} + 4 \beta_1) q^{46} + ( - 3 \beta_{3} + \beta_{2} + 5) q^{47} + (\beta_{5} - 1) q^{49} + ( - 4 \beta_{5} - \beta_{4} - \beta_1 + 4) q^{50} + (6 \beta_{5} + 5 \beta_{4} + 3 \beta_{3} - 2 \beta_{2} - \beta_1 - 2) q^{52} + (4 \beta_{3} - 4 \beta_{2} + 1) q^{53} + ( - 7 \beta_{5} - 3 \beta_{4} - 4 \beta_1 + 7) q^{55} + ( - 2 \beta_{5} - \beta_{4} - \beta_{3} + 2 \beta_{2} - 2 \beta_1) q^{56} + (8 \beta_{5} + 9 \beta_{4} + 9 \beta_{3} - 5 \beta_{2} + 5 \beta_1) q^{58} + ( - 6 \beta_{5} - 2 \beta_{4} - 2 \beta_{3} + 5 \beta_{2} - 5 \beta_1) q^{59} + ( - \beta_{5} - 2 \beta_{4} - 2 \beta_{3} - 6 \beta_{2} + 6 \beta_1) q^{61} + (6 \beta_{5} + 4 \beta_{4} + 5 \beta_1 - 6) q^{62} + ( - 6 \beta_{3} + \beta_{2} - 6) q^{64} + ( - 3 \beta_{5} + 5 \beta_{4} + 6 \beta_{3} - \beta_{2} + 3) q^{65} + ( - 6 \beta_{5} + \beta_{4} - 4 \beta_1 + 6) q^{67} + (4 \beta_{5} + 3 \beta_{4} + 2 \beta_1 - 4) q^{68} + ( - \beta_{3} + \beta_{2} - 3) q^{70} + ( - 9 \beta_{5} - 2 \beta_{4} - 2 \beta_{3} + 7 \beta_{2} - 7 \beta_1) q^{71} + ( - 2 \beta_{3} + 7 \beta_{2} - 5) q^{73} + (15 \beta_{5} + 7 \beta_{4} + 7 \beta_{3} - 5 \beta_{2} + 5 \beta_1) q^{74} + ( - 11 \beta_{5} - 12 \beta_{4} - 7 \beta_1 + 11) q^{76} + (3 \beta_{3} - \beta_{2} + 2) q^{77} + (3 \beta_{3} - 3 \beta_{2} + 3) q^{79} + (\beta_{5} + \beta_{4} + 2 \beta_1 - 1) q^{80} + (10 \beta_{5} + 6 \beta_{4} + 6 \beta_{3} - 4 \beta_{2} + 4 \beta_1) q^{82} + ( - \beta_{3} + 7 \beta_{2} - 2) q^{83} + (3 \beta_{5} + \beta_{4} + \beta_{3} - \beta_{2} + \beta_1) q^{85} + ( - 8 \beta_{3} + 2 \beta_{2} - 5) q^{86} + (5 \beta_{5} + 6 \beta_{4} - 5) q^{88} + (5 \beta_{5} - 2 \beta_1 - 5) q^{89} + (3 \beta_{5} + 2 \beta_{4} + \beta_{3} + 3 \beta_1 - 2) q^{91} + ( - 7 \beta_{3} - 2 \beta_{2} - 11) q^{92} + (2 \beta_{4} - 3 \beta_1) q^{94} + ( - 6 \beta_{5} - 5 \beta_{4} - 5 \beta_{3} - \beta_{2} + \beta_1) q^{95} + (\beta_{5} - 3 \beta_{4} - 3 \beta_{3} + 7 \beta_{2} - 7 \beta_1) q^{97} + ( - \beta_{5} - \beta_{4} - \beta_{3}) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 2 q^{2} - 3 q^{7} + 6 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 2 q^{2} - 3 q^{7} + 6 q^{8} + 7 q^{10} - 2 q^{11} - 6 q^{13} + 4 q^{14} - 2 q^{16} - 2 q^{17} + 13 q^{19} + 7 q^{20} - 13 q^{22} + 3 q^{23} - 2 q^{25} + 4 q^{26} + q^{29} - 22 q^{31} + 7 q^{32} + 12 q^{34} + 4 q^{37} - 36 q^{38} + 14 q^{40} + 2 q^{41} + 11 q^{43} + 42 q^{44} + 9 q^{46} + 38 q^{47} - 3 q^{49} + 10 q^{50} - 10 q^{53} + 14 q^{55} - 3 q^{56} + 10 q^{58} - 11 q^{59} - 7 q^{61} - 9 q^{62} - 22 q^{64} + 15 q^{67} - 7 q^{68} - 14 q^{70} - 18 q^{71} - 12 q^{73} + 33 q^{74} + 14 q^{76} + 4 q^{77} + 6 q^{79} + 20 q^{82} + 4 q^{83} + 7 q^{85} - 10 q^{86} - 9 q^{88} - 17 q^{89} - 56 q^{92} - q^{94} - 14 q^{95} + 13 q^{97} - 2 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - x^{5} + 3x^{4} + 5x^{2} - 2x + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{5} + 3\nu^{4} - 9\nu^{3} + 5\nu^{2} - 2\nu + 6 ) / 13 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -3\nu^{5} + 9\nu^{4} - 14\nu^{3} + 15\nu^{2} - 6\nu + 18 ) / 13 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -4\nu^{5} - \nu^{4} - 10\nu^{3} - 6\nu^{2} - 34\nu - 2 ) / 13 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -6\nu^{5} + 5\nu^{4} - 15\nu^{3} - 9\nu^{2} - 25\nu + 10 ) / 13 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{5} + \beta_{4} + \beta_{3} - \beta_{2} + \beta_1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} - 3\beta_{2} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 2\beta_{5} - 3\beta_{4} - 4\beta _1 - 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{5} - 4\beta_{4} - 4\beta_{3} + 9\beta_{2} - 9\beta_1 \) Copy content Toggle raw display

Character values

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

\(n\) \(92\) \(379\) \(703\)
\(\chi(n)\) \(1\) \(-\beta_{5}\) \(1\)

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
0.222521 0.385418i
0.900969 1.56052i
−0.623490 + 1.07992i
0.222521 + 0.385418i
0.900969 + 1.56052i
−0.623490 1.07992i
−1.12349 + 1.94594i 0 −1.52446 2.64044i −1.69202 0 −0.500000 0.866025i 2.35690 0 1.90097 3.29257i
568.2 −0.277479 + 0.480608i 0 0.846011 + 1.46533i −1.35690 0 −0.500000 0.866025i −2.04892 0 0.376510 0.652135i
568.3 0.400969 0.694498i 0 0.678448 + 1.17511i 3.04892 0 −0.500000 0.866025i 2.69202 0 1.22252 2.11747i
757.1 −1.12349 1.94594i 0 −1.52446 + 2.64044i −1.69202 0 −0.500000 + 0.866025i 2.35690 0 1.90097 + 3.29257i
757.2 −0.277479 0.480608i 0 0.846011 1.46533i −1.35690 0 −0.500000 + 0.866025i −2.04892 0 0.376510 + 0.652135i
757.3 0.400969 + 0.694498i 0 0.678448 1.17511i 3.04892 0 −0.500000 + 0.866025i 2.69202 0 1.22252 + 2.11747i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 568.3
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.e 6
3.b odd 2 1 273.2.k.c 6
13.c even 3 1 inner 819.2.o.e 6
39.h odd 6 1 3549.2.a.u 3
39.i odd 6 1 273.2.k.c 6
39.i odd 6 1 3549.2.a.i 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
273.2.k.c 6 3.b odd 2 1
273.2.k.c 6 39.i odd 6 1
819.2.o.e 6 1.a even 1 1 trivial
819.2.o.e 6 13.c even 3 1 inner
3549.2.a.i 3 39.i odd 6 1
3549.2.a.u 3 39.h odd 6 1

Hecke kernels

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

\( T_{2}^{6} + 2T_{2}^{5} + 5T_{2}^{4} + 3T_{2}^{2} + T_{2} + 1 \) Copy content Toggle raw display
\( T_{11}^{6} + 2T_{11}^{5} + 19T_{11}^{4} - 56T_{11}^{3} + 199T_{11}^{2} - 195T_{11} + 169 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} + 2 T^{5} + 5 T^{4} + 3 T^{2} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( T^{6} \) Copy content Toggle raw display
$5$ \( (T^{3} - 7 T - 7)^{2} \) Copy content Toggle raw display
$7$ \( (T^{2} + T + 1)^{3} \) Copy content Toggle raw display
$11$ \( T^{6} + 2 T^{5} + 19 T^{4} - 56 T^{3} + \cdots + 169 \) Copy content Toggle raw display
$13$ \( T^{6} + 6 T^{5} + 30 T^{4} + \cdots + 2197 \) Copy content Toggle raw display
$17$ \( T^{6} + 2 T^{5} + 5 T^{4} + 3 T^{2} + \cdots + 1 \) Copy content Toggle raw display
$19$ \( T^{6} - 13 T^{5} + 122 T^{4} + \cdots + 1849 \) Copy content Toggle raw display
$23$ \( T^{6} - 3 T^{5} + 55 T^{4} + \cdots + 19321 \) Copy content Toggle raw display
$29$ \( T^{6} - T^{5} + 73 T^{4} - 266 T^{3} + \cdots + 28561 \) Copy content Toggle raw display
$31$ \( (T^{3} + 11 T^{2} - 4 T - 211)^{2} \) Copy content Toggle raw display
$37$ \( T^{6} - 4 T^{5} + 125 T^{4} + \cdots + 284089 \) Copy content Toggle raw display
$41$ \( T^{6} - 2 T^{5} + 40 T^{4} + 56 T^{3} + \cdots + 64 \) Copy content Toggle raw display
$43$ \( T^{6} - 11 T^{5} + 125 T^{4} + \cdots + 44521 \) Copy content Toggle raw display
$47$ \( (T^{3} - 19 T^{2} + 104 T - 127)^{2} \) Copy content Toggle raw display
$53$ \( (T^{3} + 5 T^{2} - 29 T - 41)^{2} \) Copy content Toggle raw display
$59$ \( T^{6} + 11 T^{5} + 125 T^{4} + \cdots + 44521 \) Copy content Toggle raw display
$61$ \( T^{6} + 7 T^{5} + 154 T^{4} + \cdots + 461041 \) Copy content Toggle raw display
$67$ \( T^{6} - 15 T^{5} + 199 T^{4} + \cdots + 841 \) Copy content Toggle raw display
$71$ \( T^{6} + 18 T^{5} + 307 T^{4} + \cdots + 284089 \) Copy content Toggle raw display
$73$ \( (T^{3} + 6 T^{2} - 79 T - 377)^{2} \) Copy content Toggle raw display
$79$ \( (T^{3} - 3 T^{2} - 18 T + 27)^{2} \) Copy content Toggle raw display
$83$ \( (T^{3} - 2 T^{2} - 99 T - 13)^{2} \) Copy content Toggle raw display
$89$ \( T^{6} + 17 T^{5} + 202 T^{4} + \cdots + 16129 \) Copy content Toggle raw display
$97$ \( T^{6} - 13 T^{5} + 199 T^{4} + \cdots + 169 \) Copy content Toggle raw display
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