Properties

Label 494.2.g.a
Level $494$
Weight $2$
Character orbit 494.g
Analytic conductor $3.945$
Analytic rank $0$
Dimension $2$
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] = [494,2,Mod(191,494)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(494, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([4, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("494.191");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 494 = 2 \cdot 13 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 494.g (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: \(3.94460985985\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
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 primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \zeta_{6} + 1) q^{2} + (2 \zeta_{6} - 2) q^{3} - \zeta_{6} q^{4} - 3 q^{5} + 2 \zeta_{6} q^{6} - q^{8} - \zeta_{6} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + ( - \zeta_{6} + 1) q^{2} + (2 \zeta_{6} - 2) q^{3} - \zeta_{6} q^{4} - 3 q^{5} + 2 \zeta_{6} q^{6} - q^{8} - \zeta_{6} q^{9} + (3 \zeta_{6} - 3) q^{10} + ( - 6 \zeta_{6} + 6) q^{11} + 2 q^{12} + (\zeta_{6} + 3) q^{13} + ( - 6 \zeta_{6} + 6) q^{15} + (\zeta_{6} - 1) q^{16} - 7 \zeta_{6} q^{17} - q^{18} - \zeta_{6} q^{19} + 3 \zeta_{6} q^{20} - 6 \zeta_{6} q^{22} + ( - 4 \zeta_{6} + 4) q^{23} + ( - 2 \zeta_{6} + 2) q^{24} + 4 q^{25} + ( - 3 \zeta_{6} + 4) q^{26} - 4 q^{27} + ( - 9 \zeta_{6} + 9) q^{29} - 6 \zeta_{6} q^{30} + \zeta_{6} q^{32} + 12 \zeta_{6} q^{33} - 7 q^{34} + (\zeta_{6} - 1) q^{36} + (3 \zeta_{6} - 3) q^{37} - q^{38} + (6 \zeta_{6} - 8) q^{39} + 3 q^{40} + (5 \zeta_{6} - 5) q^{41} - 6 q^{44} + 3 \zeta_{6} q^{45} - 4 \zeta_{6} q^{46} - 2 q^{47} - 2 \zeta_{6} q^{48} + ( - 7 \zeta_{6} + 7) q^{49} + ( - 4 \zeta_{6} + 4) q^{50} + 14 q^{51} + ( - 4 \zeta_{6} + 1) q^{52} + 11 q^{53} + (4 \zeta_{6} - 4) q^{54} + (18 \zeta_{6} - 18) q^{55} + 2 q^{57} - 9 \zeta_{6} q^{58} + 10 \zeta_{6} q^{59} - 6 q^{60} - 13 \zeta_{6} q^{61} + q^{64} + ( - 3 \zeta_{6} - 9) q^{65} + 12 q^{66} + (10 \zeta_{6} - 10) q^{67} + (7 \zeta_{6} - 7) q^{68} + 8 \zeta_{6} q^{69} - 6 \zeta_{6} q^{71} + \zeta_{6} q^{72} - q^{73} + 3 \zeta_{6} q^{74} + (8 \zeta_{6} - 8) q^{75} + (\zeta_{6} - 1) q^{76} + (8 \zeta_{6} - 2) q^{78} - 10 q^{79} + ( - 3 \zeta_{6} + 3) q^{80} + ( - 11 \zeta_{6} + 11) q^{81} + 5 \zeta_{6} q^{82} - 6 q^{83} + 21 \zeta_{6} q^{85} + 18 \zeta_{6} q^{87} + (6 \zeta_{6} - 6) q^{88} + (2 \zeta_{6} - 2) q^{89} + 3 q^{90} - 4 q^{92} + (2 \zeta_{6} - 2) q^{94} + 3 \zeta_{6} q^{95} - 2 q^{96} - 18 \zeta_{6} q^{97} - 7 \zeta_{6} q^{98} - 6 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{2} - 2 q^{3} - q^{4} - 6 q^{5} + 2 q^{6} - 2 q^{8} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + q^{2} - 2 q^{3} - q^{4} - 6 q^{5} + 2 q^{6} - 2 q^{8} - q^{9} - 3 q^{10} + 6 q^{11} + 4 q^{12} + 7 q^{13} + 6 q^{15} - q^{16} - 7 q^{17} - 2 q^{18} - q^{19} + 3 q^{20} - 6 q^{22} + 4 q^{23} + 2 q^{24} + 8 q^{25} + 5 q^{26} - 8 q^{27} + 9 q^{29} - 6 q^{30} + q^{32} + 12 q^{33} - 14 q^{34} - q^{36} - 3 q^{37} - 2 q^{38} - 10 q^{39} + 6 q^{40} - 5 q^{41} - 12 q^{44} + 3 q^{45} - 4 q^{46} - 4 q^{47} - 2 q^{48} + 7 q^{49} + 4 q^{50} + 28 q^{51} - 2 q^{52} + 22 q^{53} - 4 q^{54} - 18 q^{55} + 4 q^{57} - 9 q^{58} + 10 q^{59} - 12 q^{60} - 13 q^{61} + 2 q^{64} - 21 q^{65} + 24 q^{66} - 10 q^{67} - 7 q^{68} + 8 q^{69} - 6 q^{71} + q^{72} - 2 q^{73} + 3 q^{74} - 8 q^{75} - q^{76} + 4 q^{78} - 20 q^{79} + 3 q^{80} + 11 q^{81} + 5 q^{82} - 12 q^{83} + 21 q^{85} + 18 q^{87} - 6 q^{88} - 2 q^{89} + 6 q^{90} - 8 q^{92} - 2 q^{94} + 3 q^{95} - 4 q^{96} - 18 q^{97} - 7 q^{98} - 12 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/494\mathbb{Z}\right)^\times\).

\(n\) \(287\) \(457\)
\(\chi(n)\) \(1\) \(-\zeta_{6}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
191.1
0.500000 + 0.866025i
0.500000 0.866025i
0.500000 0.866025i −1.00000 + 1.73205i −0.500000 0.866025i −3.00000 1.00000 + 1.73205i 0 −1.00000 −0.500000 0.866025i −1.50000 + 2.59808i
419.1 0.500000 + 0.866025i −1.00000 1.73205i −0.500000 + 0.866025i −3.00000 1.00000 1.73205i 0 −1.00000 −0.500000 + 0.866025i −1.50000 2.59808i
\(n\): 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 494.2.g.a 2
13.c even 3 1 inner 494.2.g.a 2
13.c even 3 1 6422.2.a.d 1
13.e even 6 1 6422.2.a.i 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
494.2.g.a 2 1.a even 1 1 trivial
494.2.g.a 2 13.c even 3 1 inner
6422.2.a.d 1 13.c even 3 1
6422.2.a.i 1 13.e even 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(494, [\chi])\):

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$3$ \( T^{2} + 2T + 4 \) Copy content Toggle raw display
$5$ \( (T + 3)^{2} \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} - 6T + 36 \) Copy content Toggle raw display
$13$ \( T^{2} - 7T + 13 \) Copy content Toggle raw display
$17$ \( T^{2} + 7T + 49 \) Copy content Toggle raw display
$19$ \( T^{2} + T + 1 \) Copy content Toggle raw display
$23$ \( T^{2} - 4T + 16 \) Copy content Toggle raw display
$29$ \( T^{2} - 9T + 81 \) Copy content Toggle raw display
$31$ \( T^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 3T + 9 \) Copy content Toggle raw display
$41$ \( T^{2} + 5T + 25 \) Copy content Toggle raw display
$43$ \( T^{2} \) Copy content Toggle raw display
$47$ \( (T + 2)^{2} \) Copy content Toggle raw display
$53$ \( (T - 11)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} - 10T + 100 \) Copy content Toggle raw display
$61$ \( T^{2} + 13T + 169 \) Copy content Toggle raw display
$67$ \( T^{2} + 10T + 100 \) Copy content Toggle raw display
$71$ \( T^{2} + 6T + 36 \) Copy content Toggle raw display
$73$ \( (T + 1)^{2} \) Copy content Toggle raw display
$79$ \( (T + 10)^{2} \) Copy content Toggle raw display
$83$ \( (T + 6)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} + 2T + 4 \) Copy content Toggle raw display
$97$ \( T^{2} + 18T + 324 \) Copy content Toggle raw display
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