Properties

Label 490.2.e.c
Level $490$
Weight $2$
Character orbit 490.e
Analytic conductor $3.913$
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] = [490,2,Mod(361,490)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(490, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([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("490.361");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 490 = 2 \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 490.e (of order \(3\), degree \(2\), not 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: \(3.91266969904\)
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: no (minimal twist has level 70)
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.

\(f(q)\) \(=\) \( q - \zeta_{6} q^{2} + (\zeta_{6} - 1) q^{4} - \zeta_{6} q^{5} + q^{8} + 3 \zeta_{6} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - \zeta_{6} q^{2} + (\zeta_{6} - 1) q^{4} - \zeta_{6} q^{5} + q^{8} + 3 \zeta_{6} q^{9} + (\zeta_{6} - 1) q^{10} + (4 \zeta_{6} - 4) q^{11} + 6 q^{13} - \zeta_{6} q^{16} + ( - 2 \zeta_{6} + 2) q^{17} + ( - 3 \zeta_{6} + 3) q^{18} + q^{20} + 4 q^{22} + (\zeta_{6} - 1) q^{25} - 6 \zeta_{6} q^{26} + 6 q^{29} + ( - 8 \zeta_{6} + 8) q^{31} + (\zeta_{6} - 1) q^{32} - 2 q^{34} - 3 q^{36} + 10 \zeta_{6} q^{37} - \zeta_{6} q^{40} - 2 q^{41} + 4 q^{43} - 4 \zeta_{6} q^{44} + ( - 3 \zeta_{6} + 3) q^{45} + 8 \zeta_{6} q^{47} + q^{50} + (6 \zeta_{6} - 6) q^{52} + ( - 2 \zeta_{6} + 2) q^{53} + 4 q^{55} - 6 \zeta_{6} q^{58} + (8 \zeta_{6} - 8) q^{59} - 14 \zeta_{6} q^{61} - 8 q^{62} + q^{64} - 6 \zeta_{6} q^{65} + ( - 12 \zeta_{6} + 12) q^{67} + 2 \zeta_{6} q^{68} - 16 q^{71} + 3 \zeta_{6} q^{72} + ( - 2 \zeta_{6} + 2) q^{73} + ( - 10 \zeta_{6} + 10) q^{74} + 8 \zeta_{6} q^{79} + (\zeta_{6} - 1) q^{80} + (9 \zeta_{6} - 9) q^{81} + 2 \zeta_{6} q^{82} - 8 q^{83} - 2 q^{85} - 4 \zeta_{6} q^{86} + (4 \zeta_{6} - 4) q^{88} + 10 \zeta_{6} q^{89} - 3 q^{90} + ( - 8 \zeta_{6} + 8) q^{94} - 2 q^{97} - 12 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{2} - q^{4} - q^{5} + 2 q^{8} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - q^{2} - q^{4} - q^{5} + 2 q^{8} + 3 q^{9} - q^{10} - 4 q^{11} + 12 q^{13} - q^{16} + 2 q^{17} + 3 q^{18} + 2 q^{20} + 8 q^{22} - q^{25} - 6 q^{26} + 12 q^{29} + 8 q^{31} - q^{32} - 4 q^{34} - 6 q^{36} + 10 q^{37} - q^{40} - 4 q^{41} + 8 q^{43} - 4 q^{44} + 3 q^{45} + 8 q^{47} + 2 q^{50} - 6 q^{52} + 2 q^{53} + 8 q^{55} - 6 q^{58} - 8 q^{59} - 14 q^{61} - 16 q^{62} + 2 q^{64} - 6 q^{65} + 12 q^{67} + 2 q^{68} - 32 q^{71} + 3 q^{72} + 2 q^{73} + 10 q^{74} + 8 q^{79} - q^{80} - 9 q^{81} + 2 q^{82} - 16 q^{83} - 4 q^{85} - 4 q^{86} - 4 q^{88} + 10 q^{89} - 6 q^{90} + 8 q^{94} - 4 q^{97} - 24 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/490\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(197\)
\(\chi(n)\) \(-\zeta_{6}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
361.1
0.500000 + 0.866025i
0.500000 0.866025i
−0.500000 0.866025i 0 −0.500000 + 0.866025i −0.500000 0.866025i 0 0 1.00000 1.50000 + 2.59808i −0.500000 + 0.866025i
471.1 −0.500000 + 0.866025i 0 −0.500000 0.866025i −0.500000 + 0.866025i 0 0 1.00000 1.50000 2.59808i −0.500000 0.866025i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 490.2.e.c 2
7.b odd 2 1 490.2.e.d 2
7.c even 3 1 490.2.a.h 1
7.c even 3 1 inner 490.2.e.c 2
7.d odd 6 1 70.2.a.a 1
7.d odd 6 1 490.2.e.d 2
21.g even 6 1 630.2.a.d 1
21.h odd 6 1 4410.2.a.b 1
28.f even 6 1 560.2.a.d 1
28.g odd 6 1 3920.2.a.t 1
35.i odd 6 1 350.2.a.b 1
35.j even 6 1 2450.2.a.l 1
35.k even 12 2 350.2.c.b 2
35.l odd 12 2 2450.2.c.k 2
56.j odd 6 1 2240.2.a.n 1
56.m even 6 1 2240.2.a.q 1
77.i even 6 1 8470.2.a.j 1
84.j odd 6 1 5040.2.a.bm 1
105.p even 6 1 3150.2.a.bj 1
105.w odd 12 2 3150.2.g.c 2
140.s even 6 1 2800.2.a.m 1
140.x odd 12 2 2800.2.g.n 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
70.2.a.a 1 7.d odd 6 1
350.2.a.b 1 35.i odd 6 1
350.2.c.b 2 35.k even 12 2
490.2.a.h 1 7.c even 3 1
490.2.e.c 2 1.a even 1 1 trivial
490.2.e.c 2 7.c even 3 1 inner
490.2.e.d 2 7.b odd 2 1
490.2.e.d 2 7.d odd 6 1
560.2.a.d 1 28.f even 6 1
630.2.a.d 1 21.g even 6 1
2240.2.a.n 1 56.j odd 6 1
2240.2.a.q 1 56.m even 6 1
2450.2.a.l 1 35.j even 6 1
2450.2.c.k 2 35.l odd 12 2
2800.2.a.m 1 140.s even 6 1
2800.2.g.n 2 140.x odd 12 2
3150.2.a.bj 1 105.p even 6 1
3150.2.g.c 2 105.w odd 12 2
3920.2.a.t 1 28.g odd 6 1
4410.2.a.b 1 21.h odd 6 1
5040.2.a.bm 1 84.j odd 6 1
8470.2.a.j 1 77.i even 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(490, [\chi])\):

\( T_{3} \) Copy content Toggle raw display
\( T_{11}^{2} + 4T_{11} + 16 \) Copy content Toggle raw display
\( T_{13} - 6 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + T + 1 \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} + T + 1 \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + 4T + 16 \) Copy content Toggle raw display
$13$ \( (T - 6)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} - 2T + 4 \) Copy content Toggle raw display
$19$ \( T^{2} \) Copy content Toggle raw display
$23$ \( T^{2} \) Copy content Toggle raw display
$29$ \( (T - 6)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} - 8T + 64 \) Copy content Toggle raw display
$37$ \( T^{2} - 10T + 100 \) Copy content Toggle raw display
$41$ \( (T + 2)^{2} \) Copy content Toggle raw display
$43$ \( (T - 4)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} - 8T + 64 \) Copy content Toggle raw display
$53$ \( T^{2} - 2T + 4 \) Copy content Toggle raw display
$59$ \( T^{2} + 8T + 64 \) Copy content Toggle raw display
$61$ \( T^{2} + 14T + 196 \) Copy content Toggle raw display
$67$ \( T^{2} - 12T + 144 \) Copy content Toggle raw display
$71$ \( (T + 16)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} - 2T + 4 \) Copy content Toggle raw display
$79$ \( T^{2} - 8T + 64 \) Copy content Toggle raw display
$83$ \( (T + 8)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} - 10T + 100 \) Copy content Toggle raw display
$97$ \( (T + 2)^{2} \) Copy content Toggle raw display
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