Properties

Label 490.6.a.m
Level $490$
Weight $6$
Character orbit 490.a
Self dual yes
Analytic conductor $78.588$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [490,6,Mod(1,490)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(490, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("490.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 490 = 2 \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 490.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: \(78.5880717084\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 70)
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)
 
\(f(q)\) \(=\) \( q + 4 q^{2} + 17 q^{3} + 16 q^{4} - 25 q^{5} + 68 q^{6} + 64 q^{8} + 46 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 4 q^{2} + 17 q^{3} + 16 q^{4} - 25 q^{5} + 68 q^{6} + 64 q^{8} + 46 q^{9} - 100 q^{10} - 715 q^{11} + 272 q^{12} - 331 q^{13} - 425 q^{15} + 256 q^{16} + 1699 q^{17} + 184 q^{18} + 1718 q^{19} - 400 q^{20} - 2860 q^{22} - 3950 q^{23} + 1088 q^{24} + 625 q^{25} - 1324 q^{26} - 3349 q^{27} + 4579 q^{29} - 1700 q^{30} - 6756 q^{31} + 1024 q^{32} - 12155 q^{33} + 6796 q^{34} + 736 q^{36} - 16518 q^{37} + 6872 q^{38} - 5627 q^{39} - 1600 q^{40} - 18876 q^{41} + 2258 q^{43} - 11440 q^{44} - 1150 q^{45} - 15800 q^{46} + 537 q^{47} + 4352 q^{48} + 2500 q^{50} + 28883 q^{51} - 5296 q^{52} - 10984 q^{53} - 13396 q^{54} + 17875 q^{55} + 29206 q^{57} + 18316 q^{58} + 25956 q^{59} - 6800 q^{60} - 39188 q^{61} - 27024 q^{62} + 4096 q^{64} + 8275 q^{65} - 48620 q^{66} + 4416 q^{67} + 27184 q^{68} - 67150 q^{69} - 31880 q^{71} + 2944 q^{72} + 5018 q^{73} - 66072 q^{74} + 10625 q^{75} + 27488 q^{76} - 22508 q^{78} - 27977 q^{79} - 6400 q^{80} - 68111 q^{81} - 75504 q^{82} - 37644 q^{83} - 42475 q^{85} + 9032 q^{86} + 77843 q^{87} - 45760 q^{88} + 17216 q^{89} - 4600 q^{90} - 63200 q^{92} - 114852 q^{93} + 2148 q^{94} - 42950 q^{95} + 17408 q^{96} + 63175 q^{97} - 32890 q^{99}+O(q^{100}) \) 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
0
4.00000 17.0000 16.0000 −25.0000 68.0000 0 64.0000 46.0000 −100.000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( -1 \)
\(5\) \( +1 \)
\(7\) \( -1 \)

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 490.6.a.m 1
7.b odd 2 1 70.6.a.e 1
21.c even 2 1 630.6.a.b 1
28.d even 2 1 560.6.a.h 1
35.c odd 2 1 350.6.a.e 1
35.f even 4 2 350.6.c.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
70.6.a.e 1 7.b odd 2 1
350.6.a.e 1 35.c odd 2 1
350.6.c.a 2 35.f even 4 2
490.6.a.m 1 1.a even 1 1 trivial
560.6.a.h 1 28.d even 2 1
630.6.a.b 1 21.c even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} - 17 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(490))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T - 4 \) Copy content Toggle raw display
$3$ \( T - 17 \) Copy content Toggle raw display
$5$ \( T + 25 \) Copy content Toggle raw display
$7$ \( T \) Copy content Toggle raw display
$11$ \( T + 715 \) Copy content Toggle raw display
$13$ \( T + 331 \) Copy content Toggle raw display
$17$ \( T - 1699 \) Copy content Toggle raw display
$19$ \( T - 1718 \) Copy content Toggle raw display
$23$ \( T + 3950 \) Copy content Toggle raw display
$29$ \( T - 4579 \) Copy content Toggle raw display
$31$ \( T + 6756 \) Copy content Toggle raw display
$37$ \( T + 16518 \) Copy content Toggle raw display
$41$ \( T + 18876 \) Copy content Toggle raw display
$43$ \( T - 2258 \) Copy content Toggle raw display
$47$ \( T - 537 \) Copy content Toggle raw display
$53$ \( T + 10984 \) Copy content Toggle raw display
$59$ \( T - 25956 \) Copy content Toggle raw display
$61$ \( T + 39188 \) Copy content Toggle raw display
$67$ \( T - 4416 \) Copy content Toggle raw display
$71$ \( T + 31880 \) Copy content Toggle raw display
$73$ \( T - 5018 \) Copy content Toggle raw display
$79$ \( T + 27977 \) Copy content Toggle raw display
$83$ \( T + 37644 \) Copy content Toggle raw display
$89$ \( T - 17216 \) Copy content Toggle raw display
$97$ \( T - 63175 \) Copy content Toggle raw display
show more
show less