Properties

Label 490.4.e.m
Level $490$
Weight $4$
Character orbit 490.e
Analytic conductor $28.911$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [490,4,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, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("490.361");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 490 = 2 \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
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: \(28.9109359028\)
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: \(\Z[a_1, a_2, a_3]\)
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.

\(f(q)\) \(=\) \( q + 2 \zeta_{6} q^{2} + (\zeta_{6} - 1) q^{3} + (4 \zeta_{6} - 4) q^{4} - 5 \zeta_{6} q^{5} - 2 q^{6} - 8 q^{8} + 26 \zeta_{6} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + 2 \zeta_{6} q^{2} + (\zeta_{6} - 1) q^{3} + (4 \zeta_{6} - 4) q^{4} - 5 \zeta_{6} q^{5} - 2 q^{6} - 8 q^{8} + 26 \zeta_{6} q^{9} + ( - 10 \zeta_{6} + 10) q^{10} + ( - 2 \zeta_{6} + 2) q^{11} - 4 \zeta_{6} q^{12} + 8 q^{13} + 5 q^{15} - 16 \zeta_{6} q^{16} + (52 \zeta_{6} - 52) q^{17} + (52 \zeta_{6} - 52) q^{18} + 26 \zeta_{6} q^{19} + 20 q^{20} + 4 q^{22} - 67 \zeta_{6} q^{23} + ( - 8 \zeta_{6} + 8) q^{24} + (25 \zeta_{6} - 25) q^{25} + 16 \zeta_{6} q^{26} - 53 q^{27} + 69 q^{29} + 10 \zeta_{6} q^{30} + (332 \zeta_{6} - 332) q^{31} + ( - 32 \zeta_{6} + 32) q^{32} + 2 \zeta_{6} q^{33} - 104 q^{34} - 104 q^{36} - 196 \zeta_{6} q^{37} + (52 \zeta_{6} - 52) q^{38} + (8 \zeta_{6} - 8) q^{39} + 40 \zeta_{6} q^{40} - 353 q^{41} - 369 q^{43} + 8 \zeta_{6} q^{44} + ( - 130 \zeta_{6} + 130) q^{45} + ( - 134 \zeta_{6} + 134) q^{46} + 88 \zeta_{6} q^{47} + 16 q^{48} - 50 q^{50} - 52 \zeta_{6} q^{51} + (32 \zeta_{6} - 32) q^{52} + (582 \zeta_{6} - 582) q^{53} - 106 \zeta_{6} q^{54} - 10 q^{55} - 26 q^{57} + 138 \zeta_{6} q^{58} + (350 \zeta_{6} - 350) q^{59} + (20 \zeta_{6} - 20) q^{60} - 467 \zeta_{6} q^{61} - 664 q^{62} + 64 q^{64} - 40 \zeta_{6} q^{65} + (4 \zeta_{6} - 4) q^{66} + (291 \zeta_{6} - 291) q^{67} - 208 \zeta_{6} q^{68} + 67 q^{69} + 770 q^{71} - 208 \zeta_{6} q^{72} + ( - 628 \zeta_{6} + 628) q^{73} + ( - 392 \zeta_{6} + 392) q^{74} - 25 \zeta_{6} q^{75} - 104 q^{76} - 16 q^{78} - 1170 \zeta_{6} q^{79} + (80 \zeta_{6} - 80) q^{80} + (649 \zeta_{6} - 649) q^{81} - 706 \zeta_{6} q^{82} - 525 q^{83} + 260 q^{85} - 738 \zeta_{6} q^{86} + (69 \zeta_{6} - 69) q^{87} + (16 \zeta_{6} - 16) q^{88} + 89 \zeta_{6} q^{89} + 260 q^{90} + 268 q^{92} - 332 \zeta_{6} q^{93} + (176 \zeta_{6} - 176) q^{94} + ( - 130 \zeta_{6} + 130) q^{95} + 32 \zeta_{6} q^{96} + 290 q^{97} + 52 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} - q^{3} - 4 q^{4} - 5 q^{5} - 4 q^{6} - 16 q^{8} + 26 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{2} - q^{3} - 4 q^{4} - 5 q^{5} - 4 q^{6} - 16 q^{8} + 26 q^{9} + 10 q^{10} + 2 q^{11} - 4 q^{12} + 16 q^{13} + 10 q^{15} - 16 q^{16} - 52 q^{17} - 52 q^{18} + 26 q^{19} + 40 q^{20} + 8 q^{22} - 67 q^{23} + 8 q^{24} - 25 q^{25} + 16 q^{26} - 106 q^{27} + 138 q^{29} + 10 q^{30} - 332 q^{31} + 32 q^{32} + 2 q^{33} - 208 q^{34} - 208 q^{36} - 196 q^{37} - 52 q^{38} - 8 q^{39} + 40 q^{40} - 706 q^{41} - 738 q^{43} + 8 q^{44} + 130 q^{45} + 134 q^{46} + 88 q^{47} + 32 q^{48} - 100 q^{50} - 52 q^{51} - 32 q^{52} - 582 q^{53} - 106 q^{54} - 20 q^{55} - 52 q^{57} + 138 q^{58} - 350 q^{59} - 20 q^{60} - 467 q^{61} - 1328 q^{62} + 128 q^{64} - 40 q^{65} - 4 q^{66} - 291 q^{67} - 208 q^{68} + 134 q^{69} + 1540 q^{71} - 208 q^{72} + 628 q^{73} + 392 q^{74} - 25 q^{75} - 208 q^{76} - 32 q^{78} - 1170 q^{79} - 80 q^{80} - 649 q^{81} - 706 q^{82} - 1050 q^{83} + 520 q^{85} - 738 q^{86} - 69 q^{87} - 16 q^{88} + 89 q^{89} + 520 q^{90} + 536 q^{92} - 332 q^{93} - 176 q^{94} + 130 q^{95} + 32 q^{96} + 580 q^{97} + 104 q^{99}+O(q^{100}) \) 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
1.00000 + 1.73205i −0.500000 + 0.866025i −2.00000 + 3.46410i −2.50000 4.33013i −2.00000 0 −8.00000 13.0000 + 22.5167i 5.00000 8.66025i
471.1 1.00000 1.73205i −0.500000 0.866025i −2.00000 3.46410i −2.50000 + 4.33013i −2.00000 0 −8.00000 13.0000 22.5167i 5.00000 + 8.66025i
\(n\): 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 490.4.e.m 2
7.b odd 2 1 70.4.e.c 2
7.c even 3 1 490.4.a.e 1
7.c even 3 1 inner 490.4.e.m 2
7.d odd 6 1 70.4.e.c 2
7.d odd 6 1 490.4.a.c 1
21.c even 2 1 630.4.k.b 2
21.g even 6 1 630.4.k.b 2
28.d even 2 1 560.4.q.d 2
28.f even 6 1 560.4.q.d 2
35.c odd 2 1 350.4.e.a 2
35.f even 4 2 350.4.j.e 4
35.i odd 6 1 350.4.e.a 2
35.i odd 6 1 2450.4.a.bg 1
35.j even 6 1 2450.4.a.be 1
35.k even 12 2 350.4.j.e 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
70.4.e.c 2 7.b odd 2 1
70.4.e.c 2 7.d odd 6 1
350.4.e.a 2 35.c odd 2 1
350.4.e.a 2 35.i odd 6 1
350.4.j.e 4 35.f even 4 2
350.4.j.e 4 35.k even 12 2
490.4.a.c 1 7.d odd 6 1
490.4.a.e 1 7.c even 3 1
490.4.e.m 2 1.a even 1 1 trivial
490.4.e.m 2 7.c even 3 1 inner
560.4.q.d 2 28.d even 2 1
560.4.q.d 2 28.f even 6 1
630.4.k.b 2 21.c even 2 1
630.4.k.b 2 21.g even 6 1
2450.4.a.be 1 35.j even 6 1
2450.4.a.bg 1 35.i odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(490, [\chi])\):

\( T_{3}^{2} + T_{3} + 1 \) Copy content Toggle raw display
\( T_{11}^{2} - 2T_{11} + 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} - 2T + 4 \) Copy content Toggle raw display
$3$ \( T^{2} + T + 1 \) Copy content Toggle raw display
$5$ \( T^{2} + 5T + 25 \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} - 2T + 4 \) Copy content Toggle raw display
$13$ \( (T - 8)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + 52T + 2704 \) Copy content Toggle raw display
$19$ \( T^{2} - 26T + 676 \) Copy content Toggle raw display
$23$ \( T^{2} + 67T + 4489 \) Copy content Toggle raw display
$29$ \( (T - 69)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} + 332T + 110224 \) Copy content Toggle raw display
$37$ \( T^{2} + 196T + 38416 \) Copy content Toggle raw display
$41$ \( (T + 353)^{2} \) Copy content Toggle raw display
$43$ \( (T + 369)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} - 88T + 7744 \) Copy content Toggle raw display
$53$ \( T^{2} + 582T + 338724 \) Copy content Toggle raw display
$59$ \( T^{2} + 350T + 122500 \) Copy content Toggle raw display
$61$ \( T^{2} + 467T + 218089 \) Copy content Toggle raw display
$67$ \( T^{2} + 291T + 84681 \) Copy content Toggle raw display
$71$ \( (T - 770)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} - 628T + 394384 \) Copy content Toggle raw display
$79$ \( T^{2} + 1170 T + 1368900 \) Copy content Toggle raw display
$83$ \( (T + 525)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} - 89T + 7921 \) Copy content Toggle raw display
$97$ \( (T - 290)^{2} \) Copy content Toggle raw display
show more
show less