Properties

Label 490.4.e.j
Level $490$
Weight $4$
Character orbit 490.e
Analytic conductor $28.911$
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] = [490,4,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: \(\Z[a_1, \ldots, a_{5}]\)
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.

\(f(q)\) \(=\) \( q + 2 \zeta_{6} q^{2} + (8 \zeta_{6} - 8) q^{3} + (4 \zeta_{6} - 4) q^{4} - 5 \zeta_{6} q^{5} - 16 q^{6} - 8 q^{8} - 37 \zeta_{6} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + 2 \zeta_{6} q^{2} + (8 \zeta_{6} - 8) q^{3} + (4 \zeta_{6} - 4) q^{4} - 5 \zeta_{6} q^{5} - 16 q^{6} - 8 q^{8} - 37 \zeta_{6} q^{9} + ( - 10 \zeta_{6} + 10) q^{10} + (68 \zeta_{6} - 68) q^{11} - 32 \zeta_{6} q^{12} - 34 q^{13} + 40 q^{15} - 16 \zeta_{6} q^{16} + ( - 74 \zeta_{6} + 74) q^{17} + ( - 74 \zeta_{6} + 74) q^{18} - 128 \zeta_{6} q^{19} + 20 q^{20} - 136 q^{22} + 80 \zeta_{6} q^{23} + ( - 64 \zeta_{6} + 64) q^{24} + (25 \zeta_{6} - 25) q^{25} - 68 \zeta_{6} q^{26} + 80 q^{27} + 286 q^{29} + 80 \zeta_{6} q^{30} + (24 \zeta_{6} - 24) q^{31} + ( - 32 \zeta_{6} + 32) q^{32} - 544 \zeta_{6} q^{33} + 148 q^{34} + 148 q^{36} - 294 \zeta_{6} q^{37} + ( - 256 \zeta_{6} + 256) q^{38} + ( - 272 \zeta_{6} + 272) q^{39} + 40 \zeta_{6} q^{40} - 66 q^{41} - 124 q^{43} - 272 \zeta_{6} q^{44} + (185 \zeta_{6} - 185) q^{45} + (160 \zeta_{6} - 160) q^{46} + 312 \zeta_{6} q^{47} + 128 q^{48} - 50 q^{50} + 592 \zeta_{6} q^{51} + ( - 136 \zeta_{6} + 136) q^{52} + ( - 34 \zeta_{6} + 34) q^{53} + 160 \zeta_{6} q^{54} + 340 q^{55} + 1024 q^{57} + 572 \zeta_{6} q^{58} + ( - 168 \zeta_{6} + 168) q^{59} + (160 \zeta_{6} - 160) q^{60} + 170 \zeta_{6} q^{61} - 48 q^{62} + 64 q^{64} + 170 \zeta_{6} q^{65} + ( - 1088 \zeta_{6} + 1088) q^{66} + (564 \zeta_{6} - 564) q^{67} + 296 \zeta_{6} q^{68} - 640 q^{69} + 616 q^{71} + 296 \zeta_{6} q^{72} + ( - 250 \zeta_{6} + 250) q^{73} + ( - 588 \zeta_{6} + 588) q^{74} - 200 \zeta_{6} q^{75} + 512 q^{76} + 544 q^{78} + 944 \zeta_{6} q^{79} + (80 \zeta_{6} - 80) q^{80} + ( - 359 \zeta_{6} + 359) q^{81} - 132 \zeta_{6} q^{82} - 672 q^{83} - 370 q^{85} - 248 \zeta_{6} q^{86} + (2288 \zeta_{6} - 2288) q^{87} + ( - 544 \zeta_{6} + 544) q^{88} - 1430 \zeta_{6} q^{89} - 370 q^{90} - 320 q^{92} - 192 \zeta_{6} q^{93} + (624 \zeta_{6} - 624) q^{94} + (640 \zeta_{6} - 640) q^{95} + 256 \zeta_{6} q^{96} + 1270 q^{97} + 2516 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} - 8 q^{3} - 4 q^{4} - 5 q^{5} - 32 q^{6} - 16 q^{8} - 37 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{2} - 8 q^{3} - 4 q^{4} - 5 q^{5} - 32 q^{6} - 16 q^{8} - 37 q^{9} + 10 q^{10} - 68 q^{11} - 32 q^{12} - 68 q^{13} + 80 q^{15} - 16 q^{16} + 74 q^{17} + 74 q^{18} - 128 q^{19} + 40 q^{20} - 272 q^{22} + 80 q^{23} + 64 q^{24} - 25 q^{25} - 68 q^{26} + 160 q^{27} + 572 q^{29} + 80 q^{30} - 24 q^{31} + 32 q^{32} - 544 q^{33} + 296 q^{34} + 296 q^{36} - 294 q^{37} + 256 q^{38} + 272 q^{39} + 40 q^{40} - 132 q^{41} - 248 q^{43} - 272 q^{44} - 185 q^{45} - 160 q^{46} + 312 q^{47} + 256 q^{48} - 100 q^{50} + 592 q^{51} + 136 q^{52} + 34 q^{53} + 160 q^{54} + 680 q^{55} + 2048 q^{57} + 572 q^{58} + 168 q^{59} - 160 q^{60} + 170 q^{61} - 96 q^{62} + 128 q^{64} + 170 q^{65} + 1088 q^{66} - 564 q^{67} + 296 q^{68} - 1280 q^{69} + 1232 q^{71} + 296 q^{72} + 250 q^{73} + 588 q^{74} - 200 q^{75} + 1024 q^{76} + 1088 q^{78} + 944 q^{79} - 80 q^{80} + 359 q^{81} - 132 q^{82} - 1344 q^{83} - 740 q^{85} - 248 q^{86} - 2288 q^{87} + 544 q^{88} - 1430 q^{89} - 740 q^{90} - 640 q^{92} - 192 q^{93} - 624 q^{94} - 640 q^{95} + 256 q^{96} + 2540 q^{97} + 5032 q^{99}+O(q^{100}) \) 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 −4.00000 + 6.92820i −2.00000 + 3.46410i −2.50000 4.33013i −16.0000 0 −8.00000 −18.5000 32.0429i 5.00000 8.66025i
471.1 1.00000 1.73205i −4.00000 6.92820i −2.00000 3.46410i −2.50000 + 4.33013i −16.0000 0 −8.00000 −18.5000 + 32.0429i 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.j 2
7.b odd 2 1 490.4.e.r 2
7.c even 3 1 490.4.a.g 1
7.c even 3 1 inner 490.4.e.j 2
7.d odd 6 1 70.4.a.a 1
7.d odd 6 1 490.4.e.r 2
21.g even 6 1 630.4.a.s 1
28.f even 6 1 560.4.a.q 1
35.i odd 6 1 350.4.a.v 1
35.j even 6 1 2450.4.a.x 1
35.k even 12 2 350.4.c.n 2
56.j odd 6 1 2240.4.a.bi 1
56.m even 6 1 2240.4.a.d 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
70.4.a.a 1 7.d odd 6 1
350.4.a.v 1 35.i odd 6 1
350.4.c.n 2 35.k even 12 2
490.4.a.g 1 7.c even 3 1
490.4.e.j 2 1.a even 1 1 trivial
490.4.e.j 2 7.c even 3 1 inner
490.4.e.r 2 7.b odd 2 1
490.4.e.r 2 7.d odd 6 1
560.4.a.q 1 28.f even 6 1
630.4.a.s 1 21.g even 6 1
2240.4.a.d 1 56.m even 6 1
2240.4.a.bi 1 56.j odd 6 1
2450.4.a.x 1 35.j even 6 1

Hecke kernels

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

\( T_{3}^{2} + 8T_{3} + 64 \) Copy content Toggle raw display
\( T_{11}^{2} + 68T_{11} + 4624 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} - 2T + 4 \) Copy content Toggle raw display
$3$ \( T^{2} + 8T + 64 \) Copy content Toggle raw display
$5$ \( T^{2} + 5T + 25 \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + 68T + 4624 \) Copy content Toggle raw display
$13$ \( (T + 34)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} - 74T + 5476 \) Copy content Toggle raw display
$19$ \( T^{2} + 128T + 16384 \) Copy content Toggle raw display
$23$ \( T^{2} - 80T + 6400 \) Copy content Toggle raw display
$29$ \( (T - 286)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} + 24T + 576 \) Copy content Toggle raw display
$37$ \( T^{2} + 294T + 86436 \) Copy content Toggle raw display
$41$ \( (T + 66)^{2} \) Copy content Toggle raw display
$43$ \( (T + 124)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} - 312T + 97344 \) Copy content Toggle raw display
$53$ \( T^{2} - 34T + 1156 \) Copy content Toggle raw display
$59$ \( T^{2} - 168T + 28224 \) Copy content Toggle raw display
$61$ \( T^{2} - 170T + 28900 \) Copy content Toggle raw display
$67$ \( T^{2} + 564T + 318096 \) Copy content Toggle raw display
$71$ \( (T - 616)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} - 250T + 62500 \) Copy content Toggle raw display
$79$ \( T^{2} - 944T + 891136 \) Copy content Toggle raw display
$83$ \( (T + 672)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} + 1430 T + 2044900 \) Copy content Toggle raw display
$97$ \( (T - 1270)^{2} \) Copy content Toggle raw display
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