Properties

Label 490.4.e.p
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: \(\Z[a_1, a_2, a_3]\)
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.

\(f(q)\) \(=\) \( q + 2 \zeta_{6} q^{2} + ( - 3 \zeta_{6} + 3) q^{3} + (4 \zeta_{6} - 4) q^{4} - 5 \zeta_{6} q^{5} + 6 q^{6} - 8 q^{8} + 18 \zeta_{6} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + 2 \zeta_{6} q^{2} + ( - 3 \zeta_{6} + 3) q^{3} + (4 \zeta_{6} - 4) q^{4} - 5 \zeta_{6} q^{5} + 6 q^{6} - 8 q^{8} + 18 \zeta_{6} q^{9} + ( - 10 \zeta_{6} + 10) q^{10} + ( - 17 \zeta_{6} + 17) q^{11} + 12 \zeta_{6} q^{12} - 81 q^{13} - 15 q^{15} - 16 \zeta_{6} q^{16} + ( - 91 \zeta_{6} + 91) q^{17} + (36 \zeta_{6} - 36) q^{18} - 102 \zeta_{6} q^{19} + 20 q^{20} + 34 q^{22} + 90 \zeta_{6} q^{23} + (24 \zeta_{6} - 24) q^{24} + (25 \zeta_{6} - 25) q^{25} - 162 \zeta_{6} q^{26} + 135 q^{27} - 129 q^{29} - 30 \zeta_{6} q^{30} + (116 \zeta_{6} - 116) q^{31} + ( - 32 \zeta_{6} + 32) q^{32} - 51 \zeta_{6} q^{33} + 182 q^{34} - 72 q^{36} - 314 \zeta_{6} q^{37} + ( - 204 \zeta_{6} + 204) q^{38} + (243 \zeta_{6} - 243) q^{39} + 40 \zeta_{6} q^{40} - 124 q^{41} - 434 q^{43} + 68 \zeta_{6} q^{44} + ( - 90 \zeta_{6} + 90) q^{45} + (180 \zeta_{6} - 180) q^{46} - 497 \zeta_{6} q^{47} - 48 q^{48} - 50 q^{50} - 273 \zeta_{6} q^{51} + ( - 324 \zeta_{6} + 324) q^{52} + ( - 584 \zeta_{6} + 584) q^{53} + 270 \zeta_{6} q^{54} - 85 q^{55} - 306 q^{57} - 258 \zeta_{6} q^{58} + ( - 332 \zeta_{6} + 332) q^{59} + ( - 60 \zeta_{6} + 60) q^{60} - 220 \zeta_{6} q^{61} - 232 q^{62} + 64 q^{64} + 405 \zeta_{6} q^{65} + ( - 102 \zeta_{6} + 102) q^{66} + (384 \zeta_{6} - 384) q^{67} + 364 \zeta_{6} q^{68} + 270 q^{69} - 664 q^{71} - 144 \zeta_{6} q^{72} + (230 \zeta_{6} - 230) q^{73} + ( - 628 \zeta_{6} + 628) q^{74} + 75 \zeta_{6} q^{75} + 408 q^{76} - 486 q^{78} - 361 \zeta_{6} q^{79} + (80 \zeta_{6} - 80) q^{80} + (81 \zeta_{6} - 81) q^{81} - 248 \zeta_{6} q^{82} + 1172 q^{83} - 455 q^{85} - 868 \zeta_{6} q^{86} + (387 \zeta_{6} - 387) q^{87} + (136 \zeta_{6} - 136) q^{88} - 40 \zeta_{6} q^{89} + 180 q^{90} - 360 q^{92} + 348 \zeta_{6} q^{93} + ( - 994 \zeta_{6} + 994) q^{94} + (510 \zeta_{6} - 510) q^{95} - 96 \zeta_{6} q^{96} - 175 q^{97} + 306 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} + 3 q^{3} - 4 q^{4} - 5 q^{5} + 12 q^{6} - 16 q^{8} + 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{2} + 3 q^{3} - 4 q^{4} - 5 q^{5} + 12 q^{6} - 16 q^{8} + 18 q^{9} + 10 q^{10} + 17 q^{11} + 12 q^{12} - 162 q^{13} - 30 q^{15} - 16 q^{16} + 91 q^{17} - 36 q^{18} - 102 q^{19} + 40 q^{20} + 68 q^{22} + 90 q^{23} - 24 q^{24} - 25 q^{25} - 162 q^{26} + 270 q^{27} - 258 q^{29} - 30 q^{30} - 116 q^{31} + 32 q^{32} - 51 q^{33} + 364 q^{34} - 144 q^{36} - 314 q^{37} + 204 q^{38} - 243 q^{39} + 40 q^{40} - 248 q^{41} - 868 q^{43} + 68 q^{44} + 90 q^{45} - 180 q^{46} - 497 q^{47} - 96 q^{48} - 100 q^{50} - 273 q^{51} + 324 q^{52} + 584 q^{53} + 270 q^{54} - 170 q^{55} - 612 q^{57} - 258 q^{58} + 332 q^{59} + 60 q^{60} - 220 q^{61} - 464 q^{62} + 128 q^{64} + 405 q^{65} + 102 q^{66} - 384 q^{67} + 364 q^{68} + 540 q^{69} - 1328 q^{71} - 144 q^{72} - 230 q^{73} + 628 q^{74} + 75 q^{75} + 816 q^{76} - 972 q^{78} - 361 q^{79} - 80 q^{80} - 81 q^{81} - 248 q^{82} + 2344 q^{83} - 910 q^{85} - 868 q^{86} - 387 q^{87} - 136 q^{88} - 40 q^{89} + 360 q^{90} - 720 q^{92} + 348 q^{93} + 994 q^{94} - 510 q^{95} - 96 q^{96} - 350 q^{97} + 612 q^{99}+O(q^{100}) \) 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 1.50000 2.59808i −2.00000 + 3.46410i −2.50000 4.33013i 6.00000 0 −8.00000 9.00000 + 15.5885i 5.00000 8.66025i
471.1 1.00000 1.73205i 1.50000 + 2.59808i −2.00000 3.46410i −2.50000 + 4.33013i 6.00000 0 −8.00000 9.00000 15.5885i 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.p 2
7.b odd 2 1 490.4.e.l 2
7.c even 3 1 70.4.a.b 1
7.c even 3 1 inner 490.4.e.p 2
7.d odd 6 1 490.4.a.f 1
7.d odd 6 1 490.4.e.l 2
21.h odd 6 1 630.4.a.m 1
28.g odd 6 1 560.4.a.k 1
35.i odd 6 1 2450.4.a.ba 1
35.j even 6 1 350.4.a.t 1
35.l odd 12 2 350.4.c.j 2
56.k odd 6 1 2240.4.a.p 1
56.p even 6 1 2240.4.a.w 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
70.4.a.b 1 7.c even 3 1
350.4.a.t 1 35.j even 6 1
350.4.c.j 2 35.l odd 12 2
490.4.a.f 1 7.d odd 6 1
490.4.e.l 2 7.b odd 2 1
490.4.e.l 2 7.d odd 6 1
490.4.e.p 2 1.a even 1 1 trivial
490.4.e.p 2 7.c even 3 1 inner
560.4.a.k 1 28.g odd 6 1
630.4.a.m 1 21.h odd 6 1
2240.4.a.p 1 56.k odd 6 1
2240.4.a.w 1 56.p even 6 1
2450.4.a.ba 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 \(S_{4}^{\mathrm{new}}(490, [\chi])\):

\( T_{3}^{2} - 3T_{3} + 9 \) Copy content Toggle raw display
\( T_{11}^{2} - 17T_{11} + 289 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} - 2T + 4 \) Copy content Toggle raw display
$3$ \( T^{2} - 3T + 9 \) Copy content Toggle raw display
$5$ \( T^{2} + 5T + 25 \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} - 17T + 289 \) Copy content Toggle raw display
$13$ \( (T + 81)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} - 91T + 8281 \) Copy content Toggle raw display
$19$ \( T^{2} + 102T + 10404 \) Copy content Toggle raw display
$23$ \( T^{2} - 90T + 8100 \) Copy content Toggle raw display
$29$ \( (T + 129)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} + 116T + 13456 \) Copy content Toggle raw display
$37$ \( T^{2} + 314T + 98596 \) Copy content Toggle raw display
$41$ \( (T + 124)^{2} \) Copy content Toggle raw display
$43$ \( (T + 434)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} + 497T + 247009 \) Copy content Toggle raw display
$53$ \( T^{2} - 584T + 341056 \) Copy content Toggle raw display
$59$ \( T^{2} - 332T + 110224 \) Copy content Toggle raw display
$61$ \( T^{2} + 220T + 48400 \) Copy content Toggle raw display
$67$ \( T^{2} + 384T + 147456 \) Copy content Toggle raw display
$71$ \( (T + 664)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 230T + 52900 \) Copy content Toggle raw display
$79$ \( T^{2} + 361T + 130321 \) Copy content Toggle raw display
$83$ \( (T - 1172)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} + 40T + 1600 \) Copy content Toggle raw display
$97$ \( (T + 175)^{2} \) Copy content Toggle raw display
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