Properties

Label 490.4.e.i
Level $490$
Weight $4$
Character orbit 490.e
Analytic conductor $28.911$
Analytic rank $1$
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: \(1\)
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 10)
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 \( q - 2 \zeta_{6} q^{2} + ( - 8 \zeta_{6} + 8) q^{3} + (4 \zeta_{6} - 4) q^{4} - 5 \zeta_{6} q^{5} - 16 q^{6} + 8 q^{8} - 37 \zeta_{6} q^{9} + (10 \zeta_{6} - 10) q^{10} + (12 \zeta_{6} - 12) q^{11} + 32 \zeta_{6} q^{12} - 58 q^{13} - 40 q^{15} - 16 \zeta_{6} q^{16} + (66 \zeta_{6} - 66) q^{17} + (74 \zeta_{6} - 74) q^{18} + 100 \zeta_{6} q^{19} + 20 q^{20} + 24 q^{22} - 132 \zeta_{6} q^{23} + ( - 64 \zeta_{6} + 64) q^{24} + (25 \zeta_{6} - 25) q^{25} + 116 \zeta_{6} q^{26} - 80 q^{27} - 90 q^{29} + 80 \zeta_{6} q^{30} + (152 \zeta_{6} - 152) q^{31} + (32 \zeta_{6} - 32) q^{32} + 96 \zeta_{6} q^{33} + 132 q^{34} + 148 q^{36} + 34 \zeta_{6} q^{37} + ( - 200 \zeta_{6} + 200) q^{38} + (464 \zeta_{6} - 464) q^{39} - 40 \zeta_{6} q^{40} - 438 q^{41} + 32 q^{43} - 48 \zeta_{6} q^{44} + (185 \zeta_{6} - 185) q^{45} + (264 \zeta_{6} - 264) q^{46} + 204 \zeta_{6} q^{47} - 128 q^{48} + 50 q^{50} + 528 \zeta_{6} q^{51} + ( - 232 \zeta_{6} + 232) q^{52} + (222 \zeta_{6} - 222) q^{53} + 160 \zeta_{6} q^{54} + 60 q^{55} + 800 q^{57} + 180 \zeta_{6} q^{58} + (420 \zeta_{6} - 420) q^{59} + ( - 160 \zeta_{6} + 160) q^{60} - 902 \zeta_{6} q^{61} + 304 q^{62} + 64 q^{64} + 290 \zeta_{6} q^{65} + ( - 192 \zeta_{6} + 192) q^{66} + ( - 1024 \zeta_{6} + 1024) q^{67} - 264 \zeta_{6} q^{68} - 1056 q^{69} + 432 q^{71} - 296 \zeta_{6} q^{72} + (362 \zeta_{6} - 362) q^{73} + ( - 68 \zeta_{6} + 68) q^{74} + 200 \zeta_{6} q^{75} - 400 q^{76} + 928 q^{78} + 160 \zeta_{6} q^{79} + (80 \zeta_{6} - 80) q^{80} + ( - 359 \zeta_{6} + 359) q^{81} + 876 \zeta_{6} q^{82} + 72 q^{83} + 330 q^{85} - 64 \zeta_{6} q^{86} + (720 \zeta_{6} - 720) q^{87} + (96 \zeta_{6} - 96) q^{88} - 810 \zeta_{6} q^{89} + 370 q^{90} + 528 q^{92} + 1216 \zeta_{6} q^{93} + ( - 408 \zeta_{6} + 408) q^{94} + ( - 500 \zeta_{6} + 500) q^{95} + 256 \zeta_{6} q^{96} + 1106 q^{97} + 444 q^{99} +O(q^{100}) \) 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 \( 2 q - 2 q^{2} + 8 q^{3} - 4 q^{4} - 5 q^{5} - 32 q^{6} + 16 q^{8} - 37 q^{9} - 10 q^{10} - 12 q^{11} + 32 q^{12} - 116 q^{13} - 80 q^{15} - 16 q^{16} - 66 q^{17} - 74 q^{18} + 100 q^{19} + 40 q^{20} + 48 q^{22} - 132 q^{23} + 64 q^{24} - 25 q^{25} + 116 q^{26} - 160 q^{27} - 180 q^{29} + 80 q^{30} - 152 q^{31} - 32 q^{32} + 96 q^{33} + 264 q^{34} + 296 q^{36} + 34 q^{37} + 200 q^{38} - 464 q^{39} - 40 q^{40} - 876 q^{41} + 64 q^{43} - 48 q^{44} - 185 q^{45} - 264 q^{46} + 204 q^{47} - 256 q^{48} + 100 q^{50} + 528 q^{51} + 232 q^{52} - 222 q^{53} + 160 q^{54} + 120 q^{55} + 1600 q^{57} + 180 q^{58} - 420 q^{59} + 160 q^{60} - 902 q^{61} + 608 q^{62} + 128 q^{64} + 290 q^{65} + 192 q^{66} + 1024 q^{67} - 264 q^{68} - 2112 q^{69} + 864 q^{71} - 296 q^{72} - 362 q^{73} + 68 q^{74} + 200 q^{75} - 800 q^{76} + 1856 q^{78} + 160 q^{79} - 80 q^{80} + 359 q^{81} + 876 q^{82} + 144 q^{83} + 660 q^{85} - 64 q^{86} - 720 q^{87} - 96 q^{88} - 810 q^{89} + 740 q^{90} + 1056 q^{92} + 1216 q^{93} + 408 q^{94} + 500 q^{95} + 256 q^{96} + 2212 q^{97} + 888 q^{99}+O(q^{100}) \) 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.i 2
7.b odd 2 1 490.4.e.a 2
7.c even 3 1 10.4.a.a 1
7.c even 3 1 inner 490.4.e.i 2
7.d odd 6 1 490.4.a.o 1
7.d odd 6 1 490.4.e.a 2
21.h odd 6 1 90.4.a.a 1
28.g odd 6 1 80.4.a.f 1
35.i odd 6 1 2450.4.a.b 1
35.j even 6 1 50.4.a.c 1
35.l odd 12 2 50.4.b.a 2
56.k odd 6 1 320.4.a.b 1
56.p even 6 1 320.4.a.m 1
63.g even 3 1 810.4.e.c 2
63.h even 3 1 810.4.e.c 2
63.j odd 6 1 810.4.e.w 2
63.n odd 6 1 810.4.e.w 2
77.h odd 6 1 1210.4.a.b 1
84.n even 6 1 720.4.a.j 1
91.r even 6 1 1690.4.a.a 1
105.o odd 6 1 450.4.a.q 1
105.x even 12 2 450.4.c.d 2
112.u odd 12 2 1280.4.d.g 2
112.w even 12 2 1280.4.d.j 2
140.p odd 6 1 400.4.a.b 1
140.w even 12 2 400.4.c.c 2
280.bf even 6 1 1600.4.a.d 1
280.bi odd 6 1 1600.4.a.bx 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
10.4.a.a 1 7.c even 3 1
50.4.a.c 1 35.j even 6 1
50.4.b.a 2 35.l odd 12 2
80.4.a.f 1 28.g odd 6 1
90.4.a.a 1 21.h odd 6 1
320.4.a.b 1 56.k odd 6 1
320.4.a.m 1 56.p even 6 1
400.4.a.b 1 140.p odd 6 1
400.4.c.c 2 140.w even 12 2
450.4.a.q 1 105.o odd 6 1
450.4.c.d 2 105.x even 12 2
490.4.a.o 1 7.d odd 6 1
490.4.e.a 2 7.b odd 2 1
490.4.e.a 2 7.d odd 6 1
490.4.e.i 2 1.a even 1 1 trivial
490.4.e.i 2 7.c even 3 1 inner
720.4.a.j 1 84.n even 6 1
810.4.e.c 2 63.g even 3 1
810.4.e.c 2 63.h even 3 1
810.4.e.w 2 63.j odd 6 1
810.4.e.w 2 63.n odd 6 1
1210.4.a.b 1 77.h odd 6 1
1280.4.d.g 2 112.u odd 12 2
1280.4.d.j 2 112.w even 12 2
1600.4.a.d 1 280.bf even 6 1
1600.4.a.bx 1 280.bi odd 6 1
1690.4.a.a 1 91.r even 6 1
2450.4.a.b 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 \(S_{4}^{\mathrm{new}}(490, [\chi])\):

\( T_{3}^{2} - 8T_{3} + 64 \) Copy content Toggle raw display
\( T_{11}^{2} + 12T_{11} + 144 \) 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
$11$ \( T^{2} + 12T + 144 \) Copy content Toggle raw display
$13$ \( (T + 58)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + 66T + 4356 \) Copy content Toggle raw display
$19$ \( T^{2} - 100T + 10000 \) Copy content Toggle raw display
$23$ \( T^{2} + 132T + 17424 \) Copy content Toggle raw display
$29$ \( (T + 90)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} + 152T + 23104 \) Copy content Toggle raw display
$37$ \( T^{2} - 34T + 1156 \) Copy content Toggle raw display
$41$ \( (T + 438)^{2} \) Copy content Toggle raw display
$43$ \( (T - 32)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} - 204T + 41616 \) Copy content Toggle raw display
$53$ \( T^{2} + 222T + 49284 \) Copy content Toggle raw display
$59$ \( T^{2} + 420T + 176400 \) Copy content Toggle raw display
$61$ \( T^{2} + 902T + 813604 \) Copy content Toggle raw display
$67$ \( T^{2} - 1024 T + 1048576 \) Copy content Toggle raw display
$71$ \( (T - 432)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 362T + 131044 \) Copy content Toggle raw display
$79$ \( T^{2} - 160T + 25600 \) Copy content Toggle raw display
$83$ \( (T - 72)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} + 810T + 656100 \) Copy content Toggle raw display
$97$ \( (T - 1106)^{2} \) Copy content Toggle raw display
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