Properties

Label 441.4.e.j
Level 441441
Weight 44
Character orbit 441.e
Analytic conductor 26.02026.020
Analytic rank 00
Dimension 22
CM discriminant -3
Inner twists 44

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [441,4,Mod(226,441)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(441, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 2]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("441.226");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: N N == 441=3272 441 = 3^{2} \cdot 7^{2}
Weight: k k == 4 4
Character orbit: [χ][\chi] == 441.e (of order 33, degree 22, 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: 26.019842312526.0198423125
Analytic rank: 00
Dimension: 22
Coefficient field: Q(3)\Q(\sqrt{-3})
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: x2x+1 x^{2} - x + 1 Copy content Toggle raw display
Coefficient ring: Z[a1,,a25]\Z[a_1, \ldots, a_{25}]
Coefficient ring index: 1 1
Twist minimal: no (minimal twist has level 9)
Sato-Tate group: U(1)[D3]\mathrm{U}(1)[D_{3}]

qq-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the qq-expansion are expressed in terms of a primitive root of unity ζ6\zeta_{6}. We also show the integral qq-expansion of the trace form.

f(q)f(q) == q+(8ζ6+8)q4+70q1364ζ6q16+56ζ6q19+(125ζ6+125)q25+(308ζ6+308)q31110ζ6q37520q43+(560ζ6+560)q52++1330q97+O(q100) q + ( - 8 \zeta_{6} + 8) q^{4} + 70 q^{13} - 64 \zeta_{6} q^{16} + 56 \zeta_{6} q^{19} + ( - 125 \zeta_{6} + 125) q^{25} + ( - 308 \zeta_{6} + 308) q^{31} - 110 \zeta_{6} q^{37} - 520 q^{43} + ( - 560 \zeta_{6} + 560) q^{52} + \cdots + 1330 q^{97} +O(q^{100}) Copy content Toggle raw display
Tr(f)(q)\operatorname{Tr}(f)(q) == 2q+8q4+140q1364q16+56q19+125q25+308q31110q371040q43+560q52+182q611024q64+880q67+1190q73+896q76884q79+2660q97+O(q100) 2 q + 8 q^{4} + 140 q^{13} - 64 q^{16} + 56 q^{19} + 125 q^{25} + 308 q^{31} - 110 q^{37} - 1040 q^{43} + 560 q^{52} + 182 q^{61} - 1024 q^{64} + 880 q^{67} + 1190 q^{73} + 896 q^{76} - 884 q^{79} + 2660 q^{97}+O(q^{100}) Copy content Toggle raw display

Character values

We give the values of χ\chi on generators for (Z/441Z)×\left(\mathbb{Z}/441\mathbb{Z}\right)^\times.

nn 199199 344344
χ(n)\chi(n) ζ6-\zeta_{6} 11

Embeddings

For each embedding ιm\iota_m of the coefficient field, the values ιm(an)\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   ιm(ν)\iota_m(\nu) a2 a_{2} a3 a_{3} a4 a_{4} a5 a_{5} a6 a_{6} a7 a_{7} a8 a_{8} a9 a_{9} a10 a_{10}
226.1
0.500000 0.866025i
0.500000 + 0.866025i
0 0 4.00000 + 6.92820i 0 0 0 0 0 0
361.1 0 0 4.00000 6.92820i 0 0 0 0 0 0
nn: e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 CM by Q(3)\Q(\sqrt{-3})
7.c even 3 1 inner
21.h odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 441.4.e.j 2
3.b odd 2 1 CM 441.4.e.j 2
7.b odd 2 1 441.4.e.i 2
7.c even 3 1 441.4.a.f 1
7.c even 3 1 inner 441.4.e.j 2
7.d odd 6 1 9.4.a.a 1
7.d odd 6 1 441.4.e.i 2
21.c even 2 1 441.4.e.i 2
21.g even 6 1 9.4.a.a 1
21.g even 6 1 441.4.e.i 2
21.h odd 6 1 441.4.a.f 1
21.h odd 6 1 inner 441.4.e.j 2
28.f even 6 1 144.4.a.d 1
35.i odd 6 1 225.4.a.d 1
35.k even 12 2 225.4.b.g 2
56.j odd 6 1 576.4.a.m 1
56.m even 6 1 576.4.a.l 1
63.i even 6 1 81.4.c.b 2
63.k odd 6 1 81.4.c.b 2
63.s even 6 1 81.4.c.b 2
63.t odd 6 1 81.4.c.b 2
77.i even 6 1 1089.4.a.g 1
84.j odd 6 1 144.4.a.d 1
91.s odd 6 1 1521.4.a.g 1
105.p even 6 1 225.4.a.d 1
105.w odd 12 2 225.4.b.g 2
168.ba even 6 1 576.4.a.m 1
168.be odd 6 1 576.4.a.l 1
231.k odd 6 1 1089.4.a.g 1
273.ba even 6 1 1521.4.a.g 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
9.4.a.a 1 7.d odd 6 1
9.4.a.a 1 21.g even 6 1
81.4.c.b 2 63.i even 6 1
81.4.c.b 2 63.k odd 6 1
81.4.c.b 2 63.s even 6 1
81.4.c.b 2 63.t odd 6 1
144.4.a.d 1 28.f even 6 1
144.4.a.d 1 84.j odd 6 1
225.4.a.d 1 35.i odd 6 1
225.4.a.d 1 105.p even 6 1
225.4.b.g 2 35.k even 12 2
225.4.b.g 2 105.w odd 12 2
441.4.a.f 1 7.c even 3 1
441.4.a.f 1 21.h odd 6 1
441.4.e.i 2 7.b odd 2 1
441.4.e.i 2 7.d odd 6 1
441.4.e.i 2 21.c even 2 1
441.4.e.i 2 21.g even 6 1
441.4.e.j 2 1.a even 1 1 trivial
441.4.e.j 2 3.b odd 2 1 CM
441.4.e.j 2 7.c even 3 1 inner
441.4.e.j 2 21.h odd 6 1 inner
576.4.a.l 1 56.m even 6 1
576.4.a.l 1 168.be odd 6 1
576.4.a.m 1 56.j odd 6 1
576.4.a.m 1 168.ba even 6 1
1089.4.a.g 1 77.i even 6 1
1089.4.a.g 1 231.k odd 6 1
1521.4.a.g 1 91.s odd 6 1
1521.4.a.g 1 273.ba even 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on S4new(441,[χ])S_{4}^{\mathrm{new}}(441, [\chi]):

T2 T_{2} Copy content Toggle raw display
T5 T_{5} Copy content Toggle raw display
T1370 T_{13} - 70 Copy content Toggle raw display

Hecke characteristic polynomials

pp Fp(T)F_p(T)
22 T2 T^{2} Copy content Toggle raw display
33 T2 T^{2} Copy content Toggle raw display
55 T2 T^{2} Copy content Toggle raw display
77 T2 T^{2} Copy content Toggle raw display
1111 T2 T^{2} Copy content Toggle raw display
1313 (T70)2 (T - 70)^{2} Copy content Toggle raw display
1717 T2 T^{2} Copy content Toggle raw display
1919 T256T+3136 T^{2} - 56T + 3136 Copy content Toggle raw display
2323 T2 T^{2} Copy content Toggle raw display
2929 T2 T^{2} Copy content Toggle raw display
3131 T2308T+94864 T^{2} - 308T + 94864 Copy content Toggle raw display
3737 T2+110T+12100 T^{2} + 110T + 12100 Copy content Toggle raw display
4141 T2 T^{2} Copy content Toggle raw display
4343 (T+520)2 (T + 520)^{2} Copy content Toggle raw display
4747 T2 T^{2} Copy content Toggle raw display
5353 T2 T^{2} Copy content Toggle raw display
5959 T2 T^{2} Copy content Toggle raw display
6161 T2182T+33124 T^{2} - 182T + 33124 Copy content Toggle raw display
6767 T2880T+774400 T^{2} - 880T + 774400 Copy content Toggle raw display
7171 T2 T^{2} Copy content Toggle raw display
7373 T21190T+1416100 T^{2} - 1190 T + 1416100 Copy content Toggle raw display
7979 T2+884T+781456 T^{2} + 884T + 781456 Copy content Toggle raw display
8383 T2 T^{2} Copy content Toggle raw display
8989 T2 T^{2} Copy content Toggle raw display
9797 (T1330)2 (T - 1330)^{2} Copy content Toggle raw display
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