Properties

Label 1638.2.c.i
Level 16381638
Weight 22
Character orbit 1638.c
Analytic conductor 13.07913.079
Analytic rank 00
Dimension 66
Inner twists 22

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1638,2,Mod(883,1638)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1638, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1638.883");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: N N == 1638=232713 1638 = 2 \cdot 3^{2} \cdot 7 \cdot 13
Weight: k k == 2 2
Character orbit: [χ][\chi] == 1638.c (of order 22, degree 11, 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: 13.079495851113.0794958511
Analytic rank: 00
Dimension: 66
Coefficient field: 6.0.30647296.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: x62x5+2x4+8x3+25x210x+2 x^{6} - 2x^{5} + 2x^{4} + 8x^{3} + 25x^{2} - 10x + 2 Copy content Toggle raw display
Coefficient ring: Z[a1,,a13]\Z[a_1, \ldots, a_{13}]
Coefficient ring index: 1 1
Twist minimal: no (minimal twist has level 182)
Sato-Tate group: SU(2)[C2]\mathrm{SU}(2)[C_{2}]

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 basis 1,β1,,β51,\beta_1,\ldots,\beta_{5} for the coefficient ring described below. We also show the integral qq-expansion of the trace form.

f(q)f(q) == q+β4q2q4+(β2β1)q5+β4q7β4q8+(β2β1)q10+(β52β4++β1)q11+(β5+β12)q13+β4q98+O(q100) q + \beta_{4} q^{2} - q^{4} + ( - \beta_{2} - \beta_1) q^{5} + \beta_{4} q^{7} - \beta_{4} q^{8} + (\beta_{2} - \beta_1) q^{10} + ( - \beta_{5} - 2 \beta_{4} + \cdots + \beta_1) q^{11} + (\beta_{5} + \beta_1 - 2) q^{13}+ \cdots - \beta_{4} q^{98}+O(q^{100}) Copy content Toggle raw display
Tr(f)(q)\operatorname{Tr}(f)(q) == 6q6q44q1010q136q14+6q16+4q17+14q22+6q2318q25+4q2616q294q35+8q38+4q40+16q436q49+10q52+36q53+40q95+O(q100) 6 q - 6 q^{4} - 4 q^{10} - 10 q^{13} - 6 q^{14} + 6 q^{16} + 4 q^{17} + 14 q^{22} + 6 q^{23} - 18 q^{25} + 4 q^{26} - 16 q^{29} - 4 q^{35} + 8 q^{38} + 4 q^{40} + 16 q^{43} - 6 q^{49} + 10 q^{52} + 36 q^{53}+ \cdots - 40 q^{95}+O(q^{100}) Copy content Toggle raw display

Basis of coefficient ring in terms of a root ν\nu of x62x5+2x4+8x3+25x210x+2 x^{6} - 2x^{5} + 2x^{4} + 8x^{3} + 25x^{2} - 10x + 2 : Copy content Toggle raw display

β1\beta_{1}== ν \nu Copy content Toggle raw display
β2\beta_{2}== (7ν518ν4+49ν3+28ν2+159ν62)/173 ( 7\nu^{5} - 18\nu^{4} + 49\nu^{3} + 28\nu^{2} + 159\nu - 62 ) / 173 Copy content Toggle raw display
β3\beta_{3}== (11ν553ν4+77ν3+44ν222ν567)/173 ( 11\nu^{5} - 53\nu^{4} + 77\nu^{3} + 44\nu^{2} - 22\nu - 567 ) / 173 Copy content Toggle raw display
β4\beta_{4}== (31ν5+55ν444ν3297ν2803ν+151)/173 ( -31\nu^{5} + 55\nu^{4} - 44\nu^{3} - 297\nu^{2} - 803\nu + 151 ) / 173 Copy content Toggle raw display
β5\beta_{5}== (100ν5+183ν4181ν3746ν22741ν+515)/173 ( -100\nu^{5} + 183\nu^{4} - 181\nu^{3} - 746\nu^{2} - 2741\nu + 515 ) / 173 Copy content Toggle raw display
ν\nu== β1 \beta_1 Copy content Toggle raw display
ν2\nu^{2}== β53β4+β2+β1 \beta_{5} - 3\beta_{4} + \beta_{2} + \beta_1 Copy content Toggle raw display
ν3\nu^{3}== β52β4β3+7β22 \beta_{5} - 2\beta_{4} - \beta_{3} + 7\beta_{2} - 2 Copy content Toggle raw display
ν4\nu^{4}== 7β3+11β211β119 -7\beta_{3} + 11\beta_{2} - 11\beta _1 - 19 Copy content Toggle raw display
ν5\nu^{5}== 11β5+26β411β355β126 -11\beta_{5} + 26\beta_{4} - 11\beta_{3} - 55\beta _1 - 26 Copy content Toggle raw display

Character values

We give the values of χ\chi on generators for (Z/1638Z)×\left(\mathbb{Z}/1638\mathbb{Z}\right)^\times.

nn 379379 703703 911911
χ(n)\chi(n) 1-1 11 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}
883.1
2.08433 + 2.08433i
0.188470 + 0.188470i
−1.27280 1.27280i
−1.27280 + 1.27280i
0.188470 0.188470i
2.08433 2.08433i
1.00000i 0 −1.00000 4.16867i 0 1.00000i 1.00000i 0 −4.16867
883.2 1.00000i 0 −1.00000 0.376939i 0 1.00000i 1.00000i 0 −0.376939
883.3 1.00000i 0 −1.00000 2.54561i 0 1.00000i 1.00000i 0 2.54561
883.4 1.00000i 0 −1.00000 2.54561i 0 1.00000i 1.00000i 0 2.54561
883.5 1.00000i 0 −1.00000 0.376939i 0 1.00000i 1.00000i 0 −0.376939
883.6 1.00000i 0 −1.00000 4.16867i 0 1.00000i 1.00000i 0 −4.16867
nn: e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 883.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1638.2.c.i 6
3.b odd 2 1 182.2.d.b 6
12.b even 2 1 1456.2.k.b 6
13.b even 2 1 inner 1638.2.c.i 6
21.c even 2 1 1274.2.d.l 6
21.g even 6 2 1274.2.n.n 12
21.h odd 6 2 1274.2.n.m 12
39.d odd 2 1 182.2.d.b 6
39.f even 4 1 2366.2.a.x 3
39.f even 4 1 2366.2.a.bc 3
156.h even 2 1 1456.2.k.b 6
273.g even 2 1 1274.2.d.l 6
273.w odd 6 2 1274.2.n.m 12
273.ba even 6 2 1274.2.n.n 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
182.2.d.b 6 3.b odd 2 1
182.2.d.b 6 39.d odd 2 1
1274.2.d.l 6 21.c even 2 1
1274.2.d.l 6 273.g even 2 1
1274.2.n.m 12 21.h odd 6 2
1274.2.n.m 12 273.w odd 6 2
1274.2.n.n 12 21.g even 6 2
1274.2.n.n 12 273.ba even 6 2
1456.2.k.b 6 12.b even 2 1
1456.2.k.b 6 156.h even 2 1
1638.2.c.i 6 1.a even 1 1 trivial
1638.2.c.i 6 13.b even 2 1 inner
2366.2.a.x 3 39.f even 4 1
2366.2.a.bc 3 39.f even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on S2new(1638,[χ])S_{2}^{\mathrm{new}}(1638, [\chi]):

T56+24T54+116T52+16 T_{5}^{6} + 24T_{5}^{4} + 116T_{5}^{2} + 16 Copy content Toggle raw display
T116+65T114+1296T112+7744 T_{11}^{6} + 65T_{11}^{4} + 1296T_{11}^{2} + 7744 Copy content Toggle raw display
T1732T17244T17+56 T_{17}^{3} - 2T_{17}^{2} - 44T_{17} + 56 Copy content Toggle raw display

Hecke characteristic polynomials

pp Fp(T)F_p(T)
22 (T2+1)3 (T^{2} + 1)^{3} Copy content Toggle raw display
33 T6 T^{6} Copy content Toggle raw display
55 T6+24T4++16 T^{6} + 24 T^{4} + \cdots + 16 Copy content Toggle raw display
77 (T2+1)3 (T^{2} + 1)^{3} Copy content Toggle raw display
1111 T6+65T4++7744 T^{6} + 65 T^{4} + \cdots + 7744 Copy content Toggle raw display
1313 T6+10T5++2197 T^{6} + 10 T^{5} + \cdots + 2197 Copy content Toggle raw display
1717 (T32T244T+56)2 (T^{3} - 2 T^{2} - 44 T + 56)^{2} Copy content Toggle raw display
1919 T6+28T4++256 T^{6} + 28 T^{4} + \cdots + 256 Copy content Toggle raw display
2323 (T33T216T+32)2 (T^{3} - 3 T^{2} - 16 T + 32)^{2} Copy content Toggle raw display
2929 (T3+8T2+176)2 (T^{3} + 8 T^{2} + \cdots - 176)^{2} Copy content Toggle raw display
3131 T6+65T4++7744 T^{6} + 65 T^{4} + \cdots + 7744 Copy content Toggle raw display
3737 T6+65T4++1936 T^{6} + 65 T^{4} + \cdots + 1936 Copy content Toggle raw display
4141 T6+113T4++1936 T^{6} + 113 T^{4} + \cdots + 1936 Copy content Toggle raw display
4343 (T38T2++128)2 (T^{3} - 8 T^{2} + \cdots + 128)^{2} Copy content Toggle raw display
4747 T6+145T4++7744 T^{6} + 145 T^{4} + \cdots + 7744 Copy content Toggle raw display
5353 (T6)6 (T - 6)^{6} Copy content Toggle raw display
5959 T6+288T4++781456 T^{6} + 288 T^{4} + \cdots + 781456 Copy content Toggle raw display
6161 (T35T22T+2)2 (T^{3} - 5 T^{2} - 2 T + 2)^{2} Copy content Toggle raw display
6767 T6+113T4++256 T^{6} + 113 T^{4} + \cdots + 256 Copy content Toggle raw display
7171 T6+336T4++802816 T^{6} + 336 T^{4} + \cdots + 802816 Copy content Toggle raw display
7373 T6+89T4++16 T^{6} + 89 T^{4} + \cdots + 16 Copy content Toggle raw display
7979 (T3+7T2++148)2 (T^{3} + 7 T^{2} + \cdots + 148)^{2} Copy content Toggle raw display
8383 T6+352T4++1567504 T^{6} + 352 T^{4} + \cdots + 1567504 Copy content Toggle raw display
8989 T6+316T4++53824 T^{6} + 316 T^{4} + \cdots + 53824 Copy content Toggle raw display
9797 T6+185T4++38416 T^{6} + 185 T^{4} + \cdots + 38416 Copy content Toggle raw display
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