Properties

Label 3.21.b.a
Level 33
Weight 2121
Character orbit 3.b
Analytic conductor 7.6057.605
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] = [3,21,Mod(2,3)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1]))
 
N = Newforms(chi, 21, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3.2");
 
S:= CuspForms(chi, 21);
 
N := Newforms(S);
 
Level: N N == 3 3
Weight: k k == 21 21
Character orbit: [χ][\chi] == 3.b (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: 7.605412953087.60541295308
Analytic rank: 00
Dimension: 66
Coefficient field: Q[x]/(x6+)\mathbb{Q}[x]/(x^{6} + \cdots)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: x6+116898x4+3059043456x2+18153107947520 x^{6} + 116898x^{4} + 3059043456x^{2} + 18153107947520 Copy content Toggle raw display
Coefficient ring: Z[a1,a2,a3]\Z[a_1, a_2, a_3]
Coefficient ring index: 21431952 2^{14}\cdot 3^{19}\cdot 5^{2}
Twist minimal: yes
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+β1q2+(β2+11β1687)q3+(β3+3β2+354200)q4+(β4+18β2+1285β1)q5+(β53β4+16066872)q6++(1784433170439β5+29 ⁣ ⁣80)q99+O(q100) q + \beta_1 q^{2} + (\beta_{2} + 11 \beta_1 - 687) q^{3} + (\beta_{3} + 3 \beta_{2} + \cdots - 354200) q^{4} + (\beta_{4} + 18 \beta_{2} + 1285 \beta_1) q^{5} + ( - \beta_{5} - 3 \beta_{4} + \cdots - 16066872) q^{6}+ \cdots + ( - 1784433170439 \beta_{5} + \cdots - 29\!\cdots\!80) q^{99}+O(q^{100}) Copy content Toggle raw display
Tr(f)(q)\operatorname{Tr}(f)(q) == 6q4122q32125200q496401232q6+559607916q71087483482q910880304480q10+63957886128q12+17847879276q13487764624960q15317472843648q16+9564155313120q18+17 ⁣ ⁣80q99+O(q100) 6 q - 4122 q^{3} - 2125200 q^{4} - 96401232 q^{6} + 559607916 q^{7} - 1087483482 q^{9} - 10880304480 q^{10} + 63957886128 q^{12} + 17847879276 q^{13} - 487764624960 q^{15} - 317472843648 q^{16} + 9564155313120 q^{18}+ \cdots - 17\!\cdots\!80 q^{99}+O(q^{100}) Copy content Toggle raw display

Basis of coefficient ring in terms of a root ν\nu of x6+116898x4+3059043456x2+18153107947520 x^{6} + 116898x^{4} + 3059043456x^{2} + 18153107947520 : Copy content Toggle raw display

β1\beta_{1}== 6ν 6\nu Copy content Toggle raw display
β2\beta_{2}== (17ν55728ν42170562ν3526987456ν255742610560ν6124748591104)/64336896 ( -17\nu^{5} - 5728\nu^{4} - 2170562\nu^{3} - 526987456\nu^{2} - 55742610560\nu - 6124748591104 ) / 64336896 Copy content Toggle raw display
β3\beta_{3}== (17ν5+5728ν4+2170562ν3+1299030208ν2+55871284352ν+36208166465536)/21445632 ( 17\nu^{5} + 5728\nu^{4} + 2170562\nu^{3} + 1299030208\nu^{2} + 55871284352\nu + 36208166465536 ) / 21445632 Copy content Toggle raw display
β4\beta_{4}== (373ν5+2864ν4+36271114ν3+263493728ν2+493897505920ν+3062374295552)/1787136 ( 373\nu^{5} + 2864\nu^{4} + 36271114\nu^{3} + 263493728\nu^{2} + 493897505920\nu + 3062374295552 ) / 1787136 Copy content Toggle raw display
β5\beta_{5}== (1343ν5+475424ν4171474398ν3+45284044352ν2++568520968810496)/21445632 ( - 1343 \nu^{5} + 475424 \nu^{4} - 171474398 \nu^{3} + 45284044352 \nu^{2} + \cdots + 568520968810496 ) / 21445632 Copy content Toggle raw display
ν\nu== (β1)/6 ( \beta_1 ) / 6 Copy content Toggle raw display
ν2\nu^{2}== (β3+3β2β11402776)/36 ( \beta_{3} + 3\beta_{2} - \beta _1 - 1402776 ) / 36 Copy content Toggle raw display
ν3\nu^{3}== (27β517β4+54β36867β21158985β1)/108 ( -27\beta_{5} - 17\beta_{4} + 54\beta_{3} - 6867\beta_{2} - 1158985\beta_1 ) / 108 Copy content Toggle raw display
ν4\nu^{4}== (416β546833β3239091β2+91761β1+45282332952)/18 ( 416\beta_{5} - 46833\beta_{3} - 239091\beta_{2} + 91761\beta _1 + 45282332952 ) / 18 Copy content Toggle raw display
ν5\nu^{5}== (1303179β5+1085281β42606358β3+336207555β2+44432621465β1)/54 ( 1303179\beta_{5} + 1085281\beta_{4} - 2606358\beta_{3} + 336207555\beta_{2} + 44432621465\beta_1 ) / 54 Copy content Toggle raw display

Character values

We give the values of χ\chi on generators for (Z/3Z)×\left(\mathbb{Z}/3\mathbb{Z}\right)^\times.

nn 22
χ(n)\chi(n) 1-1

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}
2.1
287.188i
161.041i
92.1239i
92.1239i
161.041i
287.188i
1723.13i −25942.2 53045.1i −1.92059e6 9.20415e6i −9.14036e7 + 4.47017e7i 4.08861e8 1.50260e9i −2.14079e9 + 2.75221e9i −1.58599e10
2.2 966.247i 56662.8 + 16616.6i 114943. 1.69009e7i 1.60557e7 5.47503e7i 2.16509e8 1.12425e9i 2.93456e9 + 1.88308e9i 1.63305e10
2.3 552.743i −32781.6 + 49113.6i 743051. 1.06933e7i 2.71472e7 + 1.81198e7i −3.45566e8 9.90310e8i −1.33751e9 3.22005e9i −5.91067e9
2.4 552.743i −32781.6 49113.6i 743051. 1.06933e7i 2.71472e7 1.81198e7i −3.45566e8 9.90310e8i −1.33751e9 + 3.22005e9i −5.91067e9
2.5 966.247i 56662.8 16616.6i 114943. 1.69009e7i 1.60557e7 + 5.47503e7i 2.16509e8 1.12425e9i 2.93456e9 1.88308e9i 1.63305e10
2.6 1723.13i −25942.2 + 53045.1i −1.92059e6 9.20415e6i −9.14036e7 4.47017e7i 4.08861e8 1.50260e9i −2.14079e9 2.75221e9i −1.58599e10
nn: e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2.6
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3.21.b.a 6
3.b odd 2 1 inner 3.21.b.a 6
4.b odd 2 1 48.21.e.c 6
12.b even 2 1 48.21.e.c 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3.21.b.a 6 1.a even 1 1 trivial
3.21.b.a 6 3.b odd 2 1 inner
48.21.e.c 6 4.b odd 2 1
48.21.e.c 6 12.b even 2 1

Hecke kernels

This newform subspace is the entire newspace S21new(3,[χ])S_{21}^{\mathrm{new}}(3, [\chi]).

Hecke characteristic polynomials

pp Fp(T)F_p(T)
22 T6++84 ⁣ ⁣20 T^{6} + \cdots + 84\!\cdots\!20 Copy content Toggle raw display
33 T6++42 ⁣ ⁣01 T^{6} + \cdots + 42\!\cdots\!01 Copy content Toggle raw display
55 T6++27 ⁣ ⁣00 T^{6} + \cdots + 27\!\cdots\!00 Copy content Toggle raw display
77 (T3++30 ⁣ ⁣00)2 (T^{3} + \cdots + 30\!\cdots\!00)^{2} Copy content Toggle raw display
1111 T6++72 ⁣ ⁣00 T^{6} + \cdots + 72\!\cdots\!00 Copy content Toggle raw display
1313 (T3++19 ⁣ ⁣00)2 (T^{3} + \cdots + 19\!\cdots\!00)^{2} Copy content Toggle raw display
1717 T6++21 ⁣ ⁣80 T^{6} + \cdots + 21\!\cdots\!80 Copy content Toggle raw display
1919 (T3+14 ⁣ ⁣68)2 (T^{3} + \cdots - 14\!\cdots\!68)^{2} Copy content Toggle raw display
2323 T6++24 ⁣ ⁣20 T^{6} + \cdots + 24\!\cdots\!20 Copy content Toggle raw display
2929 T6++80 ⁣ ⁣00 T^{6} + \cdots + 80\!\cdots\!00 Copy content Toggle raw display
3131 (T3+58 ⁣ ⁣88)2 (T^{3} + \cdots - 58\!\cdots\!88)^{2} Copy content Toggle raw display
3737 (T3++12 ⁣ ⁣00)2 (T^{3} + \cdots + 12\!\cdots\!00)^{2} Copy content Toggle raw display
4141 T6++47 ⁣ ⁣00 T^{6} + \cdots + 47\!\cdots\!00 Copy content Toggle raw display
4343 (T3++32 ⁣ ⁣00)2 (T^{3} + \cdots + 32\!\cdots\!00)^{2} Copy content Toggle raw display
4747 T6++15 ⁣ ⁣80 T^{6} + \cdots + 15\!\cdots\!80 Copy content Toggle raw display
5353 T6++96 ⁣ ⁣80 T^{6} + \cdots + 96\!\cdots\!80 Copy content Toggle raw display
5959 T6++12 ⁣ ⁣00 T^{6} + \cdots + 12\!\cdots\!00 Copy content Toggle raw display
6161 (T3+70 ⁣ ⁣48)2 (T^{3} + \cdots - 70\!\cdots\!48)^{2} Copy content Toggle raw display
6767 (T3++29 ⁣ ⁣00)2 (T^{3} + \cdots + 29\!\cdots\!00)^{2} Copy content Toggle raw display
7171 T6++14 ⁣ ⁣00 T^{6} + \cdots + 14\!\cdots\!00 Copy content Toggle raw display
7373 (T3+11 ⁣ ⁣00)2 (T^{3} + \cdots - 11\!\cdots\!00)^{2} Copy content Toggle raw display
7979 (T3++24 ⁣ ⁣12)2 (T^{3} + \cdots + 24\!\cdots\!12)^{2} Copy content Toggle raw display
8383 T6++26 ⁣ ⁣80 T^{6} + \cdots + 26\!\cdots\!80 Copy content Toggle raw display
8989 T6++16 ⁣ ⁣00 T^{6} + \cdots + 16\!\cdots\!00 Copy content Toggle raw display
9797 (T3++17 ⁣ ⁣00)2 (T^{3} + \cdots + 17\!\cdots\!00)^{2} Copy content Toggle raw display
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