↑

Properties

Label 19.6.a.d
Level 1919
Weight 66
Character orbit 19.a
Self dual yes
Analytic conductor 3.0473.047
Analytic rank 00
Dimension 44
CM no
Inner twists 11

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [19,6,Mod(1,19)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(19, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("19.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: N N == 19 19
Weight: k k == 6 6
Character orbit: [χ][\chi] == 19.a (trivial)

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: yes
Analytic conductor: 3.047292576453.04729257645
Analytic rank: 00
Dimension: 44
Coefficient field: Q[x]/(x4−⋯ )\mathbb{Q}[x]/(x^{4} - \cdots)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: x4−x3−84x2−154x+484 x^{4} - x^{3} - 84x^{2} - 154x + 484 Copy content Toggle raw display
Coefficient ring: Z[a1,a2,a3]\Z[a_1, a_2, a_3]
Coefficient ring index: 1 1
Twist minimal: yes
Fricke sign: −1-1
Sato-Tate group: SU(2)\mathrm{SU}(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,β2,β31,\beta_1,\beta_2,\beta_3 for the coefficient ring described below. We also show the integral qq-expansion of the trace form.

f(q)f(q) == q+(β2−β1+3)q2+(3β2+3)q3+(−β3−3β2+⋯+23)q4+(4β3−3β2+22)q5+(−3β3−15β2+⋯+81)q6+⋯+(3096β3−17271β2+⋯−62406)q99+O(q100) q + (\beta_{2} - \beta_1 + 3) q^{2} + (3 \beta_{2} + 3) q^{3} + ( - \beta_{3} - 3 \beta_{2} + \cdots + 23) q^{4} + (4 \beta_{3} - 3 \beta_{2} + 22) q^{5} + ( - 3 \beta_{3} - 15 \beta_{2} + \cdots + 81) q^{6}+ \cdots + (3096 \beta_{3} - 17271 \beta_{2} + \cdots - 62406) q^{99}+O(q^{100}) Copy content Toggle raw display
Tr⁡(f)(q)\operatorname{Tr}(f)(q) == 4q+9q2+6q3+97q4+90q5+369q6−190q7−225q8+846q9−264q10−162q11−2091q12−52q13+1395q14−1944q15−551q16−288q17−7506q18+⋯−229338q99+O(q100) 4 q + 9 q^{2} + 6 q^{3} + 97 q^{4} + 90 q^{5} + 369 q^{6} - 190 q^{7} - 225 q^{8} + 846 q^{9} - 264 q^{10} - 162 q^{11} - 2091 q^{12} - 52 q^{13} + 1395 q^{14} - 1944 q^{15} - 551 q^{16} - 288 q^{17} - 7506 q^{18}+ \cdots - 229338 q^{99}+O(q^{100}) Copy content Toggle raw display

Basis of coefficient ring in terms of a root ν\nu of x4−x3−84x2−154x+484 x^{4} - x^{3} - 84x^{2} - 154x + 484 : Copy content Toggle raw display

β1\beta_{1}== ν \nu Copy content Toggle raw display
β2\beta_{2}== (ν3−ν2−62ν−132)/22 ( \nu^{3} - \nu^{2} - 62\nu - 132 ) / 22 Copy content Toggle raw display
β3\beta_{3}== (−2ν3+13ν2+102ν−220)/11 ( -2\nu^{3} + 13\nu^{2} + 102\nu - 220 ) / 11 Copy content Toggle raw display

Display νj\nu^j in terms of βi\beta_i

ν\nu== β1 \beta_1 Copy content Toggle raw display
ν2\nu^{2}== β3+4β2+2β1+44 \beta_{3} + 4\beta_{2} + 2\beta _1 + 44 Copy content Toggle raw display
ν3\nu^{3}== β3+26β2+64β1+176 \beta_{3} + 26\beta_{2} + 64\beta _1 + 176 Copy content Toggle raw display

Display βi\beta_i in terms of νj\nu^j

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}
1.1
1.65978
10.2303
−6.51487
−4.37522
−9.25474 −28.7849 53.6502 45.0454 266.396 −170.409 −200.367 585.568 −416.883
1.2 1.84953 30.2395 −28.5793 10.2772 55.9287 19.2784 −112.043 671.426 19.0079
1.3 7.37689 −3.41393 22.4185 108.514 −25.1842 −80.6465 −70.6815 −231.345 800.499
1.4 9.02832 7.95930 49.5106 −73.8370 71.8591 41.7769 158.091 −179.649 −666.624
nn: e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

p p Sign
1919 −1 -1

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 19.6.a.d ✓ 4
3.b odd 2 1 171.6.a.i 4
4.b odd 2 1 304.6.a.l 4
5.b even 2 1 475.6.a.e 4
7.b odd 2 1 931.6.a.d 4
19.b odd 2 1 361.6.a.e 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
19.6.a.d ✓ 4 1.a even 1 1 trivial
171.6.a.i 4 3.b odd 2 1
304.6.a.l 4 4.b odd 2 1
361.6.a.e 4 19.b odd 2 1
475.6.a.e 4 5.b even 2 1
931.6.a.d 4 7.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator T24−9T23−72T22+774T2−1140 T_{2}^{4} - 9T_{2}^{3} - 72T_{2}^{2} + 774T_{2} - 1140 acting on S6new(Γ0(19))S_{6}^{\mathrm{new}}(\Gamma_0(19)). Copy content Toggle raw display

Hecke characteristic polynomials

pp Fp(T)F_p(T)
22 T4−9T3+⋯−1140 T^{4} - 9 T^{3} + \cdots - 1140 Copy content Toggle raw display
33 T4−6T3+⋯+23652 T^{4} - 6 T^{3} + \cdots + 23652 Copy content Toggle raw display
55 T4−90T3+⋯−3709248 T^{4} - 90 T^{3} + \cdots - 3709248 Copy content Toggle raw display
77 T4+190T3+⋯+11068411 T^{4} + 190 T^{3} + \cdots + 11068411 Copy content Toggle raw display
1111 T4+⋯+23023022796 T^{4} + \cdots + 23023022796 Copy content Toggle raw display
1313 T4+⋯+19147784368 T^{4} + \cdots + 19147784368 Copy content Toggle raw display
1717 T4+⋯+597258958449 T^{4} + \cdots + 597258958449 Copy content Toggle raw display
1919 (T−361)4 (T - 361)^{4} Copy content Toggle raw display
2323 T4+⋯−6908113340160 T^{4} + \cdots - 6908113340160 Copy content Toggle raw display
2929 T4+⋯+56401950031212 T^{4} + \cdots + 56401950031212 Copy content Toggle raw display
3131 T4+⋯−5096621276672 T^{4} + \cdots - 5096621276672 Copy content Toggle raw display
3737 T4+⋯+63 ⁣⋯ ⁣52 T^{4} + \cdots + 63\!\cdots\!52 Copy content Toggle raw display
4141 T4+⋯+813179406508032 T^{4} + \cdots + 813179406508032 Copy content Toggle raw display
4343 T4+⋯+13 ⁣⋯ ⁣16 T^{4} + \cdots + 13\!\cdots\!16 Copy content Toggle raw display
4747 T4+⋯−942340690593024 T^{4} + \cdots - 942340690593024 Copy content Toggle raw display
5353 T4+⋯+13 ⁣⋯ ⁣52 T^{4} + \cdots + 13\!\cdots\!52 Copy content Toggle raw display
5959 T4+⋯+19 ⁣⋯ ⁣36 T^{4} + \cdots + 19\!\cdots\!36 Copy content Toggle raw display
6161 T4+⋯−10 ⁣⋯ ⁣92 T^{4} + \cdots - 10\!\cdots\!92 Copy content Toggle raw display
6767 T4+⋯+19 ⁣⋯ ⁣12 T^{4} + \cdots + 19\!\cdots\!12 Copy content Toggle raw display
7171 T4+⋯+16 ⁣⋯ ⁣16 T^{4} + \cdots + 16\!\cdots\!16 Copy content Toggle raw display
7373 T4+⋯+12 ⁣⋯ ⁣37 T^{4} + \cdots + 12\!\cdots\!37 Copy content Toggle raw display
7979 T4+⋯−20 ⁣⋯ ⁣56 T^{4} + \cdots - 20\!\cdots\!56 Copy content Toggle raw display
8383 T4+⋯−13 ⁣⋯ ⁣16 T^{4} + \cdots - 13\!\cdots\!16 Copy content Toggle raw display
8989 T4+⋯+38 ⁣⋯ ⁣72 T^{4} + \cdots + 38\!\cdots\!72 Copy content Toggle raw display
9797 T4+⋯+42 ⁣⋯ ⁣12 T^{4} + \cdots + 42\!\cdots\!12 Copy content Toggle raw display
show more
show less