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Properties

Label 6069.2.a.ba
Level 60696069
Weight 22
Character orbit 6069.a
Self dual yes
Analytic conductor 48.46148.461
Analytic rank 00
Dimension 99
CM no
Inner twists 11

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [6069,2,Mod(1,6069)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6069, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("6069.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: N N == 6069=3⋅7⋅172 6069 = 3 \cdot 7 \cdot 17^{2}
Weight: k k == 2 2
Character orbit: [χ][\chi] == 6069.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: 48.461208986748.4612089867
Analytic rank: 00
Dimension: 99
Coefficient field: Q[x]/(x9−⋯ )\mathbb{Q}[x]/(x^{9} - \cdots)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: x9−12x7−3x6+45x5+21x4−51x3−27x2+6x+1 x^{9} - 12x^{7} - 3x^{6} + 45x^{5} + 21x^{4} - 51x^{3} - 27x^{2} + 6x + 1 Copy content Toggle raw display
Coefficient ring: Z[a1,a2]\Z[a_1, a_2]
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,…,β81,\beta_1,\ldots,\beta_{8} for the coefficient ring described below. We also show the integral qq-expansion of the trace form.

f(q)f(q) == q+β1q2−q3+(β5−β4+1)q4+(β6+β1)q5−β1q6+q7+(β6−β4−β3+⋯+1)q8+q9+(−β6+β5−2β4+⋯+2)q10+⋯+(−β8−β7+β2+2)q99+O(q100) q + \beta_1 q^{2} - q^{3} + (\beta_{5} - \beta_{4} + 1) q^{4} + (\beta_{6} + \beta_1) q^{5} - \beta_1 q^{6} + q^{7} + (\beta_{6} - \beta_{4} - \beta_{3} + \cdots + 1) q^{8} + q^{9} + ( - \beta_{6} + \beta_{5} - 2 \beta_{4} + \cdots + 2) q^{10}+ \cdots + ( - \beta_{8} - \beta_{7} + \beta_{2} + 2) q^{99}+O(q^{100}) Copy content Toggle raw display
Tr⁡(f)(q)\operatorname{Tr}(f)(q) == 9q−9q3+6q4+3q5+9q7+9q8+9q9+12q10+12q11−6q12+3q13−3q15+3q19+15q20−9q21+6q22+18q23−9q24−3q26−9q27+⋯+12q99+O(q100) 9 q - 9 q^{3} + 6 q^{4} + 3 q^{5} + 9 q^{7} + 9 q^{8} + 9 q^{9} + 12 q^{10} + 12 q^{11} - 6 q^{12} + 3 q^{13} - 3 q^{15} + 3 q^{19} + 15 q^{20} - 9 q^{21} + 6 q^{22} + 18 q^{23} - 9 q^{24} - 3 q^{26} - 9 q^{27}+ \cdots + 12 q^{99}+O(q^{100}) Copy content Toggle raw display

Basis of coefficient ring in terms of a root ν\nu of x9−12x7−3x6+45x5+21x4−51x3−27x2+6x+1 x^{9} - 12x^{7} - 3x^{6} + 45x^{5} + 21x^{4} - 51x^{3} - 27x^{2} + 6x + 1 : Copy content Toggle raw display

β1\beta_{1}== ν \nu Copy content Toggle raw display
β2\beta_{2}== −ν8+2ν7+9ν6−15ν5−22ν4+24ν3+11ν2−3ν+1 -\nu^{8} + 2\nu^{7} + 9\nu^{6} - 15\nu^{5} - 22\nu^{4} + 24\nu^{3} + 11\nu^{2} - 3\nu + 1 Copy content Toggle raw display
β3\beta_{3}== ν8−ν7−10ν6+7ν5+30ν4−9ν3−27ν2−2ν+2 \nu^{8} - \nu^{7} - 10\nu^{6} + 7\nu^{5} + 30\nu^{4} - 9\nu^{3} - 27\nu^{2} - 2\nu + 2 Copy content Toggle raw display
β4\beta_{4}== ν8−2ν7−9ν6+16ν5+22ν4−31ν3−12ν2+12ν \nu^{8} - 2\nu^{7} - 9\nu^{6} + 16\nu^{5} + 22\nu^{4} - 31\nu^{3} - 12\nu^{2} + 12\nu Copy content Toggle raw display
β5\beta_{5}== ν8−2ν7−9ν6+16ν5+22ν4−31ν3−11ν2+12ν−3 \nu^{8} - 2\nu^{7} - 9\nu^{6} + 16\nu^{5} + 22\nu^{4} - 31\nu^{3} - 11\nu^{2} + 12\nu - 3 Copy content Toggle raw display
β6\beta_{6}== 3ν8−5ν7−28ν6+38ν5+74ν4−63ν3−50ν2+8ν 3\nu^{8} - 5\nu^{7} - 28\nu^{6} + 38\nu^{5} + 74\nu^{4} - 63\nu^{3} - 50\nu^{2} + 8\nu Copy content Toggle raw display
β7\beta_{7}== 3ν8−4ν7−30ν6+29ν5+89ν4−41ν3−74ν2−3ν+3 3\nu^{8} - 4\nu^{7} - 30\nu^{6} + 29\nu^{5} + 89\nu^{4} - 41\nu^{3} - 74\nu^{2} - 3\nu + 3 Copy content Toggle raw display
β8\beta_{8}== −3ν8+5ν7+29ν6−38ν5−82ν4+63ν3+64ν2−10ν−2 -3\nu^{8} + 5\nu^{7} + 29\nu^{6} - 38\nu^{5} - 82\nu^{4} + 63\nu^{3} + 64\nu^{2} - 10\nu - 2 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}== β5−β4+3 \beta_{5} - \beta_{4} + 3 Copy content Toggle raw display
ν3\nu^{3}== β6−β4−β3+β2+5β1+1 \beta_{6} - \beta_{4} - \beta_{3} + \beta_{2} + 5\beta _1 + 1 Copy content Toggle raw display
ν4\nu^{4}== −β8−β7+5β5−6β4+β3+β1+14 -\beta_{8} - \beta_{7} + 5\beta_{5} - 6\beta_{4} + \beta_{3} + \beta _1 + 14 Copy content Toggle raw display
ν5\nu^{5}== 7β6+β5−7β4−7β3+8β2+26β1+9 7\beta_{6} + \beta_{5} - 7\beta_{4} - 7\beta_{3} + 8\beta_{2} + 26\beta _1 + 9 Copy content Toggle raw display
ν6\nu^{6}== −7β8−8β7+β6+26β5−34β4+8β3+10β1+72 -7\beta_{8} - 8\beta_{7} + \beta_{6} + 26\beta_{5} - 34\beta_{4} + 8\beta_{3} + 10\beta _1 + 72 Copy content Toggle raw display
ν7\nu^{7}== β8+42β6+10β5−43β4−40β3+50β2+140β1+62 \beta_{8} + 42\beta_{6} + 10\beta_{5} - 43\beta_{4} - 40\beta_{3} + 50\beta_{2} + 140\beta _1 + 62 Copy content Toggle raw display
ν8\nu^{8}== −39β8−50β7+12β6+140β5−190β4+51β3+3β2+75β1+387 -39\beta_{8} - 50\beta_{7} + 12\beta_{6} + 140\beta_{5} - 190\beta_{4} + 51\beta_{3} + 3\beta_{2} + 75\beta _1 + 387 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
−2.29067
−1.59403
−1.56840
−0.765541
−0.118284
0.259117
1.30928
2.35958
2.40895
−2.29067 −1.00000 3.24715 −1.24679 2.29067 1.00000 −2.85680 1.00000 2.85597
1.2 −1.59403 −1.00000 0.540946 −3.07440 1.59403 1.00000 2.32578 1.00000 4.90069
1.3 −1.56840 −1.00000 0.459871 2.88639 1.56840 1.00000 2.41553 1.00000 −4.52700
1.4 −0.765541 −1.00000 −1.41395 2.98515 0.765541 1.00000 2.61352 1.00000 −2.28526
1.5 −0.118284 −1.00000 −1.98601 −1.64632 0.118284 1.00000 0.471482 1.00000 0.194734
1.6 0.259117 −1.00000 −1.93286 −1.75189 −0.259117 1.00000 −1.01907 1.00000 −0.453943
1.7 1.30928 −1.00000 −0.285783 0.212797 −1.30928 1.00000 −2.99273 1.00000 0.278612
1.8 2.35958 −1.00000 3.56759 2.62133 −2.35958 1.00000 3.69886 1.00000 6.18522
1.9 2.40895 −1.00000 3.80304 2.01373 −2.40895 1.00000 4.34343 1.00000 4.85097
nn: e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.9
Significant digits:
Format:

Atkin-Lehner signs

p p Sign
33 +1 +1
77 −1 -1
1717 +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 6069.2.a.ba ✓ 9
17.b even 2 1 6069.2.a.bb yes 9
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6069.2.a.ba ✓ 9 1.a even 1 1 trivial
6069.2.a.bb yes 9 17.b even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on S2new(Γ0(6069))S_{2}^{\mathrm{new}}(\Gamma_0(6069)):

T29−12T27−3T26+45T25+21T24−51T23−27T22+6T2+1 T_{2}^{9} - 12T_{2}^{7} - 3T_{2}^{6} + 45T_{2}^{5} + 21T_{2}^{4} - 51T_{2}^{3} - 27T_{2}^{2} + 6T_{2} + 1 Copy content Toggle raw display
T59−3T58−18T57+51T56+111T55−261T54−330T53+456T52+423T5−107 T_{5}^{9} - 3T_{5}^{8} - 18T_{5}^{7} + 51T_{5}^{6} + 111T_{5}^{5} - 261T_{5}^{4} - 330T_{5}^{3} + 456T_{5}^{2} + 423T_{5} - 107 Copy content Toggle raw display
T119−12T118+30T117+147T116−915T115+1662T114+⋯−323 T_{11}^{9} - 12 T_{11}^{8} + 30 T_{11}^{7} + 147 T_{11}^{6} - 915 T_{11}^{5} + 1662 T_{11}^{4} + \cdots - 323 Copy content Toggle raw display
T239−18T238+90T237+90T236−1890T235+4266T234+⋯+7209 T_{23}^{9} - 18 T_{23}^{8} + 90 T_{23}^{7} + 90 T_{23}^{6} - 1890 T_{23}^{5} + 4266 T_{23}^{4} + \cdots + 7209 Copy content Toggle raw display

Hecke characteristic polynomials

pp Fp(T)F_p(T)
22 T9−12T7+⋯+1 T^{9} - 12 T^{7} + \cdots + 1 Copy content Toggle raw display
33 (T+1)9 (T + 1)^{9} Copy content Toggle raw display
55 T9−3T8+⋯−107 T^{9} - 3 T^{8} + \cdots - 107 Copy content Toggle raw display
77 (T−1)9 (T - 1)^{9} Copy content Toggle raw display
1111 T9−12T8+⋯−323 T^{9} - 12 T^{8} + \cdots - 323 Copy content Toggle raw display
1313 T9−3T8+⋯−397 T^{9} - 3 T^{8} + \cdots - 397 Copy content Toggle raw display
1717 T9 T^{9} Copy content Toggle raw display
1919 T9−3T8+⋯+17 T^{9} - 3 T^{8} + \cdots + 17 Copy content Toggle raw display
2323 T9−18T8+⋯+7209 T^{9} - 18 T^{8} + \cdots + 7209 Copy content Toggle raw display
2929 T9−6T8+⋯−60173 T^{9} - 6 T^{8} + \cdots - 60173 Copy content Toggle raw display
3131 T9−30T8+⋯−194343 T^{9} - 30 T^{8} + \cdots - 194343 Copy content Toggle raw display
3737 T9−12T8+⋯−161949 T^{9} - 12 T^{8} + \cdots - 161949 Copy content Toggle raw display
4141 T9−15T8+⋯+241361 T^{9} - 15 T^{8} + \cdots + 241361 Copy content Toggle raw display
4343 T9+12T8+⋯−86111 T^{9} + 12 T^{8} + \cdots - 86111 Copy content Toggle raw display
4747 T9−15T8+⋯+4132731 T^{9} - 15 T^{8} + \cdots + 4132731 Copy content Toggle raw display
5353 T9+6T8+⋯+44067 T^{9} + 6 T^{8} + \cdots + 44067 Copy content Toggle raw display
5959 T9−3T8+⋯−6140601 T^{9} - 3 T^{8} + \cdots - 6140601 Copy content Toggle raw display
6161 T9−3T8+⋯+507943 T^{9} - 3 T^{8} + \cdots + 507943 Copy content Toggle raw display
6767 T9+3T8+⋯−161855839 T^{9} + 3 T^{8} + \cdots - 161855839 Copy content Toggle raw display
7171 T9−12T8+⋯−23362077 T^{9} - 12 T^{8} + \cdots - 23362077 Copy content Toggle raw display
7373 T9−15T8+⋯−765216021 T^{9} - 15 T^{8} + \cdots - 765216021 Copy content Toggle raw display
7979 T9+18T8+⋯+13722047 T^{9} + 18 T^{8} + \cdots + 13722047 Copy content Toggle raw display
8383 T9−24T8+⋯−125117463 T^{9} - 24 T^{8} + \cdots - 125117463 Copy content Toggle raw display
8989 T9−252T7+⋯+9673361 T^{9} - 252 T^{7} + \cdots + 9673361 Copy content Toggle raw display
9797 T9−36T8+⋯+6040389 T^{9} - 36 T^{8} + \cdots + 6040389 Copy content Toggle raw display
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