Properties

Label 308.3.s.a
Level 308308
Weight 33
Character orbit 308.s
Analytic conductor 8.3928.392
Analytic rank 00
Dimension 88
CM discriminant -7
Inner twists 44

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [308,3,Mod(83,308)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(308, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([5, 5, 9]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("308.83");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: N N == 308=22711 308 = 2^{2} \cdot 7 \cdot 11
Weight: k k == 3 3
Character orbit: [χ][\chi] == 308.s (of order 1010, degree 44, 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: 8.392392142308.39239214230
Analytic rank: 00
Dimension: 88
Relative dimension: 22 over Q(ζ10)\Q(\zeta_{10})
Coefficient field: 8.0.37515625.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: x8x7x6+3x5x4+6x34x28x+16 x^{8} - x^{7} - x^{6} + 3x^{5} - x^{4} + 6x^{3} - 4x^{2} - 8x + 16 Copy content Toggle raw display
Coefficient ring: Z[a1,,a7]\Z[a_1, \ldots, a_{7}]
Coefficient ring index: 1 1
Twist minimal: yes
Sato-Tate group: U(1)[D10]\mathrm{U}(1)[D_{10}]

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,,β71,\beta_1,\ldots,\beta_{7} for the coefficient ring described below. We also show the integral qq-expansion of the trace form.

f(q)f(q) == q+(β5β4)q2+(β7+3β2)q47β3q7+(2β7+2β5++2)q8+9β7q9+(β7β5β3+1)q11++(72β6+63)q99+O(q100) q + (\beta_{5} - \beta_{4}) q^{2} + (\beta_{7} + 3 \beta_{2}) q^{4} - 7 \beta_{3} q^{7} + ( - 2 \beta_{7} + 2 \beta_{5} + \cdots + 2) q^{8} + 9 \beta_{7} q^{9} + (\beta_{7} - \beta_{5} - \beta_{3} + \cdots - 1) q^{11}+ \cdots + ( - 72 \beta_{6} + 63) q^{99}+O(q^{100}) Copy content Toggle raw display
Tr(f)(q)\operatorname{Tr}(f)(q) == 8q3q2q4+14q7+9q8+18q9+6q1184q14+31q16+27q18+37q2250q25+7q28228q32+9q36+114q37+116q43348q44303q46++216q99+O(q100) 8 q - 3 q^{2} - q^{4} + 14 q^{7} + 9 q^{8} + 18 q^{9} + 6 q^{11} - 84 q^{14} + 31 q^{16} + 27 q^{18} + 37 q^{22} - 50 q^{25} + 7 q^{28} - 228 q^{32} + 9 q^{36} + 114 q^{37} + 116 q^{43} - 348 q^{44} - 303 q^{46}+ \cdots + 216 q^{99}+O(q^{100}) Copy content Toggle raw display

Basis of coefficient ring in terms of a root ν\nu of x8x7x6+3x5x4+6x34x28x+16 x^{8} - x^{7} - x^{6} + 3x^{5} - x^{4} + 6x^{3} - 4x^{2} - 8x + 16 : Copy content Toggle raw display

β1\beta_{1}== ν \nu Copy content Toggle raw display
β2\beta_{2}== (ν7ν6ν5+3ν4ν3+6ν24ν8)/8 ( \nu^{7} - \nu^{6} - \nu^{5} + 3\nu^{4} - \nu^{3} + 6\nu^{2} - 4\nu - 8 ) / 8 Copy content Toggle raw display
β3\beta_{3}== (ν7ν6+3ν5ν4+3ν34ν28ν+16)/8 ( -\nu^{7} - \nu^{6} + 3\nu^{5} - \nu^{4} + 3\nu^{3} - 4\nu^{2} - 8\nu + 16 ) / 8 Copy content Toggle raw display
β4\beta_{4}== (ν73ν2)/4 ( -\nu^{7} - 3\nu^{2} ) / 4 Copy content Toggle raw display
β5\beta_{5}== (ν77ν2)/4 ( -\nu^{7} - 7\nu^{2} ) / 4 Copy content Toggle raw display
β6\beta_{6}== ν55 -\nu^{5} - 5 Copy content Toggle raw display
β7\beta_{7}== (3ν7+3ν6+3ν5ν4+3ν318ν2+12ν+24)/8 ( -3\nu^{7} + 3\nu^{6} + 3\nu^{5} - \nu^{4} + 3\nu^{3} - 18\nu^{2} + 12\nu + 24 ) / 8 Copy content Toggle raw display
ν\nu== β1 \beta_1 Copy content Toggle raw display
ν2\nu^{2}== β5+β4 -\beta_{5} + \beta_{4} Copy content Toggle raw display
ν3\nu^{3}== β7+β6β5β4+2β3+β2+β11 \beta_{7} + \beta_{6} - \beta_{5} - \beta_{4} + 2\beta_{3} + \beta_{2} + \beta _1 - 1 Copy content Toggle raw display
ν4\nu^{4}== β7+3β2 \beta_{7} + 3\beta_{2} Copy content Toggle raw display
ν5\nu^{5}== β65 -\beta_{6} - 5 Copy content Toggle raw display
ν6\nu^{6}== 2β72β52β35β12 2\beta_{7} - 2\beta_{5} - 2\beta_{3} - 5\beta _1 - 2 Copy content Toggle raw display
ν7\nu^{7}== 3β57β4 3\beta_{5} - 7\beta_{4} Copy content Toggle raw display

Character values

We give the values of χ\chi on generators for (Z/308Z)×\left(\mathbb{Z}/308\mathbb{Z}\right)^\times.

nn 4545 5757 155155
χ(n)\chi(n) 1-1 β3-\beta_{3} 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}
83.1
−1.41264 + 0.0667372i
1.10362 + 0.884319i
1.18208 + 0.776336i
−0.373058 1.36412i
−1.41264 0.0667372i
1.10362 0.884319i
1.18208 0.776336i
−0.373058 + 1.36412i
−1.99109 + 0.188551i 0 3.92890 0.750845i 0 0 5.66312 4.11450i −7.68122 + 2.23580i −2.78115 8.55951i 0
83.2 −0.435959 1.95191i 0 −3.61988 + 1.70190i 0 0 5.66312 4.11450i 4.90007 + 6.32371i −2.78115 8.55951i 0
139.1 −0.794604 1.83538i 0 −2.73721 + 2.91679i 0 0 −2.16312 6.65740i 7.52841 + 2.70611i 7.28115 + 5.29007i 0
139.2 1.72166 1.01779i 0 1.92819 3.50458i 0 0 −2.16312 6.65740i −0.247258 7.99618i 7.28115 + 5.29007i 0
167.1 −1.99109 0.188551i 0 3.92890 + 0.750845i 0 0 5.66312 + 4.11450i −7.68122 2.23580i −2.78115 + 8.55951i 0
167.2 −0.435959 + 1.95191i 0 −3.61988 1.70190i 0 0 5.66312 + 4.11450i 4.90007 6.32371i −2.78115 + 8.55951i 0
195.1 −0.794604 + 1.83538i 0 −2.73721 2.91679i 0 0 −2.16312 + 6.65740i 7.52841 2.70611i 7.28115 5.29007i 0
195.2 1.72166 + 1.01779i 0 1.92819 + 3.50458i 0 0 −2.16312 + 6.65740i −0.247258 + 7.99618i 7.28115 5.29007i 0
nn: e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 83.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 CM by Q(7)\Q(\sqrt{-7})
44.g even 10 1 inner
308.s odd 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 308.3.s.a 8
4.b odd 2 1 308.3.s.b yes 8
7.b odd 2 1 CM 308.3.s.a 8
11.d odd 10 1 308.3.s.b yes 8
28.d even 2 1 308.3.s.b yes 8
44.g even 10 1 inner 308.3.s.a 8
77.l even 10 1 308.3.s.b yes 8
308.s odd 10 1 inner 308.3.s.a 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
308.3.s.a 8 1.a even 1 1 trivial
308.3.s.a 8 7.b odd 2 1 CM
308.3.s.a 8 44.g even 10 1 inner
308.3.s.a 8 308.s odd 10 1 inner
308.3.s.b yes 8 4.b odd 2 1
308.3.s.b yes 8 11.d odd 10 1
308.3.s.b yes 8 28.d even 2 1
308.3.s.b yes 8 77.l even 10 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on S3new(308,[χ])S_{3}^{\mathrm{new}}(308, [\chi]):

T3 T_{3} Copy content Toggle raw display
T43458T4335881T432+341098T432689679 T_{43}^{4} - 58T_{43}^{3} - 5881T_{43}^{2} + 341098T_{43} - 2689679 Copy content Toggle raw display

Hecke characteristic polynomials

pp Fp(T)F_p(T)
22 T8+3T7++256 T^{8} + 3 T^{7} + \cdots + 256 Copy content Toggle raw display
33 T8 T^{8} Copy content Toggle raw display
55 T8 T^{8} Copy content Toggle raw display
77 (T47T3++2401)2 (T^{4} - 7 T^{3} + \cdots + 2401)^{2} Copy content Toggle raw display
1111 T86T7++214358881 T^{8} - 6 T^{7} + \cdots + 214358881 Copy content Toggle raw display
1313 T8 T^{8} Copy content Toggle raw display
1717 T8 T^{8} Copy content Toggle raw display
1919 T8 T^{8} Copy content Toggle raw display
2323 T8++16736855641 T^{8} + \cdots + 16736855641 Copy content Toggle raw display
2929 T8++3434605986361 T^{8} + \cdots + 3434605986361 Copy content Toggle raw display
3131 T8 T^{8} Copy content Toggle raw display
3737 T8++2470432641121 T^{8} + \cdots + 2470432641121 Copy content Toggle raw display
4141 T8 T^{8} Copy content Toggle raw display
4343 (T458T3+2689679)2 (T^{4} - 58 T^{3} + \cdots - 2689679)^{2} Copy content Toggle raw display
4747 T8 T^{8} Copy content Toggle raw display
5353 T8++15 ⁣ ⁣61 T^{8} + \cdots + 15\!\cdots\!61 Copy content Toggle raw display
5959 T8 T^{8} Copy content Toggle raw display
6161 T8 T^{8} Copy content Toggle raw display
6767 T8++703012824043321 T^{8} + \cdots + 703012824043321 Copy content Toggle raw display
7171 T8++4976517717721 T^{8} + \cdots + 4976517717721 Copy content Toggle raw display
7373 T8 T^{8} Copy content Toggle raw display
7979 T8++8422062522241 T^{8} + \cdots + 8422062522241 Copy content Toggle raw display
8383 T8 T^{8} Copy content Toggle raw display
8989 T8 T^{8} Copy content Toggle raw display
9797 T8 T^{8} Copy content Toggle raw display
show more
show less