Properties

Label 310.2.a.e
Level 310310
Weight 22
Character orbit 310.a
Self dual yes
Analytic conductor 2.4752.475
Analytic rank 00
Dimension 33
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] = [310,2,Mod(1,310)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(310, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("310.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: N N == 310=2531 310 = 2 \cdot 5 \cdot 31
Weight: k k == 2 2
Character orbit: [χ][\chi] == 310.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: 2.475362462662.47536246266
Analytic rank: 00
Dimension: 33
Coefficient field: 3.3.148.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: x3x23x+1 x^{3} - x^{2} - 3x + 1 Copy content Toggle raw display
Coefficient ring: Z[a1,a2,a3]\Z[a_1, a_2, a_3]
Coefficient ring index: 2 2
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,β21,\beta_1,\beta_2 for the coefficient ring described below. We also show the integral qq-expansion of the trace form.

f(q)f(q) == q+q2+(β1+1)q3+q4+q5+(β1+1)q6+(β2+β1)q7+q8+(β2β1+1)q9+q10+(β2+2β11)q11++(3β2+2β11)q99+O(q100) q + q^{2} + ( - \beta_1 + 1) q^{3} + q^{4} + q^{5} + ( - \beta_1 + 1) q^{6} + ( - \beta_{2} + \beta_1) q^{7} + q^{8} + (\beta_{2} - \beta_1 + 1) q^{9} + q^{10} + (\beta_{2} + 2 \beta_1 - 1) q^{11}+ \cdots + ( - 3 \beta_{2} + 2 \beta_1 - 1) q^{99}+O(q^{100}) Copy content Toggle raw display
Tr(f)(q)\operatorname{Tr}(f)(q) == 3q+3q2+2q3+3q4+3q5+2q6+3q8+3q9+3q10+2q128q13+2q15+3q16+3q18+8q19+3q2012q212q23+2q24+3q25+4q99+O(q100) 3 q + 3 q^{2} + 2 q^{3} + 3 q^{4} + 3 q^{5} + 2 q^{6} + 3 q^{8} + 3 q^{9} + 3 q^{10} + 2 q^{12} - 8 q^{13} + 2 q^{15} + 3 q^{16} + 3 q^{18} + 8 q^{19} + 3 q^{20} - 12 q^{21} - 2 q^{23} + 2 q^{24} + 3 q^{25}+ \cdots - 4 q^{99}+O(q^{100}) Copy content Toggle raw display

Basis of coefficient ring in terms of a root ν\nu of x3x23x+1 x^{3} - x^{2} - 3x + 1 : Copy content Toggle raw display

β1\beta_{1}== ν22 \nu^{2} - 2 Copy content Toggle raw display
β2\beta_{2}== ν2+2ν+2 -\nu^{2} + 2\nu + 2 Copy content Toggle raw display
ν\nu== (β2+β1)/2 ( \beta_{2} + \beta_1 ) / 2 Copy content Toggle raw display
ν2\nu^{2}== β1+2 \beta _1 + 2 Copy content Toggle raw display

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
2.17009
−1.48119
0.311108
1.00000 −1.70928 1.00000 1.00000 −1.70928 1.07838 1.00000 −0.0783777 1.00000
1.2 1.00000 0.806063 1.00000 1.00000 0.806063 3.35026 1.00000 −2.35026 1.00000
1.3 1.00000 2.90321 1.00000 1.00000 2.90321 −4.42864 1.00000 5.42864 1.00000
nn: e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

p p Sign
22 1 -1
55 1 -1
3131 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 310.2.a.e 3
3.b odd 2 1 2790.2.a.bi 3
4.b odd 2 1 2480.2.a.u 3
5.b even 2 1 1550.2.a.k 3
5.c odd 4 2 1550.2.b.j 6
8.b even 2 1 9920.2.a.bw 3
8.d odd 2 1 9920.2.a.bx 3
31.b odd 2 1 9610.2.a.u 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
310.2.a.e 3 1.a even 1 1 trivial
1550.2.a.k 3 5.b even 2 1
1550.2.b.j 6 5.c odd 4 2
2480.2.a.u 3 4.b odd 2 1
2790.2.a.bi 3 3.b odd 2 1
9610.2.a.u 3 31.b odd 2 1
9920.2.a.bw 3 8.b even 2 1
9920.2.a.bx 3 8.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator T332T324T3+4 T_{3}^{3} - 2T_{3}^{2} - 4T_{3} + 4 acting on S2new(Γ0(310))S_{2}^{\mathrm{new}}(\Gamma_0(310)). Copy content Toggle raw display

Hecke characteristic polynomials

pp Fp(T)F_p(T)
22 (T1)3 (T - 1)^{3} Copy content Toggle raw display
33 T32T2++4 T^{3} - 2 T^{2} + \cdots + 4 Copy content Toggle raw display
55 (T1)3 (T - 1)^{3} Copy content Toggle raw display
77 T316T+16 T^{3} - 16T + 16 Copy content Toggle raw display
1111 T328T52 T^{3} - 28T - 52 Copy content Toggle raw display
1313 T3+8T2++4 T^{3} + 8 T^{2} + \cdots + 4 Copy content Toggle raw display
1717 T316T+16 T^{3} - 16T + 16 Copy content Toggle raw display
1919 T38T2++160 T^{3} - 8 T^{2} + \cdots + 160 Copy content Toggle raw display
2323 T3+2T2+8 T^{3} + 2 T^{2} + \cdots - 8 Copy content Toggle raw display
2929 T3+2T2+260 T^{3} + 2 T^{2} + \cdots - 260 Copy content Toggle raw display
3131 (T1)3 (T - 1)^{3} Copy content Toggle raw display
3737 T3+8T2+92 T^{3} + 8 T^{2} + \cdots - 92 Copy content Toggle raw display
4141 T3+2T2++232 T^{3} + 2 T^{2} + \cdots + 232 Copy content Toggle raw display
4343 T3+10T2+604 T^{3} + 10 T^{2} + \cdots - 604 Copy content Toggle raw display
4747 T320T2++208 T^{3} - 20 T^{2} + \cdots + 208 Copy content Toggle raw display
5353 T3+20T2++4 T^{3} + 20 T^{2} + \cdots + 4 Copy content Toggle raw display
5959 T320T2+160 T^{3} - 20 T^{2} + \cdots - 160 Copy content Toggle raw display
6161 T3+18T2++100 T^{3} + 18 T^{2} + \cdots + 100 Copy content Toggle raw display
6767 T3+12T2+1184 T^{3} + 12 T^{2} + \cdots - 1184 Copy content Toggle raw display
7171 T38T2++128 T^{3} - 8 T^{2} + \cdots + 128 Copy content Toggle raw display
7373 T3+20T2+464 T^{3} + 20 T^{2} + \cdots - 464 Copy content Toggle raw display
7979 T3192T160 T^{3} - 192T - 160 Copy content Toggle raw display
8383 T310T2++124 T^{3} - 10 T^{2} + \cdots + 124 Copy content Toggle raw display
8989 T318T2++40 T^{3} - 18 T^{2} + \cdots + 40 Copy content Toggle raw display
9797 T3+10T2+8 T^{3} + 10 T^{2} + \cdots - 8 Copy content Toggle raw display
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