Properties

Label 74.3.g.a
Level 7474
Weight 33
Character orbit 74.g
Analytic conductor 2.0162.016
Analytic rank 00
Dimension 88
Inner twists 22

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [74,3,Mod(23,74)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(74, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([5]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("74.23");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: N N == 74=237 74 = 2 \cdot 37
Weight: k k == 3 3
Character orbit: [χ][\chi] == 74.g (of order 1212, 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: 2.016353956272.01635395627
Analytic rank: 00
Dimension: 88
Relative dimension: 22 over Q(ζ12)\Q(\zeta_{12})
Coefficient field: 8.0.303595776.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: x8+5x6+16x4+45x2+81 x^{8} + 5x^{6} + 16x^{4} + 45x^{2} + 81 Copy content Toggle raw display
Coefficient ring: Z[a1,,a5]\Z[a_1, \ldots, a_{5}]
Coefficient ring index: 1 1
Twist minimal: yes
Sato-Tate group: SU(2)[C12]\mathrm{SU}(2)[C_{12}]

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+(β7β5+2β1)q32β3q4+(2β72β6+β5++1)q5+(β7+β6+β4++β1)q6++(4β74β4+2β3+10)q99+O(q100) q + (\beta_{5} - \beta_{4}) q^{2} + ( - \beta_{7} - \beta_{5} + 2 \beta_1) q^{3} - 2 \beta_{3} q^{4} + (2 \beta_{7} - 2 \beta_{6} + \beta_{5} + \cdots + 1) q^{5} + (\beta_{7} + \beta_{6} + \beta_{4} + \cdots + \beta_1) q^{6}+ \cdots + (4 \beta_{7} - 4 \beta_{4} + 2 \beta_{3} + \cdots - 10) q^{99}+O(q^{100}) Copy content Toggle raw display
Tr(f)(q)\operatorname{Tr}(f)(q) == 8q+4q2+12q54q6+8q7+16q82q9+24q104q1210q13+16q1472q15+16q1626q17+2q1870q19+24q2030q2120q22+60q99+O(q100) 8 q + 4 q^{2} + 12 q^{5} - 4 q^{6} + 8 q^{7} + 16 q^{8} - 2 q^{9} + 24 q^{10} - 4 q^{12} - 10 q^{13} + 16 q^{14} - 72 q^{15} + 16 q^{16} - 26 q^{17} + 2 q^{18} - 70 q^{19} + 24 q^{20} - 30 q^{21} - 20 q^{22}+ \cdots - 60 q^{99}+O(q^{100}) Copy content Toggle raw display

Basis of coefficient ring in terms of a root ν\nu of x8+5x6+16x4+45x2+81 x^{8} + 5x^{6} + 16x^{4} + 45x^{2} + 81 : Copy content Toggle raw display

β1\beta_{1}== ν \nu Copy content Toggle raw display
β2\beta_{2}== (ν6+32ν4+16ν2+45)/144 ( \nu^{6} + 32\nu^{4} + 16\nu^{2} + 45 ) / 144 Copy content Toggle raw display
β3\beta_{3}== (ν7+32ν5+16ν3+45ν)/432 ( \nu^{7} + 32\nu^{5} + 16\nu^{3} + 45\nu ) / 432 Copy content Toggle raw display
β4\beta_{4}== (5ν616ν480ν2225)/144 ( -5\nu^{6} - 16\nu^{4} - 80\nu^{2} - 225 ) / 144 Copy content Toggle raw display
β5\beta_{5}== (ν7+13ν)/48 ( \nu^{7} + 13\nu ) / 48 Copy content Toggle raw display
β6\beta_{6}== (ν613)/16 ( -\nu^{6} - 13 ) / 16 Copy content Toggle raw display
β7\beta_{7}== (5ν7+16ν5+80ν3+225ν)/144 ( 5\nu^{7} + 16\nu^{5} + 80\nu^{3} + 225\nu ) / 144 Copy content Toggle raw display
ν\nu== β1 \beta_1 Copy content Toggle raw display
ν2\nu^{2}== β62β4β22 \beta_{6} - 2\beta_{4} - \beta_{2} - 2 Copy content Toggle raw display
ν3\nu^{3}== 2β73β53β32β1 2\beta_{7} - 3\beta_{5} - 3\beta_{3} - 2\beta_1 Copy content Toggle raw display
ν4\nu^{4}== β4+5β2 \beta_{4} + 5\beta_{2} Copy content Toggle raw display
ν5\nu^{5}== β7+15β3 -\beta_{7} + 15\beta_{3} Copy content Toggle raw display
ν6\nu^{6}== 16β613 -16\beta_{6} - 13 Copy content Toggle raw display
ν7\nu^{7}== 48β513β1 48\beta_{5} - 13\beta_1 Copy content Toggle raw display

Character values

We give the values of χ\chi on generators for (Z/74Z)×\left(\mathbb{Z}/74\mathbb{Z}\right)^\times.

nn 3939
χ(n)\chi(n) β3-\beta_{3}

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}
23.1
−1.26217 + 1.18614i
0.396143 1.68614i
−1.26217 1.18614i
0.396143 + 1.68614i
−0.396143 1.68614i
1.26217 + 1.18614i
−0.396143 + 1.68614i
1.26217 1.18614i
−0.366025 1.36603i −2.05446 + 1.18614i −1.73205 + 1.00000i −1.44602 + 5.39662i 2.37228 + 2.37228i 2.26217 + 3.91819i 2.00000 + 2.00000i −1.68614 + 2.92048i 7.90120
23.2 −0.366025 1.36603i 2.92048 1.68614i −1.73205 + 1.00000i 0.981918 3.66457i −3.37228 3.37228i 0.603857 + 1.04591i 2.00000 + 2.00000i 1.18614 2.05446i −5.36530
29.1 −0.366025 + 1.36603i −2.05446 1.18614i −1.73205 1.00000i −1.44602 5.39662i 2.37228 2.37228i 2.26217 3.91819i 2.00000 2.00000i −1.68614 2.92048i 7.90120
29.2 −0.366025 + 1.36603i 2.92048 + 1.68614i −1.73205 1.00000i 0.981918 + 3.66457i −3.37228 + 3.37228i 0.603857 1.04591i 2.00000 2.00000i 1.18614 + 2.05446i −5.36530
45.1 1.36603 + 0.366025i −2.92048 1.68614i 1.73205 + 1.00000i 7.76264 2.07999i −3.37228 3.37228i 1.39614 2.41819i 2.00000 + 2.00000i 1.18614 + 2.05446i 11.3653
45.2 1.36603 + 0.366025i 2.05446 + 1.18614i 1.73205 + 1.00000i −1.29854 + 0.347944i 2.37228 + 2.37228i −0.262169 + 0.454090i 2.00000 + 2.00000i −1.68614 2.92048i −1.90120
51.1 1.36603 0.366025i −2.92048 + 1.68614i 1.73205 1.00000i 7.76264 + 2.07999i −3.37228 + 3.37228i 1.39614 + 2.41819i 2.00000 2.00000i 1.18614 2.05446i 11.3653
51.2 1.36603 0.366025i 2.05446 1.18614i 1.73205 1.00000i −1.29854 0.347944i 2.37228 2.37228i −0.262169 0.454090i 2.00000 2.00000i −1.68614 + 2.92048i −1.90120
nn: e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 23.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
37.g odd 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 74.3.g.a 8
37.g odd 12 1 inner 74.3.g.a 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
74.3.g.a 8 1.a even 1 1 trivial
74.3.g.a 8 37.g odd 12 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator T3817T36+225T341088T32+4096 T_{3}^{8} - 17T_{3}^{6} + 225T_{3}^{4} - 1088T_{3}^{2} + 4096 acting on S3new(74,[χ])S_{3}^{\mathrm{new}}(74, [\chi]). Copy content Toggle raw display

Hecke characteristic polynomials

pp Fp(T)F_p(T)
22 (T42T3+2T2++4)2 (T^{4} - 2 T^{3} + 2 T^{2} + \cdots + 4)^{2} Copy content Toggle raw display
33 T817T6++4096 T^{8} - 17 T^{6} + \cdots + 4096 Copy content Toggle raw display
55 T812T7++52441 T^{8} - 12 T^{7} + \cdots + 52441 Copy content Toggle raw display
77 T88T7++64 T^{8} - 8 T^{7} + \cdots + 64 Copy content Toggle raw display
1111 T8+72T6++1024 T^{8} + 72 T^{6} + \cdots + 1024 Copy content Toggle raw display
1313 T8+10T7++37636 T^{8} + 10 T^{7} + \cdots + 37636 Copy content Toggle raw display
1717 T8+26T7++60109009 T^{8} + 26 T^{7} + \cdots + 60109009 Copy content Toggle raw display
1919 T8+70T7++4840000 T^{8} + 70 T^{7} + \cdots + 4840000 Copy content Toggle raw display
2323 T824T7++571401216 T^{8} - 24 T^{7} + \cdots + 571401216 Copy content Toggle raw display
2929 T8++1800644356 T^{8} + \cdots + 1800644356 Copy content Toggle raw display
3131 T8++227292469504 T^{8} + \cdots + 227292469504 Copy content Toggle raw display
3737 T8++3512479453921 T^{8} + \cdots + 3512479453921 Copy content Toggle raw display
4141 T8138T7++170485249 T^{8} - 138 T^{7} + \cdots + 170485249 Copy content Toggle raw display
4343 T8++1961837230336 T^{8} + \cdots + 1961837230336 Copy content Toggle raw display
4747 (T424T3++3752928)2 (T^{4} - 24 T^{3} + \cdots + 3752928)^{2} Copy content Toggle raw display
5353 T8++58441988509696 T^{8} + \cdots + 58441988509696 Copy content Toggle raw display
5959 T8++5705448177664 T^{8} + \cdots + 5705448177664 Copy content Toggle raw display
6161 T8++319111098869761 T^{8} + \cdots + 319111098869761 Copy content Toggle raw display
6767 T8++276068327012416 T^{8} + \cdots + 276068327012416 Copy content Toggle raw display
7171 T8++25732731689536 T^{8} + \cdots + 25732731689536 Copy content Toggle raw display
7373 T8++10103141959936 T^{8} + \cdots + 10103141959936 Copy content Toggle raw display
7979 T8++46 ⁣ ⁣24 T^{8} + \cdots + 46\!\cdots\!24 Copy content Toggle raw display
8383 T8++45 ⁣ ⁣36 T^{8} + \cdots + 45\!\cdots\!36 Copy content Toggle raw display
8989 T8++98438880759769 T^{8} + \cdots + 98438880759769 Copy content Toggle raw display
9797 T8++5906873324836 T^{8} + \cdots + 5906873324836 Copy content Toggle raw display
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