Properties

Label 600.3.u.d
Level 600600
Weight 33
Character orbit 600.u
Analytic conductor 16.34916.349
Analytic rank 00
Dimension 44
Inner twists 22

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [600,3,Mod(193,600)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(600, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 3]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("600.193");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: N N == 600=23352 600 = 2^{3} \cdot 3 \cdot 5^{2}
Weight: k k == 3 3
Character orbit: [χ][\chi] == 600.u (of order 44, degree 22, 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: 16.348815861616.3488158616
Analytic rank: 00
Dimension: 44
Relative dimension: 22 over Q(i)\Q(i)
Coefficient field: Q(i,6)\Q(i, \sqrt{6})
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: x4+9 x^{4} + 9 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: SU(2)[C4]\mathrm{SU}(2)[C_{4}]

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+β1q3+(β32β2+2)q7+3β2q9+(2β3+2β12)q11+(12β25β1+12)q13+(2β32β2+2)q17++(6β36β2+6β1)q99+O(q100) q + \beta_1 q^{3} + (\beta_{3} - 2 \beta_{2} + 2) q^{7} + 3 \beta_{2} q^{9} + ( - 2 \beta_{3} + 2 \beta_1 - 2) q^{11} + (12 \beta_{2} - 5 \beta_1 + 12) q^{13} + ( - 2 \beta_{3} - 2 \beta_{2} + 2) q^{17}+ \cdots + (6 \beta_{3} - 6 \beta_{2} + 6 \beta_1) q^{99}+O(q^{100}) Copy content Toggle raw display
Tr(f)(q)\operatorname{Tr}(f)(q) == 4q+8q78q11+48q13+8q1712q21+72q23+44q31+24q33+112q3732q41+104q43+80q47+24q51+104q5324q57180q61+24q63++320q97+O(q100) 4 q + 8 q^{7} - 8 q^{11} + 48 q^{13} + 8 q^{17} - 12 q^{21} + 72 q^{23} + 44 q^{31} + 24 q^{33} + 112 q^{37} - 32 q^{41} + 104 q^{43} + 80 q^{47} + 24 q^{51} + 104 q^{53} - 24 q^{57} - 180 q^{61} + 24 q^{63}+ \cdots + 320 q^{97}+O(q^{100}) Copy content Toggle raw display

Basis of coefficient ring in terms of a root ν\nu of x4+9 x^{4} + 9 : Copy content Toggle raw display

β1\beta_{1}== ν \nu Copy content Toggle raw display
β2\beta_{2}== (ν2)/3 ( \nu^{2} ) / 3 Copy content Toggle raw display
β3\beta_{3}== (ν3)/3 ( \nu^{3} ) / 3 Copy content Toggle raw display
ν\nu== β1 \beta_1 Copy content Toggle raw display
ν2\nu^{2}== 3β2 3\beta_{2} Copy content Toggle raw display
ν3\nu^{3}== 3β3 3\beta_{3} Copy content Toggle raw display

Character values

We give the values of χ\chi on generators for (Z/600Z)×\left(\mathbb{Z}/600\mathbb{Z}\right)^\times.

nn 151151 301301 401401 577577
χ(n)\chi(n) 11 11 11 β2-\beta_{2}

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}
193.1
−1.22474 1.22474i
1.22474 + 1.22474i
−1.22474 + 1.22474i
1.22474 1.22474i
0 −1.22474 1.22474i 0 0 0 3.22474 3.22474i 0 3.00000i 0
193.2 0 1.22474 + 1.22474i 0 0 0 0.775255 0.775255i 0 3.00000i 0
457.1 0 −1.22474 + 1.22474i 0 0 0 3.22474 + 3.22474i 0 3.00000i 0
457.2 0 1.22474 1.22474i 0 0 0 0.775255 + 0.775255i 0 3.00000i 0
nn: e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 600.3.u.d yes 4
3.b odd 2 1 1800.3.v.m 4
4.b odd 2 1 1200.3.bg.f 4
5.b even 2 1 600.3.u.c 4
5.c odd 4 1 600.3.u.c 4
5.c odd 4 1 inner 600.3.u.d yes 4
15.d odd 2 1 1800.3.v.l 4
15.e even 4 1 1800.3.v.l 4
15.e even 4 1 1800.3.v.m 4
20.d odd 2 1 1200.3.bg.l 4
20.e even 4 1 1200.3.bg.f 4
20.e even 4 1 1200.3.bg.l 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
600.3.u.c 4 5.b even 2 1
600.3.u.c 4 5.c odd 4 1
600.3.u.d yes 4 1.a even 1 1 trivial
600.3.u.d yes 4 5.c odd 4 1 inner
1200.3.bg.f 4 4.b odd 2 1
1200.3.bg.f 4 20.e even 4 1
1200.3.bg.l 4 20.d odd 2 1
1200.3.bg.l 4 20.e even 4 1
1800.3.v.l 4 15.d odd 2 1
1800.3.v.l 4 15.e even 4 1
1800.3.v.m 4 3.b odd 2 1
1800.3.v.m 4 15.e even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator T748T73+32T7240T7+25 T_{7}^{4} - 8T_{7}^{3} + 32T_{7}^{2} - 40T_{7} + 25 acting on S3new(600,[χ])S_{3}^{\mathrm{new}}(600, [\chi]). Copy content Toggle raw display

Hecke characteristic polynomials

pp Fp(T)F_p(T)
22 T4 T^{4} Copy content Toggle raw display
33 T4+9 T^{4} + 9 Copy content Toggle raw display
55 T4 T^{4} Copy content Toggle raw display
77 T48T3++25 T^{4} - 8 T^{3} + \cdots + 25 Copy content Toggle raw display
1111 (T2+4T20)2 (T^{2} + 4 T - 20)^{2} Copy content Toggle raw display
1313 T448T3++45369 T^{4} - 48 T^{3} + \cdots + 45369 Copy content Toggle raw display
1717 T48T3++16 T^{4} - 8 T^{3} + \cdots + 16 Copy content Toggle raw display
1919 T4+98T2+1 T^{4} + 98T^{2} + 1 Copy content Toggle raw display
2323 T472T3++291600 T^{4} - 72 T^{3} + \cdots + 291600 Copy content Toggle raw display
2929 T4+2360T2+1373584 T^{4} + 2360 T^{2} + 1373584 Copy content Toggle raw display
3131 (T222T95)2 (T^{2} - 22 T - 95)^{2} Copy content Toggle raw display
3737 T4112T3++1290496 T^{4} - 112 T^{3} + \cdots + 1290496 Copy content Toggle raw display
4141 (T2+16T+40)2 (T^{2} + 16 T + 40)^{2} Copy content Toggle raw display
4343 T4104T3++72361 T^{4} - 104 T^{3} + \cdots + 72361 Copy content Toggle raw display
4747 T480T3++44944 T^{4} - 80 T^{3} + \cdots + 44944 Copy content Toggle raw display
5353 T4104T3++2958400 T^{4} - 104 T^{3} + \cdots + 2958400 Copy content Toggle raw display
5959 T4+12296T2+37699600 T^{4} + 12296 T^{2} + 37699600 Copy content Toggle raw display
6161 (T2+90T1431)2 (T^{2} + 90 T - 1431)^{2} Copy content Toggle raw display
6767 T4264T3++64593369 T^{4} - 264 T^{3} + \cdots + 64593369 Copy content Toggle raw display
7171 (T2128T+2920)2 (T^{2} - 128 T + 2920)^{2} Copy content Toggle raw display
7373 T4112T3++135424 T^{4} - 112 T^{3} + \cdots + 135424 Copy content Toggle raw display
7979 T4+12360T2+35856144 T^{4} + 12360 T^{2} + 35856144 Copy content Toggle raw display
8383 T4+16T3++215150224 T^{4} + 16 T^{3} + \cdots + 215150224 Copy content Toggle raw display
8989 T4+10976T2+15366400 T^{4} + 10976 T^{2} + 15366400 Copy content Toggle raw display
9797 T4320T3++131721529 T^{4} - 320 T^{3} + \cdots + 131721529 Copy content Toggle raw display
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