Properties

Label 612.2.k.a
Level 612612
Weight 22
Character orbit 612.k
Analytic conductor 4.8874.887
Analytic rank 00
Dimension 22
Inner twists 22

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [612,2,Mod(217,612)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(612, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("612.217");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: N N == 612=223217 612 = 2^{2} \cdot 3^{2} \cdot 17
Weight: k k == 2 2
Character orbit: [χ][\chi] == 612.k (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: 4.886844603704.88684460370
Analytic rank: 00
Dimension: 22
Coefficient field: Q(1)\Q(\sqrt{-1})
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: x2+1 x^{2} + 1 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)[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 i=1i = \sqrt{-1}. We also show the integral qq-expansion of the trace form.

f(q)f(q) == q+(i1)q5+(2i2)q7+(4i+4)q11+(i+4)q174iq19+(6i+6)q233iq25+(5i5)q29+(6i+6)q31+4q35+(3i3)q37+(i+1)q41++(i+1)q97+O(q100) q + ( - i - 1) q^{5} + (2 i - 2) q^{7} + ( - 4 i + 4) q^{11} + (i + 4) q^{17} - 4 i q^{19} + ( - 6 i + 6) q^{23} - 3 i q^{25} + ( - 5 i - 5) q^{29} + (6 i + 6) q^{31} + 4 q^{35} + ( - 3 i - 3) q^{37} + ( - i + 1) q^{41} + \cdots + (i + 1) q^{97} +O(q^{100}) Copy content Toggle raw display
Tr(f)(q)\operatorname{Tr}(f)(q) == 2q2q54q7+8q11+8q17+12q2310q29+12q31+8q356q37+2q418q4716q5514q61+8q67+4q71+22q73+20q796q85++2q97+O(q100) 2 q - 2 q^{5} - 4 q^{7} + 8 q^{11} + 8 q^{17} + 12 q^{23} - 10 q^{29} + 12 q^{31} + 8 q^{35} - 6 q^{37} + 2 q^{41} - 8 q^{47} - 16 q^{55} - 14 q^{61} + 8 q^{67} + 4 q^{71} + 22 q^{73} + 20 q^{79} - 6 q^{85}+ \cdots + 2 q^{97}+O(q^{100}) Copy content Toggle raw display

Character values

We give the values of χ\chi on generators for (Z/612Z)×\left(\mathbb{Z}/612\mathbb{Z}\right)^\times.

nn 3737 137137 307307
χ(n)\chi(n) ii 11 11

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}
217.1
1.00000i
1.00000i
0 0 0 −1.00000 1.00000i 0 −2.00000 + 2.00000i 0 0 0
361.1 0 0 0 −1.00000 + 1.00000i 0 −2.00000 2.00000i 0 0 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
17.c even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 612.2.k.a 2
3.b odd 2 1 612.2.k.b yes 2
4.b odd 2 1 2448.2.be.f 2
12.b even 2 1 2448.2.be.k 2
17.c even 4 1 inner 612.2.k.a 2
51.f odd 4 1 612.2.k.b yes 2
68.f odd 4 1 2448.2.be.f 2
204.l even 4 1 2448.2.be.k 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
612.2.k.a 2 1.a even 1 1 trivial
612.2.k.a 2 17.c even 4 1 inner
612.2.k.b yes 2 3.b odd 2 1
612.2.k.b yes 2 51.f odd 4 1
2448.2.be.f 2 4.b odd 2 1
2448.2.be.f 2 68.f odd 4 1
2448.2.be.k 2 12.b even 2 1
2448.2.be.k 2 204.l even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on S2new(612,[χ])S_{2}^{\mathrm{new}}(612, [\chi]):

T52+2T5+2 T_{5}^{2} + 2T_{5} + 2 Copy content Toggle raw display
T72+4T7+8 T_{7}^{2} + 4T_{7} + 8 Copy content Toggle raw display

Hecke characteristic polynomials

pp Fp(T)F_p(T)
22 T2 T^{2} Copy content Toggle raw display
33 T2 T^{2} Copy content Toggle raw display
55 T2+2T+2 T^{2} + 2T + 2 Copy content Toggle raw display
77 T2+4T+8 T^{2} + 4T + 8 Copy content Toggle raw display
1111 T28T+32 T^{2} - 8T + 32 Copy content Toggle raw display
1313 T2 T^{2} Copy content Toggle raw display
1717 T28T+17 T^{2} - 8T + 17 Copy content Toggle raw display
1919 T2+16 T^{2} + 16 Copy content Toggle raw display
2323 T212T+72 T^{2} - 12T + 72 Copy content Toggle raw display
2929 T2+10T+50 T^{2} + 10T + 50 Copy content Toggle raw display
3131 T212T+72 T^{2} - 12T + 72 Copy content Toggle raw display
3737 T2+6T+18 T^{2} + 6T + 18 Copy content Toggle raw display
4141 T22T+2 T^{2} - 2T + 2 Copy content Toggle raw display
4343 T2 T^{2} Copy content Toggle raw display
4747 (T+4)2 (T + 4)^{2} Copy content Toggle raw display
5353 T2+36 T^{2} + 36 Copy content Toggle raw display
5959 T2+144 T^{2} + 144 Copy content Toggle raw display
6161 T2+14T+98 T^{2} + 14T + 98 Copy content Toggle raw display
6767 (T4)2 (T - 4)^{2} Copy content Toggle raw display
7171 T24T+8 T^{2} - 4T + 8 Copy content Toggle raw display
7373 T222T+242 T^{2} - 22T + 242 Copy content Toggle raw display
7979 T220T+200 T^{2} - 20T + 200 Copy content Toggle raw display
8383 T2+16 T^{2} + 16 Copy content Toggle raw display
8989 (T6)2 (T - 6)^{2} Copy content Toggle raw display
9797 T22T+2 T^{2} - 2T + 2 Copy content Toggle raw display
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