Properties

Label 624.2.bv.g
Level 624624
Weight 22
Character orbit 624.bv
Analytic conductor 4.9834.983
Analytic rank 00
Dimension 88
Inner twists 22

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [624,2,Mod(49,624)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(624, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("624.49");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: N N == 624=24313 624 = 2^{4} \cdot 3 \cdot 13
Weight: k k == 2 2
Character orbit: [χ][\chi] == 624.bv (of order 66, degree 22, not 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: 4.982665086134.98266508613
Analytic rank: 00
Dimension: 88
Relative dimension: 44 over Q(ζ6)\Q(\zeta_{6})
Coefficient field: 8.0.649638144.4
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: x814x6+75x4170x2+169 x^{8} - 14x^{6} + 75x^{4} - 170x^{2} + 169 Copy content Toggle raw display
Coefficient ring: Z[a1,,a7]\Z[a_1, \ldots, a_{7}]
Coefficient ring index: 24 2^{4}
Twist minimal: no (minimal twist has level 312)
Sato-Tate group: SU(2)[C6]\mathrm{SU}(2)[C_{6}]

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+(β2+1)q3+(β5+β2)q5+(β7+β6+β3++1)q7+β2q9+(β6+β5+β3++1)q11+(β7β6β5+2)q13++(β7+β6+β3)q99+O(q100) q + (\beta_{2} + 1) q^{3} + ( - \beta_{5} + \beta_{2}) q^{5} + (\beta_{7} + \beta_{6} + \beta_{3} + \cdots + 1) q^{7} + \beta_{2} q^{9} + ( - \beta_{6} + \beta_{5} + \beta_{3} + \cdots + 1) q^{11} + ( - \beta_{7} - \beta_{6} - \beta_{5} + \cdots - 2) q^{13}+ \cdots + (\beta_{7} + \beta_{6} + \cdots - \beta_{3}) q^{99}+O(q^{100}) Copy content Toggle raw display
Tr(f)(q)\operatorname{Tr}(f)(q) == 8q+4q34q9+6q116q136q15+12q176q19+2q234q258q27+6q29+6q3310q3524q41+8q436q45+18q49+24q51+66q97+O(q100) 8 q + 4 q^{3} - 4 q^{9} + 6 q^{11} - 6 q^{13} - 6 q^{15} + 12 q^{17} - 6 q^{19} + 2 q^{23} - 4 q^{25} - 8 q^{27} + 6 q^{29} + 6 q^{33} - 10 q^{35} - 24 q^{41} + 8 q^{43} - 6 q^{45} + 18 q^{49} + 24 q^{51}+ \cdots - 66 q^{97}+O(q^{100}) Copy content Toggle raw display

Basis of coefficient ring in terms of a root ν\nu of x814x6+75x4170x2+169 x^{8} - 14x^{6} + 75x^{4} - 170x^{2} + 169 : Copy content Toggle raw display

β1\beta_{1}== (ν725ν5+315ν3+182ν2740ν546)/364 ( -\nu^{7} - 25\nu^{5} + 315\nu^{3} + 182\nu^{2} - 740\nu - 546 ) / 364 Copy content Toggle raw display
β2\beta_{2}== (ν7+14ν562ν3+79ν26)/52 ( -\nu^{7} + 14\nu^{5} - 62\nu^{3} + 79\nu - 26 ) / 52 Copy content Toggle raw display
β3\beta_{3}== (7ν7+13ν6+98ν5130ν4616ν3+546ν2+1463ν845)/364 ( -7\nu^{7} + 13\nu^{6} + 98\nu^{5} - 130\nu^{4} - 616\nu^{3} + 546\nu^{2} + 1463\nu - 845 ) / 364 Copy content Toggle raw display
β4\beta_{4}== (4ν7+39ν682ν5481ν4+560ν3+1911ν21226ν2080)/364 ( 4\nu^{7} + 39\nu^{6} - 82\nu^{5} - 481\nu^{4} + 560\nu^{3} + 1911\nu^{2} - 1226\nu - 2080 ) / 364 Copy content Toggle raw display
β5\beta_{5}== (5ν713ν6+57ν5+130ν4245ν3546ν2+304ν+663)/182 ( -5\nu^{7} - 13\nu^{6} + 57\nu^{5} + 130\nu^{4} - 245\nu^{3} - 546\nu^{2} + 304\nu + 663 ) / 182 Copy content Toggle raw display
β6\beta_{6}== (11ν7+89ν5+91ν4175ν3819ν2314ν+1729)/364 ( -11\nu^{7} + 89\nu^{5} + 91\nu^{4} - 175\nu^{3} - 819\nu^{2} - 314\nu + 1729 ) / 364 Copy content Toggle raw display
β7\beta_{7}== (16ν713ν6146ν5+130ν4+420ν3546ν2+192ν+663)/364 ( 16\nu^{7} - 13\nu^{6} - 146\nu^{5} + 130\nu^{4} + 420\nu^{3} - 546\nu^{2} + 192\nu + 663 ) / 364 Copy content Toggle raw display
ν\nu== (3β7+β6+β5+β4+2β3+2β2+3β1+2)/4 ( 3\beta_{7} + \beta_{6} + \beta_{5} + \beta_{4} + 2\beta_{3} + 2\beta_{2} + 3\beta _1 + 2 ) / 4 Copy content Toggle raw display
ν2\nu^{2}== (β7β6β5β4+β1+6)/2 ( -\beta_{7} - \beta_{6} - \beta_{5} - \beta_{4} + \beta _1 + 6 ) / 2 Copy content Toggle raw display
ν3\nu^{3}== (14β7+5β6+2β5+5β4+3β3+19β2+15β1+11)/4 ( 14\beta_{7} + 5\beta_{6} + 2\beta_{5} + 5\beta_{4} + 3\beta_{3} + 19\beta_{2} + 15\beta _1 + 11 ) / 4 Copy content Toggle raw display
ν4\nu^{4}== 3β7β66β55β4+3β1+8 -3\beta_{7} - \beta_{6} - 6\beta_{5} - 5\beta_{4} + 3\beta _1 + 8 Copy content Toggle raw display
ν5\nu^{5}== (63β7+18β611β5+18β413β3+147β2+54β1+67)/4 ( 63\beta_{7} + 18\beta_{6} - 11\beta_{5} + 18\beta_{4} - 13\beta_{3} + 147\beta_{2} + 54\beta _1 + 67 ) / 4 Copy content Toggle raw display
ν6\nu^{6}== (25β7+22β699β558β4+7β3+7β2+18β1+17)/2 ( -25\beta_{7} + 22\beta_{6} - 99\beta_{5} - 58\beta_{4} + 7\beta_{3} + 7\beta_{2} + 18\beta _1 + 17 ) / 2 Copy content Toggle raw display
ν7\nu^{7}== (251β7+21β6199β5+21β4210β3+830β2+63β1+310)/4 ( 251\beta_{7} + 21\beta_{6} - 199\beta_{5} + 21\beta_{4} - 210\beta_{3} + 830\beta_{2} + 63\beta _1 + 310 ) / 4 Copy content Toggle raw display

Character values

We give the values of χ\chi on generators for (Z/624Z)×\left(\mathbb{Z}/624\mathbb{Z}\right)^\times.

nn 7979 145145 209209 469469
χ(n)\chi(n) 11 β2-\beta_{2} 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}
49.1
1.42055 0.500000i
−2.34138 + 0.500000i
−1.42055 0.500000i
2.34138 + 0.500000i
2.34138 0.500000i
−1.42055 + 0.500000i
−2.34138 0.500000i
1.42055 + 0.500000i
0 0.500000 + 0.866025i 0 1.55452i 0 2.56383 + 1.48023i 0 −0.500000 + 0.866025i 0
49.2 0 0.500000 + 0.866025i 0 0.475353i 0 −3.94508 2.27769i 0 −0.500000 + 0.866025i 0
49.3 0 0.500000 + 0.866025i 0 1.28657i 0 −1.69781 0.980228i 0 −0.500000 + 0.866025i 0
49.4 0 0.500000 + 0.866025i 0 4.20740i 0 3.07905 + 1.77769i 0 −0.500000 + 0.866025i 0
433.1 0 0.500000 0.866025i 0 4.20740i 0 3.07905 1.77769i 0 −0.500000 0.866025i 0
433.2 0 0.500000 0.866025i 0 1.28657i 0 −1.69781 + 0.980228i 0 −0.500000 0.866025i 0
433.3 0 0.500000 0.866025i 0 0.475353i 0 −3.94508 + 2.27769i 0 −0.500000 0.866025i 0
433.4 0 0.500000 0.866025i 0 1.55452i 0 2.56383 1.48023i 0 −0.500000 0.866025i 0
nn: e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 49.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.e even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 624.2.bv.g 8
3.b odd 2 1 1872.2.by.m 8
4.b odd 2 1 312.2.bf.b 8
12.b even 2 1 936.2.bi.c 8
13.e even 6 1 inner 624.2.bv.g 8
13.f odd 12 1 8112.2.a.cq 4
13.f odd 12 1 8112.2.a.cs 4
39.h odd 6 1 1872.2.by.m 8
52.i odd 6 1 312.2.bf.b 8
52.i odd 6 1 4056.2.c.p 8
52.j odd 6 1 4056.2.c.p 8
52.l even 12 1 4056.2.a.bd 4
52.l even 12 1 4056.2.a.be 4
156.r even 6 1 936.2.bi.c 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
312.2.bf.b 8 4.b odd 2 1
312.2.bf.b 8 52.i odd 6 1
624.2.bv.g 8 1.a even 1 1 trivial
624.2.bv.g 8 13.e even 6 1 inner
936.2.bi.c 8 12.b even 2 1
936.2.bi.c 8 156.r even 6 1
1872.2.by.m 8 3.b odd 2 1
1872.2.by.m 8 39.h odd 6 1
4056.2.a.bd 4 52.l even 12 1
4056.2.a.be 4 52.l even 12 1
4056.2.c.p 8 52.i odd 6 1
4056.2.c.p 8 52.j odd 6 1
8112.2.a.cq 4 13.f odd 12 1
8112.2.a.cs 4 13.f odd 12 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on S2new(624,[χ])S_{2}^{\mathrm{new}}(624, [\chi]):

T58+22T56+81T54+88T52+16 T_{5}^{8} + 22T_{5}^{6} + 81T_{5}^{4} + 88T_{5}^{2} + 16 Copy content Toggle raw display
T7823T76+435T74414T732054T72+1692T7+8836 T_{7}^{8} - 23T_{7}^{6} + 435T_{7}^{4} - 414T_{7}^{3} - 2054T_{7}^{2} + 1692T_{7} + 8836 Copy content Toggle raw display

Hecke characteristic polynomials

pp Fp(T)F_p(T)
22 T8 T^{8} Copy content Toggle raw display
33 (T2T+1)4 (T^{2} - T + 1)^{4} Copy content Toggle raw display
55 T8+22T6++16 T^{8} + 22 T^{6} + \cdots + 16 Copy content Toggle raw display
77 T823T6++8836 T^{8} - 23 T^{6} + \cdots + 8836 Copy content Toggle raw display
1111 T86T7++10816 T^{8} - 6 T^{7} + \cdots + 10816 Copy content Toggle raw display
1313 T8+6T7++28561 T^{8} + 6 T^{7} + \cdots + 28561 Copy content Toggle raw display
1717 T812T7++2304 T^{8} - 12 T^{7} + \cdots + 2304 Copy content Toggle raw display
1919 T8+6T7++141376 T^{8} + 6 T^{7} + \cdots + 141376 Copy content Toggle raw display
2323 T82T7++64 T^{8} - 2 T^{7} + \cdots + 64 Copy content Toggle raw display
2929 T86T7++1296 T^{8} - 6 T^{7} + \cdots + 1296 Copy content Toggle raw display
3131 T8+118T6++141376 T^{8} + 118 T^{6} + \cdots + 141376 Copy content Toggle raw display
3737 T841T6++16384 T^{8} - 41 T^{6} + \cdots + 16384 Copy content Toggle raw display
4141 T8+24T7++3154176 T^{8} + 24 T^{7} + \cdots + 3154176 Copy content Toggle raw display
4343 T88T7++481636 T^{8} - 8 T^{7} + \cdots + 481636 Copy content Toggle raw display
4747 T8+160T6++2096704 T^{8} + 160 T^{6} + \cdots + 2096704 Copy content Toggle raw display
5353 (T42T3++208)2 (T^{4} - 2 T^{3} + \cdots + 208)^{2} Copy content Toggle raw display
5959 T836T7++262144 T^{8} - 36 T^{7} + \cdots + 262144 Copy content Toggle raw display
6161 T82T7++42003361 T^{8} - 2 T^{7} + \cdots + 42003361 Copy content Toggle raw display
6767 T8+36T7++45796 T^{8} + 36 T^{7} + \cdots + 45796 Copy content Toggle raw display
7171 T8+6T7++1272384 T^{8} + 6 T^{7} + \cdots + 1272384 Copy content Toggle raw display
7373 T8+124T6++185761 T^{8} + 124 T^{6} + \cdots + 185761 Copy content Toggle raw display
7979 (T46T3++216)2 (T^{4} - 6 T^{3} + \cdots + 216)^{2} Copy content Toggle raw display
8383 T8+312T6++1557504 T^{8} + 312 T^{6} + \cdots + 1557504 Copy content Toggle raw display
8989 T8+12T7++589824 T^{8} + 12 T^{7} + \cdots + 589824 Copy content Toggle raw display
9797 T8+66T7++33039504 T^{8} + 66 T^{7} + \cdots + 33039504 Copy content Toggle raw display
show more
show less