Properties

Label 168.3.z.b
Level 168168
Weight 33
Character orbit 168.z
Analytic conductor 4.5784.578
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] = [168,3,Mod(73,168)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(168, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("168.73");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: N N == 168=2337 168 = 2^{3} \cdot 3 \cdot 7
Weight: k k == 3 3
Character orbit: [χ][\chi] == 168.z (of order 66, 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: 4.577668441254.57766844125
Analytic rank: 00
Dimension: 88
Relative dimension: 44 over Q(ζ6)\Q(\zeta_{6})
Coefficient field: 8.0.35911766016.9
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: x82x77x62x5+78x418x3153x2230x+529 x^{8} - 2x^{7} - 7x^{6} - 2x^{5} + 78x^{4} - 18x^{3} - 153x^{2} - 230x + 529 Copy content Toggle raw display
Coefficient ring: Z[a1,,a7]\Z[a_1, \ldots, a_{7}]
Coefficient ring index: 247 2^{4}\cdot 7
Twist minimal: yes
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+(β71)q5+(β5+β4+β3++1)q73β2q9+(2β7+β6β5+6)q11+(β7+β6+2β5+2)q13++(3β73β63β4+15)q99+O(q100) q + ( - \beta_{2} + 1) q^{3} + (\beta_{7} - 1) q^{5} + ( - \beta_{5} + \beta_{4} + \beta_{3} + \cdots + 1) q^{7} - 3 \beta_{2} q^{9} + (2 \beta_{7} + \beta_{6} - \beta_{5} + \cdots - 6) q^{11} + (\beta_{7} + \beta_{6} + 2 \beta_{5} + \cdots - 2) q^{13}+ \cdots + (3 \beta_{7} - 3 \beta_{6} - 3 \beta_{4} + \cdots - 15) q^{99}+O(q^{100}) Copy content Toggle raw display
Tr(f)(q)\operatorname{Tr}(f)(q) == 8q+12q36q5+8q7+12q922q1112q15+36q17+42q19+6q21+48q23+42q25+68q2960q3166q3312q35118q3718q39+132q99+O(q100) 8 q + 12 q^{3} - 6 q^{5} + 8 q^{7} + 12 q^{9} - 22 q^{11} - 12 q^{15} + 36 q^{17} + 42 q^{19} + 6 q^{21} + 48 q^{23} + 42 q^{25} + 68 q^{29} - 60 q^{31} - 66 q^{33} - 12 q^{35} - 118 q^{37} - 18 q^{39}+ \cdots - 132 q^{99}+O(q^{100}) Copy content Toggle raw display

Basis of coefficient ring in terms of a root ν\nu of x82x77x62x5+78x418x3153x2230x+529 x^{8} - 2x^{7} - 7x^{6} - 2x^{5} + 78x^{4} - 18x^{3} - 153x^{2} - 230x + 529 : Copy content Toggle raw display

β1\beta_{1}== (1844ν75827ν6+5814ν5+40633ν4+110804ν3500966ν2++876346)/329245 ( 1844 \nu^{7} - 5827 \nu^{6} + 5814 \nu^{5} + 40633 \nu^{4} + 110804 \nu^{3} - 500966 \nu^{2} + \cdots + 876346 ) / 329245 Copy content Toggle raw display
β2\beta_{2}== (4244ν7+873ν633756ν571462ν4+213594ν3+469674ν2+1693329)/658490 ( 4244 \nu^{7} + 873 \nu^{6} - 33756 \nu^{5} - 71462 \nu^{4} + 213594 \nu^{3} + 469674 \nu^{2} + \cdots - 1693329 ) / 658490 Copy content Toggle raw display
β3\beta_{3}== (4146ν74888ν630494ν561123ν4+340546ν3+333461ν2+2591226)/329245 ( 4146 \nu^{7} - 4888 \nu^{6} - 30494 \nu^{5} - 61123 \nu^{4} + 340546 \nu^{3} + 333461 \nu^{2} + \cdots - 2591226 ) / 329245 Copy content Toggle raw display
β4\beta_{4}== (33ν7+41ν6222ν5414ν4+488ν3+1608ν2207ν+3637)/2045 ( 33\nu^{7} + 41\nu^{6} - 222\nu^{5} - 414\nu^{4} + 488\nu^{3} + 1608\nu^{2} - 207\nu + 3637 ) / 2045 Copy content Toggle raw display
β5\beta_{5}== (6546ν7+1812ν670064ν5173218ν4+443336ν3+974856ν2+3514676)/329245 ( 6546 \nu^{7} + 1812 \nu^{6} - 70064 \nu^{5} - 173218 \nu^{4} + 443336 \nu^{3} + 974856 \nu^{2} + \cdots - 3514676 ) / 329245 Copy content Toggle raw display
β6\beta_{6}== (11533ν7+30909ν6+121832ν529696ν4997968ν3162453ν2++2603393)/329245 ( - 11533 \nu^{7} + 30909 \nu^{6} + 121832 \nu^{5} - 29696 \nu^{4} - 997968 \nu^{3} - 162453 \nu^{2} + \cdots + 2603393 ) / 329245 Copy content Toggle raw display
β7\beta_{7}== (45244ν75583ν6+215876ν5+916372ν41365974ν3++10829159)/658490 ( - 45244 \nu^{7} - 5583 \nu^{6} + 215876 \nu^{5} + 916372 \nu^{4} - 1365974 \nu^{3} + \cdots + 10829159 ) / 658490 Copy content Toggle raw display
ν\nu== (β5+2β4+4β38β22β1+2)/14 ( -\beta_{5} + 2\beta_{4} + 4\beta_{3} - 8\beta_{2} - 2\beta _1 + 2 ) / 14 Copy content Toggle raw display
ν2\nu^{2}== (5β54β4β3+30β23β1+31)/7 ( -5\beta_{5} - 4\beta_{4} - \beta_{3} + 30\beta_{2} - 3\beta _1 + 31 ) / 7 Copy content Toggle raw display
ν3\nu^{3}== (7β7+7β6+9β5+3β4+27β3+9β2+18β1+108)/14 ( 7\beta_{7} + 7\beta_{6} + 9\beta_{5} + 3\beta_{4} + 27\beta_{3} + 9\beta_{2} + 18\beta _1 + 108 ) / 14 Copy content Toggle raw display
ν4\nu^{4}== (7β733β5+10β4+20β3+121β210β1+10)/7 ( 7\beta_{7} - 33\beta_{5} + 10\beta_{4} + 20\beta_{3} + 121\beta_{2} - 10\beta _1 + 10 ) / 7 Copy content Toggle raw display
ν5\nu^{5}== (49β6108β5141β4+33β3+802β2+99β1+769)/14 ( 49\beta_{6} - 108\beta_{5} - 141\beta_{4} + 33\beta_{3} + 802\beta_{2} + 99\beta _1 + 769 ) / 14 Copy content Toggle raw display
ν6\nu^{6}== (91β7+91β6+75β5+235β4+225β3+75β2+150β1269)/7 ( 91\beta_{7} + 91\beta_{6} + 75\beta_{5} + 235\beta_{4} + 225\beta_{3} + 75\beta_{2} + 150\beta _1 - 269 ) / 7 Copy content Toggle raw display
ν7\nu^{7}== (22β7199β58β416β3+734β2+8β18)/2 ( -22\beta_{7} - 199\beta_{5} - 8\beta_{4} - 16\beta_{3} + 734\beta_{2} + 8\beta _1 - 8 ) / 2 Copy content Toggle raw display

Character values

We give the values of χ\chi on generators for (Z/168Z)×\left(\mathbb{Z}/168\mathbb{Z}\right)^\times.

nn 7373 8585 113113 127127
χ(n)\chi(n) β2-\beta_{2} 11 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}
73.1
1.83172 0.480194i
−1.90015 + 1.67440i
2.40015 0.808379i
−1.33172 + 1.34622i
1.83172 + 0.480194i
−1.90015 1.67440i
2.40015 + 0.808379i
−1.33172 1.34622i
0 1.50000 + 0.866025i 0 −6.80550 + 3.92916i 0 6.99187 0.337312i 0 1.50000 + 2.59808i 0
73.2 0 1.50000 + 0.866025i 0 −4.68140 + 2.70281i 0 −6.12873 + 3.38210i 0 1.50000 + 2.59808i 0
73.3 0 1.50000 + 0.866025i 0 3.18140 1.83678i 0 2.47188 6.54903i 0 1.50000 + 2.59808i 0
73.4 0 1.50000 + 0.866025i 0 5.30550 3.06313i 0 0.664986 + 6.96834i 0 1.50000 + 2.59808i 0
145.1 0 1.50000 0.866025i 0 −6.80550 3.92916i 0 6.99187 + 0.337312i 0 1.50000 2.59808i 0
145.2 0 1.50000 0.866025i 0 −4.68140 2.70281i 0 −6.12873 3.38210i 0 1.50000 2.59808i 0
145.3 0 1.50000 0.866025i 0 3.18140 + 1.83678i 0 2.47188 + 6.54903i 0 1.50000 2.59808i 0
145.4 0 1.50000 0.866025i 0 5.30550 + 3.06313i 0 0.664986 6.96834i 0 1.50000 2.59808i 0
nn: e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 73.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.d odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 168.3.z.b 8
3.b odd 2 1 504.3.by.c 8
4.b odd 2 1 336.3.bh.g 8
7.b odd 2 1 1176.3.z.c 8
7.c even 3 1 1176.3.f.c 8
7.c even 3 1 1176.3.z.c 8
7.d odd 6 1 inner 168.3.z.b 8
7.d odd 6 1 1176.3.f.c 8
12.b even 2 1 1008.3.cg.p 8
21.g even 6 1 504.3.by.c 8
21.g even 6 1 3528.3.f.b 8
21.h odd 6 1 3528.3.f.b 8
28.f even 6 1 336.3.bh.g 8
28.f even 6 1 2352.3.f.g 8
28.g odd 6 1 2352.3.f.g 8
84.j odd 6 1 1008.3.cg.p 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
168.3.z.b 8 1.a even 1 1 trivial
168.3.z.b 8 7.d odd 6 1 inner
336.3.bh.g 8 4.b odd 2 1
336.3.bh.g 8 28.f even 6 1
504.3.by.c 8 3.b odd 2 1
504.3.by.c 8 21.g even 6 1
1008.3.cg.p 8 12.b even 2 1
1008.3.cg.p 8 84.j odd 6 1
1176.3.f.c 8 7.c even 3 1
1176.3.f.c 8 7.d odd 6 1
1176.3.z.c 8 7.b odd 2 1
1176.3.z.c 8 7.c even 3 1
2352.3.f.g 8 28.f even 6 1
2352.3.f.g 8 28.g odd 6 1
3528.3.f.b 8 21.g even 6 1
3528.3.f.b 8 21.h odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator T58+6T5753T56390T55+2861T54+13260T5348268T52195024T5+913936 T_{5}^{8} + 6T_{5}^{7} - 53T_{5}^{6} - 390T_{5}^{5} + 2861T_{5}^{4} + 13260T_{5}^{3} - 48268T_{5}^{2} - 195024T_{5} + 913936 acting on S3new(168,[χ])S_{3}^{\mathrm{new}}(168, [\chi]). Copy content Toggle raw display

Hecke characteristic polynomials

pp Fp(T)F_p(T)
22 T8 T^{8} Copy content Toggle raw display
33 (T23T+3)4 (T^{2} - 3 T + 3)^{4} Copy content Toggle raw display
55 T8+6T7++913936 T^{8} + 6 T^{7} + \cdots + 913936 Copy content Toggle raw display
77 T88T7++5764801 T^{8} - 8 T^{7} + \cdots + 5764801 Copy content Toggle raw display
1111 T8+22T7++17875984 T^{8} + 22 T^{7} + \cdots + 17875984 Copy content Toggle raw display
1313 T8+262T6++2408704 T^{8} + 262 T^{6} + \cdots + 2408704 Copy content Toggle raw display
1717 T8++3470623744 T^{8} + \cdots + 3470623744 Copy content Toggle raw display
1919 T842T7++17272336 T^{8} - 42 T^{7} + \cdots + 17272336 Copy content Toggle raw display
2323 T8++22620160000 T^{8} + \cdots + 22620160000 Copy content Toggle raw display
2929 (T434T3++224128)2 (T^{4} - 34 T^{3} + \cdots + 224128)^{2} Copy content Toggle raw display
3131 T8++25912950625 T^{8} + \cdots + 25912950625 Copy content Toggle raw display
3737 T8++17069945104 T^{8} + \cdots + 17069945104 Copy content Toggle raw display
4141 T8++580010189056 T^{8} + \cdots + 580010189056 Copy content Toggle raw display
4343 (T4+46T3++1658308)2 (T^{4} + 46 T^{3} + \cdots + 1658308)^{2} Copy content Toggle raw display
4747 T8++52408029184 T^{8} + \cdots + 52408029184 Copy content Toggle raw display
5353 T8++1491466217536 T^{8} + \cdots + 1491466217536 Copy content Toggle raw display
5959 T8++1048985640000 T^{8} + \cdots + 1048985640000 Copy content Toggle raw display
6161 T8++43785853599744 T^{8} + \cdots + 43785853599744 Copy content Toggle raw display
6767 T8++95387087104 T^{8} + \cdots + 95387087104 Copy content Toggle raw display
7171 (T4+196T3+13209344)2 (T^{4} + 196 T^{3} + \cdots - 13209344)^{2} Copy content Toggle raw display
7373 T8++622497709609216 T^{8} + \cdots + 622497709609216 Copy content Toggle raw display
7979 T8++26 ⁣ ⁣25 T^{8} + \cdots + 26\!\cdots\!25 Copy content Toggle raw display
8383 T8++839297841424 T^{8} + \cdots + 839297841424 Copy content Toggle raw display
8989 T8++723343446016 T^{8} + \cdots + 723343446016 Copy content Toggle raw display
9797 T8++22 ⁣ ⁣16 T^{8} + \cdots + 22\!\cdots\!16 Copy content Toggle raw display
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