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Properties

Label 32.2.g.b
Level 3232
Weight 22
Character orbit 32.g
Analytic conductor 0.2560.256
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] = [32,2,Mod(5,32)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(32, base_ring=CyclotomicField(8))
 
chi = DirichletCharacter(H, H._module([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("32.5");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: N N == 32=25 32 = 2^{5}
Weight: k k == 2 2
Character orbit: [χ][\chi] == 32.g (of order 88, 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: 0.2555212864680.255521286468
Analytic rank: 00
Dimension: 88
Relative dimension: 22 over Q(ζ8)\Q(\zeta_{8})
Coefficient field: 8.0.18939904.2
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: x8−4x7+14x6−28x5+43x4−44x3+30x2−12x+2 x^{8} - 4x^{7} + 14x^{6} - 28x^{5} + 43x^{4} - 44x^{3} + 30x^{2} - 12x + 2 Copy content Toggle raw display
Coefficient ring: Z[a1,a2,a3]\Z[a_1, a_2, a_3]
Coefficient ring index: 2 2
Twist minimal: yes
Sato-Tate group: SU(2)[C8]\mathrm{SU}(2)[C_{8}]

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−β2q2+(β7+β5+β3+β1)q3+(β6−β3+β2)q4+(−β7−β6)q5+(−2β7−β5+β2+⋯−2)q6+⋯+(β6+β3−6β2+⋯+7)q99+O(q100) q - \beta_{2} q^{2} + (\beta_{7} + \beta_{5} + \beta_{3} + \beta_1) q^{3} + (\beta_{6} - \beta_{3} + \beta_{2}) q^{4} + ( - \beta_{7} - \beta_{6}) q^{5} + ( - 2 \beta_{7} - \beta_{5} + \beta_{2} + \cdots - 2) q^{6}+ \cdots + (\beta_{6} + \beta_{3} - 6 \beta_{2} + \cdots + 7) q^{99}+O(q^{100}) Copy content Toggle raw display
Tr⁡(f)(q)\operatorname{Tr}(f)(q) == 8q−4q2−4q3+4q4−8q6−8q7−4q8+4q11+12q12−8q13+12q14+20q18+4q19+4q20+4q22−8q23−8q24−8q25−20q26+8q27+⋯+28q99+O(q100) 8 q - 4 q^{2} - 4 q^{3} + 4 q^{4} - 8 q^{6} - 8 q^{7} - 4 q^{8} + 4 q^{11} + 12 q^{12} - 8 q^{13} + 12 q^{14} + 20 q^{18} + 4 q^{19} + 4 q^{20} + 4 q^{22} - 8 q^{23} - 8 q^{24} - 8 q^{25} - 20 q^{26} + 8 q^{27}+ \cdots + 28 q^{99}+O(q^{100}) Copy content Toggle raw display

Basis of coefficient ring in terms of a root ν\nu of x8−4x7+14x6−28x5+43x4−44x3+30x2−12x+2 x^{8} - 4x^{7} + 14x^{6} - 28x^{5} + 43x^{4} - 44x^{3} + 30x^{2} - 12x + 2 : Copy content Toggle raw display

β1\beta_{1}== 2ν7−7ν6+24ν5−42ν4+59ν3−48ν2+24ν−5 2\nu^{7} - 7\nu^{6} + 24\nu^{5} - 42\nu^{4} + 59\nu^{3} - 48\nu^{2} + 24\nu - 5 Copy content Toggle raw display
β2\beta_{2}== −2ν7+7ν6−24ν5+43ν4−61ν3+54ν2−29ν+8 -2\nu^{7} + 7\nu^{6} - 24\nu^{5} + 43\nu^{4} - 61\nu^{3} + 54\nu^{2} - 29\nu + 8 Copy content Toggle raw display
β3\beta_{3}== −3ν7+10ν6−35ν5+60ν4−87ν3+73ν2−42ν+11 -3\nu^{7} + 10\nu^{6} - 35\nu^{5} + 60\nu^{4} - 87\nu^{3} + 73\nu^{2} - 42\nu + 11 Copy content Toggle raw display
β4\beta_{4}== −3ν7+11ν6−38ν5+70ν4−102ν3+91ν2−53ν+13 -3\nu^{7} + 11\nu^{6} - 38\nu^{5} + 70\nu^{4} - 102\nu^{3} + 91\nu^{2} - 53\nu + 13 Copy content Toggle raw display
β5\beta_{5}== 5ν7−17ν6+60ν5−105ν4+155ν3−133ν2+77ν−19 5\nu^{7} - 17\nu^{6} + 60\nu^{5} - 105\nu^{4} + 155\nu^{3} - 133\nu^{2} + 77\nu - 19 Copy content Toggle raw display
β6\beta_{6}== −5ν7+18ν6−63ν5+115ν4−170ν3+152ν2−89ν+23 -5\nu^{7} + 18\nu^{6} - 63\nu^{5} + 115\nu^{4} - 170\nu^{3} + 152\nu^{2} - 89\nu + 23 Copy content Toggle raw display
β7\beta_{7}== −8ν7+28ν6−98ν5+175ν4−256ν3+223ν2−126ν+31 -8\nu^{7} + 28\nu^{6} - 98\nu^{5} + 175\nu^{4} - 256\nu^{3} + 223\nu^{2} - 126\nu + 31 Copy content Toggle raw display

Display νj\nu^j in terms of βi\beta_i

ν\nu== (−β7+β6−β5−β4−β3+β2−β1)/2 ( -\beta_{7} + \beta_{6} - \beta_{5} - \beta_{4} - \beta_{3} + \beta_{2} - \beta_1 ) / 2 Copy content Toggle raw display
ν2\nu^{2}== (−β7+3β6+β5−3β4+β3+β2−β1−4)/2 ( -\beta_{7} + 3\beta_{6} + \beta_{5} - 3\beta_{4} + \beta_{3} + \beta_{2} - \beta _1 - 4 ) / 2 Copy content Toggle raw display
ν3\nu^{3}== (5β7−β6+7β5−β4+5β3−3β2+3β1−2)/2 ( 5\beta_{7} - \beta_{6} + 7\beta_{5} - \beta_{4} + 5\beta_{3} - 3\beta_{2} + 3\beta _1 - 2 ) / 2 Copy content Toggle raw display
ν4\nu^{4}== (11β7−15β6+3β5+11β4−β3−5β2+9β1+14)/2 ( 11\beta_{7} - 15\beta_{6} + 3\beta_{5} + 11\beta_{4} - \beta_{3} - 5\beta_{2} + 9\beta _1 + 14 ) / 2 Copy content Toggle raw display
ν5\nu^{5}== (−13β7−11β6−29β5+17β4−23β3+13β2−3β1+18)/2 ( -13\beta_{7} - 11\beta_{6} - 29\beta_{5} + 17\beta_{4} - 23\beta_{3} + 13\beta_{2} - 3\beta _1 + 18 ) / 2 Copy content Toggle raw display
ν6\nu^{6}== (−67β7+59β6−41β5−29β4−15β3+37β2−47β1−48)/2 ( -67\beta_{7} + 59\beta_{6} - 41\beta_{5} - 29\beta_{4} - 15\beta_{3} + 37\beta_{2} - 47\beta _1 - 48 ) / 2 Copy content Toggle raw display
ν7\nu^{7}== (−7β7+113β6+97β5−105β4+91β3−31β2−39β1−122)/2 ( -7\beta_{7} + 113\beta_{6} + 97\beta_{5} - 105\beta_{4} + 91\beta_{3} - 31\beta_{2} - 39\beta _1 - 122 ) / 2 Copy content Toggle raw display

Display βi\beta_i in terms of νj\nu^j

Character values

We give the values of χ\chi on generators for (Z/32Z)×\left(\mathbb{Z}/32\mathbb{Z}\right)^\times.

nn 55 3131
χ(n)\chi(n) −β5-\beta_{5} 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}
5.1
0.500000 + 2.10607i
0.500000 − 0.691860i
0.500000 − 2.10607i
0.500000 + 0.691860i
0.500000 + 0.0297061i
0.500000 − 1.44392i
0.500000 − 0.0297061i
0.500000 + 1.44392i
−1.26330 − 0.635665i −1.07947 − 2.60607i 1.19186 + 1.60607i 0.707107 + 0.292893i −0.292893 + 3.97844i 1.68554 + 1.68554i −0.484753 − 2.78658i −3.50504 + 3.50504i −0.707107 − 0.819496i
5.2 −0.443806 + 1.34277i 0.0794708 + 0.191860i −1.60607 − 1.19186i 0.707107 + 0.292893i −0.292893 + 0.0215628i −2.27133 − 2.27133i 2.31318 − 1.62764i 2.09083 − 2.09083i −0.707107 + 0.819496i
13.1 −1.26330 + 0.635665i −1.07947 + 2.60607i 1.19186 − 1.60607i 0.707107 − 0.292893i −0.292893 − 3.97844i 1.68554 − 1.68554i −0.484753 + 2.78658i −3.50504 − 3.50504i −0.707107 + 0.819496i
13.2 −0.443806 − 1.34277i 0.0794708 − 0.191860i −1.60607 + 1.19186i 0.707107 − 0.292893i −0.292893 − 0.0215628i −2.27133 + 2.27133i 2.31318 + 1.62764i 2.09083 + 2.09083i −0.707107 − 0.819496i
21.1 −1.40426 + 0.167452i 1.27882 − 0.529706i 1.94392 − 0.470294i −0.707107 + 1.70711i −1.70711 + 0.957989i −2.74912 − 2.74912i −2.65103 + 0.985930i −0.766519 + 0.766519i 0.707107 − 2.51564i
21.2 1.11137 − 0.874559i −2.27882 + 0.943920i 0.470294 − 1.94392i −0.707107 + 1.70711i −1.70711 + 3.04201i −0.665096 − 0.665096i −1.17740 − 2.57172i 2.18073 − 2.18073i 0.707107 + 2.51564i
29.1 −1.40426 − 0.167452i 1.27882 + 0.529706i 1.94392 + 0.470294i −0.707107 − 1.70711i −1.70711 − 0.957989i −2.74912 + 2.74912i −2.65103 − 0.985930i −0.766519 − 0.766519i 0.707107 + 2.51564i
29.2 1.11137 + 0.874559i −2.27882 − 0.943920i 0.470294 + 1.94392i −0.707107 − 1.70711i −1.70711 − 3.04201i −0.665096 + 0.665096i −1.17740 + 2.57172i 2.18073 + 2.18073i 0.707107 − 2.51564i
nn: e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 5.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
32.g even 8 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 32.2.g.b ✓ 8
3.b odd 2 1 288.2.v.b 8
4.b odd 2 1 128.2.g.b 8
5.b even 2 1 800.2.y.b 8
5.c odd 4 1 800.2.ba.c 8
5.c odd 4 1 800.2.ba.d 8
8.b even 2 1 256.2.g.d 8
8.d odd 2 1 256.2.g.c 8
12.b even 2 1 1152.2.v.b 8
16.e even 4 1 512.2.g.e 8
16.e even 4 1 512.2.g.h 8
16.f odd 4 1 512.2.g.f 8
16.f odd 4 1 512.2.g.g 8
32.g even 8 1 inner 32.2.g.b ✓ 8
32.g even 8 1 256.2.g.d 8
32.g even 8 1 512.2.g.e 8
32.g even 8 1 512.2.g.h 8
32.h odd 8 1 128.2.g.b 8
32.h odd 8 1 256.2.g.c 8
32.h odd 8 1 512.2.g.f 8
32.h odd 8 1 512.2.g.g 8
64.i even 16 2 4096.2.a.k 8
64.j odd 16 2 4096.2.a.q 8
96.o even 8 1 1152.2.v.b 8
96.p odd 8 1 288.2.v.b 8
160.v odd 8 1 800.2.ba.c 8
160.z even 8 1 800.2.y.b 8
160.bb odd 8 1 800.2.ba.d 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
32.2.g.b ✓ 8 1.a even 1 1 trivial
32.2.g.b ✓ 8 32.g even 8 1 inner
128.2.g.b 8 4.b odd 2 1
128.2.g.b 8 32.h odd 8 1
256.2.g.c 8 8.d odd 2 1
256.2.g.c 8 32.h odd 8 1
256.2.g.d 8 8.b even 2 1
256.2.g.d 8 32.g even 8 1
288.2.v.b 8 3.b odd 2 1
288.2.v.b 8 96.p odd 8 1
512.2.g.e 8 16.e even 4 1
512.2.g.e 8 32.g even 8 1
512.2.g.f 8 16.f odd 4 1
512.2.g.f 8 32.h odd 8 1
512.2.g.g 8 16.f odd 4 1
512.2.g.g 8 32.h odd 8 1
512.2.g.h 8 16.e even 4 1
512.2.g.h 8 32.g even 8 1
800.2.y.b 8 5.b even 2 1
800.2.y.b 8 160.z even 8 1
800.2.ba.c 8 5.c odd 4 1
800.2.ba.c 8 160.v odd 8 1
800.2.ba.d 8 5.c odd 4 1
800.2.ba.d 8 160.bb odd 8 1
1152.2.v.b 8 12.b even 2 1
1152.2.v.b 8 96.o even 8 1
4096.2.a.k 8 64.i even 16 2
4096.2.a.q 8 64.j odd 16 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator T38+4T37+8T36−32T34−24T33+96T32−16T3+4 T_{3}^{8} + 4T_{3}^{7} + 8T_{3}^{6} - 32T_{3}^{4} - 24T_{3}^{3} + 96T_{3}^{2} - 16T_{3} + 4 acting on S2new(32,[χ])S_{2}^{\mathrm{new}}(32, [\chi]). Copy content Toggle raw display

Hecke characteristic polynomials

pp Fp(T)F_p(T)
22 T8+4T7+⋯+16 T^{8} + 4 T^{7} + \cdots + 16 Copy content Toggle raw display
33 T8+4T7+⋯+4 T^{8} + 4 T^{7} + \cdots + 4 Copy content Toggle raw display
55 (T4+2T2−4T+2)2 (T^{4} + 2 T^{2} - 4 T + 2)^{2} Copy content Toggle raw display
77 T8+8T7+⋯+784 T^{8} + 8 T^{7} + \cdots + 784 Copy content Toggle raw display
1111 T8−4T7+⋯+4 T^{8} - 4 T^{7} + \cdots + 4 Copy content Toggle raw display
1313 T8+8T7+⋯+6724 T^{8} + 8 T^{7} + \cdots + 6724 Copy content Toggle raw display
1717 T8+64T6+⋯+256 T^{8} + 64 T^{6} + \cdots + 256 Copy content Toggle raw display
1919 T8−4T7+⋯+196 T^{8} - 4 T^{7} + \cdots + 196 Copy content Toggle raw display
2323 T8+8T7+⋯+16 T^{8} + 8 T^{7} + \cdots + 16 Copy content Toggle raw display
2929 T8−12T6+⋯+188356 T^{8} - 12 T^{6} + \cdots + 188356 Copy content Toggle raw display
3131 (T2−8T+8)4 (T^{2} - 8 T + 8)^{4} Copy content Toggle raw display
3737 T8+8T7+⋯+64516 T^{8} + 8 T^{7} + \cdots + 64516 Copy content Toggle raw display
4141 T8−8T7+⋯+26896 T^{8} - 8 T^{7} + \cdots + 26896 Copy content Toggle raw display
4343 T8+12T7+⋯+31684 T^{8} + 12 T^{7} + \cdots + 31684 Copy content Toggle raw display
4747 T8+64T6+⋯+256 T^{8} + 64 T^{6} + \cdots + 256 Copy content Toggle raw display
5353 T8−8T7+⋯+158404 T^{8} - 8 T^{7} + \cdots + 158404 Copy content Toggle raw display
5959 T8+20T7+⋯+643204 T^{8} + 20 T^{7} + \cdots + 643204 Copy content Toggle raw display
6161 T8−24T7+⋯+42436 T^{8} - 24 T^{7} + \cdots + 42436 Copy content Toggle raw display
6767 T8+36T7+⋯+1285956 T^{8} + 36 T^{7} + \cdots + 1285956 Copy content Toggle raw display
7171 T8+24T7+⋯+21196816 T^{8} + 24 T^{7} + \cdots + 21196816 Copy content Toggle raw display
7373 T8+32T7+⋯+38416 T^{8} + 32 T^{7} + \cdots + 38416 Copy content Toggle raw display
7979 T8+512T6+⋯+99361024 T^{8} + 512 T^{6} + \cdots + 99361024 Copy content Toggle raw display
8383 T8−20T7+⋯+138250564 T^{8} - 20 T^{7} + \cdots + 138250564 Copy content Toggle raw display
8989 T8+16T7+⋯+17007376 T^{8} + 16 T^{7} + \cdots + 17007376 Copy content Toggle raw display
9797 (T4−16T3+⋯−992)2 (T^{4} - 16 T^{3} + \cdots - 992)^{2} Copy content Toggle raw display
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