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Properties

Label 72.4.i.a
Level 7272
Weight 44
Character orbit 72.i
Analytic conductor 4.2484.248
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] = [72,4,Mod(25,72)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(72, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 4]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("72.25");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: N N == 72=23⋅32 72 = 2^{3} \cdot 3^{2}
Weight: k k == 4 4
Character orbit: [χ][\chi] == 72.i (of order 33, 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: 4.248137520414.24813752041
Analytic rank: 00
Dimension: 88
Relative dimension: 44 over Q(ζ3)\Q(\zeta_{3})
Coefficient field: 8.0.5206055409.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: x8−3x7+x6+9x5−23x4+27x3+9x2−81x+81 x^{8} - 3x^{7} + x^{6} + 9x^{5} - 23x^{4} + 27x^{3} + 9x^{2} - 81x + 81 Copy content Toggle raw display
Coefficient ring: Z[a1,…,a23]\Z[a_1, \ldots, a_{23}]
Coefficient ring index: 26⋅34 2^{6}\cdot 3^{4}
Twist minimal: yes
Sato-Tate group: SU(2)[C3]\mathrm{SU}(2)[C_{3}]

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−β5q3+(β4−β2+2β1)q5+(β7+β6+β4+⋯+1)q7+(β7+β5+3β4+⋯−2)q9+(3β6+4β5+3β4+⋯+3)q11+⋯+(−β7−12β6+⋯−631)q99+O(q100) q - \beta_{5} q^{3} + (\beta_{4} - \beta_{2} + 2 \beta_1) q^{5} + (\beta_{7} + \beta_{6} + \beta_{4} + \cdots + 1) q^{7} + (\beta_{7} + \beta_{5} + 3 \beta_{4} + \cdots - 2) q^{9} + (3 \beta_{6} + 4 \beta_{5} + 3 \beta_{4} + \cdots + 3) q^{11}+ \cdots + ( - \beta_{7} - 12 \beta_{6} + \cdots - 631) q^{99}+O(q^{100}) Copy content Toggle raw display
Tr⁡(f)(q)\operatorname{Tr}(f)(q) == 8q−3q3−5q5+3q7−15q9+16q11+29q13+141q15−34q17−218q19+27q21+37q23+97q25−216q27−3q29+331q31+468q33−342q35+⋯−5133q99+O(q100) 8 q - 3 q^{3} - 5 q^{5} + 3 q^{7} - 15 q^{9} + 16 q^{11} + 29 q^{13} + 141 q^{15} - 34 q^{17} - 218 q^{19} + 27 q^{21} + 37 q^{23} + 97 q^{25} - 216 q^{27} - 3 q^{29} + 331 q^{31} + 468 q^{33} - 342 q^{35}+ \cdots - 5133 q^{99}+O(q^{100}) Copy content Toggle raw display

Basis of coefficient ring in terms of a root ν\nu of x8−3x7+x6+9x5−23x4+27x3+9x2−81x+81 x^{8} - 3x^{7} + x^{6} + 9x^{5} - 23x^{4} + 27x^{3} + 9x^{2} - 81x + 81 : Copy content Toggle raw display

β1\beta_{1}== (ν7+3ν6−8ν5+15ν4−14ν3−30ν2+99ν−135)/81 ( \nu^{7} + 3\nu^{6} - 8\nu^{5} + 15\nu^{4} - 14\nu^{3} - 30\nu^{2} + 99\nu - 135 ) / 81 Copy content Toggle raw display
β2\beta_{2}== (2ν7+15ν6−70ν5+66ν4−ν3−294ν2+522ν−432)/81 ( 2\nu^{7} + 15\nu^{6} - 70\nu^{5} + 66\nu^{4} - \nu^{3} - 294\nu^{2} + 522\nu - 432 ) / 81 Copy content Toggle raw display
β3\beta_{3}== (ν7−8ν5+21ν4−5ν3−24ν2+99ν−117)/9 ( \nu^{7} - 8\nu^{5} + 21\nu^{4} - 5\nu^{3} - 24\nu^{2} + 99\nu - 117 ) / 9 Copy content Toggle raw display
β4\beta_{4}== (−16ν7+24ν6−34ν5−60ν4+89ν3−150ν2+360ν−27)/81 ( -16\nu^{7} + 24\nu^{6} - 34\nu^{5} - 60\nu^{4} + 89\nu^{3} - 150\nu^{2} + 360\nu - 27 ) / 81 Copy content Toggle raw display
β5\beta_{5}== (−17ν7+12ν6+82ν5−165ν4+184ν3+6ν2−711ν+1161)/81 ( -17\nu^{7} + 12\nu^{6} + 82\nu^{5} - 165\nu^{4} + 184\nu^{3} + 6\nu^{2} - 711\nu + 1161 ) / 81 Copy content Toggle raw display
β6\beta_{6}== (22ν7−51ν6−14ν5+240ν4−362ν3+222ν2+396ν−1431)/81 ( 22\nu^{7} - 51\nu^{6} - 14\nu^{5} + 240\nu^{4} - 362\nu^{3} + 222\nu^{2} + 396\nu - 1431 ) / 81 Copy content Toggle raw display
β7\beta_{7}== (−35ν7+84ν6−44ν5−93ν4+490ν3−624ν2−63ν+837)/81 ( -35\nu^{7} + 84\nu^{6} - 44\nu^{5} - 93\nu^{4} + 490\nu^{3} - 624\nu^{2} - 63\nu + 837 ) / 81 Copy content Toggle raw display

Display νj\nu^j in terms of βi\beta_i

ν\nu== (−β7−β6+β5+β4+2β3−β2+4β1+6)/12 ( -\beta_{7} - \beta_{6} + \beta_{5} + \beta_{4} + 2\beta_{3} - \beta_{2} + 4\beta _1 + 6 ) / 12 Copy content Toggle raw display
ν2\nu^{2}== (β7−β6−5β5+β4−7β2+2β1+10)/12 ( \beta_{7} - \beta_{6} - 5\beta_{5} + \beta_{4} - 7\beta_{2} + 2\beta _1 + 10 ) / 12 Copy content Toggle raw display
ν3\nu^{3}== (−2β6−β5−β4+3β3−2β2−12β1−13)/6 ( -2\beta_{6} - \beta_{5} - \beta_{4} + 3\beta_{3} - 2\beta_{2} - 12\beta _1 - 13 ) / 6 Copy content Toggle raw display
ν4\nu^{4}== (5β7+5β6−2β5−2β4+β3−7β2+4β1+47)/12 ( 5\beta_{7} + 5\beta_{6} - 2\beta_{5} - 2\beta_{4} + \beta_{3} - 7\beta_{2} + 4\beta _1 + 47 ) / 12 Copy content Toggle raw display
ν5\nu^{5}== (−5β7−β6+22β5−2β4+13β3−7β2+86β1−7)/12 ( -5\beta_{7} - \beta_{6} + 22\beta_{5} - 2\beta_{4} + 13\beta_{3} - 7\beta_{2} + 86\beta _1 - 7 ) / 12 Copy content Toggle raw display
ν6\nu^{6}== (3β7−β6−5β5−2β4−4β3−10β2+66β1+27)/3 ( 3\beta_{7} - \beta_{6} - 5\beta_{5} - 2\beta_{4} - 4\beta_{3} - 10\beta_{2} + 66\beta _1 + 27 ) / 3 Copy content Toggle raw display
ν7\nu^{7}== (−22β7−58β6−11β5−59β4+23β3+2β2+136β1−123)/12 ( -22\beta_{7} - 58\beta_{6} - 11\beta_{5} - 59\beta_{4} + 23\beta_{3} + 2\beta_{2} + 136\beta _1 - 123 ) / 12 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/72Z)×\left(\mathbb{Z}/72\mathbb{Z}\right)^\times.

nn 3737 5555 6565
χ(n)\chi(n) 11 11 β1\beta_{1}

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}
25.1
1.70133 − 0.324778i
0.172469 − 1.72344i
−1.72895 + 0.103515i
1.35516 + 1.07868i
1.70133 + 0.324778i
0.172469 + 1.72344i
−1.72895 − 0.103515i
1.35516 − 1.07868i
0 −4.74736 − 2.11248i 0 2.99723 + 5.19136i 0 −7.78882 + 13.4906i 0 18.0749 + 20.0574i 0
25.2 0 −2.78092 + 4.38936i 0 −8.65944 − 14.9986i 0 1.18676 − 2.05553i 0 −11.5330 − 24.4129i 0
25.3 0 2.06310 − 4.76903i 0 −0.845922 − 1.46518i 0 8.57067 − 14.8448i 0 −18.4873 − 19.6779i 0
25.4 0 3.96518 + 3.35817i 0 4.00813 + 6.94228i 0 −0.468615 + 0.811666i 0 4.44536 + 26.6315i 0
49.1 0 −4.74736 + 2.11248i 0 2.99723 − 5.19136i 0 −7.78882 − 13.4906i 0 18.0749 − 20.0574i 0
49.2 0 −2.78092 − 4.38936i 0 −8.65944 + 14.9986i 0 1.18676 + 2.05553i 0 −11.5330 + 24.4129i 0
49.3 0 2.06310 + 4.76903i 0 −0.845922 + 1.46518i 0 8.57067 + 14.8448i 0 −18.4873 + 19.6779i 0
49.4 0 3.96518 − 3.35817i 0 4.00813 − 6.94228i 0 −0.468615 − 0.811666i 0 4.44536 − 26.6315i 0
nn: e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 25.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 72.4.i.a ✓ 8
3.b odd 2 1 216.4.i.a 8
4.b odd 2 1 144.4.i.e 8
9.c even 3 1 inner 72.4.i.a ✓ 8
9.c even 3 1 648.4.a.i 4
9.d odd 6 1 216.4.i.a 8
9.d odd 6 1 648.4.a.h 4
12.b even 2 1 432.4.i.e 8
36.f odd 6 1 144.4.i.e 8
36.f odd 6 1 1296.4.a.ba 4
36.h even 6 1 432.4.i.e 8
36.h even 6 1 1296.4.a.y 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
72.4.i.a ✓ 8 1.a even 1 1 trivial
72.4.i.a ✓ 8 9.c even 3 1 inner
144.4.i.e 8 4.b odd 2 1
144.4.i.e 8 36.f odd 6 1
216.4.i.a 8 3.b odd 2 1
216.4.i.a 8 9.d odd 6 1
432.4.i.e 8 12.b even 2 1
432.4.i.e 8 36.h even 6 1
648.4.a.h 4 9.d odd 6 1
648.4.a.i 4 9.c even 3 1
1296.4.a.y 4 36.h even 6 1
1296.4.a.ba 4 36.f odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator T58+5T57+214T56−1951T55+31798T54−109147T53+519121T52+708224T5+1982464 T_{5}^{8} + 5T_{5}^{7} + 214T_{5}^{6} - 1951T_{5}^{5} + 31798T_{5}^{4} - 109147T_{5}^{3} + 519121T_{5}^{2} + 708224T_{5} + 1982464 acting on S4new(72,[χ])S_{4}^{\mathrm{new}}(72, [\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 T8+3T7+⋯+531441 T^{8} + 3 T^{7} + \cdots + 531441 Copy content Toggle raw display
55 T8+5T7+⋯+1982464 T^{8} + 5 T^{7} + \cdots + 1982464 Copy content Toggle raw display
77 T8−3T7+⋯+352836 T^{8} - 3 T^{7} + \cdots + 352836 Copy content Toggle raw display
1111 T8+⋯+1515467041 T^{8} + \cdots + 1515467041 Copy content Toggle raw display
1313 T8+⋯+19954952644 T^{8} + \cdots + 19954952644 Copy content Toggle raw display
1717 (T4+17T3+⋯+2567224)2 (T^{4} + 17 T^{3} + \cdots + 2567224)^{2} Copy content Toggle raw display
1919 (T4+109T3+⋯−5081408)2 (T^{4} + 109 T^{3} + \cdots - 5081408)^{2} Copy content Toggle raw display
2323 T8+⋯+14 ⁣⋯ ⁣84 T^{8} + \cdots + 14\!\cdots\!84 Copy content Toggle raw display
2929 T8+⋯+24 ⁣⋯ ⁣84 T^{8} + \cdots + 24\!\cdots\!84 Copy content Toggle raw display
3131 T8+⋯+10 ⁣⋯ ⁣16 T^{8} + \cdots + 10\!\cdots\!16 Copy content Toggle raw display
3737 (T4+366T3+⋯−215981856)2 (T^{4} + 366 T^{3} + \cdots - 215981856)^{2} Copy content Toggle raw display
4141 T8+⋯+13 ⁣⋯ ⁣89 T^{8} + \cdots + 13\!\cdots\!89 Copy content Toggle raw display
4343 T8+⋯+32 ⁣⋯ ⁣89 T^{8} + \cdots + 32\!\cdots\!89 Copy content Toggle raw display
4747 T8+⋯+96 ⁣⋯ ⁣84 T^{8} + \cdots + 96\!\cdots\!84 Copy content Toggle raw display
5353 (T4−410T3+⋯−15963093536)2 (T^{4} - 410 T^{3} + \cdots - 15963093536)^{2} Copy content Toggle raw display
5959 T8+⋯+20 ⁣⋯ ⁣89 T^{8} + \cdots + 20\!\cdots\!89 Copy content Toggle raw display
6161 T8+⋯+23 ⁣⋯ ⁣16 T^{8} + \cdots + 23\!\cdots\!16 Copy content Toggle raw display
6767 T8+⋯+18 ⁣⋯ ⁣41 T^{8} + \cdots + 18\!\cdots\!41 Copy content Toggle raw display
7171 (T4+344T3+⋯+9762389248)2 (T^{4} + 344 T^{3} + \cdots + 9762389248)^{2} Copy content Toggle raw display
7373 (T4+1307T3+⋯−88005243128)2 (T^{4} + 1307 T^{3} + \cdots - 88005243128)^{2} Copy content Toggle raw display
7979 T8+⋯+97 ⁣⋯ ⁣36 T^{8} + \cdots + 97\!\cdots\!36 Copy content Toggle raw display
8383 T8+⋯+46 ⁣⋯ ⁣84 T^{8} + \cdots + 46\!\cdots\!84 Copy content Toggle raw display
8989 (T4−816T3+⋯+22003976592)2 (T^{4} - 816 T^{3} + \cdots + 22003976592)^{2} Copy content Toggle raw display
9797 T8+⋯+67 ⁣⋯ ⁣49 T^{8} + \cdots + 67\!\cdots\!49 Copy content Toggle raw display
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