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Properties

Label 156.2.c.c
Level 156156
Weight 22
Character orbit 156.c
Analytic conductor 1.2461.246
Analytic rank 00
Dimension 88
Inner twists 44

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [156,2,Mod(131,156)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(156, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1, 0])) N = Newforms(chi, 2, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("156.131"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: N N == 156=22⋅3⋅13 156 = 2^{2} \cdot 3 \cdot 13
Weight: k k == 2 2
Character orbit: [χ][\chi] == 156.c (of order 22, degree 11, minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: 1.245666271531.24566627153
Analytic rank: 00
Dimension: 88
Coefficient field: 8.0.121550625.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: x8−x7−4x6−9x5+23x4+18x3−16x2+8x+16 x^{8} - x^{7} - 4x^{6} - 9x^{5} + 23x^{4} + 18x^{3} - 16x^{2} + 8x + 16 Copy content Toggle raw display
Coefficient ring: Z[a1,a2,a3]\Z[a_1, a_2, a_3]
Coefficient ring index: 23 2^{3}
Twist minimal: yes
Sato-Tate group: SU(2)[C2]\mathrm{SU}(2)[C_{2}]

qq-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content 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−β5q2+(−β6+β1+1)q3+(β6−β2−β1)q4+(2β6+2β4−β2+⋯−1)q5+(β6+β4−β3−1)q6+⋯+(−β7+2β6−β5+⋯−3)q98+O(q100) q - \beta_{5} q^{2} + ( - \beta_{6} + \beta_1 + 1) q^{3} + (\beta_{6} - \beta_{2} - \beta_1) q^{4} + (2 \beta_{6} + 2 \beta_{4} - \beta_{2} + \cdots - 1) q^{5} + (\beta_{6} + \beta_{4} - \beta_{3} - 1) q^{6}+ \cdots + ( - \beta_{7} + 2 \beta_{6} - \beta_{5} + \cdots - 3) q^{98}+O(q^{100}) Copy content Toggle raw display
Tr⁡(f)(q)\operatorname{Tr}(f)(q) == 8q+4q4−5q6+2q9+6q10+15q12+8q13−28q16+3q18+22q21+5q24−36q25−30q28−25q30−6q34+q36−4q37+18q40−3q42+⋯+40q97+O(q100) 8 q + 4 q^{4} - 5 q^{6} + 2 q^{9} + 6 q^{10} + 15 q^{12} + 8 q^{13} - 28 q^{16} + 3 q^{18} + 22 q^{21} + 5 q^{24} - 36 q^{25} - 30 q^{28} - 25 q^{30} - 6 q^{34} + q^{36} - 4 q^{37} + 18 q^{40} - 3 q^{42}+ \cdots + 40 q^{97}+O(q^{100}) Copy content Toggle raw display

Basis of coefficient ring in terms of a root ν\nu of x8−x7−4x6−9x5+23x4+18x3−16x2+8x+16 x^{8} - x^{7} - 4x^{6} - 9x^{5} + 23x^{4} + 18x^{3} - 16x^{2} + 8x + 16 : Copy content Toggle raw display

β1\beta_{1}== (−19ν7+73ν6+26ν5−89ν4−1043ν3+704ν2+1696ν−520)/816 ( -19\nu^{7} + 73\nu^{6} + 26\nu^{5} - 89\nu^{4} - 1043\nu^{3} + 704\nu^{2} + 1696\nu - 520 ) / 816 Copy content Toggle raw display
β2\beta_{2}== (71ν7−33ν6−566ν5−827ν4+1811ν3+5132ν2−1320ν−920)/2448 ( 71\nu^{7} - 33\nu^{6} - 566\nu^{5} - 827\nu^{4} + 1811\nu^{3} + 5132\nu^{2} - 1320\nu - 920 ) / 2448 Copy content Toggle raw display
β3\beta_{3}== (−49ν7+165ν6+76ν5+157ν4−2323ν3+1574ν2+2316ν−296)/1224 ( -49\nu^{7} + 165\nu^{6} + 76\nu^{5} + 157\nu^{4} - 2323\nu^{3} + 1574\nu^{2} + 2316\nu - 296 ) / 1224 Copy content Toggle raw display
β4\beta_{4}== (131ν7−81ν6−530ν5−1319ν4+2603ν3+2168ν2+432ν+808)/2448 ( 131\nu^{7} - 81\nu^{6} - 530\nu^{5} - 1319\nu^{4} + 2603\nu^{3} + 2168\nu^{2} + 432\nu + 808 ) / 2448 Copy content Toggle raw display
β5\beta_{5}== (37ν7−84ν6−73ν5−181ν4+1114ν3−767ν2−606ν+1460)/612 ( 37\nu^{7} - 84\nu^{6} - 73\nu^{5} - 181\nu^{4} + 1114\nu^{3} - 767\nu^{2} - 606\nu + 1460 ) / 612 Copy content Toggle raw display
β6\beta_{6}== (20ν7−33ν6−56ν5−164ν4+536ν3+32ν2−96ν+355)/153 ( 20\nu^{7} - 33\nu^{6} - 56\nu^{5} - 164\nu^{4} + 536\nu^{3} + 32\nu^{2} - 96\nu + 355 ) / 153 Copy content Toggle raw display
β7\beta_{7}== (−205ν7+249ν6+676ν5+1681ν4−4831ν3−1858ν2+3228ν−2504)/1224 ( -205\nu^{7} + 249\nu^{6} + 676\nu^{5} + 1681\nu^{4} - 4831\nu^{3} - 1858\nu^{2} + 3228\nu - 2504 ) / 1224 Copy content Toggle raw display

Display νj\nu^j in terms of βi\beta_i

ν\nu== (β7+β6+β4+β3−β1−1)/2 ( \beta_{7} + \beta_{6} + \beta_{4} + \beta_{3} - \beta _1 - 1 ) / 2 Copy content Toggle raw display
ν2\nu^{2}== β6−β5−β4+β3−β1 \beta_{6} - \beta_{5} - \beta_{4} + \beta_{3} - \beta_1 Copy content Toggle raw display
ν3\nu^{3}== β7+β6−3β5+3β4−β3−β2−2β1+4 \beta_{7} + \beta_{6} - 3\beta_{5} + 3\beta_{4} - \beta_{3} - \beta_{2} - 2\beta _1 + 4 Copy content Toggle raw display
ν4\nu^{4}== 2β7+2β6+β5+2β4+7β3−8β1−7 2\beta_{7} + 2\beta_{6} + \beta_{5} + 2\beta_{4} + 7\beta_{3} - 8\beta _1 - 7 Copy content Toggle raw display
ν5\nu^{5}== −2β7+12β6−20β5−7β4+7β3−11β2−12β1+8 -2\beta_{7} + 12\beta_{6} - 20\beta_{5} - 7\beta_{4} + 7\beta_{3} - 11\beta_{2} - 12\beta _1 + 8 Copy content Toggle raw display
ν6\nu^{6}== 4β7−β6−28β5+44β4−4β3−20β2−24β1+39 4\beta_{7} - \beta_{6} - 28\beta_{5} + 44\beta_{4} - 4\beta_{3} - 20\beta_{2} - 24\beta _1 + 39 Copy content Toggle raw display
ν7\nu^{7}== −7β7+30β6−12β5−7β4+98β3−37β2−86β1−98 -7\beta_{7} + 30\beta_{6} - 12\beta_{5} - 7\beta_{4} + 98\beta_{3} - 37\beta_{2} - 86\beta _1 - 98 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/156Z)×\left(\mathbb{Z}/156\mathbb{Z}\right)^\times.

nn 5353 7979 145145
χ(n)\chi(n) −1-1 −1-1 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.

Copy content comment:embeddings in the coefficient field
 
Copy content 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}
131.1
−0.862555 + 0.141174i
0.553538 − 0.676408i
−0.862555 − 0.141174i
0.553538 + 0.676408i
2.25820 − 0.369600i
−1.44918 + 1.77086i
2.25820 + 0.369600i
−1.44918 − 1.77086i
−1.11803 − 0.866025i −0.586627 − 1.62968i 0.500000 + 1.93649i 3.82407i −0.755479 + 2.33008i 3.25937i 1.11803 − 2.59808i −2.31174 + 1.91203i 3.31174 − 4.27543i
131.2 −1.11803 − 0.866025i 1.70466 − 0.306808i 0.500000 + 1.93649i − 2.09201i −2.17157 − 1.13326i 0.613616i 1.11803 − 2.59808i 2.81174 − 1.04601i −1.81174 + 2.33894i
131.3 −1.11803 + 0.866025i −0.586627 + 1.62968i 0.500000 − 1.93649i − 3.82407i −0.755479 − 2.33008i − 3.25937i 1.11803 + 2.59808i −2.31174 − 1.91203i 3.31174 + 4.27543i
131.4 −1.11803 + 0.866025i 1.70466 + 0.306808i 0.500000 − 1.93649i 2.09201i −2.17157 + 1.13326i − 0.613616i 1.11803 + 2.59808i 2.81174 + 1.04601i −1.81174 − 2.33894i
131.5 1.11803 − 0.866025i −1.70466 + 0.306808i 0.500000 − 1.93649i − 2.09201i −1.64017 + 1.81930i − 0.613616i −1.11803 − 2.59808i 2.81174 − 1.04601i −1.81174 − 2.33894i
131.6 1.11803 − 0.866025i 0.586627 + 1.62968i 0.500000 − 1.93649i 3.82407i 2.06722 + 1.31401i − 3.25937i −1.11803 − 2.59808i −2.31174 + 1.91203i 3.31174 + 4.27543i
131.7 1.11803 + 0.866025i −1.70466 − 0.306808i 0.500000 + 1.93649i 2.09201i −1.64017 − 1.81930i 0.613616i −1.11803 + 2.59808i 2.81174 + 1.04601i −1.81174 + 2.33894i
131.8 1.11803 + 0.866025i 0.586627 − 1.62968i 0.500000 + 1.93649i − 3.82407i 2.06722 − 1.31401i 3.25937i −1.11803 + 2.59808i −2.31174 − 1.91203i 3.31174 − 4.27543i
nn: e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 131.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
4.b odd 2 1 inner
12.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 156.2.c.c ✓ 8
3.b odd 2 1 inner 156.2.c.c ✓ 8
4.b odd 2 1 inner 156.2.c.c ✓ 8
8.b even 2 1 2496.2.d.m 8
8.d odd 2 1 2496.2.d.m 8
12.b even 2 1 inner 156.2.c.c ✓ 8
24.f even 2 1 2496.2.d.m 8
24.h odd 2 1 2496.2.d.m 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
156.2.c.c ✓ 8 1.a even 1 1 trivial
156.2.c.c ✓ 8 3.b odd 2 1 inner
156.2.c.c ✓ 8 4.b odd 2 1 inner
156.2.c.c ✓ 8 12.b even 2 1 inner
2496.2.d.m 8 8.b even 2 1
2496.2.d.m 8 8.d odd 2 1
2496.2.d.m 8 24.f even 2 1
2496.2.d.m 8 24.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on S2new(156,[χ])S_{2}^{\mathrm{new}}(156, [\chi]):

T54+19T52+64 T_{5}^{4} + 19T_{5}^{2} + 64 Copy content Toggle raw display
T234−52T232+256 T_{23}^{4} - 52T_{23}^{2} + 256 Copy content Toggle raw display

Hecke characteristic polynomials

pp Fp(T)F_p(T)
22 (T4−T2+4)2 (T^{4} - T^{2} + 4)^{2} Copy content Toggle raw display
33 T8−T6+⋯+81 T^{8} - T^{6} + \cdots + 81 Copy content Toggle raw display
55 (T4+19T2+64)2 (T^{4} + 19 T^{2} + 64)^{2} Copy content Toggle raw display
77 (T4+11T2+4)2 (T^{4} + 11 T^{2} + 4)^{2} Copy content Toggle raw display
1111 T8 T^{8} Copy content Toggle raw display
1313 (T−1)8 (T - 1)^{8} Copy content Toggle raw display
1717 (T4+19T2+64)2 (T^{4} + 19 T^{2} + 64)^{2} Copy content Toggle raw display
1919 (T2+28)4 (T^{2} + 28)^{4} Copy content Toggle raw display
2323 (T4−52T2+256)2 (T^{4} - 52 T^{2} + 256)^{2} Copy content Toggle raw display
2929 (T2+48)4 (T^{2} + 48)^{4} Copy content Toggle raw display
3131 (T2+28)4 (T^{2} + 28)^{4} Copy content Toggle raw display
3737 (T2+T−26)4 (T^{2} + T - 26)^{4} Copy content Toggle raw display
4141 (T4+124T2+64)2 (T^{4} + 124 T^{2} + 64)^{2} Copy content Toggle raw display
4343 (T4+179T2+6724)2 (T^{4} + 179 T^{2} + 6724)^{2} Copy content Toggle raw display
4747 (T4−13T2+16)2 (T^{4} - 13 T^{2} + 16)^{2} Copy content Toggle raw display
5353 (T4+124T2+64)2 (T^{4} + 124 T^{2} + 64)^{2} Copy content Toggle raw display
5959 T8 T^{8} Copy content Toggle raw display
6161 (T+2)8 (T + 2)^{8} Copy content Toggle raw display
6767 (T2+28)4 (T^{2} + 28)^{4} Copy content Toggle raw display
7171 (T4−13T2+16)2 (T^{4} - 13 T^{2} + 16)^{2} Copy content Toggle raw display
7373 (T2+6T−96)4 (T^{2} + 6 T - 96)^{4} Copy content Toggle raw display
7979 (T4+44T2+64)2 (T^{4} + 44 T^{2} + 64)^{2} Copy content Toggle raw display
8383 (T4−208T2+4096)2 (T^{4} - 208 T^{2} + 4096)^{2} Copy content Toggle raw display
8989 (T4+220T2+1600)2 (T^{4} + 220 T^{2} + 1600)^{2} Copy content Toggle raw display
9797 (T2−10T−80)4 (T^{2} - 10 T - 80)^{4} Copy content Toggle raw display
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