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Properties

Label 1156.2.h.f
Level 11561156
Weight 22
Character orbit 1156.h
Analytic conductor 9.2319.231
Analytic rank 00
Dimension 1616
Inner twists 88

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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] = [1156,2,Mod(733,1156)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1156, base_ring=CyclotomicField(8)) chi = DirichletCharacter(H, H._module([0, 7])) 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("1156.733"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: N N == 1156=22⋅172 1156 = 2^{2} \cdot 17^{2}
Weight: k k == 2 2
Character orbit: [χ][\chi] == 1156.h (of order 88, degree 44, not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [16,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0, 0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(33)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: 9.230706473669.23070647366
Analytic rank: 00
Dimension: 1616
Relative dimension: 44 over Q(ζ8)\Q(\zeta_{8})
Coefficient field: Q(ζ48)\Q(\zeta_{48})
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: x16−x8+1 x^{16} - x^{8} + 1 Copy content Toggle raw display
Coefficient ring: Z[a1,…,a9]\Z[a_1, \ldots, a_{9}]
Coefficient ring index: 216 2^{16}
Twist minimal: no (minimal twist has level 68)
Sato-Tate group: SU(2)[C8]\mathrm{SU}(2)[C_{8}]

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,…,β151,\beta_1,\ldots,\beta_{15} for the coefficient ring described below. We also show the integral qq-expansion of the trace form.

f(q)f(q) == q+β15q3+(2β5−β3)q5−β9q7+(β13+β6)q9+(−β8−2β1)q11+(β10−2β4)q13+(−β11+6β2)q15+⋯+(β15+4β7)q99+O(q100) q + \beta_{15} q^{3} + (2 \beta_{5} - \beta_{3}) q^{5} - \beta_{9} q^{7} + (\beta_{13} + \beta_{6}) q^{9} + ( - \beta_{8} - 2 \beta_1) q^{11} + (\beta_{10} - 2 \beta_{4}) q^{13} + ( - \beta_{11} + 6 \beta_{2}) q^{15}+ \cdots + (\beta_{15} + 4 \beta_{7}) q^{99}+O(q^{100}) Copy content Toggle raw display
Tr⁡(f)(q)\operatorname{Tr}(f)(q) == 16q+96q35−128q67+O(q100) 16 q + 96 q^{35} - 128 q^{67}+O(q^{100}) Copy content Toggle raw display

Basis of coefficient ring

β1\beta_{1}== 2ζ483 2\zeta_{48}^{3} Copy content Toggle raw display
β2\beta_{2}== ζ486 \zeta_{48}^{6} Copy content Toggle raw display
β3\beta_{3}== 2ζ489 2\zeta_{48}^{9} Copy content Toggle raw display
β4\beta_{4}== ζ4812 \zeta_{48}^{12} Copy content Toggle raw display
β5\beta_{5}== ζ4813+ζ489+ζ485 \zeta_{48}^{13} + \zeta_{48}^{9} + \zeta_{48}^{5} Copy content Toggle raw display
β6\beta_{6}== −ζ4810+ζ482 -\zeta_{48}^{10} + \zeta_{48}^{2} Copy content Toggle raw display
β7\beta_{7}== −ζ4813−ζ489+ζ485+2ζ48 -\zeta_{48}^{13} - \zeta_{48}^{9} + \zeta_{48}^{5} + 2\zeta_{48} Copy content Toggle raw display
β8\beta_{8}== −ζ4815+2ζ487−ζ483 -\zeta_{48}^{15} + 2\zeta_{48}^{7} - \zeta_{48}^{3} Copy content Toggle raw display
β9\beta_{9}== −ζ4815+2ζ4811−ζ483 -\zeta_{48}^{15} + 2\zeta_{48}^{11} - \zeta_{48}^{3} Copy content Toggle raw display
β10\beta_{10}== 4ζ488−2 4\zeta_{48}^{8} - 2 Copy content Toggle raw display
β11\beta_{11}== 2ζ4810+2ζ482 2\zeta_{48}^{10} + 2\zeta_{48}^{2} Copy content Toggle raw display
β12\beta_{12}== −2ζ4812+4ζ484 -2\zeta_{48}^{12} + 4\zeta_{48}^{4} Copy content Toggle raw display
β13\beta_{13}== 4ζ4814−2ζ486 4\zeta_{48}^{14} - 2\zeta_{48}^{6} Copy content Toggle raw display
β14\beta_{14}== 2ζ4815 2\zeta_{48}^{15} Copy content Toggle raw display
β15\beta_{15}== ζ4813−ζ489−ζ485+2ζ48 \zeta_{48}^{13} - \zeta_{48}^{9} - \zeta_{48}^{5} + 2\zeta_{48} Copy content Toggle raw display

Display ζ48j\zeta_{48}^j in terms of βi\beta_i

ζ48\zeta_{48}== (β15+β7+β3)/4 ( \beta_{15} + \beta_{7} + \beta_{3} ) / 4 Copy content Toggle raw display
ζ482\zeta_{48}^{2}== (β11+2β6)/4 ( \beta_{11} + 2\beta_{6} ) / 4 Copy content Toggle raw display
ζ483\zeta_{48}^{3}== (β1)/2 ( \beta_1 ) / 2 Copy content Toggle raw display
ζ484\zeta_{48}^{4}== (β12+2β4)/4 ( \beta_{12} + 2\beta_{4} ) / 4 Copy content Toggle raw display
ζ485\zeta_{48}^{5}== (−β15+β7+2β5−β3)/4 ( -\beta_{15} + \beta_{7} + 2\beta_{5} - \beta_{3} ) / 4 Copy content Toggle raw display
ζ486\zeta_{48}^{6}== β2 \beta_{2} Copy content Toggle raw display
ζ487\zeta_{48}^{7}== (β14+2β8+β1)/4 ( \beta_{14} + 2\beta_{8} + \beta_1 ) / 4 Copy content Toggle raw display
ζ488\zeta_{48}^{8}== (β10+2)/4 ( \beta_{10} + 2 ) / 4 Copy content Toggle raw display
ζ489\zeta_{48}^{9}== (β3)/2 ( \beta_{3} ) / 2 Copy content Toggle raw display
ζ4810\zeta_{48}^{10}== (β11−2β6)/4 ( \beta_{11} - 2\beta_{6} ) / 4 Copy content Toggle raw display
ζ4811\zeta_{48}^{11}== (β14+2β9+β1)/4 ( \beta_{14} + 2\beta_{9} + \beta_1 ) / 4 Copy content Toggle raw display
ζ4812\zeta_{48}^{12}== β4 \beta_{4} Copy content Toggle raw display
ζ4813\zeta_{48}^{13}== (β15−β7+2β5−β3)/4 ( \beta_{15} - \beta_{7} + 2\beta_{5} - \beta_{3} ) / 4 Copy content Toggle raw display
ζ4814\zeta_{48}^{14}== (β13+2β2)/4 ( \beta_{13} + 2\beta_{2} ) / 4 Copy content Toggle raw display
ζ4815\zeta_{48}^{15}== (β14)/2 ( \beta_{14} ) / 2 Copy content Toggle raw display

Display βi\beta_i in terms of ζ48j\zeta_{48}^j

Character values

We give the values of χ\chi on generators for (Z/1156Z)×\left(\mathbb{Z}/1156\mathbb{Z}\right)^\times.

nn 579579 581581
χ(n)\chi(n) 11 β6\beta_{6}

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}
733.1
−0.608761 + 0.793353i
−0.991445 − 0.130526i
0.991445 + 0.130526i
0.608761 − 0.793353i
−0.608761 − 0.793353i
−0.991445 + 0.130526i
0.991445 − 0.130526i
0.608761 + 0.793353i
−0.793353 + 0.608761i
0.130526 + 0.991445i
−0.130526 − 0.991445i
0.793353 − 0.608761i
−0.793353 − 0.608761i
0.130526 − 0.991445i
−0.130526 + 0.991445i
0.793353 + 0.608761i
0 −2.52409 + 1.04551i 0 −1.32565 − 3.20041i 0 −1.04551 + 2.52409i 0 3.15660 − 3.15660i 0
733.2 0 −0.676327 + 0.280144i 0 −1.32565 − 3.20041i 0 −0.280144 + 0.676327i 0 −1.74238 + 1.74238i 0
733.3 0 0.676327 − 0.280144i 0 1.32565 + 3.20041i 0 0.280144 − 0.676327i 0 −1.74238 + 1.74238i 0
733.4 0 2.52409 − 1.04551i 0 1.32565 + 3.20041i 0 1.04551 − 2.52409i 0 3.15660 − 3.15660i 0
757.1 0 −2.52409 − 1.04551i 0 −1.32565 + 3.20041i 0 −1.04551 − 2.52409i 0 3.15660 + 3.15660i 0
757.2 0 −0.676327 − 0.280144i 0 −1.32565 + 3.20041i 0 −0.280144 − 0.676327i 0 −1.74238 − 1.74238i 0
757.3 0 0.676327 + 0.280144i 0 1.32565 − 3.20041i 0 0.280144 + 0.676327i 0 −1.74238 − 1.74238i 0
757.4 0 2.52409 + 1.04551i 0 1.32565 − 3.20041i 0 1.04551 + 2.52409i 0 3.15660 + 3.15660i 0
977.1 0 −1.04551 + 2.52409i 0 3.20041 + 1.32565i 0 2.52409 − 1.04551i 0 −3.15660 − 3.15660i 0
977.2 0 −0.280144 + 0.676327i 0 3.20041 + 1.32565i 0 0.676327 − 0.280144i 0 1.74238 + 1.74238i 0
977.3 0 0.280144 − 0.676327i 0 −3.20041 − 1.32565i 0 −0.676327 + 0.280144i 0 1.74238 + 1.74238i 0
977.4 0 1.04551 − 2.52409i 0 −3.20041 − 1.32565i 0 −2.52409 + 1.04551i 0 −3.15660 − 3.15660i 0
1001.1 0 −1.04551 − 2.52409i 0 3.20041 − 1.32565i 0 2.52409 + 1.04551i 0 −3.15660 + 3.15660i 0
1001.2 0 −0.280144 − 0.676327i 0 3.20041 − 1.32565i 0 0.676327 + 0.280144i 0 1.74238 − 1.74238i 0
1001.3 0 0.280144 + 0.676327i 0 −3.20041 + 1.32565i 0 −0.676327 − 0.280144i 0 1.74238 − 1.74238i 0
1001.4 0 1.04551 + 2.52409i 0 −3.20041 + 1.32565i 0 −2.52409 − 1.04551i 0 −3.15660 + 3.15660i 0
nn: e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 733.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
17.b even 2 1 inner
17.c even 4 2 inner
17.d even 8 4 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1156.2.h.f 16
17.b even 2 1 inner 1156.2.h.f 16
17.c even 4 2 inner 1156.2.h.f 16
17.d even 8 4 inner 1156.2.h.f 16
17.e odd 16 1 68.2.a.a ✓ 2
17.e odd 16 1 1156.2.a.a 2
17.e odd 16 2 1156.2.b.c 4
17.e odd 16 4 1156.2.e.d 8
51.i even 16 1 612.2.a.e 2
68.i even 16 1 272.2.a.e 2
68.i even 16 1 4624.2.a.x 2
85.o even 16 1 1700.2.e.c 4
85.p odd 16 1 1700.2.a.d 2
85.r even 16 1 1700.2.e.c 4
119.p even 16 1 3332.2.a.h 2
136.q odd 16 1 1088.2.a.p 2
136.s even 16 1 1088.2.a.t 2
187.m even 16 1 8228.2.a.k 2
204.t odd 16 1 2448.2.a.y 2
340.bg even 16 1 6800.2.a.bh 2
408.bg odd 16 1 9792.2.a.cs 2
408.bm even 16 1 9792.2.a.cr 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
68.2.a.a ✓ 2 17.e odd 16 1
272.2.a.e 2 68.i even 16 1
612.2.a.e 2 51.i even 16 1
1088.2.a.p 2 136.q odd 16 1
1088.2.a.t 2 136.s even 16 1
1156.2.a.a 2 17.e odd 16 1
1156.2.b.c 4 17.e odd 16 2
1156.2.e.d 8 17.e odd 16 4
1156.2.h.f 16 1.a even 1 1 trivial
1156.2.h.f 16 17.b even 2 1 inner
1156.2.h.f 16 17.c even 4 2 inner
1156.2.h.f 16 17.d even 8 4 inner
1700.2.a.d 2 85.p odd 16 1
1700.2.e.c 4 85.o even 16 1
1700.2.e.c 4 85.r even 16 1
2448.2.a.y 2 204.t odd 16 1
3332.2.a.h 2 119.p even 16 1
4624.2.a.x 2 68.i even 16 1
6800.2.a.bh 2 340.bg even 16 1
8228.2.a.k 2 187.m even 16 1
9792.2.a.cr 2 408.bm even 16 1
9792.2.a.cs 2 408.bg odd 16 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator T316+3104T38+256 T_{3}^{16} + 3104T_{3}^{8} + 256 acting on S2new(1156,[χ])S_{2}^{\mathrm{new}}(1156, [\chi]). Copy content Toggle raw display

Hecke characteristic polynomials

pp Fp(T)F_p(T)
22 T16 T^{16} Copy content Toggle raw display
33 T16+3104T8+256 T^{16} + 3104 T^{8} + 256 Copy content Toggle raw display
55 (T8+20736)2 (T^{8} + 20736)^{2} Copy content Toggle raw display
77 T16+3104T8+256 T^{16} + 3104 T^{8} + 256 Copy content Toggle raw display
1111 T16+251424T8+1679616 T^{16} + 251424 T^{8} + 1679616 Copy content Toggle raw display
1313 (T4+32T2+64)4 (T^{4} + 32 T^{2} + 64)^{4} Copy content Toggle raw display
1717 T16 T^{16} Copy content Toggle raw display
1919 (T8+896T4+4096)2 (T^{8} + 896 T^{4} + 4096)^{2} Copy content Toggle raw display
2323 T16+251424T8+1679616 T^{16} + 251424 T^{8} + 1679616 Copy content Toggle raw display
2929 (T8+20736)2 (T^{8} + 20736)^{2} Copy content Toggle raw display
3131 T16+⋯+208827064576 T^{16} + \cdots + 208827064576 Copy content Toggle raw display
3737 T16+⋯+53459728531456 T^{16} + \cdots + 53459728531456 Copy content Toggle raw display
4141 (T8+1679616)2 (T^{8} + 1679616)^{2} Copy content Toggle raw display
4343 (T8+28544T4+116985856)2 (T^{8} + 28544 T^{4} + 116985856)^{2} Copy content Toggle raw display
4747 (T2+48)8 (T^{2} + 48)^{8} Copy content Toggle raw display
5353 (T8+27936T4+20736)2 (T^{8} + 27936 T^{4} + 20736)^{2} Copy content Toggle raw display
5959 (T8+8064T4+331776)2 (T^{8} + 8064 T^{4} + 331776)^{2} Copy content Toggle raw display
6161 T16+9634304T8+65536 T^{16} + 9634304 T^{8} + 65536 Copy content Toggle raw display
6767 (T2+16T+16)8 (T^{2} + 16 T + 16)^{8} Copy content Toggle raw display
7171 T16+⋯+11019960576 T^{16} + \cdots + 11019960576 Copy content Toggle raw display
7373 (T8+256)2 (T^{8} + 256)^{2} Copy content Toggle raw display
7979 T16+⋯+54875873536 T^{16} + \cdots + 54875873536 Copy content Toggle raw display
8383 (T8+8064T4+331776)2 (T^{8} + 8064 T^{4} + 331776)^{2} Copy content Toggle raw display
8989 (T4+96T2+576)4 (T^{4} + 96 T^{2} + 576)^{4} Copy content Toggle raw display
9797 T16+⋯+14048223625216 T^{16} + \cdots + 14048223625216 Copy content Toggle raw display
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