Properties

Label 100.5.b.e
Level 100100
Weight 55
Character orbit 100.b
Analytic conductor 10.33710.337
Analytic rank 00
Dimension 88
Inner twists 44

Related objects

Downloads

Learn more

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [100,5,Mod(51,100)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(100, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0])) N = Newforms(chi, 5, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("100.51"); S:= CuspForms(chi, 5); N := Newforms(S);
 
Level: N N == 100=2252 100 = 2^{2} \cdot 5^{2}
Weight: k k == 5 5
Character orbit: [χ][\chi] == 100.b (of order 22, degree 11, minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: 10.336996308410.3369963084
Analytic rank: 00
Dimension: 88
Coefficient field: 8.0.1816805376000000.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: x86x6+31x496x2+256 x^{8} - 6x^{6} + 31x^{4} - 96x^{2} + 256 Copy content Toggle raw display
Coefficient ring: Z[a1,,a13]\Z[a_1, \ldots, a_{13}]
Coefficient ring index: 215 2^{15}
Twist minimal: no (minimal twist has level 20)
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+β2q2+β1q3+(β6+6)q4+(β5β4)q6+(2β3+6β2+β1)q7+(β7+3β3++2β1)q8+(3β63β439)q9++(648β6252β5216β4)q99+O(q100) q + \beta_{2} q^{2} + \beta_1 q^{3} + (\beta_{6} + 6) q^{4} + ( - \beta_{5} - \beta_{4}) q^{6} + (2 \beta_{3} + 6 \beta_{2} + \beta_1) q^{7} + ( - \beta_{7} + 3 \beta_{3} + \cdots + 2 \beta_1) q^{8} + ( - 3 \beta_{6} - 3 \beta_{4} - 39) q^{9}+ \cdots + (648 \beta_{6} - 252 \beta_{5} - 216 \beta_{4}) q^{99}+O(q^{100}) Copy content Toggle raw display
Tr(f)(q)\operatorname{Tr}(f)(q) == 8q+48q4312q9640q14832q16960q211920q24+2496q262704q29+2176q345712q36+1456q411920q44+7040q46+8008q49+11520q54+76800q96+O(q100) 8 q + 48 q^{4} - 312 q^{9} - 640 q^{14} - 832 q^{16} - 960 q^{21} - 1920 q^{24} + 2496 q^{26} - 2704 q^{29} + 2176 q^{34} - 5712 q^{36} + 1456 q^{41} - 1920 q^{44} + 7040 q^{46} + 8008 q^{49} + 11520 q^{54}+ \cdots - 76800 q^{96}+O(q^{100}) Copy content Toggle raw display

Basis of coefficient ring in terms of a root ν\nu of x86x6+31x496x2+256 x^{8} - 6x^{6} + 31x^{4} - 96x^{2} + 256 : Copy content Toggle raw display

β1\beta_{1}== (ν7+10ν5ν3+80ν)/32 ( \nu^{7} + 10\nu^{5} - \nu^{3} + 80\nu ) / 32 Copy content Toggle raw display
β2\beta_{2}== (ν76ν5+31ν396ν)/32 ( \nu^{7} - 6\nu^{5} + 31\nu^{3} - 96\nu ) / 32 Copy content Toggle raw display
β3\beta_{3}== (ν76ν5+31ν3+160ν)/32 ( \nu^{7} - 6\nu^{5} + 31\nu^{3} + 160\nu ) / 32 Copy content Toggle raw display
β4\beta_{4}== (ν6+6ν4+ν2+24)/4 ( -\nu^{6} + 6\nu^{4} + \nu^{2} + 24 ) / 4 Copy content Toggle raw display
β5\beta_{5}== (ν6+2ν4ν2+28)/2 ( \nu^{6} + 2\nu^{4} - \nu^{2} + 28 ) / 2 Copy content Toggle raw display
β6\beta_{6}== (ν6+6ν431ν2+72)/4 ( -\nu^{6} + 6\nu^{4} - 31\nu^{2} + 72 ) / 4 Copy content Toggle raw display
β7\beta_{7}== (3ν7+10ν513ν3+72ν)/8 ( -3\nu^{7} + 10\nu^{5} - 13\nu^{3} + 72\nu ) / 8 Copy content Toggle raw display
ν\nu== (β3β2)/8 ( \beta_{3} - \beta_{2} ) / 8 Copy content Toggle raw display
ν2\nu^{2}== (β6+β4+12)/8 ( -\beta_{6} + \beta_{4} + 12 ) / 8 Copy content Toggle raw display
ν3\nu^{3}== (β7+2β3+8β2+2β1)/8 ( \beta_{7} + 2\beta_{3} + 8\beta_{2} + 2\beta_1 ) / 8 Copy content Toggle raw display
ν4\nu^{4}== (β5+2β426)/4 ( \beta_{5} + 2\beta_{4} - 26 ) / 4 Copy content Toggle raw display
ν5\nu^{5}== (2β77β3+11β2+20β1)/8 ( 2\beta_{7} - 7\beta_{3} + 11\beta_{2} + 20\beta_1 ) / 8 Copy content Toggle raw display
ν6\nu^{6}== (β6+12β57β4108)/8 ( -\beta_{6} + 12\beta_{5} - 7\beta_{4} - 108 ) / 8 Copy content Toggle raw display
ν7\nu^{7}== (19β78β322β2+58β1)/8 ( -19\beta_{7} - 8\beta_{3} - 22\beta_{2} + 58\beta_1 ) / 8 Copy content Toggle raw display

Character values

We give the values of χ\chi on generators for (Z/100Z)×\left(\mathbb{Z}/100\mathbb{Z}\right)^\times.

nn 5151 7777
χ(n)\chi(n) 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}
51.1
1.88164 0.677813i
1.88164 + 0.677813i
1.39980 1.42849i
1.39980 + 1.42849i
−1.39980 1.42849i
−1.39980 + 1.42849i
−1.88164 0.677813i
−1.88164 + 0.677813i
−3.76328 1.35563i 13.9962i 12.3246 + 10.2032i 0 −18.9737 + 52.6718i 35.6863i −32.5490 55.1050i −114.895 0
51.2 −3.76328 + 1.35563i 13.9962i 12.3246 10.2032i 0 −18.9737 52.6718i 35.6863i −32.5490 + 55.1050i −114.895 0
51.3 −2.79959 2.85697i 6.64118i −0.324555 + 15.9967i 0 18.9737 18.5926i 39.0703i 46.6107 43.8570i 36.8947 0
51.4 −2.79959 + 2.85697i 6.64118i −0.324555 15.9967i 0 18.9737 + 18.5926i 39.0703i 46.6107 + 43.8570i 36.8947 0
51.5 2.79959 2.85697i 6.64118i −0.324555 15.9967i 0 18.9737 + 18.5926i 39.0703i −46.6107 43.8570i 36.8947 0
51.6 2.79959 + 2.85697i 6.64118i −0.324555 + 15.9967i 0 18.9737 18.5926i 39.0703i −46.6107 + 43.8570i 36.8947 0
51.7 3.76328 1.35563i 13.9962i 12.3246 10.2032i 0 −18.9737 52.6718i 35.6863i 32.5490 55.1050i −114.895 0
51.8 3.76328 + 1.35563i 13.9962i 12.3246 + 10.2032i 0 −18.9737 + 52.6718i 35.6863i 32.5490 + 55.1050i −114.895 0
nn: e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 51.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
5.b even 2 1 inner
20.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 100.5.b.e 8
4.b odd 2 1 inner 100.5.b.e 8
5.b even 2 1 inner 100.5.b.e 8
5.c odd 4 2 20.5.d.c 8
15.e even 4 2 180.5.f.g 8
20.d odd 2 1 inner 100.5.b.e 8
20.e even 4 2 20.5.d.c 8
40.i odd 4 2 320.5.h.f 8
40.k even 4 2 320.5.h.f 8
60.l odd 4 2 180.5.f.g 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
20.5.d.c 8 5.c odd 4 2
20.5.d.c 8 20.e even 4 2
100.5.b.e 8 1.a even 1 1 trivial
100.5.b.e 8 4.b odd 2 1 inner
100.5.b.e 8 5.b even 2 1 inner
100.5.b.e 8 20.d odd 2 1 inner
180.5.f.g 8 15.e even 4 2
180.5.f.g 8 60.l odd 4 2
320.5.h.f 8 40.i odd 4 2
320.5.h.f 8 40.k even 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on S5new(100,[χ])S_{5}^{\mathrm{new}}(100, [\chi]):

T34+240T32+8640 T_{3}^{4} + 240T_{3}^{2} + 8640 Copy content Toggle raw display
T13420928T132+82861056 T_{13}^{4} - 20928T_{13}^{2} + 82861056 Copy content Toggle raw display

Hecke characteristic polynomials

pp Fp(T)F_p(T)
22 T824T6++65536 T^{8} - 24 T^{6} + \cdots + 65536 Copy content Toggle raw display
33 (T4+240T2+8640)2 (T^{4} + 240 T^{2} + 8640)^{2} Copy content Toggle raw display
55 T8 T^{8} Copy content Toggle raw display
77 (T4+2800T2+1944000)2 (T^{4} + 2800 T^{2} + 1944000)^{2} Copy content Toggle raw display
1111 (T4+48000T2+552407040)2 (T^{4} + 48000 T^{2} + 552407040)^{2} Copy content Toggle raw display
1313 (T420928T2+82861056)2 (T^{4} - 20928 T^{2} + 82861056)^{2} Copy content Toggle raw display
1717 (T426368T2+16367616)2 (T^{4} - 26368 T^{2} + 16367616)^{2} Copy content Toggle raw display
1919 (T4+447360T2+10372976640)2 (T^{4} + 447360 T^{2} + 10372976640)^{2} Copy content Toggle raw display
2323 (T4+350320T2+30678152640)2 (T^{4} + 350320 T^{2} + 30678152640)^{2} Copy content Toggle raw display
2929 (T2+676T+104004)4 (T^{2} + 676 T + 104004)^{4} Copy content Toggle raw display
3131 (T4+1013760T2+1745879040)2 (T^{4} + 1013760 T^{2} + 1745879040)^{2} Copy content Toggle raw display
3737 (T43008448T2+126031666176)2 (T^{4} - 3008448 T^{2} + 126031666176)^{2} Copy content Toggle raw display
4141 (T2364T3961116)4 (T^{2} - 364 T - 3961116)^{4} Copy content Toggle raw display
4343 (T4+10034800T2+325194264000)2 (T^{4} + 10034800 T^{2} + 325194264000)^{2} Copy content Toggle raw display
4747 (T4+751600T2+49724936640)2 (T^{4} + 751600 T^{2} + 49724936640)^{2} Copy content Toggle raw display
5353 (T4++20248378776576)2 (T^{4} + \cdots + 20248378776576)^{2} Copy content Toggle raw display
5959 (T4++225559394979840)2 (T^{4} + \cdots + 225559394979840)^{2} Copy content Toggle raw display
6161 (T21644T10307356)4 (T^{2} - 1644 T - 10307356)^{4} Copy content Toggle raw display
6767 (T4++7632038946240)2 (T^{4} + \cdots + 7632038946240)^{2} Copy content Toggle raw display
7171 (T4++13443377725440)2 (T^{4} + \cdots + 13443377725440)^{2} Copy content Toggle raw display
7373 (T4++11 ⁣ ⁣16)2 (T^{4} + \cdots + 11\!\cdots\!16)^{2} Copy content Toggle raw display
7979 (T4++152966937968640)2 (T^{4} + \cdots + 152966937968640)^{2} Copy content Toggle raw display
8383 (T4++10 ⁣ ⁣40)2 (T^{4} + \cdots + 10\!\cdots\!40)^{2} Copy content Toggle raw display
8989 (T2+6916T62026236)4 (T^{2} + 6916 T - 62026236)^{4} Copy content Toggle raw display
9797 (T4++13 ⁣ ⁣76)2 (T^{4} + \cdots + 13\!\cdots\!76)^{2} Copy content Toggle raw display
show more
show less