Properties

Label 50.26.b.b
Level 5050
Weight 2626
Character orbit 50.b
Analytic conductor 197.998197.998
Analytic rank 00
Dimension 22
Inner twists 22

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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] = [50,26,Mod(49,50)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(50, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1])) N = Newforms(chi, 26, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("50.49"); S:= CuspForms(chi, 26); N := Newforms(S);
 
Level: N N == 50=252 50 = 2 \cdot 5^{2}
Weight: k k == 26 26
Character orbit: [χ][\chi] == 50.b (of order 22, degree 11, not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0,-33554432,0,1334181888] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(6)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: 197.998389976197.998389976
Analytic rank: 00
Dimension: 22
Coefficient field: Q(1)\Q(\sqrt{-1})
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: x2+1 x^{2} + 1 Copy content Toggle raw display
Coefficient ring: Z[a1,,a13]\Z[a_1, \ldots, a_{13}]
Coefficient ring index: 2 2
Twist minimal: no (minimal twist has level 10)
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 β=2i\beta = 2i. We also show the integral qq-expansion of the trace form.

f(q)f(q) == q2048βq2+81432βq316777216q4+667090944q6+8800446746βq7+34359738368βq8+820763926947q911240373835548q111366202253312βq12+92 ⁣ ⁣56q99+O(q100) q - 2048 \beta q^{2} + 81432 \beta q^{3} - 16777216 q^{4} + 667090944 q^{6} + 8800446746 \beta q^{7} + 34359738368 \beta q^{8} + 820763926947 q^{9} - 11240373835548 q^{11} - 1366202253312 \beta q^{12} + \cdots - 92\!\cdots\!56 q^{99} +O(q^{100}) Copy content Toggle raw display
Tr(f)(q)\operatorname{Tr}(f)(q) == 2q33554432q4+1334181888q6+1641527853894q922480747671096q11+144186519486464q14+562949953421312q1659 ⁣ ⁣40q1957 ⁣ ⁣76q2122 ⁣ ⁣08q24+18 ⁣ ⁣12q99+O(q100) 2 q - 33554432 q^{4} + 1334181888 q^{6} + 1641527853894 q^{9} - 22480747671096 q^{11} + 144186519486464 q^{14} + 562949953421312 q^{16} - 59\!\cdots\!40 q^{19} - 57\!\cdots\!76 q^{21} - 22\!\cdots\!08 q^{24}+ \cdots - 18\!\cdots\!12 q^{99}+O(q^{100}) Copy content Toggle raw display

Character values

We give the values of χ\chi on generators for (Z/50Z)×\left(\mathbb{Z}/50\mathbb{Z}\right)^\times.

nn 2727
χ(n)\chi(n) 1-1

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.

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}
49.1
1.00000i
1.00000i
4096.00i 162864.i −1.67772e7 0 6.67091e8 1.76009e10i 6.87195e10i 8.20764e11 0
49.2 4096.00i 162864.i −1.67772e7 0 6.67091e8 1.76009e10i 6.87195e10i 8.20764e11 0
nn: e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 50.26.b.b 2
5.b even 2 1 inner 50.26.b.b 2
5.c odd 4 1 10.26.a.a 1
5.c odd 4 1 50.26.a.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
10.26.a.a 1 5.c odd 4 1
50.26.a.a 1 5.c odd 4 1
50.26.b.b 2 1.a even 1 1 trivial
50.26.b.b 2 5.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator T32+26524682496 T_{3}^{2} + 26524682496 acting on S26new(50,[χ])S_{26}^{\mathrm{new}}(50, [\chi]). Copy content Toggle raw display

Hecke characteristic polynomials

pp Fp(T)F_p(T)
22 T2+16777216 T^{2} + 16777216 Copy content Toggle raw display
33 T2+26524682496 T^{2} + 26524682496 Copy content Toggle raw display
55 T2 T^{2} Copy content Toggle raw display
77 T2+30 ⁣ ⁣64 T^{2} + 30\!\cdots\!64 Copy content Toggle raw display
1111 (T+11240373835548)2 (T + 11240373835548)^{2} Copy content Toggle raw display
1313 T2+17 ⁣ ⁣56 T^{2} + 17\!\cdots\!56 Copy content Toggle raw display
1717 T2+12 ⁣ ⁣04 T^{2} + 12\!\cdots\!04 Copy content Toggle raw display
1919 (T+29 ⁣ ⁣20)2 (T + 29\!\cdots\!20)^{2} Copy content Toggle raw display
2323 T2+14 ⁣ ⁣16 T^{2} + 14\!\cdots\!16 Copy content Toggle raw display
2929 (T21 ⁣ ⁣70)2 (T - 21\!\cdots\!70)^{2} Copy content Toggle raw display
3131 (T42 ⁣ ⁣52)2 (T - 42\!\cdots\!52)^{2} Copy content Toggle raw display
3737 T2+58 ⁣ ⁣84 T^{2} + 58\!\cdots\!84 Copy content Toggle raw display
4141 (T+16 ⁣ ⁣98)2 (T + 16\!\cdots\!98)^{2} Copy content Toggle raw display
4343 T2+97 ⁣ ⁣36 T^{2} + 97\!\cdots\!36 Copy content Toggle raw display
4747 T2+15 ⁣ ⁣24 T^{2} + 15\!\cdots\!24 Copy content Toggle raw display
5353 T2+22 ⁣ ⁣96 T^{2} + 22\!\cdots\!96 Copy content Toggle raw display
5959 (T+12 ⁣ ⁣60)2 (T + 12\!\cdots\!60)^{2} Copy content Toggle raw display
6161 (T+20 ⁣ ⁣98)2 (T + 20\!\cdots\!98)^{2} Copy content Toggle raw display
6767 T2+37 ⁣ ⁣04 T^{2} + 37\!\cdots\!04 Copy content Toggle raw display
7171 (T13 ⁣ ⁣52)2 (T - 13\!\cdots\!52)^{2} Copy content Toggle raw display
7373 T2+50 ⁣ ⁣16 T^{2} + 50\!\cdots\!16 Copy content Toggle raw display
7979 (T+38 ⁣ ⁣80)2 (T + 38\!\cdots\!80)^{2} Copy content Toggle raw display
8383 T2+59 ⁣ ⁣76 T^{2} + 59\!\cdots\!76 Copy content Toggle raw display
8989 (T84 ⁣ ⁣10)2 (T - 84\!\cdots\!10)^{2} Copy content Toggle raw display
9797 T2+38 ⁣ ⁣24 T^{2} + 38\!\cdots\!24 Copy content Toggle raw display
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