Properties

Label 425.2.n.b
Level 425425
Weight 22
Character orbit 425.n
Analytic conductor 3.3943.394
Analytic rank 00
Dimension 44
Inner twists 22

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [425,2,Mod(49,425)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("425.49"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(425, base_ring=CyclotomicField(8)) chi = DirichletCharacter(H, H._module([4, 3])) N = Newforms(chi, 2, names="a")
 
Level: N N == 425=5217 425 = 5^{2} \cdot 17
Weight: k k == 2 2
Character orbit: [χ][\chi] == 425.n (of order 88, degree 44, not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,4,4] 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: 3.393642085903.39364208590
Analytic rank: 00
Dimension: 44
Coefficient field: Q(ζ8)\Q(\zeta_{8})
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: x4+1 x^{4} + 1 Copy content Toggle raw display
Coefficient ring: Z[a1,a2,a3]\Z[a_1, a_2, a_3]
Coefficient ring index: 1 1
Twist minimal: no (minimal twist has level 17)
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 primitive root of unity ζ8\zeta_{8}. We also show the integral qq-expansion of the trace form.

f(q)f(q) == q+(ζ82ζ8+1)q2+(ζ83ζ82++1)q3+(2ζ83+2ζ8)q4+(ζ83+ζ82++1)q6++(3ζ83+3ζ82++1)q99+O(q100) q + (\zeta_{8}^{2} - \zeta_{8} + 1) q^{2} + ( - \zeta_{8}^{3} - \zeta_{8}^{2} + \cdots + 1) q^{3} + ( - 2 \zeta_{8}^{3} + \cdots - 2 \zeta_{8}) q^{4} + ( - \zeta_{8}^{3} + \zeta_{8}^{2} + \cdots + 1) q^{6}+ \cdots + ( - 3 \zeta_{8}^{3} + 3 \zeta_{8}^{2} + \cdots + 1) q^{99}+O(q^{100}) Copy content Toggle raw display
Tr(f)(q)\operatorname{Tr}(f)(q) == 4q+4q2+4q3+4q6+4q74q88q94q1112q124q1412q16+12q178q19+4q22+12q2312q244q268q2720q28++4q99+O(q100) 4 q + 4 q^{2} + 4 q^{3} + 4 q^{6} + 4 q^{7} - 4 q^{8} - 8 q^{9} - 4 q^{11} - 12 q^{12} - 4 q^{14} - 12 q^{16} + 12 q^{17} - 8 q^{19} + 4 q^{22} + 12 q^{23} - 12 q^{24} - 4 q^{26} - 8 q^{27} - 20 q^{28}+ \cdots + 4 q^{99}+O(q^{100}) Copy content Toggle raw display

Character values

We give the values of χ\chi on generators for (Z/425Z)×\left(\mathbb{Z}/425\mathbb{Z}\right)^\times.

nn 5252 326326
χ(n)\chi(n) 1-1 ζ8\zeta_{8}

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
−0.707107 + 0.707107i
0.707107 0.707107i
0.707107 + 0.707107i
−0.707107 0.707107i
1.70711 1.70711i 1.00000 0.414214i 3.82843i 0 1.00000 2.41421i 1.00000 2.41421i −3.12132 3.12132i −1.29289 + 1.29289i 0
274.1 0.292893 0.292893i 1.00000 + 2.41421i 1.82843i 0 1.00000 + 0.414214i 1.00000 + 0.414214i 1.12132 + 1.12132i −2.70711 + 2.70711i 0
349.1 0.292893 + 0.292893i 1.00000 2.41421i 1.82843i 0 1.00000 0.414214i 1.00000 0.414214i 1.12132 1.12132i −2.70711 2.70711i 0
399.1 1.70711 + 1.70711i 1.00000 + 0.414214i 3.82843i 0 1.00000 + 2.41421i 1.00000 + 2.41421i −3.12132 + 3.12132i −1.29289 1.29289i 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
85.m even 8 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 425.2.n.b 4
5.b even 2 1 425.2.n.a 4
5.c odd 4 1 17.2.d.a 4
5.c odd 4 1 425.2.m.a 4
15.e even 4 1 153.2.l.c 4
17.d even 8 1 425.2.n.a 4
20.e even 4 1 272.2.v.d 4
35.f even 4 1 833.2.l.a 4
35.k even 12 2 833.2.v.a 8
35.l odd 12 2 833.2.v.b 8
85.f odd 4 1 289.2.d.b 4
85.g odd 4 1 289.2.d.a 4
85.i odd 4 1 289.2.d.c 4
85.k odd 8 1 17.2.d.a 4
85.k odd 8 1 289.2.d.a 4
85.m even 8 1 inner 425.2.n.b 4
85.n odd 8 1 289.2.d.b 4
85.n odd 8 1 289.2.d.c 4
85.n odd 8 1 425.2.m.a 4
85.o even 16 2 289.2.a.f 4
85.o even 16 2 289.2.b.b 4
85.r even 16 4 289.2.c.c 8
85.r even 16 2 7225.2.a.u 4
255.ba even 8 1 153.2.l.c 4
255.bc odd 16 2 2601.2.a.bb 4
340.z even 8 1 272.2.v.d 4
340.bc odd 16 2 4624.2.a.bp 4
595.bd even 8 1 833.2.l.a 4
595.ck odd 24 2 833.2.v.b 8
595.cl even 24 2 833.2.v.a 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
17.2.d.a 4 5.c odd 4 1
17.2.d.a 4 85.k odd 8 1
153.2.l.c 4 15.e even 4 1
153.2.l.c 4 255.ba even 8 1
272.2.v.d 4 20.e even 4 1
272.2.v.d 4 340.z even 8 1
289.2.a.f 4 85.o even 16 2
289.2.b.b 4 85.o even 16 2
289.2.c.c 8 85.r even 16 4
289.2.d.a 4 85.g odd 4 1
289.2.d.a 4 85.k odd 8 1
289.2.d.b 4 85.f odd 4 1
289.2.d.b 4 85.n odd 8 1
289.2.d.c 4 85.i odd 4 1
289.2.d.c 4 85.n odd 8 1
425.2.m.a 4 5.c odd 4 1
425.2.m.a 4 85.n odd 8 1
425.2.n.a 4 5.b even 2 1
425.2.n.a 4 17.d even 8 1
425.2.n.b 4 1.a even 1 1 trivial
425.2.n.b 4 85.m even 8 1 inner
833.2.l.a 4 35.f even 4 1
833.2.l.a 4 595.bd even 8 1
833.2.v.a 8 35.k even 12 2
833.2.v.a 8 595.cl even 24 2
833.2.v.b 8 35.l odd 12 2
833.2.v.b 8 595.ck odd 24 2
2601.2.a.bb 4 255.bc odd 16 2
4624.2.a.bp 4 340.bc odd 16 2
7225.2.a.u 4 85.r even 16 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator T244T23+8T224T2+1 T_{2}^{4} - 4T_{2}^{3} + 8T_{2}^{2} - 4T_{2} + 1 acting on S2new(425,[χ])S_{2}^{\mathrm{new}}(425, [\chi]). Copy content Toggle raw display

Hecke characteristic polynomials

pp Fp(T)F_p(T)
22 T44T3++1 T^{4} - 4 T^{3} + \cdots + 1 Copy content Toggle raw display
33 T44T3++8 T^{4} - 4 T^{3} + \cdots + 8 Copy content Toggle raw display
55 T4 T^{4} Copy content Toggle raw display
77 T44T3++8 T^{4} - 4 T^{3} + \cdots + 8 Copy content Toggle raw display
1111 T4+4T3++8 T^{4} + 4 T^{3} + \cdots + 8 Copy content Toggle raw display
1313 (T22)2 (T^{2} - 2)^{2} Copy content Toggle raw display
1717 (T26T+17)2 (T^{2} - 6 T + 17)^{2} Copy content Toggle raw display
1919 T4+8T3++16 T^{4} + 8 T^{3} + \cdots + 16 Copy content Toggle raw display
2323 T412T3++392 T^{4} - 12 T^{3} + \cdots + 392 Copy content Toggle raw display
2929 T44T3++2 T^{4} - 4 T^{3} + \cdots + 2 Copy content Toggle raw display
3131 T4+12T3++648 T^{4} + 12 T^{3} + \cdots + 648 Copy content Toggle raw display
3737 T4+20T3++1250 T^{4} + 20 T^{3} + \cdots + 1250 Copy content Toggle raw display
4141 T4+4T3++98 T^{4} + 4 T^{3} + \cdots + 98 Copy content Toggle raw display
4343 T48T3++16 T^{4} - 8 T^{3} + \cdots + 16 Copy content Toggle raw display
4747 (T216T+56)2 (T^{2} - 16 T + 56)^{2} Copy content Toggle raw display
5353 (T2+2T+2)2 (T^{2} + 2 T + 2)^{2} Copy content Toggle raw display
5959 T4+1296 T^{4} + 1296 Copy content Toggle raw display
6161 T4+50T2++1250 T^{4} + 50 T^{2} + \cdots + 1250 Copy content Toggle raw display
6767 T4+48T2+64 T^{4} + 48T^{2} + 64 Copy content Toggle raw display
7171 T420T3++5000 T^{4} - 20 T^{3} + \cdots + 5000 Copy content Toggle raw display
7373 T4+98T2++4802 T^{4} + 98 T^{2} + \cdots + 4802 Copy content Toggle raw display
7979 T44T3++392 T^{4} - 4 T^{3} + \cdots + 392 Copy content Toggle raw display
8383 T416T3++16 T^{4} - 16 T^{3} + \cdots + 16 Copy content Toggle raw display
8989 T4+132T2+3844 T^{4} + 132T^{2} + 3844 Copy content Toggle raw display
9797 T4+4T3++4418 T^{4} + 4 T^{3} + \cdots + 4418 Copy content Toggle raw display
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