Properties

Label 800.3.p.a
Level 800800
Weight 33
Character orbit 800.p
Analytic conductor 21.79821.798
Analytic rank 00
Dimension 22
CM discriminant -4
Inner twists 44

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [800,3,Mod(193,800)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(800, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([0, 0, 3])) N = Newforms(chi, 3, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("800.193"); S:= CuspForms(chi, 3); N := Newforms(S);
 
Level: N N == 800=2552 800 = 2^{5} \cdot 5^{2}
Weight: k k == 3 3
Character orbit: [χ][\chi] == 800.p (of order 44, degree 22, minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0,0,0,0,0,0,0,0,0,0,-14] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: 21.798421148821.7984211488
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: 1 1
Twist minimal: no (minimal twist has level 160)
Sato-Tate group: U(1)[D4]\mathrm{U}(1)[D_{4}]

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 i=1i = \sqrt{-1}. We also show the integral qq-expansion of the trace form.

f(q)f(q) == q+9iq9+(7i7)q13+(23i23)q1740iq29+(23i+23)q3780q4149iq49+(73i73)q53120q61+(7i+7)q7381q81160iq89++(137i137)q97+O(q100) q + 9 i q^{9} + (7 i - 7) q^{13} + ( - 23 i - 23) q^{17} - 40 i q^{29} + (23 i + 23) q^{37} - 80 q^{41} - 49 i q^{49} + (73 i - 73) q^{53} - 120 q^{61} + ( - 7 i + 7) q^{73} - 81 q^{81} - 160 i q^{89} + \cdots + ( - 137 i - 137) q^{97} +O(q^{100}) Copy content Toggle raw display
Tr(f)(q)\operatorname{Tr}(f)(q) == 2q14q1346q17+46q37160q41146q53240q61+14q73162q81274q97+O(q100) 2 q - 14 q^{13} - 46 q^{17} + 46 q^{37} - 160 q^{41} - 146 q^{53} - 240 q^{61} + 14 q^{73} - 162 q^{81} - 274 q^{97}+O(q^{100}) Copy content Toggle raw display

Character values

We give the values of χ\chi on generators for (Z/800Z)×\left(\mathbb{Z}/800\mathbb{Z}\right)^\times.

nn 101101 351351 577577
χ(n)\chi(n) 11 11 ii

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}
193.1
1.00000i
1.00000i
0 0 0 0 0 0 0 9.00000i 0
257.1 0 0 0 0 0 0 0 9.00000i 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
4.b odd 2 1 CM by Q(1)\Q(\sqrt{-1})
5.c odd 4 1 inner
20.e even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 800.3.p.a 2
4.b odd 2 1 CM 800.3.p.a 2
5.b even 2 1 160.3.p.b 2
5.c odd 4 1 160.3.p.b 2
5.c odd 4 1 inner 800.3.p.a 2
20.d odd 2 1 160.3.p.b 2
20.e even 4 1 160.3.p.b 2
20.e even 4 1 inner 800.3.p.a 2
40.e odd 2 1 320.3.p.d 2
40.f even 2 1 320.3.p.d 2
40.i odd 4 1 320.3.p.d 2
40.k even 4 1 320.3.p.d 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
160.3.p.b 2 5.b even 2 1
160.3.p.b 2 5.c odd 4 1
160.3.p.b 2 20.d odd 2 1
160.3.p.b 2 20.e even 4 1
320.3.p.d 2 40.e odd 2 1
320.3.p.d 2 40.f even 2 1
320.3.p.d 2 40.i odd 4 1
320.3.p.d 2 40.k even 4 1
800.3.p.a 2 1.a even 1 1 trivial
800.3.p.a 2 4.b odd 2 1 CM
800.3.p.a 2 5.c odd 4 1 inner
800.3.p.a 2 20.e even 4 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on S3new(800,[χ])S_{3}^{\mathrm{new}}(800, [\chi]):

T3 T_{3} Copy content Toggle raw display
T132+14T13+98 T_{13}^{2} + 14T_{13} + 98 Copy content Toggle raw display

Hecke characteristic polynomials

pp Fp(T)F_p(T)
22 T2 T^{2} Copy content Toggle raw display
33 T2 T^{2} Copy content Toggle raw display
55 T2 T^{2} Copy content Toggle raw display
77 T2 T^{2} Copy content Toggle raw display
1111 T2 T^{2} Copy content Toggle raw display
1313 T2+14T+98 T^{2} + 14T + 98 Copy content Toggle raw display
1717 T2+46T+1058 T^{2} + 46T + 1058 Copy content Toggle raw display
1919 T2 T^{2} Copy content Toggle raw display
2323 T2 T^{2} Copy content Toggle raw display
2929 T2+1600 T^{2} + 1600 Copy content Toggle raw display
3131 T2 T^{2} Copy content Toggle raw display
3737 T246T+1058 T^{2} - 46T + 1058 Copy content Toggle raw display
4141 (T+80)2 (T + 80)^{2} Copy content Toggle raw display
4343 T2 T^{2} Copy content Toggle raw display
4747 T2 T^{2} Copy content Toggle raw display
5353 T2+146T+10658 T^{2} + 146T + 10658 Copy content Toggle raw display
5959 T2 T^{2} Copy content Toggle raw display
6161 (T+120)2 (T + 120)^{2} Copy content Toggle raw display
6767 T2 T^{2} Copy content Toggle raw display
7171 T2 T^{2} Copy content Toggle raw display
7373 T214T+98 T^{2} - 14T + 98 Copy content Toggle raw display
7979 T2 T^{2} Copy content Toggle raw display
8383 T2 T^{2} Copy content Toggle raw display
8989 T2+25600 T^{2} + 25600 Copy content Toggle raw display
9797 T2+274T+37538 T^{2} + 274T + 37538 Copy content Toggle raw display
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