Properties

Label 576.8.a.bb
Level 576576
Weight 88
Character orbit 576.a
Self dual yes
Analytic conductor 179.934179.934
Analytic rank 11
Dimension 22
Inner twists 22

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [576,8,Mod(1,576)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(576, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 8, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("576.1"); S:= CuspForms(chi, 8); N := Newforms(S);
 
Level: N N == 576=2632 576 = 2^{6} \cdot 3^{2}
Weight: k k == 8 8
Character orbit: [χ][\chi] == 576.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0,0,-280,0,0,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: 179.933774679179.933774679
Analytic rank: 11
Dimension: 22
Coefficient field: Q(435)\Q(\sqrt{435})
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: x2435 x^{2} - 435 Copy content Toggle raw display
Coefficient ring: Z[a1,,a7]\Z[a_1, \ldots, a_{7}]
Coefficient ring index: 243 2^{4}\cdot 3
Twist minimal: no (minimal twist has level 288)
Fricke sign: 1-1
Sato-Tate group: SU(2)\mathrm{SU}(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=48435\beta = 48\sqrt{435}. We also show the integral qq-expansion of the trace form.

f(q)f(q) == q140q5+βq7+4βq11+2238q13+1144q1714βq1988βq2358525q25+41324q29183βq31140βq35+137594q37+123848q41+6619986q97+O(q100) q - 140 q^{5} + \beta q^{7} + 4 \beta q^{11} + 2238 q^{13} + 1144 q^{17} - 14 \beta q^{19} - 88 \beta q^{23} - 58525 q^{25} + 41324 q^{29} - 183 \beta q^{31} - 140 \beta q^{35} + 137594 q^{37} + 123848 q^{41} + \cdots - 6619986 q^{97} +O(q^{100}) Copy content Toggle raw display
Tr(f)(q)\operatorname{Tr}(f)(q) == 2q280q5+4476q13+2288q17117050q25+82648q29+275188q37+247696q41+357394q49+1962136q532009500q61626640q65+1207596q73+8017920q77+13239972q97+O(q100) 2 q - 280 q^{5} + 4476 q^{13} + 2288 q^{17} - 117050 q^{25} + 82648 q^{29} + 275188 q^{37} + 247696 q^{41} + 357394 q^{49} + 1962136 q^{53} - 2009500 q^{61} - 626640 q^{65} + 1207596 q^{73} + 8017920 q^{77}+ \cdots - 13239972 q^{97}+O(q^{100}) Copy content Toggle raw display

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}
1.1
−20.8567
20.8567
0 0 0 −140.000 0 −1001.12 0 0 0
1.2 0 0 0 −140.000 0 1001.12 0 0 0
nn: e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

p p Sign
22 1 -1
33 +1 +1

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 576.8.a.bb 2
3.b odd 2 1 576.8.a.br 2
4.b odd 2 1 inner 576.8.a.bb 2
8.b even 2 1 288.8.a.p yes 2
8.d odd 2 1 288.8.a.p yes 2
12.b even 2 1 576.8.a.br 2
24.f even 2 1 288.8.a.f 2
24.h odd 2 1 288.8.a.f 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
288.8.a.f 2 24.f even 2 1
288.8.a.f 2 24.h odd 2 1
288.8.a.p yes 2 8.b even 2 1
288.8.a.p yes 2 8.d odd 2 1
576.8.a.bb 2 1.a even 1 1 trivial
576.8.a.bb 2 4.b odd 2 1 inner
576.8.a.br 2 3.b odd 2 1
576.8.a.br 2 12.b even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on S8new(Γ0(576))S_{8}^{\mathrm{new}}(\Gamma_0(576)):

T5+140 T_{5} + 140 Copy content Toggle raw display
T721002240 T_{7}^{2} - 1002240 Copy content Toggle raw display
T11216035840 T_{11}^{2} - 16035840 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 (T+140)2 (T + 140)^{2} Copy content Toggle raw display
77 T21002240 T^{2} - 1002240 Copy content Toggle raw display
1111 T216035840 T^{2} - 16035840 Copy content Toggle raw display
1313 (T2238)2 (T - 2238)^{2} Copy content Toggle raw display
1717 (T1144)2 (T - 1144)^{2} Copy content Toggle raw display
1919 T2196439040 T^{2} - 196439040 Copy content Toggle raw display
2323 T27761346560 T^{2} - 7761346560 Copy content Toggle raw display
2929 (T41324)2 (T - 41324)^{2} Copy content Toggle raw display
3131 T233564015360 T^{2} - 33564015360 Copy content Toggle raw display
3737 (T137594)2 (T - 137594)^{2} Copy content Toggle raw display
4141 (T123848)2 (T - 123848)^{2} Copy content Toggle raw display
4343 T2407955778560 T^{2} - 407955778560 Copy content Toggle raw display
4747 T2603524874240 T^{2} - 603524874240 Copy content Toggle raw display
5353 (T981068)2 (T - 981068)^{2} Copy content Toggle raw display
5959 T2323346677760 T^{2} - 323346677760 Copy content Toggle raw display
6161 (T+1004750)2 (T + 1004750)^{2} Copy content Toggle raw display
6767 T21177688125440 T^{2} - 1177688125440 Copy content Toggle raw display
7171 T25469119447040 T^{2} - 5469119447040 Copy content Toggle raw display
7373 (T603798)2 (T - 603798)^{2} Copy content Toggle raw display
7979 T23367415151360 T^{2} - 3367415151360 Copy content Toggle raw display
8383 T25228341309440 T^{2} - 5228341309440 Copy content Toggle raw display
8989 (T+6185104)2 (T + 6185104)^{2} Copy content Toggle raw display
9797 (T+6619986)2 (T + 6619986)^{2} Copy content Toggle raw display
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