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Properties

Label 624.4.bc.d
Level 624624
Weight 44
Character orbit 624.bc
Analytic conductor 36.81736.817
Analytic rank 00
Dimension 2828
Inner twists 22

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [624,4,Mod(31,624)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(624, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([2, 0, 0, 3])) N = Newforms(chi, 4, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("624.31"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Level: N N == 624=24⋅3⋅13 624 = 2^{4} \cdot 3 \cdot 13
Weight: k k == 4 4
Character orbit: [χ][\chi] == 624.bc (of order 44, degree 22, minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [28,0,0,0,4,0,8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: 36.817191843636.8171918436
Analytic rank: 00
Dimension: 2828
Relative dimension: 1414 over Q(i)\Q(i)
Twist minimal: yes
Sato-Tate group: SU(2)[C4]\mathrm{SU}(2)[C_{4}]

qq-expansion

The algebraic qq-expansion of this newform has not been computed, but we have computed the trace expansion.

Tr⁡(f)(q)=\operatorname{Tr}(f)(q) = 28q+4q5+8q7−252q9−64q11−32q13+12q15−56q19+24q21−384q23−32q29+168q31−192q33+412q37−252q39+1340q41−624q43−36q45+⋯+576q99+O(q100) 28 q + 4 q^{5} + 8 q^{7} - 252 q^{9} - 64 q^{11} - 32 q^{13} + 12 q^{15} - 56 q^{19} + 24 q^{21} - 384 q^{23} - 32 q^{29} + 168 q^{31} - 192 q^{33} + 412 q^{37} - 252 q^{39} + 1340 q^{41} - 624 q^{43} - 36 q^{45}+ \cdots + 576 q^{99}+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   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}
31.1 0 3.00000i 0 −13.9138 + 13.9138i 0 3.70358 − 3.70358i 0 −9.00000 0
31.2 0 3.00000i 0 −11.5719 + 11.5719i 0 25.2676 − 25.2676i 0 −9.00000 0
31.3 0 3.00000i 0 −8.48307 + 8.48307i 0 −5.05973 + 5.05973i 0 −9.00000 0
31.4 0 3.00000i 0 −6.77018 + 6.77018i 0 −21.6794 + 21.6794i 0 −9.00000 0
31.5 0 3.00000i 0 −6.10630 + 6.10630i 0 −23.8576 + 23.8576i 0 −9.00000 0
31.6 0 3.00000i 0 −2.84589 + 2.84589i 0 9.35646 − 9.35646i 0 −9.00000 0
31.7 0 3.00000i 0 −1.88620 + 1.88620i 0 −9.36349 + 9.36349i 0 −9.00000 0
31.8 0 3.00000i 0 −0.273393 + 0.273393i 0 21.0333 − 21.0333i 0 −9.00000 0
31.9 0 3.00000i 0 5.36394 − 5.36394i 0 11.4083 − 11.4083i 0 −9.00000 0
31.10 0 3.00000i 0 7.68384 − 7.68384i 0 −9.93516 + 9.93516i 0 −9.00000 0
31.11 0 3.00000i 0 7.87509 − 7.87509i 0 15.6298 − 15.6298i 0 −9.00000 0
31.12 0 3.00000i 0 8.84364 − 8.84364i 0 3.20396 − 3.20396i 0 −9.00000 0
31.13 0 3.00000i 0 10.3664 − 10.3664i 0 −6.70327 + 6.70327i 0 −9.00000 0
31.14 0 3.00000i 0 13.7178 − 13.7178i 0 −9.00439 + 9.00439i 0 −9.00000 0
463.1 0 − 3.00000i 0 −13.9138 − 13.9138i 0 3.70358 + 3.70358i 0 −9.00000 0
463.2 0 − 3.00000i 0 −11.5719 − 11.5719i 0 25.2676 + 25.2676i 0 −9.00000 0
463.3 0 − 3.00000i 0 −8.48307 − 8.48307i 0 −5.05973 − 5.05973i 0 −9.00000 0
463.4 0 − 3.00000i 0 −6.77018 − 6.77018i 0 −21.6794 − 21.6794i 0 −9.00000 0
463.5 0 − 3.00000i 0 −6.10630 − 6.10630i 0 −23.8576 − 23.8576i 0 −9.00000 0
463.6 0 − 3.00000i 0 −2.84589 − 2.84589i 0 9.35646 + 9.35646i 0 −9.00000 0
See all 28 embeddings
nn: e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 31.14
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
52.f even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 624.4.bc.d yes 28
4.b odd 2 1 624.4.bc.c ✓ 28
13.d odd 4 1 624.4.bc.c ✓ 28
52.f even 4 1 inner 624.4.bc.d yes 28
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
624.4.bc.c ✓ 28 4.b odd 2 1
624.4.bc.c ✓ 28 13.d odd 4 1
624.4.bc.d yes 28 1.a even 1 1 trivial
624.4.bc.d yes 28 52.f 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 S4new(624,[χ])S_{4}^{\mathrm{new}}(624, [\chi]):

T528−4T527+8T526+72T525+250468T524−1122080T523+⋯+18 ⁣⋯ ⁣44 T_{5}^{28} - 4 T_{5}^{27} + 8 T_{5}^{26} + 72 T_{5}^{25} + 250468 T_{5}^{24} - 1122080 T_{5}^{23} + \cdots + 18\!\cdots\!44 Copy content Toggle raw display
T728−8T727+32T726+6464T725+2466704T724−15378944T723+⋯+39 ⁣⋯ ⁣64 T_{7}^{28} - 8 T_{7}^{27} + 32 T_{7}^{26} + 6464 T_{7}^{25} + 2466704 T_{7}^{24} - 15378944 T_{7}^{23} + \cdots + 39\!\cdots\!64 Copy content Toggle raw display