Properties

Label 465.2.q.b
Level $465$
Weight $2$
Character orbit 465.q
Analytic conductor $3.713$
Analytic rank $0$
Dimension $60$
Inner twists $4$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [465,2,Mod(304,465)] 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("465.304"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(465, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 3, 2])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 465 = 3 \cdot 5 \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 465.q (of order \(6\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [60] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(3.71304369399\)
Analytic rank: \(0\)
Dimension: \(60\)
Relative dimension: \(30\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 60 q - 68 q^{4} - 2 q^{5} + 6 q^{6} + 30 q^{9} + 2 q^{10} - 6 q^{11} + 4 q^{14} + 16 q^{15} + 20 q^{16} - 18 q^{20} + 16 q^{21} - 12 q^{24} + 18 q^{25} - 36 q^{26} - 12 q^{29} - 8 q^{30} - 2 q^{31} - 26 q^{34}+ \cdots + 6 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
304.1 − 2.71530i −0.866025 + 0.500000i −5.37283 −0.793165 − 2.09067i 1.35765 + 2.35152i −3.52866 + 2.03727i 9.15824i 0.500000 − 0.866025i −5.67678 + 2.15368i
304.2 − 2.52175i −0.866025 + 0.500000i −4.35921 1.40815 + 1.73698i 1.26087 + 2.18390i 1.27653 − 0.737003i 5.94933i 0.500000 − 0.866025i 4.38022 − 3.55101i
304.3 − 2.42055i 0.866025 − 0.500000i −3.85906 0.0792470 + 2.23466i −1.21027 − 2.09626i 3.48532 − 2.01225i 4.49995i 0.500000 − 0.866025i 5.40911 − 0.191821i
304.4 − 2.24754i 0.866025 − 0.500000i −3.05145 1.89729 − 1.18334i −1.12377 − 1.94643i −2.11510 + 1.22115i 2.36317i 0.500000 − 0.866025i −2.65961 − 4.26423i
304.5 − 2.16012i 0.866025 − 0.500000i −2.66614 −1.31012 − 1.81206i −1.08006 − 1.87072i −1.09424 + 0.631758i 1.43894i 0.500000 − 0.866025i −3.91428 + 2.83003i
304.6 − 2.11581i −0.866025 + 0.500000i −2.47665 −1.82252 − 1.29553i 1.05790 + 1.83234i 4.27194 − 2.46641i 1.00850i 0.500000 − 0.866025i −2.74110 + 3.85611i
304.7 − 1.97266i −0.866025 + 0.500000i −1.89140 1.71541 + 1.43435i 0.986332 + 1.70838i −3.55080 + 2.05006i − 0.214231i 0.500000 − 0.866025i 2.82949 − 3.38393i
304.8 − 1.56738i 0.866025 − 0.500000i −0.456691 2.09283 + 0.787438i −0.783692 − 1.35739i 0.751020 − 0.433602i − 2.41896i 0.500000 − 0.866025i 1.23422 − 3.28027i
304.9 − 1.40377i −0.866025 + 0.500000i 0.0294315 0.833968 − 2.07473i 0.701885 + 1.21570i −0.717444 + 0.414216i − 2.84885i 0.500000 − 0.866025i −2.91244 − 1.17070i
304.10 − 1.29837i −0.866025 + 0.500000i 0.314240 −1.59440 + 1.56777i 0.649184 + 1.12442i −2.02725 + 1.17043i − 3.00474i 0.500000 − 0.866025i 2.03554 + 2.07012i
304.11 − 1.09870i 0.866025 − 0.500000i 0.792852 −2.14855 + 0.619461i −0.549351 − 0.951505i 2.60166 − 1.50207i − 3.06851i 0.500000 − 0.866025i 0.680604 + 2.36062i
304.12 − 0.884829i −0.866025 + 0.500000i 1.21708 −2.18219 − 0.487898i 0.442414 + 0.766284i −2.44399 + 1.41104i − 2.84656i 0.500000 − 0.866025i −0.431706 + 1.93086i
304.13 − 0.725626i −0.866025 + 0.500000i 1.47347 −0.00238698 + 2.23607i 0.362813 + 0.628410i 3.46926 − 2.00298i − 2.52044i 0.500000 − 0.866025i 1.62255 + 0.00173205i
304.14 − 0.712440i 0.866025 − 0.500000i 1.49243 0.623543 − 2.14737i −0.356220 − 0.616991i 1.79597 − 1.03690i − 2.48815i 0.500000 − 0.866025i −1.52987 − 0.444237i
304.15 − 0.431366i 0.866025 − 0.500000i 1.81392 1.40014 + 1.74345i −0.215683 − 0.373574i −1.74686 + 1.00855i − 1.64520i 0.500000 − 0.866025i 0.752064 − 0.603970i
304.16 0.431366i −0.866025 + 0.500000i 1.81392 −2.20994 − 0.340828i −0.215683 − 0.373574i 1.74686 − 1.00855i 1.64520i 0.500000 − 0.866025i 0.147022 − 0.953292i
304.17 0.712440i −0.866025 + 0.500000i 1.49243 1.54790 − 1.61369i −0.356220 − 0.616991i −1.79597 + 1.03690i 2.48815i 0.500000 − 0.866025i 1.14966 + 1.10279i
304.18 0.725626i 0.866025 − 0.500000i 1.47347 −1.93530 + 1.12010i 0.362813 + 0.628410i −3.46926 + 2.00298i 2.52044i 0.500000 − 0.866025i −0.812774 − 1.40430i
304.19 0.884829i 0.866025 − 0.500000i 1.21708 1.51363 + 1.64588i 0.442414 + 0.766284i 2.44399 − 1.41104i 2.84656i 0.500000 − 0.866025i −1.45632 + 1.33930i
304.20 1.09870i −0.866025 + 0.500000i 0.792852 0.537806 + 2.17043i −0.549351 − 0.951505i −2.60166 + 1.50207i 3.06851i 0.500000 − 0.866025i −2.38466 + 0.590889i
See all 60 embeddings
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 304.30
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
31.c even 3 1 inner
155.j even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 465.2.q.b ✓ 60
5.b even 2 1 inner 465.2.q.b ✓ 60
31.c even 3 1 inner 465.2.q.b ✓ 60
155.j even 6 1 inner 465.2.q.b ✓ 60
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
465.2.q.b ✓ 60 1.a even 1 1 trivial
465.2.q.b ✓ 60 5.b even 2 1 inner
465.2.q.b ✓ 60 31.c even 3 1 inner
465.2.q.b ✓ 60 155.j even 6 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{30} + 47 T_{2}^{28} + 991 T_{2}^{26} + 12392 T_{2}^{24} + 102369 T_{2}^{22} + 589155 T_{2}^{20} + \cdots + 43264 \) acting on \(S_{2}^{\mathrm{new}}(465, [\chi])\). Copy content Toggle raw display