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Properties

Label 465.2.bt.a
Level $465$
Weight $2$
Character orbit 465.bt
Analytic conductor $3.713$
Analytic rank $0$
Dimension $960$
Inner twists $8$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [465,2,Mod(38,465)] 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("465.38"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(465, base_ring=CyclotomicField(60)) chi = DirichletCharacter(H, H._module([30, 45, 56])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 465 = 3 \cdot 5 \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 465.bt (of order \(60\), degree \(16\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(3.71304369399\)
Analytic rank: \(0\)
Dimension: \(960\)
Relative dimension: \(60\) over \(\Q(\zeta_{60})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{60}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 960 q - 14 q^{3} - 16 q^{6} - 60 q^{7} - 44 q^{10} - 64 q^{12} - 36 q^{13} + 4 q^{15} + 176 q^{16} - 14 q^{18} - 36 q^{21} - 88 q^{22} - 20 q^{25} - 8 q^{27} - 56 q^{28} - 200 q^{30} - 56 q^{31} + 44 q^{33}+ \cdots + 44 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
38.1 −1.25101 + 2.45524i 0.806237 + 1.53297i −3.28760 − 4.52499i −1.90733 + 1.16710i −4.77240 + 0.0617550i −2.40021 + 0.921353i 9.77942 − 1.54891i −1.69996 + 2.47187i −0.479420 − 6.14297i
38.2 −1.22442 + 2.40306i 1.60431 − 0.652834i −3.09991 − 4.26666i 2.21092 + 0.334408i −0.395550 + 4.65459i −1.18080 + 0.453265i 8.72098 − 1.38127i 2.14762 − 2.09469i −3.51069 + 4.90351i
38.3 −1.18119 + 2.31821i −0.940931 + 1.45418i −2.80333 − 3.85845i 2.22883 − 0.179772i −2.25969 − 3.89894i 3.63924 − 1.39697i 7.11645 − 1.12713i −1.22930 − 2.73657i −2.21592 + 5.37924i
38.4 −1.17572 + 2.30749i −1.23857 − 1.21076i −2.76660 − 3.80790i 0.811020 + 2.08381i 4.25004 − 1.43446i −0.469611 + 0.180267i 6.92371 − 1.09661i 0.0680993 + 2.99923i −5.76189 − 0.578561i
38.5 −1.11352 + 2.18541i −0.0291390 − 1.73181i −2.36052 − 3.24897i −1.10860 − 1.94191i 3.81715 + 1.86472i −4.25353 + 1.63278i 4.88374 − 0.773509i −2.99830 + 0.100926i 5.47832 − 0.260398i
38.6 −1.10169 + 2.16219i −1.60440 + 0.652610i −2.28579 − 3.14611i −2.18616 − 0.469781i 0.356488 − 4.18800i 0.591996 − 0.227246i 4.52712 − 0.717026i 2.14820 − 2.09409i 3.42424 − 4.20935i
38.7 −1.06358 + 2.08739i 1.35306 + 1.08131i −2.05044 − 2.82219i −0.00774343 − 2.23605i −3.69620 + 1.67431i 2.13244 − 0.818566i 3.44404 − 0.545482i 0.661550 + 2.92615i 4.67576 + 2.36206i
38.8 −0.961893 + 1.88782i −1.69628 + 0.350191i −1.46306 − 2.01373i 1.99034 − 1.01910i 0.970541 − 3.53912i −4.41291 + 1.69396i 1.02354 − 0.162113i 2.75473 − 1.18805i 0.00938435 + 4.73766i
38.9 −0.959743 + 1.88360i 1.69275 + 0.366867i −1.45128 − 1.99751i −0.878348 + 2.05633i −2.31564 + 2.83637i 3.64508 − 1.39922i 0.979401 − 0.155122i 2.73082 + 1.24203i −3.03032 − 3.62801i
38.10 −0.935774 + 1.83656i −0.0130950 − 1.73200i −1.32171 − 1.81918i 1.61285 − 1.54878i 3.19318 + 1.59671i 2.84720 − 1.09294i 0.506165 − 0.0801686i −2.99966 + 0.0453612i 1.33518 + 4.41140i
38.11 −0.918904 + 1.80345i 1.08398 − 1.35092i −1.23248 − 1.69636i −1.54898 + 1.61265i 1.44026 + 3.19627i −1.66400 + 0.638751i 0.193560 − 0.0306569i −0.649995 − 2.92874i −1.48498 − 4.27539i
38.12 −0.895320 + 1.75716i −1.08132 − 1.35305i −1.11046 − 1.52841i −2.17633 + 0.513419i 3.34566 − 0.688650i 1.94619 − 0.747072i −0.215773 + 0.0341750i −0.661483 + 2.92616i 1.04635 − 4.28384i
38.13 −0.871088 + 1.70961i −0.218968 + 1.71815i −0.988392 − 1.36040i −0.507273 − 2.17777i −2.74663 − 1.87101i −0.484283 + 0.185899i −0.603494 + 0.0955841i −2.90411 − 0.752442i 4.16501 + 1.02979i
38.14 −0.858910 + 1.68571i −0.208820 + 1.71942i −0.928308 − 1.27771i 1.05695 + 1.97049i −2.71907 − 1.82883i −2.77631 + 1.06572i −0.786070 + 0.124501i −2.91279 − 0.718096i −4.22950 + 0.0892373i
38.15 −0.743316 + 1.45884i 1.63800 + 0.563002i −0.400126 − 0.550726i 1.53289 + 1.62796i −2.03888 + 1.97109i −0.658413 + 0.252741i −2.13343 + 0.337903i 2.36606 + 1.84439i −3.51435 + 1.02615i
38.16 −0.667288 + 1.30963i −1.72812 − 0.116561i −0.0942763 − 0.129760i 1.40014 + 1.74345i 1.30581 − 2.18542i 4.04669 − 1.55338i −2.67062 + 0.422984i 2.97283 + 0.402864i −3.21756 + 0.670272i
38.17 −0.603653 + 1.18474i 1.50094 − 0.864394i 0.136370 + 0.187697i 2.06692 − 0.853147i 0.118032 + 2.30001i −1.30223 + 0.499878i −2.93127 + 0.464267i 1.50564 − 2.59481i −0.236946 + 2.96375i
38.18 −0.578427 + 1.13523i −1.70315 − 0.315097i 0.221407 + 0.304740i −0.458005 − 2.18866i 1.34285 − 1.75120i 1.07013 − 0.410783i −2.99084 + 0.473702i 2.80143 + 1.07332i 2.74955 + 0.746040i
38.19 −0.565349 + 1.10956i 1.53064 + 0.810648i 0.264067 + 0.363458i −2.18422 − 0.478739i −1.76481 + 1.24003i −3.57382 + 1.37186i −3.01248 + 0.477130i 1.68570 + 2.48162i 1.76603 − 2.15287i
38.20 −0.502407 + 0.986030i 1.37930 − 1.04764i 0.455729 + 0.627257i −1.64599 − 1.51351i 0.340033 + 1.88637i 2.38455 − 0.915342i −3.03350 + 0.480459i 0.804915 − 2.89000i 2.31932 − 0.862600i
See next 80 embeddings (of 960 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 38.60
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.c odd 4 1 inner
15.e even 4 1 inner
31.g even 15 1 inner
93.o odd 30 1 inner
155.w odd 60 1 inner
465.bt even 60 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 465.2.bt.a ✓ 960
3.b odd 2 1 inner 465.2.bt.a ✓ 960
5.c odd 4 1 inner 465.2.bt.a ✓ 960
15.e even 4 1 inner 465.2.bt.a ✓ 960
31.g even 15 1 inner 465.2.bt.a ✓ 960
93.o odd 30 1 inner 465.2.bt.a ✓ 960
155.w odd 60 1 inner 465.2.bt.a ✓ 960
465.bt even 60 1 inner 465.2.bt.a ✓ 960
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
465.2.bt.a ✓ 960 1.a even 1 1 trivial
465.2.bt.a ✓ 960 3.b odd 2 1 inner
465.2.bt.a ✓ 960 5.c odd 4 1 inner
465.2.bt.a ✓ 960 15.e even 4 1 inner
465.2.bt.a ✓ 960 31.g even 15 1 inner
465.2.bt.a ✓ 960 93.o odd 30 1 inner
465.2.bt.a ✓ 960 155.w odd 60 1 inner
465.2.bt.a ✓ 960 465.bt even 60 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(465, [\chi])\).