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Properties

Label 465.2.t.d
Level $465$
Weight $2$
Character orbit 465.t
Analytic conductor $3.713$
Analytic rank $0$
Dimension $104$
Inner twists $8$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [465,2,Mod(119,465)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(465, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([3, 3, 1])) N = Newforms(chi, 2, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("465.119"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 465 = 3 \cdot 5 \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 465.t (of order \(6\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [104,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(3.71304369399\)
Analytic rank: \(0\)
Dimension: \(104\)
Relative dimension: \(52\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 104 q + 56 q^{4} - 12 q^{6} - 2 q^{9} - 20 q^{10} - 24 q^{16} + 12 q^{19} - 6 q^{21} - 48 q^{24} + 8 q^{25} - 40 q^{31} + 108 q^{34} + 36 q^{36} + 52 q^{39} - 76 q^{40} - 4 q^{45} + 80 q^{49} - 20 q^{51}+ \cdots + 66 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
119.1 −2.66243 0.290440 + 1.70753i 5.08855 0.356449 + 2.20747i −0.773277 − 4.54617i 0.379861 + 0.219313i −8.22305 −2.83129 + 0.991867i −0.949020 − 5.87725i
119.2 −2.66243 1.62398 − 0.602235i 5.08855 1.73350 + 1.41243i −4.32374 + 1.60341i −0.379861 − 0.219313i −8.22305 2.27463 − 1.95603i −4.61534 − 3.76050i
119.3 −2.25277 −1.65302 − 0.517218i 3.07497 1.98107 + 1.03701i 3.72388 + 1.16517i 3.62956 + 2.09552i −2.42166 2.46497 + 1.70995i −4.46288 − 2.33613i
119.4 −2.25277 −1.27444 − 1.17295i 3.07497 −0.0924590 + 2.23416i 2.87101 + 2.64239i −3.62956 − 2.09552i −2.42166 0.248372 + 2.98970i 0.208289 − 5.03304i
119.5 −2.15798 −0.783102 + 1.54491i 2.65688 −0.0977344 − 2.23393i 1.68992 − 3.33389i 1.65711 + 0.956733i −1.41754 −1.77350 − 2.41965i 0.210909 + 4.82078i
119.6 −2.15798 0.946382 − 1.45064i 2.65688 −1.88577 − 1.20161i −2.04227 + 3.13046i −1.65711 − 0.956733i −1.41754 −1.20872 − 2.74572i 4.06946 + 2.59304i
119.7 −2.04620 0.485137 + 1.66272i 2.18694 2.10761 − 0.746986i −0.992687 − 3.40226i −4.09236 − 2.36273i −0.382512 −2.52928 + 1.61329i −4.31259 + 1.52848i
119.8 −2.04620 1.68253 − 0.411220i 2.18694 −1.70071 + 1.45175i −3.44279 + 0.841439i 4.09236 + 2.36273i −0.382512 2.66180 − 1.38378i 3.48000 − 2.97057i
119.9 −1.83398 −1.72961 − 0.0919572i 1.36348 −2.09097 − 0.792377i 3.17206 + 0.168647i −0.777962 − 0.449157i 1.16737 2.98309 + 0.318100i 3.83479 + 1.45320i
119.10 −1.83398 −0.944441 − 1.45191i 1.36348 0.359264 − 2.20702i 1.73209 + 2.66276i 0.777962 + 0.449157i 1.16737 −1.21606 + 2.74248i −0.658883 + 4.04762i
119.11 −1.73851 1.39172 + 1.03108i 1.02242 2.06916 − 0.847683i −2.41952 − 1.79253i 3.83754 + 2.21560i 1.69954 0.873767 + 2.86994i −3.59726 + 1.47371i
119.12 −1.73851 1.58880 + 0.689727i 1.02242 −1.76870 + 1.36811i −2.76214 − 1.19910i −3.83754 − 2.21560i 1.69954 2.04855 + 2.19167i 3.07490 − 2.37846i
119.13 −1.36470 0.0687450 + 1.73069i −0.137607 0.288745 + 2.21735i −0.0938159 − 2.36186i 1.46427 + 0.845397i 2.91718 −2.99055 + 0.237952i −0.394049 − 3.02600i
119.14 −1.36470 1.53319 − 0.805808i −0.137607 1.77591 + 1.35873i −2.09234 + 1.09968i −1.46427 − 0.845397i 2.91718 1.70135 − 2.47091i −2.42357 − 1.85426i
119.15 −1.15830 0.150473 + 1.72550i −0.658332 −2.23430 + 0.0890267i −0.174294 − 1.99866i −0.189994 − 0.109693i 3.07916 −2.95472 + 0.519284i 2.58799 − 0.103120i
119.16 −1.15830 1.56957 − 0.732437i −0.658332 1.19425 − 1.89044i −1.81803 + 0.848385i 0.189994 + 0.109693i 3.07916 1.92707 − 2.29922i −1.38330 + 2.18971i
119.17 −1.11406 −1.73171 − 0.0341087i −0.758880 1.65152 − 1.50748i 1.92923 + 0.0379990i −1.74935 − 1.00999i 3.07355 2.99767 + 0.118133i −1.83989 + 1.67941i
119.18 −1.11406 −0.895396 − 1.48265i −0.758880 −2.13127 + 0.676520i 0.997522 + 1.65176i 1.74935 + 1.00999i 3.07355 −1.39653 + 2.65513i 2.37436 − 0.753681i
119.19 −0.784965 −1.67682 + 0.433902i −1.38383 −0.314782 + 2.21380i 1.31625 − 0.340598i 2.60048 + 1.50139i 2.65619 2.62346 − 1.45515i 0.247093 − 1.73776i
119.20 −0.784965 −0.462641 − 1.66912i −1.38383 2.07460 + 0.834291i 0.363157 + 1.31020i −2.60048 − 1.50139i 2.65619 −2.57193 + 1.54441i −1.62849 − 0.654889i
See next 80 embeddings (of 104 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 119.52
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.b even 2 1 inner
15.d odd 2 1 inner
31.e odd 6 1 inner
93.g even 6 1 inner
155.i odd 6 1 inner
465.t 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.t.d ✓ 104
3.b odd 2 1 inner 465.2.t.d ✓ 104
5.b even 2 1 inner 465.2.t.d ✓ 104
15.d odd 2 1 inner 465.2.t.d ✓ 104
31.e odd 6 1 inner 465.2.t.d ✓ 104
93.g even 6 1 inner 465.2.t.d ✓ 104
155.i odd 6 1 inner 465.2.t.d ✓ 104
465.t even 6 1 inner 465.2.t.d ✓ 104
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
465.2.t.d ✓ 104 1.a even 1 1 trivial
465.2.t.d ✓ 104 3.b odd 2 1 inner
465.2.t.d ✓ 104 5.b even 2 1 inner
465.2.t.d ✓ 104 15.d odd 2 1 inner
465.2.t.d ✓ 104 31.e odd 6 1 inner
465.2.t.d ✓ 104 93.g even 6 1 inner
465.2.t.d ✓ 104 155.i odd 6 1 inner
465.2.t.d ✓ 104 465.t even 6 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{26} - 33 T_{2}^{24} + 473 T_{2}^{22} - 3876 T_{2}^{20} + 20107 T_{2}^{18} - 69071 T_{2}^{16} + \cdots - 12 \) acting on \(S_{2}^{\mathrm{new}}(465, [\chi])\). Copy content Toggle raw display