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Properties

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

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [465,2,Mod(32,465)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(465, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("465.32");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 465 = 3 \cdot 5 \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 465.k (of order \(4\), degree \(2\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
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 \(\Q(i)\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 60 q - 4 q^{6} + 16 q^{10} - 12 q^{13} - 8 q^{15} - 60 q^{16} + 34 q^{18} - 4 q^{21} + 8 q^{22} + 8 q^{25} - 6 q^{27} - 80 q^{28} - 54 q^{30} - 60 q^{31} + 10 q^{33} + 28 q^{36} - 12 q^{37} + 60 q^{40}+ \cdots + 56 q^{97}+O(q^{100}) \) 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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
32.1 −1.79339 + 1.79339i 1.53667 + 0.799162i − 4.43246i −1.30480 − 1.81590i −4.18904 + 1.32263i −1.69682 − 1.69682i 4.36234 + 4.36234i 1.72268 + 2.45609i 5.59662 + 0.916593i
32.2 −1.79199 + 1.79199i −0.946283 + 1.45071i − 4.42248i −1.54734 − 1.61423i −0.903922 − 4.29539i 1.64971 + 1.64971i 4.34106 + 4.34106i −1.20910 − 2.74556i 5.66551 + 0.119858i
32.3 −1.78894 + 1.78894i −0.177331 + 1.72295i − 4.40062i 2.22867 − 0.181744i −2.76502 − 3.39949i −2.03308 − 2.03308i 4.29457 + 4.29457i −2.93711 − 0.611065i −3.66183 + 4.31209i
32.4 −1.67772 + 1.67772i −0.586503 − 1.62973i − 3.62946i 1.63852 + 1.52159i 3.71821 + 1.75024i −2.99940 − 2.99940i 2.73377 + 2.73377i −2.31203 + 1.91168i −5.30178 + 0.196172i
32.5 −1.54899 + 1.54899i −1.05387 − 1.37454i − 2.79873i 0.879189 − 2.05597i 3.76158 + 0.496709i 0.747956 + 0.747956i 1.23723 + 1.23723i −0.778710 + 2.89717i 1.82283 + 4.54653i
32.6 −1.50242 + 1.50242i 1.66548 − 0.475577i − 2.51453i −1.96356 + 1.06978i −1.78774 + 3.21677i −0.886503 − 0.886503i 0.773042 + 0.773042i 2.54765 − 1.58413i 1.34283 − 4.55735i
32.7 −1.35339 + 1.35339i 0.765677 − 1.55362i − 1.66332i 0.838484 + 2.07291i 1.06639 + 3.13891i 1.94800 + 1.94800i −0.455658 − 0.455658i −1.82748 − 2.37914i −3.94024 − 1.67065i
32.8 −1.13436 + 1.13436i 0.328624 + 1.70059i − 0.573551i −1.95786 + 1.08018i −2.30186 − 1.55631i 1.84589 + 1.84589i −1.61811 − 1.61811i −2.78401 + 1.11771i 0.995598 − 3.44624i
32.9 −1.00031 + 1.00031i −1.58066 − 0.708168i − 0.00123647i −1.98035 − 1.03838i 2.28954 − 0.872765i −1.66917 − 1.66917i −1.99938 − 1.99938i 1.99699 + 2.23875i 3.01966 − 0.942258i
32.10 −0.814482 + 0.814482i 0.360092 − 1.69421i 0.673239i 1.17287 − 1.90378i 1.08661 + 1.67319i −0.307284 − 0.307284i −2.17730 − 2.17730i −2.74067 − 1.22014i 0.595315 + 2.50587i
32.11 −0.523880 + 0.523880i 1.58624 + 0.695595i 1.45110i 2.20857 + 0.349585i −1.19541 + 0.466589i 2.00309 + 2.00309i −1.80796 − 1.80796i 2.03230 + 2.20676i −1.34017 + 0.973885i
32.12 −0.450085 + 0.450085i −1.67224 + 0.451241i 1.59485i −1.55533 + 1.60653i 0.549552 − 0.955745i 3.64223 + 3.64223i −1.61799 − 1.61799i 2.59276 − 1.50917i −0.0230453 − 1.42311i
32.13 −0.283833 + 0.283833i 1.57294 − 0.725164i 1.83888i 0.380129 + 2.20352i −0.240626 + 0.652276i 0.900008 + 0.900008i −1.08960 − 1.08960i 1.94827 − 2.28128i −0.733324 − 0.517538i
32.14 −0.195513 + 0.195513i −1.67938 + 0.423882i 1.92355i 2.09279 − 0.787555i 0.245466 − 0.411215i −2.75743 − 2.75743i −0.767103 − 0.767103i 2.64065 − 1.42372i −0.255189 + 0.563143i
32.15 −0.150369 + 0.150369i 0.591574 + 1.62789i 1.95478i 0.684260 − 2.12880i −0.333740 − 0.155831i −0.387197 − 0.387197i −0.594677 − 0.594677i −2.30008 + 1.92604i 0.217214 + 0.422998i
32.16 0.150369 − 0.150369i −1.62789 − 0.591574i 1.95478i −0.684260 + 2.12880i −0.333740 + 0.155831i −0.387197 − 0.387197i 0.594677 + 0.594677i 2.30008 + 1.92604i 0.217214 + 0.422998i
32.17 0.195513 − 0.195513i −0.423882 + 1.67938i 1.92355i −2.09279 + 0.787555i 0.245466 + 0.411215i −2.75743 − 2.75743i 0.767103 + 0.767103i −2.64065 − 1.42372i −0.255189 + 0.563143i
32.18 0.283833 − 0.283833i 0.725164 − 1.57294i 1.83888i −0.380129 − 2.20352i −0.240626 − 0.652276i 0.900008 + 0.900008i 1.08960 + 1.08960i −1.94827 − 2.28128i −0.733324 − 0.517538i
32.19 0.450085 − 0.450085i −0.451241 + 1.67224i 1.59485i 1.55533 − 1.60653i 0.549552 + 0.955745i 3.64223 + 3.64223i 1.61799 + 1.61799i −2.59276 − 1.50917i −0.0230453 − 1.42311i
32.20 0.523880 − 0.523880i −0.695595 − 1.58624i 1.45110i −2.20857 − 0.349585i −1.19541 − 0.466589i 2.00309 + 2.00309i 1.80796 + 1.80796i −2.03230 + 2.20676i −1.34017 + 0.973885i
See all 60 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 32.30
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.c odd 4 1 inner
15.e even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 465.2.k.b ✓ 60
3.b odd 2 1 inner 465.2.k.b ✓ 60
5.c odd 4 1 inner 465.2.k.b ✓ 60
15.e even 4 1 inner 465.2.k.b ✓ 60
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
465.2.k.b ✓ 60 1.a even 1 1 trivial
465.2.k.b ✓ 60 3.b odd 2 1 inner
465.2.k.b ✓ 60 5.c odd 4 1 inner
465.2.k.b ✓ 60 15.e even 4 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{60} + 225 T_{2}^{56} + 21670 T_{2}^{52} + 1167228 T_{2}^{48} + 38564715 T_{2}^{44} + 806935409 T_{2}^{40} + \cdots + 10000 \) acting on \(S_{2}^{\mathrm{new}}(465, [\chi])\). Copy content Toggle raw display