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Properties

Label 465.2.k.a
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.

\(\operatorname{Tr}(f)(q) = \) \( 60 q - 4 q^{6} - 32 q^{10} + 4 q^{13} + 20 q^{15} - 60 q^{16} - 46 q^{18} - 4 q^{21} + 8 q^{22} - 8 q^{25} - 6 q^{27} + 112 q^{28} + 54 q^{30} + 60 q^{31} - 30 q^{33} - 4 q^{36} - 36 q^{37} - 36 q^{40}+ \cdots - 8 q^{97}+O(q^{100}) \) 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 −1.98268 + 1.98268i 1.34762 − 1.08808i − 5.86206i 1.88142 − 1.20841i −0.514593 + 4.82922i 2.34234 + 2.34234i 7.65723 + 7.65723i 0.632171 − 2.93264i −1.33436 + 6.12615i
32.2 −1.93363 + 1.93363i −1.43955 − 0.963170i − 5.47785i −1.96536 + 1.06648i 4.64597 − 0.921141i 1.12676 + 1.12676i 6.72487 + 6.72487i 1.14461 + 2.77306i 1.73810 − 5.86244i
32.3 −1.74033 + 1.74033i 1.27567 + 1.17160i − 4.05749i 0.384493 + 2.20276i −4.25907 + 0.181113i 1.30230 + 1.30230i 3.58070 + 3.58070i 0.254684 + 2.98917i −4.50267 − 3.16439i
32.4 −1.50249 + 1.50249i −1.48917 + 0.884514i − 2.51495i −0.967939 + 2.01571i 0.908494 − 3.56644i −2.29931 − 2.29931i 0.773713 + 0.773713i 1.43527 − 2.63439i −1.57427 − 4.48291i
32.5 −1.36051 + 1.36051i −1.70135 − 0.324676i − 1.70195i 1.70565 + 1.44595i 2.75642 − 1.87297i 2.86044 + 2.86044i −0.405497 − 0.405497i 2.78917 + 1.10477i −4.28776 + 0.353322i
32.6 −1.34807 + 1.34807i −1.62395 + 0.602318i − 1.63460i 1.42410 − 1.72393i 1.37723 − 3.00117i 0.523054 + 0.523054i −0.492586 − 0.492586i 2.27443 − 1.95627i 0.404203 + 4.24377i
32.7 −1.21304 + 1.21304i 1.54400 − 0.784900i − 0.942937i 2.10454 − 0.755582i −0.920817 + 2.82505i −2.81077 − 2.81077i −1.28226 − 1.28226i 1.76786 − 2.42377i −1.63634 + 3.46945i
32.8 −1.04958 + 1.04958i 1.71909 − 0.211486i − 0.203247i −0.904608 − 2.04492i −1.58236 + 2.02630i 2.89230 + 2.89230i −1.88584 − 1.88584i 2.91055 − 0.727129i 3.09577 + 1.19685i
32.9 −1.04043 + 1.04043i 0.828268 + 1.52117i − 0.164984i −0.204789 − 2.22667i −2.44443 − 0.720921i −1.82623 − 1.82623i −1.90920 − 1.90920i −1.62795 + 2.51988i 2.52976 + 2.10362i
32.10 −0.963212 + 0.963212i −0.212764 + 1.71893i 0.144444i 2.14970 + 0.615474i −1.45076 − 1.86063i 0.630920 + 0.630920i −2.06556 − 2.06556i −2.90946 − 0.731454i −2.66345 + 1.47778i
32.11 −0.618987 + 0.618987i 1.40641 + 1.01095i 1.23371i 0.181165 + 2.22872i −1.49631 + 0.244784i −3.65822 − 3.65822i −2.00162 − 2.00162i 0.955964 + 2.84361i −1.49169 − 1.26741i
32.12 −0.614519 + 0.614519i −0.407801 − 1.68336i 1.24473i −1.13325 + 1.92763i 1.28506 + 0.783854i −1.08114 − 1.08114i −1.99395 − 1.99395i −2.66740 + 1.37295i −0.488159 − 1.88097i
32.13 −0.522133 + 0.522133i 0.999094 − 1.41485i 1.45476i −2.19887 − 0.406185i 0.217082 + 1.26040i −1.35198 − 1.35198i −1.80384 − 1.80384i −1.00362 − 2.82714i 1.36018 − 0.936017i
32.14 −0.450084 + 0.450084i −1.12935 + 1.31323i 1.59485i −1.42766 − 1.72099i −0.0827594 − 1.09937i 0.166606 + 0.166606i −1.61799 − 1.61799i −0.449130 − 2.96619i 1.41716 + 0.132024i
32.15 −0.237092 + 0.237092i −0.864754 − 1.50073i 1.88757i 2.23042 + 0.158834i 0.560838 + 0.150786i 1.18294 + 1.18294i −0.921712 − 0.921712i −1.50440 + 2.59553i −0.566472 + 0.491156i
32.16 0.237092 − 0.237092i 1.50073 + 0.864754i 1.88757i −2.23042 − 0.158834i 0.560838 − 0.150786i 1.18294 + 1.18294i 0.921712 + 0.921712i 1.50440 + 2.59553i −0.566472 + 0.491156i
32.17 0.450084 − 0.450084i −1.31323 + 1.12935i 1.59485i 1.42766 + 1.72099i −0.0827594 + 1.09937i 0.166606 + 0.166606i 1.61799 + 1.61799i 0.449130 − 2.96619i 1.41716 + 0.132024i
32.18 0.522133 − 0.522133i 1.41485 − 0.999094i 1.45476i 2.19887 + 0.406185i 0.217082 − 1.26040i −1.35198 − 1.35198i 1.80384 + 1.80384i 1.00362 − 2.82714i 1.36018 − 0.936017i
32.19 0.614519 − 0.614519i 1.68336 + 0.407801i 1.24473i 1.13325 − 1.92763i 1.28506 − 0.783854i −1.08114 − 1.08114i 1.99395 + 1.99395i 2.66740 + 1.37295i −0.488159 − 1.88097i
32.20 0.618987 − 0.618987i −1.01095 − 1.40641i 1.23371i −0.181165 − 2.22872i −1.49631 − 0.244784i −3.65822 − 3.65822i 2.00162 + 2.00162i −0.955964 + 2.84361i −1.49169 − 1.26741i
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.a ✓ 60
3.b odd 2 1 inner 465.2.k.a ✓ 60
5.c odd 4 1 inner 465.2.k.a ✓ 60
15.e even 4 1 inner 465.2.k.a ✓ 60
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
465.2.k.a ✓ 60 1.a even 1 1 trivial
465.2.k.a ✓ 60 3.b odd 2 1 inner
465.2.k.a ✓ 60 5.c odd 4 1 inner
465.2.k.a ✓ 60 15.e even 4 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{60} + 225 T_{2}^{56} + 20710 T_{2}^{52} + 1021100 T_{2}^{48} + 29870203 T_{2}^{44} + \cdots + 65610000 \) acting on \(S_{2}^{\mathrm{new}}(465, [\chi])\). Copy content Toggle raw display