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Properties

Label 605.2.j.k
Level $605$
Weight $2$
Character orbit 605.j
Analytic conductor $4.831$
Analytic rank $0$
Dimension $48$
CM no
Inner twists $16$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [605,2,Mod(9,605)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(605, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([5, 6]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("605.9");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 605 = 5 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 605.j (of order \(10\), degree \(4\), not 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: \(4.83094932229\)
Analytic rank: \(0\)
Dimension: \(48\)
Relative dimension: \(12\) over \(\Q(\zeta_{10})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 48 q + 20 q^{4} + 6 q^{5} + 20 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 48 q + 20 q^{4} + 6 q^{5} + 20 q^{9} + 32 q^{14} - 20 q^{15} - 36 q^{16} - 26 q^{20} + 10 q^{25} - 20 q^{26} - 8 q^{31} + 48 q^{34} - 92 q^{36} - 72 q^{45} + 4 q^{49} + 192 q^{56} + 32 q^{59} + 92 q^{60} - 28 q^{64} + 16 q^{69} + 12 q^{70} - 112 q^{71} - 36 q^{75} + 106 q^{80} + 20 q^{81} + 56 q^{86} - 432 q^{89} - 72 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
9.1 −2.48013 − 0.805844i −1.25775 − 1.73114i 3.88364 + 2.82163i 0.0494116 − 2.23552i 1.72435 + 5.30700i −0.581248 + 0.800020i −4.29253 − 5.90817i −0.487865 + 1.50149i −1.92403 + 5.50457i
9.2 −2.48013 − 0.805844i 1.25775 + 1.73114i 3.88364 + 2.82163i −1.35398 − 1.77953i −1.72435 − 5.30700i −0.581248 + 0.800020i −4.29253 − 5.90817i −0.487865 + 1.50149i 1.92403 + 5.50457i
9.3 −1.92380 − 0.625081i −1.71325 − 2.35809i 1.69226 + 1.22950i 2.16467 + 0.560538i 1.82196 + 5.60741i −1.88754 + 2.59798i −0.109082 − 0.150138i −1.69829 + 5.22681i −3.81402 − 2.43146i
9.4 −1.92380 − 0.625081i 1.71325 + 2.35809i 1.69226 + 1.22950i −1.42178 + 1.72585i −1.82196 − 5.60741i −1.88754 + 2.59798i −0.109082 − 0.150138i −1.69829 + 5.22681i 3.81402 − 2.43146i
9.5 −0.312280 − 0.101466i −0.565450 − 0.778275i −1.53081 − 1.11220i −2.23364 + 0.104146i 0.0976102 + 0.300413i 1.92358 − 2.64758i 0.751190 + 1.03393i 0.641073 − 1.97302i 0.708089 + 0.194116i
9.6 −0.312280 − 0.101466i 0.565450 + 0.778275i −1.53081 − 1.11220i 1.86827 − 1.22865i −0.0976102 − 0.300413i 1.92358 − 2.64758i 0.751190 + 1.03393i 0.641073 − 1.97302i −0.708089 + 0.194116i
9.7 0.312280 + 0.101466i −0.565450 − 0.778275i −1.53081 − 1.11220i −2.23364 + 0.104146i −0.0976102 − 0.300413i −1.92358 + 2.64758i −0.751190 − 1.03393i 0.641073 − 1.97302i −0.708089 − 0.194116i
9.8 0.312280 + 0.101466i 0.565450 + 0.778275i −1.53081 − 1.11220i 1.86827 − 1.22865i 0.0976102 + 0.300413i −1.92358 + 2.64758i −0.751190 − 1.03393i 0.641073 − 1.97302i 0.708089 − 0.194116i
9.9 1.92380 + 0.625081i −1.71325 − 2.35809i 1.69226 + 1.22950i 2.16467 + 0.560538i −1.82196 − 5.60741i 1.88754 − 2.59798i 0.109082 + 0.150138i −1.69829 + 5.22681i 3.81402 + 2.43146i
9.10 1.92380 + 0.625081i 1.71325 + 2.35809i 1.69226 + 1.22950i −1.42178 + 1.72585i 1.82196 + 5.60741i 1.88754 − 2.59798i 0.109082 + 0.150138i −1.69829 + 5.22681i −3.81402 + 2.43146i
9.11 2.48013 + 0.805844i −1.25775 − 1.73114i 3.88364 + 2.82163i 0.0494116 − 2.23552i −1.72435 − 5.30700i 0.581248 − 0.800020i 4.29253 + 5.90817i −0.487865 + 1.50149i 1.92403 − 5.50457i
9.12 2.48013 + 0.805844i 1.25775 + 1.73114i 3.88364 + 2.82163i −1.35398 − 1.77953i 1.72435 + 5.30700i 0.581248 − 0.800020i 4.29253 + 5.90817i −0.487865 + 1.50149i −1.92403 − 5.50457i
124.1 −1.53281 − 2.10973i −2.03508 + 0.661236i −1.48342 + 4.56549i 1.27403 + 1.83762i 4.51440 + 3.27991i 0.940480 + 0.305580i 6.94547 − 2.25672i 1.27725 − 0.927974i 1.92403 − 5.50457i
124.2 −1.53281 − 2.10973i 2.03508 − 0.661236i −1.48342 + 4.56549i 2.14138 + 0.643821i −4.51440 − 3.27991i 0.940480 + 0.305580i 6.94547 − 2.25672i 1.27725 − 0.927974i −1.92403 − 5.50457i
124.3 −1.18898 − 1.63648i −2.77210 + 0.900709i −0.646385 + 1.98937i −2.08073 + 0.818876i 4.76995 + 3.46557i 3.05411 + 0.992339i 0.176498 − 0.0573478i 4.44619 − 3.23035i 3.81402 + 2.43146i
124.4 −1.18898 − 1.63648i 2.77210 − 0.900709i −0.646385 + 1.98937i 0.135816 − 2.23194i −4.76995 − 3.46557i 3.05411 + 0.992339i 0.176498 − 0.0573478i 4.44619 − 3.23035i −3.81402 + 2.43146i
124.5 −0.193000 − 0.265641i −0.914917 + 0.297274i 0.584718 − 1.79958i 1.74584 − 1.39716i 0.255547 + 0.185666i −3.11241 − 1.01128i −1.21545 + 0.394924i −1.67835 + 1.21939i −0.708089 − 0.194116i
124.6 −0.193000 − 0.265641i 0.914917 − 0.297274i 0.584718 − 1.79958i −0.789281 + 2.09214i −0.255547 − 0.185666i −3.11241 − 1.01128i −1.21545 + 0.394924i −1.67835 + 1.21939i 0.708089 − 0.194116i
124.7 0.193000 + 0.265641i −0.914917 + 0.297274i 0.584718 − 1.79958i 1.74584 − 1.39716i −0.255547 − 0.185666i 3.11241 + 1.01128i 1.21545 − 0.394924i −1.67835 + 1.21939i 0.708089 + 0.194116i
124.8 0.193000 + 0.265641i 0.914917 − 0.297274i 0.584718 − 1.79958i −0.789281 + 2.09214i 0.255547 + 0.185666i 3.11241 + 1.01128i 1.21545 − 0.394924i −1.67835 + 1.21939i −0.708089 + 0.194116i
See all 48 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 9.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
11.b odd 2 1 inner
11.c even 5 3 inner
11.d odd 10 3 inner
55.d odd 2 1 inner
55.h odd 10 3 inner
55.j even 10 3 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 605.2.j.k 48
5.b even 2 1 inner 605.2.j.k 48
11.b odd 2 1 inner 605.2.j.k 48
11.c even 5 1 605.2.b.h ✓ 12
11.c even 5 3 inner 605.2.j.k 48
11.d odd 10 1 605.2.b.h ✓ 12
11.d odd 10 3 inner 605.2.j.k 48
55.d odd 2 1 inner 605.2.j.k 48
55.h odd 10 1 605.2.b.h ✓ 12
55.h odd 10 3 inner 605.2.j.k 48
55.j even 10 1 605.2.b.h ✓ 12
55.j even 10 3 inner 605.2.j.k 48
55.k odd 20 2 3025.2.a.bo 12
55.l even 20 2 3025.2.a.bo 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
605.2.b.h ✓ 12 11.c even 5 1
605.2.b.h ✓ 12 11.d odd 10 1
605.2.b.h ✓ 12 55.h odd 10 1
605.2.b.h ✓ 12 55.j even 10 1
605.2.j.k 48 1.a even 1 1 trivial
605.2.j.k 48 5.b even 2 1 inner
605.2.j.k 48 11.b odd 2 1 inner
605.2.j.k 48 11.c even 5 3 inner
605.2.j.k 48 11.d odd 10 3 inner
605.2.j.k 48 55.d odd 2 1 inner
605.2.j.k 48 55.h odd 10 3 inner
605.2.j.k 48 55.j even 10 3 inner
3025.2.a.bo 12 55.k odd 20 2
3025.2.a.bo 12 55.l even 20 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(605, [\chi])\):

\( T_{2}^{24} - 11 T_{2}^{22} + 92 T_{2}^{20} - 696 T_{2}^{18} + 5021 T_{2}^{16} - 19632 T_{2}^{14} + \cdots + 81 \) Copy content Toggle raw display
\( T_{19}^{24} + 72 T_{19}^{22} + 3852 T_{19}^{20} + 185328 T_{19}^{18} + 8492688 T_{19}^{16} + \cdots + 228509902503936 \) Copy content Toggle raw display