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Properties

Label 605.2.g.q
Level $605$
Weight $2$
Character orbit 605.g
Analytic conductor $4.831$
Analytic rank $0$
Dimension $24$
CM no
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [605,2,Mod(81,605)]
 
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([0, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("605.81");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 605 = 5 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 605.g (of order \(5\), 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: \(24\)
Relative dimension: \(6\) over \(\Q(\zeta_{5})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 24 q - 6 q^{3} - 6 q^{4} - 6 q^{5} - 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q - 6 q^{3} - 6 q^{4} - 6 q^{5} - 12 q^{9} + 72 q^{12} + 18 q^{14} - 6 q^{15} - 18 q^{16} - 6 q^{20} + 48 q^{23} - 6 q^{25} + 36 q^{26} - 30 q^{27} + 96 q^{34} + 12 q^{36} + 30 q^{42} + 48 q^{45} - 42 q^{47} + 6 q^{48} + 24 q^{49} - 24 q^{53} - 120 q^{56} + 24 q^{58} - 18 q^{60} - 30 q^{64} + 120 q^{67} + 24 q^{69} + 18 q^{70} - 6 q^{75} - 288 q^{78} - 18 q^{80} - 30 q^{81} - 42 q^{82} + 6 q^{86} - 120 q^{89} - 36 q^{91} - 36 q^{92} + 60 q^{93} - 24 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
81.1 −2.12272 − 1.54225i 0.414045 − 1.27430i 1.50939 + 4.64542i −0.809017 + 0.587785i −2.84419 + 2.06642i −0.810808 − 2.49541i 2.33876 − 7.19797i 0.974647 + 0.708122i 2.62383
81.2 −1.11052 − 0.806841i 1.00795 − 3.10216i −0.0357685 − 0.110084i −0.809017 + 0.587785i −3.62230 + 2.63176i −0.424181 − 1.30550i −0.897462 + 2.76210i −6.18037 − 4.49030i 1.37268
81.3 −0.389057 − 0.282666i −0.494946 + 1.52329i −0.546569 − 1.68217i −0.809017 + 0.587785i 0.623145 − 0.452741i −0.148607 − 0.457364i −0.560059 + 1.72368i 0.351618 + 0.255466i 0.480901
81.4 0.389057 + 0.282666i −0.494946 + 1.52329i −0.546569 − 1.68217i −0.809017 + 0.587785i −0.623145 + 0.452741i 0.148607 + 0.457364i 0.560059 − 1.72368i 0.351618 + 0.255466i −0.480901
81.5 1.11052 + 0.806841i 1.00795 − 3.10216i −0.0357685 − 0.110084i −0.809017 + 0.587785i 3.62230 − 2.63176i 0.424181 + 1.30550i 0.897462 − 2.76210i −6.18037 − 4.49030i −1.37268
81.6 2.12272 + 1.54225i 0.414045 − 1.27430i 1.50939 + 4.64542i −0.809017 + 0.587785i 2.84419 − 2.06642i 0.810808 + 2.49541i −2.33876 + 7.19797i 0.974647 + 0.708122i −2.62383
251.1 −0.810808 + 2.49541i −1.08398 + 0.787560i −3.95163 − 2.87103i 0.309017 + 0.951057i −1.08638 − 3.34354i −2.12272 − 1.54225i 6.12296 − 4.44859i −0.372282 + 1.14577i −2.62383
251.2 −0.424181 + 1.30550i −2.63885 + 1.91724i 0.0936432 + 0.0680358i 0.309017 + 0.951057i −1.38360 − 4.25827i −1.11052 − 0.806841i −2.34959 + 1.70707i 2.36069 − 7.26546i −1.37268
251.3 −0.148607 + 0.457364i 1.29579 − 0.941443i 1.43094 + 1.03964i 0.309017 + 0.951057i 0.238020 + 0.732550i −0.389057 − 0.282666i −1.46625 + 1.06529i −0.134306 + 0.413352i −0.480901
251.4 0.148607 − 0.457364i 1.29579 − 0.941443i 1.43094 + 1.03964i 0.309017 + 0.951057i −0.238020 − 0.732550i 0.389057 + 0.282666i 1.46625 − 1.06529i −0.134306 + 0.413352i 0.480901
251.5 0.424181 − 1.30550i −2.63885 + 1.91724i 0.0936432 + 0.0680358i 0.309017 + 0.951057i 1.38360 + 4.25827i 1.11052 + 0.806841i 2.34959 − 1.70707i 2.36069 − 7.26546i 1.37268
251.6 0.810808 − 2.49541i −1.08398 + 0.787560i −3.95163 − 2.87103i 0.309017 + 0.951057i 1.08638 + 3.34354i 2.12272 + 1.54225i −6.12296 + 4.44859i −0.372282 + 1.14577i 2.62383
366.1 −2.12272 + 1.54225i 0.414045 + 1.27430i 1.50939 − 4.64542i −0.809017 − 0.587785i −2.84419 − 2.06642i −0.810808 + 2.49541i 2.33876 + 7.19797i 0.974647 − 0.708122i 2.62383
366.2 −1.11052 + 0.806841i 1.00795 + 3.10216i −0.0357685 + 0.110084i −0.809017 − 0.587785i −3.62230 − 2.63176i −0.424181 + 1.30550i −0.897462 − 2.76210i −6.18037 + 4.49030i 1.37268
366.3 −0.389057 + 0.282666i −0.494946 − 1.52329i −0.546569 + 1.68217i −0.809017 − 0.587785i 0.623145 + 0.452741i −0.148607 + 0.457364i −0.560059 − 1.72368i 0.351618 − 0.255466i 0.480901
366.4 0.389057 − 0.282666i −0.494946 − 1.52329i −0.546569 + 1.68217i −0.809017 − 0.587785i −0.623145 − 0.452741i 0.148607 − 0.457364i 0.560059 + 1.72368i 0.351618 − 0.255466i −0.480901
366.5 1.11052 − 0.806841i 1.00795 + 3.10216i −0.0357685 + 0.110084i −0.809017 − 0.587785i 3.62230 + 2.63176i 0.424181 − 1.30550i 0.897462 + 2.76210i −6.18037 + 4.49030i −1.37268
366.6 2.12272 − 1.54225i 0.414045 + 1.27430i 1.50939 − 4.64542i −0.809017 − 0.587785i 2.84419 + 2.06642i 0.810808 − 2.49541i −2.33876 − 7.19797i 0.974647 − 0.708122i −2.62383
511.1 −0.810808 − 2.49541i −1.08398 − 0.787560i −3.95163 + 2.87103i 0.309017 − 0.951057i −1.08638 + 3.34354i −2.12272 + 1.54225i 6.12296 + 4.44859i −0.372282 − 1.14577i −2.62383
511.2 −0.424181 − 1.30550i −2.63885 − 1.91724i 0.0936432 − 0.0680358i 0.309017 − 0.951057i −1.38360 + 4.25827i −1.11052 + 0.806841i −2.34959 − 1.70707i 2.36069 + 7.26546i −1.37268
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 81.6
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 605.2.g.q 24
11.b odd 2 1 inner 605.2.g.q 24
11.c even 5 1 605.2.a.m ✓ 6
11.c even 5 3 inner 605.2.g.q 24
11.d odd 10 1 605.2.a.m ✓ 6
11.d odd 10 3 inner 605.2.g.q 24
33.f even 10 1 5445.2.a.bx 6
33.h odd 10 1 5445.2.a.bx 6
44.g even 10 1 9680.2.a.cw 6
44.h odd 10 1 9680.2.a.cw 6
55.h odd 10 1 3025.2.a.bg 6
55.j even 10 1 3025.2.a.bg 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
605.2.a.m ✓ 6 11.c even 5 1
605.2.a.m ✓ 6 11.d odd 10 1
605.2.g.q 24 1.a even 1 1 trivial
605.2.g.q 24 11.b odd 2 1 inner
605.2.g.q 24 11.c even 5 3 inner
605.2.g.q 24 11.d odd 10 3 inner
3025.2.a.bg 6 55.h odd 10 1
3025.2.a.bg 6 55.j even 10 1
5445.2.a.bx 6 33.f even 10 1
5445.2.a.bx 6 33.h odd 10 1
9680.2.a.cw 6 44.g even 10 1
9680.2.a.cw 6 44.h odd 10 1

Hecke kernels

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

\( T_{2}^{24} + 9 T_{2}^{22} + 66 T_{2}^{20} + 462 T_{2}^{18} + 3195 T_{2}^{16} + 6534 T_{2}^{14} + \cdots + 81 \) Copy content Toggle raw display
\( T_{3}^{12} + 3 T_{3}^{11} + 12 T_{3}^{10} + 38 T_{3}^{9} + 129 T_{3}^{8} + 54 T_{3}^{7} + 283 T_{3}^{6} + \cdots + 2401 \) Copy content Toggle raw display