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Properties

Label 603.2.v.a
Level $603$
Weight $2$
Character orbit 603.v
Analytic conductor $4.815$
Analytic rank $0$
Dimension $240$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [603,2,Mod(8,603)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(603, base_ring=CyclotomicField(22))
 
chi = DirichletCharacter(H, H._module([11, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("603.8");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 603 = 3^{2} \cdot 67 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 603.v (of order \(22\), degree \(10\), 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: \(4.81497924188\)
Analytic rank: \(0\)
Dimension: \(240\)
Relative dimension: \(24\) over \(\Q(\zeta_{22})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{22}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 240 q - 28 q^{4}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 240 q - 28 q^{4} - 28 q^{16} - 20 q^{19} + 12 q^{22} - 24 q^{25} + 44 q^{28} - 88 q^{31} + 24 q^{37} + 32 q^{40} + 44 q^{43} - 44 q^{46} + 8 q^{49} - 220 q^{52} + 52 q^{55} - 88 q^{58} - 88 q^{61} - 148 q^{64} + 8 q^{67} - 176 q^{70} - 120 q^{73} - 64 q^{76} - 264 q^{79} + 8 q^{82} + 256 q^{88} + 256 q^{91} - 88 q^{94}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
8.1 −0.385645 + 2.68222i 0 −5.12659 − 1.50530i −2.52246 − 1.62109i 0 3.47382 + 0.499460i 3.76321 − 8.24027i 0 5.32089 − 6.14063i
8.2 −0.382519 + 2.66048i 0 −5.01283 − 1.47190i 3.20794 + 2.06162i 0 1.41064 + 0.202819i 3.60033 − 7.88361i 0 −6.71198 + 7.74603i
8.3 −0.347009 + 2.41350i 0 −3.78559 − 1.11155i 1.35822 + 0.872874i 0 −3.92298 − 0.564040i 1.97054 − 4.31487i 0 −2.57800 + 2.97517i
8.4 −0.318452 + 2.21488i 0 −2.88529 − 0.847198i −1.62085 − 1.04165i 0 0.677672 + 0.0974346i 0.936156 − 2.04990i 0 2.82330 − 3.25826i
8.5 −0.292341 + 2.03328i 0 −2.12976 − 0.625354i −0.481112 − 0.309192i 0 −1.79640 − 0.258284i 0.187454 − 0.410468i 0 0.769321 − 0.887844i
8.6 −0.222573 + 1.54803i 0 −0.427866 − 0.125633i 0.138128 + 0.0887696i 0 0.350805 + 0.0504381i −1.00966 + 2.21085i 0 −0.168161 + 0.194069i
8.7 −0.219912 + 1.52952i 0 −0.372079 − 0.109252i 2.92951 + 1.88268i 0 3.47051 + 0.498984i −1.03491 + 2.26614i 0 −3.52383 + 4.06671i
8.8 −0.197030 + 1.37037i 0 0.0798902 + 0.0234579i −2.32062 − 1.49137i 0 1.28004 + 0.184041i −1.19814 + 2.62356i 0 2.50097 − 2.88627i
8.9 −0.121792 + 0.847083i 0 1.21627 + 0.357129i −3.41791 − 2.19655i 0 −1.09217 − 0.157030i −1.16167 + 2.54370i 0 2.27694 − 2.62773i
8.10 −0.110075 + 0.765589i 0 1.34498 + 0.394921i 0.685399 + 0.440479i 0 3.44905 + 0.495899i −1.09301 + 2.39336i 0 −0.412671 + 0.476248i
8.11 −0.0591511 + 0.411405i 0 1.75323 + 0.514795i 1.42949 + 0.918678i 0 −3.87264 − 0.556801i −0.660817 + 1.44699i 0 −0.462505 + 0.533759i
8.12 −0.0114821 + 0.0798595i 0 1.91274 + 0.561631i −2.63879 − 1.69585i 0 −2.31288 − 0.332541i −0.133846 + 0.293081i 0 0.165728 − 0.191261i
8.13 0.0114821 − 0.0798595i 0 1.91274 + 0.561631i 2.63879 + 1.69585i 0 −2.31288 − 0.332541i 0.133846 − 0.293081i 0 0.165728 − 0.191261i
8.14 0.0591511 − 0.411405i 0 1.75323 + 0.514795i −1.42949 − 0.918678i 0 −3.87264 − 0.556801i 0.660817 − 1.44699i 0 −0.462505 + 0.533759i
8.15 0.110075 − 0.765589i 0 1.34498 + 0.394921i −0.685399 − 0.440479i 0 3.44905 + 0.495899i 1.09301 − 2.39336i 0 −0.412671 + 0.476248i
8.16 0.121792 − 0.847083i 0 1.21627 + 0.357129i 3.41791 + 2.19655i 0 −1.09217 − 0.157030i 1.16167 − 2.54370i 0 2.27694 − 2.62773i
8.17 0.197030 − 1.37037i 0 0.0798902 + 0.0234579i 2.32062 + 1.49137i 0 1.28004 + 0.184041i 1.19814 − 2.62356i 0 2.50097 − 2.88627i
8.18 0.219912 − 1.52952i 0 −0.372079 − 0.109252i −2.92951 − 1.88268i 0 3.47051 + 0.498984i 1.03491 − 2.26614i 0 −3.52383 + 4.06671i
8.19 0.222573 − 1.54803i 0 −0.427866 − 0.125633i −0.138128 − 0.0887696i 0 0.350805 + 0.0504381i 1.00966 − 2.21085i 0 −0.168161 + 0.194069i
8.20 0.292341 − 2.03328i 0 −2.12976 − 0.625354i 0.481112 + 0.309192i 0 −1.79640 − 0.258284i −0.187454 + 0.410468i 0 0.769321 − 0.887844i
See next 80 embeddings (of 240 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 8.24
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
67.f odd 22 1 inner
201.j even 22 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 603.2.v.a ✓ 240
3.b odd 2 1 inner 603.2.v.a ✓ 240
67.f odd 22 1 inner 603.2.v.a ✓ 240
201.j even 22 1 inner 603.2.v.a ✓ 240
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
603.2.v.a ✓ 240 1.a even 1 1 trivial
603.2.v.a ✓ 240 3.b odd 2 1 inner
603.2.v.a ✓ 240 67.f odd 22 1 inner
603.2.v.a ✓ 240 201.j even 22 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(603, [\chi])\).