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Properties

Label 603.2.f.c
Level $603$
Weight $2$
Character orbit 603.f
Analytic conductor $4.815$
Analytic rank $0$
Dimension $128$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [603,2,Mod(238,603)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(603, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([2, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("603.238");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 603 = 3^{2} \cdot 67 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 603.f (of order \(3\), 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: \(4.81497924188\)
Analytic rank: \(0\)
Dimension: \(128\)
Relative dimension: \(64\) over \(\Q(\zeta_{3})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 128 q + q^{2} - 3 q^{3} - 65 q^{4} + 5 q^{5} + 3 q^{6} - 8 q^{7} - 36 q^{8} + 5 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 128 q + q^{2} - 3 q^{3} - 65 q^{4} + 5 q^{5} + 3 q^{6} - 8 q^{7} - 36 q^{8} + 5 q^{9} - 2 q^{10} - 30 q^{11} - 10 q^{12} + 12 q^{13} - 8 q^{14} + 12 q^{15} - 59 q^{16} - q^{17} + 8 q^{18} + 8 q^{19} - 6 q^{20} + 12 q^{21} - 20 q^{22} - 18 q^{23} - 39 q^{24} - 57 q^{25} + 8 q^{26} - 24 q^{27} - 16 q^{28} - 2 q^{29} + 23 q^{30} + 16 q^{31} + 27 q^{32} - 3 q^{33} - 4 q^{34} + 11 q^{35} + 45 q^{36} + 2 q^{37} - 4 q^{38} + 8 q^{39} - 6 q^{40} - 7 q^{41} + 18 q^{42} - 11 q^{43} - 18 q^{44} + 3 q^{45} - 4 q^{46} - 24 q^{47} + 71 q^{48} + 104 q^{49} + 44 q^{50} + 42 q^{51} - 6 q^{52} - 60 q^{53} + 48 q^{54} + 10 q^{55} - 28 q^{56} + 29 q^{57} + 6 q^{59} - 7 q^{60} - 6 q^{61} - 22 q^{62} + 20 q^{63} + 56 q^{64} - 26 q^{65} + 54 q^{66} - 19 q^{68} + 8 q^{69} + 25 q^{70} + 8 q^{71} + 11 q^{72} + 22 q^{73} - 42 q^{74} - 44 q^{75} - 5 q^{76} - 70 q^{77} + 15 q^{78} - 30 q^{79} + 14 q^{80} - 63 q^{81} - 24 q^{82} - 3 q^{83} - 77 q^{84} + 24 q^{85} + 20 q^{87} - 6 q^{88} + 116 q^{89} - q^{90} - 10 q^{91} + 4 q^{92} + 26 q^{93} + 4 q^{94} - 180 q^{95} - 56 q^{96} + 19 q^{97} - 49 q^{98} + 4 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
238.1 −1.31801 + 2.28286i −0.497255 − 1.65914i −2.47429 − 4.28560i −0.680757 + 1.17911i 4.44296 + 1.05160i −3.81969 7.77252 −2.50548 + 1.65003i −1.79449 − 3.10814i
238.2 −1.29651 + 2.24561i −1.64424 − 0.544495i −2.36185 − 4.09085i 0.741692 − 1.28465i 3.35449 − 2.98639i 1.85123 7.06261 2.40705 + 1.79056i 1.92322 + 3.33111i
238.3 −1.29115 + 2.23633i 1.32753 − 1.11250i −2.33412 − 4.04282i −1.51905 + 2.63108i 0.773878 + 4.40521i 2.92470 6.89019 0.524689 − 2.95376i −3.92264 − 6.79421i
238.4 −1.27043 + 2.20045i 0.340009 − 1.69835i −2.22799 − 3.85900i 1.90103 − 3.29267i 3.30518 + 2.90581i 3.79121 6.24033 −2.76879 − 1.15491i 4.83025 + 8.36623i
238.5 −1.26375 + 2.18888i −1.33559 + 1.10282i −2.19414 − 3.80037i 0.738237 − 1.27866i −0.726081 − 4.31714i −3.84492 6.03640 0.567595 − 2.94582i 1.86590 + 3.23183i
238.6 −1.22244 + 2.11733i 0.528135 + 1.64957i −1.98873 − 3.44457i −1.52969 + 2.64950i −4.13829 − 0.898264i −2.22226 4.83464 −2.44215 + 1.74239i −3.73991 − 6.47772i
238.7 −1.18873 + 2.05893i −1.68608 − 0.396396i −1.82614 − 3.16297i −1.03610 + 1.79458i 2.82044 − 3.00033i 1.60807 3.92823 2.68574 + 1.33671i −2.46328 − 4.26653i
238.8 −1.17727 + 2.03909i 1.28378 + 1.16272i −1.77193 − 3.06907i 0.679676 − 1.17723i −3.88224 + 1.24890i 1.20666 3.63508 0.296168 + 2.98534i 1.60032 + 2.77184i
238.9 −1.10525 + 1.91435i 0.120716 + 1.72784i −1.44316 − 2.49963i 1.91520 − 3.31722i −3.44111 − 1.67860i −0.356098 1.95922 −2.97086 + 0.417157i 4.23355 + 7.33272i
238.10 −1.06030 + 1.83649i −1.26550 + 1.18259i −1.24847 − 2.16241i −1.33407 + 2.31068i −0.830016 − 3.57797i 2.27103 1.05381 0.202956 − 2.99313i −2.82903 − 4.90003i
238.11 −1.03546 + 1.79347i 1.65146 − 0.522197i −1.14435 − 1.98207i 0.450767 − 0.780752i −0.773473 + 3.50255i −0.961690 0.597864 2.45462 − 1.72477i 0.933502 + 1.61687i
238.12 −0.943426 + 1.63406i 0.958630 − 1.44258i −0.780103 − 1.35118i 0.159032 − 0.275451i 1.45286 + 2.92742i −1.45939 −0.829824 −1.16206 − 2.76579i 0.300070 + 0.519736i
238.13 −0.870770 + 1.50822i 1.58956 + 0.687973i −0.516482 − 0.894573i −0.464673 + 0.804837i −2.42175 + 1.79833i 4.53834 −1.68413 2.05339 + 2.18714i −0.809246 − 1.40166i
238.14 −0.846225 + 1.46570i −1.22862 − 1.22086i −0.432193 − 0.748581i 2.02467 − 3.50684i 2.82911 − 0.767667i −3.48396 −1.92197 0.0189988 + 2.99994i 3.42666 + 5.93514i
238.15 −0.772546 + 1.33809i −1.60578 − 0.649205i −0.193654 − 0.335419i −1.83310 + 3.17502i 2.10923 − 1.64714i −3.81842 −2.49176 2.15706 + 2.08496i −2.83230 − 4.90570i
238.16 −0.771240 + 1.33583i −0.331528 + 1.70003i −0.189624 − 0.328438i 0.689739 − 1.19466i −2.01525 − 1.75399i 4.66126 −2.49998 −2.78018 − 1.12721i 1.06391 + 1.84275i
238.17 −0.764776 + 1.32463i −1.64208 + 0.550971i −0.169766 − 0.294044i 1.10491 − 1.91376i 0.525992 − 2.59652i 0.0678727 −2.53977 2.39286 − 1.80948i 1.69002 + 2.92719i
238.18 −0.762492 + 1.32067i −0.244454 − 1.71471i −0.162787 − 0.281956i −1.67796 + 2.90631i 2.45097 + 0.984611i 1.28044 −2.55347 −2.88048 + 0.838336i −2.55886 − 4.43207i
238.19 −0.689435 + 1.19414i −1.28239 − 1.16424i 0.0493577 + 0.0854900i 0.438766 − 0.759964i 2.27440 − 0.728685i 3.67012 −2.89386 0.289073 + 2.98604i 0.605001 + 1.04789i
238.20 −0.666051 + 1.15363i 1.50889 + 0.850450i 0.112753 + 0.195295i 0.352663 − 0.610830i −1.98610 + 1.17426i −5.09129 −2.96460 1.55347 + 2.56646i 0.469783 + 0.813688i
See next 80 embeddings (of 128 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 238.64
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
603.f even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 603.2.f.c ✓ 128
9.c even 3 1 603.2.h.c yes 128
67.c even 3 1 603.2.h.c yes 128
603.f even 3 1 inner 603.2.f.c ✓ 128
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
603.2.f.c ✓ 128 1.a even 1 1 trivial
603.2.f.c ✓ 128 603.f even 3 1 inner
603.2.h.c yes 128 9.c even 3 1
603.2.h.c yes 128 67.c even 3 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(603, [\chi])\):

\( T_{2}^{128} - T_{2}^{127} + 97 T_{2}^{126} - 82 T_{2}^{125} + 4930 T_{2}^{124} - 3528 T_{2}^{123} + \cdots + 7144929 \) Copy content Toggle raw display
\( T_{5}^{128} - 5 T_{5}^{127} + 201 T_{5}^{126} - 884 T_{5}^{125} + 20749 T_{5}^{124} + \cdots + 69\!\cdots\!49 \) Copy content Toggle raw display