↑

Properties

Label 603.2.z.c
Level $603$
Weight $2$
Character orbit 603.z
Analytic conductor $4.815$
Analytic rank $0$
Dimension $100$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [603,2,Mod(10,603)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(603, base_ring=CyclotomicField(66))
 
chi = DirichletCharacter(H, H._module([0, 16]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("603.10");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 603 = 3^{2} \cdot 67 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 603.z (of order \(33\), degree \(20\), 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: \(100\)
Relative dimension: \(5\) over \(\Q(\zeta_{33})\)
Twist minimal: no (minimal twist has level 67)
Sato-Tate group: $\mathrm{SU}(2)[C_{33}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 100 q + 24 q^{2} - 18 q^{4} + 16 q^{5} - 24 q^{7} - 23 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 100 q + 24 q^{2} - 18 q^{4} + 16 q^{5} - 24 q^{7} - 23 q^{8} + 8 q^{10} + 24 q^{11} - 22 q^{13} + 32 q^{14} - 28 q^{16} - 17 q^{17} + 15 q^{20} + 49 q^{22} + 13 q^{23} - 34 q^{25} + 27 q^{26} + 22 q^{28} - 8 q^{29} + 10 q^{31} - 34 q^{32} - 50 q^{34} + q^{35} + 7 q^{37} - 50 q^{38} + 43 q^{40} + 5 q^{41} + 2 q^{43} + 19 q^{44} + 52 q^{46} + 6 q^{47} - 27 q^{49} - 134 q^{50} + 120 q^{52} + 52 q^{53} - 64 q^{55} + 124 q^{56} - 56 q^{58} - 27 q^{59} - 16 q^{61} + 74 q^{62} - 197 q^{64} + 92 q^{65} - 56 q^{67} - 16 q^{68} - 22 q^{70} + 113 q^{71} + q^{73} + 24 q^{74} - 144 q^{76} - 85 q^{77} + 36 q^{79} + 13 q^{80} - 20 q^{82} + 61 q^{83} - 6 q^{85} - 189 q^{86} + 129 q^{88} - 95 q^{89} + 42 q^{91} - 4 q^{92} + 70 q^{94} + 20 q^{95} + 53 q^{97} - q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
10.1 −1.38159 − 1.31734i 0 0.0782330 + 1.64231i 1.09559 − 2.39900i 0 −0.489299 + 2.01692i −0.444821 + 0.513350i 0 −4.67395 + 1.87117i
10.2 −0.212667 − 0.202777i 0 −0.0910553 − 1.91149i −0.243782 + 0.533809i 0 −0.729591 + 3.00742i −0.753099 + 0.869123i 0 0.160089 − 0.0640899i
10.3 0.397903 + 0.379400i 0 −0.0807812 − 1.69581i 0.210771 − 0.461524i 0 0.229068 − 0.944230i 1.33132 − 1.53642i 0 0.258968 − 0.103675i
10.4 1.70724 + 1.62785i 0 0.169609 + 3.56053i 1.68411 − 3.68769i 0 0.549774 − 2.26620i −2.41691 + 2.78926i 0 8.87819 − 3.55429i
10.5 1.86173 + 1.77515i 0 0.219695 + 4.61197i −0.754900 + 1.65300i 0 −0.561736 + 2.31551i −4.40882 + 5.08805i 0 −4.33974 + 1.73737i
19.1 −1.83066 − 0.943770i 0 1.30049 + 1.82629i 0.863926 + 0.997024i 0 −0.0491040 − 1.03082i −0.0709360 − 0.493371i 0 −0.640592 − 2.64056i
19.2 −0.322186 − 0.166098i 0 −1.08390 − 1.52212i −2.74026 − 3.16243i 0 −0.139020 − 2.91838i 0.199567 + 1.38802i 0 0.357599 + 1.47404i
19.3 0.255763 + 0.131855i 0 −1.11208 − 1.56170i 1.03286 + 1.19198i 0 0.205223 + 4.30815i −0.160414 − 1.11571i 0 0.106999 + 0.441054i
19.4 1.32946 + 0.685383i 0 0.137594 + 0.193224i 1.91440 + 2.20934i 0 −0.175175 − 3.67738i −0.375236 − 2.60982i 0 1.03087 + 4.24932i
19.5 2.42297 + 1.24913i 0 3.15035 + 4.42405i −0.493452 − 0.569474i 0 −0.0363486 − 0.763051i 1.33110 + 9.25801i 0 −0.484274 − 1.99620i
55.1 −1.18476 − 0.474308i 0 −0.268771 − 0.256273i 1.28064 − 0.823014i 0 −1.83630 + 1.44408i 1.25717 + 2.75281i 0 −1.90761 + 0.367662i
55.2 0.597030 + 0.239015i 0 −1.14815 − 1.09476i −2.47647 + 1.59153i 0 0.640337 − 0.503567i −0.958120 − 2.09799i 0 −1.85892 + 0.358278i
55.3 0.753723 + 0.301745i 0 −0.970420 − 0.925293i 0.943515 − 0.606360i 0 −2.72577 + 2.14357i −1.12676 − 2.46726i 0 0.894116 − 0.172327i
55.4 1.55359 + 0.621965i 0 0.579342 + 0.552401i 2.30812 − 1.48334i 0 0.867626 − 0.682309i −0.833879 − 1.82594i 0 4.50845 − 0.868933i
55.5 2.49451 + 0.998649i 0 3.77779 + 3.60211i −0.632765 + 0.406653i 0 3.01981 − 2.37481i 3.59404 + 7.86986i 0 −1.98454 + 0.382489i
73.1 −0.672578 + 1.94329i 0 −1.75190 − 1.37771i −0.783826 − 0.230152i 0 −4.02301 + 0.775373i 0.395684 − 0.254291i 0 0.974436 − 1.36840i
73.2 −0.260812 + 0.753567i 0 1.07227 + 0.843239i 1.62666 + 0.477630i 0 1.76797 − 0.340749i −2.25677 + 1.45034i 0 −0.784178 + 1.10122i
73.3 −0.0129395 + 0.0373862i 0 1.57088 + 1.23535i 2.74654 + 0.806457i 0 0.431636 − 0.0831910i −0.133075 + 0.0855220i 0 −0.0656892 + 0.0922475i
73.4 0.679423 − 1.96306i 0 −1.81990 − 1.43118i −2.54249 − 0.746543i 0 −3.22458 + 0.621487i −0.550889 + 0.354035i 0 −3.19294 + 4.48386i
73.5 0.781425 − 2.25778i 0 −2.91483 − 2.29225i 0.581051 + 0.170612i 0 2.97894 − 0.574144i −3.43329 + 2.20644i 0 0.839252 − 1.17856i
See all 100 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 10.5
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
67.g even 33 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 603.2.z.c 100
3.b odd 2 1 67.2.g.a ✓ 100
67.g even 33 1 inner 603.2.z.c 100
201.o odd 66 1 67.2.g.a ✓ 100
201.o odd 66 1 4489.2.a.p 50
201.p even 66 1 4489.2.a.q 50
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
67.2.g.a ✓ 100 3.b odd 2 1
67.2.g.a ✓ 100 201.o odd 66 1
603.2.z.c 100 1.a even 1 1 trivial
603.2.z.c 100 67.g even 33 1 inner
4489.2.a.p 50 201.o odd 66 1
4489.2.a.q 50 201.p even 66 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{100} - 24 T_{2}^{99} + 292 T_{2}^{98} - 2387 T_{2}^{97} + 14663 T_{2}^{96} - 71762 T_{2}^{95} + \cdots + 60466176 \) acting on \(S_{2}^{\mathrm{new}}(603, [\chi])\). Copy content Toggle raw display