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Properties

Label 603.2.z.b
Level $603$
Weight $2$
Character orbit 603.z
Analytic conductor $4.815$
Analytic rank $0$
Dimension $100$
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(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 201)
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.

\(\operatorname{Tr}(f)(q) = \) \( 100 q - 2 q^{2} + 6 q^{4} + 2 q^{5} + q^{7} + 45 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 100 q - 2 q^{2} + 6 q^{4} + 2 q^{5} + q^{7} + 45 q^{8} - 25 q^{10} - 9 q^{11} - 3 q^{13} - 22 q^{14} + 46 q^{17} - 14 q^{19} + 16 q^{20} - 17 q^{22} - 12 q^{23} - 44 q^{25} + 7 q^{26} - 90 q^{28} - 43 q^{29} - 10 q^{31} - 20 q^{32} + 20 q^{34} - 3 q^{35} + 50 q^{37} - 16 q^{38} + 55 q^{40} + 48 q^{41} - 45 q^{43} + 147 q^{44} + 47 q^{46} - 44 q^{47} - 64 q^{49} + 54 q^{50} - 34 q^{52} - 68 q^{53} - 17 q^{55} - 111 q^{56} + 88 q^{58} + 2 q^{59} + 21 q^{61} - 86 q^{62} + 19 q^{64} - 6 q^{65} + 20 q^{67} + 202 q^{68} - 20 q^{70} - q^{71} + 10 q^{73} - 67 q^{74} - 90 q^{76} + 62 q^{77} - 29 q^{79} - 199 q^{80} - 36 q^{82} - 43 q^{83} + 93 q^{85} + 83 q^{86} + 54 q^{88} - 21 q^{89} + 58 q^{91} + 192 q^{92} + 14 q^{94} + 109 q^{95} - 35 q^{97} + 15 q^{98}+O(q^{100}) \) 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 −1.66018 − 1.58298i 0 0.155214 + 3.25834i 0.764993 − 1.67510i 0 0.871467 − 3.59223i 1.89582 − 2.18789i 0 −3.92168 + 1.57000i
10.2 −0.752307 − 0.717323i 0 −0.0437507 − 0.918439i 1.22616 − 2.68491i 0 −0.560955 + 2.31229i −1.98733 + 2.29351i 0 −2.84839 + 1.14032i
10.3 −0.576780 − 0.549958i 0 −0.0649432 − 1.36333i −1.13405 + 2.48322i 0 0.467919 − 1.92879i −1.75610 + 2.02664i 0 2.01976 − 0.808591i
10.4 0.928460 + 0.885285i 0 −0.0168551 − 0.353832i −0.471271 + 1.03194i 0 0.634488 − 2.61540i 1.97780 − 2.28250i 0 −1.35112 + 0.540906i
10.5 1.45951 + 1.39164i 0 0.0983437 + 2.06449i −0.189954 + 0.415942i 0 −0.918966 + 3.78803i −0.0882556 + 0.101852i 0 −0.856081 + 0.342723i
19.1 −2.24987 − 1.15989i 0 2.55645 + 3.59004i 2.75020 + 3.17390i 0 0.118800 + 2.49393i −0.867175 − 6.03133i 0 −2.50622 − 10.3308i
19.2 −1.36337 − 0.702865i 0 0.204640 + 0.287377i −0.953826 − 1.10077i 0 0.0382883 + 0.803771i 0.359576 + 2.50090i 0 0.526720 + 2.17117i
19.3 −0.309019 − 0.159310i 0 −1.09000 − 1.53069i 0.990324 + 1.14290i 0 −0.0746351 − 1.56678i 0.191932 + 1.33492i 0 −0.123954 − 0.510945i
19.4 1.29041 + 0.665251i 0 0.0624755 + 0.0877346i −1.62071 − 1.87040i 0 0.228443 + 4.79562i −0.390970 − 2.71926i 0 −0.847092 − 3.49176i
19.5 1.46772 + 0.756664i 0 0.421557 + 0.591994i −1.22831 − 1.41754i 0 −0.160533 − 3.37000i −0.299217 − 2.08110i 0 −0.730211 − 3.00997i
55.1 −2.16787 − 0.867885i 0 2.49898 + 2.38277i 1.12708 − 0.724333i 0 −0.395971 + 0.311395i −1.40938 − 3.08612i 0 −3.07201 + 0.592082i
55.2 −1.15085 − 0.460732i 0 −0.335279 − 0.319687i −3.13533 + 2.01495i 0 −3.53898 + 2.78309i 1.26851 + 2.77764i 0 4.53666 − 0.874369i
55.3 −0.921014 − 0.368718i 0 −0.735154 − 0.700968i 0.910690 − 0.585265i 0 3.56992 − 2.80742i 1.24288 + 2.72152i 0 −1.05456 + 0.203249i
55.4 0.260772 + 0.104397i 0 −1.39036 − 1.32571i −0.528881 + 0.339891i 0 1.10467 − 0.868723i −0.457541 − 1.00188i 0 −0.173401 + 0.0334203i
55.5 2.41698 + 0.967614i 0 3.45806 + 3.29725i 0.303894 − 0.195301i 0 −3.44953 + 2.71274i 3.00455 + 6.57906i 0 0.923482 − 0.177986i
73.1 −0.904307 + 2.61282i 0 −4.43697 − 3.48927i 2.19555 + 0.644672i 0 −1.22753 + 0.236587i 8.47727 − 5.44801i 0 −3.66986 + 5.15360i
73.2 −0.624730 + 1.80504i 0 −1.29577 − 1.01901i −1.12644 − 0.330752i 0 3.13675 − 0.604558i −0.564886 + 0.363031i 0 1.30074 − 1.82663i
73.3 −0.0612751 + 0.177043i 0 1.54452 + 1.21462i −2.25262 − 0.661428i 0 −3.01227 + 0.580568i −0.624892 + 0.401594i 0 0.255130 − 0.358280i
73.4 0.446488 − 1.29004i 0 0.107246 + 0.0843391i 2.25224 + 0.661316i 0 −1.93346 + 0.372644i 2.45352 − 1.57678i 0 1.85872 − 2.61021i
73.5 0.516185 − 1.49142i 0 −0.385775 − 0.303377i −2.95304 − 0.867091i 0 3.45668 − 0.666220i 2.00377 − 1.28775i 0 −2.81751 + 3.95664i
See all 100 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 10.5
Significant digits:
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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.b 100
3.b odd 2 1 201.2.m.a ✓ 100
67.g even 33 1 inner 603.2.z.b 100
201.o odd 66 1 201.2.m.a ✓ 100
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
201.2.m.a ✓ 100 3.b odd 2 1
201.2.m.a ✓ 100 201.o odd 66 1
603.2.z.b 100 1.a even 1 1 trivial
603.2.z.b 100 67.g even 33 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{100} + 2 T_{2}^{99} - 6 T_{2}^{98} - 35 T_{2}^{97} - 43 T_{2}^{96} + 212 T_{2}^{95} + \cdots + 17161 \) acting on \(S_{2}^{\mathrm{new}}(603, [\chi])\). Copy content Toggle raw display