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Properties

Label 603.2.e.a
Level $603$
Weight $2$
Character orbit 603.e
Analytic conductor $4.815$
Analytic rank $0$
Dimension $66$
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(202,603)]
 
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, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("603.202");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 603 = 3^{2} \cdot 67 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 603.e (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: \(66\)
Relative dimension: \(33\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 66 q - 7 q^{2} - 33 q^{4} - 18 q^{5} - 3 q^{6} + 36 q^{8} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 66 q - 7 q^{2} - 33 q^{4} - 18 q^{5} - 3 q^{6} + 36 q^{8} + 4 q^{9} - 8 q^{11} + q^{12} - 7 q^{14} + 3 q^{15} - 33 q^{16} + 66 q^{17} - 11 q^{18} - 29 q^{20} + q^{21} - 17 q^{23} + 47 q^{24} - 33 q^{25} + 60 q^{26} - 21 q^{27} - 54 q^{28} - 39 q^{29} - 34 q^{30} - 53 q^{32} + 8 q^{33} - 6 q^{34} + 62 q^{35} - 35 q^{36} + 24 q^{37} - 30 q^{38} - 5 q^{39} - 6 q^{40} - 38 q^{41} + 65 q^{42} + 22 q^{44} - 9 q^{45} + 12 q^{46} - 58 q^{47} - 59 q^{48} - 33 q^{49} - 31 q^{50} + 26 q^{51} + 9 q^{52} + 128 q^{53} - 22 q^{54} - 36 q^{55} - 32 q^{56} - 34 q^{57} + 3 q^{58} - 39 q^{59} + 127 q^{60} + 138 q^{62} - 35 q^{63} + 132 q^{64} - 28 q^{65} - 94 q^{66} - 33 q^{67} - 62 q^{68} + 60 q^{69} - 6 q^{70} + 42 q^{71} - 34 q^{72} - 25 q^{74} + 55 q^{75} - 6 q^{76} - 91 q^{77} + 125 q^{78} + 116 q^{80} - 90 q^{82} - 61 q^{83} - 26 q^{84} + 15 q^{85} - 47 q^{86} - q^{87} - 12 q^{88} + 110 q^{89} - 91 q^{90} + 36 q^{91} - 41 q^{92} - 11 q^{93} - 21 q^{94} - 6 q^{95} + 80 q^{96} - 12 q^{97} + 80 q^{98} - 38 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
202.1 −1.38941 − 2.40652i 1.42799 − 0.980231i −2.86090 + 4.95522i 1.47194 − 2.54948i −4.34300 − 2.07454i 2.08373 + 3.60913i 10.3422 1.07829 − 2.79951i −8.18051
202.2 −1.37579 − 2.38294i 1.45672 + 0.936993i −2.78559 + 4.82479i −1.32441 + 2.29395i 0.228651 − 4.76039i −0.0810815 − 0.140437i 9.82639 1.24409 + 2.72988i 7.28845
202.3 −1.32675 − 2.29800i −1.72892 − 0.104057i −2.52055 + 4.36572i −1.11098 + 1.92427i 2.05473 + 4.11113i −0.853017 − 1.47747i 8.06957 2.97834 + 0.359812i 5.89596
202.4 −1.25261 − 2.16958i 0.534330 − 1.64757i −2.13804 + 3.70320i −1.48321 + 2.56899i −4.24384 + 0.904488i −1.15091 − 1.99343i 5.70208 −2.42898 − 1.76069i 7.43149
202.5 −1.14190 − 1.97784i −0.732971 + 1.56932i −1.60789 + 2.78495i −0.709671 + 1.22919i 3.94083 − 0.342311i −1.12799 − 1.95374i 2.77661 −1.92551 − 2.30053i 3.24151
202.6 −1.10414 − 1.91242i 0.561969 + 1.63835i −1.43824 + 2.49111i 0.452416 − 0.783607i 2.51273 − 2.88369i 1.84333 + 3.19274i 1.93552 −2.36838 + 1.84140i −1.99812
202.7 −1.06157 − 1.83869i 1.72426 − 0.164099i −1.25385 + 2.17173i 1.30831 − 2.26607i −2.13214 − 2.99617i −1.66212 − 2.87888i 1.07791 2.94614 − 0.565900i −5.55545
202.8 −0.884275 − 1.53161i −1.71455 + 0.245593i −0.563885 + 0.976678i −1.96647 + 3.40603i 1.89229 + 2.40885i 1.94697 + 3.37225i −1.54258 2.87937 − 0.842165i 6.95560
202.9 −0.776746 − 1.34536i −1.65560 − 0.508897i −0.206669 + 0.357961i 1.50277 − 2.60288i 0.601332 + 2.62267i −1.95952 − 3.39400i −2.46487 2.48205 + 1.68506i −4.66909
202.10 −0.764097 − 1.32345i −0.725793 − 1.57265i −0.167688 + 0.290444i −1.58216 + 2.74038i −1.52675 + 2.16221i −1.11669 − 1.93417i −2.54387 −1.94645 + 2.28284i 4.83569
202.11 −0.725966 − 1.25741i −1.21321 + 1.23617i −0.0540534 + 0.0936233i 1.12396 − 1.94676i 2.43512 + 0.628086i 1.46811 + 2.54284i −2.74690 −0.0562367 − 2.99947i −3.26383
202.12 −0.643677 − 1.11488i −0.988260 − 1.42244i 0.171359 − 0.296802i −0.0672714 + 0.116518i −0.949734 + 2.01739i 0.0335898 + 0.0581793i −3.01591 −1.04668 + 2.81149i 0.173204
202.13 −0.610365 − 1.05718i 0.644904 + 1.60751i 0.254910 − 0.441517i 0.706305 − 1.22336i 1.30581 − 1.66295i 1.05244 + 1.82287i −3.06381 −2.16820 + 2.07338i −1.72442
202.14 −0.483492 − 0.837433i 1.60455 − 0.652235i 0.532470 − 0.922266i −1.74409 + 3.02086i −1.32199 − 1.02835i −2.05816 − 3.56483i −2.96375 2.14918 − 2.09309i 3.37302
202.15 −0.425254 − 0.736562i 0.280293 − 1.70922i 0.638318 − 1.10560i −1.38444 + 2.39793i −1.37814 + 0.520400i 2.59949 + 4.50245i −2.78681 −2.84287 − 0.958165i 2.35496
202.16 −0.252979 − 0.438172i 1.33275 − 1.10624i 0.872004 − 1.51035i 0.791568 − 1.37104i −0.821881 − 0.304120i −0.347938 − 0.602647i −1.89431 0.552467 − 2.94869i −0.800999
202.17 −0.179014 − 0.310061i −1.68312 + 0.408777i 0.935908 − 1.62104i 0.529168 − 0.916546i 0.428049 + 0.448695i −0.795249 − 1.37741i −1.38622 2.66580 − 1.37604i −0.378914
202.18 0.0178129 + 0.0308529i 1.38550 + 1.03942i 0.999365 − 1.73095i −1.92812 + 3.33960i −0.00738942 + 0.0612618i −0.0309078 − 0.0535339i 0.142458 0.839205 + 2.88023i −0.137382
202.19 0.168479 + 0.291814i 0.969304 − 1.43543i 0.943230 − 1.63372i −0.371302 + 0.643113i 0.582186 + 0.0410176i 0.164135 + 0.284290i 1.30957 −1.12090 − 2.78273i −0.250226
202.20 0.172392 + 0.298592i −0.404391 + 1.68418i 0.940562 − 1.62910i 1.75478 − 3.03936i −0.572596 + 0.169592i 0.197532 + 0.342135i 1.33815 −2.67294 − 1.36214i 1.21004
See all 66 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 202.33
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.c 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.e.a ✓ 66
9.c even 3 1 inner 603.2.e.a ✓ 66
9.c even 3 1 5427.2.a.p 33
9.d odd 6 1 5427.2.a.o 33
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
603.2.e.a ✓ 66 1.a even 1 1 trivial
603.2.e.a ✓ 66 9.c even 3 1 inner
5427.2.a.o 33 9.d odd 6 1
5427.2.a.p 33 9.c even 3 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{66} + 7 T_{2}^{65} + 74 T_{2}^{64} + 373 T_{2}^{63} + 2447 T_{2}^{62} + 10210 T_{2}^{61} + \cdots + 110889 \) acting on \(S_{2}^{\mathrm{new}}(603, [\chi])\). Copy content Toggle raw display