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Properties

Label 603.2.e.b
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.

\(\operatorname{Tr}(f)(q) = \) \( 66 q + 7 q^{2} - 33 q^{4} + 18 q^{5} - 3 q^{6} - 48 q^{8} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 66 q + 7 q^{2} - 33 q^{4} + 18 q^{5} - 3 q^{6} - 48 q^{8} + 4 q^{9} + 12 q^{11} + q^{12} + 9 q^{14} + 3 q^{15} - 33 q^{16} - 62 q^{17} + 7 q^{18} + 43 q^{20} + 17 q^{21} + 19 q^{23} - 17 q^{24} - 33 q^{25} - 28 q^{26} - 3 q^{27} + 54 q^{28} + 25 q^{29} + 24 q^{30} + 45 q^{32} - 32 q^{33} - 6 q^{34} - 50 q^{35} + 53 q^{36} - 24 q^{37} + 34 q^{38} + 19 q^{39} - 6 q^{40} + 34 q^{41} - 107 q^{42} - 98 q^{44} + 9 q^{45} + 12 q^{46} + 26 q^{47} + 49 q^{48} - 33 q^{49} + 39 q^{50} - 50 q^{51} + 9 q^{52} - 104 q^{53} + 70 q^{54} + 60 q^{55} + 16 q^{56} + 6 q^{57} + 3 q^{58} + 21 q^{59} - 161 q^{60} - 54 q^{62} + q^{63} - 12 q^{64} + 52 q^{65} + 52 q^{66} + 33 q^{67} + 98 q^{68} + 2 q^{69} - 6 q^{70} - 62 q^{71} + 66 q^{72} + 27 q^{74} + 21 q^{75} - 6 q^{76} + 85 q^{77} - 107 q^{78} - 172 q^{80} + 72 q^{81} + 102 q^{82} + 71 q^{83} - 54 q^{84} - 27 q^{85} + 9 q^{86} + 3 q^{87} - 12 q^{88} - 82 q^{89} + 153 q^{90} - 60 q^{91} + 67 q^{92} - 47 q^{93} + 15 q^{94} + 58 q^{95} - 136 q^{96} - 12 q^{97} - 172 q^{98} + 96 q^{99}+O(q^{100}) \) 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 −1.29163 − 2.23717i 0.248207 + 1.71417i −2.33661 + 4.04712i 1.76796 − 3.06220i 3.51430 − 2.76935i −1.68626 − 2.92069i 6.90559 −2.87679 + 0.850939i −9.13420
202.2 −1.16403 − 2.01616i −1.21286 − 1.23651i −1.70993 + 2.96169i 0.181167 − 0.313790i −1.08120 + 3.88466i 1.22147 + 2.11564i 3.30553 −0.0579319 + 2.99944i −0.843535
202.3 −1.16110 − 2.01108i −1.69947 + 0.334372i −1.69630 + 2.93808i 1.57271 − 2.72401i 2.64570 + 3.02953i 0.347994 + 0.602743i 3.23390 2.77639 − 1.13651i −7.30427
202.4 −1.10303 − 1.91051i 0.736253 − 1.56778i −1.43335 + 2.48264i −0.220187 + 0.381374i −3.80736 + 0.322692i 0.740949 + 1.28336i 1.91201 −1.91586 − 2.30857i 0.971490
202.5 −1.07034 − 1.85388i 1.63572 + 0.569583i −1.29125 + 2.23651i −0.195185 + 0.338070i −0.694834 − 3.64208i −0.272738 − 0.472396i 1.24696 2.35115 + 1.86336i 0.835656
202.6 −0.898514 − 1.55627i 0.572028 − 1.63487i −0.614656 + 1.06462i 0.969496 − 1.67922i −3.05827 + 0.578718i −1.26856 − 2.19721i −1.38495 −2.34557 − 1.87038i −3.48442
202.7 −0.894905 − 1.55002i 1.68782 − 0.388943i −0.601709 + 1.04219i −1.22796 + 2.12689i −2.11330 − 2.26808i 1.56181 + 2.70513i −1.42573 2.69745 − 1.31293i 4.39564
202.8 −0.764755 − 1.32459i −1.25699 + 1.19163i −0.169700 + 0.293928i −0.683764 + 1.18431i 2.53971 + 0.753703i −1.32780 − 2.29981i −2.53990 0.160057 − 2.99573i 2.09165
202.9 −0.541003 − 0.937044i −0.147406 − 1.72577i 0.414632 − 0.718164i 2.12152 − 3.67458i −1.53737 + 1.07177i 2.08280 + 3.60751i −3.06128 −2.95654 + 0.508777i −4.59099
202.10 −0.531460 − 0.920515i 0.659968 + 1.60139i 0.435101 − 0.753618i 0.0404601 − 0.0700789i 1.12336 − 1.45858i −1.41390 − 2.44894i −3.05079 −2.12889 + 2.11373i −0.0860116
202.11 −0.439647 − 0.761491i 1.66257 + 0.485648i 0.613421 − 1.06248i −0.177271 + 0.307042i −0.361128 − 1.47955i 1.19424 + 2.06848i −2.83734 2.52829 + 1.61485i 0.311747
202.12 −0.343920 − 0.595686i 1.47268 + 0.911703i 0.763439 − 1.32231i 2.04306 − 3.53869i 0.0366045 − 1.19081i −0.487864 − 0.845004i −2.42592 1.33759 + 2.68530i −2.81060
202.13 −0.281257 − 0.487152i −0.338651 + 1.69862i 0.841789 − 1.45802i −1.22840 + 2.12765i 0.922735 − 0.312775i 1.06806 + 1.84994i −2.07207 −2.77063 − 1.15048i 1.38199
202.14 −0.275132 − 0.476542i −1.73126 − 0.0523493i 0.848605 − 1.46983i −1.40332 + 2.43062i 0.451378 + 0.839421i −1.08929 − 1.88671i −2.03444 2.99452 + 0.181260i 1.54439
202.15 −0.0950983 − 0.164715i −0.457819 − 1.67045i 0.981913 − 1.70072i 0.636538 − 1.10252i −0.231610 + 0.234267i −1.46668 − 2.54037i −0.753906 −2.58080 + 1.52953i −0.242135
202.16 0.139511 + 0.241640i −1.44218 + 0.959225i 0.961073 − 1.66463i −0.380308 + 0.658713i −0.432988 − 0.214667i 1.58538 + 2.74596i 1.09437 1.15977 − 2.76675i −0.212229
202.17 0.150565 + 0.260786i −1.68964 − 0.380931i 0.954661 − 1.65352i 1.88719 − 3.26872i −0.155059 − 0.497989i 1.37331 + 2.37864i 1.17721 2.70978 + 1.28727i 1.13658
202.18 0.202854 + 0.351353i 1.72071 − 0.197838i 0.917701 − 1.58950i 0.739132 − 1.28021i 0.418565 + 0.564447i 2.29620 + 3.97713i 1.55605 2.92172 − 0.680846i 0.599744
202.19 0.227213 + 0.393545i −1.40326 − 1.01531i 0.896748 − 1.55321i −1.41867 + 2.45721i 0.0807325 − 0.782938i −0.190829 − 0.330525i 1.72386 0.938274 + 2.84950i −1.28936
202.20 0.341820 + 0.592049i 1.72544 + 0.151135i 0.766318 − 1.32730i 0.565440 − 0.979371i 0.500312 + 1.07321i −2.29004 − 3.96647i 2.41505 2.95432 + 0.521550i 0.773115
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.b ✓ 66
9.c even 3 1 inner 603.2.e.b ✓ 66
9.c even 3 1 5427.2.a.n 33
9.d odd 6 1 5427.2.a.q 33
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
603.2.e.b ✓ 66 1.a even 1 1 trivial
603.2.e.b ✓ 66 9.c even 3 1 inner
5427.2.a.n 33 9.c even 3 1
5427.2.a.q 33 9.d odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{66} - 7 T_{2}^{65} + 74 T_{2}^{64} - 369 T_{2}^{63} + 2419 T_{2}^{62} - 9922 T_{2}^{61} + \cdots + 874225 \) acting on \(S_{2}^{\mathrm{new}}(603, [\chi])\). Copy content Toggle raw display