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Properties

Label 603.2.u.d
Level $603$
Weight $2$
Character orbit 603.u
Analytic conductor $4.815$
Analytic rank $0$
Dimension $50$
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(64,603)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(603, base_ring=CyclotomicField(22))
 
chi = DirichletCharacter(H, H._module([0, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("603.64");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 603 = 3^{2} \cdot 67 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 603.u (of order \(11\), degree \(10\), 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: \(50\)
Relative dimension: \(5\) over \(\Q(\zeta_{11})\)
Twist minimal: no (minimal twist has level 201)
Sato-Tate group: $\mathrm{SU}(2)[C_{11}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 50 q + 2 q^{2} - 6 q^{4} - 2 q^{5} + 2 q^{7} - 27 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 50 q + 2 q^{2} - 6 q^{4} - 2 q^{5} + 2 q^{7} - 27 q^{8} + 37 q^{10} + 15 q^{11} + 4 q^{14} + 17 q^{17} + 20 q^{19} - 4 q^{20} - q^{22} + 6 q^{23} - 13 q^{25} - 22 q^{26} - 39 q^{28} + 52 q^{29} + 16 q^{31} + 35 q^{32} - 14 q^{34} + 36 q^{35} - 68 q^{37} - 20 q^{38} - 25 q^{40} - 30 q^{41} + 33 q^{43} + 63 q^{44} - 65 q^{46} + 38 q^{47} - 29 q^{49} - 21 q^{50} - 38 q^{52} + 29 q^{53} - q^{55} - 90 q^{56} - 52 q^{58} - 35 q^{59} + 30 q^{61} + 32 q^{62} + 23 q^{64} - 6 q^{65} + 10 q^{67} - 22 q^{68} + 92 q^{70} - 2 q^{71} - 40 q^{73} + 40 q^{74} + 6 q^{76} - 86 q^{77} - 31 q^{79} - 26 q^{80} + 90 q^{82} + 16 q^{83} - 42 q^{85} - 92 q^{86} - 48 q^{88} + 12 q^{89} + 38 q^{91} + 60 q^{92} - 62 q^{94} + 29 q^{95} + 32 q^{97} - 9 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
64.1 −2.15399 − 0.632468i 0 2.55715 + 1.64338i 0.146077 + 0.319865i 0 −0.489834 − 0.143828i −1.52846 − 1.76393i 0 −0.112345 − 0.781375i
64.2 −0.907634 − 0.266505i 0 −0.929732 − 0.597503i −1.48348 − 3.24837i 0 4.49742 + 1.32056i 1.92355 + 2.21990i 0 0.480750 + 3.34369i
64.3 0.385020 + 0.113052i 0 −1.54705 − 0.994227i −0.0716395 − 0.156869i 0 −2.67716 − 0.786085i −1.00880 − 1.16422i 0 −0.00984831 − 0.0684965i
64.4 1.59954 + 0.469669i 0 0.655446 + 0.421230i −1.13272 − 2.48031i 0 0.0374006 + 0.0109818i −1.33282 − 1.53816i 0 −0.646912 − 4.49937i
64.5 1.87424 + 0.550325i 0 1.52739 + 0.981596i 1.74459 + 3.82012i 0 −0.0581100 − 0.0170626i −0.235861 − 0.272198i 0 1.16746 + 8.11989i
82.1 −1.94769 + 1.25171i 0 1.39591 − 3.05662i −0.0269454 − 0.0310967i 0 −3.56912 + 2.29374i 0.448199 + 3.11729i 0 0.0914053 + 0.0268390i
82.2 −0.729114 + 0.468573i 0 −0.518784 + 1.13598i −0.665817 − 0.768394i 0 1.30396 − 0.838004i −0.400725 − 2.78710i 0 0.845505 + 0.248263i
82.3 0.876843 − 0.563513i 0 −0.379523 + 0.831038i −2.49773 − 2.88254i 0 −1.63876 + 1.05317i 0.432190 + 3.00595i 0 −3.81447 − 1.12003i
82.4 1.00235 − 0.644173i 0 −0.241077 + 0.527885i 1.03436 + 1.19372i 0 3.81427 − 2.45128i 0.437541 + 3.04317i 0 1.80576 + 0.530219i
82.5 1.89942 − 1.22068i 0 1.28689 − 2.81791i 1.05432 + 1.21675i 0 0.374285 − 0.240538i −0.352767 − 2.45355i 0 3.48787 + 1.02413i
91.1 −0.710334 + 1.55542i 0 −0.605021 − 0.698231i 0.0752547 − 0.523408i 0 1.63245 − 3.57456i −1.76554 + 0.518410i 0 0.760660 + 0.488847i
91.2 −0.500753 + 1.09650i 0 0.358171 + 0.413351i 0.271255 − 1.88662i 0 −1.91302 + 4.18893i −2.94579 + 0.864963i 0 1.93284 + 1.24216i
91.3 −0.127669 + 0.279556i 0 1.24787 + 1.44012i −0.542214 + 3.77118i 0 0.258591 − 0.566236i −1.15167 + 0.338160i 0 −0.985035 − 0.633043i
91.4 0.373696 − 0.818280i 0 0.779789 + 0.899924i 0.297817 − 2.07136i 0 0.329681 − 0.721901i 2.75406 − 0.808665i 0 −1.58366 − 1.01776i
91.5 1.08330 − 2.37210i 0 −3.14358 − 3.62789i −0.220351 + 1.53258i 0 1.61129 − 3.52822i −7.00690 + 2.05741i 0 3.39671 + 2.18294i
226.1 −0.290637 − 2.02142i 0 −2.08270 + 0.611536i −2.61244 + 1.67891i 0 −0.145436 − 1.01153i 0.144754 + 0.316968i 0 4.15306 + 4.79289i
226.2 −0.0704781 − 0.490186i 0 1.68367 − 0.494370i 0.688821 − 0.442678i 0 0.350196 + 2.43567i −0.772445 − 1.69142i 0 −0.265542 − 0.306451i
226.3 −0.0345454 − 0.240269i 0 1.86245 − 0.546865i −0.0517218 + 0.0332396i 0 −0.587961 − 4.08936i −0.397409 − 0.870204i 0 0.00977317 + 0.0112788i
226.4 0.254047 + 1.76694i 0 −1.13854 + 0.334305i −1.11491 + 0.716511i 0 0.110811 + 0.770710i 0.603181 + 1.32078i 0 −1.54927 − 1.78795i
226.5 0.381059 + 2.65032i 0 −4.96003 + 1.45640i 2.85080 − 1.83210i 0 −0.558441 − 3.88404i −3.52537 − 7.71949i 0 5.94199 + 6.85742i
See all 50 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 64.5
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
67.e even 11 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 603.2.u.d 50
3.b odd 2 1 201.2.i.a ✓ 50
67.e even 11 1 inner 603.2.u.d 50
201.k odd 22 1 201.2.i.a ✓ 50
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
201.2.i.a ✓ 50 3.b odd 2 1
201.2.i.a ✓ 50 201.k odd 22 1
603.2.u.d 50 1.a even 1 1 trivial
603.2.u.d 50 67.e even 11 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{50} - 2 T_{2}^{49} + 10 T_{2}^{48} - 11 T_{2}^{47} + 55 T_{2}^{46} - 127 T_{2}^{45} + 457 T_{2}^{44} + \cdots + 4489 \) acting on \(S_{2}^{\mathrm{new}}(603, [\chi])\). Copy content Toggle raw display