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Properties

Label 633.2.e.b
Level $633$
Weight $2$
Character orbit 633.e
Analytic conductor $5.055$
Analytic rank $0$
Dimension $36$
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] = [633,2,Mod(196,633)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(633, base_ring=CyclotomicField(6))
 
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("633.196");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 633 = 3 \cdot 211 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 633.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: \(5.05453044795\)
Analytic rank: \(0\)
Dimension: \(36\)
Relative dimension: \(18\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 36 q - q^{2} - 18 q^{3} - 17 q^{4} + 4 q^{5} - q^{6} + q^{7} + 24 q^{8} - 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 36 q - q^{2} - 18 q^{3} - 17 q^{4} + 4 q^{5} - q^{6} + q^{7} + 24 q^{8} - 18 q^{9} + 4 q^{10} - 4 q^{11} + 34 q^{12} + 6 q^{13} + 8 q^{14} - 2 q^{15} - 11 q^{16} - 2 q^{17} + 2 q^{18} + 8 q^{19} + 5 q^{20} + q^{21} + 6 q^{22} - 12 q^{24} + 36 q^{25} - 16 q^{26} + 36 q^{27} + 22 q^{28} + 4 q^{30} + 2 q^{31} - 5 q^{32} + 2 q^{33} + 8 q^{34} - 4 q^{35} - 17 q^{36} - 14 q^{38} - 3 q^{39} - 10 q^{40} - 6 q^{41} - 16 q^{42} - 18 q^{43} + 16 q^{44} - 2 q^{45} + 11 q^{46} + 3 q^{47} - 11 q^{48} + q^{49} - 14 q^{50} - 2 q^{51} - 11 q^{52} - 12 q^{53} - q^{54} + 10 q^{55} + 11 q^{56} + 8 q^{57} - 22 q^{58} + 3 q^{59} - 10 q^{60} - 11 q^{61} + 7 q^{62} - 2 q^{63} + 48 q^{64} + 22 q^{65} + 6 q^{66} - 2 q^{67} - 10 q^{68} + 25 q^{70} - 34 q^{71} - 12 q^{72} - 9 q^{73} - 48 q^{74} - 18 q^{75} - 78 q^{76} - 2 q^{77} - 16 q^{78} + 20 q^{79} + 63 q^{80} - 18 q^{81} - 98 q^{82} - 36 q^{83} - 11 q^{84} - 63 q^{85} + 78 q^{86} - 22 q^{88} - 22 q^{89} - 8 q^{90} + q^{91} - 10 q^{92} + 2 q^{93} - 26 q^{95} + 10 q^{96} + 8 q^{97} - 54 q^{98} + 2 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
196.1 −1.33771 − 2.31698i −0.500000 − 0.866025i −2.57893 + 4.46684i −3.25944 −1.33771 + 2.31698i −0.322919 − 0.559313i 8.44861 −0.500000 + 0.866025i 4.36018 + 7.55205i
196.2 −1.29945 − 2.25072i −0.500000 − 0.866025i −2.37715 + 4.11734i 2.00628 −1.29945 + 2.25072i −0.849589 − 1.47153i 7.15814 −0.500000 + 0.866025i −2.60707 − 4.51558i
196.3 −1.08206 − 1.87418i −0.500000 − 0.866025i −1.34169 + 2.32388i −0.335529 −1.08206 + 1.87418i 0.670152 + 1.16074i 1.47893 −0.500000 + 0.866025i 0.363061 + 0.628840i
196.4 −1.01798 − 1.76320i −0.500000 − 0.866025i −1.07258 + 1.85776i −1.68022 −1.01798 + 1.76320i 1.30780 + 2.26518i 0.295525 −0.500000 + 0.866025i 1.71043 + 2.96256i
196.5 −0.907314 − 1.57151i −0.500000 − 0.866025i −0.646436 + 1.11966i 4.43619 −0.907314 + 1.57151i 1.72972 + 2.99597i −1.28317 −0.500000 + 0.866025i −4.02501 − 6.97153i
196.6 −0.701531 − 1.21509i −0.500000 − 0.866025i 0.0157096 − 0.0272097i −0.498553 −0.701531 + 1.21509i −1.94992 − 3.37735i −2.85021 −0.500000 + 0.866025i 0.349751 + 0.605786i
196.7 −0.349821 − 0.605907i −0.500000 − 0.866025i 0.755251 − 1.30813i −0.0501803 −0.349821 + 0.605907i 1.17351 + 2.03257i −2.45609 −0.500000 + 0.866025i 0.0175541 + 0.0304046i
196.8 −0.275338 − 0.476900i −0.500000 − 0.866025i 0.848377 − 1.46943i 2.35129 −0.275338 + 0.476900i −1.37889 − 2.38831i −2.03572 −0.500000 + 0.866025i −0.647399 − 1.12133i
196.9 −0.214735 − 0.371932i −0.500000 − 0.866025i 0.907778 − 1.57232i −3.85712 −0.214735 + 0.371932i 0.528467 + 0.915331i −1.63867 −0.500000 + 0.866025i 0.828259 + 1.43459i
196.10 0.234498 + 0.406162i −0.500000 − 0.866025i 0.890022 − 1.54156i 2.09043 0.234498 − 0.406162i 0.503615 + 0.872286i 1.77282 −0.500000 + 0.866025i 0.490200 + 0.849052i
196.11 0.414351 + 0.717677i −0.500000 − 0.866025i 0.656626 − 1.13731i −2.24836 0.414351 − 0.717677i 0.208618 + 0.361337i 2.74570 −0.500000 + 0.866025i −0.931612 − 1.61360i
196.12 0.433815 + 0.751390i −0.500000 − 0.866025i 0.623608 − 1.08012i −0.774154 0.433815 − 0.751390i −1.96287 − 3.39978i 2.81739 −0.500000 + 0.866025i −0.335840 − 0.581692i
196.13 0.612547 + 1.06096i −0.500000 − 0.866025i 0.249572 − 0.432272i 3.57218 0.612547 − 1.06096i 1.92595 + 3.33585i 3.06169 −0.500000 + 0.866025i 2.18813 + 3.78995i
196.14 0.702199 + 1.21624i −0.500000 − 0.866025i 0.0138326 − 0.0239587i −2.33623 0.702199 − 1.21624i 2.14704 + 3.71879i 2.84765 −0.500000 + 0.866025i −1.64050 − 2.84143i
196.15 0.850323 + 1.47280i −0.500000 − 0.866025i −0.446099 + 0.772667i 3.71509 0.850323 − 1.47280i −2.14548 − 3.71609i 1.88398 −0.500000 + 0.866025i 3.15903 + 5.47159i
196.16 0.986393 + 1.70848i −0.500000 − 0.866025i −0.945944 + 1.63842i −1.75106 0.986393 − 1.70848i −0.315641 − 0.546707i 0.213283 −0.500000 + 0.866025i −1.72724 − 2.99167i
196.17 1.12519 + 1.94889i −0.500000 − 0.866025i −1.53213 + 2.65372i 1.85498 1.12519 − 1.94889i 0.0867504 + 0.150256i −2.39499 −0.500000 + 0.866025i 2.08721 + 3.61515i
196.18 1.32662 + 2.29777i −0.500000 − 0.866025i −2.51982 + 4.36446i −1.23557 1.32662 − 2.29777i −0.856323 − 1.48319i −8.06487 −0.500000 + 0.866025i −1.63913 − 2.83906i
436.1 −1.33771 + 2.31698i −0.500000 + 0.866025i −2.57893 − 4.46684i −3.25944 −1.33771 − 2.31698i −0.322919 + 0.559313i 8.44861 −0.500000 − 0.866025i 4.36018 − 7.55205i
436.2 −1.29945 + 2.25072i −0.500000 + 0.866025i −2.37715 − 4.11734i 2.00628 −1.29945 − 2.25072i −0.849589 + 1.47153i 7.15814 −0.500000 − 0.866025i −2.60707 + 4.51558i
See all 36 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 196.18
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
211.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 633.2.e.b ✓ 36
211.c even 3 1 inner 633.2.e.b ✓ 36
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
633.2.e.b ✓ 36 1.a even 1 1 trivial
633.2.e.b ✓ 36 211.c even 3 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{36} + T_{2}^{35} + 27 T_{2}^{34} + 16 T_{2}^{33} + 418 T_{2}^{32} + 156 T_{2}^{31} + 4321 T_{2}^{30} + \cdots + 22500 \) acting on \(S_{2}^{\mathrm{new}}(633, [\chi])\). Copy content Toggle raw display