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Properties

Label 418.2.q.a
Level $418$
Weight $2$
Character orbit 418.q
Analytic conductor $3.338$
Analytic rank $0$
Dimension $60$
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] = [418,2,Mod(21,418)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(418, base_ring=CyclotomicField(18))
 
chi = DirichletCharacter(H, H._module([9, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("418.21");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 418 = 2 \cdot 11 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 418.q (of order \(18\), degree \(6\), 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: \(3.33774680449\)
Analytic rank: \(0\)
Dimension: \(60\)
Relative dimension: \(10\) over \(\Q(\zeta_{18})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{18}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 60 q - 3 q^{3} - 3 q^{6} - 18 q^{7} + 30 q^{8} - 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 60 q - 3 q^{3} - 3 q^{6} - 18 q^{7} + 30 q^{8} - 3 q^{9} + 3 q^{11} + 6 q^{13} - 12 q^{14} + 24 q^{15} - 6 q^{17} + 60 q^{18} - 30 q^{19} - 12 q^{20} + 12 q^{21} + 12 q^{22} + 3 q^{24} - 12 q^{25} + 6 q^{26} + 9 q^{27} + 6 q^{28} - 3 q^{29} - 9 q^{31} - 42 q^{33} - 6 q^{34} - 24 q^{35} - 3 q^{36} + 6 q^{38} + 15 q^{41} + 6 q^{42} - 3 q^{43} + 12 q^{44} - 48 q^{45} + 3 q^{46} + 54 q^{47} - 6 q^{48} + 6 q^{49} + 36 q^{50} - 45 q^{51} - 3 q^{52} + 24 q^{53} - 27 q^{54} + 6 q^{55} + 30 q^{57} + 24 q^{58} - 39 q^{59} + 12 q^{60} + 54 q^{61} - 66 q^{63} - 30 q^{64} - 63 q^{66} + 9 q^{67} - 27 q^{68} + 54 q^{69} - 24 q^{70} - 33 q^{71} - 6 q^{72} + 12 q^{74} + 18 q^{77} + 36 q^{79} - 93 q^{81} - 15 q^{82} - 36 q^{83} + 24 q^{84} - 60 q^{85} - 3 q^{86} + 54 q^{87} - 3 q^{88} - 3 q^{89} - 24 q^{90} - 12 q^{91} - 102 q^{93} - 12 q^{94} + 24 q^{95} - 6 q^{97} - 18 q^{98} - 78 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
21.1 −0.173648 + 0.984808i −2.10863 + 2.51297i −0.939693 − 0.342020i 1.20713 − 0.439360i −2.10863 − 2.51297i −2.67427 + 1.54399i 0.500000 − 0.866025i −1.34774 − 7.64344i 0.223069 + 1.26509i
21.2 −0.173648 + 0.984808i −1.64579 + 1.96137i −0.939693 − 0.342020i −3.44031 + 1.25217i −1.64579 − 1.96137i 0.237067 − 0.136871i 0.500000 − 0.866025i −0.617423 − 3.50158i −0.635743 − 3.60548i
21.3 −0.173648 + 0.984808i −1.49082 + 1.77669i −0.939693 − 0.342020i 1.76584 − 0.642714i −1.49082 − 1.77669i 3.47026 − 2.00356i 0.500000 − 0.866025i −0.413143 − 2.34305i 0.326315 + 1.85062i
21.4 −0.173648 + 0.984808i −0.596920 + 0.711382i −0.939693 − 0.342020i 0.587101 − 0.213687i −0.596920 − 0.711382i −1.73498 + 1.00169i 0.500000 − 0.866025i 0.371194 + 2.10515i 0.108492 + 0.615288i
21.5 −0.173648 + 0.984808i −0.138226 + 0.164732i −0.939693 − 0.342020i 3.86845 − 1.40800i −0.138226 − 0.164732i −1.22696 + 0.708384i 0.500000 − 0.866025i 0.512915 + 2.90888i 0.714861 + 4.05418i
21.6 −0.173648 + 0.984808i 0.0268646 − 0.0320160i −0.939693 − 0.342020i −0.843173 + 0.306890i 0.0268646 + 0.0320160i 2.35018 − 1.35688i 0.500000 − 0.866025i 0.520641 + 2.95270i −0.155812 − 0.883655i
21.7 −0.173648 + 0.984808i 0.144347 − 0.172026i −0.939693 − 0.342020i −2.86607 + 1.04316i 0.144347 + 0.172026i 0.416271 − 0.240334i 0.500000 − 0.866025i 0.512188 + 2.90476i −0.529628 − 3.00367i
21.8 −0.173648 + 0.984808i 0.904669 − 1.07814i −0.939693 − 0.342020i −0.316544 + 0.115212i 0.904669 + 1.07814i −4.29189 + 2.47793i 0.500000 − 0.866025i 0.176980 + 1.00370i −0.0584949 − 0.331741i
21.9 −0.173648 + 0.984808i 1.60541 − 1.91325i −0.939693 − 0.342020i 2.62065 − 0.953840i 1.60541 + 1.91325i 0.240088 − 0.138615i 0.500000 − 0.866025i −0.562250 − 3.18868i 0.484277 + 2.74647i
21.10 −0.173648 + 0.984808i 2.03306 − 2.42290i −0.939693 − 0.342020i −0.703704 + 0.256127i 2.03306 + 2.42290i −0.970561 + 0.560354i 0.500000 − 0.866025i −1.21619 − 6.89736i −0.130039 − 0.737489i
109.1 0.939693 − 0.342020i −3.24219 − 0.571685i 0.766044 − 0.642788i −2.49158 − 2.09069i −3.24219 + 0.571685i −3.04403 + 1.75747i 0.500000 − 0.866025i 7.36587 + 2.68096i −3.05638 − 1.11243i
109.2 0.939693 − 0.342020i −1.99937 − 0.352544i 0.766044 − 0.642788i 0.590160 + 0.495203i −1.99937 + 0.352544i 0.755972 − 0.436461i 0.500000 − 0.866025i 1.05413 + 0.383673i 0.723939 + 0.263492i
109.3 0.939693 − 0.342020i −1.85347 − 0.326818i 0.766044 − 0.642788i 2.99121 + 2.50992i −1.85347 + 0.326818i −4.29605 + 2.48032i 0.500000 − 0.866025i 0.509479 + 0.185435i 3.66926 + 1.33550i
109.4 0.939693 − 0.342020i −1.39303 − 0.245629i 0.766044 − 0.642788i −1.31092 − 1.09999i −1.39303 + 0.245629i 1.63419 − 0.943501i 0.500000 − 0.866025i −0.938877 − 0.341723i −1.60808 − 0.585293i
109.5 0.939693 − 0.342020i −0.274412 − 0.0483863i 0.766044 − 0.642788i 1.21736 + 1.02148i −0.274412 + 0.0483863i 2.59958 − 1.50087i 0.500000 − 0.866025i −2.74612 − 0.999505i 1.49331 + 0.543520i
109.6 0.939693 − 0.342020i 0.00614027 + 0.00108270i 0.766044 − 0.642788i −3.03422 − 2.54601i 0.00614027 − 0.00108270i −2.10812 + 1.21712i 0.500000 − 0.866025i −2.81904 − 1.02605i −3.72202 − 1.35470i
109.7 0.939693 − 0.342020i 1.58017 + 0.278627i 0.766044 − 0.642788i −2.47390 − 2.07585i 1.58017 − 0.278627i 2.13981 − 1.23542i 0.500000 − 0.866025i −0.399773 − 0.145505i −3.03469 − 1.10454i
109.8 0.939693 − 0.342020i 1.64099 + 0.289351i 0.766044 − 0.642788i 1.95336 + 1.63906i 1.64099 − 0.289351i 0.632001 − 0.364886i 0.500000 − 0.866025i −0.209943 − 0.0764129i 2.39615 + 0.872127i
109.9 0.939693 − 0.342020i 2.01819 + 0.355861i 0.766044 − 0.642788i 1.40938 + 1.18261i 2.01819 − 0.355861i −2.85785 + 1.64998i 0.500000 − 0.866025i 1.12738 + 0.410331i 1.72887 + 0.629256i
109.10 0.939693 − 0.342020i 2.84334 + 0.501357i 0.766044 − 0.642788i −0.382944 − 0.321328i 2.84334 − 0.501357i −0.682188 + 0.393861i 0.500000 − 0.866025i 5.01412 + 1.82499i −0.469750 − 0.170975i
See all 60 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 21.10
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
209.p even 18 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 418.2.q.a ✓ 60
11.b odd 2 1 418.2.q.b yes 60
19.f odd 18 1 418.2.q.b yes 60
209.p even 18 1 inner 418.2.q.a ✓ 60
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
418.2.q.a ✓ 60 1.a even 1 1 trivial
418.2.q.a ✓ 60 209.p even 18 1 inner
418.2.q.b yes 60 11.b odd 2 1
418.2.q.b yes 60 19.f odd 18 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{60} + 18 T_{7}^{59} + 54 T_{7}^{58} - 972 T_{7}^{57} - 6507 T_{7}^{56} + 28458 T_{7}^{55} + \cdots + 132936103563264 \) acting on \(S_{2}^{\mathrm{new}}(418, [\chi])\). Copy content Toggle raw display