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Properties

Label 418.2.s.b
Level $418$
Weight $2$
Character orbit 418.s
Analytic conductor $3.338$
Analytic rank $0$
Dimension $80$
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(107,418)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(418, base_ring=CyclotomicField(30))
 
chi = DirichletCharacter(H, H._module([9, 25]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("418.107");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 418 = 2 \cdot 11 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 418.s (of order \(30\), degree \(8\), 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: \(80\)
Relative dimension: \(10\) over \(\Q(\zeta_{30})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{30}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 80 q + 10 q^{2} - 3 q^{3} + 10 q^{4} - 2 q^{5} - 2 q^{6} - 10 q^{7} - 20 q^{8} - 11 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 80 q + 10 q^{2} - 3 q^{3} + 10 q^{4} - 2 q^{5} - 2 q^{6} - 10 q^{7} - 20 q^{8} - 11 q^{9} - 2 q^{10} - q^{11} - 5 q^{13} - 14 q^{14} + 3 q^{15} + 10 q^{16} + 6 q^{17} + 17 q^{18} - 13 q^{19} + 4 q^{20} - 24 q^{21} - q^{22} - 6 q^{23} + 2 q^{24} - 10 q^{26} + 45 q^{27} - 6 q^{28} - 55 q^{29} + 30 q^{30} - 40 q^{32} + 33 q^{33} + 24 q^{34} + 23 q^{35} - 16 q^{36} - 28 q^{38} - 30 q^{39} + 3 q^{40} - 16 q^{41} - 14 q^{42} - 12 q^{43} - 18 q^{44} - 12 q^{45} + 2 q^{46} - q^{47} + 3 q^{48} + 32 q^{49} - 30 q^{50} + 13 q^{51} + 5 q^{52} - 41 q^{53} + 27 q^{54} - 5 q^{55} + 97 q^{57} - 10 q^{58} - 59 q^{59} - 8 q^{60} - 38 q^{61} + 15 q^{62} - 38 q^{63} - 20 q^{64} - 8 q^{65} + 12 q^{66} - 27 q^{67} + 60 q^{69} + 7 q^{70} + 52 q^{71} - 11 q^{72} - 19 q^{73} - 12 q^{74} + 35 q^{75} + 11 q^{76} - 8 q^{77} + 6 q^{79} + 3 q^{80} - 2 q^{81} - q^{82} - 25 q^{83} - 52 q^{84} + 63 q^{85} + 37 q^{86} + 9 q^{88} - 36 q^{89} + 46 q^{90} - 32 q^{91} + 19 q^{92} + 15 q^{93} + 2 q^{94} - 39 q^{95} + 21 q^{97} + 44 q^{98} + 19 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
107.1 0.913545 + 0.406737i −2.53903 − 2.28615i 0.669131 + 0.743145i −0.201237 + 1.91464i −1.38966 − 3.12122i −2.69739 − 0.876435i 0.309017 + 0.951057i 0.906591 + 8.62564i −0.962593 + 1.66726i
107.2 0.913545 + 0.406737i −2.02949 − 1.82736i 0.669131 + 0.743145i 0.376952 − 3.58645i −1.11078 − 2.49485i 0.481898 + 0.156578i 0.309017 + 0.951057i 0.465997 + 4.43367i 1.80310 − 3.12307i
107.3 0.913545 + 0.406737i −1.55645 − 1.40143i 0.669131 + 0.743145i −0.177196 + 1.68591i −0.851871 − 1.91333i 3.09061 + 1.00420i 0.309017 + 0.951057i 0.144932 + 1.37894i −0.847599 + 1.46808i
107.4 0.913545 + 0.406737i −0.714491 − 0.643331i 0.669131 + 0.743145i 0.131360 − 1.24981i −0.391054 − 0.878321i 1.00748 + 0.327351i 0.309017 + 0.951057i −0.216962 − 2.06426i 0.628345 − 1.08833i
107.5 0.913545 + 0.406737i −0.689931 − 0.621217i 0.669131 + 0.743145i 0.105289 − 1.00176i −0.377612 − 0.848130i −4.27338 − 1.38851i 0.309017 + 0.951057i −0.223491 − 2.12637i 0.503640 − 0.872330i
107.6 0.913545 + 0.406737i 0.544898 + 0.490628i 0.669131 + 0.743145i −0.386740 + 3.67959i 0.298233 + 0.669841i −3.47915 − 1.13044i 0.309017 + 0.951057i −0.257388 − 2.44888i −1.84993 + 3.20417i
107.7 0.913545 + 0.406737i 0.772572 + 0.695627i 0.669131 + 0.743145i 0.0253771 − 0.241447i 0.422843 + 0.949720i 3.08185 + 1.00135i 0.309017 + 0.951057i −0.200615 − 1.90872i 0.121388 − 0.210251i
107.8 0.913545 + 0.406737i 0.894715 + 0.805605i 0.669131 + 0.743145i −0.249527 + 2.37409i 0.489694 + 1.09987i 1.81242 + 0.588892i 0.309017 + 0.951057i −0.162070 − 1.54199i −1.19358 + 2.06735i
107.9 0.913545 + 0.406737i 1.66168 + 1.49619i 0.669131 + 0.743145i 0.405449 − 3.85759i 0.909470 + 2.04270i −1.38944 − 0.451458i 0.309017 + 0.951057i 0.209033 + 1.98881i 1.93942 − 3.35917i
107.10 0.913545 + 0.406737i 2.17737 + 1.96051i 0.669131 + 0.743145i 0.0348758 − 0.331821i 1.19171 + 2.67664i −0.461981 − 0.150107i 0.309017 + 0.951057i 0.583745 + 5.55397i 0.166824 − 0.288948i
145.1 −0.978148 + 0.207912i −1.24883 + 2.80492i 0.913545 − 0.406737i 0.240932 − 0.267582i 0.638366 − 3.00327i 1.57046 − 2.16155i −0.809017 + 0.587785i −4.30061 − 4.77631i −0.180034 + 0.311828i
145.2 −0.978148 + 0.207912i −0.817441 + 1.83600i 0.913545 − 0.406737i 2.37377 − 2.63634i 0.417851 − 1.96584i −2.40604 + 3.31163i −0.809017 + 0.587785i −0.695302 − 0.772212i −1.77377 + 3.07226i
145.3 −0.978148 + 0.207912i −0.812291 + 1.82444i 0.913545 − 0.406737i 0.415828 − 0.461823i 0.415219 − 1.95345i 0.000562341 0 0.000773996i −0.809017 + 0.587785i −0.661358 − 0.734512i −0.310722 + 0.538187i
145.4 −0.978148 + 0.207912i −0.802803 + 1.80312i 0.913545 − 0.406737i −2.97254 + 3.30134i 0.410369 − 1.93063i 0.0503741 − 0.0693340i −0.809017 + 0.587785i −0.599373 − 0.665671i 2.22119 − 3.84722i
145.5 −0.978148 + 0.207912i −0.155948 + 0.350264i 0.913545 − 0.406737i −1.17360 + 1.30341i 0.0797158 − 0.375033i −2.57160 + 3.53951i −0.809017 + 0.587785i 1.90903 + 2.12019i 0.876959 − 1.51894i
145.6 −0.978148 + 0.207912i 0.112620 − 0.252949i 0.913545 − 0.406737i 1.24045 − 1.37766i −0.0575681 + 0.270837i −0.187685 + 0.258326i −0.809017 + 0.587785i 1.95609 + 2.17246i −0.926912 + 1.60546i
145.7 −0.978148 + 0.207912i 0.266418 − 0.598385i 0.913545 − 0.406737i 1.53333 − 1.70294i −0.136185 + 0.640700i 2.51944 − 3.46771i −0.809017 + 0.587785i 1.72031 + 1.91059i −1.14576 + 1.98452i
145.8 −0.978148 + 0.207912i 0.507299 − 1.13941i 0.913545 − 0.406737i −1.11339 + 1.23655i −0.259316 + 1.21999i −0.310569 + 0.427462i −0.809017 + 0.587785i 0.966484 + 1.07339i 0.831969 − 1.44101i
145.9 −0.978148 + 0.207912i 1.14466 − 2.57095i 0.913545 − 0.406737i 1.97776 − 2.19652i −0.585116 + 2.75275i 0.685829 − 0.943963i −0.809017 + 0.587785i −3.29213 − 3.65628i −1.47786 + 2.55972i
145.10 −0.978148 + 0.207912i 1.20179 − 2.69926i 0.913545 − 0.406737i −1.43986 + 1.59912i −0.614319 + 2.89014i 1.60553 − 2.20983i −0.809017 + 0.587785i −3.83432 − 4.25845i 1.07592 − 1.86354i
See all 80 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 107.10
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
209.t even 30 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 418.2.s.b yes 80
11.d odd 10 1 418.2.s.a ✓ 80
19.d odd 6 1 418.2.s.a ✓ 80
209.t even 30 1 inner 418.2.s.b yes 80
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
418.2.s.a ✓ 80 11.d odd 10 1
418.2.s.a ✓ 80 19.d odd 6 1
418.2.s.b yes 80 1.a even 1 1 trivial
418.2.s.b yes 80 209.t even 30 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{80} + 3 T_{3}^{79} + 25 T_{3}^{78} + 41 T_{3}^{77} + 184 T_{3}^{76} - 175 T_{3}^{75} + \cdots + 429981696 \) acting on \(S_{2}^{\mathrm{new}}(418, [\chi])\). Copy content Toggle raw display