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Properties

Label 381.3.p.a
Level $381$
Weight $3$
Character orbit 381.p
Analytic conductor $10.381$
Analytic rank $0$
Dimension $126$
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] = [381,3,Mod(28,381)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(381, base_ring=CyclotomicField(18))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("381.28");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 381 = 3 \cdot 127 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 381.p (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: \(10.3814980721\)
Analytic rank: \(0\)
Dimension: \(126\)
Relative dimension: \(21\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 126 q + 204 q^{4} + 18 q^{5} - 6 q^{7} - 30 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 126 q + 204 q^{4} + 18 q^{5} - 6 q^{7} - 30 q^{8} - 27 q^{11} + 27 q^{12} + 36 q^{13} + 105 q^{14} - 36 q^{15} + 228 q^{16} + 12 q^{17} - 9 q^{18} + 144 q^{20} + 18 q^{21} - 111 q^{22} - 9 q^{23} + 243 q^{25} - 84 q^{26} + 567 q^{27} - 66 q^{28} - 183 q^{29} - 18 q^{30} + 90 q^{31} - 90 q^{32} - 9 q^{33} + 120 q^{34} + 141 q^{35} + 54 q^{36} + 75 q^{38} - 36 q^{39} - 72 q^{40} + 9 q^{41} - 72 q^{42} + 96 q^{43} - 36 q^{44} - 54 q^{45} + 93 q^{46} + 153 q^{47} + 297 q^{48} + 414 q^{49} + 72 q^{50} + 312 q^{52} - 213 q^{53} - 126 q^{55} + 153 q^{56} + 45 q^{57} + 252 q^{58} + 276 q^{59} - 216 q^{60} + 264 q^{61} - 264 q^{62} - 522 q^{64} + 504 q^{65} + 99 q^{66} + 51 q^{67} - 21 q^{68} + 99 q^{69} - 408 q^{70} - 738 q^{71} - 36 q^{72} + 288 q^{73} + 504 q^{74} + 54 q^{75} - 60 q^{76} - 252 q^{78} - 609 q^{79} + 720 q^{80} - 168 q^{82} - 654 q^{83} - 45 q^{84} - 333 q^{85} - 1185 q^{86} - 63 q^{87} - 699 q^{88} + 135 q^{89} + 36 q^{90} + 381 q^{91} - 408 q^{92} - 216 q^{93} + 36 q^{94} - 99 q^{96} - 330 q^{97} - 768 q^{98} - 81 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
28.1 −3.98256 1.70574 + 0.300767i 11.8608 1.30381 − 0.752757i −6.79320 − 1.19782i 0.00843375 + 0.0231715i −31.3061 2.81908 + 1.02606i −5.19252 + 2.99790i
28.2 −3.40418 1.70574 + 0.300767i 7.58842 −5.22954 + 3.01928i −5.80663 − 1.02387i −0.268258 − 0.737034i −12.2156 2.81908 + 1.02606i 17.8023 − 10.2781i
28.3 −3.13374 1.70574 + 0.300767i 5.82031 1.45189 − 0.838252i −5.34533 − 0.942526i −3.68512 − 10.1248i −5.70439 2.81908 + 1.02606i −4.54986 + 2.62686i
28.4 −2.87926 1.70574 + 0.300767i 4.29012 4.34737 − 2.50995i −4.91125 − 0.865987i 1.39901 + 3.84374i −0.835319 2.81908 + 1.02606i −12.5172 + 7.22680i
28.5 −2.63208 1.70574 + 0.300767i 2.92782 −2.80870 + 1.62161i −4.48963 − 0.791643i 3.83112 + 10.5259i 2.82205 2.81908 + 1.02606i 7.39272 − 4.26819i
28.6 −2.19855 1.70574 + 0.300767i 0.833609 6.01124 − 3.47059i −3.75014 − 0.661251i −1.17684 − 3.23333i 6.96146 2.81908 + 1.02606i −13.2160 + 7.63026i
28.7 −1.93107 1.70574 + 0.300767i −0.270969 −7.28052 + 4.20341i −3.29390 − 0.580803i −1.35848 − 3.73240i 8.24754 2.81908 + 1.02606i 14.0592 − 8.11707i
28.8 −1.06705 1.70574 + 0.300767i −2.86140 −2.45713 + 1.41863i −1.82011 − 0.320934i −2.58551 − 7.10362i 7.32147 2.81908 + 1.02606i 2.62189 − 1.51375i
28.9 −0.656737 1.70574 + 0.300767i −3.56870 6.86587 − 3.96401i −1.12022 − 0.197525i 0.699651 + 1.92228i 4.97064 2.81908 + 1.02606i −4.50907 + 2.60331i
28.10 −0.525253 1.70574 + 0.300767i −3.72411 1.23704 − 0.714207i −0.895943 − 0.157979i 3.31251 + 9.10103i 4.05711 2.81908 + 1.02606i −0.649760 + 0.375139i
28.11 0.119998 1.70574 + 0.300767i −3.98560 −1.32592 + 0.765520i 0.204686 + 0.0360916i −1.13424 − 3.11630i −0.958258 2.81908 + 1.02606i −0.159108 + 0.0918611i
28.12 0.249378 1.70574 + 0.300767i −3.93781 −8.12315 + 4.68990i 0.425373 + 0.0750048i 1.05458 + 2.89743i −1.97951 2.81908 + 1.02606i −2.02573 + 1.16956i
28.13 0.994744 1.70574 + 0.300767i −3.01048 0.619705 − 0.357787i 1.69677 + 0.299187i −2.76926 − 7.60848i −6.97364 2.81908 + 1.02606i 0.616447 − 0.355906i
28.14 1.14628 1.70574 + 0.300767i −2.68605 5.77937 − 3.33672i 1.95525 + 0.344763i 3.62960 + 9.97225i −7.66407 2.81908 + 1.02606i 6.62477 − 3.82481i
28.15 1.67092 1.70574 + 0.300767i −1.20803 −2.80607 + 1.62008i 2.85015 + 0.502558i 2.02297 + 5.55805i −8.70220 2.81908 + 1.02606i −4.68871 + 2.70703i
28.16 2.12808 1.70574 + 0.300767i 0.528705 −5.60531 + 3.23623i 3.62994 + 0.640056i 1.10258 + 3.02931i −7.38718 2.81908 + 1.02606i −11.9285 + 6.88694i
28.17 2.18815 1.70574 + 0.300767i 0.788012 3.81730 − 2.20392i 3.73241 + 0.658125i −2.76737 − 7.60329i −7.02832 2.81908 + 1.02606i 8.35284 − 4.82252i
28.18 3.00836 1.70574 + 0.300767i 5.05022 6.40297 − 3.69676i 5.13147 + 0.904816i 1.74460 + 4.79324i 3.15944 2.81908 + 1.02606i 19.2624 − 11.1212i
28.19 3.36478 1.70574 + 0.300767i 7.32172 −5.16460 + 2.98179i 5.73942 + 1.01202i 2.72073 + 7.47515i 11.1768 2.81908 + 1.02606i −17.3777 + 10.0330i
28.20 3.50789 1.70574 + 0.300767i 8.30526 −1.17360 + 0.677581i 5.98353 + 1.05506i 1.39156 + 3.82328i 15.1024 2.81908 + 1.02606i −4.11687 + 2.37688i
See next 80 embeddings (of 126 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 28.21
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
127.h odd 18 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 381.3.p.a ✓ 126
127.h odd 18 1 inner 381.3.p.a ✓ 126
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
381.3.p.a ✓ 126 1.a even 1 1 trivial
381.3.p.a ✓ 126 127.h odd 18 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{63} - 177 T_{2}^{61} + 5 T_{2}^{60} + 14826 T_{2}^{59} - 828 T_{2}^{58} - 781923 T_{2}^{57} + \cdots + 166849332681 \) acting on \(S_{3}^{\mathrm{new}}(381, [\chi])\). Copy content Toggle raw display