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Properties

Label 387.2.f.d
Level $387$
Weight $2$
Character orbit 387.f
Analytic conductor $3.090$
Analytic rank $0$
Dimension $40$
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] = [387,2,Mod(130,387)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(387, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([2, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("387.130");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 387 = 3^{2} \cdot 43 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 387.f (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: \(3.09021055822\)
Analytic rank: \(0\)
Dimension: \(40\)
Relative dimension: \(20\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 40 q + 6 q^{2} + 2 q^{3} - 22 q^{4} + 17 q^{5} - 6 q^{6} - 3 q^{7} - 30 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 40 q + 6 q^{2} + 2 q^{3} - 22 q^{4} + 17 q^{5} - 6 q^{6} - 3 q^{7} - 30 q^{8} - 4 q^{10} + 10 q^{11} - 17 q^{12} + q^{13} + 10 q^{14} + 7 q^{15} - 22 q^{16} - 40 q^{17} + 3 q^{18} + 16 q^{19} + 30 q^{20} + 6 q^{21} - 15 q^{22} + 19 q^{23} + 33 q^{24} - 19 q^{25} - 50 q^{26} + 8 q^{27} - 6 q^{28} + 25 q^{29} - 54 q^{30} + 11 q^{31} + 36 q^{32} - 6 q^{33} - 9 q^{34} + 44 q^{36} + 18 q^{37} + 28 q^{38} - 35 q^{39} - 12 q^{40} + 12 q^{41} + 37 q^{42} + 20 q^{43} - 10 q^{44} - 25 q^{45} + 8 q^{46} + 38 q^{47} - 7 q^{48} - 37 q^{49} + 36 q^{50} - 4 q^{51} + 8 q^{52} - 138 q^{53} + 33 q^{54} - 18 q^{55} + 30 q^{56} - 53 q^{57} + 27 q^{58} + 31 q^{59} + 38 q^{60} - 19 q^{61} - 64 q^{62} - 35 q^{63} + 22 q^{64} + 47 q^{65} - 33 q^{66} - 9 q^{67} + 68 q^{68} + 26 q^{69} + 6 q^{70} - 42 q^{71} + 6 q^{72} - 4 q^{73} - 16 q^{74} - 56 q^{75} - 37 q^{76} + 85 q^{77} + 43 q^{78} + 4 q^{79} - 122 q^{80} - 36 q^{81} + 2 q^{82} + 19 q^{83} - 83 q^{84} + 6 q^{85} - 6 q^{86} + 49 q^{87} - 60 q^{88} - 108 q^{89} + 3 q^{90} - 6 q^{91} + 85 q^{92} - 10 q^{93} + 19 q^{94} - 11 q^{95} + 93 q^{96} - 2 q^{97} - 10 q^{98} + 19 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
130.1 −1.31834 − 2.28344i 1.31217 − 1.13058i −2.47606 + 4.28866i 1.39983 − 2.42457i −4.31150 − 1.50576i 1.99111 + 3.44870i 7.78382 0.443574 − 2.96703i −7.38182
130.2 −1.04491 − 1.80984i 1.25718 + 1.19142i −1.18368 + 2.05019i −0.136674 + 0.236726i 0.842638 − 3.52023i 1.21012 + 2.09599i 0.767710 0.161024 + 2.99568i 0.571248
130.3 −0.988037 − 1.71133i −1.72898 + 0.103053i −0.952435 + 1.64967i 0.717842 − 1.24334i 1.88466 + 2.85704i −2.37609 − 4.11551i −0.187983 2.97876 − 0.356354i −2.83702
130.4 −0.911404 − 1.57860i −0.100359 + 1.72914i −0.661315 + 1.14543i 0.216442 − 0.374888i 2.82109 − 1.41752i 0.345529 + 0.598473i −1.23472 −2.97986 − 0.347069i −0.789063
130.5 −0.875101 − 1.51572i 1.30165 − 1.14268i −0.531605 + 0.920767i 1.34854 − 2.33575i −2.87105 − 0.972975i −1.33599 − 2.31400i −1.63957 0.388575 − 2.97473i −4.72045
130.6 −0.562737 − 0.974689i −1.37645 − 1.05137i 0.366654 − 0.635063i −1.19873 + 2.07626i −0.250182 + 1.93326i 0.174845 + 0.302841i −3.07627 0.789231 + 2.89432i 2.69827
130.7 −0.391509 − 0.678114i −0.811654 + 1.53010i 0.693441 − 1.20108i 1.47053 − 2.54704i 1.35535 − 0.0486557i −1.24831 − 2.16213i −2.65199 −1.68243 − 2.48383i −2.30291
130.8 −0.110277 − 0.191005i 0.623230 − 1.61604i 0.975678 − 1.68992i −0.797629 + 1.38153i −0.377400 + 0.0591719i −0.709781 − 1.22938i −0.871488 −2.22317 − 2.01433i 0.351841
130.9 −0.0942911 − 0.163317i 0.876427 + 1.49395i 0.982218 − 1.70125i 1.89854 − 3.28836i 0.161348 − 0.284001i 2.00490 + 3.47259i −0.747622 −1.46375 + 2.61867i −0.716061
130.10 0.254302 + 0.440465i 0.0391437 − 1.73161i 0.870661 − 1.50803i 1.90300 − 3.29610i 0.772667 − 0.423111i 0.340171 + 0.589193i 1.90285 −2.99694 − 0.135563i 1.93575
130.11 0.342086 + 0.592511i 1.21379 + 1.23560i 0.765954 − 1.32667i 0.741129 − 1.28367i −0.316888 + 1.14187i −1.90727 − 3.30349i 2.41643 −0.0534343 + 2.99952i 1.01412
130.12 0.352536 + 0.610611i −1.12660 − 1.31559i 0.751436 − 1.30153i −0.464078 + 0.803807i 0.406149 − 1.15171i 2.55550 + 4.42626i 2.46978 −0.461564 + 2.96428i −0.654418
130.13 0.535034 + 0.926706i 1.69315 + 0.365031i 0.427478 − 0.740413i −1.71565 + 2.97160i 0.567616 + 1.76435i 0.0278005 + 0.0481519i 3.05500 2.73350 + 1.23610i −3.67173
130.14 0.774348 + 1.34121i −1.72435 + 0.163153i −0.199230 + 0.345076i −0.0851621 + 0.147505i −1.55407 − 2.18638i 0.931330 + 1.61311i 2.48030 2.94676 − 0.562666i −0.263781
130.15 0.856434 + 1.48339i −0.497284 − 1.65913i −0.466959 + 0.808796i 0.461242 − 0.798894i 2.03524 − 2.15860i −1.56362 − 2.70827i 1.82606 −2.50542 + 1.65012i 1.58009
130.16 1.16214 + 2.01289i −1.53664 − 0.799214i −1.70115 + 2.94647i 1.73647 − 3.00766i −0.177063 − 4.02188i −0.946105 − 1.63870i −3.25933 1.72251 + 2.45621i 8.07210
130.17 1.16730 + 2.02182i 1.71119 + 0.267981i −1.72517 + 2.98809i 1.89009 − 3.27373i 1.45566 + 3.77254i −0.823076 − 1.42561i −3.38597 2.85637 + 0.917136i 8.82518
130.18 1.18355 + 2.04996i −0.757560 + 1.55760i −1.80157 + 3.12041i −0.867534 + 1.50261i −4.08962 + 0.290518i −2.31405 − 4.00804i −3.79478 −1.85221 − 2.35994i −4.10707
130.19 1.28103 + 2.21881i 1.61972 − 0.613589i −2.28209 + 3.95269i −1.16256 + 2.01361i 3.43636 + 2.80784i −0.00975820 − 0.0169017i −6.56959 2.24702 − 1.98769i −5.95711
130.20 1.38785 + 2.40383i −0.987785 + 1.42277i −2.85226 + 4.94026i 1.14436 − 1.98209i −4.79100 − 0.399870i 2.15274 + 3.72866i −10.2826 −1.04856 − 2.81079i 6.35282
See all 40 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 130.20
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 387.2.f.d ✓ 40
3.b odd 2 1 1161.2.f.d 40
9.c even 3 1 inner 387.2.f.d ✓ 40
9.c even 3 1 3483.2.a.t 20
9.d odd 6 1 1161.2.f.d 40
9.d odd 6 1 3483.2.a.u 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
387.2.f.d ✓ 40 1.a even 1 1 trivial
387.2.f.d ✓ 40 9.c even 3 1 inner
1161.2.f.d 40 3.b odd 2 1
1161.2.f.d 40 9.d odd 6 1
3483.2.a.t 20 9.c even 3 1
3483.2.a.u 20 9.d odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{40} - 6 T_{2}^{39} + 49 T_{2}^{38} - 196 T_{2}^{37} + 1015 T_{2}^{36} - 3316 T_{2}^{35} + \cdots + 6561 \) acting on \(S_{2}^{\mathrm{new}}(387, [\chi])\). Copy content Toggle raw display