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Properties

Label 387.2.bl.a
Level $387$
Weight $2$
Character orbit 387.bl
Analytic conductor $3.090$
Analytic rank $0$
Dimension $504$
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(5,387)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(387, base_ring=CyclotomicField(42))
 
chi = DirichletCharacter(H, H._module([35, 25]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("387.5");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 387 = 3^{2} \cdot 43 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 387.bl (of order \(42\), degree \(12\), 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: \(504\)
Relative dimension: \(42\) over \(\Q(\zeta_{42})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{42}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 504 q - 21 q^{2} - 11 q^{3} + 35 q^{4} - 18 q^{5} - 4 q^{6} - 18 q^{7} - 27 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 504 q - 21 q^{2} - 11 q^{3} + 35 q^{4} - 18 q^{5} - 4 q^{6} - 18 q^{7} - 27 q^{9} - 28 q^{10} - 18 q^{11} + 11 q^{12} - 3 q^{13} - 21 q^{14} - 19 q^{15} + 35 q^{16} + 9 q^{17} + 10 q^{18} - 31 q^{19} - 27 q^{20} - 30 q^{21} - 19 q^{22} - 18 q^{23} + 22 q^{24} + 26 q^{25} + 6 q^{26} + 4 q^{27} - 12 q^{28} - 36 q^{29} + 60 q^{30} + 44 q^{31} + 9 q^{32} - 20 q^{33} + 14 q^{34} - 105 q^{35} + 6 q^{36} - 36 q^{37} - 21 q^{38} + 7 q^{39} - 13 q^{40} - 54 q^{41} - 99 q^{42} - 5 q^{43} + 42 q^{45} - 28 q^{46} - 50 q^{48} + 196 q^{49} - 114 q^{50} - 14 q^{51} - 34 q^{52} - 49 q^{54} - 7 q^{55} - 81 q^{56} - 5 q^{57} - 13 q^{58} - 126 q^{59} - 18 q^{60} + 17 q^{61} - 6 q^{62} - 10 q^{63} - 100 q^{64} - 21 q^{65} + 167 q^{66} + 9 q^{67} - 21 q^{68} + 135 q^{69} - 114 q^{70} + 24 q^{71} - 238 q^{72} - 49 q^{73} - 63 q^{74} - 37 q^{75} - 43 q^{76} - 15 q^{77} + 116 q^{78} + 5 q^{79} - 9 q^{80} - 199 q^{81} - 28 q^{82} - 21 q^{83} - 258 q^{84} - 15 q^{85} - 48 q^{86} + 168 q^{87} + 7 q^{88} - 117 q^{89} - 184 q^{90} + 36 q^{91} - 69 q^{93} - 25 q^{94} - 21 q^{95} - 62 q^{96} + 5 q^{97} - 48 q^{98} + 4 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
5.1 −2.63644 + 0.813234i −1.72096 + 0.195723i 4.63698 − 3.16144i 3.07766 + 0.463883i 4.37803 − 1.91555i 3.15931 + 1.82403i −6.21370 + 7.79173i 2.92339 − 0.673661i −8.49132 + 1.27986i
5.2 −2.57291 + 0.793639i −1.19921 + 1.24976i 4.33754 − 2.95729i −3.22793 − 0.486532i 2.09361 − 4.16726i −4.16488 − 2.40460i −5.45556 + 6.84106i −0.123787 − 2.99745i 8.69131 − 1.31000i
5.3 −2.46622 + 0.760730i 1.56492 + 0.742310i 3.85108 − 2.62562i 3.53224 + 0.532399i −4.42414 − 0.640223i −2.41526 − 1.39445i −4.28192 + 5.36936i 1.89795 + 2.32331i −9.11631 + 1.37406i
5.4 −2.33517 + 0.720303i 1.14816 − 1.29682i 3.28168 − 2.23741i −0.573863 − 0.0864960i −1.74704 + 3.85530i −1.18297 − 0.682989i −3.00437 + 3.76736i −0.363468 − 2.97790i 1.40237 − 0.211373i
5.5 −2.30831 + 0.712021i −0.704422 − 1.58234i 3.16886 − 2.16049i −2.65620 − 0.400358i 2.75268 + 3.15097i 2.24020 + 1.29338i −2.76417 + 3.46615i −2.00758 + 2.22927i 6.41642 − 0.967119i
5.6 −2.21325 + 0.682696i 0.253183 + 1.71345i 2.77991 − 1.89531i −0.591914 − 0.0892167i −1.73012 − 3.61943i 2.48824 + 1.43659i −1.97051 + 2.47094i −2.87180 + 0.867631i 1.37096 − 0.206639i
5.7 −2.06160 + 0.635920i −1.15303 − 1.29248i 2.19333 − 1.49539i 1.74290 + 0.262699i 3.19901 + 1.93135i −4.03880 − 2.33180i −0.880526 + 1.10414i −0.341032 + 2.98055i −3.76021 + 0.566761i
5.8 −1.88357 + 0.581005i 1.43309 − 0.972759i 1.55780 − 1.06209i 2.33124 + 0.351378i −2.13415 + 2.66490i 3.73396 + 2.15580i 0.140822 − 0.176586i 1.10748 − 2.78810i −4.59521 + 0.692617i
5.9 −1.77694 + 0.548114i 1.48721 + 0.887805i 1.20461 − 0.821292i −3.23873 − 0.488160i −3.12931 − 0.762415i 0.268651 + 0.155105i 0.628462 − 0.788067i 1.42360 + 2.64071i 6.02260 − 0.907762i
5.10 −1.60671 + 0.495605i −1.72272 + 0.179502i 0.683425 − 0.465952i −2.06617 − 0.311426i 2.67896 − 1.14220i 1.78133 + 1.02845i 1.22955 − 1.54180i 2.93556 − 0.618464i 3.47409 − 0.523635i
5.11 −1.48382 + 0.457699i −0.647861 − 1.60632i 0.339763 − 0.231647i 3.32484 + 0.501140i 1.69652 + 2.08697i 2.02723 + 1.17042i 1.53820 − 1.92884i −2.16055 + 2.08135i −5.16285 + 0.778174i
5.12 −1.44241 + 0.444924i 1.70835 + 0.285566i 0.230105 − 0.156883i 0.251704 + 0.0379383i −2.59119 + 0.348183i −1.62084 − 0.935790i 1.62017 − 2.03163i 2.83690 + 0.975691i −0.379940 + 0.0572667i
5.13 −1.39163 + 0.429260i −1.10601 + 1.33294i 0.0998872 − 0.0681019i 0.749286 + 0.112937i 0.966980 − 2.32973i −1.59778 − 0.922481i 1.70624 − 2.13956i −0.553469 − 2.94850i −1.09121 + 0.164473i
5.14 −0.998917 + 0.308125i 0.629871 − 1.61346i −0.749584 + 0.511058i −0.228834 − 0.0344911i −0.132041 + 1.80579i −1.66808 − 0.963065i 1.89484 − 2.37606i −2.20652 − 2.03255i 0.239213 − 0.0360556i
5.15 −0.888590 + 0.274094i 1.02351 + 1.39730i −0.938012 + 0.639526i 3.29698 + 0.496939i −1.29247 − 0.961087i 2.61252 + 1.50834i 1.81779 − 2.27944i −0.904870 + 2.86028i −3.06587 + 0.462106i
5.16 −0.774739 + 0.238975i −0.123981 + 1.72761i −1.10937 + 0.756353i −3.02020 − 0.455222i −0.316803 − 1.36807i −0.370879 − 0.214127i 1.68972 − 2.11884i −2.96926 − 0.428382i 2.44866 − 0.369075i
5.17 −0.761869 + 0.235006i −1.42994 − 0.977386i −1.12726 + 0.768553i −3.78044 − 0.569810i 1.31912 + 0.408598i −2.87373 − 1.65915i 1.67242 − 2.09714i 1.08943 + 2.79520i 3.01411 − 0.454304i
5.18 −0.651328 + 0.200908i 1.39001 − 1.03338i −1.26861 + 0.864926i −3.55421 − 0.535711i −0.697739 + 0.952333i 4.08015 + 2.35568i 1.50247 − 1.88403i 0.864259 − 2.87281i 2.42258 − 0.365146i
5.19 −0.591238 + 0.182373i 0.524247 + 1.65081i −1.33618 + 0.910989i 1.27085 + 0.191550i −0.611017 − 0.880411i −4.12632 − 2.38233i 1.39540 − 1.74977i −2.45033 + 1.73086i −0.786309 + 0.118517i
5.20 −0.369441 + 0.113958i −1.45373 − 0.941631i −1.52898 + 1.04244i 0.554558 + 0.0835861i 0.644374 + 0.182214i 0.371061 + 0.214232i 0.928177 − 1.16390i 1.22666 + 2.73775i −0.214402 + 0.0323159i
See next 80 embeddings (of 504 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 5.42
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
387.bl even 42 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 387.2.bl.a yes 504
9.d odd 6 1 387.2.bj.a ✓ 504
43.h odd 42 1 387.2.bj.a ✓ 504
387.bl even 42 1 inner 387.2.bl.a yes 504
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
387.2.bj.a ✓ 504 9.d odd 6 1
387.2.bj.a ✓ 504 43.h odd 42 1
387.2.bl.a yes 504 1.a even 1 1 trivial
387.2.bl.a yes 504 387.bl even 42 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(387, [\chi])\).