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Properties

Label 936.2.bk.c
Level $936$
Weight $2$
Character orbit 936.bk
Analytic conductor $7.474$
Analytic rank $0$
Dimension $320$
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [936,2,Mod(277,936)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(936, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 3, 4, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("936.277");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 936 = 2^{3} \cdot 3^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 936.bk (of order \(6\), 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: \(7.47399762919\)
Analytic rank: \(0\)
Dimension: \(320\)
Relative dimension: \(160\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 320 q - 3 q^{2} + q^{4} - 3 q^{6} + 10 q^{9} + 8 q^{10} - 7 q^{12} + 2 q^{14} + 24 q^{15} - 15 q^{16} + 8 q^{17} - 6 q^{18} + q^{22} + 20 q^{23} + 12 q^{24} - 144 q^{25} - 20 q^{26} - 6 q^{28} + 7 q^{30}+ \cdots - 99 q^{98}+O(q^{100}) \) Copy content Toggle raw display

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
277.1 −1.41419 + 0.00725719i −1.67323 − 0.447557i 1.99989 − 0.0205262i −0.585341 − 1.01384i 2.36952 + 0.620790i 4.60347i −2.82809 + 0.0435416i 2.59939 + 1.49773i 0.835143 + 1.42952i
277.2 −1.41214 − 0.0765910i −1.45356 − 0.941891i 1.98827 + 0.216314i −0.383122 − 0.663588i 1.98049 + 1.44141i − 3.62600i −2.79114 − 0.457749i 1.22568 + 2.73819i 0.490197 + 0.966421i
277.3 −1.41122 + 0.0919753i −0.329904 − 1.70034i 1.98308 − 0.259595i 1.44949 + 2.51060i 0.621957 + 2.36921i 1.00059i −2.77469 + 0.548739i −2.78233 + 1.12190i −2.27647 − 3.40969i
277.4 −1.41029 + 0.105221i 1.56385 − 0.744566i 1.97786 − 0.296785i −0.0509606 − 0.0882664i −2.12714 + 1.21461i − 3.21739i −2.75813 + 0.626666i 1.89124 − 2.32878i 0.0811569 + 0.119119i
277.5 −1.40701 + 0.142565i −1.35604 + 1.07757i 1.95935 − 0.401180i 0.0636918 + 0.110317i 1.75433 − 1.70948i 1.25475i −2.69963 + 0.843799i 0.677677 − 2.92246i −0.105342 − 0.146137i
277.6 −1.40548 − 0.156954i 1.70052 − 0.328998i 1.95073 + 0.441191i −0.716007 − 1.24016i −2.44168 + 0.195496i 0.965427i −2.67246 − 0.926259i 2.78352 − 1.11893i 0.811683 + 1.85540i
277.7 −1.40082 + 0.194204i −1.72956 − 0.0928202i 1.92457 − 0.544089i 2.01837 + 3.49592i 2.44082 − 0.205864i − 4.14289i −2.59030 + 1.13593i 2.98277 + 0.321077i −3.50628 − 4.50516i
277.8 −1.40017 + 0.198775i 0.954972 + 1.44500i 1.92098 − 0.556640i 1.42786 + 2.47312i −1.62436 − 1.83343i 2.15226i −2.57906 + 1.16124i −1.17606 + 2.75987i −2.49085 − 3.17898i
277.9 −1.39404 − 0.237992i −0.599465 + 1.62501i 1.88672 + 0.663544i −1.33521 − 2.31265i 1.22242 − 2.12266i 0.627165i −2.47225 − 1.37403i −2.28128 − 1.94827i 1.31095 + 3.54170i
277.10 −1.38428 − 0.289446i 1.07846 + 1.35533i 1.83244 + 0.801346i 0.255311 + 0.442211i −1.10060 − 2.18831i − 1.54657i −2.30466 − 1.63968i −0.673835 + 2.92335i −0.225425 − 0.686041i
277.11 −1.38155 − 0.302183i −1.18641 + 1.26191i 1.81737 + 0.834962i 1.67393 + 2.89933i 2.02042 − 1.38488i 0.981546i −2.25848 − 1.70272i −0.184849 − 2.99430i −1.43649 − 4.51141i
277.12 −1.38143 + 0.302754i 0.135020 + 1.72678i 1.81668 − 0.836465i −2.02734 − 3.51146i −0.709309 − 2.34454i − 2.13171i −2.25637 + 1.70552i −2.96354 + 0.466299i 3.86374 + 4.23704i
277.13 −1.36631 + 0.364950i −0.696078 − 1.58602i 1.73362 − 0.997271i 0.223235 + 0.386655i 1.52988 + 1.91297i 1.17820i −2.00472 + 1.99527i −2.03095 + 2.20799i −0.446119 − 0.446822i
277.14 −1.36380 + 0.374231i −1.67788 + 0.429783i 1.71990 − 1.02075i −1.29210 − 2.23799i 2.12746 − 1.21405i − 1.30034i −1.96361 + 2.03574i 2.63057 − 1.44225i 2.59969 + 2.56862i
277.15 −1.35348 − 0.409987i 1.70146 − 0.324069i 1.66382 + 1.10982i 1.33385 + 2.31029i −2.43576 − 0.258956i 4.70672i −1.79694 − 2.18426i 2.78996 − 1.10278i −0.858147 − 3.67379i
277.16 −1.34983 − 0.421841i 0.577388 − 1.63298i 1.64410 + 1.13883i −0.979254 − 1.69612i −1.46823 + 1.96069i 3.52394i −1.73886 − 2.23078i −2.33325 − 1.88572i 0.606338 + 2.70257i
277.17 −1.34435 − 0.439008i 0.507392 − 1.65607i 1.61454 + 1.18036i 0.331762 + 0.574628i −1.40914 + 2.00358i − 0.634652i −1.65232 − 2.29561i −2.48511 − 1.68055i −0.193737 − 0.918146i
277.18 −1.32766 + 0.487152i 1.69136 + 0.373231i 1.52537 − 1.29354i 1.73390 + 3.00321i −2.42737 + 0.328425i − 2.47808i −1.39502 + 2.46047i 2.72140 + 1.26253i −3.76505 − 3.14257i
277.19 −1.32180 + 0.502825i 0.430981 − 1.67757i 1.49433 − 1.32927i −1.84416 − 3.19418i 0.273854 + 2.43413i 2.94778i −1.30682 + 2.50843i −2.62851 − 1.44601i 4.04373 + 3.29479i
277.20 −1.32076 − 0.505570i 1.34252 + 1.09437i 1.48880 + 1.33547i −1.47285 − 2.55105i −1.21986 − 2.12413i − 0.224548i −1.29117 − 2.51652i 0.604705 + 2.93842i 0.655544 + 4.11395i
See next 80 embeddings (of 320 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 277.160
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.b even 2 1 inner
117.r even 6 1 inner
936.bk even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 936.2.bk.c ✓ 320
8.b even 2 1 inner 936.2.bk.c ✓ 320
9.c even 3 1 936.2.ca.c yes 320
13.e even 6 1 936.2.ca.c yes 320
72.n even 6 1 936.2.ca.c yes 320
104.s even 6 1 936.2.ca.c yes 320
117.r even 6 1 inner 936.2.bk.c ✓ 320
936.bk even 6 1 inner 936.2.bk.c ✓ 320
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
936.2.bk.c ✓ 320 1.a even 1 1 trivial
936.2.bk.c ✓ 320 8.b even 2 1 inner
936.2.bk.c ✓ 320 117.r even 6 1 inner
936.2.bk.c ✓ 320 936.bk even 6 1 inner
936.2.ca.c yes 320 9.c even 3 1
936.2.ca.c yes 320 13.e even 6 1
936.2.ca.c yes 320 72.n even 6 1
936.2.ca.c yes 320 104.s even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{320} + 472 T_{5}^{318} + 113938 T_{5}^{316} + 18683024 T_{5}^{314} + 2334173489 T_{5}^{312} + \cdots + 90\!\cdots\!01 \) acting on \(S_{2}^{\mathrm{new}}(936, [\chi])\). Copy content Toggle raw display