Properties

Label 270.3.q.a
Level $270$
Weight $3$
Character orbit 270.q
Analytic conductor $7.357$
Analytic rank $0$
Dimension $216$
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [270,3,Mod(7,270)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(270, base_ring=CyclotomicField(36))
 
chi = DirichletCharacter(H, H._module([32, 9]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("270.7");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 270 = 2 \cdot 3^{3} \cdot 5 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 270.q (of order \(36\), 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: \(7.35696713773\)
Analytic rank: \(0\)
Dimension: \(216\)
Relative dimension: \(18\) over \(\Q(\zeta_{36})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{36}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 216 q - 36 q^{6} + 18 q^{7} + 216 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 216 q - 36 q^{6} + 18 q^{7} + 216 q^{8} + 36 q^{11} - 18 q^{15} + 72 q^{20} + 288 q^{21} + 36 q^{22} + 108 q^{23} + 54 q^{25} - 162 q^{27} - 120 q^{30} - 432 q^{31} + 114 q^{33} - 162 q^{35} + 48 q^{36} + 108 q^{38} - 36 q^{40} + 216 q^{41} - 48 q^{42} - 216 q^{43} + 18 q^{45} + 108 q^{46} - 144 q^{47} - 72 q^{48} - 162 q^{50} + 60 q^{51} - 540 q^{53} + 324 q^{55} - 72 q^{56} - 234 q^{57} + 72 q^{58} - 96 q^{60} + 504 q^{61} - 24 q^{63} - 144 q^{66} + 72 q^{67} + 108 q^{68} + 72 q^{70} + 240 q^{72} - 216 q^{73} - 84 q^{75} + 216 q^{76} + 576 q^{77} + 168 q^{78} - 1068 q^{81} + 576 q^{83} + 576 q^{85} - 432 q^{86} - 324 q^{87} - 72 q^{88} - 606 q^{90} + 756 q^{91} - 324 q^{92} + 168 q^{93} - 468 q^{95} - 1332 q^{97} + 756 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
7.1 −1.28171 0.597672i −2.85560 0.919547i 1.28558 + 1.53209i 3.77889 3.27414i 3.11047 + 2.88531i −0.353774 4.04366i −0.732051 2.73205i 7.30887 + 5.25171i −6.80031 + 1.93797i
7.2 −1.28171 0.597672i −2.66066 1.38596i 1.28558 + 1.53209i 0.839182 + 4.92907i 2.58186 + 3.36660i −0.976758 11.1644i −0.732051 2.73205i 5.15826 + 7.37512i 1.87038 6.81921i
7.3 −1.28171 0.597672i −2.55210 + 1.57695i 1.28558 + 1.53209i −0.574327 4.96691i 4.21356 0.495878i 0.586104 + 6.69920i −0.732051 2.73205i 4.02645 8.04908i −2.23246 + 6.70941i
7.4 −1.28171 0.597672i −2.47477 1.69574i 1.28558 + 1.53209i −4.80586 1.37974i 2.15845 + 3.65255i 0.424050 + 4.84691i −0.732051 2.73205i 3.24895 + 8.39311i 5.33510 + 4.64077i
7.5 −1.28171 0.597672i −2.23717 + 1.99877i 1.28558 + 1.53209i 4.66992 + 1.78658i 4.06202 1.22475i −0.252031 2.88073i −0.732051 2.73205i 1.00984 8.94317i −4.91770 5.08096i
7.6 −1.28171 0.597672i −1.08007 2.79883i 1.28558 + 1.53209i 4.94662 0.728658i −0.288450 + 4.23282i 0.875560 + 10.0077i −0.732051 2.73205i −6.66692 + 6.04585i −6.77565 2.02253i
7.7 −1.28171 0.597672i −0.873935 + 2.86988i 1.28558 + 1.53209i −4.54722 + 2.07912i 2.83538 3.15604i −0.646339 7.38769i −0.732051 2.73205i −7.47247 5.01619i 7.07087 + 0.0529107i
7.8 −1.28171 0.597672i −0.159701 + 2.99575i 1.28558 + 1.53209i 2.13247 4.52245i 1.99517 3.74424i −0.767683 8.77465i −0.732051 2.73205i −8.94899 0.956847i −5.43616 + 4.52197i
7.9 −1.28171 0.597672i −0.0553396 2.99949i 1.28558 + 1.53209i −1.24810 + 4.84172i −1.72178 + 3.87756i 0.433256 + 4.95214i −0.732051 2.73205i −8.99388 + 0.331981i 4.49347 5.45974i
7.10 −1.28171 0.597672i 0.0131460 + 2.99997i 1.28558 + 1.53209i 3.69819 + 3.36503i 1.77615 3.85296i 0.935503 + 10.6928i −0.732051 2.73205i −8.99965 + 0.0788753i −2.72883 6.52330i
7.11 −1.28171 0.597672i 0.900393 2.86169i 1.28558 + 1.53209i −4.99985 0.0385524i −2.86440 + 3.12973i −0.623797 7.13003i −0.732051 2.73205i −7.37859 5.15330i 6.38533 + 3.03769i
7.12 −1.28171 0.597672i 1.60602 2.53391i 1.28558 + 1.53209i 4.97563 + 0.493034i −3.57291 + 2.28786i −0.621128 7.09952i −0.732051 2.73205i −3.84137 8.13903i −6.08266 3.60573i
7.13 −1.28171 0.597672i 1.96932 + 2.26313i 1.28558 + 1.53209i −3.91819 + 3.10609i −1.17150 4.07769i 0.681389 + 7.78831i −0.732051 2.73205i −1.24352 + 8.91368i 6.87842 1.63932i
7.14 −1.28171 0.597672i 2.01779 + 2.22003i 1.28558 + 1.53209i −3.55147 3.51952i −1.25938 4.05142i 0.0312991 + 0.357750i −0.732051 2.73205i −0.857053 + 8.95910i 2.44845 + 6.63363i
7.15 −1.28171 0.597672i 2.40534 1.79286i 1.28558 + 1.53209i 0.972621 4.90449i −4.15450 + 0.860332i 0.838784 + 9.58735i −0.732051 2.73205i 2.57129 8.62488i −4.17790 + 5.70484i
7.16 −1.28171 0.597672i 2.61978 + 1.46176i 1.28558 + 1.53209i 4.92278 + 0.875331i −2.48416 3.43933i −0.519758 5.94086i −0.732051 2.73205i 4.72653 + 7.65897i −5.78643 4.06414i
7.17 −1.28171 0.597672i 2.87607 0.853371i 1.28558 + 1.53209i 0.327684 + 4.98925i −4.19633 0.625169i 0.259948 + 2.97122i −0.732051 2.73205i 7.54351 4.90871i 2.56194 6.59063i
7.18 −1.28171 0.597672i 2.98164 0.331356i 1.28558 + 1.53209i −3.09252 3.92890i −4.01965 1.35734i −0.433045 4.94972i −0.732051 2.73205i 8.78041 1.97597i 1.61553 + 6.88404i
13.1 1.15846 + 0.811160i −2.99878 + 0.0854182i 0.684040 + 1.87939i −4.76648 1.51018i −3.54325 2.33354i 9.93537 4.63294i −0.732051 + 2.73205i 8.98541 0.512302i −4.29676 5.61585i
13.2 1.15846 + 0.811160i −2.96929 + 0.428152i 0.684040 + 1.87939i 0.872329 4.92332i −3.78709 1.91257i −8.20650 + 3.82675i −0.732051 + 2.73205i 8.63337 2.54261i 5.00415 4.99585i
See next 80 embeddings (of 216 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 7.18
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner
27.e even 9 1 inner
135.r odd 36 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 270.3.q.a 216
5.c odd 4 1 inner 270.3.q.a 216
27.e even 9 1 inner 270.3.q.a 216
135.r odd 36 1 inner 270.3.q.a 216
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
270.3.q.a 216 1.a even 1 1 trivial
270.3.q.a 216 5.c odd 4 1 inner
270.3.q.a 216 27.e even 9 1 inner
270.3.q.a 216 135.r odd 36 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{216} - 18 T_{7}^{215} + 162 T_{7}^{214} - 738 T_{7}^{213} + 37107 T_{7}^{212} + \cdots + 29\!\cdots\!61 \) acting on \(S_{3}^{\mathrm{new}}(270, [\chi])\). Copy content Toggle raw display