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Properties

Label 855.2.cp.a
Level $855$
Weight $2$
Character orbit 855.cp
Analytic conductor $6.827$
Analytic rank $0$
Dimension $480$
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] = [855,2,Mod(41,855)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(855, base_ring=CyclotomicField(18))
 
chi = DirichletCharacter(H, H._module([15, 0, 13]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("855.41");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 855 = 3^{2} \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 855.cp (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: \(6.82720937282\)
Analytic rank: \(0\)
Dimension: \(480\)
Relative dimension: \(80\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 480 q - 12 q^{3} - 12 q^{6} + 36 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 480 q - 12 q^{3} - 12 q^{6} + 36 q^{9} + 6 q^{13} + 6 q^{19} - 18 q^{22} + 18 q^{24} + 72 q^{26} - 18 q^{27} + 24 q^{28} - 24 q^{30} - 36 q^{34} - 36 q^{36} + 24 q^{39} - 6 q^{42} - 24 q^{43} + 198 q^{44} + 24 q^{48} - 240 q^{49} - 12 q^{51} + 48 q^{52} + 30 q^{54} - 12 q^{57} - 90 q^{59} + 42 q^{61} - 24 q^{63} - 258 q^{64} - 150 q^{66} + 120 q^{67} - 126 q^{68} - 102 q^{72} + 66 q^{73} + 252 q^{74} - 48 q^{78} - 12 q^{79} + 168 q^{81} - 18 q^{82} - 72 q^{83} + 162 q^{86} + 12 q^{87} - 108 q^{89} - 30 q^{90} - 48 q^{91} - 222 q^{96} + 108 q^{97} + 30 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
41.1 −0.470008 + 2.66555i 1.47457 − 0.908644i −5.00487 − 1.82162i −0.642788 + 0.766044i 1.72897 + 4.35762i −1.08185 + 1.87382i 4.50128 − 7.79645i 1.34873 − 2.67972i −1.73981 − 2.07343i
41.2 −0.467740 + 2.65269i −1.59309 + 0.679752i −4.93858 − 1.79749i 0.642788 − 0.766044i −1.05802 − 4.54392i 0.180721 − 0.313017i 4.38455 − 7.59427i 2.07587 − 2.16581i 1.73142 + 2.06342i
41.3 −0.459173 + 2.60410i 1.60261 + 0.656994i −4.69112 − 1.70743i 0.642788 − 0.766044i −2.44676 + 3.87169i −2.42001 + 4.19157i 3.95608 − 6.85214i 2.13672 + 2.10581i 1.69971 + 2.02563i
41.4 −0.452556 + 2.56657i 1.66849 + 0.464927i −4.50310 − 1.63899i −0.642788 + 0.766044i −1.94835 + 4.07188i 2.35016 − 4.07060i 3.63833 − 6.30178i 2.56769 + 1.55145i −1.67521 − 1.99644i
41.5 −0.448997 + 2.54639i 0.305525 + 1.70489i −4.40311 − 1.60260i 0.642788 − 0.766044i −4.47849 + 0.0124952i 1.35335 − 2.34408i 3.47215 − 6.01394i −2.81331 + 1.04177i 1.66204 + 1.98074i
41.6 −0.441407 + 2.50334i −0.276755 + 1.70980i −4.19250 − 1.52594i −0.642788 + 0.766044i −4.15805 − 1.44753i −1.24203 + 2.15126i 3.12860 − 5.41889i −2.84681 − 0.946391i −1.63394 − 1.94725i
41.7 −0.439557 + 2.49285i −1.17922 − 1.26863i −4.14173 − 1.50746i −0.642788 + 0.766044i 3.68086 − 2.38199i −0.880131 + 1.52443i 3.04711 − 5.27774i −0.218865 + 2.99201i −1.62709 − 1.93910i
41.8 −0.437801 + 2.48289i −1.15904 + 1.28710i −4.09371 − 1.48999i −0.642788 + 0.766044i −2.68830 − 3.44127i 1.56247 − 2.70627i 2.97051 − 5.14508i −0.313247 − 2.98360i −1.62059 − 1.93135i
41.9 −0.418737 + 2.37477i −1.48601 − 0.889816i −3.58483 − 1.30477i 0.642788 − 0.766044i 2.73536 − 3.15634i −1.74199 + 3.01722i 2.18822 − 3.79011i 1.41646 + 2.64455i 1.55002 + 1.84725i
41.10 −0.393532 + 2.23183i 1.72959 + 0.0923394i −2.94681 − 1.07255i 0.642788 − 0.766044i −0.886734 + 3.82381i 1.24387 − 2.15444i 1.28716 − 2.22942i 2.98295 + 0.319418i 1.45672 + 1.73606i
41.11 −0.376605 + 2.13584i 0.717034 − 1.57666i −2.54058 − 0.924694i 0.642788 − 0.766044i 3.09745 + 2.12525i −1.54923 + 2.68334i 0.763006 − 1.32157i −1.97172 − 2.26104i 1.39407 + 1.66139i
41.12 −0.368618 + 2.09054i −1.19519 − 1.25360i −2.35509 − 0.857181i −0.642788 + 0.766044i 3.06128 − 2.03649i 0.442364 − 0.766197i 0.537311 − 0.930649i −0.143045 + 2.99659i −1.36450 − 1.62615i
41.13 −0.367175 + 2.08235i 0.561751 − 1.63842i −2.32198 − 0.845131i −0.642788 + 0.766044i 3.20551 + 1.77135i 1.95850 − 3.39222i 0.497957 − 0.862488i −2.36887 − 1.84077i −1.35916 − 1.61978i
41.14 −0.356880 + 2.02397i 0.453283 + 1.67169i −2.08969 − 0.760585i 0.642788 − 0.766044i −3.54520 + 0.320838i 0.374408 − 0.648493i 0.229978 − 0.398334i −2.58907 + 1.51549i 1.32105 + 1.57437i
41.15 −0.350947 + 1.99032i −0.748360 − 1.56204i −1.95883 − 0.712955i 0.642788 − 0.766044i 3.37159 − 0.941285i 2.12995 − 3.68919i 0.0854303 − 0.147970i −1.87991 + 2.33793i 1.29909 + 1.54819i
41.16 −0.319561 + 1.81232i −1.38670 + 1.03781i −1.30300 − 0.474254i 0.642788 − 0.766044i −1.43771 − 2.84479i −0.422050 + 0.731012i −0.564390 + 0.977552i 0.845888 − 2.87828i 1.18291 + 1.40974i
41.17 −0.318914 + 1.80865i 1.36123 + 1.07100i −1.29013 − 0.469570i −0.642788 + 0.766044i −2.37118 + 2.12044i −1.21176 + 2.09884i −0.575823 + 0.997355i 0.705918 + 2.91576i −1.18051 − 1.40688i
41.18 −0.306018 + 1.73551i 1.10104 − 1.33706i −1.03898 − 0.378156i 0.642788 − 0.766044i 1.98354 + 2.32003i 0.828800 − 1.43552i −0.788047 + 1.36494i −0.575434 − 2.94430i 1.13278 + 1.34999i
41.19 −0.305514 + 1.73265i −1.71982 − 0.205446i −1.02937 − 0.374659i −0.642788 + 0.766044i 0.881396 − 2.91709i 1.40480 − 2.43319i −0.795744 + 1.37827i 2.91558 + 0.706660i −1.13091 − 1.34777i
41.20 −0.269238 + 1.52692i −0.164163 − 1.72425i −0.379624 − 0.138172i −0.642788 + 0.766044i 2.67700 + 0.213570i −2.20053 + 3.81144i −1.23729 + 2.14305i −2.94610 + 0.566117i −0.996629 − 1.18774i
See next 80 embeddings (of 480 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 41.80
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
171.x even 18 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 855.2.cp.a ✓ 480
9.d odd 6 1 855.2.dd.a yes 480
19.f odd 18 1 855.2.dd.a yes 480
171.x even 18 1 inner 855.2.cp.a ✓ 480
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
855.2.cp.a ✓ 480 1.a even 1 1 trivial
855.2.cp.a ✓ 480 171.x even 18 1 inner
855.2.dd.a yes 480 9.d odd 6 1
855.2.dd.a yes 480 19.f odd 18 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(855, [\chi])\).