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Properties

Label 855.2.dh.a
Level $855$
Weight $2$
Character orbit 855.dh
Analytic conductor $6.827$
Analytic rank $0$
Dimension $1392$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [855,2,Mod(47,855)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(855, base_ring=CyclotomicField(36))
 
chi = DirichletCharacter(H, H._module([6, 9, 16]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("855.47");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 855 = 3^{2} \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 855.dh (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: \(6.82720937282\)
Analytic rank: \(0\)
Dimension: \(1392\)
Relative dimension: \(116\) over \(\Q(\zeta_{36})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{36}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 1392 q - 18 q^{2} - 12 q^{3} - 18 q^{5} - 24 q^{6} + 6 q^{7}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 1392 q - 18 q^{2} - 12 q^{3} - 18 q^{5} - 24 q^{6} + 6 q^{7} - 24 q^{10} - 6 q^{12} - 6 q^{13} - 48 q^{15} - 12 q^{16} - 54 q^{17} - 48 q^{18} - 36 q^{20} - 24 q^{21} + 42 q^{22} - 90 q^{23} - 6 q^{25} + 144 q^{26} - 6 q^{27} - 6 q^{30} - 24 q^{31} - 18 q^{32} + 60 q^{33} - 24 q^{36} - 48 q^{37} - 18 q^{38} - 6 q^{40} - 36 q^{41} + 24 q^{42} - 6 q^{43} - 42 q^{45} - 24 q^{46} - 18 q^{47} + 42 q^{48} - 72 q^{51} - 30 q^{52} + 6 q^{55} - 72 q^{56} + 144 q^{57} - 12 q^{58} - 84 q^{60} - 12 q^{61} - 36 q^{62} + 18 q^{63} + 144 q^{66} - 6 q^{67} - 18 q^{68} - 6 q^{70} - 156 q^{72} - 24 q^{73} - 192 q^{75} - 12 q^{76} - 36 q^{77} + 30 q^{78} - 252 q^{80} - 24 q^{82} - 18 q^{83} - 6 q^{85} + 288 q^{86} + 42 q^{87} + 102 q^{88} - 24 q^{90} - 48 q^{91} + 18 q^{92} - 60 q^{93} - 18 q^{95} + 96 q^{96} - 6 q^{97} - 378 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
47.1 −2.53911 + 1.18401i −0.450623 + 1.67241i 3.75965 − 4.48058i −2.17879 + 0.502877i −0.835958 − 4.77997i 0.0501089 − 0.187009i −2.79093 + 10.4159i −2.59388 − 1.50725i 4.93678 − 3.85656i
47.2 −2.47992 + 1.15641i 1.44445 − 0.955803i 3.52715 − 4.20350i 1.92369 − 1.13992i −2.47683 + 4.04069i −0.899295 + 3.35622i −2.46970 + 9.21705i 1.17288 − 2.76122i −3.45239 + 5.05147i
47.3 −2.42264 + 1.12969i −1.56162 − 0.749237i 3.30739 − 3.94159i 0.619360 + 2.14858i 4.62964 + 0.0509808i 0.0893180 − 0.333339i −2.17612 + 8.12139i 1.87729 + 2.34004i −3.92772 − 4.50554i
47.4 −2.40044 + 1.11934i 1.37380 + 1.05483i 3.22359 − 3.84173i 0.566159 − 2.16321i −4.47844 − 0.994312i 0.246549 − 0.920132i −2.06681 + 7.71344i 0.774649 + 2.89826i 1.06234 + 5.82636i
47.5 −2.35464 + 1.09799i 1.69332 + 0.364245i 3.05319 − 3.63865i −0.957606 + 2.02064i −4.38710 + 1.00157i 0.0486856 − 0.181697i −1.84912 + 6.90101i 2.73465 + 1.23357i 0.0361794 − 5.80933i
47.6 −2.34913 + 1.09542i −1.57331 + 0.724355i 3.03289 − 3.61446i 1.69480 − 1.45865i 2.90244 − 3.42504i 0.806715 − 3.01070i −1.82361 + 6.80580i 1.95062 − 2.27927i −2.38348 + 5.28306i
47.7 −2.31350 + 1.07880i 1.11066 − 1.32907i 2.90291 − 3.45955i −1.85804 − 1.24407i −1.13571 + 4.27300i 0.504213 − 1.88175i −1.66235 + 6.20397i −0.532864 − 2.95230i 5.64068 + 0.873698i
47.8 −2.31349 + 1.07880i −0.418008 − 1.68085i 2.90287 − 3.45950i −1.75887 + 1.38072i 2.78036 + 3.43770i −1.06394 + 3.97069i −1.66230 + 6.20379i −2.65054 + 1.40522i 2.57962 − 5.09174i
47.9 −2.26383 + 1.05564i −1.72062 − 0.198661i 2.72498 − 3.24751i −1.16313 − 1.90975i 4.10491 − 1.36663i −0.872492 + 3.25619i −1.44771 + 5.40291i 2.92107 + 0.683639i 4.64914 + 3.09550i
47.10 −2.21368 + 1.03226i 0.0641665 + 1.73086i 2.54925 − 3.03808i 1.59161 + 1.57060i −1.92874 − 3.76534i 0.982655 − 3.66732i −1.24281 + 4.63822i −2.99177 + 0.222127i −5.14457 − 1.83385i
47.11 −2.15411 + 1.00448i 0.736846 + 1.56750i 2.34564 − 2.79543i 2.08760 + 0.801211i −3.16177 − 2.63642i −0.749887 + 2.79862i −1.01451 + 3.78619i −1.91411 + 2.31001i −5.30172 + 0.371049i
47.12 −2.12532 + 0.991055i −1.21817 − 1.23129i 2.24924 − 2.68054i −2.22562 + 0.215866i 3.80928 + 1.40961i 1.20348 − 4.49145i −0.909921 + 3.39587i −0.0321258 + 2.99983i 4.51624 − 2.66450i
47.13 −2.12222 + 0.989606i −0.250295 − 1.71387i 2.23891 − 2.66823i −0.424603 − 2.19538i 2.22724 + 3.38951i −0.273687 + 1.02141i −0.898853 + 3.35457i −2.87471 + 0.857946i 3.07367 + 4.23889i
47.14 −2.09659 + 0.977654i −0.578100 + 1.63273i 2.15429 − 2.56738i 1.27022 − 1.84025i −0.384206 − 3.98833i −0.921281 + 3.43827i −0.809173 + 3.01988i −2.33160 − 1.88776i −0.864004 + 5.10009i
47.15 −1.93166 + 0.900749i 1.71694 + 0.228327i 1.63439 − 1.94780i −1.90602 − 1.16922i −3.52220 + 1.10548i 0.451461 − 1.68488i −0.299354 + 1.11720i 2.89573 + 0.784046i 4.73497 + 0.541698i
47.16 −1.92613 + 0.898170i 1.52315 − 0.824630i 1.61770 − 1.92790i 1.55500 + 1.60685i −2.19313 + 2.95639i −0.428925 + 1.60077i −0.284211 + 1.06069i 1.63997 − 2.51207i −4.43837 − 1.69835i
47.17 −1.91070 + 0.890973i −0.0848870 − 1.72997i 1.57136 − 1.87267i 1.91529 − 1.15397i 1.70355 + 3.22982i 0.764645 − 2.85370i −0.242593 + 0.905371i −2.98559 + 0.293704i −2.63139 + 3.91136i
47.18 −1.90817 + 0.889793i −1.66276 + 0.484992i 1.56380 − 1.86366i −0.784840 + 2.09381i 2.74129 − 2.40496i 0.551514 − 2.05828i −0.235865 + 0.880260i 2.52957 − 1.61285i −0.365449 − 4.69368i
47.19 −1.87782 + 0.875642i 0.0262490 − 1.73185i 1.47388 − 1.75651i 1.28069 + 1.83299i 1.46719 + 3.27509i −0.285316 + 1.06481i −0.157100 + 0.586306i −2.99862 − 0.0909186i −4.00994 − 2.32059i
47.20 −1.85824 + 0.866511i −1.55461 − 0.763662i 1.41664 − 1.68828i 2.23407 − 0.0944452i 3.55056 + 0.0719772i −0.180650 + 0.674194i −0.108200 + 0.403809i 1.83364 + 2.37440i −4.06960 + 2.11135i
See next 80 embeddings (of 1392 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 47.116
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner
171.z odd 18 1 inner
855.dh even 36 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 855.2.dh.a ✓ 1392
5.c odd 4 1 inner 855.2.dh.a ✓ 1392
9.d odd 6 1 855.2.dm.a yes 1392
19.e even 9 1 855.2.dm.a yes 1392
45.l even 12 1 855.2.dm.a yes 1392
95.q odd 36 1 855.2.dm.a yes 1392
171.z odd 18 1 inner 855.2.dh.a ✓ 1392
855.dh even 36 1 inner 855.2.dh.a ✓ 1392
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
855.2.dh.a ✓ 1392 1.a even 1 1 trivial
855.2.dh.a ✓ 1392 5.c odd 4 1 inner
855.2.dh.a ✓ 1392 171.z odd 18 1 inner
855.2.dh.a ✓ 1392 855.dh even 36 1 inner
855.2.dm.a yes 1392 9.d odd 6 1
855.2.dm.a yes 1392 19.e even 9 1
855.2.dm.a yes 1392 45.l even 12 1
855.2.dm.a yes 1392 95.q odd 36 1

Hecke kernels

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