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Properties

Label 213.3.h.a
Level $213$
Weight $3$
Character orbit 213.h
Analytic conductor $5.804$
Analytic rank $0$
Dimension $184$
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [213,3,Mod(5,213)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(213, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([5, 4]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("213.5");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 213 = 3 \cdot 71 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 213.h (of order \(10\), degree \(4\), 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: \(5.80382963087\)
Analytic rank: \(0\)
Dimension: \(184\)
Relative dimension: \(46\) over \(\Q(\zeta_{10})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 184 q - q^{3} + 82 q^{4} - 20 q^{6} - 6 q^{7} - 19 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 184 q - q^{3} + 82 q^{4} - 20 q^{6} - 6 q^{7} - 19 q^{9} - 30 q^{10} + 86 q^{12} - 10 q^{13} + 85 q^{15} - 118 q^{16} - 76 q^{18} + 14 q^{19} + 45 q^{21} - 182 q^{22} - 161 q^{24} + 112 q^{25} - 121 q^{27} + 210 q^{28} - 260 q^{30} - 100 q^{31} - 52 q^{33} - 96 q^{34} - 32 q^{36} - 48 q^{37} + 102 q^{39} + 82 q^{40} - 115 q^{42} - 38 q^{43} - 434 q^{45} - 44 q^{46} + 708 q^{48} - 188 q^{49} - 98 q^{51} + 264 q^{52} - 170 q^{54} + 138 q^{55} - 260 q^{57} + 822 q^{58} - 568 q^{60} - 18 q^{61} + 433 q^{63} - 134 q^{64} - 240 q^{66} + 180 q^{67} + 571 q^{69} - 664 q^{70} + 506 q^{72} + 20 q^{73} - 405 q^{75} + 950 q^{76} + 146 q^{78} - 530 q^{79} + 373 q^{81} - 178 q^{82} + 195 q^{84} - 28 q^{85} + 207 q^{87} - 738 q^{88} + 1005 q^{90} + 760 q^{91} + 300 q^{93} - 1436 q^{94} + 783 q^{96} + 1004 q^{97} - 108 q^{99}+O(q^{100}) \) 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 −3.75237 + 1.21922i 0.592507 + 2.94091i 9.35772 − 6.79878i 0.480496 − 0.661346i −5.80892 − 10.3130i −2.29840 − 7.07374i −17.5481 + 24.1529i −8.29787 + 3.48502i −0.996673 + 3.06745i
5.2 −3.45623 + 1.12300i 2.99828 − 0.101681i 7.44833 − 5.41153i 5.10523 − 7.02675i −10.2485 + 3.71849i 4.15499 + 12.7878i −11.1217 + 15.3078i 8.97932 − 0.609738i −9.75383 + 30.0192i
5.3 −3.44584 + 1.11962i −0.256225 − 2.98904i 7.38418 − 5.36492i −1.17857 + 1.62217i 4.22950 + 10.0129i 0.748711 + 2.30429i −10.9194 + 15.0293i −8.86870 + 1.53173i 2.24496 − 6.90929i
5.4 −3.25836 + 1.05871i −2.99235 − 0.214165i 6.26001 − 4.54817i 2.93589 − 4.04090i 9.97689 − 2.47019i −0.523913 − 1.61244i −7.52711 + 10.3602i 8.90827 + 1.28171i −5.28806 + 16.2750i
5.5 −3.20118 + 1.04013i 2.81808 − 1.02879i 5.92965 − 4.30814i −2.27504 + 3.13133i −7.95114 + 6.22450i −2.25974 − 6.95476i −6.58712 + 9.06639i 6.88320 − 5.79841i 4.02585 − 12.3903i
5.6 −2.90313 + 0.943285i 1.97526 + 2.25795i 4.30232 − 3.12582i −4.92215 + 6.77476i −7.86434 − 4.69189i 1.93069 + 5.94207i −2.36474 + 3.25478i −1.19667 + 8.92009i 7.89913 − 24.3110i
5.7 −2.89274 + 0.939909i −1.71108 + 2.46418i 4.24846 − 3.08669i 0.133971 − 0.184395i 2.63361 − 8.73651i 1.49649 + 4.60571i −2.23724 + 3.07930i −3.14440 − 8.43284i −0.214229 + 0.659328i
5.8 −2.75193 + 0.894155i −2.81663 − 1.03276i 3.53752 − 2.57016i −4.43293 + 6.10140i 8.67461 + 0.323570i 2.77579 + 8.54300i −0.633737 + 0.872265i 6.86682 + 5.81780i 6.74349 − 20.7543i
5.9 −2.50963 + 0.815427i 1.02879 − 2.81808i 2.39723 − 1.74169i 5.67926 − 7.81683i −0.283943 + 7.91124i −3.02322 − 9.30451i 1.60820 − 2.21350i −6.88317 − 5.79844i −7.87876 + 24.2483i
5.10 −2.42666 + 0.788470i 0.606287 + 2.93810i 2.03093 − 1.47555i 3.15484 − 4.34227i −3.78785 − 6.65172i −0.103139 − 0.317430i 2.23409 − 3.07497i −8.26483 + 3.56266i −4.23198 + 13.0247i
5.11 −2.19634 + 0.713635i −1.97615 − 2.25718i 1.07858 − 0.783636i −0.887628 + 1.22171i 5.95110 + 3.54729i −2.83137 − 8.71405i 3.61995 − 4.98244i −1.18970 + 8.92102i 1.07768 − 3.31675i
5.12 −2.19245 + 0.712372i −2.08599 + 2.15607i 1.06331 − 0.772543i −3.25315 + 4.47758i 3.03752 − 6.21309i −3.34266 − 10.2877i 3.63911 − 5.00881i −0.297277 − 8.99509i 3.94268 − 12.1343i
5.13 −1.87013 + 0.607643i 2.97081 + 0.417484i −0.107902 + 0.0783957i −0.248441 + 0.341949i −5.80949 + 1.02444i 0.343137 + 1.05607i 4.77738 − 6.57550i 8.65141 + 2.48053i 0.256834 − 0.790453i
5.14 −1.85431 + 0.602501i 2.13843 + 2.10407i −0.160623 + 0.116700i 2.44165 − 3.36065i −5.23301 − 2.61319i −1.05309 − 3.24109i 4.81163 − 6.62264i 0.145774 + 8.99882i −2.50278 + 7.70277i
5.15 −1.82470 + 0.592882i 1.92918 − 2.29745i −0.258034 + 0.187472i −1.00000 + 1.37638i −2.15807 + 5.33593i 3.01835 + 9.28951i 4.87060 − 6.70381i −1.55651 − 8.86438i 1.00867 − 3.10437i
5.16 −1.52912 + 0.496842i −1.22500 − 2.73850i −1.14471 + 0.831679i 2.87036 − 3.95071i 3.23377 + 3.57887i 2.95618 + 9.09820i 5.11738 − 7.04347i −5.99876 + 6.70931i −2.42625 + 7.46722i
5.17 −1.15943 + 0.376721i −2.66982 + 1.36824i −2.03371 + 1.47758i 3.78040 − 5.20327i 2.58002 − 2.59215i 1.54062 + 4.74154i 4.66758 − 6.42437i 5.25584 − 7.30590i −2.42292 + 7.45698i
5.18 −1.14926 + 0.373418i 1.04084 − 2.81365i −2.05470 + 1.49283i −5.77472 + 7.94822i −0.145531 + 3.62230i −1.13560 − 3.49502i 4.64508 − 6.39340i −6.83330 − 5.85714i 3.66866 − 11.2910i
5.19 −0.717873 + 0.233251i −2.99736 − 0.125781i −2.77513 + 2.01625i −3.32573 + 4.57748i 2.18106 − 0.608843i 1.57065 + 4.83396i 3.29658 − 4.53735i 8.96836 + 0.754021i 1.31975 − 4.06178i
5.20 −0.696485 + 0.226302i 2.78411 − 1.11747i −2.80219 + 2.03591i 1.85602 − 2.55460i −1.68620 + 1.40835i −1.76955 − 5.44610i 3.21276 − 4.42198i 6.50253 − 6.22231i −0.714582 + 2.19926i
See next 80 embeddings (of 184 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 5.46
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
71.c even 5 1 inner
213.h odd 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 213.3.h.a ✓ 184
3.b odd 2 1 inner 213.3.h.a ✓ 184
71.c even 5 1 inner 213.3.h.a ✓ 184
213.h odd 10 1 inner 213.3.h.a ✓ 184
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
213.3.h.a ✓ 184 1.a even 1 1 trivial
213.3.h.a ✓ 184 3.b odd 2 1 inner
213.3.h.a ✓ 184 71.c even 5 1 inner
213.3.h.a ✓ 184 213.h odd 10 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(213, [\chi])\).