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Properties

Label 588.2.ba.a
Level $588$
Weight $2$
Character orbit 588.ba
Analytic conductor $4.695$
Analytic rank $0$
Dimension $336$
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] = [588,2,Mod(103,588)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(588, base_ring=CyclotomicField(42))
 
chi = DirichletCharacter(H, H._module([21, 0, 29]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("588.103");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 588 = 2^{2} \cdot 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 588.ba (of order \(42\), 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: \(4.69520363885\)
Analytic rank: \(0\)
Dimension: \(336\)
Relative dimension: \(28\) over \(\Q(\zeta_{42})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{42}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 336 q - 28 q^{3} + 2 q^{7} - 6 q^{8} + 28 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 336 q - 28 q^{3} + 2 q^{7} - 6 q^{8} + 28 q^{9} + 41 q^{10} + 6 q^{11} + 52 q^{14} + 20 q^{16} - 6 q^{19} - 25 q^{20} + 4 q^{21} + 6 q^{22} + 9 q^{24} - 26 q^{25} - 18 q^{26} + 56 q^{27} + 5 q^{30} + 2 q^{31} + 15 q^{32} + 6 q^{33} + 44 q^{34} - 12 q^{35} + 16 q^{37} - 100 q^{38} + 8 q^{39} - 7 q^{40} - 7 q^{42} - 53 q^{44} - 10 q^{46} + 4 q^{47} + 8 q^{48} - 4 q^{49} - 114 q^{50} - 28 q^{52} - 4 q^{53} + q^{56} - 12 q^{57} + 27 q^{58} - 10 q^{59} - 7 q^{60} + 2 q^{61} - 16 q^{62} - 12 q^{63} - 84 q^{64} - 4 q^{65} + 21 q^{66} + 42 q^{67} + 26 q^{68} - 70 q^{70} + 28 q^{71} + 15 q^{72} + 18 q^{73} + 20 q^{74} + 54 q^{75} + 49 q^{76} - 8 q^{77} - 6 q^{78} - 6 q^{79} - 40 q^{80} + 28 q^{81} - 99 q^{82} + 10 q^{83} + 54 q^{84} + 24 q^{85} - 314 q^{86} - 40 q^{88} + 20 q^{90} - 34 q^{91} + 14 q^{92} - 2 q^{93} - 152 q^{94} - 24 q^{95} - 10 q^{96} - 156 q^{98}+O(q^{100}) \) 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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
103.1 −1.40964 − 0.113611i −0.826239 − 0.563320i 1.97419 + 0.320301i 3.32240 + 0.248980i 1.10070 + 0.887950i −2.08944 − 1.62303i −2.74651 − 0.675798i 0.365341 + 0.930874i −4.65512 − 0.728433i
103.2 −1.36468 + 0.371026i −0.826239 − 0.563320i 1.72468 − 1.01266i −0.793563 − 0.0594693i 1.33655 + 0.462193i 0.251406 + 2.63378i −1.97791 + 2.02185i 0.365341 + 0.930874i 1.10502 − 0.213276i
103.3 −1.33039 − 0.479649i −0.826239 − 0.563320i 1.53987 + 1.27624i −1.47341 − 0.110417i 0.829024 + 1.14574i 2.51658 + 0.816579i −1.43648 − 2.43650i 0.365341 + 0.930874i 1.90724 + 0.853615i
103.4 −1.24022 − 0.679596i −0.826239 − 0.563320i 1.07630 + 1.68570i −2.59884 − 0.194756i 0.641889 + 1.26015i −2.28001 + 1.34222i −0.189256 − 2.82209i 0.365341 + 0.930874i 3.09079 + 2.00770i
103.5 −1.23869 + 0.682383i −0.826239 − 0.563320i 1.06871 − 1.69052i 0.961592 + 0.0720614i 1.40785 + 0.133968i 2.50588 − 0.848875i −0.170216 + 2.82330i 0.365341 + 0.930874i −1.24029 + 0.566912i
103.6 −1.15847 + 0.811146i −0.826239 − 0.563320i 0.684086 − 1.87937i 1.33423 + 0.0999870i 1.41410 − 0.0176130i −2.38917 − 1.13661i 0.731952 + 2.73208i 0.365341 + 0.930874i −1.62677 + 0.966426i
103.7 −1.08950 − 0.901663i −0.826239 − 0.563320i 0.374008 + 1.96472i 2.88553 + 0.216241i 0.392260 + 1.35872i 0.232552 + 2.63551i 1.36403 − 2.47778i 0.365341 + 0.930874i −2.94880 − 2.83737i
103.8 −0.839211 + 1.13830i −0.826239 − 0.563320i −0.591451 − 1.91055i −4.04500 − 0.303131i 1.33462 − 0.467763i −2.59666 + 0.507311i 2.67113 + 0.930102i 0.365341 + 0.930874i 3.73966 − 4.35003i
103.9 −0.706021 − 1.22537i −0.826239 − 0.563320i −1.00307 + 1.73028i −0.481855 − 0.0361100i −0.106934 + 1.41016i −1.51904 − 2.16623i 2.82842 + 0.00752107i 0.365341 + 0.930874i 0.295951 + 0.615945i
103.10 −0.677412 − 1.24142i −0.826239 − 0.563320i −1.08223 + 1.68190i −2.31910 − 0.173792i −0.139610 + 1.40731i 2.40273 − 1.10766i 2.82105 + 0.204152i 0.365341 + 0.930874i 1.35524 + 2.99670i
103.11 −0.593055 + 1.28386i −0.826239 − 0.563320i −1.29657 − 1.52279i 4.20665 + 0.315245i 1.21323 − 0.726692i 2.58503 + 0.563570i 2.72399 − 0.761513i 0.365341 + 0.930874i −2.89950 + 5.21378i
103.12 −0.476542 + 1.33151i −0.826239 − 0.563320i −1.54582 − 1.26904i −0.786913 − 0.0589710i 1.14380 − 0.831696i 0.621496 − 2.57172i 2.42638 − 1.45351i 0.365341 + 0.930874i 0.453517 − 1.01968i
103.13 −0.194188 − 1.40082i −0.826239 − 0.563320i −1.92458 + 0.544043i 1.91992 + 0.143878i −0.628664 + 1.26680i −0.854457 + 2.50398i 1.13584 + 2.59034i 0.365341 + 0.930874i −0.171278 − 2.71740i
103.14 −0.0925486 + 1.41118i −0.826239 − 0.563320i −1.98287 − 0.261206i −2.91108 − 0.218156i 0.871414 − 1.11384i 2.63369 + 0.252375i 0.552121 − 2.77402i 0.365341 + 0.930874i 0.577274 − 4.08788i
103.15 −0.0532621 + 1.41321i −0.826239 − 0.563320i −1.99433 − 0.150541i 2.71904 + 0.203764i 0.840097 − 1.13765i −2.62441 + 0.335390i 0.318968 − 2.81038i 0.365341 + 0.930874i −0.432783 + 3.83173i
103.16 0.0278575 − 1.41394i −0.826239 − 0.563320i −1.99845 − 0.0787776i 4.26189 + 0.319384i −0.819517 + 1.15256i 0.230798 − 2.63567i −0.167058 + 2.82349i 0.365341 + 0.930874i 0.570316 − 6.01715i
103.17 0.201122 − 1.39984i −0.826239 − 0.563320i −1.91910 − 0.563077i −1.87993 − 0.140882i −0.954732 + 1.04331i 1.80315 + 1.93615i −1.17419 + 2.57318i 0.365341 + 0.930874i −0.575308 + 2.60327i
103.18 0.672250 − 1.24422i −0.826239 − 0.563320i −1.09616 − 1.67285i −0.188298 − 0.0141110i −1.25633 + 0.649330i 1.32214 − 2.29171i −2.81829 + 0.239290i 0.365341 + 0.930874i −0.144141 + 0.224798i
103.19 0.700855 + 1.22833i −0.826239 − 0.563320i −1.01760 + 1.72177i −2.57232 − 0.192768i 0.112871 − 1.40970i −0.135490 − 2.64228i −2.82810 − 0.0432451i 0.365341 + 0.930874i −1.56604 − 3.29476i
103.20 0.826670 − 1.14744i −0.826239 − 0.563320i −0.633232 − 1.89711i −1.15601 − 0.0866309i −1.32940 + 0.482379i −2.18569 + 1.49090i −2.70029 − 0.841687i 0.365341 + 0.930874i −1.05504 + 1.25483i
See next 80 embeddings (of 336 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 535.28
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
196.p even 42 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 588.2.ba.a ✓ 336
4.b odd 2 1 588.2.ba.b yes 336
49.h odd 42 1 588.2.ba.b yes 336
196.p even 42 1 inner 588.2.ba.a ✓ 336
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
588.2.ba.a ✓ 336 1.a even 1 1 trivial
588.2.ba.a ✓ 336 196.p even 42 1 inner
588.2.ba.b yes 336 4.b odd 2 1
588.2.ba.b yes 336 49.h odd 42 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{11}^{336} - 6 T_{11}^{335} + 189 T_{11}^{334} - 1230 T_{11}^{333} + 16023 T_{11}^{332} - 109080 T_{11}^{331} + 693180 T_{11}^{330} - 4574286 T_{11}^{329} + 4014444 T_{11}^{328} + 13601918 T_{11}^{327} + \cdots + 41\!\cdots\!16 \) acting on \(S_{2}^{\mathrm{new}}(588, [\chi])\). Copy content Toggle raw display