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Properties

Label 588.2.x.b
Level $588$
Weight $2$
Character orbit 588.x
Analytic conductor $4.695$
Analytic rank $0$
Dimension $168$
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(55,588)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(588, base_ring=CyclotomicField(14))
 
chi = DirichletCharacter(H, H._module([7, 0, 9]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("588.55");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 588 = 2^{2} \cdot 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 588.x (of order \(14\), 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: \(4.69520363885\)
Analytic rank: \(0\)
Dimension: \(168\)
Relative dimension: \(28\) over \(\Q(\zeta_{14})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{14}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 168 q + 28 q^{3} - 2 q^{7} + 6 q^{8} - 28 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 168 q + 28 q^{3} - 2 q^{7} + 6 q^{8} - 28 q^{9} - 20 q^{10} + 14 q^{14} - 20 q^{16} - 12 q^{19} + 25 q^{20} + 2 q^{21} - 6 q^{22} - 27 q^{24} + 32 q^{25} - 6 q^{26} + 28 q^{27} + 6 q^{28} - 8 q^{30} + 4 q^{31} - 45 q^{32} - 44 q^{34} + 12 q^{35} - 10 q^{37} - 35 q^{38} - 14 q^{39} + 40 q^{40} + 7 q^{42} + 20 q^{44} + 28 q^{46} + 8 q^{47} - 8 q^{48} - 8 q^{49} + 114 q^{50} - 20 q^{52} - 8 q^{53} + 23 q^{56} + 12 q^{57} - 6 q^{58} - 20 q^{59} + 10 q^{60} - 14 q^{61} + 16 q^{62} + 12 q^{63} - 42 q^{64} - 8 q^{65} + 6 q^{66} + 16 q^{68} + 19 q^{70} - 28 q^{71} - 15 q^{72} + 22 q^{74} - 18 q^{75} - 49 q^{76} + 8 q^{77} + 6 q^{78} - 26 q^{80} - 28 q^{81} - 12 q^{82} - 10 q^{83} - 27 q^{84} - 24 q^{85} - 34 q^{86} + 94 q^{88} - 20 q^{90} + 16 q^{91} + 7 q^{92} - 4 q^{93} + 11 q^{94} + 10 q^{96} - 150 q^{98}+O(q^{100}) \) 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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
55.1 −1.41421 + 0.00366928i 0.900969 + 0.433884i 1.99997 − 0.0103783i 0.477768 − 0.992096i −1.27575 − 0.610296i 2.27188 + 1.35593i −2.82834 + 0.0220155i 0.623490 + 0.781831i −0.672024 + 1.40478i
55.2 −1.40198 + 0.185596i 0.900969 + 0.433884i 1.93111 − 0.520405i −1.40445 + 2.91638i −1.34367 − 0.441081i 0.602515 − 2.57623i −2.61079 + 1.08801i 0.623490 + 0.781831i 1.42775 − 4.34937i
55.3 −1.37146 + 0.345091i 0.900969 + 0.433884i 1.76182 − 0.946559i 1.63442 − 3.39390i −1.38538 − 0.284140i −0.364664 − 2.62050i −2.08963 + 1.90616i 0.623490 + 0.781831i −1.07034 + 5.21863i
55.4 −1.36844 − 0.356880i 0.900969 + 0.433884i 1.74527 + 0.976739i −0.0842420 + 0.174930i −1.07808 − 0.915283i −2.61856 − 0.378316i −2.03973 − 1.95946i 0.623490 + 0.781831i 0.177709 − 0.209318i
55.5 −1.25082 + 0.659893i 0.900969 + 0.433884i 1.12908 − 1.65081i −0.617993 + 1.28328i −1.41326 + 0.0518343i −2.62009 + 0.367586i −0.322916 + 2.80993i 0.623490 + 0.781831i −0.0738292 − 2.01295i
55.6 −1.14486 − 0.830241i 0.900969 + 0.433884i 0.621399 + 1.90102i −0.591080 + 1.22739i −0.671253 − 1.24476i 2.62768 − 0.308728i 0.866889 − 2.69230i 0.623490 + 0.781831i 1.69573 − 0.914448i
55.7 −1.08805 − 0.903405i 0.900969 + 0.433884i 0.367718 + 1.96591i 1.27172 − 2.64076i −0.588329 − 1.28603i −0.874558 − 2.49703i 1.37591 − 2.47121i 0.623490 + 0.781831i −3.76938 + 1.72441i
55.8 −0.974763 + 1.02462i 0.900969 + 0.433884i −0.0996732 − 1.99751i 1.10901 − 2.30289i −1.32280 + 0.500213i 0.953739 + 2.46787i 2.14384 + 1.84498i 0.623490 + 0.781831i 1.27855 + 3.38109i
55.9 −0.778541 + 1.18062i 0.900969 + 0.433884i −0.787748 − 1.83833i −1.73990 + 3.61294i −1.21369 + 0.725909i 2.50782 + 0.843102i 2.78367 + 0.501181i 0.623490 + 0.781831i −2.91094 − 4.86699i
55.10 −0.611563 − 1.27514i 0.900969 + 0.433884i −1.25198 + 1.55966i −0.776660 + 1.61275i 0.00226428 − 1.41421i −0.448708 + 2.60742i 2.75446 + 0.642622i 0.623490 + 0.781831i 2.53146 + 0.00405309i
55.11 −0.446111 + 1.34201i 0.900969 + 0.433884i −1.60197 − 1.19737i −0.0202251 + 0.0419978i −0.984208 + 1.01555i −2.50508 + 0.851222i 2.32154 − 1.61569i 0.623490 + 0.781831i −0.0473388 − 0.0458779i
55.12 −0.315691 − 1.37853i 0.900969 + 0.433884i −1.80068 + 0.870378i 0.236353 − 0.490793i 0.313693 − 1.37898i −2.32580 + 1.26121i 1.76830 + 2.20751i 0.623490 + 0.781831i −0.751186 − 0.170881i
55.13 −0.302651 + 1.38145i 0.900969 + 0.433884i −1.81681 − 0.836193i 0.0432081 − 0.0897226i −0.872067 + 1.11333i −0.103018 − 2.64374i 1.70502 − 2.25675i 0.623490 + 0.781831i 0.110870 + 0.0868444i
55.14 −0.225403 − 1.39614i 0.900969 + 0.433884i −1.89839 + 0.629386i 1.60689 − 3.33674i 0.402679 − 1.35567i 2.64502 + 0.0624089i 1.30661 + 2.50854i 0.623490 + 0.781831i −5.02073 − 1.49132i
55.15 0.148148 − 1.40643i 0.900969 + 0.433884i −1.95610 − 0.416719i −1.14386 + 2.37526i 0.743705 − 1.20287i 1.90517 − 1.83585i −0.875880 + 2.68939i 0.623490 + 0.781831i 3.17118 + 1.96066i
55.16 0.357271 − 1.36834i 0.900969 + 0.433884i −1.74472 − 0.977737i 1.10712 − 2.29896i 0.915591 − 1.07782i −2.45319 − 0.990882i −1.96121 + 2.03805i 0.623490 + 0.781831i −2.75022 − 2.33627i
55.17 0.421544 + 1.34993i 0.900969 + 0.433884i −1.64460 + 1.13811i 1.52760 − 3.17209i −0.205913 + 1.39914i 1.99561 − 1.73711i −2.22963 − 1.74033i 0.623490 + 0.781831i 4.92603 + 0.724967i
55.18 0.511445 + 1.31849i 0.900969 + 0.433884i −1.47685 + 1.34867i −1.72652 + 3.58517i −0.111277 + 1.40983i −1.74761 − 1.98642i −2.53354 − 1.25744i 0.623490 + 0.781831i −5.61004 − 0.442796i
55.19 0.696560 + 1.23077i 0.900969 + 0.433884i −1.02961 + 1.71462i −0.997385 + 2.07109i 0.0935667 + 1.41111i 0.739943 + 2.54017i −2.82749 − 0.0728789i 0.623490 + 0.781831i −3.24378 + 0.215085i
55.20 0.761615 − 1.19161i 0.900969 + 0.433884i −0.839884 − 1.81510i −1.00022 + 2.07699i 1.20321 − 0.743154i −0.0377719 + 2.64548i −2.80257 − 0.381593i 0.623490 + 0.781831i 1.71318 + 2.77375i
See next 80 embeddings (of 168 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 559.28
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
196.j even 14 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 588.2.x.b yes 168
4.b odd 2 1 588.2.x.a ✓ 168
49.f odd 14 1 588.2.x.a ✓ 168
196.j even 14 1 inner 588.2.x.b yes 168
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
588.2.x.a ✓ 168 4.b odd 2 1
588.2.x.a ✓ 168 49.f odd 14 1
588.2.x.b yes 168 1.a even 1 1 trivial
588.2.x.b yes 168 196.j even 14 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{11}^{168} - 162 T_{11}^{166} + 168 T_{11}^{165} + 15627 T_{11}^{164} - 33768 T_{11}^{163} - 1126194 T_{11}^{162} + 3686760 T_{11}^{161} + 66139323 T_{11}^{160} - 288456056 T_{11}^{159} - 3324030176 T_{11}^{158} + \cdots + 89\!\cdots\!56 \) acting on \(S_{2}^{\mathrm{new}}(588, [\chi])\). Copy content Toggle raw display