Properties

Label 667.2.x.a
Level $667$
Weight $2$
Character orbit 667.x
Analytic conductor $5.326$
Analytic rank $0$
Dimension $6960$
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] = [667,2,Mod(10,667)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(667, base_ring=CyclotomicField(308))
 
chi = DirichletCharacter(H, H._module([42, 253]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("667.10");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 667 = 23 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 667.x (of order \(308\), degree \(120\), 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: \(5.32602181482\)
Analytic rank: \(0\)
Dimension: \(6960\)
Relative dimension: \(58\) over \(\Q(\zeta_{308})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{308}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 6960 q - 112 q^{2} - 108 q^{3} - 126 q^{4} - 154 q^{5} - 126 q^{6} - 110 q^{7} - 102 q^{8} - 126 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 6960 q - 112 q^{2} - 108 q^{3} - 126 q^{4} - 154 q^{5} - 126 q^{6} - 110 q^{7} - 102 q^{8} - 126 q^{9} - 132 q^{10} - 132 q^{11} - 94 q^{12} - 126 q^{13} - 132 q^{14} - 132 q^{15} - 234 q^{16} - 132 q^{17} - 74 q^{18} - 110 q^{19} - 110 q^{20} - 198 q^{21} - 98 q^{23} - 528 q^{24} + 190 q^{25} - 50 q^{26} + 18 q^{27} - 82 q^{29} - 264 q^{30} - 92 q^{31} - 196 q^{32} - 154 q^{33} - 154 q^{34} - 70 q^{35} - 262 q^{36} - 132 q^{37} - 154 q^{38} - 168 q^{39} - 132 q^{40} - 132 q^{41} - 154 q^{42} - 44 q^{43} + 66 q^{44} - 232 q^{46} - 320 q^{47} - 14 q^{48} - 94 q^{49} + 10 q^{50} + 462 q^{51} - 282 q^{52} - 110 q^{53} - 156 q^{54} - 204 q^{55} + 110 q^{56} - 248 q^{58} - 184 q^{59} - 132 q^{60} - 132 q^{61} - 126 q^{62} - 154 q^{63} - 252 q^{64} - 110 q^{65} - 242 q^{66} - 154 q^{67} - 94 q^{69} - 420 q^{70} - 70 q^{71} + 224 q^{72} - 88 q^{73} - 148 q^{75} - 132 q^{76} + 524 q^{77} + 174 q^{78} - 132 q^{79} - 154 q^{80} + 50 q^{81} - 82 q^{82} - 242 q^{83} - 264 q^{84} - 216 q^{85} + 120 q^{87} - 440 q^{88} - 176 q^{90} + 28 q^{92} - 280 q^{93} - 86 q^{94} + 96 q^{95} - 84 q^{96} - 356 q^{98} - 396 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
10.1 −2.65729 0.809781i −0.342570 0.438674i 4.74534 + 3.18827i 1.70314 + 1.30227i 0.555077 + 1.44309i −0.549665 3.33810i −6.51978 8.00663i 0.652004 2.61000i −3.47119 4.83969i
10.2 −2.54833 0.776576i 0.0554222 + 0.0709704i 4.23079 + 2.84256i −2.60951 1.99531i −0.0861200 0.223895i 0.0248235 + 0.150752i −5.20967 6.39775i 0.725120 2.90269i 5.10038 + 7.11118i
10.3 −2.48306 0.756687i 1.06978 + 1.36989i 3.93292 + 2.64243i 1.15510 + 0.883222i −1.61974 4.21101i 0.221686 + 1.34629i −4.48807 5.51158i −0.00509350 + 0.0203895i −2.19986 3.06714i
10.4 −2.45646 0.748582i 2.01386 + 2.57883i 3.81374 + 2.56235i −0.155463 0.118871i −3.01651 7.84233i −0.639256 3.88218i −4.20716 5.16661i −1.86763 + 7.47623i 0.292903 + 0.408379i
10.5 −2.43184 0.741079i −1.78101 2.28065i 3.70457 + 2.48900i −0.499891 0.382232i 2.64099 + 6.86606i 0.676915 + 4.11088i −3.95387 4.85556i −1.30230 + 5.21316i 0.932394 + 1.29999i
10.6 −2.26057 0.688884i −0.279898 0.358420i 2.97550 + 1.99916i 3.34647 + 2.55881i 0.385818 + 1.00305i 0.738312 + 4.48375i −2.36473 2.90401i 0.676963 2.70992i −5.80220 8.08969i
10.7 −2.18547 0.666000i −1.42722 1.82761i 2.67263 + 1.79567i 1.89766 + 1.45101i 1.90195 + 4.94471i −0.270958 1.64552i −1.75979 2.16111i −0.576116 + 2.30622i −3.18091 4.43498i
10.8 −2.15111 0.655530i 1.73583 + 2.22280i 2.53747 + 1.70486i −3.19826 2.44548i −2.27686 5.91938i 0.777456 + 4.72146i −1.50091 1.84319i −1.20064 + 4.80623i 5.27673 + 7.35706i
10.9 −2.09159 0.637391i −0.992867 1.27141i 2.30839 + 1.55094i −1.59950 1.22303i 1.26629 + 3.29210i 0.0668614 + 0.406047i −1.07832 1.32424i 0.0963981 0.385886i 2.56596 + 3.57758i
10.10 −1.98974 0.606353i 0.437664 + 0.560446i 1.93131 + 1.29760i −0.227511 0.173961i −0.531010 1.38052i 0.186489 + 1.13254i −0.429147 0.527015i 0.604535 2.41998i 0.347206 + 0.484090i
10.11 −1.98560 0.605090i −0.775169 0.992635i 1.91637 + 1.28756i −1.79715 1.37415i 0.938541 + 2.44002i −0.486271 2.95311i −0.404658 0.496941i 0.342648 1.37164i 2.73693 + 3.81595i
10.12 −1.88604 0.574752i 0.762299 + 0.976154i 1.56672 + 1.05264i 2.50001 + 1.91158i −0.876682 2.27920i −0.517569 3.14318i 0.140056 + 0.171996i 0.355308 1.42232i −3.61644 5.04221i
10.13 −1.66696 0.507988i 1.27446 + 1.63200i 0.860600 + 0.578215i −2.62311 2.00571i −1.29544 3.36789i −0.605300 3.67597i 1.05986 + 1.30157i −0.312085 + 1.24929i 3.35374 + 4.67594i
10.14 −1.64944 0.502649i 1.73011 + 2.21548i 0.807886 + 0.542797i 2.52603 + 1.93147i −1.74011 4.52394i −0.0283849 0.172381i 1.11787 + 1.37280i −1.18797 + 4.75552i −3.19567 4.45555i
10.15 −1.62394 0.494880i −1.90243 2.43613i 0.732189 + 0.491938i 1.54627 + 1.18232i 1.88384 + 4.89762i −0.262699 1.59536i 1.19835 + 1.47164i −1.58843 + 6.35858i −1.92595 2.68525i
10.16 −1.37254 0.418269i −0.575339 0.736745i 0.0488291 + 0.0328070i −0.314379 0.240383i 0.481522 + 1.25186i −0.752034 4.56708i 1.75874 + 2.15982i 0.515308 2.06280i 0.330954 + 0.461432i
10.17 −1.35040 0.411519i −1.53960 1.97151i −0.00587983 0.00395050i −2.62859 2.00990i 1.26775 + 3.29590i 0.536640 + 3.25900i 1.78911 + 2.19712i −0.789426 + 3.16011i 2.72252 + 3.79587i
10.18 −1.29910 0.395886i −0.468338 0.599725i −0.129175 0.0867889i −0.0558674 0.0427178i 0.370993 + 0.964509i 0.673018 + 4.08722i 1.84852 + 2.27008i 0.586755 2.34881i 0.0556657 + 0.0776117i
10.19 −1.29780 0.395490i 1.78122 + 2.28092i −0.132239 0.0888481i 1.04153 + 0.796385i −1.40958 3.66463i 0.540900 + 3.28487i 1.84983 + 2.27169i −1.30278 + 5.21509i −1.03673 1.44546i
10.20 −1.09968 0.335116i 1.00277 + 1.28409i −0.563109 0.378338i −1.56158 1.19403i −0.672411 1.74814i 0.126397 + 0.767606i 1.94425 + 2.38764i 0.0837475 0.335245i 1.31710 + 1.83636i
See next 80 embeddings (of 6960 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 10.58
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
23.d odd 22 1 inner
29.f odd 28 1 inner
667.x even 308 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 667.2.x.a 6960
23.d odd 22 1 inner 667.2.x.a 6960
29.f odd 28 1 inner 667.2.x.a 6960
667.x even 308 1 inner 667.2.x.a 6960
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
667.2.x.a 6960 1.a even 1 1 trivial
667.2.x.a 6960 23.d odd 22 1 inner
667.2.x.a 6960 29.f odd 28 1 inner
667.2.x.a 6960 667.x even 308 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(667, [\chi])\).