Properties

Label 671.2.by.a
Level $671$
Weight $2$
Character orbit 671.by
Analytic conductor $5.358$
Analytic rank $0$
Dimension $480$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [671,2,Mod(8,671)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(671, base_ring=CyclotomicField(20))
 
chi = DirichletCharacter(H, H._module([6, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("671.8");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 671 = 11 \cdot 61 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 671.by (of order \(20\), degree \(8\), 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: \(5.35796197563\)
Analytic rank: \(0\)
Dimension: \(480\)
Relative dimension: \(60\) over \(\Q(\zeta_{20})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{20}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 480 q - 10 q^{2} - 10 q^{3} - 10 q^{4} - 10 q^{5} - 10 q^{6} - 10 q^{7} - 10 q^{8} + 110 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 480 q - 10 q^{2} - 10 q^{3} - 10 q^{4} - 10 q^{5} - 10 q^{6} - 10 q^{7} - 10 q^{8} + 110 q^{9} - 6 q^{11} - 6 q^{12} - 20 q^{13} + 24 q^{15} + 98 q^{16} - 10 q^{17} - 40 q^{18} - 10 q^{19} - 76 q^{20} - 6 q^{22} - 16 q^{23} + 70 q^{24} + 104 q^{25} - 50 q^{26} - 40 q^{27} - 10 q^{28} - 40 q^{29} + 50 q^{30} - 10 q^{31} - 70 q^{32} - 28 q^{33} - 12 q^{34} - 10 q^{35} - 10 q^{36} + 2 q^{37} - 30 q^{38} - 20 q^{39} - 60 q^{40} - 10 q^{41} - 26 q^{42} - 50 q^{45} - 10 q^{46} + 70 q^{48} + 30 q^{49} + 10 q^{50} - 40 q^{51} + 80 q^{52} + 44 q^{53} - 110 q^{54} - 100 q^{55} - 16 q^{56} + 110 q^{57} - 74 q^{58} + 10 q^{59} - 110 q^{60} + 30 q^{61} - 20 q^{62} - 10 q^{63} - 10 q^{64} + 40 q^{65} - 10 q^{66} + 36 q^{67} + 50 q^{68} - 78 q^{69} + 236 q^{70} - 28 q^{71} - 230 q^{72} + 60 q^{74} - 50 q^{75} - 40 q^{76} - 16 q^{77} - 154 q^{78} - 10 q^{79} - 50 q^{80} - 52 q^{81} + 130 q^{82} + 140 q^{84} - 110 q^{85} - 250 q^{88} + 36 q^{89} + 330 q^{90} - 36 q^{91} + 106 q^{92} + 96 q^{93} - 130 q^{94} + 70 q^{96} + 90 q^{98} + 16 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
8.1 −1.27226 2.49695i 0.229619 0.316044i −3.44055 + 4.73551i −0.373022 0.513421i −1.08128 0.171258i −0.158002 + 0.997588i 10.6658 + 1.68930i 0.879892 + 2.70803i −0.807406 + 1.58462i
8.2 −1.19854 2.35228i 2.01676 2.77583i −2.92112 + 4.02058i −1.84630 2.54121i −8.94670 1.41702i 0.360188 2.27414i 7.74356 + 1.22646i −2.71087 8.34321i −3.76476 + 7.38875i
8.3 −1.15648 2.26971i −0.804281 + 1.10700i −2.63859 + 3.63171i 2.20178 + 3.03050i 3.44270 + 0.545270i −0.640182 + 4.04195i 6.26242 + 0.991869i 0.348475 + 1.07250i 4.33205 8.50212i
8.4 −1.15551 2.26781i −0.590903 + 0.813308i −2.63221 + 3.62292i −0.341463 0.469983i 2.52722 + 0.400273i 0.492045 3.10665i 6.22986 + 0.986713i 0.614748 + 1.89200i −0.671271 + 1.31744i
8.5 −1.14775 2.25259i −1.61160 + 2.21818i −2.58126 + 3.55279i 1.58808 + 2.18581i 6.84636 + 1.08436i 0.552167 3.48625i 5.97160 + 0.945809i −1.39600 4.29645i 3.10101 6.08607i
8.6 −1.10242 2.16362i 0.888199 1.22250i −2.29036 + 3.15242i 0.977174 + 1.34496i −3.62420 0.574017i 0.241995 1.52790i 4.54880 + 0.720459i 0.221439 + 0.681519i 1.83274 3.59695i
8.7 −1.09577 2.15058i 0.591002 0.813444i −2.24869 + 3.09506i −1.43415 1.97393i −2.39698 0.379644i −0.471977 + 2.97995i 4.35236 + 0.689347i 0.614643 + 1.89168i −2.67360 + 5.24723i
8.8 −1.08112 2.12183i −1.44320 + 1.98639i −2.15774 + 2.96988i −2.06979 2.84882i 5.77505 + 0.914677i 0.0942794 0.595256i 3.93023 + 0.622487i −0.935876 2.88033i −3.80700 + 7.47165i
8.9 −0.993007 1.94889i 1.27965 1.76129i −1.63652 + 2.25248i 1.37971 + 1.89901i −4.70325 0.744922i −0.548105 + 3.46060i 1.69419 + 0.268333i −0.537581 1.65450i 2.33089 4.57463i
8.10 −0.951186 1.86681i 1.65015 2.27123i −1.40464 + 1.93333i 0.779238 + 1.07253i −5.80955 0.920142i −0.134991 + 0.852298i 0.806480 + 0.127734i −1.50846 4.64257i 1.26100 2.47486i
8.11 −0.873559 1.71446i −1.30288 + 1.79327i −1.00068 + 1.37732i −0.684503 0.942138i 4.21263 + 0.667214i −0.105751 + 0.667685i −0.565458 0.0895598i −0.591247 1.81967i −1.01730 + 1.99656i
8.12 −0.857178 1.68231i −0.511539 + 0.704073i −0.919830 + 1.26604i 0.525408 + 0.723162i 1.62295 + 0.257049i 0.291102 1.83795i −0.811382 0.128510i 0.693004 + 2.13285i 0.766212 1.50378i
8.13 −0.835532 1.63982i −0.262985 + 0.361967i −0.815339 + 1.12222i −1.83486 2.52547i 0.813295 + 0.128813i −0.270525 + 1.70803i −1.11404 0.176446i 0.865192 + 2.66279i −2.60824 + 5.11897i
8.14 −0.791651 1.55370i 0.665663 0.916207i −0.611709 + 0.841946i −1.41987 1.95429i −1.95049 0.308927i 0.725085 4.57800i −1.65219 0.261681i 0.530724 + 1.63340i −1.91234 + 3.75318i
8.15 −0.775075 1.52117i 0.0969981 0.133506i −0.537648 + 0.740009i 1.78404 + 2.45552i −0.278267 0.0440732i 0.397079 2.50706i −1.83006 0.289854i 0.918636 + 2.82727i 2.35250 4.61704i
8.16 −0.774310 1.51967i −1.70097 + 2.34118i −0.534269 + 0.735359i −0.0702390 0.0966757i 4.87490 + 0.772109i −0.473066 + 2.98682i −1.83794 0.291101i −1.66079 5.11139i −0.0925284 + 0.181597i
8.17 −0.663369 1.30194i 1.51607 2.08669i −0.0794070 + 0.109294i 0.218773 + 0.301116i −3.72246 0.589579i 0.376420 2.37662i −2.69144 0.426283i −1.12877 3.47399i 0.246906 0.484580i
8.18 −0.616127 1.20922i −1.63169 + 2.24582i 0.0929757 0.127970i 1.58212 + 2.17760i 3.72101 + 0.589351i −0.301788 + 1.90541i −2.89288 0.458188i −1.45427 4.47578i 1.65840 3.25480i
8.19 −0.555563 1.09035i 1.02315 1.40825i 0.295348 0.406511i −1.71747 2.36389i −2.10392 0.333228i 0.0499679 0.315485i −3.02466 0.479059i −0.00927322 0.0285400i −1.62332 + 3.18594i
8.20 −0.514453 1.00967i 0.385562 0.530680i 0.420797 0.579178i 0.416499 + 0.573262i −0.734165 0.116280i −0.694842 + 4.38706i −3.03972 0.481444i 0.794087 + 2.44395i 0.364536 0.715443i
See next 80 embeddings (of 480 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 8.60
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
671.by even 20 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 671.2.by.a yes 480
11.d odd 10 1 671.2.bt.a 480
61.j odd 20 1 671.2.bt.a 480
671.by even 20 1 inner 671.2.by.a yes 480
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
671.2.bt.a 480 11.d odd 10 1
671.2.bt.a 480 61.j odd 20 1
671.2.by.a yes 480 1.a even 1 1 trivial
671.2.by.a yes 480 671.by even 20 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(671, [\chi])\).