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Properties

Label 671.2.bt.a
Level $671$
Weight $2$
Character orbit 671.bt
Analytic conductor $5.358$
Analytic rank $0$
Dimension $480$
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] = [671,2,Mod(24,671)]
 
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([2, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("671.24");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 671 = 11 \cdot 61 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 671.bt (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.

\(\operatorname{Tr}(f)(q) = \) \( 480 q - 10 q^{2} - 10 q^{3} - 10 q^{5} - 30 q^{6} - 20 q^{7} + 10 q^{8} + 110 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 480 q - 10 q^{2} - 10 q^{3} - 10 q^{5} - 30 q^{6} - 20 q^{7} + 10 q^{8} + 110 q^{9} - 6 q^{11} - 6 q^{12} - 20 q^{13} - 26 q^{15} - 412 q^{16} - 10 q^{17} - 10 q^{18} - 10 q^{19} + 54 q^{20} - 6 q^{22} - 16 q^{23} + 30 q^{24} + 104 q^{25} + 30 q^{26} + 50 q^{27} - 50 q^{28} - 40 q^{29} - 10 q^{30} - 10 q^{31} + 70 q^{32} - 8 q^{33} - 12 q^{34} - 20 q^{35} - 120 q^{36} + 2 q^{37} - 30 q^{38} - 60 q^{39} + 30 q^{40} - 10 q^{41} - 36 q^{42} - 50 q^{44} - 50 q^{45} - 10 q^{46} - 70 q^{48} - 50 q^{49} + 70 q^{50} + 30 q^{51} + 80 q^{52} - 66 q^{53} + 110 q^{54} - 60 q^{55} - 16 q^{56} + 110 q^{57} + 66 q^{58} - 10 q^{59} + 110 q^{60} - 10 q^{61} - 20 q^{62} - 70 q^{63} - 40 q^{65} - 160 q^{66} + 36 q^{67} + 240 q^{68} - 48 q^{69} - 164 q^{70} + 42 q^{71} - 190 q^{72} + 10 q^{73} + 60 q^{74} + 50 q^{75} + 40 q^{76} - 16 q^{77} - 154 q^{78} - 10 q^{79} + 80 q^{80} - 132 q^{81} + 10 q^{82} - 40 q^{83} + 30 q^{84} - 10 q^{85} + 70 q^{86} - 130 q^{88} + 36 q^{89} + 100 q^{90} + 34 q^{91} + 6 q^{92} - 44 q^{93} - 190 q^{94} - 10 q^{95} + 110 q^{96} - 10 q^{97} - 90 q^{98} - 34 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
24.1 −1.97231 − 1.97231i 0.201185 − 0.276908i 5.78004i −0.786188 + 1.08210i −0.942950 + 0.149349i 0.528349 + 0.0836823i 7.45543 − 7.45543i 0.890849 + 2.74175i 3.68484 − 0.583622i
24.2 −1.90655 − 1.90655i 1.63182 − 2.24601i 5.26987i 2.36551 − 3.25584i −7.39329 + 1.17098i −3.31742 − 0.525428i 6.23417 − 6.23417i −1.45468 − 4.47703i −10.7174 + 1.69747i
24.3 −1.87923 − 1.87923i −0.632259 + 0.870229i 5.06300i 1.12324 − 1.54600i 2.82352 − 0.447201i 1.99715 + 0.316317i 5.75608 − 5.75608i 0.569503 + 1.75275i −5.01611 + 0.794473i
24.4 −1.79974 − 1.79974i 1.47800 − 2.03430i 4.47811i −1.87588 + 2.58192i −6.32122 + 1.00118i −0.0845999 − 0.0133993i 4.45994 − 4.45994i −1.02682 − 3.16023i 8.02287 − 1.27070i
24.5 −1.74227 − 1.74227i −1.98888 + 2.73745i 4.07099i 0.533210 − 0.733901i 8.23453 − 1.30422i 1.50151 + 0.237815i 3.60822 − 3.60822i −2.61097 − 8.03575i −2.20765 + 0.349657i
24.6 −1.62661 − 1.62661i 0.614683 − 0.846038i 3.29172i 0.269681 − 0.371184i −2.37602 + 0.376325i −3.92133 − 0.621078i 2.10112 − 2.10112i 0.589105 + 1.81308i −1.04244 + 0.165106i
24.7 −1.62529 − 1.62529i −0.413397 + 0.568992i 3.28310i −1.22240 + 1.68250i 1.59666 − 0.252886i −3.63999 − 0.576518i 2.08541 − 2.08541i 0.774196 + 2.38273i 4.72129 − 0.747779i
24.8 −1.61847 − 1.61847i −1.30165 + 1.79157i 3.23888i 1.44154 − 1.98411i 5.00629 − 0.792918i −4.28474 − 0.678636i 2.00510 − 2.00510i −0.588377 − 1.81084i −5.54431 + 0.878132i
24.9 −1.61168 − 1.61168i −1.04733 + 1.44152i 3.19504i −1.92750 + 2.65297i 4.01124 − 0.635318i 0.264341 + 0.0418675i 1.92602 − 1.92602i −0.0540423 − 0.166325i 7.38225 − 1.16923i
24.10 −1.58677 − 1.58677i 1.16640 − 1.60542i 3.03566i 0.713791 − 0.982450i −4.39823 + 0.696611i 4.85209 + 0.768496i 1.64335 − 1.64335i −0.289814 − 0.891954i −2.69154 + 0.426298i
24.11 −1.48741 − 1.48741i 0.466480 − 0.642054i 2.42479i 1.24388 − 1.71206i −1.64885 + 0.261152i 3.16101 + 0.500655i 0.631841 − 0.631841i 0.732421 + 2.25416i −4.39670 + 0.696369i
24.12 −1.31812 − 1.31812i −1.17193 + 1.61303i 1.47490i −0.658680 + 0.906595i 3.67092 − 0.581417i −0.0457563 − 0.00724708i −0.692149 + 0.692149i −0.301382 − 0.927557i 2.06322 − 0.326783i
24.13 −1.28778 − 1.28778i 2.00612 − 2.76119i 1.31676i 0.473616 − 0.651877i −6.13925 + 0.972361i 1.22063 + 0.193328i −0.879869 + 0.879869i −2.67259 − 8.22538i −1.44939 + 0.229560i
24.14 −1.27934 − 1.27934i 0.0168365 − 0.0231735i 1.27343i −2.14028 + 2.94584i −0.0511865 + 0.00810715i 3.52697 + 0.558617i −0.929531 + 0.929531i 0.926797 + 2.85239i 6.50689 − 1.03059i
24.15 −1.25805 − 1.25805i −0.375639 + 0.517022i 1.16539i 1.99635 − 2.74774i 1.12301 − 0.177868i 0.145802 + 0.0230928i −1.04998 + 1.04998i 0.800843 + 2.46474i −5.96832 + 0.945289i
24.16 −1.00445 − 1.00445i 1.29736 − 1.78567i 0.0178457i −0.861560 + 1.18584i −3.09676 + 0.490478i −2.64594 − 0.419075i −1.99098 + 1.99098i −0.578406 − 1.78015i 2.05651 − 0.325719i
24.17 −0.960487 − 0.960487i 1.12023 − 1.54187i − 0.154930i −2.09006 + 2.87673i −2.55692 + 0.404976i 0.620318 + 0.0982487i −2.06978 + 2.06978i −0.195387 − 0.601339i 4.77054 − 0.755579i
24.18 −0.942635 − 0.942635i −1.43822 + 1.97954i − 0.222878i 1.94159 − 2.67237i 3.22170 − 0.510268i 2.72363 + 0.431380i −2.09536 + 2.09536i −0.923054 − 2.84087i −4.34928 + 0.688858i
24.19 −0.895937 − 0.895937i −0.0916871 + 0.126196i − 0.394594i 0.0341747 − 0.0470375i 0.195210 − 0.0309182i 0.449015 + 0.0711169i −2.14541 + 2.14541i 0.919532 + 2.83003i −0.0727610 + 0.0115242i
24.20 −0.864144 − 0.864144i −1.90713 + 2.62494i − 0.506509i −1.75537 + 2.41606i 3.91636 − 0.620291i −1.08118 − 0.171241i −2.16599 + 2.16599i −2.32611 − 7.15902i 3.60472 − 0.570932i
See next 80 embeddings (of 480 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 24.60
Significant digits:
Format:

Inner twists

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

Hecke kernels

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