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Properties

Label 671.2.br.a
Level $671$
Weight $2$
Character orbit 671.br
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(145,671)]
 
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, 19]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("671.145");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 671 = 11 \cdot 61 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 671.br (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.

\(\operatorname{Tr}(f)(q) = \) \( 480 q - 10 q^{2} - 10 q^{4} - 10 q^{5} + 10 q^{6} - 10 q^{7} - 30 q^{8} - 460 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 480 q - 10 q^{2} - 10 q^{4} - 10 q^{5} + 10 q^{6} - 10 q^{7} - 30 q^{8} - 460 q^{9} - 60 q^{10} - 6 q^{11} - 6 q^{12} - 20 q^{13} + 4 q^{15} + 98 q^{16} - 30 q^{17} + 20 q^{18} - 20 q^{19} - 26 q^{20} + 60 q^{21} - 6 q^{22} + 14 q^{23} - 10 q^{24} + 94 q^{25} - 50 q^{26} - 10 q^{28} - 60 q^{29} - 60 q^{30} + 10 q^{31} - 8 q^{33} - 12 q^{34} - 10 q^{35} + 100 q^{36} + 2 q^{37} + 70 q^{39} - 10 q^{40} - 10 q^{41} + 34 q^{42} - 30 q^{43} - 40 q^{45} - 10 q^{46} - 40 q^{47} + 30 q^{48} + 30 q^{50} + 20 q^{51} + 64 q^{53} - 10 q^{54} - 60 q^{55} + 4 q^{56} - 84 q^{58} - 10 q^{59} + 100 q^{60} - 60 q^{61} + 150 q^{62} - 80 q^{63} - 50 q^{64} - 90 q^{65} + 60 q^{66} + 36 q^{67} + 70 q^{68} - 48 q^{69} + 66 q^{70} + 42 q^{71} - 20 q^{72} - 30 q^{73} - 150 q^{74} - 70 q^{75} - 40 q^{76} + 84 q^{77} + 6 q^{78} - 10 q^{79} + 408 q^{81} - 130 q^{82} - 10 q^{83} - 90 q^{84} - 10 q^{85} + 280 q^{87} - 90 q^{88} - 4 q^{89} + 40 q^{90} - 116 q^{91} + 6 q^{92} - 34 q^{93} - 10 q^{94} - 60 q^{95} - 20 q^{96} - 10 q^{97} + 20 q^{98} - 124 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
145.1 −2.77383 − 0.439331i 0.763395i 5.59899 + 1.81922i −1.98650 − 2.73419i 0.335383 − 2.11753i −0.358735 − 0.358735i −9.72678 − 4.95604i 2.41723 4.30900 + 8.45689i
145.2 −2.58945 − 0.410128i − 2.00379i 4.63492 + 1.50598i 1.15729 + 1.59288i −0.821809 + 5.18870i −2.85973 − 2.85973i −6.71228 − 3.42008i −1.01516 −2.34347 − 4.59931i
145.3 −2.57651 − 0.408080i 1.60970i 4.56978 + 1.48481i 0.730649 + 1.00565i 0.656887 − 4.14742i 1.29917 + 1.29917i −6.51958 − 3.32189i 0.408855 −1.47214 − 2.88924i
145.4 −2.56368 − 0.406048i − 1.45669i 4.50549 + 1.46392i 2.26425 + 3.11647i −0.591485 + 3.73449i 1.64439 + 1.64439i −6.33075 − 3.22568i 0.878060 −4.53937 − 8.90902i
145.5 −2.53801 − 0.401981i 2.65187i 4.37779 + 1.42243i 2.03339 + 2.79872i 1.06600 − 6.73048i −1.21488 − 1.21488i −5.95994 − 3.03674i −4.03243 −4.03572 − 7.92055i
145.6 −2.44136 − 0.386673i − 2.87668i 3.90861 + 1.26998i −0.713517 − 0.982072i −1.11233 + 7.02301i 3.37824 + 3.37824i −4.64649 − 2.36751i −5.27527 1.36221 + 2.67349i
145.7 −2.36415 − 0.374445i − 0.710920i 3.54688 + 1.15245i −1.02138 − 1.40581i −0.266200 + 1.68072i 0.375983 + 0.375983i −3.68837 − 1.87932i 2.49459 1.88830 + 3.70600i
145.8 −2.23465 − 0.353934i 3.03184i 2.96627 + 0.963800i −1.01114 − 1.39171i 1.07307 − 6.77510i 2.21960 + 2.21960i −2.25564 − 1.14931i −6.19207 1.76696 + 3.46785i
145.9 −2.13899 − 0.338783i 0.654483i 2.55841 + 0.831278i 0.881213 + 1.21289i 0.221728 − 1.39994i −3.28094 − 3.28094i −1.33157 − 0.678471i 2.57165 −1.47400 − 2.89290i
145.10 −2.06514 − 0.327086i − 1.61427i 2.25570 + 0.732921i −1.98564 − 2.73300i −0.528003 + 3.33368i −0.984325 − 0.984325i −0.692629 − 0.352912i 0.394148 3.20669 + 6.29349i
145.11 −1.98399 − 0.314233i 1.97867i 1.93537 + 0.628839i −0.205741 − 0.283179i 0.621764 − 3.92566i 0.863868 + 0.863868i −0.0625783 − 0.0318852i −0.915134 0.319205 + 0.626475i
145.12 −1.96510 − 0.311242i − 2.25965i 1.86265 + 0.605211i 0.00692469 + 0.00953101i −0.703299 + 4.44045i −1.34158 − 1.34158i 0.0735623 + 0.0374819i −2.10604 −0.0106413 − 0.0208847i
145.13 −1.72714 − 0.273552i 0.265231i 1.00607 + 0.326891i 0.901784 + 1.24120i 0.0725545 − 0.458091i 0.196613 + 0.196613i 1.46796 + 0.747961i 2.92965 −1.21797 − 2.39041i
145.14 −1.68570 − 0.266989i − 0.259685i 0.868198 + 0.282095i 2.45721 + 3.38206i −0.0693330 + 0.437752i 1.83223 + 1.83223i 1.65318 + 0.842338i 2.93256 −3.23916 − 6.35720i
145.15 −1.66363 − 0.263493i − 3.23332i 0.796118 + 0.258674i 1.63013 + 2.24368i −0.851956 + 5.37904i −0.376864 − 0.376864i 1.74528 + 0.889262i −7.45433 −2.12073 − 4.16217i
145.16 −1.51426 − 0.239836i 1.98556i 0.333355 + 0.108314i −1.93051 − 2.65712i 0.476209 − 3.00666i −1.88562 − 1.88562i 2.25326 + 1.14809i −0.942465 2.28603 + 4.48659i
145.17 −1.45026 − 0.229699i 0.910757i 0.148378 + 0.0482109i −1.55755 − 2.14379i 0.209200 − 1.32083i −1.33039 − 1.33039i 2.41248 + 1.22922i 2.17052 1.76643 + 3.46681i
145.18 −1.38429 − 0.219251i 2.55012i −0.0339128 − 0.0110189i 2.00959 + 2.76597i 0.559115 − 3.53012i −1.60538 − 1.60538i 2.54211 + 1.29527i −3.50311 −2.17543 − 4.26952i
145.19 −1.33990 − 0.212220i − 2.47084i −0.151811 − 0.0493265i 0.366790 + 0.504843i −0.524362 + 3.31069i 2.45627 + 2.45627i 2.61043 + 1.33008i −3.10507 −0.384325 − 0.754280i
145.20 −1.12668 − 0.178449i 1.16304i −0.664547 − 0.215924i −1.96364 − 2.70272i 0.207543 − 1.31037i 3.11725 + 3.11725i 2.74299 + 1.39762i 1.64734 1.73010 + 3.39552i
See next 80 embeddings (of 480 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 145.60
Significant digits:
Format:

Inner twists

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

Hecke kernels

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