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Properties

Label 671.2.cu.a
Level $671$
Weight $2$
Character orbit 671.cu
Analytic conductor $5.358$
Analytic rank $0$
Dimension $960$
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(7,671)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(671, base_ring=CyclotomicField(60))
 
chi = DirichletCharacter(H, H._module([42, 49]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("671.7");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 671 = 11 \cdot 61 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 671.cu (of order \(60\), degree \(16\), 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: \(960\)
Relative dimension: \(60\) over \(\Q(\zeta_{60})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{60}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 960 q - 20 q^{2} - 8 q^{4} - 8 q^{5} + 20 q^{6} - 20 q^{7} - 920 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 960 q - 20 q^{2} - 8 q^{4} - 8 q^{5} + 20 q^{6} - 20 q^{7} - 920 q^{9} + 60 q^{10} - 18 q^{11} + 12 q^{12} - 10 q^{13} - 18 q^{14} + 32 q^{15} - 188 q^{16} - 60 q^{17} + 40 q^{18} - 10 q^{19} + 20 q^{20} + 30 q^{21} - 18 q^{22} - 20 q^{23} - 20 q^{24} - 124 q^{25} - 64 q^{26} - 20 q^{28} + 30 q^{29} + 120 q^{30} - 28 q^{31} + 56 q^{33} - 24 q^{34} - 20 q^{35} - 136 q^{36} + 4 q^{37} - 42 q^{38} - 10 q^{39} - 20 q^{40} - 20 q^{41} + 92 q^{42} + 90 q^{43} + 18 q^{44} - 134 q^{45} - 20 q^{46} - 2 q^{47} + 102 q^{48} - 30 q^{49} + 60 q^{50} + 40 q^{51} - 30 q^{52} - 82 q^{53} - 20 q^{54} - 36 q^{55} + 20 q^{56} + 90 q^{57} - 144 q^{58} - 8 q^{59} - 10 q^{60} + 30 q^{61} - 210 q^{62} - 70 q^{63} + 20 q^{64} + 90 q^{65} + 6 q^{66} + 12 q^{67} + 140 q^{68} - 60 q^{69} - 144 q^{70} - 24 q^{71} - 160 q^{72} - 30 q^{74} + 196 q^{75} + 40 q^{76} + 30 q^{77} - 120 q^{78} - 20 q^{79} - 90 q^{80} + 720 q^{81} + 64 q^{82} - 20 q^{83} + 150 q^{84} - 20 q^{85} + 126 q^{86} - 190 q^{87} + 60 q^{88} - 92 q^{89} - 580 q^{90} - 196 q^{91} - 144 q^{92} + 160 q^{93} - 20 q^{94} + 30 q^{95} + 80 q^{96} - 56 q^{97} - 20 q^{98} - 104 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
7.1 −0.142951 + 2.72768i − 2.04236i −5.43073 − 0.570793i −2.85884 + 2.57411i 5.57091 + 0.291959i −4.54254 + 1.21717i 1.47869 − 9.33610i −1.17125 −6.61266 − 8.16596i
7.2 −0.135737 + 2.59002i 1.97899i −4.70073 − 0.494067i −2.99509 + 2.69679i −5.12563 − 0.268623i 1.88739 − 0.505724i 1.10626 − 6.98464i −0.916414 −6.57819 − 8.12338i
7.3 −0.134266 + 2.56194i 1.79609i −4.55649 − 0.478906i 2.68322 − 2.41598i −4.60148 − 0.241153i 4.64016 − 1.24333i 1.03606 − 6.54142i −0.225939 5.82935 + 7.19864i
7.4 −0.132274 + 2.52393i − 2.49815i −4.36369 − 0.458643i −1.05684 + 0.951581i 6.30515 + 0.330439i 3.71642 − 0.995812i 0.944041 − 5.96044i −3.24073 −2.26193 − 2.79326i
7.5 −0.129610 + 2.47311i − 0.279149i −4.11042 − 0.432023i −0.155490 + 0.140004i 0.690365 + 0.0361805i 2.16259 − 0.579464i 0.826370 − 5.21750i 2.92208 −0.326091 − 0.402689i
7.6 −0.126331 + 2.41054i − 2.88078i −3.80570 − 0.399996i 2.82379 − 2.54255i 6.94424 + 0.363932i −0.172679 + 0.0462692i 0.689766 − 4.35501i −5.29890 5.77219 + 7.12806i
7.7 −0.125604 + 2.39667i 1.80352i −3.73922 − 0.393008i 0.143829 − 0.129504i −4.32245 − 0.226530i −3.87144 + 1.03735i 0.660699 − 4.17149i −0.252685 0.292313 + 0.360977i
7.8 −0.124774 + 2.38082i − 0.691798i −3.66370 − 0.385071i 1.54331 − 1.38960i 1.64705 + 0.0863181i −1.73143 + 0.463935i 0.628010 − 3.96510i 2.52142 3.11583 + 3.84773i
7.9 −0.117880 + 2.24928i − 0.151290i −3.05630 − 0.321230i −0.473727 + 0.426546i 0.340292 + 0.0178339i 0.629719 − 0.168733i 0.378116 − 2.38733i 2.97711 −0.903577 − 1.11582i
7.10 −0.106766 + 2.03722i − 3.00564i −2.14983 − 0.225956i −0.425560 + 0.383176i 6.12316 + 0.320901i −1.24958 + 0.334823i 0.0515930 − 0.325746i −6.03389 −0.735178 − 0.907870i
7.11 −0.105323 + 2.00968i 3.42617i −2.03868 − 0.214274i 0.700374 − 0.630620i −6.88552 − 0.360855i 1.83596 − 0.491945i 0.0157129 − 0.0992074i −8.73867 1.19358 + 1.47395i
7.12 −0.105167 + 2.00671i 1.80609i −2.02679 − 0.213024i 3.21327 − 2.89324i −3.62431 − 0.189942i −3.78137 + 1.01322i 0.0119315 − 0.0753324i −0.261975 5.46798 + 6.75239i
7.13 −0.102348 + 1.95292i 2.33519i −1.81439 − 0.190701i −2.31739 + 2.08658i −4.56044 − 0.239003i 0.307513 − 0.0823978i −0.0537237 + 0.339198i −2.45309 −3.83776 − 4.73924i
7.14 −0.102023 + 1.94671i − 0.0203115i −1.79021 − 0.188159i −1.42312 + 1.28138i 0.0395405 + 0.00207223i −2.63327 + 0.705583i −0.0609674 + 0.384933i 2.99959 −2.34928 − 2.90112i
7.15 −0.0799319 + 1.52519i − 1.06592i −0.330775 − 0.0347658i −1.95657 + 1.76170i 1.62573 + 0.0852010i 3.79804 − 1.01768i −0.398376 + 2.51525i 1.86381 −2.53054 − 3.12495i
7.16 −0.0748236 + 1.42772i 1.79595i −0.0437390 − 0.00459715i 1.23487 − 1.11188i −2.56411 − 0.134380i 1.99643 − 0.534941i −0.437466 + 2.76205i −0.225442 1.49506 + 1.84624i
7.17 −0.0723048 + 1.37966i − 1.03452i 0.0908170 + 0.00954525i 2.45016 − 2.20614i 1.42729 + 0.0748010i 2.54159 − 0.681018i −0.451980 + 2.85369i 1.92976 2.86655 + 3.53990i
7.18 −0.0702621 + 1.34068i 1.48482i 0.196554 + 0.0206586i 0.586201 − 0.527817i −1.99067 − 0.104326i 0.329310 − 0.0882383i −0.461540 + 2.91405i 0.795320 0.666447 + 0.822994i
7.19 −0.0672046 + 1.28234i − 1.69135i 0.349164 + 0.0366986i 0.147801 − 0.133081i 2.16889 + 0.113667i −4.18041 + 1.12014i −0.472281 + 2.98186i 0.139321 0.160722 + 0.198475i
7.20 −0.0646190 + 1.23300i − 2.30205i 0.472922 + 0.0497061i 0.469937 − 0.423133i 2.83844 + 0.148756i 2.77710 − 0.744121i −0.478145 + 3.01889i −2.29944 0.491357 + 0.606776i
See next 80 embeddings (of 960 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 7.60
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
671.cu even 60 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 671.2.cu.a ✓ 960
11.d odd 10 1 671.2.dc.a yes 960
61.l odd 60 1 671.2.dc.a yes 960
671.cu even 60 1 inner 671.2.cu.a ✓ 960
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
671.2.cu.a ✓ 960 1.a even 1 1 trivial
671.2.cu.a ✓ 960 671.cu even 60 1 inner
671.2.dc.a yes 960 11.d odd 10 1
671.2.dc.a yes 960 61.l odd 60 1

Hecke kernels

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