↑

Properties

Label 671.2.dc.a
Level $671$
Weight $2$
Character orbit 671.dc
Analytic conductor $5.358$
Analytic rank $0$
Dimension $960$
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(6,671)]
 
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([54, 7]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("671.6");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 671 = 11 \cdot 61 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 671.dc (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.

\(\operatorname{Tr}(f)(q) = \) \( 960 q - 20 q^{2} - 20 q^{3} - 8 q^{4} - 18 q^{5} - 20 q^{6} - 20 q^{7} - 20 q^{8} + 220 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 960 q - 20 q^{2} - 20 q^{3} - 8 q^{4} - 18 q^{5} - 20 q^{6} - 20 q^{7} - 20 q^{8} + 220 q^{9} - 60 q^{10} - 8 q^{11} + 12 q^{12} - 10 q^{13} - 18 q^{14} - 58 q^{15} - 108 q^{16} - 30 q^{17} - 20 q^{18} - 20 q^{19} - 20 q^{20} - 30 q^{21} - 18 q^{22} - 20 q^{23} + 80 q^{24} + 416 q^{25} + 76 q^{26} - 50 q^{27} - 60 q^{28} + 10 q^{29} - 120 q^{30} - 8 q^{31} - 4 q^{33} - 24 q^{34} - 20 q^{35} - 6 q^{36} - 36 q^{37} + 28 q^{38} - 80 q^{39} - 70 q^{40} - 40 q^{41} - 28 q^{42} - 90 q^{43} + 18 q^{44} - 134 q^{45} - 10 q^{46} - 2 q^{47} + 72 q^{48} + 100 q^{49} + 30 q^{50} - 20 q^{51} - 70 q^{52} + 48 q^{53} + 20 q^{54} - 216 q^{55} + 20 q^{56} - 20 q^{57} - 64 q^{58} - 8 q^{59} + 10 q^{60} - 60 q^{61} - 210 q^{62} - 20 q^{63} - 60 q^{64} - 90 q^{65} + 186 q^{66} + 12 q^{67} + 180 q^{68} + 10 q^{69} + 146 q^{70} - 34 q^{71} + 110 q^{72} - 20 q^{73} + 120 q^{74} - 124 q^{75} - 40 q^{76} + 10 q^{77} - 120 q^{78} - 20 q^{79} - 210 q^{80} - 180 q^{81} - 16 q^{82} + 10 q^{83} + 220 q^{84} + 210 q^{85} - 364 q^{86} + 190 q^{87} + 180 q^{88} - 92 q^{89} - 230 q^{90} + 104 q^{91} + 96 q^{92} - 40 q^{93} - 180 q^{94} - 70 q^{95} - 20 q^{96} + 64 q^{97} + 20 q^{98} - 204 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
6.1 −2.81284 − 0.147415i 2.17156 − 0.705582i 5.90129 + 0.620251i 2.72461 + 1.57305i −6.21226 + 1.66457i 2.84032 + 0.148855i −10.9439 − 1.73335i 1.79077 − 1.30107i −7.43199 − 4.82639i
6.2 −2.74850 − 0.144043i −0.775665 + 0.252029i 5.54446 + 0.582747i −0.282415 − 0.163052i 2.16822 − 0.580972i −4.98173 − 0.261081i −9.71824 − 1.53922i −1.88891 + 1.37238i 0.752730 + 0.488829i
6.3 −2.57499 − 0.134950i −3.10048 + 1.00741i 4.62333 + 0.485932i 2.47030 + 1.42623i 8.11965 − 2.17565i −0.119782 − 0.00627753i −6.74591 − 1.06845i 6.17104 − 4.48352i −6.16855 − 4.00590i
6.4 −2.56445 − 0.134397i −1.15775 + 0.376175i 4.56929 + 0.480251i −1.75059 − 1.01070i 3.01954 − 0.809083i 3.73846 + 0.195925i −6.58045 − 1.04224i −1.22818 + 0.892324i 4.35346 + 2.82717i
6.5 −2.48848 − 0.130416i −0.819826 + 0.266378i 4.18648 + 0.440017i 1.51849 + 0.876700i 2.07486 − 0.555957i 1.01726 + 0.0533122i −5.43816 − 0.861320i −1.82589 + 1.32659i −3.66439 − 2.37969i
6.6 −2.48686 − 0.130331i 1.58768 − 0.515868i 4.17842 + 0.439170i −3.71491 − 2.14481i −4.01556 + 1.07597i 1.15585 + 0.0605754i −5.41469 − 0.857602i −0.172448 + 0.125290i 8.95892 + 5.81799i
6.7 −2.23303 − 0.117028i −0.211851 + 0.0688345i 2.98367 + 0.313596i −0.849713 − 0.490582i 0.481124 − 0.128917i −2.85786 − 0.149774i −2.20878 − 0.349837i −2.38691 + 1.73419i 1.84002 + 1.19492i
6.8 −2.20560 − 0.115591i 1.11197 − 0.361302i 2.86227 + 0.300837i 1.76874 + 1.02118i −2.49433 + 0.668355i 1.41323 + 0.0740642i −1.91538 − 0.303367i −1.32110 + 0.959838i −3.78310 − 2.45677i
6.9 −2.06204 − 0.108067i 3.26673 − 1.06142i 2.25128 + 0.236619i −0.148666 − 0.0858322i −6.85082 + 1.83567i −3.76245 − 0.197181i −0.537753 − 0.0851717i 7.11783 − 5.17141i 0.297279 + 0.193055i
6.10 −2.01193 − 0.105441i −3.04611 + 0.989741i 2.04772 + 0.215224i −2.06828 − 1.19412i 6.23293 − 1.67011i −3.02383 − 0.158472i −0.117394 − 0.0185935i 5.87215 − 4.26637i 4.03533 + 2.62057i
6.11 −1.98489 − 0.104024i 2.95525 − 0.960219i 1.93993 + 0.203895i −0.684600 − 0.395254i −5.96574 + 1.59851i 3.99020 + 0.209117i 0.0969418 + 0.0153541i 5.38443 − 3.91201i 1.31774 + 0.855751i
6.12 −1.98109 − 0.103825i 1.42614 − 0.463380i 1.92490 + 0.202315i 3.65472 + 2.11005i −2.87342 + 0.769929i −3.94937 − 0.206978i 0.126371 + 0.0200153i −0.607909 + 0.441672i −7.02126 − 4.55966i
6.13 −1.93406 − 0.101360i −1.88163 + 0.611379i 1.74127 + 0.183015i −2.84234 − 1.64103i 3.70115 − 0.991721i −0.892121 − 0.0467541i 0.476568 + 0.0754810i 0.739695 − 0.537420i 5.33093 + 3.46194i
6.14 −1.84653 − 0.0967726i 0.655809 − 0.213085i 1.41127 + 0.148330i 0.0696965 + 0.0402393i −1.23159 + 0.330004i −0.339531 − 0.0177940i 1.06101 + 0.168047i −2.04237 + 1.48387i −0.124803 − 0.0810478i
6.15 −1.75230 − 0.0918340i −2.46563 + 0.801132i 1.07306 + 0.112784i 0.0995429 + 0.0574711i 4.39409 − 1.17739i 4.70190 + 0.246416i 1.59623 + 0.252817i 3.01047 − 2.18723i −0.169151 − 0.109848i
6.16 −1.65497 − 0.0867333i 1.40820 − 0.457553i 0.742359 + 0.0780251i −0.249087 − 0.143811i −2.37022 + 0.635099i 2.86156 + 0.149968i 2.05186 + 0.324983i −0.653367 + 0.474699i 0.399758 + 0.259606i
6.17 −1.54467 − 0.0809527i −1.97453 + 0.641563i 0.390404 + 0.0410331i 1.90561 + 1.10021i 3.10193 − 0.831159i −0.233076 − 0.0122150i 2.45577 + 0.388955i 1.06011 − 0.770213i −2.85448 − 1.85372i
6.18 −1.31321 − 0.0688226i 1.76206 − 0.572529i −0.269251 − 0.0282994i −3.56991 − 2.06109i −2.35337 + 0.630583i −3.91849 − 0.205360i 2.94929 + 0.467121i 0.350026 − 0.254309i 4.54620 + 2.95234i
6.19 −1.29101 − 0.0676589i −1.97523 + 0.641792i −0.326917 − 0.0343603i 2.16259 + 1.24857i 2.59347 − 0.694917i −3.89004 − 0.203868i 2.97346 + 0.470949i 1.06260 − 0.772023i −2.70744 − 1.75823i
6.20 −1.25128 − 0.0655768i −0.438191 + 0.142377i −0.427642 − 0.0449470i −1.62459 − 0.937957i 0.557637 − 0.149418i 1.31577 + 0.0689565i 3.00729 + 0.476309i −2.25531 + 1.63858i 1.97131 + 1.28018i
See next 80 embeddings (of 960 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 6.60
Significant digits:
Format:

Inner twists

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

Hecke kernels

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