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Properties

Label 670.2.v.a
Level $670$
Weight $2$
Character orbit 670.v
Analytic conductor $5.350$
Analytic rank $0$
Dimension $680$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [670,2,Mod(19,670)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(670, base_ring=CyclotomicField(66))
 
chi = DirichletCharacter(H, H._module([33, 10]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("670.19");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 670 = 2 \cdot 5 \cdot 67 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 670.v (of order \(66\), degree \(20\), 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: \(5.34997693543\)
Analytic rank: \(0\)
Dimension: \(680\)
Relative dimension: \(34\) over \(\Q(\zeta_{66})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{66}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 680 q - 34 q^{4} - 4 q^{5} - 2 q^{6} + 48 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 680 q - 34 q^{4} - 4 q^{5} - 2 q^{6} + 48 q^{9} - 4 q^{11} - 4 q^{14} + 36 q^{15} + 34 q^{16} - 64 q^{19} - 2 q^{20} - 28 q^{21} + 18 q^{24} + 4 q^{25} - 6 q^{29} + 12 q^{34} - 2 q^{35} + 24 q^{36} + 122 q^{41} + 4 q^{44} + 62 q^{45} - 18 q^{46} - 32 q^{49} - 8 q^{50} + 28 q^{51} - 22 q^{54} + 164 q^{55} - 2 q^{56} + 112 q^{59} + 84 q^{60} - 442 q^{61} + 68 q^{64} + 178 q^{65} - 488 q^{66} - 46 q^{69} + 66 q^{70} - 300 q^{71} + 12 q^{74} + 48 q^{75} - 128 q^{76} + 116 q^{79} + 24 q^{80} - 412 q^{81} + 94 q^{84} - 16 q^{85} - 34 q^{86} - 148 q^{89} + 26 q^{90} - 88 q^{91} - 80 q^{94} + 100 q^{95} - 2 q^{96} + 24 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
19.1 −0.458227 + 0.888835i −0.875060 − 2.98018i −0.580057 − 0.814576i −1.15552 − 1.91436i 3.04987 + 0.587814i −0.258194 + 0.0122993i 0.989821 − 0.142315i −5.59199 + 3.59376i 2.23104 − 0.149859i
19.2 −0.458227 + 0.888835i −0.743142 − 2.53091i −0.580057 − 0.814576i 0.907283 + 2.04373i 2.59009 + 0.499199i −4.30158 + 0.204910i 0.989821 − 0.142315i −3.32948 + 2.13972i −2.23228 − 0.130066i
19.3 −0.458227 + 0.888835i −0.739818 − 2.51959i −0.580057 − 0.814576i −1.55378 + 1.60803i 2.57851 + 0.496966i 1.44340 − 0.0687575i 0.989821 − 0.142315i −3.27724 + 2.10616i −0.717287 − 2.11790i
19.4 −0.458227 + 0.888835i −0.614364 − 2.09233i −0.580057 − 0.814576i 1.94995 − 1.09440i 2.14126 + 0.412693i 4.22890 − 0.201447i 0.989821 − 0.142315i −1.47665 + 0.948983i 0.0792253 + 2.23466i
19.5 −0.458227 + 0.888835i −0.427254 − 1.45509i −0.580057 − 0.814576i −2.17783 + 0.507010i 1.48912 + 0.287004i 3.98363 − 0.189764i 0.989821 − 0.142315i 0.589009 − 0.378533i 0.547291 − 2.16806i
19.6 −0.458227 + 0.888835i −0.388079 − 1.32168i −0.580057 − 0.814576i 1.89425 − 1.18819i 1.35258 + 0.260689i −4.83752 + 0.230439i 0.989821 − 0.142315i 0.927539 − 0.596093i 0.188112 + 2.22814i
19.7 −0.458227 + 0.888835i −0.365977 − 1.24640i −0.580057 − 0.814576i −0.697586 − 2.12447i 1.27555 + 0.245842i −0.937177 + 0.0446432i 0.989821 − 0.142315i 1.10418 − 0.709611i 2.20796 + 0.353450i
19.8 −0.458227 + 0.888835i −0.245696 − 0.836763i −0.580057 − 0.814576i 1.61636 + 1.54512i 0.856329 + 0.165044i 1.19579 − 0.0569624i 0.989821 − 0.142315i 1.88396 − 1.21074i −2.11401 + 0.728661i
19.9 −0.458227 + 0.888835i 0.0150046 + 0.0511010i −0.580057 − 0.814576i −0.907915 + 2.04345i −0.0522959 − 0.0100792i −0.480436 + 0.0228860i 0.989821 − 0.142315i 2.52137 − 1.62039i −1.40026 − 1.74335i
19.10 −0.458227 + 0.888835i 0.174941 + 0.595795i −0.580057 − 0.814576i 0.488657 − 2.18202i −0.609726 − 0.117515i 3.39679 − 0.161809i 0.989821 − 0.142315i 2.19939 − 1.41346i 1.71554 + 1.43420i
19.11 −0.458227 + 0.888835i 0.248806 + 0.847354i −0.580057 − 0.814576i −2.23355 + 0.106064i −0.867168 − 0.167133i −3.44178 + 0.163952i 0.989821 − 0.142315i 1.86766 − 1.20027i 0.929199 − 2.03386i
19.12 −0.458227 + 0.888835i 0.374339 + 1.27488i −0.580057 − 0.814576i −0.813276 − 2.08293i −1.30469 − 0.251459i −1.28330 + 0.0611313i 0.989821 − 0.142315i 1.03857 − 0.667447i 2.22404 + 0.231583i
19.13 −0.458227 + 0.888835i 0.412542 + 1.40499i −0.580057 − 0.814576i 2.16096 − 0.574680i −1.43784 − 0.277121i −1.79805 + 0.0856515i 0.989821 − 0.142315i 0.719962 − 0.462691i −0.479412 + 2.18407i
19.14 −0.458227 + 0.888835i 0.635292 + 2.16361i −0.580057 − 0.814576i 1.91973 + 1.14657i −2.21420 − 0.426752i 3.52324 − 0.167833i 0.989821 − 0.142315i −1.75384 + 1.12712i −1.89879 + 1.18094i
19.15 −0.458227 + 0.888835i 0.675350 + 2.30003i −0.580057 − 0.814576i 1.25563 + 1.85024i −2.35381 − 0.453660i −3.39611 + 0.161776i 0.989821 − 0.142315i −2.31028 + 1.48473i −2.21992 + 0.268215i
19.16 −0.458227 + 0.888835i 0.751879 + 2.56067i −0.580057 − 0.814576i −1.56562 − 1.59651i −2.62054 − 0.505068i 3.80383 − 0.181199i 0.989821 − 0.142315i −3.46792 + 2.22870i 2.13644 − 0.660013i
19.17 −0.458227 + 0.888835i 0.784396 + 2.67141i −0.580057 − 0.814576i −2.11724 + 0.719224i −2.73387 − 0.526911i 0.604390 − 0.0287907i 0.989821 − 0.142315i −3.99738 + 2.56896i 0.330905 − 2.21145i
19.18 0.458227 − 0.888835i −0.784396 − 2.67141i −0.580057 − 0.814576i 1.01322 − 1.99334i −2.73387 − 0.526911i −0.604390 + 0.0287907i −0.989821 + 0.142315i −3.99738 + 2.56896i −1.30746 − 1.81398i
19.19 0.458227 − 0.888835i −0.751879 − 2.56067i −0.580057 − 0.814576i −1.35745 − 1.77689i −2.62054 − 0.505068i −3.80383 + 0.181199i −0.989821 + 0.142315i −3.46792 + 2.22870i −2.20138 + 0.392331i
19.20 0.458227 − 0.888835i −0.675350 − 2.30003i −0.580057 − 0.814576i 1.65272 + 1.50616i −2.35381 − 0.453660i 3.39611 − 0.161776i −0.989821 + 0.142315i −2.31028 + 1.48473i 2.09605 − 0.778829i
See next 80 embeddings (of 680 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 19.34
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
67.g even 33 1 inner
335.u even 66 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 670.2.v.a ✓ 680
5.b even 2 1 inner 670.2.v.a ✓ 680
67.g even 33 1 inner 670.2.v.a ✓ 680
335.u even 66 1 inner 670.2.v.a ✓ 680
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
670.2.v.a ✓ 680 1.a even 1 1 trivial
670.2.v.a ✓ 680 5.b even 2 1 inner
670.2.v.a ✓ 680 67.g even 33 1 inner
670.2.v.a ✓ 680 335.u even 66 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(670, [\chi])\).