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Properties

Label 539.2.s.d
Level $539$
Weight $2$
Character orbit 539.s
Analytic conductor $4.304$
Analytic rank $0$
Dimension $48$
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] = [539,2,Mod(19,539)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(539, base_ring=CyclotomicField(30))
 
chi = DirichletCharacter(H, H._module([25, 9]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("539.19");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 539 = 7^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 539.s (of order \(30\), degree \(8\), not 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: \(4.30393666895\)
Analytic rank: \(0\)
Dimension: \(48\)
Relative dimension: \(6\) over \(\Q(\zeta_{30})\)
Twist minimal: no (minimal twist has level 77)
Sato-Tate group: $\mathrm{SU}(2)[C_{30}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 48 q - 5 q^{2} + 9 q^{3} - 9 q^{4} + 15 q^{5} - 11 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 48 q - 5 q^{2} + 9 q^{3} - 9 q^{4} + 15 q^{5} - 11 q^{9} - q^{11} + 12 q^{12} - 27 q^{16} - 15 q^{17} + 20 q^{18} + 15 q^{19} - 76 q^{22} + 10 q^{23} - 75 q^{24} + q^{25} - 27 q^{26} - 40 q^{29} + 25 q^{30} - 9 q^{31} - 42 q^{33} - 38 q^{36} - q^{37} - 33 q^{38} - 45 q^{39} - 75 q^{40} + 30 q^{44} + 84 q^{45} - 20 q^{46} - 3 q^{47} + 30 q^{50} + 55 q^{51} + 15 q^{52} - 3 q^{53} + 60 q^{57} + 46 q^{58} + 3 q^{59} - 15 q^{60} + 30 q^{61} + 12 q^{64} + 93 q^{66} + 44 q^{67} + 75 q^{68} + 20 q^{71} - 60 q^{72} + 60 q^{73} + 45 q^{74} + 57 q^{75} + 92 q^{78} - 70 q^{79} + 75 q^{80} - 29 q^{81} + 129 q^{82} + 10 q^{85} - 62 q^{86} + 19 q^{88} - 6 q^{89} + 30 q^{92} - 92 q^{93} - 105 q^{94} + 30 q^{95} - 75 q^{96} - 12 q^{99}+O(q^{100}) \) 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 −2.06490 + 0.217030i 0.138863 + 0.125032i 2.26043 − 0.480470i 1.04356 + 2.34387i −0.313874 − 0.228043i 0 −0.613987 + 0.199497i −0.309936 − 2.94884i −2.66354 − 4.61339i
19.2 −1.30277 + 0.136927i −2.02980 − 1.82764i −0.277833 + 0.0590552i −0.166444 − 0.373838i 2.89461 + 2.10306i 0 2.84553 − 0.924570i 0.466233 + 4.43591i 0.268026 + 0.464235i
19.3 −0.476751 + 0.0501086i 1.66384 + 1.49813i −1.73151 + 0.368045i 0.547175 + 1.22898i −0.868306 − 0.630861i 0 1.71889 − 0.558501i 0.210388 + 2.00171i −0.322449 − 0.558497i
19.4 1.11135 − 0.116808i −1.15989 − 1.04437i −0.734836 + 0.156194i 1.25246 + 2.81307i −1.41104 − 1.02518i 0 −2.92398 + 0.950058i −0.0589482 − 0.560855i 1.72051 + 2.98001i
19.5 1.39661 − 0.146790i −0.151995 − 0.136856i −0.0273118 + 0.00580530i −0.618266 − 1.38865i −0.232367 − 0.168824i 0 −2.70844 + 0.880026i −0.309213 − 2.94196i −1.06732 − 1.84865i
19.6 2.05902 − 0.216412i 2.33449 + 2.10199i 2.23645 − 0.475371i −0.483200 − 1.08528i 5.26167 + 3.82283i 0 0.563952 − 0.183239i 0.717924 + 6.83059i −1.22979 − 2.13006i
68.1 −1.95877 + 1.76368i −0.650787 + 1.46169i 0.517138 − 4.92024i −0.446428 + 2.10028i −1.30322 − 4.01090i 0 4.56624 + 6.28489i 0.294372 + 0.326934i −2.82977 − 4.90131i
68.2 −1.33360 + 1.20078i 1.17499 − 2.63908i 0.127565 − 1.21370i −0.418270 + 1.96780i 1.60198 + 4.93040i 0 −0.822345 − 1.13186i −3.57672 − 3.97235i −1.80510 − 3.12652i
68.3 −1.11268 + 1.00186i −0.245339 + 0.551041i 0.0252713 − 0.240440i 0.491349 − 2.31161i −0.279082 − 0.858925i 0 −1.54736 − 2.12976i 1.76394 + 1.95905i 1.76920 + 3.06434i
68.4 −0.0202070 + 0.0181945i 0.500742 − 1.12469i −0.208980 + 1.98831i 0.240558 − 1.13174i 0.0103446 + 0.0318373i 0 −0.0639186 − 0.0879764i 0.993218 + 1.10308i 0.0157304 + 0.0272459i
68.5 0.386517 − 0.348022i −0.460952 + 1.03532i −0.180780 + 1.72001i −0.678628 + 3.19269i 0.182146 + 0.560588i 0 1.14015 + 1.56929i 1.14799 + 1.27497i 0.848825 + 1.47021i
68.6 1.75157 − 1.57712i 0.821231 − 1.84451i 0.371633 − 3.53585i 0.00617973 − 0.0290734i −1.47058 − 4.52598i 0 −2.15474 − 2.96574i −0.720422 − 0.800109i −0.0350280 − 0.0606703i
117.1 −0.842093 − 1.89137i −0.653128 − 3.07273i −1.52991 + 1.69913i −1.18148 + 0.124179i −5.26167 + 3.82283i 0 0.563952 + 0.183239i −6.27443 + 2.79356i 1.22979 + 2.13006i
117.2 −0.571183 − 1.28290i 0.0425239 + 0.200059i 0.0186834 − 0.0207501i −1.51174 + 0.158890i 0.232367 − 0.168824i 0 −2.70844 − 0.880026i 2.70242 − 1.20320i 1.06732 + 1.84865i
117.3 −0.454517 − 1.02086i 0.324506 + 1.52668i 0.502686 − 0.558290i 3.06242 − 0.321873i 1.41104 − 1.02518i 0 −2.92398 − 0.950058i 0.515189 − 0.229377i −1.72051 − 2.98001i
117.4 0.194980 + 0.437933i −0.465496 − 2.18999i 1.18449 − 1.31551i 1.33791 − 0.140620i 0.868306 − 0.630861i 0 1.71889 + 0.558501i −1.83873 + 0.818654i 0.322449 + 0.558497i
117.5 0.532803 + 1.19670i 0.567882 + 2.67168i 0.190060 − 0.211083i −0.406975 + 0.0427748i −2.89461 + 2.10306i 0 2.84553 + 0.924570i −4.07472 + 1.81418i −0.268026 − 0.464235i
117.6 0.844499 + 1.89678i −0.0388500 − 0.182775i −1.54632 + 1.71736i 2.55163 − 0.268187i 0.313874 − 0.228043i 0 −0.613987 − 0.199497i 2.70874 − 1.20601i 2.66354 + 4.61339i
129.1 −0.842093 + 1.89137i −0.653128 + 3.07273i −1.52991 − 1.69913i −1.18148 − 0.124179i −5.26167 − 3.82283i 0 0.563952 − 0.183239i −6.27443 − 2.79356i 1.22979 − 2.13006i
129.2 −0.571183 + 1.28290i 0.0425239 − 0.200059i 0.0186834 + 0.0207501i −1.51174 − 0.158890i 0.232367 + 0.168824i 0 −2.70844 + 0.880026i 2.70242 + 1.20320i 1.06732 − 1.84865i
See all 48 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 19.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.d odd 6 1 inner
11.d odd 10 1 inner
77.n even 30 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 539.2.s.d 48
7.b odd 2 1 77.2.n.a ✓ 48
7.c even 3 1 77.2.n.a ✓ 48
7.c even 3 1 539.2.m.a 48
7.d odd 6 1 539.2.m.a 48
7.d odd 6 1 inner 539.2.s.d 48
11.d odd 10 1 inner 539.2.s.d 48
21.c even 2 1 693.2.cg.a 48
21.h odd 6 1 693.2.cg.a 48
77.b even 2 1 847.2.r.c 48
77.h odd 6 1 847.2.r.c 48
77.j odd 10 1 847.2.i.b 48
77.j odd 10 1 847.2.r.a 48
77.j odd 10 1 847.2.r.c 48
77.j odd 10 1 847.2.r.d 48
77.l even 10 1 77.2.n.a ✓ 48
77.l even 10 1 847.2.i.b 48
77.l even 10 1 847.2.r.a 48
77.l even 10 1 847.2.r.d 48
77.m even 15 1 847.2.i.b 48
77.m even 15 1 847.2.r.a 48
77.m even 15 1 847.2.r.c 48
77.m even 15 1 847.2.r.d 48
77.n even 30 1 539.2.m.a 48
77.n even 30 1 inner 539.2.s.d 48
77.o odd 30 1 77.2.n.a ✓ 48
77.o odd 30 1 539.2.m.a 48
77.o odd 30 1 847.2.i.b 48
77.o odd 30 1 847.2.r.a 48
77.o odd 30 1 847.2.r.d 48
231.r odd 10 1 693.2.cg.a 48
231.be even 30 1 693.2.cg.a 48
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
77.2.n.a ✓ 48 7.b odd 2 1
77.2.n.a ✓ 48 7.c even 3 1
77.2.n.a ✓ 48 77.l even 10 1
77.2.n.a ✓ 48 77.o odd 30 1
539.2.m.a 48 7.c even 3 1
539.2.m.a 48 7.d odd 6 1
539.2.m.a 48 77.n even 30 1
539.2.m.a 48 77.o odd 30 1
539.2.s.d 48 1.a even 1 1 trivial
539.2.s.d 48 7.d odd 6 1 inner
539.2.s.d 48 11.d odd 10 1 inner
539.2.s.d 48 77.n even 30 1 inner
693.2.cg.a 48 21.c even 2 1
693.2.cg.a 48 21.h odd 6 1
693.2.cg.a 48 231.r odd 10 1
693.2.cg.a 48 231.be even 30 1
847.2.i.b 48 77.j odd 10 1
847.2.i.b 48 77.l even 10 1
847.2.i.b 48 77.m even 15 1
847.2.i.b 48 77.o odd 30 1
847.2.r.a 48 77.j odd 10 1
847.2.r.a 48 77.l even 10 1
847.2.r.a 48 77.m even 15 1
847.2.r.a 48 77.o odd 30 1
847.2.r.c 48 77.b even 2 1
847.2.r.c 48 77.h odd 6 1
847.2.r.c 48 77.j odd 10 1
847.2.r.c 48 77.m even 15 1
847.2.r.d 48 77.j odd 10 1
847.2.r.d 48 77.l even 10 1
847.2.r.d 48 77.m even 15 1
847.2.r.d 48 77.o odd 30 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{48} + 5 T_{2}^{47} + 23 T_{2}^{46} + 60 T_{2}^{45} + 142 T_{2}^{44} + 195 T_{2}^{43} + 222 T_{2}^{42} + \cdots + 1 \) acting on \(S_{2}^{\mathrm{new}}(539, [\chi])\). Copy content Toggle raw display