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Properties

Label 720.2.cm.a
Level $720$
Weight $2$
Character orbit 720.cm
Analytic conductor $5.749$
Analytic rank $0$
Dimension $560$
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] = [720,2,Mod(77,720)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(720, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([0, 9, 10, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("720.77");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 720 = 2^{4} \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 720.cm (of order \(12\), degree \(4\), 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: \(5.74922894553\)
Analytic rank: \(0\)
Dimension: \(560\)
Relative dimension: \(140\) over \(\Q(\zeta_{12})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 560 q - 6 q^{2} - 8 q^{3} - 6 q^{5} - 8 q^{6}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 560 q - 6 q^{2} - 8 q^{3} - 6 q^{5} - 8 q^{6} - 16 q^{10} - 12 q^{11} + 2 q^{12} - 4 q^{13} - 8 q^{15} - 4 q^{16} - 36 q^{18} - 6 q^{20} - 8 q^{21} - 10 q^{22} + 24 q^{24} - 8 q^{27} + 8 q^{28} + 4 q^{30} - 8 q^{31} - 66 q^{32} - 8 q^{33} + 12 q^{34} + 8 q^{36} - 16 q^{37} + 6 q^{38} + 24 q^{39} - 12 q^{40} + 14 q^{42} - 14 q^{45} - 40 q^{46} - 12 q^{47} - 38 q^{48} - 6 q^{50} - 8 q^{51} + 14 q^{52} - 24 q^{54} - 12 q^{56} - 24 q^{57} + 6 q^{58} - 16 q^{60} - 4 q^{61} + 20 q^{63} + 24 q^{64} - 12 q^{65} + 24 q^{66} - 30 q^{68} - 12 q^{69} - 6 q^{70} + 18 q^{72} - 24 q^{74} - 54 q^{75} + 8 q^{76} + 30 q^{78} - 16 q^{81} - 12 q^{83} + 12 q^{84} + 8 q^{85} - 12 q^{86} - 12 q^{87} - 42 q^{88} - 46 q^{90} - 16 q^{91} - 120 q^{92} + 4 q^{93} - 12 q^{95} + 48 q^{96} - 4 q^{97} - 32 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
77.1 −1.41417 − 0.0109957i 1.25960 + 1.18887i 1.99976 + 0.0310996i 1.57532 − 1.58694i −1.76822 − 1.69511i 4.22893 + 1.13314i −2.82766 − 0.0659689i 0.173199 + 2.99500i −2.24522 + 2.22688i
77.2 −1.41326 + 0.0519208i 1.61238 − 0.632630i 1.99461 − 0.146755i 0.330399 − 2.21152i −2.24587 + 0.977787i −3.57547 − 0.958046i −2.81128 + 0.310965i 2.19956 − 2.04008i −0.352116 + 3.14261i
77.3 −1.40778 − 0.134756i 1.27948 + 1.16745i 1.96368 + 0.379412i −1.82946 − 1.28573i −1.64390 − 1.81593i −1.09060 − 0.292224i −2.71330 − 0.798745i 0.274120 + 2.98745i 2.40221 + 2.05655i
77.4 −1.40575 − 0.154481i −0.0547502 − 1.73119i 1.95227 + 0.434323i 2.09681 − 0.776792i −0.190470 + 2.44207i 2.64311 + 0.708218i −2.67731 − 0.912138i −2.99400 + 0.189565i −3.06759 + 0.768059i
77.5 −1.40393 + 0.170197i −1.55632 − 0.760167i 1.94207 − 0.477892i 1.12540 + 1.93222i 2.31436 + 0.802343i −3.95591 − 1.05998i −2.64520 + 1.00146i 1.84429 + 2.36613i −1.90885 − 2.52117i
77.6 −1.39746 + 0.217026i 0.183300 + 1.72232i 1.90580 − 0.606570i 2.19281 − 0.437727i −0.629943 − 2.36710i −2.41467 − 0.647010i −2.53164 + 1.26127i −2.93280 + 0.631404i −2.96936 + 1.08760i
77.7 −1.39710 − 0.219335i −1.69694 + 0.346998i 1.90378 + 0.612867i −2.07771 + 0.826514i 2.44690 − 0.112594i −0.910045 − 0.243846i −2.52536 − 1.27380i 2.75918 − 1.17767i 3.08405 − 0.699009i
77.8 −1.39366 − 0.240220i 1.72701 − 0.132011i 1.88459 + 0.669572i 1.88514 + 1.20259i −2.43858 − 0.230884i −1.44560 − 0.387347i −2.46563 − 1.38587i 2.96515 − 0.455971i −2.33837 − 2.12886i
77.9 −1.38934 + 0.264092i 1.72872 + 0.107412i 1.86051 − 0.733824i −0.423799 + 2.19554i −2.43014 + 0.307308i 2.83940 + 0.760814i −2.39108 + 1.51087i 2.97693 + 0.371369i 0.00897620 − 3.16226i
77.10 −1.38896 − 0.266060i 0.346801 − 1.69698i 1.85842 + 0.739093i −0.183196 + 2.22855i −0.933190 + 2.26476i −1.37032 − 0.367175i −2.38464 − 1.52102i −2.75946 − 1.17703i 0.847380 − 3.04663i
77.11 −1.38893 + 0.266212i −1.65449 − 0.512493i 1.85826 − 0.739500i −0.907381 − 2.04369i 2.43441 + 0.271372i −0.679044 − 0.181949i −2.38414 + 1.52181i 2.47470 + 1.69583i 1.80434 + 2.59699i
77.12 −1.37882 + 0.314419i −0.854751 − 1.50645i 1.80228 − 0.867054i −2.23372 + 0.102340i 1.65220 + 1.80838i 2.49388 + 0.668233i −2.21240 + 1.76218i −1.53880 + 2.57528i 3.04772 − 0.843434i
77.13 −1.37215 + 0.342348i −0.951085 + 1.44756i 1.76560 − 0.939507i −0.442824 − 2.19178i 0.809461 − 2.31188i 3.25193 + 0.871351i −2.10102 + 1.89359i −1.19088 − 2.75351i 1.35797 + 2.85585i
77.14 −1.36540 − 0.368335i −0.295293 + 1.70669i 1.72866 + 1.00585i −2.23293 + 0.118386i 1.03183 − 2.22156i 2.12221 + 0.568645i −1.98983 − 2.01012i −2.82560 − 1.00795i 3.09246 + 0.660821i
77.15 −1.36501 + 0.369788i 0.815246 + 1.52819i 1.72651 − 1.00953i −1.56779 + 1.59437i −1.67793 − 1.78453i −3.90351 − 1.04594i −1.98340 + 2.01646i −1.67075 + 2.49171i 1.55047 − 2.75609i
77.16 −1.33899 + 0.455101i 1.25816 − 1.19039i 1.58577 − 1.21875i −2.23363 + 0.104330i −1.14291 + 2.16651i 2.66693 + 0.714601i −1.56867 + 2.35357i 0.165942 − 2.99541i 2.94332 − 1.15622i
77.17 −1.32926 + 0.482769i −1.03780 + 1.38671i 1.53387 − 1.28345i 0.432247 + 2.19389i 0.710052 − 2.34432i 0.709427 + 0.190090i −1.41930 + 2.44655i −0.845924 − 2.87827i −1.63371 − 2.70758i
77.18 −1.31064 − 0.531247i −1.66067 + 0.492103i 1.43555 + 1.39255i 1.39125 + 1.75055i 2.43797 + 0.237257i 4.31147 + 1.15526i −1.14171 − 2.58776i 2.51567 − 1.63445i −0.893451 − 3.03344i
77.19 −1.30840 − 0.536728i 1.08184 − 1.35263i 1.42385 + 1.40452i −0.871024 − 2.05945i −2.14148 + 1.18914i 2.15913 + 0.578537i −1.10912 − 2.60189i −0.659238 − 2.92667i 0.0342894 + 3.16209i
77.20 −1.30687 − 0.540447i −1.46401 + 0.925575i 1.41583 + 1.41259i 0.711505 − 2.11985i 2.41349 − 0.418392i −3.06030 − 0.820005i −1.08689 − 2.61126i 1.28662 − 2.71009i −2.07551 + 2.38584i
See next 80 embeddings (of 560 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 77.140
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.d odd 6 1 inner
80.t odd 4 1 inner
720.cm even 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 720.2.cm.a ✓ 560
5.c odd 4 1 720.2.cq.a yes 560
9.d odd 6 1 inner 720.2.cm.a ✓ 560
16.e even 4 1 720.2.cq.a yes 560
45.l even 12 1 720.2.cq.a yes 560
80.t odd 4 1 inner 720.2.cm.a ✓ 560
144.w odd 12 1 720.2.cq.a yes 560
720.cm even 12 1 inner 720.2.cm.a ✓ 560
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
720.2.cm.a ✓ 560 1.a even 1 1 trivial
720.2.cm.a ✓ 560 9.d odd 6 1 inner
720.2.cm.a ✓ 560 80.t odd 4 1 inner
720.2.cm.a ✓ 560 720.cm even 12 1 inner
720.2.cq.a yes 560 5.c odd 4 1
720.2.cq.a yes 560 16.e even 4 1
720.2.cq.a yes 560 45.l even 12 1
720.2.cq.a yes 560 144.w odd 12 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(720, [\chi])\).