Properties

Label 240.4.bc.b
Level $240$
Weight $4$
Character orbit 240.bc
Analytic conductor $14.160$
Analytic rank $0$
Dimension $72$
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] = [240,4,Mod(43,240)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(240, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 1, 0, 3]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("240.43");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 240 = 2^{4} \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 240.bc (of order \(4\), degree \(2\), 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: \(14.1604584014\)
Analytic rank: \(0\)
Dimension: \(72\)
Relative dimension: \(36\) over \(\Q(i)\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 72 q + 2 q^{2} - 2 q^{4} - 4 q^{8} - 648 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 72 q + 2 q^{2} - 2 q^{4} - 4 q^{8} - 648 q^{9} + 12 q^{12} + 110 q^{14} + 154 q^{16} + 124 q^{17} - 18 q^{18} + 12 q^{19} - 448 q^{20} - 238 q^{22} + 88 q^{23} - 126 q^{24} + 184 q^{25} - 12 q^{26} + 102 q^{28} - 462 q^{30} - 808 q^{32} + 806 q^{34} + 228 q^{35} + 18 q^{36} - 198 q^{38} - 26 q^{40} + 642 q^{42} - 432 q^{43} + 294 q^{44} - 1118 q^{46} - 80 q^{47} - 96 q^{48} - 1666 q^{50} - 372 q^{51} - 1272 q^{52} - 688 q^{55} - 286 q^{56} - 36 q^{57} - 1486 q^{58} - 688 q^{59} + 570 q^{60} - 1640 q^{61} + 836 q^{62} + 862 q^{64} - 340 q^{65} + 618 q^{66} - 1848 q^{67} + 3954 q^{68} + 264 q^{69} + 1378 q^{70} + 224 q^{71} + 36 q^{72} + 296 q^{73} + 1296 q^{74} - 552 q^{75} + 1250 q^{76} + 588 q^{78} - 928 q^{79} - 1540 q^{80} + 5832 q^{81} - 2928 q^{82} - 342 q^{84} + 3424 q^{85} + 3908 q^{86} - 2546 q^{88} + 1968 q^{89} - 848 q^{91} - 266 q^{92} + 792 q^{93} - 1406 q^{94} - 1240 q^{95} + 510 q^{96} + 1176 q^{97} + 6788 q^{98}+O(q^{100}) \) Copy content Toggle raw display

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
43.1 −2.80743 + 0.344004i 3.00000i 7.76332 1.93153i −8.98271 + 6.65664i 1.03201 + 8.42229i −25.3381 + 25.3381i −21.1305 + 8.09326i −9.00000 22.9284 21.7781i
43.2 −2.77701 + 0.536838i 3.00000i 7.42361 2.98161i 8.09867 7.70789i 1.61051 + 8.33104i −10.5146 + 10.5146i −19.0148 + 12.2653i −9.00000 −18.3522 + 25.7526i
43.3 −2.75906 0.622574i 3.00000i 7.22480 + 3.43543i −5.62604 + 9.66166i −1.86772 + 8.27717i 17.9728 17.9728i −17.7948 13.9765i −9.00000 21.5377 23.1544i
43.4 −2.73499 0.720989i 3.00000i 6.96035 + 3.94380i 10.6538 + 3.39051i −2.16297 + 8.20497i 4.24019 4.24019i −16.1931 15.8046i −9.00000 −26.6937 16.9543i
43.5 −2.72114 + 0.771611i 3.00000i 6.80923 4.19933i −7.55101 8.24514i 2.31483 + 8.16343i 21.3060 21.3060i −15.2886 + 16.6810i −9.00000 26.9094 + 16.6098i
43.6 −2.56229 1.19778i 3.00000i 5.13063 + 6.13813i −6.29244 9.24149i −3.59335 + 7.68686i −4.87036 + 4.87036i −5.79401 21.8730i −9.00000 5.05375 + 31.2163i
43.7 −2.40790 + 1.48391i 3.00000i 3.59601 7.14624i 5.93185 + 9.47698i 4.45174 + 7.22371i 2.77829 2.77829i 1.94555 + 22.5436i −9.00000 −28.3463 14.0173i
43.8 −2.09540 1.89981i 3.00000i 0.781412 + 7.96175i −5.08323 + 9.95795i −5.69944 + 6.28620i 4.30573 4.30573i 13.4885 18.1676i −9.00000 29.5697 11.2087i
43.9 −2.03968 + 1.95951i 3.00000i 0.320620 7.99357i −10.5032 + 3.83193i 5.87854 + 6.11905i −0.263728 + 0.263728i 15.0095 + 16.9326i −9.00000 13.9144 28.3970i
43.10 −1.99616 2.00384i 3.00000i −0.0307262 + 7.99994i 11.1550 0.751921i −6.01151 + 5.98847i −15.9673 + 15.9673i 16.0919 15.9076i −9.00000 −23.7739 20.8519i
43.11 −1.55489 + 2.36270i 3.00000i −3.16466 7.34744i 9.88881 5.21646i 7.08809 + 4.66466i 14.5115 14.5115i 22.2805 + 3.94729i −9.00000 −3.05106 + 31.4752i
43.12 −1.22495 2.54941i 3.00000i −4.99900 + 6.24580i −11.1782 + 0.219756i −7.64824 + 3.67485i −2.70903 + 2.70903i 22.0466 + 5.09373i −9.00000 14.2529 + 28.2286i
43.13 −1.09484 2.60794i 3.00000i −5.60265 + 5.71054i 4.50871 10.2309i −7.82381 + 3.28452i 13.9744 13.9744i 21.0267 + 8.35923i −9.00000 −31.6179 0.557207i
43.14 −0.858261 2.69507i 3.00000i −6.52678 + 4.62614i 8.57292 + 7.17671i −8.08520 + 2.57478i 10.0043 10.0043i 18.0694 + 13.6197i −9.00000 11.9839 29.2641i
43.15 −0.840991 + 2.70051i 3.00000i −6.58547 4.54220i 2.05633 + 10.9896i 8.10152 + 2.52297i −9.67693 + 9.67693i 17.8046 13.9642i −9.00000 −31.4069 3.68901i
43.16 −0.704829 + 2.73920i 3.00000i −7.00643 3.86134i −5.93312 9.47618i 8.21760 + 2.11449i −18.7256 + 18.7256i 15.5153 16.4704i −9.00000 30.1390 9.57292i
43.17 −0.548362 + 2.77476i 3.00000i −7.39860 3.04315i −11.0689 1.57442i 8.32428 + 1.64509i 21.0997 21.0997i 12.5011 18.8606i −9.00000 10.4384 29.8503i
43.18 −0.0874600 + 2.82707i 3.00000i −7.98470 0.494512i 10.8481 2.70533i 8.48122 + 0.262380i −4.78389 + 4.78389i 2.09637 22.5301i −9.00000 6.69940 + 30.9050i
43.19 0.148945 2.82450i 3.00000i −7.95563 0.841393i 2.45210 + 10.9081i −8.47351 0.446836i −14.3673 + 14.3673i −3.56147 + 22.3454i −9.00000 31.1753 5.30125i
43.20 0.305329 2.81190i 3.00000i −7.81355 1.71711i 9.00133 6.63144i −8.43570 0.915987i −20.8231 + 20.8231i −7.21404 + 21.4466i −9.00000 −15.8986 27.3356i
See all 72 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 43.36
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
80.j even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 240.4.bc.b yes 72
5.c odd 4 1 240.4.y.a 72
16.f odd 4 1 240.4.y.a 72
80.j even 4 1 inner 240.4.bc.b yes 72
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
240.4.y.a 72 5.c odd 4 1
240.4.y.a 72 16.f odd 4 1
240.4.bc.b yes 72 1.a even 1 1 trivial
240.4.bc.b yes 72 80.j even 4 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{72} - 10888 T_{7}^{69} + 5505904 T_{7}^{68} - 13062816 T_{7}^{67} + 59274272 T_{7}^{66} - 44819447360 T_{7}^{65} + 12407660711680 T_{7}^{64} - 54447132823488 T_{7}^{63} + \cdots + 92\!\cdots\!00 \) acting on \(S_{4}^{\mathrm{new}}(240, [\chi])\). Copy content Toggle raw display