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Properties

Label 240.4.y.b
Level $240$
Weight $4$
Character orbit 240.y
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(163,240)]
 
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, 3, 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("240.163");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 240 = 2^{4} \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 240.y (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.

\(\operatorname{Tr}(f)(q) = \) \( 72 q + 216 q^{3} + 10 q^{4} - 42 q^{8} + 648 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 72 q + 216 q^{3} + 10 q^{4} - 42 q^{8} + 648 q^{9} - 12 q^{10} + 30 q^{12} + 110 q^{14} - 22 q^{16} - 124 q^{17} - 12 q^{19} + 174 q^{20} + 250 q^{22} - 88 q^{23} - 126 q^{24} + 184 q^{25} + 12 q^{26} + 1944 q^{27} - 90 q^{28} - 36 q^{30} - 170 q^{32} + 186 q^{34} + 228 q^{35} + 90 q^{36} + 1258 q^{38} + 1214 q^{40} + 330 q^{42} - 1770 q^{44} + 1106 q^{46} - 896 q^{47} - 66 q^{48} - 2172 q^{50} - 372 q^{51} - 2192 q^{52} - 1112 q^{53} + 688 q^{55} + 622 q^{56} - 36 q^{57} - 1210 q^{58} + 688 q^{59} + 522 q^{60} + 2552 q^{61} - 2004 q^{62} + 634 q^{64} + 340 q^{65} + 750 q^{66} - 1470 q^{68} - 264 q^{69} - 1118 q^{70} + 224 q^{71} - 378 q^{72} - 296 q^{73} - 1040 q^{74} + 552 q^{75} - 3766 q^{76} + 36 q^{78} - 928 q^{79} - 4038 q^{80} + 5832 q^{81} - 2960 q^{82} + 2680 q^{83} - 270 q^{84} - 1844 q^{86} - 2078 q^{88} + 1968 q^{89} - 108 q^{90} - 848 q^{91} + 1666 q^{92} + 310 q^{94} + 1240 q^{95} - 510 q^{96} - 1176 q^{97} + 250 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
163.1 −2.82818 + 0.0374011i 3.00000 7.99720 − 0.211554i −1.44584 + 11.0865i −8.48454 + 0.112203i 7.76015 − 7.76015i −22.6096 + 0.897418i 9.00000 3.67445 − 31.4086i
163.2 −2.81591 − 0.265755i 3.00000 7.85875 + 1.49669i 2.98740 − 10.7738i −8.44774 − 0.797266i 6.90177 − 6.90177i −21.7318 − 6.30305i 9.00000 −11.2755 + 29.5443i
163.3 −2.81440 − 0.281351i 3.00000 7.84168 + 1.58367i −10.1697 − 4.64519i −8.44320 − 0.844054i −23.8051 + 23.8051i −21.6241 − 6.66335i 9.00000 27.3146 + 15.9347i
163.4 −2.57897 − 1.16142i 3.00000 5.30219 + 5.99056i 10.9225 − 2.38736i −7.73691 − 3.48427i −20.3288 + 20.3288i −6.71660 − 21.6076i 9.00000 −30.9415 − 6.52871i
163.5 −2.55391 − 1.21555i 3.00000 5.04487 + 6.20881i −10.9300 − 2.35285i −7.66172 − 3.64665i 23.7224 − 23.7224i −5.33701 − 21.9890i 9.00000 25.0541 + 19.2949i
163.6 −2.53693 + 1.25059i 3.00000 4.87203 − 6.34534i 4.38137 + 10.2861i −7.61079 + 3.75178i −18.0427 + 18.0427i −4.42456 + 22.1906i 9.00000 −23.9790 − 20.6158i
163.7 −2.52720 + 1.27014i 3.00000 4.77348 − 6.41981i 5.35841 − 9.81262i −7.58160 + 3.81043i 4.30891 − 4.30891i −3.90946 + 22.2871i 9.00000 −1.07836 + 31.6044i
163.8 −2.44433 + 1.42312i 3.00000 3.94945 − 6.95714i −11.1259 − 1.10167i −7.33298 + 4.26937i 2.79711 − 2.79711i 0.247124 + 22.6261i 9.00000 28.7632 − 13.1407i
163.9 −2.25699 − 1.70470i 3.00000 2.18803 + 7.69497i 8.22399 + 7.57403i −6.77098 − 5.11409i 6.62058 − 6.62058i 8.17921 − 21.0974i 9.00000 −5.65008 − 31.1139i
163.10 −1.90484 + 2.09083i 3.00000 −0.743142 − 7.96541i 10.3746 + 4.16734i −5.71453 + 6.27249i 22.3735 − 22.3735i 18.0699 + 13.6191i 9.00000 −28.4753 + 13.7535i
163.11 −1.65664 − 2.29250i 3.00000 −2.51112 + 7.59567i −7.51192 + 8.28076i −4.96991 − 6.87750i −6.35938 + 6.35938i 21.5731 − 6.82651i 9.00000 31.4282 + 3.50288i
163.12 −1.41915 − 2.44663i 3.00000 −3.97203 + 6.94428i −5.01643 − 9.99177i −4.25745 − 7.33990i −11.2717 + 11.2717i 22.6270 − 0.136854i 9.00000 −17.3271 + 26.4532i
163.13 −1.29187 + 2.51616i 3.00000 −4.66216 − 6.50110i −6.94162 − 8.76435i −3.87560 + 7.54849i −6.47277 + 6.47277i 22.3807 − 3.33220i 9.00000 31.0202 − 6.14387i
163.14 −1.06326 + 2.62097i 3.00000 −5.73896 − 5.57354i 10.9136 − 2.42765i −3.18978 + 7.86291i −14.0460 + 14.0460i 20.7101 − 9.11553i 9.00000 −5.24118 + 31.1854i
163.15 −1.02139 − 2.63757i 3.00000 −5.91351 + 5.38799i 9.39581 − 6.05960i −3.06418 − 7.91270i 19.9961 − 19.9961i 20.2512 + 10.0940i 9.00000 −25.5794 − 18.5928i
163.16 −0.773864 + 2.72050i 3.00000 −6.80227 − 4.21060i −8.59682 + 7.14805i −2.32159 + 8.16151i 3.15918 − 3.15918i 16.7190 − 15.2472i 9.00000 −12.7935 − 28.9193i
163.17 −0.375365 − 2.80341i 3.00000 −7.71820 + 2.10460i 10.3938 − 4.11941i −1.12610 − 8.41023i −9.25658 + 9.25658i 8.79721 + 20.8473i 9.00000 −15.4498 − 27.5917i
163.18 0.0922271 + 2.82692i 3.00000 −7.98299 + 0.521438i 6.69669 + 8.95290i 0.276681 + 8.48077i 10.4217 − 10.4217i −2.21031 − 22.5192i 9.00000 −24.6915 + 19.7567i
163.19 0.157618 − 2.82403i 3.00000 −7.95031 − 0.890238i −10.7828 + 2.95478i 0.472855 − 8.47210i 0.110613 − 0.110613i −3.76718 + 22.3116i 9.00000 6.64483 + 30.9168i
163.20 0.515716 + 2.78101i 3.00000 −7.46807 + 2.86842i 1.24445 − 11.1109i 1.54715 + 8.34304i −9.00178 + 9.00178i −11.8285 − 19.2895i 9.00000 31.5413 − 2.26922i
See all 72 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 163.36
Significant digits:
Format:

Inner twists

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

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{72} - 296 T_{7}^{69} + 4866416 T_{7}^{68} + 3147680 T_{7}^{67} + 43808 T_{7}^{66} + \cdots + 67\!\cdots\!00 \) acting on \(S_{4}^{\mathrm{new}}(240, [\chi])\). Copy content Toggle raw display