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Properties

Label 240.4.y.a
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.

\(\operatorname{Tr}(f)(q) = \) \( 72 q - 216 q^{3} + 2 q^{4} - 42 q^{8} + 648 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 72 q - 216 q^{3} + 2 q^{4} - 42 q^{8} + 648 q^{9} + 12 q^{10} - 6 q^{12} - 110 q^{14} + 154 q^{16} + 124 q^{17} - 12 q^{19} + 174 q^{20} - 206 q^{22} + 88 q^{23} + 126 q^{24} - 184 q^{25} - 12 q^{26} - 1944 q^{27} - 114 q^{28} - 36 q^{30} - 170 q^{32} - 806 q^{34} + 228 q^{35} + 18 q^{36} + 774 q^{38} - 386 q^{40} + 330 q^{42} - 294 q^{44} - 1118 q^{46} + 80 q^{47} - 462 q^{48} + 724 q^{50} - 372 q^{51} - 232 q^{52} + 1112 q^{53} - 688 q^{55} - 286 q^{56} + 36 q^{57} + 926 q^{58} + 688 q^{59} - 522 q^{60} - 1640 q^{61} - 604 q^{62} - 862 q^{64} - 340 q^{65} + 618 q^{66} + 6 q^{68} - 264 q^{69} - 3582 q^{70} + 224 q^{71} - 378 q^{72} - 296 q^{73} - 1296 q^{74} + 552 q^{75} + 1250 q^{76} + 36 q^{78} + 928 q^{79} - 1614 q^{80} + 5832 q^{81} - 2960 q^{82} + 2680 q^{83} + 342 q^{84} + 3908 q^{86} + 282 q^{88} - 1968 q^{89} + 108 q^{90} - 848 q^{91} + 3326 q^{92} + 1406 q^{94} + 1240 q^{95} + 510 q^{96} + 1176 q^{97} - 1514 q^{98}+O(q^{100}) \) 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 −2.82707 + 0.0874600i −3.00000 7.98470 − 0.494512i −2.70533 − 10.8481i 8.48122 − 0.262380i −4.78389 + 4.78389i −22.5301 + 2.09637i 9.00000 8.59695 + 30.4318i
163.2 −2.77476 + 0.548362i −3.00000 7.39860 − 3.04315i −1.57442 + 11.0689i 8.32428 − 1.64509i 21.0997 − 21.0997i −18.8606 + 12.5011i 9.00000 −1.70115 − 31.5770i
163.3 −2.73920 + 0.704829i −3.00000 7.00643 − 3.86134i −9.47618 + 5.93312i 8.21760 − 2.11449i −18.7256 + 18.7256i −16.4704 + 15.5153i 9.00000 21.7753 − 22.9311i
163.4 −2.70051 + 0.840991i −3.00000 6.58547 − 4.54220i 10.9896 − 2.05633i 8.10152 − 2.52297i −9.67693 + 9.67693i −13.9642 + 17.8046i 9.00000 −27.9481 + 14.7953i
163.5 −2.69764 − 0.850130i −3.00000 6.55456 + 4.58669i 10.6157 − 3.50827i 8.09293 + 2.55039i 25.0005 − 25.0005i −13.7826 − 17.9455i 9.00000 −31.6197 + 0.439373i
163.6 −2.53572 − 1.25305i −3.00000 4.85974 + 6.35476i −11.0853 − 1.45434i 7.60716 + 3.75915i 3.69418 − 3.69418i −4.36010 − 22.2034i 9.00000 26.2870 + 17.5783i
163.7 −2.45928 − 1.39711i −3.00000 4.09614 + 6.87180i −1.03010 + 11.1328i 7.37785 + 4.19134i −1.52389 + 1.52389i −0.472869 − 22.6225i 9.00000 18.0871 − 25.9395i
163.8 −2.36270 + 1.55489i −3.00000 3.16466 − 7.34744i −5.21646 − 9.88881i 7.08809 − 4.66466i 14.5115 − 14.5115i 3.94729 + 22.2805i 9.00000 27.7009 + 15.2533i
163.9 −2.22272 − 1.74915i −3.00000 1.88094 + 7.77573i 5.63719 − 9.65516i 6.66815 + 5.24745i −18.5623 + 18.5623i 9.42013 − 20.5733i 9.00000 −29.4182 + 11.6004i
163.10 −1.95951 + 2.03968i −3.00000 −0.320620 − 7.99357i 3.83193 + 10.5032i 5.87854 − 6.11905i −0.263728 + 0.263728i 16.9326 + 15.0095i 9.00000 −28.9318 − 12.7652i
163.11 −1.49689 − 2.39986i −3.00000 −3.51865 + 7.18464i 8.84524 + 6.83825i 4.49067 + 7.19958i 1.99059 − 1.99059i 22.5092 − 2.31035i 9.00000 3.17050 − 31.4634i
163.12 −1.48391 + 2.40790i −3.00000 −3.59601 − 7.14624i 9.47698 − 5.93185i 4.45174 − 7.22371i 2.77829 − 2.77829i 22.5436 + 1.94555i 9.00000 0.220326 + 31.6220i
163.13 −1.37393 − 2.47231i −3.00000 −4.22463 + 6.79356i −10.1985 − 4.58156i 4.12179 + 7.41693i −3.89626 + 3.89626i 22.6001 + 1.11072i 9.00000 2.68498 + 31.5086i
163.14 −0.771611 + 2.72114i −3.00000 −6.80923 − 4.19933i −8.24514 + 7.55101i 2.31483 − 8.16343i 21.3060 − 21.3060i 16.6810 − 15.2886i 9.00000 −14.1853 − 28.2626i
163.15 −0.756265 − 2.72545i −3.00000 −6.85613 + 4.12232i −1.38253 − 11.0945i 2.26880 + 8.17634i 18.9545 − 18.9545i 16.4202 + 15.5684i 9.00000 −29.1920 + 12.1584i
163.16 −0.536838 + 2.77701i −3.00000 −7.42361 − 2.98161i −7.70789 − 8.09867i 1.61051 − 8.33104i −10.5146 + 10.5146i 12.2653 − 19.0148i 9.00000 26.6280 − 17.0573i
163.17 −0.518338 − 2.78053i −3.00000 −7.46265 + 2.88251i −6.78809 + 8.88380i 1.55501 + 8.34158i −25.9953 + 25.9953i 11.8831 + 19.2560i 9.00000 28.2202 + 14.2696i
163.18 −0.344004 + 2.80743i −3.00000 −7.76332 − 1.93153i 6.65664 + 8.98271i 1.03201 − 8.42229i −25.3381 + 25.3381i 8.09326 − 21.1305i 9.00000 −27.5082 + 15.5980i
163.19 0.401137 − 2.79984i −3.00000 −7.67818 − 2.24624i −0.685918 + 11.1593i −1.20341 + 8.39951i 12.6294 − 12.6294i −9.36910 + 20.5966i 9.00000 30.9690 + 6.39686i
163.20 0.420600 − 2.79698i −3.00000 −7.64619 − 2.35282i 10.9490 + 2.26276i −1.26180 + 8.39094i 4.03671 − 4.03671i −9.79678 + 20.3966i 9.00000 10.9340 − 29.6723i
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.a ✓ 72
5.c odd 4 1 240.4.bc.b yes 72
16.f odd 4 1 240.4.bc.b yes 72
80.s even 4 1 inner 240.4.y.a ✓ 72
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
240.4.y.a ✓ 72 1.a even 1 1 trivial
240.4.y.a ✓ 72 80.s even 4 1 inner
240.4.bc.b yes 72 5.c odd 4 1
240.4.bc.b yes 72 16.f odd 4 1

Hecke kernels

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