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Properties

Label 141.4.e.b
Level $141$
Weight $4$
Character orbit 141.e
Analytic conductor $8.319$
Analytic rank $0$
Dimension $264$
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] = [141,4,Mod(4,141)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(141, base_ring=CyclotomicField(46))
 
chi = DirichletCharacter(H, H._module([0, 36]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("141.4");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 141 = 3 \cdot 47 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 141.e (of order \(23\), degree \(22\), 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: \(8.31926931081\)
Analytic rank: \(0\)
Dimension: \(264\)
Relative dimension: \(12\) over \(\Q(\zeta_{23})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{23}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 264 q + 2 q^{2} + 36 q^{3} - 46 q^{4} - 2 q^{5} - 6 q^{6} - 4 q^{7} + 66 q^{8} - 108 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 264 q + 2 q^{2} + 36 q^{3} - 46 q^{4} - 2 q^{5} - 6 q^{6} - 4 q^{7} + 66 q^{8} - 108 q^{9} - 702 q^{10} + 166 q^{11} + 138 q^{12} - 22 q^{13} + 36 q^{14} + 6 q^{15} - 278 q^{16} + 100 q^{17} + 18 q^{18} - 62 q^{19} - 220 q^{20} + 12 q^{21} + 426 q^{22} + 1096 q^{23} - 198 q^{24} - 260 q^{25} + 30 q^{26} + 324 q^{27} - 676 q^{28} - 470 q^{29} - 240 q^{30} + 108 q^{31} + 490 q^{32} - 912 q^{33} + 5392 q^{34} + 1966 q^{35} - 414 q^{36} + 1636 q^{37} + 1484 q^{38} + 66 q^{39} + 868 q^{40} - 1192 q^{41} - 108 q^{42} - 1016 q^{43} - 3158 q^{44} - 18 q^{45} - 1044 q^{46} - 2114 q^{47} - 12414 q^{48} - 2696 q^{49} - 2078 q^{50} - 300 q^{51} - 1826 q^{52} + 1188 q^{53} - 54 q^{54} + 286 q^{55} - 4530 q^{56} + 876 q^{57} + 13130 q^{58} + 1558 q^{59} + 660 q^{60} + 4656 q^{61} - 1470 q^{62} - 36 q^{63} - 542 q^{64} - 1508 q^{65} + 102 q^{66} + 70 q^{67} - 2304 q^{68} + 2370 q^{69} + 11676 q^{70} - 3378 q^{71} + 594 q^{72} - 816 q^{73} - 696 q^{74} + 780 q^{75} + 3619 q^{76} + 7862 q^{77} + 10053 q^{78} - 4014 q^{79} + 10200 q^{80} - 972 q^{81} + 10030 q^{82} - 4466 q^{83} + 1614 q^{84} - 1732 q^{85} - 6168 q^{86} - 936 q^{87} - 10492 q^{88} - 6232 q^{89} - 3006 q^{90} - 16880 q^{91} - 6253 q^{92} + 8232 q^{93} - 6674 q^{94} - 26364 q^{95} - 13476 q^{96} - 3350 q^{97} - 11741 q^{98} - 3060 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
4.1 −4.62246 − 1.29515i −1.38020 + 2.66366i 12.8544 + 7.81692i −5.59338 + 4.55055i 9.82974 − 10.5251i 21.9873 + 3.02208i −23.0821 − 24.7149i −5.19012 − 7.35273i 31.7489 − 13.7905i
4.2 −4.11775 − 1.15374i −1.38020 + 2.66366i 8.78940 + 5.34495i 14.7477 − 11.9981i 8.75646 − 9.37588i −7.88427 − 1.08367i −6.67529 − 7.14749i −5.19012 − 7.35273i −74.5700 + 32.3903i
4.3 −3.38850 − 0.949415i −1.38020 + 2.66366i 3.74522 + 2.27752i −14.9484 + 12.1614i 7.20571 − 7.71543i −35.1630 − 4.83304i 8.68684 + 9.30133i −5.19012 − 7.35273i 62.1989 − 27.0168i
4.4 −2.45544 − 0.687981i −1.38020 + 2.66366i −1.27950 − 0.778084i 1.48135 − 1.20517i 5.22153 − 5.59089i 8.48885 + 1.16677i 16.5305 + 17.6998i −5.19012 − 7.35273i −4.46649 + 1.94007i
4.5 −1.58271 − 0.443453i −1.38020 + 2.66366i −4.52705 − 2.75296i 5.39355 − 4.38798i 3.36565 − 3.60373i −8.35112 − 1.14784i 14.9192 + 15.9746i −5.19012 − 7.35273i −10.4823 + 4.55309i
4.6 −0.693757 − 0.194382i −1.38020 + 2.66366i −6.39184 − 3.88696i −11.8679 + 9.65529i 1.47529 − 1.57965i 7.98182 + 1.09708i 7.61293 + 8.15146i −5.19012 − 7.35273i 10.1103 − 4.39151i
4.7 1.00060 + 0.280356i −1.38020 + 2.66366i −5.91275 − 3.59562i −0.468772 + 0.381374i −2.12780 + 2.27832i 35.1726 + 4.83437i −10.5824 − 11.3310i −5.19012 − 7.35273i −0.575975 + 0.250181i
4.8 1.82864 + 0.512361i −1.38020 + 2.66366i −3.75394 − 2.28282i 12.6831 − 10.3185i −3.88863 + 4.16371i −6.65901 − 0.915260i −16.0647 − 17.2011i −5.19012 − 7.35273i 28.4797 − 12.3705i
4.9 2.27950 + 0.638686i −1.38020 + 2.66366i −2.04716 − 1.24490i −4.63169 + 3.76816i −4.84739 + 5.19029i −9.20215 − 1.26481i −16.7978 − 17.9860i −5.19012 − 7.35273i −12.9646 + 5.63132i
4.10 3.38480 + 0.948377i −1.38020 + 2.66366i 3.72212 + 2.26347i −8.82151 + 7.17683i −7.19784 + 7.70701i −14.8266 − 2.03786i −8.74222 − 9.36063i −5.19012 − 7.35273i −36.6654 + 15.9260i
4.11 4.56020 + 1.27771i −1.38020 + 2.66366i 12.3276 + 7.49656i 11.4440 − 9.31036i −9.69735 + 10.3833i 9.61079 + 1.32097i 20.7782 + 22.2480i −5.19012 − 7.35273i 64.0828 − 27.8351i
4.12 5.30076 + 1.48520i −1.38020 + 2.66366i 19.0568 + 11.5887i −10.7538 + 8.74889i −11.2721 + 12.0695i −0.884797 − 0.121613i 53.7449 + 57.5468i −5.19012 − 7.35273i −69.9973 + 30.4041i
7.1 −0.370382 + 5.41479i −2.88875 + 0.809390i −21.2573 − 2.92175i −4.44272 + 6.29390i −3.31274 − 15.9418i −3.79761 − 4.06625i 14.8600 − 71.5104i 7.68977 − 4.67626i −32.4346 − 26.3875i
7.2 −0.274049 + 4.00645i −2.88875 + 0.809390i −8.05109 − 1.10660i 6.30444 − 8.93135i −2.45113 − 11.7955i −13.9198 − 14.9045i 0.103584 − 0.498472i 7.68977 − 4.67626i 34.0553 + 27.7061i
7.3 −0.255129 + 3.72985i −2.88875 + 0.809390i −5.92120 − 0.813851i 7.78146 − 11.0238i −2.28190 − 10.9811i 19.3842 + 20.7554i −1.53886 + 7.40540i 7.68977 − 4.67626i 39.1319 + 31.8362i
7.4 −0.220599 + 3.22504i −2.88875 + 0.809390i −2.42676 − 0.333551i −11.0040 + 15.5892i −1.97306 − 9.49490i 13.7398 + 14.7117i −3.65045 + 17.5669i 7.68977 − 4.67626i −47.8483 − 38.9275i
7.5 −0.147409 + 2.15505i −2.88875 + 0.809390i 3.30298 + 0.453984i −1.77600 + 2.51602i −1.31845 − 6.34471i −13.9651 − 14.9529i −4.98111 + 23.9704i 7.68977 − 4.67626i −5.16035 − 4.19826i
7.6 −0.0283912 + 0.415065i −2.88875 + 0.809390i 7.75401 + 1.06576i −1.81465 + 2.57077i −0.253934 − 1.22200i 3.00779 + 3.22056i −1.33967 + 6.44683i 7.68977 − 4.67626i −1.01552 − 0.826185i
7.7 −0.0165233 + 0.241562i −2.88875 + 0.809390i 7.86741 + 1.08135i 2.75608 − 3.90448i −0.147786 − 0.711188i 9.13481 + 9.78100i −0.785307 + 3.77910i 7.68977 − 4.67626i 0.897636 + 0.730281i
7.8 0.114410 − 1.67262i −2.88875 + 0.809390i 5.14092 + 0.706604i −9.69845 + 13.7396i 1.02330 + 4.92438i −16.7056 − 17.8873i 4.49885 − 21.6496i 7.68977 − 4.67626i 21.8715 + 17.7938i
See next 80 embeddings (of 264 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 4.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
47.c even 23 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 141.4.e.b ✓ 264
47.c even 23 1 inner 141.4.e.b ✓ 264
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
141.4.e.b ✓ 264 1.a even 1 1 trivial
141.4.e.b ✓ 264 47.c even 23 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{264} - 2 T_{2}^{263} + 73 T_{2}^{262} - 176 T_{2}^{261} + 3458 T_{2}^{260} + \cdots + 26\!\cdots\!84 \) acting on \(S_{4}^{\mathrm{new}}(141, [\chi])\). Copy content Toggle raw display