Properties

Label 201.4.e.b
Level $201$
Weight $4$
Character orbit 201.e
Analytic conductor $11.859$
Analytic rank $0$
Dimension $36$
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] = [201,4,Mod(37,201)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(201, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 2]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("201.37");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 201 = 3 \cdot 67 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 201.e (of order \(3\), 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: \(11.8593839112\)
Analytic rank: \(0\)
Dimension: \(36\)
Relative dimension: \(18\) over \(\Q(\zeta_{3})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 36 q + 2 q^{2} - 108 q^{3} - 90 q^{4} - 4 q^{5} - 6 q^{6} + 22 q^{7} + 48 q^{8} + 324 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 36 q + 2 q^{2} - 108 q^{3} - 90 q^{4} - 4 q^{5} - 6 q^{6} + 22 q^{7} + 48 q^{8} + 324 q^{9} + 14 q^{10} - 16 q^{11} + 270 q^{12} - 46 q^{13} + 14 q^{14} + 12 q^{15} - 346 q^{16} - 8 q^{17} + 18 q^{18} - 154 q^{19} - 180 q^{20} - 66 q^{21} + 214 q^{22} - 104 q^{23} - 144 q^{24} + 1032 q^{25} - 333 q^{26} - 972 q^{27} - 473 q^{28} + 76 q^{29} - 42 q^{30} + 498 q^{31} - 285 q^{32} + 48 q^{33} + 26 q^{34} - 392 q^{35} - 810 q^{36} - 124 q^{37} + 20 q^{38} + 138 q^{39} + 638 q^{40} - 508 q^{41} - 42 q^{42} - 1400 q^{43} - 333 q^{44} - 36 q^{45} - 1372 q^{46} + 18 q^{47} + 1038 q^{48} - 238 q^{49} - 337 q^{50} + 24 q^{51} + 3640 q^{52} + 724 q^{53} - 54 q^{54} - 178 q^{55} - 829 q^{56} + 462 q^{57} - 1472 q^{58} + 720 q^{59} + 540 q^{60} + 232 q^{61} - 3882 q^{62} + 198 q^{63} + 3628 q^{64} - 1428 q^{65} - 642 q^{66} - 1164 q^{67} + 1634 q^{68} + 312 q^{69} + 2550 q^{70} + 406 q^{71} + 432 q^{72} - 2120 q^{73} + 1375 q^{74} - 3096 q^{75} + 4190 q^{76} - 800 q^{77} + 999 q^{78} + 1306 q^{79} - 1927 q^{80} + 2916 q^{81} - 794 q^{82} - 1010 q^{83} + 1419 q^{84} + 472 q^{85} + 737 q^{86} - 228 q^{87} - 1838 q^{88} + 1904 q^{89} + 126 q^{90} + 7340 q^{91} + 7368 q^{92} - 1494 q^{93} - 9862 q^{94} + 1678 q^{95} + 855 q^{96} - 2358 q^{97} - 2610 q^{98} - 144 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
37.1 −2.72652 4.72247i −3.00000 −10.8678 + 18.8236i 12.9793 8.17955 + 14.1674i −14.7538 + 25.5543i 74.9007 9.00000 −35.3882 61.2942i
37.2 −2.52313 4.37020i −3.00000 −8.73242 + 15.1250i 0.340792 7.56940 + 13.1106i 12.2513 21.2198i 47.7621 9.00000 −0.859864 1.48933i
37.3 −2.37227 4.10889i −3.00000 −7.25534 + 12.5666i −19.3810 7.11681 + 12.3267i 3.13332 5.42707i 30.8902 9.00000 45.9769 + 79.6344i
37.4 −1.83140 3.17208i −3.00000 −2.70808 + 4.69053i 12.9712 5.49421 + 9.51625i 1.64155 2.84325i −9.46413 9.00000 −23.7555 41.1456i
37.5 −1.56036 2.70262i −3.00000 −0.869446 + 1.50592i 10.0899 4.68108 + 8.10787i −2.32642 + 4.02948i −19.5392 9.00000 −15.7439 27.2693i
37.6 −1.55793 2.69841i −3.00000 −0.854281 + 1.47966i −13.0874 4.67378 + 8.09523i −12.0084 + 20.7992i −19.6032 9.00000 20.3893 + 35.3153i
37.7 −0.829978 1.43756i −3.00000 2.62227 4.54191i −7.05463 2.48993 + 4.31269i 18.3944 31.8600i −21.9854 9.00000 5.85519 + 10.1415i
37.8 −0.561904 0.973245i −3.00000 3.36853 5.83446i −9.21684 1.68571 + 2.91974i 2.79031 4.83295i −16.5616 9.00000 5.17897 + 8.97024i
37.9 0.0760461 + 0.131716i −3.00000 3.98843 6.90817i 2.82816 −0.228138 0.395147i −1.28326 + 2.22268i 2.42996 9.00000 0.215070 + 0.372513i
37.10 0.244294 + 0.423129i −3.00000 3.88064 6.72147i 20.3821 −0.732881 1.26939i 8.86889 15.3614i 7.70076 9.00000 4.97921 + 8.62424i
37.11 0.824514 + 1.42810i −3.00000 2.64035 4.57323i −20.6834 −2.47354 4.28430i 1.21474 2.10399i 21.9003 9.00000 −17.0538 29.5380i
37.12 1.20764 + 2.09170i −3.00000 1.08320 1.87615i −7.44326 −3.62293 6.27510i −10.2926 + 17.8273i 24.5547 9.00000 −8.98880 15.5691i
37.13 1.46577 + 2.53879i −3.00000 −0.296981 + 0.514386i 10.1733 −4.39732 7.61638i −10.4213 + 18.0502i 21.7111 9.00000 14.9117 + 25.8278i
37.14 1.63047 + 2.82406i −3.00000 −1.31688 + 2.28090i −0.0205579 −4.89142 8.47218i 2.93590 5.08512i 17.4990 9.00000 −0.0335191 0.0580568i
37.15 2.13007 + 3.68940i −3.00000 −5.07443 + 8.78916i 13.6463 −6.39022 11.0682i 12.7106 22.0154i −9.15443 9.00000 29.0676 + 50.3465i
37.16 2.19185 + 3.79640i −3.00000 −5.60845 + 9.71412i −15.2934 −6.57556 11.3892i 12.5287 21.7004i −14.1019 9.00000 −33.5209 58.0599i
37.17 2.50438 + 4.33772i −3.00000 −8.54385 + 14.7984i 13.1979 −7.51314 13.0131i −7.35263 + 12.7351i −45.5181 9.00000 33.0525 + 57.2486i
37.18 2.68845 + 4.65653i −3.00000 −10.4555 + 18.1094i −6.42823 −8.06534 13.9696i −7.03129 + 12.1786i −69.4209 9.00000 −17.2819 29.9332i
163.1 −2.72652 + 4.72247i −3.00000 −10.8678 18.8236i 12.9793 8.17955 14.1674i −14.7538 25.5543i 74.9007 9.00000 −35.3882 + 61.2942i
163.2 −2.52313 + 4.37020i −3.00000 −8.73242 15.1250i 0.340792 7.56940 13.1106i 12.2513 + 21.2198i 47.7621 9.00000 −0.859864 + 1.48933i
See all 36 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 37.18
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
67.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 201.4.e.b 36
67.c even 3 1 inner 201.4.e.b 36
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
201.4.e.b 36 1.a even 1 1 trivial
201.4.e.b 36 67.c even 3 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{36} - 2 T_{2}^{35} + 119 T_{2}^{34} - 230 T_{2}^{33} + 8271 T_{2}^{32} - 15533 T_{2}^{31} + 384536 T_{2}^{30} - 692967 T_{2}^{29} + 13324771 T_{2}^{28} - 22941523 T_{2}^{27} + \cdots + 152371175952384 \) acting on \(S_{4}^{\mathrm{new}}(201, [\chi])\). Copy content Toggle raw display