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Properties

Label 201.4.j.b
Level $201$
Weight $4$
Character orbit 201.j
Analytic conductor $11.859$
Analytic rank $0$
Dimension $640$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [201,4,Mod(5,201)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(201, base_ring=CyclotomicField(22))
 
chi = DirichletCharacter(H, H._module([11, 5]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("201.5");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 201 = 3 \cdot 67 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 201.j (of order \(22\), degree \(10\), 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: \(640\)
Relative dimension: \(64\) over \(\Q(\zeta_{22})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{22}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 640 q - 11 q^{3} - 290 q^{4} + 35 q^{6} - 22 q^{7} - 33 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 640 q - 11 q^{3} - 290 q^{4} + 35 q^{6} - 22 q^{7} - 33 q^{9} + 14 q^{10} - 143 q^{12} - 22 q^{13} - 31 q^{15} - 578 q^{16} - 11 q^{18} - 18 q^{19} - 1133 q^{21} - 382 q^{22} + 365 q^{24} - 2102 q^{25} - 11 q^{27} - 1870 q^{28} - 1210 q^{31} - 743 q^{33} - 22 q^{34} + 1563 q^{36} - 1048 q^{37} + 2475 q^{39} + 626 q^{40} - 11 q^{42} - 1738 q^{43} - 4521 q^{45} + 6578 q^{46} - 3531 q^{48} + 1198 q^{49} - 11 q^{51} + 10274 q^{52} + 1948 q^{54} + 3170 q^{55} - 8921 q^{57} + 418 q^{58} - 649 q^{60} - 2002 q^{61} - 5951 q^{63} - 14918 q^{64} + 2038 q^{67} - 11 q^{69} - 8734 q^{70} - 10681 q^{72} + 9870 q^{73} + 8426 q^{75} + 5230 q^{76} - 11 q^{78} + 10934 q^{79} + 2919 q^{81} + 7442 q^{82} - 4997 q^{84} - 22 q^{85} + 7480 q^{87} + 8630 q^{88} - 757 q^{90} + 13286 q^{91} - 6585 q^{93} + 7106 q^{94} + 16316 q^{96} + 2739 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
5.1 −3.59996 + 4.15458i 1.46644 + 4.98493i −3.16228 − 21.9941i 8.44887 − 2.48081i −25.9894 − 11.8531i 9.23706 + 8.00396i 65.7634 + 42.2636i −22.6991 + 14.6202i −20.1089 + 44.0324i
5.2 −3.53793 + 4.08299i −4.87975 + 1.78552i −3.01532 − 20.9720i −9.44479 + 2.77324i 9.97393 − 26.2410i −4.84382 − 4.19720i 59.9371 + 38.5193i 20.6238 − 17.4258i 22.0919 − 48.3745i
5.3 −3.44200 + 3.97228i 3.74600 − 3.60104i −2.79312 − 19.4266i 2.64211 − 0.775793i 1.41059 + 27.2749i −19.3454 − 16.7629i 51.4083 + 33.0381i 1.06506 − 26.9790i −6.01247 + 13.1655i
5.4 −3.30878 + 3.81854i −2.72588 − 4.42375i −2.49468 − 17.3509i 19.1622 − 5.62653i 25.9116 + 4.22837i −2.23659 − 1.93802i 40.5048 + 26.0309i −12.1392 + 24.1172i −41.9184 + 91.7886i
5.5 −3.23057 + 3.72827i 1.66163 − 4.92331i −2.32494 − 16.1703i −2.17468 + 0.638544i 12.9874 + 22.1001i 19.5744 + 16.9613i 34.5976 + 22.2345i −21.4780 − 16.3614i 4.64478 − 10.1707i
5.6 −3.20732 + 3.70144i 5.16210 + 0.593913i −2.27527 − 15.8248i −8.51736 + 2.50092i −18.7548 + 17.2024i 14.3505 + 12.4348i 32.9106 + 21.1503i 26.2945 + 6.13167i 18.0609 − 39.5478i
5.7 −3.03287 + 3.50012i 2.67965 + 4.45190i −1.91401 − 13.3122i −16.4593 + 4.83290i −23.7092 − 4.12294i −19.2826 − 16.7085i 21.2303 + 13.6439i −12.6389 + 23.8591i 33.0033 − 72.2672i
5.8 −3.01708 + 3.48189i −1.52278 − 4.96801i −1.88230 − 13.0917i −16.5234 + 4.85171i 21.8924 + 9.68669i −15.7610 − 13.6570i 20.2562 + 13.0179i −22.3623 + 15.1304i 32.9593 − 72.1707i
5.9 −2.89391 + 3.33975i −2.77933 + 4.39037i −1.64070 − 11.4113i 10.1030 − 2.96651i −6.61962 − 21.9876i −7.38800 − 6.40174i 13.1183 + 8.43060i −11.5507 − 24.4045i −19.3298 + 42.3263i
5.10 −2.88756 + 3.33242i −5.17395 − 0.479845i −1.62851 − 11.3266i 8.13953 − 2.38998i 16.5391 − 15.8562i 11.0496 + 9.57453i 12.7717 + 8.20789i 26.5395 + 4.96539i −15.5390 + 34.0256i
5.11 −2.83150 + 3.26773i 5.07100 + 1.13356i −1.52212 − 10.5866i 16.6951 − 4.90213i −18.0627 + 13.3610i −9.98954 − 8.65599i 9.80457 + 6.30101i 24.4301 + 11.4965i −31.2535 + 68.4356i
5.12 −2.79597 + 3.22672i −3.97281 − 3.34915i −1.45576 − 10.1250i −12.6136 + 3.70369i 21.9146 − 3.45504i 19.5475 + 16.9380i 8.00668 + 5.14558i 4.56644 + 26.6110i 23.3165 − 51.0560i
5.13 −2.41929 + 2.79201i −2.24694 + 4.68522i −0.803837 − 5.59081i −11.7964 + 3.46373i −7.64518 − 17.6084i 12.9935 + 11.2589i −7.30882 − 4.69710i −16.9025 − 21.0548i 18.8681 − 41.3155i
5.14 −2.39820 + 2.76767i −4.92182 − 1.66605i −0.770116 − 5.35628i 1.44052 − 0.422974i 16.4146 − 9.62645i −26.0066 − 22.5349i −7.97509 − 5.12528i 21.4486 + 16.4000i −2.28400 + 5.00125i
5.15 −2.31446 + 2.67103i 2.56918 + 4.51656i −0.639155 − 4.44542i −1.61194 + 0.473307i −18.0101 − 3.59106i −13.2348 − 11.4680i −10.4327 − 6.70466i −13.7987 + 23.2077i 2.46655 − 5.40098i
5.16 −2.08344 + 2.40442i 4.78249 − 2.03169i −0.301982 − 2.10033i −1.53336 + 0.450234i −5.07900 + 15.7320i −6.46221 − 5.59953i −15.7323 − 10.1106i 18.7445 − 19.4331i 2.11210 − 4.62486i
5.17 −2.00708 + 2.31629i 0.0979400 − 5.19523i −0.198329 − 1.37941i 4.53381 − 1.33125i 11.8371 + 10.6541i −2.62239 − 2.27231i −17.0337 − 10.9469i −26.9808 − 1.01764i −6.01617 + 13.1736i
5.18 −1.98603 + 2.29200i 4.03838 − 3.26978i −0.170435 − 1.18540i 19.1936 − 5.63576i −0.526014 + 15.7499i 24.7274 + 21.4264i −17.3551 − 11.1534i 5.61708 − 26.4092i −25.2020 + 55.1846i
5.19 −1.97915 + 2.28406i 1.50596 + 4.97314i −0.161385 − 1.12245i 9.47455 − 2.78198i −14.3395 − 6.40290i 23.8049 + 20.6270i −17.4567 − 11.2187i −22.4642 + 14.9787i −12.3974 + 27.1465i
5.20 −1.71809 + 1.98278i 3.65642 − 3.69196i 0.158929 + 1.10537i −18.6302 + 5.47032i 1.03829 + 13.5930i −6.55071 − 5.67622i −20.1216 − 12.9314i −0.261161 − 26.9987i 21.1619 − 46.3381i
See next 80 embeddings (of 640 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 5.64
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
67.f odd 22 1 inner
201.j even 22 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 201.4.j.b ✓ 640
3.b odd 2 1 inner 201.4.j.b ✓ 640
67.f odd 22 1 inner 201.4.j.b ✓ 640
201.j even 22 1 inner 201.4.j.b ✓ 640
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
201.4.j.b ✓ 640 1.a even 1 1 trivial
201.4.j.b ✓ 640 3.b odd 2 1 inner
201.4.j.b ✓ 640 67.f odd 22 1 inner
201.4.j.b ✓ 640 201.j even 22 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{640} + 401 T_{2}^{638} + 84201 T_{2}^{636} + 12338895 T_{2}^{634} + 1418493465 T_{2}^{632} + \cdots + 40\!\cdots\!76 \) acting on \(S_{4}^{\mathrm{new}}(201, [\chi])\). Copy content Toggle raw display