Properties

Label 201.4.d.b
Level 201
Weight 4
Character orbit 201.d
Analytic conductor 11.859
Analytic rank 0
Dimension 64
CM no
Inner twists 4

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Newspace parameters

Level: \( N \) = \( 201 = 3 \cdot 67 \)
Weight: \( k \) = \( 4 \)
Character orbit: \([\chi]\) = 201.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(11.8593839112\)
Analytic rank: \(0\)
Dimension: \(64\)
Coefficient ring index: multiple of None
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 64q + 268q^{4} - 46q^{6} + 22q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 64q + 268q^{4} - 46q^{6} + 22q^{9} - 36q^{10} + 20q^{15} + 556q^{16} + 128q^{19} + 96q^{22} - 904q^{24} + 2080q^{25} - 236q^{33} - 1574q^{36} + 1004q^{37} - 176q^{39} - 648q^{40} - 1220q^{49} + 2188q^{54} - 1344q^{55} + 550q^{60} + 4336q^{64} - 3512q^{67} + 3968q^{73} - 3316q^{76} - 1170q^{81} + 4020q^{82} - 9270q^{84} + 2436q^{88} + 746q^{90} - 3408q^{91} - 1412q^{93} - 7032q^{96} + O(q^{100}) \)

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.

Label \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
200.1 −5.43973 2.47504 4.56882i 21.5907 15.0478 −13.4636 + 24.8532i 27.3951i −73.9295 −14.7483 22.6161i −81.8562
200.2 −5.43973 2.47504 + 4.56882i 21.5907 15.0478 −13.4636 24.8532i 27.3951i −73.9295 −14.7483 + 22.6161i −81.8562
200.3 −5.08650 −0.0109234 5.19614i 17.8725 −16.6834 0.0555619 + 26.4302i 1.72249i −50.2166 −26.9998 + 0.113519i 84.8599
200.4 −5.08650 −0.0109234 + 5.19614i 17.8725 −16.6834 0.0555619 26.4302i 1.72249i −50.2166 −26.9998 0.113519i 84.8599
200.5 −4.99884 −4.36958 2.81190i 16.9884 3.29226 21.8428 + 14.0562i 9.19710i −44.9316 11.1865 + 24.5736i −16.4575
200.6 −4.99884 −4.36958 + 2.81190i 16.9884 3.29226 21.8428 14.0562i 9.19710i −44.9316 11.1865 24.5736i −16.4575
200.7 −4.88518 4.71863 2.17590i 15.8650 −9.70204 −23.0513 + 10.6297i 27.9776i −38.4219 17.5309 20.5346i 47.3962
200.8 −4.88518 4.71863 + 2.17590i 15.8650 −9.70204 −23.0513 10.6297i 27.9776i −38.4219 17.5309 + 20.5346i 47.3962
200.9 −4.16792 4.76109 2.08135i 9.37159 8.48191 −19.8439 + 8.67492i 6.56298i −5.71669 18.3359 19.8190i −35.3519
200.10 −4.16792 4.76109 + 2.08135i 9.37159 8.48191 −19.8439 8.67492i 6.56298i −5.71669 18.3359 + 19.8190i −35.3519
200.11 −3.59475 −2.38300 4.61750i 4.92223 −8.49412 8.56630 + 16.5988i 6.53043i 11.0638 −15.6426 + 22.0070i 30.5342
200.12 −3.59475 −2.38300 + 4.61750i 4.92223 −8.49412 8.56630 16.5988i 6.53043i 11.0638 −15.6426 22.0070i 30.5342
200.13 −3.43846 −4.16551 3.10621i 3.82299 21.2344 14.3229 + 10.6806i 25.8876i 14.3625 7.70288 + 25.8779i −73.0135
200.14 −3.43846 −4.16551 + 3.10621i 3.82299 21.2344 14.3229 10.6806i 25.8876i 14.3625 7.70288 25.8779i −73.0135
200.15 −3.40522 −5.19596 0.0441335i 3.59552 −14.7776 17.6934 + 0.150284i 33.3251i 14.9982 26.9961 + 0.458632i 50.3210
200.16 −3.40522 −5.19596 + 0.0441335i 3.59552 −14.7776 17.6934 0.150284i 33.3251i 14.9982 26.9961 0.458632i 50.3210
200.17 −3.37562 0.888938 5.11955i 3.39482 12.9769 −3.00072 + 17.2817i 25.4642i 15.5453 −25.4196 9.10193i −43.8050
200.18 −3.37562 0.888938 + 5.11955i 3.39482 12.9769 −3.00072 17.2817i 25.4642i 15.5453 −25.4196 + 9.10193i −43.8050
200.19 −3.33139 0.842682 5.12737i 3.09813 0.324421 −2.80730 + 17.0812i 12.9394i 16.3300 −25.5798 8.64147i −1.08077
200.20 −3.33139 0.842682 + 5.12737i 3.09813 0.324421 −2.80730 17.0812i 12.9394i 16.3300 −25.5798 + 8.64147i −1.08077
See all 64 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 200.64
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 201.4.d.b 64
3.b odd 2 1 inner 201.4.d.b 64
67.b odd 2 1 inner 201.4.d.b 64
201.d even 2 1 inner 201.4.d.b 64
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
201.4.d.b 64 1.a even 1 1 trivial
201.4.d.b 64 3.b odd 2 1 inner
201.4.d.b 64 67.b odd 2 1 inner
201.4.d.b 64 201.d even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(T_{2}^{32} - \cdots\) acting on \(S_{4}^{\mathrm{new}}(201, [\chi])\).

Hecke Characteristic Polynomials

There are no characteristic polynomials of Hecke operators in the database