Properties

Label 201.3.k.a
Level 201
Weight 3
Character orbit 201.k
Analytic conductor 5.477
Analytic rank 0
Dimension 440
CM no
Inner twists 4

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Newspace parameters

Level: \( N \) = \( 201 = 3 \cdot 67 \)
Weight: \( k \) = \( 3 \)
Character orbit: \([\chi]\) = 201.k (of order \(22\), degree \(10\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(5.47685331364\)
Analytic rank: \(0\)
Dimension: \(440\)
Relative dimension: \(44\) over \(\Q(\zeta_{22})\)
Coefficient ring index: multiple of None
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.

\(\operatorname{Tr}(f)(q) = \) \( 440q - 11q^{3} + 70q^{4} - 17q^{6} - 30q^{7} - 15q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 440q - 11q^{3} + 70q^{4} - 17q^{6} - 30q^{7} - 15q^{9} - 34q^{10} - 123q^{12} + 10q^{13} + 19q^{15} - 226q^{16} - 33q^{18} - 100q^{19} + 85q^{21} - 454q^{22} - 251q^{24} + 142q^{25} - 53q^{27} + 642q^{28} - 80q^{30} - 130q^{31} + 104q^{33} + 26q^{34} + 67q^{36} - 144q^{37} + 117q^{39} - 182q^{40} + 193q^{42} - 156q^{43} + 341q^{45} - 746q^{46} - 485q^{48} - 354q^{49} - 125q^{51} + 1130q^{52} + 272q^{54} - 590q^{55} + 15q^{57} - 274q^{58} - 217q^{60} - 1134q^{61} + 279q^{63} + 58q^{64} + 1288q^{66} - 87q^{69} + 1038q^{70} - 977q^{72} + 1208q^{73} - 751q^{75} + 1126q^{76} - 11q^{78} + 678q^{79} + 125q^{81} + 510q^{82} + 191q^{84} + 166q^{85} + 1080q^{87} + 62q^{88} - 105q^{90} + 550q^{91} + 635q^{93} + 986q^{94} - 1632q^{96} - 112q^{97} - 500q^{99} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
14.1 −3.56006 + 1.62583i 2.99912 0.0728092i 7.41130 8.55309i 2.34209 0.336742i −10.5587 + 5.13525i 0.309168 + 0.676984i −8.06833 + 27.4782i 8.98940 0.436727i −7.79051 + 5.00666i
14.2 −3.46528 + 1.58254i −2.55660 1.56965i 6.88427 7.94487i −7.01163 + 1.00812i 11.3434 + 1.39334i 3.76453 + 8.24318i −6.98975 + 23.8049i 4.07242 + 8.02592i 22.7019 14.5896i
14.3 −3.25291 + 1.48555i −2.99997 0.0132345i 5.75509 6.64173i 7.26564 1.04464i 9.77829 4.41356i −1.89172 4.14229i −4.82416 + 16.4296i 8.99965 + 0.0794063i −22.0826 + 14.1916i
14.4 −3.07669 + 1.40508i −0.235512 2.99074i 4.87231 5.62295i −1.88146 + 0.270514i 4.92681 + 8.87066i −5.50063 12.0447i −3.27825 + 11.1647i −8.88907 + 1.40871i 5.40858 3.47589i
14.5 −3.03986 + 1.38826i −1.35908 + 2.67449i 4.69406 5.41723i 1.28525 0.184791i 0.418549 10.0168i 2.83730 + 6.21282i −2.98273 + 10.1583i −5.30578 7.26971i −3.65044 + 2.34599i
14.6 −2.89089 + 1.32022i 0.926782 + 2.85326i 3.99480 4.61025i 1.94917 0.280248i −6.44616 7.02489i −0.497027 1.08834i −1.88048 + 6.40434i −7.28215 + 5.28870i −5.26484 + 3.38351i
14.7 −2.78375 + 1.27129i 0.285270 2.98641i 3.51361 4.05492i 9.82803 1.41306i 3.00248 + 8.67606i 3.12178 + 6.83574i −1.17725 + 4.00935i −8.83724 1.70386i −25.5623 + 16.4279i
14.8 −2.62930 + 1.20076i 2.11102 2.13157i 2.85196 3.29134i −4.72752 + 0.679715i −2.99101 + 8.13938i 2.36350 + 5.17534i −0.289151 + 0.984758i −0.0871716 8.99958i 11.6139 7.46381i
14.9 −2.54256 + 1.16115i 2.76891 + 1.15463i 2.49692 2.88159i −7.65897 + 1.10119i −8.38081 + 0.279405i −3.95633 8.66315i 0.147343 0.501803i 6.33368 + 6.39410i 18.1948 11.6931i
14.10 −2.43065 + 1.11004i −2.55478 + 1.57260i 2.05643 2.37324i −5.58064 + 0.802375i 4.46414 6.65836i −1.45327 3.18223i 0.647241 2.20430i 4.05385 8.03532i 12.6739 8.14504i
14.11 −1.96120 + 0.895652i −1.45460 2.62376i 0.424688 0.490116i −1.45741 + 0.209544i 5.20275 + 3.84292i 2.99885 + 6.56656i 2.03578 6.93324i −4.76826 + 7.63307i 2.67060 1.71629i
14.12 −1.74808 + 0.798320i −2.32890 1.89108i −0.200985 + 0.231949i 1.59523 0.229359i 5.58080 + 1.44655i −2.44056 5.34409i 2.33184 7.94150i 1.84760 + 8.80831i −2.60548 + 1.67444i
14.13 −1.74004 + 0.794650i 2.75393 + 1.18990i −0.223170 + 0.257552i 4.42297 0.635927i −5.73751 + 0.117936i 2.38025 + 5.21203i 2.33937 7.96717i 6.16827 + 6.55381i −7.19080 + 4.62125i
14.14 −1.64295 + 0.750310i 2.61118 1.47707i −0.483122 + 0.557552i 6.13055 0.881441i −3.18178 + 4.38595i −4.82114 10.5568i 2.41084 8.21055i 4.63653 7.71379i −9.41084 + 6.04798i
14.15 −1.30691 + 0.596845i −1.63302 + 2.51659i −1.26766 + 1.46295i 7.35419 1.05737i 0.632194 4.26362i −4.35656 9.53954i 2.40267 8.18273i −3.66648 8.21930i −8.98017 + 5.77121i
14.16 −1.16364 + 0.531418i −2.96474 + 0.458621i −1.54778 + 1.78623i 4.16270 0.598506i 3.20618 2.10919i 4.62778 + 10.1334i 2.29345 7.81077i 8.57933 2.71938i −4.52584 + 2.90858i
14.17 −1.07136 + 0.489273i 0.834932 + 2.88147i −1.71102 + 1.97463i −3.10066 + 0.445807i −2.30434 2.67858i −2.88988 6.32795i 2.19428 7.47302i −7.60578 + 4.81167i 3.10379 1.99469i
14.18 −0.912928 + 0.416920i −0.305710 + 2.98438i −1.95983 + 2.26176i −6.49262 + 0.933498i −0.965159 2.85198i 4.87674 + 10.6786i 1.97722 6.73380i −8.81308 1.82471i 5.53810 3.55912i
14.19 −0.883424 + 0.403446i 1.77485 2.41866i −2.00177 + 2.31017i −2.40583 + 0.345906i −0.592148 + 2.85276i 0.859066 + 1.88109i 1.93085 6.57586i −2.69981 8.58551i 1.98581 1.27620i
14.20 −0.456529 + 0.208490i −2.75933 1.17732i −2.45449 + 2.83264i −9.12735 + 1.31232i 1.50517 0.0378115i −1.31750 2.88491i 1.09556 3.73113i 6.22783 + 6.49724i 3.89329 2.50207i
See next 80 embeddings (of 440 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 158.44
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
67.e even 11 1 inner
201.k odd 22 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 201.3.k.a 440
3.b odd 2 1 inner 201.3.k.a 440
67.e even 11 1 inner 201.3.k.a 440
201.k odd 22 1 inner 201.3.k.a 440
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
201.3.k.a 440 1.a even 1 1 trivial
201.3.k.a 440 3.b odd 2 1 inner
201.3.k.a 440 67.e even 11 1 inner
201.3.k.a 440 201.k odd 22 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(201, [\chi])\).

Hecke Characteristic Polynomials

There are no characteristic polynomials of Hecke operators in the database