Properties

Label 201.2.p.b
Level $201$
Weight $2$
Character orbit 201.p
Analytic conductor $1.605$
Analytic rank $0$
Dimension $400$
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] = [201,2,Mod(2,201)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(201, base_ring=CyclotomicField(66))
 
chi = DirichletCharacter(H, H._module([33, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("201.2");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 201 = 3 \cdot 67 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 201.p (of order \(66\), degree \(20\), 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: \(1.60499308063\)
Analytic rank: \(0\)
Dimension: \(400\)
Relative dimension: \(20\) over \(\Q(\zeta_{66})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{66}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 400 q - 22 q^{3} - 20 q^{4} - 20 q^{6} - 38 q^{7} - 30 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 400 q - 22 q^{3} - 20 q^{4} - 20 q^{6} - 38 q^{7} - 30 q^{9} - 38 q^{10} + 2 q^{12} - 38 q^{13} + 4 q^{15} - 8 q^{16} - 28 q^{18} - 60 q^{19} - 106 q^{21} - 8 q^{22} - 62 q^{24} - 84 q^{25} - 22 q^{27} + 116 q^{28} - 90 q^{30} - 52 q^{31} - 19 q^{33} - 32 q^{34} - 24 q^{36} - 34 q^{37} + 33 q^{39} - 4 q^{40} - 22 q^{42} - 22 q^{43} - 132 q^{45} - 162 q^{46} - 54 q^{48} - 38 q^{49} - 10 q^{51} - 44 q^{52} + 101 q^{54} + 126 q^{55} + 77 q^{57} - 80 q^{60} + 146 q^{61} - 13 q^{63} + 172 q^{64} - 4 q^{67} - q^{69} + 264 q^{70} + 88 q^{72} - 6 q^{73} - 11 q^{75} + 124 q^{76} + 20 q^{78} - 246 q^{79} - 42 q^{81} - 16 q^{82} - 235 q^{84} + 34 q^{85} - 100 q^{87} + 150 q^{88} + 88 q^{90} + 120 q^{91} + 57 q^{93} - 88 q^{94} + 196 q^{96} + 24 q^{97} - 54 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2.1 −0.131747 + 2.76571i −1.12721 1.31507i −5.64086 0.538637i −0.779337 + 0.899403i 3.78560 2.94428i −0.762358 1.47877i 1.44479 10.0487i −0.458797 + 2.96471i −2.38481 2.27392i
2.2 −0.111707 + 2.34502i 1.26924 1.17858i −3.49572 0.333801i 2.49839 2.88330i 2.62201 + 3.10804i −0.0230737 0.0447567i 0.505048 3.51269i 0.221918 2.99178i 6.48232 + 6.18088i
2.3 −0.106680 + 2.23948i −0.226858 + 1.71713i −3.01294 0.287701i −0.703208 + 0.811545i −3.82127 0.691227i −1.71157 3.31998i 0.327574 2.27833i −2.89707 0.779090i −1.74242 1.66139i
2.4 −0.0942093 + 1.97770i 1.35004 1.08508i −1.91146 0.182522i −2.22606 + 2.56901i 2.01877 + 2.77219i 0.731283 + 1.41849i −0.0224982 + 0.156478i 0.645203 2.92980i −4.87100 4.64449i
2.5 −0.0888646 + 1.86550i −1.52053 + 0.829459i −1.48124 0.141441i −1.38993 + 1.60406i −1.41223 2.91025i 1.29209 + 2.50631i −0.136090 + 0.946526i 1.62400 2.52243i −2.86886 2.73545i
2.6 −0.0824371 + 1.73057i 0.744203 + 1.56402i −0.997123 0.0952137i 2.18001 2.51587i −2.76799 + 1.15896i 0.930855 + 1.80561i −0.246156 + 1.71205i −1.89232 + 2.32790i 4.17416 + 3.98006i
2.7 −0.0765690 + 1.60738i −1.57501 0.720644i −0.586870 0.0560393i 0.869862 1.00387i 1.27895 2.47647i 0.664215 + 1.28840i −0.323015 + 2.24662i 1.96134 + 2.27005i 1.54700 + 1.47507i
2.8 −0.0485141 + 1.01844i 1.68562 + 0.398362i 0.956085 + 0.0912951i −0.397858 + 0.459152i −0.487483 + 1.69737i −1.04530 2.02760i −0.429568 + 2.98771i 2.68262 + 1.34297i −0.448316 0.427468i
2.9 −0.0407297 + 0.855022i −0.210665 1.71919i 1.26154 + 0.120462i 1.16294 1.34211i 1.47853 0.110101i −1.52832 2.96453i −0.398021 + 2.76830i −2.91124 + 0.724347i 1.10017 + 1.04901i
2.10 −0.00573694 + 0.120433i −1.18335 + 1.26478i 1.97647 + 0.188730i 1.79917 2.07635i −0.145533 0.149771i 0.198599 + 0.385227i −0.0683861 + 0.475635i −0.199345 2.99337i 0.239740 + 0.228592i
2.11 0.00573694 0.120433i 0.311708 + 1.70377i 1.97647 + 0.188730i −1.79917 + 2.07635i 0.206979 0.0277656i 0.198599 + 0.385227i 0.0683861 0.475635i −2.80568 + 1.06216i 0.239740 + 0.228592i
2.12 0.0407297 0.855022i 1.10669 1.33238i 1.26154 + 0.120462i −1.16294 + 1.34211i −1.09414 1.00051i −1.52832 2.96453i 0.398021 2.76830i −0.550484 2.94906i 1.10017 + 1.04901i
2.13 0.0485141 1.01844i −1.63340 0.576190i 0.956085 + 0.0912951i 0.397858 0.459152i −0.666056 + 1.63556i −1.04530 2.02760i 0.429568 2.98771i 2.33601 + 1.88230i −0.448316 0.427468i
2.14 0.0765690 1.60738i 1.71460 + 0.245273i −0.586870 0.0560393i −0.869862 + 1.00387i 0.525532 2.73723i 0.664215 + 1.28840i 0.323015 2.24662i 2.87968 + 0.841088i 1.54700 + 1.47507i
2.15 0.0824371 1.73057i −1.47164 + 0.913392i −0.997123 0.0952137i −2.18001 + 2.51587i 1.45937 + 2.62206i 0.930855 + 1.80561i 0.246156 1.71205i 1.33143 2.68836i 4.17416 + 3.98006i
2.16 0.0888646 1.86550i 0.830708 + 1.51984i −1.48124 0.141441i 1.38993 1.60406i 2.90908 1.41462i 1.29209 + 2.50631i 0.136090 0.946526i −1.61985 + 2.52509i −2.86886 2.73545i
2.17 0.0942093 1.97770i −0.549085 1.64271i −1.91146 0.182522i 2.22606 2.56901i −3.30051 + 0.931165i 0.731283 + 1.41849i 0.0224982 0.156478i −2.39701 + 1.80398i −4.87100 4.64449i
2.18 0.106680 2.23948i −0.737505 + 1.56719i −3.01294 0.287701i 0.703208 0.811545i 3.43101 + 1.81881i −1.71157 3.31998i −0.327574 + 2.27833i −1.91217 2.31162i −1.74242 1.66139i
2.19 0.111707 2.34502i −0.430563 1.67768i −3.49572 0.333801i −2.49839 + 2.88330i −3.98230 + 0.822271i −0.0230737 0.0447567i −0.505048 + 3.51269i −2.62923 + 1.44469i 6.48232 + 6.18088i
2.20 0.131747 2.76571i 1.65925 0.496888i −5.64086 0.538637i 0.779337 0.899403i −1.15565 4.65446i −0.762358 1.47877i −1.44479 + 10.0487i 2.50620 1.64892i −2.38481 2.27392i
See next 80 embeddings (of 400 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2.20
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
67.h odd 66 1 inner
201.p even 66 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 201.2.p.b 400
3.b odd 2 1 inner 201.2.p.b 400
67.h odd 66 1 inner 201.2.p.b 400
201.p even 66 1 inner 201.2.p.b 400
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
201.2.p.b 400 1.a even 1 1 trivial
201.2.p.b 400 3.b odd 2 1 inner
201.2.p.b 400 67.h odd 66 1 inner
201.2.p.b 400 201.p even 66 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{400} - 10 T_{2}^{398} - 24 T_{2}^{396} + 926 T_{2}^{394} - 5884 T_{2}^{392} - 9646 T_{2}^{390} + 479895 T_{2}^{388} - 3178074 T_{2}^{386} - 3289561 T_{2}^{384} + 223784054 T_{2}^{382} - 1734487148 T_{2}^{380} + \cdots + 44\!\cdots\!16 \) acting on \(S_{2}^{\mathrm{new}}(201, [\chi])\). Copy content Toggle raw display