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Properties

Label 201.3.n.b
Level $201$
Weight $3$
Character orbit 201.n
Analytic conductor $5.477$
Analytic rank $0$
Dimension $240$
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] = [201,3,Mod(7,201)]
 
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([0, 23]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("201.7");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 201 = 3 \cdot 67 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 201.n (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: \(5.47685331364\)
Analytic rank: \(0\)
Dimension: \(240\)
Relative dimension: \(12\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 240 q - 34 q^{4} - 33 q^{6} - 21 q^{7} - 33 q^{8} + 72 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 240 q - 34 q^{4} - 33 q^{6} - 21 q^{7} - 33 q^{8} + 72 q^{9} + 69 q^{10} - 111 q^{11} - 3 q^{12} - 30 q^{13} - 6 q^{14} - 27 q^{15} + 98 q^{16} - 4 q^{17} + 16 q^{19} - 108 q^{20} + 21 q^{21} + 27 q^{22} + 178 q^{23} + 36 q^{24} + 222 q^{25} - 29 q^{26} - 112 q^{28} - 77 q^{29} + 90 q^{30} + 137 q^{31} + 44 q^{32} + 12 q^{33} - 72 q^{34} - 237 q^{35} + 3 q^{36} + 132 q^{37} + 210 q^{38} - 30 q^{39} + 749 q^{40} - 150 q^{41} - 132 q^{42} - 385 q^{43} + 9 q^{44} - 443 q^{46} - 166 q^{47} - 294 q^{48} - 295 q^{49} - 6 q^{50} + 276 q^{51} - 1804 q^{52} + 176 q^{53} + 199 q^{55} - 1361 q^{56} - 114 q^{57} + 968 q^{58} - 214 q^{59} - 420 q^{60} - 274 q^{61} + 334 q^{62} - 102 q^{63} + 683 q^{64} - 224 q^{65} + 47 q^{67} + 870 q^{68} + 27 q^{69} - 44 q^{70} + 271 q^{71} + 264 q^{72} + 594 q^{73} - 1289 q^{74} + 396 q^{75} + 494 q^{76} + 1360 q^{77} + 441 q^{78} + 1023 q^{79} + 15 q^{80} - 216 q^{81} - 316 q^{82} - 225 q^{83} + 1527 q^{84} - 153 q^{85} - 91 q^{86} - 1676 q^{88} + 871 q^{89} - 207 q^{90} - 692 q^{91} - 488 q^{92} - 390 q^{93} + 440 q^{94} - 531 q^{95} - 33 q^{96} + 84 q^{97} + 85 q^{98} + 36 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
7.1 −1.79502 − 3.48186i −0.487975 + 1.66189i −6.58102 + 9.24175i 1.84877 + 1.60197i 6.66240 − 1.28407i −4.70180 − 0.223974i 28.4818 + 4.09506i −2.52376 − 1.62192i 2.25925 − 9.31274i
7.2 −1.35193 − 2.62237i −0.487975 + 1.66189i −2.72891 + 3.83222i −4.43909 − 3.84649i 5.01781 − 0.967103i 1.03058 + 0.0490928i 2.05754 + 0.295830i −2.52376 − 1.62192i −4.08562 + 16.8411i
7.3 −1.08640 − 2.10732i −0.487975 + 1.66189i −0.940314 + 1.32049i 6.32185 + 5.47792i 4.03228 − 0.777157i −2.91931 − 0.139064i −5.58274 − 0.802677i −2.52376 − 1.62192i 4.67567 − 19.2734i
7.4 −0.738773 − 1.43302i −0.487975 + 1.66189i 0.812467 − 1.14095i 1.60508 + 1.39081i 2.74203 − 0.528482i −7.37536 − 0.351332i −8.61857 − 1.23916i −2.52376 − 1.62192i 0.807267 − 3.32760i
7.5 −0.577269 − 1.11975i −0.487975 + 1.66189i 1.39964 − 1.96551i −2.24142 − 1.94220i 2.14259 − 0.412950i 11.5313 + 0.549306i −7.99671 − 1.14975i −2.52376 − 1.62192i −0.880870 + 3.63099i
7.6 −0.110765 − 0.214854i −0.487975 + 1.66189i 2.28633 − 3.21071i −3.52548 − 3.05485i 0.411115 − 0.0792359i −9.28062 − 0.442091i −1.90014 − 0.273199i −2.52376 − 1.62192i −0.265847 + 1.09584i
7.7 0.193811 + 0.375940i −0.487975 + 1.66189i 2.21646 − 3.11258i 4.88626 + 4.23397i −0.719346 + 0.138643i 1.38901 + 0.0661669i 3.27433 + 0.470777i −2.52376 − 1.62192i −0.644710 + 2.65753i
7.8 0.547626 + 1.06225i −0.487975 + 1.66189i 1.49175 − 2.09488i −5.31200 − 4.60288i −2.03256 + 0.391745i −3.76569 − 0.179382i 7.77394 + 1.11772i −2.52376 − 1.62192i 1.98040 − 8.16331i
7.9 0.860039 + 1.66824i −0.487975 + 1.66189i 0.276863 − 0.388799i 0.779784 + 0.675687i −3.19211 + 0.615229i 7.09141 + 0.337805i 8.31785 + 1.19593i −2.52376 − 1.62192i −0.456565 + 1.88199i
7.10 1.46078 + 2.83351i −0.487975 + 1.66189i −3.57469 + 5.01994i 1.03546 + 0.897230i −5.42181 + 1.04497i −11.9148 − 0.567571i −6.82411 − 0.981159i −2.52376 − 1.62192i −1.02974 + 4.24463i
7.11 1.51795 + 2.94440i −0.487975 + 1.66189i −4.04513 + 5.68059i 5.90107 + 5.11330i −5.63400 + 1.08586i 7.02729 + 0.334751i −9.75051 − 1.40191i −2.52376 − 1.62192i −6.09813 + 25.1368i
7.12 1.77257 + 3.43831i −0.487975 + 1.66189i −6.35974 + 8.93100i −6.98484 − 6.05240i −6.57907 + 1.26801i 6.92425 + 0.329843i −26.6648 − 3.83382i −2.52376 − 1.62192i 8.42889 − 34.7444i
13.1 −2.23952 − 2.84778i −1.30900 − 1.13425i −2.15136 + 8.86804i −3.98657 − 6.20322i −0.298577 + 6.26791i −2.90854 + 7.26517i 16.8902 − 7.71351i 0.426945 + 2.96946i −8.73741 + 25.2451i
13.2 −2.22017 − 2.82317i −1.30900 − 1.13425i −2.09813 + 8.64859i 3.60424 + 5.60831i −0.295998 + 6.21375i 3.22377 − 8.05259i 16.0066 − 7.30997i 0.426945 + 2.96946i 7.83121 − 22.6268i
13.3 −1.77293 − 2.25446i −1.30900 − 1.13425i −0.996290 + 4.10676i 0.859686 + 1.33770i −0.236371 + 4.96203i −1.21984 + 3.04701i 0.589306 − 0.269127i 0.426945 + 2.96946i 1.49163 − 4.30977i
13.4 −1.37068 − 1.74296i −1.30900 − 1.13425i −0.216110 + 0.890817i −2.17329 − 3.38170i −0.182742 + 3.83622i 4.27625 − 10.6815i −6.21903 + 2.84014i 0.426945 + 2.96946i −2.91529 + 8.42317i
13.5 −0.827200 − 1.05187i −1.30900 − 1.13425i 0.520864 − 2.14703i 3.07093 + 4.77845i −0.110284 + 2.31515i −2.67312 + 6.67713i −7.55822 + 3.45172i 0.426945 + 2.96946i 2.48604 − 7.18295i
13.6 −0.333348 − 0.423887i −1.30900 − 1.13425i 0.874477 − 3.60464i −3.59519 − 5.59422i −0.0444427 + 0.932967i 1.22122 − 3.05045i −3.78158 + 1.72699i 0.426945 + 2.96946i −1.17287 + 3.38877i
13.7 0.127437 + 0.162049i −1.30900 − 1.13425i 0.933016 − 3.84594i 2.68870 + 4.18370i 0.0169901 − 0.356667i 1.06344 − 2.65636i 1.49223 − 0.681481i 0.426945 + 2.96946i −0.335325 + 0.968859i
13.8 0.774122 + 0.984377i −1.30900 − 1.13425i 0.573304 − 2.36319i −3.17987 − 4.94798i 0.103208 − 2.16660i −3.85408 + 9.62704i 7.32662 − 3.34595i 0.426945 + 2.96946i 2.40906 − 6.96053i
See next 80 embeddings (of 240 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 7.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
67.h odd 66 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 201.3.n.b ✓ 240
67.h odd 66 1 inner 201.3.n.b ✓ 240
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
201.3.n.b ✓ 240 1.a even 1 1 trivial
201.3.n.b ✓ 240 67.h odd 66 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{240} + 41 T_{2}^{238} + 11 T_{2}^{237} + 618 T_{2}^{236} + 183 T_{2}^{235} + \cdots + 60\!\cdots\!24 \) acting on \(S_{3}^{\mathrm{new}}(201, [\chi])\). Copy content Toggle raw display