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Properties

Label 261.3.s.b
Level $261$
Weight $3$
Character orbit 261.s
Analytic conductor $7.112$
Analytic rank $0$
Dimension $120$
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] = [261,3,Mod(10,261)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(261, base_ring=CyclotomicField(28))
 
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("261.10");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 261 = 3^{2} \cdot 29 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 261.s (of order \(28\), degree \(12\), 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: \(7.11173489980\)
Analytic rank: \(0\)
Dimension: \(120\)
Relative dimension: \(10\) over \(\Q(\zeta_{28})\)
Twist minimal: no (minimal twist has level 87)
Sato-Tate group: $\mathrm{SU}(2)[C_{28}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 120 q - 8 q^{2} + 60 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 120 q - 8 q^{2} + 60 q^{8} + 32 q^{11} + 12 q^{14} + 72 q^{16} + 8 q^{17} - 8 q^{19} + 240 q^{20} - 336 q^{22} + 132 q^{23} + 148 q^{25} + 152 q^{26} - 100 q^{29} + 172 q^{31} - 336 q^{32} + 420 q^{34} - 392 q^{35} + 188 q^{37} - 560 q^{38} + 700 q^{40} - 56 q^{41} - 320 q^{43} + 296 q^{44} - 376 q^{46} + 344 q^{47} - 556 q^{49} + 1572 q^{50} - 932 q^{52} - 56 q^{53} - 256 q^{55} + 748 q^{56} + 460 q^{58} - 160 q^{59} + 192 q^{61} - 980 q^{62} - 600 q^{65} + 560 q^{67} - 1776 q^{68} + 1052 q^{70} - 784 q^{71} - 320 q^{73} - 620 q^{74} - 536 q^{76} - 464 q^{77} - 288 q^{79} - 112 q^{82} + 368 q^{83} - 4 q^{85} + 56 q^{88} + 112 q^{89} + 420 q^{91} - 284 q^{94} + 136 q^{95} - 1324 q^{97} + 1456 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
10.1 −2.89410 + 1.81848i 0 3.33338 − 6.92184i 0.541618 + 0.123621i 0 3.21833 − 1.54986i 1.40933 + 12.5082i 0 −1.79230 + 0.627151i
10.2 −2.72695 + 1.71346i 0 2.76480 − 5.74117i 2.24899 + 0.513318i 0 −4.13335 + 1.99052i 0.855404 + 7.59192i 0 −7.01245 + 2.45376i
10.3 −1.12909 + 0.709452i 0 −0.964020 + 2.00181i −1.86346 − 0.425323i 0 5.50650 − 2.65179i −0.928933 − 8.24451i 0 2.40576 − 0.841811i
10.4 −0.689063 + 0.432967i 0 −1.44819 + 3.00719i −5.51213 − 1.25811i 0 10.2092 − 4.91649i −0.668589 − 5.93389i 0 4.34292 − 1.51965i
10.5 −0.293906 + 0.184673i 0 −1.68326 + 3.49532i 4.09256 + 0.934101i 0 −7.74224 + 3.72847i −0.306229 − 2.71785i 0 −1.37533 + 0.481249i
10.6 0.153173 − 0.0962451i 0 −1.72134 + 3.57439i −4.03972 − 0.922040i 0 −4.21732 + 2.03096i 0.161373 + 1.43223i 0 −0.707519 + 0.247572i
10.7 1.34844 − 0.847278i 0 −0.635136 + 1.31887i 8.55782 + 1.95327i 0 −3.33059 + 1.60393i 0.974241 + 8.64663i 0 13.1946 − 4.61700i
10.8 1.96972 − 1.23766i 0 0.612471 − 1.27181i 3.10034 + 0.707633i 0 5.42647 − 2.61325i 0.674177 + 5.98349i 0 6.98262 − 2.44332i
10.9 2.30831 − 1.45041i 0 1.48908 − 3.09211i −9.07338 − 2.07094i 0 −7.59823 + 3.65912i 0.173376 + 1.53876i 0 −23.9479 + 8.37974i
10.10 2.75540 − 1.73133i 0 2.85918 − 5.93716i −2.33652 − 0.533296i 0 9.68828 − 4.66563i −0.943583 − 8.37453i 0 −7.36138 + 2.57586i
19.1 −2.00834 − 3.19625i 0 −4.44708 + 9.23445i −2.63960 − 0.602470i 0 3.87010 − 1.86374i 23.4424 − 2.64133i 0 3.37555 + 9.64678i
19.2 −1.78356 − 2.83852i 0 −3.14057 + 6.52145i 2.74140 + 0.625707i 0 2.88031 − 1.38708i 10.7875 − 1.21546i 0 −3.11337 − 8.89750i
19.3 −1.21026 − 1.92611i 0 −0.509654 + 1.05831i −6.81765 − 1.55608i 0 −7.31790 + 3.52411i −6.38668 + 0.719606i 0 5.25391 + 15.0148i
19.4 −0.895405 − 1.42503i 0 0.506578 − 1.05192i −2.22038 − 0.506787i 0 −0.486644 + 0.234355i −8.64224 + 0.973747i 0 1.26595 + 3.61788i
19.5 0.285100 + 0.453733i 0 1.61094 − 3.34516i −3.27824 − 0.748238i 0 1.91324 − 0.921368i 4.10709 − 0.462758i 0 −0.595126 − 1.70077i
19.6 0.307001 + 0.488590i 0 1.59106 − 3.30388i 6.66842 + 1.52202i 0 −0.0882299 + 0.0424893i 4.39633 − 0.495347i 0 1.30357 + 3.72538i
19.7 1.05332 + 1.67635i 0 0.0348658 − 0.0723996i −7.39573 − 1.68803i 0 5.02392 − 2.41939i 8.02753 − 0.904486i 0 −4.96036 − 14.1759i
19.8 1.19609 + 1.90356i 0 −0.457390 + 0.949779i 4.88061 + 1.11397i 0 −3.36949 + 1.62266i 6.58101 − 0.741502i 0 3.71713 + 10.6229i
19.9 1.91360 + 3.04547i 0 −3.87752 + 8.05174i −7.06239 − 1.61194i 0 2.38032 − 1.14630i −17.6447 + 1.98809i 0 −8.60544 − 24.5929i
19.10 1.94439 + 3.09448i 0 −4.05961 + 8.42988i 1.11394 + 0.254250i 0 −11.8327 + 5.69831i −19.4529 + 2.19181i 0 1.37917 + 3.94144i
See next 80 embeddings (of 120 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 10.10
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
29.f odd 28 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 261.3.s.b 120
3.b odd 2 1 87.3.l.a ✓ 120
29.f odd 28 1 inner 261.3.s.b 120
87.k even 28 1 87.3.l.a ✓ 120
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
87.3.l.a ✓ 120 3.b odd 2 1
87.3.l.a ✓ 120 87.k even 28 1
261.3.s.b 120 1.a even 1 1 trivial
261.3.s.b 120 29.f odd 28 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{120} + 8 T_{2}^{119} + 32 T_{2}^{118} + 44 T_{2}^{117} - 458 T_{2}^{116} - 2880 T_{2}^{115} + \cdots + 18\!\cdots\!81 \) acting on \(S_{3}^{\mathrm{new}}(261, [\chi])\). Copy content Toggle raw display