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Properties

Label 261.3.s.a
Level $261$
Weight $3$
Character orbit 261.s
Analytic conductor $7.112$
Analytic rank $0$
Dimension $48$
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: \(48\)
Relative dimension: \(4\) over \(\Q(\zeta_{28})\)
Twist minimal: no (minimal twist has level 29)
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.

\(\operatorname{Tr}(f)(q) = \) \( 48 q + 16 q^{2} - 14 q^{4} + 14 q^{5} - 10 q^{7} - 28 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 48 q + 16 q^{2} - 14 q^{4} + 14 q^{5} - 10 q^{7} - 28 q^{8} - 20 q^{10} + 8 q^{11} - 14 q^{13} - 26 q^{14} + 18 q^{16} + 26 q^{17} + 2 q^{19} - 46 q^{20} + 154 q^{22} - 56 q^{23} - 34 q^{25} - 110 q^{26} + 170 q^{29} - 88 q^{31} + 132 q^{32} - 224 q^{34} + 210 q^{35} - 56 q^{37} + 294 q^{38} - 492 q^{40} + 34 q^{41} + 176 q^{43} - 126 q^{44} + 744 q^{46} - 208 q^{47} + 506 q^{49} - 732 q^{50} + 690 q^{52} + 14 q^{53} + 284 q^{55} - 332 q^{56} - 508 q^{58} + 44 q^{59} - 30 q^{61} + 504 q^{62} - 896 q^{64} + 554 q^{65} - 574 q^{67} + 796 q^{68} - 1066 q^{70} - 224 q^{71} - 22 q^{73} - 820 q^{74} + 514 q^{76} - 436 q^{77} + 564 q^{79} - 1162 q^{80} - 18 q^{82} + 126 q^{83} + 38 q^{85} - 384 q^{88} + 160 q^{89} - 434 q^{91} + 1022 q^{92} - 2 q^{94} + 642 q^{95} + 604 q^{97} + 102 q^{98}+O(q^{100}) \) 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 −2.05804 + 1.29315i 0 0.827740 − 1.71882i −5.87720 − 1.34143i 0 −9.36468 + 4.50979i −0.569384 − 5.05343i 0 13.8301 − 4.83938i
10.2 −1.44470 + 0.907767i 0 −0.472410 + 0.980970i 8.15497 + 1.86132i 0 7.53715 − 3.62970i −0.972146 − 8.62804i 0 −13.4711 + 4.71376i
10.3 1.29187 − 0.811733i 0 −0.725528 + 1.50658i −3.36294 − 0.767569i 0 −0.255710 + 0.123143i 0.968958 + 8.59974i 0 −4.96753 + 1.73821i
10.4 3.24379 − 2.03821i 0 4.63233 − 9.61914i 5.12808 + 1.17045i 0 −6.56728 + 3.16264i −2.86375 − 25.4165i 0 19.0200 − 6.65539i
19.1 −1.42815 − 2.27289i 0 −1.39088 + 2.88819i 6.69994 + 1.52922i 0 −4.78081 + 2.30231i −2.11889 + 0.238741i 0 −6.09279 − 17.4122i
19.2 −0.488049 − 0.776726i 0 1.37042 − 2.84571i 1.98497 + 0.453055i 0 9.56374 − 4.60566i −6.52543 + 0.735239i 0 −0.616861 − 1.76289i
19.3 0.448531 + 0.713833i 0 1.42716 − 2.96352i −4.21883 − 0.962920i 0 −10.1429 + 4.88457i 6.10659 − 0.688048i 0 −1.20491 − 3.44344i
19.4 1.63282 + 2.59861i 0 −2.35117 + 4.88225i 4.43970 + 1.01333i 0 10.7635 − 5.18344i −4.32723 + 0.487562i 0 4.61596 + 13.1917i
37.1 −1.65381 − 0.578694i 0 −0.727117 − 0.579856i 0.825315 + 1.71379i 0 −1.24782 − 1.56472i 4.59573 + 7.31406i 0 −0.373160 − 3.31188i
37.2 0.0310749 + 0.0108736i 0 −3.12648 − 2.49328i −3.79007 − 7.87017i 0 3.62612 + 4.54701i −0.140107 − 0.222980i 0 −0.0321993 − 0.285777i
37.3 2.20133 + 0.770280i 0 1.12521 + 0.897324i −2.64264 − 5.48750i 0 −3.22936 − 4.04949i −3.17747 − 5.05691i 0 −1.59042 − 14.1154i
37.4 2.32673 + 0.814157i 0 1.62348 + 1.29468i 3.83207 + 7.95737i 0 2.23777 + 2.80607i −2.52264 − 4.01476i 0 2.43763 + 21.6345i
55.1 −1.42815 + 2.27289i 0 −1.39088 − 2.88819i 6.69994 − 1.52922i 0 −4.78081 − 2.30231i −2.11889 − 0.238741i 0 −6.09279 + 17.4122i
55.2 −0.488049 + 0.776726i 0 1.37042 + 2.84571i 1.98497 − 0.453055i 0 9.56374 + 4.60566i −6.52543 − 0.735239i 0 −0.616861 + 1.76289i
55.3 0.448531 − 0.713833i 0 1.42716 + 2.96352i −4.21883 + 0.962920i 0 −10.1429 − 4.88457i 6.10659 + 0.688048i 0 −1.20491 + 3.44344i
55.4 1.63282 − 2.59861i 0 −2.35117 − 4.88225i 4.43970 − 1.01333i 0 10.7635 + 5.18344i −4.32723 − 0.487562i 0 4.61596 − 13.1917i
73.1 −2.58169 + 0.290886i 0 2.68078 − 0.611870i 2.49104 + 1.98654i 0 1.30161 − 5.70272i 3.06598 − 1.07283i 0 −7.00893 − 4.40400i
73.2 −0.415096 + 0.0467701i 0 −3.72959 + 0.851255i −0.738700 − 0.589093i 0 −0.577468 + 2.53005i 3.08546 − 1.07965i 0 0.334184 + 0.209981i
73.3 1.68783 − 0.190173i 0 −1.08711 + 0.248126i −0.141728 − 0.113024i 0 −1.55116 + 6.79606i −8.20045 + 2.86946i 0 −0.260707 − 0.163813i
73.4 3.56136 − 0.401269i 0 8.62256 − 1.96804i −5.24106 − 4.17960i 0 2.11977 − 9.28734i 16.3872 − 5.73413i 0 −20.3424 − 12.7820i
See all 48 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 10.4
Significant digits:
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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.a 48
3.b odd 2 1 29.3.f.a ✓ 48
29.f odd 28 1 inner 261.3.s.a 48
87.k even 28 1 29.3.f.a ✓ 48
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
29.3.f.a ✓ 48 3.b odd 2 1
29.3.f.a ✓ 48 87.k even 28 1
261.3.s.a 48 1.a even 1 1 trivial
261.3.s.a 48 29.f odd 28 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{48} - 16 T_{2}^{47} + 135 T_{2}^{46} - 780 T_{2}^{45} + 3312 T_{2}^{44} - 10520 T_{2}^{43} + \cdots + 352951369 \) acting on \(S_{3}^{\mathrm{new}}(261, [\chi])\). Copy content Toggle raw display