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Properties

Label 171.3.i.a
Level $171$
Weight $3$
Character orbit 171.i
Analytic conductor $4.659$
Analytic rank $0$
Dimension $76$
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] = [171,3,Mod(88,171)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(171, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([4, 5]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("171.88");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 171 = 3^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 171.i (of order \(6\), degree \(2\), 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: \(4.65941252056\)
Analytic rank: \(0\)
Dimension: \(76\)
Relative dimension: \(38\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 76 q - 3 q^{3} - 146 q^{4} + q^{5} + 7 q^{6} - 3 q^{7} - 13 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 76 q - 3 q^{3} - 146 q^{4} + q^{5} + 7 q^{6} - 3 q^{7} - 13 q^{9} - 6 q^{10} + 4 q^{11} - 15 q^{12} + 21 q^{14} - 18 q^{15} + 262 q^{16} + 25 q^{17} + 12 q^{18} - 12 q^{19} - 17 q^{20} + 24 q^{21} - 15 q^{22} + 46 q^{23} - 23 q^{24} - 149 q^{25} + 48 q^{26} - 63 q^{27} + 30 q^{28} - 30 q^{29} - 41 q^{30} + 48 q^{31} - 93 q^{33} + 15 q^{34} - 31 q^{35} - 51 q^{36} - 135 q^{38} + 28 q^{39} + 96 q^{40} + 123 q^{41} + 238 q^{42} + 182 q^{43} - 191 q^{44} - 289 q^{45} + 61 q^{47} + 123 q^{48} - 171 q^{49} + 243 q^{50} - 45 q^{51} - 42 q^{53} + 224 q^{54} + 23 q^{55} - 624 q^{56} - 133 q^{57} + 6 q^{58} - 390 q^{59} + 381 q^{60} - 6 q^{61} - 366 q^{62} + 323 q^{63} - 152 q^{64} + 582 q^{65} + 95 q^{66} - 74 q^{68} - 75 q^{69} - 150 q^{70} - 87 q^{71} + 99 q^{72} + 29 q^{73} + 252 q^{74} - 585 q^{75} - 3 q^{76} + 32 q^{77} - 216 q^{78} - 104 q^{80} - 5 q^{81} + 54 q^{82} - 23 q^{83} + 204 q^{84} + 98 q^{85} + 671 q^{87} + 132 q^{88} - 222 q^{89} + 249 q^{90} - 51 q^{91} + 694 q^{92} + 293 q^{93} + 24 q^{94} + 145 q^{95} + 147 q^{96} - 558 q^{98} - 92 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
88.1 − 3.78076i −2.81408 + 1.03968i −10.2942 −0.565984 + 0.980312i 3.93080 + 10.6394i 0.290777 − 0.503641i 23.7968i 6.83811 − 5.85151i 3.70633 + 2.13985i
88.2 − 3.77882i 1.46545 − 2.61772i −10.2795 2.67865 − 4.63956i −9.89189 − 5.53769i −5.65314 + 9.79152i 23.7291i −4.70489 − 7.67229i −17.5321 − 10.1221i
88.3 − 3.46206i 2.39274 + 1.80964i −7.98587 −3.86426 + 6.69309i 6.26507 − 8.28382i −5.09033 + 8.81670i 13.7993i 2.45044 + 8.65999i 23.1719 + 13.3783i
88.4 − 3.38557i 2.99959 − 0.0494451i −7.46206 0.678756 − 1.17564i −0.167400 − 10.1553i 5.25485 − 9.10167i 11.7210i 8.99511 − 0.296630i −3.98021 − 2.29797i
88.5 − 3.15116i −0.0857843 + 2.99877i −5.92979 −0.436422 + 0.755905i 9.44960 + 0.270320i 4.49942 − 7.79322i 6.08106i −8.98528 − 0.514495i 2.38197 + 1.37523i
88.6 − 3.03225i −1.94651 − 2.28279i −5.19456 −2.33408 + 4.04274i −6.92198 + 5.90230i −2.09721 + 3.63247i 3.62219i −1.42222 + 8.88692i 12.2586 + 7.07751i
88.7 − 2.89703i −2.37241 − 1.83621i −4.39279 4.55283 − 7.88572i −5.31956 + 6.87295i 3.39683 − 5.88348i 1.13793i 2.25666 + 8.71249i −22.8452 − 13.1897i
88.8 − 2.65629i 0.653061 − 2.92806i −3.05587 −0.157936 + 0.273553i −7.77776 − 1.73472i 2.08385 − 3.60934i − 2.50787i −8.14702 − 3.82440i 0.726637 + 0.419524i
88.9 − 2.41326i −2.25389 + 1.97989i −1.82382 3.12409 − 5.41107i 4.77800 + 5.43922i −3.26441 + 5.65412i − 5.25168i 1.16004 − 8.92493i −13.0583 − 7.53923i
88.10 − 2.07974i 2.55552 + 1.57140i −0.325323 3.09135 − 5.35437i 3.26811 − 5.31483i −2.07739 + 3.59815i − 7.64238i 4.06140 + 8.03150i −11.1357 − 6.42920i
88.11 − 1.99018i −0.443309 + 2.96707i 0.0391887 −2.19213 + 3.79688i 5.90499 + 0.882263i −1.88302 + 3.26149i − 8.03871i −8.60696 − 2.63065i 7.55647 + 4.36273i
88.12 − 1.74091i −2.95617 + 0.510929i 0.969225 −4.31209 + 7.46875i 0.889482 + 5.14644i 3.90035 − 6.75560i − 8.65098i 8.47790 − 3.02079i 13.0024 + 7.50696i
88.13 − 1.72304i 2.56260 − 1.55984i 1.03112 −1.01218 + 1.75316i −2.68767 − 4.41547i −1.46467 + 2.53688i − 8.66884i 4.13382 − 7.99447i 3.02076 + 1.74404i
88.14 − 0.985708i −0.515769 − 2.95533i 3.02838 −0.789211 + 1.36695i −2.91309 + 0.508398i 3.47559 − 6.01990i − 6.92793i −8.46796 + 3.04854i 1.34742 + 0.777931i
88.15 − 0.854005i −2.97727 − 0.368629i 3.27068 −0.519007 + 0.898946i −0.314811 + 2.54260i −6.16579 + 10.6795i − 6.20919i 8.72822 + 2.19502i 0.767704 + 0.443234i
88.16 − 0.580610i 0.720238 + 2.91226i 3.66289 2.13268 − 3.69392i 1.69089 − 0.418177i 5.37376 − 9.30762i − 4.44915i −7.96251 + 4.19504i −2.14472 − 1.23826i
88.17 − 0.515233i 2.87448 + 0.858686i 3.73454 −3.85682 + 6.68020i 0.442423 − 1.48103i 3.94745 − 6.83719i − 3.98509i 7.52532 + 4.93656i 3.44186 + 1.98716i
88.18 − 0.430289i −2.65564 − 1.39555i 3.81485 2.08594 − 3.61295i −0.600491 + 1.14269i 1.11442 − 1.93023i − 3.36264i 5.10486 + 7.41218i −1.55461 − 0.897555i
88.19 − 0.410160i 1.62411 − 2.52235i 3.83177 4.20106 − 7.27645i −1.03457 − 0.666144i −1.03558 + 1.79367i − 3.21228i −3.72454 − 8.19316i −2.98451 − 1.72311i
88.20 0.307806i 1.58044 + 2.54994i 3.90526 −0.733749 + 1.27089i −0.784889 + 0.486468i −4.17239 + 7.22680i 2.43329i −4.00443 + 8.06006i −0.391188 − 0.225853i
See all 76 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 88.38
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
171.i odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 171.3.i.a ✓ 76
3.b odd 2 1 513.3.i.a 76
9.c even 3 1 171.3.s.a yes 76
9.d odd 6 1 513.3.s.a 76
19.d odd 6 1 171.3.s.a yes 76
57.f even 6 1 513.3.s.a 76
171.i odd 6 1 inner 171.3.i.a ✓ 76
171.t even 6 1 513.3.i.a 76
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
171.3.i.a ✓ 76 1.a even 1 1 trivial
171.3.i.a ✓ 76 171.i odd 6 1 inner
171.3.s.a yes 76 9.c even 3 1
171.3.s.a yes 76 19.d odd 6 1
513.3.i.a 76 3.b odd 2 1
513.3.i.a 76 171.t even 6 1
513.3.s.a 76 9.d odd 6 1
513.3.s.a 76 57.f even 6 1

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(171, [\chi])\).