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Properties

Label 171.2.t.a
Level $171$
Weight $2$
Character orbit 171.t
Analytic conductor $1.365$
Analytic rank $0$
Dimension $36$
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] = [171,2,Mod(122,171)]
 
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([5, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("171.122");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 171 = 3^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 171.t (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: \(1.36544187456\)
Analytic rank: \(0\)
Dimension: \(36\)
Relative dimension: \(18\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 36 q - 6 q^{2} - 3 q^{3} + 30 q^{4} - 3 q^{5} - 11 q^{6} - q^{7} - 12 q^{8} + 5 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 36 q - 6 q^{2} - 3 q^{3} + 30 q^{4} - 3 q^{5} - 11 q^{6} - q^{7} - 12 q^{8} + 5 q^{9} - 6 q^{10} - 9 q^{11} - 15 q^{12} - 3 q^{14} + 18 q^{15} + 18 q^{16} + 27 q^{17} - 6 q^{18} + q^{19} + 9 q^{20} + 6 q^{21} - 6 q^{22} - 5 q^{24} + 11 q^{25} - 9 q^{27} + 2 q^{28} - 12 q^{29} - 35 q^{30} - 12 q^{31} - 30 q^{32} - 21 q^{33} - 21 q^{34} - 15 q^{35} - 3 q^{36} - 36 q^{38} - 14 q^{39} - 30 q^{40} + 18 q^{41} - 26 q^{42} - 6 q^{43} - 66 q^{44} + 29 q^{45} + 45 q^{47} - 21 q^{48} - 9 q^{49} - 3 q^{50} - 36 q^{51} + 12 q^{53} + 38 q^{54} - 7 q^{55} - 6 q^{56} + 17 q^{57} - 6 q^{58} - 9 q^{59} + 75 q^{60} + 8 q^{61} + 12 q^{62} + 41 q^{63} + 12 q^{64} + 18 q^{65} + 11 q^{66} + 60 q^{68} + 51 q^{69} - 24 q^{70} - 9 q^{71} - 81 q^{72} - 18 q^{73} + 10 q^{76} - 6 q^{77} - 18 q^{78} + 12 q^{80} + 13 q^{81} - 9 q^{82} - 9 q^{83} + 96 q^{84} - 22 q^{85} + 102 q^{86} - 19 q^{87} - 3 q^{88} + 18 q^{89} + 15 q^{90} + 27 q^{91} + 17 q^{93} - 12 q^{94} - 21 q^{95} - 9 q^{96} + 84 q^{98} - 5 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
122.1 −2.63354 1.51638 − 0.837019i 4.93552 2.90548 − 1.67748i −3.99344 + 2.20432i 1.12321 + 1.94546i −7.73082 1.59880 − 2.53847i −7.65169 + 4.41771i
122.2 −2.59353 −1.73168 + 0.0356901i 4.72638 −0.664721 + 0.383777i 4.49117 − 0.0925632i −1.86632 − 3.23257i −7.07094 2.99745 − 0.123608i 1.72397 − 0.995336i
122.3 −2.15080 −0.238300 − 1.71558i 2.62595 −2.61989 + 1.51259i 0.512537 + 3.68987i 1.86959 + 3.23822i −1.34630 −2.88643 + 0.817647i 5.63486 − 3.25329i
122.4 −2.03532 −0.453781 + 1.67155i 2.14253 −0.682489 + 0.394035i 0.923590 − 3.40214i 1.40930 + 2.44098i −0.290094 −2.58817 − 1.51704i 1.38908 − 0.801988i
122.5 −1.59456 0.781400 + 1.54577i 0.542617 3.08644 − 1.78196i −1.24599 − 2.46483i −1.39322 − 2.41313i 2.32388 −1.77883 + 2.41574i −4.92151 + 2.84144i
122.6 −1.55091 1.40355 − 1.01491i 0.405316 −1.63223 + 0.942368i −2.17678 + 1.57403i −2.41078 − 4.17560i 2.47321 0.939925 − 2.84895i 2.53144 − 1.46153i
122.7 −0.808184 −1.63815 + 0.562561i −1.34684 1.00246 − 0.578771i 1.32392 − 0.454652i −0.183020 − 0.317000i 2.70486 2.36705 − 1.84311i −0.810173 + 0.467754i
122.8 −0.691402 1.72816 − 0.116017i −1.52196 −0.0926192 + 0.0534737i −1.19485 + 0.0802142i 1.77987 + 3.08283i 2.43509 2.97308 − 0.400991i 0.0640371 − 0.0369718i
122.9 −0.676405 −1.39343 − 1.02876i −1.54248 −0.120039 + 0.0693047i 0.942525 + 0.695858i 0.877021 + 1.51904i 2.39615 0.883308 + 2.86701i 0.0811952 − 0.0468781i
122.10 −0.173194 0.875880 + 1.49427i −1.97000 −2.80886 + 1.62170i −0.151697 − 0.258798i −0.506772 − 0.877755i 0.687581 −1.46567 + 2.61760i 0.486478 − 0.280868i
122.11 0.280205 0.420272 − 1.68029i −1.92149 2.62675 − 1.51656i 0.117762 − 0.470825i 0.223737 + 0.387523i −1.09882 −2.64674 − 1.41236i 0.736029 − 0.424947i
122.12 0.675805 −0.784311 − 1.54430i −1.54329 −2.09151 + 1.20754i −0.530041 − 1.04364i −2.30620 − 3.99445i −2.39457 −1.76971 + 2.42242i −1.41346 + 0.816059i
122.13 1.24204 −1.60011 + 0.663067i −0.457348 −3.19785 + 1.84628i −1.98739 + 0.823553i 1.96257 + 3.39927i −3.05211 2.12068 − 2.12196i −3.97185 + 2.29315i
122.14 1.24245 1.64420 + 0.544601i −0.456316 1.59664 − 0.921821i 2.04284 + 0.676640i −0.955231 − 1.65451i −3.05185 2.40682 + 1.79087i 1.98375 − 1.14532i
122.15 1.75508 −1.72024 − 0.201903i 1.08031 3.41857 − 1.97371i −3.01917 − 0.354356i 0.109801 + 0.190182i −1.61413 2.91847 + 0.694645i 5.99987 − 3.46402i
122.16 1.83651 1.19363 − 1.25509i 1.37279 −1.22285 + 0.706010i 2.19211 − 2.30500i 0.660133 + 1.14338i −1.15189 −0.150515 − 2.99622i −2.24577 + 1.29660i
122.17 2.29322 −0.468871 + 1.66738i 3.25888 0.0873194 − 0.0504139i −1.07523 + 3.82368i −0.977625 − 1.69330i 2.88689 −2.56032 − 1.56357i 0.200243 − 0.115610i
122.18 2.58252 −1.03460 − 1.38910i 4.66943 −1.09060 + 0.629658i −2.67188 − 3.58739i 0.0839377 + 0.145384i 6.89386 −0.859209 + 2.87433i −2.81650 + 1.62611i
164.1 −2.63354 1.51638 + 0.837019i 4.93552 2.90548 + 1.67748i −3.99344 − 2.20432i 1.12321 − 1.94546i −7.73082 1.59880 + 2.53847i −7.65169 − 4.41771i
164.2 −2.59353 −1.73168 − 0.0356901i 4.72638 −0.664721 − 0.383777i 4.49117 + 0.0925632i −1.86632 + 3.23257i −7.07094 2.99745 + 0.123608i 1.72397 + 0.995336i
See all 36 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 122.18
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 171.2.t.a yes 36
3.b odd 2 1 513.2.t.a 36
9.c even 3 1 513.2.k.a 36
9.d odd 6 1 171.2.k.a ✓ 36
19.d odd 6 1 171.2.k.a ✓ 36
57.f even 6 1 513.2.k.a 36
171.i odd 6 1 513.2.t.a 36
171.t even 6 1 inner 171.2.t.a yes 36
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
171.2.k.a ✓ 36 9.d odd 6 1
171.2.k.a ✓ 36 19.d odd 6 1
171.2.t.a yes 36 1.a even 1 1 trivial
171.2.t.a yes 36 171.t even 6 1 inner
513.2.k.a 36 9.c even 3 1
513.2.k.a 36 57.f even 6 1
513.2.t.a 36 3.b odd 2 1
513.2.t.a 36 171.i odd 6 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(171, [\chi])\).