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Properties

Label 171.3.bf.a
Level $171$
Weight $3$
Character orbit 171.bf
Analytic conductor $4.659$
Analytic rank $0$
Dimension $228$
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(23,171)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(171, base_ring=CyclotomicField(18))
 
chi = DirichletCharacter(H, H._module([15, 2]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("171.23");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 171 = 3^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 171.bf (of order \(18\), degree \(6\), 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: \(228\)
Relative dimension: \(38\) over \(\Q(\zeta_{18})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{18}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 228 q - 9 q^{2} - 12 q^{3} - 3 q^{4} - 9 q^{5} + 6 q^{6} - 6 q^{7} - 24 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 228 q - 9 q^{2} - 12 q^{3} - 3 q^{4} - 9 q^{5} + 6 q^{6} - 6 q^{7} - 24 q^{9} - 12 q^{10} - 9 q^{11} - 3 q^{12} + 12 q^{13} - 9 q^{14} - 21 q^{15} - 27 q^{16} + 81 q^{17} - 60 q^{18} - 33 q^{19} - 18 q^{20} - 60 q^{21} + 9 q^{22} - 9 q^{23} + 345 q^{24} - 3 q^{25} + 216 q^{26} - 33 q^{27} - 36 q^{28} + 72 q^{29} - 270 q^{30} + 3 q^{31} - 153 q^{32} + 84 q^{33} - 21 q^{34} - 225 q^{35} + 6 q^{36} - 24 q^{37} + 99 q^{38} - 60 q^{39} + 48 q^{40} + 369 q^{41} - 438 q^{42} - 195 q^{43} - 441 q^{44} + 240 q^{45} - 6 q^{46} - 9 q^{47} - 630 q^{48} + 1086 q^{49} - 441 q^{50} - 81 q^{51} - 111 q^{52} - 336 q^{54} + 63 q^{55} - 459 q^{56} + 120 q^{57} - 6 q^{58} + 504 q^{59} + 225 q^{60} + 39 q^{61} + 36 q^{62} - 504 q^{63} + 372 q^{64} - 9 q^{65} + 228 q^{66} - 24 q^{67} - 120 q^{69} - 150 q^{70} - 48 q^{72} - 51 q^{73} - 990 q^{74} + 324 q^{75} - 3 q^{76} - 18 q^{77} + 141 q^{78} + 48 q^{79} + 756 q^{80} - 588 q^{81} + 132 q^{82} + 129 q^{84} - 3 q^{85} - 9 q^{86} + 453 q^{87} - 774 q^{88} + 648 q^{89} + 1515 q^{90} + 225 q^{91} + 1287 q^{92} - 387 q^{93} - 6 q^{94} - 9 q^{95} - 663 q^{96} + 267 q^{97} - 1125 q^{98} - 444 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
23.1 −3.65933 + 0.645238i −0.459116 − 2.96466i 9.21558 − 3.35420i 5.91829 − 1.04355i 3.59297 + 10.5524i 10.9088 −18.6867 + 10.7888i −8.57843 + 2.72225i −20.9836 + 7.63741i
23.2 −3.64108 + 0.642021i −2.57388 − 1.54115i 9.08651 − 3.30722i −6.42488 + 1.13288i 10.3612 + 3.95898i −7.72654 −18.1538 + 10.4811i 4.24969 + 7.93348i 22.6662 − 8.24981i
23.3 −3.62609 + 0.639378i 1.79662 + 2.40253i 8.98096 − 3.26880i −1.84348 + 0.325055i −8.05084 − 7.56307i −2.47175 −17.7208 + 10.2311i −2.54429 + 8.63288i 6.47679 − 2.35736i
23.4 −3.46435 + 0.610858i −2.11681 + 2.12582i 7.86980 − 2.86437i 9.17839 − 1.61840i 6.03480 − 8.65765i −9.01954 −13.3280 + 7.69493i −0.0382204 − 8.99992i −30.8085 + 11.2134i
23.5 −3.28012 + 0.578374i −1.93034 + 2.29647i 6.66591 − 2.42619i −4.81965 + 0.849835i 5.00352 − 8.64917i 11.2987 −8.92377 + 5.15214i −1.54759 − 8.86594i 15.3175 − 5.57512i
23.6 −3.09884 + 0.546410i 2.53490 − 1.60445i 5.54550 − 2.01840i −6.17011 + 1.08796i −6.97858 + 6.35704i 5.25884 −5.18146 + 2.99152i 3.85147 − 8.13426i 18.5257 − 6.74281i
23.7 −2.74457 + 0.483941i 0.979111 − 2.83573i 3.53968 − 1.28834i 1.23762 − 0.218226i −1.31491 + 8.25668i −12.5962 0.562706 − 0.324878i −7.08268 − 5.55298i −3.29113 + 1.19787i
23.8 −2.73684 + 0.482579i 2.99746 − 0.123538i 3.49866 − 1.27341i 7.74651 − 1.36592i −8.14395 + 1.78461i 2.71206 0.666199 − 0.384630i 8.96948 − 0.740598i −20.5418 + 7.47661i
23.9 −2.35710 + 0.415620i −2.93680 + 0.612531i 1.62440 − 0.591235i 1.35703 − 0.239281i 6.66775 − 2.66439i 3.53164 4.70804 − 2.71819i 8.24961 − 3.59776i −3.09920 + 1.12802i
23.10 −2.31732 + 0.408605i 2.50810 + 1.64603i 1.44422 − 0.525655i 1.25775 − 0.221775i −6.48464 − 2.78955i −0.0673877 5.01931 − 2.89790i 3.58115 + 8.25684i −2.82399 + 1.02785i
23.11 −1.99638 + 0.352015i −2.18466 − 2.05603i 0.102834 − 0.0374285i 2.67211 − 0.471166i 5.08516 + 3.33557i 3.51086 6.83022 − 3.94343i 0.545494 + 8.98345i −5.16869 + 1.88125i
23.12 −1.98725 + 0.350406i −0.0711634 + 2.99916i 0.0676156 − 0.0246101i −1.58861 + 0.280114i −0.909503 − 5.98501i −6.99408 6.86449 − 3.96322i −8.98987 − 0.426860i 3.05881 − 1.11332i
23.13 −1.52687 + 0.269229i −0.833307 − 2.88194i −1.49991 + 0.545922i −7.11297 + 1.25421i 2.04826 + 4.17602i 2.04957 7.51404 − 4.33823i −7.61120 + 4.80309i 10.5229 − 3.83004i
23.14 −1.24893 + 0.220221i −2.79191 + 1.09782i −2.24743 + 0.817998i −8.14879 + 1.43685i 3.24515 − 1.98595i −8.61467 7.01993 − 4.05296i 6.58956 − 6.13006i 9.86088 − 3.58907i
23.15 −1.21478 + 0.214198i −0.319844 + 2.98290i −2.32897 + 0.847674i 7.14776 − 1.26034i −0.250393 − 3.69207i 8.34631 6.92064 − 3.99563i −8.79540 − 1.90813i −8.41298 + 3.06207i
23.16 −0.817290 + 0.144110i 1.58887 + 2.54470i −3.11157 + 1.13252i −8.08346 + 1.42533i −1.66529 − 1.85078i 10.6388 5.25470 − 3.03380i −3.95097 + 8.08640i 6.40113 − 2.32982i
23.17 −0.624400 + 0.110099i 2.97879 − 0.356077i −3.38102 + 1.23059i −2.37342 + 0.418498i −1.82075 + 0.550296i −10.8393 4.17197 − 2.40869i 8.74642 − 2.12136i 1.43589 − 0.522620i
23.18 −0.461247 + 0.0813303i 2.96881 − 0.431476i −3.55264 + 1.29305i −1.75481 + 0.309421i −1.33426 + 0.440471i 3.68677 3.15593 − 1.82208i 8.62766 − 2.56194i 0.784238 − 0.285439i
23.19 −0.452066 + 0.0797114i 1.39632 − 2.65524i −3.56076 + 1.29601i 6.95951 − 1.22715i −0.419578 + 1.31165i 2.58194 3.09655 − 1.78780i −5.10056 − 7.41514i −3.04834 + 1.10950i
23.20 −0.295197 + 0.0520513i −2.80082 − 1.07489i −3.67434 + 1.33735i 7.64615 − 1.34822i 0.882745 + 0.171520i −11.2077 2.05341 − 1.18554i 6.68920 + 6.02118i −2.18695 + 0.795983i
See next 80 embeddings (of 228 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 23.38
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
171.bf odd 18 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 171.3.bf.a yes 228
9.d odd 6 1 171.3.z.a ✓ 228
19.e even 9 1 171.3.z.a ✓ 228
171.bf odd 18 1 inner 171.3.bf.a yes 228
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
171.3.z.a ✓ 228 9.d odd 6 1
171.3.z.a ✓ 228 19.e even 9 1
171.3.bf.a yes 228 1.a even 1 1 trivial
171.3.bf.a yes 228 171.bf odd 18 1 inner

Hecke kernels

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