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Properties

Label 171.3.be.a
Level $171$
Weight $3$
Character orbit 171.be
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(13,171)]
 
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([6, 5]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("171.13");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 171 = 3^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 171.be (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.

\(\operatorname{Tr}(f)(q) = \) \( 228 q - 3 q^{2} - 3 q^{4} - 3 q^{5} - 18 q^{6} - 6 q^{7} - 18 q^{8} - 24 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 228 q - 3 q^{2} - 3 q^{4} - 3 q^{5} - 18 q^{6} - 6 q^{7} - 18 q^{8} - 24 q^{9} - 12 q^{10} + 3 q^{11} - 9 q^{12} - 18 q^{13} - 51 q^{14} + 27 q^{15} - 27 q^{16} + 15 q^{17} - 144 q^{18} + 9 q^{19} - 6 q^{20} - 60 q^{21} - 15 q^{22} - 147 q^{23} - 363 q^{24} - 3 q^{25} - 78 q^{26} + 81 q^{27} - 84 q^{28} - 30 q^{29} + 24 q^{30} - 9 q^{31} + 45 q^{32} - 186 q^{33} + 15 q^{34} + 75 q^{35} + 126 q^{36} + 33 q^{38} + 36 q^{39} + 96 q^{40} + 123 q^{41} - 138 q^{42} - 195 q^{43} - 177 q^{44} - 48 q^{45} - 18 q^{46} - 165 q^{47} + 66 q^{48} + 1086 q^{49} + 423 q^{50} + 3 q^{51} - 135 q^{52} - 84 q^{53} + 228 q^{54} - 87 q^{55} - 441 q^{56} - 276 q^{57} - 6 q^{58} + 258 q^{59} + 447 q^{60} - 45 q^{61} + 216 q^{62} + 72 q^{63} + 768 q^{64} + 801 q^{65} + 1104 q^{66} + 228 q^{67} - 510 q^{68} + 342 q^{69} - 150 q^{70} + 474 q^{71} + 24 q^{72} - 267 q^{73} + 72 q^{74} - 3 q^{76} - 300 q^{77} + 207 q^{78} - 54 q^{79} - 48 q^{80} - 276 q^{81} - 156 q^{82} + 66 q^{83} - 9 q^{84} - 3 q^{85} + 321 q^{86} - 219 q^{87} + 204 q^{89} - 597 q^{90} - 543 q^{91} - 1971 q^{92} - 489 q^{93} - 1299 q^{95} - 945 q^{96} - 273 q^{97} - 945 q^{98} + 282 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
13.1 −1.28055 + 3.51829i −2.93107 − 0.639411i −7.67437 − 6.43956i 6.86740 + 2.49953i 6.00302 − 9.49354i 8.33658 19.5138 − 11.2663i 8.18231 + 3.74831i −17.5881 + 20.9607i
13.2 −1.23659 + 3.39750i 1.31521 − 2.69633i −6.94967 − 5.83146i 1.61287 + 0.587037i 7.53441 + 7.80269i −8.30749 15.8817 − 9.16928i −5.54043 − 7.09251i −3.98891 + 4.75380i
13.3 −1.23288 + 3.38730i 1.93566 + 2.29199i −6.88962 − 5.78108i −3.68277 − 1.34042i −10.1501 + 3.73090i −5.71174 15.5893 − 9.00047i −1.50648 + 8.87302i 9.08079 − 10.8221i
13.4 −1.17925 + 3.23996i 2.94950 − 0.548120i −6.04254 − 5.07029i 0.0927929 + 0.0337739i −1.70231 + 10.2026i 11.5915 11.6094 − 6.70267i 8.39913 − 3.23336i −0.218852 + 0.260818i
13.5 −1.13132 + 3.10829i −2.08972 + 2.15246i −5.31738 − 4.46181i −4.40242 − 1.60235i −4.32632 − 8.93057i 2.15664 8.42582 − 4.86465i −0.266166 − 8.99606i 9.96113 − 11.8712i
13.6 −1.05178 + 2.88974i −0.357648 + 2.97860i −4.18019 − 3.50759i 8.34566 + 3.03757i −8.23123 − 4.16635i −5.92947 3.87987 − 2.24004i −8.74418 − 2.13059i −17.5556 + 20.9219i
13.7 −0.953315 + 2.61921i −1.02673 − 2.81883i −2.88728 − 2.42272i 0.942223 + 0.342941i 8.36192 − 0.00199464i 0.0211609 −0.557408 + 0.321820i −6.89164 + 5.78838i −1.79647 + 2.14095i
13.8 −0.803590 + 2.20785i −2.58736 − 1.51840i −1.16465 − 0.977256i −6.33963 − 2.30744i 5.43158 − 4.49232i 3.33732 −5.04552 + 2.91303i 4.38889 + 7.85733i 10.1889 − 12.1427i
13.9 −0.795621 + 2.18595i 2.23028 − 2.00645i −1.08119 − 0.907228i −8.08085 − 2.94119i 2.61155 + 6.47166i −3.69912 −5.21496 + 3.01086i 0.948296 − 8.94990i 12.8586 − 15.3243i
13.10 −0.763814 + 2.09856i 1.90577 + 2.31691i −0.756373 − 0.634672i 3.21944 + 1.17178i −6.31782 + 2.22968i 8.52807 −5.82655 + 3.36396i −1.73611 + 8.83096i −4.91811 + 5.86117i
13.11 −0.648725 + 1.78236i −2.97483 + 0.387806i 0.308225 + 0.258632i 1.53225 + 0.557693i 1.23864 − 5.55379i −7.80935 −7.23144 + 4.17508i 8.69921 − 2.30732i −1.98802 + 2.36922i
13.12 −0.584386 + 1.60559i 2.78161 + 1.12367i 0.827774 + 0.694585i 0.105644 + 0.0384512i −3.42968 + 3.80947i −11.1905 −7.51783 + 4.34042i 6.47475 + 6.25121i −0.123473 + 0.147150i
13.13 −0.582021 + 1.59909i 2.63193 − 1.43977i 0.845841 + 0.709745i 6.52067 + 2.37333i 0.770476 + 5.04667i 0.121217 −7.52215 + 4.34292i 4.85414 − 7.57874i −7.59033 + 9.04580i
13.14 −0.414122 + 1.13779i 0.924960 + 2.85385i 1.94110 + 1.62878i −7.95064 − 2.89380i −3.63013 − 0.129430i 7.15820 −6.85145 + 3.95568i −7.28890 + 5.27939i 6.58508 − 7.84779i
13.15 −0.296137 + 0.813628i −2.50660 + 1.64833i 2.48988 + 2.08926i 3.68382 + 1.34080i −0.598830 − 2.52757i 9.57149 −5.43660 + 3.13882i 3.56605 − 8.26337i −2.18183 + 2.60020i
13.16 −0.280765 + 0.771396i −0.957896 + 2.84296i 2.54795 + 2.13799i −4.10900 − 1.49555i −1.92411 − 1.53712i −10.5022 −5.20830 + 3.00702i −7.16487 − 5.44652i 2.30733 − 2.74976i
13.17 −0.257519 + 0.707527i 0.533310 − 2.95222i 2.62990 + 2.20675i −1.18884 − 0.432702i 1.95143 + 1.13758i 11.6018 −4.84682 + 2.79831i −8.43116 − 3.14890i 0.612297 − 0.729707i
13.18 −0.222204 + 0.610500i −1.64520 − 2.50865i 2.74084 + 2.29984i 8.60469 + 3.13185i 1.89710 − 0.446964i −6.14847 −4.26364 + 2.46161i −3.58663 + 8.25446i −3.82399 + 4.55725i
13.19 0.0965679 − 0.265318i −2.82838 − 1.00015i 3.00311 + 2.51991i −5.05334 − 1.83926i −0.538487 + 0.653837i 0.852502 1.93665 − 1.11813i 6.99941 + 5.65758i −0.975980 + 1.16313i
13.20 0.123546 − 0.339439i −0.547910 − 2.94954i 2.96422 + 2.48728i −5.07833 − 1.84836i −1.06888 − 0.178421i −12.1239 2.46181 − 1.42133i −8.39959 + 3.23217i −1.25481 + 1.49543i
See next 80 embeddings (of 228 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 13.38
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
171.be 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.be.a yes 228
9.c even 3 1 171.3.bc.a ✓ 228
19.f odd 18 1 171.3.bc.a ✓ 228
171.be odd 18 1 inner 171.3.be.a yes 228
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
171.3.bc.a ✓ 228 9.c even 3 1
171.3.bc.a ✓ 228 19.f odd 18 1
171.3.be.a yes 228 1.a even 1 1 trivial
171.3.be.a yes 228 171.be odd 18 1 inner

Hecke kernels

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