Properties

Label 128.3.l.a
Level 128
Weight 3
Character orbit 128.l
Analytic conductor 3.488
Analytic rank 0
Dimension 496
CM No

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Newspace parameters

Level: \( N \) = \( 128 = 2^{7} \)
Weight: \( k \) = \( 3 \)
Character orbit: \([\chi]\) = 128.l (of order \(32\) and degree \(16\))

Newform invariants

Self dual: No
Analytic conductor: \(3.48774738381\)
Analytic rank: \(0\)
Dimension: \(496\)
Relative dimension: \(31\) over \(\Q(\zeta_{32})\)
Sato-Tate group: $\mathrm{SU}(2)[C_{32}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \(496q \) \(\mathstrut -\mathstrut 16q^{2} \) \(\mathstrut -\mathstrut 16q^{3} \) \(\mathstrut -\mathstrut 16q^{4} \) \(\mathstrut -\mathstrut 16q^{5} \) \(\mathstrut -\mathstrut 16q^{6} \) \(\mathstrut -\mathstrut 16q^{7} \) \(\mathstrut -\mathstrut 16q^{8} \) \(\mathstrut -\mathstrut 16q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \(496q \) \(\mathstrut -\mathstrut 16q^{2} \) \(\mathstrut -\mathstrut 16q^{3} \) \(\mathstrut -\mathstrut 16q^{4} \) \(\mathstrut -\mathstrut 16q^{5} \) \(\mathstrut -\mathstrut 16q^{6} \) \(\mathstrut -\mathstrut 16q^{7} \) \(\mathstrut -\mathstrut 16q^{8} \) \(\mathstrut -\mathstrut 16q^{9} \) \(\mathstrut -\mathstrut 16q^{10} \) \(\mathstrut -\mathstrut 16q^{11} \) \(\mathstrut -\mathstrut 16q^{12} \) \(\mathstrut -\mathstrut 16q^{13} \) \(\mathstrut -\mathstrut 16q^{14} \) \(\mathstrut -\mathstrut 16q^{15} \) \(\mathstrut -\mathstrut 16q^{16} \) \(\mathstrut -\mathstrut 16q^{17} \) \(\mathstrut -\mathstrut 16q^{18} \) \(\mathstrut -\mathstrut 16q^{19} \) \(\mathstrut -\mathstrut 16q^{20} \) \(\mathstrut -\mathstrut 16q^{21} \) \(\mathstrut -\mathstrut 16q^{22} \) \(\mathstrut -\mathstrut 16q^{23} \) \(\mathstrut -\mathstrut 16q^{24} \) \(\mathstrut -\mathstrut 16q^{25} \) \(\mathstrut -\mathstrut 16q^{26} \) \(\mathstrut -\mathstrut 16q^{27} \) \(\mathstrut -\mathstrut 16q^{28} \) \(\mathstrut -\mathstrut 16q^{29} \) \(\mathstrut -\mathstrut 16q^{30} \) \(\mathstrut -\mathstrut 16q^{31} \) \(\mathstrut -\mathstrut 16q^{32} \) \(\mathstrut -\mathstrut 16q^{33} \) \(\mathstrut -\mathstrut 16q^{34} \) \(\mathstrut -\mathstrut 16q^{35} \) \(\mathstrut -\mathstrut 16q^{36} \) \(\mathstrut -\mathstrut 16q^{37} \) \(\mathstrut -\mathstrut 16q^{38} \) \(\mathstrut -\mathstrut 16q^{39} \) \(\mathstrut -\mathstrut 16q^{40} \) \(\mathstrut -\mathstrut 16q^{41} \) \(\mathstrut -\mathstrut 16q^{42} \) \(\mathstrut -\mathstrut 16q^{43} \) \(\mathstrut -\mathstrut 16q^{44} \) \(\mathstrut -\mathstrut 16q^{45} \) \(\mathstrut -\mathstrut 16q^{46} \) \(\mathstrut -\mathstrut 16q^{47} \) \(\mathstrut -\mathstrut 16q^{48} \) \(\mathstrut -\mathstrut 16q^{49} \) \(\mathstrut -\mathstrut 640q^{50} \) \(\mathstrut -\mathstrut 16q^{51} \) \(\mathstrut -\mathstrut 1072q^{52} \) \(\mathstrut -\mathstrut 16q^{53} \) \(\mathstrut -\mathstrut 1168q^{54} \) \(\mathstrut -\mathstrut 16q^{55} \) \(\mathstrut -\mathstrut 800q^{56} \) \(\mathstrut -\mathstrut 16q^{57} \) \(\mathstrut -\mathstrut 736q^{58} \) \(\mathstrut -\mathstrut 16q^{59} \) \(\mathstrut -\mathstrut 592q^{60} \) \(\mathstrut -\mathstrut 16q^{61} \) \(\mathstrut -\mathstrut 112q^{62} \) \(\mathstrut -\mathstrut 32q^{63} \) \(\mathstrut +\mathstrut 176q^{64} \) \(\mathstrut +\mathstrut 560q^{66} \) \(\mathstrut -\mathstrut 16q^{67} \) \(\mathstrut +\mathstrut 464q^{68} \) \(\mathstrut -\mathstrut 16q^{69} \) \(\mathstrut +\mathstrut 1328q^{70} \) \(\mathstrut -\mathstrut 16q^{71} \) \(\mathstrut +\mathstrut 1280q^{72} \) \(\mathstrut -\mathstrut 16q^{73} \) \(\mathstrut +\mathstrut 1216q^{74} \) \(\mathstrut -\mathstrut 16q^{75} \) \(\mathstrut +\mathstrut 1648q^{76} \) \(\mathstrut -\mathstrut 16q^{77} \) \(\mathstrut +\mathstrut 1424q^{78} \) \(\mathstrut -\mathstrut 16q^{79} \) \(\mathstrut +\mathstrut 800q^{80} \) \(\mathstrut -\mathstrut 16q^{81} \) \(\mathstrut -\mathstrut 16q^{82} \) \(\mathstrut -\mathstrut 16q^{83} \) \(\mathstrut -\mathstrut 16q^{84} \) \(\mathstrut -\mathstrut 16q^{85} \) \(\mathstrut -\mathstrut 16q^{86} \) \(\mathstrut -\mathstrut 16q^{87} \) \(\mathstrut -\mathstrut 16q^{88} \) \(\mathstrut -\mathstrut 16q^{89} \) \(\mathstrut -\mathstrut 16q^{90} \) \(\mathstrut -\mathstrut 16q^{91} \) \(\mathstrut -\mathstrut 16q^{92} \) \(\mathstrut -\mathstrut 16q^{93} \) \(\mathstrut -\mathstrut 16q^{94} \) \(\mathstrut -\mathstrut 16q^{95} \) \(\mathstrut -\mathstrut 16q^{96} \) \(\mathstrut -\mathstrut 16q^{97} \) \(\mathstrut -\mathstrut 16q^{98} \) \(\mathstrut -\mathstrut 16q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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.

Label \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
3.1 −1.99185 0.180370i 1.05127 0.862751i 3.93493 + 0.718539i −7.90123 2.39681i −2.24958 + 1.52885i 1.29801 + 6.52556i −7.70819 2.14097i −1.39499 + 7.01311i 15.3057 + 6.19923i
3.2 −1.98170 0.269964i −2.40785 + 1.97607i 3.85424 + 1.06997i 4.69198 + 1.42330i 5.30509 3.26594i −0.744763 3.74418i −7.34908 3.16086i 0.137064 0.689066i −8.91384 4.08721i
3.3 −1.90117 + 0.620914i 0.779996 0.640126i 3.22893 2.36093i 0.179045 + 0.0543127i −1.08545 + 1.70130i −2.44195 12.2765i −4.67283 + 6.49343i −1.55718 + 7.82847i −0.374119 + 0.00791361i
3.4 −1.84253 0.777866i 1.79949 1.47680i 2.78985 + 2.86649i 5.80468 + 1.76083i −4.46437 + 1.32129i 0.867168 + 4.35955i −2.91064 7.45172i −0.698600 + 3.51210i −9.32562 7.75965i
3.5 −1.78050 + 0.910937i 3.84622 3.15651i 2.34039 3.24385i 4.46942 + 1.35578i −3.97282 + 9.12384i 1.40191 + 7.04787i −1.21212 + 7.90764i 3.07403 15.4542i −9.19284 + 1.65738i
3.6 −1.77728 + 0.917214i −4.35347 + 3.57280i 2.31744 3.26029i −6.55995 1.98994i 4.46031 10.3429i −0.342193 1.72032i −1.12835 + 7.92003i 4.43199 22.2811i 13.4841 2.48020i
3.7 −1.57051 1.23834i −3.06977 + 2.51930i 0.933028 + 3.88966i −2.31969 0.703669i 7.94087 0.155169i 0.668264 + 3.35959i 3.35139 7.26417i 1.32084 6.64029i 2.77172 + 3.97768i
3.8 −1.52087 1.29883i 3.39289 2.78447i 0.626076 + 3.95070i −2.17916 0.661042i −8.77669 0.171976i −1.02400 5.14798i 4.17911 6.82166i 2.00260 10.0677i 2.45564 + 3.83572i
3.9 −1.39308 + 1.43504i 0.447688 0.367408i −0.118660 3.99824i −2.47385 0.750435i −0.0964209 + 1.15428i 0.987061 + 4.96229i 5.90292 + 5.39958i −1.69038 + 8.49810i 4.52318 2.50465i
3.10 −1.31385 + 1.50791i −2.45029 + 2.01090i −0.547604 3.96234i 9.14318 + 2.77355i 0.187046 6.33684i 0.716485 + 3.60202i 6.69433 + 4.38017i 0.204383 1.02750i −16.1950 + 10.1431i
3.11 −0.849476 1.81063i −0.740915 + 0.608053i −2.55678 + 3.07618i −2.60449 0.790064i 1.73035 + 0.824998i −0.481425 2.42029i 7.74175 + 2.01625i −1.57659 + 7.92604i 0.781937 + 5.38692i
3.12 −0.687532 + 1.87811i 3.91403 3.21216i −3.05460 2.58252i −8.46864 2.56894i 3.34178 + 9.55944i −0.593205 2.98225i 6.95040 3.96131i 3.24584 16.3179i 10.6472 14.1388i
3.13 −0.630182 + 1.89812i −2.02704 + 1.66355i −3.20574 2.39233i −2.42484 0.735568i −1.88021 4.89591i −0.539873 2.71413i 6.56113 4.57729i −0.414318 + 2.08292i 2.92429 4.13911i
3.14 −0.494075 1.93801i 0.211016 0.173176i −3.51178 + 1.91505i 7.43419 + 2.25514i −0.439875 0.323389i 2.23153 + 11.2187i 5.44647 + 5.85969i −1.74128 + 8.75398i 0.697434 15.5218i
3.15 −0.203195 + 1.98965i 2.70039 2.21615i −3.91742 0.808573i 7.86468 + 2.38573i 3.86067 + 5.82314i −2.00235 10.0665i 2.40478 7.63001i 0.624959 3.14188i −6.34482 + 15.1632i
3.16 −0.0383871 1.99963i −4.01639 + 3.29616i −3.99705 + 0.153520i 6.07117 + 1.84167i 6.74529 + 7.90476i −2.47965 12.4660i 0.460419 + 7.98674i 3.51086 17.6503i 3.44961 12.2108i
3.17 0.0173622 1.99992i 3.90317 3.20325i −3.99940 0.0694462i 4.26052 + 1.29242i −6.33848 7.86165i −0.713727 3.58815i −0.208325 + 7.99729i 3.21811 16.1785i 2.65871 8.49828i
3.18 0.286149 + 1.97942i 0.798268 0.655122i −3.83624 + 1.13282i 0.972729 + 0.295074i 1.52519 + 1.39265i 1.76467 + 8.87160i −3.34006 7.26939i −1.54777 + 7.78114i −0.305732 + 2.00988i
3.19 0.359443 1.96744i 1.06123 0.870929i −3.74160 1.41436i −6.51407 1.97602i −1.33205 2.40095i −0.702367 3.53104i −4.12755 + 6.85298i −1.38812 + 6.97855i −6.22913 + 12.1057i
3.20 0.768183 + 1.84659i −4.58227 + 3.76057i −2.81979 + 2.83704i 2.93176 + 0.889341i −10.4642 5.57276i 1.62220 + 8.15533i −7.40496 3.02764i 5.09947 25.6368i 0.609882 + 6.09694i
See next 80 embeddings (of 496 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 123.31
Significant digits:
Format:

Inner twists

This newform does not have CM; other inner twists have not been computed.

Hecke kernels

There are no other newforms in \(S_{3}^{\mathrm{new}}(128, [\chi])\).