Properties

 Label 128.2.k.a Level 128 Weight 2 Character orbit 128.k Analytic conductor 1.022 Analytic rank 0 Dimension 240 CM no Inner twists 2

Related objects

Newspace parameters

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

Newform invariants

 Self dual: no Analytic conductor: $$1.02208514587$$ Analytic rank: $$0$$ Dimension: $$240$$ Relative dimension: $$15$$ over $$\Q(\zeta_{32})$$ Twist minimal: yes 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.

 $$\operatorname{Tr}(f)(q) =$$ $$240q - 16q^{2} - 16q^{3} - 16q^{4} - 16q^{5} - 16q^{6} - 16q^{7} - 16q^{8} - 16q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$240q - 16q^{2} - 16q^{3} - 16q^{4} - 16q^{5} - 16q^{6} - 16q^{7} - 16q^{8} - 16q^{9} - 16q^{10} - 16q^{11} - 16q^{12} - 16q^{13} - 16q^{14} - 16q^{15} - 16q^{16} - 16q^{17} - 16q^{18} - 16q^{19} - 16q^{20} - 16q^{21} - 16q^{22} - 16q^{23} - 16q^{24} - 16q^{25} - 16q^{26} - 16q^{27} - 16q^{28} - 16q^{29} - 16q^{30} - 16q^{31} - 16q^{32} - 16q^{33} - 16q^{34} - 16q^{35} - 16q^{36} - 16q^{37} - 16q^{38} - 16q^{39} - 16q^{40} - 16q^{41} - 16q^{42} - 16q^{43} - 16q^{44} - 16q^{45} - 16q^{46} - 16q^{47} - 16q^{48} - 16q^{49} + 32q^{50} - 16q^{51} + 80q^{52} - 16q^{53} + 112q^{54} - 16q^{55} + 96q^{56} - 16q^{57} + 128q^{58} - 16q^{59} + 176q^{60} - 16q^{61} + 80q^{62} + 176q^{64} + 176q^{66} - 16q^{67} + 80q^{68} - 16q^{69} + 176q^{70} - 16q^{71} + 128q^{72} - 16q^{73} + 96q^{74} - 16q^{75} + 112q^{76} - 16q^{77} + 80q^{78} - 16q^{79} + 32q^{80} - 16q^{81} - 16q^{82} - 16q^{83} - 16q^{84} - 16q^{85} - 16q^{86} - 16q^{87} - 16q^{88} - 16q^{89} - 16q^{90} - 16q^{91} - 16q^{92} - 16q^{93} - 16q^{94} - 16q^{95} - 16q^{96} - 16q^{97} - 16q^{98} - 16q^{99} + O(q^{100})$$

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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
5.1 −1.41035 0.104463i 1.66739 + 0.505799i 1.97817 + 0.294660i 0.127039 + 0.0125123i −2.29877 0.887535i 1.18113 + 1.76769i −2.75914 0.622220i 0.0299647 + 0.0200218i −0.177863 0.0309177i
5.2 −1.31795 0.512840i −1.88141 0.570718i 1.47399 + 1.35180i 2.61618 + 0.257671i 2.18691 + 1.71704i −1.78728 2.67485i −1.24939 2.53752i 0.719558 + 0.480794i −3.31585 1.68128i
5.3 −1.26050 + 0.641195i −0.417839 0.126750i 1.17774 1.61646i −2.96993 0.292513i 0.607959 0.108147i −1.08102 1.61787i −0.448081 + 2.79271i −2.33589 1.56079i 3.93117 1.53559i
5.4 −0.950249 1.04739i −1.73829 0.527306i −0.194054 + 1.99056i −2.46085 0.242372i 1.09952 + 2.32174i 2.82715 + 4.23114i 2.26930 1.68828i 0.249204 + 0.166513i 2.08456 + 2.80778i
5.5 −0.919458 + 1.07452i −2.80857 0.851969i −0.309193 1.97596i 1.35768 + 0.133720i 3.49782 2.23451i 1.67452 + 2.50610i 2.40750 + 1.48457i 4.66778 + 3.11891i −1.39201 + 1.33591i
5.6 −0.891064 + 1.09818i 1.53052 + 0.464277i −0.412009 1.95710i 3.76242 + 0.370566i −1.87365 + 1.26709i −2.04991 3.06791i 2.51638 + 1.29144i −0.367482 0.245544i −3.75950 + 3.80162i
5.7 −0.652002 1.25495i 2.77794 + 0.842679i −1.14979 + 1.63646i 0.0628209 + 0.00618731i −0.753703 4.03560i −0.619337 0.926904i 2.80333 + 0.375951i 4.51243 + 3.01511i −0.0331945 0.0828710i
5.8 −0.163059 1.40478i −0.757526 0.229793i −1.94682 + 0.458124i −1.57990 0.155607i −0.199287 + 1.10163i −2.26604 3.39136i 0.961012 + 2.66016i −1.97337 1.31856i 0.0390236 + 2.24479i
5.9 0.432421 + 1.34648i 2.51607 + 0.763243i −1.62602 + 1.16449i −1.42404 0.140256i 0.0603114 + 3.71789i −0.954234 1.42811i −2.27110 1.68586i 3.25368 + 2.17404i −0.426934 1.97810i
5.10 0.441120 + 1.34366i −0.451169 0.136861i −1.61083 + 1.18543i 2.96377 + 0.291905i −0.0151258 0.666588i 1.40525 + 2.10310i −2.30337 1.64148i −2.30959 1.54322i 0.915154 + 4.11105i
5.11 0.674225 1.24315i 0.865940 + 0.262680i −1.09084 1.67632i 1.24951 + 0.123066i 0.910389 0.899388i 0.832309 + 1.24564i −2.81939 + 0.225861i −1.81356 1.21178i 0.995438 1.47035i
5.12 0.825274 + 1.14844i −3.11090 0.943682i −0.637847 + 1.89556i −2.52957 0.249141i −1.48358 4.35149i −1.75170 2.62161i −2.70334 + 0.831825i 6.29277 + 4.20469i −1.80147 3.11068i
5.13 1.26626 + 0.629746i −0.156077 0.0473453i 1.20684 + 1.59485i −0.851642 0.0838794i −0.167818 0.158240i 1.43739 + 2.15120i 0.523829 + 2.77950i −2.47229 1.65193i −1.02558 0.642531i
5.14 1.35411 0.407902i 1.87695 + 0.569366i 1.66723 1.10469i −4.38038 0.431430i 2.77384 + 0.00537365i 0.0783406 + 0.117245i 1.80701 2.17594i 0.704347 + 0.470630i −6.10750 + 1.20256i
5.15 1.37613 0.325966i −1.74451 0.529190i 1.78749 0.897147i 2.07613 + 0.204481i −2.57317 0.159587i −0.655035 0.980329i 2.16739 1.81726i 0.268850 + 0.179640i 2.92369 0.395355i
13.1 −1.41410 + 0.0181162i 0.180024 + 0.593459i 1.99934 0.0512363i −0.372310 3.78013i −0.265323 0.835948i −0.885307 + 1.32496i −2.82634 + 0.108674i 2.17462 1.45304i 0.594964 + 5.33872i
13.2 −1.36942 0.353095i −0.855442 2.82001i 1.75065 + 0.967074i −0.0181846 0.184631i 0.175731 + 4.16385i 1.98359 2.96866i −2.05591 1.94248i −4.72629 + 3.15801i −0.0402899 + 0.259259i
13.3 −1.31306 + 0.525230i 0.580122 + 1.91241i 1.44827 1.37932i 0.217075 + 2.20400i −1.76619 2.20641i 0.814297 1.21868i −1.17721 + 2.57181i −0.826349 + 0.552149i −1.44264 2.77998i
13.4 −1.10268 0.885498i −0.152748 0.503541i 0.431788 + 1.95283i 0.326146 + 3.31141i −0.277453 + 0.690500i −2.79715 + 4.18623i 1.25311 2.53569i 2.26419 1.51288i 2.57262 3.94022i
13.5 −0.836598 1.14022i 0.187193 + 0.617093i −0.600207 + 1.90781i −0.112545 1.14269i 0.547017 0.729700i 1.96260 2.93724i 2.67746 0.911705i 2.14865 1.43568i −1.20876 + 1.08430i
See next 80 embeddings (of 240 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 125.15 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
128.k even 32 1 inner

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 128.2.k.a 240
4.b odd 2 1 512.2.k.a 240
128.k even 32 1 inner 128.2.k.a 240
128.l odd 32 1 512.2.k.a 240

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
128.2.k.a 240 1.a even 1 1 trivial
128.2.k.a 240 128.k even 32 1 inner
512.2.k.a 240 4.b odd 2 1
512.2.k.a 240 128.l odd 32 1

Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(128, [\chi])$$.

Hecke characteristic polynomials

There are no characteristic polynomials of Hecke operators in the database