# Properties

 Label 128.5.l.a Level 128 Weight 5 Character orbit 128.l Analytic conductor 13.231 Analytic rank 0 Dimension 1008 CM no Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$128 = 2^{7}$$ Weight: $$k$$ $$=$$ $$5$$ Character orbit: $$[\chi]$$ $$=$$ 128.l (of order $$32$$, degree $$16$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$13.2313552747$$ Analytic rank: $$0$$ Dimension: $$1008$$ Relative dimension: $$63$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$1008q - 16q^{2} - 16q^{3} - 16q^{4} - 16q^{5} - 16q^{6} - 16q^{7} - 16q^{8} - 16q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$1008q - 16q^{2} - 16q^{3} - 16q^{4} - 16q^{5} - 16q^{6} - 16q^{7} - 16q^{8} - 16q^{9} - 16q^{10} - 16q^{11} - 16q^{12} - 16q^{13} - 16q^{14} - 16q^{15} - 16q^{16} - 16q^{17} - 16q^{18} - 16q^{19} - 16q^{20} - 16q^{21} - 16q^{22} - 16q^{23} - 16q^{24} - 16q^{25} - 16q^{26} - 16q^{27} - 16q^{28} - 16q^{29} - 16q^{30} - 16q^{31} - 16q^{32} - 16q^{33} - 16q^{34} - 16q^{35} - 16q^{36} - 16q^{37} - 16q^{38} - 16q^{39} - 16q^{40} - 16q^{41} - 16q^{42} - 16q^{43} - 16q^{44} - 16q^{45} - 16q^{46} - 16q^{47} - 16q^{48} - 16q^{49} - 43072q^{50} - 16q^{51} - 17968q^{52} - 16q^{53} + 31088q^{54} - 16q^{55} + 49376q^{56} - 16q^{57} + 65504q^{58} - 16q^{59} + 63920q^{60} - 16q^{61} + 11792q^{62} - 32q^{63} - 24400q^{64} - 70864q^{66} - 16q^{67} - 53296q^{68} - 16q^{69} - 122320q^{70} - 16q^{71} - 81664q^{72} - 16q^{73} - 33280q^{74} - 16q^{75} + 28272q^{76} - 16q^{77} + 99344q^{78} - 16q^{79} + 105248q^{80} - 16q^{81} - 16q^{82} - 16q^{83} - 16q^{84} - 16q^{85} - 16q^{86} - 16q^{87} - 16q^{88} - 16q^{89} - 16q^{90} - 16q^{91} - 16q^{92} - 16q^{93} - 16q^{94} - 16q^{95} - 16q^{96} - 16q^{97} - 16q^{98} - 16q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
3.1 −3.98767 + 0.313821i −13.0973 + 10.7487i 15.8030 2.50283i 37.0978 + 11.2535i 48.8546 46.9724i 4.64899 + 23.3721i −62.2319 + 14.9398i 40.2030 202.114i −151.465 33.2332i
3.2 −3.95882 0.572521i −4.36463 + 3.58196i 15.3444 + 4.53301i 2.00607 + 0.608536i 19.3295 11.6815i −7.49964 37.7032i −58.1506 26.7304i −9.58275 + 48.1757i −7.59328 3.55760i
3.3 −3.93898 + 0.696008i 6.46187 5.30312i 15.0311 5.48312i −28.0844 8.51932i −21.7622 + 25.3864i 0.922571 + 4.63808i −55.3911 + 32.0597i −2.16964 + 10.9075i 116.554 + 14.0104i
3.4 −3.93799 0.701605i 11.4156 9.36851i 15.0155 + 5.52582i 39.4094 + 11.9547i −51.5273 + 28.8839i −18.6869 93.9456i −55.2539 32.2956i 26.7438 134.450i −146.806 74.7273i
3.5 −3.89182 + 0.923973i −7.47268 + 6.13267i 14.2925 7.19188i −34.4748 10.4578i 23.4159 30.7718i −4.53604 22.8042i −48.9789 + 41.1954i 2.42901 12.2115i 143.832 + 8.84616i
3.6 −3.87913 0.975872i 13.1338 10.7787i 14.0953 + 7.57107i −12.6494 3.83717i −61.4665 + 28.9949i 15.2716 + 76.7754i −47.2893 43.1245i 40.5159 203.687i 45.3243 + 27.2291i
3.7 −3.87689 + 0.984756i 3.81952 3.13460i 14.0605 7.63558i 27.8176 + 8.43838i −11.7210 + 15.9138i 7.96282 + 40.0318i −46.9918 + 43.4484i −11.0393 + 55.4983i −116.155 5.32109i
3.8 −3.86190 1.04198i 1.98252 1.62701i 13.8286 + 8.04803i −2.07933 0.630759i −9.35161 + 4.21762i −2.78288 13.9905i −45.0187 45.4898i −14.5191 + 72.9924i 7.37294 + 4.60255i
3.9 −3.81656 1.19746i −9.21046 + 7.55883i 13.1322 + 9.14031i −23.7963 7.21854i 44.2036 17.8196i 16.4786 + 82.8433i −39.1746 50.6098i 11.8943 59.7969i 82.1761 + 56.0450i
3.10 −3.38069 2.13798i −2.22632 + 1.82709i 6.85808 + 14.4557i 21.3910 + 6.48890i 11.4328 1.41700i 12.9621 + 65.1649i 7.72094 63.5326i −14.1841 + 71.3082i −58.4433 67.6706i
3.11 −3.37653 + 2.14454i −6.28903 + 5.16127i 6.80190 14.4822i 17.4865 + 5.30446i 10.1665 30.9143i −14.2328 71.5529i 8.09084 + 63.4865i −2.88917 + 14.5249i −70.4191 + 19.5897i
3.12 −3.26315 + 2.31341i 9.84830 8.08229i 5.29629 15.0980i −14.5438 4.41182i −13.4388 + 49.1569i −8.09349 40.6887i 17.6452 + 61.5195i 15.8633 79.7501i 57.6650 19.2494i
3.13 −3.25888 2.31942i 5.81494 4.77220i 5.24061 + 15.1174i −28.0812 8.51835i −30.0189 + 2.06475i −9.11012 45.7997i 17.9850 61.4210i −4.76266 + 23.9435i 71.7558 + 92.8924i
3.14 −3.22267 + 2.36947i −7.20596 + 5.91378i 4.77123 15.2720i 2.33003 + 0.706808i 9.20993 36.1325i 11.1538 + 56.0737i 20.8105 + 60.5221i 1.15079 5.78540i −9.18370 + 3.24313i
3.15 −3.09156 2.53816i −12.3327 + 10.1212i 3.11546 + 15.6938i −15.2168 4.61597i 63.8165 + 0.0121723i −11.6720 58.6789i 30.2017 56.4257i 33.8548 170.200i 35.3276 + 52.8933i
3.16 −2.96812 2.68147i −5.39383 + 4.42660i 1.61947 + 15.9178i 42.9519 + 13.0293i 27.8793 + 1.32469i −8.38667 42.1626i 37.8764 51.5886i −6.30371 + 31.6909i −92.5487 153.847i
3.17 −2.80758 + 2.84912i −0.236785 + 0.194325i −0.235026 15.9983i −27.1487 8.23546i 0.111138 1.22021i 13.1413 + 66.0660i 46.2409 + 44.2468i −15.7840 + 79.3516i 99.6858 54.2283i
3.18 −2.66833 2.97993i 8.56994 7.03317i −1.75999 + 15.9029i 16.1113 + 4.88731i −43.8258 6.77100i 8.61826 + 43.3269i 52.0858 37.1896i 8.17609 41.1040i −28.4265 61.0516i
3.19 −2.35233 + 3.23520i 1.64666 1.35138i −4.93309 15.2205i 31.2429 + 9.47742i 0.498498 + 8.50617i −6.24158 31.3785i 60.8458 + 19.8442i −14.9171 + 74.9931i −104.155 + 78.7830i
3.20 −2.15707 3.36854i −1.54548 + 1.26834i −6.69413 + 14.5323i −43.0952 13.0728i 7.60615 + 2.47011i 10.6247 + 53.4142i 63.3924 8.79776i −15.0225 + 75.5232i 48.9230 + 173.367i
See next 80 embeddings (of 1008 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 123.63 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.l odd 32 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 128.5.l.a 1008
128.l odd 32 1 inner 128.5.l.a 1008

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
128.5.l.a 1008 1.a even 1 1 trivial
128.5.l.a 1008 128.l odd 32 1 inner

## Hecke kernels

This newform subspace is the entire newspace $$S_{5}^{\mathrm{new}}(128, [\chi])$$.

## Hecke characteristic polynomials

There are no characteristic polynomials of Hecke operators in the database