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

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Newspace parameters

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

Newform invariants

Self dual: No
Analytic conductor: \(13.2313552747\)
Analytic rank: \(0\)
Dimension: \(1008\)
Relative dimension: \(63\) 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.

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

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 −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:

Inner twists

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

Hecke kernels

There are no other newforms in \(S_{5}^{\mathrm{new}}(128, [\chi])\).