Properties

Label 16.12.e.a
Level 16
Weight 12
Character orbit 16.e
Analytic conductor 12.293
Analytic rank 0
Dimension 42
CM No

Related objects

Downloads

Learn more about

Newspace parameters

Level: \( N \) = \( 16 = 2^{4} \)
Weight: \( k \) = \( 12 \)
Character orbit: \([\chi]\) = 16.e (of order \(4\) and degree \(2\))

Newform invariants

Self dual: No
Analytic conductor: \(12.293490889\)
Analytic rank: \(0\)
Dimension: \(42\)
Relative dimension: \(21\) over \(\Q(i)\)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 42q - 2q^{2} - 2q^{3} + 3080q^{4} - 2q^{5} + 30352q^{6} + 74836q^{8} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 42q - 2q^{2} - 2q^{3} + 3080q^{4} - 2q^{5} + 30352q^{6} + 74836q^{8} - 988948q^{10} - 540846q^{11} + 2300260q^{12} - 2q^{13} - 1829956q^{14} - 6075004q^{15} - 3721448q^{16} - 4q^{17} - 32219106q^{18} - 11291290q^{19} + 68655164q^{20} + 354292q^{21} - 85011388q^{22} - 66180944q^{24} + 305685384q^{26} - 66463304q^{27} - 322075880q^{28} + 77673206q^{29} + 290643636q^{30} + 343549808q^{31} + 404698568q^{32} - 4q^{33} - 882461836q^{34} - 434731684q^{35} + 1854387628q^{36} - 522762058q^{37} - 1276252832q^{38} - 483241016q^{40} + 2372079160q^{42} + 3824193658q^{43} - 2657283740q^{44} + 97301954q^{45} - 1032658964q^{46} - 4586900144q^{47} - 1058870552q^{48} - 8474257474q^{49} + 6515930886q^{50} + 7074245796q^{51} - 13915491564q^{52} - 2100608058q^{53} + 4385621536q^{54} + 12910845976q^{56} - 21491153128q^{58} + 955824746q^{59} + 25615752768q^{60} + 2150827022q^{61} - 15814519760q^{62} + 27758037828q^{63} + 9674332160q^{64} - 1884965292q^{65} + 10932459412q^{66} - 3186519018q^{67} + 13817664656q^{68} - 16193060732q^{69} - 27206453120q^{70} - 20773133628q^{72} + 14141420908q^{74} + 28890034486q^{75} - 11786004940q^{76} - 22711870540q^{77} + 37031200508q^{78} + 48011833792q^{79} - 8952586328q^{80} - 90656394430q^{81} - 81555388352q^{82} + 55713221118q^{83} + 61178517160q^{84} - 84575506252q^{85} + 10047297860q^{86} - 162855622152q^{88} + 160133627736q^{90} - 147369662716q^{91} - 169766721928q^{92} - 69689773328q^{93} + 160320530128q^{94} + 375702304500q^{95} + 230764310192q^{96} - 4q^{97} - 108887063494q^{98} - 286271331106q^{99} + O(q^{100}) \)

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 −45.0143 4.65973i 13.6026 13.6026i 2004.57 + 419.509i −3501.81 3501.81i −675.697 + 548.928i 37649.1i −88279.7 28224.7i 176777.i 141314. + 173949.i
5.2 −45.0084 + 4.71633i 396.720 396.720i 2003.51 424.549i 7616.23 + 7616.23i −15984.7 + 19726.8i 16490.4i −88172.6 + 28557.5i 137626.i −378715. 306874.i
5.3 −42.3872 + 15.8533i −435.014 + 435.014i 1545.35 1343.95i −1517.34 1517.34i 11542.6 25335.4i 59722.7i −44197.0 + 81465.1i 201326.i 88370.6 + 40261.0i
5.4 −36.4599 26.8082i −475.976 + 475.976i 610.643 + 1954.84i 2805.14 + 2805.14i 30114.1 4593.97i 76033.6i 30141.9 87643.6i 275960.i −27074.3 177476.i
5.5 −33.6365 30.2752i −4.32776 + 4.32776i 214.830 + 2036.70i 2233.99 + 2233.99i 276.594 14.5472i 71414.8i 54435.3 75011.5i 177110.i −7509.26 142778.i
5.6 −31.4901 + 32.5019i 219.312 219.312i −64.7420 2046.98i −4575.66 4575.66i 221.882 + 14034.2i 2778.29i 68569.3 + 62355.3i 80951.6i 292806. 4629.29i
5.7 −30.9382 33.0277i 585.540 585.540i −133.660 + 2043.63i −6859.07 6859.07i −37454.6 1223.52i 19914.8i 71631.8 58811.8i 508568.i −14332.4 + 438747.i
5.8 −26.9679 + 36.3419i −199.508 + 199.508i −593.467 1960.13i 8937.54 + 8937.54i −1870.19 12630.8i 55653.3i 87239.3 + 31292.8i 97540.0i −565834. + 83780.7i
5.9 −6.87277 44.7299i −291.167 + 291.167i −1953.53 + 614.837i −9753.88 9753.88i 15025.0 + 11022.8i 16447.0i 40927.8 + 83155.6i 7590.38i −369254. + 503326.i
5.10 −5.89788 44.8689i 137.041 137.041i −1978.43 + 529.262i 3899.66 + 3899.66i −6957.14 5340.63i 37938.5i 35416.0 + 85648.4i 139586.i 151974. 197973.i
5.11 −1.51469 + 45.2295i 442.035 442.035i −2043.41 137.018i 2497.66 + 2497.66i 19323.5 + 20662.6i 39392.3i 9292.38 92214.9i 213643.i −116751. + 109185.i
5.12 −0.445844 + 45.2526i −296.568 + 296.568i −2047.60 40.3512i −2867.98 2867.98i −13288.2 13552.7i 8380.90i 2738.91 92641.4i 1242.24i 131062. 128505.i
5.13 16.6669 42.0739i −491.248 + 491.248i −1492.43 1402.49i 6777.57 + 6777.57i 12481.1 + 28856.3i 47511.7i −83882.3 + 39417.0i 305503.i 398120. 172197.i
5.14 25.1148 37.6464i 367.824 367.824i −786.499 1890.96i −502.344 502.344i −4609.43 23085.1i 33063.2i −90940.4 17882.1i 93442.1i −31527.7 + 6295.18i
5.15 26.1391 + 36.9425i 255.917 255.917i −681.491 + 1931.29i −4214.49 4214.49i 16143.6 + 2764.75i 80345.3i −89160.1 + 25306.2i 46160.2i 45530.5 265857.i
5.16 30.2883 + 33.6246i −56.4628 + 56.4628i −213.234 + 2036.87i 6260.45 + 6260.45i −3608.71 188.377i 32944.0i −74947.5 + 54523.5i 170771.i −20886.8 + 400124.i
5.17 33.6326 30.2795i −145.072 + 145.072i 214.299 2036.76i −4652.81 4652.81i −486.430 + 9271.83i 43253.0i −54464.6 74990.3i 135056.i −297371. 15601.0i
5.18 40.4837 + 20.2255i −522.679 + 522.679i 1229.86 + 1637.61i −3202.37 3202.37i −31731.4 + 10588.5i 31977.4i 16667.6 + 91170.9i 369239.i −64874.2 194414.i
5.19 43.9461 + 10.8046i 221.481 221.481i 1814.52 + 949.643i −7918.20 7918.20i 12126.3 7340.22i 76836.1i 69480.6 + 61338.3i 79039.1i −262421. 433527.i
5.20 44.1073 10.1265i −179.701 + 179.701i 1842.91 893.305i 2961.29 + 2961.29i −6106.38 + 9745.87i 5179.04i 72239.7 58063.5i 112562.i 160602. + 100627.i
See all 42 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 13.21
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 \(S_{12}^{\mathrm{new}}(16, [\chi])\).