# Properties

 Label 16.14.e.a Level 16 Weight 14 Character orbit 16.e Analytic conductor 17.157 Analytic rank 0 Dimension 50 CM no Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$16 = 2^{4}$$ Weight: $$k$$ $$=$$ $$14$$ Character orbit: $$[\chi]$$ $$=$$ 16.e (of order $$4$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$17.1569486323$$ Analytic rank: $$0$$ Dimension: $$50$$ Relative dimension: $$25$$ over $$\Q(i)$$ Twist minimal: yes 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.

 $$\operatorname{Tr}(f)(q) =$$ $$50q - 2q^{2} - 2q^{3} + 360q^{4} - 2q^{5} - 255056q^{6} - 1076876q^{8} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$50q - 2q^{2} - 2q^{3} + 360q^{4} - 2q^{5} - 255056q^{6} - 1076876q^{8} + 1809804q^{10} - 4723998q^{11} - 36466556q^{12} - 2q^{13} - 32192740q^{14} + 91124996q^{15} + 49054360q^{16} - 4q^{17} - 134185218q^{18} + 422008902q^{19} - 337515748q^{20} + 3188644q^{21} + 747130500q^{22} + 4015997104q^{24} - 7112421624q^{26} - 2068699784q^{27} + 10814980184q^{28} - 3661663834q^{29} - 7391197164q^{30} - 10650044176q^{31} + 161064008q^{32} - 4q^{33} + 21518457652q^{34} + 7767977276q^{35} - 45856853012q^{36} + 21527986470q^{37} + 42250712128q^{38} + 62168140552q^{40} - 221492896520q^{42} - 18577860182q^{43} + 260848874692q^{44} + 2438217602q^{45} - 86330105076q^{46} + 215584306576q^{47} - 204935679512q^{48} - 525968913642q^{49} + 726765984390q^{50} - 551664571452q^{51} - 560232726988q^{52} + 223019793366q^{53} + 183832923424q^{54} + 377498473624q^{56} + 770591295576q^{58} + 1167423209882q^{59} - 1652643345408q^{60} + 81543039150q^{61} + 1409787089968q^{62} - 862914002556q^{63} + 1414847226240q^{64} - 27850095516q^{65} - 4165850751116q^{66} + 1390089097910q^{67} + 1936048158032q^{68} - 168685276844q^{69} - 1514955295360q^{70} - 931909002396q^{72} + 2967549191692q^{74} - 1675683188954q^{75} - 6006275999532q^{76} - 2147852144860q^{77} + 33537158300q^{78} + 8517123343488q^{79} + 3800056341544q^{80} - 9602604240358q^{81} - 1354092283296q^{82} + 2192965629438q^{83} + 6692487540136q^{84} + 2809965843748q^{85} - 2175364294396q^{86} - 2008707597832q^{88} + 1744037438232q^{90} + 3291182399236q^{91} + 5983522987064q^{92} + 3412032366928q^{93} - 10384516980592q^{94} + 7322122332660q^{95} - 22107121515856q^{96} - 4q^{97} + 27036562854554q^{98} + 19363874529854q^{99} + O(q^{100})$$

## 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 −90.1629 + 7.91484i 1174.50 1174.50i 8066.71 1427.25i −33114.8 33114.8i −96600.1 + 115192.i 548741.i −716022. + 192532.i 1.16456e6i 3.24783e6 + 2.72363e6i
5.2 −86.5802 26.3795i 913.885 913.885i 6800.25 + 4567.88i 15715.7 + 15715.7i −103232. + 55016.5i 356100.i −468268. 574875.i 76047.5i −946097. 1.77524e6i
5.3 −85.7364 29.0046i −924.404 + 924.404i 6509.46 + 4973.51i 42023.8 + 42023.8i 106067. 52443.1i 291398.i −413843. 615215.i 114722.i −2.38408e6 4.82186e6i
5.4 −85.4847 + 29.7382i −660.650 + 660.650i 6423.28 5084.32i −6982.48 6982.48i 36829.0 76122.0i 294350.i −397894. + 625648.i 721406.i 804541. + 389249.i
5.5 −81.6959 38.9586i −1316.54 + 1316.54i 5156.45 + 6365.52i −43811.8 43811.8i 158847. 56265.6i 156812.i −173269. 720925.i 1.87225e6i 1.87240e6 + 5.28609e6i
5.6 −71.4691 + 55.5353i 293.446 293.446i 2023.65 7938.12i 22716.6 + 22716.6i −4675.68 + 37268.9i 343566.i 296217. + 679714.i 1.42210e6i −2.88511e6 361960.i
5.7 −56.2421 + 70.9142i 1621.89 1621.89i −1865.65 7976.73i 8037.57 + 8037.57i 23796.5 + 206234.i 516249.i 670591. + 316327.i 3.66675e6i −1.02203e6 + 117928.i
5.8 −55.8189 71.2478i 375.766 375.766i −1960.51 + 7953.95i −11477.0 11477.0i −47747.4 5797.69i 9991.15i 676135. 304298.i 1.31192e6i −177079. + 1.45835e6i
5.9 −40.5224 + 80.9316i −1639.97 + 1639.97i −4907.86 6559.10i 9558.16 + 9558.16i −66269.9 199181.i 133901.i 729717. 131411.i 3.78468e6i −1.16088e6 + 386238.i
5.10 −30.3106 + 85.2834i −26.4760 + 26.4760i −6354.53 5169.99i −38066.2 38066.2i −1455.46 3060.47i 23704.0i 633524. 385231.i 1.59292e6i 4.40022e6 2.09260e6i
5.11 −21.6798 87.8748i −1056.56 + 1056.56i −7251.97 + 3810.23i 11923.3 + 11923.3i 115752. + 69939.2i 170872.i 492044. + 554660.i 638332.i 789265. 1.30626e6i
5.12 −18.7534 88.5455i 1610.25 1610.25i −7488.62 + 3321.06i 39780.2 + 39780.2i −172778. 112383.i 439933.i 434502. + 600803.i 3.59146e6i 2.77634e6 4.26837e6i
5.13 −0.844839 + 90.5057i 381.223 381.223i −8190.57 152.926i 29472.6 + 29472.6i 34180.8 + 34824.9i 167372.i 20760.3 741165.i 1.30366e6i −2.69234e6 + 2.64254e6i
5.14 20.0123 88.2695i 983.557 983.557i −7391.01 3532.96i −35371.2 35371.2i −67134.8 106501.i 334186.i −459764. + 581698.i 340445.i −3.83006e6 + 2.41434e6i
5.15 28.6247 85.8640i −336.873 + 336.873i −6553.25 4915.66i −6822.00 6822.00i 19282.4 + 38568.2i 509047.i −609663. + 421979.i 1.36736e6i −781042. + 390487.i
5.16 34.4647 + 83.6910i −787.355 + 787.355i −5816.37 + 5768.77i 13376.5 + 13376.5i −93030.5 38758.6i 547213.i −683254. 287958.i 354466.i −658474. + 1.58051e6i
5.17 37.4414 + 82.4023i 1317.13 1317.13i −5388.28 + 6170.52i −24836.8 24836.8i 157850. + 59219.5i 73212.9i −710210. 212974.i 1.87536e6i 1.11668e6 2.97653e6i
5.18 54.8815 71.9724i 242.259 242.259i −2168.05 7899.90i 42421.3 + 42421.3i −4140.43 30731.5i 421955.i −687560. 277518.i 1.47694e6i 5.38130e6 725019.i
5.19 58.4516 + 69.1043i −1009.54 + 1009.54i −1358.81 + 8078.52i −19923.8 19923.8i −128773. 10754.3i 484985.i −637685. + 378303.i 444038.i 212241. 2.54140e6i
5.20 64.2863 63.7124i −1618.47 + 1618.47i 73.4667 8191.67i −23191.8 23191.8i −928.945 + 207162.i 280152.i −517188. 531293.i 3.64459e6i −2.96853e6 13311.3i
See all 50 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 13.25 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
16.e even 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 16.14.e.a 50
4.b odd 2 1 64.14.e.a 50
8.b even 2 1 128.14.e.b 50
8.d odd 2 1 128.14.e.a 50
16.e even 4 1 inner 16.14.e.a 50
16.e even 4 1 128.14.e.b 50
16.f odd 4 1 64.14.e.a 50
16.f odd 4 1 128.14.e.a 50

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
16.14.e.a 50 1.a even 1 1 trivial
16.14.e.a 50 16.e even 4 1 inner
64.14.e.a 50 4.b odd 2 1
64.14.e.a 50 16.f odd 4 1
128.14.e.a 50 8.d odd 2 1
128.14.e.a 50 16.f odd 4 1
128.14.e.b 50 8.b even 2 1
128.14.e.b 50 16.e even 4 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{14}^{\mathrm{new}}(16, [\chi])$$.

## Hecke characteristic polynomials

There are no characteristic polynomials of Hecke operators in the database