# Properties

 Label 16.22.e.a Level $16$ Weight $22$ Character orbit 16.e Analytic conductor $44.716$ Analytic rank $0$ Dimension $82$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [16,22,Mod(5,16)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(16, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([0, 1]))

N = Newforms(chi, 22, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("16.5");

S:= CuspForms(chi, 22);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$16 = 2^{4}$$ Weight: $$k$$ $$=$$ $$22$$ Character orbit: $$[\chi]$$ $$=$$ 16.e (of order $$4$$, degree $$2$$, minimal)

## Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 Self dual: no Analytic conductor: $$44.7163750859$$ Analytic rank: $$0$$ Dimension: $$82$$ Relative dimension: $$41$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$82 q - 2 q^{2} - 2 q^{3} + 1604328 q^{4} - 2 q^{5} - 142760912 q^{6} - 5437750796 q^{8}+O(q^{10})$$ 82 * q - 2 * q^2 - 2 * q^3 + 1604328 * q^4 - 2 * q^5 - 142760912 * q^6 - 5437750796 * q^8 $$\operatorname{Tr}(f)(q) =$$ $$82 q - 2 q^{2} - 2 q^{3} + 1604328 q^{4} - 2 q^{5} - 142760912 q^{6} - 5437750796 q^{8} - 101897120756 q^{10} + 67333320738 q^{11} + 366511438084 q^{12} - 2 q^{13} + 3790342566044 q^{14} + 4613203124996 q^{15} + 1247007918232 q^{16} - 4 q^{17} + 24726998263422 q^{18} - 46007763621434 q^{19} - 3577794932068 q^{20} + 20920706404 q^{21} - 275054389764220 q^{22} - 568150485016400 q^{24} + 27\!\cdots\!24 q^{26}+ \cdots + 27\!\cdots\!38 q^{99}+O(q^{100})$$ 82 * q - 2 * q^2 - 2 * q^3 + 1604328 * q^4 - 2 * q^5 - 142760912 * q^6 - 5437750796 * q^8 - 101897120756 * q^10 + 67333320738 * q^11 + 366511438084 * q^12 - 2 * q^13 + 3790342566044 * q^14 + 4613203124996 * q^15 + 1247007918232 * q^16 - 4 * q^17 + 24726998263422 * q^18 - 46007763621434 * q^19 - 3577794932068 * q^20 + 20920706404 * q^21 - 275054389764220 * q^22 - 568150485016400 * q^24 + 2739706465705224 * q^26 + 1147760716556920 * q^27 - 5464734252855976 * q^28 - 2433173450216602 * q^29 + 1091847142900116 * q^30 - 9835539443769616 * q^31 + 21628424902862408 * q^32 - 4 * q^33 - 32599417569463756 * q^34 - 19253478556034884 * q^35 + 68896817700003052 * q^36 - 7245708527119386 * q^37 - 75508854642654272 * q^38 - 123452072199704568 * q^40 + 521576550189477880 * q^42 + 215332019803969386 * q^43 + 43551785926057540 * q^44 + 953653395699842 * q^45 - 217010424311981172 * q^46 + 1051982644716600976 * q^47 - 791686606171970072 * q^48 - 5585458640832840074 * q^49 + 1308434315006519430 * q^50 + 3238772068175213124 * q^51 - 3929932006670549324 * q^52 + 1889987679210147606 * q^53 + 2474850774763823392 * q^54 + 969847555025010328 * q^56 - 13207633157582037672 * q^58 + 3631979197965016922 * q^59 + 19454740864921105152 * q^60 + 10385362452353654062 * q^61 - 231078435860679632 * q^62 - 17464422381302680956 * q^63 - 28922248017206732928 * q^64 - 9364161130572538716 * q^65 - 1066134537943733516 * q^66 - 17102217044572669130 * q^67 + 1578771419266928720 * q^68 - 3951842785111207532 * q^69 + 37884335466111605120 * q^70 + 50190251198750321124 * q^72 - 146575569503312954228 * q^74 + 145960939174166395366 * q^75 + 205207294437244346196 * q^76 - 82041801652678580764 * q^77 + 18923646523591896860 * q^78 + 204448210818924484480 * q^79 - 129507348099746628056 * q^80 - 802405920297757300870 * q^81 + 830734363602759714528 * q^82 + 317463236552669319678 * q^83 - 1733590905678879817304 * q^84 + 323656241286464843748 * q^85 + 1244418239240036417540 * q^86 + 1366637891097356775928 * q^88 - 4303359727133135334888 * q^90 + 144326756239046079748 * q^91 + 3708288645268614974264 * q^92 - 669717759565183008560 * q^93 - 1146977544252955048048 * q^94 - 1411330138695550130700 * q^95 - 1490348200640250797392 * q^96 - 4 * q^97 + 7549812802452897368090 * q^98 + 2710306841526615010238 * q^99

## 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.

comment: embeddings in the coefficient field

gp: mfembed(f)

Label $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
5.1 −1439.53 + 157.777i −112866. + 112866.i 2.04736e6 454251.i −2.81638e6 2.81638e6i 1.44667e8 1.80282e8i 1.39005e9i −2.87558e9 + 9.76936e8i 1.50171e10i 4.49863e9 + 3.60991e9i
5.2 −1428.70 + 236.579i −28799.9 + 28799.9i 1.98521e6 676002.i 2.59531e7 + 2.59531e7i 3.43329e7 4.79599e7i 2.01795e8i −2.67634e9 + 1.43546e9i 8.80148e9i −4.32191e10 3.09392e10i
5.3 −1400.35 369.019i 109714. 109714.i 1.82480e6 + 1.03351e6i −1.97304e7 1.97304e7i −1.94125e8 + 1.13152e8i 1.02336e9i −2.17397e9 2.12066e9i 1.36141e10i 2.03485e10 + 3.49103e10i
5.4 −1393.10 395.509i −81293.3 + 81293.3i 1.78430e6 + 1.10197e6i −1.56158e7 1.56158e7i 1.45402e8 8.10974e7i 1.06061e9i −2.04986e9 2.24085e9i 2.75685e9i 1.55782e10 + 2.79306e10i
5.5 −1391.38 + 401.508i 72032.1 72032.1i 1.77473e6 1.11730e6i −4.01913e6 4.01913e6i −7.13027e7 + 1.29146e8i 8.04004e8i −2.02073e9 + 2.26716e9i 8.31155e7i 7.20586e9 + 3.97843e9i
5.6 −1391.35 401.620i 12781.5 12781.5i 1.77455e6 + 1.11759e6i 5.32911e6 + 5.32911e6i −2.29168e7 + 1.26502e7i 4.26451e8i −2.02018e9 2.26765e9i 1.01336e10i −5.27438e9 9.55493e9i
5.7 −1198.60 + 812.724i −21696.3 + 21696.3i 776112. 1.94825e6i −2.38583e7 2.38583e7i 8.37200e6 4.36382e7i 3.98700e8i 6.53148e8 + 2.96593e9i 9.51889e9i 4.79868e10 + 9.20627e9i
5.8 −1186.35 + 830.497i 117950. 117950.i 717703. 1.97052e6i 2.07522e7 + 2.07522e7i −4.19730e7 + 2.37888e8i 1.29970e9i 7.85064e8 + 2.93378e9i 1.73642e10i −4.18539e10 7.38473e9i
5.9 −1166.53 858.118i 132197. 132197.i 624420. + 2.00204e6i 2.49441e7 + 2.49441e7i −2.67652e8 + 4.07709e7i 1.02743e9i 9.89578e8 2.87126e9i 2.44918e10i −7.69301e9 5.05030e10i
5.10 −1134.61 + 899.894i −127691. + 127691.i 477535. 2.04206e6i 6.46851e6 + 6.46851e6i 2.99713e7 2.59787e8i 1.09329e9i 1.29582e9 + 2.74668e9i 2.21494e10i −1.31602e10 1.51828e9i
5.11 −1089.16 954.398i 17071.7 17071.7i 275400. + 2.07899e6i 3.05605e6 + 3.05605e6i −3.48871e7 + 2.30067e6i 3.52028e7i 1.68423e9 2.52720e9i 9.87747e9i −4.11849e8 6.24523e9i
5.12 −971.098 1074.30i −113951. + 113951.i −211090. + 2.08650e6i 2.44217e7 + 2.44217e7i 2.33076e8 + 1.17601e7i 1.34162e7i 2.44652e9 1.79942e9i 1.55095e10i 2.52039e9 4.99522e10i
5.13 −840.196 1179.50i −72314.6 + 72314.6i −685292. + 1.98202e6i −1.66664e7 1.66664e7i 1.46054e8 + 2.45367e7i 4.57359e8i 2.91358e9 8.56988e8i 1.53935e6i −5.65498e9 + 3.36611e10i
5.14 −810.478 + 1200.12i −26628.3 + 26628.3i −783402. 1.94533e6i 1.05871e7 + 1.05871e7i −1.03754e7 5.35386e7i 2.00853e8i 2.96956e9 + 6.36479e8i 9.04222e9i −2.12864e10 + 4.12514e9i
5.15 −687.568 1274.52i 80161.3 80161.3i −1.15165e6 + 1.75264e6i −2.92075e7 2.92075e7i −1.57284e8 4.70509e7i 9.19049e8i 3.02561e9 + 2.62745e8i 2.39133e9i −1.71434e10 + 5.73077e10i
5.16 −591.628 + 1321.79i 96038.8 96038.8i −1.39711e6 1.56402e6i −7.03831e6 7.03831e6i 7.01239e7 + 1.83762e8i 4.28554e8i 2.89387e9 9.21365e8i 7.98656e9i 1.34672e10 5.13911e9i
5.17 −410.878 1388.64i 57368.5 57368.5i −1.75951e6 + 1.14113e6i 9.04515e6 + 9.04515e6i −1.03236e8 5.60929e7i 8.24638e8i 2.30756e9 + 1.97447e9i 3.87806e9i 8.84403e9 1.62769e10i
5.18 −172.809 + 1437.81i −93060.3 + 93060.3i −2.03743e6 496931.i −1.84176e7 1.84176e7i −1.17721e8 1.49884e8i 6.32962e8i 1.06657e9 2.84355e9i 6.86010e9i 2.96637e10 2.32983e10i
5.19 −54.2492 1447.14i −21619.3 + 21619.3i −2.09127e6 + 157012.i 1.11303e7 + 1.11303e7i 3.24590e7 + 3.01133e7i 1.47909e9i 3.40668e8 + 3.01783e9i 9.52556e9i 1.55033e10 1.67109e10i
5.20 −26.6184 + 1447.91i −103140. + 103140.i −2.09573e6 77082.0i 8.68960e6 + 8.68960e6i −1.46592e8 1.52083e8i 1.15126e9i 1.67393e8 3.03238e9i 1.08153e10i −1.28131e10 + 1.23505e10i
See all 82 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 5.41 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 16.22.e.a 82
4.b odd 2 1 64.22.e.a 82
16.e even 4 1 inner 16.22.e.a 82
16.f odd 4 1 64.22.e.a 82

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
16.22.e.a 82 1.a even 1 1 trivial
16.22.e.a 82 16.e even 4 1 inner
64.22.e.a 82 4.b odd 2 1
64.22.e.a 82 16.f odd 4 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{22}^{\mathrm{new}}(16, [\chi])$$.