# Properties

 Label 16.8.e.a Level $16$ Weight $8$ Character orbit 16.e Analytic conductor $4.998$ Analytic rank $0$ Dimension $26$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [16,8,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, 8, names="a")

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

chi := DirichletCharacter("16.5");

S:= CuspForms(chi, 8);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$16 = 2^{4}$$ Weight: $$k$$ $$=$$ $$8$$ 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: $$4.99816040775$$ Analytic rank: $$0$$ Dimension: $$26$$ Relative dimension: $$13$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$26 q - 2 q^{2} - 2 q^{3} - 184 q^{4} - 2 q^{5} - 176 q^{6} - 1004 q^{8}+O(q^{10})$$ 26 * q - 2 * q^2 - 2 * q^3 - 184 * q^4 - 2 * q^5 - 176 * q^6 - 1004 * q^8 $$\operatorname{Tr}(f)(q) =$$ $$26 q - 2 q^{2} - 2 q^{3} - 184 q^{4} - 2 q^{5} - 176 q^{6} - 1004 q^{8} + 12972 q^{10} + 1202 q^{11} - 27356 q^{12} - 2 q^{13} + 22268 q^{14} - 27004 q^{15} + 13336 q^{16} - 4 q^{17} - 29346 q^{18} + 60582 q^{19} + 46204 q^{20} + 4372 q^{21} - 44348 q^{22} - 59216 q^{24} - 35960 q^{26} - 233672 q^{27} + 220312 q^{28} - 51690 q^{29} + 27636 q^{30} + 357488 q^{31} - 248632 q^{32} - 4 q^{33} + 924660 q^{34} - 252004 q^{35} - 1015508 q^{36} + 415574 q^{37} - 5472 q^{38} - 1121976 q^{40} + 124600 q^{42} + 569754 q^{43} - 758620 q^{44} + 151874 q^{45} + 2335660 q^{46} - 2076464 q^{47} + 4141288 q^{48} - 1647090 q^{49} - 787194 q^{50} + 2609508 q^{51} + 410324 q^{52} + 907814 q^{53} - 7404512 q^{54} - 4093160 q^{56} - 1860712 q^{58} - 4865142 q^{59} + 5147328 q^{60} + 2279886 q^{61} + 6274096 q^{62} + 8295108 q^{63} + 10452224 q^{64} - 1426892 q^{65} + 10937812 q^{66} - 5564458 q^{67} - 7403760 q^{68} - 4786076 q^{69} - 22140800 q^{70} - 25293948 q^{72} - 13177044 q^{74} + 6212566 q^{75} + 17814260 q^{76} + 7604308 q^{77} + 37993916 q^{78} - 9598912 q^{79} + 40047272 q^{80} - 5314414 q^{81} + 22858240 q^{82} + 4531198 q^{83} - 30159704 q^{84} + 7377748 q^{85} - 54129468 q^{86} - 59216392 q^{88} - 33515304 q^{90} + 2587652 q^{91} + 21810936 q^{92} - 14504144 q^{93} + 77196112 q^{94} - 4900620 q^{95} + 116940464 q^{96} - 4 q^{97} + 21823866 q^{98} + 18815006 q^{99}+O(q^{100})$$ 26 * q - 2 * q^2 - 2 * q^3 - 184 * q^4 - 2 * q^5 - 176 * q^6 - 1004 * q^8 + 12972 * q^10 + 1202 * q^11 - 27356 * q^12 - 2 * q^13 + 22268 * q^14 - 27004 * q^15 + 13336 * q^16 - 4 * q^17 - 29346 * q^18 + 60582 * q^19 + 46204 * q^20 + 4372 * q^21 - 44348 * q^22 - 59216 * q^24 - 35960 * q^26 - 233672 * q^27 + 220312 * q^28 - 51690 * q^29 + 27636 * q^30 + 357488 * q^31 - 248632 * q^32 - 4 * q^33 + 924660 * q^34 - 252004 * q^35 - 1015508 * q^36 + 415574 * q^37 - 5472 * q^38 - 1121976 * q^40 + 124600 * q^42 + 569754 * q^43 - 758620 * q^44 + 151874 * q^45 + 2335660 * q^46 - 2076464 * q^47 + 4141288 * q^48 - 1647090 * q^49 - 787194 * q^50 + 2609508 * q^51 + 410324 * q^52 + 907814 * q^53 - 7404512 * q^54 - 4093160 * q^56 - 1860712 * q^58 - 4865142 * q^59 + 5147328 * q^60 + 2279886 * q^61 + 6274096 * q^62 + 8295108 * q^63 + 10452224 * q^64 - 1426892 * q^65 + 10937812 * q^66 - 5564458 * q^67 - 7403760 * q^68 - 4786076 * q^69 - 22140800 * q^70 - 25293948 * q^72 - 13177044 * q^74 + 6212566 * q^75 + 17814260 * q^76 + 7604308 * q^77 + 37993916 * q^78 - 9598912 * q^79 + 40047272 * q^80 - 5314414 * q^81 + 22858240 * q^82 + 4531198 * q^83 - 30159704 * q^84 + 7377748 * q^85 - 54129468 * q^86 - 59216392 * q^88 - 33515304 * q^90 + 2587652 * q^91 + 21810936 * q^92 - 14504144 * q^93 + 77196112 * q^94 - 4900620 * q^95 + 116940464 * q^96 - 4 * q^97 + 21823866 * q^98 + 18815006 * 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 −11.0128 2.59189i −56.5688 + 56.5688i 114.564 + 57.0880i −90.1684 90.1684i 769.602 476.362i 373.897i −1113.71 925.637i 4213.06i 759.302 + 1226.72i
5.2 −10.5089 4.19081i 42.2145 42.2145i 92.8743 + 88.0816i 46.9348 + 46.9348i −620.541 + 266.716i 1384.27i −606.875 1314.86i 1377.13i −296.539 689.929i
5.3 −10.1415 + 5.01488i 8.25573 8.25573i 77.7019 101.717i 63.0500 + 63.0500i −42.3243 + 125.127i 847.676i −277.917 + 1421.24i 2050.69i −955.613 323.236i
5.4 −5.91662 9.64332i 2.27414 2.27414i −57.9872 + 114.112i −242.456 242.456i −35.3854 8.47502i 1642.72i 1443.50 115.967i 2176.66i −903.559 + 3772.59i
5.5 −3.93160 + 10.6086i −35.6625 + 35.6625i −97.0851 83.4176i 50.2015 + 50.2015i −238.119 518.540i 699.226i 1266.64 701.973i 356.628i −729.940 + 335.196i
5.6 −2.66996 10.9941i −21.1050 + 21.1050i −113.743 + 58.7079i 328.807 + 328.807i 288.382 + 175.682i 874.718i 949.131 + 1093.76i 1296.15i 2737.06 4492.86i
5.7 −2.35505 + 11.0659i 50.5872 50.5872i −116.907 52.1215i −364.370 364.370i 440.657 + 678.928i 182.346i 852.094 1170.93i 2931.14i 4890.18 3173.96i
5.8 4.26019 10.4810i 61.7547 61.7547i −91.7015 89.3019i 160.824 + 160.824i −384.163 910.337i 1202.46i −1326.64 + 580.677i 5440.30i 2370.74 1000.45i
5.9 5.18419 + 10.0561i 20.1662 20.1662i −74.2483 + 104.265i 269.345 + 269.345i 307.337 + 98.2467i 147.771i −1433.41 206.115i 1373.65i −1312.21 + 4104.88i
5.10 5.94849 9.62369i −12.2643 + 12.2643i −57.2310 114.493i −210.432 210.432i 45.0740 + 190.982i 920.183i −1442.28 130.286i 1886.17i −3276.88 + 773.380i
5.11 8.27487 + 7.71535i −37.3394 + 37.3394i 8.94683 + 127.687i −233.214 233.214i −597.065 + 20.8921i 241.506i −911.115 + 1125.62i 601.466i −130.487 3729.13i
5.12 10.5559 4.07109i −49.4488 + 49.4488i 94.8525 85.9477i 252.094 + 252.094i −320.664 + 723.285i 1482.88i 651.349 1293.41i 2703.37i 3687.36 + 1634.77i
5.13 11.3129 + 0.135147i 26.1364 26.1364i 127.963 + 3.05781i −31.6175 31.6175i 299.211 292.146i 444.381i 1447.22 + 51.8865i 820.775i −353.412 361.958i
13.1 −11.0128 + 2.59189i −56.5688 56.5688i 114.564 57.0880i −90.1684 + 90.1684i 769.602 + 476.362i 373.897i −1113.71 + 925.637i 4213.06i 759.302 1226.72i
13.2 −10.5089 + 4.19081i 42.2145 + 42.2145i 92.8743 88.0816i 46.9348 46.9348i −620.541 266.716i 1384.27i −606.875 + 1314.86i 1377.13i −296.539 + 689.929i
13.3 −10.1415 5.01488i 8.25573 + 8.25573i 77.7019 + 101.717i 63.0500 63.0500i −42.3243 125.127i 847.676i −277.917 1421.24i 2050.69i −955.613 + 323.236i
13.4 −5.91662 + 9.64332i 2.27414 + 2.27414i −57.9872 114.112i −242.456 + 242.456i −35.3854 + 8.47502i 1642.72i 1443.50 + 115.967i 2176.66i −903.559 3772.59i
13.5 −3.93160 10.6086i −35.6625 35.6625i −97.0851 + 83.4176i 50.2015 50.2015i −238.119 + 518.540i 699.226i 1266.64 + 701.973i 356.628i −729.940 335.196i
13.6 −2.66996 + 10.9941i −21.1050 21.1050i −113.743 58.7079i 328.807 328.807i 288.382 175.682i 874.718i 949.131 1093.76i 1296.15i 2737.06 + 4492.86i
13.7 −2.35505 11.0659i 50.5872 + 50.5872i −116.907 + 52.1215i −364.370 + 364.370i 440.657 678.928i 182.346i 852.094 + 1170.93i 2931.14i 4890.18 + 3173.96i
See all 26 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 5.13 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 16.8.e.a 26
4.b odd 2 1 64.8.e.a 26
8.b even 2 1 128.8.e.b 26
8.d odd 2 1 128.8.e.a 26
16.e even 4 1 inner 16.8.e.a 26
16.e even 4 1 128.8.e.b 26
16.f odd 4 1 64.8.e.a 26
16.f odd 4 1 128.8.e.a 26

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
16.8.e.a 26 1.a even 1 1 trivial
16.8.e.a 26 16.e even 4 1 inner
64.8.e.a 26 4.b odd 2 1
64.8.e.a 26 16.f odd 4 1
128.8.e.a 26 8.d odd 2 1
128.8.e.a 26 16.f odd 4 1
128.8.e.b 26 8.b even 2 1
128.8.e.b 26 16.e even 4 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{8}^{\mathrm{new}}(16, [\chi])$$.