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

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Newspace parameters

Level: \( N \) \(=\) \( 16 = 2^{4} \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 16.e (of order \(4\), degree \(2\), minimal)

Newform invariants

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.

\(\operatorname{Tr}(f)(q) = \) \( 26q - 2q^{2} - 2q^{3} - 184q^{4} - 2q^{5} - 176q^{6} - 1004q^{8} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 26q - 2q^{2} - 2q^{3} - 184q^{4} - 2q^{5} - 176q^{6} - 1004q^{8} + 12972q^{10} + 1202q^{11} - 27356q^{12} - 2q^{13} + 22268q^{14} - 27004q^{15} + 13336q^{16} - 4q^{17} - 29346q^{18} + 60582q^{19} + 46204q^{20} + 4372q^{21} - 44348q^{22} - 59216q^{24} - 35960q^{26} - 233672q^{27} + 220312q^{28} - 51690q^{29} + 27636q^{30} + 357488q^{31} - 248632q^{32} - 4q^{33} + 924660q^{34} - 252004q^{35} - 1015508q^{36} + 415574q^{37} - 5472q^{38} - 1121976q^{40} + 124600q^{42} + 569754q^{43} - 758620q^{44} + 151874q^{45} + 2335660q^{46} - 2076464q^{47} + 4141288q^{48} - 1647090q^{49} - 787194q^{50} + 2609508q^{51} + 410324q^{52} + 907814q^{53} - 7404512q^{54} - 4093160q^{56} - 1860712q^{58} - 4865142q^{59} + 5147328q^{60} + 2279886q^{61} + 6274096q^{62} + 8295108q^{63} + 10452224q^{64} - 1426892q^{65} + 10937812q^{66} - 5564458q^{67} - 7403760q^{68} - 4786076q^{69} - 22140800q^{70} - 25293948q^{72} - 13177044q^{74} + 6212566q^{75} + 17814260q^{76} + 7604308q^{77} + 37993916q^{78} - 9598912q^{79} + 40047272q^{80} - 5314414q^{81} + 22858240q^{82} + 4531198q^{83} - 30159704q^{84} + 7377748q^{85} - 54129468q^{86} - 59216392q^{88} - 33515304q^{90} + 2587652q^{91} + 21810936q^{92} - 14504144q^{93} + 77196112q^{94} - 4900620q^{95} + 116940464q^{96} - 4q^{97} + 21823866q^{98} + 18815006q^{99} + O(q^{100}) \)

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 −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 13.13
Significant digits:
Format:

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 \(S_{8}^{\mathrm{new}}(16, [\chi])\).

Hecke characteristic polynomials

There are no characteristic polynomials of Hecke operators in the database