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Properties

Label 16.8.e.a
Level $16$
Weight $8$
Character orbit 16.e
Analytic conductor $4.998$
Analytic rank $0$
Dimension $26$
Inner twists $2$

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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 algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 26 q - 2 q^{2} - 2 q^{3} - 184 q^{4} - 2 q^{5} - 176 q^{6} - 1004 q^{8} + 12972 q^{10} + 1202 q^{11} - 27356 q^{12} - 2 q^{13} + 22268 q^{14} - 27004 q^{15} + 13336 q^{16} - 4 q^{17} - 29346 q^{18} + 60582 q^{19}+ \cdots + 18815006 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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