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Properties

Label 816.2.bx.a
Level $816$
Weight $2$
Character orbit 816.bx
Analytic conductor $6.516$
Analytic rank $0$
Dimension $288$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [816,2,Mod(325,816)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(816, base_ring=CyclotomicField(8))
 
chi = DirichletCharacter(H, H._module([0, 2, 0, 7]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("816.325");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 816 = 2^{4} \cdot 3 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 816.bx (of order \(8\), degree \(4\), 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: \(6.51579280494\)
Analytic rank: \(0\)
Dimension: \(288\)
Relative dimension: \(72\) over \(\Q(\zeta_{8})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{8}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 288 q+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 288 q - 32 q^{14} + 32 q^{19} + 24 q^{22} - 16 q^{24} + 32 q^{26} - 56 q^{28} + 80 q^{32} - 32 q^{34} - 8 q^{36} + 40 q^{40} + 24 q^{42} + 8 q^{44} + 8 q^{46} - 32 q^{48} - 8 q^{54} - 128 q^{55} + 24 q^{56} - 8 q^{60} - 32 q^{61} - 32 q^{65} + 40 q^{68} - 56 q^{70} - 24 q^{76} - 24 q^{78} + 88 q^{80} + 48 q^{82} + 64 q^{86} - 40 q^{88} - 8 q^{90} + 112 q^{91} - 96 q^{94} - 48 q^{96} + 32 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
325.1 −1.41206 + 0.0780396i 0.382683 − 0.923880i 1.98782 − 0.220393i 0.880775 − 2.12638i −0.468272 + 1.33444i 1.45684 + 0.603441i −2.78972 + 0.466336i −0.707107 − 0.707107i −1.07776 + 3.07131i
325.2 −1.41165 − 0.0851331i 0.382683 − 0.923880i 1.98550 + 0.240356i −1.54809 + 3.73743i −0.618867 + 1.27161i −0.141670 − 0.0586818i −2.78237 − 0.508330i −0.707107 − 0.707107i 2.50354 − 5.14415i
325.3 −1.40708 + 0.141846i 0.382683 − 0.923880i 1.95976 − 0.399177i 1.01048 − 2.43951i −0.407419 + 1.35426i −2.90021 − 1.20131i −2.70092 + 0.839658i −0.707107 − 0.707107i −1.07579 + 3.57592i
325.4 −1.40366 − 0.172438i −0.382683 + 0.923880i 1.94053 + 0.484089i 0.00850300 − 0.0205281i 0.696470 − 1.23082i −0.0304120 − 0.0125971i −2.64037 − 1.01412i −0.707107 − 0.707107i −0.0154752 + 0.0273482i
325.5 −1.36582 + 0.366783i −0.382683 + 0.923880i 1.73094 − 1.00192i −0.210188 + 0.507438i 0.183814 − 1.40222i −2.84550 − 1.17864i −1.99667 + 2.00333i −0.707107 − 0.707107i 0.100959 − 0.770164i
325.6 −1.35273 + 0.412457i −0.382683 + 0.923880i 1.65976 − 1.11589i 0.224824 − 0.542774i 0.136607 − 1.40760i −2.69737 − 1.11729i −1.78495 + 2.19407i −0.707107 − 0.707107i −0.0802558 + 0.826957i
325.7 −1.34279 + 0.443740i −0.382683 + 0.923880i 1.60619 − 1.19170i 1.01813 − 2.45798i 0.103903 − 1.41039i 4.40107 + 1.82298i −1.62798 + 2.31294i −0.707107 − 0.707107i −0.276434 + 3.75234i
325.8 −1.33696 − 0.461019i −0.382683 + 0.923880i 1.57492 + 1.23273i −1.19041 + 2.87390i 0.937558 − 1.05877i 3.76628 + 1.56005i −1.53730 − 2.37418i −0.707107 − 0.707107i 2.91645 − 3.29349i
325.9 −1.32993 − 0.480933i 0.382683 − 0.923880i 1.53741 + 1.27921i −0.200334 + 0.483648i −0.953265 + 1.04465i 3.52856 + 1.46158i −1.42942 − 2.44065i −0.707107 − 0.707107i 0.499032 − 0.546869i
325.10 −1.30377 + 0.547892i −0.382683 + 0.923880i 1.39963 − 1.42865i −1.51484 + 3.65715i −0.00725496 − 1.41419i 0.972859 + 0.402971i −1.04205 + 2.62947i −0.707107 − 0.707107i −0.0287185 − 5.59805i
325.11 −1.27240 − 0.617244i −0.382683 + 0.923880i 1.23802 + 1.57077i 0.102939 − 0.248517i 1.05719 − 0.939338i −1.24346 − 0.515058i −0.605712 − 2.76281i −0.707107 − 0.707107i −0.284376 + 0.252676i
325.12 −1.23788 + 0.683859i 0.382683 − 0.923880i 1.06467 − 1.69307i 0.501680 − 1.21116i 0.158089 + 1.40535i 1.93415 + 0.801150i −0.160116 + 2.82389i −0.707107 − 0.707107i 0.207247 + 1.84235i
325.13 −1.22476 + 0.707087i 0.382683 − 0.923880i 1.00005 − 1.73202i −0.672379 + 1.62327i 0.184570 + 1.40212i −3.30271 − 1.36803i −0.000133961 2.82843i −0.707107 − 0.707107i −0.324291 − 2.46353i
325.14 −1.19646 − 0.753986i 0.382683 − 0.923880i 0.863010 + 1.80422i −0.236935 + 0.572010i −1.15446 + 0.816842i −4.52234 − 1.87321i 0.327805 − 2.80937i −0.707107 − 0.707107i 0.714769 − 0.505740i
325.15 −1.12878 − 0.851969i −0.382683 + 0.923880i 0.548298 + 1.92337i −1.45886 + 3.52201i 1.21908 − 0.716824i −3.93014 − 1.62792i 1.01975 − 2.63820i −0.707107 − 0.707107i 4.64738 − 2.73267i
325.16 −1.10359 + 0.884357i −0.382683 + 0.923880i 0.435827 − 1.95194i 1.67206 − 4.03670i −0.394713 − 1.35801i −1.72939 − 0.716336i 1.24523 + 2.53957i −0.707107 − 0.707107i 1.72462 + 5.93356i
325.17 −1.07255 + 0.921759i 0.382683 − 0.923880i 0.300720 − 1.97726i −0.740844 + 1.78855i 0.441148 + 1.34365i 4.41683 + 1.82951i 1.50002 + 2.39790i −0.707107 − 0.707107i −0.854026 − 2.60119i
325.18 −1.06423 − 0.931347i −0.382683 + 0.923880i 0.265187 + 1.98234i 0.749189 − 1.80870i 1.26772 − 0.626813i 2.78889 + 1.15520i 1.56403 − 2.35665i −0.707107 − 0.707107i −2.48184 + 1.22713i
325.19 −1.03747 − 0.961071i 0.382683 − 0.923880i 0.152686 + 1.99416i 1.52332 − 3.67763i −1.28494 + 0.590711i −2.42539 − 1.00463i 1.75813 − 2.21563i −0.707107 − 0.707107i −5.11486 + 2.35140i
325.20 −0.998755 − 1.00124i 0.382683 − 0.923880i −0.00497677 + 1.99999i −0.495393 + 1.19598i −1.30724 + 0.539570i 2.68565 + 1.11243i 2.00745 − 1.99252i −0.707107 − 0.707107i 1.69225 − 0.698486i
See next 80 embeddings (of 288 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 325.72
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
272.y even 8 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 816.2.bx.a yes 288
16.e even 4 1 816.2.bt.a ✓ 288
17.d even 8 1 816.2.bt.a ✓ 288
272.y even 8 1 inner 816.2.bx.a yes 288
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
816.2.bt.a ✓ 288 16.e even 4 1
816.2.bt.a ✓ 288 17.d even 8 1
816.2.bx.a yes 288 1.a even 1 1 trivial
816.2.bx.a yes 288 272.y even 8 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(816, [\chi])\).