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Properties

Label 320.2.be.a
Level $320$
Weight $2$
Character orbit 320.be
Analytic conductor $2.555$
Analytic rank $0$
Dimension $256$
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] = [320,2,Mod(21,320)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(320, base_ring=CyclotomicField(16))
 
chi = DirichletCharacter(H, H._module([0, 13, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("320.21");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 320 = 2^{6} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 320.be (of order \(16\), degree \(8\), 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: \(2.55521286468\)
Analytic rank: \(0\)
Dimension: \(256\)
Relative dimension: \(32\) over \(\Q(\zeta_{16})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{16}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 256 q+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 256 q - 16 q^{22} - 80 q^{24} - 80 q^{28} - 80 q^{32} - 80 q^{34} - 80 q^{38} - 80 q^{42} - 16 q^{44} - 32 q^{51} + 96 q^{52} + 64 q^{54} + 16 q^{56} + 144 q^{58} - 64 q^{59} + 96 q^{62} - 160 q^{63} + 96 q^{64} + 96 q^{66} - 160 q^{67} + 96 q^{68} - 64 q^{71} + 144 q^{72} + 16 q^{74} + 64 q^{76} + 48 q^{78} - 32 q^{79} - 64 q^{80} - 112 q^{84} - 208 q^{86} - 160 q^{88} - 144 q^{90} - 16 q^{94} - 272 q^{96} - 272 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
21.1 −1.41287 − 0.0617229i −0.550105 − 2.76557i 1.99238 + 0.174412i −0.831470 + 0.555570i 0.606526 + 3.94133i −0.851227 − 0.352590i −2.80420 − 0.369397i −4.57410 + 1.89466i 1.20905 − 0.733626i
21.2 −1.40597 − 0.152429i −0.310453 − 1.56075i 1.95353 + 0.428623i 0.831470 − 0.555570i 0.198585 + 2.24170i 1.48559 + 0.615351i −2.68128 − 0.900409i 0.432073 − 0.178970i −1.25371 + 0.654377i
21.3 −1.40413 + 0.168608i 0.592556 + 2.97898i 1.94314 − 0.473495i −0.831470 + 0.555570i −1.33431 − 4.08296i −0.868006 − 0.359540i −2.64858 + 0.992476i −5.75157 + 2.38238i 1.07381 − 0.920284i
21.4 −1.36558 − 0.367682i 0.0881564 + 0.443192i 1.72962 + 1.00420i 0.831470 − 0.555570i 0.0425691 − 0.637628i −4.42719 − 1.83380i −1.99271 − 2.00726i 2.58299 − 1.06991i −1.33971 + 0.452959i
21.5 −1.33651 + 0.462335i 0.414601 + 2.08434i 1.57249 − 1.23583i 0.831470 − 0.555570i −1.51778 − 2.59405i 0.306327 + 0.126885i −1.53028 + 2.37871i −1.40094 + 0.580289i −0.854404 + 1.12694i
21.6 −1.20454 + 0.740998i −0.0295152 − 0.148383i 0.901845 − 1.78513i −0.831470 + 0.555570i 0.145504 + 0.156863i −2.47972 − 1.02713i 0.236463 + 2.81853i 2.75049 − 1.13929i 0.589864 − 1.28532i
21.7 −1.12823 − 0.852698i −0.296809 − 1.49216i 0.545811 + 1.92408i −0.831470 + 0.555570i −0.937492 + 1.93659i 3.12141 + 1.29293i 1.02486 − 2.63622i 0.633198 − 0.262279i 1.41182 + 0.0821812i
21.8 −0.943132 − 1.05380i 0.424018 + 2.13168i −0.221004 + 1.98775i 0.831470 − 0.555570i 1.84647 − 2.45729i 0.717811 + 0.297327i 2.30314 − 1.64182i −1.59264 + 0.659692i −1.36965 − 0.352230i
21.9 −0.838418 + 1.13888i −0.503628 − 2.53191i −0.594109 − 1.90972i 0.831470 − 0.555570i 3.30580 + 1.54922i 3.03022 + 1.25516i 2.67306 + 0.924524i −3.38527 + 1.40223i −0.0643898 + 1.41275i
21.10 −0.826278 + 1.14772i 0.259751 + 1.30586i −0.634529 − 1.89667i −0.831470 + 0.555570i −1.71339 − 0.780880i 4.66392 + 1.93186i 2.70115 + 0.838917i 1.13384 − 0.469653i 0.0493852 − 1.41335i
21.11 −0.710489 − 1.22279i 0.0139979 + 0.0703724i −0.990412 + 1.73755i −0.831470 + 0.555570i 0.0761051 − 0.0671153i −2.30436 − 0.954495i 2.82833 − 0.0234480i 2.76688 − 1.14608i 1.27009 + 0.621983i
21.12 −0.655952 + 1.25289i −0.561689 − 2.82380i −1.13945 − 1.64367i −0.831470 + 0.555570i 3.90635 + 1.14855i −2.37084 − 0.982034i 2.80676 − 0.349441i −4.88673 + 2.02415i −0.150663 − 1.40617i
21.13 −0.485032 − 1.32844i −0.163098 − 0.819950i −1.52949 + 1.28867i 0.831470 − 0.555570i −1.01014 + 0.614368i 3.40804 + 1.41166i 2.45377 + 1.40678i 2.12592 − 0.880586i −1.14133 − 0.835085i
21.14 −0.426397 + 1.34840i 0.0112611 + 0.0566134i −1.63637 − 1.14991i 0.831470 − 0.555570i −0.0811393 − 0.00895530i −0.767422 − 0.317877i 2.24828 − 1.71617i 2.76856 − 1.14678i 0.394595 + 1.35805i
21.15 −0.349830 + 1.37026i 0.490400 + 2.46541i −1.75524 − 0.958718i −0.831470 + 0.555570i −3.54981 − 0.190497i −4.35441 − 1.80366i 1.92773 − 2.06975i −3.06610 + 1.27002i −0.470404 − 1.33369i
21.16 −0.0198245 − 1.41407i 0.188389 + 0.947096i −1.99921 + 0.0560666i −0.831470 + 0.555570i 1.33553 − 0.285172i 0.919468 + 0.380856i 0.118916 + 2.82593i 1.91014 − 0.791205i 0.802101 + 1.16475i
21.17 0.0428197 − 1.41357i −0.436584 − 2.19486i −1.99633 − 0.121057i 0.831470 − 0.555570i −3.12127 + 0.523157i −1.44001 − 0.596473i −0.256604 + 2.81676i −1.85515 + 0.768430i −0.749731 − 1.19913i
21.18 0.117925 + 1.40929i 0.670382 + 3.37024i −1.97219 + 0.332380i 0.831470 − 0.555570i −4.67059 + 1.34220i 3.53499 + 1.46424i −0.700989 − 2.74019i −8.13747 + 3.37065i 0.881010 + 1.10626i
21.19 0.296452 + 1.38279i −0.352426 − 1.77176i −1.82423 + 0.819863i −0.831470 + 0.555570i 2.34551 − 1.01257i 2.65351 + 1.09912i −1.67450 − 2.27949i −0.243307 + 0.100781i −1.01473 − 0.985051i
21.20 0.399774 + 1.35653i −0.513414 − 2.58111i −1.68036 + 1.08461i 0.831470 − 0.555570i 3.29611 − 1.72832i −3.65525 − 1.51405i −2.14308 − 1.84586i −3.62688 + 1.50230i 1.08605 + 0.905813i
See next 80 embeddings (of 256 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 21.32
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
64.i even 16 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 320.2.be.a ✓ 256
64.i even 16 1 inner 320.2.be.a ✓ 256
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
320.2.be.a ✓ 256 1.a even 1 1 trivial
320.2.be.a ✓ 256 64.i even 16 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(320, [\chi])\).