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Properties

Label 480.2.by.a
Level $480$
Weight $2$
Character orbit 480.by
Analytic conductor $3.833$
Analytic rank $0$
Dimension $192$
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [480,2,Mod(109,480)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(480, base_ring=CyclotomicField(8))
 
chi = DirichletCharacter(H, H._module([0, 7, 0, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("480.109");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 480 = 2^{5} \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 480.by (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: \(3.83281929702\)
Analytic rank: \(0\)
Dimension: \(192\)
Relative dimension: \(48\) over \(\Q(\zeta_{8})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{8}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 192 q+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 192 q + 32 q^{14} + 8 q^{24} - 80 q^{26} + 96 q^{31} - 48 q^{35} + 72 q^{40} - 24 q^{50} + 16 q^{51} - 8 q^{54} - 32 q^{55} + 96 q^{56} - 64 q^{59} - 48 q^{60} - 32 q^{61} - 96 q^{64} + 48 q^{66} + 32 q^{69} - 120 q^{70} + 64 q^{71} - 272 q^{74} - 32 q^{75} - 56 q^{76} - 104 q^{80} - 64 q^{86} - 200 q^{94} + 80 q^{96}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
109.1 −1.41404 − 0.0221331i −0.923880 − 0.382683i 1.99902 + 0.0625943i −0.856252 − 2.06563i 1.29793 + 0.561578i 0.957969 + 0.957969i −2.82531 − 0.132755i 0.707107 + 0.707107i 1.16506 + 2.93984i
109.2 −1.41322 + 0.0530630i 0.923880 + 0.382683i 1.99437 − 0.149979i 2.22994 − 0.165446i −1.32595 − 0.491791i −1.89462 − 1.89462i −2.81052 + 0.317781i 0.707107 + 0.707107i −3.14261 + 0.352139i
109.3 −1.39385 + 0.239138i 0.923880 + 0.382683i 1.88563 − 0.666645i 1.05060 − 1.97389i −1.37926 − 0.312468i 2.49225 + 2.49225i −2.46886 + 1.38013i 0.707107 + 0.707107i −0.992338 + 3.00254i
109.4 −1.36569 + 0.367268i −0.923880 − 0.382683i 1.73023 − 1.00315i 0.644000 + 2.14132i 1.40228 + 0.183316i 2.22105 + 2.22105i −1.99453 + 2.00545i 0.707107 + 0.707107i −1.66595 − 2.68787i
109.5 −1.36536 − 0.368505i 0.923880 + 0.382683i 1.72841 + 1.00628i −2.18123 − 0.492191i −1.12041 − 0.862954i 0.199762 + 0.199762i −1.98908 − 2.01086i 0.707107 + 0.707107i 2.79678 + 1.47581i
109.6 −1.36505 + 0.369659i −0.923880 − 0.382683i 1.72670 − 1.00920i −1.79936 + 1.32751i 1.40260 + 0.180860i −3.29270 − 3.29270i −1.98397 + 2.01590i 0.707107 + 0.707107i 1.96549 − 2.47727i
109.7 −1.25910 − 0.643945i −0.923880 − 0.382683i 1.17067 + 1.62158i 1.94143 + 1.10943i 0.916831 + 1.07676i −1.41157 − 1.41157i −0.429782 − 2.79558i 0.707107 + 0.707107i −1.73005 − 2.64706i
109.8 −1.24780 − 0.665583i 0.923880 + 0.382683i 1.11400 + 1.66103i 0.414699 + 2.19728i −0.898107 − 1.09243i 3.54290 + 3.54290i −0.284493 − 2.81408i 0.707107 + 0.707107i 0.945010 − 3.01777i
109.9 −1.20372 + 0.742325i 0.923880 + 0.382683i 0.897907 − 1.78711i −1.61314 − 1.54848i −1.39617 + 0.225174i −2.53105 − 2.53105i 0.245785 + 2.81773i 0.707107 + 0.707107i 3.09125 + 0.666468i
109.10 −1.17012 − 0.794237i −0.923880 − 0.382683i 0.738376 + 1.85871i −1.36493 + 1.77115i 0.777111 + 1.18157i −0.0567794 − 0.0567794i 0.612265 − 2.76136i 0.707107 + 0.707107i 3.00385 − 0.988379i
109.11 −1.10932 + 0.877164i −0.923880 − 0.382683i 0.461166 − 1.94611i 1.89656 − 1.18450i 1.36055 − 0.385877i −0.282057 − 0.282057i 1.19547 + 2.56336i 0.707107 + 0.707107i −1.06488 + 2.97759i
109.12 −1.04798 + 0.949600i 0.923880 + 0.382683i 0.196522 − 1.99032i 1.67205 + 1.48467i −1.33160 + 0.476271i −1.15754 − 1.15754i 1.68406 + 2.27243i 0.707107 + 0.707107i −3.16212 + 0.0318746i
109.13 −0.887597 − 1.10099i −0.923880 − 0.382683i −0.424341 + 1.95447i −1.23432 − 1.86453i 0.398704 + 1.35685i 1.12884 + 1.12884i 2.52848 − 1.26758i 0.707107 + 0.707107i −0.957240 + 3.01392i
109.14 −0.844718 − 1.13422i 0.923880 + 0.382683i −0.572903 + 1.91619i 2.15748 + 0.587617i −0.346371 − 1.37114i −0.421861 − 0.421861i 2.65732 − 0.968843i 0.707107 + 0.707107i −1.15597 − 2.94342i
109.15 −0.818722 − 1.15312i 0.923880 + 0.382683i −0.659388 + 1.88818i −1.33049 + 1.79717i −0.315119 − 1.37866i −2.87719 − 2.87719i 2.71716 − 0.785535i 0.707107 + 0.707107i 3.16165 + 0.0628379i
109.16 −0.736509 + 1.20729i −0.923880 − 0.382683i −0.915108 − 1.77836i −1.06394 − 1.96673i 1.14246 − 0.833543i −1.42928 − 1.42928i 2.82099 + 0.204978i 0.707107 + 0.707107i 3.15802 + 0.164031i
109.17 −0.666736 + 1.24718i 0.923880 + 0.382683i −1.11093 − 1.66308i −1.35821 − 1.77631i −1.09326 + 0.897097i 1.72385 + 1.72385i 2.81486 − 0.276692i 0.707107 + 0.707107i 3.12095 − 0.509601i
109.18 −0.465976 + 1.33524i 0.923880 + 0.382683i −1.56573 − 1.24438i −1.09220 + 1.95118i −0.941480 + 1.05528i 1.39293 + 1.39293i 2.39114 − 1.51078i 0.707107 + 0.707107i −2.09636 − 2.36755i
109.19 −0.452730 − 1.33979i −0.923880 − 0.382683i −1.59007 + 1.21313i 1.59364 − 1.56854i −0.0944472 + 1.41106i −2.14270 − 2.14270i 2.34521 + 1.58114i 0.707107 + 0.707107i −2.82300 − 1.42502i
109.20 −0.346994 + 1.37098i −0.923880 − 0.382683i −1.75919 − 0.951447i 1.86332 + 1.23614i 0.845234 − 1.13383i 1.95278 + 1.95278i 1.91485 − 2.08167i 0.707107 + 0.707107i −2.34129 + 2.12565i
See next 80 embeddings (of 192 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 109.48
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
32.g even 8 1 inner
160.z even 8 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 480.2.by.a ✓ 192
5.b even 2 1 inner 480.2.by.a ✓ 192
32.g even 8 1 inner 480.2.by.a ✓ 192
160.z even 8 1 inner 480.2.by.a ✓ 192
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
480.2.by.a ✓ 192 1.a even 1 1 trivial
480.2.by.a ✓ 192 5.b even 2 1 inner
480.2.by.a ✓ 192 32.g even 8 1 inner
480.2.by.a ✓ 192 160.z even 8 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(480, [\chi])\).