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Properties

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

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [480,2,Mod(163,480)]
 
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([4, 3, 0, 6]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("480.163");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 480 = 2^{5} \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 480.cc (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.

\(\operatorname{Tr}(f)(q) = \) \( 192 q - 24 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 192 q - 24 q^{8} - 24 q^{10} + 16 q^{12} + 16 q^{19} + 16 q^{22} - 40 q^{32} + 8 q^{34} + 48 q^{35} - 56 q^{38} - 40 q^{40} + 40 q^{42} - 32 q^{43} + 32 q^{44} + 32 q^{48} + 192 q^{49} - 16 q^{51} - 88 q^{52} - 32 q^{55} - 96 q^{56} + 32 q^{58} + 32 q^{61} + 72 q^{62} + 96 q^{64} - 48 q^{66} + 48 q^{67} - 56 q^{68} - 72 q^{70} - 64 q^{71} - 72 q^{76} - 48 q^{78} - 168 q^{80} - 120 q^{82} - 80 q^{83} - 64 q^{86} - 72 q^{88} + 16 q^{90} - 72 q^{92} - 64 q^{94} - 112 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
163.1 −1.41421 + 0.00298787i −0.923880 + 0.382683i 1.99998 − 0.00845096i −1.46720 + 1.68740i 1.30542 − 0.543955i 1.00081 −2.82837 + 0.0179271i 0.707107 − 0.707107i 2.06989 − 2.39072i
163.2 −1.41289 − 0.0610723i −0.923880 + 0.382683i 1.99254 + 0.172577i 2.13212 + 0.673850i 1.32872 − 0.484268i −3.68417 −2.80471 − 0.365523i 0.707107 − 0.707107i −2.97130 − 1.08229i
163.3 −1.41205 + 0.0782370i 0.923880 − 0.382683i 1.98776 − 0.220949i −1.50497 + 1.65380i −1.27462 + 0.612649i 0.278743 −2.78952 + 0.467507i 0.707107 − 0.707107i 1.99570 − 2.45299i
163.4 −1.37703 − 0.322153i 0.923880 − 0.382683i 1.79244 + 0.887229i 1.71797 + 1.43128i −1.39549 + 0.229337i 2.41213 −2.18242 − 1.79918i 0.707107 − 0.707107i −1.90461 − 2.52437i
163.5 −1.36817 − 0.357911i −0.923880 + 0.382683i 1.74380 + 0.979369i 0.869621 − 2.06004i 1.40099 − 0.192911i 4.51841 −2.03529 − 1.96407i 0.707107 − 0.707107i −1.92710 + 2.50724i
163.6 −1.34267 + 0.444125i 0.923880 − 0.382683i 1.60551 − 1.19262i −1.74678 − 1.39598i −1.07050 + 0.924134i 4.97048 −1.62598 + 2.31434i 0.707107 − 0.707107i 2.96533 + 1.09854i
163.7 −1.31601 − 0.517799i 0.923880 − 0.382683i 1.46377 + 1.36286i −1.60648 − 1.55539i −1.41399 + 0.0252318i −2.67298 −1.22065 − 2.55147i 0.707107 − 0.707107i 1.30876 + 2.87874i
163.8 −1.29907 + 0.558936i 0.923880 − 0.382683i 1.37518 − 1.45220i 1.07117 − 1.96280i −0.986292 + 1.01352i −3.77805 −0.974778 + 2.65515i 0.707107 − 0.707107i −0.294444 + 3.14854i
163.9 −1.23883 + 0.682130i 0.923880 − 0.382683i 1.06940 − 1.69009i 0.164937 + 2.22998i −0.883489 + 1.10429i −3.87441 −0.171941 + 2.82320i 0.707107 − 0.707107i −1.72546 − 2.65005i
163.10 −1.14061 + 0.836072i −0.923880 + 0.382683i 0.601968 − 1.90726i 1.36685 + 1.76967i 0.733832 − 1.20892i 2.20092 0.907997 + 2.67872i 0.707107 − 0.707107i −3.03861 − 0.875706i
163.11 −1.02543 − 0.973911i 0.923880 − 0.382683i 0.102996 + 1.99735i 2.20164 − 0.390853i −1.32007 − 0.507363i −1.83648 1.83962 − 2.14844i 0.707107 − 0.707107i −2.63828 − 1.74341i
163.12 −1.00995 + 0.989948i 0.923880 − 0.382683i 0.0400049 − 1.99960i 2.13730 − 0.657215i −0.554237 + 1.30108i 3.93782 1.93910 + 2.05910i 0.707107 − 0.707107i −1.50796 + 2.77958i
163.13 −0.996170 − 1.00382i −0.923880 + 0.382683i −0.0152903 + 1.99994i 1.92456 + 1.13846i 1.30448 + 0.546187i 2.39666 2.02280 − 1.97693i 0.707107 − 0.707107i −0.774387 − 3.06599i
163.14 −0.957087 + 1.04115i −0.923880 + 0.382683i −0.167968 − 1.99293i −2.15114 + 0.610391i 0.485804 − 1.32815i −5.03321 2.23569 + 1.73253i 0.707107 − 0.707107i 1.42333 − 2.82385i
163.15 −0.895327 + 1.09471i −0.923880 + 0.382683i −0.396780 − 1.96025i 0.883893 − 2.05396i 0.408247 − 1.35401i −0.703513 2.50115 + 1.32070i 0.707107 − 0.707107i 1.45711 + 2.80657i
163.16 −0.861793 − 1.12130i −0.923880 + 0.382683i −0.514627 + 1.93266i 0.662814 − 2.13557i 1.22530 + 0.706152i −3.61201 2.61059 − 1.08850i 0.707107 − 0.707107i −2.96583 + 1.09721i
163.17 −0.655602 − 1.25307i 0.923880 − 0.382683i −1.14037 + 1.64303i −2.19437 + 0.429797i −1.08523 − 0.906798i −1.85821 2.80646 + 0.351792i 0.707107 − 0.707107i 1.97720 + 2.46793i
163.18 −0.531005 − 1.31074i −0.923880 + 0.382683i −1.43607 + 1.39202i −0.474561 + 2.18513i 0.992182 + 1.00776i −1.76292 2.58713 + 1.14314i 0.707107 − 0.707107i 3.11613 − 0.538290i
163.19 −0.435433 + 1.34551i 0.923880 − 0.382683i −1.62080 − 1.17176i 1.86300 + 1.23662i 0.112617 + 1.40972i −1.18050 2.28236 − 1.67058i 0.707107 − 0.707107i −2.47510 + 1.96822i
163.20 −0.388900 + 1.35969i −0.923880 + 0.382683i −1.69751 − 1.05757i −2.18315 − 0.483570i −0.161034 − 1.40502i 4.14760 2.09813 − 1.89680i 0.707107 − 0.707107i 1.50653 − 2.78035i
See next 80 embeddings (of 192 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 163.48
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
160.ba 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.cc.a yes 192
5.c odd 4 1 480.2.bo.a ✓ 192
32.h odd 8 1 480.2.bo.a ✓ 192
160.ba even 8 1 inner 480.2.cc.a yes 192
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
480.2.bo.a ✓ 192 5.c odd 4 1
480.2.bo.a ✓ 192 32.h odd 8 1
480.2.cc.a yes 192 1.a even 1 1 trivial
480.2.cc.a yes 192 160.ba even 8 1 inner

Hecke kernels

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