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Properties

Label 480.2.bo.a
Level $480$
Weight $2$
Character orbit 480.bo
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(43,480)]
 
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, 5, 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("480.43");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 480 = 2^{5} \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 480.bo (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
\(\operatorname{Tr}(f)(q) = \) \( 192 q + 24 q^{8} + 24 q^{10} + 16 q^{12} - 16 q^{19} + 16 q^{22} + 40 q^{32} - 8 q^{34} + 48 q^{35} + 56 q^{38} - 40 q^{40} - 40 q^{42} + 32 q^{43} - 32 q^{44} - 32 q^{48} - 192 q^{49} - 16 q^{51} + 88 q^{52} + 32 q^{55} - 96 q^{56} + 32 q^{58} + 32 q^{61} - 72 q^{62} - 96 q^{64} - 48 q^{66} - 48 q^{67} - 56 q^{68} - 72 q^{70} - 64 q^{71} - 72 q^{76} + 48 q^{78} + 8 q^{80} - 120 q^{82} + 80 q^{83} - 64 q^{86} - 104 q^{88} - 16 q^{90} - 72 q^{92} + 64 q^{94} + 64 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
43.1 −1.41307 + 0.0569637i 0.382683 − 0.923880i 1.99351 − 0.160987i −2.17585 − 0.515441i −0.488129 + 1.32730i 0.870718i −2.80779 + 0.341043i −0.707107 − 0.707107i 3.10398 + 0.604408i
43.2 −1.36357 + 0.375058i −0.382683 + 0.923880i 1.71866 − 1.02284i 1.63916 + 1.52091i 0.175308 − 1.40331i 0.653119i −1.95990 + 2.03931i −0.707107 − 0.707107i −2.80554 − 1.45909i
43.3 −1.35969 + 0.388900i 0.382683 − 0.923880i 1.69751 − 1.05757i 1.20179 + 1.88566i −0.161034 + 1.40502i − 4.14760i −1.89680 + 2.09813i −0.707107 − 0.707107i −2.36739 − 2.09654i
43.4 −1.34870 − 0.425463i −0.382683 + 0.923880i 1.63796 + 1.14764i −1.50298 + 1.65562i 0.909200 − 1.08322i − 3.14559i −1.72083 − 2.24471i −0.707107 − 0.707107i 2.73146 − 1.59346i
43.5 −1.34551 + 0.435433i −0.382683 + 0.923880i 1.62080 − 1.17176i −0.442914 − 2.19176i 0.112617 − 1.40972i 1.18050i −1.67058 + 2.28236i −0.707107 − 0.707107i 1.55031 + 2.75618i
43.6 −1.32399 − 0.497042i −0.382683 + 0.923880i 1.50590 + 1.31616i 2.18428 + 0.478463i 0.965876 − 1.03300i − 0.314594i −1.33961 − 2.49107i −0.707107 − 0.707107i −2.65415 − 1.71916i
43.7 −1.25848 − 0.645158i 0.382683 − 0.923880i 1.16754 + 1.62384i −0.432254 + 2.19389i −1.07765 + 0.915793i − 0.389799i −0.421700 − 2.79681i −0.707107 − 0.707107i 1.95939 − 2.48210i
43.8 −1.21572 − 0.722511i 0.382683 − 0.923880i 0.955956 + 1.75674i 0.341956 − 2.20977i −1.13275 + 0.846687i − 4.62151i 0.107090 − 2.82640i −0.707107 − 0.707107i −2.01230 + 2.43939i
43.9 −1.14806 − 0.825814i 0.382683 − 0.923880i 0.636062 + 1.89616i −1.00788 − 1.99604i −1.20229 + 0.744639i 4.38174i 0.835643 − 2.70217i −0.707107 − 0.707107i −0.491262 + 3.12389i
43.10 −1.09471 + 0.895327i 0.382683 − 0.923880i 0.396780 − 1.96025i −2.07737 + 0.827359i 0.408247 + 1.35401i 0.703513i 1.32070 + 2.50115i −0.707107 − 0.707107i 1.53336 − 2.76565i
43.11 −1.06970 − 0.925058i −0.382683 + 0.923880i 0.288536 + 1.97908i −1.94905 − 1.09599i 1.26400 − 0.634274i 0.930209i 1.52211 − 2.38394i −0.707107 − 0.707107i 1.07106 + 2.97537i
43.12 −1.04115 + 0.957087i 0.382683 − 0.923880i 0.167968 − 1.99293i 1.95270 + 1.08948i 0.485804 + 1.32815i 5.03321i 1.73253 + 2.23569i −0.707107 − 0.707107i −3.07577 + 0.734601i
43.13 −0.989948 + 1.00995i −0.382683 + 0.923880i −0.0400049 − 1.99960i −1.97602 − 1.04658i −0.554237 − 1.30108i − 3.93782i 2.05910 + 1.93910i −0.707107 − 0.707107i 3.01316 − 0.959628i
43.14 −0.836072 + 1.14061i 0.382683 − 0.923880i −0.601968 − 1.90726i 0.284832 − 2.21785i 0.733832 + 1.20892i − 2.20092i 2.67872 + 0.907997i −0.707107 − 0.707107i 2.29156 + 2.17917i
43.15 −0.764012 − 1.19008i −0.382683 + 0.923880i −0.832571 + 1.81847i 0.831682 − 2.07565i 1.39186 − 0.250432i − 2.51949i 2.80021 − 0.398507i −0.707107 − 0.707107i −3.10560 + 0.596052i
43.16 −0.726530 − 1.21332i 0.382683 − 0.923880i −0.944307 + 1.76303i 2.22740 + 0.196657i −1.39900 + 0.206908i 3.01860i 2.82520 − 0.135147i −0.707107 − 0.707107i −1.37967 − 2.84544i
43.17 −0.705649 − 1.22559i 0.382683 − 0.923880i −1.00412 + 1.72967i −1.25922 + 1.84780i −1.40233 + 0.182923i − 0.827911i 2.82841 + 0.0100976i −0.707107 − 0.707107i 3.15320 + 0.239380i
43.18 −0.682130 + 1.23883i −0.382683 + 0.923880i −1.06940 − 1.69009i 1.46020 − 1.69346i −0.883489 − 1.10429i 3.87441i 2.82320 − 0.171941i −0.707107 − 0.707107i 1.10186 + 2.96410i
43.19 −0.558936 + 1.29907i −0.382683 + 0.923880i −1.37518 − 1.45220i −2.14534 + 0.630482i −0.986292 − 1.01352i 3.77805i 2.65515 − 0.974778i −0.707107 − 0.707107i 0.380066 − 3.13936i
43.20 −0.444125 + 1.34267i −0.382683 + 0.923880i −1.60551 − 1.19262i 0.248056 + 2.22227i −1.07050 − 0.924134i − 4.97048i 2.31434 − 1.62598i −0.707107 − 0.707107i −3.09393 − 0.653908i
See next 80 embeddings (of 192 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 43.48
Significant digits:
Format:

Inner twists

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

Hecke kernels

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