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Properties

Label 2280.2.r.b
Level $2280$
Weight $2$
Character orbit 2280.r
Analytic conductor $18.206$
Analytic rank $0$
Dimension $40$
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] = [2280,2,Mod(1481,2280)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2280, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2280.1481");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2280 = 2^{3} \cdot 3 \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2280.r (of order \(2\), degree \(1\), 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: \(18.2058916609\)
Analytic rank: \(0\)
Dimension: \(40\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 40 q + 2 q^{3} + 4 q^{7} - 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 40 q + 2 q^{3} + 4 q^{7} - 6 q^{9} - 4 q^{19} + 6 q^{21} - 40 q^{25} + 20 q^{27} + 12 q^{29} + 8 q^{33} - 6 q^{39} + 8 q^{41} - 40 q^{43} + 28 q^{49} - 22 q^{51} + 28 q^{53} + 2 q^{57} + 20 q^{59} + 8 q^{61} + 6 q^{63} + 8 q^{65} + 30 q^{69} - 40 q^{71} + 36 q^{73} - 2 q^{75} - 14 q^{81} - 6 q^{87} - 48 q^{89} - 4 q^{95} - 16 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1481.1 0 −1.70128 − 0.325034i 0 − 1.00000i 0 −0.146424 0 2.78871 + 1.10595i 0
1481.2 0 −1.70128 + 0.325034i 0 1.00000i 0 −0.146424 0 2.78871 − 1.10595i 0
1481.3 0 −1.69792 − 0.342174i 0 − 1.00000i 0 1.09585 0 2.76583 + 1.16197i 0
1481.4 0 −1.69792 + 0.342174i 0 1.00000i 0 1.09585 0 2.76583 − 1.16197i 0
1481.5 0 −1.45029 − 0.946924i 0 1.00000i 0 3.14372 0 1.20667 + 2.74662i 0
1481.6 0 −1.45029 + 0.946924i 0 − 1.00000i 0 3.14372 0 1.20667 − 2.74662i 0
1481.7 0 −1.19153 − 1.25709i 0 1.00000i 0 −0.464718 0 −0.160527 + 2.99570i 0
1481.8 0 −1.19153 + 1.25709i 0 − 1.00000i 0 −0.464718 0 −0.160527 − 2.99570i 0
1481.9 0 −1.16437 − 1.28228i 0 1.00000i 0 −3.17896 0 −0.288498 + 2.98610i 0
1481.10 0 −1.16437 + 1.28228i 0 − 1.00000i 0 −3.17896 0 −0.288498 − 2.98610i 0
1481.11 0 −1.13508 − 1.30828i 0 − 1.00000i 0 −2.58104 0 −0.423200 + 2.97000i 0
1481.12 0 −1.13508 + 1.30828i 0 1.00000i 0 −2.58104 0 −0.423200 − 2.97000i 0
1481.13 0 −0.802168 − 1.53510i 0 − 1.00000i 0 −1.63635 0 −1.71305 + 2.46281i 0
1481.14 0 −0.802168 + 1.53510i 0 1.00000i 0 −1.63635 0 −1.71305 − 2.46281i 0
1481.15 0 −0.386711 − 1.68833i 0 − 1.00000i 0 5.10451 0 −2.70091 + 1.30579i 0
1481.16 0 −0.386711 + 1.68833i 0 1.00000i 0 5.10451 0 −2.70091 − 1.30579i 0
1481.17 0 −0.262534 − 1.71204i 0 1.00000i 0 −4.09845 0 −2.86215 + 0.898937i 0
1481.18 0 −0.262534 + 1.71204i 0 − 1.00000i 0 −4.09845 0 −2.86215 − 0.898937i 0
1481.19 0 −0.249833 − 1.71394i 0 1.00000i 0 1.56704 0 −2.87517 + 0.856396i 0
1481.20 0 −0.249833 + 1.71394i 0 − 1.00000i 0 1.56704 0 −2.87517 − 0.856396i 0
See all 40 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1481.40
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
57.d even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2280.2.r.b yes 40
3.b odd 2 1 2280.2.r.a ✓ 40
19.b odd 2 1 2280.2.r.a ✓ 40
57.d even 2 1 inner 2280.2.r.b yes 40
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2280.2.r.a ✓ 40 3.b odd 2 1
2280.2.r.a ✓ 40 19.b odd 2 1
2280.2.r.b yes 40 1.a even 1 1 trivial
2280.2.r.b yes 40 57.d even 2 1 inner