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Properties

Label 1380.2.p.a
Level $1380$
Weight $2$
Character orbit 1380.p
Analytic conductor $11.019$
Analytic rank $0$
Dimension $48$
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] = [1380,2,Mod(91,1380)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1380, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 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("1380.91");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1380 = 2^{2} \cdot 3 \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1380.p (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: \(11.0193554789\)
Analytic rank: \(0\)
Dimension: \(48\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 48 q - 4 q^{2} - 2 q^{4} - 2 q^{6} - 4 q^{8} - 48 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 48 q - 4 q^{2} - 2 q^{4} - 2 q^{6} - 4 q^{8} - 48 q^{9} - 2 q^{10} + 20 q^{14} - 48 q^{15} - 6 q^{16} + 4 q^{18} + 16 q^{19} + 28 q^{22} + 4 q^{23} + 2 q^{24} - 48 q^{25} - 20 q^{26} + 32 q^{29} + 4 q^{30} + 16 q^{32} - 28 q^{34} + 2 q^{36} + 2 q^{40} - 8 q^{41} + 14 q^{46} + 16 q^{48} + 40 q^{49} + 4 q^{50} + 16 q^{51} - 16 q^{52} + 2 q^{54} + 40 q^{56} - 8 q^{58} + 2 q^{60} + 24 q^{62} - 26 q^{64} - 48 q^{67} - 44 q^{68} - 8 q^{69} + 4 q^{72} + 20 q^{74} - 64 q^{76} + 32 q^{77} - 64 q^{79} + 16 q^{80} + 48 q^{81} - 20 q^{82} + 16 q^{85} - 40 q^{86} + 2 q^{90} - 4 q^{92} - 32 q^{94} - 2 q^{96} + 24 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
91.1 −1.41370 − 0.0379392i 1.00000i 1.99712 + 0.107270i 1.00000i 0.0379392 − 1.41370i −1.92200 −2.81927 − 0.227417i −1.00000 0.0379392 − 1.41370i
91.2 −1.41370 + 0.0379392i − 1.00000i 1.99712 − 0.107270i − 1.00000i 0.0379392 + 1.41370i −1.92200 −2.81927 + 0.227417i −1.00000 0.0379392 + 1.41370i
91.3 −1.39049 − 0.257970i − 1.00000i 1.86690 + 0.717408i − 1.00000i −0.257970 + 1.39049i 2.59830 −2.41083 − 1.47915i −1.00000 −0.257970 + 1.39049i
91.4 −1.39049 + 0.257970i 1.00000i 1.86690 − 0.717408i 1.00000i −0.257970 − 1.39049i 2.59830 −2.41083 + 1.47915i −1.00000 −0.257970 − 1.39049i
91.5 −1.32949 − 0.482147i 1.00000i 1.53507 + 1.28202i 1.00000i 0.482147 − 1.32949i −4.27008 −1.42273 − 2.44455i −1.00000 0.482147 − 1.32949i
91.6 −1.32949 + 0.482147i − 1.00000i 1.53507 − 1.28202i − 1.00000i 0.482147 + 1.32949i −4.27008 −1.42273 + 2.44455i −1.00000 0.482147 + 1.32949i
91.7 −1.25058 − 0.660333i − 1.00000i 1.12792 + 1.65160i − 1.00000i −0.660333 + 1.25058i −3.91467 −0.319949 − 2.81027i −1.00000 −0.660333 + 1.25058i
91.8 −1.25058 + 0.660333i 1.00000i 1.12792 − 1.65160i 1.00000i −0.660333 − 1.25058i −3.91467 −0.319949 + 2.81027i −1.00000 −0.660333 − 1.25058i
91.9 −1.18761 − 0.767836i − 1.00000i 0.820857 + 1.82379i − 1.00000i −0.767836 + 1.18761i 0.597446 0.425506 − 2.79624i −1.00000 −0.767836 + 1.18761i
91.10 −1.18761 + 0.767836i 1.00000i 0.820857 − 1.82379i 1.00000i −0.767836 − 1.18761i 0.597446 0.425506 + 2.79624i −1.00000 −0.767836 − 1.18761i
91.11 −1.17369 − 0.788958i 1.00000i 0.755090 + 1.85198i 1.00000i 0.788958 − 1.17369i −0.567108 0.574895 − 2.76939i −1.00000 0.788958 − 1.17369i
91.12 −1.17369 + 0.788958i − 1.00000i 0.755090 − 1.85198i − 1.00000i 0.788958 + 1.17369i −0.567108 0.574895 + 2.76939i −1.00000 0.788958 + 1.17369i
91.13 −1.01915 − 0.980473i 1.00000i 0.0773439 + 1.99850i 1.00000i 0.980473 − 1.01915i 2.01213 1.88065 − 2.11261i −1.00000 0.980473 − 1.01915i
91.14 −1.01915 + 0.980473i − 1.00000i 0.0773439 − 1.99850i − 1.00000i 0.980473 + 1.01915i 2.01213 1.88065 + 2.11261i −1.00000 0.980473 + 1.01915i
91.15 −0.952347 − 1.04548i − 1.00000i −0.186069 + 1.99133i − 1.00000i −1.04548 + 0.952347i 3.49678 2.25910 − 1.70190i −1.00000 −1.04548 + 0.952347i
91.16 −0.952347 + 1.04548i 1.00000i −0.186069 − 1.99133i 1.00000i −1.04548 − 0.952347i 3.49678 2.25910 + 1.70190i −1.00000 −1.04548 − 0.952347i
91.17 −0.698077 − 1.22991i − 1.00000i −1.02538 + 1.71715i − 1.00000i −1.22991 + 0.698077i 0.344556 2.82774 + 0.0624257i −1.00000 −1.22991 + 0.698077i
91.18 −0.698077 + 1.22991i 1.00000i −1.02538 − 1.71715i 1.00000i −1.22991 − 0.698077i 0.344556 2.82774 − 0.0624257i −1.00000 −1.22991 − 0.698077i
91.19 −0.651202 − 1.25536i 1.00000i −1.15187 + 1.63499i 1.00000i 1.25536 − 0.651202i 2.14037 2.80261 + 0.381312i −1.00000 1.25536 − 0.651202i
91.20 −0.651202 + 1.25536i − 1.00000i −1.15187 − 1.63499i − 1.00000i 1.25536 + 0.651202i 2.14037 2.80261 − 0.381312i −1.00000 1.25536 + 0.651202i
See all 48 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 91.48
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
92.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1380.2.p.a ✓ 48
4.b odd 2 1 1380.2.p.b yes 48
23.b odd 2 1 1380.2.p.b yes 48
92.b even 2 1 inner 1380.2.p.a ✓ 48
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1380.2.p.a ✓ 48 1.a even 1 1 trivial
1380.2.p.a ✓ 48 92.b even 2 1 inner
1380.2.p.b yes 48 4.b odd 2 1
1380.2.p.b yes 48 23.b odd 2 1