Properties

Label 1380.2.n.a
Level $1380$
Weight $2$
Character orbit 1380.n
Analytic conductor $11.019$
Analytic rank $0$
Dimension $48$
CM no
Inner twists $8$

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Newspace parameters

Level: \( N \) \(=\) \( 1380 = 2^{2} \cdot 3 \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1380.n (of order \(2\), degree \(1\), minimal)

Newform invariants

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.

\(\operatorname{Tr}(f)(q) = \) \( 48q + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 48q + 16q^{31} + 24q^{39} + 112q^{49} + 8q^{55} - 16q^{69} + 20q^{75} - 8q^{81} + 32q^{85} + O(q^{100}) \)

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.

Label \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
689.1 0 −1.71333 0.253996i 0 −0.923440 2.03648i 0 −2.63922 0 2.87097 + 0.870355i 0
689.2 0 −1.71333 0.253996i 0 0.923440 + 2.03648i 0 2.63922 0 2.87097 + 0.870355i 0
689.3 0 −1.71333 + 0.253996i 0 −0.923440 + 2.03648i 0 −2.63922 0 2.87097 0.870355i 0
689.4 0 −1.71333 + 0.253996i 0 0.923440 2.03648i 0 2.63922 0 2.87097 0.870355i 0
689.5 0 −1.63297 0.577419i 0 −2.15297 + 0.603919i 0 1.63712 0 2.33317 + 1.88581i 0
689.6 0 −1.63297 0.577419i 0 2.15297 0.603919i 0 −1.63712 0 2.33317 + 1.88581i 0
689.7 0 −1.63297 + 0.577419i 0 −2.15297 0.603919i 0 1.63712 0 2.33317 1.88581i 0
689.8 0 −1.63297 + 0.577419i 0 2.15297 + 0.603919i 0 −1.63712 0 2.33317 1.88581i 0
689.9 0 −1.35581 1.07786i 0 −1.70581 1.44575i 0 −5.00303 0 0.676420 + 2.92275i 0
689.10 0 −1.35581 1.07786i 0 1.70581 + 1.44575i 0 5.00303 0 0.676420 + 2.92275i 0
689.11 0 −1.35581 + 1.07786i 0 −1.70581 + 1.44575i 0 −5.00303 0 0.676420 2.92275i 0
689.12 0 −1.35581 + 1.07786i 0 1.70581 1.44575i 0 5.00303 0 0.676420 2.92275i 0
689.13 0 −0.888021 1.48708i 0 −1.29815 + 1.82066i 0 0.470527 0 −1.42284 + 2.64112i 0
689.14 0 −0.888021 1.48708i 0 1.29815 1.82066i 0 −0.470527 0 −1.42284 + 2.64112i 0
689.15 0 −0.888021 + 1.48708i 0 −1.29815 1.82066i 0 0.470527 0 −1.42284 2.64112i 0
689.16 0 −0.888021 + 1.48708i 0 1.29815 + 1.82066i 0 −0.470527 0 −1.42284 2.64112i 0
689.17 0 −0.779801 1.54658i 0 −0.268100 2.21994i 0 2.49920 0 −1.78382 + 2.41205i 0
689.18 0 −0.779801 1.54658i 0 0.268100 + 2.21994i 0 −2.49920 0 −1.78382 + 2.41205i 0
689.19 0 −0.779801 + 1.54658i 0 −0.268100 + 2.21994i 0 2.49920 0 −1.78382 2.41205i 0
689.20 0 −0.779801 + 1.54658i 0 0.268100 2.21994i 0 −2.49920 0 −1.78382 2.41205i 0
See all 48 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 689.48
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.b even 2 1 inner
15.d odd 2 1 inner
23.b odd 2 1 inner
69.c even 2 1 inner
115.c odd 2 1 inner
345.h 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.n.a 48
3.b odd 2 1 inner 1380.2.n.a 48
5.b even 2 1 inner 1380.2.n.a 48
15.d odd 2 1 inner 1380.2.n.a 48
23.b odd 2 1 inner 1380.2.n.a 48
69.c even 2 1 inner 1380.2.n.a 48
115.c odd 2 1 inner 1380.2.n.a 48
345.h even 2 1 inner 1380.2.n.a 48
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1380.2.n.a 48 1.a even 1 1 trivial
1380.2.n.a 48 3.b odd 2 1 inner
1380.2.n.a 48 5.b even 2 1 inner
1380.2.n.a 48 15.d odd 2 1 inner
1380.2.n.a 48 23.b odd 2 1 inner
1380.2.n.a 48 69.c even 2 1 inner
1380.2.n.a 48 115.c odd 2 1 inner
1380.2.n.a 48 345.h even 2 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(1380, [\chi])\).