# 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 $2$

# Related objects

## Newspace parameters

 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

 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.

 $$\operatorname{Tr}(f)(q) =$$ $$48q - 4q^{2} - 2q^{4} - 2q^{6} - 4q^{8} - 48q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$48q - 4q^{2} - 2q^{4} - 2q^{6} - 4q^{8} - 48q^{9} - 2q^{10} + 20q^{14} - 48q^{15} - 6q^{16} + 4q^{18} + 16q^{19} + 28q^{22} + 4q^{23} + 2q^{24} - 48q^{25} - 20q^{26} + 32q^{29} + 4q^{30} + 16q^{32} - 28q^{34} + 2q^{36} + 2q^{40} - 8q^{41} + 14q^{46} + 16q^{48} + 40q^{49} + 4q^{50} + 16q^{51} - 16q^{52} + 2q^{54} + 40q^{56} - 8q^{58} + 2q^{60} + 24q^{62} - 26q^{64} - 48q^{67} - 44q^{68} - 8q^{69} + 4q^{72} + 20q^{74} - 64q^{76} + 32q^{77} - 64q^{79} + 16q^{80} + 48q^{81} - 20q^{82} + 16q^{85} - 40q^{86} + 2q^{90} - 4q^{92} - 32q^{94} - 2q^{96} + 24q^{98} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
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: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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