# Properties

 Label 1380.1.bn.d Level $1380$ Weight $1$ Character orbit 1380.bn Analytic conductor $0.689$ Analytic rank $0$ Dimension $20$ Projective image $D_{22}$ CM discriminant -20 Inner twists $8$

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## Newspace parameters

 Level: $$N$$ $$=$$ $$1380 = 2^{2} \cdot 3 \cdot 5 \cdot 23$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 1380.bn (of order $$22$$, degree $$10$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.688709717434$$ Analytic rank: $$0$$ Dimension: $$20$$ Relative dimension: $$2$$ over $$\Q(\zeta_{22})$$ Coefficient field: $$\Q(\zeta_{44})$$ Defining polynomial: $$x^{20} - x^{18} + x^{16} - x^{14} + x^{12} - x^{10} + x^{8} - x^{6} + x^{4} - x^{2} + 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image: $$D_{22}$$ Projective field: Galois closure of $$\mathbb{Q}[x]/(x^{22} - \cdots)$$

## $q$-expansion

The $$q$$-expansion and trace form are shown below.

 $$f(q)$$ $$=$$ $$q + \zeta_{44}^{21} q^{2} -\zeta_{44}^{15} q^{3} -\zeta_{44}^{20} q^{4} -\zeta_{44}^{16} q^{5} + \zeta_{44}^{14} q^{6} + ( \zeta_{44}^{3} + \zeta_{44}^{11} ) q^{7} + \zeta_{44}^{19} q^{8} -\zeta_{44}^{8} q^{9} +O(q^{10})$$ $$q + \zeta_{44}^{21} q^{2} -\zeta_{44}^{15} q^{3} -\zeta_{44}^{20} q^{4} -\zeta_{44}^{16} q^{5} + \zeta_{44}^{14} q^{6} + ( \zeta_{44}^{3} + \zeta_{44}^{11} ) q^{7} + \zeta_{44}^{19} q^{8} -\zeta_{44}^{8} q^{9} + \zeta_{44}^{15} q^{10} -\zeta_{44}^{13} q^{12} + ( -\zeta_{44}^{2} - \zeta_{44}^{10} ) q^{14} -\zeta_{44}^{9} q^{15} -\zeta_{44}^{18} q^{16} + \zeta_{44}^{7} q^{18} -\zeta_{44}^{14} q^{20} + ( \zeta_{44}^{4} - \zeta_{44}^{18} ) q^{21} -\zeta_{44}^{21} q^{23} + \zeta_{44}^{12} q^{24} -\zeta_{44}^{10} q^{25} -\zeta_{44} q^{27} + ( \zeta_{44} + \zeta_{44}^{9} ) q^{28} + ( \zeta_{44}^{8} + \zeta_{44}^{18} ) q^{29} + \zeta_{44}^{8} q^{30} + \zeta_{44}^{17} q^{32} + ( \zeta_{44}^{5} - \zeta_{44}^{19} ) q^{35} -\zeta_{44}^{6} q^{36} + \zeta_{44}^{13} q^{40} + ( \zeta_{44}^{12} - \zeta_{44}^{20} ) q^{41} + ( -\zeta_{44}^{3} + \zeta_{44}^{17} ) q^{42} + ( -\zeta_{44} - \zeta_{44}^{5} ) q^{43} -\zeta_{44}^{2} q^{45} + \zeta_{44}^{20} q^{46} + ( -\zeta_{44}^{5} - \zeta_{44}^{17} ) q^{47} -\zeta_{44}^{11} q^{48} + ( -1 + \zeta_{44}^{6} + \zeta_{44}^{14} ) q^{49} + \zeta_{44}^{9} q^{50} + q^{54} + ( -1 - \zeta_{44}^{8} ) q^{56} + ( -\zeta_{44}^{7} - \zeta_{44}^{17} ) q^{58} -\zeta_{44}^{7} q^{60} + ( -\zeta_{44}^{6} + \zeta_{44}^{10} ) q^{61} + ( -\zeta_{44}^{11} - \zeta_{44}^{19} ) q^{63} -\zeta_{44}^{16} q^{64} + ( -\zeta_{44}^{7} + \zeta_{44}^{13} ) q^{67} -\zeta_{44}^{14} q^{69} + ( -\zeta_{44}^{4} + \zeta_{44}^{18} ) q^{70} + \zeta_{44}^{5} q^{72} -\zeta_{44}^{3} q^{75} -\zeta_{44}^{12} q^{80} + \zeta_{44}^{16} q^{81} + ( -\zeta_{44}^{11} + \zeta_{44}^{19} ) q^{82} + ( -\zeta_{44}^{3} - \zeta_{44}^{9} ) q^{83} + ( \zeta_{44}^{2} - \zeta_{44}^{16} ) q^{84} + ( 1 + \zeta_{44}^{4} ) q^{86} + ( \zeta_{44} + \zeta_{44}^{11} ) q^{87} + ( \zeta_{44}^{2} - \zeta_{44}^{4} ) q^{89} + \zeta_{44} q^{90} -\zeta_{44}^{19} q^{92} + ( \zeta_{44}^{4} + \zeta_{44}^{16} ) q^{94} + \zeta_{44}^{10} q^{96} + ( -\zeta_{44}^{5} - \zeta_{44}^{13} - \zeta_{44}^{21} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$20q + 2q^{4} + 2q^{5} + 2q^{6} + 2q^{9} + O(q^{10})$$ $$20q + 2q^{4} + 2q^{5} + 2q^{6} + 2q^{9} - 4q^{14} - 2q^{16} - 2q^{20} - 4q^{21} - 2q^{24} - 2q^{25} - 2q^{30} - 2q^{36} - 2q^{45} - 2q^{46} - 16q^{49} + 20q^{54} - 18q^{56} + 2q^{64} - 2q^{69} + 4q^{70} + 2q^{80} - 2q^{81} + 4q^{84} + 18q^{86} + 4q^{89} - 4q^{94} + 2q^{96} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1380\mathbb{Z}\right)^\times$$.

 $$n$$ $$277$$ $$461$$ $$691$$ $$1201$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$-1$$ $$\zeta_{44}^{10}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
359.1
 0.281733 − 0.959493i −0.281733 + 0.959493i 0.281733 + 0.959493i −0.281733 − 0.959493i 0.909632 + 0.415415i −0.909632 − 0.415415i 0.755750 + 0.654861i −0.755750 − 0.654861i 0.989821 − 0.142315i −0.989821 + 0.142315i 0.989821 + 0.142315i −0.989821 − 0.142315i 0.540641 − 0.841254i −0.540641 + 0.841254i 0.755750 − 0.654861i −0.755750 + 0.654861i 0.540641 + 0.841254i −0.540641 − 0.841254i 0.909632 − 0.415415i −0.909632 + 0.415415i
−0.281733 0.959493i −0.909632 + 0.415415i −0.841254 + 0.540641i 0.142315 + 0.989821i 0.654861 + 0.755750i −0.755750 0.345139i 0.755750 + 0.654861i 0.654861 0.755750i 0.909632 0.415415i
359.2 0.281733 + 0.959493i 0.909632 0.415415i −0.841254 + 0.540641i 0.142315 + 0.989821i 0.654861 + 0.755750i 0.755750 + 0.345139i −0.755750 0.654861i 0.654861 0.755750i −0.909632 + 0.415415i
419.1 −0.281733 + 0.959493i −0.909632 0.415415i −0.841254 0.540641i 0.142315 0.989821i 0.654861 0.755750i −0.755750 + 0.345139i 0.755750 0.654861i 0.654861 + 0.755750i 0.909632 + 0.415415i
419.2 0.281733 0.959493i 0.909632 + 0.415415i −0.841254 0.540641i 0.142315 0.989821i 0.654861 0.755750i 0.755750 0.345139i −0.755750 + 0.654861i 0.654861 + 0.755750i −0.909632 0.415415i
479.1 −0.909632 + 0.415415i −0.989821 0.142315i 0.654861 0.755750i −0.841254 0.540641i 0.959493 0.281733i 0.281733 0.0405070i −0.281733 + 0.959493i 0.959493 + 0.281733i 0.989821 + 0.142315i
479.2 0.909632 0.415415i 0.989821 + 0.142315i 0.654861 0.755750i −0.841254 0.540641i 0.959493 0.281733i −0.281733 + 0.0405070i 0.281733 0.959493i 0.959493 + 0.281733i −0.989821 0.142315i
539.1 −0.755750 + 0.654861i 0.281733 + 0.959493i 0.142315 0.989821i −0.415415 + 0.909632i −0.841254 0.540641i −0.540641 + 1.84125i 0.540641 + 0.841254i −0.841254 + 0.540641i −0.281733 0.959493i
539.2 0.755750 0.654861i −0.281733 0.959493i 0.142315 0.989821i −0.415415 + 0.909632i −0.841254 0.540641i 0.540641 1.84125i −0.540641 0.841254i −0.841254 + 0.540641i 0.281733 + 0.959493i
659.1 −0.989821 0.142315i 0.540641 + 0.841254i 0.959493 + 0.281733i 0.654861 + 0.755750i −0.415415 0.909632i 0.909632 1.41542i −0.909632 0.415415i −0.415415 + 0.909632i −0.540641 0.841254i
659.2 0.989821 + 0.142315i −0.540641 0.841254i 0.959493 + 0.281733i 0.654861 + 0.755750i −0.415415 0.909632i −0.909632 + 1.41542i 0.909632 + 0.415415i −0.415415 + 0.909632i 0.540641 + 0.841254i
779.1 −0.989821 + 0.142315i 0.540641 0.841254i 0.959493 0.281733i 0.654861 0.755750i −0.415415 + 0.909632i 0.909632 + 1.41542i −0.909632 + 0.415415i −0.415415 0.909632i −0.540641 + 0.841254i
779.2 0.989821 0.142315i −0.540641 + 0.841254i 0.959493 0.281733i 0.654861 0.755750i −0.415415 + 0.909632i −0.909632 1.41542i 0.909632 0.415415i −0.415415 0.909632i 0.540641 0.841254i
839.1 −0.540641 0.841254i 0.755750 + 0.654861i −0.415415 + 0.909632i 0.959493 0.281733i 0.142315 0.989821i −0.989821 + 0.857685i 0.989821 0.142315i 0.142315 + 0.989821i −0.755750 0.654861i
839.2 0.540641 + 0.841254i −0.755750 0.654861i −0.415415 + 0.909632i 0.959493 0.281733i 0.142315 0.989821i 0.989821 0.857685i −0.989821 + 0.142315i 0.142315 + 0.989821i 0.755750 + 0.654861i
1019.1 −0.755750 0.654861i 0.281733 0.959493i 0.142315 + 0.989821i −0.415415 0.909632i −0.841254 + 0.540641i −0.540641 1.84125i 0.540641 0.841254i −0.841254 0.540641i −0.281733 + 0.959493i
1019.2 0.755750 + 0.654861i −0.281733 + 0.959493i 0.142315 + 0.989821i −0.415415 0.909632i −0.841254 + 0.540641i 0.540641 + 1.84125i −0.540641 + 0.841254i −0.841254 0.540641i 0.281733 0.959493i
1079.1 −0.540641 + 0.841254i 0.755750 0.654861i −0.415415 0.909632i 0.959493 + 0.281733i 0.142315 + 0.989821i −0.989821 0.857685i 0.989821 + 0.142315i 0.142315 0.989821i −0.755750 + 0.654861i
1079.2 0.540641 0.841254i −0.755750 + 0.654861i −0.415415 0.909632i 0.959493 + 0.281733i 0.142315 + 0.989821i 0.989821 + 0.857685i −0.989821 0.142315i 0.142315 0.989821i 0.755750 0.654861i
1259.1 −0.909632 0.415415i −0.989821 + 0.142315i 0.654861 + 0.755750i −0.841254 + 0.540641i 0.959493 + 0.281733i 0.281733 + 0.0405070i −0.281733 0.959493i 0.959493 0.281733i 0.989821 0.142315i
1259.2 0.909632 + 0.415415i 0.989821 0.142315i 0.654861 + 0.755750i −0.841254 + 0.540641i 0.959493 + 0.281733i −0.281733 0.0405070i 0.281733 + 0.959493i 0.959493 0.281733i −0.989821 + 0.142315i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1259.2 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
20.d odd 2 1 CM by $$\Q(\sqrt{-5})$$
4.b odd 2 1 inner
5.b even 2 1 inner
69.g even 22 1 inner
276.j odd 22 1 inner
345.n even 22 1 inner
1380.bn odd 22 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1380.1.bn.d yes 20
3.b odd 2 1 1380.1.bn.a 20
4.b odd 2 1 inner 1380.1.bn.d yes 20
5.b even 2 1 inner 1380.1.bn.d yes 20
12.b even 2 1 1380.1.bn.a 20
15.d odd 2 1 1380.1.bn.a 20
20.d odd 2 1 CM 1380.1.bn.d yes 20
23.d odd 22 1 1380.1.bn.a 20
60.h even 2 1 1380.1.bn.a 20
69.g even 22 1 inner 1380.1.bn.d yes 20
92.h even 22 1 1380.1.bn.a 20
115.i odd 22 1 1380.1.bn.a 20
276.j odd 22 1 inner 1380.1.bn.d yes 20
345.n even 22 1 inner 1380.1.bn.d yes 20
460.o even 22 1 1380.1.bn.a 20
1380.bn odd 22 1 inner 1380.1.bn.d yes 20

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1380.1.bn.a 20 3.b odd 2 1
1380.1.bn.a 20 12.b even 2 1
1380.1.bn.a 20 15.d odd 2 1
1380.1.bn.a 20 23.d odd 22 1
1380.1.bn.a 20 60.h even 2 1
1380.1.bn.a 20 92.h even 22 1
1380.1.bn.a 20 115.i odd 22 1
1380.1.bn.a 20 460.o even 22 1
1380.1.bn.d yes 20 1.a even 1 1 trivial
1380.1.bn.d yes 20 4.b odd 2 1 inner
1380.1.bn.d yes 20 5.b even 2 1 inner
1380.1.bn.d yes 20 20.d odd 2 1 CM
1380.1.bn.d yes 20 69.g even 22 1 inner
1380.1.bn.d yes 20 276.j odd 22 1 inner
1380.1.bn.d yes 20 345.n even 22 1 inner
1380.1.bn.d yes 20 1380.bn odd 22 1 inner

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{1}^{\mathrm{new}}(1380, [\chi])$$:

 $$T_{7}^{20} + \cdots$$ $$T_{19}$$ $$T_{29}^{10} - 11 T_{29}^{7} + 33 T_{29}^{4} + 11 T_{29}^{3} - 22 T_{29} + 11$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} - T^{14} + T^{16} - T^{18} + T^{20}$$
$3$ $$1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} - T^{14} + T^{16} - T^{18} + T^{20}$$
$5$ $$( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} )^{2}$$
$7$ $$1 - 25 T^{2} + 185 T^{4} - 236 T^{6} + 224 T^{8} + 54 T^{10} + 102 T^{12} + 57 T^{14} + 27 T^{16} + 7 T^{18} + T^{20}$$
$11$ $$T^{20}$$
$13$ $$T^{20}$$
$17$ $$T^{20}$$
$19$ $$T^{20}$$
$23$ $$1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} - T^{14} + T^{16} - T^{18} + T^{20}$$
$29$ $$( 11 - 22 T + 11 T^{3} + 33 T^{4} - 11 T^{7} + T^{10} )^{2}$$
$31$ $$T^{20}$$
$37$ $$T^{20}$$
$41$ $$( 11 + 22 T - 11 T^{3} + 33 T^{4} + 11 T^{7} + T^{10} )^{2}$$
$43$ $$1 + 8 T^{2} + 130 T^{4} - 335 T^{6} + 125 T^{8} + 120 T^{10} + 36 T^{12} - 9 T^{14} + 16 T^{16} - 4 T^{18} + T^{20}$$
$47$ $$( 1 + 15 T^{2} + 35 T^{4} + 28 T^{6} + 9 T^{8} + T^{10} )^{2}$$
$53$ $$T^{20}$$
$59$ $$T^{20}$$
$61$ $$( 11 + 22 T - 11 T^{3} + 33 T^{4} + 11 T^{7} + T^{10} )^{2}$$
$67$ $$1 + 8 T^{2} + 130 T^{4} - 335 T^{6} + 125 T^{8} + 120 T^{10} + 36 T^{12} - 9 T^{14} + 16 T^{16} - 4 T^{18} + T^{20}$$
$71$ $$T^{20}$$
$73$ $$T^{20}$$
$79$ $$T^{20}$$
$83$ $$121 - 121 T^{2} + 847 T^{4} + 1573 T^{6} + 1452 T^{8} + 462 T^{10} + 22 T^{12} + T^{20}$$
$89$ $$( 1 - 6 T + 25 T^{2} - 51 T^{3} + 53 T^{4} - 32 T^{5} + 16 T^{6} - 8 T^{7} + 4 T^{8} - 2 T^{9} + T^{10} )^{2}$$
$97$ $$T^{20}$$
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