# Properties

 Label 1127.1.x.a Level $1127$ Weight $1$ Character orbit 1127.x Analytic conductor $0.562$ Analytic rank $0$ Dimension $20$ Projective image $D_{22}$ CM discriminant -7 Inner twists $8$

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## Newspace parameters

 Level: $$N$$ $$=$$ $$1127 = 7^{2} \cdot 23$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 1127.x (of order $$66$$, degree $$20$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.562446269237$$ Analytic rank: $$0$$ Dimension: $$20$$ Coefficient field: $$\Q(\zeta_{33})$$ Defining polynomial: $$x^{20} - x^{19} + x^{17} - x^{16} + x^{14} - x^{13} + x^{11} - x^{10} + x^{9} - x^{7} + x^{6} - x^{4} + x^{3} - x + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{4}]$$ 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.

 $$f(q)$$ $$=$$ $$q + ( \zeta_{66}^{5} + \zeta_{66}^{29} ) q^{2} + ( -\zeta_{66} + \zeta_{66}^{10} - \zeta_{66}^{25} ) q^{4} + ( -\zeta_{66}^{6} + \zeta_{66}^{15} + \zeta_{66}^{21} - \zeta_{66}^{30} ) q^{8} -\zeta_{66}^{8} q^{9} +O(q^{10})$$ $$q + ( \zeta_{66}^{5} + \zeta_{66}^{29} ) q^{2} + ( -\zeta_{66} + \zeta_{66}^{10} - \zeta_{66}^{25} ) q^{4} + ( -\zeta_{66}^{6} + \zeta_{66}^{15} + \zeta_{66}^{21} - \zeta_{66}^{30} ) q^{8} -\zeta_{66}^{8} q^{9} + ( \zeta_{66} - \zeta_{66}^{31} ) q^{11} + ( \zeta_{66}^{2} - \zeta_{66}^{11} - \zeta_{66}^{17} + \zeta_{66}^{20} + \zeta_{66}^{26} ) q^{16} + ( \zeta_{66}^{4} - \zeta_{66}^{13} ) q^{18} + ( \zeta_{66}^{3} + \zeta_{66}^{6} + \zeta_{66}^{27} + \zeta_{66}^{30} ) q^{22} + \zeta_{66}^{14} q^{23} + \zeta_{66}^{28} q^{25} + ( -\zeta_{66}^{15} - \zeta_{66}^{27} ) q^{29} + ( \zeta_{66}^{7} + \zeta_{66}^{13} - \zeta_{66}^{16} - \zeta_{66}^{22} + \zeta_{66}^{25} + \zeta_{66}^{31} ) q^{32} + ( -1 + \zeta_{66}^{9} - \zeta_{66}^{18} ) q^{36} + ( \zeta_{66}^{17} + \zeta_{66}^{32} ) q^{37} + ( -\zeta_{66}^{12} + \zeta_{66}^{18} ) q^{43} + ( -\zeta_{66}^{2} + \zeta_{66}^{8} + \zeta_{66}^{11} - \zeta_{66}^{23} - \zeta_{66}^{26} + \zeta_{66}^{32} ) q^{44} + ( -\zeta_{66}^{10} + \zeta_{66}^{19} ) q^{46} + ( -1 - \zeta_{66}^{24} ) q^{50} + ( \zeta_{66}^{7} - \zeta_{66}^{19} ) q^{53} + ( \zeta_{66}^{11} - \zeta_{66}^{20} + \zeta_{66}^{23} - \zeta_{66}^{32} ) q^{58} + ( -\zeta_{66}^{3} - \zeta_{66}^{9} + \zeta_{66}^{12} + \zeta_{66}^{18} - \zeta_{66}^{21} - \zeta_{66}^{27} + \zeta_{66}^{30} ) q^{64} + ( -\zeta_{66}^{7} - \zeta_{66}^{16} ) q^{67} + ( -\zeta_{66}^{21} + \zeta_{66}^{24} ) q^{71} + ( -\zeta_{66}^{5} + \zeta_{66}^{14} - \zeta_{66}^{23} - \zeta_{66}^{29} ) q^{72} + ( -\zeta_{66}^{4} - \zeta_{66}^{13} + \zeta_{66}^{22} - \zeta_{66}^{28} ) q^{74} + ( -\zeta_{66}^{14} + \zeta_{66}^{26} ) q^{79} + \zeta_{66}^{16} q^{81} + ( \zeta_{66}^{8} - \zeta_{66}^{14} - \zeta_{66}^{17} + \zeta_{66}^{23} ) q^{86} + ( -\zeta_{66}^{4} - \zeta_{66}^{7} + \zeta_{66}^{13} + \zeta_{66}^{16} + \zeta_{66}^{19} + \zeta_{66}^{22} - \zeta_{66}^{28} - \zeta_{66}^{31} ) q^{88} + ( \zeta_{66}^{6} - \zeta_{66}^{15} + \zeta_{66}^{24} ) q^{92} + ( -\zeta_{66}^{6} - \zeta_{66}^{9} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$20q - 2q^{2} + 3q^{4} + 8q^{8} - q^{9} + O(q^{10})$$ $$20q - 2q^{2} + 3q^{4} + 8q^{8} - q^{9} - 6q^{16} + 2q^{18} + q^{23} + q^{25} - 4q^{29} + 5q^{32} - 16q^{36} + 11q^{44} - 2q^{46} - 18q^{50} + 7q^{58} - 14q^{64} - 4q^{71} + 4q^{72} - 11q^{74} + q^{81} - 11q^{88} - 6q^{92} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1127\mathbb{Z}\right)^\times$$.

 $$n$$ $$346$$ $$442$$ $$\chi(n)$$ $$-\zeta_{66}^{11}$$ $$-\zeta_{66}^{6}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
30.1
 −0.786053 + 0.618159i −0.995472 − 0.0950560i 0.928368 − 0.371662i 0.928368 + 0.371662i 0.981929 + 0.189251i −0.786053 − 0.618159i 0.0475819 − 0.998867i −0.327068 − 0.945001i −0.995472 + 0.0950560i −0.888835 − 0.458227i −0.888835 + 0.458227i −0.327068 + 0.945001i 0.723734 + 0.690079i 0.580057 + 0.814576i 0.0475819 + 0.998867i 0.981929 − 0.189251i 0.235759 − 0.971812i 0.235759 + 0.971812i 0.580057 − 0.814576i 0.723734 − 0.690079i
−0.0930932 0.268975i 0 0.722372 0.568079i 0 0 0 −0.459493 0.295298i −0.580057 0.814576i 0
67.1 −0.0395325 + 0.829889i 0 0.308319 + 0.0294409i 0 0 0 −0.154861 + 1.07708i −0.723734 0.690079i 0
79.1 0.279486 0.0538665i 0 −0.853157 + 0.341553i 0 0 0 −0.459493 + 0.295298i 0.995472 + 0.0950560i 0
214.1 0.279486 + 0.0538665i 0 −0.853157 0.341553i 0 0 0 −0.459493 0.295298i 0.995472 0.0950560i 0
226.1 −1.30379 0.124497i 0 0.702443 + 0.135385i 0 0 0 0.357685 + 0.105026i −0.0475819 0.998867i 0
263.1 −0.0930932 + 0.268975i 0 0.722372 + 0.568079i 0 0 0 −0.459493 + 0.295298i −0.580057 + 0.814576i 0
373.1 −1.21769 + 1.16106i 0 0.0871144 1.82876i 0 0 0 0.915415 + 1.05645i −0.928368 0.371662i 0
410.1 0.759713 1.06687i 0 −0.233975 0.676026i 0 0 0 0.357685 + 0.105026i 0.888835 + 0.458227i 0
471.1 −0.0395325 0.829889i 0 0.308319 0.0294409i 0 0 0 −0.154861 1.07708i −0.723734 + 0.690079i 0
520.1 −0.396666 + 1.63508i 0 −1.62731 0.838935i 0 0 0 0.915415 1.05645i 0.786053 + 0.618159i 0
557.1 −0.396666 1.63508i 0 −1.62731 + 0.838935i 0 0 0 0.915415 + 1.05645i 0.786053 0.618159i 0
569.1 0.759713 + 1.06687i 0 −0.233975 + 0.676026i 0 0 0 0.357685 0.105026i 0.888835 0.458227i 0
618.1 1.78153 + 0.713215i 0 1.94142 + 1.85114i 0 0 0 1.34125 + 2.93694i −0.981929 + 0.189251i 0
655.1 0.738471 + 0.380708i 0 −0.179656 0.252292i 0 0 0 −0.154861 1.07708i −0.235759 0.971812i 0
704.1 −1.21769 1.16106i 0 0.0871144 + 1.82876i 0 0 0 0.915415 1.05645i −0.928368 + 0.371662i 0
753.1 −1.30379 + 0.124497i 0 0.702443 0.135385i 0 0 0 0.357685 0.105026i −0.0475819 + 0.998867i 0
802.1 −1.50842 + 1.18624i 0 0.632425 2.60689i 0 0 0 1.34125 + 2.93694i 0.327068 0.945001i 0
912.1 −1.50842 1.18624i 0 0.632425 + 2.60689i 0 0 0 1.34125 2.93694i 0.327068 + 0.945001i 0
1010.1 0.738471 0.380708i 0 −0.179656 + 0.252292i 0 0 0 −0.154861 + 1.07708i −0.235759 + 0.971812i 0
1096.1 1.78153 0.713215i 0 1.94142 1.85114i 0 0 0 1.34125 2.93694i −0.981929 0.189251i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1096.1 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 CM by $$\Q(\sqrt{-7})$$
7.c even 3 1 inner
7.d odd 6 1 inner
23.d odd 22 1 inner
161.k even 22 1 inner
161.o even 66 1 inner
161.p odd 66 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1127.1.x.a 20
7.b odd 2 1 CM 1127.1.x.a 20
7.c even 3 1 1127.1.o.a 10
7.c even 3 1 inner 1127.1.x.a 20
7.d odd 6 1 1127.1.o.a 10
7.d odd 6 1 inner 1127.1.x.a 20
23.d odd 22 1 inner 1127.1.x.a 20
161.k even 22 1 inner 1127.1.x.a 20
161.o even 66 1 1127.1.o.a 10
161.o even 66 1 inner 1127.1.x.a 20
161.p odd 66 1 1127.1.o.a 10
161.p odd 66 1 inner 1127.1.x.a 20

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1127.1.o.a 10 7.c even 3 1
1127.1.o.a 10 7.d odd 6 1
1127.1.o.a 10 161.o even 66 1
1127.1.o.a 10 161.p odd 66 1
1127.1.x.a 20 1.a even 1 1 trivial
1127.1.x.a 20 7.b odd 2 1 CM
1127.1.x.a 20 7.c even 3 1 inner
1127.1.x.a 20 7.d odd 6 1 inner
1127.1.x.a 20 23.d odd 22 1 inner
1127.1.x.a 20 161.k even 22 1 inner
1127.1.x.a 20 161.o even 66 1 inner
1127.1.x.a 20 161.p odd 66 1 inner

## Hecke kernels

This newform subspace is the entire newspace $$S_{1}^{\mathrm{new}}(1127, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - 5 T + 11 T^{2} - 62 T^{3} + 178 T^{4} - 77 T^{5} + 49 T^{6} + 19 T^{7} - 154 T^{8} + 130 T^{9} + 164 T^{10} + 49 T^{11} + 11 T^{12} + 40 T^{13} + 42 T^{14} + 11 T^{15} - 5 T^{16} - 8 T^{17} + 2 T^{19} + T^{20}$$
$3$ $$T^{20}$$
$5$ $$T^{20}$$
$7$ $$T^{20}$$
$11$ $$121 + 242 T + 484 T^{2} + 242 T^{3} - 121 T^{4} - 1452 T^{5} + 121 T^{6} - 242 T^{7} + 1573 T^{8} + 110 T^{10} - 770 T^{11} - 11 T^{13} + 187 T^{14} - 22 T^{17} + T^{20}$$
$13$ $$T^{20}$$
$17$ $$T^{20}$$
$19$ $$T^{20}$$
$23$ $$1 - T + T^{3} - T^{4} + T^{6} - T^{7} + T^{9} - T^{10} + T^{11} - T^{13} + T^{14} - T^{16} + T^{17} - T^{19} + T^{20}$$
$29$ $$( 1 - 5 T + 3 T^{2} + 7 T^{3} + 20 T^{4} + 10 T^{5} + 16 T^{6} + 8 T^{7} + 4 T^{8} + 2 T^{9} + T^{10} )^{2}$$
$31$ $$T^{20}$$
$37$ $$121 + 242 T + 484 T^{2} + 242 T^{3} - 121 T^{4} - 1452 T^{5} + 121 T^{6} - 242 T^{7} + 1573 T^{8} + 110 T^{10} - 770 T^{11} - 11 T^{13} + 187 T^{14} - 22 T^{17} + T^{20}$$
$41$ $$T^{20}$$
$43$ $$( 11 + 33 T + 22 T^{2} - 11 T^{5} + 11 T^{6} + T^{10} )^{2}$$
$47$ $$T^{20}$$
$53$ $$121 + 242 T + 484 T^{2} + 242 T^{3} - 121 T^{4} - 1452 T^{5} + 121 T^{6} - 242 T^{7} + 1573 T^{8} + 110 T^{10} - 770 T^{11} - 11 T^{13} + 187 T^{14} - 22 T^{17} + T^{20}$$
$59$ $$T^{20}$$
$61$ $$T^{20}$$
$67$ $$121 + 363 T + 847 T^{2} + 726 T^{3} + 484 T^{4} - 121 T^{5} + 605 T^{6} + 847 T^{7} - 242 T^{8} + 110 T^{10} - 187 T^{11} + 99 T^{12} - 11 T^{15} - 11 T^{16} + T^{20}$$
$71$ $$( 1 + 6 T + 14 T^{2} + 7 T^{3} + 9 T^{4} - 12 T^{5} - 6 T^{6} - 3 T^{7} + 4 T^{8} + 2 T^{9} + T^{10} )^{2}$$
$73$ $$T^{20}$$
$79$ $$121 - 242 T + 484 T^{2} - 242 T^{3} - 121 T^{4} + 1452 T^{5} + 121 T^{6} + 242 T^{7} + 1573 T^{8} + 110 T^{10} + 770 T^{11} + 11 T^{13} + 187 T^{14} + 22 T^{17} + T^{20}$$
$83$ $$T^{20}$$
$89$ $$T^{20}$$
$97$ $$T^{20}$$
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