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:

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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