Properties

Label 1449.1.bq.b
Level $1449$
Weight $1$
Character orbit 1449.bq
Analytic conductor $0.723$
Analytic rank $0$
Dimension $20$
Projective image $D_{22}$
CM discriminant -7
Inner twists $8$

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Newspace parameters

Level: \( N \) \(=\) \( 1449 = 3^{2} \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1449.bq (of order \(22\), degree \(10\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.723145203305\)
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: \(\Z[a_1, \ldots, a_{23}]\)
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.

\(f(q)\) \(=\) \( q + ( -\zeta_{44}^{5} - \zeta_{44}^{7} ) q^{2} + ( \zeta_{44}^{10} + \zeta_{44}^{12} + \zeta_{44}^{14} ) q^{4} + \zeta_{44}^{2} q^{7} + ( -\zeta_{44}^{15} - \zeta_{44}^{17} - \zeta_{44}^{19} - \zeta_{44}^{21} ) q^{8} +O(q^{10})\) \( q + ( -\zeta_{44}^{5} - \zeta_{44}^{7} ) q^{2} + ( \zeta_{44}^{10} + \zeta_{44}^{12} + \zeta_{44}^{14} ) q^{4} + \zeta_{44}^{2} q^{7} + ( -\zeta_{44}^{15} - \zeta_{44}^{17} - \zeta_{44}^{19} - \zeta_{44}^{21} ) q^{8} + ( -\zeta_{44} + \zeta_{44}^{9} ) q^{11} + ( -\zeta_{44}^{7} - \zeta_{44}^{9} ) q^{14} + ( -1 - \zeta_{44}^{2} - \zeta_{44}^{4} - \zeta_{44}^{6} + \zeta_{44}^{20} ) q^{16} + ( \zeta_{44}^{6} + \zeta_{44}^{8} - \zeta_{44}^{14} - \zeta_{44}^{16} ) q^{22} + \zeta_{44}^{3} q^{23} -\zeta_{44}^{6} q^{25} + ( \zeta_{44}^{12} + \zeta_{44}^{14} + \zeta_{44}^{16} ) q^{28} + ( \zeta_{44}^{5} + \zeta_{44}^{15} ) q^{29} + ( \zeta_{44}^{3} + \zeta_{44}^{5} + \zeta_{44}^{7} + \zeta_{44}^{9} + \zeta_{44}^{11} + \zeta_{44}^{13} ) q^{32} + ( \zeta_{44}^{6} + \zeta_{44}^{10} ) q^{37} + ( -\zeta_{44}^{12} + \zeta_{44}^{18} ) q^{43} + ( -\zeta_{44} - \zeta_{44}^{11} - \zeta_{44}^{13} - \zeta_{44}^{15} + \zeta_{44}^{19} + \zeta_{44}^{21} ) q^{44} + ( -\zeta_{44}^{8} - \zeta_{44}^{10} ) q^{46} + \zeta_{44}^{4} q^{49} + ( \zeta_{44}^{11} + \zeta_{44}^{13} ) q^{50} + ( \zeta_{44}^{7} - \zeta_{44}^{19} ) q^{53} + ( \zeta_{44} - \zeta_{44}^{17} - \zeta_{44}^{19} - \zeta_{44}^{21} ) q^{56} + ( 1 - \zeta_{44}^{10} - \zeta_{44}^{12} - \zeta_{44}^{20} ) q^{58} + ( -\zeta_{44}^{8} - \zeta_{44}^{10} - \zeta_{44}^{12} - \zeta_{44}^{14} - \zeta_{44}^{16} - \zeta_{44}^{18} - \zeta_{44}^{20} ) q^{64} + ( \zeta_{44}^{16} - \zeta_{44}^{18} ) q^{67} + ( -\zeta_{44}^{13} + \zeta_{44}^{21} ) q^{71} + ( -\zeta_{44}^{11} - \zeta_{44}^{13} - \zeta_{44}^{15} - \zeta_{44}^{17} ) q^{74} + ( -\zeta_{44}^{3} + \zeta_{44}^{11} ) q^{77} + ( \zeta_{44}^{4} - \zeta_{44}^{14} ) q^{79} + ( \zeta_{44} + \zeta_{44}^{3} + \zeta_{44}^{17} + \zeta_{44}^{19} ) q^{86} + ( -1 + \zeta_{44}^{2} + \zeta_{44}^{4} + \zeta_{44}^{6} + \zeta_{44}^{8} + \zeta_{44}^{16} + \zeta_{44}^{18} + \zeta_{44}^{20} ) q^{88} + ( \zeta_{44}^{13} + \zeta_{44}^{15} + \zeta_{44}^{17} ) q^{92} + ( -\zeta_{44}^{9} - \zeta_{44}^{11} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20q + 2q^{4} + 2q^{7} + O(q^{10}) \) \( 20q + 2q^{4} + 2q^{7} - 24q^{16} - 2q^{25} - 2q^{28} + 4q^{37} + 4q^{43} - 2q^{49} + 22q^{58} + 2q^{64} - 4q^{67} - 4q^{79} - 22q^{88} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1449\mathbb{Z}\right)^\times\).

\(n\) \(442\) \(829\) \(1289\)
\(\chi(n)\) \(-\zeta_{44}^{6}\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
55.1
0.540641 + 0.841254i
−0.540641 0.841254i
0.281733 + 0.959493i
−0.281733 0.959493i
0.281733 0.959493i
−0.281733 + 0.959493i
−0.909632 + 0.415415i
0.909632 0.415415i
0.989821 + 0.142315i
−0.989821 0.142315i
0.540641 0.841254i
−0.540641 + 0.841254i
−0.909632 0.415415i
0.909632 + 0.415415i
−0.755750 + 0.654861i
0.755750 0.654861i
0.989821 0.142315i
−0.989821 + 0.142315i
−0.755750 0.654861i
0.755750 + 0.654861i
−1.03748 + 0.304632i 0 0.142315 0.0914602i 0 0 −0.415415 + 0.909632i 0.588302 0.678936i 0 0
55.2 1.03748 0.304632i 0 0.142315 0.0914602i 0 0 −0.415415 + 0.909632i −0.588302 + 0.678936i 0 0
118.1 −0.0801894 0.557730i 0 0.654861 0.192284i 0 0 −0.841254 + 0.540641i −0.393828 0.862362i 0 0
118.2 0.0801894 + 0.557730i 0 0.654861 0.192284i 0 0 −0.841254 + 0.540641i 0.393828 + 0.862362i 0 0
307.1 −0.0801894 + 0.557730i 0 0.654861 + 0.192284i 0 0 −0.841254 0.540641i −0.393828 + 0.862362i 0 0
307.2 0.0801894 0.557730i 0 0.654861 + 0.192284i 0 0 −0.841254 0.540641i 0.393828 0.862362i 0 0
370.1 −1.53046 0.983568i 0 0.959493 + 2.10100i 0 0 0.654861 0.755750i 0.339098 2.35848i 0 0
370.2 1.53046 + 0.983568i 0 0.959493 + 2.10100i 0 0 0.654861 0.755750i −0.339098 + 2.35848i 0 0
496.1 −1.29639 1.49611i 0 −0.415415 + 2.88927i 0 0 0.959493 + 0.281733i 3.19584 2.05384i 0 0
496.2 1.29639 + 1.49611i 0 −0.415415 + 2.88927i 0 0 0.959493 + 0.281733i −3.19584 + 2.05384i 0 0
685.1 −1.03748 0.304632i 0 0.142315 + 0.0914602i 0 0 −0.415415 0.909632i 0.588302 + 0.678936i 0 0
685.2 1.03748 + 0.304632i 0 0.142315 + 0.0914602i 0 0 −0.415415 0.909632i −0.588302 0.678936i 0 0
748.1 −1.53046 + 0.983568i 0 0.959493 2.10100i 0 0 0.654861 + 0.755750i 0.339098 + 2.35848i 0 0
748.2 1.53046 0.983568i 0 0.959493 2.10100i 0 0 0.654861 + 0.755750i −0.339098 2.35848i 0 0
811.1 −0.627899 + 1.37491i 0 −0.841254 0.970858i 0 0 0.142315 0.989821i 0.412791 0.121206i 0 0
811.2 0.627899 1.37491i 0 −0.841254 0.970858i 0 0 0.142315 0.989821i −0.412791 + 0.121206i 0 0
1189.1 −1.29639 + 1.49611i 0 −0.415415 2.88927i 0 0 0.959493 0.281733i 3.19584 + 2.05384i 0 0
1189.2 1.29639 1.49611i 0 −0.415415 2.88927i 0 0 0.959493 0.281733i −3.19584 2.05384i 0 0
1315.1 −0.627899 1.37491i 0 −0.841254 + 0.970858i 0 0 0.142315 + 0.989821i 0.412791 + 0.121206i 0 0
1315.2 0.627899 + 1.37491i 0 −0.841254 + 0.970858i 0 0 0.142315 + 0.989821i −0.412791 0.121206i 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1315.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 CM by \(\Q(\sqrt{-7}) \)
3.b odd 2 1 inner
21.c even 2 1 inner
23.c even 11 1 inner
69.h odd 22 1 inner
161.l odd 22 1 inner
483.v even 22 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1449.1.bq.b 20
3.b odd 2 1 inner 1449.1.bq.b 20
7.b odd 2 1 CM 1449.1.bq.b 20
21.c even 2 1 inner 1449.1.bq.b 20
23.c even 11 1 inner 1449.1.bq.b 20
69.h odd 22 1 inner 1449.1.bq.b 20
161.l odd 22 1 inner 1449.1.bq.b 20
483.v even 22 1 inner 1449.1.bq.b 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1449.1.bq.b 20 1.a even 1 1 trivial
1449.1.bq.b 20 3.b odd 2 1 inner
1449.1.bq.b 20 7.b odd 2 1 CM
1449.1.bq.b 20 21.c even 2 1 inner
1449.1.bq.b 20 23.c even 11 1 inner
1449.1.bq.b 20 69.h odd 22 1 inner
1449.1.bq.b 20 161.l odd 22 1 inner
1449.1.bq.b 20 483.v even 22 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(T_{2}^{20} + \cdots\) acting on \(S_{1}^{\mathrm{new}}(1449, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 121 + 605 T^{2} + 484 T^{4} - 968 T^{6} + 484 T^{8} + 99 T^{10} + 165 T^{12} + 22 T^{16} + T^{20} \)
$3$ \( T^{20} \)
$5$ \( T^{20} \)
$7$ \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} )^{2} \)
$11$ \( 121 + 484 T^{2} + 1210 T^{4} + 121 T^{6} + 605 T^{8} - 264 T^{10} + 55 T^{14} + T^{20} \)
$13$ \( T^{20} \)
$17$ \( T^{20} \)
$19$ \( T^{20} \)
$23$ \( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} - T^{14} + T^{16} - T^{18} + T^{20} \)
$29$ \( 121 - 121 T^{2} + 847 T^{4} + 1573 T^{6} + 1452 T^{8} + 462 T^{10} + 22 T^{12} + T^{20} \)
$31$ \( T^{20} \)
$37$ \( ( 1 - 6 T + 14 T^{2} - 7 T^{3} + 9 T^{4} + 12 T^{5} - 6 T^{6} + 3 T^{7} + 4 T^{8} - 2 T^{9} + T^{10} )^{2} \)
$41$ \( T^{20} \)
$43$ \( ( 1 + 5 T + 14 T^{2} + 4 T^{3} - 2 T^{4} + T^{5} + 5 T^{6} - 8 T^{7} + 4 T^{8} - 2 T^{9} + T^{10} )^{2} \)
$47$ \( T^{20} \)
$53$ \( 121 + 484 T^{2} + 1210 T^{4} + 121 T^{6} + 605 T^{8} - 264 T^{10} + 55 T^{14} + T^{20} \)
$59$ \( T^{20} \)
$61$ \( T^{20} \)
$67$ \( ( 1 - 5 T + 14 T^{2} - 4 T^{3} - 2 T^{4} - T^{5} + 5 T^{6} + 8 T^{7} + 4 T^{8} + 2 T^{9} + T^{10} )^{2} \)
$71$ \( 121 + 484 T^{2} + 1210 T^{4} + 121 T^{6} + 605 T^{8} - 264 T^{10} + 55 T^{14} + T^{20} \)
$73$ \( T^{20} \)
$79$ \( ( 1 + 6 T + 14 T^{2} + 7 T^{3} + 9 T^{4} - 12 T^{5} - 6 T^{6} - 3 T^{7} + 4 T^{8} + 2 T^{9} + T^{10} )^{2} \)
$83$ \( T^{20} \)
$89$ \( T^{20} \)
$97$ \( T^{20} \)
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