Properties

Label 1840.1.bq.a
Level $1840$
Weight $1$
Character orbit 1840.bq
Analytic conductor $0.918$
Analytic rank $0$
Dimension $20$
Projective image $D_{22}$
CM discriminant -20
Inner twists $8$

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Newspace parameters

Level: \( N \) \(=\) \( 1840 = 2^{4} \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1840.bq (of order \(22\), degree \(10\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.918279623245\)
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.

\(f(q)\) \(=\) \( q + ( -\zeta_{44}^{7} - \zeta_{44}^{9} ) q^{3} + \zeta_{44}^{6} q^{5} + ( \zeta_{44}^{11} - \zeta_{44}^{19} ) q^{7} + ( \zeta_{44}^{14} + \zeta_{44}^{16} + \zeta_{44}^{18} ) q^{9} +O(q^{10})\) \( q + ( -\zeta_{44}^{7} - \zeta_{44}^{9} ) q^{3} + \zeta_{44}^{6} q^{5} + ( \zeta_{44}^{11} - \zeta_{44}^{19} ) q^{7} + ( \zeta_{44}^{14} + \zeta_{44}^{16} + \zeta_{44}^{18} ) q^{9} + ( -\zeta_{44}^{13} - \zeta_{44}^{15} ) q^{15} + ( -\zeta_{44}^{4} - \zeta_{44}^{6} - \zeta_{44}^{18} - \zeta_{44}^{20} ) q^{21} + \zeta_{44} q^{23} + \zeta_{44}^{12} q^{25} + ( \zeta_{44} + \zeta_{44}^{3} + \zeta_{44}^{5} - \zeta_{44}^{21} ) q^{27} + ( \zeta_{44}^{4} - \zeta_{44}^{14} ) q^{29} + ( \zeta_{44}^{3} + \zeta_{44}^{17} ) q^{35} + ( \zeta_{44}^{2} + \zeta_{44}^{10} ) q^{41} + ( -\zeta_{44}^{17} + \zeta_{44}^{21} ) q^{43} + ( -1 - \zeta_{44}^{2} + \zeta_{44}^{20} ) q^{45} + ( -\zeta_{44}^{5} + \zeta_{44}^{17} ) q^{47} + ( -1 + \zeta_{44}^{8} - \zeta_{44}^{16} ) q^{49} + ( -\zeta_{44}^{12} - \zeta_{44}^{16} ) q^{61} + ( -\zeta_{44}^{3} - \zeta_{44}^{5} - \zeta_{44}^{7} + \zeta_{44}^{11} + \zeta_{44}^{13} + \zeta_{44}^{15} ) q^{63} + ( -\zeta_{44}^{9} - \zeta_{44}^{15} ) q^{67} + ( -\zeta_{44}^{8} - \zeta_{44}^{10} ) q^{69} + ( -\zeta_{44}^{19} - \zeta_{44}^{21} ) q^{75} + ( -\zeta_{44}^{6} - \zeta_{44}^{8} - \zeta_{44}^{10} - \zeta_{44}^{12} - \zeta_{44}^{14} ) q^{81} + ( \zeta_{44}^{13} + \zeta_{44}^{19} ) q^{83} + ( -\zeta_{44} - \zeta_{44}^{11} - \zeta_{44}^{13} + \zeta_{44}^{21} ) q^{87} + ( -\zeta_{44}^{18} + \zeta_{44}^{20} ) q^{89} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20q + 2q^{5} + 2q^{9} + O(q^{10}) \) \( 20q + 2q^{5} + 2q^{9} - 2q^{25} - 4q^{29} + 4q^{41} - 24q^{45} - 20q^{49} + 4q^{61} - 2q^{81} - 4q^{89} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1840\mathbb{Z}\right)^\times\).

\(n\) \(737\) \(1151\) \(1201\) \(1381\)
\(\chi(n)\) \(-1\) \(-1\) \(\zeta_{44}^{12}\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
239.1
−0.281733 0.959493i
0.281733 + 0.959493i
0.989821 + 0.142315i
−0.989821 0.142315i
−0.281733 + 0.959493i
0.281733 0.959493i
−0.909632 0.415415i
0.909632 + 0.415415i
0.755750 + 0.654861i
−0.755750 0.654861i
−0.909632 + 0.415415i
0.909632 0.415415i
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 −0.368991 0.425839i 0 0.142315 + 0.989821i 0 0.755750 1.65486i 0 0.0971309 0.675560i 0
239.2 0 0.368991 + 0.425839i 0 0.142315 + 0.989821i 0 −0.755750 + 1.65486i 0 0.0971309 0.675560i 0
399.1 0 −0.822373 1.80075i 0 0.654861 + 0.755750i 0 0.909632 + 0.584585i 0 −1.91153 + 2.20602i 0
399.2 0 0.822373 + 1.80075i 0 0.654861 + 0.755750i 0 −0.909632 0.584585i 0 −1.91153 + 2.20602i 0
639.1 0 −0.368991 + 0.425839i 0 0.142315 0.989821i 0 0.755750 + 1.65486i 0 0.0971309 + 0.675560i 0
639.2 0 0.368991 0.425839i 0 0.142315 0.989821i 0 −0.755750 1.65486i 0 0.0971309 + 0.675560i 0
719.1 0 −1.74557 0.512546i 0 −0.841254 + 0.540641i 0 −0.281733 + 1.95949i 0 1.94306 + 1.24873i 0
719.2 0 1.74557 + 0.512546i 0 −0.841254 + 0.540641i 0 0.281733 1.95949i 0 1.94306 + 1.24873i 0
959.1 0 −1.27155 + 0.817178i 0 −0.415415 0.909632i 0 −0.540641 + 0.158746i 0 0.533654 1.16854i 0
959.2 0 1.27155 0.817178i 0 −0.415415 0.909632i 0 0.540641 0.158746i 0 0.533654 1.16854i 0
1039.1 0 −1.74557 + 0.512546i 0 −0.841254 0.540641i 0 −0.281733 1.95949i 0 1.94306 1.24873i 0
1039.2 0 1.74557 0.512546i 0 −0.841254 0.540641i 0 0.281733 + 1.95949i 0 1.94306 1.24873i 0
1199.1 0 −0.822373 + 1.80075i 0 0.654861 0.755750i 0 0.909632 0.584585i 0 −1.91153 2.20602i 0
1199.2 0 0.822373 1.80075i 0 0.654861 0.755750i 0 −0.909632 + 0.584585i 0 −1.91153 2.20602i 0
1359.1 0 −0.153882 1.07028i 0 0.959493 + 0.281733i 0 0.989821 1.14231i 0 −0.162317 + 0.0476607i 0
1359.2 0 0.153882 + 1.07028i 0 0.959493 + 0.281733i 0 −0.989821 + 1.14231i 0 −0.162317 + 0.0476607i 0
1439.1 0 −1.27155 0.817178i 0 −0.415415 + 0.909632i 0 −0.540641 0.158746i 0 0.533654 + 1.16854i 0
1439.2 0 1.27155 + 0.817178i 0 −0.415415 + 0.909632i 0 0.540641 + 0.158746i 0 0.533654 + 1.16854i 0
1599.1 0 −0.153882 + 1.07028i 0 0.959493 0.281733i 0 0.989821 + 1.14231i 0 −0.162317 0.0476607i 0
1599.2 0 0.153882 1.07028i 0 0.959493 0.281733i 0 −0.989821 1.14231i 0 −0.162317 0.0476607i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1599.2
Significant digits:
Format:

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
23.c even 11 1 inner
92.g odd 22 1 inner
115.j even 22 1 inner
460.n odd 22 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1840.1.bq.a 20
4.b odd 2 1 inner 1840.1.bq.a 20
5.b even 2 1 inner 1840.1.bq.a 20
20.d odd 2 1 CM 1840.1.bq.a 20
23.c even 11 1 inner 1840.1.bq.a 20
92.g odd 22 1 inner 1840.1.bq.a 20
115.j even 22 1 inner 1840.1.bq.a 20
460.n odd 22 1 inner 1840.1.bq.a 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1840.1.bq.a 20 1.a even 1 1 trivial
1840.1.bq.a 20 4.b odd 2 1 inner
1840.1.bq.a 20 5.b even 2 1 inner
1840.1.bq.a 20 20.d odd 2 1 CM
1840.1.bq.a 20 23.c even 11 1 inner
1840.1.bq.a 20 92.g odd 22 1 inner
1840.1.bq.a 20 115.j even 22 1 inner
1840.1.bq.a 20 460.n odd 22 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(1840, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{20} \)
$3$ \( 121 + 242 T^{2} + 1331 T^{4} + 1331 T^{6} + 121 T^{8} - 22 T^{10} + 154 T^{12} - 22 T^{14} + T^{20} \)
$5$ \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} )^{2} \)
$7$ \( 121 - 605 T^{2} + 1089 T^{4} + 484 T^{8} + 462 T^{10} + 330 T^{12} + 165 T^{14} + 55 T^{16} + 11 T^{18} + T^{20} \)
$11$ \( T^{20} \)
$13$ \( T^{20} \)
$17$ \( T^{20} \)
$19$ \( T^{20} \)
$23$ \( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} - T^{14} + T^{16} - T^{18} + T^{20} \)
$29$ \( ( 1 + 6 T + 14 T^{2} + 7 T^{3} + 9 T^{4} - 12 T^{5} - 6 T^{6} - 3 T^{7} + 4 T^{8} + 2 T^{9} + T^{10} )^{2} \)
$31$ \( T^{20} \)
$37$ \( T^{20} \)
$41$ \( ( 1 - 6 T + 14 T^{2} - 7 T^{3} + 9 T^{4} + 12 T^{5} - 6 T^{6} + 3 T^{7} + 4 T^{8} - 2 T^{9} + T^{10} )^{2} \)
$43$ \( 121 + 484 T^{2} + 1210 T^{4} + 121 T^{6} + 605 T^{8} - 264 T^{10} + 55 T^{14} + T^{20} \)
$47$ \( ( -11 + 55 T^{2} - 77 T^{4} + 44 T^{6} - 11 T^{8} + T^{10} )^{2} \)
$53$ \( T^{20} \)
$59$ \( T^{20} \)
$61$ \( ( 1 - 6 T + 14 T^{2} - 7 T^{3} + 9 T^{4} + 12 T^{5} - 6 T^{6} + 3 T^{7} + 4 T^{8} - 2 T^{9} + T^{10} )^{2} \)
$67$ \( 121 + 484 T^{2} + 1210 T^{4} + 121 T^{6} + 605 T^{8} - 264 T^{10} + 55 T^{14} + T^{20} \)
$71$ \( T^{20} \)
$73$ \( T^{20} \)
$79$ \( T^{20} \)
$83$ \( 121 - 121 T^{2} + 847 T^{4} + 1573 T^{6} + 1452 T^{8} + 462 T^{10} + 22 T^{12} + T^{20} \)
$89$ \( ( 1 + 6 T + 25 T^{2} + 51 T^{3} + 53 T^{4} + 32 T^{5} + 16 T^{6} + 8 T^{7} + 4 T^{8} + 2 T^{9} + T^{10} )^{2} \)
$97$ \( T^{20} \)
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