Properties

Label 1288.1.bi.e
Level $1288$
Weight $1$
Character orbit 1288.bi
Analytic conductor $0.643$
Analytic rank $0$
Dimension $20$
Projective image $D_{22}$
CM discriminant -56
Inner twists $8$

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Newspace parameters

Level: \( N \) \(=\) \( 1288 = 2^{3} \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1288.bi (of order \(22\), degree \(10\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.642795736271\)
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: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 11 \)
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.

\(f(q)\) \(=\) \( q + \zeta_{44}^{18} q^{2} + ( \zeta_{44}^{11} + \zeta_{44}^{13} ) q^{3} -\zeta_{44}^{14} q^{4} + ( -\zeta_{44}^{3} + \zeta_{44}^{15} ) q^{5} + ( -\zeta_{44}^{7} - \zeta_{44}^{9} ) q^{6} + \zeta_{44}^{6} q^{7} + \zeta_{44}^{10} q^{8} + ( -1 - \zeta_{44}^{2} - \zeta_{44}^{4} ) q^{9} +O(q^{10})\) \( q + \zeta_{44}^{18} q^{2} + ( \zeta_{44}^{11} + \zeta_{44}^{13} ) q^{3} -\zeta_{44}^{14} q^{4} + ( -\zeta_{44}^{3} + \zeta_{44}^{15} ) q^{5} + ( -\zeta_{44}^{7} - \zeta_{44}^{9} ) q^{6} + \zeta_{44}^{6} q^{7} + \zeta_{44}^{10} q^{8} + ( -1 - \zeta_{44}^{2} - \zeta_{44}^{4} ) q^{9} + ( -\zeta_{44}^{11} - \zeta_{44}^{21} ) q^{10} + ( \zeta_{44}^{3} + \zeta_{44}^{5} ) q^{12} + ( -\zeta_{44} + \zeta_{44}^{9} ) q^{13} -\zeta_{44}^{2} q^{14} + ( -\zeta_{44}^{4} - \zeta_{44}^{6} - \zeta_{44}^{14} - \zeta_{44}^{16} ) q^{15} -\zeta_{44}^{6} q^{16} + ( 1 - \zeta_{44}^{18} - \zeta_{44}^{20} ) q^{18} + ( \zeta_{44} - \zeta_{44}^{5} ) q^{19} + ( \zeta_{44}^{7} + \zeta_{44}^{17} ) q^{20} + ( \zeta_{44}^{17} + \zeta_{44}^{19} ) q^{21} + \zeta_{44}^{16} q^{23} + ( -\zeta_{44} + \zeta_{44}^{21} ) q^{24} + ( \zeta_{44}^{6} - \zeta_{44}^{8} - \zeta_{44}^{18} ) q^{25} + ( -\zeta_{44}^{5} - \zeta_{44}^{19} ) q^{26} + ( -\zeta_{44}^{11} - \zeta_{44}^{13} - \zeta_{44}^{15} - \zeta_{44}^{17} ) q^{27} -\zeta_{44}^{20} q^{28} + ( 1 + \zeta_{44}^{2} + \zeta_{44}^{10} + \zeta_{44}^{12} ) q^{30} + \zeta_{44}^{2} q^{32} + ( -\zeta_{44}^{9} + \zeta_{44}^{21} ) q^{35} + ( \zeta_{44}^{14} + \zeta_{44}^{16} + \zeta_{44}^{18} ) q^{36} + ( \zeta_{44} + \zeta_{44}^{19} ) q^{38} + ( -1 - \zeta_{44}^{12} - \zeta_{44}^{14} + \zeta_{44}^{20} ) q^{39} + ( -\zeta_{44}^{3} - \zeta_{44}^{13} ) q^{40} + ( -\zeta_{44}^{13} - \zeta_{44}^{15} ) q^{42} + ( \zeta_{44}^{3} + \zeta_{44}^{5} + \zeta_{44}^{7} - \zeta_{44}^{15} - \zeta_{44}^{17} - \zeta_{44}^{19} ) q^{45} -\zeta_{44}^{12} q^{46} + ( -\zeta_{44}^{17} - \zeta_{44}^{19} ) q^{48} + \zeta_{44}^{12} q^{49} + ( -\zeta_{44}^{2} + \zeta_{44}^{4} + \zeta_{44}^{14} ) q^{50} + ( \zeta_{44} + \zeta_{44}^{15} ) q^{52} + ( \zeta_{44}^{7} + \zeta_{44}^{9} + \zeta_{44}^{11} + \zeta_{44}^{13} ) q^{54} + \zeta_{44}^{16} q^{56} + ( \zeta_{44}^{12} + \zeta_{44}^{14} - \zeta_{44}^{16} - \zeta_{44}^{18} ) q^{57} + ( \zeta_{44}^{3} - \zeta_{44}^{7} ) q^{59} + ( -\zeta_{44}^{6} - \zeta_{44}^{8} + \zeta_{44}^{18} + \zeta_{44}^{20} ) q^{60} + ( -\zeta_{44}^{6} - \zeta_{44}^{8} - \zeta_{44}^{10} ) q^{63} + \zeta_{44}^{20} q^{64} + ( -\zeta_{44}^{2} + \zeta_{44}^{4} - \zeta_{44}^{12} - \zeta_{44}^{16} ) q^{65} + ( -\zeta_{44}^{5} - \zeta_{44}^{7} ) q^{69} + ( \zeta_{44}^{5} - \zeta_{44}^{17} ) q^{70} + ( -1 + \zeta_{44}^{14} ) q^{71} + ( -\zeta_{44}^{10} - \zeta_{44}^{12} - \zeta_{44}^{14} ) q^{72} + ( \zeta_{44}^{7} + \zeta_{44}^{9} + \zeta_{44}^{17} - \zeta_{44}^{21} ) q^{75} + ( -\zeta_{44}^{15} + \zeta_{44}^{19} ) q^{76} + ( \zeta_{44}^{8} + \zeta_{44}^{10} - \zeta_{44}^{16} - \zeta_{44}^{18} ) q^{78} + ( \zeta_{44}^{2} - \zeta_{44}^{8} ) q^{79} + ( \zeta_{44}^{9} - \zeta_{44}^{21} ) q^{80} + ( 1 + \zeta_{44}^{2} + \zeta_{44}^{4} + \zeta_{44}^{6} + \zeta_{44}^{8} ) q^{81} + ( \zeta_{44}^{5} - \zeta_{44}^{21} ) q^{83} + ( \zeta_{44}^{9} + \zeta_{44}^{11} ) q^{84} + ( -\zeta_{44} - \zeta_{44}^{3} + \zeta_{44}^{11} + \zeta_{44}^{13} + \zeta_{44}^{15} + \zeta_{44}^{21} ) q^{90} + ( -\zeta_{44}^{7} + \zeta_{44}^{15} ) q^{91} + \zeta_{44}^{8} q^{92} + ( -\zeta_{44}^{4} + \zeta_{44}^{8} + \zeta_{44}^{16} - \zeta_{44}^{20} ) q^{95} + ( \zeta_{44}^{13} + \zeta_{44}^{15} ) q^{96} -\zeta_{44}^{8} q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20q + 2q^{2} - 2q^{4} + 2q^{7} + 2q^{8} - 20q^{9} + O(q^{10}) \) \( 20q + 2q^{2} - 2q^{4} + 2q^{7} + 2q^{8} - 20q^{9} - 2q^{14} - 2q^{16} + 20q^{18} - 2q^{23} + 2q^{25} + 2q^{28} + 22q^{30} + 2q^{32} + 2q^{36} - 22q^{39} + 2q^{46} - 2q^{49} - 2q^{50} - 2q^{56} - 2q^{63} - 2q^{64} - 18q^{71} - 2q^{72} + 4q^{79} + 20q^{81} - 2q^{92} + 2q^{98} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1288\mathbb{Z}\right)^\times\).

\(n\) \(185\) \(281\) \(645\) \(967\)
\(\chi(n)\) \(-1\) \(-\zeta_{44}^{18}\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
13.1
−0.540641 + 0.841254i
0.540641 0.841254i
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
0.755750 0.654861i
−0.755750 + 0.654861i
0.281733 0.959493i
−0.281733 + 0.959493i
−0.540641 0.841254i
0.540641 + 0.841254i
−0.909632 0.415415i
0.909632 + 0.415415i
0.654861 + 0.755750i −0.909632 0.584585i −0.142315 + 0.989821i −0.234072 + 0.512546i −0.153882 1.07028i 0.959493 + 0.281733i −0.841254 + 0.540641i 0.0702757 + 0.153882i −0.540641 + 0.158746i
13.2 0.654861 + 0.755750i 0.909632 + 0.584585i −0.142315 + 0.989821i 0.234072 0.512546i 0.153882 + 1.07028i 0.959493 + 0.281733i −0.841254 + 0.540641i 0.0702757 + 0.153882i 0.540641 0.158746i
349.1 −0.415415 0.909632i −0.540641 + 0.158746i −0.654861 + 0.755750i 1.66538 + 1.07028i 0.368991 + 0.425839i 0.142315 + 0.989821i 0.959493 + 0.281733i −0.574161 + 0.368991i 0.281733 1.95949i
349.2 −0.415415 0.909632i 0.540641 0.158746i −0.654861 + 0.755750i −1.66538 1.07028i −0.368991 0.425839i 0.142315 + 0.989821i 0.959493 + 0.281733i −0.574161 + 0.368991i −0.281733 + 1.95949i
629.1 0.142315 0.989821i −0.755750 1.65486i −0.959493 0.281733i −0.708089 0.817178i −1.74557 + 0.512546i −0.841254 0.540641i −0.415415 + 0.909632i −1.51255 + 1.74557i −0.909632 + 0.584585i
629.2 0.142315 0.989821i 0.755750 + 1.65486i −0.959493 0.281733i 0.708089 + 0.817178i 1.74557 0.512546i −0.841254 0.540641i −0.415415 + 0.909632i −1.51255 + 1.74557i 0.909632 0.584585i
685.1 0.959493 + 0.281733i −0.989821 + 1.14231i 0.841254 + 0.540641i 0.258908 1.80075i −1.27155 + 0.817178i −0.415415 0.909632i 0.654861 + 0.755750i −0.182822 1.27155i 0.755750 1.65486i
685.2 0.959493 + 0.281733i 0.989821 1.14231i 0.841254 + 0.540641i −0.258908 + 1.80075i 1.27155 0.817178i −0.415415 0.909632i 0.654861 + 0.755750i −0.182822 1.27155i −0.755750 + 1.65486i
853.1 −0.841254 0.540641i −0.281733 1.95949i 0.415415 + 0.909632i −1.45027 0.425839i −0.822373 + 1.80075i 0.654861 0.755750i 0.142315 0.989821i −2.80075 + 0.822373i 0.989821 + 1.14231i
853.2 −0.841254 0.540641i 0.281733 + 1.95949i 0.415415 + 0.909632i 1.45027 + 0.425839i 0.822373 1.80075i 0.654861 0.755750i 0.142315 0.989821i −2.80075 + 0.822373i −0.989821 1.14231i
909.1 −0.841254 + 0.540641i −0.281733 + 1.95949i 0.415415 0.909632i −1.45027 + 0.425839i −0.822373 1.80075i 0.654861 + 0.755750i 0.142315 + 0.989821i −2.80075 0.822373i 0.989821 1.14231i
909.2 −0.841254 + 0.540641i 0.281733 1.95949i 0.415415 0.909632i 1.45027 0.425839i 0.822373 + 1.80075i 0.654861 + 0.755750i 0.142315 + 0.989821i −2.80075 0.822373i −0.989821 + 1.14231i
1021.1 0.959493 0.281733i −0.989821 1.14231i 0.841254 0.540641i 0.258908 + 1.80075i −1.27155 0.817178i −0.415415 + 0.909632i 0.654861 0.755750i −0.182822 + 1.27155i 0.755750 + 1.65486i
1021.2 0.959493 0.281733i 0.989821 + 1.14231i 0.841254 0.540641i −0.258908 1.80075i 1.27155 + 0.817178i −0.415415 + 0.909632i 0.654861 0.755750i −0.182822 + 1.27155i −0.755750 1.65486i
1133.1 −0.415415 + 0.909632i −0.540641 0.158746i −0.654861 0.755750i 1.66538 1.07028i 0.368991 0.425839i 0.142315 0.989821i 0.959493 0.281733i −0.574161 0.368991i 0.281733 + 1.95949i
1133.2 −0.415415 + 0.909632i 0.540641 + 0.158746i −0.654861 0.755750i −1.66538 + 1.07028i −0.368991 + 0.425839i 0.142315 0.989821i 0.959493 0.281733i −0.574161 0.368991i −0.281733 1.95949i
1189.1 0.654861 0.755750i −0.909632 + 0.584585i −0.142315 0.989821i −0.234072 0.512546i −0.153882 + 1.07028i 0.959493 0.281733i −0.841254 0.540641i 0.0702757 0.153882i −0.540641 0.158746i
1189.2 0.654861 0.755750i 0.909632 0.584585i −0.142315 0.989821i 0.234072 + 0.512546i 0.153882 1.07028i 0.959493 0.281733i −0.841254 0.540641i 0.0702757 0.153882i 0.540641 + 0.158746i
1245.1 0.142315 + 0.989821i −0.755750 + 1.65486i −0.959493 + 0.281733i −0.708089 + 0.817178i −1.74557 0.512546i −0.841254 + 0.540641i −0.415415 0.909632i −1.51255 1.74557i −0.909632 0.584585i
1245.2 0.142315 + 0.989821i 0.755750 1.65486i −0.959493 + 0.281733i 0.708089 0.817178i 1.74557 + 0.512546i −0.841254 + 0.540641i −0.415415 0.909632i −1.51255 1.74557i 0.909632 + 0.584585i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1245.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
56.h odd 2 1 CM by \(\Q(\sqrt{-14}) \)
7.b odd 2 1 inner
8.b even 2 1 inner
23.c even 11 1 inner
161.l odd 22 1 inner
184.p even 22 1 inner
1288.bi odd 22 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1288.1.bi.e 20
7.b odd 2 1 inner 1288.1.bi.e 20
8.b even 2 1 inner 1288.1.bi.e 20
23.c even 11 1 inner 1288.1.bi.e 20
56.h odd 2 1 CM 1288.1.bi.e 20
161.l odd 22 1 inner 1288.1.bi.e 20
184.p even 22 1 inner 1288.1.bi.e 20
1288.bi odd 22 1 inner 1288.1.bi.e 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1288.1.bi.e 20 1.a even 1 1 trivial
1288.1.bi.e 20 7.b odd 2 1 inner
1288.1.bi.e 20 8.b even 2 1 inner
1288.1.bi.e 20 23.c even 11 1 inner
1288.1.bi.e 20 56.h odd 2 1 CM
1288.1.bi.e 20 161.l odd 22 1 inner
1288.1.bi.e 20 184.p even 22 1 inner
1288.1.bi.e 20 1288.bi 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 \(S_{1}^{\mathrm{new}}(1288, [\chi])\):

\(T_{3}^{20} + \cdots\)
\( T_{11} \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} )^{2} \)
$3$ \( 121 - 605 T^{2} + 1089 T^{4} + 484 T^{8} + 462 T^{10} + 330 T^{12} + 165 T^{14} + 55 T^{16} + 11 T^{18} + T^{20} \)
$5$ \( 121 + 484 T^{2} + 1210 T^{4} + 121 T^{6} + 605 T^{8} - 264 T^{10} + 55 T^{14} + T^{20} \)
$7$ \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} )^{2} \)
$11$ \( T^{20} \)
$13$ \( 121 + 484 T^{2} + 1210 T^{4} + 121 T^{6} + 605 T^{8} - 264 T^{10} + 55 T^{14} + T^{20} \)
$17$ \( T^{20} \)
$19$ \( 121 + 484 T^{2} + 1210 T^{4} + 121 T^{6} + 605 T^{8} - 264 T^{10} + 55 T^{14} + T^{20} \)
$23$ \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} )^{2} \)
$29$ \( T^{20} \)
$31$ \( T^{20} \)
$37$ \( T^{20} \)
$41$ \( T^{20} \)
$43$ \( T^{20} \)
$47$ \( T^{20} \)
$53$ \( T^{20} \)
$59$ \( 121 + 242 T^{2} + 1331 T^{4} + 1331 T^{6} + 121 T^{8} - 22 T^{10} + 154 T^{12} - 22 T^{14} + T^{20} \)
$61$ \( T^{20} \)
$67$ \( T^{20} \)
$71$ \( ( 1 + 5 T + 25 T^{2} + 70 T^{3} + 130 T^{4} + 166 T^{5} + 148 T^{6} + 91 T^{7} + 37 T^{8} + 9 T^{9} + T^{10} )^{2} \)
$73$ \( T^{20} \)
$79$ \( ( 1 + 5 T + 3 T^{2} - 7 T^{3} + 20 T^{4} - 10 T^{5} + 16 T^{6} - 8 T^{7} + 4 T^{8} - 2 T^{9} + T^{10} )^{2} \)
$83$ \( 121 + 242 T^{2} + 1331 T^{4} + 1331 T^{6} + 121 T^{8} - 22 T^{10} + 154 T^{12} - 22 T^{14} + T^{20} \)
$89$ \( T^{20} \)
$97$ \( T^{20} \)
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