Properties

Label 869.1.j.a
Level $869$
Weight $1$
Character orbit 869.j
Analytic conductor $0.434$
Analytic rank $0$
Dimension $20$
Projective image $D_{25}$
CM discriminant -79
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 869 = 11 \cdot 79 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 869.j (of order \(10\), degree \(4\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.433687495978\)
Analytic rank: \(0\)
Dimension: \(20\)
Relative dimension: \(5\) over \(\Q(\zeta_{10})\)
Coefficient field: \(\Q(\zeta_{50})\)
Defining polynomial: \(x^{20} - x^{15} + x^{10} - x^{5} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{25}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{25} - \cdots)\)

$q$-expansion

The \(q\)-expansion and trace form are shown below.

\(f(q)\) \(=\) \( q + ( -\zeta_{50}^{9} - \zeta_{50}^{21} ) q^{2} + ( -\zeta_{50}^{5} - \zeta_{50}^{17} + \zeta_{50}^{18} ) q^{4} + ( -\zeta_{50}^{7} - \zeta_{50}^{13} ) q^{5} + ( -\zeta_{50} + \zeta_{50}^{2} - \zeta_{50}^{13} + \zeta_{50}^{14} ) q^{8} -\zeta_{50}^{15} q^{9} +O(q^{10})\) \( q + ( -\zeta_{50}^{9} - \zeta_{50}^{21} ) q^{2} + ( -\zeta_{50}^{5} - \zeta_{50}^{17} + \zeta_{50}^{18} ) q^{4} + ( -\zeta_{50}^{7} - \zeta_{50}^{13} ) q^{5} + ( -\zeta_{50} + \zeta_{50}^{2} - \zeta_{50}^{13} + \zeta_{50}^{14} ) q^{8} -\zeta_{50}^{15} q^{9} + ( -\zeta_{50}^{3} - \zeta_{50}^{9} + \zeta_{50}^{16} + \zeta_{50}^{22} ) q^{10} + \zeta_{50}^{6} q^{11} + ( \zeta_{50}^{14} + \zeta_{50}^{16} ) q^{13} + ( -\zeta_{50}^{9} + \zeta_{50}^{10} - \zeta_{50}^{11} + \zeta_{50}^{22} - \zeta_{50}^{23} ) q^{16} + ( -\zeta_{50}^{11} + \zeta_{50}^{24} ) q^{18} + ( -\zeta_{50}^{3} + \zeta_{50}^{12} ) q^{19} + ( 1 - \zeta_{50}^{5} + \zeta_{50}^{6} + \zeta_{50}^{12} + \zeta_{50}^{18} + \zeta_{50}^{24} ) q^{20} + ( \zeta_{50}^{2} - \zeta_{50}^{15} ) q^{22} + ( \zeta_{50}^{8} - \zeta_{50}^{17} ) q^{23} + ( -\zeta_{50} + \zeta_{50}^{14} + \zeta_{50}^{20} ) q^{25} + ( 1 + \zeta_{50}^{10} + \zeta_{50}^{12} - \zeta_{50}^{23} ) q^{26} + ( -\zeta_{50}^{11} - \zeta_{50}^{19} ) q^{31} + ( -\zeta_{50}^{5} + \zeta_{50}^{6} - \zeta_{50}^{7} + \zeta_{50}^{18} - \zeta_{50}^{19} + \zeta_{50}^{20} ) q^{32} + ( -\zeta_{50}^{7} + \zeta_{50}^{8} + \zeta_{50}^{20} ) q^{36} + ( \zeta_{50}^{8} + \zeta_{50}^{12} - \zeta_{50}^{21} + \zeta_{50}^{24} ) q^{38} + ( -\zeta_{50} + \zeta_{50}^{2} + \zeta_{50}^{8} - \zeta_{50}^{9} + \zeta_{50}^{14} - \zeta_{50}^{15} + \zeta_{50}^{20} - \zeta_{50}^{21} ) q^{40} + ( -\zeta_{50}^{11} - \zeta_{50}^{23} + \zeta_{50}^{24} ) q^{44} + ( -\zeta_{50}^{3} + \zeta_{50}^{22} ) q^{45} + ( -\zeta_{50} + \zeta_{50}^{4} - \zeta_{50}^{13} - \zeta_{50}^{17} ) q^{46} + \zeta_{50}^{10} q^{49} + ( \zeta_{50}^{4} + 2 \zeta_{50}^{10} + \zeta_{50}^{16} + \zeta_{50}^{22} - \zeta_{50}^{23} ) q^{50} + ( \zeta_{50}^{6} - \zeta_{50}^{7} + \zeta_{50}^{8} - \zeta_{50}^{9} - \zeta_{50}^{19} - \zeta_{50}^{21} ) q^{52} + ( -\zeta_{50}^{13} - \zeta_{50}^{19} ) q^{55} + ( -\zeta_{50}^{3} - \zeta_{50}^{7} - \zeta_{50}^{15} + \zeta_{50}^{20} ) q^{62} + ( -\zeta_{50} + \zeta_{50}^{2} - \zeta_{50}^{3} + \zeta_{50}^{4} + \zeta_{50}^{14} - \zeta_{50}^{15} + \zeta_{50}^{16} ) q^{64} + ( \zeta_{50}^{2} + \zeta_{50}^{4} - \zeta_{50}^{21} - \zeta_{50}^{23} ) q^{65} + ( \zeta_{50}^{2} - \zeta_{50}^{23} ) q^{67} + ( -\zeta_{50}^{3} + \zeta_{50}^{4} + \zeta_{50}^{16} - \zeta_{50}^{17} ) q^{72} + ( -\zeta_{50}^{17} + \zeta_{50}^{18} ) q^{73} + ( \zeta_{50}^{4} - \zeta_{50}^{5} + \zeta_{50}^{8} - \zeta_{50}^{17} + \zeta_{50}^{20} - \zeta_{50}^{21} ) q^{76} -\zeta_{50}^{15} q^{79} + ( \zeta_{50}^{4} - \zeta_{50}^{5} + \zeta_{50}^{10} - \zeta_{50}^{11} + \zeta_{50}^{16} - \zeta_{50}^{17} + \zeta_{50}^{18} + \zeta_{50}^{22} - \zeta_{50}^{23} + \zeta_{50}^{24} ) q^{80} -\zeta_{50}^{5} q^{81} + ( 1 + \zeta_{50}^{20} ) q^{83} + ( -\zeta_{50}^{7} + \zeta_{50}^{8} - \zeta_{50}^{19} + \zeta_{50}^{20} ) q^{88} + ( -\zeta_{50}^{11} + \zeta_{50}^{14} ) q^{89} + ( \zeta_{50}^{6} + \zeta_{50}^{12} + \zeta_{50}^{18} + \zeta_{50}^{24} ) q^{90} + ( 1 - \zeta_{50} - \zeta_{50}^{9} + \zeta_{50}^{10} - \zeta_{50}^{13} + \zeta_{50}^{22} ) q^{92} + ( 1 + \zeta_{50}^{10} + \zeta_{50}^{16} - \zeta_{50}^{19} ) q^{95} + ( \zeta_{50}^{8} + \zeta_{50}^{22} ) q^{97} + ( \zeta_{50}^{6} - \zeta_{50}^{19} ) q^{98} -\zeta_{50}^{21} q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20q - 5q^{4} - 5q^{9} + O(q^{10}) \) \( 20q - 5q^{4} - 5q^{9} - 5q^{16} + 15q^{20} - 5q^{22} - 5q^{25} + 15q^{26} - 10q^{32} - 5q^{36} - 10q^{40} - 5q^{49} - 10q^{50} - 10q^{62} - 5q^{64} - 10q^{76} - 5q^{79} - 10q^{80} - 5q^{81} + 15q^{83} - 5q^{88} + 15q^{92} + 15q^{95} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/869\mathbb{Z}\right)^\times\).

\(n\) \(475\) \(793\)
\(\chi(n)\) \(-\zeta_{50}^{5}\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
157.1
−0.535827 0.844328i
0.637424 0.770513i
−0.968583 + 0.248690i
0.929776 + 0.368125i
−0.0627905 + 0.998027i
−0.876307 0.481754i
0.187381 0.982287i
−0.728969 + 0.684547i
0.992115 0.125333i
0.425779 + 0.904827i
−0.876307 + 0.481754i
0.187381 + 0.982287i
−0.728969 0.684547i
0.992115 + 0.125333i
0.425779 0.904827i
−0.535827 + 0.844328i
0.637424 + 0.770513i
−0.968583 0.248690i
0.929776 0.368125i
−0.0627905 0.998027i
−1.56720 + 1.13864i 0 0.850604 2.61789i 1.60528 + 1.16630i 0 0 1.04914 + 3.22894i −0.809017 + 0.587785i −3.84378
157.2 −0.866986 + 0.629902i 0 0.0458709 0.141176i −1.41789 1.03016i 0 0 −0.282001 0.867911i −0.809017 + 0.587785i 1.87819
157.3 −0.101597 + 0.0738147i 0 −0.304144 + 0.936058i −1.17950 0.856954i 0 0 −0.0770013 0.236986i −0.809017 + 0.587785i 0.183089
157.4 1.03137 0.749337i 0 0.193209 0.594636i 0.688925 + 0.500534i 0 0 0.147638 + 0.454382i −0.809017 + 0.587785i 1.08561
157.5 1.50441 1.09302i 0 0.759544 2.33764i 0.303189 + 0.220280i 0 0 −0.837780 2.57842i −0.809017 + 0.587785i 0.696891
236.1 −0.613161 1.88711i 0 −2.37622 + 1.72642i 0.0388067 0.119435i 0 0 3.10969 + 2.25932i 0.309017 + 0.951057i −0.249182
236.2 −0.263146 0.809880i 0 0.222357 0.161552i 0.331159 1.01920i 0 0 −0.878275 0.638104i 0.309017 + 0.951057i −0.912576
236.3 −0.115808 0.356420i 0 0.695393 0.505233i −0.393950 + 1.21245i 0 0 −0.563797 0.409622i 0.309017 + 0.951057i 0.477765
236.4 0.450527 + 1.38658i 0 −0.910614 + 0.661600i −0.574633 + 1.76854i 0 0 −0.148122 0.107617i 0.309017 + 0.951057i −2.71111
236.5 0.541587 + 1.66683i 0 −1.67600 + 1.21769i 0.598617 1.84235i 0 0 −1.51949 1.10397i 0.309017 + 0.951057i 3.39510
394.1 −0.613161 + 1.88711i 0 −2.37622 1.72642i 0.0388067 + 0.119435i 0 0 3.10969 2.25932i 0.309017 0.951057i −0.249182
394.2 −0.263146 + 0.809880i 0 0.222357 + 0.161552i 0.331159 + 1.01920i 0 0 −0.878275 + 0.638104i 0.309017 0.951057i −0.912576
394.3 −0.115808 + 0.356420i 0 0.695393 + 0.505233i −0.393950 1.21245i 0 0 −0.563797 + 0.409622i 0.309017 0.951057i 0.477765
394.4 0.450527 1.38658i 0 −0.910614 0.661600i −0.574633 1.76854i 0 0 −0.148122 + 0.107617i 0.309017 0.951057i −2.71111
394.5 0.541587 1.66683i 0 −1.67600 1.21769i 0.598617 + 1.84235i 0 0 −1.51949 + 1.10397i 0.309017 0.951057i 3.39510
631.1 −1.56720 1.13864i 0 0.850604 + 2.61789i 1.60528 1.16630i 0 0 1.04914 3.22894i −0.809017 0.587785i −3.84378
631.2 −0.866986 0.629902i 0 0.0458709 + 0.141176i −1.41789 + 1.03016i 0 0 −0.282001 + 0.867911i −0.809017 0.587785i 1.87819
631.3 −0.101597 0.0738147i 0 −0.304144 0.936058i −1.17950 + 0.856954i 0 0 −0.0770013 + 0.236986i −0.809017 0.587785i 0.183089
631.4 1.03137 + 0.749337i 0 0.193209 + 0.594636i 0.688925 0.500534i 0 0 0.147638 0.454382i −0.809017 0.587785i 1.08561
631.5 1.50441 + 1.09302i 0 0.759544 + 2.33764i 0.303189 0.220280i 0 0 −0.837780 + 2.57842i −0.809017 0.587785i 0.696891
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 631.5
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
79.b odd 2 1 CM by \(\Q(\sqrt{-79}) \)
11.c even 5 1 inner
869.j odd 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 869.1.j.a 20
11.c even 5 1 inner 869.1.j.a 20
79.b odd 2 1 CM 869.1.j.a 20
869.j odd 10 1 inner 869.1.j.a 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
869.1.j.a 20 1.a even 1 1 trivial
869.1.j.a 20 11.c even 5 1 inner
869.1.j.a 20 79.b odd 2 1 CM
869.1.j.a 20 869.j odd 10 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(869, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + 15 T + 100 T^{2} + 260 T^{3} + 675 T^{4} + 498 T^{5} + 715 T^{6} + 150 T^{7} + 485 T^{8} + 200 T^{9} + 374 T^{10} + 5 T^{11} + 275 T^{12} - 5 T^{13} + 75 T^{14} + 2 T^{15} + 20 T^{16} + 5 T^{18} + T^{20} \)
$3$ \( T^{20} \)
$5$ \( 1 - 10 T + 100 T^{2} - 390 T^{3} + 800 T^{4} - 752 T^{5} + 740 T^{6} - 475 T^{7} + 510 T^{8} - 175 T^{9} + 374 T^{10} + 30 T^{11} + 275 T^{12} + 20 T^{13} + 75 T^{14} + 2 T^{15} + 20 T^{16} + 5 T^{18} + T^{20} \)
$7$ \( T^{20} \)
$11$ \( 1 + T^{5} + T^{10} + T^{15} + T^{20} \)
$13$ \( 1 + 15 T + 100 T^{2} + 260 T^{3} + 675 T^{4} + 498 T^{5} + 715 T^{6} + 150 T^{7} + 485 T^{8} + 200 T^{9} + 374 T^{10} + 5 T^{11} + 275 T^{12} - 5 T^{13} + 75 T^{14} + 2 T^{15} + 20 T^{16} + 5 T^{18} + T^{20} \)
$17$ \( T^{20} \)
$19$ \( 1 + 15 T + 100 T^{2} + 260 T^{3} + 675 T^{4} + 498 T^{5} + 715 T^{6} + 150 T^{7} + 485 T^{8} + 200 T^{9} + 374 T^{10} + 5 T^{11} + 275 T^{12} - 5 T^{13} + 75 T^{14} + 2 T^{15} + 20 T^{16} + 5 T^{18} + T^{20} \)
$23$ \( ( -1 + 5 T + 25 T^{2} - 5 T^{3} - 50 T^{4} + T^{5} + 35 T^{6} - 10 T^{8} + T^{10} )^{2} \)
$29$ \( T^{20} \)
$31$ \( 1 + 15 T + 100 T^{2} + 260 T^{3} + 675 T^{4} + 498 T^{5} + 715 T^{6} + 150 T^{7} + 485 T^{8} + 200 T^{9} + 374 T^{10} + 5 T^{11} + 275 T^{12} - 5 T^{13} + 75 T^{14} + 2 T^{15} + 20 T^{16} + 5 T^{18} + T^{20} \)
$37$ \( T^{20} \)
$41$ \( T^{20} \)
$43$ \( T^{20} \)
$47$ \( T^{20} \)
$53$ \( T^{20} \)
$59$ \( T^{20} \)
$61$ \( T^{20} \)
$67$ \( ( -1 + 5 T + 25 T^{2} - 5 T^{3} - 50 T^{4} + T^{5} + 35 T^{6} - 10 T^{8} + T^{10} )^{2} \)
$71$ \( T^{20} \)
$73$ \( 1 + 15 T + 100 T^{2} + 260 T^{3} + 675 T^{4} + 498 T^{5} + 715 T^{6} + 150 T^{7} + 485 T^{8} + 200 T^{9} + 374 T^{10} + 5 T^{11} + 275 T^{12} - 5 T^{13} + 75 T^{14} + 2 T^{15} + 20 T^{16} + 5 T^{18} + T^{20} \)
$79$ \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{5} \)
$83$ \( ( 1 - 2 T + 4 T^{2} - 3 T^{3} + T^{4} )^{5} \)
$89$ \( ( -1 + 5 T + 25 T^{2} - 5 T^{3} - 50 T^{4} + T^{5} + 35 T^{6} - 10 T^{8} + T^{10} )^{2} \)
$97$ \( 1 - 10 T + 100 T^{2} - 390 T^{3} + 800 T^{4} - 752 T^{5} + 740 T^{6} - 475 T^{7} + 510 T^{8} - 175 T^{9} + 374 T^{10} + 30 T^{11} + 275 T^{12} + 20 T^{13} + 75 T^{14} + 2 T^{15} + 20 T^{16} + 5 T^{18} + T^{20} \)
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