Properties

Label 1359.1.bp.a
Level $1359$
Weight $1$
Character orbit 1359.bp
Analytic conductor $0.678$
Analytic rank $0$
Dimension $20$
Projective image $D_{50}$
CM discriminant -3
Inner twists $4$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 1359 = 3^{2} \cdot 151 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1359.bp (of order \(50\), degree \(20\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.678229352168\)
Analytic rank: \(0\)
Dimension: \(20\)
Coefficient field: \(\Q(\zeta_{50})\)
Defining polynomial: \(x^{20} - x^{15} + x^{10} - x^{5} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{50}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{50} - \cdots)\)

$q$-expansion

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

\(f(q)\) \(=\) \( q -\zeta_{50}^{20} q^{4} + ( \zeta_{50}^{8} - \zeta_{50}^{14} ) q^{7} +O(q^{10})\) \( q -\zeta_{50}^{20} q^{4} + ( \zeta_{50}^{8} - \zeta_{50}^{14} ) q^{7} + ( \zeta_{50}^{7} + \zeta_{50}^{24} ) q^{13} -\zeta_{50}^{15} q^{16} + ( \zeta_{50}^{4} - \zeta_{50}^{11} ) q^{19} + \zeta_{50}^{17} q^{25} + ( \zeta_{50}^{3} - \zeta_{50}^{9} ) q^{28} + ( \zeta_{50}^{10} - \zeta_{50}^{13} ) q^{31} + ( -\zeta_{50} - \zeta_{50}^{5} ) q^{37} + ( \zeta_{50} - \zeta_{50}^{2} ) q^{43} + ( -\zeta_{50}^{3} + \zeta_{50}^{16} - \zeta_{50}^{22} ) q^{49} + ( \zeta_{50}^{2} + \zeta_{50}^{19} ) q^{52} + ( -\zeta_{50}^{12} - \zeta_{50}^{17} ) q^{61} -\zeta_{50}^{10} q^{64} + ( \zeta_{50}^{11} + \zeta_{50}^{22} ) q^{67} + ( \zeta_{50}^{6} - \zeta_{50}^{16} ) q^{73} + ( -\zeta_{50}^{6} - \zeta_{50}^{24} ) q^{76} + ( \zeta_{50}^{9} + \zeta_{50}^{18} ) q^{79} + ( -\zeta_{50}^{7} + \zeta_{50}^{13} + \zeta_{50}^{15} - \zeta_{50}^{21} ) q^{91} + ( \zeta_{50}^{14} - \zeta_{50}^{19} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20q + 5q^{4} + O(q^{10}) \) \( 20q + 5q^{4} - 5q^{16} - 5q^{31} - 5q^{37} + 5q^{64} + 5q^{91} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1359\mathbb{Z}\right)^\times\).

\(n\) \(605\) \(1063\)
\(\chi(n)\) \(1\) \(-\zeta_{50}^{16}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
28.1
0.425779 + 0.904827i
−0.876307 0.481754i
0.929776 0.368125i
−0.968583 0.248690i
−0.0627905 0.998027i
0.637424 0.770513i
0.992115 0.125333i
−0.0627905 + 0.998027i
0.187381 0.982287i
−0.728969 + 0.684547i
0.425779 0.904827i
0.187381 + 0.982287i
0.992115 + 0.125333i
0.929776 + 0.368125i
−0.728969 0.684547i
−0.535827 + 0.844328i
−0.876307 + 0.481754i
0.637424 + 0.770513i
−0.968583 + 0.248690i
−0.535827 0.844328i
0 0 0.809017 + 0.587785i 0 0 0.0623382 + 0.493458i 0 0 0
73.1 0 0 0.809017 + 0.587785i 0 0 −1.36639 1.45506i 0 0 0
154.1 0 0 −0.309017 + 0.951057i 0 0 −1.52794 0.969661i 0 0 0
208.1 0 0 −0.309017 + 0.951057i 0 0 0.503997 + 1.27295i 0 0 0
343.1 0 0 −0.309017 + 0.951057i 0 0 1.51373 1.25227i 0 0 0
424.1 0 0 −0.309017 0.951057i 0 0 −0.239615 0.933237i 0 0 0
433.1 0 0 0.809017 + 0.587785i 0 0 0.723208 + 0.137959i 0 0 0
523.1 0 0 −0.309017 0.951057i 0 0 1.51373 + 1.25227i 0 0 0
532.1 0 0 0.809017 + 0.587785i 0 0 −0.813516 + 1.47978i 0 0 0
595.1 0 0 0.809017 + 0.587785i 0 0 1.39436 0.656137i 0 0 0
631.1 0 0 0.809017 0.587785i 0 0 0.0623382 0.493458i 0 0 0
820.1 0 0 0.809017 0.587785i 0 0 −0.813516 1.47978i 0 0 0
838.1 0 0 0.809017 0.587785i 0 0 0.723208 0.137959i 0 0 0
856.1 0 0 −0.309017 0.951057i 0 0 −1.52794 + 0.969661i 0 0 0
973.1 0 0 0.809017 0.587785i 0 0 1.39436 + 0.656137i 0 0 0
1081.1 0 0 −0.309017 + 0.951057i 0 0 −0.250172 + 0.0157395i 0 0 0
1117.1 0 0 0.809017 0.587785i 0 0 −1.36639 + 1.45506i 0 0 0
1234.1 0 0 −0.309017 + 0.951057i 0 0 −0.239615 + 0.933237i 0 0 0
1261.1 0 0 −0.309017 0.951057i 0 0 0.503997 1.27295i 0 0 0
1315.1 0 0 −0.309017 0.951057i 0 0 −0.250172 0.0157395i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1315.1
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 CM by \(\Q(\sqrt{-3}) \)
151.j odd 50 1 inner
453.s even 50 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1359.1.bp.a 20
3.b odd 2 1 CM 1359.1.bp.a 20
151.j odd 50 1 inner 1359.1.bp.a 20
453.s even 50 1 inner 1359.1.bp.a 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1359.1.bp.a 20 1.a even 1 1 trivial
1359.1.bp.a 20 3.b odd 2 1 CM
1359.1.bp.a 20 151.j odd 50 1 inner
1359.1.bp.a 20 453.s even 50 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(1359, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{20} \)
$3$ \( T^{20} \)
$5$ \( T^{20} \)
$7$ \( 5 + 25 T + 125 T^{4} - 620 T^{5} + 625 T^{6} - 625 T^{7} + 875 T^{8} - 375 T^{9} + 250 T^{10} - 125 T^{11} + 125 T^{12} - 75 T^{13} + 25 T^{14} + 20 T^{16} - 5 T^{17} + T^{20} \)
$11$ \( T^{20} \)
$13$ \( 5 - 50 T + 225 T^{2} - 500 T^{3} + 625 T^{4} - 620 T^{5} + 575 T^{6} - 475 T^{7} + 375 T^{8} + 125 T^{9} + 250 T^{10} - 125 T^{11} + 50 T^{12} - 75 T^{13} + 25 T^{14} - 5 T^{16} - 5 T^{17} + T^{20} \)
$17$ \( T^{20} \)
$19$ \( 1 - 10 T + 100 T^{2} - 390 T^{3} + 800 T^{4} - 752 T^{5} + 740 T^{6} - 475 T^{7} + 510 T^{8} - 175 T^{9} + 374 T^{10} + 30 T^{11} + 275 T^{12} + 20 T^{13} + 75 T^{14} + 2 T^{15} + 20 T^{16} + 5 T^{18} + T^{20} \)
$23$ \( T^{20} \)
$29$ \( T^{20} \)
$31$ \( 1 - 10 T + 70 T^{2} - 45 T^{3} - 340 T^{4} - 252 T^{5} + 665 T^{6} + 1520 T^{7} + 1655 T^{8} + 1235 T^{9} + 629 T^{10} + 205 T^{11} + 165 T^{12} + 185 T^{13} + 170 T^{14} + 122 T^{15} + 70 T^{16} + 35 T^{17} + 15 T^{18} + 5 T^{19} + T^{20} \)
$37$ \( 1 + 15 T + 120 T^{2} + 380 T^{3} + 435 T^{4} - 252 T^{5} - 735 T^{6} - 905 T^{7} - 645 T^{8} + 85 T^{9} + 629 T^{10} + 730 T^{11} + 540 T^{12} + 310 T^{13} + 195 T^{14} + 122 T^{15} + 70 T^{16} + 35 T^{17} + 15 T^{18} + 5 T^{19} + T^{20} \)
$41$ \( T^{20} \)
$43$ \( 1 - 10 T + 110 T^{2} - 575 T^{3} + 1850 T^{4} - 3628 T^{5} + 4160 T^{6} - 2685 T^{7} + 1050 T^{8} - 525 T^{9} + 379 T^{10} - 145 T^{11} + 45 T^{12} - 50 T^{13} + 25 T^{14} - 2 T^{15} + 5 T^{16} - 5 T^{17} + T^{20} \)
$47$ \( T^{20} \)
$53$ \( T^{20} \)
$59$ \( T^{20} \)
$61$ \( 3125 - 1875 T^{5} + 750 T^{10} - 50 T^{15} + T^{20} \)
$67$ \( 5 + 50 T + 300 T^{2} + 1125 T^{3} + 2750 T^{4} + 4380 T^{5} + 4400 T^{6} + 2525 T^{7} + 500 T^{8} - 375 T^{9} - 375 T^{10} - 125 T^{11} + 25 T^{12} + 50 T^{13} + 25 T^{14} - 5 T^{16} - 5 T^{17} + T^{20} \)
$71$ \( T^{20} \)
$73$ \( 3125 - 1875 T^{5} + 750 T^{10} - 50 T^{15} + T^{20} \)
$79$ \( 5 + 50 T + 300 T^{2} + 1125 T^{3} + 2750 T^{4} + 4380 T^{5} + 4400 T^{6} + 2525 T^{7} + 500 T^{8} - 375 T^{9} - 375 T^{10} - 125 T^{11} + 25 T^{12} + 50 T^{13} + 25 T^{14} - 5 T^{16} - 5 T^{17} + T^{20} \)
$83$ \( T^{20} \)
$89$ \( T^{20} \)
$97$ \( 1 + 18 T^{5} + 124 T^{10} + 7 T^{15} + T^{20} \)
show more
show less