# 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

## 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: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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

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