# Properties

 Label 580.1.y.c Level $580$ Weight $1$ Character orbit 580.y Analytic conductor $0.289$ Analytic rank $0$ Dimension $12$ Projective image $D_{14}$ CM discriminant -20 Inner twists $8$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$580 = 2^{2} \cdot 5 \cdot 29$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 580.y (of order $$14$$, degree $$6$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.289457707327$$ Analytic rank: $$0$$ Dimension: $$12$$ Relative dimension: $$2$$ over $$\Q(\zeta_{14})$$ Coefficient field: $$\Q(\zeta_{28})$$ Defining polynomial: $$x^{12} - x^{10} + x^{8} - x^{6} + x^{4} - x^{2} + 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image: $$D_{14}$$ Projective field: Galois closure of $$\mathbb{Q}[x]/(x^{14} + \cdots)$$

## $q$-expansion

The $$q$$-expansion and trace form are shown below.

 $$f(q)$$ $$=$$ $$q + \zeta_{28}^{9} q^{2} + ( \zeta_{28} + \zeta_{28}^{5} ) q^{3} -\zeta_{28}^{4} q^{4} -\zeta_{28}^{2} q^{5} + ( -1 + \zeta_{28}^{10} ) q^{6} + ( \zeta_{28}^{7} + \zeta_{28}^{13} ) q^{7} -\zeta_{28}^{13} q^{8} + ( \zeta_{28}^{2} + \zeta_{28}^{6} + \zeta_{28}^{10} ) q^{9} +O(q^{10})$$ $$q + \zeta_{28}^{9} q^{2} + ( \zeta_{28} + \zeta_{28}^{5} ) q^{3} -\zeta_{28}^{4} q^{4} -\zeta_{28}^{2} q^{5} + ( -1 + \zeta_{28}^{10} ) q^{6} + ( \zeta_{28}^{7} + \zeta_{28}^{13} ) q^{7} -\zeta_{28}^{13} q^{8} + ( \zeta_{28}^{2} + \zeta_{28}^{6} + \zeta_{28}^{10} ) q^{9} -\zeta_{28}^{11} q^{10} + ( -\zeta_{28}^{5} - \zeta_{28}^{9} ) q^{12} + ( -\zeta_{28}^{2} - \zeta_{28}^{8} ) q^{14} + ( -\zeta_{28}^{3} - \zeta_{28}^{7} ) q^{15} + \zeta_{28}^{8} q^{16} + ( -\zeta_{28} - \zeta_{28}^{5} + \zeta_{28}^{11} ) q^{18} + \zeta_{28}^{6} q^{20} + ( -1 - \zeta_{28}^{4} + \zeta_{28}^{8} + \zeta_{28}^{12} ) q^{21} + ( \zeta_{28}^{3} - \zeta_{28}^{7} ) q^{23} + ( 1 + \zeta_{28}^{4} ) q^{24} + \zeta_{28}^{4} q^{25} + ( -\zeta_{28} + \zeta_{28}^{3} + \zeta_{28}^{7} + \zeta_{28}^{11} ) q^{27} + ( \zeta_{28}^{3} - \zeta_{28}^{11} ) q^{28} -\zeta_{28}^{10} q^{29} + ( \zeta_{28}^{2} - \zeta_{28}^{12} ) q^{30} -\zeta_{28}^{3} q^{32} + ( \zeta_{28} - \zeta_{28}^{9} ) q^{35} + ( 1 - \zeta_{28}^{6} - \zeta_{28}^{10} ) q^{36} -\zeta_{28} q^{40} + ( -\zeta_{28}^{6} - \zeta_{28}^{8} ) q^{41} + ( -\zeta_{28}^{3} - \zeta_{28}^{7} - \zeta_{28}^{9} - \zeta_{28}^{13} ) q^{42} + ( \zeta_{28}^{11} - \zeta_{28}^{13} ) q^{43} + ( -\zeta_{28}^{4} - \zeta_{28}^{8} - \zeta_{28}^{12} ) q^{45} + ( \zeta_{28}^{2} + \zeta_{28}^{12} ) q^{46} + ( \zeta_{28}^{5} - \zeta_{28}^{11} ) q^{47} + ( \zeta_{28}^{9} + \zeta_{28}^{13} ) q^{48} + ( -1 - \zeta_{28}^{6} - \zeta_{28}^{12} ) q^{49} + \zeta_{28}^{13} q^{50} + ( -\zeta_{28}^{2} - \zeta_{28}^{6} - \zeta_{28}^{10} + \zeta_{28}^{12} ) q^{54} + ( \zeta_{28}^{6} + \zeta_{28}^{12} ) q^{56} + \zeta_{28}^{5} q^{58} + ( \zeta_{28}^{7} + \zeta_{28}^{11} ) q^{60} + ( -\zeta_{28} - \zeta_{28}^{3} - \zeta_{28}^{5} + \zeta_{28}^{13} ) q^{63} -\zeta_{28}^{12} q^{64} + ( \zeta_{28}^{4} - \zeta_{28}^{12} ) q^{69} + ( \zeta_{28}^{4} + \zeta_{28}^{10} ) q^{70} + ( \zeta_{28} + \zeta_{28}^{5} + \zeta_{28}^{9} ) q^{72} + ( \zeta_{28}^{5} + \zeta_{28}^{9} ) q^{75} -\zeta_{28}^{10} q^{80} + ( -\zeta_{28}^{2} + \zeta_{28}^{4} - \zeta_{28}^{6} + \zeta_{28}^{8} + \zeta_{28}^{12} ) q^{81} + ( \zeta_{28} + \zeta_{28}^{3} ) q^{82} + ( -\zeta_{28}^{3} - \zeta_{28}^{5} ) q^{83} + ( \zeta_{28}^{2} + \zeta_{28}^{4} + \zeta_{28}^{8} - \zeta_{28}^{12} ) q^{84} + ( -\zeta_{28}^{6} + \zeta_{28}^{8} ) q^{86} + ( \zeta_{28} - \zeta_{28}^{11} ) q^{87} + ( \zeta_{28}^{6} + \zeta_{28}^{12} ) q^{89} + ( \zeta_{28}^{3} + \zeta_{28}^{7} - \zeta_{28}^{13} ) q^{90} + ( -\zeta_{28}^{7} + \zeta_{28}^{11} ) q^{92} + ( -1 + \zeta_{28}^{6} ) q^{94} + ( -\zeta_{28}^{4} - \zeta_{28}^{8} ) q^{96} + ( \zeta_{28} + \zeta_{28}^{7} - \zeta_{28}^{9} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$12q + 2q^{4} - 2q^{5} - 10q^{6} + 6q^{9} + O(q^{10})$$ $$12q + 2q^{4} - 2q^{5} - 10q^{6} + 6q^{9} - 2q^{16} + 2q^{20} - 14q^{21} + 10q^{24} - 2q^{25} - 2q^{29} + 4q^{30} + 8q^{36} + 6q^{45} - 12q^{49} - 8q^{54} + 2q^{64} - 2q^{80} - 10q^{81} - 4q^{86} - 10q^{94} + 4q^{96} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/580\mathbb{Z}\right)^\times$$.

 $$n$$ $$117$$ $$291$$ $$321$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$-\zeta_{28}^{4}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
179.1
 0.433884 − 0.900969i −0.433884 + 0.900969i 0.974928 − 0.222521i −0.974928 + 0.222521i −0.781831 − 0.623490i 0.781831 + 0.623490i 0.974928 + 0.222521i −0.974928 − 0.222521i −0.781831 + 0.623490i 0.781831 − 0.623490i 0.433884 + 0.900969i −0.433884 − 0.900969i
−0.781831 + 0.623490i 1.21572 0.277479i 0.222521 0.974928i 0.623490 + 0.781831i −0.777479 + 0.974928i −0.433884 1.90097i 0.433884 + 0.900969i 0.500000 0.240787i −0.974928 0.222521i
179.2 0.781831 0.623490i −1.21572 + 0.277479i 0.222521 0.974928i 0.623490 + 0.781831i −0.777479 + 0.974928i 0.433884 + 1.90097i −0.433884 0.900969i 0.500000 0.240787i 0.974928 + 0.222521i
299.1 −0.433884 0.900969i 1.40881 1.12349i −0.623490 + 0.781831i −0.900969 + 0.433884i −1.62349 0.781831i −0.974928 1.22252i 0.974928 + 0.222521i 0.500000 2.19064i 0.781831 + 0.623490i
299.2 0.433884 + 0.900969i −1.40881 + 1.12349i −0.623490 + 0.781831i −0.900969 + 0.433884i −1.62349 0.781831i 0.974928 + 1.22252i −0.974928 0.222521i 0.500000 2.19064i −0.781831 0.623490i
399.1 −0.974928 + 0.222521i 0.193096 0.400969i 0.900969 0.433884i −0.222521 0.974928i −0.0990311 + 0.433884i 0.781831 + 0.376510i −0.781831 + 0.623490i 0.500000 + 0.626980i 0.433884 + 0.900969i
399.2 0.974928 0.222521i −0.193096 + 0.400969i 0.900969 0.433884i −0.222521 0.974928i −0.0990311 + 0.433884i −0.781831 0.376510i 0.781831 0.623490i 0.500000 + 0.626980i −0.433884 0.900969i
419.1 −0.433884 + 0.900969i 1.40881 + 1.12349i −0.623490 0.781831i −0.900969 0.433884i −1.62349 + 0.781831i −0.974928 + 1.22252i 0.974928 0.222521i 0.500000 + 2.19064i 0.781831 0.623490i
419.2 0.433884 0.900969i −1.40881 1.12349i −0.623490 0.781831i −0.900969 0.433884i −1.62349 + 0.781831i 0.974928 1.22252i −0.974928 + 0.222521i 0.500000 + 2.19064i −0.781831 + 0.623490i
439.1 −0.974928 0.222521i 0.193096 + 0.400969i 0.900969 + 0.433884i −0.222521 + 0.974928i −0.0990311 0.433884i 0.781831 0.376510i −0.781831 0.623490i 0.500000 0.626980i 0.433884 0.900969i
439.2 0.974928 + 0.222521i −0.193096 0.400969i 0.900969 + 0.433884i −0.222521 + 0.974928i −0.0990311 0.433884i −0.781831 + 0.376510i 0.781831 + 0.623490i 0.500000 0.626980i −0.433884 + 0.900969i
499.1 −0.781831 0.623490i 1.21572 + 0.277479i 0.222521 + 0.974928i 0.623490 0.781831i −0.777479 0.974928i −0.433884 + 1.90097i 0.433884 0.900969i 0.500000 + 0.240787i −0.974928 + 0.222521i
499.2 0.781831 + 0.623490i −1.21572 0.277479i 0.222521 + 0.974928i 0.623490 0.781831i −0.777479 0.974928i 0.433884 1.90097i −0.433884 + 0.900969i 0.500000 + 0.240787i 0.974928 0.222521i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 499.2 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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
29.e even 14 1 inner
116.h odd 14 1 inner
145.l even 14 1 inner
580.y odd 14 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 580.1.y.c 12
4.b odd 2 1 inner 580.1.y.c 12
5.b even 2 1 inner 580.1.y.c 12
5.c odd 4 1 2900.1.be.a 6
5.c odd 4 1 2900.1.be.b 6
20.d odd 2 1 CM 580.1.y.c 12
20.e even 4 1 2900.1.be.a 6
20.e even 4 1 2900.1.be.b 6
29.e even 14 1 inner 580.1.y.c 12
116.h odd 14 1 inner 580.1.y.c 12
145.l even 14 1 inner 580.1.y.c 12
145.q odd 28 1 2900.1.be.a 6
145.q odd 28 1 2900.1.be.b 6
580.y odd 14 1 inner 580.1.y.c 12
580.bh even 28 1 2900.1.be.a 6
580.bh even 28 1 2900.1.be.b 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
580.1.y.c 12 1.a even 1 1 trivial
580.1.y.c 12 4.b odd 2 1 inner
580.1.y.c 12 5.b even 2 1 inner
580.1.y.c 12 20.d odd 2 1 CM
580.1.y.c 12 29.e even 14 1 inner
580.1.y.c 12 116.h odd 14 1 inner
580.1.y.c 12 145.l even 14 1 inner
580.1.y.c 12 580.y odd 14 1 inner
2900.1.be.a 6 5.c odd 4 1
2900.1.be.a 6 20.e even 4 1
2900.1.be.a 6 145.q odd 28 1
2900.1.be.a 6 580.bh even 28 1
2900.1.be.b 6 5.c odd 4 1
2900.1.be.b 6 20.e even 4 1
2900.1.be.b 6 145.q odd 28 1
2900.1.be.b 6 580.bh even 28 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{1}^{\mathrm{new}}(580, [\chi])$$:

 $$T_{3}^{12} - 4 T_{3}^{10} + 16 T_{3}^{8} - 29 T_{3}^{6} + 18 T_{3}^{4} + 5 T_{3}^{2} + 1$$ $$T_{13}$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12}$$
$3$ $$1 + 5 T^{2} + 18 T^{4} - 29 T^{6} + 16 T^{8} - 4 T^{10} + T^{12}$$
$5$ $$( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2}$$
$7$ $$49 - 49 T^{2} + 49 T^{4} + 35 T^{6} + 21 T^{8} + 7 T^{10} + T^{12}$$
$11$ $$T^{12}$$
$13$ $$T^{12}$$
$17$ $$T^{12}$$
$19$ $$T^{12}$$
$23$ $$49 - 49 T^{2} + 49 T^{4} + 35 T^{6} + 21 T^{8} + 7 T^{10} + T^{12}$$
$29$ $$( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2}$$
$31$ $$T^{12}$$
$37$ $$T^{12}$$
$41$ $$( 7 + 14 T^{2} + 7 T^{4} + T^{6} )^{2}$$
$43$ $$1 + 5 T^{2} + 18 T^{4} - 29 T^{6} + 16 T^{8} - 4 T^{10} + T^{12}$$
$47$ $$1 + 5 T^{2} + 18 T^{4} - 29 T^{6} + 16 T^{8} - 4 T^{10} + T^{12}$$
$53$ $$T^{12}$$
$59$ $$T^{12}$$
$61$ $$T^{12}$$
$67$ $$T^{12}$$
$71$ $$T^{12}$$
$73$ $$T^{12}$$
$79$ $$T^{12}$$
$83$ $$49 + 98 T^{2} + 49 T^{4} - 14 T^{6} + 14 T^{8} + T^{12}$$
$89$ $$( 7 + 14 T + 7 T^{2} + T^{6} )^{2}$$
$97$ $$T^{12}$$