Properties

Label 585.1.o.a
Level $585$
Weight $1$
Character orbit 585.o
Analytic conductor $0.292$
Analytic rank $0$
Dimension $8$
Projective image $D_{8}$
CM discriminant -39
Inner twists $8$

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Newspace parameters

Level: \( N \) \(=\) \( 585 = 3^{2} \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 585.o (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.291953032390\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(i)\)
Coefficient field: \(\Q(\zeta_{16})\)
Defining polynomial: \( x^{8} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{8}\)
Projective field: Galois closure of 8.2.12049171875.2

$q$-expansion

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

\(f(q)\) \(=\) \( q + ( - \zeta_{16}^{7} + \zeta_{16}^{5}) q^{2} + ( - \zeta_{16}^{6} + \zeta_{16}^{4} - \zeta_{16}^{2}) q^{4} + \zeta_{16}^{3} q^{5} + ( - \zeta_{16}^{7} - \zeta_{16}^{5} - \zeta_{16}^{3} + \zeta_{16}) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \zeta_{16}^{7} + \zeta_{16}^{5}) q^{2} + ( - \zeta_{16}^{6} + \zeta_{16}^{4} - \zeta_{16}^{2}) q^{4} + \zeta_{16}^{3} q^{5} + ( - \zeta_{16}^{7} - \zeta_{16}^{5} - \zeta_{16}^{3} + \zeta_{16}) q^{8} + (\zeta_{16}^{2} - 1) q^{10} + (\zeta_{16}^{7} + \zeta_{16}) q^{11} - \zeta_{16}^{2} q^{13} + ( - \zeta_{16}^{6} - \zeta_{16}^{2} + 1) q^{16} + (\zeta_{16}^{7} - \zeta_{16}^{5} + \zeta_{16}) q^{20} + (\zeta_{16}^{6} - \zeta_{16}^{4} + 1) q^{22} + \zeta_{16}^{6} q^{25} + ( - \zeta_{16}^{7} - \zeta_{16}) q^{26} + (\zeta_{16}^{7} - \zeta_{16}^{5} - \zeta_{16}^{3} - \zeta_{16}) q^{32} + (\zeta_{16}^{6} - \zeta_{16}^{4} + \zeta_{16}^{2} + 1) q^{40} + (\zeta_{16}^{5} + \zeta_{16}^{3}) q^{41} + ( - \zeta_{16}^{4} - 1) q^{43} + ( - \zeta_{16}^{7} + \zeta_{16}^{5} - \zeta_{16}^{3} + \zeta_{16}) q^{44} + ( - \zeta_{16}^{3} - \zeta_{16}) q^{47} - \zeta_{16}^{4} q^{49} + (\zeta_{16}^{5} - \zeta_{16}^{3}) q^{50} + ( - \zeta_{16}^{6} + \zeta_{16}^{4} - 1) q^{52} + (\zeta_{16}^{4} - \zeta_{16}^{2}) q^{55} + (\zeta_{16}^{7} - \zeta_{16}) q^{59} + ( - \zeta_{16}^{6} + \zeta_{16}^{2}) q^{61} + ( - \zeta_{16}^{4} - \zeta_{16}^{2}) q^{64} - \zeta_{16}^{5} q^{65} + ( - \zeta_{16}^{5} - \zeta_{16}^{3}) q^{71} + (\zeta_{16}^{6} + \zeta_{16}^{2}) q^{79} + (\zeta_{16}^{5} - \zeta_{16}^{3} + \zeta_{16}) q^{80} + (\zeta_{16}^{4} - 1) q^{82} + (\zeta_{16}^{7} + \zeta_{16}^{5}) q^{83} + (\zeta_{16}^{7} - \zeta_{16}^{5} - \zeta_{16}^{3} + \zeta_{16}) q^{86} + (\zeta_{16}^{6} + \zeta_{16}^{4} - \zeta_{16}^{2} + 2) q^{88} + ( - \zeta_{16}^{5} + \zeta_{16}^{3}) q^{89} + ( - \zeta_{16}^{6} - \zeta_{16}^{2}) q^{94} + ( - \zeta_{16}^{3} + \zeta_{16}) q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 8 q^{10} - 8 q^{16} + 8 q^{22} + 8 q^{40} - 8 q^{43} - 8 q^{52} - 8 q^{82} + 16 q^{88}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/585\mathbb{Z}\right)^\times\).

\(n\) \(326\) \(352\) \(496\)
\(\chi(n)\) \(1\) \(-\zeta_{16}^{4}\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
298.1
−0.382683 + 0.923880i
−0.923880 0.382683i
0.923880 + 0.382683i
0.382683 0.923880i
−0.382683 0.923880i
−0.923880 + 0.382683i
0.923880 0.382683i
0.382683 + 0.923880i
−1.30656 1.30656i 0 2.41421i 0.923880 0.382683i 0 0 1.84776 1.84776i 0 −1.70711 0.707107i
298.2 −0.541196 0.541196i 0 0.414214i −0.382683 0.923880i 0 0 −0.765367 + 0.765367i 0 −0.292893 + 0.707107i
298.3 0.541196 + 0.541196i 0 0.414214i 0.382683 + 0.923880i 0 0 0.765367 0.765367i 0 −0.292893 + 0.707107i
298.4 1.30656 + 1.30656i 0 2.41421i −0.923880 + 0.382683i 0 0 −1.84776 + 1.84776i 0 −1.70711 0.707107i
532.1 −1.30656 + 1.30656i 0 2.41421i 0.923880 + 0.382683i 0 0 1.84776 + 1.84776i 0 −1.70711 + 0.707107i
532.2 −0.541196 + 0.541196i 0 0.414214i −0.382683 + 0.923880i 0 0 −0.765367 0.765367i 0 −0.292893 0.707107i
532.3 0.541196 0.541196i 0 0.414214i 0.382683 0.923880i 0 0 0.765367 + 0.765367i 0 −0.292893 0.707107i
532.4 1.30656 1.30656i 0 2.41421i −0.923880 0.382683i 0 0 −1.84776 1.84776i 0 −1.70711 + 0.707107i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 532.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
39.d odd 2 1 CM by \(\Q(\sqrt{-39}) \)
3.b odd 2 1 inner
5.c odd 4 1 inner
13.b even 2 1 inner
15.e even 4 1 inner
65.h odd 4 1 inner
195.s even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 585.1.o.a 8
3.b odd 2 1 inner 585.1.o.a 8
5.b even 2 1 2925.1.o.b 8
5.c odd 4 1 inner 585.1.o.a 8
5.c odd 4 1 2925.1.o.b 8
13.b even 2 1 inner 585.1.o.a 8
15.d odd 2 1 2925.1.o.b 8
15.e even 4 1 inner 585.1.o.a 8
15.e even 4 1 2925.1.o.b 8
39.d odd 2 1 CM 585.1.o.a 8
65.d even 2 1 2925.1.o.b 8
65.h odd 4 1 inner 585.1.o.a 8
65.h odd 4 1 2925.1.o.b 8
195.e odd 2 1 2925.1.o.b 8
195.s even 4 1 inner 585.1.o.a 8
195.s even 4 1 2925.1.o.b 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
585.1.o.a 8 1.a even 1 1 trivial
585.1.o.a 8 3.b odd 2 1 inner
585.1.o.a 8 5.c odd 4 1 inner
585.1.o.a 8 13.b even 2 1 inner
585.1.o.a 8 15.e even 4 1 inner
585.1.o.a 8 39.d odd 2 1 CM
585.1.o.a 8 65.h odd 4 1 inner
585.1.o.a 8 195.s even 4 1 inner
2925.1.o.b 8 5.b even 2 1
2925.1.o.b 8 5.c odd 4 1
2925.1.o.b 8 15.d odd 2 1
2925.1.o.b 8 15.e even 4 1
2925.1.o.b 8 65.d even 2 1
2925.1.o.b 8 65.h odd 4 1
2925.1.o.b 8 195.e odd 2 1
2925.1.o.b 8 195.s even 4 1

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(585, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} + 12T^{4} + 4 \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( T^{8} + 1 \) Copy content Toggle raw display
$7$ \( T^{8} \) Copy content Toggle raw display
$11$ \( (T^{4} + 4 T^{2} + 2)^{2} \) Copy content Toggle raw display
$13$ \( (T^{4} + 1)^{2} \) Copy content Toggle raw display
$17$ \( T^{8} \) Copy content Toggle raw display
$19$ \( T^{8} \) Copy content Toggle raw display
$23$ \( T^{8} \) Copy content Toggle raw display
$29$ \( T^{8} \) Copy content Toggle raw display
$31$ \( T^{8} \) Copy content Toggle raw display
$37$ \( T^{8} \) Copy content Toggle raw display
$41$ \( (T^{4} + 4 T^{2} + 2)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} + 2 T + 2)^{4} \) Copy content Toggle raw display
$47$ \( T^{8} + 12T^{4} + 4 \) Copy content Toggle raw display
$53$ \( T^{8} \) Copy content Toggle raw display
$59$ \( (T^{4} - 4 T^{2} + 2)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} - 2)^{4} \) Copy content Toggle raw display
$67$ \( T^{8} \) Copy content Toggle raw display
$71$ \( (T^{4} + 4 T^{2} + 2)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} \) Copy content Toggle raw display
$79$ \( (T^{2} + 2)^{4} \) Copy content Toggle raw display
$83$ \( T^{8} + 12T^{4} + 4 \) Copy content Toggle raw display
$89$ \( (T^{4} - 4 T^{2} + 2)^{2} \) Copy content Toggle raw display
$97$ \( T^{8} \) Copy content Toggle raw display
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