Properties

Label 585.2.j.e
Level $585$
Weight $2$
Character orbit 585.j
Analytic conductor $4.671$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(4.67124851824\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{5})\)
Defining polynomial: \( x^{4} - x^{3} + 2x^{2} + x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 65)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + ( - \beta_{3} + \beta_{2} + \beta_1) q^{4} - q^{5} + (3 \beta_{3} - 2 \beta_{2} - 2 \beta_1) q^{7} + ( - 2 \beta_{2} - 1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + ( - \beta_{3} + \beta_{2} + \beta_1) q^{4} - q^{5} + (3 \beta_{3} - 2 \beta_{2} - 2 \beta_1) q^{7} + ( - 2 \beta_{2} - 1) q^{8} - \beta_1 q^{10} + (\beta_{3} + 2 \beta_1 + 1) q^{11} + (4 \beta_{3} + 1) q^{13} + (\beta_{2} + 2) q^{14} + 3 \beta_1 q^{16} + (\beta_{3} - 4 \beta_{2} - 4 \beta_1) q^{17} + (3 \beta_{3} - 2 \beta_{2} - 2 \beta_1) q^{19} + (\beta_{3} - \beta_{2} - \beta_1) q^{20} + (2 \beta_{3} + 3 \beta_{2} + 3 \beta_1) q^{22} + (5 \beta_{3} + 2 \beta_1 + 5) q^{23} + q^{25} + (4 \beta_{2} + \beta_1) q^{26} + (5 \beta_{3} - 3 \beta_1 + 5) q^{28} + ( - 5 \beta_{3} + 4 \beta_1 - 5) q^{29} + (5 \beta_{3} - \beta_{2} - \beta_1) q^{32} + ( - 3 \beta_{2} + 4) q^{34} + ( - 3 \beta_{3} + 2 \beta_{2} + 2 \beta_1) q^{35} + ( - 3 \beta_{3} - 3) q^{37} + (\beta_{2} + 2) q^{38} + (2 \beta_{2} + 1) q^{40} + (7 \beta_{3} - 8 \beta_1 + 7) q^{41} + ( - 5 \beta_{3} + 2 \beta_{2} + 2 \beta_1) q^{43} + (\beta_{2} - 1) q^{44} + (2 \beta_{3} + 7 \beta_{2} + 7 \beta_1) q^{46} + 8 \beta_{2} q^{47} + ( - 6 \beta_{3} + 8 \beta_1 - 6) q^{49} + \beta_1 q^{50} + (3 \beta_{3} + \beta_{2} - 3 \beta_1 + 4) q^{52} - 6 q^{53} + ( - \beta_{3} - 2 \beta_1 - 1) q^{55} + ( - 7 \beta_{3} + 4 \beta_{2} + 4 \beta_1) q^{56} + (4 \beta_{3} - \beta_{2} - \beta_1) q^{58} + ( - 3 \beta_{3} - 6 \beta_{2} - 6 \beta_1) q^{59} + (7 \beta_{3} - 12 \beta_{2} - 12 \beta_1) q^{61} + ( - 2 \beta_{2} + 1) q^{64} + ( - 4 \beta_{3} - 1) q^{65} + ( - \beta_{3} - 6 \beta_1 - 1) q^{67} + (5 \beta_{3} - \beta_1 + 5) q^{68} + ( - \beta_{2} - 2) q^{70} + (5 \beta_{3} - 2 \beta_{2} - 2 \beta_1) q^{71} - 6 q^{73} + ( - 3 \beta_{2} - 3 \beta_1) q^{74} + (5 \beta_{3} - 3 \beta_1 + 5) q^{76} + q^{77} - 3 \beta_1 q^{80} + ( - 8 \beta_{3} - \beta_{2} - \beta_1) q^{82} + ( - 8 \beta_{2} - 4) q^{83} + ( - \beta_{3} + 4 \beta_{2} + 4 \beta_1) q^{85} + ( - 3 \beta_{2} - 2) q^{86} + (3 \beta_{3} + 4 \beta_1 + 3) q^{88} + ( - 9 \beta_{3} - 9) q^{89} + ( - 9 \beta_{3} - 2 \beta_{2} + 6 \beta_1 - 12) q^{91} + (5 \beta_{2} + 3) q^{92} + ( - 8 \beta_{3} - 8 \beta_1 - 8) q^{94} + ( - 3 \beta_{3} + 2 \beta_{2} + 2 \beta_1) q^{95} + ( - \beta_{3} + 4 \beta_{2} + 4 \beta_1) q^{97} + (8 \beta_{3} + 2 \beta_{2} + 2 \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + q^{2} + q^{4} - 4 q^{5} - 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + q^{2} + q^{4} - 4 q^{5} - 4 q^{7} - q^{10} + 4 q^{11} - 4 q^{13} + 6 q^{14} + 3 q^{16} + 2 q^{17} - 4 q^{19} - q^{20} - 7 q^{22} + 12 q^{23} + 4 q^{25} - 7 q^{26} + 7 q^{28} - 6 q^{29} - 9 q^{32} + 22 q^{34} + 4 q^{35} - 6 q^{37} + 6 q^{38} + 6 q^{41} + 8 q^{43} - 6 q^{44} - 11 q^{46} - 16 q^{47} - 4 q^{49} + q^{50} + 5 q^{52} - 24 q^{53} - 4 q^{55} + 10 q^{56} - 7 q^{58} + 12 q^{59} - 2 q^{61} + 8 q^{64} + 4 q^{65} - 8 q^{67} + 9 q^{68} - 6 q^{70} - 8 q^{71} - 24 q^{73} + 3 q^{74} + 7 q^{76} + 4 q^{77} - 3 q^{80} + 17 q^{82} - 2 q^{85} - 2 q^{86} + 10 q^{88} - 18 q^{89} - 20 q^{91} + 2 q^{92} - 24 q^{94} + 4 q^{95} - 2 q^{97} - 18 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - x^{3} + 2x^{2} + x + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} + 1 ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{3} + 2\nu^{2} - 2\nu - 1 ) / 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} + \beta_{2} + \beta_1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{2} - 1 \) Copy content Toggle raw display

Character values

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

\(n\) \(326\) \(352\) \(496\)
\(\chi(n)\) \(1\) \(1\) \(\beta_{3}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
406.1
−0.309017 0.535233i
0.809017 + 1.40126i
−0.309017 + 0.535233i
0.809017 1.40126i
−0.309017 0.535233i 0 0.809017 1.40126i −1.00000 0 −2.11803 + 3.66854i −2.23607 0 0.309017 + 0.535233i
406.2 0.809017 + 1.40126i 0 −0.309017 + 0.535233i −1.00000 0 0.118034 0.204441i 2.23607 0 −0.809017 1.40126i
451.1 −0.309017 + 0.535233i 0 0.809017 + 1.40126i −1.00000 0 −2.11803 3.66854i −2.23607 0 0.309017 0.535233i
451.2 0.809017 1.40126i 0 −0.309017 0.535233i −1.00000 0 0.118034 + 0.204441i 2.23607 0 −0.809017 + 1.40126i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 585.2.j.e 4
3.b odd 2 1 65.2.e.a 4
12.b even 2 1 1040.2.q.n 4
13.c even 3 1 inner 585.2.j.e 4
13.c even 3 1 7605.2.a.ba 2
13.e even 6 1 7605.2.a.bf 2
15.d odd 2 1 325.2.e.b 4
15.e even 4 2 325.2.o.a 8
39.d odd 2 1 845.2.e.g 4
39.f even 4 2 845.2.m.e 8
39.h odd 6 1 845.2.a.b 2
39.h odd 6 1 845.2.e.g 4
39.i odd 6 1 65.2.e.a 4
39.i odd 6 1 845.2.a.e 2
39.k even 12 2 845.2.c.c 4
39.k even 12 2 845.2.m.e 8
156.p even 6 1 1040.2.q.n 4
195.x odd 6 1 325.2.e.b 4
195.x odd 6 1 4225.2.a.u 2
195.y odd 6 1 4225.2.a.y 2
195.bl even 12 2 325.2.o.a 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
65.2.e.a 4 3.b odd 2 1
65.2.e.a 4 39.i odd 6 1
325.2.e.b 4 15.d odd 2 1
325.2.e.b 4 195.x odd 6 1
325.2.o.a 8 15.e even 4 2
325.2.o.a 8 195.bl even 12 2
585.2.j.e 4 1.a even 1 1 trivial
585.2.j.e 4 13.c even 3 1 inner
845.2.a.b 2 39.h odd 6 1
845.2.a.e 2 39.i odd 6 1
845.2.c.c 4 39.k even 12 2
845.2.e.g 4 39.d odd 2 1
845.2.e.g 4 39.h odd 6 1
845.2.m.e 8 39.f even 4 2
845.2.m.e 8 39.k even 12 2
1040.2.q.n 4 12.b even 2 1
1040.2.q.n 4 156.p even 6 1
4225.2.a.u 2 195.x odd 6 1
4225.2.a.y 2 195.y odd 6 1
7605.2.a.ba 2 13.c even 3 1
7605.2.a.bf 2 13.e even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{4} - T_{2}^{3} + 2T_{2}^{2} + T_{2} + 1 \) acting on \(S_{2}^{\mathrm{new}}(585, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} - T^{3} + 2 T^{2} + T + 1 \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( (T + 1)^{4} \) Copy content Toggle raw display
$7$ \( T^{4} + 4 T^{3} + 17 T^{2} - 4 T + 1 \) Copy content Toggle raw display
$11$ \( T^{4} - 4 T^{3} + 17 T^{2} + 4 T + 1 \) Copy content Toggle raw display
$13$ \( (T^{2} + 2 T + 13)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} - 2 T^{3} + 23 T^{2} + 38 T + 361 \) Copy content Toggle raw display
$19$ \( T^{4} + 4 T^{3} + 17 T^{2} - 4 T + 1 \) Copy content Toggle raw display
$23$ \( T^{4} - 12 T^{3} + 113 T^{2} + \cdots + 961 \) Copy content Toggle raw display
$29$ \( T^{4} + 6 T^{3} + 47 T^{2} - 66 T + 121 \) Copy content Toggle raw display
$31$ \( T^{4} \) Copy content Toggle raw display
$37$ \( (T^{2} + 3 T + 9)^{2} \) Copy content Toggle raw display
$41$ \( T^{4} - 6 T^{3} + 107 T^{2} + \cdots + 5041 \) Copy content Toggle raw display
$43$ \( T^{4} - 8 T^{3} + 53 T^{2} - 88 T + 121 \) Copy content Toggle raw display
$47$ \( (T^{2} + 8 T - 64)^{2} \) Copy content Toggle raw display
$53$ \( (T + 6)^{4} \) Copy content Toggle raw display
$59$ \( T^{4} - 12 T^{3} + 153 T^{2} + \cdots + 81 \) Copy content Toggle raw display
$61$ \( T^{4} + 2 T^{3} + 183 T^{2} + \cdots + 32041 \) Copy content Toggle raw display
$67$ \( T^{4} + 8 T^{3} + 93 T^{2} - 232 T + 841 \) Copy content Toggle raw display
$71$ \( T^{4} + 8 T^{3} + 53 T^{2} + 88 T + 121 \) Copy content Toggle raw display
$73$ \( (T + 6)^{4} \) Copy content Toggle raw display
$79$ \( T^{4} \) Copy content Toggle raw display
$83$ \( (T^{2} - 80)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} + 9 T + 81)^{2} \) Copy content Toggle raw display
$97$ \( T^{4} + 2 T^{3} + 23 T^{2} - 38 T + 361 \) Copy content Toggle raw display
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