Properties

Label 585.2.i.b
Level $585$
Weight $2$
Character orbit 585.i
Analytic conductor $4.671$
Analytic rank $0$
Dimension $2$
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.i (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(4.67124851824\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \zeta_{6} + 1) q^{2} + (2 \zeta_{6} - 1) q^{3} + \zeta_{6} q^{4} + \zeta_{6} q^{5} + (\zeta_{6} + 1) q^{6} + (2 \zeta_{6} - 2) q^{7} + 3 q^{8} - 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \zeta_{6} + 1) q^{2} + (2 \zeta_{6} - 1) q^{3} + \zeta_{6} q^{4} + \zeta_{6} q^{5} + (\zeta_{6} + 1) q^{6} + (2 \zeta_{6} - 2) q^{7} + 3 q^{8} - 3 q^{9} + q^{10} + ( - \zeta_{6} + 1) q^{11} + (\zeta_{6} - 2) q^{12} + \zeta_{6} q^{13} + 2 \zeta_{6} q^{14} + (\zeta_{6} - 2) q^{15} + ( - \zeta_{6} + 1) q^{16} + 2 q^{17} + (3 \zeta_{6} - 3) q^{18} - 3 q^{19} + (\zeta_{6} - 1) q^{20} + ( - 2 \zeta_{6} - 2) q^{21} - \zeta_{6} q^{22} + (6 \zeta_{6} - 3) q^{24} + (\zeta_{6} - 1) q^{25} + q^{26} + ( - 6 \zeta_{6} + 3) q^{27} - 2 q^{28} + (5 \zeta_{6} - 5) q^{29} + (2 \zeta_{6} - 1) q^{30} + \zeta_{6} q^{31} + 5 \zeta_{6} q^{32} + (\zeta_{6} + 1) q^{33} + ( - 2 \zeta_{6} + 2) q^{34} - 2 q^{35} - 3 \zeta_{6} q^{36} - 5 q^{37} + (3 \zeta_{6} - 3) q^{38} + (\zeta_{6} - 2) q^{39} + 3 \zeta_{6} q^{40} + (2 \zeta_{6} - 4) q^{42} + ( - 8 \zeta_{6} + 8) q^{43} + q^{44} - 3 \zeta_{6} q^{45} + ( - 2 \zeta_{6} + 2) q^{47} + (\zeta_{6} + 1) q^{48} + 3 \zeta_{6} q^{49} + \zeta_{6} q^{50} + (4 \zeta_{6} - 2) q^{51} + (\zeta_{6} - 1) q^{52} + 14 q^{53} + ( - 3 \zeta_{6} - 3) q^{54} + q^{55} + (6 \zeta_{6} - 6) q^{56} + ( - 6 \zeta_{6} + 3) q^{57} + 5 \zeta_{6} q^{58} + 9 \zeta_{6} q^{59} + ( - \zeta_{6} - 1) q^{60} + ( - \zeta_{6} + 1) q^{61} + q^{62} + ( - 6 \zeta_{6} + 6) q^{63} + 7 q^{64} + (\zeta_{6} - 1) q^{65} + ( - \zeta_{6} + 2) q^{66} - 14 \zeta_{6} q^{67} + 2 \zeta_{6} q^{68} + (2 \zeta_{6} - 2) q^{70} - 9 q^{72} + 6 q^{73} + (5 \zeta_{6} - 5) q^{74} + ( - \zeta_{6} - 1) q^{75} - 3 \zeta_{6} q^{76} + 2 \zeta_{6} q^{77} + (2 \zeta_{6} - 1) q^{78} + ( - 12 \zeta_{6} + 12) q^{79} + q^{80} + 9 q^{81} + ( - 10 \zeta_{6} + 10) q^{83} + ( - 4 \zeta_{6} + 2) q^{84} + 2 \zeta_{6} q^{85} - 8 \zeta_{6} q^{86} + ( - 5 \zeta_{6} - 5) q^{87} + ( - 3 \zeta_{6} + 3) q^{88} + 2 q^{89} - 3 q^{90} - 2 q^{91} + (\zeta_{6} - 2) q^{93} - 2 \zeta_{6} q^{94} - 3 \zeta_{6} q^{95} + (5 \zeta_{6} - 10) q^{96} + (\zeta_{6} - 1) q^{97} + 3 q^{98} + (3 \zeta_{6} - 3) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{2} + q^{4} + q^{5} + 3 q^{6} - 2 q^{7} + 6 q^{8} - 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + q^{2} + q^{4} + q^{5} + 3 q^{6} - 2 q^{7} + 6 q^{8} - 6 q^{9} + 2 q^{10} + q^{11} - 3 q^{12} + q^{13} + 2 q^{14} - 3 q^{15} + q^{16} + 4 q^{17} - 3 q^{18} - 6 q^{19} - q^{20} - 6 q^{21} - q^{22} - q^{25} + 2 q^{26} - 4 q^{28} - 5 q^{29} + q^{31} + 5 q^{32} + 3 q^{33} + 2 q^{34} - 4 q^{35} - 3 q^{36} - 10 q^{37} - 3 q^{38} - 3 q^{39} + 3 q^{40} - 6 q^{42} + 8 q^{43} + 2 q^{44} - 3 q^{45} + 2 q^{47} + 3 q^{48} + 3 q^{49} + q^{50} - q^{52} + 28 q^{53} - 9 q^{54} + 2 q^{55} - 6 q^{56} + 5 q^{58} + 9 q^{59} - 3 q^{60} + q^{61} + 2 q^{62} + 6 q^{63} + 14 q^{64} - q^{65} + 3 q^{66} - 14 q^{67} + 2 q^{68} - 2 q^{70} - 18 q^{72} + 12 q^{73} - 5 q^{74} - 3 q^{75} - 3 q^{76} + 2 q^{77} + 12 q^{79} + 2 q^{80} + 18 q^{81} + 10 q^{83} + 2 q^{85} - 8 q^{86} - 15 q^{87} + 3 q^{88} + 4 q^{89} - 6 q^{90} - 4 q^{91} - 3 q^{93} - 2 q^{94} - 3 q^{95} - 15 q^{96} - q^{97} + 6 q^{98} - 3 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(326\) \(352\) \(496\)
\(\chi(n)\) \(-\zeta_{6}\) \(1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
196.1
0.500000 + 0.866025i
0.500000 0.866025i
0.500000 0.866025i 1.73205i 0.500000 + 0.866025i 0.500000 + 0.866025i 1.50000 + 0.866025i −1.00000 + 1.73205i 3.00000 −3.00000 1.00000
391.1 0.500000 + 0.866025i 1.73205i 0.500000 0.866025i 0.500000 0.866025i 1.50000 0.866025i −1.00000 1.73205i 3.00000 −3.00000 1.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.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.i.b 2
3.b odd 2 1 1755.2.i.b 2
9.c even 3 1 inner 585.2.i.b 2
9.c even 3 1 5265.2.a.e 1
9.d odd 6 1 1755.2.i.b 2
9.d odd 6 1 5265.2.a.m 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
585.2.i.b 2 1.a even 1 1 trivial
585.2.i.b 2 9.c even 3 1 inner
1755.2.i.b 2 3.b odd 2 1
1755.2.i.b 2 9.d odd 6 1
5265.2.a.e 1 9.c even 3 1
5265.2.a.m 1 9.d odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(585, [\chi])\):

\( T_{2}^{2} - T_{2} + 1 \) Copy content Toggle raw display
\( T_{7}^{2} + 2T_{7} + 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$3$ \( T^{2} + 3 \) Copy content Toggle raw display
$5$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$7$ \( T^{2} + 2T + 4 \) Copy content Toggle raw display
$11$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$13$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$17$ \( (T - 2)^{2} \) Copy content Toggle raw display
$19$ \( (T + 3)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} \) Copy content Toggle raw display
$29$ \( T^{2} + 5T + 25 \) Copy content Toggle raw display
$31$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$37$ \( (T + 5)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} \) Copy content Toggle raw display
$43$ \( T^{2} - 8T + 64 \) Copy content Toggle raw display
$47$ \( T^{2} - 2T + 4 \) Copy content Toggle raw display
$53$ \( (T - 14)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} - 9T + 81 \) Copy content Toggle raw display
$61$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$67$ \( T^{2} + 14T + 196 \) Copy content Toggle raw display
$71$ \( T^{2} \) Copy content Toggle raw display
$73$ \( (T - 6)^{2} \) Copy content Toggle raw display
$79$ \( T^{2} - 12T + 144 \) Copy content Toggle raw display
$83$ \( T^{2} - 10T + 100 \) Copy content Toggle raw display
$89$ \( (T - 2)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + T + 1 \) Copy content Toggle raw display
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