Properties

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

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Newspace parameters

Level: \( N \) \(=\) \( 585 = 3^{2} \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 585.bt (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(4.67124851824\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{12})\)
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{12}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 2 \zeta_{12} q^{2} + ( - \zeta_{12}^{2} + 2) q^{3} + 2 \zeta_{12}^{2} q^{4} + ( - \zeta_{12}^{3} + \zeta_{12}) q^{5} + ( - 2 \zeta_{12}^{3} + 4 \zeta_{12}) q^{6} + 2 \zeta_{12} q^{7} + ( - 3 \zeta_{12}^{2} + 3) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 2 \zeta_{12} q^{2} + ( - \zeta_{12}^{2} + 2) q^{3} + 2 \zeta_{12}^{2} q^{4} + ( - \zeta_{12}^{3} + \zeta_{12}) q^{5} + ( - 2 \zeta_{12}^{3} + 4 \zeta_{12}) q^{6} + 2 \zeta_{12} q^{7} + ( - 3 \zeta_{12}^{2} + 3) q^{9} + 2 q^{10} - 6 \zeta_{12} q^{11} + (2 \zeta_{12}^{2} + 2) q^{12} + (2 \zeta_{12}^{3} + 3 \zeta_{12}^{2} - 2 \zeta_{12}) q^{13} + 4 \zeta_{12}^{2} q^{14} + ( - 2 \zeta_{12}^{3} + \zeta_{12}) q^{15} + ( - 4 \zeta_{12}^{2} + 4) q^{16} - q^{17} + ( - 6 \zeta_{12}^{3} + 6 \zeta_{12}) q^{18} + 8 \zeta_{12}^{3} q^{19} + 2 \zeta_{12} q^{20} + ( - 2 \zeta_{12}^{3} + 4 \zeta_{12}) q^{21} - 12 \zeta_{12}^{2} q^{22} - \zeta_{12}^{2} q^{23} + ( - \zeta_{12}^{2} + 1) q^{25} + (6 \zeta_{12}^{3} - 4) q^{26} + ( - 6 \zeta_{12}^{2} + 3) q^{27} + 4 \zeta_{12}^{3} q^{28} + (2 \zeta_{12}^{2} - 2) q^{29} + ( - 2 \zeta_{12}^{2} + 4) q^{30} + (8 \zeta_{12}^{3} - 8 \zeta_{12}) q^{31} + ( - 8 \zeta_{12}^{3} + 8 \zeta_{12}) q^{32} + (6 \zeta_{12}^{3} - 12 \zeta_{12}) q^{33} - 2 \zeta_{12} q^{34} + 2 q^{35} + 6 q^{36} - 2 \zeta_{12}^{3} q^{37} + (16 \zeta_{12}^{2} - 16) q^{38} + (4 \zeta_{12}^{3} + 3 \zeta_{12}^{2} - 2 \zeta_{12} + 3) q^{39} + (2 \zeta_{12}^{3} - 2 \zeta_{12}) q^{41} + (4 \zeta_{12}^{2} + 4) q^{42} + (5 \zeta_{12}^{2} - 5) q^{43} - 12 \zeta_{12}^{3} q^{44} - 3 \zeta_{12}^{3} q^{45} - 2 \zeta_{12}^{3} q^{46} + 10 \zeta_{12} q^{47} + ( - 8 \zeta_{12}^{2} + 4) q^{48} - 3 \zeta_{12}^{2} q^{49} + ( - 2 \zeta_{12}^{3} + 2 \zeta_{12}) q^{50} + (\zeta_{12}^{2} - 2) q^{51} + (6 \zeta_{12}^{2} - 4 \zeta_{12} - 6) q^{52} + 9 q^{53} + ( - 12 \zeta_{12}^{3} + 6 \zeta_{12}) q^{54} - 6 q^{55} + (8 \zeta_{12}^{3} + 8 \zeta_{12}) q^{57} + (4 \zeta_{12}^{3} - 4 \zeta_{12}) q^{58} + (6 \zeta_{12}^{3} - 6 \zeta_{12}) q^{59} + ( - 2 \zeta_{12}^{3} + 4 \zeta_{12}) q^{60} + ( - 5 \zeta_{12}^{2} + 5) q^{61} - 16 q^{62} + ( - 6 \zeta_{12}^{3} + 6 \zeta_{12}) q^{63} + 8 q^{64} + (2 \zeta_{12}^{2} + 3 \zeta_{12} - 2) q^{65} + ( - 12 \zeta_{12}^{2} - 12) q^{66} - 2 \zeta_{12}^{2} q^{68} + ( - \zeta_{12}^{2} - 1) q^{69} + 4 \zeta_{12} q^{70} - 10 \zeta_{12}^{3} q^{71} + 4 \zeta_{12}^{3} q^{73} + ( - 4 \zeta_{12}^{2} + 4) q^{74} + ( - 2 \zeta_{12}^{2} + 1) q^{75} + (16 \zeta_{12}^{3} - 16 \zeta_{12}) q^{76} - 12 \zeta_{12}^{2} q^{77} + (6 \zeta_{12}^{3} + 4 \zeta_{12}^{2} + 6 \zeta_{12} - 8) q^{78} + (\zeta_{12}^{2} - 1) q^{79} - 4 \zeta_{12}^{3} q^{80} - 9 \zeta_{12}^{2} q^{81} - 4 q^{82} + (4 \zeta_{12}^{3} + 4 \zeta_{12}) q^{84} + (\zeta_{12}^{3} - \zeta_{12}) q^{85} + (10 \zeta_{12}^{3} - 10 \zeta_{12}) q^{86} + (4 \zeta_{12}^{2} - 2) q^{87} + ( - 6 \zeta_{12}^{2} + 6) q^{90} + (6 \zeta_{12}^{3} - 4) q^{91} + ( - 2 \zeta_{12}^{2} + 2) q^{92} + (16 \zeta_{12}^{3} - 8 \zeta_{12}) q^{93} + 20 \zeta_{12}^{2} q^{94} + 8 \zeta_{12}^{2} q^{95} + ( - 16 \zeta_{12}^{3} + 8 \zeta_{12}) q^{96} - 10 \zeta_{12} q^{97} - 6 \zeta_{12}^{3} q^{98} + (18 \zeta_{12}^{3} - 18 \zeta_{12}) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 6 q^{3} + 4 q^{4} + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 6 q^{3} + 4 q^{4} + 6 q^{9} + 8 q^{10} + 12 q^{12} + 6 q^{13} + 8 q^{14} + 8 q^{16} - 4 q^{17} - 24 q^{22} - 2 q^{23} + 2 q^{25} - 16 q^{26} - 4 q^{29} + 12 q^{30} + 8 q^{35} + 24 q^{36} - 32 q^{38} + 18 q^{39} + 24 q^{42} - 10 q^{43} - 6 q^{49} - 6 q^{51} - 12 q^{52} + 36 q^{53} - 24 q^{55} + 10 q^{61} - 64 q^{62} + 32 q^{64} - 4 q^{65} - 72 q^{66} - 4 q^{68} - 6 q^{69} + 8 q^{74} - 24 q^{77} - 24 q^{78} - 2 q^{79} - 18 q^{81} - 16 q^{82} + 12 q^{90} - 16 q^{91} + 4 q^{92} + 40 q^{94} + 16 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(326\) \(352\) \(496\)
\(\chi(n)\) \(-\zeta_{12}^{2}\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
376.1
−0.866025 0.500000i
0.866025 + 0.500000i
−0.866025 + 0.500000i
0.866025 0.500000i
−1.73205 1.00000i 1.50000 0.866025i 1.00000 + 1.73205i −0.866025 + 0.500000i −3.46410 −1.73205 1.00000i 0 1.50000 2.59808i 2.00000
376.2 1.73205 + 1.00000i 1.50000 0.866025i 1.00000 + 1.73205i 0.866025 0.500000i 3.46410 1.73205 + 1.00000i 0 1.50000 2.59808i 2.00000
571.1 −1.73205 + 1.00000i 1.50000 + 0.866025i 1.00000 1.73205i −0.866025 0.500000i −3.46410 −1.73205 + 1.00000i 0 1.50000 + 2.59808i 2.00000
571.2 1.73205 1.00000i 1.50000 + 0.866025i 1.00000 1.73205i 0.866025 + 0.500000i 3.46410 1.73205 1.00000i 0 1.50000 + 2.59808i 2.00000
\(n\): 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
13.b even 2 1 inner
117.t even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 585.2.bt.a 4
9.c even 3 1 inner 585.2.bt.a 4
13.b even 2 1 inner 585.2.bt.a 4
117.t even 6 1 inner 585.2.bt.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
585.2.bt.a 4 1.a even 1 1 trivial
585.2.bt.a 4 9.c even 3 1 inner
585.2.bt.a 4 13.b even 2 1 inner
585.2.bt.a 4 117.t even 6 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{4} - 4T_{2}^{2} + 16 \) acting on \(S_{2}^{\mathrm{new}}(585, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} - 4T^{2} + 16 \) Copy content Toggle raw display
$3$ \( (T^{2} - 3 T + 3)^{2} \) Copy content Toggle raw display
$5$ \( T^{4} - T^{2} + 1 \) Copy content Toggle raw display
$7$ \( T^{4} - 4T^{2} + 16 \) Copy content Toggle raw display
$11$ \( T^{4} - 36T^{2} + 1296 \) Copy content Toggle raw display
$13$ \( T^{4} - 6 T^{3} + 23 T^{2} - 78 T + 169 \) Copy content Toggle raw display
$17$ \( (T + 1)^{4} \) Copy content Toggle raw display
$19$ \( (T^{2} + 64)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} + T + 1)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} + 2 T + 4)^{2} \) Copy content Toggle raw display
$31$ \( T^{4} - 64T^{2} + 4096 \) Copy content Toggle raw display
$37$ \( (T^{2} + 4)^{2} \) Copy content Toggle raw display
$41$ \( T^{4} - 4T^{2} + 16 \) Copy content Toggle raw display
$43$ \( (T^{2} + 5 T + 25)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} - 100 T^{2} + 10000 \) Copy content Toggle raw display
$53$ \( (T - 9)^{4} \) Copy content Toggle raw display
$59$ \( T^{4} - 36T^{2} + 1296 \) Copy content Toggle raw display
$61$ \( (T^{2} - 5 T + 25)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} \) Copy content Toggle raw display
$71$ \( (T^{2} + 100)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} + 16)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} + T + 1)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} \) Copy content Toggle raw display
$89$ \( T^{4} \) Copy content Toggle raw display
$97$ \( T^{4} - 100 T^{2} + 10000 \) Copy content Toggle raw display
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