Properties

Label 585.2.b.g
Level $585$
Weight $2$
Character orbit 585.b
Analytic conductor $4.671$
Analytic rank $0$
Dimension $6$
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.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(4.67124851824\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.5089536.1
Defining polynomial: \( x^{6} - 2x^{5} + 2x^{4} + 2x^{3} + 16x^{2} - 24x + 18 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 65)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{5} q^{2} + (\beta_{3} - 2) q^{4} + \beta_{4} q^{5} + ( - \beta_{4} - \beta_{2}) q^{7} + (\beta_{5} + \beta_{4} + \beta_{2}) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{5} q^{2} + (\beta_{3} - 2) q^{4} + \beta_{4} q^{5} + ( - \beta_{4} - \beta_{2}) q^{7} + (\beta_{5} + \beta_{4} + \beta_{2}) q^{8} - \beta_1 q^{10} + ( - \beta_{5} + 2 \beta_{4} - \beta_{2}) q^{11} + ( - \beta_{5} + \beta_{4} + \beta_{2} + \beta_1 - 1) q^{13} + (\beta_{3} + 2 \beta_1 - 1) q^{14} + ( - 2 \beta_1 + 1) q^{16} + (2 \beta_{3} - 2) q^{17} + (\beta_{5} + 2 \beta_{4} - \beta_{2}) q^{19} + ( - 2 \beta_{4} - \beta_{2}) q^{20} + (2 \beta_{3} - \beta_1 - 5) q^{22} + ( - \beta_1 - 3) q^{23} - q^{25} + (\beta_{5} + 4 \beta_{4} + \beta_{2} - 2 \beta_1 - 3) q^{26} + (2 \beta_{5} + 7 \beta_{4} + \beta_{2}) q^{28} + ( - \beta_{3} + 1) q^{29} + ( - \beta_{5} + 4 \beta_{4} + \beta_{2}) q^{31} + (\beta_{5} - 6 \beta_{4}) q^{32} + (4 \beta_{5} + 2 \beta_{4} + 2 \beta_{2}) q^{34} + ( - \beta_{3} + 1) q^{35} + ( - 7 \beta_{4} - \beta_{2}) q^{37} + ( - \beta_1 + 3) q^{38} + (\beta_{3} + \beta_1 - 1) q^{40} + ( - 2 \beta_{5} + 2 \beta_{4} + 2 \beta_{2}) q^{41} + (2 \beta_{3} - \beta_1 + 3) q^{43} + (5 \beta_{5} + 2 \beta_{4} - \beta_{2}) q^{44} + (3 \beta_{5} - 4 \beta_{4} - \beta_{2}) q^{46} + ( - \beta_{4} - \beta_{2}) q^{47} + (2 \beta_1 + 1) q^{49} + \beta_{5} q^{50} + (\beta_{5} - 6 \beta_{4} - 2 \beta_{3} - 3 \beta_1 + 3) q^{52} + (2 \beta_{3} + 2 \beta_1 - 2) q^{53} + ( - \beta_{3} - \beta_1 - 2) q^{55} + ( - \beta_{3} - 4 \beta_1 + 7) q^{56} + ( - 2 \beta_{5} - \beta_{4} - \beta_{2}) q^{58} + ( - \beta_{5} - 4 \beta_{4} - \beta_{2}) q^{59} + ( - \beta_{3} + 2 \beta_1 - 1) q^{61} + ( - 5 \beta_1 - 3) q^{62} + ( - \beta_{3} + 2 \beta_1 + 6) q^{64} + ( - \beta_{5} - \beta_{4} + \beta_{3} - \beta_1 - 1) q^{65} + ( - 2 \beta_{5} + 7 \beta_{4} + \beta_{2}) q^{67} + ( - 2 \beta_{3} - 4 \beta_1 + 14) q^{68} + ( - 2 \beta_{5} - \beta_{4} - \beta_{2}) q^{70} + ( - \beta_{5} - 2 \beta_{4} + \beta_{2}) q^{71} + ( - 4 \beta_{5} - 5 \beta_{4} + \beta_{2}) q^{73} + (\beta_{3} + 8 \beta_1 - 1) q^{74} + ( - \beta_{5} - 3 \beta_{2}) q^{76} + ( - 2 \beta_{3} + 4 \beta_1 - 4) q^{77} + ( - 2 \beta_{3} + 4 \beta_1 + 6) q^{79} + (2 \beta_{5} + \beta_{4}) q^{80} + ( - 4 \beta_1 - 6) q^{82} + ( - 2 \beta_{5} - 5 \beta_{4} + \beta_{2}) q^{83} + ( - 2 \beta_{4} - 2 \beta_{2}) q^{85} + ( - \beta_{5} - 2 \beta_{4} + \beta_{2}) q^{86} + ( - 3 \beta_1 + 9) q^{88} + (4 \beta_{5} - 2 \beta_{4} - 2 \beta_{2}) q^{89} + (2 \beta_{5} + 2 \beta_{4} + \beta_{3} + 2 \beta_{2} + 5) q^{91} + ( - 2 \beta_{3} + 3 \beta_1 + 5) q^{92} + (\beta_{3} + 2 \beta_1 - 1) q^{94} + ( - \beta_{3} + \beta_1 - 2) q^{95} + ( - 4 \beta_{4} + 2 \beta_{2}) q^{97} + ( - \beta_{5} + 8 \beta_{4} + 2 \beta_{2}) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 10 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 10 q^{4} + 2 q^{10} - 8 q^{13} - 8 q^{14} + 10 q^{16} - 8 q^{17} - 24 q^{22} - 16 q^{23} - 6 q^{25} - 14 q^{26} + 4 q^{29} + 4 q^{35} + 20 q^{38} - 6 q^{40} + 24 q^{43} + 2 q^{49} + 20 q^{52} - 12 q^{53} - 12 q^{55} + 48 q^{56} - 12 q^{61} - 8 q^{62} + 30 q^{64} - 2 q^{65} + 88 q^{68} - 20 q^{74} - 36 q^{77} + 24 q^{79} - 28 q^{82} + 60 q^{88} + 32 q^{91} + 20 q^{92} - 8 q^{94} - 16 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - 2x^{5} + 2x^{4} + 2x^{3} + 16x^{2} - 24x + 18 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -\nu^{5} + 24\nu^{4} - 6\nu^{3} - \nu^{2} + 6\nu + 285 ) / 131 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -5\nu^{5} - 11\nu^{4} + 101\nu^{3} - 136\nu^{2} + 292\nu - 147 ) / 393 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 6\nu^{5} - 13\nu^{4} + 36\nu^{3} + 6\nu^{2} - 36\nu - 7 ) / 131 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 23\nu^{5} - 28\nu^{4} + 7\nu^{3} + 154\nu^{2} + 386\nu - 267 ) / 393 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -23\nu^{5} + 28\nu^{4} - 7\nu^{3} - 23\nu^{2} - 386\nu + 267 ) / 131 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{4} - \beta_{3} + \beta_{2} + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{5} + 3\beta_{4} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{5} + 2\beta_{4} + 2\beta_{3} + 2\beta_{2} + \beta _1 - 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{3} + 6\beta _1 - 13 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -7\beta_{5} - 12\beta_{4} + 9\beta_{3} - 9\beta_{2} + 7\beta _1 - 12 \) Copy content Toggle raw display

Character values

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

\(n\) \(326\) \(352\) \(496\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
181.1
1.66044 + 1.66044i
0.675970 0.675970i
−1.33641 1.33641i
−1.33641 + 1.33641i
0.675970 + 0.675970i
1.66044 1.66044i
2.51414i 0 −4.32088 1.00000i 0 3.32088i 5.83502i 0 2.51414
181.2 2.08613i 0 −2.35194 1.00000i 0 1.35194i 0.734191i 0 −2.08613
181.3 0.571993i 0 1.67282 1.00000i 0 2.67282i 2.10083i 0 0.571993
181.4 0.571993i 0 1.67282 1.00000i 0 2.67282i 2.10083i 0 0.571993
181.5 2.08613i 0 −2.35194 1.00000i 0 1.35194i 0.734191i 0 −2.08613
181.6 2.51414i 0 −4.32088 1.00000i 0 3.32088i 5.83502i 0 2.51414
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 181.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 585.2.b.g 6
3.b odd 2 1 65.2.c.a 6
12.b even 2 1 1040.2.k.d 6
13.b even 2 1 inner 585.2.b.g 6
13.d odd 4 1 7605.2.a.bs 3
13.d odd 4 1 7605.2.a.cc 3
15.d odd 2 1 325.2.c.g 6
15.e even 4 1 325.2.d.e 6
15.e even 4 1 325.2.d.f 6
39.d odd 2 1 65.2.c.a 6
39.f even 4 1 845.2.a.i 3
39.f even 4 1 845.2.a.k 3
39.h odd 6 2 845.2.m.h 12
39.i odd 6 2 845.2.m.h 12
39.k even 12 2 845.2.e.i 6
39.k even 12 2 845.2.e.k 6
156.h even 2 1 1040.2.k.d 6
195.e odd 2 1 325.2.c.g 6
195.n even 4 1 4225.2.a.bc 3
195.n even 4 1 4225.2.a.be 3
195.s even 4 1 325.2.d.e 6
195.s even 4 1 325.2.d.f 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
65.2.c.a 6 3.b odd 2 1
65.2.c.a 6 39.d odd 2 1
325.2.c.g 6 15.d odd 2 1
325.2.c.g 6 195.e odd 2 1
325.2.d.e 6 15.e even 4 1
325.2.d.e 6 195.s even 4 1
325.2.d.f 6 15.e even 4 1
325.2.d.f 6 195.s even 4 1
585.2.b.g 6 1.a even 1 1 trivial
585.2.b.g 6 13.b even 2 1 inner
845.2.a.i 3 39.f even 4 1
845.2.a.k 3 39.f even 4 1
845.2.e.i 6 39.k even 12 2
845.2.e.k 6 39.k even 12 2
845.2.m.h 12 39.h odd 6 2
845.2.m.h 12 39.i odd 6 2
1040.2.k.d 6 12.b even 2 1
1040.2.k.d 6 156.h even 2 1
4225.2.a.bc 3 195.n even 4 1
4225.2.a.be 3 195.n even 4 1
7605.2.a.bs 3 13.d odd 4 1
7605.2.a.cc 3 13.d odd 4 1

Hecke kernels

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

\( T_{2}^{6} + 11T_{2}^{4} + 31T_{2}^{2} + 9 \) Copy content Toggle raw display
\( T_{7}^{6} + 20T_{7}^{4} + 112T_{7}^{2} + 144 \) Copy content Toggle raw display
\( T_{17}^{3} + 4T_{17}^{2} - 32T_{17} - 96 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} + 11 T^{4} + 31 T^{2} + 9 \) Copy content Toggle raw display
$3$ \( T^{6} \) Copy content Toggle raw display
$5$ \( (T^{2} + 1)^{3} \) Copy content Toggle raw display
$7$ \( T^{6} + 20 T^{4} + 112 T^{2} + \cdots + 144 \) Copy content Toggle raw display
$11$ \( T^{6} + 48 T^{4} + 684 T^{2} + \cdots + 2916 \) Copy content Toggle raw display
$13$ \( T^{6} + 8 T^{5} + 39 T^{4} + \cdots + 2197 \) Copy content Toggle raw display
$17$ \( (T^{3} + 4 T^{2} - 32 T - 96)^{2} \) Copy content Toggle raw display
$19$ \( T^{6} + 44 T^{4} + 196 T^{2} + \cdots + 36 \) Copy content Toggle raw display
$23$ \( (T^{3} + 8 T^{2} + 16 T + 6)^{2} \) Copy content Toggle raw display
$29$ \( (T^{3} - 2 T^{2} - 8 T + 12)^{2} \) Copy content Toggle raw display
$31$ \( T^{6} + 56 T^{4} + 604 T^{2} + \cdots + 36 \) Copy content Toggle raw display
$37$ \( T^{6} + 152 T^{4} + 6256 T^{2} + \cdots + 51984 \) Copy content Toggle raw display
$41$ \( T^{6} + 92 T^{4} + 2224 T^{2} + \cdots + 5184 \) Copy content Toggle raw display
$43$ \( (T^{3} - 12 T^{2} + 12 T - 2)^{2} \) Copy content Toggle raw display
$47$ \( T^{6} + 20 T^{4} + 112 T^{2} + \cdots + 144 \) Copy content Toggle raw display
$53$ \( (T^{3} + 6 T^{2} - 60 T + 72)^{2} \) Copy content Toggle raw display
$59$ \( T^{6} + 84 T^{4} + 468 T^{2} + \cdots + 324 \) Copy content Toggle raw display
$61$ \( (T^{3} + 6 T^{2} - 12 T - 76)^{2} \) Copy content Toggle raw display
$67$ \( T^{6} + 156 T^{4} + 3168 T^{2} + \cdots + 11664 \) Copy content Toggle raw display
$71$ \( T^{6} + 44 T^{4} + 196 T^{2} + \cdots + 36 \) Copy content Toggle raw display
$73$ \( T^{6} + 296 T^{4} + 23344 T^{2} + \cdots + 266256 \) Copy content Toggle raw display
$79$ \( (T^{3} - 12 T^{2} - 48 T + 32)^{2} \) Copy content Toggle raw display
$83$ \( T^{6} + 156 T^{4} + 5760 T^{2} + \cdots + 1296 \) Copy content Toggle raw display
$89$ \( T^{6} + 192 T^{4} + 9216 T^{2} + \cdots + 82944 \) Copy content Toggle raw display
$97$ \( T^{6} + 140 T^{4} + 1456 T^{2} + \cdots + 576 \) Copy content Toggle raw display
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