Properties

Label 585.2.c.c
Level $585$
Weight $2$
Character orbit 585.c
Analytic conductor $4.671$
Analytic rank $0$
Dimension $10$
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.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(4.67124851824\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} + \cdots)\)
Defining polynomial: \(x^{10} + 16 x^{8} + 90 x^{6} + 208 x^{4} + 169 x^{2} + 16\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 195)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{10} + 16 x^{8} + 90 x^{6} + 208 x^{4} + 169 x^{2} + 16\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{5} + 8 \nu^{3} + 13 \nu \)\()/4\)
\(\beta_{3}\)\(=\)\((\)\( \nu^{6} + 8 \nu^{4} + 13 \nu^{2} \)\()/4\)
\(\beta_{4}\)\(=\)\((\)\( \nu^{8} + 13 \nu^{6} + 53 \nu^{4} + 4 \nu^{3} + 65 \nu^{2} + 20 \nu \)\()/8\)
\(\beta_{5}\)\(=\)\((\)\( \nu^{7} + 11 \nu^{5} + 33 \nu^{3} + 19 \nu \)\()/4\)
\(\beta_{6}\)\(=\)\((\)\( -\nu^{8} - 13 \nu^{6} - 53 \nu^{4} + 4 \nu^{3} - 65 \nu^{2} + 20 \nu \)\()/8\)
\(\beta_{7}\)\(=\)\((\)\( \nu^{8} + 13 \nu^{6} + 53 \nu^{4} + 69 \nu^{2} + 12 \)\()/4\)
\(\beta_{8}\)\(=\)\((\)\( \nu^{9} + 15 \nu^{7} + 2 \nu^{6} + 77 \nu^{5} + 20 \nu^{4} + 155 \nu^{3} + 54 \nu^{2} + 96 \nu + 32 \)\()/8\)
\(\beta_{9}\)\(=\)\((\)\( -\nu^{9} - 15 \nu^{7} + 2 \nu^{6} - 77 \nu^{5} + 20 \nu^{4} - 155 \nu^{3} + 54 \nu^{2} - 96 \nu + 32 \)\()/8\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{7} + \beta_{6} - \beta_{4} - 3\)
\(\nu^{3}\)\(=\)\(\beta_{6} + \beta_{4} - 5 \beta_{1}\)
\(\nu^{4}\)\(=\)\(\beta_{9} + \beta_{8} - 7 \beta_{7} - 7 \beta_{6} + 7 \beta_{4} - 2 \beta_{3} + 13\)
\(\nu^{5}\)\(=\)\(-8 \beta_{6} - 8 \beta_{4} + 4 \beta_{2} + 27 \beta_{1}\)
\(\nu^{6}\)\(=\)\(-8 \beta_{9} - 8 \beta_{8} + 43 \beta_{7} + 43 \beta_{6} - 43 \beta_{4} + 20 \beta_{3} - 65\)
\(\nu^{7}\)\(=\)\(55 \beta_{6} + 4 \beta_{5} + 55 \beta_{4} - 44 \beta_{2} - 151 \beta_{1}\)
\(\nu^{8}\)\(=\)\(51 \beta_{9} + 51 \beta_{8} - 253 \beta_{7} - 257 \beta_{6} + 257 \beta_{4} - 154 \beta_{3} + 351\)
\(\nu^{9}\)\(=\)\(-4 \beta_{9} + 4 \beta_{8} - 364 \beta_{6} - 60 \beta_{5} - 364 \beta_{4} + 352 \beta_{2} + 865 \beta_{1}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
469.1
2.47948i
2.26036i
1.77159i
1.22308i
0.329386i
0.329386i
1.22308i
1.77159i
2.26036i
2.47948i
2.47948i 0 −4.14785 1.72481 1.42303i 0 0.949959i 5.32555i 0 −3.52839 4.27665i
469.2 2.26036i 0 −3.10922 2.23266 0.123438i 0 4.96953i 2.50723i 0 −0.279015 5.04661i
469.3 1.77159i 0 −1.13853 −1.51036 + 1.64888i 0 0.437634i 1.52618i 0 2.92114 + 2.67573i
469.4 1.22308i 0 0.504085 0.638796 + 2.14288i 0 4.18388i 3.06269i 0 2.62091 0.781296i
469.5 0.329386i 0 1.89150 −2.08591 + 0.805596i 0 3.70203i 1.28181i 0 0.265352 + 0.687069i
469.6 0.329386i 0 1.89150 −2.08591 0.805596i 0 3.70203i 1.28181i 0 0.265352 0.687069i
469.7 1.22308i 0 0.504085 0.638796 2.14288i 0 4.18388i 3.06269i 0 2.62091 + 0.781296i
469.8 1.77159i 0 −1.13853 −1.51036 1.64888i 0 0.437634i 1.52618i 0 2.92114 2.67573i
469.9 2.26036i 0 −3.10922 2.23266 + 0.123438i 0 4.96953i 2.50723i 0 −0.279015 + 5.04661i
469.10 2.47948i 0 −4.14785 1.72481 + 1.42303i 0 0.949959i 5.32555i 0 −3.52839 + 4.27665i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 469.10
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.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.c.c 10
3.b odd 2 1 195.2.c.b 10
5.b even 2 1 inner 585.2.c.c 10
5.c odd 4 1 2925.2.a.bl 5
5.c odd 4 1 2925.2.a.bm 5
12.b even 2 1 3120.2.l.p 10
15.d odd 2 1 195.2.c.b 10
15.e even 4 1 975.2.a.r 5
15.e even 4 1 975.2.a.s 5
60.h even 2 1 3120.2.l.p 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
195.2.c.b 10 3.b odd 2 1
195.2.c.b 10 15.d odd 2 1
585.2.c.c 10 1.a even 1 1 trivial
585.2.c.c 10 5.b even 2 1 inner
975.2.a.r 5 15.e even 4 1
975.2.a.s 5 15.e even 4 1
2925.2.a.bl 5 5.c odd 4 1
2925.2.a.bm 5 5.c odd 4 1
3120.2.l.p 10 12.b even 2 1
3120.2.l.p 10 60.h even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{10} + 16 T_{2}^{8} + 90 T_{2}^{6} + 208 T_{2}^{4} + 169 T_{2}^{2} + 16 \) acting on \(S_{2}^{\mathrm{new}}(585, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 16 + 169 T^{2} + 208 T^{4} + 90 T^{6} + 16 T^{8} + T^{10} \)
$3$ \( T^{10} \)
$5$ \( 3125 - 1250 T - 375 T^{2} + 200 T^{3} + 50 T^{4} - 68 T^{5} + 10 T^{6} + 8 T^{7} - 3 T^{8} - 2 T^{9} + T^{10} \)
$7$ \( 1024 + 6656 T^{2} + 7040 T^{4} + 1072 T^{6} + 57 T^{8} + T^{10} \)
$11$ \( ( 452 + 156 T - 100 T^{2} - 24 T^{3} + 5 T^{4} + T^{5} )^{2} \)
$13$ \( ( 1 + T^{2} )^{5} \)
$17$ \( 6400 + 34816 T^{2} + 15456 T^{4} + 1744 T^{6} + 73 T^{8} + T^{10} \)
$19$ \( ( 1280 - 608 T - 448 T^{2} - 40 T^{3} + 8 T^{4} + T^{5} )^{2} \)
$23$ \( 102400 + 915456 T^{2} + 153472 T^{4} + 8096 T^{6} + 161 T^{8} + T^{10} \)
$29$ \( ( -640 + 144 T + 192 T^{2} - 24 T^{3} - 8 T^{4} + T^{5} )^{2} \)
$31$ \( ( 128 - 1376 T + 736 T^{2} - 40 T^{3} - 12 T^{4} + T^{5} )^{2} \)
$37$ \( 350464 + 381440 T^{2} + 121824 T^{4} + 11312 T^{6} + 209 T^{8} + T^{10} \)
$41$ \( ( 196 - 84 T - 204 T^{2} - 48 T^{3} + 5 T^{4} + T^{5} )^{2} \)
$43$ \( 200704 + 258304 T^{2} + 64768 T^{4} + 4960 T^{6} + 128 T^{8} + T^{10} \)
$47$ \( 64 + 389264 T^{2} + 167552 T^{4} + 10360 T^{6} + 204 T^{8} + T^{10} \)
$53$ \( 256 + 8704 T^{2} + 9440 T^{4} + 2608 T^{6} + 217 T^{8} + T^{10} \)
$59$ \( ( -400 + 3620 T + 472 T^{2} - 108 T^{3} - 8 T^{4} + T^{5} )^{2} \)
$61$ \( ( -256 + 128 T^{2} - 13 T^{4} + T^{5} )^{2} \)
$67$ \( 15745024 + 6358016 T^{2} + 777728 T^{4} + 30336 T^{6} + 320 T^{8} + T^{10} \)
$71$ \( ( 28868 + 5580 T - 1220 T^{2} - 208 T^{3} + 5 T^{4} + T^{5} )^{2} \)
$73$ \( 223323136 + 53617920 T^{2} + 3358336 T^{4} + 62112 T^{6} + 436 T^{8} + T^{10} \)
$79$ \( ( 400 - 112 T - 728 T^{2} - 248 T^{3} + T^{4} + T^{5} )^{2} \)
$83$ \( 3685946944 + 316718992 T^{2} + 9439488 T^{4} + 116152 T^{6} + 588 T^{8} + T^{10} \)
$89$ \( ( 4900 + 5020 T + 1108 T^{2} - 56 T^{3} - 19 T^{4} + T^{5} )^{2} \)
$97$ \( 1260534016 + 187275776 T^{2} + 8686176 T^{4} + 130032 T^{6} + 649 T^{8} + T^{10} \)
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