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: \( x^{10} + 16x^{8} + 90x^{6} + 208x^{4} + 169x^{2} + 16 \) Copy content Toggle raw display
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (\beta_{7} + \beta_{6} - \beta_{4} - 1) q^{4} + ( - \beta_{7} - \beta_{6}) q^{5} + ( - \beta_{9} + \beta_{8} - \beta_{6} - \beta_{5} - \beta_{4}) q^{7} + (\beta_{6} + \beta_{4} - \beta_1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + (\beta_{7} + \beta_{6} - \beta_{4} - 1) q^{4} + ( - \beta_{7} - \beta_{6}) q^{5} + ( - \beta_{9} + \beta_{8} - \beta_{6} - \beta_{5} - \beta_{4}) q^{7} + (\beta_{6} + \beta_{4} - \beta_1) q^{8} + ( - \beta_{8} + \beta_{7} + \beta_{6} + \beta_{5} - \beta_{4} + \beta_{3} + \beta_{2} + \beta_1 + 1) q^{10} + (\beta_{9} + \beta_{8} - \beta_{7} - 2) q^{11} - \beta_{2} q^{13} + ( - 2 \beta_{7} + 2 \beta_{3} + 2) q^{14} + (\beta_{9} + \beta_{8} - \beta_{7} - \beta_{6} + \beta_{4} - 2 \beta_{3} - 1) q^{16} + ( - \beta_{9} + \beta_{8} - \beta_{6} - \beta_{5} - \beta_{4} + 2 \beta_1) q^{17} + ( - 2 \beta_{7} + 2 \beta_{3} - 2) q^{19} + ( - \beta_{9} - \beta_{8} + 2 \beta_{7} + 2 \beta_{6} - \beta_{4} + \beta_{3} - 2 \beta_{2} + \cdots - 2) q^{20}+ \cdots + ( - 4 \beta_{9} + 4 \beta_{8} - 8 \beta_{6} - 12 \beta_{5} - 8 \beta_{4} - \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 12 q^{4} + 2 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 10 q - 12 q^{4} + 2 q^{5} + 4 q^{10} - 10 q^{11} + 24 q^{14} - 16 q^{19} - 32 q^{20} + 10 q^{25} + 16 q^{29} + 24 q^{31} - 40 q^{34} - 12 q^{35} + 36 q^{40} - 10 q^{41} + 36 q^{44} - 24 q^{46} - 44 q^{49} - 40 q^{50} + 2 q^{55} + 16 q^{59} + 26 q^{61} + 32 q^{64} + 56 q^{70} - 10 q^{71} + 24 q^{74} - 2 q^{79} + 12 q^{80} - 4 q^{85} + 38 q^{89} + 10 q^{91} + 24 q^{94} + 48 q^{95}+O(q^{100}) \) Copy content Toggle raw display

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

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{5} + 8\nu^{3} + 13\nu ) / 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{6} + 8\nu^{4} + 13\nu^{2} ) / 4 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{8} + 13\nu^{6} + 53\nu^{4} + 4\nu^{3} + 65\nu^{2} + 20\nu ) / 8 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{7} + 11\nu^{5} + 33\nu^{3} + 19\nu ) / 4 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -\nu^{8} - 13\nu^{6} - 53\nu^{4} + 4\nu^{3} - 65\nu^{2} + 20\nu ) / 8 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( \nu^{8} + 13\nu^{6} + 53\nu^{4} + 69\nu^{2} + 12 ) / 4 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( \nu^{9} + 15\nu^{7} + 2\nu^{6} + 77\nu^{5} + 20\nu^{4} + 155\nu^{3} + 54\nu^{2} + 96\nu + 32 ) / 8 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( -\nu^{9} - 15\nu^{7} + 2\nu^{6} - 77\nu^{5} + 20\nu^{4} - 155\nu^{3} + 54\nu^{2} - 96\nu + 32 ) / 8 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{7} + \beta_{6} - \beta_{4} - 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{6} + \beta_{4} - 5\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{9} + \beta_{8} - 7\beta_{7} - 7\beta_{6} + 7\beta_{4} - 2\beta_{3} + 13 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -8\beta_{6} - 8\beta_{4} + 4\beta_{2} + 27\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -8\beta_{9} - 8\beta_{8} + 43\beta_{7} + 43\beta_{6} - 43\beta_{4} + 20\beta_{3} - 65 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 55\beta_{6} + 4\beta_{5} + 55\beta_{4} - 44\beta_{2} - 151\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 51\beta_{9} + 51\beta_{8} - 253\beta_{7} - 257\beta_{6} + 257\beta_{4} - 154\beta_{3} + 351 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( -4\beta_{9} + 4\beta_{8} - 364\beta_{6} - 60\beta_{5} - 364\beta_{4} + 352\beta_{2} + 865\beta_1 \) Copy content Toggle raw display

Character values

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

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

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 \( T_{2}^{10} + 16T_{2}^{8} + 90T_{2}^{6} + 208T_{2}^{4} + 169T_{2}^{2} + 16 \) acting on \(S_{2}^{\mathrm{new}}(585, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} + 16 T^{8} + 90 T^{6} + 208 T^{4} + \cdots + 16 \) Copy content Toggle raw display
$3$ \( T^{10} \) Copy content Toggle raw display
$5$ \( T^{10} - 2 T^{9} - 3 T^{8} + 8 T^{7} + \cdots + 3125 \) Copy content Toggle raw display
$7$ \( T^{10} + 57 T^{8} + 1072 T^{6} + \cdots + 1024 \) Copy content Toggle raw display
$11$ \( (T^{5} + 5 T^{4} - 24 T^{3} - 100 T^{2} + \cdots + 452)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} + 1)^{5} \) Copy content Toggle raw display
$17$ \( T^{10} + 73 T^{8} + 1744 T^{6} + \cdots + 6400 \) Copy content Toggle raw display
$19$ \( (T^{5} + 8 T^{4} - 40 T^{3} - 448 T^{2} + \cdots + 1280)^{2} \) Copy content Toggle raw display
$23$ \( T^{10} + 161 T^{8} + 8096 T^{6} + \cdots + 102400 \) Copy content Toggle raw display
$29$ \( (T^{5} - 8 T^{4} - 24 T^{3} + 192 T^{2} + \cdots - 640)^{2} \) Copy content Toggle raw display
$31$ \( (T^{5} - 12 T^{4} - 40 T^{3} + 736 T^{2} + \cdots + 128)^{2} \) Copy content Toggle raw display
$37$ \( T^{10} + 209 T^{8} + 11312 T^{6} + \cdots + 350464 \) Copy content Toggle raw display
$41$ \( (T^{5} + 5 T^{4} - 48 T^{3} - 204 T^{2} + \cdots + 196)^{2} \) Copy content Toggle raw display
$43$ \( T^{10} + 128 T^{8} + 4960 T^{6} + \cdots + 200704 \) Copy content Toggle raw display
$47$ \( T^{10} + 204 T^{8} + 10360 T^{6} + \cdots + 64 \) Copy content Toggle raw display
$53$ \( T^{10} + 217 T^{8} + 2608 T^{6} + \cdots + 256 \) Copy content Toggle raw display
$59$ \( (T^{5} - 8 T^{4} - 108 T^{3} + 472 T^{2} + \cdots - 400)^{2} \) Copy content Toggle raw display
$61$ \( (T^{5} - 13 T^{4} + 128 T^{2} - 256)^{2} \) Copy content Toggle raw display
$67$ \( T^{10} + 320 T^{8} + \cdots + 15745024 \) Copy content Toggle raw display
$71$ \( (T^{5} + 5 T^{4} - 208 T^{3} - 1220 T^{2} + \cdots + 28868)^{2} \) Copy content Toggle raw display
$73$ \( T^{10} + 436 T^{8} + \cdots + 223323136 \) Copy content Toggle raw display
$79$ \( (T^{5} + T^{4} - 248 T^{3} - 728 T^{2} + \cdots + 400)^{2} \) Copy content Toggle raw display
$83$ \( T^{10} + 588 T^{8} + \cdots + 3685946944 \) Copy content Toggle raw display
$89$ \( (T^{5} - 19 T^{4} - 56 T^{3} + 1108 T^{2} + \cdots + 4900)^{2} \) Copy content Toggle raw display
$97$ \( T^{10} + 649 T^{8} + \cdots + 1260534016 \) Copy content Toggle raw display
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