Properties

Label 585.2.h.g
Level $585$
Weight $2$
Character orbit 585.h
Analytic conductor $4.671$
Analytic rank $0$
Dimension $12$
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.h (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(4.67124851824\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: 12.0.2593100598870016.1
Defining polynomial: \(x^{12} - 4 x^{10} + 9 x^{8} - 16 x^{6} + 36 x^{4} - 64 x^{2} + 64\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{9} \)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{10} q^{2} + ( 1 + \beta_{5} - \beta_{11} ) q^{4} + ( -\beta_{1} - \beta_{10} ) q^{5} + ( -\beta_{5} + \beta_{6} ) q^{7} + ( -\beta_{1} + \beta_{2} - \beta_{5} + \beta_{6} + \beta_{10} ) q^{8} +O(q^{10})\) \( q + \beta_{10} q^{2} + ( 1 + \beta_{5} - \beta_{11} ) q^{4} + ( -\beta_{1} - \beta_{10} ) q^{5} + ( -\beta_{5} + \beta_{6} ) q^{7} + ( -\beta_{1} + \beta_{2} - \beta_{5} + \beta_{6} + \beta_{10} ) q^{8} + ( -2 - \beta_{8} + \beta_{11} ) q^{10} + \beta_{3} q^{11} + ( \beta_{6} + \beta_{9} + \beta_{10} ) q^{13} + ( -2 - \beta_{5} + \beta_{6} - 2 \beta_{7} ) q^{14} + ( 1 - \beta_{5} + \beta_{6} - 2 \beta_{7} ) q^{16} + ( \beta_{5} - \beta_{6} - 2 \beta_{8} ) q^{17} + ( \beta_{1} + \beta_{2} + \beta_{3} ) q^{19} + ( \beta_{1} - 3 \beta_{2} - \beta_{4} + \beta_{5} - \beta_{6} - \beta_{10} ) q^{20} + ( -\beta_{5} + \beta_{8} - \beta_{9} ) q^{22} + ( -\beta_{5} + \beta_{6} + \beta_{8} - \beta_{9} + \beta_{11} ) q^{23} + ( -2 \beta_{5} - \beta_{7} + \beta_{8} ) q^{25} + ( 2 + \beta_{1} - \beta_{3} - \beta_{4} + \beta_{5} - \beta_{7} - \beta_{11} ) q^{26} + ( -2 \beta_{1} + 2 \beta_{2} - \beta_{5} + \beta_{6} ) q^{28} + ( 2 - 2 \beta_{5} + 2 \beta_{11} ) q^{29} + ( -\beta_{1} - \beta_{2} + 3 \beta_{3} ) q^{31} + ( -\beta_{5} + \beta_{6} + \beta_{10} ) q^{32} + ( -2 \beta_{1} - 4 \beta_{2} - 2 \beta_{4} + \beta_{5} - \beta_{6} ) q^{34} + ( 3 \beta_{5} - \beta_{6} + 2 \beta_{7} + 2 \beta_{9} - 2 \beta_{11} ) q^{35} + ( 2 \beta_{1} - 2 \beta_{2} + \beta_{5} - \beta_{6} - 2 \beta_{10} ) q^{37} + ( -2 \beta_{5} + 3 \beta_{8} - \beta_{9} - \beta_{11} ) q^{38} + ( -1 + \beta_{5} - \beta_{6} + \beta_{7} - 2 \beta_{8} + 2 \beta_{9} + \beta_{11} ) q^{40} + ( -\beta_{1} - \beta_{2} + 2 \beta_{3} ) q^{41} + ( -\beta_{8} - \beta_{9} + \beta_{11} ) q^{43} + ( 2 \beta_{2} - \beta_{3} + 2 \beta_{4} - \beta_{5} + \beta_{6} ) q^{44} + ( -\beta_{1} + \beta_{2} + \beta_{3} + 2 \beta_{4} - \beta_{5} + \beta_{6} ) q^{46} + ( \beta_{1} - \beta_{2} ) q^{47} + ( 5 + 2 \beta_{5} - 2 \beta_{6} + 4 \beta_{7} ) q^{49} + ( 2 \beta_{1} + 3 \beta_{2} + \beta_{4} - \beta_{5} + \beta_{6} - \beta_{10} ) q^{50} + ( -2 \beta_{1} + 2 \beta_{2} - 2 \beta_{8} + \beta_{9} + 3 \beta_{10} - \beta_{11} ) q^{52} + ( \beta_{5} + \beta_{6} + 2 \beta_{9} ) q^{53} + ( 1 + 2 \beta_{6} - \beta_{7} + \beta_{9} + \beta_{11} ) q^{55} + ( 6 + 3 \beta_{5} - \beta_{6} + 2 \beta_{7} - 2 \beta_{11} ) q^{56} + ( 2 \beta_{1} - 2 \beta_{2} + 2 \beta_{5} - 2 \beta_{6} - 2 \beta_{10} ) q^{58} + ( 2 \beta_{1} + 2 \beta_{2} + \beta_{3} ) q^{59} + ( \beta_{5} + \beta_{6} - 2 \beta_{7} - 2 \beta_{11} ) q^{61} + ( -2 \beta_{5} + \beta_{8} - 3 \beta_{9} + \beta_{11} ) q^{62} + ( -1 + 2 \beta_{5} - \beta_{6} + 2 \beta_{7} - \beta_{11} ) q^{64} + ( -2 - \beta_{2} + 2 \beta_{3} + 3 \beta_{5} - 2 \beta_{6} + \beta_{7} - \beta_{8} + \beta_{9} ) q^{65} + ( -4 \beta_{1} + 4 \beta_{2} - \beta_{5} + \beta_{6} - 4 \beta_{10} ) q^{67} + ( 5 \beta_{5} + \beta_{6} - 6 \beta_{8} + 4 \beta_{9} + 2 \beta_{11} ) q^{68} + ( 4 \beta_{1} - 2 \beta_{3} - 2 \beta_{4} + 3 \beta_{5} - 3 \beta_{6} ) q^{70} + ( -2 \beta_{1} + 2 \beta_{2} - \beta_{3} + 4 \beta_{4} - 2 \beta_{5} + 2 \beta_{6} ) q^{71} + ( -\beta_{5} + \beta_{6} + 2 \beta_{10} ) q^{73} + ( -8 - 3 \beta_{5} - \beta_{6} + 2 \beta_{7} + 4 \beta_{11} ) q^{74} + ( \beta_{1} + 5 \beta_{2} - \beta_{3} + 4 \beta_{4} - 2 \beta_{5} + 2 \beta_{6} ) q^{76} + ( -2 \beta_{6} - 2 \beta_{11} ) q^{77} + ( -6 - \beta_{5} + \beta_{6} - 2 \beta_{7} ) q^{79} + ( \beta_{1} - 2 \beta_{3} - 2 \beta_{4} + 3 \beta_{5} - 3 \beta_{6} - 3 \beta_{10} ) q^{80} + ( -\beta_{5} - 2 \beta_{9} + \beta_{11} ) q^{82} + ( -3 \beta_{1} + 3 \beta_{2} - 2 \beta_{5} + 2 \beta_{6} - 4 \beta_{10} ) q^{83} + ( 2 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} + 2 \beta_{4} - \beta_{5} + \beta_{6} - 4 \beta_{10} ) q^{85} + ( -3 \beta_{1} - 3 \beta_{2} + \beta_{3} ) q^{86} + ( -2 \beta_{5} + \beta_{6} + 3 \beta_{8} - \beta_{9} ) q^{88} + ( -\beta_{1} - \beta_{2} - 2 \beta_{3} ) q^{89} + ( 4 - 4 \beta_{1} - 2 \beta_{2} + 2 \beta_{4} - \beta_{5} + \beta_{6} ) q^{91} + ( -3 \beta_{5} - \beta_{6} + 3 \beta_{8} - 3 \beta_{9} - \beta_{11} ) q^{92} + ( -2 - \beta_{5} + \beta_{11} ) q^{94} + ( 6 + 2 \beta_{5} + 2 \beta_{6} - \beta_{8} + \beta_{9} + \beta_{11} ) q^{95} + ( 4 \beta_{1} - 4 \beta_{2} + \beta_{5} - \beta_{6} + 6 \beta_{10} ) q^{97} + ( 4 \beta_{1} - 4 \beta_{2} + 6 \beta_{5} - 6 \beta_{6} + \beta_{10} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 12 q^{4} + O(q^{10}) \) \( 12 q + 12 q^{4} - 24 q^{10} - 16 q^{14} + 20 q^{16} + 4 q^{25} + 28 q^{26} + 24 q^{29} - 8 q^{35} - 16 q^{40} + 44 q^{49} + 16 q^{55} + 64 q^{56} + 8 q^{61} - 20 q^{64} - 28 q^{65} - 104 q^{74} - 64 q^{79} + 48 q^{91} - 24 q^{94} + 72 q^{95} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{12} - 4 x^{10} + 9 x^{8} - 16 x^{6} + 36 x^{4} - 64 x^{2} + 64\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( -\nu^{11} - 4 \nu^{9} - \nu^{7} + 8 \nu^{5} - 12 \nu^{3} - 112 \nu \)\()/80\)
\(\beta_{2}\)\(=\)\((\)\( -3 \nu^{11} + 8 \nu^{9} - 43 \nu^{7} + 44 \nu^{5} - 76 \nu^{3} + 144 \nu \)\()/160\)
\(\beta_{3}\)\(=\)\((\)\( \nu^{11} - 8 \nu^{9} + 9 \nu^{7} - 20 \nu^{5} + 52 \nu^{3} - 80 \nu \)\()/32\)
\(\beta_{4}\)\(=\)\((\)\( -\nu^{11} + \nu^{9} - \nu^{7} - 7 \nu^{5} - 12 \nu^{3} - 12 \nu \)\()/20\)
\(\beta_{5}\)\(=\)\((\)\( \nu^{11} - 2 \nu^{10} - \nu^{9} + 2 \nu^{8} + \nu^{7} - 2 \nu^{6} - 13 \nu^{5} + 26 \nu^{4} + 32 \nu^{3} - 24 \nu^{2} - 28 \nu + 16 \)\()/40\)
\(\beta_{6}\)\(=\)\((\)\( -\nu^{11} - 2 \nu^{10} + \nu^{9} + 2 \nu^{8} - \nu^{7} - 2 \nu^{6} + 13 \nu^{5} + 26 \nu^{4} - 32 \nu^{3} - 24 \nu^{2} + 28 \nu + 16 \)\()/40\)
\(\beta_{7}\)\(=\)\((\)\( -2 \nu^{11} + 5 \nu^{10} + 2 \nu^{9} - 20 \nu^{8} - 2 \nu^{7} + 45 \nu^{6} + 26 \nu^{5} - 80 \nu^{4} - 64 \nu^{3} + 100 \nu^{2} + 56 \nu - 240 \)\()/80\)
\(\beta_{8}\)\(=\)\((\)\( \nu^{11} - 3 \nu^{10} - \nu^{9} - 2 \nu^{8} + \nu^{7} - 3 \nu^{6} - 13 \nu^{5} - 6 \nu^{4} + 32 \nu^{3} + 4 \nu^{2} - 28 \nu - 56 \)\()/40\)
\(\beta_{9}\)\(=\)\((\)\( 9 \nu^{10} - 24 \nu^{8} + 49 \nu^{6} - 132 \nu^{4} + 308 \nu^{2} - 352 \)\()/80\)
\(\beta_{10}\)\(=\)\((\)\( -3 \nu^{11} + 8 \nu^{9} - 13 \nu^{7} + 24 \nu^{5} - 46 \nu^{3} + 84 \nu \)\()/40\)
\(\beta_{11}\)\(=\)\((\)\( 2 \nu^{11} + 11 \nu^{10} - 2 \nu^{9} - 36 \nu^{8} + 2 \nu^{7} + 51 \nu^{6} - 26 \nu^{5} - 88 \nu^{4} + 64 \nu^{3} + 252 \nu^{2} - 56 \nu - 368 \)\()/80\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{6} - \beta_{5} + \beta_{3} + \beta_{2} - 3 \beta_{1}\)\()/4\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{9} - \beta_{7} + \beta_{6} + 1\)\()/2\)
\(\nu^{3}\)\(=\)\((\)\(2 \beta_{10} - \beta_{6} + \beta_{5} - \beta_{4} + \beta_{3} - \beta_{2}\)\()/2\)
\(\nu^{4}\)\(=\)\((\)\(\beta_{11} - \beta_{8} - \beta_{7} + 2 \beta_{6} + \beta_{5} - 1\)\()/2\)
\(\nu^{5}\)\(=\)\((\)\(2 \beta_{10} - 3 \beta_{4} - 2 \beta_{2} + 3 \beta_{1}\)\()/2\)
\(\nu^{6}\)\(=\)\((\)\(-3 \beta_{11} + 2 \beta_{9} - 3 \beta_{8} + 5 \beta_{7} - 2 \beta_{6} + 9 \beta_{5} + 3\)\()/2\)
\(\nu^{7}\)\(=\)\((\)\(2 \beta_{10} + 2 \beta_{6} - 2 \beta_{5} - \beta_{4} - 10 \beta_{2} - \beta_{1}\)\()/2\)
\(\nu^{8}\)\(=\)\((\)\(-9 \beta_{11} + 8 \beta_{9} - 7 \beta_{8} + \beta_{7} - 6 \beta_{6} + 11 \beta_{5} - 15\)\()/2\)
\(\nu^{9}\)\(=\)\((\)\(6 \beta_{10} - 10 \beta_{6} + 10 \beta_{5} - \beta_{4} - 10 \beta_{3} - 16 \beta_{2} + 7 \beta_{1}\)\()/2\)
\(\nu^{10}\)\(=\)\((\)\(7 \beta_{11} - 6 \beta_{9} - 17 \beta_{8} - 5 \beta_{7} - 10 \beta_{6} - 5 \beta_{5} - 27\)\()/2\)
\(\nu^{11}\)\(=\)\((\)\(-34 \beta_{10} - 6 \beta_{6} + 6 \beta_{5} - 7 \beta_{4} - 28 \beta_{3} + 14 \beta_{2} + 5 \beta_{1}\)\()/2\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
64.1
−1.37820 0.317122i
−1.37820 + 0.317122i
0.721581 1.21627i
0.721581 + 1.21627i
1.25694 + 0.648161i
1.25694 0.648161i
−1.25694 + 0.648161i
−1.25694 0.648161i
−0.721581 1.21627i
−0.721581 + 1.21627i
1.37820 0.317122i
1.37820 + 0.317122i
−2.51912 0 4.34596 1.45804 1.69532i 0 −1.26849 −5.90976 0 −3.67298 + 4.27072i
64.2 −2.51912 0 4.34596 1.45804 + 1.69532i 0 −1.26849 −5.90976 0 −3.67298 4.27072i
64.3 −1.61036 0 0.593272 1.11567 1.93785i 0 4.86509 2.26534 0 −1.79664 + 3.12065i
64.4 −1.61036 0 0.593272 1.11567 + 1.93785i 0 4.86509 2.26534 0 −1.79664 3.12065i
64.5 −0.246506 0 −1.93923 2.15160 0.608775i 0 −2.59264 0.971044 0 −0.530383 + 0.150067i
64.6 −0.246506 0 −1.93923 2.15160 + 0.608775i 0 −2.59264 0.971044 0 −0.530383 0.150067i
64.7 0.246506 0 −1.93923 −2.15160 0.608775i 0 2.59264 −0.971044 0 −0.530383 0.150067i
64.8 0.246506 0 −1.93923 −2.15160 + 0.608775i 0 2.59264 −0.971044 0 −0.530383 + 0.150067i
64.9 1.61036 0 0.593272 −1.11567 1.93785i 0 −4.86509 −2.26534 0 −1.79664 3.12065i
64.10 1.61036 0 0.593272 −1.11567 + 1.93785i 0 −4.86509 −2.26534 0 −1.79664 + 3.12065i
64.11 2.51912 0 4.34596 −1.45804 1.69532i 0 1.26849 5.90976 0 −3.67298 4.27072i
64.12 2.51912 0 4.34596 −1.45804 + 1.69532i 0 1.26849 5.90976 0 −3.67298 + 4.27072i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 64.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
13.b even 2 1 inner
65.d 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.h.g 12
3.b odd 2 1 195.2.h.c 12
5.b even 2 1 inner 585.2.h.g 12
12.b even 2 1 3120.2.r.n 12
13.b even 2 1 inner 585.2.h.g 12
15.d odd 2 1 195.2.h.c 12
15.e even 4 1 975.2.b.j 6
15.e even 4 1 975.2.b.l 6
39.d odd 2 1 195.2.h.c 12
60.h even 2 1 3120.2.r.n 12
65.d even 2 1 inner 585.2.h.g 12
156.h even 2 1 3120.2.r.n 12
195.e odd 2 1 195.2.h.c 12
195.s even 4 1 975.2.b.j 6
195.s even 4 1 975.2.b.l 6
780.d even 2 1 3120.2.r.n 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
195.2.h.c 12 3.b odd 2 1
195.2.h.c 12 15.d odd 2 1
195.2.h.c 12 39.d odd 2 1
195.2.h.c 12 195.e odd 2 1
585.2.h.g 12 1.a even 1 1 trivial
585.2.h.g 12 5.b even 2 1 inner
585.2.h.g 12 13.b even 2 1 inner
585.2.h.g 12 65.d even 2 1 inner
975.2.b.j 6 15.e even 4 1
975.2.b.j 6 195.s even 4 1
975.2.b.l 6 15.e even 4 1
975.2.b.l 6 195.s even 4 1
3120.2.r.n 12 12.b even 2 1
3120.2.r.n 12 60.h even 2 1
3120.2.r.n 12 156.h even 2 1
3120.2.r.n 12 780.d even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{6} - 9 T_{2}^{4} + 17 T_{2}^{2} - 1 \) acting on \(S_{2}^{\mathrm{new}}(585, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( -1 + 17 T^{2} - 9 T^{4} + T^{6} )^{2} \)
$3$ \( T^{12} \)
$5$ \( 15625 - 1250 T^{2} + 675 T^{4} - 164 T^{6} + 27 T^{8} - 2 T^{10} + T^{12} \)
$7$ \( ( -256 + 208 T^{2} - 32 T^{4} + T^{6} )^{2} \)
$11$ \( ( 64 + 84 T^{2} + 20 T^{4} + T^{6} )^{2} \)
$13$ \( 4826809 + 742586 T^{2} + 57967 T^{4} + 3500 T^{6} + 343 T^{8} + 26 T^{10} + T^{12} \)
$17$ \( ( 16384 + 2192 T^{2} + 88 T^{4} + T^{6} )^{2} \)
$19$ \( ( 6400 + 1232 T^{2} + 64 T^{4} + T^{6} )^{2} \)
$23$ \( ( 16 + T^{2} )^{6} \)
$29$ \( ( 104 - 28 T - 6 T^{2} + T^{3} )^{4} \)
$31$ \( ( 43264 + 6224 T^{2} + 160 T^{4} + T^{6} )^{2} \)
$37$ \( ( -256 + 2304 T^{2} - 132 T^{4} + T^{6} )^{2} \)
$41$ \( ( 1024 + 1460 T^{2} + 76 T^{4} + T^{6} )^{2} \)
$43$ \( ( 256 + 656 T^{2} + 72 T^{4} + T^{6} )^{2} \)
$47$ \( ( -64 + 84 T^{2} - 20 T^{4} + T^{6} )^{2} \)
$53$ \( ( 25600 + 3216 T^{2} + 104 T^{4} + T^{6} )^{2} \)
$59$ \( ( 141376 + 8500 T^{2} + 164 T^{4} + T^{6} )^{2} \)
$61$ \( ( -256 - 96 T - 2 T^{2} + T^{3} )^{4} \)
$67$ \( ( -1024 + 10064 T^{2} - 240 T^{4} + T^{6} )^{2} \)
$71$ \( ( 3356224 + 70484 T^{2} + 468 T^{4} + T^{6} )^{2} \)
$73$ \( ( -1024 + 512 T^{2} - 52 T^{4} + T^{6} )^{2} \)
$79$ \( ( 32 + 52 T + 16 T^{2} + T^{3} )^{4} \)
$83$ \( ( -12544 + 9428 T^{2} - 228 T^{4} + T^{6} )^{2} \)
$89$ \( ( 16384 + 5044 T^{2} + 140 T^{4} + T^{6} )^{2} \)
$97$ \( ( -640000 + 31232 T^{2} - 340 T^{4} + T^{6} )^{2} \)
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