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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: \( x^{12} - 4x^{10} + 9x^{8} - 16x^{6} + 36x^{4} - 64x^{2} + 64 \) Copy content Toggle raw display
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.

\(f(q)\) \(=\) \( q + \beta_{10} q^{2} + ( - \beta_{11} + \beta_{5} + 1) q^{4} + ( - \beta_{10} - \beta_1) q^{5} + (\beta_{6} - \beta_{5}) q^{7} + (\beta_{10} + \beta_{6} - \beta_{5} + \beta_{2} - \beta_1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{10} q^{2} + ( - \beta_{11} + \beta_{5} + 1) q^{4} + ( - \beta_{10} - \beta_1) q^{5} + (\beta_{6} - \beta_{5}) q^{7} + (\beta_{10} + \beta_{6} - \beta_{5} + \beta_{2} - \beta_1) q^{8} + (\beta_{11} - \beta_{8} - 2) q^{10} + \beta_{3} q^{11} + (\beta_{10} + \beta_{9} + \beta_{6}) q^{13} + ( - 2 \beta_{7} + \beta_{6} - \beta_{5} - 2) q^{14} + ( - 2 \beta_{7} + \beta_{6} - \beta_{5} + 1) q^{16} + ( - 2 \beta_{8} - \beta_{6} + \beta_{5}) q^{17} + (\beta_{3} + \beta_{2} + \beta_1) q^{19} + ( - \beta_{10} - \beta_{6} + \beta_{5} - \beta_{4} - 3 \beta_{2} + \beta_1) q^{20} + ( - \beta_{9} + \beta_{8} - \beta_{5}) q^{22} + (\beta_{11} - \beta_{9} + \beta_{8} + \beta_{6} - \beta_{5}) q^{23} + (\beta_{8} - \beta_{7} - 2 \beta_{5}) q^{25} + ( - \beta_{11} - \beta_{7} + \beta_{5} - \beta_{4} - \beta_{3} + \beta_1 + 2) q^{26} + (\beta_{6} - \beta_{5} + 2 \beta_{2} - 2 \beta_1) q^{28} + (2 \beta_{11} - 2 \beta_{5} + 2) q^{29} + (3 \beta_{3} - \beta_{2} - \beta_1) q^{31} + (\beta_{10} + \beta_{6} - \beta_{5}) q^{32} + ( - \beta_{6} + \beta_{5} - 2 \beta_{4} - 4 \beta_{2} - 2 \beta_1) q^{34} + ( - 2 \beta_{11} + 2 \beta_{9} + 2 \beta_{7} - \beta_{6} + 3 \beta_{5}) q^{35} + ( - 2 \beta_{10} - \beta_{6} + \beta_{5} - 2 \beta_{2} + 2 \beta_1) q^{37} + ( - \beta_{11} - \beta_{9} + 3 \beta_{8} - 2 \beta_{5}) q^{38} + (\beta_{11} + 2 \beta_{9} - 2 \beta_{8} + \beta_{7} - \beta_{6} + \beta_{5} - 1) q^{40} + (2 \beta_{3} - \beta_{2} - \beta_1) q^{41} + (\beta_{11} - \beta_{9} - \beta_{8}) q^{43} + (\beta_{6} - \beta_{5} + 2 \beta_{4} - \beta_{3} + 2 \beta_{2}) q^{44} + (\beta_{6} - \beta_{5} + 2 \beta_{4} + \beta_{3} + \beta_{2} - \beta_1) q^{46} + ( - \beta_{2} + \beta_1) q^{47} + (4 \beta_{7} - 2 \beta_{6} + 2 \beta_{5} + 5) q^{49} + ( - \beta_{10} + \beta_{6} - \beta_{5} + \beta_{4} + 3 \beta_{2} + 2 \beta_1) q^{50} + ( - \beta_{11} + 3 \beta_{10} + \beta_{9} - 2 \beta_{8} + 2 \beta_{2} - 2 \beta_1) q^{52} + (2 \beta_{9} + \beta_{6} + \beta_{5}) q^{53} + (\beta_{11} + \beta_{9} - \beta_{7} + 2 \beta_{6} + 1) q^{55} + ( - 2 \beta_{11} + 2 \beta_{7} - \beta_{6} + 3 \beta_{5} + 6) q^{56} + ( - 2 \beta_{10} - 2 \beta_{6} + 2 \beta_{5} - 2 \beta_{2} + 2 \beta_1) q^{58} + (\beta_{3} + 2 \beta_{2} + 2 \beta_1) q^{59} + ( - 2 \beta_{11} - 2 \beta_{7} + \beta_{6} + \beta_{5}) q^{61} + (\beta_{11} - 3 \beta_{9} + \beta_{8} - 2 \beta_{5}) q^{62} + ( - \beta_{11} + 2 \beta_{7} - \beta_{6} + 2 \beta_{5} - 1) q^{64} + (\beta_{9} - \beta_{8} + \beta_{7} - 2 \beta_{6} + 3 \beta_{5} + 2 \beta_{3} - \beta_{2} - 2) q^{65} + ( - 4 \beta_{10} + \beta_{6} - \beta_{5} + 4 \beta_{2} - 4 \beta_1) q^{67} + (2 \beta_{11} + 4 \beta_{9} - 6 \beta_{8} + \beta_{6} + 5 \beta_{5}) q^{68} + ( - 3 \beta_{6} + 3 \beta_{5} - 2 \beta_{4} - 2 \beta_{3} + 4 \beta_1) q^{70} + (2 \beta_{6} - 2 \beta_{5} + 4 \beta_{4} - \beta_{3} + 2 \beta_{2} - 2 \beta_1) q^{71} + (2 \beta_{10} + \beta_{6} - \beta_{5}) q^{73} + (4 \beta_{11} + 2 \beta_{7} - \beta_{6} - 3 \beta_{5} - 8) q^{74} + (2 \beta_{6} - 2 \beta_{5} + 4 \beta_{4} - \beta_{3} + 5 \beta_{2} + \beta_1) q^{76} + ( - 2 \beta_{11} - 2 \beta_{6}) q^{77} + ( - 2 \beta_{7} + \beta_{6} - \beta_{5} - 6) q^{79} + ( - 3 \beta_{10} - 3 \beta_{6} + 3 \beta_{5} - 2 \beta_{4} - 2 \beta_{3} + \beta_1) q^{80} + (\beta_{11} - 2 \beta_{9} - \beta_{5}) q^{82} + ( - 4 \beta_{10} + 2 \beta_{6} - 2 \beta_{5} + 3 \beta_{2} - 3 \beta_1) q^{83} + ( - 4 \beta_{10} + \beta_{6} - \beta_{5} + 2 \beta_{4} - 2 \beta_{3} + 2 \beta_{2} + 2 \beta_1) q^{85} + (\beta_{3} - 3 \beta_{2} - 3 \beta_1) q^{86} + ( - \beta_{9} + 3 \beta_{8} + \beta_{6} - 2 \beta_{5}) q^{88} + ( - 2 \beta_{3} - \beta_{2} - \beta_1) q^{89} + (\beta_{6} - \beta_{5} + 2 \beta_{4} - 2 \beta_{2} - 4 \beta_1 + 4) q^{91} + ( - \beta_{11} - 3 \beta_{9} + 3 \beta_{8} - \beta_{6} - 3 \beta_{5}) q^{92} + (\beta_{11} - \beta_{5} - 2) q^{94} + (\beta_{11} + \beta_{9} - \beta_{8} + 2 \beta_{6} + 2 \beta_{5} + 6) q^{95} + (6 \beta_{10} - \beta_{6} + \beta_{5} - 4 \beta_{2} + 4 \beta_1) q^{97} + (\beta_{10} - 6 \beta_{6} + 6 \beta_{5} - 4 \beta_{2} + 4 \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 12 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 12 q^{4} - 24 q^{10} - 16 q^{14} + 20 q^{16} + 4 q^{25} + 28 q^{26} + 24 q^{29} - 8 q^{35} - 16 q^{40} + 44 q^{49} + 16 q^{55} + 64 q^{56} + 8 q^{61} - 20 q^{64} - 28 q^{65} - 104 q^{74} - 64 q^{79} + 48 q^{91} - 24 q^{94} + 72 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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

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

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( ( \beta_{6} - \beta_{5} + \beta_{3} + \beta_{2} - 3\beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{9} - \beta_{7} + \beta_{6} + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 2\beta_{10} - \beta_{6} + \beta_{5} - \beta_{4} + \beta_{3} - \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( \beta_{11} - \beta_{8} - \beta_{7} + 2\beta_{6} + \beta_{5} - 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 2\beta_{10} - 3\beta_{4} - 2\beta_{2} + 3\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -3\beta_{11} + 2\beta_{9} - 3\beta_{8} + 5\beta_{7} - 2\beta_{6} + 9\beta_{5} + 3 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 2\beta_{10} + 2\beta_{6} - 2\beta_{5} - \beta_{4} - 10\beta_{2} - \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( -9\beta_{11} + 8\beta_{9} - 7\beta_{8} + \beta_{7} - 6\beta_{6} + 11\beta_{5} - 15 ) / 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 6\beta_{10} - 10\beta_{6} + 10\beta_{5} - \beta_{4} - 10\beta_{3} - 16\beta_{2} + 7\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 7\beta_{11} - 6\beta_{9} - 17\beta_{8} - 5\beta_{7} - 10\beta_{6} - 5\beta_{5} - 27 ) / 2 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( -34\beta_{10} - 6\beta_{6} + 6\beta_{5} - 7\beta_{4} - 28\beta_{3} + 14\beta_{2} + 5\beta_1 ) / 2 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

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 \( T_{2}^{6} - 9T_{2}^{4} + 17T_{2}^{2} - 1 \) acting on \(S_{2}^{\mathrm{new}}(585, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

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