Properties

Label 592.2.bc.b
Level $592$
Weight $2$
Character orbit 592.bc
Analytic conductor $4.727$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 592 = 2^{4} \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 592.bc (of order \(9\), degree \(6\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(4.72714379966\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: \(\Q(\zeta_{18})\)
Defining polynomial: \(x^{6} - x^{3} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 74)
Sato-Tate group: $\mathrm{SU}(2)[C_{9}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{18}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( \zeta_{18} + \zeta_{18}^{3} ) q^{3} + ( 1 + \zeta_{18} + 2 \zeta_{18}^{2} - \zeta_{18}^{3} - 2 \zeta_{18}^{5} ) q^{5} + ( -1 - \zeta_{18} + \zeta_{18}^{2} + \zeta_{18}^{4} - \zeta_{18}^{5} ) q^{7} + ( -1 + \zeta_{18}^{2} + \zeta_{18}^{3} - \zeta_{18}^{4} ) q^{9} +O(q^{10})\) \( q + ( \zeta_{18} + \zeta_{18}^{3} ) q^{3} + ( 1 + \zeta_{18} + 2 \zeta_{18}^{2} - \zeta_{18}^{3} - 2 \zeta_{18}^{5} ) q^{5} + ( -1 - \zeta_{18} + \zeta_{18}^{2} + \zeta_{18}^{4} - \zeta_{18}^{5} ) q^{7} + ( -1 + \zeta_{18}^{2} + \zeta_{18}^{3} - \zeta_{18}^{4} ) q^{9} + ( -\zeta_{18} + \zeta_{18}^{2} + 3 \zeta_{18}^{3} + \zeta_{18}^{4} - \zeta_{18}^{5} ) q^{11} + ( -1 - \zeta_{18} + 2 \zeta_{18}^{3} + 2 \zeta_{18}^{4} + 3 \zeta_{18}^{5} ) q^{13} + ( 3 + \zeta_{18} + 3 \zeta_{18}^{2} ) q^{15} -3 \zeta_{18}^{4} q^{17} + ( -3 \zeta_{18} + \zeta_{18}^{2} - 3 \zeta_{18}^{3} ) q^{19} + ( 1 - 2 \zeta_{18} - \zeta_{18}^{3} + \zeta_{18}^{5} ) q^{21} + ( 5 - 2 \zeta_{18} + 3 \zeta_{18}^{2} - 5 \zeta_{18}^{3} - \zeta_{18}^{4} - \zeta_{18}^{5} ) q^{23} + ( 4 + \zeta_{18} + \zeta_{18}^{2} - \zeta_{18}^{3} - \zeta_{18}^{4} - 4 \zeta_{18}^{5} ) q^{25} + ( -1 + 3 \zeta_{18} + \zeta_{18}^{3} - 3 \zeta_{18}^{4} - 3 \zeta_{18}^{5} ) q^{27} + ( -4 \zeta_{18} + 5 \zeta_{18}^{2} + 5 \zeta_{18}^{4} - 4 \zeta_{18}^{5} ) q^{29} + ( 3 - \zeta_{18} - \zeta_{18}^{2} + \zeta_{18}^{5} ) q^{31} + ( -2 - \zeta_{18} + 3 \zeta_{18}^{3} + 3 \zeta_{18}^{4} + \zeta_{18}^{5} ) q^{33} + ( -2 + \zeta_{18} - 3 \zeta_{18}^{2} + 3 \zeta_{18}^{3} - \zeta_{18}^{4} + 2 \zeta_{18}^{5} ) q^{35} + ( -1 - 2 \zeta_{18} - 4 \zeta_{18}^{2} + 5 \zeta_{18}^{3} + 2 \zeta_{18}^{4} + 2 \zeta_{18}^{5} ) q^{37} + ( -5 - 3 \zeta_{18} - 4 \zeta_{18}^{2} + 4 \zeta_{18}^{3} + 3 \zeta_{18}^{4} + 5 \zeta_{18}^{5} ) q^{39} + ( 4 + 2 \zeta_{18} - 6 \zeta_{18}^{3} - 6 \zeta_{18}^{4} + 2 \zeta_{18}^{5} ) q^{41} + ( -2 - \zeta_{18} - \zeta_{18}^{2} - 5 \zeta_{18}^{4} + 6 \zeta_{18}^{5} ) q^{43} + ( \zeta_{18}^{2} + \zeta_{18}^{4} ) q^{45} + ( 1 - 4 \zeta_{18}^{2} - \zeta_{18}^{3} + 4 \zeta_{18}^{4} + 4 \zeta_{18}^{5} ) q^{47} + ( -1 - 4 \zeta_{18} - 2 \zeta_{18}^{2} + 2 \zeta_{18}^{3} + 4 \zeta_{18}^{4} + \zeta_{18}^{5} ) q^{49} + ( 3 \zeta_{18} - 3 \zeta_{18}^{4} - 3 \zeta_{18}^{5} ) q^{51} + ( -5 + 3 \zeta_{18} - \zeta_{18}^{2} + 4 \zeta_{18}^{3} - 4 \zeta_{18}^{5} ) q^{53} + ( 2 + 2 \zeta_{18} + 5 \zeta_{18}^{2} + 2 \zeta_{18}^{3} + 2 \zeta_{18}^{4} ) q^{55} + ( 3 - 3 \zeta_{18}^{2} - 2 \zeta_{18}^{3} - 6 \zeta_{18}^{4} + \zeta_{18}^{5} ) q^{57} + ( -3 + \zeta_{18} + 5 \zeta_{18}^{2} + 8 \zeta_{18}^{3} - 8 \zeta_{18}^{5} ) q^{59} + ( 4 \zeta_{18} - 4 \zeta_{18}^{3} - 4 \zeta_{18}^{4} + 4 \zeta_{18}^{5} ) q^{61} + ( \zeta_{18} - 2 \zeta_{18}^{3} + \zeta_{18}^{5} ) q^{63} + ( -4 + 4 \zeta_{18}^{2} + 8 \zeta_{18}^{3} + 9 \zeta_{18}^{4} + 4 \zeta_{18}^{5} ) q^{65} + ( 1 + \zeta_{18} + 7 \zeta_{18}^{2} - \zeta_{18}^{3} - 7 \zeta_{18}^{5} ) q^{67} + ( 6 + 6 \zeta_{18} - \zeta_{18}^{2} + 2 \zeta_{18}^{3} - 8 \zeta_{18}^{4} + \zeta_{18}^{5} ) q^{69} + ( 4 - 6 \zeta_{18} - 8 \zeta_{18}^{2} - 6 \zeta_{18}^{3} + 4 \zeta_{18}^{4} ) q^{71} + ( -6 - \zeta_{18} - \zeta_{18}^{2} - 2 \zeta_{18}^{4} + 3 \zeta_{18}^{5} ) q^{73} + ( 5 + 5 \zeta_{18} + 5 \zeta_{18}^{2} - \zeta_{18}^{4} - 4 \zeta_{18}^{5} ) q^{75} + ( -2 - \zeta_{18} + 2 \zeta_{18}^{2} - \zeta_{18}^{3} - 2 \zeta_{18}^{4} ) q^{77} + ( -6 - 6 \zeta_{18} - \zeta_{18}^{2} + 2 \zeta_{18}^{3} + 4 \zeta_{18}^{4} + \zeta_{18}^{5} ) q^{79} + ( 5 + 5 \zeta_{18} + 3 \zeta_{18}^{2} - 6 \zeta_{18}^{3} + \zeta_{18}^{4} - 3 \zeta_{18}^{5} ) q^{81} + ( 1 - \zeta_{18}^{2} - 4 \zeta_{18}^{3} + 3 \zeta_{18}^{4} - 3 \zeta_{18}^{5} ) q^{83} + ( -3 \zeta_{18} - 6 \zeta_{18}^{3} - 3 \zeta_{18}^{5} ) q^{85} + ( 4 - 5 \zeta_{18} + \zeta_{18}^{3} + \zeta_{18}^{4} + 6 \zeta_{18}^{5} ) q^{87} + ( -8 - \zeta_{18} - 3 \zeta_{18}^{2} + 5 \zeta_{18}^{3} - 5 \zeta_{18}^{5} ) q^{89} + ( -3 \zeta_{18}^{3} - 3 \zeta_{18}^{5} ) q^{91} + ( -1 + 3 \zeta_{18} - 2 \zeta_{18}^{2} + 3 \zeta_{18}^{3} - \zeta_{18}^{4} ) q^{93} + ( -9 - \zeta_{18} - 8 \zeta_{18}^{2} + \zeta_{18}^{3} - \zeta_{18}^{5} ) q^{95} + ( 14 + 2 \zeta_{18}^{2} - 14 \zeta_{18}^{3} - 2 \zeta_{18}^{4} - 2 \zeta_{18}^{5} ) q^{97} + ( -4 + 4 \zeta_{18} + \zeta_{18}^{2} - \zeta_{18}^{3} - 4 \zeta_{18}^{4} + 4 \zeta_{18}^{5} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q + 3q^{3} + 3q^{5} - 6q^{7} - 3q^{9} + O(q^{10}) \) \( 6q + 3q^{3} + 3q^{5} - 6q^{7} - 3q^{9} + 9q^{11} + 18q^{15} - 9q^{19} + 3q^{21} + 15q^{23} + 21q^{25} - 3q^{27} + 18q^{31} - 3q^{33} - 3q^{35} + 9q^{37} - 18q^{39} + 6q^{41} - 12q^{43} + 3q^{47} - 18q^{53} + 18q^{55} + 12q^{57} + 6q^{59} - 12q^{61} - 6q^{63} + 3q^{67} + 42q^{69} + 6q^{71} - 36q^{73} + 30q^{75} - 15q^{77} - 30q^{79} + 12q^{81} - 6q^{83} - 18q^{85} + 27q^{87} - 33q^{89} - 9q^{91} + 3q^{93} - 51q^{95} + 42q^{97} - 27q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/592\mathbb{Z}\right)^\times\).

\(n\) \(113\) \(149\) \(223\)
\(\chi(n)\) \(-\zeta_{18}^{5}\) \(1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
33.1
−0.766044 0.642788i
0.939693 + 0.342020i
−0.173648 + 0.984808i
0.939693 0.342020i
−0.766044 + 0.642788i
−0.173648 0.984808i
0 −0.266044 1.50881i 0 −1.79813 + 1.50881i 0 −1.93969 + 1.62760i 0 0.613341 0.223238i 0
49.1 0 1.43969 + 1.20805i 0 3.31908 1.20805i 0 −0.826352 + 0.300767i 0 0.0923963 + 0.524005i 0
81.1 0 0.326352 + 0.118782i 0 −0.0209445 0.118782i 0 −0.233956 1.32683i 0 −2.20574 1.85083i 0
145.1 0 1.43969 1.20805i 0 3.31908 + 1.20805i 0 −0.826352 0.300767i 0 0.0923963 0.524005i 0
305.1 0 −0.266044 + 1.50881i 0 −1.79813 1.50881i 0 −1.93969 1.62760i 0 0.613341 + 0.223238i 0
497.1 0 0.326352 0.118782i 0 −0.0209445 + 0.118782i 0 −0.233956 + 1.32683i 0 −2.20574 + 1.85083i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 497.1
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
37.f even 9 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 592.2.bc.b 6
4.b odd 2 1 74.2.f.a 6
12.b even 2 1 666.2.x.c 6
37.f even 9 1 inner 592.2.bc.b 6
148.o odd 18 1 2738.2.a.p 3
148.p odd 18 1 74.2.f.a 6
148.p odd 18 1 2738.2.a.m 3
444.z even 18 1 666.2.x.c 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
74.2.f.a 6 4.b odd 2 1
74.2.f.a 6 148.p odd 18 1
592.2.bc.b 6 1.a even 1 1 trivial
592.2.bc.b 6 37.f even 9 1 inner
666.2.x.c 6 12.b even 2 1
666.2.x.c 6 444.z even 18 1
2738.2.a.m 3 148.p odd 18 1
2738.2.a.p 3 148.o odd 18 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{6} - 3 T_{3}^{5} + 6 T_{3}^{4} - 8 T_{3}^{3} + 12 T_{3}^{2} - 6 T_{3} + 1 \) acting on \(S_{2}^{\mathrm{new}}(592, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \)
$3$ \( 1 - 6 T + 12 T^{2} - 8 T^{3} + 6 T^{4} - 3 T^{5} + T^{6} \)
$5$ \( 1 + 3 T + 69 T^{2} + 8 T^{3} - 6 T^{4} - 3 T^{5} + T^{6} \)
$7$ \( 9 + 27 T + 36 T^{2} + 30 T^{3} + 18 T^{4} + 6 T^{5} + T^{6} \)
$11$ \( 289 - 408 T + 423 T^{2} - 182 T^{3} + 57 T^{4} - 9 T^{5} + T^{6} \)
$13$ \( 81 + 81 T - 90 T^{3} + 36 T^{4} + T^{6} \)
$17$ \( 729 - 27 T^{3} + T^{6} \)
$19$ \( 2809 + 3339 T + 1530 T^{2} + 352 T^{3} + 63 T^{4} + 9 T^{5} + T^{6} \)
$23$ \( 9 - 162 T + 2871 T^{2} - 804 T^{3} + 171 T^{4} - 15 T^{5} + T^{6} \)
$29$ \( 29241 - 10773 T + 3969 T^{2} - 342 T^{3} + 63 T^{4} + T^{6} \)
$31$ \( ( -17 + 24 T - 9 T^{2} + T^{3} )^{2} \)
$37$ \( 50653 - 12321 T + 1998 T^{2} - 305 T^{3} + 54 T^{4} - 9 T^{5} + T^{6} \)
$41$ \( 207936 - 16416 T - 288 T^{2} - 192 T^{3} + 36 T^{4} - 6 T^{5} + T^{6} \)
$43$ \( ( -467 - 81 T + 6 T^{2} + T^{3} )^{2} \)
$47$ \( 289 + 765 T + 1974 T^{2} + 169 T^{3} + 54 T^{4} - 3 T^{5} + T^{6} \)
$53$ \( 81 + 567 T + 1134 T^{2} + 72 T^{3} + 144 T^{4} + 18 T^{5} + T^{6} \)
$59$ \( 687241 - 62175 T - 4134 T^{2} - 260 T^{3} + 120 T^{4} - 6 T^{5} + T^{6} \)
$61$ \( 4096 - 3072 T + 1536 T^{2} - 512 T^{3} + 48 T^{4} + 12 T^{5} + T^{6} \)
$67$ \( 103041 + 5778 T + 2439 T^{2} - 267 T^{3} - 36 T^{4} - 3 T^{5} + T^{6} \)
$71$ \( 207936 - 32832 T + 30816 T^{2} - 624 T^{3} - 144 T^{4} - 6 T^{5} + T^{6} \)
$73$ \( ( 53 + 87 T + 18 T^{2} + T^{3} )^{2} \)
$79$ \( 45369 + 26838 T + 9756 T^{2} + 2265 T^{3} + 360 T^{4} + 30 T^{5} + T^{6} \)
$83$ \( 32041 + 2685 T + 1218 T^{2} + 28 T^{3} - 12 T^{4} + 6 T^{5} + T^{6} \)
$89$ \( 687241 + 203934 T + 33171 T^{2} + 4661 T^{3} + 516 T^{4} + 33 T^{5} + T^{6} \)
$97$ \( 6594624 - 1479168 T + 223920 T^{2} - 19056 T^{3} + 1188 T^{4} - 42 T^{5} + T^{6} \)
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