Properties

Label 592.3.k.c
Level $592$
Weight $3$
Character orbit 592.k
Analytic conductor $16.131$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 592 = 2^{4} \cdot 37 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 592.k (of order \(4\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(16.1308316501\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 74)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -4 i q^{3} + ( 3 - 3 i ) q^{5} + 4 q^{7} -7 q^{9} +O(q^{10})\) \( q -4 i q^{3} + ( 3 - 3 i ) q^{5} + 4 q^{7} -7 q^{9} -4 i q^{11} + ( 3 - 3 i ) q^{13} + ( -12 - 12 i ) q^{15} + ( 23 - 23 i ) q^{17} + ( -10 + 10 i ) q^{19} -16 i q^{21} + ( 10 - 10 i ) q^{23} + 7 i q^{25} -8 i q^{27} + ( -19 - 19 i ) q^{29} + ( 18 + 18 i ) q^{31} -16 q^{33} + ( 12 - 12 i ) q^{35} -37 i q^{37} + ( -12 - 12 i ) q^{39} + 74 i q^{41} + ( -42 + 42 i ) q^{43} + ( -21 + 21 i ) q^{45} + 44 q^{47} -33 q^{49} + ( -92 - 92 i ) q^{51} -80 q^{53} + ( -12 - 12 i ) q^{55} + ( 40 + 40 i ) q^{57} + ( 54 - 54 i ) q^{59} + ( -3 - 3 i ) q^{61} -28 q^{63} -18 i q^{65} + 12 i q^{67} + ( -40 - 40 i ) q^{69} -124 q^{71} -10 i q^{73} + 28 q^{75} -16 i q^{77} + ( -14 + 14 i ) q^{79} -95 q^{81} + 64 q^{83} -138 i q^{85} + ( -76 + 76 i ) q^{87} + ( 17 + 17 i ) q^{89} + ( 12 - 12 i ) q^{91} + ( 72 - 72 i ) q^{93} + 60 i q^{95} + ( 129 - 129 i ) q^{97} + 28 i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 6 q^{5} + 8 q^{7} - 14 q^{9} + O(q^{10}) \) \( 2 q + 6 q^{5} + 8 q^{7} - 14 q^{9} + 6 q^{13} - 24 q^{15} + 46 q^{17} - 20 q^{19} + 20 q^{23} - 38 q^{29} + 36 q^{31} - 32 q^{33} + 24 q^{35} - 24 q^{39} - 84 q^{43} - 42 q^{45} + 88 q^{47} - 66 q^{49} - 184 q^{51} - 160 q^{53} - 24 q^{55} + 80 q^{57} + 108 q^{59} - 6 q^{61} - 56 q^{63} - 80 q^{69} - 248 q^{71} + 56 q^{75} - 28 q^{79} - 190 q^{81} + 128 q^{83} - 152 q^{87} + 34 q^{89} + 24 q^{91} + 144 q^{93} + 258 q^{97} + O(q^{100}) \)

Character values

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

\(n\) \(113\) \(149\) \(223\)
\(\chi(n)\) \(i\) \(1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
401.1
1.00000i
1.00000i
0 4.00000i 0 3.00000 3.00000i 0 4.00000 0 −7.00000 0
561.1 0 4.00000i 0 3.00000 + 3.00000i 0 4.00000 0 −7.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
37.d odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 592.3.k.c 2
4.b odd 2 1 74.3.d.b 2
12.b even 2 1 666.3.i.a 2
37.d odd 4 1 inner 592.3.k.c 2
148.g even 4 1 74.3.d.b 2
444.j odd 4 1 666.3.i.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
74.3.d.b 2 4.b odd 2 1
74.3.d.b 2 148.g even 4 1
592.3.k.c 2 1.a even 1 1 trivial
592.3.k.c 2 37.d odd 4 1 inner
666.3.i.a 2 12.b even 2 1
666.3.i.a 2 444.j odd 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(592, [\chi])\):

\( T_{3}^{2} + 16 \)
\( T_{5}^{2} - 6 T_{5} + 18 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( 16 + T^{2} \)
$5$ \( 18 - 6 T + T^{2} \)
$7$ \( ( -4 + T )^{2} \)
$11$ \( 16 + T^{2} \)
$13$ \( 18 - 6 T + T^{2} \)
$17$ \( 1058 - 46 T + T^{2} \)
$19$ \( 200 + 20 T + T^{2} \)
$23$ \( 200 - 20 T + T^{2} \)
$29$ \( 722 + 38 T + T^{2} \)
$31$ \( 648 - 36 T + T^{2} \)
$37$ \( 1369 + T^{2} \)
$41$ \( 5476 + T^{2} \)
$43$ \( 3528 + 84 T + T^{2} \)
$47$ \( ( -44 + T )^{2} \)
$53$ \( ( 80 + T )^{2} \)
$59$ \( 5832 - 108 T + T^{2} \)
$61$ \( 18 + 6 T + T^{2} \)
$67$ \( 144 + T^{2} \)
$71$ \( ( 124 + T )^{2} \)
$73$ \( 100 + T^{2} \)
$79$ \( 392 + 28 T + T^{2} \)
$83$ \( ( -64 + T )^{2} \)
$89$ \( 578 - 34 T + T^{2} \)
$97$ \( 33282 - 258 T + T^{2} \)
show more
show less