# 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

## 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: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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

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