# Properties

 Label 4368.2.h.k Level $4368$ Weight $2$ Character orbit 4368.h Analytic conductor $34.879$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$4368 = 2^{4} \cdot 3 \cdot 7 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4368.h (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$34.8786556029$$ 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 546) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + q^{3} + i q^{5} + i q^{7} + q^{9} +O(q^{10})$$ $$q + q^{3} + i q^{5} + i q^{7} + q^{9} -i q^{11} + ( 3 + 2 i ) q^{13} + i q^{15} + q^{17} + i q^{19} + i q^{21} -3 q^{23} + 4 q^{25} + q^{27} + 9 q^{29} + 4 i q^{31} -i q^{33} - q^{35} -9 i q^{37} + ( 3 + 2 i ) q^{39} + 8 i q^{41} -7 q^{43} + i q^{45} -8 i q^{47} - q^{49} + q^{51} -10 q^{53} + q^{55} + i q^{57} + 6 i q^{59} + 11 q^{61} + i q^{63} + ( -2 + 3 i ) q^{65} + 12 i q^{67} -3 q^{69} -6 i q^{71} + 11 i q^{73} + 4 q^{75} + q^{77} + 12 q^{79} + q^{81} -6 i q^{83} + i q^{85} + 9 q^{87} + 12 i q^{89} + ( -2 + 3 i ) q^{91} + 4 i q^{93} - q^{95} -2 i q^{97} -i q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 2q^{3} + 2q^{9} + O(q^{10})$$ $$2q + 2q^{3} + 2q^{9} + 6q^{13} + 2q^{17} - 6q^{23} + 8q^{25} + 2q^{27} + 18q^{29} - 2q^{35} + 6q^{39} - 14q^{43} - 2q^{49} + 2q^{51} - 20q^{53} + 2q^{55} + 22q^{61} - 4q^{65} - 6q^{69} + 8q^{75} + 2q^{77} + 24q^{79} + 2q^{81} + 18q^{87} - 4q^{91} - 2q^{95} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/4368\mathbb{Z}\right)^\times$$.

 $$n$$ $$1093$$ $$1249$$ $$1457$$ $$2017$$ $$3823$$ $$\chi(n)$$ $$1$$ $$1$$ $$1$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
337.1
 − 1.00000i 1.00000i
0 1.00000 0 1.00000i 0 1.00000i 0 1.00000 0
337.2 0 1.00000 0 1.00000i 0 1.00000i 0 1.00000 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
13.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 4368.2.h.k 2
4.b odd 2 1 546.2.c.a 2
12.b even 2 1 1638.2.c.f 2
13.b even 2 1 inner 4368.2.h.k 2
28.d even 2 1 3822.2.c.e 2
52.b odd 2 1 546.2.c.a 2
52.f even 4 1 7098.2.a.d 1
52.f even 4 1 7098.2.a.s 1
156.h even 2 1 1638.2.c.f 2
364.h even 2 1 3822.2.c.e 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
546.2.c.a 2 4.b odd 2 1
546.2.c.a 2 52.b odd 2 1
1638.2.c.f 2 12.b even 2 1
1638.2.c.f 2 156.h even 2 1
3822.2.c.e 2 28.d even 2 1
3822.2.c.e 2 364.h even 2 1
4368.2.h.k 2 1.a even 1 1 trivial
4368.2.h.k 2 13.b even 2 1 inner
7098.2.a.d 1 52.f even 4 1
7098.2.a.s 1 52.f even 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(4368, [\chi])$$:

 $$T_{5}^{2} + 1$$ $$T_{11}^{2} + 1$$ $$T_{17} - 1$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$( -1 + T )^{2}$$
$5$ $$1 + T^{2}$$
$7$ $$1 + T^{2}$$
$11$ $$1 + T^{2}$$
$13$ $$13 - 6 T + T^{2}$$
$17$ $$( -1 + T )^{2}$$
$19$ $$1 + T^{2}$$
$23$ $$( 3 + T )^{2}$$
$29$ $$( -9 + T )^{2}$$
$31$ $$16 + T^{2}$$
$37$ $$81 + T^{2}$$
$41$ $$64 + T^{2}$$
$43$ $$( 7 + T )^{2}$$
$47$ $$64 + T^{2}$$
$53$ $$( 10 + T )^{2}$$
$59$ $$36 + T^{2}$$
$61$ $$( -11 + T )^{2}$$
$67$ $$144 + T^{2}$$
$71$ $$36 + T^{2}$$
$73$ $$121 + T^{2}$$
$79$ $$( -12 + T )^{2}$$
$83$ $$36 + T^{2}$$
$89$ $$144 + T^{2}$$
$97$ $$4 + T^{2}$$