# Properties

 Label 4368.2.h.j 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: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 1092) 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.

 $$f(q)$$ $$=$$ $$q + q^{3} + 2 i q^{5} - i q^{7} + q^{9} +O(q^{10})$$ q + q^3 + 2*i * q^5 - i * q^7 + q^9 $$q + q^{3} + 2 i q^{5} - i q^{7} + q^{9} - 2 i q^{11} + ( - 2 i + 3) q^{13} + 2 i q^{15} - 2 q^{17} - 4 i q^{19} - i q^{21} + q^{25} + q^{27} - 6 q^{29} - 4 i q^{31} - 2 i q^{33} + 2 q^{35} + ( - 2 i + 3) q^{39} - 2 i q^{41} + 8 q^{43} + 2 i q^{45} - 10 i q^{47} - q^{49} - 2 q^{51} - 10 q^{53} + 4 q^{55} - 4 i q^{57} - 6 i q^{59} + 2 q^{61} - i q^{63} + (6 i + 4) q^{65} - 6 i q^{71} + 4 i q^{73} + q^{75} - 2 q^{77} + q^{81} - 6 i q^{83} - 4 i q^{85} - 6 q^{87} + 6 i q^{89} + ( - 3 i - 2) q^{91} - 4 i q^{93} + 8 q^{95} - 4 i q^{97} - 2 i q^{99} +O(q^{100})$$ q + q^3 + 2*i * q^5 - i * q^7 + q^9 - 2*i * q^11 + (-2*i + 3) * q^13 + 2*i * q^15 - 2 * q^17 - 4*i * q^19 - i * q^21 + q^25 + q^27 - 6 * q^29 - 4*i * q^31 - 2*i * q^33 + 2 * q^35 + (-2*i + 3) * q^39 - 2*i * q^41 + 8 * q^43 + 2*i * q^45 - 10*i * q^47 - q^49 - 2 * q^51 - 10 * q^53 + 4 * q^55 - 4*i * q^57 - 6*i * q^59 + 2 * q^61 - i * q^63 + (6*i + 4) * q^65 - 6*i * q^71 + 4*i * q^73 + q^75 - 2 * q^77 + q^81 - 6*i * q^83 - 4*i * q^85 - 6 * q^87 + 6*i * q^89 + (-3*i - 2) * q^91 - 4*i * q^93 + 8 * q^95 - 4*i * q^97 - 2*i * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{3} + 2 q^{9}+O(q^{10})$$ 2 * q + 2 * q^3 + 2 * q^9 $$2 q + 2 q^{3} + 2 q^{9} + 6 q^{13} - 4 q^{17} + 2 q^{25} + 2 q^{27} - 12 q^{29} + 4 q^{35} + 6 q^{39} + 16 q^{43} - 2 q^{49} - 4 q^{51} - 20 q^{53} + 8 q^{55} + 4 q^{61} + 8 q^{65} + 2 q^{75} - 4 q^{77} + 2 q^{81} - 12 q^{87} - 4 q^{91} + 16 q^{95}+O(q^{100})$$ 2 * q + 2 * q^3 + 2 * q^9 + 6 * q^13 - 4 * q^17 + 2 * q^25 + 2 * q^27 - 12 * q^29 + 4 * q^35 + 6 * q^39 + 16 * q^43 - 2 * q^49 - 4 * q^51 - 20 * q^53 + 8 * q^55 + 4 * q^61 + 8 * q^65 + 2 * q^75 - 4 * q^77 + 2 * q^81 - 12 * q^87 - 4 * q^91 + 16 * q^95

## 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 2.00000i 0 1.00000i 0 1.00000 0
337.2 0 1.00000 0 2.00000i 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.j 2
4.b odd 2 1 1092.2.e.a 2
12.b even 2 1 3276.2.e.d 2
13.b even 2 1 inner 4368.2.h.j 2
28.d even 2 1 7644.2.e.f 2
52.b odd 2 1 1092.2.e.a 2
156.h even 2 1 3276.2.e.d 2
364.h even 2 1 7644.2.e.f 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1092.2.e.a 2 4.b odd 2 1
1092.2.e.a 2 52.b odd 2 1
3276.2.e.d 2 12.b even 2 1
3276.2.e.d 2 156.h even 2 1
4368.2.h.j 2 1.a even 1 1 trivial
4368.2.h.j 2 13.b even 2 1 inner
7644.2.e.f 2 28.d even 2 1
7644.2.e.f 2 364.h even 2 1

## Hecke kernels

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

 $$T_{5}^{2} + 4$$ T5^2 + 4 $$T_{11}^{2} + 4$$ T11^2 + 4 $$T_{17} + 2$$ T17 + 2

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$(T - 1)^{2}$$
$5$ $$T^{2} + 4$$
$7$ $$T^{2} + 1$$
$11$ $$T^{2} + 4$$
$13$ $$T^{2} - 6T + 13$$
$17$ $$(T + 2)^{2}$$
$19$ $$T^{2} + 16$$
$23$ $$T^{2}$$
$29$ $$(T + 6)^{2}$$
$31$ $$T^{2} + 16$$
$37$ $$T^{2}$$
$41$ $$T^{2} + 4$$
$43$ $$(T - 8)^{2}$$
$47$ $$T^{2} + 100$$
$53$ $$(T + 10)^{2}$$
$59$ $$T^{2} + 36$$
$61$ $$(T - 2)^{2}$$
$67$ $$T^{2}$$
$71$ $$T^{2} + 36$$
$73$ $$T^{2} + 16$$
$79$ $$T^{2}$$
$83$ $$T^{2} + 36$$
$89$ $$T^{2} + 36$$
$97$ $$T^{2} + 16$$