# Properties

 Label 4368.2.h.h 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: $$\Z[a_1, \ldots, a_{5}]$$ 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 $$q - q^{3} + i q^{5} - i q^{7} + q^{9} + 3 i q^{11} + (2 i + 3) q^{13} - i q^{15} + 7 q^{17} + 3 i q^{19} + i q^{21} - q^{23} + 4 q^{25} - q^{27} - q^{29} - 8 i q^{31} - 3 i q^{33} + q^{35} + i q^{37} + ( - 2 i - 3) q^{39} - 4 i q^{41} + 5 q^{43} + i q^{45} - q^{49} - 7 q^{51} - 6 q^{53} - 3 q^{55} - 3 i q^{57} + 10 i q^{59} - 13 q^{61} - i q^{63} + (3 i - 2) q^{65} - 8 i q^{67} + q^{69} + 6 i q^{71} + 13 i q^{73} - 4 q^{75} + 3 q^{77} + 12 q^{79} + q^{81} + 2 i q^{83} + 7 i q^{85} + q^{87} - 12 i q^{89} + ( - 3 i + 2) q^{91} + 8 i q^{93} - 3 q^{95} - 6 i q^{97} + 3 i q^{99} +O(q^{100})$$ q - q^3 + i * q^5 - i * q^7 + q^9 + 3*i * q^11 + (2*i + 3) * q^13 - i * q^15 + 7 * q^17 + 3*i * q^19 + i * q^21 - q^23 + 4 * q^25 - q^27 - q^29 - 8*i * q^31 - 3*i * q^33 + q^35 + i * q^37 + (-2*i - 3) * q^39 - 4*i * q^41 + 5 * q^43 + i * q^45 - q^49 - 7 * q^51 - 6 * q^53 - 3 * q^55 - 3*i * q^57 + 10*i * q^59 - 13 * q^61 - i * q^63 + (3*i - 2) * q^65 - 8*i * q^67 + q^69 + 6*i * q^71 + 13*i * q^73 - 4 * q^75 + 3 * q^77 + 12 * q^79 + q^81 + 2*i * q^83 + 7*i * q^85 + q^87 - 12*i * q^89 + (-3*i + 2) * q^91 + 8*i * q^93 - 3 * q^95 - 6*i * q^97 + 3*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 $$2 q - 2 q^{3} + 2 q^{9} + 6 q^{13} + 14 q^{17} - 2 q^{23} + 8 q^{25} - 2 q^{27} - 2 q^{29} + 2 q^{35} - 6 q^{39} + 10 q^{43} - 2 q^{49} - 14 q^{51} - 12 q^{53} - 6 q^{55} - 26 q^{61} - 4 q^{65} + 2 q^{69} - 8 q^{75} + 6 q^{77} + 24 q^{79} + 2 q^{81} + 2 q^{87} + 4 q^{91} - 6 q^{95}+O(q^{100})$$ 2 * q - 2 * q^3 + 2 * q^9 + 6 * q^13 + 14 * q^17 - 2 * q^23 + 8 * q^25 - 2 * q^27 - 2 * q^29 + 2 * q^35 - 6 * q^39 + 10 * q^43 - 2 * q^49 - 14 * q^51 - 12 * q^53 - 6 * q^55 - 26 * q^61 - 4 * q^65 + 2 * q^69 - 8 * q^75 + 6 * q^77 + 24 * q^79 + 2 * q^81 + 2 * q^87 + 4 * q^91 - 6 * 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 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.h 2
4.b odd 2 1 546.2.c.c 2
12.b even 2 1 1638.2.c.e 2
13.b even 2 1 inner 4368.2.h.h 2
28.d even 2 1 3822.2.c.b 2
52.b odd 2 1 546.2.c.c 2
52.f even 4 1 7098.2.a.o 1
52.f even 4 1 7098.2.a.y 1
156.h even 2 1 1638.2.c.e 2
364.h even 2 1 3822.2.c.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
546.2.c.c 2 4.b odd 2 1
546.2.c.c 2 52.b odd 2 1
1638.2.c.e 2 12.b even 2 1
1638.2.c.e 2 156.h even 2 1
3822.2.c.b 2 28.d even 2 1
3822.2.c.b 2 364.h even 2 1
4368.2.h.h 2 1.a even 1 1 trivial
4368.2.h.h 2 13.b even 2 1 inner
7098.2.a.o 1 52.f even 4 1
7098.2.a.y 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 $$S_{2}^{\mathrm{new}}(4368, [\chi])$$:

 $$T_{5}^{2} + 1$$ T5^2 + 1 $$T_{11}^{2} + 9$$ T11^2 + 9 $$T_{17} - 7$$ T17 - 7

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$(T + 1)^{2}$$
$5$ $$T^{2} + 1$$
$7$ $$T^{2} + 1$$
$11$ $$T^{2} + 9$$
$13$ $$T^{2} - 6T + 13$$
$17$ $$(T - 7)^{2}$$
$19$ $$T^{2} + 9$$
$23$ $$(T + 1)^{2}$$
$29$ $$(T + 1)^{2}$$
$31$ $$T^{2} + 64$$
$37$ $$T^{2} + 1$$
$41$ $$T^{2} + 16$$
$43$ $$(T - 5)^{2}$$
$47$ $$T^{2}$$
$53$ $$(T + 6)^{2}$$
$59$ $$T^{2} + 100$$
$61$ $$(T + 13)^{2}$$
$67$ $$T^{2} + 64$$
$71$ $$T^{2} + 36$$
$73$ $$T^{2} + 169$$
$79$ $$(T - 12)^{2}$$
$83$ $$T^{2} + 4$$
$89$ $$T^{2} + 144$$
$97$ $$T^{2} + 36$$