# Properties

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

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## 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 273) 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.

 $$f(q)$$ $$=$$ $$q - q^{3} + 3 i q^{5} + i q^{7} + q^{9}+O(q^{10})$$ q - q^3 + 3*i * q^5 + i * q^7 + q^9 $$q - q^{3} + 3 i q^{5} + i q^{7} + q^{9} + ( - 3 i + 2) q^{13} - 3 i q^{15} - 2 q^{17} - i q^{19} - i q^{21} + q^{23} - 4 q^{25} - q^{27} + 5 q^{29} - 5 i q^{31} - 3 q^{35} - 8 i q^{37} + (3 i - 2) q^{39} - 10 i q^{41} - 9 q^{43} + 3 i q^{45} - 7 i q^{47} - q^{49} + 2 q^{51} + 9 q^{53} + i q^{57} + 4 i q^{59} - 8 q^{61} + i q^{63} + (6 i + 9) q^{65} - 2 i q^{67} - q^{69} + 9 i q^{73} + 4 q^{75} - 15 q^{79} + q^{81} - 9 i q^{83} - 6 i q^{85} - 5 q^{87} - 9 i q^{89} + (2 i + 3) q^{91} + 5 i q^{93} + 3 q^{95} - 13 i q^{97} +O(q^{100})$$ q - q^3 + 3*i * q^5 + i * q^7 + q^9 + (-3*i + 2) * q^13 - 3*i * q^15 - 2 * q^17 - i * q^19 - i * q^21 + q^23 - 4 * q^25 - q^27 + 5 * q^29 - 5*i * q^31 - 3 * q^35 - 8*i * q^37 + (3*i - 2) * q^39 - 10*i * q^41 - 9 * q^43 + 3*i * q^45 - 7*i * q^47 - q^49 + 2 * q^51 + 9 * q^53 + i * q^57 + 4*i * q^59 - 8 * q^61 + i * q^63 + (6*i + 9) * q^65 - 2*i * q^67 - q^69 + 9*i * q^73 + 4 * q^75 - 15 * q^79 + q^81 - 9*i * q^83 - 6*i * q^85 - 5 * q^87 - 9*i * q^89 + (2*i + 3) * q^91 + 5*i * q^93 + 3 * q^95 - 13*i * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{3} + 2 q^{9}+O(q^{10})$$ 2 * q - 2 * q^3 + 2 * q^9 $$2 q - 2 q^{3} + 2 q^{9} + 4 q^{13} - 4 q^{17} + 2 q^{23} - 8 q^{25} - 2 q^{27} + 10 q^{29} - 6 q^{35} - 4 q^{39} - 18 q^{43} - 2 q^{49} + 4 q^{51} + 18 q^{53} - 16 q^{61} + 18 q^{65} - 2 q^{69} + 8 q^{75} - 30 q^{79} + 2 q^{81} - 10 q^{87} + 6 q^{91} + 6 q^{95}+O(q^{100})$$ 2 * q - 2 * q^3 + 2 * q^9 + 4 * q^13 - 4 * q^17 + 2 * q^23 - 8 * q^25 - 2 * q^27 + 10 * q^29 - 6 * q^35 - 4 * q^39 - 18 * q^43 - 2 * q^49 + 4 * q^51 + 18 * q^53 - 16 * q^61 + 18 * q^65 - 2 * q^69 + 8 * q^75 - 30 * q^79 + 2 * q^81 - 10 * q^87 + 6 * 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 3.00000i 0 1.00000i 0 1.00000 0
337.2 0 −1.00000 0 3.00000i 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.e 2
4.b odd 2 1 273.2.c.a 2
12.b even 2 1 819.2.c.a 2
13.b even 2 1 inner 4368.2.h.e 2
28.d even 2 1 1911.2.c.a 2
52.b odd 2 1 273.2.c.a 2
52.f even 4 1 3549.2.a.a 1
52.f even 4 1 3549.2.a.e 1
156.h even 2 1 819.2.c.a 2
364.h even 2 1 1911.2.c.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
273.2.c.a 2 4.b odd 2 1
273.2.c.a 2 52.b odd 2 1
819.2.c.a 2 12.b even 2 1
819.2.c.a 2 156.h even 2 1
1911.2.c.a 2 28.d even 2 1
1911.2.c.a 2 364.h even 2 1
3549.2.a.a 1 52.f even 4 1
3549.2.a.e 1 52.f even 4 1
4368.2.h.e 2 1.a even 1 1 trivial
4368.2.h.e 2 13.b even 2 1 inner

## Hecke kernels

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

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$(T + 1)^{2}$$
$5$ $$T^{2} + 9$$
$7$ $$T^{2} + 1$$
$11$ $$T^{2}$$
$13$ $$T^{2} - 4T + 13$$
$17$ $$(T + 2)^{2}$$
$19$ $$T^{2} + 1$$
$23$ $$(T - 1)^{2}$$
$29$ $$(T - 5)^{2}$$
$31$ $$T^{2} + 25$$
$37$ $$T^{2} + 64$$
$41$ $$T^{2} + 100$$
$43$ $$(T + 9)^{2}$$
$47$ $$T^{2} + 49$$
$53$ $$(T - 9)^{2}$$
$59$ $$T^{2} + 16$$
$61$ $$(T + 8)^{2}$$
$67$ $$T^{2} + 4$$
$71$ $$T^{2}$$
$73$ $$T^{2} + 81$$
$79$ $$(T + 15)^{2}$$
$83$ $$T^{2} + 81$$
$89$ $$T^{2} + 81$$
$97$ $$T^{2} + 169$$
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