# Properties

 Label 4368.2.h.n Level $4368$ Weight $2$ Character orbit 4368.h Analytic conductor $34.879$ Analytic rank $0$ Dimension $4$ 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: $$4$$ Coefficient field: $$\Q(i, \sqrt{17})$$ Defining polynomial: $$x^{4} + 9x^{2} + 16$$ x^4 + 9*x^2 + 16 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ 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 a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + q^{3} + ( - \beta_{2} + \beta_1) q^{5} - \beta_{2} q^{7} + q^{9}+O(q^{10})$$ q + q^3 + (-b2 + b1) * q^5 - b2 * q^7 + q^9 $$q + q^{3} + ( - \beta_{2} + \beta_1) q^{5} - \beta_{2} q^{7} + q^{9} + (3 \beta_{2} - \beta_1) q^{11} + ( - \beta_{3} - 2 \beta_{2} - \beta_1 + 2) q^{13} + ( - \beta_{2} + \beta_1) q^{15} + (3 \beta_{3} - 2) q^{17} + (\beta_{2} + \beta_1) q^{19} - \beta_{2} q^{21} + ( - 3 \beta_{3} + 2) q^{23} + (3 \beta_{3} - 3) q^{25} + q^{27} - \beta_{3} q^{29} + (4 \beta_{2} + 4 \beta_1) q^{31} + (3 \beta_{2} - \beta_1) q^{33} + (\beta_{3} - 2) q^{35} + ( - 3 \beta_{2} + 3 \beta_1) q^{37} + ( - \beta_{3} - 2 \beta_{2} - \beta_1 + 2) q^{39} - 4 \beta_{2} q^{41} + (\beta_{3} + 8) q^{43} + ( - \beta_{2} + \beta_1) q^{45} + 4 \beta_1 q^{47} - q^{49} + (3 \beta_{3} - 2) q^{51} + (2 \beta_{3} + 8) q^{53} + ( - 5 \beta_{3} + 12) q^{55} + (\beta_{2} + \beta_1) q^{57} + ( - 6 \beta_{2} - 4 \beta_1) q^{59} + \beta_{3} q^{61} - \beta_{2} q^{63} + ( - 5 \beta_{2} + 3 \beta_1 + 2) q^{65} + (4 \beta_{2} + 2 \beta_1) q^{67} + ( - 3 \beta_{3} + 2) q^{69} + ( - 6 \beta_{2} - 6 \beta_1) q^{71} + ( - 9 \beta_{2} + \beta_1) q^{73} + (3 \beta_{3} - 3) q^{75} + ( - \beta_{3} + 4) q^{77} - 16 q^{79} + q^{81} - 2 \beta_{2} q^{83} + (11 \beta_{2} - 5 \beta_1) q^{85} - \beta_{3} q^{87} + 8 \beta_{2} q^{89} + ( - \beta_{3} - \beta_{2} + \beta_1 - 1) q^{91} + (4 \beta_{2} + 4 \beta_1) q^{93} + (\beta_{3} - 4) q^{95} - 10 \beta_{2} q^{97} + (3 \beta_{2} - \beta_1) q^{99}+O(q^{100})$$ q + q^3 + (-b2 + b1) * q^5 - b2 * q^7 + q^9 + (3*b2 - b1) * q^11 + (-b3 - 2*b2 - b1 + 2) * q^13 + (-b2 + b1) * q^15 + (3*b3 - 2) * q^17 + (b2 + b1) * q^19 - b2 * q^21 + (-3*b3 + 2) * q^23 + (3*b3 - 3) * q^25 + q^27 - b3 * q^29 + (4*b2 + 4*b1) * q^31 + (3*b2 - b1) * q^33 + (b3 - 2) * q^35 + (-3*b2 + 3*b1) * q^37 + (-b3 - 2*b2 - b1 + 2) * q^39 - 4*b2 * q^41 + (b3 + 8) * q^43 + (-b2 + b1) * q^45 + 4*b1 * q^47 - q^49 + (3*b3 - 2) * q^51 + (2*b3 + 8) * q^53 + (-5*b3 + 12) * q^55 + (b2 + b1) * q^57 + (-6*b2 - 4*b1) * q^59 + b3 * q^61 - b2 * q^63 + (-5*b2 + 3*b1 + 2) * q^65 + (4*b2 + 2*b1) * q^67 + (-3*b3 + 2) * q^69 + (-6*b2 - 6*b1) * q^71 + (-9*b2 + b1) * q^73 + (3*b3 - 3) * q^75 + (-b3 + 4) * q^77 - 16 * q^79 + q^81 - 2*b2 * q^83 + (11*b2 - 5*b1) * q^85 - b3 * q^87 + 8*b2 * q^89 + (-b3 - b2 + b1 - 1) * q^91 + (4*b2 + 4*b1) * q^93 + (b3 - 4) * q^95 - 10*b2 * q^97 + (3*b2 - b1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 4 q^{3} + 4 q^{9}+O(q^{10})$$ 4 * q + 4 * q^3 + 4 * q^9 $$4 q + 4 q^{3} + 4 q^{9} + 6 q^{13} - 2 q^{17} + 2 q^{23} - 6 q^{25} + 4 q^{27} - 2 q^{29} - 6 q^{35} + 6 q^{39} + 34 q^{43} - 4 q^{49} - 2 q^{51} + 36 q^{53} + 38 q^{55} + 2 q^{61} + 8 q^{65} + 2 q^{69} - 6 q^{75} + 14 q^{77} - 64 q^{79} + 4 q^{81} - 2 q^{87} - 6 q^{91} - 14 q^{95}+O(q^{100})$$ 4 * q + 4 * q^3 + 4 * q^9 + 6 * q^13 - 2 * q^17 + 2 * q^23 - 6 * q^25 + 4 * q^27 - 2 * q^29 - 6 * q^35 + 6 * q^39 + 34 * q^43 - 4 * q^49 - 2 * q^51 + 36 * q^53 + 38 * q^55 + 2 * q^61 + 8 * q^65 + 2 * q^69 - 6 * q^75 + 14 * q^77 - 64 * q^79 + 4 * q^81 - 2 * q^87 - 6 * q^91 - 14 * q^95

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 9x^{2} + 16$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{3} + 5\nu ) / 4$$ (v^3 + 5*v) / 4 $$\beta_{3}$$ $$=$$ $$\nu^{2} + 5$$ v^2 + 5
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{3} - 5$$ b3 - 5 $$\nu^{3}$$ $$=$$ $$4\beta_{2} - 5\beta_1$$ 4*b2 - 5*b1

## 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
 − 2.56155i − 1.56155i 1.56155i 2.56155i
0 1.00000 0 3.56155i 0 1.00000i 0 1.00000 0
337.2 0 1.00000 0 0.561553i 0 1.00000i 0 1.00000 0
337.3 0 1.00000 0 0.561553i 0 1.00000i 0 1.00000 0
337.4 0 1.00000 0 3.56155i 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.n 4
4.b odd 2 1 546.2.c.e 4
12.b even 2 1 1638.2.c.h 4
13.b even 2 1 inner 4368.2.h.n 4
28.d even 2 1 3822.2.c.h 4
52.b odd 2 1 546.2.c.e 4
52.f even 4 1 7098.2.a.bg 2
52.f even 4 1 7098.2.a.bv 2
156.h even 2 1 1638.2.c.h 4
364.h even 2 1 3822.2.c.h 4

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

 $$T_{5}^{4} + 13T_{5}^{2} + 4$$ T5^4 + 13*T5^2 + 4 $$T_{11}^{4} + 33T_{11}^{2} + 64$$ T11^4 + 33*T11^2 + 64 $$T_{17}^{2} + T_{17} - 38$$ T17^2 + T17 - 38

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$(T - 1)^{4}$$
$5$ $$T^{4} + 13T^{2} + 4$$
$7$ $$(T^{2} + 1)^{2}$$
$11$ $$T^{4} + 33T^{2} + 64$$
$13$ $$T^{4} - 6 T^{3} + 18 T^{2} - 78 T + 169$$
$17$ $$(T^{2} + T - 38)^{2}$$
$19$ $$T^{4} + 9T^{2} + 16$$
$23$ $$(T^{2} - T - 38)^{2}$$
$29$ $$(T^{2} + T - 4)^{2}$$
$31$ $$T^{4} + 144T^{2} + 4096$$
$37$ $$T^{4} + 117T^{2} + 324$$
$41$ $$(T^{2} + 16)^{2}$$
$43$ $$(T^{2} - 17 T + 68)^{2}$$
$47$ $$T^{4} + 144T^{2} + 4096$$
$53$ $$(T^{2} - 18 T + 64)^{2}$$
$59$ $$T^{4} + 168T^{2} + 2704$$
$61$ $$(T^{2} - T - 4)^{2}$$
$67$ $$T^{4} + 52T^{2} + 64$$
$71$ $$T^{4} + 324 T^{2} + 20736$$
$73$ $$T^{4} + 189T^{2} + 7396$$
$79$ $$(T + 16)^{4}$$
$83$ $$(T^{2} + 4)^{2}$$
$89$ $$(T^{2} + 64)^{2}$$
$97$ $$(T^{2} + 100)^{2}$$