# Properties

 Label 1344.3.d.c Level $1344$ Weight $3$ Character orbit 1344.d Analytic conductor $36.621$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1344 = 2^{6} \cdot 3 \cdot 7$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 1344.d (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$36.6213475300$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\sqrt{-2}, \sqrt{7})$$ Defining polynomial: $$x^{4} + 8x^{2} + 9$$ x^4 + 8*x^2 + 9 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 42) 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.

 $$f(q)$$ $$=$$ $$q + ( - \beta_{3} - \beta_1) q^{3} + (\beta_{2} - 2 \beta_1) q^{5} - \beta_{3} q^{7} + (2 \beta_{2} + 5) q^{9}+O(q^{10})$$ q + (-b3 - b1) * q^3 + (b2 - 2*b1) * q^5 - b3 * q^7 + (2*b2 + 5) * q^9 $$q + ( - \beta_{3} - \beta_1) q^{3} + (\beta_{2} - 2 \beta_1) q^{5} - \beta_{3} q^{7} + (2 \beta_{2} + 5) q^{9} + (2 \beta_{2} + 5 \beta_1) q^{11} + (4 \beta_{3} - 10) q^{13} + (2 \beta_{3} + 2 \beta_{2} - 7 \beta_1 - 4) q^{15} + ( - 5 \beta_{2} - 2 \beta_1) q^{17} + 16 q^{19} + (\beta_{2} + 7) q^{21} + (10 \beta_{2} + \beta_1) q^{23} + (8 \beta_{3} + 3) q^{25} + ( - \beta_{3} - 19 \beta_1) q^{27} + ( - 2 \beta_{2} - 20 \beta_1) q^{29} + ( - 10 \beta_{3} - 32) q^{31} + (4 \beta_{3} - 5 \beta_{2} - 14 \beta_1 + 10) q^{33} + (2 \beta_{2} - 7 \beta_1) q^{35} - 20 q^{37} + (10 \beta_{3} - 4 \beta_{2} + 10 \beta_1 - 28) q^{39} + (9 \beta_{2} - 30 \beta_1) q^{41} + (12 \beta_{3} - 20) q^{43} + (8 \beta_{3} + 5 \beta_{2} - 10 \beta_1 - 28) q^{45} - 6 \beta_1 q^{47} + 7 q^{49} + ( - 10 \beta_{3} + 2 \beta_{2} + 35 \beta_1 - 4) q^{51} + 36 \beta_1 q^{53} + ( - 2 \beta_{3} - 8) q^{55} + ( - 16 \beta_{3} - 16 \beta_1) q^{57} + ( - 8 \beta_{2} - 20 \beta_1) q^{59} + ( - 20 \beta_{3} + 14) q^{61} + ( - 5 \beta_{3} - 14 \beta_1) q^{63} + ( - 18 \beta_{2} + 48 \beta_1) q^{65} + ( - 4 \beta_{3} - 60) q^{67} + (20 \beta_{3} - \beta_{2} - 70 \beta_1 + 2) q^{69} + ( - 14 \beta_{2} + 25 \beta_1) q^{71} + (16 \beta_{3} + 30) q^{73} + ( - 3 \beta_{3} - 8 \beta_{2} - 3 \beta_1 - 56) q^{75} + ( - 5 \beta_{2} - 14 \beta_1) q^{77} + (20 \beta_{3} - 32) q^{79} + (20 \beta_{2} - 31) q^{81} + (20 \beta_{2} + 50 \beta_1) q^{83} + ( - 16 \beta_{3} + 62) q^{85} + ( - 4 \beta_{3} + 20 \beta_{2} + 14 \beta_1 - 40) q^{87} + (3 \beta_{2} + 30 \beta_1) q^{89} + (10 \beta_{3} - 28) q^{91} + (32 \beta_{3} + 10 \beta_{2} + 32 \beta_1 + 70) q^{93} + (16 \beta_{2} - 32 \beta_1) q^{95} + ( - 8 \beta_{3} - 90) q^{97} + ( - 20 \beta_{3} + 10 \beta_{2} + 25 \beta_1 - 56) q^{99}+O(q^{100})$$ q + (-b3 - b1) * q^3 + (b2 - 2*b1) * q^5 - b3 * q^7 + (2*b2 + 5) * q^9 + (2*b2 + 5*b1) * q^11 + (4*b3 - 10) * q^13 + (2*b3 + 2*b2 - 7*b1 - 4) * q^15 + (-5*b2 - 2*b1) * q^17 + 16 * q^19 + (b2 + 7) * q^21 + (10*b2 + b1) * q^23 + (8*b3 + 3) * q^25 + (-b3 - 19*b1) * q^27 + (-2*b2 - 20*b1) * q^29 + (-10*b3 - 32) * q^31 + (4*b3 - 5*b2 - 14*b1 + 10) * q^33 + (2*b2 - 7*b1) * q^35 - 20 * q^37 + (10*b3 - 4*b2 + 10*b1 - 28) * q^39 + (9*b2 - 30*b1) * q^41 + (12*b3 - 20) * q^43 + (8*b3 + 5*b2 - 10*b1 - 28) * q^45 - 6*b1 * q^47 + 7 * q^49 + (-10*b3 + 2*b2 + 35*b1 - 4) * q^51 + 36*b1 * q^53 + (-2*b3 - 8) * q^55 + (-16*b3 - 16*b1) * q^57 + (-8*b2 - 20*b1) * q^59 + (-20*b3 + 14) * q^61 + (-5*b3 - 14*b1) * q^63 + (-18*b2 + 48*b1) * q^65 + (-4*b3 - 60) * q^67 + (20*b3 - b2 - 70*b1 + 2) * q^69 + (-14*b2 + 25*b1) * q^71 + (16*b3 + 30) * q^73 + (-3*b3 - 8*b2 - 3*b1 - 56) * q^75 + (-5*b2 - 14*b1) * q^77 + (20*b3 - 32) * q^79 + (20*b2 - 31) * q^81 + (20*b2 + 50*b1) * q^83 + (-16*b3 + 62) * q^85 + (-4*b3 + 20*b2 + 14*b1 - 40) * q^87 + (3*b2 + 30*b1) * q^89 + (10*b3 - 28) * q^91 + (32*b3 + 10*b2 + 32*b1 + 70) * q^93 + (16*b2 - 32*b1) * q^95 + (-8*b3 - 90) * q^97 + (-20*b3 + 10*b2 + 25*b1 - 56) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 20 q^{9}+O(q^{10})$$ 4 * q + 20 * q^9 $$4 q + 20 q^{9} - 40 q^{13} - 16 q^{15} + 64 q^{19} + 28 q^{21} + 12 q^{25} - 128 q^{31} + 40 q^{33} - 80 q^{37} - 112 q^{39} - 80 q^{43} - 112 q^{45} + 28 q^{49} - 16 q^{51} - 32 q^{55} + 56 q^{61} - 240 q^{67} + 8 q^{69} + 120 q^{73} - 224 q^{75} - 128 q^{79} - 124 q^{81} + 248 q^{85} - 160 q^{87} - 112 q^{91} + 280 q^{93} - 360 q^{97} - 224 q^{99}+O(q^{100})$$ 4 * q + 20 * q^9 - 40 * q^13 - 16 * q^15 + 64 * q^19 + 28 * q^21 + 12 * q^25 - 128 * q^31 + 40 * q^33 - 80 * q^37 - 112 * q^39 - 80 * q^43 - 112 * q^45 + 28 * q^49 - 16 * q^51 - 32 * q^55 + 56 * q^61 - 240 * q^67 + 8 * q^69 + 120 * q^73 - 224 * q^75 - 128 * q^79 - 124 * q^81 + 248 * q^85 - 160 * q^87 - 112 * q^91 + 280 * q^93 - 360 * q^97 - 224 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 8x^{2} + 9$$ :

 $$\beta_{1}$$ $$=$$ $$( \nu^{3} + 5\nu ) / 3$$ (v^3 + 5*v) / 3 $$\beta_{2}$$ $$=$$ $$( \nu^{3} + 11\nu ) / 3$$ (v^3 + 11*v) / 3 $$\beta_{3}$$ $$=$$ $$\nu^{2} + 4$$ v^2 + 4
 $$\nu$$ $$=$$ $$( \beta_{2} - \beta_1 ) / 2$$ (b2 - b1) / 2 $$\nu^{2}$$ $$=$$ $$\beta_{3} - 4$$ b3 - 4 $$\nu^{3}$$ $$=$$ $$( -5\beta_{2} + 11\beta_1 ) / 2$$ (-5*b2 + 11*b1) / 2

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1344\mathbb{Z}\right)^\times$$.

 $$n$$ $$127$$ $$449$$ $$577$$ $$1093$$ $$\chi(n)$$ $$1$$ $$-1$$ $$1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
449.1
 1.16372i − 1.16372i − 2.57794i 2.57794i
0 −2.64575 1.41421i 0 0.913230i 0 −2.64575 0 5.00000 + 7.48331i 0
449.2 0 −2.64575 + 1.41421i 0 0.913230i 0 −2.64575 0 5.00000 7.48331i 0
449.3 0 2.64575 1.41421i 0 6.57008i 0 2.64575 0 5.00000 7.48331i 0
449.4 0 2.64575 + 1.41421i 0 6.57008i 0 2.64575 0 5.00000 + 7.48331i 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
3.b odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1344.3.d.c 4
3.b odd 2 1 inner 1344.3.d.c 4
4.b odd 2 1 1344.3.d.e 4
8.b even 2 1 42.3.b.a 4
8.d odd 2 1 336.3.d.b 4
12.b even 2 1 1344.3.d.e 4
24.f even 2 1 336.3.d.b 4
24.h odd 2 1 42.3.b.a 4
40.f even 2 1 1050.3.e.a 4
40.i odd 4 2 1050.3.c.a 8
56.h odd 2 1 294.3.b.h 4
56.j odd 6 2 294.3.h.g 8
56.p even 6 2 294.3.h.d 8
72.j odd 6 2 1134.3.q.a 8
72.n even 6 2 1134.3.q.a 8
120.i odd 2 1 1050.3.e.a 4
120.w even 4 2 1050.3.c.a 8
168.i even 2 1 294.3.b.h 4
168.s odd 6 2 294.3.h.d 8
168.ba even 6 2 294.3.h.g 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
42.3.b.a 4 8.b even 2 1
42.3.b.a 4 24.h odd 2 1
294.3.b.h 4 56.h odd 2 1
294.3.b.h 4 168.i even 2 1
294.3.h.d 8 56.p even 6 2
294.3.h.d 8 168.s odd 6 2
294.3.h.g 8 56.j odd 6 2
294.3.h.g 8 168.ba even 6 2
336.3.d.b 4 8.d odd 2 1
336.3.d.b 4 24.f even 2 1
1050.3.c.a 8 40.i odd 4 2
1050.3.c.a 8 120.w even 4 2
1050.3.e.a 4 40.f even 2 1
1050.3.e.a 4 120.i odd 2 1
1134.3.q.a 8 72.j odd 6 2
1134.3.q.a 8 72.n even 6 2
1344.3.d.c 4 1.a even 1 1 trivial
1344.3.d.c 4 3.b odd 2 1 inner
1344.3.d.e 4 4.b odd 2 1
1344.3.d.e 4 12.b even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{3}^{\mathrm{new}}(1344, [\chi])$$:

 $$T_{5}^{4} + 44T_{5}^{2} + 36$$ T5^4 + 44*T5^2 + 36 $$T_{19} - 16$$ T19 - 16

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4} - 10T^{2} + 81$$
$5$ $$T^{4} + 44T^{2} + 36$$
$7$ $$(T^{2} - 7)^{2}$$
$11$ $$T^{4} + 212T^{2} + 36$$
$13$ $$(T^{2} + 20 T - 12)^{2}$$
$17$ $$T^{4} + 716 T^{2} + 116964$$
$19$ $$(T - 16)^{4}$$
$23$ $$T^{4} + 2804 T^{2} + \cdots + 1954404$$
$29$ $$T^{4} + 1712 T^{2} + 553536$$
$31$ $$(T^{2} + 64 T + 324)^{2}$$
$37$ $$(T + 20)^{4}$$
$41$ $$T^{4} + 5868 T^{2} + 443556$$
$43$ $$(T^{2} + 40 T - 608)^{2}$$
$47$ $$(T^{2} + 72)^{2}$$
$53$ $$(T^{2} + 2592)^{2}$$
$59$ $$T^{4} + 3392 T^{2} + 9216$$
$61$ $$(T^{2} - 28 T - 2604)^{2}$$
$67$ $$(T^{2} + 120 T + 3488)^{2}$$
$71$ $$T^{4} + 7988 T^{2} + \cdots + 2232036$$
$73$ $$(T^{2} - 60 T - 892)^{2}$$
$79$ $$(T^{2} + 64 T - 1776)^{2}$$
$83$ $$T^{4} + 21200 T^{2} + \cdots + 360000$$
$89$ $$T^{4} + 3852 T^{2} + \cdots + 2802276$$
$97$ $$(T^{2} + 180 T + 7652)^{2}$$