# Properties

 Label 1344.3.f.e Level $1344$ Weight $3$ Character orbit 1344.f 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.f (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$36.6213475300$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\sqrt{2}, \sqrt{-3})$$ Defining polynomial: $$x^{4} + 2x^{2} + 4$$ x^4 + 2*x^2 + 4 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{2}\cdot 3$$ 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.

 $$f(q)$$ $$=$$ $$q - \beta_{2} q^{3} + (2 \beta_{2} - \beta_1) q^{5} + (\beta_{3} - \beta_{2} - 2 \beta_1 - 2) q^{7} - 3 q^{9}+O(q^{10})$$ q - b2 * q^3 + (2*b2 - b1) * q^5 + (b3 - b2 - 2*b1 - 2) * q^7 - 3 * q^9 $$q - \beta_{2} q^{3} + (2 \beta_{2} - \beta_1) q^{5} + (\beta_{3} - \beta_{2} - 2 \beta_1 - 2) q^{7} - 3 q^{9} + ( - \beta_{3} - 6) q^{11} + ( - 8 \beta_{2} + 2 \beta_1) q^{13} + ( - \beta_{3} + 6) q^{15} + ( - 2 \beta_{2} - 11 \beta_1) q^{17} + ( - 2 \beta_{2} + 8 \beta_1) q^{19} + ( - 2 \beta_{3} + 2 \beta_{2} - 3 \beta_1 - 3) q^{21} + (3 \beta_{3} - 6) q^{23} + (4 \beta_{3} + 7) q^{25} + 3 \beta_{2} q^{27} - 30 q^{29} + (12 \beta_{2} + 12 \beta_1) q^{31} + (6 \beta_{2} + 3 \beta_1) q^{33} + (3 \beta_{3} - 10 \beta_{2} + 8 \beta_1 - 6) q^{35} + ( - 12 \beta_{3} + 20) q^{37} + (2 \beta_{3} - 24) q^{39} + (14 \beta_{2} - 7 \beta_1) q^{41} + ( - 10 \beta_{3} - 32) q^{43} + ( - 6 \beta_{2} + 3 \beta_1) q^{45} + ( - 28 \beta_{2} - 4 \beta_1) q^{47} + ( - 8 \beta_{3} - 20 \beta_{2} + 2 \beta_1 - 5) q^{49} + ( - 11 \beta_{3} - 6) q^{51} + (4 \beta_{3} + 54) q^{53} - 6 \beta_{2} q^{55} + (8 \beta_{3} - 6) q^{57} + ( - 28 \beta_{2} + 20 \beta_1) q^{59} + ( - 4 \beta_{2} + 4 \beta_1) q^{61} + ( - 3 \beta_{3} + 3 \beta_{2} + 6 \beta_1 + 6) q^{63} + ( - 12 \beta_{3} + 60) q^{65} + ( - 4 \beta_{3} + 44) q^{67} + (6 \beta_{2} - 9 \beta_1) q^{69} + ( - 19 \beta_{3} + 30) q^{71} + (4 \beta_{2} + 26 \beta_1) q^{73} + ( - 7 \beta_{2} - 12 \beta_1) q^{75} + ( - 4 \beta_{3} + 18 \beta_{2} + 15 \beta_1 - 6) q^{77} + ( - 24 \beta_{3} - 32) q^{79} + 9 q^{81} + (32 \beta_{2} + 20 \beta_1) q^{83} + (20 \beta_{3} - 54) q^{85} + 30 \beta_{2} q^{87} + ( - 54 \beta_{2} - 21 \beta_1) q^{89} + ( - 14 \beta_{3} + 28 \beta_{2} - 28 \beta_1) q^{91} + (12 \beta_{3} + 36) q^{93} + ( - 18 \beta_{3} + 60) q^{95} + (44 \beta_{2} + 10 \beta_1) q^{97} + (3 \beta_{3} + 18) q^{99}+O(q^{100})$$ q - b2 * q^3 + (2*b2 - b1) * q^5 + (b3 - b2 - 2*b1 - 2) * q^7 - 3 * q^9 + (-b3 - 6) * q^11 + (-8*b2 + 2*b1) * q^13 + (-b3 + 6) * q^15 + (-2*b2 - 11*b1) * q^17 + (-2*b2 + 8*b1) * q^19 + (-2*b3 + 2*b2 - 3*b1 - 3) * q^21 + (3*b3 - 6) * q^23 + (4*b3 + 7) * q^25 + 3*b2 * q^27 - 30 * q^29 + (12*b2 + 12*b1) * q^31 + (6*b2 + 3*b1) * q^33 + (3*b3 - 10*b2 + 8*b1 - 6) * q^35 + (-12*b3 + 20) * q^37 + (2*b3 - 24) * q^39 + (14*b2 - 7*b1) * q^41 + (-10*b3 - 32) * q^43 + (-6*b2 + 3*b1) * q^45 + (-28*b2 - 4*b1) * q^47 + (-8*b3 - 20*b2 + 2*b1 - 5) * q^49 + (-11*b3 - 6) * q^51 + (4*b3 + 54) * q^53 - 6*b2 * q^55 + (8*b3 - 6) * q^57 + (-28*b2 + 20*b1) * q^59 + (-4*b2 + 4*b1) * q^61 + (-3*b3 + 3*b2 + 6*b1 + 6) * q^63 + (-12*b3 + 60) * q^65 + (-4*b3 + 44) * q^67 + (6*b2 - 9*b1) * q^69 + (-19*b3 + 30) * q^71 + (4*b2 + 26*b1) * q^73 + (-7*b2 - 12*b1) * q^75 + (-4*b3 + 18*b2 + 15*b1 - 6) * q^77 + (-24*b3 - 32) * q^79 + 9 * q^81 + (32*b2 + 20*b1) * q^83 + (20*b3 - 54) * q^85 + 30*b2 * q^87 + (-54*b2 - 21*b1) * q^89 + (-14*b3 + 28*b2 - 28*b1) * q^91 + (12*b3 + 36) * q^93 + (-18*b3 + 60) * q^95 + (44*b2 + 10*b1) * q^97 + (3*b3 + 18) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 8 q^{7} - 12 q^{9}+O(q^{10})$$ 4 * q - 8 * q^7 - 12 * q^9 $$4 q - 8 q^{7} - 12 q^{9} - 24 q^{11} + 24 q^{15} - 12 q^{21} - 24 q^{23} + 28 q^{25} - 120 q^{29} - 24 q^{35} + 80 q^{37} - 96 q^{39} - 128 q^{43} - 20 q^{49} - 24 q^{51} + 216 q^{53} - 24 q^{57} + 24 q^{63} + 240 q^{65} + 176 q^{67} + 120 q^{71} - 24 q^{77} - 128 q^{79} + 36 q^{81} - 216 q^{85} + 144 q^{93} + 240 q^{95} + 72 q^{99}+O(q^{100})$$ 4 * q - 8 * q^7 - 12 * q^9 - 24 * q^11 + 24 * q^15 - 12 * q^21 - 24 * q^23 + 28 * q^25 - 120 * q^29 - 24 * q^35 + 80 * q^37 - 96 * q^39 - 128 * q^43 - 20 * q^49 - 24 * q^51 + 216 * q^53 - 24 * q^57 + 24 * q^63 + 240 * q^65 + 176 * q^67 + 120 * q^71 - 24 * q^77 - 128 * q^79 + 36 * q^81 - 216 * q^85 + 144 * q^93 + 240 * q^95 + 72 * q^99

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

 $$\beta_{1}$$ $$=$$ $$( \nu^{3} + 4\nu ) / 2$$ (v^3 + 4*v) / 2 $$\beta_{2}$$ $$=$$ $$\nu^{2} + 1$$ v^2 + 1 $$\beta_{3}$$ $$=$$ $$( -3\nu^{3} ) / 2$$ (-3*v^3) / 2
 $$\nu$$ $$=$$ $$( \beta_{3} + 3\beta_1 ) / 6$$ (b3 + 3*b1) / 6 $$\nu^{2}$$ $$=$$ $$\beta_{2} - 1$$ b2 - 1 $$\nu^{3}$$ $$=$$ $$( -2\beta_{3} ) / 3$$ (-2*b3) / 3

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
769.1
 0.707107 + 1.22474i −0.707107 − 1.22474i −0.707107 + 1.22474i 0.707107 − 1.22474i
0 1.73205i 0 1.01461i 0 2.24264 6.63103i 0 −3.00000 0
769.2 0 1.73205i 0 5.91359i 0 −6.24264 + 3.16693i 0 −3.00000 0
769.3 0 1.73205i 0 5.91359i 0 −6.24264 3.16693i 0 −3.00000 0
769.4 0 1.73205i 0 1.01461i 0 2.24264 + 6.63103i 0 −3.00000 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
7.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.f.e 4
4.b odd 2 1 1344.3.f.f 4
7.b odd 2 1 inner 1344.3.f.e 4
8.b even 2 1 336.3.f.c 4
8.d odd 2 1 42.3.c.a 4
24.f even 2 1 126.3.c.b 4
24.h odd 2 1 1008.3.f.g 4
28.d even 2 1 1344.3.f.f 4
40.e odd 2 1 1050.3.f.a 4
40.k even 4 2 1050.3.h.a 8
56.e even 2 1 42.3.c.a 4
56.h odd 2 1 336.3.f.c 4
56.k odd 6 1 294.3.g.b 4
56.k odd 6 1 294.3.g.c 4
56.m even 6 1 294.3.g.b 4
56.m even 6 1 294.3.g.c 4
168.e odd 2 1 126.3.c.b 4
168.i even 2 1 1008.3.f.g 4
168.v even 6 1 882.3.n.a 4
168.v even 6 1 882.3.n.d 4
168.be odd 6 1 882.3.n.a 4
168.be odd 6 1 882.3.n.d 4
280.n even 2 1 1050.3.f.a 4
280.y odd 4 2 1050.3.h.a 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
42.3.c.a 4 8.d odd 2 1
42.3.c.a 4 56.e even 2 1
126.3.c.b 4 24.f even 2 1
126.3.c.b 4 168.e odd 2 1
294.3.g.b 4 56.k odd 6 1
294.3.g.b 4 56.m even 6 1
294.3.g.c 4 56.k odd 6 1
294.3.g.c 4 56.m even 6 1
336.3.f.c 4 8.b even 2 1
336.3.f.c 4 56.h odd 2 1
882.3.n.a 4 168.v even 6 1
882.3.n.a 4 168.be odd 6 1
882.3.n.d 4 168.v even 6 1
882.3.n.d 4 168.be odd 6 1
1008.3.f.g 4 24.h odd 2 1
1008.3.f.g 4 168.i even 2 1
1050.3.f.a 4 40.e odd 2 1
1050.3.f.a 4 280.n even 2 1
1050.3.h.a 8 40.k even 4 2
1050.3.h.a 8 280.y odd 4 2
1344.3.f.e 4 1.a even 1 1 trivial
1344.3.f.e 4 7.b odd 2 1 inner
1344.3.f.f 4 4.b odd 2 1
1344.3.f.f 4 28.d even 2 1

## Hecke kernels

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

 $$T_{5}^{4} + 36T_{5}^{2} + 36$$ T5^4 + 36*T5^2 + 36 $$T_{11}^{2} + 12T_{11} + 18$$ T11^2 + 12*T11 + 18

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$(T^{2} + 3)^{2}$$
$5$ $$T^{4} + 36T^{2} + 36$$
$7$ $$T^{4} + 8 T^{3} + 42 T^{2} + \cdots + 2401$$
$11$ $$(T^{2} + 12 T + 18)^{2}$$
$13$ $$T^{4} + 432 T^{2} + 28224$$
$17$ $$T^{4} + 1476 T^{2} + 509796$$
$19$ $$T^{4} + 792 T^{2} + 138384$$
$23$ $$(T^{2} + 12 T - 126)^{2}$$
$29$ $$(T + 30)^{4}$$
$31$ $$T^{4} + 2592 T^{2} + 186624$$
$37$ $$(T^{2} - 40 T - 2192)^{2}$$
$41$ $$T^{4} + 1764 T^{2} + 86436$$
$43$ $$(T^{2} + 64 T - 776)^{2}$$
$47$ $$T^{4} + 4896 T^{2} + \cdots + 5089536$$
$53$ $$(T^{2} - 108 T + 2628)^{2}$$
$59$ $$T^{4} + 9504 T^{2} + 2304$$
$61$ $$T^{4} + 288T^{2} + 2304$$
$67$ $$(T^{2} - 88 T + 1648)^{2}$$
$71$ $$(T^{2} - 60 T - 5598)^{2}$$
$73$ $$T^{4} + 8208 T^{2} + \cdots + 16064064$$
$79$ $$(T^{2} + 64 T - 9344)^{2}$$
$83$ $$T^{4} + 10944 T^{2} + \cdots + 451584$$
$89$ $$T^{4} + 22788 T^{2} + \cdots + 37234404$$
$97$ $$T^{4} + 12816 T^{2} + \cdots + 27123264$$