# Properties

 Label 1344.3.d.f 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: 4.0.65856.1 Defining polynomial: $$x^{4} + 14 x^{2} + 21$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 21) 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.

 $$f(q)$$ $$=$$ $$q + ( \beta_{1} + \beta_{2} - \beta_{3} ) q^{3} + ( 1 - \beta_{2} + 2 \beta_{3} ) q^{5} -\beta_{2} q^{7} + ( -6 - \beta_{1} + 2 \beta_{2} - 2 \beta_{3} ) q^{9} +O(q^{10})$$ $$q + ( \beta_{1} + \beta_{2} - \beta_{3} ) q^{3} + ( 1 - \beta_{2} + 2 \beta_{3} ) q^{5} -\beta_{2} q^{7} + ( -6 - \beta_{1} + 2 \beta_{2} - 2 \beta_{3} ) q^{9} -2 \beta_{1} q^{11} + ( 9 - \beta_{2} ) q^{13} + ( 9 - \beta_{1} + 2 \beta_{2} + 4 \beta_{3} ) q^{15} + ( 2 - 2 \beta_{1} - 2 \beta_{2} + 4 \beta_{3} ) q^{17} + ( -3 + 5 \beta_{2} ) q^{19} + ( -3 + 2 \beta_{1} - \beta_{2} + \beta_{3} ) q^{21} + ( -2 + 8 \beta_{1} + 2 \beta_{2} - 4 \beta_{3} ) q^{23} + ( -3 - 10 \beta_{2} ) q^{25} + ( -3 - 5 \beta_{1} - 8 \beta_{2} - \beta_{3} ) q^{27} + ( 2 + 2 \beta_{1} - 2 \beta_{2} + 4 \beta_{3} ) q^{29} + ( 34 + 2 \beta_{2} ) q^{31} + ( 6 + 2 \beta_{1} - 4 \beta_{2} - 2 \beta_{3} ) q^{33} + ( -3 + 2 \beta_{1} + 3 \beta_{2} - 6 \beta_{3} ) q^{35} + ( -4 - 14 \beta_{2} ) q^{37} + ( -3 + 11 \beta_{1} + 8 \beta_{2} - 8 \beta_{3} ) q^{39} + ( 8 - 22 \beta_{1} - 8 \beta_{2} + 16 \beta_{3} ) q^{41} + ( 40 + 6 \beta_{2} ) q^{43} + ( 33 - 2 \beta_{1} + 13 \beta_{2} - 4 \beta_{3} ) q^{45} + ( 4 + 8 \beta_{1} - 4 \beta_{2} + 8 \beta_{3} ) q^{47} + 7 q^{49} + ( 24 + 6 \beta_{3} ) q^{51} + ( 16 - 10 \beta_{1} - 16 \beta_{2} + 32 \beta_{3} ) q^{53} + ( 14 + 2 \beta_{2} ) q^{55} + ( 15 - 13 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} ) q^{57} + ( 1 + 26 \beta_{1} - \beta_{2} + 2 \beta_{3} ) q^{59} + ( 39 - 7 \beta_{2} ) q^{61} + ( -3 - 5 \beta_{1} + \beta_{2} + 8 \beta_{3} ) q^{63} + ( 6 + 2 \beta_{1} - 6 \beta_{2} + 12 \beta_{3} ) q^{65} + ( 6 + 8 \beta_{2} ) q^{67} + ( -42 - 6 \beta_{1} + 12 \beta_{2} ) q^{69} + ( 6 + 18 \beta_{1} - 6 \beta_{2} + 12 \beta_{3} ) q^{71} + ( -8 + 26 \beta_{2} ) q^{73} + ( -30 + 17 \beta_{1} - 13 \beta_{2} + 13 \beta_{3} ) q^{75} + ( 2 - 6 \beta_{1} - 2 \beta_{2} + 4 \beta_{3} ) q^{77} + ( 32 - 36 \beta_{2} ) q^{79} + ( -15 + 20 \beta_{1} - 22 \beta_{2} + 4 \beta_{3} ) q^{81} + ( 9 + 18 \beta_{1} - 9 \beta_{2} + 18 \beta_{3} ) q^{83} + ( -42 - 18 \beta_{2} ) q^{85} + ( 12 - 4 \beta_{1} + 8 \beta_{2} + 10 \beta_{3} ) q^{87} + ( -16 + 42 \beta_{1} + 16 \beta_{2} - 32 \beta_{3} ) q^{89} + ( 7 - 9 \beta_{2} ) q^{91} + ( 6 + 30 \beta_{1} + 36 \beta_{2} - 36 \beta_{3} ) q^{93} + ( 12 - 10 \beta_{1} - 12 \beta_{2} + 24 \beta_{3} ) q^{95} + ( -2 + 8 \beta_{2} ) q^{97} + ( -30 + 16 \beta_{1} + 4 \beta_{2} - 4 \beta_{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 2q^{3} - 20q^{9} + O(q^{10})$$ $$4q + 2q^{3} - 20q^{9} + 36q^{13} + 28q^{15} - 12q^{19} - 14q^{21} - 12q^{25} - 10q^{27} + 136q^{31} + 28q^{33} - 16q^{37} + 4q^{39} + 160q^{43} + 140q^{45} + 28q^{49} + 84q^{51} + 56q^{55} + 64q^{57} + 156q^{61} - 28q^{63} + 24q^{67} - 168q^{69} - 32q^{73} - 146q^{75} + 128q^{79} - 68q^{81} - 168q^{85} + 28q^{87} + 28q^{91} + 96q^{93} - 8q^{97} - 112q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 14 x^{2} + 21$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{2} + 7$$$$)/2$$ $$\beta_{3}$$ $$=$$ $$($$$$\nu^{3} + \nu^{2} + 13 \nu + 5$$$$)/4$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$2 \beta_{2} - 7$$ $$\nu^{3}$$ $$=$$ $$4 \beta_{3} - 2 \beta_{2} - 13 \beta_{1} + 2$$

## 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
 − 3.50592i 3.50592i 1.30710i − 1.30710i
0 −0.822876 2.88494i 0 1.24197i 0 2.64575 0 −7.64575 + 4.74789i 0
449.2 0 −0.822876 + 2.88494i 0 1.24197i 0 2.64575 0 −7.64575 4.74789i 0
449.3 0 1.82288 2.38267i 0 7.37953i 0 −2.64575 0 −2.35425 8.68663i 0
449.4 0 1.82288 + 2.38267i 0 7.37953i 0 −2.64575 0 −2.35425 + 8.68663i 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.f 4
3.b odd 2 1 inner 1344.3.d.f 4
4.b odd 2 1 1344.3.d.b 4
8.b even 2 1 21.3.b.a 4
8.d odd 2 1 336.3.d.c 4
12.b even 2 1 1344.3.d.b 4
24.f even 2 1 336.3.d.c 4
24.h odd 2 1 21.3.b.a 4
40.f even 2 1 525.3.c.a 4
40.i odd 4 2 525.3.f.a 8
56.h odd 2 1 147.3.b.f 4
56.j odd 6 2 147.3.h.c 8
56.p even 6 2 147.3.h.e 8
72.j odd 6 2 567.3.r.c 8
72.n even 6 2 567.3.r.c 8
120.i odd 2 1 525.3.c.a 4
120.w even 4 2 525.3.f.a 8
168.i even 2 1 147.3.b.f 4
168.s odd 6 2 147.3.h.e 8
168.ba even 6 2 147.3.h.c 8

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

## Hecke kernels

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

 $$T_{5}^{4} + 56 T_{5}^{2} + 84$$ $$T_{19}^{2} + 6 T_{19} - 166$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ $$1 - 2 T + 12 T^{2} - 18 T^{3} + 81 T^{4}$$
$5$ $$1 - 44 T^{2} + 1034 T^{4} - 27500 T^{6} + 390625 T^{8}$$
$7$ $$( 1 - 7 T^{2} )^{2}$$
$11$ $$1 - 428 T^{2} + 74630 T^{4} - 6266348 T^{6} + 214358881 T^{8}$$
$13$ $$( 1 - 18 T + 412 T^{2} - 3042 T^{3} + 28561 T^{4} )^{2}$$
$17$ $$1 - 988 T^{2} + 407046 T^{4} - 82518748 T^{6} + 6975757441 T^{8}$$
$19$ $$( 1 + 6 T + 556 T^{2} + 2166 T^{3} + 130321 T^{4} )^{2}$$
$23$ $$1 - 1444 T^{2} + 980166 T^{4} - 404090404 T^{6} + 78310985281 T^{8}$$
$29$ $$1 - 2972 T^{2} + 3611558 T^{4} - 2102039132 T^{6} + 500246412961 T^{8}$$
$31$ $$( 1 - 68 T + 3050 T^{2} - 65348 T^{3} + 923521 T^{4} )^{2}$$
$37$ $$( 1 + 8 T + 1382 T^{2} + 10952 T^{3} + 1874161 T^{4} )^{2}$$
$41$ $$1 - 1292 T^{2} + 2832038 T^{4} - 3650883212 T^{6} + 7984925229121 T^{8}$$
$43$ $$( 1 - 80 T + 5046 T^{2} - 147920 T^{3} + 3418801 T^{4} )^{2}$$
$47$ $$1 - 6148 T^{2} + 19144326 T^{4} - 30000278788 T^{6} + 23811286661761 T^{8}$$
$53$ $$1 + 20 T^{2} - 13350138 T^{4} + 157809620 T^{6} + 62259690411361 T^{8}$$
$59$ $$1 - 3676 T^{2} + 15964266 T^{4} - 44543419036 T^{6} + 146830437604321 T^{8}$$
$61$ $$( 1 - 78 T + 8620 T^{2} - 290238 T^{3} + 13845841 T^{4} )^{2}$$
$67$ $$( 1 - 12 T + 8566 T^{2} - 53868 T^{3} + 20151121 T^{4} )^{2}$$
$71$ $$1 - 10588 T^{2} + 78813510 T^{4} - 269058878428 T^{6} + 645753531245761 T^{8}$$
$73$ $$( 1 + 16 T + 5990 T^{2} + 85264 T^{3} + 28398241 T^{4} )^{2}$$
$79$ $$( 1 - 64 T + 4434 T^{2} - 399424 T^{3} + 38950081 T^{4} )^{2}$$
$83$ $$1 - 13948 T^{2} + 141899946 T^{4} - 661948661308 T^{6} + 2252292232139041 T^{8}$$
$89$ $$1 - 11468 T^{2} + 120945830 T^{4} - 719528019788 T^{6} + 3936588805702081 T^{8}$$
$97$ $$( 1 + 4 T + 18374 T^{2} + 37636 T^{3} + 88529281 T^{4} )^{2}$$