# Properties

 Label 1344.4.c.a Level $1344$ Weight $4$ Character orbit 1344.c Analytic conductor $79.299$ Analytic rank $0$ Dimension $6$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1344 = 2^{6} \cdot 3 \cdot 7$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 1344.c (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$79.2985670477$$ Analytic rank: $$0$$ Dimension: $$6$$ Coefficient field: $$\mathbb{Q}[x]/(x^{6} - \cdots)$$ Defining polynomial: $$x^{6} - 2 x^{5} + 2 x^{4} + 722 x^{3} + 11881 x^{2} + 54936 x + 127008$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{4}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{5}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 3 \beta_{1} q^{3} + ( \beta_{1} - \beta_{3} ) q^{5} -7 q^{7} -9 q^{9} +O(q^{10})$$ $$q + 3 \beta_{1} q^{3} + ( \beta_{1} - \beta_{3} ) q^{5} -7 q^{7} -9 q^{9} + ( -9 \beta_{1} + 2 \beta_{3} + \beta_{5} ) q^{11} + ( -2 \beta_{1} - \beta_{3} - \beta_{5} ) q^{13} + ( -3 + 3 \beta_{2} ) q^{15} + ( -3 - 2 \beta_{2} - \beta_{4} ) q^{17} + ( -22 \beta_{1} + 5 \beta_{3} - \beta_{5} ) q^{19} -21 \beta_{1} q^{21} + ( 9 + 3 \beta_{4} ) q^{23} + ( -25 - 5 \beta_{2} + \beta_{4} ) q^{25} -27 \beta_{1} q^{27} + ( -18 \beta_{1} + \beta_{3} - \beta_{5} ) q^{29} + ( 92 - 3 \beta_{2} - 3 \beta_{4} ) q^{31} + ( 27 - 6 \beta_{2} - 3 \beta_{4} ) q^{33} + ( -7 \beta_{1} + 7 \beta_{3} ) q^{35} + ( -40 \beta_{1} + \beta_{3} + \beta_{5} ) q^{37} + ( 6 + 3 \beta_{2} + 3 \beta_{4} ) q^{39} + ( -53 - 18 \beta_{2} + 5 \beta_{4} ) q^{41} + ( -46 \beta_{1} + 10 \beta_{3} - 2 \beta_{5} ) q^{43} + ( -9 \beta_{1} + 9 \beta_{3} ) q^{45} + ( 260 - 8 \beta_{2} ) q^{47} + 49 q^{49} + ( -9 \beta_{1} - 6 \beta_{3} - 3 \beta_{5} ) q^{51} + ( -70 \beta_{1} + 5 \beta_{3} + 11 \beta_{5} ) q^{53} + ( 272 - 11 \beta_{2} - 11 \beta_{4} ) q^{55} + ( 66 - 15 \beta_{2} + 3 \beta_{4} ) q^{57} + ( 164 \beta_{1} - 2 \beta_{3} + 6 \beta_{5} ) q^{59} + ( 164 \beta_{1} - 11 \beta_{3} - 5 \beta_{5} ) q^{61} + 63 q^{63} + ( -112 + 6 \beta_{2} + 10 \beta_{4} ) q^{65} + ( -92 \beta_{1} + 42 \beta_{3} ) q^{67} + ( 27 \beta_{1} + 9 \beta_{5} ) q^{69} + ( 435 + 26 \beta_{2} - 5 \beta_{4} ) q^{71} + ( 88 + 28 \beta_{2} - 14 \beta_{4} ) q^{73} + ( -75 \beta_{1} - 15 \beta_{3} + 3 \beta_{5} ) q^{75} + ( 63 \beta_{1} - 14 \beta_{3} - 7 \beta_{5} ) q^{77} + ( 582 - 16 \beta_{2} + 2 \beta_{4} ) q^{79} + 81 q^{81} + ( 238 \beta_{1} - 2 \beta_{3} - 8 \beta_{5} ) q^{83} + ( 260 \beta_{1} + \beta_{3} - 11 \beta_{5} ) q^{85} + ( 54 - 3 \beta_{2} + 3 \beta_{4} ) q^{87} + ( 5 - 14 \beta_{2} - 21 \beta_{4} ) q^{89} + ( 14 \beta_{1} + 7 \beta_{3} + 7 \beta_{5} ) q^{91} + ( 276 \beta_{1} - 9 \beta_{3} - 9 \beta_{5} ) q^{93} + ( 802 + 22 \beta_{2} + 4 \beta_{4} ) q^{95} + ( 280 + 62 \beta_{2} - 4 \beta_{4} ) q^{97} + ( 81 \beta_{1} - 18 \beta_{3} - 9 \beta_{5} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6q - 42q^{7} - 54q^{9} + O(q^{10})$$ $$6q - 42q^{7} - 54q^{9} - 24q^{15} - 16q^{17} + 60q^{23} - 138q^{25} + 552q^{31} + 168q^{33} + 36q^{39} - 272q^{41} + 1576q^{47} + 294q^{49} + 1632q^{55} + 432q^{57} + 378q^{63} - 664q^{65} + 2548q^{71} + 444q^{73} + 3528q^{79} + 486q^{81} + 336q^{87} + 16q^{89} + 4776q^{95} + 1548q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} - 2 x^{5} + 2 x^{4} + 722 x^{3} + 11881 x^{2} + 54936 x + 127008$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-5869 \nu^{5} + 25724 \nu^{4} - 85196 \nu^{3} - 2684972 \nu^{2} - 64680643 \nu - 172672164$$$$)/ 142698276$$ $$\beta_{2}$$ $$=$$ $$($$$$-111 \nu^{5} + 583 \nu^{4} - 12321 \nu^{3} - 40071 \nu^{2} - 55944 \nu - 7048478$$$$)/1132526$$ $$\beta_{3}$$ $$=$$ $$($$$$19855 \nu^{5} - 99182 \nu^{4} + 1637642 \nu^{3} + 7733918 \nu^{2} + 357126139 \nu + 918082116$$$$)/ 142698276$$ $$\beta_{4}$$ $$=$$ $$($$$$-1721 \nu^{5} + 29445 \nu^{4} - 191031 \nu^{3} - 621281 \nu^{2} - 867384 \nu + 78246822$$$$)/1132526$$ $$\beta_{5}$$ $$=$$ $$($$$$-233885 \nu^{5} + 988654 \nu^{4} + 653126 \nu^{3} - 206179030 \nu^{2} - 2162653733 \nu - 5879361852$$$$)/47566092$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{3} + \beta_{2} + \beta_{1} + 1$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$($$$$-\beta_{5} + 9 \beta_{3} + 150 \beta_{1}$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$($$$$-\beta_{5} + \beta_{4} + 118 \beta_{3} - 118 \beta_{2} + 763 \beta_{1} - 763$$$$)/2$$ $$\nu^{4}$$ $$=$$ $$($$$$111 \beta_{4} - 1721 \beta_{2} - 18380$$$$)/2$$ $$\nu^{5}$$ $$=$$ $$($$$$472 \beta_{5} + 472 \beta_{4} - 16851 \beta_{3} - 16851 \beta_{2} - 139347 \beta_{1} - 139347$$$$)/2$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
673.1
 −4.87742 + 4.87742i −2.93235 + 2.93235i 8.80977 − 8.80977i 8.80977 + 8.80977i −2.93235 − 2.93235i −4.87742 − 4.87742i
0 3.00000i 0 11.7548i 0 −7.00000 0 −9.00000 0
673.2 0 3.00000i 0 7.86469i 0 −7.00000 0 −9.00000 0
673.3 0 3.00000i 0 15.6195i 0 −7.00000 0 −9.00000 0
673.4 0 3.00000i 0 15.6195i 0 −7.00000 0 −9.00000 0
673.5 0 3.00000i 0 7.86469i 0 −7.00000 0 −9.00000 0
673.6 0 3.00000i 0 11.7548i 0 −7.00000 0 −9.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 673.6 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1344.4.c.a 6
4.b odd 2 1 1344.4.c.d yes 6
8.b even 2 1 inner 1344.4.c.a 6
8.d odd 2 1 1344.4.c.d yes 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1344.4.c.a 6 1.a even 1 1 trivial
1344.4.c.a 6 8.b even 2 1 inner
1344.4.c.d yes 6 4.b odd 2 1
1344.4.c.d yes 6 8.d odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(1344, [\chi])$$:

 $$T_{5}^{6} + 444 T_{5}^{4} + 57348 T_{5}^{2} + 2085136$$ $$T_{23}^{3} - 30 T_{23}^{2} - 21186 T_{23} - 470448$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{6}$$
$3$ $$( 9 + T^{2} )^{3}$$
$5$ $$2085136 + 57348 T^{2} + 444 T^{4} + T^{6}$$
$7$ $$( 7 + T )^{6}$$
$11$ $$906973456 + 6187140 T^{2} + 6396 T^{4} + T^{6}$$
$13$ $$369869824 + 6460800 T^{2} + 5028 T^{4} + T^{6}$$
$17$ $$( -66092 - 3046 T + 8 T^{2} + T^{3} )^{2}$$
$19$ $$71892624384 + 82646928 T^{2} + 18456 T^{4} + T^{6}$$
$23$ $$( -470448 - 21186 T - 30 T^{2} + T^{3} )^{2}$$
$29$ $$97140736 + 1651728 T^{2} + 6456 T^{4} + T^{6}$$
$31$ $$( 913888 + 2820 T - 276 T^{2} + T^{3} )^{2}$$
$37$ $$488233216 + 10556304 T^{2} + 9816 T^{4} + T^{6}$$
$41$ $$( -9550804 - 133462 T + 136 T^{2} + T^{3} )^{2}$$
$43$ $$4829129310784 + 1332972336 T^{2} + 74412 T^{4} + T^{6}$$
$47$ $$( -13844416 + 192944 T - 788 T^{2} + T^{3} )^{2}$$
$53$ $$924071477319936 + 88056208656 T^{2} + 591720 T^{4} + T^{6}$$
$59$ $$4716055035904 + 2193475584 T^{2} + 259344 T^{4} + T^{6}$$
$61$ $$268125124864 + 519241152 T^{2} + 244260 T^{4} + T^{6}$$
$67$ $$15703315888888384 + 204446401008 T^{2} + 807516 T^{4} + T^{6}$$
$71$ $$( 4238528 + 320246 T - 1274 T^{2} + T^{3} )^{2}$$
$73$ $$( 226444392 - 662124 T - 222 T^{2} + T^{3} )^{2}$$
$79$ $$( -157706752 + 968376 T - 1764 T^{2} + T^{3} )^{2}$$
$83$ $$964272413085696 + 44195674176 T^{2} + 471264 T^{4} + T^{6}$$
$89$ $$( 690068 - 1066774 T - 8 T^{2} + T^{3} )^{2}$$
$97$ $$( -107008376 - 706092 T - 774 T^{2} + T^{3} )^{2}$$