# Properties

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

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## 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: 6.0.14024243776.1 Defining polynomial: $$x^{6} - 2 x^{5} - 17 x^{4} - 164 x^{3} + 299 x^{2} + 2466 x + 13042$$ 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.

 $$f(q)$$ $$=$$ $$q -3 \beta_{1} q^{3} + ( \beta_{1} - \beta_{2} ) q^{5} -7 q^{7} -9 q^{9} +O(q^{10})$$ $$q -3 \beta_{1} q^{3} + ( \beta_{1} - \beta_{2} ) q^{5} -7 q^{7} -9 q^{9} + ( 23 \beta_{1} - 2 \beta_{2} - 3 \beta_{4} ) q^{11} + ( 18 \beta_{1} - 5 \beta_{2} - \beta_{4} ) q^{13} + ( 3 - 3 \beta_{5} ) q^{15} + ( -9 + \beta_{3} + 10 \beta_{5} ) q^{17} + ( 14 \beta_{1} - 7 \beta_{2} + 7 \beta_{4} ) q^{19} + 21 \beta_{1} q^{21} + ( 35 + \beta_{3} + 8 \beta_{5} ) q^{23} + ( 43 + 3 \beta_{3} + 9 \beta_{5} ) q^{25} + 27 \beta_{1} q^{27} + ( -30 \beta_{1} + 13 \beta_{2} - 17 \beta_{4} ) q^{29} + ( -20 + 19 \beta_{3} + 11 \beta_{5} ) q^{31} + ( 69 - 9 \beta_{3} - 6 \beta_{5} ) q^{33} + ( -7 \beta_{1} + 7 \beta_{2} ) q^{35} + ( -212 \beta_{1} + 9 \beta_{2} + 21 \beta_{4} ) q^{37} + ( 54 - 3 \beta_{3} - 15 \beta_{5} ) q^{39} + ( -39 - \beta_{3} - 10 \beta_{5} ) q^{41} + ( -6 \beta_{1} - 22 \beta_{2} - 26 \beta_{4} ) q^{43} + ( -9 \beta_{1} + 9 \beta_{2} ) q^{45} + ( 72 - 40 \beta_{3} - 28 \beta_{5} ) q^{47} + 49 q^{49} + ( 27 \beta_{1} - 30 \beta_{2} - 3 \beta_{4} ) q^{51} + ( -390 \beta_{1} - 15 \beta_{2} + 15 \beta_{4} ) q^{53} + ( -104 - 9 \beta_{3} + 39 \beta_{5} ) q^{55} + ( 42 + 21 \beta_{3} - 21 \beta_{5} ) q^{57} + ( -32 \beta_{1} + 6 \beta_{2} + 58 \beta_{4} ) q^{59} + ( -400 \beta_{1} - 35 \beta_{2} - \beta_{4} ) q^{61} + 63 q^{63} + ( -396 + 10 \beta_{3} + 58 \beta_{5} ) q^{65} + ( -20 \beta_{1} + 78 \beta_{2} + 12 \beta_{4} ) q^{67} + ( -105 \beta_{1} - 24 \beta_{2} - 3 \beta_{4} ) q^{69} + ( 177 + 29 \beta_{3} - 22 \beta_{5} ) q^{71} + ( -12 - 54 \beta_{3} + 48 \beta_{5} ) q^{73} + ( -129 \beta_{1} - 27 \beta_{2} - 9 \beta_{4} ) q^{75} + ( -161 \beta_{1} + 14 \beta_{2} + 21 \beta_{4} ) q^{77} + ( -246 - 14 \beta_{3} - 76 \beta_{5} ) q^{79} + 81 q^{81} + ( 34 \beta_{1} - 74 \beta_{2} - 76 \beta_{4} ) q^{83} + ( -792 \beta_{1} + 89 \beta_{2} + 25 \beta_{4} ) q^{85} + ( -90 - 51 \beta_{3} + 39 \beta_{5} ) q^{87} + ( 399 - 7 \beta_{3} - 70 \beta_{5} ) q^{89} + ( -126 \beta_{1} + 35 \beta_{2} + 7 \beta_{4} ) q^{91} + ( 60 \beta_{1} - 33 \beta_{2} - 57 \beta_{4} ) q^{93} + ( -770 + 56 \beta_{3} + 70 \beta_{5} ) q^{95} + ( -412 + 72 \beta_{3} + 6 \beta_{5} ) q^{97} + ( -207 \beta_{1} + 18 \beta_{2} + 27 \beta_{4} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6q - 42q^{7} - 54q^{9} + O(q^{10})$$ $$6q - 42q^{7} - 54q^{9} + 24q^{15} - 72q^{17} + 196q^{23} + 246q^{25} - 104q^{31} + 408q^{33} + 348q^{39} - 216q^{41} + 408q^{47} + 294q^{49} - 720q^{55} + 336q^{57} + 378q^{63} - 2472q^{65} + 1164q^{71} - 276q^{73} - 1352q^{79} + 486q^{81} - 720q^{87} + 2520q^{89} - 4648q^{95} - 2340q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} - 2 x^{5} - 17 x^{4} - 164 x^{3} + 299 x^{2} + 2466 x + 13042$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$4 \nu^{5} - 54 \nu^{4} + 553 \nu^{3} - 453 \nu^{2} - 157 \nu - 47393$$$$)/39375$$ $$\beta_{2}$$ $$=$$ $$($$$$-11 \nu^{5} + 236 \nu^{4} - 427 \nu^{3} + 2952 \nu^{2} - 31287 \nu + 6037$$$$)/13125$$ $$\beta_{3}$$ $$=$$ $$($$$$-3 \nu^{5} + 28 \nu^{4} + 54 \nu^{3} - 529 \nu^{2} - 4726 \nu - 1699$$$$)/1875$$ $$\beta_{4}$$ $$=$$ $$($$$$-31 \nu^{5} - 194 \nu^{4} + 1183 \nu^{3} + 4692 \nu^{2} + 123 \nu - 88273$$$$)/13125$$ $$\beta_{5}$$ $$=$$ $$($$$$-9 \nu^{5} + 34 \nu^{4} + 162 \nu^{3} + 563 \nu^{2} - 1678 \nu - 17197$$$$)/1875$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{5} - \beta_{4} - \beta_{3} - \beta_{2} + 2$$$$)/4$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{5} + \beta_{4} - 7 \beta_{3} + 7 \beta_{2} + 18 \beta_{1} + 28$$$$)/4$$ $$\nu^{3}$$ $$=$$ $$($$$$8 \beta_{5} - 7 \beta_{4} - 5 \beta_{3} + 2 \beta_{2} + 153 \beta_{1} + 205$$$$)/2$$ $$\nu^{4}$$ $$=$$ $$($$$$143 \beta_{5} - 207 \beta_{4} - 101 \beta_{3} + 51 \beta_{2} + 774 \beta_{1} + 736$$$$)/4$$ $$\nu^{5}$$ $$=$$ $$($$$$-129 \beta_{5} - 785 \beta_{4} - 813 \beta_{3} + 889 \beta_{2} + 9558 \beta_{1} + 3896$$$$)/4$$

## 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
 5.82579 + 1.30833i −1.73614 − 4.18492i −3.08965 + 2.87659i −3.08965 − 2.87659i −1.73614 + 4.18492i 5.82579 − 1.30833i
0 3.00000i 0 7.03493i 0 −7.00000 0 −9.00000 0
673.2 0 3.00000i 0 2.89754i 0 −7.00000 0 −9.00000 0
673.3 0 3.00000i 0 13.9325i 0 −7.00000 0 −9.00000 0
673.4 0 3.00000i 0 13.9325i 0 −7.00000 0 −9.00000 0
673.5 0 3.00000i 0 2.89754i 0 −7.00000 0 −9.00000 0
673.6 0 3.00000i 0 7.03493i 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.b 6
4.b odd 2 1 1344.4.c.c yes 6
8.b even 2 1 inner 1344.4.c.b 6
8.d odd 2 1 1344.4.c.c yes 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1344.4.c.b 6 1.a even 1 1 trivial
1344.4.c.b 6 8.b even 2 1 inner
1344.4.c.c yes 6 4.b odd 2 1
1344.4.c.c 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 $$S_{4}^{\mathrm{new}}(1344, [\chi])$$:

 $$T_{5}^{6} + 252 T_{5}^{4} + 11652 T_{5}^{2} + 80656$$ $$T_{23}^{3} - 98 T_{23}^{2} - 4266 T_{23} + 455616$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{6}$$
$3$ $$( 9 + T^{2} )^{3}$$
$5$ $$80656 + 11652 T^{2} + 252 T^{4} + T^{6}$$
$7$ $$( 7 + T )^{6}$$
$11$ $$2361571216 + 6611172 T^{2} + 4716 T^{4} + T^{6}$$
$13$ $$107495424 + 1894912 T^{2} + 6884 T^{4} + T^{6}$$
$17$ $$( 340452 - 11326 T + 36 T^{2} + T^{3} )^{2}$$
$19$ $$1405192126464 + 423536400 T^{2} + 37240 T^{4} + T^{6}$$
$23$ $$( 455616 - 4266 T - 98 T^{2} + T^{3} )^{2}$$
$29$ $$173862579004416 + 9898410640 T^{2} + 178520 T^{4} + T^{6}$$
$31$ $$( -6626656 - 60268 T + 52 T^{2} + T^{3} )^{2}$$
$37$ $$134116781574400 + 17858360976 T^{2} + 272664 T^{4} + T^{6}$$
$41$ $$( -856452 - 7870 T + 108 T^{2} + T^{3} )^{2}$$
$43$ $$277856627817024 + 16860993328 T^{2} + 265772 T^{4} + T^{6}$$
$47$ $$( 78456384 - 273328 T - 204 T^{2} + T^{3} )^{2}$$
$53$ $$1004763204000000 + 50298570000 T^{2} + 599400 T^{4} + T^{6}$$
$59$ $$222264684199936 + 308422990848 T^{2} + 1110288 T^{4} + T^{6}$$
$61$ $$352990747650304 + 136560775872 T^{2} + 750468 T^{4} + T^{6}$$
$67$ $$39337457673745984 + 439811966448 T^{2} + 1415580 T^{4} + T^{6}$$
$71$ $$( 89664720 - 138514 T - 582 T^{2} + T^{3} )^{2}$$
$73$ $$( -401113080 - 970956 T + 138 T^{2} + T^{3} )^{2}$$
$79$ $$( -347261568 - 514184 T + 676 T^{2} + T^{3} )^{2}$$
$83$ $$236495066255917056 + 1639169810496 T^{2} + 2493472 T^{4} + T^{6}$$
$89$ $$( 3313380 - 46942 T - 1260 T^{2} + T^{3} )^{2}$$
$97$ $$( -273572696 - 404076 T + 1170 T^{2} + T^{3} )^{2}$$
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