# Properties

 Label 1344.4.b.f Level $1344$ Weight $4$ Character orbit 1344.b Analytic conductor $79.299$ Analytic rank $0$ Dimension $8$ 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.b (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$79.2985670477$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: $$\mathbb{Q}[x]/(x^{8} + \cdots)$$ Defining polynomial: $$x^{8} + 158 x^{6} + 8461 x^{4} + 180672 x^{2} + 1232100$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{15}\cdot 3^{3}$$ Twist minimal: no (minimal twist has level 336) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 3 q^{3} + \beta_{5} q^{5} + \beta_{3} q^{7} + 9 q^{9} +O(q^{10})$$ $$q + 3 q^{3} + \beta_{5} q^{5} + \beta_{3} q^{7} + 9 q^{9} + ( -\beta_{1} - \beta_{2} - \beta_{3} + \beta_{5} ) q^{11} + ( -2 \beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} + \beta_{5} ) q^{13} + 3 \beta_{5} q^{15} + ( -2 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} + 3 \beta_{5} ) q^{17} + ( 8 - \beta_{2} + \beta_{3} + \beta_{7} ) q^{19} + 3 \beta_{3} q^{21} + ( -2 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} - 3 \beta_{4} - 2 \beta_{5} ) q^{23} + ( -77 - 5 \beta_{2} + 5 \beta_{3} - \beta_{6} + \beta_{7} ) q^{25} + 27 q^{27} + ( -32 + 2 \beta_{2} - 2 \beta_{3} - \beta_{6} - \beta_{7} ) q^{29} + ( 40 + 2 \beta_{6} - \beta_{7} ) q^{31} + ( -3 \beta_{1} - 3 \beta_{2} - 3 \beta_{3} + 3 \beta_{5} ) q^{33} + ( -74 + 6 \beta_{1} + 2 \beta_{2} + 4 \beta_{3} + 3 \beta_{4} - 10 \beta_{5} - \beta_{6} + 2 \beta_{7} ) q^{35} + ( -52 + 3 \beta_{2} - 3 \beta_{3} + \beta_{6} + 3 \beta_{7} ) q^{37} + ( -6 \beta_{1} - 3 \beta_{2} - 3 \beta_{3} - 3 \beta_{4} + 3 \beta_{5} ) q^{39} + ( -2 \beta_{1} - 8 \beta_{2} - 8 \beta_{3} - 6 \beta_{4} + \beta_{5} ) q^{41} + ( -7 \beta_{1} - \beta_{2} - \beta_{3} + 6 \beta_{4} - 6 \beta_{5} ) q^{43} + 9 \beta_{5} q^{45} + ( -90 - 12 \beta_{2} + 12 \beta_{3} + \beta_{6} ) q^{47} + ( 1 + 9 \beta_{1} - 4 \beta_{2} + 2 \beta_{3} + \beta_{4} - \beta_{5} + 2 \beta_{6} + 3 \beta_{7} ) q^{49} + ( -6 \beta_{1} - 6 \beta_{2} - 6 \beta_{3} + 9 \beta_{5} ) q^{51} + ( -38 + 2 \beta_{2} - 2 \beta_{3} - 2 \beta_{6} + 5 \beta_{7} ) q^{53} + ( -52 - 5 \beta_{2} + 5 \beta_{3} - 5 \beta_{7} ) q^{55} + ( 24 - 3 \beta_{2} + 3 \beta_{3} + 3 \beta_{7} ) q^{57} + ( 218 + 10 \beta_{2} - 10 \beta_{3} + \beta_{6} + 4 \beta_{7} ) q^{59} + ( -4 \beta_{1} + 3 \beta_{2} + 3 \beta_{3} - \beta_{4} + 7 \beta_{5} ) q^{61} + 9 \beta_{3} q^{63} + ( -82 - 20 \beta_{2} + 20 \beta_{3} - 3 \beta_{6} - 8 \beta_{7} ) q^{65} + ( 13 \beta_{1} - 9 \beta_{2} - 9 \beta_{3} + 6 \beta_{4} ) q^{67} + ( -6 \beta_{1} - 6 \beta_{2} - 6 \beta_{3} - 9 \beta_{4} - 6 \beta_{5} ) q^{69} + ( 30 \beta_{1} - 6 \beta_{2} - 6 \beta_{3} - 3 \beta_{4} + 32 \beta_{5} ) q^{71} + ( -28 \beta_{1} - 18 \beta_{2} - 18 \beta_{3} + 2 \beta_{4} + 16 \beta_{5} ) q^{73} + ( -231 - 15 \beta_{2} + 15 \beta_{3} - 3 \beta_{6} + 3 \beta_{7} ) q^{75} + ( 258 - 2 \beta_{1} + 4 \beta_{2} + 4 \beta_{3} + 6 \beta_{4} - 13 \beta_{5} - 2 \beta_{6} - 3 \beta_{7} ) q^{77} + ( 4 \beta_{1} - \beta_{2} - \beta_{3} - 6 \beta_{4} - 24 \beta_{5} ) q^{79} + 81 q^{81} + ( -210 - 5 \beta_{6} ) q^{83} + ( -306 - 15 \beta_{2} + 15 \beta_{3} - \beta_{6} - 9 \beta_{7} ) q^{85} + ( -96 + 6 \beta_{2} - 6 \beta_{3} - 3 \beta_{6} - 3 \beta_{7} ) q^{87} + ( 38 \beta_{1} - 4 \beta_{2} - 4 \beta_{3} - 6 \beta_{4} + 19 \beta_{5} ) q^{89} + ( 112 + 35 \beta_{1} + 8 \beta_{3} - 42 \beta_{5} - 7 \beta_{7} ) q^{91} + ( 120 + 6 \beta_{6} - 3 \beta_{7} ) q^{93} + ( -54 \beta_{1} - 30 \beta_{2} - 30 \beta_{3} - 6 \beta_{4} - 4 \beta_{5} ) q^{95} + ( -22 \beta_{1} - 14 \beta_{2} - 14 \beta_{3} + 10 \beta_{4} - 40 \beta_{5} ) q^{97} + ( -9 \beta_{1} - 9 \beta_{2} - 9 \beta_{3} + 9 \beta_{5} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q + 24q^{3} - 4q^{7} + 72q^{9} + O(q^{10})$$ $$8q + 24q^{3} - 4q^{7} + 72q^{9} + 56q^{19} - 12q^{21} - 656q^{25} + 216q^{27} - 240q^{29} + 320q^{31} - 600q^{35} - 392q^{37} - 816q^{47} - 16q^{49} - 288q^{53} - 456q^{55} + 168q^{57} + 1824q^{59} - 36q^{63} - 816q^{65} - 1968q^{75} + 2064q^{77} + 648q^{81} - 1680q^{83} - 2568q^{85} - 720q^{87} + 864q^{91} + 960q^{93} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} + 158 x^{6} + 8461 x^{4} + 180672 x^{2} + 1232100$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-2 \nu^{7} - 316 \nu^{5} - 14702 \nu^{3} - 185964 \nu$$$$)/11655$$ $$\beta_{2}$$ $$=$$ $$($$$$-169 \nu^{7} + 740 \nu^{6} - 20597 \nu^{5} + 88060 \nu^{4} - 690094 \nu^{3} + 2786840 \nu^{2} - 5860488 \nu + 21107760$$$$)/93240$$ $$\beta_{3}$$ $$=$$ $$($$$$-169 \nu^{7} - 740 \nu^{6} - 20597 \nu^{5} - 88060 \nu^{4} - 690094 \nu^{3} - 2786840 \nu^{2} - 5860488 \nu - 21107760$$$$)/93240$$ $$\beta_{4}$$ $$=$$ $$($$$$251 \nu^{7} + 31333 \nu^{5} + 1117496 \nu^{3} + 11580252 \nu$$$$)/93240$$ $$\beta_{5}$$ $$=$$ $$($$$$-257 \nu^{7} - 31171 \nu^{5} - 1027292 \nu^{3} - 8481804 \nu$$$$)/93240$$ $$\beta_{6}$$ $$=$$ $$($$$$\nu^{6} + 119 \nu^{4} + 3892 \nu^{2} + 33438$$$$)/21$$ $$\beta_{7}$$ $$=$$ $$($$$$5 \nu^{6} + 613 \nu^{4} + 20630 \nu^{2} + 177216$$$$)/63$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$-6 \beta_{5} + 2 \beta_{4} + 6 \beta_{3} + 6 \beta_{2} + \beta_{1}$$$$)/24$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{6} + 3 \beta_{3} - 3 \beta_{2} - 234$$$$)/6$$ $$\nu^{3}$$ $$=$$ $$($$$$396 \beta_{5} - 44 \beta_{4} - 336 \beta_{3} - 336 \beta_{2} + 47 \beta_{1}$$$$)/24$$ $$\nu^{4}$$ $$=$$ $$($$$$21 \beta_{7} - 100 \beta_{6} - 195 \beta_{3} + 195 \beta_{2} + 11868$$$$)/6$$ $$\nu^{5}$$ $$=$$ $$($$$$-26136 \beta_{5} + 752 \beta_{4} + 20892 \beta_{3} + 20892 \beta_{2} - 9737 \beta_{1}$$$$)/24$$ $$\nu^{6}$$ $$=$$ $$($$$$-2499 \beta_{7} + 8134 \beta_{6} + 11529 \beta_{3} - 11529 \beta_{2} - 702192$$$$)/6$$ $$\nu^{7}$$ $$=$$ $$($$$$1776384 \beta_{5} + 18664 \beta_{4} - 1388892 \beta_{3} - 1388892 \beta_{2} + 960107 \beta_{1}$$$$)/24$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
895.1
 6.15149i − 5.92762i 8.49618i − 3.58293i 3.58293i − 8.49618i 5.92762i − 6.15149i
0 3.00000 0 20.9291i 0 −17.6950 5.46693i 0 9.00000 0
895.2 0 3.00000 0 17.7376i 0 2.09706 18.4011i 0 9.00000 0
895.3 0 3.00000 0 7.52281i 0 −4.86375 + 17.8702i 0 9.00000 0
895.4 0 3.00000 0 4.33129i 0 18.4617 + 1.47188i 0 9.00000 0
895.5 0 3.00000 0 4.33129i 0 18.4617 1.47188i 0 9.00000 0
895.6 0 3.00000 0 7.52281i 0 −4.86375 17.8702i 0 9.00000 0
895.7 0 3.00000 0 17.7376i 0 2.09706 + 18.4011i 0 9.00000 0
895.8 0 3.00000 0 20.9291i 0 −17.6950 + 5.46693i 0 9.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 895.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
28.d 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.b.f 8
4.b odd 2 1 1344.4.b.e 8
7.b odd 2 1 1344.4.b.e 8
8.b even 2 1 336.4.b.e 8
8.d odd 2 1 336.4.b.f yes 8
24.f even 2 1 1008.4.b.k 8
24.h odd 2 1 1008.4.b.i 8
28.d even 2 1 inner 1344.4.b.f 8
56.e even 2 1 336.4.b.e 8
56.h odd 2 1 336.4.b.f yes 8
168.e odd 2 1 1008.4.b.i 8
168.i even 2 1 1008.4.b.k 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
336.4.b.e 8 8.b even 2 1
336.4.b.e 8 56.e even 2 1
336.4.b.f yes 8 8.d odd 2 1
336.4.b.f yes 8 56.h odd 2 1
1008.4.b.i 8 24.h odd 2 1
1008.4.b.i 8 168.e odd 2 1
1008.4.b.k 8 24.f even 2 1
1008.4.b.k 8 168.i even 2 1
1344.4.b.e 8 4.b odd 2 1
1344.4.b.e 8 7.b odd 2 1
1344.4.b.f 8 1.a even 1 1 trivial
1344.4.b.f 8 28.d even 2 1 inner

## Hecke kernels

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

 $$T_{5}^{8} + 828 T_{5}^{6} + 195588 T_{5}^{4} + 11183616 T_{5}^{2} + 146313216$$ $$T_{19}^{4} - 28 T_{19}^{3} - 10380 T_{19}^{2} + 500320 T_{19} - 5935552$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$( -3 + T )^{8}$$
$5$ $$146313216 + 11183616 T^{2} + 195588 T^{4} + 828 T^{6} + T^{8}$$
$7$ $$13841287201 + 161414428 T + 1882384 T^{2} - 1046836 T^{3} - 171010 T^{4} - 3052 T^{5} + 16 T^{6} + 4 T^{7} + T^{8}$$
$11$ $$238089347136 + 1446701472 T^{2} + 3169908 T^{4} + 2964 T^{6} + T^{8}$$
$13$ $$624529833984 + 22553911296 T^{2} + 27819072 T^{4} + 9840 T^{6} + T^{8}$$
$17$ $$104135983727616 + 138739858560 T^{2} + 66615300 T^{4} + 13596 T^{6} + T^{8}$$
$19$ $$( -5935552 + 500320 T - 10380 T^{2} - 28 T^{3} + T^{4} )^{2}$$
$23$ $$5818468754516544 + 6470457571872 T^{2} + 1320235956 T^{4} + 67860 T^{6} + T^{8}$$
$29$ $$( 242576208 - 2067552 T - 31896 T^{2} + 120 T^{3} + T^{4} )^{2}$$
$31$ $$( -1109014528 + 20486656 T - 82992 T^{2} - 160 T^{3} + T^{4} )^{2}$$
$37$ $$( 300652096 - 14480672 T - 131916 T^{2} + 196 T^{3} + T^{4} )^{2}$$
$41$ $$11069541602331780096 + 1065438661373568 T^{2} + 31341780036 T^{4} + 313596 T^{6} + T^{8}$$
$43$ $$246975952593429504 + 691288901694720 T^{2} + 30124053072 T^{4} + 362184 T^{6} + T^{8}$$
$47$ $$( -1755758592 - 65028096 T - 148464 T^{2} + 408 T^{3} + T^{4} )^{2}$$
$53$ $$( -460467504 + 29521152 T - 323352 T^{2} + 144 T^{3} + T^{4} )^{2}$$
$59$ $$( -4942861056 + 76273920 T - 68832 T^{2} - 912 T^{3} + T^{4} )^{2}$$
$61$ $$999247734374400 + 37564216639488 T^{2} + 3961377792 T^{4} + 120576 T^{6} + T^{8}$$
$67$ $$89609869372701081600 + 9293053582559232 T^{2} + 156216900240 T^{4} + 731928 T^{6} + T^{8}$$
$71$ $$72\!\cdots\!04$$$$+ 137280907575717408 T^{2} + 757865651508 T^{4} + 1517652 T^{6} + T^{8}$$
$73$ $$86\!\cdots\!36$$$$+ 127143686540648448 T^{2} + 654236122176 T^{4} + 1372656 T^{6} + T^{8}$$
$79$ $$38\!\cdots\!04$$$$+ 14624360187411456 T^{2} + 166115326608 T^{4} + 712920 T^{6} + T^{8}$$
$83$ $$( -57542400000 - 333504000 T - 284400 T^{2} + 840 T^{3} + T^{4} )^{2}$$
$89$ $$38\!\cdots\!00$$$$+ 161959725986951808 T^{2} + 908640274500 T^{4} + 1673724 T^{6} + T^{8}$$
$97$ $$20\!\cdots\!56$$$$+ 1930242581274820608 T^{2} + 4981287716928 T^{4} + 4045296 T^{6} + T^{8}$$