# Properties

 Label 7168.2.a.bf Level $7168$ Weight $2$ Character orbit 7168.a Self dual yes Analytic conductor $57.237$ Analytic rank $0$ Dimension $8$ CM no Inner twists $1$

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## Newspace parameters

 Level: $$N$$ $$=$$ $$7168 = 2^{10} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 7168.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$57.2367681689$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: 8.8.9433055232.1 Defining polynomial: $$x^{8} - 4 x^{7} - 6 x^{6} + 32 x^{5} + 9 x^{4} - 76 x^{3} - 4 x^{2} + 48 x - 8$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 1792) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-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.

 $$f(q)$$ $$=$$ $$q -\beta_{5} q^{3} + ( 1 + \beta_{2} ) q^{5} + q^{7} + ( 2 + \beta_{1} + \beta_{2} - \beta_{3} + \beta_{6} ) q^{9} +O(q^{10})$$ $$q -\beta_{5} q^{3} + ( 1 + \beta_{2} ) q^{5} + q^{7} + ( 2 + \beta_{1} + \beta_{2} - \beta_{3} + \beta_{6} ) q^{9} + ( 1 + \beta_{3} + \beta_{4} - \beta_{5} - \beta_{7} ) q^{11} + ( 2 + \beta_{2} - \beta_{3} - \beta_{4} + \beta_{5} - \beta_{6} ) q^{13} + ( -1 - \beta_{4} - 2 \beta_{5} - \beta_{6} - \beta_{7} ) q^{15} + ( \beta_{2} - \beta_{4} - 2 \beta_{6} + \beta_{7} ) q^{17} + ( \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} - \beta_{7} ) q^{19} -\beta_{5} q^{21} + ( 1 + \beta_{1} - 2 \beta_{2} + \beta_{3} + 3 \beta_{4} - 2 \beta_{5} + \beta_{6} - \beta_{7} ) q^{23} + ( 1 + 2 \beta_{2} + \beta_{3} - 2 \beta_{4} + \beta_{5} - \beta_{7} ) q^{25} + ( -2 - \beta_{1} - \beta_{2} - 3 \beta_{3} - \beta_{4} - 2 \beta_{6} + \beta_{7} ) q^{27} + ( 1 + 2 \beta_{1} + \beta_{2} - 2 \beta_{4} - \beta_{5} + \beta_{6} - \beta_{7} ) q^{29} + ( -1 - \beta_{1} + 3 \beta_{4} - \beta_{5} - \beta_{6} ) q^{31} + ( -\beta_{1} + \beta_{3} + 2 \beta_{4} - 2 \beta_{5} - 2 \beta_{7} ) q^{33} + ( 1 + \beta_{2} ) q^{35} + ( 2 + \beta_{1} + 2 \beta_{4} + 2 \beta_{7} ) q^{37} + ( -2 + \beta_{1} + \beta_{2} + 2 \beta_{3} - \beta_{4} - 3 \beta_{5} ) q^{39} + ( -2 - \beta_{1} + \beta_{2} - 3 \beta_{3} - \beta_{4} + 2 \beta_{5} - 2 \beta_{6} + \beta_{7} ) q^{41} + ( -\beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} + 2 \beta_{5} + \beta_{6} ) q^{43} + ( 8 + 2 \beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} + \beta_{5} + 3 \beta_{6} ) q^{45} + ( 3 - \beta_{1} - \beta_{4} - \beta_{5} + \beta_{6} ) q^{47} + q^{49} + ( 5 + 3 \beta_{1} + 3 \beta_{2} - 4 \beta_{4} + \beta_{5} + \beta_{6} + \beta_{7} ) q^{51} + ( 6 + \beta_{1} - 2 \beta_{2} - 2 \beta_{3} + 2 \beta_{6} ) q^{53} + ( 2 + 2 \beta_{2} + 2 \beta_{3} - 2 \beta_{5} - 2 \beta_{7} ) q^{55} + ( -2 \beta_{1} - \beta_{3} + 2 \beta_{4} + \beta_{5} + \beta_{7} ) q^{57} + ( -\beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} + 3 \beta_{5} - \beta_{7} ) q^{59} + ( -1 - 2 \beta_{1} + \beta_{2} - 2 \beta_{3} - 2 \beta_{4} + 2 \beta_{5} - 2 \beta_{6} + 2 \beta_{7} ) q^{61} + ( 2 + \beta_{1} + \beta_{2} - \beta_{3} + \beta_{6} ) q^{63} + ( 5 - 3 \beta_{1} + 3 \beta_{2} + \beta_{3} - 2 \beta_{4} + 2 \beta_{5} - \beta_{6} + 2 \beta_{7} ) q^{65} + ( 6 - \beta_{1} - \beta_{2} - 3 \beta_{3} + \beta_{4} + 3 \beta_{6} + 2 \beta_{7} ) q^{67} + ( 1 - 2 \beta_{1} - 2 \beta_{2} - 3 \beta_{3} + 5 \beta_{4} - \beta_{5} + \beta_{6} ) q^{69} + ( -1 - \beta_{1} - \beta_{2} - 2 \beta_{3} - 2 \beta_{4} + \beta_{5} + \beta_{6} + \beta_{7} ) q^{71} + ( 2 + 3 \beta_{1} - 3 \beta_{3} + 2 \beta_{4} - 2 \beta_{5} + 2 \beta_{6} - 2 \beta_{7} ) q^{73} + ( -6 - 3 \beta_{1} + \beta_{2} + 3 \beta_{3} - 3 \beta_{4} - \beta_{5} - 4 \beta_{6} - \beta_{7} ) q^{75} + ( 1 + \beta_{3} + \beta_{4} - \beta_{5} - \beta_{7} ) q^{77} + ( 2 + \beta_{1} + \beta_{3} - 2 \beta_{4} + 2 \beta_{6} + 2 \beta_{7} ) q^{79} + ( 5 + 4 \beta_{1} + 5 \beta_{3} + 2 \beta_{4} + \beta_{5} + 4 \beta_{6} + \beta_{7} ) q^{81} + ( 2 \beta_{1} + 2 \beta_{3} + 4 \beta_{4} - \beta_{5} + 2 \beta_{7} ) q^{83} + ( 1 - 3 \beta_{1} + \beta_{2} - 6 \beta_{4} + \beta_{5} - 3 \beta_{6} + \beta_{7} ) q^{85} + ( 3 - 3 \beta_{1} + 2 \beta_{2} - 4 \beta_{3} - 3 \beta_{4} + \beta_{5} - 3 \beta_{6} + 4 \beta_{7} ) q^{87} + ( 2 + \beta_{1} + \beta_{3} + 2 \beta_{4} ) q^{89} + ( 2 + \beta_{2} - \beta_{3} - \beta_{4} + \beta_{5} - \beta_{6} ) q^{91} + ( 1 + 3 \beta_{1} - \beta_{2} + 2 \beta_{3} + 4 \beta_{4} - 3 \beta_{5} + 3 \beta_{6} - 5 \beta_{7} ) q^{93} + ( -4 + \beta_{1} - \beta_{2} + 2 \beta_{3} + 3 \beta_{4} - 3 \beta_{5} + 2 \beta_{6} ) q^{95} + ( -2 + \beta_{2} - 3 \beta_{4} + 2 \beta_{6} + 3 \beta_{7} ) q^{97} + ( -1 + \beta_{3} + 3 \beta_{4} - \beta_{5} + 2 \beta_{6} - 3 \beta_{7} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q + 8q^{5} + 8q^{7} + 12q^{9} + O(q^{10})$$ $$8q + 8q^{5} + 8q^{7} + 12q^{9} + 12q^{11} + 20q^{13} + 4q^{17} + 4q^{19} + 8q^{23} + 12q^{25} - 12q^{27} + 8q^{29} - 4q^{31} + 8q^{33} + 8q^{35} + 8q^{37} - 16q^{39} - 12q^{41} - 4q^{43} + 52q^{45} + 20q^{47} + 8q^{49} + 32q^{51} + 40q^{53} + 24q^{55} - 4q^{57} + 4q^{59} - 8q^{61} + 12q^{63} + 36q^{65} + 28q^{67} + 4q^{69} - 16q^{71} + 16q^{73} - 28q^{75} + 12q^{77} + 20q^{81} - 8q^{83} + 16q^{85} + 20q^{87} + 16q^{89} + 20q^{91} + 16q^{93} - 40q^{95} - 36q^{97} - 4q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 4 x^{7} - 6 x^{6} + 32 x^{5} + 9 x^{4} - 76 x^{3} - 4 x^{2} + 48 x - 8$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$\nu^{4} - 2 \nu^{3} - 5 \nu^{2} + 6 \nu + 4$$$$)/2$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{7} - 4 \nu^{6} - 2 \nu^{5} + 20 \nu^{4} - 15 \nu^{3} - 8 \nu^{2} + 32 \nu - 20$$$$)/4$$ $$\beta_{3}$$ $$=$$ $$($$$$\nu^{6} - 3 \nu^{5} - 6 \nu^{4} + 17 \nu^{3} + 9 \nu^{2} - 18 \nu$$$$)/2$$ $$\beta_{4}$$ $$=$$ $$($$$$-\nu^{6} + 3 \nu^{5} + 6 \nu^{4} - 17 \nu^{3} - 7 \nu^{2} + 16 \nu - 6$$$$)/2$$ $$\beta_{5}$$ $$=$$ $$($$$$\nu^{7} - 4 \nu^{6} - 4 \nu^{5} + 26 \nu^{4} - \nu^{3} - 42 \nu^{2} + 8$$$$)/4$$ $$\beta_{6}$$ $$=$$ $$($$$$-\nu^{7} + 4 \nu^{6} + 4 \nu^{5} - 26 \nu^{4} + 5 \nu^{3} + 38 \nu^{2} - 16 \nu - 8$$$$)/4$$ $$\beta_{7}$$ $$=$$ $$($$$$-\nu^{7} + 4 \nu^{6} + 6 \nu^{5} - 30 \nu^{4} - 13 \nu^{3} + 62 \nu^{2} + 20 \nu - 24$$$$)/4$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$-\beta_{7} + \beta_{6} - \beta_{5} + \beta_{2} + \beta_{1} + 1$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$($$$$-\beta_{7} + \beta_{6} - \beta_{5} + 2 \beta_{4} + 2 \beta_{3} + \beta_{2} + \beta_{1} + 7$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$($$$$-5 \beta_{7} + 7 \beta_{6} - 3 \beta_{5} + 2 \beta_{4} + 2 \beta_{3} + 5 \beta_{2} + 5 \beta_{1} + 11$$$$)/2$$ $$\nu^{4}$$ $$=$$ $$($$$$-9 \beta_{7} + 13 \beta_{6} - 5 \beta_{5} + 14 \beta_{4} + 14 \beta_{3} + 9 \beta_{2} + 13 \beta_{1} + 43$$$$)/2$$ $$\nu^{5}$$ $$=$$ $$($$$$-29 \beta_{7} + 55 \beta_{6} - 7 \beta_{5} + 22 \beta_{4} + 22 \beta_{3} + 33 \beta_{2} + 41 \beta_{1} + 99$$$$)/2$$ $$\nu^{6}$$ $$=$$ $$($$$$-65 \beta_{7} + 133 \beta_{6} - 9 \beta_{5} + 98 \beta_{4} + 102 \beta_{3} + 77 \beta_{2} + 125 \beta_{1} + 323$$$$)/2$$ $$\nu^{7}$$ $$=$$ $$($$$$-189 \beta_{7} + 463 \beta_{6} + 29 \beta_{5} + 202 \beta_{4} + 218 \beta_{3} + 253 \beta_{2} + 373 \beta_{1} + 859$$$$)/2$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 3.02908 −1.38887 0.177920 −2.02908 −1.21236 2.21236 2.38887 0.822080
0 −3.42180 0 3.59402 0 1.00000 0 8.70871 0
1.2 0 −2.09974 0 2.59647 0 1.00000 0 1.40890 0
1.3 0 −1.67251 0 −2.65618 0 1.00000 0 −0.202696 0
1.4 0 −0.242103 0 −0.379610 0 1.00000 0 −2.94139 0
1.5 0 0.463900 0 −1.98268 0 1.00000 0 −2.78480 0
1.6 0 1.78579 0 4.18248 0 1.00000 0 0.189043 0
1.7 0 2.29954 0 1.65321 0 1.00000 0 2.28788 0
1.8 0 2.88693 0 0.992279 0 1.00000 0 5.33435 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$1$$
$$7$$ $$-1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 7168.2.a.bf 8
4.b odd 2 1 7168.2.a.be 8
8.b even 2 1 7168.2.a.bb 8
8.d odd 2 1 7168.2.a.ba 8
32.g even 8 2 1792.2.m.e 16
32.g even 8 2 1792.2.m.h yes 16
32.h odd 8 2 1792.2.m.f yes 16
32.h odd 8 2 1792.2.m.g yes 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1792.2.m.e 16 32.g even 8 2
1792.2.m.f yes 16 32.h odd 8 2
1792.2.m.g yes 16 32.h odd 8 2
1792.2.m.h yes 16 32.g even 8 2
7168.2.a.ba 8 8.d odd 2 1
7168.2.a.bb 8 8.b even 2 1
7168.2.a.be 8 4.b odd 2 1
7168.2.a.bf 8 1.a even 1 1 trivial

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(7168))$$:

 $$T_{3}^{8} - 18 T_{3}^{6} + 4 T_{3}^{5} + 94 T_{3}^{4} - 24 T_{3}^{3} - 152 T_{3}^{2} + 32 T_{3} + 16$$ $$T_{5}^{8} - \cdots$$ $$T_{11}^{8} - \cdots$$ $$T_{13}^{8} - 20 T_{13}^{7} + 138 T_{13}^{6} - 284 T_{13}^{5} - 882 T_{13}^{4} + 4960 T_{13}^{3} - 6784 T_{13}^{2} + 4096$$ $$T_{17}^{8} - \cdots$$ $$T_{23}^{8} - \cdots$$ $$T_{31}^{8} + \cdots$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$16 + 32 T - 152 T^{2} - 24 T^{3} + 94 T^{4} + 4 T^{5} - 18 T^{6} + T^{8}$$
$5$ $$-128 - 128 T + 512 T^{2} - 176 T^{3} - 162 T^{4} + 84 T^{5} + 6 T^{6} - 8 T^{7} + T^{8}$$
$7$ $$( -1 + T )^{8}$$
$11$ $$64 - 128 T - 192 T^{2} + 640 T^{3} - 492 T^{4} + 88 T^{5} + 32 T^{6} - 12 T^{7} + T^{8}$$
$13$ $$4096 - 6784 T^{2} + 4960 T^{3} - 882 T^{4} - 284 T^{5} + 138 T^{6} - 20 T^{7} + T^{8}$$
$17$ $$116608 + 15872 T - 46272 T^{2} - 4992 T^{3} + 3528 T^{4} + 272 T^{5} - 100 T^{6} - 4 T^{7} + T^{8}$$
$19$ $$16 + 160 T - 1528 T^{2} - 584 T^{3} + 526 T^{4} + 140 T^{5} - 46 T^{6} - 4 T^{7} + T^{8}$$
$23$ $$-70592 + 56064 T + 12800 T^{2} - 15680 T^{3} + 1428 T^{4} + 688 T^{5} - 84 T^{6} - 8 T^{7} + T^{8}$$
$29$ $$91792 + 112704 T + 2816 T^{2} - 21728 T^{3} + 1560 T^{4} + 880 T^{5} - 96 T^{6} - 8 T^{7} + T^{8}$$
$31$ $$-251648 - 366848 T - 59584 T^{2} + 25792 T^{3} + 5304 T^{4} - 576 T^{5} - 132 T^{6} + 4 T^{7} + T^{8}$$
$37$ $$695056 + 197056 T - 172960 T^{2} - 40544 T^{3} + 9240 T^{4} + 1104 T^{5} - 168 T^{6} - 8 T^{7} + T^{8}$$
$41$ $$-9344 - 2048 T + 15360 T^{2} + 2560 T^{3} - 4440 T^{4} - 1424 T^{5} - 76 T^{6} + 12 T^{7} + T^{8}$$
$43$ $$45952 - 51840 T - 22624 T^{2} + 12064 T^{3} + 3012 T^{4} - 536 T^{5} - 120 T^{6} + 4 T^{7} + T^{8}$$
$47$ $$-18176 + 24832 T + 32832 T^{2} - 2880 T^{3} - 4680 T^{4} + 784 T^{5} + 68 T^{6} - 20 T^{7} + T^{8}$$
$53$ $$365584 - 127808 T - 504800 T^{2} + 228896 T^{3} - 28392 T^{4} - 1136 T^{5} + 520 T^{6} - 40 T^{7} + T^{8}$$
$59$ $$407824 + 240928 T - 107960 T^{2} - 46072 T^{3} + 10254 T^{4} + 964 T^{5} - 206 T^{6} - 4 T^{7} + T^{8}$$
$61$ $$-98432 - 84992 T + 16480 T^{2} + 21344 T^{3} + 990 T^{4} - 1124 T^{5} - 122 T^{6} + 8 T^{7} + T^{8}$$
$67$ $$-51056384 + 6458368 T + 2259520 T^{2} - 323776 T^{3} - 27900 T^{4} + 5272 T^{5} + 16 T^{6} - 28 T^{7} + T^{8}$$
$71$ $$-8192 - 45056 T + 35328 T^{2} + 14592 T^{3} - 2544 T^{4} - 1088 T^{5} - 16 T^{6} + 16 T^{7} + T^{8}$$
$73$ $$7618816 + 2791424 T - 384640 T^{2} - 162304 T^{3} + 10656 T^{4} + 2880 T^{5} - 168 T^{6} - 16 T^{7} + T^{8}$$
$79$ $$-4822784 - 3951616 T - 633344 T^{2} + 107520 T^{3} + 25504 T^{4} - 704 T^{5} - 288 T^{6} + T^{8}$$
$83$ $$-20041712 - 10520992 T - 569112 T^{2} + 398808 T^{3} + 41022 T^{4} - 3652 T^{5} - 418 T^{6} + 8 T^{7} + T^{8}$$
$89$ $$( 16 + 32 T - 16 T^{2} - 8 T^{3} + T^{4} )^{2}$$
$97$ $$-2571392 + 3808512 T + 3034816 T^{2} + 99840 T^{3} - 84856 T^{4} - 8256 T^{5} + 116 T^{6} + 36 T^{7} + T^{8}$$
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