# Properties

 Label 8016.2.a.bb Level $8016$ Weight $2$ Character orbit 8016.a Self dual yes Analytic conductor $64.008$ Analytic rank $0$ Dimension $9$ CM no Inner twists $1$

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## Newspace parameters

 Level: $$N$$ $$=$$ $$8016 = 2^{4} \cdot 3 \cdot 167$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 8016.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$64.0080822603$$ Analytic rank: $$0$$ Dimension: $$9$$ Coefficient field: $$\mathbb{Q}[x]/(x^{9} - \cdots)$$ Defining polynomial: $$x^{9} - 29 x^{7} - 7 x^{6} + 266 x^{5} + 69 x^{4} - 901 x^{3} - 199 x^{2} + 875 x + 391$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 2004) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{8}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - q^{3} + ( 1 - \beta_{1} ) q^{5} + \beta_{6} q^{7} + q^{9} +O(q^{10})$$ $$q - q^{3} + ( 1 - \beta_{1} ) q^{5} + \beta_{6} q^{7} + q^{9} + ( -1 + \beta_{5} ) q^{11} + ( 1 - \beta_{4} + \beta_{6} + \beta_{7} ) q^{13} + ( -1 + \beta_{1} ) q^{15} + ( 1 - \beta_{2} + \beta_{3} + \beta_{8} ) q^{17} + ( \beta_{2} + \beta_{7} ) q^{19} -\beta_{6} q^{21} + ( -2 - \beta_{1} - \beta_{2} - \beta_{5} + \beta_{8} ) q^{23} + ( 3 - 2 \beta_{1} + 2 \beta_{2} + \beta_{4} ) q^{25} - q^{27} + ( \beta_{1} - 2 \beta_{3} + \beta_{5} + \beta_{7} - \beta_{8} ) q^{29} + ( -1 + \beta_{2} - \beta_{4} + \beta_{5} + \beta_{7} ) q^{31} + ( 1 - \beta_{5} ) q^{33} + ( -2 \beta_{2} + 2 \beta_{3} - \beta_{5} + 3 \beta_{6} + \beta_{7} ) q^{35} + ( 2 - \beta_{1} + \beta_{2} + 2 \beta_{4} - \beta_{7} + \beta_{8} ) q^{37} + ( -1 + \beta_{4} - \beta_{6} - \beta_{7} ) q^{39} + ( 2 + \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} + \beta_{5} ) q^{41} + ( 2 + \beta_{2} - \beta_{4} - \beta_{5} - \beta_{6} - \beta_{7} + \beta_{8} ) q^{43} + ( 1 - \beta_{1} ) q^{45} + ( -4 - 2 \beta_{1} - 2 \beta_{3} + \beta_{4} - \beta_{6} ) q^{47} + ( 2 - \beta_{4} + 2 \beta_{6} + \beta_{7} - 2 \beta_{8} ) q^{49} + ( -1 + \beta_{2} - \beta_{3} - \beta_{8} ) q^{51} + ( 2 + \beta_{2} - \beta_{3} + \beta_{4} + \beta_{5} - \beta_{6} - 2 \beta_{7} + \beta_{8} ) q^{53} + ( -1 + 2 \beta_{1} - \beta_{3} - \beta_{4} + 2 \beta_{5} + \beta_{6} + 2 \beta_{7} - 2 \beta_{8} ) q^{55} + ( -\beta_{2} - \beta_{7} ) q^{57} + ( 1 - \beta_{1} - \beta_{2} + 2 \beta_{3} - \beta_{4} - 2 \beta_{7} + \beta_{8} ) q^{59} + ( 3 + \beta_{1} - 2 \beta_{2} + 2 \beta_{3} - 2 \beta_{4} + 2 \beta_{6} - \beta_{7} + \beta_{8} ) q^{61} + \beta_{6} q^{63} + ( 2 - 3 \beta_{2} + 2 \beta_{3} - 2 \beta_{4} + 3 \beta_{6} + \beta_{7} ) q^{65} + ( -\beta_{1} - 2 \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} - \beta_{7} ) q^{67} + ( 2 + \beta_{1} + \beta_{2} + \beta_{5} - \beta_{8} ) q^{69} + ( -2 - \beta_{1} + 2 \beta_{2} - 2 \beta_{3} - 2 \beta_{4} + 2 \beta_{5} - \beta_{6} - \beta_{8} ) q^{71} + ( -3 + \beta_{2} - 2 \beta_{3} - \beta_{4} + 2 \beta_{5} + 2 \beta_{7} ) q^{73} + ( -3 + 2 \beta_{1} - 2 \beta_{2} - \beta_{4} ) q^{75} + ( 2 - 2 \beta_{2} + 2 \beta_{3} - 3 \beta_{5} + \beta_{6} + \beta_{7} ) q^{77} + ( 1 - \beta_{1} + \beta_{2} + \beta_{3} + 2 \beta_{4} ) q^{79} + q^{81} + ( -1 - \beta_{1} + \beta_{4} + \beta_{5} - \beta_{6} - \beta_{8} ) q^{83} + ( 1 - 2 \beta_{2} - 2 \beta_{4} - \beta_{5} + 2 \beta_{6} ) q^{85} + ( -\beta_{1} + 2 \beta_{3} - \beta_{5} - \beta_{7} + \beta_{8} ) q^{87} + ( 4 + \beta_{2} + 2 \beta_{4} - \beta_{7} ) q^{89} + ( 4 - \beta_{1} + 2 \beta_{4} + 2 \beta_{6} - \beta_{8} ) q^{91} + ( 1 - \beta_{2} + \beta_{4} - \beta_{5} - \beta_{7} ) q^{93} + ( -1 - \beta_{1} - 2 \beta_{2} - \beta_{4} + \beta_{6} - \beta_{7} - \beta_{8} ) q^{95} + ( 4 - \beta_{1} + 5 \beta_{2} - 2 \beta_{3} + 2 \beta_{4} - \beta_{6} - \beta_{7} - \beta_{8} ) q^{97} + ( -1 + \beta_{5} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$9q - 9q^{3} + 9q^{5} - 2q^{7} + 9q^{9} + O(q^{10})$$ $$9q - 9q^{3} + 9q^{5} - 2q^{7} + 9q^{9} - 7q^{11} + 6q^{13} - 9q^{15} + 7q^{17} - 2q^{19} + 2q^{21} - 19q^{23} + 22q^{25} - 9q^{27} + 13q^{29} - 12q^{31} + 7q^{33} - 4q^{35} + 15q^{37} - 6q^{39} + 18q^{41} + 6q^{43} + 9q^{45} - 25q^{47} + 19q^{49} - 7q^{51} + 17q^{53} + 3q^{55} + 2q^{57} - 3q^{59} + 14q^{61} - 2q^{63} + 14q^{65} + 4q^{67} + 19q^{69} - 17q^{71} - 20q^{73} - 22q^{75} + 14q^{77} + 8q^{79} + 9q^{81} + q^{83} + 5q^{85} - 13q^{87} + 36q^{89} + 41q^{91} + 12q^{93} - 5q^{95} + 31q^{97} - 7q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{9} - 29 x^{7} - 7 x^{6} + 266 x^{5} + 69 x^{4} - 901 x^{3} - 199 x^{2} + 875 x + 391$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$-1006 \nu^{8} + 2871 \nu^{7} + 22263 \nu^{6} - 53775 \nu^{5} - 157887 \nu^{4} + 312589 \nu^{3} + 454514 \nu^{2} - 550699 \nu - 575149$$$$)/51607$$ $$\beta_{3}$$ $$=$$ $$($$$$1976 \nu^{8} - 9538 \nu^{7} - 25159 \nu^{6} + 160208 \nu^{5} - 1570 \nu^{4} - 711563 \nu^{3} + 508013 \nu^{2} + 557721 \nu - 255465$$$$)/51607$$ $$\beta_{4}$$ $$=$$ $$($$$$2012 \nu^{8} - 5742 \nu^{7} - 44526 \nu^{6} + 107550 \nu^{5} + 315774 \nu^{4} - 625178 \nu^{3} - 857421 \nu^{2} + 1101398 \nu + 789049$$$$)/51607$$ $$\beta_{5}$$ $$=$$ $$($$$$2980 \nu^{8} - 18354 \nu^{7} - 43479 \nu^{6} + 348793 \nu^{5} + 150155 \nu^{4} - 1846062 \nu^{3} - 203222 \nu^{2} + 2331119 \nu + 1064431$$$$)/51607$$ $$\beta_{6}$$ $$=$$ $$($$$$3832 \nu^{8} - 14527 \nu^{7} - 71773 \nu^{6} + 255110 \nu^{5} + 401245 \nu^{4} - 1246613 \nu^{3} - 814593 \nu^{2} + 1505084 \nu + 875617$$$$)/51607$$ $$\beta_{7}$$ $$=$$ $$($$$$3879 \nu^{8} - 3837 \nu^{7} - 87023 \nu^{6} + 28674 \nu^{5} + 546052 \nu^{4} + 57429 \nu^{3} - 948686 \nu^{2} - 418506 \nu + 172141$$$$)/51607$$ $$\beta_{8}$$ $$=$$ $$($$$$-3886 \nu^{8} + 8833 \nu^{7} + 75020 \nu^{6} - 124516 \nu^{5} - 361191 \nu^{4} + 420341 \nu^{3} + 179180 \nu^{2} - 75695 \nu + 292783$$$$)/51607$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{4} + 2 \beta_{2} + 7$$ $$\nu^{3}$$ $$=$$ $$\beta_{8} + 2 \beta_{7} - \beta_{6} + 2 \beta_{5} - \beta_{3} + 4 \beta_{2} + 10 \beta_{1} + 3$$ $$\nu^{4}$$ $$=$$ $$5 \beta_{8} + 4 \beta_{7} + \beta_{5} + 14 \beta_{4} + 2 \beta_{3} + 31 \beta_{2} + 5 \beta_{1} + 79$$ $$\nu^{5}$$ $$=$$ $$22 \beta_{8} + 38 \beta_{7} - 23 \beta_{6} + 36 \beta_{5} + 17 \beta_{4} - 12 \beta_{3} + 91 \beta_{2} + 123 \beta_{1} + 91$$ $$\nu^{6}$$ $$=$$ $$115 \beta_{8} + 102 \beta_{7} - 21 \beta_{6} + 38 \beta_{5} + 207 \beta_{4} + 46 \beta_{3} + 486 \beta_{2} + 163 \beta_{1} + 1059$$ $$\nu^{7}$$ $$=$$ $$431 \beta_{8} + 648 \beta_{7} - 397 \beta_{6} + 555 \beta_{5} + 464 \beta_{4} - 97 \beta_{3} + 1703 \beta_{2} + 1714 \beta_{1} + 2087$$ $$\nu^{8}$$ $$=$$ $$2125 \beta_{8} + 2069 \beta_{7} - 679 \beta_{6} + 965 \beta_{5} + 3251 \beta_{4} + 758 \beta_{3} + 7981 \beta_{2} + 3699 \beta_{1} + 15652$$

## 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
 4.16840 2.69402 1.79204 1.61974 −0.529979 −0.907808 −2.42964 −3.19525 −3.21153
0 −1.00000 0 −3.16840 0 0.230890 0 1.00000 0
1.2 0 −1.00000 0 −1.69402 0 4.12928 0 1.00000 0
1.3 0 −1.00000 0 −0.792043 0 −3.80237 0 1.00000 0
1.4 0 −1.00000 0 −0.619742 0 −1.05844 0 1.00000 0
1.5 0 −1.00000 0 1.52998 0 1.05249 0 1.00000 0
1.6 0 −1.00000 0 1.90781 0 −2.81337 0 1.00000 0
1.7 0 −1.00000 0 3.42964 0 −3.44225 0 1.00000 0
1.8 0 −1.00000 0 4.19525 0 −1.43344 0 1.00000 0
1.9 0 −1.00000 0 4.21153 0 5.13720 0 1.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.9 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$3$$ $$1$$
$$167$$ $$1$$

## 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 8016.2.a.bb 9
4.b odd 2 1 2004.2.a.d 9
12.b even 2 1 6012.2.a.h 9

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2004.2.a.d 9 4.b odd 2 1
6012.2.a.h 9 12.b even 2 1
8016.2.a.bb 9 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(8016))$$:

 $$T_{5}^{9} - \cdots$$ $$T_{7}^{9} + \cdots$$ $$T_{11}^{9} + \cdots$$ $$T_{13}^{9} - \cdots$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{9}$$
$3$ $$( 1 + T )^{9}$$
$5$ $$-466 - 856 T + 506 T^{2} + 964 T^{3} - 405 T^{4} - 259 T^{5} + 126 T^{6} + 7 T^{7} - 9 T^{8} + T^{9}$$
$7$ $$288 - 912 T - 1640 T^{2} + 480 T^{3} + 1365 T^{4} + 351 T^{5} - 109 T^{6} - 39 T^{7} + 2 T^{8} + T^{9}$$
$11$ $$-36 - 384 T - 1108 T^{2} - 520 T^{3} + 989 T^{4} + 213 T^{5} - 168 T^{6} - 23 T^{7} + 7 T^{8} + T^{9}$$
$13$ $$-5520 - 5712 T + 9600 T^{2} + 1820 T^{3} - 3243 T^{4} + 90 T^{5} + 334 T^{6} - 45 T^{7} - 6 T^{8} + T^{9}$$
$17$ $$-1562 - 8736 T + 11546 T^{2} + 7872 T^{3} - 5653 T^{4} - 263 T^{5} + 540 T^{6} - 58 T^{7} - 7 T^{8} + T^{9}$$
$19$ $$-64 + 64 T + 448 T^{2} - 700 T^{3} - 223 T^{4} + 684 T^{5} - 134 T^{6} - 69 T^{7} + 2 T^{8} + T^{9}$$
$23$ $$288512 + 786560 T + 615744 T^{2} + 164976 T^{3} - 4015 T^{4} - 9127 T^{5} - 1212 T^{6} + 47 T^{7} + 19 T^{8} + T^{9}$$
$29$ $$-272720 + 173296 T + 488440 T^{2} - 11912 T^{3} - 52505 T^{4} + 1713 T^{5} + 1686 T^{6} - 107 T^{7} - 13 T^{8} + T^{9}$$
$31$ $$280000 - 178400 T - 131888 T^{2} + 32480 T^{3} + 18931 T^{4} - 746 T^{5} - 895 T^{6} - 42 T^{7} + 12 T^{8} + T^{9}$$
$37$ $$-5712 - 434496 T - 544344 T^{2} + 450848 T^{3} - 81337 T^{4} - 4701 T^{5} + 2302 T^{6} - 91 T^{7} - 15 T^{8} + T^{9}$$
$41$ $$11178 - 6318 T - 18522 T^{2} + 11862 T^{3} + 6255 T^{4} - 5714 T^{5} + 1079 T^{6} + 22 T^{7} - 18 T^{8} + T^{9}$$
$43$ $$8589994 + 7610890 T + 142622 T^{2} - 886908 T^{3} - 70579 T^{4} + 26210 T^{5} + 1389 T^{6} - 290 T^{7} - 6 T^{8} + T^{9}$$
$47$ $$-51047340 - 22532892 T + 556332 T^{2} + 1431344 T^{3} + 131535 T^{4} - 23500 T^{5} - 3713 T^{6} + 25 T^{7} + 25 T^{8} + T^{9}$$
$53$ $$12400550 - 7077610 T - 987166 T^{2} + 1221572 T^{3} - 198307 T^{4} - 9120 T^{5} + 4129 T^{6} - 159 T^{7} - 17 T^{8} + T^{9}$$
$59$ $$-881152 + 13443712 T - 1538720 T^{2} - 1035632 T^{3} + 68223 T^{4} + 27296 T^{5} - 829 T^{6} - 283 T^{7} + 3 T^{8} + T^{9}$$
$61$ $$-4149788 + 5249068 T + 5792052 T^{2} - 172628 T^{3} - 289205 T^{4} + 12968 T^{5} + 4170 T^{6} - 263 T^{7} - 14 T^{8} + T^{9}$$
$67$ $$-150554 + 260346 T + 729620 T^{2} - 155954 T^{3} - 50139 T^{4} + 10100 T^{5} + 972 T^{6} - 201 T^{7} - 4 T^{8} + T^{9}$$
$71$ $$-343184800 - 207451056 T - 31230720 T^{2} + 2518796 T^{3} + 858773 T^{4} + 17563 T^{5} - 6815 T^{6} - 320 T^{7} + 17 T^{8} + T^{9}$$
$73$ $$45115056 - 5442336 T - 6872328 T^{2} + 357948 T^{3} + 312735 T^{4} + 1642 T^{5} - 4594 T^{6} - 159 T^{7} + 20 T^{8} + T^{9}$$
$79$ $$-2054818 - 3887890 T + 2543762 T^{2} - 136474 T^{3} - 128245 T^{4} + 14886 T^{5} + 1819 T^{6} - 240 T^{7} - 8 T^{8} + T^{9}$$
$83$ $$3972576 + 1719024 T - 894312 T^{2} - 334868 T^{3} + 40425 T^{4} + 16090 T^{5} - 497 T^{6} - 259 T^{7} - T^{8} + T^{9}$$
$89$ $$-93648 - 152784 T + 1378944 T^{2} - 1004108 T^{3} + 278945 T^{4} - 30616 T^{5} - 510 T^{6} + 425 T^{7} - 36 T^{8} + T^{9}$$
$97$ $$-481925792 - 18464048 T + 55502112 T^{2} + 1278656 T^{3} - 1237713 T^{4} - 4645 T^{5} + 10451 T^{6} - 160 T^{7} - 31 T^{8} + T^{9}$$
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