# Properties

 Label 8016.2.a.v Level $8016$ Weight $2$ Character orbit 8016.a Self dual yes Analytic conductor $64.008$ Analytic rank $0$ Dimension $7$ CM no Inner twists $1$

# Related objects

## 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: $$7$$ Coefficient field: $$\mathbb{Q}[x]/(x^{7} - \cdots)$$ Defining polynomial: $$x^{7} - 2 x^{6} - 20 x^{5} + 2 x^{4} + 87 x^{3} + 46 x^{2} - 48 x - 2$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 1002) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{6}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - q^{3} + ( 1 - \beta_{1} ) q^{5} + ( -1 + \beta_{3} ) q^{7} + q^{9} +O(q^{10})$$ $$q - q^{3} + ( 1 - \beta_{1} ) q^{5} + ( -1 + \beta_{3} ) q^{7} + q^{9} + ( -1 - \beta_{2} - \beta_{5} + \beta_{6} ) q^{11} + ( \beta_{1} - \beta_{2} + \beta_{3} - \beta_{5} ) q^{13} + ( -1 + \beta_{1} ) q^{15} + ( \beta_{1} + \beta_{3} - \beta_{5} + \beta_{6} ) q^{17} + ( 1 + \beta_{2} - \beta_{5} - \beta_{6} ) q^{19} + ( 1 - \beta_{3} ) q^{21} + ( -\beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} ) q^{23} + ( 1 + \beta_{1} - \beta_{4} - \beta_{5} + \beta_{6} ) q^{25} - q^{27} + ( -1 - \beta_{2} - 2 \beta_{4} + \beta_{5} + \beta_{6} ) q^{29} + ( -1 - \beta_{1} - \beta_{2} - \beta_{4} ) q^{31} + ( 1 + \beta_{2} + \beta_{5} - \beta_{6} ) q^{33} + ( 1 + 2 \beta_{1} + \beta_{3} - \beta_{4} + \beta_{6} ) q^{35} + ( -1 + \beta_{1} - \beta_{2} + 2 \beta_{3} + \beta_{4} - \beta_{5} - \beta_{6} ) q^{37} + ( -\beta_{1} + \beta_{2} - \beta_{3} + \beta_{5} ) q^{39} + ( -2 - \beta_{2} + 2 \beta_{3} + \beta_{4} - \beta_{5} - \beta_{6} ) q^{41} + ( -\beta_{1} + \beta_{3} + \beta_{4} + \beta_{6} ) q^{43} + ( 1 - \beta_{1} ) q^{45} + ( 1 + \beta_{3} + 2 \beta_{4} - 2 \beta_{5} ) q^{47} + ( 2 - \beta_{1} + \beta_{2} - 2 \beta_{3} + \beta_{4} ) q^{49} + ( -\beta_{1} - \beta_{3} + \beta_{5} - \beta_{6} ) q^{51} + ( 1 - 2 \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} ) q^{53} + ( -2 - \beta_{1} - 3 \beta_{2} + \beta_{3} + \beta_{4} - 2 \beta_{5} + 2 \beta_{6} ) q^{55} + ( -1 - \beta_{2} + \beta_{5} + \beta_{6} ) q^{57} + ( 3 - \beta_{1} + \beta_{2} + \beta_{4} + \beta_{5} + \beta_{6} ) q^{59} + ( 3 + \beta_{1} + 2 \beta_{2} - \beta_{3} - \beta_{4} + \beta_{5} - \beta_{6} ) q^{61} + ( -1 + \beta_{3} ) q^{63} + ( -2 - \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} - 2 \beta_{5} ) q^{65} + ( 3 + 3 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} - 2 \beta_{5} - 2 \beta_{6} ) q^{67} + ( \beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} ) q^{69} + ( 4 - \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} + 2 \beta_{5} ) q^{71} + ( 4 - \beta_{1} + 3 \beta_{2} - \beta_{3} + \beta_{4} + 2 \beta_{5} - 2 \beta_{6} ) q^{73} + ( -1 - \beta_{1} + \beta_{4} + \beta_{5} - \beta_{6} ) q^{75} + ( -4 - \beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} + 2 \beta_{5} + 2 \beta_{6} ) q^{77} + ( -\beta_{2} + 2 \beta_{3} - \beta_{4} + \beta_{5} - \beta_{6} ) q^{79} + q^{81} + ( -1 + 2 \beta_{1} - 2 \beta_{2} + 3 \beta_{3} - \beta_{5} + \beta_{6} ) q^{83} + ( -3 - 4 \beta_{1} - 3 \beta_{2} + 2 \beta_{3} + \beta_{5} + \beta_{6} ) q^{85} + ( 1 + \beta_{2} + 2 \beta_{4} - \beta_{5} - \beta_{6} ) q^{87} + ( -1 - 2 \beta_{1} - 3 \beta_{3} - 2 \beta_{5} + 2 \beta_{6} ) q^{89} + ( 2 - 3 \beta_{1} + 3 \beta_{2} - 3 \beta_{3} + \beta_{4} + 4 \beta_{5} ) q^{91} + ( 1 + \beta_{1} + \beta_{2} + \beta_{4} ) q^{93} + ( 6 - \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} - 4 \beta_{6} ) q^{95} + ( -2 + \beta_{1} - 2 \beta_{2} + 2 \beta_{3} - \beta_{4} - \beta_{5} + \beta_{6} ) q^{97} + ( -1 - \beta_{2} - \beta_{5} + \beta_{6} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$7q - 7q^{3} + 5q^{5} - 7q^{7} + 7q^{9} + O(q^{10})$$ $$7q - 7q^{3} + 5q^{5} - 7q^{7} + 7q^{9} + 6q^{13} - 5q^{15} + 6q^{17} + 2q^{19} + 7q^{21} + 12q^{25} - 7q^{27} - 4q^{29} - 7q^{31} + 13q^{35} - 3q^{37} - 6q^{39} - 12q^{41} + 2q^{43} + 5q^{45} + 11q^{47} + 10q^{49} - 6q^{51} + q^{53} + 2q^{55} - 2q^{57} + 19q^{59} + 12q^{61} - 7q^{63} - 10q^{65} + 17q^{67} + 20q^{71} + 10q^{73} - 12q^{75} - 24q^{77} - 2q^{79} + 7q^{81} + 7q^{83} - 18q^{85} + 4q^{87} - 3q^{89} - 4q^{91} + 7q^{93} + 24q^{95} - 3q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{7} - 2 x^{6} - 20 x^{5} + 2 x^{4} + 87 x^{3} + 46 x^{2} - 48 x - 2$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$5 \nu^{6} - 21 \nu^{5} - 33 \nu^{4} + 41 \nu^{3} + 12 \nu^{2} + 162 \nu - 14$$$$)/104$$ $$\beta_{3}$$ $$=$$ $$($$$$-3 \nu^{6} + 23 \nu^{5} - \nu^{4} - 243 \nu^{3} + 128 \nu^{2} + 610 \nu - 158$$$$)/52$$ $$\beta_{4}$$ $$=$$ $$($$$$-25 \nu^{6} + 105 \nu^{5} + 269 \nu^{4} - 621 \nu^{3} - 892 \nu^{2} + 542 \nu + 486$$$$)/104$$ $$\beta_{5}$$ $$=$$ $$($$$$-17 \nu^{6} + 61 \nu^{5} + 237 \nu^{4} - 389 \nu^{3} - 800 \nu^{2} + 406 \nu + 162$$$$)/52$$ $$\beta_{6}$$ $$=$$ $$($$$$-59 \nu^{6} + 227 \nu^{5} + 743 \nu^{4} - 1399 \nu^{3} - 2388 \nu^{2} + 1042 \nu + 290$$$$)/104$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{6} - \beta_{5} - \beta_{4} + 3 \beta_{1} + 5$$ $$\nu^{3}$$ $$=$$ $$4 \beta_{6} - 5 \beta_{5} - 2 \beta_{4} - \beta_{3} + 2 \beta_{2} + 18 \beta_{1} + 11$$ $$\nu^{4}$$ $$=$$ $$24 \beta_{6} - 28 \beta_{5} - 15 \beta_{4} - 4 \beta_{3} + 13 \beta_{2} + 83 \beta_{1} + 80$$ $$\nu^{5}$$ $$=$$ $$119 \beta_{6} - 148 \beta_{5} - 59 \beta_{4} - 24 \beta_{3} + 74 \beta_{2} + 437 \beta_{1} + 342$$ $$\nu^{6}$$ $$=$$ $$623 \beta_{6} - 763 \beta_{5} - 328 \beta_{4} - 119 \beta_{3} + 401 \beta_{2} + 2196 \beta_{1} + 1865$$

## 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
 5.12652 2.30288 0.561628 −0.0402333 −1.55552 −2.09849 −2.29679
0 −1.00000 0 −4.12652 0 −3.19234 0 1.00000 0
1.2 0 −1.00000 0 −1.30288 0 −1.53952 0 1.00000 0
1.3 0 −1.00000 0 0.438372 0 2.51945 0 1.00000 0
1.4 0 −1.00000 0 1.04023 0 −4.50614 0 1.00000 0
1.5 0 −1.00000 0 2.55552 0 −3.69934 0 1.00000 0
1.6 0 −1.00000 0 3.09849 0 2.06938 0 1.00000 0
1.7 0 −1.00000 0 3.29679 0 1.34851 0 1.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.7 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.v 7
4.b odd 2 1 1002.2.a.k 7
12.b even 2 1 3006.2.a.u 7

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1002.2.a.k 7 4.b odd 2 1
3006.2.a.u 7 12.b even 2 1
8016.2.a.v 7 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(8016))$$:

 $$T_{5}^{7} - 5 T_{5}^{6} - 11 T_{5}^{5} + 93 T_{5}^{4} - 110 T_{5}^{3} - 110 T_{5}^{2} + 208 T_{5} - 64$$ $$T_{7}^{7} + 7 T_{7}^{6} - 5 T_{7}^{5} - 99 T_{7}^{4} - 28 T_{7}^{3} + 448 T_{7}^{2} + 96 T_{7} - 576$$ $$T_{11}^{7} - 48 T_{11}^{5} - 12 T_{11}^{4} + 624 T_{11}^{3} + 368 T_{11}^{2} - 1344 T_{11} - 1152$$ $$T_{13}^{7} - 6 T_{13}^{6} - 30 T_{13}^{5} + 244 T_{13}^{4} - 288 T_{13}^{3} - 552 T_{13}^{2} + 1088 T_{13} - 384$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ $$( 1 + T )^{7}$$
$5$ $$1 - 5 T + 24 T^{2} - 57 T^{3} + 140 T^{4} - 125 T^{5} + 183 T^{6} + 286 T^{7} + 915 T^{8} - 3125 T^{9} + 17500 T^{10} - 35625 T^{11} + 75000 T^{12} - 78125 T^{13} + 78125 T^{14}$$
$7$ $$1 + 7 T + 44 T^{2} + 195 T^{3} + 826 T^{4} + 2821 T^{5} + 9063 T^{6} + 24610 T^{7} + 63441 T^{8} + 138229 T^{9} + 283318 T^{10} + 468195 T^{11} + 739508 T^{12} + 823543 T^{13} + 823543 T^{14}$$
$11$ $$1 + 29 T^{2} - 12 T^{3} + 525 T^{4} - 160 T^{5} + 7753 T^{6} - 1768 T^{7} + 85283 T^{8} - 19360 T^{9} + 698775 T^{10} - 175692 T^{11} + 4670479 T^{12} + 19487171 T^{14}$$
$13$ $$1 - 6 T + 61 T^{2} - 224 T^{3} + 1311 T^{4} - 3074 T^{5} + 16051 T^{6} - 30960 T^{7} + 208663 T^{8} - 519506 T^{9} + 2880267 T^{10} - 6397664 T^{11} + 22648873 T^{12} - 28960854 T^{13} + 62748517 T^{14}$$
$17$ $$1 - 6 T + 63 T^{2} - 428 T^{3} + 2417 T^{4} - 13658 T^{5} + 63663 T^{6} - 277752 T^{7} + 1082271 T^{8} - 3947162 T^{9} + 11874721 T^{10} - 35746988 T^{11} + 89450991 T^{12} - 144825414 T^{13} + 410338673 T^{14}$$
$19$ $$1 - 2 T + 17 T^{2} - 184 T^{3} + 497 T^{4} - 4174 T^{5} + 22745 T^{6} - 45392 T^{7} + 432155 T^{8} - 1506814 T^{9} + 3408923 T^{10} - 23979064 T^{11} + 42093683 T^{12} - 94091762 T^{13} + 893871739 T^{14}$$
$23$ $$1 + 69 T^{2} + 140 T^{3} + 2337 T^{4} + 9584 T^{5} + 59309 T^{6} + 291720 T^{7} + 1364107 T^{8} + 5069936 T^{9} + 28434279 T^{10} + 39177740 T^{11} + 444107667 T^{12} + 3404825447 T^{14}$$
$29$ $$1 + 4 T + 43 T^{2} + 300 T^{3} + 2397 T^{4} + 13692 T^{5} + 90543 T^{6} + 417320 T^{7} + 2625747 T^{8} + 11514972 T^{9} + 58460433 T^{10} + 212184300 T^{11} + 881979407 T^{12} + 2379293284 T^{13} + 17249876309 T^{14}$$
$31$ $$1 + 7 T + 148 T^{2} + 931 T^{3} + 11198 T^{4} + 58509 T^{5} + 519739 T^{6} + 2272786 T^{7} + 16111909 T^{8} + 56227149 T^{9} + 333599618 T^{10} + 859798051 T^{11} + 4237114348 T^{12} + 6212525767 T^{13} + 27512614111 T^{14}$$
$37$ $$1 + 3 T + 114 T^{2} + 211 T^{3} + 8476 T^{4} + 16119 T^{5} + 421173 T^{6} + 585550 T^{7} + 15583401 T^{8} + 22066911 T^{9} + 429334828 T^{10} + 395447971 T^{11} + 7905211098 T^{12} + 7697179227 T^{13} + 94931877133 T^{14}$$
$41$ $$1 + 12 T + 183 T^{2} + 1700 T^{3} + 17105 T^{4} + 131452 T^{5} + 1040039 T^{6} + 6545832 T^{7} + 42641599 T^{8} + 220970812 T^{9} + 1178893705 T^{10} + 4803793700 T^{11} + 21201684783 T^{12} + 57001250892 T^{13} + 194754273881 T^{14}$$
$43$ $$1 - 2 T + 167 T^{2} - 660 T^{3} + 14931 T^{4} - 67206 T^{5} + 899789 T^{6} - 3782856 T^{7} + 38690927 T^{8} - 124263894 T^{9} + 1187119017 T^{10} - 2256408660 T^{11} + 24550409981 T^{12} - 12642726098 T^{13} + 271818611107 T^{14}$$
$47$ $$1 - 11 T + 192 T^{2} - 1847 T^{3} + 17634 T^{4} - 150193 T^{5} + 1120915 T^{6} - 8253386 T^{7} + 52683005 T^{8} - 331776337 T^{9} + 1830814782 T^{10} - 9012770807 T^{11} + 44034241344 T^{12} - 118571368619 T^{13} + 506623120463 T^{14}$$
$53$ $$1 - T + 270 T^{2} - 231 T^{3} + 34242 T^{4} - 26021 T^{5} + 2690283 T^{6} - 1753150 T^{7} + 142584999 T^{8} - 73092989 T^{9} + 5097846234 T^{10} - 1822701111 T^{11} + 112912783110 T^{12} - 22164361129 T^{13} + 1174711139837 T^{14}$$
$59$ $$1 - 19 T + 364 T^{2} - 4667 T^{3} + 57934 T^{4} - 561883 T^{5} + 5308991 T^{6} - 41425142 T^{7} + 313230469 T^{8} - 1955914723 T^{9} + 11898426986 T^{10} - 56551723787 T^{11} + 260232444836 T^{12} - 801430139179 T^{13} + 2488651484819 T^{14}$$
$61$ $$1 - 12 T + 295 T^{2} - 3288 T^{3} + 43865 T^{4} - 410260 T^{5} + 4116935 T^{6} - 31032016 T^{7} + 251133035 T^{8} - 1526577460 T^{9} + 9956521565 T^{10} - 45525125208 T^{11} + 249155908795 T^{12} - 618244492332 T^{13} + 3142742836021 T^{14}$$
$67$ $$1 - 17 T + 98 T^{2} - 137 T^{3} + 4010 T^{4} - 81897 T^{5} + 561533 T^{6} - 2400242 T^{7} + 37622711 T^{8} - 367635633 T^{9} + 1206059630 T^{10} - 2760703577 T^{11} + 132312260486 T^{12} - 1537792496873 T^{13} + 6060711605323 T^{14}$$
$71$ $$1 - 20 T + 529 T^{2} - 7100 T^{3} + 107685 T^{4} - 1088764 T^{5} + 12096037 T^{6} - 97336680 T^{7} + 858818627 T^{8} - 5488459324 T^{9} + 38541646035 T^{10} - 180422935100 T^{11} + 954437326679 T^{12} - 2562005678420 T^{13} + 9095120158391 T^{14}$$
$73$ $$1 - 10 T + 331 T^{2} - 3088 T^{3} + 57889 T^{4} - 453094 T^{5} + 6297923 T^{6} - 41415872 T^{7} + 459748379 T^{8} - 2414537926 T^{9} + 22519805113 T^{10} - 87693768208 T^{11} + 686186697283 T^{12} - 1513342262890 T^{13} + 11047398519097 T^{14}$$
$79$ $$1 + 2 T + 293 T^{2} + 672 T^{3} + 48493 T^{4} + 101038 T^{5} + 5408089 T^{6} + 10050656 T^{7} + 427239031 T^{8} + 630578158 T^{9} + 23908940227 T^{10} + 26174454432 T^{11} + 901577524907 T^{12} + 486174911042 T^{13} + 19203908986159 T^{14}$$
$83$ $$1 - 7 T + 406 T^{2} - 2165 T^{3} + 78228 T^{4} - 332467 T^{5} + 9448267 T^{6} - 33037258 T^{7} + 784206161 T^{8} - 2290365163 T^{9} + 44729753436 T^{10} - 102747264965 T^{11} + 1599250501058 T^{12} - 2288582613583 T^{13} + 27136050989627 T^{14}$$
$89$ $$1 + 3 T + 142 T^{2} + 1563 T^{3} + 15208 T^{4} + 131465 T^{5} + 2000929 T^{6} + 8498354 T^{7} + 178082681 T^{8} + 1041334265 T^{9} + 10721168552 T^{10} + 98066122683 T^{11} + 792936441758 T^{12} + 1490943872883 T^{13} + 44231334895529 T^{14}$$
$97$ $$1 + 3 T + 542 T^{2} + 1923 T^{3} + 132516 T^{4} + 492061 T^{5} + 19451309 T^{6} + 64748442 T^{7} + 1886776973 T^{8} + 4629801949 T^{9} + 120943775268 T^{10} + 170241807363 T^{11} + 4654338419294 T^{12} + 2498916014787 T^{13} + 80798284478113 T^{14}$$