# Properties

 Label 4016.2.a.g Level 4016 Weight 2 Character orbit 4016.a Self dual yes Analytic conductor 32.068 Analytic rank 0 Dimension 7 CM no Inner twists 1

# Related objects

## Newspace parameters

 Level: $$N$$ = $$4016 = 2^{4} \cdot 251$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 4016.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$32.0679214517$$ Analytic rank: $$0$$ Dimension: $$7$$ Coefficient field: $$\mathbb{Q}[x]/(x^{7} - \cdots)$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 1004) 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.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{3} + \beta_{5} q^{5} + ( 1 + \beta_{6} ) q^{7} + ( -1 + \beta_{1} + \beta_{3} + \beta_{4} ) q^{9} +O(q^{10})$$ $$q + \beta_{1} q^{3} + \beta_{5} q^{5} + ( 1 + \beta_{6} ) q^{7} + ( -1 + \beta_{1} + \beta_{3} + \beta_{4} ) q^{9} + ( 1 - \beta_{1} + \beta_{3} + \beta_{4} + 2 \beta_{5} - \beta_{6} ) q^{11} + ( -1 + 2 \beta_{1} + \beta_{2} - \beta_{5} + \beta_{6} ) q^{13} + ( 1 - \beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} + \beta_{5} - \beta_{6} ) q^{15} + ( -2 + 2 \beta_{1} - \beta_{2} - \beta_{3} - \beta_{5} + \beta_{6} ) q^{17} + ( 2 + \beta_{2} + \beta_{4} ) q^{19} + ( -1 + \beta_{1} + \beta_{2} + \beta_{3} ) q^{21} + ( 1 + \beta_{1} - 2 \beta_{2} - 2 \beta_{3} - \beta_{4} + \beta_{6} ) q^{23} + ( -1 - \beta_{3} ) q^{25} + ( 1 + \beta_{2} + \beta_{3} - \beta_{5} + \beta_{6} ) q^{27} + ( -\beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} + 2 \beta_{5} - 2 \beta_{6} ) q^{29} + ( 3 + \beta_{1} - \beta_{2} - \beta_{3} - 2 \beta_{4} - \beta_{5} ) q^{31} + ( \beta_{1} + 2 \beta_{2} + \beta_{5} - \beta_{6} ) q^{33} + ( 2 - \beta_{2} - \beta_{3} - 2 \beta_{4} + 2 \beta_{6} ) q^{35} + ( -\beta_{1} - 3 \beta_{2} + \beta_{4} + \beta_{5} + \beta_{6} ) q^{37} + ( 2 + \beta_{2} + 4 \beta_{3} + 2 \beta_{4} + \beta_{5} ) q^{39} + ( -3 \beta_{1} + \beta_{2} + 3 \beta_{3} + 4 \beta_{5} - 2 \beta_{6} ) q^{41} + ( 3 + \beta_{1} - \beta_{2} - 2 \beta_{3} - \beta_{5} ) q^{43} + ( -1 + 2 \beta_{2} + \beta_{3} - \beta_{5} - \beta_{6} ) q^{45} + ( 2 \beta_{1} + 3 \beta_{2} + \beta_{3} + 3 \beta_{4} + \beta_{5} - \beta_{6} ) q^{47} + ( -\beta_{1} - \beta_{2} - \beta_{3} - 2 \beta_{4} + \beta_{5} + 3 \beta_{6} ) q^{49} + ( 2 + \beta_{1} - 2 \beta_{2} - \beta_{3} - 2 \beta_{5} + \beta_{6} ) q^{51} + ( \beta_{1} - \beta_{2} - 2 \beta_{3} - \beta_{4} - \beta_{5} + \beta_{6} ) q^{53} + ( 3 + 2 \beta_{1} + \beta_{2} - 2 \beta_{3} - \beta_{6} ) q^{55} + ( -1 + \beta_{1} + \beta_{2} + \beta_{3} + 2 \beta_{5} - \beta_{6} ) q^{57} + ( 2 - \beta_{1} - \beta_{2} + \beta_{3} - 3 \beta_{4} + 2 \beta_{5} ) q^{59} + ( 2 \beta_{1} - 2 \beta_{2} - 5 \beta_{3} - 3 \beta_{5} + \beta_{6} ) q^{61} + ( -1 + 2 \beta_{2} + 4 \beta_{3} + 2 \beta_{4} + \beta_{5} - 3 \beta_{6} ) q^{63} + ( 1 - 2 \beta_{1} - \beta_{2} + 2 \beta_{3} + \beta_{5} ) q^{65} + ( 4 + \beta_{1} - \beta_{2} - \beta_{3} - 3 \beta_{4} - 3 \beta_{5} + 2 \beta_{6} ) q^{67} + ( 2 + \beta_{1} - 3 \beta_{2} - 3 \beta_{3} - 2 \beta_{5} ) q^{69} + ( 3 - 2 \beta_{2} - \beta_{4} + 2 \beta_{5} - \beta_{6} ) q^{71} + ( 1 - 2 \beta_{1} + 2 \beta_{2} - 3 \beta_{4} - \beta_{5} + 2 \beta_{6} ) q^{73} + ( -3 \beta_{1} - \beta_{2} - \beta_{3} + \beta_{5} - \beta_{6} ) q^{75} + ( 1 - \beta_{1} - \beta_{2} + \beta_{3} + \beta_{6} ) q^{77} + ( 4 - \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} + \beta_{5} - 2 \beta_{6} ) q^{79} + ( 1 - \beta_{1} + 2 \beta_{2} - 3 \beta_{4} + \beta_{6} ) q^{81} + ( -2 + 3 \beta_{1} - \beta_{2} - 4 \beta_{3} - \beta_{4} - \beta_{5} + \beta_{6} ) q^{83} + ( 1 - 3 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} - \beta_{6} ) q^{85} + ( 1 + 2 \beta_{1} - 3 \beta_{3} - \beta_{4} - \beta_{5} ) q^{87} + ( 1 - \beta_{1} + 2 \beta_{2} - 3 \beta_{3} - \beta_{4} - \beta_{5} - 4 \beta_{6} ) q^{89} + ( 2 + 2 \beta_{1} + 2 \beta_{2} + \beta_{3} - \beta_{4} ) q^{91} + ( 3 + 3 \beta_{1} - 3 \beta_{2} - \beta_{3} + \beta_{4} - 2 \beta_{5} + \beta_{6} ) q^{93} + ( 1 - 2 \beta_{2} - \beta_{4} + 4 \beta_{5} - \beta_{6} ) q^{95} + ( -3 + 5 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} + \beta_{4} - 3 \beta_{5} + 2 \beta_{6} ) q^{97} + ( 1 - \beta_{1} + 2 \beta_{2} + 2 \beta_{3} + \beta_{4} - \beta_{5} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$7q + 3q^{3} - 2q^{5} + 6q^{7} + O(q^{10})$$ $$7q + 3q^{3} - 2q^{5} + 6q^{7} + 5q^{11} - q^{13} + 6q^{15} - 8q^{17} + 15q^{19} - 3q^{21} + 5q^{23} - 9q^{25} + 9q^{27} + 21q^{31} + 7q^{35} - q^{37} + 23q^{39} - 10q^{41} + 23q^{43} - 4q^{45} + 10q^{47} - 13q^{49} + 20q^{51} - q^{53} + 23q^{55} - 6q^{57} + 4q^{59} + 3q^{61} + 4q^{63} + 4q^{65} + 28q^{67} + 18q^{69} + 18q^{71} - 7q^{73} - 11q^{75} + 6q^{77} + 30q^{79} - 5q^{81} - 13q^{83} + q^{85} + 7q^{87} + 18q^{91} + 36q^{93} - 2q^{97} + 10q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{7} - 3 x^{6} - 6 x^{5} + 18 x^{4} + 8 x^{3} - 17 x^{2} - 9 x - 1$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{6} - 2 \nu^{5} - 8 \nu^{4} + 13 \nu^{3} + 15 \nu^{2} - 17 \nu - 5$$$$)/3$$ $$\beta_{3}$$ $$=$$ $$-\nu^{6} + 3 \nu^{5} + 6 \nu^{4} - 18 \nu^{3} - 7 \nu^{2} + 17 \nu + 5$$ $$\beta_{4}$$ $$=$$ $$\nu^{6} - 3 \nu^{5} - 6 \nu^{4} + 18 \nu^{3} + 8 \nu^{2} - 18 \nu - 7$$ $$\beta_{5}$$ $$=$$ $$($$$$5 \nu^{6} - 16 \nu^{5} - 25 \nu^{4} + 92 \nu^{3} + 9 \nu^{2} - 73 \nu - 16$$$$)/3$$ $$\beta_{6}$$ $$=$$ $$($$$$7 \nu^{6} - 23 \nu^{5} - 35 \nu^{4} + 136 \nu^{3} + 15 \nu^{2} - 125 \nu - 29$$$$)/3$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{4} + \beta_{3} + \beta_{1} + 2$$ $$\nu^{3}$$ $$=$$ $$\beta_{6} - \beta_{5} + \beta_{3} + \beta_{2} + 6 \beta_{1} + 1$$ $$\nu^{4}$$ $$=$$ $$\beta_{6} + 6 \beta_{4} + 9 \beta_{3} + 2 \beta_{2} + 8 \beta_{1} + 10$$ $$\nu^{5}$$ $$=$$ $$7 \beta_{6} - 5 \beta_{5} + 4 \beta_{4} + 16 \beta_{3} + 12 \beta_{2} + 38 \beta_{1} + 9$$ $$\nu^{6}$$ $$=$$ $$9 \beta_{6} + 3 \beta_{5} + 41 \beta_{4} + 76 \beta_{3} + 30 \beta_{2} + 64 \beta_{1} + 60$$

## 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
 −2.18229 −0.844838 −0.358013 −0.164919 1.40474 2.29157 2.85375
0 −2.18229 0 −1.66714 0 1.81734 0 1.76240 0
1.2 0 −0.844838 0 −2.47034 0 0.972158 0 −2.28625 0
1.3 0 −0.358013 0 2.25358 0 4.66949 0 −2.87183 0
1.4 0 −0.164919 0 −1.38175 0 −1.87004 0 −2.97280 0
1.5 0 1.40474 0 2.59588 0 −1.10443 0 −1.02671 0
1.6 0 2.29157 0 −1.78468 0 −0.671050 0 2.25130 0
1.7 0 2.85375 0 0.454452 0 2.18653 0 5.14389 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

## 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 4016.2.a.g 7
4.b odd 2 1 1004.2.a.a 7
12.b even 2 1 9036.2.a.i 7

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1004.2.a.a 7 4.b odd 2 1
4016.2.a.g 7 1.a even 1 1 trivial
9036.2.a.i 7 12.b even 2 1

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$251$$ $$1$$

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{7} - 3 T_{3}^{6} - 6 T_{3}^{5} + 18 T_{3}^{4} + 8 T_{3}^{3} - 17 T_{3}^{2} - 9 T_{3} - 1$$ acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(4016))$$.

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ 
$3$ $$1 - 3 T + 15 T^{2} - 36 T^{3} + 107 T^{4} - 206 T^{5} + 468 T^{6} - 751 T^{7} + 1404 T^{8} - 1854 T^{9} + 2889 T^{10} - 2916 T^{11} + 3645 T^{12} - 2187 T^{13} + 2187 T^{14}$$
$5$ $$1 + 2 T + 24 T^{2} + 35 T^{3} + 276 T^{4} + 331 T^{5} + 2035 T^{6} + 2033 T^{7} + 10175 T^{8} + 8275 T^{9} + 34500 T^{10} + 21875 T^{11} + 75000 T^{12} + 31250 T^{13} + 78125 T^{14}$$
$7$ $$1 - 6 T + 49 T^{2} - 217 T^{3} + 1012 T^{4} - 3486 T^{5} + 11665 T^{6} - 31629 T^{7} + 81655 T^{8} - 170814 T^{9} + 347116 T^{10} - 521017 T^{11} + 823543 T^{12} - 705894 T^{13} + 823543 T^{14}$$
$11$ $$1 - 5 T + 57 T^{2} - 246 T^{3} + 1548 T^{4} - 5621 T^{5} + 25904 T^{6} - 77368 T^{7} + 284944 T^{8} - 680141 T^{9} + 2060388 T^{10} - 3601686 T^{11} + 9179907 T^{12} - 8857805 T^{13} + 19487171 T^{14}$$
$13$ $$1 + T + 49 T^{2} + 10 T^{3} + 1145 T^{4} - 534 T^{5} + 18318 T^{6} - 12855 T^{7} + 238134 T^{8} - 90246 T^{9} + 2515565 T^{10} + 285610 T^{11} + 18193357 T^{12} + 4826809 T^{13} + 62748517 T^{14}$$
$17$ $$1 + 8 T + 112 T^{2} + 599 T^{3} + 4772 T^{4} + 19059 T^{5} + 115579 T^{6} + 380413 T^{7} + 1964843 T^{8} + 5508051 T^{9} + 23444836 T^{10} + 50029079 T^{11} + 159023984 T^{12} + 193100552 T^{13} + 410338673 T^{14}$$
$19$ $$1 - 15 T + 198 T^{2} - 1705 T^{3} + 13153 T^{4} - 79977 T^{5} + 441300 T^{6} - 2015782 T^{7} + 8384700 T^{8} - 28871697 T^{9} + 90216427 T^{10} - 222197305 T^{11} + 490267602 T^{12} - 705688215 T^{13} + 893871739 T^{14}$$
$23$ $$1 - 5 T + 114 T^{2} - 562 T^{3} + 6547 T^{4} - 28211 T^{5} + 231026 T^{6} - 828297 T^{7} + 5313598 T^{8} - 14923619 T^{9} + 79657349 T^{10} - 157270642 T^{11} + 733743102 T^{12} - 740179445 T^{13} + 3404825447 T^{14}$$
$29$ $$1 + 132 T^{2} + 90 T^{3} + 8557 T^{4} + 8078 T^{5} + 357562 T^{6} + 322880 T^{7} + 10369298 T^{8} + 6793598 T^{9} + 208696673 T^{10} + 63655290 T^{11} + 2707471668 T^{12} + 17249876309 T^{14}$$
$31$ $$1 - 21 T + 323 T^{2} - 3712 T^{3} + 34723 T^{4} - 275114 T^{5} + 1883364 T^{6} - 11173393 T^{7} + 58384284 T^{8} - 264384554 T^{9} + 1034432893 T^{10} - 3428109952 T^{11} + 9247215773 T^{12} - 18637577301 T^{13} + 27512614111 T^{14}$$
$37$ $$1 + T + 107 T^{2} - 80 T^{3} + 5714 T^{4} - 14045 T^{5} + 214630 T^{6} - 801168 T^{7} + 7941310 T^{8} - 19227605 T^{9} + 289431242 T^{10} - 149932880 T^{11} + 7419803399 T^{12} + 2565726409 T^{13} + 94931877133 T^{14}$$
$41$ $$1 + 10 T + 183 T^{2} + 1925 T^{3} + 18522 T^{4} + 156974 T^{5} + 1223611 T^{6} + 7778621 T^{7} + 50168051 T^{8} + 263873294 T^{9} + 1276554762 T^{10} + 5439589925 T^{11} + 21201684783 T^{12} + 47501042410 T^{13} + 194754273881 T^{14}$$
$43$ $$1 - 23 T + 479 T^{2} - 6374 T^{3} + 76418 T^{4} - 709137 T^{5} + 5980176 T^{6} - 41069820 T^{7} + 257147568 T^{8} - 1311194313 T^{9} + 6075765926 T^{10} - 21791437574 T^{11} + 70417044197 T^{12} - 145391350127 T^{13} + 271818611107 T^{14}$$
$47$ $$1 - 10 T + 175 T^{2} - 1405 T^{3} + 16332 T^{4} - 102894 T^{5} + 981162 T^{6} - 5586582 T^{7} + 46114614 T^{8} - 227292846 T^{9} + 1695637236 T^{10} - 6855951805 T^{11} + 40135376225 T^{12} - 107792153290 T^{13} + 506623120463 T^{14}$$
$53$ $$1 + T + 341 T^{2} + 296 T^{3} + 51270 T^{4} + 37529 T^{5} + 4404456 T^{6} + 2612924 T^{7} + 233436168 T^{8} + 105418961 T^{9} + 7632923790 T^{10} + 2335582376 T^{11} + 142604663113 T^{12} + 22164361129 T^{13} + 1174711139837 T^{14}$$
$59$ $$1 - 4 T + 192 T^{2} - 766 T^{3} + 17341 T^{4} - 80260 T^{5} + 1108838 T^{6} - 5770956 T^{7} + 65421442 T^{8} - 279385060 T^{9} + 3561477239 T^{10} - 9281898526 T^{11} + 137265465408 T^{12} - 168722134564 T^{13} + 2488651484819 T^{14}$$
$61$ $$1 - 3 T + 162 T^{2} + 61 T^{3} + 11799 T^{4} + 53561 T^{5} + 692826 T^{6} + 4721098 T^{7} + 42262386 T^{8} + 199300481 T^{9} + 2678148819 T^{10} + 844596301 T^{11} + 136824600762 T^{12} - 154561123083 T^{13} + 3142742836021 T^{14}$$
$67$ $$1 - 28 T + 648 T^{2} - 9931 T^{3} + 135284 T^{4} - 1472821 T^{5} + 14791793 T^{6} - 125329735 T^{7} + 991050131 T^{8} - 6611493469 T^{9} + 40688421692 T^{10} - 200120782651 T^{11} + 874881069336 T^{12} - 2532834700732 T^{13} + 6060711605323 T^{14}$$
$71$ $$1 - 18 T + 509 T^{2} - 6143 T^{3} + 99682 T^{4} - 906794 T^{5} + 10907742 T^{6} - 79739874 T^{7} + 774449682 T^{8} - 4571148554 T^{9} + 35677284302 T^{10} - 156103956383 T^{11} + 918352739659 T^{12} - 2305805110578 T^{13} + 9095120158391 T^{14}$$
$73$ $$1 + 7 T + 277 T^{2} + 2034 T^{3} + 44471 T^{4} + 290456 T^{5} + 4664818 T^{6} + 26135103 T^{7} + 340531714 T^{8} + 1547840024 T^{9} + 17299975007 T^{10} + 57762022194 T^{11} + 574240831261 T^{12} + 1059339584023 T^{13} + 11047398519097 T^{14}$$
$79$ $$1 - 30 T + 864 T^{2} - 15277 T^{3} + 251870 T^{4} - 3126597 T^{5} + 36178295 T^{6} - 332923983 T^{7} + 2858085305 T^{8} - 19513091877 T^{9} + 124181732930 T^{10} - 595040387437 T^{11} + 2658576728736 T^{12} - 7292623665630 T^{13} + 19203908986159 T^{14}$$
$83$ $$1 + 13 T + 485 T^{2} + 5360 T^{3} + 109330 T^{4} + 996823 T^{5} + 14416686 T^{6} + 106582888 T^{7} + 1196584938 T^{8} + 6867113647 T^{9} + 62513472710 T^{10} + 254376600560 T^{11} + 1910434711855 T^{12} + 4250224853797 T^{13} + 27136050989627 T^{14}$$
$89$ $$1 + 109 T^{2} - 211 T^{3} + 18778 T^{4} + 41946 T^{5} + 1597555 T^{6} + 597701 T^{7} + 142182395 T^{8} + 332254266 T^{9} + 13237907882 T^{10} - 13238612851 T^{11} + 608662479941 T^{12} + 44231334895529 T^{14}$$
$97$ $$1 + 2 T + 455 T^{2} - 86 T^{3} + 89144 T^{4} - 192034 T^{5} + 10939992 T^{6} - 31982204 T^{7} + 1061179224 T^{8} - 1806847906 T^{9} + 81359321912 T^{10} - 7613518166 T^{11} + 3907239816935 T^{12} + 1665944009858 T^{13} + 80798284478113 T^{14}$$