# Properties

 Label 4018.2.a.bq Level 4018 Weight 2 Character orbit 4018.a Self dual yes Analytic conductor 32.084 Analytic rank 0 Dimension 6 CM no Inner twists 1

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$4018 = 2 \cdot 7^{2} \cdot 41$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4018.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$32.0838915322$$ Analytic rank: $$0$$ Dimension: $$6$$ Coefficient field: 6.6.5163008.1 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{5}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + q^{2} + ( 1 + \beta_{2} + \beta_{3} - \beta_{4} ) q^{3} + q^{4} + ( 2 + \beta_{4} ) q^{5} + ( 1 + \beta_{2} + \beta_{3} - \beta_{4} ) q^{6} + q^{8} + ( 4 - 2 \beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} - \beta_{5} ) q^{9} +O(q^{10})$$ $$q + q^{2} + ( 1 + \beta_{2} + \beta_{3} - \beta_{4} ) q^{3} + q^{4} + ( 2 + \beta_{4} ) q^{5} + ( 1 + \beta_{2} + \beta_{3} - \beta_{4} ) q^{6} + q^{8} + ( 4 - 2 \beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} - \beta_{5} ) q^{9} + ( 2 + \beta_{4} ) q^{10} + ( -1 + 2 \beta_{1} - \beta_{2} ) q^{11} + ( 1 + \beta_{2} + \beta_{3} - \beta_{4} ) q^{12} + ( 1 - \beta_{2} + \beta_{3} - \beta_{4} ) q^{13} + ( \beta_{1} + 3 \beta_{2} + 4 \beta_{3} ) q^{15} + q^{16} + ( 3 - \beta_{1} + \beta_{4} - 2 \beta_{5} ) q^{17} + ( 4 - 2 \beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} - \beta_{5} ) q^{18} + ( 1 + 2 \beta_{1} - \beta_{2} - \beta_{3} + \beta_{4} ) q^{19} + ( 2 + \beta_{4} ) q^{20} + ( -1 + 2 \beta_{1} - \beta_{2} ) q^{22} + ( 2 + \beta_{1} + \beta_{2} - 2 \beta_{4} + \beta_{5} ) q^{23} + ( 1 + \beta_{2} + \beta_{3} - \beta_{4} ) q^{24} + ( 3 - \beta_{1} + \beta_{2} + 2 \beta_{4} - \beta_{5} ) q^{25} + ( 1 - \beta_{2} + \beta_{3} - \beta_{4} ) q^{26} + ( 6 + 4 \beta_{2} + 4 \beta_{3} - 2 \beta_{4} - 4 \beta_{5} ) q^{27} + ( \beta_{1} + \beta_{2} + 2 \beta_{3} + \beta_{4} + \beta_{5} ) q^{29} + ( \beta_{1} + 3 \beta_{2} + 4 \beta_{3} ) q^{30} + ( -1 - \beta_{1} - 2 \beta_{2} - 3 \beta_{3} - \beta_{4} + 3 \beta_{5} ) q^{31} + q^{32} + ( -4 - 2 \beta_{1} - 4 \beta_{2} - 7 \beta_{3} + 4 \beta_{5} ) q^{33} + ( 3 - \beta_{1} + \beta_{4} - 2 \beta_{5} ) q^{34} + ( 4 - 2 \beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} - \beta_{5} ) q^{36} + ( -4 - 2 \beta_{1} - 2 \beta_{2} + 2 \beta_{5} ) q^{37} + ( 1 + 2 \beta_{1} - \beta_{2} - \beta_{3} + \beta_{4} ) q^{38} + ( 1 - 2 \beta_{1} - \beta_{2} - 3 \beta_{3} - 3 \beta_{4} + 3 \beta_{5} ) q^{39} + ( 2 + \beta_{4} ) q^{40} - q^{41} + ( 1 + \beta_{2} - 3 \beta_{3} - \beta_{4} - \beta_{5} ) q^{43} + ( -1 + 2 \beta_{1} - \beta_{2} ) q^{44} + ( 7 - 5 \beta_{1} + 2 \beta_{2} + 4 \beta_{3} + 4 \beta_{4} - \beta_{5} ) q^{45} + ( 2 + \beta_{1} + \beta_{2} - 2 \beta_{4} + \beta_{5} ) q^{46} + ( -3 + \beta_{1} - 4 \beta_{3} - \beta_{4} ) q^{47} + ( 1 + \beta_{2} + \beta_{3} - \beta_{4} ) q^{48} + ( 3 - \beta_{1} + \beta_{2} + 2 \beta_{4} - \beta_{5} ) q^{50} + ( -1 + 2 \beta_{1} + 7 \beta_{2} + 5 \beta_{3} - 3 \beta_{4} - 3 \beta_{5} ) q^{51} + ( 1 - \beta_{2} + \beta_{3} - \beta_{4} ) q^{52} + ( -3 - 3 \beta_{1} - 4 \beta_{2} - 4 \beta_{3} + 2 \beta_{4} + 2 \beta_{5} ) q^{53} + ( 6 + 4 \beta_{2} + 4 \beta_{3} - 2 \beta_{4} - 4 \beta_{5} ) q^{54} + ( -5 + 6 \beta_{1} - 3 \beta_{2} - \beta_{3} - \beta_{4} ) q^{55} + ( -5 - \beta_{2} - 3 \beta_{3} - \beta_{4} + 3 \beta_{5} ) q^{57} + ( \beta_{1} + \beta_{2} + 2 \beta_{3} + \beta_{4} + \beta_{5} ) q^{58} + ( 2 - 2 \beta_{2} - 6 \beta_{3} + \beta_{4} + 6 \beta_{5} ) q^{59} + ( \beta_{1} + 3 \beta_{2} + 4 \beta_{3} ) q^{60} + ( 5 - \beta_{1} - 2 \beta_{2} - \beta_{3} + 2 \beta_{4} + \beta_{5} ) q^{61} + ( -1 - \beta_{1} - 2 \beta_{2} - 3 \beta_{3} - \beta_{4} + 3 \beta_{5} ) q^{62} + q^{64} + ( -2 + \beta_{1} - 3 \beta_{2} + 2 \beta_{3} ) q^{65} + ( -4 - 2 \beta_{1} - 4 \beta_{2} - 7 \beta_{3} + 4 \beta_{5} ) q^{66} + ( -3 + 3 \beta_{1} + 4 \beta_{2} + 2 \beta_{4} - 2 \beta_{5} ) q^{67} + ( 3 - \beta_{1} + \beta_{4} - 2 \beta_{5} ) q^{68} + ( 10 - 3 \beta_{1} - \beta_{2} - \beta_{3} - 4 \beta_{4} ) q^{69} + ( 3 - 3 \beta_{1} + 2 \beta_{3} - \beta_{4} - 3 \beta_{5} ) q^{71} + ( 4 - 2 \beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} - \beta_{5} ) q^{72} + ( 3 - \beta_{1} - \beta_{3} + \beta_{4} + \beta_{5} ) q^{73} + ( -4 - 2 \beta_{1} - 2 \beta_{2} + 2 \beta_{5} ) q^{74} + ( 1 + 3 \beta_{1} + 8 \beta_{2} + 10 \beta_{3} + \beta_{4} - 4 \beta_{5} ) q^{75} + ( 1 + 2 \beta_{1} - \beta_{2} - \beta_{3} + \beta_{4} ) q^{76} + ( 1 - 2 \beta_{1} - \beta_{2} - 3 \beta_{3} - 3 \beta_{4} + 3 \beta_{5} ) q^{78} + ( 2 \beta_{3} - 2 \beta_{4} + 2 \beta_{5} ) q^{79} + ( 2 + \beta_{4} ) q^{80} + ( 10 + 9 \beta_{2} + 3 \beta_{3} - 3 \beta_{4} - 5 \beta_{5} ) q^{81} - q^{82} + ( 4 + 4 \beta_{1} + 2 \beta_{3} - 3 \beta_{4} - 2 \beta_{5} ) q^{83} + ( 9 - 4 \beta_{1} - \beta_{2} + 5 \beta_{4} - 3 \beta_{5} ) q^{85} + ( 1 + \beta_{2} - 3 \beta_{3} - \beta_{4} - \beta_{5} ) q^{86} + ( 4 - 2 \beta_{1} + 3 \beta_{3} + 6 \beta_{4} + 2 \beta_{5} ) q^{87} + ( -1 + 2 \beta_{1} - \beta_{2} ) q^{88} + ( 2 - 3 \beta_{1} - \beta_{2} - \beta_{3} + 2 \beta_{4} - 2 \beta_{5} ) q^{89} + ( 7 - 5 \beta_{1} + 2 \beta_{2} + 4 \beta_{3} + 4 \beta_{4} - \beta_{5} ) q^{90} + ( 2 + \beta_{1} + \beta_{2} - 2 \beta_{4} + \beta_{5} ) q^{92} + ( -5 + 3 \beta_{1} - 6 \beta_{2} - 2 \beta_{3} - 3 \beta_{4} + 3 \beta_{5} ) q^{93} + ( -3 + \beta_{1} - 4 \beta_{3} - \beta_{4} ) q^{94} + ( 2 + 5 \beta_{1} - 3 \beta_{2} - 4 \beta_{3} + 2 \beta_{4} ) q^{95} + ( 1 + \beta_{2} + \beta_{3} - \beta_{4} ) q^{96} + ( 8 \beta_{1} + 4 \beta_{2} + 4 \beta_{3} + 2 \beta_{4} - 4 \beta_{5} ) q^{97} + ( -16 + 3 \beta_{1} - 7 \beta_{2} - 4 \beta_{3} - 3 \beta_{4} + 3 \beta_{5} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6q + 6q^{2} + 4q^{3} + 6q^{4} + 12q^{5} + 4q^{6} + 6q^{8} + 18q^{9} + O(q^{10})$$ $$6q + 6q^{2} + 4q^{3} + 6q^{4} + 12q^{5} + 4q^{6} + 6q^{8} + 18q^{9} + 12q^{10} + 4q^{12} + 8q^{13} - 4q^{15} + 6q^{16} + 16q^{17} + 18q^{18} + 12q^{19} + 12q^{20} + 12q^{23} + 4q^{24} + 14q^{25} + 8q^{26} + 28q^{27} - 4q^{30} - 4q^{31} + 6q^{32} - 20q^{33} + 16q^{34} + 18q^{36} - 24q^{37} + 12q^{38} + 4q^{39} + 12q^{40} - 6q^{41} + 4q^{43} + 28q^{45} + 12q^{46} - 16q^{47} + 4q^{48} + 14q^{50} - 16q^{51} + 8q^{52} - 16q^{53} + 28q^{54} - 12q^{55} - 28q^{57} + 16q^{59} - 4q^{60} + 32q^{61} - 4q^{62} + 6q^{64} - 4q^{65} - 20q^{66} - 20q^{67} + 16q^{68} + 56q^{69} + 12q^{71} + 18q^{72} + 16q^{73} - 24q^{74} - 4q^{75} + 12q^{76} + 4q^{78} + 12q^{80} + 42q^{81} - 6q^{82} + 32q^{83} + 48q^{85} + 4q^{86} + 20q^{87} + 8q^{89} + 28q^{90} + 12q^{92} - 12q^{93} - 16q^{94} + 28q^{95} + 4q^{96} + 8q^{97} - 76q^{99} + O(q^{100})$$

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

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{3} - \nu^{2} - 4 \nu + 1$$ $$\beta_{3}$$ $$=$$ $$\nu^{5} - 2 \nu^{4} - 5 \nu^{3} + 7 \nu^{2} + 6 \nu - 3$$ $$\beta_{4}$$ $$=$$ $$\nu^{5} - 2 \nu^{4} - 5 \nu^{3} + 8 \nu^{2} + 5 \nu - 5$$ $$\beta_{5}$$ $$=$$ $$2 \nu^{5} - 3 \nu^{4} - 11 \nu^{3} + 10 \nu^{2} + 13 \nu - 5$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{4} - \beta_{3} + \beta_{1} + 2$$ $$\nu^{3}$$ $$=$$ $$\beta_{4} - \beta_{3} + \beta_{2} + 5 \beta_{1} + 1$$ $$\nu^{4}$$ $$=$$ $$\beta_{5} + 5 \beta_{4} - 7 \beta_{3} + \beta_{2} + 8 \beta_{1} + 8$$ $$\nu^{5}$$ $$=$$ $$2 \beta_{5} + 8 \beta_{4} - 11 \beta_{3} + 7 \beta_{2} + 28 \beta_{1} + 10$$

## 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
 −1.78566 1.60043 2.56754 0.545336 0.218114 −1.14577
1.00000 −2.71387 1.00000 3.56002 −2.71387 0 1.00000 4.36512 3.56002
1.2 1.00000 −1.82475 1.00000 2.37517 −1.82475 0 1.00000 0.329701 2.37517
1.3 1.00000 0.0387751 1.00000 2.61052 0.0387751 0 1.00000 −2.99850 2.61052
1.4 1.00000 1.93139 1.00000 1.16627 1.93139 0 1.00000 0.730254 1.16627
1.5 1.00000 3.26089 1.00000 −1.58475 3.26089 0 1.00000 7.63338 −1.58475
1.6 1.00000 3.30757 1.00000 3.87277 3.30757 0 1.00000 7.94005 3.87277
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.6 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 4018.2.a.bq yes 6
7.b odd 2 1 4018.2.a.bp 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4018.2.a.bp 6 7.b odd 2 1
4018.2.a.bq yes 6 1.a even 1 1 trivial

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$7$$ $$1$$
$$41$$ $$1$$

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(4018))$$:

 $$T_{3}^{6} - 4 T_{3}^{5} - 10 T_{3}^{4} + 44 T_{3}^{3} + 20 T_{3}^{2} - 104 T_{3} + 4$$ $$T_{5}^{6} - 12 T_{5}^{5} + 50 T_{5}^{4} - 68 T_{5}^{3} - 68 T_{5}^{2} + 248 T_{5} - 158$$ $$T_{11}^{6} - 50 T_{11}^{4} + 8 T_{11}^{3} + 498 T_{11}^{2} - 300 T_{11} - 398$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 - T )^{6}$$
$3$ $$1 - 4 T + 8 T^{2} - 16 T^{3} + 35 T^{4} - 68 T^{5} + 124 T^{6} - 204 T^{7} + 315 T^{8} - 432 T^{9} + 648 T^{10} - 972 T^{11} + 729 T^{12}$$
$5$ $$1 - 12 T + 80 T^{2} - 368 T^{3} + 1307 T^{4} - 3772 T^{5} + 9162 T^{6} - 18860 T^{7} + 32675 T^{8} - 46000 T^{9} + 50000 T^{10} - 37500 T^{11} + 15625 T^{12}$$
$7$ 1
$11$ $$1 + 16 T^{2} + 8 T^{3} + 113 T^{4} - 36 T^{5} + 878 T^{6} - 396 T^{7} + 13673 T^{8} + 10648 T^{9} + 234256 T^{10} + 1771561 T^{12}$$
$13$ $$1 - 8 T + 72 T^{2} - 404 T^{3} + 2275 T^{4} - 9484 T^{5} + 38732 T^{6} - 123292 T^{7} + 384475 T^{8} - 887588 T^{9} + 2056392 T^{10} - 2970344 T^{11} + 4826809 T^{12}$$
$17$ $$1 - 16 T + 156 T^{2} - 1092 T^{3} + 6311 T^{4} - 31084 T^{5} + 136444 T^{6} - 528428 T^{7} + 1823879 T^{8} - 5364996 T^{9} + 13029276 T^{10} - 22717712 T^{11} + 24137569 T^{12}$$
$19$ $$1 - 12 T + 128 T^{2} - 952 T^{3} + 6203 T^{4} - 32708 T^{5} + 157212 T^{6} - 621452 T^{7} + 2239283 T^{8} - 6529768 T^{9} + 16681088 T^{10} - 29713188 T^{11} + 47045881 T^{12}$$
$23$ $$1 - 12 T + 134 T^{2} - 992 T^{3} + 7183 T^{4} - 40036 T^{5} + 214808 T^{6} - 920828 T^{7} + 3799807 T^{8} - 12069664 T^{9} + 37498694 T^{10} - 77236116 T^{11} + 148035889 T^{12}$$
$29$ $$1 + 96 T^{2} + 48 T^{3} + 4853 T^{4} + 3300 T^{5} + 165406 T^{6} + 95700 T^{7} + 4081373 T^{8} + 1170672 T^{9} + 67898976 T^{10} + 594823321 T^{12}$$
$31$ $$1 + 4 T + 96 T^{2} + 492 T^{3} + 5139 T^{4} + 23896 T^{5} + 194656 T^{6} + 740776 T^{7} + 4938579 T^{8} + 14657172 T^{9} + 88658016 T^{10} + 114516604 T^{11} + 887503681 T^{12}$$
$37$ $$1 + 24 T + 398 T^{2} + 4600 T^{3} + 43895 T^{4} + 339152 T^{5} + 2260068 T^{6} + 12548624 T^{7} + 60092255 T^{8} + 233003800 T^{9} + 745916078 T^{10} + 1664254968 T^{11} + 2565726409 T^{12}$$
$41$ $$( 1 + T )^{6}$$
$43$ $$1 - 4 T + 150 T^{2} - 708 T^{3} + 11959 T^{4} - 55888 T^{5} + 614084 T^{6} - 2403184 T^{7} + 22112191 T^{8} - 56290956 T^{9} + 512820150 T^{10} - 588033772 T^{11} + 6321363049 T^{12}$$
$47$ $$1 + 16 T + 256 T^{2} + 2652 T^{3} + 27239 T^{4} + 215852 T^{5} + 1650068 T^{6} + 10145044 T^{7} + 60170951 T^{8} + 275338596 T^{9} + 1249198336 T^{10} + 3669520112 T^{11} + 10779215329 T^{12}$$
$53$ $$1 + 16 T + 234 T^{2} + 1596 T^{3} + 10025 T^{4} + 6328 T^{5} + 37970 T^{6} + 335384 T^{7} + 28160225 T^{8} + 237607692 T^{9} + 1846372554 T^{10} + 6691127888 T^{11} + 22164361129 T^{12}$$
$59$ $$1 - 16 T + 120 T^{2} - 1140 T^{3} + 13555 T^{4} - 89804 T^{5} + 500490 T^{6} - 5298436 T^{7} + 47184955 T^{8} - 234132060 T^{9} + 1454083320 T^{10} - 11438788784 T^{11} + 42180533641 T^{12}$$
$61$ $$1 - 32 T + 726 T^{2} - 11300 T^{3} + 143695 T^{4} - 1456668 T^{5} + 12552758 T^{6} - 88856748 T^{7} + 534689095 T^{8} - 2564885300 T^{9} + 10052080566 T^{10} - 27027081632 T^{11} + 51520374361 T^{12}$$
$67$ $$1 + 20 T + 362 T^{2} + 4304 T^{3} + 50513 T^{4} + 485936 T^{5} + 4399546 T^{6} + 32557712 T^{7} + 226752857 T^{8} + 1294483952 T^{9} + 7294705802 T^{10} + 27002502140 T^{11} + 90458382169 T^{12}$$
$71$ $$1 - 12 T + 334 T^{2} - 2596 T^{3} + 45983 T^{4} - 276920 T^{5} + 3969444 T^{6} - 19661320 T^{7} + 231800303 T^{8} - 929136956 T^{9} + 8487501454 T^{10} - 21650752212 T^{11} + 128100283921 T^{12}$$
$73$ $$1 - 16 T + 512 T^{2} - 5856 T^{3} + 101027 T^{4} - 855728 T^{5} + 10072096 T^{6} - 62468144 T^{7} + 538372883 T^{8} - 2278083552 T^{9} + 14539899392 T^{10} - 33169145488 T^{11} + 151334226289 T^{12}$$
$79$ $$1 + 386 T^{2} - 288 T^{3} + 65775 T^{4} - 67872 T^{5} + 6560348 T^{6} - 5361888 T^{7} + 410501775 T^{8} - 141995232 T^{9} + 15034731266 T^{10} + 243087455521 T^{12}$$
$83$ $$1 - 32 T + 632 T^{2} - 9580 T^{3} + 121787 T^{4} - 1345308 T^{5} + 13167754 T^{6} - 111660564 T^{7} + 838990643 T^{8} - 5477719460 T^{9} + 29993658872 T^{10} - 126049300576 T^{11} + 326940373369 T^{12}$$
$89$ $$1 - 8 T + 406 T^{2} - 3056 T^{3} + 76583 T^{4} - 508264 T^{5} + 8599604 T^{6} - 45235496 T^{7} + 606613943 T^{8} - 2154385264 T^{9} + 25473349846 T^{10} - 44672475592 T^{11} + 496981290961 T^{12}$$
$97$ $$1 - 8 T + 142 T^{2} - 584 T^{3} + 14639 T^{4} - 14000 T^{5} + 886340 T^{6} - 1358000 T^{7} + 137738351 T^{8} - 533001032 T^{9} + 12571157902 T^{10} - 68698722056 T^{11} + 832972004929 T^{12}$$