# Properties

 Label 4018.2.a.bn 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: 6.6.52046292.1 Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 574) 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.

 $$f(q)$$ $$=$$ $$q - q^{2} + \beta_{3} q^{3} + q^{4} + \beta_{4} q^{5} -\beta_{3} q^{6} - q^{8} + ( 1 - \beta_{3} - \beta_{4} + \beta_{5} ) q^{9} +O(q^{10})$$ $$q - q^{2} + \beta_{3} q^{3} + q^{4} + \beta_{4} q^{5} -\beta_{3} q^{6} - q^{8} + ( 1 - \beta_{3} - \beta_{4} + \beta_{5} ) q^{9} -\beta_{4} q^{10} + ( 1 + \beta_{2} + \beta_{5} ) q^{11} + \beta_{3} q^{12} + ( -1 - \beta_{1} - \beta_{2} + \beta_{4} ) q^{13} + ( 2 \beta_{1} - 2 \beta_{3} + \beta_{5} ) q^{15} + q^{16} + ( \beta_{3} - \beta_{5} ) q^{17} + ( -1 + \beta_{3} + \beta_{4} - \beta_{5} ) q^{18} + ( \beta_{3} + \beta_{4} - 2 \beta_{5} ) q^{19} + \beta_{4} q^{20} + ( -1 - \beta_{2} - \beta_{5} ) q^{22} + ( 3 - \beta_{2} - \beta_{4} ) q^{23} -\beta_{3} q^{24} + ( -\beta_{2} + \beta_{3} + \beta_{5} ) q^{25} + ( 1 + \beta_{1} + \beta_{2} - \beta_{4} ) q^{26} + ( -2 - 2 \beta_{1} + \beta_{3} + \beta_{4} - \beta_{5} ) q^{27} + ( 1 + \beta_{1} + \beta_{2} - \beta_{3} ) q^{29} + ( -2 \beta_{1} + 2 \beta_{3} - \beta_{5} ) q^{30} + ( \beta_{3} + \beta_{4} + \beta_{5} ) q^{31} - q^{32} + ( 3 + 2 \beta_{1} - \beta_{2} - \beta_{4} + 2 \beta_{5} ) q^{33} + ( -\beta_{3} + \beta_{5} ) q^{34} + ( 1 - \beta_{3} - \beta_{4} + \beta_{5} ) q^{36} + ( 1 + 3 \beta_{2} - \beta_{3} ) q^{37} + ( -\beta_{3} - \beta_{4} + 2 \beta_{5} ) q^{38} + ( -2 - \beta_{3} ) q^{39} -\beta_{4} q^{40} - q^{41} + ( 3 - 2 \beta_{1} + 2 \beta_{5} ) q^{43} + ( 1 + \beta_{2} + \beta_{5} ) q^{44} + ( -4 + 2 \beta_{2} + \beta_{4} - \beta_{5} ) q^{45} + ( -3 + \beta_{2} + \beta_{4} ) q^{46} + ( 3 - 2 \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} ) q^{47} + \beta_{3} q^{48} + ( \beta_{2} - \beta_{3} - \beta_{5} ) q^{50} + ( 2 - \beta_{3} - \beta_{4} ) q^{51} + ( -1 - \beta_{1} - \beta_{2} + \beta_{4} ) q^{52} + ( 3 - 2 \beta_{1} - \beta_{2} + 3 \beta_{3} + \beta_{4} + \beta_{5} ) q^{53} + ( 2 + 2 \beta_{1} - \beta_{3} - \beta_{4} + \beta_{5} ) q^{54} + ( 2 + 4 \beta_{1} + 2 \beta_{2} - 3 \beta_{3} - \beta_{4} + 3 \beta_{5} ) q^{55} + ( 2 \beta_{1} - 3 \beta_{3} - \beta_{4} ) q^{57} + ( -1 - \beta_{1} - \beta_{2} + \beta_{3} ) q^{58} + ( -4 + 2 \beta_{1} - 3 \beta_{2} - \beta_{3} + 2 \beta_{5} ) q^{59} + ( 2 \beta_{1} - 2 \beta_{3} + \beta_{5} ) q^{60} + ( 4 + 2 \beta_{3} + \beta_{5} ) q^{61} + ( -\beta_{3} - \beta_{4} - \beta_{5} ) q^{62} + q^{64} + ( 2 - 2 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} + \beta_{4} - 2 \beta_{5} ) q^{65} + ( -3 - 2 \beta_{1} + \beta_{2} + \beta_{4} - 2 \beta_{5} ) q^{66} + ( -1 + 3 \beta_{2} + 2 \beta_{3} - 2 \beta_{4} - 2 \beta_{5} ) q^{67} + ( \beta_{3} - \beta_{5} ) q^{68} + ( -1 - 4 \beta_{1} + \beta_{2} + 6 \beta_{3} + \beta_{4} - 2 \beta_{5} ) q^{69} + ( 2 - 5 \beta_{1} + 4 \beta_{3} + 2 \beta_{4} - 4 \beta_{5} ) q^{71} + ( -1 + \beta_{3} + \beta_{4} - \beta_{5} ) q^{72} + ( 1 + 2 \beta_{1} + 2 \beta_{2} + \beta_{3} + \beta_{5} ) q^{73} + ( -1 - 3 \beta_{2} + \beta_{3} ) q^{74} + ( 5 - 2 \beta_{1} + \beta_{2} + \beta_{5} ) q^{75} + ( \beta_{3} + \beta_{4} - 2 \beta_{5} ) q^{76} + ( 2 + \beta_{3} ) q^{78} + ( 2 + 2 \beta_{1} - 2 \beta_{2} + 3 \beta_{3} - \beta_{5} ) q^{79} + \beta_{4} q^{80} + ( -3 + 2 \beta_{1} - 2 \beta_{2} - 2 \beta_{5} ) q^{81} + q^{82} + ( -2 + 2 \beta_{1} + \beta_{2} - 2 \beta_{3} + 2 \beta_{5} ) q^{83} + ( -1 - \beta_{2} - \beta_{3} ) q^{85} + ( -3 + 2 \beta_{1} - 2 \beta_{5} ) q^{86} + ( -2 + 2 \beta_{1} + \beta_{4} ) q^{87} + ( -1 - \beta_{2} - \beta_{5} ) q^{88} + ( 4 - 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} + \beta_{4} ) q^{89} + ( 4 - 2 \beta_{2} - \beta_{4} + \beta_{5} ) q^{90} + ( 3 - \beta_{2} - \beta_{4} ) q^{92} + ( 6 + 2 \beta_{1} - 3 \beta_{3} - \beta_{4} + 3 \beta_{5} ) q^{93} + ( -3 + 2 \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} ) q^{94} + ( 3 - 2 \beta_{1} - 3 \beta_{2} + \beta_{3} ) q^{95} -\beta_{3} q^{96} + ( -1 + 2 \beta_{1} + \beta_{2} + \beta_{4} - 3 \beta_{5} ) q^{97} + ( 2 - 4 \beta_{1} + 4 \beta_{3} + 3 \beta_{4} - 3 \beta_{5} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6q - 6q^{2} - q^{3} + 6q^{4} + q^{6} - 6q^{8} + 5q^{9} + O(q^{10})$$ $$6q - 6q^{2} - q^{3} + 6q^{4} + q^{6} - 6q^{8} + 5q^{9} + q^{11} - q^{12} - 4q^{13} + 2q^{15} + 6q^{16} + q^{17} - 5q^{18} + 3q^{19} - q^{22} + 21q^{23} + q^{24} + 4q^{26} - 13q^{27} + 5q^{29} - 2q^{30} - 3q^{31} - 6q^{32} + 19q^{33} - q^{34} + 5q^{36} - 2q^{37} - 3q^{38} - 11q^{39} - 6q^{41} + 12q^{43} + q^{44} - 28q^{45} - 21q^{46} + 18q^{47} - q^{48} + 13q^{51} - 4q^{52} + 14q^{53} + 13q^{54} + 7q^{55} + 5q^{57} - 5q^{58} - 16q^{59} + 2q^{60} + 20q^{61} + 3q^{62} + 6q^{64} + 18q^{65} - 19q^{66} - 13q^{67} + q^{68} - 15q^{69} + 11q^{71} - 5q^{72} - q^{73} + 2q^{74} + 23q^{75} + 3q^{76} + 11q^{78} + 19q^{79} - 6q^{81} + 6q^{82} - 15q^{83} - 2q^{85} - 12q^{86} - 10q^{87} - q^{88} + 14q^{89} + 28q^{90} + 21q^{92} + 35q^{93} - 18q^{94} + 24q^{95} + q^{96} - q^{97} + 10q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} - x^{5} - 10 x^{4} + 9 x^{3} + 24 x^{2} - 18 x + 1$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} - 4$$ $$\beta_{3}$$ $$=$$ $$\nu^{3} - 5 \nu$$ $$\beta_{4}$$ $$=$$ $$\nu^{4} + \nu^{3} - 6 \nu^{2} - 5 \nu + 3$$ $$\beta_{5}$$ $$=$$ $$\nu^{5} + \nu^{4} - 7 \nu^{3} - 5 \nu^{2} + 8 \nu - 2$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} + 4$$ $$\nu^{3}$$ $$=$$ $$\beta_{3} + 5 \beta_{1}$$ $$\nu^{4}$$ $$=$$ $$\beta_{4} - \beta_{3} + 6 \beta_{2} + 21$$ $$\nu^{5}$$ $$=$$ $$\beta_{5} - \beta_{4} + 8 \beta_{3} - \beta_{2} + 27 \beta_{1} + 1$$

## 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
 0.643548 −2.43859 0.0605469 2.28320 2.38442 −1.93313
−1.00000 −2.95121 1.00000 −2.26461 2.95121 0 −1.00000 5.70965 2.26461
1.2 −1.00000 −2.30861 1.00000 0.374437 2.30861 0 −1.00000 2.32969 −0.374437
1.3 −1.00000 −0.302513 1.00000 2.67551 0.302513 0 −1.00000 −2.90849 −2.67551
1.4 −1.00000 0.486321 1.00000 −0.616311 −0.486321 0 −1.00000 −2.76349 0.616311
1.5 −1.00000 1.63447 1.00000 2.84627 −1.63447 0 −1.00000 −0.328504 −2.84627
1.6 −1.00000 2.44154 1.00000 −3.01529 −2.44154 0 −1.00000 2.96114 3.01529
 $$n$$: 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.bn 6
7.b odd 2 1 4018.2.a.bo 6
7.d odd 6 2 574.2.e.g 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
574.2.e.g 12 7.d odd 6 2
4018.2.a.bn 6 1.a even 1 1 trivial
4018.2.a.bo 6 7.b odd 2 1

## 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(4018))$$:

 $$T_{3}^{6} + T_{3}^{5} - 11 T_{3}^{4} - 5 T_{3}^{3} + 30 T_{3}^{2} - 4 T_{3} - 4$$ $$T_{5}^{6} - 15 T_{5}^{4} - T_{5}^{3} + 56 T_{5}^{2} + 12 T_{5} - 12$$ $$T_{11}^{6} - T_{11}^{5} - 38 T_{11}^{4} + 7 T_{11}^{3} + 320 T_{11}^{2} - 48 T_{11} - 684$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T )^{6}$$
$3$ $$1 + T + 7 T^{2} + 10 T^{3} + 33 T^{4} + 41 T^{5} + 122 T^{6} + 123 T^{7} + 297 T^{8} + 270 T^{9} + 567 T^{10} + 243 T^{11} + 729 T^{12}$$
$5$ $$1 + 15 T^{2} - T^{3} + 131 T^{4} - 3 T^{5} + 798 T^{6} - 15 T^{7} + 3275 T^{8} - 125 T^{9} + 9375 T^{10} + 15625 T^{12}$$
$7$ 1
$11$ $$1 - T + 28 T^{2} - 48 T^{3} + 463 T^{4} - 1027 T^{5} + 5388 T^{6} - 11297 T^{7} + 56023 T^{8} - 63888 T^{9} + 409948 T^{10} - 161051 T^{11} + 1771561 T^{12}$$
$13$ $$1 + 4 T + 53 T^{2} + 130 T^{3} + 1064 T^{4} + 1632 T^{5} + 14139 T^{6} + 21216 T^{7} + 179816 T^{8} + 285610 T^{9} + 1513733 T^{10} + 1485172 T^{11} + 4826809 T^{12}$$
$17$ $$1 - T + 86 T^{2} - 90 T^{3} + 3291 T^{4} - 3133 T^{5} + 71976 T^{6} - 53261 T^{7} + 951099 T^{8} - 442170 T^{9} + 7182806 T^{10} - 1419857 T^{11} + 24137569 T^{12}$$
$19$ $$1 - 3 T + 58 T^{2} - 272 T^{3} + 2021 T^{4} - 8421 T^{5} + 49416 T^{6} - 159999 T^{7} + 729581 T^{8} - 1865648 T^{9} + 7558618 T^{10} - 7428297 T^{11} + 47045881 T^{12}$$
$23$ $$1 - 21 T + 286 T^{2} - 2780 T^{3} + 21501 T^{4} - 135231 T^{5} + 710192 T^{6} - 3110313 T^{7} + 11374029 T^{8} - 33824260 T^{9} + 80034526 T^{10} - 135163203 T^{11} + 148035889 T^{12}$$
$29$ $$1 - 5 T + 158 T^{2} - 624 T^{3} + 10767 T^{4} - 33725 T^{5} + 407883 T^{6} - 978025 T^{7} + 9055047 T^{8} - 15218736 T^{9} + 111750398 T^{10} - 102555745 T^{11} + 594823321 T^{12}$$
$31$ $$1 + 3 T + 133 T^{2} + 238 T^{3} + 7871 T^{4} + 8247 T^{5} + 291654 T^{6} + 255657 T^{7} + 7564031 T^{8} + 7090258 T^{9} + 122828293 T^{10} + 85887453 T^{11} + 887503681 T^{12}$$
$37$ $$1 + 2 T + 81 T^{2} + 293 T^{3} + 4297 T^{4} + 15765 T^{5} + 170250 T^{6} + 583305 T^{7} + 5882593 T^{8} + 14841329 T^{9} + 151807041 T^{10} + 138687914 T^{11} + 2565726409 T^{12}$$
$41$ $$( 1 + T )^{6}$$
$43$ $$1 - 12 T + 207 T^{2} - 1548 T^{3} + 15734 T^{4} - 87500 T^{5} + 747019 T^{6} - 3762500 T^{7} + 29092166 T^{8} - 123076836 T^{9} + 707691807 T^{10} - 1764101316 T^{11} + 6321363049 T^{12}$$
$47$ $$1 - 18 T + 367 T^{2} - 4187 T^{3} + 48285 T^{4} - 390765 T^{5} + 3124970 T^{6} - 18365955 T^{7} + 106661565 T^{8} - 434706901 T^{9} + 1790842927 T^{10} - 4128210126 T^{11} + 10779215329 T^{12}$$
$53$ $$1 - 14 T + 243 T^{2} - 2555 T^{3} + 28319 T^{4} - 233939 T^{5} + 1903710 T^{6} - 12398767 T^{7} + 79548071 T^{8} - 380380735 T^{9} + 1917386883 T^{10} - 5854736902 T^{11} + 22164361129 T^{12}$$
$59$ $$1 + 16 T + 238 T^{2} + 2547 T^{3} + 26935 T^{4} + 227428 T^{5} + 1910883 T^{6} + 13418252 T^{7} + 93760735 T^{8} + 523100313 T^{9} + 2883931918 T^{10} + 11438788784 T^{11} + 42180533641 T^{12}$$
$61$ $$1 - 20 T + 453 T^{2} - 5909 T^{3} + 76465 T^{4} - 709515 T^{5} + 6412098 T^{6} - 43280415 T^{7} + 284526265 T^{8} - 1341230729 T^{9} + 6272165973 T^{10} - 16891926020 T^{11} + 51520374361 T^{12}$$
$67$ $$1 + 13 T + 176 T^{2} + 1888 T^{3} + 17413 T^{4} + 153731 T^{5} + 1305512 T^{6} + 10299977 T^{7} + 78166957 T^{8} + 567840544 T^{9} + 3546597296 T^{10} + 17551626391 T^{11} + 90458382169 T^{12}$$
$71$ $$1 - 11 T + 136 T^{2} + 162 T^{3} + 2821 T^{4} + 3931 T^{5} + 799059 T^{6} + 279101 T^{7} + 14220661 T^{8} + 57981582 T^{9} + 3455988616 T^{10} - 19846522861 T^{11} + 128100283921 T^{12}$$
$73$ $$1 + T + 254 T^{2} - 330 T^{3} + 30443 T^{4} - 87177 T^{5} + 2509711 T^{6} - 6363921 T^{7} + 162230747 T^{8} - 128375610 T^{9} + 7213153214 T^{10} + 2073071593 T^{11} + 151334226289 T^{12}$$
$79$ $$1 - 19 T + 440 T^{2} - 4604 T^{3} + 60189 T^{4} - 431041 T^{5} + 4937144 T^{6} - 34052239 T^{7} + 375639549 T^{8} - 2269951556 T^{9} + 17138035640 T^{10} - 58464071581 T^{11} + 243087455521 T^{12}$$
$83$ $$1 + 15 T + 500 T^{2} + 5412 T^{3} + 100709 T^{4} + 829239 T^{5} + 10974575 T^{6} + 68826837 T^{7} + 693784301 T^{8} + 3094511244 T^{9} + 23729160500 T^{10} + 59085609645 T^{11} + 326940373369 T^{12}$$
$89$ $$1 - 14 T + 431 T^{2} - 3891 T^{3} + 73571 T^{4} - 493163 T^{5} + 7716806 T^{6} - 43891507 T^{7} + 582755891 T^{8} - 2743034379 T^{9} + 27041905871 T^{10} - 78176832286 T^{11} + 496981290961 T^{12}$$
$97$ $$1 + T + 368 T^{2} - 414 T^{3} + 58751 T^{4} - 162591 T^{5} + 6301840 T^{6} - 15771327 T^{7} + 552788159 T^{8} - 377846622 T^{9} + 32578775408 T^{10} + 8587340257 T^{11} + 832972004929 T^{12}$$