# Properties

 Label 18.3.d.a Level 18 Weight 3 Character orbit 18.d Analytic conductor 0.490 Analytic rank 0 Dimension 4 CM no Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$18 = 2 \cdot 3^{2}$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 18.d (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.490464475849$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\sqrt{-2}, \sqrt{-3})$$ Defining polynomial: $$x^{4} - 2 x^{2} + 4$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{2} + ( 1 - 2 \beta_{1} - 2 \beta_{2} + \beta_{3} ) q^{3} + 2 \beta_{2} q^{4} + ( -6 + 3 \beta_{2} ) q^{5} + ( -2 + \beta_{1} - 2 \beta_{2} - 2 \beta_{3} ) q^{6} + ( 1 + 3 \beta_{1} - \beta_{2} - 6 \beta_{3} ) q^{7} + 2 \beta_{3} q^{8} + ( 3 + 6 \beta_{3} ) q^{9} +O(q^{10})$$ $$q + \beta_{1} q^{2} + ( 1 - 2 \beta_{1} - 2 \beta_{2} + \beta_{3} ) q^{3} + 2 \beta_{2} q^{4} + ( -6 + 3 \beta_{2} ) q^{5} + ( -2 + \beta_{1} - 2 \beta_{2} - 2 \beta_{3} ) q^{6} + ( 1 + 3 \beta_{1} - \beta_{2} - 6 \beta_{3} ) q^{7} + 2 \beta_{3} q^{8} + ( 3 + 6 \beta_{3} ) q^{9} + ( -6 \beta_{1} + 3 \beta_{3} ) q^{10} + ( 3 - 3 \beta_{1} + 3 \beta_{2} ) q^{11} + ( 4 - 2 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} ) q^{12} + ( 6 \beta_{1} - 5 \beta_{2} + 6 \beta_{3} ) q^{13} + ( 12 + \beta_{1} - 6 \beta_{2} - \beta_{3} ) q^{14} + ( 9 \beta_{1} + 9 \beta_{2} - 9 \beta_{3} ) q^{15} + ( -4 + 4 \beta_{2} ) q^{16} + ( 6 - 12 \beta_{2} - 6 \beta_{3} ) q^{17} + ( -12 + 3 \beta_{1} + 12 \beta_{2} ) q^{18} + ( -10 - 12 \beta_{1} + 6 \beta_{3} ) q^{19} + ( -6 - 6 \beta_{2} ) q^{20} + ( -19 - 10 \beta_{1} + 17 \beta_{2} + 2 \beta_{3} ) q^{21} + ( 3 \beta_{1} - 6 \beta_{2} + 3 \beta_{3} ) q^{22} + ( 6 - 3 \beta_{1} - 3 \beta_{2} + 3 \beta_{3} ) q^{23} + ( 4 + 4 \beta_{1} - 8 \beta_{2} - 2 \beta_{3} ) q^{24} + ( 2 - 2 \beta_{2} ) q^{25} + ( -12 + 24 \beta_{2} - 5 \beta_{3} ) q^{26} + ( 15 + 6 \beta_{1} - 30 \beta_{2} - 3 \beta_{3} ) q^{27} + ( 2 + 12 \beta_{1} - 6 \beta_{3} ) q^{28} + ( 3 + 6 \beta_{1} + 3 \beta_{2} ) q^{29} + ( 18 + 9 \beta_{3} ) q^{30} + ( -9 \beta_{1} + 19 \beta_{2} - 9 \beta_{3} ) q^{31} + ( -4 \beta_{1} + 4 \beta_{3} ) q^{32} + ( 15 - 12 \beta_{1} - 3 \beta_{2} + 6 \beta_{3} ) q^{33} + ( 12 + 6 \beta_{1} - 12 \beta_{2} - 12 \beta_{3} ) q^{34} + ( -3 + 6 \beta_{2} + 27 \beta_{3} ) q^{35} + ( -12 \beta_{1} + 6 \beta_{2} + 12 \beta_{3} ) q^{36} + ( 32 + 12 \beta_{1} - 6 \beta_{3} ) q^{37} + ( -12 - 10 \beta_{1} - 12 \beta_{2} ) q^{38} + ( -10 + 23 \beta_{1} - 31 \beta_{2} - 13 \beta_{3} ) q^{39} + ( -6 \beta_{1} - 6 \beta_{3} ) q^{40} + ( -42 + 18 \beta_{1} + 21 \beta_{2} - 18 \beta_{3} ) q^{41} + ( -4 - 19 \beta_{1} - 16 \beta_{2} + 17 \beta_{3} ) q^{42} + ( -23 - 9 \beta_{1} + 23 \beta_{2} + 18 \beta_{3} ) q^{43} + ( -6 + 12 \beta_{2} - 6 \beta_{3} ) q^{44} + ( -18 - 18 \beta_{1} + 9 \beta_{2} - 18 \beta_{3} ) q^{45} + ( -6 + 6 \beta_{1} - 3 \beta_{3} ) q^{46} + ( 9 + 21 \beta_{1} + 9 \beta_{2} ) q^{47} + ( 4 + 4 \beta_{1} + 4 \beta_{2} - 8 \beta_{3} ) q^{48} + ( -6 \beta_{1} - 6 \beta_{2} - 6 \beta_{3} ) q^{49} + ( 2 \beta_{1} - 2 \beta_{3} ) q^{50} + ( -30 - 12 \beta_{1} + 24 \beta_{2} + 24 \beta_{3} ) q^{51} + ( 10 - 12 \beta_{1} - 10 \beta_{2} + 24 \beta_{3} ) q^{52} + ( 30 - 60 \beta_{2} - 30 \beta_{3} ) q^{53} + ( 6 + 15 \beta_{1} + 6 \beta_{2} - 30 \beta_{3} ) q^{54} + ( -27 + 18 \beta_{1} - 9 \beta_{3} ) q^{55} + ( 12 + 2 \beta_{1} + 12 \beta_{2} ) q^{56} + ( 26 + 20 \beta_{1} + 20 \beta_{2} + 8 \beta_{3} ) q^{57} + ( 3 \beta_{1} + 12 \beta_{2} + 3 \beta_{3} ) q^{58} + ( 42 - 39 \beta_{1} - 21 \beta_{2} + 39 \beta_{3} ) q^{59} + ( -18 + 18 \beta_{1} + 18 \beta_{2} ) q^{60} + ( 31 - 18 \beta_{1} - 31 \beta_{2} + 36 \beta_{3} ) q^{61} + ( 18 - 36 \beta_{2} + 19 \beta_{3} ) q^{62} + ( 39 + 15 \beta_{1} + 33 \beta_{2} - 18 \beta_{3} ) q^{63} -8 q^{64} + ( 15 - 54 \beta_{1} + 15 \beta_{2} ) q^{65} + ( -12 + 15 \beta_{1} - 12 \beta_{2} - 3 \beta_{3} ) q^{66} + ( -9 \beta_{1} - 53 \beta_{2} - 9 \beta_{3} ) q^{67} + ( 24 + 12 \beta_{1} - 12 \beta_{2} - 12 \beta_{3} ) q^{68} + ( 12 - 6 \beta_{1} - 15 \beta_{2} + 12 \beta_{3} ) q^{69} + ( -54 - 3 \beta_{1} + 54 \beta_{2} + 6 \beta_{3} ) q^{70} + ( -30 + 60 \beta_{2} - 24 \beta_{3} ) q^{71} + ( -24 + 6 \beta_{3} ) q^{72} + ( -52 + 36 \beta_{1} - 18 \beta_{3} ) q^{73} + ( 12 + 32 \beta_{1} + 12 \beta_{2} ) q^{74} + ( -2 - 2 \beta_{1} - 2 \beta_{2} + 4 \beta_{3} ) q^{75} + ( -12 \beta_{1} - 20 \beta_{2} - 12 \beta_{3} ) q^{76} + ( -30 + 24 \beta_{1} + 15 \beta_{2} - 24 \beta_{3} ) q^{77} + ( 26 - 10 \beta_{1} + 20 \beta_{2} - 31 \beta_{3} ) q^{78} + ( 7 + 15 \beta_{1} - 7 \beta_{2} - 30 \beta_{3} ) q^{79} + ( 12 - 24 \beta_{2} ) q^{80} + ( -63 + 36 \beta_{3} ) q^{81} + ( 36 - 42 \beta_{1} + 21 \beta_{3} ) q^{82} + ( -63 - 15 \beta_{1} - 63 \beta_{2} ) q^{83} + ( -34 - 4 \beta_{1} - 4 \beta_{2} - 16 \beta_{3} ) q^{84} + ( 18 \beta_{1} + 54 \beta_{2} + 18 \beta_{3} ) q^{85} + ( -36 - 23 \beta_{1} + 18 \beta_{2} + 23 \beta_{3} ) q^{86} + ( -3 - 3 \beta_{1} - 21 \beta_{2} - 12 \beta_{3} ) q^{87} + ( 12 - 6 \beta_{1} - 12 \beta_{2} + 12 \beta_{3} ) q^{88} + ( -30 + 60 \beta_{2} + 66 \beta_{3} ) q^{89} + ( 36 - 18 \beta_{1} - 72 \beta_{2} + 9 \beta_{3} ) q^{90} + ( 103 - 18 \beta_{1} + 9 \beta_{3} ) q^{91} + ( 6 - 6 \beta_{1} + 6 \beta_{2} ) q^{92} + ( 38 - 46 \beta_{1} + 35 \beta_{2} + 8 \beta_{3} ) q^{93} + ( 9 \beta_{1} + 42 \beta_{2} + 9 \beta_{3} ) q^{94} + ( 60 + 54 \beta_{1} - 30 \beta_{2} - 54 \beta_{3} ) q^{95} + ( 16 + 4 \beta_{1} - 8 \beta_{2} + 4 \beta_{3} ) q^{96} + ( 7 + 42 \beta_{1} - 7 \beta_{2} - 84 \beta_{3} ) q^{97} + ( 12 - 24 \beta_{2} - 6 \beta_{3} ) q^{98} + ( 45 - 27 \beta_{1} - 27 \beta_{2} + 36 \beta_{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 4q^{4} - 18q^{5} - 12q^{6} + 2q^{7} + 12q^{9} + O(q^{10})$$ $$4q + 4q^{4} - 18q^{5} - 12q^{6} + 2q^{7} + 12q^{9} + 18q^{11} + 12q^{12} - 10q^{13} + 36q^{14} + 18q^{15} - 8q^{16} - 24q^{18} - 40q^{19} - 36q^{20} - 42q^{21} - 12q^{22} + 18q^{23} + 4q^{25} + 8q^{28} + 18q^{29} + 72q^{30} + 38q^{31} + 54q^{33} + 24q^{34} + 12q^{36} + 128q^{37} - 72q^{38} - 102q^{39} - 126q^{41} - 48q^{42} - 46q^{43} - 54q^{45} - 24q^{46} + 54q^{47} + 24q^{48} - 12q^{49} - 72q^{51} + 20q^{52} + 36q^{54} - 108q^{55} + 72q^{56} + 144q^{57} + 24q^{58} + 126q^{59} - 36q^{60} + 62q^{61} + 222q^{63} - 32q^{64} + 90q^{65} - 72q^{66} - 106q^{67} + 72q^{68} + 18q^{69} - 108q^{70} - 96q^{72} - 208q^{73} + 72q^{74} - 12q^{75} - 40q^{76} - 90q^{77} + 144q^{78} + 14q^{79} - 252q^{81} + 144q^{82} - 378q^{83} - 144q^{84} + 108q^{85} - 108q^{86} - 54q^{87} + 24q^{88} + 412q^{91} + 36q^{92} + 222q^{93} + 84q^{94} + 180q^{95} + 48q^{96} + 14q^{97} + 126q^{99} + O(q^{100})$$

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

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2}$$$$/2$$ $$\beta_{3}$$ $$=$$ $$\nu^{3}$$$$/2$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$2 \beta_{2}$$ $$\nu^{3}$$ $$=$$ $$2 \beta_{3}$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/18\mathbb{Z}\right)^\times$$.

 $$n$$ $$11$$ $$\chi(n)$$ $$\beta_{2}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
5.1
 −1.22474 + 0.707107i 1.22474 − 0.707107i −1.22474 − 0.707107i 1.22474 + 0.707107i
−1.22474 + 0.707107i 2.44949 + 1.73205i 1.00000 1.73205i −4.50000 2.59808i −4.22474 0.389270i −3.17423 5.49794i 2.82843i 3.00000 + 8.48528i 7.34847
5.2 1.22474 0.707107i −2.44949 + 1.73205i 1.00000 1.73205i −4.50000 2.59808i −1.77526 + 3.85337i 4.17423 + 7.22999i 2.82843i 3.00000 8.48528i −7.34847
11.1 −1.22474 0.707107i 2.44949 1.73205i 1.00000 + 1.73205i −4.50000 + 2.59808i −4.22474 + 0.389270i −3.17423 + 5.49794i 2.82843i 3.00000 8.48528i 7.34847
11.2 1.22474 + 0.707107i −2.44949 1.73205i 1.00000 + 1.73205i −4.50000 + 2.59808i −1.77526 3.85337i 4.17423 7.22999i 2.82843i 3.00000 + 8.48528i −7.34847
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.d odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 18.3.d.a 4
3.b odd 2 1 54.3.d.a 4
4.b odd 2 1 144.3.q.c 4
5.b even 2 1 450.3.i.b 4
5.c odd 4 2 450.3.k.a 8
8.b even 2 1 576.3.q.f 4
8.d odd 2 1 576.3.q.e 4
9.c even 3 1 54.3.d.a 4
9.c even 3 1 162.3.b.a 4
9.d odd 6 1 inner 18.3.d.a 4
9.d odd 6 1 162.3.b.a 4
12.b even 2 1 432.3.q.d 4
15.d odd 2 1 1350.3.i.b 4
15.e even 4 2 1350.3.k.a 8
24.f even 2 1 1728.3.q.c 4
24.h odd 2 1 1728.3.q.d 4
36.f odd 6 1 432.3.q.d 4
36.f odd 6 1 1296.3.e.g 4
36.h even 6 1 144.3.q.c 4
36.h even 6 1 1296.3.e.g 4
45.h odd 6 1 450.3.i.b 4
45.j even 6 1 1350.3.i.b 4
45.k odd 12 2 1350.3.k.a 8
45.l even 12 2 450.3.k.a 8
72.j odd 6 1 576.3.q.f 4
72.l even 6 1 576.3.q.e 4
72.n even 6 1 1728.3.q.d 4
72.p odd 6 1 1728.3.q.c 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
18.3.d.a 4 1.a even 1 1 trivial
18.3.d.a 4 9.d odd 6 1 inner
54.3.d.a 4 3.b odd 2 1
54.3.d.a 4 9.c even 3 1
144.3.q.c 4 4.b odd 2 1
144.3.q.c 4 36.h even 6 1
162.3.b.a 4 9.c even 3 1
162.3.b.a 4 9.d odd 6 1
432.3.q.d 4 12.b even 2 1
432.3.q.d 4 36.f odd 6 1
450.3.i.b 4 5.b even 2 1
450.3.i.b 4 45.h odd 6 1
450.3.k.a 8 5.c odd 4 2
450.3.k.a 8 45.l even 12 2
576.3.q.e 4 8.d odd 2 1
576.3.q.e 4 72.l even 6 1
576.3.q.f 4 8.b even 2 1
576.3.q.f 4 72.j odd 6 1
1296.3.e.g 4 36.f odd 6 1
1296.3.e.g 4 36.h even 6 1
1350.3.i.b 4 15.d odd 2 1
1350.3.i.b 4 45.j even 6 1
1350.3.k.a 8 15.e even 4 2
1350.3.k.a 8 45.k odd 12 2
1728.3.q.c 4 24.f even 2 1
1728.3.q.c 4 72.p odd 6 1
1728.3.q.d 4 24.h odd 2 1
1728.3.q.d 4 72.n even 6 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{3}^{\mathrm{new}}(18, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - 2 T^{2} + 4 T^{4}$$
$3$ $$1 - 6 T^{2} + 81 T^{4}$$
$5$ $$( 1 + 9 T + 52 T^{2} + 225 T^{3} + 625 T^{4} )^{2}$$
$7$ $$1 - 2 T - 41 T^{2} + 106 T^{3} - 572 T^{4} + 5194 T^{5} - 98441 T^{6} - 235298 T^{7} + 5764801 T^{8}$$
$11$ $$1 - 18 T + 359 T^{2} - 4518 T^{3} + 61428 T^{4} - 546678 T^{5} + 5256119 T^{6} - 31888098 T^{7} + 214358881 T^{8}$$
$13$ $$1 + 10 T - 47 T^{2} - 1910 T^{3} - 23852 T^{4} - 322790 T^{5} - 1342367 T^{6} + 48268090 T^{7} + 815730721 T^{8}$$
$17$ $$1 - 796 T^{2} + 294342 T^{4} - 66482716 T^{6} + 6975757441 T^{8}$$
$19$ $$( 1 + 20 T + 606 T^{2} + 7220 T^{3} + 130321 T^{4} )^{2}$$
$23$ $$1 - 18 T + 1175 T^{2} - 19206 T^{3} + 915780 T^{4} - 10159974 T^{5} + 328813175 T^{6} - 2664646002 T^{7} + 78310985281 T^{8}$$
$29$ $$1 - 18 T + 1745 T^{2} - 29466 T^{3} + 2063316 T^{4} - 24780906 T^{5} + 1234205345 T^{6} - 10706819778 T^{7} + 500246412961 T^{8}$$
$31$ $$1 - 38 T - 353 T^{2} + 4750 T^{3} + 918004 T^{4} + 4564750 T^{5} - 326002913 T^{6} - 33725139878 T^{7} + 852891037441 T^{8}$$
$37$ $$( 1 - 64 T + 3546 T^{2} - 87616 T^{3} + 1874161 T^{4} )^{2}$$
$41$ $$1 + 126 T + 9329 T^{2} + 508662 T^{3} + 22367460 T^{4} + 855060822 T^{5} + 26361524369 T^{6} + 598513134366 T^{7} + 7984925229121 T^{8}$$
$43$ $$1 + 46 T - 1625 T^{2} + 1978 T^{3} + 6663796 T^{4} + 3657322 T^{5} - 5555551625 T^{6} + 290782700254 T^{7} + 11688200277601 T^{8}$$
$47$ $$1 - 54 T + 4751 T^{2} - 204066 T^{3} + 11548308 T^{4} - 450781794 T^{5} + 23183364431 T^{6} - 582077627766 T^{7} + 23811286661761 T^{8}$$
$53$ $$1 - 2236 T^{2} - 2409114 T^{4} - 17643115516 T^{6} + 62259690411361 T^{8}$$
$59$ $$1 - 126 T + 10535 T^{2} - 660618 T^{3} + 33793140 T^{4} - 2299611258 T^{5} + 127656398135 T^{6} - 5314747238766 T^{7} + 146830437604321 T^{8}$$
$61$ $$1 - 62 T - 2615 T^{2} + 60946 T^{3} + 13569316 T^{4} + 226780066 T^{5} - 36206874215 T^{6} - 3194263210382 T^{7} + 191707312997281 T^{8}$$
$67$ $$1 + 106 T - 65 T^{2} + 246238 T^{3} + 57123076 T^{4} + 1105362382 T^{5} - 1309822865 T^{6} + 9588588509914 T^{7} + 406067677556641 T^{8}$$
$71$ $$1 - 12460 T^{2} + 77194662 T^{4} - 316629545260 T^{6} + 645753531245761 T^{8}$$
$73$ $$( 1 + 104 T + 11418 T^{2} + 554216 T^{3} + 28398241 T^{4} )^{2}$$
$79$ $$1 - 14 T - 10985 T^{2} + 18214 T^{3} + 84841444 T^{4} + 113673574 T^{5} - 427866639785 T^{6} - 3403224377294 T^{7} + 1517108809906561 T^{8}$$
$83$ $$1 + 378 T + 72863 T^{2} + 9538830 T^{3} + 917456196 T^{4} + 65712999870 T^{5} + 3457955643023 T^{6} + 123583461133482 T^{7} + 2252292232139041 T^{8}$$
$89$ $$1 - 8860 T^{2} + 51019782 T^{4} - 555896255260 T^{6} + 3936588805702081 T^{8}$$
$97$ $$1 - 14 T - 8087 T^{2} + 147490 T^{3} - 21765356 T^{4} + 1387733410 T^{5} - 715936295447 T^{6} - 11661608069006 T^{7} + 7837433594376961 T^{8}$$