# Properties

 Label 18.4.c.b Level 18 Weight 4 Character orbit 18.c Analytic conductor 1.062 Analytic rank 0 Dimension 4 CM No Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ = $$18 = 2 \cdot 3^{2}$$ Weight: $$k$$ = $$4$$ Character orbit: $$[\chi]$$ = 18.c (of order $$3$$ and degree $$2$$)

## Newform invariants

 Self dual: No Analytic conductor: $$1.0620343801$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\Q(\sqrt{-3}, \sqrt{-35})$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$3$$ Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q -2 \beta_{1} q^{2} + ( -\beta_{1} - \beta_{3} ) q^{3} + ( -4 - 4 \beta_{1} ) q^{4} + ( 6 + 6 \beta_{1} + 3 \beta_{3} ) q^{5} + ( -2 - 2 \beta_{1} - 2 \beta_{2} ) q^{6} + ( 8 \beta_{1} + 3 \beta_{2} - 3 \beta_{3} ) q^{7} -8 q^{8} + ( -1 + 25 \beta_{1} - \beta_{2} - \beta_{3} ) q^{9} +O(q^{10})$$ $$q -2 \beta_{1} q^{2} + ( -\beta_{1} - \beta_{3} ) q^{3} + ( -4 - 4 \beta_{1} ) q^{4} + ( 6 + 6 \beta_{1} + 3 \beta_{3} ) q^{5} + ( -2 - 2 \beta_{1} - 2 \beta_{2} ) q^{6} + ( 8 \beta_{1} + 3 \beta_{2} - 3 \beta_{3} ) q^{7} -8 q^{8} + ( -1 + 25 \beta_{1} - \beta_{2} - \beta_{3} ) q^{9} + ( 12 + 6 \beta_{2} ) q^{10} + ( -9 \beta_{1} - 6 \beta_{2} + 6 \beta_{3} ) q^{11} + ( -4 - 4 \beta_{2} + 4 \beta_{3} ) q^{12} + ( -32 - 32 \beta_{1} - 3 \beta_{3} ) q^{13} + ( 16 + 16 \beta_{1} - 6 \beta_{3} ) q^{14} + ( 6 - 78 \beta_{1} + 6 \beta_{2} - 3 \beta_{3} ) q^{15} + 16 \beta_{1} q^{16} + ( 3 + 3 \beta_{2} ) q^{17} + ( 50 + 52 \beta_{1} - 4 \beta_{2} + 2 \beta_{3} ) q^{18} + ( 59 - 15 \beta_{2} ) q^{19} + ( -24 \beta_{1} + 12 \beta_{2} - 12 \beta_{3} ) q^{20} + ( -70 + 8 \beta_{1} + 11 \beta_{2} - 3 \beta_{3} ) q^{21} + ( -18 - 18 \beta_{1} + 12 \beta_{3} ) q^{22} + ( -36 - 36 \beta_{1} - 3 \beta_{3} ) q^{23} + ( 8 \beta_{1} + 8 \beta_{3} ) q^{24} + ( 145 \beta_{1} - 27 \beta_{2} + 27 \beta_{3} ) q^{25} + ( -64 - 6 \beta_{2} ) q^{26} + ( 51 + 78 \beta_{1} + 24 \beta_{2} ) q^{27} + ( 32 - 12 \beta_{2} ) q^{28} + ( 108 \beta_{1} + 21 \beta_{2} - 21 \beta_{3} ) q^{29} + ( -156 - 168 \beta_{1} + 6 \beta_{2} - 12 \beta_{3} ) q^{30} + ( -110 - 110 \beta_{1} - 9 \beta_{3} ) q^{31} + ( 32 + 32 \beta_{1} ) q^{32} + ( 147 - 9 \beta_{1} - 15 \beta_{2} + 6 \beta_{3} ) q^{33} + ( -6 \beta_{1} + 6 \beta_{2} - 6 \beta_{3} ) q^{34} + ( 186 - 15 \beta_{2} ) q^{35} + ( 104 + 4 \beta_{1} - 4 \beta_{2} + 8 \beta_{3} ) q^{36} + ( 152 + 42 \beta_{2} ) q^{37} + ( -118 \beta_{1} - 30 \beta_{2} + 30 \beta_{3} ) q^{38} + ( -32 + 78 \beta_{1} - 32 \beta_{2} + 29 \beta_{3} ) q^{39} + ( -48 - 48 \beta_{1} - 24 \beta_{3} ) q^{40} + ( -237 - 237 \beta_{1} - 6 \beta_{3} ) q^{41} + ( 16 + 156 \beta_{1} + 16 \beta_{2} - 22 \beta_{3} ) q^{42} + ( -79 \beta_{1} + 72 \beta_{2} - 72 \beta_{3} ) q^{43} + ( -36 + 24 \beta_{2} ) q^{44} + ( -234 - 162 \beta_{1} - 72 \beta_{2} - 9 \beta_{3} ) q^{45} + ( -72 - 6 \beta_{2} ) q^{46} + ( 264 \beta_{1} - 45 \beta_{2} + 45 \beta_{3} ) q^{47} + ( 16 + 16 \beta_{1} + 16 \beta_{2} ) q^{48} + ( 45 + 45 \beta_{1} + 57 \beta_{3} ) q^{49} + ( 290 + 290 \beta_{1} + 54 \beta_{3} ) q^{50} + ( -78 - 81 \beta_{1} + 3 \beta_{2} - 3 \beta_{3} ) q^{51} + ( 128 \beta_{1} - 12 \beta_{2} + 12 \beta_{3} ) q^{52} + ( 96 + 42 \beta_{2} ) q^{53} + ( 156 + 54 \beta_{1} + 48 \beta_{2} - 48 \beta_{3} ) q^{54} + ( -414 + 9 \beta_{2} ) q^{55} + ( -64 \beta_{1} - 24 \beta_{2} + 24 \beta_{3} ) q^{56} + ( 390 + 331 \beta_{1} - 15 \beta_{2} - 59 \beta_{3} ) q^{57} + ( 216 + 216 \beta_{1} - 42 \beta_{3} ) q^{58} + ( -81 - 81 \beta_{1} + 6 \beta_{3} ) q^{59} + ( -336 - 24 \beta_{1} - 12 \beta_{2} - 12 \beta_{3} ) q^{60} + ( -502 \beta_{1} - 45 \beta_{2} + 45 \beta_{3} ) q^{61} + ( -220 - 18 \beta_{2} ) q^{62} + ( -278 - 130 \beta_{1} + 19 \beta_{2} + 67 \beta_{3} ) q^{63} + 64 q^{64} + ( -426 \beta_{1} + 105 \beta_{2} - 105 \beta_{3} ) q^{65} + ( -18 - 312 \beta_{1} - 18 \beta_{2} + 30 \beta_{3} ) q^{66} + ( 565 + 565 \beta_{1} - 36 \beta_{3} ) q^{67} + ( -12 - 12 \beta_{1} - 12 \beta_{3} ) q^{68} + ( -36 + 78 \beta_{1} - 36 \beta_{2} + 33 \beta_{3} ) q^{69} + ( -372 \beta_{1} - 30 \beta_{2} + 30 \beta_{3} ) q^{70} + ( -204 - 96 \beta_{2} ) q^{71} + ( 8 - 200 \beta_{1} + 8 \beta_{2} + 8 \beta_{3} ) q^{72} + ( -187 - 63 \beta_{2} ) q^{73} + ( -304 \beta_{1} + 84 \beta_{2} - 84 \beta_{3} ) q^{74} + ( 847 + 145 \beta_{1} + 118 \beta_{2} + 27 \beta_{3} ) q^{75} + ( -236 - 236 \beta_{1} + 60 \beta_{3} ) q^{76} + ( 540 + 540 \beta_{1} - 93 \beta_{3} ) q^{77} + ( 156 + 220 \beta_{1} - 6 \beta_{2} + 64 \beta_{3} ) q^{78} + ( 194 \beta_{1} - 39 \beta_{2} + 39 \beta_{3} ) q^{79} + ( -96 - 48 \beta_{2} ) q^{80} + ( -546 - 597 \beta_{1} + 102 \beta_{2} - 51 \beta_{3} ) q^{81} + ( -474 - 12 \beta_{2} ) q^{82} + ( 642 \beta_{1} - 63 \beta_{2} + 63 \beta_{3} ) q^{83} + ( 312 + 280 \beta_{1} - 12 \beta_{2} - 32 \beta_{3} ) q^{84} + ( 252 + 252 \beta_{1} + 18 \beta_{3} ) q^{85} + ( -158 - 158 \beta_{1} - 144 \beta_{3} ) q^{86} + ( -438 + 108 \beta_{1} + 129 \beta_{2} - 21 \beta_{3} ) q^{87} + ( 72 \beta_{1} + 48 \beta_{2} - 48 \beta_{3} ) q^{88} + ( -186 + 120 \beta_{2} ) q^{89} + ( -324 + 144 \beta_{1} - 162 \beta_{2} + 144 \beta_{3} ) q^{90} + ( 22 - 63 \beta_{2} ) q^{91} + ( 144 \beta_{1} - 12 \beta_{2} + 12 \beta_{3} ) q^{92} + ( -110 + 234 \beta_{1} - 110 \beta_{2} + 101 \beta_{3} ) q^{93} + ( 528 + 528 \beta_{1} + 90 \beta_{3} ) q^{94} + ( -816 - 816 \beta_{1} + 132 \beta_{3} ) q^{95} + ( 32 + 32 \beta_{2} - 32 \beta_{3} ) q^{96} + ( -31 \beta_{1} - 66 \beta_{2} + 66 \beta_{3} ) q^{97} + ( 90 + 114 \beta_{2} ) q^{98} + ( 381 + 78 \beta_{1} - 24 \beta_{2} - 141 \beta_{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 4q^{2} + 3q^{3} - 8q^{4} + 9q^{5} - 19q^{7} - 32q^{8} - 51q^{9} + O(q^{10})$$ $$4q + 4q^{2} + 3q^{3} - 8q^{4} + 9q^{5} - 19q^{7} - 32q^{8} - 51q^{9} + 36q^{10} + 24q^{11} - 12q^{12} - 61q^{13} + 38q^{14} + 171q^{15} - 32q^{16} + 6q^{17} + 102q^{18} + 266q^{19} + 36q^{20} - 315q^{21} - 48q^{22} - 69q^{23} - 24q^{24} - 263q^{25} - 244q^{26} + 152q^{28} - 237q^{29} - 288q^{30} - 211q^{31} + 64q^{32} + 630q^{33} + 6q^{34} + 774q^{35} + 408q^{36} + 524q^{37} + 266q^{38} - 249q^{39} - 72q^{40} - 468q^{41} - 258q^{42} + 86q^{43} - 192q^{44} - 459q^{45} - 276q^{46} - 483q^{47} + 33q^{49} + 526q^{50} - 153q^{51} - 244q^{52} + 300q^{53} + 468q^{54} - 1674q^{55} + 152q^{56} + 987q^{57} + 474q^{58} - 168q^{59} - 1260q^{60} + 1049q^{61} - 844q^{62} - 957q^{63} + 256q^{64} + 747q^{65} + 558q^{66} + 1166q^{67} - 12q^{68} - 261q^{69} + 774q^{70} - 624q^{71} + 408q^{72} - 622q^{73} + 524q^{74} + 2835q^{75} - 532q^{76} + 1173q^{77} + 132q^{78} - 349q^{79} - 288q^{80} - 1143q^{81} - 1872q^{82} - 1221q^{83} + 744q^{84} + 486q^{85} - 172q^{86} - 2205q^{87} - 192q^{88} - 984q^{89} - 1404q^{90} + 214q^{91} - 276q^{92} - 789q^{93} + 966q^{94} - 1764q^{95} + 96q^{96} + 128q^{97} + 132q^{98} + 1557q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - x^{3} - 8 x^{2} - 9 x + 81$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$\nu^{3} + 8 \nu^{2} - 8 \nu - 81$$$$)/72$$ $$\beta_{2}$$ $$=$$ $$($$$$-\nu^{3} + \nu^{2} + 17 \nu$$$$)/9$$ $$\beta_{3}$$ $$=$$ $$($$$$\nu^{3} + 8 \nu - 17$$$$)/8$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{3} + \beta_{2} - \beta_{1} + 1$$$$)/3$$ $$\nu^{2}$$ $$=$$ $$($$$$-\beta_{3} + 2 \beta_{2} + 25 \beta_{1} + 26$$$$)/3$$ $$\nu^{3}$$ $$=$$ $$($$$$16 \beta_{3} - 8 \beta_{2} + 8 \beta_{1} + 43$$$$)/3$$

## Character Values

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

 $$n$$ $$11$$ $$\chi(n)$$ $$-1 - \beta_{1}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
7.1
 2.81174 + 1.04601i −2.31174 − 1.91203i 2.81174 − 1.04601i −2.31174 + 1.91203i
1.00000 1.73205i −1.81174 4.87007i −2.00000 3.46410i 9.93521 + 17.2083i −10.2470 1.73205i 2.93521 5.08394i −8.00000 −20.4352 + 17.6466i 39.7409
7.2 1.00000 1.73205i 3.31174 + 4.00405i −2.00000 3.46410i −5.43521 9.41407i 10.2470 1.73205i −12.4352 + 21.5384i −8.00000 −5.06479 + 26.5207i −21.7409
13.1 1.00000 + 1.73205i −1.81174 + 4.87007i −2.00000 + 3.46410i 9.93521 17.2083i −10.2470 + 1.73205i 2.93521 + 5.08394i −8.00000 −20.4352 17.6466i 39.7409
13.2 1.00000 + 1.73205i 3.31174 4.00405i −2.00000 + 3.46410i −5.43521 + 9.41407i 10.2470 + 1.73205i −12.4352 21.5384i −8.00000 −5.06479 26.5207i −21.7409
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
9.c Even 1 yes

## Hecke kernels

This newform can be constructed as the kernel of the linear operator $$T_{5}^{4} - 9 T_{5}^{3} + 297 T_{5}^{2} + 1944 T_{5} + 46656$$ acting on $$S_{4}^{\mathrm{new}}(18, [\chi])$$.