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.

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

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

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

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 $$T_{5}^{4}$$ $$\mathstrut -\mathstrut 9 T_{5}^{3}$$ $$\mathstrut +\mathstrut 297 T_{5}^{2}$$ $$\mathstrut +\mathstrut 1944 T_{5}$$ $$\mathstrut +\mathstrut 46656$$ acting on $$S_{4}^{\mathrm{new}}(18, [\chi])$$.