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 $$6$$ and degree $$2$$)

Newform invariants

 Self dual: No Analytic conductor: $$0.490464475849$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\sqrt{-2}, \sqrt{-3})$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ 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.

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

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

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

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. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
9.d Odd 1 yes

Hecke kernels

There are no other newforms in $$S_{3}^{\mathrm{new}}(18, [\chi])$$.