# Properties

 Label 18.4.c.a Level 18 Weight 4 Character orbit 18.c Analytic conductor 1.062 Analytic rank 0 Dimension 2 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: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q$$ $$-2 \zeta_{6} q^{2}$$ $$+ ( 3 - 6 \zeta_{6} ) q^{3}$$ $$+ ( -4 + 4 \zeta_{6} ) q^{4}$$ $$+ ( 9 - 9 \zeta_{6} ) q^{5}$$ $$+ ( -12 + 6 \zeta_{6} ) q^{6}$$ $$+ 31 \zeta_{6} q^{7}$$ $$+ 8 q^{8}$$ $$-27 q^{9}$$ $$+O(q^{10})$$ $$q$$ $$-2 \zeta_{6} q^{2}$$ $$+ ( 3 - 6 \zeta_{6} ) q^{3}$$ $$+ ( -4 + 4 \zeta_{6} ) q^{4}$$ $$+ ( 9 - 9 \zeta_{6} ) q^{5}$$ $$+ ( -12 + 6 \zeta_{6} ) q^{6}$$ $$+ 31 \zeta_{6} q^{7}$$ $$+ 8 q^{8}$$ $$-27 q^{9}$$ $$-18 q^{10}$$ $$+ 15 \zeta_{6} q^{11}$$ $$+ ( 12 + 12 \zeta_{6} ) q^{12}$$ $$+ ( 37 - 37 \zeta_{6} ) q^{13}$$ $$+ ( 62 - 62 \zeta_{6} ) q^{14}$$ $$+ ( -27 - 27 \zeta_{6} ) q^{15}$$ $$-16 \zeta_{6} q^{16}$$ $$-42 q^{17}$$ $$+ 54 \zeta_{6} q^{18}$$ $$-28 q^{19}$$ $$+ 36 \zeta_{6} q^{20}$$ $$+ ( 186 - 93 \zeta_{6} ) q^{21}$$ $$+ ( 30 - 30 \zeta_{6} ) q^{22}$$ $$+ ( -195 + 195 \zeta_{6} ) q^{23}$$ $$+ ( 24 - 48 \zeta_{6} ) q^{24}$$ $$+ 44 \zeta_{6} q^{25}$$ $$-74 q^{26}$$ $$+ ( -81 + 162 \zeta_{6} ) q^{27}$$ $$-124 q^{28}$$ $$-111 \zeta_{6} q^{29}$$ $$+ ( -54 + 108 \zeta_{6} ) q^{30}$$ $$+ ( 205 - 205 \zeta_{6} ) q^{31}$$ $$+ ( -32 + 32 \zeta_{6} ) q^{32}$$ $$+ ( 90 - 45 \zeta_{6} ) q^{33}$$ $$+ 84 \zeta_{6} q^{34}$$ $$+ 279 q^{35}$$ $$+ ( 108 - 108 \zeta_{6} ) q^{36}$$ $$-166 q^{37}$$ $$+ 56 \zeta_{6} q^{38}$$ $$+ ( -111 - 111 \zeta_{6} ) q^{39}$$ $$+ ( 72 - 72 \zeta_{6} ) q^{40}$$ $$+ ( 261 - 261 \zeta_{6} ) q^{41}$$ $$+ ( -186 - 186 \zeta_{6} ) q^{42}$$ $$+ 43 \zeta_{6} q^{43}$$ $$-60 q^{44}$$ $$+ ( -243 + 243 \zeta_{6} ) q^{45}$$ $$+ 390 q^{46}$$ $$-177 \zeta_{6} q^{47}$$ $$+ ( -96 + 48 \zeta_{6} ) q^{48}$$ $$+ ( -618 + 618 \zeta_{6} ) q^{49}$$ $$+ ( 88 - 88 \zeta_{6} ) q^{50}$$ $$+ ( -126 + 252 \zeta_{6} ) q^{51}$$ $$+ 148 \zeta_{6} q^{52}$$ $$+ 114 q^{53}$$ $$+ ( 324 - 162 \zeta_{6} ) q^{54}$$ $$+ 135 q^{55}$$ $$+ 248 \zeta_{6} q^{56}$$ $$+ ( -84 + 168 \zeta_{6} ) q^{57}$$ $$+ ( -222 + 222 \zeta_{6} ) q^{58}$$ $$+ ( -159 + 159 \zeta_{6} ) q^{59}$$ $$+ ( 216 - 108 \zeta_{6} ) q^{60}$$ $$-191 \zeta_{6} q^{61}$$ $$-410 q^{62}$$ $$-837 \zeta_{6} q^{63}$$ $$+ 64 q^{64}$$ $$-333 \zeta_{6} q^{65}$$ $$+ ( -90 - 90 \zeta_{6} ) q^{66}$$ $$+ ( 421 - 421 \zeta_{6} ) q^{67}$$ $$+ ( 168 - 168 \zeta_{6} ) q^{68}$$ $$+ ( 585 + 585 \zeta_{6} ) q^{69}$$ $$-558 \zeta_{6} q^{70}$$ $$+ 156 q^{71}$$ $$-216 q^{72}$$ $$+ 182 q^{73}$$ $$+ 332 \zeta_{6} q^{74}$$ $$+ ( 264 - 132 \zeta_{6} ) q^{75}$$ $$+ ( 112 - 112 \zeta_{6} ) q^{76}$$ $$+ ( -465 + 465 \zeta_{6} ) q^{77}$$ $$+ ( -222 + 444 \zeta_{6} ) q^{78}$$ $$-1133 \zeta_{6} q^{79}$$ $$-144 q^{80}$$ $$+ 729 q^{81}$$ $$-522 q^{82}$$ $$+ 1083 \zeta_{6} q^{83}$$ $$+ ( -372 + 744 \zeta_{6} ) q^{84}$$ $$+ ( -378 + 378 \zeta_{6} ) q^{85}$$ $$+ ( 86 - 86 \zeta_{6} ) q^{86}$$ $$+ ( -666 + 333 \zeta_{6} ) q^{87}$$ $$+ 120 \zeta_{6} q^{88}$$ $$-1050 q^{89}$$ $$+ 486 q^{90}$$ $$+ 1147 q^{91}$$ $$-780 \zeta_{6} q^{92}$$ $$+ ( -615 - 615 \zeta_{6} ) q^{93}$$ $$+ ( -354 + 354 \zeta_{6} ) q^{94}$$ $$+ ( -252 + 252 \zeta_{6} ) q^{95}$$ $$+ ( 96 + 96 \zeta_{6} ) q^{96}$$ $$+ 901 \zeta_{6} q^{97}$$ $$+ 1236 q^{98}$$ $$-405 \zeta_{6} q^{99}$$ $$+O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q$$ $$\mathstrut -\mathstrut 2q^{2}$$ $$\mathstrut -\mathstrut 4q^{4}$$ $$\mathstrut +\mathstrut 9q^{5}$$ $$\mathstrut -\mathstrut 18q^{6}$$ $$\mathstrut +\mathstrut 31q^{7}$$ $$\mathstrut +\mathstrut 16q^{8}$$ $$\mathstrut -\mathstrut 54q^{9}$$ $$\mathstrut +\mathstrut O(q^{10})$$ $$2q$$ $$\mathstrut -\mathstrut 2q^{2}$$ $$\mathstrut -\mathstrut 4q^{4}$$ $$\mathstrut +\mathstrut 9q^{5}$$ $$\mathstrut -\mathstrut 18q^{6}$$ $$\mathstrut +\mathstrut 31q^{7}$$ $$\mathstrut +\mathstrut 16q^{8}$$ $$\mathstrut -\mathstrut 54q^{9}$$ $$\mathstrut -\mathstrut 36q^{10}$$ $$\mathstrut +\mathstrut 15q^{11}$$ $$\mathstrut +\mathstrut 36q^{12}$$ $$\mathstrut +\mathstrut 37q^{13}$$ $$\mathstrut +\mathstrut 62q^{14}$$ $$\mathstrut -\mathstrut 81q^{15}$$ $$\mathstrut -\mathstrut 16q^{16}$$ $$\mathstrut -\mathstrut 84q^{17}$$ $$\mathstrut +\mathstrut 54q^{18}$$ $$\mathstrut -\mathstrut 56q^{19}$$ $$\mathstrut +\mathstrut 36q^{20}$$ $$\mathstrut +\mathstrut 279q^{21}$$ $$\mathstrut +\mathstrut 30q^{22}$$ $$\mathstrut -\mathstrut 195q^{23}$$ $$\mathstrut +\mathstrut 44q^{25}$$ $$\mathstrut -\mathstrut 148q^{26}$$ $$\mathstrut -\mathstrut 248q^{28}$$ $$\mathstrut -\mathstrut 111q^{29}$$ $$\mathstrut +\mathstrut 205q^{31}$$ $$\mathstrut -\mathstrut 32q^{32}$$ $$\mathstrut +\mathstrut 135q^{33}$$ $$\mathstrut +\mathstrut 84q^{34}$$ $$\mathstrut +\mathstrut 558q^{35}$$ $$\mathstrut +\mathstrut 108q^{36}$$ $$\mathstrut -\mathstrut 332q^{37}$$ $$\mathstrut +\mathstrut 56q^{38}$$ $$\mathstrut -\mathstrut 333q^{39}$$ $$\mathstrut +\mathstrut 72q^{40}$$ $$\mathstrut +\mathstrut 261q^{41}$$ $$\mathstrut -\mathstrut 558q^{42}$$ $$\mathstrut +\mathstrut 43q^{43}$$ $$\mathstrut -\mathstrut 120q^{44}$$ $$\mathstrut -\mathstrut 243q^{45}$$ $$\mathstrut +\mathstrut 780q^{46}$$ $$\mathstrut -\mathstrut 177q^{47}$$ $$\mathstrut -\mathstrut 144q^{48}$$ $$\mathstrut -\mathstrut 618q^{49}$$ $$\mathstrut +\mathstrut 88q^{50}$$ $$\mathstrut +\mathstrut 148q^{52}$$ $$\mathstrut +\mathstrut 228q^{53}$$ $$\mathstrut +\mathstrut 486q^{54}$$ $$\mathstrut +\mathstrut 270q^{55}$$ $$\mathstrut +\mathstrut 248q^{56}$$ $$\mathstrut -\mathstrut 222q^{58}$$ $$\mathstrut -\mathstrut 159q^{59}$$ $$\mathstrut +\mathstrut 324q^{60}$$ $$\mathstrut -\mathstrut 191q^{61}$$ $$\mathstrut -\mathstrut 820q^{62}$$ $$\mathstrut -\mathstrut 837q^{63}$$ $$\mathstrut +\mathstrut 128q^{64}$$ $$\mathstrut -\mathstrut 333q^{65}$$ $$\mathstrut -\mathstrut 270q^{66}$$ $$\mathstrut +\mathstrut 421q^{67}$$ $$\mathstrut +\mathstrut 168q^{68}$$ $$\mathstrut +\mathstrut 1755q^{69}$$ $$\mathstrut -\mathstrut 558q^{70}$$ $$\mathstrut +\mathstrut 312q^{71}$$ $$\mathstrut -\mathstrut 432q^{72}$$ $$\mathstrut +\mathstrut 364q^{73}$$ $$\mathstrut +\mathstrut 332q^{74}$$ $$\mathstrut +\mathstrut 396q^{75}$$ $$\mathstrut +\mathstrut 112q^{76}$$ $$\mathstrut -\mathstrut 465q^{77}$$ $$\mathstrut -\mathstrut 1133q^{79}$$ $$\mathstrut -\mathstrut 288q^{80}$$ $$\mathstrut +\mathstrut 1458q^{81}$$ $$\mathstrut -\mathstrut 1044q^{82}$$ $$\mathstrut +\mathstrut 1083q^{83}$$ $$\mathstrut -\mathstrut 378q^{85}$$ $$\mathstrut +\mathstrut 86q^{86}$$ $$\mathstrut -\mathstrut 999q^{87}$$ $$\mathstrut +\mathstrut 120q^{88}$$ $$\mathstrut -\mathstrut 2100q^{89}$$ $$\mathstrut +\mathstrut 972q^{90}$$ $$\mathstrut +\mathstrut 2294q^{91}$$ $$\mathstrut -\mathstrut 780q^{92}$$ $$\mathstrut -\mathstrut 1845q^{93}$$ $$\mathstrut -\mathstrut 354q^{94}$$ $$\mathstrut -\mathstrut 252q^{95}$$ $$\mathstrut +\mathstrut 288q^{96}$$ $$\mathstrut +\mathstrut 901q^{97}$$ $$\mathstrut +\mathstrut 2472q^{98}$$ $$\mathstrut -\mathstrut 405q^{99}$$ $$\mathstrut +\mathstrut O(q^{100})$$

## Character Values

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

 $$n$$ $$11$$ $$\chi(n)$$ $$-1 + \zeta_{6}$$

## 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
 0.5 − 0.866025i 0.5 + 0.866025i
−1.00000 + 1.73205i 5.19615i −2.00000 3.46410i 4.50000 + 7.79423i −9.00000 5.19615i 15.5000 26.8468i 8.00000 −27.0000 −18.0000
13.1 −1.00000 1.73205i 5.19615i −2.00000 + 3.46410i 4.50000 7.79423i −9.00000 + 5.19615i 15.5000 + 26.8468i 8.00000 −27.0000 −18.0000
 $$n$$: 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 $$T_{5}^{2}$$ $$\mathstrut -\mathstrut 9 T_{5}$$ $$\mathstrut +\mathstrut 81$$ acting on $$S_{4}^{\mathrm{new}}(18, [\chi])$$.