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

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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
0.500000 0.866025i
0.500000 + 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:

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 \(S_{4}^{\mathrm{new}}(18, [\chi])\).