Properties

Label 18.2.c.a
Level 18
Weight 2
Character orbit 18.c
Analytic conductor 0.144
Analytic rank 0
Dimension 2
CM No
Inner twists 2

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Newspace parameters

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

Newform invariants

Self dual: No
Analytic conductor: \(0.143730723638\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Coefficient ring: \(\Z[a_1, a_2]\)
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.

\(f(q)\) \(=\) \( q\) \( -\zeta_{6} q^{2} \) \( + ( -2 + \zeta_{6} ) q^{3} \) \( + ( -1 + \zeta_{6} ) q^{4} \) \( + ( 1 + \zeta_{6} ) q^{6} \) \( -2 \zeta_{6} q^{7} \) \(+ q^{8}\) \( + ( 3 - 3 \zeta_{6} ) q^{9} \) \(+O(q^{10})\) \( q\) \( -\zeta_{6} q^{2} \) \( + ( -2 + \zeta_{6} ) q^{3} \) \( + ( -1 + \zeta_{6} ) q^{4} \) \( + ( 1 + \zeta_{6} ) q^{6} \) \( -2 \zeta_{6} q^{7} \) \(+ q^{8}\) \( + ( 3 - 3 \zeta_{6} ) q^{9} \) \( + 3 \zeta_{6} q^{11} \) \( + ( 1 - 2 \zeta_{6} ) q^{12} \) \( + ( -2 + 2 \zeta_{6} ) q^{13} \) \( + ( -2 + 2 \zeta_{6} ) q^{14} \) \( -\zeta_{6} q^{16} \) \( -3 q^{17} \) \( -3 q^{18} \) \(- q^{19}\) \( + ( 2 + 2 \zeta_{6} ) q^{21} \) \( + ( 3 - 3 \zeta_{6} ) q^{22} \) \( + ( 6 - 6 \zeta_{6} ) q^{23} \) \( + ( -2 + \zeta_{6} ) q^{24} \) \( + 5 \zeta_{6} q^{25} \) \( + 2 q^{26} \) \( + ( -3 + 6 \zeta_{6} ) q^{27} \) \( + 2 q^{28} \) \( -6 \zeta_{6} q^{29} \) \( + ( 4 - 4 \zeta_{6} ) q^{31} \) \( + ( -1 + \zeta_{6} ) q^{32} \) \( + ( -3 - 3 \zeta_{6} ) q^{33} \) \( + 3 \zeta_{6} q^{34} \) \( + 3 \zeta_{6} q^{36} \) \( -4 q^{37} \) \( + \zeta_{6} q^{38} \) \( + ( 2 - 4 \zeta_{6} ) q^{39} \) \( + ( -9 + 9 \zeta_{6} ) q^{41} \) \( + ( 2 - 4 \zeta_{6} ) q^{42} \) \( + \zeta_{6} q^{43} \) \( -3 q^{44} \) \( -6 q^{46} \) \( + 6 \zeta_{6} q^{47} \) \( + ( 1 + \zeta_{6} ) q^{48} \) \( + ( 3 - 3 \zeta_{6} ) q^{49} \) \( + ( 5 - 5 \zeta_{6} ) q^{50} \) \( + ( 6 - 3 \zeta_{6} ) q^{51} \) \( -2 \zeta_{6} q^{52} \) \( + 12 q^{53} \) \( + ( 6 - 3 \zeta_{6} ) q^{54} \) \( -2 \zeta_{6} q^{56} \) \( + ( 2 - \zeta_{6} ) q^{57} \) \( + ( -6 + 6 \zeta_{6} ) q^{58} \) \( + ( -3 + 3 \zeta_{6} ) q^{59} \) \( -8 \zeta_{6} q^{61} \) \( -4 q^{62} \) \( -6 q^{63} \) \(+ q^{64}\) \( + ( -3 + 6 \zeta_{6} ) q^{66} \) \( + ( -5 + 5 \zeta_{6} ) q^{67} \) \( + ( 3 - 3 \zeta_{6} ) q^{68} \) \( + ( -6 + 12 \zeta_{6} ) q^{69} \) \( -12 q^{71} \) \( + ( 3 - 3 \zeta_{6} ) q^{72} \) \( + 11 q^{73} \) \( + 4 \zeta_{6} q^{74} \) \( + ( -5 - 5 \zeta_{6} ) q^{75} \) \( + ( 1 - \zeta_{6} ) q^{76} \) \( + ( 6 - 6 \zeta_{6} ) q^{77} \) \( + ( -4 + 2 \zeta_{6} ) q^{78} \) \( + 4 \zeta_{6} q^{79} \) \( -9 \zeta_{6} q^{81} \) \( + 9 q^{82} \) \( -12 \zeta_{6} q^{83} \) \( + ( -4 + 2 \zeta_{6} ) q^{84} \) \( + ( 1 - \zeta_{6} ) q^{86} \) \( + ( 6 + 6 \zeta_{6} ) q^{87} \) \( + 3 \zeta_{6} q^{88} \) \( + 6 q^{89} \) \( + 4 q^{91} \) \( + 6 \zeta_{6} q^{92} \) \( + ( -4 + 8 \zeta_{6} ) q^{93} \) \( + ( 6 - 6 \zeta_{6} ) q^{94} \) \( + ( 1 - 2 \zeta_{6} ) q^{96} \) \( -5 \zeta_{6} q^{97} \) \( -3 q^{98} \) \( + 9 q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(2q \) \(\mathstrut -\mathstrut q^{2} \) \(\mathstrut -\mathstrut 3q^{3} \) \(\mathstrut -\mathstrut q^{4} \) \(\mathstrut +\mathstrut 3q^{6} \) \(\mathstrut -\mathstrut 2q^{7} \) \(\mathstrut +\mathstrut 2q^{8} \) \(\mathstrut +\mathstrut 3q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(2q \) \(\mathstrut -\mathstrut q^{2} \) \(\mathstrut -\mathstrut 3q^{3} \) \(\mathstrut -\mathstrut q^{4} \) \(\mathstrut +\mathstrut 3q^{6} \) \(\mathstrut -\mathstrut 2q^{7} \) \(\mathstrut +\mathstrut 2q^{8} \) \(\mathstrut +\mathstrut 3q^{9} \) \(\mathstrut +\mathstrut 3q^{11} \) \(\mathstrut -\mathstrut 2q^{13} \) \(\mathstrut -\mathstrut 2q^{14} \) \(\mathstrut -\mathstrut q^{16} \) \(\mathstrut -\mathstrut 6q^{17} \) \(\mathstrut -\mathstrut 6q^{18} \) \(\mathstrut -\mathstrut 2q^{19} \) \(\mathstrut +\mathstrut 6q^{21} \) \(\mathstrut +\mathstrut 3q^{22} \) \(\mathstrut +\mathstrut 6q^{23} \) \(\mathstrut -\mathstrut 3q^{24} \) \(\mathstrut +\mathstrut 5q^{25} \) \(\mathstrut +\mathstrut 4q^{26} \) \(\mathstrut +\mathstrut 4q^{28} \) \(\mathstrut -\mathstrut 6q^{29} \) \(\mathstrut +\mathstrut 4q^{31} \) \(\mathstrut -\mathstrut q^{32} \) \(\mathstrut -\mathstrut 9q^{33} \) \(\mathstrut +\mathstrut 3q^{34} \) \(\mathstrut +\mathstrut 3q^{36} \) \(\mathstrut -\mathstrut 8q^{37} \) \(\mathstrut +\mathstrut q^{38} \) \(\mathstrut -\mathstrut 9q^{41} \) \(\mathstrut +\mathstrut q^{43} \) \(\mathstrut -\mathstrut 6q^{44} \) \(\mathstrut -\mathstrut 12q^{46} \) \(\mathstrut +\mathstrut 6q^{47} \) \(\mathstrut +\mathstrut 3q^{48} \) \(\mathstrut +\mathstrut 3q^{49} \) \(\mathstrut +\mathstrut 5q^{50} \) \(\mathstrut +\mathstrut 9q^{51} \) \(\mathstrut -\mathstrut 2q^{52} \) \(\mathstrut +\mathstrut 24q^{53} \) \(\mathstrut +\mathstrut 9q^{54} \) \(\mathstrut -\mathstrut 2q^{56} \) \(\mathstrut +\mathstrut 3q^{57} \) \(\mathstrut -\mathstrut 6q^{58} \) \(\mathstrut -\mathstrut 3q^{59} \) \(\mathstrut -\mathstrut 8q^{61} \) \(\mathstrut -\mathstrut 8q^{62} \) \(\mathstrut -\mathstrut 12q^{63} \) \(\mathstrut +\mathstrut 2q^{64} \) \(\mathstrut -\mathstrut 5q^{67} \) \(\mathstrut +\mathstrut 3q^{68} \) \(\mathstrut -\mathstrut 24q^{71} \) \(\mathstrut +\mathstrut 3q^{72} \) \(\mathstrut +\mathstrut 22q^{73} \) \(\mathstrut +\mathstrut 4q^{74} \) \(\mathstrut -\mathstrut 15q^{75} \) \(\mathstrut +\mathstrut q^{76} \) \(\mathstrut +\mathstrut 6q^{77} \) \(\mathstrut -\mathstrut 6q^{78} \) \(\mathstrut +\mathstrut 4q^{79} \) \(\mathstrut -\mathstrut 9q^{81} \) \(\mathstrut +\mathstrut 18q^{82} \) \(\mathstrut -\mathstrut 12q^{83} \) \(\mathstrut -\mathstrut 6q^{84} \) \(\mathstrut +\mathstrut q^{86} \) \(\mathstrut +\mathstrut 18q^{87} \) \(\mathstrut +\mathstrut 3q^{88} \) \(\mathstrut +\mathstrut 12q^{89} \) \(\mathstrut +\mathstrut 8q^{91} \) \(\mathstrut +\mathstrut 6q^{92} \) \(\mathstrut +\mathstrut 6q^{94} \) \(\mathstrut -\mathstrut 5q^{97} \) \(\mathstrut -\mathstrut 6q^{98} \) \(\mathstrut +\mathstrut 18q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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
−0.500000 + 0.866025i −1.50000 0.866025i −0.500000 0.866025i 0 1.50000 0.866025i −1.00000 + 1.73205i 1.00000 1.50000 + 2.59808i 0
13.1 −0.500000 0.866025i −1.50000 + 0.866025i −0.500000 + 0.866025i 0 1.50000 + 0.866025i −1.00000 1.73205i 1.00000 1.50000 2.59808i 0
\(n\): 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

There are no other newforms in \(S_{2}^{\mathrm{new}}(18, [\chi])\).