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.

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

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 \(S_{2}^{\mathrm{new}}(18, [\chi])\).