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

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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:

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 \(S_{3}^{\mathrm{new}}(18, [\chi])\).