Properties

Label 18.4.c.b
Level 18
Weight 4
Character orbit 18.c
Analytic conductor 1.062
Analytic rank 0
Dimension 4
CM No
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 18 = 2 \cdot 3^{2} \)
Weight: \( k \) = \( 4 \)
Character orbit: \([\chi]\) = 18.c (of order \(3\) and degree \(2\))

Newform invariants

Self dual: No
Analytic conductor: \(1.0620343801\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{-35})\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 3 \)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

\(f(q)\) \(=\) \( q\) \( -2 \beta_{1} q^{2} \) \( + ( -\beta_{1} - \beta_{3} ) q^{3} \) \( + ( -4 - 4 \beta_{1} ) q^{4} \) \( + ( 6 + 6 \beta_{1} + 3 \beta_{3} ) q^{5} \) \( + ( -2 - 2 \beta_{1} - 2 \beta_{2} ) q^{6} \) \( + ( 8 \beta_{1} + 3 \beta_{2} - 3 \beta_{3} ) q^{7} \) \( -8 q^{8} \) \( + ( -1 + 25 \beta_{1} - \beta_{2} - \beta_{3} ) q^{9} \) \(+O(q^{10})\) \( q\) \( -2 \beta_{1} q^{2} \) \( + ( -\beta_{1} - \beta_{3} ) q^{3} \) \( + ( -4 - 4 \beta_{1} ) q^{4} \) \( + ( 6 + 6 \beta_{1} + 3 \beta_{3} ) q^{5} \) \( + ( -2 - 2 \beta_{1} - 2 \beta_{2} ) q^{6} \) \( + ( 8 \beta_{1} + 3 \beta_{2} - 3 \beta_{3} ) q^{7} \) \( -8 q^{8} \) \( + ( -1 + 25 \beta_{1} - \beta_{2} - \beta_{3} ) q^{9} \) \( + ( 12 + 6 \beta_{2} ) q^{10} \) \( + ( -9 \beta_{1} - 6 \beta_{2} + 6 \beta_{3} ) q^{11} \) \( + ( -4 - 4 \beta_{2} + 4 \beta_{3} ) q^{12} \) \( + ( -32 - 32 \beta_{1} - 3 \beta_{3} ) q^{13} \) \( + ( 16 + 16 \beta_{1} - 6 \beta_{3} ) q^{14} \) \( + ( 6 - 78 \beta_{1} + 6 \beta_{2} - 3 \beta_{3} ) q^{15} \) \( + 16 \beta_{1} q^{16} \) \( + ( 3 + 3 \beta_{2} ) q^{17} \) \( + ( 50 + 52 \beta_{1} - 4 \beta_{2} + 2 \beta_{3} ) q^{18} \) \( + ( 59 - 15 \beta_{2} ) q^{19} \) \( + ( -24 \beta_{1} + 12 \beta_{2} - 12 \beta_{3} ) q^{20} \) \( + ( -70 + 8 \beta_{1} + 11 \beta_{2} - 3 \beta_{3} ) q^{21} \) \( + ( -18 - 18 \beta_{1} + 12 \beta_{3} ) q^{22} \) \( + ( -36 - 36 \beta_{1} - 3 \beta_{3} ) q^{23} \) \( + ( 8 \beta_{1} + 8 \beta_{3} ) q^{24} \) \( + ( 145 \beta_{1} - 27 \beta_{2} + 27 \beta_{3} ) q^{25} \) \( + ( -64 - 6 \beta_{2} ) q^{26} \) \( + ( 51 + 78 \beta_{1} + 24 \beta_{2} ) q^{27} \) \( + ( 32 - 12 \beta_{2} ) q^{28} \) \( + ( 108 \beta_{1} + 21 \beta_{2} - 21 \beta_{3} ) q^{29} \) \( + ( -156 - 168 \beta_{1} + 6 \beta_{2} - 12 \beta_{3} ) q^{30} \) \( + ( -110 - 110 \beta_{1} - 9 \beta_{3} ) q^{31} \) \( + ( 32 + 32 \beta_{1} ) q^{32} \) \( + ( 147 - 9 \beta_{1} - 15 \beta_{2} + 6 \beta_{3} ) q^{33} \) \( + ( -6 \beta_{1} + 6 \beta_{2} - 6 \beta_{3} ) q^{34} \) \( + ( 186 - 15 \beta_{2} ) q^{35} \) \( + ( 104 + 4 \beta_{1} - 4 \beta_{2} + 8 \beta_{3} ) q^{36} \) \( + ( 152 + 42 \beta_{2} ) q^{37} \) \( + ( -118 \beta_{1} - 30 \beta_{2} + 30 \beta_{3} ) q^{38} \) \( + ( -32 + 78 \beta_{1} - 32 \beta_{2} + 29 \beta_{3} ) q^{39} \) \( + ( -48 - 48 \beta_{1} - 24 \beta_{3} ) q^{40} \) \( + ( -237 - 237 \beta_{1} - 6 \beta_{3} ) q^{41} \) \( + ( 16 + 156 \beta_{1} + 16 \beta_{2} - 22 \beta_{3} ) q^{42} \) \( + ( -79 \beta_{1} + 72 \beta_{2} - 72 \beta_{3} ) q^{43} \) \( + ( -36 + 24 \beta_{2} ) q^{44} \) \( + ( -234 - 162 \beta_{1} - 72 \beta_{2} - 9 \beta_{3} ) q^{45} \) \( + ( -72 - 6 \beta_{2} ) q^{46} \) \( + ( 264 \beta_{1} - 45 \beta_{2} + 45 \beta_{3} ) q^{47} \) \( + ( 16 + 16 \beta_{1} + 16 \beta_{2} ) q^{48} \) \( + ( 45 + 45 \beta_{1} + 57 \beta_{3} ) q^{49} \) \( + ( 290 + 290 \beta_{1} + 54 \beta_{3} ) q^{50} \) \( + ( -78 - 81 \beta_{1} + 3 \beta_{2} - 3 \beta_{3} ) q^{51} \) \( + ( 128 \beta_{1} - 12 \beta_{2} + 12 \beta_{3} ) q^{52} \) \( + ( 96 + 42 \beta_{2} ) q^{53} \) \( + ( 156 + 54 \beta_{1} + 48 \beta_{2} - 48 \beta_{3} ) q^{54} \) \( + ( -414 + 9 \beta_{2} ) q^{55} \) \( + ( -64 \beta_{1} - 24 \beta_{2} + 24 \beta_{3} ) q^{56} \) \( + ( 390 + 331 \beta_{1} - 15 \beta_{2} - 59 \beta_{3} ) q^{57} \) \( + ( 216 + 216 \beta_{1} - 42 \beta_{3} ) q^{58} \) \( + ( -81 - 81 \beta_{1} + 6 \beta_{3} ) q^{59} \) \( + ( -336 - 24 \beta_{1} - 12 \beta_{2} - 12 \beta_{3} ) q^{60} \) \( + ( -502 \beta_{1} - 45 \beta_{2} + 45 \beta_{3} ) q^{61} \) \( + ( -220 - 18 \beta_{2} ) q^{62} \) \( + ( -278 - 130 \beta_{1} + 19 \beta_{2} + 67 \beta_{3} ) q^{63} \) \( + 64 q^{64} \) \( + ( -426 \beta_{1} + 105 \beta_{2} - 105 \beta_{3} ) q^{65} \) \( + ( -18 - 312 \beta_{1} - 18 \beta_{2} + 30 \beta_{3} ) q^{66} \) \( + ( 565 + 565 \beta_{1} - 36 \beta_{3} ) q^{67} \) \( + ( -12 - 12 \beta_{1} - 12 \beta_{3} ) q^{68} \) \( + ( -36 + 78 \beta_{1} - 36 \beta_{2} + 33 \beta_{3} ) q^{69} \) \( + ( -372 \beta_{1} - 30 \beta_{2} + 30 \beta_{3} ) q^{70} \) \( + ( -204 - 96 \beta_{2} ) q^{71} \) \( + ( 8 - 200 \beta_{1} + 8 \beta_{2} + 8 \beta_{3} ) q^{72} \) \( + ( -187 - 63 \beta_{2} ) q^{73} \) \( + ( -304 \beta_{1} + 84 \beta_{2} - 84 \beta_{3} ) q^{74} \) \( + ( 847 + 145 \beta_{1} + 118 \beta_{2} + 27 \beta_{3} ) q^{75} \) \( + ( -236 - 236 \beta_{1} + 60 \beta_{3} ) q^{76} \) \( + ( 540 + 540 \beta_{1} - 93 \beta_{3} ) q^{77} \) \( + ( 156 + 220 \beta_{1} - 6 \beta_{2} + 64 \beta_{3} ) q^{78} \) \( + ( 194 \beta_{1} - 39 \beta_{2} + 39 \beta_{3} ) q^{79} \) \( + ( -96 - 48 \beta_{2} ) q^{80} \) \( + ( -546 - 597 \beta_{1} + 102 \beta_{2} - 51 \beta_{3} ) q^{81} \) \( + ( -474 - 12 \beta_{2} ) q^{82} \) \( + ( 642 \beta_{1} - 63 \beta_{2} + 63 \beta_{3} ) q^{83} \) \( + ( 312 + 280 \beta_{1} - 12 \beta_{2} - 32 \beta_{3} ) q^{84} \) \( + ( 252 + 252 \beta_{1} + 18 \beta_{3} ) q^{85} \) \( + ( -158 - 158 \beta_{1} - 144 \beta_{3} ) q^{86} \) \( + ( -438 + 108 \beta_{1} + 129 \beta_{2} - 21 \beta_{3} ) q^{87} \) \( + ( 72 \beta_{1} + 48 \beta_{2} - 48 \beta_{3} ) q^{88} \) \( + ( -186 + 120 \beta_{2} ) q^{89} \) \( + ( -324 + 144 \beta_{1} - 162 \beta_{2} + 144 \beta_{3} ) q^{90} \) \( + ( 22 - 63 \beta_{2} ) q^{91} \) \( + ( 144 \beta_{1} - 12 \beta_{2} + 12 \beta_{3} ) q^{92} \) \( + ( -110 + 234 \beta_{1} - 110 \beta_{2} + 101 \beta_{3} ) q^{93} \) \( + ( 528 + 528 \beta_{1} + 90 \beta_{3} ) q^{94} \) \( + ( -816 - 816 \beta_{1} + 132 \beta_{3} ) q^{95} \) \( + ( 32 + 32 \beta_{2} - 32 \beta_{3} ) q^{96} \) \( + ( -31 \beta_{1} - 66 \beta_{2} + 66 \beta_{3} ) q^{97} \) \( + ( 90 + 114 \beta_{2} ) q^{98} \) \( + ( 381 + 78 \beta_{1} - 24 \beta_{2} - 141 \beta_{3} ) q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(4q \) \(\mathstrut +\mathstrut 4q^{2} \) \(\mathstrut +\mathstrut 3q^{3} \) \(\mathstrut -\mathstrut 8q^{4} \) \(\mathstrut +\mathstrut 9q^{5} \) \(\mathstrut -\mathstrut 19q^{7} \) \(\mathstrut -\mathstrut 32q^{8} \) \(\mathstrut -\mathstrut 51q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(4q \) \(\mathstrut +\mathstrut 4q^{2} \) \(\mathstrut +\mathstrut 3q^{3} \) \(\mathstrut -\mathstrut 8q^{4} \) \(\mathstrut +\mathstrut 9q^{5} \) \(\mathstrut -\mathstrut 19q^{7} \) \(\mathstrut -\mathstrut 32q^{8} \) \(\mathstrut -\mathstrut 51q^{9} \) \(\mathstrut +\mathstrut 36q^{10} \) \(\mathstrut +\mathstrut 24q^{11} \) \(\mathstrut -\mathstrut 12q^{12} \) \(\mathstrut -\mathstrut 61q^{13} \) \(\mathstrut +\mathstrut 38q^{14} \) \(\mathstrut +\mathstrut 171q^{15} \) \(\mathstrut -\mathstrut 32q^{16} \) \(\mathstrut +\mathstrut 6q^{17} \) \(\mathstrut +\mathstrut 102q^{18} \) \(\mathstrut +\mathstrut 266q^{19} \) \(\mathstrut +\mathstrut 36q^{20} \) \(\mathstrut -\mathstrut 315q^{21} \) \(\mathstrut -\mathstrut 48q^{22} \) \(\mathstrut -\mathstrut 69q^{23} \) \(\mathstrut -\mathstrut 24q^{24} \) \(\mathstrut -\mathstrut 263q^{25} \) \(\mathstrut -\mathstrut 244q^{26} \) \(\mathstrut +\mathstrut 152q^{28} \) \(\mathstrut -\mathstrut 237q^{29} \) \(\mathstrut -\mathstrut 288q^{30} \) \(\mathstrut -\mathstrut 211q^{31} \) \(\mathstrut +\mathstrut 64q^{32} \) \(\mathstrut +\mathstrut 630q^{33} \) \(\mathstrut +\mathstrut 6q^{34} \) \(\mathstrut +\mathstrut 774q^{35} \) \(\mathstrut +\mathstrut 408q^{36} \) \(\mathstrut +\mathstrut 524q^{37} \) \(\mathstrut +\mathstrut 266q^{38} \) \(\mathstrut -\mathstrut 249q^{39} \) \(\mathstrut -\mathstrut 72q^{40} \) \(\mathstrut -\mathstrut 468q^{41} \) \(\mathstrut -\mathstrut 258q^{42} \) \(\mathstrut +\mathstrut 86q^{43} \) \(\mathstrut -\mathstrut 192q^{44} \) \(\mathstrut -\mathstrut 459q^{45} \) \(\mathstrut -\mathstrut 276q^{46} \) \(\mathstrut -\mathstrut 483q^{47} \) \(\mathstrut +\mathstrut 33q^{49} \) \(\mathstrut +\mathstrut 526q^{50} \) \(\mathstrut -\mathstrut 153q^{51} \) \(\mathstrut -\mathstrut 244q^{52} \) \(\mathstrut +\mathstrut 300q^{53} \) \(\mathstrut +\mathstrut 468q^{54} \) \(\mathstrut -\mathstrut 1674q^{55} \) \(\mathstrut +\mathstrut 152q^{56} \) \(\mathstrut +\mathstrut 987q^{57} \) \(\mathstrut +\mathstrut 474q^{58} \) \(\mathstrut -\mathstrut 168q^{59} \) \(\mathstrut -\mathstrut 1260q^{60} \) \(\mathstrut +\mathstrut 1049q^{61} \) \(\mathstrut -\mathstrut 844q^{62} \) \(\mathstrut -\mathstrut 957q^{63} \) \(\mathstrut +\mathstrut 256q^{64} \) \(\mathstrut +\mathstrut 747q^{65} \) \(\mathstrut +\mathstrut 558q^{66} \) \(\mathstrut +\mathstrut 1166q^{67} \) \(\mathstrut -\mathstrut 12q^{68} \) \(\mathstrut -\mathstrut 261q^{69} \) \(\mathstrut +\mathstrut 774q^{70} \) \(\mathstrut -\mathstrut 624q^{71} \) \(\mathstrut +\mathstrut 408q^{72} \) \(\mathstrut -\mathstrut 622q^{73} \) \(\mathstrut +\mathstrut 524q^{74} \) \(\mathstrut +\mathstrut 2835q^{75} \) \(\mathstrut -\mathstrut 532q^{76} \) \(\mathstrut +\mathstrut 1173q^{77} \) \(\mathstrut +\mathstrut 132q^{78} \) \(\mathstrut -\mathstrut 349q^{79} \) \(\mathstrut -\mathstrut 288q^{80} \) \(\mathstrut -\mathstrut 1143q^{81} \) \(\mathstrut -\mathstrut 1872q^{82} \) \(\mathstrut -\mathstrut 1221q^{83} \) \(\mathstrut +\mathstrut 744q^{84} \) \(\mathstrut +\mathstrut 486q^{85} \) \(\mathstrut -\mathstrut 172q^{86} \) \(\mathstrut -\mathstrut 2205q^{87} \) \(\mathstrut -\mathstrut 192q^{88} \) \(\mathstrut -\mathstrut 984q^{89} \) \(\mathstrut -\mathstrut 1404q^{90} \) \(\mathstrut +\mathstrut 214q^{91} \) \(\mathstrut -\mathstrut 276q^{92} \) \(\mathstrut -\mathstrut 789q^{93} \) \(\mathstrut +\mathstrut 966q^{94} \) \(\mathstrut -\mathstrut 1764q^{95} \) \(\mathstrut +\mathstrut 96q^{96} \) \(\mathstrut +\mathstrut 128q^{97} \) \(\mathstrut +\mathstrut 132q^{98} \) \(\mathstrut +\mathstrut 1557q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4}\mathstrut -\mathstrut \) \(x^{3}\mathstrut -\mathstrut \) \(8\) \(x^{2}\mathstrut -\mathstrut \) \(9\) \(x\mathstrut +\mathstrut \) \(81\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( \nu^{3} + 8 \nu^{2} - 8 \nu - 81 \)\()/72\)
\(\beta_{2}\)\(=\)\((\)\( -\nu^{3} + \nu^{2} + 17 \nu \)\()/9\)
\(\beta_{3}\)\(=\)\((\)\( \nu^{3} + 8 \nu - 17 \)\()/8\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{3}\mathstrut +\mathstrut \) \(\beta_{2}\mathstrut -\mathstrut \) \(\beta_{1}\mathstrut +\mathstrut \) \(1\)\()/3\)
\(\nu^{2}\)\(=\)\((\)\(-\)\(\beta_{3}\mathstrut +\mathstrut \) \(2\) \(\beta_{2}\mathstrut +\mathstrut \) \(25\) \(\beta_{1}\mathstrut +\mathstrut \) \(26\)\()/3\)
\(\nu^{3}\)\(=\)\((\)\(16\) \(\beta_{3}\mathstrut -\mathstrut \) \(8\) \(\beta_{2}\mathstrut +\mathstrut \) \(8\) \(\beta_{1}\mathstrut +\mathstrut \) \(43\)\()/3\)

Character Values

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

\(n\) \(11\)
\(\chi(n)\) \(-1 - \beta_{1}\)

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
2.81174 + 1.04601i
−2.31174 1.91203i
2.81174 1.04601i
−2.31174 + 1.91203i
1.00000 1.73205i −1.81174 4.87007i −2.00000 3.46410i 9.93521 + 17.2083i −10.2470 1.73205i 2.93521 5.08394i −8.00000 −20.4352 + 17.6466i 39.7409
7.2 1.00000 1.73205i 3.31174 + 4.00405i −2.00000 3.46410i −5.43521 9.41407i 10.2470 1.73205i −12.4352 + 21.5384i −8.00000 −5.06479 + 26.5207i −21.7409
13.1 1.00000 + 1.73205i −1.81174 + 4.87007i −2.00000 + 3.46410i 9.93521 17.2083i −10.2470 + 1.73205i 2.93521 + 5.08394i −8.00000 −20.4352 17.6466i 39.7409
13.2 1.00000 + 1.73205i 3.31174 4.00405i −2.00000 + 3.46410i −5.43521 + 9.41407i 10.2470 + 1.73205i −12.4352 21.5384i −8.00000 −5.06479 26.5207i −21.7409
\(n\): 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 \(T_{5}^{4} \) \(\mathstrut -\mathstrut 9 T_{5}^{3} \) \(\mathstrut +\mathstrut 297 T_{5}^{2} \) \(\mathstrut +\mathstrut 1944 T_{5} \) \(\mathstrut +\mathstrut 46656 \) acting on \(S_{4}^{\mathrm{new}}(18, [\chi])\).