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 \(3\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.06203438010\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
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})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{2} - 4q^{4} + 9q^{5} - 18q^{6} + 31q^{7} + 16q^{8} - 54q^{9} + O(q^{10}) \) \( 2q - 2q^{2} - 4q^{4} + 9q^{5} - 18q^{6} + 31q^{7} + 16q^{8} - 54q^{9} - 36q^{10} + 15q^{11} + 36q^{12} + 37q^{13} + 62q^{14} - 81q^{15} - 16q^{16} - 84q^{17} + 54q^{18} - 56q^{19} + 36q^{20} + 279q^{21} + 30q^{22} - 195q^{23} + 44q^{25} - 148q^{26} - 248q^{28} - 111q^{29} + 205q^{31} - 32q^{32} + 135q^{33} + 84q^{34} + 558q^{35} + 108q^{36} - 332q^{37} + 56q^{38} - 333q^{39} + 72q^{40} + 261q^{41} - 558q^{42} + 43q^{43} - 120q^{44} - 243q^{45} + 780q^{46} - 177q^{47} - 144q^{48} - 618q^{49} + 88q^{50} + 148q^{52} + 228q^{53} + 486q^{54} + 270q^{55} + 248q^{56} - 222q^{58} - 159q^{59} + 324q^{60} - 191q^{61} - 820q^{62} - 837q^{63} + 128q^{64} - 333q^{65} - 270q^{66} + 421q^{67} + 168q^{68} + 1755q^{69} - 558q^{70} + 312q^{71} - 432q^{72} + 364q^{73} + 332q^{74} + 396q^{75} + 112q^{76} - 465q^{77} - 1133q^{79} - 288q^{80} + 1458q^{81} - 1044q^{82} + 1083q^{83} - 378q^{85} + 86q^{86} - 999q^{87} + 120q^{88} - 2100q^{89} + 972q^{90} + 2294q^{91} - 780q^{92} - 1845q^{93} - 354q^{94} - 252q^{95} + 288q^{96} + 901q^{97} + 2472q^{98} - 405q^{99} + O(q^{100}) \)

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
−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 Parity Ord Mult Type
1.a even 1 1 trivial
9.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 18.4.c.a 2
3.b odd 2 1 54.4.c.a 2
4.b odd 2 1 144.4.i.a 2
9.c even 3 1 inner 18.4.c.a 2
9.c even 3 1 162.4.a.d 1
9.d odd 6 1 54.4.c.a 2
9.d odd 6 1 162.4.a.a 1
12.b even 2 1 432.4.i.a 2
36.f odd 6 1 144.4.i.a 2
36.f odd 6 1 1296.4.a.b 1
36.h even 6 1 432.4.i.a 2
36.h even 6 1 1296.4.a.g 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
18.4.c.a 2 1.a even 1 1 trivial
18.4.c.a 2 9.c even 3 1 inner
54.4.c.a 2 3.b odd 2 1
54.4.c.a 2 9.d odd 6 1
144.4.i.a 2 4.b odd 2 1
144.4.i.a 2 36.f odd 6 1
162.4.a.a 1 9.d odd 6 1
162.4.a.d 1 9.c even 3 1
432.4.i.a 2 12.b even 2 1
432.4.i.a 2 36.h even 6 1
1296.4.a.b 1 36.f odd 6 1
1296.4.a.g 1 36.h even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} - 9 T_{5} + 81 \) acting on \(S_{4}^{\mathrm{new}}(18, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + 2 T + 4 T^{2} \)
$3$ \( 1 + 27 T^{2} \)
$5$ \( 1 - 9 T - 44 T^{2} - 1125 T^{3} + 15625 T^{4} \)
$7$ \( 1 - 31 T + 618 T^{2} - 10633 T^{3} + 117649 T^{4} \)
$11$ \( 1 - 15 T - 1106 T^{2} - 19965 T^{3} + 1771561 T^{4} \)
$13$ \( 1 - 37 T - 828 T^{2} - 81289 T^{3} + 4826809 T^{4} \)
$17$ \( ( 1 + 42 T + 4913 T^{2} )^{2} \)
$19$ \( ( 1 + 28 T + 6859 T^{2} )^{2} \)
$23$ \( 1 + 195 T + 25858 T^{2} + 2372565 T^{3} + 148035889 T^{4} \)
$29$ \( 1 + 111 T - 12068 T^{2} + 2707179 T^{3} + 594823321 T^{4} \)
$31$ \( 1 - 205 T + 12234 T^{2} - 6107155 T^{3} + 887503681 T^{4} \)
$37$ \( ( 1 + 166 T + 50653 T^{2} )^{2} \)
$41$ \( 1 - 261 T - 800 T^{2} - 17988381 T^{3} + 4750104241 T^{4} \)
$43$ \( 1 - 43 T - 77658 T^{2} - 3418801 T^{3} + 6321363049 T^{4} \)
$47$ \( 1 + 177 T - 72494 T^{2} + 18376671 T^{3} + 10779215329 T^{4} \)
$53$ \( ( 1 - 114 T + 148877 T^{2} )^{2} \)
$59$ \( 1 + 159 T - 180098 T^{2} + 32655261 T^{3} + 42180533641 T^{4} \)
$61$ \( 1 + 191 T - 190500 T^{2} + 43353371 T^{3} + 51520374361 T^{4} \)
$67$ \( 1 - 421 T - 123522 T^{2} - 126621223 T^{3} + 90458382169 T^{4} \)
$71$ \( ( 1 - 156 T + 357911 T^{2} )^{2} \)
$73$ \( ( 1 - 182 T + 389017 T^{2} )^{2} \)
$79$ \( 1 + 1133 T + 790650 T^{2} + 558613187 T^{3} + 243087455521 T^{4} \)
$83$ \( 1 - 1083 T + 601102 T^{2} - 619245321 T^{3} + 326940373369 T^{4} \)
$89$ \( ( 1 + 1050 T + 704969 T^{2} )^{2} \)
$97$ \( 1 - 901 T - 100872 T^{2} - 822318373 T^{3} + 832972004929 T^{4} \)
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