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: \(\Q(\sqrt{-3}) \)
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
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.

\(f(q)\) \(=\) \( q - 2 \zeta_{6} q^{2} + ( - 6 \zeta_{6} + 3) q^{3} + (4 \zeta_{6} - 4) q^{4} + ( - 9 \zeta_{6} + 9) q^{5} + (6 \zeta_{6} - 12) q^{6} + 31 \zeta_{6} q^{7} + 8 q^{8} - 27 q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - 2 \zeta_{6} q^{2} + ( - 6 \zeta_{6} + 3) q^{3} + (4 \zeta_{6} - 4) q^{4} + ( - 9 \zeta_{6} + 9) q^{5} + (6 \zeta_{6} - 12) q^{6} + 31 \zeta_{6} q^{7} + 8 q^{8} - 27 q^{9} - 18 q^{10} + 15 \zeta_{6} q^{11} + (12 \zeta_{6} + 12) q^{12} + ( - 37 \zeta_{6} + 37) q^{13} + ( - 62 \zeta_{6} + 62) q^{14} + ( - 27 \zeta_{6} - 27) q^{15} - 16 \zeta_{6} q^{16} - 42 q^{17} + 54 \zeta_{6} q^{18} - 28 q^{19} + 36 \zeta_{6} q^{20} + ( - 93 \zeta_{6} + 186) q^{21} + ( - 30 \zeta_{6} + 30) q^{22} + (195 \zeta_{6} - 195) q^{23} + ( - 48 \zeta_{6} + 24) q^{24} + 44 \zeta_{6} q^{25} - 74 q^{26} + (162 \zeta_{6} - 81) q^{27} - 124 q^{28} - 111 \zeta_{6} q^{29} + (108 \zeta_{6} - 54) q^{30} + ( - 205 \zeta_{6} + 205) q^{31} + (32 \zeta_{6} - 32) q^{32} + ( - 45 \zeta_{6} + 90) q^{33} + 84 \zeta_{6} q^{34} + 279 q^{35} + ( - 108 \zeta_{6} + 108) q^{36} - 166 q^{37} + 56 \zeta_{6} q^{38} + ( - 111 \zeta_{6} - 111) q^{39} + ( - 72 \zeta_{6} + 72) q^{40} + ( - 261 \zeta_{6} + 261) q^{41} + ( - 186 \zeta_{6} - 186) q^{42} + 43 \zeta_{6} q^{43} - 60 q^{44} + (243 \zeta_{6} - 243) q^{45} + 390 q^{46} - 177 \zeta_{6} q^{47} + (48 \zeta_{6} - 96) q^{48} + (618 \zeta_{6} - 618) q^{49} + ( - 88 \zeta_{6} + 88) q^{50} + (252 \zeta_{6} - 126) q^{51} + 148 \zeta_{6} q^{52} + 114 q^{53} + ( - 162 \zeta_{6} + 324) q^{54} + 135 q^{55} + 248 \zeta_{6} q^{56} + (168 \zeta_{6} - 84) q^{57} + (222 \zeta_{6} - 222) q^{58} + (159 \zeta_{6} - 159) q^{59} + ( - 108 \zeta_{6} + 216) q^{60} - 191 \zeta_{6} q^{61} - 410 q^{62} - 837 \zeta_{6} q^{63} + 64 q^{64} - 333 \zeta_{6} q^{65} + ( - 90 \zeta_{6} - 90) q^{66} + ( - 421 \zeta_{6} + 421) q^{67} + ( - 168 \zeta_{6} + 168) q^{68} + (585 \zeta_{6} + 585) q^{69} - 558 \zeta_{6} q^{70} + 156 q^{71} - 216 q^{72} + 182 q^{73} + 332 \zeta_{6} q^{74} + ( - 132 \zeta_{6} + 264) q^{75} + ( - 112 \zeta_{6} + 112) q^{76} + (465 \zeta_{6} - 465) q^{77} + (444 \zeta_{6} - 222) q^{78} - 1133 \zeta_{6} q^{79} - 144 q^{80} + 729 q^{81} - 522 q^{82} + 1083 \zeta_{6} q^{83} + (744 \zeta_{6} - 372) q^{84} + (378 \zeta_{6} - 378) q^{85} + ( - 86 \zeta_{6} + 86) q^{86} + (333 \zeta_{6} - 666) q^{87} + 120 \zeta_{6} q^{88} - 1050 q^{89} + 486 q^{90} + 1147 q^{91} - 780 \zeta_{6} q^{92} + ( - 615 \zeta_{6} - 615) q^{93} + (354 \zeta_{6} - 354) q^{94} + (252 \zeta_{6} - 252) q^{95} + (96 \zeta_{6} + 96) q^{96} + 901 \zeta_{6} q^{97} + 1236 q^{98} - 405 \zeta_{6} q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} - 4 q^{4} + 9 q^{5} - 18 q^{6} + 31 q^{7} + 16 q^{8} - 54 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{2} - 4 q^{4} + 9 q^{5} - 18 q^{6} + 31 q^{7} + 16 q^{8} - 54 q^{9} - 36 q^{10} + 15 q^{11} + 36 q^{12} + 37 q^{13} + 62 q^{14} - 81 q^{15} - 16 q^{16} - 84 q^{17} + 54 q^{18} - 56 q^{19} + 36 q^{20} + 279 q^{21} + 30 q^{22} - 195 q^{23} + 44 q^{25} - 148 q^{26} - 248 q^{28} - 111 q^{29} + 205 q^{31} - 32 q^{32} + 135 q^{33} + 84 q^{34} + 558 q^{35} + 108 q^{36} - 332 q^{37} + 56 q^{38} - 333 q^{39} + 72 q^{40} + 261 q^{41} - 558 q^{42} + 43 q^{43} - 120 q^{44} - 243 q^{45} + 780 q^{46} - 177 q^{47} - 144 q^{48} - 618 q^{49} + 88 q^{50} + 148 q^{52} + 228 q^{53} + 486 q^{54} + 270 q^{55} + 248 q^{56} - 222 q^{58} - 159 q^{59} + 324 q^{60} - 191 q^{61} - 820 q^{62} - 837 q^{63} + 128 q^{64} - 333 q^{65} - 270 q^{66} + 421 q^{67} + 168 q^{68} + 1755 q^{69} - 558 q^{70} + 312 q^{71} - 432 q^{72} + 364 q^{73} + 332 q^{74} + 396 q^{75} + 112 q^{76} - 465 q^{77} - 1133 q^{79} - 288 q^{80} + 1458 q^{81} - 1044 q^{82} + 1083 q^{83} - 378 q^{85} + 86 q^{86} - 999 q^{87} + 120 q^{88} - 2100 q^{89} + 972 q^{90} + 2294 q^{91} - 780 q^{92} - 1845 q^{93} - 354 q^{94} - 252 q^{95} + 288 q^{96} + 901 q^{97} + 2472 q^{98} - 405 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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 \( T_{5}^{2} - 9T_{5} + 81 \) acting on \(S_{4}^{\mathrm{new}}(18, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 2T + 4 \) Copy content Toggle raw display
$3$ \( T^{2} + 27 \) Copy content Toggle raw display
$5$ \( T^{2} - 9T + 81 \) Copy content Toggle raw display
$7$ \( T^{2} - 31T + 961 \) Copy content Toggle raw display
$11$ \( T^{2} - 15T + 225 \) Copy content Toggle raw display
$13$ \( T^{2} - 37T + 1369 \) Copy content Toggle raw display
$17$ \( (T + 42)^{2} \) Copy content Toggle raw display
$19$ \( (T + 28)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 195T + 38025 \) Copy content Toggle raw display
$29$ \( T^{2} + 111T + 12321 \) Copy content Toggle raw display
$31$ \( T^{2} - 205T + 42025 \) Copy content Toggle raw display
$37$ \( (T + 166)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} - 261T + 68121 \) Copy content Toggle raw display
$43$ \( T^{2} - 43T + 1849 \) Copy content Toggle raw display
$47$ \( T^{2} + 177T + 31329 \) Copy content Toggle raw display
$53$ \( (T - 114)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} + 159T + 25281 \) Copy content Toggle raw display
$61$ \( T^{2} + 191T + 36481 \) Copy content Toggle raw display
$67$ \( T^{2} - 421T + 177241 \) Copy content Toggle raw display
$71$ \( (T - 156)^{2} \) Copy content Toggle raw display
$73$ \( (T - 182)^{2} \) Copy content Toggle raw display
$79$ \( T^{2} + 1133 T + 1283689 \) Copy content Toggle raw display
$83$ \( T^{2} - 1083 T + 1172889 \) Copy content Toggle raw display
$89$ \( (T + 1050)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} - 901T + 811801 \) Copy content Toggle raw display
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