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

Newform invariants

Self dual: no
Analytic conductor: \(1.06203438010\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{-35})\)
Defining polynomial: \( x^{4} - x^{3} - 8x^{2} - 9x + 81 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 3 \)
Twist minimal: yes
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.

\(f(q)\) \(=\) \( q - 2 \beta_1 q^{2} + ( - \beta_{3} - \beta_1) q^{3} + ( - 4 \beta_1 - 4) q^{4} + (3 \beta_{3} + 6 \beta_1 + 6) q^{5} + ( - 2 \beta_{2} - 2 \beta_1 - 2) q^{6} + ( - 3 \beta_{3} + 3 \beta_{2} + 8 \beta_1) q^{7} - 8 q^{8} + ( - \beta_{3} - \beta_{2} + 25 \beta_1 - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 2 \beta_1 q^{2} + ( - \beta_{3} - \beta_1) q^{3} + ( - 4 \beta_1 - 4) q^{4} + (3 \beta_{3} + 6 \beta_1 + 6) q^{5} + ( - 2 \beta_{2} - 2 \beta_1 - 2) q^{6} + ( - 3 \beta_{3} + 3 \beta_{2} + 8 \beta_1) q^{7} - 8 q^{8} + ( - \beta_{3} - \beta_{2} + 25 \beta_1 - 1) q^{9} + (6 \beta_{2} + 12) q^{10} + (6 \beta_{3} - 6 \beta_{2} - 9 \beta_1) q^{11} + (4 \beta_{3} - 4 \beta_{2} - 4) q^{12} + ( - 3 \beta_{3} - 32 \beta_1 - 32) q^{13} + ( - 6 \beta_{3} + 16 \beta_1 + 16) q^{14} + ( - 3 \beta_{3} + 6 \beta_{2} - 78 \beta_1 + 6) q^{15} + 16 \beta_1 q^{16} + (3 \beta_{2} + 3) q^{17} + (2 \beta_{3} - 4 \beta_{2} + 52 \beta_1 + 50) q^{18} + ( - 15 \beta_{2} + 59) q^{19} + ( - 12 \beta_{3} + 12 \beta_{2} - 24 \beta_1) q^{20} + ( - 3 \beta_{3} + 11 \beta_{2} + 8 \beta_1 - 70) q^{21} + (12 \beta_{3} - 18 \beta_1 - 18) q^{22} + ( - 3 \beta_{3} - 36 \beta_1 - 36) q^{23} + (8 \beta_{3} + 8 \beta_1) q^{24} + (27 \beta_{3} - 27 \beta_{2} + 145 \beta_1) q^{25} + ( - 6 \beta_{2} - 64) q^{26} + (24 \beta_{2} + 78 \beta_1 + 51) q^{27} + ( - 12 \beta_{2} + 32) q^{28} + ( - 21 \beta_{3} + 21 \beta_{2} + 108 \beta_1) q^{29} + ( - 12 \beta_{3} + 6 \beta_{2} - 168 \beta_1 - 156) q^{30} + ( - 9 \beta_{3} - 110 \beta_1 - 110) q^{31} + (32 \beta_1 + 32) q^{32} + (6 \beta_{3} - 15 \beta_{2} - 9 \beta_1 + 147) q^{33} + ( - 6 \beta_{3} + 6 \beta_{2} - 6 \beta_1) q^{34} + ( - 15 \beta_{2} + 186) q^{35} + (8 \beta_{3} - 4 \beta_{2} + 4 \beta_1 + 104) q^{36} + (42 \beta_{2} + 152) q^{37} + (30 \beta_{3} - 30 \beta_{2} - 118 \beta_1) q^{38} + (29 \beta_{3} - 32 \beta_{2} + 78 \beta_1 - 32) q^{39} + ( - 24 \beta_{3} - 48 \beta_1 - 48) q^{40} + ( - 6 \beta_{3} - 237 \beta_1 - 237) q^{41} + ( - 22 \beta_{3} + 16 \beta_{2} + 156 \beta_1 + 16) q^{42} + ( - 72 \beta_{3} + 72 \beta_{2} - 79 \beta_1) q^{43} + (24 \beta_{2} - 36) q^{44} + ( - 9 \beta_{3} - 72 \beta_{2} - 162 \beta_1 - 234) q^{45} + ( - 6 \beta_{2} - 72) q^{46} + (45 \beta_{3} - 45 \beta_{2} + 264 \beta_1) q^{47} + (16 \beta_{2} + 16 \beta_1 + 16) q^{48} + (57 \beta_{3} + 45 \beta_1 + 45) q^{49} + (54 \beta_{3} + 290 \beta_1 + 290) q^{50} + ( - 3 \beta_{3} + 3 \beta_{2} - 81 \beta_1 - 78) q^{51} + (12 \beta_{3} - 12 \beta_{2} + 128 \beta_1) q^{52} + (42 \beta_{2} + 96) q^{53} + ( - 48 \beta_{3} + 48 \beta_{2} + 54 \beta_1 + 156) q^{54} + (9 \beta_{2} - 414) q^{55} + (24 \beta_{3} - 24 \beta_{2} - 64 \beta_1) q^{56} + ( - 59 \beta_{3} - 15 \beta_{2} + 331 \beta_1 + 390) q^{57} + ( - 42 \beta_{3} + 216 \beta_1 + 216) q^{58} + (6 \beta_{3} - 81 \beta_1 - 81) q^{59} + ( - 12 \beta_{3} - 12 \beta_{2} - 24 \beta_1 - 336) q^{60} + (45 \beta_{3} - 45 \beta_{2} - 502 \beta_1) q^{61} + ( - 18 \beta_{2} - 220) q^{62} + (67 \beta_{3} + 19 \beta_{2} - 130 \beta_1 - 278) q^{63} + 64 q^{64} + ( - 105 \beta_{3} + 105 \beta_{2} - 426 \beta_1) q^{65} + (30 \beta_{3} - 18 \beta_{2} - 312 \beta_1 - 18) q^{66} + ( - 36 \beta_{3} + 565 \beta_1 + 565) q^{67} + ( - 12 \beta_{3} - 12 \beta_1 - 12) q^{68} + (33 \beta_{3} - 36 \beta_{2} + 78 \beta_1 - 36) q^{69} + (30 \beta_{3} - 30 \beta_{2} - 372 \beta_1) q^{70} + ( - 96 \beta_{2} - 204) q^{71} + (8 \beta_{3} + 8 \beta_{2} - 200 \beta_1 + 8) q^{72} + ( - 63 \beta_{2} - 187) q^{73} + ( - 84 \beta_{3} + 84 \beta_{2} - 304 \beta_1) q^{74} + (27 \beta_{3} + 118 \beta_{2} + 145 \beta_1 + 847) q^{75} + (60 \beta_{3} - 236 \beta_1 - 236) q^{76} + ( - 93 \beta_{3} + 540 \beta_1 + 540) q^{77} + (64 \beta_{3} - 6 \beta_{2} + 220 \beta_1 + 156) q^{78} + (39 \beta_{3} - 39 \beta_{2} + 194 \beta_1) q^{79} + ( - 48 \beta_{2} - 96) q^{80} + ( - 51 \beta_{3} + 102 \beta_{2} - 597 \beta_1 - 546) q^{81} + ( - 12 \beta_{2} - 474) q^{82} + (63 \beta_{3} - 63 \beta_{2} + 642 \beta_1) q^{83} + ( - 32 \beta_{3} - 12 \beta_{2} + 280 \beta_1 + 312) q^{84} + (18 \beta_{3} + 252 \beta_1 + 252) q^{85} + ( - 144 \beta_{3} - 158 \beta_1 - 158) q^{86} + ( - 21 \beta_{3} + 129 \beta_{2} + 108 \beta_1 - 438) q^{87} + ( - 48 \beta_{3} + 48 \beta_{2} + 72 \beta_1) q^{88} + (120 \beta_{2} - 186) q^{89} + (144 \beta_{3} - 162 \beta_{2} + 144 \beta_1 - 324) q^{90} + ( - 63 \beta_{2} + 22) q^{91} + (12 \beta_{3} - 12 \beta_{2} + 144 \beta_1) q^{92} + (101 \beta_{3} - 110 \beta_{2} + 234 \beta_1 - 110) q^{93} + (90 \beta_{3} + 528 \beta_1 + 528) q^{94} + (132 \beta_{3} - 816 \beta_1 - 816) q^{95} + ( - 32 \beta_{3} + 32 \beta_{2} + 32) q^{96} + (66 \beta_{3} - 66 \beta_{2} - 31 \beta_1) q^{97} + (114 \beta_{2} + 90) q^{98} + ( - 141 \beta_{3} - 24 \beta_{2} + 78 \beta_1 + 381) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{2} + 3 q^{3} - 8 q^{4} + 9 q^{5} - 19 q^{7} - 32 q^{8} - 51 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 4 q^{2} + 3 q^{3} - 8 q^{4} + 9 q^{5} - 19 q^{7} - 32 q^{8} - 51 q^{9} + 36 q^{10} + 24 q^{11} - 12 q^{12} - 61 q^{13} + 38 q^{14} + 171 q^{15} - 32 q^{16} + 6 q^{17} + 102 q^{18} + 266 q^{19} + 36 q^{20} - 315 q^{21} - 48 q^{22} - 69 q^{23} - 24 q^{24} - 263 q^{25} - 244 q^{26} + 152 q^{28} - 237 q^{29} - 288 q^{30} - 211 q^{31} + 64 q^{32} + 630 q^{33} + 6 q^{34} + 774 q^{35} + 408 q^{36} + 524 q^{37} + 266 q^{38} - 249 q^{39} - 72 q^{40} - 468 q^{41} - 258 q^{42} + 86 q^{43} - 192 q^{44} - 459 q^{45} - 276 q^{46} - 483 q^{47} + 33 q^{49} + 526 q^{50} - 153 q^{51} - 244 q^{52} + 300 q^{53} + 468 q^{54} - 1674 q^{55} + 152 q^{56} + 987 q^{57} + 474 q^{58} - 168 q^{59} - 1260 q^{60} + 1049 q^{61} - 844 q^{62} - 957 q^{63} + 256 q^{64} + 747 q^{65} + 558 q^{66} + 1166 q^{67} - 12 q^{68} - 261 q^{69} + 774 q^{70} - 624 q^{71} + 408 q^{72} - 622 q^{73} + 524 q^{74} + 2835 q^{75} - 532 q^{76} + 1173 q^{77} + 132 q^{78} - 349 q^{79} - 288 q^{80} - 1143 q^{81} - 1872 q^{82} - 1221 q^{83} + 744 q^{84} + 486 q^{85} - 172 q^{86} - 2205 q^{87} - 192 q^{88} - 984 q^{89} - 1404 q^{90} + 214 q^{91} - 276 q^{92} - 789 q^{93} + 966 q^{94} - 1764 q^{95} + 96 q^{96} + 128 q^{97} + 132 q^{98} + 1557 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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

\(\beta_{1}\)\(=\) \( ( \nu^{3} + 8\nu^{2} - 8\nu - 81 ) / 72 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{3} + \nu^{2} + 17\nu ) / 9 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} + 8\nu - 17 ) / 8 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{3} + \beta_{2} - \beta _1 + 1 ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{3} + 2\beta_{2} + 25\beta _1 + 26 ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 16\beta_{3} - 8\beta_{2} + 8\beta _1 + 43 ) / 3 \) Copy content Toggle raw display

Character values

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

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

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

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{4} - 9T_{5}^{3} + 297T_{5}^{2} + 1944T_{5} + 46656 \) acting on \(S_{4}^{\mathrm{new}}(18, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} - 2 T + 4)^{2} \) Copy content Toggle raw display
$3$ \( T^{4} - 3 T^{3} + 30 T^{2} - 81 T + 729 \) Copy content Toggle raw display
$5$ \( T^{4} - 9 T^{3} + 297 T^{2} + \cdots + 46656 \) Copy content Toggle raw display
$7$ \( T^{4} + 19 T^{3} + 507 T^{2} + \cdots + 21316 \) Copy content Toggle raw display
$11$ \( T^{4} - 24 T^{3} + 1377 T^{2} + \cdots + 641601 \) Copy content Toggle raw display
$13$ \( T^{4} + 61 T^{3} + 3027 T^{2} + \cdots + 481636 \) Copy content Toggle raw display
$17$ \( (T^{2} - 3 T - 234)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} - 133 T - 1484)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} + 69 T^{3} + 3807 T^{2} + \cdots + 910116 \) Copy content Toggle raw display
$29$ \( T^{4} + 237 T^{3} + 53703 T^{2} + \cdots + 6081156 \) Copy content Toggle raw display
$31$ \( T^{4} + 211 T^{3} + \cdots + 81072016 \) Copy content Toggle raw display
$37$ \( (T^{2} - 262 T - 29144)^{2} \) Copy content Toggle raw display
$41$ \( T^{4} + 468 T^{3} + \cdots + 2895623721 \) Copy content Toggle raw display
$43$ \( T^{4} - 86 T^{3} + \cdots + 18017961361 \) Copy content Toggle raw display
$47$ \( T^{4} + 483 T^{3} + \cdots + 26687556 \) Copy content Toggle raw display
$53$ \( (T^{2} - 150 T - 40680)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} + 168 T^{3} + \cdots + 37344321 \) Copy content Toggle raw display
$61$ \( T^{4} - 1049 T^{3} + \cdots + 49259139136 \) Copy content Toggle raw display
$67$ \( T^{4} - 1166 T^{3} + \cdots + 93555845161 \) Copy content Toggle raw display
$71$ \( (T^{2} + 312 T - 217584)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} + 311 T - 80006)^{2} \) Copy content Toggle raw display
$79$ \( T^{4} + 349 T^{3} + \cdots + 89794576 \) Copy content Toggle raw display
$83$ \( T^{4} + 1221 T^{3} + \cdots + 72105138576 \) Copy content Toggle raw display
$89$ \( (T^{2} + 492 T - 317484)^{2} \) Copy content Toggle raw display
$97$ \( T^{4} - 128 T^{3} + \cdots + 12154842001 \) Copy content Toggle raw display
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