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: \(\Q(\sqrt{-3}, \sqrt{-35})\)
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.

\(f(q)\) \(=\) \( q -2 \beta_{1} q^{2} + ( -\beta_{1} - \beta_{3} ) q^{3} + ( -4 - 4 \beta_{1} ) q^{4} + ( 6 + 6 \beta_{1} + 3 \beta_{3} ) q^{5} + ( -2 - 2 \beta_{1} - 2 \beta_{2} ) q^{6} + ( 8 \beta_{1} + 3 \beta_{2} - 3 \beta_{3} ) q^{7} -8 q^{8} + ( -1 + 25 \beta_{1} - \beta_{2} - \beta_{3} ) q^{9} +O(q^{10})\) \( 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})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 4q^{2} + 3q^{3} - 8q^{4} + 9q^{5} - 19q^{7} - 32q^{8} - 51q^{9} + O(q^{10}) \) \( 4q + 4q^{2} + 3q^{3} - 8q^{4} + 9q^{5} - 19q^{7} - 32q^{8} - 51q^{9} + 36q^{10} + 24q^{11} - 12q^{12} - 61q^{13} + 38q^{14} + 171q^{15} - 32q^{16} + 6q^{17} + 102q^{18} + 266q^{19} + 36q^{20} - 315q^{21} - 48q^{22} - 69q^{23} - 24q^{24} - 263q^{25} - 244q^{26} + 152q^{28} - 237q^{29} - 288q^{30} - 211q^{31} + 64q^{32} + 630q^{33} + 6q^{34} + 774q^{35} + 408q^{36} + 524q^{37} + 266q^{38} - 249q^{39} - 72q^{40} - 468q^{41} - 258q^{42} + 86q^{43} - 192q^{44} - 459q^{45} - 276q^{46} - 483q^{47} + 33q^{49} + 526q^{50} - 153q^{51} - 244q^{52} + 300q^{53} + 468q^{54} - 1674q^{55} + 152q^{56} + 987q^{57} + 474q^{58} - 168q^{59} - 1260q^{60} + 1049q^{61} - 844q^{62} - 957q^{63} + 256q^{64} + 747q^{65} + 558q^{66} + 1166q^{67} - 12q^{68} - 261q^{69} + 774q^{70} - 624q^{71} + 408q^{72} - 622q^{73} + 524q^{74} + 2835q^{75} - 532q^{76} + 1173q^{77} + 132q^{78} - 349q^{79} - 288q^{80} - 1143q^{81} - 1872q^{82} - 1221q^{83} + 744q^{84} + 486q^{85} - 172q^{86} - 2205q^{87} - 192q^{88} - 984q^{89} - 1404q^{90} + 214q^{91} - 276q^{92} - 789q^{93} + 966q^{94} - 1764q^{95} + 96q^{96} + 128q^{97} + 132q^{98} + 1557q^{99} + O(q^{100}) \)

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

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

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 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 \( T_{5}^{4} - 9 T_{5}^{3} + 297 T_{5}^{2} + 1944 T_{5} + 46656 \) acting on \(S_{4}^{\mathrm{new}}(18, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 - 2 T + 4 T^{2} )^{2} \)
$3$ \( 1 - 3 T + 30 T^{2} - 81 T^{3} + 729 T^{4} \)
$5$ \( 1 - 9 T + 47 T^{2} + 1944 T^{3} - 24594 T^{4} + 243000 T^{5} + 734375 T^{6} - 17578125 T^{7} + 244140625 T^{8} \)
$7$ \( 1 + 19 T - 179 T^{2} - 2774 T^{3} + 50128 T^{4} - 951482 T^{5} - 21059171 T^{6} + 766718533 T^{7} + 13841287201 T^{8} \)
$11$ \( 1 - 24 T - 1285 T^{2} + 19224 T^{3} + 925104 T^{4} + 25587144 T^{5} - 2276455885 T^{6} - 56590744584 T^{7} + 3138428376721 T^{8} \)
$13$ \( 1 + 61 T - 1367 T^{2} + 42334 T^{3} + 12885898 T^{4} + 93007798 T^{5} - 6598247903 T^{6} + 646874461753 T^{7} + 23298085122481 T^{8} \)
$17$ \( ( 1 - 3 T + 9592 T^{2} - 14739 T^{3} + 24137569 T^{4} )^{2} \)
$19$ \( ( 1 - 133 T + 12234 T^{2} - 912247 T^{3} + 47045881 T^{4} )^{2} \)
$23$ \( 1 + 69 T - 20527 T^{2} + 65826 T^{3} + 433519968 T^{4} + 800904942 T^{5} - 3038732693503 T^{6} + 124279533640947 T^{7} + 21914624432020321 T^{8} \)
$29$ \( 1 + 237 T + 4925 T^{2} + 584442 T^{3} + 661218474 T^{4} + 14253955938 T^{5} + 2929504855925 T^{6} + 3438193596280953 T^{7} + 353814783205469041 T^{8} \)
$31$ \( 1 + 211 T - 24065 T^{2} + 1899844 T^{3} + 2490210604 T^{4} + 56598252604 T^{5} - 21357776083265 T^{6} + 5578760275901581 T^{7} + 787662783788549761 T^{8} \)
$37$ \( ( 1 - 262 T + 72162 T^{2} - 13271086 T^{3} + 2565726409 T^{4} )^{2} \)
$41$ \( 1 + 468 T + 27371 T^{2} + 25183548 T^{3} + 16885415064 T^{4} + 1735675311708 T^{5} + 130015103180411 T^{6} + 153214745296373748 T^{7} + 22563490300366186081 T^{8} \)
$43$ \( 1 - 86 T - 17387 T^{2} + 11543866 T^{3} - 6295199732 T^{4} + 917818154062 T^{5} - 109909539332963 T^{6} - 43222964626568498 T^{7} + 39959630797262576401 T^{8} \)
$47$ \( 1 + 483 T + 20477 T^{2} + 2495178 T^{3} + 10288968168 T^{4} + 259056865494 T^{5} + 220725992291933 T^{6} + 540540018508636461 T^{7} + \)\(11\!\cdots\!41\)\( T^{8} \)
$53$ \( ( 1 - 150 T + 257074 T^{2} - 22331550 T^{3} + 22164361129 T^{4} )^{2} \)
$59$ \( 1 + 168 T - 388645 T^{2} + 1026648 T^{3} + 125802612624 T^{4} + 210851939592 T^{5} - 16393253496906445 T^{6} + 1455383297534029752 T^{7} + \)\(17\!\cdots\!81\)\( T^{8} \)
$61$ \( 1 - 1049 T + 424495 T^{2} - 232819256 T^{3} + 155558427094 T^{4} - 52845547546136 T^{5} + 21870141314372695 T^{6} - 12267159251383013909 T^{7} + \)\(26\!\cdots\!21\)\( T^{8} \)
$67$ \( 1 - 1166 T + 452161 T^{2} - 356643254 T^{3} + 324003162628 T^{4} - 107265095002802 T^{5} + 40901752539917209 T^{6} - 31722819106079908202 T^{7} + \)\(81\!\cdots\!61\)\( T^{8} \)
$71$ \( ( 1 + 312 T + 498238 T^{2} + 111668232 T^{3} + 128100283921 T^{4} )^{2} \)
$73$ \( ( 1 + 311 T + 698028 T^{2} + 120984287 T^{3} + 151334226289 T^{4} )^{2} \)
$79$ \( 1 + 349 T - 854801 T^{2} - 3307124 T^{3} + 650611367644 T^{4} - 1630541109836 T^{5} - 207791400066806321 T^{6} + 41828206997933793331 T^{7} + \)\(59\!\cdots\!41\)\( T^{8} \)
$83$ \( 1 + 1221 T + 78743 T^{2} + 327867804 T^{3} + 814636885368 T^{4} + 187470548045748 T^{5} + 25744265820195167 T^{6} + \)\(22\!\cdots\!63\)\( T^{7} + \)\(10\!\cdots\!61\)\( T^{8} \)
$89$ \( ( 1 + 492 T + 1092454 T^{2} + 346844748 T^{3} + 496981290961 T^{4} )^{2} \)
$97$ \( 1 - 128 T - 1698713 T^{2} + 14111872 T^{3} + 2093632480048 T^{4} + 12879524553856 T^{5} - 1414980373408956377 T^{6} - 97309575507784347776 T^{7} + \)\(69\!\cdots\!41\)\( T^{8} \)
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