Properties

Label 18.6.c.a
Level $18$
Weight $6$
Character orbit 18.c
Analytic conductor $2.887$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 18 = 2 \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 18.c (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(2.88690875663\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{-3})\)
Defining polynomial: \( x^{4} - 2x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2}\cdot 3^{3} \)
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.

\(f(q)\) \(=\) \( q + 4 \beta_1 q^{2} + (\beta_{3} + 6 \beta_1 - 3) q^{3} + (16 \beta_1 - 16) q^{4} + (2 \beta_{3} - 2 \beta_{2} + 27 \beta_1 - 27) q^{5} + (4 \beta_{2} + 12 \beta_1 - 24) q^{6} + (\beta_{2} + 37 \beta_1) q^{7} - 64 q^{8} + ( - 6 \beta_{3} + 12 \beta_{2} + 189) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 4 \beta_1 q^{2} + (\beta_{3} + 6 \beta_1 - 3) q^{3} + (16 \beta_1 - 16) q^{4} + (2 \beta_{3} - 2 \beta_{2} + 27 \beta_1 - 27) q^{5} + (4 \beta_{2} + 12 \beta_1 - 24) q^{6} + (\beta_{2} + 37 \beta_1) q^{7} - 64 q^{8} + ( - 6 \beta_{3} + 12 \beta_{2} + 189) q^{9} + (8 \beta_{3} - 108) q^{10} + ( - 43 \beta_{2} - 39 \beta_1) q^{11} + ( - 16 \beta_{3} + 16 \beta_{2} - 48 \beta_1 - 48) q^{12} + (20 \beta_{3} - 20 \beta_{2} - 553 \beta_1 + 553) q^{13} + ( - 4 \beta_{3} + 4 \beta_{2} + 148 \beta_1 - 148) q^{14} + ( - 21 \beta_{3} + 33 \beta_{2} - 513 \beta_1 + 351) q^{15} - 256 \beta_1 q^{16} + ( - 94 \beta_{3} + 246) q^{17} + ( - 48 \beta_{3} + 24 \beta_{2} + 756 \beta_1) q^{18} + (124 \beta_{3} - 820) q^{19} + (32 \beta_{2} - 432 \beta_1) q^{20} + ( - 6 \beta_{3} + 40 \beta_{2} + 327 \beta_1 - 222) q^{21} + (172 \beta_{3} - 172 \beta_{2} - 156 \beta_1 + 156) q^{22} + (25 \beta_{3} - 25 \beta_{2} + 2769 \beta_1 - 2769) q^{23} + ( - 64 \beta_{3} - 384 \beta_1 + 192) q^{24} + (108 \beta_{2} + 1532 \beta_1) q^{25} + (80 \beta_{3} + 2212) q^{26} + (135 \beta_{3} + 3726 \beta_1 - 1863) q^{27} + ( - 16 \beta_{3} - 592) q^{28} + ( - 124 \beta_{2} - 1947 \beta_1) q^{29} + ( - 132 \beta_{3} + 48 \beta_{2} - 648 \beta_1 + 2052) q^{30} + ( - 435 \beta_{3} + 435 \beta_{2} - 2359 \beta_1 + 2359) q^{31} + ( - 1024 \beta_1 + 1024) q^{32} + (258 \beta_{3} - 168 \beta_{2} - 9405 \beta_1 + 234) q^{33} + ( - 376 \beta_{2} + 984 \beta_1) q^{34} + (47 \beta_{3} - 567) q^{35} + ( - 96 \beta_{3} - 96 \beta_{2} + 3024 \beta_1 - 3024) q^{36} + ( - 826 \beta_{3} - 2398) q^{37} + (496 \beta_{2} - 3280 \beta_1) q^{38} + (613 \beta_{3} - 493 \beta_{2} - 2661 \beta_1 + 5979) q^{39} + ( - 128 \beta_{3} + 128 \beta_{2} - 1728 \beta_1 + 1728) q^{40} + (14 \beta_{3} - 14 \beta_{2} - 7677 \beta_1 + 7677) q^{41} + ( - 160 \beta_{3} + 136 \beta_{2} + 420 \beta_1 - 1308) q^{42} + (267 \beta_{2} + 16429 \beta_1) q^{43} + (688 \beta_{3} + 624) q^{44} + (216 \beta_{3} - 540 \beta_{2} + 7695 \beta_1 - 2511) q^{45} + (100 \beta_{3} - 11076) q^{46} + (249 \beta_{2} + 12477 \beta_1) q^{47} + ( - 256 \beta_{2} - 768 \beta_1 + 1536) q^{48} + ( - 74 \beta_{3} + 74 \beta_{2} - 15222 \beta_1 + 15222) q^{49} + ( - 432 \beta_{3} + 432 \beta_{2} + 6128 \beta_1 - 6128) q^{50} + (528 \beta_{3} - 564 \beta_{2} + 1476 \beta_1 - 21042) q^{51} + (320 \beta_{2} + 8848 \beta_1) q^{52} + (178 \beta_{3} - 8166) q^{53} + (540 \beta_{2} + 7452 \beta_1 - 14904) q^{54} + (1083 \beta_{3} - 17523) q^{55} + ( - 64 \beta_{2} - 2368 \beta_1) q^{56} + ( - 1192 \beta_{3} + 744 \beta_{2} - 4920 \beta_1 + 29244) q^{57} + (496 \beta_{3} - 496 \beta_{2} - 7788 \beta_1 + 7788) q^{58} + ( - 311 \beta_{3} + 311 \beta_{2} - 10983 \beta_1 + 10983) q^{59} + ( - 192 \beta_{3} - 336 \beta_{2} + 5616 \beta_1 + 2592) q^{60} + ( - 708 \beta_{2} + 1525 \beta_1) q^{61} + ( - 1740 \beta_{3} + 9436) q^{62} + ( - 444 \beta_{3} + 411 \beta_{2} + 8289 \beta_1 - 2592) q^{63} + 4096 q^{64} + ( - 566 \beta_{2} + 6291 \beta_1) q^{65} + (672 \beta_{3} + 360 \beta_{2} - 36684 \beta_1 + 37620) q^{66} + (1671 \beta_{3} - 1671 \beta_{2} - 18379 \beta_1 + 18379) q^{67} + (1504 \beta_{3} - 1504 \beta_{2} + 3936 \beta_1 - 3936) q^{68} + ( - 2694 \beta_{3} + 2844 \beta_{2} - 13707 \beta_1 - 2907) q^{69} + (188 \beta_{2} - 2268 \beta_1) q^{70} + ( - 1798 \beta_{3} - 36924) q^{71} + (384 \beta_{3} - 768 \beta_{2} - 12096) q^{72} + (870 \beta_{3} - 25594) q^{73} + ( - 3304 \beta_{2} - 9592 \beta_1) q^{74} + ( - 648 \beta_{3} + 1856 \beta_{2} + 27924 \beta_1 - 9192) q^{75} + ( - 1984 \beta_{3} + 1984 \beta_{2} - 13120 \beta_1 + 13120) q^{76} + (1630 \beta_{3} - 1630 \beta_{2} - 10731 \beta_1 + 10731) q^{77} + (1972 \beta_{3} + 480 \beta_{2} + 13272 \beta_1 + 10644) q^{78} + (707 \beta_{2} - 7463 \beta_1) q^{79} + ( - 512 \beta_{3} + 6912) q^{80} + ( - 2268 \beta_{3} + 4536 \beta_{2} + 12393) q^{81} + (56 \beta_{3} + 30708) q^{82} + (147 \beta_{2} + 45381 \beta_1) q^{83} + ( - 544 \beta_{3} - 96 \beta_{2} - 3552 \beta_1 - 1680) q^{84} + (3030 \beta_{3} - 3030 \beta_{2} + 47250 \beta_1 - 47250) q^{85} + ( - 1068 \beta_{3} + 1068 \beta_{2} + 65716 \beta_1 - 65716) q^{86} + (744 \beta_{3} - 2319 \beta_{2} - 32625 \beta_1 + 11682) q^{87} + (2752 \beta_{2} + 2496 \beta_1) q^{88} + (6086 \beta_{3} - 4650) q^{89} + (2160 \beta_{3} - 1296 \beta_{2} + 20736 \beta_1 - 30780) q^{90} + (1293 \beta_{3} + 24781) q^{91} + (400 \beta_{2} - 44304 \beta_1) q^{92} + (1054 \beta_{3} - 3664 \beta_{2} + 101037 \beta_1 - 86883) q^{93} + ( - 996 \beta_{3} + 996 \beta_{2} + 49908 \beta_1 - 49908) q^{94} + ( - 4988 \beta_{3} + 4988 \beta_{2} - 75708 \beta_1 + 75708) q^{95} + (1024 \beta_{3} - 1024 \beta_{2} + 3072 \beta_1 + 3072) q^{96} + ( - 6574 \beta_{2} - 15131 \beta_1) q^{97} + ( - 296 \beta_{3} + 60888) q^{98} + (468 \beta_{3} - 8361 \beta_{2} - 63099 \beta_1 + 111456) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 8 q^{2} - 32 q^{4} - 54 q^{5} - 72 q^{6} + 74 q^{7} - 256 q^{8} + 756 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 8 q^{2} - 32 q^{4} - 54 q^{5} - 72 q^{6} + 74 q^{7} - 256 q^{8} + 756 q^{9} - 432 q^{10} - 78 q^{11} - 288 q^{12} + 1106 q^{13} - 296 q^{14} + 378 q^{15} - 512 q^{16} + 984 q^{17} + 1512 q^{18} - 3280 q^{19} - 864 q^{20} - 234 q^{21} + 312 q^{22} - 5538 q^{23} + 3064 q^{25} + 8848 q^{26} - 2368 q^{28} - 3894 q^{29} + 6912 q^{30} + 4718 q^{31} + 2048 q^{32} - 17874 q^{33} + 1968 q^{34} - 2268 q^{35} - 6048 q^{36} - 9592 q^{37} - 6560 q^{38} + 18594 q^{39} + 3456 q^{40} + 15354 q^{41} - 4392 q^{42} + 32858 q^{43} + 2496 q^{44} + 5346 q^{45} - 44304 q^{46} + 24954 q^{47} + 4608 q^{48} + 30444 q^{49} - 12256 q^{50} - 81216 q^{51} + 17696 q^{52} - 32664 q^{53} - 44712 q^{54} - 70092 q^{55} - 4736 q^{56} + 107136 q^{57} + 15576 q^{58} + 21966 q^{59} + 21600 q^{60} + 3050 q^{61} + 37744 q^{62} + 6210 q^{63} + 16384 q^{64} + 12582 q^{65} + 77112 q^{66} + 36758 q^{67} - 7872 q^{68} - 39042 q^{69} - 4536 q^{70} - 147696 q^{71} - 48384 q^{72} - 102376 q^{73} - 19184 q^{74} + 19080 q^{75} + 26240 q^{76} + 21462 q^{77} + 69120 q^{78} - 14926 q^{79} + 27648 q^{80} + 49572 q^{81} + 122832 q^{82} + 90762 q^{83} - 13824 q^{84} - 94500 q^{85} - 131432 q^{86} - 18522 q^{87} + 4992 q^{88} - 18600 q^{89} - 81648 q^{90} + 99124 q^{91} - 88608 q^{92} - 145458 q^{93} - 99816 q^{94} + 151416 q^{95} + 18432 q^{96} - 30262 q^{97} + 243552 q^{98} + 319626 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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

\(\beta_{1}\)\(=\) \( ( \nu^{2} ) / 2 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( 3\nu^{3} + 6\nu \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -3\nu^{3} + 12\nu \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{3} + \beta_{2} ) / 18 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 2\beta_1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -\beta_{3} + 2\beta_{2} ) / 9 \) 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
−1.22474 + 0.707107i
1.22474 0.707107i
−1.22474 0.707107i
1.22474 + 0.707107i
2.00000 3.46410i −14.6969 5.19615i −8.00000 13.8564i −28.1969 48.8385i −47.3939 + 40.5194i 11.1515 19.3150i −64.0000 189.000 + 152.735i −225.576
7.2 2.00000 3.46410i 14.6969 5.19615i −8.00000 13.8564i 1.19694 + 2.07316i 11.3939 61.3040i 25.8485 44.7709i −64.0000 189.000 152.735i 9.57551
13.1 2.00000 + 3.46410i −14.6969 + 5.19615i −8.00000 + 13.8564i −28.1969 + 48.8385i −47.3939 40.5194i 11.1515 + 19.3150i −64.0000 189.000 152.735i −225.576
13.2 2.00000 + 3.46410i 14.6969 + 5.19615i −8.00000 + 13.8564i 1.19694 2.07316i 11.3939 + 61.3040i 25.8485 + 44.7709i −64.0000 189.000 + 152.735i 9.57551
\(n\): 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 18.6.c.a 4
3.b odd 2 1 54.6.c.a 4
4.b odd 2 1 144.6.i.a 4
9.c even 3 1 inner 18.6.c.a 4
9.c even 3 1 162.6.a.e 2
9.d odd 6 1 54.6.c.a 4
9.d odd 6 1 162.6.a.f 2
12.b even 2 1 432.6.i.a 4
36.f odd 6 1 144.6.i.a 4
36.h even 6 1 432.6.i.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
18.6.c.a 4 1.a even 1 1 trivial
18.6.c.a 4 9.c even 3 1 inner
54.6.c.a 4 3.b odd 2 1
54.6.c.a 4 9.d odd 6 1
144.6.i.a 4 4.b odd 2 1
144.6.i.a 4 36.f odd 6 1
162.6.a.e 2 9.c even 3 1
162.6.a.f 2 9.d odd 6 1
432.6.i.a 4 12.b even 2 1
432.6.i.a 4 36.h even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{4} + 54T_{5}^{3} + 3051T_{5}^{2} - 7290T_{5} + 18225 \) acting on \(S_{6}^{\mathrm{new}}(18, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} - 4 T + 16)^{2} \) Copy content Toggle raw display
$3$ \( T^{4} - 378 T^{2} + 59049 \) Copy content Toggle raw display
$5$ \( T^{4} + 54 T^{3} + 3051 T^{2} + \cdots + 18225 \) Copy content Toggle raw display
$7$ \( T^{4} - 74 T^{3} + 4323 T^{2} + \cdots + 1329409 \) Copy content Toggle raw display
$11$ \( T^{4} + 78 T^{3} + \cdots + 158294966769 \) Copy content Toggle raw display
$13$ \( T^{4} - 1106 T^{3} + \cdots + 48140309281 \) Copy content Toggle raw display
$17$ \( (T^{2} - 492 T - 1848060)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} + 1640 T - 2648816)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} + 5538 T^{3} + \cdots + 56736462234321 \) Copy content Toggle raw display
$29$ \( T^{4} + 3894 T^{3} + \cdots + 220517585649 \) Copy content Toggle raw display
$31$ \( T^{4} - 4718 T^{3} + \cdots + 12\!\cdots\!61 \) Copy content Toggle raw display
$37$ \( (T^{2} + 4796 T - 141621212)^{2} \) Copy content Toggle raw display
$41$ \( T^{4} - 15354 T^{3} + \cdots + 34\!\cdots\!49 \) Copy content Toggle raw display
$43$ \( T^{4} - 32858 T^{3} + \cdots + 64\!\cdots\!89 \) Copy content Toggle raw display
$47$ \( T^{4} - 24954 T^{3} + \cdots + 20\!\cdots\!69 \) Copy content Toggle raw display
$53$ \( (T^{2} + 16332 T + 59839812)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} - 21966 T^{3} + \cdots + 99\!\cdots\!09 \) Copy content Toggle raw display
$61$ \( T^{4} - 3050 T^{3} + \cdots + 11\!\cdots\!01 \) Copy content Toggle raw display
$67$ \( T^{4} - 36758 T^{3} + \cdots + 70\!\cdots\!25 \) Copy content Toggle raw display
$71$ \( (T^{2} + 73848 T + 665096112)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} + 51188 T + 491562436)^{2} \) Copy content Toggle raw display
$79$ \( T^{4} + 14926 T^{3} + \cdots + 27\!\cdots\!25 \) Copy content Toggle raw display
$83$ \( T^{4} - 90762 T^{3} + \cdots + 42\!\cdots\!89 \) Copy content Toggle raw display
$89$ \( (T^{2} + 9300 T - 7978887036)^{2} \) Copy content Toggle raw display
$97$ \( T^{4} + 30262 T^{3} + \cdots + 82\!\cdots\!25 \) Copy content Toggle raw display
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