Properties

Label 18.3.d.a
Level $18$
Weight $3$
Character orbit 18.d
Analytic conductor $0.490$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(0.490464475849\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
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: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (\beta_{3} - 2 \beta_{2} - 2 \beta_1 + 1) q^{3} + 2 \beta_{2} q^{4} + (3 \beta_{2} - 6) q^{5} + ( - 2 \beta_{3} - 2 \beta_{2} + \beta_1 - 2) q^{6} + ( - 6 \beta_{3} - \beta_{2} + 3 \beta_1 + 1) q^{7} + 2 \beta_{3} q^{8} + (6 \beta_{3} + 3) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + (\beta_{3} - 2 \beta_{2} - 2 \beta_1 + 1) q^{3} + 2 \beta_{2} q^{4} + (3 \beta_{2} - 6) q^{5} + ( - 2 \beta_{3} - 2 \beta_{2} + \beta_1 - 2) q^{6} + ( - 6 \beta_{3} - \beta_{2} + 3 \beta_1 + 1) q^{7} + 2 \beta_{3} q^{8} + (6 \beta_{3} + 3) q^{9} + (3 \beta_{3} - 6 \beta_1) q^{10} + (3 \beta_{2} - 3 \beta_1 + 3) q^{11} + ( - 2 \beta_{3} - 2 \beta_{2} - 2 \beta_1 + 4) q^{12} + (6 \beta_{3} - 5 \beta_{2} + 6 \beta_1) q^{13} + ( - \beta_{3} - 6 \beta_{2} + \beta_1 + 12) q^{14} + ( - 9 \beta_{3} + 9 \beta_{2} + 9 \beta_1) q^{15} + (4 \beta_{2} - 4) q^{16} + ( - 6 \beta_{3} - 12 \beta_{2} + 6) q^{17} + (12 \beta_{2} + 3 \beta_1 - 12) q^{18} + (6 \beta_{3} - 12 \beta_1 - 10) q^{19} + ( - 6 \beta_{2} - 6) q^{20} + (2 \beta_{3} + 17 \beta_{2} - 10 \beta_1 - 19) q^{21} + (3 \beta_{3} - 6 \beta_{2} + 3 \beta_1) q^{22} + (3 \beta_{3} - 3 \beta_{2} - 3 \beta_1 + 6) q^{23} + ( - 2 \beta_{3} - 8 \beta_{2} + 4 \beta_1 + 4) q^{24} + ( - 2 \beta_{2} + 2) q^{25} + ( - 5 \beta_{3} + 24 \beta_{2} - 12) q^{26} + ( - 3 \beta_{3} - 30 \beta_{2} + 6 \beta_1 + 15) q^{27} + ( - 6 \beta_{3} + 12 \beta_1 + 2) q^{28} + (3 \beta_{2} + 6 \beta_1 + 3) q^{29} + (9 \beta_{3} + 18) q^{30} + ( - 9 \beta_{3} + 19 \beta_{2} - 9 \beta_1) q^{31} + (4 \beta_{3} - 4 \beta_1) q^{32} + (6 \beta_{3} - 3 \beta_{2} - 12 \beta_1 + 15) q^{33} + ( - 12 \beta_{3} - 12 \beta_{2} + 6 \beta_1 + 12) q^{34} + (27 \beta_{3} + 6 \beta_{2} - 3) q^{35} + (12 \beta_{3} + 6 \beta_{2} - 12 \beta_1) q^{36} + ( - 6 \beta_{3} + 12 \beta_1 + 32) q^{37} + ( - 12 \beta_{2} - 10 \beta_1 - 12) q^{38} + ( - 13 \beta_{3} - 31 \beta_{2} + 23 \beta_1 - 10) q^{39} + ( - 6 \beta_{3} - 6 \beta_1) q^{40} + ( - 18 \beta_{3} + 21 \beta_{2} + 18 \beta_1 - 42) q^{41} + (17 \beta_{3} - 16 \beta_{2} - 19 \beta_1 - 4) q^{42} + (18 \beta_{3} + 23 \beta_{2} - 9 \beta_1 - 23) q^{43} + ( - 6 \beta_{3} + 12 \beta_{2} - 6) q^{44} + ( - 18 \beta_{3} + 9 \beta_{2} - 18 \beta_1 - 18) q^{45} + ( - 3 \beta_{3} + 6 \beta_1 - 6) q^{46} + (9 \beta_{2} + 21 \beta_1 + 9) q^{47} + ( - 8 \beta_{3} + 4 \beta_{2} + 4 \beta_1 + 4) q^{48} + ( - 6 \beta_{3} - 6 \beta_{2} - 6 \beta_1) q^{49} + ( - 2 \beta_{3} + 2 \beta_1) q^{50} + (24 \beta_{3} + 24 \beta_{2} - 12 \beta_1 - 30) q^{51} + (24 \beta_{3} - 10 \beta_{2} - 12 \beta_1 + 10) q^{52} + ( - 30 \beta_{3} - 60 \beta_{2} + 30) q^{53} + ( - 30 \beta_{3} + 6 \beta_{2} + 15 \beta_1 + 6) q^{54} + ( - 9 \beta_{3} + 18 \beta_1 - 27) q^{55} + (12 \beta_{2} + 2 \beta_1 + 12) q^{56} + (8 \beta_{3} + 20 \beta_{2} + 20 \beta_1 + 26) q^{57} + (3 \beta_{3} + 12 \beta_{2} + 3 \beta_1) q^{58} + (39 \beta_{3} - 21 \beta_{2} - 39 \beta_1 + 42) q^{59} + (18 \beta_{2} + 18 \beta_1 - 18) q^{60} + (36 \beta_{3} - 31 \beta_{2} - 18 \beta_1 + 31) q^{61} + (19 \beta_{3} - 36 \beta_{2} + 18) q^{62} + ( - 18 \beta_{3} + 33 \beta_{2} + 15 \beta_1 + 39) q^{63} - 8 q^{64} + (15 \beta_{2} - 54 \beta_1 + 15) q^{65} + ( - 3 \beta_{3} - 12 \beta_{2} + 15 \beta_1 - 12) q^{66} + ( - 9 \beta_{3} - 53 \beta_{2} - 9 \beta_1) q^{67} + ( - 12 \beta_{3} - 12 \beta_{2} + 12 \beta_1 + 24) q^{68} + (12 \beta_{3} - 15 \beta_{2} - 6 \beta_1 + 12) q^{69} + (6 \beta_{3} + 54 \beta_{2} - 3 \beta_1 - 54) q^{70} + ( - 24 \beta_{3} + 60 \beta_{2} - 30) q^{71} + (6 \beta_{3} - 24) q^{72} + ( - 18 \beta_{3} + 36 \beta_1 - 52) q^{73} + (12 \beta_{2} + 32 \beta_1 + 12) q^{74} + (4 \beta_{3} - 2 \beta_{2} - 2 \beta_1 - 2) q^{75} + ( - 12 \beta_{3} - 20 \beta_{2} - 12 \beta_1) q^{76} + ( - 24 \beta_{3} + 15 \beta_{2} + 24 \beta_1 - 30) q^{77} + ( - 31 \beta_{3} + 20 \beta_{2} - 10 \beta_1 + 26) q^{78} + ( - 30 \beta_{3} - 7 \beta_{2} + 15 \beta_1 + 7) q^{79} + ( - 24 \beta_{2} + 12) q^{80} + (36 \beta_{3} - 63) q^{81} + (21 \beta_{3} - 42 \beta_1 + 36) q^{82} + ( - 63 \beta_{2} - 15 \beta_1 - 63) q^{83} + ( - 16 \beta_{3} - 4 \beta_{2} - 4 \beta_1 - 34) q^{84} + (18 \beta_{3} + 54 \beta_{2} + 18 \beta_1) q^{85} + (23 \beta_{3} + 18 \beta_{2} - 23 \beta_1 - 36) q^{86} + ( - 12 \beta_{3} - 21 \beta_{2} - 3 \beta_1 - 3) q^{87} + (12 \beta_{3} - 12 \beta_{2} - 6 \beta_1 + 12) q^{88} + (66 \beta_{3} + 60 \beta_{2} - 30) q^{89} + (9 \beta_{3} - 72 \beta_{2} - 18 \beta_1 + 36) q^{90} + (9 \beta_{3} - 18 \beta_1 + 103) q^{91} + (6 \beta_{2} - 6 \beta_1 + 6) q^{92} + (8 \beta_{3} + 35 \beta_{2} - 46 \beta_1 + 38) q^{93} + (9 \beta_{3} + 42 \beta_{2} + 9 \beta_1) q^{94} + ( - 54 \beta_{3} - 30 \beta_{2} + 54 \beta_1 + 60) q^{95} + (4 \beta_{3} - 8 \beta_{2} + 4 \beta_1 + 16) q^{96} + ( - 84 \beta_{3} - 7 \beta_{2} + 42 \beta_1 + 7) q^{97} + ( - 6 \beta_{3} - 24 \beta_{2} + 12) q^{98} + (36 \beta_{3} - 27 \beta_{2} - 27 \beta_1 + 45) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{4} - 18 q^{5} - 12 q^{6} + 2 q^{7} + 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 4 q^{4} - 18 q^{5} - 12 q^{6} + 2 q^{7} + 12 q^{9} + 18 q^{11} + 12 q^{12} - 10 q^{13} + 36 q^{14} + 18 q^{15} - 8 q^{16} - 24 q^{18} - 40 q^{19} - 36 q^{20} - 42 q^{21} - 12 q^{22} + 18 q^{23} + 4 q^{25} + 8 q^{28} + 18 q^{29} + 72 q^{30} + 38 q^{31} + 54 q^{33} + 24 q^{34} + 12 q^{36} + 128 q^{37} - 72 q^{38} - 102 q^{39} - 126 q^{41} - 48 q^{42} - 46 q^{43} - 54 q^{45} - 24 q^{46} + 54 q^{47} + 24 q^{48} - 12 q^{49} - 72 q^{51} + 20 q^{52} + 36 q^{54} - 108 q^{55} + 72 q^{56} + 144 q^{57} + 24 q^{58} + 126 q^{59} - 36 q^{60} + 62 q^{61} + 222 q^{63} - 32 q^{64} + 90 q^{65} - 72 q^{66} - 106 q^{67} + 72 q^{68} + 18 q^{69} - 108 q^{70} - 96 q^{72} - 208 q^{73} + 72 q^{74} - 12 q^{75} - 40 q^{76} - 90 q^{77} + 144 q^{78} + 14 q^{79} - 252 q^{81} + 144 q^{82} - 378 q^{83} - 144 q^{84} + 108 q^{85} - 108 q^{86} - 54 q^{87} + 24 q^{88} + 412 q^{91} + 36 q^{92} + 222 q^{93} + 84 q^{94} + 180 q^{95} + 48 q^{96} + 14 q^{97} + 126 q^{99}+O(q^{100}) \) 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

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{2} ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} ) / 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 2\beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{3} \) Copy content Toggle raw display

Character values

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

\(n\) \(11\)
\(\chi(n)\) \(\beta_{2}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
5.1
−1.22474 + 0.707107i
1.22474 0.707107i
−1.22474 0.707107i
1.22474 + 0.707107i
−1.22474 + 0.707107i 2.44949 + 1.73205i 1.00000 1.73205i −4.50000 2.59808i −4.22474 0.389270i −3.17423 5.49794i 2.82843i 3.00000 + 8.48528i 7.34847
5.2 1.22474 0.707107i −2.44949 + 1.73205i 1.00000 1.73205i −4.50000 2.59808i −1.77526 + 3.85337i 4.17423 + 7.22999i 2.82843i 3.00000 8.48528i −7.34847
11.1 −1.22474 0.707107i 2.44949 1.73205i 1.00000 + 1.73205i −4.50000 + 2.59808i −4.22474 + 0.389270i −3.17423 + 5.49794i 2.82843i 3.00000 8.48528i 7.34847
11.2 1.22474 + 0.707107i −2.44949 1.73205i 1.00000 + 1.73205i −4.50000 + 2.59808i −1.77526 3.85337i 4.17423 7.22999i 2.82843i 3.00000 + 8.48528i −7.34847
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.d odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 18.3.d.a 4
3.b odd 2 1 54.3.d.a 4
4.b odd 2 1 144.3.q.c 4
5.b even 2 1 450.3.i.b 4
5.c odd 4 2 450.3.k.a 8
8.b even 2 1 576.3.q.f 4
8.d odd 2 1 576.3.q.e 4
9.c even 3 1 54.3.d.a 4
9.c even 3 1 162.3.b.a 4
9.d odd 6 1 inner 18.3.d.a 4
9.d odd 6 1 162.3.b.a 4
12.b even 2 1 432.3.q.d 4
15.d odd 2 1 1350.3.i.b 4
15.e even 4 2 1350.3.k.a 8
24.f even 2 1 1728.3.q.c 4
24.h odd 2 1 1728.3.q.d 4
36.f odd 6 1 432.3.q.d 4
36.f odd 6 1 1296.3.e.g 4
36.h even 6 1 144.3.q.c 4
36.h even 6 1 1296.3.e.g 4
45.h odd 6 1 450.3.i.b 4
45.j even 6 1 1350.3.i.b 4
45.k odd 12 2 1350.3.k.a 8
45.l even 12 2 450.3.k.a 8
72.j odd 6 1 576.3.q.f 4
72.l even 6 1 576.3.q.e 4
72.n even 6 1 1728.3.q.d 4
72.p odd 6 1 1728.3.q.c 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
18.3.d.a 4 1.a even 1 1 trivial
18.3.d.a 4 9.d odd 6 1 inner
54.3.d.a 4 3.b odd 2 1
54.3.d.a 4 9.c even 3 1
144.3.q.c 4 4.b odd 2 1
144.3.q.c 4 36.h even 6 1
162.3.b.a 4 9.c even 3 1
162.3.b.a 4 9.d odd 6 1
432.3.q.d 4 12.b even 2 1
432.3.q.d 4 36.f odd 6 1
450.3.i.b 4 5.b even 2 1
450.3.i.b 4 45.h odd 6 1
450.3.k.a 8 5.c odd 4 2
450.3.k.a 8 45.l even 12 2
576.3.q.e 4 8.d odd 2 1
576.3.q.e 4 72.l even 6 1
576.3.q.f 4 8.b even 2 1
576.3.q.f 4 72.j odd 6 1
1296.3.e.g 4 36.f odd 6 1
1296.3.e.g 4 36.h even 6 1
1350.3.i.b 4 15.d odd 2 1
1350.3.i.b 4 45.j even 6 1
1350.3.k.a 8 15.e even 4 2
1350.3.k.a 8 45.k odd 12 2
1728.3.q.c 4 24.f even 2 1
1728.3.q.c 4 72.p odd 6 1
1728.3.q.d 4 24.h odd 2 1
1728.3.q.d 4 72.n even 6 1

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(18, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} - 2T^{2} + 4 \) Copy content Toggle raw display
$3$ \( T^{4} - 6T^{2} + 81 \) Copy content Toggle raw display
$5$ \( (T^{2} + 9 T + 27)^{2} \) Copy content Toggle raw display
$7$ \( T^{4} - 2 T^{3} + 57 T^{2} + \cdots + 2809 \) Copy content Toggle raw display
$11$ \( T^{4} - 18 T^{3} + 117 T^{2} + \cdots + 81 \) Copy content Toggle raw display
$13$ \( T^{4} + 10 T^{3} + 291 T^{2} + \cdots + 36481 \) Copy content Toggle raw display
$17$ \( T^{4} + 360T^{2} + 1296 \) Copy content Toggle raw display
$19$ \( (T^{2} + 20 T - 116)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} - 18 T^{3} + 117 T^{2} + \cdots + 81 \) Copy content Toggle raw display
$29$ \( T^{4} - 18 T^{3} + 63 T^{2} + \cdots + 2025 \) Copy content Toggle raw display
$31$ \( T^{4} - 38 T^{3} + 1569 T^{2} + \cdots + 15625 \) Copy content Toggle raw display
$37$ \( (T^{2} - 64 T + 808)^{2} \) Copy content Toggle raw display
$41$ \( T^{4} + 126 T^{3} + 5967 T^{2} + \cdots + 455625 \) Copy content Toggle raw display
$43$ \( T^{4} + 46 T^{3} + 2073 T^{2} + \cdots + 1849 \) Copy content Toggle raw display
$47$ \( T^{4} - 54 T^{3} + 333 T^{2} + \cdots + 408321 \) Copy content Toggle raw display
$53$ \( T^{4} + 9000 T^{2} + 810000 \) Copy content Toggle raw display
$59$ \( T^{4} - 126 T^{3} + 3573 T^{2} + \cdots + 2954961 \) Copy content Toggle raw display
$61$ \( T^{4} - 62 T^{3} + 4827 T^{2} + \cdots + 966289 \) Copy content Toggle raw display
$67$ \( T^{4} + 106 T^{3} + 8913 T^{2} + \cdots + 5396329 \) Copy content Toggle raw display
$71$ \( T^{4} + 7704 T^{2} + \cdots + 2396304 \) Copy content Toggle raw display
$73$ \( (T^{2} + 104 T + 760)^{2} \) Copy content Toggle raw display
$79$ \( T^{4} - 14 T^{3} + 1497 T^{2} + \cdots + 1692601 \) Copy content Toggle raw display
$83$ \( T^{4} + 378 T^{3} + \cdots + 131262849 \) Copy content Toggle raw display
$89$ \( T^{4} + 22824 T^{2} + \cdots + 36144144 \) Copy content Toggle raw display
$97$ \( T^{4} - 14 T^{3} + \cdots + 110986225 \) Copy content Toggle raw display
show more
show less