Properties

Label 18.6.a.b
Level $18$
Weight $6$
Character orbit 18.a
Self dual yes
Analytic conductor $2.887$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 18 = 2 \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 18.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(2.88690875663\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 6)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - 4 q^{2} + 16 q^{4} + 66 q^{5} + 176 q^{7} - 64 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - 4 q^{2} + 16 q^{4} + 66 q^{5} + 176 q^{7} - 64 q^{8} - 264 q^{10} + 60 q^{11} - 658 q^{13} - 704 q^{14} + 256 q^{16} + 414 q^{17} + 956 q^{19} + 1056 q^{20} - 240 q^{22} - 600 q^{23} + 1231 q^{25} + 2632 q^{26} + 2816 q^{28} - 5574 q^{29} - 3592 q^{31} - 1024 q^{32} - 1656 q^{34} + 11616 q^{35} - 8458 q^{37} - 3824 q^{38} - 4224 q^{40} - 19194 q^{41} + 13316 q^{43} + 960 q^{44} + 2400 q^{46} + 19680 q^{47} + 14169 q^{49} - 4924 q^{50} - 10528 q^{52} + 31266 q^{53} + 3960 q^{55} - 11264 q^{56} + 22296 q^{58} - 26340 q^{59} - 31090 q^{61} + 14368 q^{62} + 4096 q^{64} - 43428 q^{65} - 16804 q^{67} + 6624 q^{68} - 46464 q^{70} - 6120 q^{71} - 25558 q^{73} + 33832 q^{74} + 15296 q^{76} + 10560 q^{77} + 74408 q^{79} + 16896 q^{80} + 76776 q^{82} + 6468 q^{83} + 27324 q^{85} - 53264 q^{86} - 3840 q^{88} + 32742 q^{89} - 115808 q^{91} - 9600 q^{92} - 78720 q^{94} + 63096 q^{95} + 166082 q^{97} - 56676 q^{98}+O(q^{100}) \) Copy content Toggle raw display

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
0
−4.00000 0 16.0000 66.0000 0 176.000 −64.0000 0 −264.000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(3\) \(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 18.6.a.b 1
3.b odd 2 1 6.6.a.a 1
4.b odd 2 1 144.6.a.j 1
5.b even 2 1 450.6.a.m 1
5.c odd 4 2 450.6.c.j 2
7.b odd 2 1 882.6.a.a 1
8.b even 2 1 576.6.a.j 1
8.d odd 2 1 576.6.a.i 1
9.c even 3 2 162.6.c.h 2
9.d odd 6 2 162.6.c.e 2
12.b even 2 1 48.6.a.c 1
15.d odd 2 1 150.6.a.d 1
15.e even 4 2 150.6.c.b 2
21.c even 2 1 294.6.a.m 1
21.g even 6 2 294.6.e.a 2
21.h odd 6 2 294.6.e.g 2
24.f even 2 1 192.6.a.g 1
24.h odd 2 1 192.6.a.o 1
33.d even 2 1 726.6.a.a 1
39.d odd 2 1 1014.6.a.c 1
48.i odd 4 2 768.6.d.c 2
48.k even 4 2 768.6.d.p 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6.6.a.a 1 3.b odd 2 1
18.6.a.b 1 1.a even 1 1 trivial
48.6.a.c 1 12.b even 2 1
144.6.a.j 1 4.b odd 2 1
150.6.a.d 1 15.d odd 2 1
150.6.c.b 2 15.e even 4 2
162.6.c.e 2 9.d odd 6 2
162.6.c.h 2 9.c even 3 2
192.6.a.g 1 24.f even 2 1
192.6.a.o 1 24.h odd 2 1
294.6.a.m 1 21.c even 2 1
294.6.e.a 2 21.g even 6 2
294.6.e.g 2 21.h odd 6 2
450.6.a.m 1 5.b even 2 1
450.6.c.j 2 5.c odd 4 2
576.6.a.i 1 8.d odd 2 1
576.6.a.j 1 8.b even 2 1
726.6.a.a 1 33.d even 2 1
768.6.d.c 2 48.i odd 4 2
768.6.d.p 2 48.k even 4 2
882.6.a.a 1 7.b odd 2 1
1014.6.a.c 1 39.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5} - 66 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(18))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T + 4 \) Copy content Toggle raw display
$3$ \( T \) Copy content Toggle raw display
$5$ \( T - 66 \) Copy content Toggle raw display
$7$ \( T - 176 \) Copy content Toggle raw display
$11$ \( T - 60 \) Copy content Toggle raw display
$13$ \( T + 658 \) Copy content Toggle raw display
$17$ \( T - 414 \) Copy content Toggle raw display
$19$ \( T - 956 \) Copy content Toggle raw display
$23$ \( T + 600 \) Copy content Toggle raw display
$29$ \( T + 5574 \) Copy content Toggle raw display
$31$ \( T + 3592 \) Copy content Toggle raw display
$37$ \( T + 8458 \) Copy content Toggle raw display
$41$ \( T + 19194 \) Copy content Toggle raw display
$43$ \( T - 13316 \) Copy content Toggle raw display
$47$ \( T - 19680 \) Copy content Toggle raw display
$53$ \( T - 31266 \) Copy content Toggle raw display
$59$ \( T + 26340 \) Copy content Toggle raw display
$61$ \( T + 31090 \) Copy content Toggle raw display
$67$ \( T + 16804 \) Copy content Toggle raw display
$71$ \( T + 6120 \) Copy content Toggle raw display
$73$ \( T + 25558 \) Copy content Toggle raw display
$79$ \( T - 74408 \) Copy content Toggle raw display
$83$ \( T - 6468 \) Copy content Toggle raw display
$89$ \( T - 32742 \) Copy content Toggle raw display
$97$ \( T - 166082 \) Copy content Toggle raw display
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