Properties

Label 18.2.c.a
Level 18
Weight 2
Character orbit 18.c
Analytic conductor 0.144
Analytic rank 0
Dimension 2
CM no
Inner twists 2

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Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(0.143730723638\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\zeta_{6} q^{2} + ( -2 + \zeta_{6} ) q^{3} + ( -1 + \zeta_{6} ) q^{4} + ( 1 + \zeta_{6} ) q^{6} -2 \zeta_{6} q^{7} + q^{8} + ( 3 - 3 \zeta_{6} ) q^{9} +O(q^{10})\) \( q -\zeta_{6} q^{2} + ( -2 + \zeta_{6} ) q^{3} + ( -1 + \zeta_{6} ) q^{4} + ( 1 + \zeta_{6} ) q^{6} -2 \zeta_{6} q^{7} + q^{8} + ( 3 - 3 \zeta_{6} ) q^{9} + 3 \zeta_{6} q^{11} + ( 1 - 2 \zeta_{6} ) q^{12} + ( -2 + 2 \zeta_{6} ) q^{13} + ( -2 + 2 \zeta_{6} ) q^{14} -\zeta_{6} q^{16} -3 q^{17} -3 q^{18} - q^{19} + ( 2 + 2 \zeta_{6} ) q^{21} + ( 3 - 3 \zeta_{6} ) q^{22} + ( 6 - 6 \zeta_{6} ) q^{23} + ( -2 + \zeta_{6} ) q^{24} + 5 \zeta_{6} q^{25} + 2 q^{26} + ( -3 + 6 \zeta_{6} ) q^{27} + 2 q^{28} -6 \zeta_{6} q^{29} + ( 4 - 4 \zeta_{6} ) q^{31} + ( -1 + \zeta_{6} ) q^{32} + ( -3 - 3 \zeta_{6} ) q^{33} + 3 \zeta_{6} q^{34} + 3 \zeta_{6} q^{36} -4 q^{37} + \zeta_{6} q^{38} + ( 2 - 4 \zeta_{6} ) q^{39} + ( -9 + 9 \zeta_{6} ) q^{41} + ( 2 - 4 \zeta_{6} ) q^{42} + \zeta_{6} q^{43} -3 q^{44} -6 q^{46} + 6 \zeta_{6} q^{47} + ( 1 + \zeta_{6} ) q^{48} + ( 3 - 3 \zeta_{6} ) q^{49} + ( 5 - 5 \zeta_{6} ) q^{50} + ( 6 - 3 \zeta_{6} ) q^{51} -2 \zeta_{6} q^{52} + 12 q^{53} + ( 6 - 3 \zeta_{6} ) q^{54} -2 \zeta_{6} q^{56} + ( 2 - \zeta_{6} ) q^{57} + ( -6 + 6 \zeta_{6} ) q^{58} + ( -3 + 3 \zeta_{6} ) q^{59} -8 \zeta_{6} q^{61} -4 q^{62} -6 q^{63} + q^{64} + ( -3 + 6 \zeta_{6} ) q^{66} + ( -5 + 5 \zeta_{6} ) q^{67} + ( 3 - 3 \zeta_{6} ) q^{68} + ( -6 + 12 \zeta_{6} ) q^{69} -12 q^{71} + ( 3 - 3 \zeta_{6} ) q^{72} + 11 q^{73} + 4 \zeta_{6} q^{74} + ( -5 - 5 \zeta_{6} ) q^{75} + ( 1 - \zeta_{6} ) q^{76} + ( 6 - 6 \zeta_{6} ) q^{77} + ( -4 + 2 \zeta_{6} ) q^{78} + 4 \zeta_{6} q^{79} -9 \zeta_{6} q^{81} + 9 q^{82} -12 \zeta_{6} q^{83} + ( -4 + 2 \zeta_{6} ) q^{84} + ( 1 - \zeta_{6} ) q^{86} + ( 6 + 6 \zeta_{6} ) q^{87} + 3 \zeta_{6} q^{88} + 6 q^{89} + 4 q^{91} + 6 \zeta_{6} q^{92} + ( -4 + 8 \zeta_{6} ) q^{93} + ( 6 - 6 \zeta_{6} ) q^{94} + ( 1 - 2 \zeta_{6} ) q^{96} -5 \zeta_{6} q^{97} -3 q^{98} + 9 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - q^{2} - 3q^{3} - q^{4} + 3q^{6} - 2q^{7} + 2q^{8} + 3q^{9} + O(q^{10}) \) \( 2q - q^{2} - 3q^{3} - q^{4} + 3q^{6} - 2q^{7} + 2q^{8} + 3q^{9} + 3q^{11} - 2q^{13} - 2q^{14} - q^{16} - 6q^{17} - 6q^{18} - 2q^{19} + 6q^{21} + 3q^{22} + 6q^{23} - 3q^{24} + 5q^{25} + 4q^{26} + 4q^{28} - 6q^{29} + 4q^{31} - q^{32} - 9q^{33} + 3q^{34} + 3q^{36} - 8q^{37} + q^{38} - 9q^{41} + q^{43} - 6q^{44} - 12q^{46} + 6q^{47} + 3q^{48} + 3q^{49} + 5q^{50} + 9q^{51} - 2q^{52} + 24q^{53} + 9q^{54} - 2q^{56} + 3q^{57} - 6q^{58} - 3q^{59} - 8q^{61} - 8q^{62} - 12q^{63} + 2q^{64} - 5q^{67} + 3q^{68} - 24q^{71} + 3q^{72} + 22q^{73} + 4q^{74} - 15q^{75} + q^{76} + 6q^{77} - 6q^{78} + 4q^{79} - 9q^{81} + 18q^{82} - 12q^{83} - 6q^{84} + q^{86} + 18q^{87} + 3q^{88} + 12q^{89} + 8q^{91} + 6q^{92} + 6q^{94} - 5q^{97} - 6q^{98} + 18q^{99} + O(q^{100}) \)

Character values

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

\(n\) \(11\)
\(\chi(n)\) \(-1 + \zeta_{6}\)

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
0.500000 0.866025i
0.500000 + 0.866025i
−0.500000 + 0.866025i −1.50000 0.866025i −0.500000 0.866025i 0 1.50000 0.866025i −1.00000 + 1.73205i 1.00000 1.50000 + 2.59808i 0
13.1 −0.500000 0.866025i −1.50000 + 0.866025i −0.500000 + 0.866025i 0 1.50000 + 0.866025i −1.00000 1.73205i 1.00000 1.50000 2.59808i 0
\(n\): 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 18.2.c.a 2
3.b odd 2 1 54.2.c.a 2
4.b odd 2 1 144.2.i.c 2
5.b even 2 1 450.2.e.i 2
5.c odd 4 2 450.2.j.e 4
7.b odd 2 1 882.2.f.d 2
7.c even 3 1 882.2.e.i 2
7.c even 3 1 882.2.h.c 2
7.d odd 6 1 882.2.e.g 2
7.d odd 6 1 882.2.h.b 2
8.b even 2 1 576.2.i.g 2
8.d odd 2 1 576.2.i.a 2
9.c even 3 1 inner 18.2.c.a 2
9.c even 3 1 162.2.a.c 1
9.d odd 6 1 54.2.c.a 2
9.d odd 6 1 162.2.a.b 1
12.b even 2 1 432.2.i.b 2
15.d odd 2 1 1350.2.e.c 2
15.e even 4 2 1350.2.j.a 4
21.c even 2 1 2646.2.f.g 2
21.g even 6 1 2646.2.e.c 2
21.g even 6 1 2646.2.h.i 2
21.h odd 6 1 2646.2.e.b 2
21.h odd 6 1 2646.2.h.h 2
24.f even 2 1 1728.2.i.f 2
24.h odd 2 1 1728.2.i.e 2
36.f odd 6 1 144.2.i.c 2
36.f odd 6 1 1296.2.a.g 1
36.h even 6 1 432.2.i.b 2
36.h even 6 1 1296.2.a.f 1
45.h odd 6 1 1350.2.e.c 2
45.h odd 6 1 4050.2.a.v 1
45.j even 6 1 450.2.e.i 2
45.j even 6 1 4050.2.a.c 1
45.k odd 12 2 450.2.j.e 4
45.k odd 12 2 4050.2.c.c 2
45.l even 12 2 1350.2.j.a 4
45.l even 12 2 4050.2.c.r 2
63.g even 3 1 882.2.e.i 2
63.h even 3 1 882.2.h.c 2
63.i even 6 1 2646.2.h.i 2
63.j odd 6 1 2646.2.h.h 2
63.k odd 6 1 882.2.e.g 2
63.l odd 6 1 882.2.f.d 2
63.l odd 6 1 7938.2.a.x 1
63.n odd 6 1 2646.2.e.b 2
63.o even 6 1 2646.2.f.g 2
63.o even 6 1 7938.2.a.i 1
63.s even 6 1 2646.2.e.c 2
63.t odd 6 1 882.2.h.b 2
72.j odd 6 1 1728.2.i.e 2
72.j odd 6 1 5184.2.a.q 1
72.l even 6 1 1728.2.i.f 2
72.l even 6 1 5184.2.a.p 1
72.n even 6 1 576.2.i.g 2
72.n even 6 1 5184.2.a.r 1
72.p odd 6 1 576.2.i.a 2
72.p odd 6 1 5184.2.a.o 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
18.2.c.a 2 1.a even 1 1 trivial
18.2.c.a 2 9.c even 3 1 inner
54.2.c.a 2 3.b odd 2 1
54.2.c.a 2 9.d odd 6 1
144.2.i.c 2 4.b odd 2 1
144.2.i.c 2 36.f odd 6 1
162.2.a.b 1 9.d odd 6 1
162.2.a.c 1 9.c even 3 1
432.2.i.b 2 12.b even 2 1
432.2.i.b 2 36.h even 6 1
450.2.e.i 2 5.b even 2 1
450.2.e.i 2 45.j even 6 1
450.2.j.e 4 5.c odd 4 2
450.2.j.e 4 45.k odd 12 2
576.2.i.a 2 8.d odd 2 1
576.2.i.a 2 72.p odd 6 1
576.2.i.g 2 8.b even 2 1
576.2.i.g 2 72.n even 6 1
882.2.e.g 2 7.d odd 6 1
882.2.e.g 2 63.k odd 6 1
882.2.e.i 2 7.c even 3 1
882.2.e.i 2 63.g even 3 1
882.2.f.d 2 7.b odd 2 1
882.2.f.d 2 63.l odd 6 1
882.2.h.b 2 7.d odd 6 1
882.2.h.b 2 63.t odd 6 1
882.2.h.c 2 7.c even 3 1
882.2.h.c 2 63.h even 3 1
1296.2.a.f 1 36.h even 6 1
1296.2.a.g 1 36.f odd 6 1
1350.2.e.c 2 15.d odd 2 1
1350.2.e.c 2 45.h odd 6 1
1350.2.j.a 4 15.e even 4 2
1350.2.j.a 4 45.l even 12 2
1728.2.i.e 2 24.h odd 2 1
1728.2.i.e 2 72.j odd 6 1
1728.2.i.f 2 24.f even 2 1
1728.2.i.f 2 72.l even 6 1
2646.2.e.b 2 21.h odd 6 1
2646.2.e.b 2 63.n odd 6 1
2646.2.e.c 2 21.g even 6 1
2646.2.e.c 2 63.s even 6 1
2646.2.f.g 2 21.c even 2 1
2646.2.f.g 2 63.o even 6 1
2646.2.h.h 2 21.h odd 6 1
2646.2.h.h 2 63.j odd 6 1
2646.2.h.i 2 21.g even 6 1
2646.2.h.i 2 63.i even 6 1
4050.2.a.c 1 45.j even 6 1
4050.2.a.v 1 45.h odd 6 1
4050.2.c.c 2 45.k odd 12 2
4050.2.c.r 2 45.l even 12 2
5184.2.a.o 1 72.p odd 6 1
5184.2.a.p 1 72.l even 6 1
5184.2.a.q 1 72.j odd 6 1
5184.2.a.r 1 72.n even 6 1
7938.2.a.i 1 63.o even 6 1
7938.2.a.x 1 63.l odd 6 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(18, [\chi])\).

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 + T + T^{2} \)
$3$ \( 1 + 3 T + 3 T^{2} \)
$5$ \( 1 - 5 T^{2} + 25 T^{4} \)
$7$ \( 1 + 2 T - 3 T^{2} + 14 T^{3} + 49 T^{4} \)
$11$ \( 1 - 3 T - 2 T^{2} - 33 T^{3} + 121 T^{4} \)
$13$ \( ( 1 - 5 T + 13 T^{2} )( 1 + 7 T + 13 T^{2} ) \)
$17$ \( ( 1 + 3 T + 17 T^{2} )^{2} \)
$19$ \( ( 1 + T + 19 T^{2} )^{2} \)
$23$ \( 1 - 6 T + 13 T^{2} - 138 T^{3} + 529 T^{4} \)
$29$ \( 1 + 6 T + 7 T^{2} + 174 T^{3} + 841 T^{4} \)
$31$ \( ( 1 - 11 T + 31 T^{2} )( 1 + 7 T + 31 T^{2} ) \)
$37$ \( ( 1 + 4 T + 37 T^{2} )^{2} \)
$41$ \( 1 + 9 T + 40 T^{2} + 369 T^{3} + 1681 T^{4} \)
$43$ \( 1 - T - 42 T^{2} - 43 T^{3} + 1849 T^{4} \)
$47$ \( 1 - 6 T - 11 T^{2} - 282 T^{3} + 2209 T^{4} \)
$53$ \( ( 1 - 12 T + 53 T^{2} )^{2} \)
$59$ \( 1 + 3 T - 50 T^{2} + 177 T^{3} + 3481 T^{4} \)
$61$ \( 1 + 8 T + 3 T^{2} + 488 T^{3} + 3721 T^{4} \)
$67$ \( ( 1 - 11 T + 67 T^{2} )( 1 + 16 T + 67 T^{2} ) \)
$71$ \( ( 1 + 12 T + 71 T^{2} )^{2} \)
$73$ \( ( 1 - 11 T + 73 T^{2} )^{2} \)
$79$ \( ( 1 - 17 T + 79 T^{2} )( 1 + 13 T + 79 T^{2} ) \)
$83$ \( 1 + 12 T + 61 T^{2} + 996 T^{3} + 6889 T^{4} \)
$89$ \( ( 1 - 6 T + 89 T^{2} )^{2} \)
$97$ \( ( 1 - 14 T + 97 T^{2} )( 1 + 19 T + 97 T^{2} ) \)
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