Properties

Label 18.6.a.a
Level 18
Weight 6
Character orbit 18.a
Self dual yes
Analytic conductor 2.887
Analytic rank 1
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: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - 4q^{2} + 16q^{4} - 96q^{5} - 148q^{7} - 64q^{8} + O(q^{10}) \) \( q - 4q^{2} + 16q^{4} - 96q^{5} - 148q^{7} - 64q^{8} + 384q^{10} + 384q^{11} - 334q^{13} + 592q^{14} + 256q^{16} + 576q^{17} - 664q^{19} - 1536q^{20} - 1536q^{22} - 3840q^{23} + 6091q^{25} + 1336q^{26} - 2368q^{28} + 96q^{29} - 4564q^{31} - 1024q^{32} - 2304q^{34} + 14208q^{35} + 5798q^{37} + 2656q^{38} + 6144q^{40} - 6720q^{41} - 14872q^{43} + 6144q^{44} + 15360q^{46} - 19200q^{47} + 5097q^{49} - 24364q^{50} - 5344q^{52} + 7776q^{53} - 36864q^{55} + 9472q^{56} - 384q^{58} - 13056q^{59} + 42782q^{61} + 18256q^{62} + 4096q^{64} + 32064q^{65} + 36656q^{67} + 9216q^{68} - 56832q^{70} + 64512q^{71} - 16810q^{73} - 23192q^{74} - 10624q^{76} - 56832q^{77} + 28076q^{79} - 24576q^{80} + 26880q^{82} - 66432q^{83} - 55296q^{85} + 59488q^{86} - 24576q^{88} - 81792q^{89} + 49432q^{91} - 61440q^{92} + 76800q^{94} + 63744q^{95} - 29938q^{97} - 20388q^{98} + O(q^{100}) \)

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 −96.0000 0 −148.000 −64.0000 0 384.000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

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.a 1
3.b odd 2 1 18.6.a.c yes 1
4.b odd 2 1 144.6.a.a 1
5.b even 2 1 450.6.a.v 1
5.c odd 4 2 450.6.c.m 2
7.b odd 2 1 882.6.a.k 1
8.b even 2 1 576.6.a.bh 1
8.d odd 2 1 576.6.a.bi 1
9.c even 3 2 162.6.c.l 2
9.d odd 6 2 162.6.c.a 2
12.b even 2 1 144.6.a.l 1
15.d odd 2 1 450.6.a.k 1
15.e even 4 2 450.6.c.c 2
21.c even 2 1 882.6.a.l 1
24.f even 2 1 576.6.a.b 1
24.h odd 2 1 576.6.a.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
18.6.a.a 1 1.a even 1 1 trivial
18.6.a.c yes 1 3.b odd 2 1
144.6.a.a 1 4.b odd 2 1
144.6.a.l 1 12.b even 2 1
162.6.c.a 2 9.d odd 6 2
162.6.c.l 2 9.c even 3 2
450.6.a.k 1 15.d odd 2 1
450.6.a.v 1 5.b even 2 1
450.6.c.c 2 15.e even 4 2
450.6.c.m 2 5.c odd 4 2
576.6.a.a 1 24.h odd 2 1
576.6.a.b 1 24.f even 2 1
576.6.a.bh 1 8.b even 2 1
576.6.a.bi 1 8.d odd 2 1
882.6.a.k 1 7.b odd 2 1
882.6.a.l 1 21.c even 2 1

Atkin-Lehner signs

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

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5} + 96 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(18))\).

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 + 4 T \)
$3$ \( \)
$5$ \( 1 + 96 T + 3125 T^{2} \)
$7$ \( 1 + 148 T + 16807 T^{2} \)
$11$ \( 1 - 384 T + 161051 T^{2} \)
$13$ \( 1 + 334 T + 371293 T^{2} \)
$17$ \( 1 - 576 T + 1419857 T^{2} \)
$19$ \( 1 + 664 T + 2476099 T^{2} \)
$23$ \( 1 + 3840 T + 6436343 T^{2} \)
$29$ \( 1 - 96 T + 20511149 T^{2} \)
$31$ \( 1 + 4564 T + 28629151 T^{2} \)
$37$ \( 1 - 5798 T + 69343957 T^{2} \)
$41$ \( 1 + 6720 T + 115856201 T^{2} \)
$43$ \( 1 + 14872 T + 147008443 T^{2} \)
$47$ \( 1 + 19200 T + 229345007 T^{2} \)
$53$ \( 1 - 7776 T + 418195493 T^{2} \)
$59$ \( 1 + 13056 T + 714924299 T^{2} \)
$61$ \( 1 - 42782 T + 844596301 T^{2} \)
$67$ \( 1 - 36656 T + 1350125107 T^{2} \)
$71$ \( 1 - 64512 T + 1804229351 T^{2} \)
$73$ \( 1 + 16810 T + 2073071593 T^{2} \)
$79$ \( 1 - 28076 T + 3077056399 T^{2} \)
$83$ \( 1 + 66432 T + 3939040643 T^{2} \)
$89$ \( 1 + 81792 T + 5584059449 T^{2} \)
$97$ \( 1 + 29938 T + 8587340257 T^{2} \)
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