LMFDB - Newform orbit 162.4.a.a

Newspace parameters

Level:	$ N $	$=$	$ 162 = 2 \cdot 3^{4} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	162.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	$9.55830942093$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 18)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

$q$-expansion

$f(q)$

$=$

$ q - 2 q^{2} + 4 q^{4} + 9 q^{5} - 31 q^{7} - 8 q^{8}+O(q^{10}) $

$ q - 2 q^{2} + 4 q^{4} + 9 q^{5} - 31 q^{7} - 8 q^{8} - 18 q^{10} + 15 q^{11} - 37 q^{13} + 62 q^{14} + 16 q^{16} + 42 q^{17} - 28 q^{19} + 36 q^{20} - 30 q^{22} - 195 q^{23} - 44 q^{25} + 74 q^{26} - 124 q^{28} - 111 q^{29} - 205 q^{31} - 32 q^{32} - 84 q^{34} - 279 q^{35} - 166 q^{37} + 56 q^{38} - 72 q^{40} + 261 q^{41} - 43 q^{43} + 60 q^{44} + 390 q^{46} - 177 q^{47} + 618 q^{49} + 88 q^{50} - 148 q^{52} - 114 q^{53} + 135 q^{55} + 248 q^{56} + 222 q^{58} - 159 q^{59} + 191 q^{61} + 410 q^{62} + 64 q^{64} - 333 q^{65} - 421 q^{67} + 168 q^{68} + 558 q^{70} - 156 q^{71} + 182 q^{73} + 332 q^{74} - 112 q^{76} - 465 q^{77} + 1133 q^{79} + 144 q^{80} - 522 q^{82} + 1083 q^{83} + 378 q^{85} + 86 q^{86} - 120 q^{88} + 1050 q^{89} + 1147 q^{91} - 780 q^{92} + 354 q^{94} - 252 q^{95} - 901 q^{97} - 1236 q^{98}+O(q^{100}) $

Display 100 coefficients 10 coefficients

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

−2.00000

4.00000

9.00000

−31.0000

−8.00000

−18.0000

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	162.4.a.a		1
3.b	odd	2	1		162.4.a.d		1
4.b	odd	2	1		1296.4.a.g		1
9.c	even	3	2		54.4.c.a		2
9.d	odd	6	2		18.4.c.a	✓	2
12.b	even	2	1		1296.4.a.b		1
36.f	odd	6	2		432.4.i.a		2
36.h	even	6	2		144.4.i.a		2

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
18.4.c.a	✓	2	9.d	odd	6	2
54.4.c.a		2	9.c	even	3	2
144.4.i.a		2	36.h	even	6	2
162.4.a.a		1	1.a	even	1	1	trivial
162.4.a.d		1	3.b	odd	2	1
432.4.i.a		2	36.f	odd	6	2
1296.4.a.b		1	12.b	even	2	1
1296.4.a.g		1	4.b	odd	2	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{5} - 9 $ T5 - 9 acting on $S_{4}^{\mathrm{new}}(\Gamma_0(162))$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T + 2 $ T + 2
$3$	$ T $ T
$5$	$ T - 9 $ T - 9
$7$	$ T + 31 $ T + 31
$11$	$ T - 15 $ T - 15
$13$	$ T + 37 $ T + 37
$17$	$ T - 42 $ T - 42
$19$	$ T + 28 $ T + 28
$23$	$ T + 195 $ T + 195
$29$	$ T + 111 $ T + 111
$31$	$ T + 205 $ T + 205
$37$	$ T + 166 $ T + 166
$41$	$ T - 261 $ T - 261
$43$	$ T + 43 $ T + 43
$47$	$ T + 177 $ T + 177
$53$	$ T + 114 $ T + 114
$59$	$ T + 159 $ T + 159
$61$	$ T - 191 $ T - 191
$67$	$ T + 421 $ T + 421
$71$	$ T + 156 $ T + 156
$73$	$ T - 182 $ T - 182
$79$	$ T - 1133 $ T - 1133
$83$	$ T - 1083 $ T - 1083
$89$	$ T - 1050 $ T - 1050
$97$	$ T + 901 $ T + 901
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Classical	Maass
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Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 162 = 2 \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	162.a (trivial)

\(n\):		e.g. 2-40 or 990-1000
Significant digits:
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