Newform orbit 468.4.a.g

• Label: 468.4.a.g
• Level: $468$
• Weight: $4$
• Character orbit: 468.a
• Self dual: yes
• Analytic conductor: $27.613$
• Analytic rank: $0$
• Dimension: $2$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	$N$	$=$	$468 = 2^{2} \cdot 3^{2} \cdot 13$
Weight:	$k$	$=$	$4$
Character orbit:	$[\chi]$	$=$	468.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	$27.6128938827$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{22})$
Defining polynomial:	$x^{2} - 22$
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$2$
Twist minimal:	no (minimal twist has level 156)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

$q$-expansion

Coefficients of the $q$-expansion are expressed in terms of $\beta = 2\sqrt{22}$. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	q + b * q^5 + (3*b + 4) * q^7 + (2*b + 30) * q^11 - 13 * q^13 + (6*b + 54) * q^17 + (-3*b - 108) * q^19 + (-8*b + 108) * q^23 - 37 * q^25 + (-20*b + 54) * q^29 + (15*b + 40) * q^31 + (4*b + 264) * q^35 + (-18*b + 54) * q^37 + (-15*b + 24) * q^41 + (-54*b + 4) * q^43 + (44*b + 114) * q^47 + (24*b + 465) * q^49 + (-12*b - 270) * q^53 + (30*b + 176) * q^55 + (40*b + 426) * q^59 + (12*b + 154) * q^61 - 13*b * q^65 + (39*b - 152) * q^67 + (-82*b + 114) * q^71 + (6*b + 710) * q^73 + (98*b + 648) * q^77 + (-60*b + 248) * q^79 + (-12*b - 618) * q^83 + (54*b + 528) * q^85 + (63*b + 708) * q^89 + (-39*b - 52) * q^91 + (-108*b - 264) * q^95 + (-30*b + 302) * q^97
$\operatorname{Tr}(f)(q)$	$=$	2 * q + 8 * q^7 + 60 * q^11 - 26 * q^13 + 108 * q^17 - 216 * q^19 + 216 * q^23 - 74 * q^25 + 108 * q^29 + 80 * q^31 + 528 * q^35 + 108 * q^37 + 48 * q^41 + 8 * q^43 + 228 * q^47 + 930 * q^49 - 540 * q^53 + 352 * q^55 + 852 * q^59 + 308 * q^61 - 304 * q^67 + 228 * q^71 + 1420 * q^73 + 1296 * q^77 + 496 * q^79 - 1236 * q^83 + 1056 * q^85 + 1416 * q^89 - 104 * q^91 - 528 * q^95 + 604 * q^97

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

−4.69042
4.69042

0

0

0

−9.38083

0

−24.1425

0

0

0

1.2

0

0

0

9.38083

0

32.1425

0

0

0

Atkin-Lehner signs

$p$	Sign
$2$	$-1$
$3$	$-1$
$13$	$+1$

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	468.4.a.g		2
3.b	odd	2	1		156.4.a.c	✓	2
4.b	odd	2	1		1872.4.a.y		2
12.b	even	2	1		624.4.a.p		2
24.f	even	2	1		2496.4.a.w		2
24.h	odd	2	1		2496.4.a.bf		2
39.d	odd	2	1		2028.4.a.d		2
39.f	even	4	2		2028.4.b.e		4

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
156.4.a.c	✓	2	3.b	odd	2	1
468.4.a.g		2	1.a	even	1	1	trivial
624.4.a.p		2	12.b	even	2	1
1872.4.a.y		2	4.b	odd	2	1
2028.4.a.d		2	39.d	odd	2	1
2028.4.b.e		4	39.f	even	4	2
2496.4.a.w		2	24.f	even	2	1
2496.4.a.bf		2	24.h	odd	2	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $T_{5}^{2} - 88$ acting on $S_{4}^{\mathrm{new}}(\Gamma_0(468))$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{2}$
$3$	$T^{2}$
$5$	$T^{2} - 88$
$7$	$T^{2} - 8T - 776$
$11$	$T^{2} - 60T + 548$