Newform orbit 432.8.a.i

• Label: 432.8.a.i
• Level: $432$
• Weight: $8$
• Character orbit: 432.a
• Self dual: yes
• Analytic conductor: $134.950$
• Analytic rank: $1$
• Dimension: $2$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	$N$	$=$	$432 = 2^{4} \cdot 3^{3}$
Weight:	$k$	$=$	$8$
Character orbit:	$[\chi]$	$=$	432.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	$134.950331009$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{1289})$
Defining polynomial:	$x^{2} - x - 322$
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$2\cdot 3^{2}$
Twist minimal:	no (minimal twist has level 108)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

$q$-expansion

Coefficients of the $q$-expansion are expressed in terms of $\beta = 9\sqrt{1289}$. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	q + (-b - 162) * q^5 + (3*b + 490) * q^7 + (4*b + 405) * q^11 + (-24*b - 3940) * q^13 + (40*b + 10368) * q^17 + (-30*b - 5228) * q^19 + (-40*b - 30294) * q^23 + (324*b + 52528) * q^25 + (-622*b - 16200) * q^29 + (405*b + 72994) * q^31 + (-976*b - 392607) * q^35 + (-192*b - 5230) * q^37 + (410*b + 542700) * q^41 + (1782*b + 303220) * q^43 + (840*b - 747954) * q^47 + (2940*b + 356238) * q^49 + (-1375*b + 1039878) * q^53 + (-1053*b - 483246) * q^55 + (2000*b - 1537380) * q^59 + (2520*b + 300560) * q^61 + (7828*b + 3144096) * q^65 + (2358*b + 1214620) * q^67 + (4472*b - 1600560) * q^71 + (-6732*b + 548765) * q^73 + (3175*b + 1451358) * q^77 + (1860*b + 756580) * q^79 + (-4740*b - 1880091) * q^83 + (-16848*b - 5855976) * q^85 + (-15746*b - 5392980) * q^89 + (-23580*b - 9448048) * q^91 + (10088*b + 3979206) * q^95 + (18552*b + 1494455) * q^97
$\operatorname{Tr}(f)(q)$	$=$	2 * q - 324 * q^5 + 980 * q^7 + 810 * q^11 - 7880 * q^13 + 20736 * q^17 - 10456 * q^19 - 60588 * q^23 + 105056 * q^25 - 32400 * q^29 + 145988 * q^31 - 785214 * q^35 - 10460 * q^37 + 1085400 * q^41 + 606440 * q^43 - 1495908 * q^47 + 712476 * q^49 + 2079756 * q^53 - 966492 * q^55 - 3074760 * q^59 + 601120 * q^61 + 6288192 * q^65 + 2429240 * q^67 - 3201120 * q^71 + 1097530 * q^73 + 2902716 * q^77 + 1513160 * q^79 - 3760182 * q^83 - 11711952 * q^85 - 10785960 * q^89 - 18896096 * q^91 + 7958412 * q^95 + 2988910 * 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

18.4513
−17.4513

0

0

0

−485.124

0

1459.37

0

0

0

1.2

0

0

0

161.124

0

−479.371

0

0

0

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	432.8.a.i		2
3.b	odd	2	1		432.8.a.r		2
4.b	odd	2	1		108.8.a.b	✓	2
12.b	even	2	1		108.8.a.e	yes	2
36.f	odd	6	2		324.8.e.j		4
36.h	even	6	2		324.8.e.g		4

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
108.8.a.b	✓	2	4.b	odd	2	1
108.8.a.e	yes	2	12.b	even	2	1
324.8.e.g		4	36.h	even	6	2
324.8.e.j		4	36.f	odd	6	2
432.8.a.i		2	1.a	even	1	1	trivial
432.8.a.r		2	3.b	odd	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{8}^{\mathrm{new}}(\Gamma_0(432))$:

$T_{5}^{2} + 324T_{5} - 78165$

$T_{7}^{2} - 980T_{7} - 699581$

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{2}$
$3$	$T^{2}$
$5$	$T^{2} + 324T - 78165$
$7$	$T^{2} - 980T - 699581$
$11$	$T^{2} - 810 T - 1506519$