Newform orbit 400.6.a.o

• Label: 400.6.a.o
• Level: $400$
• Weight: $6$
• Character orbit: 400.a
• Self dual: yes
• Analytic conductor: $64.154$
• Analytic rank: $1$
• Dimension: $2$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	$N$	$=$	$400 = 2^{4} \cdot 5^{2}$
Weight:	$k$	$=$	$6$
Character orbit:	$[\chi]$	$=$	400.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	$64.1535279252$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{241})$
Defining polynomial:	$x^{2} - x - 60$
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$2$
Twist minimal:	no (minimal twist has level 25)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

$q$-expansion

Coefficients of the $q$-expansion are expressed in terms of $\beta = \sqrt{241}$. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	q + (-b - 10) * q^3 + (-2*b - 100) * q^7 + (20*b + 98) * q^9 + (-25*b + 98) * q^11 + (16*b - 180) * q^13 + (-68*b + 745) * q^17 + (35*b + 1590) * q^19 + (120*b + 1482) * q^21 + (-6*b - 780) * q^23 + (-55*b - 3370) * q^27 + (-40*b - 1960) * q^29 + (550*b + 548) * q^31 + (152*b + 5045) * q^33 + (192*b + 1010) * q^37 + (20*b - 2056) * q^39 + (200*b + 13877) * q^41 + (1064*b + 1500) * q^43 + (-772*b - 12880) * q^47 + (400*b - 5843) * q^49 + (-65*b + 8938) * q^51 + (376*b - 13490) * q^53 + (-1940*b - 24335) * q^57 + (980*b - 5980) * q^59 + (1000*b - 12198) * q^61 + (-2196*b - 19440) * q^63 + (793*b + 20030) * q^67 + (840*b + 9246) * q^69 + (-500*b + 43648) * q^71 + (556*b - 35145) * q^73 + (2304*b + 2250) * q^77 + (-2510*b - 32740) * q^79 + (-940*b + 23141) * q^81 + (429*b - 46290) * q^83 + (2360*b + 29240) * q^87 + (-5220*b - 36405) * q^89 + (-1240*b + 10288) * q^91 + (-6048*b - 138030) * q^93 + (5472*b - 63070) * q^97 + (-490*b - 110896) * q^99
$\operatorname{Tr}(f)(q)$	$=$	2 * q - 20 * q^3 - 200 * q^7 + 196 * q^9 + 196 * q^11 - 360 * q^13 + 1490 * q^17 + 3180 * q^19 + 2964 * q^21 - 1560 * q^23 - 6740 * q^27 - 3920 * q^29 + 1096 * q^31 + 10090 * q^33 + 2020 * q^37 - 4112 * q^39 + 27754 * q^41 + 3000 * q^43 - 25760 * q^47 - 11686 * q^49 + 17876 * q^51 - 26980 * q^53 - 48670 * q^57 - 11960 * q^59 - 24396 * q^61 - 38880 * q^63 + 40060 * q^67 + 18492 * q^69 + 87296 * q^71 - 70290 * q^73 + 4500 * q^77 - 65480 * q^79 + 46282 * q^81 - 92580 * q^83 + 58480 * q^87 - 72810 * q^89 + 20576 * q^91 - 276060 * q^93 - 126140 * q^97 - 221792 * q^99

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

8.26209
−7.26209

0

−25.5242

0

0

0

−131.048

0

408.483

0

1.2

0

5.52417

0

0

0

−68.9517

0

−212.483

0

Atkin-Lehner signs

$p$	Sign
$2$	$-1$
$5$	$-1$

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	400.6.a.o		2
4.b	odd	2	1		25.6.a.d	yes	2
5.b	even	2	1		400.6.a.w		2
5.c	odd	4	2		400.6.c.n		4
12.b	even	2	1		225.6.a.l		2
20.d	odd	2	1		25.6.a.b	✓	2
20.e	even	4	2		25.6.b.b		4
60.h	even	2	1		225.6.a.s		2
60.l	odd	4	2		225.6.b.i		4

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
25.6.a.b	✓	2	20.d	odd	2	1
25.6.a.d	yes	2	4.b	odd	2	1
25.6.b.b		4	20.e	even	4	2
225.6.a.l		2	12.b	even	2	1
225.6.a.s		2	60.h	even	2	1
225.6.b.i		4	60.l	odd	4	2
400.6.a.o		2	1.a	even	1	1	trivial
400.6.a.w		2	5.b	even	2	1
400.6.c.n		4	5.c	odd	4	2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $T_{3}^{2} + 20T_{3} - 141$ acting on $S_{6}^{\mathrm{new}}(\Gamma_0(400))$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{2}$
$3$	$T^{2} + 20T - 141$
$5$	$T^{2}$
$7$	$T^{2} + 200T + 9036$
$11$	$T^{2} - 196T - 141021$