Newform orbit 700.4.a.p

• Label: 700.4.a.p
• Level: $700$
• Weight: $4$
• Character orbit: 700.a
• Self dual: yes
• Analytic conductor: $41.301$
• Analytic rank: $0$
• Dimension: $2$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	$N$	$=$	$700 = 2^{2} \cdot 5^{2} \cdot 7$
Weight:	$k$	$=$	$4$
Character orbit:	$[\chi]$	$=$	700.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	$41.3013370040$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{6})$
Defining polynomial:	$x^{2} - 6$
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 140)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

$q$-expansion

Coefficients of the $q$-expansion are expressed in terms of $\beta = \sqrt{6}$. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	q + b * q^3 - 7 * q^7 - 21 * q^9 + (14*b + 24) * q^11 + (23*b + 14) * q^13 + (-24*b + 14) * q^17 + (7*b - 44) * q^19 - 7*b * q^21 + (-2*b - 14) * q^23 - 48*b * q^27 + (-14*b - 70) * q^29 + (-14*b + 52) * q^31 + (24*b + 84) * q^33 + (48*b + 182) * q^37 + (14*b + 138) * q^39 + (154*b + 90) * q^41 + (-94*b + 196) * q^43 + (70*b + 28) * q^47 + 49 * q^49 + (14*b - 144) * q^51 + (50*b + 462) * q^53 + (-44*b + 42) * q^57 + (161*b + 284) * q^59 + (-161*b - 234) * q^61 + 147 * q^63 + (288*b + 112) * q^67 + (-14*b - 12) * q^69 + (-14*b - 486) * q^71 + (-98*b + 658) * q^73 + (-98*b - 168) * q^77 + (-350*b + 482) * q^79 + 279 * q^81 + (-287*b + 336) * q^83 + (-70*b - 84) * q^87 + (168*b - 826) * q^89 + (-161*b - 98) * q^91 + (52*b - 84) * q^93 + (20*b + 714) * q^97 + (-294*b - 504) * q^99
$\operatorname{Tr}(f)(q)$	$=$	2 * q - 14 * q^7 - 42 * q^9 + 48 * q^11 + 28 * q^13 + 28 * q^17 - 88 * q^19 - 28 * q^23 - 140 * q^29 + 104 * q^31 + 168 * q^33 + 364 * q^37 + 276 * q^39 + 180 * q^41 + 392 * q^43 + 56 * q^47 + 98 * q^49 - 288 * q^51 + 924 * q^53 + 84 * q^57 + 568 * q^59 - 468 * q^61 + 294 * q^63 + 224 * q^67 - 24 * q^69 - 972 * q^71 + 1316 * q^73 - 336 * q^77 + 964 * q^79 + 558 * q^81 + 672 * q^83 - 168 * q^87 - 1652 * q^89 - 196 * q^91 - 168 * q^93 + 1428 * q^97 - 1008 * 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

−2.44949
2.44949

0

−2.44949

0

0

0

−7.00000

0

−21.0000

0

1.2

0

2.44949

0

0

0

−7.00000

0

−21.0000

0

Atkin-Lehner signs

$p$	Sign
$2$	$-1$
$5$	$-1$
$7$	$+1$

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	700.4.a.p		2
5.b	even	2	1		700.4.a.q		2
5.c	odd	4	2		140.4.e.c	✓	4
15.e	even	4	2		1260.4.k.d		4
20.e	even	4	2		560.4.g.d		4
35.f	even	4	2		980.4.e.d		4

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
140.4.e.c	✓	4	5.c	odd	4	2
560.4.g.d		4	20.e	even	4	2
700.4.a.p		2	1.a	even	1	1	trivial
700.4.a.q		2	5.b	even	2	1
980.4.e.d		4	35.f	even	4	2
1260.4.k.d		4	15.e	even	4	2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{4}^{\mathrm{new}}(\Gamma_0(700))$:

$T_{3}^{2} - 6$

$T_{11}^{2} - 48T_{11} - 600$

$T_{13}^{2} - 28T_{13} - 2978$

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{2}$
$3$	$T^{2} - 6$
$5$	$T^{2}$
$7$	$(T + 7)^{2}$
$11$	$T^{2} - 48T - 600$