Newform orbit 1050.4.g.d

• Label: 1050.4.g.d
• Level: $1050$
• Weight: $4$
• Character orbit: 1050.g
• Analytic conductor: $61.952$
• Analytic rank: $0$
• Dimension: $2$
• Inner twists: $2$

Newspace parameters

Level:	$N$	$=$	$1050 = 2 \cdot 3 \cdot 5^{2} \cdot 7$
Weight:	$k$	$=$	$4$
Character orbit:	$[\chi]$	$=$	1050.g (of order $2$, degree $1$, not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	$61.9520055060$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{-1})$
Defining polynomial:	$x^{2} + 1$
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 210)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the $q$-expansion are expressed in terms of $i = \sqrt{-1}$. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	q + 2*i * q^2 + 3*i * q^3 - 4 * q^4 - 6 * q^6 - 7*i * q^7 - 8*i * q^8 - 9 * q^9 - 12*i * q^12 + 26*i * q^13 + 14 * q^14 + 16 * q^16 - 18*i * q^17 - 18*i * q^18 - 92 * q^19 + 21 * q^21 + 24 * q^24 - 52 * q^26 - 27*i * q^27 + 28*i * q^28 + 6 * q^29 - 4 * q^31 + 32*i * q^32 + 36 * q^34 + 36 * q^36 - 410*i * q^37 - 184*i * q^38 - 78 * q^39 + 174 * q^41 + 42*i * q^42 + 248*i * q^43 - 420*i * q^47 + 48*i * q^48 - 49 * q^49 + 54 * q^51 - 104*i * q^52 + 102*i * q^53 + 54 * q^54 - 56 * q^56 - 276*i * q^57 + 12*i * q^58 + 588 * q^59 + 650 * q^61 - 8*i * q^62 + 63*i * q^63 - 64 * q^64 - 152*i * q^67 + 72*i * q^68 - 168 * q^71 + 72*i * q^72 - 610*i * q^73 + 820 * q^74 + 368 * q^76 - 156*i * q^78 + 1048 * q^79 + 81 * q^81 + 348*i * q^82 - 684*i * q^83 - 84 * q^84 - 496 * q^86 + 18*i * q^87 + 834 * q^89 + 182 * q^91 - 12*i * q^93 + 840 * q^94 - 96 * q^96 - 110*i * q^97 - 98*i * q^98
$\operatorname{Tr}(f)(q)$	$=$	2 * q - 8 * q^4 - 12 * q^6 - 18 * q^9 + 28 * q^14 + 32 * q^16 - 184 * q^19 + 42 * q^21 + 48 * q^24 - 104 * q^26 + 12 * q^29 - 8 * q^31 + 72 * q^34 + 72 * q^36 - 156 * q^39 + 348 * q^41 - 98 * q^49 + 108 * q^51 + 108 * q^54 - 112 * q^56 + 1176 * q^59 + 1300 * q^61 - 128 * q^64 - 336 * q^71 + 1640 * q^74 + 736 * q^76 + 2096 * q^79 + 162 * q^81 - 168 * q^84 - 992 * q^86 + 1668 * q^89 + 364 * q^91 + 1680 * q^94 - 192 * q^96

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1050\mathbb{Z}\right)^\times$.

$n$	$127$	$451$	$701$
$\chi(n)$	$-1$	$1$	$1$

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  

$\iota_m(\nu)$

$a_{2}$

$a_{3}$

$a_{4}$

$a_{5}$

$a_{6}$

$a_{7}$

$a_{8}$

$a_{9}$

$a_{10}$

799.1

	−	1.00000i
		1.00000i

−

2.00000i

−

3.00000i

−4.00000

0

−6.00000

7.00000i

8.00000i

−9.00000

0

799.2

2.00000i

3.00000i

−4.00000

0

−6.00000

−

7.00000i

−

8.00000i

−9.00000

0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
5.b	even	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1050.4.g.d		2
5.b	even	2	1	inner	1050.4.g.d		2
5.c	odd	4	1		210.4.a.e	✓	1
5.c	odd	4	1		1050.4.a.n		1
15.e	even	4	1		630.4.a.w		1
20.e	even	4	1		1680.4.a.a		1
35.f	even	4	1		1470.4.a.g		1

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
210.4.a.e	✓	1	5.c	odd	4	1
630.4.a.w		1	15.e	even	4	1
1050.4.a.n		1	5.c	odd	4	1
1050.4.g.d		2	1.a	even	1	1	trivial
1050.4.g.d		2	5.b	even	2	1	inner
1470.4.a.g		1	35.f	even	4	1
1680.4.a.a		1	20.e	even	4	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{4}^{\mathrm{new}}(1050, [\chi])$:

$T_{11}$

$T_{13}^{2} + 676$

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{2} + 4$
$3$	$T^{2} + 9$
$5$	$T^{2}$
$7$	$T^{2} + 49$
$11$	$T^{2}$