Newform orbit 200.6.c.d

• Label: 200.6.c.d
• Level: $200$
• Weight: $6$
• Character orbit: 200.c
• Analytic conductor: $32.077$
• Analytic rank: $0$
• Dimension: $2$
• Inner twists: $2$

Newspace parameters

Level:	$N$	$=$	$200 = 2^{3} \cdot 5^{2}$
Weight:	$k$	$=$	$6$
Character orbit:	$[\chi]$	$=$	200.c (of order $2$, degree $1$, not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	$32.0767639626$
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:	$2$
Twist minimal:	no (minimal twist has level 40)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

$f(q)$	$=$	q + b * q^3 - 31*b * q^7 + 239 * q^9 - 144 * q^11 + 327*b * q^13 - 595*b * q^17 - 556 * q^19 + 124 * q^21 - 1091*b * q^23 + 482*b * q^27 + 1578 * q^29 + 9660 * q^31 - 144*b * q^33 - 1767*b * q^37 - 1308 * q^39 + 7462 * q^41 + 3557*b * q^43 - 14147*b * q^47 + 12963 * q^49 + 2380 * q^51 + 6523*b * q^53 - 556*b * q^57 + 37092 * q^59 + 39570 * q^61 - 7409*b * q^63 - 28367*b * q^67 + 4364 * q^69 + 45588 * q^71 - 5921*b * q^73 + 4464*b * q^77 - 94216 * q^79 + 56149 * q^81 + 15741*b * q^83 + 1578*b * q^87 + 94054 * q^89 + 40548 * q^91 + 9660*b * q^93 + 11857*b * q^97 - 34416 * q^99
$\operatorname{Tr}(f)(q)$	$=$	2 * q + 478 * q^9 - 288 * q^11 - 1112 * q^19 + 248 * q^21 + 3156 * q^29 + 19320 * q^31 - 2616 * q^39 + 14924 * q^41 + 25926 * q^49 + 4760 * q^51 + 74184 * q^59 + 79140 * q^61 + 8728 * q^69 + 91176 * q^71 - 188432 * q^79 + 112298 * q^81 + 188108 * q^89 + 81096 * q^91 - 68832 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/200\mathbb{Z}\right)^\times$.

$n$	$101$	$151$	$177$
$\chi(n)$	$1$	$1$	$-1$

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  

$\iota_m(\nu)$

$a_{2}$

$a_{3}$

$a_{4}$

$a_{5}$

$a_{6}$

$a_{7}$

$a_{8}$

$a_{9}$

$a_{10}$

49.1

	−	1.00000i
		1.00000i

0

−

2.00000i

0

0

0

62.0000i

0

239.000

0

49.2

0

2.00000i

0

0

0

−

62.0000i

0

239.000

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	200.6.c.d		2
4.b	odd	2	1		400.6.c.k		2
5.b	even	2	1	inner	200.6.c.d		2
5.c	odd	4	1		40.6.a.c	✓	1
5.c	odd	4	1		200.6.a.b		1
15.e	even	4	1		360.6.a.f		1
20.d	odd	2	1		400.6.c.k		2
20.e	even	4	1		80.6.a.d		1
20.e	even	4	1		400.6.a.h		1
40.i	odd	4	1		320.6.a.i		1
40.k	even	4	1		320.6.a.h		1
60.l	odd	4	1		720.6.a.t		1

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
40.6.a.c	✓	1	5.c	odd	4	1
80.6.a.d		1	20.e	even	4	1
200.6.a.b		1	5.c	odd	4	1
200.6.c.d		2	1.a	even	1	1	trivial
200.6.c.d		2	5.b	even	2	1	inner
320.6.a.h		1	40.k	even	4	1
320.6.a.i		1	40.i	odd	4	1
360.6.a.f		1	15.e	even	4	1
400.6.a.h		1	20.e	even	4	1
400.6.c.k		2	4.b	odd	2	1
400.6.c.k		2	20.d	odd	2	1
720.6.a.t		1	60.l	odd	4	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $T_{3}^{2} + 4$ acting on $S_{6}^{\mathrm{new}}(200, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{2}$
$3$	$T^{2} + 4$
$5$	$T^{2}$
$7$	$T^{2} + 3844$
$11$	$(T + 144)^{2}$