Newform orbit 704.5.h.f

• Label: 704.5.h.f
• Level: $704$
• Weight: $5$
• Character orbit: 704.h
• Analytic conductor: $72.772$
• Analytic rank: $0$
• Dimension: $2$
• Inner twists: $2$

Newspace parameters

Level:	$N$	$=$	$704 = 2^{6} \cdot 11$
Weight:	$k$	$=$	$5$
Character orbit:	$[\chi]$	$=$	704.h (of order $2$, degree $1$, not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	$72.7724540110$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{-30})$
Defining polynomial:	$x^{2} + 30$
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$2$
Twist minimal:	no (minimal twist has level 11)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the $q$-expansion are expressed in terms of $\beta = 2\sqrt{-30}$. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	q + 3 * q^3 - 31 * q^5 + 5*b * q^7 - 72 * q^9 + (-11*b - 11) * q^11 + 17*b * q^13 - 93 * q^15 + 21*b * q^17 + 9*b * q^19 + 15*b * q^21 + 277 * q^23 + 336 * q^25 - 459 * q^27 + 116*b * q^29 - 1363 * q^31 + (-33*b - 33) * q^33 - 155*b * q^35 - 167 * q^37 + 51*b * q^39 - 97*b * q^41 - 110*b * q^43 + 2232 * q^45 + 1702 * q^47 - 599 * q^49 + 63*b * q^51 - 4522 * q^53 + (341*b + 341) * q^55 + 27*b * q^57 + 2363 * q^59 - 362*b * q^61 - 360*b * q^63 - 527*b * q^65 + 2803 * q^67 + 831 * q^69 + 3397 * q^71 - 303*b * q^73 + 1008 * q^75 + (-55*b + 6600) * q^77 - 556*b * q^79 + 4455 * q^81 - 76*b * q^83 - 651*b * q^85 + 348*b * q^87 - 4673 * q^89 - 10200 * q^91 - 4089 * q^93 - 279*b * q^95 + 4247 * q^97 + (792*b + 792) * q^99
$\operatorname{Tr}(f)(q)$	$=$	2 * q + 6 * q^3 - 62 * q^5 - 144 * q^9 - 22 * q^11 - 186 * q^15 + 554 * q^23 + 672 * q^25 - 918 * q^27 - 2726 * q^31 - 66 * q^33 - 334 * q^37 + 4464 * q^45 + 3404 * q^47 - 1198 * q^49 - 9044 * q^53 + 682 * q^55 + 4726 * q^59 + 5606 * q^67 + 1662 * q^69 + 6794 * q^71 + 2016 * q^75 + 13200 * q^77 + 8910 * q^81 - 9346 * q^89 - 20400 * q^91 - 8178 * q^93 + 8494 * q^97 + 1584 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/704\mathbb{Z}\right)^\times$.

$n$	$133$	$321$	$639$
$\chi(n)$	$1$	$-1$	$1$

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  

$\iota_m(\nu)$

$a_{2}$

$a_{3}$

$a_{4}$

$a_{5}$

$a_{6}$

$a_{7}$

$a_{8}$

$a_{9}$

$a_{10}$

65.1

	−	5.47723i
		5.47723i

0

3.00000

0

−31.0000

0

−

54.7723i

0

−72.0000

0

65.2

0

3.00000

0

−31.0000

0

54.7723i

0

−72.0000

0

Inner twists

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

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	704.5.h.f		2
4.b	odd	2	1		704.5.h.d		2
8.b	even	2	1		11.5.b.b	✓	2
8.d	odd	2	1		176.5.h.c		2
11.b	odd	2	1	inner	704.5.h.f		2
24.h	odd	2	1		99.5.c.b		2
40.f	even	2	1		275.5.c.e		2
40.i	odd	4	2		275.5.d.b		4
44.c	even	2	1		704.5.h.d		2
88.b	odd	2	1		11.5.b.b	✓	2
88.g	even	2	1		176.5.h.c		2
88.o	even	10	4		121.5.d.b		8
88.p	odd	10	4		121.5.d.b		8
264.m	even	2	1		99.5.c.b		2
440.o	odd	2	1		275.5.c.e		2
440.t	even	4	2		275.5.d.b		4

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
11.5.b.b	✓	2	8.b	even	2	1
11.5.b.b	✓	2	88.b	odd	2	1
99.5.c.b		2	24.h	odd	2	1
99.5.c.b		2	264.m	even	2	1
121.5.d.b		8	88.o	even	10	4
121.5.d.b		8	88.p	odd	10	4
176.5.h.c		2	8.d	odd	2	1
176.5.h.c		2	88.g	even	2	1
275.5.c.e		2	40.f	even	2	1
275.5.c.e		2	440.o	odd	2	1
275.5.d.b		4	40.i	odd	4	2
275.5.d.b		4	440.t	even	4	2
704.5.h.d		2	4.b	odd	2	1
704.5.h.d		2	44.c	even	2	1
704.5.h.f		2	1.a	even	1	1	trivial
704.5.h.f		2	11.b	odd	2	1	inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{5}^{\mathrm{new}}(704, [\chi])$:

$T_{3} - 3$

$T_{5} + 31$

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{2}$
$3$	$(T - 3)^{2}$
$5$	$(T + 31)^{2}$
$7$	$T^{2} + 3000$
$11$	$T^{2} + 22T + 14641$