Newform orbit 704.4.e.b

• Label: 704.4.e.b
• Level: $704$
• Weight: $4$
• Character orbit: 704.e
• Analytic conductor: $41.537$
• Analytic rank: $0$
• Dimension: $2$
• Inner twists: $2$

Newspace parameters

Level:	$N$	$=$	$704 = 2^{6} \cdot 11$
Weight:	$k$	$=$	$4$
Character orbit:	$[\chi]$	$=$	704.e (of order $2$, degree $1$, not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	$41.5373446440$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{-19})$
Defining polynomial:	$x^{2} - x + 5$
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$2$
Twist minimal:	no (minimal twist has level 176)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the $q$-expansion are expressed in terms of $\beta = \sqrt{-19}$. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	q - b * q^3 - 7 * q^5 + 20 * q^7 + 8 * q^9 + (-7*b + 20) * q^11 + 20*b * q^13 + 7*b * q^15 + 20*b * q^17 - 140 * q^19 - 20*b * q^21 + 21*b * q^23 - 76 * q^25 - 35*b * q^27 + 41*b * q^31 + (-20*b - 133) * q^33 - 140 * q^35 - 91 * q^37 + 380 * q^39 - 20*b * q^41 + 280 * q^43 - 56 * q^45 - 22*b * q^47 + 57 * q^49 + 380 * q^51 - 462 * q^53 + (49*b - 140) * q^55 + 140*b * q^57 + 183*b * q^59 + 120*b * q^61 + 160 * q^63 - 140*b * q^65 + 147*b * q^67 + 399 * q^69 + 105*b * q^71 + 20*b * q^73 + 76*b * q^75 + (-140*b + 400) * q^77 - 560 * q^79 - 449 * q^81 + 1120 * q^83 - 140*b * q^85 + 399 * q^89 + 400*b * q^91 + 779 * q^93 + 980 * q^95 + 959 * q^97 + (-56*b + 160) * q^99
$\operatorname{Tr}(f)(q)$	$=$	2 * q - 14 * q^5 + 40 * q^7 + 16 * q^9 + 40 * q^11 - 280 * q^19 - 152 * q^25 - 266 * q^33 - 280 * q^35 - 182 * q^37 + 760 * q^39 + 560 * q^43 - 112 * q^45 + 114 * q^49 + 760 * q^51 - 924 * q^53 - 280 * q^55 + 320 * q^63 + 798 * q^69 + 800 * q^77 - 1120 * q^79 - 898 * q^81 + 2240 * q^83 + 798 * q^89 + 1558 * q^93 + 1960 * q^95 + 1918 * q^97 + 320 * 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  

$\iota_m(\nu)$

$a_{2}$

$a_{3}$

$a_{4}$

$a_{5}$

$a_{6}$

$a_{7}$

$a_{8}$

$a_{9}$

$a_{10}$

703.1

0.500000	+	2.17945i
0.500000	−	2.17945i

0

−

4.35890i

0

−7.00000

0

20.0000

0

8.00000

0

703.2

0

4.35890i

0

−7.00000

0

20.0000

0

8.00000

0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
44.c	even	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	704.4.e.b		2
4.b	odd	2	1		704.4.e.a		2
8.b	even	2	1		176.4.e.c	yes	2
8.d	odd	2	1		176.4.e.b	✓	2
11.b	odd	2	1		704.4.e.a		2
24.f	even	2	1		1584.4.o.a		2
24.h	odd	2	1		1584.4.o.b		2
44.c	even	2	1	inner	704.4.e.b		2
88.b	odd	2	1		176.4.e.b	✓	2
88.g	even	2	1		176.4.e.c	yes	2
264.m	even	2	1		1584.4.o.a		2
264.p	odd	2	1		1584.4.o.b		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
176.4.e.b	✓	2	8.d	odd	2	1
176.4.e.b	✓	2	88.b	odd	2	1
176.4.e.c	yes	2	8.b	even	2	1
176.4.e.c	yes	2	88.g	even	2	1
704.4.e.a		2	4.b	odd	2	1
704.4.e.a		2	11.b	odd	2	1
704.4.e.b		2	1.a	even	1	1	trivial
704.4.e.b		2	44.c	even	2	1	inner
1584.4.o.a		2	24.f	even	2	1
1584.4.o.a		2	264.m	even	2	1
1584.4.o.b		2	24.h	odd	2	1
1584.4.o.b		2	264.p	odd	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{4}^{\mathrm{new}}(704, [\chi])$:

$T_{3}^{2} + 19$

$T_{7} - 20$

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{2}$
$3$	$T^{2} + 19$
$5$	$(T + 7)^{2}$
$7$	$(T - 20)^{2}$
$11$	$T^{2} - 40T + 1331$