Newform orbit 176.4.e.a

• Label: 176.4.e.a
• Level: $176$
• Weight: $4$
• Character orbit: 176.e
• Analytic conductor: $10.384$
• Analytic rank: $0$
• Dimension: $2$
• CM discriminant: -11
• Inner twists: $4$

Newspace parameters

Level:	$N$	$=$	$176 = 2^{4} \cdot 11$
Weight:	$k$	$=$	$4$
Character orbit:	$[\chi]$	$=$	176.e (of order $2$, degree $1$, minimal)

Newform invariants

Self dual:	no
Analytic conductor:	$10.3843361610$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{-11})$
Defining polynomial:	$x^{2} - x + 3$
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$2$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

$q$-expansion

Coefficients of the $q$-expansion are expressed in terms of $\beta = \sqrt{-11}$. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	q - 2*b * q^3 - 18 * q^5 - 17 * q^9 + 11*b * q^11 + 36*b * q^15 + 58*b * q^23 + 199 * q^25 - 20*b * q^27 + 18*b * q^31 + 242 * q^33 - 434 * q^37 + 306 * q^45 + 194*b * q^47 - 343 * q^49 - 738 * q^53 - 198*b * q^55 + 166*b * q^59 - 306*b * q^67 + 1276 * q^69 - 310*b * q^71 - 398*b * q^75 - 899 * q^81 + 1674 * q^89 + 396 * q^93 + 34 * q^97 - 187*b * q^99
$\operatorname{Tr}(f)(q)$	$=$	$2 q - 36 q^{5} - 34 q^{9} + 398 q^{25} + 484 q^{33} - 868 q^{37} + 612 q^{45} - 686 q^{49} - 1476 q^{53} + 2552 q^{69} - 1798 q^{81} + 3348 q^{89} + 792 q^{93} + 68 q^{97}+O(q^{100})$

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/176\mathbb{Z}\right)^\times$.

$n$	$111$	$133$	$145$
$\chi(n)$	$-1$	$1$	$-1$

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  

$\iota_m(\nu)$

$a_{2}$

$a_{3}$

$a_{4}$

$a_{5}$

$a_{6}$

$a_{7}$

$a_{8}$

$a_{9}$

$a_{10}$

175.1

0.500000	+	1.65831i
0.500000	−	1.65831i

0

−

6.63325i

0

−18.0000

0

0

0

−17.0000

0

175.2

0

6.63325i

0

−18.0000

0

0

0

−17.0000

0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
11.b	odd	2	1	CM by $\Q(\sqrt{-11})$
4.b	odd	2	1	inner
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	176.4.e.a	✓	2
3.b	odd	2	1		1584.4.o.c		2
4.b	odd	2	1	inner	176.4.e.a	✓	2
8.b	even	2	1		704.4.e.c		2
8.d	odd	2	1		704.4.e.c		2
11.b	odd	2	1	CM	176.4.e.a	✓	2
12.b	even	2	1		1584.4.o.c		2
33.d	even	2	1		1584.4.o.c		2
44.c	even	2	1	inner	176.4.e.a	✓	2
88.b	odd	2	1		704.4.e.c		2
88.g	even	2	1		704.4.e.c		2
132.d	odd	2	1		1584.4.o.c		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
176.4.e.a	✓	2	1.a	even	1	1	trivial
176.4.e.a	✓	2	4.b	odd	2	1	inner
176.4.e.a	✓	2	11.b	odd	2	1	CM
176.4.e.a	✓	2	44.c	even	2	1	inner
704.4.e.c		2	8.b	even	2	1
704.4.e.c		2	8.d	odd	2	1
704.4.e.c		2	88.b	odd	2	1
704.4.e.c		2	88.g	even	2	1
1584.4.o.c		2	3.b	odd	2	1
1584.4.o.c		2	12.b	even	2	1
1584.4.o.c		2	33.d	even	2	1
1584.4.o.c		2	132.d	odd	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{4}^{\mathrm{new}}(176, [\chi])$:

$T_{3}^{2} + 44$

$T_{7}$

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{2}$
$3$	$T^{2} + 44$
$5$	$(T + 18)^{2}$
$7$	$T^{2}$
$11$	$T^{2} + 1331$