Newform orbit 3344.2.a.j

• Label: 3344.2.a.j
• Level: $3344$
• Weight: $2$
• Character orbit: 3344.a
• Self dual: yes
• Analytic conductor: $26.702$
• Analytic rank: $1$
• Dimension: $2$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	$N$	$=$	$3344 = 2^{4} \cdot 11 \cdot 19$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	3344.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	$26.7019744359$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{13})$
Defining polynomial:	$x^{2} - x - 3$
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 1672)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

$q$-expansion

Coefficients of the $q$-expansion are expressed in terms of $\beta = \frac{1}{2}(1 + \sqrt{13})$. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	q - b * q^3 - b * q^5 + (b - 1) * q^7 + b * q^9 + q^11 + (-b - 3) * q^13 + (b + 3) * q^15 + 2*b * q^17 + q^19 - 3 * q^21 + (-2*b + 2) * q^23 + (b - 2) * q^25 + (2*b - 3) * q^27 + (b - 8) * q^29 + (3*b + 1) * q^31 - b * q^33 - 3 * q^35 + 8 * q^37 + (4*b + 3) * q^39 + (-b - 5) * q^41 + b * q^43 + (-b - 3) * q^45 + (4*b + 2) * q^47 + (-b - 3) * q^49 + (-2*b - 6) * q^51 + (2*b + 2) * q^53 - b * q^55 - b * q^57 + 4 * q^59 - 2 * q^61 + 3 * q^63 + (4*b + 3) * q^65 + (-5*b - 1) * q^67 + 6 * q^69 + (3*b + 6) * q^71 + (2*b - 14) * q^73 + (b - 3) * q^75 + (b - 1) * q^77 + (6*b - 2) * q^79 + (-2*b - 6) * q^81 + (-3*b - 4) * q^83 + (-2*b - 6) * q^85 + (7*b - 3) * q^87 + 6*b * q^89 - 3*b * q^91 + (-4*b - 9) * q^93 - b * q^95 + (8*b - 4) * q^97 + b * q^99
$\operatorname{Tr}(f)(q)$	$=$	2 * q - q^3 - q^5 - q^7 + q^9 + 2 * q^11 - 7 * q^13 + 7 * q^15 + 2 * q^17 + 2 * q^19 - 6 * q^21 + 2 * q^23 - 3 * q^25 - 4 * q^27 - 15 * q^29 + 5 * q^31 - q^33 - 6 * q^35 + 16 * q^37 + 10 * q^39 - 11 * q^41 + q^43 - 7 * q^45 + 8 * q^47 - 7 * q^49 - 14 * q^51 + 6 * q^53 - q^55 - q^57 + 8 * q^59 - 4 * q^61 + 6 * q^63 + 10 * q^65 - 7 * q^67 + 12 * q^69 + 15 * q^71 - 26 * q^73 - 5 * q^75 - q^77 + 2 * q^79 - 14 * q^81 - 11 * q^83 - 14 * q^85 + q^87 + 6 * q^89 - 3 * q^91 - 22 * q^93 - q^95 + q^99

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  

$\iota_m(\nu)$

$a_{2}$

$a_{3}$

$a_{4}$

$a_{5}$

$a_{6}$

$a_{7}$

$a_{8}$

$a_{9}$

$a_{10}$

1.1

2.30278
−1.30278

0

−2.30278

0

−2.30278

0

1.30278

0

2.30278

0

1.2

0

1.30278

0

1.30278

0

−2.30278

0

−1.30278

0

Atkin-Lehner signs

$p$	Sign
$2$	$+1$
$11$	$-1$
$19$	$-1$

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	3344.2.a.j		2
4.b	odd	2	1		1672.2.a.d	✓	2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
1672.2.a.d	✓	2	4.b	odd	2	1
3344.2.a.j		2	1.a	even	1	1	trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(\Gamma_0(3344))$:

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

$T_{5}^{2} + T_{5} - 3$

$T_{7}^{2} + T_{7} - 3$

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{2}$
$3$	$T^{2} + T - 3$
$5$	$T^{2} + T - 3$
$7$	$T^{2} + T - 3$
$11$	$(T - 1)^{2}$