Newform orbit 3344.2.a.n

• Label: 3344.2.a.n
• Level: $3344$
• Weight: $2$
• Character orbit: 3344.a
• Self dual: yes
• Analytic conductor: $26.702$
• Analytic rank: $0$
• 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:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{2}) \)
Defining polynomial:	\( x^{2} - 2 \)
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 209)
Fricke sign:	\(-1\)
Sato-Tate group:	$\mathrm{SU}(2)$

$q$-expansion

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

\(f(q)\)	\(=\)	q + (b + 1) * q^3 - q^5 + (b + 2) * q^7 + 2*b * q^9 + q^11 + (3*b - 2) * q^13 + (-b - 1) * q^15 + (b + 2) * q^17 + q^19 + (3*b + 4) * q^21 + 3 * q^23 - 4 * q^25 + (-b + 1) * q^27 + (-3*b - 2) * q^29 + (b + 5) * q^31 + (b + 1) * q^33 + (-b - 2) * q^35 + (5*b + 3) * q^37 + (b + 4) * q^39 + (-4*b + 4) * q^41 + (4*b - 6) * q^43 - 2*b * q^45 + (-2*b - 6) * q^47 + (4*b - 1) * q^49 + (3*b + 4) * q^51 + (-6*b + 4) * q^53 - q^55 + (b + 1) * q^57 + (-b + 3) * q^59 + (-5*b - 4) * q^61 + (4*b + 4) * q^63 + (-3*b + 2) * q^65 + (b + 9) * q^67 + (3*b + 3) * q^69 + (b + 11) * q^71 + (-6*b + 4) * q^73 + (-4*b - 4) * q^75 + (b + 2) * q^77 + (-b + 16) * q^79 + (-6*b - 1) * q^81 + (-b - 2) * q^83 + (-b - 2) * q^85 + (-5*b - 8) * q^87 + (7*b - 5) * q^89 + (4*b + 2) * q^91 + (6*b + 7) * q^93 - q^95 + (b + 1) * q^97 + 2*b * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	2 * q + 2 * q^3 - 2 * q^5 + 4 * q^7 + 2 * q^11 - 4 * q^13 - 2 * q^15 + 4 * q^17 + 2 * q^19 + 8 * q^21 + 6 * q^23 - 8 * q^25 + 2 * q^27 - 4 * q^29 + 10 * q^31 + 2 * q^33 - 4 * q^35 + 6 * q^37 + 8 * q^39 + 8 * q^41 - 12 * q^43 - 12 * q^47 - 2 * q^49 + 8 * q^51 + 8 * q^53 - 2 * q^55 + 2 * q^57 + 6 * q^59 - 8 * q^61 + 8 * q^63 + 4 * q^65 + 18 * q^67 + 6 * q^69 + 22 * q^71 + 8 * q^73 - 8 * q^75 + 4 * q^77 + 32 * q^79 - 2 * q^81 - 4 * q^83 - 4 * q^85 - 16 * q^87 - 10 * q^89 + 4 * q^91 + 14 * q^93 - 2 * q^95 + 2 * q^97

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

−1.41421
1.41421

0

−0.414214

0

−1.00000

0

0.585786

0

−2.82843

0

1.2

0

2.41421

0

−1.00000

0

3.41421

0

2.82843

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.n		2
4.b	odd	2	1		209.2.a.b	✓	2
12.b	even	2	1		1881.2.a.d		2
20.d	odd	2	1		5225.2.a.f		2
44.c	even	2	1		2299.2.a.f		2
76.d	even	2	1		3971.2.a.d		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
209.2.a.b	✓	2	4.b	odd	2	1
1881.2.a.d		2	12.b	even	2	1
2299.2.a.f		2	44.c	even	2	1
3344.2.a.n		2	1.a	even	1	1	trivial
3971.2.a.d		2	76.d	even	2	1
5225.2.a.f		2	20.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(3344))\):

\( T_{3}^{2} - 2T_{3} - 1 \)

\( T_{5} + 1 \)

\( T_{7}^{2} - 4T_{7} + 2 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{2} \)
$3$	\( T^{2} - 2T - 1 \)
$5$	\( (T + 1)^{2} \)
$7$	\( T^{2} - 4T + 2 \)
$11$	\( (T - 1)^{2} \)