Newform orbit 6975.2.a.t

• Label: 6975.2.a.t
• Level: $6975$
• Weight: $2$
• Character orbit: 6975.a
• Self dual: yes
• Analytic conductor: $55.696$
• Analytic rank: $0$
• Dimension: $2$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 6975 = 3^{2} \cdot 5^{2} \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6975.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(55.6956554098\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{5}) \)
Defining polynomial:	\( x^{2} - x - 1 \)
Coefficient ring:	\(\Z[a_1, a_2]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 93)
Fricke sign:	\(-1\)
Sato-Tate group:	$\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(1 + \sqrt{5})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	q + (-b - 1) * q^2 + 3*b * q^4 + (2*b + 1) * q^7 + (-4*b - 1) * q^8 + (2*b + 2) * q^11 + 2*b * q^13 + (-5*b - 3) * q^14 + (3*b + 5) * q^16 + (4*b - 4) * q^17 + (2*b - 5) * q^19 + (-6*b - 4) * q^22 + 2*b * q^23 + (-4*b - 2) * q^26 + (9*b + 6) * q^28 + (2*b - 2) * q^29 - q^31 + (-3*b - 6) * q^32 - 4*b * q^34 + (-6*b + 2) * q^37 + (b + 3) * q^38 + (6*b - 3) * q^41 + (-6*b + 6) * q^43 + (12*b + 6) * q^44 + (-4*b - 2) * q^46 + 4*b * q^47 + (8*b - 2) * q^49 + (6*b + 6) * q^52 + (-8*b + 4) * q^53 + (-14*b - 9) * q^56 - 2*b * q^58 + 3 * q^59 + 8 * q^61 + (b + 1) * q^62 + (6*b - 1) * q^64 + 12 * q^67 + 12 * q^68 - 9 * q^71 + (2*b - 2) * q^73 + (10*b + 4) * q^74 + (-9*b + 6) * q^76 + (10*b + 6) * q^77 + (-4*b + 6) * q^79 + (-9*b - 3) * q^82 + (4*b - 14) * q^83 + 6*b * q^86 + (-18*b - 10) * q^88 + (8*b - 2) * q^89 + (6*b + 4) * q^91 + (6*b + 6) * q^92 + (-8*b - 4) * q^94 - 9 * q^97 + (-14*b - 6) * q^98
\(\operatorname{Tr}(f)(q)\)	\(=\)	2 * q - 3 * q^2 + 3 * q^4 + 4 * q^7 - 6 * q^8 + 6 * q^11 + 2 * q^13 - 11 * q^14 + 13 * q^16 - 4 * q^17 - 8 * q^19 - 14 * q^22 + 2 * q^23 - 8 * q^26 + 21 * q^28 - 2 * q^29 - 2 * q^31 - 15 * q^32 - 4 * q^34 - 2 * q^37 + 7 * q^38 + 6 * q^43 + 24 * q^44 - 8 * q^46 + 4 * q^47 + 4 * q^49 + 18 * q^52 - 32 * q^56 - 2 * q^58 + 6 * q^59 + 16 * q^61 + 3 * q^62 + 4 * q^64 + 24 * q^67 + 24 * q^68 - 18 * q^71 - 2 * q^73 + 18 * q^74 + 3 * q^76 + 22 * q^77 + 8 * q^79 - 15 * q^82 - 24 * q^83 + 6 * q^86 - 38 * q^88 + 4 * q^89 + 14 * q^91 + 18 * q^92 - 16 * q^94 - 18 * q^97 - 26 * q^98

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

1.61803
−0.618034

−2.61803

0

4.85410

0

0

4.23607

−7.47214

0

0

1.2

−0.381966

0

−1.85410

0

0

−0.236068

1.47214

0

0

Atkin-Lehner signs

\( p \)	Sign
\(3\)	\( -1 \)
\(5\)	\( +1 \)
\(31\)	\( +1 \)

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	6975.2.a.t		2
3.b	odd	2	1		2325.2.a.o		2
5.b	even	2	1		279.2.a.b		2
15.d	odd	2	1		93.2.a.a	✓	2
15.e	even	4	2		2325.2.c.h		4
20.d	odd	2	1		4464.2.a.bn		2
60.h	even	2	1		1488.2.a.q		2
105.g	even	2	1		4557.2.a.p		2
120.i	odd	2	1		5952.2.a.bv		2
120.m	even	2	1		5952.2.a.bo		2
155.c	odd	2	1		8649.2.a.m		2
465.g	even	2	1		2883.2.a.a		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
93.2.a.a	✓	2	15.d	odd	2	1
279.2.a.b		2	5.b	even	2	1
1488.2.a.q		2	60.h	even	2	1
2325.2.a.o		2	3.b	odd	2	1
2325.2.c.h		4	15.e	even	4	2
2883.2.a.a		2	465.g	even	2	1
4464.2.a.bn		2	20.d	odd	2	1
4557.2.a.p		2	105.g	even	2	1
5952.2.a.bo		2	120.m	even	2	1
5952.2.a.bv		2	120.i	odd	2	1
6975.2.a.t		2	1.a	even	1	1	trivial
8649.2.a.m		2	155.c	odd	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6975))\):

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

\( T_{7}^{2} - 4T_{7} - 1 \)

\( T_{11}^{2} - 6T_{11} + 4 \)

\( T_{13}^{2} - 2T_{13} - 4 \)

\( T_{17}^{2} + 4T_{17} - 16 \)

\( T_{29}^{2} + 2T_{29} - 4 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{2} + 3T + 1 \)
$3$	\( T^{2} \)
$5$	\( T^{2} \)
$7$	\( T^{2} - 4T - 1 \)
$11$	\( T^{2} - 6T + 4 \)