Newform orbit 2142.4.a.l

• Label: 2142.4.a.l
• Level: $2142$
• Weight: $4$
• Character orbit: 2142.a
• Self dual: yes
• Analytic conductor: $126.382$
• Analytic rank: $0$
• Dimension: $2$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 2142 = 2 \cdot 3^{2} \cdot 7 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2142.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(126.382091232\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{93}) \)
Defining polynomial:	\( x^{2} - x - 23 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 238)
Fricke sign:	\(+1\)
Sato-Tate group:	$\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(1 + \sqrt{93})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	q + 2 * q^2 + 4 * q^4 + (-b - 4) * q^5 - 7 * q^7 + 8 * q^8 + (-2*b - 8) * q^10 + (8*b - 2) * q^11 + (-8*b - 10) * q^13 - 14 * q^14 + 16 * q^16 - 17 * q^17 + (-4*b + 8) * q^19 + (-4*b - 16) * q^20 + (16*b - 4) * q^22 + (2*b - 26) * q^23 + (9*b - 86) * q^25 + (-16*b - 20) * q^26 - 28 * q^28 + (-28*b + 118) * q^29 + (-27*b + 108) * q^31 + 32 * q^32 - 34 * q^34 + (7*b + 28) * q^35 + (32*b - 176) * q^37 + (-8*b + 16) * q^38 + (-8*b - 32) * q^40 + (7*b + 225) * q^41 + (69*b - 69) * q^43 + (32*b - 8) * q^44 + (4*b - 52) * q^46 + (-6*b + 390) * q^47 + 49 * q^49 + (18*b - 172) * q^50 + (-32*b - 40) * q^52 + (15*b + 358) * q^53 + (-38*b - 176) * q^55 - 56 * q^56 + (-56*b + 236) * q^58 - 148 * q^59 + (-69*b - 135) * q^61 + (-54*b + 216) * q^62 + 64 * q^64 + (50*b + 224) * q^65 + (-49*b + 258) * q^67 - 68 * q^68 + (14*b + 56) * q^70 + (-16*b - 100) * q^71 + (-21*b + 421) * q^73 + (64*b - 352) * q^74 + (-16*b + 32) * q^76 + (-56*b + 14) * q^77 + (-78*b + 54) * q^79 + (-16*b - 64) * q^80 + (14*b + 450) * q^82 + (-158*b + 258) * q^83 + (17*b + 68) * q^85 + (138*b - 138) * q^86 + (64*b - 16) * q^88 + (-38*b + 478) * q^89 + (56*b + 70) * q^91 + (8*b - 104) * q^92 + (-12*b + 780) * q^94 + (12*b + 60) * q^95 + (109*b - 512) * q^97 + 98 * q^98
\(\operatorname{Tr}(f)(q)\)	\(=\)	2 * q + 4 * q^2 + 8 * q^4 - 9 * q^5 - 14 * q^7 + 16 * q^8 - 18 * q^10 + 4 * q^11 - 28 * q^13 - 28 * q^14 + 32 * q^16 - 34 * q^17 + 12 * q^19 - 36 * q^20 + 8 * q^22 - 50 * q^23 - 163 * q^25 - 56 * q^26 - 56 * q^28 + 208 * q^29 + 189 * q^31 + 64 * q^32 - 68 * q^34 + 63 * q^35 - 320 * q^37 + 24 * q^38 - 72 * q^40 + 457 * q^41 - 69 * q^43 + 16 * q^44 - 100 * q^46 + 774 * q^47 + 98 * q^49 - 326 * q^50 - 112 * q^52 + 731 * q^53 - 390 * q^55 - 112 * q^56 + 416 * q^58 - 296 * q^59 - 339 * q^61 + 378 * q^62 + 128 * q^64 + 498 * q^65 + 467 * q^67 - 136 * q^68 + 126 * q^70 - 216 * q^71 + 821 * q^73 - 640 * q^74 + 48 * q^76 - 28 * q^77 + 30 * q^79 - 144 * q^80 + 914 * q^82 + 358 * q^83 + 153 * q^85 - 138 * q^86 + 32 * q^88 + 918 * q^89 + 196 * q^91 - 200 * q^92 + 1548 * q^94 + 132 * q^95 - 915 * q^97 + 196 * 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

5.32183
−4.32183

2.00000

0

4.00000

−9.32183

0

−7.00000

8.00000

0

−18.6437

1.2

2.00000

0

4.00000

0.321825

0

−7.00000

8.00000

0

0.643651

Atkin-Lehner signs

\( p \)	Sign
\(2\)	\( -1 \)
\(3\)	\( -1 \)
\(7\)	\( +1 \)
\(17\)	\( +1 \)

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	2142.4.a.l		2
3.b	odd	2	1		238.4.a.a	✓	2
12.b	even	2	1		1904.4.a.b		2
21.c	even	2	1		1666.4.a.c		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
238.4.a.a	✓	2	3.b	odd	2	1
1666.4.a.c		2	21.c	even	2	1
1904.4.a.b		2	12.b	even	2	1
2142.4.a.l		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 \(S_{4}^{\mathrm{new}}(\Gamma_0(2142))\):

\( T_{5}^{2} + 9T_{5} - 3 \)

\( T_{11}^{2} - 4T_{11} - 1484 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( (T - 2)^{2} \)
$3$	\( T^{2} \)
$5$	\( T^{2} + 9T - 3 \)
$7$	\( (T + 7)^{2} \)
$11$	\( T^{2} - 4T - 1484 \)