Newform orbit 2166.4.a.j

• Label: 2166.4.a.j
• Level: $2166$
• Weight: $4$
• Character orbit: 2166.a
• Self dual: yes
• Analytic conductor: $127.798$
• Analytic rank: $1$
• Dimension: $2$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 2166 = 2 \cdot 3 \cdot 19^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2166.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(127.798137072\)
Analytic rank:	\(1\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{5}) \)
Defining polynomial:	\( x^{2} - x - 1 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
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 - 2 * q^2 - 3 * q^3 + 4 * q^4 + (-5*b - 4) * q^5 + 6 * q^6 + (7*b - 12) * q^7 - 8 * q^8 + 9 * q^9 + (10*b + 8) * q^10 + (8*b + 3) * q^11 - 12 * q^12 + (5*b - 12) * q^13 + (-14*b + 24) * q^14 + (15*b + 12) * q^15 + 16 * q^16 + (-34*b + 11) * q^17 - 18 * q^18 + (-20*b - 16) * q^20 + (-21*b + 36) * q^21 + (-16*b - 6) * q^22 + (16*b - 35) * q^23 + 24 * q^24 + (65*b - 84) * q^25 + (-10*b + 24) * q^26 - 27 * q^27 + (28*b - 48) * q^28 + (86*b + 22) * q^29 + (-30*b - 24) * q^30 + (-161*b + 200) * q^31 - 32 * q^32 + (-24*b - 9) * q^33 + (68*b - 22) * q^34 + (-3*b + 13) * q^35 + 36 * q^36 + (-160*b + 154) * q^37 + (-15*b + 36) * q^39 + (40*b + 32) * q^40 + (-75*b - 128) * q^41 + (42*b - 72) * q^42 + (94*b - 159) * q^43 + (32*b + 12) * q^44 + (-45*b - 36) * q^45 + (-32*b + 70) * q^46 + (-19*b + 176) * q^47 - 48 * q^48 + (-119*b - 150) * q^49 + (-130*b + 168) * q^50 + (102*b - 33) * q^51 + (20*b - 48) * q^52 + (332*b + 217) * q^53 + 54 * q^54 + (-87*b - 52) * q^55 + (-56*b + 96) * q^56 + (-172*b - 44) * q^58 + (424*b + 18) * q^59 + (60*b + 48) * q^60 + (-250*b + 372) * q^61 + (322*b - 400) * q^62 + (63*b - 108) * q^63 + 64 * q^64 + (15*b + 23) * q^65 + (48*b + 18) * q^66 + (-129*b + 131) * q^67 + (-136*b + 44) * q^68 + (-48*b + 105) * q^69 + (6*b - 26) * q^70 + (-245*b + 352) * q^71 - 72 * q^72 + (74*b + 161) * q^73 + (320*b - 308) * q^74 + (-195*b + 252) * q^75 + (-19*b + 20) * q^77 + (30*b - 72) * q^78 + (-250*b - 375) * q^79 + (-80*b - 64) * q^80 + 81 * q^81 + (150*b + 256) * q^82 + (-212*b + 69) * q^83 + (-84*b + 144) * q^84 + (251*b + 126) * q^85 + (-188*b + 318) * q^86 + (-258*b - 66) * q^87 + (-64*b - 24) * q^88 + (372*b + 404) * q^89 + (90*b + 72) * q^90 + (-109*b + 179) * q^91 + (64*b - 140) * q^92 + (483*b - 600) * q^93 + (38*b - 352) * q^94 + 96 * q^96 + (-122*b + 1030) * q^97 + (238*b + 300) * q^98 + (72*b + 27) * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	2 * q - 4 * q^2 - 6 * q^3 + 8 * q^4 - 13 * q^5 + 12 * q^6 - 17 * q^7 - 16 * q^8 + 18 * q^9 + 26 * q^10 + 14 * q^11 - 24 * q^12 - 19 * q^13 + 34 * q^14 + 39 * q^15 + 32 * q^16 - 12 * q^17 - 36 * q^18 - 52 * q^20 + 51 * q^21 - 28 * q^22 - 54 * q^23 + 48 * q^24 - 103 * q^25 + 38 * q^26 - 54 * q^27 - 68 * q^28 + 130 * q^29 - 78 * q^30 + 239 * q^31 - 64 * q^32 - 42 * q^33 + 24 * q^34 + 23 * q^35 + 72 * q^36 + 148 * q^37 + 57 * q^39 + 104 * q^40 - 331 * q^41 - 102 * q^42 - 224 * q^43 + 56 * q^44 - 117 * q^45 + 108 * q^46 + 333 * q^47 - 96 * q^48 - 419 * q^49 + 206 * q^50 + 36 * q^51 - 76 * q^52 + 766 * q^53 + 108 * q^54 - 191 * q^55 + 136 * q^56 - 260 * q^58 + 460 * q^59 + 156 * q^60 + 494 * q^61 - 478 * q^62 - 153 * q^63 + 128 * q^64 + 61 * q^65 + 84 * q^66 + 133 * q^67 - 48 * q^68 + 162 * q^69 - 46 * q^70 + 459 * q^71 - 144 * q^72 + 396 * q^73 - 296 * q^74 + 309 * q^75 + 21 * q^77 - 114 * q^78 - 1000 * q^79 - 208 * q^80 + 162 * q^81 + 662 * q^82 - 74 * q^83 + 204 * q^84 + 503 * q^85 + 448 * q^86 - 390 * q^87 - 112 * q^88 + 1180 * q^89 + 234 * q^90 + 249 * q^91 - 216 * q^92 - 717 * q^93 - 666 * q^94 + 192 * q^96 + 1938 * q^97 + 838 * q^98 + 126 * 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

1.61803
−0.618034

−2.00000

−3.00000

4.00000

−12.0902

6.00000

−0.673762

−8.00000

9.00000

24.1803

1.2

−2.00000

−3.00000

4.00000

−0.909830

6.00000

−16.3262

−8.00000

9.00000

1.81966

Atkin-Lehner signs

\( p \)	Sign
\(2\)	\( +1 \)
\(3\)	\( +1 \)
\(19\)	\( -1 \)

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	2166.4.a.j	✓	2
19.b	odd	2	1		2166.4.a.p	yes	2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
2166.4.a.j	✓	2	1.a	even	1	1	trivial
2166.4.a.p	yes	2	19.b	odd	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(2166))\):

\( T_{5}^{2} + 13T_{5} + 11 \)

\( T_{13}^{2} + 19T_{13} + 59 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( (T + 2)^{2} \)
$3$	\( (T + 3)^{2} \)
$5$	\( T^{2} + 13T + 11 \)
$7$	\( T^{2} + 17T + 11 \)
$11$	\( T^{2} - 14T - 31 \)