Newform orbit 114.4.a.f

• Label: 114.4.a.f
• Level: $114$
• Weight: $4$
• Character orbit: 114.a
• Self dual: yes
• Analytic conductor: $6.726$
• Analytic rank: $0$
• Dimension: $2$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 114 = 2 \cdot 3 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	114.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(6.72621774065\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{273}) \)
Defining polynomial:	\( x^{2} - x - 68 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{5}]\)
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 \(\beta = \frac{1}{2}(1 + \sqrt{273})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	q + 2 * q^2 + 3 * q^3 + 4 * q^4 + (-b + 6) * q^5 + 6 * q^6 + (b + 4) * q^7 + 8 * q^8 + 9 * q^9 + (-2*b + 12) * q^10 + (7*b - 4) * q^11 + 12 * q^12 + (-4*b + 10) * q^13 + (2*b + 8) * q^14 + (-3*b + 18) * q^15 + 16 * q^16 + (-3*b + 6) * q^17 + 18 * q^18 - 19 * q^19 + (-4*b + 24) * q^20 + (3*b + 12) * q^21 + (14*b - 8) * q^22 - 4*b * q^23 + 24 * q^24 + (-11*b - 21) * q^25 + (-8*b + 20) * q^26 + 27 * q^27 + (4*b + 16) * q^28 + (-8*b - 6) * q^29 + (-6*b + 36) * q^30 + (26*b - 132) * q^31 + 32 * q^32 + (21*b - 12) * q^33 + (-6*b + 12) * q^34 + (b - 44) * q^35 + 36 * q^36 + (-10*b - 126) * q^37 - 38 * q^38 + (-12*b + 30) * q^39 + (-8*b + 48) * q^40 + (42*b - 82) * q^41 + (6*b + 24) * q^42 + (-27*b - 200) * q^43 + (28*b - 16) * q^44 + (-9*b + 54) * q^45 - 8*b * q^46 + (-43*b + 84) * q^47 + 48 * q^48 + (9*b - 259) * q^49 + (-22*b - 42) * q^50 + (-9*b + 18) * q^51 + (-16*b + 40) * q^52 + (-24*b + 82) * q^53 + 54 * q^54 + (39*b - 500) * q^55 + (8*b + 32) * q^56 - 57 * q^57 + (-16*b - 12) * q^58 + (56*b + 20) * q^59 + (-12*b + 72) * q^60 + (5*b - 278) * q^61 + (52*b - 264) * q^62 + (9*b + 36) * q^63 + 64 * q^64 + (-30*b + 332) * q^65 + (42*b - 24) * q^66 + (56*b - 484) * q^67 + (-12*b + 24) * q^68 - 12*b * q^69 + (2*b - 88) * q^70 + (-64*b + 384) * q^71 + 72 * q^72 + (-13*b - 154) * q^73 + (-20*b - 252) * q^74 + (-33*b - 63) * q^75 - 76 * q^76 + (31*b + 460) * q^77 + (-24*b + 60) * q^78 + (-104*b - 204) * q^79 + (-16*b + 96) * q^80 + 81 * q^81 + (84*b - 164) * q^82 + (84*b + 552) * q^83 + (12*b + 48) * q^84 + (-21*b + 240) * q^85 + (-54*b - 400) * q^86 + (-24*b - 18) * q^87 + (56*b - 32) * q^88 + (4*b + 886) * q^89 + (-18*b + 108) * q^90 + (-10*b - 232) * q^91 - 16*b * q^92 + (78*b - 396) * q^93 + (-86*b + 168) * q^94 + (19*b - 114) * q^95 + 96 * q^96 + (8*b + 882) * q^97 + (18*b - 518) * q^98 + (63*b - 36) * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	2 * q + 4 * q^2 + 6 * q^3 + 8 * q^4 + 11 * q^5 + 12 * q^6 + 9 * q^7 + 16 * q^8 + 18 * q^9 + 22 * q^10 - q^11 + 24 * q^12 + 16 * q^13 + 18 * q^14 + 33 * q^15 + 32 * q^16 + 9 * q^17 + 36 * q^18 - 38 * q^19 + 44 * q^20 + 27 * q^21 - 2 * q^22 - 4 * q^23 + 48 * q^24 - 53 * q^25 + 32 * q^26 + 54 * q^27 + 36 * q^28 - 20 * q^29 + 66 * q^30 - 238 * q^31 + 64 * q^32 - 3 * q^33 + 18 * q^34 - 87 * q^35 + 72 * q^36 - 262 * q^37 - 76 * q^38 + 48 * q^39 + 88 * q^40 - 122 * q^41 + 54 * q^42 - 427 * q^43 - 4 * q^44 + 99 * q^45 - 8 * q^46 + 125 * q^47 + 96 * q^48 - 509 * q^49 - 106 * q^50 + 27 * q^51 + 64 * q^52 + 140 * q^53 + 108 * q^54 - 961 * q^55 + 72 * q^56 - 114 * q^57 - 40 * q^58 + 96 * q^59 + 132 * q^60 - 551 * q^61 - 476 * q^62 + 81 * q^63 + 128 * q^64 + 634 * q^65 - 6 * q^66 - 912 * q^67 + 36 * q^68 - 12 * q^69 - 174 * q^70 + 704 * q^71 + 144 * q^72 - 321 * q^73 - 524 * q^74 - 159 * q^75 - 152 * q^76 + 951 * q^77 + 96 * q^78 - 512 * q^79 + 176 * q^80 + 162 * q^81 - 244 * q^82 + 1188 * q^83 + 108 * q^84 + 459 * q^85 - 854 * q^86 - 60 * q^87 - 8 * q^88 + 1776 * q^89 + 198 * q^90 - 474 * q^91 - 16 * q^92 - 714 * q^93 + 250 * q^94 - 209 * q^95 + 192 * q^96 + 1772 * q^97 - 1018 * q^98 - 9 * 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

8.76136
−7.76136

2.00000

3.00000

4.00000

−2.76136

6.00000

12.7614

8.00000

9.00000

−5.52271

1.2

2.00000

3.00000

4.00000

13.7614

6.00000

−3.76136

8.00000

9.00000

27.5227

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	114.4.a.f	✓	2
3.b	odd	2	1		342.4.a.g		2
4.b	odd	2	1		912.4.a.j		2
19.b	odd	2	1		2166.4.a.l		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
114.4.a.f	✓	2	1.a	even	1	1	trivial
342.4.a.g		2	3.b	odd	2	1
912.4.a.j		2	4.b	odd	2	1
2166.4.a.l		2	19.b	odd	2	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} - 11T_{5} - 38 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(114))\).

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( (T - 2)^{2} \)
$3$	\( (T - 3)^{2} \)
$5$	\( T^{2} - 11T - 38 \)
$7$	\( T^{2} - 9T - 48 \)
$11$	\( T^{2} + T - 3344 \)