Newform orbit 46.4.a.d

• Label: 46.4.a.d
• Level: $46$
• Weight: $4$
• Character orbit: 46.a
• Self dual: yes
• Analytic conductor: $2.714$
• Analytic rank: $0$
• Dimension: $2$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 46 = 2 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	46.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(2.71408786026\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{73}) \)
Defining polynomial:	\( x^{2} - x - 18 \)
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
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 \(\beta = \frac{1}{2}(1 + \sqrt{73})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	q + 2 * q^2 + (-b + 2) * q^3 + 4 * q^4 + (2*b + 4) * q^5 + (-2*b + 4) * q^6 + (4*b + 4) * q^7 + 8 * q^8 + (-3*b - 5) * q^9 + (4*b + 8) * q^10 - 6 * q^11 + (-4*b + 8) * q^12 + (-11*b - 20) * q^13 + (8*b + 8) * q^14 + (-2*b - 28) * q^15 + 16 * q^16 + (-6*b - 30) * q^17 + (-6*b - 10) * q^18 + (20*b - 18) * q^19 + (8*b + 16) * q^20 - 64 * q^21 - 12 * q^22 + 23 * q^23 + (-8*b + 16) * q^24 + (20*b - 37) * q^25 + (-22*b - 40) * q^26 + (29*b - 10) * q^27 + (16*b + 16) * q^28 + (-33*b - 12) * q^29 + (-4*b - 56) * q^30 + (-69*b + 26) * q^31 + 32 * q^32 + (6*b - 12) * q^33 + (-12*b - 60) * q^34 + (32*b + 160) * q^35 + (-12*b - 20) * q^36 + (38*b + 84) * q^37 + (40*b - 36) * q^38 + (9*b + 158) * q^39 + (16*b + 32) * q^40 + (29*b + 172) * q^41 - 128 * q^42 + (-82*b + 198) * q^43 - 24 * q^44 + (-28*b - 128) * q^45 + 46 * q^46 + (-25*b + 442) * q^47 + (-16*b + 32) * q^48 + (48*b - 39) * q^49 + (40*b - 74) * q^50 + (24*b + 48) * q^51 + (-44*b - 80) * q^52 + (-38*b + 44) * q^53 + (58*b - 20) * q^54 + (-12*b - 24) * q^55 + (32*b + 32) * q^56 + (38*b - 396) * q^57 + (-66*b - 24) * q^58 + (60*b + 276) * q^59 + (-8*b - 112) * q^60 + (58*b - 560) * q^61 + (-138*b + 52) * q^62 + (-44*b - 236) * q^63 + 64 * q^64 + (-106*b - 476) * q^65 + (12*b - 24) * q^66 + (56*b - 450) * q^67 + (-24*b - 120) * q^68 + (-23*b + 46) * q^69 + (64*b + 320) * q^70 + (-21*b + 210) * q^71 + (-24*b - 40) * q^72 + (3*b - 640) * q^73 + (76*b + 168) * q^74 + (57*b - 434) * q^75 + (80*b - 72) * q^76 + (-24*b - 24) * q^77 + (18*b + 316) * q^78 + (26*b + 48) * q^79 + (32*b + 64) * q^80 + (120*b - 407) * q^81 + (58*b + 344) * q^82 + (22*b + 890) * q^83 - 256 * q^84 + (-96*b - 336) * q^85 + (-164*b + 396) * q^86 + (-21*b + 570) * q^87 - 48 * q^88 + (18*b + 1014) * q^89 + (-56*b - 256) * q^90 + (-168*b - 872) * q^91 + 92 * q^92 + (-95*b + 1294) * q^93 + (-50*b + 884) * q^94 + (84*b + 648) * q^95 + (-32*b + 64) * q^96 + (-274*b - 318) * q^97 + (96*b - 78) * q^98 + (18*b + 30) * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	2 * q + 4 * q^2 + 3 * q^3 + 8 * q^4 + 10 * q^5 + 6 * q^6 + 12 * q^7 + 16 * q^8 - 13 * q^9 + 20 * q^10 - 12 * q^11 + 12 * q^12 - 51 * q^13 + 24 * q^14 - 58 * q^15 + 32 * q^16 - 66 * q^17 - 26 * q^18 - 16 * q^19 + 40 * q^20 - 128 * q^21 - 24 * q^22 + 46 * q^23 + 24 * q^24 - 54 * q^25 - 102 * q^26 + 9 * q^27 + 48 * q^28 - 57 * q^29 - 116 * q^30 - 17 * q^31 + 64 * q^32 - 18 * q^33 - 132 * q^34 + 352 * q^35 - 52 * q^36 + 206 * q^37 - 32 * q^38 + 325 * q^39 + 80 * q^40 + 373 * q^41 - 256 * q^42 + 314 * q^43 - 48 * q^44 - 284 * q^45 + 92 * q^46 + 859 * q^47 + 48 * q^48 - 30 * q^49 - 108 * q^50 + 120 * q^51 - 204 * q^52 + 50 * q^53 + 18 * q^54 - 60 * q^55 + 96 * q^56 - 754 * q^57 - 114 * q^58 + 612 * q^59 - 232 * q^60 - 1062 * q^61 - 34 * q^62 - 516 * q^63 + 128 * q^64 - 1058 * q^65 - 36 * q^66 - 844 * q^67 - 264 * q^68 + 69 * q^69 + 704 * q^70 + 399 * q^71 - 104 * q^72 - 1277 * q^73 + 412 * q^74 - 811 * q^75 - 64 * q^76 - 72 * q^77 + 650 * q^78 + 122 * q^79 + 160 * q^80 - 694 * q^81 + 746 * q^82 + 1802 * q^83 - 512 * q^84 - 768 * q^85 + 628 * q^86 + 1119 * q^87 - 96 * q^88 + 2046 * q^89 - 568 * q^90 - 1912 * q^91 + 184 * q^92 + 2493 * q^93 + 1718 * q^94 + 1380 * q^95 + 96 * q^96 - 910 * q^97 - 60 * q^98 + 78 * 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

4.77200
−3.77200

2.00000

−2.77200

4.00000

13.5440

−5.54400

23.0880

8.00000

−19.3160

27.0880

1.2

2.00000

5.77200

4.00000

−3.54400

11.5440

−11.0880

8.00000

6.31601

−7.08801

Atkin-Lehner signs

\( p \)	Sign
\(2\)	\( -1 \)
\(23\)	\( -1 \)

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	46.4.a.d	✓	2
3.b	odd	2	1		414.4.a.f		2
4.b	odd	2	1		368.4.a.f		2
5.b	even	2	1		1150.4.a.j		2
5.c	odd	4	2		1150.4.b.j		4
7.b	odd	2	1		2254.4.a.f		2
8.b	even	2	1		1472.4.a.k		2
8.d	odd	2	1		1472.4.a.n		2
23.b	odd	2	1		1058.4.a.j		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
46.4.a.d	✓	2	1.a	even	1	1	trivial
368.4.a.f		2	4.b	odd	2	1
414.4.a.f		2	3.b	odd	2	1
1058.4.a.j		2	23.b	odd	2	1
1150.4.a.j		2	5.b	even	2	1
1150.4.b.j		4	5.c	odd	4	2
1472.4.a.k		2	8.b	even	2	1
1472.4.a.n		2	8.d	odd	2	1
2254.4.a.f		2	7.b	odd	2	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} - 3T_{3} - 16 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(46))\).

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( (T - 2)^{2} \)
$3$	\( T^{2} - 3T - 16 \)
$5$	\( T^{2} - 10T - 48 \)
$7$	\( T^{2} - 12T - 256 \)
$11$	\( (T + 6)^{2} \)