Newform orbit 1050.4.a.be

• Label: 1050.4.a.be
• Level: $1050$
• Weight: $4$
• Character orbit: 1050.a
• Self dual: yes
• Analytic conductor: $61.952$
• Analytic rank: $0$
• Dimension: $2$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 1050 = 2 \cdot 3 \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1050.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(61.9520055060\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{6001}) \)
Defining polynomial:	\( x^{2} - x - 1500 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{11}]\)
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 \(\beta = \frac{1}{2}(1 + \sqrt{6001})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	q + 2 * q^2 - 3 * q^3 + 4 * q^4 - 6 * q^6 - 7 * q^7 + 8 * q^8 + 9 * q^9 + (-b + 26) * q^11 - 12 * q^12 + (b - 27) * q^13 - 14 * q^14 + 16 * q^16 + (3*b - 11) * q^17 + 18 * q^18 + (-2*b + 24) * q^19 + 21 * q^21 + (-2*b + 52) * q^22 + (-4*b - 21) * q^23 - 24 * q^24 + (2*b - 54) * q^26 - 27 * q^27 - 28 * q^28 + (2*b + 207) * q^29 + (b - 115) * q^31 + 32 * q^32 + (3*b - 78) * q^33 + (6*b - 22) * q^34 + 36 * q^36 + (-5*b + 110) * q^37 + (-4*b + 48) * q^38 + (-3*b + 81) * q^39 + (5*b - 41) * q^41 + 42 * q^42 + (4*b + 133) * q^43 + (-4*b + 104) * q^44 + (-8*b - 42) * q^46 + (-10*b - 200) * q^47 - 48 * q^48 + 49 * q^49 + (-9*b + 33) * q^51 + (4*b - 108) * q^52 + (5*b + 353) * q^53 - 54 * q^54 - 56 * q^56 + (6*b - 72) * q^57 + (4*b + 414) * q^58 + (b + 701) * q^59 + (19*b + 1) * q^61 + (2*b - 230) * q^62 - 63 * q^63 + 64 * q^64 + (6*b - 156) * q^66 + (15*b - 432) * q^67 + (12*b - 44) * q^68 + (12*b + 63) * q^69 + (-11*b + 328) * q^71 + 72 * q^72 + (-10*b + 230) * q^73 + (-10*b + 220) * q^74 + (-8*b + 96) * q^76 + (7*b - 182) * q^77 + (-6*b + 162) * q^78 + (-3*b + 370) * q^79 + 81 * q^81 + (10*b - 82) * q^82 + (-19*b + 613) * q^83 + 84 * q^84 + (8*b + 266) * q^86 + (-6*b - 621) * q^87 + (-8*b + 208) * q^88 + 566 * q^89 + (-7*b + 189) * q^91 + (-16*b - 84) * q^92 + (-3*b + 345) * q^93 + (-20*b - 400) * q^94 - 96 * q^96 + (-12*b - 358) * q^97 + 98 * q^98 + (-9*b + 234) * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	2 * q + 4 * q^2 - 6 * q^3 + 8 * q^4 - 12 * q^6 - 14 * q^7 + 16 * q^8 + 18 * q^9 + 51 * q^11 - 24 * q^12 - 53 * q^13 - 28 * q^14 + 32 * q^16 - 19 * q^17 + 36 * q^18 + 46 * q^19 + 42 * q^21 + 102 * q^22 - 46 * q^23 - 48 * q^24 - 106 * q^26 - 54 * q^27 - 56 * q^28 + 416 * q^29 - 229 * q^31 + 64 * q^32 - 153 * q^33 - 38 * q^34 + 72 * q^36 + 215 * q^37 + 92 * q^38 + 159 * q^39 - 77 * q^41 + 84 * q^42 + 270 * q^43 + 204 * q^44 - 92 * q^46 - 410 * q^47 - 96 * q^48 + 98 * q^49 + 57 * q^51 - 212 * q^52 + 711 * q^53 - 108 * q^54 - 112 * q^56 - 138 * q^57 + 832 * q^58 + 1403 * q^59 + 21 * q^61 - 458 * q^62 - 126 * q^63 + 128 * q^64 - 306 * q^66 - 849 * q^67 - 76 * q^68 + 138 * q^69 + 645 * q^71 + 144 * q^72 + 450 * q^73 + 430 * q^74 + 184 * q^76 - 357 * q^77 + 318 * q^78 + 737 * q^79 + 162 * q^81 - 154 * q^82 + 1207 * q^83 + 168 * q^84 + 540 * q^86 - 1248 * q^87 + 408 * q^88 + 1132 * q^89 + 371 * q^91 - 184 * q^92 + 687 * q^93 - 820 * q^94 - 192 * q^96 - 728 * q^97 + 196 * q^98 + 459 * 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

39.2331
−38.2331

2.00000

−3.00000

4.00000

0

−6.00000

−7.00000

8.00000

9.00000

0

1.2

2.00000

−3.00000

4.00000

0

−6.00000

−7.00000

8.00000

9.00000

0

Atkin-Lehner signs

\( p \)	Sign
\(2\)	\( -1 \)
\(3\)	\( +1 \)
\(5\)	\( -1 \)
\(7\)	\( +1 \)

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	1050.4.a.be	yes	2
5.b	even	2	1		1050.4.a.bd	✓	2
5.c	odd	4	2		1050.4.g.t		4

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
1050.4.a.bd	✓	2	5.b	even	2	1
1050.4.a.be	yes	2	1.a	even	1	1	trivial
1050.4.g.t		4	5.c	odd	4	2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1050))\):

\( T_{11}^{2} - 51T_{11} - 850 \)

\( T_{13}^{2} + 53T_{13} - 798 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( (T - 2)^{2} \)
$3$	\( (T + 3)^{2} \)
$5$	\( T^{2} \)
$7$	\( (T + 7)^{2} \)
$11$	\( T^{2} - 51T - 850 \)