Newform orbit 840.4.a.o

• Label: 840.4.a.o
• Level: $840$
• Weight: $4$
• Character orbit: 840.a
• Self dual: yes
• Analytic conductor: $49.562$
• Analytic rank: $0$
• Dimension: $2$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 840 = 2^{3} \cdot 3 \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	840.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(49.5616044048\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{2881}) \)
Defining polynomial:	\( x^{2} - x - 720 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index:	\( 2 \)
Twist minimal:	yes
Fricke sign:	\(+1\)
Sato-Tate group:	$\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{2881}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	q + 3 * q^3 + 5 * q^5 - 7 * q^7 + 9 * q^9 + (-b - 11) * q^11 + (b - 11) * q^13 + 15 * q^15 - 14 * q^17 + (b + 107) * q^19 - 21 * q^21 + (-2*b - 70) * q^23 + 25 * q^25 + 27 * q^27 + 294 * q^29 + (-3*b + 67) * q^31 + (-3*b - 33) * q^33 - 35 * q^35 + (6*b + 88) * q^37 + (3*b - 33) * q^39 + (2*b + 120) * q^41 + (8*b - 4) * q^43 + 45 * q^45 + 96 * q^47 + 49 * q^49 - 42 * q^51 + (-5*b + 339) * q^53 + (-5*b - 55) * q^55 + (3*b + 321) * q^57 + (-6*b + 266) * q^59 + (-12*b + 114) * q^61 - 63 * q^63 + (5*b - 55) * q^65 + (8*b + 100) * q^67 + (-6*b - 210) * q^69 + (5*b + 403) * q^71 + (-13*b - 313) * q^73 + 75 * q^75 + (7*b + 77) * q^77 + (-4*b - 236) * q^79 + 81 * q^81 + (12*b + 176) * q^83 - 70 * q^85 + 882 * q^87 + (4*b - 890) * q^89 + (-7*b + 77) * q^91 + (-9*b + 201) * q^93 + (5*b + 535) * q^95 + (5*b + 509) * q^97 + (-9*b - 99) * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	2 * q + 6 * q^3 + 10 * q^5 - 14 * q^7 + 18 * q^9 - 22 * q^11 - 22 * q^13 + 30 * q^15 - 28 * q^17 + 214 * q^19 - 42 * q^21 - 140 * q^23 + 50 * q^25 + 54 * q^27 + 588 * q^29 + 134 * q^31 - 66 * q^33 - 70 * q^35 + 176 * q^37 - 66 * q^39 + 240 * q^41 - 8 * q^43 + 90 * q^45 + 192 * q^47 + 98 * q^49 - 84 * q^51 + 678 * q^53 - 110 * q^55 + 642 * q^57 + 532 * q^59 + 228 * q^61 - 126 * q^63 - 110 * q^65 + 200 * q^67 - 420 * q^69 + 806 * q^71 - 626 * q^73 + 150 * q^75 + 154 * q^77 - 472 * q^79 + 162 * q^81 + 352 * q^83 - 140 * q^85 + 1764 * q^87 - 1780 * q^89 + 154 * q^91 + 402 * q^93 + 1070 * q^95 + 1018 * q^97 - 198 * 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

27.3375
−26.3375

0

3.00000

0

5.00000

0

−7.00000

0

9.00000

0

1.2

0

3.00000

0

5.00000

0

−7.00000

0

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	840.4.a.o	✓	2
4.b	odd	2	1		1680.4.a.bf		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
840.4.a.o	✓	2	1.a	even	1	1	trivial
1680.4.a.bf		2	4.b	odd	2	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{11}^{2} + 22T_{11} - 2760 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(840))\).

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{2} \)
$3$	\( (T - 3)^{2} \)
$5$	\( (T - 5)^{2} \)
$7$	\( (T + 7)^{2} \)
$11$	\( T^{2} + 22T - 2760 \)