Newform orbit 840.4.a.m

• Label: 840.4.a.m
• Level: $840$
• Weight: $4$
• Character orbit: 840.a
• Self dual: yes
• Analytic conductor: $49.562$
• Analytic rank: $1$
• 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:	\(1\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{21}) \)
Defining polynomial:	\( x^{2} - x - 5 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index:	\( 2^{2} \)
Twist minimal:	yes
Fricke sign:	\(-1\)
Sato-Tate group:	$\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 2\sqrt{21}\). 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 + (3*b + 14) * q^11 + (b - 8) * q^13 - 15 * q^15 + (-6*b - 6) * q^17 + (-11*b + 6) * q^19 - 21 * q^21 + (-10*b - 72) * q^23 + 25 * q^25 + 27 * q^27 + (10*b - 6) * q^29 + (27*b - 42) * q^31 + (9*b + 42) * q^33 + 35 * q^35 + (4*b - 138) * q^37 + (3*b - 24) * q^39 + (12*b + 46) * q^41 + (8*b - 60) * q^43 - 45 * q^45 + (-8*b - 240) * q^47 + 49 * q^49 + (-18*b - 18) * q^51 + (11*b - 324) * q^53 + (-15*b - 70) * q^55 + (-33*b + 18) * q^57 + (-46*b - 392) * q^59 + (-38*b - 334) * q^61 - 63 * q^63 + (-5*b + 40) * q^65 + (74*b - 184) * q^67 + (-30*b - 216) * q^69 + (59*b - 14) * q^71 + (-57*b - 560) * q^73 + 75 * q^75 + (-21*b - 98) * q^77 + (-22*b - 444) * q^79 + 81 * q^81 + (-34*b - 600) * q^83 + (30*b + 30) * q^85 + (30*b - 18) * q^87 + (140*b + 94) * q^89 + (-7*b + 56) * q^91 + (81*b - 126) * q^93 + (55*b - 30) * q^95 + (49*b - 548) * q^97 + (27*b + 126) * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	2 * q + 6 * q^3 - 10 * q^5 - 14 * q^7 + 18 * q^9 + 28 * q^11 - 16 * q^13 - 30 * q^15 - 12 * q^17 + 12 * q^19 - 42 * q^21 - 144 * q^23 + 50 * q^25 + 54 * q^27 - 12 * q^29 - 84 * q^31 + 84 * q^33 + 70 * q^35 - 276 * q^37 - 48 * q^39 + 92 * q^41 - 120 * q^43 - 90 * q^45 - 480 * q^47 + 98 * q^49 - 36 * q^51 - 648 * q^53 - 140 * q^55 + 36 * q^57 - 784 * q^59 - 668 * q^61 - 126 * q^63 + 80 * q^65 - 368 * q^67 - 432 * q^69 - 28 * q^71 - 1120 * q^73 + 150 * q^75 - 196 * q^77 - 888 * q^79 + 162 * q^81 - 1200 * q^83 + 60 * q^85 - 36 * q^87 + 188 * q^89 + 112 * q^91 - 252 * q^93 - 60 * q^95 - 1096 * q^97 + 252 * 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

−1.79129
2.79129

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.m	✓	2
4.b	odd	2	1		1680.4.a.ba		2

    

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

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{11}^{2} - 28T_{11} - 560 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(840))\).

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{2} \)
$3$	\( (T - 3)^{2} \)
$5$	\( (T + 5)^{2} \)
$7$	\( (T + 7)^{2} \)
$11$	\( T^{2} - 28T - 560 \)