Newform orbit 840.4.a.i

• Label: 840.4.a.i
• 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:	\(\Q(\sqrt{6}) \)
Defining polynomial:	\( x^{2} - 6 \)
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 \(\beta = 4\sqrt{6}\). 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 + (4*b + 8) * q^11 + (-5*b - 6) * q^13 - 15 * q^15 + (5*b - 62) * q^17 + (-14*b + 16) * q^19 + 21 * q^21 + (b + 136) * q^23 + 25 * q^25 - 27 * q^27 + (23*b - 66) * q^29 + (-15*b - 140) * q^31 + (-12*b - 24) * q^33 - 35 * q^35 + (21*b - 146) * q^37 + (15*b + 18) * q^39 + (-4*b - 254) * q^41 + (13*b + 228) * q^43 + 45 * q^45 + (-3*b + 528) * q^47 + 49 * q^49 + (-15*b + 186) * q^51 + (-46*b + 26) * q^53 + (20*b + 40) * q^55 + (42*b - 48) * q^57 + (6*b - 220) * q^59 + (-35*b - 378) * q^61 - 63 * q^63 + (-25*b - 30) * q^65 + (45*b - 124) * q^67 + (-3*b - 408) * q^69 + (-15*b - 244) * q^71 + (-51*b - 546) * q^73 - 75 * q^75 + (-28*b - 56) * q^77 + (-36*b - 616) * q^79 + 81 * q^81 + (86*b - 388) * q^83 + (25*b - 310) * q^85 + (-69*b + 198) * q^87 + (-28*b - 78) * q^89 + (35*b + 42) * q^91 + (45*b + 420) * q^93 + (-70*b + 80) * q^95 + (91*b + 78) * q^97 + (36*b + 72) * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	2 * q - 6 * q^3 + 10 * q^5 - 14 * q^7 + 18 * q^9 + 16 * q^11 - 12 * q^13 - 30 * q^15 - 124 * q^17 + 32 * q^19 + 42 * q^21 + 272 * q^23 + 50 * q^25 - 54 * q^27 - 132 * q^29 - 280 * q^31 - 48 * q^33 - 70 * q^35 - 292 * q^37 + 36 * q^39 - 508 * q^41 + 456 * q^43 + 90 * q^45 + 1056 * q^47 + 98 * q^49 + 372 * q^51 + 52 * q^53 + 80 * q^55 - 96 * q^57 - 440 * q^59 - 756 * q^61 - 126 * q^63 - 60 * q^65 - 248 * q^67 - 816 * q^69 - 488 * q^71 - 1092 * q^73 - 150 * q^75 - 112 * q^77 - 1232 * q^79 + 162 * q^81 - 776 * q^83 - 620 * q^85 + 396 * q^87 - 156 * q^89 + 84 * q^91 + 840 * q^93 + 160 * q^95 + 156 * q^97 + 144 * 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

−2.44949
2.44949

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.i	✓	2
4.b	odd	2	1		1680.4.a.bn		2

    

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

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{11}^{2} - 16T_{11} - 1472 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(840))\).

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{2} \)
$3$	\( (T + 3)^{2} \)
$5$	\( (T - 5)^{2} \)
$7$	\( (T + 7)^{2} \)
$11$	\( T^{2} - 16T - 1472 \)