Newform orbit 1710.4.a.n

• Label: 1710.4.a.n
• Level: $1710$
• Weight: $4$
• Character orbit: 1710.a
• Self dual: yes
• Analytic conductor: $100.893$
• Analytic rank: $0$
• Dimension: $2$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 1710 = 2 \cdot 3^{2} \cdot 5 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1710.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(100.893266110\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{26}) \)
Defining polynomial:	\( x^{2} - 26 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 570)
Fricke sign:	\(+1\)
Sato-Tate group:	$\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{26}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	q - 2 * q^2 + 4 * q^4 + 5 * q^5 + (b + 22) * q^7 - 8 * q^8 - 10 * q^10 + (3*b - 4) * q^11 + (-9*b + 28) * q^13 + (-2*b - 44) * q^14 + 16 * q^16 + (-10*b - 16) * q^17 - 19 * q^19 + 20 * q^20 + (-6*b + 8) * q^22 + (-42*b + 6) * q^23 + 25 * q^25 + (18*b - 56) * q^26 + (4*b + 88) * q^28 + (33*b - 2) * q^29 + (26*b + 38) * q^31 - 32 * q^32 + (20*b + 32) * q^34 + (5*b + 110) * q^35 + (-33*b + 108) * q^37 + 38 * q^38 - 40 * q^40 + (37*b - 10) * q^41 + (-23*b + 434) * q^43 + (12*b - 16) * q^44 + (84*b - 12) * q^46 + (58*b + 122) * q^47 + (44*b + 167) * q^49 - 50 * q^50 + (-36*b + 112) * q^52 + (-58*b - 260) * q^53 + (15*b - 20) * q^55 + (-8*b - 176) * q^56 + (-66*b + 4) * q^58 + (70*b + 280) * q^59 + (-56*b + 16) * q^61 + (-52*b - 76) * q^62 + 64 * q^64 + (-45*b + 140) * q^65 + (88*b + 544) * q^67 + (-40*b - 64) * q^68 + (-10*b - 220) * q^70 + (-110*b - 272) * q^71 + (12*b + 354) * q^73 + (66*b - 216) * q^74 - 76 * q^76 + (62*b - 10) * q^77 + (-4*b + 296) * q^79 + 80 * q^80 + (-74*b + 20) * q^82 + (88*b - 94) * q^83 + (-50*b - 80) * q^85 + (46*b - 868) * q^86 + (-24*b + 32) * q^88 + (-117*b + 778) * q^89 + (-170*b + 382) * q^91 + (-168*b + 24) * q^92 + (-116*b - 244) * q^94 - 95 * q^95 + (255*b + 156) * q^97 + (-88*b - 334) * q^98
\(\operatorname{Tr}(f)(q)\)	\(=\)	2 * q - 4 * q^2 + 8 * q^4 + 10 * q^5 + 44 * q^7 - 16 * q^8 - 20 * q^10 - 8 * q^11 + 56 * q^13 - 88 * q^14 + 32 * q^16 - 32 * q^17 - 38 * q^19 + 40 * q^20 + 16 * q^22 + 12 * q^23 + 50 * q^25 - 112 * q^26 + 176 * q^28 - 4 * q^29 + 76 * q^31 - 64 * q^32 + 64 * q^34 + 220 * q^35 + 216 * q^37 + 76 * q^38 - 80 * q^40 - 20 * q^41 + 868 * q^43 - 32 * q^44 - 24 * q^46 + 244 * q^47 + 334 * q^49 - 100 * q^50 + 224 * q^52 - 520 * q^53 - 40 * q^55 - 352 * q^56 + 8 * q^58 + 560 * q^59 + 32 * q^61 - 152 * q^62 + 128 * q^64 + 280 * q^65 + 1088 * q^67 - 128 * q^68 - 440 * q^70 - 544 * q^71 + 708 * q^73 - 432 * q^74 - 152 * q^76 - 20 * q^77 + 592 * q^79 + 160 * q^80 + 40 * q^82 - 188 * q^83 - 160 * q^85 - 1736 * q^86 + 64 * q^88 + 1556 * q^89 + 764 * q^91 + 48 * q^92 - 488 * q^94 - 190 * q^95 + 312 * q^97 - 668 * q^98

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

−5.09902
5.09902

−2.00000

0

4.00000

5.00000

0

16.9010

−8.00000

0

−10.0000

1.2

−2.00000

0

4.00000

5.00000

0

27.0990

−8.00000

0

−10.0000

Atkin-Lehner signs

\( p \)	Sign
\(2\)	\( +1 \)
\(3\)	\( -1 \)
\(5\)	\( -1 \)
\(19\)	\( +1 \)

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	1710.4.a.n		2
3.b	odd	2	1		570.4.a.o	✓	2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
570.4.a.o	✓	2	3.b	odd	2	1
1710.4.a.n		2	1.a	even	1	1	trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1710))\):

\( T_{7}^{2} - 44T_{7} + 458 \)

\( T_{11}^{2} + 8T_{11} - 218 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( (T + 2)^{2} \)
$3$	\( T^{2} \)
$5$	\( (T - 5)^{2} \)
$7$	\( T^{2} - 44T + 458 \)
$11$	\( T^{2} + 8T - 218 \)