Newform orbit 1710.4.a.k

• Label: 1710.4.a.k
• 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:	\(\Q(\sqrt{5}) \)
Defining polynomial:	\( x^{2} - x - 1 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index:	\( 2 \)
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 \(\beta = \sqrt{5}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	q - 2 * q^2 + 4 * q^4 + 5 * q^5 + (-b - 5) * q^7 - 8 * q^8 - 10 * q^10 + (3*b - 1) * q^11 + (21*b + 31) * q^13 + (2*b + 10) * q^14 + 16 * q^16 + (34*b + 32) * q^17 - 19 * q^19 + 20 * q^20 + (-6*b + 2) * q^22 + (-18*b + 42) * q^23 + 25 * q^25 + (-42*b - 62) * q^26 + (-4*b - 20) * q^28 + (27*b + 133) * q^29 + (64*b - 76) * q^31 - 32 * q^32 + (-68*b - 64) * q^34 + (-5*b - 25) * q^35 + (75*b - 195) * q^37 + 38 * q^38 - 40 * q^40 + (29*b + 215) * q^41 + (-181*b - 73) * q^43 + (12*b - 4) * q^44 + (36*b - 84) * q^46 + (110*b - 22) * q^47 + (10*b - 313) * q^49 - 50 * q^50 + (84*b + 124) * q^52 + (-134*b + 460) * q^53 + (15*b - 5) * q^55 + (8*b + 40) * q^56 + (-54*b - 266) * q^58 + (-286*b + 82) * q^59 + (-322*b - 56) * q^61 + (-128*b + 152) * q^62 + 64 * q^64 + (105*b + 155) * q^65 + (-202*b - 14) * q^67 + (136*b + 128) * q^68 + (10*b + 50) * q^70 + (-172*b + 412) * q^71 + (450*b + 12) * q^73 + (-150*b + 390) * q^74 - 76 * q^76 + (-14*b - 10) * q^77 + (364*b - 196) * q^79 + 80 * q^80 + (-58*b - 430) * q^82 + (152*b + 104) * q^83 + (170*b + 160) * q^85 + (362*b + 146) * q^86 + (-24*b + 8) * q^88 + (501*b - 41) * q^89 + (-136*b - 260) * q^91 + (-72*b + 168) * q^92 + (-220*b + 44) * q^94 - 95 * q^95 + (-357*b + 645) * q^97 + (-20*b + 626) * q^98
\(\operatorname{Tr}(f)(q)\)	\(=\)	2 * q - 4 * q^2 + 8 * q^4 + 10 * q^5 - 10 * q^7 - 16 * q^8 - 20 * q^10 - 2 * q^11 + 62 * q^13 + 20 * q^14 + 32 * q^16 + 64 * q^17 - 38 * q^19 + 40 * q^20 + 4 * q^22 + 84 * q^23 + 50 * q^25 - 124 * q^26 - 40 * q^28 + 266 * q^29 - 152 * q^31 - 64 * q^32 - 128 * q^34 - 50 * q^35 - 390 * q^37 + 76 * q^38 - 80 * q^40 + 430 * q^41 - 146 * q^43 - 8 * q^44 - 168 * q^46 - 44 * q^47 - 626 * q^49 - 100 * q^50 + 248 * q^52 + 920 * q^53 - 10 * q^55 + 80 * q^56 - 532 * q^58 + 164 * q^59 - 112 * q^61 + 304 * q^62 + 128 * q^64 + 310 * q^65 - 28 * q^67 + 256 * q^68 + 100 * q^70 + 824 * q^71 + 24 * q^73 + 780 * q^74 - 152 * q^76 - 20 * q^77 - 392 * q^79 + 160 * q^80 - 860 * q^82 + 208 * q^83 + 320 * q^85 + 292 * q^86 + 16 * q^88 - 82 * q^89 - 520 * q^91 + 336 * q^92 + 88 * q^94 - 190 * q^95 + 1290 * q^97 + 1252 * 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

1.61803
−0.618034

−2.00000

0

4.00000

5.00000

0

−7.23607

−8.00000

0

−10.0000

1.2

−2.00000

0

4.00000

5.00000

0

−2.76393

−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.k		2
3.b	odd	2	1		570.4.a.m	✓	2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
570.4.a.m	✓	2	3.b	odd	2	1
1710.4.a.k		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 \(S_{4}^{\mathrm{new}}(\Gamma_0(1710))\):

\( T_{7}^{2} + 10T_{7} + 20 \)

\( T_{11}^{2} + 2T_{11} - 44 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( (T + 2)^{2} \)
$3$	\( T^{2} \)
$5$	\( (T - 5)^{2} \)
$7$	\( T^{2} + 10T + 20 \)
$11$	\( T^{2} + 2T - 44 \)