Newform orbit 1710.4.a.l

• Label: 1710.4.a.l
• Level: $1710$
• Weight: $4$
• Character orbit: 1710.a
• Self dual: yes
• Analytic conductor: $100.893$
• Analytic rank: $1$
• 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:	\(1\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{34}) \)
Defining polynomial:	\( x^{2} - 34 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 190)
Fricke sign:	\(-1\)
Sato-Tate group:	$\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{34}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	q - 2 * q^2 + 4 * q^4 + 5 * q^5 + (2*b + 12) * q^7 - 8 * q^8 - 10 * q^10 + (2*b - 26) * q^11 + (-5*b + 22) * q^13 + (-4*b - 24) * q^14 + 16 * q^16 + (2*b - 50) * q^17 + 19 * q^19 + 20 * q^20 + (-4*b + 52) * q^22 + (8*b - 8) * q^23 + 25 * q^25 + (10*b - 44) * q^26 + (8*b + 48) * q^28 + (-18*b + 6) * q^29 + (-20*b - 8) * q^31 - 32 * q^32 + (-4*b + 100) * q^34 + (10*b + 60) * q^35 + (-31*b + 18) * q^37 - 38 * q^38 - 40 * q^40 + (26*b - 74) * q^41 + (-26*b - 236) * q^43 + (8*b - 104) * q^44 + (-16*b + 16) * q^46 + (26*b - 188) * q^47 + (48*b - 63) * q^49 - 50 * q^50 + (-20*b + 88) * q^52 + (-47*b - 58) * q^53 + (10*b - 130) * q^55 + (-16*b - 96) * q^56 + (36*b - 12) * q^58 + (-82*b - 56) * q^59 + (36*b - 340) * q^61 + (40*b + 16) * q^62 + 64 * q^64 + (-25*b + 110) * q^65 + (-95*b - 104) * q^67 + (8*b - 200) * q^68 + (-20*b - 120) * q^70 + (-20*b + 308) * q^71 + (14*b - 666) * q^73 + (62*b - 36) * q^74 + 76 * q^76 + (-28*b - 176) * q^77 + (-94*b + 108) * q^79 + 80 * q^80 + (-52*b + 148) * q^82 + (184*b + 20) * q^83 + (10*b - 250) * q^85 + (52*b + 472) * q^86 + (-16*b + 208) * q^88 + (26*b - 242) * q^89 + (-16*b - 76) * q^91 + (32*b - 32) * q^92 + (-52*b + 376) * q^94 + 95 * q^95 + (39*b + 278) * q^97 + (-96*b + 126) * q^98
\(\operatorname{Tr}(f)(q)\)	\(=\)	2 * q - 4 * q^2 + 8 * q^4 + 10 * q^5 + 24 * q^7 - 16 * q^8 - 20 * q^10 - 52 * q^11 + 44 * q^13 - 48 * q^14 + 32 * q^16 - 100 * q^17 + 38 * q^19 + 40 * q^20 + 104 * q^22 - 16 * q^23 + 50 * q^25 - 88 * q^26 + 96 * q^28 + 12 * q^29 - 16 * q^31 - 64 * q^32 + 200 * q^34 + 120 * q^35 + 36 * q^37 - 76 * q^38 - 80 * q^40 - 148 * q^41 - 472 * q^43 - 208 * q^44 + 32 * q^46 - 376 * q^47 - 126 * q^49 - 100 * q^50 + 176 * q^52 - 116 * q^53 - 260 * q^55 - 192 * q^56 - 24 * q^58 - 112 * q^59 - 680 * q^61 + 32 * q^62 + 128 * q^64 + 220 * q^65 - 208 * q^67 - 400 * q^68 - 240 * q^70 + 616 * q^71 - 1332 * q^73 - 72 * q^74 + 152 * q^76 - 352 * q^77 + 216 * q^79 + 160 * q^80 + 296 * q^82 + 40 * q^83 - 500 * q^85 + 944 * q^86 + 416 * q^88 - 484 * q^89 - 152 * q^91 - 64 * q^92 + 752 * q^94 + 190 * q^95 + 556 * q^97 + 252 * 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

−5.83095
5.83095

−2.00000

0

4.00000

5.00000

0

0.338096

−8.00000

0

−10.0000

1.2

−2.00000

0

4.00000

5.00000

0

23.6619

−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.l		2
3.b	odd	2	1		190.4.a.f	✓	2
12.b	even	2	1		1520.4.a.l		2
15.d	odd	2	1		950.4.a.f		2
15.e	even	4	2		950.4.b.h		4

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
190.4.a.f	✓	2	3.b	odd	2	1
950.4.a.f		2	15.d	odd	2	1
950.4.b.h		4	15.e	even	4	2
1520.4.a.l		2	12.b	even	2	1
1710.4.a.l		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 \(S_{4}^{\mathrm{new}}(\Gamma_0(1710))\):

\( T_{7}^{2} - 24T_{7} + 8 \)

\( T_{11}^{2} + 52T_{11} + 540 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( (T + 2)^{2} \)
$3$	\( T^{2} \)
$5$	\( (T - 5)^{2} \)
$7$	\( T^{2} - 24T + 8 \)
$11$	\( T^{2} + 52T + 540 \)