Newform orbit 1690.4.a.s

• Label: 1690.4.a.s
• Level: $1690$
• Weight: $4$
• Character orbit: 1690.a
• Self dual: yes
• Analytic conductor: $99.713$
• Analytic rank: $0$
• Dimension: $2$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 1690 = 2 \cdot 5 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1690.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(99.7132279097\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{19}) \)
Defining polynomial:	\( x^{2} - 19 \)
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 130)
Fricke sign:	\(+1\)
Sato-Tate group:	$\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{19}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	q + 2 * q^2 + (b + 1) * q^3 + 4 * q^4 + 5 * q^5 + (2*b + 2) * q^6 + 6*b * q^7 + 8 * q^8 + (2*b - 7) * q^9 + 10 * q^10 + (7*b + 25) * q^11 + (4*b + 4) * q^12 + 12*b * q^14 + (5*b + 5) * q^15 + 16 * q^16 + (14*b - 44) * q^17 + (4*b - 14) * q^18 + (-15*b + 47) * q^19 + 20 * q^20 + (6*b + 114) * q^21 + (14*b + 50) * q^22 + (3*b - 61) * q^23 + (8*b + 8) * q^24 + 25 * q^25 + (-32*b + 4) * q^27 + 24*b * q^28 + (22*b - 130) * q^29 + (10*b + 10) * q^30 + (-b + 201) * q^31 + 32 * q^32 + (32*b + 158) * q^33 + (28*b - 88) * q^34 + 30*b * q^35 + (8*b - 28) * q^36 + (32*b - 160) * q^37 + (-30*b + 94) * q^38 + 40 * q^40 + (-34*b + 268) * q^41 + (12*b + 228) * q^42 + (89*b - 31) * q^43 + (28*b + 100) * q^44 + (10*b - 35) * q^45 + (6*b - 122) * q^46 + (10*b - 376) * q^47 + (16*b + 16) * q^48 + 341 * q^49 + 50 * q^50 + (-30*b + 222) * q^51 + (-122*b + 124) * q^53 + (-64*b + 8) * q^54 + (35*b + 125) * q^55 + 48*b * q^56 + (32*b - 238) * q^57 + (44*b - 260) * q^58 + (-13*b + 165) * q^59 + (20*b + 20) * q^60 + (-66*b + 310) * q^61 + (-2*b + 402) * q^62 + (-42*b + 228) * q^63 + 64 * q^64 + (64*b + 316) * q^66 + (-124*b - 230) * q^67 + (56*b - 176) * q^68 + (-58*b - 4) * q^69 + 60*b * q^70 + (13*b + 367) * q^71 + (16*b - 56) * q^72 + (-164*b - 228) * q^73 + (64*b - 320) * q^74 + (25*b + 25) * q^75 + (-60*b + 188) * q^76 + (150*b + 798) * q^77 + (-118*b - 462) * q^79 + 80 * q^80 + (-82*b - 415) * q^81 + (-68*b + 536) * q^82 + (6*b + 1152) * q^83 + (24*b + 456) * q^84 + (70*b - 220) * q^85 + (178*b - 62) * q^86 + (-108*b + 288) * q^87 + (56*b + 200) * q^88 + (60*b + 374) * q^89 + (20*b - 70) * q^90 + (12*b - 244) * q^92 + (200*b + 182) * q^93 + (20*b - 752) * q^94 + (-75*b + 235) * q^95 + (32*b + 32) * q^96 + (-184*b - 962) * q^97 + 682 * q^98 + (b + 91) * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	2 * q + 4 * q^2 + 2 * q^3 + 8 * q^4 + 10 * q^5 + 4 * q^6 + 16 * q^8 - 14 * q^9 + 20 * q^10 + 50 * q^11 + 8 * q^12 + 10 * q^15 + 32 * q^16 - 88 * q^17 - 28 * q^18 + 94 * q^19 + 40 * q^20 + 228 * q^21 + 100 * q^22 - 122 * q^23 + 16 * q^24 + 50 * q^25 + 8 * q^27 - 260 * q^29 + 20 * q^30 + 402 * q^31 + 64 * q^32 + 316 * q^33 - 176 * q^34 - 56 * q^36 - 320 * q^37 + 188 * q^38 + 80 * q^40 + 536 * q^41 + 456 * q^42 - 62 * q^43 + 200 * q^44 - 70 * q^45 - 244 * q^46 - 752 * q^47 + 32 * q^48 + 682 * q^49 + 100 * q^50 + 444 * q^51 + 248 * q^53 + 16 * q^54 + 250 * q^55 - 476 * q^57 - 520 * q^58 + 330 * q^59 + 40 * q^60 + 620 * q^61 + 804 * q^62 + 456 * q^63 + 128 * q^64 + 632 * q^66 - 460 * q^67 - 352 * q^68 - 8 * q^69 + 734 * q^71 - 112 * q^72 - 456 * q^73 - 640 * q^74 + 50 * q^75 + 376 * q^76 + 1596 * q^77 - 924 * q^79 + 160 * q^80 - 830 * q^81 + 1072 * q^82 + 2304 * q^83 + 912 * q^84 - 440 * q^85 - 124 * q^86 + 576 * q^87 + 400 * q^88 + 748 * q^89 - 140 * q^90 - 488 * q^92 + 364 * q^93 - 1504 * q^94 + 470 * q^95 + 64 * q^96 - 1924 * q^97 + 1364 * q^98 + 182 * 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

−4.35890
4.35890

2.00000

−3.35890

4.00000

5.00000

−6.71780

−26.1534

8.00000

−15.7178

10.0000

1.2

2.00000

5.35890

4.00000

5.00000

10.7178

26.1534

8.00000

1.71780

10.0000

Atkin-Lehner signs

\( p \)	Sign
\(2\)	\( -1 \)
\(5\)	\( -1 \)
\(13\)	\( +1 \)

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	1690.4.a.s		2
13.b	even	2	1		130.4.a.d	✓	2
39.d	odd	2	1		1170.4.a.ba		2
52.b	odd	2	1		1040.4.a.i		2
65.d	even	2	1		650.4.a.r		2
65.h	odd	4	2		650.4.b.l		4

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
130.4.a.d	✓	2	13.b	even	2	1
650.4.a.r		2	65.d	even	2	1
650.4.b.l		4	65.h	odd	4	2
1040.4.a.i		2	52.b	odd	2	1
1170.4.a.ba		2	39.d	odd	2	1
1690.4.a.s		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 \(S_{4}^{\mathrm{new}}(\Gamma_0(1690))\):

\( T_{3}^{2} - 2T_{3} - 18 \)

\( T_{7}^{2} - 684 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( (T - 2)^{2} \)
$3$	\( T^{2} - 2T - 18 \)
$5$	\( (T - 5)^{2} \)
$7$	\( T^{2} - 684 \)
$11$	\( T^{2} - 50T - 306 \)