Newform orbit 1040.4.a.i

• Label: 1040.4.a.i
• Level: $1040$
• Weight: $4$
• Character orbit: 1040.a
• Self dual: yes
• Analytic conductor: $61.362$
• Analytic rank: $0$
• Dimension: $2$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 1040 = 2^{4} \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1040.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(61.3619864060\)
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 + (b - 1) * q^3 - 5 * q^5 - 6*b * q^7 + (-2*b - 7) * q^9 + (-7*b + 25) * q^11 + 13 * q^13 + (-5*b + 5) * q^15 + (-14*b - 44) * q^17 + (15*b + 47) * q^19 + (6*b - 114) * q^21 + (3*b + 61) * q^23 + 25 * q^25 + (-32*b - 4) * q^27 + (-22*b - 130) * q^29 + (b + 201) * q^31 + (32*b - 158) * q^33 + 30*b * q^35 + (32*b + 160) * q^37 + (13*b - 13) * q^39 + (-34*b - 268) * q^41 + (89*b + 31) * q^43 + (10*b + 35) * q^45 + (-10*b - 376) * q^47 + 341 * q^49 + (-30*b - 222) * q^51 + (122*b + 124) * q^53 + (35*b - 125) * q^55 + (32*b + 238) * q^57 + (13*b + 165) * q^59 + (66*b + 310) * q^61 + (42*b + 228) * q^63 - 65 * q^65 + (124*b - 230) * q^67 + (58*b - 4) * q^69 + (-13*b + 367) * q^71 + (-164*b + 228) * q^73 + (25*b - 25) * q^75 + (-150*b + 798) * q^77 + (-118*b + 462) * q^79 + (82*b - 415) * q^81 + (-6*b + 1152) * q^83 + (70*b + 220) * q^85 + (-108*b - 288) * q^87 + (60*b - 374) * q^89 - 78*b * q^91 + (200*b - 182) * q^93 + (-75*b - 235) * q^95 + (-184*b + 962) * q^97 + (-b + 91) * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	2 * q - 2 * q^3 - 10 * q^5 - 14 * q^9 + 50 * q^11 + 26 * q^13 + 10 * q^15 - 88 * q^17 + 94 * q^19 - 228 * q^21 + 122 * q^23 + 50 * q^25 - 8 * q^27 - 260 * q^29 + 402 * q^31 - 316 * q^33 + 320 * q^37 - 26 * q^39 - 536 * q^41 + 62 * q^43 + 70 * q^45 - 752 * q^47 + 682 * q^49 - 444 * q^51 + 248 * q^53 - 250 * q^55 + 476 * q^57 + 330 * q^59 + 620 * q^61 + 456 * q^63 - 130 * q^65 - 460 * q^67 - 8 * q^69 + 734 * q^71 + 456 * q^73 - 50 * q^75 + 1596 * q^77 + 924 * q^79 - 830 * q^81 + 2304 * q^83 + 440 * q^85 - 576 * q^87 - 748 * q^89 - 364 * q^93 - 470 * q^95 + 1924 * q^97 + 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

0

−5.35890

0

−5.00000

0

26.1534

0

1.71780

0

1.2

0

3.35890

0

−5.00000

0

−26.1534

0

−15.7178

0

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	1040.4.a.i		2
4.b	odd	2	1		130.4.a.d	✓	2
12.b	even	2	1		1170.4.a.ba		2
20.d	odd	2	1		650.4.a.r		2
20.e	even	4	2		650.4.b.l		4
52.b	odd	2	1		1690.4.a.s		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
130.4.a.d	✓	2	4.b	odd	2	1
650.4.a.r		2	20.d	odd	2	1
650.4.b.l		4	20.e	even	4	2
1040.4.a.i		2	1.a	even	1	1	trivial
1170.4.a.ba		2	12.b	even	2	1
1690.4.a.s		2	52.b	odd	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1040))\):

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

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

Hecke characteristic polynomials

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