Newform orbit 1360.4.a.m

• Label: 1360.4.a.m
• Level: $1360$
• Weight: $4$
• Character orbit: 1360.a
• Self dual: yes
• Analytic conductor: $80.243$
• Analytic rank: $0$
• Dimension: $2$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 1360 = 2^{4} \cdot 5 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1360.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(80.2425976078\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{3}) \)
Defining polynomial:	\( x^{2} - 3 \)
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 85)
Fricke sign:	\(+1\)
Sato-Tate group:	$\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{3}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	q + (b + 1) * q^3 + 5 * q^5 + (-9*b - 1) * q^7 + (2*b - 23) * q^9 + (13*b + 19) * q^11 + (28*b - 40) * q^13 + (5*b + 5) * q^15 - 17 * q^17 + (6*b + 90) * q^19 + (-10*b - 28) * q^21 + (101*b + 21) * q^23 + 25 * q^25 + (-48*b - 44) * q^27 + (-46*b - 108) * q^29 + (-141*b - 35) * q^31 + (32*b + 58) * q^33 + (-45*b - 5) * q^35 + (-106*b + 176) * q^37 + (-12*b + 44) * q^39 + (-142*b + 172) * q^41 + (166*b + 44) * q^43 + (10*b - 115) * q^45 + (8*b + 174) * q^47 + (18*b - 99) * q^49 + (-17*b - 17) * q^51 + (-64*b - 82) * q^53 + (65*b + 95) * q^55 + (96*b + 108) * q^57 + (-66*b + 718) * q^59 + (268*b - 38) * q^61 + (205*b - 31) * q^63 + (140*b - 200) * q^65 + (-308*b + 22) * q^67 + (122*b + 324) * q^69 + (-119*b + 755) * q^71 + (18*b + 544) * q^73 + (25*b + 25) * q^75 + (-184*b - 370) * q^77 + (437*b + 479) * q^79 + (-146*b + 433) * q^81 + (-238*b - 124) * q^83 - 85 * q^85 + (-154*b - 246) * q^87 + (742*b - 102) * q^89 + (332*b - 716) * q^91 + (-176*b - 458) * q^93 + (30*b + 450) * q^95 + (436*b + 506) * q^97 + (-261*b - 359) * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	2 * q + 2 * q^3 + 10 * q^5 - 2 * q^7 - 46 * q^9 + 38 * q^11 - 80 * q^13 + 10 * q^15 - 34 * q^17 + 180 * q^19 - 56 * q^21 + 42 * q^23 + 50 * q^25 - 88 * q^27 - 216 * q^29 - 70 * q^31 + 116 * q^33 - 10 * q^35 + 352 * q^37 + 88 * q^39 + 344 * q^41 + 88 * q^43 - 230 * q^45 + 348 * q^47 - 198 * q^49 - 34 * q^51 - 164 * q^53 + 190 * q^55 + 216 * q^57 + 1436 * q^59 - 76 * q^61 - 62 * q^63 - 400 * q^65 + 44 * q^67 + 648 * q^69 + 1510 * q^71 + 1088 * q^73 + 50 * q^75 - 740 * q^77 + 958 * q^79 + 866 * q^81 - 248 * q^83 - 170 * q^85 - 492 * q^87 - 204 * q^89 - 1432 * q^91 - 916 * q^93 + 900 * q^95 + 1012 * q^97 - 718 * 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

−1.73205
1.73205

0

−0.732051

0

5.00000

0

14.5885

0

−26.4641

0

1.2

0

2.73205

0

5.00000

0

−16.5885

0

−19.5359

0

Atkin-Lehner signs

\( p \)	Sign
\(2\)	\( -1 \)
\(5\)	\( -1 \)
\(17\)	\( +1 \)

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	1360.4.a.m		2
4.b	odd	2	1		85.4.a.d	✓	2
12.b	even	2	1		765.4.a.i		2
20.d	odd	2	1		425.4.a.e		2
20.e	even	4	2		425.4.b.e		4
68.d	odd	2	1		1445.4.a.i		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
85.4.a.d	✓	2	4.b	odd	2	1
425.4.a.e		2	20.d	odd	2	1
425.4.b.e		4	20.e	even	4	2
765.4.a.i		2	12.b	even	2	1
1360.4.a.m		2	1.a	even	1	1	trivial
1445.4.a.i		2	68.d	odd	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1360))\):

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

\( T_{7}^{2} + 2T_{7} - 242 \)

\( T_{11}^{2} - 38T_{11} - 146 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{2} \)
$3$	\( T^{2} - 2T - 2 \)
$5$	\( (T - 5)^{2} \)
$7$	\( T^{2} + 2T - 242 \)
$11$	\( T^{2} - 38T - 146 \)