Newform orbit 1520.4.a.m

• Label: 1520.4.a.m
• Level: $1520$
• Weight: $4$
• Character orbit: 1520.a
• Self dual: yes
• Analytic conductor: $89.683$
• Analytic rank: $1$
• Dimension: $2$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 1520 = 2^{4} \cdot 5 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1520.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(89.6829032087\)
Analytic rank:	\(1\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{177}) \)
Defining polynomial:	\( x^{2} - x - 44 \)
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 760)
Fricke sign:	\(-1\)
Sato-Tate group:	$\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(1 + \sqrt{177})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	q + (-b + 3) * q^3 - 5 * q^5 + (-3*b - 7) * q^7 + (-5*b + 26) * q^9 + (4*b - 4) * q^11 + (-b - 73) * q^13 + (5*b - 15) * q^15 + (13*b + 1) * q^17 + 19 * q^19 + (b + 111) * q^21 + (29*b - 3) * q^23 + 25 * q^25 + (-9*b + 217) * q^27 + (-27*b + 93) * q^29 + (34*b - 126) * q^31 + (12*b - 188) * q^33 + (15*b + 35) * q^35 + (4*b + 182) * q^37 + (71*b - 175) * q^39 + (-4*b - 150) * q^41 + (28*b + 34) * q^43 + (25*b - 130) * q^45 + (-22*b - 280) * q^47 + (51*b + 102) * q^49 + (25*b - 569) * q^51 + (27*b - 77) * q^53 + (-20*b + 20) * q^55 + (-19*b + 57) * q^57 + (39*b + 289) * q^59 + (-2*b - 844) * q^61 + (-28*b + 478) * q^63 + (5*b + 365) * q^65 + (-77*b + 51) * q^67 + (61*b - 1285) * q^69 + (-12*b - 784) * q^71 + (63*b + 79) * q^73 + (-25*b + 75) * q^75 + (-28*b - 500) * q^77 + (-2*b + 1026) * q^79 + (-100*b + 345) * q^81 + (-76*b + 938) * q^83 + (-65*b - 5) * q^85 + (-147*b + 1467) * q^87 + (-136*b - 430) * q^89 + (229*b + 643) * q^91 + (194*b - 1874) * q^93 - 95 * q^95 + (-20*b + 838) * q^97 + (104*b - 984) * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	2 * q + 5 * q^3 - 10 * q^5 - 17 * q^7 + 47 * q^9 - 4 * q^11 - 147 * q^13 - 25 * q^15 + 15 * q^17 + 38 * q^19 + 223 * q^21 + 23 * q^23 + 50 * q^25 + 425 * q^27 + 159 * q^29 - 218 * q^31 - 364 * q^33 + 85 * q^35 + 368 * q^37 - 279 * q^39 - 304 * q^41 + 96 * q^43 - 235 * q^45 - 582 * q^47 + 255 * q^49 - 1113 * q^51 - 127 * q^53 + 20 * q^55 + 95 * q^57 + 617 * q^59 - 1690 * q^61 + 928 * q^63 + 735 * q^65 + 25 * q^67 - 2509 * q^69 - 1580 * q^71 + 221 * q^73 + 125 * q^75 - 1028 * q^77 + 2050 * q^79 + 590 * q^81 + 1800 * q^83 - 75 * q^85 + 2787 * q^87 - 996 * q^89 + 1515 * q^91 - 3554 * q^93 - 190 * q^95 + 1656 * q^97 - 1864 * 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

7.15207
−6.15207

0

−4.15207

0

−5.00000

0

−28.4562

0

−9.76034

0

1.2

0

9.15207

0

−5.00000

0

11.4562

0

56.7603

0

Atkin-Lehner signs

\( p \)	Sign
\(2\)	\( +1 \)
\(5\)	\( +1 \)
\(19\)	\( -1 \)

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	1520.4.a.m		2
4.b	odd	2	1		760.4.a.b	✓	2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
760.4.a.b	✓	2	4.b	odd	2	1
1520.4.a.m		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 \(S_{4}^{\mathrm{new}}(\Gamma_0(1520))\):

\( T_{3}^{2} - 5T_{3} - 38 \)

\( T_{7}^{2} + 17T_{7} - 326 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{2} \)
$3$	\( T^{2} - 5T - 38 \)
$5$	\( (T + 5)^{2} \)
$7$	\( T^{2} + 17T - 326 \)
$11$	\( T^{2} + 4T - 704 \)