Newform orbit 220.4.a.d

• Label: 220.4.a.d
• Level: $220$
• Weight: $4$
• Character orbit: 220.a
• Self dual: yes
• Analytic conductor: $12.980$
• Analytic rank: $1$
• Dimension: $2$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 220 = 2^{2} \cdot 5 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	220.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(12.9804202013\)
Analytic rank:	\(1\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{97}) \)
Defining polynomial:	\( x^{2} - x - 24 \)
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Fricke sign:	\(-1\)
Sato-Tate group:	$\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(1 + \sqrt{97})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	q + (-b - 4) * q^3 + 5 * q^5 + (b - 8) * q^7 + (9*b + 13) * q^9 + 11 * q^11 + (12*b - 10) * q^13 + (-5*b - 20) * q^15 + (-11*b - 22) * q^17 + (-5*b - 44) * q^19 + (3*b + 8) * q^21 + (-2*b - 112) * q^23 + 25 * q^25 + (-31*b - 160) * q^27 + (-3*b + 30) * q^29 + (41*b - 120) * q^31 + (-11*b - 44) * q^33 + (5*b - 40) * q^35 + (-27*b - 202) * q^37 + (-50*b - 248) * q^39 + (12*b + 90) * q^41 + (-16*b - 372) * q^43 + (45*b + 65) * q^45 + (-2*b - 88) * q^47 + (-15*b - 255) * q^49 + (77*b + 352) * q^51 + (-15*b - 90) * q^53 + 55 * q^55 + (69*b + 296) * q^57 + (-74*b - 76) * q^59 + (-77*b + 430) * q^61 + (-50*b + 112) * q^63 + (60*b - 50) * q^65 + (20*b + 324) * q^67 + (122*b + 496) * q^69 + (91*b + 176) * q^71 + (112*b + 346) * q^73 + (-25*b - 100) * q^75 + (11*b - 88) * q^77 + (-38*b - 128) * q^79 + (72*b + 1033) * q^81 + (-82*b - 716) * q^83 + (-55*b - 110) * q^85 + (-15*b - 48) * q^87 + (71*b + 370) * q^89 + (-94*b + 368) * q^91 + (-85*b - 504) * q^93 + (-25*b - 220) * q^95 + (-298*b + 18) * q^97 + (99*b + 143) * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	2 * q - 9 * q^3 + 10 * q^5 - 15 * q^7 + 35 * q^9 + 22 * q^11 - 8 * q^13 - 45 * q^15 - 55 * q^17 - 93 * q^19 + 19 * q^21 - 226 * q^23 + 50 * q^25 - 351 * q^27 + 57 * q^29 - 199 * q^31 - 99 * q^33 - 75 * q^35 - 431 * q^37 - 546 * q^39 + 192 * q^41 - 760 * q^43 + 175 * q^45 - 178 * q^47 - 525 * q^49 + 781 * q^51 - 195 * q^53 + 110 * q^55 + 661 * q^57 - 226 * q^59 + 783 * q^61 + 174 * q^63 - 40 * q^65 + 668 * q^67 + 1114 * q^69 + 443 * q^71 + 804 * q^73 - 225 * q^75 - 165 * q^77 - 294 * q^79 + 2138 * q^81 - 1514 * q^83 - 275 * q^85 - 111 * q^87 + 811 * q^89 + 642 * q^91 - 1093 * q^93 - 465 * q^95 - 262 * q^97 + 385 * 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

5.42443
−4.42443

0

−9.42443

0

5.00000

0

−2.57557

0

61.8199

0

1.2

0

0.424429

0

5.00000

0

−12.4244

0

−26.8199

0

Atkin-Lehner signs

\( p \)	Sign
\(2\)	\( -1 \)
\(5\)	\( -1 \)
\(11\)	\( -1 \)

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	220.4.a.d	✓	2
3.b	odd	2	1		1980.4.a.f		2
4.b	odd	2	1		880.4.a.u		2
5.b	even	2	1		1100.4.a.h		2
5.c	odd	4	2		1100.4.b.f		4
11.b	odd	2	1		2420.4.a.g		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
220.4.a.d	✓	2	1.a	even	1	1	trivial
880.4.a.u		2	4.b	odd	2	1
1100.4.a.h		2	5.b	even	2	1
1100.4.b.f		4	5.c	odd	4	2
1980.4.a.f		2	3.b	odd	2	1
2420.4.a.g		2	11.b	odd	2	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} + 9T_{3} - 4 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(220))\).

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{2} \)
$3$	\( T^{2} + 9T - 4 \)
$5$	\( (T - 5)^{2} \)
$7$	\( T^{2} + 15T + 32 \)
$11$	\( (T - 11)^{2} \)