Newform orbit 552.4.a.c

• Label: 552.4.a.c
• Level: $552$
• Weight: $4$
• Character orbit: 552.a
• Self dual: yes
• Analytic conductor: $32.569$
• Analytic rank: $0$
• Dimension: $2$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 552 = 2^{3} \cdot 3 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	552.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(32.5690543232\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{2}) \)
Defining polynomial:	\( x^{2} - 2 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index:	\( 2 \)
Twist minimal:	yes
Fricke sign:	\(+1\)
Sato-Tate group:	$\mathrm{SU}(2)$

$q$-expansion

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

\(f(q)\)	\(=\)	q - 3 * q^3 + (4*b - 4) * q^5 + (9*b + 2) * q^7 + 9 * q^9 + (7*b + 20) * q^11 + (14*b + 18) * q^13 + (-12*b + 12) * q^15 + (19*b - 64) * q^17 + (16*b - 98) * q^19 + (-27*b - 6) * q^21 + 23 * q^23 + (-32*b + 19) * q^25 - 27 * q^27 + (58*b - 38) * q^29 + (-22*b - 84) * q^31 + (-21*b - 60) * q^33 + (-28*b + 280) * q^35 + (-27*b - 14) * q^37 + (-42*b - 54) * q^39 + (-30*b + 146) * q^41 + (38*b + 390) * q^43 + (36*b - 36) * q^45 + (-40*b + 216) * q^47 + (36*b + 309) * q^49 + (-57*b + 192) * q^51 + (-190*b - 184) * q^53 + (52*b + 144) * q^55 + (-48*b + 294) * q^57 + (90*b - 20) * q^59 + (-81*b - 278) * q^61 + (81*b + 18) * q^63 + (16*b + 376) * q^65 + (102*b - 230) * q^67 - 69 * q^69 + (34*b + 1056) * q^71 + (40*b - 774) * q^73 + (96*b - 57) * q^75 + (194*b + 544) * q^77 + (243*b + 710) * q^79 + 81 * q^81 + (-67*b + 108) * q^83 + (-332*b + 864) * q^85 + (-174*b + 114) * q^87 + (131*b - 556) * q^89 + (190*b + 1044) * q^91 + (66*b + 252) * q^93 + (-456*b + 904) * q^95 + (232*b + 174) * q^97 + (63*b + 180) * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	2 * q - 6 * q^3 - 8 * q^5 + 4 * q^7 + 18 * q^9 + 40 * q^11 + 36 * q^13 + 24 * q^15 - 128 * q^17 - 196 * q^19 - 12 * q^21 + 46 * q^23 + 38 * q^25 - 54 * q^27 - 76 * q^29 - 168 * q^31 - 120 * q^33 + 560 * q^35 - 28 * q^37 - 108 * q^39 + 292 * q^41 + 780 * q^43 - 72 * q^45 + 432 * q^47 + 618 * q^49 + 384 * q^51 - 368 * q^53 + 288 * q^55 + 588 * q^57 - 40 * q^59 - 556 * q^61 + 36 * q^63 + 752 * q^65 - 460 * q^67 - 138 * q^69 + 2112 * q^71 - 1548 * q^73 - 114 * q^75 + 1088 * q^77 + 1420 * q^79 + 162 * q^81 + 216 * q^83 + 1728 * q^85 + 228 * q^87 - 1112 * q^89 + 2088 * q^91 + 504 * q^93 + 1808 * q^95 + 348 * q^97 + 360 * 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

−1.41421
1.41421

0

−3.00000

0

−15.3137

0

−23.4558

0

9.00000

0

1.2

0

−3.00000

0

7.31371

0

27.4558

0

9.00000

0

Atkin-Lehner signs

\( p \)	Sign
\(2\)	\( -1 \)
\(3\)	\( +1 \)
\(23\)	\( -1 \)

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	552.4.a.c	✓	2
3.b	odd	2	1		1656.4.a.g		2
4.b	odd	2	1		1104.4.a.m		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
552.4.a.c	✓	2	1.a	even	1	1	trivial
1104.4.a.m		2	4.b	odd	2	1
1656.4.a.g		2	3.b	odd	2	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} + 8T_{5} - 112 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(552))\).

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{2} \)
$3$	\( (T + 3)^{2} \)
$5$	\( T^{2} + 8T - 112 \)
$7$	\( T^{2} - 4T - 644 \)
$11$	\( T^{2} - 40T + 8 \)