Newform orbit 528.4.a.q

• Label: 528.4.a.q
• Level: $528$
• Weight: $4$
• Character orbit: 528.a
• Self dual: yes
• Analytic conductor: $31.153$
• Analytic rank: $0$
• Dimension: $2$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 528 = 2^{4} \cdot 3 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	528.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(31.1530084830\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{137}) \)
Defining polynomial:	\( x^{2} - x - 34 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index:	\( 2 \)
Twist minimal:	no (minimal twist has level 264)
Fricke sign:	\(+1\)
Sato-Tate group:	$\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{137}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	q + 3 * q^3 + (-b - 3) * q^5 + (-2*b - 8) * q^7 + 9 * q^9 + 11 * q^11 + (b + 11) * q^13 + (-3*b - 9) * q^15 + (-3*b - 11) * q^17 + (-3*b + 31) * q^19 + (-6*b - 24) * q^21 + (5*b + 105) * q^23 + (6*b + 21) * q^25 + 27 * q^27 + (-7*b - 107) * q^29 + (4*b + 164) * q^31 + 33 * q^33 + (14*b + 298) * q^35 + (2*b - 100) * q^37 + (3*b + 33) * q^39 + (29*b + 57) * q^41 + (17*b + 303) * q^43 + (-9*b - 27) * q^45 + (-21*b + 191) * q^47 + (32*b + 269) * q^49 + (-9*b - 33) * q^51 + (29*b - 1) * q^53 + (-11*b - 33) * q^55 + (-9*b + 93) * q^57 + (2*b + 542) * q^59 + (-55*b - 177) * q^61 + (-18*b - 72) * q^63 + (-14*b - 170) * q^65 + (8*b - 228) * q^67 + (15*b + 315) * q^69 + (-27*b + 17) * q^71 + (-70*b - 56) * q^73 + (18*b + 63) * q^75 + (-22*b - 88) * q^77 + (20*b - 410) * q^79 + 81 * q^81 + (82*b + 506) * q^83 + (20*b + 444) * q^85 + (-21*b - 321) * q^87 + (-12*b + 518) * q^89 + (-30*b - 362) * q^91 + (12*b + 492) * q^93 + (-22*b + 318) * q^95 + (16*b + 654) * q^97 + 99 * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	2 * q + 6 * q^3 - 6 * q^5 - 16 * q^7 + 18 * q^9 + 22 * q^11 + 22 * q^13 - 18 * q^15 - 22 * q^17 + 62 * q^19 - 48 * q^21 + 210 * q^23 + 42 * q^25 + 54 * q^27 - 214 * q^29 + 328 * q^31 + 66 * q^33 + 596 * q^35 - 200 * q^37 + 66 * q^39 + 114 * q^41 + 606 * q^43 - 54 * q^45 + 382 * q^47 + 538 * q^49 - 66 * q^51 - 2 * q^53 - 66 * q^55 + 186 * q^57 + 1084 * q^59 - 354 * q^61 - 144 * q^63 - 340 * q^65 - 456 * q^67 + 630 * q^69 + 34 * q^71 - 112 * q^73 + 126 * q^75 - 176 * q^77 - 820 * q^79 + 162 * q^81 + 1012 * q^83 + 888 * q^85 - 642 * q^87 + 1036 * q^89 - 724 * q^91 + 984 * q^93 + 636 * q^95 + 1308 * q^97 + 198 * 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

6.35235
−5.35235

0

3.00000

0

−14.7047

0

−31.4094

0

9.00000

0

1.2

0

3.00000

0

8.70470

0

15.4094

0

9.00000

0

Atkin-Lehner signs

\( p \)	Sign
\(2\)	\( +1 \)
\(3\)	\( -1 \)
\(11\)	\( -1 \)

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	528.4.a.q		2
3.b	odd	2	1		1584.4.a.be		2
4.b	odd	2	1		264.4.a.f	✓	2
8.b	even	2	1		2112.4.a.bd		2
8.d	odd	2	1		2112.4.a.bm		2
12.b	even	2	1		792.4.a.i		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
264.4.a.f	✓	2	4.b	odd	2	1
528.4.a.q		2	1.a	even	1	1	trivial
792.4.a.i		2	12.b	even	2	1
1584.4.a.be		2	3.b	odd	2	1
2112.4.a.bd		2	8.b	even	2	1
2112.4.a.bm		2	8.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 \(S_{4}^{\mathrm{new}}(\Gamma_0(528))\):

\( T_{5}^{2} + 6T_{5} - 128 \)

\( T_{7}^{2} + 16T_{7} - 484 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{2} \)
$3$	\( (T - 3)^{2} \)
$5$	\( T^{2} + 6T - 128 \)
$7$	\( T^{2} + 16T - 484 \)
$11$	\( (T - 11)^{2} \)