Newform orbit 882.4.a.z

• Label: 882.4.a.z
• Level: $882$
• Weight: $4$
• Character orbit: 882.a
• Self dual: yes
• Analytic conductor: $52.040$
• Analytic rank: $0$
• Dimension: $2$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 882 = 2 \cdot 3^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	882.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(52.0396846251\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{1345}) \)
Defining polynomial:	\( x^{2} - x - 336 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 42)
Fricke sign:	\(+1\)
Sato-Tate group:	$\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(1 + \sqrt{1345})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	q - 2 * q^2 + 4 * q^4 + (-b + 3) * q^5 - 8 * q^8 + (2*b - 6) * q^10 + (-b - 33) * q^11 + (-b - 20) * q^13 + 16 * q^16 + (4*b - 48) * q^17 + (-3*b - 20) * q^19 + (-4*b + 12) * q^20 + (2*b + 66) * q^22 + (-4*b - 72) * q^23 + (-5*b + 220) * q^25 + (2*b + 40) * q^26 + (-11*b - 33) * q^29 + (2*b + 259) * q^31 - 32 * q^32 + (-8*b + 96) * q^34 + (-9*b + 8) * q^37 + (6*b + 40) * q^38 + (8*b - 24) * q^40 + (6*b - 216) * q^41 + (-15*b - 46) * q^43 + (-4*b - 132) * q^44 + (8*b + 144) * q^46 + (-12*b + 294) * q^47 + (10*b - 440) * q^50 + (-4*b - 80) * q^52 + (-3*b + 123) * q^53 + (31*b + 237) * q^55 + (22*b + 66) * q^58 + (-25*b + 9) * q^59 + (-4*b - 110) * q^61 + (-4*b - 518) * q^62 + 64 * q^64 + (18*b + 276) * q^65 + (11*b + 338) * q^67 + (16*b - 192) * q^68 + (20*b - 246) * q^71 + (-35*b + 478) * q^73 + (18*b - 16) * q^74 + (-12*b - 80) * q^76 + (-8*b - 259) * q^79 + (-16*b + 48) * q^80 + (-12*b + 432) * q^82 + (25*b - 123) * q^83 + (56*b - 1488) * q^85 + (30*b + 92) * q^86 + (8*b + 264) * q^88 + (42*b + 366) * q^89 + (-16*b - 288) * q^92 + (24*b - 588) * q^94 + (14*b + 948) * q^95 + (-35*b - 959) * q^97
\(\operatorname{Tr}(f)(q)\)	\(=\)	2 * q - 4 * q^2 + 8 * q^4 + 5 * q^5 - 16 * q^8 - 10 * q^10 - 67 * q^11 - 41 * q^13 + 32 * q^16 - 92 * q^17 - 43 * q^19 + 20 * q^20 + 134 * q^22 - 148 * q^23 + 435 * q^25 + 82 * q^26 - 77 * q^29 + 520 * q^31 - 64 * q^32 + 184 * q^34 + 7 * q^37 + 86 * q^38 - 40 * q^40 - 426 * q^41 - 107 * q^43 - 268 * q^44 + 296 * q^46 + 576 * q^47 - 870 * q^50 - 164 * q^52 + 243 * q^53 + 505 * q^55 + 154 * q^58 - 7 * q^59 - 224 * q^61 - 1040 * q^62 + 128 * q^64 + 570 * q^65 + 687 * q^67 - 368 * q^68 - 472 * q^71 + 921 * q^73 - 14 * q^74 - 172 * q^76 - 526 * q^79 + 80 * q^80 + 852 * q^82 - 221 * q^83 - 2920 * q^85 + 214 * q^86 + 536 * q^88 + 774 * q^89 - 592 * q^92 - 1152 * q^94 + 1910 * q^95 - 1953 * q^97

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

18.8371
−17.8371

−2.00000

0

4.00000

−15.8371

0

0

−8.00000

0

31.6742

1.2

−2.00000

0

4.00000

20.8371

0

0

−8.00000

0

−41.6742

Atkin-Lehner signs

\( p \)	Sign
\(2\)	\( +1 \)
\(3\)	\( -1 \)
\(7\)	\( -1 \)

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	882.4.a.z		2
3.b	odd	2	1		294.4.a.m		2
7.b	odd	2	1		882.4.a.v		2
7.c	even	3	2		882.4.g.bf		4
7.d	odd	6	2		126.4.g.g		4
12.b	even	2	1		2352.4.a.ca		2
21.c	even	2	1		294.4.a.n		2
21.g	even	6	2		42.4.e.c	✓	4
21.h	odd	6	2		294.4.e.l		4
84.h	odd	2	1		2352.4.a.bq		2
84.j	odd	6	2		336.4.q.j		4

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
42.4.e.c	✓	4	21.g	even	6	2
126.4.g.g		4	7.d	odd	6	2
294.4.a.m		2	3.b	odd	2	1
294.4.a.n		2	21.c	even	2	1
294.4.e.l		4	21.h	odd	6	2
336.4.q.j		4	84.j	odd	6	2
882.4.a.v		2	7.b	odd	2	1
882.4.a.z		2	1.a	even	1	1	trivial
882.4.g.bf		4	7.c	even	3	2
2352.4.a.bq		2	84.h	odd	2	1
2352.4.a.ca		2	12.b	even	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(882))\):

\( T_{5}^{2} - 5T_{5} - 330 \)

\( T_{11}^{2} + 67T_{11} + 786 \)

\( T_{13}^{2} + 41T_{13} + 84 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( (T + 2)^{2} \)
$3$	\( T^{2} \)
$5$	\( T^{2} - 5T - 330 \)
$7$	\( T^{2} \)
$11$	\( T^{2} + 67T + 786 \)