Newform orbit 525.4.a.l

• Label: 525.4.a.l
• Level: $525$
• Weight: $4$
• Character orbit: 525.a
• Self dual: yes
• Analytic conductor: $30.976$
• Analytic rank: $0$
• Dimension: $2$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 525 = 3 \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	525.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(30.9760027530\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{2}) \)
Defining polynomial:	\( x^{2} - 2 \)
Coefficient ring:	\(\Z[a_1, a_2]\)
Coefficient ring index:	\( 2 \)
Twist minimal:	no (minimal twist has level 105)
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 + (b + 1) * q^2 + 3 * q^3 + (2*b + 1) * q^4 + (3*b + 3) * q^6 + 7 * q^7 + (-5*b + 9) * q^8 + 9 * q^9 + (20*b - 8) * q^11 + (6*b + 3) * q^12 + (2*b + 38) * q^13 + (7*b + 7) * q^14 + (-12*b - 39) * q^16 + (2*b + 62) * q^17 + (9*b + 9) * q^18 + (-16*b - 48) * q^19 + 21 * q^21 + (12*b + 152) * q^22 + (34*b + 8) * q^23 + (-15*b + 27) * q^24 + (40*b + 54) * q^26 + 27 * q^27 + (14*b + 7) * q^28 + (-54*b + 94) * q^29 + (18*b - 60) * q^31 + (-11*b - 207) * q^32 + (60*b - 24) * q^33 + (64*b + 78) * q^34 + (18*b + 9) * q^36 + (66*b + 66) * q^37 + (-64*b - 176) * q^38 + (6*b + 114) * q^39 + (80*b + 50) * q^41 + (21*b + 21) * q^42 + (-62*b + 268) * q^43 + (4*b + 312) * q^44 + (42*b + 280) * q^46 + (42*b + 464) * q^47 + (-36*b - 117) * q^48 + 49 * q^49 + (6*b + 186) * q^51 + (78*b + 70) * q^52 + (-64*b - 442) * q^53 + (27*b + 27) * q^54 + (-35*b + 63) * q^56 + (-48*b - 144) * q^57 + (40*b - 338) * q^58 + (-204*b + 52) * q^59 + (-42*b - 234) * q^61 + (-42*b + 84) * q^62 + 63 * q^63 + (-122*b + 17) * q^64 + (36*b + 456) * q^66 + (-38*b + 844) * q^67 + (126*b + 94) * q^68 + (102*b + 24) * q^69 + (-150*b - 68) * q^71 + (-45*b + 81) * q^72 + (-322*b - 254) * q^73 + (132*b + 594) * q^74 + (-112*b - 304) * q^76 + (140*b - 56) * q^77 + (120*b + 162) * q^78 + (-232*b - 216) * q^79 + 81 * q^81 + (130*b + 690) * q^82 + (84*b + 292) * q^83 + (42*b + 21) * q^84 + (206*b - 228) * q^86 + (-162*b + 282) * q^87 + (220*b - 872) * q^88 + (112*b - 702) * q^89 + (14*b + 266) * q^91 + (50*b + 552) * q^92 + (54*b - 180) * q^93 + (506*b + 800) * q^94 + (-33*b - 621) * q^96 + (-46*b + 594) * q^97 + (49*b + 49) * q^98 + (180*b - 72) * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	2 * q + 2 * q^2 + 6 * q^3 + 2 * q^4 + 6 * q^6 + 14 * q^7 + 18 * q^8 + 18 * q^9 - 16 * q^11 + 6 * q^12 + 76 * q^13 + 14 * q^14 - 78 * q^16 + 124 * q^17 + 18 * q^18 - 96 * q^19 + 42 * q^21 + 304 * q^22 + 16 * q^23 + 54 * q^24 + 108 * q^26 + 54 * q^27 + 14 * q^28 + 188 * q^29 - 120 * q^31 - 414 * q^32 - 48 * q^33 + 156 * q^34 + 18 * q^36 + 132 * q^37 - 352 * q^38 + 228 * q^39 + 100 * q^41 + 42 * q^42 + 536 * q^43 + 624 * q^44 + 560 * q^46 + 928 * q^47 - 234 * q^48 + 98 * q^49 + 372 * q^51 + 140 * q^52 - 884 * q^53 + 54 * q^54 + 126 * q^56 - 288 * q^57 - 676 * q^58 + 104 * q^59 - 468 * q^61 + 168 * q^62 + 126 * q^63 + 34 * q^64 + 912 * q^66 + 1688 * q^67 + 188 * q^68 + 48 * q^69 - 136 * q^71 + 162 * q^72 - 508 * q^73 + 1188 * q^74 - 608 * q^76 - 112 * q^77 + 324 * q^78 - 432 * q^79 + 162 * q^81 + 1380 * q^82 + 584 * q^83 + 42 * q^84 - 456 * q^86 + 564 * q^87 - 1744 * q^88 - 1404 * q^89 + 532 * q^91 + 1104 * q^92 - 360 * q^93 + 1600 * q^94 - 1242 * q^96 + 1188 * q^97 + 98 * q^98 - 144 * 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

−1.82843

3.00000

−4.65685

0

−5.48528

7.00000

23.1421

9.00000

0

1.2

3.82843

3.00000

6.65685

0

11.4853

7.00000

−5.14214

9.00000

0

Atkin-Lehner signs

\( p \)	Sign
\(3\)	\( -1 \)
\(5\)	\( +1 \)
\(7\)	\( -1 \)

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	525.4.a.l		2
3.b	odd	2	1		1575.4.a.q		2
5.b	even	2	1		105.4.a.e	✓	2
5.c	odd	4	2		525.4.d.l		4
15.d	odd	2	1		315.4.a.k		2
20.d	odd	2	1		1680.4.a.bo		2
35.c	odd	2	1		735.4.a.o		2
105.g	even	2	1		2205.4.a.bb		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
105.4.a.e	✓	2	5.b	even	2	1
315.4.a.k		2	15.d	odd	2	1
525.4.a.l		2	1.a	even	1	1	trivial
525.4.d.l		4	5.c	odd	4	2
735.4.a.o		2	35.c	odd	2	1
1575.4.a.q		2	3.b	odd	2	1
1680.4.a.bo		2	20.d	odd	2	1
2205.4.a.bb		2	105.g	even	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(525))\):

\( T_{2}^{2} - 2T_{2} - 7 \)

\( T_{11}^{2} + 16T_{11} - 3136 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{2} - 2T - 7 \)
$3$	\( (T - 3)^{2} \)
$5$	\( T^{2} \)
$7$	\( (T - 7)^{2} \)
$11$	\( T^{2} + 16T - 3136 \)