Newform orbit 525.4.a.i

• Label: 525.4.a.i
• 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:	\(\Q(\sqrt{41}) \)
Defining polynomial:	\( x^{2} - x - 10 \)
Coefficient ring:	\(\Z[a_1, a_2]\)
Coefficient ring index:	\( 1 \)
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 \(\beta = \frac{1}{2}(1 + \sqrt{41})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	q + (-b - 1) * q^2 - 3 * q^3 + (3*b + 3) * q^4 + (3*b + 3) * q^6 - 7 * q^7 + (-b - 25) * q^8 + 9 * q^9 + (-2*b + 32) * q^11 + (-9*b - 9) * q^12 + (10*b - 2) * q^13 + (7*b + 7) * q^14 + (3*b + 11) * q^16 + (12*b - 26) * q^17 + (-9*b - 9) * q^18 + (-2*b - 60) * q^19 + 21 * q^21 + (-28*b - 12) * q^22 + (48*b - 32) * q^23 + (3*b + 75) * q^24 + (-18*b - 98) * q^26 - 27 * q^27 + (-21*b - 21) * q^28 + (12*b + 170) * q^29 + (-38*b + 52) * q^31 + (-9*b + 159) * q^32 + (6*b - 96) * q^33 + (2*b - 94) * q^34 + (27*b + 27) * q^36 + (-80*b + 134) * q^37 + (64*b + 80) * q^38 + (-30*b + 6) * q^39 + (-108*b + 62) * q^41 + (-21*b - 21) * q^42 + (-100*b + 248) * q^43 + (84*b + 36) * q^44 + (-64*b - 448) * q^46 + (-140*b + 164) * q^47 + (-9*b - 33) * q^48 + 49 * q^49 + (-36*b + 78) * q^51 + (54*b + 294) * q^52 + (-58*b - 462) * q^53 + (27*b + 27) * q^54 + (7*b + 175) * q^56 + (6*b + 180) * q^57 + (-194*b - 290) * q^58 + (76*b + 220) * q^59 + (-84*b - 398) * q^61 + (24*b + 328) * q^62 - 63 * q^63 + (-165*b - 157) * q^64 + (84*b + 36) * q^66 + (228*b + 64) * q^67 + (-6*b + 282) * q^68 + (-144*b + 96) * q^69 + (86*b + 112) * q^71 + (-9*b - 225) * q^72 + (38*b - 182) * q^73 + (26*b + 666) * q^74 + (-192*b - 240) * q^76 + (14*b - 224) * q^77 + (54*b + 294) * q^78 + (-8*b + 920) * q^79 + 81 * q^81 + (154*b + 1018) * q^82 + (224*b + 228) * q^83 + (63*b + 63) * q^84 + (-48*b + 752) * q^86 + (-36*b - 510) * q^87 + (20*b - 780) * q^88 + (336*b + 230) * q^89 + (-70*b + 14) * q^91 + (192*b + 1344) * q^92 + (114*b - 156) * q^93 + (116*b + 1236) * q^94 + (27*b - 477) * q^96 + (-278*b + 474) * q^97 + (-49*b - 49) * q^98 + (-18*b + 288) * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	2 * q - 3 * q^2 - 6 * q^3 + 9 * q^4 + 9 * q^6 - 14 * q^7 - 51 * q^8 + 18 * q^9 + 62 * q^11 - 27 * q^12 + 6 * q^13 + 21 * q^14 + 25 * q^16 - 40 * q^17 - 27 * q^18 - 122 * q^19 + 42 * q^21 - 52 * q^22 - 16 * q^23 + 153 * q^24 - 214 * q^26 - 54 * q^27 - 63 * q^28 + 352 * q^29 + 66 * q^31 + 309 * q^32 - 186 * q^33 - 186 * q^34 + 81 * q^36 + 188 * q^37 + 224 * q^38 - 18 * q^39 + 16 * q^41 - 63 * q^42 + 396 * q^43 + 156 * q^44 - 960 * q^46 + 188 * q^47 - 75 * q^48 + 98 * q^49 + 120 * q^51 + 642 * q^52 - 982 * q^53 + 81 * q^54 + 357 * q^56 + 366 * q^57 - 774 * q^58 + 516 * q^59 - 880 * q^61 + 680 * q^62 - 126 * q^63 - 479 * q^64 + 156 * q^66 + 356 * q^67 + 558 * q^68 + 48 * q^69 + 310 * q^71 - 459 * q^72 - 326 * q^73 + 1358 * q^74 - 672 * q^76 - 434 * q^77 + 642 * q^78 + 1832 * q^79 + 162 * q^81 + 2190 * q^82 + 680 * q^83 + 189 * q^84 + 1456 * q^86 - 1056 * q^87 - 1540 * q^88 + 796 * q^89 - 42 * q^91 + 2880 * q^92 - 198 * q^93 + 2588 * q^94 - 927 * q^96 + 670 * q^97 - 147 * q^98 + 558 * 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

3.70156
−2.70156

−4.70156

−3.00000

14.1047

0

14.1047

−7.00000

−28.7016

9.00000

0

1.2

1.70156

−3.00000

−5.10469

0

−5.10469

−7.00000

−22.2984

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.i		2
3.b	odd	2	1		1575.4.a.y		2
5.b	even	2	1		105.4.a.g	✓	2
5.c	odd	4	2		525.4.d.j		4
15.d	odd	2	1		315.4.a.g		2
20.d	odd	2	1		1680.4.a.y		2
35.c	odd	2	1		735.4.a.q		2
105.g	even	2	1		2205.4.a.v		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
105.4.a.g	✓	2	5.b	even	2	1
315.4.a.g		2	15.d	odd	2	1
525.4.a.i		2	1.a	even	1	1	trivial
525.4.d.j		4	5.c	odd	4	2
735.4.a.q		2	35.c	odd	2	1
1575.4.a.y		2	3.b	odd	2	1
1680.4.a.y		2	20.d	odd	2	1
2205.4.a.v		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 \(S_{4}^{\mathrm{new}}(\Gamma_0(525))\):

\( T_{2}^{2} + 3T_{2} - 8 \)

\( T_{11}^{2} - 62T_{11} + 920 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{2} + 3T - 8 \)
$3$	\( (T + 3)^{2} \)
$5$	\( T^{2} \)
$7$	\( (T + 7)^{2} \)
$11$	\( T^{2} - 62T + 920 \)