Newform orbit 525.4.a.k

• Label: 525.4.a.k
• Level: $525$
• Weight: $4$
• Character orbit: 525.a
• Self dual: yes
• Analytic conductor: $30.976$
• Analytic rank: $1$
• 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:	\(1\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{65}) \)
Defining polynomial:	\( x^{2} - x - 16 \)
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 \(\beta = \frac{1}{2}(1 + \sqrt{65})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	q - b * q^2 - 3 * q^3 + (b + 8) * q^4 + 3*b * q^6 + 7 * q^7 + (-b - 16) * q^8 + 9 * q^9 + (-2*b - 10) * q^11 + (-3*b - 24) * q^12 + (-2*b + 12) * q^13 - 7*b * q^14 + (9*b - 48) * q^16 + (16*b - 66) * q^17 - 9*b * q^18 + (-14*b + 58) * q^19 - 21 * q^21 + (12*b + 32) * q^22 + (20*b - 140) * q^23 + (3*b + 48) * q^24 + (-10*b + 32) * q^26 - 27 * q^27 + (7*b + 56) * q^28 + (-48*b - 74) * q^29 + (42*b + 54) * q^31 + (47*b - 16) * q^32 + (6*b + 30) * q^33 + (50*b - 256) * q^34 + (9*b + 72) * q^36 + (36*b + 30) * q^37 + (-44*b + 224) * q^38 + (6*b - 36) * q^39 + (100*b - 138) * q^41 + 21*b * q^42 + (32*b + 156) * q^43 + (-28*b - 112) * q^44 + (120*b - 320) * q^46 + (48*b - 304) * q^47 + (-27*b + 144) * q^48 + 49 * q^49 + (-48*b + 198) * q^51 + (-6*b + 64) * q^52 + (-86*b - 120) * q^53 + 27*b * q^54 + (-7*b - 112) * q^56 + (42*b - 174) * q^57 + (122*b + 768) * q^58 + (84*b - 464) * q^59 + (24*b - 114) * q^61 + (-96*b - 672) * q^62 + 63 * q^63 + (-103*b - 368) * q^64 + (-36*b - 96) * q^66 + (-64*b + 84) * q^67 + (78*b - 272) * q^68 + (-60*b + 420) * q^69 + (42*b + 814) * q^71 + (-9*b - 144) * q^72 + (202*b + 92) * q^73 + (-66*b - 576) * q^74 + (-68*b + 240) * q^76 + (-14*b - 70) * q^77 + (30*b - 96) * q^78 + (-104*b - 392) * q^79 + 81 * q^81 + (38*b - 1600) * q^82 + (-216*b - 356) * q^83 + (-21*b - 168) * q^84 + (-188*b - 512) * q^86 + (144*b + 222) * q^87 + (44*b + 192) * q^88 + (8*b + 290) * q^89 + (-14*b + 84) * q^91 + (40*b - 800) * q^92 + (-126*b - 162) * q^93 + (256*b - 768) * q^94 + (-141*b + 48) * q^96 + (-314*b - 104) * q^97 - 49*b * q^98 + (-18*b - 90) * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	2 * q - q^2 - 6 * q^3 + 17 * q^4 + 3 * q^6 + 14 * q^7 - 33 * q^8 + 18 * q^9 - 22 * q^11 - 51 * q^12 + 22 * q^13 - 7 * q^14 - 87 * q^16 - 116 * q^17 - 9 * q^18 + 102 * q^19 - 42 * q^21 + 76 * q^22 - 260 * q^23 + 99 * q^24 + 54 * q^26 - 54 * q^27 + 119 * q^28 - 196 * q^29 + 150 * q^31 + 15 * q^32 + 66 * q^33 - 462 * q^34 + 153 * q^36 + 96 * q^37 + 404 * q^38 - 66 * q^39 - 176 * q^41 + 21 * q^42 + 344 * q^43 - 252 * q^44 - 520 * q^46 - 560 * q^47 + 261 * q^48 + 98 * q^49 + 348 * q^51 + 122 * q^52 - 326 * q^53 + 27 * q^54 - 231 * q^56 - 306 * q^57 + 1658 * q^58 - 844 * q^59 - 204 * q^61 - 1440 * q^62 + 126 * q^63 - 839 * q^64 - 228 * q^66 + 104 * q^67 - 466 * q^68 + 780 * q^69 + 1670 * q^71 - 297 * q^72 + 386 * q^73 - 1218 * q^74 + 412 * q^76 - 154 * q^77 - 162 * q^78 - 888 * q^79 + 162 * q^81 - 3162 * q^82 - 928 * q^83 - 357 * q^84 - 1212 * q^86 + 588 * q^87 + 428 * q^88 + 588 * q^89 + 154 * q^91 - 1560 * q^92 - 450 * q^93 - 1280 * q^94 - 45 * q^96 - 522 * q^97 - 49 * q^98 - 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

4.53113
−3.53113

−4.53113

−3.00000

12.5311

0

13.5934

7.00000

−20.5311

9.00000

0

1.2

3.53113

−3.00000

4.46887

0

−10.5934

7.00000

−12.4689

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.k		2
3.b	odd	2	1		1575.4.a.w		2
5.b	even	2	1		105.4.a.f	✓	2
5.c	odd	4	2		525.4.d.h		4
15.d	odd	2	1		315.4.a.i		2
20.d	odd	2	1		1680.4.a.bg		2
35.c	odd	2	1		735.4.a.p		2
105.g	even	2	1		2205.4.a.z		2

    

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

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

\( T_{11}^{2} + 22T_{11} + 56 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{2} + T - 16 \)
$3$	\( (T + 3)^{2} \)
$5$	\( T^{2} \)
$7$	\( (T - 7)^{2} \)
$11$	\( T^{2} + 22T + 56 \)