Newform orbit 525.4.a.m

• Label: 525.4.a.m
• 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:	\(\Q(\sqrt{17}) \)
Defining polynomial:	\( x^{2} - x - 4 \)
Coefficient ring:	\(\Z[a_1, a_2]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Fricke sign:	\(+1\)
Sato-Tate group:	$\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(1 + \sqrt{17})\). 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 + (-5*b + 1) * q^8 + 9 * q^9 + (5*b - 18) * q^11 + (-9*b + 9) * q^12 + (17*b + 11) * q^13 + (-7*b - 7) * q^14 + (-33*b + 5) * q^16 + (27*b - 53) * q^17 + (9*b + 9) * q^18 + (56*b - 56) * q^19 + 21 * q^21 + (-8*b + 2) * q^22 + (24*b + 115) * q^23 + (15*b - 3) * q^24 + (45*b + 79) * q^26 - 27 * q^27 + (-21*b + 21) * q^28 + (84*b - 73) * q^29 + (-13*b - 61) * q^31 + (-21*b - 135) * q^32 + (-15*b + 54) * q^33 + (b + 55) * q^34 + (27*b - 27) * q^36 + (-19*b + 66) * q^37 + (56*b + 168) * q^38 + (-51*b - 33) * q^39 + (45*b + 95) * q^41 + (21*b + 21) * q^42 + (58*b + 373) * q^43 + (-54*b + 114) * q^44 + (163*b + 211) * q^46 + (-88*b + 120) * q^47 + (99*b - 15) * q^48 + 49 * q^49 + (-81*b + 159) * q^51 + (33*b + 171) * q^52 + (-89*b + 119) * q^53 + (-27*b - 27) * q^54 + (35*b - 7) * q^56 + (-168*b + 168) * q^57 + (95*b + 263) * q^58 + (209*b - 325) * q^59 + (-273*b + 25) * q^61 + (-87*b - 113) * q^62 - 63 * q^63 + (87*b - 259) * q^64 + (24*b - 6) * q^66 + (-123*b + 640) * q^67 + (-159*b + 483) * q^68 + (-72*b - 345) * q^69 + (235*b + 192) * q^71 + (-45*b + 9) * q^72 + (88*b + 90) * q^73 + (28*b - 10) * q^74 + (-168*b + 840) * q^76 + (-35*b + 126) * q^77 + (-135*b - 237) * q^78 + (-103*b - 162) * q^79 + 81 * q^81 + (185*b + 275) * q^82 + (-431*b + 821) * q^83 + (63*b - 63) * q^84 + (489*b + 605) * q^86 + (-252*b + 219) * q^87 + (70*b - 118) * q^88 + (-522*b + 494) * q^89 + (-119*b - 77) * q^91 + (345*b - 57) * q^92 + (39*b + 183) * q^93 + (-56*b - 232) * q^94 + (63*b + 405) * q^96 + (704*b - 266) * q^97 + (49*b + 49) * q^98 + (45*b - 162) * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	2 * q + 3 * q^2 - 6 * q^3 - 3 * q^4 - 9 * q^6 - 14 * q^7 - 3 * q^8 + 18 * q^9 - 31 * q^11 + 9 * q^12 + 39 * q^13 - 21 * q^14 - 23 * q^16 - 79 * q^17 + 27 * q^18 - 56 * q^19 + 42 * q^21 - 4 * q^22 + 254 * q^23 + 9 * q^24 + 203 * q^26 - 54 * q^27 + 21 * q^28 - 62 * q^29 - 135 * q^31 - 291 * q^32 + 93 * q^33 + 111 * q^34 - 27 * q^36 + 113 * q^37 + 392 * q^38 - 117 * q^39 + 235 * q^41 + 63 * q^42 + 804 * q^43 + 174 * q^44 + 585 * q^46 + 152 * q^47 + 69 * q^48 + 98 * q^49 + 237 * q^51 + 375 * q^52 + 149 * q^53 - 81 * q^54 + 21 * q^56 + 168 * q^57 + 621 * q^58 - 441 * q^59 - 223 * q^61 - 313 * q^62 - 126 * q^63 - 431 * q^64 + 12 * q^66 + 1157 * q^67 + 807 * q^68 - 762 * q^69 + 619 * q^71 - 27 * q^72 + 268 * q^73 + 8 * q^74 + 1512 * q^76 + 217 * q^77 - 609 * q^78 - 427 * q^79 + 162 * q^81 + 735 * q^82 + 1211 * q^83 - 63 * q^84 + 1699 * q^86 + 186 * q^87 - 166 * q^88 + 466 * q^89 - 273 * q^91 + 231 * q^92 + 405 * q^93 - 520 * q^94 + 873 * q^96 + 172 * q^97 + 147 * q^98 - 279 * 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

−1.56155
2.56155

−0.561553

−3.00000

−7.68466

0

1.68466

−7.00000

8.80776

9.00000

0

1.2

3.56155

−3.00000

4.68466

0

−10.6847

−7.00000

−11.8078

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.m	yes	2
3.b	odd	2	1		1575.4.a.o		2
5.b	even	2	1		525.4.a.j	✓	2
5.c	odd	4	2		525.4.d.m		4
15.d	odd	2	1		1575.4.a.x		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
525.4.a.j	✓	2	5.b	even	2	1
525.4.a.m	yes	2	1.a	even	1	1	trivial
525.4.d.m		4	5.c	odd	4	2
1575.4.a.o		2	3.b	odd	2	1
1575.4.a.x		2	15.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 \(S_{4}^{\mathrm{new}}(\Gamma_0(525))\):

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

\( T_{11}^{2} + 31T_{11} + 134 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{2} - 3T - 2 \)
$3$	\( (T + 3)^{2} \)
$5$	\( T^{2} \)
$7$	\( (T + 7)^{2} \)
$11$	\( T^{2} + 31T + 134 \)