Newform orbit 1575.4.a.o

• Label: 1575.4.a.o
• Level: $1575$
• Weight: $4$
• Character orbit: 1575.a
• Self dual: yes
• Analytic conductor: $92.928$
• Analytic rank: $1$
• Dimension: $2$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 1575 = 3^{2} \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1575.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(92.9280082590\)
Analytic rank:	\(1\)
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:	no (minimal twist has level 525)
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*b - 3) * q^4 - 7 * q^7 + (5*b - 1) * q^8 + (-5*b + 18) * q^11 + (17*b + 11) * q^13 + (7*b + 7) * q^14 + (-33*b + 5) * q^16 + (-27*b + 53) * q^17 + (56*b - 56) * q^19 + (-8*b + 2) * q^22 + (-24*b - 115) * q^23 + (-45*b - 79) * q^26 + (-21*b + 21) * q^28 + (-84*b + 73) * q^29 + (-13*b - 61) * q^31 + (21*b + 135) * q^32 + (b + 55) * q^34 + (-19*b + 66) * q^37 + (-56*b - 168) * q^38 + (-45*b - 95) * q^41 + (58*b + 373) * q^43 + (54*b - 114) * q^44 + (163*b + 211) * q^46 + (88*b - 120) * q^47 + 49 * q^49 + (33*b + 171) * q^52 + (89*b - 119) * q^53 + (-35*b + 7) * q^56 + (95*b + 263) * q^58 + (-209*b + 325) * q^59 + (-273*b + 25) * q^61 + (87*b + 113) * q^62 + (87*b - 259) * q^64 + (-123*b + 640) * q^67 + (159*b - 483) * q^68 + (-235*b - 192) * q^71 + (88*b + 90) * q^73 + (-28*b + 10) * q^74 + (-168*b + 840) * q^76 + (35*b - 126) * q^77 + (-103*b - 162) * q^79 + (185*b + 275) * q^82 + (431*b - 821) * q^83 + (-489*b - 605) * q^86 + (70*b - 118) * q^88 + (522*b - 494) * q^89 + (-119*b - 77) * q^91 + (-345*b + 57) * q^92 + (-56*b - 232) * q^94 + (704*b - 266) * q^97 + (-49*b - 49) * q^98
\(\operatorname{Tr}(f)(q)\)	\(=\)	2 * q - 3 * q^2 - 3 * q^4 - 14 * q^7 + 3 * q^8 + 31 * q^11 + 39 * q^13 + 21 * q^14 - 23 * q^16 + 79 * q^17 - 56 * q^19 - 4 * q^22 - 254 * q^23 - 203 * q^26 + 21 * q^28 + 62 * q^29 - 135 * q^31 + 291 * q^32 + 111 * q^34 + 113 * q^37 - 392 * q^38 - 235 * q^41 + 804 * q^43 - 174 * q^44 + 585 * q^46 - 152 * q^47 + 98 * q^49 + 375 * q^52 - 149 * q^53 - 21 * q^56 + 621 * q^58 + 441 * q^59 - 223 * q^61 + 313 * q^62 - 431 * q^64 + 1157 * q^67 - 807 * q^68 - 619 * q^71 + 268 * q^73 - 8 * q^74 + 1512 * q^76 - 217 * q^77 - 427 * q^79 + 735 * q^82 - 1211 * q^83 - 1699 * q^86 - 166 * q^88 - 466 * q^89 - 273 * q^91 - 231 * q^92 - 520 * q^94 + 172 * q^97 - 147 * q^98

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

2.56155
−1.56155

−3.56155

0

4.68466

0

0

−7.00000

11.8078

0

0

1.2

0.561553

0

−7.68466

0

0

−7.00000

−8.80776

0

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

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
525.4.a.j	✓	2	15.d	odd	2	1
525.4.a.m	yes	2	3.b	odd	2	1
525.4.d.m		4	15.e	even	4	2
1575.4.a.o		2	1.a	even	1	1	trivial
1575.4.a.x		2	5.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 \(S_{4}^{\mathrm{new}}(\Gamma_0(1575))\):

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

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

\( T_{13}^{2} - 39T_{13} - 848 \)

Hecke characteristic polynomials

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