Newform orbit 1575.4.a.m

• Label: 1575.4.a.m
• Level: $1575$
• Weight: $4$
• Character orbit: 1575.a
• Self dual: yes
• Analytic conductor: $92.928$
• Analytic rank: $0$
• 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:	\(0\)
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:	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 \(\beta = \frac{1}{2}(1 + \sqrt{17})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	q + (-b - 3) * q^2 + (7*b + 5) * q^4 + 7 * q^7 + (-25*b - 19) * q^8 + (10*b + 8) * q^11 + (22*b - 18) * q^13 + (-7*b - 21) * q^14 + (63*b + 117) * q^16 + (28*b - 6) * q^17 + (-26*b + 100) * q^19 + (-48*b - 64) * q^22 + (56*b + 64) * q^23 + (-70*b - 34) * q^26 + (49*b + 35) * q^28 + (84*b - 26) * q^29 + (18*b + 156) * q^31 + (-169*b - 451) * q^32 + (-106*b - 94) * q^34 + (-24*b + 78) * q^37 + (4*b - 196) * q^38 + (-140*b - 30) * q^41 + (68*b - 216) * q^43 + (176*b + 320) * q^44 + (-288*b - 416) * q^46 + (108*b + 92) * q^47 + 49 * q^49 + (138*b + 526) * q^52 + (214*b - 90) * q^53 + (-175*b - 133) * q^56 + (-310*b - 258) * q^58 + (-36*b - 164) * q^59 + (-252*b + 522) * q^61 + (-228*b - 540) * q^62 + (623*b + 1093) * q^64 + (-28*b + 408) * q^67 + (294*b + 754) * q^68 + (330*b - 392) * q^71 + (178*b - 478) * q^73 + (18*b - 138) * q^74 + (388*b - 228) * q^76 + (70*b + 56) * q^77 + (88*b + 160) * q^79 + (590*b + 650) * q^82 + (-264*b + 700) * q^83 + (-56*b + 376) * q^86 + (-640*b - 1152) * q^88 + (728*b - 382) * q^89 + (154*b - 126) * q^91 + (1120*b + 1888) * q^92 + (-524*b - 708) * q^94 + (-146*b + 322) * q^97 + (-49*b - 147) * q^98
\(\operatorname{Tr}(f)(q)\)	\(=\)	2 * q - 7 * q^2 + 17 * q^4 + 14 * q^7 - 63 * q^8 + 26 * q^11 - 14 * q^13 - 49 * q^14 + 297 * q^16 + 16 * q^17 + 174 * q^19 - 176 * q^22 + 184 * q^23 - 138 * q^26 + 119 * q^28 + 32 * q^29 + 330 * q^31 - 1071 * q^32 - 294 * q^34 + 132 * q^37 - 388 * q^38 - 200 * q^41 - 364 * q^43 + 816 * q^44 - 1120 * q^46 + 292 * q^47 + 98 * q^49 + 1190 * q^52 + 34 * q^53 - 441 * q^56 - 826 * q^58 - 364 * q^59 + 792 * q^61 - 1308 * q^62 + 2809 * q^64 + 788 * q^67 + 1802 * q^68 - 454 * q^71 - 778 * q^73 - 258 * q^74 - 68 * q^76 + 182 * q^77 + 408 * q^79 + 1890 * q^82 + 1136 * q^83 + 696 * q^86 - 2944 * q^88 - 36 * q^89 - 98 * q^91 + 4896 * q^92 - 1940 * q^94 + 498 * q^97 - 343 * 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

−5.56155

0

22.9309

0

0

7.00000

−83.0388

0

0

1.2

−1.43845

0

−5.93087

0

0

7.00000

20.0388

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.m		2
3.b	odd	2	1		525.4.a.p		2
5.b	even	2	1		315.4.a.m		2
15.d	odd	2	1		105.4.a.c	✓	2
15.e	even	4	2		525.4.d.i		4
35.c	odd	2	1		2205.4.a.bh		2
60.h	even	2	1		1680.4.a.bk		2
105.g	even	2	1		735.4.a.k		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
105.4.a.c	✓	2	15.d	odd	2	1
315.4.a.m		2	5.b	even	2	1
525.4.a.p		2	3.b	odd	2	1
525.4.d.i		4	15.e	even	4	2
735.4.a.k		2	105.g	even	2	1
1575.4.a.m		2	1.a	even	1	1	trivial
1680.4.a.bk		2	60.h	even	2	1
2205.4.a.bh		2	35.c	odd	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1575))\):

\( T_{2}^{2} + 7T_{2} + 8 \)

\( T_{11}^{2} - 26T_{11} - 256 \)

\( T_{13}^{2} + 14T_{13} - 2008 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{2} + 7T + 8 \)
$3$	\( T^{2} \)
$5$	\( T^{2} \)
$7$	\( (T - 7)^{2} \)
$11$	\( T^{2} - 26T - 256 \)