Newform orbit 1575.4.a.u

• Label: 1575.4.a.u
• 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 315)
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 * q^2 + (b - 4) * q^4 + 7 * q^7 + (-11*b + 4) * q^8 + (-4*b + 4) * q^11 + 22*b * q^13 + 7*b * q^14 + (-15*b - 12) * q^16 + (26*b - 42) * q^17 + (-44*b + 22) * q^19 - 16 * q^22 + (-14*b - 34) * q^23 + (22*b + 88) * q^26 + (7*b - 28) * q^28 + (30*b + 152) * q^29 + (18*b - 114) * q^31 + (61*b - 92) * q^32 + (-16*b + 104) * q^34 + (30*b - 18) * q^37 + (-22*b - 176) * q^38 + (-52*b + 114) * q^41 + (-166*b + 60) * q^43 + (16*b - 32) * q^44 + (-48*b - 56) * q^46 + (30*b - 272) * q^47 + 49 * q^49 + (-66*b + 88) * q^52 + (-52*b - 378) * q^53 + (-77*b + 28) * q^56 + (182*b + 120) * q^58 + 284 * q^59 + (90*b - 354) * q^61 + (-96*b + 72) * q^62 + (89*b + 340) * q^64 + (98*b - 396) * q^67 + (-120*b + 272) * q^68 + (-162*b + 488) * q^71 + (-290*b + 104) * q^73 + (12*b + 120) * q^74 + (154*b - 264) * q^76 + (-28*b + 28) * q^77 + (328*b + 136) * q^79 + (62*b - 208) * q^82 + (-300*b + 284) * q^83 + (-106*b - 664) * q^86 + (-16*b + 192) * q^88 + (-476*b + 202) * q^89 + 154*b * q^91 + (8*b + 80) * q^92 + (-242*b + 120) * q^94 + (-482*b - 572) * q^97 + 49*b * q^98
\(\operatorname{Tr}(f)(q)\)	\(=\)	2 * q + q^2 - 7 * q^4 + 14 * q^7 - 3 * q^8 + 4 * q^11 + 22 * q^13 + 7 * q^14 - 39 * q^16 - 58 * q^17 - 32 * q^22 - 82 * q^23 + 198 * q^26 - 49 * q^28 + 334 * q^29 - 210 * q^31 - 123 * q^32 + 192 * q^34 - 6 * q^37 - 374 * q^38 + 176 * q^41 - 46 * q^43 - 48 * q^44 - 160 * q^46 - 514 * q^47 + 98 * q^49 + 110 * q^52 - 808 * q^53 - 21 * q^56 + 422 * q^58 + 568 * q^59 - 618 * q^61 + 48 * q^62 + 769 * q^64 - 694 * q^67 + 424 * q^68 + 814 * q^71 - 82 * q^73 + 252 * q^74 - 374 * q^76 + 28 * q^77 + 600 * q^79 - 354 * q^82 + 268 * q^83 - 1434 * q^86 + 368 * q^88 - 72 * q^89 + 154 * q^91 + 168 * q^92 - 2 * q^94 - 1626 * q^97 + 49 * 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

−1.56155
2.56155

−1.56155

0

−5.56155

0

0

7.00000

21.1771

0

0

1.2

2.56155

0

−1.43845

0

0

7.00000

−24.1771

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.u		2
3.b	odd	2	1		1575.4.a.r		2
5.b	even	2	1		315.4.a.h	✓	2
15.d	odd	2	1		315.4.a.j	yes	2
35.c	odd	2	1		2205.4.a.y		2
105.g	even	2	1		2205.4.a.ba		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
315.4.a.h	✓	2	5.b	even	2	1
315.4.a.j	yes	2	15.d	odd	2	1
1575.4.a.r		2	3.b	odd	2	1
1575.4.a.u		2	1.a	even	1	1	trivial
2205.4.a.y		2	35.c	odd	2	1
2205.4.a.ba		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 \(S_{4}^{\mathrm{new}}(\Gamma_0(1575))\):

\( T_{2}^{2} - T_{2} - 4 \)

\( T_{11}^{2} - 4T_{11} - 64 \)

\( T_{13}^{2} - 22T_{13} - 1936 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{2} - T - 4 \)
$3$	\( T^{2} \)
$5$	\( T^{2} \)
$7$	\( (T - 7)^{2} \)
$11$	\( T^{2} - 4T - 64 \)