Newform orbit 2352.4.a.bw

• Label: 2352.4.a.bw
• Level: $2352$
• Weight: $4$
• Character orbit: 2352.a
• Self dual: yes
• Analytic conductor: $138.772$
• Analytic rank: $0$
• Dimension: $2$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 2352 = 2^{4} \cdot 3 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2352.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(138.772492334\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{2}) \)
Defining polynomial:	\( x^{2} - 2 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 294)
Fricke sign:	\(+1\)
Sato-Tate group:	$\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	q + 3 * q^3 + (b - 6) * q^5 + 9 * q^9 + (6*b - 2) * q^11 + (-15*b - 24) * q^13 + (3*b - 18) * q^15 + (-11*b - 66) * q^17 + (-46*b + 60) * q^19 + (-102*b + 38) * q^23 + (-12*b - 87) * q^25 + 27 * q^27 + (-150*b - 56) * q^29 + (54*b + 216) * q^31 + (18*b - 6) * q^33 + (180*b - 140) * q^37 + (-45*b - 72) * q^39 + (127*b - 18) * q^41 + (288*b + 64) * q^43 + (9*b - 54) * q^45 + (338*b - 132) * q^47 + (-33*b - 198) * q^51 + (192*b + 134) * q^53 + (-38*b + 24) * q^55 + (-138*b + 180) * q^57 + (-298*b + 168) * q^59 + (353*b + 252) * q^61 + (66*b + 114) * q^65 + (-144*b + 192) * q^67 + (-306*b + 114) * q^69 + (342*b + 198) * q^71 + (-595*b + 156) * q^73 + (-36*b - 261) * q^75 + (-300*b + 424) * q^79 + 81 * q^81 + (-80*b - 324) * q^83 + 374 * q^85 + (-450*b - 168) * q^87 + (-175*b + 306) * q^89 + (162*b + 648) * q^93 + (336*b - 452) * q^95 + (133*b + 1092) * q^97 + (54*b - 18) * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	2 * q + 6 * q^3 - 12 * q^5 + 18 * q^9 - 4 * q^11 - 48 * q^13 - 36 * q^15 - 132 * q^17 + 120 * q^19 + 76 * q^23 - 174 * q^25 + 54 * q^27 - 112 * q^29 + 432 * q^31 - 12 * q^33 - 280 * q^37 - 144 * q^39 - 36 * q^41 + 128 * q^43 - 108 * q^45 - 264 * q^47 - 396 * q^51 + 268 * q^53 + 48 * q^55 + 360 * q^57 + 336 * q^59 + 504 * q^61 + 228 * q^65 + 384 * q^67 + 228 * q^69 + 396 * q^71 + 312 * q^73 - 522 * q^75 + 848 * q^79 + 162 * q^81 - 648 * q^83 + 748 * q^85 - 336 * q^87 + 612 * q^89 + 1296 * q^93 - 904 * q^95 + 2184 * q^97 - 36 * 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

−1.41421
1.41421

0

3.00000

0

−7.41421

0

0

0

9.00000

0

1.2

0

3.00000

0

−4.58579

0

0

0

9.00000

0

Atkin-Lehner signs

\( p \)	Sign
\(2\)	\( -1 \)
\(3\)	\( -1 \)
\(7\)	\( +1 \)

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	2352.4.a.bw		2
4.b	odd	2	1		294.4.a.l	✓	2
7.b	odd	2	1		2352.4.a.bu		2
12.b	even	2	1		882.4.a.bb		2
28.d	even	2	1		294.4.a.o	yes	2
28.f	even	6	2		294.4.e.k		4
28.g	odd	6	2		294.4.e.m		4
84.h	odd	2	1		882.4.a.t		2
84.j	odd	6	2		882.4.g.bk		4
84.n	even	6	2		882.4.g.be		4

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
294.4.a.l	✓	2	4.b	odd	2	1
294.4.a.o	yes	2	28.d	even	2	1
294.4.e.k		4	28.f	even	6	2
294.4.e.m		4	28.g	odd	6	2
882.4.a.t		2	84.h	odd	2	1
882.4.a.bb		2	12.b	even	2	1
882.4.g.be		4	84.n	even	6	2
882.4.g.bk		4	84.j	odd	6	2
2352.4.a.bu		2	7.b	odd	2	1
2352.4.a.bw		2	1.a	even	1	1	trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(2352))\):

\( T_{5}^{2} + 12T_{5} + 34 \)

\( T_{11}^{2} + 4T_{11} - 68 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{2} \)
$3$	\( (T - 3)^{2} \)
$5$	\( T^{2} + 12T + 34 \)
$7$	\( T^{2} \)
$11$	\( T^{2} + 4T - 68 \)