Newform orbit 2352.4.a.ce

• Label: 2352.4.a.ce
• 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:	$\Q(\sqrt{137})$
Defining polynomial:	$x^{2} - x - 34$
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$2$
Twist minimal:	no (minimal twist has level 1176)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

$q$-expansion

Coefficients of the $q$-expansion are expressed in terms of $\beta = \sqrt{137}$. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	q + 3 * q^3 + (-b + 9) * q^5 + 9 * q^9 + (-3*b - 29) * q^11 + (4*b + 24) * q^13 + (-3*b + 27) * q^15 + (-b - 3) * q^17 + (-10*b - 42) * q^19 + (15*b - 3) * q^23 + (-18*b + 93) * q^25 + 27 * q^27 + (6*b - 52) * q^29 + (-2*b - 126) * q^31 + (-9*b - 87) * q^33 + (-6*b + 112) * q^37 + (12*b + 72) * q^39 + (-23*b + 219) * q^41 + 388 * q^43 + (-9*b + 81) * q^45 + (32*b + 120) * q^47 + (-3*b - 9) * q^51 + (-6*b + 540) * q^53 + (2*b + 150) * q^55 + (-30*b - 126) * q^57 + (-16*b - 156) * q^59 + (22*b - 330) * q^61 + (12*b - 332) * q^65 + (-42*b + 198) * q^67 + (45*b - 9) * q^69 + (-51*b - 33) * q^71 + (18*b + 486) * q^73 + (-54*b + 279) * q^75 + (18*b + 430) * q^79 + 81 * q^81 + (16*b - 1260) * q^83 + (-6*b + 110) * q^85 + (18*b - 156) * q^87 + (-15*b + 843) * q^89 + (-6*b - 378) * q^93 + (-48*b + 992) * q^95 + (22*b + 1050) * q^97 + (-27*b - 261) * q^99
$\operatorname{Tr}(f)(q)$	$=$	2 * q + 6 * q^3 + 18 * q^5 + 18 * q^9 - 58 * q^11 + 48 * q^13 + 54 * q^15 - 6 * q^17 - 84 * q^19 - 6 * q^23 + 186 * q^25 + 54 * q^27 - 104 * q^29 - 252 * q^31 - 174 * q^33 + 224 * q^37 + 144 * q^39 + 438 * q^41 + 776 * q^43 + 162 * q^45 + 240 * q^47 - 18 * q^51 + 1080 * q^53 + 300 * q^55 - 252 * q^57 - 312 * q^59 - 660 * q^61 - 664 * q^65 + 396 * q^67 - 18 * q^69 - 66 * q^71 + 972 * q^73 + 558 * q^75 + 860 * q^79 + 162 * q^81 - 2520 * q^83 + 220 * q^85 - 312 * q^87 + 1686 * q^89 - 756 * q^93 + 1984 * q^95 + 2100 * q^97 - 522 * 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

6.35235
−5.35235

0

3.00000

0

−2.70470

0

0

0

9.00000

0

1.2

0

3.00000

0

20.7047

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.ce		2
4.b	odd	2	1		1176.4.a.s	✓	2
7.b	odd	2	1		2352.4.a.bm		2
28.d	even	2	1		1176.4.a.t	yes	2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
1176.4.a.s	✓	2	4.b	odd	2	1
1176.4.a.t	yes	2	28.d	even	2	1
2352.4.a.bm		2	7.b	odd	2	1
2352.4.a.ce		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 $S_{4}^{\mathrm{new}}(\Gamma_0(2352))$:

$T_{5}^{2} - 18T_{5} - 56$

$T_{11}^{2} + 58T_{11} - 392$

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{2}$
$3$	$(T - 3)^{2}$
$5$	$T^{2} - 18T - 56$
$7$	$T^{2}$
$11$	$T^{2} + 58T - 392$