Newform orbit 114.6.a.f

• Label: 114.6.a.f
• Level: $114$
• Weight: $6$
• Character orbit: 114.a
• Self dual: yes
• Analytic conductor: $18.284$
• Analytic rank: $0$
• Dimension: $2$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	$N$	$=$	$114 = 2 \cdot 3 \cdot 19$
Weight:	$k$	$=$	$6$
Character orbit:	$[\chi]$	$=$	114.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	$18.2837554587$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{4089})$
Defining polynomial:	$x^{2} - x - 1022$
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$1$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

$q$-expansion

Coefficients of the $q$-expansion are expressed in terms of $\beta = \frac{1}{2}(1 + \sqrt{4089})$. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	q - 4 * q^2 + 9 * q^3 + 16 * q^4 + (-b - 24) * q^5 - 36 * q^6 + (-5*b - 50) * q^7 - 64 * q^8 + 81 * q^9 + (4*b + 96) * q^10 + (b + 362) * q^11 + 144 * q^12 + (20*b - 38) * q^13 + (20*b + 200) * q^14 + (-9*b - 216) * q^15 + 256 * q^16 + (-39*b + 780) * q^17 - 324 * q^18 - 361 * q^19 + (-16*b - 384) * q^20 + (-45*b - 450) * q^21 + (-4*b - 1448) * q^22 + (116*b + 792) * q^23 - 576 * q^24 + (49*b - 1527) * q^25 + (-80*b + 152) * q^26 + 729 * q^27 + (-80*b - 800) * q^28 + (16*b + 6354) * q^29 + (36*b + 864) * q^30 + (-130*b + 4344) * q^31 - 1024 * q^32 + (9*b + 3258) * q^33 + (156*b - 3120) * q^34 + (175*b + 6310) * q^35 + 1296 * q^36 + (-202*b + 6318) * q^37 + 1444 * q^38 + (180*b - 342) * q^39 + (64*b + 1536) * q^40 + (-270*b + 10250) * q^41 + (180*b + 1800) * q^42 + (-21*b + 2458) * q^43 + (16*b + 5792) * q^44 + (-81*b - 1944) * q^45 + (-464*b - 3168) * q^46 + (119*b + 10470) * q^47 + 2304 * q^48 + (525*b + 11243) * q^49 + (-196*b + 6108) * q^50 + (-351*b + 7020) * q^51 + (320*b - 608) * q^52 + (-168*b + 12682) * q^53 - 2916 * q^54 + (-387*b - 9710) * q^55 + (320*b + 3200) * q^56 - 3249 * q^57 + (-64*b - 25416) * q^58 + (32*b + 764) * q^59 + (-144*b - 3456) * q^60 + (1493*b - 1808) * q^61 + (520*b - 17376) * q^62 + (-405*b - 4050) * q^63 + 4096 * q^64 + (-462*b - 19528) * q^65 + (-36*b - 13032) * q^66 + (32*b - 23500) * q^67 + (-624*b + 12480) * q^68 + (1044*b + 7128) * q^69 + (-700*b - 25240) * q^70 + (-1480*b + 30768) * q^71 - 5184 * q^72 + (-505*b - 28336) * q^73 + (808*b - 25272) * q^74 + (441*b - 13743) * q^75 - 5776 * q^76 + (-1865*b - 23210) * q^77 + (-720*b + 1368) * q^78 + (-320*b - 49524) * q^79 + (-256*b - 6144) * q^80 + 6561 * q^81 + (1080*b - 41000) * q^82 + (3372*b - 12576) * q^83 + (-720*b - 7200) * q^84 + (195*b + 21138) * q^85 + (84*b - 9832) * q^86 + (144*b + 57186) * q^87 + (-64*b - 23168) * q^88 + (2932*b - 7394) * q^89 + (324*b + 7776) * q^90 + (-910*b - 100300) * q^91 + (1856*b + 12672) * q^92 + (-1170*b + 39096) * q^93 + (-476*b - 41880) * q^94 + (361*b + 8664) * q^95 - 9216 * q^96 + (-4336*b - 36366) * q^97 + (-2100*b - 44972) * q^98 + (81*b + 29322) * q^99
$\operatorname{Tr}(f)(q)$	$=$	2 * q - 8 * q^2 + 18 * q^3 + 32 * q^4 - 49 * q^5 - 72 * q^6 - 105 * q^7 - 128 * q^8 + 162 * q^9 + 196 * q^10 + 725 * q^11 + 288 * q^12 - 56 * q^13 + 420 * q^14 - 441 * q^15 + 512 * q^16 + 1521 * q^17 - 648 * q^18 - 722 * q^19 - 784 * q^20 - 945 * q^21 - 2900 * q^22 + 1700 * q^23 - 1152 * q^24 - 3005 * q^25 + 224 * q^26 + 1458 * q^27 - 1680 * q^28 + 12724 * q^29 + 1764 * q^30 + 8558 * q^31 - 2048 * q^32 + 6525 * q^33 - 6084 * q^34 + 12795 * q^35 + 2592 * q^36 + 12434 * q^37 + 2888 * q^38 - 504 * q^39 + 3136 * q^40 + 20230 * q^41 + 3780 * q^42 + 4895 * q^43 + 11600 * q^44 - 3969 * q^45 - 6800 * q^46 + 21059 * q^47 + 4608 * q^48 + 23011 * q^49 + 12020 * q^50 + 13689 * q^51 - 896 * q^52 + 25196 * q^53 - 5832 * q^54 - 19807 * q^55 + 6720 * q^56 - 6498 * q^57 - 50896 * q^58 + 1560 * q^59 - 7056 * q^60 - 2123 * q^61 - 34232 * q^62 - 8505 * q^63 + 8192 * q^64 - 39518 * q^65 - 26100 * q^66 - 46968 * q^67 + 24336 * q^68 + 15300 * q^69 - 51180 * q^70 + 60056 * q^71 - 10368 * q^72 - 57177 * q^73 - 49736 * q^74 - 27045 * q^75 - 11552 * q^76 - 48285 * q^77 + 2016 * q^78 - 99368 * q^79 - 12544 * q^80 + 13122 * q^81 - 80920 * q^82 - 21780 * q^83 - 15120 * q^84 + 42471 * q^85 - 19580 * q^86 + 114516 * q^87 - 46400 * q^88 - 11856 * q^89 + 15876 * q^90 - 201510 * q^91 + 27200 * q^92 + 77022 * q^93 - 84236 * q^94 + 17689 * q^95 - 18432 * q^96 - 77068 * q^97 - 92044 * q^98 + 58725 * 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

32.4726
−31.4726

−4.00000

9.00000

16.0000

−56.4726

−36.0000

−212.363

−64.0000

81.0000

225.891

1.2

−4.00000

9.00000

16.0000

7.47264

−36.0000

107.363

−64.0000

81.0000

−29.8906

Atkin-Lehner signs

$p$	Sign
$2$	$+1$
$3$	$-1$
$19$	$+1$

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	114.6.a.f	✓	2
3.b	odd	2	1		342.6.a.j		2
4.b	odd	2	1		912.6.a.h		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
114.6.a.f	✓	2	1.a	even	1	1	trivial
342.6.a.j		2	3.b	odd	2	1
912.6.a.h		2	4.b	odd	2	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $T_{5}^{2} + 49T_{5} - 422$ acting on $S_{6}^{\mathrm{new}}(\Gamma_0(114))$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$(T + 4)^{2}$
$3$	$(T - 9)^{2}$
$5$	$T^{2} + 49T - 422$
$7$	$T^{2} + 105T - 22800$
$11$	$T^{2} - 725T + 130384$