Newform orbit 114.6.a.g

• Label: 114.6.a.g
• 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:	$\Q(\sqrt{2441})$
Defining polynomial:	$x^{2} - x - 610$
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$3$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

$q$-expansion

Coefficients of the $q$-expansion are expressed in terms of $\beta = \frac{1}{2}(-1 + 3\sqrt{2441})$. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	q + 4 * q^2 - 9 * q^3 + 16 * q^4 + (-b - 3) * q^5 - 36 * q^6 + (-b - 53) * q^7 + 64 * q^8 + 81 * q^9 + (-4*b - 12) * q^10 + (7*b + 229) * q^11 - 144 * q^12 + (4*b + 550) * q^13 + (-4*b - 212) * q^14 + (9*b + 27) * q^15 + 256 * q^16 + (-3*b + 1527) * q^17 + 324 * q^18 - 361 * q^19 + (-16*b - 48) * q^20 + (9*b + 477) * q^21 + (28*b + 916) * q^22 + (-22*b - 204) * q^23 - 576 * q^24 + (5*b + 2376) * q^25 + (16*b + 2200) * q^26 - 729 * q^27 + (-16*b - 848) * q^28 + (-98*b + 1674) * q^29 + (36*b + 108) * q^30 + (34*b + 7698) * q^31 + 1024 * q^32 + (-63*b - 2061) * q^33 + (-12*b + 6108) * q^34 + (55*b + 5651) * q^35 + 1296 * q^36 + (10*b + 2592) * q^37 - 1444 * q^38 + (-36*b - 4950) * q^39 + (-64*b - 192) * q^40 + (96*b - 1016) * q^41 + (36*b + 1908) * q^42 + (27*b - 65) * q^43 + (112*b + 3664) * q^44 + (-81*b - 243) * q^45 + (-88*b - 816) * q^46 + (149*b - 597) * q^47 - 2304 * q^48 + (105*b - 8506) * q^49 + (20*b + 9504) * q^50 + (27*b - 13743) * q^51 + (64*b + 8800) * q^52 + (6*b + 2666) * q^53 - 2916 * q^54 + (-243*b - 39131) * q^55 + (-64*b - 3392) * q^56 + 3249 * q^57 + (-392*b + 6696) * q^58 + (-4*b - 2900) * q^59 + (144*b + 432) * q^60 + (229*b + 619) * q^61 + (136*b + 30792) * q^62 + (-81*b - 4293) * q^63 + 4096 * q^64 + (-558*b - 23618) * q^65 + (-252*b - 8244) * q^66 + (-128*b + 22796) * q^67 + (-48*b + 24432) * q^68 + (198*b + 1836) * q^69 + (220*b + 22604) * q^70 + (428*b - 25224) * q^71 + 5184 * q^72 + (415*b + 48161) * q^73 + (40*b + 10368) * q^74 + (-45*b - 21384) * q^75 - 5776 * q^76 + (-593*b - 50581) * q^77 + (-144*b - 19800) * q^78 + (596*b - 2268) * q^79 + (-256*b - 768) * q^80 + 6561 * q^81 + (384*b - 4064) * q^82 + (294*b - 30684) * q^83 + (144*b + 7632) * q^84 + (-1521*b + 11895) * q^85 + (108*b - 260) * q^86 + (882*b - 15066) * q^87 + (448*b + 14656) * q^88 + (58*b - 334) * q^89 + (-324*b - 972) * q^90 + (-758*b - 51118) * q^91 + (-352*b - 3264) * q^92 + (-306*b - 69282) * q^93 + (596*b - 2388) * q^94 + (361*b + 1083) * q^95 - 9216 * q^96 + (1444*b + 38610) * q^97 + (420*b - 34024) * q^98 + (567*b + 18549) * q^99
$\operatorname{Tr}(f)(q)$	$=$	2 * q + 8 * q^2 - 18 * q^3 + 32 * q^4 - 5 * q^5 - 72 * q^6 - 105 * q^7 + 128 * q^8 + 162 * q^9 - 20 * q^10 + 451 * q^11 - 288 * q^12 + 1096 * q^13 - 420 * q^14 + 45 * q^15 + 512 * q^16 + 3057 * q^17 + 648 * q^18 - 722 * q^19 - 80 * q^20 + 945 * q^21 + 1804 * q^22 - 386 * q^23 - 1152 * q^24 + 4747 * q^25 + 4384 * q^26 - 1458 * q^27 - 1680 * q^28 + 3446 * q^29 + 180 * q^30 + 15362 * q^31 + 2048 * q^32 - 4059 * q^33 + 12228 * q^34 + 11247 * q^35 + 2592 * q^36 + 5174 * q^37 - 2888 * q^38 - 9864 * q^39 - 320 * q^40 - 2128 * q^41 + 3780 * q^42 - 157 * q^43 + 7216 * q^44 - 405 * q^45 - 1544 * q^46 - 1343 * q^47 - 4608 * q^48 - 17117 * q^49 + 18988 * q^50 - 27513 * q^51 + 17536 * q^52 + 5326 * q^53 - 5832 * q^54 - 78019 * q^55 - 6720 * q^56 + 6498 * q^57 + 13784 * q^58 - 5796 * q^59 + 720 * q^60 + 1009 * q^61 + 61448 * q^62 - 8505 * q^63 + 8192 * q^64 - 46678 * q^65 - 16236 * q^66 + 45720 * q^67 + 48912 * q^68 + 3474 * q^69 + 44988 * q^70 - 50876 * q^71 + 10368 * q^72 + 95907 * q^73 + 20696 * q^74 - 42723 * q^75 - 11552 * q^76 - 100569 * q^77 - 39456 * q^78 - 5132 * q^79 - 1280 * q^80 + 13122 * q^81 - 8512 * q^82 - 61662 * q^83 + 15120 * q^84 + 25311 * q^85 - 628 * q^86 - 31014 * q^87 + 28864 * q^88 - 726 * q^89 - 1620 * q^90 - 101478 * q^91 - 6176 * q^92 - 138258 * q^93 - 5372 * q^94 + 1805 * q^95 - 18432 * q^96 + 75776 * q^97 - 68468 * q^98 + 36531 * 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

25.2032
−24.2032

4.00000

−9.00000

16.0000

−76.6097

−36.0000

−126.610

64.0000

81.0000

−306.439

1.2

4.00000

−9.00000

16.0000

71.6097

−36.0000

21.6097

64.0000

81.0000

286.439

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.g	✓	2
3.b	odd	2	1		342.6.a.g		2
4.b	odd	2	1		912.6.a.j		2

    

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

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $T_{5}^{2} + 5T_{5} - 5486$ acting on $S_{6}^{\mathrm{new}}(\Gamma_0(114))$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$(T - 4)^{2}$
$3$	$(T + 9)^{2}$
$5$	$T^{2} + 5T - 5486$
$7$	$T^{2} + 105T - 2736$
$11$	$T^{2} - 451T - 218270$