Embedded newform 114.12.e.b.7.6

• Label: 114.12.e.b.7.6
• Level: $114$
• Weight: $12$
• Character: 114.7
• Analytic conductor: $87.591$
• Analytic rank: $0$
• Dimension: $18$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 114 = 2 \cdot 3 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 12 \)
Character orbit:	\([\chi]\)	\(=\)	114.e (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(87.5911225838\)
Analytic rank:	\(0\)
Dimension:	\(18\)
Relative dimension:	\(9\) over \(\Q(\zeta_{3})\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{18} - \cdots)\)
Defining polynomial:	x^18 - 5*x^17 + 294880699*x^16 - 502930193282*x^15 + 58010769323549578*x^14 - 107962469874566484896*x^13 + 6462950999566890012014308*x^12 - 11896428734707375935312399680*x^11 + 519285520583573846691128750232193*x^10 - 692884029364083538053156632692698801*x^9 + 24689309396155847956017564465034753902837*x^8 - 8973166541658131218277079733218668822870496*x^7 + 820043514499299080450708516880176278207316065440*x^6 + 287778341167683358358263946237064064438256890402164*x^5 + 13945925406732094645473488744875796514712230817314612762*x^4 + 31434839715775176789744984590696756110658152748821063150434*x^3 + 139902872770278728043786434425265070357181675994273961821746647*x^2 + 124933492983128017052411326931038150648544220156661860270854588847*x + 105517350362802459566297751633435756408365362459904278213342175579601
Coefficient ring:	\(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index:	\( 2^{34}\cdot 3^{13}\cdot 19^{2} \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			7.6
Root			\(-1455.26 + 2520.58i\) of defining polynomial
Character	\(\chi\)	\(=\)	114.7
Dual form			114.12.e.b.49.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(16.0000 + 27.7128i) q^{2} +(121.500 + 210.444i) q^{3} +(-512.000 + 886.810i) q^{4} +(2038.76 + 3531.24i) q^{5} +(-3888.00 + 6734.21i) q^{6} -33266.6 q^{7} -32768.0 q^{8} +(-29524.5 + 51137.9i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 18 q + 288 q^{2} + 2187 q^{3} - 9216 q^{4} + 10498 q^{5} - 69984 q^{6} + 33594 q^{7} - 589824 q^{8} - 531441 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/114\mathbb{Z}\right)^\times\).

\(n\)	\(77\)	\(97\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{1}{3}\right)\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

(See \(a_n\) instead)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	16.0000	+	27.7128i	0.353553	+	0.612372i
							\(3\)	121.500	+	210.444i	0.288675	+	0.500000i
\(5\)	2038.76	+	3531.24i	0.291764	+	0.505349i								0.974227	−	0.225571i	\(-0.0724246\pi\)
							−0.682463	+	0.730920i	\(0.739091\pi\)
\(7\)	−33266.6			−0.748116			−0.374058	−	0.927405i	\(-0.622034\pi\)
							−0.374058	+	0.927405i	\(0.622034\pi\)
\(11\)	493715.			0.924307			0.462154	−	0.886800i	\(-0.347077\pi\)
							0.462154	+	0.886800i	\(0.347077\pi\)
\(13\)	738318.	−	1.27880e6i	0.551512	−	0.955247i	−0.446654	−	0.894707i	\(-0.647384\pi\)
							0.998166	−	0.0605398i	\(-0.0192822\pi\)
\(17\)	−2.73745e6	−	4.74140e6i	−0.467602	−	0.809911i	0.531712	−	0.846925i	\(-0.321549\pi\)
							−0.999315	+	0.0370140i	\(0.988215\pi\)
\(19\)	−8.12221e6	−	7.10774e6i	−0.752539	−	0.658547i
							\(23\)	−707808.	+	1.22596e6i	−0.0229304	+	0.0397167i	−0.877263	−	0.480010i	\(-0.840633\pi\)
0.854332	+	0.519727i	\(0.173966\pi\)
\(29\)	4.99878e7	−	8.65815e7i	0.452559	−	0.783855i	−0.545985	−	0.837795i	\(-0.683844\pi\)
							0.998544	+	0.0539395i	\(0.0171778\pi\)
\(31\)	9.00639e7			0.565017			0.282509	−	0.959265i	\(-0.408833\pi\)
							0.282509	+	0.959265i	\(0.408833\pi\)
\(37\)	−2.32040e7			−0.0550115			−0.0275057	−	0.999622i	\(-0.508756\pi\)
							−0.0275057	+	0.999622i	\(0.508756\pi\)
\(41\)	−3.48347e8	−	6.03354e8i	−0.469570	−	0.813320i	0.529824	−	0.848107i	\(-0.322258\pi\)
							−0.999395	+	0.0347877i	\(0.988924\pi\)
\(43\)	3.61198e8	+	6.25613e8i	0.374687	+	0.648977i	0.990280	−	0.139087i	\(-0.0444168\pi\)
							−0.615593	+	0.788064i	\(0.711083\pi\)
\(47\)	−3.15627e8	+	5.46681e8i	−0.200741	+	0.347693i	−0.948767	−	0.315976i	\(-0.897668\pi\)
							0.748027	+	0.663669i	\(0.231002\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	114.12.e.b.7.6	✓	18
19.11	even	3	inner	114.12.e.b.49.6	yes	18

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
114.12.e.b.7.6	✓	18	1.1	even	1	trivial
114.12.e.b.49.6	yes	18	19.11	even	3	inner