Embedded newform 7.14.c.a.4.6

• Label: 7.14.c.a.4.6
• Level: $7$
• Weight: $14$
• Character: 7.4
• Analytic conductor: $7.506$
• Analytic rank: $0$
• Dimension: $16$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 7 \)
Weight:	\( k \)	\(=\)	\( 14 \)
Character orbit:	\([\chi]\)	\(=\)	7.c (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(7.50616502663\)
Analytic rank:	\(0\)
Dimension:	\(16\)
Relative dimension:	\(8\) over \(\Q(\zeta_{3})\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Defining polynomial:	x^16 - x^15 + 12266*x^14 - 55087*x^13 + 107030285*x^12 - 640627660*x^11 + 438282746176*x^10 - 5859940491984*x^9 + 1300952456341392*x^8 - 14674000482091008*x^7 + 1878469861868220672*x^6 - 14476511842718976000*x^5 + 1885010114115081494784*x^4 - 9244597941717380241408*x^3 + 656480392494410296000512*x^2 + 3883492623114966780739584*x + 127864623964939499445682176
Coefficient ring:	\(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index:	\( 2^{34}\cdot 3^{7}\cdot 7^{11} \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			4.6
Root			\(-18.3288 - 31.7465i\) of defining polynomial
Character	\(\chi\)	\(=\)	7.4
Dual form			7.14.c.a.2.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(36.6577 + 63.4930i) q^{2} +(306.948 - 531.649i) q^{3} +(1408.43 - 2439.47i) q^{4} +(-16706.4 - 28936.3i) q^{5} +45008.0 q^{6} +(-98806.7 - 295172. i) q^{7} +807118. q^{8} +(608728. + 1.05435e6i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 2 q^{2} + 728 q^{3} - 32588 q^{4} + 46928 q^{5} - 30212 q^{6} + 257992 q^{7} - 1093248 q^{8} - 2910808 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/7\mathbb{Z}\right)^\times\).

\(n\)	\(3\)
\(\chi(n)\)	\(e\left(\frac{2}{3}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	36.6577	+	63.4930i	0.405014	+	0.701505i	0.994323	−	0.106403i	\(-0.0339333\pi\)
							−0.589309	+	0.807908i	\(0.700600\pi\)
\(3\)	306.948	−	531.649i	0.243095	−	0.421053i	−0.718499	−	0.695528i	\(-0.755171\pi\)
							0.961594	+	0.274475i	\(0.0885039\pi\)
\(5\)	−16706.4	−	28936.3i	−0.478165	−	0.828206i	0.521522	−	0.853238i	\(-0.325365\pi\)
							−0.999687	+	0.0250322i	\(0.992031\pi\)
\(7\)	−98806.7	−	295172.i	−0.317431	−	0.948281i
							\(11\)	3.83866e6	−	6.64876e6i	0.653322	−	1.13159i	−0.328989	−	0.944334i	\(-0.606708\pi\)
0.982312	−	0.187254i	\(-0.0599587\pi\)
\(13\)	−1.65980e6			−0.0953726			−0.0476863	−	0.998862i	\(-0.515185\pi\)
							−0.0476863	+	0.998862i	\(0.515185\pi\)
\(17\)	−7.20703e7	+	1.24829e8i	−0.724167	+	1.25429i	0.235150	+	0.971959i	\(0.424442\pi\)
							−0.959316	+	0.282334i	\(0.908891\pi\)
\(19\)	−3.66670e7	−	6.35091e7i	−0.178804	−	0.309697i	0.762667	−	0.646791i	\(-0.223889\pi\)
							−0.941471	+	0.337094i	\(0.890556\pi\)
\(23\)	−1.39288e8	−	2.41254e8i	−0.196193	−	0.339816i	0.751098	−	0.660191i	\(-0.229525\pi\)
							−0.947291	+	0.320375i	\(0.896191\pi\)
\(29\)	2.55113e9			0.796426			0.398213	−	0.917293i	\(-0.369630\pi\)
							0.398213	+	0.917293i	\(0.369630\pi\)
\(31\)	−3.23131e9	+	5.59679e9i	−0.653924	+	1.13263i	0.328239	+	0.944595i	\(0.393545\pi\)
							−0.982162	+	0.188034i	\(0.939788\pi\)
\(37\)	1.39483e10	+	2.41592e10i	0.893740	+	1.54800i	0.835357	+	0.549708i	\(0.185261\pi\)
							0.0583830	+	0.998294i	\(0.481406\pi\)
\(41\)	1.89636e10			0.623484			0.311742	−	0.950167i	\(-0.399088\pi\)
							0.311742	+	0.950167i	\(0.399088\pi\)
\(43\)	2.29633e10			0.553973			0.276987	−	0.960874i	\(-0.410664\pi\)
							0.276987	+	0.960874i	\(0.410664\pi\)
\(47\)	−4.03227e8	−	6.98410e8i	−0.00545650	−	0.00945093i	0.863284	−	0.504718i	\(-0.168404\pi\)
							−0.868741	+	0.495267i	\(0.835070\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7.14.c.a.4.6	yes	16
3.2	odd	2		63.14.e.c.46.3		16
7.2	even	3	inner	7.14.c.a.2.6	✓	16
7.3	odd	6		49.14.a.f.1.3		8
7.4	even	3		49.14.a.e.1.3		8
7.5	odd	6		49.14.c.g.30.6		16
7.6	odd	2		49.14.c.g.18.6		16
21.2	odd	6		63.14.e.c.37.3		16

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
7.14.c.a.2.6	✓	16	7.2	even	3	inner
7.14.c.a.4.6	yes	16	1.1	even	1	trivial
49.14.a.e.1.3		8	7.4	even	3
49.14.a.f.1.3		8	7.3	odd	6
49.14.c.g.18.6		16	7.6	odd	2
49.14.c.g.30.6		16	7.5	odd	6
63.14.e.c.37.3		16	21.2	odd	6
63.14.e.c.46.3		16	3.2	odd	2