Embedded newform 7.18.c.a.4.8

• Label: 7.18.c.a.4.8
• Level: $7$
• Weight: $18$
• Character: 7.4
• Analytic conductor: $12.826$
• Analytic rank: $0$
• Dimension: $20$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 7 \)
Weight:	\( k \)	\(=\)	\( 18 \)
Character orbit:	\([\chi]\)	\(=\)	7.c (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(12.8255461141\)
Analytic rank:	\(0\)
Dimension:	\(20\)
Relative dimension:	\(10\) over \(\Q(\zeta_{3})\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{20} - \cdots)\)
Defining polynomial:	x^20 - 5*x^19 + 231900*x^18 + 2573595*x^17 + 36222331443*x^16 + 522056283042*x^15 + 3095689791534860*x^14 + 63517964980290950*x^13 + 191169147185579182773*x^12 + 4661871029431062841865*x^11 + 6728981840934861993437384*x^10 + 251477264592720071695768485*x^9 + 169785400547236219777139622573*x^8 + 7884822396560931315753101896470*x^7 + 2084722393634413874384133666310116*x^6 + 170299785457406646702831504431152090*x^5 + 19136307004107023234373216890155693651*x^4 + 919913048844099315146584754515161210807*x^3 + 39214983331357496369283116903046958144404*x^2 + 515992296986415442815810669806567791765095*x + 5495215718943433371703477919782244002731225
Coefficient ring:	\(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index:	\( 2^{50}\cdot 3^{18}\cdot 7^{19} \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			4.8
Root			\(90.6104 + 156.942i\) of defining polynomial
Character	\(\chi\)	\(=\)	7.4
Dual form			7.18.c.a.2.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(194.221 + 336.400i) q^{2} +(1157.82 - 2005.40i) q^{3} +(-9907.51 + 17160.3i) q^{4} +(92672.9 + 160514. i) q^{5} +899489. q^{6} +(-2.00510e6 + 1.51199e7i) q^{7} +4.32169e7 q^{8} +(6.18890e7 + 1.07195e8i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 270 q^{2} + 6560 q^{3} - 551660 q^{4} + 1089798 q^{5} - 1687556 q^{6} + 14395360 q^{7} - 237823200 q^{8} - 498883228 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	194.221	+	336.400i	0.536464	+	0.929184i	0.999091	+	0.0426304i	\(0.0135738\pi\)
							−0.462626	+	0.886553i	\(0.653093\pi\)
\(3\)	1157.82	−	2005.40i	0.101885	−	0.176470i	−0.810576	−	0.585633i	\(-0.800846\pi\)
							0.912461	+	0.409163i	\(0.134179\pi\)
\(5\)	92672.9	+	160514.i	0.106098	+	0.183767i	0.914186	−	0.405294i	\(-0.132831\pi\)
							−0.808088	+	0.589062i	\(0.799498\pi\)
\(7\)	−2.00510e6	+	1.51199e7i	−0.131463	+	0.991321i
							\(11\)	−1.35144e8	+	2.34077e8i	−0.190090	+	0.329246i	−0.945280	−	0.326260i	\(-0.894211\pi\)
0.755190	+	0.655506i	\(0.227545\pi\)
\(13\)	−1.73012e9			−0.588243			−0.294121	−	0.955768i	\(-0.595027\pi\)
							−0.294121	+	0.955768i	\(0.595027\pi\)
\(17\)	−1.04737e10	+	1.81410e10i	−0.364154	+	0.630733i	−0.988640	−	0.150303i	\(-0.951975\pi\)
							0.624486	+	0.781036i	\(0.285308\pi\)
\(19\)	3.83789e10	+	6.64742e10i	0.518426	+	0.897940i	0.999771	+	0.0214086i	\(0.00681508\pi\)
							−0.481345	+	0.876531i	\(0.659852\pi\)
\(23\)	−1.71776e11	−	2.97525e11i	−0.457378	−	0.792202i	0.541443	−	0.840737i	\(-0.317878\pi\)
							−0.998821	+	0.0485349i	\(0.984545\pi\)
\(29\)	2.49396e12			0.925779			0.462889	−	0.886416i	\(-0.346813\pi\)
							0.462889	+	0.886416i	\(0.346813\pi\)
\(31\)	6.41787e11	−	1.11161e12i	0.135150	−	0.234087i	−0.790505	−	0.612456i	\(-0.790182\pi\)
							0.925655	+	0.378369i	\(0.123515\pi\)
\(37\)	3.56484e12	+	6.17449e12i	0.166850	+	0.288992i	0.937311	−	0.348495i	\(-0.113307\pi\)
							−0.770461	+	0.637487i	\(0.779974\pi\)
\(41\)	2.38867e13			0.467189			0.233595	−	0.972334i	\(-0.424951\pi\)
							0.233595	+	0.972334i	\(0.424951\pi\)
\(43\)	5.28752e13			0.689875			0.344938	−	0.938626i	\(-0.387900\pi\)
							0.344938	+	0.938626i	\(0.387900\pi\)
\(47\)	−1.58058e14	−	2.73764e14i	−0.968244	−	1.67705i	−0.700636	−	0.713519i	\(-0.747100\pi\)
							−0.267608	−	0.963528i	\(-0.586233\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7.18.c.a.4.8	yes	20
7.2	even	3	inner	7.18.c.a.2.8	✓	20
7.3	odd	6		49.18.a.g.1.3		10
7.4	even	3		49.18.a.f.1.3		10

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
7.18.c.a.2.8	✓	20	7.2	even	3	inner
7.18.c.a.4.8	yes	20	1.1	even	1	trivial
49.18.a.f.1.3		10	7.4	even	3
49.18.a.g.1.3		10	7.3	odd	6