Embedded newform 7.18.c.a.4.6

• Label: 7.18.c.a.4.6
• 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			4.6
Root			\(-7.21985 - 12.5051i\) of defining polynomial
Character	\(\chi\)	\(=\)	7.4
Dual form			7.18.c.a.2.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.43970 - 2.49364i) q^{2} +(3646.82 - 6316.47i) q^{3} +(65531.9 - 113505. i) q^{4} +(804501. + 1.39344e6i) q^{5} -21001.3 q^{6} +(-6.26914e6 - 1.39043e7i) q^{7} -754795. q^{8} +(3.79715e7 + 6.57686e7i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 270 q^{2} + 6560 q^{3} - 551660 q^{4} + 1089798 q^{5} - 1687556 q^{6} + 14395360 q^{7} - 237823200 q^{8} - 498883228 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.43970	−	2.49364i	−0.00397665	−	0.00688777i	0.864030	−	0.503440i	\(-0.167932\pi\)
							−0.868007	+	0.496552i	\(0.834599\pi\)
\(3\)	3646.82	−	6316.47i	0.320910	−	0.555832i	−0.659766	−	0.751471i	\(-0.729345\pi\)
							0.980676	+	0.195639i	\(0.0626780\pi\)
\(5\)	804501.	+	1.39344e6i	0.921046	+	1.59530i	0.797800	+	0.602922i	\(0.205997\pi\)
							0.123246	+	0.992376i	\(0.460669\pi\)
\(7\)	−6.26914e6	−	1.39043e7i	−0.411031	−	0.911621i
							\(11\)	2.17631e8	−	3.76948e8i	0.306114	−	0.530205i	−0.671395	−	0.741100i	\(-0.734304\pi\)
0.977509	+	0.210895i	\(0.0676377\pi\)
\(13\)	3.39665e9			1.15487			0.577433	−	0.816438i	\(-0.304054\pi\)
							0.577433	+	0.816438i	\(0.304054\pi\)
\(17\)	1.99795e10	−	3.46056e10i	0.694656	−	1.20318i	−0.275641	−	0.961261i	\(-0.588890\pi\)
							0.970297	−	0.241918i	\(-0.0777766\pi\)
\(19\)	1.46561e10	+	2.53851e10i	0.197976	+	0.342905i	0.947872	−	0.318651i	\(-0.103230\pi\)
							−0.749896	+	0.661556i	\(0.769896\pi\)
\(23\)	−2.44628e11	−	4.23709e11i	−0.651359	−	1.12819i	−0.982793	−	0.184708i	\(-0.940866\pi\)
							0.331435	−	0.943478i	\(-0.392467\pi\)
\(29\)	−3.11829e11			−0.115753			−0.0578766	−	0.998324i	\(-0.518433\pi\)
							−0.0578766	+	0.998324i	\(0.518433\pi\)
\(31\)	−2.10217e12	+	3.64106e12i	−0.442683	+	0.766749i	−0.997888	−	0.0649646i	\(-0.979307\pi\)
							0.555205	+	0.831714i	\(0.312640\pi\)
\(37\)	8.99033e12	+	1.55717e13i	0.420786	+	0.728822i	0.996017	−	0.0891690i	\(-0.0284211\pi\)
							−0.575231	+	0.817991i	\(0.695088\pi\)
\(41\)	−4.81124e13			−0.941011			−0.470505	−	0.882397i	\(-0.655928\pi\)
							−0.470505	+	0.882397i	\(0.655928\pi\)
\(43\)	4.11144e13			0.536429			0.268214	−	0.963359i	\(-0.413566\pi\)
							0.268214	+	0.963359i	\(0.413566\pi\)
\(47\)	7.90090e13	+	1.36848e14i	0.483999	+	0.838311i	0.999831	−	0.0183786i	\(-0.00585042\pi\)
							−0.515832	+	0.856690i	\(0.672517\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7.18.c.a.4.6	yes	20
7.2	even	3	inner	7.18.c.a.2.6	✓	20
7.3	odd	6		49.18.a.g.1.5		10
7.4	even	3		49.18.a.f.1.5		10

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
7.18.c.a.2.6	✓	20	7.2	even	3	inner
7.18.c.a.4.6	yes	20	1.1	even	1	trivial
49.18.a.f.1.5		10	7.4	even	3
49.18.a.g.1.5		10	7.3	odd	6