Embedded newform 7.18.c.a.2.5

• Label: 7.18.c.a.2.5
• Level: $7$
• Weight: $18$
• Character: 7.2
• 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			2.5
Root			\(-28.2933 + 49.0055i\) of defining polynomial
Character	\(\chi\)	\(=\)	7.2
Dual form			7.18.c.a.4.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-43.5867 + 75.4943i) q^{2} +(5626.01 + 9744.54i) q^{3} +(61736.4 + 106931. i) q^{4} +(-472563. + 818503. i) q^{5} -980877. q^{6} +(1.52373e7 + 674362. i) q^{7} -2.21895e7 q^{8} +(1.26603e6 - 2.19283e6i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 270 q^{2} + 6560 q^{3} - 551660 q^{4} + 1089798 q^{5} - 1687556 q^{6} + 14395360 q^{7} - 237823200 q^{8} - 498883228 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−43.5867	+	75.4943i	−0.120392	+	0.208526i	−0.919922	−	0.392100i	\(-0.871749\pi\)
							0.799530	+	0.600626i	\(0.205082\pi\)
\(3\)	5626.01	+	9744.54i	0.495074	+	0.857493i	0.999984	−	0.00567879i	\(-0.00180763\pi\)
							−0.504910	+	0.863172i	\(0.668474\pi\)
\(5\)	−472563.	+	818503.i	−0.541021	+	0.937077i	0.457824	+	0.889043i	\(0.348629\pi\)
							−0.998846	+	0.0480338i	\(0.984704\pi\)
\(7\)	1.52373e7	+	674362.i	0.999022	+	0.0442140i
							\(11\)	4.92773e7	+	8.53508e7i	0.0693121	+	0.120052i	0.898599	−	0.438771i	\(-0.144586\pi\)
−0.829287	+	0.558824i	\(0.811253\pi\)
\(13\)	−1.90989e9			−0.649367			−0.324684	−	0.945823i	\(-0.605258\pi\)
							−0.324684	+	0.945823i	\(0.605258\pi\)
\(17\)	−6.75628e9	−	1.17022e10i	−0.234905	−	0.406867i	0.724340	−	0.689443i	\(-0.242144\pi\)
							−0.959245	+	0.282576i	\(0.908811\pi\)
\(19\)	−4.54384e10	+	7.87017e10i	−0.613787	+	1.06311i	0.376809	+	0.926291i	\(0.377021\pi\)
							−0.990596	+	0.136820i	\(0.956312\pi\)
\(23\)	2.91834e11	−	5.05471e11i	0.777050	−	1.34589i	−0.156585	−	0.987664i	\(-0.550049\pi\)
							0.933635	−	0.358225i	\(-0.116618\pi\)
\(29\)	1.73142e12			0.642717			0.321358	−	0.946958i	\(-0.395861\pi\)
							0.321358	+	0.946958i	\(0.395861\pi\)
\(31\)	−2.04474e12	−	3.54159e12i	−0.430589	−	0.745802i	0.566335	−	0.824175i	\(-0.308361\pi\)
							−0.996924	+	0.0783733i	\(0.975027\pi\)
\(37\)	−1.95468e13	+	3.38561e13i	−0.914875	+	1.58461i	−0.107791	+	0.994174i	\(0.534378\pi\)
							−0.807084	+	0.590437i	\(0.798956\pi\)
\(41\)	−1.74354e13			−0.341012			−0.170506	−	0.985357i	\(-0.554540\pi\)
							−0.170506	+	0.985357i	\(0.554540\pi\)
\(43\)	1.09302e14			1.42608			0.713042	−	0.701121i	\(-0.247317\pi\)
							0.713042	+	0.701121i	\(0.247317\pi\)
\(47\)	9.14036e13	−	1.58316e14i	0.559927	−	0.969822i	−0.437575	−	0.899182i	\(-0.644163\pi\)
							0.997502	−	0.0706399i	\(-0.0225041\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7.18.c.a.2.5	✓	20
7.2	even	3		49.18.a.f.1.6		10
7.4	even	3	inner	7.18.c.a.4.5	yes	20
7.5	odd	6		49.18.a.g.1.6		10

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
7.18.c.a.2.5	✓	20	1.1	even	1	trivial
7.18.c.a.4.5	yes	20	7.4	even	3	inner
49.18.a.f.1.6		10	7.2	even	3
49.18.a.g.1.6		10	7.5	odd	6