Embedded newform 7.12.c.a.2.5

• Label: 7.12.c.a.2.5
• Level: $7$
• Weight: $12$
• Character: 7.2
• Analytic conductor: $5.378$
• Analytic rank: $0$
• Dimension: $12$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 7 \)
Weight:	\( k \)	\(=\)	\( 12 \)
Character orbit:	\([\chi]\)	\(=\)	7.c (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.37840226392\)
Analytic rank:	\(0\)
Dimension:	\(12\)
Relative dimension:	\(6\) over \(\Q(\zeta_{3})\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Defining polynomial:	x^12 - x^11 + 1846*x^10 + 9475*x^9 + 2735534*x^8 + 11305015*x^7 + 1247863105*x^6 + 8299724444*x^5 + 456730004116*x^4 + 1967575859568*x^3 + 7281700309008*x^2 + 5939576563968*x + 4089842896896
Coefficient ring:	\(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index:	\( 2^{20}\cdot 3^{3}\cdot 7^{6} \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			2.5
Root			\(-10.4934 + 18.1751i\) of defining polynomial
Character	\(\chi\)	\(=\)	7.2
Dual form			7.12.c.a.4.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(22.9868 - 39.8144i) q^{2} +(311.801 + 540.056i) q^{3} +(-32.7899 - 56.7937i) q^{4} +(-5294.67 + 9170.64i) q^{5} +28669.3 q^{6} +(23661.4 - 37649.3i) q^{7} +91139.2 q^{8} +(-105867. + 183367. i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 22 q^{2} - 244 q^{3} - 2556 q^{4} - 8782 q^{5} + 38140 q^{6} - 504 q^{7} + 97008 q^{8} - 172348 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	22.9868	−	39.8144i	0.507942	−	0.879782i	−0.492015	−	0.870586i	\(-0.663740\pi\)
							0.999958	−	0.00919535i	\(-0.00292701\pi\)
\(3\)	311.801	+	540.056i	0.740818	+	1.28313i	0.952124	+	0.305714i	\(0.0988950\pi\)
							−0.211306	+	0.977420i	\(0.567772\pi\)
\(5\)	−5294.67	+	9170.64i	−0.757712	+	1.31240i	0.186303	+	0.982492i	\(0.440349\pi\)
							−0.944015	+	0.329903i	\(0.892984\pi\)
\(7\)	23661.4	−	37649.3i	0.532109	−	0.846676i
							\(11\)	−340592.	−	589922.i	−0.637638	−	1.10442i	−0.985950	−	0.167042i	\(-0.946578\pi\)
0.348312	−	0.937379i	\(-0.386755\pi\)
\(13\)	821058.			0.613318			0.306659	−	0.951820i	\(-0.400789\pi\)
							0.306659	+	0.951820i	\(0.400789\pi\)
\(17\)	2.07057e6	+	3.58634e6i	0.353689	+	0.612608i	0.986893	−	0.161378i	\(-0.0515937\pi\)
							−0.633204	+	0.773985i	\(0.718260\pi\)
\(19\)	548471.	−	949979.i	0.0508170	−	0.0880176i	−0.839498	−	0.543363i	\(-0.817151\pi\)
							0.890315	+	0.455345i	\(0.150484\pi\)
\(23\)	5.52784e6	−	9.57451e6i	0.179082	−	0.310180i	−0.762484	−	0.647007i	\(-0.776020\pi\)
							0.941566	+	0.336827i	\(0.109354\pi\)
\(29\)	−1.24344e7			−0.112573			−0.0562866	−	0.998415i	\(-0.517926\pi\)
							−0.0562866	+	0.998415i	\(0.517926\pi\)
\(31\)	−1.22720e8	−	2.12556e8i	−0.769882	−	1.33348i	−0.937626	−	0.347645i	\(-0.886982\pi\)
							0.167744	−	0.985831i	\(-0.446352\pi\)
\(37\)	7.61936e6	−	1.31971e7i	0.0180638	−	0.0312874i	−0.856852	−	0.515562i	\(-0.827583\pi\)
							0.874916	+	0.484275i	\(0.160916\pi\)
\(41\)	−3.80543e8			−0.512971			−0.256485	−	0.966548i	\(-0.582565\pi\)
							−0.256485	+	0.966548i	\(0.582565\pi\)
\(43\)	9.14092e6			0.00948230			0.00474115	−	0.999989i	\(-0.498491\pi\)
							0.00474115	+	0.999989i	\(0.498491\pi\)
\(47\)	−7.65016e8	+	1.32505e9i	−0.486556	+	0.842739i	−0.999881	−	0.0154553i	\(-0.995080\pi\)
							0.513325	+	0.858194i	\(0.328414\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7.12.c.a.2.5	✓	12
3.2	odd	2		63.12.e.b.37.2		12
4.3	odd	2		112.12.i.c.65.1		12
7.2	even	3		49.12.a.g.1.2		6
7.3	odd	6		49.12.c.i.18.5		12
7.4	even	3	inner	7.12.c.a.4.5	yes	12
7.5	odd	6		49.12.a.f.1.2		6
7.6	odd	2		49.12.c.i.30.5		12
21.11	odd	6		63.12.e.b.46.2		12
28.11	odd	6		112.12.i.c.81.1		12

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
7.12.c.a.2.5	✓	12	1.1	even	1	trivial
7.12.c.a.4.5	yes	12	7.4	even	3	inner
49.12.a.f.1.2		6	7.5	odd	6
49.12.a.g.1.2		6	7.2	even	3
49.12.c.i.18.5		12	7.3	odd	6
49.12.c.i.30.5		12	7.6	odd	2
63.12.e.b.37.2		12	3.2	odd	2
63.12.e.b.46.2		12	21.11	odd	6
112.12.i.c.65.1		12	4.3	odd	2
112.12.i.c.81.1		12	28.11	odd	6