Embedded newform 7.12.c.a.2.1

• Label: 7.12.c.a.2.1
• 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			2.1
Root			\(19.2454 - 33.3340i\) of defining polynomial
Character	\(\chi\)	\(=\)	7.2
Dual form			7.12.c.a.4.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-36.4908 + 63.2040i) q^{2} +(-167.747 - 290.547i) q^{3} +(-1639.16 - 2839.11i) q^{4} +(-539.162 + 933.856i) q^{5} +24484.9 q^{6} +(44465.4 - 388.777i) q^{7} +89790.9 q^{8} +(32295.3 - 55937.0i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 22 q^{2} - 244 q^{3} - 2556 q^{4} - 8782 q^{5} + 38140 q^{6} - 504 q^{7} + 97008 q^{8} - 172348 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−36.4908	+	63.2040i	−0.806341	+	1.39662i	0.109041	+	0.994037i	\(0.465222\pi\)
							−0.915382	+	0.402586i	\(0.868111\pi\)
\(3\)	−167.747	−	290.547i	−0.398555	−	0.690318i	0.594993	−	0.803731i	\(-0.297155\pi\)
							−0.993548	+	0.113413i	\(0.963822\pi\)
\(5\)	−539.162	+	933.856i	−0.0771585	+	0.133643i	−0.902023	−	0.431688i	\(-0.857918\pi\)
							0.824864	+	0.565331i	\(0.191251\pi\)
\(7\)	44465.4	−	388.777i	0.999962	−	0.00874301i
							\(11\)	−413574.	−	716332.i	−0.774273	−	1.34108i	−0.935202	−	0.354114i	\(-0.884783\pi\)
0.160930	−	0.986966i	\(-0.448551\pi\)
\(13\)	1.32230e6			0.987735			0.493867	−	0.869537i	\(-0.335583\pi\)
							0.493867	+	0.869537i	\(0.335583\pi\)
\(17\)	−4.99342e6	−	8.64886e6i	−0.852961	−	1.47737i	−0.878523	−	0.477699i	\(-0.841471\pi\)
							0.0255621	−	0.999673i	\(-0.491862\pi\)
\(19\)	−2.37271e6	+	4.10965e6i	−0.219836	+	0.380768i	−0.954758	−	0.297385i	\(-0.903886\pi\)
							0.734921	+	0.678152i	\(0.237219\pi\)
\(23\)	1.93826e6	−	3.35717e6i	0.0627928	−	0.108760i	−0.832920	−	0.553393i	\(-0.813333\pi\)
							0.895713	+	0.444633i	\(0.146666\pi\)
\(29\)	1.61895e7			0.146569			0.0732847	−	0.997311i	\(-0.476652\pi\)
							0.0732847	+	0.997311i	\(0.476652\pi\)
\(31\)	−8.44561e7	−	1.46282e8i	−0.529836	−	0.917703i	−0.999394	−	0.0348015i	\(-0.988920\pi\)
							0.469558	−	0.882902i	\(-0.344413\pi\)
\(37\)	2.41761e8	−	4.18743e8i	0.573162	−	0.992746i	−0.423076	−	0.906094i	\(-0.639050\pi\)
							0.996239	−	0.0866522i	\(-0.0276169\pi\)
\(41\)	−3.92340e8			−0.528873			−0.264437	−	0.964403i	\(-0.585186\pi\)
							−0.264437	+	0.964403i	\(0.585186\pi\)
\(43\)	−3.00642e8			−0.311870			−0.155935	−	0.987767i	\(-0.549839\pi\)
							−0.155935	+	0.987767i	\(0.549839\pi\)
\(47\)	9.27181e8	−	1.60592e9i	0.589694	−	1.02138i	−0.404579	−	0.914503i	\(-0.632582\pi\)
							0.994272	−	0.106876i	\(-0.0340848\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7.12.c.a.2.1	✓	12
3.2	odd	2		63.12.e.b.37.6		12
4.3	odd	2		112.12.i.c.65.5		12
7.2	even	3		49.12.a.g.1.6		6
7.3	odd	6		49.12.c.i.18.1		12
7.4	even	3	inner	7.12.c.a.4.1	yes	12
7.5	odd	6		49.12.a.f.1.6		6
7.6	odd	2		49.12.c.i.30.1		12
21.11	odd	6		63.12.e.b.46.6		12
28.11	odd	6		112.12.i.c.81.5		12

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
7.12.c.a.2.1	✓	12	1.1	even	1	trivial
7.12.c.a.4.1	yes	12	7.4	even	3	inner
49.12.a.f.1.6		6	7.5	odd	6
49.12.a.g.1.6		6	7.2	even	3
49.12.c.i.18.1		12	7.3	odd	6
49.12.c.i.30.1		12	7.6	odd	2
63.12.e.b.37.6		12	3.2	odd	2
63.12.e.b.46.6		12	21.11	odd	6
112.12.i.c.65.5		12	4.3	odd	2
112.12.i.c.81.5		12	28.11	odd	6