Embedded newform 11.12.c.a.4.4

• Label: 11.12.c.a.4.4
• Level: $11$
• Weight: $12$
• Character: 11.4
• Analytic conductor: $8.452$
• Analytic rank: $0$
• Dimension: $40$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 11 \)
Weight:	\( k \)	\(=\)	\( 12 \)
Character orbit:	\([\chi]\)	\(=\)	11.c (of order \(5\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(8.45177498616\)
Analytic rank:	\(0\)
Dimension:	\(40\)
Relative dimension:	\(10\) over \(\Q(\zeta_{5})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{5}]$

Embedding invariants

Embedding label			4.4
Character	\(\chi\)	\(=\)	11.4
Dual form			11.12.c.a.3.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-32.2868 - 23.4578i) q^{2} +(-12.6213 + 38.8444i) q^{3} +(-140.693 - 433.009i) q^{4} +(-5225.04 + 3796.22i) q^{5} +(1318.70 - 958.095i) q^{6} +(-10421.9 - 32075.4i) q^{7} +(-30871.8 + 95013.6i) q^{8} +(141965. + 103144. i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q + 11 q^{2} - 276 q^{3} - 6137 q^{4} + 6038 q^{5} - 30371 q^{6} + 55440 q^{7} - 347809 q^{8} - 240812 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/11\mathbb{Z}\right)^\times\).

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{1}{5}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−32.2868	−	23.4578i	−0.713445	−	0.518348i	0.170838	−	0.985299i	\(-0.445353\pi\)
							−0.884283	+	0.466951i	\(0.845353\pi\)
\(3\)	−12.6213	+	38.8444i	−0.0299873	+	0.0922914i	−0.964930	−	0.262507i	\(-0.915451\pi\)
							0.934943	+	0.354799i	\(0.115451\pi\)
\(5\)	−5225.04	+	3796.22i	−0.747747	+	0.543270i	−0.895128	−	0.445810i	\(-0.852916\pi\)
							0.147380	+	0.989080i	\(0.452916\pi\)
\(7\)	−10421.9	−	32075.4i	−0.234374	−	0.721329i	−0.997204	−	0.0747296i	\(-0.976191\pi\)
							0.762830	−	0.646599i	\(-0.223809\pi\)
\(11\)	534118.	+	5398.75i	0.999949	+	0.0101073i
							\(13\)	1.51212e6	+	1.09862e6i	1.12953	+	0.820649i	0.985626	−	0.168943i	\(-0.0540353\pi\)
0.143901	+	0.989592i	\(0.454035\pi\)
\(17\)	−7.07508e6	+	5.14035e6i	−1.20854	+	0.878058i	−0.995098	−	0.0988967i	\(-0.968469\pi\)
							−0.213446	+	0.976955i	\(0.568469\pi\)
\(19\)	1.12278e6	−	3.45556e6i	0.104028	−	0.320165i	−0.885473	−	0.464691i	\(-0.846166\pi\)
							0.989501	+	0.144525i	\(0.0461656\pi\)
\(23\)	3.57846e6			0.115929			0.0579646	−	0.998319i	\(-0.481539\pi\)
							0.0579646	+	0.998319i	\(0.481539\pi\)
\(29\)	4.87068e7	+	1.49904e8i	0.440961	+	1.35714i	0.886853	+	0.462052i	\(0.152887\pi\)
							−0.445892	+	0.895087i	\(0.647113\pi\)
\(31\)	−7.75532e7	−	5.63457e7i	−0.486531	−	0.353486i	0.317318	−	0.948319i	\(-0.397218\pi\)
							−0.803849	+	0.594834i	\(0.797218\pi\)
\(37\)	1.22645e8	+	3.77462e8i	0.290763	+	0.894877i	0.984612	+	0.174756i	\(0.0559137\pi\)
							−0.693849	+	0.720121i	\(0.744086\pi\)
\(41\)	1.29621e8	−	3.98933e8i	0.174729	−	0.537760i	−0.824892	−	0.565290i	\(-0.808764\pi\)
							0.999621	+	0.0275300i	\(0.00876416\pi\)
\(43\)	−1.37574e9			−1.42712			−0.713559	−	0.700595i	\(-0.752918\pi\)
							−0.713559	+	0.700595i	\(0.752918\pi\)
\(47\)	1.45109e7	−	4.46598e7i	0.00922901	−	0.0284040i	−0.946336	−	0.323185i	\(-0.895247\pi\)
							0.955565	+	0.294781i	\(0.0952465\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	11.12.c.a.4.4	yes	40
3.2	odd	2		99.12.f.a.37.7		40
11.3	even	5	inner	11.12.c.a.3.4	✓	40
11.5	even	5		121.12.a.j.1.13		20
11.6	odd	10		121.12.a.h.1.8		20
33.14	odd	10		99.12.f.a.91.7		40

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
11.12.c.a.3.4	✓	40	11.3	even	5	inner
11.12.c.a.4.4	yes	40	1.1	even	1	trivial
99.12.f.a.37.7		40	3.2	odd	2
99.12.f.a.91.7		40	33.14	odd	10
121.12.a.h.1.8		20	11.6	odd	10
121.12.a.j.1.13		20	11.5	even	5