Embedded newform 11.14.c.a.3.10

• Label: 11.14.c.a.3.10
• Level: $11$
• Weight: $14$
• Character: 11.3
• Analytic conductor: $11.795$
• Analytic rank: $0$
• Dimension: $48$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 11 \)
Weight:	\( k \)	\(=\)	\( 14 \)
Character orbit:	\([\chi]\)	\(=\)	11.c (of order \(5\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(11.7954021847\)
Analytic rank:	\(0\)
Dimension:	\(48\)
Relative dimension:	\(12\) over \(\Q(\zeta_{5})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{5}]$

Embedding invariants

Embedding label			3.10
Character	\(\chi\)	\(=\)	11.3
Dual form			11.14.c.a.4.10

$q$-expansion

\(f(q)\)	\(=\)	\(q+(96.0816 - 69.8074i) q^{2} +(738.641 + 2273.30i) q^{3} +(1827.14 - 5623.35i) q^{4} +(-7207.48 - 5236.54i) q^{5} +(229663. + 166860. i) q^{6} +(-816.456 + 2512.79i) q^{7} +(83648.5 + 257444. i) q^{8} +(-3.33248e6 + 2.42119e6i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q + 59 q^{2} - 864 q^{3} - 44697 q^{4} - 52240 q^{5} - 130115 q^{6} + 390696 q^{7} + 876527 q^{8} - 3520244 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{4}{5}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	96.0816	−	69.8074i	1.06156	−	0.771270i	0.0871851	−	0.996192i	\(-0.472213\pi\)
							0.974377	+	0.224922i	\(0.0722128\pi\)
\(3\)	738.641	+	2273.30i	0.584986	+	1.80040i	0.599328	+	0.800503i	\(0.295434\pi\)
							−0.0143427	+	0.999897i	\(0.504566\pi\)
\(5\)	−7207.48	−	5236.54i	−0.206290	−	0.149879i	0.479843	−	0.877354i	\(-0.340693\pi\)
							−0.686134	+	0.727476i	\(0.740693\pi\)
\(7\)	−816.456	+	2512.79i	−0.00262298	+	0.00807272i	−0.952359	−	0.304978i	\(-0.901351\pi\)
							0.949736	+	0.313051i	\(0.101351\pi\)
\(11\)	1.29290e6	+	5.73159e6i	0.220045	+	0.975490i
							\(13\)	1.24940e7	−	9.07739e6i	0.717907	−	0.521590i	−0.167808	−	0.985820i	\(-0.553669\pi\)
0.885715	+	0.464230i	\(0.153669\pi\)
\(17\)	2.05570e6	+	1.49356e6i	0.0206558	+	0.0150073i	0.598065	−	0.801447i	\(-0.295936\pi\)
							−0.577409	+	0.816455i	\(0.695936\pi\)
\(19\)	−6.63791e7	−	2.04294e8i	−0.323693	−	0.996224i	−0.972027	−	0.234868i	\(-0.924534\pi\)
							0.648335	−	0.761356i	\(-0.275466\pi\)
\(23\)	1.08387e9			1.52667			0.763335	−	0.646002i	\(-0.223560\pi\)
							0.763335	+	0.646002i	\(0.223560\pi\)
\(29\)	−9.29556e8	+	2.86088e9i	−0.290194	+	0.893125i	0.694599	+	0.719397i	\(0.255582\pi\)
							−0.984794	+	0.173729i	\(0.944418\pi\)
\(31\)	6.34199e9	−	4.60773e9i	1.28344	−	0.932472i	0.283786	−	0.958888i	\(-0.408409\pi\)
							0.999651	+	0.0264161i	\(0.00840947\pi\)
\(37\)	1.26126e9	−	3.88176e9i	0.0808152	−	0.248724i	−0.902483	−	0.430726i	\(-0.858258\pi\)
							0.983298	+	0.182002i	\(0.0582577\pi\)
\(41\)	−1.24152e9	−	3.82100e9i	−0.0408186	−	0.125627i	0.928571	−	0.371156i	\(-0.121038\pi\)
							−0.969389	+	0.245529i	\(0.921038\pi\)
\(43\)	−6.97765e10			−1.68331			−0.841655	−	0.540016i	\(-0.818418\pi\)
							−0.841655	+	0.540016i	\(0.818418\pi\)
\(47\)	−5.12166e9	−	1.57629e10i	−0.0693067	−	0.213304i	0.910404	−	0.413720i	\(-0.135771\pi\)
							−0.979711	+	0.200416i	\(0.935771\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	11.14.c.a.3.10	✓	48
11.2	odd	10		121.14.a.i.1.20		24
11.4	even	5	inner	11.14.c.a.4.10	yes	48
11.9	even	5		121.14.a.h.1.5		24

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
11.14.c.a.3.10	✓	48	1.1	even	1	trivial
11.14.c.a.4.10	yes	48	11.4	even	5	inner
121.14.a.h.1.5		24	11.9	even	5
121.14.a.i.1.20		24	11.2	odd	10