Embedded newform 121.3.h.a.2.5

• Label: 121.3.h.a.2.5
• Level: $121$
• Weight: $3$
• Character: 121.2
• Analytic conductor: $3.297$
• Analytic rank: $0$
• Dimension: $840$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 121 = 11^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	121.h (of order \(110\), degree \(40\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.29701119876\)
Analytic rank:	\(0\)
Dimension:	\(840\)
Relative dimension:	\(21\) over \(\Q(\zeta_{110})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{110}]$

Embedding invariants

Embedding label			2.5
Character	\(\chi\)	\(=\)	121.2
Dual form			121.3.h.a.61.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-3.00938 - 0.0859711i) q^{2} +(-0.0156601 + 0.0113778i) q^{3} +(5.05550 + 0.289084i) q^{4} +(0.517797 - 0.858670i) q^{5} +(0.0481054 - 0.0328937i) q^{6} +(-3.09405 - 0.626936i) q^{7} +(-3.19080 - 0.274058i) q^{8} +(-2.78104 + 8.55915i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 840 q - 39 q^{2} - 34 q^{3} - 75 q^{4} - 43 q^{5} - 15 q^{6} - 54 q^{7} - 59 q^{8} - 594 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/121\mathbb{Z}\right)^\times\).

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{1}{110}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−3.00938	−	0.0859711i	−1.50469	−	0.0429855i	−0.733543	−	0.679643i	\(-0.762135\pi\)
							−0.771147	+	0.636657i	\(0.780317\pi\)
\(3\)	−0.0156601	+	0.0113778i	−0.00522005	+	0.00379258i	−0.590392	−	0.807116i	\(-0.701027\pi\)
							0.585172	+	0.810909i	\(0.301027\pi\)
\(5\)	0.517797	−	0.858670i	0.103559	−	0.171734i	−0.800253	−	0.599662i	\(-0.795302\pi\)
							0.903813	+	0.427928i	\(0.140756\pi\)
\(7\)	−3.09405	−	0.626936i	−0.442007	−	0.0895623i	−0.0275280	−	0.999621i	\(-0.508764\pi\)
							−0.414479	+	0.910059i	\(0.636036\pi\)
\(11\)	−1.89134	−	10.8362i	−0.171940	−	0.985107i
							\(13\)	18.9598	−	4.98366i	1.45845	−	0.383358i	0.562232	−	0.826979i	\(-0.309943\pi\)
0.896214	+	0.443621i	\(0.146306\pi\)
\(17\)	−3.85898	−	22.2989i	−0.226999	−	1.31170i	−0.848879	−	0.528588i	\(-0.822722\pi\)
							0.621880	−	0.783112i	\(-0.286369\pi\)
\(19\)	23.8798	−	23.2074i	1.25683	−	1.22144i	0.292646	−	0.956221i	\(-0.405464\pi\)
							0.964188	−	0.265220i	\(-0.0854447\pi\)
\(23\)	11.2994	−	24.7422i	0.491278	−	1.07575i	−0.487928	−	0.872884i	\(-0.662247\pi\)
							0.979207	−	0.202866i	\(-0.0650254\pi\)
\(29\)	−26.2629	−	13.8575i	−0.905616	−	0.477844i	−0.0519405	−	0.998650i	\(-0.516541\pi\)
							−0.853675	+	0.520806i	\(0.825632\pi\)
\(31\)	7.92300	−	6.47861i	0.255581	−	0.208987i	−0.496538	−	0.868015i	\(-0.665396\pi\)
							0.752119	+	0.659028i	\(0.229032\pi\)
\(37\)	9.33858	+	23.9859i	0.252394	+	0.648269i	0.999872	−	0.0160182i	\(-0.00509898\pi\)
							−0.747478	+	0.664287i	\(0.768735\pi\)
\(41\)	−14.7692	−	11.3887i	−0.360223	−	0.277774i	0.414532	−	0.910035i	\(-0.363945\pi\)
							−0.774756	+	0.632261i	\(0.782127\pi\)
\(43\)	20.2803	+	17.5730i	0.471636	+	0.408675i	0.857983	−	0.513679i	\(-0.171718\pi\)
							−0.386347	+	0.922354i	\(0.626263\pi\)
\(47\)	−55.9927	+	19.9782i	−1.19133	+	0.425067i	−0.855972	−	0.517022i	\(-0.827040\pi\)
							−0.335362	+	0.942089i	\(0.608859\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	121.3.h.a.2.5	✓	840
121.61	odd	110	inner	121.3.h.a.61.5	yes	840

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
121.3.h.a.2.5	✓	840	1.1	even	1	trivial
121.3.h.a.61.5	yes	840	121.61	odd	110	inner