Embedded newform 111.3.l.b.82.2

• Label: 111.3.l.b.82.2
• Level: $111$
• Weight: $3$
• Character: 111.82
• Analytic conductor: $3.025$
• Analytic rank: $0$
• Dimension: $24$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 111 = 3 \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	111.l (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.02453093440\)
Analytic rank:	\(0\)
Dimension:	\(24\)
Relative dimension:	\(6\) over \(\Q(\zeta_{12})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label			82.2
Character	\(\chi\)	\(=\)	111.82
Dual form			111.3.l.b.88.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.51039 - 0.672656i) q^{2} +(1.50000 + 0.866025i) q^{3} +(2.38548 + 1.37726i) q^{4} +(-2.86514 + 0.767711i) q^{5} +(-3.18304 - 3.18304i) q^{6} +(4.00759 - 6.94135i) q^{7} +(2.28887 + 2.28887i) q^{8} +(1.50000 + 2.59808i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - 4 q^{2} + 36 q^{3} + 18 q^{4} + 8 q^{5} - 6 q^{6} - 36 q^{8} + 36 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/111\mathbb{Z}\right)^\times\).

\(n\)	\(38\)	\(76\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{1}{12}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.51039	−	0.672656i	−1.25519	−	0.336328i	−0.430853	−	0.902422i	\(-0.641787\pi\)
							−0.824341	+	0.566094i	\(0.808454\pi\)
\(3\)	1.50000	+	0.866025i	0.500000	+	0.288675i
							\(5\)	−2.86514	+	0.767711i	−0.573028	+	0.153542i	−0.533685	−	0.845683i	\(-0.679193\pi\)
−0.0393428	+	0.999226i	\(0.512526\pi\)
\(7\)	4.00759	−	6.94135i	0.572513	−	0.991622i	−0.423794	−	0.905759i	\(-0.639302\pi\)
							0.996307	−	0.0858633i	\(-0.0273648\pi\)
\(11\)			1.33278i			0.121162i	0.998163	+	0.0605808i	\(0.0192953\pi\)
							−0.998163	+	0.0605808i	\(0.980705\pi\)
\(13\)	16.6263	−	4.45501i	1.27895	−	0.342693i	0.445495	−	0.895284i	\(-0.353028\pi\)
							0.833452	+	0.552592i	\(0.186361\pi\)
\(17\)	6.93909	−	25.8970i	0.408182	−	1.52336i	−0.389929	−	0.920845i	\(-0.627500\pi\)
							0.798111	−	0.602511i	\(-0.205833\pi\)
\(19\)	29.7902	−	7.98225i	1.56790	−	0.420119i	0.632749	−	0.774357i	\(-0.281927\pi\)
							0.935155	+	0.354239i	\(0.115260\pi\)
\(23\)	5.77984	+	5.77984i	0.251297	+	0.251297i	0.821502	−	0.570205i	\(-0.193136\pi\)
							−0.570205	+	0.821502i	\(0.693136\pi\)
\(29\)	10.5071	−	10.5071i	0.362316	−	0.362316i	−0.502349	−	0.864665i	\(-0.667531\pi\)
							0.864665	+	0.502349i	\(0.167531\pi\)
\(31\)	−10.1956	+	10.1956i	−0.328890	+	0.328890i	−0.852164	−	0.523274i	\(-0.824710\pi\)
							0.523274	+	0.852164i	\(0.324710\pi\)
\(37\)	−29.9155	+	21.7729i	−0.808528	+	0.588458i
							\(41\)	−9.03829	−	5.21826i	−0.220446	−	0.127275i	0.385711	−	0.922620i	\(-0.373956\pi\)
−0.606157	+	0.795345i	\(0.707290\pi\)
\(43\)	27.8062	+	27.8062i	0.646656	+	0.646656i	0.952183	−	0.305527i	\(-0.0988327\pi\)
							−0.305527	+	0.952183i	\(0.598833\pi\)
\(47\)	−41.4628			−0.882188			−0.441094	−	0.897461i	\(-0.645410\pi\)
							−0.441094	+	0.897461i	\(0.645410\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	111.3.l.b.82.2	✓	24
3.2	odd	2		333.3.bb.b.82.5		24
37.14	odd	12	inner	111.3.l.b.88.2	yes	24
111.14	even	12		333.3.bb.b.199.5		24

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
111.3.l.b.82.2	✓	24	1.1	even	1	trivial
111.3.l.b.88.2	yes	24	37.14	odd	12	inner
333.3.bb.b.82.5		24	3.2	odd	2
333.3.bb.b.199.5		24	111.14	even	12