Embedded newform 29.6.e.a.22.7

• Label: 29.6.e.a.22.7
• Level: $29$
• Weight: $6$
• Character: 29.22
• Analytic conductor: $4.651$
• Analytic rank: $0$
• Dimension: $72$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 29 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	29.e (of order \(14\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.65113077458\)
Analytic rank:	\(0\)
Dimension:	\(72\)
Relative dimension:	\(12\) over \(\Q(\zeta_{14})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{14}]$

Embedding invariants

Embedding label			22.7
Character	\(\chi\)	\(=\)	29.22
Dual form			29.6.e.a.4.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.435057 - 0.0992990i) q^{2} +(6.82610 - 14.1745i) q^{3} +(-28.6516 + 13.7979i) q^{4} +(-13.0294 - 57.0855i) q^{5} +(1.56223 - 6.84456i) q^{6} +(-84.1454 - 40.5223i) q^{7} +(-22.2594 + 17.7513i) q^{8} +(-2.81388 - 3.52849i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 72 q - 7 q^{2} - 7 q^{3} + 165 q^{4} - 53 q^{5} - 311 q^{6} - 27 q^{7} - 1498 q^{8} + 1567 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/29\mathbb{Z}\right)^\times\).

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{13}{14}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.435057	−	0.0992990i	0.0769080	−	0.0175538i	−0.183894	−	0.982946i	\(-0.558870\pi\)
							0.260802	+	0.965392i	\(0.416013\pi\)
\(3\)	6.82610	−	14.1745i	0.437894	−	0.909297i	−0.558896	−	0.829238i	\(-0.688775\pi\)
							0.996790	−	0.0800591i	\(-0.0255109\pi\)
\(5\)	−13.0294	−	57.0855i	−0.233077	−	1.02118i	−0.947071	−	0.321024i	\(-0.895973\pi\)
							0.713994	−	0.700152i	\(-0.246884\pi\)
\(7\)	−84.1454	−	40.5223i	−0.649060	−	0.312571i	0.0802290	−	0.996776i	\(-0.474435\pi\)
							−0.729289	+	0.684205i	\(0.760149\pi\)
\(11\)	−463.074	−	369.289i	−1.15390	−	0.920205i	−0.156182	−	0.987728i	\(-0.549919\pi\)
							−0.997718	+	0.0675235i	\(0.978490\pi\)
\(13\)	682.917	−	856.350i	1.12075	−	1.40538i	0.217609	−	0.976036i	\(-0.430174\pi\)
							0.903142	−	0.429342i	\(-0.141254\pi\)
\(17\)			1247.48i			1.04692i	0.852051	+	0.523459i	\(0.175359\pi\)
							−0.852051	+	0.523459i	\(0.824641\pi\)
\(19\)	1051.17	+	2182.78i	0.668020	+	1.38716i	0.909070	+	0.416644i	\(0.136794\pi\)
							−0.241050	+	0.970513i	\(0.577492\pi\)
\(23\)	301.154	−	1319.44i	0.118705	−	0.520080i	−0.880256	−	0.474500i	\(-0.842629\pi\)
							0.998961	−	0.0455808i	\(-0.0145138\pi\)
\(29\)	−1255.97	−	4351.28i	−0.277323	−	0.960777i
							\(31\)	−1912.85	+	436.596i	−0.357501	+	0.0815973i	−0.397499	−	0.917602i	\(-0.630122\pi\)
0.0399982	+	0.999200i	\(0.487265\pi\)
\(37\)	−1201.46	+	958.132i	−0.144280	+	0.115059i	−0.692970	−	0.720967i	\(-0.743698\pi\)
							0.548690	+	0.836026i	\(0.315127\pi\)
\(41\)		−	12669.8i		−	1.17709i	−0.808464	−	0.588545i	\(-0.799701\pi\)
							0.808464	−	0.588545i	\(-0.200299\pi\)
\(43\)	−7790.27	−	1778.08i	−0.642512	−	0.146649i	−0.111163	−	0.993802i	\(-0.535458\pi\)
							−0.531349	+	0.847153i	\(0.678315\pi\)
\(47\)	−240.649	−	191.911i	−0.0158905	−	0.0126723i	0.615512	−	0.788127i	\(-0.288949\pi\)
							−0.631403	+	0.775455i	\(0.717521\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	29.6.e.a.22.7	yes	72
29.2	odd	28		841.6.a.l.1.38		72
29.4	even	14	inner	29.6.e.a.4.7	✓	72
29.27	odd	28		841.6.a.l.1.35		72

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
29.6.e.a.4.7	✓	72	29.4	even	14	inner
29.6.e.a.22.7	yes	72	1.1	even	1	trivial
841.6.a.l.1.35		72	29.27	odd	28
841.6.a.l.1.38		72	29.2	odd	28