Embedded newform 349.2.e.a.123.7

• Label: 349.2.e.a.123.7
• Level: $349$
• Weight: $2$
• Character: 349.123
• Analytic conductor: $2.787$
• Analytic rank: $0$
• Dimension: $58$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 349 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	349.e (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.78677903054\)
Analytic rank:	\(0\)
Dimension:	\(58\)
Relative dimension:	\(29\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			123.7
Character	\(\chi\)	\(=\)	349.123
Dual form			349.2.e.a.227.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.34862 + 0.778626i) q^{2} +(1.56432 - 2.70948i) q^{3} +(0.212518 - 0.368092i) q^{4} +(-1.71886 + 2.97716i) q^{5} +4.87208i q^{6} +(0.0378395 - 0.0218466i) q^{7} -2.45262i q^{8} +(-3.39418 - 5.87890i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 58 q - 3 q^{2} + 27 q^{4} + 2 q^{5} - 29 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/349\mathbb{Z}\right)^\times\).

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{5}{6}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.34862	+	0.778626i	−0.953619	+	0.550572i	−0.894203	−	0.447661i	\(-0.852257\pi\)
							−0.0594154	+	0.998233i	\(0.518924\pi\)
\(3\)	1.56432	−	2.70948i	0.903159	−	1.56432i	0.0797905	−	0.996812i	\(-0.474575\pi\)
							0.823369	−	0.567506i	\(-0.192092\pi\)
\(5\)	−1.71886	+	2.97716i	−0.768698	+	1.33142i	0.169571	+	0.985518i	\(0.445762\pi\)
							−0.938269	+	0.345907i	\(0.887571\pi\)
\(7\)	0.0378395	−	0.0218466i	0.0143020	−	0.00825725i	−0.492832	−	0.870125i	\(-0.664038\pi\)
							0.507134	+	0.861867i	\(0.330705\pi\)
\(11\)		−	5.81943i		−	1.75462i	−0.479920	−	0.877312i	\(-0.659334\pi\)
							0.479920	−	0.877312i	\(-0.340666\pi\)
\(13\)	0.607751	−	0.350885i	0.168560	−	0.0973181i	−0.413347	−	0.910574i	\(-0.635640\pi\)
							0.581907	+	0.813256i	\(0.302307\pi\)
\(17\)	1.42655			0.345990			0.172995	−	0.984923i	\(-0.444656\pi\)
							0.172995	+	0.984923i	\(0.444656\pi\)
\(19\)	2.16840	−	3.75578i	0.497466	−	0.861636i	−0.502530	−	0.864560i	\(-0.667597\pi\)
							0.999996	+	0.00292405i	\(0.000930756\pi\)
\(23\)	−3.24640	−	5.62292i	−0.676920	−	1.17246i	−0.975904	−	0.218202i	\(-0.929981\pi\)
							0.298983	−	0.954258i	\(-0.403352\pi\)
\(29\)	0.358257	−	0.620519i	0.0665266	−	0.115227i	−0.830844	−	0.556506i	\(-0.812142\pi\)
							0.897370	+	0.441279i	\(0.145475\pi\)
\(31\)	8.37278			1.50380			0.751898	−	0.659280i	\(-0.229139\pi\)
							0.751898	+	0.659280i	\(0.229139\pi\)
\(37\)	0.874031			0.143690			0.0718449	−	0.997416i	\(-0.477111\pi\)
							0.0718449	+	0.997416i	\(0.477111\pi\)
\(41\)	−10.8746			−1.69833			−0.849165	−	0.528128i	\(-0.822894\pi\)
							−0.849165	+	0.528128i	\(0.822894\pi\)
\(43\)	−3.09729	−	1.78822i	−0.472332	−	0.272701i	0.244884	−	0.969552i	\(-0.421250\pi\)
							−0.717215	+	0.696852i	\(0.754584\pi\)
\(47\)			7.39390i			1.07851i	0.842142	+	0.539255i	\(0.181294\pi\)
							−0.842142	+	0.539255i	\(0.818706\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	349.2.e.a.123.7	✓	58
349.227	even	6	inner	349.2.e.a.227.7	yes	58

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
349.2.e.a.123.7	✓	58	1.1	even	1	trivial
349.2.e.a.227.7	yes	58	349.227	even	6	inner