Embedded newform 605.2.o.a.34.8

• Label: 605.2.o.a.34.8
• Level: $605$
• Weight: $2$
• Character: 605.34
• Analytic conductor: $4.831$
• Analytic rank: $0$
• Dimension: $640$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 605 = 5 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	605.o (of order \(22\), degree \(10\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.83094932229\)
Analytic rank:	\(0\)
Dimension:	\(640\)
Relative dimension:	\(64\) over \(\Q(\zeta_{22})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{22}]$

Embedding invariants

Embedding label			34.8
Character	\(\chi\)	\(=\)	605.34
Dual form			605.2.o.a.89.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.21266 + 0.318132i) q^{2} +2.50454i q^{3} +(2.87566 - 0.844370i) q^{4} +(-1.55949 + 1.60249i) q^{5} +(-0.796776 - 5.54170i) q^{6} +(-1.48610 + 2.31242i) q^{7} +(-2.02743 + 0.925896i) q^{8} -3.27274 q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 640 q + 44 q^{4} - 7 q^{5} + 14 q^{6} - 644 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/605\mathbb{Z}\right)^\times\).

\(n\)	\(122\)	\(486\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{5}{11}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.21266	+	0.318132i	−1.56459	+	0.224953i	−0.869485	−	0.493959i	\(-0.835549\pi\)
							−0.695100	+	0.718913i	\(0.744640\pi\)
\(3\)			2.50454i			1.44600i	0.690848	+	0.723000i	\(0.257237\pi\)
							−0.690848	+	0.723000i	\(0.742763\pi\)
\(5\)	−1.55949	+	1.60249i	−0.697426	+	0.716657i
							\(7\)	−1.48610	+	2.31242i	−0.561693	+	0.874012i	−0.999688	−	0.0249654i	\(-0.992052\pi\)
0.437995	+	0.898977i	\(0.355689\pi\)
\(11\)	3.10190	+	1.17398i	0.935257	+	0.353970i
							\(13\)	1.70624	+	5.81092i	0.473226	+	1.61166i	0.757445	+	0.652899i	\(0.226448\pi\)
−0.284219	+	0.958759i	\(0.591734\pi\)
\(17\)	−1.43389	+	1.24247i	−0.347769	+	0.301344i	−0.811176	−	0.584802i	\(-0.801172\pi\)
							0.463407	+	0.886146i	\(0.346627\pi\)
\(19\)	−4.63401	+	5.34793i	−1.06312	+	1.22690i	−0.0901547	+	0.995928i	\(0.528736\pi\)
							−0.972960	+	0.230972i	\(0.925809\pi\)
\(23\)	−1.99799	−	3.10893i	−0.416609	−	0.648257i	0.568000	−	0.823029i	\(-0.307718\pi\)
							−0.984609	+	0.174772i	\(0.944081\pi\)
\(29\)	−0.452588	+	0.522315i	−0.0840436	+	0.0969914i	−0.796214	−	0.605015i	\(-0.793167\pi\)
							0.712170	+	0.702007i	\(0.247712\pi\)
\(31\)	10.0046	+	2.93762i	1.79688	+	0.527612i	0.997333	−	0.0729879i	\(-0.0232534\pi\)
							0.799550	+	0.600600i	\(0.205072\pi\)
\(37\)	−0.216309	+	0.736681i	−0.0355610	+	0.121110i	−0.975361	−	0.220616i	\(-0.929193\pi\)
							0.939800	+	0.341726i	\(0.111011\pi\)
\(41\)	0.717811	+	4.99249i	0.112103	+	0.779695i	0.965868	+	0.259036i	\(0.0834048\pi\)
							−0.853764	+	0.520660i	\(0.825686\pi\)
\(43\)	7.58059	−	3.46194i	1.15603	−	0.527941i	0.257251	−	0.966345i	\(-0.417183\pi\)
							0.898778	+	0.438404i	\(0.144456\pi\)
\(47\)	−3.90735	−	0.561792i	−0.569945	−	0.0819457i	−0.148683	−	0.988885i	\(-0.547504\pi\)
							−0.421262	+	0.906939i	\(0.638413\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	605.2.o.a.34.8	✓	640
5.4	even	2	inner	605.2.o.a.34.57	yes	640
121.89	even	11	inner	605.2.o.a.89.57	yes	640
605.89	even	22	inner	605.2.o.a.89.8	yes	640

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
605.2.o.a.34.8	✓	640	1.1	even	1	trivial
605.2.o.a.34.57	yes	640	5.4	even	2	inner
605.2.o.a.89.8	yes	640	605.89	even	22	inner
605.2.o.a.89.57	yes	640	121.89	even	11	inner