Embedded newform 310.2.w.a.177.7

• Label: 310.2.w.a.177.7
• Level: $310$
• Weight: $2$
• Character: 310.177
• Analytic conductor: $2.475$
• Analytic rank: $0$
• Dimension: $256$
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 310 = 2 \cdot 5 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	310.w (of order \(60\), degree \(16\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.47536246266\)
Analytic rank:	\(0\)
Dimension:	\(256\)
Relative dimension:	\(16\) over \(\Q(\zeta_{60})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{60}]$

Embedding invariants

Embedding label			177.7
Character	\(\chi\)	\(=\)	310.177
Dual form			310.2.w.a.303.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.891007 + 0.453990i) q^{2} +(1.36712 + 2.10518i) q^{3} +(0.587785 - 0.809017i) q^{4} +(-0.467866 - 2.18657i) q^{5} +(-2.17384 - 1.25507i) q^{6} +(4.10013 + 1.57389i) q^{7} +(-0.156434 + 0.987688i) q^{8} +(-1.34255 + 3.01542i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 256 q - 24 q^{7}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/310\mathbb{Z}\right)^\times\).

\(n\)	\(187\)	\(251\)
\(\chi(n)\)	\(e\left(\frac{1}{4}\right)\)	\(e\left(\frac{17}{30}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.891007	+	0.453990i	−0.630037	+	0.321020i
							\(3\)	1.36712	+	2.10518i	0.789306	+	1.21543i	0.972627	+	0.232372i	\(0.0746487\pi\)
−0.183321	+	0.983053i	\(0.558685\pi\)
\(5\)	−0.467866	−	2.18657i	−0.209236	−	0.977865i
							\(7\)	4.10013	+	1.57389i	1.54970	+	0.594875i	0.974748	−	0.223309i	\(-0.0716859\pi\)
0.574956	+	0.818185i	\(0.305019\pi\)
\(11\)	−1.47638	−	0.155174i	−0.445145	−	0.0467866i	−0.120695	−	0.992690i	\(-0.538512\pi\)
							−0.324450	+	0.945903i	\(0.605179\pi\)
\(13\)	0.758207	−	0.0397359i	0.210289	−	0.0110208i	0.0530993	−	0.998589i	\(-0.483090\pi\)
							0.157189	+	0.987568i	\(0.449757\pi\)
\(17\)	4.07297	+	3.29823i	0.987841	+	0.799938i	0.979790	−	0.200030i	\(-0.0641039\pi\)
							0.00805124	+	0.999968i	\(0.497437\pi\)
\(19\)	4.43562	−	3.99385i	1.01760	−	0.916252i	0.0210954	−	0.999777i	\(-0.493285\pi\)
							0.996506	+	0.0835254i	\(0.0266180\pi\)
\(23\)	−8.28120	−	1.31161i	−1.72675	−	0.273490i	−0.787398	−	0.616445i	\(-0.788572\pi\)
							−0.939352	+	0.342954i	\(0.888572\pi\)
\(29\)	−3.01883	+	9.29099i	−0.560582	+	1.72529i	0.120145	+	0.992756i	\(0.461664\pi\)
							−0.680727	+	0.732537i	\(0.738336\pi\)
\(31\)	−5.46096	+	1.08530i	−0.980818	+	0.194926i
							\(37\)	−1.07842	−	4.02472i	−0.177291	−	0.661659i	−0.996150	−	0.0876644i	\(-0.972060\pi\)
0.818859	−	0.573995i	\(-0.194607\pi\)
\(41\)	−0.824797	+	0.175316i	−0.128812	+	0.0273798i	−0.271866	−	0.962335i	\(-0.587641\pi\)
							0.143055	+	0.989715i	\(0.454308\pi\)
\(43\)	−0.133449	+	2.54636i	−0.0203508	+	0.388316i	0.969398	+	0.245495i	\(0.0789505\pi\)
							−0.989749	+	0.142821i	\(0.954383\pi\)
\(47\)	3.28768	−	6.45243i	0.479557	−	0.941184i	−0.516816	−	0.856096i	\(-0.672883\pi\)
							0.996373	−	0.0850874i	\(-0.0271170\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	310.2.w.a.177.7	yes	256
5.3	odd	4	inner	310.2.w.a.53.10	✓	256
31.24	odd	30	inner	310.2.w.a.117.10	yes	256
155.148	even	60	inner	310.2.w.a.303.7	yes	256

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
310.2.w.a.53.10	✓	256	5.3	odd	4	inner
310.2.w.a.117.10	yes	256	31.24	odd	30	inner
310.2.w.a.177.7	yes	256	1.1	even	1	trivial
310.2.w.a.303.7	yes	256	155.148	even	60	inner