Embedded newform 310.2.w.a.117.10

• Label: 310.2.w.a.117.10
• Level: $310$
• Weight: $2$
• Character: 310.117
• Analytic conductor: $2.475$
• Analytic rank: $0$
• Dimension: $256$
• CM: no
• 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			117.10
Character	\(\chi\)	\(=\)	310.117
Dual form			310.2.w.a.53.10

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.453990 - 0.891007i) q^{2} +(-2.10518 - 1.36712i) q^{3} +(-0.587785 - 0.809017i) q^{4} +(-1.65969 + 1.49847i) q^{5} +(-2.17384 + 1.25507i) q^{6} +(1.57389 + 4.10013i) q^{7} +(-0.987688 + 0.156434i) q^{8} +(1.34255 + 3.01542i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 256 q - 24 q^{7}+O(q^{10}) \)

Character values

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

\(n\)	\(187\)	\(251\)
\(\chi(n)\)	\(e\left(\frac{1}{4}\right)\)	\(e\left(\frac{13}{30}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.453990	−	0.891007i	0.321020	−	0.630037i
							\(3\)	−2.10518	−	1.36712i	−1.21543	−	0.789306i	−0.232372	−	0.972627i	\(-0.574649\pi\)
−0.983053	+	0.183321i	\(0.941315\pi\)
\(5\)	−1.65969	+	1.49847i	−0.742238	+	0.670136i
							\(7\)	1.57389	+	4.10013i	0.594875	+	1.54970i	0.818185	+	0.574956i	\(0.194981\pi\)
−0.223309	+	0.974748i	\(0.571686\pi\)
\(11\)	−1.47638	+	0.155174i	−0.445145	+	0.0467866i	−0.324450	−	0.945903i	\(-0.605179\pi\)
							−0.120695	+	0.992690i	\(0.538512\pi\)
\(13\)	0.0397359	−	0.758207i	0.0110208	−	0.210289i	−0.987568	−	0.157189i	\(-0.949757\pi\)
							0.998589	−	0.0530993i	\(-0.0169100\pi\)
\(17\)	3.29823	+	4.07297i	0.799938	+	0.987841i	0.999968	+	0.00805124i	\(0.00256282\pi\)
							−0.200030	+	0.979790i	\(0.564104\pi\)
\(19\)	−4.43562	−	3.99385i	−1.01760	−	0.916252i	−0.0210954	−	0.999777i	\(-0.506715\pi\)
							−0.996506	+	0.0835254i	\(0.973382\pi\)
\(23\)	1.31161	+	8.28120i	0.273490	+	1.72675i	0.616445	+	0.787398i	\(0.288572\pi\)
							−0.342954	+	0.939352i	\(0.611428\pi\)
\(29\)	3.01883	+	9.29099i	0.560582	+	1.72529i	0.680727	+	0.732537i	\(0.261664\pi\)
							−0.120145	+	0.992756i	\(0.538336\pi\)
\(31\)	−5.46096	−	1.08530i	−0.980818	−	0.194926i
							\(37\)	−4.02472	−	1.07842i	−0.661659	−	0.177291i	−0.0876644	−	0.996150i	\(-0.527940\pi\)
−0.573995	+	0.818859i	\(0.694607\pi\)
\(41\)	−0.824797	−	0.175316i	−0.128812	−	0.0273798i	0.143055	−	0.989715i	\(-0.454308\pi\)
							−0.271866	+	0.962335i	\(0.587641\pi\)
\(43\)	−2.54636	+	0.133449i	−0.388316	+	0.0203508i	−0.245495	−	0.969398i	\(-0.578951\pi\)
							−0.142821	+	0.989749i	\(0.545617\pi\)
\(47\)	−6.45243	+	3.28768i	−0.941184	+	0.479557i	−0.856096	−	0.516816i	\(-0.827117\pi\)
							−0.0850874	+	0.996373i	\(0.527117\pi\)

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
310.2.w.a.53.10	✓	256	155.53	even	60	inner
310.2.w.a.117.10	yes	256	1.1	even	1	trivial
310.2.w.a.177.7	yes	256	31.22	odd	30	inner
310.2.w.a.303.7	yes	256	5.3	odd	4	inner