Embedded newform 300.2.n.a.191.7

• Label: 300.2.n.a.191.7
• Level: $300$
• Weight: $2$
• Character: 300.191
• Analytic conductor: $2.396$
• Analytic rank: $0$
• Dimension: $224$
• CM: no
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 300 = 2^{2} \cdot 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	300.n (of order \(10\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.39551206064\)
Analytic rank:	\(0\)
Dimension:	\(224\)
Relative dimension:	\(56\) over \(\Q(\zeta_{10})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{10}]$

Embedding invariants

Embedding label			191.7
Character	\(\chi\)	\(=\)	300.191
Dual form			300.2.n.a.11.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.29913 + 0.558792i) q^{2} +(1.06350 + 1.36711i) q^{3} +(1.37550 - 1.45189i) q^{4} +(1.51671 + 1.64305i) q^{5} +(-2.14555 - 1.18178i) q^{6} +1.88972i q^{7} +(-0.975659 + 2.65482i) q^{8} +(-0.737954 + 2.90782i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 224 q - 6 q^{4} + q^{6} - 6 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/300\mathbb{Z}\right)^\times\).

\(n\)	\(101\)	\(151\)	\(277\)
\(\chi(n)\)	\(-1\)	\(-1\)	\(e\left(\frac{1}{5}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.29913	+	0.558792i	−0.918627	+	0.395125i
							\(3\)	1.06350	+	1.36711i	0.614010	+	0.789299i
\(5\)	1.51671	+	1.64305i	0.678292	+	0.734793i
							\(7\)			1.88972i			0.714245i	0.934057	+	0.357123i	\(0.116242\pi\)
−0.934057	+	0.357123i	\(0.883758\pi\)
\(11\)	−3.00660	−	2.18442i	−0.906523	−	0.658628i	0.0336099	−	0.999435i	\(-0.489300\pi\)
							−0.940133	+	0.340807i	\(0.889300\pi\)
\(13\)	1.87647	−	1.36333i	0.520438	−	0.378120i	−0.296331	−	0.955085i	\(-0.595763\pi\)
							0.816769	+	0.576965i	\(0.195763\pi\)
\(17\)	5.42228	+	1.76180i	1.31510	+	0.427300i	0.880808	−	0.473474i	\(-0.157000\pi\)
							0.434287	+	0.900774i	\(0.357000\pi\)
\(19\)	0.894947	+	0.290786i	0.205315	+	0.0667108i	0.409869	−	0.912144i	\(-0.365574\pi\)
							−0.204554	+	0.978855i	\(0.565574\pi\)
\(23\)	−4.58028	−	3.32777i	−0.955054	−	0.693887i	−0.00305715	−	0.999995i	\(-0.500973\pi\)
							−0.951997	+	0.306108i	\(0.900973\pi\)
\(29\)	−4.43170	+	1.43995i	−0.822947	+	0.267392i	−0.690072	−	0.723741i	\(-0.742421\pi\)
							−0.132875	+	0.991133i	\(0.542421\pi\)
\(31\)	0.681127	+	0.221312i	0.122334	+	0.0397488i	0.369544	−	0.929213i	\(-0.379514\pi\)
							−0.247210	+	0.968962i	\(0.579514\pi\)
\(37\)	3.22240	−	2.34121i	0.529759	−	0.384893i	−0.290508	−	0.956872i	\(-0.593824\pi\)
							0.820268	+	0.571980i	\(0.193824\pi\)
\(41\)	−4.79706	−	6.60258i	−0.749174	−	1.03115i	−0.998038	−	0.0626117i	\(-0.980057\pi\)
							0.248863	−	0.968539i	\(-0.419943\pi\)
\(43\)			1.32683i			0.202340i	0.994869	+	0.101170i	\(0.0322586\pi\)
							−0.994869	+	0.101170i	\(0.967741\pi\)
\(47\)	2.90966	+	8.95502i	0.424418	+	1.30622i	0.903551	+	0.428481i	\(0.140951\pi\)
							−0.479133	+	0.877742i	\(0.659049\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	300.2.n.a.191.7	yes	224
3.2	odd	2	inner	300.2.n.a.191.50	yes	224
4.3	odd	2	inner	300.2.n.a.191.27	yes	224
12.11	even	2	inner	300.2.n.a.191.30	yes	224
25.11	even	5	inner	300.2.n.a.11.30	yes	224
75.11	odd	10	inner	300.2.n.a.11.27	yes	224
100.11	odd	10	inner	300.2.n.a.11.50	yes	224
300.11	even	10	inner	300.2.n.a.11.7	✓	224

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
300.2.n.a.11.7	✓	224	300.11	even	10	inner
300.2.n.a.11.27	yes	224	75.11	odd	10	inner
300.2.n.a.11.30	yes	224	25.11	even	5	inner
300.2.n.a.11.50	yes	224	100.11	odd	10	inner
300.2.n.a.191.7	yes	224	1.1	even	1	trivial
300.2.n.a.191.27	yes	224	4.3	odd	2	inner
300.2.n.a.191.30	yes	224	12.11	even	2	inner
300.2.n.a.191.50	yes	224	3.2	odd	2	inner