Embedded newform 950.2.r.b.11.25

• Label: 950.2.r.b.11.25
• Level: $950$
• Weight: $2$
• Character: 950.11
• Analytic conductor: $7.586$
• Analytic rank: $0$
• Dimension: $200$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 950 = 2 \cdot 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	950.r (of order \(15\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(7.58578819202\)
Analytic rank:	\(0\)
Dimension:	\(200\)
Relative dimension:	\(25\) over \(\Q(\zeta_{15})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{15}]$

Embedding invariants

Embedding label			11.25
Character	\(\chi\)	\(=\)	950.11
Dual form			950.2.r.b.691.25

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.104528 - 0.994522i) q^{2} +(2.12066 + 2.35523i) q^{3} +(-0.978148 + 0.207912i) q^{4} +(-1.19464 + 1.89020i) q^{5} +(2.12066 - 2.35523i) q^{6} -4.64425 q^{7} +(0.309017 + 0.951057i) q^{8} +(-0.736326 + 7.00568i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 200 q + 25 q^{2} + 25 q^{4} + q^{5} + 28 q^{7} - 50 q^{8} + 13 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/950\mathbb{Z}\right)^\times\).

\(n\)	\(77\)	\(401\)
\(\chi(n)\)	\(e\left(\frac{4}{5}\right)\)	\(e\left(\frac{2}{3}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.104528	−	0.994522i	−0.0739128	−	0.703233i
							\(3\)	2.12066	+	2.35523i	1.22436	+	1.35979i	0.912192	+	0.409764i	\(0.134389\pi\)
0.312169	+	0.950027i	\(0.398944\pi\)
\(5\)	−1.19464	+	1.89020i	−0.534257	+	0.845322i
							\(7\)	−4.64425			−1.75536			−0.877680	−	0.479247i	\(-0.840910\pi\)
−0.877680	+	0.479247i	\(0.840910\pi\)
\(11\)	1.47946	−	1.07489i	0.446075	−	0.324092i	−0.341969	−	0.939711i	\(-0.611094\pi\)
							0.788044	+	0.615619i	\(0.211094\pi\)
\(13\)	−0.0544061	+	0.517640i	−0.0150895	+	0.143567i	−0.999473	−	0.0324721i	\(-0.989662\pi\)
							0.984383	+	0.176040i	\(0.0563287\pi\)
\(17\)	0.146176	+	0.0310707i	0.0354529	+	0.00753575i	0.225604	−	0.974219i	\(-0.427564\pi\)
							−0.190151	+	0.981755i	\(0.560898\pi\)
\(19\)	0.547363	−	4.32440i	0.125574	−	0.992084i
							\(23\)	−2.60724	−	1.16082i	−0.543648	−	0.242048i	0.116490	−	0.993192i	\(-0.462836\pi\)
−0.660138	+	0.751144i	\(0.729502\pi\)
\(29\)	−9.74158	+	2.07064i	−1.80897	+	0.384508i	−0.983640	−	0.180147i	\(-0.942343\pi\)
							−0.825327	+	0.564655i	\(0.809009\pi\)
\(31\)	0.664045	+	2.04372i	0.119266	+	0.367063i	0.992813	−	0.119677i	\(-0.0381859\pi\)
							−0.873547	+	0.486740i	\(0.838186\pi\)
\(37\)	7.26100	+	5.27542i	1.19370	+	0.867274i	0.993650	−	0.112511i	\(-0.0358894\pi\)
							0.200050	+	0.979786i	\(0.435889\pi\)
\(41\)	0.509670	−	0.226920i	0.0795971	−	0.0354389i	−0.366552	−	0.930398i	\(-0.619462\pi\)
							0.446149	+	0.894959i	\(0.352795\pi\)
\(43\)	−4.77073	+	8.26315i	−0.727530	+	1.26012i	0.230394	+	0.973097i	\(0.425998\pi\)
							−0.957924	+	0.287021i	\(0.907335\pi\)
\(47\)	−7.74366	+	1.64597i	−1.12953	+	0.240089i	−0.734519	−	0.678588i	\(-0.762592\pi\)
							−0.395010	+	0.918677i	\(0.629259\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	950.2.r.b.11.25	✓	200
19.7	even	3	inner	950.2.r.b.311.1	yes	200
25.16	even	5	inner	950.2.r.b.391.1	yes	200
475.216	even	15	inner	950.2.r.b.691.25	yes	200

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
950.2.r.b.11.25	✓	200	1.1	even	1	trivial
950.2.r.b.311.1	yes	200	19.7	even	3	inner
950.2.r.b.391.1	yes	200	25.16	even	5	inner
950.2.r.b.691.25	yes	200	475.216	even	15	inner