Embedded newform 950.2.bi.a.117.35

• Label: 950.2.bi.a.117.35
• Level: $950$
• Weight: $2$
• Character: 950.117
• Analytic conductor: $7.586$
• Analytic rank: $0$
• Dimension: $2400$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 950 = 2 \cdot 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	950.bi (of order \(180\), degree \(48\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(7.58578819202\)
Analytic rank:	\(0\)
Dimension:	\(2400\)
Relative dimension:	\(50\) over \(\Q(\zeta_{180})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{180}]$

Embedding invariants

Embedding label			117.35
Character	\(\chi\)	\(=\)	950.117
Dual form			950.2.bi.a.203.35

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.390731 + 0.920505i) q^{2} +(-1.09904 + 0.213631i) q^{3} +(-0.694658 + 0.719340i) q^{4} +(1.58129 + 1.58099i) q^{5} +(-0.626075 - 0.928195i) q^{6} +(-1.16483 + 4.34720i) q^{7} +(-0.933580 - 0.358368i) q^{8} +(-1.61931 + 0.654244i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2400 q + 12 q^{7}+O(q^{10}) \)

Character values

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

\(n\)	\(77\)	\(401\)
\(\chi(n)\)	\(e\left(\frac{13}{20}\right)\)	\(e\left(\frac{13}{18}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.390731	+	0.920505i	0.276289	+	0.650895i
							\(3\)	−1.09904	+	0.213631i	−0.634528	+	0.123340i	−0.497832	−	0.867273i	\(-0.665871\pi\)
−0.136696	+	0.990613i	\(0.543648\pi\)
\(5\)	1.58129	+	1.58099i	0.707173	+	0.707040i
							\(7\)	−1.16483	+	4.34720i	−0.440264	+	1.64309i	0.287882	+	0.957666i	\(0.407049\pi\)
−0.728146	+	0.685422i	\(0.759618\pi\)
\(11\)	−0.164230	−	0.0349081i	−0.0495171	−	0.0105252i	0.183087	−	0.983097i	\(-0.441391\pi\)
							−0.232604	+	0.972572i	\(0.574724\pi\)
\(13\)	−3.94160	+	0.483968i	−1.09320	+	0.134229i	−0.648326	−	0.761363i	\(-0.724531\pi\)
							−0.444878	+	0.895591i	\(0.646753\pi\)
\(17\)	3.61409	−	6.01485i	0.876545	−	1.45882i	−0.0119497	−	0.999929i	\(-0.503804\pi\)
							0.888495	−	0.458887i	\(-0.151752\pi\)
\(19\)	0.268465	+	4.35062i	0.0615902	+	0.998102i
							\(23\)	1.26852	−	4.14914i	0.264504	−	0.865155i	−0.720623	−	0.693327i	\(-0.756144\pi\)
0.985127	−	0.171827i	\(-0.0549671\pi\)
\(29\)	0.135035	−	0.541595i	0.0250754	−	0.100572i	−0.956454	−	0.291883i	\(-0.905718\pi\)
							0.981529	+	0.191311i	\(0.0612739\pi\)
\(31\)	−5.53120	−	0.581353i	−0.993433	−	0.104414i	−0.406159	−	0.913802i	\(-0.633132\pi\)
							−0.587274	+	0.809388i	\(0.699799\pi\)
\(37\)	−0.541401	−	1.06256i	−0.0890058	−	0.174684i	0.842212	−	0.539146i	\(-0.181253\pi\)
							−0.931218	+	0.364462i	\(0.881253\pi\)
\(41\)	−3.66158	+	0.127865i	−0.571843	+	0.0199692i	−0.319362	−	0.947633i	\(-0.603469\pi\)
							−0.252481	+	0.967602i	\(0.581246\pi\)
\(43\)	6.71656	+	9.59224i	1.02427	+	1.46280i	0.880999	+	0.473119i	\(0.156872\pi\)
							0.143267	+	0.989684i	\(0.454239\pi\)
\(47\)	5.05458	−	3.03710i	0.737286	−	0.443006i	−0.0963310	−	0.995349i	\(-0.530711\pi\)
							0.833617	+	0.552343i	\(0.186266\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	950.2.bi.a.117.35	yes	2400
19.13	odd	18	inner	950.2.bi.a.317.16	yes	2400
25.3	odd	20	inner	950.2.bi.a.3.16	✓	2400
475.203	even	180	inner	950.2.bi.a.203.35	yes	2400

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
950.2.bi.a.3.16	✓	2400	25.3	odd	20	inner
950.2.bi.a.117.35	yes	2400	1.1	even	1	trivial
950.2.bi.a.203.35	yes	2400	475.203	even	180	inner
950.2.bi.a.317.16	yes	2400	19.13	odd	18	inner