Embedded newform 751.2.l.a.8.2

• Label: 751.2.l.a.8.2
• Level: $751$
• Weight: $2$
• Character: 751.8
• Analytic conductor: $5.997$
• Analytic rank: $0$
• Dimension: $6100$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 751 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	751.l (of order \(125\), degree \(100\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.99676519180\)
Analytic rank:	\(0\)
Dimension:	\(6100\)
Relative dimension:	\(61\) over \(\Q(\zeta_{125})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{125}]$

Embedding invariants

Embedding label			8.2
Character	\(\chi\)	\(=\)	751.8
Dual form			751.2.l.a.94.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.05161 - 1.74113i) q^{2} +(0.0379170 - 0.0669635i) q^{3} +(0.852323 + 5.17086i) q^{4} +(-1.47049 + 1.97137i) q^{5} +(-0.194383 + 0.0713650i) q^{6} +(1.51628 + 1.65589i) q^{7} +(4.48594 - 7.47768i) q^{8} +(1.54027 + 2.56750i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6100 q - 75 q^{2} - 100 q^{3} - 75 q^{4} - 100 q^{5} - 100 q^{6} - 100 q^{7} - 75 q^{8} - 100 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/751\mathbb{Z}\right)^\times\).

\(n\)	\(3\)
\(\chi(n)\)	\(e\left(\frac{83}{125}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.05161	−	1.74113i	−1.45071	−	1.23116i	−0.924707	−	0.380679i	\(-0.875690\pi\)
							−0.526003	−	0.850483i	\(-0.676310\pi\)
\(3\)	0.0379170	−	0.0669635i	0.0218914	−	0.0386614i	−0.859023	−	0.511936i	\(-0.828928\pi\)
							0.880915	+	0.473275i	\(0.156928\pi\)
\(5\)	−1.47049	+	1.97137i	−0.657622	+	0.881625i	−0.998284	−	0.0585607i	\(-0.981349\pi\)
							0.340662	+	0.940186i	\(0.389349\pi\)
\(7\)	1.51628	+	1.65589i	0.573099	+	0.625867i	0.954381	−	0.298592i	\(-0.0965171\pi\)
							−0.381281	+	0.924459i	\(0.624517\pi\)
\(11\)	−1.92847	+	1.81096i	−0.581457	+	0.546024i	−0.918478	−	0.395473i	\(-0.870581\pi\)
							0.337021	+	0.941497i	\(0.390581\pi\)
\(13\)	2.61303	+	3.89741i	0.724724	+	1.08095i	0.993059	+	0.117621i	\(0.0375267\pi\)
							−0.268335	+	0.963326i	\(0.586473\pi\)
\(17\)	−0.706283	−	5.07689i	−0.171299	−	1.23133i	−0.861518	−	0.507727i	\(-0.830486\pi\)
							0.690219	−	0.723600i	\(-0.257514\pi\)
\(19\)	−3.01182	+	0.457645i	−0.690958	+	0.104991i	−0.486234	−	0.873829i	\(-0.661630\pi\)
							−0.204724	+	0.978820i	\(0.565630\pi\)
\(23\)	1.71324	+	1.69184i	0.357235	+	0.352774i	0.858768	−	0.512365i	\(-0.171230\pi\)
							−0.501533	+	0.865139i	\(0.667230\pi\)
\(29\)	−4.62513	−	1.96706i	−0.858866	−	0.365275i	−0.0832869	−	0.996526i	\(-0.526542\pi\)
							−0.775579	+	0.631251i	\(0.782542\pi\)
\(31\)	−2.34461	+	0.539572i	−0.421105	+	0.0969101i	−0.429526	−	0.903054i	\(-0.641319\pi\)
							0.00842138	+	0.999965i	\(0.497319\pi\)
\(37\)	3.20548	−	9.08222i	0.526978	−	1.49311i	−0.312609	−	0.949882i	\(-0.601203\pi\)
							0.839587	−	0.543225i	\(-0.182797\pi\)
\(41\)	−5.26231	+	4.94164i	−0.821835	+	0.771755i	−0.976341	−	0.216237i	\(-0.930621\pi\)
							0.154506	+	0.987992i	\(0.450621\pi\)
\(43\)	−6.16163	+	8.70882i	−0.939639	+	1.32808i	0.00500540	+	0.999987i	\(0.498407\pi\)
							−0.944645	+	0.328095i	\(0.893593\pi\)
\(47\)	4.53490	+	1.53674i	0.661482	+	0.224157i	0.631479	−	0.775393i	\(-0.282448\pi\)
							0.0300034	+	0.999550i	\(0.490448\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	751.2.l.a.8.2	✓	6100
751.94	even	125	inner	751.2.l.a.94.2	yes	6100

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
751.2.l.a.8.2	✓	6100	1.1	even	1	trivial
751.2.l.a.94.2	yes	6100	751.94	even	125	inner