Embedded newform 507.2.x.b.2.5

• Label: 507.2.x.b.2.5
• Level: $507$
• Weight: $2$
• Character: 507.2
• Analytic conductor: $4.048$
• Analytic rank: $0$
• Dimension: $2784$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 507 = 3 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	507.x (of order \(156\), degree \(48\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.04841538248\)
Analytic rank:	\(0\)
Dimension:	\(2784\)
Relative dimension:	\(58\) over \(\Q(\zeta_{156})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{156}]$

Embedding invariants

Embedding label			2.5
Character	\(\chi\)	\(=\)	507.2
Dual form			507.2.x.b.254.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.0502666 + 2.49572i) q^{2} +(-1.68648 - 0.394699i) q^{3} +(-4.22771 - 0.170371i) q^{4} +(0.127318 - 0.694751i) q^{5} +(1.06983 - 4.18914i) q^{6} +(1.03440 - 1.56510i) q^{7} +(0.336274 - 5.55927i) q^{8} +(2.68843 + 1.33130i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2784 q - 50 q^{3} - 92 q^{4} - 50 q^{6} - 100 q^{7} - 56 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/507\mathbb{Z}\right)^\times\).

\(n\)	\(170\)	\(340\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{1}{156}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.0502666	+	2.49572i	−0.0355439	+	1.76474i	0.452353	+	0.891839i	\(0.350585\pi\)
							−0.487897	+	0.872901i	\(0.662236\pi\)
\(3\)	−1.68648	−	0.394699i	−0.973689	−	0.227879i
							\(5\)	0.127318	−	0.694751i	0.0569383	−	0.310702i	−0.942805	−	0.333344i	\(-0.891823\pi\)
0.999744	+	0.0226416i	\(0.00720765\pi\)
\(7\)	1.03440	−	1.56510i	0.390968	−	0.591551i	−0.584580	−	0.811336i	\(-0.698741\pi\)
							0.975548	+	0.219785i	\(0.0705356\pi\)
\(11\)	0.769753	+	0.423983i	0.232089	+	0.127836i	0.594251	−	0.804279i	\(-0.297448\pi\)
							−0.362162	+	0.932115i	\(0.617961\pi\)
\(13\)	2.68607	−	2.40521i	0.744982	−	0.667085i
							\(17\)	1.53945	+	7.54074i	0.373372	+	1.82890i	0.538498	+	0.842627i	\(0.318992\pi\)
−0.165126	+	0.986273i	\(0.552803\pi\)
\(19\)	−1.79517	−	6.69966i	−0.411840	−	1.53701i	−0.791082	−	0.611710i	\(-0.790482\pi\)
							0.379242	−	0.925298i	\(-0.376185\pi\)
\(23\)	1.01189	+	1.75264i	0.210993	+	0.365450i	0.952025	−	0.306019i	\(-0.0989970\pi\)
							−0.741033	+	0.671469i	\(0.765664\pi\)
\(29\)	−0.369966	+	0.355357i	−0.0687010	+	0.0659882i	−0.726289	−	0.687390i	\(-0.758756\pi\)
							0.657588	+	0.753378i	\(0.271577\pi\)
\(31\)	8.26255	+	3.71867i	1.48400	+	0.667893i	0.980575	−	0.196147i	\(-0.0628430\pi\)
							0.503423	+	0.864040i	\(0.332074\pi\)
\(37\)	−8.81196	+	0.890306i	−1.44868	+	0.146365i	−0.793252	−	0.608894i	\(-0.791614\pi\)
							−0.655426	+	0.755259i	\(0.727511\pi\)
\(41\)	2.52747	+	0.358673i	0.394724	+	0.0560154i	0.335045	−	0.942202i	\(-0.391249\pi\)
							0.0596794	+	0.998218i	\(0.480992\pi\)
\(43\)	2.92953	−	2.39189i	0.446749	−	0.364759i	−0.382204	−	0.924078i	\(-0.624835\pi\)
							0.828953	+	0.559319i	\(0.188937\pi\)
\(47\)	8.20533	−	2.55688i	1.19687	−	0.372960i	0.366631	−	0.930366i	\(-0.380511\pi\)
							0.830240	+	0.557406i	\(0.188203\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	507.2.x.b.2.5	✓	2784
3.2	odd	2	inner	507.2.x.b.2.54	yes	2784
169.85	odd	156	inner	507.2.x.b.254.54	yes	2784
507.254	even	156	inner	507.2.x.b.254.5	yes	2784

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
507.2.x.b.2.5	✓	2784	1.1	even	1	trivial
507.2.x.b.2.54	yes	2784	3.2	odd	2	inner
507.2.x.b.254.5	yes	2784	507.254	even	156	inner
507.2.x.b.254.54	yes	2784	169.85	odd	156	inner