Embedded newform 315.2.z.b.104.4

• Label: 315.2.z.b.104.4
• Level: $315$
• Weight: $2$
• Character: 315.104
• Analytic conductor: $2.515$
• Analytic rank: $0$
• Dimension: $80$
• CM: no
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 315 = 3^{2} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	315.z (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.51528766367\)
Analytic rank:	\(0\)
Dimension:	\(80\)
Relative dimension:	\(40\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			104.4
Character	\(\chi\)	\(=\)	315.104
Dual form			315.2.z.b.209.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.19580 - 2.07119i) q^{2} +(0.122865 - 1.72769i) q^{3} +(-1.85987 + 3.22139i) q^{4} +(-0.810771 + 2.08390i) q^{5} +(-3.72528 + 1.81149i) q^{6} +(-2.57040 + 0.626932i) q^{7} +4.11294 q^{8} +(-2.96981 - 0.424546i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 80 q - 52 q^{4} - 4 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/315\mathbb{Z}\right)^\times\).

\(n\)	\(127\)	\(136\)	\(281\)
\(\chi(n)\)	\(-1\)	\(-1\)	\(e\left(\frac{5}{6}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.19580	−	2.07119i	−0.845558	−	1.46455i	−0.885136	−	0.465333i	\(-0.845935\pi\)
							0.0395778	−	0.999216i	\(-0.487399\pi\)
\(3\)	0.122865	−	1.72769i	0.0709364	−	0.997481i
							\(5\)	−0.810771	+	2.08390i	−0.362588	+	0.931950i
\(7\)	−2.57040	+	0.626932i	−0.971520	+	0.236958i
							\(11\)	3.27627	−	1.89156i	0.987833	−	0.570326i	0.0832072	−	0.996532i	\(-0.473484\pi\)
0.904626	+	0.426207i	\(0.140150\pi\)
\(13\)	−2.95115	+	5.11155i	−0.818503	+	1.41769i	0.0882821	+	0.996096i	\(0.471862\pi\)
							−0.906785	+	0.421593i	\(0.861471\pi\)
\(17\)			1.69241i			0.410470i	0.978713	+	0.205235i	\(0.0657958\pi\)
							−0.978713	+	0.205235i	\(0.934204\pi\)
\(19\)			4.29448i			0.985222i	0.870250	+	0.492611i	\(0.163957\pi\)
							−0.870250	+	0.492611i	\(0.836043\pi\)
\(23\)	3.30241	−	5.71995i	0.688601	−	1.19269i	−0.283690	−	0.958916i	\(-0.591559\pi\)
							0.972291	−	0.233775i	\(-0.0751080\pi\)
\(29\)	−6.21072	+	3.58576i	−1.15330	+	0.665859i	−0.949690	−	0.313192i	\(-0.898602\pi\)
							−0.203613	+	0.979052i	\(0.565268\pi\)
\(31\)	−3.85236	−	2.22416i	−0.691904	−	0.399471i	0.112421	−	0.993661i	\(-0.464140\pi\)
							−0.804325	+	0.594190i	\(0.797473\pi\)
\(37\)			4.73927i			0.779132i	0.920999	+	0.389566i	\(0.127375\pi\)
							−0.920999	+	0.389566i	\(0.872625\pi\)
\(41\)	−0.696581	+	1.20651i	−0.108788	+	0.188426i	−0.915279	−	0.402820i	\(-0.868030\pi\)
							0.806492	+	0.591245i	\(0.201364\pi\)
\(43\)	−4.60768	+	2.66025i	−0.702665	+	0.405684i	−0.808339	−	0.588717i	\(-0.799633\pi\)
							0.105674	+	0.994401i	\(0.466300\pi\)
\(47\)	−8.81141	+	5.08727i	−1.28528	+	0.742055i	−0.977808	−	0.209503i	\(-0.932815\pi\)
							−0.307469	+	0.951558i	\(0.599482\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	315.2.z.b.104.4	yes	80
3.2	odd	2		945.2.z.b.314.38		80
5.4	even	2	inner	315.2.z.b.104.37	yes	80
7.6	odd	2	inner	315.2.z.b.104.3	✓	80
9.2	odd	6	inner	315.2.z.b.209.38	yes	80
9.7	even	3		945.2.z.b.629.3		80
15.14	odd	2		945.2.z.b.314.4		80
21.20	even	2		945.2.z.b.314.37		80
35.34	odd	2	inner	315.2.z.b.104.38	yes	80
45.29	odd	6	inner	315.2.z.b.209.3	yes	80
45.34	even	6		945.2.z.b.629.37		80
63.20	even	6	inner	315.2.z.b.209.37	yes	80
63.34	odd	6		945.2.z.b.629.4		80
105.104	even	2		945.2.z.b.314.3		80
315.34	odd	6		945.2.z.b.629.38		80
315.209	even	6	inner	315.2.z.b.209.4	yes	80

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
315.2.z.b.104.3	✓	80	7.6	odd	2	inner
315.2.z.b.104.4	yes	80	1.1	even	1	trivial
315.2.z.b.104.37	yes	80	5.4	even	2	inner
315.2.z.b.104.38	yes	80	35.34	odd	2	inner
315.2.z.b.209.3	yes	80	45.29	odd	6	inner
315.2.z.b.209.4	yes	80	315.209	even	6	inner
315.2.z.b.209.37	yes	80	63.20	even	6	inner
315.2.z.b.209.38	yes	80	9.2	odd	6	inner
945.2.z.b.314.3		80	105.104	even	2
945.2.z.b.314.4		80	15.14	odd	2
945.2.z.b.314.37		80	21.20	even	2
945.2.z.b.314.38		80	3.2	odd	2
945.2.z.b.629.3		80	9.7	even	3
945.2.z.b.629.4		80	63.34	odd	6
945.2.z.b.629.37		80	45.34	even	6
945.2.z.b.629.38		80	315.34	odd	6