Embedded newform 10.9.c.b.3.1

• Label: 10.9.c.b.3.1
• Level: $10$
• Weight: $9$
• Character: 10.3
• Analytic conductor: $4.074$
• Analytic rank: $0$
• Dimension: $4$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 10 = 2 \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 9 \)
Character orbit:	\([\chi]\)	\(=\)	10.c (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.07378610061\)
Analytic rank:	\(0\)
Dimension:	\(4\)
Relative dimension:	\(2\) over \(\Q(i)\)
Coefficient field:	\(\Q(i, \sqrt{601})\)
Defining polynomial:	\( x^{4} + 301x^{2} + 22500 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index:	\( 2\cdot 5^{2} \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			3.1
Root			\(-11.7577i\) of defining polynomial
Character	\(\chi\)	\(=\)	10.3
Dual form			10.9.c.b.7.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(8.00000 - 8.00000i) q^{2} +(-39.7883 - 39.7883i) q^{3} -128.000i q^{4} +(-401.365 - 479.094i) q^{5} -636.612 q^{6} +(144.447 - 144.447i) q^{7} +(-1024.00 - 1024.00i) q^{8} -3394.79i q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 32 q^{2} + 86 q^{3} - 870 q^{5} + 1376 q^{6} + 5726 q^{7} - 4096 q^{8}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/10\mathbb{Z}\right)^\times\).

\(n\)	\(7\)
\(\chi(n)\)	\(e\left(\frac{3}{4}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	8.00000	−	8.00000i	0.500000	−	0.500000i
							\(3\)	−39.7883	−	39.7883i	−0.491213	−	0.491213i	0.417475	−	0.908688i	\(-0.362915\pi\)
−0.908688	+	0.417475i	\(0.862915\pi\)
\(5\)	−401.365	−	479.094i	−0.642184	−	0.766551i
							\(7\)	144.447	−	144.447i	0.0601610	−	0.0601610i	−0.676386	−	0.736547i	\(-0.736455\pi\)
0.736547	+	0.676386i	\(0.236455\pi\)
\(11\)	13600.3			0.928918			0.464459	−	0.885595i	\(-0.346249\pi\)
							0.464459	+	0.885595i	\(0.346249\pi\)
\(13\)	30883.7	+	30883.7i	1.08132	+	1.08132i	0.996386	+	0.0849366i	\(0.0270688\pi\)
							0.0849366	+	0.996386i	\(0.472931\pi\)
\(17\)	4949.02	−	4949.02i	0.0592549	−	0.0592549i	−0.676858	−	0.736113i	\(-0.736659\pi\)
							0.736113	+	0.676858i	\(0.236659\pi\)
\(19\)		−	176579.i		−	1.35495i	−0.735546	−	0.677475i	\(-0.763074\pi\)
							0.735546	−	0.677475i	\(-0.236926\pi\)
\(23\)	−229482.	−	229482.i	−0.820043	−	0.820043i	0.166071	−	0.986114i	\(-0.446892\pi\)
							−0.986114	+	0.166071i	\(0.946892\pi\)
\(29\)		−	144167.i		−	0.203833i	−0.994793	−	0.101917i	\(-0.967503\pi\)
							0.994793	−	0.101917i	\(-0.0324975\pi\)
\(31\)	1.50971e6			1.63473			0.817365	−	0.576120i	\(-0.195434\pi\)
							0.817365	+	0.576120i	\(0.195434\pi\)
\(37\)	−1.60715e6	+	1.60715e6i	−0.857532	+	0.857532i	−0.991047	−	0.133515i	\(-0.957374\pi\)
							0.133515	+	0.991047i	\(0.457374\pi\)
\(41\)	3.62276e6			1.28205			0.641023	−	0.767522i	\(-0.278510\pi\)
							0.641023	+	0.767522i	\(0.278510\pi\)
\(43\)	−575576.	−	575576.i	−0.168356	−	0.168356i	0.617900	−	0.786256i	\(-0.287983\pi\)
							−0.786256	+	0.617900i	\(0.787983\pi\)
\(47\)	3.11186e6	−	3.11186e6i	0.637718	−	0.637718i	−0.312274	−	0.949992i	\(-0.601091\pi\)
							0.949992	+	0.312274i	\(0.101091\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	10.9.c.b.3.1	✓	4
3.2	odd	2		90.9.g.a.73.2		4
4.3	odd	2		80.9.p.a.33.2		4
5.2	odd	4	inner	10.9.c.b.7.1	yes	4
5.3	odd	4		50.9.c.b.7.2		4
5.4	even	2		50.9.c.b.43.2		4
15.2	even	4		90.9.g.a.37.2		4
20.7	even	4		80.9.p.a.17.2		4

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
10.9.c.b.3.1	✓	4	1.1	even	1	trivial
10.9.c.b.7.1	yes	4	5.2	odd	4	inner
50.9.c.b.7.2		4	5.3	odd	4
50.9.c.b.43.2		4	5.4	even	2
80.9.p.a.17.2		4	20.7	even	4
80.9.p.a.33.2		4	4.3	odd	2
90.9.g.a.37.2		4	15.2	even	4
90.9.g.a.73.2		4	3.2	odd	2