Embedded newform 224.3.v.b.69.13

• Label: 224.3.v.b.69.13
• Level: $224$
• Weight: $3$
• Character: 224.69
• Analytic conductor: $6.104$
• Analytic rank: $0$
• Dimension: $240$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 224 = 2^{5} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	224.v (of order \(8\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(6.10355792167\)
Analytic rank:	\(0\)
Dimension:	\(240\)
Relative dimension:	\(60\) over \(\Q(\zeta_{8})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{8}]$

Embedding invariants

Embedding label			69.13
Character	\(\chi\)	\(=\)	224.69
Dual form			224.3.v.b.13.13

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.46638 - 1.36005i) q^{2} +(-1.65755 + 0.686579i) q^{3} +(0.300514 + 3.98870i) q^{4} +(0.205220 - 0.495445i) q^{5} +(3.36437 + 1.24757i) q^{6} +(-0.193440 - 6.99733i) q^{7} +(4.98417 - 6.25764i) q^{8} +(-4.08789 + 4.08789i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 240 q - 8 q^{2} - 8 q^{4} - 4 q^{7} - 8 q^{8} - 8 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/224\mathbb{Z}\right)^\times\).

\(n\)	\(127\)	\(129\)	\(197\)
\(\chi(n)\)	\(1\)	\(-1\)	\(e\left(\frac{1}{8}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.46638	−	1.36005i	−0.733188	−	0.680026i
							\(3\)	−1.65755	+	0.686579i	−0.552516	+	0.228860i	−0.641433	−	0.767179i	\(-0.721660\pi\)
0.0889164	+	0.996039i	\(0.471660\pi\)
\(5\)	0.205220	−	0.495445i	0.0410440	−	0.0990890i	−0.902028	−	0.431677i	\(-0.857922\pi\)
							0.943072	+	0.332588i	\(0.107922\pi\)
\(7\)	−0.193440	−	6.99733i	−0.0276343	−	0.999618i
							\(11\)	−3.44988	+	8.32874i	−0.313625	+	0.757158i	0.685940	+	0.727658i	\(0.259391\pi\)
−0.999565	+	0.0294996i	\(0.990609\pi\)
\(13\)	−1.31375	−	3.17168i	−0.101058	−	0.243976i	0.865262	−	0.501321i	\(-0.167152\pi\)
							−0.966320	+	0.257345i	\(0.917152\pi\)
\(17\)	28.1178			1.65399			0.826995	−	0.562210i	\(-0.190049\pi\)
							0.826995	+	0.562210i	\(0.190049\pi\)
\(19\)	12.1154	+	29.2491i	0.637651	+	1.53943i	0.829801	+	0.558060i	\(0.188454\pi\)
							−0.192149	+	0.981366i	\(0.561546\pi\)
\(23\)	−11.0067	+	11.0067i	−0.478553	+	0.478553i	−0.904669	−	0.426115i	\(-0.859882\pi\)
							0.426115	+	0.904669i	\(0.359882\pi\)
\(29\)	6.55469	+	15.8244i	0.226024	+	0.545670i	0.995687	−	0.0927808i	\(-0.0295756\pi\)
							−0.769663	+	0.638451i	\(0.779576\pi\)
\(31\)		−	31.5457i		−	1.01760i	−0.860883	−	0.508802i	\(-0.830088\pi\)
							0.860883	−	0.508802i	\(-0.169912\pi\)
\(37\)	39.9500	+	16.5478i	1.07973	+	0.447238i	0.850414	−	0.526114i	\(-0.176352\pi\)
							0.229314	+	0.973352i	\(0.426352\pi\)
\(41\)	53.7822	+	53.7822i	1.31176	+	1.31176i	0.920116	+	0.391645i	\(0.128094\pi\)
							0.391645	+	0.920116i	\(0.371906\pi\)
\(43\)	8.17552	−	19.7375i	0.190128	−	0.459011i	−0.799855	−	0.600193i	\(-0.795090\pi\)
							0.989984	+	0.141182i	\(0.0450904\pi\)
\(47\)	−30.9029			−0.657508			−0.328754	−	0.944416i	\(-0.606629\pi\)
							−0.328754	+	0.944416i	\(0.606629\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	224.3.v.b.69.13	yes	240
7.6	odd	2	inner	224.3.v.b.69.14	yes	240
32.13	even	8	inner	224.3.v.b.13.14	yes	240
224.13	odd	8	inner	224.3.v.b.13.13	✓	240

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
224.3.v.b.13.13	✓	240	224.13	odd	8	inner
224.3.v.b.13.14	yes	240	32.13	even	8	inner
224.3.v.b.69.13	yes	240	1.1	even	1	trivial
224.3.v.b.69.14	yes	240	7.6	odd	2	inner