Embedded newform 430.2.x.a.3.14

• Label: 430.2.x.a.3.14
• Level: $430$
• Weight: $2$
• Character: 430.3
• Analytic conductor: $3.434$
• Analytic rank: $0$
• Dimension: $528$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 430 = 2 \cdot 5 \cdot 43 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	430.x (of order \(84\), degree \(24\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.43356728692\)
Analytic rank:	\(0\)
Dimension:	\(528\)
Relative dimension:	\(22\) over \(\Q(\zeta_{84})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{84}]$

Embedding invariants

Embedding label			3.14
Character	\(\chi\)	\(=\)	430.3
Dual form			430.2.x.a.287.14

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.330279 + 0.943883i) q^{2} +(-1.26939 - 1.47505i) q^{3} +(-0.781831 + 0.623490i) q^{4} +(-0.345539 + 2.20921i) q^{5} +(0.973028 - 1.68533i) q^{6} +(-0.0286434 + 0.106899i) q^{7} +(-0.846724 - 0.532032i) q^{8} +(-0.117316 + 0.778339i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 528 q - 4 q^{6} - 12 q^{7}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/430\mathbb{Z}\right)^\times\).

\(n\)	\(87\)	\(261\)
\(\chi(n)\)	\(e\left(\frac{3}{4}\right)\)	\(e\left(\frac{1}{42}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.330279	+	0.943883i	0.233543	+	0.667426i
							\(3\)	−1.26939	−	1.47505i	−0.732881	−	0.851623i	0.260621	−	0.965441i	\(-0.416073\pi\)
−0.993502	+	0.113818i	\(0.963692\pi\)
\(5\)	−0.345539	+	2.20921i	−0.154530	+	0.987988i
							\(7\)	−0.0286434	+	0.106899i	−0.0108262	+	0.0404039i	−0.971128	−	0.238560i	\(-0.923324\pi\)
0.960301	+	0.278964i	\(0.0899912\pi\)
\(11\)	−0.175744	+	0.220376i	−0.0529887	+	0.0664458i	−0.807620	−	0.589704i	\(-0.799245\pi\)
							0.754631	+	0.656150i	\(0.227816\pi\)
\(13\)	−5.40356	−	0.202187i	−1.49868	−	0.0560766i	−0.724601	−	0.689168i	\(-0.757976\pi\)
							−0.774077	+	0.633092i	\(0.781786\pi\)
\(17\)	−2.89301	+	5.47385i	−0.701658	+	1.32760i	0.233147	+	0.972442i	\(0.425098\pi\)
							−0.934805	+	0.355161i	\(0.884426\pi\)
\(19\)	−3.36064	+	0.506534i	−0.770983	+	0.116207i	−0.522741	−	0.852491i	\(-0.675091\pi\)
							−0.248241	+	0.968698i	\(0.579853\pi\)
\(23\)	0.459756	−	1.05377i	0.0958657	−	0.219727i	−0.862021	−	0.506872i	\(-0.830802\pi\)
							0.957887	+	0.287146i	\(0.0927064\pi\)
\(29\)	0.214288	+	2.85948i	0.0397923	+	0.530992i	0.981108	+	0.193463i	\(0.0619719\pi\)
							−0.941315	+	0.337529i	\(0.890409\pi\)
\(31\)	2.28210	−	1.55591i	0.409876	−	0.279449i	−0.340778	−	0.940144i	\(-0.610691\pi\)
							0.750655	+	0.660695i	\(0.229738\pi\)
\(37\)	−3.03461	+	0.813123i	−0.498888	+	0.133677i	−0.499484	−	0.866323i	\(-0.666477\pi\)
							0.000596106		1.00000i	\(0.499810\pi\)
\(41\)	−9.71099	−	4.67656i	−1.51660	−	0.730357i	−0.523993	−	0.851722i	\(-0.675558\pi\)
							−0.992608	+	0.121366i	\(0.961273\pi\)
\(43\)	−5.10522	+	4.11542i	−0.778539	+	0.627596i
							\(47\)	4.32386	+	0.487182i	0.630700	+	0.0710629i	0.421527	−	0.906816i	\(-0.361494\pi\)
0.209173	+	0.977879i	\(0.432923\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	430.2.x.a.3.14	✓	528
5.2	odd	4	inner	430.2.x.a.347.3	yes	528
43.29	odd	42	inner	430.2.x.a.373.3	yes	528
215.72	even	84	inner	430.2.x.a.287.14	yes	528

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
430.2.x.a.3.14	✓	528	1.1	even	1	trivial
430.2.x.a.287.14	yes	528	215.72	even	84	inner
430.2.x.a.347.3	yes	528	5.2	odd	4	inner
430.2.x.a.373.3	yes	528	43.29	odd	42	inner