Embedded newform 432.2.v.a.251.7

• Label: 432.2.v.a.251.7
• Level: $432$
• Weight: $2$
• Character: 432.251
• Analytic conductor: $3.450$
• Analytic rank: $0$
• Dimension: $88$
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 432 = 2^{4} \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	432.v (of order \(12\), degree \(4\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.44953736732\)
Analytic rank:	\(0\)
Dimension:	\(88\)
Relative dimension:	\(22\) over \(\Q(\zeta_{12})\)
Twist minimal:	no (minimal twist has level 144)
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label			251.7
Character	\(\chi\)	\(=\)	432.251
Dual form			432.2.v.a.179.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.717593 - 1.21863i) q^{2} +(-0.970121 + 1.74896i) q^{4} +(-1.20583 - 0.323102i) q^{5} +(0.140266 - 0.242948i) q^{7} +(2.82749 - 0.0728225i) q^{8} +(0.471556 + 1.70132i) q^{10} +(3.07444 - 0.823794i) q^{11} +(2.76539 + 0.740984i) q^{13} +(-0.396717 + 0.00340517i) q^{14} +(-2.11773 - 3.39341i) q^{16} -3.72031i q^{17} +(-4.10860 - 4.10860i) q^{19} +(1.73490 - 1.79551i) q^{20} +(-3.21010 - 3.15546i) q^{22} +(1.57595 - 0.909876i) q^{23} +(-2.98049 - 1.72078i) q^{25} +(-1.08144 - 3.90172i) q^{26} +(0.288831 + 0.481009i) q^{28} +(3.83006 - 1.02626i) q^{29} +(8.81101 - 5.08704i) q^{31} +(-2.61564 + 5.01582i) q^{32} +(-4.53369 + 2.66967i) q^{34} +(-0.247635 + 0.247635i) q^{35} +(1.76964 + 1.76964i) q^{37} +(-2.05856 + 7.95516i) q^{38} +(-3.43301 - 0.825757i) q^{40} +(-2.66819 - 4.62144i) q^{41} +(-1.84748 - 6.89490i) q^{43} +(-1.54180 + 6.17626i) q^{44} +(-2.23969 - 1.26758i) q^{46} +(5.48486 - 9.50006i) q^{47} +(3.46065 + 5.99402i) q^{49} +(0.0417747 + 4.86693i) q^{50} +(-3.97872 + 4.11772i) q^{52} +(-8.58403 + 8.58403i) q^{53} -3.97344 q^{55} +(0.378909 - 0.697147i) q^{56} +(-3.99906 - 3.93099i) q^{58} +(-1.44299 + 5.38532i) q^{59} +(1.66977 + 6.23168i) q^{61} +(-12.5219 - 7.08695i) q^{62} +(7.98939 - 0.411810i) q^{64} +(-3.09519 - 1.78701i) q^{65} +(-1.00652 + 3.75640i) q^{67} +(6.50668 + 3.60915i) q^{68} +(0.479476 + 0.124074i) q^{70} -10.6808i q^{71} +9.30419i q^{73} +(0.886658 - 3.42643i) q^{74} +(11.1716 - 3.19994i) q^{76} +(0.231101 - 0.862479i) q^{77} +(8.70990 + 5.02866i) q^{79} +(1.45721 + 4.77613i) q^{80} +(-3.71715 + 6.56784i) q^{82} +(-0.588691 - 2.19703i) q^{83} +(-1.20204 + 4.48608i) q^{85} +(-7.07659 + 7.19913i) q^{86} +(8.63296 - 2.55316i) q^{88} +1.87637 q^{89} +(0.567911 - 0.567911i) q^{91} +(0.0624738 + 3.63897i) q^{92} +(-15.5130 + 0.133153i) q^{94} +(3.62679 + 6.28179i) q^{95} +(9.19070 - 15.9188i) q^{97} +(4.82116 - 8.51852i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	88 * q + 6 * q^2 - 2 * q^4 + 6 * q^5 - 4 * q^7 - 8 * q^10 + 6 * q^11 - 2 * q^13 + 6 * q^14 - 2 * q^16 - 8 * q^19 + 48 * q^20 - 2 * q^22 + 12 * q^23 + 8 * q^28 + 6 * q^29 + 6 * q^32 + 2 * q^34 - 8 * q^37 + 6 * q^38 - 2 * q^40 - 2 * q^43 - 40 * q^46 - 24 * q^49 - 72 * q^50 - 2 * q^52 - 16 * q^55 - 36 * q^56 + 16 * q^58 + 42 * q^59 - 2 * q^61 - 44 * q^64 + 12 * q^65 - 2 * q^67 - 96 * q^68 - 16 * q^70 - 78 * q^74 - 14 * q^76 + 6 * q^77 - 36 * q^82 - 54 * q^83 + 8 * q^85 - 54 * q^86 + 22 * q^88 + 20 * q^91 - 108 * q^92 + 6 * q^94 - 4 * q^97

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/432\mathbb{Z}\right)^\times\).

\(n\)	\(271\)	\(325\)	\(353\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{1}{4}\right)\)	\(e\left(\frac{1}{6}\right)\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\). You can download additional coefficients here.

Currently showing only \(a_p\); display all \(a_n\)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.717593	−	1.21863i	−0.507415	−	0.861702i
							\(3\)		0			0
\(5\)	−1.20583	−	0.323102i	−0.539266	−	0.144496i								−0.0211020	−	0.999777i	\(-0.506717\pi\)
							−0.518164	+	0.855282i	\(0.673384\pi\)
\(7\)	0.140266	−	0.242948i	0.0530156	−	0.0918256i	−0.838300	−	0.545210i	\(-0.816450\pi\)
							0.891315	+	0.453384i	\(0.149783\pi\)
\(11\)	3.07444	−	0.823794i	0.926979	−	0.248383i	0.236413	−	0.971653i	\(-0.424028\pi\)
							0.690566	+	0.723269i	\(0.257361\pi\)
\(13\)	2.76539	+	0.740984i	0.766982	+	0.205512i	0.621038	−	0.783781i	\(-0.286711\pi\)
							0.145944	+	0.989293i	\(0.453378\pi\)
\(17\)		−	3.72031i		−	0.902309i	−0.892446	−	0.451154i	\(-0.851012\pi\)
							0.892446	−	0.451154i	\(-0.148988\pi\)
\(19\)	−4.10860	−	4.10860i	−0.942577	−	0.942577i	0.0558614	−	0.998439i	\(-0.482209\pi\)
							−0.998439	+	0.0558614i	\(0.982209\pi\)
\(23\)	1.57595	−	0.909876i	0.328609	−	0.189722i	−0.326615	−	0.945158i	\(-0.605908\pi\)
							0.655223	+	0.755435i	\(0.272575\pi\)
\(29\)	3.83006	−	1.02626i	0.711224	−	0.190572i	0.114972	−	0.993369i	\(-0.463322\pi\)
							0.596253	+	0.802797i	\(0.296656\pi\)
\(31\)	8.81101	−	5.08704i	1.58250	−	0.913659i	0.588012	−	0.808852i	\(-0.299911\pi\)
							0.994493	−	0.104807i	\(-0.0334225\pi\)
\(37\)	1.76964	+	1.76964i	0.290928	+	0.290928i	0.837447	−	0.546519i	\(-0.184047\pi\)
							−0.546519	+	0.837447i	\(0.684047\pi\)
\(41\)	−2.66819	−	4.62144i	−0.416701	−	0.721747i	0.578904	−	0.815395i	\(-0.303480\pi\)
							−0.995605	+	0.0936482i	\(0.970147\pi\)
\(43\)	−1.84748	−	6.89490i	−0.281738	−	1.05146i	−0.951190	−	0.308606i	\(-0.900138\pi\)
							0.669452	−	0.742856i	\(-0.266529\pi\)
\(47\)	5.48486	−	9.50006i	0.800049	−	1.38573i	−0.119534	−	0.992830i	\(-0.538140\pi\)
							0.919583	−	0.392896i	\(-0.128527\pi\)
\(53\)	−8.58403	+	8.58403i	−1.17911	+	1.17911i	−0.199135	+	0.979972i	\(0.563813\pi\)
							−0.979972	+	0.199135i	\(0.936187\pi\)
\(59\)	−1.44299	+	5.38532i	−0.187862	+	0.701109i	0.806138	+	0.591727i	\(0.201554\pi\)
							−0.994000	+	0.109382i	\(0.965113\pi\)
\(61\)	1.66977	+	6.23168i	0.213793	+	0.797885i	0.986588	+	0.163230i	\(0.0521911\pi\)
							−0.772796	+	0.634655i	\(0.781142\pi\)
\(67\)	−1.00652	+	3.75640i	−0.122967	+	0.458918i	−0.999759	−	0.0219514i	\(-0.993012\pi\)
							0.876792	+	0.480869i	\(0.159679\pi\)
\(71\)		−	10.6808i		−	1.26758i	−0.773506	−	0.633789i	\(-0.781499\pi\)
							0.773506	−	0.633789i	\(-0.218501\pi\)
\(73\)			9.30419i			1.08897i	0.838770	+	0.544487i	\(0.183275\pi\)
							−0.838770	+	0.544487i	\(0.816725\pi\)
\(79\)	8.70990	+	5.02866i	0.979940	+	0.565769i	0.902252	−	0.431209i	\(-0.141913\pi\)
							0.0776882	+	0.996978i	\(0.475246\pi\)
\(83\)	−0.588691	−	2.19703i	−0.0646173	−	0.241155i	0.926062	−	0.377372i	\(-0.123172\pi\)
							−0.990679	+	0.136217i	\(0.956506\pi\)
\(89\)	1.87637			0.198895			0.0994475	−	0.995043i	\(-0.468292\pi\)
							0.0994475	+	0.995043i	\(0.468292\pi\)
\(97\)	9.19070	−	15.9188i	0.933175	−	1.61631i	0.155317	−	0.987865i	\(-0.450360\pi\)
							0.777857	−	0.628441i	\(-0.216307\pi\)

Currently showing only \(a_p\); display all \(a_n\)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	432.2.v.a.251.7		88
3.2	odd	2		144.2.u.a.11.16	✓	88
4.3	odd	2		1728.2.z.a.143.7		88
9.4	even	3		144.2.u.a.59.22	yes	88
9.5	odd	6	inner	432.2.v.a.395.1		88
12.11	even	2		576.2.y.a.335.6		88
16.3	odd	4	inner	432.2.v.a.35.1		88
16.13	even	4		1728.2.z.a.1007.7		88
36.23	even	6		1728.2.z.a.719.7		88
36.31	odd	6		576.2.y.a.527.17		88
48.29	odd	4		576.2.y.a.47.17		88
48.35	even	4		144.2.u.a.83.22	yes	88
144.13	even	12		576.2.y.a.239.6		88
144.67	odd	12		144.2.u.a.131.16	yes	88
144.77	odd	12		1728.2.z.a.1583.7		88
144.131	even	12	inner	432.2.v.a.179.7		88

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
144.2.u.a.11.16	✓	88	3.2	odd	2
144.2.u.a.59.22	yes	88	9.4	even	3
144.2.u.a.83.22	yes	88	48.35	even	4
144.2.u.a.131.16	yes	88	144.67	odd	12
432.2.v.a.35.1		88	16.3	odd	4	inner
432.2.v.a.179.7		88	144.131	even	12	inner
432.2.v.a.251.7		88	1.1	even	1	trivial
432.2.v.a.395.1		88	9.5	odd	6	inner
576.2.y.a.47.17		88	48.29	odd	4
576.2.y.a.239.6		88	144.13	even	12
576.2.y.a.335.6		88	12.11	even	2
576.2.y.a.527.17		88	36.31	odd	6
1728.2.z.a.143.7		88	4.3	odd	2
1728.2.z.a.719.7		88	36.23	even	6
1728.2.z.a.1007.7		88	16.13	even	4
1728.2.z.a.1583.7		88	144.77	odd	12