Embedded newform 74.6.c.b.47.4

• Label: 74.6.c.b.47.4
• Level: $74$
• Weight: $6$
• Character: 74.47
• Analytic conductor: $11.868$
• Analytic rank: $0$
• Dimension: $16$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 74 = 2 \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	74.c (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(11.8684026662\)
Analytic rank:	\(0\)
Dimension:	\(16\)
Relative dimension:	\(8\) over \(\Q(\zeta_{3})\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{16} + \cdots)\)
Defining polynomial:	x^16 + 1317*x^14 - 2712*x^13 + 1206654*x^12 - 2683704*x^11 + 572597881*x^10 - 1472178324*x^9 + 197936317446*x^8 - 538366712748*x^7 + 33466187997381*x^6 - 131063526625884*x^5 + 4041550479749833*x^4 - 11057529820868736*x^3 + 29709621477714456*x^2 - 4628457422701056*x + 683643854869056
Coefficient ring:	\(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index:	\( 2^{10}\cdot 3 \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			47.4
Root			\(1.38002 + 2.39026i\) of defining polynomial
Character	\(\chi\)	\(=\)	74.47
Dual form			74.6.c.b.63.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(2.00000 - 3.46410i) q^{2} +(-1.88002 - 3.25629i) q^{3} +(-8.00000 - 13.8564i) q^{4} +(7.78936 + 13.4916i) q^{5} -15.0401 q^{6} +(-28.3238 - 49.0583i) q^{7} -64.0000 q^{8} +(114.431 - 198.200i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 32 q^{2} - 8 q^{3} - 128 q^{4} + 70 q^{5} - 64 q^{6} - 32 q^{7} - 1024 q^{8} - 698 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/74\mathbb{Z}\right)^\times\).

\(n\)	\(39\)
\(\chi(n)\)	\(e\left(\frac{2}{3}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	2.00000	−	3.46410i	0.353553	−	0.612372i
							\(3\)	−1.88002	−	3.25629i	−0.120603	−	0.208891i	0.799403	−	0.600796i	\(-0.205150\pi\)
−0.920006	+	0.391905i	\(0.871816\pi\)
\(5\)	7.78936	+	13.4916i	0.139340	+	0.241344i	0.927247	−	0.374450i	\(-0.122168\pi\)
							−0.787907	+	0.615795i	\(0.788835\pi\)
\(7\)	−28.3238	−	49.0583i	−0.218477	−	0.378414i	0.735865	−	0.677128i	\(-0.236776\pi\)
							−0.954343	+	0.298714i	\(0.903442\pi\)
\(11\)	−674.932			−1.68181			−0.840907	−	0.541180i	\(-0.817978\pi\)
							−0.840907	+	0.541180i	\(0.817978\pi\)
\(13\)	−195.754	−	339.055i	−0.321256	−	0.556432i	0.659491	−	0.751712i	\(-0.270772\pi\)
							−0.980748	+	0.195280i	\(0.937438\pi\)
\(17\)	−745.244	+	1290.80i	−0.625427	+	1.08327i	0.363031	+	0.931777i	\(0.381742\pi\)
							−0.988458	+	0.151494i	\(0.951592\pi\)
\(19\)	−142.329	−	246.521i	−0.0904502	−	0.156664i	0.817250	−	0.576283i	\(-0.195497\pi\)
							−0.907701	+	0.419618i	\(0.862164\pi\)
\(23\)	−2913.01			−1.14821			−0.574106	−	0.818781i	\(-0.694650\pi\)
							−0.574106	+	0.818781i	\(0.694650\pi\)
\(29\)	4007.08			0.884776			0.442388	−	0.896824i	\(-0.354132\pi\)
							0.442388	+	0.896824i	\(0.354132\pi\)
\(31\)	−529.770			−0.0990109			−0.0495055	−	0.998774i	\(-0.515765\pi\)
							−0.0495055	+	0.998774i	\(0.515765\pi\)
\(37\)	−8112.59	−	1878.80i	−0.974216	−	0.225619i
							\(41\)	−8640.01	−	14964.9i	−0.802703	−	1.39032i	−0.917831	−	0.396972i	\(-0.870061\pi\)
0.115128	−	0.993351i	\(-0.463272\pi\)
\(43\)	12551.9			1.03523			0.517617	−	0.855612i	\(-0.326819\pi\)
							0.517617	+	0.855612i	\(0.326819\pi\)
\(47\)	15727.0			1.03849			0.519243	−	0.854627i	\(-0.326214\pi\)
							0.519243	+	0.854627i	\(0.326214\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	74.6.c.b.47.4	✓	16
37.26	even	3	inner	74.6.c.b.63.4	yes	16

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
74.6.c.b.47.4	✓	16	1.1	even	1	trivial
74.6.c.b.63.4	yes	16	37.26	even	3	inner