Embedded newform 21.30.e.a.4.15

• Label: 21.30.e.a.4.15
• Level: $21$
• Weight: $30$
• Character: 21.4
• Analytic conductor: $111.884$
• Analytic rank: $0$
• Dimension: $38$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 21 = 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 30 \)
Character orbit:	\([\chi]\)	\(=\)	21.e (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(111.883889004\)
Analytic rank:	\(0\)
Dimension:	\(38\)
Relative dimension:	\(19\) over \(\Q(\zeta_{3})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			4.15
Character	\(\chi\)	\(=\)	21.4
Dual form			21.30.e.a.16.15

$q$-expansion

\(f(q)\)	\(=\)	\(q+(14667.6 + 25405.0i) q^{2} +(2.39148e6 - 4.14217e6i) q^{3} +(-1.61841e8 + 2.80316e8i) q^{4} +(-8.78577e9 - 1.52174e10i) q^{5} +1.40309e11 q^{6} +(1.75086e12 - 3.92932e11i) q^{7} +6.25395e12 q^{8} +(-1.14384e13 - 1.98119e13i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 38 q + 25025 q^{2} + 90876411 q^{3} - 5448583733 q^{4} - 2101591778 q^{5} + 239387598450 q^{6} - 1147384421099 q^{7} - 53840580588510 q^{8} - 434659056644259 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/21\mathbb{Z}\right)^\times\).

\(n\)	\(8\)	\(10\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{2}{3}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	14667.6	+	25405.0i	0.633029	+	1.09644i	0.986929	+	0.161155i	\(0.0515220\pi\)
							−0.353900	+	0.935283i	\(0.615145\pi\)
\(3\)	2.39148e6	−	4.14217e6i	0.288675	−	0.500000i
							\(5\)	−8.78577e9	−	1.52174e10i	−0.643746	−	1.11500i	−0.984590	−	0.174881i	\(-0.944046\pi\)
0.340844	−	0.940120i	\(-0.389287\pi\)
\(7\)	1.75086e12	−	3.92932e11i	0.975730	−	0.218976i
							\(11\)	−3.68015e14	+	6.37421e14i	−0.292194	+	0.506095i	−0.974328	−	0.225132i	\(-0.927719\pi\)
0.682134	+	0.731227i	\(0.261052\pi\)
\(13\)	1.33998e16			0.943885			0.471942	−	0.881629i	\(-0.343553\pi\)
							0.471942	+	0.881629i	\(0.343553\pi\)
\(17\)	6.08879e17	−	1.05461e18i	0.877044	−	1.51909i	0.0224757	−	0.999747i	\(-0.492845\pi\)
							0.854569	−	0.519338i	\(-0.173821\pi\)
\(19\)	1.08381e18	+	1.87722e18i	0.311191	+	0.538999i	0.978621	−	0.205674i	\(-0.0659387\pi\)
							−0.667429	+	0.744673i	\(0.732605\pi\)
\(23\)	3.17136e19	+	5.49295e19i	0.570415	+	0.987988i	0.996523	+	0.0833160i	\(0.0265511\pi\)
							−0.426108	+	0.904672i	\(0.640116\pi\)
\(29\)	−2.64699e21			−1.65189			−0.825947	−	0.563748i	\(-0.809359\pi\)
							−0.825947	+	0.563748i	\(0.809359\pi\)
\(31\)	3.52204e20	−	6.10036e20i	0.0835699	−	0.144747i	−0.821211	−	0.570625i	\(-0.806701\pi\)
							0.904781	+	0.425877i	\(0.140034\pi\)
\(37\)	−2.76359e22	−	4.78667e22i	−0.504136	−	0.873190i	−0.999989	−	0.00478288i	\(-0.998478\pi\)
							0.495852	−	0.868407i	\(-0.334856\pi\)
\(41\)	−3.34472e21			−0.0137719			−0.00688596	−	0.999976i	\(-0.502192\pi\)
							−0.00688596	+	0.999976i	\(0.502192\pi\)
\(43\)	6.46493e23			1.33435			0.667177	−	0.744899i	\(-0.267502\pi\)
							0.667177	+	0.744899i	\(0.267502\pi\)
\(47\)	1.48989e24	+	2.58056e24i	0.846708	+	1.46654i	0.884129	+	0.467242i	\(0.154752\pi\)
							−0.0374209	+	0.999300i	\(0.511914\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	21.30.e.a.4.15	✓	38
7.2	even	3	inner	21.30.e.a.16.15	yes	38

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
21.30.e.a.4.15	✓	38	1.1	even	1	trivial
21.30.e.a.16.15	yes	38	7.2	even	3	inner