Embedded newform 41.9.h.a.6.12

• Label: 41.9.h.a.6.12
• Level: $41$
• Weight: $9$
• Character: 41.6
• Analytic conductor: $16.703$
• Analytic rank: $0$
• Dimension: $432$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 41 \)
Weight:	\( k \)	\(=\)	\( 9 \)
Character orbit:	\([\chi]\)	\(=\)	41.h (of order \(40\), degree \(16\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(16.7025230125\)
Analytic rank:	\(0\)
Dimension:	\(432\)
Relative dimension:	\(27\) over \(\Q(\zeta_{40})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{40}]$

Embedding invariants

Embedding label			6.12
Character	\(\chi\)	\(=\)	41.6
Dual form			41.9.h.a.7.12

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-3.68742 + 7.23697i) q^{2} +(37.9696 + 91.6667i) q^{3} +(111.696 + 153.737i) q^{4} +(-132.649 + 837.515i) q^{5} +(-803.399 - 63.2289i) q^{6} +(-3765.52 + 296.353i) q^{7} +(-3578.15 + 566.724i) q^{8} +(-2321.77 + 2321.77i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 432 q - 16 q^{2} - 240 q^{3} - 20 q^{4} - 16 q^{5} - 6484 q^{6} - 16 q^{7} - 1040 q^{8} + 21520 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/41\mathbb{Z}\right)^\times\).

\(n\)	\(6\)
\(\chi(n)\)	\(e\left(\frac{1}{40}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−3.68742	+	7.23697i	−0.230464	+	0.452310i	−0.977059	−	0.212968i	\(-0.931687\pi\)
							0.746596	+	0.665278i	\(0.231687\pi\)
\(3\)	37.9696	+	91.6667i	0.468761	+	1.13169i	0.964705	+	0.263334i	\(0.0848220\pi\)
							−0.495944	+	0.868354i	\(0.665178\pi\)
\(5\)	−132.649	+	837.515i	−0.212239	+	1.34002i	0.619561	+	0.784949i	\(0.287311\pi\)
							−0.831800	+	0.555076i	\(0.812689\pi\)
\(7\)	−3765.52	+	296.353i	−1.56831	+	0.123429i	−0.832603	−	0.553870i	\(-0.813150\pi\)
							−0.735708	+	0.677299i	\(0.763150\pi\)
\(11\)	25405.0	+	6099.20i	1.73520	+	0.416584i	0.973008	−	0.230771i	\(-0.0741249\pi\)
							0.762189	+	0.647355i	\(0.224125\pi\)
\(13\)	10083.8	−	8612.37i	0.353062	−	0.301543i	−0.455174	−	0.890403i	\(-0.650423\pi\)
							0.808235	+	0.588860i	\(0.200423\pi\)
\(17\)	73223.7	−	44871.6i	0.876710	−	0.537249i	−0.00980288	−	0.999952i	\(-0.503120\pi\)
							0.886513	+	0.462703i	\(0.153120\pi\)
\(19\)	40143.3	+	34285.6i	0.308034	+	0.263086i	0.790004	−	0.613102i	\(-0.210078\pi\)
							−0.481970	+	0.876188i	\(0.660078\pi\)
\(23\)	296096.	−	96207.3i	1.05809	−	0.343793i	0.272249	−	0.962227i	\(-0.412233\pi\)
							0.785837	+	0.618434i	\(0.212233\pi\)
\(29\)	−729602.	−	447101.i	−1.03156	−	0.632140i	−0.0997214	−	0.995015i	\(-0.531795\pi\)
							−0.931838	+	0.362875i	\(0.881795\pi\)
\(31\)	−890383.	+	1.22551e6i	−0.964117	+	1.32699i	−0.0191543	+	0.999817i	\(0.506097\pi\)
							−0.944963	+	0.327177i	\(0.893903\pi\)
\(37\)	807707.	−	586834.i	0.430970	−	0.313118i	−0.351067	−	0.936351i	\(-0.614181\pi\)
							0.782037	+	0.623232i	\(0.214181\pi\)
\(41\)	−107676.	+	2.82371e6i	−0.0381053	+	0.999274i
							\(43\)	657720.	−	1.29085e6i	0.192383	−	0.377573i	−0.774585	−	0.632469i	\(-0.782041\pi\)
0.966968	+	0.254896i	\(0.0820414\pi\)
\(47\)	−7.11096e6	−	559644.i	−1.45726	−	0.114689i	−0.675081	−	0.737744i	\(-0.735891\pi\)
							−0.782178	+	0.623055i	\(0.785891\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	41.9.h.a.6.12	✓	432
41.7	odd	40	inner	41.9.h.a.7.12	yes	432

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
41.9.h.a.6.12	✓	432	1.1	even	1	trivial
41.9.h.a.7.12	yes	432	41.7	odd	40	inner