Embedded newform 400.2.bm.a.363.50

• Label: 400.2.bm.a.363.50
• Level: $400$
• Weight: $2$
• Character: 400.363
• Analytic conductor: $3.194$
• Analytic rank: $0$
• Dimension: $464$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 400 = 2^{4} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	400.bm (of order \(20\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.19401608085\)
Analytic rank:	\(0\)
Dimension:	\(464\)
Relative dimension:	\(58\) over \(\Q(\zeta_{20})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{20}]$

Embedding invariants

Embedding label			363.50
Character	\(\chi\)	\(=\)	400.363
Dual form			400.2.bm.a.227.50

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.32306 + 0.499525i) q^{2} +(-2.67086 + 0.867815i) q^{3} +(1.50095 + 1.32180i) q^{4} +(2.16115 - 0.573972i) q^{5} +(-3.96719 - 0.185994i) q^{6} +(3.07706 - 3.07706i) q^{7} +(1.32557 + 2.49857i) q^{8} +(3.95334 - 2.87227i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 464 q - 8 q^{2} - 10 q^{3} - 10 q^{4} - 8 q^{5} - 6 q^{6} - 16 q^{7} - 2 q^{8} + 108 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/400\mathbb{Z}\right)^\times\).

\(n\)	\(101\)	\(177\)	\(351\)
\(\chi(n)\)	\(e\left(\frac{1}{4}\right)\)	\(e\left(\frac{19}{20}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.32306	+	0.499525i	0.935541	+	0.353217i
							\(3\)	−2.67086	+	0.867815i	−1.54202	+	0.501033i	−0.951933	−	0.306306i	\(-0.900907\pi\)
−0.590088	+	0.807339i	\(0.700907\pi\)
\(5\)	2.16115	−	0.573972i	0.966494	−	0.256688i
							\(7\)	3.07706	−	3.07706i	1.16302	−	1.16302i	0.179207	−	0.983811i	\(-0.442647\pi\)
0.983811	−	0.179207i	\(-0.0573531\pi\)
\(11\)	−2.14769	+	0.340160i	−0.647552	+	0.102562i	−0.471567	−	0.881830i	\(-0.656311\pi\)
							−0.175986	+	0.984393i	\(0.556311\pi\)
\(13\)	0.756417	−	0.549569i	0.209792	−	0.152423i	−0.477927	−	0.878399i	\(-0.658612\pi\)
							0.687720	+	0.725976i	\(0.258612\pi\)
\(17\)	2.82182	+	5.53813i	0.684391	+	1.34319i	0.927730	+	0.373252i	\(0.121757\pi\)
							−0.243339	+	0.969941i	\(0.578243\pi\)
\(19\)	−2.30884	−	4.53136i	−0.529685	−	1.03957i	−0.988527	−	0.151047i	\(-0.951735\pi\)
							0.458842	−	0.888518i	\(-0.348265\pi\)
\(23\)	1.10051	+	6.94835i	0.229472	+	1.44883i	0.786116	+	0.618079i	\(0.212089\pi\)
							−0.556643	+	0.830752i	\(0.687911\pi\)
\(29\)	−0.330058	+	0.647775i	−0.0612902	+	0.120289i	−0.919619	−	0.392811i	\(-0.871503\pi\)
							0.858329	+	0.513099i	\(0.171503\pi\)
\(31\)	1.32044	+	0.429037i	0.237158	+	0.0770573i	0.425185	−	0.905107i	\(-0.360209\pi\)
							−0.188027	+	0.982164i	\(0.560209\pi\)
\(37\)	−0.950919	+	0.690883i	−0.156330	+	0.113580i	−0.663200	−	0.748442i	\(-0.730802\pi\)
							0.506870	+	0.862022i	\(0.330802\pi\)
\(41\)	−6.35396	−	8.74548i	−0.992322	−	1.36581i	−0.929919	−	0.367763i	\(-0.880124\pi\)
							−0.0624027	−	0.998051i	\(-0.519876\pi\)
\(43\)	−12.3840			−1.88854			−0.944272	−	0.329166i	\(-0.893233\pi\)
							−0.944272	+	0.329166i	\(0.893233\pi\)
\(47\)	0.702652	−	1.37903i	0.102492	−	0.201152i	−0.834066	−	0.551665i	\(-0.813993\pi\)
							0.936558	+	0.350513i	\(0.113993\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	400.2.bm.a.363.50	yes	464
16.3	odd	4		400.2.bd.a.163.46	yes	464
25.2	odd	20		400.2.bd.a.27.46	✓	464
400.227	even	20	inner	400.2.bm.a.227.50	yes	464

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
400.2.bd.a.27.46	✓	464	25.2	odd	20
400.2.bd.a.163.46	yes	464	16.3	odd	4
400.2.bm.a.227.50	yes	464	400.227	even	20	inner
400.2.bm.a.363.50	yes	464	1.1	even	1	trivial