Embedded newform 25.9.f.a.13.18

• Label: 25.9.f.a.13.18
• Level: $25$
• Weight: $9$
• Character: 25.13
• Analytic conductor: $10.184$
• Analytic rank: $0$
• Dimension: $152$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 25 = 5^{2} \)
Weight:	\( k \)	\(=\)	\( 9 \)
Character orbit:	\([\chi]\)	\(=\)	25.f (of order \(20\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(10.1844652515\)
Analytic rank:	\(0\)
Dimension:	\(152\)
Relative dimension:	\(19\) over \(\Q(\zeta_{20})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{20}]$

Embedding invariants

Embedding label			13.18
Character	\(\chi\)	\(=\)	25.13
Dual form			25.9.f.a.2.18

$q$-expansion

\(f(q)\)	\(=\)	\(q+(26.2450 - 4.15680i) q^{2} +(39.3019 - 77.1344i) q^{3} +(428.052 - 139.082i) q^{4} +(614.814 - 112.380i) q^{5} +(710.848 - 2187.76i) q^{6} +(-2106.70 + 2106.70i) q^{7} +(4595.04 - 2341.29i) q^{8} +(-548.613 - 755.101i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 152 q - 8 q^{2} + 62 q^{3} - 10 q^{4} - 230 q^{5} - 6 q^{6} + 2342 q^{7} + 8210 q^{8} - 10 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/25\mathbb{Z}\right)^\times\).

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{19}{20}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	26.2450	−	4.15680i	1.64031	−	0.259800i	0.732992	−	0.680237i	\(-0.238123\pi\)
							0.907322	+	0.420437i	\(0.138123\pi\)
\(3\)	39.3019	−	77.1344i	0.485209	−	0.952276i	−0.510512	−	0.859870i	\(-0.670544\pi\)
							0.995721	−	0.0924060i	\(-0.0294558\pi\)
\(5\)	614.814	−	112.380i	0.983702	−	0.179809i
							\(7\)	−2106.70	+	2106.70i	−0.877427	+	0.877427i	−0.993268	−	0.115841i	\(-0.963044\pi\)
0.115841	+	0.993268i	\(0.463044\pi\)
\(11\)	−18936.2	−	13757.9i	−1.29337	−	0.939685i	−0.293498	−	0.955960i	\(-0.594819\pi\)
							−0.999868	+	0.0162746i	\(0.994819\pi\)
\(13\)	29486.0	+	4670.12i	1.03239	+	0.163514i	0.649563	−	0.760308i	\(-0.274952\pi\)
							0.382823	+	0.923822i	\(0.374952\pi\)
\(17\)	43555.6	+	85482.7i	0.521493	+	1.02349i	0.990137	+	0.140100i	\(0.0447426\pi\)
							−0.468644	+	0.883387i	\(0.655257\pi\)
\(19\)	−76800.7	−	24954.0i	−0.589319	−	0.191481i	−0.000848001	−	1.00000i	\(-0.500270\pi\)
							−0.588471	+	0.808518i	\(0.700270\pi\)
\(23\)	−2133.80	−	13472.3i	−0.00762506	−	0.0481427i	0.983582	−	0.180459i	\(-0.0577584\pi\)
							−0.991208	+	0.132317i	\(0.957758\pi\)
\(29\)	−774551.	+	251667.i	−1.09511	+	0.355823i	−0.800219	−	0.599707i	\(-0.795284\pi\)
							−0.294892	+	0.955531i	\(0.595284\pi\)
\(31\)	74455.5	−	229151.i	0.0806214	−	0.248127i	−0.902619	−	0.430440i	\(-0.858358\pi\)
							0.983241	+	0.182313i	\(0.0583584\pi\)
\(37\)	−425205.	+	2.68464e6i	−0.226877	+	1.43245i	0.566673	+	0.823943i	\(0.308230\pi\)
							−0.793551	+	0.608504i	\(0.791770\pi\)
\(41\)	1.39526e6	−	1.01372e6i	0.493766	−	0.358742i	−0.312865	−	0.949798i	\(-0.601289\pi\)
							0.806631	+	0.591056i	\(0.201289\pi\)
\(43\)	−3.27807e6	−	3.27807e6i	−0.958836	−	0.958836i	0.0403499	−	0.999186i	\(-0.487153\pi\)
							−0.999186	+	0.0403499i	\(0.987153\pi\)
\(47\)	3.34505e6	+	1.70439e6i	0.685505	+	0.349282i	0.761808	−	0.647802i	\(-0.224312\pi\)
							−0.0763034	+	0.997085i	\(0.524312\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	25.9.f.a.13.18	yes	152
25.2	odd	20	inner	25.9.f.a.2.18	✓	152

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
25.9.f.a.2.18	✓	152	25.2	odd	20	inner
25.9.f.a.13.18	yes	152	1.1	even	1	trivial