Embedded newform 925.2.bb.e.226.9

• Label: 925.2.bb.e.226.9
• Level: $925$
• Weight: $2$
• Character: 925.226
• Analytic conductor: $7.386$
• Analytic rank: $0$
• Dimension: $96$
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 925 = 5^{2} \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	925.bb (of order \(18\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(7.38616218697\)
Analytic rank:	\(0\)
Dimension:	\(96\)
Relative dimension:	\(16\) over \(\Q(\zeta_{18})\)
Twist minimal:	no (minimal twist has level 185)
Sato-Tate group:	$\mathrm{SU}(2)[C_{18}]$

Embedding invariants

Embedding label			226.9
Character	\(\chi\)	\(=\)	925.226
Dual form			925.2.bb.e.176.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.183007 + 0.0322691i) q^{2} +(0.0999019 + 0.566572i) q^{3} +(-1.84693 - 0.672229i) q^{4} +0.106911i q^{6} +(-1.58789 + 1.33240i) q^{7} +(-0.638178 - 0.368452i) q^{8} +(2.50805 - 0.912857i) q^{9} +(0.238483 - 0.413065i) q^{11} +(0.196354 - 1.11358i) q^{12} +(0.469995 - 1.29130i) q^{13} +(-0.333590 + 0.192598i) q^{14} +(2.90637 + 2.43873i) q^{16} +(1.33624 + 3.67129i) q^{17} +(0.488450 - 0.0861268i) q^{18} +(-2.47154 + 0.435800i) q^{19} +(-0.913531 - 0.766543i) q^{21} +(0.0569735 - 0.0678984i) q^{22} +(-3.34387 + 1.93058i) q^{23} +(0.145000 - 0.398383i) q^{24} +(0.127682 - 0.221151i) q^{26} +(1.63073 + 2.82450i) q^{27} +(3.82840 - 1.39342i) q^{28} +(6.06811 + 3.50343i) q^{29} +1.94048i q^{31} +(1.40054 + 1.66910i) q^{32} +(0.257856 + 0.0938520i) q^{33} +(0.126072 + 0.714992i) q^{34} -5.24586 q^{36} +(1.22985 + 5.95714i) q^{37} -0.466374 q^{38} +(0.778569 + 0.137283i) q^{39} +(-5.79284 - 2.10842i) q^{41} +(-0.142447 - 0.169762i) q^{42} +10.6440i q^{43} +(-0.718138 + 0.602589i) q^{44} +(-0.674251 + 0.245407i) q^{46} +(5.11213 + 8.85447i) q^{47} +(-1.09137 + 1.89030i) q^{48} +(-0.469429 + 2.66227i) q^{49} +(-1.94656 + 1.12385i) q^{51} +(-1.73610 + 2.06901i) q^{52} +(4.25872 + 3.57349i) q^{53} +(0.207291 + 0.569527i) q^{54} +(1.50428 - 0.265245i) q^{56} +(-0.493824 - 1.35677i) q^{57} +(0.997457 + 0.836966i) q^{58} +(1.93057 - 2.30076i) q^{59} +(4.47819 - 12.3037i) q^{61} +(-0.0626176 + 0.355122i) q^{62} +(-2.76622 + 4.79123i) q^{63} +(-3.59155 - 6.22074i) q^{64} +(0.0441611 + 0.0254964i) q^{66} +(-1.71265 + 1.43708i) q^{67} -7.67889i q^{68} +(-1.42787 - 1.70167i) q^{69} +(1.33200 + 7.55415i) q^{71} +(-1.93693 - 0.341533i) q^{72} +0.472707 q^{73} +(0.0328408 + 1.12989i) q^{74} +(4.85774 + 0.856550i) q^{76} +(0.171682 + 0.973655i) q^{77} +(0.138054 + 0.0502475i) q^{78} +(-5.91818 - 7.05302i) q^{79} +(4.69638 - 3.94073i) q^{81} +(-0.992096 - 0.572787i) q^{82} +(7.36374 - 2.68018i) q^{83} +(1.17194 + 2.02986i) q^{84} +(-0.343474 + 1.94794i) q^{86} +(-1.37873 + 3.78802i) q^{87} +(-0.304390 + 0.175740i) q^{88} +(-5.21281 + 6.21239i) q^{89} +(0.974225 + 2.67666i) q^{91} +(7.47370 - 1.31782i) q^{92} +(-1.09942 + 0.193858i) q^{93} +(0.649831 + 1.78540i) q^{94} +(-0.805746 + 0.960251i) q^{96} +(12.0364 - 6.94919i) q^{97} +(-0.171818 + 0.472066i) q^{98} +(0.221060 - 1.25369i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	96 * q + 12 * q^4 + 6 * q^9 - 30 * q^11 + 36 * q^14 + 18 * q^19 - 24 * q^21 - 96 * q^24 + 48 * q^26 + 18 * q^29 + 54 * q^34 + 24 * q^36 + 36 * q^39 + 72 * q^41 + 84 * q^44 - 18 * q^46 + 6 * q^49 - 18 * q^51 - 24 * q^54 + 90 * q^56 - 18 * q^59 - 144 * q^61 - 66 * q^64 - 36 * q^66 + 6 * q^69 - 30 * q^71 + 6 * q^74 - 54 * q^76 - 60 * q^79 - 18 * q^81 - 48 * q^84 - 72 * q^86 - 108 * q^89 - 54 * q^91 - 6 * q^94 + 24 * q^96 + 24 * q^99

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/925\mathbb{Z}\right)^\times\).

\(n\)	\(76\)	\(852\)
\(\chi(n)\)	\(e\left(\frac{1}{18}\right)\)	\(1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\). You can download additional coefficients here.

Currently showing only \(a_p\); display all \(a_n\)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.183007	+	0.0322691i	0.129406	+	0.0228177i	0.237976	−	0.971271i	\(-0.423516\pi\)
							−0.108570	+	0.994089i	\(0.534627\pi\)
\(3\)	0.0999019	+	0.566572i	0.0576784	+	0.327110i	0.999970	−	0.00768141i	\(-0.00244509\pi\)
							−0.942292	+	0.334792i	\(0.891334\pi\)
\(5\)		0			0
							\(7\)	−1.58789	+	1.33240i	−0.600165	+	0.503598i	−0.891499	−	0.453023i	\(-0.850345\pi\)
0.291334	+	0.956621i	\(0.405901\pi\)
\(11\)	0.238483	−	0.413065i	0.0719055	−	0.124544i	−0.827831	−	0.560978i	\(-0.810425\pi\)
							0.899736	+	0.436434i	\(0.143759\pi\)
\(13\)	0.469995	−	1.29130i	0.130353	−	0.358143i	−0.857296	−	0.514824i	\(-0.827857\pi\)
							0.987649	+	0.156681i	\(0.0500795\pi\)
\(17\)	1.33624	+	3.67129i	0.324086	+	0.890419i	0.989576	+	0.144012i	\(0.0460004\pi\)
							−0.665490	+	0.746407i	\(0.731777\pi\)
\(19\)	−2.47154	+	0.435800i	−0.567011	+	0.0999793i	−0.449803	−	0.893128i	\(-0.648506\pi\)
							−0.117208	+	0.993107i	\(0.537395\pi\)
\(23\)	−3.34387	+	1.93058i	−0.697245	+	0.402555i	−0.806320	−	0.591479i	\(-0.798544\pi\)
							0.109075	+	0.994033i	\(0.465211\pi\)
\(29\)	6.06811	+	3.50343i	1.12682	+	0.650570i	0.943133	−	0.332415i	\(-0.107863\pi\)
							0.183687	+	0.982985i	\(0.441197\pi\)
\(31\)			1.94048i			0.348520i	0.984700	+	0.174260i	\(0.0557534\pi\)
							−0.984700	+	0.174260i	\(0.944247\pi\)
\(37\)	1.22985	+	5.95714i	0.202187	+	0.979347i
							\(41\)	−5.79284	−	2.10842i	−0.904690	−	0.329280i	−0.152559	−	0.988294i	\(-0.548752\pi\)
−0.752131	+	0.659014i	\(0.770974\pi\)
\(43\)			10.6440i			1.62320i	0.584213	+	0.811601i	\(0.301403\pi\)
							−0.584213	+	0.811601i	\(0.698597\pi\)
\(47\)	5.11213	+	8.85447i	0.745681	+	1.29156i	0.949876	+	0.312626i	\(0.101209\pi\)
							−0.204196	+	0.978930i	\(0.565458\pi\)
\(53\)	4.25872	+	3.57349i	0.584980	+	0.490857i	0.886578	−	0.462578i	\(-0.153076\pi\)
							−0.301598	+	0.953435i	\(0.597520\pi\)
\(59\)	1.93057	−	2.30076i	0.251339	−	0.299534i	−0.625592	−	0.780150i	\(-0.715143\pi\)
							0.876931	+	0.480616i	\(0.159587\pi\)
\(61\)	4.47819	−	12.3037i	0.573373	−	1.57533i	−0.225764	−	0.974182i	\(-0.572488\pi\)
							0.799137	−	0.601149i	\(-0.205290\pi\)
\(67\)	−1.71265	+	1.43708i	−0.209233	+	0.175567i	−0.741382	−	0.671084i	\(-0.765829\pi\)
							0.532149	+	0.846651i	\(0.321385\pi\)
\(71\)	1.33200	+	7.55415i	0.158079	+	0.896513i	0.955916	+	0.293639i	\(0.0948665\pi\)
							−0.797837	+	0.602873i	\(0.794022\pi\)
\(73\)	0.472707			0.0553262			0.0276631	−	0.999617i	\(-0.491193\pi\)
							0.0276631	+	0.999617i	\(0.491193\pi\)
\(79\)	−5.91818	−	7.05302i	−0.665848	−	0.793526i	0.322365	−	0.946616i	\(-0.395522\pi\)
							−0.988212	+	0.153089i	\(0.951078\pi\)
\(83\)	7.36374	−	2.68018i	0.808275	−	0.294188i	0.0953641	−	0.995442i	\(-0.469598\pi\)
							0.712911	+	0.701254i	\(0.247376\pi\)
\(89\)	−5.21281	+	6.21239i	−0.552557	+	0.658512i	−0.967954	−	0.251128i	\(-0.919198\pi\)
							0.415397	+	0.909640i	\(0.363643\pi\)
\(97\)	12.0364	−	6.94919i	1.22211	−	0.705584i	0.256740	−	0.966480i	\(-0.417352\pi\)
							0.965367	+	0.260897i	\(0.0840182\pi\)

Currently showing only \(a_p\); display all \(a_n\)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	925.2.bb.e.226.9		96
5.2	odd	4		185.2.v.a.4.8	✓	96
5.3	odd	4		185.2.v.a.4.9	yes	96
5.4	even	2	inner	925.2.bb.e.226.8		96
37.28	even	18	inner	925.2.bb.e.176.9		96
185.28	odd	36		185.2.v.a.139.8	yes	96
185.102	odd	36		185.2.v.a.139.9	yes	96
185.139	even	18	inner	925.2.bb.e.176.8		96

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
185.2.v.a.4.8	✓	96	5.2	odd	4
185.2.v.a.4.9	yes	96	5.3	odd	4
185.2.v.a.139.8	yes	96	185.28	odd	36
185.2.v.a.139.9	yes	96	185.102	odd	36
925.2.bb.e.176.8		96	185.139	even	18	inner
925.2.bb.e.176.9		96	37.28	even	18	inner
925.2.bb.e.226.8		96	5.4	even	2	inner
925.2.bb.e.226.9		96	1.1	even	1	trivial