Embedded newform 459.2.ba.a.25.8

• Label: 459.2.ba.a.25.8
• Level: $459$
• Weight: $2$
• Character: 459.25
• Analytic conductor: $3.665$
• Analytic rank: $0$
• Dimension: $1248$
• Inner twists: $4$

Newspace parameters

Level:	$N$	$=$	$459 = 3^{3} \cdot 17$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	459.ba (of order $72$, degree $24$, minimal)

Newform invariants

Self dual:	no
Analytic conductor:	$3.66513345278$
Analytic rank:	$0$
Dimension:	$1248$
Relative dimension:	$52$ over $\Q(\zeta_{72})$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{72}]$

Embedding invariants

Embedding label			25.8
Character	$\chi$	$=$	459.25
Dual form			459.2.ba.a.202.8

$q$-expansion

$f(q)$	$=$	$q+(-1.22861 - 1.75463i) q^{2} +(0.00134192 + 1.73205i) q^{3} +(-0.885220 + 2.43212i) q^{4} +(2.58809 - 0.816022i) q^{5} +(3.03746 - 2.13036i) q^{6} +(2.95845 - 2.71092i) q^{7} +(1.21702 - 0.326100i) q^{8} +(-3.00000 + 0.00464853i) q^{9} +(-4.61156 - 3.53858i) q^{10} +(-0.525873 + 1.66786i) q^{11} +(-4.21375 - 1.52998i) q^{12} +(1.62518 - 0.286563i) q^{13} +(-8.39145 - 1.86034i) q^{14} +(1.41686 + 4.48161i) q^{15} +(1.89794 + 1.59256i) q^{16} +(1.02412 - 3.99389i) q^{17} +(3.69397 + 5.25818i) q^{18} +(1.64676 + 6.14578i) q^{19} +(-0.306365 + 7.01691i) q^{20} +(4.69942 + 5.12055i) q^{21} +(3.57257 - 1.12643i) q^{22} +(-0.211394 - 4.84172i) q^{23} +(0.566455 + 2.10751i) q^{24} +(1.93656 - 1.35599i) q^{25} +(-2.49952 - 2.49952i) q^{26} +(-0.0120772 - 5.19614i) q^{27} +(3.97441 + 9.59508i) q^{28} +(-2.02494 + 3.17851i) q^{29} +(6.12280 - 7.99221i) q^{30} +(4.03251 - 4.40072i) q^{31} +(0.682162 - 7.79714i) q^{32} +(-2.88952 - 0.908600i) q^{33} +(-8.26606 + 3.10997i) q^{34} +(5.44457 - 9.43027i) q^{35} +(2.64435 - 7.30047i) q^{36} +(5.95541 - 0.784045i) q^{37} +(8.76037 - 10.4402i) q^{38} +(0.498523 + 2.81451i) q^{39} +(2.88366 - 1.83709i) q^{40} +(-4.02756 - 6.32200i) q^{41} +(3.21094 - 14.5369i) q^{42} +(0.461280 + 5.27246i) q^{43} +(-3.59092 - 2.75541i) q^{44} +(-7.76047 + 2.46009i) q^{45} +(-8.23572 + 6.31949i) q^{46} +(2.23480 + 6.14007i) q^{47} +(-2.75585 + 3.28947i) q^{48} +(0.793252 - 9.06692i) q^{49} +(-4.75853 - 1.73196i) q^{50} +(6.91900 + 1.76847i) q^{51} +(-0.741686 + 4.20631i) q^{52} +(1.89823 - 1.89823i) q^{53} +(-9.10248 + 6.40520i) q^{54} +4.74569i q^{55} +(2.71647 - 4.26400i) q^{56} +(-10.6426 + 2.86051i) q^{57} +(8.06497 - 0.352124i) q^{58} +(7.01689 + 0.613899i) q^{59} +(-12.1541 - 0.521223i) q^{60} +(3.79828 - 3.48048i) q^{61} +(-12.6760 - 1.66883i) q^{62} +(-8.86275 + 8.14651i) q^{63} +(-10.2279 + 5.90509i) q^{64} +(3.97227 - 2.06783i) q^{65} +(1.95582 + 6.18635i) q^{66} +(-2.11370 - 11.9874i) q^{67} +(8.80706 + 6.02626i) q^{68} +(8.38581 - 0.372642i) q^{69} +(-23.2359 + 2.03288i) q^{70} +(-7.26864 + 0.956935i) q^{71} +(-3.64954 + 0.983956i) q^{72} +(-12.0577 + 9.25223i) q^{73} +(-8.69257 - 9.48627i) q^{74} +(2.35124 + 3.35239i) q^{75} +(-16.4050 - 1.43525i) q^{76} +(2.96566 + 6.35988i) q^{77} +(4.32594 - 4.33265i) q^{78} +(7.83817 - 1.73768i) q^{79} +(6.21161 + 2.57293i) q^{80} +(8.99996 - 0.0278911i) q^{81} +(-6.14451 + 14.8341i) q^{82} +(3.56519 + 5.09161i) q^{83} +(-16.6138 + 6.89676i) q^{84} +(-0.608586 - 11.1723i) q^{85} +(8.68450 - 7.28716i) q^{86} +(-5.50806 - 3.50302i) q^{87} +(-0.0961110 + 2.20130i) q^{88} +(-4.39501 - 2.53746i) q^{89} +(13.8511 + 10.5943i) q^{90} +(4.03117 - 5.25352i) q^{91} +(11.9628 + 3.77185i) q^{92} +(7.62767 + 6.97861i) q^{93} +(8.02787 - 11.4650i) q^{94} +(9.27704 + 14.5620i) q^{95} +(13.5060 + 1.17107i) q^{96} +(-0.246246 - 0.473034i) q^{97} +(-16.8837 + 9.74781i) q^{98} +(1.56986 - 5.00601i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	1248 * q - 24 * q^2 - 24 * q^3 - 24 * q^5 - 24 * q^6 - 24 * q^7 - 12 * q^8 - 24 * q^9 - 12 * q^10 - 36 * q^11 + 12 * q^12 - 24 * q^14 - 24 * q^15 - 48 * q^16 - 12 * q^17 - 48 * q^18 - 12 * q^19 - 24 * q^20 - 24 * q^22 + 48 * q^23 - 84 * q^24 - 24 * q^25 - 48 * q^26 - 24 * q^27 - 48 * q^28 - 24 * q^29 - 24 * q^31 - 72 * q^32 - 48 * q^33 - 24 * q^34 - 216 * q^35 - 24 * q^36 - 12 * q^37 - 24 * q^39 - 72 * q^40 - 24 * q^41 + 96 * q^42 + 12 * q^43 - 12 * q^44 - 24 * q^45 - 12 * q^46 - 24 * q^48 - 24 * q^49 - 48 * q^50 - 24 * q^51 - 48 * q^52 - 48 * q^53 - 96 * q^54 + 96 * q^56 - 60 * q^57 - 24 * q^58 + 144 * q^59 - 480 * q^60 - 24 * q^61 - 12 * q^62 + 120 * q^63 - 72 * q^65 - 204 * q^66 - 48 * q^67 - 72 * q^68 + 96 * q^69 + 36 * q^71 - 12 * q^73 - 192 * q^74 - 24 * q^75 - 72 * q^76 - 60 * q^77 + 324 * q^78 - 24 * q^79 - 48 * q^80 - 48 * q^82 - 144 * q^83 - 48 * q^84 - 24 * q^85 + 72 * q^86 - 24 * q^87 + 192 * q^88 - 252 * q^90 - 12 * q^91 - 24 * q^92 - 24 * q^93 - 24 * q^94 + 192 * q^95 + 288 * q^96 - 132 * q^97 - 168 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/459\mathbb{Z}\right)^\times$.

$n$	$137$	$190$
$\chi(n)$	$e\left(\frac{5}{9}\right)$	$e\left(\frac{5}{8}\right)$

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)

$p$	$a_p$			$a_p / p^{(k-1)/2}$			$\alpha_p$			$\theta_p$
$2$	−1.22861	−	1.75463i	−0.868756	−	1.24071i	−0.969325	−	0.245782i	$-0.920955\pi$
							0.100569	−	0.994930i	$-0.467934\pi$
$3$	0.00134192	+	1.73205i	0.000774755	+	1.00000i
							$5$	2.58809	−	0.816022i	1.15743	−	0.364936i	0.339697	−	0.940535i	$-0.389676\pi$
0.817732	+	0.575599i	$0.195231\pi$
$7$	2.95845	−	2.71092i	1.11819	−	1.02463i	0.118693	−	0.992931i	$-0.462130\pi$
							0.999497	−	0.0317016i	$-0.0100926\pi$
$11$	−0.525873	+	1.66786i	−0.158557	+	0.502878i	−0.999254	−	0.0386283i	$-0.987701\pi$
							0.840697	+	0.541506i	$0.182146\pi$
$13$	1.62518	−	0.286563i	0.450744	−	0.0794783i	0.0563321	−	0.998412i	$-0.482059\pi$
							0.394412	+	0.918934i	$0.370948\pi$
$17$	1.02412	−	3.99389i	0.248386	−	0.968661i
							$19$	1.64676	+	6.14578i	0.377792	+	1.40994i	0.849222	+	0.528035i	$0.177071\pi$
−0.471430	+	0.881903i	$0.656262\pi$
$23$	−0.211394	−	4.84172i	−0.0440787	−	1.00957i	−0.884182	−	0.467142i	$-0.845284\pi$
							0.840104	−	0.542426i	$-0.182494\pi$
$29$	−2.02494	+	3.17851i	−0.376021	+	0.590235i	−0.978083	−	0.208214i	$-0.933235\pi$
							0.602062	+	0.798449i	$0.294346\pi$
$31$	4.03251	−	4.40072i	0.724261	−	0.790392i	−0.260283	−	0.965532i	$-0.583816\pi$
							0.984543	+	0.175141i	$0.0560380\pi$
$37$	5.95541	−	0.784045i	0.979063	−	0.128896i	0.376026	−	0.926609i	$-0.377290\pi$
							0.603037	+	0.797713i	$0.293957\pi$
$41$	−4.02756	−	6.32200i	−0.628999	−	0.987331i	−0.998315	−	0.0580272i	$-0.981519\pi$
							0.369316	−	0.929304i	$-0.379592\pi$
$43$	0.461280	+	5.27246i	0.0703446	+	0.804042i	0.946645	+	0.322278i	$0.104448\pi$
							−0.876301	+	0.481765i	$0.839996\pi$
$47$	2.23480	+	6.14007i	0.325979	+	0.895621i	0.989118	+	0.147124i	$0.0470017\pi$
							−0.663139	+	0.748496i	$0.730776\pi$

(See $a_n$ instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	459.2.ba.a.25.8	✓	1248
17.15	even	8	inner	459.2.ba.a.457.45	yes	1248
27.13	even	9	inner	459.2.ba.a.229.45	yes	1248
459.202	even	72	inner	459.2.ba.a.202.8	yes	1248

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
459.2.ba.a.25.8	✓	1248	1.1	even	1	trivial
459.2.ba.a.202.8	yes	1248	459.202	even	72	inner
459.2.ba.a.229.45	yes	1248	27.13	even	9	inner
459.2.ba.a.457.45	yes	1248	17.15	even	8	inner