Embedded newform 270.3.q.b.103.14

• Label: 270.3.q.b.103.14
• Level: $270$
• Weight: $3$
• Character: 270.103
• Analytic conductor: $7.357$
• Analytic rank: $0$
• Dimension: $216$
• Inner twists: $4$

Newspace parameters

Level:	$N$	$=$	$270 = 2 \cdot 3^{3} \cdot 5$
Weight:	$k$	$=$	$3$
Character orbit:	$[\chi]$	$=$	270.q (of order $36$, degree $12$, minimal)

Newform invariants

Self dual:	no
Analytic conductor:	$7.35696713773$
Analytic rank:	$0$
Dimension:	$216$
Relative dimension:	$18$ over $\Q(\zeta_{36})$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{36}]$

Embedding invariants

Embedding label			103.14
Character	$\chi$	$=$	270.103
Dual form			270.3.q.b.97.14

$q$-expansion

$f(q)$	$=$	$q+(-0.123257 + 1.40883i) q^{2} +(2.33981 - 1.87758i) q^{3} +(-1.96962 - 0.347296i) q^{4} +(4.99461 + 0.232069i) q^{5} +(2.35680 + 3.52782i) q^{6} +(-10.5915 - 7.41628i) q^{7} +(0.732051 - 2.73205i) q^{8} +(1.94940 - 8.78634i) q^{9} +(-0.942566 + 7.00796i) q^{10} +(18.1011 + 6.58827i) q^{11} +(-5.26060 + 2.88550i) q^{12} +(-0.453792 - 5.18687i) q^{13} +(11.7538 - 14.0076i) q^{14} +(12.1222 - 8.83478i) q^{15} +(3.75877 + 1.36808i) q^{16} +(-5.66531 - 21.1432i) q^{17} +(12.1382 + 3.82935i) q^{18} +(8.60490 - 4.96804i) q^{19} +(-9.75687 - 2.19170i) q^{20} +(-38.7068 + 2.53378i) q^{21} +(-11.5129 + 24.6894i) q^{22} +(-12.2632 + 8.58680i) q^{23} +(-3.41678 - 7.76696i) q^{24} +(24.8923 + 2.31819i) q^{25} +7.36336 q^{26} +(-11.9358 - 24.2185i) q^{27} +(18.2856 + 18.2856i) q^{28} +(11.3766 + 13.5581i) q^{29} +(10.9526 + 18.1670i) q^{30} +(3.94528 - 22.3748i) q^{31} +(-2.39069 + 5.12685i) q^{32} +(54.7231 - 18.5710i) q^{33} +(30.4856 - 5.37543i) q^{34} +(-51.1795 - 39.4994i) q^{35} +(-6.89103 + 16.6287i) q^{36} +(12.3371 + 46.0426i) q^{37} +(5.93852 + 12.7352i) q^{38} +(-10.8005 - 11.2842i) q^{39} +(4.29033 - 13.4756i) q^{40} +(-14.2759 - 11.9789i) q^{41} +(1.20120 - 54.8437i) q^{42} +(-9.53985 - 20.4583i) q^{43} +(-33.3642 - 19.2628i) q^{44} +(11.7755 - 43.4320i) q^{45} +(-10.5858 - 18.3352i) q^{46} +(40.1970 + 28.1462i) q^{47} +(11.3635 - 3.85634i) q^{48} +(40.4206 + 111.055i) q^{49} +(-6.33408 + 34.7833i) q^{50} +(-52.9538 - 38.8340i) q^{51} +(-0.907585 + 10.3737i) q^{52} +(46.1290 + 46.1290i) q^{53} +(35.5910 - 13.8305i) q^{54} +(88.8792 + 37.1066i) q^{55} +(-28.0152 + 23.5075i) q^{56} +(10.8059 - 27.7806i) q^{57} +(-20.5034 + 14.3566i) q^{58} +(-25.7670 - 70.7942i) q^{59} +(-26.9443 + 13.1911i) q^{60} +(15.7907 + 89.5537i) q^{61} +(31.0361 + 8.31609i) q^{62} +(-85.8091 + 78.6036i) q^{63} +(-6.92820 - 4.00000i) q^{64} +(-1.06280 - 26.0117i) q^{65} +(19.4184 + 79.3847i) q^{66} +(-72.2944 + 6.32494i) q^{67} +(3.81552 + 43.6116i) q^{68} +(-12.5712 + 43.1166i) q^{69} +(61.9562 - 67.2348i) q^{70} +(-54.4225 + 94.2626i) q^{71} +(-22.5777 - 11.7579i) q^{72} +(0.988798 - 3.69024i) q^{73} +(-66.3870 + 11.7058i) q^{74} +(62.5957 - 41.3131i) q^{75} +(-18.6737 + 6.79668i) q^{76} +(-142.858 - 204.023i) q^{77} +(17.2288 - 13.8253i) q^{78} +(12.7894 + 15.2418i) q^{79} +(18.4561 + 7.70532i) q^{80} +(-73.3997 - 34.2562i) q^{81} +(18.6359 - 18.6359i) q^{82} +(-106.494 - 9.31701i) q^{83} +(77.1175 + 8.45215i) q^{84} +(-23.3893 - 106.917i) q^{85} +(29.9981 - 10.9184i) q^{86} +(52.0756 + 10.3629i) q^{87} +(31.2504 - 44.6302i) q^{88} +(46.0129 - 26.5656i) q^{89} +(59.7369 + 21.9430i) q^{90} +(-33.6609 + 58.3024i) q^{91} +(27.1360 - 12.6537i) q^{92} +(-32.7793 - 59.7604i) q^{93} +(-44.6079 + 53.1616i) q^{94} +(44.1311 - 22.8165i) q^{95} +(4.03231 + 16.4846i) q^{96} +(-61.6919 + 28.7674i) q^{97} +(-161.439 + 43.2576i) q^{98} +(93.1732 - 146.200i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	216 * q + 12 * q^6 - 18 * q^7 - 216 * q^8 + 36 * q^11 + 18 * q^15 + 72 * q^20 - 288 * q^21 - 36 * q^22 + 36 * q^23 - 90 * q^25 + 66 * q^27 - 168 * q^30 + 432 * q^31 + 18 * q^33 - 162 * q^35 + 48 * q^36 + 108 * q^38 + 36 * q^40 + 216 * q^41 + 144 * q^42 + 216 * q^43 - 78 * q^45 - 108 * q^46 + 108 * q^47 + 24 * q^48 - 126 * q^50 - 108 * q^51 - 540 * q^53 - 324 * q^55 - 72 * q^56 - 234 * q^57 - 72 * q^58 + 96 * q^60 - 216 * q^61 + 108 * q^63 - 144 * q^65 + 144 * q^66 + 864 * q^67 + 108 * q^68 - 72 * q^70 + 144 * q^72 + 216 * q^73 + 588 * q^75 - 216 * q^76 + 576 * q^77 + 216 * q^78 + 1044 * q^81 - 216 * q^83 - 576 * q^85 - 432 * q^86 - 900 * q^87 + 72 * q^88 + 606 * q^90 - 756 * q^91 - 252 * q^92 - 120 * q^93 + 108 * q^95 + 108 * q^97 - 756 * q^98

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/270\mathbb{Z}\right)^\times$.

$n$	$191$	$217$
$\chi(n)$	$e\left(\frac{7}{9}\right)$	$e\left(\frac{3}{4}\right)$

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)

$p$	$a_p$			$a_p / p^{(k-1)/2}$			$\alpha_p$			$\theta_p$
$2$	−0.123257	+	1.40883i	−0.0616284	+	0.704416i
							$3$	2.33981	−	1.87758i	0.779936	−	0.625859i
$5$	4.99461	+	0.232069i	0.998922	+	0.0464138i
							$7$	−10.5915	−	7.41628i	−1.51308	−	1.05947i	−0.976475	−	0.215633i	$-0.930819\pi$
−0.536602	−	0.843835i	$-0.680293\pi$
$11$	18.1011	+	6.58827i	1.64556	+	0.598934i	0.987998	−	0.154469i	$-0.0493667\pi$
							0.657559	+	0.753403i	$0.271589\pi$
$13$	−0.453792	−	5.18687i	−0.0349071	−	0.398990i	−0.993469	−	0.114105i	$-0.963600\pi$
							0.958562	−	0.284885i	$-0.0919555\pi$
$17$	−5.66531	−	21.1432i	−0.333254	−	1.24372i	−0.905750	−	0.423812i	$-0.860692\pi$
							0.572496	−	0.819907i	$-0.305975\pi$
$19$	8.60490	−	4.96804i	0.452889	−	0.261476i	−0.256160	−	0.966634i	$-0.582457\pi$
							0.709050	+	0.705158i	$0.249124\pi$
$23$	−12.2632	+	8.58680i	−0.533184	+	0.373339i	−0.808943	−	0.587888i	$-0.799960\pi$
							0.275759	+	0.961227i	$0.411071\pi$
$29$	11.3766	+	13.5581i	0.392298	+	0.467522i	0.925655	−	0.378367i	$-0.123514\pi$
							−0.533358	+	0.845890i	$0.679070\pi$
$31$	3.94528	−	22.3748i	0.127267	−	0.721768i	−0.852668	−	0.522453i	$-0.825017\pi$
							0.979935	−	0.199315i	$-0.0638718\pi$
$37$	12.3371	+	46.0426i	0.333435	+	1.24440i	0.905556	+	0.424226i	$0.139454\pi$
							−0.572121	+	0.820169i	$0.693879\pi$
$41$	−14.2759	−	11.9789i	−0.348193	−	0.292169i	0.451871	−	0.892083i	$-0.350757\pi$
							−0.800064	+	0.599915i	$0.795201\pi$
$43$	−9.53985	−	20.4583i	−0.221857	−	0.475774i	0.763602	−	0.645687i	$-0.223429\pi$
							−0.985459	+	0.169913i	$0.945651\pi$
$47$	40.1970	+	28.1462i	0.855255	+	0.598856i	0.916844	−	0.399246i	$-0.130728\pi$
							−0.0615890	+	0.998102i	$0.519617\pi$

(See $a_n$ instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	270.3.q.b.103.14	yes	216
5.2	odd	4	inner	270.3.q.b.157.5	yes	216
27.16	even	9	inner	270.3.q.b.43.5	✓	216
135.97	odd	36	inner	270.3.q.b.97.14	yes	216

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
270.3.q.b.43.5	✓	216	27.16	even	9	inner
270.3.q.b.97.14	yes	216	135.97	odd	36	inner
270.3.q.b.103.14	yes	216	1.1	even	1	trivial
270.3.q.b.157.5	yes	216	5.2	odd	4	inner