Embedded newform 270.3.q.a.103.3

• Label: 270.3.q.a.103.3
• 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			103.3
Character	$\chi$	$=$	270.103
Dual form			270.3.q.a.97.3

$q$-expansion

$f(q)$	$=$	$q+(0.123257 - 1.40883i) q^{2} +(-2.55916 - 1.56548i) q^{3} +(-1.96962 - 0.347296i) q^{4} +(-4.02280 + 2.96936i) q^{5} +(-2.52093 + 3.41247i) q^{6} +(2.35838 + 1.65135i) q^{7} +(-0.732051 + 2.73205i) q^{8} +(4.09857 + 8.01260i) q^{9} +(3.68749 + 6.03344i) q^{10} +(5.11381 + 1.86128i) q^{11} +(4.49687 + 3.97217i) q^{12} +(-1.40437 - 16.0520i) q^{13} +(2.61716 - 3.11901i) q^{14} +(14.9434 - 1.30146i) q^{15} +(3.75877 + 1.36808i) q^{16} +(7.49195 + 27.9603i) q^{17} +(11.7936 - 4.78659i) q^{18} +(0.970790 - 0.560486i) q^{19} +(8.95462 - 4.45139i) q^{20} +(-3.45030 - 7.91805i) q^{21} +(3.25254 - 6.97509i) q^{22} +(34.3534 - 24.0545i) q^{23} +(6.15039 - 5.84574i) q^{24} +(7.36583 - 23.8903i) q^{25} -22.7876 q^{26} +(2.05464 - 26.9217i) q^{27} +(-4.07158 - 4.07158i) q^{28} +(3.65957 + 4.36130i) q^{29} +(0.00834613 - 21.2132i) q^{30} +(-8.44013 + 47.8664i) q^{31} +(2.39069 - 5.12685i) q^{32} +(-10.1733 - 12.7689i) q^{33} +(40.3149 - 7.10860i) q^{34} +(-14.3907 + 0.359800i) q^{35} +(-5.28986 - 17.2052i) q^{36} +(-2.23659 - 8.34706i) q^{37} +(-0.669974 - 1.43676i) q^{38} +(-21.5350 + 43.2780i) q^{39} +(-5.16754 - 13.1642i) q^{40} +(50.5697 + 42.4330i) q^{41} +(-11.5805 + 3.88494i) q^{42} +(10.5200 + 22.5603i) q^{43} +(-9.42583 - 5.44201i) q^{44} +(-40.2800 - 20.0629i) q^{45} +(-29.6544 - 51.3630i) q^{46} +(50.5606 + 35.4029i) q^{47} +(-7.47759 - 9.38540i) q^{48} +(-13.9240 - 38.2559i) q^{49} +(-32.7495 - 13.3219i) q^{50} +(24.5982 - 83.2834i) q^{51} +(-2.80873 + 32.1039i) q^{52} +(-20.4015 - 20.4015i) q^{53} +(-37.6749 - 6.21292i) q^{54} +(-26.0986 + 7.69720i) q^{55} +(-6.23803 + 5.23433i) q^{56} +(-3.36183 - 0.0853767i) q^{57} +(6.59541 - 4.61815i) q^{58} +(32.9877 + 90.6329i) q^{59} +(-29.8848 - 2.62643i) q^{60} +(7.17732 + 40.7046i) q^{61} +(66.3954 + 17.7906i) q^{62} +(-3.56565 + 25.6649i) q^{63} +(-6.92820 - 4.00000i) q^{64} +(53.3135 + 60.4038i) q^{65} +(-19.2431 + 12.7586i) q^{66} +(83.9811 - 7.34739i) q^{67} +(-5.04574 - 57.6731i) q^{68} +(-125.572 + 7.77985i) q^{69} +(-1.26686 + 20.3185i) q^{70} +(-41.0311 + 71.0679i) q^{71} +(-24.8912 + 5.33188i) q^{72} +(4.21248 - 15.7212i) q^{73} +(-12.0353 + 2.12215i) q^{74} +(-56.2500 + 49.6079i) q^{75} +(-2.10674 + 0.766789i) q^{76} +(8.98667 + 12.8343i) q^{77} +(58.3171 + 35.6735i) q^{78} +(-33.9874 - 40.5046i) q^{79} +(-19.1831 + 5.65762i) q^{80} +(-47.4034 + 65.6804i) q^{81} +(66.0140 - 66.0140i) q^{82} +(7.66773 + 0.670839i) q^{83} +(4.04586 + 16.7938i) q^{84} +(-113.163 - 90.2326i) q^{85} +(33.0803 - 12.0402i) q^{86} +(-2.53789 - 16.8902i) q^{87} +(-8.82867 + 12.6087i) q^{88} +(-67.5355 + 38.9917i) q^{89} +(-33.2301 + 54.2749i) q^{90} +(23.1954 - 40.1757i) q^{91} +(-76.0169 + 35.4473i) q^{92} +(96.5332 - 109.285i) q^{93} +(56.1087 - 66.8678i) q^{94} +(-2.24101 + 5.13734i) q^{95} +(-14.1441 + 9.37785i) q^{96} +(86.1483 - 40.1716i) q^{97} +(-55.6124 + 14.9013i) q^{98} +(6.04568 + 48.6035i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	216 * q - 36 * q^6 + 18 * q^7 + 216 * q^8 + 36 * q^11 - 18 * q^15 + 72 * q^20 + 288 * q^21 + 36 * q^22 + 108 * q^23 + 54 * q^25 - 162 * q^27 - 120 * q^30 - 432 * q^31 + 114 * q^33 - 162 * q^35 + 48 * q^36 + 108 * q^38 - 36 * q^40 + 216 * q^41 - 48 * q^42 - 216 * q^43 + 18 * q^45 + 108 * q^46 - 144 * q^47 - 72 * q^48 - 162 * q^50 + 60 * q^51 - 540 * q^53 + 324 * q^55 - 72 * q^56 - 234 * q^57 + 72 * q^58 - 96 * q^60 + 504 * q^61 - 24 * q^63 - 144 * q^66 + 72 * q^67 + 108 * q^68 + 72 * q^70 + 240 * q^72 - 216 * q^73 - 84 * q^75 + 216 * q^76 + 576 * q^77 + 168 * q^78 - 1068 * q^81 + 576 * q^83 + 576 * q^85 - 432 * q^86 - 324 * q^87 - 72 * q^88 - 606 * q^90 + 756 * q^91 - 324 * q^92 + 168 * q^93 - 468 * q^95 - 1332 * 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
							$3$	−2.55916	−	1.56548i	−0.853052	−	0.521825i
$5$	−4.02280	+	2.96936i	−0.804560	+	0.593871i
							$7$	2.35838	+	1.65135i	0.336911	+	0.235907i	0.729772	−	0.683690i	$-0.239626\pi$
−0.392862	+	0.919598i	$0.628515\pi$
$11$	5.11381	+	1.86128i	0.464892	+	0.169207i	0.563837	−	0.825886i	$-0.309325\pi$
							−0.0989449	+	0.995093i	$0.531547\pi$
$13$	−1.40437	−	16.0520i	−0.108028	−	1.23477i	−0.835833	−	0.548983i	$-0.815015\pi$
							0.727805	−	0.685784i	$-0.240540\pi$
$17$	7.49195	+	27.9603i	0.440703	+	1.64473i	0.727039	+	0.686597i	$0.240896\pi$
							−0.286336	+	0.958129i	$0.592437\pi$
$19$	0.970790	−	0.560486i	0.0510942	−	0.0294992i	−0.474235	−	0.880398i	$-0.657275\pi$
							0.525329	+	0.850899i	$0.323942\pi$
$23$	34.3534	−	24.0545i	1.49362	−	1.04585i	0.511141	−	0.859497i	$-0.329223\pi$
							0.982483	−	0.186350i	$-0.0596659\pi$
$29$	3.65957	+	4.36130i	0.126192	+	0.150390i	0.825441	−	0.564489i	$-0.190927\pi$
							−0.699249	+	0.714878i	$0.746482\pi$
$31$	−8.44013	+	47.8664i	−0.272262	+	1.54408i	0.475264	+	0.879843i	$0.342353\pi$
							−0.747526	+	0.664233i	$0.768758\pi$
$37$	−2.23659	−	8.34706i	−0.0604484	−	0.225596i	0.929093	−	0.369846i	$-0.120590\pi$
							−0.989541	+	0.144250i	$0.953923\pi$
$41$	50.5697	+	42.4330i	1.23341	+	1.03495i	0.998010	+	0.0630520i	$0.0200834\pi$
							0.235397	+	0.971899i	$0.424361\pi$
$43$	10.5200	+	22.5603i	0.244652	+	0.524657i	0.989838	−	0.142199i	$-0.0454173\pi$
							−0.745186	+	0.666856i	$0.767640\pi$
$47$	50.5606	+	35.4029i	1.07576	+	0.753254i	0.970478	−	0.241189i	$-0.0775373\pi$
							0.105280	+	0.994443i	$0.466426\pi$

(See $a_n$ instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	270.3.q.a.103.3	yes	216
5.2	odd	4	inner	270.3.q.a.157.7	yes	216
27.16	even	9	inner	270.3.q.a.43.7	✓	216
135.97	odd	36	inner	270.3.q.a.97.3	yes	216

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
270.3.q.a.43.7	✓	216	27.16	even	9	inner
270.3.q.a.97.3	yes	216	135.97	odd	36	inner
270.3.q.a.103.3	yes	216	1.1	even	1	trivial
270.3.q.a.157.7	yes	216	5.2	odd	4	inner