Embedded newform 207.2.m.a.4.15

• Label: 207.2.m.a.4.15
• Level: $207$
• Weight: $2$
• Character: 207.4
• Analytic conductor: $1.653$
• Analytic rank: $0$
• Dimension: $440$
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 207 = 3^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	207.m (of order \(33\), degree \(20\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.65290332184\)
Analytic rank:	\(0\)
Dimension:	\(440\)
Relative dimension:	\(22\) over \(\Q(\zeta_{33})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{33}]$

Embedding invariants

Embedding label			4.15
Character	\(\chi\)	\(=\)	207.4
Dual form			207.2.m.a.52.15

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.680467 - 0.955583i) q^{2} +(0.453546 - 1.67161i) q^{3} +(0.204034 + 0.589516i) q^{4} +(0.783941 - 0.404150i) q^{5} +(-1.28874 - 1.57088i) q^{6} +(0.987285 - 0.776409i) q^{7} +(2.95334 + 0.867179i) q^{8} +(-2.58859 - 1.51631i) q^{9} +(0.147248 - 1.02413i) q^{10} +(-5.40376 + 0.515997i) q^{11} +(1.07798 - 0.0736927i) q^{12} +(3.11925 + 2.45300i) q^{13} +(-0.0701082 - 1.47175i) q^{14} +(-0.320029 - 1.49375i) q^{15} +(1.85759 - 1.46083i) q^{16} +(-1.39651 - 1.61165i) q^{17} +(-3.21041 + 1.44181i) q^{18} +(-2.07689 + 2.39686i) q^{19} +(0.398203 + 0.379686i) q^{20} +(-0.850078 - 2.00250i) q^{21} +(-3.18401 + 5.51486i) q^{22} +(2.05109 - 4.33509i) q^{23} +(2.78907 - 4.54354i) q^{24} +(-2.44906 + 3.43922i) q^{25} +(4.46659 - 1.31151i) q^{26} +(-3.70873 + 3.63941i) q^{27} +(0.659145 + 0.423607i) q^{28} +(-0.580942 + 1.67852i) q^{29} +(-1.64517 - 0.710632i) q^{30} +(-4.17742 + 3.98316i) q^{31} +(0.161007 + 3.37995i) q^{32} +(-1.58831 + 9.26704i) q^{33} +(-2.49035 + 0.237799i) q^{34} +(0.460188 - 1.00767i) q^{35} +(0.365729 - 1.83540i) q^{36} +(8.16473 - 5.24715i) q^{37} +(0.877141 + 3.61562i) q^{38} +(5.51520 - 4.10163i) q^{39} +(2.66572 - 0.513775i) q^{40} +(5.10932 - 2.63404i) q^{41} +(-2.49200 - 0.550314i) q^{42} +(2.70102 + 2.57542i) q^{43} +(-1.40674 - 3.08033i) q^{44} +(-2.64212 - 0.142518i) q^{45} +(-2.74684 - 4.90988i) q^{46} +(4.83497 + 8.37442i) q^{47} +(-1.59943 - 3.76773i) q^{48} +(-1.27839 + 5.26961i) q^{49} +(1.61996 + 4.68055i) q^{50} +(-3.32745 + 1.60346i) q^{51} +(-0.809655 + 2.33934i) q^{52} +(-0.421087 - 2.92873i) q^{53} +(0.954090 + 6.02050i) q^{54} +(-4.02769 + 2.58844i) q^{55} +(3.58907 - 1.43685i) q^{56} +(3.06466 + 4.55885i) q^{57} +(1.20865 + 1.69732i) q^{58} +(-4.34326 - 3.41558i) q^{59} +(0.815292 - 0.493437i) q^{60} +(-1.19045 - 4.90712i) q^{61} +(0.963643 + 6.70228i) q^{62} +(-3.73295 + 0.512778i) q^{63} +(7.31546 + 4.70136i) q^{64} +(3.43669 + 0.662367i) q^{65} +(7.77463 + 7.82368i) q^{66} +(-7.52989 - 0.719017i) q^{67} +(0.665162 - 1.15210i) q^{68} +(-6.31634 - 5.39480i) q^{69} +(-0.649769 - 1.12543i) q^{70} +(-5.07489 - 11.1125i) q^{71} +(-6.33008 - 6.72295i) q^{72} +(1.23411 - 1.42424i) q^{73} +(0.541743 - 11.3726i) q^{74} +(4.63829 + 5.65373i) q^{75} +(-1.83674 - 0.735321i) q^{76} +(-4.93443 + 4.70497i) q^{77} +(-0.166534 - 8.06126i) q^{78} +(-2.14010 + 0.856767i) q^{79} +(0.865850 - 1.89595i) q^{80} +(4.40162 + 7.85021i) q^{81} +(0.959685 - 6.67476i) q^{82} +(-14.0362 - 7.23618i) q^{83} +(1.00706 - 0.909712i) q^{84} +(-1.74613 - 0.699045i) q^{85} +(4.29898 - 0.828561i) q^{86} +(2.54236 + 1.73240i) q^{87} +(-16.4066 - 3.16212i) q^{88} +(-2.53090 + 0.743140i) q^{89} +(-1.93406 + 2.42778i) q^{90} +4.98412 q^{91} +(2.97410 + 0.324649i) q^{92} +(4.76366 + 8.78959i) q^{93} +(11.2925 + 1.07830i) q^{94} +(-0.659470 + 2.71837i) q^{95} +(5.72299 + 1.26382i) q^{96} +(0.546376 - 11.4699i) q^{97} +(4.16564 + 4.80740i) q^{98} +(14.7706 + 6.85807i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	440 * q - 9 * q^2 - 20 * q^3 + 11 * q^4 - 11 * q^5 - 17 * q^6 - 9 * q^7 - 50 * q^8 - 24 * q^9 - 44 * q^10 - 17 * q^11 - 30 * q^12 - 9 * q^13 - 21 * q^14 - 34 * q^15 + 11 * q^16 - 20 * q^17 + 3 * q^18 - 36 * q^19 - 25 * q^20 + 3 * q^21 - 28 * q^22 - 5 * q^23 - 172 * q^24 + 11 * q^25 - 48 * q^26 + 31 * q^27 - 60 * q^28 + 9 * q^29 + 12 * q^30 - 3 * q^31 - 3 * q^32 - 44 * q^33 - 17 * q^34 - 56 * q^35 + 47 * q^36 - 48 * q^37 + 3 * q^38 - 64 * q^39 + 55 * q^40 - 9 * q^41 - 80 * q^42 - 9 * q^43 + 24 * q^44 - 56 * q^45 - 44 * q^46 - 12 * q^47 + 4 * q^48 - 53 * q^49 + 27 * q^50 + 14 * q^51 - 12 * q^52 - 84 * q^53 - 18 * q^54 - 68 * q^55 - 125 * q^56 + 27 * q^57 - 8 * q^58 + 81 * q^59 + 88 * q^60 + 3 * q^61 + 58 * q^62 + 43 * q^63 - 70 * q^64 - 84 * q^65 + 35 * q^66 - 3 * q^67 + 194 * q^68 + 33 * q^69 - 10 * q^70 + 86 * q^71 + 182 * q^72 - 36 * q^73 - 57 * q^74 + 55 * q^75 - 21 * q^76 + 91 * q^77 + 191 * q^78 - 9 * q^79 + 220 * q^80 - 228 * q^81 - 62 * q^82 - 68 * q^83 + 144 * q^84 + q^85 - 53 * q^86 + 160 * q^87 + 83 * q^88 - 28 * q^89 - 314 * q^90 - 108 * q^91 + q^92 + 132 * q^93 + 48 * q^94 - 3 * q^95 + 174 * q^96 - 36 * q^97 - 156 * q^98 - 220 * q^99

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/207\mathbb{Z}\right)^\times\).

\(n\)	\(28\)	\(47\)
\(\chi(n)\)	\(e\left(\frac{2}{11}\right)\)	\(e\left(\frac{1}{3}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.680467	−	0.955583i	0.481163	−	0.675699i	−0.500628	−	0.865662i	\(-0.666898\pi\)
							0.981791	+	0.189964i	\(0.0608370\pi\)
\(3\)	0.453546	−	1.67161i	0.261855	−	0.965107i
							\(5\)	0.783941	−	0.404150i	0.350589	−	0.180741i	−0.273932	−	0.961749i	\(-0.588324\pi\)
0.624521	+	0.781008i	\(0.285294\pi\)
\(7\)	0.987285	−	0.776409i	0.373159	−	0.293455i	−0.413913	−	0.910316i	\(-0.635838\pi\)
							0.787072	+	0.616861i	\(0.211596\pi\)
\(11\)	−5.40376	+	0.515997i	−1.62930	+	0.155579i	−0.869278	−	0.494323i	\(-0.835416\pi\)
							−0.760018	+	0.649902i	\(0.774810\pi\)
\(13\)	3.11925	+	2.45300i	0.865124	+	0.680341i	0.948706	−	0.316160i	\(-0.102394\pi\)
							−0.0835820	+	0.996501i	\(0.526636\pi\)
\(17\)	−1.39651	−	1.61165i	−0.338703	−	0.390884i	0.560690	−	0.828026i	\(-0.310536\pi\)
							−0.899392	+	0.437142i	\(0.855991\pi\)
\(19\)	−2.07689	+	2.39686i	−0.476471	+	0.549877i	−0.942200	−	0.335050i	\(-0.891247\pi\)
							0.465729	+	0.884927i	\(0.345792\pi\)
\(23\)	2.05109	−	4.33509i	0.427682	−	0.903929i
							\(29\)	−0.580942	+	1.67852i	−0.107878	+	0.311693i	−0.986000	−	0.166743i	\(-0.946675\pi\)
0.878122	+	0.478436i	\(0.158796\pi\)
\(31\)	−4.17742	+	3.98316i	−0.750287	+	0.715397i	−0.965280	−	0.261218i	\(-0.915876\pi\)
							0.214993	+	0.976616i	\(0.431027\pi\)
\(37\)	8.16473	−	5.24715i	1.34227	−	0.862626i	0.345159	−	0.938544i	\(-0.387825\pi\)
							0.997114	+	0.0759178i	\(0.0241886\pi\)
\(41\)	5.10932	−	2.63404i	0.797942	−	0.411368i	−0.0104988	−	0.999945i	\(-0.503342\pi\)
							0.808441	+	0.588577i	\(0.200312\pi\)
\(43\)	2.70102	+	2.57542i	0.411902	+	0.392748i	0.867499	−	0.497438i	\(-0.165726\pi\)
							−0.455597	+	0.890186i	\(0.650574\pi\)
\(47\)	4.83497	+	8.37442i	0.705253	+	1.22153i	0.966600	+	0.256290i	\(0.0825000\pi\)
							−0.261347	+	0.965245i	\(0.584167\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	207.2.m.a.4.15	✓	440
3.2	odd	2		621.2.q.a.280.8		440
9.2	odd	6		621.2.q.a.73.15		440
9.7	even	3	inner	207.2.m.a.142.8	yes	440
23.6	even	11	inner	207.2.m.a.121.8	yes	440
69.29	odd	22		621.2.q.a.604.15		440
207.29	odd	66		621.2.q.a.397.8		440
207.52	even	33	inner	207.2.m.a.52.15	yes	440

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
207.2.m.a.4.15	✓	440	1.1	even	1	trivial
207.2.m.a.52.15	yes	440	207.52	even	33	inner
207.2.m.a.121.8	yes	440	23.6	even	11	inner
207.2.m.a.142.8	yes	440	9.7	even	3	inner
621.2.q.a.73.15		440	9.2	odd	6
621.2.q.a.280.8		440	3.2	odd	2
621.2.q.a.397.8		440	207.29	odd	66
621.2.q.a.604.15		440	69.29	odd	22