Embedded newform 261.2.x.a.101.8

• Label: 261.2.x.a.101.8
• Level: $261$
• Weight: $2$
• Character: 261.101
• Analytic conductor: $2.084$
• Analytic rank: $0$
• Dimension: $672$
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 261 = 3^{2} \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	261.x (of order \(84\), degree \(24\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.08409549276\)
Analytic rank:	\(0\)
Dimension:	\(672\)
Relative dimension:	\(28\) over \(\Q(\zeta_{84})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{84}]$

Embedding invariants

Embedding label			101.8
Character	\(\chi\)	\(=\)	261.101
Dual form			261.2.x.a.230.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.571642 + 1.31022i) q^{2} +(1.68128 + 0.416278i) q^{3} +(-0.0295503 - 0.0318477i) q^{4} +(2.61481 + 0.394119i) q^{5} +(-1.50651 + 1.96489i) q^{6} +(-1.32416 - 1.22864i) q^{7} +(-2.63993 + 0.923750i) q^{8} +(2.65343 + 1.39976i) q^{9} +(-2.01111 + 3.20067i) q^{10} +(-0.291375 + 0.250749i) q^{11} +(-0.0364250 - 0.0658461i) q^{12} +(1.83426 - 2.69036i) q^{13} +(2.36673 - 1.03259i) q^{14} +(4.23217 + 1.75111i) q^{15} +(0.305273 - 4.07358i) q^{16} +(0.359515 + 0.359515i) q^{17} +(-3.35080 + 2.67640i) q^{18} +(-6.57779 - 4.13310i) q^{19} +(-0.0647166 - 0.0949219i) q^{20} +(-1.71483 - 2.61691i) q^{21} +(-0.161973 - 0.525104i) q^{22} +(0.0597689 + 0.0234575i) q^{23} +(-4.82300 + 0.454143i) q^{24} +(1.90402 + 0.587312i) q^{25} +(2.47642 + 3.94120i) q^{26} +(3.87847 + 3.45796i) q^{27} +0.0784782i q^{28} +(2.80031 + 4.59981i) q^{29} +(-4.71362 + 4.54405i) q^{30} +(-7.73493 + 5.70864i) q^{31} +(0.217246 + 0.114818i) q^{32} +(-0.594266 + 0.300287i) q^{33} +(-0.676557 + 0.265529i) q^{34} +(-2.97819 - 3.73453i) q^{35} +(-0.0338305 - 0.125869i) q^{36} +(-1.91078 - 5.46069i) q^{37} +(9.17541 - 6.25569i) q^{38} +(4.20384 - 3.75970i) q^{39} +(-7.26696 + 1.37498i) q^{40} +(-1.60421 - 0.429846i) q^{41} +(4.40899 - 0.750866i) q^{42} +(1.21456 - 1.64567i) q^{43} +(0.0165960 + 0.00186992i) q^{44} +(6.38652 + 4.70587i) q^{45} +(-0.0649009 + 0.0649009i) q^{46} +(-0.331957 - 0.385741i) q^{47} +(2.20899 - 6.72176i) q^{48} +(-0.279270 - 3.72659i) q^{49} +(-1.85792 + 2.15895i) q^{50} +(0.454788 + 0.754104i) q^{51} +(-0.139885 + 0.0210842i) q^{52} +(1.68105 + 1.34059i) q^{53} +(-6.74777 + 3.10493i) q^{54} +(-0.860715 + 0.540823i) q^{55} +(4.63064 + 2.02033i) q^{56} +(-9.33861 - 9.68710i) q^{57} +(-7.62753 + 1.03957i) q^{58} +(3.87386 - 2.23658i) q^{59} +(-0.0692931 - 0.186531i) q^{60} +(9.16119 - 0.342788i) q^{61} +(-3.05795 - 13.3977i) q^{62} +(-1.79376 - 5.11361i) q^{63} +(6.11295 - 4.87491i) q^{64} +(5.85654 - 6.31185i) q^{65} +(-0.0537336 - 0.950274i) q^{66} +(13.5780 - 1.01753i) q^{67} +(0.000825933 - 0.0220735i) q^{68} +(0.0907235 + 0.0643192i) q^{69} +(6.59551 - 1.76726i) q^{70} +(-14.0494 + 6.76584i) q^{71} +(-8.29788 - 1.24416i) q^{72} +(-0.264280 + 2.34555i) q^{73} +(8.24697 + 0.618025i) q^{74} +(2.95671 + 1.78004i) q^{75} +(0.0627463 + 0.331622i) q^{76} +(0.693908 + 0.0259642i) q^{77} +(2.52293 + 7.65715i) q^{78} +(1.45117 - 7.66960i) q^{79} +(2.40370 - 10.5313i) q^{80} +(5.08134 + 7.42832i) q^{81} +(1.48022 - 1.85614i) q^{82} +(0.0736019 - 0.238612i) q^{83} +(-0.0326687 + 0.131944i) q^{84} +(0.798370 + 1.08175i) q^{85} +(1.46189 + 2.53208i) q^{86} +(2.79331 + 8.89929i) q^{87} +(0.537580 - 0.931116i) q^{88} +(8.94513 - 1.00787i) q^{89} +(-9.81652 + 5.67766i) q^{90} +(-5.73433 + 1.30882i) q^{91} +(-0.00101912 - 0.00259668i) q^{92} +(-15.3810 + 6.37796i) q^{93} +(0.695166 - 0.214430i) q^{94} +(-15.5707 - 13.3997i) q^{95} +(0.317455 + 0.283475i) q^{96} +(5.87063 + 11.1078i) q^{97} +(5.04229 + 1.76437i) q^{98} +(-1.12413 + 0.257487i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	672 * q - 36 * q^2 - 24 * q^3 - 14 * q^4 - 42 * q^5 - 28 * q^6 - 10 * q^7 - 28 * q^9 - 56 * q^10 - 48 * q^11 - 10 * q^12 - 14 * q^13 - 24 * q^14 - 20 * q^15 - 54 * q^16 - 50 * q^18 - 48 * q^19 - 30 * q^20 - 80 * q^21 - 14 * q^22 - 30 * q^23 - 16 * q^24 + 30 * q^25 + 36 * q^27 - 168 * q^30 - 12 * q^31 + 24 * q^32 - 70 * q^33 - 14 * q^34 + 128 * q^36 - 48 * q^37 - 42 * q^38 - 44 * q^39 - 2 * q^40 - 24 * q^41 - 28 * q^42 - 12 * q^43 + 24 * q^45 - 64 * q^46 - 42 * q^47 - 28 * q^48 + 22 * q^49 - 66 * q^50 - 28 * q^51 + 22 * q^52 - 36 * q^54 + 76 * q^55 - 42 * q^56 - 42 * q^58 - 132 * q^59 - 20 * q^60 - 28 * q^63 - 308 * q^64 - 66 * q^65 - 46 * q^66 - 14 * q^67 + 60 * q^68 - 64 * q^69 - 14 * q^70 + 56 * q^72 - 48 * q^73 + 66 * q^74 + 24 * q^75 - 28 * q^76 + 30 * q^77 + 72 * q^78 - 12 * q^79 + 140 * q^81 - 136 * q^82 + 246 * q^83 + 238 * q^84 + 34 * q^85 + 118 * q^87 + 8 * q^88 - 48 * q^90 - 56 * q^91 + 462 * q^92 + 28 * q^93 + 26 * q^94 - 246 * q^95 + 126 * q^96 - 36 * q^97 - 16 * q^99

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/261\mathbb{Z}\right)^\times\).

\(n\)	\(118\)	\(146\)
\(\chi(n)\)	\(e\left(\frac{13}{28}\right)\)	\(e\left(\frac{1}{6}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.571642	+	1.31022i	−0.404212	+	0.926464i	0.588781	+	0.808292i	\(0.299608\pi\)
							−0.992993	+	0.118171i	\(0.962297\pi\)
\(3\)	1.68128	+	0.416278i	0.970689	+	0.240338i
							\(5\)	2.61481	+	0.394119i	1.16938	+	0.176255i	0.704885	−	0.709322i	\(-0.250999\pi\)
0.464492	+	0.885577i	\(0.346237\pi\)
\(7\)	−1.32416	−	1.22864i	−0.500485	−	0.464383i	0.389071	−	0.921208i	\(-0.372796\pi\)
							−0.889557	+	0.456825i	\(0.848987\pi\)
\(11\)	−0.291375	+	0.250749i	−0.0878530	+	0.0756036i	−0.695118	−	0.718896i	\(-0.744648\pi\)
							0.607265	+	0.794500i	\(0.292267\pi\)
\(13\)	1.83426	−	2.69036i	0.508731	−	0.746171i	−0.482844	−	0.875706i	\(-0.660396\pi\)
							0.991575	+	0.129535i	\(0.0413485\pi\)
\(17\)	0.359515	+	0.359515i	0.0871952	+	0.0871952i	0.749359	−	0.662164i	\(-0.230362\pi\)
							−0.662164	+	0.749359i	\(0.730362\pi\)
\(19\)	−6.57779	−	4.13310i	−1.50905	−	0.948198i	−0.995954	−	0.0898667i	\(-0.971356\pi\)
							−0.513095	−	0.858332i	\(-0.671501\pi\)
\(23\)	0.0597689	+	0.0234575i	0.0124627	+	0.00489124i	0.371564	−	0.928407i	\(-0.378822\pi\)
							−0.359102	+	0.933299i	\(0.616917\pi\)
\(29\)	2.80031	+	4.59981i	0.520004	+	0.854164i
							\(31\)	−7.73493	+	5.70864i	−1.38924	+	1.02530i	−0.394933	+	0.918710i	\(0.629232\pi\)
−0.994303	+	0.106592i	\(0.966006\pi\)
\(37\)	−1.91078	−	5.46069i	−0.314130	−	0.897732i	−0.987387	−	0.158324i	\(-0.949391\pi\)
							0.673257	−	0.739408i	\(-0.264895\pi\)
\(41\)	−1.60421	−	0.429846i	−0.250535	−	0.0671306i	0.131366	−	0.991334i	\(-0.458064\pi\)
							−0.381901	+	0.924203i	\(0.624730\pi\)
\(43\)	1.21456	−	1.64567i	0.185219	−	0.250963i	−0.702143	−	0.712036i	\(-0.747773\pi\)
							0.887362	+	0.461073i	\(0.152535\pi\)
\(47\)	−0.331957	−	0.385741i	−0.0484209	−	0.0562661i	0.733239	−	0.679971i	\(-0.238008\pi\)
							−0.781660	+	0.623705i	\(0.785627\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	261.2.x.a.101.8	yes	672
3.2	odd	2		783.2.be.a.710.21		672
9.4	even	3		783.2.be.a.449.8		672
9.5	odd	6	inner	261.2.x.a.14.21	✓	672
29.27	odd	28	inner	261.2.x.a.56.21	yes	672
87.56	even	28		783.2.be.a.143.8		672
261.85	odd	84		783.2.be.a.665.21		672
261.230	even	84	inner	261.2.x.a.230.8	yes	672

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
261.2.x.a.14.21	✓	672	9.5	odd	6	inner
261.2.x.a.56.21	yes	672	29.27	odd	28	inner
261.2.x.a.101.8	yes	672	1.1	even	1	trivial
261.2.x.a.230.8	yes	672	261.230	even	84	inner
783.2.be.a.143.8		672	87.56	even	28
783.2.be.a.449.8		672	9.4	even	3
783.2.be.a.665.21		672	261.85	odd	84
783.2.be.a.710.21		672	3.2	odd	2