# Properties

 Label 825.2.a.n.1.3 Level $825$ Weight $2$ Character 825.1 Self dual yes Analytic conductor $6.588$ Analytic rank $0$ Dimension $3$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$825 = 3 \cdot 5^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 825.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$6.58765816676$$ Analytic rank: $$0$$ Dimension: $$3$$ Coefficient field: 3.3.148.1 Defining polynomial: $$x^{3} - x^{2} - 3x + 1$$ x^3 - x^2 - 3*x + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 165) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.3 Root $$-1.48119$$ of defining polynomial Character $$\chi$$ $$=$$ 825.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+2.67513 q^{2} -1.00000 q^{3} +5.15633 q^{4} -2.67513 q^{6} +2.80606 q^{7} +8.44358 q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q+2.67513 q^{2} -1.00000 q^{3} +5.15633 q^{4} -2.67513 q^{6} +2.80606 q^{7} +8.44358 q^{8} +1.00000 q^{9} -1.00000 q^{11} -5.15633 q^{12} -5.11871 q^{13} +7.50659 q^{14} +12.2750 q^{16} +4.54420 q^{17} +2.67513 q^{18} -4.57452 q^{19} -2.80606 q^{21} -2.67513 q^{22} +4.00000 q^{23} -8.44358 q^{24} -13.6932 q^{26} -1.00000 q^{27} +14.4690 q^{28} -2.38787 q^{29} -0.962389 q^{31} +15.9502 q^{32} +1.00000 q^{33} +12.1563 q^{34} +5.15633 q^{36} +1.61213 q^{37} -12.2374 q^{38} +5.11871 q^{39} -2.38787 q^{41} -7.50659 q^{42} +2.80606 q^{43} -5.15633 q^{44} +10.7005 q^{46} -4.31265 q^{47} -12.2750 q^{48} +0.873992 q^{49} -4.54420 q^{51} -26.3938 q^{52} -6.57452 q^{53} -2.67513 q^{54} +23.6932 q^{56} +4.57452 q^{57} -6.38787 q^{58} -13.2750 q^{59} +7.92478 q^{61} -2.57452 q^{62} +2.80606 q^{63} +18.1187 q^{64} +2.67513 q^{66} +10.7005 q^{67} +23.4314 q^{68} -4.00000 q^{69} -7.35026 q^{71} +8.44358 q^{72} +6.41819 q^{73} +4.31265 q^{74} -23.5877 q^{76} -2.80606 q^{77} +13.6932 q^{78} +1.35026 q^{79} +1.00000 q^{81} -6.38787 q^{82} -0.806063 q^{83} -14.4690 q^{84} +7.50659 q^{86} +2.38787 q^{87} -8.44358 q^{88} -2.96239 q^{89} -14.3634 q^{91} +20.6253 q^{92} +0.962389 q^{93} -11.5369 q^{94} -15.9502 q^{96} -9.92478 q^{97} +2.33804 q^{98} -1.00000 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q + 3 q^{2} - 3 q^{3} + 5 q^{4} - 3 q^{6} + 8 q^{7} + 9 q^{8} + 3 q^{9}+O(q^{10})$$ 3 * q + 3 * q^2 - 3 * q^3 + 5 * q^4 - 3 * q^6 + 8 * q^7 + 9 * q^8 + 3 * q^9 $$3 q + 3 q^{2} - 3 q^{3} + 5 q^{4} - 3 q^{6} + 8 q^{7} + 9 q^{8} + 3 q^{9} - 3 q^{11} - 5 q^{12} + 6 q^{13} + 2 q^{14} + 5 q^{16} + 4 q^{17} + 3 q^{18} - 2 q^{19} - 8 q^{21} - 3 q^{22} + 12 q^{23} - 9 q^{24} - 8 q^{26} - 3 q^{27} + 12 q^{28} - 8 q^{29} + 8 q^{31} + 11 q^{32} + 3 q^{33} + 26 q^{34} + 5 q^{36} + 4 q^{37} + 6 q^{38} - 6 q^{39} - 8 q^{41} - 2 q^{42} + 8 q^{43} - 5 q^{44} + 12 q^{46} + 8 q^{47} - 5 q^{48} + 11 q^{49} - 4 q^{51} - 26 q^{52} - 8 q^{53} - 3 q^{54} + 38 q^{56} + 2 q^{57} - 20 q^{58} - 8 q^{59} + 2 q^{61} + 4 q^{62} + 8 q^{63} + 33 q^{64} + 3 q^{66} + 12 q^{67} + 28 q^{68} - 12 q^{69} - 12 q^{71} + 9 q^{72} + 18 q^{73} - 8 q^{74} - 18 q^{76} - 8 q^{77} + 8 q^{78} - 6 q^{79} + 3 q^{81} - 20 q^{82} - 2 q^{83} - 12 q^{84} + 2 q^{86} + 8 q^{87} - 9 q^{88} + 2 q^{89} + 8 q^{91} + 20 q^{92} - 8 q^{93} - 12 q^{94} - 11 q^{96} - 8 q^{97} - 29 q^{98} - 3 q^{99}+O(q^{100})$$ 3 * q + 3 * q^2 - 3 * q^3 + 5 * q^4 - 3 * q^6 + 8 * q^7 + 9 * q^8 + 3 * q^9 - 3 * q^11 - 5 * q^12 + 6 * q^13 + 2 * q^14 + 5 * q^16 + 4 * q^17 + 3 * q^18 - 2 * q^19 - 8 * q^21 - 3 * q^22 + 12 * q^23 - 9 * q^24 - 8 * q^26 - 3 * q^27 + 12 * q^28 - 8 * q^29 + 8 * q^31 + 11 * q^32 + 3 * q^33 + 26 * q^34 + 5 * q^36 + 4 * q^37 + 6 * q^38 - 6 * q^39 - 8 * q^41 - 2 * q^42 + 8 * q^43 - 5 * q^44 + 12 * q^46 + 8 * q^47 - 5 * q^48 + 11 * q^49 - 4 * q^51 - 26 * q^52 - 8 * q^53 - 3 * q^54 + 38 * q^56 + 2 * q^57 - 20 * q^58 - 8 * q^59 + 2 * q^61 + 4 * q^62 + 8 * q^63 + 33 * q^64 + 3 * q^66 + 12 * q^67 + 28 * q^68 - 12 * q^69 - 12 * q^71 + 9 * q^72 + 18 * q^73 - 8 * q^74 - 18 * q^76 - 8 * q^77 + 8 * q^78 - 6 * q^79 + 3 * q^81 - 20 * q^82 - 2 * q^83 - 12 * q^84 + 2 * q^86 + 8 * q^87 - 9 * q^88 + 2 * q^89 + 8 * q^91 + 20 * q^92 - 8 * q^93 - 12 * q^94 - 11 * q^96 - 8 * q^97 - 29 * q^98 - 3 * q^99

## Coefficient data

For each $$n$$ we display the coefficients of the $$q$$-expansion $$a_n$$, the Satake parameters $$\alpha_p$$, and the Satake angles $$\theta_p = \textrm{Arg}(\alpha_p)$$.

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 2.67513 1.89160 0.945802 0.324745i $$-0.105279\pi$$
0.945802 + 0.324745i $$0.105279\pi$$
$$3$$ −1.00000 −0.577350
$$4$$ 5.15633 2.57816
$$5$$ 0 0
$$6$$ −2.67513 −1.09212
$$7$$ 2.80606 1.06059 0.530296 0.847812i $$-0.322081\pi$$
0.530296 + 0.847812i $$0.322081\pi$$
$$8$$ 8.44358 2.98526
$$9$$ 1.00000 0.333333
$$10$$ 0 0
$$11$$ −1.00000 −0.301511
$$12$$ −5.15633 −1.48850
$$13$$ −5.11871 −1.41968 −0.709838 0.704365i $$-0.751232\pi$$
−0.709838 + 0.704365i $$0.751232\pi$$
$$14$$ 7.50659 2.00622
$$15$$ 0 0
$$16$$ 12.2750 3.06876
$$17$$ 4.54420 1.10213 0.551065 0.834462i $$-0.314222\pi$$
0.551065 + 0.834462i $$0.314222\pi$$
$$18$$ 2.67513 0.630534
$$19$$ −4.57452 −1.04947 −0.524733 0.851267i $$-0.675835\pi$$
−0.524733 + 0.851267i $$0.675835\pi$$
$$20$$ 0 0
$$21$$ −2.80606 −0.612333
$$22$$ −2.67513 −0.570340
$$23$$ 4.00000 0.834058 0.417029 0.908893i $$-0.363071\pi$$
0.417029 + 0.908893i $$0.363071\pi$$
$$24$$ −8.44358 −1.72354
$$25$$ 0 0
$$26$$ −13.6932 −2.68546
$$27$$ −1.00000 −0.192450
$$28$$ 14.4690 2.73438
$$29$$ −2.38787 −0.443417 −0.221708 0.975113i $$-0.571163\pi$$
−0.221708 + 0.975113i $$0.571163\pi$$
$$30$$ 0 0
$$31$$ −0.962389 −0.172850 −0.0864250 0.996258i $$-0.527544\pi$$
−0.0864250 + 0.996258i $$0.527544\pi$$
$$32$$ 15.9502 2.81962
$$33$$ 1.00000 0.174078
$$34$$ 12.1563 2.08479
$$35$$ 0 0
$$36$$ 5.15633 0.859388
$$37$$ 1.61213 0.265032 0.132516 0.991181i $$-0.457694\pi$$
0.132516 + 0.991181i $$0.457694\pi$$
$$38$$ −12.2374 −1.98517
$$39$$ 5.11871 0.819650
$$40$$ 0 0
$$41$$ −2.38787 −0.372923 −0.186462 0.982462i $$-0.559702\pi$$
−0.186462 + 0.982462i $$0.559702\pi$$
$$42$$ −7.50659 −1.15829
$$43$$ 2.80606 0.427921 0.213960 0.976842i $$-0.431364\pi$$
0.213960 + 0.976842i $$0.431364\pi$$
$$44$$ −5.15633 −0.777345
$$45$$ 0 0
$$46$$ 10.7005 1.57771
$$47$$ −4.31265 −0.629065 −0.314532 0.949247i $$-0.601848\pi$$
−0.314532 + 0.949247i $$0.601848\pi$$
$$48$$ −12.2750 −1.77175
$$49$$ 0.873992 0.124856
$$50$$ 0 0
$$51$$ −4.54420 −0.636315
$$52$$ −26.3938 −3.66015
$$53$$ −6.57452 −0.903079 −0.451540 0.892251i $$-0.649125\pi$$
−0.451540 + 0.892251i $$0.649125\pi$$
$$54$$ −2.67513 −0.364039
$$55$$ 0 0
$$56$$ 23.6932 3.16614
$$57$$ 4.57452 0.605909
$$58$$ −6.38787 −0.838769
$$59$$ −13.2750 −1.72826 −0.864131 0.503266i $$-0.832132\pi$$
−0.864131 + 0.503266i $$0.832132\pi$$
$$60$$ 0 0
$$61$$ 7.92478 1.01466 0.507332 0.861751i $$-0.330632\pi$$
0.507332 + 0.861751i $$0.330632\pi$$
$$62$$ −2.57452 −0.326964
$$63$$ 2.80606 0.353531
$$64$$ 18.1187 2.26484
$$65$$ 0 0
$$66$$ 2.67513 0.329286
$$67$$ 10.7005 1.30728 0.653639 0.756807i $$-0.273242\pi$$
0.653639 + 0.756807i $$0.273242\pi$$
$$68$$ 23.4314 2.84147
$$69$$ −4.00000 −0.481543
$$70$$ 0 0
$$71$$ −7.35026 −0.872316 −0.436158 0.899870i $$-0.643661\pi$$
−0.436158 + 0.899870i $$0.643661\pi$$
$$72$$ 8.44358 0.995086
$$73$$ 6.41819 0.751192 0.375596 0.926783i $$-0.377438\pi$$
0.375596 + 0.926783i $$0.377438\pi$$
$$74$$ 4.31265 0.501335
$$75$$ 0 0
$$76$$ −23.5877 −2.70569
$$77$$ −2.80606 −0.319781
$$78$$ 13.6932 1.55045
$$79$$ 1.35026 0.151916 0.0759582 0.997111i $$-0.475798\pi$$
0.0759582 + 0.997111i $$0.475798\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ −6.38787 −0.705423
$$83$$ −0.806063 −0.0884770 −0.0442385 0.999021i $$-0.514086\pi$$
−0.0442385 + 0.999021i $$0.514086\pi$$
$$84$$ −14.4690 −1.57869
$$85$$ 0 0
$$86$$ 7.50659 0.809456
$$87$$ 2.38787 0.256007
$$88$$ −8.44358 −0.900089
$$89$$ −2.96239 −0.314013 −0.157006 0.987598i $$-0.550184\pi$$
−0.157006 + 0.987598i $$0.550184\pi$$
$$90$$ 0 0
$$91$$ −14.3634 −1.50570
$$92$$ 20.6253 2.15034
$$93$$ 0.962389 0.0997950
$$94$$ −11.5369 −1.18994
$$95$$ 0 0
$$96$$ −15.9502 −1.62791
$$97$$ −9.92478 −1.00771 −0.503854 0.863789i $$-0.668085\pi$$
−0.503854 + 0.863789i $$0.668085\pi$$
$$98$$ 2.33804 0.236178
$$99$$ −1.00000 −0.100504
$$100$$ 0 0
$$101$$ −13.6121 −1.35446 −0.677229 0.735773i $$-0.736819\pi$$
−0.677229 + 0.735773i $$0.736819\pi$$
$$102$$ −12.1563 −1.20366
$$103$$ 16.3127 1.60733 0.803667 0.595080i $$-0.202880\pi$$
0.803667 + 0.595080i $$0.202880\pi$$
$$104$$ −43.2203 −4.23810
$$105$$ 0 0
$$106$$ −17.5877 −1.70827
$$107$$ −9.43136 −0.911764 −0.455882 0.890040i $$-0.650676\pi$$
−0.455882 + 0.890040i $$0.650676\pi$$
$$108$$ −5.15633 −0.496168
$$109$$ −15.4010 −1.47515 −0.737576 0.675264i $$-0.764030\pi$$
−0.737576 + 0.675264i $$0.764030\pi$$
$$110$$ 0 0
$$111$$ −1.61213 −0.153016
$$112$$ 34.4445 3.25470
$$113$$ −13.7381 −1.29238 −0.646188 0.763179i $$-0.723638\pi$$
−0.646188 + 0.763179i $$0.723638\pi$$
$$114$$ 12.2374 1.14614
$$115$$ 0 0
$$116$$ −12.3127 −1.14320
$$117$$ −5.11871 −0.473225
$$118$$ −35.5125 −3.26919
$$119$$ 12.7513 1.16891
$$120$$ 0 0
$$121$$ 1.00000 0.0909091
$$122$$ 21.1998 1.91934
$$123$$ 2.38787 0.215307
$$124$$ −4.96239 −0.445636
$$125$$ 0 0
$$126$$ 7.50659 0.668740
$$127$$ 12.4182 1.10194 0.550968 0.834526i $$-0.314259\pi$$
0.550968 + 0.834526i $$0.314259\pi$$
$$128$$ 16.5696 1.46456
$$129$$ −2.80606 −0.247060
$$130$$ 0 0
$$131$$ 5.92478 0.517650 0.258825 0.965924i $$-0.416665\pi$$
0.258825 + 0.965924i $$0.416665\pi$$
$$132$$ 5.15633 0.448800
$$133$$ −12.8364 −1.11306
$$134$$ 28.6253 2.47285
$$135$$ 0 0
$$136$$ 38.3693 3.29014
$$137$$ 9.79877 0.837165 0.418583 0.908179i $$-0.362527\pi$$
0.418583 + 0.908179i $$0.362527\pi$$
$$138$$ −10.7005 −0.910889
$$139$$ −2.12601 −0.180326 −0.0901628 0.995927i $$-0.528739\pi$$
−0.0901628 + 0.995927i $$0.528739\pi$$
$$140$$ 0 0
$$141$$ 4.31265 0.363191
$$142$$ −19.6629 −1.65007
$$143$$ 5.11871 0.428048
$$144$$ 12.2750 1.02292
$$145$$ 0 0
$$146$$ 17.1695 1.42096
$$147$$ −0.873992 −0.0720856
$$148$$ 8.31265 0.683296
$$149$$ 8.31265 0.680999 0.340499 0.940245i $$-0.389404\pi$$
0.340499 + 0.940245i $$0.389404\pi$$
$$150$$ 0 0
$$151$$ −15.2750 −1.24307 −0.621533 0.783388i $$-0.713490\pi$$
−0.621533 + 0.783388i $$0.713490\pi$$
$$152$$ −38.6253 −3.13293
$$153$$ 4.54420 0.367377
$$154$$ −7.50659 −0.604898
$$155$$ 0 0
$$156$$ 26.3938 2.11319
$$157$$ 7.01317 0.559712 0.279856 0.960042i $$-0.409713\pi$$
0.279856 + 0.960042i $$0.409713\pi$$
$$158$$ 3.61213 0.287365
$$159$$ 6.57452 0.521393
$$160$$ 0 0
$$161$$ 11.2243 0.884595
$$162$$ 2.67513 0.210178
$$163$$ 6.76116 0.529575 0.264787 0.964307i $$-0.414698\pi$$
0.264787 + 0.964307i $$0.414698\pi$$
$$164$$ −12.3127 −0.961456
$$165$$ 0 0
$$166$$ −2.15633 −0.167363
$$167$$ −11.8192 −0.914600 −0.457300 0.889312i $$-0.651183\pi$$
−0.457300 + 0.889312i $$0.651183\pi$$
$$168$$ −23.6932 −1.82797
$$169$$ 13.2012 1.01548
$$170$$ 0 0
$$171$$ −4.57452 −0.349822
$$172$$ 14.4690 1.10325
$$173$$ 6.99271 0.531646 0.265823 0.964022i $$-0.414356\pi$$
0.265823 + 0.964022i $$0.414356\pi$$
$$174$$ 6.38787 0.484263
$$175$$ 0 0
$$176$$ −12.2750 −0.925266
$$177$$ 13.2750 0.997813
$$178$$ −7.92478 −0.593987
$$179$$ −2.70052 −0.201847 −0.100923 0.994894i $$-0.532180\pi$$
−0.100923 + 0.994894i $$0.532180\pi$$
$$180$$ 0 0
$$181$$ −21.1998 −1.57577 −0.787885 0.615822i $$-0.788824\pi$$
−0.787885 + 0.615822i $$0.788824\pi$$
$$182$$ −38.4241 −2.84818
$$183$$ −7.92478 −0.585816
$$184$$ 33.7743 2.48988
$$185$$ 0 0
$$186$$ 2.57452 0.188773
$$187$$ −4.54420 −0.332305
$$188$$ −22.2374 −1.62183
$$189$$ −2.80606 −0.204111
$$190$$ 0 0
$$191$$ 13.1490 0.951430 0.475715 0.879599i $$-0.342189\pi$$
0.475715 + 0.879599i $$0.342189\pi$$
$$192$$ −18.1187 −1.30761
$$193$$ 5.89446 0.424293 0.212146 0.977238i $$-0.431955\pi$$
0.212146 + 0.977238i $$0.431955\pi$$
$$194$$ −26.5501 −1.90618
$$195$$ 0 0
$$196$$ 4.50659 0.321899
$$197$$ 20.3938 1.45299 0.726497 0.687169i $$-0.241147\pi$$
0.726497 + 0.687169i $$0.241147\pi$$
$$198$$ −2.67513 −0.190113
$$199$$ 17.4010 1.23353 0.616764 0.787148i $$-0.288443\pi$$
0.616764 + 0.787148i $$0.288443\pi$$
$$200$$ 0 0
$$201$$ −10.7005 −0.754757
$$202$$ −36.4142 −2.56210
$$203$$ −6.70052 −0.470285
$$204$$ −23.4314 −1.64052
$$205$$ 0 0
$$206$$ 43.6385 3.04044
$$207$$ 4.00000 0.278019
$$208$$ −62.8324 −4.35664
$$209$$ 4.57452 0.316426
$$210$$ 0 0
$$211$$ 18.1260 1.24785 0.623923 0.781486i $$-0.285538\pi$$
0.623923 + 0.781486i $$0.285538\pi$$
$$212$$ −33.9003 −2.32828
$$213$$ 7.35026 0.503632
$$214$$ −25.2301 −1.72470
$$215$$ 0 0
$$216$$ −8.44358 −0.574513
$$217$$ −2.70052 −0.183323
$$218$$ −41.1998 −2.79040
$$219$$ −6.41819 −0.433701
$$220$$ 0 0
$$221$$ −23.2605 −1.56467
$$222$$ −4.31265 −0.289446
$$223$$ 23.6385 1.58295 0.791475 0.611202i $$-0.209314\pi$$
0.791475 + 0.611202i $$0.209314\pi$$
$$224$$ 44.7572 2.99047
$$225$$ 0 0
$$226$$ −36.7513 −2.44466
$$227$$ −1.26916 −0.0842371 −0.0421185 0.999113i $$-0.513411\pi$$
−0.0421185 + 0.999113i $$0.513411\pi$$
$$228$$ 23.5877 1.56213
$$229$$ 16.1768 1.06899 0.534496 0.845171i $$-0.320501\pi$$
0.534496 + 0.845171i $$0.320501\pi$$
$$230$$ 0 0
$$231$$ 2.80606 0.184625
$$232$$ −20.1622 −1.32371
$$233$$ −18.4690 −1.20994 −0.604971 0.796247i $$-0.706815\pi$$
−0.604971 + 0.796247i $$0.706815\pi$$
$$234$$ −13.6932 −0.895154
$$235$$ 0 0
$$236$$ −68.4504 −4.45574
$$237$$ −1.35026 −0.0877089
$$238$$ 34.1114 2.21111
$$239$$ 19.3258 1.25008 0.625042 0.780591i $$-0.285082\pi$$
0.625042 + 0.780591i $$0.285082\pi$$
$$240$$ 0 0
$$241$$ 28.5501 1.83907 0.919536 0.393006i $$-0.128565\pi$$
0.919536 + 0.393006i $$0.128565\pi$$
$$242$$ 2.67513 0.171964
$$243$$ −1.00000 −0.0641500
$$244$$ 40.8627 2.61597
$$245$$ 0 0
$$246$$ 6.38787 0.407276
$$247$$ 23.4156 1.48990
$$248$$ −8.12601 −0.516002
$$249$$ 0.806063 0.0510822
$$250$$ 0 0
$$251$$ 23.1998 1.46436 0.732180 0.681112i $$-0.238503\pi$$
0.732180 + 0.681112i $$0.238503\pi$$
$$252$$ 14.4690 0.911460
$$253$$ −4.00000 −0.251478
$$254$$ 33.2203 2.08443
$$255$$ 0 0
$$256$$ 8.08840 0.505525
$$257$$ −10.8872 −0.679123 −0.339561 0.940584i $$-0.610279\pi$$
−0.339561 + 0.940584i $$0.610279\pi$$
$$258$$ −7.50659 −0.467340
$$259$$ 4.52373 0.281091
$$260$$ 0 0
$$261$$ −2.38787 −0.147806
$$262$$ 15.8496 0.979189
$$263$$ 9.11871 0.562284 0.281142 0.959666i $$-0.409287\pi$$
0.281142 + 0.959666i $$0.409287\pi$$
$$264$$ 8.44358 0.519667
$$265$$ 0 0
$$266$$ −34.3390 −2.10546
$$267$$ 2.96239 0.181295
$$268$$ 55.1754 3.37037
$$269$$ −11.2995 −0.688941 −0.344471 0.938797i $$-0.611942\pi$$
−0.344471 + 0.938797i $$0.611942\pi$$
$$270$$ 0 0
$$271$$ 25.1998 1.53078 0.765390 0.643567i $$-0.222546\pi$$
0.765390 + 0.643567i $$0.222546\pi$$
$$272$$ 55.7802 3.38217
$$273$$ 14.3634 0.869315
$$274$$ 26.2130 1.58358
$$275$$ 0 0
$$276$$ −20.6253 −1.24150
$$277$$ −2.41819 −0.145295 −0.0726475 0.997358i $$-0.523145\pi$$
−0.0726475 + 0.997358i $$0.523145\pi$$
$$278$$ −5.68735 −0.341105
$$279$$ −0.962389 −0.0576167
$$280$$ 0 0
$$281$$ 30.4894 1.81885 0.909424 0.415870i $$-0.136523\pi$$
0.909424 + 0.415870i $$0.136523\pi$$
$$282$$ 11.5369 0.687013
$$283$$ 8.35756 0.496805 0.248403 0.968657i $$-0.420094\pi$$
0.248403 + 0.968657i $$0.420094\pi$$
$$284$$ −37.9003 −2.24897
$$285$$ 0 0
$$286$$ 13.6932 0.809698
$$287$$ −6.70052 −0.395519
$$288$$ 15.9502 0.939873
$$289$$ 3.64974 0.214690
$$290$$ 0 0
$$291$$ 9.92478 0.581801
$$292$$ 33.0943 1.93670
$$293$$ −23.0943 −1.34918 −0.674591 0.738192i $$-0.735680\pi$$
−0.674591 + 0.738192i $$0.735680\pi$$
$$294$$ −2.33804 −0.136357
$$295$$ 0 0
$$296$$ 13.6121 0.791189
$$297$$ 1.00000 0.0580259
$$298$$ 22.2374 1.28818
$$299$$ −20.4749 −1.18409
$$300$$ 0 0
$$301$$ 7.87399 0.453849
$$302$$ −40.8627 −2.35139
$$303$$ 13.6121 0.781996
$$304$$ −56.1524 −3.22056
$$305$$ 0 0
$$306$$ 12.1563 0.694931
$$307$$ 2.65562 0.151564 0.0757821 0.997124i $$-0.475855\pi$$
0.0757821 + 0.997124i $$0.475855\pi$$
$$308$$ −14.4690 −0.824446
$$309$$ −16.3127 −0.927994
$$310$$ 0 0
$$311$$ −15.9756 −0.905891 −0.452946 0.891538i $$-0.649627\pi$$
−0.452946 + 0.891538i $$0.649627\pi$$
$$312$$ 43.2203 2.44687
$$313$$ −0.0606343 −0.00342726 −0.00171363 0.999999i $$-0.500545\pi$$
−0.00171363 + 0.999999i $$0.500545\pi$$
$$314$$ 18.7612 1.05875
$$315$$ 0 0
$$316$$ 6.96239 0.391665
$$317$$ 3.81336 0.214180 0.107090 0.994249i $$-0.465847\pi$$
0.107090 + 0.994249i $$0.465847\pi$$
$$318$$ 17.5877 0.986269
$$319$$ 2.38787 0.133695
$$320$$ 0 0
$$321$$ 9.43136 0.526407
$$322$$ 30.0263 1.67330
$$323$$ −20.7875 −1.15665
$$324$$ 5.15633 0.286463
$$325$$ 0 0
$$326$$ 18.0870 1.00175
$$327$$ 15.4010 0.851680
$$328$$ −20.1622 −1.11327
$$329$$ −12.1016 −0.667181
$$330$$ 0 0
$$331$$ 24.2882 1.33500 0.667500 0.744609i $$-0.267364\pi$$
0.667500 + 0.744609i $$0.267364\pi$$
$$332$$ −4.15633 −0.228108
$$333$$ 1.61213 0.0883440
$$334$$ −31.6180 −1.73006
$$335$$ 0 0
$$336$$ −34.4445 −1.87910
$$337$$ −22.5804 −1.23003 −0.615016 0.788514i $$-0.710851\pi$$
−0.615016 + 0.788514i $$0.710851\pi$$
$$338$$ 35.3150 1.92088
$$339$$ 13.7381 0.746153
$$340$$ 0 0
$$341$$ 0.962389 0.0521163
$$342$$ −12.2374 −0.661724
$$343$$ −17.1900 −0.928171
$$344$$ 23.6932 1.27745
$$345$$ 0 0
$$346$$ 18.7064 1.00566
$$347$$ 20.1925 1.08399 0.541996 0.840381i $$-0.317669\pi$$
0.541996 + 0.840381i $$0.317669\pi$$
$$348$$ 12.3127 0.660027
$$349$$ −27.2506 −1.45869 −0.729346 0.684145i $$-0.760175\pi$$
−0.729346 + 0.684145i $$0.760175\pi$$
$$350$$ 0 0
$$351$$ 5.11871 0.273217
$$352$$ −15.9502 −0.850147
$$353$$ 5.02302 0.267349 0.133674 0.991025i $$-0.457322\pi$$
0.133674 + 0.991025i $$0.457322\pi$$
$$354$$ 35.5125 1.88747
$$355$$ 0 0
$$356$$ −15.2750 −0.809575
$$357$$ −12.7513 −0.674871
$$358$$ −7.22425 −0.381814
$$359$$ −18.1768 −0.959334 −0.479667 0.877450i $$-0.659243\pi$$
−0.479667 + 0.877450i $$0.659243\pi$$
$$360$$ 0 0
$$361$$ 1.92619 0.101379
$$362$$ −56.7123 −2.98073
$$363$$ −1.00000 −0.0524864
$$364$$ −74.0625 −3.88193
$$365$$ 0 0
$$366$$ −21.1998 −1.10813
$$367$$ −8.56467 −0.447072 −0.223536 0.974696i $$-0.571760\pi$$
−0.223536 + 0.974696i $$0.571760\pi$$
$$368$$ 49.1002 2.55952
$$369$$ −2.38787 −0.124308
$$370$$ 0 0
$$371$$ −18.4485 −0.957799
$$372$$ 4.96239 0.257288
$$373$$ 7.81924 0.404865 0.202432 0.979296i $$-0.435115\pi$$
0.202432 + 0.979296i $$0.435115\pi$$
$$374$$ −12.1563 −0.628589
$$375$$ 0 0
$$376$$ −36.4142 −1.87792
$$377$$ 12.2228 0.629508
$$378$$ −7.50659 −0.386097
$$379$$ 1.14903 0.0590218 0.0295109 0.999564i $$-0.490605\pi$$
0.0295109 + 0.999564i $$0.490605\pi$$
$$380$$ 0 0
$$381$$ −12.4182 −0.636203
$$382$$ 35.1754 1.79973
$$383$$ 34.0870 1.74176 0.870882 0.491493i $$-0.163549\pi$$
0.870882 + 0.491493i $$0.163549\pi$$
$$384$$ −16.5696 −0.845563
$$385$$ 0 0
$$386$$ 15.7685 0.802593
$$387$$ 2.80606 0.142640
$$388$$ −51.1754 −2.59804
$$389$$ 23.7743 1.20541 0.602703 0.797965i $$-0.294090\pi$$
0.602703 + 0.797965i $$0.294090\pi$$
$$390$$ 0 0
$$391$$ 18.1768 0.919240
$$392$$ 7.37962 0.372727
$$393$$ −5.92478 −0.298865
$$394$$ 54.5560 2.74849
$$395$$ 0 0
$$396$$ −5.15633 −0.259115
$$397$$ −4.15045 −0.208305 −0.104152 0.994561i $$-0.533213\pi$$
−0.104152 + 0.994561i $$0.533213\pi$$
$$398$$ 46.5501 2.33334
$$399$$ 12.8364 0.642623
$$400$$ 0 0
$$401$$ −33.9149 −1.69363 −0.846815 0.531887i $$-0.821483\pi$$
−0.846815 + 0.531887i $$0.821483\pi$$
$$402$$ −28.6253 −1.42770
$$403$$ 4.92619 0.245391
$$404$$ −70.1886 −3.49201
$$405$$ 0 0
$$406$$ −17.9248 −0.889592
$$407$$ −1.61213 −0.0799102
$$408$$ −38.3693 −1.89956
$$409$$ 8.55008 0.422774 0.211387 0.977402i $$-0.432202\pi$$
0.211387 + 0.977402i $$0.432202\pi$$
$$410$$ 0 0
$$411$$ −9.79877 −0.483338
$$412$$ 84.1133 4.14397
$$413$$ −37.2506 −1.83298
$$414$$ 10.7005 0.525902
$$415$$ 0 0
$$416$$ −81.6444 −4.00294
$$417$$ 2.12601 0.104111
$$418$$ 12.2374 0.598552
$$419$$ 13.2995 0.649722 0.324861 0.945762i $$-0.394682\pi$$
0.324861 + 0.945762i $$0.394682\pi$$
$$420$$ 0 0
$$421$$ 33.3014 1.62301 0.811505 0.584345i $$-0.198649\pi$$
0.811505 + 0.584345i $$0.198649\pi$$
$$422$$ 48.4894 2.36043
$$423$$ −4.31265 −0.209688
$$424$$ −55.5125 −2.69592
$$425$$ 0 0
$$426$$ 19.6629 0.952671
$$427$$ 22.2374 1.07614
$$428$$ −48.6312 −2.35068
$$429$$ −5.11871 −0.247134
$$430$$ 0 0
$$431$$ 2.07522 0.0999600 0.0499800 0.998750i $$-0.484084\pi$$
0.0499800 + 0.998750i $$0.484084\pi$$
$$432$$ −12.2750 −0.590583
$$433$$ −37.4010 −1.79738 −0.898690 0.438585i $$-0.855480\pi$$
−0.898690 + 0.438585i $$0.855480\pi$$
$$434$$ −7.22425 −0.346775
$$435$$ 0 0
$$436$$ −79.4128 −3.80318
$$437$$ −18.2981 −0.875315
$$438$$ −17.1695 −0.820390
$$439$$ −6.02444 −0.287531 −0.143765 0.989612i $$-0.545921\pi$$
−0.143765 + 0.989612i $$0.545921\pi$$
$$440$$ 0 0
$$441$$ 0.873992 0.0416187
$$442$$ −62.2247 −2.95973
$$443$$ −10.1359 −0.481569 −0.240785 0.970579i $$-0.577405\pi$$
−0.240785 + 0.970579i $$0.577405\pi$$
$$444$$ −8.31265 −0.394501
$$445$$ 0 0
$$446$$ 63.2360 2.99431
$$447$$ −8.31265 −0.393175
$$448$$ 50.8423 2.40207
$$449$$ −11.4861 −0.542063 −0.271032 0.962570i $$-0.587365\pi$$
−0.271032 + 0.962570i $$0.587365\pi$$
$$450$$ 0 0
$$451$$ 2.38787 0.112441
$$452$$ −70.8383 −3.33195
$$453$$ 15.2750 0.717684
$$454$$ −3.39517 −0.159343
$$455$$ 0 0
$$456$$ 38.6253 1.80880
$$457$$ −15.0435 −0.703705 −0.351852 0.936055i $$-0.614448\pi$$
−0.351852 + 0.936055i $$0.614448\pi$$
$$458$$ 43.2750 2.02211
$$459$$ −4.54420 −0.212105
$$460$$ 0 0
$$461$$ −26.2374 −1.22200 −0.610999 0.791631i $$-0.709232\pi$$
−0.610999 + 0.791631i $$0.709232\pi$$
$$462$$ 7.50659 0.349238
$$463$$ −5.46168 −0.253826 −0.126913 0.991914i $$-0.540507\pi$$
−0.126913 + 0.991914i $$0.540507\pi$$
$$464$$ −29.3112 −1.36074
$$465$$ 0 0
$$466$$ −49.4069 −2.28873
$$467$$ −2.70052 −0.124965 −0.0624827 0.998046i $$-0.519902\pi$$
−0.0624827 + 0.998046i $$0.519902\pi$$
$$468$$ −26.3938 −1.22005
$$469$$ 30.0263 1.38649
$$470$$ 0 0
$$471$$ −7.01317 −0.323150
$$472$$ −112.089 −5.15931
$$473$$ −2.80606 −0.129023
$$474$$ −3.61213 −0.165910
$$475$$ 0 0
$$476$$ 65.7499 3.01364
$$477$$ −6.57452 −0.301026
$$478$$ 51.6991 2.36466
$$479$$ 16.7757 0.766503 0.383252 0.923644i $$-0.374804\pi$$
0.383252 + 0.923644i $$0.374804\pi$$
$$480$$ 0 0
$$481$$ −8.25202 −0.376260
$$482$$ 76.3752 3.47879
$$483$$ −11.2243 −0.510721
$$484$$ 5.15633 0.234378
$$485$$ 0 0
$$486$$ −2.67513 −0.121346
$$487$$ −34.3996 −1.55880 −0.779398 0.626529i $$-0.784475\pi$$
−0.779398 + 0.626529i $$0.784475\pi$$
$$488$$ 66.9135 3.02903
$$489$$ −6.76116 −0.305750
$$490$$ 0 0
$$491$$ 14.9525 0.674799 0.337399 0.941362i $$-0.390453\pi$$
0.337399 + 0.941362i $$0.390453\pi$$
$$492$$ 12.3127 0.555097
$$493$$ −10.8510 −0.488703
$$494$$ 62.6399 2.81830
$$495$$ 0 0
$$496$$ −11.8134 −0.530435
$$497$$ −20.6253 −0.925171
$$498$$ 2.15633 0.0966272
$$499$$ −2.85097 −0.127627 −0.0638135 0.997962i $$-0.520326\pi$$
−0.0638135 + 0.997962i $$0.520326\pi$$
$$500$$ 0 0
$$501$$ 11.8192 0.528045
$$502$$ 62.0625 2.76999
$$503$$ 1.48024 0.0660006 0.0330003 0.999455i $$-0.489494\pi$$
0.0330003 + 0.999455i $$0.489494\pi$$
$$504$$ 23.6932 1.05538
$$505$$ 0 0
$$506$$ −10.7005 −0.475696
$$507$$ −13.2012 −0.586287
$$508$$ 64.0322 2.84097
$$509$$ −23.2995 −1.03273 −0.516366 0.856368i $$-0.672715\pi$$
−0.516366 + 0.856368i $$0.672715\pi$$
$$510$$ 0 0
$$511$$ 18.0098 0.796709
$$512$$ −11.5017 −0.508306
$$513$$ 4.57452 0.201970
$$514$$ −29.1246 −1.28463
$$515$$ 0 0
$$516$$ −14.4690 −0.636961
$$517$$ 4.31265 0.189670
$$518$$ 12.1016 0.531712
$$519$$ −6.99271 −0.306946
$$520$$ 0 0
$$521$$ −6.81194 −0.298437 −0.149218 0.988804i $$-0.547676\pi$$
−0.149218 + 0.988804i $$0.547676\pi$$
$$522$$ −6.38787 −0.279590
$$523$$ −16.0567 −0.702109 −0.351054 0.936355i $$-0.614177\pi$$
−0.351054 + 0.936355i $$0.614177\pi$$
$$524$$ 30.5501 1.33459
$$525$$ 0 0
$$526$$ 24.3938 1.06362
$$527$$ −4.37328 −0.190503
$$528$$ 12.2750 0.534203
$$529$$ −7.00000 −0.304348
$$530$$ 0 0
$$531$$ −13.2750 −0.576088
$$532$$ −66.1886 −2.86964
$$533$$ 12.2228 0.529430
$$534$$ 7.92478 0.342939
$$535$$ 0 0
$$536$$ 90.3508 3.90256
$$537$$ 2.70052 0.116536
$$538$$ −30.2276 −1.30320
$$539$$ −0.873992 −0.0376455
$$540$$ 0 0
$$541$$ 6.62530 0.284844 0.142422 0.989806i $$-0.454511\pi$$
0.142422 + 0.989806i $$0.454511\pi$$
$$542$$ 67.4128 2.89563
$$543$$ 21.1998 0.909771
$$544$$ 72.4807 3.10759
$$545$$ 0 0
$$546$$ 38.4241 1.64440
$$547$$ −5.75860 −0.246220 −0.123110 0.992393i $$-0.539287\pi$$
−0.123110 + 0.992393i $$0.539287\pi$$
$$548$$ 50.5256 2.15835
$$549$$ 7.92478 0.338221
$$550$$ 0 0
$$551$$ 10.9234 0.465351
$$552$$ −33.7743 −1.43753
$$553$$ 3.78892 0.161121
$$554$$ −6.46898 −0.274840
$$555$$ 0 0
$$556$$ −10.9624 −0.464909
$$557$$ 19.8700 0.841920 0.420960 0.907079i $$-0.361693\pi$$
0.420960 + 0.907079i $$0.361693\pi$$
$$558$$ −2.57452 −0.108988
$$559$$ −14.3634 −0.607509
$$560$$ 0 0
$$561$$ 4.54420 0.191856
$$562$$ 81.5633 3.44054
$$563$$ 5.83383 0.245866 0.122933 0.992415i $$-0.460770\pi$$
0.122933 + 0.992415i $$0.460770\pi$$
$$564$$ 22.2374 0.936365
$$565$$ 0 0
$$566$$ 22.3576 0.939758
$$567$$ 2.80606 0.117844
$$568$$ −62.0625 −2.60409
$$569$$ 46.7123 1.95828 0.979140 0.203185i $$-0.0651293\pi$$
0.979140 + 0.203185i $$0.0651293\pi$$
$$570$$ 0 0
$$571$$ −24.4241 −1.02212 −0.511058 0.859546i $$-0.670746\pi$$
−0.511058 + 0.859546i $$0.670746\pi$$
$$572$$ 26.3938 1.10358
$$573$$ −13.1490 −0.549309
$$574$$ −17.9248 −0.748166
$$575$$ 0 0
$$576$$ 18.1187 0.754946
$$577$$ 16.5647 0.689596 0.344798 0.938677i $$-0.387947\pi$$
0.344798 + 0.938677i $$0.387947\pi$$
$$578$$ 9.76353 0.406109
$$579$$ −5.89446 −0.244965
$$580$$ 0 0
$$581$$ −2.26187 −0.0938380
$$582$$ 26.5501 1.10054
$$583$$ 6.57452 0.272289
$$584$$ 54.1925 2.24250
$$585$$ 0 0
$$586$$ −61.7802 −2.55212
$$587$$ 27.3258 1.12786 0.563929 0.825823i $$-0.309289\pi$$
0.563929 + 0.825823i $$0.309289\pi$$
$$588$$ −4.50659 −0.185849
$$589$$ 4.40246 0.181400
$$590$$ 0 0
$$591$$ −20.3938 −0.838887
$$592$$ 19.7889 0.813320
$$593$$ −12.6048 −0.517618 −0.258809 0.965928i $$-0.583330\pi$$
−0.258809 + 0.965928i $$0.583330\pi$$
$$594$$ 2.67513 0.109762
$$595$$ 0 0
$$596$$ 42.8627 1.75573
$$597$$ −17.4010 −0.712177
$$598$$ −54.7729 −2.23983
$$599$$ 10.5990 0.433061 0.216531 0.976276i $$-0.430526\pi$$
0.216531 + 0.976276i $$0.430526\pi$$
$$600$$ 0 0
$$601$$ −22.4749 −0.916768 −0.458384 0.888754i $$-0.651572\pi$$
−0.458384 + 0.888754i $$0.651572\pi$$
$$602$$ 21.0640 0.858503
$$603$$ 10.7005 0.435759
$$604$$ −78.7631 −3.20482
$$605$$ 0 0
$$606$$ 36.4142 1.47923
$$607$$ 33.5183 1.36047 0.680234 0.732995i $$-0.261878\pi$$
0.680234 + 0.732995i $$0.261878\pi$$
$$608$$ −72.9643 −2.95909
$$609$$ 6.70052 0.271519
$$610$$ 0 0
$$611$$ 22.0752 0.893068
$$612$$ 23.4314 0.947157
$$613$$ 11.5672 0.467196 0.233598 0.972333i $$-0.424950\pi$$
0.233598 + 0.972333i $$0.424950\pi$$
$$614$$ 7.10413 0.286699
$$615$$ 0 0
$$616$$ −23.6932 −0.954627
$$617$$ −33.3357 −1.34204 −0.671022 0.741438i $$-0.734144\pi$$
−0.671022 + 0.741438i $$0.734144\pi$$
$$618$$ −43.6385 −1.75540
$$619$$ 4.43866 0.178405 0.0892024 0.996014i $$-0.471568\pi$$
0.0892024 + 0.996014i $$0.471568\pi$$
$$620$$ 0 0
$$621$$ −4.00000 −0.160514
$$622$$ −42.7367 −1.71359
$$623$$ −8.31265 −0.333039
$$624$$ 62.8324 2.51531
$$625$$ 0 0
$$626$$ −0.162205 −0.00648301
$$627$$ −4.57452 −0.182689
$$628$$ 36.1622 1.44303
$$629$$ 7.32582 0.292100
$$630$$ 0 0
$$631$$ −39.8496 −1.58639 −0.793193 0.608971i $$-0.791583\pi$$
−0.793193 + 0.608971i $$0.791583\pi$$
$$632$$ 11.4010 0.453509
$$633$$ −18.1260 −0.720444
$$634$$ 10.2012 0.405143
$$635$$ 0 0
$$636$$ 33.9003 1.34424
$$637$$ −4.47371 −0.177255
$$638$$ 6.38787 0.252898
$$639$$ −7.35026 −0.290772
$$640$$ 0 0
$$641$$ −3.88858 −0.153590 −0.0767948 0.997047i $$-0.524469\pi$$
−0.0767948 + 0.997047i $$0.524469\pi$$
$$642$$ 25.2301 0.995754
$$643$$ 40.9380 1.61444 0.807218 0.590254i $$-0.200972\pi$$
0.807218 + 0.590254i $$0.200972\pi$$
$$644$$ 57.8759 2.28063
$$645$$ 0 0
$$646$$ −55.6093 −2.18792
$$647$$ 1.76257 0.0692939 0.0346469 0.999400i $$-0.488969\pi$$
0.0346469 + 0.999400i $$0.488969\pi$$
$$648$$ 8.44358 0.331695
$$649$$ 13.2750 0.521091
$$650$$ 0 0
$$651$$ 2.70052 0.105842
$$652$$ 34.8627 1.36533
$$653$$ −7.03761 −0.275403 −0.137702 0.990474i $$-0.543971\pi$$
−0.137702 + 0.990474i $$0.543971\pi$$
$$654$$ 41.1998 1.61104
$$655$$ 0 0
$$656$$ −29.3112 −1.14441
$$657$$ 6.41819 0.250397
$$658$$ −32.3733 −1.26204
$$659$$ −12.6253 −0.491812 −0.245906 0.969294i $$-0.579085\pi$$
−0.245906 + 0.969294i $$0.579085\pi$$
$$660$$ 0 0
$$661$$ 13.2243 0.514364 0.257182 0.966363i $$-0.417206\pi$$
0.257182 + 0.966363i $$0.417206\pi$$
$$662$$ 64.9741 2.52529
$$663$$ 23.2605 0.903361
$$664$$ −6.80606 −0.264126
$$665$$ 0 0
$$666$$ 4.31265 0.167112
$$667$$ −9.55149 −0.369835
$$668$$ −60.9438 −2.35799
$$669$$ −23.6385 −0.913916
$$670$$ 0 0
$$671$$ −7.92478 −0.305933
$$672$$ −44.7572 −1.72655
$$673$$ 18.2677 0.704170 0.352085 0.935968i $$-0.385473\pi$$
0.352085 + 0.935968i $$0.385473\pi$$
$$674$$ −60.4055 −2.32673
$$675$$ 0 0
$$676$$ 68.0698 2.61807
$$677$$ −33.0191 −1.26903 −0.634513 0.772912i $$-0.718799\pi$$
−0.634513 + 0.772912i $$0.718799\pi$$
$$678$$ 36.7513 1.41143
$$679$$ −27.8496 −1.06877
$$680$$ 0 0
$$681$$ 1.26916 0.0486343
$$682$$ 2.57452 0.0985833
$$683$$ 30.8627 1.18093 0.590465 0.807063i $$-0.298944\pi$$
0.590465 + 0.807063i $$0.298944\pi$$
$$684$$ −23.5877 −0.901898
$$685$$ 0 0
$$686$$ −45.9854 −1.75573
$$687$$ −16.1768 −0.617183
$$688$$ 34.4445 1.31319
$$689$$ 33.6531 1.28208
$$690$$ 0 0
$$691$$ 27.6991 1.05372 0.526862 0.849951i $$-0.323369\pi$$
0.526862 + 0.849951i $$0.323369\pi$$
$$692$$ 36.0567 1.37067
$$693$$ −2.80606 −0.106594
$$694$$ 54.0176 2.05048
$$695$$ 0 0
$$696$$ 20.1622 0.764246
$$697$$ −10.8510 −0.411010
$$698$$ −72.8989 −2.75927
$$699$$ 18.4690 0.698561
$$700$$ 0 0
$$701$$ −38.9643 −1.47166 −0.735831 0.677166i $$-0.763208\pi$$
−0.735831 + 0.677166i $$0.763208\pi$$
$$702$$ 13.6932 0.516818
$$703$$ −7.37470 −0.278142
$$704$$ −18.1187 −0.682875
$$705$$ 0 0
$$706$$ 13.4372 0.505717
$$707$$ −38.1965 −1.43653
$$708$$ 68.4504 2.57252
$$709$$ −4.32250 −0.162335 −0.0811674 0.996700i $$-0.525865\pi$$
−0.0811674 + 0.996700i $$0.525865\pi$$
$$710$$ 0 0
$$711$$ 1.35026 0.0506388
$$712$$ −25.0132 −0.937408
$$713$$ −3.84955 −0.144167
$$714$$ −34.1114 −1.27659
$$715$$ 0 0
$$716$$ −13.9248 −0.520393
$$717$$ −19.3258 −0.721736
$$718$$ −48.6253 −1.81468
$$719$$ −38.3996 −1.43206 −0.716032 0.698067i $$-0.754044\pi$$
−0.716032 + 0.698067i $$0.754044\pi$$
$$720$$ 0 0
$$721$$ 45.7743 1.70473
$$722$$ 5.15282 0.191768
$$723$$ −28.5501 −1.06179
$$724$$ −109.313 −4.06259
$$725$$ 0 0
$$726$$ −2.67513 −0.0992834
$$727$$ −21.6728 −0.803798 −0.401899 0.915684i $$-0.631650\pi$$
−0.401899 + 0.915684i $$0.631650\pi$$
$$728$$ −121.279 −4.49489
$$729$$ 1.00000 0.0370370
$$730$$ 0 0
$$731$$ 12.7513 0.471624
$$732$$ −40.8627 −1.51033
$$733$$ 25.0698 0.925976 0.462988 0.886365i $$-0.346777\pi$$
0.462988 + 0.886365i $$0.346777\pi$$
$$734$$ −22.9116 −0.845683
$$735$$ 0 0
$$736$$ 63.8007 2.35172
$$737$$ −10.7005 −0.394159
$$738$$ −6.38787 −0.235141
$$739$$ −18.9018 −0.695312 −0.347656 0.937622i $$-0.613022\pi$$
−0.347656 + 0.937622i $$0.613022\pi$$
$$740$$ 0 0
$$741$$ −23.4156 −0.860195
$$742$$ −49.3522 −1.81178
$$743$$ 43.6688 1.60205 0.801026 0.598629i $$-0.204288\pi$$
0.801026 + 0.598629i $$0.204288\pi$$
$$744$$ 8.12601 0.297914
$$745$$ 0 0
$$746$$ 20.9175 0.765843
$$747$$ −0.806063 −0.0294923
$$748$$ −23.4314 −0.856736
$$749$$ −26.4650 −0.967010
$$750$$ 0 0
$$751$$ −47.0541 −1.71703 −0.858514 0.512789i $$-0.828612\pi$$
−0.858514 + 0.512789i $$0.828612\pi$$
$$752$$ −52.9380 −1.93045
$$753$$ −23.1998 −0.845448
$$754$$ 32.6977 1.19078
$$755$$ 0 0
$$756$$ −14.4690 −0.526232
$$757$$ −26.9116 −0.978119 −0.489059 0.872250i $$-0.662660\pi$$
−0.489059 + 0.872250i $$0.662660\pi$$
$$758$$ 3.07381 0.111646
$$759$$ 4.00000 0.145191
$$760$$ 0 0
$$761$$ 16.6859 0.604865 0.302432 0.953171i $$-0.402201\pi$$
0.302432 + 0.953171i $$0.402201\pi$$
$$762$$ −33.2203 −1.20344
$$763$$ −43.2163 −1.56454
$$764$$ 67.8007 2.45294
$$765$$ 0 0
$$766$$ 91.1871 3.29473
$$767$$ 67.9511 2.45357
$$768$$ −8.08840 −0.291865
$$769$$ 23.2995 0.840201 0.420100 0.907478i $$-0.361995\pi$$
0.420100 + 0.907478i $$0.361995\pi$$
$$770$$ 0 0
$$771$$ 10.8872 0.392092
$$772$$ 30.3938 1.09390
$$773$$ −1.63656 −0.0588631 −0.0294316 0.999567i $$-0.509370\pi$$
−0.0294316 + 0.999567i $$0.509370\pi$$
$$774$$ 7.50659 0.269819
$$775$$ 0 0
$$776$$ −83.8007 −3.00827
$$777$$ −4.52373 −0.162288
$$778$$ 63.5994 2.28015
$$779$$ 10.9234 0.391370
$$780$$ 0 0
$$781$$ 7.35026 0.263013
$$782$$ 48.6253 1.73884
$$783$$ 2.38787 0.0853356
$$784$$ 10.7283 0.383153
$$785$$ 0 0
$$786$$ −15.8496 −0.565335
$$787$$ −2.09095 −0.0745344 −0.0372672 0.999305i $$-0.511865\pi$$
−0.0372672 + 0.999305i $$0.511865\pi$$
$$788$$ 105.157 3.74606
$$789$$ −9.11871 −0.324635
$$790$$ 0 0
$$791$$ −38.5501 −1.37068
$$792$$ −8.44358 −0.300030
$$793$$ −40.5647 −1.44049
$$794$$ −11.1030 −0.394030
$$795$$ 0 0
$$796$$ 89.7255 3.18023
$$797$$ −26.8872 −0.952392 −0.476196 0.879339i $$-0.657985\pi$$
−0.476196 + 0.879339i $$0.657985\pi$$
$$798$$ 34.3390 1.21559
$$799$$ −19.5975 −0.693311
$$800$$ 0 0
$$801$$ −2.96239 −0.104671
$$802$$ −90.7269 −3.20368
$$803$$ −6.41819 −0.226493
$$804$$ −55.1754 −1.94589
$$805$$ 0 0
$$806$$ 13.1782 0.464183
$$807$$ 11.2995 0.397760
$$808$$ −114.935 −4.04340
$$809$$ −45.4128 −1.59663 −0.798315 0.602241i $$-0.794275\pi$$
−0.798315 + 0.602241i $$0.794275\pi$$
$$810$$ 0 0
$$811$$ 34.3488 1.20615 0.603076 0.797684i $$-0.293942\pi$$
0.603076 + 0.797684i $$0.293942\pi$$
$$812$$ −34.5501 −1.21247
$$813$$ −25.1998 −0.883796
$$814$$ −4.31265 −0.151158
$$815$$ 0 0
$$816$$ −55.7802 −1.95270
$$817$$ −12.8364 −0.449088
$$818$$ 22.8726 0.799721
$$819$$ −14.3634 −0.501899
$$820$$ 0 0
$$821$$ 18.7612 0.654769 0.327384 0.944891i $$-0.393833\pi$$
0.327384 + 0.944891i $$0.393833\pi$$
$$822$$ −26.2130 −0.914283
$$823$$ −33.0592 −1.15237 −0.576186 0.817319i $$-0.695460\pi$$
−0.576186 + 0.817319i $$0.695460\pi$$
$$824$$ 137.737 4.79830
$$825$$ 0 0
$$826$$ −99.6502 −3.46728
$$827$$ −10.8813 −0.378379 −0.189190 0.981941i $$-0.560586\pi$$
−0.189190 + 0.981941i $$0.560586\pi$$
$$828$$ 20.6253 0.716779
$$829$$ 42.7221 1.48380 0.741900 0.670510i $$-0.233925\pi$$
0.741900 + 0.670510i $$0.233925\pi$$
$$830$$ 0 0
$$831$$ 2.41819 0.0838861
$$832$$ −92.7445 −3.21534
$$833$$ 3.97159 0.137608
$$834$$ 5.68735 0.196937
$$835$$ 0 0
$$836$$ 23.5877 0.815797
$$837$$ 0.962389 0.0332650
$$838$$ 35.5778 1.22902
$$839$$ −21.1735 −0.730989 −0.365495 0.930813i $$-0.619100\pi$$
−0.365495 + 0.930813i $$0.619100\pi$$
$$840$$ 0 0
$$841$$ −23.2981 −0.803381
$$842$$ 89.0856 3.07009
$$843$$ −30.4894 −1.05011
$$844$$ 93.4636 3.21715
$$845$$ 0 0
$$846$$ −11.5369 −0.396647
$$847$$ 2.80606 0.0964175
$$848$$ −80.7024 −2.77133
$$849$$ −8.35756 −0.286831
$$850$$ 0 0
$$851$$ 6.44851 0.221052
$$852$$ 37.9003 1.29844
$$853$$ 54.4709 1.86505 0.932524 0.361109i $$-0.117602\pi$$
0.932524 + 0.361109i $$0.117602\pi$$
$$854$$ 59.4880 2.03564
$$855$$ 0 0
$$856$$ −79.6345 −2.72185
$$857$$ 38.5705 1.31754 0.658772 0.752342i $$-0.271076\pi$$
0.658772 + 0.752342i $$0.271076\pi$$
$$858$$ −13.6932 −0.467479
$$859$$ −18.5139 −0.631685 −0.315843 0.948812i $$-0.602287\pi$$
−0.315843 + 0.948812i $$0.602287\pi$$
$$860$$ 0 0
$$861$$ 6.70052 0.228353
$$862$$ 5.55149 0.189085
$$863$$ 22.5383 0.767213 0.383607 0.923497i $$-0.374682\pi$$
0.383607 + 0.923497i $$0.374682\pi$$
$$864$$ −15.9502 −0.542636
$$865$$ 0 0
$$866$$ −100.053 −3.39993
$$867$$ −3.64974 −0.123952
$$868$$ −13.9248 −0.472638
$$869$$ −1.35026 −0.0458045
$$870$$ 0 0
$$871$$ −54.7729 −1.85591
$$872$$ −130.040 −4.40371
$$873$$ −9.92478 −0.335903
$$874$$ −48.9497 −1.65575
$$875$$ 0 0
$$876$$ −33.0943 −1.11815
$$877$$ 18.9419 0.639623 0.319812 0.947481i $$-0.396380\pi$$
0.319812 + 0.947481i $$0.396380\pi$$
$$878$$ −16.1162 −0.543894
$$879$$ 23.0943 0.778951
$$880$$ 0 0
$$881$$ −20.0263 −0.674705 −0.337352 0.941378i $$-0.609531\pi$$
−0.337352 + 0.941378i $$0.609531\pi$$
$$882$$ 2.33804 0.0787260
$$883$$ 22.2981 0.750390 0.375195 0.926946i $$-0.377576\pi$$
0.375195 + 0.926946i $$0.377576\pi$$
$$884$$ −119.938 −4.03397
$$885$$ 0 0
$$886$$ −27.1147 −0.910938
$$887$$ 3.10413 0.104226 0.0521132 0.998641i $$-0.483404\pi$$
0.0521132 + 0.998641i $$0.483404\pi$$
$$888$$ −13.6121 −0.456793
$$889$$ 34.8462 1.16871
$$890$$ 0 0
$$891$$ −1.00000 −0.0335013
$$892$$ 121.888 4.08110
$$893$$ 19.7283 0.660182
$$894$$ −22.2374 −0.743731
$$895$$ 0 0
$$896$$ 46.4953 1.55330
$$897$$ 20.4749 0.683636
$$898$$ −30.7269 −1.02537
$$899$$ 2.29806 0.0766447
$$900$$ 0 0
$$901$$ −29.8759 −0.995311
$$902$$ 6.38787 0.212693
$$903$$ −7.87399 −0.262030
$$904$$ −115.999 −3.85807
$$905$$ 0 0
$$906$$ 40.8627 1.35757
$$907$$ 10.8627 0.360691 0.180345 0.983603i $$-0.442278\pi$$
0.180345 + 0.983603i $$0.442278\pi$$
$$908$$ −6.54420 −0.217177
$$909$$ −13.6121 −0.451486
$$910$$ 0 0
$$911$$ −5.82321 −0.192931 −0.0964657 0.995336i $$-0.530754\pi$$
−0.0964657 + 0.995336i $$0.530754\pi$$
$$912$$ 56.1524 1.85939
$$913$$ 0.806063 0.0266768
$$914$$ −40.2433 −1.33113
$$915$$ 0 0
$$916$$ 83.4128 2.75604
$$917$$ 16.6253 0.549016
$$918$$ −12.1563 −0.401219
$$919$$ −31.2750 −1.03167 −0.515834 0.856688i $$-0.672518\pi$$
−0.515834 + 0.856688i $$0.672518\pi$$
$$920$$ 0 0
$$921$$ −2.65562 −0.0875056
$$922$$ −70.1886 −2.31154
$$923$$ 37.6239 1.23841
$$924$$ 14.4690 0.475994
$$925$$ 0 0
$$926$$ −14.6107 −0.480138
$$927$$ 16.3127 0.535778
$$928$$ −38.0870 −1.25027
$$929$$ −21.2243 −0.696345 −0.348173 0.937430i $$-0.613198\pi$$
−0.348173 + 0.937430i $$0.613198\pi$$
$$930$$ 0 0
$$931$$ −3.99809 −0.131032
$$932$$ −95.2320 −3.11943
$$933$$ 15.9756 0.523016
$$934$$ −7.22425 −0.236385
$$935$$ 0 0
$$936$$ −43.2203 −1.41270
$$937$$ 16.4328 0.536835 0.268418 0.963303i $$-0.413499\pi$$
0.268418 + 0.963303i $$0.413499\pi$$
$$938$$ 80.3244 2.62268
$$939$$ 0.0606343 0.00197873
$$940$$ 0 0
$$941$$ 5.86414 0.191166 0.0955828 0.995421i $$-0.469529\pi$$
0.0955828 + 0.995421i $$0.469529\pi$$
$$942$$ −18.7612 −0.611272
$$943$$ −9.55149 −0.311039
$$944$$ −162.952 −5.30362
$$945$$ 0 0
$$946$$ −7.50659 −0.244060
$$947$$ −38.7415 −1.25893 −0.629464 0.777030i $$-0.716726\pi$$
−0.629464 + 0.777030i $$0.716726\pi$$
$$948$$ −6.96239 −0.226128
$$949$$ −32.8529 −1.06645
$$950$$ 0 0
$$951$$ −3.81336 −0.123657
$$952$$ 107.667 3.48950
$$953$$ −6.09569 −0.197459 −0.0987294 0.995114i $$-0.531478\pi$$
−0.0987294 + 0.995114i $$0.531478\pi$$
$$954$$ −17.5877 −0.569422
$$955$$ 0 0
$$956$$ 99.6502 3.22292
$$957$$ −2.38787 −0.0771890
$$958$$ 44.8773 1.44992
$$959$$ 27.4960 0.887891
$$960$$ 0 0
$$961$$ −30.0738 −0.970123
$$962$$ −22.0752 −0.711734
$$963$$ −9.43136 −0.303921
$$964$$ 147.213 4.74143
$$965$$ 0 0
$$966$$ −30.0263 −0.966082
$$967$$ 37.9697 1.22102 0.610511 0.792008i $$-0.290964\pi$$
0.610511 + 0.792008i $$0.290964\pi$$
$$968$$ 8.44358 0.271387
$$969$$ 20.7875 0.667791
$$970$$ 0 0
$$971$$ −60.8773 −1.95365 −0.976823 0.214049i $$-0.931335\pi$$
−0.976823 + 0.214049i $$0.931335\pi$$
$$972$$ −5.15633 −0.165389
$$973$$ −5.96571 −0.191252
$$974$$ −92.0235 −2.94862
$$975$$ 0 0
$$976$$ 97.2769 3.11376
$$977$$ 21.3963 0.684529 0.342264 0.939604i $$-0.388806\pi$$
0.342264 + 0.939604i $$0.388806\pi$$
$$978$$ −18.0870 −0.578358
$$979$$ 2.96239 0.0946784
$$980$$ 0 0
$$981$$ −15.4010 −0.491718
$$982$$ 40.0000 1.27645
$$983$$ 18.8919 0.602558 0.301279 0.953536i $$-0.402586\pi$$
0.301279 + 0.953536i $$0.402586\pi$$
$$984$$ 20.1622 0.642748
$$985$$ 0 0
$$986$$ −29.0278 −0.924432
$$987$$ 12.1016 0.385197
$$988$$ 120.739 3.84121
$$989$$ 11.2243 0.356911
$$990$$ 0 0
$$991$$ −1.33567 −0.0424291 −0.0212145 0.999775i $$-0.506753\pi$$
−0.0212145 + 0.999775i $$0.506753\pi$$
$$992$$ −15.3503 −0.487371
$$993$$ −24.2882 −0.770763
$$994$$ −55.1754 −1.75006
$$995$$ 0 0
$$996$$ 4.15633 0.131698
$$997$$ −48.9076 −1.54892 −0.774460 0.632623i $$-0.781978\pi$$
−0.774460 + 0.632623i $$0.781978\pi$$
$$998$$ −7.62672 −0.241419
$$999$$ −1.61213 −0.0510054
Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000

## Twists

By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 825.2.a.n.1.3 3
3.2 odd 2 2475.2.a.y.1.1 3
5.2 odd 4 165.2.c.a.34.6 yes 6
5.3 odd 4 165.2.c.a.34.1 6
5.4 even 2 825.2.a.h.1.1 3
11.10 odd 2 9075.2.a.cc.1.1 3
15.2 even 4 495.2.c.d.199.1 6
15.8 even 4 495.2.c.d.199.6 6
15.14 odd 2 2475.2.a.be.1.3 3
20.3 even 4 2640.2.d.i.529.4 6
20.7 even 4 2640.2.d.i.529.1 6
55.32 even 4 1815.2.c.d.364.1 6
55.43 even 4 1815.2.c.d.364.6 6
55.54 odd 2 9075.2.a.ck.1.3 3

By twisted newform
Twist Min Dim Char Parity Ord Type
165.2.c.a.34.1 6 5.3 odd 4
165.2.c.a.34.6 yes 6 5.2 odd 4
495.2.c.d.199.1 6 15.2 even 4
495.2.c.d.199.6 6 15.8 even 4
825.2.a.h.1.1 3 5.4 even 2
825.2.a.n.1.3 3 1.1 even 1 trivial
1815.2.c.d.364.1 6 55.32 even 4
1815.2.c.d.364.6 6 55.43 even 4
2475.2.a.y.1.1 3 3.2 odd 2
2475.2.a.be.1.3 3 15.14 odd 2
2640.2.d.i.529.1 6 20.7 even 4
2640.2.d.i.529.4 6 20.3 even 4
9075.2.a.cc.1.1 3 11.10 odd 2
9075.2.a.ck.1.3 3 55.54 odd 2