# Properties

 Label 2450.2.a.bs.1.2 Level $2450$ Weight $2$ Character 2450.1 Self dual yes Analytic conductor $19.563$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2450 = 2 \cdot 5^{2} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2450.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$19.5633484952$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\zeta_{8})^+$$ Defining polynomial: $$x^{2} - 2$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 490) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$1.41421$$ of defining polynomial Character $$\chi$$ $$=$$ 2450.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+1.00000 q^{2} +3.41421 q^{3} +1.00000 q^{4} +3.41421 q^{6} +1.00000 q^{8} +8.65685 q^{9} +O(q^{10})$$ $$q+1.00000 q^{2} +3.41421 q^{3} +1.00000 q^{4} +3.41421 q^{6} +1.00000 q^{8} +8.65685 q^{9} -0.828427 q^{11} +3.41421 q^{12} -4.82843 q^{13} +1.00000 q^{16} +2.58579 q^{17} +8.65685 q^{18} -0.585786 q^{19} -0.828427 q^{22} +1.17157 q^{23} +3.41421 q^{24} -4.82843 q^{26} +19.3137 q^{27} -4.82843 q^{29} +2.82843 q^{31} +1.00000 q^{32} -2.82843 q^{33} +2.58579 q^{34} +8.65685 q^{36} +7.65685 q^{37} -0.585786 q^{38} -16.4853 q^{39} +3.07107 q^{41} +8.82843 q^{43} -0.828427 q^{44} +1.17157 q^{46} +5.17157 q^{47} +3.41421 q^{48} +8.82843 q^{51} -4.82843 q^{52} -6.48528 q^{53} +19.3137 q^{54} -2.00000 q^{57} -4.82843 q^{58} -8.58579 q^{59} -9.31371 q^{61} +2.82843 q^{62} +1.00000 q^{64} -2.82843 q^{66} -1.65685 q^{67} +2.58579 q^{68} +4.00000 q^{69} -4.48528 q^{71} +8.65685 q^{72} -9.41421 q^{73} +7.65685 q^{74} -0.585786 q^{76} -16.4853 q^{78} -6.82843 q^{79} +39.9706 q^{81} +3.07107 q^{82} -2.24264 q^{83} +8.82843 q^{86} -16.4853 q^{87} -0.828427 q^{88} -12.7279 q^{89} +1.17157 q^{92} +9.65685 q^{93} +5.17157 q^{94} +3.41421 q^{96} -7.75736 q^{97} -7.17157 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 2q^{2} + 4q^{3} + 2q^{4} + 4q^{6} + 2q^{8} + 6q^{9} + O(q^{10})$$ $$2q + 2q^{2} + 4q^{3} + 2q^{4} + 4q^{6} + 2q^{8} + 6q^{9} + 4q^{11} + 4q^{12} - 4q^{13} + 2q^{16} + 8q^{17} + 6q^{18} - 4q^{19} + 4q^{22} + 8q^{23} + 4q^{24} - 4q^{26} + 16q^{27} - 4q^{29} + 2q^{32} + 8q^{34} + 6q^{36} + 4q^{37} - 4q^{38} - 16q^{39} - 8q^{41} + 12q^{43} + 4q^{44} + 8q^{46} + 16q^{47} + 4q^{48} + 12q^{51} - 4q^{52} + 4q^{53} + 16q^{54} - 4q^{57} - 4q^{58} - 20q^{59} + 4q^{61} + 2q^{64} + 8q^{67} + 8q^{68} + 8q^{69} + 8q^{71} + 6q^{72} - 16q^{73} + 4q^{74} - 4q^{76} - 16q^{78} - 8q^{79} + 46q^{81} - 8q^{82} + 4q^{83} + 12q^{86} - 16q^{87} + 4q^{88} + 8q^{92} + 8q^{93} + 16q^{94} + 4q^{96} - 24q^{97} - 20q^{99} + O(q^{100})$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 1.00000 0.707107
$$3$$ 3.41421 1.97120 0.985599 0.169102i $$-0.0540867\pi$$
0.985599 + 0.169102i $$0.0540867\pi$$
$$4$$ 1.00000 0.500000
$$5$$ 0 0
$$6$$ 3.41421 1.39385
$$7$$ 0 0
$$8$$ 1.00000 0.353553
$$9$$ 8.65685 2.88562
$$10$$ 0 0
$$11$$ −0.828427 −0.249780 −0.124890 0.992171i $$-0.539858\pi$$
−0.124890 + 0.992171i $$0.539858\pi$$
$$12$$ 3.41421 0.985599
$$13$$ −4.82843 −1.33916 −0.669582 0.742738i $$-0.733527\pi$$
−0.669582 + 0.742738i $$0.733527\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ 2.58579 0.627145 0.313573 0.949564i $$-0.398474\pi$$
0.313573 + 0.949564i $$0.398474\pi$$
$$18$$ 8.65685 2.04044
$$19$$ −0.585786 −0.134389 −0.0671943 0.997740i $$-0.521405\pi$$
−0.0671943 + 0.997740i $$0.521405\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ −0.828427 −0.176621
$$23$$ 1.17157 0.244290 0.122145 0.992512i $$-0.461023\pi$$
0.122145 + 0.992512i $$0.461023\pi$$
$$24$$ 3.41421 0.696923
$$25$$ 0 0
$$26$$ −4.82843 −0.946932
$$27$$ 19.3137 3.71692
$$28$$ 0 0
$$29$$ −4.82843 −0.896616 −0.448308 0.893879i $$-0.647973\pi$$
−0.448308 + 0.893879i $$0.647973\pi$$
$$30$$ 0 0
$$31$$ 2.82843 0.508001 0.254000 0.967204i $$-0.418254\pi$$
0.254000 + 0.967204i $$0.418254\pi$$
$$32$$ 1.00000 0.176777
$$33$$ −2.82843 −0.492366
$$34$$ 2.58579 0.443459
$$35$$ 0 0
$$36$$ 8.65685 1.44281
$$37$$ 7.65685 1.25878 0.629390 0.777090i $$-0.283305\pi$$
0.629390 + 0.777090i $$0.283305\pi$$
$$38$$ −0.585786 −0.0950271
$$39$$ −16.4853 −2.63976
$$40$$ 0 0
$$41$$ 3.07107 0.479620 0.239810 0.970820i $$-0.422915\pi$$
0.239810 + 0.970820i $$0.422915\pi$$
$$42$$ 0 0
$$43$$ 8.82843 1.34632 0.673161 0.739496i $$-0.264936\pi$$
0.673161 + 0.739496i $$0.264936\pi$$
$$44$$ −0.828427 −0.124890
$$45$$ 0 0
$$46$$ 1.17157 0.172739
$$47$$ 5.17157 0.754351 0.377176 0.926142i $$-0.376895\pi$$
0.377176 + 0.926142i $$0.376895\pi$$
$$48$$ 3.41421 0.492799
$$49$$ 0 0
$$50$$ 0 0
$$51$$ 8.82843 1.23623
$$52$$ −4.82843 −0.669582
$$53$$ −6.48528 −0.890822 −0.445411 0.895326i $$-0.646942\pi$$
−0.445411 + 0.895326i $$0.646942\pi$$
$$54$$ 19.3137 2.62826
$$55$$ 0 0
$$56$$ 0 0
$$57$$ −2.00000 −0.264906
$$58$$ −4.82843 −0.634004
$$59$$ −8.58579 −1.11777 −0.558887 0.829244i $$-0.688771\pi$$
−0.558887 + 0.829244i $$0.688771\pi$$
$$60$$ 0 0
$$61$$ −9.31371 −1.19250 −0.596249 0.802799i $$-0.703343\pi$$
−0.596249 + 0.802799i $$0.703343\pi$$
$$62$$ 2.82843 0.359211
$$63$$ 0 0
$$64$$ 1.00000 0.125000
$$65$$ 0 0
$$66$$ −2.82843 −0.348155
$$67$$ −1.65685 −0.202417 −0.101208 0.994865i $$-0.532271\pi$$
−0.101208 + 0.994865i $$0.532271\pi$$
$$68$$ 2.58579 0.313573
$$69$$ 4.00000 0.481543
$$70$$ 0 0
$$71$$ −4.48528 −0.532305 −0.266152 0.963931i $$-0.585752\pi$$
−0.266152 + 0.963931i $$0.585752\pi$$
$$72$$ 8.65685 1.02022
$$73$$ −9.41421 −1.10185 −0.550925 0.834555i $$-0.685725\pi$$
−0.550925 + 0.834555i $$0.685725\pi$$
$$74$$ 7.65685 0.890091
$$75$$ 0 0
$$76$$ −0.585786 −0.0671943
$$77$$ 0 0
$$78$$ −16.4853 −1.86659
$$79$$ −6.82843 −0.768258 −0.384129 0.923279i $$-0.625498\pi$$
−0.384129 + 0.923279i $$0.625498\pi$$
$$80$$ 0 0
$$81$$ 39.9706 4.44117
$$82$$ 3.07107 0.339143
$$83$$ −2.24264 −0.246162 −0.123081 0.992397i $$-0.539277\pi$$
−0.123081 + 0.992397i $$0.539277\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 8.82843 0.951994
$$87$$ −16.4853 −1.76741
$$88$$ −0.828427 −0.0883106
$$89$$ −12.7279 −1.34916 −0.674579 0.738203i $$-0.735675\pi$$
−0.674579 + 0.738203i $$0.735675\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 1.17157 0.122145
$$93$$ 9.65685 1.00137
$$94$$ 5.17157 0.533407
$$95$$ 0 0
$$96$$ 3.41421 0.348462
$$97$$ −7.75736 −0.787641 −0.393820 0.919187i $$-0.628847\pi$$
−0.393820 + 0.919187i $$0.628847\pi$$
$$98$$ 0 0
$$99$$ −7.17157 −0.720770
$$100$$ 0 0
$$101$$ −13.3137 −1.32476 −0.662382 0.749166i $$-0.730454\pi$$
−0.662382 + 0.749166i $$0.730454\pi$$
$$102$$ 8.82843 0.874145
$$103$$ −14.8284 −1.46109 −0.730544 0.682865i $$-0.760734\pi$$
−0.730544 + 0.682865i $$0.760734\pi$$
$$104$$ −4.82843 −0.473466
$$105$$ 0 0
$$106$$ −6.48528 −0.629906
$$107$$ −9.65685 −0.933563 −0.466782 0.884373i $$-0.654587\pi$$
−0.466782 + 0.884373i $$0.654587\pi$$
$$108$$ 19.3137 1.85846
$$109$$ 2.48528 0.238047 0.119023 0.992891i $$-0.462024\pi$$
0.119023 + 0.992891i $$0.462024\pi$$
$$110$$ 0 0
$$111$$ 26.1421 2.48130
$$112$$ 0 0
$$113$$ 15.3137 1.44059 0.720296 0.693667i $$-0.244006\pi$$
0.720296 + 0.693667i $$0.244006\pi$$
$$114$$ −2.00000 −0.187317
$$115$$ 0 0
$$116$$ −4.82843 −0.448308
$$117$$ −41.7990 −3.86432
$$118$$ −8.58579 −0.790386
$$119$$ 0 0
$$120$$ 0 0
$$121$$ −10.3137 −0.937610
$$122$$ −9.31371 −0.843224
$$123$$ 10.4853 0.945426
$$124$$ 2.82843 0.254000
$$125$$ 0 0
$$126$$ 0 0
$$127$$ −2.82843 −0.250982 −0.125491 0.992095i $$-0.540051\pi$$
−0.125491 + 0.992095i $$0.540051\pi$$
$$128$$ 1.00000 0.0883883
$$129$$ 30.1421 2.65387
$$130$$ 0 0
$$131$$ 6.24264 0.545422 0.272711 0.962096i $$-0.412080\pi$$
0.272711 + 0.962096i $$0.412080\pi$$
$$132$$ −2.82843 −0.246183
$$133$$ 0 0
$$134$$ −1.65685 −0.143130
$$135$$ 0 0
$$136$$ 2.58579 0.221729
$$137$$ 16.0000 1.36697 0.683486 0.729964i $$-0.260463\pi$$
0.683486 + 0.729964i $$0.260463\pi$$
$$138$$ 4.00000 0.340503
$$139$$ −19.8995 −1.68785 −0.843927 0.536459i $$-0.819762\pi$$
−0.843927 + 0.536459i $$0.819762\pi$$
$$140$$ 0 0
$$141$$ 17.6569 1.48698
$$142$$ −4.48528 −0.376396
$$143$$ 4.00000 0.334497
$$144$$ 8.65685 0.721405
$$145$$ 0 0
$$146$$ −9.41421 −0.779126
$$147$$ 0 0
$$148$$ 7.65685 0.629390
$$149$$ −6.00000 −0.491539 −0.245770 0.969328i $$-0.579041\pi$$
−0.245770 + 0.969328i $$0.579041\pi$$
$$150$$ 0 0
$$151$$ −11.3137 −0.920697 −0.460348 0.887738i $$-0.652275\pi$$
−0.460348 + 0.887738i $$0.652275\pi$$
$$152$$ −0.585786 −0.0475136
$$153$$ 22.3848 1.80970
$$154$$ 0 0
$$155$$ 0 0
$$156$$ −16.4853 −1.31988
$$157$$ 6.48528 0.517582 0.258791 0.965933i $$-0.416676\pi$$
0.258791 + 0.965933i $$0.416676\pi$$
$$158$$ −6.82843 −0.543240
$$159$$ −22.1421 −1.75599
$$160$$ 0 0
$$161$$ 0 0
$$162$$ 39.9706 3.14038
$$163$$ 20.1421 1.57765 0.788827 0.614615i $$-0.210689\pi$$
0.788827 + 0.614615i $$0.210689\pi$$
$$164$$ 3.07107 0.239810
$$165$$ 0 0
$$166$$ −2.24264 −0.174063
$$167$$ 15.7990 1.22256 0.611281 0.791413i $$-0.290654\pi$$
0.611281 + 0.791413i $$0.290654\pi$$
$$168$$ 0 0
$$169$$ 10.3137 0.793362
$$170$$ 0 0
$$171$$ −5.07107 −0.387794
$$172$$ 8.82843 0.673161
$$173$$ −8.82843 −0.671213 −0.335606 0.942002i $$-0.608941\pi$$
−0.335606 + 0.942002i $$0.608941\pi$$
$$174$$ −16.4853 −1.24975
$$175$$ 0 0
$$176$$ −0.828427 −0.0624450
$$177$$ −29.3137 −2.20335
$$178$$ −12.7279 −0.953998
$$179$$ 4.00000 0.298974 0.149487 0.988764i $$-0.452238\pi$$
0.149487 + 0.988764i $$0.452238\pi$$
$$180$$ 0 0
$$181$$ 2.48528 0.184730 0.0923648 0.995725i $$-0.470557\pi$$
0.0923648 + 0.995725i $$0.470557\pi$$
$$182$$ 0 0
$$183$$ −31.7990 −2.35065
$$184$$ 1.17157 0.0863695
$$185$$ 0 0
$$186$$ 9.65685 0.708075
$$187$$ −2.14214 −0.156648
$$188$$ 5.17157 0.377176
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 10.1421 0.733859 0.366930 0.930249i $$-0.380409\pi$$
0.366930 + 0.930249i $$0.380409\pi$$
$$192$$ 3.41421 0.246400
$$193$$ −5.65685 −0.407189 −0.203595 0.979055i $$-0.565262\pi$$
−0.203595 + 0.979055i $$0.565262\pi$$
$$194$$ −7.75736 −0.556946
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 25.7990 1.83810 0.919051 0.394139i $$-0.128957\pi$$
0.919051 + 0.394139i $$0.128957\pi$$
$$198$$ −7.17157 −0.509661
$$199$$ −16.4853 −1.16861 −0.584305 0.811534i $$-0.698633\pi$$
−0.584305 + 0.811534i $$0.698633\pi$$
$$200$$ 0 0
$$201$$ −5.65685 −0.399004
$$202$$ −13.3137 −0.936749
$$203$$ 0 0
$$204$$ 8.82843 0.618114
$$205$$ 0 0
$$206$$ −14.8284 −1.03315
$$207$$ 10.1421 0.704927
$$208$$ −4.82843 −0.334791
$$209$$ 0.485281 0.0335676
$$210$$ 0 0
$$211$$ 18.6274 1.28236 0.641182 0.767389i $$-0.278444\pi$$
0.641182 + 0.767389i $$0.278444\pi$$
$$212$$ −6.48528 −0.445411
$$213$$ −15.3137 −1.04928
$$214$$ −9.65685 −0.660129
$$215$$ 0 0
$$216$$ 19.3137 1.31413
$$217$$ 0 0
$$218$$ 2.48528 0.168324
$$219$$ −32.1421 −2.17196
$$220$$ 0 0
$$221$$ −12.4853 −0.839851
$$222$$ 26.1421 1.75455
$$223$$ −7.31371 −0.489762 −0.244881 0.969553i $$-0.578749\pi$$
−0.244881 + 0.969553i $$0.578749\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 15.3137 1.01865
$$227$$ 18.2426 1.21081 0.605403 0.795919i $$-0.293012\pi$$
0.605403 + 0.795919i $$0.293012\pi$$
$$228$$ −2.00000 −0.132453
$$229$$ 16.1421 1.06670 0.533351 0.845894i $$-0.320932\pi$$
0.533351 + 0.845894i $$0.320932\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ −4.82843 −0.317002
$$233$$ −23.3137 −1.52733 −0.763666 0.645612i $$-0.776603\pi$$
−0.763666 + 0.645612i $$0.776603\pi$$
$$234$$ −41.7990 −2.73249
$$235$$ 0 0
$$236$$ −8.58579 −0.558887
$$237$$ −23.3137 −1.51439
$$238$$ 0 0
$$239$$ 1.65685 0.107173 0.0535865 0.998563i $$-0.482935\pi$$
0.0535865 + 0.998563i $$0.482935\pi$$
$$240$$ 0 0
$$241$$ 13.4142 0.864085 0.432043 0.901853i $$-0.357793\pi$$
0.432043 + 0.901853i $$0.357793\pi$$
$$242$$ −10.3137 −0.662990
$$243$$ 78.5269 5.03750
$$244$$ −9.31371 −0.596249
$$245$$ 0 0
$$246$$ 10.4853 0.668517
$$247$$ 2.82843 0.179969
$$248$$ 2.82843 0.179605
$$249$$ −7.65685 −0.485233
$$250$$ 0 0
$$251$$ −0.585786 −0.0369745 −0.0184873 0.999829i $$-0.505885\pi$$
−0.0184873 + 0.999829i $$0.505885\pi$$
$$252$$ 0 0
$$253$$ −0.970563 −0.0610188
$$254$$ −2.82843 −0.177471
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ −9.89949 −0.617514 −0.308757 0.951141i $$-0.599913\pi$$
−0.308757 + 0.951141i $$0.599913\pi$$
$$258$$ 30.1421 1.87657
$$259$$ 0 0
$$260$$ 0 0
$$261$$ −41.7990 −2.58729
$$262$$ 6.24264 0.385672
$$263$$ −28.0000 −1.72655 −0.863277 0.504730i $$-0.831592\pi$$
−0.863277 + 0.504730i $$0.831592\pi$$
$$264$$ −2.82843 −0.174078
$$265$$ 0 0
$$266$$ 0 0
$$267$$ −43.4558 −2.65945
$$268$$ −1.65685 −0.101208
$$269$$ −18.4853 −1.12707 −0.563534 0.826093i $$-0.690559\pi$$
−0.563534 + 0.826093i $$0.690559\pi$$
$$270$$ 0 0
$$271$$ −12.0000 −0.728948 −0.364474 0.931214i $$-0.618751\pi$$
−0.364474 + 0.931214i $$0.618751\pi$$
$$272$$ 2.58579 0.156786
$$273$$ 0 0
$$274$$ 16.0000 0.966595
$$275$$ 0 0
$$276$$ 4.00000 0.240772
$$277$$ 8.14214 0.489214 0.244607 0.969622i $$-0.421341\pi$$
0.244607 + 0.969622i $$0.421341\pi$$
$$278$$ −19.8995 −1.19349
$$279$$ 24.4853 1.46590
$$280$$ 0 0
$$281$$ 8.00000 0.477240 0.238620 0.971113i $$-0.423305\pi$$
0.238620 + 0.971113i $$0.423305\pi$$
$$282$$ 17.6569 1.05145
$$283$$ −2.24264 −0.133311 −0.0666556 0.997776i $$-0.521233\pi$$
−0.0666556 + 0.997776i $$0.521233\pi$$
$$284$$ −4.48528 −0.266152
$$285$$ 0 0
$$286$$ 4.00000 0.236525
$$287$$ 0 0
$$288$$ 8.65685 0.510110
$$289$$ −10.3137 −0.606689
$$290$$ 0 0
$$291$$ −26.4853 −1.55259
$$292$$ −9.41421 −0.550925
$$293$$ −8.34315 −0.487412 −0.243706 0.969849i $$-0.578363\pi$$
−0.243706 + 0.969849i $$0.578363\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 7.65685 0.445046
$$297$$ −16.0000 −0.928414
$$298$$ −6.00000 −0.347571
$$299$$ −5.65685 −0.327144
$$300$$ 0 0
$$301$$ 0 0
$$302$$ −11.3137 −0.651031
$$303$$ −45.4558 −2.61137
$$304$$ −0.585786 −0.0335972
$$305$$ 0 0
$$306$$ 22.3848 1.27965
$$307$$ 14.9289 0.852039 0.426020 0.904714i $$-0.359915\pi$$
0.426020 + 0.904714i $$0.359915\pi$$
$$308$$ 0 0
$$309$$ −50.6274 −2.88009
$$310$$ 0 0
$$311$$ −4.00000 −0.226819 −0.113410 0.993548i $$-0.536177\pi$$
−0.113410 + 0.993548i $$0.536177\pi$$
$$312$$ −16.4853 −0.933295
$$313$$ 14.3848 0.813076 0.406538 0.913634i $$-0.366736\pi$$
0.406538 + 0.913634i $$0.366736\pi$$
$$314$$ 6.48528 0.365986
$$315$$ 0 0
$$316$$ −6.82843 −0.384129
$$317$$ −10.4853 −0.588912 −0.294456 0.955665i $$-0.595138\pi$$
−0.294456 + 0.955665i $$0.595138\pi$$
$$318$$ −22.1421 −1.24167
$$319$$ 4.00000 0.223957
$$320$$ 0 0
$$321$$ −32.9706 −1.84024
$$322$$ 0 0
$$323$$ −1.51472 −0.0842812
$$324$$ 39.9706 2.22059
$$325$$ 0 0
$$326$$ 20.1421 1.11557
$$327$$ 8.48528 0.469237
$$328$$ 3.07107 0.169571
$$329$$ 0 0
$$330$$ 0 0
$$331$$ 33.7990 1.85776 0.928880 0.370380i $$-0.120773\pi$$
0.928880 + 0.370380i $$0.120773\pi$$
$$332$$ −2.24264 −0.123081
$$333$$ 66.2843 3.63236
$$334$$ 15.7990 0.864482
$$335$$ 0 0
$$336$$ 0 0
$$337$$ −6.00000 −0.326841 −0.163420 0.986557i $$-0.552253\pi$$
−0.163420 + 0.986557i $$0.552253\pi$$
$$338$$ 10.3137 0.560992
$$339$$ 52.2843 2.83969
$$340$$ 0 0
$$341$$ −2.34315 −0.126888
$$342$$ −5.07107 −0.274212
$$343$$ 0 0
$$344$$ 8.82843 0.475997
$$345$$ 0 0
$$346$$ −8.82843 −0.474619
$$347$$ 3.17157 0.170259 0.0851295 0.996370i $$-0.472870\pi$$
0.0851295 + 0.996370i $$0.472870\pi$$
$$348$$ −16.4853 −0.883704
$$349$$ −2.48528 −0.133034 −0.0665170 0.997785i $$-0.521189\pi$$
−0.0665170 + 0.997785i $$0.521189\pi$$
$$350$$ 0 0
$$351$$ −93.2548 −4.97757
$$352$$ −0.828427 −0.0441553
$$353$$ −2.38478 −0.126929 −0.0634644 0.997984i $$-0.520215\pi$$
−0.0634644 + 0.997984i $$0.520215\pi$$
$$354$$ −29.3137 −1.55801
$$355$$ 0 0
$$356$$ −12.7279 −0.674579
$$357$$ 0 0
$$358$$ 4.00000 0.211407
$$359$$ 28.2843 1.49279 0.746393 0.665505i $$-0.231784\pi$$
0.746393 + 0.665505i $$0.231784\pi$$
$$360$$ 0 0
$$361$$ −18.6569 −0.981940
$$362$$ 2.48528 0.130623
$$363$$ −35.2132 −1.84821
$$364$$ 0 0
$$365$$ 0 0
$$366$$ −31.7990 −1.66216
$$367$$ 24.9706 1.30345 0.651726 0.758454i $$-0.274045\pi$$
0.651726 + 0.758454i $$0.274045\pi$$
$$368$$ 1.17157 0.0610725
$$369$$ 26.5858 1.38400
$$370$$ 0 0
$$371$$ 0 0
$$372$$ 9.65685 0.500685
$$373$$ 30.4853 1.57847 0.789234 0.614093i $$-0.210478\pi$$
0.789234 + 0.614093i $$0.210478\pi$$
$$374$$ −2.14214 −0.110767
$$375$$ 0 0
$$376$$ 5.17157 0.266704
$$377$$ 23.3137 1.20072
$$378$$ 0 0
$$379$$ 34.4853 1.77139 0.885695 0.464268i $$-0.153682\pi$$
0.885695 + 0.464268i $$0.153682\pi$$
$$380$$ 0 0
$$381$$ −9.65685 −0.494736
$$382$$ 10.1421 0.518917
$$383$$ −32.4853 −1.65992 −0.829960 0.557823i $$-0.811637\pi$$
−0.829960 + 0.557823i $$0.811637\pi$$
$$384$$ 3.41421 0.174231
$$385$$ 0 0
$$386$$ −5.65685 −0.287926
$$387$$ 76.4264 3.88497
$$388$$ −7.75736 −0.393820
$$389$$ 28.1421 1.42686 0.713431 0.700725i $$-0.247140\pi$$
0.713431 + 0.700725i $$0.247140\pi$$
$$390$$ 0 0
$$391$$ 3.02944 0.153205
$$392$$ 0 0
$$393$$ 21.3137 1.07513
$$394$$ 25.7990 1.29973
$$395$$ 0 0
$$396$$ −7.17157 −0.360385
$$397$$ 33.7990 1.69632 0.848161 0.529738i $$-0.177710\pi$$
0.848161 + 0.529738i $$0.177710\pi$$
$$398$$ −16.4853 −0.826332
$$399$$ 0 0
$$400$$ 0 0
$$401$$ −6.00000 −0.299626 −0.149813 0.988714i $$-0.547867\pi$$
−0.149813 + 0.988714i $$0.547867\pi$$
$$402$$ −5.65685 −0.282138
$$403$$ −13.6569 −0.680296
$$404$$ −13.3137 −0.662382
$$405$$ 0 0
$$406$$ 0 0
$$407$$ −6.34315 −0.314418
$$408$$ 8.82843 0.437072
$$409$$ 10.5858 0.523433 0.261717 0.965145i $$-0.415711\pi$$
0.261717 + 0.965145i $$0.415711\pi$$
$$410$$ 0 0
$$411$$ 54.6274 2.69457
$$412$$ −14.8284 −0.730544
$$413$$ 0 0
$$414$$ 10.1421 0.498459
$$415$$ 0 0
$$416$$ −4.82843 −0.236733
$$417$$ −67.9411 −3.32709
$$418$$ 0.485281 0.0237359
$$419$$ 20.8701 1.01957 0.509785 0.860302i $$-0.329725\pi$$
0.509785 + 0.860302i $$0.329725\pi$$
$$420$$ 0 0
$$421$$ 17.3137 0.843819 0.421909 0.906638i $$-0.361360\pi$$
0.421909 + 0.906638i $$0.361360\pi$$
$$422$$ 18.6274 0.906768
$$423$$ 44.7696 2.17677
$$424$$ −6.48528 −0.314953
$$425$$ 0 0
$$426$$ −15.3137 −0.741952
$$427$$ 0 0
$$428$$ −9.65685 −0.466782
$$429$$ 13.6569 0.659359
$$430$$ 0 0
$$431$$ −22.3431 −1.07623 −0.538116 0.842871i $$-0.680864\pi$$
−0.538116 + 0.842871i $$0.680864\pi$$
$$432$$ 19.3137 0.929231
$$433$$ 10.5858 0.508720 0.254360 0.967110i $$-0.418135\pi$$
0.254360 + 0.967110i $$0.418135\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 2.48528 0.119023
$$437$$ −0.686292 −0.0328298
$$438$$ −32.1421 −1.53581
$$439$$ 24.9706 1.19178 0.595890 0.803066i $$-0.296799\pi$$
0.595890 + 0.803066i $$0.296799\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ −12.4853 −0.593864
$$443$$ −3.02944 −0.143933 −0.0719665 0.997407i $$-0.522927\pi$$
−0.0719665 + 0.997407i $$0.522927\pi$$
$$444$$ 26.1421 1.24065
$$445$$ 0 0
$$446$$ −7.31371 −0.346314
$$447$$ −20.4853 −0.968921
$$448$$ 0 0
$$449$$ −16.6274 −0.784696 −0.392348 0.919817i $$-0.628337\pi$$
−0.392348 + 0.919817i $$0.628337\pi$$
$$450$$ 0 0
$$451$$ −2.54416 −0.119800
$$452$$ 15.3137 0.720296
$$453$$ −38.6274 −1.81487
$$454$$ 18.2426 0.856170
$$455$$ 0 0
$$456$$ −2.00000 −0.0936586
$$457$$ −21.6569 −1.01306 −0.506532 0.862221i $$-0.669073\pi$$
−0.506532 + 0.862221i $$0.669073\pi$$
$$458$$ 16.1421 0.754272
$$459$$ 49.9411 2.33105
$$460$$ 0 0
$$461$$ 12.8284 0.597479 0.298740 0.954335i $$-0.403434\pi$$
0.298740 + 0.954335i $$0.403434\pi$$
$$462$$ 0 0
$$463$$ 16.9706 0.788689 0.394344 0.918963i $$-0.370972\pi$$
0.394344 + 0.918963i $$0.370972\pi$$
$$464$$ −4.82843 −0.224154
$$465$$ 0 0
$$466$$ −23.3137 −1.07999
$$467$$ −15.8995 −0.735741 −0.367870 0.929877i $$-0.619913\pi$$
−0.367870 + 0.929877i $$0.619913\pi$$
$$468$$ −41.7990 −1.93216
$$469$$ 0 0
$$470$$ 0 0
$$471$$ 22.1421 1.02026
$$472$$ −8.58579 −0.395193
$$473$$ −7.31371 −0.336285
$$474$$ −23.3137 −1.07083
$$475$$ 0 0
$$476$$ 0 0
$$477$$ −56.1421 −2.57057
$$478$$ 1.65685 0.0757827
$$479$$ 17.1716 0.784589 0.392295 0.919840i $$-0.371681\pi$$
0.392295 + 0.919840i $$0.371681\pi$$
$$480$$ 0 0
$$481$$ −36.9706 −1.68571
$$482$$ 13.4142 0.611001
$$483$$ 0 0
$$484$$ −10.3137 −0.468805
$$485$$ 0 0
$$486$$ 78.5269 3.56205
$$487$$ −31.7990 −1.44095 −0.720475 0.693481i $$-0.756076\pi$$
−0.720475 + 0.693481i $$0.756076\pi$$
$$488$$ −9.31371 −0.421612
$$489$$ 68.7696 3.10987
$$490$$ 0 0
$$491$$ 32.2843 1.45697 0.728484 0.685062i $$-0.240225\pi$$
0.728484 + 0.685062i $$0.240225\pi$$
$$492$$ 10.4853 0.472713
$$493$$ −12.4853 −0.562309
$$494$$ 2.82843 0.127257
$$495$$ 0 0
$$496$$ 2.82843 0.127000
$$497$$ 0 0
$$498$$ −7.65685 −0.343112
$$499$$ 30.3431 1.35835 0.679173 0.733978i $$-0.262339\pi$$
0.679173 + 0.733978i $$0.262339\pi$$
$$500$$ 0 0
$$501$$ 53.9411 2.40991
$$502$$ −0.585786 −0.0261449
$$503$$ 17.6569 0.787280 0.393640 0.919265i $$-0.371216\pi$$
0.393640 + 0.919265i $$0.371216\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ −0.970563 −0.0431468
$$507$$ 35.2132 1.56387
$$508$$ −2.82843 −0.125491
$$509$$ −5.79899 −0.257036 −0.128518 0.991707i $$-0.541022\pi$$
−0.128518 + 0.991707i $$0.541022\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ 1.00000 0.0441942
$$513$$ −11.3137 −0.499512
$$514$$ −9.89949 −0.436648
$$515$$ 0 0
$$516$$ 30.1421 1.32693
$$517$$ −4.28427 −0.188422
$$518$$ 0 0
$$519$$ −30.1421 −1.32309
$$520$$ 0 0
$$521$$ −19.0711 −0.835519 −0.417759 0.908558i $$-0.637184\pi$$
−0.417759 + 0.908558i $$0.637184\pi$$
$$522$$ −41.7990 −1.82949
$$523$$ 23.8995 1.04505 0.522526 0.852623i $$-0.324990\pi$$
0.522526 + 0.852623i $$0.324990\pi$$
$$524$$ 6.24264 0.272711
$$525$$ 0 0
$$526$$ −28.0000 −1.22086
$$527$$ 7.31371 0.318590
$$528$$ −2.82843 −0.123091
$$529$$ −21.6274 −0.940322
$$530$$ 0 0
$$531$$ −74.3259 −3.22547
$$532$$ 0 0
$$533$$ −14.8284 −0.642290
$$534$$ −43.4558 −1.88052
$$535$$ 0 0
$$536$$ −1.65685 −0.0715652
$$537$$ 13.6569 0.589337
$$538$$ −18.4853 −0.796957
$$539$$ 0 0
$$540$$ 0 0
$$541$$ −14.9706 −0.643635 −0.321817 0.946802i $$-0.604294\pi$$
−0.321817 + 0.946802i $$0.604294\pi$$
$$542$$ −12.0000 −0.515444
$$543$$ 8.48528 0.364138
$$544$$ 2.58579 0.110865
$$545$$ 0 0
$$546$$ 0 0
$$547$$ 10.4853 0.448318 0.224159 0.974553i $$-0.428036\pi$$
0.224159 + 0.974553i $$0.428036\pi$$
$$548$$ 16.0000 0.683486
$$549$$ −80.6274 −3.44109
$$550$$ 0 0
$$551$$ 2.82843 0.120495
$$552$$ 4.00000 0.170251
$$553$$ 0 0
$$554$$ 8.14214 0.345926
$$555$$ 0 0
$$556$$ −19.8995 −0.843927
$$557$$ −15.1716 −0.642840 −0.321420 0.946937i $$-0.604160\pi$$
−0.321420 + 0.946937i $$0.604160\pi$$
$$558$$ 24.4853 1.03654
$$559$$ −42.6274 −1.80295
$$560$$ 0 0
$$561$$ −7.31371 −0.308785
$$562$$ 8.00000 0.337460
$$563$$ −36.5858 −1.54191 −0.770954 0.636891i $$-0.780220\pi$$
−0.770954 + 0.636891i $$0.780220\pi$$
$$564$$ 17.6569 0.743488
$$565$$ 0 0
$$566$$ −2.24264 −0.0942652
$$567$$ 0 0
$$568$$ −4.48528 −0.188198
$$569$$ −29.3137 −1.22889 −0.614447 0.788958i $$-0.710621\pi$$
−0.614447 + 0.788958i $$0.710621\pi$$
$$570$$ 0 0
$$571$$ −2.20101 −0.0921094 −0.0460547 0.998939i $$-0.514665\pi$$
−0.0460547 + 0.998939i $$0.514665\pi$$
$$572$$ 4.00000 0.167248
$$573$$ 34.6274 1.44658
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 8.65685 0.360702
$$577$$ 6.10051 0.253967 0.126984 0.991905i $$-0.459470\pi$$
0.126984 + 0.991905i $$0.459470\pi$$
$$578$$ −10.3137 −0.428994
$$579$$ −19.3137 −0.802650
$$580$$ 0 0
$$581$$ 0 0
$$582$$ −26.4853 −1.09785
$$583$$ 5.37258 0.222510
$$584$$ −9.41421 −0.389563
$$585$$ 0 0
$$586$$ −8.34315 −0.344652
$$587$$ 17.0711 0.704598 0.352299 0.935887i $$-0.385400\pi$$
0.352299 + 0.935887i $$0.385400\pi$$
$$588$$ 0 0
$$589$$ −1.65685 −0.0682695
$$590$$ 0 0
$$591$$ 88.0833 3.62326
$$592$$ 7.65685 0.314695
$$593$$ 3.27208 0.134368 0.0671841 0.997741i $$-0.478599\pi$$
0.0671841 + 0.997741i $$0.478599\pi$$
$$594$$ −16.0000 −0.656488
$$595$$ 0 0
$$596$$ −6.00000 −0.245770
$$597$$ −56.2843 −2.30356
$$598$$ −5.65685 −0.231326
$$599$$ −10.8284 −0.442438 −0.221219 0.975224i $$-0.571003\pi$$
−0.221219 + 0.975224i $$0.571003\pi$$
$$600$$ 0 0
$$601$$ −6.58579 −0.268640 −0.134320 0.990938i $$-0.542885\pi$$
−0.134320 + 0.990938i $$0.542885\pi$$
$$602$$ 0 0
$$603$$ −14.3431 −0.584098
$$604$$ −11.3137 −0.460348
$$605$$ 0 0
$$606$$ −45.4558 −1.84652
$$607$$ 16.2843 0.660958 0.330479 0.943813i $$-0.392790\pi$$
0.330479 + 0.943813i $$0.392790\pi$$
$$608$$ −0.585786 −0.0237568
$$609$$ 0 0
$$610$$ 0 0
$$611$$ −24.9706 −1.01020
$$612$$ 22.3848 0.904851
$$613$$ 12.3431 0.498535 0.249267 0.968435i $$-0.419810\pi$$
0.249267 + 0.968435i $$0.419810\pi$$
$$614$$ 14.9289 0.602483
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 33.3137 1.34116 0.670580 0.741837i $$-0.266045\pi$$
0.670580 + 0.741837i $$0.266045\pi$$
$$618$$ −50.6274 −2.03653
$$619$$ −29.0711 −1.16846 −0.584232 0.811586i $$-0.698604\pi$$
−0.584232 + 0.811586i $$0.698604\pi$$
$$620$$ 0 0
$$621$$ 22.6274 0.908007
$$622$$ −4.00000 −0.160385
$$623$$ 0 0
$$624$$ −16.4853 −0.659939
$$625$$ 0 0
$$626$$ 14.3848 0.574931
$$627$$ 1.65685 0.0661684
$$628$$ 6.48528 0.258791
$$629$$ 19.7990 0.789437
$$630$$ 0 0
$$631$$ −12.4853 −0.497031 −0.248516 0.968628i $$-0.579943\pi$$
−0.248516 + 0.968628i $$0.579943\pi$$
$$632$$ −6.82843 −0.271620
$$633$$ 63.5980 2.52779
$$634$$ −10.4853 −0.416424
$$635$$ 0 0
$$636$$ −22.1421 −0.877993
$$637$$ 0 0
$$638$$ 4.00000 0.158362
$$639$$ −38.8284 −1.53603
$$640$$ 0 0
$$641$$ −24.6274 −0.972724 −0.486362 0.873757i $$-0.661676\pi$$
−0.486362 + 0.873757i $$0.661676\pi$$
$$642$$ −32.9706 −1.30124
$$643$$ 4.78680 0.188773 0.0943864 0.995536i $$-0.469911\pi$$
0.0943864 + 0.995536i $$0.469911\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ −1.51472 −0.0595958
$$647$$ −23.1127 −0.908654 −0.454327 0.890835i $$-0.650120\pi$$
−0.454327 + 0.890835i $$0.650120\pi$$
$$648$$ 39.9706 1.57019
$$649$$ 7.11270 0.279198
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 20.1421 0.788827
$$653$$ −4.34315 −0.169960 −0.0849802 0.996383i $$-0.527083\pi$$
−0.0849802 + 0.996383i $$0.527083\pi$$
$$654$$ 8.48528 0.331801
$$655$$ 0 0
$$656$$ 3.07107 0.119905
$$657$$ −81.4975 −3.17952
$$658$$ 0 0
$$659$$ −27.1716 −1.05845 −0.529227 0.848480i $$-0.677518\pi$$
−0.529227 + 0.848480i $$0.677518\pi$$
$$660$$ 0 0
$$661$$ 38.2843 1.48909 0.744543 0.667575i $$-0.232668\pi$$
0.744543 + 0.667575i $$0.232668\pi$$
$$662$$ 33.7990 1.31364
$$663$$ −42.6274 −1.65551
$$664$$ −2.24264 −0.0870313
$$665$$ 0 0
$$666$$ 66.2843 2.56846
$$667$$ −5.65685 −0.219034
$$668$$ 15.7990 0.611281
$$669$$ −24.9706 −0.965418
$$670$$ 0 0
$$671$$ 7.71573 0.297862
$$672$$ 0 0
$$673$$ 48.0000 1.85026 0.925132 0.379646i $$-0.123954\pi$$
0.925132 + 0.379646i $$0.123954\pi$$
$$674$$ −6.00000 −0.231111
$$675$$ 0 0
$$676$$ 10.3137 0.396681
$$677$$ 39.4558 1.51641 0.758206 0.652015i $$-0.226076\pi$$
0.758206 + 0.652015i $$0.226076\pi$$
$$678$$ 52.2843 2.00797
$$679$$ 0 0
$$680$$ 0 0
$$681$$ 62.2843 2.38674
$$682$$ −2.34315 −0.0897237
$$683$$ −33.6569 −1.28784 −0.643922 0.765091i $$-0.722694\pi$$
−0.643922 + 0.765091i $$0.722694\pi$$
$$684$$ −5.07107 −0.193897
$$685$$ 0 0
$$686$$ 0 0
$$687$$ 55.1127 2.10268
$$688$$ 8.82843 0.336581
$$689$$ 31.3137 1.19296
$$690$$ 0 0
$$691$$ −1.75736 −0.0668531 −0.0334265 0.999441i $$-0.510642\pi$$
−0.0334265 + 0.999441i $$0.510642\pi$$
$$692$$ −8.82843 −0.335606
$$693$$ 0 0
$$694$$ 3.17157 0.120391
$$695$$ 0 0
$$696$$ −16.4853 −0.624873
$$697$$ 7.94113 0.300792
$$698$$ −2.48528 −0.0940693
$$699$$ −79.5980 −3.01067
$$700$$ 0 0
$$701$$ 2.48528 0.0938678 0.0469339 0.998898i $$-0.485055\pi$$
0.0469339 + 0.998898i $$0.485055\pi$$
$$702$$ −93.2548 −3.51968
$$703$$ −4.48528 −0.169166
$$704$$ −0.828427 −0.0312225
$$705$$ 0 0
$$706$$ −2.38478 −0.0897522
$$707$$ 0 0
$$708$$ −29.3137 −1.10168
$$709$$ 45.1127 1.69424 0.847121 0.531399i $$-0.178334\pi$$
0.847121 + 0.531399i $$0.178334\pi$$
$$710$$ 0 0
$$711$$ −59.1127 −2.21690
$$712$$ −12.7279 −0.476999
$$713$$ 3.31371 0.124099
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 4.00000 0.149487
$$717$$ 5.65685 0.211259
$$718$$ 28.2843 1.05556
$$719$$ −41.4558 −1.54604 −0.773021 0.634380i $$-0.781255\pi$$
−0.773021 + 0.634380i $$0.781255\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ −18.6569 −0.694336
$$723$$ 45.7990 1.70328
$$724$$ 2.48528 0.0923648
$$725$$ 0 0
$$726$$ −35.2132 −1.30688
$$727$$ −3.51472 −0.130354 −0.0651768 0.997874i $$-0.520761\pi$$
−0.0651768 + 0.997874i $$0.520761\pi$$
$$728$$ 0 0
$$729$$ 148.196 5.48874
$$730$$ 0 0
$$731$$ 22.8284 0.844340
$$732$$ −31.7990 −1.17532
$$733$$ 34.0000 1.25582 0.627909 0.778287i $$-0.283911\pi$$
0.627909 + 0.778287i $$0.283911\pi$$
$$734$$ 24.9706 0.921680
$$735$$ 0 0
$$736$$ 1.17157 0.0431847
$$737$$ 1.37258 0.0505597
$$738$$ 26.5858 0.978636
$$739$$ 3.17157 0.116668 0.0583341 0.998297i $$-0.481421\pi$$
0.0583341 + 0.998297i $$0.481421\pi$$
$$740$$ 0 0
$$741$$ 9.65685 0.354753
$$742$$ 0 0
$$743$$ −51.7990 −1.90032 −0.950160 0.311762i $$-0.899081\pi$$
−0.950160 + 0.311762i $$0.899081\pi$$
$$744$$ 9.65685 0.354037
$$745$$ 0 0
$$746$$ 30.4853 1.11615
$$747$$ −19.4142 −0.710329
$$748$$ −2.14214 −0.0783242
$$749$$ 0 0
$$750$$ 0 0
$$751$$ −39.3137 −1.43458 −0.717289 0.696776i $$-0.754617\pi$$
−0.717289 + 0.696776i $$0.754617\pi$$
$$752$$ 5.17157 0.188588
$$753$$ −2.00000 −0.0728841
$$754$$ 23.3137 0.849035
$$755$$ 0 0
$$756$$ 0 0
$$757$$ −3.65685 −0.132911 −0.0664553 0.997789i $$-0.521169\pi$$
−0.0664553 + 0.997789i $$0.521169\pi$$
$$758$$ 34.4853 1.25256
$$759$$ −3.31371 −0.120280
$$760$$ 0 0
$$761$$ −22.3848 −0.811448 −0.405724 0.913996i $$-0.632980\pi$$
−0.405724 + 0.913996i $$0.632980\pi$$
$$762$$ −9.65685 −0.349831
$$763$$ 0 0
$$764$$ 10.1421 0.366930
$$765$$ 0 0
$$766$$ −32.4853 −1.17374
$$767$$ 41.4558 1.49688
$$768$$ 3.41421 0.123200
$$769$$ −19.5563 −0.705220 −0.352610 0.935770i $$-0.614706\pi$$
−0.352610 + 0.935770i $$0.614706\pi$$
$$770$$ 0 0
$$771$$ −33.7990 −1.21724
$$772$$ −5.65685 −0.203595
$$773$$ −2.00000 −0.0719350 −0.0359675 0.999353i $$-0.511451\pi$$
−0.0359675 + 0.999353i $$0.511451\pi$$
$$774$$ 76.4264 2.74709
$$775$$ 0 0
$$776$$ −7.75736 −0.278473
$$777$$ 0 0
$$778$$ 28.1421 1.00894
$$779$$ −1.79899 −0.0644555
$$780$$ 0 0
$$781$$ 3.71573 0.132959
$$782$$ 3.02944 0.108332
$$783$$ −93.2548 −3.33266
$$784$$ 0 0
$$785$$ 0 0
$$786$$ 21.3137 0.760235
$$787$$ −1.27208 −0.0453447 −0.0226723 0.999743i $$-0.507217\pi$$
−0.0226723 + 0.999743i $$0.507217\pi$$
$$788$$ 25.7990 0.919051
$$789$$ −95.5980 −3.40338
$$790$$ 0 0
$$791$$ 0 0
$$792$$ −7.17157 −0.254831
$$793$$ 44.9706 1.59695
$$794$$ 33.7990 1.19948
$$795$$ 0 0
$$796$$ −16.4853 −0.584305
$$797$$ −41.7990 −1.48060 −0.740298 0.672279i $$-0.765316\pi$$
−0.740298 + 0.672279i $$0.765316\pi$$
$$798$$ 0 0
$$799$$ 13.3726 0.473088
$$800$$ 0 0
$$801$$ −110.184 −3.89315
$$802$$ −6.00000 −0.211867
$$803$$ 7.79899 0.275220
$$804$$ −5.65685 −0.199502
$$805$$ 0 0
$$806$$ −13.6569 −0.481042
$$807$$ −63.1127 −2.22167
$$808$$ −13.3137 −0.468375
$$809$$ 3.02944 0.106509 0.0532547 0.998581i $$-0.483040\pi$$
0.0532547 + 0.998581i $$0.483040\pi$$
$$810$$ 0 0
$$811$$ −32.5858 −1.14424 −0.572121 0.820169i $$-0.693879\pi$$
−0.572121 + 0.820169i $$0.693879\pi$$
$$812$$ 0 0
$$813$$ −40.9706 −1.43690
$$814$$ −6.34315 −0.222327
$$815$$ 0 0
$$816$$ 8.82843 0.309057
$$817$$ −5.17157 −0.180930
$$818$$ 10.5858 0.370123
$$819$$ 0 0
$$820$$ 0 0
$$821$$ 17.3137 0.604253 0.302126 0.953268i $$-0.402304\pi$$
0.302126 + 0.953268i $$0.402304\pi$$
$$822$$ 54.6274 1.90535
$$823$$ −20.2843 −0.707065 −0.353533 0.935422i $$-0.615020\pi$$
−0.353533 + 0.935422i $$0.615020\pi$$
$$824$$ −14.8284 −0.516573
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 5.37258 0.186823 0.0934115 0.995628i $$-0.470223\pi$$
0.0934115 + 0.995628i $$0.470223\pi$$
$$828$$ 10.1421 0.352464
$$829$$ 5.02944 0.174680 0.0873398 0.996179i $$-0.472163\pi$$
0.0873398 + 0.996179i $$0.472163\pi$$
$$830$$ 0 0
$$831$$ 27.7990 0.964336
$$832$$ −4.82843 −0.167396
$$833$$ 0 0
$$834$$ −67.9411 −2.35261
$$835$$ 0 0
$$836$$ 0.485281 0.0167838
$$837$$ 54.6274 1.88820
$$838$$ 20.8701 0.720944
$$839$$ 42.1421 1.45491 0.727454 0.686156i $$-0.240703\pi$$
0.727454 + 0.686156i $$0.240703\pi$$
$$840$$ 0 0
$$841$$ −5.68629 −0.196079
$$842$$ 17.3137 0.596670
$$843$$ 27.3137 0.940734
$$844$$ 18.6274 0.641182
$$845$$ 0 0
$$846$$ 44.7696 1.53921
$$847$$ 0 0
$$848$$ −6.48528 −0.222705
$$849$$ −7.65685 −0.262783
$$850$$ 0 0
$$851$$ 8.97056 0.307507
$$852$$ −15.3137 −0.524639
$$853$$ −43.1716 −1.47817 −0.739083 0.673614i $$-0.764741\pi$$
−0.739083 + 0.673614i $$0.764741\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ −9.65685 −0.330064
$$857$$ −4.92893 −0.168369 −0.0841846 0.996450i $$-0.526829\pi$$
−0.0841846 + 0.996450i $$0.526829\pi$$
$$858$$ 13.6569 0.466237
$$859$$ 7.21320 0.246111 0.123056 0.992400i $$-0.460731\pi$$
0.123056 + 0.992400i $$0.460731\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ −22.3431 −0.761011
$$863$$ 4.97056 0.169200 0.0846000 0.996415i $$-0.473039\pi$$
0.0846000 + 0.996415i $$0.473039\pi$$
$$864$$ 19.3137 0.657066
$$865$$ 0 0
$$866$$ 10.5858 0.359720
$$867$$ −35.2132 −1.19590
$$868$$ 0 0
$$869$$ 5.65685 0.191896
$$870$$ 0 0
$$871$$ 8.00000 0.271070
$$872$$ 2.48528 0.0841622
$$873$$ −67.1543 −2.27283
$$874$$ −0.686292 −0.0232142
$$875$$ 0 0
$$876$$ −32.1421 −1.08598
$$877$$ 30.2843 1.02263 0.511314 0.859394i $$-0.329159\pi$$
0.511314 + 0.859394i $$0.329159\pi$$
$$878$$ 24.9706 0.842716
$$879$$ −28.4853 −0.960785
$$880$$ 0 0
$$881$$ 2.38478 0.0803452 0.0401726 0.999193i $$-0.487209\pi$$
0.0401726 + 0.999193i $$0.487209\pi$$
$$882$$ 0 0
$$883$$ 41.6569 1.40186 0.700932 0.713228i $$-0.252767\pi$$
0.700932 + 0.713228i $$0.252767\pi$$
$$884$$ −12.4853 −0.419925
$$885$$ 0 0
$$886$$ −3.02944 −0.101776
$$887$$ −55.1127 −1.85050 −0.925252 0.379354i $$-0.876146\pi$$
−0.925252 + 0.379354i $$0.876146\pi$$
$$888$$ 26.1421 0.877273
$$889$$ 0 0
$$890$$ 0 0
$$891$$ −33.1127 −1.10932
$$892$$ −7.31371 −0.244881
$$893$$ −3.02944 −0.101376
$$894$$ −20.4853 −0.685130
$$895$$ 0 0
$$896$$ 0 0
$$897$$ −19.3137 −0.644866
$$898$$ −16.6274 −0.554864
$$899$$ −13.6569 −0.455482
$$900$$ 0 0
$$901$$ −16.7696 −0.558675
$$902$$ −2.54416 −0.0847111
$$903$$ 0 0
$$904$$ 15.3137 0.509326
$$905$$ 0 0
$$906$$ −38.6274 −1.28331
$$907$$ −0.284271 −0.00943907 −0.00471954 0.999989i $$-0.501502\pi$$
−0.00471954 + 0.999989i $$0.501502\pi$$
$$908$$ 18.2426 0.605403
$$909$$ −115.255 −3.82276
$$910$$ 0 0
$$911$$ 36.2843 1.20215 0.601076 0.799192i $$-0.294739\pi$$
0.601076 + 0.799192i $$0.294739\pi$$
$$912$$ −2.00000 −0.0662266
$$913$$ 1.85786 0.0614863
$$914$$ −21.6569 −0.716345
$$915$$ 0 0
$$916$$ 16.1421 0.533351
$$917$$ 0 0
$$918$$ 49.9411 1.64830
$$919$$ 15.5147 0.511783 0.255892 0.966705i $$-0.417631\pi$$
0.255892 + 0.966705i $$0.417631\pi$$
$$920$$ 0 0
$$921$$ 50.9706 1.67954
$$922$$ 12.8284 0.422482
$$923$$ 21.6569 0.712844
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 16.9706 0.557687
$$927$$ −128.368 −4.21614
$$928$$ −4.82843 −0.158501
$$929$$ 17.2132 0.564747 0.282373 0.959305i $$-0.408878\pi$$
0.282373 + 0.959305i $$0.408878\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ −23.3137 −0.763666
$$933$$ −13.6569 −0.447105
$$934$$ −15.8995 −0.520247
$$935$$ 0 0
$$936$$ −41.7990 −1.36624
$$937$$ 20.2426 0.661298 0.330649 0.943754i $$-0.392732\pi$$
0.330649 + 0.943754i $$0.392732\pi$$
$$938$$ 0 0
$$939$$ 49.1127 1.60273
$$940$$ 0 0
$$941$$ 50.0000 1.62995 0.814977 0.579494i $$-0.196750\pi$$
0.814977 + 0.579494i $$0.196750\pi$$
$$942$$ 22.1421 0.721430
$$943$$ 3.59798 0.117166
$$944$$ −8.58579 −0.279444
$$945$$ 0 0
$$946$$ −7.31371 −0.237789
$$947$$ 4.82843 0.156903 0.0784514 0.996918i $$-0.475002\pi$$
0.0784514 + 0.996918i $$0.475002\pi$$
$$948$$ −23.3137 −0.757194
$$949$$ 45.4558 1.47556
$$950$$ 0 0
$$951$$ −35.7990 −1.16086
$$952$$ 0 0
$$953$$ −0.343146 −0.0111156 −0.00555779 0.999985i $$-0.501769\pi$$
−0.00555779 + 0.999985i $$0.501769\pi$$
$$954$$ −56.1421 −1.81767
$$955$$ 0 0
$$956$$ 1.65685 0.0535865
$$957$$ 13.6569 0.441463
$$958$$ 17.1716 0.554788
$$959$$ 0 0
$$960$$ 0 0
$$961$$ −23.0000 −0.741935
$$962$$ −36.9706 −1.19198
$$963$$ −83.5980 −2.69391
$$964$$ 13.4142 0.432043
$$965$$ 0 0
$$966$$ 0 0
$$967$$ 37.4558 1.20450 0.602249 0.798308i $$-0.294271\pi$$
0.602249 + 0.798308i $$0.294271\pi$$
$$968$$ −10.3137 −0.331495
$$969$$ −5.17157 −0.166135
$$970$$ 0 0
$$971$$ 33.3553 1.07042 0.535212 0.844718i $$-0.320232\pi$$
0.535212 + 0.844718i $$0.320232\pi$$
$$972$$ 78.5269 2.51875
$$973$$ 0 0
$$974$$ −31.7990 −1.01891
$$975$$ 0 0
$$976$$ −9.31371 −0.298125
$$977$$ 12.6863 0.405870 0.202935 0.979192i $$-0.434952\pi$$
0.202935 + 0.979192i $$0.434952\pi$$
$$978$$ 68.7696 2.19901
$$979$$ 10.5442 0.336993
$$980$$ 0 0
$$981$$ 21.5147 0.686912
$$982$$ 32.2843 1.03023
$$983$$ −12.2010 −0.389152 −0.194576 0.980887i $$-0.562333\pi$$
−0.194576 + 0.980887i $$0.562333\pi$$
$$984$$ 10.4853 0.334259
$$985$$ 0 0
$$986$$ −12.4853 −0.397612
$$987$$ 0 0
$$988$$ 2.82843 0.0899843
$$989$$ 10.3431 0.328893
$$990$$ 0 0
$$991$$ 44.7696 1.42215 0.711076 0.703115i $$-0.248208\pi$$
0.711076 + 0.703115i $$0.248208\pi$$
$$992$$ 2.82843 0.0898027
$$993$$ 115.397 3.66201
$$994$$ 0 0
$$995$$ 0 0
$$996$$ −7.65685 −0.242617
$$997$$ −18.2843 −0.579069 −0.289534 0.957168i $$-0.593500\pi$$
−0.289534 + 0.957168i $$0.593500\pi$$
$$998$$ 30.3431 0.960495
$$999$$ 147.882 4.67879
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 2450.2.a.bs.1.2 2
5.2 odd 4 2450.2.c.w.99.3 4
5.3 odd 4 2450.2.c.w.99.2 4
5.4 even 2 490.2.a.l.1.1 2
7.6 odd 2 2450.2.a.bn.1.1 2
15.14 odd 2 4410.2.a.by.1.2 2
20.19 odd 2 3920.2.a.ca.1.2 2
35.4 even 6 490.2.e.j.471.2 4
35.9 even 6 490.2.e.j.361.2 4
35.13 even 4 2450.2.c.t.99.1 4
35.19 odd 6 490.2.e.i.361.1 4
35.24 odd 6 490.2.e.i.471.1 4
35.27 even 4 2450.2.c.t.99.4 4
35.34 odd 2 490.2.a.m.1.2 yes 2
105.104 even 2 4410.2.a.bt.1.2 2
140.139 even 2 3920.2.a.bm.1.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
490.2.a.l.1.1 2 5.4 even 2
490.2.a.m.1.2 yes 2 35.34 odd 2
490.2.e.i.361.1 4 35.19 odd 6
490.2.e.i.471.1 4 35.24 odd 6
490.2.e.j.361.2 4 35.9 even 6
490.2.e.j.471.2 4 35.4 even 6
2450.2.a.bn.1.1 2 7.6 odd 2
2450.2.a.bs.1.2 2 1.1 even 1 trivial
2450.2.c.t.99.1 4 35.13 even 4
2450.2.c.t.99.4 4 35.27 even 4
2450.2.c.w.99.2 4 5.3 odd 4
2450.2.c.w.99.3 4 5.2 odd 4
3920.2.a.bm.1.1 2 140.139 even 2
3920.2.a.ca.1.2 2 20.19 odd 2
4410.2.a.bt.1.2 2 105.104 even 2
4410.2.a.by.1.2 2 15.14 odd 2