Properties

 Label 7605.2.a.bd.1.1 Level $7605$ Weight $2$ Character 7605.1 Self dual yes Analytic conductor $60.726$ Analytic rank $1$ Dimension $2$ CM no Inner twists $1$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$7605 = 3^{2} \cdot 5 \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 7605.a (trivial)

Newform invariants

 Self dual: yes Analytic conductor: $$60.7262307372$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{17})$$ Defining polynomial: $$x^{2} - x - 4$$ x^2 - x - 4 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 585) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

 Embedding label 1.1 Root $$2.56155$$ of defining polynomial Character $$\chi$$ $$=$$ 7605.1

$q$-expansion

 $$f(q)$$ $$=$$ $$q-2.56155 q^{2} +4.56155 q^{4} -1.00000 q^{5} +0.438447 q^{7} -6.56155 q^{8} +O(q^{10})$$ $$q-2.56155 q^{2} +4.56155 q^{4} -1.00000 q^{5} +0.438447 q^{7} -6.56155 q^{8} +2.56155 q^{10} +1.56155 q^{11} -1.12311 q^{14} +7.68466 q^{16} +1.56155 q^{17} +5.12311 q^{19} -4.56155 q^{20} -4.00000 q^{22} +2.43845 q^{23} +1.00000 q^{25} +2.00000 q^{28} -7.12311 q^{29} -6.00000 q^{31} -6.56155 q^{32} -4.00000 q^{34} -0.438447 q^{35} +10.6847 q^{37} -13.1231 q^{38} +6.56155 q^{40} -3.56155 q^{41} +3.12311 q^{43} +7.12311 q^{44} -6.24621 q^{46} -11.1231 q^{47} -6.80776 q^{49} -2.56155 q^{50} -4.68466 q^{53} -1.56155 q^{55} -2.87689 q^{56} +18.2462 q^{58} -12.0000 q^{59} -6.68466 q^{61} +15.3693 q^{62} +1.43845 q^{64} +11.3693 q^{67} +7.12311 q^{68} +1.12311 q^{70} -10.4384 q^{71} +6.00000 q^{73} -27.3693 q^{74} +23.3693 q^{76} +0.684658 q^{77} +4.68466 q^{79} -7.68466 q^{80} +9.12311 q^{82} +16.4924 q^{83} -1.56155 q^{85} -8.00000 q^{86} -10.2462 q^{88} -10.6847 q^{89} +11.1231 q^{92} +28.4924 q^{94} -5.12311 q^{95} -16.9309 q^{97} +17.4384 q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - q^{2} + 5 q^{4} - 2 q^{5} + 5 q^{7} - 9 q^{8}+O(q^{10})$$ 2 * q - q^2 + 5 * q^4 - 2 * q^5 + 5 * q^7 - 9 * q^8 $$2 q - q^{2} + 5 q^{4} - 2 q^{5} + 5 q^{7} - 9 q^{8} + q^{10} - q^{11} + 6 q^{14} + 3 q^{16} - q^{17} + 2 q^{19} - 5 q^{20} - 8 q^{22} + 9 q^{23} + 2 q^{25} + 4 q^{28} - 6 q^{29} - 12 q^{31} - 9 q^{32} - 8 q^{34} - 5 q^{35} + 9 q^{37} - 18 q^{38} + 9 q^{40} - 3 q^{41} - 2 q^{43} + 6 q^{44} + 4 q^{46} - 14 q^{47} + 7 q^{49} - q^{50} + 3 q^{53} + q^{55} - 14 q^{56} + 20 q^{58} - 24 q^{59} - q^{61} + 6 q^{62} + 7 q^{64} - 2 q^{67} + 6 q^{68} - 6 q^{70} - 25 q^{71} + 12 q^{73} - 30 q^{74} + 22 q^{76} - 11 q^{77} - 3 q^{79} - 3 q^{80} + 10 q^{82} + q^{85} - 16 q^{86} - 4 q^{88} - 9 q^{89} + 14 q^{92} + 24 q^{94} - 2 q^{95} - 5 q^{97} + 39 q^{98}+O(q^{100})$$ 2 * q - q^2 + 5 * q^4 - 2 * q^5 + 5 * q^7 - 9 * q^8 + q^10 - q^11 + 6 * q^14 + 3 * q^16 - q^17 + 2 * q^19 - 5 * q^20 - 8 * q^22 + 9 * q^23 + 2 * q^25 + 4 * q^28 - 6 * q^29 - 12 * q^31 - 9 * q^32 - 8 * q^34 - 5 * q^35 + 9 * q^37 - 18 * q^38 + 9 * q^40 - 3 * q^41 - 2 * q^43 + 6 * q^44 + 4 * q^46 - 14 * q^47 + 7 * q^49 - q^50 + 3 * q^53 + q^55 - 14 * q^56 + 20 * q^58 - 24 * q^59 - q^61 + 6 * q^62 + 7 * q^64 - 2 * q^67 + 6 * q^68 - 6 * q^70 - 25 * q^71 + 12 * q^73 - 30 * q^74 + 22 * q^76 - 11 * q^77 - 3 * q^79 - 3 * q^80 + 10 * q^82 + q^85 - 16 * q^86 - 4 * q^88 - 9 * q^89 + 14 * q^92 + 24 * q^94 - 2 * q^95 - 5 * q^97 + 39 * q^98

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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −2.56155 −1.81129 −0.905646 0.424035i $$-0.860613\pi$$
−0.905646 + 0.424035i $$0.860613\pi$$
$$3$$ 0 0
$$4$$ 4.56155 2.28078
$$5$$ −1.00000 −0.447214
$$6$$ 0 0
$$7$$ 0.438447 0.165717 0.0828587 0.996561i $$-0.473595\pi$$
0.0828587 + 0.996561i $$0.473595\pi$$
$$8$$ −6.56155 −2.31986
$$9$$ 0 0
$$10$$ 2.56155 0.810034
$$11$$ 1.56155 0.470826 0.235413 0.971895i $$-0.424356\pi$$
0.235413 + 0.971895i $$0.424356\pi$$
$$12$$ 0 0
$$13$$ 0 0
$$14$$ −1.12311 −0.300163
$$15$$ 0 0
$$16$$ 7.68466 1.92116
$$17$$ 1.56155 0.378732 0.189366 0.981907i $$-0.439357\pi$$
0.189366 + 0.981907i $$0.439357\pi$$
$$18$$ 0 0
$$19$$ 5.12311 1.17532 0.587661 0.809108i $$-0.300049\pi$$
0.587661 + 0.809108i $$0.300049\pi$$
$$20$$ −4.56155 −1.01999
$$21$$ 0 0
$$22$$ −4.00000 −0.852803
$$23$$ 2.43845 0.508451 0.254226 0.967145i $$-0.418179\pi$$
0.254226 + 0.967145i $$0.418179\pi$$
$$24$$ 0 0
$$25$$ 1.00000 0.200000
$$26$$ 0 0
$$27$$ 0 0
$$28$$ 2.00000 0.377964
$$29$$ −7.12311 −1.32273 −0.661364 0.750065i $$-0.730022\pi$$
−0.661364 + 0.750065i $$0.730022\pi$$
$$30$$ 0 0
$$31$$ −6.00000 −1.07763 −0.538816 0.842424i $$-0.681128\pi$$
−0.538816 + 0.842424i $$0.681128\pi$$
$$32$$ −6.56155 −1.15993
$$33$$ 0 0
$$34$$ −4.00000 −0.685994
$$35$$ −0.438447 −0.0741111
$$36$$ 0 0
$$37$$ 10.6847 1.75655 0.878274 0.478159i $$-0.158696\pi$$
0.878274 + 0.478159i $$0.158696\pi$$
$$38$$ −13.1231 −2.12885
$$39$$ 0 0
$$40$$ 6.56155 1.03747
$$41$$ −3.56155 −0.556221 −0.278111 0.960549i $$-0.589708\pi$$
−0.278111 + 0.960549i $$0.589708\pi$$
$$42$$ 0 0
$$43$$ 3.12311 0.476269 0.238135 0.971232i $$-0.423464\pi$$
0.238135 + 0.971232i $$0.423464\pi$$
$$44$$ 7.12311 1.07385
$$45$$ 0 0
$$46$$ −6.24621 −0.920954
$$47$$ −11.1231 −1.62247 −0.811236 0.584719i $$-0.801205\pi$$
−0.811236 + 0.584719i $$0.801205\pi$$
$$48$$ 0 0
$$49$$ −6.80776 −0.972538
$$50$$ −2.56155 −0.362258
$$51$$ 0 0
$$52$$ 0 0
$$53$$ −4.68466 −0.643487 −0.321744 0.946827i $$-0.604269\pi$$
−0.321744 + 0.946827i $$0.604269\pi$$
$$54$$ 0 0
$$55$$ −1.56155 −0.210560
$$56$$ −2.87689 −0.384441
$$57$$ 0 0
$$58$$ 18.2462 2.39584
$$59$$ −12.0000 −1.56227 −0.781133 0.624364i $$-0.785358\pi$$
−0.781133 + 0.624364i $$0.785358\pi$$
$$60$$ 0 0
$$61$$ −6.68466 −0.855883 −0.427941 0.903806i $$-0.640761\pi$$
−0.427941 + 0.903806i $$0.640761\pi$$
$$62$$ 15.3693 1.95191
$$63$$ 0 0
$$64$$ 1.43845 0.179806
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 11.3693 1.38898 0.694492 0.719501i $$-0.255629\pi$$
0.694492 + 0.719501i $$0.255629\pi$$
$$68$$ 7.12311 0.863803
$$69$$ 0 0
$$70$$ 1.12311 0.134237
$$71$$ −10.4384 −1.23882 −0.619408 0.785069i $$-0.712627\pi$$
−0.619408 + 0.785069i $$0.712627\pi$$
$$72$$ 0 0
$$73$$ 6.00000 0.702247 0.351123 0.936329i $$-0.385800\pi$$
0.351123 + 0.936329i $$0.385800\pi$$
$$74$$ −27.3693 −3.18162
$$75$$ 0 0
$$76$$ 23.3693 2.68064
$$77$$ 0.684658 0.0780241
$$78$$ 0 0
$$79$$ 4.68466 0.527065 0.263533 0.964650i $$-0.415112\pi$$
0.263533 + 0.964650i $$0.415112\pi$$
$$80$$ −7.68466 −0.859171
$$81$$ 0 0
$$82$$ 9.12311 1.00748
$$83$$ 16.4924 1.81028 0.905139 0.425115i $$-0.139766\pi$$
0.905139 + 0.425115i $$0.139766\pi$$
$$84$$ 0 0
$$85$$ −1.56155 −0.169374
$$86$$ −8.00000 −0.862662
$$87$$ 0 0
$$88$$ −10.2462 −1.09225
$$89$$ −10.6847 −1.13257 −0.566286 0.824209i $$-0.691620\pi$$
−0.566286 + 0.824209i $$0.691620\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 11.1231 1.15966
$$93$$ 0 0
$$94$$ 28.4924 2.93877
$$95$$ −5.12311 −0.525620
$$96$$ 0 0
$$97$$ −16.9309 −1.71907 −0.859535 0.511077i $$-0.829247\pi$$
−0.859535 + 0.511077i $$0.829247\pi$$
$$98$$ 17.4384 1.76155
$$99$$ 0 0
$$100$$ 4.56155 0.456155
$$101$$ 10.2462 1.01954 0.509768 0.860312i $$-0.329731\pi$$
0.509768 + 0.860312i $$0.329731\pi$$
$$102$$ 0 0
$$103$$ −15.1231 −1.49012 −0.745062 0.666995i $$-0.767580\pi$$
−0.745062 + 0.666995i $$0.767580\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 12.0000 1.16554
$$107$$ 10.9309 1.05673 0.528364 0.849018i $$-0.322806\pi$$
0.528364 + 0.849018i $$0.322806\pi$$
$$108$$ 0 0
$$109$$ 2.00000 0.191565 0.0957826 0.995402i $$-0.469465\pi$$
0.0957826 + 0.995402i $$0.469465\pi$$
$$110$$ 4.00000 0.381385
$$111$$ 0 0
$$112$$ 3.36932 0.318371
$$113$$ −4.87689 −0.458780 −0.229390 0.973335i $$-0.573673\pi$$
−0.229390 + 0.973335i $$0.573673\pi$$
$$114$$ 0 0
$$115$$ −2.43845 −0.227386
$$116$$ −32.4924 −3.01685
$$117$$ 0 0
$$118$$ 30.7386 2.82972
$$119$$ 0.684658 0.0627625
$$120$$ 0 0
$$121$$ −8.56155 −0.778323
$$122$$ 17.1231 1.55025
$$123$$ 0 0
$$124$$ −27.3693 −2.45784
$$125$$ −1.00000 −0.0894427
$$126$$ 0 0
$$127$$ 1.75379 0.155624 0.0778118 0.996968i $$-0.475207\pi$$
0.0778118 + 0.996968i $$0.475207\pi$$
$$128$$ 9.43845 0.834249
$$129$$ 0 0
$$130$$ 0 0
$$131$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$132$$ 0 0
$$133$$ 2.24621 0.194771
$$134$$ −29.1231 −2.51585
$$135$$ 0 0
$$136$$ −10.2462 −0.878605
$$137$$ −1.12311 −0.0959534 −0.0479767 0.998848i $$-0.515277\pi$$
−0.0479767 + 0.998848i $$0.515277\pi$$
$$138$$ 0 0
$$139$$ 3.31534 0.281204 0.140602 0.990066i $$-0.455096\pi$$
0.140602 + 0.990066i $$0.455096\pi$$
$$140$$ −2.00000 −0.169031
$$141$$ 0 0
$$142$$ 26.7386 2.24386
$$143$$ 0 0
$$144$$ 0 0
$$145$$ 7.12311 0.591542
$$146$$ −15.3693 −1.27197
$$147$$ 0 0
$$148$$ 48.7386 4.00629
$$149$$ 17.8078 1.45887 0.729434 0.684051i $$-0.239783\pi$$
0.729434 + 0.684051i $$0.239783\pi$$
$$150$$ 0 0
$$151$$ −11.3693 −0.925222 −0.462611 0.886561i $$-0.653087\pi$$
−0.462611 + 0.886561i $$0.653087\pi$$
$$152$$ −33.6155 −2.72658
$$153$$ 0 0
$$154$$ −1.75379 −0.141324
$$155$$ 6.00000 0.481932
$$156$$ 0 0
$$157$$ 3.36932 0.268901 0.134450 0.990920i $$-0.457073\pi$$
0.134450 + 0.990920i $$0.457073\pi$$
$$158$$ −12.0000 −0.954669
$$159$$ 0 0
$$160$$ 6.56155 0.518736
$$161$$ 1.06913 0.0842593
$$162$$ 0 0
$$163$$ −16.0540 −1.25744 −0.628722 0.777630i $$-0.716422\pi$$
−0.628722 + 0.777630i $$0.716422\pi$$
$$164$$ −16.2462 −1.26862
$$165$$ 0 0
$$166$$ −42.2462 −3.27894
$$167$$ −4.87689 −0.377385 −0.188693 0.982036i $$-0.560425\pi$$
−0.188693 + 0.982036i $$0.560425\pi$$
$$168$$ 0 0
$$169$$ 0 0
$$170$$ 4.00000 0.306786
$$171$$ 0 0
$$172$$ 14.2462 1.08626
$$173$$ 12.8769 0.979012 0.489506 0.872000i $$-0.337177\pi$$
0.489506 + 0.872000i $$0.337177\pi$$
$$174$$ 0 0
$$175$$ 0.438447 0.0331435
$$176$$ 12.0000 0.904534
$$177$$ 0 0
$$178$$ 27.3693 2.05142
$$179$$ −4.87689 −0.364516 −0.182258 0.983251i $$-0.558341\pi$$
−0.182258 + 0.983251i $$0.558341\pi$$
$$180$$ 0 0
$$181$$ 13.3153 0.989722 0.494861 0.868972i $$-0.335219\pi$$
0.494861 + 0.868972i $$0.335219\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ −16.0000 −1.17954
$$185$$ −10.6847 −0.785552
$$186$$ 0 0
$$187$$ 2.43845 0.178317
$$188$$ −50.7386 −3.70050
$$189$$ 0 0
$$190$$ 13.1231 0.952050
$$191$$ 19.6155 1.41933 0.709665 0.704539i $$-0.248846\pi$$
0.709665 + 0.704539i $$0.248846\pi$$
$$192$$ 0 0
$$193$$ −19.5616 −1.40807 −0.704036 0.710165i $$-0.748621\pi$$
−0.704036 + 0.710165i $$0.748621\pi$$
$$194$$ 43.3693 3.11374
$$195$$ 0 0
$$196$$ −31.0540 −2.21814
$$197$$ 3.36932 0.240054 0.120027 0.992771i $$-0.461702\pi$$
0.120027 + 0.992771i $$0.461702\pi$$
$$198$$ 0 0
$$199$$ −8.00000 −0.567105 −0.283552 0.958957i $$-0.591513\pi$$
−0.283552 + 0.958957i $$0.591513\pi$$
$$200$$ −6.56155 −0.463972
$$201$$ 0 0
$$202$$ −26.2462 −1.84668
$$203$$ −3.12311 −0.219199
$$204$$ 0 0
$$205$$ 3.56155 0.248750
$$206$$ 38.7386 2.69905
$$207$$ 0 0
$$208$$ 0 0
$$209$$ 8.00000 0.553372
$$210$$ 0 0
$$211$$ 6.24621 0.430007 0.215003 0.976613i $$-0.431024\pi$$
0.215003 + 0.976613i $$0.431024\pi$$
$$212$$ −21.3693 −1.46765
$$213$$ 0 0
$$214$$ −28.0000 −1.91404
$$215$$ −3.12311 −0.212994
$$216$$ 0 0
$$217$$ −2.63068 −0.178582
$$218$$ −5.12311 −0.346980
$$219$$ 0 0
$$220$$ −7.12311 −0.480240
$$221$$ 0 0
$$222$$ 0 0
$$223$$ 15.3693 1.02921 0.514603 0.857429i $$-0.327939\pi$$
0.514603 + 0.857429i $$0.327939\pi$$
$$224$$ −2.87689 −0.192221
$$225$$ 0 0
$$226$$ 12.4924 0.830984
$$227$$ −5.75379 −0.381892 −0.190946 0.981601i $$-0.561156\pi$$
−0.190946 + 0.981601i $$0.561156\pi$$
$$228$$ 0 0
$$229$$ −17.1231 −1.13153 −0.565763 0.824568i $$-0.691418\pi$$
−0.565763 + 0.824568i $$0.691418\pi$$
$$230$$ 6.24621 0.411863
$$231$$ 0 0
$$232$$ 46.7386 3.06854
$$233$$ −27.8078 −1.82175 −0.910874 0.412685i $$-0.864591\pi$$
−0.910874 + 0.412685i $$0.864591\pi$$
$$234$$ 0 0
$$235$$ 11.1231 0.725591
$$236$$ −54.7386 −3.56318
$$237$$ 0 0
$$238$$ −1.75379 −0.113681
$$239$$ −22.9309 −1.48327 −0.741637 0.670801i $$-0.765950\pi$$
−0.741637 + 0.670801i $$0.765950\pi$$
$$240$$ 0 0
$$241$$ −24.7386 −1.59356 −0.796778 0.604272i $$-0.793464\pi$$
−0.796778 + 0.604272i $$0.793464\pi$$
$$242$$ 21.9309 1.40977
$$243$$ 0 0
$$244$$ −30.4924 −1.95208
$$245$$ 6.80776 0.434932
$$246$$ 0 0
$$247$$ 0 0
$$248$$ 39.3693 2.49995
$$249$$ 0 0
$$250$$ 2.56155 0.162007
$$251$$ 26.2462 1.65665 0.828323 0.560251i $$-0.189295\pi$$
0.828323 + 0.560251i $$0.189295\pi$$
$$252$$ 0 0
$$253$$ 3.80776 0.239392
$$254$$ −4.49242 −0.281880
$$255$$ 0 0
$$256$$ −27.0540 −1.69087
$$257$$ −12.8769 −0.803239 −0.401619 0.915807i $$-0.631552\pi$$
−0.401619 + 0.915807i $$0.631552\pi$$
$$258$$ 0 0
$$259$$ 4.68466 0.291091
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 0 0
$$263$$ −2.24621 −0.138507 −0.0692537 0.997599i $$-0.522062\pi$$
−0.0692537 + 0.997599i $$0.522062\pi$$
$$264$$ 0 0
$$265$$ 4.68466 0.287776
$$266$$ −5.75379 −0.352787
$$267$$ 0 0
$$268$$ 51.8617 3.16796
$$269$$ −0.876894 −0.0534652 −0.0267326 0.999643i $$-0.508510\pi$$
−0.0267326 + 0.999643i $$0.508510\pi$$
$$270$$ 0 0
$$271$$ 19.3693 1.17660 0.588301 0.808642i $$-0.299797\pi$$
0.588301 + 0.808642i $$0.299797\pi$$
$$272$$ 12.0000 0.727607
$$273$$ 0 0
$$274$$ 2.87689 0.173800
$$275$$ 1.56155 0.0941652
$$276$$ 0 0
$$277$$ −12.2462 −0.735804 −0.367902 0.929865i $$-0.619924\pi$$
−0.367902 + 0.929865i $$0.619924\pi$$
$$278$$ −8.49242 −0.509342
$$279$$ 0 0
$$280$$ 2.87689 0.171927
$$281$$ −4.24621 −0.253308 −0.126654 0.991947i $$-0.540424\pi$$
−0.126654 + 0.991947i $$0.540424\pi$$
$$282$$ 0 0
$$283$$ −4.00000 −0.237775 −0.118888 0.992908i $$-0.537933\pi$$
−0.118888 + 0.992908i $$0.537933\pi$$
$$284$$ −47.6155 −2.82546
$$285$$ 0 0
$$286$$ 0 0
$$287$$ −1.56155 −0.0921755
$$288$$ 0 0
$$289$$ −14.5616 −0.856562
$$290$$ −18.2462 −1.07145
$$291$$ 0 0
$$292$$ 27.3693 1.60167
$$293$$ 20.2462 1.18280 0.591398 0.806380i $$-0.298576\pi$$
0.591398 + 0.806380i $$0.298576\pi$$
$$294$$ 0 0
$$295$$ 12.0000 0.698667
$$296$$ −70.1080 −4.07494
$$297$$ 0 0
$$298$$ −45.6155 −2.64244
$$299$$ 0 0
$$300$$ 0 0
$$301$$ 1.36932 0.0789261
$$302$$ 29.1231 1.67585
$$303$$ 0 0
$$304$$ 39.3693 2.25799
$$305$$ 6.68466 0.382762
$$306$$ 0 0
$$307$$ −7.56155 −0.431561 −0.215780 0.976442i $$-0.569230\pi$$
−0.215780 + 0.976442i $$0.569230\pi$$
$$308$$ 3.12311 0.177955
$$309$$ 0 0
$$310$$ −15.3693 −0.872919
$$311$$ 2.63068 0.149172 0.0745862 0.997215i $$-0.476236\pi$$
0.0745862 + 0.997215i $$0.476236\pi$$
$$312$$ 0 0
$$313$$ 29.1231 1.64614 0.823068 0.567943i $$-0.192261\pi$$
0.823068 + 0.567943i $$0.192261\pi$$
$$314$$ −8.63068 −0.487058
$$315$$ 0 0
$$316$$ 21.3693 1.20212
$$317$$ −1.50758 −0.0846740 −0.0423370 0.999103i $$-0.513480\pi$$
−0.0423370 + 0.999103i $$0.513480\pi$$
$$318$$ 0 0
$$319$$ −11.1231 −0.622774
$$320$$ −1.43845 −0.0804116
$$321$$ 0 0
$$322$$ −2.73863 −0.152618
$$323$$ 8.00000 0.445132
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 41.1231 2.27760
$$327$$ 0 0
$$328$$ 23.3693 1.29035
$$329$$ −4.87689 −0.268872
$$330$$ 0 0
$$331$$ −29.1231 −1.60075 −0.800375 0.599499i $$-0.795366\pi$$
−0.800375 + 0.599499i $$0.795366\pi$$
$$332$$ 75.2311 4.12884
$$333$$ 0 0
$$334$$ 12.4924 0.683555
$$335$$ −11.3693 −0.621172
$$336$$ 0 0
$$337$$ −30.4924 −1.66103 −0.830514 0.556998i $$-0.811953\pi$$
−0.830514 + 0.556998i $$0.811953\pi$$
$$338$$ 0 0
$$339$$ 0 0
$$340$$ −7.12311 −0.386305
$$341$$ −9.36932 −0.507377
$$342$$ 0 0
$$343$$ −6.05398 −0.326884
$$344$$ −20.4924 −1.10488
$$345$$ 0 0
$$346$$ −32.9848 −1.77328
$$347$$ −26.0540 −1.39865 −0.699325 0.714804i $$-0.746516\pi$$
−0.699325 + 0.714804i $$0.746516\pi$$
$$348$$ 0 0
$$349$$ 23.3693 1.25093 0.625465 0.780252i $$-0.284909\pi$$
0.625465 + 0.780252i $$0.284909\pi$$
$$350$$ −1.12311 −0.0600325
$$351$$ 0 0
$$352$$ −10.2462 −0.546125
$$353$$ 22.4924 1.19715 0.598575 0.801066i $$-0.295734\pi$$
0.598575 + 0.801066i $$0.295734\pi$$
$$354$$ 0 0
$$355$$ 10.4384 0.554015
$$356$$ −48.7386 −2.58314
$$357$$ 0 0
$$358$$ 12.4924 0.660245
$$359$$ 14.2462 0.751886 0.375943 0.926643i $$-0.377319\pi$$
0.375943 + 0.926643i $$0.377319\pi$$
$$360$$ 0 0
$$361$$ 7.24621 0.381380
$$362$$ −34.1080 −1.79267
$$363$$ 0 0
$$364$$ 0 0
$$365$$ −6.00000 −0.314054
$$366$$ 0 0
$$367$$ −1.75379 −0.0915470 −0.0457735 0.998952i $$-0.514575\pi$$
−0.0457735 + 0.998952i $$0.514575\pi$$
$$368$$ 18.7386 0.976819
$$369$$ 0 0
$$370$$ 27.3693 1.42286
$$371$$ −2.05398 −0.106637
$$372$$ 0 0
$$373$$ 12.2462 0.634085 0.317042 0.948411i $$-0.397310\pi$$
0.317042 + 0.948411i $$0.397310\pi$$
$$374$$ −6.24621 −0.322984
$$375$$ 0 0
$$376$$ 72.9848 3.76391
$$377$$ 0 0
$$378$$ 0 0
$$379$$ 30.4924 1.56629 0.783145 0.621839i $$-0.213614\pi$$
0.783145 + 0.621839i $$0.213614\pi$$
$$380$$ −23.3693 −1.19882
$$381$$ 0 0
$$382$$ −50.2462 −2.57082
$$383$$ −3.50758 −0.179229 −0.0896144 0.995977i $$-0.528563\pi$$
−0.0896144 + 0.995977i $$0.528563\pi$$
$$384$$ 0 0
$$385$$ −0.684658 −0.0348934
$$386$$ 50.1080 2.55043
$$387$$ 0 0
$$388$$ −77.2311 −3.92081
$$389$$ 37.8617 1.91967 0.959833 0.280571i $$-0.0905239\pi$$
0.959833 + 0.280571i $$0.0905239\pi$$
$$390$$ 0 0
$$391$$ 3.80776 0.192567
$$392$$ 44.6695 2.25615
$$393$$ 0 0
$$394$$ −8.63068 −0.434808
$$395$$ −4.68466 −0.235711
$$396$$ 0 0
$$397$$ 4.43845 0.222759 0.111380 0.993778i $$-0.464473\pi$$
0.111380 + 0.993778i $$0.464473\pi$$
$$398$$ 20.4924 1.02719
$$399$$ 0 0
$$400$$ 7.68466 0.384233
$$401$$ −3.75379 −0.187455 −0.0937276 0.995598i $$-0.529878\pi$$
−0.0937276 + 0.995598i $$0.529878\pi$$
$$402$$ 0 0
$$403$$ 0 0
$$404$$ 46.7386 2.32533
$$405$$ 0 0
$$406$$ 8.00000 0.397033
$$407$$ 16.6847 0.827028
$$408$$ 0 0
$$409$$ 6.87689 0.340041 0.170020 0.985441i $$-0.445617\pi$$
0.170020 + 0.985441i $$0.445617\pi$$
$$410$$ −9.12311 −0.450558
$$411$$ 0 0
$$412$$ −68.9848 −3.39864
$$413$$ −5.26137 −0.258895
$$414$$ 0 0
$$415$$ −16.4924 −0.809581
$$416$$ 0 0
$$417$$ 0 0
$$418$$ −20.4924 −1.00232
$$419$$ 7.61553 0.372043 0.186021 0.982546i $$-0.440441\pi$$
0.186021 + 0.982546i $$0.440441\pi$$
$$420$$ 0 0
$$421$$ −39.3693 −1.91874 −0.959372 0.282146i $$-0.908954\pi$$
−0.959372 + 0.282146i $$0.908954\pi$$
$$422$$ −16.0000 −0.778868
$$423$$ 0 0
$$424$$ 30.7386 1.49280
$$425$$ 1.56155 0.0757464
$$426$$ 0 0
$$427$$ −2.93087 −0.141835
$$428$$ 49.8617 2.41016
$$429$$ 0 0
$$430$$ 8.00000 0.385794
$$431$$ 3.50758 0.168954 0.0844770 0.996425i $$-0.473078\pi$$
0.0844770 + 0.996425i $$0.473078\pi$$
$$432$$ 0 0
$$433$$ −9.61553 −0.462093 −0.231046 0.972943i $$-0.574215\pi$$
−0.231046 + 0.972943i $$0.574215\pi$$
$$434$$ 6.73863 0.323465
$$435$$ 0 0
$$436$$ 9.12311 0.436918
$$437$$ 12.4924 0.597594
$$438$$ 0 0
$$439$$ 22.0540 1.05258 0.526289 0.850306i $$-0.323583\pi$$
0.526289 + 0.850306i $$0.323583\pi$$
$$440$$ 10.2462 0.488469
$$441$$ 0 0
$$442$$ 0 0
$$443$$ −7.80776 −0.370958 −0.185479 0.982648i $$-0.559384\pi$$
−0.185479 + 0.982648i $$0.559384\pi$$
$$444$$ 0 0
$$445$$ 10.6847 0.506501
$$446$$ −39.3693 −1.86419
$$447$$ 0 0
$$448$$ 0.630683 0.0297970
$$449$$ 35.1771 1.66011 0.830055 0.557682i $$-0.188309\pi$$
0.830055 + 0.557682i $$0.188309\pi$$
$$450$$ 0 0
$$451$$ −5.56155 −0.261883
$$452$$ −22.2462 −1.04637
$$453$$ 0 0
$$454$$ 14.7386 0.691718
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 13.3153 0.622865 0.311433 0.950268i $$-0.399191\pi$$
0.311433 + 0.950268i $$0.399191\pi$$
$$458$$ 43.8617 2.04952
$$459$$ 0 0
$$460$$ −11.1231 −0.518617
$$461$$ −8.05398 −0.375111 −0.187556 0.982254i $$-0.560056\pi$$
−0.187556 + 0.982254i $$0.560056\pi$$
$$462$$ 0 0
$$463$$ 28.9309 1.34453 0.672266 0.740310i $$-0.265321\pi$$
0.672266 + 0.740310i $$0.265321\pi$$
$$464$$ −54.7386 −2.54118
$$465$$ 0 0
$$466$$ 71.2311 3.29971
$$467$$ 6.93087 0.320722 0.160361 0.987058i $$-0.448734\pi$$
0.160361 + 0.987058i $$0.448734\pi$$
$$468$$ 0 0
$$469$$ 4.98485 0.230179
$$470$$ −28.4924 −1.31426
$$471$$ 0 0
$$472$$ 78.7386 3.62424
$$473$$ 4.87689 0.224240
$$474$$ 0 0
$$475$$ 5.12311 0.235064
$$476$$ 3.12311 0.143147
$$477$$ 0 0
$$478$$ 58.7386 2.68664
$$479$$ −26.0540 −1.19044 −0.595218 0.803564i $$-0.702934\pi$$
−0.595218 + 0.803564i $$0.702934\pi$$
$$480$$ 0 0
$$481$$ 0 0
$$482$$ 63.3693 2.88639
$$483$$ 0 0
$$484$$ −39.0540 −1.77518
$$485$$ 16.9309 0.768791
$$486$$ 0 0
$$487$$ 32.0540 1.45250 0.726252 0.687428i $$-0.241260\pi$$
0.726252 + 0.687428i $$0.241260\pi$$
$$488$$ 43.8617 1.98553
$$489$$ 0 0
$$490$$ −17.4384 −0.787789
$$491$$ −27.6155 −1.24627 −0.623136 0.782114i $$-0.714142\pi$$
−0.623136 + 0.782114i $$0.714142\pi$$
$$492$$ 0 0
$$493$$ −11.1231 −0.500959
$$494$$ 0 0
$$495$$ 0 0
$$496$$ −46.1080 −2.07031
$$497$$ −4.57671 −0.205293
$$498$$ 0 0
$$499$$ −34.9848 −1.56614 −0.783068 0.621936i $$-0.786347\pi$$
−0.783068 + 0.621936i $$0.786347\pi$$
$$500$$ −4.56155 −0.203999
$$501$$ 0 0
$$502$$ −67.2311 −3.00067
$$503$$ 13.7538 0.613251 0.306626 0.951830i $$-0.400800\pi$$
0.306626 + 0.951830i $$0.400800\pi$$
$$504$$ 0 0
$$505$$ −10.2462 −0.455950
$$506$$ −9.75379 −0.433609
$$507$$ 0 0
$$508$$ 8.00000 0.354943
$$509$$ 23.5616 1.04435 0.522174 0.852839i $$-0.325121\pi$$
0.522174 + 0.852839i $$0.325121\pi$$
$$510$$ 0 0
$$511$$ 2.63068 0.116375
$$512$$ 50.4233 2.22842
$$513$$ 0 0
$$514$$ 32.9848 1.45490
$$515$$ 15.1231 0.666404
$$516$$ 0 0
$$517$$ −17.3693 −0.763902
$$518$$ −12.0000 −0.527250
$$519$$ 0 0
$$520$$ 0 0
$$521$$ −25.8617 −1.13302 −0.566512 0.824054i $$-0.691707\pi$$
−0.566512 + 0.824054i $$0.691707\pi$$
$$522$$ 0 0
$$523$$ −30.7386 −1.34411 −0.672053 0.740503i $$-0.734587\pi$$
−0.672053 + 0.740503i $$0.734587\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 5.75379 0.250877
$$527$$ −9.36932 −0.408134
$$528$$ 0 0
$$529$$ −17.0540 −0.741477
$$530$$ −12.0000 −0.521247
$$531$$ 0 0
$$532$$ 10.2462 0.444230
$$533$$ 0 0
$$534$$ 0 0
$$535$$ −10.9309 −0.472583
$$536$$ −74.6004 −3.22225
$$537$$ 0 0
$$538$$ 2.24621 0.0968410
$$539$$ −10.6307 −0.457896
$$540$$ 0 0
$$541$$ −18.8769 −0.811581 −0.405791 0.913966i $$-0.633004\pi$$
−0.405791 + 0.913966i $$0.633004\pi$$
$$542$$ −49.6155 −2.13117
$$543$$ 0 0
$$544$$ −10.2462 −0.439303
$$545$$ −2.00000 −0.0856706
$$546$$ 0 0
$$547$$ −5.36932 −0.229575 −0.114788 0.993390i $$-0.536619\pi$$
−0.114788 + 0.993390i $$0.536619\pi$$
$$548$$ −5.12311 −0.218848
$$549$$ 0 0
$$550$$ −4.00000 −0.170561
$$551$$ −36.4924 −1.55463
$$552$$ 0 0
$$553$$ 2.05398 0.0873439
$$554$$ 31.3693 1.33275
$$555$$ 0 0
$$556$$ 15.1231 0.641363
$$557$$ −6.49242 −0.275093 −0.137546 0.990495i $$-0.543922\pi$$
−0.137546 + 0.990495i $$0.543922\pi$$
$$558$$ 0 0
$$559$$ 0 0
$$560$$ −3.36932 −0.142380
$$561$$ 0 0
$$562$$ 10.8769 0.458814
$$563$$ −19.3153 −0.814045 −0.407022 0.913418i $$-0.633433\pi$$
−0.407022 + 0.913418i $$0.633433\pi$$
$$564$$ 0 0
$$565$$ 4.87689 0.205172
$$566$$ 10.2462 0.430680
$$567$$ 0 0
$$568$$ 68.4924 2.87388
$$569$$ −32.8769 −1.37827 −0.689136 0.724632i $$-0.742010\pi$$
−0.689136 + 0.724632i $$0.742010\pi$$
$$570$$ 0 0
$$571$$ −22.0540 −0.922930 −0.461465 0.887158i $$-0.652676\pi$$
−0.461465 + 0.887158i $$0.652676\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 4.00000 0.166957
$$575$$ 2.43845 0.101690
$$576$$ 0 0
$$577$$ −24.4384 −1.01739 −0.508693 0.860948i $$-0.669871\pi$$
−0.508693 + 0.860948i $$0.669871\pi$$
$$578$$ 37.3002 1.55148
$$579$$ 0 0
$$580$$ 32.4924 1.34917
$$581$$ 7.23106 0.299995
$$582$$ 0 0
$$583$$ −7.31534 −0.302970
$$584$$ −39.3693 −1.62911
$$585$$ 0 0
$$586$$ −51.8617 −2.14239
$$587$$ 32.4924 1.34111 0.670553 0.741862i $$-0.266057\pi$$
0.670553 + 0.741862i $$0.266057\pi$$
$$588$$ 0 0
$$589$$ −30.7386 −1.26656
$$590$$ −30.7386 −1.26549
$$591$$ 0 0
$$592$$ 82.1080 3.37462
$$593$$ −24.2462 −0.995673 −0.497836 0.867271i $$-0.665872\pi$$
−0.497836 + 0.867271i $$0.665872\pi$$
$$594$$ 0 0
$$595$$ −0.684658 −0.0280683
$$596$$ 81.2311 3.32735
$$597$$ 0 0
$$598$$ 0 0
$$599$$ −9.36932 −0.382820 −0.191410 0.981510i $$-0.561306\pi$$
−0.191410 + 0.981510i $$0.561306\pi$$
$$600$$ 0 0
$$601$$ −28.5464 −1.16443 −0.582216 0.813034i $$-0.697814\pi$$
−0.582216 + 0.813034i $$0.697814\pi$$
$$602$$ −3.50758 −0.142958
$$603$$ 0 0
$$604$$ −51.8617 −2.11022
$$605$$ 8.56155 0.348077
$$606$$ 0 0
$$607$$ −24.0000 −0.974130 −0.487065 0.873366i $$-0.661933\pi$$
−0.487065 + 0.873366i $$0.661933\pi$$
$$608$$ −33.6155 −1.36329
$$609$$ 0 0
$$610$$ −17.1231 −0.693294
$$611$$ 0 0
$$612$$ 0 0
$$613$$ 32.5464 1.31454 0.657268 0.753657i $$-0.271712\pi$$
0.657268 + 0.753657i $$0.271712\pi$$
$$614$$ 19.3693 0.781682
$$615$$ 0 0
$$616$$ −4.49242 −0.181005
$$617$$ −0.738634 −0.0297363 −0.0148681 0.999889i $$-0.504733\pi$$
−0.0148681 + 0.999889i $$0.504733\pi$$
$$618$$ 0 0
$$619$$ 8.63068 0.346896 0.173448 0.984843i $$-0.444509\pi$$
0.173448 + 0.984843i $$0.444509\pi$$
$$620$$ 27.3693 1.09918
$$621$$ 0 0
$$622$$ −6.73863 −0.270195
$$623$$ −4.68466 −0.187687
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ −74.6004 −2.98163
$$627$$ 0 0
$$628$$ 15.3693 0.613303
$$629$$ 16.6847 0.665261
$$630$$ 0 0
$$631$$ 32.7386 1.30330 0.651652 0.758518i $$-0.274076\pi$$
0.651652 + 0.758518i $$0.274076\pi$$
$$632$$ −30.7386 −1.22272
$$633$$ 0 0
$$634$$ 3.86174 0.153369
$$635$$ −1.75379 −0.0695970
$$636$$ 0 0
$$637$$ 0 0
$$638$$ 28.4924 1.12803
$$639$$ 0 0
$$640$$ −9.43845 −0.373087
$$641$$ −32.9848 −1.30282 −0.651412 0.758725i $$-0.725823\pi$$
−0.651412 + 0.758725i $$0.725823\pi$$
$$642$$ 0 0
$$643$$ 10.6847 0.421362 0.210681 0.977555i $$-0.432432\pi$$
0.210681 + 0.977555i $$0.432432\pi$$
$$644$$ 4.87689 0.192177
$$645$$ 0 0
$$646$$ −20.4924 −0.806264
$$647$$ 31.8078 1.25049 0.625246 0.780428i $$-0.284999\pi$$
0.625246 + 0.780428i $$0.284999\pi$$
$$648$$ 0 0
$$649$$ −18.7386 −0.735556
$$650$$ 0 0
$$651$$ 0 0
$$652$$ −73.2311 −2.86795
$$653$$ −33.3693 −1.30584 −0.652921 0.757426i $$-0.726457\pi$$
−0.652921 + 0.757426i $$0.726457\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ −27.3693 −1.06859
$$657$$ 0 0
$$658$$ 12.4924 0.487005
$$659$$ 0.876894 0.0341590 0.0170795 0.999854i $$-0.494563\pi$$
0.0170795 + 0.999854i $$0.494563\pi$$
$$660$$ 0 0
$$661$$ −6.49242 −0.252526 −0.126263 0.991997i $$-0.540298\pi$$
−0.126263 + 0.991997i $$0.540298\pi$$
$$662$$ 74.6004 2.89943
$$663$$ 0 0
$$664$$ −108.216 −4.19959
$$665$$ −2.24621 −0.0871043
$$666$$ 0 0
$$667$$ −17.3693 −0.672543
$$668$$ −22.2462 −0.860732
$$669$$ 0 0
$$670$$ 29.1231 1.12512
$$671$$ −10.4384 −0.402972
$$672$$ 0 0
$$673$$ 16.7386 0.645227 0.322613 0.946531i $$-0.395439\pi$$
0.322613 + 0.946531i $$0.395439\pi$$
$$674$$ 78.1080 3.00861
$$675$$ 0 0
$$676$$ 0 0
$$677$$ −14.4384 −0.554915 −0.277457 0.960738i $$-0.589492\pi$$
−0.277457 + 0.960738i $$0.589492\pi$$
$$678$$ 0 0
$$679$$ −7.42329 −0.284880
$$680$$ 10.2462 0.392924
$$681$$ 0 0
$$682$$ 24.0000 0.919007
$$683$$ −32.4924 −1.24329 −0.621644 0.783300i $$-0.713535\pi$$
−0.621644 + 0.783300i $$0.713535\pi$$
$$684$$ 0 0
$$685$$ 1.12311 0.0429117
$$686$$ 15.5076 0.592082
$$687$$ 0 0
$$688$$ 24.0000 0.914991
$$689$$ 0 0
$$690$$ 0 0
$$691$$ 21.6155 0.822293 0.411147 0.911569i $$-0.365128\pi$$
0.411147 + 0.911569i $$0.365128\pi$$
$$692$$ 58.7386 2.23291
$$693$$ 0 0
$$694$$ 66.7386 2.53336
$$695$$ −3.31534 −0.125758
$$696$$ 0 0
$$697$$ −5.56155 −0.210659
$$698$$ −59.8617 −2.26580
$$699$$ 0 0
$$700$$ 2.00000 0.0755929
$$701$$ −48.9848 −1.85013 −0.925066 0.379806i $$-0.875991\pi$$
−0.925066 + 0.379806i $$0.875991\pi$$
$$702$$ 0 0
$$703$$ 54.7386 2.06451
$$704$$ 2.24621 0.0846573
$$705$$ 0 0
$$706$$ −57.6155 −2.16839
$$707$$ 4.49242 0.168955
$$708$$ 0 0
$$709$$ −9.12311 −0.342625 −0.171313 0.985217i $$-0.554801\pi$$
−0.171313 + 0.985217i $$0.554801\pi$$
$$710$$ −26.7386 −1.00348
$$711$$ 0 0
$$712$$ 70.1080 2.62741
$$713$$ −14.6307 −0.547923
$$714$$ 0 0
$$715$$ 0 0
$$716$$ −22.2462 −0.831380
$$717$$ 0 0
$$718$$ −36.4924 −1.36189
$$719$$ 36.0000 1.34257 0.671287 0.741198i $$-0.265742\pi$$
0.671287 + 0.741198i $$0.265742\pi$$
$$720$$ 0 0
$$721$$ −6.63068 −0.246940
$$722$$ −18.5616 −0.690789
$$723$$ 0 0
$$724$$ 60.7386 2.25733
$$725$$ −7.12311 −0.264546
$$726$$ 0 0
$$727$$ 12.8769 0.477578 0.238789 0.971072i $$-0.423250\pi$$
0.238789 + 0.971072i $$0.423250\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ 15.3693 0.568844
$$731$$ 4.87689 0.180378
$$732$$ 0 0
$$733$$ 16.4384 0.607168 0.303584 0.952805i $$-0.401817\pi$$
0.303584 + 0.952805i $$0.401817\pi$$
$$734$$ 4.49242 0.165818
$$735$$ 0 0
$$736$$ −16.0000 −0.589768
$$737$$ 17.7538 0.653969
$$738$$ 0 0
$$739$$ −48.7386 −1.79288 −0.896440 0.443166i $$-0.853855\pi$$
−0.896440 + 0.443166i $$0.853855\pi$$
$$740$$ −48.7386 −1.79167
$$741$$ 0 0
$$742$$ 5.26137 0.193151
$$743$$ −18.7386 −0.687454 −0.343727 0.939070i $$-0.611689\pi$$
−0.343727 + 0.939070i $$0.611689\pi$$
$$744$$ 0 0
$$745$$ −17.8078 −0.652426
$$746$$ −31.3693 −1.14851
$$747$$ 0 0
$$748$$ 11.1231 0.406701
$$749$$ 4.79261 0.175118
$$750$$ 0 0
$$751$$ 14.0540 0.512837 0.256418 0.966566i $$-0.417458\pi$$
0.256418 + 0.966566i $$0.417458\pi$$
$$752$$ −85.4773 −3.11704
$$753$$ 0 0
$$754$$ 0 0
$$755$$ 11.3693 0.413772
$$756$$ 0 0
$$757$$ −14.4924 −0.526736 −0.263368 0.964695i $$-0.584833\pi$$
−0.263368 + 0.964695i $$0.584833\pi$$
$$758$$ −78.1080 −2.83701
$$759$$ 0 0
$$760$$ 33.6155 1.21936
$$761$$ −45.2311 −1.63962 −0.819812 0.572632i $$-0.805922\pi$$
−0.819812 + 0.572632i $$0.805922\pi$$
$$762$$ 0 0
$$763$$ 0.876894 0.0317457
$$764$$ 89.4773 3.23717
$$765$$ 0 0
$$766$$ 8.98485 0.324636
$$767$$ 0 0
$$768$$ 0 0
$$769$$ 30.0000 1.08183 0.540914 0.841078i $$-0.318079\pi$$
0.540914 + 0.841078i $$0.318079\pi$$
$$770$$ 1.75379 0.0632022
$$771$$ 0 0
$$772$$ −89.2311 −3.21150
$$773$$ 9.12311 0.328135 0.164068 0.986449i $$-0.447538\pi$$
0.164068 + 0.986449i $$0.447538\pi$$
$$774$$ 0 0
$$775$$ −6.00000 −0.215526
$$776$$ 111.093 3.98800
$$777$$ 0 0
$$778$$ −96.9848 −3.47708
$$779$$ −18.2462 −0.653738
$$780$$ 0 0
$$781$$ −16.3002 −0.583267
$$782$$ −9.75379 −0.348795
$$783$$ 0 0
$$784$$ −52.3153 −1.86841
$$785$$ −3.36932 −0.120256
$$786$$ 0 0
$$787$$ −11.3693 −0.405272 −0.202636 0.979254i $$-0.564951\pi$$
−0.202636 + 0.979254i $$0.564951\pi$$
$$788$$ 15.3693 0.547509
$$789$$ 0 0
$$790$$ 12.0000 0.426941
$$791$$ −2.13826 −0.0760278
$$792$$ 0 0
$$793$$ 0 0
$$794$$ −11.3693 −0.403482
$$795$$ 0 0
$$796$$ −36.4924 −1.29344
$$797$$ 4.68466 0.165939 0.0829696 0.996552i $$-0.473560\pi$$
0.0829696 + 0.996552i $$0.473560\pi$$
$$798$$ 0 0
$$799$$ −17.3693 −0.614482
$$800$$ −6.56155 −0.231986
$$801$$ 0 0
$$802$$ 9.61553 0.339536
$$803$$ 9.36932 0.330636
$$804$$ 0 0
$$805$$ −1.06913 −0.0376819
$$806$$ 0 0
$$807$$ 0 0
$$808$$ −67.2311 −2.36518
$$809$$ 7.50758 0.263952 0.131976 0.991253i $$-0.457868\pi$$
0.131976 + 0.991253i $$0.457868\pi$$
$$810$$ 0 0
$$811$$ −4.24621 −0.149105 −0.0745523 0.997217i $$-0.523753\pi$$
−0.0745523 + 0.997217i $$0.523753\pi$$
$$812$$ −14.2462 −0.499944
$$813$$ 0 0
$$814$$ −42.7386 −1.49799
$$815$$ 16.0540 0.562346
$$816$$ 0 0
$$817$$ 16.0000 0.559769
$$818$$ −17.6155 −0.615912
$$819$$ 0 0
$$820$$ 16.2462 0.567342
$$821$$ −48.5464 −1.69428 −0.847140 0.531369i $$-0.821678\pi$$
−0.847140 + 0.531369i $$0.821678\pi$$
$$822$$ 0 0
$$823$$ −29.7538 −1.03715 −0.518576 0.855032i $$-0.673538\pi$$
−0.518576 + 0.855032i $$0.673538\pi$$
$$824$$ 99.2311 3.45688
$$825$$ 0 0
$$826$$ 13.4773 0.468934
$$827$$ −7.12311 −0.247695 −0.123847 0.992301i $$-0.539523\pi$$
−0.123847 + 0.992301i $$0.539523\pi$$
$$828$$ 0 0
$$829$$ 0.738634 0.0256538 0.0128269 0.999918i $$-0.495917\pi$$
0.0128269 + 0.999918i $$0.495917\pi$$
$$830$$ 42.2462 1.46639
$$831$$ 0 0
$$832$$ 0 0
$$833$$ −10.6307 −0.368331
$$834$$ 0 0
$$835$$ 4.87689 0.168772
$$836$$ 36.4924 1.26212
$$837$$ 0 0
$$838$$ −19.5076 −0.673878
$$839$$ 44.7926 1.54641 0.773206 0.634155i $$-0.218652\pi$$
0.773206 + 0.634155i $$0.218652\pi$$
$$840$$ 0 0
$$841$$ 21.7386 0.749608
$$842$$ 100.847 3.47540
$$843$$ 0 0
$$844$$ 28.4924 0.980750
$$845$$ 0 0
$$846$$ 0 0
$$847$$ −3.75379 −0.128982
$$848$$ −36.0000 −1.23625
$$849$$ 0 0
$$850$$ −4.00000 −0.137199
$$851$$ 26.0540 0.893119
$$852$$ 0 0
$$853$$ −41.4233 −1.41831 −0.709153 0.705054i $$-0.750923\pi$$
−0.709153 + 0.705054i $$0.750923\pi$$
$$854$$ 7.50758 0.256904
$$855$$ 0 0
$$856$$ −71.7235 −2.45146
$$857$$ 2.43845 0.0832958 0.0416479 0.999132i $$-0.486739\pi$$
0.0416479 + 0.999132i $$0.486739\pi$$
$$858$$ 0 0
$$859$$ 3.80776 0.129919 0.0649596 0.997888i $$-0.479308\pi$$
0.0649596 + 0.997888i $$0.479308\pi$$
$$860$$ −14.2462 −0.485792
$$861$$ 0 0
$$862$$ −8.98485 −0.306025
$$863$$ 9.36932 0.318935 0.159468 0.987203i $$-0.449022\pi$$
0.159468 + 0.987203i $$0.449022\pi$$
$$864$$ 0 0
$$865$$ −12.8769 −0.437828
$$866$$ 24.6307 0.836985
$$867$$ 0 0
$$868$$ −12.0000 −0.407307
$$869$$ 7.31534 0.248156
$$870$$ 0 0
$$871$$ 0 0
$$872$$ −13.1231 −0.444404
$$873$$ 0 0
$$874$$ −32.0000 −1.08242
$$875$$ −0.438447 −0.0148222
$$876$$ 0 0
$$877$$ −46.9848 −1.58657 −0.793283 0.608853i $$-0.791630\pi$$
−0.793283 + 0.608853i $$0.791630\pi$$
$$878$$ −56.4924 −1.90653
$$879$$ 0 0
$$880$$ −12.0000 −0.404520
$$881$$ −21.3693 −0.719951 −0.359975 0.932962i $$-0.617215\pi$$
−0.359975 + 0.932962i $$0.617215\pi$$
$$882$$ 0 0
$$883$$ −56.1080 −1.88818 −0.944091 0.329684i $$-0.893058\pi$$
−0.944091 + 0.329684i $$0.893058\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 20.0000 0.671913
$$887$$ 5.56155 0.186739 0.0933693 0.995632i $$-0.470236\pi$$
0.0933693 + 0.995632i $$0.470236\pi$$
$$888$$ 0 0
$$889$$ 0.768944 0.0257895
$$890$$ −27.3693 −0.917422
$$891$$ 0 0
$$892$$ 70.1080 2.34739
$$893$$ −56.9848 −1.90693
$$894$$ 0 0
$$895$$ 4.87689 0.163017
$$896$$ 4.13826 0.138250
$$897$$ 0 0
$$898$$ −90.1080 −3.00694
$$899$$ 42.7386 1.42541
$$900$$ 0 0
$$901$$ −7.31534 −0.243709
$$902$$ 14.2462 0.474347
$$903$$ 0 0
$$904$$ 32.0000 1.06430
$$905$$ −13.3153 −0.442617
$$906$$ 0 0
$$907$$ −2.63068 −0.0873504 −0.0436752 0.999046i $$-0.513907\pi$$
−0.0436752 + 0.999046i $$0.513907\pi$$
$$908$$ −26.2462 −0.871011
$$909$$ 0 0
$$910$$ 0 0
$$911$$ −12.0000 −0.397578 −0.198789 0.980042i $$-0.563701\pi$$
−0.198789 + 0.980042i $$0.563701\pi$$
$$912$$ 0 0
$$913$$ 25.7538 0.852326
$$914$$ −34.1080 −1.12819
$$915$$ 0 0
$$916$$ −78.1080 −2.58076
$$917$$ 0 0
$$918$$ 0 0
$$919$$ 1.94602 0.0641934 0.0320967 0.999485i $$-0.489782\pi$$
0.0320967 + 0.999485i $$0.489782\pi$$
$$920$$ 16.0000 0.527504
$$921$$ 0 0
$$922$$ 20.6307 0.679435
$$923$$ 0 0
$$924$$ 0 0
$$925$$ 10.6847 0.351309
$$926$$ −74.1080 −2.43534
$$927$$ 0 0
$$928$$ 46.7386 1.53427
$$929$$ −39.6695 −1.30151 −0.650757 0.759286i $$-0.725548\pi$$
−0.650757 + 0.759286i $$0.725548\pi$$
$$930$$ 0 0
$$931$$ −34.8769 −1.14304
$$932$$ −126.847 −4.15500
$$933$$ 0 0
$$934$$ −17.7538 −0.580922
$$935$$ −2.43845 −0.0797458
$$936$$ 0 0
$$937$$ −27.3693 −0.894117 −0.447058 0.894505i $$-0.647528\pi$$
−0.447058 + 0.894505i $$0.647528\pi$$
$$938$$ −12.7689 −0.416921
$$939$$ 0 0
$$940$$ 50.7386 1.65491
$$941$$ 41.8078 1.36289 0.681447 0.731867i $$-0.261351\pi$$
0.681447 + 0.731867i $$0.261351\pi$$
$$942$$ 0 0
$$943$$ −8.68466 −0.282811
$$944$$ −92.2159 −3.00137
$$945$$ 0 0
$$946$$ −12.4924 −0.406164
$$947$$ −60.6004 −1.96925 −0.984624 0.174688i $$-0.944108\pi$$
−0.984624 + 0.174688i $$0.944108\pi$$
$$948$$ 0 0
$$949$$ 0 0
$$950$$ −13.1231 −0.425770
$$951$$ 0 0
$$952$$ −4.49242 −0.145600
$$953$$ 29.5616 0.957593 0.478796 0.877926i $$-0.341073\pi$$
0.478796 + 0.877926i $$0.341073\pi$$
$$954$$ 0 0
$$955$$ −19.6155 −0.634744
$$956$$ −104.600 −3.38302
$$957$$ 0 0
$$958$$ 66.7386 2.15623
$$959$$ −0.492423 −0.0159012
$$960$$ 0 0
$$961$$ 5.00000 0.161290
$$962$$ 0 0
$$963$$ 0 0
$$964$$ −112.847 −3.63454
$$965$$ 19.5616 0.629709
$$966$$ 0 0
$$967$$ 7.36932 0.236981 0.118491 0.992955i $$-0.462194\pi$$
0.118491 + 0.992955i $$0.462194\pi$$
$$968$$ 56.1771 1.80560
$$969$$ 0 0
$$970$$ −43.3693 −1.39250
$$971$$ −24.4924 −0.785999 −0.393000 0.919539i $$-0.628563\pi$$
−0.393000 + 0.919539i $$0.628563\pi$$
$$972$$ 0 0
$$973$$ 1.45360 0.0466003
$$974$$ −82.1080 −2.63091
$$975$$ 0 0
$$976$$ −51.3693 −1.64429
$$977$$ 43.4773 1.39096 0.695481 0.718545i $$-0.255192\pi$$
0.695481 + 0.718545i $$0.255192\pi$$
$$978$$ 0 0
$$979$$ −16.6847 −0.533244
$$980$$ 31.0540 0.991983
$$981$$ 0 0
$$982$$ 70.7386 2.25736
$$983$$ −42.7386 −1.36315 −0.681575 0.731748i $$-0.738705\pi$$
−0.681575 + 0.731748i $$0.738705\pi$$
$$984$$ 0 0
$$985$$ −3.36932 −0.107355
$$986$$ 28.4924 0.907384
$$987$$ 0 0
$$988$$ 0 0
$$989$$ 7.61553 0.242160
$$990$$ 0 0
$$991$$ 10.0540 0.319375 0.159688 0.987168i $$-0.448951\pi$$
0.159688 + 0.987168i $$0.448951\pi$$
$$992$$ 39.3693 1.24998
$$993$$ 0 0
$$994$$ 11.7235 0.371846
$$995$$ 8.00000 0.253617
$$996$$ 0 0
$$997$$ 28.2462 0.894566 0.447283 0.894392i $$-0.352392\pi$$
0.447283 + 0.894392i $$0.352392\pi$$
$$998$$ 89.6155 2.83673
$$999$$ 0 0
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 7605.2.a.bd.1.1 2
3.2 odd 2 7605.2.a.bi.1.2 2
13.12 even 2 585.2.a.l.1.2 yes 2
39.38 odd 2 585.2.a.j.1.1 2
52.51 odd 2 9360.2.a.cw.1.1 2
65.12 odd 4 2925.2.c.p.2224.4 4
65.38 odd 4 2925.2.c.p.2224.1 4
65.64 even 2 2925.2.a.x.1.1 2
156.155 even 2 9360.2.a.cl.1.1 2
195.38 even 4 2925.2.c.o.2224.4 4
195.77 even 4 2925.2.c.o.2224.1 4
195.194 odd 2 2925.2.a.bc.1.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
585.2.a.j.1.1 2 39.38 odd 2
585.2.a.l.1.2 yes 2 13.12 even 2
2925.2.a.x.1.1 2 65.64 even 2
2925.2.a.bc.1.2 2 195.194 odd 2
2925.2.c.o.2224.1 4 195.77 even 4
2925.2.c.o.2224.4 4 195.38 even 4
2925.2.c.p.2224.1 4 65.38 odd 4
2925.2.c.p.2224.4 4 65.12 odd 4
7605.2.a.bd.1.1 2 1.1 even 1 trivial
7605.2.a.bi.1.2 2 3.2 odd 2
9360.2.a.cl.1.1 2 156.155 even 2
9360.2.a.cw.1.1 2 52.51 odd 2