# Properties

 Label 9075.2.a.db.1.4 Level $9075$ Weight $2$ Character 9075.1 Self dual yes Analytic conductor $72.464$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$9075 = 3 \cdot 5^{2} \cdot 11^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 9075.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$72.4642398343$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\sqrt{3}, \sqrt{11})$$ Defining polynomial: $$x^{4} - 7 x^{2} + 4$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.4 Root $$2.52434$$ of defining polynomial Character $$\chi$$ $$=$$ 9075.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+2.52434 q^{2} +1.00000 q^{3} +4.37228 q^{4} +2.52434 q^{6} +0.792287 q^{7} +5.98844 q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q+2.52434 q^{2} +1.00000 q^{3} +4.37228 q^{4} +2.52434 q^{6} +0.792287 q^{7} +5.98844 q^{8} +1.00000 q^{9} +4.37228 q^{12} -0.147477 q^{13} +2.00000 q^{14} +6.37228 q^{16} +6.63325 q^{17} +2.52434 q^{18} -4.40387 q^{19} +0.792287 q^{21} -8.00000 q^{23} +5.98844 q^{24} -0.372281 q^{26} +1.00000 q^{27} +3.46410 q^{28} +10.0974 q^{29} +2.37228 q^{31} +4.10891 q^{32} +16.7446 q^{34} +4.37228 q^{36} -5.00000 q^{37} -11.1168 q^{38} -0.147477 q^{39} +6.92820 q^{41} +2.00000 q^{42} +9.94987 q^{43} -20.1947 q^{46} +8.74456 q^{47} +6.37228 q^{48} -6.37228 q^{49} +6.63325 q^{51} -0.644810 q^{52} +1.25544 q^{53} +2.52434 q^{54} +4.74456 q^{56} -4.40387 q^{57} +25.4891 q^{58} +4.00000 q^{59} +5.98844 q^{61} +5.98844 q^{62} +0.792287 q^{63} -2.37228 q^{64} +11.1168 q^{67} +29.0024 q^{68} -8.00000 q^{69} +10.7446 q^{71} +5.98844 q^{72} -5.19615 q^{73} -12.6217 q^{74} -19.2549 q^{76} -0.372281 q^{78} -6.78073 q^{79} +1.00000 q^{81} +17.4891 q^{82} +8.51278 q^{83} +3.46410 q^{84} +25.1168 q^{86} +10.0974 q^{87} +5.48913 q^{89} -0.116844 q^{91} -34.9783 q^{92} +2.37228 q^{93} +22.0742 q^{94} +4.10891 q^{96} +9.37228 q^{97} -16.0858 q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 4 q^{3} + 6 q^{4} + 4 q^{9} + O(q^{10})$$ $$4 q + 4 q^{3} + 6 q^{4} + 4 q^{9} + 6 q^{12} + 8 q^{14} + 14 q^{16} - 32 q^{23} + 10 q^{26} + 4 q^{27} - 2 q^{31} + 44 q^{34} + 6 q^{36} - 20 q^{37} - 10 q^{38} + 8 q^{42} + 12 q^{47} + 14 q^{48} - 14 q^{49} + 28 q^{53} - 4 q^{56} + 56 q^{58} + 16 q^{59} + 2 q^{64} + 10 q^{67} - 32 q^{69} + 20 q^{71} + 10 q^{78} + 4 q^{81} + 24 q^{82} + 66 q^{86} - 24 q^{89} + 34 q^{91} - 48 q^{92} - 2 q^{93} + 26 q^{97} + O(q^{100})$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 2.52434 1.78498 0.892488 0.451071i $$-0.148958\pi$$
0.892488 + 0.451071i $$0.148958\pi$$
$$3$$ 1.00000 0.577350
$$4$$ 4.37228 2.18614
$$5$$ 0 0
$$6$$ 2.52434 1.03056
$$7$$ 0.792287 0.299456 0.149728 0.988727i $$-0.452160\pi$$
0.149728 + 0.988727i $$0.452160\pi$$
$$8$$ 5.98844 2.11723
$$9$$ 1.00000 0.333333
$$10$$ 0 0
$$11$$ 0 0
$$12$$ 4.37228 1.26217
$$13$$ −0.147477 −0.0409027 −0.0204514 0.999791i $$-0.506510\pi$$
−0.0204514 + 0.999791i $$0.506510\pi$$
$$14$$ 2.00000 0.534522
$$15$$ 0 0
$$16$$ 6.37228 1.59307
$$17$$ 6.63325 1.60880 0.804400 0.594089i $$-0.202487\pi$$
0.804400 + 0.594089i $$0.202487\pi$$
$$18$$ 2.52434 0.594992
$$19$$ −4.40387 −1.01032 −0.505158 0.863027i $$-0.668566\pi$$
−0.505158 + 0.863027i $$0.668566\pi$$
$$20$$ 0 0
$$21$$ 0.792287 0.172891
$$22$$ 0 0
$$23$$ −8.00000 −1.66812 −0.834058 0.551677i $$-0.813988\pi$$
−0.834058 + 0.551677i $$0.813988\pi$$
$$24$$ 5.98844 1.22239
$$25$$ 0 0
$$26$$ −0.372281 −0.0730104
$$27$$ 1.00000 0.192450
$$28$$ 3.46410 0.654654
$$29$$ 10.0974 1.87503 0.937516 0.347943i $$-0.113120\pi$$
0.937516 + 0.347943i $$0.113120\pi$$
$$30$$ 0 0
$$31$$ 2.37228 0.426074 0.213037 0.977044i $$-0.431664\pi$$
0.213037 + 0.977044i $$0.431664\pi$$
$$32$$ 4.10891 0.726360
$$33$$ 0 0
$$34$$ 16.7446 2.87167
$$35$$ 0 0
$$36$$ 4.37228 0.728714
$$37$$ −5.00000 −0.821995 −0.410997 0.911636i $$-0.634819\pi$$
−0.410997 + 0.911636i $$0.634819\pi$$
$$38$$ −11.1168 −1.80339
$$39$$ −0.147477 −0.0236152
$$40$$ 0 0
$$41$$ 6.92820 1.08200 0.541002 0.841021i $$-0.318045\pi$$
0.541002 + 0.841021i $$0.318045\pi$$
$$42$$ 2.00000 0.308607
$$43$$ 9.94987 1.51734 0.758671 0.651474i $$-0.225849\pi$$
0.758671 + 0.651474i $$0.225849\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ −20.1947 −2.97755
$$47$$ 8.74456 1.27553 0.637763 0.770233i $$-0.279860\pi$$
0.637763 + 0.770233i $$0.279860\pi$$
$$48$$ 6.37228 0.919760
$$49$$ −6.37228 −0.910326
$$50$$ 0 0
$$51$$ 6.63325 0.928841
$$52$$ −0.644810 −0.0894191
$$53$$ 1.25544 0.172448 0.0862238 0.996276i $$-0.472520\pi$$
0.0862238 + 0.996276i $$0.472520\pi$$
$$54$$ 2.52434 0.343519
$$55$$ 0 0
$$56$$ 4.74456 0.634019
$$57$$ −4.40387 −0.583306
$$58$$ 25.4891 3.34689
$$59$$ 4.00000 0.520756 0.260378 0.965507i $$-0.416153\pi$$
0.260378 + 0.965507i $$0.416153\pi$$
$$60$$ 0 0
$$61$$ 5.98844 0.766741 0.383371 0.923595i $$-0.374763\pi$$
0.383371 + 0.923595i $$0.374763\pi$$
$$62$$ 5.98844 0.760533
$$63$$ 0.792287 0.0998188
$$64$$ −2.37228 −0.296535
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 11.1168 1.35814 0.679069 0.734074i $$-0.262384\pi$$
0.679069 + 0.734074i $$0.262384\pi$$
$$68$$ 29.0024 3.51706
$$69$$ −8.00000 −0.963087
$$70$$ 0 0
$$71$$ 10.7446 1.27514 0.637572 0.770390i $$-0.279939\pi$$
0.637572 + 0.770390i $$0.279939\pi$$
$$72$$ 5.98844 0.705744
$$73$$ −5.19615 −0.608164 −0.304082 0.952646i $$-0.598350\pi$$
−0.304082 + 0.952646i $$0.598350\pi$$
$$74$$ −12.6217 −1.46724
$$75$$ 0 0
$$76$$ −19.2549 −2.20869
$$77$$ 0 0
$$78$$ −0.372281 −0.0421526
$$79$$ −6.78073 −0.762891 −0.381446 0.924391i $$-0.624574\pi$$
−0.381446 + 0.924391i $$0.624574\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 17.4891 1.93135
$$83$$ 8.51278 0.934399 0.467199 0.884152i $$-0.345263\pi$$
0.467199 + 0.884152i $$0.345263\pi$$
$$84$$ 3.46410 0.377964
$$85$$ 0 0
$$86$$ 25.1168 2.70842
$$87$$ 10.0974 1.08255
$$88$$ 0 0
$$89$$ 5.48913 0.581846 0.290923 0.956746i $$-0.406038\pi$$
0.290923 + 0.956746i $$0.406038\pi$$
$$90$$ 0 0
$$91$$ −0.116844 −0.0122486
$$92$$ −34.9783 −3.64673
$$93$$ 2.37228 0.245994
$$94$$ 22.0742 2.27678
$$95$$ 0 0
$$96$$ 4.10891 0.419364
$$97$$ 9.37228 0.951611 0.475805 0.879551i $$-0.342157\pi$$
0.475805 + 0.879551i $$0.342157\pi$$
$$98$$ −16.0858 −1.62491
$$99$$ 0 0
$$100$$ 0 0
$$101$$ −13.2665 −1.32007 −0.660033 0.751237i $$-0.729458\pi$$
−0.660033 + 0.751237i $$0.729458\pi$$
$$102$$ 16.7446 1.65796
$$103$$ −3.74456 −0.368963 −0.184481 0.982836i $$-0.559061\pi$$
−0.184481 + 0.982836i $$0.559061\pi$$
$$104$$ −0.883156 −0.0866006
$$105$$ 0 0
$$106$$ 3.16915 0.307815
$$107$$ −8.21782 −0.794447 −0.397223 0.917722i $$-0.630026\pi$$
−0.397223 + 0.917722i $$0.630026\pi$$
$$108$$ 4.37228 0.420723
$$109$$ −2.67181 −0.255913 −0.127957 0.991780i $$-0.540842\pi$$
−0.127957 + 0.991780i $$0.540842\pi$$
$$110$$ 0 0
$$111$$ −5.00000 −0.474579
$$112$$ 5.04868 0.477055
$$113$$ −16.2337 −1.52714 −0.763568 0.645727i $$-0.776554\pi$$
−0.763568 + 0.645727i $$0.776554\pi$$
$$114$$ −11.1168 −1.04119
$$115$$ 0 0
$$116$$ 44.1485 4.09908
$$117$$ −0.147477 −0.0136342
$$118$$ 10.0974 0.929537
$$119$$ 5.25544 0.481765
$$120$$ 0 0
$$121$$ 0 0
$$122$$ 15.1168 1.36861
$$123$$ 6.92820 0.624695
$$124$$ 10.3723 0.931458
$$125$$ 0 0
$$126$$ 2.00000 0.178174
$$127$$ −10.2448 −0.909081 −0.454541 0.890726i $$-0.650197\pi$$
−0.454541 + 0.890726i $$0.650197\pi$$
$$128$$ −14.2063 −1.25567
$$129$$ 9.94987 0.876038
$$130$$ 0 0
$$131$$ −18.6101 −1.62597 −0.812987 0.582282i $$-0.802160\pi$$
−0.812987 + 0.582282i $$0.802160\pi$$
$$132$$ 0 0
$$133$$ −3.48913 −0.302546
$$134$$ 28.0627 2.42425
$$135$$ 0 0
$$136$$ 39.7228 3.40620
$$137$$ 0.744563 0.0636123 0.0318061 0.999494i $$-0.489874\pi$$
0.0318061 + 0.999494i $$0.489874\pi$$
$$138$$ −20.1947 −1.71909
$$139$$ −10.3923 −0.881464 −0.440732 0.897639i $$-0.645281\pi$$
−0.440732 + 0.897639i $$0.645281\pi$$
$$140$$ 0 0
$$141$$ 8.74456 0.736425
$$142$$ 27.1229 2.27610
$$143$$ 0 0
$$144$$ 6.37228 0.531023
$$145$$ 0 0
$$146$$ −13.1168 −1.08556
$$147$$ −6.37228 −0.525577
$$148$$ −21.8614 −1.79700
$$149$$ −1.28962 −0.105650 −0.0528249 0.998604i $$-0.516823\pi$$
−0.0528249 + 0.998604i $$0.516823\pi$$
$$150$$ 0 0
$$151$$ −23.8063 −1.93733 −0.968664 0.248375i $$-0.920103\pi$$
−0.968664 + 0.248375i $$0.920103\pi$$
$$152$$ −26.3723 −2.13907
$$153$$ 6.63325 0.536266
$$154$$ 0 0
$$155$$ 0 0
$$156$$ −0.644810 −0.0516261
$$157$$ −8.37228 −0.668181 −0.334090 0.942541i $$-0.608429\pi$$
−0.334090 + 0.942541i $$0.608429\pi$$
$$158$$ −17.1168 −1.36174
$$159$$ 1.25544 0.0995627
$$160$$ 0 0
$$161$$ −6.33830 −0.499528
$$162$$ 2.52434 0.198331
$$163$$ −14.3723 −1.12572 −0.562862 0.826551i $$-0.690300\pi$$
−0.562862 + 0.826551i $$0.690300\pi$$
$$164$$ 30.2921 2.36541
$$165$$ 0 0
$$166$$ 21.4891 1.66788
$$167$$ 5.04868 0.390678 0.195339 0.980736i $$-0.437419\pi$$
0.195339 + 0.980736i $$0.437419\pi$$
$$168$$ 4.74456 0.366051
$$169$$ −12.9783 −0.998327
$$170$$ 0 0
$$171$$ −4.40387 −0.336772
$$172$$ 43.5036 3.31712
$$173$$ −6.92820 −0.526742 −0.263371 0.964695i $$-0.584834\pi$$
−0.263371 + 0.964695i $$0.584834\pi$$
$$174$$ 25.4891 1.93233
$$175$$ 0 0
$$176$$ 0 0
$$177$$ 4.00000 0.300658
$$178$$ 13.8564 1.03858
$$179$$ 19.4891 1.45669 0.728343 0.685213i $$-0.240291\pi$$
0.728343 + 0.685213i $$0.240291\pi$$
$$180$$ 0 0
$$181$$ 13.8614 1.03031 0.515155 0.857097i $$-0.327734\pi$$
0.515155 + 0.857097i $$0.327734\pi$$
$$182$$ −0.294954 −0.0218634
$$183$$ 5.98844 0.442678
$$184$$ −47.9075 −3.53179
$$185$$ 0 0
$$186$$ 5.98844 0.439094
$$187$$ 0 0
$$188$$ 38.2337 2.78848
$$189$$ 0.792287 0.0576304
$$190$$ 0 0
$$191$$ −17.4891 −1.26547 −0.632734 0.774369i $$-0.718067\pi$$
−0.632734 + 0.774369i $$0.718067\pi$$
$$192$$ −2.37228 −0.171205
$$193$$ −21.1345 −1.52129 −0.760646 0.649167i $$-0.775118\pi$$
−0.760646 + 0.649167i $$0.775118\pi$$
$$194$$ 23.6588 1.69860
$$195$$ 0 0
$$196$$ −27.8614 −1.99010
$$197$$ 8.51278 0.606510 0.303255 0.952909i $$-0.401927\pi$$
0.303255 + 0.952909i $$0.401927\pi$$
$$198$$ 0 0
$$199$$ 16.8614 1.19527 0.597637 0.801767i $$-0.296107\pi$$
0.597637 + 0.801767i $$0.296107\pi$$
$$200$$ 0 0
$$201$$ 11.1168 0.784122
$$202$$ −33.4891 −2.35629
$$203$$ 8.00000 0.561490
$$204$$ 29.0024 2.03058
$$205$$ 0 0
$$206$$ −9.45254 −0.658590
$$207$$ −8.00000 −0.556038
$$208$$ −0.939764 −0.0651609
$$209$$ 0 0
$$210$$ 0 0
$$211$$ 4.55134 0.313327 0.156664 0.987652i $$-0.449926\pi$$
0.156664 + 0.987652i $$0.449926\pi$$
$$212$$ 5.48913 0.376995
$$213$$ 10.7446 0.736205
$$214$$ −20.7446 −1.41807
$$215$$ 0 0
$$216$$ 5.98844 0.407462
$$217$$ 1.87953 0.127591
$$218$$ −6.74456 −0.456799
$$219$$ −5.19615 −0.351123
$$220$$ 0 0
$$221$$ −0.978251 −0.0658043
$$222$$ −12.6217 −0.847112
$$223$$ 23.8614 1.59788 0.798939 0.601412i $$-0.205395\pi$$
0.798939 + 0.601412i $$0.205395\pi$$
$$224$$ 3.25544 0.217513
$$225$$ 0 0
$$226$$ −40.9793 −2.72590
$$227$$ −17.3205 −1.14960 −0.574801 0.818293i $$-0.694921\pi$$
−0.574801 + 0.818293i $$0.694921\pi$$
$$228$$ −19.2549 −1.27519
$$229$$ −10.4891 −0.693141 −0.346570 0.938024i $$-0.612654\pi$$
−0.346570 + 0.938024i $$0.612654\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 60.4674 3.96988
$$233$$ 15.1460 0.992249 0.496125 0.868251i $$-0.334756\pi$$
0.496125 + 0.868251i $$0.334756\pi$$
$$234$$ −0.372281 −0.0243368
$$235$$ 0 0
$$236$$ 17.4891 1.13845
$$237$$ −6.78073 −0.440456
$$238$$ 13.2665 0.859939
$$239$$ 13.2665 0.858138 0.429069 0.903272i $$-0.358842\pi$$
0.429069 + 0.903272i $$0.358842\pi$$
$$240$$ 0 0
$$241$$ −14.0039 −0.902069 −0.451035 0.892506i $$-0.648945\pi$$
−0.451035 + 0.892506i $$0.648945\pi$$
$$242$$ 0 0
$$243$$ 1.00000 0.0641500
$$244$$ 26.1831 1.67620
$$245$$ 0 0
$$246$$ 17.4891 1.11507
$$247$$ 0.649468 0.0413247
$$248$$ 14.2063 0.902099
$$249$$ 8.51278 0.539475
$$250$$ 0 0
$$251$$ −0.510875 −0.0322461 −0.0161231 0.999870i $$-0.505132\pi$$
−0.0161231 + 0.999870i $$0.505132\pi$$
$$252$$ 3.46410 0.218218
$$253$$ 0 0
$$254$$ −25.8614 −1.62269
$$255$$ 0 0
$$256$$ −31.1168 −1.94480
$$257$$ −17.4891 −1.09094 −0.545471 0.838130i $$-0.683649\pi$$
−0.545471 + 0.838130i $$0.683649\pi$$
$$258$$ 25.1168 1.56371
$$259$$ −3.96143 −0.246152
$$260$$ 0 0
$$261$$ 10.0974 0.625010
$$262$$ −46.9783 −2.90233
$$263$$ −5.04868 −0.311315 −0.155657 0.987811i $$-0.549750\pi$$
−0.155657 + 0.987811i $$0.549750\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ −8.80773 −0.540037
$$267$$ 5.48913 0.335929
$$268$$ 48.6060 2.96908
$$269$$ −2.74456 −0.167339 −0.0836695 0.996494i $$-0.526664\pi$$
−0.0836695 + 0.996494i $$0.526664\pi$$
$$270$$ 0 0
$$271$$ −10.3923 −0.631288 −0.315644 0.948878i $$-0.602220\pi$$
−0.315644 + 0.948878i $$0.602220\pi$$
$$272$$ 42.2689 2.56293
$$273$$ −0.116844 −0.00707172
$$274$$ 1.87953 0.113546
$$275$$ 0 0
$$276$$ −34.9783 −2.10544
$$277$$ 20.5446 1.23440 0.617201 0.786805i $$-0.288266\pi$$
0.617201 + 0.786805i $$0.288266\pi$$
$$278$$ −26.2337 −1.57339
$$279$$ 2.37228 0.142025
$$280$$ 0 0
$$281$$ −8.51278 −0.507830 −0.253915 0.967227i $$-0.581718\pi$$
−0.253915 + 0.967227i $$0.581718\pi$$
$$282$$ 22.0742 1.31450
$$283$$ −28.3576 −1.68569 −0.842843 0.538160i $$-0.819120\pi$$
−0.842843 + 0.538160i $$0.819120\pi$$
$$284$$ 46.9783 2.78765
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 5.48913 0.324013
$$288$$ 4.10891 0.242120
$$289$$ 27.0000 1.58824
$$290$$ 0 0
$$291$$ 9.37228 0.549413
$$292$$ −22.7190 −1.32953
$$293$$ −32.7615 −1.91395 −0.956973 0.290176i $$-0.906286\pi$$
−0.956973 + 0.290176i $$0.906286\pi$$
$$294$$ −16.0858 −0.938142
$$295$$ 0 0
$$296$$ −29.9422 −1.74035
$$297$$ 0 0
$$298$$ −3.25544 −0.188582
$$299$$ 1.17981 0.0682304
$$300$$ 0 0
$$301$$ 7.88316 0.454378
$$302$$ −60.0951 −3.45808
$$303$$ −13.2665 −0.762140
$$304$$ −28.0627 −1.60950
$$305$$ 0 0
$$306$$ 16.7446 0.957223
$$307$$ −26.4781 −1.51118 −0.755592 0.655042i $$-0.772651\pi$$
−0.755592 + 0.655042i $$0.772651\pi$$
$$308$$ 0 0
$$309$$ −3.74456 −0.213021
$$310$$ 0 0
$$311$$ −10.2337 −0.580299 −0.290150 0.956981i $$-0.593705\pi$$
−0.290150 + 0.956981i $$0.593705\pi$$
$$312$$ −0.883156 −0.0499989
$$313$$ 25.9783 1.46838 0.734189 0.678945i $$-0.237563\pi$$
0.734189 + 0.678945i $$0.237563\pi$$
$$314$$ −21.1345 −1.19269
$$315$$ 0 0
$$316$$ −29.6472 −1.66779
$$317$$ 18.9783 1.06592 0.532962 0.846139i $$-0.321079\pi$$
0.532962 + 0.846139i $$0.321079\pi$$
$$318$$ 3.16915 0.177717
$$319$$ 0 0
$$320$$ 0 0
$$321$$ −8.21782 −0.458674
$$322$$ −16.0000 −0.891645
$$323$$ −29.2119 −1.62540
$$324$$ 4.37228 0.242905
$$325$$ 0 0
$$326$$ −36.2805 −2.00939
$$327$$ −2.67181 −0.147752
$$328$$ 41.4891 2.29085
$$329$$ 6.92820 0.381964
$$330$$ 0 0
$$331$$ 1.23369 0.0678096 0.0339048 0.999425i $$-0.489206\pi$$
0.0339048 + 0.999425i $$0.489206\pi$$
$$332$$ 37.2203 2.04273
$$333$$ −5.00000 −0.273998
$$334$$ 12.7446 0.697351
$$335$$ 0 0
$$336$$ 5.04868 0.275428
$$337$$ 10.5947 0.577129 0.288565 0.957460i $$-0.406822\pi$$
0.288565 + 0.957460i $$0.406822\pi$$
$$338$$ −32.7615 −1.78199
$$339$$ −16.2337 −0.881693
$$340$$ 0 0
$$341$$ 0 0
$$342$$ −11.1168 −0.601130
$$343$$ −10.5947 −0.572059
$$344$$ 59.5842 3.21257
$$345$$ 0 0
$$346$$ −17.4891 −0.940221
$$347$$ −2.17448 −0.116732 −0.0583661 0.998295i $$-0.518589\pi$$
−0.0583661 + 0.998295i $$0.518589\pi$$
$$348$$ 44.1485 2.36661
$$349$$ 8.66025 0.463573 0.231786 0.972767i $$-0.425543\pi$$
0.231786 + 0.972767i $$0.425543\pi$$
$$350$$ 0 0
$$351$$ −0.147477 −0.00787173
$$352$$ 0 0
$$353$$ 16.2337 0.864032 0.432016 0.901866i $$-0.357802\pi$$
0.432016 + 0.901866i $$0.357802\pi$$
$$354$$ 10.0974 0.536668
$$355$$ 0 0
$$356$$ 24.0000 1.27200
$$357$$ 5.25544 0.278147
$$358$$ 49.1971 2.60015
$$359$$ 6.63325 0.350090 0.175045 0.984560i $$-0.443993\pi$$
0.175045 + 0.984560i $$0.443993\pi$$
$$360$$ 0 0
$$361$$ 0.394031 0.0207385
$$362$$ 34.9909 1.83908
$$363$$ 0 0
$$364$$ −0.510875 −0.0267771
$$365$$ 0 0
$$366$$ 15.1168 0.790170
$$367$$ 15.0000 0.782994 0.391497 0.920179i $$-0.371957\pi$$
0.391497 + 0.920179i $$0.371957\pi$$
$$368$$ −50.9783 −2.65743
$$369$$ 6.92820 0.360668
$$370$$ 0 0
$$371$$ 0.994667 0.0516405
$$372$$ 10.3723 0.537778
$$373$$ 26.0357 1.34808 0.674038 0.738697i $$-0.264558\pi$$
0.674038 + 0.738697i $$0.264558\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 52.3663 2.70058
$$377$$ −1.48913 −0.0766939
$$378$$ 2.00000 0.102869
$$379$$ −5.00000 −0.256833 −0.128416 0.991720i $$-0.540989\pi$$
−0.128416 + 0.991720i $$0.540989\pi$$
$$380$$ 0 0
$$381$$ −10.2448 −0.524858
$$382$$ −44.1485 −2.25883
$$383$$ 10.2337 0.522917 0.261459 0.965215i $$-0.415797\pi$$
0.261459 + 0.965215i $$0.415797\pi$$
$$384$$ −14.2063 −0.724960
$$385$$ 0 0
$$386$$ −53.3505 −2.71547
$$387$$ 9.94987 0.505781
$$388$$ 40.9783 2.08036
$$389$$ 32.7446 1.66022 0.830108 0.557603i $$-0.188279\pi$$
0.830108 + 0.557603i $$0.188279\pi$$
$$390$$ 0 0
$$391$$ −53.0660 −2.68366
$$392$$ −38.1600 −1.92737
$$393$$ −18.6101 −0.938757
$$394$$ 21.4891 1.08261
$$395$$ 0 0
$$396$$ 0 0
$$397$$ 2.62772 0.131881 0.0659407 0.997824i $$-0.478995\pi$$
0.0659407 + 0.997824i $$0.478995\pi$$
$$398$$ 42.5639 2.13353
$$399$$ −3.48913 −0.174675
$$400$$ 0 0
$$401$$ −30.9783 −1.54698 −0.773490 0.633808i $$-0.781491\pi$$
−0.773490 + 0.633808i $$0.781491\pi$$
$$402$$ 28.0627 1.39964
$$403$$ −0.349857 −0.0174276
$$404$$ −58.0049 −2.88585
$$405$$ 0 0
$$406$$ 20.1947 1.00225
$$407$$ 0 0
$$408$$ 39.7228 1.96657
$$409$$ 38.3075 1.89418 0.947092 0.320962i $$-0.104006\pi$$
0.947092 + 0.320962i $$0.104006\pi$$
$$410$$ 0 0
$$411$$ 0.744563 0.0367266
$$412$$ −16.3723 −0.806604
$$413$$ 3.16915 0.155944
$$414$$ −20.1947 −0.992515
$$415$$ 0 0
$$416$$ −0.605969 −0.0297101
$$417$$ −10.3923 −0.508913
$$418$$ 0 0
$$419$$ −2.23369 −0.109123 −0.0545614 0.998510i $$-0.517376\pi$$
−0.0545614 + 0.998510i $$0.517376\pi$$
$$420$$ 0 0
$$421$$ 18.0000 0.877266 0.438633 0.898666i $$-0.355463\pi$$
0.438633 + 0.898666i $$0.355463\pi$$
$$422$$ 11.4891 0.559282
$$423$$ 8.74456 0.425175
$$424$$ 7.51811 0.365112
$$425$$ 0 0
$$426$$ 27.1229 1.31411
$$427$$ 4.74456 0.229605
$$428$$ −35.9306 −1.73677
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 23.6588 1.13960 0.569802 0.821782i $$-0.307020\pi$$
0.569802 + 0.821782i $$0.307020\pi$$
$$432$$ 6.37228 0.306587
$$433$$ −40.0951 −1.92685 −0.963424 0.267983i $$-0.913643\pi$$
−0.963424 + 0.267983i $$0.913643\pi$$
$$434$$ 4.74456 0.227746
$$435$$ 0 0
$$436$$ −11.6819 −0.559463
$$437$$ 35.2309 1.68532
$$438$$ −13.1168 −0.626747
$$439$$ 7.13058 0.340324 0.170162 0.985416i $$-0.445571\pi$$
0.170162 + 0.985416i $$0.445571\pi$$
$$440$$ 0 0
$$441$$ −6.37228 −0.303442
$$442$$ −2.46943 −0.117459
$$443$$ −9.25544 −0.439739 −0.219870 0.975529i $$-0.570563\pi$$
−0.219870 + 0.975529i $$0.570563\pi$$
$$444$$ −21.8614 −1.03750
$$445$$ 0 0
$$446$$ 60.2343 2.85217
$$447$$ −1.28962 −0.0609969
$$448$$ −1.87953 −0.0887993
$$449$$ 12.7446 0.601453 0.300727 0.953710i $$-0.402771\pi$$
0.300727 + 0.953710i $$0.402771\pi$$
$$450$$ 0 0
$$451$$ 0 0
$$452$$ −70.9783 −3.33854
$$453$$ −23.8063 −1.11852
$$454$$ −43.7228 −2.05201
$$455$$ 0 0
$$456$$ −26.3723 −1.23500
$$457$$ −19.6048 −0.917074 −0.458537 0.888675i $$-0.651626\pi$$
−0.458537 + 0.888675i $$0.651626\pi$$
$$458$$ −26.4781 −1.23724
$$459$$ 6.63325 0.309614
$$460$$ 0 0
$$461$$ 1.87953 0.0875383 0.0437692 0.999042i $$-0.486063\pi$$
0.0437692 + 0.999042i $$0.486063\pi$$
$$462$$ 0 0
$$463$$ −26.9783 −1.25379 −0.626893 0.779106i $$-0.715674\pi$$
−0.626893 + 0.779106i $$0.715674\pi$$
$$464$$ 64.3432 2.98706
$$465$$ 0 0
$$466$$ 38.2337 1.77114
$$467$$ 1.48913 0.0689085 0.0344543 0.999406i $$-0.489031\pi$$
0.0344543 + 0.999406i $$0.489031\pi$$
$$468$$ −0.644810 −0.0298064
$$469$$ 8.80773 0.406703
$$470$$ 0 0
$$471$$ −8.37228 −0.385774
$$472$$ 23.9538 1.10256
$$473$$ 0 0
$$474$$ −17.1168 −0.786203
$$475$$ 0 0
$$476$$ 22.9783 1.05321
$$477$$ 1.25544 0.0574825
$$478$$ 33.4891 1.53176
$$479$$ 2.87419 0.131325 0.0656626 0.997842i $$-0.479084\pi$$
0.0656626 + 0.997842i $$0.479084\pi$$
$$480$$ 0 0
$$481$$ 0.737384 0.0336218
$$482$$ −35.3505 −1.61017
$$483$$ −6.33830 −0.288402
$$484$$ 0 0
$$485$$ 0 0
$$486$$ 2.52434 0.114506
$$487$$ 35.9783 1.63033 0.815165 0.579229i $$-0.196646\pi$$
0.815165 + 0.579229i $$0.196646\pi$$
$$488$$ 35.8614 1.62337
$$489$$ −14.3723 −0.649937
$$490$$ 0 0
$$491$$ −6.63325 −0.299354 −0.149677 0.988735i $$-0.547823\pi$$
−0.149677 + 0.988735i $$0.547823\pi$$
$$492$$ 30.2921 1.36567
$$493$$ 66.9783 3.01655
$$494$$ 1.63948 0.0737636
$$495$$ 0 0
$$496$$ 15.1168 0.678766
$$497$$ 8.51278 0.381850
$$498$$ 21.4891 0.962951
$$499$$ −14.1168 −0.631957 −0.315978 0.948766i $$-0.602333\pi$$
−0.315978 + 0.948766i $$0.602333\pi$$
$$500$$ 0 0
$$501$$ 5.04868 0.225558
$$502$$ −1.28962 −0.0575586
$$503$$ −3.16915 −0.141305 −0.0706527 0.997501i $$-0.522508\pi$$
−0.0706527 + 0.997501i $$0.522508\pi$$
$$504$$ 4.74456 0.211340
$$505$$ 0 0
$$506$$ 0 0
$$507$$ −12.9783 −0.576384
$$508$$ −44.7933 −1.98738
$$509$$ −30.9783 −1.37309 −0.686543 0.727089i $$-0.740873\pi$$
−0.686543 + 0.727089i $$0.740873\pi$$
$$510$$ 0 0
$$511$$ −4.11684 −0.182118
$$512$$ −50.1369 −2.21576
$$513$$ −4.40387 −0.194435
$$514$$ −44.1485 −1.94731
$$515$$ 0 0
$$516$$ 43.5036 1.91514
$$517$$ 0 0
$$518$$ −10.0000 −0.439375
$$519$$ −6.92820 −0.304114
$$520$$ 0 0
$$521$$ −36.7446 −1.60981 −0.804904 0.593405i $$-0.797783\pi$$
−0.804904 + 0.593405i $$0.797783\pi$$
$$522$$ 25.4891 1.11563
$$523$$ 27.0680 1.18360 0.591801 0.806084i $$-0.298417\pi$$
0.591801 + 0.806084i $$0.298417\pi$$
$$524$$ −81.3687 −3.55461
$$525$$ 0 0
$$526$$ −12.7446 −0.555689
$$527$$ 15.7359 0.685468
$$528$$ 0 0
$$529$$ 41.0000 1.78261
$$530$$ 0 0
$$531$$ 4.00000 0.173585
$$532$$ −15.2554 −0.661407
$$533$$ −1.02175 −0.0442569
$$534$$ 13.8564 0.599625
$$535$$ 0 0
$$536$$ 66.5725 2.87550
$$537$$ 19.4891 0.841018
$$538$$ −6.92820 −0.298696
$$539$$ 0 0
$$540$$ 0 0
$$541$$ 23.5113 1.01083 0.505415 0.862876i $$-0.331339\pi$$
0.505415 + 0.862876i $$0.331339\pi$$
$$542$$ −26.2337 −1.12683
$$543$$ 13.8614 0.594850
$$544$$ 27.2554 1.16857
$$545$$ 0 0
$$546$$ −0.294954 −0.0126229
$$547$$ 0.884861 0.0378339 0.0189170 0.999821i $$-0.493978\pi$$
0.0189170 + 0.999821i $$0.493978\pi$$
$$548$$ 3.25544 0.139065
$$549$$ 5.98844 0.255580
$$550$$ 0 0
$$551$$ −44.4674 −1.89437
$$552$$ −47.9075 −2.03908
$$553$$ −5.37228 −0.228453
$$554$$ 51.8614 2.20338
$$555$$ 0 0
$$556$$ −45.4381 −1.92700
$$557$$ −6.33830 −0.268562 −0.134281 0.990943i $$-0.542873\pi$$
−0.134281 + 0.990943i $$0.542873\pi$$
$$558$$ 5.98844 0.253511
$$559$$ −1.46738 −0.0620634
$$560$$ 0 0
$$561$$ 0 0
$$562$$ −21.4891 −0.906464
$$563$$ −43.8535 −1.84820 −0.924102 0.382145i $$-0.875186\pi$$
−0.924102 + 0.382145i $$0.875186\pi$$
$$564$$ 38.2337 1.60993
$$565$$ 0 0
$$566$$ −71.5842 −3.00891
$$567$$ 0.792287 0.0332729
$$568$$ 64.3432 2.69978
$$569$$ 23.0689 0.967098 0.483549 0.875317i $$-0.339347\pi$$
0.483549 + 0.875317i $$0.339347\pi$$
$$570$$ 0 0
$$571$$ −3.37153 −0.141094 −0.0705470 0.997508i $$-0.522474\pi$$
−0.0705470 + 0.997508i $$0.522474\pi$$
$$572$$ 0 0
$$573$$ −17.4891 −0.730619
$$574$$ 13.8564 0.578355
$$575$$ 0 0
$$576$$ −2.37228 −0.0988451
$$577$$ 24.0951 1.00309 0.501546 0.865131i $$-0.332765\pi$$
0.501546 + 0.865131i $$0.332765\pi$$
$$578$$ 68.1571 2.83496
$$579$$ −21.1345 −0.878318
$$580$$ 0 0
$$581$$ 6.74456 0.279812
$$582$$ 23.6588 0.980689
$$583$$ 0 0
$$584$$ −31.1168 −1.28762
$$585$$ 0 0
$$586$$ −82.7011 −3.41635
$$587$$ −34.7446 −1.43406 −0.717031 0.697041i $$-0.754499\pi$$
−0.717031 + 0.697041i $$0.754499\pi$$
$$588$$ −27.8614 −1.14899
$$589$$ −10.4472 −0.430470
$$590$$ 0 0
$$591$$ 8.51278 0.350169
$$592$$ −31.8614 −1.30950
$$593$$ 24.5437 1.00789 0.503944 0.863736i $$-0.331882\pi$$
0.503944 + 0.863736i $$0.331882\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ −5.63858 −0.230965
$$597$$ 16.8614 0.690091
$$598$$ 2.97825 0.121790
$$599$$ −33.2554 −1.35878 −0.679390 0.733777i $$-0.737756\pi$$
−0.679390 + 0.733777i $$0.737756\pi$$
$$600$$ 0 0
$$601$$ −11.0371 −0.450213 −0.225107 0.974334i $$-0.572273\pi$$
−0.225107 + 0.974334i $$0.572273\pi$$
$$602$$ 19.8997 0.811053
$$603$$ 11.1168 0.452713
$$604$$ −104.088 −4.23527
$$605$$ 0 0
$$606$$ −33.4891 −1.36040
$$607$$ −33.7562 −1.37012 −0.685060 0.728487i $$-0.740224\pi$$
−0.685060 + 0.728487i $$0.740224\pi$$
$$608$$ −18.0951 −0.733853
$$609$$ 8.00000 0.324176
$$610$$ 0 0
$$611$$ −1.28962 −0.0521725
$$612$$ 29.0024 1.17235
$$613$$ 14.9985 0.605786 0.302893 0.953025i $$-0.402048\pi$$
0.302893 + 0.953025i $$0.402048\pi$$
$$614$$ −66.8397 −2.69743
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 2.00000 0.0805170 0.0402585 0.999189i $$-0.487182\pi$$
0.0402585 + 0.999189i $$0.487182\pi$$
$$618$$ −9.45254 −0.380237
$$619$$ −48.4891 −1.94894 −0.974471 0.224512i $$-0.927921\pi$$
−0.974471 + 0.224512i $$0.927921\pi$$
$$620$$ 0 0
$$621$$ −8.00000 −0.321029
$$622$$ −25.8333 −1.03582
$$623$$ 4.34896 0.174238
$$624$$ −0.939764 −0.0376207
$$625$$ 0 0
$$626$$ 65.5779 2.62102
$$627$$ 0 0
$$628$$ −36.6060 −1.46074
$$629$$ −33.1662 −1.32242
$$630$$ 0 0
$$631$$ −36.4891 −1.45261 −0.726305 0.687373i $$-0.758764\pi$$
−0.726305 + 0.687373i $$0.758764\pi$$
$$632$$ −40.6060 −1.61522
$$633$$ 4.55134 0.180900
$$634$$ 47.9075 1.90265
$$635$$ 0 0
$$636$$ 5.48913 0.217658
$$637$$ 0.939764 0.0372348
$$638$$ 0 0
$$639$$ 10.7446 0.425048
$$640$$ 0 0
$$641$$ 28.9783 1.14457 0.572286 0.820054i $$-0.306057\pi$$
0.572286 + 0.820054i $$0.306057\pi$$
$$642$$ −20.7446 −0.818723
$$643$$ 9.23369 0.364141 0.182071 0.983285i $$-0.441720\pi$$
0.182071 + 0.983285i $$0.441720\pi$$
$$644$$ −27.7128 −1.09204
$$645$$ 0 0
$$646$$ −73.7408 −2.90129
$$647$$ −2.00000 −0.0786281 −0.0393141 0.999227i $$-0.512517\pi$$
−0.0393141 + 0.999227i $$0.512517\pi$$
$$648$$ 5.98844 0.235248
$$649$$ 0 0
$$650$$ 0 0
$$651$$ 1.87953 0.0736645
$$652$$ −62.8397 −2.46099
$$653$$ −45.4891 −1.78013 −0.890064 0.455837i $$-0.849340\pi$$
−0.890064 + 0.455837i $$0.849340\pi$$
$$654$$ −6.74456 −0.263733
$$655$$ 0 0
$$656$$ 44.1485 1.72371
$$657$$ −5.19615 −0.202721
$$658$$ 17.4891 0.681797
$$659$$ −12.2718 −0.478043 −0.239021 0.971014i $$-0.576827\pi$$
−0.239021 + 0.971014i $$0.576827\pi$$
$$660$$ 0 0
$$661$$ −3.00000 −0.116686 −0.0583432 0.998297i $$-0.518582\pi$$
−0.0583432 + 0.998297i $$0.518582\pi$$
$$662$$ 3.11425 0.121039
$$663$$ −0.978251 −0.0379921
$$664$$ 50.9783 1.97834
$$665$$ 0 0
$$666$$ −12.6217 −0.489081
$$667$$ −80.7788 −3.12777
$$668$$ 22.0742 0.854078
$$669$$ 23.8614 0.922535
$$670$$ 0 0
$$671$$ 0 0
$$672$$ 3.25544 0.125581
$$673$$ 28.8550 1.11228 0.556138 0.831090i $$-0.312282\pi$$
0.556138 + 0.831090i $$0.312282\pi$$
$$674$$ 26.7446 1.03016
$$675$$ 0 0
$$676$$ −56.7446 −2.18248
$$677$$ −10.9822 −0.422081 −0.211040 0.977477i $$-0.567685\pi$$
−0.211040 + 0.977477i $$0.567685\pi$$
$$678$$ −40.9793 −1.57380
$$679$$ 7.42554 0.284966
$$680$$ 0 0
$$681$$ −17.3205 −0.663723
$$682$$ 0 0
$$683$$ −18.0000 −0.688751 −0.344375 0.938832i $$-0.611909\pi$$
−0.344375 + 0.938832i $$0.611909\pi$$
$$684$$ −19.2549 −0.736231
$$685$$ 0 0
$$686$$ −26.7446 −1.02111
$$687$$ −10.4891 −0.400185
$$688$$ 63.4034 2.41723
$$689$$ −0.185148 −0.00705357
$$690$$ 0 0
$$691$$ −2.25544 −0.0858009 −0.0429004 0.999079i $$-0.513660\pi$$
−0.0429004 + 0.999079i $$0.513660\pi$$
$$692$$ −30.2921 −1.15153
$$693$$ 0 0
$$694$$ −5.48913 −0.208364
$$695$$ 0 0
$$696$$ 60.4674 2.29201
$$697$$ 45.9565 1.74073
$$698$$ 21.8614 0.827466
$$699$$ 15.1460 0.572875
$$700$$ 0 0
$$701$$ −4.05401 −0.153118 −0.0765589 0.997065i $$-0.524393\pi$$
−0.0765589 + 0.997065i $$0.524393\pi$$
$$702$$ −0.372281 −0.0140509
$$703$$ 22.0193 0.830475
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 40.9793 1.54228
$$707$$ −10.5109 −0.395302
$$708$$ 17.4891 0.657282
$$709$$ 31.0000 1.16423 0.582115 0.813107i $$-0.302225\pi$$
0.582115 + 0.813107i $$0.302225\pi$$
$$710$$ 0 0
$$711$$ −6.78073 −0.254297
$$712$$ 32.8713 1.23190
$$713$$ −18.9783 −0.710741
$$714$$ 13.2665 0.496486
$$715$$ 0 0
$$716$$ 85.2119 3.18452
$$717$$ 13.2665 0.495446
$$718$$ 16.7446 0.624902
$$719$$ 14.0000 0.522112 0.261056 0.965324i $$-0.415929\pi$$
0.261056 + 0.965324i $$0.415929\pi$$
$$720$$ 0 0
$$721$$ −2.96677 −0.110488
$$722$$ 0.994667 0.0370177
$$723$$ −14.0039 −0.520810
$$724$$ 60.6060 2.25240
$$725$$ 0 0
$$726$$ 0 0
$$727$$ −21.4674 −0.796181 −0.398090 0.917346i $$-0.630327\pi$$
−0.398090 + 0.917346i $$0.630327\pi$$
$$728$$ −0.699713 −0.0259331
$$729$$ 1.00000 0.0370370
$$730$$ 0 0
$$731$$ 66.0000 2.44110
$$732$$ 26.1831 0.967757
$$733$$ −13.8564 −0.511798 −0.255899 0.966704i $$-0.582371\pi$$
−0.255899 + 0.966704i $$0.582371\pi$$
$$734$$ 37.8651 1.39763
$$735$$ 0 0
$$736$$ −32.8713 −1.21165
$$737$$ 0 0
$$738$$ 17.4891 0.643784
$$739$$ 33.6087 1.23632 0.618158 0.786054i $$-0.287879\pi$$
0.618158 + 0.786054i $$0.287879\pi$$
$$740$$ 0 0
$$741$$ 0.649468 0.0238588
$$742$$ 2.51087 0.0921771
$$743$$ −41.9740 −1.53988 −0.769938 0.638119i $$-0.779713\pi$$
−0.769938 + 0.638119i $$0.779713\pi$$
$$744$$ 14.2063 0.520827
$$745$$ 0 0
$$746$$ 65.7228 2.40628
$$747$$ 8.51278 0.311466
$$748$$ 0 0
$$749$$ −6.51087 −0.237902
$$750$$ 0 0
$$751$$ 29.2337 1.06675 0.533376 0.845878i $$-0.320923\pi$$
0.533376 + 0.845878i $$0.320923\pi$$
$$752$$ 55.7228 2.03200
$$753$$ −0.510875 −0.0186173
$$754$$ −3.75906 −0.136897
$$755$$ 0 0
$$756$$ 3.46410 0.125988
$$757$$ −32.3505 −1.17580 −0.587900 0.808934i $$-0.700045\pi$$
−0.587900 + 0.808934i $$0.700045\pi$$
$$758$$ −12.6217 −0.458440
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 18.6101 0.674617 0.337308 0.941394i $$-0.390484\pi$$
0.337308 + 0.941394i $$0.390484\pi$$
$$762$$ −25.8614 −0.936860
$$763$$ −2.11684 −0.0766349
$$764$$ −76.4674 −2.76649
$$765$$ 0 0
$$766$$ 25.8333 0.933395
$$767$$ −0.589907 −0.0213003
$$768$$ −31.1168 −1.12283
$$769$$ 29.3523 1.05847 0.529235 0.848475i $$-0.322479\pi$$
0.529235 + 0.848475i $$0.322479\pi$$
$$770$$ 0 0
$$771$$ −17.4891 −0.629855
$$772$$ −92.4058 −3.32576
$$773$$ −0.510875 −0.0183749 −0.00918744 0.999958i $$-0.502924\pi$$
−0.00918744 + 0.999958i $$0.502924\pi$$
$$774$$ 25.1168 0.902806
$$775$$ 0 0
$$776$$ 56.1253 2.01478
$$777$$ −3.96143 −0.142116
$$778$$ 82.6583 2.96344
$$779$$ −30.5109 −1.09317
$$780$$ 0 0
$$781$$ 0 0
$$782$$ −133.957 −4.79027
$$783$$ 10.0974 0.360850
$$784$$ −40.6060 −1.45021
$$785$$ 0 0
$$786$$ −46.9783 −1.67566
$$787$$ −13.4140 −0.478157 −0.239078 0.971000i $$-0.576845\pi$$
−0.239078 + 0.971000i $$0.576845\pi$$
$$788$$ 37.2203 1.32592
$$789$$ −5.04868 −0.179738
$$790$$ 0 0
$$791$$ −12.8617 −0.457311
$$792$$ 0 0
$$793$$ −0.883156 −0.0313618
$$794$$ 6.63325 0.235405
$$795$$ 0 0
$$796$$ 73.7228 2.61304
$$797$$ 29.7228 1.05284 0.526418 0.850226i $$-0.323535\pi$$
0.526418 + 0.850226i $$0.323535\pi$$
$$798$$ −8.80773 −0.311790
$$799$$ 58.0049 2.05206
$$800$$ 0 0
$$801$$ 5.48913 0.193949
$$802$$ −78.1996 −2.76132
$$803$$ 0 0
$$804$$ 48.6060 1.71420
$$805$$ 0 0
$$806$$ −0.883156 −0.0311078
$$807$$ −2.74456 −0.0966132
$$808$$ −79.4456 −2.79489
$$809$$ −20.4897 −0.720378 −0.360189 0.932879i $$-0.617288\pi$$
−0.360189 + 0.932879i $$0.617288\pi$$
$$810$$ 0 0
$$811$$ −4.95610 −0.174032 −0.0870161 0.996207i $$-0.527733\pi$$
−0.0870161 + 0.996207i $$0.527733\pi$$
$$812$$ 34.9783 1.22750
$$813$$ −10.3923 −0.364474
$$814$$ 0 0
$$815$$ 0 0
$$816$$ 42.2689 1.47971
$$817$$ −43.8179 −1.53299
$$818$$ 96.7011 3.38107
$$819$$ −0.116844 −0.00408286
$$820$$ 0 0
$$821$$ −46.0280 −1.60639 −0.803194 0.595718i $$-0.796868\pi$$
−0.803194 + 0.595718i $$0.796868\pi$$
$$822$$ 1.87953 0.0655561
$$823$$ −2.35053 −0.0819344 −0.0409672 0.999160i $$-0.513044\pi$$
−0.0409672 + 0.999160i $$0.513044\pi$$
$$824$$ −22.4241 −0.781180
$$825$$ 0 0
$$826$$ 8.00000 0.278356
$$827$$ −16.1407 −0.561267 −0.280633 0.959815i $$-0.590545\pi$$
−0.280633 + 0.959815i $$0.590545\pi$$
$$828$$ −34.9783 −1.21558
$$829$$ 41.4674 1.44022 0.720111 0.693859i $$-0.244091\pi$$
0.720111 + 0.693859i $$0.244091\pi$$
$$830$$ 0 0
$$831$$ 20.5446 0.712683
$$832$$ 0.349857 0.0121291
$$833$$ −42.2689 −1.46453
$$834$$ −26.2337 −0.908398
$$835$$ 0 0
$$836$$ 0 0
$$837$$ 2.37228 0.0819980
$$838$$ −5.63858 −0.194782
$$839$$ 23.2554 0.802867 0.401433 0.915888i $$-0.368512\pi$$
0.401433 + 0.915888i $$0.368512\pi$$
$$840$$ 0 0
$$841$$ 72.9565 2.51574
$$842$$ 45.4381 1.56590
$$843$$ −8.51278 −0.293196
$$844$$ 19.8997 0.684978
$$845$$ 0 0
$$846$$ 22.0742 0.758928
$$847$$ 0 0
$$848$$ 8.00000 0.274721
$$849$$ −28.3576 −0.973231
$$850$$ 0 0
$$851$$ 40.0000 1.37118
$$852$$ 46.9783 1.60945
$$853$$ 10.4472 0.357706 0.178853 0.983876i $$-0.442761\pi$$
0.178853 + 0.983876i $$0.442761\pi$$
$$854$$ 11.9769 0.409840
$$855$$ 0 0
$$856$$ −49.2119 −1.68203
$$857$$ −42.5639 −1.45395 −0.726977 0.686661i $$-0.759075\pi$$
−0.726977 + 0.686661i $$0.759075\pi$$
$$858$$ 0 0
$$859$$ −10.2554 −0.349911 −0.174956 0.984576i $$-0.555978\pi$$
−0.174956 + 0.984576i $$0.555978\pi$$
$$860$$ 0 0
$$861$$ 5.48913 0.187069
$$862$$ 59.7228 2.03417
$$863$$ 20.2337 0.688763 0.344381 0.938830i $$-0.388089\pi$$
0.344381 + 0.938830i $$0.388089\pi$$
$$864$$ 4.10891 0.139788
$$865$$ 0 0
$$866$$ −101.214 −3.43938
$$867$$ 27.0000 0.916968
$$868$$ 8.21782 0.278931
$$869$$ 0 0
$$870$$ 0 0
$$871$$ −1.63948 −0.0555516
$$872$$ −16.0000 −0.541828
$$873$$ 9.37228 0.317204
$$874$$ 88.9348 3.00826
$$875$$ 0 0
$$876$$ −22.7190 −0.767605
$$877$$ −46.3778 −1.56607 −0.783034 0.621979i $$-0.786329\pi$$
−0.783034 + 0.621979i $$0.786329\pi$$
$$878$$ 18.0000 0.607471
$$879$$ −32.7615 −1.10502
$$880$$ 0 0
$$881$$ 46.4674 1.56553 0.782763 0.622320i $$-0.213810\pi$$
0.782763 + 0.622320i $$0.213810\pi$$
$$882$$ −16.0858 −0.541637
$$883$$ −11.1386 −0.374844 −0.187422 0.982280i $$-0.560013\pi$$
−0.187422 + 0.982280i $$0.560013\pi$$
$$884$$ −4.27719 −0.143857
$$885$$ 0 0
$$886$$ −23.3639 −0.784924
$$887$$ 21.4843 0.721373 0.360686 0.932687i $$-0.382542\pi$$
0.360686 + 0.932687i $$0.382542\pi$$
$$888$$ −29.9422 −1.00479
$$889$$ −8.11684 −0.272230
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 104.329 3.49319
$$893$$ −38.5099 −1.28868
$$894$$ −3.25544 −0.108878
$$895$$ 0 0
$$896$$ −11.2554 −0.376018
$$897$$ 1.17981 0.0393929
$$898$$ 32.1716 1.07358
$$899$$ 23.9538 0.798903
$$900$$ 0 0
$$901$$ 8.32763 0.277434
$$902$$ 0 0
$$903$$ 7.88316 0.262335
$$904$$ −97.2145 −3.23330
$$905$$ 0 0
$$906$$ −60.0951 −1.99653
$$907$$ 19.2337 0.638644 0.319322 0.947646i $$-0.396545\pi$$
0.319322 + 0.947646i $$0.396545\pi$$
$$908$$ −75.7301 −2.51319
$$909$$ −13.2665 −0.440022
$$910$$ 0 0
$$911$$ 4.51087 0.149452 0.0747260 0.997204i $$-0.476192\pi$$
0.0747260 + 0.997204i $$0.476192\pi$$
$$912$$ −28.0627 −0.929248
$$913$$ 0 0
$$914$$ −49.4891 −1.63695
$$915$$ 0 0
$$916$$ −45.8614 −1.51530
$$917$$ −14.7446 −0.486908
$$918$$ 16.7446 0.552653
$$919$$ 10.8896 0.359216 0.179608 0.983738i $$-0.442517\pi$$
0.179608 + 0.983738i $$0.442517\pi$$
$$920$$ 0 0
$$921$$ −26.4781 −0.872483
$$922$$ 4.74456 0.156254
$$923$$ −1.58457 −0.0521569
$$924$$ 0 0
$$925$$ 0 0
$$926$$ −68.1022 −2.23798
$$927$$ −3.74456 −0.122988
$$928$$ 41.4891 1.36195
$$929$$ 13.2554 0.434897 0.217448 0.976072i $$-0.430227\pi$$
0.217448 + 0.976072i $$0.430227\pi$$
$$930$$ 0 0
$$931$$ 28.0627 0.919717
$$932$$ 66.2227 2.16920
$$933$$ −10.2337 −0.335036
$$934$$ 3.75906 0.123000
$$935$$ 0 0
$$936$$ −0.883156 −0.0288669
$$937$$ −14.9436 −0.488188 −0.244094 0.969752i $$-0.578490\pi$$
−0.244094 + 0.969752i $$0.578490\pi$$
$$938$$ 22.2337 0.725956
$$939$$ 25.9783 0.847768
$$940$$ 0 0
$$941$$ 3.75906 0.122542 0.0612708 0.998121i $$-0.480485\pi$$
0.0612708 + 0.998121i $$0.480485\pi$$
$$942$$ −21.1345 −0.688598
$$943$$ −55.4256 −1.80491
$$944$$ 25.4891 0.829600
$$945$$ 0 0
$$946$$ 0 0
$$947$$ −21.4891 −0.698303 −0.349151 0.937066i $$-0.613530\pi$$
−0.349151 + 0.937066i $$0.613530\pi$$
$$948$$ −29.6472 −0.962898
$$949$$ 0.766312 0.0248755
$$950$$ 0 0
$$951$$ 18.9783 0.615412
$$952$$ 31.4719 1.02001
$$953$$ −25.2434 −0.817713 −0.408857 0.912599i $$-0.634072\pi$$
−0.408857 + 0.912599i $$0.634072\pi$$
$$954$$ 3.16915 0.102605
$$955$$ 0 0
$$956$$ 58.0049 1.87601
$$957$$ 0 0
$$958$$ 7.25544 0.234413
$$959$$ 0.589907 0.0190491
$$960$$ 0 0
$$961$$ −25.3723 −0.818461
$$962$$ 1.86141 0.0600142
$$963$$ −8.21782 −0.264816
$$964$$ −61.2289 −1.97205
$$965$$ 0 0
$$966$$ −16.0000 −0.514792
$$967$$ 7.96054 0.255994 0.127997 0.991775i $$-0.459145\pi$$
0.127997 + 0.991775i $$0.459145\pi$$
$$968$$ 0 0
$$969$$ −29.2119 −0.938423
$$970$$ 0 0
$$971$$ 49.4891 1.58818 0.794091 0.607799i $$-0.207947\pi$$
0.794091 + 0.607799i $$0.207947\pi$$
$$972$$ 4.37228 0.140241
$$973$$ −8.23369 −0.263960
$$974$$ 90.8213 2.91010
$$975$$ 0 0
$$976$$ 38.1600 1.22147
$$977$$ −38.7446 −1.23955 −0.619774 0.784780i $$-0.712776\pi$$
−0.619774 + 0.784780i $$0.712776\pi$$
$$978$$ −36.2805 −1.16012
$$979$$ 0 0
$$980$$ 0 0
$$981$$ −2.67181 −0.0853045
$$982$$ −16.7446 −0.534340
$$983$$ 37.4891 1.19572 0.597859 0.801602i $$-0.296018\pi$$
0.597859 + 0.801602i $$0.296018\pi$$
$$984$$ 41.4891 1.32263
$$985$$ 0 0
$$986$$ 169.076 5.38447
$$987$$ 6.92820 0.220527
$$988$$ 2.83966 0.0903415
$$989$$ −79.5990 −2.53110
$$990$$ 0 0
$$991$$ 46.9565 1.49162 0.745811 0.666157i $$-0.232062\pi$$
0.745811 + 0.666157i $$0.232062\pi$$
$$992$$ 9.74749 0.309483
$$993$$ 1.23369 0.0391499
$$994$$ 21.4891 0.681594
$$995$$ 0 0
$$996$$ 37.2203 1.17937
$$997$$ −0.552236 −0.0174895 −0.00874475 0.999962i $$-0.502784\pi$$
−0.00874475 + 0.999962i $$0.502784\pi$$
$$998$$ −35.6357 −1.12803
$$999$$ −5.00000 −0.158193
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 9075.2.a.db.1.4 yes 4
5.4 even 2 9075.2.a.cw.1.1 4
11.10 odd 2 inner 9075.2.a.db.1.1 yes 4
55.54 odd 2 9075.2.a.cw.1.4 yes 4

By twisted newform
Twist Min Dim Char Parity Ord Type
9075.2.a.cw.1.1 4 5.4 even 2
9075.2.a.cw.1.4 yes 4 55.54 odd 2
9075.2.a.db.1.1 yes 4 11.10 odd 2 inner
9075.2.a.db.1.4 yes 4 1.1 even 1 trivial