# Properties

 Label 5625.2.a.f.1.2 Level $5625$ Weight $2$ Character 5625.1 Self dual yes Analytic conductor $44.916$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$5625 = 3^{2} \cdot 5^{4}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 5625.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$44.9158511370$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\zeta_{10})^+$$ Defining polynomial: $$x^{2} - x - 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 25) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$1.61803$$ of defining polynomial Character $$\chi$$ $$=$$ 5625.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+1.61803 q^{2} +0.618034 q^{4} +0.618034 q^{7} -2.23607 q^{8} +O(q^{10})$$ $$q+1.61803 q^{2} +0.618034 q^{4} +0.618034 q^{7} -2.23607 q^{8} +5.23607 q^{11} +1.85410 q^{13} +1.00000 q^{14} -4.85410 q^{16} +5.23607 q^{17} +0.854102 q^{19} +8.47214 q^{22} -3.76393 q^{23} +3.00000 q^{26} +0.381966 q^{28} +3.61803 q^{29} -3.00000 q^{31} -3.38197 q^{32} +8.47214 q^{34} -0.236068 q^{37} +1.38197 q^{38} +0.763932 q^{41} -4.85410 q^{43} +3.23607 q^{44} -6.09017 q^{46} -0.618034 q^{47} -6.61803 q^{49} +1.14590 q^{52} +3.47214 q^{53} -1.38197 q^{56} +5.85410 q^{58} +10.8541 q^{59} +8.70820 q^{61} -4.85410 q^{62} +4.23607 q^{64} +4.76393 q^{67} +3.23607 q^{68} +6.61803 q^{71} -9.00000 q^{73} -0.381966 q^{74} +0.527864 q^{76} +3.23607 q^{77} -8.09017 q^{79} +1.23607 q^{82} +6.23607 q^{83} -7.85410 q^{86} -11.7082 q^{88} +8.94427 q^{89} +1.14590 q^{91} -2.32624 q^{92} -1.00000 q^{94} -3.85410 q^{97} -10.7082 q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + q^{2} - q^{4} - q^{7} + O(q^{10})$$ $$2 q + q^{2} - q^{4} - q^{7} + 6 q^{11} - 3 q^{13} + 2 q^{14} - 3 q^{16} + 6 q^{17} - 5 q^{19} + 8 q^{22} - 12 q^{23} + 6 q^{26} + 3 q^{28} + 5 q^{29} - 6 q^{31} - 9 q^{32} + 8 q^{34} + 4 q^{37} + 5 q^{38} + 6 q^{41} - 3 q^{43} + 2 q^{44} - q^{46} + q^{47} - 11 q^{49} + 9 q^{52} - 2 q^{53} - 5 q^{56} + 5 q^{58} + 15 q^{59} + 4 q^{61} - 3 q^{62} + 4 q^{64} + 14 q^{67} + 2 q^{68} + 11 q^{71} - 18 q^{73} - 3 q^{74} + 10 q^{76} + 2 q^{77} - 5 q^{79} - 2 q^{82} + 8 q^{83} - 9 q^{86} - 10 q^{88} + 9 q^{91} + 11 q^{92} - 2 q^{94} - q^{97} - 8 q^{98} + O(q^{100})$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 1.61803 1.14412 0.572061 0.820211i $$-0.306144\pi$$
0.572061 + 0.820211i $$0.306144\pi$$
$$3$$ 0 0
$$4$$ 0.618034 0.309017
$$5$$ 0 0
$$6$$ 0 0
$$7$$ 0.618034 0.233595 0.116797 0.993156i $$-0.462737\pi$$
0.116797 + 0.993156i $$0.462737\pi$$
$$8$$ −2.23607 −0.790569
$$9$$ 0 0
$$10$$ 0 0
$$11$$ 5.23607 1.57873 0.789367 0.613922i $$-0.210409\pi$$
0.789367 + 0.613922i $$0.210409\pi$$
$$12$$ 0 0
$$13$$ 1.85410 0.514235 0.257118 0.966380i $$-0.417227\pi$$
0.257118 + 0.966380i $$0.417227\pi$$
$$14$$ 1.00000 0.267261
$$15$$ 0 0
$$16$$ −4.85410 −1.21353
$$17$$ 5.23607 1.26993 0.634967 0.772540i $$-0.281014\pi$$
0.634967 + 0.772540i $$0.281014\pi$$
$$18$$ 0 0
$$19$$ 0.854102 0.195944 0.0979722 0.995189i $$-0.468764\pi$$
0.0979722 + 0.995189i $$0.468764\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 8.47214 1.80627
$$23$$ −3.76393 −0.784834 −0.392417 0.919787i $$-0.628361\pi$$
−0.392417 + 0.919787i $$0.628361\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ 3.00000 0.588348
$$27$$ 0 0
$$28$$ 0.381966 0.0721848
$$29$$ 3.61803 0.671852 0.335926 0.941888i $$-0.390951\pi$$
0.335926 + 0.941888i $$0.390951\pi$$
$$30$$ 0 0
$$31$$ −3.00000 −0.538816 −0.269408 0.963026i $$-0.586828\pi$$
−0.269408 + 0.963026i $$0.586828\pi$$
$$32$$ −3.38197 −0.597853
$$33$$ 0 0
$$34$$ 8.47214 1.45296
$$35$$ 0 0
$$36$$ 0 0
$$37$$ −0.236068 −0.0388093 −0.0194047 0.999812i $$-0.506177\pi$$
−0.0194047 + 0.999812i $$0.506177\pi$$
$$38$$ 1.38197 0.224184
$$39$$ 0 0
$$40$$ 0 0
$$41$$ 0.763932 0.119306 0.0596531 0.998219i $$-0.481001\pi$$
0.0596531 + 0.998219i $$0.481001\pi$$
$$42$$ 0 0
$$43$$ −4.85410 −0.740244 −0.370122 0.928983i $$-0.620684\pi$$
−0.370122 + 0.928983i $$0.620684\pi$$
$$44$$ 3.23607 0.487856
$$45$$ 0 0
$$46$$ −6.09017 −0.897947
$$47$$ −0.618034 −0.0901495 −0.0450748 0.998984i $$-0.514353\pi$$
−0.0450748 + 0.998984i $$0.514353\pi$$
$$48$$ 0 0
$$49$$ −6.61803 −0.945433
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 1.14590 0.158907
$$53$$ 3.47214 0.476935 0.238467 0.971151i $$-0.423355\pi$$
0.238467 + 0.971151i $$0.423355\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ −1.38197 −0.184673
$$57$$ 0 0
$$58$$ 5.85410 0.768681
$$59$$ 10.8541 1.41308 0.706542 0.707671i $$-0.250254\pi$$
0.706542 + 0.707671i $$0.250254\pi$$
$$60$$ 0 0
$$61$$ 8.70820 1.11497 0.557486 0.830187i $$-0.311766\pi$$
0.557486 + 0.830187i $$0.311766\pi$$
$$62$$ −4.85410 −0.616472
$$63$$ 0 0
$$64$$ 4.23607 0.529508
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 4.76393 0.582007 0.291003 0.956722i $$-0.406011\pi$$
0.291003 + 0.956722i $$0.406011\pi$$
$$68$$ 3.23607 0.392431
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 6.61803 0.785416 0.392708 0.919663i $$-0.371538\pi$$
0.392708 + 0.919663i $$0.371538\pi$$
$$72$$ 0 0
$$73$$ −9.00000 −1.05337 −0.526685 0.850060i $$-0.676565\pi$$
−0.526685 + 0.850060i $$0.676565\pi$$
$$74$$ −0.381966 −0.0444026
$$75$$ 0 0
$$76$$ 0.527864 0.0605502
$$77$$ 3.23607 0.368784
$$78$$ 0 0
$$79$$ −8.09017 −0.910215 −0.455108 0.890436i $$-0.650399\pi$$
−0.455108 + 0.890436i $$0.650399\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ 1.23607 0.136501
$$83$$ 6.23607 0.684497 0.342249 0.939609i $$-0.388811\pi$$
0.342249 + 0.939609i $$0.388811\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ −7.85410 −0.846930
$$87$$ 0 0
$$88$$ −11.7082 −1.24810
$$89$$ 8.94427 0.948091 0.474045 0.880500i $$-0.342793\pi$$
0.474045 + 0.880500i $$0.342793\pi$$
$$90$$ 0 0
$$91$$ 1.14590 0.120123
$$92$$ −2.32624 −0.242527
$$93$$ 0 0
$$94$$ −1.00000 −0.103142
$$95$$ 0 0
$$96$$ 0 0
$$97$$ −3.85410 −0.391325 −0.195662 0.980671i $$-0.562686\pi$$
−0.195662 + 0.980671i $$0.562686\pi$$
$$98$$ −10.7082 −1.08169
$$99$$ 0 0
$$100$$ 0 0
$$101$$ −1.47214 −0.146483 −0.0732415 0.997314i $$-0.523334\pi$$
−0.0732415 + 0.997314i $$0.523334\pi$$
$$102$$ 0 0
$$103$$ 8.56231 0.843669 0.421835 0.906673i $$-0.361386\pi$$
0.421835 + 0.906673i $$0.361386\pi$$
$$104$$ −4.14590 −0.406539
$$105$$ 0 0
$$106$$ 5.61803 0.545672
$$107$$ 16.4164 1.58703 0.793517 0.608548i $$-0.208248\pi$$
0.793517 + 0.608548i $$0.208248\pi$$
$$108$$ 0 0
$$109$$ 10.0000 0.957826 0.478913 0.877862i $$-0.341031\pi$$
0.478913 + 0.877862i $$0.341031\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ −3.00000 −0.283473
$$113$$ −16.8541 −1.58550 −0.792750 0.609547i $$-0.791352\pi$$
−0.792750 + 0.609547i $$0.791352\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 2.23607 0.207614
$$117$$ 0 0
$$118$$ 17.5623 1.61674
$$119$$ 3.23607 0.296650
$$120$$ 0 0
$$121$$ 16.4164 1.49240
$$122$$ 14.0902 1.27566
$$123$$ 0 0
$$124$$ −1.85410 −0.166503
$$125$$ 0 0
$$126$$ 0 0
$$127$$ 19.8885 1.76482 0.882411 0.470479i $$-0.155919\pi$$
0.882411 + 0.470479i $$0.155919\pi$$
$$128$$ 13.6180 1.20368
$$129$$ 0 0
$$130$$ 0 0
$$131$$ −6.79837 −0.593977 −0.296988 0.954881i $$-0.595982\pi$$
−0.296988 + 0.954881i $$0.595982\pi$$
$$132$$ 0 0
$$133$$ 0.527864 0.0457716
$$134$$ 7.70820 0.665887
$$135$$ 0 0
$$136$$ −11.7082 −1.00397
$$137$$ 11.9443 1.02047 0.510234 0.860036i $$-0.329559\pi$$
0.510234 + 0.860036i $$0.329559\pi$$
$$138$$ 0 0
$$139$$ 5.00000 0.424094 0.212047 0.977259i $$-0.431987\pi$$
0.212047 + 0.977259i $$0.431987\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 10.7082 0.898613
$$143$$ 9.70820 0.811841
$$144$$ 0 0
$$145$$ 0 0
$$146$$ −14.5623 −1.20519
$$147$$ 0 0
$$148$$ −0.145898 −0.0119927
$$149$$ 3.94427 0.323127 0.161564 0.986862i $$-0.448346\pi$$
0.161564 + 0.986862i $$0.448346\pi$$
$$150$$ 0 0
$$151$$ 14.5623 1.18506 0.592532 0.805547i $$-0.298128\pi$$
0.592532 + 0.805547i $$0.298128\pi$$
$$152$$ −1.90983 −0.154908
$$153$$ 0 0
$$154$$ 5.23607 0.421934
$$155$$ 0 0
$$156$$ 0 0
$$157$$ 13.1803 1.05191 0.525953 0.850514i $$-0.323709\pi$$
0.525953 + 0.850514i $$0.323709\pi$$
$$158$$ −13.0902 −1.04140
$$159$$ 0 0
$$160$$ 0 0
$$161$$ −2.32624 −0.183333
$$162$$ 0 0
$$163$$ 11.0000 0.861586 0.430793 0.902451i $$-0.358234\pi$$
0.430793 + 0.902451i $$0.358234\pi$$
$$164$$ 0.472136 0.0368676
$$165$$ 0 0
$$166$$ 10.0902 0.783149
$$167$$ −14.5623 −1.12687 −0.563433 0.826162i $$-0.690520\pi$$
−0.563433 + 0.826162i $$0.690520\pi$$
$$168$$ 0 0
$$169$$ −9.56231 −0.735562
$$170$$ 0 0
$$171$$ 0 0
$$172$$ −3.00000 −0.228748
$$173$$ −18.8885 −1.43607 −0.718035 0.696007i $$-0.754958\pi$$
−0.718035 + 0.696007i $$0.754958\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ −25.4164 −1.91583
$$177$$ 0 0
$$178$$ 14.4721 1.08473
$$179$$ 0.527864 0.0394544 0.0197272 0.999805i $$-0.493720\pi$$
0.0197272 + 0.999805i $$0.493720\pi$$
$$180$$ 0 0
$$181$$ 0.291796 0.0216890 0.0108445 0.999941i $$-0.496548\pi$$
0.0108445 + 0.999941i $$0.496548\pi$$
$$182$$ 1.85410 0.137435
$$183$$ 0 0
$$184$$ 8.41641 0.620466
$$185$$ 0 0
$$186$$ 0 0
$$187$$ 27.4164 2.00489
$$188$$ −0.381966 −0.0278577
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 1.81966 0.131666 0.0658330 0.997831i $$-0.479030\pi$$
0.0658330 + 0.997831i $$0.479030\pi$$
$$192$$ 0 0
$$193$$ 7.70820 0.554849 0.277424 0.960747i $$-0.410519\pi$$
0.277424 + 0.960747i $$0.410519\pi$$
$$194$$ −6.23607 −0.447724
$$195$$ 0 0
$$196$$ −4.09017 −0.292155
$$197$$ −3.70820 −0.264199 −0.132099 0.991236i $$-0.542172\pi$$
−0.132099 + 0.991236i $$0.542172\pi$$
$$198$$ 0 0
$$199$$ −17.5623 −1.24496 −0.622479 0.782636i $$-0.713875\pi$$
−0.622479 + 0.782636i $$0.713875\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ −2.38197 −0.167595
$$203$$ 2.23607 0.156941
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 13.8541 0.965261
$$207$$ 0 0
$$208$$ −9.00000 −0.624038
$$209$$ 4.47214 0.309344
$$210$$ 0 0
$$211$$ −9.18034 −0.632001 −0.316000 0.948759i $$-0.602340\pi$$
−0.316000 + 0.948759i $$0.602340\pi$$
$$212$$ 2.14590 0.147381
$$213$$ 0 0
$$214$$ 26.5623 1.81576
$$215$$ 0 0
$$216$$ 0 0
$$217$$ −1.85410 −0.125865
$$218$$ 16.1803 1.09587
$$219$$ 0 0
$$220$$ 0 0
$$221$$ 9.70820 0.653044
$$222$$ 0 0
$$223$$ −0.180340 −0.0120765 −0.00603823 0.999982i $$-0.501922\pi$$
−0.00603823 + 0.999982i $$0.501922\pi$$
$$224$$ −2.09017 −0.139655
$$225$$ 0 0
$$226$$ −27.2705 −1.81401
$$227$$ −14.7639 −0.979917 −0.489958 0.871746i $$-0.662988\pi$$
−0.489958 + 0.871746i $$0.662988\pi$$
$$228$$ 0 0
$$229$$ 21.7082 1.43452 0.717259 0.696806i $$-0.245396\pi$$
0.717259 + 0.696806i $$0.245396\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ −8.09017 −0.531146
$$233$$ 2.94427 0.192886 0.0964428 0.995339i $$-0.469254\pi$$
0.0964428 + 0.995339i $$0.469254\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 6.70820 0.436667
$$237$$ 0 0
$$238$$ 5.23607 0.339404
$$239$$ 20.5279 1.32784 0.663919 0.747805i $$-0.268892\pi$$
0.663919 + 0.747805i $$0.268892\pi$$
$$240$$ 0 0
$$241$$ 2.52786 0.162834 0.0814170 0.996680i $$-0.474055\pi$$
0.0814170 + 0.996680i $$0.474055\pi$$
$$242$$ 26.5623 1.70749
$$243$$ 0 0
$$244$$ 5.38197 0.344545
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 1.58359 0.100762
$$248$$ 6.70820 0.425971
$$249$$ 0 0
$$250$$ 0 0
$$251$$ 29.1803 1.84185 0.920923 0.389744i $$-0.127436\pi$$
0.920923 + 0.389744i $$0.127436\pi$$
$$252$$ 0 0
$$253$$ −19.7082 −1.23904
$$254$$ 32.1803 2.01917
$$255$$ 0 0
$$256$$ 13.5623 0.847644
$$257$$ −22.8541 −1.42560 −0.712800 0.701367i $$-0.752573\pi$$
−0.712800 + 0.701367i $$0.752573\pi$$
$$258$$ 0 0
$$259$$ −0.145898 −0.00906566
$$260$$ 0 0
$$261$$ 0 0
$$262$$ −11.0000 −0.679582
$$263$$ 10.9098 0.672729 0.336364 0.941732i $$-0.390803\pi$$
0.336364 + 0.941732i $$0.390803\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 0.854102 0.0523684
$$267$$ 0 0
$$268$$ 2.94427 0.179850
$$269$$ 12.7639 0.778231 0.389115 0.921189i $$-0.372781\pi$$
0.389115 + 0.921189i $$0.372781\pi$$
$$270$$ 0 0
$$271$$ −8.00000 −0.485965 −0.242983 0.970031i $$-0.578126\pi$$
−0.242983 + 0.970031i $$0.578126\pi$$
$$272$$ −25.4164 −1.54110
$$273$$ 0 0
$$274$$ 19.3262 1.16754
$$275$$ 0 0
$$276$$ 0 0
$$277$$ −24.7082 −1.48457 −0.742286 0.670083i $$-0.766258\pi$$
−0.742286 + 0.670083i $$0.766258\pi$$
$$278$$ 8.09017 0.485216
$$279$$ 0 0
$$280$$ 0 0
$$281$$ −10.0902 −0.601929 −0.300965 0.953635i $$-0.597309\pi$$
−0.300965 + 0.953635i $$0.597309\pi$$
$$282$$ 0 0
$$283$$ −29.8541 −1.77464 −0.887321 0.461152i $$-0.847436\pi$$
−0.887321 + 0.461152i $$0.847436\pi$$
$$284$$ 4.09017 0.242707
$$285$$ 0 0
$$286$$ 15.7082 0.928846
$$287$$ 0.472136 0.0278693
$$288$$ 0 0
$$289$$ 10.4164 0.612730
$$290$$ 0 0
$$291$$ 0 0
$$292$$ −5.56231 −0.325509
$$293$$ 19.5279 1.14083 0.570415 0.821357i $$-0.306782\pi$$
0.570415 + 0.821357i $$0.306782\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 0.527864 0.0306815
$$297$$ 0 0
$$298$$ 6.38197 0.369697
$$299$$ −6.97871 −0.403589
$$300$$ 0 0
$$301$$ −3.00000 −0.172917
$$302$$ 23.5623 1.35586
$$303$$ 0 0
$$304$$ −4.14590 −0.237784
$$305$$ 0 0
$$306$$ 0 0
$$307$$ 9.23607 0.527130 0.263565 0.964642i $$-0.415102\pi$$
0.263565 + 0.964642i $$0.415102\pi$$
$$308$$ 2.00000 0.113961
$$309$$ 0 0
$$310$$ 0 0
$$311$$ −8.50658 −0.482364 −0.241182 0.970480i $$-0.577535\pi$$
−0.241182 + 0.970480i $$0.577535\pi$$
$$312$$ 0 0
$$313$$ −16.7639 −0.947553 −0.473777 0.880645i $$-0.657110\pi$$
−0.473777 + 0.880645i $$0.657110\pi$$
$$314$$ 21.3262 1.20351
$$315$$ 0 0
$$316$$ −5.00000 −0.281272
$$317$$ −7.65248 −0.429806 −0.214903 0.976635i $$-0.568944\pi$$
−0.214903 + 0.976635i $$0.568944\pi$$
$$318$$ 0 0
$$319$$ 18.9443 1.06068
$$320$$ 0 0
$$321$$ 0 0
$$322$$ −3.76393 −0.209756
$$323$$ 4.47214 0.248836
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 17.7984 0.985761
$$327$$ 0 0
$$328$$ −1.70820 −0.0943198
$$329$$ −0.381966 −0.0210585
$$330$$ 0 0
$$331$$ −23.1246 −1.27104 −0.635522 0.772083i $$-0.719215\pi$$
−0.635522 + 0.772083i $$0.719215\pi$$
$$332$$ 3.85410 0.211521
$$333$$ 0 0
$$334$$ −23.5623 −1.28927
$$335$$ 0 0
$$336$$ 0 0
$$337$$ 7.85410 0.427840 0.213920 0.976851i $$-0.431377\pi$$
0.213920 + 0.976851i $$0.431377\pi$$
$$338$$ −15.4721 −0.841573
$$339$$ 0 0
$$340$$ 0 0
$$341$$ −15.7082 −0.850647
$$342$$ 0 0
$$343$$ −8.41641 −0.454443
$$344$$ 10.8541 0.585214
$$345$$ 0 0
$$346$$ −30.5623 −1.64304
$$347$$ 19.9098 1.06882 0.534408 0.845227i $$-0.320535\pi$$
0.534408 + 0.845227i $$0.320535\pi$$
$$348$$ 0 0
$$349$$ 21.7082 1.16201 0.581007 0.813899i $$-0.302659\pi$$
0.581007 + 0.813899i $$0.302659\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ −17.7082 −0.943850
$$353$$ −12.9098 −0.687121 −0.343560 0.939131i $$-0.611633\pi$$
−0.343560 + 0.939131i $$0.611633\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 5.52786 0.292976
$$357$$ 0 0
$$358$$ 0.854102 0.0451407
$$359$$ −13.7426 −0.725309 −0.362655 0.931924i $$-0.618130\pi$$
−0.362655 + 0.931924i $$0.618130\pi$$
$$360$$ 0 0
$$361$$ −18.2705 −0.961606
$$362$$ 0.472136 0.0248149
$$363$$ 0 0
$$364$$ 0.708204 0.0371200
$$365$$ 0 0
$$366$$ 0 0
$$367$$ −25.5623 −1.33434 −0.667171 0.744905i $$-0.732495\pi$$
−0.667171 + 0.744905i $$0.732495\pi$$
$$368$$ 18.2705 0.952416
$$369$$ 0 0
$$370$$ 0 0
$$371$$ 2.14590 0.111409
$$372$$ 0 0
$$373$$ −28.2705 −1.46379 −0.731896 0.681417i $$-0.761364\pi$$
−0.731896 + 0.681417i $$0.761364\pi$$
$$374$$ 44.3607 2.29384
$$375$$ 0 0
$$376$$ 1.38197 0.0712695
$$377$$ 6.70820 0.345490
$$378$$ 0 0
$$379$$ 14.5967 0.749785 0.374892 0.927068i $$-0.377680\pi$$
0.374892 + 0.927068i $$0.377680\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 2.94427 0.150642
$$383$$ −33.3607 −1.70465 −0.852326 0.523012i $$-0.824808\pi$$
−0.852326 + 0.523012i $$0.824808\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 12.4721 0.634815
$$387$$ 0 0
$$388$$ −2.38197 −0.120926
$$389$$ −15.0000 −0.760530 −0.380265 0.924878i $$-0.624167\pi$$
−0.380265 + 0.924878i $$0.624167\pi$$
$$390$$ 0 0
$$391$$ −19.7082 −0.996687
$$392$$ 14.7984 0.747431
$$393$$ 0 0
$$394$$ −6.00000 −0.302276
$$395$$ 0 0
$$396$$ 0 0
$$397$$ 29.0344 1.45720 0.728598 0.684941i $$-0.240172\pi$$
0.728598 + 0.684941i $$0.240172\pi$$
$$398$$ −28.4164 −1.42439
$$399$$ 0 0
$$400$$ 0 0
$$401$$ −26.5967 −1.32818 −0.664089 0.747653i $$-0.731180\pi$$
−0.664089 + 0.747653i $$0.731180\pi$$
$$402$$ 0 0
$$403$$ −5.56231 −0.277078
$$404$$ −0.909830 −0.0452657
$$405$$ 0 0
$$406$$ 3.61803 0.179560
$$407$$ −1.23607 −0.0612696
$$408$$ 0 0
$$409$$ 1.58359 0.0783036 0.0391518 0.999233i $$-0.487534\pi$$
0.0391518 + 0.999233i $$0.487534\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 5.29180 0.260708
$$413$$ 6.70820 0.330089
$$414$$ 0 0
$$415$$ 0 0
$$416$$ −6.27051 −0.307437
$$417$$ 0 0
$$418$$ 7.23607 0.353928
$$419$$ 9.47214 0.462744 0.231372 0.972865i $$-0.425679\pi$$
0.231372 + 0.972865i $$0.425679\pi$$
$$420$$ 0 0
$$421$$ 32.0000 1.55958 0.779792 0.626038i $$-0.215325\pi$$
0.779792 + 0.626038i $$0.215325\pi$$
$$422$$ −14.8541 −0.723086
$$423$$ 0 0
$$424$$ −7.76393 −0.377050
$$425$$ 0 0
$$426$$ 0 0
$$427$$ 5.38197 0.260452
$$428$$ 10.1459 0.490420
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 29.8328 1.43700 0.718498 0.695529i $$-0.244830\pi$$
0.718498 + 0.695529i $$0.244830\pi$$
$$432$$ 0 0
$$433$$ 26.8541 1.29053 0.645263 0.763961i $$-0.276748\pi$$
0.645263 + 0.763961i $$0.276748\pi$$
$$434$$ −3.00000 −0.144005
$$435$$ 0 0
$$436$$ 6.18034 0.295985
$$437$$ −3.21478 −0.153784
$$438$$ 0 0
$$439$$ −40.9787 −1.95581 −0.977904 0.209056i $$-0.932961\pi$$
−0.977904 + 0.209056i $$0.932961\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 15.7082 0.747163
$$443$$ −29.9443 −1.42270 −0.711348 0.702840i $$-0.751915\pi$$
−0.711348 + 0.702840i $$0.751915\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ −0.291796 −0.0138169
$$447$$ 0 0
$$448$$ 2.61803 0.123690
$$449$$ −4.67376 −0.220568 −0.110284 0.993900i $$-0.535176\pi$$
−0.110284 + 0.993900i $$0.535176\pi$$
$$450$$ 0 0
$$451$$ 4.00000 0.188353
$$452$$ −10.4164 −0.489947
$$453$$ 0 0
$$454$$ −23.8885 −1.12114
$$455$$ 0 0
$$456$$ 0 0
$$457$$ −21.4164 −1.00182 −0.500909 0.865500i $$-0.667001\pi$$
−0.500909 + 0.865500i $$0.667001\pi$$
$$458$$ 35.1246 1.64127
$$459$$ 0 0
$$460$$ 0 0
$$461$$ −0.819660 −0.0381754 −0.0190877 0.999818i $$-0.506076\pi$$
−0.0190877 + 0.999818i $$0.506076\pi$$
$$462$$ 0 0
$$463$$ −24.1246 −1.12117 −0.560583 0.828098i $$-0.689423\pi$$
−0.560583 + 0.828098i $$0.689423\pi$$
$$464$$ −17.5623 −0.815310
$$465$$ 0 0
$$466$$ 4.76393 0.220685
$$467$$ −27.4508 −1.27027 −0.635137 0.772400i $$-0.719056\pi$$
−0.635137 + 0.772400i $$0.719056\pi$$
$$468$$ 0 0
$$469$$ 2.94427 0.135954
$$470$$ 0 0
$$471$$ 0 0
$$472$$ −24.2705 −1.11714
$$473$$ −25.4164 −1.16865
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 2.00000 0.0916698
$$477$$ 0 0
$$478$$ 33.2148 1.51921
$$479$$ 10.8541 0.495937 0.247968 0.968768i $$-0.420237\pi$$
0.247968 + 0.968768i $$0.420237\pi$$
$$480$$ 0 0
$$481$$ −0.437694 −0.0199571
$$482$$ 4.09017 0.186302
$$483$$ 0 0
$$484$$ 10.1459 0.461177
$$485$$ 0 0
$$486$$ 0 0
$$487$$ −36.4164 −1.65018 −0.825092 0.564998i $$-0.808877\pi$$
−0.825092 + 0.564998i $$0.808877\pi$$
$$488$$ −19.4721 −0.881462
$$489$$ 0 0
$$490$$ 0 0
$$491$$ 43.2492 1.95181 0.975905 0.218196i $$-0.0700171\pi$$
0.975905 + 0.218196i $$0.0700171\pi$$
$$492$$ 0 0
$$493$$ 18.9443 0.853207
$$494$$ 2.56231 0.115284
$$495$$ 0 0
$$496$$ 14.5623 0.653867
$$497$$ 4.09017 0.183469
$$498$$ 0 0
$$499$$ 7.56231 0.338535 0.169268 0.985570i $$-0.445860\pi$$
0.169268 + 0.985570i $$0.445860\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 47.2148 2.10730
$$503$$ 37.4164 1.66832 0.834158 0.551526i $$-0.185954\pi$$
0.834158 + 0.551526i $$0.185954\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ −31.8885 −1.41762
$$507$$ 0 0
$$508$$ 12.2918 0.545360
$$509$$ 20.3262 0.900945 0.450472 0.892790i $$-0.351256\pi$$
0.450472 + 0.892790i $$0.351256\pi$$
$$510$$ 0 0
$$511$$ −5.56231 −0.246062
$$512$$ −5.29180 −0.233867
$$513$$ 0 0
$$514$$ −36.9787 −1.63106
$$515$$ 0 0
$$516$$ 0 0
$$517$$ −3.23607 −0.142322
$$518$$ −0.236068 −0.0103722
$$519$$ 0 0
$$520$$ 0 0
$$521$$ −29.3607 −1.28631 −0.643157 0.765734i $$-0.722376\pi$$
−0.643157 + 0.765734i $$0.722376\pi$$
$$522$$ 0 0
$$523$$ −13.1459 −0.574830 −0.287415 0.957806i $$-0.592796\pi$$
−0.287415 + 0.957806i $$0.592796\pi$$
$$524$$ −4.20163 −0.183549
$$525$$ 0 0
$$526$$ 17.6525 0.769685
$$527$$ −15.7082 −0.684260
$$528$$ 0 0
$$529$$ −8.83282 −0.384035
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 0.326238 0.0141442
$$533$$ 1.41641 0.0613514
$$534$$ 0 0
$$535$$ 0 0
$$536$$ −10.6525 −0.460117
$$537$$ 0 0
$$538$$ 20.6525 0.890391
$$539$$ −34.6525 −1.49259
$$540$$ 0 0
$$541$$ 27.1246 1.16618 0.583089 0.812408i $$-0.301844\pi$$
0.583089 + 0.812408i $$0.301844\pi$$
$$542$$ −12.9443 −0.556004
$$543$$ 0 0
$$544$$ −17.7082 −0.759233
$$545$$ 0 0
$$546$$ 0 0
$$547$$ −21.2918 −0.910371 −0.455186 0.890397i $$-0.650427\pi$$
−0.455186 + 0.890397i $$0.650427\pi$$
$$548$$ 7.38197 0.315342
$$549$$ 0 0
$$550$$ 0 0
$$551$$ 3.09017 0.131646
$$552$$ 0 0
$$553$$ −5.00000 −0.212622
$$554$$ −39.9787 −1.69853
$$555$$ 0 0
$$556$$ 3.09017 0.131052
$$557$$ −4.76393 −0.201854 −0.100927 0.994894i $$-0.532181\pi$$
−0.100927 + 0.994894i $$0.532181\pi$$
$$558$$ 0 0
$$559$$ −9.00000 −0.380659
$$560$$ 0 0
$$561$$ 0 0
$$562$$ −16.3262 −0.688681
$$563$$ −7.38197 −0.311113 −0.155556 0.987827i $$-0.549717\pi$$
−0.155556 + 0.987827i $$0.549717\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ −48.3050 −2.03041
$$567$$ 0 0
$$568$$ −14.7984 −0.620926
$$569$$ −20.5279 −0.860573 −0.430286 0.902692i $$-0.641587\pi$$
−0.430286 + 0.902692i $$0.641587\pi$$
$$570$$ 0 0
$$571$$ −8.12461 −0.340004 −0.170002 0.985444i $$-0.554377\pi$$
−0.170002 + 0.985444i $$0.554377\pi$$
$$572$$ 6.00000 0.250873
$$573$$ 0 0
$$574$$ 0.763932 0.0318859
$$575$$ 0 0
$$576$$ 0 0
$$577$$ −33.7771 −1.40616 −0.703079 0.711111i $$-0.748192\pi$$
−0.703079 + 0.711111i $$0.748192\pi$$
$$578$$ 16.8541 0.701038
$$579$$ 0 0
$$580$$ 0 0
$$581$$ 3.85410 0.159895
$$582$$ 0 0
$$583$$ 18.1803 0.752953
$$584$$ 20.1246 0.832762
$$585$$ 0 0
$$586$$ 31.5967 1.30525
$$587$$ −5.29180 −0.218416 −0.109208 0.994019i $$-0.534831\pi$$
−0.109208 + 0.994019i $$0.534831\pi$$
$$588$$ 0 0
$$589$$ −2.56231 −0.105578
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 1.14590 0.0470961
$$593$$ 10.9098 0.448013 0.224007 0.974588i $$-0.428086\pi$$
0.224007 + 0.974588i $$0.428086\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 2.43769 0.0998518
$$597$$ 0 0
$$598$$ −11.2918 −0.461756
$$599$$ 9.47214 0.387021 0.193510 0.981098i $$-0.438013\pi$$
0.193510 + 0.981098i $$0.438013\pi$$
$$600$$ 0 0
$$601$$ 2.72949 0.111338 0.0556691 0.998449i $$-0.482271\pi$$
0.0556691 + 0.998449i $$0.482271\pi$$
$$602$$ −4.85410 −0.197838
$$603$$ 0 0
$$604$$ 9.00000 0.366205
$$605$$ 0 0
$$606$$ 0 0
$$607$$ −35.5623 −1.44343 −0.721715 0.692191i $$-0.756646\pi$$
−0.721715 + 0.692191i $$0.756646\pi$$
$$608$$ −2.88854 −0.117146
$$609$$ 0 0
$$610$$ 0 0
$$611$$ −1.14590 −0.0463581
$$612$$ 0 0
$$613$$ −14.9787 −0.604985 −0.302492 0.953152i $$-0.597819\pi$$
−0.302492 + 0.953152i $$0.597819\pi$$
$$614$$ 14.9443 0.603102
$$615$$ 0 0
$$616$$ −7.23607 −0.291549
$$617$$ −14.2361 −0.573123 −0.286561 0.958062i $$-0.592512\pi$$
−0.286561 + 0.958062i $$0.592512\pi$$
$$618$$ 0 0
$$619$$ 30.5279 1.22702 0.613509 0.789688i $$-0.289757\pi$$
0.613509 + 0.789688i $$0.289757\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ −13.7639 −0.551883
$$623$$ 5.52786 0.221469
$$624$$ 0 0
$$625$$ 0 0
$$626$$ −27.1246 −1.08412
$$627$$ 0 0
$$628$$ 8.14590 0.325057
$$629$$ −1.23607 −0.0492853
$$630$$ 0 0
$$631$$ −10.2361 −0.407491 −0.203746 0.979024i $$-0.565312\pi$$
−0.203746 + 0.979024i $$0.565312\pi$$
$$632$$ 18.0902 0.719588
$$633$$ 0 0
$$634$$ −12.3820 −0.491751
$$635$$ 0 0
$$636$$ 0 0
$$637$$ −12.2705 −0.486175
$$638$$ 30.6525 1.21354
$$639$$ 0 0
$$640$$ 0 0
$$641$$ 1.09017 0.0430591 0.0215296 0.999768i $$-0.493146\pi$$
0.0215296 + 0.999768i $$0.493146\pi$$
$$642$$ 0 0
$$643$$ −30.8328 −1.21593 −0.607964 0.793965i $$-0.708013\pi$$
−0.607964 + 0.793965i $$0.708013\pi$$
$$644$$ −1.43769 −0.0566531
$$645$$ 0 0
$$646$$ 7.23607 0.284699
$$647$$ 36.5410 1.43658 0.718288 0.695746i $$-0.244926\pi$$
0.718288 + 0.695746i $$0.244926\pi$$
$$648$$ 0 0
$$649$$ 56.8328 2.23088
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 6.79837 0.266245
$$653$$ −19.0902 −0.747056 −0.373528 0.927619i $$-0.621852\pi$$
−0.373528 + 0.927619i $$0.621852\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ −3.70820 −0.144781
$$657$$ 0 0
$$658$$ −0.618034 −0.0240935
$$659$$ −15.5279 −0.604880 −0.302440 0.953168i $$-0.597801\pi$$
−0.302440 + 0.953168i $$0.597801\pi$$
$$660$$ 0 0
$$661$$ 19.6869 0.765732 0.382866 0.923804i $$-0.374937\pi$$
0.382866 + 0.923804i $$0.374937\pi$$
$$662$$ −37.4164 −1.45423
$$663$$ 0 0
$$664$$ −13.9443 −0.541143
$$665$$ 0 0
$$666$$ 0 0
$$667$$ −13.6180 −0.527292
$$668$$ −9.00000 −0.348220
$$669$$ 0 0
$$670$$ 0 0
$$671$$ 45.5967 1.76024
$$672$$ 0 0
$$673$$ 12.1803 0.469518 0.234759 0.972054i $$-0.424570\pi$$
0.234759 + 0.972054i $$0.424570\pi$$
$$674$$ 12.7082 0.489502
$$675$$ 0 0
$$676$$ −5.90983 −0.227301
$$677$$ −10.6180 −0.408084 −0.204042 0.978962i $$-0.565408\pi$$
−0.204042 + 0.978962i $$0.565408\pi$$
$$678$$ 0 0
$$679$$ −2.38197 −0.0914115
$$680$$ 0 0
$$681$$ 0 0
$$682$$ −25.4164 −0.973245
$$683$$ 13.4721 0.515497 0.257748 0.966212i $$-0.417019\pi$$
0.257748 + 0.966212i $$0.417019\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ −13.6180 −0.519939
$$687$$ 0 0
$$688$$ 23.5623 0.898304
$$689$$ 6.43769 0.245257
$$690$$ 0 0
$$691$$ 36.2705 1.37980 0.689898 0.723907i $$-0.257656\pi$$
0.689898 + 0.723907i $$0.257656\pi$$
$$692$$ −11.6738 −0.443770
$$693$$ 0 0
$$694$$ 32.2148 1.22286
$$695$$ 0 0
$$696$$ 0 0
$$697$$ 4.00000 0.151511
$$698$$ 35.1246 1.32949
$$699$$ 0 0
$$700$$ 0 0
$$701$$ 41.0132 1.54905 0.774523 0.632546i $$-0.217990\pi$$
0.774523 + 0.632546i $$0.217990\pi$$
$$702$$ 0 0
$$703$$ −0.201626 −0.00760447
$$704$$ 22.1803 0.835953
$$705$$ 0 0
$$706$$ −20.8885 −0.786151
$$707$$ −0.909830 −0.0342177
$$708$$ 0 0
$$709$$ −33.5410 −1.25966 −0.629830 0.776733i $$-0.716875\pi$$
−0.629830 + 0.776733i $$0.716875\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ −20.0000 −0.749532
$$713$$ 11.2918 0.422881
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0.326238 0.0121921
$$717$$ 0 0
$$718$$ −22.2361 −0.829843
$$719$$ 23.2918 0.868637 0.434319 0.900759i $$-0.356989\pi$$
0.434319 + 0.900759i $$0.356989\pi$$
$$720$$ 0 0
$$721$$ 5.29180 0.197077
$$722$$ −29.5623 −1.10020
$$723$$ 0 0
$$724$$ 0.180340 0.00670228
$$725$$ 0 0
$$726$$ 0 0
$$727$$ 24.5623 0.910966 0.455483 0.890245i $$-0.349467\pi$$
0.455483 + 0.890245i $$0.349467\pi$$
$$728$$ −2.56231 −0.0949654
$$729$$ 0 0
$$730$$ 0 0
$$731$$ −25.4164 −0.940060
$$732$$ 0 0
$$733$$ −19.9787 −0.737931 −0.368965 0.929443i $$-0.620288\pi$$
−0.368965 + 0.929443i $$0.620288\pi$$
$$734$$ −41.3607 −1.52665
$$735$$ 0 0
$$736$$ 12.7295 0.469215
$$737$$ 24.9443 0.918834
$$738$$ 0 0
$$739$$ 15.9787 0.587786 0.293893 0.955838i $$-0.405049\pi$$
0.293893 + 0.955838i $$0.405049\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 3.47214 0.127466
$$743$$ −28.3607 −1.04045 −0.520226 0.854029i $$-0.674152\pi$$
−0.520226 + 0.854029i $$0.674152\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ −45.7426 −1.67476
$$747$$ 0 0
$$748$$ 16.9443 0.619544
$$749$$ 10.1459 0.370723
$$750$$ 0 0
$$751$$ −5.11146 −0.186520 −0.0932598 0.995642i $$-0.529729\pi$$
−0.0932598 + 0.995642i $$0.529729\pi$$
$$752$$ 3.00000 0.109399
$$753$$ 0 0
$$754$$ 10.8541 0.395283
$$755$$ 0 0
$$756$$ 0 0
$$757$$ 30.4164 1.10550 0.552752 0.833346i $$-0.313578\pi$$
0.552752 + 0.833346i $$0.313578\pi$$
$$758$$ 23.6180 0.857846
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 18.4508 0.668843 0.334421 0.942424i $$-0.391459\pi$$
0.334421 + 0.942424i $$0.391459\pi$$
$$762$$ 0 0
$$763$$ 6.18034 0.223743
$$764$$ 1.12461 0.0406870
$$765$$ 0 0
$$766$$ −53.9787 −1.95033
$$767$$ 20.1246 0.726658
$$768$$ 0 0
$$769$$ 13.4164 0.483808 0.241904 0.970300i $$-0.422228\pi$$
0.241904 + 0.970300i $$0.422228\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 4.76393 0.171458
$$773$$ 36.1591 1.30055 0.650275 0.759699i $$-0.274654\pi$$
0.650275 + 0.759699i $$0.274654\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 8.61803 0.309369
$$777$$ 0 0
$$778$$ −24.2705 −0.870140
$$779$$ 0.652476 0.0233774
$$780$$ 0 0
$$781$$ 34.6525 1.23996
$$782$$ −31.8885 −1.14033
$$783$$ 0 0
$$784$$ 32.1246 1.14731
$$785$$ 0 0
$$786$$ 0 0
$$787$$ −11.8197 −0.421325 −0.210663 0.977559i $$-0.567562\pi$$
−0.210663 + 0.977559i $$0.567562\pi$$
$$788$$ −2.29180 −0.0816419
$$789$$ 0 0
$$790$$ 0 0
$$791$$ −10.4164 −0.370365
$$792$$ 0 0
$$793$$ 16.1459 0.573358
$$794$$ 46.9787 1.66721
$$795$$ 0 0
$$796$$ −10.8541 −0.384713
$$797$$ −9.76393 −0.345856 −0.172928 0.984934i $$-0.555323\pi$$
−0.172928 + 0.984934i $$0.555323\pi$$
$$798$$ 0 0
$$799$$ −3.23607 −0.114484
$$800$$ 0 0
$$801$$ 0 0
$$802$$ −43.0344 −1.51960
$$803$$ −47.1246 −1.66299
$$804$$ 0 0
$$805$$ 0 0
$$806$$ −9.00000 −0.317011
$$807$$ 0 0
$$808$$ 3.29180 0.115805
$$809$$ −30.9787 −1.08915 −0.544577 0.838711i $$-0.683310\pi$$
−0.544577 + 0.838711i $$0.683310\pi$$
$$810$$ 0 0
$$811$$ −14.7082 −0.516475 −0.258237 0.966081i $$-0.583142\pi$$
−0.258237 + 0.966081i $$0.583142\pi$$
$$812$$ 1.38197 0.0484975
$$813$$ 0 0
$$814$$ −2.00000 −0.0701000
$$815$$ 0 0
$$816$$ 0 0
$$817$$ −4.14590 −0.145047
$$818$$ 2.56231 0.0895889
$$819$$ 0 0
$$820$$ 0 0
$$821$$ 40.6869 1.41998 0.709992 0.704210i $$-0.248699\pi$$
0.709992 + 0.704210i $$0.248699\pi$$
$$822$$ 0 0
$$823$$ 47.7082 1.66300 0.831502 0.555522i $$-0.187482\pi$$
0.831502 + 0.555522i $$0.187482\pi$$
$$824$$ −19.1459 −0.666979
$$825$$ 0 0
$$826$$ 10.8541 0.377663
$$827$$ 0.965558 0.0335757 0.0167879 0.999859i $$-0.494656\pi$$
0.0167879 + 0.999859i $$0.494656\pi$$
$$828$$ 0 0
$$829$$ −35.8541 −1.24526 −0.622632 0.782515i $$-0.713937\pi$$
−0.622632 + 0.782515i $$0.713937\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 7.85410 0.272292
$$833$$ −34.6525 −1.20064
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 2.76393 0.0955926
$$837$$ 0 0
$$838$$ 15.3262 0.529436
$$839$$ −10.8541 −0.374725 −0.187363 0.982291i $$-0.559994\pi$$
−0.187363 + 0.982291i $$0.559994\pi$$
$$840$$ 0 0
$$841$$ −15.9098 −0.548615
$$842$$ 51.7771 1.78436
$$843$$ 0 0
$$844$$ −5.67376 −0.195299
$$845$$ 0 0
$$846$$ 0 0
$$847$$ 10.1459 0.348617
$$848$$ −16.8541 −0.578772
$$849$$ 0 0
$$850$$ 0 0
$$851$$ 0.888544 0.0304589
$$852$$ 0 0
$$853$$ −15.3050 −0.524032 −0.262016 0.965064i $$-0.584387\pi$$
−0.262016 + 0.965064i $$0.584387\pi$$
$$854$$ 8.70820 0.297989
$$855$$ 0 0
$$856$$ −36.7082 −1.25466
$$857$$ −19.6869 −0.672492 −0.336246 0.941774i $$-0.609157\pi$$
−0.336246 + 0.941774i $$0.609157\pi$$
$$858$$ 0 0
$$859$$ −1.58359 −0.0540315 −0.0270157 0.999635i $$-0.508600\pi$$
−0.0270157 + 0.999635i $$0.508600\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 48.2705 1.64410
$$863$$ 21.4377 0.729748 0.364874 0.931057i $$-0.381112\pi$$
0.364874 + 0.931057i $$0.381112\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 43.4508 1.47652
$$867$$ 0 0
$$868$$ −1.14590 −0.0388943
$$869$$ −42.3607 −1.43699
$$870$$ 0 0
$$871$$ 8.83282 0.299289
$$872$$ −22.3607 −0.757228
$$873$$ 0 0
$$874$$ −5.20163 −0.175948
$$875$$ 0 0
$$876$$ 0 0
$$877$$ −36.5410 −1.23390 −0.616951 0.787001i $$-0.711632\pi$$
−0.616951 + 0.787001i $$0.711632\pi$$
$$878$$ −66.3050 −2.23768
$$879$$ 0 0
$$880$$ 0 0
$$881$$ 40.3607 1.35979 0.679893 0.733311i $$-0.262026\pi$$
0.679893 + 0.733311i $$0.262026\pi$$
$$882$$ 0 0
$$883$$ −20.5836 −0.692693 −0.346347 0.938107i $$-0.612578\pi$$
−0.346347 + 0.938107i $$0.612578\pi$$
$$884$$ 6.00000 0.201802
$$885$$ 0 0
$$886$$ −48.4508 −1.62774
$$887$$ −29.8885 −1.00356 −0.501780 0.864996i $$-0.667321\pi$$
−0.501780 + 0.864996i $$0.667321\pi$$
$$888$$ 0 0
$$889$$ 12.2918 0.412254
$$890$$ 0 0
$$891$$ 0 0
$$892$$ −0.111456 −0.00373183
$$893$$ −0.527864 −0.0176643
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 8.41641 0.281172
$$897$$ 0 0
$$898$$ −7.56231 −0.252357
$$899$$ −10.8541 −0.362005
$$900$$ 0 0
$$901$$ 18.1803 0.605675
$$902$$ 6.47214 0.215499
$$903$$ 0 0
$$904$$ 37.6869 1.25345
$$905$$ 0 0
$$906$$ 0 0
$$907$$ −33.2492 −1.10402 −0.552011 0.833837i $$-0.686139\pi$$
−0.552011 + 0.833837i $$0.686139\pi$$
$$908$$ −9.12461 −0.302811
$$909$$ 0 0
$$910$$ 0 0
$$911$$ 40.2361 1.33308 0.666540 0.745469i $$-0.267774\pi$$
0.666540 + 0.745469i $$0.267774\pi$$
$$912$$ 0 0
$$913$$ 32.6525 1.08064
$$914$$ −34.6525 −1.14620
$$915$$ 0 0
$$916$$ 13.4164 0.443291
$$917$$ −4.20163 −0.138750
$$918$$ 0 0
$$919$$ −53.2148 −1.75539 −0.877697 0.479216i $$-0.840921\pi$$
−0.877697 + 0.479216i $$0.840921\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ −1.32624 −0.0436773
$$923$$ 12.2705 0.403889
$$924$$ 0 0
$$925$$ 0 0
$$926$$ −39.0344 −1.28275
$$927$$ 0 0
$$928$$ −12.2361 −0.401669
$$929$$ −41.6312 −1.36588 −0.682938 0.730477i $$-0.739298\pi$$
−0.682938 + 0.730477i $$0.739298\pi$$
$$930$$ 0 0
$$931$$ −5.65248 −0.185252
$$932$$ 1.81966 0.0596049
$$933$$ 0 0
$$934$$ −44.4164 −1.45335
$$935$$ 0 0
$$936$$ 0 0
$$937$$ 17.7295 0.579197 0.289599 0.957148i $$-0.406478\pi$$
0.289599 + 0.957148i $$0.406478\pi$$
$$938$$ 4.76393 0.155548
$$939$$ 0 0
$$940$$ 0 0
$$941$$ 46.4164 1.51313 0.756566 0.653918i $$-0.226876\pi$$
0.756566 + 0.653918i $$0.226876\pi$$
$$942$$ 0 0
$$943$$ −2.87539 −0.0936355
$$944$$ −52.6869 −1.71481
$$945$$ 0 0
$$946$$ −41.1246 −1.33708
$$947$$ −2.65248 −0.0861939 −0.0430969 0.999071i $$-0.513722\pi$$
−0.0430969 + 0.999071i $$0.513722\pi$$
$$948$$ 0 0
$$949$$ −16.6869 −0.541680
$$950$$ 0 0
$$951$$ 0 0
$$952$$ −7.23607 −0.234522
$$953$$ 7.74265 0.250809 0.125404 0.992106i $$-0.459977\pi$$
0.125404 + 0.992106i $$0.459977\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 12.6869 0.410324
$$957$$ 0 0
$$958$$ 17.5623 0.567412
$$959$$ 7.38197 0.238376
$$960$$ 0 0
$$961$$ −22.0000 −0.709677
$$962$$ −0.708204 −0.0228334
$$963$$ 0 0
$$964$$ 1.56231 0.0503185
$$965$$ 0 0
$$966$$ 0 0
$$967$$ 4.11146 0.132216 0.0661078 0.997812i $$-0.478942\pi$$
0.0661078 + 0.997812i $$0.478942\pi$$
$$968$$ −36.7082 −1.17985
$$969$$ 0 0
$$970$$ 0 0
$$971$$ −5.61803 −0.180291 −0.0901456 0.995929i $$-0.528733\pi$$
−0.0901456 + 0.995929i $$0.528733\pi$$
$$972$$ 0 0
$$973$$ 3.09017 0.0990663
$$974$$ −58.9230 −1.88801
$$975$$ 0 0
$$976$$ −42.2705 −1.35305
$$977$$ 2.34752 0.0751040 0.0375520 0.999295i $$-0.488044\pi$$
0.0375520 + 0.999295i $$0.488044\pi$$
$$978$$ 0 0
$$979$$ 46.8328 1.49678
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 69.9787 2.23311
$$983$$ −9.61803 −0.306768 −0.153384 0.988167i $$-0.549017\pi$$
−0.153384 + 0.988167i $$0.549017\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 30.6525 0.976174
$$987$$ 0 0
$$988$$ 0.978714 0.0311370
$$989$$ 18.2705 0.580968
$$990$$ 0 0
$$991$$ −15.3607 −0.487948 −0.243974 0.969782i $$-0.578451\pi$$
−0.243974 + 0.969782i $$0.578451\pi$$
$$992$$ 10.1459 0.322133
$$993$$ 0 0
$$994$$ 6.61803 0.209911
$$995$$ 0 0
$$996$$ 0 0
$$997$$ 24.8885 0.788228 0.394114 0.919062i $$-0.371051\pi$$
0.394114 + 0.919062i $$0.371051\pi$$
$$998$$ 12.2361 0.387326
$$999$$ 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 5625.2.a.f.1.2 2
3.2 odd 2 625.2.a.b.1.1 2
5.4 even 2 5625.2.a.d.1.1 2
12.11 even 2 10000.2.a.c.1.1 2
15.2 even 4 625.2.b.a.624.1 4
15.8 even 4 625.2.b.a.624.4 4
15.14 odd 2 625.2.a.c.1.2 2
25.11 even 5 225.2.h.b.46.1 4
25.16 even 5 225.2.h.b.181.1 4
60.59 even 2 10000.2.a.l.1.2 2
75.2 even 20 125.2.e.a.24.1 8
75.8 even 20 625.2.e.c.374.1 8
75.11 odd 10 25.2.d.a.21.1 yes 4
75.14 odd 10 125.2.d.a.101.1 4
75.17 even 20 625.2.e.c.374.2 8
75.23 even 20 125.2.e.a.24.2 8
75.29 odd 10 625.2.d.b.376.1 4
75.38 even 20 125.2.e.a.99.1 8
75.41 odd 10 25.2.d.a.6.1 4
75.44 odd 10 625.2.d.b.251.1 4
75.47 even 20 625.2.e.c.249.1 8
75.53 even 20 625.2.e.c.249.2 8
75.56 odd 10 625.2.d.h.251.1 4
75.59 odd 10 125.2.d.a.26.1 4
75.62 even 20 125.2.e.a.99.2 8
75.71 odd 10 625.2.d.h.376.1 4
300.11 even 10 400.2.u.b.321.1 4
300.191 even 10 400.2.u.b.81.1 4

By twisted newform
Twist Min Dim Char Parity Ord Type
25.2.d.a.6.1 4 75.41 odd 10
25.2.d.a.21.1 yes 4 75.11 odd 10
125.2.d.a.26.1 4 75.59 odd 10
125.2.d.a.101.1 4 75.14 odd 10
125.2.e.a.24.1 8 75.2 even 20
125.2.e.a.24.2 8 75.23 even 20
125.2.e.a.99.1 8 75.38 even 20
125.2.e.a.99.2 8 75.62 even 20
225.2.h.b.46.1 4 25.11 even 5
225.2.h.b.181.1 4 25.16 even 5
400.2.u.b.81.1 4 300.191 even 10
400.2.u.b.321.1 4 300.11 even 10
625.2.a.b.1.1 2 3.2 odd 2
625.2.a.c.1.2 2 15.14 odd 2
625.2.b.a.624.1 4 15.2 even 4
625.2.b.a.624.4 4 15.8 even 4
625.2.d.b.251.1 4 75.44 odd 10
625.2.d.b.376.1 4 75.29 odd 10
625.2.d.h.251.1 4 75.56 odd 10
625.2.d.h.376.1 4 75.71 odd 10
625.2.e.c.249.1 8 75.47 even 20
625.2.e.c.249.2 8 75.53 even 20
625.2.e.c.374.1 8 75.8 even 20
625.2.e.c.374.2 8 75.17 even 20
5625.2.a.d.1.1 2 5.4 even 2
5625.2.a.f.1.2 2 1.1 even 1 trivial
10000.2.a.c.1.1 2 12.11 even 2
10000.2.a.l.1.2 2 60.59 even 2