# Properties

 Label 5550.2.a.bo.1.1 Level $5550$ Weight $2$ Character 5550.1 Self dual yes Analytic conductor $44.317$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$5550 = 2 \cdot 3 \cdot 5^{2} \cdot 37$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 5550.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$44.3169731218$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 1110) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 5550.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} +1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} +1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +5.00000 q^{11} +1.00000 q^{12} +1.00000 q^{14} +1.00000 q^{16} +1.00000 q^{17} +1.00000 q^{18} +1.00000 q^{21} +5.00000 q^{22} +4.00000 q^{23} +1.00000 q^{24} +1.00000 q^{27} +1.00000 q^{28} -3.00000 q^{29} +1.00000 q^{31} +1.00000 q^{32} +5.00000 q^{33} +1.00000 q^{34} +1.00000 q^{36} -1.00000 q^{37} -1.00000 q^{41} +1.00000 q^{42} +7.00000 q^{43} +5.00000 q^{44} +4.00000 q^{46} -4.00000 q^{47} +1.00000 q^{48} -6.00000 q^{49} +1.00000 q^{51} +3.00000 q^{53} +1.00000 q^{54} +1.00000 q^{56} -3.00000 q^{58} -8.00000 q^{59} +5.00000 q^{61} +1.00000 q^{62} +1.00000 q^{63} +1.00000 q^{64} +5.00000 q^{66} -4.00000 q^{67} +1.00000 q^{68} +4.00000 q^{69} -6.00000 q^{71} +1.00000 q^{72} +10.0000 q^{73} -1.00000 q^{74} +5.00000 q^{77} +1.00000 q^{81} -1.00000 q^{82} -8.00000 q^{83} +1.00000 q^{84} +7.00000 q^{86} -3.00000 q^{87} +5.00000 q^{88} +8.00000 q^{89} +4.00000 q^{92} +1.00000 q^{93} -4.00000 q^{94} +1.00000 q^{96} -1.00000 q^{97} -6.00000 q^{98} +5.00000 q^{99} +O(q^{100})$$

## Coefficient data

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

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 1.00000 0.707107
$$3$$ 1.00000 0.577350
$$4$$ 1.00000 0.500000
$$5$$ 0 0
$$6$$ 1.00000 0.408248
$$7$$ 1.00000 0.377964 0.188982 0.981981i $$-0.439481\pi$$
0.188982 + 0.981981i $$0.439481\pi$$
$$8$$ 1.00000 0.353553
$$9$$ 1.00000 0.333333
$$10$$ 0 0
$$11$$ 5.00000 1.50756 0.753778 0.657129i $$-0.228229\pi$$
0.753778 + 0.657129i $$0.228229\pi$$
$$12$$ 1.00000 0.288675
$$13$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$14$$ 1.00000 0.267261
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ 1.00000 0.242536 0.121268 0.992620i $$-0.461304\pi$$
0.121268 + 0.992620i $$0.461304\pi$$
$$18$$ 1.00000 0.235702
$$19$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$20$$ 0 0
$$21$$ 1.00000 0.218218
$$22$$ 5.00000 1.06600
$$23$$ 4.00000 0.834058 0.417029 0.908893i $$-0.363071\pi$$
0.417029 + 0.908893i $$0.363071\pi$$
$$24$$ 1.00000 0.204124
$$25$$ 0 0
$$26$$ 0 0
$$27$$ 1.00000 0.192450
$$28$$ 1.00000 0.188982
$$29$$ −3.00000 −0.557086 −0.278543 0.960424i $$-0.589851\pi$$
−0.278543 + 0.960424i $$0.589851\pi$$
$$30$$ 0 0
$$31$$ 1.00000 0.179605 0.0898027 0.995960i $$-0.471376\pi$$
0.0898027 + 0.995960i $$0.471376\pi$$
$$32$$ 1.00000 0.176777
$$33$$ 5.00000 0.870388
$$34$$ 1.00000 0.171499
$$35$$ 0 0
$$36$$ 1.00000 0.166667
$$37$$ −1.00000 −0.164399
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 0 0
$$41$$ −1.00000 −0.156174 −0.0780869 0.996947i $$-0.524881\pi$$
−0.0780869 + 0.996947i $$0.524881\pi$$
$$42$$ 1.00000 0.154303
$$43$$ 7.00000 1.06749 0.533745 0.845645i $$-0.320784\pi$$
0.533745 + 0.845645i $$0.320784\pi$$
$$44$$ 5.00000 0.753778
$$45$$ 0 0
$$46$$ 4.00000 0.589768
$$47$$ −4.00000 −0.583460 −0.291730 0.956501i $$-0.594231\pi$$
−0.291730 + 0.956501i $$0.594231\pi$$
$$48$$ 1.00000 0.144338
$$49$$ −6.00000 −0.857143
$$50$$ 0 0
$$51$$ 1.00000 0.140028
$$52$$ 0 0
$$53$$ 3.00000 0.412082 0.206041 0.978543i $$-0.433942\pi$$
0.206041 + 0.978543i $$0.433942\pi$$
$$54$$ 1.00000 0.136083
$$55$$ 0 0
$$56$$ 1.00000 0.133631
$$57$$ 0 0
$$58$$ −3.00000 −0.393919
$$59$$ −8.00000 −1.04151 −0.520756 0.853706i $$-0.674350\pi$$
−0.520756 + 0.853706i $$0.674350\pi$$
$$60$$ 0 0
$$61$$ 5.00000 0.640184 0.320092 0.947386i $$-0.396286\pi$$
0.320092 + 0.947386i $$0.396286\pi$$
$$62$$ 1.00000 0.127000
$$63$$ 1.00000 0.125988
$$64$$ 1.00000 0.125000
$$65$$ 0 0
$$66$$ 5.00000 0.615457
$$67$$ −4.00000 −0.488678 −0.244339 0.969690i $$-0.578571\pi$$
−0.244339 + 0.969690i $$0.578571\pi$$
$$68$$ 1.00000 0.121268
$$69$$ 4.00000 0.481543
$$70$$ 0 0
$$71$$ −6.00000 −0.712069 −0.356034 0.934473i $$-0.615871\pi$$
−0.356034 + 0.934473i $$0.615871\pi$$
$$72$$ 1.00000 0.117851
$$73$$ 10.0000 1.17041 0.585206 0.810885i $$-0.301014\pi$$
0.585206 + 0.810885i $$0.301014\pi$$
$$74$$ −1.00000 −0.116248
$$75$$ 0 0
$$76$$ 0 0
$$77$$ 5.00000 0.569803
$$78$$ 0 0
$$79$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ −1.00000 −0.110432
$$83$$ −8.00000 −0.878114 −0.439057 0.898459i $$-0.644687\pi$$
−0.439057 + 0.898459i $$0.644687\pi$$
$$84$$ 1.00000 0.109109
$$85$$ 0 0
$$86$$ 7.00000 0.754829
$$87$$ −3.00000 −0.321634
$$88$$ 5.00000 0.533002
$$89$$ 8.00000 0.847998 0.423999 0.905663i $$-0.360626\pi$$
0.423999 + 0.905663i $$0.360626\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 4.00000 0.417029
$$93$$ 1.00000 0.103695
$$94$$ −4.00000 −0.412568
$$95$$ 0 0
$$96$$ 1.00000 0.102062
$$97$$ −1.00000 −0.101535 −0.0507673 0.998711i $$-0.516167\pi$$
−0.0507673 + 0.998711i $$0.516167\pi$$
$$98$$ −6.00000 −0.606092
$$99$$ 5.00000 0.502519
$$100$$ 0 0
$$101$$ 10.0000 0.995037 0.497519 0.867453i $$-0.334245\pi$$
0.497519 + 0.867453i $$0.334245\pi$$
$$102$$ 1.00000 0.0990148
$$103$$ 8.00000 0.788263 0.394132 0.919054i $$-0.371045\pi$$
0.394132 + 0.919054i $$0.371045\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 3.00000 0.291386
$$107$$ 6.00000 0.580042 0.290021 0.957020i $$-0.406338\pi$$
0.290021 + 0.957020i $$0.406338\pi$$
$$108$$ 1.00000 0.0962250
$$109$$ 1.00000 0.0957826 0.0478913 0.998853i $$-0.484750\pi$$
0.0478913 + 0.998853i $$0.484750\pi$$
$$110$$ 0 0
$$111$$ −1.00000 −0.0949158
$$112$$ 1.00000 0.0944911
$$113$$ −21.0000 −1.97551 −0.987757 0.156001i $$-0.950140\pi$$
−0.987757 + 0.156001i $$0.950140\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ −3.00000 −0.278543
$$117$$ 0 0
$$118$$ −8.00000 −0.736460
$$119$$ 1.00000 0.0916698
$$120$$ 0 0
$$121$$ 14.0000 1.27273
$$122$$ 5.00000 0.452679
$$123$$ −1.00000 −0.0901670
$$124$$ 1.00000 0.0898027
$$125$$ 0 0
$$126$$ 1.00000 0.0890871
$$127$$ −8.00000 −0.709885 −0.354943 0.934888i $$-0.615500\pi$$
−0.354943 + 0.934888i $$0.615500\pi$$
$$128$$ 1.00000 0.0883883
$$129$$ 7.00000 0.616316
$$130$$ 0 0
$$131$$ −6.00000 −0.524222 −0.262111 0.965038i $$-0.584419\pi$$
−0.262111 + 0.965038i $$0.584419\pi$$
$$132$$ 5.00000 0.435194
$$133$$ 0 0
$$134$$ −4.00000 −0.345547
$$135$$ 0 0
$$136$$ 1.00000 0.0857493
$$137$$ 18.0000 1.53784 0.768922 0.639343i $$-0.220793\pi$$
0.768922 + 0.639343i $$0.220793\pi$$
$$138$$ 4.00000 0.340503
$$139$$ −11.0000 −0.933008 −0.466504 0.884519i $$-0.654487\pi$$
−0.466504 + 0.884519i $$0.654487\pi$$
$$140$$ 0 0
$$141$$ −4.00000 −0.336861
$$142$$ −6.00000 −0.503509
$$143$$ 0 0
$$144$$ 1.00000 0.0833333
$$145$$ 0 0
$$146$$ 10.0000 0.827606
$$147$$ −6.00000 −0.494872
$$148$$ −1.00000 −0.0821995
$$149$$ 4.00000 0.327693 0.163846 0.986486i $$-0.447610\pi$$
0.163846 + 0.986486i $$0.447610\pi$$
$$150$$ 0 0
$$151$$ −2.00000 −0.162758 −0.0813788 0.996683i $$-0.525932\pi$$
−0.0813788 + 0.996683i $$0.525932\pi$$
$$152$$ 0 0
$$153$$ 1.00000 0.0808452
$$154$$ 5.00000 0.402911
$$155$$ 0 0
$$156$$ 0 0
$$157$$ 23.0000 1.83560 0.917800 0.397043i $$-0.129964\pi$$
0.917800 + 0.397043i $$0.129964\pi$$
$$158$$ 0 0
$$159$$ 3.00000 0.237915
$$160$$ 0 0
$$161$$ 4.00000 0.315244
$$162$$ 1.00000 0.0785674
$$163$$ −1.00000 −0.0783260 −0.0391630 0.999233i $$-0.512469\pi$$
−0.0391630 + 0.999233i $$0.512469\pi$$
$$164$$ −1.00000 −0.0780869
$$165$$ 0 0
$$166$$ −8.00000 −0.620920
$$167$$ −6.00000 −0.464294 −0.232147 0.972681i $$-0.574575\pi$$
−0.232147 + 0.972681i $$0.574575\pi$$
$$168$$ 1.00000 0.0771517
$$169$$ −13.0000 −1.00000
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 7.00000 0.533745
$$173$$ −13.0000 −0.988372 −0.494186 0.869356i $$-0.664534\pi$$
−0.494186 + 0.869356i $$0.664534\pi$$
$$174$$ −3.00000 −0.227429
$$175$$ 0 0
$$176$$ 5.00000 0.376889
$$177$$ −8.00000 −0.601317
$$178$$ 8.00000 0.599625
$$179$$ 10.0000 0.747435 0.373718 0.927543i $$-0.378083\pi$$
0.373718 + 0.927543i $$0.378083\pi$$
$$180$$ 0 0
$$181$$ −16.0000 −1.18927 −0.594635 0.803996i $$-0.702704\pi$$
−0.594635 + 0.803996i $$0.702704\pi$$
$$182$$ 0 0
$$183$$ 5.00000 0.369611
$$184$$ 4.00000 0.294884
$$185$$ 0 0
$$186$$ 1.00000 0.0733236
$$187$$ 5.00000 0.365636
$$188$$ −4.00000 −0.291730
$$189$$ 1.00000 0.0727393
$$190$$ 0 0
$$191$$ 3.00000 0.217072 0.108536 0.994092i $$-0.465384\pi$$
0.108536 + 0.994092i $$0.465384\pi$$
$$192$$ 1.00000 0.0721688
$$193$$ 10.0000 0.719816 0.359908 0.932988i $$-0.382808\pi$$
0.359908 + 0.932988i $$0.382808\pi$$
$$194$$ −1.00000 −0.0717958
$$195$$ 0 0
$$196$$ −6.00000 −0.428571
$$197$$ 26.0000 1.85242 0.926212 0.377004i $$-0.123046\pi$$
0.926212 + 0.377004i $$0.123046\pi$$
$$198$$ 5.00000 0.355335
$$199$$ 24.0000 1.70131 0.850657 0.525720i $$-0.176204\pi$$
0.850657 + 0.525720i $$0.176204\pi$$
$$200$$ 0 0
$$201$$ −4.00000 −0.282138
$$202$$ 10.0000 0.703598
$$203$$ −3.00000 −0.210559
$$204$$ 1.00000 0.0700140
$$205$$ 0 0
$$206$$ 8.00000 0.557386
$$207$$ 4.00000 0.278019
$$208$$ 0 0
$$209$$ 0 0
$$210$$ 0 0
$$211$$ −21.0000 −1.44570 −0.722850 0.691005i $$-0.757168\pi$$
−0.722850 + 0.691005i $$0.757168\pi$$
$$212$$ 3.00000 0.206041
$$213$$ −6.00000 −0.411113
$$214$$ 6.00000 0.410152
$$215$$ 0 0
$$216$$ 1.00000 0.0680414
$$217$$ 1.00000 0.0678844
$$218$$ 1.00000 0.0677285
$$219$$ 10.0000 0.675737
$$220$$ 0 0
$$221$$ 0 0
$$222$$ −1.00000 −0.0671156
$$223$$ −5.00000 −0.334825 −0.167412 0.985887i $$-0.553541\pi$$
−0.167412 + 0.985887i $$0.553541\pi$$
$$224$$ 1.00000 0.0668153
$$225$$ 0 0
$$226$$ −21.0000 −1.39690
$$227$$ 1.00000 0.0663723 0.0331862 0.999449i $$-0.489435\pi$$
0.0331862 + 0.999449i $$0.489435\pi$$
$$228$$ 0 0
$$229$$ −16.0000 −1.05731 −0.528655 0.848837i $$-0.677303\pi$$
−0.528655 + 0.848837i $$0.677303\pi$$
$$230$$ 0 0
$$231$$ 5.00000 0.328976
$$232$$ −3.00000 −0.196960
$$233$$ −10.0000 −0.655122 −0.327561 0.944830i $$-0.606227\pi$$
−0.327561 + 0.944830i $$0.606227\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ −8.00000 −0.520756
$$237$$ 0 0
$$238$$ 1.00000 0.0648204
$$239$$ 11.0000 0.711531 0.355765 0.934575i $$-0.384220\pi$$
0.355765 + 0.934575i $$0.384220\pi$$
$$240$$ 0 0
$$241$$ −2.00000 −0.128831 −0.0644157 0.997923i $$-0.520518\pi$$
−0.0644157 + 0.997923i $$0.520518\pi$$
$$242$$ 14.0000 0.899954
$$243$$ 1.00000 0.0641500
$$244$$ 5.00000 0.320092
$$245$$ 0 0
$$246$$ −1.00000 −0.0637577
$$247$$ 0 0
$$248$$ 1.00000 0.0635001
$$249$$ −8.00000 −0.506979
$$250$$ 0 0
$$251$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$252$$ 1.00000 0.0629941
$$253$$ 20.0000 1.25739
$$254$$ −8.00000 −0.501965
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ 6.00000 0.374270 0.187135 0.982334i $$-0.440080\pi$$
0.187135 + 0.982334i $$0.440080\pi$$
$$258$$ 7.00000 0.435801
$$259$$ −1.00000 −0.0621370
$$260$$ 0 0
$$261$$ −3.00000 −0.185695
$$262$$ −6.00000 −0.370681
$$263$$ 19.0000 1.17159 0.585795 0.810459i $$-0.300782\pi$$
0.585795 + 0.810459i $$0.300782\pi$$
$$264$$ 5.00000 0.307729
$$265$$ 0 0
$$266$$ 0 0
$$267$$ 8.00000 0.489592
$$268$$ −4.00000 −0.244339
$$269$$ −12.0000 −0.731653 −0.365826 0.930683i $$-0.619214\pi$$
−0.365826 + 0.930683i $$0.619214\pi$$
$$270$$ 0 0
$$271$$ −14.0000 −0.850439 −0.425220 0.905090i $$-0.639803\pi$$
−0.425220 + 0.905090i $$0.639803\pi$$
$$272$$ 1.00000 0.0606339
$$273$$ 0 0
$$274$$ 18.0000 1.08742
$$275$$ 0 0
$$276$$ 4.00000 0.240772
$$277$$ −8.00000 −0.480673 −0.240337 0.970690i $$-0.577258\pi$$
−0.240337 + 0.970690i $$0.577258\pi$$
$$278$$ −11.0000 −0.659736
$$279$$ 1.00000 0.0598684
$$280$$ 0 0
$$281$$ 18.0000 1.07379 0.536895 0.843649i $$-0.319597\pi$$
0.536895 + 0.843649i $$0.319597\pi$$
$$282$$ −4.00000 −0.238197
$$283$$ −16.0000 −0.951101 −0.475551 0.879688i $$-0.657751\pi$$
−0.475551 + 0.879688i $$0.657751\pi$$
$$284$$ −6.00000 −0.356034
$$285$$ 0 0
$$286$$ 0 0
$$287$$ −1.00000 −0.0590281
$$288$$ 1.00000 0.0589256
$$289$$ −16.0000 −0.941176
$$290$$ 0 0
$$291$$ −1.00000 −0.0586210
$$292$$ 10.0000 0.585206
$$293$$ 9.00000 0.525786 0.262893 0.964825i $$-0.415323\pi$$
0.262893 + 0.964825i $$0.415323\pi$$
$$294$$ −6.00000 −0.349927
$$295$$ 0 0
$$296$$ −1.00000 −0.0581238
$$297$$ 5.00000 0.290129
$$298$$ 4.00000 0.231714
$$299$$ 0 0
$$300$$ 0 0
$$301$$ 7.00000 0.403473
$$302$$ −2.00000 −0.115087
$$303$$ 10.0000 0.574485
$$304$$ 0 0
$$305$$ 0 0
$$306$$ 1.00000 0.0571662
$$307$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$308$$ 5.00000 0.284901
$$309$$ 8.00000 0.455104
$$310$$ 0 0
$$311$$ 15.0000 0.850572 0.425286 0.905059i $$-0.360174\pi$$
0.425286 + 0.905059i $$0.360174\pi$$
$$312$$ 0 0
$$313$$ 10.0000 0.565233 0.282617 0.959233i $$-0.408798\pi$$
0.282617 + 0.959233i $$0.408798\pi$$
$$314$$ 23.0000 1.29797
$$315$$ 0 0
$$316$$ 0 0
$$317$$ −5.00000 −0.280828 −0.140414 0.990093i $$-0.544843\pi$$
−0.140414 + 0.990093i $$0.544843\pi$$
$$318$$ 3.00000 0.168232
$$319$$ −15.0000 −0.839839
$$320$$ 0 0
$$321$$ 6.00000 0.334887
$$322$$ 4.00000 0.222911
$$323$$ 0 0
$$324$$ 1.00000 0.0555556
$$325$$ 0 0
$$326$$ −1.00000 −0.0553849
$$327$$ 1.00000 0.0553001
$$328$$ −1.00000 −0.0552158
$$329$$ −4.00000 −0.220527
$$330$$ 0 0
$$331$$ −26.0000 −1.42909 −0.714545 0.699590i $$-0.753366\pi$$
−0.714545 + 0.699590i $$0.753366\pi$$
$$332$$ −8.00000 −0.439057
$$333$$ −1.00000 −0.0547997
$$334$$ −6.00000 −0.328305
$$335$$ 0 0
$$336$$ 1.00000 0.0545545
$$337$$ 16.0000 0.871576 0.435788 0.900049i $$-0.356470\pi$$
0.435788 + 0.900049i $$0.356470\pi$$
$$338$$ −13.0000 −0.707107
$$339$$ −21.0000 −1.14056
$$340$$ 0 0
$$341$$ 5.00000 0.270765
$$342$$ 0 0
$$343$$ −13.0000 −0.701934
$$344$$ 7.00000 0.377415
$$345$$ 0 0
$$346$$ −13.0000 −0.698884
$$347$$ 12.0000 0.644194 0.322097 0.946707i $$-0.395612\pi$$
0.322097 + 0.946707i $$0.395612\pi$$
$$348$$ −3.00000 −0.160817
$$349$$ −2.00000 −0.107058 −0.0535288 0.998566i $$-0.517047\pi$$
−0.0535288 + 0.998566i $$0.517047\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 5.00000 0.266501
$$353$$ −31.0000 −1.64996 −0.824982 0.565159i $$-0.808815\pi$$
−0.824982 + 0.565159i $$0.808815\pi$$
$$354$$ −8.00000 −0.425195
$$355$$ 0 0
$$356$$ 8.00000 0.423999
$$357$$ 1.00000 0.0529256
$$358$$ 10.0000 0.528516
$$359$$ 6.00000 0.316668 0.158334 0.987386i $$-0.449388\pi$$
0.158334 + 0.987386i $$0.449388\pi$$
$$360$$ 0 0
$$361$$ −19.0000 −1.00000
$$362$$ −16.0000 −0.840941
$$363$$ 14.0000 0.734809
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 5.00000 0.261354
$$367$$ 13.0000 0.678594 0.339297 0.940679i $$-0.389811\pi$$
0.339297 + 0.940679i $$0.389811\pi$$
$$368$$ 4.00000 0.208514
$$369$$ −1.00000 −0.0520579
$$370$$ 0 0
$$371$$ 3.00000 0.155752
$$372$$ 1.00000 0.0518476
$$373$$ 6.00000 0.310668 0.155334 0.987862i $$-0.450355\pi$$
0.155334 + 0.987862i $$0.450355\pi$$
$$374$$ 5.00000 0.258544
$$375$$ 0 0
$$376$$ −4.00000 −0.206284
$$377$$ 0 0
$$378$$ 1.00000 0.0514344
$$379$$ 20.0000 1.02733 0.513665 0.857991i $$-0.328287\pi$$
0.513665 + 0.857991i $$0.328287\pi$$
$$380$$ 0 0
$$381$$ −8.00000 −0.409852
$$382$$ 3.00000 0.153493
$$383$$ 8.00000 0.408781 0.204390 0.978889i $$-0.434479\pi$$
0.204390 + 0.978889i $$0.434479\pi$$
$$384$$ 1.00000 0.0510310
$$385$$ 0 0
$$386$$ 10.0000 0.508987
$$387$$ 7.00000 0.355830
$$388$$ −1.00000 −0.0507673
$$389$$ 23.0000 1.16615 0.583073 0.812420i $$-0.301850\pi$$
0.583073 + 0.812420i $$0.301850\pi$$
$$390$$ 0 0
$$391$$ 4.00000 0.202289
$$392$$ −6.00000 −0.303046
$$393$$ −6.00000 −0.302660
$$394$$ 26.0000 1.30986
$$395$$ 0 0
$$396$$ 5.00000 0.251259
$$397$$ 22.0000 1.10415 0.552074 0.833795i $$-0.313837\pi$$
0.552074 + 0.833795i $$0.313837\pi$$
$$398$$ 24.0000 1.20301
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 10.0000 0.499376 0.249688 0.968326i $$-0.419672\pi$$
0.249688 + 0.968326i $$0.419672\pi$$
$$402$$ −4.00000 −0.199502
$$403$$ 0 0
$$404$$ 10.0000 0.497519
$$405$$ 0 0
$$406$$ −3.00000 −0.148888
$$407$$ −5.00000 −0.247841
$$408$$ 1.00000 0.0495074
$$409$$ −40.0000 −1.97787 −0.988936 0.148340i $$-0.952607\pi$$
−0.988936 + 0.148340i $$0.952607\pi$$
$$410$$ 0 0
$$411$$ 18.0000 0.887875
$$412$$ 8.00000 0.394132
$$413$$ −8.00000 −0.393654
$$414$$ 4.00000 0.196589
$$415$$ 0 0
$$416$$ 0 0
$$417$$ −11.0000 −0.538672
$$418$$ 0 0
$$419$$ 20.0000 0.977064 0.488532 0.872546i $$-0.337533\pi$$
0.488532 + 0.872546i $$0.337533\pi$$
$$420$$ 0 0
$$421$$ −38.0000 −1.85201 −0.926003 0.377515i $$-0.876779\pi$$
−0.926003 + 0.377515i $$0.876779\pi$$
$$422$$ −21.0000 −1.02226
$$423$$ −4.00000 −0.194487
$$424$$ 3.00000 0.145693
$$425$$ 0 0
$$426$$ −6.00000 −0.290701
$$427$$ 5.00000 0.241967
$$428$$ 6.00000 0.290021
$$429$$ 0 0
$$430$$ 0 0
$$431$$ −13.0000 −0.626188 −0.313094 0.949722i $$-0.601365\pi$$
−0.313094 + 0.949722i $$0.601365\pi$$
$$432$$ 1.00000 0.0481125
$$433$$ −20.0000 −0.961139 −0.480569 0.876957i $$-0.659570\pi$$
−0.480569 + 0.876957i $$0.659570\pi$$
$$434$$ 1.00000 0.0480015
$$435$$ 0 0
$$436$$ 1.00000 0.0478913
$$437$$ 0 0
$$438$$ 10.0000 0.477818
$$439$$ −33.0000 −1.57500 −0.787502 0.616312i $$-0.788626\pi$$
−0.787502 + 0.616312i $$0.788626\pi$$
$$440$$ 0 0
$$441$$ −6.00000 −0.285714
$$442$$ 0 0
$$443$$ −6.00000 −0.285069 −0.142534 0.989790i $$-0.545525\pi$$
−0.142534 + 0.989790i $$0.545525\pi$$
$$444$$ −1.00000 −0.0474579
$$445$$ 0 0
$$446$$ −5.00000 −0.236757
$$447$$ 4.00000 0.189194
$$448$$ 1.00000 0.0472456
$$449$$ −24.0000 −1.13263 −0.566315 0.824189i $$-0.691631\pi$$
−0.566315 + 0.824189i $$0.691631\pi$$
$$450$$ 0 0
$$451$$ −5.00000 −0.235441
$$452$$ −21.0000 −0.987757
$$453$$ −2.00000 −0.0939682
$$454$$ 1.00000 0.0469323
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 19.0000 0.888783 0.444391 0.895833i $$-0.353420\pi$$
0.444391 + 0.895833i $$0.353420\pi$$
$$458$$ −16.0000 −0.747631
$$459$$ 1.00000 0.0466760
$$460$$ 0 0
$$461$$ 3.00000 0.139724 0.0698620 0.997557i $$-0.477744\pi$$
0.0698620 + 0.997557i $$0.477744\pi$$
$$462$$ 5.00000 0.232621
$$463$$ −24.0000 −1.11537 −0.557687 0.830051i $$-0.688311\pi$$
−0.557687 + 0.830051i $$0.688311\pi$$
$$464$$ −3.00000 −0.139272
$$465$$ 0 0
$$466$$ −10.0000 −0.463241
$$467$$ 27.0000 1.24941 0.624705 0.780860i $$-0.285219\pi$$
0.624705 + 0.780860i $$0.285219\pi$$
$$468$$ 0 0
$$469$$ −4.00000 −0.184703
$$470$$ 0 0
$$471$$ 23.0000 1.05978
$$472$$ −8.00000 −0.368230
$$473$$ 35.0000 1.60930
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 1.00000 0.0458349
$$477$$ 3.00000 0.137361
$$478$$ 11.0000 0.503128
$$479$$ −28.0000 −1.27935 −0.639676 0.768644i $$-0.720932\pi$$
−0.639676 + 0.768644i $$0.720932\pi$$
$$480$$ 0 0
$$481$$ 0 0
$$482$$ −2.00000 −0.0910975
$$483$$ 4.00000 0.182006
$$484$$ 14.0000 0.636364
$$485$$ 0 0
$$486$$ 1.00000 0.0453609
$$487$$ −12.0000 −0.543772 −0.271886 0.962329i $$-0.587647\pi$$
−0.271886 + 0.962329i $$0.587647\pi$$
$$488$$ 5.00000 0.226339
$$489$$ −1.00000 −0.0452216
$$490$$ 0 0
$$491$$ 12.0000 0.541552 0.270776 0.962642i $$-0.412720\pi$$
0.270776 + 0.962642i $$0.412720\pi$$
$$492$$ −1.00000 −0.0450835
$$493$$ −3.00000 −0.135113
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 1.00000 0.0449013
$$497$$ −6.00000 −0.269137
$$498$$ −8.00000 −0.358489
$$499$$ −30.0000 −1.34298 −0.671492 0.741012i $$-0.734346\pi$$
−0.671492 + 0.741012i $$0.734346\pi$$
$$500$$ 0 0
$$501$$ −6.00000 −0.268060
$$502$$ 0 0
$$503$$ 6.00000 0.267527 0.133763 0.991013i $$-0.457294\pi$$
0.133763 + 0.991013i $$0.457294\pi$$
$$504$$ 1.00000 0.0445435
$$505$$ 0 0
$$506$$ 20.0000 0.889108
$$507$$ −13.0000 −0.577350
$$508$$ −8.00000 −0.354943
$$509$$ −42.0000 −1.86162 −0.930809 0.365507i $$-0.880896\pi$$
−0.930809 + 0.365507i $$0.880896\pi$$
$$510$$ 0 0
$$511$$ 10.0000 0.442374
$$512$$ 1.00000 0.0441942
$$513$$ 0 0
$$514$$ 6.00000 0.264649
$$515$$ 0 0
$$516$$ 7.00000 0.308158
$$517$$ −20.0000 −0.879599
$$518$$ −1.00000 −0.0439375
$$519$$ −13.0000 −0.570637
$$520$$ 0 0
$$521$$ −35.0000 −1.53338 −0.766689 0.642019i $$-0.778097\pi$$
−0.766689 + 0.642019i $$0.778097\pi$$
$$522$$ −3.00000 −0.131306
$$523$$ 20.0000 0.874539 0.437269 0.899331i $$-0.355946\pi$$
0.437269 + 0.899331i $$0.355946\pi$$
$$524$$ −6.00000 −0.262111
$$525$$ 0 0
$$526$$ 19.0000 0.828439
$$527$$ 1.00000 0.0435607
$$528$$ 5.00000 0.217597
$$529$$ −7.00000 −0.304348
$$530$$ 0 0
$$531$$ −8.00000 −0.347170
$$532$$ 0 0
$$533$$ 0 0
$$534$$ 8.00000 0.346194
$$535$$ 0 0
$$536$$ −4.00000 −0.172774
$$537$$ 10.0000 0.431532
$$538$$ −12.0000 −0.517357
$$539$$ −30.0000 −1.29219
$$540$$ 0 0
$$541$$ 34.0000 1.46177 0.730887 0.682498i $$-0.239107\pi$$
0.730887 + 0.682498i $$0.239107\pi$$
$$542$$ −14.0000 −0.601351
$$543$$ −16.0000 −0.686626
$$544$$ 1.00000 0.0428746
$$545$$ 0 0
$$546$$ 0 0
$$547$$ −41.0000 −1.75303 −0.876517 0.481371i $$-0.840139\pi$$
−0.876517 + 0.481371i $$0.840139\pi$$
$$548$$ 18.0000 0.768922
$$549$$ 5.00000 0.213395
$$550$$ 0 0
$$551$$ 0 0
$$552$$ 4.00000 0.170251
$$553$$ 0 0
$$554$$ −8.00000 −0.339887
$$555$$ 0 0
$$556$$ −11.0000 −0.466504
$$557$$ 6.00000 0.254228 0.127114 0.991888i $$-0.459429\pi$$
0.127114 + 0.991888i $$0.459429\pi$$
$$558$$ 1.00000 0.0423334
$$559$$ 0 0
$$560$$ 0 0
$$561$$ 5.00000 0.211100
$$562$$ 18.0000 0.759284
$$563$$ −19.0000 −0.800755 −0.400377 0.916350i $$-0.631121\pi$$
−0.400377 + 0.916350i $$0.631121\pi$$
$$564$$ −4.00000 −0.168430
$$565$$ 0 0
$$566$$ −16.0000 −0.672530
$$567$$ 1.00000 0.0419961
$$568$$ −6.00000 −0.251754
$$569$$ −10.0000 −0.419222 −0.209611 0.977785i $$-0.567220\pi$$
−0.209611 + 0.977785i $$0.567220\pi$$
$$570$$ 0 0
$$571$$ −29.0000 −1.21361 −0.606806 0.794850i $$-0.707550\pi$$
−0.606806 + 0.794850i $$0.707550\pi$$
$$572$$ 0 0
$$573$$ 3.00000 0.125327
$$574$$ −1.00000 −0.0417392
$$575$$ 0 0
$$576$$ 1.00000 0.0416667
$$577$$ 22.0000 0.915872 0.457936 0.888985i $$-0.348589\pi$$
0.457936 + 0.888985i $$0.348589\pi$$
$$578$$ −16.0000 −0.665512
$$579$$ 10.0000 0.415586
$$580$$ 0 0
$$581$$ −8.00000 −0.331896
$$582$$ −1.00000 −0.0414513
$$583$$ 15.0000 0.621237
$$584$$ 10.0000 0.413803
$$585$$ 0 0
$$586$$ 9.00000 0.371787
$$587$$ −33.0000 −1.36206 −0.681028 0.732257i $$-0.738467\pi$$
−0.681028 + 0.732257i $$0.738467\pi$$
$$588$$ −6.00000 −0.247436
$$589$$ 0 0
$$590$$ 0 0
$$591$$ 26.0000 1.06950
$$592$$ −1.00000 −0.0410997
$$593$$ −24.0000 −0.985562 −0.492781 0.870153i $$-0.664020\pi$$
−0.492781 + 0.870153i $$0.664020\pi$$
$$594$$ 5.00000 0.205152
$$595$$ 0 0
$$596$$ 4.00000 0.163846
$$597$$ 24.0000 0.982255
$$598$$ 0 0
$$599$$ 24.0000 0.980613 0.490307 0.871550i $$-0.336885\pi$$
0.490307 + 0.871550i $$0.336885\pi$$
$$600$$ 0 0
$$601$$ −7.00000 −0.285536 −0.142768 0.989756i $$-0.545600\pi$$
−0.142768 + 0.989756i $$0.545600\pi$$
$$602$$ 7.00000 0.285299
$$603$$ −4.00000 −0.162893
$$604$$ −2.00000 −0.0813788
$$605$$ 0 0
$$606$$ 10.0000 0.406222
$$607$$ −10.0000 −0.405887 −0.202944 0.979190i $$-0.565051\pi$$
−0.202944 + 0.979190i $$0.565051\pi$$
$$608$$ 0 0
$$609$$ −3.00000 −0.121566
$$610$$ 0 0
$$611$$ 0 0
$$612$$ 1.00000 0.0404226
$$613$$ 37.0000 1.49442 0.747208 0.664590i $$-0.231394\pi$$
0.747208 + 0.664590i $$0.231394\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 5.00000 0.201456
$$617$$ 34.0000 1.36879 0.684394 0.729112i $$-0.260067\pi$$
0.684394 + 0.729112i $$0.260067\pi$$
$$618$$ 8.00000 0.321807
$$619$$ −31.0000 −1.24600 −0.622998 0.782224i $$-0.714085\pi$$
−0.622998 + 0.782224i $$0.714085\pi$$
$$620$$ 0 0
$$621$$ 4.00000 0.160514
$$622$$ 15.0000 0.601445
$$623$$ 8.00000 0.320513
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 10.0000 0.399680
$$627$$ 0 0
$$628$$ 23.0000 0.917800
$$629$$ −1.00000 −0.0398726
$$630$$ 0 0
$$631$$ 5.00000 0.199047 0.0995234 0.995035i $$-0.468268\pi$$
0.0995234 + 0.995035i $$0.468268\pi$$
$$632$$ 0 0
$$633$$ −21.0000 −0.834675
$$634$$ −5.00000 −0.198575
$$635$$ 0 0
$$636$$ 3.00000 0.118958
$$637$$ 0 0
$$638$$ −15.0000 −0.593856
$$639$$ −6.00000 −0.237356
$$640$$ 0 0
$$641$$ −37.0000 −1.46141 −0.730706 0.682692i $$-0.760809\pi$$
−0.730706 + 0.682692i $$0.760809\pi$$
$$642$$ 6.00000 0.236801
$$643$$ −21.0000 −0.828159 −0.414080 0.910241i $$-0.635896\pi$$
−0.414080 + 0.910241i $$0.635896\pi$$
$$644$$ 4.00000 0.157622
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 42.0000 1.65119 0.825595 0.564263i $$-0.190840\pi$$
0.825595 + 0.564263i $$0.190840\pi$$
$$648$$ 1.00000 0.0392837
$$649$$ −40.0000 −1.57014
$$650$$ 0 0
$$651$$ 1.00000 0.0391931
$$652$$ −1.00000 −0.0391630
$$653$$ 10.0000 0.391330 0.195665 0.980671i $$-0.437313\pi$$
0.195665 + 0.980671i $$0.437313\pi$$
$$654$$ 1.00000 0.0391031
$$655$$ 0 0
$$656$$ −1.00000 −0.0390434
$$657$$ 10.0000 0.390137
$$658$$ −4.00000 −0.155936
$$659$$ −32.0000 −1.24654 −0.623272 0.782006i $$-0.714197\pi$$
−0.623272 + 0.782006i $$0.714197\pi$$
$$660$$ 0 0
$$661$$ −13.0000 −0.505641 −0.252821 0.967513i $$-0.581358\pi$$
−0.252821 + 0.967513i $$0.581358\pi$$
$$662$$ −26.0000 −1.01052
$$663$$ 0 0
$$664$$ −8.00000 −0.310460
$$665$$ 0 0
$$666$$ −1.00000 −0.0387492
$$667$$ −12.0000 −0.464642
$$668$$ −6.00000 −0.232147
$$669$$ −5.00000 −0.193311
$$670$$ 0 0
$$671$$ 25.0000 0.965114
$$672$$ 1.00000 0.0385758
$$673$$ 36.0000 1.38770 0.693849 0.720121i $$-0.255914\pi$$
0.693849 + 0.720121i $$0.255914\pi$$
$$674$$ 16.0000 0.616297
$$675$$ 0 0
$$676$$ −13.0000 −0.500000
$$677$$ −2.00000 −0.0768662 −0.0384331 0.999261i $$-0.512237\pi$$
−0.0384331 + 0.999261i $$0.512237\pi$$
$$678$$ −21.0000 −0.806500
$$679$$ −1.00000 −0.0383765
$$680$$ 0 0
$$681$$ 1.00000 0.0383201
$$682$$ 5.00000 0.191460
$$683$$ 37.0000 1.41577 0.707883 0.706330i $$-0.249650\pi$$
0.707883 + 0.706330i $$0.249650\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ −13.0000 −0.496342
$$687$$ −16.0000 −0.610438
$$688$$ 7.00000 0.266872
$$689$$ 0 0
$$690$$ 0 0
$$691$$ −5.00000 −0.190209 −0.0951045 0.995467i $$-0.530319\pi$$
−0.0951045 + 0.995467i $$0.530319\pi$$
$$692$$ −13.0000 −0.494186
$$693$$ 5.00000 0.189934
$$694$$ 12.0000 0.455514
$$695$$ 0 0
$$696$$ −3.00000 −0.113715
$$697$$ −1.00000 −0.0378777
$$698$$ −2.00000 −0.0757011
$$699$$ −10.0000 −0.378235
$$700$$ 0 0
$$701$$ 26.0000 0.982006 0.491003 0.871158i $$-0.336630\pi$$
0.491003 + 0.871158i $$0.336630\pi$$
$$702$$ 0 0
$$703$$ 0 0
$$704$$ 5.00000 0.188445
$$705$$ 0 0
$$706$$ −31.0000 −1.16670
$$707$$ 10.0000 0.376089
$$708$$ −8.00000 −0.300658
$$709$$ 9.00000 0.338002 0.169001 0.985616i $$-0.445946\pi$$
0.169001 + 0.985616i $$0.445946\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 8.00000 0.299813
$$713$$ 4.00000 0.149801
$$714$$ 1.00000 0.0374241
$$715$$ 0 0
$$716$$ 10.0000 0.373718
$$717$$ 11.0000 0.410803
$$718$$ 6.00000 0.223918
$$719$$ −28.0000 −1.04422 −0.522112 0.852877i $$-0.674856\pi$$
−0.522112 + 0.852877i $$0.674856\pi$$
$$720$$ 0 0
$$721$$ 8.00000 0.297936
$$722$$ −19.0000 −0.707107
$$723$$ −2.00000 −0.0743808
$$724$$ −16.0000 −0.594635
$$725$$ 0 0
$$726$$ 14.0000 0.519589
$$727$$ −42.0000 −1.55769 −0.778847 0.627214i $$-0.784195\pi$$
−0.778847 + 0.627214i $$0.784195\pi$$
$$728$$ 0 0
$$729$$ 1.00000 0.0370370
$$730$$ 0 0
$$731$$ 7.00000 0.258904
$$732$$ 5.00000 0.184805
$$733$$ −19.0000 −0.701781 −0.350891 0.936416i $$-0.614121\pi$$
−0.350891 + 0.936416i $$0.614121\pi$$
$$734$$ 13.0000 0.479839
$$735$$ 0 0
$$736$$ 4.00000 0.147442
$$737$$ −20.0000 −0.736709
$$738$$ −1.00000 −0.0368105
$$739$$ 37.0000 1.36107 0.680534 0.732717i $$-0.261748\pi$$
0.680534 + 0.732717i $$0.261748\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 3.00000 0.110133
$$743$$ 15.0000 0.550297 0.275148 0.961402i $$-0.411273\pi$$
0.275148 + 0.961402i $$0.411273\pi$$
$$744$$ 1.00000 0.0366618
$$745$$ 0 0
$$746$$ 6.00000 0.219676
$$747$$ −8.00000 −0.292705
$$748$$ 5.00000 0.182818
$$749$$ 6.00000 0.219235
$$750$$ 0 0
$$751$$ 20.0000 0.729810 0.364905 0.931045i $$-0.381101\pi$$
0.364905 + 0.931045i $$0.381101\pi$$
$$752$$ −4.00000 −0.145865
$$753$$ 0 0
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 1.00000 0.0363696
$$757$$ −20.0000 −0.726912 −0.363456 0.931611i $$-0.618403\pi$$
−0.363456 + 0.931611i $$0.618403\pi$$
$$758$$ 20.0000 0.726433
$$759$$ 20.0000 0.725954
$$760$$ 0 0
$$761$$ −21.0000 −0.761249 −0.380625 0.924730i $$-0.624291\pi$$
−0.380625 + 0.924730i $$0.624291\pi$$
$$762$$ −8.00000 −0.289809
$$763$$ 1.00000 0.0362024
$$764$$ 3.00000 0.108536
$$765$$ 0 0
$$766$$ 8.00000 0.289052
$$767$$ 0 0
$$768$$ 1.00000 0.0360844
$$769$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$770$$ 0 0
$$771$$ 6.00000 0.216085
$$772$$ 10.0000 0.359908
$$773$$ −27.0000 −0.971123 −0.485561 0.874203i $$-0.661385\pi$$
−0.485561 + 0.874203i $$0.661385\pi$$
$$774$$ 7.00000 0.251610
$$775$$ 0 0
$$776$$ −1.00000 −0.0358979
$$777$$ −1.00000 −0.0358748
$$778$$ 23.0000 0.824590
$$779$$ 0 0
$$780$$ 0 0
$$781$$ −30.0000 −1.07348
$$782$$ 4.00000 0.143040
$$783$$ −3.00000 −0.107211
$$784$$ −6.00000 −0.214286
$$785$$ 0 0
$$786$$ −6.00000 −0.214013
$$787$$ 10.0000 0.356462 0.178231 0.983989i $$-0.442963\pi$$
0.178231 + 0.983989i $$0.442963\pi$$
$$788$$ 26.0000 0.926212
$$789$$ 19.0000 0.676418
$$790$$ 0 0
$$791$$ −21.0000 −0.746674
$$792$$ 5.00000 0.177667
$$793$$ 0 0
$$794$$ 22.0000 0.780751
$$795$$ 0 0
$$796$$ 24.0000 0.850657
$$797$$ −8.00000 −0.283375 −0.141687 0.989911i $$-0.545253\pi$$
−0.141687 + 0.989911i $$0.545253\pi$$
$$798$$ 0 0
$$799$$ −4.00000 −0.141510
$$800$$ 0 0
$$801$$ 8.00000 0.282666
$$802$$ 10.0000 0.353112
$$803$$ 50.0000 1.76446
$$804$$ −4.00000 −0.141069
$$805$$ 0 0
$$806$$ 0 0
$$807$$ −12.0000 −0.422420
$$808$$ 10.0000 0.351799
$$809$$ 48.0000 1.68759 0.843795 0.536666i $$-0.180316\pi$$
0.843795 + 0.536666i $$0.180316\pi$$
$$810$$ 0 0
$$811$$ −8.00000 −0.280918 −0.140459 0.990086i $$-0.544858\pi$$
−0.140459 + 0.990086i $$0.544858\pi$$
$$812$$ −3.00000 −0.105279
$$813$$ −14.0000 −0.491001
$$814$$ −5.00000 −0.175250
$$815$$ 0 0
$$816$$ 1.00000 0.0350070
$$817$$ 0 0
$$818$$ −40.0000 −1.39857
$$819$$ 0 0
$$820$$ 0 0
$$821$$ −50.0000 −1.74501 −0.872506 0.488603i $$-0.837507\pi$$
−0.872506 + 0.488603i $$0.837507\pi$$
$$822$$ 18.0000 0.627822
$$823$$ −24.0000 −0.836587 −0.418294 0.908312i $$-0.637372\pi$$
−0.418294 + 0.908312i $$0.637372\pi$$
$$824$$ 8.00000 0.278693
$$825$$ 0 0
$$826$$ −8.00000 −0.278356
$$827$$ −35.0000 −1.21707 −0.608535 0.793527i $$-0.708242\pi$$
−0.608535 + 0.793527i $$0.708242\pi$$
$$828$$ 4.00000 0.139010
$$829$$ 21.0000 0.729360 0.364680 0.931133i $$-0.381178\pi$$
0.364680 + 0.931133i $$0.381178\pi$$
$$830$$ 0 0
$$831$$ −8.00000 −0.277517
$$832$$ 0 0
$$833$$ −6.00000 −0.207888
$$834$$ −11.0000 −0.380899
$$835$$ 0 0
$$836$$ 0 0
$$837$$ 1.00000 0.0345651
$$838$$ 20.0000 0.690889
$$839$$ 40.0000 1.38095 0.690477 0.723355i $$-0.257401\pi$$
0.690477 + 0.723355i $$0.257401\pi$$
$$840$$ 0 0
$$841$$ −20.0000 −0.689655
$$842$$ −38.0000 −1.30957
$$843$$ 18.0000 0.619953
$$844$$ −21.0000 −0.722850
$$845$$ 0 0
$$846$$ −4.00000 −0.137523
$$847$$ 14.0000 0.481046
$$848$$ 3.00000 0.103020
$$849$$ −16.0000 −0.549119
$$850$$ 0 0
$$851$$ −4.00000 −0.137118
$$852$$ −6.00000 −0.205557
$$853$$ 2.00000 0.0684787 0.0342393 0.999414i $$-0.489099\pi$$
0.0342393 + 0.999414i $$0.489099\pi$$
$$854$$ 5.00000 0.171096
$$855$$ 0 0
$$856$$ 6.00000 0.205076
$$857$$ 27.0000 0.922302 0.461151 0.887322i $$-0.347437\pi$$
0.461151 + 0.887322i $$0.347437\pi$$
$$858$$ 0 0
$$859$$ 8.00000 0.272956 0.136478 0.990643i $$-0.456422\pi$$
0.136478 + 0.990643i $$0.456422\pi$$
$$860$$ 0 0
$$861$$ −1.00000 −0.0340799
$$862$$ −13.0000 −0.442782
$$863$$ 21.0000 0.714848 0.357424 0.933942i $$-0.383655\pi$$
0.357424 + 0.933942i $$0.383655\pi$$
$$864$$ 1.00000 0.0340207
$$865$$ 0 0
$$866$$ −20.0000 −0.679628
$$867$$ −16.0000 −0.543388
$$868$$ 1.00000 0.0339422
$$869$$ 0 0
$$870$$ 0 0
$$871$$ 0 0
$$872$$ 1.00000 0.0338643
$$873$$ −1.00000 −0.0338449
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 10.0000 0.337869
$$877$$ −13.0000 −0.438979 −0.219489 0.975615i $$-0.570439\pi$$
−0.219489 + 0.975615i $$0.570439\pi$$
$$878$$ −33.0000 −1.11370
$$879$$ 9.00000 0.303562
$$880$$ 0 0
$$881$$ −45.0000 −1.51609 −0.758044 0.652203i $$-0.773845\pi$$
−0.758044 + 0.652203i $$0.773845\pi$$
$$882$$ −6.00000 −0.202031
$$883$$ 33.0000 1.11054 0.555269 0.831671i $$-0.312615\pi$$
0.555269 + 0.831671i $$0.312615\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ −6.00000 −0.201574
$$887$$ −45.0000 −1.51095 −0.755476 0.655176i $$-0.772594\pi$$
−0.755476 + 0.655176i $$0.772594\pi$$
$$888$$ −1.00000 −0.0335578
$$889$$ −8.00000 −0.268311
$$890$$ 0 0
$$891$$ 5.00000 0.167506
$$892$$ −5.00000 −0.167412
$$893$$ 0 0
$$894$$ 4.00000 0.133780
$$895$$ 0 0
$$896$$ 1.00000 0.0334077
$$897$$ 0 0
$$898$$ −24.0000 −0.800890
$$899$$ −3.00000 −0.100056
$$900$$ 0 0
$$901$$ 3.00000 0.0999445
$$902$$ −5.00000 −0.166482
$$903$$ 7.00000 0.232945
$$904$$ −21.0000 −0.698450
$$905$$ 0 0
$$906$$ −2.00000 −0.0664455
$$907$$ 36.0000 1.19536 0.597680 0.801735i $$-0.296089\pi$$
0.597680 + 0.801735i $$0.296089\pi$$
$$908$$ 1.00000 0.0331862
$$909$$ 10.0000 0.331679
$$910$$ 0 0
$$911$$ 12.0000 0.397578 0.198789 0.980042i $$-0.436299\pi$$
0.198789 + 0.980042i $$0.436299\pi$$
$$912$$ 0 0
$$913$$ −40.0000 −1.32381
$$914$$ 19.0000 0.628464
$$915$$ 0 0
$$916$$ −16.0000 −0.528655
$$917$$ −6.00000 −0.198137
$$918$$ 1.00000 0.0330049
$$919$$ 48.0000 1.58337 0.791687 0.610927i $$-0.209203\pi$$
0.791687 + 0.610927i $$0.209203\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 3.00000 0.0987997
$$923$$ 0 0
$$924$$ 5.00000 0.164488
$$925$$ 0 0
$$926$$ −24.0000 −0.788689
$$927$$ 8.00000 0.262754
$$928$$ −3.00000 −0.0984798
$$929$$ −3.00000 −0.0984268 −0.0492134 0.998788i $$-0.515671\pi$$
−0.0492134 + 0.998788i $$0.515671\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ −10.0000 −0.327561
$$933$$ 15.0000 0.491078
$$934$$ 27.0000 0.883467
$$935$$ 0 0
$$936$$ 0 0
$$937$$ −16.0000 −0.522697 −0.261349 0.965244i $$-0.584167\pi$$
−0.261349 + 0.965244i $$0.584167\pi$$
$$938$$ −4.00000 −0.130605
$$939$$ 10.0000 0.326338
$$940$$ 0 0
$$941$$ −22.0000 −0.717180 −0.358590 0.933495i $$-0.616742\pi$$
−0.358590 + 0.933495i $$0.616742\pi$$
$$942$$ 23.0000 0.749380
$$943$$ −4.00000 −0.130258
$$944$$ −8.00000 −0.260378
$$945$$ 0 0
$$946$$ 35.0000 1.13795
$$947$$ −27.0000 −0.877382 −0.438691 0.898638i $$-0.644558\pi$$
−0.438691 + 0.898638i $$0.644558\pi$$
$$948$$ 0 0
$$949$$ 0 0
$$950$$ 0 0
$$951$$ −5.00000 −0.162136
$$952$$ 1.00000 0.0324102
$$953$$ 14.0000 0.453504 0.226752 0.973952i $$-0.427189\pi$$
0.226752 + 0.973952i $$0.427189\pi$$
$$954$$ 3.00000 0.0971286
$$955$$ 0 0
$$956$$ 11.0000 0.355765
$$957$$ −15.0000 −0.484881
$$958$$ −28.0000 −0.904639
$$959$$ 18.0000 0.581250
$$960$$ 0 0
$$961$$ −30.0000 −0.967742
$$962$$ 0 0
$$963$$ 6.00000 0.193347
$$964$$ −2.00000 −0.0644157
$$965$$ 0 0
$$966$$ 4.00000 0.128698
$$967$$ −10.0000 −0.321578 −0.160789 0.986989i $$-0.551404\pi$$
−0.160789 + 0.986989i $$0.551404\pi$$
$$968$$ 14.0000 0.449977
$$969$$ 0 0
$$970$$ 0 0
$$971$$ 39.0000 1.25157 0.625785 0.779996i $$-0.284779\pi$$
0.625785 + 0.779996i $$0.284779\pi$$
$$972$$ 1.00000 0.0320750
$$973$$ −11.0000 −0.352644
$$974$$ −12.0000 −0.384505
$$975$$ 0 0
$$976$$ 5.00000 0.160046
$$977$$ −3.00000 −0.0959785 −0.0479893 0.998848i $$-0.515281\pi$$
−0.0479893 + 0.998848i $$0.515281\pi$$
$$978$$ −1.00000 −0.0319765
$$979$$ 40.0000 1.27841
$$980$$ 0 0
$$981$$ 1.00000 0.0319275
$$982$$ 12.0000 0.382935
$$983$$ −7.00000 −0.223265 −0.111633 0.993750i $$-0.535608\pi$$
−0.111633 + 0.993750i $$0.535608\pi$$
$$984$$ −1.00000 −0.0318788
$$985$$ 0 0
$$986$$ −3.00000 −0.0955395
$$987$$ −4.00000 −0.127321
$$988$$ 0 0
$$989$$ 28.0000 0.890348
$$990$$ 0 0
$$991$$ −5.00000 −0.158830 −0.0794151 0.996842i $$-0.525305\pi$$
−0.0794151 + 0.996842i $$0.525305\pi$$
$$992$$ 1.00000 0.0317500
$$993$$ −26.0000 −0.825085
$$994$$ −6.00000 −0.190308
$$995$$ 0 0
$$996$$ −8.00000 −0.253490
$$997$$ −24.0000 −0.760088 −0.380044 0.924968i $$-0.624091\pi$$
−0.380044 + 0.924968i $$0.624091\pi$$
$$998$$ −30.0000 −0.949633
$$999$$ −1.00000 −0.0316386
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 5550.2.a.bo.1.1 1
5.2 odd 4 1110.2.d.c.889.2 yes 2
5.3 odd 4 1110.2.d.c.889.1 2
5.4 even 2 5550.2.a.d.1.1 1
15.2 even 4 3330.2.d.d.1999.1 2
15.8 even 4 3330.2.d.d.1999.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
1110.2.d.c.889.1 2 5.3 odd 4
1110.2.d.c.889.2 yes 2 5.2 odd 4
3330.2.d.d.1999.1 2 15.2 even 4
3330.2.d.d.1999.2 2 15.8 even 4
5550.2.a.d.1.1 1 5.4 even 2
5550.2.a.bo.1.1 1 1.1 even 1 trivial