# Properties

 Label 3025.2.a.y.1.3 Level $3025$ Weight $2$ Character 3025.1 Self dual yes Analytic conductor $24.155$ Analytic rank $1$ Dimension $4$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$3025 = 5^{2} \cdot 11^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3025.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$24.1547466114$$ Analytic rank: $$1$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{24})^+$$ Defining polynomial: $$x^{4} - 4x^{2} + 1$$ x^4 - 4*x^2 + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 605) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.3 Root $$0.517638$$ of defining polynomial Character $$\chi$$ $$=$$ 3025.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+0.517638 q^{2} -1.93185 q^{3} -1.73205 q^{4} -1.00000 q^{6} -3.34607 q^{7} -1.93185 q^{8} +0.732051 q^{9} +O(q^{10})$$ $$q+0.517638 q^{2} -1.93185 q^{3} -1.73205 q^{4} -1.00000 q^{6} -3.34607 q^{7} -1.93185 q^{8} +0.732051 q^{9} +3.34607 q^{12} +4.24264 q^{13} -1.73205 q^{14} +2.46410 q^{16} +3.86370 q^{17} +0.378937 q^{18} -4.19615 q^{19} +6.46410 q^{21} +3.48477 q^{23} +3.73205 q^{24} +2.19615 q^{26} +4.38134 q^{27} +5.79555 q^{28} +6.92820 q^{29} -8.73205 q^{31} +5.13922 q^{32} +2.00000 q^{34} -1.26795 q^{36} +1.79315 q^{37} -2.17209 q^{38} -8.19615 q^{39} +1.73205 q^{41} +3.34607 q^{42} -6.45189 q^{43} +1.80385 q^{46} +11.4524 q^{47} -4.76028 q^{48} +4.19615 q^{49} -7.46410 q^{51} -7.34847 q^{52} +2.17209 q^{53} +2.26795 q^{54} +6.46410 q^{56} +8.10634 q^{57} +3.58630 q^{58} -1.26795 q^{59} -11.7321 q^{61} -4.52004 q^{62} -2.44949 q^{63} -2.26795 q^{64} +2.20925 q^{67} -6.69213 q^{68} -6.73205 q^{69} -8.19615 q^{71} -1.41421 q^{72} -4.89898 q^{73} +0.928203 q^{74} +7.26795 q^{76} -4.24264 q^{78} +1.46410 q^{79} -10.6603 q^{81} +0.896575 q^{82} -9.89949 q^{83} -11.1962 q^{84} -3.33975 q^{86} -13.3843 q^{87} +0.464102 q^{89} -14.1962 q^{91} -6.03579 q^{92} +16.8690 q^{93} +5.92820 q^{94} -9.92820 q^{96} -9.14162 q^{97} +2.17209 q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 4 q^{6} - 4 q^{9}+O(q^{10})$$ 4 * q - 4 * q^6 - 4 * q^9 $$4 q - 4 q^{6} - 4 q^{9} - 4 q^{16} + 4 q^{19} + 12 q^{21} + 8 q^{24} - 12 q^{26} - 28 q^{31} + 8 q^{34} - 12 q^{36} - 12 q^{39} + 28 q^{46} - 4 q^{49} - 16 q^{51} + 16 q^{54} + 12 q^{56} - 12 q^{59} - 40 q^{61} - 16 q^{64} - 20 q^{69} - 12 q^{71} - 24 q^{74} + 36 q^{76} - 8 q^{79} - 8 q^{81} - 24 q^{84} - 48 q^{86} - 12 q^{89} - 36 q^{91} - 4 q^{94} - 12 q^{96}+O(q^{100})$$ 4 * q - 4 * q^6 - 4 * q^9 - 4 * q^16 + 4 * q^19 + 12 * q^21 + 8 * q^24 - 12 * q^26 - 28 * q^31 + 8 * q^34 - 12 * q^36 - 12 * q^39 + 28 * q^46 - 4 * q^49 - 16 * q^51 + 16 * q^54 + 12 * q^56 - 12 * q^59 - 40 * q^61 - 16 * q^64 - 20 * q^69 - 12 * q^71 - 24 * q^74 + 36 * q^76 - 8 * q^79 - 8 * q^81 - 24 * q^84 - 48 * q^86 - 12 * q^89 - 36 * q^91 - 4 * q^94 - 12 * q^96

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0.517638 0.366025 0.183013 0.983111i $$-0.441415\pi$$
0.183013 + 0.983111i $$0.441415\pi$$
$$3$$ −1.93185 −1.11536 −0.557678 0.830058i $$-0.688307\pi$$
−0.557678 + 0.830058i $$0.688307\pi$$
$$4$$ −1.73205 −0.866025
$$5$$ 0 0
$$6$$ −1.00000 −0.408248
$$7$$ −3.34607 −1.26469 −0.632347 0.774685i $$-0.717908\pi$$
−0.632347 + 0.774685i $$0.717908\pi$$
$$8$$ −1.93185 −0.683013
$$9$$ 0.732051 0.244017
$$10$$ 0 0
$$11$$ 0 0
$$12$$ 3.34607 0.965926
$$13$$ 4.24264 1.17670 0.588348 0.808608i $$-0.299778\pi$$
0.588348 + 0.808608i $$0.299778\pi$$
$$14$$ −1.73205 −0.462910
$$15$$ 0 0
$$16$$ 2.46410 0.616025
$$17$$ 3.86370 0.937086 0.468543 0.883441i $$-0.344779\pi$$
0.468543 + 0.883441i $$0.344779\pi$$
$$18$$ 0.378937 0.0893164
$$19$$ −4.19615 −0.962663 −0.481332 0.876539i $$-0.659847\pi$$
−0.481332 + 0.876539i $$0.659847\pi$$
$$20$$ 0 0
$$21$$ 6.46410 1.41058
$$22$$ 0 0
$$23$$ 3.48477 0.726624 0.363312 0.931668i $$-0.381646\pi$$
0.363312 + 0.931668i $$0.381646\pi$$
$$24$$ 3.73205 0.761802
$$25$$ 0 0
$$26$$ 2.19615 0.430701
$$27$$ 4.38134 0.843190
$$28$$ 5.79555 1.09526
$$29$$ 6.92820 1.28654 0.643268 0.765641i $$-0.277578\pi$$
0.643268 + 0.765641i $$0.277578\pi$$
$$30$$ 0 0
$$31$$ −8.73205 −1.56832 −0.784161 0.620557i $$-0.786907\pi$$
−0.784161 + 0.620557i $$0.786907\pi$$
$$32$$ 5.13922 0.908494
$$33$$ 0 0
$$34$$ 2.00000 0.342997
$$35$$ 0 0
$$36$$ −1.26795 −0.211325
$$37$$ 1.79315 0.294792 0.147396 0.989078i $$-0.452911\pi$$
0.147396 + 0.989078i $$0.452911\pi$$
$$38$$ −2.17209 −0.352359
$$39$$ −8.19615 −1.31243
$$40$$ 0 0
$$41$$ 1.73205 0.270501 0.135250 0.990811i $$-0.456816\pi$$
0.135250 + 0.990811i $$0.456816\pi$$
$$42$$ 3.34607 0.516309
$$43$$ −6.45189 −0.983905 −0.491952 0.870622i $$-0.663717\pi$$
−0.491952 + 0.870622i $$0.663717\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 1.80385 0.265963
$$47$$ 11.4524 1.67051 0.835253 0.549866i $$-0.185321\pi$$
0.835253 + 0.549866i $$0.185321\pi$$
$$48$$ −4.76028 −0.687087
$$49$$ 4.19615 0.599450
$$50$$ 0 0
$$51$$ −7.46410 −1.04518
$$52$$ −7.34847 −1.01905
$$53$$ 2.17209 0.298359 0.149180 0.988810i $$-0.452337\pi$$
0.149180 + 0.988810i $$0.452337\pi$$
$$54$$ 2.26795 0.308629
$$55$$ 0 0
$$56$$ 6.46410 0.863802
$$57$$ 8.10634 1.07371
$$58$$ 3.58630 0.470905
$$59$$ −1.26795 −0.165073 −0.0825365 0.996588i $$-0.526302\pi$$
−0.0825365 + 0.996588i $$0.526302\pi$$
$$60$$ 0 0
$$61$$ −11.7321 −1.50214 −0.751068 0.660225i $$-0.770461\pi$$
−0.751068 + 0.660225i $$0.770461\pi$$
$$62$$ −4.52004 −0.574046
$$63$$ −2.44949 −0.308607
$$64$$ −2.26795 −0.283494
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 2.20925 0.269903 0.134952 0.990852i $$-0.456912\pi$$
0.134952 + 0.990852i $$0.456912\pi$$
$$68$$ −6.69213 −0.811540
$$69$$ −6.73205 −0.810444
$$70$$ 0 0
$$71$$ −8.19615 −0.972704 −0.486352 0.873763i $$-0.661673\pi$$
−0.486352 + 0.873763i $$0.661673\pi$$
$$72$$ −1.41421 −0.166667
$$73$$ −4.89898 −0.573382 −0.286691 0.958023i $$-0.592555\pi$$
−0.286691 + 0.958023i $$0.592555\pi$$
$$74$$ 0.928203 0.107901
$$75$$ 0 0
$$76$$ 7.26795 0.833691
$$77$$ 0 0
$$78$$ −4.24264 −0.480384
$$79$$ 1.46410 0.164724 0.0823622 0.996602i $$-0.473754\pi$$
0.0823622 + 0.996602i $$0.473754\pi$$
$$80$$ 0 0
$$81$$ −10.6603 −1.18447
$$82$$ 0.896575 0.0990102
$$83$$ −9.89949 −1.08661 −0.543305 0.839535i $$-0.682827\pi$$
−0.543305 + 0.839535i $$0.682827\pi$$
$$84$$ −11.1962 −1.22160
$$85$$ 0 0
$$86$$ −3.33975 −0.360134
$$87$$ −13.3843 −1.43494
$$88$$ 0 0
$$89$$ 0.464102 0.0491947 0.0245973 0.999697i $$-0.492170\pi$$
0.0245973 + 0.999697i $$0.492170\pi$$
$$90$$ 0 0
$$91$$ −14.1962 −1.48816
$$92$$ −6.03579 −0.629275
$$93$$ 16.8690 1.74924
$$94$$ 5.92820 0.611447
$$95$$ 0 0
$$96$$ −9.92820 −1.01329
$$97$$ −9.14162 −0.928191 −0.464095 0.885785i $$-0.653621\pi$$
−0.464095 + 0.885785i $$0.653621\pi$$
$$98$$ 2.17209 0.219414
$$99$$ 0 0
$$100$$ 0 0
$$101$$ 19.3923 1.92961 0.964803 0.262973i $$-0.0847030\pi$$
0.964803 + 0.262973i $$0.0847030\pi$$
$$102$$ −3.86370 −0.382564
$$103$$ 4.24264 0.418040 0.209020 0.977911i $$-0.432973\pi$$
0.209020 + 0.977911i $$0.432973\pi$$
$$104$$ −8.19615 −0.803699
$$105$$ 0 0
$$106$$ 1.12436 0.109207
$$107$$ 2.96713 0.286843 0.143422 0.989662i $$-0.454190\pi$$
0.143422 + 0.989662i $$0.454190\pi$$
$$108$$ −7.58871 −0.730224
$$109$$ 9.19615 0.880832 0.440416 0.897794i $$-0.354831\pi$$
0.440416 + 0.897794i $$0.354831\pi$$
$$110$$ 0 0
$$111$$ −3.46410 −0.328798
$$112$$ −8.24504 −0.779083
$$113$$ 2.82843 0.266076 0.133038 0.991111i $$-0.457527\pi$$
0.133038 + 0.991111i $$0.457527\pi$$
$$114$$ 4.19615 0.393006
$$115$$ 0 0
$$116$$ −12.0000 −1.11417
$$117$$ 3.10583 0.287134
$$118$$ −0.656339 −0.0604209
$$119$$ −12.9282 −1.18513
$$120$$ 0 0
$$121$$ 0 0
$$122$$ −6.07296 −0.549820
$$123$$ −3.34607 −0.301705
$$124$$ 15.1244 1.35821
$$125$$ 0 0
$$126$$ −1.26795 −0.112958
$$127$$ −0.896575 −0.0795582 −0.0397791 0.999208i $$-0.512665\pi$$
−0.0397791 + 0.999208i $$0.512665\pi$$
$$128$$ −11.4524 −1.01226
$$129$$ 12.4641 1.09740
$$130$$ 0 0
$$131$$ −3.12436 −0.272976 −0.136488 0.990642i $$-0.543582\pi$$
−0.136488 + 0.990642i $$0.543582\pi$$
$$132$$ 0 0
$$133$$ 14.0406 1.21747
$$134$$ 1.14359 0.0987914
$$135$$ 0 0
$$136$$ −7.46410 −0.640041
$$137$$ −8.86422 −0.757321 −0.378661 0.925536i $$-0.623615\pi$$
−0.378661 + 0.925536i $$0.623615\pi$$
$$138$$ −3.48477 −0.296643
$$139$$ −14.5885 −1.23738 −0.618688 0.785636i $$-0.712336\pi$$
−0.618688 + 0.785636i $$0.712336\pi$$
$$140$$ 0 0
$$141$$ −22.1244 −1.86321
$$142$$ −4.24264 −0.356034
$$143$$ 0 0
$$144$$ 1.80385 0.150321
$$145$$ 0 0
$$146$$ −2.53590 −0.209872
$$147$$ −8.10634 −0.668600
$$148$$ −3.10583 −0.255298
$$149$$ −10.2679 −0.841183 −0.420592 0.907250i $$-0.638177\pi$$
−0.420592 + 0.907250i $$0.638177\pi$$
$$150$$ 0 0
$$151$$ 4.19615 0.341478 0.170739 0.985316i $$-0.445384\pi$$
0.170739 + 0.985316i $$0.445384\pi$$
$$152$$ 8.10634 0.657511
$$153$$ 2.82843 0.228665
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 14.1962 1.13660
$$157$$ −21.8695 −1.74538 −0.872690 0.488275i $$-0.837626\pi$$
−0.872690 + 0.488275i $$0.837626\pi$$
$$158$$ 0.757875 0.0602933
$$159$$ −4.19615 −0.332777
$$160$$ 0 0
$$161$$ −11.6603 −0.918957
$$162$$ −5.51815 −0.433547
$$163$$ −5.13922 −0.402534 −0.201267 0.979536i $$-0.564506\pi$$
−0.201267 + 0.979536i $$0.564506\pi$$
$$164$$ −3.00000 −0.234261
$$165$$ 0 0
$$166$$ −5.12436 −0.397727
$$167$$ 0.517638 0.0400560 0.0200280 0.999799i $$-0.493624\pi$$
0.0200280 + 0.999799i $$0.493624\pi$$
$$168$$ −12.4877 −0.963446
$$169$$ 5.00000 0.384615
$$170$$ 0 0
$$171$$ −3.07180 −0.234906
$$172$$ 11.1750 0.852086
$$173$$ 21.7680 1.65499 0.827495 0.561472i $$-0.189765\pi$$
0.827495 + 0.561472i $$0.189765\pi$$
$$174$$ −6.92820 −0.525226
$$175$$ 0 0
$$176$$ 0 0
$$177$$ 2.44949 0.184115
$$178$$ 0.240237 0.0180065
$$179$$ −15.4641 −1.15584 −0.577921 0.816093i $$-0.696136\pi$$
−0.577921 + 0.816093i $$0.696136\pi$$
$$180$$ 0 0
$$181$$ −17.3923 −1.29276 −0.646380 0.763016i $$-0.723718\pi$$
−0.646380 + 0.763016i $$0.723718\pi$$
$$182$$ −7.34847 −0.544705
$$183$$ 22.6646 1.67541
$$184$$ −6.73205 −0.496293
$$185$$ 0 0
$$186$$ 8.73205 0.640265
$$187$$ 0 0
$$188$$ −19.8362 −1.44670
$$189$$ −14.6603 −1.06638
$$190$$ 0 0
$$191$$ −7.26795 −0.525890 −0.262945 0.964811i $$-0.584694\pi$$
−0.262945 + 0.964811i $$0.584694\pi$$
$$192$$ 4.38134 0.316196
$$193$$ 15.1774 1.09249 0.546247 0.837624i $$-0.316056\pi$$
0.546247 + 0.837624i $$0.316056\pi$$
$$194$$ −4.73205 −0.339741
$$195$$ 0 0
$$196$$ −7.26795 −0.519139
$$197$$ 17.2480 1.22887 0.614433 0.788969i $$-0.289385\pi$$
0.614433 + 0.788969i $$0.289385\pi$$
$$198$$ 0 0
$$199$$ −2.00000 −0.141776 −0.0708881 0.997484i $$-0.522583\pi$$
−0.0708881 + 0.997484i $$0.522583\pi$$
$$200$$ 0 0
$$201$$ −4.26795 −0.301038
$$202$$ 10.0382 0.706285
$$203$$ −23.1822 −1.62707
$$204$$ 12.9282 0.905155
$$205$$ 0 0
$$206$$ 2.19615 0.153013
$$207$$ 2.55103 0.177309
$$208$$ 10.4543 0.724875
$$209$$ 0 0
$$210$$ 0 0
$$211$$ −13.1244 −0.903518 −0.451759 0.892140i $$-0.649203\pi$$
−0.451759 + 0.892140i $$0.649203\pi$$
$$212$$ −3.76217 −0.258387
$$213$$ 15.8338 1.08491
$$214$$ 1.53590 0.104992
$$215$$ 0 0
$$216$$ −8.46410 −0.575909
$$217$$ 29.2180 1.98345
$$218$$ 4.76028 0.322407
$$219$$ 9.46410 0.639525
$$220$$ 0 0
$$221$$ 16.3923 1.10267
$$222$$ −1.79315 −0.120348
$$223$$ 1.55291 0.103991 0.0519954 0.998647i $$-0.483442\pi$$
0.0519954 + 0.998647i $$0.483442\pi$$
$$224$$ −17.1962 −1.14897
$$225$$ 0 0
$$226$$ 1.46410 0.0973906
$$227$$ 20.1136 1.33498 0.667492 0.744617i $$-0.267368\pi$$
0.667492 + 0.744617i $$0.267368\pi$$
$$228$$ −14.0406 −0.929861
$$229$$ −4.07180 −0.269072 −0.134536 0.990909i $$-0.542954\pi$$
−0.134536 + 0.990909i $$0.542954\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ −13.3843 −0.878720
$$233$$ 1.69161 0.110821 0.0554107 0.998464i $$-0.482353\pi$$
0.0554107 + 0.998464i $$0.482353\pi$$
$$234$$ 1.60770 0.105098
$$235$$ 0 0
$$236$$ 2.19615 0.142957
$$237$$ −2.82843 −0.183726
$$238$$ −6.69213 −0.433786
$$239$$ 5.66025 0.366131 0.183066 0.983101i $$-0.441398\pi$$
0.183066 + 0.983101i $$0.441398\pi$$
$$240$$ 0 0
$$241$$ 14.1244 0.909830 0.454915 0.890535i $$-0.349670\pi$$
0.454915 + 0.890535i $$0.349670\pi$$
$$242$$ 0 0
$$243$$ 7.45001 0.477918
$$244$$ 20.3205 1.30089
$$245$$ 0 0
$$246$$ −1.73205 −0.110432
$$247$$ −17.8028 −1.13276
$$248$$ 16.8690 1.07118
$$249$$ 19.1244 1.21196
$$250$$ 0 0
$$251$$ −3.80385 −0.240097 −0.120048 0.992768i $$-0.538305\pi$$
−0.120048 + 0.992768i $$0.538305\pi$$
$$252$$ 4.24264 0.267261
$$253$$ 0 0
$$254$$ −0.464102 −0.0291203
$$255$$ 0 0
$$256$$ −1.39230 −0.0870191
$$257$$ 14.7985 0.923103 0.461552 0.887113i $$-0.347293\pi$$
0.461552 + 0.887113i $$0.347293\pi$$
$$258$$ 6.45189 0.401677
$$259$$ −6.00000 −0.372822
$$260$$ 0 0
$$261$$ 5.07180 0.313936
$$262$$ −1.61729 −0.0999162
$$263$$ −6.79367 −0.418915 −0.209458 0.977818i $$-0.567170\pi$$
−0.209458 + 0.977818i $$0.567170\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 7.26795 0.445627
$$267$$ −0.896575 −0.0548695
$$268$$ −3.82654 −0.233743
$$269$$ −16.2679 −0.991874 −0.495937 0.868358i $$-0.665175\pi$$
−0.495937 + 0.868358i $$0.665175\pi$$
$$270$$ 0 0
$$271$$ 11.4641 0.696395 0.348197 0.937421i $$-0.386794\pi$$
0.348197 + 0.937421i $$0.386794\pi$$
$$272$$ 9.52056 0.577269
$$273$$ 27.4249 1.65983
$$274$$ −4.58846 −0.277199
$$275$$ 0 0
$$276$$ 11.6603 0.701865
$$277$$ −31.1870 −1.87385 −0.936923 0.349535i $$-0.886340\pi$$
−0.936923 + 0.349535i $$0.886340\pi$$
$$278$$ −7.55154 −0.452911
$$279$$ −6.39230 −0.382697
$$280$$ 0 0
$$281$$ −17.3205 −1.03325 −0.516627 0.856210i $$-0.672813\pi$$
−0.516627 + 0.856210i $$0.672813\pi$$
$$282$$ −11.4524 −0.681981
$$283$$ −8.06918 −0.479663 −0.239831 0.970815i $$-0.577092\pi$$
−0.239831 + 0.970815i $$0.577092\pi$$
$$284$$ 14.1962 0.842387
$$285$$ 0 0
$$286$$ 0 0
$$287$$ −5.79555 −0.342101
$$288$$ 3.76217 0.221688
$$289$$ −2.07180 −0.121870
$$290$$ 0 0
$$291$$ 17.6603 1.03526
$$292$$ 8.48528 0.496564
$$293$$ 15.0759 0.880742 0.440371 0.897816i $$-0.354847\pi$$
0.440371 + 0.897816i $$0.354847\pi$$
$$294$$ −4.19615 −0.244725
$$295$$ 0 0
$$296$$ −3.46410 −0.201347
$$297$$ 0 0
$$298$$ −5.31508 −0.307894
$$299$$ 14.7846 0.855016
$$300$$ 0 0
$$301$$ 21.5885 1.24434
$$302$$ 2.17209 0.124990
$$303$$ −37.4631 −2.15220
$$304$$ −10.3397 −0.593025
$$305$$ 0 0
$$306$$ 1.46410 0.0836971
$$307$$ −7.82894 −0.446821 −0.223411 0.974724i $$-0.571719\pi$$
−0.223411 + 0.974724i $$0.571719\pi$$
$$308$$ 0 0
$$309$$ −8.19615 −0.466263
$$310$$ 0 0
$$311$$ 10.0526 0.570028 0.285014 0.958523i $$-0.408002\pi$$
0.285014 + 0.958523i $$0.408002\pi$$
$$312$$ 15.8338 0.896410
$$313$$ −17.1464 −0.969173 −0.484587 0.874743i $$-0.661030\pi$$
−0.484587 + 0.874743i $$0.661030\pi$$
$$314$$ −11.3205 −0.638853
$$315$$ 0 0
$$316$$ −2.53590 −0.142655
$$317$$ −12.6264 −0.709168 −0.354584 0.935024i $$-0.615378\pi$$
−0.354584 + 0.935024i $$0.615378\pi$$
$$318$$ −2.17209 −0.121805
$$319$$ 0 0
$$320$$ 0 0
$$321$$ −5.73205 −0.319932
$$322$$ −6.03579 −0.336362
$$323$$ −16.2127 −0.902098
$$324$$ 18.4641 1.02578
$$325$$ 0 0
$$326$$ −2.66025 −0.147338
$$327$$ −17.7656 −0.982440
$$328$$ −3.34607 −0.184756
$$329$$ −38.3205 −2.11268
$$330$$ 0 0
$$331$$ −18.1962 −1.00015 −0.500075 0.865982i $$-0.666694\pi$$
−0.500075 + 0.865982i $$0.666694\pi$$
$$332$$ 17.1464 0.941033
$$333$$ 1.31268 0.0719343
$$334$$ 0.267949 0.0146615
$$335$$ 0 0
$$336$$ 15.9282 0.868955
$$337$$ −17.6269 −0.960199 −0.480099 0.877214i $$-0.659399\pi$$
−0.480099 + 0.877214i $$0.659399\pi$$
$$338$$ 2.58819 0.140779
$$339$$ −5.46410 −0.296769
$$340$$ 0 0
$$341$$ 0 0
$$342$$ −1.59008 −0.0859816
$$343$$ 9.38186 0.506573
$$344$$ 12.4641 0.672019
$$345$$ 0 0
$$346$$ 11.2679 0.605769
$$347$$ −4.38134 −0.235203 −0.117601 0.993061i $$-0.537521\pi$$
−0.117601 + 0.993061i $$0.537521\pi$$
$$348$$ 23.1822 1.24270
$$349$$ −7.32051 −0.391858 −0.195929 0.980618i $$-0.562772\pi$$
−0.195929 + 0.980618i $$0.562772\pi$$
$$350$$ 0 0
$$351$$ 18.5885 0.992178
$$352$$ 0 0
$$353$$ −13.0053 −0.692204 −0.346102 0.938197i $$-0.612495\pi$$
−0.346102 + 0.938197i $$0.612495\pi$$
$$354$$ 1.26795 0.0673907
$$355$$ 0 0
$$356$$ −0.803848 −0.0426038
$$357$$ 24.9754 1.32184
$$358$$ −8.00481 −0.423067
$$359$$ 35.6603 1.88208 0.941038 0.338301i $$-0.109852\pi$$
0.941038 + 0.338301i $$0.109852\pi$$
$$360$$ 0 0
$$361$$ −1.39230 −0.0732792
$$362$$ −9.00292 −0.473183
$$363$$ 0 0
$$364$$ 24.5885 1.28879
$$365$$ 0 0
$$366$$ 11.7321 0.613244
$$367$$ −16.9062 −0.882496 −0.441248 0.897385i $$-0.645464\pi$$
−0.441248 + 0.897385i $$0.645464\pi$$
$$368$$ 8.58682 0.447619
$$369$$ 1.26795 0.0660068
$$370$$ 0 0
$$371$$ −7.26795 −0.377333
$$372$$ −29.2180 −1.51488
$$373$$ 0.175865 0.00910597 0.00455298 0.999990i $$-0.498551\pi$$
0.00455298 + 0.999990i $$0.498551\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ −22.1244 −1.14098
$$377$$ 29.3939 1.51386
$$378$$ −7.58871 −0.390321
$$379$$ 1.46410 0.0752058 0.0376029 0.999293i $$-0.488028\pi$$
0.0376029 + 0.999293i $$0.488028\pi$$
$$380$$ 0 0
$$381$$ 1.73205 0.0887357
$$382$$ −3.76217 −0.192489
$$383$$ −38.4612 −1.96527 −0.982637 0.185539i $$-0.940597\pi$$
−0.982637 + 0.185539i $$0.940597\pi$$
$$384$$ 22.1244 1.12903
$$385$$ 0 0
$$386$$ 7.85641 0.399881
$$387$$ −4.72311 −0.240089
$$388$$ 15.8338 0.803837
$$389$$ 6.12436 0.310517 0.155259 0.987874i $$-0.450379\pi$$
0.155259 + 0.987874i $$0.450379\pi$$
$$390$$ 0 0
$$391$$ 13.4641 0.680909
$$392$$ −8.10634 −0.409432
$$393$$ 6.03579 0.304465
$$394$$ 8.92820 0.449796
$$395$$ 0 0
$$396$$ 0 0
$$397$$ 32.9802 1.65523 0.827614 0.561298i $$-0.189698\pi$$
0.827614 + 0.561298i $$0.189698\pi$$
$$398$$ −1.03528 −0.0518937
$$399$$ −27.1244 −1.35792
$$400$$ 0 0
$$401$$ 2.32051 0.115881 0.0579403 0.998320i $$-0.481547\pi$$
0.0579403 + 0.998320i $$0.481547\pi$$
$$402$$ −2.20925 −0.110188
$$403$$ −37.0470 −1.84544
$$404$$ −33.5885 −1.67109
$$405$$ 0 0
$$406$$ −12.0000 −0.595550
$$407$$ 0 0
$$408$$ 14.4195 0.713873
$$409$$ −26.1244 −1.29177 −0.645883 0.763436i $$-0.723511\pi$$
−0.645883 + 0.763436i $$0.723511\pi$$
$$410$$ 0 0
$$411$$ 17.1244 0.844682
$$412$$ −7.34847 −0.362033
$$413$$ 4.24264 0.208767
$$414$$ 1.32051 0.0648994
$$415$$ 0 0
$$416$$ 21.8038 1.06902
$$417$$ 28.1827 1.38011
$$418$$ 0 0
$$419$$ −12.3397 −0.602836 −0.301418 0.953492i $$-0.597460\pi$$
−0.301418 + 0.953492i $$0.597460\pi$$
$$420$$ 0 0
$$421$$ −17.0526 −0.831091 −0.415545 0.909572i $$-0.636409\pi$$
−0.415545 + 0.909572i $$0.636409\pi$$
$$422$$ −6.79367 −0.330711
$$423$$ 8.38375 0.407632
$$424$$ −4.19615 −0.203783
$$425$$ 0 0
$$426$$ 8.19615 0.397105
$$427$$ 39.2562 1.89974
$$428$$ −5.13922 −0.248413
$$429$$ 0 0
$$430$$ 0 0
$$431$$ −5.07180 −0.244300 −0.122150 0.992512i $$-0.538979\pi$$
−0.122150 + 0.992512i $$0.538979\pi$$
$$432$$ 10.7961 0.519426
$$433$$ −28.7375 −1.38104 −0.690519 0.723314i $$-0.742618\pi$$
−0.690519 + 0.723314i $$0.742618\pi$$
$$434$$ 15.1244 0.725992
$$435$$ 0 0
$$436$$ −15.9282 −0.762823
$$437$$ −14.6226 −0.699494
$$438$$ 4.89898 0.234082
$$439$$ 28.2487 1.34824 0.674119 0.738623i $$-0.264524\pi$$
0.674119 + 0.738623i $$0.264524\pi$$
$$440$$ 0 0
$$441$$ 3.07180 0.146276
$$442$$ 8.48528 0.403604
$$443$$ 27.5636 1.30958 0.654792 0.755809i $$-0.272756\pi$$
0.654792 + 0.755809i $$0.272756\pi$$
$$444$$ 6.00000 0.284747
$$445$$ 0 0
$$446$$ 0.803848 0.0380633
$$447$$ 19.8362 0.938218
$$448$$ 7.58871 0.358533
$$449$$ 34.5167 1.62894 0.814471 0.580204i $$-0.197027\pi$$
0.814471 + 0.580204i $$0.197027\pi$$
$$450$$ 0 0
$$451$$ 0 0
$$452$$ −4.89898 −0.230429
$$453$$ −8.10634 −0.380869
$$454$$ 10.4115 0.488638
$$455$$ 0 0
$$456$$ −15.6603 −0.733359
$$457$$ −21.2132 −0.992312 −0.496156 0.868233i $$-0.665256\pi$$
−0.496156 + 0.868233i $$0.665256\pi$$
$$458$$ −2.10772 −0.0984872
$$459$$ 16.9282 0.790141
$$460$$ 0 0
$$461$$ −33.0000 −1.53696 −0.768482 0.639872i $$-0.778987\pi$$
−0.768482 + 0.639872i $$0.778987\pi$$
$$462$$ 0 0
$$463$$ −20.3166 −0.944194 −0.472097 0.881547i $$-0.656503\pi$$
−0.472097 + 0.881547i $$0.656503\pi$$
$$464$$ 17.0718 0.792538
$$465$$ 0 0
$$466$$ 0.875644 0.0405634
$$467$$ −0.619174 −0.0286520 −0.0143260 0.999897i $$-0.504560\pi$$
−0.0143260 + 0.999897i $$0.504560\pi$$
$$468$$ −5.37945 −0.248665
$$469$$ −7.39230 −0.341345
$$470$$ 0 0
$$471$$ 42.2487 1.94672
$$472$$ 2.44949 0.112747
$$473$$ 0 0
$$474$$ −1.46410 −0.0672484
$$475$$ 0 0
$$476$$ 22.3923 1.02635
$$477$$ 1.59008 0.0728047
$$478$$ 2.92996 0.134013
$$479$$ 2.53590 0.115868 0.0579341 0.998320i $$-0.481549\pi$$
0.0579341 + 0.998320i $$0.481549\pi$$
$$480$$ 0 0
$$481$$ 7.60770 0.346881
$$482$$ 7.31130 0.333021
$$483$$ 22.5259 1.02496
$$484$$ 0 0
$$485$$ 0 0
$$486$$ 3.85641 0.174930
$$487$$ 12.7279 0.576757 0.288379 0.957516i $$-0.406884\pi$$
0.288379 + 0.957516i $$0.406884\pi$$
$$488$$ 22.6646 1.02598
$$489$$ 9.92820 0.448969
$$490$$ 0 0
$$491$$ 39.7128 1.79221 0.896107 0.443838i $$-0.146383\pi$$
0.896107 + 0.443838i $$0.146383\pi$$
$$492$$ 5.79555 0.261284
$$493$$ 26.7685 1.20559
$$494$$ −9.21539 −0.414620
$$495$$ 0 0
$$496$$ −21.5167 −0.966127
$$497$$ 27.4249 1.23017
$$498$$ 9.89949 0.443607
$$499$$ 17.6077 0.788229 0.394114 0.919061i $$-0.371051\pi$$
0.394114 + 0.919061i $$0.371051\pi$$
$$500$$ 0 0
$$501$$ −1.00000 −0.0446767
$$502$$ −1.96902 −0.0878815
$$503$$ −7.38563 −0.329309 −0.164655 0.986351i $$-0.552651\pi$$
−0.164655 + 0.986351i $$0.552651\pi$$
$$504$$ 4.73205 0.210782
$$505$$ 0 0
$$506$$ 0 0
$$507$$ −9.65926 −0.428983
$$508$$ 1.55291 0.0688994
$$509$$ −8.07180 −0.357776 −0.178888 0.983869i $$-0.557250\pi$$
−0.178888 + 0.983869i $$0.557250\pi$$
$$510$$ 0 0
$$511$$ 16.3923 0.725153
$$512$$ 22.1841 0.980408
$$513$$ −18.3848 −0.811708
$$514$$ 7.66025 0.337879
$$515$$ 0 0
$$516$$ −21.5885 −0.950379
$$517$$ 0 0
$$518$$ −3.10583 −0.136462
$$519$$ −42.0526 −1.84590
$$520$$ 0 0
$$521$$ −9.24871 −0.405193 −0.202597 0.979262i $$-0.564938\pi$$
−0.202597 + 0.979262i $$0.564938\pi$$
$$522$$ 2.62536 0.114909
$$523$$ 25.6317 1.12080 0.560398 0.828223i $$-0.310648\pi$$
0.560398 + 0.828223i $$0.310648\pi$$
$$524$$ 5.41154 0.236404
$$525$$ 0 0
$$526$$ −3.51666 −0.153334
$$527$$ −33.7381 −1.46965
$$528$$ 0 0
$$529$$ −10.8564 −0.472018
$$530$$ 0 0
$$531$$ −0.928203 −0.0402806
$$532$$ −24.3190 −1.05436
$$533$$ 7.34847 0.318298
$$534$$ −0.464102 −0.0200836
$$535$$ 0 0
$$536$$ −4.26795 −0.184347
$$537$$ 29.8744 1.28917
$$538$$ −8.42091 −0.363051
$$539$$ 0 0
$$540$$ 0 0
$$541$$ −2.60770 −0.112114 −0.0560568 0.998428i $$-0.517853\pi$$
−0.0560568 + 0.998428i $$0.517853\pi$$
$$542$$ 5.93426 0.254898
$$543$$ 33.5994 1.44189
$$544$$ 19.8564 0.851336
$$545$$ 0 0
$$546$$ 14.1962 0.607539
$$547$$ −28.7375 −1.22873 −0.614364 0.789023i $$-0.710587\pi$$
−0.614364 + 0.789023i $$0.710587\pi$$
$$548$$ 15.3533 0.655859
$$549$$ −8.58846 −0.366546
$$550$$ 0 0
$$551$$ −29.0718 −1.23850
$$552$$ 13.0053 0.553543
$$553$$ −4.89898 −0.208326
$$554$$ −16.1436 −0.685876
$$555$$ 0 0
$$556$$ 25.2679 1.07160
$$557$$ −37.6018 −1.59324 −0.796619 0.604482i $$-0.793380\pi$$
−0.796619 + 0.604482i $$0.793380\pi$$
$$558$$ −3.30890 −0.140077
$$559$$ −27.3731 −1.15776
$$560$$ 0 0
$$561$$ 0 0
$$562$$ −8.96575 −0.378198
$$563$$ 15.9725 0.673159 0.336579 0.941655i $$-0.390730\pi$$
0.336579 + 0.941655i $$0.390730\pi$$
$$564$$ 38.3205 1.61358
$$565$$ 0 0
$$566$$ −4.17691 −0.175569
$$567$$ 35.6699 1.49800
$$568$$ 15.8338 0.664369
$$569$$ 30.1244 1.26288 0.631439 0.775425i $$-0.282464\pi$$
0.631439 + 0.775425i $$0.282464\pi$$
$$570$$ 0 0
$$571$$ −19.7128 −0.824956 −0.412478 0.910968i $$-0.635337\pi$$
−0.412478 + 0.910968i $$0.635337\pi$$
$$572$$ 0 0
$$573$$ 14.0406 0.586554
$$574$$ −3.00000 −0.125218
$$575$$ 0 0
$$576$$ −1.66025 −0.0691773
$$577$$ −3.76217 −0.156621 −0.0783105 0.996929i $$-0.524953\pi$$
−0.0783105 + 0.996929i $$0.524953\pi$$
$$578$$ −1.07244 −0.0446077
$$579$$ −29.3205 −1.21852
$$580$$ 0 0
$$581$$ 33.1244 1.37423
$$582$$ 9.14162 0.378932
$$583$$ 0 0
$$584$$ 9.46410 0.391627
$$585$$ 0 0
$$586$$ 7.80385 0.322374
$$587$$ −20.3910 −0.841625 −0.420812 0.907148i $$-0.638255\pi$$
−0.420812 + 0.907148i $$0.638255\pi$$
$$588$$ 14.0406 0.579025
$$589$$ 36.6410 1.50977
$$590$$ 0 0
$$591$$ −33.3205 −1.37062
$$592$$ 4.41851 0.181599
$$593$$ 0.859411 0.0352918 0.0176459 0.999844i $$-0.494383\pi$$
0.0176459 + 0.999844i $$0.494383\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 17.7846 0.728486
$$597$$ 3.86370 0.158131
$$598$$ 7.65308 0.312958
$$599$$ −38.4449 −1.57081 −0.785407 0.618979i $$-0.787546\pi$$
−0.785407 + 0.618979i $$0.787546\pi$$
$$600$$ 0 0
$$601$$ 20.0000 0.815817 0.407909 0.913023i $$-0.366258\pi$$
0.407909 + 0.913023i $$0.366258\pi$$
$$602$$ 11.1750 0.455459
$$603$$ 1.61729 0.0658610
$$604$$ −7.26795 −0.295729
$$605$$ 0 0
$$606$$ −19.3923 −0.787759
$$607$$ 36.3906 1.47705 0.738525 0.674226i $$-0.235523\pi$$
0.738525 + 0.674226i $$0.235523\pi$$
$$608$$ −21.5649 −0.874574
$$609$$ 44.7846 1.81476
$$610$$ 0 0
$$611$$ 48.5885 1.96568
$$612$$ −4.89898 −0.198030
$$613$$ 1.79315 0.0724247 0.0362123 0.999344i $$-0.488471\pi$$
0.0362123 + 0.999344i $$0.488471\pi$$
$$614$$ −4.05256 −0.163548
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 13.0053 0.523575 0.261787 0.965126i $$-0.415688\pi$$
0.261787 + 0.965126i $$0.415688\pi$$
$$618$$ −4.24264 −0.170664
$$619$$ 12.7846 0.513857 0.256928 0.966430i $$-0.417290\pi$$
0.256928 + 0.966430i $$0.417290\pi$$
$$620$$ 0 0
$$621$$ 15.2679 0.612682
$$622$$ 5.20359 0.208645
$$623$$ −1.55291 −0.0622162
$$624$$ −20.1962 −0.808493
$$625$$ 0 0
$$626$$ −8.87564 −0.354742
$$627$$ 0 0
$$628$$ 37.8792 1.51154
$$629$$ 6.92820 0.276246
$$630$$ 0 0
$$631$$ −15.3205 −0.609900 −0.304950 0.952368i $$-0.598640\pi$$
−0.304950 + 0.952368i $$0.598640\pi$$
$$632$$ −2.82843 −0.112509
$$633$$ 25.3543 1.00774
$$634$$ −6.53590 −0.259574
$$635$$ 0 0
$$636$$ 7.26795 0.288193
$$637$$ 17.8028 0.705371
$$638$$ 0 0
$$639$$ −6.00000 −0.237356
$$640$$ 0 0
$$641$$ −7.85641 −0.310309 −0.155155 0.987890i $$-0.549588\pi$$
−0.155155 + 0.987890i $$0.549588\pi$$
$$642$$ −2.96713 −0.117103
$$643$$ −10.5187 −0.414816 −0.207408 0.978255i $$-0.566503\pi$$
−0.207408 + 0.978255i $$0.566503\pi$$
$$644$$ 20.1962 0.795840
$$645$$ 0 0
$$646$$ −8.39230 −0.330191
$$647$$ −37.3615 −1.46883 −0.734416 0.678699i $$-0.762544\pi$$
−0.734416 + 0.678699i $$0.762544\pi$$
$$648$$ 20.5940 0.809010
$$649$$ 0 0
$$650$$ 0 0
$$651$$ −56.4449 −2.21225
$$652$$ 8.90138 0.348605
$$653$$ 12.1459 0.475306 0.237653 0.971350i $$-0.423622\pi$$
0.237653 + 0.971350i $$0.423622\pi$$
$$654$$ −9.19615 −0.359598
$$655$$ 0 0
$$656$$ 4.26795 0.166635
$$657$$ −3.58630 −0.139915
$$658$$ −19.8362 −0.773294
$$659$$ −23.3205 −0.908438 −0.454219 0.890890i $$-0.650082\pi$$
−0.454219 + 0.890890i $$0.650082\pi$$
$$660$$ 0 0
$$661$$ 29.5885 1.15086 0.575429 0.817852i $$-0.304835\pi$$
0.575429 + 0.817852i $$0.304835\pi$$
$$662$$ −9.41902 −0.366081
$$663$$ −31.6675 −1.22986
$$664$$ 19.1244 0.742169
$$665$$ 0 0
$$666$$ 0.679492 0.0263298
$$667$$ 24.1432 0.934827
$$668$$ −0.896575 −0.0346895
$$669$$ −3.00000 −0.115987
$$670$$ 0 0
$$671$$ 0 0
$$672$$ 33.2204 1.28151
$$673$$ −40.8091 −1.57308 −0.786538 0.617542i $$-0.788129\pi$$
−0.786538 + 0.617542i $$0.788129\pi$$
$$674$$ −9.12436 −0.351457
$$675$$ 0 0
$$676$$ −8.66025 −0.333087
$$677$$ 20.3538 0.782260 0.391130 0.920335i $$-0.372084\pi$$
0.391130 + 0.920335i $$0.372084\pi$$
$$678$$ −2.82843 −0.108625
$$679$$ 30.5885 1.17388
$$680$$ 0 0
$$681$$ −38.8564 −1.48898
$$682$$ 0 0
$$683$$ 29.8372 1.14169 0.570844 0.821058i $$-0.306616\pi$$
0.570844 + 0.821058i $$0.306616\pi$$
$$684$$ 5.32051 0.203435
$$685$$ 0 0
$$686$$ 4.85641 0.185418
$$687$$ 7.86611 0.300111
$$688$$ −15.8981 −0.606110
$$689$$ 9.21539 0.351078
$$690$$ 0 0
$$691$$ 37.9090 1.44213 0.721063 0.692870i $$-0.243654\pi$$
0.721063 + 0.692870i $$0.243654\pi$$
$$692$$ −37.7033 −1.43326
$$693$$ 0 0
$$694$$ −2.26795 −0.0860902
$$695$$ 0 0
$$696$$ 25.8564 0.980085
$$697$$ 6.69213 0.253483
$$698$$ −3.78937 −0.143430
$$699$$ −3.26795 −0.123605
$$700$$ 0 0
$$701$$ 37.8564 1.42982 0.714908 0.699218i $$-0.246468\pi$$
0.714908 + 0.699218i $$0.246468\pi$$
$$702$$ 9.62209 0.363163
$$703$$ −7.52433 −0.283786
$$704$$ 0 0
$$705$$ 0 0
$$706$$ −6.73205 −0.253364
$$707$$ −64.8879 −2.44036
$$708$$ −4.24264 −0.159448
$$709$$ 2.60770 0.0979340 0.0489670 0.998800i $$-0.484407\pi$$
0.0489670 + 0.998800i $$0.484407\pi$$
$$710$$ 0 0
$$711$$ 1.07180 0.0401955
$$712$$ −0.896575 −0.0336006
$$713$$ −30.4292 −1.13958
$$714$$ 12.9282 0.483826
$$715$$ 0 0
$$716$$ 26.7846 1.00099
$$717$$ −10.9348 −0.408367
$$718$$ 18.4591 0.688888
$$719$$ −50.1962 −1.87200 −0.936000 0.351999i $$-0.885502\pi$$
−0.936000 + 0.351999i $$0.885502\pi$$
$$720$$ 0 0
$$721$$ −14.1962 −0.528692
$$722$$ −0.720710 −0.0268220
$$723$$ −27.2862 −1.01478
$$724$$ 30.1244 1.11956
$$725$$ 0 0
$$726$$ 0 0
$$727$$ −8.24504 −0.305792 −0.152896 0.988242i $$-0.548860\pi$$
−0.152896 + 0.988242i $$0.548860\pi$$
$$728$$ 27.4249 1.01643
$$729$$ 17.5885 0.651424
$$730$$ 0 0
$$731$$ −24.9282 −0.922003
$$732$$ −39.2562 −1.45095
$$733$$ −9.31749 −0.344149 −0.172075 0.985084i $$-0.555047\pi$$
−0.172075 + 0.985084i $$0.555047\pi$$
$$734$$ −8.75129 −0.323016
$$735$$ 0 0
$$736$$ 17.9090 0.660133
$$737$$ 0 0
$$738$$ 0.656339 0.0241602
$$739$$ −38.9282 −1.43200 −0.715999 0.698102i $$-0.754028\pi$$
−0.715999 + 0.698102i $$0.754028\pi$$
$$740$$ 0 0
$$741$$ 34.3923 1.26343
$$742$$ −3.76217 −0.138114
$$743$$ 43.3973 1.59209 0.796046 0.605235i $$-0.206921\pi$$
0.796046 + 0.605235i $$0.206921\pi$$
$$744$$ −32.5885 −1.19475
$$745$$ 0 0
$$746$$ 0.0910347 0.00333302
$$747$$ −7.24693 −0.265151
$$748$$ 0 0
$$749$$ −9.92820 −0.362769
$$750$$ 0 0
$$751$$ 48.6410 1.77494 0.887468 0.460869i $$-0.152462\pi$$
0.887468 + 0.460869i $$0.152462\pi$$
$$752$$ 28.2199 1.02907
$$753$$ 7.34847 0.267793
$$754$$ 15.2154 0.554112
$$755$$ 0 0
$$756$$ 25.3923 0.923509
$$757$$ −49.7749 −1.80910 −0.904549 0.426369i $$-0.859793\pi$$
−0.904549 + 0.426369i $$0.859793\pi$$
$$758$$ 0.757875 0.0275273
$$759$$ 0 0
$$760$$ 0 0
$$761$$ −19.8564 −0.719794 −0.359897 0.932992i $$-0.617188\pi$$
−0.359897 + 0.932992i $$0.617188\pi$$
$$762$$ 0.896575 0.0324795
$$763$$ −30.7709 −1.11398
$$764$$ 12.5885 0.455434
$$765$$ 0 0
$$766$$ −19.9090 −0.719340
$$767$$ −5.37945 −0.194241
$$768$$ 2.68973 0.0970571
$$769$$ −9.85641 −0.355431 −0.177716 0.984082i $$-0.556871\pi$$
−0.177716 + 0.984082i $$0.556871\pi$$
$$770$$ 0 0
$$771$$ −28.5885 −1.02959
$$772$$ −26.2880 −0.946128
$$773$$ −12.0444 −0.433206 −0.216603 0.976260i $$-0.569498\pi$$
−0.216603 + 0.976260i $$0.569498\pi$$
$$774$$ −2.44486 −0.0878788
$$775$$ 0 0
$$776$$ 17.6603 0.633966
$$777$$ 11.5911 0.415829
$$778$$ 3.17020 0.113657
$$779$$ −7.26795 −0.260401
$$780$$ 0 0
$$781$$ 0 0
$$782$$ 6.96953 0.249230
$$783$$ 30.3548 1.08479
$$784$$ 10.3397 0.369277
$$785$$ 0 0
$$786$$ 3.12436 0.111442
$$787$$ −17.3867 −0.619768 −0.309884 0.950774i $$-0.600290\pi$$
−0.309884 + 0.950774i $$0.600290\pi$$
$$788$$ −29.8744 −1.06423
$$789$$ 13.1244 0.467239
$$790$$ 0 0
$$791$$ −9.46410 −0.336505
$$792$$ 0 0
$$793$$ −49.7749 −1.76756
$$794$$ 17.0718 0.605855
$$795$$ 0 0
$$796$$ 3.46410 0.122782
$$797$$ 36.1875 1.28183 0.640914 0.767612i $$-0.278555\pi$$
0.640914 + 0.767612i $$0.278555\pi$$
$$798$$ −14.0406 −0.497032
$$799$$ 44.2487 1.56541
$$800$$ 0 0
$$801$$ 0.339746 0.0120043
$$802$$ 1.20118 0.0424153
$$803$$ 0 0
$$804$$ 7.39230 0.260706
$$805$$ 0 0
$$806$$ −19.1769 −0.675478
$$807$$ 31.4273 1.10629
$$808$$ −37.4631 −1.31795
$$809$$ −14.5359 −0.511055 −0.255527 0.966802i $$-0.582249\pi$$
−0.255527 + 0.966802i $$0.582249\pi$$
$$810$$ 0 0
$$811$$ 18.9808 0.666505 0.333252 0.942838i $$-0.391854\pi$$
0.333252 + 0.942838i $$0.391854\pi$$
$$812$$ 40.1528 1.40909
$$813$$ −22.1469 −0.776727
$$814$$ 0 0
$$815$$ 0 0
$$816$$ −18.3923 −0.643859
$$817$$ 27.0731 0.947169
$$818$$ −13.5230 −0.472819
$$819$$ −10.3923 −0.363137
$$820$$ 0 0
$$821$$ 29.1962 1.01895 0.509476 0.860485i $$-0.329839\pi$$
0.509476 + 0.860485i $$0.329839\pi$$
$$822$$ 8.86422 0.309175
$$823$$ −6.75650 −0.235517 −0.117758 0.993042i $$-0.537571\pi$$
−0.117758 + 0.993042i $$0.537571\pi$$
$$824$$ −8.19615 −0.285526
$$825$$ 0 0
$$826$$ 2.19615 0.0764139
$$827$$ −34.0798 −1.18507 −0.592536 0.805544i $$-0.701873\pi$$
−0.592536 + 0.805544i $$0.701873\pi$$
$$828$$ −4.41851 −0.153554
$$829$$ −9.98076 −0.346646 −0.173323 0.984865i $$-0.555450\pi$$
−0.173323 + 0.984865i $$0.555450\pi$$
$$830$$ 0 0
$$831$$ 60.2487 2.09000
$$832$$ −9.62209 −0.333586
$$833$$ 16.2127 0.561736
$$834$$ 14.5885 0.505157
$$835$$ 0 0
$$836$$ 0 0
$$837$$ −38.2581 −1.32239
$$838$$ −6.38752 −0.220653
$$839$$ −38.7846 −1.33899 −0.669497 0.742815i $$-0.733490\pi$$
−0.669497 + 0.742815i $$0.733490\pi$$
$$840$$ 0 0
$$841$$ 19.0000 0.655172
$$842$$ −8.82705 −0.304200
$$843$$ 33.4607 1.15245
$$844$$ 22.7321 0.782469
$$845$$ 0 0
$$846$$ 4.33975 0.149204
$$847$$ 0 0
$$848$$ 5.35225 0.183797
$$849$$ 15.5885 0.534994
$$850$$ 0 0
$$851$$ 6.24871 0.214203
$$852$$ −27.4249 −0.939560
$$853$$ −46.3644 −1.58749 −0.793744 0.608252i $$-0.791871\pi$$
−0.793744 + 0.608252i $$0.791871\pi$$
$$854$$ 20.3205 0.695353
$$855$$ 0 0
$$856$$ −5.73205 −0.195917
$$857$$ 29.3195 1.00154 0.500768 0.865581i $$-0.333051\pi$$
0.500768 + 0.865581i $$0.333051\pi$$
$$858$$ 0 0
$$859$$ −22.1962 −0.757323 −0.378661 0.925535i $$-0.623616\pi$$
−0.378661 + 0.925535i $$0.623616\pi$$
$$860$$ 0 0
$$861$$ 11.1962 0.381564
$$862$$ −2.62536 −0.0894200
$$863$$ 17.1093 0.582406 0.291203 0.956661i $$-0.405944\pi$$
0.291203 + 0.956661i $$0.405944\pi$$
$$864$$ 22.5167 0.766032
$$865$$ 0 0
$$866$$ −14.8756 −0.505495
$$867$$ 4.00240 0.135929
$$868$$ −50.6071 −1.71772
$$869$$ 0 0
$$870$$ 0 0
$$871$$ 9.37307 0.317594
$$872$$ −17.7656 −0.601619
$$873$$ −6.69213 −0.226494
$$874$$ −7.56922 −0.256033
$$875$$ 0 0
$$876$$ −16.3923 −0.553845
$$877$$ 11.8957 0.401690 0.200845 0.979623i $$-0.435631\pi$$
0.200845 + 0.979623i $$0.435631\pi$$
$$878$$ 14.6226 0.493489
$$879$$ −29.1244 −0.982340
$$880$$ 0 0
$$881$$ −26.9090 −0.906586 −0.453293 0.891362i $$-0.649751\pi$$
−0.453293 + 0.891362i $$0.649751\pi$$
$$882$$ 1.59008 0.0535407
$$883$$ −21.5649 −0.725718 −0.362859 0.931844i $$-0.618199\pi$$
−0.362859 + 0.931844i $$0.618199\pi$$
$$884$$ −28.3923 −0.954937
$$885$$ 0 0
$$886$$ 14.2679 0.479341
$$887$$ 15.6950 0.526988 0.263494 0.964661i $$-0.415125\pi$$
0.263494 + 0.964661i $$0.415125\pi$$
$$888$$ 6.69213 0.224573
$$889$$ 3.00000 0.100617
$$890$$ 0 0
$$891$$ 0 0
$$892$$ −2.68973 −0.0900587
$$893$$ −48.0561 −1.60813
$$894$$ 10.2679 0.343412
$$895$$ 0 0
$$896$$ 38.3205 1.28020
$$897$$ −28.5617 −0.953646
$$898$$ 17.8671 0.596234
$$899$$ −60.4974 −2.01770
$$900$$ 0 0
$$901$$ 8.39230 0.279588
$$902$$ 0 0
$$903$$ −41.7057 −1.38788
$$904$$ −5.46410 −0.181733
$$905$$ 0 0
$$906$$ −4.19615 −0.139408
$$907$$ −30.1146 −0.999938 −0.499969 0.866043i $$-0.666655\pi$$
−0.499969 + 0.866043i $$0.666655\pi$$
$$908$$ −34.8377 −1.15613
$$909$$ 14.1962 0.470857
$$910$$ 0 0
$$911$$ −30.2487 −1.00218 −0.501092 0.865394i $$-0.667068\pi$$
−0.501092 + 0.865394i $$0.667068\pi$$
$$912$$ 19.9749 0.661434
$$913$$ 0 0
$$914$$ −10.9808 −0.363211
$$915$$ 0 0
$$916$$ 7.05256 0.233023
$$917$$ 10.4543 0.345231
$$918$$ 8.76268 0.289212
$$919$$ −18.9808 −0.626118 −0.313059 0.949734i $$-0.601354\pi$$
−0.313059 + 0.949734i $$0.601354\pi$$
$$920$$ 0 0
$$921$$ 15.1244 0.498364
$$922$$ −17.0821 −0.562568
$$923$$ −34.7733 −1.14458
$$924$$ 0 0
$$925$$ 0 0
$$926$$ −10.5167 −0.345599
$$927$$ 3.10583 0.102009
$$928$$ 35.6055 1.16881
$$929$$ −43.8564 −1.43888 −0.719441 0.694554i $$-0.755602\pi$$
−0.719441 + 0.694554i $$0.755602\pi$$
$$930$$ 0 0
$$931$$ −17.6077 −0.577069
$$932$$ −2.92996 −0.0959741
$$933$$ −19.4201 −0.635784
$$934$$ −0.320508 −0.0104873
$$935$$ 0 0
$$936$$ −6.00000 −0.196116
$$937$$ −31.1870 −1.01884 −0.509418 0.860519i $$-0.670139\pi$$
−0.509418 + 0.860519i $$0.670139\pi$$
$$938$$ −3.82654 −0.124941
$$939$$ 33.1244 1.08097
$$940$$ 0 0
$$941$$ −27.0000 −0.880175 −0.440087 0.897955i $$-0.645053\pi$$
−0.440087 + 0.897955i $$0.645053\pi$$
$$942$$ 21.8695 0.712548
$$943$$ 6.03579 0.196552
$$944$$ −3.12436 −0.101689
$$945$$ 0 0
$$946$$ 0 0
$$947$$ −19.9749 −0.649096 −0.324548 0.945869i $$-0.605212\pi$$
−0.324548 + 0.945869i $$0.605212\pi$$
$$948$$ 4.89898 0.159111
$$949$$ −20.7846 −0.674697
$$950$$ 0 0
$$951$$ 24.3923 0.790975
$$952$$ 24.9754 0.809456
$$953$$ 16.6932 0.540745 0.270372 0.962756i $$-0.412853\pi$$
0.270372 + 0.962756i $$0.412853\pi$$
$$954$$ 0.823085 0.0266484
$$955$$ 0 0
$$956$$ −9.80385 −0.317079
$$957$$ 0 0
$$958$$ 1.31268 0.0424107
$$959$$ 29.6603 0.957780
$$960$$ 0 0
$$961$$ 45.2487 1.45964
$$962$$ 3.93803 0.126967
$$963$$ 2.17209 0.0699946
$$964$$ −24.4641 −0.787936
$$965$$ 0 0
$$966$$ 11.6603 0.375163
$$967$$ 4.24264 0.136434 0.0682171 0.997671i $$-0.478269\pi$$
0.0682171 + 0.997671i $$0.478269\pi$$
$$968$$ 0 0
$$969$$ 31.3205 1.00616
$$970$$ 0 0
$$971$$ −24.9282 −0.799984 −0.399992 0.916519i $$-0.630987\pi$$
−0.399992 + 0.916519i $$0.630987\pi$$
$$972$$ −12.9038 −0.413889
$$973$$ 48.8139 1.56490
$$974$$ 6.58846 0.211108
$$975$$ 0 0
$$976$$ −28.9090 −0.925353
$$977$$ −29.1165 −0.931519 −0.465759 0.884911i $$-0.654219\pi$$
−0.465759 + 0.884911i $$0.654219\pi$$
$$978$$ 5.13922 0.164334
$$979$$ 0 0
$$980$$ 0 0
$$981$$ 6.73205 0.214938
$$982$$ 20.5569 0.655996
$$983$$ −56.0237 −1.78688 −0.893439 0.449184i $$-0.851715\pi$$
−0.893439 + 0.449184i $$0.851715\pi$$
$$984$$ 6.46410 0.206068
$$985$$ 0 0
$$986$$ 13.8564 0.441278
$$987$$ 74.0295 2.35639
$$988$$ 30.8353 0.981001
$$989$$ −22.4833 −0.714929
$$990$$ 0 0
$$991$$ 19.9090 0.632429 0.316215 0.948688i $$-0.397588\pi$$
0.316215 + 0.948688i $$0.397588\pi$$
$$992$$ −44.8759 −1.42481
$$993$$ 35.1523 1.11552
$$994$$ 14.1962 0.450275
$$995$$ 0 0
$$996$$ −33.1244 −1.04959
$$997$$ −34.1170 −1.08050 −0.540248 0.841506i $$-0.681670\pi$$
−0.540248 + 0.841506i $$0.681670\pi$$
$$998$$ 9.11441 0.288512
$$999$$ 7.85641 0.248566
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 3025.2.a.y.1.3 4
5.2 odd 4 605.2.b.d.364.3 yes 4
5.3 odd 4 605.2.b.d.364.2 4
5.4 even 2 inner 3025.2.a.y.1.2 4
11.10 odd 2 3025.2.a.z.1.2 4
55.2 even 20 605.2.j.e.444.2 16
55.3 odd 20 605.2.j.f.9.2 16
55.7 even 20 605.2.j.e.269.3 16
55.8 even 20 605.2.j.e.9.3 16
55.13 even 20 605.2.j.e.444.3 16
55.17 even 20 605.2.j.e.124.3 16
55.18 even 20 605.2.j.e.269.2 16
55.27 odd 20 605.2.j.f.124.2 16
55.28 even 20 605.2.j.e.124.2 16
55.32 even 4 605.2.b.e.364.2 yes 4
55.37 odd 20 605.2.j.f.269.2 16
55.38 odd 20 605.2.j.f.124.3 16
55.42 odd 20 605.2.j.f.444.3 16
55.43 even 4 605.2.b.e.364.3 yes 4
55.47 odd 20 605.2.j.f.9.3 16
55.48 odd 20 605.2.j.f.269.3 16
55.52 even 20 605.2.j.e.9.2 16
55.53 odd 20 605.2.j.f.444.2 16
55.54 odd 2 3025.2.a.z.1.3 4

By twisted newform
Twist Min Dim Char Parity Ord Type
605.2.b.d.364.2 4 5.3 odd 4
605.2.b.d.364.3 yes 4 5.2 odd 4
605.2.b.e.364.2 yes 4 55.32 even 4
605.2.b.e.364.3 yes 4 55.43 even 4
605.2.j.e.9.2 16 55.52 even 20
605.2.j.e.9.3 16 55.8 even 20
605.2.j.e.124.2 16 55.28 even 20
605.2.j.e.124.3 16 55.17 even 20
605.2.j.e.269.2 16 55.18 even 20
605.2.j.e.269.3 16 55.7 even 20
605.2.j.e.444.2 16 55.2 even 20
605.2.j.e.444.3 16 55.13 even 20
605.2.j.f.9.2 16 55.3 odd 20
605.2.j.f.9.3 16 55.47 odd 20
605.2.j.f.124.2 16 55.27 odd 20
605.2.j.f.124.3 16 55.38 odd 20
605.2.j.f.269.2 16 55.37 odd 20
605.2.j.f.269.3 16 55.48 odd 20
605.2.j.f.444.2 16 55.53 odd 20
605.2.j.f.444.3 16 55.42 odd 20
3025.2.a.y.1.2 4 5.4 even 2 inner
3025.2.a.y.1.3 4 1.1 even 1 trivial
3025.2.a.z.1.2 4 11.10 odd 2
3025.2.a.z.1.3 4 55.54 odd 2