# Properties

 Label 5445.2.a.cb.1.3 Level $5445$ Weight $2$ Character 5445.1 Self dual yes Analytic conductor $43.479$ Analytic rank $1$ Dimension $8$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$5445 = 3^{2} \cdot 5 \cdot 11^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 5445.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$43.4785439006$$ Analytic rank: $$1$$ Dimension: $$8$$ Coefficient field: $$\mathbb{Q}[x]/(x^{8} - \cdots)$$ Defining polynomial: $$x^{8} - 4 x^{7} - 3 x^{6} + 22 x^{5} - 3 x^{4} - 32 x^{3} + 9 x^{2} + 8 x + 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 495) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.3 Root $$-0.226007$$ of defining polynomial Character $$\chi$$ $$=$$ 5445.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-1.22601 q^{2} -0.496906 q^{4} +1.00000 q^{5} +0.451695 q^{7} +3.06123 q^{8} +O(q^{10})$$ $$q-1.22601 q^{2} -0.496906 q^{4} +1.00000 q^{5} +0.451695 q^{7} +3.06123 q^{8} -1.22601 q^{10} +4.84034 q^{13} -0.553781 q^{14} -2.75927 q^{16} +0.740078 q^{17} -6.80375 q^{19} -0.496906 q^{20} +0.00634166 q^{23} +1.00000 q^{25} -5.93429 q^{26} -0.224450 q^{28} -0.323900 q^{29} -5.60583 q^{31} -2.73956 q^{32} -0.907341 q^{34} +0.451695 q^{35} +7.36894 q^{37} +8.34144 q^{38} +3.06123 q^{40} -10.9705 q^{41} +1.80668 q^{43} -0.00777492 q^{46} +1.67948 q^{47} -6.79597 q^{49} -1.22601 q^{50} -2.40520 q^{52} -8.63948 q^{53} +1.38274 q^{56} +0.397104 q^{58} +1.50054 q^{59} -13.0454 q^{61} +6.87279 q^{62} +8.87727 q^{64} +4.84034 q^{65} +9.60773 q^{67} -0.367750 q^{68} -0.553781 q^{70} -11.4126 q^{71} -10.2399 q^{73} -9.03438 q^{74} +3.38083 q^{76} +2.09965 q^{79} -2.75927 q^{80} +13.4499 q^{82} -15.8945 q^{83} +0.740078 q^{85} -2.21500 q^{86} +9.36925 q^{89} +2.18636 q^{91} -0.00315121 q^{92} -2.05906 q^{94} -6.80375 q^{95} +5.87983 q^{97} +8.33191 q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q - 4 q^{2} + 6 q^{4} + 8 q^{5} - 8 q^{7} - 12 q^{8} + O(q^{10})$$ $$8 q - 4 q^{2} + 6 q^{4} + 8 q^{5} - 8 q^{7} - 12 q^{8} - 4 q^{10} - 6 q^{13} + 14 q^{14} + 14 q^{16} - 8 q^{17} + 2 q^{19} + 6 q^{20} + 4 q^{23} + 8 q^{25} - 2 q^{26} - 24 q^{28} - 22 q^{29} + 10 q^{31} - 28 q^{32} - 2 q^{34} - 8 q^{35} - 14 q^{37} - 20 q^{38} - 12 q^{40} - 22 q^{41} - 14 q^{43} + 2 q^{46} + 10 q^{47} - 4 q^{50} + 10 q^{52} - 18 q^{53} + 34 q^{56} + 12 q^{58} + 2 q^{59} + 14 q^{61} - 30 q^{62} + 30 q^{64} - 6 q^{65} + 10 q^{67} - 6 q^{68} + 14 q^{70} - 2 q^{71} - 16 q^{73} - 24 q^{74} + 22 q^{76} + 16 q^{79} + 14 q^{80} + 10 q^{82} - 46 q^{83} - 8 q^{85} - 28 q^{86} + 38 q^{89} + 8 q^{91} - 24 q^{92} + 10 q^{94} + 2 q^{95} - 4 q^{97} - 4 q^{98} + O(q^{100})$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −1.22601 −0.866918 −0.433459 0.901173i $$-0.642707\pi$$
−0.433459 + 0.901173i $$0.642707\pi$$
$$3$$ 0 0
$$4$$ −0.496906 −0.248453
$$5$$ 1.00000 0.447214
$$6$$ 0 0
$$7$$ 0.451695 0.170725 0.0853623 0.996350i $$-0.472795\pi$$
0.0853623 + 0.996350i $$0.472795\pi$$
$$8$$ 3.06123 1.08231
$$9$$ 0 0
$$10$$ −1.22601 −0.387698
$$11$$ 0 0
$$12$$ 0 0
$$13$$ 4.84034 1.34247 0.671235 0.741245i $$-0.265764\pi$$
0.671235 + 0.741245i $$0.265764\pi$$
$$14$$ −0.553781 −0.148004
$$15$$ 0 0
$$16$$ −2.75927 −0.689818
$$17$$ 0.740078 0.179495 0.0897477 0.995965i $$-0.471394\pi$$
0.0897477 + 0.995965i $$0.471394\pi$$
$$18$$ 0 0
$$19$$ −6.80375 −1.56089 −0.780443 0.625227i $$-0.785007\pi$$
−0.780443 + 0.625227i $$0.785007\pi$$
$$20$$ −0.496906 −0.111112
$$21$$ 0 0
$$22$$ 0 0
$$23$$ 0.00634166 0.00132233 0.000661164 1.00000i $$-0.499790\pi$$
0.000661164 1.00000i $$0.499790\pi$$
$$24$$ 0 0
$$25$$ 1.00000 0.200000
$$26$$ −5.93429 −1.16381
$$27$$ 0 0
$$28$$ −0.224450 −0.0424171
$$29$$ −0.323900 −0.0601468 −0.0300734 0.999548i $$-0.509574\pi$$
−0.0300734 + 0.999548i $$0.509574\pi$$
$$30$$ 0 0
$$31$$ −5.60583 −1.00684 −0.503419 0.864043i $$-0.667925\pi$$
−0.503419 + 0.864043i $$0.667925\pi$$
$$32$$ −2.73956 −0.484291
$$33$$ 0 0
$$34$$ −0.907341 −0.155608
$$35$$ 0.451695 0.0763504
$$36$$ 0 0
$$37$$ 7.36894 1.21145 0.605723 0.795675i $$-0.292884\pi$$
0.605723 + 0.795675i $$0.292884\pi$$
$$38$$ 8.34144 1.35316
$$39$$ 0 0
$$40$$ 3.06123 0.484022
$$41$$ −10.9705 −1.71330 −0.856649 0.515900i $$-0.827457\pi$$
−0.856649 + 0.515900i $$0.827457\pi$$
$$42$$ 0 0
$$43$$ 1.80668 0.275516 0.137758 0.990466i $$-0.456010\pi$$
0.137758 + 0.990466i $$0.456010\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ −0.00777492 −0.00114635
$$47$$ 1.67948 0.244978 0.122489 0.992470i $$-0.460912\pi$$
0.122489 + 0.992470i $$0.460912\pi$$
$$48$$ 0 0
$$49$$ −6.79597 −0.970853
$$50$$ −1.22601 −0.173384
$$51$$ 0 0
$$52$$ −2.40520 −0.333541
$$53$$ −8.63948 −1.18672 −0.593362 0.804936i $$-0.702200\pi$$
−0.593362 + 0.804936i $$0.702200\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 1.38274 0.184776
$$57$$ 0 0
$$58$$ 0.397104 0.0521423
$$59$$ 1.50054 0.195353 0.0976766 0.995218i $$-0.468859\pi$$
0.0976766 + 0.995218i $$0.468859\pi$$
$$60$$ 0 0
$$61$$ −13.0454 −1.67029 −0.835147 0.550027i $$-0.814617\pi$$
−0.835147 + 0.550027i $$0.814617\pi$$
$$62$$ 6.87279 0.872845
$$63$$ 0 0
$$64$$ 8.87727 1.10966
$$65$$ 4.84034 0.600371
$$66$$ 0 0
$$67$$ 9.60773 1.17377 0.586885 0.809670i $$-0.300354\pi$$
0.586885 + 0.809670i $$0.300354\pi$$
$$68$$ −0.367750 −0.0445962
$$69$$ 0 0
$$70$$ −0.553781 −0.0661895
$$71$$ −11.4126 −1.35442 −0.677211 0.735789i $$-0.736812\pi$$
−0.677211 + 0.735789i $$0.736812\pi$$
$$72$$ 0 0
$$73$$ −10.2399 −1.19849 −0.599243 0.800567i $$-0.704532\pi$$
−0.599243 + 0.800567i $$0.704532\pi$$
$$74$$ −9.03438 −1.05023
$$75$$ 0 0
$$76$$ 3.38083 0.387807
$$77$$ 0 0
$$78$$ 0 0
$$79$$ 2.09965 0.236229 0.118115 0.993000i $$-0.462315\pi$$
0.118115 + 0.993000i $$0.462315\pi$$
$$80$$ −2.75927 −0.308496
$$81$$ 0 0
$$82$$ 13.4499 1.48529
$$83$$ −15.8945 −1.74465 −0.872323 0.488929i $$-0.837388\pi$$
−0.872323 + 0.488929i $$0.837388\pi$$
$$84$$ 0 0
$$85$$ 0.740078 0.0802727
$$86$$ −2.21500 −0.238850
$$87$$ 0 0
$$88$$ 0 0
$$89$$ 9.36925 0.993138 0.496569 0.867997i $$-0.334593\pi$$
0.496569 + 0.867997i $$0.334593\pi$$
$$90$$ 0 0
$$91$$ 2.18636 0.229193
$$92$$ −0.00315121 −0.000328536 0
$$93$$ 0 0
$$94$$ −2.05906 −0.212375
$$95$$ −6.80375 −0.698050
$$96$$ 0 0
$$97$$ 5.87983 0.597007 0.298503 0.954409i $$-0.403513\pi$$
0.298503 + 0.954409i $$0.403513\pi$$
$$98$$ 8.33191 0.841650
$$99$$ 0 0
$$100$$ −0.496906 −0.0496906
$$101$$ −2.20797 −0.219701 −0.109851 0.993948i $$-0.535037\pi$$
−0.109851 + 0.993948i $$0.535037\pi$$
$$102$$ 0 0
$$103$$ 9.67943 0.953743 0.476871 0.878973i $$-0.341771\pi$$
0.476871 + 0.878973i $$0.341771\pi$$
$$104$$ 14.8174 1.45296
$$105$$ 0 0
$$106$$ 10.5921 1.02879
$$107$$ −11.7564 −1.13654 −0.568269 0.822843i $$-0.692387\pi$$
−0.568269 + 0.822843i $$0.692387\pi$$
$$108$$ 0 0
$$109$$ 12.5091 1.19816 0.599078 0.800691i $$-0.295534\pi$$
0.599078 + 0.800691i $$0.295534\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ −1.24635 −0.117769
$$113$$ −9.90608 −0.931885 −0.465943 0.884815i $$-0.654285\pi$$
−0.465943 + 0.884815i $$0.654285\pi$$
$$114$$ 0 0
$$115$$ 0.00634166 0.000591363 0
$$116$$ 0.160948 0.0149437
$$117$$ 0 0
$$118$$ −1.83967 −0.169355
$$119$$ 0.334290 0.0306443
$$120$$ 0 0
$$121$$ 0 0
$$122$$ 15.9938 1.44801
$$123$$ 0 0
$$124$$ 2.78557 0.250152
$$125$$ 1.00000 0.0894427
$$126$$ 0 0
$$127$$ 0.561561 0.0498304 0.0249152 0.999690i $$-0.492068\pi$$
0.0249152 + 0.999690i $$0.492068\pi$$
$$128$$ −5.40447 −0.477692
$$129$$ 0 0
$$130$$ −5.93429 −0.520472
$$131$$ 3.03500 0.265170 0.132585 0.991172i $$-0.457672\pi$$
0.132585 + 0.991172i $$0.457672\pi$$
$$132$$ 0 0
$$133$$ −3.07322 −0.266482
$$134$$ −11.7791 −1.01756
$$135$$ 0 0
$$136$$ 2.26555 0.194269
$$137$$ 20.7004 1.76855 0.884275 0.466966i $$-0.154653\pi$$
0.884275 + 0.466966i $$0.154653\pi$$
$$138$$ 0 0
$$139$$ −3.54873 −0.300999 −0.150500 0.988610i $$-0.548088\pi$$
−0.150500 + 0.988610i $$0.548088\pi$$
$$140$$ −0.224450 −0.0189695
$$141$$ 0 0
$$142$$ 13.9919 1.17417
$$143$$ 0 0
$$144$$ 0 0
$$145$$ −0.323900 −0.0268985
$$146$$ 12.5542 1.03899
$$147$$ 0 0
$$148$$ −3.66168 −0.300988
$$149$$ 6.30459 0.516492 0.258246 0.966079i $$-0.416855\pi$$
0.258246 + 0.966079i $$0.416855\pi$$
$$150$$ 0 0
$$151$$ 3.68432 0.299826 0.149913 0.988699i $$-0.452101\pi$$
0.149913 + 0.988699i $$0.452101\pi$$
$$152$$ −20.8278 −1.68936
$$153$$ 0 0
$$154$$ 0 0
$$155$$ −5.60583 −0.450271
$$156$$ 0 0
$$157$$ 2.82869 0.225754 0.112877 0.993609i $$-0.463993\pi$$
0.112877 + 0.993609i $$0.463993\pi$$
$$158$$ −2.57419 −0.204792
$$159$$ 0 0
$$160$$ −2.73956 −0.216582
$$161$$ 0.00286450 0.000225754 0
$$162$$ 0 0
$$163$$ 8.85141 0.693296 0.346648 0.937995i $$-0.387320\pi$$
0.346648 + 0.937995i $$0.387320\pi$$
$$164$$ 5.45129 0.425674
$$165$$ 0 0
$$166$$ 19.4868 1.51247
$$167$$ −19.6842 −1.52321 −0.761605 0.648042i $$-0.775588\pi$$
−0.761605 + 0.648042i $$0.775588\pi$$
$$168$$ 0 0
$$169$$ 10.4289 0.802224
$$170$$ −0.907341 −0.0695899
$$171$$ 0 0
$$172$$ −0.897750 −0.0684528
$$173$$ −2.91136 −0.221347 −0.110673 0.993857i $$-0.535301\pi$$
−0.110673 + 0.993857i $$0.535301\pi$$
$$174$$ 0 0
$$175$$ 0.451695 0.0341449
$$176$$ 0 0
$$177$$ 0 0
$$178$$ −11.4868 −0.860969
$$179$$ −6.29309 −0.470368 −0.235184 0.971951i $$-0.575569\pi$$
−0.235184 + 0.971951i $$0.575569\pi$$
$$180$$ 0 0
$$181$$ −11.1642 −0.829826 −0.414913 0.909861i $$-0.636188\pi$$
−0.414913 + 0.909861i $$0.636188\pi$$
$$182$$ −2.68049 −0.198691
$$183$$ 0 0
$$184$$ 0.0194132 0.00143116
$$185$$ 7.36894 0.541776
$$186$$ 0 0
$$187$$ 0 0
$$188$$ −0.834545 −0.0608655
$$189$$ 0 0
$$190$$ 8.34144 0.605152
$$191$$ −20.3946 −1.47570 −0.737851 0.674964i $$-0.764159\pi$$
−0.737851 + 0.674964i $$0.764159\pi$$
$$192$$ 0 0
$$193$$ 2.97433 0.214097 0.107049 0.994254i $$-0.465860\pi$$
0.107049 + 0.994254i $$0.465860\pi$$
$$194$$ −7.20872 −0.517556
$$195$$ 0 0
$$196$$ 3.37696 0.241212
$$197$$ −5.20127 −0.370575 −0.185288 0.982684i $$-0.559322\pi$$
−0.185288 + 0.982684i $$0.559322\pi$$
$$198$$ 0 0
$$199$$ 8.10264 0.574381 0.287191 0.957873i $$-0.407279\pi$$
0.287191 + 0.957873i $$0.407279\pi$$
$$200$$ 3.06123 0.216461
$$201$$ 0 0
$$202$$ 2.70699 0.190463
$$203$$ −0.146304 −0.0102685
$$204$$ 0 0
$$205$$ −10.9705 −0.766210
$$206$$ −11.8671 −0.826817
$$207$$ 0 0
$$208$$ −13.3558 −0.926059
$$209$$ 0 0
$$210$$ 0 0
$$211$$ 1.31335 0.0904146 0.0452073 0.998978i $$-0.485605\pi$$
0.0452073 + 0.998978i $$0.485605\pi$$
$$212$$ 4.29301 0.294845
$$213$$ 0 0
$$214$$ 14.4135 0.985286
$$215$$ 1.80668 0.123214
$$216$$ 0 0
$$217$$ −2.53213 −0.171892
$$218$$ −15.3363 −1.03870
$$219$$ 0 0
$$220$$ 0 0
$$221$$ 3.58223 0.240967
$$222$$ 0 0
$$223$$ 28.6347 1.91752 0.958761 0.284214i $$-0.0917325\pi$$
0.958761 + 0.284214i $$0.0917325\pi$$
$$224$$ −1.23745 −0.0826804
$$225$$ 0 0
$$226$$ 12.1449 0.807868
$$227$$ 7.86603 0.522087 0.261043 0.965327i $$-0.415933\pi$$
0.261043 + 0.965327i $$0.415933\pi$$
$$228$$ 0 0
$$229$$ −16.1510 −1.06729 −0.533644 0.845710i $$-0.679178\pi$$
−0.533644 + 0.845710i $$0.679178\pi$$
$$230$$ −0.00777492 −0.000512663 0
$$231$$ 0 0
$$232$$ −0.991532 −0.0650973
$$233$$ −18.4345 −1.20768 −0.603841 0.797105i $$-0.706364\pi$$
−0.603841 + 0.797105i $$0.706364\pi$$
$$234$$ 0 0
$$235$$ 1.67948 0.109557
$$236$$ −0.745626 −0.0485361
$$237$$ 0 0
$$238$$ −0.409841 −0.0265661
$$239$$ −22.3638 −1.44660 −0.723298 0.690536i $$-0.757375\pi$$
−0.723298 + 0.690536i $$0.757375\pi$$
$$240$$ 0 0
$$241$$ 13.2213 0.851662 0.425831 0.904803i $$-0.359982\pi$$
0.425831 + 0.904803i $$0.359982\pi$$
$$242$$ 0 0
$$243$$ 0 0
$$244$$ 6.48235 0.414990
$$245$$ −6.79597 −0.434179
$$246$$ 0 0
$$247$$ −32.9325 −2.09544
$$248$$ −17.1607 −1.08971
$$249$$ 0 0
$$250$$ −1.22601 −0.0775395
$$251$$ −21.0032 −1.32571 −0.662854 0.748749i $$-0.730655\pi$$
−0.662854 + 0.748749i $$0.730655\pi$$
$$252$$ 0 0
$$253$$ 0 0
$$254$$ −0.688477 −0.0431989
$$255$$ 0 0
$$256$$ −11.1286 −0.695539
$$257$$ −3.36772 −0.210073 −0.105036 0.994468i $$-0.533496\pi$$
−0.105036 + 0.994468i $$0.533496\pi$$
$$258$$ 0 0
$$259$$ 3.32852 0.206824
$$260$$ −2.40520 −0.149164
$$261$$ 0 0
$$262$$ −3.72094 −0.229880
$$263$$ −21.0450 −1.29769 −0.648845 0.760920i $$-0.724748\pi$$
−0.648845 + 0.760920i $$0.724748\pi$$
$$264$$ 0 0
$$265$$ −8.63948 −0.530719
$$266$$ 3.76779 0.231018
$$267$$ 0 0
$$268$$ −4.77414 −0.291627
$$269$$ 23.3816 1.42560 0.712800 0.701367i $$-0.247427\pi$$
0.712800 + 0.701367i $$0.247427\pi$$
$$270$$ 0 0
$$271$$ 20.5474 1.24817 0.624084 0.781357i $$-0.285472\pi$$
0.624084 + 0.781357i $$0.285472\pi$$
$$272$$ −2.04208 −0.123819
$$273$$ 0 0
$$274$$ −25.3788 −1.53319
$$275$$ 0 0
$$276$$ 0 0
$$277$$ 19.5818 1.17656 0.588280 0.808658i $$-0.299805\pi$$
0.588280 + 0.808658i $$0.299805\pi$$
$$278$$ 4.35077 0.260942
$$279$$ 0 0
$$280$$ 1.38274 0.0826345
$$281$$ −13.7926 −0.822800 −0.411400 0.911455i $$-0.634960\pi$$
−0.411400 + 0.911455i $$0.634960\pi$$
$$282$$ 0 0
$$283$$ −31.5177 −1.87353 −0.936765 0.349958i $$-0.886196\pi$$
−0.936765 + 0.349958i $$0.886196\pi$$
$$284$$ 5.67098 0.336511
$$285$$ 0 0
$$286$$ 0 0
$$287$$ −4.95530 −0.292502
$$288$$ 0 0
$$289$$ −16.4523 −0.967781
$$290$$ 0.397104 0.0233188
$$291$$ 0 0
$$292$$ 5.08826 0.297768
$$293$$ 23.3131 1.36196 0.680982 0.732300i $$-0.261553\pi$$
0.680982 + 0.732300i $$0.261553\pi$$
$$294$$ 0 0
$$295$$ 1.50054 0.0873646
$$296$$ 22.5580 1.31116
$$297$$ 0 0
$$298$$ −7.72948 −0.447757
$$299$$ 0.0306958 0.00177518
$$300$$ 0 0
$$301$$ 0.816068 0.0470374
$$302$$ −4.51700 −0.259924
$$303$$ 0 0
$$304$$ 18.7734 1.07673
$$305$$ −13.0454 −0.746978
$$306$$ 0 0
$$307$$ 10.0938 0.576083 0.288042 0.957618i $$-0.406996\pi$$
0.288042 + 0.957618i $$0.406996\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 6.87279 0.390348
$$311$$ −13.3254 −0.755614 −0.377807 0.925884i $$-0.623322\pi$$
−0.377807 + 0.925884i $$0.623322\pi$$
$$312$$ 0 0
$$313$$ 23.1682 1.30955 0.654773 0.755825i $$-0.272764\pi$$
0.654773 + 0.755825i $$0.272764\pi$$
$$314$$ −3.46799 −0.195710
$$315$$ 0 0
$$316$$ −1.04333 −0.0586920
$$317$$ 6.93189 0.389334 0.194667 0.980869i $$-0.437637\pi$$
0.194667 + 0.980869i $$0.437637\pi$$
$$318$$ 0 0
$$319$$ 0 0
$$320$$ 8.87727 0.496254
$$321$$ 0 0
$$322$$ −0.00351189 −0.000195710 0
$$323$$ −5.03530 −0.280172
$$324$$ 0 0
$$325$$ 4.84034 0.268494
$$326$$ −10.8519 −0.601031
$$327$$ 0 0
$$328$$ −33.5830 −1.85431
$$329$$ 0.758613 0.0418237
$$330$$ 0 0
$$331$$ −10.5717 −0.581075 −0.290538 0.956864i $$-0.593834\pi$$
−0.290538 + 0.956864i $$0.593834\pi$$
$$332$$ 7.89807 0.433463
$$333$$ 0 0
$$334$$ 24.1330 1.32050
$$335$$ 9.60773 0.524926
$$336$$ 0 0
$$337$$ 6.34668 0.345726 0.172863 0.984946i $$-0.444698\pi$$
0.172863 + 0.984946i $$0.444698\pi$$
$$338$$ −12.7859 −0.695463
$$339$$ 0 0
$$340$$ −0.367750 −0.0199440
$$341$$ 0 0
$$342$$ 0 0
$$343$$ −6.23157 −0.336473
$$344$$ 5.53065 0.298193
$$345$$ 0 0
$$346$$ 3.56935 0.191890
$$347$$ −25.2417 −1.35505 −0.677524 0.735501i $$-0.736947\pi$$
−0.677524 + 0.735501i $$0.736947\pi$$
$$348$$ 0 0
$$349$$ 11.1035 0.594359 0.297179 0.954822i $$-0.403954\pi$$
0.297179 + 0.954822i $$0.403954\pi$$
$$350$$ −0.553781 −0.0296009
$$351$$ 0 0
$$352$$ 0 0
$$353$$ 18.8552 1.00356 0.501780 0.864996i $$-0.332679\pi$$
0.501780 + 0.864996i $$0.332679\pi$$
$$354$$ 0 0
$$355$$ −11.4126 −0.605716
$$356$$ −4.65564 −0.246748
$$357$$ 0 0
$$358$$ 7.71537 0.407770
$$359$$ −21.2928 −1.12379 −0.561895 0.827208i $$-0.689928\pi$$
−0.561895 + 0.827208i $$0.689928\pi$$
$$360$$ 0 0
$$361$$ 27.2910 1.43637
$$362$$ 13.6873 0.719391
$$363$$ 0 0
$$364$$ −1.08642 −0.0569437
$$365$$ −10.2399 −0.535980
$$366$$ 0 0
$$367$$ −31.2857 −1.63310 −0.816550 0.577275i $$-0.804116\pi$$
−0.816550 + 0.577275i $$0.804116\pi$$
$$368$$ −0.0174984 −0.000912165 0
$$369$$ 0 0
$$370$$ −9.03438 −0.469675
$$371$$ −3.90241 −0.202603
$$372$$ 0 0
$$373$$ −9.39368 −0.486386 −0.243193 0.969978i $$-0.578195\pi$$
−0.243193 + 0.969978i $$0.578195\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 5.14127 0.265141
$$377$$ −1.56779 −0.0807452
$$378$$ 0 0
$$379$$ −8.19727 −0.421065 −0.210533 0.977587i $$-0.567520\pi$$
−0.210533 + 0.977587i $$0.567520\pi$$
$$380$$ 3.38083 0.173433
$$381$$ 0 0
$$382$$ 25.0039 1.27931
$$383$$ −33.6830 −1.72112 −0.860560 0.509349i $$-0.829886\pi$$
−0.860560 + 0.509349i $$0.829886\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ −3.64656 −0.185605
$$387$$ 0 0
$$388$$ −2.92173 −0.148328
$$389$$ 32.9338 1.66981 0.834905 0.550395i $$-0.185523\pi$$
0.834905 + 0.550395i $$0.185523\pi$$
$$390$$ 0 0
$$391$$ 0.00469332 0.000237352 0
$$392$$ −20.8040 −1.05076
$$393$$ 0 0
$$394$$ 6.37680 0.321259
$$395$$ 2.09965 0.105645
$$396$$ 0 0
$$397$$ −15.3865 −0.772228 −0.386114 0.922451i $$-0.626183\pi$$
−0.386114 + 0.922451i $$0.626183\pi$$
$$398$$ −9.93390 −0.497941
$$399$$ 0 0
$$400$$ −2.75927 −0.137964
$$401$$ 6.64295 0.331733 0.165866 0.986148i $$-0.446958\pi$$
0.165866 + 0.986148i $$0.446958\pi$$
$$402$$ 0 0
$$403$$ −27.1341 −1.35165
$$404$$ 1.09715 0.0545855
$$405$$ 0 0
$$406$$ 0.179370 0.00890198
$$407$$ 0 0
$$408$$ 0 0
$$409$$ −24.4850 −1.21071 −0.605353 0.795957i $$-0.706968\pi$$
−0.605353 + 0.795957i $$0.706968\pi$$
$$410$$ 13.4499 0.664241
$$411$$ 0 0
$$412$$ −4.80977 −0.236960
$$413$$ 0.677784 0.0333516
$$414$$ 0 0
$$415$$ −15.8945 −0.780230
$$416$$ −13.2604 −0.650146
$$417$$ 0 0
$$418$$ 0 0
$$419$$ 20.8656 1.01935 0.509676 0.860367i $$-0.329765\pi$$
0.509676 + 0.860367i $$0.329765\pi$$
$$420$$ 0 0
$$421$$ −16.9311 −0.825172 −0.412586 0.910919i $$-0.635374\pi$$
−0.412586 + 0.910919i $$0.635374\pi$$
$$422$$ −1.61017 −0.0783821
$$423$$ 0 0
$$424$$ −26.4474 −1.28440
$$425$$ 0.740078 0.0358991
$$426$$ 0 0
$$427$$ −5.89255 −0.285160
$$428$$ 5.84186 0.282377
$$429$$ 0 0
$$430$$ −2.21500 −0.106817
$$431$$ −26.1647 −1.26031 −0.630154 0.776470i $$-0.717008\pi$$
−0.630154 + 0.776470i $$0.717008\pi$$
$$432$$ 0 0
$$433$$ −25.0593 −1.20427 −0.602137 0.798393i $$-0.705684\pi$$
−0.602137 + 0.798393i $$0.705684\pi$$
$$434$$ 3.10440 0.149016
$$435$$ 0 0
$$436$$ −6.21586 −0.297686
$$437$$ −0.0431470 −0.00206400
$$438$$ 0 0
$$439$$ −5.11234 −0.243999 −0.121999 0.992530i $$-0.538931\pi$$
−0.121999 + 0.992530i $$0.538931\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ −4.39184 −0.208899
$$443$$ 28.0773 1.33399 0.666995 0.745062i $$-0.267580\pi$$
0.666995 + 0.745062i $$0.267580\pi$$
$$444$$ 0 0
$$445$$ 9.36925 0.444145
$$446$$ −35.1064 −1.66233
$$447$$ 0 0
$$448$$ 4.00982 0.189446
$$449$$ 36.9951 1.74591 0.872953 0.487804i $$-0.162202\pi$$
0.872953 + 0.487804i $$0.162202\pi$$
$$450$$ 0 0
$$451$$ 0 0
$$452$$ 4.92239 0.231530
$$453$$ 0 0
$$454$$ −9.64381 −0.452606
$$455$$ 2.18636 0.102498
$$456$$ 0 0
$$457$$ −36.8462 −1.72359 −0.861795 0.507256i $$-0.830660\pi$$
−0.861795 + 0.507256i $$0.830660\pi$$
$$458$$ 19.8012 0.925250
$$459$$ 0 0
$$460$$ −0.00315121 −0.000146926 0
$$461$$ −23.0013 −1.07128 −0.535640 0.844447i $$-0.679929\pi$$
−0.535640 + 0.844447i $$0.679929\pi$$
$$462$$ 0 0
$$463$$ −36.1571 −1.68036 −0.840181 0.542306i $$-0.817551\pi$$
−0.840181 + 0.542306i $$0.817551\pi$$
$$464$$ 0.893729 0.0414903
$$465$$ 0 0
$$466$$ 22.6008 1.04696
$$467$$ −2.26498 −0.104811 −0.0524053 0.998626i $$-0.516689\pi$$
−0.0524053 + 0.998626i $$0.516689\pi$$
$$468$$ 0 0
$$469$$ 4.33976 0.200392
$$470$$ −2.05906 −0.0949772
$$471$$ 0 0
$$472$$ 4.59348 0.211432
$$473$$ 0 0
$$474$$ 0 0
$$475$$ −6.80375 −0.312177
$$476$$ −0.166111 −0.00761367
$$477$$ 0 0
$$478$$ 27.4182 1.25408
$$479$$ 30.5308 1.39499 0.697495 0.716590i $$-0.254298\pi$$
0.697495 + 0.716590i $$0.254298\pi$$
$$480$$ 0 0
$$481$$ 35.6682 1.62633
$$482$$ −16.2095 −0.738321
$$483$$ 0 0
$$484$$ 0 0
$$485$$ 5.87983 0.266989
$$486$$ 0 0
$$487$$ −16.1543 −0.732021 −0.366010 0.930611i $$-0.619277\pi$$
−0.366010 + 0.930611i $$0.619277\pi$$
$$488$$ −39.9349 −1.80777
$$489$$ 0 0
$$490$$ 8.33191 0.376397
$$491$$ 11.6766 0.526960 0.263480 0.964665i $$-0.415130\pi$$
0.263480 + 0.964665i $$0.415130\pi$$
$$492$$ 0 0
$$493$$ −0.239712 −0.0107961
$$494$$ 40.3754 1.81658
$$495$$ 0 0
$$496$$ 15.4680 0.694534
$$497$$ −5.15500 −0.231233
$$498$$ 0 0
$$499$$ −40.9366 −1.83257 −0.916287 0.400522i $$-0.868829\pi$$
−0.916287 + 0.400522i $$0.868829\pi$$
$$500$$ −0.496906 −0.0222223
$$501$$ 0 0
$$502$$ 25.7500 1.14928
$$503$$ 0.424478 0.0189266 0.00946328 0.999955i $$-0.496988\pi$$
0.00946328 + 0.999955i $$0.496988\pi$$
$$504$$ 0 0
$$505$$ −2.20797 −0.0982533
$$506$$ 0 0
$$507$$ 0 0
$$508$$ −0.279043 −0.0123805
$$509$$ −19.8053 −0.877854 −0.438927 0.898523i $$-0.644641\pi$$
−0.438927 + 0.898523i $$0.644641\pi$$
$$510$$ 0 0
$$511$$ −4.62530 −0.204611
$$512$$ 24.4527 1.08067
$$513$$ 0 0
$$514$$ 4.12885 0.182116
$$515$$ 9.67943 0.426527
$$516$$ 0 0
$$517$$ 0 0
$$518$$ −4.08078 −0.179299
$$519$$ 0 0
$$520$$ 14.8174 0.649785
$$521$$ 36.8095 1.61265 0.806327 0.591470i $$-0.201452\pi$$
0.806327 + 0.591470i $$0.201452\pi$$
$$522$$ 0 0
$$523$$ 10.5015 0.459198 0.229599 0.973285i $$-0.426258\pi$$
0.229599 + 0.973285i $$0.426258\pi$$
$$524$$ −1.50811 −0.0658822
$$525$$ 0 0
$$526$$ 25.8013 1.12499
$$527$$ −4.14875 −0.180723
$$528$$ 0 0
$$529$$ −23.0000 −0.999998
$$530$$ 10.5921 0.460090
$$531$$ 0 0
$$532$$ 1.52710 0.0662083
$$533$$ −53.1008 −2.30005
$$534$$ 0 0
$$535$$ −11.7564 −0.508276
$$536$$ 29.4114 1.27038
$$537$$ 0 0
$$538$$ −28.6660 −1.23588
$$539$$ 0 0
$$540$$ 0 0
$$541$$ −0.654613 −0.0281440 −0.0140720 0.999901i $$-0.504479\pi$$
−0.0140720 + 0.999901i $$0.504479\pi$$
$$542$$ −25.1913 −1.08206
$$543$$ 0 0
$$544$$ −2.02749 −0.0869280
$$545$$ 12.5091 0.535832
$$546$$ 0 0
$$547$$ 13.4159 0.573621 0.286810 0.957987i $$-0.407405\pi$$
0.286810 + 0.957987i $$0.407405\pi$$
$$548$$ −10.2861 −0.439402
$$549$$ 0 0
$$550$$ 0 0
$$551$$ 2.20374 0.0938823
$$552$$ 0 0
$$553$$ 0.948403 0.0403302
$$554$$ −24.0075 −1.01998
$$555$$ 0 0
$$556$$ 1.76339 0.0747843
$$557$$ 13.2421 0.561087 0.280544 0.959841i $$-0.409485\pi$$
0.280544 + 0.959841i $$0.409485\pi$$
$$558$$ 0 0
$$559$$ 8.74494 0.369872
$$560$$ −1.24635 −0.0526679
$$561$$ 0 0
$$562$$ 16.9099 0.713300
$$563$$ 21.3619 0.900297 0.450149 0.892954i $$-0.351371\pi$$
0.450149 + 0.892954i $$0.351371\pi$$
$$564$$ 0 0
$$565$$ −9.90608 −0.416752
$$566$$ 38.6409 1.62420
$$567$$ 0 0
$$568$$ −34.9364 −1.46590
$$569$$ −45.3375 −1.90065 −0.950323 0.311267i $$-0.899247\pi$$
−0.950323 + 0.311267i $$0.899247\pi$$
$$570$$ 0 0
$$571$$ 25.1544 1.05268 0.526339 0.850275i $$-0.323564\pi$$
0.526339 + 0.850275i $$0.323564\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 6.07523 0.253575
$$575$$ 0.00634166 0.000264465 0
$$576$$ 0 0
$$577$$ 7.62653 0.317496 0.158748 0.987319i $$-0.449254\pi$$
0.158748 + 0.987319i $$0.449254\pi$$
$$578$$ 20.1706 0.838987
$$579$$ 0 0
$$580$$ 0.160948 0.00668301
$$581$$ −7.17946 −0.297854
$$582$$ 0 0
$$583$$ 0 0
$$584$$ −31.3466 −1.29713
$$585$$ 0 0
$$586$$ −28.5820 −1.18071
$$587$$ −7.88825 −0.325583 −0.162791 0.986660i $$-0.552050\pi$$
−0.162791 + 0.986660i $$0.552050\pi$$
$$588$$ 0 0
$$589$$ 38.1407 1.57156
$$590$$ −1.83967 −0.0757379
$$591$$ 0 0
$$592$$ −20.3329 −0.835678
$$593$$ −8.68507 −0.356653 −0.178327 0.983971i $$-0.557068\pi$$
−0.178327 + 0.983971i $$0.557068\pi$$
$$594$$ 0 0
$$595$$ 0.334290 0.0137045
$$596$$ −3.13279 −0.128324
$$597$$ 0 0
$$598$$ −0.0376333 −0.00153894
$$599$$ 14.0016 0.572088 0.286044 0.958216i $$-0.407660\pi$$
0.286044 + 0.958216i $$0.407660\pi$$
$$600$$ 0 0
$$601$$ 10.2936 0.419886 0.209943 0.977714i $$-0.432672\pi$$
0.209943 + 0.977714i $$0.432672\pi$$
$$602$$ −1.00050 −0.0407775
$$603$$ 0 0
$$604$$ −1.83076 −0.0744927
$$605$$ 0 0
$$606$$ 0 0
$$607$$ −27.4756 −1.11520 −0.557599 0.830110i $$-0.688277\pi$$
−0.557599 + 0.830110i $$0.688277\pi$$
$$608$$ 18.6393 0.755923
$$609$$ 0 0
$$610$$ 15.9938 0.647569
$$611$$ 8.12926 0.328875
$$612$$ 0 0
$$613$$ 10.8580 0.438550 0.219275 0.975663i $$-0.429631\pi$$
0.219275 + 0.975663i $$0.429631\pi$$
$$614$$ −12.3751 −0.499417
$$615$$ 0 0
$$616$$ 0 0
$$617$$ −18.2404 −0.734330 −0.367165 0.930156i $$-0.619671\pi$$
−0.367165 + 0.930156i $$0.619671\pi$$
$$618$$ 0 0
$$619$$ 19.6799 0.791004 0.395502 0.918465i $$-0.370571\pi$$
0.395502 + 0.918465i $$0.370571\pi$$
$$620$$ 2.78557 0.111871
$$621$$ 0 0
$$622$$ 16.3370 0.655056
$$623$$ 4.23204 0.169553
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ −28.4044 −1.13527
$$627$$ 0 0
$$628$$ −1.40559 −0.0560893
$$629$$ 5.45359 0.217449
$$630$$ 0 0
$$631$$ −30.4249 −1.21120 −0.605598 0.795771i $$-0.707066\pi$$
−0.605598 + 0.795771i $$0.707066\pi$$
$$632$$ 6.42751 0.255673
$$633$$ 0 0
$$634$$ −8.49854 −0.337520
$$635$$ 0.561561 0.0222849
$$636$$ 0 0
$$637$$ −32.8948 −1.30334
$$638$$ 0 0
$$639$$ 0 0
$$640$$ −5.40447 −0.213630
$$641$$ −37.3528 −1.47535 −0.737674 0.675157i $$-0.764076\pi$$
−0.737674 + 0.675157i $$0.764076\pi$$
$$642$$ 0 0
$$643$$ 1.06494 0.0419971 0.0209986 0.999780i $$-0.493315\pi$$
0.0209986 + 0.999780i $$0.493315\pi$$
$$644$$ −0.00142339 −5.60893e−5 0
$$645$$ 0 0
$$646$$ 6.17332 0.242886
$$647$$ 15.8049 0.621354 0.310677 0.950516i $$-0.399444\pi$$
0.310677 + 0.950516i $$0.399444\pi$$
$$648$$ 0 0
$$649$$ 0 0
$$650$$ −5.93429 −0.232762
$$651$$ 0 0
$$652$$ −4.39832 −0.172252
$$653$$ −35.3513 −1.38340 −0.691701 0.722184i $$-0.743138\pi$$
−0.691701 + 0.722184i $$0.743138\pi$$
$$654$$ 0 0
$$655$$ 3.03500 0.118587
$$656$$ 30.2705 1.18186
$$657$$ 0 0
$$658$$ −0.930065 −0.0362577
$$659$$ −28.4474 −1.10815 −0.554077 0.832465i $$-0.686929\pi$$
−0.554077 + 0.832465i $$0.686929\pi$$
$$660$$ 0 0
$$661$$ 39.5989 1.54022 0.770110 0.637911i $$-0.220201\pi$$
0.770110 + 0.637911i $$0.220201\pi$$
$$662$$ 12.9610 0.503745
$$663$$ 0 0
$$664$$ −48.6566 −1.88824
$$665$$ −3.07322 −0.119174
$$666$$ 0 0
$$667$$ −0.00205407 −7.95337e−5 0
$$668$$ 9.78121 0.378446
$$669$$ 0 0
$$670$$ −11.7791 −0.455068
$$671$$ 0 0
$$672$$ 0 0
$$673$$ 17.5621 0.676969 0.338484 0.940972i $$-0.390086\pi$$
0.338484 + 0.940972i $$0.390086\pi$$
$$674$$ −7.78108 −0.299716
$$675$$ 0 0
$$676$$ −5.18220 −0.199315
$$677$$ −45.0690 −1.73214 −0.866071 0.499922i $$-0.833362\pi$$
−0.866071 + 0.499922i $$0.833362\pi$$
$$678$$ 0 0
$$679$$ 2.65589 0.101924
$$680$$ 2.26555 0.0868797
$$681$$ 0 0
$$682$$ 0 0
$$683$$ −15.7677 −0.603334 −0.301667 0.953413i $$-0.597543\pi$$
−0.301667 + 0.953413i $$0.597543\pi$$
$$684$$ 0 0
$$685$$ 20.7004 0.790920
$$686$$ 7.63995 0.291695
$$687$$ 0 0
$$688$$ −4.98512 −0.190056
$$689$$ −41.8180 −1.59314
$$690$$ 0 0
$$691$$ 2.96619 0.112839 0.0564197 0.998407i $$-0.482032\pi$$
0.0564197 + 0.998407i $$0.482032\pi$$
$$692$$ 1.44668 0.0549944
$$693$$ 0 0
$$694$$ 30.9465 1.17471
$$695$$ −3.54873 −0.134611
$$696$$ 0 0
$$697$$ −8.11899 −0.307529
$$698$$ −13.6130 −0.515260
$$699$$ 0 0
$$700$$ −0.224450 −0.00848342
$$701$$ 37.1301 1.40238 0.701192 0.712973i $$-0.252652\pi$$
0.701192 + 0.712973i $$0.252652\pi$$
$$702$$ 0 0
$$703$$ −50.1364 −1.89093
$$704$$ 0 0
$$705$$ 0 0
$$706$$ −23.1166 −0.870003
$$707$$ −0.997329 −0.0375084
$$708$$ 0 0
$$709$$ −31.7505 −1.19241 −0.596207 0.802831i $$-0.703326\pi$$
−0.596207 + 0.802831i $$0.703326\pi$$
$$710$$ 13.9919 0.525106
$$711$$ 0 0
$$712$$ 28.6814 1.07488
$$713$$ −0.0355503 −0.00133137
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 3.12708 0.116864
$$717$$ 0 0
$$718$$ 26.1051 0.974234
$$719$$ 31.8772 1.18882 0.594409 0.804163i $$-0.297386\pi$$
0.594409 + 0.804163i $$0.297386\pi$$
$$720$$ 0 0
$$721$$ 4.37215 0.162827
$$722$$ −33.4589 −1.24521
$$723$$ 0 0
$$724$$ 5.54755 0.206173
$$725$$ −0.323900 −0.0120294
$$726$$ 0 0
$$727$$ 41.4129 1.53592 0.767959 0.640499i $$-0.221272\pi$$
0.767959 + 0.640499i $$0.221272\pi$$
$$728$$ 6.69294 0.248057
$$729$$ 0 0
$$730$$ 12.5542 0.464650
$$731$$ 1.33708 0.0494538
$$732$$ 0 0
$$733$$ −47.6920 −1.76155 −0.880773 0.473540i $$-0.842976\pi$$
−0.880773 + 0.473540i $$0.842976\pi$$
$$734$$ 38.3565 1.41576
$$735$$ 0 0
$$736$$ −0.0173734 −0.000640391 0
$$737$$ 0 0
$$738$$ 0 0
$$739$$ 12.4567 0.458228 0.229114 0.973400i $$-0.426417\pi$$
0.229114 + 0.973400i $$0.426417\pi$$
$$740$$ −3.66168 −0.134606
$$741$$ 0 0
$$742$$ 4.78438 0.175640
$$743$$ −27.3723 −1.00419 −0.502095 0.864812i $$-0.667437\pi$$
−0.502095 + 0.864812i $$0.667437\pi$$
$$744$$ 0 0
$$745$$ 6.30459 0.230982
$$746$$ 11.5167 0.421657
$$747$$ 0 0
$$748$$ 0 0
$$749$$ −5.31033 −0.194035
$$750$$ 0 0
$$751$$ −41.8725 −1.52795 −0.763975 0.645246i $$-0.776755\pi$$
−0.763975 + 0.645246i $$0.776755\pi$$
$$752$$ −4.63414 −0.168990
$$753$$ 0 0
$$754$$ 1.92212 0.0699995
$$755$$ 3.68432 0.134086
$$756$$ 0 0
$$757$$ 16.1192 0.585863 0.292932 0.956133i $$-0.405369\pi$$
0.292932 + 0.956133i $$0.405369\pi$$
$$758$$ 10.0499 0.365029
$$759$$ 0 0
$$760$$ −20.8278 −0.755504
$$761$$ 10.9239 0.395990 0.197995 0.980203i $$-0.436557\pi$$
0.197995 + 0.980203i $$0.436557\pi$$
$$762$$ 0 0
$$763$$ 5.65030 0.204555
$$764$$ 10.1342 0.366643
$$765$$ 0 0
$$766$$ 41.2956 1.49207
$$767$$ 7.26311 0.262256
$$768$$ 0 0
$$769$$ 34.2074 1.23355 0.616775 0.787139i $$-0.288439\pi$$
0.616775 + 0.787139i $$0.288439\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ −1.47797 −0.0531932
$$773$$ 8.63688 0.310647 0.155323 0.987864i $$-0.450358\pi$$
0.155323 + 0.987864i $$0.450358\pi$$
$$774$$ 0 0
$$775$$ −5.60583 −0.201367
$$776$$ 17.9995 0.646144
$$777$$ 0 0
$$778$$ −40.3771 −1.44759
$$779$$ 74.6402 2.67426
$$780$$ 0 0
$$781$$ 0 0
$$782$$ −0.00575405 −0.000205764 0
$$783$$ 0 0
$$784$$ 18.7519 0.669712
$$785$$ 2.82869 0.100960
$$786$$ 0 0
$$787$$ 23.1043 0.823581 0.411790 0.911279i $$-0.364904\pi$$
0.411790 + 0.911279i $$0.364904\pi$$
$$788$$ 2.58455 0.0920707
$$789$$ 0 0
$$790$$ −2.57419 −0.0915856
$$791$$ −4.47453 −0.159096
$$792$$ 0 0
$$793$$ −63.1443 −2.24232
$$794$$ 18.8640 0.669458
$$795$$ 0 0
$$796$$ −4.02626 −0.142707
$$797$$ −39.7333 −1.40743 −0.703714 0.710484i $$-0.748476\pi$$
−0.703714 + 0.710484i $$0.748476\pi$$
$$798$$ 0 0
$$799$$ 1.24295 0.0439723
$$800$$ −2.73956 −0.0968582
$$801$$ 0 0
$$802$$ −8.14430 −0.287585
$$803$$ 0 0
$$804$$ 0 0
$$805$$ 0.00286450 0.000100960 0
$$806$$ 33.2667 1.17177
$$807$$ 0 0
$$808$$ −6.75909 −0.237784
$$809$$ 31.0140 1.09039 0.545197 0.838308i $$-0.316455\pi$$
0.545197 + 0.838308i $$0.316455\pi$$
$$810$$ 0 0
$$811$$ −7.45922 −0.261929 −0.130964 0.991387i $$-0.541807\pi$$
−0.130964 + 0.991387i $$0.541807\pi$$
$$812$$ 0.0726995 0.00255125
$$813$$ 0 0
$$814$$ 0 0
$$815$$ 8.85141 0.310051
$$816$$ 0 0
$$817$$ −12.2922 −0.430049
$$818$$ 30.0188 1.04958
$$819$$ 0 0
$$820$$ 5.45129 0.190367
$$821$$ 12.6556 0.441683 0.220841 0.975310i $$-0.429120\pi$$
0.220841 + 0.975310i $$0.429120\pi$$
$$822$$ 0 0
$$823$$ −27.2890 −0.951234 −0.475617 0.879652i $$-0.657775\pi$$
−0.475617 + 0.879652i $$0.657775\pi$$
$$824$$ 29.6309 1.03224
$$825$$ 0 0
$$826$$ −0.830969 −0.0289131
$$827$$ 16.9544 0.589563 0.294781 0.955565i $$-0.404753\pi$$
0.294781 + 0.955565i $$0.404753\pi$$
$$828$$ 0 0
$$829$$ 19.7794 0.686967 0.343483 0.939159i $$-0.388393\pi$$
0.343483 + 0.939159i $$0.388393\pi$$
$$830$$ 19.4868 0.676395
$$831$$ 0 0
$$832$$ 42.9690 1.48968
$$833$$ −5.02955 −0.174264
$$834$$ 0 0
$$835$$ −19.6842 −0.681200
$$836$$ 0 0
$$837$$ 0 0
$$838$$ −25.5814 −0.883694
$$839$$ −4.97974 −0.171920 −0.0859598 0.996299i $$-0.527396\pi$$
−0.0859598 + 0.996299i $$0.527396\pi$$
$$840$$ 0 0
$$841$$ −28.8951 −0.996382
$$842$$ 20.7577 0.715357
$$843$$ 0 0
$$844$$ −0.652611 −0.0224638
$$845$$ 10.4289 0.358766
$$846$$ 0 0
$$847$$ 0 0
$$848$$ 23.8387 0.818623
$$849$$ 0 0
$$850$$ −0.907341 −0.0311215
$$851$$ 0.0467313 0.00160193
$$852$$ 0 0
$$853$$ 23.2464 0.795943 0.397971 0.917398i $$-0.369714\pi$$
0.397971 + 0.917398i $$0.369714\pi$$
$$854$$ 7.22430 0.247211
$$855$$ 0 0
$$856$$ −35.9891 −1.23008
$$857$$ 30.6976 1.04861 0.524306 0.851530i $$-0.324325\pi$$
0.524306 + 0.851530i $$0.324325\pi$$
$$858$$ 0 0
$$859$$ −11.1830 −0.381560 −0.190780 0.981633i $$-0.561102\pi$$
−0.190780 + 0.981633i $$0.561102\pi$$
$$860$$ −0.897750 −0.0306130
$$861$$ 0 0
$$862$$ 32.0781 1.09258
$$863$$ −3.66335 −0.124702 −0.0623510 0.998054i $$-0.519860\pi$$
−0.0623510 + 0.998054i $$0.519860\pi$$
$$864$$ 0 0
$$865$$ −2.91136 −0.0989894
$$866$$ 30.7229 1.04401
$$867$$ 0 0
$$868$$ 1.25823 0.0427071
$$869$$ 0 0
$$870$$ 0 0
$$871$$ 46.5047 1.57575
$$872$$ 38.2932 1.29677
$$873$$ 0 0
$$874$$ 0.0528986 0.00178932
$$875$$ 0.451695 0.0152701
$$876$$ 0 0
$$877$$ −7.36271 −0.248621 −0.124310 0.992243i $$-0.539672\pi$$
−0.124310 + 0.992243i $$0.539672\pi$$
$$878$$ 6.26776 0.211527
$$879$$ 0 0
$$880$$ 0 0
$$881$$ 6.07165 0.204559 0.102280 0.994756i $$-0.467386\pi$$
0.102280 + 0.994756i $$0.467386\pi$$
$$882$$ 0 0
$$883$$ 31.1767 1.04918 0.524589 0.851356i $$-0.324219\pi$$
0.524589 + 0.851356i $$0.324219\pi$$
$$884$$ −1.78003 −0.0598690
$$885$$ 0 0
$$886$$ −34.4229 −1.15646
$$887$$ 5.68999 0.191051 0.0955255 0.995427i $$-0.469547\pi$$
0.0955255 + 0.995427i $$0.469547\pi$$
$$888$$ 0 0
$$889$$ 0.253654 0.00850729
$$890$$ −11.4868 −0.385037
$$891$$ 0 0
$$892$$ −14.2288 −0.476414
$$893$$ −11.4268 −0.382382
$$894$$ 0 0
$$895$$ −6.29309 −0.210355
$$896$$ −2.44117 −0.0815538
$$897$$ 0 0
$$898$$ −45.3563 −1.51356
$$899$$ 1.81573 0.0605580
$$900$$ 0 0
$$901$$ −6.39389 −0.213011
$$902$$ 0 0
$$903$$ 0 0
$$904$$ −30.3247 −1.00859
$$905$$ −11.1642 −0.371109
$$906$$ 0 0
$$907$$ 42.4917 1.41091 0.705456 0.708754i $$-0.250742\pi$$
0.705456 + 0.708754i $$0.250742\pi$$
$$908$$ −3.90868 −0.129714
$$909$$ 0 0
$$910$$ −2.68049 −0.0888574
$$911$$ −2.14405 −0.0710356 −0.0355178 0.999369i $$-0.511308\pi$$
−0.0355178 + 0.999369i $$0.511308\pi$$
$$912$$ 0 0
$$913$$ 0 0
$$914$$ 45.1737 1.49421
$$915$$ 0 0
$$916$$ 8.02553 0.265171
$$917$$ 1.37090 0.0452710
$$918$$ 0 0
$$919$$ −16.0683 −0.530044 −0.265022 0.964242i $$-0.585379\pi$$
−0.265022 + 0.964242i $$0.585379\pi$$
$$920$$ 0.0194132 0.000640036 0
$$921$$ 0 0
$$922$$ 28.1998 0.928711
$$923$$ −55.2407 −1.81827
$$924$$ 0 0
$$925$$ 7.36894 0.242289
$$926$$ 44.3289 1.45674
$$927$$ 0 0
$$928$$ 0.887346 0.0291286
$$929$$ −37.4770 −1.22958 −0.614791 0.788690i $$-0.710760\pi$$
−0.614791 + 0.788690i $$0.710760\pi$$
$$930$$ 0 0
$$931$$ 46.2381 1.51539
$$932$$ 9.16021 0.300053
$$933$$ 0 0
$$934$$ 2.77688 0.0908621
$$935$$ 0 0
$$936$$ 0 0
$$937$$ 26.0054 0.849560 0.424780 0.905297i $$-0.360351\pi$$
0.424780 + 0.905297i $$0.360351\pi$$
$$938$$ −5.32058 −0.173723
$$939$$ 0 0
$$940$$ −0.834545 −0.0272199
$$941$$ −8.58482 −0.279857 −0.139929 0.990162i $$-0.544687\pi$$
−0.139929 + 0.990162i $$0.544687\pi$$
$$942$$ 0 0
$$943$$ −0.0695709 −0.00226554
$$944$$ −4.14038 −0.134758
$$945$$ 0 0
$$946$$ 0 0
$$947$$ −36.0304 −1.17083 −0.585416 0.810733i $$-0.699069\pi$$
−0.585416 + 0.810733i $$0.699069\pi$$
$$948$$ 0 0
$$949$$ −49.5645 −1.60893
$$950$$ 8.34144 0.270632
$$951$$ 0 0
$$952$$ 1.02334 0.0331665
$$953$$ 43.8525 1.42052 0.710261 0.703938i $$-0.248577\pi$$
0.710261 + 0.703938i $$0.248577\pi$$
$$954$$ 0 0
$$955$$ −20.3946 −0.659954
$$956$$ 11.1127 0.359412
$$957$$ 0 0
$$958$$ −37.4310 −1.20934
$$959$$ 9.35025 0.301935
$$960$$ 0 0
$$961$$ 0.425347 0.0137209
$$962$$ −43.7295 −1.40990
$$963$$ 0 0
$$964$$ −6.56977 −0.211598
$$965$$ 2.97433 0.0957472
$$966$$ 0 0
$$967$$ 48.1271 1.54766 0.773831 0.633392i $$-0.218338\pi$$
0.773831 + 0.633392i $$0.218338\pi$$
$$968$$ 0 0
$$969$$ 0 0
$$970$$ −7.20872 −0.231458
$$971$$ 31.7334 1.01837 0.509187 0.860656i $$-0.329946\pi$$
0.509187 + 0.860656i $$0.329946\pi$$
$$972$$ 0 0
$$973$$ −1.60294 −0.0513880
$$974$$ 19.8053 0.634602
$$975$$ 0 0
$$976$$ 35.9958 1.15220
$$977$$ 43.1280 1.37979 0.689893 0.723911i $$-0.257658\pi$$
0.689893 + 0.723911i $$0.257658\pi$$
$$978$$ 0 0
$$979$$ 0 0
$$980$$ 3.37696 0.107873
$$981$$ 0 0
$$982$$ −14.3156 −0.456831
$$983$$ 44.6922 1.42546 0.712729 0.701439i $$-0.247459\pi$$
0.712729 + 0.701439i $$0.247459\pi$$
$$984$$ 0 0
$$985$$ −5.20127 −0.165726
$$986$$ 0.293888 0.00935931
$$987$$ 0 0
$$988$$ 16.3644 0.520619
$$989$$ 0.0114573 0.000364322 0
$$990$$ 0 0
$$991$$ 13.7657 0.437282 0.218641 0.975805i $$-0.429838\pi$$
0.218641 + 0.975805i $$0.429838\pi$$
$$992$$ 15.3575 0.487602
$$993$$ 0 0
$$994$$ 6.32007 0.200460
$$995$$ 8.10264 0.256871
$$996$$ 0 0
$$997$$ −4.89609 −0.155061 −0.0775303 0.996990i $$-0.524703\pi$$
−0.0775303 + 0.996990i $$0.524703\pi$$
$$998$$ 50.1886 1.58869
$$999$$ 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 5445.2.a.cb.1.3 8
3.2 odd 2 5445.2.a.cc.1.6 8
11.2 odd 10 495.2.n.g.136.2 yes 16
11.6 odd 10 495.2.n.g.91.2 16
11.10 odd 2 5445.2.a.cd.1.6 8
33.2 even 10 495.2.n.h.136.3 yes 16
33.17 even 10 495.2.n.h.91.3 yes 16
33.32 even 2 5445.2.a.ca.1.3 8

By twisted newform
Twist Min Dim Char Parity Ord Type
495.2.n.g.91.2 16 11.6 odd 10
495.2.n.g.136.2 yes 16 11.2 odd 10
495.2.n.h.91.3 yes 16 33.17 even 10
495.2.n.h.136.3 yes 16 33.2 even 10
5445.2.a.ca.1.3 8 33.32 even 2
5445.2.a.cb.1.3 8 1.1 even 1 trivial
5445.2.a.cc.1.6 8 3.2 odd 2
5445.2.a.cd.1.6 8 11.10 odd 2