Properties

 Label 4600.2.e.p.4049.5 Level $4600$ Weight $2$ Character 4600.4049 Analytic conductor $36.731$ Analytic rank $0$ Dimension $6$ CM no Inner twists $2$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$4600 = 2^{3} \cdot 5^{2} \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4600.e (of order $$2$$, degree $$1$$, not minimal)

Newform invariants

 Self dual: no Analytic conductor: $$36.7311849298$$ Analytic rank: $$0$$ Dimension: $$6$$ Coefficient field: 6.0.431642176.2 Defining polynomial: $$x^{6} + 19x^{4} + 97x^{2} + 64$$ x^6 + 19*x^4 + 97*x^2 + 64 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 920) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

 Embedding label 4049.5 Root $$2.95759i$$ of defining polynomial Character $$\chi$$ $$=$$ 4600.4049 Dual form 4600.2.e.p.4049.2

$q$-expansion

 $$f(q)$$ $$=$$ $$q+2.95759i q^{3} +3.95759i q^{7} -5.74732 q^{9} +O(q^{10})$$ $$q+2.95759i q^{3} +3.95759i q^{7} -5.74732 q^{9} -0.957587 q^{11} +2.74732i q^{13} +5.74732i q^{17} +6.74732 q^{19} -11.7049 q^{21} -1.00000i q^{23} -8.12544i q^{27} -5.21027 q^{29} -5.95759 q^{31} -2.83215i q^{33} +9.12544i q^{37} -8.12544 q^{39} +0.252679 q^{41} -8.00000i q^{43} +5.49464i q^{47} -8.66249 q^{49} -16.9982 q^{51} +7.12544i q^{53} +19.9558i q^{57} +4.78973 q^{59} +12.4522 q^{61} -22.7455i q^{63} +9.12544i q^{67} +2.95759 q^{69} +1.66249 q^{71} -12.3357i q^{73} -3.78973i q^{77} -11.8303 q^{79} +6.78973 q^{81} -0.704908i q^{83} -15.4098i q^{87} +15.8303 q^{89} -10.8728 q^{91} -17.6201i q^{93} -10.8728i q^{97} +5.50356 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q - 20 q^{9}+O(q^{10})$$ 6 * q - 20 * q^9 $$6 q - 20 q^{9} + 14 q^{11} + 26 q^{19} - 36 q^{21} - 26 q^{29} - 16 q^{31} - 4 q^{39} + 16 q^{41} + 2 q^{49} + 2 q^{51} + 34 q^{59} + 26 q^{61} - 2 q^{69} - 44 q^{71} + 8 q^{79} + 46 q^{81} + 16 q^{89} - 6 q^{91} - 42 q^{99}+O(q^{100})$$ 6 * q - 20 * q^9 + 14 * q^11 + 26 * q^19 - 36 * q^21 - 26 * q^29 - 16 * q^31 - 4 * q^39 + 16 * q^41 + 2 * q^49 + 2 * q^51 + 34 * q^59 + 26 * q^61 - 2 * q^69 - 44 * q^71 + 8 * q^79 + 46 * q^81 + 16 * q^89 - 6 * q^91 - 42 * q^99

Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/4600\mathbb{Z}\right)^\times$$.

 $$n$$ $$1151$$ $$1201$$ $$2301$$ $$2577$$ $$\chi(n)$$ $$1$$ $$1$$ $$1$$ $$-1$$

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 0
$$3$$ 2.95759i 1.70756i 0.520631 + 0.853782i $$0.325697\pi$$
−0.520631 + 0.853782i $$0.674303\pi$$
$$4$$ 0 0
$$5$$ 0 0
$$6$$ 0 0
$$7$$ 3.95759i 1.49583i 0.663796 + 0.747914i $$0.268944\pi$$
−0.663796 + 0.747914i $$0.731056\pi$$
$$8$$ 0 0
$$9$$ −5.74732 −1.91577
$$10$$ 0 0
$$11$$ −0.957587 −0.288723 −0.144362 0.989525i $$-0.546113\pi$$
−0.144362 + 0.989525i $$0.546113\pi$$
$$12$$ 0 0
$$13$$ 2.74732i 0.761970i 0.924581 + 0.380985i $$0.124415\pi$$
−0.924581 + 0.380985i $$0.875585\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ 5.74732i 1.39393i 0.717105 + 0.696965i $$0.245467\pi$$
−0.717105 + 0.696965i $$0.754533\pi$$
$$18$$ 0 0
$$19$$ 6.74732 1.54794 0.773971 0.633221i $$-0.218268\pi$$
0.773971 + 0.633221i $$0.218268\pi$$
$$20$$ 0 0
$$21$$ −11.7049 −2.55422
$$22$$ 0 0
$$23$$ − 1.00000i − 0.208514i
$$24$$ 0 0
$$25$$ 0 0
$$26$$ 0 0
$$27$$ − 8.12544i − 1.56374i
$$28$$ 0 0
$$29$$ −5.21027 −0.967522 −0.483761 0.875200i $$-0.660730\pi$$
−0.483761 + 0.875200i $$0.660730\pi$$
$$30$$ 0 0
$$31$$ −5.95759 −1.07001 −0.535007 0.844848i $$-0.679691\pi$$
−0.535007 + 0.844848i $$0.679691\pi$$
$$32$$ 0 0
$$33$$ − 2.83215i − 0.493013i
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 9.12544i 1.50021i 0.661317 + 0.750107i $$0.269998\pi$$
−0.661317 + 0.750107i $$0.730002\pi$$
$$38$$ 0 0
$$39$$ −8.12544 −1.30111
$$40$$ 0 0
$$41$$ 0.252679 0.0394619 0.0197309 0.999805i $$-0.493719\pi$$
0.0197309 + 0.999805i $$0.493719\pi$$
$$42$$ 0 0
$$43$$ − 8.00000i − 1.21999i −0.792406 0.609994i $$-0.791172\pi$$
0.792406 0.609994i $$-0.208828\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ 5.49464i 0.801476i 0.916193 + 0.400738i $$0.131246\pi$$
−0.916193 + 0.400738i $$0.868754\pi$$
$$48$$ 0 0
$$49$$ −8.66249 −1.23750
$$50$$ 0 0
$$51$$ −16.9982 −2.38022
$$52$$ 0 0
$$53$$ 7.12544i 0.978754i 0.872072 + 0.489377i $$0.162776\pi$$
−0.872072 + 0.489377i $$0.837224\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ 19.9558i 2.64321i
$$58$$ 0 0
$$59$$ 4.78973 0.623570 0.311785 0.950153i $$-0.399073\pi$$
0.311785 + 0.950153i $$0.399073\pi$$
$$60$$ 0 0
$$61$$ 12.4522 1.59434 0.797172 0.603752i $$-0.206328\pi$$
0.797172 + 0.603752i $$0.206328\pi$$
$$62$$ 0 0
$$63$$ − 22.7455i − 2.86567i
$$64$$ 0 0
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 9.12544i 1.11485i 0.830227 + 0.557425i $$0.188211\pi$$
−0.830227 + 0.557425i $$0.811789\pi$$
$$68$$ 0 0
$$69$$ 2.95759 0.356052
$$70$$ 0 0
$$71$$ 1.66249 0.197302 0.0986509 0.995122i $$-0.468547\pi$$
0.0986509 + 0.995122i $$0.468547\pi$$
$$72$$ 0 0
$$73$$ − 12.3357i − 1.44379i −0.692005 0.721893i $$-0.743272\pi$$
0.692005 0.721893i $$-0.256728\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ − 3.78973i − 0.431880i
$$78$$ 0 0
$$79$$ −11.8303 −1.33102 −0.665509 0.746390i $$-0.731786\pi$$
−0.665509 + 0.746390i $$0.731786\pi$$
$$80$$ 0 0
$$81$$ 6.78973 0.754415
$$82$$ 0 0
$$83$$ − 0.704908i − 0.0773737i −0.999251 0.0386868i $$-0.987683\pi$$
0.999251 0.0386868i $$-0.0123175\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 0 0
$$87$$ − 15.4098i − 1.65211i
$$88$$ 0 0
$$89$$ 15.8303 1.67801 0.839007 0.544121i $$-0.183137\pi$$
0.839007 + 0.544121i $$0.183137\pi$$
$$90$$ 0 0
$$91$$ −10.8728 −1.13978
$$92$$ 0 0
$$93$$ − 17.6201i − 1.82712i
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 0 0
$$97$$ − 10.8728i − 1.10396i −0.833857 0.551981i $$-0.813872\pi$$
0.833857 0.551981i $$-0.186128\pi$$
$$98$$ 0 0
$$99$$ 5.50356 0.553129
$$100$$ 0 0
$$101$$ 9.21027 0.916456 0.458228 0.888835i $$-0.348484\pi$$
0.458228 + 0.888835i $$0.348484\pi$$
$$102$$ 0 0
$$103$$ − 12.4522i − 1.22695i −0.789712 0.613477i $$-0.789770\pi$$
0.789712 0.613477i $$-0.210230\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 3.63080i 0.351003i 0.984479 + 0.175501i $$0.0561546\pi$$
−0.984479 + 0.175501i $$0.943845\pi$$
$$108$$ 0 0
$$109$$ −16.6625 −1.59598 −0.797989 0.602672i $$-0.794103\pi$$
−0.797989 + 0.602672i $$0.794103\pi$$
$$110$$ 0 0
$$111$$ −26.9893 −2.56171
$$112$$ 0 0
$$113$$ 18.1147i 1.70409i 0.523469 + 0.852045i $$0.324638\pi$$
−0.523469 + 0.852045i $$0.675362\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 0 0
$$117$$ − 15.7897i − 1.45976i
$$118$$ 0 0
$$119$$ −22.7455 −2.08508
$$120$$ 0 0
$$121$$ −10.0830 −0.916639
$$122$$ 0 0
$$123$$ 0.747321i 0.0673836i
$$124$$ 0 0
$$125$$ 0 0
$$126$$ 0 0
$$127$$ − 5.91517i − 0.524887i −0.964947 0.262443i $$-0.915472\pi$$
0.964947 0.262443i $$-0.0845283\pi$$
$$128$$ 0 0
$$129$$ 23.6607 2.08321
$$130$$ 0 0
$$131$$ 3.40982 0.297917 0.148958 0.988843i $$-0.452408\pi$$
0.148958 + 0.988843i $$0.452408\pi$$
$$132$$ 0 0
$$133$$ 26.7031i 2.31545i
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ − 1.67321i − 0.142952i −0.997442 0.0714761i $$-0.977229\pi$$
0.997442 0.0714761i $$-0.0227710\pi$$
$$138$$ 0 0
$$139$$ −8.62008 −0.731146 −0.365573 0.930783i $$-0.619127\pi$$
−0.365573 + 0.930783i $$0.619127\pi$$
$$140$$ 0 0
$$141$$ −16.2509 −1.36857
$$142$$ 0 0
$$143$$ − 2.63080i − 0.219998i
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 0 0
$$147$$ − 25.6201i − 2.11311i
$$148$$ 0 0
$$149$$ 23.0830 1.89104 0.945518 0.325571i $$-0.105556\pi$$
0.945518 + 0.325571i $$0.105556\pi$$
$$150$$ 0 0
$$151$$ 10.7879 0.877910 0.438955 0.898509i $$-0.355349\pi$$
0.438955 + 0.898509i $$0.355349\pi$$
$$152$$ 0 0
$$153$$ − 33.0317i − 2.67045i
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ 8.19955i 0.654395i 0.944956 + 0.327198i $$0.106104\pi$$
−0.944956 + 0.327198i $$0.893896\pi$$
$$158$$ 0 0
$$159$$ −21.0741 −1.67129
$$160$$ 0 0
$$161$$ 3.95759 0.311902
$$162$$ 0 0
$$163$$ − 16.6625i − 1.30511i −0.757743 0.652554i $$-0.773698\pi$$
0.757743 0.652554i $$-0.226302\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ − 8.00000i − 0.619059i −0.950890 0.309529i $$-0.899829\pi$$
0.950890 0.309529i $$-0.100171\pi$$
$$168$$ 0 0
$$169$$ 5.45223 0.419402
$$170$$ 0 0
$$171$$ −38.7790 −2.96551
$$172$$ 0 0
$$173$$ − 8.87276i − 0.674584i −0.941400 0.337292i $$-0.890489\pi$$
0.941400 0.337292i $$-0.109511\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ 14.1661i 1.06479i
$$178$$ 0 0
$$179$$ −1.49464 −0.111715 −0.0558574 0.998439i $$-0.517789\pi$$
−0.0558574 + 0.998439i $$0.517789\pi$$
$$180$$ 0 0
$$181$$ −12.5777 −0.934891 −0.467445 0.884022i $$-0.654826\pi$$
−0.467445 + 0.884022i $$0.654826\pi$$
$$182$$ 0 0
$$183$$ 36.8285i 2.72244i
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 0 0
$$187$$ − 5.50356i − 0.402460i
$$188$$ 0 0
$$189$$ 32.1571 2.33909
$$190$$ 0 0
$$191$$ 16.9045 1.22316 0.611582 0.791181i $$-0.290534\pi$$
0.611582 + 0.791181i $$0.290534\pi$$
$$192$$ 0 0
$$193$$ − 9.91517i − 0.713710i −0.934160 0.356855i $$-0.883849\pi$$
0.934160 0.356855i $$-0.116151\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 4.53705i 0.323252i 0.986852 + 0.161626i $$0.0516738\pi$$
−0.986852 + 0.161626i $$0.948326\pi$$
$$198$$ 0 0
$$199$$ 14.0000 0.992434 0.496217 0.868199i $$-0.334722\pi$$
0.496217 + 0.868199i $$0.334722\pi$$
$$200$$ 0 0
$$201$$ −26.9893 −1.90368
$$202$$ 0 0
$$203$$ − 20.6201i − 1.44725i
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 0 0
$$207$$ 5.74732i 0.399466i
$$208$$ 0 0
$$209$$ −6.46115 −0.446927
$$210$$ 0 0
$$211$$ 8.70491 0.599271 0.299635 0.954054i $$-0.403135\pi$$
0.299635 + 0.954054i $$0.403135\pi$$
$$212$$ 0 0
$$213$$ 4.91697i 0.336905i
$$214$$ 0 0
$$215$$ 0 0
$$216$$ 0 0
$$217$$ − 23.5777i − 1.60056i
$$218$$ 0 0
$$219$$ 36.4839 2.46536
$$220$$ 0 0
$$221$$ −15.7897 −1.06213
$$222$$ 0 0
$$223$$ 5.40982i 0.362268i 0.983458 + 0.181134i $$0.0579768\pi$$
−0.983458 + 0.181134i $$0.942023\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ − 17.3250i − 1.14990i −0.818189 0.574950i $$-0.805022\pi$$
0.818189 0.574950i $$-0.194978\pi$$
$$228$$ 0 0
$$229$$ 11.4098 0.753982 0.376991 0.926217i $$-0.376959\pi$$
0.376991 + 0.926217i $$0.376959\pi$$
$$230$$ 0 0
$$231$$ 11.2085 0.737463
$$232$$ 0 0
$$233$$ − 2.08483i − 0.136581i −0.997665 0.0682907i $$-0.978245\pi$$
0.997665 0.0682907i $$-0.0217546\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 0 0
$$237$$ − 34.9893i − 2.27280i
$$238$$ 0 0
$$239$$ 18.5353 1.19895 0.599473 0.800395i $$-0.295377\pi$$
0.599473 + 0.800395i $$0.295377\pi$$
$$240$$ 0 0
$$241$$ 10.2509 0.660317 0.330159 0.943925i $$-0.392898\pi$$
0.330159 + 0.943925i $$0.392898\pi$$
$$242$$ 0 0
$$243$$ − 4.29509i − 0.275530i
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 18.5371i 1.17948i
$$248$$ 0 0
$$249$$ 2.08483 0.132120
$$250$$ 0 0
$$251$$ −17.2527 −1.08898 −0.544490 0.838768i $$-0.683277\pi$$
−0.544490 + 0.838768i $$0.683277\pi$$
$$252$$ 0 0
$$253$$ 0.957587i 0.0602030i
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ 28.9045i 1.80301i 0.432768 + 0.901505i $$0.357537\pi$$
−0.432768 + 0.901505i $$0.642463\pi$$
$$258$$ 0 0
$$259$$ −36.1147 −2.24406
$$260$$ 0 0
$$261$$ 29.9451 1.85355
$$262$$ 0 0
$$263$$ − 5.24196i − 0.323233i −0.986854 0.161617i $$-0.948329\pi$$
0.986854 0.161617i $$-0.0516708\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 0 0
$$267$$ 46.8196i 2.86531i
$$268$$ 0 0
$$269$$ 5.37992 0.328019 0.164010 0.986459i $$-0.447557\pi$$
0.164010 + 0.986459i $$0.447557\pi$$
$$270$$ 0 0
$$271$$ −11.5371 −0.700826 −0.350413 0.936595i $$-0.613959\pi$$
−0.350413 + 0.936595i $$0.613959\pi$$
$$272$$ 0 0
$$273$$ − 32.1571i − 1.94624i
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 0 0
$$277$$ 5.83035i 0.350312i 0.984541 + 0.175156i $$0.0560429\pi$$
−0.984541 + 0.175156i $$0.943957\pi$$
$$278$$ 0 0
$$279$$ 34.2402 2.04990
$$280$$ 0 0
$$281$$ 11.0741 0.660626 0.330313 0.943871i $$-0.392846\pi$$
0.330313 + 0.943871i $$0.392846\pi$$
$$282$$ 0 0
$$283$$ 3.86384i 0.229682i 0.993384 + 0.114841i $$0.0366358\pi$$
−0.993384 + 0.114841i $$0.963364\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 1.00000i 0.0590281i
$$288$$ 0 0
$$289$$ −16.0317 −0.943041
$$290$$ 0 0
$$291$$ 32.1571 1.88508
$$292$$ 0 0
$$293$$ 1.54597i 0.0903167i 0.998980 + 0.0451583i $$0.0143792\pi$$
−0.998980 + 0.0451583i $$0.985621\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 0 0
$$297$$ 7.78082i 0.451489i
$$298$$ 0 0
$$299$$ 2.74732 0.158882
$$300$$ 0 0
$$301$$ 31.6607 1.82489
$$302$$ 0 0
$$303$$ 27.2402i 1.56491i
$$304$$ 0 0
$$305$$ 0 0
$$306$$ 0 0
$$307$$ − 6.45223i − 0.368248i −0.982903 0.184124i $$-0.941055\pi$$
0.982903 0.184124i $$-0.0589448\pi$$
$$308$$ 0 0
$$309$$ 36.8285 2.09510
$$310$$ 0 0
$$311$$ 22.8196 1.29398 0.646991 0.762497i $$-0.276027\pi$$
0.646991 + 0.762497i $$0.276027\pi$$
$$312$$ 0 0
$$313$$ 16.8620i 0.953099i 0.879148 + 0.476550i $$0.158113\pi$$
−0.879148 + 0.476550i $$0.841887\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ − 1.33751i − 0.0751218i −0.999294 0.0375609i $$-0.988041\pi$$
0.999294 0.0375609i $$-0.0119588\pi$$
$$318$$ 0 0
$$319$$ 4.98928 0.279346
$$320$$ 0 0
$$321$$ −10.7384 −0.599359
$$322$$ 0 0
$$323$$ 38.7790i 2.15772i
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 0 0
$$327$$ − 49.2808i − 2.72523i
$$328$$ 0 0
$$329$$ −21.7455 −1.19887
$$330$$ 0 0
$$331$$ −32.2808 −1.77431 −0.887156 0.461470i $$-0.847322\pi$$
−0.887156 + 0.461470i $$0.847322\pi$$
$$332$$ 0 0
$$333$$ − 52.4468i − 2.87407i
$$334$$ 0 0
$$335$$ 0 0
$$336$$ 0 0
$$337$$ − 4.83215i − 0.263224i −0.991301 0.131612i $$-0.957985\pi$$
0.991301 0.131612i $$-0.0420153\pi$$
$$338$$ 0 0
$$339$$ −53.5759 −2.90984
$$340$$ 0 0
$$341$$ 5.70491 0.308938
$$342$$ 0 0
$$343$$ − 6.57947i − 0.355258i
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ − 34.6183i − 1.85841i −0.369569 0.929203i $$-0.620495\pi$$
0.369569 0.929203i $$-0.379505\pi$$
$$348$$ 0 0
$$349$$ −26.9558 −1.44291 −0.721455 0.692461i $$-0.756526\pi$$
−0.721455 + 0.692461i $$0.756526\pi$$
$$350$$ 0 0
$$351$$ 22.3232 1.19152
$$352$$ 0 0
$$353$$ 8.00000i 0.425797i 0.977074 + 0.212899i $$0.0682904\pi$$
−0.977074 + 0.212899i $$0.931710\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0 0
$$357$$ − 67.2719i − 3.56040i
$$358$$ 0 0
$$359$$ −0.420532 −0.0221949 −0.0110974 0.999938i $$-0.503532\pi$$
−0.0110974 + 0.999938i $$0.503532\pi$$
$$360$$ 0 0
$$361$$ 26.5263 1.39612
$$362$$ 0 0
$$363$$ − 29.8214i − 1.56522i
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 0 0
$$367$$ 11.7156i 0.611551i 0.952104 + 0.305775i $$0.0989156\pi$$
−0.952104 + 0.305775i $$0.901084\pi$$
$$368$$ 0 0
$$369$$ −1.45223 −0.0756000
$$370$$ 0 0
$$371$$ −28.1995 −1.46405
$$372$$ 0 0
$$373$$ 28.1661i 1.45838i 0.684310 + 0.729192i $$0.260104\pi$$
−0.684310 + 0.729192i $$0.739896\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ − 14.3143i − 0.737223i
$$378$$ 0 0
$$379$$ −27.3781 −1.40632 −0.703160 0.711032i $$-0.748228\pi$$
−0.703160 + 0.711032i $$0.748228\pi$$
$$380$$ 0 0
$$381$$ 17.4946 0.896278
$$382$$ 0 0
$$383$$ 31.7754i 1.62365i 0.583902 + 0.811824i $$0.301525\pi$$
−0.583902 + 0.811824i $$0.698475\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ 45.9786i 2.33722i
$$388$$ 0 0
$$389$$ 18.7031 0.948285 0.474143 0.880448i $$-0.342758\pi$$
0.474143 + 0.880448i $$0.342758\pi$$
$$390$$ 0 0
$$391$$ 5.74732 0.290655
$$392$$ 0 0
$$393$$ 10.0848i 0.508712i
$$394$$ 0 0
$$395$$ 0 0
$$396$$ 0 0
$$397$$ − 39.6924i − 1.99210i −0.0887714 0.996052i $$-0.528294\pi$$
0.0887714 0.996052i $$-0.471706\pi$$
$$398$$ 0 0
$$399$$ −78.9768 −3.95378
$$400$$ 0 0
$$401$$ −11.3250 −0.565543 −0.282771 0.959187i $$-0.591254\pi$$
−0.282771 + 0.959187i $$0.591254\pi$$
$$402$$ 0 0
$$403$$ − 16.3674i − 0.815318i
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ − 8.73840i − 0.433147i
$$408$$ 0 0
$$409$$ −9.45223 −0.467383 −0.233691 0.972311i $$-0.575081\pi$$
−0.233691 + 0.972311i $$0.575081\pi$$
$$410$$ 0 0
$$411$$ 4.94867 0.244100
$$412$$ 0 0
$$413$$ 18.9558i 0.932753i
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 0 0
$$417$$ − 25.4946i − 1.24848i
$$418$$ 0 0
$$419$$ 33.2402 1.62389 0.811944 0.583735i $$-0.198409\pi$$
0.811944 + 0.583735i $$0.198409\pi$$
$$420$$ 0 0
$$421$$ −38.8285 −1.89239 −0.946194 0.323600i $$-0.895107\pi$$
−0.946194 + 0.323600i $$0.895107\pi$$
$$422$$ 0 0
$$423$$ − 31.5795i − 1.53545i
$$424$$ 0 0
$$425$$ 0 0
$$426$$ 0 0
$$427$$ 49.2808i 2.38486i
$$428$$ 0 0
$$429$$ 7.78082 0.375661
$$430$$ 0 0
$$431$$ −12.4839 −0.601329 −0.300665 0.953730i $$-0.597209\pi$$
−0.300665 + 0.953730i $$0.597209\pi$$
$$432$$ 0 0
$$433$$ 5.78793i 0.278150i 0.990282 + 0.139075i $$0.0444130\pi$$
−0.990282 + 0.139075i $$0.955587\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ − 6.74732i − 0.322768i
$$438$$ 0 0
$$439$$ 11.9241 0.569106 0.284553 0.958660i $$-0.408155\pi$$
0.284553 + 0.958660i $$0.408155\pi$$
$$440$$ 0 0
$$441$$ 49.7861 2.37077
$$442$$ 0 0
$$443$$ − 39.3973i − 1.87182i −0.352236 0.935911i $$-0.614579\pi$$
0.352236 0.935911i $$-0.385421\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 0 0
$$447$$ 68.2701i 3.22906i
$$448$$ 0 0
$$449$$ −21.3674 −1.00839 −0.504195 0.863590i $$-0.668211\pi$$
−0.504195 + 0.863590i $$0.668211\pi$$
$$450$$ 0 0
$$451$$ −0.241962 −0.0113936
$$452$$ 0 0
$$453$$ 31.9063i 1.49909i
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ − 20.3656i − 0.952663i −0.879266 0.476331i $$-0.841966\pi$$
0.879266 0.476331i $$-0.158034\pi$$
$$458$$ 0 0
$$459$$ 46.6995 2.17975
$$460$$ 0 0
$$461$$ −27.4946 −1.28055 −0.640277 0.768144i $$-0.721180\pi$$
−0.640277 + 0.768144i $$0.721180\pi$$
$$462$$ 0 0
$$463$$ 16.8196i 0.781675i 0.920460 + 0.390837i $$0.127814\pi$$
−0.920460 + 0.390837i $$0.872186\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ − 24.4504i − 1.13143i −0.824601 0.565715i $$-0.808600\pi$$
0.824601 0.565715i $$-0.191400\pi$$
$$468$$ 0 0
$$469$$ −36.1147 −1.66762
$$470$$ 0 0
$$471$$ −24.2509 −1.11742
$$472$$ 0 0
$$473$$ 7.66070i 0.352239i
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 0 0
$$477$$ − 40.9522i − 1.87507i
$$478$$ 0 0
$$479$$ −10.5688 −0.482899 −0.241449 0.970413i $$-0.577623\pi$$
−0.241449 + 0.970413i $$0.577623\pi$$
$$480$$ 0 0
$$481$$ −25.0705 −1.14312
$$482$$ 0 0
$$483$$ 11.7049i 0.532592i
$$484$$ 0 0
$$485$$ 0 0
$$486$$ 0 0
$$487$$ − 13.0741i − 0.592444i −0.955119 0.296222i $$-0.904273\pi$$
0.955119 0.296222i $$-0.0957269\pi$$
$$488$$ 0 0
$$489$$ 49.2808 2.22855
$$490$$ 0 0
$$491$$ 2.22098 0.100232 0.0501158 0.998743i $$-0.484041\pi$$
0.0501158 + 0.998743i $$0.484041\pi$$
$$492$$ 0 0
$$493$$ − 29.9451i − 1.34866i
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 6.57947i 0.295129i
$$498$$ 0 0
$$499$$ −2.87456 −0.128683 −0.0643415 0.997928i $$-0.520495\pi$$
−0.0643415 + 0.997928i $$0.520495\pi$$
$$500$$ 0 0
$$501$$ 23.6607 1.05708
$$502$$ 0 0
$$503$$ − 9.36740i − 0.417672i −0.977951 0.208836i $$-0.933033\pi$$
0.977951 0.208836i $$-0.0669675\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 0 0
$$507$$ 16.1254i 0.716156i
$$508$$ 0 0
$$509$$ 40.5866 1.79897 0.899484 0.436953i $$-0.143942\pi$$
0.899484 + 0.436953i $$0.143942\pi$$
$$510$$ 0 0
$$511$$ 48.8196 2.15965
$$512$$ 0 0
$$513$$ − 54.8250i − 2.42058i
$$514$$ 0 0
$$515$$ 0 0
$$516$$ 0 0
$$517$$ − 5.26160i − 0.231405i
$$518$$ 0 0
$$519$$ 26.2420 1.15189
$$520$$ 0 0
$$521$$ 34.6714 1.51898 0.759491 0.650518i $$-0.225448\pi$$
0.759491 + 0.650518i $$0.225448\pi$$
$$522$$ 0 0
$$523$$ − 35.4098i − 1.54836i −0.632964 0.774182i $$-0.718162\pi$$
0.632964 0.774182i $$-0.281838\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ − 34.2402i − 1.49152i
$$528$$ 0 0
$$529$$ −1.00000 −0.0434783
$$530$$ 0 0
$$531$$ −27.5281 −1.19462
$$532$$ 0 0
$$533$$ 0.694191i 0.0300687i
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 0 0
$$537$$ − 4.42053i − 0.190760i
$$538$$ 0 0
$$539$$ 8.29509 0.357295
$$540$$ 0 0
$$541$$ 37.0705 1.59379 0.796893 0.604121i $$-0.206475\pi$$
0.796893 + 0.604121i $$0.206475\pi$$
$$542$$ 0 0
$$543$$ − 37.1995i − 1.59639i
$$544$$ 0 0
$$545$$ 0 0
$$546$$ 0 0
$$547$$ 21.4187i 0.915799i 0.889004 + 0.457899i $$0.151398\pi$$
−0.889004 + 0.457899i $$0.848602\pi$$
$$548$$ 0 0
$$549$$ −71.5670 −3.05440
$$550$$ 0 0
$$551$$ −35.1553 −1.49767
$$552$$ 0 0
$$553$$ − 46.8196i − 1.99097i
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ 20.9558i 0.887925i 0.896045 + 0.443963i $$0.146428\pi$$
−0.896045 + 0.443963i $$0.853572\pi$$
$$558$$ 0 0
$$559$$ 21.9786 0.929594
$$560$$ 0 0
$$561$$ 16.2773 0.687226
$$562$$ 0 0
$$563$$ − 6.70491i − 0.282578i −0.989968 0.141289i $$-0.954875\pi$$
0.989968 0.141289i $$-0.0451247\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 0 0
$$567$$ 26.8710i 1.12847i
$$568$$ 0 0
$$569$$ −6.00000 −0.251533 −0.125767 0.992060i $$-0.540139\pi$$
−0.125767 + 0.992060i $$0.540139\pi$$
$$570$$ 0 0
$$571$$ 13.2933 0.556307 0.278154 0.960537i $$-0.410278\pi$$
0.278154 + 0.960537i $$0.410278\pi$$
$$572$$ 0 0
$$573$$ 49.9964i 2.08863i
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 0 0
$$577$$ 23.2402i 0.967501i 0.875206 + 0.483750i $$0.160726\pi$$
−0.875206 + 0.483750i $$0.839274\pi$$
$$578$$ 0 0
$$579$$ 29.3250 1.21870
$$580$$ 0 0
$$581$$ 2.78973 0.115738
$$582$$ 0 0
$$583$$ − 6.82323i − 0.282589i
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ 17.6112i 0.726891i 0.931616 + 0.363445i $$0.118400\pi$$
−0.931616 + 0.363445i $$0.881600\pi$$
$$588$$ 0 0
$$589$$ −40.1978 −1.65632
$$590$$ 0 0
$$591$$ −13.4187 −0.551973
$$592$$ 0 0
$$593$$ 41.5759i 1.70732i 0.520834 + 0.853658i $$0.325621\pi$$
−0.520834 + 0.853658i $$0.674379\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ 41.4062i 1.69464i
$$598$$ 0 0
$$599$$ −17.7138 −0.723767 −0.361884 0.932223i $$-0.617866\pi$$
−0.361884 + 0.932223i $$0.617866\pi$$
$$600$$ 0 0
$$601$$ −46.7772 −1.90808 −0.954041 0.299675i $$-0.903122\pi$$
−0.954041 + 0.299675i $$0.903122\pi$$
$$602$$ 0 0
$$603$$ − 52.4468i − 2.13580i
$$604$$ 0 0
$$605$$ 0 0
$$606$$ 0 0
$$607$$ 42.9893i 1.74488i 0.488720 + 0.872441i $$0.337464\pi$$
−0.488720 + 0.872441i $$0.662536\pi$$
$$608$$ 0 0
$$609$$ 60.9857 2.47126
$$610$$ 0 0
$$611$$ −15.0955 −0.610700
$$612$$ 0 0
$$613$$ 32.6500i 1.31872i 0.751827 + 0.659360i $$0.229173\pi$$
−0.751827 + 0.659360i $$0.770827\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ − 11.3232i − 0.455854i −0.973678 0.227927i $$-0.926805\pi$$
0.973678 0.227927i $$-0.0731948\pi$$
$$618$$ 0 0
$$619$$ 10.6625 0.428562 0.214281 0.976772i $$-0.431259\pi$$
0.214281 + 0.976772i $$0.431259\pi$$
$$620$$ 0 0
$$621$$ −8.12544 −0.326063
$$622$$ 0 0
$$623$$ 62.6500i 2.51002i
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 0 0
$$627$$ − 19.1094i − 0.763156i
$$628$$ 0 0
$$629$$ −52.4468 −2.09119
$$630$$ 0 0
$$631$$ 5.24376 0.208751 0.104375 0.994538i $$-0.466716\pi$$
0.104375 + 0.994538i $$0.466716\pi$$
$$632$$ 0 0
$$633$$ 25.7455i 1.02329i
$$634$$ 0 0
$$635$$ 0 0
$$636$$ 0 0
$$637$$ − 23.7987i − 0.942937i
$$638$$ 0 0
$$639$$ −9.55489 −0.377986
$$640$$ 0 0
$$641$$ 30.4205 1.20154 0.600769 0.799422i $$-0.294861\pi$$
0.600769 + 0.799422i $$0.294861\pi$$
$$642$$ 0 0
$$643$$ 9.37992i 0.369908i 0.982747 + 0.184954i $$0.0592136\pi$$
−0.982747 + 0.184954i $$0.940786\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ − 18.6714i − 0.734049i −0.930211 0.367024i $$-0.880377\pi$$
0.930211 0.367024i $$-0.119623\pi$$
$$648$$ 0 0
$$649$$ −4.58659 −0.180039
$$650$$ 0 0
$$651$$ 69.7330 2.73305
$$652$$ 0 0
$$653$$ − 12.0723i − 0.472426i −0.971701 0.236213i $$-0.924094\pi$$
0.971701 0.236213i $$-0.0759063\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 0 0
$$657$$ 70.8973i 2.76597i
$$658$$ 0 0
$$659$$ −21.2402 −0.827399 −0.413700 0.910413i $$-0.635764\pi$$
−0.413700 + 0.910413i $$0.635764\pi$$
$$660$$ 0 0
$$661$$ −26.9362 −1.04769 −0.523847 0.851812i $$-0.675504\pi$$
−0.523847 + 0.851812i $$0.675504\pi$$
$$662$$ 0 0
$$663$$ − 46.6995i − 1.81366i
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 5.21027i 0.201742i
$$668$$ 0 0
$$669$$ −16.0000 −0.618596
$$670$$ 0 0
$$671$$ −11.9241 −0.460324
$$672$$ 0 0
$$673$$ 11.4098i 0.439816i 0.975521 + 0.219908i $$0.0705757\pi$$
−0.975521 + 0.219908i $$0.929424\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ − 11.8638i − 0.455965i −0.973665 0.227982i $$-0.926787\pi$$
0.973665 0.227982i $$-0.0732128\pi$$
$$678$$ 0 0
$$679$$ 43.0299 1.65134
$$680$$ 0 0
$$681$$ 51.2402 1.96353
$$682$$ 0 0
$$683$$ − 10.0723i − 0.385406i −0.981257 0.192703i $$-0.938275\pi$$
0.981257 0.192703i $$-0.0617254\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 0 0
$$687$$ 33.7455i 1.28747i
$$688$$ 0 0
$$689$$ −19.5759 −0.745781
$$690$$ 0 0
$$691$$ −45.9964 −1.74979 −0.874893 0.484317i $$-0.839068\pi$$
−0.874893 + 0.484317i $$0.839068\pi$$
$$692$$ 0 0
$$693$$ 21.7808i 0.827385i
$$694$$ 0 0
$$695$$ 0 0
$$696$$ 0 0
$$697$$ 1.45223i 0.0550071i
$$698$$ 0 0
$$699$$ 6.16605 0.233222
$$700$$ 0 0
$$701$$ −21.2312 −0.801893 −0.400947 0.916101i $$-0.631319\pi$$
−0.400947 + 0.916101i $$0.631319\pi$$
$$702$$ 0 0
$$703$$ 61.5723i 2.32224i
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ 36.4504i 1.37086i
$$708$$ 0 0
$$709$$ −31.3781 −1.17843 −0.589215 0.807976i $$-0.700563\pi$$
−0.589215 + 0.807976i $$0.700563\pi$$
$$710$$ 0 0
$$711$$ 67.9928 2.54993
$$712$$ 0 0
$$713$$ 5.95759i 0.223113i
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ 54.8196i 2.04728i
$$718$$ 0 0
$$719$$ 20.4629 0.763139 0.381570 0.924340i $$-0.375384\pi$$
0.381570 + 0.924340i $$0.375384\pi$$
$$720$$ 0 0
$$721$$ 49.2808 1.83531
$$722$$ 0 0
$$723$$ 30.3179i 1.12753i
$$724$$ 0 0
$$725$$ 0 0
$$726$$ 0 0
$$727$$ 14.9875i 0.555855i 0.960602 + 0.277928i $$0.0896475\pi$$
−0.960602 + 0.277928i $$0.910352\pi$$
$$728$$ 0 0
$$729$$ 33.0723 1.22490
$$730$$ 0 0
$$731$$ 45.9786 1.70058
$$732$$ 0 0
$$733$$ 22.2844i 0.823092i 0.911389 + 0.411546i $$0.135011\pi$$
−0.911389 + 0.411546i $$0.864989\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ − 8.73840i − 0.321883i
$$738$$ 0 0
$$739$$ −30.9344 −1.13794 −0.568969 0.822359i $$-0.692658\pi$$
−0.568969 + 0.822359i $$0.692658\pi$$
$$740$$ 0 0
$$741$$ −54.8250 −2.01404
$$742$$ 0 0
$$743$$ 44.8285i 1.64460i 0.569054 + 0.822300i $$0.307309\pi$$
−0.569054 + 0.822300i $$0.692691\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 0 0
$$747$$ 4.05133i 0.148230i
$$748$$ 0 0
$$749$$ −14.3692 −0.525039
$$750$$ 0 0
$$751$$ 32.1661 1.17376 0.586878 0.809675i $$-0.300357\pi$$
0.586878 + 0.809675i $$0.300357\pi$$
$$752$$ 0 0
$$753$$ − 51.0263i − 1.85950i
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 0 0
$$757$$ 40.6835i 1.47867i 0.673340 + 0.739333i $$0.264859\pi$$
−0.673340 + 0.739333i $$0.735141\pi$$
$$758$$ 0 0
$$759$$ −2.83215 −0.102800
$$760$$ 0 0
$$761$$ 28.4415 1.03100 0.515502 0.856888i $$-0.327605\pi$$
0.515502 + 0.856888i $$0.327605\pi$$
$$762$$ 0 0
$$763$$ − 65.9433i − 2.38731i
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ 13.1589i 0.475142i
$$768$$ 0 0
$$769$$ 40.4205 1.45760 0.728801 0.684726i $$-0.240078\pi$$
0.728801 + 0.684726i $$0.240078\pi$$
$$770$$ 0 0
$$771$$ −85.4874 −3.07876
$$772$$ 0 0
$$773$$ − 24.5688i − 0.883677i −0.897095 0.441838i $$-0.854327\pi$$
0.897095 0.441838i $$-0.145673\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 0 0
$$777$$ − 106.812i − 3.83187i
$$778$$ 0 0
$$779$$ 1.70491 0.0610847
$$780$$ 0 0
$$781$$ −1.59198 −0.0569656
$$782$$ 0 0
$$783$$ 42.3357i 1.51295i
$$784$$ 0 0
$$785$$ 0 0
$$786$$ 0 0
$$787$$ − 11.7790i − 0.419877i −0.977715 0.209938i $$-0.932674\pi$$
0.977715 0.209938i $$-0.0673263\pi$$
$$788$$ 0 0
$$789$$ 15.5036 0.551941
$$790$$ 0 0
$$791$$ −71.6906 −2.54902
$$792$$ 0 0
$$793$$ 34.2103i 1.21484i
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ − 13.8602i − 0.490955i −0.969402 0.245478i $$-0.921055\pi$$
0.969402 0.245478i $$-0.0789448\pi$$
$$798$$ 0 0
$$799$$ −31.5795 −1.11720
$$800$$ 0 0
$$801$$ −90.9821 −3.21469
$$802$$ 0 0
$$803$$ 11.8125i 0.416854i
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ 15.9116i 0.560114i
$$808$$ 0 0
$$809$$ 52.0830 1.83114 0.915571 0.402157i $$-0.131739\pi$$
0.915571 + 0.402157i $$0.131739\pi$$
$$810$$ 0 0
$$811$$ −40.7013 −1.42922 −0.714608 0.699525i $$-0.753395\pi$$
−0.714608 + 0.699525i $$0.753395\pi$$
$$812$$ 0 0
$$813$$ − 34.1218i − 1.19671i
$$814$$ 0 0
$$815$$ 0 0
$$816$$ 0 0
$$817$$ − 53.9786i − 1.88847i
$$818$$ 0 0
$$819$$ 62.4892 2.18355
$$820$$ 0 0
$$821$$ 10.0000 0.349002 0.174501 0.984657i $$-0.444169\pi$$
0.174501 + 0.984657i $$0.444169\pi$$
$$822$$ 0 0
$$823$$ 42.9009i 1.49543i 0.664020 + 0.747715i $$0.268849\pi$$
−0.664020 + 0.747715i $$0.731151\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 39.4397i 1.37145i 0.727859 + 0.685727i $$0.240515\pi$$
−0.727859 + 0.685727i $$0.759485\pi$$
$$828$$ 0 0
$$829$$ −29.5460 −1.02617 −0.513087 0.858337i $$-0.671498\pi$$
−0.513087 + 0.858337i $$0.671498\pi$$
$$830$$ 0 0
$$831$$ −17.2438 −0.598179
$$832$$ 0 0
$$833$$ − 49.7861i − 1.72499i
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 0 0
$$837$$ 48.4080i 1.67323i
$$838$$ 0 0
$$839$$ 1.09554 0.0378223 0.0189112 0.999821i $$-0.493980\pi$$
0.0189112 + 0.999821i $$0.493980\pi$$
$$840$$ 0 0
$$841$$ −1.85313 −0.0639009
$$842$$ 0 0
$$843$$ 32.7526i 1.12806i
$$844$$ 0 0
$$845$$ 0 0
$$846$$ 0 0
$$847$$ − 39.9045i − 1.37113i
$$848$$ 0 0
$$849$$ −11.4277 −0.392196
$$850$$ 0 0
$$851$$ 9.12544 0.312816
$$852$$ 0 0
$$853$$ 10.0937i 0.345603i 0.984957 + 0.172802i $$0.0552819\pi$$
−0.984957 + 0.172802i $$0.944718\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ 41.8303i 1.42890i 0.699688 + 0.714449i $$0.253322\pi$$
−0.699688 + 0.714449i $$0.746678\pi$$
$$858$$ 0 0
$$859$$ 3.35848 0.114590 0.0572950 0.998357i $$-0.481752\pi$$
0.0572950 + 0.998357i $$0.481752\pi$$
$$860$$ 0 0
$$861$$ −2.95759 −0.100794
$$862$$ 0 0
$$863$$ 3.83395i 0.130509i 0.997869 + 0.0652545i $$0.0207859\pi$$
−0.997869 + 0.0652545i $$0.979214\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 0 0
$$867$$ − 47.4151i − 1.61030i
$$868$$ 0 0
$$869$$ 11.3286 0.384296
$$870$$ 0 0
$$871$$ −25.0705 −0.849482
$$872$$ 0 0
$$873$$ 62.4892i 2.11494i
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 15.5263i 0.524287i 0.965029 + 0.262144i $$0.0844294\pi$$
−0.965029 + 0.262144i $$0.915571\pi$$
$$878$$ 0 0
$$879$$ −4.57235 −0.154221
$$880$$ 0 0
$$881$$ 26.9893 0.909292 0.454646 0.890672i $$-0.349766\pi$$
0.454646 + 0.890672i $$0.349766\pi$$
$$882$$ 0 0
$$883$$ − 21.5884i − 0.726507i −0.931690 0.363254i $$-0.881666\pi$$
0.931690 0.363254i $$-0.118334\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ 49.4098i 1.65902i 0.558492 + 0.829510i $$0.311380\pi$$
−0.558492 + 0.829510i $$0.688620\pi$$
$$888$$ 0 0
$$889$$ 23.4098 0.785140
$$890$$ 0 0
$$891$$ −6.50176 −0.217817
$$892$$ 0 0
$$893$$ 37.0741i 1.24064i
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 0 0
$$897$$ 8.12544i 0.271301i
$$898$$ 0 0
$$899$$ 31.0406 1.03526
$$900$$ 0 0
$$901$$ −40.9522 −1.36432
$$902$$ 0 0
$$903$$ 93.6393i 3.11612i
$$904$$ 0 0
$$905$$ 0 0
$$906$$ 0 0
$$907$$ − 9.54957i − 0.317088i −0.987352 0.158544i $$-0.949320\pi$$
0.987352 0.158544i $$-0.0506800\pi$$
$$908$$ 0 0
$$909$$ −52.9344 −1.75572
$$910$$ 0 0
$$911$$ −47.3036 −1.56724 −0.783618 0.621243i $$-0.786628\pi$$
−0.783618 + 0.621243i $$0.786628\pi$$
$$912$$ 0 0
$$913$$ 0.675011i 0.0223396i
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ 13.4946i 0.445632i
$$918$$ 0 0
$$919$$ −8.00000 −0.263896 −0.131948 0.991257i $$-0.542123\pi$$
−0.131948 + 0.991257i $$0.542123\pi$$
$$920$$ 0 0
$$921$$ 19.0830 0.628807
$$922$$ 0 0
$$923$$ 4.56741i 0.150338i
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 0 0
$$927$$ 71.5670i 2.35057i
$$928$$ 0 0
$$929$$ 19.6942 0.646145 0.323073 0.946374i $$-0.395284\pi$$
0.323073 + 0.946374i $$0.395284\pi$$
$$930$$ 0 0
$$931$$ −58.4486 −1.91558
$$932$$ 0 0
$$933$$ 67.4910i 2.20956i
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ − 31.8214i − 1.03956i −0.854300 0.519780i $$-0.826014\pi$$
0.854300 0.519780i $$-0.173986\pi$$
$$938$$ 0 0
$$939$$ −49.8710 −1.62748
$$940$$ 0 0
$$941$$ −2.36740 −0.0771751 −0.0385876 0.999255i $$-0.512286\pi$$
−0.0385876 + 0.999255i $$0.512286\pi$$
$$942$$ 0 0
$$943$$ − 0.252679i − 0.00822837i
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ − 9.90266i − 0.321793i −0.986971 0.160897i $$-0.948561\pi$$
0.986971 0.160897i $$-0.0514386\pi$$
$$948$$ 0 0
$$949$$ 33.8901 1.10012
$$950$$ 0 0
$$951$$ 3.95579 0.128275
$$952$$ 0 0
$$953$$ 31.5442i 1.02182i 0.859635 + 0.510908i $$0.170691\pi$$
−0.859635 + 0.510908i $$0.829309\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 0 0
$$957$$ 14.7562i 0.477001i
$$958$$ 0 0
$$959$$ 6.62188 0.213832
$$960$$ 0 0
$$961$$ 4.49284 0.144930
$$962$$ 0 0
$$963$$ − 20.8674i − 0.672441i
$$964$$ 0 0
$$965$$ 0 0
$$966$$ 0 0
$$967$$ − 15.5973i − 0.501575i −0.968042 0.250788i $$-0.919310\pi$$
0.968042 0.250788i $$-0.0806896\pi$$
$$968$$ 0 0
$$969$$ −114.692 −3.68445
$$970$$ 0 0
$$971$$ −47.1022 −1.51158 −0.755791 0.654813i $$-0.772747\pi$$
−0.755791 + 0.654813i $$0.772747\pi$$
$$972$$ 0 0
$$973$$ − 34.1147i − 1.09367i
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ 28.9469i 0.926092i 0.886334 + 0.463046i $$0.153244\pi$$
−0.886334 + 0.463046i $$0.846756\pi$$
$$978$$ 0 0
$$979$$ −15.1589 −0.484482
$$980$$ 0 0
$$981$$ 95.7647 3.05753
$$982$$ 0 0
$$983$$ 36.2312i 1.15560i 0.816179 + 0.577799i $$0.196088\pi$$
−0.816179 + 0.577799i $$0.803912\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 0 0
$$987$$ − 64.3143i − 2.04715i
$$988$$ 0 0
$$989$$ −8.00000 −0.254385
$$990$$ 0 0
$$991$$ −35.1750 −1.11737 −0.558685 0.829380i $$-0.688694\pi$$
−0.558685 + 0.829380i $$0.688694\pi$$
$$992$$ 0 0
$$993$$ − 95.4732i − 3.02975i
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 0 0
$$997$$ 19.2402i 0.609342i 0.952458 + 0.304671i $$0.0985465\pi$$
−0.952458 + 0.304671i $$0.901453\pi$$
$$998$$ 0 0
$$999$$ 74.1482 2.34595
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 4600.2.e.p.4049.5 6
5.2 odd 4 4600.2.a.x.1.3 3
5.3 odd 4 920.2.a.h.1.1 3
5.4 even 2 inner 4600.2.e.p.4049.2 6
15.8 even 4 8280.2.a.bj.1.3 3
20.3 even 4 1840.2.a.s.1.3 3
20.7 even 4 9200.2.a.ce.1.1 3
40.3 even 4 7360.2.a.cc.1.1 3
40.13 odd 4 7360.2.a.by.1.3 3

By twisted newform
Twist Min Dim Char Parity Ord Type
920.2.a.h.1.1 3 5.3 odd 4
1840.2.a.s.1.3 3 20.3 even 4
4600.2.a.x.1.3 3 5.2 odd 4
4600.2.e.p.4049.2 6 5.4 even 2 inner
4600.2.e.p.4049.5 6 1.1 even 1 trivial
7360.2.a.by.1.3 3 40.13 odd 4
7360.2.a.cc.1.1 3 40.3 even 4
8280.2.a.bj.1.3 3 15.8 even 4
9200.2.a.ce.1.1 3 20.7 even 4