# Properties

 Label 6012.2.a.a.1.1 Level 6012 Weight 2 Character 6012.1 Self dual yes Analytic conductor 48.006 Analytic rank 0 Dimension 2 CM no Inner twists 1

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$6012 = 2^{2} \cdot 3^{2} \cdot 167$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 6012.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$48.0060616952$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{13})$$ Defining polynomial: $$x^{2} - x - 3$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 668) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$2.30278$$ of defining polynomial Character $$\chi$$ $$=$$ 6012.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+3.00000 q^{5} -0.302776 q^{7} +O(q^{10})$$ $$q+3.00000 q^{5} -0.302776 q^{7} +2.69722 q^{13} -2.30278 q^{17} +2.00000 q^{19} +2.30278 q^{23} +4.00000 q^{25} -7.60555 q^{29} +6.60555 q^{31} -0.908327 q^{35} +0.394449 q^{37} +6.21110 q^{41} +9.60555 q^{43} -1.60555 q^{47} -6.90833 q^{49} +4.60555 q^{53} -7.81665 q^{59} +6.81665 q^{61} +8.09167 q^{65} +8.21110 q^{67} -3.90833 q^{71} -3.09167 q^{73} +6.39445 q^{79} +1.60555 q^{83} -6.90833 q^{85} +13.8167 q^{89} -0.816654 q^{91} +6.00000 q^{95} +4.51388 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 6q^{5} + 3q^{7} + O(q^{10})$$ $$2q + 6q^{5} + 3q^{7} + 9q^{13} - q^{17} + 4q^{19} + q^{23} + 8q^{25} - 8q^{29} + 6q^{31} + 9q^{35} + 8q^{37} - 2q^{41} + 12q^{43} + 4q^{47} - 3q^{49} + 2q^{53} + 6q^{59} - 8q^{61} + 27q^{65} + 2q^{67} + 3q^{71} - 17q^{73} + 20q^{79} - 4q^{83} - 3q^{85} + 6q^{89} + 20q^{91} + 12q^{95} - 9q^{97} + O(q^{100})$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0 0
$$3$$ 0 0
$$4$$ 0 0
$$5$$ 3.00000 1.34164 0.670820 0.741620i $$-0.265942\pi$$
0.670820 + 0.741620i $$0.265942\pi$$
$$6$$ 0 0
$$7$$ −0.302776 −0.114438 −0.0572192 0.998362i $$-0.518223\pi$$
−0.0572192 + 0.998362i $$0.518223\pi$$
$$8$$ 0 0
$$9$$ 0 0
$$10$$ 0 0
$$11$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$12$$ 0 0
$$13$$ 2.69722 0.748075 0.374038 0.927413i $$-0.377973\pi$$
0.374038 + 0.927413i $$0.377973\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ −2.30278 −0.558505 −0.279253 0.960218i $$-0.590087\pi$$
−0.279253 + 0.960218i $$0.590087\pi$$
$$18$$ 0 0
$$19$$ 2.00000 0.458831 0.229416 0.973329i $$-0.426318\pi$$
0.229416 + 0.973329i $$0.426318\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ 2.30278 0.480162 0.240081 0.970753i $$-0.422826\pi$$
0.240081 + 0.970753i $$0.422826\pi$$
$$24$$ 0 0
$$25$$ 4.00000 0.800000
$$26$$ 0 0
$$27$$ 0 0
$$28$$ 0 0
$$29$$ −7.60555 −1.41232 −0.706158 0.708055i $$-0.749573\pi$$
−0.706158 + 0.708055i $$0.749573\pi$$
$$30$$ 0 0
$$31$$ 6.60555 1.18639 0.593196 0.805058i $$-0.297866\pi$$
0.593196 + 0.805058i $$0.297866\pi$$
$$32$$ 0 0
$$33$$ 0 0
$$34$$ 0 0
$$35$$ −0.908327 −0.153535
$$36$$ 0 0
$$37$$ 0.394449 0.0648470 0.0324235 0.999474i $$-0.489677\pi$$
0.0324235 + 0.999474i $$0.489677\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 0 0
$$41$$ 6.21110 0.970011 0.485006 0.874511i $$-0.338818\pi$$
0.485006 + 0.874511i $$0.338818\pi$$
$$42$$ 0 0
$$43$$ 9.60555 1.46483 0.732416 0.680857i $$-0.238392\pi$$
0.732416 + 0.680857i $$0.238392\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ −1.60555 −0.234194 −0.117097 0.993120i $$-0.537359\pi$$
−0.117097 + 0.993120i $$0.537359\pi$$
$$48$$ 0 0
$$49$$ −6.90833 −0.986904
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 0 0
$$53$$ 4.60555 0.632621 0.316311 0.948656i $$-0.397556\pi$$
0.316311 + 0.948656i $$0.397556\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 0 0
$$59$$ −7.81665 −1.01764 −0.508821 0.860872i $$-0.669918\pi$$
−0.508821 + 0.860872i $$0.669918\pi$$
$$60$$ 0 0
$$61$$ 6.81665 0.872783 0.436392 0.899757i $$-0.356256\pi$$
0.436392 + 0.899757i $$0.356256\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ 8.09167 1.00365
$$66$$ 0 0
$$67$$ 8.21110 1.00315 0.501573 0.865115i $$-0.332755\pi$$
0.501573 + 0.865115i $$0.332755\pi$$
$$68$$ 0 0
$$69$$ 0 0
$$70$$ 0 0
$$71$$ −3.90833 −0.463833 −0.231917 0.972736i $$-0.574500\pi$$
−0.231917 + 0.972736i $$0.574500\pi$$
$$72$$ 0 0
$$73$$ −3.09167 −0.361853 −0.180926 0.983497i $$-0.557910\pi$$
−0.180926 + 0.983497i $$0.557910\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ 0 0
$$78$$ 0 0
$$79$$ 6.39445 0.719432 0.359716 0.933062i $$-0.382874\pi$$
0.359716 + 0.933062i $$0.382874\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ 0 0
$$83$$ 1.60555 0.176232 0.0881161 0.996110i $$-0.471915\pi$$
0.0881161 + 0.996110i $$0.471915\pi$$
$$84$$ 0 0
$$85$$ −6.90833 −0.749313
$$86$$ 0 0
$$87$$ 0 0
$$88$$ 0 0
$$89$$ 13.8167 1.46456 0.732281 0.681002i $$-0.238456\pi$$
0.732281 + 0.681002i $$0.238456\pi$$
$$90$$ 0 0
$$91$$ −0.816654 −0.0856086
$$92$$ 0 0
$$93$$ 0 0
$$94$$ 0 0
$$95$$ 6.00000 0.615587
$$96$$ 0 0
$$97$$ 4.51388 0.458315 0.229157 0.973389i $$-0.426403\pi$$
0.229157 + 0.973389i $$0.426403\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 0 0
$$101$$ 10.8167 1.07630 0.538149 0.842850i $$-0.319124\pi$$
0.538149 + 0.842850i $$0.319124\pi$$
$$102$$ 0 0
$$103$$ −3.30278 −0.325432 −0.162716 0.986673i $$-0.552025\pi$$
−0.162716 + 0.986673i $$0.552025\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ −7.60555 −0.735256 −0.367628 0.929973i $$-0.619830\pi$$
−0.367628 + 0.929973i $$0.619830\pi$$
$$108$$ 0 0
$$109$$ 2.00000 0.191565 0.0957826 0.995402i $$-0.469465\pi$$
0.0957826 + 0.995402i $$0.469465\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 0 0
$$113$$ −9.42221 −0.886366 −0.443183 0.896431i $$-0.646151\pi$$
−0.443183 + 0.896431i $$0.646151\pi$$
$$114$$ 0 0
$$115$$ 6.90833 0.644205
$$116$$ 0 0
$$117$$ 0 0
$$118$$ 0 0
$$119$$ 0.697224 0.0639145
$$120$$ 0 0
$$121$$ −11.0000 −1.00000
$$122$$ 0 0
$$123$$ 0 0
$$124$$ 0 0
$$125$$ −3.00000 −0.268328
$$126$$ 0 0
$$127$$ 14.2111 1.26103 0.630516 0.776176i $$-0.282843\pi$$
0.630516 + 0.776176i $$0.282843\pi$$
$$128$$ 0 0
$$129$$ 0 0
$$130$$ 0 0
$$131$$ 20.0278 1.74983 0.874917 0.484274i $$-0.160916\pi$$
0.874917 + 0.484274i $$0.160916\pi$$
$$132$$ 0 0
$$133$$ −0.605551 −0.0525080
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ 2.78890 0.238272 0.119136 0.992878i $$-0.461988\pi$$
0.119136 + 0.992878i $$0.461988\pi$$
$$138$$ 0 0
$$139$$ 5.90833 0.501138 0.250569 0.968099i $$-0.419382\pi$$
0.250569 + 0.968099i $$0.419382\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 0 0
$$143$$ 0 0
$$144$$ 0 0
$$145$$ −22.8167 −1.89482
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ −17.5139 −1.43479 −0.717396 0.696665i $$-0.754666\pi$$
−0.717396 + 0.696665i $$0.754666\pi$$
$$150$$ 0 0
$$151$$ 19.5139 1.58802 0.794008 0.607907i $$-0.207991\pi$$
0.794008 + 0.607907i $$0.207991\pi$$
$$152$$ 0 0
$$153$$ 0 0
$$154$$ 0 0
$$155$$ 19.8167 1.59171
$$156$$ 0 0
$$157$$ 0.816654 0.0651761 0.0325880 0.999469i $$-0.489625\pi$$
0.0325880 + 0.999469i $$0.489625\pi$$
$$158$$ 0 0
$$159$$ 0 0
$$160$$ 0 0
$$161$$ −0.697224 −0.0549490
$$162$$ 0 0
$$163$$ −5.39445 −0.422526 −0.211263 0.977429i $$-0.567758\pi$$
−0.211263 + 0.977429i $$0.567758\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 1.00000 0.0773823
$$168$$ 0 0
$$169$$ −5.72498 −0.440383
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 0 0
$$173$$ −5.78890 −0.440122 −0.220061 0.975486i $$-0.570626\pi$$
−0.220061 + 0.975486i $$0.570626\pi$$
$$174$$ 0 0
$$175$$ −1.21110 −0.0915507
$$176$$ 0 0
$$177$$ 0 0
$$178$$ 0 0
$$179$$ −0.908327 −0.0678915 −0.0339458 0.999424i $$-0.510807\pi$$
−0.0339458 + 0.999424i $$0.510807\pi$$
$$180$$ 0 0
$$181$$ −7.21110 −0.535997 −0.267999 0.963419i $$-0.586362\pi$$
−0.267999 + 0.963419i $$0.586362\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 0 0
$$185$$ 1.18335 0.0870013
$$186$$ 0 0
$$187$$ 0 0
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 11.3028 0.817840 0.408920 0.912570i $$-0.365905\pi$$
0.408920 + 0.912570i $$0.365905\pi$$
$$192$$ 0 0
$$193$$ 5.90833 0.425291 0.212645 0.977129i $$-0.431792\pi$$
0.212645 + 0.977129i $$0.431792\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ −3.69722 −0.263416 −0.131708 0.991289i $$-0.542046\pi$$
−0.131708 + 0.991289i $$0.542046\pi$$
$$198$$ 0 0
$$199$$ −10.6972 −0.758306 −0.379153 0.925334i $$-0.623785\pi$$
−0.379153 + 0.925334i $$0.623785\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 0 0
$$203$$ 2.30278 0.161623
$$204$$ 0 0
$$205$$ 18.6333 1.30141
$$206$$ 0 0
$$207$$ 0 0
$$208$$ 0 0
$$209$$ 0 0
$$210$$ 0 0
$$211$$ 13.5139 0.930334 0.465167 0.885223i $$-0.345994\pi$$
0.465167 + 0.885223i $$0.345994\pi$$
$$212$$ 0 0
$$213$$ 0 0
$$214$$ 0 0
$$215$$ 28.8167 1.96528
$$216$$ 0 0
$$217$$ −2.00000 −0.135769
$$218$$ 0 0
$$219$$ 0 0
$$220$$ 0 0
$$221$$ −6.21110 −0.417804
$$222$$ 0 0
$$223$$ −0.513878 −0.0344118 −0.0172059 0.999852i $$-0.505477\pi$$
−0.0172059 + 0.999852i $$0.505477\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ −1.18335 −0.0785414 −0.0392707 0.999229i $$-0.512503\pi$$
−0.0392707 + 0.999229i $$0.512503\pi$$
$$228$$ 0 0
$$229$$ 8.69722 0.574729 0.287364 0.957821i $$-0.407221\pi$$
0.287364 + 0.957821i $$0.407221\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ −6.90833 −0.452580 −0.226290 0.974060i $$-0.572660\pi$$
−0.226290 + 0.974060i $$0.572660\pi$$
$$234$$ 0 0
$$235$$ −4.81665 −0.314204
$$236$$ 0 0
$$237$$ 0 0
$$238$$ 0 0
$$239$$ −11.0917 −0.717461 −0.358730 0.933441i $$-0.616790\pi$$
−0.358730 + 0.933441i $$0.616790\pi$$
$$240$$ 0 0
$$241$$ 2.69722 0.173743 0.0868717 0.996220i $$-0.472313\pi$$
0.0868717 + 0.996220i $$0.472313\pi$$
$$242$$ 0 0
$$243$$ 0 0
$$244$$ 0 0
$$245$$ −20.7250 −1.32407
$$246$$ 0 0
$$247$$ 5.39445 0.343241
$$248$$ 0 0
$$249$$ 0 0
$$250$$ 0 0
$$251$$ 10.3944 0.656092 0.328046 0.944662i $$-0.393610\pi$$
0.328046 + 0.944662i $$0.393610\pi$$
$$252$$ 0 0
$$253$$ 0 0
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ −19.1194 −1.19264 −0.596319 0.802748i $$-0.703371\pi$$
−0.596319 + 0.802748i $$0.703371\pi$$
$$258$$ 0 0
$$259$$ −0.119429 −0.00742099
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 0 0
$$263$$ −12.2111 −0.752969 −0.376484 0.926423i $$-0.622867\pi$$
−0.376484 + 0.926423i $$0.622867\pi$$
$$264$$ 0 0
$$265$$ 13.8167 0.848750
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 0 0
$$269$$ −5.30278 −0.323316 −0.161658 0.986847i $$-0.551684\pi$$
−0.161658 + 0.986847i $$0.551684\pi$$
$$270$$ 0 0
$$271$$ 14.4222 0.876087 0.438043 0.898954i $$-0.355672\pi$$
0.438043 + 0.898954i $$0.355672\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 0 0
$$277$$ 25.5139 1.53298 0.766490 0.642256i $$-0.222001\pi$$
0.766490 + 0.642256i $$0.222001\pi$$
$$278$$ 0 0
$$279$$ 0 0
$$280$$ 0 0
$$281$$ 15.4222 0.920012 0.460006 0.887916i $$-0.347847\pi$$
0.460006 + 0.887916i $$0.347847\pi$$
$$282$$ 0 0
$$283$$ −10.4222 −0.619536 −0.309768 0.950812i $$-0.600251\pi$$
−0.309768 + 0.950812i $$0.600251\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ −1.88057 −0.111007
$$288$$ 0 0
$$289$$ −11.6972 −0.688072
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 0 0
$$293$$ −18.0000 −1.05157 −0.525786 0.850617i $$-0.676229\pi$$
−0.525786 + 0.850617i $$0.676229\pi$$
$$294$$ 0 0
$$295$$ −23.4500 −1.36531
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ 6.21110 0.359197
$$300$$ 0 0
$$301$$ −2.90833 −0.167633
$$302$$ 0 0
$$303$$ 0 0
$$304$$ 0 0
$$305$$ 20.4500 1.17096
$$306$$ 0 0
$$307$$ 25.9361 1.48025 0.740125 0.672469i $$-0.234766\pi$$
0.740125 + 0.672469i $$0.234766\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 0 0
$$311$$ −6.21110 −0.352199 −0.176100 0.984372i $$-0.556348\pi$$
−0.176100 + 0.984372i $$0.556348\pi$$
$$312$$ 0 0
$$313$$ 10.5139 0.594280 0.297140 0.954834i $$-0.403967\pi$$
0.297140 + 0.954834i $$0.403967\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ −28.5416 −1.60306 −0.801529 0.597956i $$-0.795980\pi$$
−0.801529 + 0.597956i $$0.795980\pi$$
$$318$$ 0 0
$$319$$ 0 0
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 0 0
$$323$$ −4.60555 −0.256260
$$324$$ 0 0
$$325$$ 10.7889 0.598460
$$326$$ 0 0
$$327$$ 0 0
$$328$$ 0 0
$$329$$ 0.486122 0.0268008
$$330$$ 0 0
$$331$$ 9.18335 0.504762 0.252381 0.967628i $$-0.418786\pi$$
0.252381 + 0.967628i $$0.418786\pi$$
$$332$$ 0 0
$$333$$ 0 0
$$334$$ 0 0
$$335$$ 24.6333 1.34586
$$336$$ 0 0
$$337$$ 27.6056 1.50377 0.751885 0.659294i $$-0.229145\pi$$
0.751885 + 0.659294i $$0.229145\pi$$
$$338$$ 0 0
$$339$$ 0 0
$$340$$ 0 0
$$341$$ 0 0
$$342$$ 0 0
$$343$$ 4.21110 0.227378
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ −21.9083 −1.17610 −0.588050 0.808824i $$-0.700104\pi$$
−0.588050 + 0.808824i $$0.700104\pi$$
$$348$$ 0 0
$$349$$ −36.4500 −1.95112 −0.975561 0.219729i $$-0.929483\pi$$
−0.975561 + 0.219729i $$0.929483\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 0 0
$$353$$ 4.18335 0.222657 0.111329 0.993784i $$-0.464489\pi$$
0.111329 + 0.993784i $$0.464489\pi$$
$$354$$ 0 0
$$355$$ −11.7250 −0.622297
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ −1.18335 −0.0624546 −0.0312273 0.999512i $$-0.509942\pi$$
−0.0312273 + 0.999512i $$0.509942\pi$$
$$360$$ 0 0
$$361$$ −15.0000 −0.789474
$$362$$ 0 0
$$363$$ 0 0
$$364$$ 0 0
$$365$$ −9.27502 −0.485477
$$366$$ 0 0
$$367$$ 18.8167 0.982221 0.491111 0.871097i $$-0.336591\pi$$
0.491111 + 0.871097i $$0.336591\pi$$
$$368$$ 0 0
$$369$$ 0 0
$$370$$ 0 0
$$371$$ −1.39445 −0.0723962
$$372$$ 0 0
$$373$$ −16.4861 −0.853619 −0.426810 0.904342i $$-0.640363\pi$$
−0.426810 + 0.904342i $$0.640363\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ −20.5139 −1.05652
$$378$$ 0 0
$$379$$ −0.0277564 −0.00142575 −0.000712875 1.00000i $$-0.500227\pi$$
−0.000712875 1.00000i $$0.500227\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 0 0
$$383$$ 27.8444 1.42278 0.711391 0.702796i $$-0.248065\pi$$
0.711391 + 0.702796i $$0.248065\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ 0 0
$$388$$ 0 0
$$389$$ 4.81665 0.244214 0.122107 0.992517i $$-0.461035\pi$$
0.122107 + 0.992517i $$0.461035\pi$$
$$390$$ 0 0
$$391$$ −5.30278 −0.268173
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 0 0
$$395$$ 19.1833 0.965219
$$396$$ 0 0
$$397$$ 3.11943 0.156560 0.0782798 0.996931i $$-0.475057\pi$$
0.0782798 + 0.996931i $$0.475057\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 21.6972 1.08351 0.541754 0.840537i $$-0.317760\pi$$
0.541754 + 0.840537i $$0.317760\pi$$
$$402$$ 0 0
$$403$$ 17.8167 0.887511
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 0 0
$$408$$ 0 0
$$409$$ 19.0917 0.944022 0.472011 0.881593i $$-0.343528\pi$$
0.472011 + 0.881593i $$0.343528\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 0 0
$$413$$ 2.36669 0.116457
$$414$$ 0 0
$$415$$ 4.81665 0.236440
$$416$$ 0 0
$$417$$ 0 0
$$418$$ 0 0
$$419$$ 12.2111 0.596551 0.298276 0.954480i $$-0.403589\pi$$
0.298276 + 0.954480i $$0.403589\pi$$
$$420$$ 0 0
$$421$$ 12.6056 0.614357 0.307178 0.951652i $$-0.400615\pi$$
0.307178 + 0.951652i $$0.400615\pi$$
$$422$$ 0 0
$$423$$ 0 0
$$424$$ 0 0
$$425$$ −9.21110 −0.446804
$$426$$ 0 0
$$427$$ −2.06392 −0.0998799
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ −17.0278 −0.820198 −0.410099 0.912041i $$-0.634506\pi$$
−0.410099 + 0.912041i $$0.634506\pi$$
$$432$$ 0 0
$$433$$ −29.3305 −1.40954 −0.704768 0.709438i $$-0.748949\pi$$
−0.704768 + 0.709438i $$0.748949\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ 4.60555 0.220313
$$438$$ 0 0
$$439$$ −22.2111 −1.06008 −0.530039 0.847973i $$-0.677823\pi$$
−0.530039 + 0.847973i $$0.677823\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ −0.275019 −0.0130666 −0.00653328 0.999979i $$-0.502080\pi$$
−0.00653328 + 0.999979i $$0.502080\pi$$
$$444$$ 0 0
$$445$$ 41.4500 1.96492
$$446$$ 0 0
$$447$$ 0 0
$$448$$ 0 0
$$449$$ −3.97224 −0.187462 −0.0937309 0.995598i $$-0.529879\pi$$
−0.0937309 + 0.995598i $$0.529879\pi$$
$$450$$ 0 0
$$451$$ 0 0
$$452$$ 0 0
$$453$$ 0 0
$$454$$ 0 0
$$455$$ −2.44996 −0.114856
$$456$$ 0 0
$$457$$ 6.88057 0.321860 0.160930 0.986966i $$-0.448551\pi$$
0.160930 + 0.986966i $$0.448551\pi$$
$$458$$ 0 0
$$459$$ 0 0
$$460$$ 0 0
$$461$$ 21.6972 1.01054 0.505270 0.862961i $$-0.331393\pi$$
0.505270 + 0.862961i $$0.331393\pi$$
$$462$$ 0 0
$$463$$ 1.78890 0.0831371 0.0415686 0.999136i $$-0.486765\pi$$
0.0415686 + 0.999136i $$0.486765\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ 13.1194 0.607095 0.303547 0.952816i $$-0.401829\pi$$
0.303547 + 0.952816i $$0.401829\pi$$
$$468$$ 0 0
$$469$$ −2.48612 −0.114798
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 0 0
$$473$$ 0 0
$$474$$ 0 0
$$475$$ 8.00000 0.367065
$$476$$ 0 0
$$477$$ 0 0
$$478$$ 0 0
$$479$$ −3.63331 −0.166010 −0.0830050 0.996549i $$-0.526452\pi$$
−0.0830050 + 0.996549i $$0.526452\pi$$
$$480$$ 0 0
$$481$$ 1.06392 0.0485104
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ 13.5416 0.614894
$$486$$ 0 0
$$487$$ 30.8167 1.39644 0.698218 0.715885i $$-0.253977\pi$$
0.698218 + 0.715885i $$0.253977\pi$$
$$488$$ 0 0
$$489$$ 0 0
$$490$$ 0 0
$$491$$ −0.633308 −0.0285808 −0.0142904 0.999898i $$-0.504549\pi$$
−0.0142904 + 0.999898i $$0.504549\pi$$
$$492$$ 0 0
$$493$$ 17.5139 0.788785
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 1.18335 0.0530803
$$498$$ 0 0
$$499$$ 0.669468 0.0299695 0.0149848 0.999888i $$-0.495230\pi$$
0.0149848 + 0.999888i $$0.495230\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 0 0
$$503$$ 0.486122 0.0216751 0.0108376 0.999941i $$-0.496550\pi$$
0.0108376 + 0.999941i $$0.496550\pi$$
$$504$$ 0 0
$$505$$ 32.4500 1.44400
$$506$$ 0 0
$$507$$ 0 0
$$508$$ 0 0
$$509$$ −24.9083 −1.10404 −0.552021 0.833830i $$-0.686143\pi$$
−0.552021 + 0.833830i $$0.686143\pi$$
$$510$$ 0 0
$$511$$ 0.936083 0.0414099
$$512$$ 0 0
$$513$$ 0 0
$$514$$ 0 0
$$515$$ −9.90833 −0.436613
$$516$$ 0 0
$$517$$ 0 0
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 0 0
$$521$$ 43.8167 1.91964 0.959821 0.280612i $$-0.0905375\pi$$
0.959821 + 0.280612i $$0.0905375\pi$$
$$522$$ 0 0
$$523$$ −8.81665 −0.385525 −0.192763 0.981245i $$-0.561745\pi$$
−0.192763 + 0.981245i $$0.561745\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ −15.2111 −0.662606
$$528$$ 0 0
$$529$$ −17.6972 −0.769445
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 0 0
$$533$$ 16.7527 0.725642
$$534$$ 0 0
$$535$$ −22.8167 −0.986450
$$536$$ 0 0
$$537$$ 0 0
$$538$$ 0 0
$$539$$ 0 0
$$540$$ 0 0
$$541$$ −35.6056 −1.53080 −0.765401 0.643554i $$-0.777459\pi$$
−0.765401 + 0.643554i $$0.777459\pi$$
$$542$$ 0 0
$$543$$ 0 0
$$544$$ 0 0
$$545$$ 6.00000 0.257012
$$546$$ 0 0
$$547$$ −34.3583 −1.46905 −0.734527 0.678579i $$-0.762596\pi$$
−0.734527 + 0.678579i $$0.762596\pi$$
$$548$$ 0 0
$$549$$ 0 0
$$550$$ 0 0
$$551$$ −15.2111 −0.648015
$$552$$ 0 0
$$553$$ −1.93608 −0.0823306
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ −10.8167 −0.458316 −0.229158 0.973389i $$-0.573597\pi$$
−0.229158 + 0.973389i $$0.573597\pi$$
$$558$$ 0 0
$$559$$ 25.9083 1.09581
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ −23.5139 −0.990992 −0.495496 0.868610i $$-0.665014\pi$$
−0.495496 + 0.868610i $$0.665014\pi$$
$$564$$ 0 0
$$565$$ −28.2666 −1.18919
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ −36.2111 −1.51805 −0.759024 0.651062i $$-0.774324\pi$$
−0.759024 + 0.651062i $$0.774324\pi$$
$$570$$ 0 0
$$571$$ −14.5416 −0.608548 −0.304274 0.952584i $$-0.598414\pi$$
−0.304274 + 0.952584i $$0.598414\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 0 0
$$575$$ 9.21110 0.384130
$$576$$ 0 0
$$577$$ −24.5139 −1.02053 −0.510263 0.860018i $$-0.670452\pi$$
−0.510263 + 0.860018i $$0.670452\pi$$
$$578$$ 0 0
$$579$$ 0 0
$$580$$ 0 0
$$581$$ −0.486122 −0.0201677
$$582$$ 0 0
$$583$$ 0 0
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ 42.9083 1.77102 0.885508 0.464624i $$-0.153810\pi$$
0.885508 + 0.464624i $$0.153810\pi$$
$$588$$ 0 0
$$589$$ 13.2111 0.544354
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 0 0
$$593$$ 28.3944 1.16602 0.583010 0.812465i $$-0.301875\pi$$
0.583010 + 0.812465i $$0.301875\pi$$
$$594$$ 0 0
$$595$$ 2.09167 0.0857502
$$596$$ 0 0
$$597$$ 0 0
$$598$$ 0 0
$$599$$ 26.0917 1.06608 0.533038 0.846091i $$-0.321050\pi$$
0.533038 + 0.846091i $$0.321050\pi$$
$$600$$ 0 0
$$601$$ 24.5416 1.00107 0.500537 0.865715i $$-0.333136\pi$$
0.500537 + 0.865715i $$0.333136\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ 0 0
$$605$$ −33.0000 −1.34164
$$606$$ 0 0
$$607$$ 13.7250 0.557080 0.278540 0.960425i $$-0.410150\pi$$
0.278540 + 0.960425i $$0.410150\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ −4.33053 −0.175195
$$612$$ 0 0
$$613$$ −38.3305 −1.54816 −0.774078 0.633090i $$-0.781786\pi$$
−0.774078 + 0.633090i $$0.781786\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 38.4500 1.54794 0.773969 0.633224i $$-0.218269\pi$$
0.773969 + 0.633224i $$0.218269\pi$$
$$618$$ 0 0
$$619$$ 25.0278 1.00595 0.502975 0.864301i $$-0.332239\pi$$
0.502975 + 0.864301i $$0.332239\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 0 0
$$623$$ −4.18335 −0.167602
$$624$$ 0 0
$$625$$ −29.0000 −1.16000
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ −0.908327 −0.0362174
$$630$$ 0 0
$$631$$ −33.9361 −1.35097 −0.675487 0.737372i $$-0.736067\pi$$
−0.675487 + 0.737372i $$0.736067\pi$$
$$632$$ 0 0
$$633$$ 0 0
$$634$$ 0 0
$$635$$ 42.6333 1.69185
$$636$$ 0 0
$$637$$ −18.6333 −0.738279
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 0 0
$$641$$ 11.3028 0.446433 0.223216 0.974769i $$-0.428344\pi$$
0.223216 + 0.974769i $$0.428344\pi$$
$$642$$ 0 0
$$643$$ −12.5139 −0.493499 −0.246750 0.969079i $$-0.579363\pi$$
−0.246750 + 0.969079i $$0.579363\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 35.9361 1.41279 0.706397 0.707816i $$-0.250320\pi$$
0.706397 + 0.707816i $$0.250320\pi$$
$$648$$ 0 0
$$649$$ 0 0
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ 13.1194 0.513403 0.256701 0.966491i $$-0.417364\pi$$
0.256701 + 0.966491i $$0.417364\pi$$
$$654$$ 0 0
$$655$$ 60.0833 2.34765
$$656$$ 0 0
$$657$$ 0 0
$$658$$ 0 0
$$659$$ 19.8167 0.771947 0.385974 0.922510i $$-0.373866\pi$$
0.385974 + 0.922510i $$0.373866\pi$$
$$660$$ 0 0
$$661$$ −38.1194 −1.48267 −0.741337 0.671133i $$-0.765808\pi$$
−0.741337 + 0.671133i $$0.765808\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ −1.81665 −0.0704468
$$666$$ 0 0
$$667$$ −17.5139 −0.678140
$$668$$ 0 0
$$669$$ 0 0
$$670$$ 0 0
$$671$$ 0 0
$$672$$ 0 0
$$673$$ 41.6333 1.60485 0.802423 0.596756i $$-0.203544\pi$$
0.802423 + 0.596756i $$0.203544\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ 13.8167 0.531017 0.265509 0.964108i $$-0.414460\pi$$
0.265509 + 0.964108i $$0.414460\pi$$
$$678$$ 0 0
$$679$$ −1.36669 −0.0524488
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 0 0
$$683$$ 49.5416 1.89566 0.947829 0.318779i $$-0.103273\pi$$
0.947829 + 0.318779i $$0.103273\pi$$
$$684$$ 0 0
$$685$$ 8.36669 0.319675
$$686$$ 0 0
$$687$$ 0 0
$$688$$ 0 0
$$689$$ 12.4222 0.473248
$$690$$ 0 0
$$691$$ −42.4500 −1.61487 −0.807436 0.589955i $$-0.799146\pi$$
−0.807436 + 0.589955i $$0.799146\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ 17.7250 0.672347
$$696$$ 0 0
$$697$$ −14.3028 −0.541756
$$698$$ 0 0
$$699$$ 0 0
$$700$$ 0 0
$$701$$ −8.02776 −0.303204 −0.151602 0.988442i $$-0.548443\pi$$
−0.151602 + 0.988442i $$0.548443\pi$$
$$702$$ 0 0
$$703$$ 0.788897 0.0297538
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ −3.27502 −0.123170
$$708$$ 0 0
$$709$$ 50.1472 1.88332 0.941659 0.336570i $$-0.109267\pi$$
0.941659 + 0.336570i $$0.109267\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 0 0
$$713$$ 15.2111 0.569660
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ 0 0
$$718$$ 0 0
$$719$$ 22.8806 0.853301 0.426651 0.904417i $$-0.359693\pi$$
0.426651 + 0.904417i $$0.359693\pi$$
$$720$$ 0 0
$$721$$ 1.00000 0.0372419
$$722$$ 0 0
$$723$$ 0 0
$$724$$ 0 0
$$725$$ −30.4222 −1.12985
$$726$$ 0 0
$$727$$ 28.7250 1.06535 0.532675 0.846320i $$-0.321187\pi$$
0.532675 + 0.846320i $$0.321187\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ 0 0
$$731$$ −22.1194 −0.818117
$$732$$ 0 0
$$733$$ 14.0000 0.517102 0.258551 0.965998i $$-0.416755\pi$$
0.258551 + 0.965998i $$0.416755\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 0 0
$$738$$ 0 0
$$739$$ −3.93608 −0.144791 −0.0723956 0.997376i $$-0.523064\pi$$
−0.0723956 + 0.997376i $$0.523064\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ −52.1194 −1.91208 −0.956038 0.293242i $$-0.905266\pi$$
−0.956038 + 0.293242i $$0.905266\pi$$
$$744$$ 0 0
$$745$$ −52.5416 −1.92498
$$746$$ 0 0
$$747$$ 0 0
$$748$$ 0 0
$$749$$ 2.30278 0.0841416
$$750$$ 0 0
$$751$$ −23.6056 −0.861379 −0.430689 0.902500i $$-0.641730\pi$$
−0.430689 + 0.902500i $$0.641730\pi$$
$$752$$ 0 0
$$753$$ 0 0
$$754$$ 0 0
$$755$$ 58.5416 2.13055
$$756$$ 0 0
$$757$$ 22.7250 0.825953 0.412977 0.910742i $$-0.364489\pi$$
0.412977 + 0.910742i $$0.364489\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 3.69722 0.134024 0.0670121 0.997752i $$-0.478653\pi$$
0.0670121 + 0.997752i $$0.478653\pi$$
$$762$$ 0 0
$$763$$ −0.605551 −0.0219224
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ −21.0833 −0.761273
$$768$$ 0 0
$$769$$ −14.8167 −0.534302 −0.267151 0.963655i $$-0.586082\pi$$
−0.267151 + 0.963655i $$0.586082\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 0 0
$$773$$ −10.6056 −0.381455 −0.190728 0.981643i $$-0.561085\pi$$
−0.190728 + 0.981643i $$0.561085\pi$$
$$774$$ 0 0
$$775$$ 26.4222 0.949114
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ 12.4222 0.445072
$$780$$ 0 0
$$781$$ 0 0
$$782$$ 0 0
$$783$$ 0 0
$$784$$ 0 0
$$785$$ 2.44996 0.0874429
$$786$$ 0 0
$$787$$ 43.4500 1.54882 0.774412 0.632682i $$-0.218046\pi$$
0.774412 + 0.632682i $$0.218046\pi$$
$$788$$ 0 0
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 2.85281 0.101434
$$792$$ 0 0
$$793$$ 18.3860 0.652908
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ −36.6333 −1.29762 −0.648809 0.760951i $$-0.724733\pi$$
−0.648809 + 0.760951i $$0.724733\pi$$
$$798$$ 0 0
$$799$$ 3.69722 0.130798
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 0 0
$$803$$ 0 0
$$804$$ 0 0
$$805$$ −2.09167 −0.0737218
$$806$$ 0 0
$$807$$ 0 0
$$808$$ 0 0
$$809$$ −23.7250 −0.834126 −0.417063 0.908878i $$-0.636941\pi$$
−0.417063 + 0.908878i $$0.636941\pi$$
$$810$$ 0 0
$$811$$ −37.8444 −1.32890 −0.664448 0.747334i $$-0.731333\pi$$
−0.664448 + 0.747334i $$0.731333\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ 0 0
$$815$$ −16.1833 −0.566878
$$816$$ 0 0
$$817$$ 19.2111 0.672111
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ −14.7250 −0.513905 −0.256953 0.966424i $$-0.582718\pi$$
−0.256953 + 0.966424i $$0.582718\pi$$
$$822$$ 0 0
$$823$$ −21.3028 −0.742568 −0.371284 0.928519i $$-0.621082\pi$$
−0.371284 + 0.928519i $$0.621082\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ −1.45837 −0.0507123 −0.0253562 0.999678i $$-0.508072\pi$$
−0.0253562 + 0.999678i $$0.508072\pi$$
$$828$$ 0 0
$$829$$ 37.8722 1.31535 0.657677 0.753300i $$-0.271539\pi$$
0.657677 + 0.753300i $$0.271539\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 0 0
$$833$$ 15.9083 0.551191
$$834$$ 0 0
$$835$$ 3.00000 0.103819
$$836$$ 0 0
$$837$$ 0 0
$$838$$ 0 0
$$839$$ −40.3944 −1.39457 −0.697286 0.716793i $$-0.745609\pi$$
−0.697286 + 0.716793i $$0.745609\pi$$
$$840$$ 0 0
$$841$$ 28.8444 0.994635
$$842$$ 0 0
$$843$$ 0 0
$$844$$ 0 0
$$845$$ −17.1749 −0.590836
$$846$$ 0 0
$$847$$ 3.33053 0.114438
$$848$$ 0 0
$$849$$ 0 0
$$850$$ 0 0
$$851$$ 0.908327 0.0311370
$$852$$ 0 0
$$853$$ 40.7250 1.39440 0.697198 0.716878i $$-0.254430\pi$$
0.697198 + 0.716878i $$0.254430\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ −25.5416 −0.872486 −0.436243 0.899829i $$-0.643691\pi$$
−0.436243 + 0.899829i $$0.643691\pi$$
$$858$$ 0 0
$$859$$ −41.3944 −1.41236 −0.706180 0.708032i $$-0.749583\pi$$
−0.706180 + 0.708032i $$0.749583\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ 11.5139 0.391937 0.195968 0.980610i $$-0.437215\pi$$
0.195968 + 0.980610i $$0.437215\pi$$
$$864$$ 0 0
$$865$$ −17.3667 −0.590485
$$866$$ 0 0
$$867$$ 0 0
$$868$$ 0 0
$$869$$ 0 0
$$870$$ 0 0
$$871$$ 22.1472 0.750429
$$872$$ 0 0
$$873$$ 0 0
$$874$$ 0 0
$$875$$ 0.908327 0.0307071
$$876$$ 0 0
$$877$$ 19.5139 0.658937 0.329468 0.944167i $$-0.393130\pi$$
0.329468 + 0.944167i $$0.393130\pi$$
$$878$$ 0 0
$$879$$ 0 0
$$880$$ 0 0
$$881$$ 14.7889 0.498251 0.249125 0.968471i $$-0.419857\pi$$
0.249125 + 0.968471i $$0.419857\pi$$
$$882$$ 0 0
$$883$$ −27.9361 −0.940124 −0.470062 0.882633i $$-0.655768\pi$$
−0.470062 + 0.882633i $$0.655768\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ −10.8167 −0.363188 −0.181594 0.983374i $$-0.558126\pi$$
−0.181594 + 0.983374i $$0.558126\pi$$
$$888$$ 0 0
$$889$$ −4.30278 −0.144310
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 0 0
$$893$$ −3.21110 −0.107455
$$894$$ 0 0
$$895$$ −2.72498 −0.0910861
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 0 0
$$899$$ −50.2389 −1.67556
$$900$$ 0 0
$$901$$ −10.6056 −0.353322
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ −21.6333 −0.719115
$$906$$ 0 0
$$907$$ −4.63331 −0.153846 −0.0769232 0.997037i $$-0.524510\pi$$
−0.0769232 + 0.997037i $$0.524510\pi$$
$$908$$ 0 0
$$909$$ 0 0
$$910$$ 0 0
$$911$$ −7.18335 −0.237995 −0.118997 0.992895i $$-0.537968\pi$$
−0.118997 + 0.992895i $$0.537968\pi$$
$$912$$ 0 0
$$913$$ 0 0
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ −6.06392 −0.200248
$$918$$ 0 0
$$919$$ −0.0916731 −0.00302402 −0.00151201 0.999999i $$-0.500481\pi$$
−0.00151201 + 0.999999i $$0.500481\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 0 0
$$923$$ −10.5416 −0.346982
$$924$$ 0 0
$$925$$ 1.57779 0.0518776
$$926$$ 0 0
$$927$$ 0 0
$$928$$ 0 0
$$929$$ −15.4861 −0.508083 −0.254042 0.967193i $$-0.581760\pi$$
−0.254042 + 0.967193i $$0.581760\pi$$
$$930$$ 0 0
$$931$$ −13.8167 −0.452823
$$932$$ 0 0
$$933$$ 0 0
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ −51.4500 −1.68080 −0.840398 0.541969i $$-0.817679\pi$$
−0.840398 + 0.541969i $$0.817679\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ 0 0
$$941$$ −59.2389 −1.93113 −0.965566 0.260159i $$-0.916225\pi$$
−0.965566 + 0.260159i $$0.916225\pi$$
$$942$$ 0 0
$$943$$ 14.3028 0.465762
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ −14.0917 −0.457918 −0.228959 0.973436i $$-0.573532\pi$$
−0.228959 + 0.973436i $$0.573532\pi$$
$$948$$ 0 0
$$949$$ −8.33894 −0.270693
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 0 0
$$953$$ −42.3583 −1.37212 −0.686060 0.727545i $$-0.740661\pi$$
−0.686060 + 0.727545i $$0.740661\pi$$
$$954$$ 0 0
$$955$$ 33.9083 1.09725
$$956$$ 0 0
$$957$$ 0 0
$$958$$ 0 0
$$959$$ −0.844410 −0.0272674
$$960$$ 0 0
$$961$$ 12.6333 0.407526
$$962$$ 0 0
$$963$$ 0 0
$$964$$ 0 0
$$965$$ 17.7250 0.570587
$$966$$ 0 0
$$967$$ −25.4861 −0.819578 −0.409789 0.912180i $$-0.634398\pi$$
−0.409789 + 0.912180i $$0.634398\pi$$
$$968$$ 0 0
$$969$$ 0 0
$$970$$ 0 0
$$971$$ −49.4777 −1.58782 −0.793908 0.608038i $$-0.791957\pi$$
−0.793908 + 0.608038i $$0.791957\pi$$
$$972$$ 0 0
$$973$$ −1.78890 −0.0573494
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ 42.5694 1.36192 0.680958 0.732323i $$-0.261564\pi$$
0.680958 + 0.732323i $$0.261564\pi$$
$$978$$ 0 0
$$979$$ 0 0
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 0 0
$$983$$ 31.2666 0.997250 0.498625 0.866818i $$-0.333838\pi$$
0.498625 + 0.866818i $$0.333838\pi$$
$$984$$ 0 0
$$985$$ −11.0917 −0.353410
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ 22.1194 0.703357
$$990$$ 0 0
$$991$$ −19.6333 −0.623673 −0.311836 0.950136i $$-0.600944\pi$$
−0.311836 + 0.950136i $$0.600944\pi$$
$$992$$ 0 0
$$993$$ 0 0
$$994$$ 0 0
$$995$$ −32.0917 −1.01737
$$996$$ 0 0
$$997$$ −24.7889 −0.785072 −0.392536 0.919737i $$-0.628402\pi$$
−0.392536 + 0.919737i $$0.628402\pi$$
$$998$$ 0 0
$$999$$ 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 6012.2.a.a.1.1 2
3.2 odd 2 668.2.a.a.1.1 2
12.11 even 2 2672.2.a.c.1.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
668.2.a.a.1.1 2 3.2 odd 2
2672.2.a.c.1.2 2 12.11 even 2
6012.2.a.a.1.1 2 1.1 even 1 trivial