# Properties

 Label 7920.2.a.cj.1.1 Level $7920$ Weight $2$ Character 7920.1 Self dual yes Analytic conductor $63.242$ Analytic rank $0$ Dimension $3$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$7920 = 2^{4} \cdot 3^{2} \cdot 5 \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 7920.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$63.2415184009$$ Analytic rank: $$0$$ Dimension: $$3$$ Coefficient field: 3.3.148.1 Defining polynomial: $$x^{3} - x^{2} - 3x + 1$$ x^3 - x^2 - 3*x + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 165) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$-1.48119$$ of defining polynomial Character $$\chi$$ $$=$$ 7920.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-1.00000 q^{5} -3.35026 q^{7} +O(q^{10})$$ $$q-1.00000 q^{5} -3.35026 q^{7} +1.00000 q^{11} +2.96239 q^{13} +4.57452 q^{17} +4.31265 q^{19} -6.70052 q^{23} +1.00000 q^{25} +3.61213 q^{29} -9.92478 q^{31} +3.35026 q^{35} -2.00000 q^{37} +4.38787 q^{41} +9.27504 q^{43} -9.92478 q^{47} +4.22425 q^{49} -4.70052 q^{53} -1.00000 q^{55} +10.7005 q^{59} -8.70052 q^{61} -2.96239 q^{65} -5.92478 q^{67} +9.92478 q^{71} -7.73813 q^{73} -3.35026 q^{77} -11.5369 q^{79} +10.8872 q^{83} -4.57452 q^{85} +2.77575 q^{89} -9.92478 q^{91} -4.31265 q^{95} +0.0752228 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q - 3 q^{5}+O(q^{10})$$ 3 * q - 3 * q^5 $$3 q - 3 q^{5} + 3 q^{11} - 2 q^{13} + 2 q^{17} - 8 q^{19} + 3 q^{25} + 10 q^{29} - 8 q^{31} - 6 q^{37} + 14 q^{41} - 4 q^{43} - 8 q^{47} + 11 q^{49} + 6 q^{53} - 3 q^{55} + 12 q^{59} - 6 q^{61} + 2 q^{65} + 4 q^{67} + 8 q^{71} - 14 q^{73} - 12 q^{79} - 2 q^{85} + 10 q^{89} - 8 q^{91} + 8 q^{95} + 22 q^{97}+O(q^{100})$$ 3 * q - 3 * q^5 + 3 * q^11 - 2 * q^13 + 2 * q^17 - 8 * q^19 + 3 * q^25 + 10 * q^29 - 8 * q^31 - 6 * q^37 + 14 * q^41 - 4 * q^43 - 8 * q^47 + 11 * q^49 + 6 * q^53 - 3 * q^55 + 12 * q^59 - 6 * q^61 + 2 * q^65 + 4 * q^67 + 8 * q^71 - 14 * q^73 - 12 * q^79 - 2 * q^85 + 10 * q^89 - 8 * q^91 + 8 * q^95 + 22 * q^97

## 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
$$5$$ −1.00000 −0.447214
$$6$$ 0 0
$$7$$ −3.35026 −1.26628 −0.633140 0.774037i $$-0.718234\pi$$
−0.633140 + 0.774037i $$0.718234\pi$$
$$8$$ 0 0
$$9$$ 0 0
$$10$$ 0 0
$$11$$ 1.00000 0.301511
$$12$$ 0 0
$$13$$ 2.96239 0.821619 0.410809 0.911721i $$-0.365246\pi$$
0.410809 + 0.911721i $$0.365246\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ 4.57452 1.10948 0.554741 0.832023i $$-0.312817\pi$$
0.554741 + 0.832023i $$0.312817\pi$$
$$18$$ 0 0
$$19$$ 4.31265 0.989390 0.494695 0.869067i $$-0.335280\pi$$
0.494695 + 0.869067i $$0.335280\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ −6.70052 −1.39716 −0.698578 0.715534i $$-0.746183\pi$$
−0.698578 + 0.715534i $$0.746183\pi$$
$$24$$ 0 0
$$25$$ 1.00000 0.200000
$$26$$ 0 0
$$27$$ 0 0
$$28$$ 0 0
$$29$$ 3.61213 0.670755 0.335378 0.942084i $$-0.391136\pi$$
0.335378 + 0.942084i $$0.391136\pi$$
$$30$$ 0 0
$$31$$ −9.92478 −1.78254 −0.891271 0.453470i $$-0.850186\pi$$
−0.891271 + 0.453470i $$0.850186\pi$$
$$32$$ 0 0
$$33$$ 0 0
$$34$$ 0 0
$$35$$ 3.35026 0.566298
$$36$$ 0 0
$$37$$ −2.00000 −0.328798 −0.164399 0.986394i $$-0.552568\pi$$
−0.164399 + 0.986394i $$0.552568\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 0 0
$$41$$ 4.38787 0.685271 0.342635 0.939468i $$-0.388680\pi$$
0.342635 + 0.939468i $$0.388680\pi$$
$$42$$ 0 0
$$43$$ 9.27504 1.41443 0.707215 0.706998i $$-0.249951\pi$$
0.707215 + 0.706998i $$0.249951\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ −9.92478 −1.44768 −0.723839 0.689969i $$-0.757624\pi$$
−0.723839 + 0.689969i $$0.757624\pi$$
$$48$$ 0 0
$$49$$ 4.22425 0.603465
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 0 0
$$53$$ −4.70052 −0.645667 −0.322833 0.946456i $$-0.604635\pi$$
−0.322833 + 0.946456i $$0.604635\pi$$
$$54$$ 0 0
$$55$$ −1.00000 −0.134840
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 0 0
$$59$$ 10.7005 1.39309 0.696545 0.717513i $$-0.254720\pi$$
0.696545 + 0.717513i $$0.254720\pi$$
$$60$$ 0 0
$$61$$ −8.70052 −1.11399 −0.556994 0.830517i $$-0.688045\pi$$
−0.556994 + 0.830517i $$0.688045\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ −2.96239 −0.367439
$$66$$ 0 0
$$67$$ −5.92478 −0.723827 −0.361913 0.932212i $$-0.617876\pi$$
−0.361913 + 0.932212i $$0.617876\pi$$
$$68$$ 0 0
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 9.92478 1.17785 0.588927 0.808186i $$-0.299550\pi$$
0.588927 + 0.808186i $$0.299550\pi$$
$$72$$ 0 0
$$73$$ −7.73813 −0.905680 −0.452840 0.891592i $$-0.649589\pi$$
−0.452840 + 0.891592i $$0.649589\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ −3.35026 −0.381798
$$78$$ 0 0
$$79$$ −11.5369 −1.29800 −0.649002 0.760787i $$-0.724813\pi$$
−0.649002 + 0.760787i $$0.724813\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ 0 0
$$83$$ 10.8872 1.19502 0.597511 0.801861i $$-0.296156\pi$$
0.597511 + 0.801861i $$0.296156\pi$$
$$84$$ 0 0
$$85$$ −4.57452 −0.496176
$$86$$ 0 0
$$87$$ 0 0
$$88$$ 0 0
$$89$$ 2.77575 0.294229 0.147114 0.989120i $$-0.453001\pi$$
0.147114 + 0.989120i $$0.453001\pi$$
$$90$$ 0 0
$$91$$ −9.92478 −1.04040
$$92$$ 0 0
$$93$$ 0 0
$$94$$ 0 0
$$95$$ −4.31265 −0.442469
$$96$$ 0 0
$$97$$ 0.0752228 0.00763772 0.00381886 0.999993i $$-0.498784\pi$$
0.00381886 + 0.999993i $$0.498784\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 0 0
$$101$$ 15.0884 1.50135 0.750676 0.660671i $$-0.229728\pi$$
0.750676 + 0.660671i $$0.229728\pi$$
$$102$$ 0 0
$$103$$ 3.22425 0.317695 0.158848 0.987303i $$-0.449222\pi$$
0.158848 + 0.987303i $$0.449222\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ −0.962389 −0.0930376 −0.0465188 0.998917i $$-0.514813\pi$$
−0.0465188 + 0.998917i $$0.514813\pi$$
$$108$$ 0 0
$$109$$ 11.4010 1.09202 0.546011 0.837778i $$-0.316146\pi$$
0.546011 + 0.837778i $$0.316146\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 0 0
$$113$$ 6.00000 0.564433 0.282216 0.959351i $$-0.408930\pi$$
0.282216 + 0.959351i $$0.408930\pi$$
$$114$$ 0 0
$$115$$ 6.70052 0.624827
$$116$$ 0 0
$$117$$ 0 0
$$118$$ 0 0
$$119$$ −15.3258 −1.40492
$$120$$ 0 0
$$121$$ 1.00000 0.0909091
$$122$$ 0 0
$$123$$ 0 0
$$124$$ 0 0
$$125$$ −1.00000 −0.0894427
$$126$$ 0 0
$$127$$ 14.5745 1.29328 0.646640 0.762796i $$-0.276174\pi$$
0.646640 + 0.762796i $$0.276174\pi$$
$$128$$ 0 0
$$129$$ 0 0
$$130$$ 0 0
$$131$$ −5.92478 −0.517650 −0.258825 0.965924i $$-0.583335\pi$$
−0.258825 + 0.965924i $$0.583335\pi$$
$$132$$ 0 0
$$133$$ −14.4485 −1.25284
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ −13.8496 −1.18325 −0.591624 0.806214i $$-0.701513\pi$$
−0.591624 + 0.806214i $$0.701513\pi$$
$$138$$ 0 0
$$139$$ −13.6121 −1.15457 −0.577283 0.816544i $$-0.695887\pi$$
−0.577283 + 0.816544i $$0.695887\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 0 0
$$143$$ 2.96239 0.247727
$$144$$ 0 0
$$145$$ −3.61213 −0.299971
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ −1.53690 −0.125908 −0.0629540 0.998016i $$-0.520052\pi$$
−0.0629540 + 0.998016i $$0.520052\pi$$
$$150$$ 0 0
$$151$$ 6.76116 0.550215 0.275108 0.961413i $$-0.411287\pi$$
0.275108 + 0.961413i $$0.411287\pi$$
$$152$$ 0 0
$$153$$ 0 0
$$154$$ 0 0
$$155$$ 9.92478 0.797177
$$156$$ 0 0
$$157$$ −5.47627 −0.437054 −0.218527 0.975831i $$-0.570125\pi$$
−0.218527 + 0.975831i $$0.570125\pi$$
$$158$$ 0 0
$$159$$ 0 0
$$160$$ 0 0
$$161$$ 22.4485 1.76919
$$162$$ 0 0
$$163$$ −12.6253 −0.988890 −0.494445 0.869209i $$-0.664629\pi$$
−0.494445 + 0.869209i $$0.664629\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 18.3634 1.42101 0.710503 0.703695i $$-0.248468\pi$$
0.710503 + 0.703695i $$0.248468\pi$$
$$168$$ 0 0
$$169$$ −4.22425 −0.324943
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 0 0
$$173$$ 8.57452 0.651908 0.325954 0.945386i $$-0.394314\pi$$
0.325954 + 0.945386i $$0.394314\pi$$
$$174$$ 0 0
$$175$$ −3.35026 −0.253256
$$176$$ 0 0
$$177$$ 0 0
$$178$$ 0 0
$$179$$ 14.1768 1.05962 0.529812 0.848115i $$-0.322263\pi$$
0.529812 + 0.848115i $$0.322263\pi$$
$$180$$ 0 0
$$181$$ −5.22425 −0.388316 −0.194158 0.980970i $$-0.562197\pi$$
−0.194158 + 0.980970i $$0.562197\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 0 0
$$185$$ 2.00000 0.147043
$$186$$ 0 0
$$187$$ 4.57452 0.334522
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ −16.6253 −1.20296 −0.601482 0.798886i $$-0.705423\pi$$
−0.601482 + 0.798886i $$0.705423\pi$$
$$192$$ 0 0
$$193$$ −16.3634 −1.17787 −0.588933 0.808182i $$-0.700452\pi$$
−0.588933 + 0.808182i $$0.700452\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 20.4241 1.45515 0.727577 0.686026i $$-0.240646\pi$$
0.727577 + 0.686026i $$0.240646\pi$$
$$198$$ 0 0
$$199$$ 8.62530 0.611431 0.305716 0.952123i $$-0.401104\pi$$
0.305716 + 0.952123i $$0.401104\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 0 0
$$203$$ −12.1016 −0.849364
$$204$$ 0 0
$$205$$ −4.38787 −0.306462
$$206$$ 0 0
$$207$$ 0 0
$$208$$ 0 0
$$209$$ 4.31265 0.298312
$$210$$ 0 0
$$211$$ −9.08840 −0.625671 −0.312836 0.949807i $$-0.601279\pi$$
−0.312836 + 0.949807i $$0.601279\pi$$
$$212$$ 0 0
$$213$$ 0 0
$$214$$ 0 0
$$215$$ −9.27504 −0.632552
$$216$$ 0 0
$$217$$ 33.2506 2.25720
$$218$$ 0 0
$$219$$ 0 0
$$220$$ 0 0
$$221$$ 13.5515 0.911572
$$222$$ 0 0
$$223$$ 6.70052 0.448700 0.224350 0.974509i $$-0.427974\pi$$
0.224350 + 0.974509i $$0.427974\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ 16.9624 1.12583 0.562917 0.826514i $$-0.309679\pi$$
0.562917 + 0.826514i $$0.309679\pi$$
$$228$$ 0 0
$$229$$ 25.8496 1.70819 0.854093 0.520120i $$-0.174113\pi$$
0.854093 + 0.520120i $$0.174113\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ 19.2750 1.26275 0.631375 0.775478i $$-0.282491\pi$$
0.631375 + 0.775478i $$0.282491\pi$$
$$234$$ 0 0
$$235$$ 9.92478 0.647421
$$236$$ 0 0
$$237$$ 0 0
$$238$$ 0 0
$$239$$ 26.5501 1.71738 0.858691 0.512494i $$-0.171278\pi$$
0.858691 + 0.512494i $$0.171278\pi$$
$$240$$ 0 0
$$241$$ 28.5501 1.83907 0.919536 0.393006i $$-0.128565\pi$$
0.919536 + 0.393006i $$0.128565\pi$$
$$242$$ 0 0
$$243$$ 0 0
$$244$$ 0 0
$$245$$ −4.22425 −0.269878
$$246$$ 0 0
$$247$$ 12.7757 0.812901
$$248$$ 0 0
$$249$$ 0 0
$$250$$ 0 0
$$251$$ 29.9248 1.88884 0.944418 0.328748i $$-0.106627\pi$$
0.944418 + 0.328748i $$0.106627\pi$$
$$252$$ 0 0
$$253$$ −6.70052 −0.421258
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ −8.70052 −0.542724 −0.271362 0.962477i $$-0.587474\pi$$
−0.271362 + 0.962477i $$0.587474\pi$$
$$258$$ 0 0
$$259$$ 6.70052 0.416350
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 0 0
$$263$$ 12.2882 0.757724 0.378862 0.925453i $$-0.376316\pi$$
0.378862 + 0.925453i $$0.376316\pi$$
$$264$$ 0 0
$$265$$ 4.70052 0.288751
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 0 0
$$269$$ 5.84955 0.356654 0.178327 0.983971i $$-0.442932\pi$$
0.178327 + 0.983971i $$0.442932\pi$$
$$270$$ 0 0
$$271$$ 5.08840 0.309098 0.154549 0.987985i $$-0.450608\pi$$
0.154549 + 0.987985i $$0.450608\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ 1.00000 0.0603023
$$276$$ 0 0
$$277$$ 1.41090 0.0847725 0.0423863 0.999101i $$-0.486504\pi$$
0.0423863 + 0.999101i $$0.486504\pi$$
$$278$$ 0 0
$$279$$ 0 0
$$280$$ 0 0
$$281$$ 4.38787 0.261759 0.130879 0.991398i $$-0.458220\pi$$
0.130879 + 0.991398i $$0.458220\pi$$
$$282$$ 0 0
$$283$$ −26.5745 −1.57969 −0.789845 0.613306i $$-0.789839\pi$$
−0.789845 + 0.613306i $$0.789839\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ −14.7005 −0.867744
$$288$$ 0 0
$$289$$ 3.92619 0.230952
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 0 0
$$293$$ 3.42548 0.200119 0.100059 0.994981i $$-0.468097\pi$$
0.100059 + 0.994981i $$0.468097\pi$$
$$294$$ 0 0
$$295$$ −10.7005 −0.623009
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ −19.8496 −1.14793
$$300$$ 0 0
$$301$$ −31.0738 −1.79106
$$302$$ 0 0
$$303$$ 0 0
$$304$$ 0 0
$$305$$ 8.70052 0.498191
$$306$$ 0 0
$$307$$ 16.6497 0.950251 0.475125 0.879918i $$-0.342403\pi$$
0.475125 + 0.879918i $$0.342403\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 0 0
$$311$$ 32.9986 1.87118 0.935589 0.353091i $$-0.114869\pi$$
0.935589 + 0.353091i $$0.114869\pi$$
$$312$$ 0 0
$$313$$ 15.4010 0.870519 0.435259 0.900305i $$-0.356657\pi$$
0.435259 + 0.900305i $$0.356657\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ −2.15045 −0.120781 −0.0603905 0.998175i $$-0.519235\pi$$
−0.0603905 + 0.998175i $$0.519235\pi$$
$$318$$ 0 0
$$319$$ 3.61213 0.202240
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 0 0
$$323$$ 19.7283 1.09771
$$324$$ 0 0
$$325$$ 2.96239 0.164324
$$326$$ 0 0
$$327$$ 0 0
$$328$$ 0 0
$$329$$ 33.2506 1.83316
$$330$$ 0 0
$$331$$ 14.5501 0.799745 0.399872 0.916571i $$-0.369054\pi$$
0.399872 + 0.916571i $$0.369054\pi$$
$$332$$ 0 0
$$333$$ 0 0
$$334$$ 0 0
$$335$$ 5.92478 0.323705
$$336$$ 0 0
$$337$$ 16.2619 0.885840 0.442920 0.896561i $$-0.353943\pi$$
0.442920 + 0.896561i $$0.353943\pi$$
$$338$$ 0 0
$$339$$ 0 0
$$340$$ 0 0
$$341$$ −9.92478 −0.537457
$$342$$ 0 0
$$343$$ 9.29948 0.502125
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ −0.962389 −0.0516637 −0.0258319 0.999666i $$-0.508223\pi$$
−0.0258319 + 0.999666i $$0.508223\pi$$
$$348$$ 0 0
$$349$$ 20.7005 1.10807 0.554037 0.832492i $$-0.313087\pi$$
0.554037 + 0.832492i $$0.313087\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 0 0
$$353$$ −20.5501 −1.09377 −0.546885 0.837208i $$-0.684187\pi$$
−0.546885 + 0.837208i $$0.684187\pi$$
$$354$$ 0 0
$$355$$ −9.92478 −0.526752
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ 17.9248 0.946034 0.473017 0.881053i $$-0.343165\pi$$
0.473017 + 0.881053i $$0.343165\pi$$
$$360$$ 0 0
$$361$$ −0.401047 −0.0211077
$$362$$ 0 0
$$363$$ 0 0
$$364$$ 0 0
$$365$$ 7.73813 0.405032
$$366$$ 0 0
$$367$$ 29.6531 1.54788 0.773939 0.633261i $$-0.218284\pi$$
0.773939 + 0.633261i $$0.218284\pi$$
$$368$$ 0 0
$$369$$ 0 0
$$370$$ 0 0
$$371$$ 15.7480 0.817595
$$372$$ 0 0
$$373$$ −9.13918 −0.473209 −0.236604 0.971606i $$-0.576035\pi$$
−0.236604 + 0.971606i $$0.576035\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 10.7005 0.551105
$$378$$ 0 0
$$379$$ 20.0000 1.02733 0.513665 0.857991i $$-0.328287\pi$$
0.513665 + 0.857991i $$0.328287\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 0 0
$$383$$ −34.9234 −1.78450 −0.892250 0.451541i $$-0.850874\pi$$
−0.892250 + 0.451541i $$0.850874\pi$$
$$384$$ 0 0
$$385$$ 3.35026 0.170745
$$386$$ 0 0
$$387$$ 0 0
$$388$$ 0 0
$$389$$ −2.77575 −0.140736 −0.0703680 0.997521i $$-0.522417\pi$$
−0.0703680 + 0.997521i $$0.522417\pi$$
$$390$$ 0 0
$$391$$ −30.6516 −1.55012
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 0 0
$$395$$ 11.5369 0.580485
$$396$$ 0 0
$$397$$ −19.9248 −0.999996 −0.499998 0.866027i $$-0.666666\pi$$
−0.499998 + 0.866027i $$0.666666\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ −2.00000 −0.0998752 −0.0499376 0.998752i $$-0.515902\pi$$
−0.0499376 + 0.998752i $$0.515902\pi$$
$$402$$ 0 0
$$403$$ −29.4010 −1.46457
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ −2.00000 −0.0991363
$$408$$ 0 0
$$409$$ −13.0738 −0.646458 −0.323229 0.946321i $$-0.604768\pi$$
−0.323229 + 0.946321i $$0.604768\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 0 0
$$413$$ −35.8496 −1.76404
$$414$$ 0 0
$$415$$ −10.8872 −0.534430
$$416$$ 0 0
$$417$$ 0 0
$$418$$ 0 0
$$419$$ 7.22425 0.352928 0.176464 0.984307i $$-0.443534\pi$$
0.176464 + 0.984307i $$0.443534\pi$$
$$420$$ 0 0
$$421$$ 30.6253 1.49259 0.746293 0.665618i $$-0.231832\pi$$
0.746293 + 0.665618i $$0.231832\pi$$
$$422$$ 0 0
$$423$$ 0 0
$$424$$ 0 0
$$425$$ 4.57452 0.221897
$$426$$ 0 0
$$427$$ 29.1490 1.41062
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ −33.8759 −1.63174 −0.815872 0.578232i $$-0.803743\pi$$
−0.815872 + 0.578232i $$0.803743\pi$$
$$432$$ 0 0
$$433$$ −9.47627 −0.455400 −0.227700 0.973731i $$-0.573121\pi$$
−0.227700 + 0.973731i $$0.573121\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ −28.8970 −1.38233
$$438$$ 0 0
$$439$$ 29.4617 1.40613 0.703065 0.711126i $$-0.251814\pi$$
0.703065 + 0.711126i $$0.251814\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ −19.0738 −0.906224 −0.453112 0.891454i $$-0.649686\pi$$
−0.453112 + 0.891454i $$0.649686\pi$$
$$444$$ 0 0
$$445$$ −2.77575 −0.131583
$$446$$ 0 0
$$447$$ 0 0
$$448$$ 0 0
$$449$$ −35.8759 −1.69309 −0.846544 0.532318i $$-0.821321\pi$$
−0.846544 + 0.532318i $$0.821321\pi$$
$$450$$ 0 0
$$451$$ 4.38787 0.206617
$$452$$ 0 0
$$453$$ 0 0
$$454$$ 0 0
$$455$$ 9.92478 0.465281
$$456$$ 0 0
$$457$$ 5.28963 0.247438 0.123719 0.992317i $$-0.460518\pi$$
0.123719 + 0.992317i $$0.460518\pi$$
$$458$$ 0 0
$$459$$ 0 0
$$460$$ 0 0
$$461$$ −36.3390 −1.69248 −0.846238 0.532805i $$-0.821138\pi$$
−0.846238 + 0.532805i $$0.821138\pi$$
$$462$$ 0 0
$$463$$ −10.5501 −0.490304 −0.245152 0.969485i $$-0.578838\pi$$
−0.245152 + 0.969485i $$0.578838\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ 18.7005 0.865357 0.432679 0.901548i $$-0.357569\pi$$
0.432679 + 0.901548i $$0.357569\pi$$
$$468$$ 0 0
$$469$$ 19.8496 0.916567
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 0 0
$$473$$ 9.27504 0.426467
$$474$$ 0 0
$$475$$ 4.31265 0.197878
$$476$$ 0 0
$$477$$ 0 0
$$478$$ 0 0
$$479$$ −9.29948 −0.424904 −0.212452 0.977172i $$-0.568145\pi$$
−0.212452 + 0.977172i $$0.568145\pi$$
$$480$$ 0 0
$$481$$ −5.92478 −0.270147
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ −0.0752228 −0.00341569
$$486$$ 0 0
$$487$$ 35.4763 1.60758 0.803792 0.594911i $$-0.202813\pi$$
0.803792 + 0.594911i $$0.202813\pi$$
$$488$$ 0 0
$$489$$ 0 0
$$490$$ 0 0
$$491$$ 24.7757 1.11811 0.559057 0.829129i $$-0.311163\pi$$
0.559057 + 0.829129i $$0.311163\pi$$
$$492$$ 0 0
$$493$$ 16.5237 0.744191
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ −33.2506 −1.49149
$$498$$ 0 0
$$499$$ −14.1768 −0.634640 −0.317320 0.948318i $$-0.602783\pi$$
−0.317320 + 0.948318i $$0.602783\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 0 0
$$503$$ −8.43866 −0.376261 −0.188131 0.982144i $$-0.560243\pi$$
−0.188131 + 0.982144i $$0.560243\pi$$
$$504$$ 0 0
$$505$$ −15.0884 −0.671425
$$506$$ 0 0
$$507$$ 0 0
$$508$$ 0 0
$$509$$ −1.10299 −0.0488890 −0.0244445 0.999701i $$-0.507782\pi$$
−0.0244445 + 0.999701i $$0.507782\pi$$
$$510$$ 0 0
$$511$$ 25.9248 1.14684
$$512$$ 0 0
$$513$$ 0 0
$$514$$ 0 0
$$515$$ −3.22425 −0.142078
$$516$$ 0 0
$$517$$ −9.92478 −0.436491
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 0 0
$$521$$ 12.4485 0.545379 0.272690 0.962102i $$-0.412087\pi$$
0.272690 + 0.962102i $$0.412087\pi$$
$$522$$ 0 0
$$523$$ −30.0508 −1.31403 −0.657015 0.753878i $$-0.728181\pi$$
−0.657015 + 0.753878i $$0.728181\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ −45.4010 −1.97770
$$528$$ 0 0
$$529$$ 21.8970 0.952044
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 0 0
$$533$$ 12.9986 0.563031
$$534$$ 0 0
$$535$$ 0.962389 0.0416077
$$536$$ 0 0
$$537$$ 0 0
$$538$$ 0 0
$$539$$ 4.22425 0.181951
$$540$$ 0 0
$$541$$ −18.0000 −0.773880 −0.386940 0.922105i $$-0.626468\pi$$
−0.386940 + 0.922105i $$0.626468\pi$$
$$542$$ 0 0
$$543$$ 0 0
$$544$$ 0 0
$$545$$ −11.4010 −0.488367
$$546$$ 0 0
$$547$$ 14.3028 0.611544 0.305772 0.952105i $$-0.401086\pi$$
0.305772 + 0.952105i $$0.401086\pi$$
$$548$$ 0 0
$$549$$ 0 0
$$550$$ 0 0
$$551$$ 15.5778 0.663638
$$552$$ 0 0
$$553$$ 38.6516 1.64364
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ 11.7988 0.499930 0.249965 0.968255i $$-0.419581\pi$$
0.249965 + 0.968255i $$0.419581\pi$$
$$558$$ 0 0
$$559$$ 27.4763 1.16212
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ 30.4847 1.28478 0.642389 0.766379i $$-0.277944\pi$$
0.642389 + 0.766379i $$0.277944\pi$$
$$564$$ 0 0
$$565$$ −6.00000 −0.252422
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ 27.0884 1.13560 0.567802 0.823165i $$-0.307794\pi$$
0.567802 + 0.823165i $$0.307794\pi$$
$$570$$ 0 0
$$571$$ −7.28489 −0.304863 −0.152432 0.988314i $$-0.548710\pi$$
−0.152432 + 0.988314i $$0.548710\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 0 0
$$575$$ −6.70052 −0.279431
$$576$$ 0 0
$$577$$ −31.6239 −1.31652 −0.658260 0.752791i $$-0.728707\pi$$
−0.658260 + 0.752791i $$0.728707\pi$$
$$578$$ 0 0
$$579$$ 0 0
$$580$$ 0 0
$$581$$ −36.4749 −1.51323
$$582$$ 0 0
$$583$$ −4.70052 −0.194676
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ 33.1490 1.36821 0.684103 0.729385i $$-0.260194\pi$$
0.684103 + 0.729385i $$0.260194\pi$$
$$588$$ 0 0
$$589$$ −42.8021 −1.76363
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 0 0
$$593$$ −34.4993 −1.41672 −0.708358 0.705853i $$-0.750564\pi$$
−0.708358 + 0.705853i $$0.750564\pi$$
$$594$$ 0 0
$$595$$ 15.3258 0.628298
$$596$$ 0 0
$$597$$ 0 0
$$598$$ 0 0
$$599$$ −14.4485 −0.590350 −0.295175 0.955443i $$-0.595378\pi$$
−0.295175 + 0.955443i $$0.595378\pi$$
$$600$$ 0 0
$$601$$ −15.9248 −0.649585 −0.324793 0.945785i $$-0.605295\pi$$
−0.324793 + 0.945785i $$0.605295\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ 0 0
$$605$$ −1.00000 −0.0406558
$$606$$ 0 0
$$607$$ 14.5745 0.591561 0.295781 0.955256i $$-0.404420\pi$$
0.295781 + 0.955256i $$0.404420\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ −29.4010 −1.18944
$$612$$ 0 0
$$613$$ 16.4123 0.662887 0.331443 0.943475i $$-0.392464\pi$$
0.331443 + 0.943475i $$0.392464\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 17.8496 0.718596 0.359298 0.933223i $$-0.383016\pi$$
0.359298 + 0.933223i $$0.383016\pi$$
$$618$$ 0 0
$$619$$ 0.402462 0.0161763 0.00808815 0.999967i $$-0.497425\pi$$
0.00808815 + 0.999967i $$0.497425\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 0 0
$$623$$ −9.29948 −0.372576
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ −9.14903 −0.364796
$$630$$ 0 0
$$631$$ 38.0263 1.51380 0.756902 0.653528i $$-0.226712\pi$$
0.756902 + 0.653528i $$0.226712\pi$$
$$632$$ 0 0
$$633$$ 0 0
$$634$$ 0 0
$$635$$ −14.5745 −0.578372
$$636$$ 0 0
$$637$$ 12.5139 0.495818
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 0 0
$$641$$ 28.0263 1.10697 0.553487 0.832858i $$-0.313297\pi$$
0.553487 + 0.832858i $$0.313297\pi$$
$$642$$ 0 0
$$643$$ 4.62530 0.182404 0.0912020 0.995832i $$-0.470929\pi$$
0.0912020 + 0.995832i $$0.470929\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 23.5778 0.926941 0.463470 0.886112i $$-0.346604\pi$$
0.463470 + 0.886112i $$0.346604\pi$$
$$648$$ 0 0
$$649$$ 10.7005 0.420032
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ 2.25202 0.0881282 0.0440641 0.999029i $$-0.485969\pi$$
0.0440641 + 0.999029i $$0.485969\pi$$
$$654$$ 0 0
$$655$$ 5.92478 0.231500
$$656$$ 0 0
$$657$$ 0 0
$$658$$ 0 0
$$659$$ −41.4010 −1.61276 −0.806378 0.591401i $$-0.798575\pi$$
−0.806378 + 0.591401i $$0.798575\pi$$
$$660$$ 0 0
$$661$$ 3.40105 0.132285 0.0661427 0.997810i $$-0.478931\pi$$
0.0661427 + 0.997810i $$0.478931\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ 14.4485 0.560289
$$666$$ 0 0
$$667$$ −24.2031 −0.937149
$$668$$ 0 0
$$669$$ 0 0
$$670$$ 0 0
$$671$$ −8.70052 −0.335880
$$672$$ 0 0
$$673$$ 0.887166 0.0341977 0.0170989 0.999854i $$-0.494557\pi$$
0.0170989 + 0.999854i $$0.494557\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ −18.9018 −0.726453 −0.363227 0.931701i $$-0.618325\pi$$
−0.363227 + 0.931701i $$0.618325\pi$$
$$678$$ 0 0
$$679$$ −0.252016 −0.00967149
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 0 0
$$683$$ −20.8773 −0.798848 −0.399424 0.916766i $$-0.630790\pi$$
−0.399424 + 0.916766i $$0.630790\pi$$
$$684$$ 0 0
$$685$$ 13.8496 0.529164
$$686$$ 0 0
$$687$$ 0 0
$$688$$ 0 0
$$689$$ −13.9248 −0.530492
$$690$$ 0 0
$$691$$ 2.44851 0.0931456 0.0465728 0.998915i $$-0.485170\pi$$
0.0465728 + 0.998915i $$0.485170\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ 13.6121 0.516337
$$696$$ 0 0
$$697$$ 20.0724 0.760296
$$698$$ 0 0
$$699$$ 0 0
$$700$$ 0 0
$$701$$ 2.98683 0.112811 0.0564054 0.998408i $$-0.482036\pi$$
0.0564054 + 0.998408i $$0.482036\pi$$
$$702$$ 0 0
$$703$$ −8.62530 −0.325309
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ −50.5501 −1.90113
$$708$$ 0 0
$$709$$ 24.1768 0.907979 0.453989 0.891007i $$-0.350000\pi$$
0.453989 + 0.891007i $$0.350000\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 0 0
$$713$$ 66.5012 2.49049
$$714$$ 0 0
$$715$$ −2.96239 −0.110787
$$716$$ 0 0
$$717$$ 0 0
$$718$$ 0 0
$$719$$ −30.0263 −1.11979 −0.559897 0.828562i $$-0.689159\pi$$
−0.559897 + 0.828562i $$0.689159\pi$$
$$720$$ 0 0
$$721$$ −10.8021 −0.402291
$$722$$ 0 0
$$723$$ 0 0
$$724$$ 0 0
$$725$$ 3.61213 0.134151
$$726$$ 0 0
$$727$$ −14.9525 −0.554559 −0.277279 0.960789i $$-0.589433\pi$$
−0.277279 + 0.960789i $$0.589433\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ 0 0
$$731$$ 42.4288 1.56929
$$732$$ 0 0
$$733$$ −19.1128 −0.705949 −0.352974 0.935633i $$-0.614830\pi$$
−0.352974 + 0.935633i $$0.614830\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ −5.92478 −0.218242
$$738$$ 0 0
$$739$$ 3.31406 0.121910 0.0609549 0.998141i $$-0.480585\pi$$
0.0609549 + 0.998141i $$0.480585\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ −34.9887 −1.28361 −0.641806 0.766867i $$-0.721815\pi$$
−0.641806 + 0.766867i $$0.721815\pi$$
$$744$$ 0 0
$$745$$ 1.53690 0.0563078
$$746$$ 0 0
$$747$$ 0 0
$$748$$ 0 0
$$749$$ 3.22425 0.117812
$$750$$ 0 0
$$751$$ 26.9234 0.982447 0.491224 0.871033i $$-0.336550\pi$$
0.491224 + 0.871033i $$0.336550\pi$$
$$752$$ 0 0
$$753$$ 0 0
$$754$$ 0 0
$$755$$ −6.76116 −0.246064
$$756$$ 0 0
$$757$$ 15.9248 0.578796 0.289398 0.957209i $$-0.406545\pi$$
0.289398 + 0.957209i $$0.406545\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 30.9380 1.12150 0.560750 0.827985i $$-0.310513\pi$$
0.560750 + 0.827985i $$0.310513\pi$$
$$762$$ 0 0
$$763$$ −38.1965 −1.38281
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ 31.6991 1.14459
$$768$$ 0 0
$$769$$ 9.32582 0.336298 0.168149 0.985762i $$-0.446221\pi$$
0.168149 + 0.985762i $$0.446221\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 0 0
$$773$$ −44.7005 −1.60777 −0.803883 0.594787i $$-0.797236\pi$$
−0.803883 + 0.594787i $$0.797236\pi$$
$$774$$ 0 0
$$775$$ −9.92478 −0.356509
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ 18.9234 0.678000
$$780$$ 0 0
$$781$$ 9.92478 0.355136
$$782$$ 0 0
$$783$$ 0 0
$$784$$ 0 0
$$785$$ 5.47627 0.195456
$$786$$ 0 0
$$787$$ −21.6775 −0.772719 −0.386360 0.922348i $$-0.626268\pi$$
−0.386360 + 0.922348i $$0.626268\pi$$
$$788$$ 0 0
$$789$$ 0 0
$$790$$ 0 0
$$791$$ −20.1016 −0.714730
$$792$$ 0 0
$$793$$ −25.7743 −0.915273
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ −22.7466 −0.805725 −0.402862 0.915261i $$-0.631985\pi$$
−0.402862 + 0.915261i $$0.631985\pi$$
$$798$$ 0 0
$$799$$ −45.4010 −1.60617
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 0 0
$$803$$ −7.73813 −0.273073
$$804$$ 0 0
$$805$$ −22.4485 −0.791206
$$806$$ 0 0
$$807$$ 0 0
$$808$$ 0 0
$$809$$ 23.6121 0.830158 0.415079 0.909785i $$-0.363754\pi$$
0.415079 + 0.909785i $$0.363754\pi$$
$$810$$ 0 0
$$811$$ 26.0870 0.916038 0.458019 0.888942i $$-0.348559\pi$$
0.458019 + 0.888942i $$0.348559\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ 0 0
$$815$$ 12.6253 0.442245
$$816$$ 0 0
$$817$$ 40.0000 1.39942
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ 54.4142 1.89907 0.949535 0.313662i $$-0.101556\pi$$
0.949535 + 0.313662i $$0.101556\pi$$
$$822$$ 0 0
$$823$$ −0.121269 −0.00422716 −0.00211358 0.999998i $$-0.500673\pi$$
−0.00211358 + 0.999998i $$0.500673\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ −18.2130 −0.633328 −0.316664 0.948538i $$-0.602563\pi$$
−0.316664 + 0.948538i $$0.602563\pi$$
$$828$$ 0 0
$$829$$ 13.0738 0.454072 0.227036 0.973886i $$-0.427096\pi$$
0.227036 + 0.973886i $$0.427096\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 0 0
$$833$$ 19.3239 0.669534
$$834$$ 0 0
$$835$$ −18.3634 −0.635493
$$836$$ 0 0
$$837$$ 0 0
$$838$$ 0 0
$$839$$ −26.5501 −0.916610 −0.458305 0.888795i $$-0.651543\pi$$
−0.458305 + 0.888795i $$0.651543\pi$$
$$840$$ 0 0
$$841$$ −15.9525 −0.550088
$$842$$ 0 0
$$843$$ 0 0
$$844$$ 0 0
$$845$$ 4.22425 0.145319
$$846$$ 0 0
$$847$$ −3.35026 −0.115116
$$848$$ 0 0
$$849$$ 0 0
$$850$$ 0 0
$$851$$ 13.4010 0.459382
$$852$$ 0 0
$$853$$ 40.6155 1.39065 0.695323 0.718697i $$-0.255261\pi$$
0.695323 + 0.718697i $$0.255261\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ −20.1721 −0.689064 −0.344532 0.938775i $$-0.611962\pi$$
−0.344532 + 0.938775i $$0.611962\pi$$
$$858$$ 0 0
$$859$$ −21.8035 −0.743926 −0.371963 0.928248i $$-0.621315\pi$$
−0.371963 + 0.928248i $$0.621315\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ 35.4274 1.20596 0.602981 0.797755i $$-0.293979\pi$$
0.602981 + 0.797755i $$0.293979\pi$$
$$864$$ 0 0
$$865$$ −8.57452 −0.291542
$$866$$ 0 0
$$867$$ 0 0
$$868$$ 0 0
$$869$$ −11.5369 −0.391363
$$870$$ 0 0
$$871$$ −17.5515 −0.594710
$$872$$ 0 0
$$873$$ 0 0
$$874$$ 0 0
$$875$$ 3.35026 0.113260
$$876$$ 0 0
$$877$$ −14.0362 −0.473969 −0.236984 0.971513i $$-0.576159\pi$$
−0.236984 + 0.971513i $$0.576159\pi$$
$$878$$ 0 0
$$879$$ 0 0
$$880$$ 0 0
$$881$$ 21.0738 0.709995 0.354997 0.934867i $$-0.384482\pi$$
0.354997 + 0.934867i $$0.384482\pi$$
$$882$$ 0 0
$$883$$ 42.1476 1.41838 0.709190 0.705017i $$-0.249061\pi$$
0.709190 + 0.705017i $$0.249061\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ 6.93604 0.232889 0.116445 0.993197i $$-0.462850\pi$$
0.116445 + 0.993197i $$0.462850\pi$$
$$888$$ 0 0
$$889$$ −48.8284 −1.63765
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 0 0
$$893$$ −42.8021 −1.43232
$$894$$ 0 0
$$895$$ −14.1768 −0.473878
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 0 0
$$899$$ −35.8496 −1.19565
$$900$$ 0 0
$$901$$ −21.5026 −0.716356
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ 5.22425 0.173660
$$906$$ 0 0
$$907$$ −53.2017 −1.76653 −0.883267 0.468870i $$-0.844661\pi$$
−0.883267 + 0.468870i $$0.844661\pi$$
$$908$$ 0 0
$$909$$ 0 0
$$910$$ 0 0
$$911$$ 36.4749 1.20847 0.604233 0.796808i $$-0.293480\pi$$
0.604233 + 0.796808i $$0.293480\pi$$
$$912$$ 0 0
$$913$$ 10.8872 0.360313
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ 19.8496 0.655490
$$918$$ 0 0
$$919$$ −9.73340 −0.321075 −0.160538 0.987030i $$-0.551323\pi$$
−0.160538 + 0.987030i $$0.551323\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 0 0
$$923$$ 29.4010 0.967747
$$924$$ 0 0
$$925$$ −2.00000 −0.0657596
$$926$$ 0 0
$$927$$ 0 0
$$928$$ 0 0
$$929$$ 24.1768 0.793215 0.396607 0.917988i $$-0.370187\pi$$
0.396607 + 0.917988i $$0.370187\pi$$
$$930$$ 0 0
$$931$$ 18.2177 0.597062
$$932$$ 0 0
$$933$$ 0 0
$$934$$ 0 0
$$935$$ −4.57452 −0.149603
$$936$$ 0 0
$$937$$ −7.48612 −0.244561 −0.122280 0.992496i $$-0.539021\pi$$
−0.122280 + 0.992496i $$0.539021\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ 0 0
$$941$$ 21.2360 0.692274 0.346137 0.938184i $$-0.387493\pi$$
0.346137 + 0.938184i $$0.387493\pi$$
$$942$$ 0 0
$$943$$ −29.4010 −0.957430
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ −15.4763 −0.502911 −0.251456 0.967869i $$-0.580909\pi$$
−0.251456 + 0.967869i $$0.580909\pi$$
$$948$$ 0 0
$$949$$ −22.9234 −0.744124
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 0 0
$$953$$ 32.0508 1.03823 0.519113 0.854705i $$-0.326262\pi$$
0.519113 + 0.854705i $$0.326262\pi$$
$$954$$ 0 0
$$955$$ 16.6253 0.537982
$$956$$ 0 0
$$957$$ 0 0
$$958$$ 0 0
$$959$$ 46.3996 1.49832
$$960$$ 0 0
$$961$$ 67.5012 2.17746
$$962$$ 0 0
$$963$$ 0 0
$$964$$ 0 0
$$965$$ 16.3634 0.526758
$$966$$ 0 0
$$967$$ 17.3766 0.558794 0.279397 0.960176i $$-0.409865\pi$$
0.279397 + 0.960176i $$0.409865\pi$$
$$968$$ 0 0
$$969$$ 0 0
$$970$$ 0 0
$$971$$ 36.2031 1.16181 0.580907 0.813970i $$-0.302698\pi$$
0.580907 + 0.813970i $$0.302698\pi$$
$$972$$ 0 0
$$973$$ 45.6042 1.46200
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ 28.1476 0.900522 0.450261 0.892897i $$-0.351331\pi$$
0.450261 + 0.892897i $$0.351331\pi$$
$$978$$ 0 0
$$979$$ 2.77575 0.0887132
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 0 0
$$983$$ 7.07381 0.225619 0.112810 0.993617i $$-0.464015\pi$$
0.112810 + 0.993617i $$0.464015\pi$$
$$984$$ 0 0
$$985$$ −20.4241 −0.650765
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ −62.1476 −1.97618
$$990$$ 0 0
$$991$$ −44.4260 −1.41124 −0.705619 0.708592i $$-0.749331\pi$$
−0.705619 + 0.708592i $$0.749331\pi$$
$$992$$ 0 0
$$993$$ 0 0
$$994$$ 0 0
$$995$$ −8.62530 −0.273440
$$996$$ 0 0
$$997$$ −28.4847 −0.902120 −0.451060 0.892494i $$-0.648954\pi$$
−0.451060 + 0.892494i $$0.648954\pi$$
$$998$$ 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 7920.2.a.cj.1.1 3
3.2 odd 2 2640.2.a.be.1.1 3
4.3 odd 2 495.2.a.e.1.2 3
12.11 even 2 165.2.a.c.1.2 3
20.3 even 4 2475.2.c.r.199.3 6
20.7 even 4 2475.2.c.r.199.4 6
20.19 odd 2 2475.2.a.bb.1.2 3
44.43 even 2 5445.2.a.z.1.2 3
60.23 odd 4 825.2.c.g.199.4 6
60.47 odd 4 825.2.c.g.199.3 6
60.59 even 2 825.2.a.k.1.2 3
84.83 odd 2 8085.2.a.bk.1.2 3
132.131 odd 2 1815.2.a.m.1.2 3
660.659 odd 2 9075.2.a.cf.1.2 3

By twisted newform
Twist Min Dim Char Parity Ord Type
165.2.a.c.1.2 3 12.11 even 2
495.2.a.e.1.2 3 4.3 odd 2
825.2.a.k.1.2 3 60.59 even 2
825.2.c.g.199.3 6 60.47 odd 4
825.2.c.g.199.4 6 60.23 odd 4
1815.2.a.m.1.2 3 132.131 odd 2
2475.2.a.bb.1.2 3 20.19 odd 2
2475.2.c.r.199.3 6 20.3 even 4
2475.2.c.r.199.4 6 20.7 even 4
2640.2.a.be.1.1 3 3.2 odd 2
5445.2.a.z.1.2 3 44.43 even 2
7920.2.a.cj.1.1 3 1.1 even 1 trivial
8085.2.a.bk.1.2 3 84.83 odd 2
9075.2.a.cf.1.2 3 660.659 odd 2