# Properties

 Label 6930.2.a.bv.1.1 Level $6930$ Weight $2$ Character 6930.1 Self dual yes Analytic conductor $55.336$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$6930 = 2 \cdot 3^{2} \cdot 5 \cdot 7 \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 6930.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$55.3363286007$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\zeta_{12})^+$$ Defining polynomial: $$x^{2} - 3$$ Coefficient ring: $$\Z[a_1, \ldots, a_{19}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 770) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$-1.73205$$ of defining polynomial Character $$\chi$$ $$=$$ 6930.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{7} -1.00000 q^{8} +O(q^{10})$$ $$q-1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{7} -1.00000 q^{8} -1.00000 q^{10} -1.00000 q^{11} -1.46410 q^{13} -1.00000 q^{14} +1.00000 q^{16} +3.46410 q^{17} +6.73205 q^{19} +1.00000 q^{20} +1.00000 q^{22} +8.19615 q^{23} +1.00000 q^{25} +1.46410 q^{26} +1.00000 q^{28} +4.73205 q^{29} +2.00000 q^{31} -1.00000 q^{32} -3.46410 q^{34} +1.00000 q^{35} +0.732051 q^{37} -6.73205 q^{38} -1.00000 q^{40} +2.19615 q^{41} +2.00000 q^{43} -1.00000 q^{44} -8.19615 q^{46} -6.92820 q^{47} +1.00000 q^{49} -1.00000 q^{50} -1.46410 q^{52} +7.26795 q^{53} -1.00000 q^{55} -1.00000 q^{56} -4.73205 q^{58} -6.92820 q^{59} -4.92820 q^{61} -2.00000 q^{62} +1.00000 q^{64} -1.46410 q^{65} -4.00000 q^{67} +3.46410 q^{68} -1.00000 q^{70} -9.46410 q^{71} -14.3923 q^{73} -0.732051 q^{74} +6.73205 q^{76} -1.00000 q^{77} -12.1962 q^{79} +1.00000 q^{80} -2.19615 q^{82} +16.3923 q^{83} +3.46410 q^{85} -2.00000 q^{86} +1.00000 q^{88} -3.46410 q^{89} -1.46410 q^{91} +8.19615 q^{92} +6.92820 q^{94} +6.73205 q^{95} +14.5885 q^{97} -1.00000 q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{2} + 2 q^{4} + 2 q^{5} + 2 q^{7} - 2 q^{8} + O(q^{10})$$ $$2 q - 2 q^{2} + 2 q^{4} + 2 q^{5} + 2 q^{7} - 2 q^{8} - 2 q^{10} - 2 q^{11} + 4 q^{13} - 2 q^{14} + 2 q^{16} + 10 q^{19} + 2 q^{20} + 2 q^{22} + 6 q^{23} + 2 q^{25} - 4 q^{26} + 2 q^{28} + 6 q^{29} + 4 q^{31} - 2 q^{32} + 2 q^{35} - 2 q^{37} - 10 q^{38} - 2 q^{40} - 6 q^{41} + 4 q^{43} - 2 q^{44} - 6 q^{46} + 2 q^{49} - 2 q^{50} + 4 q^{52} + 18 q^{53} - 2 q^{55} - 2 q^{56} - 6 q^{58} + 4 q^{61} - 4 q^{62} + 2 q^{64} + 4 q^{65} - 8 q^{67} - 2 q^{70} - 12 q^{71} - 8 q^{73} + 2 q^{74} + 10 q^{76} - 2 q^{77} - 14 q^{79} + 2 q^{80} + 6 q^{82} + 12 q^{83} - 4 q^{86} + 2 q^{88} + 4 q^{91} + 6 q^{92} + 10 q^{95} - 2 q^{97} - 2 q^{98} + O(q^{100})$$

## Coefficient data

For each $$n$$ we display the coefficients of the $$q$$-expansion $$a_n$$, the Satake parameters $$\alpha_p$$, and the Satake angles $$\theta_p = \textrm{Arg}(\alpha_p)$$.

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −1.00000 −0.707107
$$3$$ 0 0
$$4$$ 1.00000 0.500000
$$5$$ 1.00000 0.447214
$$6$$ 0 0
$$7$$ 1.00000 0.377964
$$8$$ −1.00000 −0.353553
$$9$$ 0 0
$$10$$ −1.00000 −0.316228
$$11$$ −1.00000 −0.301511
$$12$$ 0 0
$$13$$ −1.46410 −0.406069 −0.203034 0.979172i $$-0.565080\pi$$
−0.203034 + 0.979172i $$0.565080\pi$$
$$14$$ −1.00000 −0.267261
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ 3.46410 0.840168 0.420084 0.907485i $$-0.362001\pi$$
0.420084 + 0.907485i $$0.362001\pi$$
$$18$$ 0 0
$$19$$ 6.73205 1.54444 0.772219 0.635356i $$-0.219147\pi$$
0.772219 + 0.635356i $$0.219147\pi$$
$$20$$ 1.00000 0.223607
$$21$$ 0 0
$$22$$ 1.00000 0.213201
$$23$$ 8.19615 1.70902 0.854508 0.519438i $$-0.173859\pi$$
0.854508 + 0.519438i $$0.173859\pi$$
$$24$$ 0 0
$$25$$ 1.00000 0.200000
$$26$$ 1.46410 0.287134
$$27$$ 0 0
$$28$$ 1.00000 0.188982
$$29$$ 4.73205 0.878720 0.439360 0.898311i $$-0.355205\pi$$
0.439360 + 0.898311i $$0.355205\pi$$
$$30$$ 0 0
$$31$$ 2.00000 0.359211 0.179605 0.983739i $$-0.442518\pi$$
0.179605 + 0.983739i $$0.442518\pi$$
$$32$$ −1.00000 −0.176777
$$33$$ 0 0
$$34$$ −3.46410 −0.594089
$$35$$ 1.00000 0.169031
$$36$$ 0 0
$$37$$ 0.732051 0.120348 0.0601742 0.998188i $$-0.480834\pi$$
0.0601742 + 0.998188i $$0.480834\pi$$
$$38$$ −6.73205 −1.09208
$$39$$ 0 0
$$40$$ −1.00000 −0.158114
$$41$$ 2.19615 0.342981 0.171491 0.985186i $$-0.445142\pi$$
0.171491 + 0.985186i $$0.445142\pi$$
$$42$$ 0 0
$$43$$ 2.00000 0.304997 0.152499 0.988304i $$-0.451268\pi$$
0.152499 + 0.988304i $$0.451268\pi$$
$$44$$ −1.00000 −0.150756
$$45$$ 0 0
$$46$$ −8.19615 −1.20846
$$47$$ −6.92820 −1.01058 −0.505291 0.862949i $$-0.668615\pi$$
−0.505291 + 0.862949i $$0.668615\pi$$
$$48$$ 0 0
$$49$$ 1.00000 0.142857
$$50$$ −1.00000 −0.141421
$$51$$ 0 0
$$52$$ −1.46410 −0.203034
$$53$$ 7.26795 0.998330 0.499165 0.866507i $$-0.333640\pi$$
0.499165 + 0.866507i $$0.333640\pi$$
$$54$$ 0 0
$$55$$ −1.00000 −0.134840
$$56$$ −1.00000 −0.133631
$$57$$ 0 0
$$58$$ −4.73205 −0.621349
$$59$$ −6.92820 −0.901975 −0.450988 0.892530i $$-0.648928\pi$$
−0.450988 + 0.892530i $$0.648928\pi$$
$$60$$ 0 0
$$61$$ −4.92820 −0.630992 −0.315496 0.948927i $$-0.602171\pi$$
−0.315496 + 0.948927i $$0.602171\pi$$
$$62$$ −2.00000 −0.254000
$$63$$ 0 0
$$64$$ 1.00000 0.125000
$$65$$ −1.46410 −0.181599
$$66$$ 0 0
$$67$$ −4.00000 −0.488678 −0.244339 0.969690i $$-0.578571\pi$$
−0.244339 + 0.969690i $$0.578571\pi$$
$$68$$ 3.46410 0.420084
$$69$$ 0 0
$$70$$ −1.00000 −0.119523
$$71$$ −9.46410 −1.12318 −0.561591 0.827415i $$-0.689811\pi$$
−0.561591 + 0.827415i $$0.689811\pi$$
$$72$$ 0 0
$$73$$ −14.3923 −1.68449 −0.842246 0.539093i $$-0.818767\pi$$
−0.842246 + 0.539093i $$0.818767\pi$$
$$74$$ −0.732051 −0.0850992
$$75$$ 0 0
$$76$$ 6.73205 0.772219
$$77$$ −1.00000 −0.113961
$$78$$ 0 0
$$79$$ −12.1962 −1.37217 −0.686087 0.727519i $$-0.740673\pi$$
−0.686087 + 0.727519i $$0.740673\pi$$
$$80$$ 1.00000 0.111803
$$81$$ 0 0
$$82$$ −2.19615 −0.242524
$$83$$ 16.3923 1.79929 0.899645 0.436623i $$-0.143826\pi$$
0.899645 + 0.436623i $$0.143826\pi$$
$$84$$ 0 0
$$85$$ 3.46410 0.375735
$$86$$ −2.00000 −0.215666
$$87$$ 0 0
$$88$$ 1.00000 0.106600
$$89$$ −3.46410 −0.367194 −0.183597 0.983002i $$-0.558774\pi$$
−0.183597 + 0.983002i $$0.558774\pi$$
$$90$$ 0 0
$$91$$ −1.46410 −0.153480
$$92$$ 8.19615 0.854508
$$93$$ 0 0
$$94$$ 6.92820 0.714590
$$95$$ 6.73205 0.690694
$$96$$ 0 0
$$97$$ 14.5885 1.48123 0.740617 0.671928i $$-0.234533\pi$$
0.740617 + 0.671928i $$0.234533\pi$$
$$98$$ −1.00000 −0.101015
$$99$$ 0 0
$$100$$ 1.00000 0.100000
$$101$$ 7.85641 0.781742 0.390871 0.920446i $$-0.372174\pi$$
0.390871 + 0.920446i $$0.372174\pi$$
$$102$$ 0 0
$$103$$ 12.3923 1.22105 0.610525 0.791997i $$-0.290958\pi$$
0.610525 + 0.791997i $$0.290958\pi$$
$$104$$ 1.46410 0.143567
$$105$$ 0 0
$$106$$ −7.26795 −0.705926
$$107$$ −7.85641 −0.759507 −0.379754 0.925088i $$-0.623991\pi$$
−0.379754 + 0.925088i $$0.623991\pi$$
$$108$$ 0 0
$$109$$ −15.6603 −1.49998 −0.749990 0.661449i $$-0.769942\pi$$
−0.749990 + 0.661449i $$0.769942\pi$$
$$110$$ 1.00000 0.0953463
$$111$$ 0 0
$$112$$ 1.00000 0.0944911
$$113$$ −7.85641 −0.739069 −0.369534 0.929217i $$-0.620483\pi$$
−0.369534 + 0.929217i $$0.620483\pi$$
$$114$$ 0 0
$$115$$ 8.19615 0.764295
$$116$$ 4.73205 0.439360
$$117$$ 0 0
$$118$$ 6.92820 0.637793
$$119$$ 3.46410 0.317554
$$120$$ 0 0
$$121$$ 1.00000 0.0909091
$$122$$ 4.92820 0.446179
$$123$$ 0 0
$$124$$ 2.00000 0.179605
$$125$$ 1.00000 0.0894427
$$126$$ 0 0
$$127$$ 1.07180 0.0951066 0.0475533 0.998869i $$-0.484858\pi$$
0.0475533 + 0.998869i $$0.484858\pi$$
$$128$$ −1.00000 −0.0883883
$$129$$ 0 0
$$130$$ 1.46410 0.128410
$$131$$ −5.66025 −0.494539 −0.247269 0.968947i $$-0.579533\pi$$
−0.247269 + 0.968947i $$0.579533\pi$$
$$132$$ 0 0
$$133$$ 6.73205 0.583743
$$134$$ 4.00000 0.345547
$$135$$ 0 0
$$136$$ −3.46410 −0.297044
$$137$$ 0.928203 0.0793018 0.0396509 0.999214i $$-0.487375\pi$$
0.0396509 + 0.999214i $$0.487375\pi$$
$$138$$ 0 0
$$139$$ 13.6603 1.15865 0.579324 0.815097i $$-0.303317\pi$$
0.579324 + 0.815097i $$0.303317\pi$$
$$140$$ 1.00000 0.0845154
$$141$$ 0 0
$$142$$ 9.46410 0.794210
$$143$$ 1.46410 0.122434
$$144$$ 0 0
$$145$$ 4.73205 0.392975
$$146$$ 14.3923 1.19112
$$147$$ 0 0
$$148$$ 0.732051 0.0601742
$$149$$ 7.26795 0.595414 0.297707 0.954657i $$-0.403778\pi$$
0.297707 + 0.954657i $$0.403778\pi$$
$$150$$ 0 0
$$151$$ 11.1244 0.905287 0.452644 0.891692i $$-0.350481\pi$$
0.452644 + 0.891692i $$0.350481\pi$$
$$152$$ −6.73205 −0.546041
$$153$$ 0 0
$$154$$ 1.00000 0.0805823
$$155$$ 2.00000 0.160644
$$156$$ 0 0
$$157$$ 4.53590 0.362004 0.181002 0.983483i $$-0.442066\pi$$
0.181002 + 0.983483i $$0.442066\pi$$
$$158$$ 12.1962 0.970274
$$159$$ 0 0
$$160$$ −1.00000 −0.0790569
$$161$$ 8.19615 0.645947
$$162$$ 0 0
$$163$$ −4.00000 −0.313304 −0.156652 0.987654i $$-0.550070\pi$$
−0.156652 + 0.987654i $$0.550070\pi$$
$$164$$ 2.19615 0.171491
$$165$$ 0 0
$$166$$ −16.3923 −1.27229
$$167$$ −13.8564 −1.07224 −0.536120 0.844141i $$-0.680111\pi$$
−0.536120 + 0.844141i $$0.680111\pi$$
$$168$$ 0 0
$$169$$ −10.8564 −0.835108
$$170$$ −3.46410 −0.265684
$$171$$ 0 0
$$172$$ 2.00000 0.152499
$$173$$ −0.928203 −0.0705700 −0.0352850 0.999377i $$-0.511234\pi$$
−0.0352850 + 0.999377i $$0.511234\pi$$
$$174$$ 0 0
$$175$$ 1.00000 0.0755929
$$176$$ −1.00000 −0.0753778
$$177$$ 0 0
$$178$$ 3.46410 0.259645
$$179$$ 6.00000 0.448461 0.224231 0.974536i $$-0.428013\pi$$
0.224231 + 0.974536i $$0.428013\pi$$
$$180$$ 0 0
$$181$$ 14.0000 1.04061 0.520306 0.853980i $$-0.325818\pi$$
0.520306 + 0.853980i $$0.325818\pi$$
$$182$$ 1.46410 0.108526
$$183$$ 0 0
$$184$$ −8.19615 −0.604228
$$185$$ 0.732051 0.0538214
$$186$$ 0 0
$$187$$ −3.46410 −0.253320
$$188$$ −6.92820 −0.505291
$$189$$ 0 0
$$190$$ −6.73205 −0.488394
$$191$$ 12.0000 0.868290 0.434145 0.900843i $$-0.357051\pi$$
0.434145 + 0.900843i $$0.357051\pi$$
$$192$$ 0 0
$$193$$ 12.3923 0.892018 0.446009 0.895029i $$-0.352845\pi$$
0.446009 + 0.895029i $$0.352845\pi$$
$$194$$ −14.5885 −1.04739
$$195$$ 0 0
$$196$$ 1.00000 0.0714286
$$197$$ 24.2487 1.72765 0.863825 0.503793i $$-0.168062\pi$$
0.863825 + 0.503793i $$0.168062\pi$$
$$198$$ 0 0
$$199$$ 2.92820 0.207575 0.103787 0.994600i $$-0.466904\pi$$
0.103787 + 0.994600i $$0.466904\pi$$
$$200$$ −1.00000 −0.0707107
$$201$$ 0 0
$$202$$ −7.85641 −0.552775
$$203$$ 4.73205 0.332125
$$204$$ 0 0
$$205$$ 2.19615 0.153386
$$206$$ −12.3923 −0.863413
$$207$$ 0 0
$$208$$ −1.46410 −0.101517
$$209$$ −6.73205 −0.465666
$$210$$ 0 0
$$211$$ 26.9282 1.85381 0.926907 0.375291i $$-0.122457\pi$$
0.926907 + 0.375291i $$0.122457\pi$$
$$212$$ 7.26795 0.499165
$$213$$ 0 0
$$214$$ 7.85641 0.537053
$$215$$ 2.00000 0.136399
$$216$$ 0 0
$$217$$ 2.00000 0.135769
$$218$$ 15.6603 1.06065
$$219$$ 0 0
$$220$$ −1.00000 −0.0674200
$$221$$ −5.07180 −0.341166
$$222$$ 0 0
$$223$$ −25.4641 −1.70520 −0.852601 0.522562i $$-0.824976\pi$$
−0.852601 + 0.522562i $$0.824976\pi$$
$$224$$ −1.00000 −0.0668153
$$225$$ 0 0
$$226$$ 7.85641 0.522600
$$227$$ −6.92820 −0.459841 −0.229920 0.973209i $$-0.573847\pi$$
−0.229920 + 0.973209i $$0.573847\pi$$
$$228$$ 0 0
$$229$$ 24.3923 1.61189 0.805944 0.591991i $$-0.201658\pi$$
0.805944 + 0.591991i $$0.201658\pi$$
$$230$$ −8.19615 −0.540438
$$231$$ 0 0
$$232$$ −4.73205 −0.310674
$$233$$ 7.85641 0.514690 0.257345 0.966320i $$-0.417152\pi$$
0.257345 + 0.966320i $$0.417152\pi$$
$$234$$ 0 0
$$235$$ −6.92820 −0.451946
$$236$$ −6.92820 −0.450988
$$237$$ 0 0
$$238$$ −3.46410 −0.224544
$$239$$ −1.26795 −0.0820168 −0.0410084 0.999159i $$-0.513057\pi$$
−0.0410084 + 0.999159i $$0.513057\pi$$
$$240$$ 0 0
$$241$$ 3.26795 0.210507 0.105254 0.994445i $$-0.466435\pi$$
0.105254 + 0.994445i $$0.466435\pi$$
$$242$$ −1.00000 −0.0642824
$$243$$ 0 0
$$244$$ −4.92820 −0.315496
$$245$$ 1.00000 0.0638877
$$246$$ 0 0
$$247$$ −9.85641 −0.627148
$$248$$ −2.00000 −0.127000
$$249$$ 0 0
$$250$$ −1.00000 −0.0632456
$$251$$ −12.0000 −0.757433 −0.378717 0.925513i $$-0.623635\pi$$
−0.378717 + 0.925513i $$0.623635\pi$$
$$252$$ 0 0
$$253$$ −8.19615 −0.515288
$$254$$ −1.07180 −0.0672505
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ 23.6603 1.47589 0.737943 0.674863i $$-0.235797\pi$$
0.737943 + 0.674863i $$0.235797\pi$$
$$258$$ 0 0
$$259$$ 0.732051 0.0454874
$$260$$ −1.46410 −0.0907997
$$261$$ 0 0
$$262$$ 5.66025 0.349692
$$263$$ 24.0000 1.47990 0.739952 0.672660i $$-0.234848\pi$$
0.739952 + 0.672660i $$0.234848\pi$$
$$264$$ 0 0
$$265$$ 7.26795 0.446467
$$266$$ −6.73205 −0.412769
$$267$$ 0 0
$$268$$ −4.00000 −0.244339
$$269$$ −28.3923 −1.73111 −0.865555 0.500814i $$-0.833034\pi$$
−0.865555 + 0.500814i $$0.833034\pi$$
$$270$$ 0 0
$$271$$ 0.392305 0.0238308 0.0119154 0.999929i $$-0.496207\pi$$
0.0119154 + 0.999929i $$0.496207\pi$$
$$272$$ 3.46410 0.210042
$$273$$ 0 0
$$274$$ −0.928203 −0.0560748
$$275$$ −1.00000 −0.0603023
$$276$$ 0 0
$$277$$ 7.07180 0.424903 0.212452 0.977172i $$-0.431855\pi$$
0.212452 + 0.977172i $$0.431855\pi$$
$$278$$ −13.6603 −0.819288
$$279$$ 0 0
$$280$$ −1.00000 −0.0597614
$$281$$ −22.3923 −1.33581 −0.667906 0.744245i $$-0.732809\pi$$
−0.667906 + 0.744245i $$0.732809\pi$$
$$282$$ 0 0
$$283$$ −31.7128 −1.88513 −0.942566 0.334021i $$-0.891594\pi$$
−0.942566 + 0.334021i $$0.891594\pi$$
$$284$$ −9.46410 −0.561591
$$285$$ 0 0
$$286$$ −1.46410 −0.0865741
$$287$$ 2.19615 0.129635
$$288$$ 0 0
$$289$$ −5.00000 −0.294118
$$290$$ −4.73205 −0.277876
$$291$$ 0 0
$$292$$ −14.3923 −0.842246
$$293$$ −21.4641 −1.25395 −0.626973 0.779041i $$-0.715706\pi$$
−0.626973 + 0.779041i $$0.715706\pi$$
$$294$$ 0 0
$$295$$ −6.92820 −0.403376
$$296$$ −0.732051 −0.0425496
$$297$$ 0 0
$$298$$ −7.26795 −0.421021
$$299$$ −12.0000 −0.693978
$$300$$ 0 0
$$301$$ 2.00000 0.115278
$$302$$ −11.1244 −0.640135
$$303$$ 0 0
$$304$$ 6.73205 0.386110
$$305$$ −4.92820 −0.282188
$$306$$ 0 0
$$307$$ 24.3923 1.39214 0.696071 0.717973i $$-0.254930\pi$$
0.696071 + 0.717973i $$0.254930\pi$$
$$308$$ −1.00000 −0.0569803
$$309$$ 0 0
$$310$$ −2.00000 −0.113592
$$311$$ −24.9282 −1.41355 −0.706774 0.707439i $$-0.749850\pi$$
−0.706774 + 0.707439i $$0.749850\pi$$
$$312$$ 0 0
$$313$$ 22.1962 1.25460 0.627300 0.778777i $$-0.284160\pi$$
0.627300 + 0.778777i $$0.284160\pi$$
$$314$$ −4.53590 −0.255976
$$315$$ 0 0
$$316$$ −12.1962 −0.686087
$$317$$ −30.5885 −1.71802 −0.859009 0.511960i $$-0.828920\pi$$
−0.859009 + 0.511960i $$0.828920\pi$$
$$318$$ 0 0
$$319$$ −4.73205 −0.264944
$$320$$ 1.00000 0.0559017
$$321$$ 0 0
$$322$$ −8.19615 −0.456754
$$323$$ 23.3205 1.29759
$$324$$ 0 0
$$325$$ −1.46410 −0.0812137
$$326$$ 4.00000 0.221540
$$327$$ 0 0
$$328$$ −2.19615 −0.121262
$$329$$ −6.92820 −0.381964
$$330$$ 0 0
$$331$$ −18.7846 −1.03250 −0.516248 0.856439i $$-0.672672\pi$$
−0.516248 + 0.856439i $$0.672672\pi$$
$$332$$ 16.3923 0.899645
$$333$$ 0 0
$$334$$ 13.8564 0.758189
$$335$$ −4.00000 −0.218543
$$336$$ 0 0
$$337$$ 22.7846 1.24116 0.620578 0.784144i $$-0.286898\pi$$
0.620578 + 0.784144i $$0.286898\pi$$
$$338$$ 10.8564 0.590511
$$339$$ 0 0
$$340$$ 3.46410 0.187867
$$341$$ −2.00000 −0.108306
$$342$$ 0 0
$$343$$ 1.00000 0.0539949
$$344$$ −2.00000 −0.107833
$$345$$ 0 0
$$346$$ 0.928203 0.0499005
$$347$$ 23.0718 1.23856 0.619279 0.785171i $$-0.287425\pi$$
0.619279 + 0.785171i $$0.287425\pi$$
$$348$$ 0 0
$$349$$ −5.60770 −0.300173 −0.150087 0.988673i $$-0.547955\pi$$
−0.150087 + 0.988673i $$0.547955\pi$$
$$350$$ −1.00000 −0.0534522
$$351$$ 0 0
$$352$$ 1.00000 0.0533002
$$353$$ −14.1962 −0.755585 −0.377792 0.925890i $$-0.623317\pi$$
−0.377792 + 0.925890i $$0.623317\pi$$
$$354$$ 0 0
$$355$$ −9.46410 −0.502302
$$356$$ −3.46410 −0.183597
$$357$$ 0 0
$$358$$ −6.00000 −0.317110
$$359$$ 34.0526 1.79723 0.898613 0.438743i $$-0.144576\pi$$
0.898613 + 0.438743i $$0.144576\pi$$
$$360$$ 0 0
$$361$$ 26.3205 1.38529
$$362$$ −14.0000 −0.735824
$$363$$ 0 0
$$364$$ −1.46410 −0.0767398
$$365$$ −14.3923 −0.753328
$$366$$ 0 0
$$367$$ 12.3923 0.646873 0.323437 0.946250i $$-0.395162\pi$$
0.323437 + 0.946250i $$0.395162\pi$$
$$368$$ 8.19615 0.427254
$$369$$ 0 0
$$370$$ −0.732051 −0.0380575
$$371$$ 7.26795 0.377333
$$372$$ 0 0
$$373$$ 18.3923 0.952317 0.476159 0.879359i $$-0.342029\pi$$
0.476159 + 0.879359i $$0.342029\pi$$
$$374$$ 3.46410 0.179124
$$375$$ 0 0
$$376$$ 6.92820 0.357295
$$377$$ −6.92820 −0.356821
$$378$$ 0 0
$$379$$ 6.14359 0.315575 0.157788 0.987473i $$-0.449564\pi$$
0.157788 + 0.987473i $$0.449564\pi$$
$$380$$ 6.73205 0.345347
$$381$$ 0 0
$$382$$ −12.0000 −0.613973
$$383$$ 33.4641 1.70994 0.854968 0.518681i $$-0.173577\pi$$
0.854968 + 0.518681i $$0.173577\pi$$
$$384$$ 0 0
$$385$$ −1.00000 −0.0509647
$$386$$ −12.3923 −0.630752
$$387$$ 0 0
$$388$$ 14.5885 0.740617
$$389$$ 1.60770 0.0815134 0.0407567 0.999169i $$-0.487023\pi$$
0.0407567 + 0.999169i $$0.487023\pi$$
$$390$$ 0 0
$$391$$ 28.3923 1.43586
$$392$$ −1.00000 −0.0505076
$$393$$ 0 0
$$394$$ −24.2487 −1.22163
$$395$$ −12.1962 −0.613655
$$396$$ 0 0
$$397$$ 30.3923 1.52535 0.762673 0.646784i $$-0.223887\pi$$
0.762673 + 0.646784i $$0.223887\pi$$
$$398$$ −2.92820 −0.146778
$$399$$ 0 0
$$400$$ 1.00000 0.0500000
$$401$$ −2.53590 −0.126637 −0.0633184 0.997993i $$-0.520168\pi$$
−0.0633184 + 0.997993i $$0.520168\pi$$
$$402$$ 0 0
$$403$$ −2.92820 −0.145864
$$404$$ 7.85641 0.390871
$$405$$ 0 0
$$406$$ −4.73205 −0.234848
$$407$$ −0.732051 −0.0362864
$$408$$ 0 0
$$409$$ 10.8756 0.537766 0.268883 0.963173i $$-0.413346\pi$$
0.268883 + 0.963173i $$0.413346\pi$$
$$410$$ −2.19615 −0.108460
$$411$$ 0 0
$$412$$ 12.3923 0.610525
$$413$$ −6.92820 −0.340915
$$414$$ 0 0
$$415$$ 16.3923 0.804667
$$416$$ 1.46410 0.0717835
$$417$$ 0 0
$$418$$ 6.73205 0.329275
$$419$$ −30.9282 −1.51094 −0.755471 0.655182i $$-0.772592\pi$$
−0.755471 + 0.655182i $$0.772592\pi$$
$$420$$ 0 0
$$421$$ −35.8564 −1.74753 −0.873767 0.486344i $$-0.838330\pi$$
−0.873767 + 0.486344i $$0.838330\pi$$
$$422$$ −26.9282 −1.31084
$$423$$ 0 0
$$424$$ −7.26795 −0.352963
$$425$$ 3.46410 0.168034
$$426$$ 0 0
$$427$$ −4.92820 −0.238492
$$428$$ −7.85641 −0.379754
$$429$$ 0 0
$$430$$ −2.00000 −0.0964486
$$431$$ −3.12436 −0.150495 −0.0752475 0.997165i $$-0.523975\pi$$
−0.0752475 + 0.997165i $$0.523975\pi$$
$$432$$ 0 0
$$433$$ −18.1962 −0.874451 −0.437226 0.899352i $$-0.644039\pi$$
−0.437226 + 0.899352i $$0.644039\pi$$
$$434$$ −2.00000 −0.0960031
$$435$$ 0 0
$$436$$ −15.6603 −0.749990
$$437$$ 55.1769 2.63947
$$438$$ 0 0
$$439$$ 26.2487 1.25278 0.626391 0.779509i $$-0.284531\pi$$
0.626391 + 0.779509i $$0.284531\pi$$
$$440$$ 1.00000 0.0476731
$$441$$ 0 0
$$442$$ 5.07180 0.241241
$$443$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$444$$ 0 0
$$445$$ −3.46410 −0.164214
$$446$$ 25.4641 1.20576
$$447$$ 0 0
$$448$$ 1.00000 0.0472456
$$449$$ −40.3923 −1.90623 −0.953115 0.302607i $$-0.902143\pi$$
−0.953115 + 0.302607i $$0.902143\pi$$
$$450$$ 0 0
$$451$$ −2.19615 −0.103413
$$452$$ −7.85641 −0.369534
$$453$$ 0 0
$$454$$ 6.92820 0.325157
$$455$$ −1.46410 −0.0686381
$$456$$ 0 0
$$457$$ 2.00000 0.0935561 0.0467780 0.998905i $$-0.485105\pi$$
0.0467780 + 0.998905i $$0.485105\pi$$
$$458$$ −24.3923 −1.13978
$$459$$ 0 0
$$460$$ 8.19615 0.382148
$$461$$ −22.3923 −1.04291 −0.521457 0.853278i $$-0.674611\pi$$
−0.521457 + 0.853278i $$0.674611\pi$$
$$462$$ 0 0
$$463$$ 27.5167 1.27881 0.639404 0.768871i $$-0.279181\pi$$
0.639404 + 0.768871i $$0.279181\pi$$
$$464$$ 4.73205 0.219680
$$465$$ 0 0
$$466$$ −7.85641 −0.363941
$$467$$ 34.0526 1.57576 0.787882 0.615826i $$-0.211178\pi$$
0.787882 + 0.615826i $$0.211178\pi$$
$$468$$ 0 0
$$469$$ −4.00000 −0.184703
$$470$$ 6.92820 0.319574
$$471$$ 0 0
$$472$$ 6.92820 0.318896
$$473$$ −2.00000 −0.0919601
$$474$$ 0 0
$$475$$ 6.73205 0.308888
$$476$$ 3.46410 0.158777
$$477$$ 0 0
$$478$$ 1.26795 0.0579946
$$479$$ 32.7846 1.49797 0.748984 0.662589i $$-0.230542\pi$$
0.748984 + 0.662589i $$0.230542\pi$$
$$480$$ 0 0
$$481$$ −1.07180 −0.0488697
$$482$$ −3.26795 −0.148851
$$483$$ 0 0
$$484$$ 1.00000 0.0454545
$$485$$ 14.5885 0.662428
$$486$$ 0 0
$$487$$ 4.19615 0.190146 0.0950729 0.995470i $$-0.469692\pi$$
0.0950729 + 0.995470i $$0.469692\pi$$
$$488$$ 4.92820 0.223089
$$489$$ 0 0
$$490$$ −1.00000 −0.0451754
$$491$$ 27.7128 1.25066 0.625331 0.780360i $$-0.284964\pi$$
0.625331 + 0.780360i $$0.284964\pi$$
$$492$$ 0 0
$$493$$ 16.3923 0.738272
$$494$$ 9.85641 0.443461
$$495$$ 0 0
$$496$$ 2.00000 0.0898027
$$497$$ −9.46410 −0.424523
$$498$$ 0 0
$$499$$ 12.1436 0.543622 0.271811 0.962351i $$-0.412377\pi$$
0.271811 + 0.962351i $$0.412377\pi$$
$$500$$ 1.00000 0.0447214
$$501$$ 0 0
$$502$$ 12.0000 0.535586
$$503$$ −8.78461 −0.391686 −0.195843 0.980635i $$-0.562744\pi$$
−0.195843 + 0.980635i $$0.562744\pi$$
$$504$$ 0 0
$$505$$ 7.85641 0.349605
$$506$$ 8.19615 0.364363
$$507$$ 0 0
$$508$$ 1.07180 0.0475533
$$509$$ −11.0718 −0.490749 −0.245374 0.969428i $$-0.578911\pi$$
−0.245374 + 0.969428i $$0.578911\pi$$
$$510$$ 0 0
$$511$$ −14.3923 −0.636678
$$512$$ −1.00000 −0.0441942
$$513$$ 0 0
$$514$$ −23.6603 −1.04361
$$515$$ 12.3923 0.546070
$$516$$ 0 0
$$517$$ 6.92820 0.304702
$$518$$ −0.732051 −0.0321645
$$519$$ 0 0
$$520$$ 1.46410 0.0642051
$$521$$ −10.3923 −0.455295 −0.227648 0.973744i $$-0.573103\pi$$
−0.227648 + 0.973744i $$0.573103\pi$$
$$522$$ 0 0
$$523$$ −29.1769 −1.27582 −0.637909 0.770112i $$-0.720200\pi$$
−0.637909 + 0.770112i $$0.720200\pi$$
$$524$$ −5.66025 −0.247269
$$525$$ 0 0
$$526$$ −24.0000 −1.04645
$$527$$ 6.92820 0.301797
$$528$$ 0 0
$$529$$ 44.1769 1.92074
$$530$$ −7.26795 −0.315700
$$531$$ 0 0
$$532$$ 6.73205 0.291871
$$533$$ −3.21539 −0.139274
$$534$$ 0 0
$$535$$ −7.85641 −0.339662
$$536$$ 4.00000 0.172774
$$537$$ 0 0
$$538$$ 28.3923 1.22408
$$539$$ −1.00000 −0.0430730
$$540$$ 0 0
$$541$$ −20.7321 −0.891340 −0.445670 0.895197i $$-0.647035\pi$$
−0.445670 + 0.895197i $$0.647035\pi$$
$$542$$ −0.392305 −0.0168509
$$543$$ 0 0
$$544$$ −3.46410 −0.148522
$$545$$ −15.6603 −0.670812
$$546$$ 0 0
$$547$$ 28.7846 1.23074 0.615371 0.788238i $$-0.289006\pi$$
0.615371 + 0.788238i $$0.289006\pi$$
$$548$$ 0.928203 0.0396509
$$549$$ 0 0
$$550$$ 1.00000 0.0426401
$$551$$ 31.8564 1.35713
$$552$$ 0 0
$$553$$ −12.1962 −0.518633
$$554$$ −7.07180 −0.300452
$$555$$ 0 0
$$556$$ 13.6603 0.579324
$$557$$ 25.6077 1.08503 0.542516 0.840045i $$-0.317472\pi$$
0.542516 + 0.840045i $$0.317472\pi$$
$$558$$ 0 0
$$559$$ −2.92820 −0.123850
$$560$$ 1.00000 0.0422577
$$561$$ 0 0
$$562$$ 22.3923 0.944562
$$563$$ −5.07180 −0.213751 −0.106875 0.994272i $$-0.534085\pi$$
−0.106875 + 0.994272i $$0.534085\pi$$
$$564$$ 0 0
$$565$$ −7.85641 −0.330522
$$566$$ 31.7128 1.33299
$$567$$ 0 0
$$568$$ 9.46410 0.397105
$$569$$ 6.00000 0.251533 0.125767 0.992060i $$-0.459861\pi$$
0.125767 + 0.992060i $$0.459861\pi$$
$$570$$ 0 0
$$571$$ 3.60770 0.150977 0.0754887 0.997147i $$-0.475948\pi$$
0.0754887 + 0.997147i $$0.475948\pi$$
$$572$$ 1.46410 0.0612172
$$573$$ 0 0
$$574$$ −2.19615 −0.0916656
$$575$$ 8.19615 0.341803
$$576$$ 0 0
$$577$$ −34.5885 −1.43994 −0.719968 0.694007i $$-0.755844\pi$$
−0.719968 + 0.694007i $$0.755844\pi$$
$$578$$ 5.00000 0.207973
$$579$$ 0 0
$$580$$ 4.73205 0.196488
$$581$$ 16.3923 0.680067
$$582$$ 0 0
$$583$$ −7.26795 −0.301008
$$584$$ 14.3923 0.595558
$$585$$ 0 0
$$586$$ 21.4641 0.886674
$$587$$ 11.4115 0.471005 0.235502 0.971874i $$-0.424326\pi$$
0.235502 + 0.971874i $$0.424326\pi$$
$$588$$ 0 0
$$589$$ 13.4641 0.554779
$$590$$ 6.92820 0.285230
$$591$$ 0 0
$$592$$ 0.732051 0.0300871
$$593$$ −0.248711 −0.0102133 −0.00510667 0.999987i $$-0.501626\pi$$
−0.00510667 + 0.999987i $$0.501626\pi$$
$$594$$ 0 0
$$595$$ 3.46410 0.142014
$$596$$ 7.26795 0.297707
$$597$$ 0 0
$$598$$ 12.0000 0.490716
$$599$$ −37.1769 −1.51901 −0.759504 0.650503i $$-0.774558\pi$$
−0.759504 + 0.650503i $$0.774558\pi$$
$$600$$ 0 0
$$601$$ −5.51666 −0.225029 −0.112515 0.993650i $$-0.535891\pi$$
−0.112515 + 0.993650i $$0.535891\pi$$
$$602$$ −2.00000 −0.0815139
$$603$$ 0 0
$$604$$ 11.1244 0.452644
$$605$$ 1.00000 0.0406558
$$606$$ 0 0
$$607$$ 20.9282 0.849450 0.424725 0.905323i $$-0.360371\pi$$
0.424725 + 0.905323i $$0.360371\pi$$
$$608$$ −6.73205 −0.273021
$$609$$ 0 0
$$610$$ 4.92820 0.199537
$$611$$ 10.1436 0.410366
$$612$$ 0 0
$$613$$ −35.1769 −1.42078 −0.710391 0.703807i $$-0.751482\pi$$
−0.710391 + 0.703807i $$0.751482\pi$$
$$614$$ −24.3923 −0.984393
$$615$$ 0 0
$$616$$ 1.00000 0.0402911
$$617$$ −17.3205 −0.697297 −0.348649 0.937253i $$-0.613359\pi$$
−0.348649 + 0.937253i $$0.613359\pi$$
$$618$$ 0 0
$$619$$ 28.7846 1.15695 0.578476 0.815700i $$-0.303648\pi$$
0.578476 + 0.815700i $$0.303648\pi$$
$$620$$ 2.00000 0.0803219
$$621$$ 0 0
$$622$$ 24.9282 0.999530
$$623$$ −3.46410 −0.138786
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ −22.1962 −0.887137
$$627$$ 0 0
$$628$$ 4.53590 0.181002
$$629$$ 2.53590 0.101113
$$630$$ 0 0
$$631$$ −46.9282 −1.86818 −0.934091 0.357035i $$-0.883788\pi$$
−0.934091 + 0.357035i $$0.883788\pi$$
$$632$$ 12.1962 0.485137
$$633$$ 0 0
$$634$$ 30.5885 1.21482
$$635$$ 1.07180 0.0425330
$$636$$ 0 0
$$637$$ −1.46410 −0.0580098
$$638$$ 4.73205 0.187344
$$639$$ 0 0
$$640$$ −1.00000 −0.0395285
$$641$$ 35.5692 1.40490 0.702450 0.711733i $$-0.252090\pi$$
0.702450 + 0.711733i $$0.252090\pi$$
$$642$$ 0 0
$$643$$ 33.2679 1.31196 0.655980 0.754778i $$-0.272256\pi$$
0.655980 + 0.754778i $$0.272256\pi$$
$$644$$ 8.19615 0.322974
$$645$$ 0 0
$$646$$ −23.3205 −0.917533
$$647$$ 26.5359 1.04323 0.521617 0.853180i $$-0.325329\pi$$
0.521617 + 0.853180i $$0.325329\pi$$
$$648$$ 0 0
$$649$$ 6.92820 0.271956
$$650$$ 1.46410 0.0574268
$$651$$ 0 0
$$652$$ −4.00000 −0.156652
$$653$$ −11.6603 −0.456301 −0.228151 0.973626i $$-0.573268\pi$$
−0.228151 + 0.973626i $$0.573268\pi$$
$$654$$ 0 0
$$655$$ −5.66025 −0.221164
$$656$$ 2.19615 0.0857453
$$657$$ 0 0
$$658$$ 6.92820 0.270089
$$659$$ −30.2487 −1.17832 −0.589161 0.808015i $$-0.700542\pi$$
−0.589161 + 0.808015i $$0.700542\pi$$
$$660$$ 0 0
$$661$$ −46.0000 −1.78919 −0.894596 0.446875i $$-0.852537\pi$$
−0.894596 + 0.446875i $$0.852537\pi$$
$$662$$ 18.7846 0.730085
$$663$$ 0 0
$$664$$ −16.3923 −0.636145
$$665$$ 6.73205 0.261058
$$666$$ 0 0
$$667$$ 38.7846 1.50175
$$668$$ −13.8564 −0.536120
$$669$$ 0 0
$$670$$ 4.00000 0.154533
$$671$$ 4.92820 0.190251
$$672$$ 0 0
$$673$$ 2.00000 0.0770943 0.0385472 0.999257i $$-0.487727\pi$$
0.0385472 + 0.999257i $$0.487727\pi$$
$$674$$ −22.7846 −0.877630
$$675$$ 0 0
$$676$$ −10.8564 −0.417554
$$677$$ 4.14359 0.159251 0.0796256 0.996825i $$-0.474628\pi$$
0.0796256 + 0.996825i $$0.474628\pi$$
$$678$$ 0 0
$$679$$ 14.5885 0.559854
$$680$$ −3.46410 −0.132842
$$681$$ 0 0
$$682$$ 2.00000 0.0765840
$$683$$ −6.24871 −0.239100 −0.119550 0.992828i $$-0.538145\pi$$
−0.119550 + 0.992828i $$0.538145\pi$$
$$684$$ 0 0
$$685$$ 0.928203 0.0354648
$$686$$ −1.00000 −0.0381802
$$687$$ 0 0
$$688$$ 2.00000 0.0762493
$$689$$ −10.6410 −0.405390
$$690$$ 0 0
$$691$$ −13.4641 −0.512199 −0.256099 0.966650i $$-0.582437\pi$$
−0.256099 + 0.966650i $$0.582437\pi$$
$$692$$ −0.928203 −0.0352850
$$693$$ 0 0
$$694$$ −23.0718 −0.875793
$$695$$ 13.6603 0.518163
$$696$$ 0 0
$$697$$ 7.60770 0.288162
$$698$$ 5.60770 0.212254
$$699$$ 0 0
$$700$$ 1.00000 0.0377964
$$701$$ −41.9090 −1.58288 −0.791440 0.611247i $$-0.790668\pi$$
−0.791440 + 0.611247i $$0.790668\pi$$
$$702$$ 0 0
$$703$$ 4.92820 0.185871
$$704$$ −1.00000 −0.0376889
$$705$$ 0 0
$$706$$ 14.1962 0.534279
$$707$$ 7.85641 0.295471
$$708$$ 0 0
$$709$$ 25.3205 0.950932 0.475466 0.879734i $$-0.342280\pi$$
0.475466 + 0.879734i $$0.342280\pi$$
$$710$$ 9.46410 0.355181
$$711$$ 0 0
$$712$$ 3.46410 0.129823
$$713$$ 16.3923 0.613897
$$714$$ 0 0
$$715$$ 1.46410 0.0547543
$$716$$ 6.00000 0.224231
$$717$$ 0 0
$$718$$ −34.0526 −1.27083
$$719$$ 27.7128 1.03351 0.516757 0.856132i $$-0.327139\pi$$
0.516757 + 0.856132i $$0.327139\pi$$
$$720$$ 0 0
$$721$$ 12.3923 0.461514
$$722$$ −26.3205 −0.979548
$$723$$ 0 0
$$724$$ 14.0000 0.520306
$$725$$ 4.73205 0.175744
$$726$$ 0 0
$$727$$ −4.00000 −0.148352 −0.0741759 0.997245i $$-0.523633\pi$$
−0.0741759 + 0.997245i $$0.523633\pi$$
$$728$$ 1.46410 0.0542632
$$729$$ 0 0
$$730$$ 14.3923 0.532683
$$731$$ 6.92820 0.256249
$$732$$ 0 0
$$733$$ 26.0000 0.960332 0.480166 0.877178i $$-0.340576\pi$$
0.480166 + 0.877178i $$0.340576\pi$$
$$734$$ −12.3923 −0.457408
$$735$$ 0 0
$$736$$ −8.19615 −0.302114
$$737$$ 4.00000 0.147342
$$738$$ 0 0
$$739$$ 11.7128 0.430863 0.215431 0.976519i $$-0.430884\pi$$
0.215431 + 0.976519i $$0.430884\pi$$
$$740$$ 0.732051 0.0269107
$$741$$ 0 0
$$742$$ −7.26795 −0.266815
$$743$$ 32.7846 1.20275 0.601375 0.798967i $$-0.294620\pi$$
0.601375 + 0.798967i $$0.294620\pi$$
$$744$$ 0 0
$$745$$ 7.26795 0.266277
$$746$$ −18.3923 −0.673390
$$747$$ 0 0
$$748$$ −3.46410 −0.126660
$$749$$ −7.85641 −0.287067
$$750$$ 0 0
$$751$$ −11.6077 −0.423571 −0.211785 0.977316i $$-0.567928\pi$$
−0.211785 + 0.977316i $$0.567928\pi$$
$$752$$ −6.92820 −0.252646
$$753$$ 0 0
$$754$$ 6.92820 0.252310
$$755$$ 11.1244 0.404857
$$756$$ 0 0
$$757$$ −43.3731 −1.57642 −0.788210 0.615406i $$-0.788992\pi$$
−0.788210 + 0.615406i $$0.788992\pi$$
$$758$$ −6.14359 −0.223145
$$759$$ 0 0
$$760$$ −6.73205 −0.244197
$$761$$ 12.3397 0.447315 0.223658 0.974668i $$-0.428200\pi$$
0.223658 + 0.974668i $$0.428200\pi$$
$$762$$ 0 0
$$763$$ −15.6603 −0.566939
$$764$$ 12.0000 0.434145
$$765$$ 0 0
$$766$$ −33.4641 −1.20911
$$767$$ 10.1436 0.366264
$$768$$ 0 0
$$769$$ −18.1962 −0.656170 −0.328085 0.944648i $$-0.606403\pi$$
−0.328085 + 0.944648i $$0.606403\pi$$
$$770$$ 1.00000 0.0360375
$$771$$ 0 0
$$772$$ 12.3923 0.446009
$$773$$ 0.928203 0.0333851 0.0166926 0.999861i $$-0.494686\pi$$
0.0166926 + 0.999861i $$0.494686\pi$$
$$774$$ 0 0
$$775$$ 2.00000 0.0718421
$$776$$ −14.5885 −0.523695
$$777$$ 0 0
$$778$$ −1.60770 −0.0576387
$$779$$ 14.7846 0.529714
$$780$$ 0 0
$$781$$ 9.46410 0.338652
$$782$$ −28.3923 −1.01531
$$783$$ 0 0
$$784$$ 1.00000 0.0357143
$$785$$ 4.53590 0.161893
$$786$$ 0 0
$$787$$ 18.1436 0.646749 0.323375 0.946271i $$-0.395183\pi$$
0.323375 + 0.946271i $$0.395183\pi$$
$$788$$ 24.2487 0.863825
$$789$$ 0 0
$$790$$ 12.1962 0.433920
$$791$$ −7.85641 −0.279342
$$792$$ 0 0
$$793$$ 7.21539 0.256226
$$794$$ −30.3923 −1.07858
$$795$$ 0 0
$$796$$ 2.92820 0.103787
$$797$$ 25.6077 0.907071 0.453536 0.891238i $$-0.350163\pi$$
0.453536 + 0.891238i $$0.350163\pi$$
$$798$$ 0 0
$$799$$ −24.0000 −0.849059
$$800$$ −1.00000 −0.0353553
$$801$$ 0 0
$$802$$ 2.53590 0.0895457
$$803$$ 14.3923 0.507893
$$804$$ 0 0
$$805$$ 8.19615 0.288876
$$806$$ 2.92820 0.103142
$$807$$ 0 0
$$808$$ −7.85641 −0.276387
$$809$$ 31.1769 1.09612 0.548061 0.836438i $$-0.315366\pi$$
0.548061 + 0.836438i $$0.315366\pi$$
$$810$$ 0 0
$$811$$ −43.1244 −1.51430 −0.757150 0.653241i $$-0.773409\pi$$
−0.757150 + 0.653241i $$0.773409\pi$$
$$812$$ 4.73205 0.166062
$$813$$ 0 0
$$814$$ 0.732051 0.0256584
$$815$$ −4.00000 −0.140114
$$816$$ 0 0
$$817$$ 13.4641 0.471049
$$818$$ −10.8756 −0.380258
$$819$$ 0 0
$$820$$ 2.19615 0.0766930
$$821$$ −17.9090 −0.625027 −0.312514 0.949913i $$-0.601171\pi$$
−0.312514 + 0.949913i $$0.601171\pi$$
$$822$$ 0 0
$$823$$ −0.875644 −0.0305230 −0.0152615 0.999884i $$-0.504858\pi$$
−0.0152615 + 0.999884i $$0.504858\pi$$
$$824$$ −12.3923 −0.431706
$$825$$ 0 0
$$826$$ 6.92820 0.241063
$$827$$ 25.8564 0.899115 0.449558 0.893251i $$-0.351582\pi$$
0.449558 + 0.893251i $$0.351582\pi$$
$$828$$ 0 0
$$829$$ 2.24871 0.0781010 0.0390505 0.999237i $$-0.487567\pi$$
0.0390505 + 0.999237i $$0.487567\pi$$
$$830$$ −16.3923 −0.568985
$$831$$ 0 0
$$832$$ −1.46410 −0.0507586
$$833$$ 3.46410 0.120024
$$834$$ 0 0
$$835$$ −13.8564 −0.479521
$$836$$ −6.73205 −0.232833
$$837$$ 0 0
$$838$$ 30.9282 1.06840
$$839$$ 31.8564 1.09981 0.549903 0.835229i $$-0.314665\pi$$
0.549903 + 0.835229i $$0.314665\pi$$
$$840$$ 0 0
$$841$$ −6.60770 −0.227852
$$842$$ 35.8564 1.23569
$$843$$ 0 0
$$844$$ 26.9282 0.926907
$$845$$ −10.8564 −0.373472
$$846$$ 0 0
$$847$$ 1.00000 0.0343604
$$848$$ 7.26795 0.249582
$$849$$ 0 0
$$850$$ −3.46410 −0.118818
$$851$$ 6.00000 0.205677
$$852$$ 0 0
$$853$$ −54.7846 −1.87579 −0.937895 0.346920i $$-0.887227\pi$$
−0.937895 + 0.346920i $$0.887227\pi$$
$$854$$ 4.92820 0.168640
$$855$$ 0 0
$$856$$ 7.85641 0.268526
$$857$$ 27.4641 0.938156 0.469078 0.883157i $$-0.344586\pi$$
0.469078 + 0.883157i $$0.344586\pi$$
$$858$$ 0 0
$$859$$ −12.7846 −0.436205 −0.218103 0.975926i $$-0.569987\pi$$
−0.218103 + 0.975926i $$0.569987\pi$$
$$860$$ 2.00000 0.0681994
$$861$$ 0 0
$$862$$ 3.12436 0.106416
$$863$$ 7.51666 0.255870 0.127935 0.991783i $$-0.459165\pi$$
0.127935 + 0.991783i $$0.459165\pi$$
$$864$$ 0 0
$$865$$ −0.928203 −0.0315599
$$866$$ 18.1962 0.618330
$$867$$ 0 0
$$868$$ 2.00000 0.0678844
$$869$$ 12.1962 0.413726
$$870$$ 0 0
$$871$$ 5.85641 0.198437
$$872$$ 15.6603 0.530323
$$873$$ 0 0
$$874$$ −55.1769 −1.86639
$$875$$ 1.00000 0.0338062
$$876$$ 0 0
$$877$$ −21.3205 −0.719942 −0.359971 0.932963i $$-0.617213\pi$$
−0.359971 + 0.932963i $$0.617213\pi$$
$$878$$ −26.2487 −0.885851
$$879$$ 0 0
$$880$$ −1.00000 −0.0337100
$$881$$ −11.0718 −0.373018 −0.186509 0.982453i $$-0.559717\pi$$
−0.186509 + 0.982453i $$0.559717\pi$$
$$882$$ 0 0
$$883$$ −35.6077 −1.19829 −0.599147 0.800639i $$-0.704494\pi$$
−0.599147 + 0.800639i $$0.704494\pi$$
$$884$$ −5.07180 −0.170583
$$885$$ 0 0
$$886$$ 0 0
$$887$$ −13.8564 −0.465253 −0.232626 0.972566i $$-0.574732\pi$$
−0.232626 + 0.972566i $$0.574732\pi$$
$$888$$ 0 0
$$889$$ 1.07180 0.0359469
$$890$$ 3.46410 0.116117
$$891$$ 0 0
$$892$$ −25.4641 −0.852601
$$893$$ −46.6410 −1.56078
$$894$$ 0 0
$$895$$ 6.00000 0.200558
$$896$$ −1.00000 −0.0334077
$$897$$ 0 0
$$898$$ 40.3923 1.34791
$$899$$ 9.46410 0.315645
$$900$$ 0 0
$$901$$ 25.1769 0.838765
$$902$$ 2.19615 0.0731239
$$903$$ 0 0
$$904$$ 7.85641 0.261300
$$905$$ 14.0000 0.465376
$$906$$ 0 0
$$907$$ −31.0333 −1.03044 −0.515222 0.857057i $$-0.672291\pi$$
−0.515222 + 0.857057i $$0.672291\pi$$
$$908$$ −6.92820 −0.229920
$$909$$ 0 0
$$910$$ 1.46410 0.0485345
$$911$$ −9.46410 −0.313560 −0.156780 0.987634i $$-0.550111\pi$$
−0.156780 + 0.987634i $$0.550111\pi$$
$$912$$ 0 0
$$913$$ −16.3923 −0.542506
$$914$$ −2.00000 −0.0661541
$$915$$ 0 0
$$916$$ 24.3923 0.805944
$$917$$ −5.66025 −0.186918
$$918$$ 0 0
$$919$$ −25.3731 −0.836980 −0.418490 0.908221i $$-0.637441\pi$$
−0.418490 + 0.908221i $$0.637441\pi$$
$$920$$ −8.19615 −0.270219
$$921$$ 0 0
$$922$$ 22.3923 0.737451
$$923$$ 13.8564 0.456089
$$924$$ 0 0
$$925$$ 0.732051 0.0240697
$$926$$ −27.5167 −0.904254
$$927$$ 0 0
$$928$$ −4.73205 −0.155337
$$929$$ 25.6077 0.840161 0.420081 0.907487i $$-0.362002\pi$$
0.420081 + 0.907487i $$0.362002\pi$$
$$930$$ 0 0
$$931$$ 6.73205 0.220634
$$932$$ 7.85641 0.257345
$$933$$ 0 0
$$934$$ −34.0526 −1.11423
$$935$$ −3.46410 −0.113288
$$936$$ 0 0
$$937$$ −1.21539 −0.0397051 −0.0198525 0.999803i $$-0.506320\pi$$
−0.0198525 + 0.999803i $$0.506320\pi$$
$$938$$ 4.00000 0.130605
$$939$$ 0 0
$$940$$ −6.92820 −0.225973
$$941$$ 14.7846 0.481965 0.240982 0.970530i $$-0.422530\pi$$
0.240982 + 0.970530i $$0.422530\pi$$
$$942$$ 0 0
$$943$$ 18.0000 0.586161
$$944$$ −6.92820 −0.225494
$$945$$ 0 0
$$946$$ 2.00000 0.0650256
$$947$$ −47.3205 −1.53771 −0.768855 0.639423i $$-0.779173\pi$$
−0.768855 + 0.639423i $$0.779173\pi$$
$$948$$ 0 0
$$949$$ 21.0718 0.684019
$$950$$ −6.73205 −0.218417
$$951$$ 0 0
$$952$$ −3.46410 −0.112272
$$953$$ −26.5359 −0.859582 −0.429791 0.902928i $$-0.641413\pi$$
−0.429791 + 0.902928i $$0.641413\pi$$
$$954$$ 0 0
$$955$$ 12.0000 0.388311
$$956$$ −1.26795 −0.0410084
$$957$$ 0 0
$$958$$ −32.7846 −1.05922
$$959$$ 0.928203 0.0299732
$$960$$ 0 0
$$961$$ −27.0000 −0.870968
$$962$$ 1.07180 0.0345561
$$963$$ 0 0
$$964$$ 3.26795 0.105254
$$965$$ 12.3923 0.398922
$$966$$ 0 0
$$967$$ 26.9282 0.865953 0.432976 0.901405i $$-0.357463\pi$$
0.432976 + 0.901405i $$0.357463\pi$$
$$968$$ −1.00000 −0.0321412
$$969$$ 0 0
$$970$$ −14.5885 −0.468407
$$971$$ −42.9282 −1.37763 −0.688816 0.724936i $$-0.741869\pi$$
−0.688816 + 0.724936i $$0.741869\pi$$
$$972$$ 0 0
$$973$$ 13.6603 0.437928
$$974$$ −4.19615 −0.134453
$$975$$ 0 0
$$976$$ −4.92820 −0.157748
$$977$$ 33.7128 1.07857 0.539284 0.842124i $$-0.318695\pi$$
0.539284 + 0.842124i $$0.318695\pi$$
$$978$$ 0 0
$$979$$ 3.46410 0.110713
$$980$$ 1.00000 0.0319438
$$981$$ 0 0
$$982$$ −27.7128 −0.884351
$$983$$ 37.1769 1.18576 0.592880 0.805291i $$-0.297991\pi$$
0.592880 + 0.805291i $$0.297991\pi$$
$$984$$ 0 0
$$985$$ 24.2487 0.772628
$$986$$ −16.3923 −0.522037
$$987$$ 0 0
$$988$$ −9.85641 −0.313574
$$989$$ 16.3923 0.521245
$$990$$ 0 0
$$991$$ −4.00000 −0.127064 −0.0635321 0.997980i $$-0.520237\pi$$
−0.0635321 + 0.997980i $$0.520237\pi$$
$$992$$ −2.00000 −0.0635001
$$993$$ 0 0
$$994$$ 9.46410 0.300183
$$995$$ 2.92820 0.0928303
$$996$$ 0 0
$$997$$ 17.7128 0.560970 0.280485 0.959858i $$-0.409505\pi$$
0.280485 + 0.959858i $$0.409505\pi$$
$$998$$ −12.1436 −0.384399
$$999$$ 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 6930.2.a.bv.1.1 2
3.2 odd 2 770.2.a.j.1.2 2
12.11 even 2 6160.2.a.t.1.1 2
15.2 even 4 3850.2.c.x.1849.3 4
15.8 even 4 3850.2.c.x.1849.2 4
15.14 odd 2 3850.2.a.bd.1.1 2
21.20 even 2 5390.2.a.bs.1.1 2
33.32 even 2 8470.2.a.br.1.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
770.2.a.j.1.2 2 3.2 odd 2
3850.2.a.bd.1.1 2 15.14 odd 2
3850.2.c.x.1849.2 4 15.8 even 4
3850.2.c.x.1849.3 4 15.2 even 4
5390.2.a.bs.1.1 2 21.20 even 2
6160.2.a.t.1.1 2 12.11 even 2
6930.2.a.bv.1.1 2 1.1 even 1 trivial
8470.2.a.br.1.2 2 33.32 even 2