# Properties

 Label 3920.2.a.bq.1.1 Level $3920$ Weight $2$ Character 3920.1 Self dual yes Analytic conductor $31.301$ Analytic rank $1$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$3920 = 2^{4} \cdot 5 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3920.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$31.3013575923$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\zeta_{8})^+$$ Defining polynomial: $$x^{2} - 2$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 35) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$-1.41421$$ of defining polynomial Character $$\chi$$ $$=$$ 3920.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-2.41421 q^{3} +1.00000 q^{5} +2.82843 q^{9} +O(q^{10})$$ $$q-2.41421 q^{3} +1.00000 q^{5} +2.82843 q^{9} -4.82843 q^{11} +0.828427 q^{13} -2.41421 q^{15} -0.828427 q^{17} +2.82843 q^{19} +2.41421 q^{23} +1.00000 q^{25} +0.414214 q^{27} -1.00000 q^{29} +6.00000 q^{31} +11.6569 q^{33} -2.00000 q^{39} -2.17157 q^{41} -6.41421 q^{43} +2.82843 q^{45} -2.00000 q^{47} +2.00000 q^{51} -6.82843 q^{53} -4.82843 q^{55} -6.82843 q^{57} +12.4853 q^{59} -11.4853 q^{61} +0.828427 q^{65} -12.4142 q^{67} -5.82843 q^{69} +12.4853 q^{71} +4.82843 q^{73} -2.41421 q^{75} -9.17157 q^{79} -9.48528 q^{81} +11.7279 q^{83} -0.828427 q^{85} +2.41421 q^{87} +2.65685 q^{89} -14.4853 q^{93} +2.82843 q^{95} +0.343146 q^{97} -13.6569 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{3} + 2q^{5} + O(q^{10})$$ $$2q - 2q^{3} + 2q^{5} - 4q^{11} - 4q^{13} - 2q^{15} + 4q^{17} + 2q^{23} + 2q^{25} - 2q^{27} - 2q^{29} + 12q^{31} + 12q^{33} - 4q^{39} - 10q^{41} - 10q^{43} - 4q^{47} + 4q^{51} - 8q^{53} - 4q^{55} - 8q^{57} + 8q^{59} - 6q^{61} - 4q^{65} - 22q^{67} - 6q^{69} + 8q^{71} + 4q^{73} - 2q^{75} - 24q^{79} - 2q^{81} - 2q^{83} + 4q^{85} + 2q^{87} - 6q^{89} - 12q^{93} + 12q^{97} - 16q^{99} + O(q^{100})$$

## 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
$$3$$ −2.41421 −1.39385 −0.696923 0.717146i $$-0.745448\pi$$
−0.696923 + 0.717146i $$0.745448\pi$$
$$4$$ 0 0
$$5$$ 1.00000 0.447214
$$6$$ 0 0
$$7$$ 0 0
$$8$$ 0 0
$$9$$ 2.82843 0.942809
$$10$$ 0 0
$$11$$ −4.82843 −1.45583 −0.727913 0.685670i $$-0.759509\pi$$
−0.727913 + 0.685670i $$0.759509\pi$$
$$12$$ 0 0
$$13$$ 0.828427 0.229764 0.114882 0.993379i $$-0.463351\pi$$
0.114882 + 0.993379i $$0.463351\pi$$
$$14$$ 0 0
$$15$$ −2.41421 −0.623347
$$16$$ 0 0
$$17$$ −0.828427 −0.200923 −0.100462 0.994941i $$-0.532032\pi$$
−0.100462 + 0.994941i $$0.532032\pi$$
$$18$$ 0 0
$$19$$ 2.82843 0.648886 0.324443 0.945905i $$-0.394823\pi$$
0.324443 + 0.945905i $$0.394823\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ 2.41421 0.503398 0.251699 0.967806i $$-0.419011\pi$$
0.251699 + 0.967806i $$0.419011\pi$$
$$24$$ 0 0
$$25$$ 1.00000 0.200000
$$26$$ 0 0
$$27$$ 0.414214 0.0797154
$$28$$ 0 0
$$29$$ −1.00000 −0.185695 −0.0928477 0.995680i $$-0.529597\pi$$
−0.0928477 + 0.995680i $$0.529597\pi$$
$$30$$ 0 0
$$31$$ 6.00000 1.07763 0.538816 0.842424i $$-0.318872\pi$$
0.538816 + 0.842424i $$0.318872\pi$$
$$32$$ 0 0
$$33$$ 11.6569 2.02920
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$38$$ 0 0
$$39$$ −2.00000 −0.320256
$$40$$ 0 0
$$41$$ −2.17157 −0.339143 −0.169571 0.985518i $$-0.554238\pi$$
−0.169571 + 0.985518i $$0.554238\pi$$
$$42$$ 0 0
$$43$$ −6.41421 −0.978158 −0.489079 0.872239i $$-0.662667\pi$$
−0.489079 + 0.872239i $$0.662667\pi$$
$$44$$ 0 0
$$45$$ 2.82843 0.421637
$$46$$ 0 0
$$47$$ −2.00000 −0.291730 −0.145865 0.989305i $$-0.546597\pi$$
−0.145865 + 0.989305i $$0.546597\pi$$
$$48$$ 0 0
$$49$$ 0 0
$$50$$ 0 0
$$51$$ 2.00000 0.280056
$$52$$ 0 0
$$53$$ −6.82843 −0.937957 −0.468978 0.883210i $$-0.655378\pi$$
−0.468978 + 0.883210i $$0.655378\pi$$
$$54$$ 0 0
$$55$$ −4.82843 −0.651065
$$56$$ 0 0
$$57$$ −6.82843 −0.904447
$$58$$ 0 0
$$59$$ 12.4853 1.62545 0.812723 0.582651i $$-0.197984\pi$$
0.812723 + 0.582651i $$0.197984\pi$$
$$60$$ 0 0
$$61$$ −11.4853 −1.47054 −0.735270 0.677775i $$-0.762945\pi$$
−0.735270 + 0.677775i $$0.762945\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ 0.828427 0.102754
$$66$$ 0 0
$$67$$ −12.4142 −1.51664 −0.758319 0.651884i $$-0.773979\pi$$
−0.758319 + 0.651884i $$0.773979\pi$$
$$68$$ 0 0
$$69$$ −5.82843 −0.701660
$$70$$ 0 0
$$71$$ 12.4853 1.48173 0.740865 0.671654i $$-0.234416\pi$$
0.740865 + 0.671654i $$0.234416\pi$$
$$72$$ 0 0
$$73$$ 4.82843 0.565125 0.282562 0.959249i $$-0.408816\pi$$
0.282562 + 0.959249i $$0.408816\pi$$
$$74$$ 0 0
$$75$$ −2.41421 −0.278769
$$76$$ 0 0
$$77$$ 0 0
$$78$$ 0 0
$$79$$ −9.17157 −1.03188 −0.515941 0.856624i $$-0.672558\pi$$
−0.515941 + 0.856624i $$0.672558\pi$$
$$80$$ 0 0
$$81$$ −9.48528 −1.05392
$$82$$ 0 0
$$83$$ 11.7279 1.28731 0.643653 0.765317i $$-0.277418\pi$$
0.643653 + 0.765317i $$0.277418\pi$$
$$84$$ 0 0
$$85$$ −0.828427 −0.0898555
$$86$$ 0 0
$$87$$ 2.41421 0.258831
$$88$$ 0 0
$$89$$ 2.65685 0.281626 0.140813 0.990036i $$-0.455028\pi$$
0.140813 + 0.990036i $$0.455028\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 0 0
$$93$$ −14.4853 −1.50205
$$94$$ 0 0
$$95$$ 2.82843 0.290191
$$96$$ 0 0
$$97$$ 0.343146 0.0348412 0.0174206 0.999848i $$-0.494455\pi$$
0.0174206 + 0.999848i $$0.494455\pi$$
$$98$$ 0 0
$$99$$ −13.6569 −1.37257
$$100$$ 0 0
$$101$$ 12.3137 1.22526 0.612630 0.790370i $$-0.290112\pi$$
0.612630 + 0.790370i $$0.290112\pi$$
$$102$$ 0 0
$$103$$ −0.414214 −0.0408137 −0.0204068 0.999792i $$-0.506496\pi$$
−0.0204068 + 0.999792i $$0.506496\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ −2.75736 −0.266564 −0.133282 0.991078i $$-0.542552\pi$$
−0.133282 + 0.991078i $$0.542552\pi$$
$$108$$ 0 0
$$109$$ 3.48528 0.333829 0.166915 0.985971i $$-0.446620\pi$$
0.166915 + 0.985971i $$0.446620\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 0 0
$$113$$ 12.4853 1.17452 0.587258 0.809400i $$-0.300207\pi$$
0.587258 + 0.809400i $$0.300207\pi$$
$$114$$ 0 0
$$115$$ 2.41421 0.225127
$$116$$ 0 0
$$117$$ 2.34315 0.216624
$$118$$ 0 0
$$119$$ 0 0
$$120$$ 0 0
$$121$$ 12.3137 1.11943
$$122$$ 0 0
$$123$$ 5.24264 0.472713
$$124$$ 0 0
$$125$$ 1.00000 0.0894427
$$126$$ 0 0
$$127$$ −13.3137 −1.18140 −0.590700 0.806891i $$-0.701148\pi$$
−0.590700 + 0.806891i $$0.701148\pi$$
$$128$$ 0 0
$$129$$ 15.4853 1.36340
$$130$$ 0 0
$$131$$ −3.31371 −0.289520 −0.144760 0.989467i $$-0.546241\pi$$
−0.144760 + 0.989467i $$0.546241\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ 0 0
$$135$$ 0.414214 0.0356498
$$136$$ 0 0
$$137$$ 1.65685 0.141555 0.0707773 0.997492i $$-0.477452\pi$$
0.0707773 + 0.997492i $$0.477452\pi$$
$$138$$ 0 0
$$139$$ −12.1421 −1.02988 −0.514941 0.857225i $$-0.672186\pi$$
−0.514941 + 0.857225i $$0.672186\pi$$
$$140$$ 0 0
$$141$$ 4.82843 0.406627
$$142$$ 0 0
$$143$$ −4.00000 −0.334497
$$144$$ 0 0
$$145$$ −1.00000 −0.0830455
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ −7.82843 −0.641330 −0.320665 0.947193i $$-0.603906\pi$$
−0.320665 + 0.947193i $$0.603906\pi$$
$$150$$ 0 0
$$151$$ −0.343146 −0.0279248 −0.0139624 0.999903i $$-0.504445\pi$$
−0.0139624 + 0.999903i $$0.504445\pi$$
$$152$$ 0 0
$$153$$ −2.34315 −0.189432
$$154$$ 0 0
$$155$$ 6.00000 0.481932
$$156$$ 0 0
$$157$$ −5.31371 −0.424080 −0.212040 0.977261i $$-0.568011\pi$$
−0.212040 + 0.977261i $$0.568011\pi$$
$$158$$ 0 0
$$159$$ 16.4853 1.30737
$$160$$ 0 0
$$161$$ 0 0
$$162$$ 0 0
$$163$$ −23.6569 −1.85295 −0.926474 0.376359i $$-0.877176\pi$$
−0.926474 + 0.376359i $$0.877176\pi$$
$$164$$ 0 0
$$165$$ 11.6569 0.907485
$$166$$ 0 0
$$167$$ −19.5858 −1.51559 −0.757797 0.652491i $$-0.773724\pi$$
−0.757797 + 0.652491i $$0.773724\pi$$
$$168$$ 0 0
$$169$$ −12.3137 −0.947208
$$170$$ 0 0
$$171$$ 8.00000 0.611775
$$172$$ 0 0
$$173$$ −19.3137 −1.46839 −0.734197 0.678936i $$-0.762441\pi$$
−0.734197 + 0.678936i $$0.762441\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ −30.1421 −2.26562
$$178$$ 0 0
$$179$$ 10.0000 0.747435 0.373718 0.927543i $$-0.378083\pi$$
0.373718 + 0.927543i $$0.378083\pi$$
$$180$$ 0 0
$$181$$ −8.65685 −0.643459 −0.321729 0.946832i $$-0.604264\pi$$
−0.321729 + 0.946832i $$0.604264\pi$$
$$182$$ 0 0
$$183$$ 27.7279 2.04971
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 0 0
$$187$$ 4.00000 0.292509
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 7.17157 0.518917 0.259458 0.965754i $$-0.416456\pi$$
0.259458 + 0.965754i $$0.416456\pi$$
$$192$$ 0 0
$$193$$ −2.00000 −0.143963 −0.0719816 0.997406i $$-0.522932\pi$$
−0.0719816 + 0.997406i $$0.522932\pi$$
$$194$$ 0 0
$$195$$ −2.00000 −0.143223
$$196$$ 0 0
$$197$$ −23.6569 −1.68548 −0.842741 0.538320i $$-0.819059\pi$$
−0.842741 + 0.538320i $$0.819059\pi$$
$$198$$ 0 0
$$199$$ −1.65685 −0.117451 −0.0587256 0.998274i $$-0.518704\pi$$
−0.0587256 + 0.998274i $$0.518704\pi$$
$$200$$ 0 0
$$201$$ 29.9706 2.11396
$$202$$ 0 0
$$203$$ 0 0
$$204$$ 0 0
$$205$$ −2.17157 −0.151669
$$206$$ 0 0
$$207$$ 6.82843 0.474608
$$208$$ 0 0
$$209$$ −13.6569 −0.944664
$$210$$ 0 0
$$211$$ −3.51472 −0.241963 −0.120982 0.992655i $$-0.538604\pi$$
−0.120982 + 0.992655i $$0.538604\pi$$
$$212$$ 0 0
$$213$$ −30.1421 −2.06531
$$214$$ 0 0
$$215$$ −6.41421 −0.437446
$$216$$ 0 0
$$217$$ 0 0
$$218$$ 0 0
$$219$$ −11.6569 −0.787697
$$220$$ 0 0
$$221$$ −0.686292 −0.0461650
$$222$$ 0 0
$$223$$ −11.6569 −0.780601 −0.390300 0.920688i $$-0.627629\pi$$
−0.390300 + 0.920688i $$0.627629\pi$$
$$224$$ 0 0
$$225$$ 2.82843 0.188562
$$226$$ 0 0
$$227$$ −26.9706 −1.79010 −0.895050 0.445967i $$-0.852860\pi$$
−0.895050 + 0.445967i $$0.852860\pi$$
$$228$$ 0 0
$$229$$ −0.343146 −0.0226757 −0.0113379 0.999936i $$-0.503609\pi$$
−0.0113379 + 0.999936i $$0.503609\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ −11.1716 −0.731874 −0.365937 0.930640i $$-0.619251\pi$$
−0.365937 + 0.930640i $$0.619251\pi$$
$$234$$ 0 0
$$235$$ −2.00000 −0.130466
$$236$$ 0 0
$$237$$ 22.1421 1.43829
$$238$$ 0 0
$$239$$ −1.31371 −0.0849767 −0.0424884 0.999097i $$-0.513529\pi$$
−0.0424884 + 0.999097i $$0.513529\pi$$
$$240$$ 0 0
$$241$$ 16.3431 1.05275 0.526377 0.850251i $$-0.323550\pi$$
0.526377 + 0.850251i $$0.323550\pi$$
$$242$$ 0 0
$$243$$ 21.6569 1.38929
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 2.34315 0.149091
$$248$$ 0 0
$$249$$ −28.3137 −1.79431
$$250$$ 0 0
$$251$$ −13.3137 −0.840354 −0.420177 0.907442i $$-0.638032\pi$$
−0.420177 + 0.907442i $$0.638032\pi$$
$$252$$ 0 0
$$253$$ −11.6569 −0.732860
$$254$$ 0 0
$$255$$ 2.00000 0.125245
$$256$$ 0 0
$$257$$ 17.6569 1.10140 0.550702 0.834702i $$-0.314360\pi$$
0.550702 + 0.834702i $$0.314360\pi$$
$$258$$ 0 0
$$259$$ 0 0
$$260$$ 0 0
$$261$$ −2.82843 −0.175075
$$262$$ 0 0
$$263$$ −19.0416 −1.17416 −0.587079 0.809530i $$-0.699722\pi$$
−0.587079 + 0.809530i $$0.699722\pi$$
$$264$$ 0 0
$$265$$ −6.82843 −0.419467
$$266$$ 0 0
$$267$$ −6.41421 −0.392543
$$268$$ 0 0
$$269$$ −30.4558 −1.85693 −0.928463 0.371425i $$-0.878869\pi$$
−0.928463 + 0.371425i $$0.878869\pi$$
$$270$$ 0 0
$$271$$ 0.485281 0.0294787 0.0147394 0.999891i $$-0.495308\pi$$
0.0147394 + 0.999891i $$0.495308\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ −4.82843 −0.291165
$$276$$ 0 0
$$277$$ −12.1421 −0.729550 −0.364775 0.931096i $$-0.618854\pi$$
−0.364775 + 0.931096i $$0.618854\pi$$
$$278$$ 0 0
$$279$$ 16.9706 1.01600
$$280$$ 0 0
$$281$$ 26.2843 1.56799 0.783994 0.620768i $$-0.213179\pi$$
0.783994 + 0.620768i $$0.213179\pi$$
$$282$$ 0 0
$$283$$ −14.0000 −0.832214 −0.416107 0.909316i $$-0.636606\pi$$
−0.416107 + 0.909316i $$0.636606\pi$$
$$284$$ 0 0
$$285$$ −6.82843 −0.404481
$$286$$ 0 0
$$287$$ 0 0
$$288$$ 0 0
$$289$$ −16.3137 −0.959630
$$290$$ 0 0
$$291$$ −0.828427 −0.0485633
$$292$$ 0 0
$$293$$ −16.0000 −0.934730 −0.467365 0.884064i $$-0.654797\pi$$
−0.467365 + 0.884064i $$0.654797\pi$$
$$294$$ 0 0
$$295$$ 12.4853 0.726921
$$296$$ 0 0
$$297$$ −2.00000 −0.116052
$$298$$ 0 0
$$299$$ 2.00000 0.115663
$$300$$ 0 0
$$301$$ 0 0
$$302$$ 0 0
$$303$$ −29.7279 −1.70782
$$304$$ 0 0
$$305$$ −11.4853 −0.657645
$$306$$ 0 0
$$307$$ 13.2426 0.755797 0.377899 0.925847i $$-0.376647\pi$$
0.377899 + 0.925847i $$0.376647\pi$$
$$308$$ 0 0
$$309$$ 1.00000 0.0568880
$$310$$ 0 0
$$311$$ 18.8284 1.06766 0.533831 0.845591i $$-0.320752\pi$$
0.533831 + 0.845591i $$0.320752\pi$$
$$312$$ 0 0
$$313$$ −17.6569 −0.998024 −0.499012 0.866595i $$-0.666304\pi$$
−0.499012 + 0.866595i $$0.666304\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ 25.7990 1.44902 0.724508 0.689267i $$-0.242067\pi$$
0.724508 + 0.689267i $$0.242067\pi$$
$$318$$ 0 0
$$319$$ 4.82843 0.270340
$$320$$ 0 0
$$321$$ 6.65685 0.371549
$$322$$ 0 0
$$323$$ −2.34315 −0.130376
$$324$$ 0 0
$$325$$ 0.828427 0.0459529
$$326$$ 0 0
$$327$$ −8.41421 −0.465307
$$328$$ 0 0
$$329$$ 0 0
$$330$$ 0 0
$$331$$ 10.9706 0.602997 0.301498 0.953467i $$-0.402513\pi$$
0.301498 + 0.953467i $$0.402513\pi$$
$$332$$ 0 0
$$333$$ 0 0
$$334$$ 0 0
$$335$$ −12.4142 −0.678261
$$336$$ 0 0
$$337$$ 14.8284 0.807756 0.403878 0.914813i $$-0.367662\pi$$
0.403878 + 0.914813i $$0.367662\pi$$
$$338$$ 0 0
$$339$$ −30.1421 −1.63710
$$340$$ 0 0
$$341$$ −28.9706 −1.56884
$$342$$ 0 0
$$343$$ 0 0
$$344$$ 0 0
$$345$$ −5.82843 −0.313792
$$346$$ 0 0
$$347$$ −22.0711 −1.18484 −0.592418 0.805630i $$-0.701827\pi$$
−0.592418 + 0.805630i $$0.701827\pi$$
$$348$$ 0 0
$$349$$ −26.6569 −1.42691 −0.713454 0.700702i $$-0.752870\pi$$
−0.713454 + 0.700702i $$0.752870\pi$$
$$350$$ 0 0
$$351$$ 0.343146 0.0183158
$$352$$ 0 0
$$353$$ −21.1716 −1.12685 −0.563425 0.826168i $$-0.690516\pi$$
−0.563425 + 0.826168i $$0.690516\pi$$
$$354$$ 0 0
$$355$$ 12.4853 0.662650
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ 10.0000 0.527780 0.263890 0.964553i $$-0.414994\pi$$
0.263890 + 0.964553i $$0.414994\pi$$
$$360$$ 0 0
$$361$$ −11.0000 −0.578947
$$362$$ 0 0
$$363$$ −29.7279 −1.56031
$$364$$ 0 0
$$365$$ 4.82843 0.252731
$$366$$ 0 0
$$367$$ 11.2426 0.586861 0.293431 0.955980i $$-0.405203\pi$$
0.293431 + 0.955980i $$0.405203\pi$$
$$368$$ 0 0
$$369$$ −6.14214 −0.319747
$$370$$ 0 0
$$371$$ 0 0
$$372$$ 0 0
$$373$$ 12.9706 0.671590 0.335795 0.941935i $$-0.390995\pi$$
0.335795 + 0.941935i $$0.390995\pi$$
$$374$$ 0 0
$$375$$ −2.41421 −0.124669
$$376$$ 0 0
$$377$$ −0.828427 −0.0426662
$$378$$ 0 0
$$379$$ −21.1716 −1.08751 −0.543755 0.839244i $$-0.682998\pi$$
−0.543755 + 0.839244i $$0.682998\pi$$
$$380$$ 0 0
$$381$$ 32.1421 1.64669
$$382$$ 0 0
$$383$$ −16.8995 −0.863524 −0.431762 0.901988i $$-0.642108\pi$$
−0.431762 + 0.901988i $$0.642108\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ −18.1421 −0.922217
$$388$$ 0 0
$$389$$ 12.3431 0.625822 0.312911 0.949782i $$-0.398696\pi$$
0.312911 + 0.949782i $$0.398696\pi$$
$$390$$ 0 0
$$391$$ −2.00000 −0.101144
$$392$$ 0 0
$$393$$ 8.00000 0.403547
$$394$$ 0 0
$$395$$ −9.17157 −0.461472
$$396$$ 0 0
$$397$$ 28.6274 1.43677 0.718384 0.695646i $$-0.244882\pi$$
0.718384 + 0.695646i $$0.244882\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 7.68629 0.383835 0.191918 0.981411i $$-0.438529\pi$$
0.191918 + 0.981411i $$0.438529\pi$$
$$402$$ 0 0
$$403$$ 4.97056 0.247601
$$404$$ 0 0
$$405$$ −9.48528 −0.471327
$$406$$ 0 0
$$407$$ 0 0
$$408$$ 0 0
$$409$$ 24.7990 1.22623 0.613116 0.789993i $$-0.289916\pi$$
0.613116 + 0.789993i $$0.289916\pi$$
$$410$$ 0 0
$$411$$ −4.00000 −0.197305
$$412$$ 0 0
$$413$$ 0 0
$$414$$ 0 0
$$415$$ 11.7279 0.575701
$$416$$ 0 0
$$417$$ 29.3137 1.43550
$$418$$ 0 0
$$419$$ 23.3137 1.13895 0.569475 0.822009i $$-0.307147\pi$$
0.569475 + 0.822009i $$0.307147\pi$$
$$420$$ 0 0
$$421$$ −3.48528 −0.169862 −0.0849311 0.996387i $$-0.527067\pi$$
−0.0849311 + 0.996387i $$0.527067\pi$$
$$422$$ 0 0
$$423$$ −5.65685 −0.275046
$$424$$ 0 0
$$425$$ −0.828427 −0.0401846
$$426$$ 0 0
$$427$$ 0 0
$$428$$ 0 0
$$429$$ 9.65685 0.466237
$$430$$ 0 0
$$431$$ 21.7990 1.05002 0.525010 0.851096i $$-0.324062\pi$$
0.525010 + 0.851096i $$0.324062\pi$$
$$432$$ 0 0
$$433$$ −31.7990 −1.52816 −0.764081 0.645120i $$-0.776807\pi$$
−0.764081 + 0.645120i $$0.776807\pi$$
$$434$$ 0 0
$$435$$ 2.41421 0.115753
$$436$$ 0 0
$$437$$ 6.82843 0.326648
$$438$$ 0 0
$$439$$ −33.9411 −1.61992 −0.809961 0.586484i $$-0.800512\pi$$
−0.809961 + 0.586484i $$0.800512\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ 12.2132 0.580267 0.290133 0.956986i $$-0.406300\pi$$
0.290133 + 0.956986i $$0.406300\pi$$
$$444$$ 0 0
$$445$$ 2.65685 0.125947
$$446$$ 0 0
$$447$$ 18.8995 0.893915
$$448$$ 0 0
$$449$$ −1.82843 −0.0862888 −0.0431444 0.999069i $$-0.513738\pi$$
−0.0431444 + 0.999069i $$0.513738\pi$$
$$450$$ 0 0
$$451$$ 10.4853 0.493733
$$452$$ 0 0
$$453$$ 0.828427 0.0389229
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ −32.2843 −1.51019 −0.755097 0.655613i $$-0.772410\pi$$
−0.755097 + 0.655613i $$0.772410\pi$$
$$458$$ 0 0
$$459$$ −0.343146 −0.0160167
$$460$$ 0 0
$$461$$ 18.6863 0.870307 0.435154 0.900356i $$-0.356694\pi$$
0.435154 + 0.900356i $$0.356694\pi$$
$$462$$ 0 0
$$463$$ −11.0416 −0.513148 −0.256574 0.966525i $$-0.582594\pi$$
−0.256574 + 0.966525i $$0.582594\pi$$
$$464$$ 0 0
$$465$$ −14.4853 −0.671739
$$466$$ 0 0
$$467$$ 22.8995 1.05966 0.529831 0.848103i $$-0.322255\pi$$
0.529831 + 0.848103i $$0.322255\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ 0 0
$$471$$ 12.8284 0.591103
$$472$$ 0 0
$$473$$ 30.9706 1.42403
$$474$$ 0 0
$$475$$ 2.82843 0.129777
$$476$$ 0 0
$$477$$ −19.3137 −0.884314
$$478$$ 0 0
$$479$$ 24.3431 1.11227 0.556133 0.831093i $$-0.312284\pi$$
0.556133 + 0.831093i $$0.312284\pi$$
$$480$$ 0 0
$$481$$ 0 0
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ 0.343146 0.0155814
$$486$$ 0 0
$$487$$ −15.6569 −0.709480 −0.354740 0.934965i $$-0.615431\pi$$
−0.354740 + 0.934965i $$0.615431\pi$$
$$488$$ 0 0
$$489$$ 57.1127 2.58273
$$490$$ 0 0
$$491$$ 13.3137 0.600839 0.300420 0.953807i $$-0.402873\pi$$
0.300420 + 0.953807i $$0.402873\pi$$
$$492$$ 0 0
$$493$$ 0.828427 0.0373105
$$494$$ 0 0
$$495$$ −13.6569 −0.613830
$$496$$ 0 0
$$497$$ 0 0
$$498$$ 0 0
$$499$$ −4.82843 −0.216150 −0.108075 0.994143i $$-0.534469\pi$$
−0.108075 + 0.994143i $$0.534469\pi$$
$$500$$ 0 0
$$501$$ 47.2843 2.11251
$$502$$ 0 0
$$503$$ −37.8701 −1.68854 −0.844271 0.535916i $$-0.819966\pi$$
−0.844271 + 0.535916i $$0.819966\pi$$
$$504$$ 0 0
$$505$$ 12.3137 0.547953
$$506$$ 0 0
$$507$$ 29.7279 1.32026
$$508$$ 0 0
$$509$$ 24.6569 1.09290 0.546448 0.837493i $$-0.315980\pi$$
0.546448 + 0.837493i $$0.315980\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ 0 0
$$513$$ 1.17157 0.0517262
$$514$$ 0 0
$$515$$ −0.414214 −0.0182524
$$516$$ 0 0
$$517$$ 9.65685 0.424708
$$518$$ 0 0
$$519$$ 46.6274 2.04672
$$520$$ 0 0
$$521$$ −18.9706 −0.831115 −0.415558 0.909567i $$-0.636414\pi$$
−0.415558 + 0.909567i $$0.636414\pi$$
$$522$$ 0 0
$$523$$ −24.3431 −1.06445 −0.532226 0.846602i $$-0.678644\pi$$
−0.532226 + 0.846602i $$0.678644\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ −4.97056 −0.216521
$$528$$ 0 0
$$529$$ −17.1716 −0.746590
$$530$$ 0 0
$$531$$ 35.3137 1.53248
$$532$$ 0 0
$$533$$ −1.79899 −0.0779229
$$534$$ 0 0
$$535$$ −2.75736 −0.119211
$$536$$ 0 0
$$537$$ −24.1421 −1.04181
$$538$$ 0 0
$$539$$ 0 0
$$540$$ 0 0
$$541$$ 18.6569 0.802121 0.401060 0.916052i $$-0.368642\pi$$
0.401060 + 0.916052i $$0.368642\pi$$
$$542$$ 0 0
$$543$$ 20.8995 0.896883
$$544$$ 0 0
$$545$$ 3.48528 0.149293
$$546$$ 0 0
$$547$$ −5.10051 −0.218082 −0.109041 0.994037i $$-0.534778\pi$$
−0.109041 + 0.994037i $$0.534778\pi$$
$$548$$ 0 0
$$549$$ −32.4853 −1.38644
$$550$$ 0 0
$$551$$ −2.82843 −0.120495
$$552$$ 0 0
$$553$$ 0 0
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ −34.2843 −1.45267 −0.726336 0.687340i $$-0.758778\pi$$
−0.726336 + 0.687340i $$0.758778\pi$$
$$558$$ 0 0
$$559$$ −5.31371 −0.224746
$$560$$ 0 0
$$561$$ −9.65685 −0.407713
$$562$$ 0 0
$$563$$ −16.2721 −0.685786 −0.342893 0.939374i $$-0.611407\pi$$
−0.342893 + 0.939374i $$0.611407\pi$$
$$564$$ 0 0
$$565$$ 12.4853 0.525260
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ −3.65685 −0.153303 −0.0766517 0.997058i $$-0.524423\pi$$
−0.0766517 + 0.997058i $$0.524423\pi$$
$$570$$ 0 0
$$571$$ −14.8284 −0.620550 −0.310275 0.950647i $$-0.600421\pi$$
−0.310275 + 0.950647i $$0.600421\pi$$
$$572$$ 0 0
$$573$$ −17.3137 −0.723291
$$574$$ 0 0
$$575$$ 2.41421 0.100680
$$576$$ 0 0
$$577$$ 23.9411 0.996682 0.498341 0.866981i $$-0.333943\pi$$
0.498341 + 0.866981i $$0.333943\pi$$
$$578$$ 0 0
$$579$$ 4.82843 0.200663
$$580$$ 0 0
$$581$$ 0 0
$$582$$ 0 0
$$583$$ 32.9706 1.36550
$$584$$ 0 0
$$585$$ 2.34315 0.0968772
$$586$$ 0 0
$$587$$ −22.2843 −0.919770 −0.459885 0.887978i $$-0.652109\pi$$
−0.459885 + 0.887978i $$0.652109\pi$$
$$588$$ 0 0
$$589$$ 16.9706 0.699260
$$590$$ 0 0
$$591$$ 57.1127 2.34930
$$592$$ 0 0
$$593$$ 43.7990 1.79861 0.899304 0.437323i $$-0.144073\pi$$
0.899304 + 0.437323i $$0.144073\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ 4.00000 0.163709
$$598$$ 0 0
$$599$$ −17.6569 −0.721440 −0.360720 0.932674i $$-0.617469\pi$$
−0.360720 + 0.932674i $$0.617469\pi$$
$$600$$ 0 0
$$601$$ 8.34315 0.340324 0.170162 0.985416i $$-0.445571\pi$$
0.170162 + 0.985416i $$0.445571\pi$$
$$602$$ 0 0
$$603$$ −35.1127 −1.42990
$$604$$ 0 0
$$605$$ 12.3137 0.500623
$$606$$ 0 0
$$607$$ 4.21320 0.171009 0.0855043 0.996338i $$-0.472750\pi$$
0.0855043 + 0.996338i $$0.472750\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ −1.65685 −0.0670291
$$612$$ 0 0
$$613$$ −15.4558 −0.624256 −0.312128 0.950040i $$-0.601042\pi$$
−0.312128 + 0.950040i $$0.601042\pi$$
$$614$$ 0 0
$$615$$ 5.24264 0.211404
$$616$$ 0 0
$$617$$ −11.3137 −0.455473 −0.227736 0.973723i $$-0.573132\pi$$
−0.227736 + 0.973723i $$0.573132\pi$$
$$618$$ 0 0
$$619$$ −42.4853 −1.70763 −0.853814 0.520578i $$-0.825716\pi$$
−0.853814 + 0.520578i $$0.825716\pi$$
$$620$$ 0 0
$$621$$ 1.00000 0.0401286
$$622$$ 0 0
$$623$$ 0 0
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ 0 0
$$627$$ 32.9706 1.31672
$$628$$ 0 0
$$629$$ 0 0
$$630$$ 0 0
$$631$$ −8.14214 −0.324133 −0.162067 0.986780i $$-0.551816\pi$$
−0.162067 + 0.986780i $$0.551816\pi$$
$$632$$ 0 0
$$633$$ 8.48528 0.337260
$$634$$ 0 0
$$635$$ −13.3137 −0.528338
$$636$$ 0 0
$$637$$ 0 0
$$638$$ 0 0
$$639$$ 35.3137 1.39699
$$640$$ 0 0
$$641$$ 14.5147 0.573297 0.286648 0.958036i $$-0.407459\pi$$
0.286648 + 0.958036i $$0.407459\pi$$
$$642$$ 0 0
$$643$$ 30.2843 1.19430 0.597148 0.802131i $$-0.296301\pi$$
0.597148 + 0.802131i $$0.296301\pi$$
$$644$$ 0 0
$$645$$ 15.4853 0.609732
$$646$$ 0 0
$$647$$ −17.0416 −0.669976 −0.334988 0.942222i $$-0.608732\pi$$
−0.334988 + 0.942222i $$0.608732\pi$$
$$648$$ 0 0
$$649$$ −60.2843 −2.36636
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ −24.8284 −0.971611 −0.485806 0.874067i $$-0.661474\pi$$
−0.485806 + 0.874067i $$0.661474\pi$$
$$654$$ 0 0
$$655$$ −3.31371 −0.129477
$$656$$ 0 0
$$657$$ 13.6569 0.532805
$$658$$ 0 0
$$659$$ −26.8284 −1.04509 −0.522544 0.852613i $$-0.675017\pi$$
−0.522544 + 0.852613i $$0.675017\pi$$
$$660$$ 0 0
$$661$$ 26.1716 1.01796 0.508978 0.860779i $$-0.330023\pi$$
0.508978 + 0.860779i $$0.330023\pi$$
$$662$$ 0 0
$$663$$ 1.65685 0.0643469
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ −2.41421 −0.0934787
$$668$$ 0 0
$$669$$ 28.1421 1.08804
$$670$$ 0 0
$$671$$ 55.4558 2.14085
$$672$$ 0 0
$$673$$ 18.3431 0.707076 0.353538 0.935420i $$-0.384978\pi$$
0.353538 + 0.935420i $$0.384978\pi$$
$$674$$ 0 0
$$675$$ 0.414214 0.0159431
$$676$$ 0 0
$$677$$ 0.142136 0.00546272 0.00273136 0.999996i $$-0.499131\pi$$
0.00273136 + 0.999996i $$0.499131\pi$$
$$678$$ 0 0
$$679$$ 0 0
$$680$$ 0 0
$$681$$ 65.1127 2.49512
$$682$$ 0 0
$$683$$ 43.2426 1.65463 0.827317 0.561736i $$-0.189866\pi$$
0.827317 + 0.561736i $$0.189866\pi$$
$$684$$ 0 0
$$685$$ 1.65685 0.0633051
$$686$$ 0 0
$$687$$ 0.828427 0.0316065
$$688$$ 0 0
$$689$$ −5.65685 −0.215509
$$690$$ 0 0
$$691$$ 4.82843 0.183682 0.0918410 0.995774i $$-0.470725\pi$$
0.0918410 + 0.995774i $$0.470725\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ −12.1421 −0.460577
$$696$$ 0 0
$$697$$ 1.79899 0.0681416
$$698$$ 0 0
$$699$$ 26.9706 1.02012
$$700$$ 0 0
$$701$$ −42.7990 −1.61650 −0.808248 0.588843i $$-0.799584\pi$$
−0.808248 + 0.588843i $$0.799584\pi$$
$$702$$ 0 0
$$703$$ 0 0
$$704$$ 0 0
$$705$$ 4.82843 0.181849
$$706$$ 0 0
$$707$$ 0 0
$$708$$ 0 0
$$709$$ 38.3137 1.43890 0.719451 0.694543i $$-0.244394\pi$$
0.719451 + 0.694543i $$0.244394\pi$$
$$710$$ 0 0
$$711$$ −25.9411 −0.972868
$$712$$ 0 0
$$713$$ 14.4853 0.542478
$$714$$ 0 0
$$715$$ −4.00000 −0.149592
$$716$$ 0 0
$$717$$ 3.17157 0.118445
$$718$$ 0 0
$$719$$ 41.1127 1.53324 0.766622 0.642098i $$-0.221936\pi$$
0.766622 + 0.642098i $$0.221936\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ 0 0
$$723$$ −39.4558 −1.46738
$$724$$ 0 0
$$725$$ −1.00000 −0.0371391
$$726$$ 0 0
$$727$$ 40.4142 1.49888 0.749440 0.662072i $$-0.230323\pi$$
0.749440 + 0.662072i $$0.230323\pi$$
$$728$$ 0 0
$$729$$ −23.8284 −0.882534
$$730$$ 0 0
$$731$$ 5.31371 0.196535
$$732$$ 0 0
$$733$$ −22.0000 −0.812589 −0.406294 0.913742i $$-0.633179\pi$$
−0.406294 + 0.913742i $$0.633179\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 59.9411 2.20796
$$738$$ 0 0
$$739$$ −41.1127 −1.51236 −0.756178 0.654367i $$-0.772935\pi$$
−0.756178 + 0.654367i $$0.772935\pi$$
$$740$$ 0 0
$$741$$ −5.65685 −0.207810
$$742$$ 0 0
$$743$$ 1.92893 0.0707657 0.0353828 0.999374i $$-0.488735\pi$$
0.0353828 + 0.999374i $$0.488735\pi$$
$$744$$ 0 0
$$745$$ −7.82843 −0.286811
$$746$$ 0 0
$$747$$ 33.1716 1.21368
$$748$$ 0 0
$$749$$ 0 0
$$750$$ 0 0
$$751$$ 41.6569 1.52008 0.760040 0.649876i $$-0.225179\pi$$
0.760040 + 0.649876i $$0.225179\pi$$
$$752$$ 0 0
$$753$$ 32.1421 1.17132
$$754$$ 0 0
$$755$$ −0.343146 −0.0124884
$$756$$ 0 0
$$757$$ 19.4558 0.707135 0.353567 0.935409i $$-0.384969\pi$$
0.353567 + 0.935409i $$0.384969\pi$$
$$758$$ 0 0
$$759$$ 28.1421 1.02149
$$760$$ 0 0
$$761$$ −13.3137 −0.482622 −0.241311 0.970448i $$-0.577577\pi$$
−0.241311 + 0.970448i $$0.577577\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ 0 0
$$765$$ −2.34315 −0.0847166
$$766$$ 0 0
$$767$$ 10.3431 0.373469
$$768$$ 0 0
$$769$$ 44.6274 1.60931 0.804653 0.593745i $$-0.202351\pi$$
0.804653 + 0.593745i $$0.202351\pi$$
$$770$$ 0 0
$$771$$ −42.6274 −1.53519
$$772$$ 0 0
$$773$$ 25.1127 0.903241 0.451620 0.892210i $$-0.350846\pi$$
0.451620 + 0.892210i $$0.350846\pi$$
$$774$$ 0 0
$$775$$ 6.00000 0.215526
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ −6.14214 −0.220065
$$780$$ 0 0
$$781$$ −60.2843 −2.15714
$$782$$ 0 0
$$783$$ −0.414214 −0.0148028
$$784$$ 0 0
$$785$$ −5.31371 −0.189654
$$786$$ 0 0
$$787$$ 28.5563 1.01792 0.508962 0.860789i $$-0.330029\pi$$
0.508962 + 0.860789i $$0.330029\pi$$
$$788$$ 0 0
$$789$$ 45.9706 1.63660
$$790$$ 0 0
$$791$$ 0 0
$$792$$ 0 0
$$793$$ −9.51472 −0.337878
$$794$$ 0 0
$$795$$ 16.4853 0.584673
$$796$$ 0 0
$$797$$ −8.00000 −0.283375 −0.141687 0.989911i $$-0.545253\pi$$
−0.141687 + 0.989911i $$0.545253\pi$$
$$798$$ 0 0
$$799$$ 1.65685 0.0586153
$$800$$ 0 0
$$801$$ 7.51472 0.265520
$$802$$ 0 0
$$803$$ −23.3137 −0.822723
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ 73.5269 2.58827
$$808$$ 0 0
$$809$$ −9.62742 −0.338482 −0.169241 0.985575i $$-0.554132\pi$$
−0.169241 + 0.985575i $$0.554132\pi$$
$$810$$ 0 0
$$811$$ −24.6274 −0.864786 −0.432393 0.901685i $$-0.642331\pi$$
−0.432393 + 0.901685i $$0.642331\pi$$
$$812$$ 0 0
$$813$$ −1.17157 −0.0410889
$$814$$ 0 0
$$815$$ −23.6569 −0.828663
$$816$$ 0 0
$$817$$ −18.1421 −0.634713
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ −19.9411 −0.695950 −0.347975 0.937504i $$-0.613131\pi$$
−0.347975 + 0.937504i $$0.613131\pi$$
$$822$$ 0 0
$$823$$ 12.0711 0.420771 0.210385 0.977619i $$-0.432528\pi$$
0.210385 + 0.977619i $$0.432528\pi$$
$$824$$ 0 0
$$825$$ 11.6569 0.405840
$$826$$ 0 0
$$827$$ −16.2132 −0.563788 −0.281894 0.959446i $$-0.590963\pi$$
−0.281894 + 0.959446i $$0.590963\pi$$
$$828$$ 0 0
$$829$$ 6.68629 0.232225 0.116112 0.993236i $$-0.462957\pi$$
0.116112 + 0.993236i $$0.462957\pi$$
$$830$$ 0 0
$$831$$ 29.3137 1.01688
$$832$$ 0 0
$$833$$ 0 0
$$834$$ 0 0
$$835$$ −19.5858 −0.677794
$$836$$ 0 0
$$837$$ 2.48528 0.0859039
$$838$$ 0 0
$$839$$ −20.8284 −0.719077 −0.359539 0.933130i $$-0.617066\pi$$
−0.359539 + 0.933130i $$0.617066\pi$$
$$840$$ 0 0
$$841$$ −28.0000 −0.965517
$$842$$ 0 0
$$843$$ −63.4558 −2.18554
$$844$$ 0 0
$$845$$ −12.3137 −0.423604
$$846$$ 0 0
$$847$$ 0 0
$$848$$ 0 0
$$849$$ 33.7990 1.15998
$$850$$ 0 0
$$851$$ 0 0
$$852$$ 0 0
$$853$$ 53.4558 1.83029 0.915147 0.403121i $$-0.132075\pi$$
0.915147 + 0.403121i $$0.132075\pi$$
$$854$$ 0 0
$$855$$ 8.00000 0.273594
$$856$$ 0 0
$$857$$ −22.2843 −0.761216 −0.380608 0.924736i $$-0.624285\pi$$
−0.380608 + 0.924736i $$0.624285\pi$$
$$858$$ 0 0
$$859$$ −46.6274 −1.59091 −0.795453 0.606015i $$-0.792767\pi$$
−0.795453 + 0.606015i $$0.792767\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ 16.5563 0.563585 0.281792 0.959475i $$-0.409071\pi$$
0.281792 + 0.959475i $$0.409071\pi$$
$$864$$ 0 0
$$865$$ −19.3137 −0.656686
$$866$$ 0 0
$$867$$ 39.3848 1.33758
$$868$$ 0 0
$$869$$ 44.2843 1.50224
$$870$$ 0 0
$$871$$ −10.2843 −0.348469
$$872$$ 0 0
$$873$$ 0.970563 0.0328486
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 30.8284 1.04100 0.520501 0.853861i $$-0.325745\pi$$
0.520501 + 0.853861i $$0.325745\pi$$
$$878$$ 0 0
$$879$$ 38.6274 1.30287
$$880$$ 0 0
$$881$$ −3.82843 −0.128983 −0.0644915 0.997918i $$-0.520543\pi$$
−0.0644915 + 0.997918i $$0.520543\pi$$
$$882$$ 0 0
$$883$$ 38.2843 1.28837 0.644184 0.764870i $$-0.277197\pi$$
0.644184 + 0.764870i $$0.277197\pi$$
$$884$$ 0 0
$$885$$ −30.1421 −1.01322
$$886$$ 0 0
$$887$$ −44.0711 −1.47976 −0.739881 0.672738i $$-0.765118\pi$$
−0.739881 + 0.672738i $$0.765118\pi$$
$$888$$ 0 0
$$889$$ 0 0
$$890$$ 0 0
$$891$$ 45.7990 1.53432
$$892$$ 0 0
$$893$$ −5.65685 −0.189299
$$894$$ 0 0
$$895$$ 10.0000 0.334263
$$896$$ 0 0
$$897$$ −4.82843 −0.161216
$$898$$ 0 0
$$899$$ −6.00000 −0.200111
$$900$$ 0 0
$$901$$ 5.65685 0.188457
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ −8.65685 −0.287764
$$906$$ 0 0
$$907$$ −28.2132 −0.936804 −0.468402 0.883515i $$-0.655170\pi$$
−0.468402 + 0.883515i $$0.655170\pi$$
$$908$$ 0 0
$$909$$ 34.8284 1.15519
$$910$$ 0 0
$$911$$ 49.7990 1.64991 0.824957 0.565195i $$-0.191199\pi$$
0.824957 + 0.565195i $$0.191199\pi$$
$$912$$ 0 0
$$913$$ −56.6274 −1.87409
$$914$$ 0 0
$$915$$ 27.7279 0.916657
$$916$$ 0 0
$$917$$ 0 0
$$918$$ 0 0
$$919$$ −19.1127 −0.630470 −0.315235 0.949014i $$-0.602083\pi$$
−0.315235 + 0.949014i $$0.602083\pi$$
$$920$$ 0 0
$$921$$ −31.9706 −1.05347
$$922$$ 0 0
$$923$$ 10.3431 0.340449
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 0 0
$$927$$ −1.17157 −0.0384795
$$928$$ 0 0
$$929$$ −11.4853 −0.376820 −0.188410 0.982090i $$-0.560333\pi$$
−0.188410 + 0.982090i $$0.560333\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ 0 0
$$933$$ −45.4558 −1.48816
$$934$$ 0 0
$$935$$ 4.00000 0.130814
$$936$$ 0 0
$$937$$ −10.6274 −0.347183 −0.173591 0.984818i $$-0.555537\pi$$
−0.173591 + 0.984818i $$0.555537\pi$$
$$938$$ 0 0
$$939$$ 42.6274 1.39109
$$940$$ 0 0
$$941$$ 10.2843 0.335258 0.167629 0.985850i $$-0.446389\pi$$
0.167629 + 0.985850i $$0.446389\pi$$
$$942$$ 0 0
$$943$$ −5.24264 −0.170724
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 43.1838 1.40328 0.701642 0.712530i $$-0.252451\pi$$
0.701642 + 0.712530i $$0.252451\pi$$
$$948$$ 0 0
$$949$$ 4.00000 0.129845
$$950$$ 0 0
$$951$$ −62.2843 −2.01971
$$952$$ 0 0
$$953$$ −2.34315 −0.0759019 −0.0379510 0.999280i $$-0.512083\pi$$
−0.0379510 + 0.999280i $$0.512083\pi$$
$$954$$ 0 0
$$955$$ 7.17157 0.232067
$$956$$ 0 0
$$957$$ −11.6569 −0.376813
$$958$$ 0 0
$$959$$ 0 0
$$960$$ 0 0
$$961$$ 5.00000 0.161290
$$962$$ 0 0
$$963$$ −7.79899 −0.251319
$$964$$ 0 0
$$965$$ −2.00000 −0.0643823
$$966$$ 0 0
$$967$$ 27.5269 0.885206 0.442603 0.896718i $$-0.354055\pi$$
0.442603 + 0.896718i $$0.354055\pi$$
$$968$$ 0 0
$$969$$ 5.65685 0.181724
$$970$$ 0 0
$$971$$ 24.0000 0.770197 0.385098 0.922876i $$-0.374168\pi$$
0.385098 + 0.922876i $$0.374168\pi$$
$$972$$ 0 0
$$973$$ 0 0
$$974$$ 0 0
$$975$$ −2.00000 −0.0640513
$$976$$ 0 0
$$977$$ 21.3137 0.681886 0.340943 0.940084i $$-0.389254\pi$$
0.340943 + 0.940084i $$0.389254\pi$$
$$978$$ 0 0
$$979$$ −12.8284 −0.409998
$$980$$ 0 0
$$981$$ 9.85786 0.314737
$$982$$ 0 0
$$983$$ 14.2132 0.453331 0.226665 0.973973i $$-0.427218\pi$$
0.226665 + 0.973973i $$0.427218\pi$$
$$984$$ 0 0
$$985$$ −23.6569 −0.753770
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ −15.4853 −0.492403
$$990$$ 0 0
$$991$$ 15.6569 0.497356 0.248678 0.968586i $$-0.420004\pi$$
0.248678 + 0.968586i $$0.420004\pi$$
$$992$$ 0 0
$$993$$ −26.4853 −0.840485
$$994$$ 0 0
$$995$$ −1.65685 −0.0525258
$$996$$ 0 0
$$997$$ −17.4558 −0.552832 −0.276416 0.961038i $$-0.589147\pi$$
−0.276416 + 0.961038i $$0.589147\pi$$
$$998$$ 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 3920.2.a.bq.1.1 2
4.3 odd 2 245.2.a.h.1.1 2
7.2 even 3 560.2.q.k.81.2 4
7.4 even 3 560.2.q.k.401.2 4
7.6 odd 2 3920.2.a.bv.1.2 2
12.11 even 2 2205.2.a.n.1.2 2
20.3 even 4 1225.2.b.g.99.3 4
20.7 even 4 1225.2.b.g.99.2 4
20.19 odd 2 1225.2.a.k.1.2 2
28.3 even 6 245.2.e.e.226.2 4
28.11 odd 6 35.2.e.a.16.2 yes 4
28.19 even 6 245.2.e.e.116.2 4
28.23 odd 6 35.2.e.a.11.2 4
28.27 even 2 245.2.a.g.1.1 2
84.11 even 6 315.2.j.e.226.1 4
84.23 even 6 315.2.j.e.46.1 4
84.83 odd 2 2205.2.a.q.1.2 2
140.23 even 12 175.2.k.a.74.3 8
140.27 odd 4 1225.2.b.h.99.2 4
140.39 odd 6 175.2.e.c.51.1 4
140.67 even 12 175.2.k.a.149.3 8
140.79 odd 6 175.2.e.c.151.1 4
140.83 odd 4 1225.2.b.h.99.3 4
140.107 even 12 175.2.k.a.74.2 8
140.123 even 12 175.2.k.a.149.2 8
140.139 even 2 1225.2.a.m.1.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
35.2.e.a.11.2 4 28.23 odd 6
35.2.e.a.16.2 yes 4 28.11 odd 6
175.2.e.c.51.1 4 140.39 odd 6
175.2.e.c.151.1 4 140.79 odd 6
175.2.k.a.74.2 8 140.107 even 12
175.2.k.a.74.3 8 140.23 even 12
175.2.k.a.149.2 8 140.123 even 12
175.2.k.a.149.3 8 140.67 even 12
245.2.a.g.1.1 2 28.27 even 2
245.2.a.h.1.1 2 4.3 odd 2
245.2.e.e.116.2 4 28.19 even 6
245.2.e.e.226.2 4 28.3 even 6
315.2.j.e.46.1 4 84.23 even 6
315.2.j.e.226.1 4 84.11 even 6
560.2.q.k.81.2 4 7.2 even 3
560.2.q.k.401.2 4 7.4 even 3
1225.2.a.k.1.2 2 20.19 odd 2
1225.2.a.m.1.2 2 140.139 even 2
1225.2.b.g.99.2 4 20.7 even 4
1225.2.b.g.99.3 4 20.3 even 4
1225.2.b.h.99.2 4 140.27 odd 4
1225.2.b.h.99.3 4 140.83 odd 4
2205.2.a.n.1.2 2 12.11 even 2
2205.2.a.q.1.2 2 84.83 odd 2
3920.2.a.bq.1.1 2 1.1 even 1 trivial
3920.2.a.bv.1.2 2 7.6 odd 2