# Properties

 Label 3450.2.a.br.1.2 Level $3450$ Weight $2$ Character 3450.1 Self dual yes Analytic conductor $27.548$ Analytic rank $1$ Dimension $3$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$3450 = 2 \cdot 3 \cdot 5^{2} \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3450.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$27.5483886973$$ Analytic rank: $$1$$ Dimension: $$3$$ Coefficient field: 3.3.568.1 Defining polynomial: $$x^{3} - x^{2} - 6 x - 2$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 690) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$-1.76156$$ of defining polynomial Character $$\chi$$ $$=$$ 3450.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} -2.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} -2.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} -0.864641 q^{11} -1.00000 q^{12} -5.52311 q^{13} -2.00000 q^{14} +1.00000 q^{16} +3.52311 q^{17} +1.00000 q^{18} +8.11704 q^{19} +2.00000 q^{21} -0.864641 q^{22} -1.00000 q^{23} -1.00000 q^{24} -5.52311 q^{26} -1.00000 q^{27} -2.00000 q^{28} -2.00000 q^{29} -3.25240 q^{31} +1.00000 q^{32} +0.864641 q^{33} +3.52311 q^{34} +1.00000 q^{36} -5.13536 q^{37} +8.11704 q^{38} +5.52311 q^{39} +3.52311 q^{41} +2.00000 q^{42} -1.13536 q^{43} -0.864641 q^{44} -1.00000 q^{46} -5.52311 q^{47} -1.00000 q^{48} -3.00000 q^{49} -3.52311 q^{51} -5.52311 q^{52} -1.34153 q^{53} -1.00000 q^{54} -2.00000 q^{56} -8.11704 q^{57} -2.00000 q^{58} -10.9817 q^{59} -5.91087 q^{61} -3.25240 q^{62} -2.00000 q^{63} +1.00000 q^{64} +0.864641 q^{66} +5.91087 q^{67} +3.52311 q^{68} +1.00000 q^{69} -5.79383 q^{71} +1.00000 q^{72} -15.2524 q^{73} -5.13536 q^{74} +8.11704 q^{76} +1.72928 q^{77} +5.52311 q^{78} -3.79383 q^{79} +1.00000 q^{81} +3.52311 q^{82} -8.32320 q^{83} +2.00000 q^{84} -1.13536 q^{86} +2.00000 q^{87} -0.864641 q^{88} +4.77551 q^{89} +11.0462 q^{91} -1.00000 q^{92} +3.25240 q^{93} -5.52311 q^{94} -1.00000 q^{96} +1.04623 q^{97} -3.00000 q^{98} -0.864641 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3q + 3q^{2} - 3q^{3} + 3q^{4} - 3q^{6} - 6q^{7} + 3q^{8} + 3q^{9} + O(q^{10})$$ $$3q + 3q^{2} - 3q^{3} + 3q^{4} - 3q^{6} - 6q^{7} + 3q^{8} + 3q^{9} - 3q^{12} - 4q^{13} - 6q^{14} + 3q^{16} - 2q^{17} + 3q^{18} + 4q^{19} + 6q^{21} - 3q^{23} - 3q^{24} - 4q^{26} - 3q^{27} - 6q^{28} - 6q^{29} + 8q^{31} + 3q^{32} - 2q^{34} + 3q^{36} - 18q^{37} + 4q^{38} + 4q^{39} - 2q^{41} + 6q^{42} - 6q^{43} - 3q^{46} - 4q^{47} - 3q^{48} - 9q^{49} + 2q^{51} - 4q^{52} - 14q^{53} - 3q^{54} - 6q^{56} - 4q^{57} - 6q^{58} - 10q^{59} + 10q^{61} + 8q^{62} - 6q^{63} + 3q^{64} - 10q^{67} - 2q^{68} + 3q^{69} - 10q^{71} + 3q^{72} - 28q^{73} - 18q^{74} + 4q^{76} + 4q^{78} - 4q^{79} + 3q^{81} - 2q^{82} - 12q^{83} + 6q^{84} - 6q^{86} + 6q^{87} - 16q^{89} + 8q^{91} - 3q^{92} - 8q^{93} - 4q^{94} - 3q^{96} - 22q^{97} - 9q^{98} + O(q^{100})$$

## 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
$$3$$ −1.00000 −0.577350
$$4$$ 1.00000 0.500000
$$5$$ 0 0
$$6$$ −1.00000 −0.408248
$$7$$ −2.00000 −0.755929 −0.377964 0.925820i $$-0.623376\pi$$
−0.377964 + 0.925820i $$0.623376\pi$$
$$8$$ 1.00000 0.353553
$$9$$ 1.00000 0.333333
$$10$$ 0 0
$$11$$ −0.864641 −0.260699 −0.130350 0.991468i $$-0.541610\pi$$
−0.130350 + 0.991468i $$0.541610\pi$$
$$12$$ −1.00000 −0.288675
$$13$$ −5.52311 −1.53184 −0.765918 0.642938i $$-0.777715\pi$$
−0.765918 + 0.642938i $$0.777715\pi$$
$$14$$ −2.00000 −0.534522
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ 3.52311 0.854481 0.427240 0.904138i $$-0.359486\pi$$
0.427240 + 0.904138i $$0.359486\pi$$
$$18$$ 1.00000 0.235702
$$19$$ 8.11704 1.86218 0.931088 0.364795i $$-0.118861\pi$$
0.931088 + 0.364795i $$0.118861\pi$$
$$20$$ 0 0
$$21$$ 2.00000 0.436436
$$22$$ −0.864641 −0.184342
$$23$$ −1.00000 −0.208514
$$24$$ −1.00000 −0.204124
$$25$$ 0 0
$$26$$ −5.52311 −1.08317
$$27$$ −1.00000 −0.192450
$$28$$ −2.00000 −0.377964
$$29$$ −2.00000 −0.371391 −0.185695 0.982607i $$-0.559454\pi$$
−0.185695 + 0.982607i $$0.559454\pi$$
$$30$$ 0 0
$$31$$ −3.25240 −0.584148 −0.292074 0.956396i $$-0.594345\pi$$
−0.292074 + 0.956396i $$0.594345\pi$$
$$32$$ 1.00000 0.176777
$$33$$ 0.864641 0.150515
$$34$$ 3.52311 0.604209
$$35$$ 0 0
$$36$$ 1.00000 0.166667
$$37$$ −5.13536 −0.844248 −0.422124 0.906538i $$-0.638715\pi$$
−0.422124 + 0.906538i $$0.638715\pi$$
$$38$$ 8.11704 1.31676
$$39$$ 5.52311 0.884406
$$40$$ 0 0
$$41$$ 3.52311 0.550218 0.275109 0.961413i $$-0.411286\pi$$
0.275109 + 0.961413i $$0.411286\pi$$
$$42$$ 2.00000 0.308607
$$43$$ −1.13536 −0.173141 −0.0865703 0.996246i $$-0.527591\pi$$
−0.0865703 + 0.996246i $$0.527591\pi$$
$$44$$ −0.864641 −0.130350
$$45$$ 0 0
$$46$$ −1.00000 −0.147442
$$47$$ −5.52311 −0.805629 −0.402815 0.915282i $$-0.631968\pi$$
−0.402815 + 0.915282i $$0.631968\pi$$
$$48$$ −1.00000 −0.144338
$$49$$ −3.00000 −0.428571
$$50$$ 0 0
$$51$$ −3.52311 −0.493335
$$52$$ −5.52311 −0.765918
$$53$$ −1.34153 −0.184273 −0.0921364 0.995746i $$-0.529370\pi$$
−0.0921364 + 0.995746i $$0.529370\pi$$
$$54$$ −1.00000 −0.136083
$$55$$ 0 0
$$56$$ −2.00000 −0.267261
$$57$$ −8.11704 −1.07513
$$58$$ −2.00000 −0.262613
$$59$$ −10.9817 −1.42969 −0.714846 0.699282i $$-0.753503\pi$$
−0.714846 + 0.699282i $$0.753503\pi$$
$$60$$ 0 0
$$61$$ −5.91087 −0.756809 −0.378405 0.925640i $$-0.623527\pi$$
−0.378405 + 0.925640i $$0.623527\pi$$
$$62$$ −3.25240 −0.413055
$$63$$ −2.00000 −0.251976
$$64$$ 1.00000 0.125000
$$65$$ 0 0
$$66$$ 0.864641 0.106430
$$67$$ 5.91087 0.722128 0.361064 0.932541i $$-0.382414\pi$$
0.361064 + 0.932541i $$0.382414\pi$$
$$68$$ 3.52311 0.427240
$$69$$ 1.00000 0.120386
$$70$$ 0 0
$$71$$ −5.79383 −0.687601 −0.343801 0.939043i $$-0.611714\pi$$
−0.343801 + 0.939043i $$0.611714\pi$$
$$72$$ 1.00000 0.117851
$$73$$ −15.2524 −1.78516 −0.892579 0.450891i $$-0.851106\pi$$
−0.892579 + 0.450891i $$0.851106\pi$$
$$74$$ −5.13536 −0.596973
$$75$$ 0 0
$$76$$ 8.11704 0.931088
$$77$$ 1.72928 0.197070
$$78$$ 5.52311 0.625370
$$79$$ −3.79383 −0.426840 −0.213420 0.976961i $$-0.568460\pi$$
−0.213420 + 0.976961i $$0.568460\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 3.52311 0.389063
$$83$$ −8.32320 −0.913590 −0.456795 0.889572i $$-0.651003\pi$$
−0.456795 + 0.889572i $$0.651003\pi$$
$$84$$ 2.00000 0.218218
$$85$$ 0 0
$$86$$ −1.13536 −0.122429
$$87$$ 2.00000 0.214423
$$88$$ −0.864641 −0.0921710
$$89$$ 4.77551 0.506203 0.253102 0.967440i $$-0.418549\pi$$
0.253102 + 0.967440i $$0.418549\pi$$
$$90$$ 0 0
$$91$$ 11.0462 1.15796
$$92$$ −1.00000 −0.104257
$$93$$ 3.25240 0.337258
$$94$$ −5.52311 −0.569666
$$95$$ 0 0
$$96$$ −1.00000 −0.102062
$$97$$ 1.04623 0.106228 0.0531142 0.998588i $$-0.483085\pi$$
0.0531142 + 0.998588i $$0.483085\pi$$
$$98$$ −3.00000 −0.303046
$$99$$ −0.864641 −0.0868997
$$100$$ 0 0
$$101$$ 9.04623 0.900133 0.450067 0.892995i $$-0.351400\pi$$
0.450067 + 0.892995i $$0.351400\pi$$
$$102$$ −3.52311 −0.348840
$$103$$ −11.3169 −1.11509 −0.557546 0.830146i $$-0.688257\pi$$
−0.557546 + 0.830146i $$0.688257\pi$$
$$104$$ −5.52311 −0.541586
$$105$$ 0 0
$$106$$ −1.34153 −0.130301
$$107$$ −5.64015 −0.545254 −0.272627 0.962120i $$-0.587892\pi$$
−0.272627 + 0.962120i $$0.587892\pi$$
$$108$$ −1.00000 −0.0962250
$$109$$ −12.1816 −1.16678 −0.583392 0.812191i $$-0.698275\pi$$
−0.583392 + 0.812191i $$0.698275\pi$$
$$110$$ 0 0
$$111$$ 5.13536 0.487427
$$112$$ −2.00000 −0.188982
$$113$$ −5.79383 −0.545038 −0.272519 0.962150i $$-0.587857\pi$$
−0.272519 + 0.962150i $$0.587857\pi$$
$$114$$ −8.11704 −0.760230
$$115$$ 0 0
$$116$$ −2.00000 −0.185695
$$117$$ −5.52311 −0.510612
$$118$$ −10.9817 −1.01095
$$119$$ −7.04623 −0.645927
$$120$$ 0 0
$$121$$ −10.2524 −0.932036
$$122$$ −5.91087 −0.535145
$$123$$ −3.52311 −0.317669
$$124$$ −3.25240 −0.292074
$$125$$ 0 0
$$126$$ −2.00000 −0.178174
$$127$$ −0.747604 −0.0663391 −0.0331696 0.999450i $$-0.510560\pi$$
−0.0331696 + 0.999450i $$0.510560\pi$$
$$128$$ 1.00000 0.0883883
$$129$$ 1.13536 0.0999628
$$130$$ 0 0
$$131$$ 19.7572 1.72619 0.863097 0.505039i $$-0.168522\pi$$
0.863097 + 0.505039i $$0.168522\pi$$
$$132$$ 0.864641 0.0752573
$$133$$ −16.2341 −1.40767
$$134$$ 5.91087 0.510621
$$135$$ 0 0
$$136$$ 3.52311 0.302105
$$137$$ −0.476886 −0.0407431 −0.0203715 0.999792i $$-0.506485\pi$$
−0.0203715 + 0.999792i $$0.506485\pi$$
$$138$$ 1.00000 0.0851257
$$139$$ −3.45856 −0.293352 −0.146676 0.989185i $$-0.546857\pi$$
−0.146676 + 0.989185i $$0.546857\pi$$
$$140$$ 0 0
$$141$$ 5.52311 0.465130
$$142$$ −5.79383 −0.486208
$$143$$ 4.77551 0.399348
$$144$$ 1.00000 0.0833333
$$145$$ 0 0
$$146$$ −15.2524 −1.26230
$$147$$ 3.00000 0.247436
$$148$$ −5.13536 −0.422124
$$149$$ −8.86464 −0.726220 −0.363110 0.931746i $$-0.618285\pi$$
−0.363110 + 0.931746i $$0.618285\pi$$
$$150$$ 0 0
$$151$$ −6.71096 −0.546130 −0.273065 0.961996i $$-0.588037\pi$$
−0.273065 + 0.961996i $$0.588037\pi$$
$$152$$ 8.11704 0.658379
$$153$$ 3.52311 0.284827
$$154$$ 1.72928 0.139349
$$155$$ 0 0
$$156$$ 5.52311 0.442203
$$157$$ 14.6864 1.17210 0.586050 0.810275i $$-0.300682\pi$$
0.586050 + 0.810275i $$0.300682\pi$$
$$158$$ −3.79383 −0.301821
$$159$$ 1.34153 0.106390
$$160$$ 0 0
$$161$$ 2.00000 0.157622
$$162$$ 1.00000 0.0785674
$$163$$ 23.2803 1.82345 0.911727 0.410797i $$-0.134749\pi$$
0.911727 + 0.410797i $$0.134749\pi$$
$$164$$ 3.52311 0.275109
$$165$$ 0 0
$$166$$ −8.32320 −0.646006
$$167$$ 5.93545 0.459299 0.229649 0.973273i $$-0.426242\pi$$
0.229649 + 0.973273i $$0.426242\pi$$
$$168$$ 2.00000 0.154303
$$169$$ 17.5048 1.34652
$$170$$ 0 0
$$171$$ 8.11704 0.620725
$$172$$ −1.13536 −0.0865703
$$173$$ −25.8217 −1.96319 −0.981595 0.190973i $$-0.938836\pi$$
−0.981595 + 0.190973i $$0.938836\pi$$
$$174$$ 2.00000 0.151620
$$175$$ 0 0
$$176$$ −0.864641 −0.0651748
$$177$$ 10.9817 0.825433
$$178$$ 4.77551 0.357940
$$179$$ −23.3449 −1.74488 −0.872438 0.488725i $$-0.837462\pi$$
−0.872438 + 0.488725i $$0.837462\pi$$
$$180$$ 0 0
$$181$$ 13.3694 0.993742 0.496871 0.867824i $$-0.334482\pi$$
0.496871 + 0.867824i $$0.334482\pi$$
$$182$$ 11.0462 0.818801
$$183$$ 5.91087 0.436944
$$184$$ −1.00000 −0.0737210
$$185$$ 0 0
$$186$$ 3.25240 0.238477
$$187$$ −3.04623 −0.222762
$$188$$ −5.52311 −0.402815
$$189$$ 2.00000 0.145479
$$190$$ 0 0
$$191$$ 12.2341 0.885227 0.442613 0.896713i $$-0.354051\pi$$
0.442613 + 0.896713i $$0.354051\pi$$
$$192$$ −1.00000 −0.0721688
$$193$$ −7.45856 −0.536879 −0.268440 0.963297i $$-0.586508\pi$$
−0.268440 + 0.963297i $$0.586508\pi$$
$$194$$ 1.04623 0.0751148
$$195$$ 0 0
$$196$$ −3.00000 −0.214286
$$197$$ −2.00000 −0.142494 −0.0712470 0.997459i $$-0.522698\pi$$
−0.0712470 + 0.997459i $$0.522698\pi$$
$$198$$ −0.864641 −0.0614474
$$199$$ 22.8401 1.61909 0.809545 0.587059i $$-0.199714\pi$$
0.809545 + 0.587059i $$0.199714\pi$$
$$200$$ 0 0
$$201$$ −5.91087 −0.416921
$$202$$ 9.04623 0.636490
$$203$$ 4.00000 0.280745
$$204$$ −3.52311 −0.246667
$$205$$ 0 0
$$206$$ −11.3169 −0.788489
$$207$$ −1.00000 −0.0695048
$$208$$ −5.52311 −0.382959
$$209$$ −7.01832 −0.485467
$$210$$ 0 0
$$211$$ 24.2341 1.66834 0.834171 0.551506i $$-0.185946\pi$$
0.834171 + 0.551506i $$0.185946\pi$$
$$212$$ −1.34153 −0.0921364
$$213$$ 5.79383 0.396987
$$214$$ −5.64015 −0.385553
$$215$$ 0 0
$$216$$ −1.00000 −0.0680414
$$217$$ 6.50479 0.441574
$$218$$ −12.1816 −0.825041
$$219$$ 15.2524 1.03066
$$220$$ 0 0
$$221$$ −19.4586 −1.30892
$$222$$ 5.13536 0.344663
$$223$$ −13.9634 −0.935055 −0.467528 0.883978i $$-0.654855\pi$$
−0.467528 + 0.883978i $$0.654855\pi$$
$$224$$ −2.00000 −0.133631
$$225$$ 0 0
$$226$$ −5.79383 −0.385400
$$227$$ 4.45231 0.295510 0.147755 0.989024i $$-0.452795\pi$$
0.147755 + 0.989024i $$0.452795\pi$$
$$228$$ −8.11704 −0.537564
$$229$$ −4.05249 −0.267796 −0.133898 0.990995i $$-0.542749\pi$$
−0.133898 + 0.990995i $$0.542749\pi$$
$$230$$ 0 0
$$231$$ −1.72928 −0.113778
$$232$$ −2.00000 −0.131306
$$233$$ 24.5327 1.60719 0.803595 0.595176i $$-0.202918\pi$$
0.803595 + 0.595176i $$0.202918\pi$$
$$234$$ −5.52311 −0.361057
$$235$$ 0 0
$$236$$ −10.9817 −0.714846
$$237$$ 3.79383 0.246436
$$238$$ −7.04623 −0.456739
$$239$$ 10.7755 0.697010 0.348505 0.937307i $$-0.386689\pi$$
0.348505 + 0.937307i $$0.386689\pi$$
$$240$$ 0 0
$$241$$ −10.7755 −0.694112 −0.347056 0.937844i $$-0.612819\pi$$
−0.347056 + 0.937844i $$0.612819\pi$$
$$242$$ −10.2524 −0.659049
$$243$$ −1.00000 −0.0641500
$$244$$ −5.91087 −0.378405
$$245$$ 0 0
$$246$$ −3.52311 −0.224626
$$247$$ −44.8313 −2.85255
$$248$$ −3.25240 −0.206527
$$249$$ 8.32320 0.527462
$$250$$ 0 0
$$251$$ −3.91087 −0.246852 −0.123426 0.992354i $$-0.539388\pi$$
−0.123426 + 0.992354i $$0.539388\pi$$
$$252$$ −2.00000 −0.125988
$$253$$ 0.864641 0.0543595
$$254$$ −0.747604 −0.0469088
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ −28.0925 −1.75236 −0.876180 0.481985i $$-0.839916\pi$$
−0.876180 + 0.481985i $$0.839916\pi$$
$$258$$ 1.13536 0.0706844
$$259$$ 10.2707 0.638191
$$260$$ 0 0
$$261$$ −2.00000 −0.123797
$$262$$ 19.7572 1.22060
$$263$$ −29.5510 −1.82219 −0.911097 0.412192i $$-0.864763\pi$$
−0.911097 + 0.412192i $$0.864763\pi$$
$$264$$ 0.864641 0.0532150
$$265$$ 0 0
$$266$$ −16.2341 −0.995375
$$267$$ −4.77551 −0.292256
$$268$$ 5.91087 0.361064
$$269$$ −17.2803 −1.05360 −0.526799 0.849990i $$-0.676608\pi$$
−0.526799 + 0.849990i $$0.676608\pi$$
$$270$$ 0 0
$$271$$ 9.96336 0.605231 0.302615 0.953113i $$-0.402140\pi$$
0.302615 + 0.953113i $$0.402140\pi$$
$$272$$ 3.52311 0.213620
$$273$$ −11.0462 −0.668548
$$274$$ −0.476886 −0.0288097
$$275$$ 0 0
$$276$$ 1.00000 0.0601929
$$277$$ −27.0741 −1.62673 −0.813364 0.581756i $$-0.802366\pi$$
−0.813364 + 0.581756i $$0.802366\pi$$
$$278$$ −3.45856 −0.207431
$$279$$ −3.25240 −0.194716
$$280$$ 0 0
$$281$$ −27.2803 −1.62741 −0.813703 0.581281i $$-0.802552\pi$$
−0.813703 + 0.581281i $$0.802552\pi$$
$$282$$ 5.52311 0.328897
$$283$$ 30.9205 1.83803 0.919015 0.394222i $$-0.128986\pi$$
0.919015 + 0.394222i $$0.128986\pi$$
$$284$$ −5.79383 −0.343801
$$285$$ 0 0
$$286$$ 4.77551 0.282382
$$287$$ −7.04623 −0.415926
$$288$$ 1.00000 0.0589256
$$289$$ −4.58767 −0.269863
$$290$$ 0 0
$$291$$ −1.04623 −0.0613310
$$292$$ −15.2524 −0.892579
$$293$$ −11.0708 −0.646764 −0.323382 0.946269i $$-0.604820\pi$$
−0.323382 + 0.946269i $$0.604820\pi$$
$$294$$ 3.00000 0.174964
$$295$$ 0 0
$$296$$ −5.13536 −0.298487
$$297$$ 0.864641 0.0501716
$$298$$ −8.86464 −0.513515
$$299$$ 5.52311 0.319410
$$300$$ 0 0
$$301$$ 2.27072 0.130882
$$302$$ −6.71096 −0.386172
$$303$$ −9.04623 −0.519692
$$304$$ 8.11704 0.465544
$$305$$ 0 0
$$306$$ 3.52311 0.201403
$$307$$ −4.23407 −0.241651 −0.120826 0.992674i $$-0.538554\pi$$
−0.120826 + 0.992674i $$0.538554\pi$$
$$308$$ 1.72928 0.0985350
$$309$$ 11.3169 0.643799
$$310$$ 0 0
$$311$$ −13.2803 −0.753057 −0.376528 0.926405i $$-0.622882\pi$$
−0.376528 + 0.926405i $$0.622882\pi$$
$$312$$ 5.52311 0.312685
$$313$$ 31.9634 1.80668 0.903338 0.428930i $$-0.141109\pi$$
0.903338 + 0.428930i $$0.141109\pi$$
$$314$$ 14.6864 0.828800
$$315$$ 0 0
$$316$$ −3.79383 −0.213420
$$317$$ 13.2803 0.745896 0.372948 0.927852i $$-0.378347\pi$$
0.372948 + 0.927852i $$0.378347\pi$$
$$318$$ 1.34153 0.0752291
$$319$$ 1.72928 0.0968212
$$320$$ 0 0
$$321$$ 5.64015 0.314803
$$322$$ 2.00000 0.111456
$$323$$ 28.5972 1.59119
$$324$$ 1.00000 0.0555556
$$325$$ 0 0
$$326$$ 23.2803 1.28938
$$327$$ 12.1816 0.673643
$$328$$ 3.52311 0.194531
$$329$$ 11.0462 0.608998
$$330$$ 0 0
$$331$$ −11.8217 −0.649782 −0.324891 0.945752i $$-0.605328\pi$$
−0.324891 + 0.945752i $$0.605328\pi$$
$$332$$ −8.32320 −0.456795
$$333$$ −5.13536 −0.281416
$$334$$ 5.93545 0.324773
$$335$$ 0 0
$$336$$ 2.00000 0.109109
$$337$$ −13.2803 −0.723424 −0.361712 0.932290i $$-0.617808\pi$$
−0.361712 + 0.932290i $$0.617808\pi$$
$$338$$ 17.5048 0.952135
$$339$$ 5.79383 0.314678
$$340$$ 0 0
$$341$$ 2.81215 0.152287
$$342$$ 8.11704 0.438919
$$343$$ 20.0000 1.07990
$$344$$ −1.13536 −0.0612145
$$345$$ 0 0
$$346$$ −25.8217 −1.38819
$$347$$ 24.3632 1.30788 0.653942 0.756545i $$-0.273114\pi$$
0.653942 + 0.756545i $$0.273114\pi$$
$$348$$ 2.00000 0.107211
$$349$$ 3.49521 0.187094 0.0935471 0.995615i $$-0.470179\pi$$
0.0935471 + 0.995615i $$0.470179\pi$$
$$350$$ 0 0
$$351$$ 5.52311 0.294802
$$352$$ −0.864641 −0.0460855
$$353$$ −31.5789 −1.68078 −0.840388 0.541985i $$-0.817673\pi$$
−0.840388 + 0.541985i $$0.817673\pi$$
$$354$$ 10.9817 0.583670
$$355$$ 0 0
$$356$$ 4.77551 0.253102
$$357$$ 7.04623 0.372926
$$358$$ −23.3449 −1.23381
$$359$$ −21.1878 −1.11825 −0.559126 0.829083i $$-0.688863\pi$$
−0.559126 + 0.829083i $$0.688863\pi$$
$$360$$ 0 0
$$361$$ 46.8863 2.46770
$$362$$ 13.3694 0.702682
$$363$$ 10.2524 0.538111
$$364$$ 11.0462 0.578980
$$365$$ 0 0
$$366$$ 5.91087 0.308966
$$367$$ −13.0462 −0.681008 −0.340504 0.940243i $$-0.610598\pi$$
−0.340504 + 0.940243i $$0.610598\pi$$
$$368$$ −1.00000 −0.0521286
$$369$$ 3.52311 0.183406
$$370$$ 0 0
$$371$$ 2.68305 0.139297
$$372$$ 3.25240 0.168629
$$373$$ −18.8646 −0.976774 −0.488387 0.872627i $$-0.662415\pi$$
−0.488387 + 0.872627i $$0.662415\pi$$
$$374$$ −3.04623 −0.157517
$$375$$ 0 0
$$376$$ −5.52311 −0.284833
$$377$$ 11.0462 0.568910
$$378$$ 2.00000 0.102869
$$379$$ 26.7509 1.37410 0.687052 0.726609i $$-0.258905\pi$$
0.687052 + 0.726609i $$0.258905\pi$$
$$380$$ 0 0
$$381$$ 0.747604 0.0383009
$$382$$ 12.2341 0.625950
$$383$$ 22.5048 1.14994 0.574971 0.818174i $$-0.305013\pi$$
0.574971 + 0.818174i $$0.305013\pi$$
$$384$$ −1.00000 −0.0510310
$$385$$ 0 0
$$386$$ −7.45856 −0.379631
$$387$$ −1.13536 −0.0577135
$$388$$ 1.04623 0.0531142
$$389$$ −7.67680 −0.389229 −0.194614 0.980880i $$-0.562346\pi$$
−0.194614 + 0.980880i $$0.562346\pi$$
$$390$$ 0 0
$$391$$ −3.52311 −0.178172
$$392$$ −3.00000 −0.151523
$$393$$ −19.7572 −0.996618
$$394$$ −2.00000 −0.100759
$$395$$ 0 0
$$396$$ −0.864641 −0.0434498
$$397$$ 31.7938 1.59569 0.797843 0.602865i $$-0.205974\pi$$
0.797843 + 0.602865i $$0.205974\pi$$
$$398$$ 22.8401 1.14487
$$399$$ 16.2341 0.812720
$$400$$ 0 0
$$401$$ −18.0925 −0.903494 −0.451747 0.892146i $$-0.649199\pi$$
−0.451747 + 0.892146i $$0.649199\pi$$
$$402$$ −5.91087 −0.294807
$$403$$ 17.9634 0.894818
$$404$$ 9.04623 0.450067
$$405$$ 0 0
$$406$$ 4.00000 0.198517
$$407$$ 4.44024 0.220095
$$408$$ −3.52311 −0.174420
$$409$$ 2.00000 0.0988936 0.0494468 0.998777i $$-0.484254\pi$$
0.0494468 + 0.998777i $$0.484254\pi$$
$$410$$ 0 0
$$411$$ 0.476886 0.0235230
$$412$$ −11.3169 −0.557546
$$413$$ 21.9634 1.08075
$$414$$ −1.00000 −0.0491473
$$415$$ 0 0
$$416$$ −5.52311 −0.270793
$$417$$ 3.45856 0.169367
$$418$$ −7.01832 −0.343277
$$419$$ 20.4523 0.999161 0.499580 0.866268i $$-0.333488\pi$$
0.499580 + 0.866268i $$0.333488\pi$$
$$420$$ 0 0
$$421$$ −12.9571 −0.631490 −0.315745 0.948844i $$-0.602254\pi$$
−0.315745 + 0.948844i $$0.602254\pi$$
$$422$$ 24.2341 1.17970
$$423$$ −5.52311 −0.268543
$$424$$ −1.34153 −0.0651503
$$425$$ 0 0
$$426$$ 5.79383 0.280712
$$427$$ 11.8217 0.572094
$$428$$ −5.64015 −0.272627
$$429$$ −4.77551 −0.230564
$$430$$ 0 0
$$431$$ 33.3728 1.60751 0.803755 0.594961i $$-0.202832\pi$$
0.803755 + 0.594961i $$0.202832\pi$$
$$432$$ −1.00000 −0.0481125
$$433$$ −19.0096 −0.913542 −0.456771 0.889584i $$-0.650994\pi$$
−0.456771 + 0.889584i $$0.650994\pi$$
$$434$$ 6.50479 0.312240
$$435$$ 0 0
$$436$$ −12.1816 −0.583392
$$437$$ −8.11704 −0.388291
$$438$$ 15.2524 0.728788
$$439$$ 32.5972 1.55578 0.777891 0.628399i $$-0.216290\pi$$
0.777891 + 0.628399i $$0.216290\pi$$
$$440$$ 0 0
$$441$$ −3.00000 −0.142857
$$442$$ −19.4586 −0.925549
$$443$$ −4.00000 −0.190046 −0.0950229 0.995475i $$-0.530292\pi$$
−0.0950229 + 0.995475i $$0.530292\pi$$
$$444$$ 5.13536 0.243713
$$445$$ 0 0
$$446$$ −13.9634 −0.661184
$$447$$ 8.86464 0.419283
$$448$$ −2.00000 −0.0944911
$$449$$ 39.5789 1.86785 0.933923 0.357475i $$-0.116362\pi$$
0.933923 + 0.357475i $$0.116362\pi$$
$$450$$ 0 0
$$451$$ −3.04623 −0.143441
$$452$$ −5.79383 −0.272519
$$453$$ 6.71096 0.315308
$$454$$ 4.45231 0.208957
$$455$$ 0 0
$$456$$ −8.11704 −0.380115
$$457$$ −24.5048 −1.14629 −0.573143 0.819455i $$-0.694276\pi$$
−0.573143 + 0.819455i $$0.694276\pi$$
$$458$$ −4.05249 −0.189360
$$459$$ −3.52311 −0.164445
$$460$$ 0 0
$$461$$ 18.2341 0.849245 0.424623 0.905370i $$-0.360407\pi$$
0.424623 + 0.905370i $$0.360407\pi$$
$$462$$ −1.72928 −0.0804535
$$463$$ 4.20617 0.195477 0.0977386 0.995212i $$-0.468839\pi$$
0.0977386 + 0.995212i $$0.468839\pi$$
$$464$$ −2.00000 −0.0928477
$$465$$ 0 0
$$466$$ 24.5327 1.13646
$$467$$ −19.1912 −0.888062 −0.444031 0.896012i $$-0.646452\pi$$
−0.444031 + 0.896012i $$0.646452\pi$$
$$468$$ −5.52311 −0.255306
$$469$$ −11.8217 −0.545877
$$470$$ 0 0
$$471$$ −14.6864 −0.676713
$$472$$ −10.9817 −0.505473
$$473$$ 0.981678 0.0451376
$$474$$ 3.79383 0.174257
$$475$$ 0 0
$$476$$ −7.04623 −0.322963
$$477$$ −1.34153 −0.0614243
$$478$$ 10.7755 0.492860
$$479$$ −25.6801 −1.17335 −0.586677 0.809821i $$-0.699564\pi$$
−0.586677 + 0.809821i $$0.699564\pi$$
$$480$$ 0 0
$$481$$ 28.3632 1.29325
$$482$$ −10.7755 −0.490811
$$483$$ −2.00000 −0.0910032
$$484$$ −10.2524 −0.466018
$$485$$ 0 0
$$486$$ −1.00000 −0.0453609
$$487$$ −8.74760 −0.396392 −0.198196 0.980162i $$-0.563508\pi$$
−0.198196 + 0.980162i $$0.563508\pi$$
$$488$$ −5.91087 −0.267572
$$489$$ −23.2803 −1.05277
$$490$$ 0 0
$$491$$ 23.8863 1.07797 0.538987 0.842314i $$-0.318807\pi$$
0.538987 + 0.842314i $$0.318807\pi$$
$$492$$ −3.52311 −0.158834
$$493$$ −7.04623 −0.317346
$$494$$ −44.8313 −2.01706
$$495$$ 0 0
$$496$$ −3.25240 −0.146037
$$497$$ 11.5877 0.519778
$$498$$ 8.32320 0.372972
$$499$$ 29.7293 1.33087 0.665433 0.746458i $$-0.268247\pi$$
0.665433 + 0.746458i $$0.268247\pi$$
$$500$$ 0 0
$$501$$ −5.93545 −0.265176
$$502$$ −3.91087 −0.174551
$$503$$ −0.775511 −0.0345783 −0.0172892 0.999851i $$-0.505504\pi$$
−0.0172892 + 0.999851i $$0.505504\pi$$
$$504$$ −2.00000 −0.0890871
$$505$$ 0 0
$$506$$ 0.864641 0.0384380
$$507$$ −17.5048 −0.777415
$$508$$ −0.747604 −0.0331696
$$509$$ −26.0000 −1.15243 −0.576215 0.817298i $$-0.695471\pi$$
−0.576215 + 0.817298i $$0.695471\pi$$
$$510$$ 0 0
$$511$$ 30.5048 1.34945
$$512$$ 1.00000 0.0441942
$$513$$ −8.11704 −0.358376
$$514$$ −28.0925 −1.23911
$$515$$ 0 0
$$516$$ 1.13536 0.0499814
$$517$$ 4.77551 0.210027
$$518$$ 10.2707 0.451269
$$519$$ 25.8217 1.13345
$$520$$ 0 0
$$521$$ 13.4219 0.588025 0.294012 0.955802i $$-0.405009\pi$$
0.294012 + 0.955802i $$0.405009\pi$$
$$522$$ −2.00000 −0.0875376
$$523$$ 41.9667 1.83507 0.917537 0.397649i $$-0.130174\pi$$
0.917537 + 0.397649i $$0.130174\pi$$
$$524$$ 19.7572 0.863097
$$525$$ 0 0
$$526$$ −29.5510 −1.28849
$$527$$ −11.4586 −0.499143
$$528$$ 0.864641 0.0376287
$$529$$ 1.00000 0.0434783
$$530$$ 0 0
$$531$$ −10.9817 −0.476564
$$532$$ −16.2341 −0.703836
$$533$$ −19.4586 −0.842844
$$534$$ −4.77551 −0.206657
$$535$$ 0 0
$$536$$ 5.91087 0.255311
$$537$$ 23.3449 1.00740
$$538$$ −17.2803 −0.745007
$$539$$ 2.59392 0.111728
$$540$$ 0 0
$$541$$ −1.58767 −0.0682591 −0.0341295 0.999417i $$-0.510866\pi$$
−0.0341295 + 0.999417i $$0.510866\pi$$
$$542$$ 9.96336 0.427963
$$543$$ −13.3694 −0.573737
$$544$$ 3.52311 0.151052
$$545$$ 0 0
$$546$$ −11.0462 −0.472735
$$547$$ 20.6464 0.882777 0.441388 0.897316i $$-0.354486\pi$$
0.441388 + 0.897316i $$0.354486\pi$$
$$548$$ −0.476886 −0.0203715
$$549$$ −5.91087 −0.252270
$$550$$ 0 0
$$551$$ −16.2341 −0.691595
$$552$$ 1.00000 0.0425628
$$553$$ 7.58767 0.322660
$$554$$ −27.0741 −1.15027
$$555$$ 0 0
$$556$$ −3.45856 −0.146676
$$557$$ −9.47063 −0.401283 −0.200642 0.979665i $$-0.564303\pi$$
−0.200642 + 0.979665i $$0.564303\pi$$
$$558$$ −3.25240 −0.137685
$$559$$ 6.27072 0.265223
$$560$$ 0 0
$$561$$ 3.04623 0.128612
$$562$$ −27.2803 −1.15075
$$563$$ −1.51105 −0.0636832 −0.0318416 0.999493i $$-0.510137\pi$$
−0.0318416 + 0.999493i $$0.510137\pi$$
$$564$$ 5.52311 0.232565
$$565$$ 0 0
$$566$$ 30.9205 1.29968
$$567$$ −2.00000 −0.0839921
$$568$$ −5.79383 −0.243104
$$569$$ −20.0000 −0.838444 −0.419222 0.907884i $$-0.637697\pi$$
−0.419222 + 0.907884i $$0.637697\pi$$
$$570$$ 0 0
$$571$$ −35.1633 −1.47154 −0.735768 0.677233i $$-0.763179\pi$$
−0.735768 + 0.677233i $$0.763179\pi$$
$$572$$ 4.77551 0.199674
$$573$$ −12.2341 −0.511086
$$574$$ −7.04623 −0.294104
$$575$$ 0 0
$$576$$ 1.00000 0.0416667
$$577$$ 22.7110 0.945470 0.472735 0.881205i $$-0.343267\pi$$
0.472735 + 0.881205i $$0.343267\pi$$
$$578$$ −4.58767 −0.190822
$$579$$ 7.45856 0.309967
$$580$$ 0 0
$$581$$ 16.6464 0.690609
$$582$$ −1.04623 −0.0433676
$$583$$ 1.15994 0.0480398
$$584$$ −15.2524 −0.631149
$$585$$ 0 0
$$586$$ −11.0708 −0.457331
$$587$$ −3.76593 −0.155436 −0.0777182 0.996975i $$-0.524763\pi$$
−0.0777182 + 0.996975i $$0.524763\pi$$
$$588$$ 3.00000 0.123718
$$589$$ −26.3998 −1.08779
$$590$$ 0 0
$$591$$ 2.00000 0.0822690
$$592$$ −5.13536 −0.211062
$$593$$ −1.04623 −0.0429635 −0.0214817 0.999769i $$-0.506838\pi$$
−0.0214817 + 0.999769i $$0.506838\pi$$
$$594$$ 0.864641 0.0354766
$$595$$ 0 0
$$596$$ −8.86464 −0.363110
$$597$$ −22.8401 −0.934781
$$598$$ 5.52311 0.225857
$$599$$ −21.2803 −0.869490 −0.434745 0.900554i $$-0.643161\pi$$
−0.434745 + 0.900554i $$0.643161\pi$$
$$600$$ 0 0
$$601$$ 30.8034 1.25650 0.628249 0.778012i $$-0.283772\pi$$
0.628249 + 0.778012i $$0.283772\pi$$
$$602$$ 2.27072 0.0925476
$$603$$ 5.91087 0.240709
$$604$$ −6.71096 −0.273065
$$605$$ 0 0
$$606$$ −9.04623 −0.367478
$$607$$ 22.0925 0.896705 0.448353 0.893857i $$-0.352011\pi$$
0.448353 + 0.893857i $$0.352011\pi$$
$$608$$ 8.11704 0.329189
$$609$$ −4.00000 −0.162088
$$610$$ 0 0
$$611$$ 30.5048 1.23409
$$612$$ 3.52311 0.142413
$$613$$ 41.4985 1.67611 0.838055 0.545586i $$-0.183693\pi$$
0.838055 + 0.545586i $$0.183693\pi$$
$$614$$ −4.23407 −0.170873
$$615$$ 0 0
$$616$$ 1.72928 0.0696747
$$617$$ 30.9325 1.24530 0.622648 0.782502i $$-0.286057\pi$$
0.622648 + 0.782502i $$0.286057\pi$$
$$618$$ 11.3169 0.455234
$$619$$ 14.5660 0.585458 0.292729 0.956196i $$-0.405437\pi$$
0.292729 + 0.956196i $$0.405437\pi$$
$$620$$ 0 0
$$621$$ 1.00000 0.0401286
$$622$$ −13.2803 −0.532492
$$623$$ −9.55102 −0.382654
$$624$$ 5.52311 0.221102
$$625$$ 0 0
$$626$$ 31.9634 1.27751
$$627$$ 7.01832 0.280285
$$628$$ 14.6864 0.586050
$$629$$ −18.0925 −0.721394
$$630$$ 0 0
$$631$$ −38.4036 −1.52882 −0.764412 0.644729i $$-0.776970\pi$$
−0.764412 + 0.644729i $$0.776970\pi$$
$$632$$ −3.79383 −0.150911
$$633$$ −24.2341 −0.963218
$$634$$ 13.2803 0.527428
$$635$$ 0 0
$$636$$ 1.34153 0.0531950
$$637$$ 16.5693 0.656501
$$638$$ 1.72928 0.0684629
$$639$$ −5.79383 −0.229200
$$640$$ 0 0
$$641$$ −31.9267 −1.26103 −0.630515 0.776177i $$-0.717156\pi$$
−0.630515 + 0.776177i $$0.717156\pi$$
$$642$$ 5.64015 0.222599
$$643$$ 4.72302 0.186258 0.0931289 0.995654i $$-0.470313\pi$$
0.0931289 + 0.995654i $$0.470313\pi$$
$$644$$ 2.00000 0.0788110
$$645$$ 0 0
$$646$$ 28.5972 1.12514
$$647$$ 43.4586 1.70853 0.854266 0.519836i $$-0.174007\pi$$
0.854266 + 0.519836i $$0.174007\pi$$
$$648$$ 1.00000 0.0392837
$$649$$ 9.49521 0.372720
$$650$$ 0 0
$$651$$ −6.50479 −0.254943
$$652$$ 23.2803 0.911727
$$653$$ −2.36318 −0.0924782 −0.0462391 0.998930i $$-0.514724\pi$$
−0.0462391 + 0.998930i $$0.514724\pi$$
$$654$$ 12.1816 0.476338
$$655$$ 0 0
$$656$$ 3.52311 0.137555
$$657$$ −15.2524 −0.595053
$$658$$ 11.0462 0.430627
$$659$$ −22.4157 −0.873190 −0.436595 0.899658i $$-0.643816\pi$$
−0.436595 + 0.899658i $$0.643816\pi$$
$$660$$ 0 0
$$661$$ −14.9937 −0.583189 −0.291594 0.956542i $$-0.594186\pi$$
−0.291594 + 0.956542i $$0.594186\pi$$
$$662$$ −11.8217 −0.459465
$$663$$ 19.4586 0.755708
$$664$$ −8.32320 −0.323003
$$665$$ 0 0
$$666$$ −5.13536 −0.198991
$$667$$ 2.00000 0.0774403
$$668$$ 5.93545 0.229649
$$669$$ 13.9634 0.539855
$$670$$ 0 0
$$671$$ 5.11078 0.197299
$$672$$ 2.00000 0.0771517
$$673$$ 49.2158 1.89713 0.948564 0.316586i $$-0.102536\pi$$
0.948564 + 0.316586i $$0.102536\pi$$
$$674$$ −13.2803 −0.511538
$$675$$ 0 0
$$676$$ 17.5048 0.673261
$$677$$ 5.34153 0.205292 0.102646 0.994718i $$-0.467269\pi$$
0.102646 + 0.994718i $$0.467269\pi$$
$$678$$ 5.79383 0.222511
$$679$$ −2.09246 −0.0803011
$$680$$ 0 0
$$681$$ −4.45231 −0.170613
$$682$$ 2.81215 0.107683
$$683$$ −27.1512 −1.03891 −0.519456 0.854497i $$-0.673865\pi$$
−0.519456 + 0.854497i $$0.673865\pi$$
$$684$$ 8.11704 0.310363
$$685$$ 0 0
$$686$$ 20.0000 0.763604
$$687$$ 4.05249 0.154612
$$688$$ −1.13536 −0.0432852
$$689$$ 7.40940 0.282276
$$690$$ 0 0
$$691$$ 12.1050 0.460495 0.230247 0.973132i $$-0.426046\pi$$
0.230247 + 0.973132i $$0.426046\pi$$
$$692$$ −25.8217 −0.981595
$$693$$ 1.72928 0.0656900
$$694$$ 24.3632 0.924814
$$695$$ 0 0
$$696$$ 2.00000 0.0758098
$$697$$ 12.4123 0.470151
$$698$$ 3.49521 0.132296
$$699$$ −24.5327 −0.927912
$$700$$ 0 0
$$701$$ 2.59392 0.0979711 0.0489856 0.998799i $$-0.484401\pi$$
0.0489856 + 0.998799i $$0.484401\pi$$
$$702$$ 5.52311 0.208457
$$703$$ −41.6839 −1.57214
$$704$$ −0.864641 −0.0325874
$$705$$ 0 0
$$706$$ −31.5789 −1.18849
$$707$$ −18.0925 −0.680437
$$708$$ 10.9817 0.412717
$$709$$ 4.00333 0.150348 0.0751741 0.997170i $$-0.476049\pi$$
0.0751741 + 0.997170i $$0.476049\pi$$
$$710$$ 0 0
$$711$$ −3.79383 −0.142280
$$712$$ 4.77551 0.178970
$$713$$ 3.25240 0.121803
$$714$$ 7.04623 0.263698
$$715$$ 0 0
$$716$$ −23.3449 −0.872438
$$717$$ −10.7755 −0.402419
$$718$$ −21.1878 −0.790723
$$719$$ 20.2707 0.755970 0.377985 0.925812i $$-0.376617\pi$$
0.377985 + 0.925812i $$0.376617\pi$$
$$720$$ 0 0
$$721$$ 22.6339 0.842930
$$722$$ 46.8863 1.74493
$$723$$ 10.7755 0.400746
$$724$$ 13.3694 0.496871
$$725$$ 0 0
$$726$$ 10.2524 0.380502
$$727$$ 32.6339 1.21032 0.605162 0.796102i $$-0.293108\pi$$
0.605162 + 0.796102i $$0.293108\pi$$
$$728$$ 11.0462 0.409400
$$729$$ 1.00000 0.0370370
$$730$$ 0 0
$$731$$ −4.00000 −0.147945
$$732$$ 5.91087 0.218472
$$733$$ 1.13536 0.0419354 0.0209677 0.999780i $$-0.493325\pi$$
0.0209677 + 0.999780i $$0.493325\pi$$
$$734$$ −13.0462 −0.481545
$$735$$ 0 0
$$736$$ −1.00000 −0.0368605
$$737$$ −5.11078 −0.188258
$$738$$ 3.52311 0.129688
$$739$$ −29.4219 −1.08230 −0.541151 0.840925i $$-0.682011\pi$$
−0.541151 + 0.840925i $$0.682011\pi$$
$$740$$ 0 0
$$741$$ 44.8313 1.64692
$$742$$ 2.68305 0.0984980
$$743$$ 27.4586 1.00736 0.503678 0.863891i $$-0.331980\pi$$
0.503678 + 0.863891i $$0.331980\pi$$
$$744$$ 3.25240 0.119239
$$745$$ 0 0
$$746$$ −18.8646 −0.690684
$$747$$ −8.32320 −0.304530
$$748$$ −3.04623 −0.111381
$$749$$ 11.2803 0.412173
$$750$$ 0 0
$$751$$ 4.02791 0.146980 0.0734902 0.997296i $$-0.476586\pi$$
0.0734902 + 0.997296i $$0.476586\pi$$
$$752$$ −5.52311 −0.201407
$$753$$ 3.91087 0.142520
$$754$$ 11.0462 0.402280
$$755$$ 0 0
$$756$$ 2.00000 0.0727393
$$757$$ 31.5110 1.14529 0.572644 0.819804i $$-0.305918\pi$$
0.572644 + 0.819804i $$0.305918\pi$$
$$758$$ 26.7509 0.971638
$$759$$ −0.864641 −0.0313845
$$760$$ 0 0
$$761$$ −19.9634 −0.723671 −0.361836 0.932242i $$-0.617850\pi$$
−0.361836 + 0.932242i $$0.617850\pi$$
$$762$$ 0.747604 0.0270828
$$763$$ 24.3632 0.882006
$$764$$ 12.2341 0.442613
$$765$$ 0 0
$$766$$ 22.5048 0.813131
$$767$$ 60.6531 2.19006
$$768$$ −1.00000 −0.0360844
$$769$$ −46.4190 −1.67391 −0.836956 0.547271i $$-0.815667\pi$$
−0.836956 + 0.547271i $$0.815667\pi$$
$$770$$ 0 0
$$771$$ 28.0925 1.01173
$$772$$ −7.45856 −0.268440
$$773$$ 29.0342 1.04429 0.522143 0.852858i $$-0.325133\pi$$
0.522143 + 0.852858i $$0.325133\pi$$
$$774$$ −1.13536 −0.0408096
$$775$$ 0 0
$$776$$ 1.04623 0.0375574
$$777$$ −10.2707 −0.368460
$$778$$ −7.67680 −0.275226
$$779$$ 28.5972 1.02460
$$780$$ 0 0
$$781$$ 5.00958 0.179257
$$782$$ −3.52311 −0.125986
$$783$$ 2.00000 0.0714742
$$784$$ −3.00000 −0.107143
$$785$$ 0 0
$$786$$ −19.7572 −0.704716
$$787$$ −35.7693 −1.27504 −0.637518 0.770435i $$-0.720039\pi$$
−0.637518 + 0.770435i $$0.720039\pi$$
$$788$$ −2.00000 −0.0712470
$$789$$ 29.5510 1.05204
$$790$$ 0 0
$$791$$ 11.5877 0.412010
$$792$$ −0.864641 −0.0307237
$$793$$ 32.6464 1.15931
$$794$$ 31.7938 1.12832
$$795$$ 0 0
$$796$$ 22.8401 0.809545
$$797$$ 22.8367 0.808919 0.404459 0.914556i $$-0.367460\pi$$
0.404459 + 0.914556i $$0.367460\pi$$
$$798$$ 16.2341 0.574680
$$799$$ −19.4586 −0.688394
$$800$$ 0 0
$$801$$ 4.77551 0.168734
$$802$$ −18.0925 −0.638867
$$803$$ 13.1878 0.465389
$$804$$ −5.91087 −0.208460
$$805$$ 0 0
$$806$$ 17.9634 0.632732
$$807$$ 17.2803 0.608295
$$808$$ 9.04623 0.318245
$$809$$ −36.9171 −1.29794 −0.648969 0.760815i $$-0.724799\pi$$
−0.648969 + 0.760815i $$0.724799\pi$$
$$810$$ 0 0
$$811$$ 51.8217 1.81971 0.909854 0.414929i $$-0.136194\pi$$
0.909854 + 0.414929i $$0.136194\pi$$
$$812$$ 4.00000 0.140372
$$813$$ −9.96336 −0.349430
$$814$$ 4.44024 0.155630
$$815$$ 0 0
$$816$$ −3.52311 −0.123334
$$817$$ −9.21575 −0.322418
$$818$$ 2.00000 0.0699284
$$819$$ 11.0462 0.385986
$$820$$ 0 0
$$821$$ −17.8709 −0.623699 −0.311849 0.950132i $$-0.600948\pi$$
−0.311849 + 0.950132i $$0.600948\pi$$
$$822$$ 0.476886 0.0166333
$$823$$ −9.00958 −0.314054 −0.157027 0.987594i $$-0.550191\pi$$
−0.157027 + 0.987594i $$0.550191\pi$$
$$824$$ −11.3169 −0.394245
$$825$$ 0 0
$$826$$ 21.9634 0.764203
$$827$$ −42.1083 −1.46425 −0.732125 0.681171i $$-0.761471\pi$$
−0.732125 + 0.681171i $$0.761471\pi$$
$$828$$ −1.00000 −0.0347524
$$829$$ −35.5510 −1.23474 −0.617369 0.786674i $$-0.711801\pi$$
−0.617369 + 0.786674i $$0.711801\pi$$
$$830$$ 0 0
$$831$$ 27.0741 0.939191
$$832$$ −5.52311 −0.191480
$$833$$ −10.5693 −0.366206
$$834$$ 3.45856 0.119760
$$835$$ 0 0
$$836$$ −7.01832 −0.242734
$$837$$ 3.25240 0.112419
$$838$$ 20.4523 0.706513
$$839$$ −25.6801 −0.886576 −0.443288 0.896379i $$-0.646188\pi$$
−0.443288 + 0.896379i $$0.646188\pi$$
$$840$$ 0 0
$$841$$ −25.0000 −0.862069
$$842$$ −12.9571 −0.446531
$$843$$ 27.2803 0.939584
$$844$$ 24.2341 0.834171
$$845$$ 0 0
$$846$$ −5.52311 −0.189889
$$847$$ 20.5048 0.704553
$$848$$ −1.34153 −0.0460682
$$849$$ −30.9205 −1.06119
$$850$$ 0 0
$$851$$ 5.13536 0.176038
$$852$$ 5.79383 0.198493
$$853$$ 11.6647 0.399393 0.199696 0.979858i $$-0.436004\pi$$
0.199696 + 0.979858i $$0.436004\pi$$
$$854$$ 11.8217 0.404532
$$855$$ 0 0
$$856$$ −5.64015 −0.192776
$$857$$ −41.5423 −1.41906 −0.709529 0.704677i $$-0.751092\pi$$
−0.709529 + 0.704677i $$0.751092\pi$$
$$858$$ −4.77551 −0.163033
$$859$$ 21.5510 0.735311 0.367656 0.929962i $$-0.380161\pi$$
0.367656 + 0.929962i $$0.380161\pi$$
$$860$$ 0 0
$$861$$ 7.04623 0.240135
$$862$$ 33.3728 1.13668
$$863$$ 6.91713 0.235462 0.117731 0.993046i $$-0.462438\pi$$
0.117731 + 0.993046i $$0.462438\pi$$
$$864$$ −1.00000 −0.0340207
$$865$$ 0 0
$$866$$ −19.0096 −0.645972
$$867$$ 4.58767 0.155805
$$868$$ 6.50479 0.220787
$$869$$ 3.28030 0.111277
$$870$$ 0 0
$$871$$ −32.6464 −1.10618
$$872$$ −12.1816 −0.412521
$$873$$ 1.04623 0.0354095
$$874$$ −8.11704 −0.274563
$$875$$ 0 0
$$876$$ 15.2524 0.515331
$$877$$ −2.53270 −0.0855232 −0.0427616 0.999085i $$-0.513616\pi$$
−0.0427616 + 0.999085i $$0.513616\pi$$
$$878$$ 32.5972 1.10010
$$879$$ 11.0708 0.373409
$$880$$ 0 0
$$881$$ −10.8680 −0.366151 −0.183076 0.983099i $$-0.558605\pi$$
−0.183076 + 0.983099i $$0.558605\pi$$
$$882$$ −3.00000 −0.101015
$$883$$ −40.4190 −1.36021 −0.680104 0.733116i $$-0.738065\pi$$
−0.680104 + 0.733116i $$0.738065\pi$$
$$884$$ −19.4586 −0.654462
$$885$$ 0 0
$$886$$ −4.00000 −0.134383
$$887$$ 7.48647 0.251371 0.125686 0.992070i $$-0.459887\pi$$
0.125686 + 0.992070i $$0.459887\pi$$
$$888$$ 5.13536 0.172331
$$889$$ 1.49521 0.0501477
$$890$$ 0 0
$$891$$ −0.864641 −0.0289666
$$892$$ −13.9634 −0.467528
$$893$$ −44.8313 −1.50022
$$894$$ 8.86464 0.296478
$$895$$ 0 0
$$896$$ −2.00000 −0.0668153
$$897$$ −5.52311 −0.184411
$$898$$ 39.5789 1.32077
$$899$$ 6.50479 0.216947
$$900$$ 0 0
$$901$$ −4.72635 −0.157458
$$902$$ −3.04623 −0.101428
$$903$$ −2.27072 −0.0755648
$$904$$ −5.79383 −0.192700
$$905$$ 0 0
$$906$$ 6.71096 0.222957
$$907$$ 22.6864 0.753289 0.376644 0.926358i $$-0.377078\pi$$
0.376644 + 0.926358i $$0.377078\pi$$
$$908$$ 4.45231 0.147755
$$909$$ 9.04623 0.300044
$$910$$ 0 0
$$911$$ −35.8217 −1.18683 −0.593414 0.804898i $$-0.702220\pi$$
−0.593414 + 0.804898i $$0.702220\pi$$
$$912$$ −8.11704 −0.268782
$$913$$ 7.19658 0.238172
$$914$$ −24.5048 −0.810546
$$915$$ 0 0
$$916$$ −4.05249 −0.133898
$$917$$ −39.5144 −1.30488
$$918$$ −3.52311 −0.116280
$$919$$ −13.2890 −0.438365 −0.219182 0.975684i $$-0.570339\pi$$
−0.219182 + 0.975684i $$0.570339\pi$$
$$920$$ 0 0
$$921$$ 4.23407 0.139517
$$922$$ 18.2341 0.600507
$$923$$ 32.0000 1.05329
$$924$$ −1.72928 −0.0568892
$$925$$ 0 0
$$926$$ 4.20617 0.138223
$$927$$ −11.3169 −0.371697
$$928$$ −2.00000 −0.0656532
$$929$$ 41.1299 1.34943 0.674715 0.738078i $$-0.264267\pi$$
0.674715 + 0.738078i $$0.264267\pi$$
$$930$$ 0 0
$$931$$ −24.3511 −0.798075
$$932$$ 24.5327 0.803595
$$933$$ 13.2803 0.434778
$$934$$ −19.1912 −0.627954
$$935$$ 0 0
$$936$$ −5.52311 −0.180529
$$937$$ −22.7755 −0.744043 −0.372022 0.928224i $$-0.621335\pi$$
−0.372022 + 0.928224i $$0.621335\pi$$
$$938$$ −11.8217 −0.385993
$$939$$ −31.9634 −1.04308
$$940$$ 0 0
$$941$$ −54.7788 −1.78574 −0.892870 0.450315i $$-0.851312\pi$$
−0.892870 + 0.450315i $$0.851312\pi$$
$$942$$ −14.6864 −0.478508
$$943$$ −3.52311 −0.114728
$$944$$ −10.9817 −0.357423
$$945$$ 0 0
$$946$$ 0.981678 0.0319171
$$947$$ 11.4094 0.370756 0.185378 0.982667i $$-0.440649\pi$$
0.185378 + 0.982667i $$0.440649\pi$$
$$948$$ 3.79383 0.123218
$$949$$ 84.2407 2.73457
$$950$$ 0 0
$$951$$ −13.2803 −0.430643
$$952$$ −7.04623 −0.228370
$$953$$ 41.6714 1.34987 0.674934 0.737878i $$-0.264172\pi$$
0.674934 + 0.737878i $$0.264172\pi$$
$$954$$ −1.34153 −0.0434335
$$955$$ 0 0
$$956$$ 10.7755 0.348505
$$957$$ −1.72928 −0.0558997
$$958$$ −25.6801 −0.829687
$$959$$ 0.953771 0.0307989
$$960$$ 0 0
$$961$$ −20.4219 −0.658772
$$962$$ 28.3632 0.914465
$$963$$ −5.64015 −0.181751
$$964$$ −10.7755 −0.347056
$$965$$ 0 0
$$966$$ −2.00000 −0.0643489
$$967$$ 3.17533 0.102112 0.0510559 0.998696i $$-0.483741\pi$$
0.0510559 + 0.998696i $$0.483741\pi$$
$$968$$ −10.2524 −0.329524
$$969$$ −28.5972 −0.918676
$$970$$ 0 0
$$971$$ −35.7326 −1.14671 −0.573357 0.819306i $$-0.694359\pi$$
−0.573357 + 0.819306i $$0.694359\pi$$
$$972$$ −1.00000 −0.0320750
$$973$$ 6.91713 0.221753
$$974$$ −8.74760 −0.280291
$$975$$ 0 0
$$976$$ −5.91087 −0.189202
$$977$$ 2.02791 0.0648785 0.0324392 0.999474i $$-0.489672\pi$$
0.0324392 + 0.999474i $$0.489672\pi$$
$$978$$ −23.2803 −0.744422
$$979$$ −4.12910 −0.131967
$$980$$ 0 0
$$981$$ −12.1816 −0.388928
$$982$$ 23.8863 0.762242
$$983$$ −1.67347 −0.0533754 −0.0266877 0.999644i $$-0.508496\pi$$
−0.0266877 + 0.999644i $$0.508496\pi$$
$$984$$ −3.52311 −0.112313
$$985$$ 0 0
$$986$$ −7.04623 −0.224398
$$987$$ −11.0462 −0.351605
$$988$$ −44.8313 −1.42627
$$989$$ 1.13536 0.0361023
$$990$$ 0 0
$$991$$ −55.8496 −1.77412 −0.887061 0.461652i $$-0.847257\pi$$
−0.887061 + 0.461652i $$0.847257\pi$$
$$992$$ −3.25240 −0.103264
$$993$$ 11.8217 0.375152
$$994$$ 11.5877 0.367538
$$995$$ 0 0
$$996$$ 8.32320 0.263731
$$997$$ 44.2062 1.40002 0.700012 0.714131i $$-0.253178\pi$$
0.700012 + 0.714131i $$0.253178\pi$$
$$998$$ 29.7293 0.941064
$$999$$ 5.13536 0.162476
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 3450.2.a.br.1.2 3
5.2 odd 4 690.2.d.d.139.5 yes 6
5.3 odd 4 690.2.d.d.139.2 6
5.4 even 2 3450.2.a.bq.1.2 3
15.2 even 4 2070.2.d.d.829.2 6
15.8 even 4 2070.2.d.d.829.5 6

By twisted newform
Twist Min Dim Char Parity Ord Type
690.2.d.d.139.2 6 5.3 odd 4
690.2.d.d.139.5 yes 6 5.2 odd 4
2070.2.d.d.829.2 6 15.2 even 4
2070.2.d.d.829.5 6 15.8 even 4
3450.2.a.bq.1.2 3 5.4 even 2
3450.2.a.br.1.2 3 1.1 even 1 trivial