# Properties

 Label 3450.2.a.bh.1.2 Level $3450$ Weight $2$ Character 3450.1 Self dual yes Analytic conductor $27.548$ Analytic rank $0$ Dimension $2$ 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: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{73})$$ Defining polynomial: $$x^{2} - x - 18$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$-3.77200$$ of defining polynomial Character $$\chi$$ $$=$$ 3450.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} -3.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} -3.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +5.77200 q^{11} +1.00000 q^{12} -3.77200 q^{13} +3.00000 q^{14} +1.00000 q^{16} +0.772002 q^{17} -1.00000 q^{18} +7.77200 q^{19} -3.00000 q^{21} -5.77200 q^{22} +1.00000 q^{23} -1.00000 q^{24} +3.77200 q^{26} +1.00000 q^{27} -3.00000 q^{28} +3.00000 q^{29} -9.54400 q^{31} -1.00000 q^{32} +5.77200 q^{33} -0.772002 q^{34} +1.00000 q^{36} +6.77200 q^{37} -7.77200 q^{38} -3.77200 q^{39} -5.77200 q^{41} +3.00000 q^{42} +7.77200 q^{43} +5.77200 q^{44} -1.00000 q^{46} -8.77200 q^{47} +1.00000 q^{48} +2.00000 q^{49} +0.772002 q^{51} -3.77200 q^{52} +4.00000 q^{53} -1.00000 q^{54} +3.00000 q^{56} +7.77200 q^{57} -3.00000 q^{58} +6.00000 q^{59} -9.54400 q^{61} +9.54400 q^{62} -3.00000 q^{63} +1.00000 q^{64} -5.77200 q^{66} -9.54400 q^{67} +0.772002 q^{68} +1.00000 q^{69} -4.77200 q^{71} -1.00000 q^{72} +6.54400 q^{73} -6.77200 q^{74} +7.77200 q^{76} -17.3160 q^{77} +3.77200 q^{78} +2.22800 q^{79} +1.00000 q^{81} +5.77200 q^{82} -1.00000 q^{83} -3.00000 q^{84} -7.77200 q^{86} +3.00000 q^{87} -5.77200 q^{88} +16.7720 q^{89} +11.3160 q^{91} +1.00000 q^{92} -9.54400 q^{93} +8.77200 q^{94} -1.00000 q^{96} +17.5440 q^{97} -2.00000 q^{98} +5.77200 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{2} + 2q^{3} + 2q^{4} - 2q^{6} - 6q^{7} - 2q^{8} + 2q^{9} + O(q^{10})$$ $$2q - 2q^{2} + 2q^{3} + 2q^{4} - 2q^{6} - 6q^{7} - 2q^{8} + 2q^{9} + 3q^{11} + 2q^{12} + q^{13} + 6q^{14} + 2q^{16} - 7q^{17} - 2q^{18} + 7q^{19} - 6q^{21} - 3q^{22} + 2q^{23} - 2q^{24} - q^{26} + 2q^{27} - 6q^{28} + 6q^{29} - 2q^{31} - 2q^{32} + 3q^{33} + 7q^{34} + 2q^{36} + 5q^{37} - 7q^{38} + q^{39} - 3q^{41} + 6q^{42} + 7q^{43} + 3q^{44} - 2q^{46} - 9q^{47} + 2q^{48} + 4q^{49} - 7q^{51} + q^{52} + 8q^{53} - 2q^{54} + 6q^{56} + 7q^{57} - 6q^{58} + 12q^{59} - 2q^{61} + 2q^{62} - 6q^{63} + 2q^{64} - 3q^{66} - 2q^{67} - 7q^{68} + 2q^{69} - q^{71} - 2q^{72} - 4q^{73} - 5q^{74} + 7q^{76} - 9q^{77} - q^{78} + 13q^{79} + 2q^{81} + 3q^{82} - 2q^{83} - 6q^{84} - 7q^{86} + 6q^{87} - 3q^{88} + 25q^{89} - 3q^{91} + 2q^{92} - 2q^{93} + 9q^{94} - 2q^{96} + 18q^{97} - 4q^{98} + 3q^{99} + O(q^{100})$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −1.00000 −0.707107
$$3$$ 1.00000 0.577350
$$4$$ 1.00000 0.500000
$$5$$ 0 0
$$6$$ −1.00000 −0.408248
$$7$$ −3.00000 −1.13389 −0.566947 0.823754i $$-0.691875\pi$$
−0.566947 + 0.823754i $$0.691875\pi$$
$$8$$ −1.00000 −0.353553
$$9$$ 1.00000 0.333333
$$10$$ 0 0
$$11$$ 5.77200 1.74032 0.870162 0.492766i $$-0.164014\pi$$
0.870162 + 0.492766i $$0.164014\pi$$
$$12$$ 1.00000 0.288675
$$13$$ −3.77200 −1.04617 −0.523083 0.852282i $$-0.675218\pi$$
−0.523083 + 0.852282i $$0.675218\pi$$
$$14$$ 3.00000 0.801784
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ 0.772002 0.187238 0.0936190 0.995608i $$-0.470156\pi$$
0.0936190 + 0.995608i $$0.470156\pi$$
$$18$$ −1.00000 −0.235702
$$19$$ 7.77200 1.78302 0.891510 0.453001i $$-0.149647\pi$$
0.891510 + 0.453001i $$0.149647\pi$$
$$20$$ 0 0
$$21$$ −3.00000 −0.654654
$$22$$ −5.77200 −1.23059
$$23$$ 1.00000 0.208514
$$24$$ −1.00000 −0.204124
$$25$$ 0 0
$$26$$ 3.77200 0.739750
$$27$$ 1.00000 0.192450
$$28$$ −3.00000 −0.566947
$$29$$ 3.00000 0.557086 0.278543 0.960424i $$-0.410149\pi$$
0.278543 + 0.960424i $$0.410149\pi$$
$$30$$ 0 0
$$31$$ −9.54400 −1.71415 −0.857077 0.515189i $$-0.827722\pi$$
−0.857077 + 0.515189i $$0.827722\pi$$
$$32$$ −1.00000 −0.176777
$$33$$ 5.77200 1.00478
$$34$$ −0.772002 −0.132397
$$35$$ 0 0
$$36$$ 1.00000 0.166667
$$37$$ 6.77200 1.11331 0.556655 0.830744i $$-0.312084\pi$$
0.556655 + 0.830744i $$0.312084\pi$$
$$38$$ −7.77200 −1.26079
$$39$$ −3.77200 −0.604004
$$40$$ 0 0
$$41$$ −5.77200 −0.901435 −0.450718 0.892667i $$-0.648832\pi$$
−0.450718 + 0.892667i $$0.648832\pi$$
$$42$$ 3.00000 0.462910
$$43$$ 7.77200 1.18522 0.592610 0.805490i $$-0.298098\pi$$
0.592610 + 0.805490i $$0.298098\pi$$
$$44$$ 5.77200 0.870162
$$45$$ 0 0
$$46$$ −1.00000 −0.147442
$$47$$ −8.77200 −1.27953 −0.639764 0.768571i $$-0.720968\pi$$
−0.639764 + 0.768571i $$0.720968\pi$$
$$48$$ 1.00000 0.144338
$$49$$ 2.00000 0.285714
$$50$$ 0 0
$$51$$ 0.772002 0.108102
$$52$$ −3.77200 −0.523083
$$53$$ 4.00000 0.549442 0.274721 0.961524i $$-0.411414\pi$$
0.274721 + 0.961524i $$0.411414\pi$$
$$54$$ −1.00000 −0.136083
$$55$$ 0 0
$$56$$ 3.00000 0.400892
$$57$$ 7.77200 1.02943
$$58$$ −3.00000 −0.393919
$$59$$ 6.00000 0.781133 0.390567 0.920575i $$-0.372279\pi$$
0.390567 + 0.920575i $$0.372279\pi$$
$$60$$ 0 0
$$61$$ −9.54400 −1.22198 −0.610992 0.791637i $$-0.709229\pi$$
−0.610992 + 0.791637i $$0.709229\pi$$
$$62$$ 9.54400 1.21209
$$63$$ −3.00000 −0.377964
$$64$$ 1.00000 0.125000
$$65$$ 0 0
$$66$$ −5.77200 −0.710484
$$67$$ −9.54400 −1.16599 −0.582993 0.812477i $$-0.698118\pi$$
−0.582993 + 0.812477i $$0.698118\pi$$
$$68$$ 0.772002 0.0936190
$$69$$ 1.00000 0.120386
$$70$$ 0 0
$$71$$ −4.77200 −0.566332 −0.283166 0.959071i $$-0.591385\pi$$
−0.283166 + 0.959071i $$0.591385\pi$$
$$72$$ −1.00000 −0.117851
$$73$$ 6.54400 0.765918 0.382959 0.923765i $$-0.374905\pi$$
0.382959 + 0.923765i $$0.374905\pi$$
$$74$$ −6.77200 −0.787229
$$75$$ 0 0
$$76$$ 7.77200 0.891510
$$77$$ −17.3160 −1.97334
$$78$$ 3.77200 0.427095
$$79$$ 2.22800 0.250669 0.125335 0.992115i $$-0.460000\pi$$
0.125335 + 0.992115i $$0.460000\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 5.77200 0.637411
$$83$$ −1.00000 −0.109764 −0.0548821 0.998493i $$-0.517478\pi$$
−0.0548821 + 0.998493i $$0.517478\pi$$
$$84$$ −3.00000 −0.327327
$$85$$ 0 0
$$86$$ −7.77200 −0.838077
$$87$$ 3.00000 0.321634
$$88$$ −5.77200 −0.615297
$$89$$ 16.7720 1.77783 0.888914 0.458073i $$-0.151460\pi$$
0.888914 + 0.458073i $$0.151460\pi$$
$$90$$ 0 0
$$91$$ 11.3160 1.18624
$$92$$ 1.00000 0.104257
$$93$$ −9.54400 −0.989667
$$94$$ 8.77200 0.904763
$$95$$ 0 0
$$96$$ −1.00000 −0.102062
$$97$$ 17.5440 1.78132 0.890662 0.454666i $$-0.150241\pi$$
0.890662 + 0.454666i $$0.150241\pi$$
$$98$$ −2.00000 −0.202031
$$99$$ 5.77200 0.580108
$$100$$ 0 0
$$101$$ 14.3160 1.42450 0.712248 0.701928i $$-0.247677\pi$$
0.712248 + 0.701928i $$0.247677\pi$$
$$102$$ −0.772002 −0.0764396
$$103$$ 11.0000 1.08386 0.541931 0.840423i $$-0.317693\pi$$
0.541931 + 0.840423i $$0.317693\pi$$
$$104$$ 3.77200 0.369875
$$105$$ 0 0
$$106$$ −4.00000 −0.388514
$$107$$ −4.00000 −0.386695 −0.193347 0.981130i $$-0.561934\pi$$
−0.193347 + 0.981130i $$0.561934\pi$$
$$108$$ 1.00000 0.0962250
$$109$$ 14.7720 1.41490 0.707451 0.706763i $$-0.249845\pi$$
0.707451 + 0.706763i $$0.249845\pi$$
$$110$$ 0 0
$$111$$ 6.77200 0.642770
$$112$$ −3.00000 −0.283473
$$113$$ 4.77200 0.448912 0.224456 0.974484i $$-0.427939\pi$$
0.224456 + 0.974484i $$0.427939\pi$$
$$114$$ −7.77200 −0.727915
$$115$$ 0 0
$$116$$ 3.00000 0.278543
$$117$$ −3.77200 −0.348722
$$118$$ −6.00000 −0.552345
$$119$$ −2.31601 −0.212308
$$120$$ 0 0
$$121$$ 22.3160 2.02873
$$122$$ 9.54400 0.864073
$$123$$ −5.77200 −0.520444
$$124$$ −9.54400 −0.857077
$$125$$ 0 0
$$126$$ 3.00000 0.267261
$$127$$ 4.45600 0.395406 0.197703 0.980262i $$-0.436652\pi$$
0.197703 + 0.980262i $$0.436652\pi$$
$$128$$ −1.00000 −0.0883883
$$129$$ 7.77200 0.684287
$$130$$ 0 0
$$131$$ −19.0880 −1.66773 −0.833863 0.551971i $$-0.813876\pi$$
−0.833863 + 0.551971i $$0.813876\pi$$
$$132$$ 5.77200 0.502388
$$133$$ −23.3160 −2.02175
$$134$$ 9.54400 0.824476
$$135$$ 0 0
$$136$$ −0.772002 −0.0661986
$$137$$ 14.7720 1.26206 0.631029 0.775760i $$-0.282633\pi$$
0.631029 + 0.775760i $$0.282633\pi$$
$$138$$ −1.00000 −0.0851257
$$139$$ −0.772002 −0.0654803 −0.0327402 0.999464i $$-0.510423\pi$$
−0.0327402 + 0.999464i $$0.510423\pi$$
$$140$$ 0 0
$$141$$ −8.77200 −0.738736
$$142$$ 4.77200 0.400458
$$143$$ −21.7720 −1.82067
$$144$$ 1.00000 0.0833333
$$145$$ 0 0
$$146$$ −6.54400 −0.541586
$$147$$ 2.00000 0.164957
$$148$$ 6.77200 0.556655
$$149$$ −11.0880 −0.908365 −0.454182 0.890909i $$-0.650069\pi$$
−0.454182 + 0.890909i $$0.650069\pi$$
$$150$$ 0 0
$$151$$ 23.5440 1.91598 0.957992 0.286795i $$-0.0925899\pi$$
0.957992 + 0.286795i $$0.0925899\pi$$
$$152$$ −7.77200 −0.630393
$$153$$ 0.772002 0.0624127
$$154$$ 17.3160 1.39536
$$155$$ 0 0
$$156$$ −3.77200 −0.302002
$$157$$ 13.5440 1.08093 0.540465 0.841367i $$-0.318248\pi$$
0.540465 + 0.841367i $$0.318248\pi$$
$$158$$ −2.22800 −0.177250
$$159$$ 4.00000 0.317221
$$160$$ 0 0
$$161$$ −3.00000 −0.236433
$$162$$ −1.00000 −0.0785674
$$163$$ 16.0000 1.25322 0.626608 0.779334i $$-0.284443\pi$$
0.626608 + 0.779334i $$0.284443\pi$$
$$164$$ −5.77200 −0.450718
$$165$$ 0 0
$$166$$ 1.00000 0.0776151
$$167$$ 16.3160 1.26257 0.631285 0.775551i $$-0.282528\pi$$
0.631285 + 0.775551i $$0.282528\pi$$
$$168$$ 3.00000 0.231455
$$169$$ 1.22800 0.0944614
$$170$$ 0 0
$$171$$ 7.77200 0.594340
$$172$$ 7.77200 0.592610
$$173$$ −15.7720 −1.19912 −0.599562 0.800329i $$-0.704658\pi$$
−0.599562 + 0.800329i $$0.704658\pi$$
$$174$$ −3.00000 −0.227429
$$175$$ 0 0
$$176$$ 5.77200 0.435081
$$177$$ 6.00000 0.450988
$$178$$ −16.7720 −1.25711
$$179$$ 7.54400 0.563865 0.281933 0.959434i $$-0.409025\pi$$
0.281933 + 0.959434i $$0.409025\pi$$
$$180$$ 0 0
$$181$$ 14.3160 1.06410 0.532050 0.846713i $$-0.321422\pi$$
0.532050 + 0.846713i $$0.321422\pi$$
$$182$$ −11.3160 −0.838798
$$183$$ −9.54400 −0.705513
$$184$$ −1.00000 −0.0737210
$$185$$ 0 0
$$186$$ 9.54400 0.699800
$$187$$ 4.45600 0.325855
$$188$$ −8.77200 −0.639764
$$189$$ −3.00000 −0.218218
$$190$$ 0 0
$$191$$ −7.31601 −0.529368 −0.264684 0.964335i $$-0.585268\pi$$
−0.264684 + 0.964335i $$0.585268\pi$$
$$192$$ 1.00000 0.0721688
$$193$$ 4.77200 0.343496 0.171748 0.985141i $$-0.445058\pi$$
0.171748 + 0.985141i $$0.445058\pi$$
$$194$$ −17.5440 −1.25959
$$195$$ 0 0
$$196$$ 2.00000 0.142857
$$197$$ −13.0000 −0.926212 −0.463106 0.886303i $$-0.653265\pi$$
−0.463106 + 0.886303i $$0.653265\pi$$
$$198$$ −5.77200 −0.410198
$$199$$ 12.0880 0.856896 0.428448 0.903566i $$-0.359060\pi$$
0.428448 + 0.903566i $$0.359060\pi$$
$$200$$ 0 0
$$201$$ −9.54400 −0.673182
$$202$$ −14.3160 −1.00727
$$203$$ −9.00000 −0.631676
$$204$$ 0.772002 0.0540509
$$205$$ 0 0
$$206$$ −11.0000 −0.766406
$$207$$ 1.00000 0.0695048
$$208$$ −3.77200 −0.261541
$$209$$ 44.8600 3.10303
$$210$$ 0 0
$$211$$ −10.3160 −0.710183 −0.355092 0.934832i $$-0.615550\pi$$
−0.355092 + 0.934832i $$0.615550\pi$$
$$212$$ 4.00000 0.274721
$$213$$ −4.77200 −0.326972
$$214$$ 4.00000 0.273434
$$215$$ 0 0
$$216$$ −1.00000 −0.0680414
$$217$$ 28.6320 1.94367
$$218$$ −14.7720 −1.00049
$$219$$ 6.54400 0.442203
$$220$$ 0 0
$$221$$ −2.91199 −0.195882
$$222$$ −6.77200 −0.454507
$$223$$ 10.0000 0.669650 0.334825 0.942280i $$-0.391323\pi$$
0.334825 + 0.942280i $$0.391323\pi$$
$$224$$ 3.00000 0.200446
$$225$$ 0 0
$$226$$ −4.77200 −0.317429
$$227$$ −10.7720 −0.714963 −0.357481 0.933920i $$-0.616364\pi$$
−0.357481 + 0.933920i $$0.616364\pi$$
$$228$$ 7.77200 0.514713
$$229$$ −6.45600 −0.426624 −0.213312 0.976984i $$-0.568425\pi$$
−0.213312 + 0.976984i $$0.568425\pi$$
$$230$$ 0 0
$$231$$ −17.3160 −1.13931
$$232$$ −3.00000 −0.196960
$$233$$ −2.68399 −0.175834 −0.0879172 0.996128i $$-0.528021\pi$$
−0.0879172 + 0.996128i $$0.528021\pi$$
$$234$$ 3.77200 0.246583
$$235$$ 0 0
$$236$$ 6.00000 0.390567
$$237$$ 2.22800 0.144724
$$238$$ 2.31601 0.150124
$$239$$ 22.7720 1.47300 0.736499 0.676438i $$-0.236478\pi$$
0.736499 + 0.676438i $$0.236478\pi$$
$$240$$ 0 0
$$241$$ −26.6320 −1.71552 −0.857759 0.514051i $$-0.828144\pi$$
−0.857759 + 0.514051i $$0.828144\pi$$
$$242$$ −22.3160 −1.43453
$$243$$ 1.00000 0.0641500
$$244$$ −9.54400 −0.610992
$$245$$ 0 0
$$246$$ 5.77200 0.368009
$$247$$ −29.3160 −1.86533
$$248$$ 9.54400 0.606045
$$249$$ −1.00000 −0.0633724
$$250$$ 0 0
$$251$$ −3.68399 −0.232532 −0.116266 0.993218i $$-0.537092\pi$$
−0.116266 + 0.993218i $$0.537092\pi$$
$$252$$ −3.00000 −0.188982
$$253$$ 5.77200 0.362883
$$254$$ −4.45600 −0.279594
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ −9.08801 −0.566894 −0.283447 0.958988i $$-0.591478\pi$$
−0.283447 + 0.958988i $$0.591478\pi$$
$$258$$ −7.77200 −0.483864
$$259$$ −20.3160 −1.26238
$$260$$ 0 0
$$261$$ 3.00000 0.185695
$$262$$ 19.0880 1.17926
$$263$$ −6.45600 −0.398094 −0.199047 0.979990i $$-0.563785\pi$$
−0.199047 + 0.979990i $$0.563785\pi$$
$$264$$ −5.77200 −0.355242
$$265$$ 0 0
$$266$$ 23.3160 1.42960
$$267$$ 16.7720 1.02643
$$268$$ −9.54400 −0.582993
$$269$$ −26.8600 −1.63768 −0.818842 0.574019i $$-0.805383\pi$$
−0.818842 + 0.574019i $$0.805383\pi$$
$$270$$ 0 0
$$271$$ 16.0000 0.971931 0.485965 0.873978i $$-0.338468\pi$$
0.485965 + 0.873978i $$0.338468\pi$$
$$272$$ 0.772002 0.0468095
$$273$$ 11.3160 0.684876
$$274$$ −14.7720 −0.892409
$$275$$ 0 0
$$276$$ 1.00000 0.0601929
$$277$$ −8.22800 −0.494372 −0.247186 0.968968i $$-0.579506\pi$$
−0.247186 + 0.968968i $$0.579506\pi$$
$$278$$ 0.772002 0.0463016
$$279$$ −9.54400 −0.571385
$$280$$ 0 0
$$281$$ 15.8600 0.946129 0.473064 0.881028i $$-0.343148\pi$$
0.473064 + 0.881028i $$0.343148\pi$$
$$282$$ 8.77200 0.522365
$$283$$ 28.0000 1.66443 0.832214 0.554455i $$-0.187073\pi$$
0.832214 + 0.554455i $$0.187073\pi$$
$$284$$ −4.77200 −0.283166
$$285$$ 0 0
$$286$$ 21.7720 1.28741
$$287$$ 17.3160 1.02213
$$288$$ −1.00000 −0.0589256
$$289$$ −16.4040 −0.964942
$$290$$ 0 0
$$291$$ 17.5440 1.02845
$$292$$ 6.54400 0.382959
$$293$$ −9.54400 −0.557567 −0.278783 0.960354i $$-0.589931\pi$$
−0.278783 + 0.960354i $$0.589931\pi$$
$$294$$ −2.00000 −0.116642
$$295$$ 0 0
$$296$$ −6.77200 −0.393615
$$297$$ 5.77200 0.334926
$$298$$ 11.0880 0.642311
$$299$$ −3.77200 −0.218141
$$300$$ 0 0
$$301$$ −23.3160 −1.34391
$$302$$ −23.5440 −1.35481
$$303$$ 14.3160 0.822433
$$304$$ 7.77200 0.445755
$$305$$ 0 0
$$306$$ −0.772002 −0.0441324
$$307$$ −20.3160 −1.15950 −0.579748 0.814796i $$-0.696849\pi$$
−0.579748 + 0.814796i $$0.696849\pi$$
$$308$$ −17.3160 −0.986671
$$309$$ 11.0000 0.625768
$$310$$ 0 0
$$311$$ −17.8600 −1.01275 −0.506374 0.862314i $$-0.669015\pi$$
−0.506374 + 0.862314i $$0.669015\pi$$
$$312$$ 3.77200 0.213548
$$313$$ −8.00000 −0.452187 −0.226093 0.974106i $$-0.572595\pi$$
−0.226093 + 0.974106i $$0.572595\pi$$
$$314$$ −13.5440 −0.764332
$$315$$ 0 0
$$316$$ 2.22800 0.125335
$$317$$ −9.00000 −0.505490 −0.252745 0.967533i $$-0.581333\pi$$
−0.252745 + 0.967533i $$0.581333\pi$$
$$318$$ −4.00000 −0.224309
$$319$$ 17.3160 0.969510
$$320$$ 0 0
$$321$$ −4.00000 −0.223258
$$322$$ 3.00000 0.167183
$$323$$ 6.00000 0.333849
$$324$$ 1.00000 0.0555556
$$325$$ 0 0
$$326$$ −16.0000 −0.886158
$$327$$ 14.7720 0.816894
$$328$$ 5.77200 0.318705
$$329$$ 26.3160 1.45085
$$330$$ 0 0
$$331$$ 34.7720 1.91124 0.955621 0.294599i $$-0.0951860\pi$$
0.955621 + 0.294599i $$0.0951860\pi$$
$$332$$ −1.00000 −0.0548821
$$333$$ 6.77200 0.371103
$$334$$ −16.3160 −0.892772
$$335$$ 0 0
$$336$$ −3.00000 −0.163663
$$337$$ 4.00000 0.217894 0.108947 0.994048i $$-0.465252\pi$$
0.108947 + 0.994048i $$0.465252\pi$$
$$338$$ −1.22800 −0.0667943
$$339$$ 4.77200 0.259180
$$340$$ 0 0
$$341$$ −55.0880 −2.98318
$$342$$ −7.77200 −0.420262
$$343$$ 15.0000 0.809924
$$344$$ −7.77200 −0.419038
$$345$$ 0 0
$$346$$ 15.7720 0.847908
$$347$$ 9.54400 0.512349 0.256174 0.966631i $$-0.417538\pi$$
0.256174 + 0.966631i $$0.417538\pi$$
$$348$$ 3.00000 0.160817
$$349$$ −0.227998 −0.0122045 −0.00610223 0.999981i $$-0.501942\pi$$
−0.00610223 + 0.999981i $$0.501942\pi$$
$$350$$ 0 0
$$351$$ −3.77200 −0.201335
$$352$$ −5.77200 −0.307649
$$353$$ −18.8600 −1.00382 −0.501909 0.864921i $$-0.667369\pi$$
−0.501909 + 0.864921i $$0.667369\pi$$
$$354$$ −6.00000 −0.318896
$$355$$ 0 0
$$356$$ 16.7720 0.888914
$$357$$ −2.31601 −0.122576
$$358$$ −7.54400 −0.398713
$$359$$ −10.2280 −0.539813 −0.269907 0.962887i $$-0.586993\pi$$
−0.269907 + 0.962887i $$0.586993\pi$$
$$360$$ 0 0
$$361$$ 41.4040 2.17916
$$362$$ −14.3160 −0.752433
$$363$$ 22.3160 1.17129
$$364$$ 11.3160 0.593120
$$365$$ 0 0
$$366$$ 9.54400 0.498873
$$367$$ 33.3160 1.73908 0.869541 0.493861i $$-0.164415\pi$$
0.869541 + 0.493861i $$0.164415\pi$$
$$368$$ 1.00000 0.0521286
$$369$$ −5.77200 −0.300478
$$370$$ 0 0
$$371$$ −12.0000 −0.623009
$$372$$ −9.54400 −0.494834
$$373$$ −12.7720 −0.661309 −0.330655 0.943752i $$-0.607270\pi$$
−0.330655 + 0.943752i $$0.607270\pi$$
$$374$$ −4.45600 −0.230414
$$375$$ 0 0
$$376$$ 8.77200 0.452381
$$377$$ −11.3160 −0.582804
$$378$$ 3.00000 0.154303
$$379$$ −23.0880 −1.18595 −0.592976 0.805220i $$-0.702047\pi$$
−0.592976 + 0.805220i $$0.702047\pi$$
$$380$$ 0 0
$$381$$ 4.45600 0.228288
$$382$$ 7.31601 0.374319
$$383$$ 13.7720 0.703716 0.351858 0.936053i $$-0.385550\pi$$
0.351858 + 0.936053i $$0.385550\pi$$
$$384$$ −1.00000 −0.0510310
$$385$$ 0 0
$$386$$ −4.77200 −0.242889
$$387$$ 7.77200 0.395073
$$388$$ 17.5440 0.890662
$$389$$ 18.0000 0.912636 0.456318 0.889817i $$-0.349168\pi$$
0.456318 + 0.889817i $$0.349168\pi$$
$$390$$ 0 0
$$391$$ 0.772002 0.0390418
$$392$$ −2.00000 −0.101015
$$393$$ −19.0880 −0.962863
$$394$$ 13.0000 0.654931
$$395$$ 0 0
$$396$$ 5.77200 0.290054
$$397$$ −14.6320 −0.734360 −0.367180 0.930150i $$-0.619677\pi$$
−0.367180 + 0.930150i $$0.619677\pi$$
$$398$$ −12.0880 −0.605917
$$399$$ −23.3160 −1.16726
$$400$$ 0 0
$$401$$ −28.6320 −1.42981 −0.714907 0.699219i $$-0.753531\pi$$
−0.714907 + 0.699219i $$0.753531\pi$$
$$402$$ 9.54400 0.476012
$$403$$ 36.0000 1.79329
$$404$$ 14.3160 0.712248
$$405$$ 0 0
$$406$$ 9.00000 0.446663
$$407$$ 39.0880 1.93752
$$408$$ −0.772002 −0.0382198
$$409$$ −23.0000 −1.13728 −0.568638 0.822588i $$-0.692530\pi$$
−0.568638 + 0.822588i $$0.692530\pi$$
$$410$$ 0 0
$$411$$ 14.7720 0.728649
$$412$$ 11.0000 0.541931
$$413$$ −18.0000 −0.885722
$$414$$ −1.00000 −0.0491473
$$415$$ 0 0
$$416$$ 3.77200 0.184938
$$417$$ −0.772002 −0.0378051
$$418$$ −44.8600 −2.19417
$$419$$ 25.6320 1.25221 0.626103 0.779740i $$-0.284649\pi$$
0.626103 + 0.779740i $$0.284649\pi$$
$$420$$ 0 0
$$421$$ 22.0000 1.07221 0.536107 0.844150i $$-0.319894\pi$$
0.536107 + 0.844150i $$0.319894\pi$$
$$422$$ 10.3160 0.502175
$$423$$ −8.77200 −0.426509
$$424$$ −4.00000 −0.194257
$$425$$ 0 0
$$426$$ 4.77200 0.231204
$$427$$ 28.6320 1.38560
$$428$$ −4.00000 −0.193347
$$429$$ −21.7720 −1.05116
$$430$$ 0 0
$$431$$ 24.6320 1.18648 0.593241 0.805025i $$-0.297848\pi$$
0.593241 + 0.805025i $$0.297848\pi$$
$$432$$ 1.00000 0.0481125
$$433$$ 6.45600 0.310255 0.155128 0.987894i $$-0.450421\pi$$
0.155128 + 0.987894i $$0.450421\pi$$
$$434$$ −28.6320 −1.37438
$$435$$ 0 0
$$436$$ 14.7720 0.707451
$$437$$ 7.77200 0.371785
$$438$$ −6.54400 −0.312685
$$439$$ −12.0000 −0.572729 −0.286364 0.958121i $$-0.592447\pi$$
−0.286364 + 0.958121i $$0.592447\pi$$
$$440$$ 0 0
$$441$$ 2.00000 0.0952381
$$442$$ 2.91199 0.138509
$$443$$ 24.0000 1.14027 0.570137 0.821549i $$-0.306890\pi$$
0.570137 + 0.821549i $$0.306890\pi$$
$$444$$ 6.77200 0.321385
$$445$$ 0 0
$$446$$ −10.0000 −0.473514
$$447$$ −11.0880 −0.524445
$$448$$ −3.00000 −0.141737
$$449$$ −2.00000 −0.0943858 −0.0471929 0.998886i $$-0.515028\pi$$
−0.0471929 + 0.998886i $$0.515028\pi$$
$$450$$ 0 0
$$451$$ −33.3160 −1.56879
$$452$$ 4.77200 0.224456
$$453$$ 23.5440 1.10619
$$454$$ 10.7720 0.505555
$$455$$ 0 0
$$456$$ −7.77200 −0.363957
$$457$$ −30.6320 −1.43291 −0.716453 0.697636i $$-0.754235\pi$$
−0.716453 + 0.697636i $$0.754235\pi$$
$$458$$ 6.45600 0.301669
$$459$$ 0.772002 0.0360340
$$460$$ 0 0
$$461$$ 12.5440 0.584232 0.292116 0.956383i $$-0.405641\pi$$
0.292116 + 0.956383i $$0.405641\pi$$
$$462$$ 17.3160 0.805613
$$463$$ 0.911993 0.0423839 0.0211919 0.999775i $$-0.493254\pi$$
0.0211919 + 0.999775i $$0.493254\pi$$
$$464$$ 3.00000 0.139272
$$465$$ 0 0
$$466$$ 2.68399 0.124334
$$467$$ 18.5440 0.858114 0.429057 0.903277i $$-0.358846\pi$$
0.429057 + 0.903277i $$0.358846\pi$$
$$468$$ −3.77200 −0.174361
$$469$$ 28.6320 1.32210
$$470$$ 0 0
$$471$$ 13.5440 0.624075
$$472$$ −6.00000 −0.276172
$$473$$ 44.8600 2.06267
$$474$$ −2.22800 −0.102335
$$475$$ 0 0
$$476$$ −2.31601 −0.106154
$$477$$ 4.00000 0.183147
$$478$$ −22.7720 −1.04157
$$479$$ −35.3160 −1.61363 −0.806815 0.590805i $$-0.798810\pi$$
−0.806815 + 0.590805i $$0.798810\pi$$
$$480$$ 0 0
$$481$$ −25.5440 −1.16471
$$482$$ 26.6320 1.21305
$$483$$ −3.00000 −0.136505
$$484$$ 22.3160 1.01436
$$485$$ 0 0
$$486$$ −1.00000 −0.0453609
$$487$$ 12.4560 0.564435 0.282218 0.959350i $$-0.408930\pi$$
0.282218 + 0.959350i $$0.408930\pi$$
$$488$$ 9.54400 0.432037
$$489$$ 16.0000 0.723545
$$490$$ 0 0
$$491$$ −10.4560 −0.471873 −0.235936 0.971769i $$-0.575816\pi$$
−0.235936 + 0.971769i $$0.575816\pi$$
$$492$$ −5.77200 −0.260222
$$493$$ 2.31601 0.104308
$$494$$ 29.3160 1.31899
$$495$$ 0 0
$$496$$ −9.54400 −0.428538
$$497$$ 14.3160 0.642161
$$498$$ 1.00000 0.0448111
$$499$$ −9.68399 −0.433515 −0.216758 0.976225i $$-0.569548\pi$$
−0.216758 + 0.976225i $$0.569548\pi$$
$$500$$ 0 0
$$501$$ 16.3160 0.728945
$$502$$ 3.68399 0.164425
$$503$$ −39.9480 −1.78119 −0.890597 0.454793i $$-0.849713\pi$$
−0.890597 + 0.454793i $$0.849713\pi$$
$$504$$ 3.00000 0.133631
$$505$$ 0 0
$$506$$ −5.77200 −0.256597
$$507$$ 1.22800 0.0545373
$$508$$ 4.45600 0.197703
$$509$$ 27.8600 1.23487 0.617437 0.786621i $$-0.288171\pi$$
0.617437 + 0.786621i $$0.288171\pi$$
$$510$$ 0 0
$$511$$ −19.6320 −0.868469
$$512$$ −1.00000 −0.0441942
$$513$$ 7.77200 0.343142
$$514$$ 9.08801 0.400855
$$515$$ 0 0
$$516$$ 7.77200 0.342143
$$517$$ −50.6320 −2.22679
$$518$$ 20.3160 0.892634
$$519$$ −15.7720 −0.692314
$$520$$ 0 0
$$521$$ 6.77200 0.296687 0.148343 0.988936i $$-0.452606\pi$$
0.148343 + 0.988936i $$0.452606\pi$$
$$522$$ −3.00000 −0.131306
$$523$$ −13.3160 −0.582268 −0.291134 0.956682i $$-0.594033\pi$$
−0.291134 + 0.956682i $$0.594033\pi$$
$$524$$ −19.0880 −0.833863
$$525$$ 0 0
$$526$$ 6.45600 0.281495
$$527$$ −7.36799 −0.320955
$$528$$ 5.77200 0.251194
$$529$$ 1.00000 0.0434783
$$530$$ 0 0
$$531$$ 6.00000 0.260378
$$532$$ −23.3160 −1.01088
$$533$$ 21.7720 0.943050
$$534$$ −16.7720 −0.725796
$$535$$ 0 0
$$536$$ 9.54400 0.412238
$$537$$ 7.54400 0.325548
$$538$$ 26.8600 1.15802
$$539$$ 11.5440 0.497235
$$540$$ 0 0
$$541$$ −10.6840 −0.459341 −0.229670 0.973268i $$-0.573765\pi$$
−0.229670 + 0.973268i $$0.573765\pi$$
$$542$$ −16.0000 −0.687259
$$543$$ 14.3160 0.614359
$$544$$ −0.772002 −0.0330993
$$545$$ 0 0
$$546$$ −11.3160 −0.484280
$$547$$ 11.8600 0.507097 0.253549 0.967323i $$-0.418402\pi$$
0.253549 + 0.967323i $$0.418402\pi$$
$$548$$ 14.7720 0.631029
$$549$$ −9.54400 −0.407328
$$550$$ 0 0
$$551$$ 23.3160 0.993295
$$552$$ −1.00000 −0.0425628
$$553$$ −6.68399 −0.284232
$$554$$ 8.22800 0.349574
$$555$$ 0 0
$$556$$ −0.772002 −0.0327402
$$557$$ −21.0880 −0.893528 −0.446764 0.894652i $$-0.647424\pi$$
−0.446764 + 0.894652i $$0.647424\pi$$
$$558$$ 9.54400 0.404030
$$559$$ −29.3160 −1.23993
$$560$$ 0 0
$$561$$ 4.45600 0.188132
$$562$$ −15.8600 −0.669014
$$563$$ 43.9480 1.85219 0.926094 0.377293i $$-0.123145\pi$$
0.926094 + 0.377293i $$0.123145\pi$$
$$564$$ −8.77200 −0.369368
$$565$$ 0 0
$$566$$ −28.0000 −1.17693
$$567$$ −3.00000 −0.125988
$$568$$ 4.77200 0.200229
$$569$$ 30.0000 1.25767 0.628833 0.777541i $$-0.283533\pi$$
0.628833 + 0.777541i $$0.283533\pi$$
$$570$$ 0 0
$$571$$ 12.6320 0.528633 0.264317 0.964436i $$-0.414854\pi$$
0.264317 + 0.964436i $$0.414854\pi$$
$$572$$ −21.7720 −0.910333
$$573$$ −7.31601 −0.305631
$$574$$ −17.3160 −0.722756
$$575$$ 0 0
$$576$$ 1.00000 0.0416667
$$577$$ −31.3160 −1.30370 −0.651851 0.758347i $$-0.726007\pi$$
−0.651851 + 0.758347i $$0.726007\pi$$
$$578$$ 16.4040 0.682317
$$579$$ 4.77200 0.198318
$$580$$ 0 0
$$581$$ 3.00000 0.124461
$$582$$ −17.5440 −0.727222
$$583$$ 23.0880 0.956208
$$584$$ −6.54400 −0.270793
$$585$$ 0 0
$$586$$ 9.54400 0.394259
$$587$$ 24.4560 1.00941 0.504703 0.863293i $$-0.331602\pi$$
0.504703 + 0.863293i $$0.331602\pi$$
$$588$$ 2.00000 0.0824786
$$589$$ −74.1760 −3.05637
$$590$$ 0 0
$$591$$ −13.0000 −0.534749
$$592$$ 6.77200 0.278328
$$593$$ 41.3160 1.69664 0.848322 0.529480i $$-0.177613\pi$$
0.848322 + 0.529480i $$0.177613\pi$$
$$594$$ −5.77200 −0.236828
$$595$$ 0 0
$$596$$ −11.0880 −0.454182
$$597$$ 12.0880 0.494729
$$598$$ 3.77200 0.154249
$$599$$ 8.00000 0.326871 0.163436 0.986554i $$-0.447742\pi$$
0.163436 + 0.986554i $$0.447742\pi$$
$$600$$ 0 0
$$601$$ −32.7720 −1.33680 −0.668399 0.743803i $$-0.733020\pi$$
−0.668399 + 0.743803i $$0.733020\pi$$
$$602$$ 23.3160 0.950289
$$603$$ −9.54400 −0.388662
$$604$$ 23.5440 0.957992
$$605$$ 0 0
$$606$$ −14.3160 −0.581548
$$607$$ 5.08801 0.206516 0.103258 0.994655i $$-0.467073\pi$$
0.103258 + 0.994655i $$0.467073\pi$$
$$608$$ −7.77200 −0.315196
$$609$$ −9.00000 −0.364698
$$610$$ 0 0
$$611$$ 33.0880 1.33860
$$612$$ 0.772002 0.0312063
$$613$$ 20.7720 0.838973 0.419487 0.907762i $$-0.362210\pi$$
0.419487 + 0.907762i $$0.362210\pi$$
$$614$$ 20.3160 0.819887
$$615$$ 0 0
$$616$$ 17.3160 0.697682
$$617$$ −18.0000 −0.724653 −0.362326 0.932051i $$-0.618017\pi$$
−0.362326 + 0.932051i $$0.618017\pi$$
$$618$$ −11.0000 −0.442485
$$619$$ −0.911993 −0.0366561 −0.0183280 0.999832i $$-0.505834\pi$$
−0.0183280 + 0.999832i $$0.505834\pi$$
$$620$$ 0 0
$$621$$ 1.00000 0.0401286
$$622$$ 17.8600 0.716121
$$623$$ −50.3160 −2.01587
$$624$$ −3.77200 −0.151001
$$625$$ 0 0
$$626$$ 8.00000 0.319744
$$627$$ 44.8600 1.79154
$$628$$ 13.5440 0.540465
$$629$$ 5.22800 0.208454
$$630$$ 0 0
$$631$$ 14.0880 0.560835 0.280417 0.959878i $$-0.409527\pi$$
0.280417 + 0.959878i $$0.409527\pi$$
$$632$$ −2.22800 −0.0886250
$$633$$ −10.3160 −0.410024
$$634$$ 9.00000 0.357436
$$635$$ 0 0
$$636$$ 4.00000 0.158610
$$637$$ −7.54400 −0.298904
$$638$$ −17.3160 −0.685547
$$639$$ −4.77200 −0.188777
$$640$$ 0 0
$$641$$ −17.8600 −0.705428 −0.352714 0.935731i $$-0.614741\pi$$
−0.352714 + 0.935731i $$0.614741\pi$$
$$642$$ 4.00000 0.157867
$$643$$ −19.3160 −0.761749 −0.380874 0.924627i $$-0.624377\pi$$
−0.380874 + 0.924627i $$0.624377\pi$$
$$644$$ −3.00000 −0.118217
$$645$$ 0 0
$$646$$ −6.00000 −0.236067
$$647$$ −30.7720 −1.20977 −0.604886 0.796312i $$-0.706781\pi$$
−0.604886 + 0.796312i $$0.706781\pi$$
$$648$$ −1.00000 −0.0392837
$$649$$ 34.6320 1.35943
$$650$$ 0 0
$$651$$ 28.6320 1.12218
$$652$$ 16.0000 0.626608
$$653$$ 23.4560 0.917904 0.458952 0.888461i $$-0.348225\pi$$
0.458952 + 0.888461i $$0.348225\pi$$
$$654$$ −14.7720 −0.577631
$$655$$ 0 0
$$656$$ −5.77200 −0.225359
$$657$$ 6.54400 0.255306
$$658$$ −26.3160 −1.02590
$$659$$ −43.6320 −1.69966 −0.849831 0.527055i $$-0.823296\pi$$
−0.849831 + 0.527055i $$0.823296\pi$$
$$660$$ 0 0
$$661$$ −17.8600 −0.694674 −0.347337 0.937740i $$-0.612914\pi$$
−0.347337 + 0.937740i $$0.612914\pi$$
$$662$$ −34.7720 −1.35145
$$663$$ −2.91199 −0.113092
$$664$$ 1.00000 0.0388075
$$665$$ 0 0
$$666$$ −6.77200 −0.262410
$$667$$ 3.00000 0.116160
$$668$$ 16.3160 0.631285
$$669$$ 10.0000 0.386622
$$670$$ 0 0
$$671$$ −55.0880 −2.12665
$$672$$ 3.00000 0.115728
$$673$$ −26.5440 −1.02320 −0.511598 0.859225i $$-0.670946\pi$$
−0.511598 + 0.859225i $$0.670946\pi$$
$$674$$ −4.00000 −0.154074
$$675$$ 0 0
$$676$$ 1.22800 0.0472307
$$677$$ −48.0000 −1.84479 −0.922395 0.386248i $$-0.873771\pi$$
−0.922395 + 0.386248i $$0.873771\pi$$
$$678$$ −4.77200 −0.183268
$$679$$ −52.6320 −2.01983
$$680$$ 0 0
$$681$$ −10.7720 −0.412784
$$682$$ 55.0880 2.10943
$$683$$ −26.6320 −1.01905 −0.509523 0.860457i $$-0.670178\pi$$
−0.509523 + 0.860457i $$0.670178\pi$$
$$684$$ 7.77200 0.297170
$$685$$ 0 0
$$686$$ −15.0000 −0.572703
$$687$$ −6.45600 −0.246312
$$688$$ 7.77200 0.296305
$$689$$ −15.0880 −0.574807
$$690$$ 0 0
$$691$$ −39.2280 −1.49230 −0.746152 0.665776i $$-0.768101\pi$$
−0.746152 + 0.665776i $$0.768101\pi$$
$$692$$ −15.7720 −0.599562
$$693$$ −17.3160 −0.657781
$$694$$ −9.54400 −0.362285
$$695$$ 0 0
$$696$$ −3.00000 −0.113715
$$697$$ −4.45600 −0.168783
$$698$$ 0.227998 0.00862986
$$699$$ −2.68399 −0.101518
$$700$$ 0 0
$$701$$ 1.54400 0.0583162 0.0291581 0.999575i $$-0.490717\pi$$
0.0291581 + 0.999575i $$0.490717\pi$$
$$702$$ 3.77200 0.142365
$$703$$ 52.6320 1.98505
$$704$$ 5.77200 0.217541
$$705$$ 0 0
$$706$$ 18.8600 0.709806
$$707$$ −42.9480 −1.61523
$$708$$ 6.00000 0.225494
$$709$$ 24.3160 0.913207 0.456603 0.889670i $$-0.349066\pi$$
0.456603 + 0.889670i $$0.349066\pi$$
$$710$$ 0 0
$$711$$ 2.22800 0.0835565
$$712$$ −16.7720 −0.628557
$$713$$ −9.54400 −0.357426
$$714$$ 2.31601 0.0866743
$$715$$ 0 0
$$716$$ 7.54400 0.281933
$$717$$ 22.7720 0.850436
$$718$$ 10.2280 0.381705
$$719$$ −30.3160 −1.13060 −0.565298 0.824887i $$-0.691239\pi$$
−0.565298 + 0.824887i $$0.691239\pi$$
$$720$$ 0 0
$$721$$ −33.0000 −1.22898
$$722$$ −41.4040 −1.54090
$$723$$ −26.6320 −0.990455
$$724$$ 14.3160 0.532050
$$725$$ 0 0
$$726$$ −22.3160 −0.828225
$$727$$ 8.00000 0.296704 0.148352 0.988935i $$-0.452603\pi$$
0.148352 + 0.988935i $$0.452603\pi$$
$$728$$ −11.3160 −0.419399
$$729$$ 1.00000 0.0370370
$$730$$ 0 0
$$731$$ 6.00000 0.221918
$$732$$ −9.54400 −0.352757
$$733$$ 17.2280 0.636331 0.318165 0.948035i $$-0.396933\pi$$
0.318165 + 0.948035i $$0.396933\pi$$
$$734$$ −33.3160 −1.22972
$$735$$ 0 0
$$736$$ −1.00000 −0.0368605
$$737$$ −55.0880 −2.02919
$$738$$ 5.77200 0.212470
$$739$$ 17.2280 0.633742 0.316871 0.948469i $$-0.397368\pi$$
0.316871 + 0.948469i $$0.397368\pi$$
$$740$$ 0 0
$$741$$ −29.3160 −1.07695
$$742$$ 12.0000 0.440534
$$743$$ 20.2280 0.742093 0.371047 0.928614i $$-0.378999\pi$$
0.371047 + 0.928614i $$0.378999\pi$$
$$744$$ 9.54400 0.349900
$$745$$ 0 0
$$746$$ 12.7720 0.467616
$$747$$ −1.00000 −0.0365881
$$748$$ 4.45600 0.162927
$$749$$ 12.0000 0.438470
$$750$$ 0 0
$$751$$ −26.5440 −0.968604 −0.484302 0.874901i $$-0.660926\pi$$
−0.484302 + 0.874901i $$0.660926\pi$$
$$752$$ −8.77200 −0.319882
$$753$$ −3.68399 −0.134252
$$754$$ 11.3160 0.412105
$$755$$ 0 0
$$756$$ −3.00000 −0.109109
$$757$$ −41.2280 −1.49846 −0.749229 0.662312i $$-0.769576\pi$$
−0.749229 + 0.662312i $$0.769576\pi$$
$$758$$ 23.0880 0.838594
$$759$$ 5.77200 0.209510
$$760$$ 0 0
$$761$$ −14.8600 −0.538675 −0.269337 0.963046i $$-0.586805\pi$$
−0.269337 + 0.963046i $$0.586805\pi$$
$$762$$ −4.45600 −0.161424
$$763$$ −44.3160 −1.60435
$$764$$ −7.31601 −0.264684
$$765$$ 0 0
$$766$$ −13.7720 −0.497603
$$767$$ −22.6320 −0.817195
$$768$$ 1.00000 0.0360844
$$769$$ −9.08801 −0.327722 −0.163861 0.986483i $$-0.552395\pi$$
−0.163861 + 0.986483i $$0.552395\pi$$
$$770$$ 0 0
$$771$$ −9.08801 −0.327297
$$772$$ 4.77200 0.171748
$$773$$ −14.4560 −0.519946 −0.259973 0.965616i $$-0.583714\pi$$
−0.259973 + 0.965616i $$0.583714\pi$$
$$774$$ −7.77200 −0.279359
$$775$$ 0 0
$$776$$ −17.5440 −0.629793
$$777$$ −20.3160 −0.728833
$$778$$ −18.0000 −0.645331
$$779$$ −44.8600 −1.60728
$$780$$ 0 0
$$781$$ −27.5440 −0.985602
$$782$$ −0.772002 −0.0276067
$$783$$ 3.00000 0.107211
$$784$$ 2.00000 0.0714286
$$785$$ 0 0
$$786$$ 19.0880 0.680847
$$787$$ 21.7720 0.776088 0.388044 0.921641i $$-0.373151\pi$$
0.388044 + 0.921641i $$0.373151\pi$$
$$788$$ −13.0000 −0.463106
$$789$$ −6.45600 −0.229840
$$790$$ 0 0
$$791$$ −14.3160 −0.509019
$$792$$ −5.77200 −0.205099
$$793$$ 36.0000 1.27840
$$794$$ 14.6320 0.519271
$$795$$ 0 0
$$796$$ 12.0880 0.428448
$$797$$ −2.91199 −0.103148 −0.0515740 0.998669i $$-0.516424\pi$$
−0.0515740 + 0.998669i $$0.516424\pi$$
$$798$$ 23.3160 0.825378
$$799$$ −6.77200 −0.239576
$$800$$ 0 0
$$801$$ 16.7720 0.592610
$$802$$ 28.6320 1.01103
$$803$$ 37.7720 1.33294
$$804$$ −9.54400 −0.336591
$$805$$ 0 0
$$806$$ −36.0000 −1.26805
$$807$$ −26.8600 −0.945517
$$808$$ −14.3160 −0.503635
$$809$$ 47.3160 1.66354 0.831771 0.555119i $$-0.187327\pi$$
0.831771 + 0.555119i $$0.187327\pi$$
$$810$$ 0 0
$$811$$ −38.6320 −1.35655 −0.678277 0.734807i $$-0.737273\pi$$
−0.678277 + 0.734807i $$0.737273\pi$$
$$812$$ −9.00000 −0.315838
$$813$$ 16.0000 0.561144
$$814$$ −39.0880 −1.37003
$$815$$ 0 0
$$816$$ 0.772002 0.0270255
$$817$$ 60.4040 2.11327
$$818$$ 23.0000 0.804176
$$819$$ 11.3160 0.395413
$$820$$ 0 0
$$821$$ −12.2280 −0.426760 −0.213380 0.976969i $$-0.568447\pi$$
−0.213380 + 0.976969i $$0.568447\pi$$
$$822$$ −14.7720 −0.515233
$$823$$ −23.5440 −0.820692 −0.410346 0.911930i $$-0.634592\pi$$
−0.410346 + 0.911930i $$0.634592\pi$$
$$824$$ −11.0000 −0.383203
$$825$$ 0 0
$$826$$ 18.0000 0.626300
$$827$$ −51.6320 −1.79542 −0.897710 0.440586i $$-0.854771\pi$$
−0.897710 + 0.440586i $$0.854771\pi$$
$$828$$ 1.00000 0.0347524
$$829$$ −38.4040 −1.33383 −0.666913 0.745135i $$-0.732385\pi$$
−0.666913 + 0.745135i $$0.732385\pi$$
$$830$$ 0 0
$$831$$ −8.22800 −0.285426
$$832$$ −3.77200 −0.130771
$$833$$ 1.54400 0.0534966
$$834$$ 0.772002 0.0267322
$$835$$ 0 0
$$836$$ 44.8600 1.55152
$$837$$ −9.54400 −0.329889
$$838$$ −25.6320 −0.885443
$$839$$ 32.4040 1.11871 0.559355 0.828928i $$-0.311049\pi$$
0.559355 + 0.828928i $$0.311049\pi$$
$$840$$ 0 0
$$841$$ −20.0000 −0.689655
$$842$$ −22.0000 −0.758170
$$843$$ 15.8600 0.546248
$$844$$ −10.3160 −0.355092
$$845$$ 0 0
$$846$$ 8.77200 0.301588
$$847$$ −66.9480 −2.30036
$$848$$ 4.00000 0.137361
$$849$$ 28.0000 0.960958
$$850$$ 0 0
$$851$$ 6.77200 0.232141
$$852$$ −4.77200 −0.163486
$$853$$ 8.86001 0.303361 0.151680 0.988430i $$-0.451532\pi$$
0.151680 + 0.988430i $$0.451532\pi$$
$$854$$ −28.6320 −0.979767
$$855$$ 0 0
$$856$$ 4.00000 0.136717
$$857$$ 6.00000 0.204956 0.102478 0.994735i $$-0.467323\pi$$
0.102478 + 0.994735i $$0.467323\pi$$
$$858$$ 21.7720 0.743284
$$859$$ −24.4560 −0.834428 −0.417214 0.908808i $$-0.636993\pi$$
−0.417214 + 0.908808i $$0.636993\pi$$
$$860$$ 0 0
$$861$$ 17.3160 0.590128
$$862$$ −24.6320 −0.838970
$$863$$ 2.31601 0.0788377 0.0394189 0.999223i $$-0.487449\pi$$
0.0394189 + 0.999223i $$0.487449\pi$$
$$864$$ −1.00000 −0.0340207
$$865$$ 0 0
$$866$$ −6.45600 −0.219384
$$867$$ −16.4040 −0.557109
$$868$$ 28.6320 0.971834
$$869$$ 12.8600 0.436246
$$870$$ 0 0
$$871$$ 36.0000 1.21981
$$872$$ −14.7720 −0.500243
$$873$$ 17.5440 0.593775
$$874$$ −7.77200 −0.262892
$$875$$ 0 0
$$876$$ 6.54400 0.221101
$$877$$ 47.2640 1.59599 0.797996 0.602662i $$-0.205893\pi$$
0.797996 + 0.602662i $$0.205893\pi$$
$$878$$ 12.0000 0.404980
$$879$$ −9.54400 −0.321911
$$880$$ 0 0
$$881$$ 10.4560 0.352271 0.176136 0.984366i $$-0.443640\pi$$
0.176136 + 0.984366i $$0.443640\pi$$
$$882$$ −2.00000 −0.0673435
$$883$$ −10.7720 −0.362507 −0.181253 0.983436i $$-0.558015\pi$$
−0.181253 + 0.983436i $$0.558015\pi$$
$$884$$ −2.91199 −0.0979409
$$885$$ 0 0
$$886$$ −24.0000 −0.806296
$$887$$ −46.9480 −1.57636 −0.788180 0.615445i $$-0.788976\pi$$
−0.788180 + 0.615445i $$0.788976\pi$$
$$888$$ −6.77200 −0.227254
$$889$$ −13.3680 −0.448348
$$890$$ 0 0
$$891$$ 5.77200 0.193369
$$892$$ 10.0000 0.334825
$$893$$ −68.1760 −2.28142
$$894$$ 11.0880 0.370838
$$895$$ 0 0
$$896$$ 3.00000 0.100223
$$897$$ −3.77200 −0.125943
$$898$$ 2.00000 0.0667409
$$899$$ −28.6320 −0.954931
$$900$$ 0 0
$$901$$ 3.08801 0.102876
$$902$$ 33.3160 1.10930
$$903$$ −23.3160 −0.775908
$$904$$ −4.77200 −0.158714
$$905$$ 0 0
$$906$$ −23.5440 −0.782197
$$907$$ 34.4040 1.14237 0.571183 0.820823i $$-0.306485\pi$$
0.571183 + 0.820823i $$0.306485\pi$$
$$908$$ −10.7720 −0.357481
$$909$$ 14.3160 0.474832
$$910$$ 0 0
$$911$$ 5.77200 0.191235 0.0956175 0.995418i $$-0.469517\pi$$
0.0956175 + 0.995418i $$0.469517\pi$$
$$912$$ 7.77200 0.257357
$$913$$ −5.77200 −0.191025
$$914$$ 30.6320 1.01322
$$915$$ 0 0
$$916$$ −6.45600 −0.213312
$$917$$ 57.2640 1.89102
$$918$$ −0.772002 −0.0254799
$$919$$ 20.3160 0.670163 0.335082 0.942189i $$-0.391236\pi$$
0.335082 + 0.942189i $$0.391236\pi$$
$$920$$ 0 0
$$921$$ −20.3160 −0.669435
$$922$$ −12.5440 −0.413115
$$923$$ 18.0000 0.592477
$$924$$ −17.3160 −0.569655
$$925$$ 0 0
$$926$$ −0.911993 −0.0299699
$$927$$ 11.0000 0.361287
$$928$$ −3.00000 −0.0984798
$$929$$ 24.6840 0.809856 0.404928 0.914349i $$-0.367297\pi$$
0.404928 + 0.914349i $$0.367297\pi$$
$$930$$ 0 0
$$931$$ 15.5440 0.509434
$$932$$ −2.68399 −0.0879172
$$933$$ −17.8600 −0.584710
$$934$$ −18.5440 −0.606778
$$935$$ 0 0
$$936$$ 3.77200 0.123292
$$937$$ 8.63201 0.281996 0.140998 0.990010i $$-0.454969\pi$$
0.140998 + 0.990010i $$0.454969\pi$$
$$938$$ −28.6320 −0.934868
$$939$$ −8.00000 −0.261070
$$940$$ 0 0
$$941$$ 20.6320 0.672584 0.336292 0.941758i $$-0.390827\pi$$
0.336292 + 0.941758i $$0.390827\pi$$
$$942$$ −13.5440 −0.441287
$$943$$ −5.77200 −0.187962
$$944$$ 6.00000 0.195283
$$945$$ 0 0
$$946$$ −44.8600 −1.45852
$$947$$ 31.0880 1.01022 0.505112 0.863054i $$-0.331451\pi$$
0.505112 + 0.863054i $$0.331451\pi$$
$$948$$ 2.22800 0.0723620
$$949$$ −24.6840 −0.801276
$$950$$ 0 0
$$951$$ −9.00000 −0.291845
$$952$$ 2.31601 0.0750622
$$953$$ −46.7720 −1.51509 −0.757547 0.652781i $$-0.773602\pi$$
−0.757547 + 0.652781i $$0.773602\pi$$
$$954$$ −4.00000 −0.129505
$$955$$ 0 0
$$956$$ 22.7720 0.736499
$$957$$ 17.3160 0.559747
$$958$$ 35.3160 1.14101
$$959$$ −44.3160 −1.43104
$$960$$ 0 0
$$961$$ 60.0880 1.93832
$$962$$ 25.5440 0.823572
$$963$$ −4.00000 −0.128898
$$964$$ −26.6320 −0.857759
$$965$$ 0 0
$$966$$ 3.00000 0.0965234
$$967$$ 8.63201 0.277587 0.138793 0.990321i $$-0.455678\pi$$
0.138793 + 0.990321i $$0.455678\pi$$
$$968$$ −22.3160 −0.717264
$$969$$ 6.00000 0.192748
$$970$$ 0 0
$$971$$ −31.6320 −1.01512 −0.507560 0.861617i $$-0.669452\pi$$
−0.507560 + 0.861617i $$0.669452\pi$$
$$972$$ 1.00000 0.0320750
$$973$$ 2.31601 0.0742477
$$974$$ −12.4560 −0.399116
$$975$$ 0 0
$$976$$ −9.54400 −0.305496
$$977$$ −60.3160 −1.92968 −0.964840 0.262838i $$-0.915342\pi$$
−0.964840 + 0.262838i $$0.915342\pi$$
$$978$$ −16.0000 −0.511624
$$979$$ 96.8080 3.09400
$$980$$ 0 0
$$981$$ 14.7720 0.471634
$$982$$ 10.4560 0.333664
$$983$$ 8.40401 0.268046 0.134023 0.990978i $$-0.457210\pi$$
0.134023 + 0.990978i $$0.457210\pi$$
$$984$$ 5.77200 0.184005
$$985$$ 0 0
$$986$$ −2.31601 −0.0737566
$$987$$ 26.3160 0.837648
$$988$$ −29.3160 −0.932666
$$989$$ 7.77200 0.247135
$$990$$ 0 0
$$991$$ 30.1760 0.958573 0.479286 0.877659i $$-0.340896\pi$$
0.479286 + 0.877659i $$0.340896\pi$$
$$992$$ 9.54400 0.303022
$$993$$ 34.7720 1.10346
$$994$$ −14.3160 −0.454076
$$995$$ 0 0
$$996$$ −1.00000 −0.0316862
$$997$$ −0.227998 −0.00722077 −0.00361039 0.999993i $$-0.501149\pi$$
−0.00361039 + 0.999993i $$0.501149\pi$$
$$998$$ 9.68399 0.306541
$$999$$ 6.77200 0.214257
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.bh.1.2 2
5.2 odd 4 3450.2.d.w.2899.2 4
5.3 odd 4 3450.2.d.w.2899.4 4
5.4 even 2 3450.2.a.bj.1.2 yes 2

By twisted newform
Twist Min Dim Char Parity Ord Type
3450.2.a.bh.1.2 2 1.1 even 1 trivial
3450.2.a.bj.1.2 yes 2 5.4 even 2
3450.2.d.w.2899.2 4 5.2 odd 4
3450.2.d.w.2899.4 4 5.3 odd 4