# Properties

 Label 3450.2.d.b.2899.2 Level $3450$ Weight $2$ Character 3450.2899 Analytic conductor $27.548$ Analytic rank $1$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$3450 = 2 \cdot 3 \cdot 5^{2} \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3450.d (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$27.5483886973$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(i)$$ Defining polynomial: $$x^{2} + 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 2899.2 Root $$1.00000i$$ of defining polynomial Character $$\chi$$ $$=$$ 3450.2899 Dual form 3450.2.d.b.2899.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{6} +3.00000i q^{7} -1.00000i q^{8} -1.00000 q^{9} +O(q^{10})$$ $$q+1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{6} +3.00000i q^{7} -1.00000i q^{8} -1.00000 q^{9} -3.00000 q^{11} -1.00000i q^{12} +3.00000i q^{13} -3.00000 q^{14} +1.00000 q^{16} -4.00000i q^{17} -1.00000i q^{18} +3.00000 q^{19} -3.00000 q^{21} -3.00000i q^{22} +1.00000i q^{23} +1.00000 q^{24} -3.00000 q^{26} -1.00000i q^{27} -3.00000i q^{28} -3.00000 q^{29} -10.0000 q^{31} +1.00000i q^{32} -3.00000i q^{33} +4.00000 q^{34} +1.00000 q^{36} -8.00000i q^{37} +3.00000i q^{38} -3.00000 q^{39} -9.00000 q^{41} -3.00000i q^{42} -1.00000i q^{43} +3.00000 q^{44} -1.00000 q^{46} +2.00000i q^{47} +1.00000i q^{48} -2.00000 q^{49} +4.00000 q^{51} -3.00000i q^{52} +6.00000i q^{53} +1.00000 q^{54} +3.00000 q^{56} +3.00000i q^{57} -3.00000i q^{58} -8.00000 q^{59} +12.0000 q^{61} -10.0000i q^{62} -3.00000i q^{63} -1.00000 q^{64} +3.00000 q^{66} +8.00000i q^{67} +4.00000i q^{68} -1.00000 q^{69} -14.0000 q^{71} +1.00000i q^{72} -13.0000i q^{73} +8.00000 q^{74} -3.00000 q^{76} -9.00000i q^{77} -3.00000i q^{78} +17.0000 q^{79} +1.00000 q^{81} -9.00000i q^{82} -1.00000i q^{83} +3.00000 q^{84} +1.00000 q^{86} -3.00000i q^{87} +3.00000i q^{88} +6.00000 q^{89} -9.00000 q^{91} -1.00000i q^{92} -10.0000i q^{93} -2.00000 q^{94} -1.00000 q^{96} +6.00000i q^{97} -2.00000i q^{98} +3.00000 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{4} - 2q^{6} - 2q^{9} + O(q^{10})$$ $$2q - 2q^{4} - 2q^{6} - 2q^{9} - 6q^{11} - 6q^{14} + 2q^{16} + 6q^{19} - 6q^{21} + 2q^{24} - 6q^{26} - 6q^{29} - 20q^{31} + 8q^{34} + 2q^{36} - 6q^{39} - 18q^{41} + 6q^{44} - 2q^{46} - 4q^{49} + 8q^{51} + 2q^{54} + 6q^{56} - 16q^{59} + 24q^{61} - 2q^{64} + 6q^{66} - 2q^{69} - 28q^{71} + 16q^{74} - 6q^{76} + 34q^{79} + 2q^{81} + 6q^{84} + 2q^{86} + 12q^{89} - 18q^{91} - 4q^{94} - 2q^{96} + 6q^{99} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/3450\mathbb{Z}\right)^\times$$.

 $$n$$ $$277$$ $$1151$$ $$1201$$ $$\chi(n)$$ $$-1$$ $$1$$ $$1$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 1.00000i 0.707107i
$$3$$ 1.00000i 0.577350i
$$4$$ −1.00000 −0.500000
$$5$$ 0 0
$$6$$ −1.00000 −0.408248
$$7$$ 3.00000i 1.13389i 0.823754 + 0.566947i $$0.191875\pi$$
−0.823754 + 0.566947i $$0.808125\pi$$
$$8$$ − 1.00000i − 0.353553i
$$9$$ −1.00000 −0.333333
$$10$$ 0 0
$$11$$ −3.00000 −0.904534 −0.452267 0.891883i $$-0.649385\pi$$
−0.452267 + 0.891883i $$0.649385\pi$$
$$12$$ − 1.00000i − 0.288675i
$$13$$ 3.00000i 0.832050i 0.909353 + 0.416025i $$0.136577\pi$$
−0.909353 + 0.416025i $$0.863423\pi$$
$$14$$ −3.00000 −0.801784
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ − 4.00000i − 0.970143i −0.874475 0.485071i $$-0.838794\pi$$
0.874475 0.485071i $$-0.161206\pi$$
$$18$$ − 1.00000i − 0.235702i
$$19$$ 3.00000 0.688247 0.344124 0.938924i $$-0.388176\pi$$
0.344124 + 0.938924i $$0.388176\pi$$
$$20$$ 0 0
$$21$$ −3.00000 −0.654654
$$22$$ − 3.00000i − 0.639602i
$$23$$ 1.00000i 0.208514i
$$24$$ 1.00000 0.204124
$$25$$ 0 0
$$26$$ −3.00000 −0.588348
$$27$$ − 1.00000i − 0.192450i
$$28$$ − 3.00000i − 0.566947i
$$29$$ −3.00000 −0.557086 −0.278543 0.960424i $$-0.589851\pi$$
−0.278543 + 0.960424i $$0.589851\pi$$
$$30$$ 0 0
$$31$$ −10.0000 −1.79605 −0.898027 0.439941i $$-0.854999\pi$$
−0.898027 + 0.439941i $$0.854999\pi$$
$$32$$ 1.00000i 0.176777i
$$33$$ − 3.00000i − 0.522233i
$$34$$ 4.00000 0.685994
$$35$$ 0 0
$$36$$ 1.00000 0.166667
$$37$$ − 8.00000i − 1.31519i −0.753371 0.657596i $$-0.771573\pi$$
0.753371 0.657596i $$-0.228427\pi$$
$$38$$ 3.00000i 0.486664i
$$39$$ −3.00000 −0.480384
$$40$$ 0 0
$$41$$ −9.00000 −1.40556 −0.702782 0.711405i $$-0.748059\pi$$
−0.702782 + 0.711405i $$0.748059\pi$$
$$42$$ − 3.00000i − 0.462910i
$$43$$ − 1.00000i − 0.152499i −0.997089 0.0762493i $$-0.975706\pi$$
0.997089 0.0762493i $$-0.0242945\pi$$
$$44$$ 3.00000 0.452267
$$45$$ 0 0
$$46$$ −1.00000 −0.147442
$$47$$ 2.00000i 0.291730i 0.989305 + 0.145865i $$0.0465965\pi$$
−0.989305 + 0.145865i $$0.953403\pi$$
$$48$$ 1.00000i 0.144338i
$$49$$ −2.00000 −0.285714
$$50$$ 0 0
$$51$$ 4.00000 0.560112
$$52$$ − 3.00000i − 0.416025i
$$53$$ 6.00000i 0.824163i 0.911147 + 0.412082i $$0.135198\pi$$
−0.911147 + 0.412082i $$0.864802\pi$$
$$54$$ 1.00000 0.136083
$$55$$ 0 0
$$56$$ 3.00000 0.400892
$$57$$ 3.00000i 0.397360i
$$58$$ − 3.00000i − 0.393919i
$$59$$ −8.00000 −1.04151 −0.520756 0.853706i $$-0.674350\pi$$
−0.520756 + 0.853706i $$0.674350\pi$$
$$60$$ 0 0
$$61$$ 12.0000 1.53644 0.768221 0.640184i $$-0.221142\pi$$
0.768221 + 0.640184i $$0.221142\pi$$
$$62$$ − 10.0000i − 1.27000i
$$63$$ − 3.00000i − 0.377964i
$$64$$ −1.00000 −0.125000
$$65$$ 0 0
$$66$$ 3.00000 0.369274
$$67$$ 8.00000i 0.977356i 0.872464 + 0.488678i $$0.162521\pi$$
−0.872464 + 0.488678i $$0.837479\pi$$
$$68$$ 4.00000i 0.485071i
$$69$$ −1.00000 −0.120386
$$70$$ 0 0
$$71$$ −14.0000 −1.66149 −0.830747 0.556650i $$-0.812086\pi$$
−0.830747 + 0.556650i $$0.812086\pi$$
$$72$$ 1.00000i 0.117851i
$$73$$ − 13.0000i − 1.52153i −0.649025 0.760767i $$-0.724823\pi$$
0.649025 0.760767i $$-0.275177\pi$$
$$74$$ 8.00000 0.929981
$$75$$ 0 0
$$76$$ −3.00000 −0.344124
$$77$$ − 9.00000i − 1.02565i
$$78$$ − 3.00000i − 0.339683i
$$79$$ 17.0000 1.91265 0.956325 0.292306i $$-0.0944227\pi$$
0.956325 + 0.292306i $$0.0944227\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ − 9.00000i − 0.993884i
$$83$$ − 1.00000i − 0.109764i −0.998493 0.0548821i $$-0.982522\pi$$
0.998493 0.0548821i $$-0.0174783\pi$$
$$84$$ 3.00000 0.327327
$$85$$ 0 0
$$86$$ 1.00000 0.107833
$$87$$ − 3.00000i − 0.321634i
$$88$$ 3.00000i 0.319801i
$$89$$ 6.00000 0.635999 0.317999 0.948091i $$-0.396989\pi$$
0.317999 + 0.948091i $$0.396989\pi$$
$$90$$ 0 0
$$91$$ −9.00000 −0.943456
$$92$$ − 1.00000i − 0.104257i
$$93$$ − 10.0000i − 1.03695i
$$94$$ −2.00000 −0.206284
$$95$$ 0 0
$$96$$ −1.00000 −0.102062
$$97$$ 6.00000i 0.609208i 0.952479 + 0.304604i $$0.0985241\pi$$
−0.952479 + 0.304604i $$0.901476\pi$$
$$98$$ − 2.00000i − 0.202031i
$$99$$ 3.00000 0.301511
$$100$$ 0 0
$$101$$ −6.00000 −0.597022 −0.298511 0.954406i $$-0.596490\pi$$
−0.298511 + 0.954406i $$0.596490\pi$$
$$102$$ 4.00000i 0.396059i
$$103$$ − 13.0000i − 1.28093i −0.767988 0.640464i $$-0.778742\pi$$
0.767988 0.640464i $$-0.221258\pi$$
$$104$$ 3.00000 0.294174
$$105$$ 0 0
$$106$$ −6.00000 −0.582772
$$107$$ − 12.0000i − 1.16008i −0.814587 0.580042i $$-0.803036\pi$$
0.814587 0.580042i $$-0.196964\pi$$
$$108$$ 1.00000i 0.0962250i
$$109$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$110$$ 0 0
$$111$$ 8.00000 0.759326
$$112$$ 3.00000i 0.283473i
$$113$$ − 6.00000i − 0.564433i −0.959351 0.282216i $$-0.908930\pi$$
0.959351 0.282216i $$-0.0910696\pi$$
$$114$$ −3.00000 −0.280976
$$115$$ 0 0
$$116$$ 3.00000 0.278543
$$117$$ − 3.00000i − 0.277350i
$$118$$ − 8.00000i − 0.736460i
$$119$$ 12.0000 1.10004
$$120$$ 0 0
$$121$$ −2.00000 −0.181818
$$122$$ 12.0000i 1.08643i
$$123$$ − 9.00000i − 0.811503i
$$124$$ 10.0000 0.898027
$$125$$ 0 0
$$126$$ 3.00000 0.267261
$$127$$ 16.0000i 1.41977i 0.704317 + 0.709885i $$0.251253\pi$$
−0.704317 + 0.709885i $$0.748747\pi$$
$$128$$ − 1.00000i − 0.0883883i
$$129$$ 1.00000 0.0880451
$$130$$ 0 0
$$131$$ 10.0000 0.873704 0.436852 0.899533i $$-0.356093\pi$$
0.436852 + 0.899533i $$0.356093\pi$$
$$132$$ 3.00000i 0.261116i
$$133$$ 9.00000i 0.780399i
$$134$$ −8.00000 −0.691095
$$135$$ 0 0
$$136$$ −4.00000 −0.342997
$$137$$ − 18.0000i − 1.53784i −0.639343 0.768922i $$-0.720793\pi$$
0.639343 0.768922i $$-0.279207\pi$$
$$138$$ − 1.00000i − 0.0851257i
$$139$$ −8.00000 −0.678551 −0.339276 0.940687i $$-0.610182\pi$$
−0.339276 + 0.940687i $$0.610182\pi$$
$$140$$ 0 0
$$141$$ −2.00000 −0.168430
$$142$$ − 14.0000i − 1.17485i
$$143$$ − 9.00000i − 0.752618i
$$144$$ −1.00000 −0.0833333
$$145$$ 0 0
$$146$$ 13.0000 1.07589
$$147$$ − 2.00000i − 0.164957i
$$148$$ 8.00000i 0.657596i
$$149$$ 6.00000 0.491539 0.245770 0.969328i $$-0.420959\pi$$
0.245770 + 0.969328i $$0.420959\pi$$
$$150$$ 0 0
$$151$$ −22.0000 −1.79033 −0.895167 0.445730i $$-0.852944\pi$$
−0.895167 + 0.445730i $$0.852944\pi$$
$$152$$ − 3.00000i − 0.243332i
$$153$$ 4.00000i 0.323381i
$$154$$ 9.00000 0.725241
$$155$$ 0 0
$$156$$ 3.00000 0.240192
$$157$$ 6.00000i 0.478852i 0.970915 + 0.239426i $$0.0769593\pi$$
−0.970915 + 0.239426i $$0.923041\pi$$
$$158$$ 17.0000i 1.35245i
$$159$$ −6.00000 −0.475831
$$160$$ 0 0
$$161$$ −3.00000 −0.236433
$$162$$ 1.00000i 0.0785674i
$$163$$ − 6.00000i − 0.469956i −0.972001 0.234978i $$-0.924498\pi$$
0.972001 0.234978i $$-0.0755019\pi$$
$$164$$ 9.00000 0.702782
$$165$$ 0 0
$$166$$ 1.00000 0.0776151
$$167$$ 2.00000i 0.154765i 0.997001 + 0.0773823i $$0.0246562\pi$$
−0.997001 + 0.0773823i $$0.975344\pi$$
$$168$$ 3.00000i 0.231455i
$$169$$ 4.00000 0.307692
$$170$$ 0 0
$$171$$ −3.00000 −0.229416
$$172$$ 1.00000i 0.0762493i
$$173$$ − 15.0000i − 1.14043i −0.821496 0.570214i $$-0.806860\pi$$
0.821496 0.570214i $$-0.193140\pi$$
$$174$$ 3.00000 0.227429
$$175$$ 0 0
$$176$$ −3.00000 −0.226134
$$177$$ − 8.00000i − 0.601317i
$$178$$ 6.00000i 0.449719i
$$179$$ 10.0000 0.747435 0.373718 0.927543i $$-0.378083\pi$$
0.373718 + 0.927543i $$0.378083\pi$$
$$180$$ 0 0
$$181$$ −16.0000 −1.18927 −0.594635 0.803996i $$-0.702704\pi$$
−0.594635 + 0.803996i $$0.702704\pi$$
$$182$$ − 9.00000i − 0.667124i
$$183$$ 12.0000i 0.887066i
$$184$$ 1.00000 0.0737210
$$185$$ 0 0
$$186$$ 10.0000 0.733236
$$187$$ 12.0000i 0.877527i
$$188$$ − 2.00000i − 0.145865i
$$189$$ 3.00000 0.218218
$$190$$ 0 0
$$191$$ −15.0000 −1.08536 −0.542681 0.839939i $$-0.682591\pi$$
−0.542681 + 0.839939i $$0.682591\pi$$
$$192$$ − 1.00000i − 0.0721688i
$$193$$ 6.00000i 0.431889i 0.976406 + 0.215945i $$0.0692831\pi$$
−0.976406 + 0.215945i $$0.930717\pi$$
$$194$$ −6.00000 −0.430775
$$195$$ 0 0
$$196$$ 2.00000 0.142857
$$197$$ − 17.0000i − 1.21120i −0.795769 0.605600i $$-0.792933\pi$$
0.795769 0.605600i $$-0.207067\pi$$
$$198$$ 3.00000i 0.213201i
$$199$$ −11.0000 −0.779769 −0.389885 0.920864i $$-0.627485\pi$$
−0.389885 + 0.920864i $$0.627485\pi$$
$$200$$ 0 0
$$201$$ −8.00000 −0.564276
$$202$$ − 6.00000i − 0.422159i
$$203$$ − 9.00000i − 0.631676i
$$204$$ −4.00000 −0.280056
$$205$$ 0 0
$$206$$ 13.0000 0.905753
$$207$$ − 1.00000i − 0.0695048i
$$208$$ 3.00000i 0.208013i
$$209$$ −9.00000 −0.622543
$$210$$ 0 0
$$211$$ 22.0000 1.51454 0.757271 0.653101i $$-0.226532\pi$$
0.757271 + 0.653101i $$0.226532\pi$$
$$212$$ − 6.00000i − 0.412082i
$$213$$ − 14.0000i − 0.959264i
$$214$$ 12.0000 0.820303
$$215$$ 0 0
$$216$$ −1.00000 −0.0680414
$$217$$ − 30.0000i − 2.03653i
$$218$$ 0 0
$$219$$ 13.0000 0.878459
$$220$$ 0 0
$$221$$ 12.0000 0.807207
$$222$$ 8.00000i 0.536925i
$$223$$ 6.00000i 0.401790i 0.979613 + 0.200895i $$0.0643850\pi$$
−0.979613 + 0.200895i $$0.935615\pi$$
$$224$$ −3.00000 −0.200446
$$225$$ 0 0
$$226$$ 6.00000 0.399114
$$227$$ − 12.0000i − 0.796468i −0.917284 0.398234i $$-0.869623\pi$$
0.917284 0.398234i $$-0.130377\pi$$
$$228$$ − 3.00000i − 0.198680i
$$229$$ −6.00000 −0.396491 −0.198246 0.980152i $$-0.563524\pi$$
−0.198246 + 0.980152i $$0.563524\pi$$
$$230$$ 0 0
$$231$$ 9.00000 0.592157
$$232$$ 3.00000i 0.196960i
$$233$$ 1.00000i 0.0655122i 0.999463 + 0.0327561i $$0.0104285\pi$$
−0.999463 + 0.0327561i $$0.989572\pi$$
$$234$$ 3.00000 0.196116
$$235$$ 0 0
$$236$$ 8.00000 0.520756
$$237$$ 17.0000i 1.10427i
$$238$$ 12.0000i 0.777844i
$$239$$ 10.0000 0.646846 0.323423 0.946254i $$-0.395166\pi$$
0.323423 + 0.946254i $$0.395166\pi$$
$$240$$ 0 0
$$241$$ −10.0000 −0.644157 −0.322078 0.946713i $$-0.604381\pi$$
−0.322078 + 0.946713i $$0.604381\pi$$
$$242$$ − 2.00000i − 0.128565i
$$243$$ 1.00000i 0.0641500i
$$244$$ −12.0000 −0.768221
$$245$$ 0 0
$$246$$ 9.00000 0.573819
$$247$$ 9.00000i 0.572656i
$$248$$ 10.0000i 0.635001i
$$249$$ 1.00000 0.0633724
$$250$$ 0 0
$$251$$ −16.0000 −1.00991 −0.504956 0.863145i $$-0.668491\pi$$
−0.504956 + 0.863145i $$0.668491\pi$$
$$252$$ 3.00000i 0.188982i
$$253$$ − 3.00000i − 0.188608i
$$254$$ −16.0000 −1.00393
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ − 18.0000i − 1.12281i −0.827541 0.561405i $$-0.810261\pi$$
0.827541 0.561405i $$-0.189739\pi$$
$$258$$ 1.00000i 0.0622573i
$$259$$ 24.0000 1.49129
$$260$$ 0 0
$$261$$ 3.00000 0.185695
$$262$$ 10.0000i 0.617802i
$$263$$ 8.00000i 0.493301i 0.969104 + 0.246651i $$0.0793300\pi$$
−0.969104 + 0.246651i $$0.920670\pi$$
$$264$$ −3.00000 −0.184637
$$265$$ 0 0
$$266$$ −9.00000 −0.551825
$$267$$ 6.00000i 0.367194i
$$268$$ − 8.00000i − 0.488678i
$$269$$ 5.00000 0.304855 0.152428 0.988315i $$-0.451291\pi$$
0.152428 + 0.988315i $$0.451291\pi$$
$$270$$ 0 0
$$271$$ 22.0000 1.33640 0.668202 0.743980i $$-0.267064\pi$$
0.668202 + 0.743980i $$0.267064\pi$$
$$272$$ − 4.00000i − 0.242536i
$$273$$ − 9.00000i − 0.544705i
$$274$$ 18.0000 1.08742
$$275$$ 0 0
$$276$$ 1.00000 0.0601929
$$277$$ − 13.0000i − 0.781094i −0.920583 0.390547i $$-0.872286\pi$$
0.920583 0.390547i $$-0.127714\pi$$
$$278$$ − 8.00000i − 0.479808i
$$279$$ 10.0000 0.598684
$$280$$ 0 0
$$281$$ −10.0000 −0.596550 −0.298275 0.954480i $$-0.596411\pi$$
−0.298275 + 0.954480i $$0.596411\pi$$
$$282$$ − 2.00000i − 0.119098i
$$283$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$284$$ 14.0000 0.830747
$$285$$ 0 0
$$286$$ 9.00000 0.532181
$$287$$ − 27.0000i − 1.59376i
$$288$$ − 1.00000i − 0.0589256i
$$289$$ 1.00000 0.0588235
$$290$$ 0 0
$$291$$ −6.00000 −0.351726
$$292$$ 13.0000i 0.760767i
$$293$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$294$$ 2.00000 0.116642
$$295$$ 0 0
$$296$$ −8.00000 −0.464991
$$297$$ 3.00000i 0.174078i
$$298$$ 6.00000i 0.347571i
$$299$$ −3.00000 −0.173494
$$300$$ 0 0
$$301$$ 3.00000 0.172917
$$302$$ − 22.0000i − 1.26596i
$$303$$ − 6.00000i − 0.344691i
$$304$$ 3.00000 0.172062
$$305$$ 0 0
$$306$$ −4.00000 −0.228665
$$307$$ 8.00000i 0.456584i 0.973593 + 0.228292i $$0.0733141\pi$$
−0.973593 + 0.228292i $$0.926686\pi$$
$$308$$ 9.00000i 0.512823i
$$309$$ 13.0000 0.739544
$$310$$ 0 0
$$311$$ 24.0000 1.36092 0.680458 0.732787i $$-0.261781\pi$$
0.680458 + 0.732787i $$0.261781\pi$$
$$312$$ 3.00000i 0.169842i
$$313$$ − 24.0000i − 1.35656i −0.734803 0.678280i $$-0.762726\pi$$
0.734803 0.678280i $$-0.237274\pi$$
$$314$$ −6.00000 −0.338600
$$315$$ 0 0
$$316$$ −17.0000 −0.956325
$$317$$ 31.0000i 1.74113i 0.492050 + 0.870567i $$0.336248\pi$$
−0.492050 + 0.870567i $$0.663752\pi$$
$$318$$ − 6.00000i − 0.336463i
$$319$$ 9.00000 0.503903
$$320$$ 0 0
$$321$$ 12.0000 0.669775
$$322$$ − 3.00000i − 0.167183i
$$323$$ − 12.0000i − 0.667698i
$$324$$ −1.00000 −0.0555556
$$325$$ 0 0
$$326$$ 6.00000 0.332309
$$327$$ 0 0
$$328$$ 9.00000i 0.496942i
$$329$$ −6.00000 −0.330791
$$330$$ 0 0
$$331$$ −26.0000 −1.42909 −0.714545 0.699590i $$-0.753366\pi$$
−0.714545 + 0.699590i $$0.753366\pi$$
$$332$$ 1.00000i 0.0548821i
$$333$$ 8.00000i 0.438397i
$$334$$ −2.00000 −0.109435
$$335$$ 0 0
$$336$$ −3.00000 −0.163663
$$337$$ 14.0000i 0.762629i 0.924445 + 0.381314i $$0.124528\pi$$
−0.924445 + 0.381314i $$0.875472\pi$$
$$338$$ 4.00000i 0.217571i
$$339$$ 6.00000 0.325875
$$340$$ 0 0
$$341$$ 30.0000 1.62459
$$342$$ − 3.00000i − 0.162221i
$$343$$ 15.0000i 0.809924i
$$344$$ −1.00000 −0.0539164
$$345$$ 0 0
$$346$$ 15.0000 0.806405
$$347$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$348$$ 3.00000i 0.160817i
$$349$$ 5.00000 0.267644 0.133822 0.991005i $$-0.457275\pi$$
0.133822 + 0.991005i $$0.457275\pi$$
$$350$$ 0 0
$$351$$ 3.00000 0.160128
$$352$$ − 3.00000i − 0.159901i
$$353$$ − 9.00000i − 0.479022i −0.970894 0.239511i $$-0.923013\pi$$
0.970894 0.239511i $$-0.0769871\pi$$
$$354$$ 8.00000 0.425195
$$355$$ 0 0
$$356$$ −6.00000 −0.317999
$$357$$ 12.0000i 0.635107i
$$358$$ 10.0000i 0.528516i
$$359$$ −31.0000 −1.63612 −0.818059 0.575135i $$-0.804950\pi$$
−0.818059 + 0.575135i $$0.804950\pi$$
$$360$$ 0 0
$$361$$ −10.0000 −0.526316
$$362$$ − 16.0000i − 0.840941i
$$363$$ − 2.00000i − 0.104973i
$$364$$ 9.00000 0.471728
$$365$$ 0 0
$$366$$ −12.0000 −0.627250
$$367$$ 27.0000i 1.40939i 0.709511 + 0.704694i $$0.248916\pi$$
−0.709511 + 0.704694i $$0.751084\pi$$
$$368$$ 1.00000i 0.0521286i
$$369$$ 9.00000 0.468521
$$370$$ 0 0
$$371$$ −18.0000 −0.934513
$$372$$ 10.0000i 0.518476i
$$373$$ 16.0000i 0.828449i 0.910175 + 0.414224i $$0.135947\pi$$
−0.910175 + 0.414224i $$0.864053\pi$$
$$374$$ −12.0000 −0.620505
$$375$$ 0 0
$$376$$ 2.00000 0.103142
$$377$$ − 9.00000i − 0.463524i
$$378$$ 3.00000i 0.154303i
$$379$$ −12.0000 −0.616399 −0.308199 0.951322i $$-0.599726\pi$$
−0.308199 + 0.951322i $$0.599726\pi$$
$$380$$ 0 0
$$381$$ −16.0000 −0.819705
$$382$$ − 15.0000i − 0.767467i
$$383$$ 29.0000i 1.48183i 0.671598 + 0.740915i $$0.265608\pi$$
−0.671598 + 0.740915i $$0.734392\pi$$
$$384$$ 1.00000 0.0510310
$$385$$ 0 0
$$386$$ −6.00000 −0.305392
$$387$$ 1.00000i 0.0508329i
$$388$$ − 6.00000i − 0.304604i
$$389$$ −38.0000 −1.92668 −0.963338 0.268290i $$-0.913542\pi$$
−0.963338 + 0.268290i $$0.913542\pi$$
$$390$$ 0 0
$$391$$ 4.00000 0.202289
$$392$$ 2.00000i 0.101015i
$$393$$ 10.0000i 0.504433i
$$394$$ 17.0000 0.856448
$$395$$ 0 0
$$396$$ −3.00000 −0.150756
$$397$$ 30.0000i 1.50566i 0.658217 + 0.752828i $$0.271311\pi$$
−0.658217 + 0.752828i $$0.728689\pi$$
$$398$$ − 11.0000i − 0.551380i
$$399$$ −9.00000 −0.450564
$$400$$ 0 0
$$401$$ −28.0000 −1.39825 −0.699127 0.714998i $$-0.746428\pi$$
−0.699127 + 0.714998i $$0.746428\pi$$
$$402$$ − 8.00000i − 0.399004i
$$403$$ − 30.0000i − 1.49441i
$$404$$ 6.00000 0.298511
$$405$$ 0 0
$$406$$ 9.00000 0.446663
$$407$$ 24.0000i 1.18964i
$$408$$ − 4.00000i − 0.198030i
$$409$$ 23.0000 1.13728 0.568638 0.822588i $$-0.307470\pi$$
0.568638 + 0.822588i $$0.307470\pi$$
$$410$$ 0 0
$$411$$ 18.0000 0.887875
$$412$$ 13.0000i 0.640464i
$$413$$ − 24.0000i − 1.18096i
$$414$$ 1.00000 0.0491473
$$415$$ 0 0
$$416$$ −3.00000 −0.147087
$$417$$ − 8.00000i − 0.391762i
$$418$$ − 9.00000i − 0.440204i
$$419$$ −19.0000 −0.928211 −0.464105 0.885780i $$-0.653624\pi$$
−0.464105 + 0.885780i $$0.653624\pi$$
$$420$$ 0 0
$$421$$ −10.0000 −0.487370 −0.243685 0.969854i $$-0.578356\pi$$
−0.243685 + 0.969854i $$0.578356\pi$$
$$422$$ 22.0000i 1.07094i
$$423$$ − 2.00000i − 0.0972433i
$$424$$ 6.00000 0.291386
$$425$$ 0 0
$$426$$ 14.0000 0.678302
$$427$$ 36.0000i 1.74216i
$$428$$ 12.0000i 0.580042i
$$429$$ 9.00000 0.434524
$$430$$ 0 0
$$431$$ 16.0000 0.770693 0.385346 0.922772i $$-0.374082\pi$$
0.385346 + 0.922772i $$0.374082\pi$$
$$432$$ − 1.00000i − 0.0481125i
$$433$$ − 4.00000i − 0.192228i −0.995370 0.0961139i $$-0.969359\pi$$
0.995370 0.0961139i $$-0.0306413\pi$$
$$434$$ 30.0000 1.44005
$$435$$ 0 0
$$436$$ 0 0
$$437$$ 3.00000i 0.143509i
$$438$$ 13.0000i 0.621164i
$$439$$ 4.00000 0.190910 0.0954548 0.995434i $$-0.469569\pi$$
0.0954548 + 0.995434i $$0.469569\pi$$
$$440$$ 0 0
$$441$$ 2.00000 0.0952381
$$442$$ 12.0000i 0.570782i
$$443$$ − 26.0000i − 1.23530i −0.786454 0.617649i $$-0.788085\pi$$
0.786454 0.617649i $$-0.211915\pi$$
$$444$$ −8.00000 −0.379663
$$445$$ 0 0
$$446$$ −6.00000 −0.284108
$$447$$ 6.00000i 0.283790i
$$448$$ − 3.00000i − 0.141737i
$$449$$ 26.0000 1.22702 0.613508 0.789689i $$-0.289758\pi$$
0.613508 + 0.789689i $$0.289758\pi$$
$$450$$ 0 0
$$451$$ 27.0000 1.27138
$$452$$ 6.00000i 0.282216i
$$453$$ − 22.0000i − 1.03365i
$$454$$ 12.0000 0.563188
$$455$$ 0 0
$$456$$ 3.00000 0.140488
$$457$$ 22.0000i 1.02912i 0.857455 + 0.514558i $$0.172044\pi$$
−0.857455 + 0.514558i $$0.827956\pi$$
$$458$$ − 6.00000i − 0.280362i
$$459$$ −4.00000 −0.186704
$$460$$ 0 0
$$461$$ −9.00000 −0.419172 −0.209586 0.977790i $$-0.567212\pi$$
−0.209586 + 0.977790i $$0.567212\pi$$
$$462$$ 9.00000i 0.418718i
$$463$$ 4.00000i 0.185896i 0.995671 + 0.0929479i $$0.0296290\pi$$
−0.995671 + 0.0929479i $$0.970371\pi$$
$$464$$ −3.00000 −0.139272
$$465$$ 0 0
$$466$$ −1.00000 −0.0463241
$$467$$ 11.0000i 0.509019i 0.967070 + 0.254510i $$0.0819141\pi$$
−0.967070 + 0.254510i $$0.918086\pi$$
$$468$$ 3.00000i 0.138675i
$$469$$ −24.0000 −1.10822
$$470$$ 0 0
$$471$$ −6.00000 −0.276465
$$472$$ 8.00000i 0.368230i
$$473$$ 3.00000i 0.137940i
$$474$$ −17.0000 −0.780836
$$475$$ 0 0
$$476$$ −12.0000 −0.550019
$$477$$ − 6.00000i − 0.274721i
$$478$$ 10.0000i 0.457389i
$$479$$ −33.0000 −1.50781 −0.753904 0.656984i $$-0.771832\pi$$
−0.753904 + 0.656984i $$0.771832\pi$$
$$480$$ 0 0
$$481$$ 24.0000 1.09431
$$482$$ − 10.0000i − 0.455488i
$$483$$ − 3.00000i − 0.136505i
$$484$$ 2.00000 0.0909091
$$485$$ 0 0
$$486$$ −1.00000 −0.0453609
$$487$$ − 2.00000i − 0.0906287i −0.998973 0.0453143i $$-0.985571\pi$$
0.998973 0.0453143i $$-0.0144289\pi$$
$$488$$ − 12.0000i − 0.543214i
$$489$$ 6.00000 0.271329
$$490$$ 0 0
$$491$$ 12.0000 0.541552 0.270776 0.962642i $$-0.412720\pi$$
0.270776 + 0.962642i $$0.412720\pi$$
$$492$$ 9.00000i 0.405751i
$$493$$ 12.0000i 0.540453i
$$494$$ −9.00000 −0.404929
$$495$$ 0 0
$$496$$ −10.0000 −0.449013
$$497$$ − 42.0000i − 1.88396i
$$498$$ 1.00000i 0.0448111i
$$499$$ −34.0000 −1.52205 −0.761025 0.648723i $$-0.775303\pi$$
−0.761025 + 0.648723i $$0.775303\pi$$
$$500$$ 0 0
$$501$$ −2.00000 −0.0893534
$$502$$ − 16.0000i − 0.714115i
$$503$$ 35.0000i 1.56057i 0.625422 + 0.780286i $$0.284927\pi$$
−0.625422 + 0.780286i $$0.715073\pi$$
$$504$$ −3.00000 −0.133631
$$505$$ 0 0
$$506$$ 3.00000 0.133366
$$507$$ 4.00000i 0.177646i
$$508$$ − 16.0000i − 0.709885i
$$509$$ −34.0000 −1.50702 −0.753512 0.657434i $$-0.771642\pi$$
−0.753512 + 0.657434i $$0.771642\pi$$
$$510$$ 0 0
$$511$$ 39.0000 1.72526
$$512$$ 1.00000i 0.0441942i
$$513$$ − 3.00000i − 0.132453i
$$514$$ 18.0000 0.793946
$$515$$ 0 0
$$516$$ −1.00000 −0.0440225
$$517$$ − 6.00000i − 0.263880i
$$518$$ 24.0000i 1.05450i
$$519$$ 15.0000 0.658427
$$520$$ 0 0
$$521$$ 4.00000 0.175243 0.0876216 0.996154i $$-0.472073\pi$$
0.0876216 + 0.996154i $$0.472073\pi$$
$$522$$ 3.00000i 0.131306i
$$523$$ − 11.0000i − 0.480996i −0.970650 0.240498i $$-0.922689\pi$$
0.970650 0.240498i $$-0.0773108\pi$$
$$524$$ −10.0000 −0.436852
$$525$$ 0 0
$$526$$ −8.00000 −0.348817
$$527$$ 40.0000i 1.74243i
$$528$$ − 3.00000i − 0.130558i
$$529$$ −1.00000 −0.0434783
$$530$$ 0 0
$$531$$ 8.00000 0.347170
$$532$$ − 9.00000i − 0.390199i
$$533$$ − 27.0000i − 1.16950i
$$534$$ −6.00000 −0.259645
$$535$$ 0 0
$$536$$ 8.00000 0.345547
$$537$$ 10.0000i 0.431532i
$$538$$ 5.00000i 0.215565i
$$539$$ 6.00000 0.258438
$$540$$ 0 0
$$541$$ −13.0000 −0.558914 −0.279457 0.960158i $$-0.590154\pi$$
−0.279457 + 0.960158i $$0.590154\pi$$
$$542$$ 22.0000i 0.944981i
$$543$$ − 16.0000i − 0.686626i
$$544$$ 4.00000 0.171499
$$545$$ 0 0
$$546$$ 9.00000 0.385164
$$547$$ 22.0000i 0.940652i 0.882493 + 0.470326i $$0.155864\pi$$
−0.882493 + 0.470326i $$0.844136\pi$$
$$548$$ 18.0000i 0.768922i
$$549$$ −12.0000 −0.512148
$$550$$ 0 0
$$551$$ −9.00000 −0.383413
$$552$$ 1.00000i 0.0425628i
$$553$$ 51.0000i 2.16874i
$$554$$ 13.0000 0.552317
$$555$$ 0 0
$$556$$ 8.00000 0.339276
$$557$$ − 32.0000i − 1.35588i −0.735116 0.677942i $$-0.762872\pi$$
0.735116 0.677942i $$-0.237128\pi$$
$$558$$ 10.0000i 0.423334i
$$559$$ 3.00000 0.126886
$$560$$ 0 0
$$561$$ −12.0000 −0.506640
$$562$$ − 10.0000i − 0.421825i
$$563$$ − 5.00000i − 0.210725i −0.994434 0.105362i $$-0.966400\pi$$
0.994434 0.105362i $$-0.0336003\pi$$
$$564$$ 2.00000 0.0842152
$$565$$ 0 0
$$566$$ 0 0
$$567$$ 3.00000i 0.125988i
$$568$$ 14.0000i 0.587427i
$$569$$ 20.0000 0.838444 0.419222 0.907884i $$-0.362303\pi$$
0.419222 + 0.907884i $$0.362303\pi$$
$$570$$ 0 0
$$571$$ −4.00000 −0.167395 −0.0836974 0.996491i $$-0.526673\pi$$
−0.0836974 + 0.996491i $$0.526673\pi$$
$$572$$ 9.00000i 0.376309i
$$573$$ − 15.0000i − 0.626634i
$$574$$ 27.0000 1.12696
$$575$$ 0 0
$$576$$ 1.00000 0.0416667
$$577$$ − 7.00000i − 0.291414i −0.989328 0.145707i $$-0.953454\pi$$
0.989328 0.145707i $$-0.0465456\pi$$
$$578$$ 1.00000i 0.0415945i
$$579$$ −6.00000 −0.249351
$$580$$ 0 0
$$581$$ 3.00000 0.124461
$$582$$ − 6.00000i − 0.248708i
$$583$$ − 18.0000i − 0.745484i
$$584$$ −13.0000 −0.537944
$$585$$ 0 0
$$586$$ 0 0
$$587$$ 24.0000i 0.990586i 0.868726 + 0.495293i $$0.164939\pi$$
−0.868726 + 0.495293i $$0.835061\pi$$
$$588$$ 2.00000i 0.0824786i
$$589$$ −30.0000 −1.23613
$$590$$ 0 0
$$591$$ 17.0000 0.699287
$$592$$ − 8.00000i − 0.328798i
$$593$$ 21.0000i 0.862367i 0.902264 + 0.431183i $$0.141904\pi$$
−0.902264 + 0.431183i $$0.858096\pi$$
$$594$$ −3.00000 −0.123091
$$595$$ 0 0
$$596$$ −6.00000 −0.245770
$$597$$ − 11.0000i − 0.450200i
$$598$$ − 3.00000i − 0.122679i
$$599$$ −6.00000 −0.245153 −0.122577 0.992459i $$-0.539116\pi$$
−0.122577 + 0.992459i $$0.539116\pi$$
$$600$$ 0 0
$$601$$ −14.0000 −0.571072 −0.285536 0.958368i $$-0.592172\pi$$
−0.285536 + 0.958368i $$0.592172\pi$$
$$602$$ 3.00000i 0.122271i
$$603$$ − 8.00000i − 0.325785i
$$604$$ 22.0000 0.895167
$$605$$ 0 0
$$606$$ 6.00000 0.243733
$$607$$ 2.00000i 0.0811775i 0.999176 + 0.0405887i $$0.0129233\pi$$
−0.999176 + 0.0405887i $$0.987077\pi$$
$$608$$ 3.00000i 0.121666i
$$609$$ 9.00000 0.364698
$$610$$ 0 0
$$611$$ −6.00000 −0.242734
$$612$$ − 4.00000i − 0.161690i
$$613$$ − 20.0000i − 0.807792i −0.914805 0.403896i $$-0.867656\pi$$
0.914805 0.403896i $$-0.132344\pi$$
$$614$$ −8.00000 −0.322854
$$615$$ 0 0
$$616$$ −9.00000 −0.362620
$$617$$ − 16.0000i − 0.644136i −0.946717 0.322068i $$-0.895622\pi$$
0.946717 0.322068i $$-0.104378\pi$$
$$618$$ 13.0000i 0.522937i
$$619$$ −20.0000 −0.803868 −0.401934 0.915669i $$-0.631662\pi$$
−0.401934 + 0.915669i $$0.631662\pi$$
$$620$$ 0 0
$$621$$ 1.00000 0.0401286
$$622$$ 24.0000i 0.962312i
$$623$$ 18.0000i 0.721155i
$$624$$ −3.00000 −0.120096
$$625$$ 0 0
$$626$$ 24.0000 0.959233
$$627$$ − 9.00000i − 0.359425i
$$628$$ − 6.00000i − 0.239426i
$$629$$ −32.0000 −1.27592
$$630$$ 0 0
$$631$$ 5.00000 0.199047 0.0995234 0.995035i $$-0.468268\pi$$
0.0995234 + 0.995035i $$0.468268\pi$$
$$632$$ − 17.0000i − 0.676224i
$$633$$ 22.0000i 0.874421i
$$634$$ −31.0000 −1.23117
$$635$$ 0 0
$$636$$ 6.00000 0.237915
$$637$$ − 6.00000i − 0.237729i
$$638$$ 9.00000i 0.356313i
$$639$$ 14.0000 0.553831
$$640$$ 0 0
$$641$$ −16.0000 −0.631962 −0.315981 0.948766i $$-0.602334\pi$$
−0.315981 + 0.948766i $$0.602334\pi$$
$$642$$ 12.0000i 0.473602i
$$643$$ 31.0000i 1.22252i 0.791430 + 0.611260i $$0.209337\pi$$
−0.791430 + 0.611260i $$0.790663\pi$$
$$644$$ 3.00000 0.118217
$$645$$ 0 0
$$646$$ 12.0000 0.472134
$$647$$ − 18.0000i − 0.707653i −0.935311 0.353827i $$-0.884880\pi$$
0.935311 0.353827i $$-0.115120\pi$$
$$648$$ − 1.00000i − 0.0392837i
$$649$$ 24.0000 0.942082
$$650$$ 0 0
$$651$$ 30.0000 1.17579
$$652$$ 6.00000i 0.234978i
$$653$$ 3.00000i 0.117399i 0.998276 + 0.0586995i $$0.0186954\pi$$
−0.998276 + 0.0586995i $$0.981305\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ −9.00000 −0.351391
$$657$$ 13.0000i 0.507178i
$$658$$ − 6.00000i − 0.233904i
$$659$$ 33.0000 1.28550 0.642749 0.766077i $$-0.277794\pi$$
0.642749 + 0.766077i $$0.277794\pi$$
$$660$$ 0 0
$$661$$ −10.0000 −0.388955 −0.194477 0.980907i $$-0.562301\pi$$
−0.194477 + 0.980907i $$0.562301\pi$$
$$662$$ − 26.0000i − 1.01052i
$$663$$ 12.0000i 0.466041i
$$664$$ −1.00000 −0.0388075
$$665$$ 0 0
$$666$$ −8.00000 −0.309994
$$667$$ − 3.00000i − 0.116160i
$$668$$ − 2.00000i − 0.0773823i
$$669$$ −6.00000 −0.231973
$$670$$ 0 0
$$671$$ −36.0000 −1.38976
$$672$$ − 3.00000i − 0.115728i
$$673$$ − 3.00000i − 0.115642i −0.998327 0.0578208i $$-0.981585\pi$$
0.998327 0.0578208i $$-0.0184152\pi$$
$$674$$ −14.0000 −0.539260
$$675$$ 0 0
$$676$$ −4.00000 −0.153846
$$677$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$678$$ 6.00000i 0.230429i
$$679$$ −18.0000 −0.690777
$$680$$ 0 0
$$681$$ 12.0000 0.459841
$$682$$ 30.0000i 1.14876i
$$683$$ − 38.0000i − 1.45403i −0.686622 0.727015i $$-0.740907\pi$$
0.686622 0.727015i $$-0.259093\pi$$
$$684$$ 3.00000 0.114708
$$685$$ 0 0
$$686$$ −15.0000 −0.572703
$$687$$ − 6.00000i − 0.228914i
$$688$$ − 1.00000i − 0.0381246i
$$689$$ −18.0000 −0.685745
$$690$$ 0 0
$$691$$ −48.0000 −1.82601 −0.913003 0.407953i $$-0.866243\pi$$
−0.913003 + 0.407953i $$0.866243\pi$$
$$692$$ 15.0000i 0.570214i
$$693$$ 9.00000i 0.341882i
$$694$$ 0 0
$$695$$ 0 0
$$696$$ −3.00000 −0.113715
$$697$$ 36.0000i 1.36360i
$$698$$ 5.00000i 0.189253i
$$699$$ −1.00000 −0.0378235
$$700$$ 0 0
$$701$$ −36.0000 −1.35970 −0.679851 0.733351i $$-0.737955\pi$$
−0.679851 + 0.733351i $$0.737955\pi$$
$$702$$ 3.00000i 0.113228i
$$703$$ − 24.0000i − 0.905177i
$$704$$ 3.00000 0.113067
$$705$$ 0 0
$$706$$ 9.00000 0.338719
$$707$$ − 18.0000i − 0.676960i
$$708$$ 8.00000i 0.300658i
$$709$$ −26.0000 −0.976450 −0.488225 0.872718i $$-0.662356\pi$$
−0.488225 + 0.872718i $$0.662356\pi$$
$$710$$ 0 0
$$711$$ −17.0000 −0.637550
$$712$$ − 6.00000i − 0.224860i
$$713$$ − 10.0000i − 0.374503i
$$714$$ −12.0000 −0.449089
$$715$$ 0 0
$$716$$ −10.0000 −0.373718
$$717$$ 10.0000i 0.373457i
$$718$$ − 31.0000i − 1.15691i
$$719$$ −16.0000 −0.596699 −0.298350 0.954457i $$-0.596436\pi$$
−0.298350 + 0.954457i $$0.596436\pi$$
$$720$$ 0 0
$$721$$ 39.0000 1.45244
$$722$$ − 10.0000i − 0.372161i
$$723$$ − 10.0000i − 0.371904i
$$724$$ 16.0000 0.594635
$$725$$ 0 0
$$726$$ 2.00000 0.0742270
$$727$$ − 16.0000i − 0.593407i −0.954970 0.296704i $$-0.904113\pi$$
0.954970 0.296704i $$-0.0958873\pi$$
$$728$$ 9.00000i 0.333562i
$$729$$ −1.00000 −0.0370370
$$730$$ 0 0
$$731$$ −4.00000 −0.147945
$$732$$ − 12.0000i − 0.443533i
$$733$$ − 14.0000i − 0.517102i −0.965998 0.258551i $$-0.916755\pi$$
0.965998 0.258551i $$-0.0832450\pi$$
$$734$$ −27.0000 −0.996588
$$735$$ 0 0
$$736$$ −1.00000 −0.0368605
$$737$$ − 24.0000i − 0.884051i
$$738$$ 9.00000i 0.331295i
$$739$$ −50.0000 −1.83928 −0.919640 0.392763i $$-0.871519\pi$$
−0.919640 + 0.392763i $$0.871519\pi$$
$$740$$ 0 0
$$741$$ −9.00000 −0.330623
$$742$$ − 18.0000i − 0.660801i
$$743$$ 29.0000i 1.06391i 0.846774 + 0.531953i $$0.178542\pi$$
−0.846774 + 0.531953i $$0.821458\pi$$
$$744$$ −10.0000 −0.366618
$$745$$ 0 0
$$746$$ −16.0000 −0.585802
$$747$$ 1.00000i 0.0365881i
$$748$$ − 12.0000i − 0.438763i
$$749$$ 36.0000 1.31541
$$750$$ 0 0
$$751$$ 45.0000 1.64207 0.821037 0.570875i $$-0.193396\pi$$
0.821037 + 0.570875i $$0.193396\pi$$
$$752$$ 2.00000i 0.0729325i
$$753$$ − 16.0000i − 0.583072i
$$754$$ 9.00000 0.327761
$$755$$ 0 0
$$756$$ −3.00000 −0.109109
$$757$$ − 20.0000i − 0.726912i −0.931611 0.363456i $$-0.881597\pi$$
0.931611 0.363456i $$-0.118403\pi$$
$$758$$ − 12.0000i − 0.435860i
$$759$$ 3.00000 0.108893
$$760$$ 0 0
$$761$$ −27.0000 −0.978749 −0.489375 0.872074i $$-0.662775\pi$$
−0.489375 + 0.872074i $$0.662775\pi$$
$$762$$ − 16.0000i − 0.579619i
$$763$$ 0 0
$$764$$ 15.0000 0.542681
$$765$$ 0 0
$$766$$ −29.0000 −1.04781
$$767$$ − 24.0000i − 0.866590i
$$768$$ 1.00000i 0.0360844i
$$769$$ −36.0000 −1.29819 −0.649097 0.760706i $$-0.724853\pi$$
−0.649097 + 0.760706i $$0.724853\pi$$
$$770$$ 0 0
$$771$$ 18.0000 0.648254
$$772$$ − 6.00000i − 0.215945i
$$773$$ 40.0000i 1.43870i 0.694648 + 0.719350i $$0.255560\pi$$
−0.694648 + 0.719350i $$0.744440\pi$$
$$774$$ −1.00000 −0.0359443
$$775$$ 0 0
$$776$$ 6.00000 0.215387
$$777$$ 24.0000i 0.860995i
$$778$$ − 38.0000i − 1.36237i
$$779$$ −27.0000 −0.967375
$$780$$ 0 0
$$781$$ 42.0000 1.50288
$$782$$ 4.00000i 0.143040i
$$783$$ 3.00000i 0.107211i
$$784$$ −2.00000 −0.0714286
$$785$$ 0 0
$$786$$ −10.0000 −0.356688
$$787$$ − 5.00000i − 0.178231i −0.996021 0.0891154i $$-0.971596\pi$$
0.996021 0.0891154i $$-0.0284040\pi$$
$$788$$ 17.0000i 0.605600i
$$789$$ −8.00000 −0.284808
$$790$$ 0 0
$$791$$ 18.0000 0.640006
$$792$$ − 3.00000i − 0.106600i
$$793$$ 36.0000i 1.27840i
$$794$$ −30.0000 −1.06466
$$795$$ 0 0
$$796$$ 11.0000 0.389885
$$797$$ 24.0000i 0.850124i 0.905164 + 0.425062i $$0.139748\pi$$
−0.905164 + 0.425062i $$0.860252\pi$$
$$798$$ − 9.00000i − 0.318597i
$$799$$ 8.00000 0.283020
$$800$$ 0 0
$$801$$ −6.00000 −0.212000
$$802$$ − 28.0000i − 0.988714i
$$803$$ 39.0000i 1.37628i
$$804$$ 8.00000 0.282138
$$805$$ 0 0
$$806$$ 30.0000 1.05670
$$807$$ 5.00000i 0.176008i
$$808$$ 6.00000i 0.211079i
$$809$$ −21.0000 −0.738321 −0.369160 0.929366i $$-0.620355\pi$$
−0.369160 + 0.929366i $$0.620355\pi$$
$$810$$ 0 0
$$811$$ 16.0000 0.561836 0.280918 0.959732i $$-0.409361\pi$$
0.280918 + 0.959732i $$0.409361\pi$$
$$812$$ 9.00000i 0.315838i
$$813$$ 22.0000i 0.771574i
$$814$$ −24.0000 −0.841200
$$815$$ 0 0
$$816$$ 4.00000 0.140028
$$817$$ − 3.00000i − 0.104957i
$$818$$ 23.0000i 0.804176i
$$819$$ 9.00000 0.314485
$$820$$ 0 0
$$821$$ −55.0000 −1.91951 −0.959757 0.280833i $$-0.909389\pi$$
−0.959757 + 0.280833i $$0.909389\pi$$
$$822$$ 18.0000i 0.627822i
$$823$$ 16.0000i 0.557725i 0.960331 + 0.278862i $$0.0899574\pi$$
−0.960331 + 0.278862i $$0.910043\pi$$
$$824$$ −13.0000 −0.452876
$$825$$ 0 0
$$826$$ 24.0000 0.835067
$$827$$ − 9.00000i − 0.312961i −0.987681 0.156480i $$-0.949985\pi$$
0.987681 0.156480i $$-0.0500148\pi$$
$$828$$ 1.00000i 0.0347524i
$$829$$ −11.0000 −0.382046 −0.191023 0.981586i $$-0.561180\pi$$
−0.191023 + 0.981586i $$0.561180\pi$$
$$830$$ 0 0
$$831$$ 13.0000 0.450965
$$832$$ − 3.00000i − 0.104006i
$$833$$ 8.00000i 0.277184i
$$834$$ 8.00000 0.277017
$$835$$ 0 0
$$836$$ 9.00000 0.311272
$$837$$ 10.0000i 0.345651i
$$838$$ − 19.0000i − 0.656344i
$$839$$ −33.0000 −1.13929 −0.569643 0.821892i $$-0.692919\pi$$
−0.569643 + 0.821892i $$0.692919\pi$$
$$840$$ 0 0
$$841$$ −20.0000 −0.689655
$$842$$ − 10.0000i − 0.344623i
$$843$$ − 10.0000i − 0.344418i
$$844$$ −22.0000 −0.757271
$$845$$ 0 0
$$846$$ 2.00000 0.0687614
$$847$$ − 6.00000i − 0.206162i
$$848$$ 6.00000i 0.206041i
$$849$$ 0 0
$$850$$ 0 0
$$851$$ 8.00000 0.274236
$$852$$ 14.0000i 0.479632i
$$853$$ 23.0000i 0.787505i 0.919216 + 0.393753i $$0.128823\pi$$
−0.919216 + 0.393753i $$0.871177\pi$$
$$854$$ −36.0000 −1.23189
$$855$$ 0 0
$$856$$ −12.0000 −0.410152
$$857$$ 22.0000i 0.751506i 0.926720 + 0.375753i $$0.122616\pi$$
−0.926720 + 0.375753i $$0.877384\pi$$
$$858$$ 9.00000i 0.307255i
$$859$$ −20.0000 −0.682391 −0.341196 0.939992i $$-0.610832\pi$$
−0.341196 + 0.939992i $$0.610832\pi$$
$$860$$ 0 0
$$861$$ 27.0000 0.920158
$$862$$ 16.0000i 0.544962i
$$863$$ − 54.0000i − 1.83818i −0.394046 0.919091i $$-0.628925\pi$$
0.394046 0.919091i $$-0.371075\pi$$
$$864$$ 1.00000 0.0340207
$$865$$ 0 0
$$866$$ 4.00000 0.135926
$$867$$ 1.00000i 0.0339618i
$$868$$ 30.0000i 1.01827i
$$869$$ −51.0000 −1.73006
$$870$$ 0 0
$$871$$ −24.0000 −0.813209
$$872$$ 0 0
$$873$$ − 6.00000i − 0.203069i
$$874$$ −3.00000 −0.101477
$$875$$ 0 0
$$876$$ −13.0000 −0.439229
$$877$$ − 22.0000i − 0.742887i −0.928456 0.371444i $$-0.878863\pi$$
0.928456 0.371444i $$-0.121137\pi$$
$$878$$ 4.00000i 0.134993i
$$879$$ 0 0
$$880$$ 0 0
$$881$$ −40.0000 −1.34763 −0.673817 0.738898i $$-0.735346\pi$$
−0.673817 + 0.738898i $$0.735346\pi$$
$$882$$ 2.00000i 0.0673435i
$$883$$ 48.0000i 1.61533i 0.589643 + 0.807664i $$0.299269\pi$$
−0.589643 + 0.807664i $$0.700731\pi$$
$$884$$ −12.0000 −0.403604
$$885$$ 0 0
$$886$$ 26.0000 0.873487
$$887$$ − 8.00000i − 0.268614i −0.990940 0.134307i $$-0.957119\pi$$
0.990940 0.134307i $$-0.0428808\pi$$
$$888$$ − 8.00000i − 0.268462i
$$889$$ −48.0000 −1.60987
$$890$$ 0 0
$$891$$ −3.00000 −0.100504
$$892$$ − 6.00000i − 0.200895i
$$893$$ 6.00000i 0.200782i
$$894$$ −6.00000 −0.200670
$$895$$ 0 0
$$896$$ 3.00000 0.100223
$$897$$ − 3.00000i − 0.100167i
$$898$$ 26.0000i 0.867631i
$$899$$ 30.0000 1.00056
$$900$$ 0 0
$$901$$ 24.0000 0.799556
$$902$$ 27.0000i 0.899002i
$$903$$ 3.00000i 0.0998337i
$$904$$ −6.00000 −0.199557
$$905$$ 0 0
$$906$$ 22.0000 0.730901
$$907$$ − 1.00000i − 0.0332045i −0.999862 0.0166022i $$-0.994715\pi$$
0.999862 0.0166022i $$-0.00528490\pi$$
$$908$$ 12.0000i 0.398234i
$$909$$ 6.00000 0.199007
$$910$$ 0 0
$$911$$ 55.0000 1.82223 0.911116 0.412151i $$-0.135222\pi$$
0.911116 + 0.412151i $$0.135222\pi$$
$$912$$ 3.00000i 0.0993399i
$$913$$ 3.00000i 0.0992855i
$$914$$ −22.0000 −0.727695
$$915$$ 0 0
$$916$$ 6.00000 0.198246
$$917$$ 30.0000i 0.990687i
$$918$$ − 4.00000i − 0.132020i
$$919$$ −32.0000 −1.05558 −0.527791 0.849374i $$-0.676980\pi$$
−0.527791 + 0.849374i $$0.676980\pi$$
$$920$$ 0 0
$$921$$ −8.00000 −0.263609
$$922$$ − 9.00000i − 0.296399i
$$923$$ − 42.0000i − 1.38245i
$$924$$ −9.00000 −0.296078
$$925$$ 0 0
$$926$$ −4.00000 −0.131448
$$927$$ 13.0000i 0.426976i
$$928$$ − 3.00000i − 0.0984798i
$$929$$ 41.0000 1.34517 0.672583 0.740022i $$-0.265185\pi$$
0.672583 + 0.740022i $$0.265185\pi$$
$$930$$ 0 0
$$931$$ −6.00000 −0.196642
$$932$$ − 1.00000i − 0.0327561i
$$933$$ 24.0000i 0.785725i
$$934$$ −11.0000 −0.359931
$$935$$ 0 0
$$936$$ −3.00000 −0.0980581
$$937$$ − 22.0000i − 0.718709i −0.933201 0.359354i $$-0.882997\pi$$
0.933201 0.359354i $$-0.117003\pi$$
$$938$$ − 24.0000i − 0.783628i
$$939$$ 24.0000 0.783210
$$940$$ 0 0
$$941$$ −6.00000 −0.195594 −0.0977972 0.995206i $$-0.531180\pi$$
−0.0977972 + 0.995206i $$0.531180\pi$$
$$942$$ − 6.00000i − 0.195491i
$$943$$ − 9.00000i − 0.293080i
$$944$$ −8.00000 −0.260378
$$945$$ 0 0
$$946$$ −3.00000 −0.0975384
$$947$$ 2.00000i 0.0649913i 0.999472 + 0.0324956i $$0.0103455\pi$$
−0.999472 + 0.0324956i $$0.989654\pi$$
$$948$$ − 17.0000i − 0.552134i
$$949$$ 39.0000 1.26599
$$950$$ 0 0
$$951$$ −31.0000 −1.00524
$$952$$ − 12.0000i − 0.388922i
$$953$$ 24.0000i 0.777436i 0.921357 + 0.388718i $$0.127082\pi$$
−0.921357 + 0.388718i $$0.872918\pi$$
$$954$$ 6.00000 0.194257
$$955$$ 0 0
$$956$$ −10.0000 −0.323423
$$957$$ 9.00000i 0.290929i
$$958$$ − 33.0000i − 1.06618i
$$959$$ 54.0000 1.74375
$$960$$ 0 0
$$961$$ 69.0000 2.22581
$$962$$ 24.0000i 0.773791i
$$963$$ 12.0000i 0.386695i
$$964$$ 10.0000 0.322078
$$965$$ 0 0
$$966$$ 3.00000 0.0965234
$$967$$ − 44.0000i − 1.41494i −0.706741 0.707472i $$-0.749835\pi$$
0.706741 0.707472i $$-0.250165\pi$$
$$968$$ 2.00000i 0.0642824i
$$969$$ 12.0000 0.385496
$$970$$ 0 0
$$971$$ 43.0000 1.37994 0.689968 0.723840i $$-0.257625\pi$$
0.689968 + 0.723840i $$0.257625\pi$$
$$972$$ − 1.00000i − 0.0320750i
$$973$$ − 24.0000i − 0.769405i
$$974$$ 2.00000 0.0640841
$$975$$ 0 0
$$976$$ 12.0000 0.384111
$$977$$ 38.0000i 1.21573i 0.794041 + 0.607864i $$0.207973\pi$$
−0.794041 + 0.607864i $$0.792027\pi$$
$$978$$ 6.00000i 0.191859i
$$979$$ −18.0000 −0.575282
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 12.0000i 0.382935i
$$983$$ 39.0000i 1.24391i 0.783054 + 0.621953i $$0.213661\pi$$
−0.783054 + 0.621953i $$0.786339\pi$$
$$984$$ −9.00000 −0.286910
$$985$$ 0 0
$$986$$ −12.0000 −0.382158
$$987$$ − 6.00000i − 0.190982i
$$988$$ − 9.00000i − 0.286328i
$$989$$ 1.00000 0.0317982
$$990$$ 0 0
$$991$$ −26.0000 −0.825917 −0.412959 0.910750i $$-0.635505\pi$$
−0.412959 + 0.910750i $$0.635505\pi$$
$$992$$ − 10.0000i − 0.317500i
$$993$$ − 26.0000i − 0.825085i
$$994$$ 42.0000 1.33216
$$995$$ 0 0
$$996$$ −1.00000 −0.0316862
$$997$$ − 49.0000i − 1.55185i −0.630828 0.775923i $$-0.717285\pi$$
0.630828 0.775923i $$-0.282715\pi$$
$$998$$ − 34.0000i − 1.07625i
$$999$$ −8.00000 −0.253109
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.d.b.2899.2 2
5.2 odd 4 3450.2.a.h.1.1 1
5.3 odd 4 3450.2.a.s.1.1 yes 1
5.4 even 2 inner 3450.2.d.b.2899.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
3450.2.a.h.1.1 1 5.2 odd 4
3450.2.a.s.1.1 yes 1 5.3 odd 4
3450.2.d.b.2899.1 2 5.4 even 2 inner
3450.2.d.b.2899.2 2 1.1 even 1 trivial