# Properties

 Label 3450.2.d.m.2899.1 Level $3450$ Weight $2$ Character 3450.2899 Analytic conductor $27.548$ Analytic rank $0$ 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: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(i)$$ Defining polynomial: $$x^{2} + 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 690) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

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

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} +1.00000i q^{8} -1.00000 q^{9} +O(q^{10})$$ $$q-1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} +1.00000i q^{8} -1.00000 q^{9} -2.00000 q^{11} -1.00000i q^{12} +4.00000i q^{13} +1.00000 q^{16} -6.00000i q^{17} +1.00000i q^{18} +8.00000 q^{19} +2.00000i q^{22} -1.00000i q^{23} -1.00000 q^{24} +4.00000 q^{26} -1.00000i q^{27} -4.00000 q^{29} -1.00000i q^{32} -2.00000i q^{33} -6.00000 q^{34} +1.00000 q^{36} +2.00000i q^{37} -8.00000i q^{38} -4.00000 q^{39} -2.00000 q^{41} +2.00000i q^{43} +2.00000 q^{44} -1.00000 q^{46} +12.0000i q^{47} +1.00000i q^{48} +7.00000 q^{49} +6.00000 q^{51} -4.00000i q^{52} -6.00000i q^{53} -1.00000 q^{54} +8.00000i q^{57} +4.00000i q^{58} -8.00000 q^{59} +2.00000 q^{61} -1.00000 q^{64} -2.00000 q^{66} +6.00000i q^{67} +6.00000i q^{68} +1.00000 q^{69} +10.0000 q^{71} -1.00000i q^{72} -2.00000i q^{73} +2.00000 q^{74} -8.00000 q^{76} +4.00000i q^{78} +8.00000 q^{79} +1.00000 q^{81} +2.00000i q^{82} +8.00000i q^{83} +2.00000 q^{86} -4.00000i q^{87} -2.00000i q^{88} +12.0000 q^{89} +1.00000i q^{92} +12.0000 q^{94} +1.00000 q^{96} +16.0000i q^{97} -7.00000i q^{98} +2.00000 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{4} + 2q^{6} - 2q^{9} + O(q^{10})$$ $$2q - 2q^{4} + 2q^{6} - 2q^{9} - 4q^{11} + 2q^{16} + 16q^{19} - 2q^{24} + 8q^{26} - 8q^{29} - 12q^{34} + 2q^{36} - 8q^{39} - 4q^{41} + 4q^{44} - 2q^{46} + 14q^{49} + 12q^{51} - 2q^{54} - 16q^{59} + 4q^{61} - 2q^{64} - 4q^{66} + 2q^{69} + 20q^{71} + 4q^{74} - 16q^{76} + 16q^{79} + 2q^{81} + 4q^{86} + 24q^{89} + 24q^{94} + 2q^{96} + 4q^{99} + O(q^{100})$$

## 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
$$7$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$8$$ 1.00000i 0.353553i
$$9$$ −1.00000 −0.333333
$$10$$ 0 0
$$11$$ −2.00000 −0.603023 −0.301511 0.953463i $$-0.597491\pi$$
−0.301511 + 0.953463i $$0.597491\pi$$
$$12$$ − 1.00000i − 0.288675i
$$13$$ 4.00000i 1.10940i 0.832050 + 0.554700i $$0.187167\pi$$
−0.832050 + 0.554700i $$0.812833\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ − 6.00000i − 1.45521i −0.685994 0.727607i $$-0.740633\pi$$
0.685994 0.727607i $$-0.259367\pi$$
$$18$$ 1.00000i 0.235702i
$$19$$ 8.00000 1.83533 0.917663 0.397360i $$-0.130073\pi$$
0.917663 + 0.397360i $$0.130073\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 2.00000i 0.426401i
$$23$$ − 1.00000i − 0.208514i
$$24$$ −1.00000 −0.204124
$$25$$ 0 0
$$26$$ 4.00000 0.784465
$$27$$ − 1.00000i − 0.192450i
$$28$$ 0 0
$$29$$ −4.00000 −0.742781 −0.371391 0.928477i $$-0.621119\pi$$
−0.371391 + 0.928477i $$0.621119\pi$$
$$30$$ 0 0
$$31$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$32$$ − 1.00000i − 0.176777i
$$33$$ − 2.00000i − 0.348155i
$$34$$ −6.00000 −1.02899
$$35$$ 0 0
$$36$$ 1.00000 0.166667
$$37$$ 2.00000i 0.328798i 0.986394 + 0.164399i $$0.0525685\pi$$
−0.986394 + 0.164399i $$0.947432\pi$$
$$38$$ − 8.00000i − 1.29777i
$$39$$ −4.00000 −0.640513
$$40$$ 0 0
$$41$$ −2.00000 −0.312348 −0.156174 0.987730i $$-0.549916\pi$$
−0.156174 + 0.987730i $$0.549916\pi$$
$$42$$ 0 0
$$43$$ 2.00000i 0.304997i 0.988304 + 0.152499i $$0.0487319\pi$$
−0.988304 + 0.152499i $$0.951268\pi$$
$$44$$ 2.00000 0.301511
$$45$$ 0 0
$$46$$ −1.00000 −0.147442
$$47$$ 12.0000i 1.75038i 0.483779 + 0.875190i $$0.339264\pi$$
−0.483779 + 0.875190i $$0.660736\pi$$
$$48$$ 1.00000i 0.144338i
$$49$$ 7.00000 1.00000
$$50$$ 0 0
$$51$$ 6.00000 0.840168
$$52$$ − 4.00000i − 0.554700i
$$53$$ − 6.00000i − 0.824163i −0.911147 0.412082i $$-0.864802\pi$$
0.911147 0.412082i $$-0.135198\pi$$
$$54$$ −1.00000 −0.136083
$$55$$ 0 0
$$56$$ 0 0
$$57$$ 8.00000i 1.05963i
$$58$$ 4.00000i 0.525226i
$$59$$ −8.00000 −1.04151 −0.520756 0.853706i $$-0.674350\pi$$
−0.520756 + 0.853706i $$0.674350\pi$$
$$60$$ 0 0
$$61$$ 2.00000 0.256074 0.128037 0.991769i $$-0.459132\pi$$
0.128037 + 0.991769i $$0.459132\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ −1.00000 −0.125000
$$65$$ 0 0
$$66$$ −2.00000 −0.246183
$$67$$ 6.00000i 0.733017i 0.930415 + 0.366508i $$0.119447\pi$$
−0.930415 + 0.366508i $$0.880553\pi$$
$$68$$ 6.00000i 0.727607i
$$69$$ 1.00000 0.120386
$$70$$ 0 0
$$71$$ 10.0000 1.18678 0.593391 0.804914i $$-0.297789\pi$$
0.593391 + 0.804914i $$0.297789\pi$$
$$72$$ − 1.00000i − 0.117851i
$$73$$ − 2.00000i − 0.234082i −0.993127 0.117041i $$-0.962659\pi$$
0.993127 0.117041i $$-0.0373409\pi$$
$$74$$ 2.00000 0.232495
$$75$$ 0 0
$$76$$ −8.00000 −0.917663
$$77$$ 0 0
$$78$$ 4.00000i 0.452911i
$$79$$ 8.00000 0.900070 0.450035 0.893011i $$-0.351411\pi$$
0.450035 + 0.893011i $$0.351411\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 2.00000i 0.220863i
$$83$$ 8.00000i 0.878114i 0.898459 + 0.439057i $$0.144687\pi$$
−0.898459 + 0.439057i $$0.855313\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 2.00000 0.215666
$$87$$ − 4.00000i − 0.428845i
$$88$$ − 2.00000i − 0.213201i
$$89$$ 12.0000 1.27200 0.635999 0.771690i $$-0.280588\pi$$
0.635999 + 0.771690i $$0.280588\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 1.00000i 0.104257i
$$93$$ 0 0
$$94$$ 12.0000 1.23771
$$95$$ 0 0
$$96$$ 1.00000 0.102062
$$97$$ 16.0000i 1.62455i 0.583272 + 0.812277i $$0.301772\pi$$
−0.583272 + 0.812277i $$0.698228\pi$$
$$98$$ − 7.00000i − 0.707107i
$$99$$ 2.00000 0.201008
$$100$$ 0 0
$$101$$ 12.0000 1.19404 0.597022 0.802225i $$-0.296350\pi$$
0.597022 + 0.802225i $$0.296350\pi$$
$$102$$ − 6.00000i − 0.594089i
$$103$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$104$$ −4.00000 −0.392232
$$105$$ 0 0
$$106$$ −6.00000 −0.582772
$$107$$ 12.0000i 1.16008i 0.814587 + 0.580042i $$0.196964\pi$$
−0.814587 + 0.580042i $$0.803036\pi$$
$$108$$ 1.00000i 0.0962250i
$$109$$ 10.0000 0.957826 0.478913 0.877862i $$-0.341031\pi$$
0.478913 + 0.877862i $$0.341031\pi$$
$$110$$ 0 0
$$111$$ −2.00000 −0.189832
$$112$$ 0 0
$$113$$ 18.0000i 1.69330i 0.532152 + 0.846649i $$0.321383\pi$$
−0.532152 + 0.846649i $$0.678617\pi$$
$$114$$ 8.00000 0.749269
$$115$$ 0 0
$$116$$ 4.00000 0.371391
$$117$$ − 4.00000i − 0.369800i
$$118$$ 8.00000i 0.736460i
$$119$$ 0 0
$$120$$ 0 0
$$121$$ −7.00000 −0.636364
$$122$$ − 2.00000i − 0.181071i
$$123$$ − 2.00000i − 0.180334i
$$124$$ 0 0
$$125$$ 0 0
$$126$$ 0 0
$$127$$ − 2.00000i − 0.177471i −0.996055 0.0887357i $$-0.971717\pi$$
0.996055 0.0887357i $$-0.0282826\pi$$
$$128$$ 1.00000i 0.0883883i
$$129$$ −2.00000 −0.176090
$$130$$ 0 0
$$131$$ −4.00000 −0.349482 −0.174741 0.984614i $$-0.555909\pi$$
−0.174741 + 0.984614i $$0.555909\pi$$
$$132$$ 2.00000i 0.174078i
$$133$$ 0 0
$$134$$ 6.00000 0.518321
$$135$$ 0 0
$$136$$ 6.00000 0.514496
$$137$$ − 2.00000i − 0.170872i −0.996344 0.0854358i $$-0.972772\pi$$
0.996344 0.0854358i $$-0.0272282\pi$$
$$138$$ − 1.00000i − 0.0851257i
$$139$$ −4.00000 −0.339276 −0.169638 0.985506i $$-0.554260\pi$$
−0.169638 + 0.985506i $$0.554260\pi$$
$$140$$ 0 0
$$141$$ −12.0000 −1.01058
$$142$$ − 10.0000i − 0.839181i
$$143$$ − 8.00000i − 0.668994i
$$144$$ −1.00000 −0.0833333
$$145$$ 0 0
$$146$$ −2.00000 −0.165521
$$147$$ 7.00000i 0.577350i
$$148$$ − 2.00000i − 0.164399i
$$149$$ −14.0000 −1.14692 −0.573462 0.819232i $$-0.694400\pi$$
−0.573462 + 0.819232i $$0.694400\pi$$
$$150$$ 0 0
$$151$$ −4.00000 −0.325515 −0.162758 0.986666i $$-0.552039\pi$$
−0.162758 + 0.986666i $$0.552039\pi$$
$$152$$ 8.00000i 0.648886i
$$153$$ 6.00000i 0.485071i
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 4.00000 0.320256
$$157$$ 10.0000i 0.798087i 0.916932 + 0.399043i $$0.130658\pi$$
−0.916932 + 0.399043i $$0.869342\pi$$
$$158$$ − 8.00000i − 0.636446i
$$159$$ 6.00000 0.475831
$$160$$ 0 0
$$161$$ 0 0
$$162$$ − 1.00000i − 0.0785674i
$$163$$ − 16.0000i − 1.25322i −0.779334 0.626608i $$-0.784443\pi$$
0.779334 0.626608i $$-0.215557\pi$$
$$164$$ 2.00000 0.156174
$$165$$ 0 0
$$166$$ 8.00000 0.620920
$$167$$ 16.0000i 1.23812i 0.785345 + 0.619059i $$0.212486\pi$$
−0.785345 + 0.619059i $$0.787514\pi$$
$$168$$ 0 0
$$169$$ −3.00000 −0.230769
$$170$$ 0 0
$$171$$ −8.00000 −0.611775
$$172$$ − 2.00000i − 0.152499i
$$173$$ − 10.0000i − 0.760286i −0.924928 0.380143i $$-0.875875\pi$$
0.924928 0.380143i $$-0.124125\pi$$
$$174$$ −4.00000 −0.303239
$$175$$ 0 0
$$176$$ −2.00000 −0.150756
$$177$$ − 8.00000i − 0.601317i
$$178$$ − 12.0000i − 0.899438i
$$179$$ −16.0000 −1.19590 −0.597948 0.801535i $$-0.704017\pi$$
−0.597948 + 0.801535i $$0.704017\pi$$
$$180$$ 0 0
$$181$$ 10.0000 0.743294 0.371647 0.928374i $$-0.378793\pi$$
0.371647 + 0.928374i $$0.378793\pi$$
$$182$$ 0 0
$$183$$ 2.00000i 0.147844i
$$184$$ 1.00000 0.0737210
$$185$$ 0 0
$$186$$ 0 0
$$187$$ 12.0000i 0.877527i
$$188$$ − 12.0000i − 0.875190i
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 24.0000 1.73658 0.868290 0.496058i $$-0.165220\pi$$
0.868290 + 0.496058i $$0.165220\pi$$
$$192$$ − 1.00000i − 0.0721688i
$$193$$ 26.0000i 1.87152i 0.352636 + 0.935760i $$0.385285\pi$$
−0.352636 + 0.935760i $$0.614715\pi$$
$$194$$ 16.0000 1.14873
$$195$$ 0 0
$$196$$ −7.00000 −0.500000
$$197$$ 18.0000i 1.28245i 0.767354 + 0.641223i $$0.221573\pi$$
−0.767354 + 0.641223i $$0.778427\pi$$
$$198$$ − 2.00000i − 0.142134i
$$199$$ 16.0000 1.13421 0.567105 0.823646i $$-0.308063\pi$$
0.567105 + 0.823646i $$0.308063\pi$$
$$200$$ 0 0
$$201$$ −6.00000 −0.423207
$$202$$ − 12.0000i − 0.844317i
$$203$$ 0 0
$$204$$ −6.00000 −0.420084
$$205$$ 0 0
$$206$$ 0 0
$$207$$ 1.00000i 0.0695048i
$$208$$ 4.00000i 0.277350i
$$209$$ −16.0000 −1.10674
$$210$$ 0 0
$$211$$ 4.00000 0.275371 0.137686 0.990476i $$-0.456034\pi$$
0.137686 + 0.990476i $$0.456034\pi$$
$$212$$ 6.00000i 0.412082i
$$213$$ 10.0000i 0.685189i
$$214$$ 12.0000 0.820303
$$215$$ 0 0
$$216$$ 1.00000 0.0680414
$$217$$ 0 0
$$218$$ − 10.0000i − 0.677285i
$$219$$ 2.00000 0.135147
$$220$$ 0 0
$$221$$ 24.0000 1.61441
$$222$$ 2.00000i 0.134231i
$$223$$ − 2.00000i − 0.133930i −0.997755 0.0669650i $$-0.978668\pi$$
0.997755 0.0669650i $$-0.0213316\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 18.0000 1.19734
$$227$$ − 28.0000i − 1.85843i −0.369546 0.929213i $$-0.620487\pi$$
0.369546 0.929213i $$-0.379513\pi$$
$$228$$ − 8.00000i − 0.529813i
$$229$$ 10.0000 0.660819 0.330409 0.943838i $$-0.392813\pi$$
0.330409 + 0.943838i $$0.392813\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ − 4.00000i − 0.262613i
$$233$$ 6.00000i 0.393073i 0.980497 + 0.196537i $$0.0629694\pi$$
−0.980497 + 0.196537i $$0.937031\pi$$
$$234$$ −4.00000 −0.261488
$$235$$ 0 0
$$236$$ 8.00000 0.520756
$$237$$ 8.00000i 0.519656i
$$238$$ 0 0
$$239$$ 14.0000 0.905585 0.452792 0.891616i $$-0.350428\pi$$
0.452792 + 0.891616i $$0.350428\pi$$
$$240$$ 0 0
$$241$$ −6.00000 −0.386494 −0.193247 0.981150i $$-0.561902\pi$$
−0.193247 + 0.981150i $$0.561902\pi$$
$$242$$ 7.00000i 0.449977i
$$243$$ 1.00000i 0.0641500i
$$244$$ −2.00000 −0.128037
$$245$$ 0 0
$$246$$ −2.00000 −0.127515
$$247$$ 32.0000i 2.03611i
$$248$$ 0 0
$$249$$ −8.00000 −0.506979
$$250$$ 0 0
$$251$$ 2.00000 0.126239 0.0631194 0.998006i $$-0.479895\pi$$
0.0631194 + 0.998006i $$0.479895\pi$$
$$252$$ 0 0
$$253$$ 2.00000i 0.125739i
$$254$$ −2.00000 −0.125491
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ − 22.0000i − 1.37232i −0.727450 0.686161i $$-0.759294\pi$$
0.727450 0.686161i $$-0.240706\pi$$
$$258$$ 2.00000i 0.124515i
$$259$$ 0 0
$$260$$ 0 0
$$261$$ 4.00000 0.247594
$$262$$ 4.00000i 0.247121i
$$263$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$264$$ 2.00000 0.123091
$$265$$ 0 0
$$266$$ 0 0
$$267$$ 12.0000i 0.734388i
$$268$$ − 6.00000i − 0.366508i
$$269$$ −8.00000 −0.487769 −0.243884 0.969804i $$-0.578422\pi$$
−0.243884 + 0.969804i $$0.578422\pi$$
$$270$$ 0 0
$$271$$ 24.0000 1.45790 0.728948 0.684569i $$-0.240010\pi$$
0.728948 + 0.684569i $$0.240010\pi$$
$$272$$ − 6.00000i − 0.363803i
$$273$$ 0 0
$$274$$ −2.00000 −0.120824
$$275$$ 0 0
$$276$$ −1.00000 −0.0601929
$$277$$ 28.0000i 1.68236i 0.540758 + 0.841178i $$0.318138\pi$$
−0.540758 + 0.841178i $$0.681862\pi$$
$$278$$ 4.00000i 0.239904i
$$279$$ 0 0
$$280$$ 0 0
$$281$$ 12.0000 0.715860 0.357930 0.933748i $$-0.383483\pi$$
0.357930 + 0.933748i $$0.383483\pi$$
$$282$$ 12.0000i 0.714590i
$$283$$ 14.0000i 0.832214i 0.909316 + 0.416107i $$0.136606\pi$$
−0.909316 + 0.416107i $$0.863394\pi$$
$$284$$ −10.0000 −0.593391
$$285$$ 0 0
$$286$$ −8.00000 −0.473050
$$287$$ 0 0
$$288$$ 1.00000i 0.0589256i
$$289$$ −19.0000 −1.11765
$$290$$ 0 0
$$291$$ −16.0000 −0.937937
$$292$$ 2.00000i 0.117041i
$$293$$ − 14.0000i − 0.817889i −0.912559 0.408944i $$-0.865897\pi$$
0.912559 0.408944i $$-0.134103\pi$$
$$294$$ 7.00000 0.408248
$$295$$ 0 0
$$296$$ −2.00000 −0.116248
$$297$$ 2.00000i 0.116052i
$$298$$ 14.0000i 0.810998i
$$299$$ 4.00000 0.231326
$$300$$ 0 0
$$301$$ 0 0
$$302$$ 4.00000i 0.230174i
$$303$$ 12.0000i 0.689382i
$$304$$ 8.00000 0.458831
$$305$$ 0 0
$$306$$ 6.00000 0.342997
$$307$$ − 28.0000i − 1.59804i −0.601302 0.799022i $$-0.705351\pi$$
0.601302 0.799022i $$-0.294649\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 0 0
$$311$$ 26.0000 1.47432 0.737162 0.675716i $$-0.236165\pi$$
0.737162 + 0.675716i $$0.236165\pi$$
$$312$$ − 4.00000i − 0.226455i
$$313$$ 28.0000i 1.58265i 0.611393 + 0.791327i $$0.290609\pi$$
−0.611393 + 0.791327i $$0.709391\pi$$
$$314$$ 10.0000 0.564333
$$315$$ 0 0
$$316$$ −8.00000 −0.450035
$$317$$ − 18.0000i − 1.01098i −0.862832 0.505490i $$-0.831312\pi$$
0.862832 0.505490i $$-0.168688\pi$$
$$318$$ − 6.00000i − 0.336463i
$$319$$ 8.00000 0.447914
$$320$$ 0 0
$$321$$ −12.0000 −0.669775
$$322$$ 0 0
$$323$$ − 48.0000i − 2.67079i
$$324$$ −1.00000 −0.0555556
$$325$$ 0 0
$$326$$ −16.0000 −0.886158
$$327$$ 10.0000i 0.553001i
$$328$$ − 2.00000i − 0.110432i
$$329$$ 0 0
$$330$$ 0 0
$$331$$ 28.0000 1.53902 0.769510 0.638635i $$-0.220501\pi$$
0.769510 + 0.638635i $$0.220501\pi$$
$$332$$ − 8.00000i − 0.439057i
$$333$$ − 2.00000i − 0.109599i
$$334$$ 16.0000 0.875481
$$335$$ 0 0
$$336$$ 0 0
$$337$$ 28.0000i 1.52526i 0.646837 + 0.762629i $$0.276092\pi$$
−0.646837 + 0.762629i $$0.723908\pi$$
$$338$$ 3.00000i 0.163178i
$$339$$ −18.0000 −0.977626
$$340$$ 0 0
$$341$$ 0 0
$$342$$ 8.00000i 0.432590i
$$343$$ 0 0
$$344$$ −2.00000 −0.107833
$$345$$ 0 0
$$346$$ −10.0000 −0.537603
$$347$$ 12.0000i 0.644194i 0.946707 + 0.322097i $$0.104388\pi$$
−0.946707 + 0.322097i $$0.895612\pi$$
$$348$$ 4.00000i 0.214423i
$$349$$ −26.0000 −1.39175 −0.695874 0.718164i $$-0.744983\pi$$
−0.695874 + 0.718164i $$0.744983\pi$$
$$350$$ 0 0
$$351$$ 4.00000 0.213504
$$352$$ 2.00000i 0.106600i
$$353$$ 10.0000i 0.532246i 0.963939 + 0.266123i $$0.0857428\pi$$
−0.963939 + 0.266123i $$0.914257\pi$$
$$354$$ −8.00000 −0.425195
$$355$$ 0 0
$$356$$ −12.0000 −0.635999
$$357$$ 0 0
$$358$$ 16.0000i 0.845626i
$$359$$ 16.0000 0.844448 0.422224 0.906492i $$-0.361250\pi$$
0.422224 + 0.906492i $$0.361250\pi$$
$$360$$ 0 0
$$361$$ 45.0000 2.36842
$$362$$ − 10.0000i − 0.525588i
$$363$$ − 7.00000i − 0.367405i
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 2.00000 0.104542
$$367$$ − 16.0000i − 0.835193i −0.908633 0.417597i $$-0.862873\pi$$
0.908633 0.417597i $$-0.137127\pi$$
$$368$$ − 1.00000i − 0.0521286i
$$369$$ 2.00000 0.104116
$$370$$ 0 0
$$371$$ 0 0
$$372$$ 0 0
$$373$$ 6.00000i 0.310668i 0.987862 + 0.155334i $$0.0496454\pi$$
−0.987862 + 0.155334i $$0.950355\pi$$
$$374$$ 12.0000 0.620505
$$375$$ 0 0
$$376$$ −12.0000 −0.618853
$$377$$ − 16.0000i − 0.824042i
$$378$$ 0 0
$$379$$ −12.0000 −0.616399 −0.308199 0.951322i $$-0.599726\pi$$
−0.308199 + 0.951322i $$0.599726\pi$$
$$380$$ 0 0
$$381$$ 2.00000 0.102463
$$382$$ − 24.0000i − 1.22795i
$$383$$ − 32.0000i − 1.63512i −0.575841 0.817562i $$-0.695325\pi$$
0.575841 0.817562i $$-0.304675\pi$$
$$384$$ −1.00000 −0.0510310
$$385$$ 0 0
$$386$$ 26.0000 1.32337
$$387$$ − 2.00000i − 0.101666i
$$388$$ − 16.0000i − 0.812277i
$$389$$ −26.0000 −1.31825 −0.659126 0.752032i $$-0.729074\pi$$
−0.659126 + 0.752032i $$0.729074\pi$$
$$390$$ 0 0
$$391$$ −6.00000 −0.303433
$$392$$ 7.00000i 0.353553i
$$393$$ − 4.00000i − 0.201773i
$$394$$ 18.0000 0.906827
$$395$$ 0 0
$$396$$ −2.00000 −0.100504
$$397$$ − 28.0000i − 1.40528i −0.711546 0.702640i $$-0.752005\pi$$
0.711546 0.702640i $$-0.247995\pi$$
$$398$$ − 16.0000i − 0.802008i
$$399$$ 0 0
$$400$$ 0 0
$$401$$ −8.00000 −0.399501 −0.199750 0.979847i $$-0.564013\pi$$
−0.199750 + 0.979847i $$0.564013\pi$$
$$402$$ 6.00000i 0.299253i
$$403$$ 0 0
$$404$$ −12.0000 −0.597022
$$405$$ 0 0
$$406$$ 0 0
$$407$$ − 4.00000i − 0.198273i
$$408$$ 6.00000i 0.297044i
$$409$$ 10.0000 0.494468 0.247234 0.968956i $$-0.420478\pi$$
0.247234 + 0.968956i $$0.420478\pi$$
$$410$$ 0 0
$$411$$ 2.00000 0.0986527
$$412$$ 0 0
$$413$$ 0 0
$$414$$ 1.00000 0.0491473
$$415$$ 0 0
$$416$$ 4.00000 0.196116
$$417$$ − 4.00000i − 0.195881i
$$418$$ 16.0000i 0.782586i
$$419$$ −30.0000 −1.46560 −0.732798 0.680446i $$-0.761786\pi$$
−0.732798 + 0.680446i $$0.761786\pi$$
$$420$$ 0 0
$$421$$ −14.0000 −0.682318 −0.341159 0.940006i $$-0.610819\pi$$
−0.341159 + 0.940006i $$0.610819\pi$$
$$422$$ − 4.00000i − 0.194717i
$$423$$ − 12.0000i − 0.583460i
$$424$$ 6.00000 0.291386
$$425$$ 0 0
$$426$$ 10.0000 0.484502
$$427$$ 0 0
$$428$$ − 12.0000i − 0.580042i
$$429$$ 8.00000 0.386244
$$430$$ 0 0
$$431$$ −8.00000 −0.385346 −0.192673 0.981263i $$-0.561716\pi$$
−0.192673 + 0.981263i $$0.561716\pi$$
$$432$$ − 1.00000i − 0.0481125i
$$433$$ 16.0000i 0.768911i 0.923144 + 0.384455i $$0.125611\pi$$
−0.923144 + 0.384455i $$0.874389\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ −10.0000 −0.478913
$$437$$ − 8.00000i − 0.382692i
$$438$$ − 2.00000i − 0.0955637i
$$439$$ −28.0000 −1.33637 −0.668184 0.743996i $$-0.732928\pi$$
−0.668184 + 0.743996i $$0.732928\pi$$
$$440$$ 0 0
$$441$$ −7.00000 −0.333333
$$442$$ − 24.0000i − 1.14156i
$$443$$ − 4.00000i − 0.190046i −0.995475 0.0950229i $$-0.969708\pi$$
0.995475 0.0950229i $$-0.0302924\pi$$
$$444$$ 2.00000 0.0949158
$$445$$ 0 0
$$446$$ −2.00000 −0.0947027
$$447$$ − 14.0000i − 0.662177i
$$448$$ 0 0
$$449$$ −30.0000 −1.41579 −0.707894 0.706319i $$-0.750354\pi$$
−0.707894 + 0.706319i $$0.750354\pi$$
$$450$$ 0 0
$$451$$ 4.00000 0.188353
$$452$$ − 18.0000i − 0.846649i
$$453$$ − 4.00000i − 0.187936i
$$454$$ −28.0000 −1.31411
$$455$$ 0 0
$$456$$ −8.00000 −0.374634
$$457$$ 8.00000i 0.374224i 0.982339 + 0.187112i $$0.0599128\pi$$
−0.982339 + 0.187112i $$0.940087\pi$$
$$458$$ − 10.0000i − 0.467269i
$$459$$ −6.00000 −0.280056
$$460$$ 0 0
$$461$$ −12.0000 −0.558896 −0.279448 0.960161i $$-0.590151\pi$$
−0.279448 + 0.960161i $$0.590151\pi$$
$$462$$ 0 0
$$463$$ − 26.0000i − 1.20832i −0.796862 0.604161i $$-0.793508\pi$$
0.796862 0.604161i $$-0.206492\pi$$
$$464$$ −4.00000 −0.185695
$$465$$ 0 0
$$466$$ 6.00000 0.277945
$$467$$ 24.0000i 1.11059i 0.831654 + 0.555294i $$0.187394\pi$$
−0.831654 + 0.555294i $$0.812606\pi$$
$$468$$ 4.00000i 0.184900i
$$469$$ 0 0
$$470$$ 0 0
$$471$$ −10.0000 −0.460776
$$472$$ − 8.00000i − 0.368230i
$$473$$ − 4.00000i − 0.183920i
$$474$$ 8.00000 0.367452
$$475$$ 0 0
$$476$$ 0 0
$$477$$ 6.00000i 0.274721i
$$478$$ − 14.0000i − 0.640345i
$$479$$ 16.0000 0.731059 0.365529 0.930800i $$-0.380888\pi$$
0.365529 + 0.930800i $$0.380888\pi$$
$$480$$ 0 0
$$481$$ −8.00000 −0.364769
$$482$$ 6.00000i 0.273293i
$$483$$ 0 0
$$484$$ 7.00000 0.318182
$$485$$ 0 0
$$486$$ 1.00000 0.0453609
$$487$$ 26.0000i 1.17817i 0.808070 + 0.589086i $$0.200512\pi$$
−0.808070 + 0.589086i $$0.799488\pi$$
$$488$$ 2.00000i 0.0905357i
$$489$$ 16.0000 0.723545
$$490$$ 0 0
$$491$$ 20.0000 0.902587 0.451294 0.892375i $$-0.350963\pi$$
0.451294 + 0.892375i $$0.350963\pi$$
$$492$$ 2.00000i 0.0901670i
$$493$$ 24.0000i 1.08091i
$$494$$ 32.0000 1.43975
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 0 0
$$498$$ 8.00000i 0.358489i
$$499$$ 4.00000 0.179065 0.0895323 0.995984i $$-0.471463\pi$$
0.0895323 + 0.995984i $$0.471463\pi$$
$$500$$ 0 0
$$501$$ −16.0000 −0.714827
$$502$$ − 2.00000i − 0.0892644i
$$503$$ − 16.0000i − 0.713405i −0.934218 0.356702i $$-0.883901\pi$$
0.934218 0.356702i $$-0.116099\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 2.00000 0.0889108
$$507$$ − 3.00000i − 0.133235i
$$508$$ 2.00000i 0.0887357i
$$509$$ −20.0000 −0.886484 −0.443242 0.896402i $$-0.646172\pi$$
−0.443242 + 0.896402i $$0.646172\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ − 1.00000i − 0.0441942i
$$513$$ − 8.00000i − 0.353209i
$$514$$ −22.0000 −0.970378
$$515$$ 0 0
$$516$$ 2.00000 0.0880451
$$517$$ − 24.0000i − 1.05552i
$$518$$ 0 0
$$519$$ 10.0000 0.438951
$$520$$ 0 0
$$521$$ −20.0000 −0.876216 −0.438108 0.898922i $$-0.644351\pi$$
−0.438108 + 0.898922i $$0.644351\pi$$
$$522$$ − 4.00000i − 0.175075i
$$523$$ 30.0000i 1.31181i 0.754844 + 0.655904i $$0.227712\pi$$
−0.754844 + 0.655904i $$0.772288\pi$$
$$524$$ 4.00000 0.174741
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 0 0
$$528$$ − 2.00000i − 0.0870388i
$$529$$ −1.00000 −0.0434783
$$530$$ 0 0
$$531$$ 8.00000 0.347170
$$532$$ 0 0
$$533$$ − 8.00000i − 0.346518i
$$534$$ 12.0000 0.519291
$$535$$ 0 0
$$536$$ −6.00000 −0.259161
$$537$$ − 16.0000i − 0.690451i
$$538$$ 8.00000i 0.344904i
$$539$$ −14.0000 −0.603023
$$540$$ 0 0
$$541$$ −22.0000 −0.945854 −0.472927 0.881102i $$-0.656803\pi$$
−0.472927 + 0.881102i $$0.656803\pi$$
$$542$$ − 24.0000i − 1.03089i
$$543$$ 10.0000i 0.429141i
$$544$$ −6.00000 −0.257248
$$545$$ 0 0
$$546$$ 0 0
$$547$$ − 4.00000i − 0.171028i −0.996337 0.0855138i $$-0.972747\pi$$
0.996337 0.0855138i $$-0.0272532\pi$$
$$548$$ 2.00000i 0.0854358i
$$549$$ −2.00000 −0.0853579
$$550$$ 0 0
$$551$$ −32.0000 −1.36325
$$552$$ 1.00000i 0.0425628i
$$553$$ 0 0
$$554$$ 28.0000 1.18961
$$555$$ 0 0
$$556$$ 4.00000 0.169638
$$557$$ − 38.0000i − 1.61011i −0.593199 0.805056i $$-0.702135\pi$$
0.593199 0.805056i $$-0.297865\pi$$
$$558$$ 0 0
$$559$$ −8.00000 −0.338364
$$560$$ 0 0
$$561$$ −12.0000 −0.506640
$$562$$ − 12.0000i − 0.506189i
$$563$$ − 24.0000i − 1.01148i −0.862686 0.505740i $$-0.831220\pi$$
0.862686 0.505740i $$-0.168780\pi$$
$$564$$ 12.0000 0.505291
$$565$$ 0 0
$$566$$ 14.0000 0.588464
$$567$$ 0 0
$$568$$ 10.0000i 0.419591i
$$569$$ 24.0000 1.00613 0.503066 0.864248i $$-0.332205\pi$$
0.503066 + 0.864248i $$0.332205\pi$$
$$570$$ 0 0
$$571$$ 20.0000 0.836974 0.418487 0.908223i $$-0.362561\pi$$
0.418487 + 0.908223i $$0.362561\pi$$
$$572$$ 8.00000i 0.334497i
$$573$$ 24.0000i 1.00261i
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 1.00000 0.0416667
$$577$$ − 38.0000i − 1.58196i −0.611842 0.790980i $$-0.709571\pi$$
0.611842 0.790980i $$-0.290429\pi$$
$$578$$ 19.0000i 0.790296i
$$579$$ −26.0000 −1.08052
$$580$$ 0 0
$$581$$ 0 0
$$582$$ 16.0000i 0.663221i
$$583$$ 12.0000i 0.496989i
$$584$$ 2.00000 0.0827606
$$585$$ 0 0
$$586$$ −14.0000 −0.578335
$$587$$ − 4.00000i − 0.165098i −0.996587 0.0825488i $$-0.973694\pi$$
0.996587 0.0825488i $$-0.0263060\pi$$
$$588$$ − 7.00000i − 0.288675i
$$589$$ 0 0
$$590$$ 0 0
$$591$$ −18.0000 −0.740421
$$592$$ 2.00000i 0.0821995i
$$593$$ 6.00000i 0.246390i 0.992382 + 0.123195i $$0.0393141\pi$$
−0.992382 + 0.123195i $$0.960686\pi$$
$$594$$ 2.00000 0.0820610
$$595$$ 0 0
$$596$$ 14.0000 0.573462
$$597$$ 16.0000i 0.654836i
$$598$$ − 4.00000i − 0.163572i
$$599$$ 34.0000 1.38920 0.694601 0.719395i $$-0.255581\pi$$
0.694601 + 0.719395i $$0.255581\pi$$
$$600$$ 0 0
$$601$$ −26.0000 −1.06056 −0.530281 0.847822i $$-0.677914\pi$$
−0.530281 + 0.847822i $$0.677914\pi$$
$$602$$ 0 0
$$603$$ − 6.00000i − 0.244339i
$$604$$ 4.00000 0.162758
$$605$$ 0 0
$$606$$ 12.0000 0.487467
$$607$$ 10.0000i 0.405887i 0.979190 + 0.202944i $$0.0650509\pi$$
−0.979190 + 0.202944i $$0.934949\pi$$
$$608$$ − 8.00000i − 0.324443i
$$609$$ 0 0
$$610$$ 0 0
$$611$$ −48.0000 −1.94187
$$612$$ − 6.00000i − 0.242536i
$$613$$ − 30.0000i − 1.21169i −0.795583 0.605844i $$-0.792835\pi$$
0.795583 0.605844i $$-0.207165\pi$$
$$614$$ −28.0000 −1.12999
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 18.0000i 0.724653i 0.932051 + 0.362326i $$0.118017\pi$$
−0.932051 + 0.362326i $$0.881983\pi$$
$$618$$ 0 0
$$619$$ −32.0000 −1.28619 −0.643094 0.765787i $$-0.722350\pi$$
−0.643094 + 0.765787i $$0.722350\pi$$
$$620$$ 0 0
$$621$$ −1.00000 −0.0401286
$$622$$ − 26.0000i − 1.04251i
$$623$$ 0 0
$$624$$ −4.00000 −0.160128
$$625$$ 0 0
$$626$$ 28.0000 1.11911
$$627$$ − 16.0000i − 0.638978i
$$628$$ − 10.0000i − 0.399043i
$$629$$ 12.0000 0.478471
$$630$$ 0 0
$$631$$ −32.0000 −1.27390 −0.636950 0.770905i $$-0.719804\pi$$
−0.636950 + 0.770905i $$0.719804\pi$$
$$632$$ 8.00000i 0.318223i
$$633$$ 4.00000i 0.158986i
$$634$$ −18.0000 −0.714871
$$635$$ 0 0
$$636$$ −6.00000 −0.237915
$$637$$ 28.0000i 1.10940i
$$638$$ − 8.00000i − 0.316723i
$$639$$ −10.0000 −0.395594
$$640$$ 0 0
$$641$$ −4.00000 −0.157991 −0.0789953 0.996875i $$-0.525171\pi$$
−0.0789953 + 0.996875i $$0.525171\pi$$
$$642$$ 12.0000i 0.473602i
$$643$$ − 10.0000i − 0.394362i −0.980367 0.197181i $$-0.936821\pi$$
0.980367 0.197181i $$-0.0631786\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ −48.0000 −1.88853
$$647$$ 28.0000i 1.10079i 0.834903 + 0.550397i $$0.185524\pi$$
−0.834903 + 0.550397i $$0.814476\pi$$
$$648$$ 1.00000i 0.0392837i
$$649$$ 16.0000 0.628055
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 16.0000i 0.626608i
$$653$$ 2.00000i 0.0782660i 0.999234 + 0.0391330i $$0.0124596\pi$$
−0.999234 + 0.0391330i $$0.987540\pi$$
$$654$$ 10.0000 0.391031
$$655$$ 0 0
$$656$$ −2.00000 −0.0780869
$$657$$ 2.00000i 0.0780274i
$$658$$ 0 0
$$659$$ 26.0000 1.01282 0.506408 0.862294i $$-0.330973\pi$$
0.506408 + 0.862294i $$0.330973\pi$$
$$660$$ 0 0
$$661$$ −50.0000 −1.94477 −0.972387 0.233373i $$-0.925024\pi$$
−0.972387 + 0.233373i $$0.925024\pi$$
$$662$$ − 28.0000i − 1.08825i
$$663$$ 24.0000i 0.932083i
$$664$$ −8.00000 −0.310460
$$665$$ 0 0
$$666$$ −2.00000 −0.0774984
$$667$$ 4.00000i 0.154881i
$$668$$ − 16.0000i − 0.619059i
$$669$$ 2.00000 0.0773245
$$670$$ 0 0
$$671$$ −4.00000 −0.154418
$$672$$ 0 0
$$673$$ − 10.0000i − 0.385472i −0.981251 0.192736i $$-0.938264\pi$$
0.981251 0.192736i $$-0.0617360\pi$$
$$674$$ 28.0000 1.07852
$$675$$ 0 0
$$676$$ 3.00000 0.115385
$$677$$ 10.0000i 0.384331i 0.981363 + 0.192166i $$0.0615511\pi$$
−0.981363 + 0.192166i $$0.938449\pi$$
$$678$$ 18.0000i 0.691286i
$$679$$ 0 0
$$680$$ 0 0
$$681$$ 28.0000 1.07296
$$682$$ 0 0
$$683$$ − 12.0000i − 0.459167i −0.973289 0.229584i $$-0.926264\pi$$
0.973289 0.229584i $$-0.0737364\pi$$
$$684$$ 8.00000 0.305888
$$685$$ 0 0
$$686$$ 0 0
$$687$$ 10.0000i 0.381524i
$$688$$ 2.00000i 0.0762493i
$$689$$ 24.0000 0.914327
$$690$$ 0 0
$$691$$ 20.0000 0.760836 0.380418 0.924815i $$-0.375780\pi$$
0.380418 + 0.924815i $$0.375780\pi$$
$$692$$ 10.0000i 0.380143i
$$693$$ 0 0
$$694$$ 12.0000 0.455514
$$695$$ 0 0
$$696$$ 4.00000 0.151620
$$697$$ 12.0000i 0.454532i
$$698$$ 26.0000i 0.984115i
$$699$$ −6.00000 −0.226941
$$700$$ 0 0
$$701$$ −14.0000 −0.528773 −0.264386 0.964417i $$-0.585169\pi$$
−0.264386 + 0.964417i $$0.585169\pi$$
$$702$$ − 4.00000i − 0.150970i
$$703$$ 16.0000i 0.603451i
$$704$$ 2.00000 0.0753778
$$705$$ 0 0
$$706$$ 10.0000 0.376355
$$707$$ 0 0
$$708$$ 8.00000i 0.300658i
$$709$$ −38.0000 −1.42712 −0.713560 0.700594i $$-0.752918\pi$$
−0.713560 + 0.700594i $$0.752918\pi$$
$$710$$ 0 0
$$711$$ −8.00000 −0.300023
$$712$$ 12.0000i 0.449719i
$$713$$ 0 0
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 16.0000 0.597948
$$717$$ 14.0000i 0.522840i
$$718$$ − 16.0000i − 0.597115i
$$719$$ 2.00000 0.0745874 0.0372937 0.999304i $$-0.488126\pi$$
0.0372937 + 0.999304i $$0.488126\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ − 45.0000i − 1.67473i
$$723$$ − 6.00000i − 0.223142i
$$724$$ −10.0000 −0.371647
$$725$$ 0 0
$$726$$ −7.00000 −0.259794
$$727$$ − 28.0000i − 1.03846i −0.854634 0.519231i $$-0.826218\pi$$
0.854634 0.519231i $$-0.173782\pi$$
$$728$$ 0 0
$$729$$ −1.00000 −0.0370370
$$730$$ 0 0
$$731$$ 12.0000 0.443836
$$732$$ − 2.00000i − 0.0739221i
$$733$$ − 42.0000i − 1.55131i −0.631160 0.775653i $$-0.717421\pi$$
0.631160 0.775653i $$-0.282579\pi$$
$$734$$ −16.0000 −0.590571
$$735$$ 0 0
$$736$$ −1.00000 −0.0368605
$$737$$ − 12.0000i − 0.442026i
$$738$$ − 2.00000i − 0.0736210i
$$739$$ 36.0000 1.32428 0.662141 0.749380i $$-0.269648\pi$$
0.662141 + 0.749380i $$0.269648\pi$$
$$740$$ 0 0
$$741$$ −32.0000 −1.17555
$$742$$ 0 0
$$743$$ − 24.0000i − 0.880475i −0.897881 0.440237i $$-0.854894\pi$$
0.897881 0.440237i $$-0.145106\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 6.00000 0.219676
$$747$$ − 8.00000i − 0.292705i
$$748$$ − 12.0000i − 0.438763i
$$749$$ 0 0
$$750$$ 0 0
$$751$$ 32.0000 1.16770 0.583848 0.811863i $$-0.301546\pi$$
0.583848 + 0.811863i $$0.301546\pi$$
$$752$$ 12.0000i 0.437595i
$$753$$ 2.00000i 0.0728841i
$$754$$ −16.0000 −0.582686
$$755$$ 0 0
$$756$$ 0 0
$$757$$ 6.00000i 0.218074i 0.994038 + 0.109037i $$0.0347767\pi$$
−0.994038 + 0.109037i $$0.965223\pi$$
$$758$$ 12.0000i 0.435860i
$$759$$ −2.00000 −0.0725954
$$760$$ 0 0
$$761$$ −22.0000 −0.797499 −0.398750 0.917060i $$-0.630556\pi$$
−0.398750 + 0.917060i $$0.630556\pi$$
$$762$$ − 2.00000i − 0.0724524i
$$763$$ 0 0
$$764$$ −24.0000 −0.868290
$$765$$ 0 0
$$766$$ −32.0000 −1.15621
$$767$$ − 32.0000i − 1.15545i
$$768$$ 1.00000i 0.0360844i
$$769$$ 2.00000 0.0721218 0.0360609 0.999350i $$-0.488519\pi$$
0.0360609 + 0.999350i $$0.488519\pi$$
$$770$$ 0 0
$$771$$ 22.0000 0.792311
$$772$$ − 26.0000i − 0.935760i
$$773$$ − 6.00000i − 0.215805i −0.994161 0.107903i $$-0.965587\pi$$
0.994161 0.107903i $$-0.0344134\pi$$
$$774$$ −2.00000 −0.0718885
$$775$$ 0 0
$$776$$ −16.0000 −0.574367
$$777$$ 0 0
$$778$$ 26.0000i 0.932145i
$$779$$ −16.0000 −0.573259
$$780$$ 0 0
$$781$$ −20.0000 −0.715656
$$782$$ 6.00000i 0.214560i
$$783$$ 4.00000i 0.142948i
$$784$$ 7.00000 0.250000
$$785$$ 0 0
$$786$$ −4.00000 −0.142675
$$787$$ − 22.0000i − 0.784215i −0.919919 0.392108i $$-0.871746\pi$$
0.919919 0.392108i $$-0.128254\pi$$
$$788$$ − 18.0000i − 0.641223i
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 0 0
$$792$$ 2.00000i 0.0710669i
$$793$$ 8.00000i 0.284088i
$$794$$ −28.0000 −0.993683
$$795$$ 0 0
$$796$$ −16.0000 −0.567105
$$797$$ − 14.0000i − 0.495905i −0.968772 0.247953i $$-0.920242\pi$$
0.968772 0.247953i $$-0.0797578\pi$$
$$798$$ 0 0
$$799$$ 72.0000 2.54718
$$800$$ 0 0
$$801$$ −12.0000 −0.423999
$$802$$ 8.00000i 0.282490i
$$803$$ 4.00000i 0.141157i
$$804$$ 6.00000 0.211604
$$805$$ 0 0
$$806$$ 0 0
$$807$$ − 8.00000i − 0.281613i
$$808$$ 12.0000i 0.422159i
$$809$$ 46.0000 1.61727 0.808637 0.588308i $$-0.200206\pi$$
0.808637 + 0.588308i $$0.200206\pi$$
$$810$$ 0 0
$$811$$ 4.00000 0.140459 0.0702295 0.997531i $$-0.477627\pi$$
0.0702295 + 0.997531i $$0.477627\pi$$
$$812$$ 0 0
$$813$$ 24.0000i 0.841717i
$$814$$ −4.00000 −0.140200
$$815$$ 0 0
$$816$$ 6.00000 0.210042
$$817$$ 16.0000i 0.559769i
$$818$$ − 10.0000i − 0.349642i
$$819$$ 0 0
$$820$$ 0 0
$$821$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$822$$ − 2.00000i − 0.0697580i
$$823$$ − 22.0000i − 0.766872i −0.923567 0.383436i $$-0.874741\pi$$
0.923567 0.383436i $$-0.125259\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ − 12.0000i − 0.417281i −0.977992 0.208640i $$-0.933096\pi$$
0.977992 0.208640i $$-0.0669038\pi$$
$$828$$ − 1.00000i − 0.0347524i
$$829$$ 38.0000 1.31979 0.659897 0.751356i $$-0.270600\pi$$
0.659897 + 0.751356i $$0.270600\pi$$
$$830$$ 0 0
$$831$$ −28.0000 −0.971309
$$832$$ − 4.00000i − 0.138675i
$$833$$ − 42.0000i − 1.45521i
$$834$$ −4.00000 −0.138509
$$835$$ 0 0
$$836$$ 16.0000 0.553372
$$837$$ 0 0
$$838$$ 30.0000i 1.03633i
$$839$$ 12.0000 0.414286 0.207143 0.978311i $$-0.433583\pi$$
0.207143 + 0.978311i $$0.433583\pi$$
$$840$$ 0 0
$$841$$ −13.0000 −0.448276
$$842$$ 14.0000i 0.482472i
$$843$$ 12.0000i 0.413302i
$$844$$ −4.00000 −0.137686
$$845$$ 0 0
$$846$$ −12.0000 −0.412568
$$847$$ 0 0
$$848$$ − 6.00000i − 0.206041i
$$849$$ −14.0000 −0.480479
$$850$$ 0 0
$$851$$ 2.00000 0.0685591
$$852$$ − 10.0000i − 0.342594i
$$853$$ 36.0000i 1.23262i 0.787505 + 0.616308i $$0.211372\pi$$
−0.787505 + 0.616308i $$0.788628\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ −12.0000 −0.410152
$$857$$ − 18.0000i − 0.614868i −0.951569 0.307434i $$-0.900530\pi$$
0.951569 0.307434i $$-0.0994704\pi$$
$$858$$ − 8.00000i − 0.273115i
$$859$$ 28.0000 0.955348 0.477674 0.878537i $$-0.341480\pi$$
0.477674 + 0.878537i $$0.341480\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 8.00000i 0.272481i
$$863$$ − 20.0000i − 0.680808i −0.940279 0.340404i $$-0.889436\pi$$
0.940279 0.340404i $$-0.110564\pi$$
$$864$$ −1.00000 −0.0340207
$$865$$ 0 0
$$866$$ 16.0000 0.543702
$$867$$ − 19.0000i − 0.645274i
$$868$$ 0 0
$$869$$ −16.0000 −0.542763
$$870$$ 0 0
$$871$$ −24.0000 −0.813209
$$872$$ 10.0000i 0.338643i
$$873$$ − 16.0000i − 0.541518i
$$874$$ −8.00000 −0.270604
$$875$$ 0 0
$$876$$ −2.00000 −0.0675737
$$877$$ 20.0000i 0.675352i 0.941262 + 0.337676i $$0.109641\pi$$
−0.941262 + 0.337676i $$0.890359\pi$$
$$878$$ 28.0000i 0.944954i
$$879$$ 14.0000 0.472208
$$880$$ 0 0
$$881$$ 4.00000 0.134763 0.0673817 0.997727i $$-0.478535\pi$$
0.0673817 + 0.997727i $$0.478535\pi$$
$$882$$ 7.00000i 0.235702i
$$883$$ 32.0000i 1.07689i 0.842662 + 0.538443i $$0.180987\pi$$
−0.842662 + 0.538443i $$0.819013\pi$$
$$884$$ −24.0000 −0.807207
$$885$$ 0 0
$$886$$ −4.00000 −0.134383
$$887$$ − 36.0000i − 1.20876i −0.796696 0.604381i $$-0.793421\pi$$
0.796696 0.604381i $$-0.206579\pi$$
$$888$$ − 2.00000i − 0.0671156i
$$889$$ 0 0
$$890$$ 0 0
$$891$$ −2.00000 −0.0670025
$$892$$ 2.00000i 0.0669650i
$$893$$ 96.0000i 3.21252i
$$894$$ −14.0000 −0.468230
$$895$$ 0 0
$$896$$ 0 0
$$897$$ 4.00000i 0.133556i
$$898$$ 30.0000i 1.00111i
$$899$$ 0 0
$$900$$ 0 0
$$901$$ −36.0000 −1.19933
$$902$$ − 4.00000i − 0.133185i
$$903$$ 0 0
$$904$$ −18.0000 −0.598671
$$905$$ 0 0
$$906$$ −4.00000 −0.132891
$$907$$ 22.0000i 0.730498i 0.930910 + 0.365249i $$0.119016\pi$$
−0.930910 + 0.365249i $$0.880984\pi$$
$$908$$ 28.0000i 0.929213i
$$909$$ −12.0000 −0.398015
$$910$$ 0 0
$$911$$ −4.00000 −0.132526 −0.0662630 0.997802i $$-0.521108\pi$$
−0.0662630 + 0.997802i $$0.521108\pi$$
$$912$$ 8.00000i 0.264906i
$$913$$ − 16.0000i − 0.529523i
$$914$$ 8.00000 0.264616
$$915$$ 0 0
$$916$$ −10.0000 −0.330409
$$917$$ 0 0
$$918$$ 6.00000i 0.198030i
$$919$$ −40.0000 −1.31948 −0.659739 0.751495i $$-0.729333\pi$$
−0.659739 + 0.751495i $$0.729333\pi$$
$$920$$ 0 0
$$921$$ 28.0000 0.922631
$$922$$ 12.0000i 0.395199i
$$923$$ 40.0000i 1.31662i
$$924$$ 0 0
$$925$$ 0 0
$$926$$ −26.0000 −0.854413
$$927$$ 0 0
$$928$$ 4.00000i 0.131306i
$$929$$ −30.0000 −0.984268 −0.492134 0.870519i $$-0.663783\pi$$
−0.492134 + 0.870519i $$0.663783\pi$$
$$930$$ 0 0
$$931$$ 56.0000 1.83533
$$932$$ − 6.00000i − 0.196537i
$$933$$ 26.0000i 0.851202i
$$934$$ 24.0000 0.785304
$$935$$ 0 0
$$936$$ 4.00000 0.130744
$$937$$ − 32.0000i − 1.04539i −0.852518 0.522697i $$-0.824926\pi$$
0.852518 0.522697i $$-0.175074\pi$$
$$938$$ 0 0
$$939$$ −28.0000 −0.913745
$$940$$ 0 0
$$941$$ 38.0000 1.23876 0.619382 0.785090i $$-0.287383\pi$$
0.619382 + 0.785090i $$0.287383\pi$$
$$942$$ 10.0000i 0.325818i
$$943$$ 2.00000i 0.0651290i
$$944$$ −8.00000 −0.260378
$$945$$ 0 0
$$946$$ −4.00000 −0.130051
$$947$$ − 4.00000i − 0.129983i −0.997886 0.0649913i $$-0.979298\pi$$
0.997886 0.0649913i $$-0.0207020\pi$$
$$948$$ − 8.00000i − 0.259828i
$$949$$ 8.00000 0.259691
$$950$$ 0 0
$$951$$ 18.0000 0.583690
$$952$$ 0 0
$$953$$ 26.0000i 0.842223i 0.907009 + 0.421111i $$0.138360\pi$$
−0.907009 + 0.421111i $$0.861640\pi$$
$$954$$ 6.00000 0.194257
$$955$$ 0 0
$$956$$ −14.0000 −0.452792
$$957$$ 8.00000i 0.258603i
$$958$$ − 16.0000i − 0.516937i
$$959$$ 0 0
$$960$$ 0 0
$$961$$ −31.0000 −1.00000
$$962$$ 8.00000i 0.257930i
$$963$$ − 12.0000i − 0.386695i
$$964$$ 6.00000 0.193247
$$965$$ 0 0
$$966$$ 0 0
$$967$$ 42.0000i 1.35063i 0.737530 + 0.675314i $$0.235992\pi$$
−0.737530 + 0.675314i $$0.764008\pi$$
$$968$$ − 7.00000i − 0.224989i
$$969$$ 48.0000 1.54198
$$970$$ 0 0
$$971$$ −14.0000 −0.449281 −0.224641 0.974442i $$-0.572121\pi$$
−0.224641 + 0.974442i $$0.572121\pi$$
$$972$$ − 1.00000i − 0.0320750i
$$973$$ 0 0
$$974$$ 26.0000 0.833094
$$975$$ 0 0
$$976$$ 2.00000 0.0640184
$$977$$ 42.0000i 1.34370i 0.740688 + 0.671850i $$0.234500\pi$$
−0.740688 + 0.671850i $$0.765500\pi$$
$$978$$ − 16.0000i − 0.511624i
$$979$$ −24.0000 −0.767043
$$980$$ 0 0
$$981$$ −10.0000 −0.319275
$$982$$ − 20.0000i − 0.638226i
$$983$$ 48.0000i 1.53096i 0.643458 + 0.765481i $$0.277499\pi$$
−0.643458 + 0.765481i $$0.722501\pi$$
$$984$$ 2.00000 0.0637577
$$985$$ 0 0
$$986$$ 24.0000 0.764316
$$987$$ 0 0
$$988$$ − 32.0000i − 1.01806i
$$989$$ 2.00000 0.0635963
$$990$$ 0 0
$$991$$ 12.0000 0.381193 0.190596 0.981669i $$-0.438958\pi$$
0.190596 + 0.981669i $$0.438958\pi$$
$$992$$ 0 0
$$993$$ 28.0000i 0.888553i
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 8.00000 0.253490
$$997$$ 4.00000i 0.126681i 0.997992 + 0.0633406i $$0.0201755\pi$$
−0.997992 + 0.0633406i $$0.979825\pi$$
$$998$$ − 4.00000i − 0.126618i
$$999$$ 2.00000 0.0632772
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.m.2899.1 2
5.2 odd 4 690.2.a.j.1.1 1
5.3 odd 4 3450.2.a.b.1.1 1
5.4 even 2 inner 3450.2.d.m.2899.2 2
15.2 even 4 2070.2.a.c.1.1 1
20.7 even 4 5520.2.a.l.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
690.2.a.j.1.1 1 5.2 odd 4
2070.2.a.c.1.1 1 15.2 even 4
3450.2.a.b.1.1 1 5.3 odd 4
3450.2.d.m.2899.1 2 1.1 even 1 trivial
3450.2.d.m.2899.2 2 5.4 even 2 inner
5520.2.a.l.1.1 1 20.7 even 4