# Properties

 Label 3675.2.a.e.1.1 Level $3675$ Weight $2$ Character 3675.1 Self dual yes Analytic conductor $29.345$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

# Learn more

## Newspace parameters

 Level: $$N$$ $$=$$ $$3675 = 3 \cdot 5^{2} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3675.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$29.3450227428$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 525) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 3675.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-1.00000 q^{2} -1.00000 q^{3} -1.00000 q^{4} +1.00000 q^{6} +3.00000 q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q-1.00000 q^{2} -1.00000 q^{3} -1.00000 q^{4} +1.00000 q^{6} +3.00000 q^{8} +1.00000 q^{9} +1.00000 q^{12} +3.00000 q^{13} -1.00000 q^{16} -2.00000 q^{17} -1.00000 q^{18} +1.00000 q^{19} -2.00000 q^{23} -3.00000 q^{24} -3.00000 q^{26} -1.00000 q^{27} -8.00000 q^{29} -8.00000 q^{31} -5.00000 q^{32} +2.00000 q^{34} -1.00000 q^{36} -7.00000 q^{37} -1.00000 q^{38} -3.00000 q^{39} +8.00000 q^{43} +2.00000 q^{46} +10.0000 q^{47} +1.00000 q^{48} +2.00000 q^{51} -3.00000 q^{52} +14.0000 q^{53} +1.00000 q^{54} -1.00000 q^{57} +8.00000 q^{58} +10.0000 q^{59} +7.00000 q^{61} +8.00000 q^{62} +7.00000 q^{64} +5.00000 q^{67} +2.00000 q^{68} +2.00000 q^{69} -12.0000 q^{71} +3.00000 q^{72} +11.0000 q^{73} +7.00000 q^{74} -1.00000 q^{76} +3.00000 q^{78} -7.00000 q^{79} +1.00000 q^{81} -14.0000 q^{83} -8.00000 q^{86} +8.00000 q^{87} -6.00000 q^{89} +2.00000 q^{92} +8.00000 q^{93} -10.0000 q^{94} +5.00000 q^{96} -9.00000 q^{97} +O(q^{100})$$

## 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 −0.353553 0.935414i $$-0.615027\pi$$
−0.353553 + 0.935414i $$0.615027\pi$$
$$3$$ −1.00000 −0.577350
$$4$$ −1.00000 −0.500000
$$5$$ 0 0
$$6$$ 1.00000 0.408248
$$7$$ 0 0
$$8$$ 3.00000 1.06066
$$9$$ 1.00000 0.333333
$$10$$ 0 0
$$11$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$12$$ 1.00000 0.288675
$$13$$ 3.00000 0.832050 0.416025 0.909353i $$-0.363423\pi$$
0.416025 + 0.909353i $$0.363423\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ −1.00000 −0.250000
$$17$$ −2.00000 −0.485071 −0.242536 0.970143i $$-0.577979\pi$$
−0.242536 + 0.970143i $$0.577979\pi$$
$$18$$ −1.00000 −0.235702
$$19$$ 1.00000 0.229416 0.114708 0.993399i $$-0.463407\pi$$
0.114708 + 0.993399i $$0.463407\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ −2.00000 −0.417029 −0.208514 0.978019i $$-0.566863\pi$$
−0.208514 + 0.978019i $$0.566863\pi$$
$$24$$ −3.00000 −0.612372
$$25$$ 0 0
$$26$$ −3.00000 −0.588348
$$27$$ −1.00000 −0.192450
$$28$$ 0 0
$$29$$ −8.00000 −1.48556 −0.742781 0.669534i $$-0.766494\pi$$
−0.742781 + 0.669534i $$0.766494\pi$$
$$30$$ 0 0
$$31$$ −8.00000 −1.43684 −0.718421 0.695608i $$-0.755135\pi$$
−0.718421 + 0.695608i $$0.755135\pi$$
$$32$$ −5.00000 −0.883883
$$33$$ 0 0
$$34$$ 2.00000 0.342997
$$35$$ 0 0
$$36$$ −1.00000 −0.166667
$$37$$ −7.00000 −1.15079 −0.575396 0.817875i $$-0.695152\pi$$
−0.575396 + 0.817875i $$0.695152\pi$$
$$38$$ −1.00000 −0.162221
$$39$$ −3.00000 −0.480384
$$40$$ 0 0
$$41$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$42$$ 0 0
$$43$$ 8.00000 1.21999 0.609994 0.792406i $$-0.291172\pi$$
0.609994 + 0.792406i $$0.291172\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 2.00000 0.294884
$$47$$ 10.0000 1.45865 0.729325 0.684167i $$-0.239834\pi$$
0.729325 + 0.684167i $$0.239834\pi$$
$$48$$ 1.00000 0.144338
$$49$$ 0 0
$$50$$ 0 0
$$51$$ 2.00000 0.280056
$$52$$ −3.00000 −0.416025
$$53$$ 14.0000 1.92305 0.961524 0.274721i $$-0.0885855\pi$$
0.961524 + 0.274721i $$0.0885855\pi$$
$$54$$ 1.00000 0.136083
$$55$$ 0 0
$$56$$ 0 0
$$57$$ −1.00000 −0.132453
$$58$$ 8.00000 1.05045
$$59$$ 10.0000 1.30189 0.650945 0.759125i $$-0.274373\pi$$
0.650945 + 0.759125i $$0.274373\pi$$
$$60$$ 0 0
$$61$$ 7.00000 0.896258 0.448129 0.893969i $$-0.352090\pi$$
0.448129 + 0.893969i $$0.352090\pi$$
$$62$$ 8.00000 1.01600
$$63$$ 0 0
$$64$$ 7.00000 0.875000
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 5.00000 0.610847 0.305424 0.952217i $$-0.401202\pi$$
0.305424 + 0.952217i $$0.401202\pi$$
$$68$$ 2.00000 0.242536
$$69$$ 2.00000 0.240772
$$70$$ 0 0
$$71$$ −12.0000 −1.42414 −0.712069 0.702109i $$-0.752242\pi$$
−0.712069 + 0.702109i $$0.752242\pi$$
$$72$$ 3.00000 0.353553
$$73$$ 11.0000 1.28745 0.643726 0.765256i $$-0.277388\pi$$
0.643726 + 0.765256i $$0.277388\pi$$
$$74$$ 7.00000 0.813733
$$75$$ 0 0
$$76$$ −1.00000 −0.114708
$$77$$ 0 0
$$78$$ 3.00000 0.339683
$$79$$ −7.00000 −0.787562 −0.393781 0.919204i $$-0.628833\pi$$
−0.393781 + 0.919204i $$0.628833\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 0 0
$$83$$ −14.0000 −1.53670 −0.768350 0.640030i $$-0.778922\pi$$
−0.768350 + 0.640030i $$0.778922\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ −8.00000 −0.862662
$$87$$ 8.00000 0.857690
$$88$$ 0 0
$$89$$ −6.00000 −0.635999 −0.317999 0.948091i $$-0.603011\pi$$
−0.317999 + 0.948091i $$0.603011\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 2.00000 0.208514
$$93$$ 8.00000 0.829561
$$94$$ −10.0000 −1.03142
$$95$$ 0 0
$$96$$ 5.00000 0.510310
$$97$$ −9.00000 −0.913812 −0.456906 0.889515i $$-0.651042\pi$$
−0.456906 + 0.889515i $$0.651042\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 0 0
$$101$$ −16.0000 −1.59206 −0.796030 0.605257i $$-0.793070\pi$$
−0.796030 + 0.605257i $$0.793070\pi$$
$$102$$ −2.00000 −0.198030
$$103$$ 13.0000 1.28093 0.640464 0.767988i $$-0.278742\pi$$
0.640464 + 0.767988i $$0.278742\pi$$
$$104$$ 9.00000 0.882523
$$105$$ 0 0
$$106$$ −14.0000 −1.35980
$$107$$ −6.00000 −0.580042 −0.290021 0.957020i $$-0.593662\pi$$
−0.290021 + 0.957020i $$0.593662\pi$$
$$108$$ 1.00000 0.0962250
$$109$$ −15.0000 −1.43674 −0.718370 0.695662i $$-0.755111\pi$$
−0.718370 + 0.695662i $$0.755111\pi$$
$$110$$ 0 0
$$111$$ 7.00000 0.664411
$$112$$ 0 0
$$113$$ 6.00000 0.564433 0.282216 0.959351i $$-0.408930\pi$$
0.282216 + 0.959351i $$0.408930\pi$$
$$114$$ 1.00000 0.0936586
$$115$$ 0 0
$$116$$ 8.00000 0.742781
$$117$$ 3.00000 0.277350
$$118$$ −10.0000 −0.920575
$$119$$ 0 0
$$120$$ 0 0
$$121$$ −11.0000 −1.00000
$$122$$ −7.00000 −0.633750
$$123$$ 0 0
$$124$$ 8.00000 0.718421
$$125$$ 0 0
$$126$$ 0 0
$$127$$ −11.0000 −0.976092 −0.488046 0.872818i $$-0.662290\pi$$
−0.488046 + 0.872818i $$0.662290\pi$$
$$128$$ 3.00000 0.265165
$$129$$ −8.00000 −0.704361
$$130$$ 0 0
$$131$$ 14.0000 1.22319 0.611593 0.791173i $$-0.290529\pi$$
0.611593 + 0.791173i $$0.290529\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ −5.00000 −0.431934
$$135$$ 0 0
$$136$$ −6.00000 −0.514496
$$137$$ 10.0000 0.854358 0.427179 0.904167i $$-0.359507\pi$$
0.427179 + 0.904167i $$0.359507\pi$$
$$138$$ −2.00000 −0.170251
$$139$$ 3.00000 0.254457 0.127228 0.991873i $$-0.459392\pi$$
0.127228 + 0.991873i $$0.459392\pi$$
$$140$$ 0 0
$$141$$ −10.0000 −0.842152
$$142$$ 12.0000 1.00702
$$143$$ 0 0
$$144$$ −1.00000 −0.0833333
$$145$$ 0 0
$$146$$ −11.0000 −0.910366
$$147$$ 0 0
$$148$$ 7.00000 0.575396
$$149$$ −4.00000 −0.327693 −0.163846 0.986486i $$-0.552390\pi$$
−0.163846 + 0.986486i $$0.552390\pi$$
$$150$$ 0 0
$$151$$ −1.00000 −0.0813788 −0.0406894 0.999172i $$-0.512955\pi$$
−0.0406894 + 0.999172i $$0.512955\pi$$
$$152$$ 3.00000 0.243332
$$153$$ −2.00000 −0.161690
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 3.00000 0.240192
$$157$$ −7.00000 −0.558661 −0.279330 0.960195i $$-0.590112\pi$$
−0.279330 + 0.960195i $$0.590112\pi$$
$$158$$ 7.00000 0.556890
$$159$$ −14.0000 −1.11027
$$160$$ 0 0
$$161$$ 0 0
$$162$$ −1.00000 −0.0785674
$$163$$ 9.00000 0.704934 0.352467 0.935824i $$-0.385343\pi$$
0.352467 + 0.935824i $$0.385343\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 14.0000 1.08661
$$167$$ −2.00000 −0.154765 −0.0773823 0.997001i $$-0.524656\pi$$
−0.0773823 + 0.997001i $$0.524656\pi$$
$$168$$ 0 0
$$169$$ −4.00000 −0.307692
$$170$$ 0 0
$$171$$ 1.00000 0.0764719
$$172$$ −8.00000 −0.609994
$$173$$ −18.0000 −1.36851 −0.684257 0.729241i $$-0.739873\pi$$
−0.684257 + 0.729241i $$0.739873\pi$$
$$174$$ −8.00000 −0.606478
$$175$$ 0 0
$$176$$ 0 0
$$177$$ −10.0000 −0.751646
$$178$$ 6.00000 0.449719
$$179$$ −12.0000 −0.896922 −0.448461 0.893802i $$-0.648028\pi$$
−0.448461 + 0.893802i $$0.648028\pi$$
$$180$$ 0 0
$$181$$ 22.0000 1.63525 0.817624 0.575753i $$-0.195291\pi$$
0.817624 + 0.575753i $$0.195291\pi$$
$$182$$ 0 0
$$183$$ −7.00000 −0.517455
$$184$$ −6.00000 −0.442326
$$185$$ 0 0
$$186$$ −8.00000 −0.586588
$$187$$ 0 0
$$188$$ −10.0000 −0.729325
$$189$$ 0 0
$$190$$ 0 0
$$191$$ −8.00000 −0.578860 −0.289430 0.957199i $$-0.593466\pi$$
−0.289430 + 0.957199i $$0.593466\pi$$
$$192$$ −7.00000 −0.505181
$$193$$ −6.00000 −0.431889 −0.215945 0.976406i $$-0.569283\pi$$
−0.215945 + 0.976406i $$0.569283\pi$$
$$194$$ 9.00000 0.646162
$$195$$ 0 0
$$196$$ 0 0
$$197$$ −12.0000 −0.854965 −0.427482 0.904024i $$-0.640599\pi$$
−0.427482 + 0.904024i $$0.640599\pi$$
$$198$$ 0 0
$$199$$ 11.0000 0.779769 0.389885 0.920864i $$-0.372515\pi$$
0.389885 + 0.920864i $$0.372515\pi$$
$$200$$ 0 0
$$201$$ −5.00000 −0.352673
$$202$$ 16.0000 1.12576
$$203$$ 0 0
$$204$$ −2.00000 −0.140028
$$205$$ 0 0
$$206$$ −13.0000 −0.905753
$$207$$ −2.00000 −0.139010
$$208$$ −3.00000 −0.208013
$$209$$ 0 0
$$210$$ 0 0
$$211$$ −5.00000 −0.344214 −0.172107 0.985078i $$-0.555058\pi$$
−0.172107 + 0.985078i $$0.555058\pi$$
$$212$$ −14.0000 −0.961524
$$213$$ 12.0000 0.822226
$$214$$ 6.00000 0.410152
$$215$$ 0 0
$$216$$ −3.00000 −0.204124
$$217$$ 0 0
$$218$$ 15.0000 1.01593
$$219$$ −11.0000 −0.743311
$$220$$ 0 0
$$221$$ −6.00000 −0.403604
$$222$$ −7.00000 −0.469809
$$223$$ 15.0000 1.00447 0.502237 0.864730i $$-0.332510\pi$$
0.502237 + 0.864730i $$0.332510\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ −6.00000 −0.399114
$$227$$ −20.0000 −1.32745 −0.663723 0.747978i $$-0.731025\pi$$
−0.663723 + 0.747978i $$0.731025\pi$$
$$228$$ 1.00000 0.0662266
$$229$$ 13.0000 0.859064 0.429532 0.903052i $$-0.358679\pi$$
0.429532 + 0.903052i $$0.358679\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ −24.0000 −1.57568
$$233$$ −24.0000 −1.57229 −0.786146 0.618041i $$-0.787927\pi$$
−0.786146 + 0.618041i $$0.787927\pi$$
$$234$$ −3.00000 −0.196116
$$235$$ 0 0
$$236$$ −10.0000 −0.650945
$$237$$ 7.00000 0.454699
$$238$$ 0 0
$$239$$ −28.0000 −1.81117 −0.905585 0.424165i $$-0.860568\pi$$
−0.905585 + 0.424165i $$0.860568\pi$$
$$240$$ 0 0
$$241$$ 21.0000 1.35273 0.676364 0.736567i $$-0.263554\pi$$
0.676364 + 0.736567i $$0.263554\pi$$
$$242$$ 11.0000 0.707107
$$243$$ −1.00000 −0.0641500
$$244$$ −7.00000 −0.448129
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 3.00000 0.190885
$$248$$ −24.0000 −1.52400
$$249$$ 14.0000 0.887214
$$250$$ 0 0
$$251$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$252$$ 0 0
$$253$$ 0 0
$$254$$ 11.0000 0.690201
$$255$$ 0 0
$$256$$ −17.0000 −1.06250
$$257$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$258$$ 8.00000 0.498058
$$259$$ 0 0
$$260$$ 0 0
$$261$$ −8.00000 −0.495188
$$262$$ −14.0000 −0.864923
$$263$$ −16.0000 −0.986602 −0.493301 0.869859i $$-0.664210\pi$$
−0.493301 + 0.869859i $$0.664210\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 0 0
$$267$$ 6.00000 0.367194
$$268$$ −5.00000 −0.305424
$$269$$ −16.0000 −0.975537 −0.487769 0.872973i $$-0.662189\pi$$
−0.487769 + 0.872973i $$0.662189\pi$$
$$270$$ 0 0
$$271$$ −12.0000 −0.728948 −0.364474 0.931214i $$-0.618751\pi$$
−0.364474 + 0.931214i $$0.618751\pi$$
$$272$$ 2.00000 0.121268
$$273$$ 0 0
$$274$$ −10.0000 −0.604122
$$275$$ 0 0
$$276$$ −2.00000 −0.120386
$$277$$ −5.00000 −0.300421 −0.150210 0.988654i $$-0.547995\pi$$
−0.150210 + 0.988654i $$0.547995\pi$$
$$278$$ −3.00000 −0.179928
$$279$$ −8.00000 −0.478947
$$280$$ 0 0
$$281$$ 6.00000 0.357930 0.178965 0.983855i $$-0.442725\pi$$
0.178965 + 0.983855i $$0.442725\pi$$
$$282$$ 10.0000 0.595491
$$283$$ 19.0000 1.12943 0.564716 0.825285i $$-0.308986\pi$$
0.564716 + 0.825285i $$0.308986\pi$$
$$284$$ 12.0000 0.712069
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 0 0
$$288$$ −5.00000 −0.294628
$$289$$ −13.0000 −0.764706
$$290$$ 0 0
$$291$$ 9.00000 0.527589
$$292$$ −11.0000 −0.643726
$$293$$ 16.0000 0.934730 0.467365 0.884064i $$-0.345203\pi$$
0.467365 + 0.884064i $$0.345203\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ −21.0000 −1.22060
$$297$$ 0 0
$$298$$ 4.00000 0.231714
$$299$$ −6.00000 −0.346989
$$300$$ 0 0
$$301$$ 0 0
$$302$$ 1.00000 0.0575435
$$303$$ 16.0000 0.919176
$$304$$ −1.00000 −0.0573539
$$305$$ 0 0
$$306$$ 2.00000 0.114332
$$307$$ 12.0000 0.684876 0.342438 0.939540i $$-0.388747\pi$$
0.342438 + 0.939540i $$0.388747\pi$$
$$308$$ 0 0
$$309$$ −13.0000 −0.739544
$$310$$ 0 0
$$311$$ −30.0000 −1.70114 −0.850572 0.525859i $$-0.823744\pi$$
−0.850572 + 0.525859i $$0.823744\pi$$
$$312$$ −9.00000 −0.509525
$$313$$ −14.0000 −0.791327 −0.395663 0.918396i $$-0.629485\pi$$
−0.395663 + 0.918396i $$0.629485\pi$$
$$314$$ 7.00000 0.395033
$$315$$ 0 0
$$316$$ 7.00000 0.393781
$$317$$ −14.0000 −0.786318 −0.393159 0.919470i $$-0.628618\pi$$
−0.393159 + 0.919470i $$0.628618\pi$$
$$318$$ 14.0000 0.785081
$$319$$ 0 0
$$320$$ 0 0
$$321$$ 6.00000 0.334887
$$322$$ 0 0
$$323$$ −2.00000 −0.111283
$$324$$ −1.00000 −0.0555556
$$325$$ 0 0
$$326$$ −9.00000 −0.498464
$$327$$ 15.0000 0.829502
$$328$$ 0 0
$$329$$ 0 0
$$330$$ 0 0
$$331$$ 21.0000 1.15426 0.577132 0.816651i $$-0.304172\pi$$
0.577132 + 0.816651i $$0.304172\pi$$
$$332$$ 14.0000 0.768350
$$333$$ −7.00000 −0.383598
$$334$$ 2.00000 0.109435
$$335$$ 0 0
$$336$$ 0 0
$$337$$ 2.00000 0.108947 0.0544735 0.998515i $$-0.482652\pi$$
0.0544735 + 0.998515i $$0.482652\pi$$
$$338$$ 4.00000 0.217571
$$339$$ −6.00000 −0.325875
$$340$$ 0 0
$$341$$ 0 0
$$342$$ −1.00000 −0.0540738
$$343$$ 0 0
$$344$$ 24.0000 1.29399
$$345$$ 0 0
$$346$$ 18.0000 0.967686
$$347$$ −10.0000 −0.536828 −0.268414 0.963304i $$-0.586500\pi$$
−0.268414 + 0.963304i $$0.586500\pi$$
$$348$$ −8.00000 −0.428845
$$349$$ −22.0000 −1.17763 −0.588817 0.808267i $$-0.700406\pi$$
−0.588817 + 0.808267i $$0.700406\pi$$
$$350$$ 0 0
$$351$$ −3.00000 −0.160128
$$352$$ 0 0
$$353$$ −24.0000 −1.27739 −0.638696 0.769460i $$-0.720526\pi$$
−0.638696 + 0.769460i $$0.720526\pi$$
$$354$$ 10.0000 0.531494
$$355$$ 0 0
$$356$$ 6.00000 0.317999
$$357$$ 0 0
$$358$$ 12.0000 0.634220
$$359$$ 30.0000 1.58334 0.791670 0.610949i $$-0.209212\pi$$
0.791670 + 0.610949i $$0.209212\pi$$
$$360$$ 0 0
$$361$$ −18.0000 −0.947368
$$362$$ −22.0000 −1.15629
$$363$$ 11.0000 0.577350
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 7.00000 0.365896
$$367$$ 8.00000 0.417597 0.208798 0.977959i $$-0.433045\pi$$
0.208798 + 0.977959i $$0.433045\pi$$
$$368$$ 2.00000 0.104257
$$369$$ 0 0
$$370$$ 0 0
$$371$$ 0 0
$$372$$ −8.00000 −0.414781
$$373$$ −11.0000 −0.569558 −0.284779 0.958593i $$-0.591920\pi$$
−0.284779 + 0.958593i $$0.591920\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 30.0000 1.54713
$$377$$ −24.0000 −1.23606
$$378$$ 0 0
$$379$$ 35.0000 1.79783 0.898915 0.438124i $$-0.144357\pi$$
0.898915 + 0.438124i $$0.144357\pi$$
$$380$$ 0 0
$$381$$ 11.0000 0.563547
$$382$$ 8.00000 0.409316
$$383$$ −10.0000 −0.510976 −0.255488 0.966812i $$-0.582236\pi$$
−0.255488 + 0.966812i $$0.582236\pi$$
$$384$$ −3.00000 −0.153093
$$385$$ 0 0
$$386$$ 6.00000 0.305392
$$387$$ 8.00000 0.406663
$$388$$ 9.00000 0.456906
$$389$$ −36.0000 −1.82527 −0.912636 0.408773i $$-0.865957\pi$$
−0.912636 + 0.408773i $$0.865957\pi$$
$$390$$ 0 0
$$391$$ 4.00000 0.202289
$$392$$ 0 0
$$393$$ −14.0000 −0.706207
$$394$$ 12.0000 0.604551
$$395$$ 0 0
$$396$$ 0 0
$$397$$ −14.0000 −0.702640 −0.351320 0.936255i $$-0.614267\pi$$
−0.351320 + 0.936255i $$0.614267\pi$$
$$398$$ −11.0000 −0.551380
$$399$$ 0 0
$$400$$ 0 0
$$401$$ −12.0000 −0.599251 −0.299626 0.954057i $$-0.596862\pi$$
−0.299626 + 0.954057i $$0.596862\pi$$
$$402$$ 5.00000 0.249377
$$403$$ −24.0000 −1.19553
$$404$$ 16.0000 0.796030
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 0 0
$$408$$ 6.00000 0.297044
$$409$$ 5.00000 0.247234 0.123617 0.992330i $$-0.460551\pi$$
0.123617 + 0.992330i $$0.460551\pi$$
$$410$$ 0 0
$$411$$ −10.0000 −0.493264
$$412$$ −13.0000 −0.640464
$$413$$ 0 0
$$414$$ 2.00000 0.0982946
$$415$$ 0 0
$$416$$ −15.0000 −0.735436
$$417$$ −3.00000 −0.146911
$$418$$ 0 0
$$419$$ 24.0000 1.17248 0.586238 0.810139i $$-0.300608\pi$$
0.586238 + 0.810139i $$0.300608\pi$$
$$420$$ 0 0
$$421$$ −35.0000 −1.70580 −0.852898 0.522078i $$-0.825157\pi$$
−0.852898 + 0.522078i $$0.825157\pi$$
$$422$$ 5.00000 0.243396
$$423$$ 10.0000 0.486217
$$424$$ 42.0000 2.03970
$$425$$ 0 0
$$426$$ −12.0000 −0.581402
$$427$$ 0 0
$$428$$ 6.00000 0.290021
$$429$$ 0 0
$$430$$ 0 0
$$431$$ −2.00000 −0.0963366 −0.0481683 0.998839i $$-0.515338\pi$$
−0.0481683 + 0.998839i $$0.515338\pi$$
$$432$$ 1.00000 0.0481125
$$433$$ 2.00000 0.0961139 0.0480569 0.998845i $$-0.484697\pi$$
0.0480569 + 0.998845i $$0.484697\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 15.0000 0.718370
$$437$$ −2.00000 −0.0956730
$$438$$ 11.0000 0.525600
$$439$$ −11.0000 −0.525001 −0.262501 0.964932i $$-0.584547\pi$$
−0.262501 + 0.964932i $$0.584547\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 6.00000 0.285391
$$443$$ −6.00000 −0.285069 −0.142534 0.989790i $$-0.545525\pi$$
−0.142534 + 0.989790i $$0.545525\pi$$
$$444$$ −7.00000 −0.332205
$$445$$ 0 0
$$446$$ −15.0000 −0.710271
$$447$$ 4.00000 0.189194
$$448$$ 0 0
$$449$$ 6.00000 0.283158 0.141579 0.989927i $$-0.454782\pi$$
0.141579 + 0.989927i $$0.454782\pi$$
$$450$$ 0 0
$$451$$ 0 0
$$452$$ −6.00000 −0.282216
$$453$$ 1.00000 0.0469841
$$454$$ 20.0000 0.938647
$$455$$ 0 0
$$456$$ −3.00000 −0.140488
$$457$$ 3.00000 0.140334 0.0701670 0.997535i $$-0.477647\pi$$
0.0701670 + 0.997535i $$0.477647\pi$$
$$458$$ −13.0000 −0.607450
$$459$$ 2.00000 0.0933520
$$460$$ 0 0
$$461$$ −2.00000 −0.0931493 −0.0465746 0.998915i $$-0.514831\pi$$
−0.0465746 + 0.998915i $$0.514831\pi$$
$$462$$ 0 0
$$463$$ −33.0000 −1.53364 −0.766820 0.641862i $$-0.778162\pi$$
−0.766820 + 0.641862i $$0.778162\pi$$
$$464$$ 8.00000 0.371391
$$465$$ 0 0
$$466$$ 24.0000 1.11178
$$467$$ 2.00000 0.0925490 0.0462745 0.998929i $$-0.485265\pi$$
0.0462745 + 0.998929i $$0.485265\pi$$
$$468$$ −3.00000 −0.138675
$$469$$ 0 0
$$470$$ 0 0
$$471$$ 7.00000 0.322543
$$472$$ 30.0000 1.38086
$$473$$ 0 0
$$474$$ −7.00000 −0.321521
$$475$$ 0 0
$$476$$ 0 0
$$477$$ 14.0000 0.641016
$$478$$ 28.0000 1.28069
$$479$$ 2.00000 0.0913823 0.0456912 0.998956i $$-0.485451\pi$$
0.0456912 + 0.998956i $$0.485451\pi$$
$$480$$ 0 0
$$481$$ −21.0000 −0.957518
$$482$$ −21.0000 −0.956524
$$483$$ 0 0
$$484$$ 11.0000 0.500000
$$485$$ 0 0
$$486$$ 1.00000 0.0453609
$$487$$ −8.00000 −0.362515 −0.181257 0.983436i $$-0.558017\pi$$
−0.181257 + 0.983436i $$0.558017\pi$$
$$488$$ 21.0000 0.950625
$$489$$ −9.00000 −0.406994
$$490$$ 0 0
$$491$$ −6.00000 −0.270776 −0.135388 0.990793i $$-0.543228\pi$$
−0.135388 + 0.990793i $$0.543228\pi$$
$$492$$ 0 0
$$493$$ 16.0000 0.720604
$$494$$ −3.00000 −0.134976
$$495$$ 0 0
$$496$$ 8.00000 0.359211
$$497$$ 0 0
$$498$$ −14.0000 −0.627355
$$499$$ −19.0000 −0.850557 −0.425278 0.905063i $$-0.639824\pi$$
−0.425278 + 0.905063i $$0.639824\pi$$
$$500$$ 0 0
$$501$$ 2.00000 0.0893534
$$502$$ 0 0
$$503$$ −40.0000 −1.78351 −0.891756 0.452517i $$-0.850526\pi$$
−0.891756 + 0.452517i $$0.850526\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 0 0
$$507$$ 4.00000 0.177646
$$508$$ 11.0000 0.488046
$$509$$ −8.00000 −0.354594 −0.177297 0.984157i $$-0.556735\pi$$
−0.177297 + 0.984157i $$0.556735\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ 11.0000 0.486136
$$513$$ −1.00000 −0.0441511
$$514$$ 0 0
$$515$$ 0 0
$$516$$ 8.00000 0.352180
$$517$$ 0 0
$$518$$ 0 0
$$519$$ 18.0000 0.790112
$$520$$ 0 0
$$521$$ 34.0000 1.48957 0.744784 0.667306i $$-0.232553\pi$$
0.744784 + 0.667306i $$0.232553\pi$$
$$522$$ 8.00000 0.350150
$$523$$ 16.0000 0.699631 0.349816 0.936819i $$-0.386244\pi$$
0.349816 + 0.936819i $$0.386244\pi$$
$$524$$ −14.0000 −0.611593
$$525$$ 0 0
$$526$$ 16.0000 0.697633
$$527$$ 16.0000 0.696971
$$528$$ 0 0
$$529$$ −19.0000 −0.826087
$$530$$ 0 0
$$531$$ 10.0000 0.433963
$$532$$ 0 0
$$533$$ 0 0
$$534$$ −6.00000 −0.259645
$$535$$ 0 0
$$536$$ 15.0000 0.647901
$$537$$ 12.0000 0.517838
$$538$$ 16.0000 0.689809
$$539$$ 0 0
$$540$$ 0 0
$$541$$ −3.00000 −0.128980 −0.0644900 0.997918i $$-0.520542\pi$$
−0.0644900 + 0.997918i $$0.520542\pi$$
$$542$$ 12.0000 0.515444
$$543$$ −22.0000 −0.944110
$$544$$ 10.0000 0.428746
$$545$$ 0 0
$$546$$ 0 0
$$547$$ 12.0000 0.513083 0.256541 0.966533i $$-0.417417\pi$$
0.256541 + 0.966533i $$0.417417\pi$$
$$548$$ −10.0000 −0.427179
$$549$$ 7.00000 0.298753
$$550$$ 0 0
$$551$$ −8.00000 −0.340811
$$552$$ 6.00000 0.255377
$$553$$ 0 0
$$554$$ 5.00000 0.212430
$$555$$ 0 0
$$556$$ −3.00000 −0.127228
$$557$$ 4.00000 0.169485 0.0847427 0.996403i $$-0.472993\pi$$
0.0847427 + 0.996403i $$0.472993\pi$$
$$558$$ 8.00000 0.338667
$$559$$ 24.0000 1.01509
$$560$$ 0 0
$$561$$ 0 0
$$562$$ −6.00000 −0.253095
$$563$$ −28.0000 −1.18006 −0.590030 0.807382i $$-0.700884\pi$$
−0.590030 + 0.807382i $$0.700884\pi$$
$$564$$ 10.0000 0.421076
$$565$$ 0 0
$$566$$ −19.0000 −0.798630
$$567$$ 0 0
$$568$$ −36.0000 −1.51053
$$569$$ 12.0000 0.503066 0.251533 0.967849i $$-0.419065\pi$$
0.251533 + 0.967849i $$0.419065\pi$$
$$570$$ 0 0
$$571$$ −3.00000 −0.125546 −0.0627730 0.998028i $$-0.519994\pi$$
−0.0627730 + 0.998028i $$0.519994\pi$$
$$572$$ 0 0
$$573$$ 8.00000 0.334205
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 7.00000 0.291667
$$577$$ 2.00000 0.0832611 0.0416305 0.999133i $$-0.486745\pi$$
0.0416305 + 0.999133i $$0.486745\pi$$
$$578$$ 13.0000 0.540729
$$579$$ 6.00000 0.249351
$$580$$ 0 0
$$581$$ 0 0
$$582$$ −9.00000 −0.373062
$$583$$ 0 0
$$584$$ 33.0000 1.36555
$$585$$ 0 0
$$586$$ −16.0000 −0.660954
$$587$$ 6.00000 0.247647 0.123823 0.992304i $$-0.460484\pi$$
0.123823 + 0.992304i $$0.460484\pi$$
$$588$$ 0 0
$$589$$ −8.00000 −0.329634
$$590$$ 0 0
$$591$$ 12.0000 0.493614
$$592$$ 7.00000 0.287698
$$593$$ −42.0000 −1.72473 −0.862367 0.506284i $$-0.831019\pi$$
−0.862367 + 0.506284i $$0.831019\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 4.00000 0.163846
$$597$$ −11.0000 −0.450200
$$598$$ 6.00000 0.245358
$$599$$ 26.0000 1.06233 0.531166 0.847268i $$-0.321754\pi$$
0.531166 + 0.847268i $$0.321754\pi$$
$$600$$ 0 0
$$601$$ −45.0000 −1.83559 −0.917794 0.397057i $$-0.870032\pi$$
−0.917794 + 0.397057i $$0.870032\pi$$
$$602$$ 0 0
$$603$$ 5.00000 0.203616
$$604$$ 1.00000 0.0406894
$$605$$ 0 0
$$606$$ −16.0000 −0.649956
$$607$$ 43.0000 1.74532 0.872658 0.488332i $$-0.162394\pi$$
0.872658 + 0.488332i $$0.162394\pi$$
$$608$$ −5.00000 −0.202777
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 30.0000 1.21367
$$612$$ 2.00000 0.0808452
$$613$$ −6.00000 −0.242338 −0.121169 0.992632i $$-0.538664\pi$$
−0.121169 + 0.992632i $$0.538664\pi$$
$$614$$ −12.0000 −0.484281
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 24.0000 0.966204 0.483102 0.875564i $$-0.339510\pi$$
0.483102 + 0.875564i $$0.339510\pi$$
$$618$$ 13.0000 0.522937
$$619$$ −12.0000 −0.482321 −0.241160 0.970485i $$-0.577528\pi$$
−0.241160 + 0.970485i $$0.577528\pi$$
$$620$$ 0 0
$$621$$ 2.00000 0.0802572
$$622$$ 30.0000 1.20289
$$623$$ 0 0
$$624$$ 3.00000 0.120096
$$625$$ 0 0
$$626$$ 14.0000 0.559553
$$627$$ 0 0
$$628$$ 7.00000 0.279330
$$629$$ 14.0000 0.558217
$$630$$ 0 0
$$631$$ 45.0000 1.79142 0.895711 0.444637i $$-0.146667\pi$$
0.895711 + 0.444637i $$0.146667\pi$$
$$632$$ −21.0000 −0.835335
$$633$$ 5.00000 0.198732
$$634$$ 14.0000 0.556011
$$635$$ 0 0
$$636$$ 14.0000 0.555136
$$637$$ 0 0
$$638$$ 0 0
$$639$$ −12.0000 −0.474713
$$640$$ 0 0
$$641$$ 24.0000 0.947943 0.473972 0.880540i $$-0.342820\pi$$
0.473972 + 0.880540i $$0.342820\pi$$
$$642$$ −6.00000 −0.236801
$$643$$ 29.0000 1.14365 0.571824 0.820376i $$-0.306236\pi$$
0.571824 + 0.820376i $$0.306236\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 2.00000 0.0786889
$$647$$ −12.0000 −0.471769 −0.235884 0.971781i $$-0.575799\pi$$
−0.235884 + 0.971781i $$0.575799\pi$$
$$648$$ 3.00000 0.117851
$$649$$ 0 0
$$650$$ 0 0
$$651$$ 0 0
$$652$$ −9.00000 −0.352467
$$653$$ −44.0000 −1.72185 −0.860927 0.508729i $$-0.830115\pi$$
−0.860927 + 0.508729i $$0.830115\pi$$
$$654$$ −15.0000 −0.586546
$$655$$ 0 0
$$656$$ 0 0
$$657$$ 11.0000 0.429151
$$658$$ 0 0
$$659$$ −24.0000 −0.934907 −0.467454 0.884018i $$-0.654829\pi$$
−0.467454 + 0.884018i $$0.654829\pi$$
$$660$$ 0 0
$$661$$ 11.0000 0.427850 0.213925 0.976850i $$-0.431375\pi$$
0.213925 + 0.976850i $$0.431375\pi$$
$$662$$ −21.0000 −0.816188
$$663$$ 6.00000 0.233021
$$664$$ −42.0000 −1.62992
$$665$$ 0 0
$$666$$ 7.00000 0.271244
$$667$$ 16.0000 0.619522
$$668$$ 2.00000 0.0773823
$$669$$ −15.0000 −0.579934
$$670$$ 0 0
$$671$$ 0 0
$$672$$ 0 0
$$673$$ 1.00000 0.0385472 0.0192736 0.999814i $$-0.493865\pi$$
0.0192736 + 0.999814i $$0.493865\pi$$
$$674$$ −2.00000 −0.0770371
$$675$$ 0 0
$$676$$ 4.00000 0.153846
$$677$$ 14.0000 0.538064 0.269032 0.963131i $$-0.413296\pi$$
0.269032 + 0.963131i $$0.413296\pi$$
$$678$$ 6.00000 0.230429
$$679$$ 0 0
$$680$$ 0 0
$$681$$ 20.0000 0.766402
$$682$$ 0 0
$$683$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$684$$ −1.00000 −0.0382360
$$685$$ 0 0
$$686$$ 0 0
$$687$$ −13.0000 −0.495981
$$688$$ −8.00000 −0.304997
$$689$$ 42.0000 1.60007
$$690$$ 0 0
$$691$$ −15.0000 −0.570627 −0.285313 0.958434i $$-0.592098\pi$$
−0.285313 + 0.958434i $$0.592098\pi$$
$$692$$ 18.0000 0.684257
$$693$$ 0 0
$$694$$ 10.0000 0.379595
$$695$$ 0 0
$$696$$ 24.0000 0.909718
$$697$$ 0 0
$$698$$ 22.0000 0.832712
$$699$$ 24.0000 0.907763
$$700$$ 0 0
$$701$$ 34.0000 1.28416 0.642081 0.766637i $$-0.278071\pi$$
0.642081 + 0.766637i $$0.278071\pi$$
$$702$$ 3.00000 0.113228
$$703$$ −7.00000 −0.264010
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 24.0000 0.903252
$$707$$ 0 0
$$708$$ 10.0000 0.375823
$$709$$ 1.00000 0.0375558 0.0187779 0.999824i $$-0.494022\pi$$
0.0187779 + 0.999824i $$0.494022\pi$$
$$710$$ 0 0
$$711$$ −7.00000 −0.262521
$$712$$ −18.0000 −0.674579
$$713$$ 16.0000 0.599205
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 12.0000 0.448461
$$717$$ 28.0000 1.04568
$$718$$ −30.0000 −1.11959
$$719$$ −4.00000 −0.149175 −0.0745874 0.997214i $$-0.523764\pi$$
−0.0745874 + 0.997214i $$0.523764\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ 18.0000 0.669891
$$723$$ −21.0000 −0.780998
$$724$$ −22.0000 −0.817624
$$725$$ 0 0
$$726$$ −11.0000 −0.408248
$$727$$ −43.0000 −1.59478 −0.797391 0.603463i $$-0.793787\pi$$
−0.797391 + 0.603463i $$0.793787\pi$$
$$728$$ 0 0
$$729$$ 1.00000 0.0370370
$$730$$ 0 0
$$731$$ −16.0000 −0.591781
$$732$$ 7.00000 0.258727
$$733$$ 31.0000 1.14501 0.572506 0.819901i $$-0.305971\pi$$
0.572506 + 0.819901i $$0.305971\pi$$
$$734$$ −8.00000 −0.295285
$$735$$ 0 0
$$736$$ 10.0000 0.368605
$$737$$ 0 0
$$738$$ 0 0
$$739$$ −49.0000 −1.80249 −0.901247 0.433306i $$-0.857347\pi$$
−0.901247 + 0.433306i $$0.857347\pi$$
$$740$$ 0 0
$$741$$ −3.00000 −0.110208
$$742$$ 0 0
$$743$$ −6.00000 −0.220119 −0.110059 0.993925i $$-0.535104\pi$$
−0.110059 + 0.993925i $$0.535104\pi$$
$$744$$ 24.0000 0.879883
$$745$$ 0 0
$$746$$ 11.0000 0.402739
$$747$$ −14.0000 −0.512233
$$748$$ 0 0
$$749$$ 0 0
$$750$$ 0 0
$$751$$ 23.0000 0.839282 0.419641 0.907690i $$-0.362156\pi$$
0.419641 + 0.907690i $$0.362156\pi$$
$$752$$ −10.0000 −0.364662
$$753$$ 0 0
$$754$$ 24.0000 0.874028
$$755$$ 0 0
$$756$$ 0 0
$$757$$ 43.0000 1.56286 0.781431 0.623992i $$-0.214490\pi$$
0.781431 + 0.623992i $$0.214490\pi$$
$$758$$ −35.0000 −1.27126
$$759$$ 0 0
$$760$$ 0 0
$$761$$ −6.00000 −0.217500 −0.108750 0.994069i $$-0.534685\pi$$
−0.108750 + 0.994069i $$0.534685\pi$$
$$762$$ −11.0000 −0.398488
$$763$$ 0 0
$$764$$ 8.00000 0.289430
$$765$$ 0 0
$$766$$ 10.0000 0.361315
$$767$$ 30.0000 1.08324
$$768$$ 17.0000 0.613435
$$769$$ −10.0000 −0.360609 −0.180305 0.983611i $$-0.557708\pi$$
−0.180305 + 0.983611i $$0.557708\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 6.00000 0.215945
$$773$$ −48.0000 −1.72644 −0.863220 0.504828i $$-0.831556\pi$$
−0.863220 + 0.504828i $$0.831556\pi$$
$$774$$ −8.00000 −0.287554
$$775$$ 0 0
$$776$$ −27.0000 −0.969244
$$777$$ 0 0
$$778$$ 36.0000 1.29066
$$779$$ 0 0
$$780$$ 0 0
$$781$$ 0 0
$$782$$ −4.00000 −0.143040
$$783$$ 8.00000 0.285897
$$784$$ 0 0
$$785$$ 0 0
$$786$$ 14.0000 0.499363
$$787$$ 17.0000 0.605985 0.302992 0.952993i $$-0.402014\pi$$
0.302992 + 0.952993i $$0.402014\pi$$
$$788$$ 12.0000 0.427482
$$789$$ 16.0000 0.569615
$$790$$ 0 0
$$791$$ 0 0
$$792$$ 0 0
$$793$$ 21.0000 0.745732
$$794$$ 14.0000 0.496841
$$795$$ 0 0
$$796$$ −11.0000 −0.389885
$$797$$ −30.0000 −1.06265 −0.531327 0.847167i $$-0.678307\pi$$
−0.531327 + 0.847167i $$0.678307\pi$$
$$798$$ 0 0
$$799$$ −20.0000 −0.707549
$$800$$ 0 0
$$801$$ −6.00000 −0.212000
$$802$$ 12.0000 0.423735
$$803$$ 0 0
$$804$$ 5.00000 0.176336
$$805$$ 0 0
$$806$$ 24.0000 0.845364
$$807$$ 16.0000 0.563227
$$808$$ −48.0000 −1.68863
$$809$$ −6.00000 −0.210949 −0.105474 0.994422i $$-0.533636\pi$$
−0.105474 + 0.994422i $$0.533636\pi$$
$$810$$ 0 0
$$811$$ 3.00000 0.105344 0.0526721 0.998612i $$-0.483226\pi$$
0.0526721 + 0.998612i $$0.483226\pi$$
$$812$$ 0 0
$$813$$ 12.0000 0.420858
$$814$$ 0 0
$$815$$ 0 0
$$816$$ −2.00000 −0.0700140
$$817$$ 8.00000 0.279885
$$818$$ −5.00000 −0.174821
$$819$$ 0 0
$$820$$ 0 0
$$821$$ −48.0000 −1.67521 −0.837606 0.546275i $$-0.816045\pi$$
−0.837606 + 0.546275i $$0.816045\pi$$
$$822$$ 10.0000 0.348790
$$823$$ −5.00000 −0.174289 −0.0871445 0.996196i $$-0.527774\pi$$
−0.0871445 + 0.996196i $$0.527774\pi$$
$$824$$ 39.0000 1.35863
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 24.0000 0.834562 0.417281 0.908778i $$-0.362983\pi$$
0.417281 + 0.908778i $$0.362983\pi$$
$$828$$ 2.00000 0.0695048
$$829$$ −15.0000 −0.520972 −0.260486 0.965478i $$-0.583883\pi$$
−0.260486 + 0.965478i $$0.583883\pi$$
$$830$$ 0 0
$$831$$ 5.00000 0.173448
$$832$$ 21.0000 0.728044
$$833$$ 0 0
$$834$$ 3.00000 0.103882
$$835$$ 0 0
$$836$$ 0 0
$$837$$ 8.00000 0.276520
$$838$$ −24.0000 −0.829066
$$839$$ −42.0000 −1.45000 −0.725001 0.688748i $$-0.758161\pi$$
−0.725001 + 0.688748i $$0.758161\pi$$
$$840$$ 0 0
$$841$$ 35.0000 1.20690
$$842$$ 35.0000 1.20618
$$843$$ −6.00000 −0.206651
$$844$$ 5.00000 0.172107
$$845$$ 0 0
$$846$$ −10.0000 −0.343807
$$847$$ 0 0
$$848$$ −14.0000 −0.480762
$$849$$ −19.0000 −0.652078
$$850$$ 0 0
$$851$$ 14.0000 0.479914
$$852$$ −12.0000 −0.411113
$$853$$ 6.00000 0.205436 0.102718 0.994711i $$-0.467246\pi$$
0.102718 + 0.994711i $$0.467246\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ −18.0000 −0.615227
$$857$$ −18.0000 −0.614868 −0.307434 0.951569i $$-0.599470\pi$$
−0.307434 + 0.951569i $$0.599470\pi$$
$$858$$ 0 0
$$859$$ −52.0000 −1.77422 −0.887109 0.461561i $$-0.847290\pi$$
−0.887109 + 0.461561i $$0.847290\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 2.00000 0.0681203
$$863$$ −6.00000 −0.204242 −0.102121 0.994772i $$-0.532563\pi$$
−0.102121 + 0.994772i $$0.532563\pi$$
$$864$$ 5.00000 0.170103
$$865$$ 0 0
$$866$$ −2.00000 −0.0679628
$$867$$ 13.0000 0.441503
$$868$$ 0 0
$$869$$ 0 0
$$870$$ 0 0
$$871$$ 15.0000 0.508256
$$872$$ −45.0000 −1.52389
$$873$$ −9.00000 −0.304604
$$874$$ 2.00000 0.0676510
$$875$$ 0 0
$$876$$ 11.0000 0.371656
$$877$$ −23.0000 −0.776655 −0.388327 0.921521i $$-0.626947\pi$$
−0.388327 + 0.921521i $$0.626947\pi$$
$$878$$ 11.0000 0.371232
$$879$$ −16.0000 −0.539667
$$880$$ 0 0
$$881$$ 12.0000 0.404290 0.202145 0.979356i $$-0.435209\pi$$
0.202145 + 0.979356i $$0.435209\pi$$
$$882$$ 0 0
$$883$$ −31.0000 −1.04323 −0.521617 0.853180i $$-0.674671\pi$$
−0.521617 + 0.853180i $$0.674671\pi$$
$$884$$ 6.00000 0.201802
$$885$$ 0 0
$$886$$ 6.00000 0.201574
$$887$$ −58.0000 −1.94745 −0.973725 0.227728i $$-0.926870\pi$$
−0.973725 + 0.227728i $$0.926870\pi$$
$$888$$ 21.0000 0.704714
$$889$$ 0 0
$$890$$ 0 0
$$891$$ 0 0
$$892$$ −15.0000 −0.502237
$$893$$ 10.0000 0.334637
$$894$$ −4.00000 −0.133780
$$895$$ 0 0
$$896$$ 0 0
$$897$$ 6.00000 0.200334
$$898$$ −6.00000 −0.200223
$$899$$ 64.0000 2.13452
$$900$$ 0 0
$$901$$ −28.0000 −0.932815
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 18.0000 0.598671
$$905$$ 0 0
$$906$$ −1.00000 −0.0332228
$$907$$ −1.00000 −0.0332045 −0.0166022 0.999862i $$-0.505285\pi$$
−0.0166022 + 0.999862i $$0.505285\pi$$
$$908$$ 20.0000 0.663723
$$909$$ −16.0000 −0.530687
$$910$$ 0 0
$$911$$ 54.0000 1.78910 0.894550 0.446968i $$-0.147496\pi$$
0.894550 + 0.446968i $$0.147496\pi$$
$$912$$ 1.00000 0.0331133
$$913$$ 0 0
$$914$$ −3.00000 −0.0992312
$$915$$ 0 0
$$916$$ −13.0000 −0.429532
$$917$$ 0 0
$$918$$ −2.00000 −0.0660098
$$919$$ 24.0000 0.791687 0.395843 0.918318i $$-0.370452\pi$$
0.395843 + 0.918318i $$0.370452\pi$$
$$920$$ 0 0
$$921$$ −12.0000 −0.395413
$$922$$ 2.00000 0.0658665
$$923$$ −36.0000 −1.18495
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 33.0000 1.08445
$$927$$ 13.0000 0.426976
$$928$$ 40.0000 1.31306
$$929$$ −44.0000 −1.44359 −0.721797 0.692105i $$-0.756683\pi$$
−0.721797 + 0.692105i $$0.756683\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ 24.0000 0.786146
$$933$$ 30.0000 0.982156
$$934$$ −2.00000 −0.0654420
$$935$$ 0 0
$$936$$ 9.00000 0.294174
$$937$$ −22.0000 −0.718709 −0.359354 0.933201i $$-0.617003\pi$$
−0.359354 + 0.933201i $$0.617003\pi$$
$$938$$ 0 0
$$939$$ 14.0000 0.456873
$$940$$ 0 0
$$941$$ 42.0000 1.36916 0.684580 0.728937i $$-0.259985\pi$$
0.684580 + 0.728937i $$0.259985\pi$$
$$942$$ −7.00000 −0.228072
$$943$$ 0 0
$$944$$ −10.0000 −0.325472
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 22.0000 0.714904 0.357452 0.933932i $$-0.383646\pi$$
0.357452 + 0.933932i $$0.383646\pi$$
$$948$$ −7.00000 −0.227349
$$949$$ 33.0000 1.07123
$$950$$ 0 0
$$951$$ 14.0000 0.453981
$$952$$ 0 0
$$953$$ 20.0000 0.647864 0.323932 0.946080i $$-0.394995\pi$$
0.323932 + 0.946080i $$0.394995\pi$$
$$954$$ −14.0000 −0.453267
$$955$$ 0 0
$$956$$ 28.0000 0.905585
$$957$$ 0 0
$$958$$ −2.00000 −0.0646171
$$959$$ 0 0
$$960$$ 0 0
$$961$$ 33.0000 1.06452
$$962$$ 21.0000 0.677067
$$963$$ −6.00000 −0.193347
$$964$$ −21.0000 −0.676364
$$965$$ 0 0
$$966$$ 0 0
$$967$$ 23.0000 0.739630 0.369815 0.929105i $$-0.379421\pi$$
0.369815 + 0.929105i $$0.379421\pi$$
$$968$$ −33.0000 −1.06066
$$969$$ 2.00000 0.0642493
$$970$$ 0 0
$$971$$ 8.00000 0.256732 0.128366 0.991727i $$-0.459027\pi$$
0.128366 + 0.991727i $$0.459027\pi$$
$$972$$ 1.00000 0.0320750
$$973$$ 0 0
$$974$$ 8.00000 0.256337
$$975$$ 0 0
$$976$$ −7.00000 −0.224065
$$977$$ 40.0000 1.27971 0.639857 0.768494i $$-0.278994\pi$$
0.639857 + 0.768494i $$0.278994\pi$$
$$978$$ 9.00000 0.287788
$$979$$ 0 0
$$980$$ 0 0
$$981$$ −15.0000 −0.478913
$$982$$ 6.00000 0.191468
$$983$$ 22.0000 0.701691 0.350846 0.936433i $$-0.385894\pi$$
0.350846 + 0.936433i $$0.385894\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ −16.0000 −0.509544
$$987$$ 0 0
$$988$$ −3.00000 −0.0954427
$$989$$ −16.0000 −0.508770
$$990$$ 0 0
$$991$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$992$$ 40.0000 1.27000
$$993$$ −21.0000 −0.666415
$$994$$ 0 0
$$995$$ 0 0
$$996$$ −14.0000 −0.443607
$$997$$ 37.0000 1.17180 0.585901 0.810383i $$-0.300741\pi$$
0.585901 + 0.810383i $$0.300741\pi$$
$$998$$ 19.0000 0.601434
$$999$$ 7.00000 0.221470
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 3675.2.a.e.1.1 1
5.4 even 2 3675.2.a.m.1.1 1
7.2 even 3 525.2.i.d.151.1 yes 2
7.4 even 3 525.2.i.d.226.1 yes 2
7.6 odd 2 3675.2.a.g.1.1 1
35.2 odd 12 525.2.r.c.424.1 4
35.4 even 6 525.2.i.b.226.1 yes 2
35.9 even 6 525.2.i.b.151.1 2
35.18 odd 12 525.2.r.c.499.1 4
35.23 odd 12 525.2.r.c.424.2 4
35.32 odd 12 525.2.r.c.499.2 4
35.34 odd 2 3675.2.a.k.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
525.2.i.b.151.1 2 35.9 even 6
525.2.i.b.226.1 yes 2 35.4 even 6
525.2.i.d.151.1 yes 2 7.2 even 3
525.2.i.d.226.1 yes 2 7.4 even 3
525.2.r.c.424.1 4 35.2 odd 12
525.2.r.c.424.2 4 35.23 odd 12
525.2.r.c.499.1 4 35.18 odd 12
525.2.r.c.499.2 4 35.32 odd 12
3675.2.a.e.1.1 1 1.1 even 1 trivial
3675.2.a.g.1.1 1 7.6 odd 2
3675.2.a.k.1.1 1 35.34 odd 2
3675.2.a.m.1.1 1 5.4 even 2