# Properties

 Label 245.6.a.a.1.1 Level $245$ Weight $6$ Character 245.1 Self dual yes Analytic conductor $39.294$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$245 = 5 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 245.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$39.2940358542$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 35) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 245.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-8.00000 q^{2} -1.00000 q^{3} +32.0000 q^{4} -25.0000 q^{5} +8.00000 q^{6} -242.000 q^{9} +O(q^{10})$$ $$q-8.00000 q^{2} -1.00000 q^{3} +32.0000 q^{4} -25.0000 q^{5} +8.00000 q^{6} -242.000 q^{9} +200.000 q^{10} -453.000 q^{11} -32.0000 q^{12} +969.000 q^{13} +25.0000 q^{15} -1024.00 q^{16} -1637.00 q^{17} +1936.00 q^{18} +1550.00 q^{19} -800.000 q^{20} +3624.00 q^{22} -1654.00 q^{23} +625.000 q^{25} -7752.00 q^{26} +485.000 q^{27} -4985.00 q^{29} -200.000 q^{30} -1192.00 q^{31} +8192.00 q^{32} +453.000 q^{33} +13096.0 q^{34} -7744.00 q^{36} -11018.0 q^{37} -12400.0 q^{38} -969.000 q^{39} +1728.00 q^{41} -10814.0 q^{43} -14496.0 q^{44} +6050.00 q^{45} +13232.0 q^{46} -26237.0 q^{47} +1024.00 q^{48} -5000.00 q^{50} +1637.00 q^{51} +31008.0 q^{52} +25936.0 q^{53} -3880.00 q^{54} +11325.0 q^{55} -1550.00 q^{57} +39880.0 q^{58} +4580.00 q^{59} +800.000 q^{60} +12488.0 q^{61} +9536.00 q^{62} -32768.0 q^{64} -24225.0 q^{65} -3624.00 q^{66} -15848.0 q^{67} -52384.0 q^{68} +1654.00 q^{69} +51792.0 q^{71} -4846.00 q^{73} +88144.0 q^{74} -625.000 q^{75} +49600.0 q^{76} +7752.00 q^{78} +62765.0 q^{79} +25600.0 q^{80} +58321.0 q^{81} -13824.0 q^{82} +23644.0 q^{83} +40925.0 q^{85} +86512.0 q^{86} +4985.00 q^{87} +147300. q^{89} -48400.0 q^{90} -52928.0 q^{92} +1192.00 q^{93} +209896. q^{94} -38750.0 q^{95} -8192.00 q^{96} +8343.00 q^{97} +109626. q^{99} +O(q^{100})$$

## Coefficient data

For each $$n$$ we display the coefficients of the $$q$$-expansion $$a_n$$, the Satake parameters $$\alpha_p$$, and the Satake angles $$\theta_p = \textrm{Arg}(\alpha_p)$$.

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −8.00000 −1.41421 −0.707107 0.707107i $$-0.750000\pi$$
−0.707107 + 0.707107i $$0.750000\pi$$
$$3$$ −1.00000 −0.0641500 −0.0320750 0.999485i $$-0.510212\pi$$
−0.0320750 + 0.999485i $$0.510212\pi$$
$$4$$ 32.0000 1.00000
$$5$$ −25.0000 −0.447214
$$6$$ 8.00000 0.0907218
$$7$$ 0 0
$$8$$ 0 0
$$9$$ −242.000 −0.995885
$$10$$ 200.000 0.632456
$$11$$ −453.000 −1.12880 −0.564399 0.825502i $$-0.690892\pi$$
−0.564399 + 0.825502i $$0.690892\pi$$
$$12$$ −32.0000 −0.0641500
$$13$$ 969.000 1.59025 0.795125 0.606446i $$-0.207405\pi$$
0.795125 + 0.606446i $$0.207405\pi$$
$$14$$ 0 0
$$15$$ 25.0000 0.0286888
$$16$$ −1024.00 −1.00000
$$17$$ −1637.00 −1.37381 −0.686905 0.726748i $$-0.741031\pi$$
−0.686905 + 0.726748i $$0.741031\pi$$
$$18$$ 1936.00 1.40839
$$19$$ 1550.00 0.985026 0.492513 0.870305i $$-0.336078\pi$$
0.492513 + 0.870305i $$0.336078\pi$$
$$20$$ −800.000 −0.447214
$$21$$ 0 0
$$22$$ 3624.00 1.59636
$$23$$ −1654.00 −0.651952 −0.325976 0.945378i $$-0.605693\pi$$
−0.325976 + 0.945378i $$0.605693\pi$$
$$24$$ 0 0
$$25$$ 625.000 0.200000
$$26$$ −7752.00 −2.24895
$$27$$ 485.000 0.128036
$$28$$ 0 0
$$29$$ −4985.00 −1.10070 −0.550352 0.834933i $$-0.685506\pi$$
−0.550352 + 0.834933i $$0.685506\pi$$
$$30$$ −200.000 −0.0405720
$$31$$ −1192.00 −0.222778 −0.111389 0.993777i $$-0.535530\pi$$
−0.111389 + 0.993777i $$0.535530\pi$$
$$32$$ 8192.00 1.41421
$$33$$ 453.000 0.0724125
$$34$$ 13096.0 1.94286
$$35$$ 0 0
$$36$$ −7744.00 −0.995885
$$37$$ −11018.0 −1.32312 −0.661559 0.749893i $$-0.730105\pi$$
−0.661559 + 0.749893i $$0.730105\pi$$
$$38$$ −12400.0 −1.39304
$$39$$ −969.000 −0.102015
$$40$$ 0 0
$$41$$ 1728.00 0.160540 0.0802702 0.996773i $$-0.474422\pi$$
0.0802702 + 0.996773i $$0.474422\pi$$
$$42$$ 0 0
$$43$$ −10814.0 −0.891898 −0.445949 0.895058i $$-0.647134\pi$$
−0.445949 + 0.895058i $$0.647134\pi$$
$$44$$ −14496.0 −1.12880
$$45$$ 6050.00 0.445373
$$46$$ 13232.0 0.922000
$$47$$ −26237.0 −1.73249 −0.866243 0.499624i $$-0.833472\pi$$
−0.866243 + 0.499624i $$0.833472\pi$$
$$48$$ 1024.00 0.0641500
$$49$$ 0 0
$$50$$ −5000.00 −0.282843
$$51$$ 1637.00 0.0881299
$$52$$ 31008.0 1.59025
$$53$$ 25936.0 1.26827 0.634137 0.773220i $$-0.281355\pi$$
0.634137 + 0.773220i $$0.281355\pi$$
$$54$$ −3880.00 −0.181070
$$55$$ 11325.0 0.504814
$$56$$ 0 0
$$57$$ −1550.00 −0.0631894
$$58$$ 39880.0 1.55663
$$59$$ 4580.00 0.171291 0.0856457 0.996326i $$-0.472705\pi$$
0.0856457 + 0.996326i $$0.472705\pi$$
$$60$$ 800.000 0.0286888
$$61$$ 12488.0 0.429703 0.214851 0.976647i $$-0.431073\pi$$
0.214851 + 0.976647i $$0.431073\pi$$
$$62$$ 9536.00 0.315055
$$63$$ 0 0
$$64$$ −32768.0 −1.00000
$$65$$ −24225.0 −0.711181
$$66$$ −3624.00 −0.102407
$$67$$ −15848.0 −0.431308 −0.215654 0.976470i $$-0.569188\pi$$
−0.215654 + 0.976470i $$0.569188\pi$$
$$68$$ −52384.0 −1.37381
$$69$$ 1654.00 0.0418228
$$70$$ 0 0
$$71$$ 51792.0 1.21932 0.609659 0.792664i $$-0.291306\pi$$
0.609659 + 0.792664i $$0.291306\pi$$
$$72$$ 0 0
$$73$$ −4846.00 −0.106433 −0.0532165 0.998583i $$-0.516947\pi$$
−0.0532165 + 0.998583i $$0.516947\pi$$
$$74$$ 88144.0 1.87117
$$75$$ −625.000 −0.0128300
$$76$$ 49600.0 0.985026
$$77$$ 0 0
$$78$$ 7752.00 0.144270
$$79$$ 62765.0 1.13149 0.565744 0.824581i $$-0.308589\pi$$
0.565744 + 0.824581i $$0.308589\pi$$
$$80$$ 25600.0 0.447214
$$81$$ 58321.0 0.987671
$$82$$ −13824.0 −0.227038
$$83$$ 23644.0 0.376726 0.188363 0.982099i $$-0.439682\pi$$
0.188363 + 0.982099i $$0.439682\pi$$
$$84$$ 0 0
$$85$$ 40925.0 0.614386
$$86$$ 86512.0 1.26133
$$87$$ 4985.00 0.0706101
$$88$$ 0 0
$$89$$ 147300. 1.97119 0.985593 0.169133i $$-0.0540967\pi$$
0.985593 + 0.169133i $$0.0540967\pi$$
$$90$$ −48400.0 −0.629853
$$91$$ 0 0
$$92$$ −52928.0 −0.651952
$$93$$ 1192.00 0.0142912
$$94$$ 209896. 2.45010
$$95$$ −38750.0 −0.440517
$$96$$ −8192.00 −0.0907218
$$97$$ 8343.00 0.0900312 0.0450156 0.998986i $$-0.485666\pi$$
0.0450156 + 0.998986i $$0.485666\pi$$
$$98$$ 0 0
$$99$$ 109626. 1.12415
$$100$$ 20000.0 0.200000
$$101$$ 11878.0 0.115862 0.0579308 0.998321i $$-0.481550\pi$$
0.0579308 + 0.998321i $$0.481550\pi$$
$$102$$ −13096.0 −0.124634
$$103$$ 132439. 1.23005 0.615025 0.788508i $$-0.289146\pi$$
0.615025 + 0.788508i $$0.289146\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ −207488. −1.79361
$$107$$ 136842. 1.15547 0.577737 0.816223i $$-0.303936\pi$$
0.577737 + 0.816223i $$0.303936\pi$$
$$108$$ 15520.0 0.128036
$$109$$ 109485. 0.882650 0.441325 0.897347i $$-0.354509\pi$$
0.441325 + 0.897347i $$0.354509\pi$$
$$110$$ −90600.0 −0.713915
$$111$$ 11018.0 0.0848780
$$112$$ 0 0
$$113$$ −200934. −1.48033 −0.740163 0.672428i $$-0.765252\pi$$
−0.740163 + 0.672428i $$0.765252\pi$$
$$114$$ 12400.0 0.0893634
$$115$$ 41350.0 0.291562
$$116$$ −159520. −1.10070
$$117$$ −234498. −1.58371
$$118$$ −36640.0 −0.242243
$$119$$ 0 0
$$120$$ 0 0
$$121$$ 44158.0 0.274186
$$122$$ −99904.0 −0.607692
$$123$$ −1728.00 −0.0102987
$$124$$ −38144.0 −0.222778
$$125$$ −15625.0 −0.0894427
$$126$$ 0 0
$$127$$ 330692. 1.81934 0.909671 0.415329i $$-0.136334\pi$$
0.909671 + 0.415329i $$0.136334\pi$$
$$128$$ 0 0
$$129$$ 10814.0 0.0572153
$$130$$ 193800. 1.00576
$$131$$ −43982.0 −0.223922 −0.111961 0.993713i $$-0.535713\pi$$
−0.111961 + 0.993713i $$0.535713\pi$$
$$132$$ 14496.0 0.0724125
$$133$$ 0 0
$$134$$ 126784. 0.609962
$$135$$ −12125.0 −0.0572595
$$136$$ 0 0
$$137$$ −99748.0 −0.454049 −0.227025 0.973889i $$-0.572900\pi$$
−0.227025 + 0.973889i $$0.572900\pi$$
$$138$$ −13232.0 −0.0591463
$$139$$ −258930. −1.13670 −0.568349 0.822787i $$-0.692418\pi$$
−0.568349 + 0.822787i $$0.692418\pi$$
$$140$$ 0 0
$$141$$ 26237.0 0.111139
$$142$$ −414336. −1.72438
$$143$$ −438957. −1.79507
$$144$$ 247808. 0.995885
$$145$$ 124625. 0.492249
$$146$$ 38768.0 0.150519
$$147$$ 0 0
$$148$$ −352576. −1.32312
$$149$$ −498430. −1.83924 −0.919620 0.392809i $$-0.871503\pi$$
−0.919620 + 0.392809i $$0.871503\pi$$
$$150$$ 5000.00 0.0181444
$$151$$ −245803. −0.877293 −0.438647 0.898660i $$-0.644542\pi$$
−0.438647 + 0.898660i $$0.644542\pi$$
$$152$$ 0 0
$$153$$ 396154. 1.36816
$$154$$ 0 0
$$155$$ 29800.0 0.0996293
$$156$$ −31008.0 −0.102015
$$157$$ 85478.0 0.276761 0.138381 0.990379i $$-0.455810\pi$$
0.138381 + 0.990379i $$0.455810\pi$$
$$158$$ −502120. −1.60017
$$159$$ −25936.0 −0.0813599
$$160$$ −204800. −0.632456
$$161$$ 0 0
$$162$$ −466568. −1.39678
$$163$$ 193026. 0.569045 0.284523 0.958669i $$-0.408165\pi$$
0.284523 + 0.958669i $$0.408165\pi$$
$$164$$ 55296.0 0.160540
$$165$$ −11325.0 −0.0323838
$$166$$ −189152. −0.532771
$$167$$ 157783. 0.437793 0.218897 0.975748i $$-0.429754\pi$$
0.218897 + 0.975748i $$0.429754\pi$$
$$168$$ 0 0
$$169$$ 567668. 1.52889
$$170$$ −327400. −0.868873
$$171$$ −375100. −0.980972
$$172$$ −346048. −0.891898
$$173$$ 265659. 0.674853 0.337427 0.941352i $$-0.390444\pi$$
0.337427 + 0.941352i $$0.390444\pi$$
$$174$$ −39880.0 −0.0998578
$$175$$ 0 0
$$176$$ 463872. 1.12880
$$177$$ −4580.00 −0.0109883
$$178$$ −1.17840e6 −2.78768
$$179$$ 183660. 0.428432 0.214216 0.976786i $$-0.431280\pi$$
0.214216 + 0.976786i $$0.431280\pi$$
$$180$$ 193600. 0.445373
$$181$$ 635048. 1.44082 0.720411 0.693548i $$-0.243953\pi$$
0.720411 + 0.693548i $$0.243953\pi$$
$$182$$ 0 0
$$183$$ −12488.0 −0.0275655
$$184$$ 0 0
$$185$$ 275450. 0.591716
$$186$$ −9536.00 −0.0202108
$$187$$ 741561. 1.55075
$$188$$ −839584. −1.73249
$$189$$ 0 0
$$190$$ 310000. 0.622985
$$191$$ −226613. −0.449471 −0.224735 0.974420i $$-0.572152\pi$$
−0.224735 + 0.974420i $$0.572152\pi$$
$$192$$ 32768.0 0.0641500
$$193$$ 46476.0 0.0898122 0.0449061 0.998991i $$-0.485701\pi$$
0.0449061 + 0.998991i $$0.485701\pi$$
$$194$$ −66744.0 −0.127323
$$195$$ 24225.0 0.0456223
$$196$$ 0 0
$$197$$ 204972. 0.376295 0.188148 0.982141i $$-0.439752\pi$$
0.188148 + 0.982141i $$0.439752\pi$$
$$198$$ −877008. −1.58979
$$199$$ 953020. 1.70596 0.852981 0.521942i $$-0.174792\pi$$
0.852981 + 0.521942i $$0.174792\pi$$
$$200$$ 0 0
$$201$$ 15848.0 0.0276684
$$202$$ −95024.0 −0.163853
$$203$$ 0 0
$$204$$ 52384.0 0.0881299
$$205$$ −43200.0 −0.0717958
$$206$$ −1.05951e6 −1.73955
$$207$$ 400268. 0.649270
$$208$$ −992256. −1.59025
$$209$$ −702150. −1.11190
$$210$$ 0 0
$$211$$ −223523. −0.345634 −0.172817 0.984954i $$-0.555287\pi$$
−0.172817 + 0.984954i $$0.555287\pi$$
$$212$$ 829952. 1.26827
$$213$$ −51792.0 −0.0782193
$$214$$ −1.09474e6 −1.63409
$$215$$ 270350. 0.398869
$$216$$ 0 0
$$217$$ 0 0
$$218$$ −875880. −1.24826
$$219$$ 4846.00 0.00682768
$$220$$ 362400. 0.504814
$$221$$ −1.58625e6 −2.18470
$$222$$ −88144.0 −0.120036
$$223$$ −1.01480e6 −1.36653 −0.683264 0.730171i $$-0.739440\pi$$
−0.683264 + 0.730171i $$0.739440\pi$$
$$224$$ 0 0
$$225$$ −151250. −0.199177
$$226$$ 1.60747e6 2.09350
$$227$$ −999797. −1.28780 −0.643898 0.765111i $$-0.722684\pi$$
−0.643898 + 0.765111i $$0.722684\pi$$
$$228$$ −49600.0 −0.0631894
$$229$$ 851120. 1.07251 0.536256 0.844055i $$-0.319838\pi$$
0.536256 + 0.844055i $$0.319838\pi$$
$$230$$ −330800. −0.412331
$$231$$ 0 0
$$232$$ 0 0
$$233$$ 1.09270e6 1.31859 0.659295 0.751885i $$-0.270855\pi$$
0.659295 + 0.751885i $$0.270855\pi$$
$$234$$ 1.87598e6 2.23970
$$235$$ 655925. 0.774791
$$236$$ 146560. 0.171291
$$237$$ −62765.0 −0.0725850
$$238$$ 0 0
$$239$$ 765905. 0.867322 0.433661 0.901076i $$-0.357222\pi$$
0.433661 + 0.901076i $$0.357222\pi$$
$$240$$ −25600.0 −0.0286888
$$241$$ 1.21094e6 1.34301 0.671505 0.741000i $$-0.265648\pi$$
0.671505 + 0.741000i $$0.265648\pi$$
$$242$$ −353264. −0.387758
$$243$$ −176176. −0.191395
$$244$$ 399616. 0.429703
$$245$$ 0 0
$$246$$ 13824.0 0.0145645
$$247$$ 1.50195e6 1.56644
$$248$$ 0 0
$$249$$ −23644.0 −0.0241670
$$250$$ 125000. 0.126491
$$251$$ −278262. −0.278785 −0.139393 0.990237i $$-0.544515\pi$$
−0.139393 + 0.990237i $$0.544515\pi$$
$$252$$ 0 0
$$253$$ 749262. 0.735923
$$254$$ −2.64554e6 −2.57294
$$255$$ −40925.0 −0.0394129
$$256$$ 1.04858e6 1.00000
$$257$$ 352998. 0.333380 0.166690 0.986009i $$-0.446692\pi$$
0.166690 + 0.986009i $$0.446692\pi$$
$$258$$ −86512.0 −0.0809146
$$259$$ 0 0
$$260$$ −775200. −0.711181
$$261$$ 1.20637e6 1.09617
$$262$$ 351856. 0.316674
$$263$$ −1.55809e6 −1.38901 −0.694503 0.719490i $$-0.744376\pi$$
−0.694503 + 0.719490i $$0.744376\pi$$
$$264$$ 0 0
$$265$$ −648400. −0.567190
$$266$$ 0 0
$$267$$ −147300. −0.126452
$$268$$ −507136. −0.431308
$$269$$ 1.21963e6 1.02766 0.513828 0.857893i $$-0.328227\pi$$
0.513828 + 0.857893i $$0.328227\pi$$
$$270$$ 97000.0 0.0809771
$$271$$ −405792. −0.335645 −0.167823 0.985817i $$-0.553674\pi$$
−0.167823 + 0.985817i $$0.553674\pi$$
$$272$$ 1.67629e6 1.37381
$$273$$ 0 0
$$274$$ 797984. 0.642122
$$275$$ −283125. −0.225760
$$276$$ 52928.0 0.0418228
$$277$$ 652442. 0.510908 0.255454 0.966821i $$-0.417775\pi$$
0.255454 + 0.966821i $$0.417775\pi$$
$$278$$ 2.07144e6 1.60753
$$279$$ 288464. 0.221861
$$280$$ 0 0
$$281$$ 118827. 0.0897737 0.0448869 0.998992i $$-0.485707\pi$$
0.0448869 + 0.998992i $$0.485707\pi$$
$$282$$ −209896. −0.157174
$$283$$ −1.48801e6 −1.10443 −0.552217 0.833700i $$-0.686218\pi$$
−0.552217 + 0.833700i $$0.686218\pi$$
$$284$$ 1.65734e6 1.21932
$$285$$ 38750.0 0.0282592
$$286$$ 3.51166e6 2.53862
$$287$$ 0 0
$$288$$ −1.98246e6 −1.40839
$$289$$ 1.25991e6 0.887351
$$290$$ −997000. −0.696146
$$291$$ −8343.00 −0.00577550
$$292$$ −155072. −0.106433
$$293$$ −1.89580e6 −1.29010 −0.645050 0.764140i $$-0.723164\pi$$
−0.645050 + 0.764140i $$0.723164\pi$$
$$294$$ 0 0
$$295$$ −114500. −0.0766038
$$296$$ 0 0
$$297$$ −219705. −0.144527
$$298$$ 3.98744e6 2.60108
$$299$$ −1.60273e6 −1.03677
$$300$$ −20000.0 −0.0128300
$$301$$ 0 0
$$302$$ 1.96642e6 1.24068
$$303$$ −11878.0 −0.00743253
$$304$$ −1.58720e6 −0.985026
$$305$$ −312200. −0.192169
$$306$$ −3.16923e6 −1.93486
$$307$$ 821853. 0.497678 0.248839 0.968545i $$-0.419951\pi$$
0.248839 + 0.968545i $$0.419951\pi$$
$$308$$ 0 0
$$309$$ −132439. −0.0789078
$$310$$ −238400. −0.140897
$$311$$ 2.09600e6 1.22882 0.614412 0.788985i $$-0.289393\pi$$
0.614412 + 0.788985i $$0.289393\pi$$
$$312$$ 0 0
$$313$$ −394571. −0.227648 −0.113824 0.993501i $$-0.536310\pi$$
−0.113824 + 0.993501i $$0.536310\pi$$
$$314$$ −683824. −0.391399
$$315$$ 0 0
$$316$$ 2.00848e6 1.13149
$$317$$ 321422. 0.179650 0.0898250 0.995958i $$-0.471369\pi$$
0.0898250 + 0.995958i $$0.471369\pi$$
$$318$$ 207488. 0.115060
$$319$$ 2.25820e6 1.24247
$$320$$ 819200. 0.447214
$$321$$ −136842. −0.0741237
$$322$$ 0 0
$$323$$ −2.53735e6 −1.35324
$$324$$ 1.86627e6 0.987671
$$325$$ 605625. 0.318050
$$326$$ −1.54421e6 −0.804752
$$327$$ −109485. −0.0566220
$$328$$ 0 0
$$329$$ 0 0
$$330$$ 90600.0 0.0457977
$$331$$ −2.23259e6 −1.12005 −0.560027 0.828475i $$-0.689209\pi$$
−0.560027 + 0.828475i $$0.689209\pi$$
$$332$$ 756608. 0.376726
$$333$$ 2.66636e6 1.31767
$$334$$ −1.26226e6 −0.619133
$$335$$ 396200. 0.192887
$$336$$ 0 0
$$337$$ −3.65656e6 −1.75387 −0.876936 0.480608i $$-0.840416\pi$$
−0.876936 + 0.480608i $$0.840416\pi$$
$$338$$ −4.54134e6 −2.16218
$$339$$ 200934. 0.0949629
$$340$$ 1.30960e6 0.614386
$$341$$ 539976. 0.251471
$$342$$ 3.00080e6 1.38730
$$343$$ 0 0
$$344$$ 0 0
$$345$$ −41350.0 −0.0187037
$$346$$ −2.12527e6 −0.954386
$$347$$ −1.88962e6 −0.842462 −0.421231 0.906953i $$-0.638402\pi$$
−0.421231 + 0.906953i $$0.638402\pi$$
$$348$$ 159520. 0.0706101
$$349$$ 2.69329e6 1.18364 0.591820 0.806070i $$-0.298410\pi$$
0.591820 + 0.806070i $$0.298410\pi$$
$$350$$ 0 0
$$351$$ 469965. 0.203609
$$352$$ −3.71098e6 −1.59636
$$353$$ 1.57468e6 0.672598 0.336299 0.941755i $$-0.390825\pi$$
0.336299 + 0.941755i $$0.390825\pi$$
$$354$$ 36640.0 0.0155399
$$355$$ −1.29480e6 −0.545295
$$356$$ 4.71360e6 1.97119
$$357$$ 0 0
$$358$$ −1.46928e6 −0.605894
$$359$$ 4.05576e6 1.66087 0.830436 0.557114i $$-0.188091\pi$$
0.830436 + 0.557114i $$0.188091\pi$$
$$360$$ 0 0
$$361$$ −73599.0 −0.0297238
$$362$$ −5.08038e6 −2.03763
$$363$$ −44158.0 −0.0175891
$$364$$ 0 0
$$365$$ 121150. 0.0475983
$$366$$ 99904.0 0.0389834
$$367$$ 4.90628e6 1.90146 0.950731 0.310018i $$-0.100335\pi$$
0.950731 + 0.310018i $$0.100335\pi$$
$$368$$ 1.69370e6 0.651952
$$369$$ −418176. −0.159880
$$370$$ −2.20360e6 −0.836813
$$371$$ 0 0
$$372$$ 38144.0 0.0142912
$$373$$ −3.45336e6 −1.28520 −0.642599 0.766202i $$-0.722144\pi$$
−0.642599 + 0.766202i $$0.722144\pi$$
$$374$$ −5.93249e6 −2.19310
$$375$$ 15625.0 0.00573775
$$376$$ 0 0
$$377$$ −4.83046e6 −1.75039
$$378$$ 0 0
$$379$$ −4.23466e6 −1.51433 −0.757165 0.653224i $$-0.773416\pi$$
−0.757165 + 0.653224i $$0.773416\pi$$
$$380$$ −1.24000e6 −0.440517
$$381$$ −330692. −0.116711
$$382$$ 1.81290e6 0.635648
$$383$$ 1.86460e6 0.649516 0.324758 0.945797i $$-0.394717\pi$$
0.324758 + 0.945797i $$0.394717\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ −371808. −0.127014
$$387$$ 2.61699e6 0.888228
$$388$$ 266976. 0.0900312
$$389$$ −4.81502e6 −1.61333 −0.806666 0.591008i $$-0.798730\pi$$
−0.806666 + 0.591008i $$0.798730\pi$$
$$390$$ −193800. −0.0645197
$$391$$ 2.70760e6 0.895658
$$392$$ 0 0
$$393$$ 43982.0 0.0143646
$$394$$ −1.63978e6 −0.532162
$$395$$ −1.56912e6 −0.506017
$$396$$ 3.50803e6 1.12415
$$397$$ −1.21376e6 −0.386505 −0.193253 0.981149i $$-0.561904\pi$$
−0.193253 + 0.981149i $$0.561904\pi$$
$$398$$ −7.62416e6 −2.41259
$$399$$ 0 0
$$400$$ −640000. −0.200000
$$401$$ 5.90442e6 1.83365 0.916824 0.399291i $$-0.130744\pi$$
0.916824 + 0.399291i $$0.130744\pi$$
$$402$$ −126784. −0.0391291
$$403$$ −1.15505e6 −0.354272
$$404$$ 380096. 0.115862
$$405$$ −1.45802e6 −0.441700
$$406$$ 0 0
$$407$$ 4.99115e6 1.49353
$$408$$ 0 0
$$409$$ −4.84289e6 −1.43152 −0.715758 0.698348i $$-0.753919\pi$$
−0.715758 + 0.698348i $$0.753919\pi$$
$$410$$ 345600. 0.101535
$$411$$ 99748.0 0.0291273
$$412$$ 4.23805e6 1.23005
$$413$$ 0 0
$$414$$ −3.20214e6 −0.918206
$$415$$ −591100. −0.168477
$$416$$ 7.93805e6 2.24895
$$417$$ 258930. 0.0729193
$$418$$ 5.61720e6 1.57246
$$419$$ −270360. −0.0752328 −0.0376164 0.999292i $$-0.511977\pi$$
−0.0376164 + 0.999292i $$0.511977\pi$$
$$420$$ 0 0
$$421$$ 3.13648e6 0.862456 0.431228 0.902243i $$-0.358080\pi$$
0.431228 + 0.902243i $$0.358080\pi$$
$$422$$ 1.78818e6 0.488800
$$423$$ 6.34935e6 1.72536
$$424$$ 0 0
$$425$$ −1.02312e6 −0.274762
$$426$$ 414336. 0.110619
$$427$$ 0 0
$$428$$ 4.37894e6 1.15547
$$429$$ 438957. 0.115154
$$430$$ −2.16280e6 −0.564086
$$431$$ −1.87703e6 −0.486719 −0.243360 0.969936i $$-0.578250\pi$$
−0.243360 + 0.969936i $$0.578250\pi$$
$$432$$ −496640. −0.128036
$$433$$ −3.20357e6 −0.821134 −0.410567 0.911830i $$-0.634669\pi$$
−0.410567 + 0.911830i $$0.634669\pi$$
$$434$$ 0 0
$$435$$ −124625. −0.0315778
$$436$$ 3.50352e6 0.882650
$$437$$ −2.56370e6 −0.642190
$$438$$ −38768.0 −0.00965580
$$439$$ 6.27209e6 1.55328 0.776642 0.629942i $$-0.216921\pi$$
0.776642 + 0.629942i $$0.216921\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 1.26900e7 3.08963
$$443$$ 724986. 0.175517 0.0877587 0.996142i $$-0.472030\pi$$
0.0877587 + 0.996142i $$0.472030\pi$$
$$444$$ 352576. 0.0848780
$$445$$ −3.68250e6 −0.881541
$$446$$ 8.11841e6 1.93256
$$447$$ 498430. 0.117987
$$448$$ 0 0
$$449$$ −875985. −0.205060 −0.102530 0.994730i $$-0.532694\pi$$
−0.102530 + 0.994730i $$0.532694\pi$$
$$450$$ 1.21000e6 0.281679
$$451$$ −782784. −0.181218
$$452$$ −6.42989e6 −1.48033
$$453$$ 245803. 0.0562784
$$454$$ 7.99838e6 1.82122
$$455$$ 0 0
$$456$$ 0 0
$$457$$ −832668. −0.186501 −0.0932505 0.995643i $$-0.529726\pi$$
−0.0932505 + 0.995643i $$0.529726\pi$$
$$458$$ −6.80896e6 −1.51676
$$459$$ −793945. −0.175897
$$460$$ 1.32320e6 0.291562
$$461$$ −5.92115e6 −1.29764 −0.648820 0.760942i $$-0.724737\pi$$
−0.648820 + 0.760942i $$0.724737\pi$$
$$462$$ 0 0
$$463$$ 682776. 0.148022 0.0740109 0.997257i $$-0.476420\pi$$
0.0740109 + 0.997257i $$0.476420\pi$$
$$464$$ 5.10464e6 1.10070
$$465$$ −29800.0 −0.00639122
$$466$$ −8.74157e6 −1.86477
$$467$$ 5.41667e6 1.14932 0.574659 0.818393i $$-0.305135\pi$$
0.574659 + 0.818393i $$0.305135\pi$$
$$468$$ −7.50394e6 −1.58371
$$469$$ 0 0
$$470$$ −5.24740e6 −1.09572
$$471$$ −85478.0 −0.0177542
$$472$$ 0 0
$$473$$ 4.89874e6 1.00677
$$474$$ 502120. 0.102651
$$475$$ 968750. 0.197005
$$476$$ 0 0
$$477$$ −6.27651e6 −1.26306
$$478$$ −6.12724e6 −1.22658
$$479$$ −1.98599e6 −0.395493 −0.197746 0.980253i $$-0.563362\pi$$
−0.197746 + 0.980253i $$0.563362\pi$$
$$480$$ 204800. 0.0405720
$$481$$ −1.06764e7 −2.10409
$$482$$ −9.68750e6 −1.89930
$$483$$ 0 0
$$484$$ 1.41306e6 0.274186
$$485$$ −208575. −0.0402632
$$486$$ 1.40941e6 0.270674
$$487$$ −1.06974e6 −0.204388 −0.102194 0.994764i $$-0.532586\pi$$
−0.102194 + 0.994764i $$0.532586\pi$$
$$488$$ 0 0
$$489$$ −193026. −0.0365043
$$490$$ 0 0
$$491$$ 4.59246e6 0.859689 0.429844 0.902903i $$-0.358568\pi$$
0.429844 + 0.902903i $$0.358568\pi$$
$$492$$ −55296.0 −0.0102987
$$493$$ 8.16045e6 1.51216
$$494$$ −1.20156e7 −2.21528
$$495$$ −2.74065e6 −0.502737
$$496$$ 1.22061e6 0.222778
$$497$$ 0 0
$$498$$ 189152. 0.0341773
$$499$$ 1.96066e6 0.352492 0.176246 0.984346i $$-0.443605\pi$$
0.176246 + 0.984346i $$0.443605\pi$$
$$500$$ −500000. −0.0894427
$$501$$ −157783. −0.0280844
$$502$$ 2.22610e6 0.394262
$$503$$ −3.51483e6 −0.619419 −0.309709 0.950831i $$-0.600232\pi$$
−0.309709 + 0.950831i $$0.600232\pi$$
$$504$$ 0 0
$$505$$ −296950. −0.0518149
$$506$$ −5.99410e6 −1.04075
$$507$$ −567668. −0.0980787
$$508$$ 1.05821e7 1.81934
$$509$$ 1.45211e6 0.248431 0.124215 0.992255i $$-0.460359\pi$$
0.124215 + 0.992255i $$0.460359\pi$$
$$510$$ 327400. 0.0557382
$$511$$ 0 0
$$512$$ −8.38861e6 −1.41421
$$513$$ 751750. 0.126119
$$514$$ −2.82398e6 −0.471470
$$515$$ −3.31098e6 −0.550095
$$516$$ 346048. 0.0572153
$$517$$ 1.18854e7 1.95563
$$518$$ 0 0
$$519$$ −265659. −0.0432918
$$520$$ 0 0
$$521$$ 4.24240e6 0.684726 0.342363 0.939568i $$-0.388773\pi$$
0.342363 + 0.939568i $$0.388773\pi$$
$$522$$ −9.65096e6 −1.55022
$$523$$ 7.56012e6 1.20858 0.604289 0.796765i $$-0.293457\pi$$
0.604289 + 0.796765i $$0.293457\pi$$
$$524$$ −1.40742e6 −0.223922
$$525$$ 0 0
$$526$$ 1.24648e7 1.96435
$$527$$ 1.95130e6 0.306054
$$528$$ −463872. −0.0724125
$$529$$ −3.70063e6 −0.574958
$$530$$ 5.18720e6 0.802127
$$531$$ −1.10836e6 −0.170586
$$532$$ 0 0
$$533$$ 1.67443e6 0.255299
$$534$$ 1.17840e6 0.178830
$$535$$ −3.42105e6 −0.516743
$$536$$ 0 0
$$537$$ −183660. −0.0274839
$$538$$ −9.75704e6 −1.45332
$$539$$ 0 0
$$540$$ −388000. −0.0572595
$$541$$ 1.24065e6 0.182245 0.0911224 0.995840i $$-0.470955\pi$$
0.0911224 + 0.995840i $$0.470955\pi$$
$$542$$ 3.24634e6 0.474674
$$543$$ −635048. −0.0924287
$$544$$ −1.34103e7 −1.94286
$$545$$ −2.73712e6 −0.394733
$$546$$ 0 0
$$547$$ −1.85057e6 −0.264446 −0.132223 0.991220i $$-0.542211\pi$$
−0.132223 + 0.991220i $$0.542211\pi$$
$$548$$ −3.19194e6 −0.454049
$$549$$ −3.02210e6 −0.427935
$$550$$ 2.26500e6 0.319272
$$551$$ −7.72675e6 −1.08422
$$552$$ 0 0
$$553$$ 0 0
$$554$$ −5.21954e6 −0.722533
$$555$$ −275450. −0.0379586
$$556$$ −8.28576e6 −1.13670
$$557$$ 7.77555e6 1.06192 0.530962 0.847396i $$-0.321831\pi$$
0.530962 + 0.847396i $$0.321831\pi$$
$$558$$ −2.30771e6 −0.313759
$$559$$ −1.04788e7 −1.41834
$$560$$ 0 0
$$561$$ −741561. −0.0994809
$$562$$ −950616. −0.126959
$$563$$ −8.37716e6 −1.11385 −0.556924 0.830564i $$-0.688018\pi$$
−0.556924 + 0.830564i $$0.688018\pi$$
$$564$$ 839584. 0.111139
$$565$$ 5.02335e6 0.662022
$$566$$ 1.19041e7 1.56191
$$567$$ 0 0
$$568$$ 0 0
$$569$$ −6.15591e6 −0.797098 −0.398549 0.917147i $$-0.630486\pi$$
−0.398549 + 0.917147i $$0.630486\pi$$
$$570$$ −310000. −0.0399645
$$571$$ 7.21513e6 0.926092 0.463046 0.886334i $$-0.346757\pi$$
0.463046 + 0.886334i $$0.346757\pi$$
$$572$$ −1.40466e7 −1.79507
$$573$$ 226613. 0.0288336
$$574$$ 0 0
$$575$$ −1.03375e6 −0.130390
$$576$$ 7.92986e6 0.995885
$$577$$ −1.36699e7 −1.70933 −0.854666 0.519177i $$-0.826238\pi$$
−0.854666 + 0.519177i $$0.826238\pi$$
$$578$$ −1.00793e7 −1.25490
$$579$$ −46476.0 −0.00576146
$$580$$ 3.98800e6 0.492249
$$581$$ 0 0
$$582$$ 66744.0 0.00816779
$$583$$ −1.17490e7 −1.43163
$$584$$ 0 0
$$585$$ 5.86245e6 0.708255
$$586$$ 1.51664e7 1.82448
$$587$$ 1.00686e7 1.20608 0.603040 0.797711i $$-0.293956\pi$$
0.603040 + 0.797711i $$0.293956\pi$$
$$588$$ 0 0
$$589$$ −1.84760e6 −0.219442
$$590$$ 916000. 0.108334
$$591$$ −204972. −0.0241394
$$592$$ 1.12824e7 1.32312
$$593$$ −9.80615e6 −1.14515 −0.572574 0.819853i $$-0.694055\pi$$
−0.572574 + 0.819853i $$0.694055\pi$$
$$594$$ 1.75764e6 0.204392
$$595$$ 0 0
$$596$$ −1.59498e7 −1.83924
$$597$$ −953020. −0.109438
$$598$$ 1.28218e7 1.46621
$$599$$ 8.26257e6 0.940911 0.470455 0.882424i $$-0.344090\pi$$
0.470455 + 0.882424i $$0.344090\pi$$
$$600$$ 0 0
$$601$$ 3.59492e6 0.405978 0.202989 0.979181i $$-0.434934\pi$$
0.202989 + 0.979181i $$0.434934\pi$$
$$602$$ 0 0
$$603$$ 3.83522e6 0.429533
$$604$$ −7.86570e6 −0.877293
$$605$$ −1.10395e6 −0.122620
$$606$$ 95024.0 0.0105112
$$607$$ 1.32969e7 1.46480 0.732401 0.680873i $$-0.238400\pi$$
0.732401 + 0.680873i $$0.238400\pi$$
$$608$$ 1.26976e7 1.39304
$$609$$ 0 0
$$610$$ 2.49760e6 0.271768
$$611$$ −2.54237e7 −2.75508
$$612$$ 1.26769e7 1.36816
$$613$$ 2.50327e6 0.269064 0.134532 0.990909i $$-0.457047\pi$$
0.134532 + 0.990909i $$0.457047\pi$$
$$614$$ −6.57482e6 −0.703823
$$615$$ 43200.0 0.00460570
$$616$$ 0 0
$$617$$ 1.88254e6 0.199082 0.0995409 0.995033i $$-0.468263\pi$$
0.0995409 + 0.995033i $$0.468263\pi$$
$$618$$ 1.05951e6 0.111592
$$619$$ −8.21487e6 −0.861736 −0.430868 0.902415i $$-0.641792\pi$$
−0.430868 + 0.902415i $$0.641792\pi$$
$$620$$ 953600. 0.0996293
$$621$$ −802190. −0.0834734
$$622$$ −1.67680e7 −1.73782
$$623$$ 0 0
$$624$$ 992256. 0.102015
$$625$$ 390625. 0.0400000
$$626$$ 3.15657e6 0.321943
$$627$$ 702150. 0.0713282
$$628$$ 2.73530e6 0.276761
$$629$$ 1.80365e7 1.81771
$$630$$ 0 0
$$631$$ 1.61155e7 1.61128 0.805638 0.592408i $$-0.201823\pi$$
0.805638 + 0.592408i $$0.201823\pi$$
$$632$$ 0 0
$$633$$ 223523. 0.0221724
$$634$$ −2.57138e6 −0.254064
$$635$$ −8.26730e6 −0.813635
$$636$$ −829952. −0.0813599
$$637$$ 0 0
$$638$$ −1.80656e7 −1.75712
$$639$$ −1.25337e7 −1.21430
$$640$$ 0 0
$$641$$ −8.50544e6 −0.817620 −0.408810 0.912619i $$-0.634056\pi$$
−0.408810 + 0.912619i $$0.634056\pi$$
$$642$$ 1.09474e6 0.104827
$$643$$ 1.32191e7 1.26088 0.630440 0.776238i $$-0.282874\pi$$
0.630440 + 0.776238i $$0.282874\pi$$
$$644$$ 0 0
$$645$$ −270350. −0.0255875
$$646$$ 2.02988e7 1.91377
$$647$$ −1.89115e6 −0.177609 −0.0888047 0.996049i $$-0.528305\pi$$
−0.0888047 + 0.996049i $$0.528305\pi$$
$$648$$ 0 0
$$649$$ −2.07474e6 −0.193353
$$650$$ −4.84500e6 −0.449791
$$651$$ 0 0
$$652$$ 6.17683e6 0.569045
$$653$$ 4.90587e6 0.450228 0.225114 0.974332i $$-0.427725\pi$$
0.225114 + 0.974332i $$0.427725\pi$$
$$654$$ 875880. 0.0800756
$$655$$ 1.09955e6 0.100141
$$656$$ −1.76947e6 −0.160540
$$657$$ 1.17273e6 0.105995
$$658$$ 0 0
$$659$$ 1.36367e7 1.22319 0.611597 0.791169i $$-0.290527\pi$$
0.611597 + 0.791169i $$0.290527\pi$$
$$660$$ −362400. −0.0323838
$$661$$ 2.22345e6 0.197935 0.0989677 0.995091i $$-0.468446\pi$$
0.0989677 + 0.995091i $$0.468446\pi$$
$$662$$ 1.78607e7 1.58399
$$663$$ 1.58625e6 0.140149
$$664$$ 0 0
$$665$$ 0 0
$$666$$ −2.13308e7 −1.86347
$$667$$ 8.24519e6 0.717606
$$668$$ 5.04906e6 0.437793
$$669$$ 1.01480e6 0.0876629
$$670$$ −3.16960e6 −0.272783
$$671$$ −5.65706e6 −0.485048
$$672$$ 0 0
$$673$$ 4.88484e6 0.415731 0.207865 0.978157i $$-0.433348\pi$$
0.207865 + 0.978157i $$0.433348\pi$$
$$674$$ 2.92525e7 2.48035
$$675$$ 303125. 0.0256072
$$676$$ 1.81654e7 1.52889
$$677$$ 1.98785e7 1.66691 0.833453 0.552590i $$-0.186360\pi$$
0.833453 + 0.552590i $$0.186360\pi$$
$$678$$ −1.60747e6 −0.134298
$$679$$ 0 0
$$680$$ 0 0
$$681$$ 999797. 0.0826122
$$682$$ −4.31981e6 −0.355634
$$683$$ 4.27870e6 0.350962 0.175481 0.984483i $$-0.443852\pi$$
0.175481 + 0.984483i $$0.443852\pi$$
$$684$$ −1.20032e7 −0.980972
$$685$$ 2.49370e6 0.203057
$$686$$ 0 0
$$687$$ −851120. −0.0688017
$$688$$ 1.10735e7 0.891898
$$689$$ 2.51320e7 2.01687
$$690$$ 330800. 0.0264510
$$691$$ −9.48925e6 −0.756026 −0.378013 0.925800i $$-0.623393\pi$$
−0.378013 + 0.925800i $$0.623393\pi$$
$$692$$ 8.50109e6 0.674853
$$693$$ 0 0
$$694$$ 1.51169e7 1.19142
$$695$$ 6.47325e6 0.508347
$$696$$ 0 0
$$697$$ −2.82874e6 −0.220552
$$698$$ −2.15463e7 −1.67392
$$699$$ −1.09270e6 −0.0845875
$$700$$ 0 0
$$701$$ −5.86385e6 −0.450700 −0.225350 0.974278i $$-0.572353\pi$$
−0.225350 + 0.974278i $$0.572353\pi$$
$$702$$ −3.75972e6 −0.287947
$$703$$ −1.70779e7 −1.30331
$$704$$ 1.48439e7 1.12880
$$705$$ −655925. −0.0497029
$$706$$ −1.25974e7 −0.951197
$$707$$ 0 0
$$708$$ −146560. −0.0109883
$$709$$ −2.66670e6 −0.199232 −0.0996161 0.995026i $$-0.531761\pi$$
−0.0996161 + 0.995026i $$0.531761\pi$$
$$710$$ 1.03584e7 0.771164
$$711$$ −1.51891e7 −1.12683
$$712$$ 0 0
$$713$$ 1.97157e6 0.145241
$$714$$ 0 0
$$715$$ 1.09739e7 0.802781
$$716$$ 5.87712e6 0.428432
$$717$$ −765905. −0.0556387
$$718$$ −3.24461e7 −2.34883
$$719$$ 4.46629e6 0.322199 0.161100 0.986938i $$-0.448496\pi$$
0.161100 + 0.986938i $$0.448496\pi$$
$$720$$ −6.19520e6 −0.445373
$$721$$ 0 0
$$722$$ 588792. 0.0420358
$$723$$ −1.21094e6 −0.0861541
$$724$$ 2.03215e7 1.44082
$$725$$ −3.11562e6 −0.220141
$$726$$ 353264. 0.0248747
$$727$$ 7.47757e6 0.524716 0.262358 0.964971i $$-0.415500\pi$$
0.262358 + 0.964971i $$0.415500\pi$$
$$728$$ 0 0
$$729$$ −1.39958e7 −0.975393
$$730$$ −969200. −0.0673141
$$731$$ 1.77025e7 1.22530
$$732$$ −399616. −0.0275655
$$733$$ 4.39751e6 0.302306 0.151153 0.988510i $$-0.451701\pi$$
0.151153 + 0.988510i $$0.451701\pi$$
$$734$$ −3.92503e7 −2.68907
$$735$$ 0 0
$$736$$ −1.35496e7 −0.922000
$$737$$ 7.17914e6 0.486860
$$738$$ 3.34541e6 0.226104
$$739$$ 2.84036e7 1.91321 0.956603 0.291395i $$-0.0941195\pi$$
0.956603 + 0.291395i $$0.0941195\pi$$
$$740$$ 8.81440e6 0.591716
$$741$$ −1.50195e6 −0.100487
$$742$$ 0 0
$$743$$ −1.96012e7 −1.30260 −0.651299 0.758821i $$-0.725776\pi$$
−0.651299 + 0.758821i $$0.725776\pi$$
$$744$$ 0 0
$$745$$ 1.24608e7 0.822533
$$746$$ 2.76269e7 1.81755
$$747$$ −5.72185e6 −0.375176
$$748$$ 2.37300e7 1.55075
$$749$$ 0 0
$$750$$ −125000. −0.00811441
$$751$$ −2.60344e6 −0.168441 −0.0842206 0.996447i $$-0.526840\pi$$
−0.0842206 + 0.996447i $$0.526840\pi$$
$$752$$ 2.68667e7 1.73249
$$753$$ 278262. 0.0178841
$$754$$ 3.86437e7 2.47543
$$755$$ 6.14508e6 0.392337
$$756$$ 0 0
$$757$$ −2.98869e7 −1.89558 −0.947789 0.318899i $$-0.896687\pi$$
−0.947789 + 0.318899i $$0.896687\pi$$
$$758$$ 3.38773e7 2.14159
$$759$$ −749262. −0.0472095
$$760$$ 0 0
$$761$$ −1.21470e7 −0.760338 −0.380169 0.924917i $$-0.624134\pi$$
−0.380169 + 0.924917i $$0.624134\pi$$
$$762$$ 2.64554e6 0.165054
$$763$$ 0 0
$$764$$ −7.25162e6 −0.449471
$$765$$ −9.90385e6 −0.611858
$$766$$ −1.49168e7 −0.918554
$$767$$ 4.43802e6 0.272396
$$768$$ −1.04858e6 −0.0641500
$$769$$ −4.53845e6 −0.276753 −0.138376 0.990380i $$-0.544188\pi$$
−0.138376 + 0.990380i $$0.544188\pi$$
$$770$$ 0 0
$$771$$ −352998. −0.0213863
$$772$$ 1.48723e6 0.0898122
$$773$$ −1.93330e7 −1.16372 −0.581861 0.813288i $$-0.697675\pi$$
−0.581861 + 0.813288i $$0.697675\pi$$
$$774$$ −2.09359e7 −1.25614
$$775$$ −745000. −0.0445556
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 3.85201e7 2.28160
$$779$$ 2.67840e6 0.158136
$$780$$ 775200. 0.0456223
$$781$$ −2.34618e7 −1.37636
$$782$$ −2.16608e7 −1.26665
$$783$$ −2.41772e6 −0.140930
$$784$$ 0 0
$$785$$ −2.13695e6 −0.123771
$$786$$ −351856. −0.0203146
$$787$$ −1.66392e7 −0.957627 −0.478814 0.877917i $$-0.658933\pi$$
−0.478814 + 0.877917i $$0.658933\pi$$
$$788$$ 6.55910e6 0.376295
$$789$$ 1.55809e6 0.0891048
$$790$$ 1.25530e7 0.715616
$$791$$ 0 0
$$792$$ 0 0
$$793$$ 1.21009e7 0.683335
$$794$$ 9.71006e6 0.546601
$$795$$ 648400. 0.0363852
$$796$$ 3.04966e7 1.70596
$$797$$ −1.80409e7 −1.00603 −0.503017 0.864276i $$-0.667777\pi$$
−0.503017 + 0.864276i $$0.667777\pi$$
$$798$$ 0 0
$$799$$ 4.29500e7 2.38010
$$800$$ 5.12000e6 0.282843
$$801$$ −3.56466e7 −1.96307
$$802$$ −4.72353e7 −2.59317
$$803$$ 2.19524e6 0.120141
$$804$$ 507136. 0.0276684
$$805$$ 0 0
$$806$$ 9.24038e6 0.501017
$$807$$ −1.21963e6 −0.0659241
$$808$$ 0 0
$$809$$ 2.33891e7 1.25644 0.628220 0.778036i $$-0.283784\pi$$
0.628220 + 0.778036i $$0.283784\pi$$
$$810$$ 1.16642e7 0.624658
$$811$$ 2.29037e7 1.22279 0.611397 0.791324i $$-0.290608\pi$$
0.611397 + 0.791324i $$0.290608\pi$$
$$812$$ 0 0
$$813$$ 405792. 0.0215316
$$814$$ −3.99292e7 −2.11218
$$815$$ −4.82565e6 −0.254485
$$816$$ −1.67629e6 −0.0881299
$$817$$ −1.67617e7 −0.878543
$$818$$ 3.87431e7 2.02447
$$819$$ 0 0
$$820$$ −1.38240e6 −0.0717958
$$821$$ −1.80745e7 −0.935853 −0.467926 0.883767i $$-0.654999\pi$$
−0.467926 + 0.883767i $$0.654999\pi$$
$$822$$ −797984. −0.0411922
$$823$$ 1.17989e7 0.607216 0.303608 0.952797i $$-0.401809\pi$$
0.303608 + 0.952797i $$0.401809\pi$$
$$824$$ 0 0
$$825$$ 283125. 0.0144825
$$826$$ 0 0
$$827$$ 2.57650e6 0.130999 0.0654993 0.997853i $$-0.479136\pi$$
0.0654993 + 0.997853i $$0.479136\pi$$
$$828$$ 1.28086e7 0.649270
$$829$$ 3.84340e7 1.94236 0.971178 0.238356i $$-0.0766084\pi$$
0.971178 + 0.238356i $$0.0766084\pi$$
$$830$$ 4.72880e6 0.238263
$$831$$ −652442. −0.0327747
$$832$$ −3.17522e7 −1.59025
$$833$$ 0 0
$$834$$ −2.07144e6 −0.103123
$$835$$ −3.94458e6 −0.195787
$$836$$ −2.24688e7 −1.11190
$$837$$ −578120. −0.0285236
$$838$$ 2.16288e6 0.106395
$$839$$ 1.24222e7 0.609247 0.304623 0.952473i $$-0.401469\pi$$
0.304623 + 0.952473i $$0.401469\pi$$
$$840$$ 0 0
$$841$$ 4.33908e6 0.211547
$$842$$ −2.50918e7 −1.21970
$$843$$ −118827. −0.00575899
$$844$$ −7.15274e6 −0.345634
$$845$$ −1.41917e7 −0.683743
$$846$$ −5.07948e7 −2.44002
$$847$$ 0 0
$$848$$ −2.65585e7 −1.26827
$$849$$ 1.48801e6 0.0708495
$$850$$ 8.18500e6 0.388572
$$851$$ 1.82238e7 0.862610
$$852$$ −1.65734e6 −0.0782193
$$853$$ −7.92067e6 −0.372726 −0.186363 0.982481i $$-0.559670\pi$$
−0.186363 + 0.982481i $$0.559670\pi$$
$$854$$ 0 0
$$855$$ 9.37750e6 0.438704
$$856$$ 0 0
$$857$$ −1.48983e7 −0.692924 −0.346462 0.938064i $$-0.612617\pi$$
−0.346462 + 0.938064i $$0.612617\pi$$
$$858$$ −3.51166e6 −0.162852
$$859$$ 1.38740e7 0.641534 0.320767 0.947158i $$-0.396059\pi$$
0.320767 + 0.947158i $$0.396059\pi$$
$$860$$ 8.65120e6 0.398869
$$861$$ 0 0
$$862$$ 1.50163e7 0.688325
$$863$$ 1.25500e7 0.573610 0.286805 0.957989i $$-0.407407\pi$$
0.286805 + 0.957989i $$0.407407\pi$$
$$864$$ 3.97312e6 0.181070
$$865$$ −6.64147e6 −0.301804
$$866$$ 2.56285e7 1.16126
$$867$$ −1.25991e6 −0.0569236
$$868$$ 0 0
$$869$$ −2.84325e7 −1.27722
$$870$$ 997000. 0.0446578
$$871$$ −1.53567e7 −0.685887
$$872$$ 0 0
$$873$$ −2.01901e6 −0.0896607
$$874$$ 2.05096e7 0.908194
$$875$$ 0 0
$$876$$ 155072. 0.00682768
$$877$$ −2.86002e7 −1.25565 −0.627827 0.778353i $$-0.716056\pi$$
−0.627827 + 0.778353i $$0.716056\pi$$
$$878$$ −5.01767e7 −2.19668
$$879$$ 1.89580e6 0.0827600
$$880$$ −1.15968e7 −0.504814
$$881$$ −4.09608e7 −1.77799 −0.888993 0.457922i $$-0.848594\pi$$
−0.888993 + 0.457922i $$0.848594\pi$$
$$882$$ 0 0
$$883$$ 1.30504e7 0.563279 0.281639 0.959520i $$-0.409122\pi$$
0.281639 + 0.959520i $$0.409122\pi$$
$$884$$ −5.07601e7 −2.18470
$$885$$ 114500. 0.00491414
$$886$$ −5.79989e6 −0.248219
$$887$$ −2.53595e7 −1.08226 −0.541129 0.840939i $$-0.682003\pi$$
−0.541129 + 0.840939i $$0.682003\pi$$
$$888$$ 0 0
$$889$$ 0 0
$$890$$ 2.94600e7 1.24669
$$891$$ −2.64194e7 −1.11488
$$892$$ −3.24736e7 −1.36653
$$893$$ −4.06674e7 −1.70654
$$894$$ −3.98744e6 −0.166859
$$895$$ −4.59150e6 −0.191601
$$896$$ 0 0
$$897$$ 1.60273e6 0.0665087
$$898$$ 7.00788e6 0.289999
$$899$$ 5.94212e6 0.245212
$$900$$ −4.84000e6 −0.199177
$$901$$ −4.24572e7 −1.74237
$$902$$ 6.26227e6 0.256281
$$903$$ 0 0
$$904$$ 0 0
$$905$$ −1.58762e7 −0.644355
$$906$$ −1.96642e6 −0.0795897
$$907$$ 1.98595e7 0.801585 0.400793 0.916169i $$-0.368735\pi$$
0.400793 + 0.916169i $$0.368735\pi$$
$$908$$ −3.19935e7 −1.28780
$$909$$ −2.87448e6 −0.115385
$$910$$ 0 0
$$911$$ −1.99344e7 −0.795808 −0.397904 0.917427i $$-0.630262\pi$$
−0.397904 + 0.917427i $$0.630262\pi$$
$$912$$ 1.58720e6 0.0631894
$$913$$ −1.07107e7 −0.425248
$$914$$ 6.66134e6 0.263752
$$915$$ 312200. 0.0123276
$$916$$ 2.72358e7 1.07251
$$917$$ 0 0
$$918$$ 6.35156e6 0.248756
$$919$$ −1.10695e7 −0.432355 −0.216178 0.976354i $$-0.569359\pi$$
−0.216178 + 0.976354i $$0.569359\pi$$
$$920$$ 0 0
$$921$$ −821853. −0.0319260
$$922$$ 4.73692e7 1.83514
$$923$$ 5.01864e7 1.93902
$$924$$ 0 0
$$925$$ −6.88625e6 −0.264624
$$926$$ −5.46221e6 −0.209334
$$927$$ −3.20502e7 −1.22499
$$928$$ −4.08371e7 −1.55663
$$929$$ 3.25682e7 1.23810 0.619048 0.785353i $$-0.287519\pi$$
0.619048 + 0.785353i $$0.287519\pi$$
$$930$$ 238400. 0.00903855
$$931$$ 0 0
$$932$$ 3.49663e7 1.31859
$$933$$ −2.09600e6 −0.0788291
$$934$$ −4.33334e7 −1.62538
$$935$$ −1.85390e7 −0.693518
$$936$$ 0 0
$$937$$ −3.15690e7 −1.17466 −0.587329 0.809348i $$-0.699821\pi$$
−0.587329 + 0.809348i $$0.699821\pi$$
$$938$$ 0 0
$$939$$ 394571. 0.0146036
$$940$$ 2.09896e7 0.774791
$$941$$ 3.67997e7 1.35479 0.677393 0.735622i $$-0.263110\pi$$
0.677393 + 0.735622i $$0.263110\pi$$
$$942$$ 683824. 0.0251083
$$943$$ −2.85811e6 −0.104665
$$944$$ −4.68992e6 −0.171291
$$945$$ 0 0
$$946$$ −3.91899e7 −1.42379
$$947$$ 1.88453e7 0.682853 0.341426 0.939909i $$-0.389090\pi$$
0.341426 + 0.939909i $$0.389090\pi$$
$$948$$ −2.00848e6 −0.0725850
$$949$$ −4.69577e6 −0.169255
$$950$$ −7.75000e6 −0.278607
$$951$$ −321422. −0.0115246
$$952$$ 0 0
$$953$$ −1.25120e7 −0.446265 −0.223133 0.974788i $$-0.571628\pi$$
−0.223133 + 0.974788i $$0.571628\pi$$
$$954$$ 5.02121e7 1.78623
$$955$$ 5.66532e6 0.201009
$$956$$ 2.45090e7 0.867322
$$957$$ −2.25820e6 −0.0797046
$$958$$ 1.58879e7 0.559311
$$959$$ 0 0
$$960$$ −819200. −0.0286888
$$961$$ −2.72083e7 −0.950370
$$962$$ 8.54115e7 2.97563
$$963$$ −3.31158e7 −1.15072
$$964$$ 3.87500e7 1.34301
$$965$$ −1.16190e6 −0.0401652
$$966$$ 0 0
$$967$$ −3.42344e7 −1.17733 −0.588663 0.808379i $$-0.700346\pi$$
−0.588663 + 0.808379i $$0.700346\pi$$
$$968$$ 0 0
$$969$$ 2.53735e6 0.0868102
$$970$$ 1.66860e6 0.0569407
$$971$$ 2.62027e7 0.891864 0.445932 0.895067i $$-0.352872\pi$$
0.445932 + 0.895067i $$0.352872\pi$$
$$972$$ −5.63763e6 −0.191395
$$973$$ 0 0
$$974$$ 8.55790e6 0.289048
$$975$$ −605625. −0.0204029
$$976$$ −1.27877e7 −0.429703
$$977$$ −8.01114e6 −0.268508 −0.134254 0.990947i $$-0.542864\pi$$
−0.134254 + 0.990947i $$0.542864\pi$$
$$978$$ 1.54421e6 0.0516248
$$979$$ −6.67269e7 −2.22507
$$980$$ 0 0
$$981$$ −2.64954e7 −0.879017
$$982$$ −3.67397e7 −1.21578
$$983$$ 4.60126e7 1.51877 0.759387 0.650639i $$-0.225499\pi$$
0.759387 + 0.650639i $$0.225499\pi$$
$$984$$ 0 0
$$985$$ −5.12430e6 −0.168284
$$986$$ −6.52836e7 −2.13851
$$987$$ 0 0
$$988$$ 4.80624e7 1.56644
$$989$$ 1.78864e7 0.581475
$$990$$ 2.19252e7 0.710977
$$991$$ −3.75828e7 −1.21564 −0.607821 0.794074i $$-0.707956\pi$$
−0.607821 + 0.794074i $$0.707956\pi$$
$$992$$ −9.76486e6 −0.315055
$$993$$ 2.23259e6 0.0718514
$$994$$ 0 0
$$995$$ −2.38255e7 −0.762929
$$996$$ −756608. −0.0241670
$$997$$ −2.22066e7 −0.707529 −0.353765 0.935334i $$-0.615099\pi$$
−0.353765 + 0.935334i $$0.615099\pi$$
$$998$$ −1.56852e7 −0.498500
$$999$$ −5.34373e6 −0.169407
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 245.6.a.a.1.1 1
7.6 odd 2 35.6.a.a.1.1 1
21.20 even 2 315.6.a.a.1.1 1
28.27 even 2 560.6.a.c.1.1 1
35.13 even 4 175.6.b.b.99.2 2
35.27 even 4 175.6.b.b.99.1 2
35.34 odd 2 175.6.a.a.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
35.6.a.a.1.1 1 7.6 odd 2
175.6.a.a.1.1 1 35.34 odd 2
175.6.b.b.99.1 2 35.27 even 4
175.6.b.b.99.2 2 35.13 even 4
245.6.a.a.1.1 1 1.1 even 1 trivial
315.6.a.a.1.1 1 21.20 even 2
560.6.a.c.1.1 1 28.27 even 2