Properties

 Label 1950.2.e.n.1249.1 Level $1950$ Weight $2$ Character 1950.1249 Analytic conductor $15.571$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$1950 = 2 \cdot 3 \cdot 5^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1950.e (of order $$2$$, degree $$1$$, not minimal)

Newform invariants

 Self dual: no Analytic conductor: $$15.5708283941$$ 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: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

 Embedding label 1249.1 Root $$1.00000i$$ of defining polynomial Character $$\chi$$ $$=$$ 1950.1249 Dual form 1950.2.e.n.1249.2

$q$-expansion

 $$f(q)$$ $$=$$ $$q-1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} +4.00000i q^{7} +1.00000i q^{8} -1.00000 q^{9} +O(q^{10})$$ $$q-1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} +4.00000i q^{7} +1.00000i q^{8} -1.00000 q^{9} +4.00000 q^{11} -1.00000i q^{12} +1.00000i q^{13} +4.00000 q^{14} +1.00000 q^{16} +4.00000i q^{17} +1.00000i q^{18} -7.00000 q^{19} -4.00000 q^{21} -4.00000i q^{22} +4.00000i q^{23} -1.00000 q^{24} +1.00000 q^{26} -1.00000i q^{27} -4.00000i q^{28} -5.00000 q^{29} +4.00000 q^{31} -1.00000i q^{32} +4.00000i q^{33} +4.00000 q^{34} +1.00000 q^{36} -9.00000i q^{37} +7.00000i q^{38} -1.00000 q^{39} -5.00000 q^{41} +4.00000i q^{42} -10.0000i q^{43} -4.00000 q^{44} +4.00000 q^{46} -3.00000i q^{47} +1.00000i q^{48} -9.00000 q^{49} -4.00000 q^{51} -1.00000i q^{52} +9.00000i q^{53} -1.00000 q^{54} -4.00000 q^{56} -7.00000i q^{57} +5.00000i q^{58} +6.00000 q^{59} +4.00000 q^{61} -4.00000i q^{62} -4.00000i q^{63} -1.00000 q^{64} +4.00000 q^{66} +7.00000i q^{67} -4.00000i q^{68} -4.00000 q^{69} -15.0000 q^{71} -1.00000i q^{72} +12.0000i q^{73} -9.00000 q^{74} +7.00000 q^{76} +16.0000i q^{77} +1.00000i q^{78} -7.00000 q^{79} +1.00000 q^{81} +5.00000i q^{82} +6.00000i q^{83} +4.00000 q^{84} -10.0000 q^{86} -5.00000i q^{87} +4.00000i q^{88} -14.0000 q^{89} -4.00000 q^{91} -4.00000i q^{92} +4.00000i q^{93} -3.00000 q^{94} +1.00000 q^{96} +16.0000i q^{97} +9.00000i q^{98} -4.00000 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{4} + 2q^{6} - 2q^{9} + O(q^{10})$$ $$2q - 2q^{4} + 2q^{6} - 2q^{9} + 8q^{11} + 8q^{14} + 2q^{16} - 14q^{19} - 8q^{21} - 2q^{24} + 2q^{26} - 10q^{29} + 8q^{31} + 8q^{34} + 2q^{36} - 2q^{39} - 10q^{41} - 8q^{44} + 8q^{46} - 18q^{49} - 8q^{51} - 2q^{54} - 8q^{56} + 12q^{59} + 8q^{61} - 2q^{64} + 8q^{66} - 8q^{69} - 30q^{71} - 18q^{74} + 14q^{76} - 14q^{79} + 2q^{81} + 8q^{84} - 20q^{86} - 28q^{89} - 8q^{91} - 6q^{94} + 2q^{96} - 8q^{99} + O(q^{100})$$

Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1950\mathbb{Z}\right)^\times$$.

 $$n$$ $$301$$ $$1301$$ $$1327$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1$$

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
$$7$$ 4.00000i 1.51186i 0.654654 + 0.755929i $$0.272814\pi$$
−0.654654 + 0.755929i $$0.727186\pi$$
$$8$$ 1.00000i 0.353553i
$$9$$ −1.00000 −0.333333
$$10$$ 0 0
$$11$$ 4.00000 1.20605 0.603023 0.797724i $$-0.293963\pi$$
0.603023 + 0.797724i $$0.293963\pi$$
$$12$$ − 1.00000i − 0.288675i
$$13$$ 1.00000i 0.277350i
$$14$$ 4.00000 1.06904
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ 4.00000i 0.970143i 0.874475 + 0.485071i $$0.161206\pi$$
−0.874475 + 0.485071i $$0.838794\pi$$
$$18$$ 1.00000i 0.235702i
$$19$$ −7.00000 −1.60591 −0.802955 0.596040i $$-0.796740\pi$$
−0.802955 + 0.596040i $$0.796740\pi$$
$$20$$ 0 0
$$21$$ −4.00000 −0.872872
$$22$$ − 4.00000i − 0.852803i
$$23$$ 4.00000i 0.834058i 0.908893 + 0.417029i $$0.136929\pi$$
−0.908893 + 0.417029i $$0.863071\pi$$
$$24$$ −1.00000 −0.204124
$$25$$ 0 0
$$26$$ 1.00000 0.196116
$$27$$ − 1.00000i − 0.192450i
$$28$$ − 4.00000i − 0.755929i
$$29$$ −5.00000 −0.928477 −0.464238 0.885710i $$-0.653672\pi$$
−0.464238 + 0.885710i $$0.653672\pi$$
$$30$$ 0 0
$$31$$ 4.00000 0.718421 0.359211 0.933257i $$-0.383046\pi$$
0.359211 + 0.933257i $$0.383046\pi$$
$$32$$ − 1.00000i − 0.176777i
$$33$$ 4.00000i 0.696311i
$$34$$ 4.00000 0.685994
$$35$$ 0 0
$$36$$ 1.00000 0.166667
$$37$$ − 9.00000i − 1.47959i −0.672832 0.739795i $$-0.734922\pi$$
0.672832 0.739795i $$-0.265078\pi$$
$$38$$ 7.00000i 1.13555i
$$39$$ −1.00000 −0.160128
$$40$$ 0 0
$$41$$ −5.00000 −0.780869 −0.390434 0.920631i $$-0.627675\pi$$
−0.390434 + 0.920631i $$0.627675\pi$$
$$42$$ 4.00000i 0.617213i
$$43$$ − 10.0000i − 1.52499i −0.646997 0.762493i $$-0.723975\pi$$
0.646997 0.762493i $$-0.276025\pi$$
$$44$$ −4.00000 −0.603023
$$45$$ 0 0
$$46$$ 4.00000 0.589768
$$47$$ − 3.00000i − 0.437595i −0.975770 0.218797i $$-0.929787\pi$$
0.975770 0.218797i $$-0.0702134\pi$$
$$48$$ 1.00000i 0.144338i
$$49$$ −9.00000 −1.28571
$$50$$ 0 0
$$51$$ −4.00000 −0.560112
$$52$$ − 1.00000i − 0.138675i
$$53$$ 9.00000i 1.23625i 0.786082 + 0.618123i $$0.212106\pi$$
−0.786082 + 0.618123i $$0.787894\pi$$
$$54$$ −1.00000 −0.136083
$$55$$ 0 0
$$56$$ −4.00000 −0.534522
$$57$$ − 7.00000i − 0.927173i
$$58$$ 5.00000i 0.656532i
$$59$$ 6.00000 0.781133 0.390567 0.920575i $$-0.372279\pi$$
0.390567 + 0.920575i $$0.372279\pi$$
$$60$$ 0 0
$$61$$ 4.00000 0.512148 0.256074 0.966657i $$-0.417571\pi$$
0.256074 + 0.966657i $$0.417571\pi$$
$$62$$ − 4.00000i − 0.508001i
$$63$$ − 4.00000i − 0.503953i
$$64$$ −1.00000 −0.125000
$$65$$ 0 0
$$66$$ 4.00000 0.492366
$$67$$ 7.00000i 0.855186i 0.903971 + 0.427593i $$0.140638\pi$$
−0.903971 + 0.427593i $$0.859362\pi$$
$$68$$ − 4.00000i − 0.485071i
$$69$$ −4.00000 −0.481543
$$70$$ 0 0
$$71$$ −15.0000 −1.78017 −0.890086 0.455792i $$-0.849356\pi$$
−0.890086 + 0.455792i $$0.849356\pi$$
$$72$$ − 1.00000i − 0.117851i
$$73$$ 12.0000i 1.40449i 0.711934 + 0.702247i $$0.247820\pi$$
−0.711934 + 0.702247i $$0.752180\pi$$
$$74$$ −9.00000 −1.04623
$$75$$ 0 0
$$76$$ 7.00000 0.802955
$$77$$ 16.0000i 1.82337i
$$78$$ 1.00000i 0.113228i
$$79$$ −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
$$82$$ 5.00000i 0.552158i
$$83$$ 6.00000i 0.658586i 0.944228 + 0.329293i $$0.106810\pi$$
−0.944228 + 0.329293i $$0.893190\pi$$
$$84$$ 4.00000 0.436436
$$85$$ 0 0
$$86$$ −10.0000 −1.07833
$$87$$ − 5.00000i − 0.536056i
$$88$$ 4.00000i 0.426401i
$$89$$ −14.0000 −1.48400 −0.741999 0.670402i $$-0.766122\pi$$
−0.741999 + 0.670402i $$0.766122\pi$$
$$90$$ 0 0
$$91$$ −4.00000 −0.419314
$$92$$ − 4.00000i − 0.417029i
$$93$$ 4.00000i 0.414781i
$$94$$ −3.00000 −0.309426
$$95$$ 0 0
$$96$$ 1.00000 0.102062
$$97$$ 16.0000i 1.62455i 0.583272 + 0.812277i $$0.301772\pi$$
−0.583272 + 0.812277i $$0.698228\pi$$
$$98$$ 9.00000i 0.909137i
$$99$$ −4.00000 −0.402015
$$100$$ 0 0
$$101$$ −10.0000 −0.995037 −0.497519 0.867453i $$-0.665755\pi$$
−0.497519 + 0.867453i $$0.665755\pi$$
$$102$$ 4.00000i 0.396059i
$$103$$ 8.00000i 0.788263i 0.919054 + 0.394132i $$0.128955\pi$$
−0.919054 + 0.394132i $$0.871045\pi$$
$$104$$ −1.00000 −0.0980581
$$105$$ 0 0
$$106$$ 9.00000 0.874157
$$107$$ − 5.00000i − 0.483368i −0.970355 0.241684i $$-0.922300\pi$$
0.970355 0.241684i $$-0.0776998\pi$$
$$108$$ 1.00000i 0.0962250i
$$109$$ 11.0000 1.05361 0.526804 0.849987i $$-0.323390\pi$$
0.526804 + 0.849987i $$0.323390\pi$$
$$110$$ 0 0
$$111$$ 9.00000 0.854242
$$112$$ 4.00000i 0.377964i
$$113$$ − 14.0000i − 1.31701i −0.752577 0.658505i $$-0.771189\pi$$
0.752577 0.658505i $$-0.228811\pi$$
$$114$$ −7.00000 −0.655610
$$115$$ 0 0
$$116$$ 5.00000 0.464238
$$117$$ − 1.00000i − 0.0924500i
$$118$$ − 6.00000i − 0.552345i
$$119$$ −16.0000 −1.46672
$$120$$ 0 0
$$121$$ 5.00000 0.454545
$$122$$ − 4.00000i − 0.362143i
$$123$$ − 5.00000i − 0.450835i
$$124$$ −4.00000 −0.359211
$$125$$ 0 0
$$126$$ −4.00000 −0.356348
$$127$$ 5.00000i 0.443678i 0.975083 + 0.221839i $$0.0712060\pi$$
−0.975083 + 0.221839i $$0.928794\pi$$
$$128$$ 1.00000i 0.0883883i
$$129$$ 10.0000 0.880451
$$130$$ 0 0
$$131$$ 17.0000 1.48530 0.742648 0.669681i $$-0.233569\pi$$
0.742648 + 0.669681i $$0.233569\pi$$
$$132$$ − 4.00000i − 0.348155i
$$133$$ − 28.0000i − 2.42791i
$$134$$ 7.00000 0.604708
$$135$$ 0 0
$$136$$ −4.00000 −0.342997
$$137$$ 19.0000i 1.62328i 0.584158 + 0.811640i $$0.301425\pi$$
−0.584158 + 0.811640i $$0.698575\pi$$
$$138$$ 4.00000i 0.340503i
$$139$$ 4.00000 0.339276 0.169638 0.985506i $$-0.445740\pi$$
0.169638 + 0.985506i $$0.445740\pi$$
$$140$$ 0 0
$$141$$ 3.00000 0.252646
$$142$$ 15.0000i 1.25877i
$$143$$ 4.00000i 0.334497i
$$144$$ −1.00000 −0.0833333
$$145$$ 0 0
$$146$$ 12.0000 0.993127
$$147$$ − 9.00000i − 0.742307i
$$148$$ 9.00000i 0.739795i
$$149$$ 4.00000 0.327693 0.163846 0.986486i $$-0.447610\pi$$
0.163846 + 0.986486i $$0.447610\pi$$
$$150$$ 0 0
$$151$$ −4.00000 −0.325515 −0.162758 0.986666i $$-0.552039\pi$$
−0.162758 + 0.986666i $$0.552039\pi$$
$$152$$ − 7.00000i − 0.567775i
$$153$$ − 4.00000i − 0.323381i
$$154$$ 16.0000 1.28932
$$155$$ 0 0
$$156$$ 1.00000 0.0800641
$$157$$ 14.0000i 1.11732i 0.829396 + 0.558661i $$0.188685\pi$$
−0.829396 + 0.558661i $$0.811315\pi$$
$$158$$ 7.00000i 0.556890i
$$159$$ −9.00000 −0.713746
$$160$$ 0 0
$$161$$ −16.0000 −1.26098
$$162$$ − 1.00000i − 0.0785674i
$$163$$ − 24.0000i − 1.87983i −0.341415 0.939913i $$-0.610906\pi$$
0.341415 0.939913i $$-0.389094\pi$$
$$164$$ 5.00000 0.390434
$$165$$ 0 0
$$166$$ 6.00000 0.465690
$$167$$ − 9.00000i − 0.696441i −0.937413 0.348220i $$-0.886786\pi$$
0.937413 0.348220i $$-0.113214\pi$$
$$168$$ − 4.00000i − 0.308607i
$$169$$ −1.00000 −0.0769231
$$170$$ 0 0
$$171$$ 7.00000 0.535303
$$172$$ 10.0000i 0.762493i
$$173$$ 13.0000i 0.988372i 0.869356 + 0.494186i $$0.164534\pi$$
−0.869356 + 0.494186i $$0.835466\pi$$
$$174$$ −5.00000 −0.379049
$$175$$ 0 0
$$176$$ 4.00000 0.301511
$$177$$ 6.00000i 0.450988i
$$178$$ 14.0000i 1.04934i
$$179$$ 4.00000 0.298974 0.149487 0.988764i $$-0.452238\pi$$
0.149487 + 0.988764i $$0.452238\pi$$
$$180$$ 0 0
$$181$$ −4.00000 −0.297318 −0.148659 0.988889i $$-0.547496\pi$$
−0.148659 + 0.988889i $$0.547496\pi$$
$$182$$ 4.00000i 0.296500i
$$183$$ 4.00000i 0.295689i
$$184$$ −4.00000 −0.294884
$$185$$ 0 0
$$186$$ 4.00000 0.293294
$$187$$ 16.0000i 1.17004i
$$188$$ 3.00000i 0.218797i
$$189$$ 4.00000 0.290957
$$190$$ 0 0
$$191$$ 10.0000 0.723575 0.361787 0.932261i $$-0.382167\pi$$
0.361787 + 0.932261i $$0.382167\pi$$
$$192$$ − 1.00000i − 0.0721688i
$$193$$ − 12.0000i − 0.863779i −0.901927 0.431889i $$-0.857847\pi$$
0.901927 0.431889i $$-0.142153\pi$$
$$194$$ 16.0000 1.14873
$$195$$ 0 0
$$196$$ 9.00000 0.642857
$$197$$ 16.0000i 1.13995i 0.821661 + 0.569976i $$0.193048\pi$$
−0.821661 + 0.569976i $$0.806952\pi$$
$$198$$ 4.00000i 0.284268i
$$199$$ 3.00000 0.212664 0.106332 0.994331i $$-0.466089\pi$$
0.106332 + 0.994331i $$0.466089\pi$$
$$200$$ 0 0
$$201$$ −7.00000 −0.493742
$$202$$ 10.0000i 0.703598i
$$203$$ − 20.0000i − 1.40372i
$$204$$ 4.00000 0.280056
$$205$$ 0 0
$$206$$ 8.00000 0.557386
$$207$$ − 4.00000i − 0.278019i
$$208$$ 1.00000i 0.0693375i
$$209$$ −28.0000 −1.93680
$$210$$ 0 0
$$211$$ −16.0000 −1.10149 −0.550743 0.834675i $$-0.685655\pi$$
−0.550743 + 0.834675i $$0.685655\pi$$
$$212$$ − 9.00000i − 0.618123i
$$213$$ − 15.0000i − 1.02778i
$$214$$ −5.00000 −0.341793
$$215$$ 0 0
$$216$$ 1.00000 0.0680414
$$217$$ 16.0000i 1.08615i
$$218$$ − 11.0000i − 0.745014i
$$219$$ −12.0000 −0.810885
$$220$$ 0 0
$$221$$ −4.00000 −0.269069
$$222$$ − 9.00000i − 0.604040i
$$223$$ − 2.00000i − 0.133930i −0.997755 0.0669650i $$-0.978668\pi$$
0.997755 0.0669650i $$-0.0213316\pi$$
$$224$$ 4.00000 0.267261
$$225$$ 0 0
$$226$$ −14.0000 −0.931266
$$227$$ − 6.00000i − 0.398234i −0.979976 0.199117i $$-0.936193\pi$$
0.979976 0.199117i $$-0.0638074\pi$$
$$228$$ 7.00000i 0.463586i
$$229$$ 17.0000 1.12339 0.561696 0.827344i $$-0.310149\pi$$
0.561696 + 0.827344i $$0.310149\pi$$
$$230$$ 0 0
$$231$$ −16.0000 −1.05272
$$232$$ − 5.00000i − 0.328266i
$$233$$ 8.00000i 0.524097i 0.965055 + 0.262049i $$0.0843981\pi$$
−0.965055 + 0.262049i $$0.915602\pi$$
$$234$$ −1.00000 −0.0653720
$$235$$ 0 0
$$236$$ −6.00000 −0.390567
$$237$$ − 7.00000i − 0.454699i
$$238$$ 16.0000i 1.03713i
$$239$$ −16.0000 −1.03495 −0.517477 0.855697i $$-0.673129\pi$$
−0.517477 + 0.855697i $$0.673129\pi$$
$$240$$ 0 0
$$241$$ 8.00000 0.515325 0.257663 0.966235i $$-0.417048\pi$$
0.257663 + 0.966235i $$0.417048\pi$$
$$242$$ − 5.00000i − 0.321412i
$$243$$ 1.00000i 0.0641500i
$$244$$ −4.00000 −0.256074
$$245$$ 0 0
$$246$$ −5.00000 −0.318788
$$247$$ − 7.00000i − 0.445399i
$$248$$ 4.00000i 0.254000i
$$249$$ −6.00000 −0.380235
$$250$$ 0 0
$$251$$ −5.00000 −0.315597 −0.157799 0.987471i $$-0.550440\pi$$
−0.157799 + 0.987471i $$0.550440\pi$$
$$252$$ 4.00000i 0.251976i
$$253$$ 16.0000i 1.00591i
$$254$$ 5.00000 0.313728
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ − 8.00000i − 0.499026i −0.968371 0.249513i $$-0.919729\pi$$
0.968371 0.249513i $$-0.0802706\pi$$
$$258$$ − 10.0000i − 0.622573i
$$259$$ 36.0000 2.23693
$$260$$ 0 0
$$261$$ 5.00000 0.309492
$$262$$ − 17.0000i − 1.05026i
$$263$$ 30.0000i 1.84988i 0.380114 + 0.924940i $$0.375885\pi$$
−0.380114 + 0.924940i $$0.624115\pi$$
$$264$$ −4.00000 −0.246183
$$265$$ 0 0
$$266$$ −28.0000 −1.71679
$$267$$ − 14.0000i − 0.856786i
$$268$$ − 7.00000i − 0.427593i
$$269$$ −5.00000 −0.304855 −0.152428 0.988315i $$-0.548709\pi$$
−0.152428 + 0.988315i $$0.548709\pi$$
$$270$$ 0 0
$$271$$ 32.0000 1.94386 0.971931 0.235267i $$-0.0755965\pi$$
0.971931 + 0.235267i $$0.0755965\pi$$
$$272$$ 4.00000i 0.242536i
$$273$$ − 4.00000i − 0.242091i
$$274$$ 19.0000 1.14783
$$275$$ 0 0
$$276$$ 4.00000 0.240772
$$277$$ − 26.0000i − 1.56219i −0.624413 0.781094i $$-0.714662\pi$$
0.624413 0.781094i $$-0.285338\pi$$
$$278$$ − 4.00000i − 0.239904i
$$279$$ −4.00000 −0.239474
$$280$$ 0 0
$$281$$ 17.0000 1.01413 0.507067 0.861906i $$-0.330729\pi$$
0.507067 + 0.861906i $$0.330729\pi$$
$$282$$ − 3.00000i − 0.178647i
$$283$$ 8.00000i 0.475551i 0.971320 + 0.237775i $$0.0764182\pi$$
−0.971320 + 0.237775i $$0.923582\pi$$
$$284$$ 15.0000 0.890086
$$285$$ 0 0
$$286$$ 4.00000 0.236525
$$287$$ − 20.0000i − 1.18056i
$$288$$ 1.00000i 0.0589256i
$$289$$ 1.00000 0.0588235
$$290$$ 0 0
$$291$$ −16.0000 −0.937937
$$292$$ − 12.0000i − 0.702247i
$$293$$ − 12.0000i − 0.701047i −0.936554 0.350524i $$-0.886004\pi$$
0.936554 0.350524i $$-0.113996\pi$$
$$294$$ −9.00000 −0.524891
$$295$$ 0 0
$$296$$ 9.00000 0.523114
$$297$$ − 4.00000i − 0.232104i
$$298$$ − 4.00000i − 0.231714i
$$299$$ −4.00000 −0.231326
$$300$$ 0 0
$$301$$ 40.0000 2.30556
$$302$$ 4.00000i 0.230174i
$$303$$ − 10.0000i − 0.574485i
$$304$$ −7.00000 −0.401478
$$305$$ 0 0
$$306$$ −4.00000 −0.228665
$$307$$ − 21.0000i − 1.19853i −0.800549 0.599267i $$-0.795459\pi$$
0.800549 0.599267i $$-0.204541\pi$$
$$308$$ − 16.0000i − 0.911685i
$$309$$ −8.00000 −0.455104
$$310$$ 0 0
$$311$$ 2.00000 0.113410 0.0567048 0.998391i $$-0.481941\pi$$
0.0567048 + 0.998391i $$0.481941\pi$$
$$312$$ − 1.00000i − 0.0566139i
$$313$$ − 23.0000i − 1.30004i −0.759918 0.650018i $$-0.774761\pi$$
0.759918 0.650018i $$-0.225239\pi$$
$$314$$ 14.0000 0.790066
$$315$$ 0 0
$$316$$ 7.00000 0.393781
$$317$$ 28.0000i 1.57264i 0.617822 + 0.786318i $$0.288015\pi$$
−0.617822 + 0.786318i $$0.711985\pi$$
$$318$$ 9.00000i 0.504695i
$$319$$ −20.0000 −1.11979
$$320$$ 0 0
$$321$$ 5.00000 0.279073
$$322$$ 16.0000i 0.891645i
$$323$$ − 28.0000i − 1.55796i
$$324$$ −1.00000 −0.0555556
$$325$$ 0 0
$$326$$ −24.0000 −1.32924
$$327$$ 11.0000i 0.608301i
$$328$$ − 5.00000i − 0.276079i
$$329$$ 12.0000 0.661581
$$330$$ 0 0
$$331$$ −28.0000 −1.53902 −0.769510 0.638635i $$-0.779499\pi$$
−0.769510 + 0.638635i $$0.779499\pi$$
$$332$$ − 6.00000i − 0.329293i
$$333$$ 9.00000i 0.493197i
$$334$$ −9.00000 −0.492458
$$335$$ 0 0
$$336$$ −4.00000 −0.218218
$$337$$ 34.0000i 1.85210i 0.377403 + 0.926049i $$0.376817\pi$$
−0.377403 + 0.926049i $$0.623183\pi$$
$$338$$ 1.00000i 0.0543928i
$$339$$ 14.0000 0.760376
$$340$$ 0 0
$$341$$ 16.0000 0.866449
$$342$$ − 7.00000i − 0.378517i
$$343$$ − 8.00000i − 0.431959i
$$344$$ 10.0000 0.539164
$$345$$ 0 0
$$346$$ 13.0000 0.698884
$$347$$ − 11.0000i − 0.590511i −0.955418 0.295255i $$-0.904595\pi$$
0.955418 0.295255i $$-0.0954048\pi$$
$$348$$ 5.00000i 0.268028i
$$349$$ 34.0000 1.81998 0.909989 0.414632i $$-0.136090\pi$$
0.909989 + 0.414632i $$0.136090\pi$$
$$350$$ 0 0
$$351$$ 1.00000 0.0533761
$$352$$ − 4.00000i − 0.213201i
$$353$$ − 21.0000i − 1.11772i −0.829263 0.558859i $$-0.811239\pi$$
0.829263 0.558859i $$-0.188761\pi$$
$$354$$ 6.00000 0.318896
$$355$$ 0 0
$$356$$ 14.0000 0.741999
$$357$$ − 16.0000i − 0.846810i
$$358$$ − 4.00000i − 0.211407i
$$359$$ 15.0000 0.791670 0.395835 0.918322i $$-0.370455\pi$$
0.395835 + 0.918322i $$0.370455\pi$$
$$360$$ 0 0
$$361$$ 30.0000 1.57895
$$362$$ 4.00000i 0.210235i
$$363$$ 5.00000i 0.262432i
$$364$$ 4.00000 0.209657
$$365$$ 0 0
$$366$$ 4.00000 0.209083
$$367$$ − 7.00000i − 0.365397i −0.983169 0.182699i $$-0.941517\pi$$
0.983169 0.182699i $$-0.0584832\pi$$
$$368$$ 4.00000i 0.208514i
$$369$$ 5.00000 0.260290
$$370$$ 0 0
$$371$$ −36.0000 −1.86903
$$372$$ − 4.00000i − 0.207390i
$$373$$ 22.0000i 1.13912i 0.821951 + 0.569558i $$0.192886\pi$$
−0.821951 + 0.569558i $$0.807114\pi$$
$$374$$ 16.0000 0.827340
$$375$$ 0 0
$$376$$ 3.00000 0.154713
$$377$$ − 5.00000i − 0.257513i
$$378$$ − 4.00000i − 0.205738i
$$379$$ 20.0000 1.02733 0.513665 0.857991i $$-0.328287\pi$$
0.513665 + 0.857991i $$0.328287\pi$$
$$380$$ 0 0
$$381$$ −5.00000 −0.256158
$$382$$ − 10.0000i − 0.511645i
$$383$$ 5.00000i 0.255488i 0.991807 + 0.127744i $$0.0407736\pi$$
−0.991807 + 0.127744i $$0.959226\pi$$
$$384$$ −1.00000 −0.0510310
$$385$$ 0 0
$$386$$ −12.0000 −0.610784
$$387$$ 10.0000i 0.508329i
$$388$$ − 16.0000i − 0.812277i
$$389$$ −5.00000 −0.253510 −0.126755 0.991934i $$-0.540456\pi$$
−0.126755 + 0.991934i $$0.540456\pi$$
$$390$$ 0 0
$$391$$ −16.0000 −0.809155
$$392$$ − 9.00000i − 0.454569i
$$393$$ 17.0000i 0.857537i
$$394$$ 16.0000 0.806068
$$395$$ 0 0
$$396$$ 4.00000 0.201008
$$397$$ 15.0000i 0.752828i 0.926451 + 0.376414i $$0.122843\pi$$
−0.926451 + 0.376414i $$0.877157\pi$$
$$398$$ − 3.00000i − 0.150376i
$$399$$ 28.0000 1.40175
$$400$$ 0 0
$$401$$ −30.0000 −1.49813 −0.749064 0.662497i $$-0.769497\pi$$
−0.749064 + 0.662497i $$0.769497\pi$$
$$402$$ 7.00000i 0.349128i
$$403$$ 4.00000i 0.199254i
$$404$$ 10.0000 0.497519
$$405$$ 0 0
$$406$$ −20.0000 −0.992583
$$407$$ − 36.0000i − 1.78445i
$$408$$ − 4.00000i − 0.198030i
$$409$$ 24.0000 1.18672 0.593362 0.804936i $$-0.297800\pi$$
0.593362 + 0.804936i $$0.297800\pi$$
$$410$$ 0 0
$$411$$ −19.0000 −0.937201
$$412$$ − 8.00000i − 0.394132i
$$413$$ 24.0000i 1.18096i
$$414$$ −4.00000 −0.196589
$$415$$ 0 0
$$416$$ 1.00000 0.0490290
$$417$$ 4.00000i 0.195881i
$$418$$ 28.0000i 1.36952i
$$419$$ −23.0000 −1.12362 −0.561812 0.827265i $$-0.689895\pi$$
−0.561812 + 0.827265i $$0.689895\pi$$
$$420$$ 0 0
$$421$$ 14.0000 0.682318 0.341159 0.940006i $$-0.389181\pi$$
0.341159 + 0.940006i $$0.389181\pi$$
$$422$$ 16.0000i 0.778868i
$$423$$ 3.00000i 0.145865i
$$424$$ −9.00000 −0.437079
$$425$$ 0 0
$$426$$ −15.0000 −0.726752
$$427$$ 16.0000i 0.774294i
$$428$$ 5.00000i 0.241684i
$$429$$ −4.00000 −0.193122
$$430$$ 0 0
$$431$$ 19.0000 0.915198 0.457599 0.889159i $$-0.348710\pi$$
0.457599 + 0.889159i $$0.348710\pi$$
$$432$$ − 1.00000i − 0.0481125i
$$433$$ − 19.0000i − 0.913082i −0.889702 0.456541i $$-0.849088\pi$$
0.889702 0.456541i $$-0.150912\pi$$
$$434$$ 16.0000 0.768025
$$435$$ 0 0
$$436$$ −11.0000 −0.526804
$$437$$ − 28.0000i − 1.33942i
$$438$$ 12.0000i 0.573382i
$$439$$ 7.00000 0.334092 0.167046 0.985949i $$-0.446577\pi$$
0.167046 + 0.985949i $$0.446577\pi$$
$$440$$ 0 0
$$441$$ 9.00000 0.428571
$$442$$ 4.00000i 0.190261i
$$443$$ 25.0000i 1.18779i 0.804544 + 0.593893i $$0.202410\pi$$
−0.804544 + 0.593893i $$0.797590\pi$$
$$444$$ −9.00000 −0.427121
$$445$$ 0 0
$$446$$ −2.00000 −0.0947027
$$447$$ 4.00000i 0.189194i
$$448$$ − 4.00000i − 0.188982i
$$449$$ 27.0000 1.27421 0.637104 0.770778i $$-0.280132\pi$$
0.637104 + 0.770778i $$0.280132\pi$$
$$450$$ 0 0
$$451$$ −20.0000 −0.941763
$$452$$ 14.0000i 0.658505i
$$453$$ − 4.00000i − 0.187936i
$$454$$ −6.00000 −0.281594
$$455$$ 0 0
$$456$$ 7.00000 0.327805
$$457$$ − 4.00000i − 0.187112i −0.995614 0.0935561i $$-0.970177\pi$$
0.995614 0.0935561i $$-0.0298234\pi$$
$$458$$ − 17.0000i − 0.794358i
$$459$$ 4.00000 0.186704
$$460$$ 0 0
$$461$$ −24.0000 −1.11779 −0.558896 0.829238i $$-0.688775\pi$$
−0.558896 + 0.829238i $$0.688775\pi$$
$$462$$ 16.0000i 0.744387i
$$463$$ − 26.0000i − 1.20832i −0.796862 0.604161i $$-0.793508\pi$$
0.796862 0.604161i $$-0.206492\pi$$
$$464$$ −5.00000 −0.232119
$$465$$ 0 0
$$466$$ 8.00000 0.370593
$$467$$ − 3.00000i − 0.138823i −0.997588 0.0694117i $$-0.977888\pi$$
0.997588 0.0694117i $$-0.0221122\pi$$
$$468$$ 1.00000i 0.0462250i
$$469$$ −28.0000 −1.29292
$$470$$ 0 0
$$471$$ −14.0000 −0.645086
$$472$$ 6.00000i 0.276172i
$$473$$ − 40.0000i − 1.83920i
$$474$$ −7.00000 −0.321521
$$475$$ 0 0
$$476$$ 16.0000 0.733359
$$477$$ − 9.00000i − 0.412082i
$$478$$ 16.0000i 0.731823i
$$479$$ −15.0000 −0.685367 −0.342684 0.939451i $$-0.611336\pi$$
−0.342684 + 0.939451i $$0.611336\pi$$
$$480$$ 0 0
$$481$$ 9.00000 0.410365
$$482$$ − 8.00000i − 0.364390i
$$483$$ − 16.0000i − 0.728025i
$$484$$ −5.00000 −0.227273
$$485$$ 0 0
$$486$$ 1.00000 0.0453609
$$487$$ − 22.0000i − 0.996915i −0.866914 0.498458i $$-0.833900\pi$$
0.866914 0.498458i $$-0.166100\pi$$
$$488$$ 4.00000i 0.181071i
$$489$$ 24.0000 1.08532
$$490$$ 0 0
$$491$$ −40.0000 −1.80517 −0.902587 0.430507i $$-0.858335\pi$$
−0.902587 + 0.430507i $$0.858335\pi$$
$$492$$ 5.00000i 0.225417i
$$493$$ − 20.0000i − 0.900755i
$$494$$ −7.00000 −0.314945
$$495$$ 0 0
$$496$$ 4.00000 0.179605
$$497$$ − 60.0000i − 2.69137i
$$498$$ 6.00000i 0.268866i
$$499$$ 23.0000 1.02962 0.514811 0.857304i $$-0.327862\pi$$
0.514811 + 0.857304i $$0.327862\pi$$
$$500$$ 0 0
$$501$$ 9.00000 0.402090
$$502$$ 5.00000i 0.223161i
$$503$$ 36.0000i 1.60516i 0.596544 + 0.802580i $$0.296540\pi$$
−0.596544 + 0.802580i $$0.703460\pi$$
$$504$$ 4.00000 0.178174
$$505$$ 0 0
$$506$$ 16.0000 0.711287
$$507$$ − 1.00000i − 0.0444116i
$$508$$ − 5.00000i − 0.221839i
$$509$$ 4.00000 0.177297 0.0886484 0.996063i $$-0.471745\pi$$
0.0886484 + 0.996063i $$0.471745\pi$$
$$510$$ 0 0
$$511$$ −48.0000 −2.12339
$$512$$ − 1.00000i − 0.0441942i
$$513$$ 7.00000i 0.309058i
$$514$$ −8.00000 −0.352865
$$515$$ 0 0
$$516$$ −10.0000 −0.440225
$$517$$ − 12.0000i − 0.527759i
$$518$$ − 36.0000i − 1.58175i
$$519$$ −13.0000 −0.570637
$$520$$ 0 0
$$521$$ 36.0000 1.57719 0.788594 0.614914i $$-0.210809\pi$$
0.788594 + 0.614914i $$0.210809\pi$$
$$522$$ − 5.00000i − 0.218844i
$$523$$ − 14.0000i − 0.612177i −0.952003 0.306089i $$-0.900980\pi$$
0.952003 0.306089i $$-0.0990204\pi$$
$$524$$ −17.0000 −0.742648
$$525$$ 0 0
$$526$$ 30.0000 1.30806
$$527$$ 16.0000i 0.696971i
$$528$$ 4.00000i 0.174078i
$$529$$ 7.00000 0.304348
$$530$$ 0 0
$$531$$ −6.00000 −0.260378
$$532$$ 28.0000i 1.21395i
$$533$$ − 5.00000i − 0.216574i
$$534$$ −14.0000 −0.605839
$$535$$ 0 0
$$536$$ −7.00000 −0.302354
$$537$$ 4.00000i 0.172613i
$$538$$ 5.00000i 0.215565i
$$539$$ −36.0000 −1.55063
$$540$$ 0 0
$$541$$ 18.0000 0.773880 0.386940 0.922105i $$-0.373532\pi$$
0.386940 + 0.922105i $$0.373532\pi$$
$$542$$ − 32.0000i − 1.37452i
$$543$$ − 4.00000i − 0.171656i
$$544$$ 4.00000 0.171499
$$545$$ 0 0
$$546$$ −4.00000 −0.171184
$$547$$ − 6.00000i − 0.256541i −0.991739 0.128271i $$-0.959057\pi$$
0.991739 0.128271i $$-0.0409426\pi$$
$$548$$ − 19.0000i − 0.811640i
$$549$$ −4.00000 −0.170716
$$550$$ 0 0
$$551$$ 35.0000 1.49105
$$552$$ − 4.00000i − 0.170251i
$$553$$ − 28.0000i − 1.19068i
$$554$$ −26.0000 −1.10463
$$555$$ 0 0
$$556$$ −4.00000 −0.169638
$$557$$ 38.0000i 1.61011i 0.593199 + 0.805056i $$0.297865\pi$$
−0.593199 + 0.805056i $$0.702135\pi$$
$$558$$ 4.00000i 0.169334i
$$559$$ 10.0000 0.422955
$$560$$ 0 0
$$561$$ −16.0000 −0.675521
$$562$$ − 17.0000i − 0.717102i
$$563$$ 39.0000i 1.64365i 0.569737 + 0.821827i $$0.307045\pi$$
−0.569737 + 0.821827i $$0.692955\pi$$
$$564$$ −3.00000 −0.126323
$$565$$ 0 0
$$566$$ 8.00000 0.336265
$$567$$ 4.00000i 0.167984i
$$568$$ − 15.0000i − 0.629386i
$$569$$ −8.00000 −0.335377 −0.167689 0.985840i $$-0.553630\pi$$
−0.167689 + 0.985840i $$0.553630\pi$$
$$570$$ 0 0
$$571$$ −28.0000 −1.17176 −0.585882 0.810397i $$-0.699252\pi$$
−0.585882 + 0.810397i $$0.699252\pi$$
$$572$$ − 4.00000i − 0.167248i
$$573$$ 10.0000i 0.417756i
$$574$$ −20.0000 −0.834784
$$575$$ 0 0
$$576$$ 1.00000 0.0416667
$$577$$ 8.00000i 0.333044i 0.986038 + 0.166522i $$0.0532537\pi$$
−0.986038 + 0.166522i $$0.946746\pi$$
$$578$$ − 1.00000i − 0.0415945i
$$579$$ 12.0000 0.498703
$$580$$ 0 0
$$581$$ −24.0000 −0.995688
$$582$$ 16.0000i 0.663221i
$$583$$ 36.0000i 1.49097i
$$584$$ −12.0000 −0.496564
$$585$$ 0 0
$$586$$ −12.0000 −0.495715
$$587$$ − 12.0000i − 0.495293i −0.968850 0.247647i $$-0.920343\pi$$
0.968850 0.247647i $$-0.0796572\pi$$
$$588$$ 9.00000i 0.371154i
$$589$$ −28.0000 −1.15372
$$590$$ 0 0
$$591$$ −16.0000 −0.658152
$$592$$ − 9.00000i − 0.369898i
$$593$$ 29.0000i 1.19089i 0.803397 + 0.595444i $$0.203024\pi$$
−0.803397 + 0.595444i $$0.796976\pi$$
$$594$$ −4.00000 −0.164122
$$595$$ 0 0
$$596$$ −4.00000 −0.163846
$$597$$ 3.00000i 0.122782i
$$598$$ 4.00000i 0.163572i
$$599$$ 12.0000 0.490307 0.245153 0.969484i $$-0.421162\pi$$
0.245153 + 0.969484i $$0.421162\pi$$
$$600$$ 0 0
$$601$$ −3.00000 −0.122373 −0.0611863 0.998126i $$-0.519488\pi$$
−0.0611863 + 0.998126i $$0.519488\pi$$
$$602$$ − 40.0000i − 1.63028i
$$603$$ − 7.00000i − 0.285062i
$$604$$ 4.00000 0.162758
$$605$$ 0 0
$$606$$ −10.0000 −0.406222
$$607$$ 3.00000i 0.121766i 0.998145 + 0.0608831i $$0.0193917\pi$$
−0.998145 + 0.0608831i $$0.980608\pi$$
$$608$$ 7.00000i 0.283887i
$$609$$ 20.0000 0.810441
$$610$$ 0 0
$$611$$ 3.00000 0.121367
$$612$$ 4.00000i 0.161690i
$$613$$ 2.00000i 0.0807792i 0.999184 + 0.0403896i $$0.0128599\pi$$
−0.999184 + 0.0403896i $$0.987140\pi$$
$$614$$ −21.0000 −0.847491
$$615$$ 0 0
$$616$$ −16.0000 −0.644658
$$617$$ − 7.00000i − 0.281809i −0.990023 0.140905i $$-0.954999\pi$$
0.990023 0.140905i $$-0.0450011\pi$$
$$618$$ 8.00000i 0.321807i
$$619$$ −8.00000 −0.321547 −0.160774 0.986991i $$-0.551399\pi$$
−0.160774 + 0.986991i $$0.551399\pi$$
$$620$$ 0 0
$$621$$ 4.00000 0.160514
$$622$$ − 2.00000i − 0.0801927i
$$623$$ − 56.0000i − 2.24359i
$$624$$ −1.00000 −0.0400320
$$625$$ 0 0
$$626$$ −23.0000 −0.919265
$$627$$ − 28.0000i − 1.11821i
$$628$$ − 14.0000i − 0.558661i
$$629$$ 36.0000 1.43541
$$630$$ 0 0
$$631$$ 40.0000 1.59237 0.796187 0.605050i $$-0.206847\pi$$
0.796187 + 0.605050i $$0.206847\pi$$
$$632$$ − 7.00000i − 0.278445i
$$633$$ − 16.0000i − 0.635943i
$$634$$ 28.0000 1.11202
$$635$$ 0 0
$$636$$ 9.00000 0.356873
$$637$$ − 9.00000i − 0.356593i
$$638$$ 20.0000i 0.791808i
$$639$$ 15.0000 0.593391
$$640$$ 0 0
$$641$$ −18.0000 −0.710957 −0.355479 0.934684i $$-0.615682\pi$$
−0.355479 + 0.934684i $$0.615682\pi$$
$$642$$ − 5.00000i − 0.197334i
$$643$$ − 5.00000i − 0.197181i −0.995128 0.0985904i $$-0.968567\pi$$
0.995128 0.0985904i $$-0.0314334\pi$$
$$644$$ 16.0000 0.630488
$$645$$ 0 0
$$646$$ −28.0000 −1.10165
$$647$$ 36.0000i 1.41531i 0.706560 + 0.707653i $$0.250246\pi$$
−0.706560 + 0.707653i $$0.749754\pi$$
$$648$$ 1.00000i 0.0392837i
$$649$$ 24.0000 0.942082
$$650$$ 0 0
$$651$$ −16.0000 −0.627089
$$652$$ 24.0000i 0.939913i
$$653$$ − 10.0000i − 0.391330i −0.980671 0.195665i $$-0.937313\pi$$
0.980671 0.195665i $$-0.0626866\pi$$
$$654$$ 11.0000 0.430134
$$655$$ 0 0
$$656$$ −5.00000 −0.195217
$$657$$ − 12.0000i − 0.468165i
$$658$$ − 12.0000i − 0.467809i
$$659$$ −33.0000 −1.28550 −0.642749 0.766077i $$-0.722206\pi$$
−0.642749 + 0.766077i $$0.722206\pi$$
$$660$$ 0 0
$$661$$ −7.00000 −0.272268 −0.136134 0.990690i $$-0.543468\pi$$
−0.136134 + 0.990690i $$0.543468\pi$$
$$662$$ 28.0000i 1.08825i
$$663$$ − 4.00000i − 0.155347i
$$664$$ −6.00000 −0.232845
$$665$$ 0 0
$$666$$ 9.00000 0.348743
$$667$$ − 20.0000i − 0.774403i
$$668$$ 9.00000i 0.348220i
$$669$$ 2.00000 0.0773245
$$670$$ 0 0
$$671$$ 16.0000 0.617673
$$672$$ 4.00000i 0.154303i
$$673$$ 19.0000i 0.732396i 0.930537 + 0.366198i $$0.119341\pi$$
−0.930537 + 0.366198i $$0.880659\pi$$
$$674$$ 34.0000 1.30963
$$675$$ 0 0
$$676$$ 1.00000 0.0384615
$$677$$ 14.0000i 0.538064i 0.963131 + 0.269032i $$0.0867037\pi$$
−0.963131 + 0.269032i $$0.913296\pi$$
$$678$$ − 14.0000i − 0.537667i
$$679$$ −64.0000 −2.45609
$$680$$ 0 0
$$681$$ 6.00000 0.229920
$$682$$ − 16.0000i − 0.612672i
$$683$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$684$$ −7.00000 −0.267652
$$685$$ 0 0
$$686$$ −8.00000 −0.305441
$$687$$ 17.0000i 0.648590i
$$688$$ − 10.0000i − 0.381246i
$$689$$ −9.00000 −0.342873
$$690$$ 0 0
$$691$$ 3.00000 0.114125 0.0570627 0.998371i $$-0.481827\pi$$
0.0570627 + 0.998371i $$0.481827\pi$$
$$692$$ − 13.0000i − 0.494186i
$$693$$ − 16.0000i − 0.607790i
$$694$$ −11.0000 −0.417554
$$695$$ 0 0
$$696$$ 5.00000 0.189525
$$697$$ − 20.0000i − 0.757554i
$$698$$ − 34.0000i − 1.28692i
$$699$$ −8.00000 −0.302588
$$700$$ 0 0
$$701$$ 34.0000 1.28416 0.642081 0.766637i $$-0.278071\pi$$
0.642081 + 0.766637i $$0.278071\pi$$
$$702$$ − 1.00000i − 0.0377426i
$$703$$ 63.0000i 2.37609i
$$704$$ −4.00000 −0.150756
$$705$$ 0 0
$$706$$ −21.0000 −0.790345
$$707$$ − 40.0000i − 1.50435i
$$708$$ − 6.00000i − 0.225494i
$$709$$ 38.0000 1.42712 0.713560 0.700594i $$-0.247082\pi$$
0.713560 + 0.700594i $$0.247082\pi$$
$$710$$ 0 0
$$711$$ 7.00000 0.262521
$$712$$ − 14.0000i − 0.524672i
$$713$$ 16.0000i 0.599205i
$$714$$ −16.0000 −0.598785
$$715$$ 0 0
$$716$$ −4.00000 −0.149487
$$717$$ − 16.0000i − 0.597531i
$$718$$ − 15.0000i − 0.559795i
$$719$$ 12.0000 0.447524 0.223762 0.974644i $$-0.428166\pi$$
0.223762 + 0.974644i $$0.428166\pi$$
$$720$$ 0 0
$$721$$ −32.0000 −1.19174
$$722$$ − 30.0000i − 1.11648i
$$723$$ 8.00000i 0.297523i
$$724$$ 4.00000 0.148659
$$725$$ 0 0
$$726$$ 5.00000 0.185567
$$727$$ − 28.0000i − 1.03846i −0.854634 0.519231i $$-0.826218\pi$$
0.854634 0.519231i $$-0.173782\pi$$
$$728$$ − 4.00000i − 0.148250i
$$729$$ −1.00000 −0.0370370
$$730$$ 0 0
$$731$$ 40.0000 1.47945
$$732$$ − 4.00000i − 0.147844i
$$733$$ − 3.00000i − 0.110808i −0.998464 0.0554038i $$-0.982355\pi$$
0.998464 0.0554038i $$-0.0176446\pi$$
$$734$$ −7.00000 −0.258375
$$735$$ 0 0
$$736$$ 4.00000 0.147442
$$737$$ 28.0000i 1.03139i
$$738$$ − 5.00000i − 0.184053i
$$739$$ 37.0000 1.36107 0.680534 0.732717i $$-0.261748\pi$$
0.680534 + 0.732717i $$0.261748\pi$$
$$740$$ 0 0
$$741$$ 7.00000 0.257151
$$742$$ 36.0000i 1.32160i
$$743$$ − 21.0000i − 0.770415i −0.922830 0.385208i $$-0.874130\pi$$
0.922830 0.385208i $$-0.125870\pi$$
$$744$$ −4.00000 −0.146647
$$745$$ 0 0
$$746$$ 22.0000 0.805477
$$747$$ − 6.00000i − 0.219529i
$$748$$ − 16.0000i − 0.585018i
$$749$$ 20.0000 0.730784
$$750$$ 0 0
$$751$$ −23.0000 −0.839282 −0.419641 0.907690i $$-0.637844\pi$$
−0.419641 + 0.907690i $$0.637844\pi$$
$$752$$ − 3.00000i − 0.109399i
$$753$$ − 5.00000i − 0.182210i
$$754$$ −5.00000 −0.182089
$$755$$ 0 0
$$756$$ −4.00000 −0.145479
$$757$$ − 8.00000i − 0.290765i −0.989376 0.145382i $$-0.953559\pi$$
0.989376 0.145382i $$-0.0464413\pi$$
$$758$$ − 20.0000i − 0.726433i
$$759$$ −16.0000 −0.580763
$$760$$ 0 0
$$761$$ 35.0000 1.26875 0.634375 0.773026i $$-0.281258\pi$$
0.634375 + 0.773026i $$0.281258\pi$$
$$762$$ 5.00000i 0.181131i
$$763$$ 44.0000i 1.59291i
$$764$$ −10.0000 −0.361787
$$765$$ 0 0
$$766$$ 5.00000 0.180657
$$767$$ 6.00000i 0.216647i
$$768$$ 1.00000i 0.0360844i
$$769$$ −40.0000 −1.44244 −0.721218 0.692708i $$-0.756418\pi$$
−0.721218 + 0.692708i $$0.756418\pi$$
$$770$$ 0 0
$$771$$ 8.00000 0.288113
$$772$$ 12.0000i 0.431889i
$$773$$ − 8.00000i − 0.287740i −0.989597 0.143870i $$-0.954045\pi$$
0.989597 0.143870i $$-0.0459547\pi$$
$$774$$ 10.0000 0.359443
$$775$$ 0 0
$$776$$ −16.0000 −0.574367
$$777$$ 36.0000i 1.29149i
$$778$$ 5.00000i 0.179259i
$$779$$ 35.0000 1.25401
$$780$$ 0 0
$$781$$ −60.0000 −2.14697
$$782$$ 16.0000i 0.572159i
$$783$$ 5.00000i 0.178685i
$$784$$ −9.00000 −0.321429
$$785$$ 0 0
$$786$$ 17.0000 0.606370
$$787$$ 32.0000i 1.14068i 0.821410 + 0.570338i $$0.193188\pi$$
−0.821410 + 0.570338i $$0.806812\pi$$
$$788$$ − 16.0000i − 0.569976i
$$789$$ −30.0000 −1.06803
$$790$$ 0 0
$$791$$ 56.0000 1.99113
$$792$$ − 4.00000i − 0.142134i
$$793$$ 4.00000i 0.142044i
$$794$$ 15.0000 0.532330
$$795$$ 0 0
$$796$$ −3.00000 −0.106332
$$797$$ 30.0000i 1.06265i 0.847167 + 0.531327i $$0.178307\pi$$
−0.847167 + 0.531327i $$0.821693\pi$$
$$798$$ − 28.0000i − 0.991189i
$$799$$ 12.0000 0.424529
$$800$$ 0 0
$$801$$ 14.0000 0.494666
$$802$$ 30.0000i 1.05934i
$$803$$ 48.0000i 1.69388i
$$804$$ 7.00000 0.246871
$$805$$ 0 0
$$806$$ 4.00000 0.140894
$$807$$ − 5.00000i − 0.176008i
$$808$$ − 10.0000i − 0.351799i
$$809$$ −18.0000 −0.632846 −0.316423 0.948618i $$-0.602482\pi$$
−0.316423 + 0.948618i $$0.602482\pi$$
$$810$$ 0 0
$$811$$ 20.0000 0.702295 0.351147 0.936320i $$-0.385792\pi$$
0.351147 + 0.936320i $$0.385792\pi$$
$$812$$ 20.0000i 0.701862i
$$813$$ 32.0000i 1.12229i
$$814$$ −36.0000 −1.26180
$$815$$ 0 0
$$816$$ −4.00000 −0.140028
$$817$$ 70.0000i 2.44899i
$$818$$ − 24.0000i − 0.839140i
$$819$$ 4.00000 0.139771
$$820$$ 0 0
$$821$$ 42.0000 1.46581 0.732905 0.680331i $$-0.238164\pi$$
0.732905 + 0.680331i $$0.238164\pi$$
$$822$$ 19.0000i 0.662701i
$$823$$ − 19.0000i − 0.662298i −0.943578 0.331149i $$-0.892564\pi$$
0.943578 0.331149i $$-0.107436\pi$$
$$824$$ −8.00000 −0.278693
$$825$$ 0 0
$$826$$ 24.0000 0.835067
$$827$$ 4.00000i 0.139094i 0.997579 + 0.0695468i $$0.0221553\pi$$
−0.997579 + 0.0695468i $$0.977845\pi$$
$$828$$ 4.00000i 0.139010i
$$829$$ 34.0000 1.18087 0.590434 0.807086i $$-0.298956\pi$$
0.590434 + 0.807086i $$0.298956\pi$$
$$830$$ 0 0
$$831$$ 26.0000 0.901930
$$832$$ − 1.00000i − 0.0346688i
$$833$$ − 36.0000i − 1.24733i
$$834$$ 4.00000 0.138509
$$835$$ 0 0
$$836$$ 28.0000 0.968400
$$837$$ − 4.00000i − 0.138260i
$$838$$ 23.0000i 0.794522i
$$839$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$840$$ 0 0
$$841$$ −4.00000 −0.137931
$$842$$ − 14.0000i − 0.482472i
$$843$$ 17.0000i 0.585511i
$$844$$ 16.0000 0.550743
$$845$$ 0 0
$$846$$ 3.00000 0.103142
$$847$$ 20.0000i 0.687208i
$$848$$ 9.00000i 0.309061i
$$849$$ −8.00000 −0.274559
$$850$$ 0 0
$$851$$ 36.0000 1.23406
$$852$$ 15.0000i 0.513892i
$$853$$ 19.0000i 0.650548i 0.945620 + 0.325274i $$0.105456\pi$$
−0.945620 + 0.325274i $$0.894544\pi$$
$$854$$ 16.0000 0.547509
$$855$$ 0 0
$$856$$ 5.00000 0.170896
$$857$$ 50.0000i 1.70797i 0.520300 + 0.853984i $$0.325820\pi$$
−0.520300 + 0.853984i $$0.674180\pi$$
$$858$$ 4.00000i 0.136558i
$$859$$ 2.00000 0.0682391 0.0341196 0.999418i $$-0.489137\pi$$
0.0341196 + 0.999418i $$0.489137\pi$$
$$860$$ 0 0
$$861$$ 20.0000 0.681598
$$862$$ − 19.0000i − 0.647143i
$$863$$ 39.0000i 1.32758i 0.747921 + 0.663788i $$0.231052\pi$$
−0.747921 + 0.663788i $$0.768948\pi$$
$$864$$ −1.00000 −0.0340207
$$865$$ 0 0
$$866$$ −19.0000 −0.645646
$$867$$ 1.00000i 0.0339618i
$$868$$ − 16.0000i − 0.543075i
$$869$$ −28.0000 −0.949835
$$870$$ 0 0
$$871$$ −7.00000 −0.237186
$$872$$ 11.0000i 0.372507i
$$873$$ − 16.0000i − 0.541518i
$$874$$ −28.0000 −0.947114
$$875$$ 0 0
$$876$$ 12.0000 0.405442
$$877$$ − 43.0000i − 1.45201i −0.687691 0.726003i $$-0.741376\pi$$
0.687691 0.726003i $$-0.258624\pi$$
$$878$$ − 7.00000i − 0.236239i
$$879$$ 12.0000 0.404750
$$880$$ 0 0
$$881$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$882$$ − 9.00000i − 0.303046i
$$883$$ 40.0000i 1.34611i 0.739594 + 0.673054i $$0.235018\pi$$
−0.739594 + 0.673054i $$0.764982\pi$$
$$884$$ 4.00000 0.134535
$$885$$ 0 0
$$886$$ 25.0000 0.839891
$$887$$ 36.0000i 1.20876i 0.796696 + 0.604381i $$0.206579\pi$$
−0.796696 + 0.604381i $$0.793421\pi$$
$$888$$ 9.00000i 0.302020i
$$889$$ −20.0000 −0.670778
$$890$$ 0 0
$$891$$ 4.00000 0.134005
$$892$$ 2.00000i 0.0669650i
$$893$$ 21.0000i 0.702738i
$$894$$ 4.00000 0.133780
$$895$$ 0 0
$$896$$ −4.00000 −0.133631
$$897$$ − 4.00000i − 0.133556i
$$898$$ − 27.0000i − 0.901002i
$$899$$ −20.0000 −0.667037
$$900$$ 0 0
$$901$$ −36.0000 −1.19933
$$902$$ 20.0000i 0.665927i
$$903$$ 40.0000i 1.33112i
$$904$$ 14.0000 0.465633
$$905$$ 0 0
$$906$$ −4.00000 −0.132891
$$907$$ − 50.0000i − 1.66022i −0.557598 0.830111i $$-0.688277\pi$$
0.557598 0.830111i $$-0.311723\pi$$
$$908$$ 6.00000i 0.199117i
$$909$$ 10.0000 0.331679
$$910$$ 0 0
$$911$$ −18.0000 −0.596367 −0.298183 0.954509i $$-0.596381\pi$$
−0.298183 + 0.954509i $$0.596381\pi$$
$$912$$ − 7.00000i − 0.231793i
$$913$$ 24.0000i 0.794284i
$$914$$ −4.00000 −0.132308
$$915$$ 0 0
$$916$$ −17.0000 −0.561696
$$917$$ 68.0000i 2.24556i
$$918$$ − 4.00000i − 0.132020i
$$919$$ −29.0000 −0.956622 −0.478311 0.878191i $$-0.658751\pi$$
−0.478311 + 0.878191i $$0.658751\pi$$
$$920$$ 0 0
$$921$$ 21.0000 0.691974
$$922$$ 24.0000i 0.790398i
$$923$$ − 15.0000i − 0.493731i
$$924$$ 16.0000 0.526361
$$925$$ 0 0
$$926$$ −26.0000 −0.854413
$$927$$ − 8.00000i − 0.262754i
$$928$$ 5.00000i 0.164133i
$$929$$ −11.0000 −0.360898 −0.180449 0.983584i $$-0.557755\pi$$
−0.180449 + 0.983584i $$0.557755\pi$$
$$930$$ 0 0
$$931$$ 63.0000 2.06474
$$932$$ − 8.00000i − 0.262049i
$$933$$ 2.00000i 0.0654771i
$$934$$ −3.00000 −0.0981630
$$935$$ 0 0
$$936$$ 1.00000 0.0326860
$$937$$ 22.0000i 0.718709i 0.933201 + 0.359354i $$0.117003\pi$$
−0.933201 + 0.359354i $$0.882997\pi$$
$$938$$ 28.0000i 0.914232i
$$939$$ 23.0000 0.750577
$$940$$ 0 0
$$941$$ 52.0000 1.69515 0.847576 0.530674i $$-0.178061\pi$$
0.847576 + 0.530674i $$0.178061\pi$$
$$942$$ 14.0000i 0.456145i
$$943$$ − 20.0000i − 0.651290i
$$944$$ 6.00000 0.195283
$$945$$ 0 0
$$946$$ −40.0000 −1.30051
$$947$$ 8.00000i 0.259965i 0.991516 + 0.129983i $$0.0414921\pi$$
−0.991516 + 0.129983i $$0.958508\pi$$
$$948$$ 7.00000i 0.227349i
$$949$$ −12.0000 −0.389536
$$950$$ 0 0
$$951$$ −28.0000 −0.907962
$$952$$ − 16.0000i − 0.518563i
$$953$$ 36.0000i 1.16615i 0.812417 + 0.583077i $$0.198151\pi$$
−0.812417 + 0.583077i $$0.801849\pi$$
$$954$$ −9.00000 −0.291386
$$955$$ 0 0
$$956$$ 16.0000 0.517477
$$957$$ − 20.0000i − 0.646508i
$$958$$ 15.0000i 0.484628i
$$959$$ −76.0000 −2.45417
$$960$$ 0 0
$$961$$ −15.0000 −0.483871
$$962$$ − 9.00000i − 0.290172i
$$963$$ 5.00000i 0.161123i
$$964$$ −8.00000 −0.257663
$$965$$ 0 0
$$966$$ −16.0000 −0.514792
$$967$$ 44.0000i 1.41494i 0.706741 + 0.707472i $$0.250165\pi$$
−0.706741 + 0.707472i $$0.749835\pi$$
$$968$$ 5.00000i 0.160706i
$$969$$ 28.0000 0.899490
$$970$$ 0 0
$$971$$ −21.0000 −0.673922 −0.336961 0.941519i $$-0.609399\pi$$
−0.336961 + 0.941519i $$0.609399\pi$$
$$972$$ − 1.00000i − 0.0320750i
$$973$$ 16.0000i 0.512936i
$$974$$ −22.0000 −0.704925
$$975$$ 0 0
$$976$$ 4.00000 0.128037
$$977$$ − 18.0000i − 0.575871i −0.957650 0.287936i $$-0.907031\pi$$
0.957650 0.287936i $$-0.0929689\pi$$
$$978$$ − 24.0000i − 0.767435i
$$979$$ −56.0000 −1.78977
$$980$$ 0 0
$$981$$ −11.0000 −0.351203
$$982$$ 40.0000i 1.27645i
$$983$$ − 48.0000i − 1.53096i −0.643458 0.765481i $$-0.722501\pi$$
0.643458 0.765481i $$-0.277499\pi$$
$$984$$ 5.00000 0.159394
$$985$$ 0 0
$$986$$ −20.0000 −0.636930
$$987$$ 12.0000i 0.381964i
$$988$$ 7.00000i 0.222700i
$$989$$ 40.0000 1.27193
$$990$$ 0 0
$$991$$ 13.0000 0.412959 0.206479 0.978451i $$-0.433799\pi$$
0.206479 + 0.978451i $$0.433799\pi$$
$$992$$ − 4.00000i − 0.127000i
$$993$$ − 28.0000i − 0.888553i
$$994$$ −60.0000 −1.90308
$$995$$ 0 0
$$996$$ 6.00000 0.190117
$$997$$ 10.0000i 0.316703i 0.987383 + 0.158352i $$0.0506179\pi$$
−0.987383 + 0.158352i $$0.949382\pi$$
$$998$$ − 23.0000i − 0.728052i
$$999$$ −9.00000 −0.284747
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 1950.2.e.n.1249.1 2
3.2 odd 2 5850.2.e.c.5149.2 2
5.2 odd 4 1950.2.a.x.1.1 yes 1
5.3 odd 4 1950.2.a.e.1.1 1
5.4 even 2 inner 1950.2.e.n.1249.2 2
15.2 even 4 5850.2.a.a.1.1 1
15.8 even 4 5850.2.a.by.1.1 1
15.14 odd 2 5850.2.e.c.5149.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
1950.2.a.e.1.1 1 5.3 odd 4
1950.2.a.x.1.1 yes 1 5.2 odd 4
1950.2.e.n.1249.1 2 1.1 even 1 trivial
1950.2.e.n.1249.2 2 5.4 even 2 inner
5850.2.a.a.1.1 1 15.2 even 4
5850.2.a.by.1.1 1 15.8 even 4
5850.2.e.c.5149.1 2 15.14 odd 2
5850.2.e.c.5149.2 2 3.2 odd 2