# Properties

 Label 1950.2.e.f.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: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 390) 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.f.1249.2

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{6} -2.00000i q^{7} +1.00000i q^{8} -1.00000 q^{9} +O(q^{10})$$ $$q-1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{6} -2.00000i q^{7} +1.00000i q^{8} -1.00000 q^{9} +4.00000 q^{11} +1.00000i q^{12} -1.00000i q^{13} -2.00000 q^{14} +1.00000 q^{16} -8.00000i q^{17} +1.00000i q^{18} +6.00000 q^{19} -2.00000 q^{21} -4.00000i q^{22} +6.00000i q^{23} +1.00000 q^{24} -1.00000 q^{26} +1.00000i q^{27} +2.00000i q^{28} +4.00000 q^{29} -1.00000i q^{32} -4.00000i q^{33} -8.00000 q^{34} +1.00000 q^{36} +2.00000i q^{37} -6.00000i q^{38} -1.00000 q^{39} -2.00000 q^{41} +2.00000i q^{42} -4.00000i q^{43} -4.00000 q^{44} +6.00000 q^{46} -1.00000i q^{48} +3.00000 q^{49} -8.00000 q^{51} +1.00000i q^{52} -10.0000i q^{53} +1.00000 q^{54} +2.00000 q^{56} -6.00000i q^{57} -4.00000i q^{58} -4.00000 q^{59} -10.0000 q^{61} +2.00000i q^{63} -1.00000 q^{64} -4.00000 q^{66} -12.0000i q^{67} +8.00000i q^{68} +6.00000 q^{69} -8.00000 q^{71} -1.00000i q^{72} -8.00000i q^{73} +2.00000 q^{74} -6.00000 q^{76} -8.00000i q^{77} +1.00000i q^{78} -8.00000 q^{79} +1.00000 q^{81} +2.00000i q^{82} +12.0000i q^{83} +2.00000 q^{84} -4.00000 q^{86} -4.00000i q^{87} +4.00000i q^{88} +14.0000 q^{89} -2.00000 q^{91} -6.00000i q^{92} -1.00000 q^{96} +16.0000i q^{97} -3.00000i q^{98} -4.00000 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{4} - 2 q^{6} - 2 q^{9}+O(q^{10})$$ 2 * q - 2 * q^4 - 2 * q^6 - 2 * q^9 $$2 q - 2 q^{4} - 2 q^{6} - 2 q^{9} + 8 q^{11} - 4 q^{14} + 2 q^{16} + 12 q^{19} - 4 q^{21} + 2 q^{24} - 2 q^{26} + 8 q^{29} - 16 q^{34} + 2 q^{36} - 2 q^{39} - 4 q^{41} - 8 q^{44} + 12 q^{46} + 6 q^{49} - 16 q^{51} + 2 q^{54} + 4 q^{56} - 8 q^{59} - 20 q^{61} - 2 q^{64} - 8 q^{66} + 12 q^{69} - 16 q^{71} + 4 q^{74} - 12 q^{76} - 16 q^{79} + 2 q^{81} + 4 q^{84} - 8 q^{86} + 28 q^{89} - 4 q^{91} - 2 q^{96} - 8 q^{99}+O(q^{100})$$ 2 * q - 2 * q^4 - 2 * q^6 - 2 * q^9 + 8 * q^11 - 4 * q^14 + 2 * q^16 + 12 * q^19 - 4 * q^21 + 2 * q^24 - 2 * q^26 + 8 * q^29 - 16 * q^34 + 2 * q^36 - 2 * q^39 - 4 * q^41 - 8 * q^44 + 12 * q^46 + 6 * q^49 - 16 * q^51 + 2 * q^54 + 4 * q^56 - 8 * q^59 - 20 * q^61 - 2 * q^64 - 8 * q^66 + 12 * q^69 - 16 * q^71 + 4 * q^74 - 12 * q^76 - 16 * q^79 + 2 * q^81 + 4 * q^84 - 8 * q^86 + 28 * q^89 - 4 * q^91 - 2 * q^96 - 8 * q^99

## 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
$$7$$ − 2.00000i − 0.755929i −0.925820 0.377964i $$-0.876624\pi$$
0.925820 0.377964i $$-0.123376\pi$$
$$8$$ 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
$$14$$ −2.00000 −0.534522
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ − 8.00000i − 1.94029i −0.242536 0.970143i $$-0.577979\pi$$
0.242536 0.970143i $$-0.422021\pi$$
$$18$$ 1.00000i 0.235702i
$$19$$ 6.00000 1.37649 0.688247 0.725476i $$-0.258380\pi$$
0.688247 + 0.725476i $$0.258380\pi$$
$$20$$ 0 0
$$21$$ −2.00000 −0.436436
$$22$$ − 4.00000i − 0.852803i
$$23$$ 6.00000i 1.25109i 0.780189 + 0.625543i $$0.215123\pi$$
−0.780189 + 0.625543i $$0.784877\pi$$
$$24$$ 1.00000 0.204124
$$25$$ 0 0
$$26$$ −1.00000 −0.196116
$$27$$ 1.00000i 0.192450i
$$28$$ 2.00000i 0.377964i
$$29$$ 4.00000 0.742781 0.371391 0.928477i $$-0.378881\pi$$
0.371391 + 0.928477i $$0.378881\pi$$
$$30$$ 0 0
$$31$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$32$$ − 1.00000i − 0.176777i
$$33$$ − 4.00000i − 0.696311i
$$34$$ −8.00000 −1.37199
$$35$$ 0 0
$$36$$ 1.00000 0.166667
$$37$$ 2.00000i 0.328798i 0.986394 + 0.164399i $$0.0525685\pi$$
−0.986394 + 0.164399i $$0.947432\pi$$
$$38$$ − 6.00000i − 0.973329i
$$39$$ −1.00000 −0.160128
$$40$$ 0 0
$$41$$ −2.00000 −0.312348 −0.156174 0.987730i $$-0.549916\pi$$
−0.156174 + 0.987730i $$0.549916\pi$$
$$42$$ 2.00000i 0.308607i
$$43$$ − 4.00000i − 0.609994i −0.952353 0.304997i $$-0.901344\pi$$
0.952353 0.304997i $$-0.0986555\pi$$
$$44$$ −4.00000 −0.603023
$$45$$ 0 0
$$46$$ 6.00000 0.884652
$$47$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$48$$ − 1.00000i − 0.144338i
$$49$$ 3.00000 0.428571
$$50$$ 0 0
$$51$$ −8.00000 −1.12022
$$52$$ 1.00000i 0.138675i
$$53$$ − 10.0000i − 1.37361i −0.726844 0.686803i $$-0.759014\pi$$
0.726844 0.686803i $$-0.240986\pi$$
$$54$$ 1.00000 0.136083
$$55$$ 0 0
$$56$$ 2.00000 0.267261
$$57$$ − 6.00000i − 0.794719i
$$58$$ − 4.00000i − 0.525226i
$$59$$ −4.00000 −0.520756 −0.260378 0.965507i $$-0.583847\pi$$
−0.260378 + 0.965507i $$0.583847\pi$$
$$60$$ 0 0
$$61$$ −10.0000 −1.28037 −0.640184 0.768221i $$-0.721142\pi$$
−0.640184 + 0.768221i $$0.721142\pi$$
$$62$$ 0 0
$$63$$ 2.00000i 0.251976i
$$64$$ −1.00000 −0.125000
$$65$$ 0 0
$$66$$ −4.00000 −0.492366
$$67$$ − 12.0000i − 1.46603i −0.680211 0.733017i $$-0.738112\pi$$
0.680211 0.733017i $$-0.261888\pi$$
$$68$$ 8.00000i 0.970143i
$$69$$ 6.00000 0.722315
$$70$$ 0 0
$$71$$ −8.00000 −0.949425 −0.474713 0.880141i $$-0.657448\pi$$
−0.474713 + 0.880141i $$0.657448\pi$$
$$72$$ − 1.00000i − 0.117851i
$$73$$ − 8.00000i − 0.936329i −0.883641 0.468165i $$-0.844915\pi$$
0.883641 0.468165i $$-0.155085\pi$$
$$74$$ 2.00000 0.232495
$$75$$ 0 0
$$76$$ −6.00000 −0.688247
$$77$$ − 8.00000i − 0.911685i
$$78$$ 1.00000i 0.113228i
$$79$$ −8.00000 −0.900070 −0.450035 0.893011i $$-0.648589\pi$$
−0.450035 + 0.893011i $$0.648589\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 2.00000i 0.220863i
$$83$$ 12.0000i 1.31717i 0.752506 + 0.658586i $$0.228845\pi$$
−0.752506 + 0.658586i $$0.771155\pi$$
$$84$$ 2.00000 0.218218
$$85$$ 0 0
$$86$$ −4.00000 −0.431331
$$87$$ − 4.00000i − 0.428845i
$$88$$ 4.00000i 0.426401i
$$89$$ 14.0000 1.48400 0.741999 0.670402i $$-0.233878\pi$$
0.741999 + 0.670402i $$0.233878\pi$$
$$90$$ 0 0
$$91$$ −2.00000 −0.209657
$$92$$ − 6.00000i − 0.625543i
$$93$$ 0 0
$$94$$ 0 0
$$95$$ 0 0
$$96$$ −1.00000 −0.102062
$$97$$ 16.0000i 1.62455i 0.583272 + 0.812277i $$0.301772\pi$$
−0.583272 + 0.812277i $$0.698228\pi$$
$$98$$ − 3.00000i − 0.303046i
$$99$$ −4.00000 −0.402015
$$100$$ 0 0
$$101$$ −16.0000 −1.59206 −0.796030 0.605257i $$-0.793070\pi$$
−0.796030 + 0.605257i $$0.793070\pi$$
$$102$$ 8.00000i 0.792118i
$$103$$ − 12.0000i − 1.18240i −0.806527 0.591198i $$-0.798655\pi$$
0.806527 0.591198i $$-0.201345\pi$$
$$104$$ 1.00000 0.0980581
$$105$$ 0 0
$$106$$ −10.0000 −0.971286
$$107$$ − 12.0000i − 1.16008i −0.814587 0.580042i $$-0.803036\pi$$
0.814587 0.580042i $$-0.196964\pi$$
$$108$$ − 1.00000i − 0.0962250i
$$109$$ −12.0000 −1.14939 −0.574696 0.818367i $$-0.694880\pi$$
−0.574696 + 0.818367i $$0.694880\pi$$
$$110$$ 0 0
$$111$$ 2.00000 0.189832
$$112$$ − 2.00000i − 0.188982i
$$113$$ 20.0000i 1.88144i 0.339182 + 0.940721i $$0.389850\pi$$
−0.339182 + 0.940721i $$0.610150\pi$$
$$114$$ −6.00000 −0.561951
$$115$$ 0 0
$$116$$ −4.00000 −0.371391
$$117$$ 1.00000i 0.0924500i
$$118$$ 4.00000i 0.368230i
$$119$$ −16.0000 −1.46672
$$120$$ 0 0
$$121$$ 5.00000 0.454545
$$122$$ 10.0000i 0.905357i
$$123$$ 2.00000i 0.180334i
$$124$$ 0 0
$$125$$ 0 0
$$126$$ 2.00000 0.178174
$$127$$ − 4.00000i − 0.354943i −0.984126 0.177471i $$-0.943208\pi$$
0.984126 0.177471i $$-0.0567917\pi$$
$$128$$ 1.00000i 0.0883883i
$$129$$ −4.00000 −0.352180
$$130$$ 0 0
$$131$$ 10.0000 0.873704 0.436852 0.899533i $$-0.356093\pi$$
0.436852 + 0.899533i $$0.356093\pi$$
$$132$$ 4.00000i 0.348155i
$$133$$ − 12.0000i − 1.04053i
$$134$$ −12.0000 −1.03664
$$135$$ 0 0
$$136$$ 8.00000 0.685994
$$137$$ 6.00000i 0.512615i 0.966595 + 0.256307i $$0.0825059\pi$$
−0.966595 + 0.256307i $$0.917494\pi$$
$$138$$ − 6.00000i − 0.510754i
$$139$$ 8.00000 0.678551 0.339276 0.940687i $$-0.389818\pi$$
0.339276 + 0.940687i $$0.389818\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 8.00000i 0.671345i
$$143$$ − 4.00000i − 0.334497i
$$144$$ −1.00000 −0.0833333
$$145$$ 0 0
$$146$$ −8.00000 −0.662085
$$147$$ − 3.00000i − 0.247436i
$$148$$ − 2.00000i − 0.164399i
$$149$$ 10.0000 0.819232 0.409616 0.912258i $$-0.365663\pi$$
0.409616 + 0.912258i $$0.365663\pi$$
$$150$$ 0 0
$$151$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$152$$ 6.00000i 0.486664i
$$153$$ 8.00000i 0.646762i
$$154$$ −8.00000 −0.644658
$$155$$ 0 0
$$156$$ 1.00000 0.0800641
$$157$$ − 22.0000i − 1.75579i −0.478852 0.877896i $$-0.658947\pi$$
0.478852 0.877896i $$-0.341053\pi$$
$$158$$ 8.00000i 0.636446i
$$159$$ −10.0000 −0.793052
$$160$$ 0 0
$$161$$ 12.0000 0.945732
$$162$$ − 1.00000i − 0.0785674i
$$163$$ − 16.0000i − 1.25322i −0.779334 0.626608i $$-0.784443\pi$$
0.779334 0.626608i $$-0.215557\pi$$
$$164$$ 2.00000 0.156174
$$165$$ 0 0
$$166$$ 12.0000 0.931381
$$167$$ 4.00000i 0.309529i 0.987951 + 0.154765i $$0.0494619\pi$$
−0.987951 + 0.154765i $$0.950538\pi$$
$$168$$ − 2.00000i − 0.154303i
$$169$$ −1.00000 −0.0769231
$$170$$ 0 0
$$171$$ −6.00000 −0.458831
$$172$$ 4.00000i 0.304997i
$$173$$ 22.0000i 1.67263i 0.548250 + 0.836315i $$0.315294\pi$$
−0.548250 + 0.836315i $$0.684706\pi$$
$$174$$ −4.00000 −0.303239
$$175$$ 0 0
$$176$$ 4.00000 0.301511
$$177$$ 4.00000i 0.300658i
$$178$$ − 14.0000i − 1.04934i
$$179$$ 10.0000 0.747435 0.373718 0.927543i $$-0.378083\pi$$
0.373718 + 0.927543i $$0.378083\pi$$
$$180$$ 0 0
$$181$$ 10.0000 0.743294 0.371647 0.928374i $$-0.378793\pi$$
0.371647 + 0.928374i $$0.378793\pi$$
$$182$$ 2.00000i 0.148250i
$$183$$ 10.0000i 0.739221i
$$184$$ −6.00000 −0.442326
$$185$$ 0 0
$$186$$ 0 0
$$187$$ − 32.0000i − 2.34007i
$$188$$ 0 0
$$189$$ 2.00000 0.145479
$$190$$ 0 0
$$191$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$192$$ 1.00000i 0.0721688i
$$193$$ − 4.00000i − 0.287926i −0.989583 0.143963i $$-0.954015\pi$$
0.989583 0.143963i $$-0.0459847\pi$$
$$194$$ 16.0000 1.14873
$$195$$ 0 0
$$196$$ −3.00000 −0.214286
$$197$$ 18.0000i 1.28245i 0.767354 + 0.641223i $$0.221573\pi$$
−0.767354 + 0.641223i $$0.778427\pi$$
$$198$$ 4.00000i 0.284268i
$$199$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$200$$ 0 0
$$201$$ −12.0000 −0.846415
$$202$$ 16.0000i 1.12576i
$$203$$ − 8.00000i − 0.561490i
$$204$$ 8.00000 0.560112
$$205$$ 0 0
$$206$$ −12.0000 −0.836080
$$207$$ − 6.00000i − 0.417029i
$$208$$ − 1.00000i − 0.0693375i
$$209$$ 24.0000 1.66011
$$210$$ 0 0
$$211$$ −20.0000 −1.37686 −0.688428 0.725304i $$-0.741699\pi$$
−0.688428 + 0.725304i $$0.741699\pi$$
$$212$$ 10.0000i 0.686803i
$$213$$ 8.00000i 0.548151i
$$214$$ −12.0000 −0.820303
$$215$$ 0 0
$$216$$ −1.00000 −0.0680414
$$217$$ 0 0
$$218$$ 12.0000i 0.812743i
$$219$$ −8.00000 −0.540590
$$220$$ 0 0
$$221$$ −8.00000 −0.538138
$$222$$ − 2.00000i − 0.134231i
$$223$$ − 2.00000i − 0.133930i −0.997755 0.0669650i $$-0.978668\pi$$
0.997755 0.0669650i $$-0.0213316\pi$$
$$224$$ −2.00000 −0.133631
$$225$$ 0 0
$$226$$ 20.0000 1.33038
$$227$$ 4.00000i 0.265489i 0.991150 + 0.132745i $$0.0423790\pi$$
−0.991150 + 0.132745i $$0.957621\pi$$
$$228$$ 6.00000i 0.397360i
$$229$$ −4.00000 −0.264327 −0.132164 0.991228i $$-0.542192\pi$$
−0.132164 + 0.991228i $$0.542192\pi$$
$$230$$ 0 0
$$231$$ −8.00000 −0.526361
$$232$$ 4.00000i 0.262613i
$$233$$ 24.0000i 1.57229i 0.618041 + 0.786146i $$0.287927\pi$$
−0.618041 + 0.786146i $$0.712073\pi$$
$$234$$ 1.00000 0.0653720
$$235$$ 0 0
$$236$$ 4.00000 0.260378
$$237$$ 8.00000i 0.519656i
$$238$$ 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
$$241$$ 2.00000 0.128831 0.0644157 0.997923i $$-0.479482\pi$$
0.0644157 + 0.997923i $$0.479482\pi$$
$$242$$ − 5.00000i − 0.321412i
$$243$$ − 1.00000i − 0.0641500i
$$244$$ 10.0000 0.640184
$$245$$ 0 0
$$246$$ 2.00000 0.127515
$$247$$ − 6.00000i − 0.381771i
$$248$$ 0 0
$$249$$ 12.0000 0.760469
$$250$$ 0 0
$$251$$ 6.00000 0.378717 0.189358 0.981908i $$-0.439359\pi$$
0.189358 + 0.981908i $$0.439359\pi$$
$$252$$ − 2.00000i − 0.125988i
$$253$$ 24.0000i 1.50887i
$$254$$ −4.00000 −0.250982
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ − 12.0000i − 0.748539i −0.927320 0.374270i $$-0.877893\pi$$
0.927320 0.374270i $$-0.122107\pi$$
$$258$$ 4.00000i 0.249029i
$$259$$ 4.00000 0.248548
$$260$$ 0 0
$$261$$ −4.00000 −0.247594
$$262$$ − 10.0000i − 0.617802i
$$263$$ 2.00000i 0.123325i 0.998097 + 0.0616626i $$0.0196403\pi$$
−0.998097 + 0.0616626i $$0.980360\pi$$
$$264$$ 4.00000 0.246183
$$265$$ 0 0
$$266$$ −12.0000 −0.735767
$$267$$ − 14.0000i − 0.856786i
$$268$$ 12.0000i 0.733017i
$$269$$ 24.0000 1.46331 0.731653 0.681677i $$-0.238749\pi$$
0.731653 + 0.681677i $$0.238749\pi$$
$$270$$ 0 0
$$271$$ 16.0000 0.971931 0.485965 0.873978i $$-0.338468\pi$$
0.485965 + 0.873978i $$0.338468\pi$$
$$272$$ − 8.00000i − 0.485071i
$$273$$ 2.00000i 0.121046i
$$274$$ 6.00000 0.362473
$$275$$ 0 0
$$276$$ −6.00000 −0.361158
$$277$$ 10.0000i 0.600842i 0.953807 + 0.300421i $$0.0971271\pi$$
−0.953807 + 0.300421i $$0.902873\pi$$
$$278$$ − 8.00000i − 0.479808i
$$279$$ 0 0
$$280$$ 0 0
$$281$$ −6.00000 −0.357930 −0.178965 0.983855i $$-0.557275\pi$$
−0.178965 + 0.983855i $$0.557275\pi$$
$$282$$ 0 0
$$283$$ − 4.00000i − 0.237775i −0.992908 0.118888i $$-0.962067\pi$$
0.992908 0.118888i $$-0.0379328\pi$$
$$284$$ 8.00000 0.474713
$$285$$ 0 0
$$286$$ −4.00000 −0.236525
$$287$$ 4.00000i 0.236113i
$$288$$ 1.00000i 0.0589256i
$$289$$ −47.0000 −2.76471
$$290$$ 0 0
$$291$$ 16.0000 0.937937
$$292$$ 8.00000i 0.468165i
$$293$$ − 26.0000i − 1.51894i −0.650545 0.759468i $$-0.725459\pi$$
0.650545 0.759468i $$-0.274541\pi$$
$$294$$ −3.00000 −0.174964
$$295$$ 0 0
$$296$$ −2.00000 −0.116248
$$297$$ 4.00000i 0.232104i
$$298$$ − 10.0000i − 0.579284i
$$299$$ 6.00000 0.346989
$$300$$ 0 0
$$301$$ −8.00000 −0.461112
$$302$$ 0 0
$$303$$ 16.0000i 0.919176i
$$304$$ 6.00000 0.344124
$$305$$ 0 0
$$306$$ 8.00000 0.457330
$$307$$ − 12.0000i − 0.684876i −0.939540 0.342438i $$-0.888747\pi$$
0.939540 0.342438i $$-0.111253\pi$$
$$308$$ 8.00000i 0.455842i
$$309$$ −12.0000 −0.682656
$$310$$ 0 0
$$311$$ 24.0000 1.36092 0.680458 0.732787i $$-0.261781\pi$$
0.680458 + 0.732787i $$0.261781\pi$$
$$312$$ − 1.00000i − 0.0566139i
$$313$$ 6.00000i 0.339140i 0.985518 + 0.169570i $$0.0542379\pi$$
−0.985518 + 0.169570i $$0.945762\pi$$
$$314$$ −22.0000 −1.24153
$$315$$ 0 0
$$316$$ 8.00000 0.450035
$$317$$ 30.0000i 1.68497i 0.538721 + 0.842484i $$0.318908\pi$$
−0.538721 + 0.842484i $$0.681092\pi$$
$$318$$ 10.0000i 0.560772i
$$319$$ 16.0000 0.895828
$$320$$ 0 0
$$321$$ −12.0000 −0.669775
$$322$$ − 12.0000i − 0.668734i
$$323$$ − 48.0000i − 2.67079i
$$324$$ −1.00000 −0.0555556
$$325$$ 0 0
$$326$$ −16.0000 −0.886158
$$327$$ 12.0000i 0.663602i
$$328$$ − 2.00000i − 0.110432i
$$329$$ 0 0
$$330$$ 0 0
$$331$$ 10.0000 0.549650 0.274825 0.961494i $$-0.411380\pi$$
0.274825 + 0.961494i $$0.411380\pi$$
$$332$$ − 12.0000i − 0.658586i
$$333$$ − 2.00000i − 0.109599i
$$334$$ 4.00000 0.218870
$$335$$ 0 0
$$336$$ −2.00000 −0.109109
$$337$$ − 14.0000i − 0.762629i −0.924445 0.381314i $$-0.875472\pi$$
0.924445 0.381314i $$-0.124528\pi$$
$$338$$ 1.00000i 0.0543928i
$$339$$ 20.0000 1.08625
$$340$$ 0 0
$$341$$ 0 0
$$342$$ 6.00000i 0.324443i
$$343$$ − 20.0000i − 1.07990i
$$344$$ 4.00000 0.215666
$$345$$ 0 0
$$346$$ 22.0000 1.18273
$$347$$ 24.0000i 1.28839i 0.764862 + 0.644194i $$0.222807\pi$$
−0.764862 + 0.644194i $$0.777193\pi$$
$$348$$ 4.00000i 0.214423i
$$349$$ 28.0000 1.49881 0.749403 0.662114i $$-0.230341\pi$$
0.749403 + 0.662114i $$0.230341\pi$$
$$350$$ 0 0
$$351$$ 1.00000 0.0533761
$$352$$ − 4.00000i − 0.213201i
$$353$$ − 14.0000i − 0.745145i −0.928003 0.372572i $$-0.878476\pi$$
0.928003 0.372572i $$-0.121524\pi$$
$$354$$ 4.00000 0.212598
$$355$$ 0 0
$$356$$ −14.0000 −0.741999
$$357$$ 16.0000i 0.846810i
$$358$$ − 10.0000i − 0.528516i
$$359$$ 24.0000 1.26667 0.633336 0.773877i $$-0.281685\pi$$
0.633336 + 0.773877i $$0.281685\pi$$
$$360$$ 0 0
$$361$$ 17.0000 0.894737
$$362$$ − 10.0000i − 0.525588i
$$363$$ − 5.00000i − 0.262432i
$$364$$ 2.00000 0.104828
$$365$$ 0 0
$$366$$ 10.0000 0.522708
$$367$$ 36.0000i 1.87918i 0.342296 + 0.939592i $$0.388796\pi$$
−0.342296 + 0.939592i $$0.611204\pi$$
$$368$$ 6.00000i 0.312772i
$$369$$ 2.00000 0.104116
$$370$$ 0 0
$$371$$ −20.0000 −1.03835
$$372$$ 0 0
$$373$$ − 2.00000i − 0.103556i −0.998659 0.0517780i $$-0.983511\pi$$
0.998659 0.0517780i $$-0.0164888\pi$$
$$374$$ −32.0000 −1.65468
$$375$$ 0 0
$$376$$ 0 0
$$377$$ − 4.00000i − 0.206010i
$$378$$ − 2.00000i − 0.102869i
$$379$$ 18.0000 0.924598 0.462299 0.886724i $$-0.347025\pi$$
0.462299 + 0.886724i $$0.347025\pi$$
$$380$$ 0 0
$$381$$ −4.00000 −0.204926
$$382$$ 0 0
$$383$$ − 12.0000i − 0.613171i −0.951843 0.306586i $$-0.900813\pi$$
0.951843 0.306586i $$-0.0991866\pi$$
$$384$$ 1.00000 0.0510310
$$385$$ 0 0
$$386$$ −4.00000 −0.203595
$$387$$ 4.00000i 0.203331i
$$388$$ − 16.0000i − 0.812277i
$$389$$ −8.00000 −0.405616 −0.202808 0.979219i $$-0.565007\pi$$
−0.202808 + 0.979219i $$0.565007\pi$$
$$390$$ 0 0
$$391$$ 48.0000 2.42746
$$392$$ 3.00000i 0.151523i
$$393$$ − 10.0000i − 0.504433i
$$394$$ 18.0000 0.906827
$$395$$ 0 0
$$396$$ 4.00000 0.201008
$$397$$ − 14.0000i − 0.702640i −0.936255 0.351320i $$-0.885733\pi$$
0.936255 0.351320i $$-0.114267\pi$$
$$398$$ 0 0
$$399$$ −12.0000 −0.600751
$$400$$ 0 0
$$401$$ 30.0000 1.49813 0.749064 0.662497i $$-0.230503\pi$$
0.749064 + 0.662497i $$0.230503\pi$$
$$402$$ 12.0000i 0.598506i
$$403$$ 0 0
$$404$$ 16.0000 0.796030
$$405$$ 0 0
$$406$$ −8.00000 −0.397033
$$407$$ 8.00000i 0.396545i
$$408$$ − 8.00000i − 0.396059i
$$409$$ 26.0000 1.28562 0.642809 0.766027i $$-0.277769\pi$$
0.642809 + 0.766027i $$0.277769\pi$$
$$410$$ 0 0
$$411$$ 6.00000 0.295958
$$412$$ 12.0000i 0.591198i
$$413$$ 8.00000i 0.393654i
$$414$$ −6.00000 −0.294884
$$415$$ 0 0
$$416$$ −1.00000 −0.0490290
$$417$$ − 8.00000i − 0.391762i
$$418$$ − 24.0000i − 1.17388i
$$419$$ −10.0000 −0.488532 −0.244266 0.969708i $$-0.578547\pi$$
−0.244266 + 0.969708i $$0.578547\pi$$
$$420$$ 0 0
$$421$$ −8.00000 −0.389896 −0.194948 0.980814i $$-0.562454\pi$$
−0.194948 + 0.980814i $$0.562454\pi$$
$$422$$ 20.0000i 0.973585i
$$423$$ 0 0
$$424$$ 10.0000 0.485643
$$425$$ 0 0
$$426$$ 8.00000 0.387601
$$427$$ 20.0000i 0.967868i
$$428$$ 12.0000i 0.580042i
$$429$$ −4.00000 −0.193122
$$430$$ 0 0
$$431$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$432$$ 1.00000i 0.0481125i
$$433$$ − 2.00000i − 0.0961139i −0.998845 0.0480569i $$-0.984697\pi$$
0.998845 0.0480569i $$-0.0153029\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 12.0000 0.574696
$$437$$ 36.0000i 1.72211i
$$438$$ 8.00000i 0.382255i
$$439$$ 32.0000 1.52728 0.763638 0.645644i $$-0.223411\pi$$
0.763638 + 0.645644i $$0.223411\pi$$
$$440$$ 0 0
$$441$$ −3.00000 −0.142857
$$442$$ 8.00000i 0.380521i
$$443$$ 16.0000i 0.760183i 0.924949 + 0.380091i $$0.124107\pi$$
−0.924949 + 0.380091i $$0.875893\pi$$
$$444$$ −2.00000 −0.0949158
$$445$$ 0 0
$$446$$ −2.00000 −0.0947027
$$447$$ − 10.0000i − 0.472984i
$$448$$ 2.00000i 0.0944911i
$$449$$ 6.00000 0.283158 0.141579 0.989927i $$-0.454782\pi$$
0.141579 + 0.989927i $$0.454782\pi$$
$$450$$ 0 0
$$451$$ −8.00000 −0.376705
$$452$$ − 20.0000i − 0.940721i
$$453$$ 0 0
$$454$$ 4.00000 0.187729
$$455$$ 0 0
$$456$$ 6.00000 0.280976
$$457$$ − 8.00000i − 0.374224i −0.982339 0.187112i $$-0.940087\pi$$
0.982339 0.187112i $$-0.0599128\pi$$
$$458$$ 4.00000i 0.186908i
$$459$$ 8.00000 0.373408
$$460$$ 0 0
$$461$$ −6.00000 −0.279448 −0.139724 0.990190i $$-0.544622\pi$$
−0.139724 + 0.990190i $$0.544622\pi$$
$$462$$ 8.00000i 0.372194i
$$463$$ − 26.0000i − 1.20832i −0.796862 0.604161i $$-0.793508\pi$$
0.796862 0.604161i $$-0.206492\pi$$
$$464$$ 4.00000 0.185695
$$465$$ 0 0
$$466$$ 24.0000 1.11178
$$467$$ − 28.0000i − 1.29569i −0.761774 0.647843i $$-0.775671\pi$$
0.761774 0.647843i $$-0.224329\pi$$
$$468$$ − 1.00000i − 0.0462250i
$$469$$ −24.0000 −1.10822
$$470$$ 0 0
$$471$$ −22.0000 −1.01371
$$472$$ − 4.00000i − 0.184115i
$$473$$ − 16.0000i − 0.735681i
$$474$$ 8.00000 0.367452
$$475$$ 0 0
$$476$$ 16.0000 0.733359
$$477$$ 10.0000i 0.457869i
$$478$$ 16.0000i 0.731823i
$$479$$ −32.0000 −1.46212 −0.731059 0.682315i $$-0.760973\pi$$
−0.731059 + 0.682315i $$0.760973\pi$$
$$480$$ 0 0
$$481$$ 2.00000 0.0911922
$$482$$ − 2.00000i − 0.0910975i
$$483$$ − 12.0000i − 0.546019i
$$484$$ −5.00000 −0.227273
$$485$$ 0 0
$$486$$ −1.00000 −0.0453609
$$487$$ − 26.0000i − 1.17817i −0.808070 0.589086i $$-0.799488\pi$$
0.808070 0.589086i $$-0.200512\pi$$
$$488$$ − 10.0000i − 0.452679i
$$489$$ −16.0000 −0.723545
$$490$$ 0 0
$$491$$ 42.0000 1.89543 0.947717 0.319113i $$-0.103385\pi$$
0.947717 + 0.319113i $$0.103385\pi$$
$$492$$ − 2.00000i − 0.0901670i
$$493$$ − 32.0000i − 1.44121i
$$494$$ −6.00000 −0.269953
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 16.0000i 0.717698i
$$498$$ − 12.0000i − 0.537733i
$$499$$ −38.0000 −1.70111 −0.850557 0.525883i $$-0.823735\pi$$
−0.850557 + 0.525883i $$0.823735\pi$$
$$500$$ 0 0
$$501$$ 4.00000 0.178707
$$502$$ − 6.00000i − 0.267793i
$$503$$ 10.0000i 0.445878i 0.974832 + 0.222939i $$0.0715651\pi$$
−0.974832 + 0.222939i $$0.928435\pi$$
$$504$$ −2.00000 −0.0890871
$$505$$ 0 0
$$506$$ 24.0000 1.06693
$$507$$ 1.00000i 0.0444116i
$$508$$ 4.00000i 0.177471i
$$509$$ −18.0000 −0.797836 −0.398918 0.916987i $$-0.630614\pi$$
−0.398918 + 0.916987i $$0.630614\pi$$
$$510$$ 0 0
$$511$$ −16.0000 −0.707798
$$512$$ − 1.00000i − 0.0441942i
$$513$$ 6.00000i 0.264906i
$$514$$ −12.0000 −0.529297
$$515$$ 0 0
$$516$$ 4.00000 0.176090
$$517$$ 0 0
$$518$$ − 4.00000i − 0.175750i
$$519$$ 22.0000 0.965693
$$520$$ 0 0
$$521$$ −26.0000 −1.13908 −0.569540 0.821963i $$-0.692879\pi$$
−0.569540 + 0.821963i $$0.692879\pi$$
$$522$$ 4.00000i 0.175075i
$$523$$ 36.0000i 1.57417i 0.616844 + 0.787085i $$0.288411\pi$$
−0.616844 + 0.787085i $$0.711589\pi$$
$$524$$ −10.0000 −0.436852
$$525$$ 0 0
$$526$$ 2.00000 0.0872041
$$527$$ 0 0
$$528$$ − 4.00000i − 0.174078i
$$529$$ −13.0000 −0.565217
$$530$$ 0 0
$$531$$ 4.00000 0.173585
$$532$$ 12.0000i 0.520266i
$$533$$ 2.00000i 0.0866296i
$$534$$ −14.0000 −0.605839
$$535$$ 0 0
$$536$$ 12.0000 0.518321
$$537$$ − 10.0000i − 0.431532i
$$538$$ − 24.0000i − 1.03471i
$$539$$ 12.0000 0.516877
$$540$$ 0 0
$$541$$ −8.00000 −0.343947 −0.171973 0.985102i $$-0.555014\pi$$
−0.171973 + 0.985102i $$0.555014\pi$$
$$542$$ − 16.0000i − 0.687259i
$$543$$ − 10.0000i − 0.429141i
$$544$$ −8.00000 −0.342997
$$545$$ 0 0
$$546$$ 2.00000 0.0855921
$$547$$ 20.0000i 0.855138i 0.903983 + 0.427569i $$0.140630\pi$$
−0.903983 + 0.427569i $$0.859370\pi$$
$$548$$ − 6.00000i − 0.256307i
$$549$$ 10.0000 0.426790
$$550$$ 0 0
$$551$$ 24.0000 1.02243
$$552$$ 6.00000i 0.255377i
$$553$$ 16.0000i 0.680389i
$$554$$ 10.0000 0.424859
$$555$$ 0 0
$$556$$ −8.00000 −0.339276
$$557$$ − 2.00000i − 0.0847427i −0.999102 0.0423714i $$-0.986509\pi$$
0.999102 0.0423714i $$-0.0134913\pi$$
$$558$$ 0 0
$$559$$ −4.00000 −0.169182
$$560$$ 0 0
$$561$$ −32.0000 −1.35104
$$562$$ 6.00000i 0.253095i
$$563$$ 24.0000i 1.01148i 0.862686 + 0.505740i $$0.168780\pi$$
−0.862686 + 0.505740i $$0.831220\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ −4.00000 −0.168133
$$567$$ − 2.00000i − 0.0839921i
$$568$$ − 8.00000i − 0.335673i
$$569$$ −6.00000 −0.251533 −0.125767 0.992060i $$-0.540139\pi$$
−0.125767 + 0.992060i $$0.540139\pi$$
$$570$$ 0 0
$$571$$ 40.0000 1.67395 0.836974 0.547243i $$-0.184323\pi$$
0.836974 + 0.547243i $$0.184323\pi$$
$$572$$ 4.00000i 0.167248i
$$573$$ 0 0
$$574$$ 4.00000 0.166957
$$575$$ 0 0
$$576$$ 1.00000 0.0416667
$$577$$ − 12.0000i − 0.499567i −0.968302 0.249783i $$-0.919641\pi$$
0.968302 0.249783i $$-0.0803594\pi$$
$$578$$ 47.0000i 1.95494i
$$579$$ −4.00000 −0.166234
$$580$$ 0 0
$$581$$ 24.0000 0.995688
$$582$$ − 16.0000i − 0.663221i
$$583$$ − 40.0000i − 1.65663i
$$584$$ 8.00000 0.331042
$$585$$ 0 0
$$586$$ −26.0000 −1.07405
$$587$$ − 36.0000i − 1.48588i −0.669359 0.742940i $$-0.733431\pi$$
0.669359 0.742940i $$-0.266569\pi$$
$$588$$ 3.00000i 0.123718i
$$589$$ 0 0
$$590$$ 0 0
$$591$$ 18.0000 0.740421
$$592$$ 2.00000i 0.0821995i
$$593$$ 22.0000i 0.903432i 0.892162 + 0.451716i $$0.149188\pi$$
−0.892162 + 0.451716i $$0.850812\pi$$
$$594$$ 4.00000 0.164122
$$595$$ 0 0
$$596$$ −10.0000 −0.409616
$$597$$ 0 0
$$598$$ − 6.00000i − 0.245358i
$$599$$ −24.0000 −0.980613 −0.490307 0.871550i $$-0.663115\pi$$
−0.490307 + 0.871550i $$0.663115\pi$$
$$600$$ 0 0
$$601$$ 22.0000 0.897399 0.448699 0.893683i $$-0.351887\pi$$
0.448699 + 0.893683i $$0.351887\pi$$
$$602$$ 8.00000i 0.326056i
$$603$$ 12.0000i 0.488678i
$$604$$ 0 0
$$605$$ 0 0
$$606$$ 16.0000 0.649956
$$607$$ 28.0000i 1.13648i 0.822861 + 0.568242i $$0.192376\pi$$
−0.822861 + 0.568242i $$0.807624\pi$$
$$608$$ − 6.00000i − 0.243332i
$$609$$ −8.00000 −0.324176
$$610$$ 0 0
$$611$$ 0 0
$$612$$ − 8.00000i − 0.323381i
$$613$$ − 10.0000i − 0.403896i −0.979396 0.201948i $$-0.935273\pi$$
0.979396 0.201948i $$-0.0647272\pi$$
$$614$$ −12.0000 −0.484281
$$615$$ 0 0
$$616$$ 8.00000 0.322329
$$617$$ 6.00000i 0.241551i 0.992680 + 0.120775i $$0.0385381\pi$$
−0.992680 + 0.120775i $$0.961462\pi$$
$$618$$ 12.0000i 0.482711i
$$619$$ 10.0000 0.401934 0.200967 0.979598i $$-0.435592\pi$$
0.200967 + 0.979598i $$0.435592\pi$$
$$620$$ 0 0
$$621$$ −6.00000 −0.240772
$$622$$ − 24.0000i − 0.962312i
$$623$$ − 28.0000i − 1.12180i
$$624$$ −1.00000 −0.0400320
$$625$$ 0 0
$$626$$ 6.00000 0.239808
$$627$$ − 24.0000i − 0.958468i
$$628$$ 22.0000i 0.877896i
$$629$$ 16.0000 0.637962
$$630$$ 0 0
$$631$$ 12.0000 0.477712 0.238856 0.971055i $$-0.423228\pi$$
0.238856 + 0.971055i $$0.423228\pi$$
$$632$$ − 8.00000i − 0.318223i
$$633$$ 20.0000i 0.794929i
$$634$$ 30.0000 1.19145
$$635$$ 0 0
$$636$$ 10.0000 0.396526
$$637$$ − 3.00000i − 0.118864i
$$638$$ − 16.0000i − 0.633446i
$$639$$ 8.00000 0.316475
$$640$$ 0 0
$$641$$ 18.0000 0.710957 0.355479 0.934684i $$-0.384318\pi$$
0.355479 + 0.934684i $$0.384318\pi$$
$$642$$ 12.0000i 0.473602i
$$643$$ 44.0000i 1.73519i 0.497271 + 0.867595i $$0.334335\pi$$
−0.497271 + 0.867595i $$0.665665\pi$$
$$644$$ −12.0000 −0.472866
$$645$$ 0 0
$$646$$ −48.0000 −1.88853
$$647$$ 30.0000i 1.17942i 0.807614 + 0.589711i $$0.200758\pi$$
−0.807614 + 0.589711i $$0.799242\pi$$
$$648$$ 1.00000i 0.0392837i
$$649$$ −16.0000 −0.628055
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 16.0000i 0.626608i
$$653$$ 42.0000i 1.64359i 0.569785 + 0.821794i $$0.307026\pi$$
−0.569785 + 0.821794i $$0.692974\pi$$
$$654$$ 12.0000 0.469237
$$655$$ 0 0
$$656$$ −2.00000 −0.0780869
$$657$$ 8.00000i 0.312110i
$$658$$ 0 0
$$659$$ −2.00000 −0.0779089 −0.0389545 0.999241i $$-0.512403\pi$$
−0.0389545 + 0.999241i $$0.512403\pi$$
$$660$$ 0 0
$$661$$ 48.0000 1.86698 0.933492 0.358599i $$-0.116745\pi$$
0.933492 + 0.358599i $$0.116745\pi$$
$$662$$ − 10.0000i − 0.388661i
$$663$$ 8.00000i 0.310694i
$$664$$ −12.0000 −0.465690
$$665$$ 0 0
$$666$$ −2.00000 −0.0774984
$$667$$ 24.0000i 0.929284i
$$668$$ − 4.00000i − 0.154765i
$$669$$ −2.00000 −0.0773245
$$670$$ 0 0
$$671$$ −40.0000 −1.54418
$$672$$ 2.00000i 0.0771517i
$$673$$ 26.0000i 1.00223i 0.865382 + 0.501113i $$0.167076\pi$$
−0.865382 + 0.501113i $$0.832924\pi$$
$$674$$ −14.0000 −0.539260
$$675$$ 0 0
$$676$$ 1.00000 0.0384615
$$677$$ 30.0000i 1.15299i 0.817099 + 0.576497i $$0.195581\pi$$
−0.817099 + 0.576497i $$0.804419\pi$$
$$678$$ − 20.0000i − 0.768095i
$$679$$ 32.0000 1.22805
$$680$$ 0 0
$$681$$ 4.00000 0.153280
$$682$$ 0 0
$$683$$ − 44.0000i − 1.68361i −0.539779 0.841807i $$-0.681492\pi$$
0.539779 0.841807i $$-0.318508\pi$$
$$684$$ 6.00000 0.229416
$$685$$ 0 0
$$686$$ −20.0000 −0.763604
$$687$$ 4.00000i 0.152610i
$$688$$ − 4.00000i − 0.152499i
$$689$$ −10.0000 −0.380970
$$690$$ 0 0
$$691$$ −14.0000 −0.532585 −0.266293 0.963892i $$-0.585799\pi$$
−0.266293 + 0.963892i $$0.585799\pi$$
$$692$$ − 22.0000i − 0.836315i
$$693$$ 8.00000i 0.303895i
$$694$$ 24.0000 0.911028
$$695$$ 0 0
$$696$$ 4.00000 0.151620
$$697$$ 16.0000i 0.606043i
$$698$$ − 28.0000i − 1.05982i
$$699$$ 24.0000 0.907763
$$700$$ 0 0
$$701$$ 32.0000 1.20862 0.604312 0.796748i $$-0.293448\pi$$
0.604312 + 0.796748i $$0.293448\pi$$
$$702$$ − 1.00000i − 0.0377426i
$$703$$ 12.0000i 0.452589i
$$704$$ −4.00000 −0.150756
$$705$$ 0 0
$$706$$ −14.0000 −0.526897
$$707$$ 32.0000i 1.20348i
$$708$$ − 4.00000i − 0.150329i
$$709$$ 4.00000 0.150223 0.0751116 0.997175i $$-0.476069\pi$$
0.0751116 + 0.997175i $$0.476069\pi$$
$$710$$ 0 0
$$711$$ 8.00000 0.300023
$$712$$ 14.0000i 0.524672i
$$713$$ 0 0
$$714$$ 16.0000 0.598785
$$715$$ 0 0
$$716$$ −10.0000 −0.373718
$$717$$ 16.0000i 0.597531i
$$718$$ − 24.0000i − 0.895672i
$$719$$ −36.0000 −1.34257 −0.671287 0.741198i $$-0.734258\pi$$
−0.671287 + 0.741198i $$0.734258\pi$$
$$720$$ 0 0
$$721$$ −24.0000 −0.893807
$$722$$ − 17.0000i − 0.632674i
$$723$$ − 2.00000i − 0.0743808i
$$724$$ −10.0000 −0.371647
$$725$$ 0 0
$$726$$ −5.00000 −0.185567
$$727$$ − 16.0000i − 0.593407i −0.954970 0.296704i $$-0.904113\pi$$
0.954970 0.296704i $$-0.0958873\pi$$
$$728$$ − 2.00000i − 0.0741249i
$$729$$ −1.00000 −0.0370370
$$730$$ 0 0
$$731$$ −32.0000 −1.18356
$$732$$ − 10.0000i − 0.369611i
$$733$$ 38.0000i 1.40356i 0.712393 + 0.701781i $$0.247612\pi$$
−0.712393 + 0.701781i $$0.752388\pi$$
$$734$$ 36.0000 1.32878
$$735$$ 0 0
$$736$$ 6.00000 0.221163
$$737$$ − 48.0000i − 1.76810i
$$738$$ − 2.00000i − 0.0736210i
$$739$$ −30.0000 −1.10357 −0.551784 0.833987i $$-0.686053\pi$$
−0.551784 + 0.833987i $$0.686053\pi$$
$$740$$ 0 0
$$741$$ −6.00000 −0.220416
$$742$$ 20.0000i 0.734223i
$$743$$ 12.0000i 0.440237i 0.975473 + 0.220119i $$0.0706445\pi$$
−0.975473 + 0.220119i $$0.929356\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ −2.00000 −0.0732252
$$747$$ − 12.0000i − 0.439057i
$$748$$ 32.0000i 1.17004i
$$749$$ −24.0000 −0.876941
$$750$$ 0 0
$$751$$ −16.0000 −0.583848 −0.291924 0.956441i $$-0.594295\pi$$
−0.291924 + 0.956441i $$0.594295\pi$$
$$752$$ 0 0
$$753$$ − 6.00000i − 0.218652i
$$754$$ −4.00000 −0.145671
$$755$$ 0 0
$$756$$ −2.00000 −0.0727393
$$757$$ 6.00000i 0.218074i 0.994038 + 0.109037i $$0.0347767\pi$$
−0.994038 + 0.109037i $$0.965223\pi$$
$$758$$ − 18.0000i − 0.653789i
$$759$$ 24.0000 0.871145
$$760$$ 0 0
$$761$$ 22.0000 0.797499 0.398750 0.917060i $$-0.369444\pi$$
0.398750 + 0.917060i $$0.369444\pi$$
$$762$$ 4.00000i 0.144905i
$$763$$ 24.0000i 0.868858i
$$764$$ 0 0
$$765$$ 0 0
$$766$$ −12.0000 −0.433578
$$767$$ 4.00000i 0.144432i
$$768$$ − 1.00000i − 0.0360844i
$$769$$ −10.0000 −0.360609 −0.180305 0.983611i $$-0.557708\pi$$
−0.180305 + 0.983611i $$0.557708\pi$$
$$770$$ 0 0
$$771$$ −12.0000 −0.432169
$$772$$ 4.00000i 0.143963i
$$773$$ 38.0000i 1.36677i 0.730061 + 0.683383i $$0.239492\pi$$
−0.730061 + 0.683383i $$0.760508\pi$$
$$774$$ 4.00000 0.143777
$$775$$ 0 0
$$776$$ −16.0000 −0.574367
$$777$$ − 4.00000i − 0.143499i
$$778$$ 8.00000i 0.286814i
$$779$$ −12.0000 −0.429945
$$780$$ 0 0
$$781$$ −32.0000 −1.14505
$$782$$ − 48.0000i − 1.71648i
$$783$$ 4.00000i 0.142948i
$$784$$ 3.00000 0.107143
$$785$$ 0 0
$$786$$ −10.0000 −0.356688
$$787$$ 32.0000i 1.14068i 0.821410 + 0.570338i $$0.193188\pi$$
−0.821410 + 0.570338i $$0.806812\pi$$
$$788$$ − 18.0000i − 0.641223i
$$789$$ 2.00000 0.0712019
$$790$$ 0 0
$$791$$ 40.0000 1.42224
$$792$$ − 4.00000i − 0.142134i
$$793$$ 10.0000i 0.355110i
$$794$$ −14.0000 −0.496841
$$795$$ 0 0
$$796$$ 0 0
$$797$$ 10.0000i 0.354218i 0.984191 + 0.177109i $$0.0566745\pi$$
−0.984191 + 0.177109i $$0.943325\pi$$
$$798$$ 12.0000i 0.424795i
$$799$$ 0 0
$$800$$ 0 0
$$801$$ −14.0000 −0.494666
$$802$$ − 30.0000i − 1.05934i
$$803$$ − 32.0000i − 1.12926i
$$804$$ 12.0000 0.423207
$$805$$ 0 0
$$806$$ 0 0
$$807$$ − 24.0000i − 0.844840i
$$808$$ − 16.0000i − 0.562878i
$$809$$ −2.00000 −0.0703163 −0.0351581 0.999382i $$-0.511193\pi$$
−0.0351581 + 0.999382i $$0.511193\pi$$
$$810$$ 0 0
$$811$$ −38.0000 −1.33436 −0.667180 0.744896i $$-0.732499\pi$$
−0.667180 + 0.744896i $$0.732499\pi$$
$$812$$ 8.00000i 0.280745i
$$813$$ − 16.0000i − 0.561144i
$$814$$ 8.00000 0.280400
$$815$$ 0 0
$$816$$ −8.00000 −0.280056
$$817$$ − 24.0000i − 0.839654i
$$818$$ − 26.0000i − 0.909069i
$$819$$ 2.00000 0.0698857
$$820$$ 0 0
$$821$$ −10.0000 −0.349002 −0.174501 0.984657i $$-0.555831\pi$$
−0.174501 + 0.984657i $$0.555831\pi$$
$$822$$ − 6.00000i − 0.209274i
$$823$$ − 52.0000i − 1.81261i −0.422628 0.906303i $$-0.638892\pi$$
0.422628 0.906303i $$-0.361108\pi$$
$$824$$ 12.0000 0.418040
$$825$$ 0 0
$$826$$ 8.00000 0.278356
$$827$$ 12.0000i 0.417281i 0.977992 + 0.208640i $$0.0669038\pi$$
−0.977992 + 0.208640i $$0.933096\pi$$
$$828$$ 6.00000i 0.208514i
$$829$$ 10.0000 0.347314 0.173657 0.984806i $$-0.444442\pi$$
0.173657 + 0.984806i $$0.444442\pi$$
$$830$$ 0 0
$$831$$ 10.0000 0.346896
$$832$$ 1.00000i 0.0346688i
$$833$$ − 24.0000i − 0.831551i
$$834$$ −8.00000 −0.277017
$$835$$ 0 0
$$836$$ −24.0000 −0.830057
$$837$$ 0 0
$$838$$ 10.0000i 0.345444i
$$839$$ 40.0000 1.38095 0.690477 0.723355i $$-0.257401\pi$$
0.690477 + 0.723355i $$0.257401\pi$$
$$840$$ 0 0
$$841$$ −13.0000 −0.448276
$$842$$ 8.00000i 0.275698i
$$843$$ 6.00000i 0.206651i
$$844$$ 20.0000 0.688428
$$845$$ 0 0
$$846$$ 0 0
$$847$$ − 10.0000i − 0.343604i
$$848$$ − 10.0000i − 0.343401i
$$849$$ −4.00000 −0.137280
$$850$$ 0 0
$$851$$ −12.0000 −0.411355
$$852$$ − 8.00000i − 0.274075i
$$853$$ 46.0000i 1.57501i 0.616308 + 0.787505i $$0.288628\pi$$
−0.616308 + 0.787505i $$0.711372\pi$$
$$854$$ 20.0000 0.684386
$$855$$ 0 0
$$856$$ 12.0000 0.410152
$$857$$ − 4.00000i − 0.136637i −0.997664 0.0683187i $$-0.978237\pi$$
0.997664 0.0683187i $$-0.0217635\pi$$
$$858$$ 4.00000i 0.136558i
$$859$$ −36.0000 −1.22830 −0.614152 0.789188i $$-0.710502\pi$$
−0.614152 + 0.789188i $$0.710502\pi$$
$$860$$ 0 0
$$861$$ 4.00000 0.136320
$$862$$ 0 0
$$863$$ − 12.0000i − 0.408485i −0.978920 0.204242i $$-0.934527\pi$$
0.978920 0.204242i $$-0.0654731\pi$$
$$864$$ 1.00000 0.0340207
$$865$$ 0 0
$$866$$ −2.00000 −0.0679628
$$867$$ 47.0000i 1.59620i
$$868$$ 0 0
$$869$$ −32.0000 −1.08553
$$870$$ 0 0
$$871$$ −12.0000 −0.406604
$$872$$ − 12.0000i − 0.406371i
$$873$$ − 16.0000i − 0.541518i
$$874$$ 36.0000 1.21772
$$875$$ 0 0
$$876$$ 8.00000 0.270295
$$877$$ 14.0000i 0.472746i 0.971662 + 0.236373i $$0.0759588\pi$$
−0.971662 + 0.236373i $$0.924041\pi$$
$$878$$ − 32.0000i − 1.07995i
$$879$$ −26.0000 −0.876958
$$880$$ 0 0
$$881$$ −18.0000 −0.606435 −0.303218 0.952921i $$-0.598061\pi$$
−0.303218 + 0.952921i $$0.598061\pi$$
$$882$$ 3.00000i 0.101015i
$$883$$ − 28.0000i − 0.942275i −0.882060 0.471138i $$-0.843844\pi$$
0.882060 0.471138i $$-0.156156\pi$$
$$884$$ 8.00000 0.269069
$$885$$ 0 0
$$886$$ 16.0000 0.537531
$$887$$ 2.00000i 0.0671534i 0.999436 + 0.0335767i $$0.0106898\pi$$
−0.999436 + 0.0335767i $$0.989310\pi$$
$$888$$ 2.00000i 0.0671156i
$$889$$ −8.00000 −0.268311
$$890$$ 0 0
$$891$$ 4.00000 0.134005
$$892$$ 2.00000i 0.0669650i
$$893$$ 0 0
$$894$$ −10.0000 −0.334450
$$895$$ 0 0
$$896$$ 2.00000 0.0668153
$$897$$ − 6.00000i − 0.200334i
$$898$$ − 6.00000i − 0.200223i
$$899$$ 0 0
$$900$$ 0 0
$$901$$ −80.0000 −2.66519
$$902$$ 8.00000i 0.266371i
$$903$$ 8.00000i 0.266223i
$$904$$ −20.0000 −0.665190
$$905$$ 0 0
$$906$$ 0 0
$$907$$ − 52.0000i − 1.72663i −0.504664 0.863316i $$-0.668384\pi$$
0.504664 0.863316i $$-0.331616\pi$$
$$908$$ − 4.00000i − 0.132745i
$$909$$ 16.0000 0.530687
$$910$$ 0 0
$$911$$ −20.0000 −0.662630 −0.331315 0.943520i $$-0.607492\pi$$
−0.331315 + 0.943520i $$0.607492\pi$$
$$912$$ − 6.00000i − 0.198680i
$$913$$ 48.0000i 1.58857i
$$914$$ −8.00000 −0.264616
$$915$$ 0 0
$$916$$ 4.00000 0.132164
$$917$$ − 20.0000i − 0.660458i
$$918$$ − 8.00000i − 0.264039i
$$919$$ −40.0000 −1.31948 −0.659739 0.751495i $$-0.729333\pi$$
−0.659739 + 0.751495i $$0.729333\pi$$
$$920$$ 0 0
$$921$$ −12.0000 −0.395413
$$922$$ 6.00000i 0.197599i
$$923$$ 8.00000i 0.263323i
$$924$$ 8.00000 0.263181
$$925$$ 0 0
$$926$$ −26.0000 −0.854413
$$927$$ 12.0000i 0.394132i
$$928$$ − 4.00000i − 0.131306i
$$929$$ 30.0000 0.984268 0.492134 0.870519i $$-0.336217\pi$$
0.492134 + 0.870519i $$0.336217\pi$$
$$930$$ 0 0
$$931$$ 18.0000 0.589926
$$932$$ − 24.0000i − 0.786146i
$$933$$ − 24.0000i − 0.785725i
$$934$$ −28.0000 −0.916188
$$935$$ 0 0
$$936$$ −1.00000 −0.0326860
$$937$$ − 18.0000i − 0.588034i −0.955800 0.294017i $$-0.905008\pi$$
0.955800 0.294017i $$-0.0949923\pi$$
$$938$$ 24.0000i 0.783628i
$$939$$ 6.00000 0.195803
$$940$$ 0 0
$$941$$ −34.0000 −1.10837 −0.554184 0.832394i $$-0.686970\pi$$
−0.554184 + 0.832394i $$0.686970\pi$$
$$942$$ 22.0000i 0.716799i
$$943$$ − 12.0000i − 0.390774i
$$944$$ −4.00000 −0.130189
$$945$$ 0 0
$$946$$ −16.0000 −0.520205
$$947$$ − 4.00000i − 0.129983i −0.997886 0.0649913i $$-0.979298\pi$$
0.997886 0.0649913i $$-0.0207020\pi$$
$$948$$ − 8.00000i − 0.259828i
$$949$$ −8.00000 −0.259691
$$950$$ 0 0
$$951$$ 30.0000 0.972817
$$952$$ − 16.0000i − 0.518563i
$$953$$ 24.0000i 0.777436i 0.921357 + 0.388718i $$0.127082\pi$$
−0.921357 + 0.388718i $$0.872918\pi$$
$$954$$ 10.0000 0.323762
$$955$$ 0 0
$$956$$ 16.0000 0.517477
$$957$$ − 16.0000i − 0.517207i
$$958$$ 32.0000i 1.03387i
$$959$$ 12.0000 0.387500
$$960$$ 0 0
$$961$$ −31.0000 −1.00000
$$962$$ − 2.00000i − 0.0644826i
$$963$$ 12.0000i 0.386695i
$$964$$ −2.00000 −0.0644157
$$965$$ 0 0
$$966$$ −12.0000 −0.386094
$$967$$ 14.0000i 0.450210i 0.974335 + 0.225105i $$0.0722725\pi$$
−0.974335 + 0.225105i $$0.927728\pi$$
$$968$$ 5.00000i 0.160706i
$$969$$ −48.0000 −1.54198
$$970$$ 0 0
$$971$$ 38.0000 1.21948 0.609739 0.792602i $$-0.291274\pi$$
0.609739 + 0.792602i $$0.291274\pi$$
$$972$$ 1.00000i 0.0320750i
$$973$$ − 16.0000i − 0.512936i
$$974$$ −26.0000 −0.833094
$$975$$ 0 0
$$976$$ −10.0000 −0.320092
$$977$$ − 30.0000i − 0.959785i −0.877327 0.479893i $$-0.840676\pi$$
0.877327 0.479893i $$-0.159324\pi$$
$$978$$ 16.0000i 0.511624i
$$979$$ 56.0000 1.78977
$$980$$ 0 0
$$981$$ 12.0000 0.383131
$$982$$ − 42.0000i − 1.34027i
$$983$$ − 52.0000i − 1.65854i −0.558846 0.829271i $$-0.688756\pi$$
0.558846 0.829271i $$-0.311244\pi$$
$$984$$ −2.00000 −0.0637577
$$985$$ 0 0
$$986$$ −32.0000 −1.01909
$$987$$ 0 0
$$988$$ 6.00000i 0.190885i
$$989$$ 24.0000 0.763156
$$990$$ 0 0
$$991$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$992$$ 0 0
$$993$$ − 10.0000i − 0.317340i
$$994$$ 16.0000 0.507489
$$995$$ 0 0
$$996$$ −12.0000 −0.380235
$$997$$ 26.0000i 0.823428i 0.911313 + 0.411714i $$0.135070\pi$$
−0.911313 + 0.411714i $$0.864930\pi$$
$$998$$ 38.0000i 1.20287i
$$999$$ −2.00000 −0.0632772
Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000

## Twists

By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 1950.2.e.f.1249.1 2
3.2 odd 2 5850.2.e.i.5149.2 2
5.2 odd 4 390.2.a.e.1.1 1
5.3 odd 4 1950.2.a.h.1.1 1
5.4 even 2 inner 1950.2.e.f.1249.2 2
15.2 even 4 1170.2.a.e.1.1 1
15.8 even 4 5850.2.a.bi.1.1 1
15.14 odd 2 5850.2.e.i.5149.1 2
20.7 even 4 3120.2.a.o.1.1 1
60.47 odd 4 9360.2.a.bh.1.1 1
65.12 odd 4 5070.2.a.e.1.1 1
65.47 even 4 5070.2.b.e.1351.2 2
65.57 even 4 5070.2.b.e.1351.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
390.2.a.e.1.1 1 5.2 odd 4
1170.2.a.e.1.1 1 15.2 even 4
1950.2.a.h.1.1 1 5.3 odd 4
1950.2.e.f.1249.1 2 1.1 even 1 trivial
1950.2.e.f.1249.2 2 5.4 even 2 inner
3120.2.a.o.1.1 1 20.7 even 4
5070.2.a.e.1.1 1 65.12 odd 4
5070.2.b.e.1351.1 2 65.57 even 4
5070.2.b.e.1351.2 2 65.47 even 4
5850.2.a.bi.1.1 1 15.8 even 4
5850.2.e.i.5149.1 2 15.14 odd 2
5850.2.e.i.5149.2 2 3.2 odd 2
9360.2.a.bh.1.1 1 60.47 odd 4