# Properties

 Label 1850.2.b.e.149.1 Level $1850$ Weight $2$ Character 1850.149 Analytic conductor $14.772$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1850 = 2 \cdot 5^{2} \cdot 37$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1850.b (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$14.7723243739$$ 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: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 149.1 Root $$1.00000i$$ of defining polynomial Character $$\chi$$ $$=$$ 1850.149 Dual form 1850.2.b.e.149.2

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} +4.00000i q^{7} +1.00000i q^{8} +2.00000 q^{9} +O(q^{10})$$ $$q-1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} +4.00000i q^{7} +1.00000i q^{8} +2.00000 q^{9} +3.00000 q^{11} -1.00000i q^{12} -6.00000i q^{13} +4.00000 q^{14} +1.00000 q^{16} +3.00000i q^{17} -2.00000i q^{18} +3.00000 q^{19} -4.00000 q^{21} -3.00000i q^{22} -2.00000i q^{23} -1.00000 q^{24} -6.00000 q^{26} +5.00000i q^{27} -4.00000i q^{28} -1.00000i q^{32} +3.00000i q^{33} +3.00000 q^{34} -2.00000 q^{36} -1.00000i q^{37} -3.00000i q^{38} +6.00000 q^{39} -3.00000 q^{41} +4.00000i q^{42} +4.00000i q^{43} -3.00000 q^{44} -2.00000 q^{46} +4.00000i q^{47} +1.00000i q^{48} -9.00000 q^{49} -3.00000 q^{51} +6.00000i q^{52} +2.00000i q^{53} +5.00000 q^{54} -4.00000 q^{56} +3.00000i q^{57} +12.0000 q^{59} +12.0000 q^{61} +8.00000i q^{63} -1.00000 q^{64} +3.00000 q^{66} +9.00000i q^{67} -3.00000i q^{68} +2.00000 q^{69} -2.00000 q^{71} +2.00000i q^{72} +9.00000i q^{73} -1.00000 q^{74} -3.00000 q^{76} +12.0000i q^{77} -6.00000i q^{78} +2.00000 q^{79} +1.00000 q^{81} +3.00000i q^{82} +7.00000i q^{83} +4.00000 q^{84} +4.00000 q^{86} +3.00000i q^{88} +3.00000 q^{89} +24.0000 q^{91} +2.00000i q^{92} +4.00000 q^{94} +1.00000 q^{96} +2.00000i q^{97} +9.00000i q^{98} +6.00000 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{4} + 2 q^{6} + 4 q^{9}+O(q^{10})$$ 2 * q - 2 * q^4 + 2 * q^6 + 4 * q^9 $$2 q - 2 q^{4} + 2 q^{6} + 4 q^{9} + 6 q^{11} + 8 q^{14} + 2 q^{16} + 6 q^{19} - 8 q^{21} - 2 q^{24} - 12 q^{26} + 6 q^{34} - 4 q^{36} + 12 q^{39} - 6 q^{41} - 6 q^{44} - 4 q^{46} - 18 q^{49} - 6 q^{51} + 10 q^{54} - 8 q^{56} + 24 q^{59} + 24 q^{61} - 2 q^{64} + 6 q^{66} + 4 q^{69} - 4 q^{71} - 2 q^{74} - 6 q^{76} + 4 q^{79} + 2 q^{81} + 8 q^{84} + 8 q^{86} + 6 q^{89} + 48 q^{91} + 8 q^{94} + 2 q^{96} + 12 q^{99}+O(q^{100})$$ 2 * q - 2 * q^4 + 2 * q^6 + 4 * q^9 + 6 * q^11 + 8 * q^14 + 2 * q^16 + 6 * q^19 - 8 * q^21 - 2 * q^24 - 12 * q^26 + 6 * q^34 - 4 * q^36 + 12 * q^39 - 6 * q^41 - 6 * q^44 - 4 * q^46 - 18 * q^49 - 6 * q^51 + 10 * q^54 - 8 * q^56 + 24 * q^59 + 24 * q^61 - 2 * q^64 + 6 * q^66 + 4 * q^69 - 4 * q^71 - 2 * q^74 - 6 * q^76 + 4 * q^79 + 2 * q^81 + 8 * q^84 + 8 * q^86 + 6 * q^89 + 48 * q^91 + 8 * q^94 + 2 * q^96 + 12 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1850\mathbb{Z}\right)^\times$$.

 $$n$$ $$1001$$ $$1777$$ $$\chi(n)$$ $$1$$ $$-1$$

## 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 0.957427 + 0.288675i $$0.0932147\pi$$
−0.957427 + 0.288675i $$0.906785\pi$$
$$4$$ −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
$$9$$ 2.00000 0.666667
$$10$$ 0 0
$$11$$ 3.00000 0.904534 0.452267 0.891883i $$-0.350615\pi$$
0.452267 + 0.891883i $$0.350615\pi$$
$$12$$ − 1.00000i − 0.288675i
$$13$$ − 6.00000i − 1.66410i −0.554700 0.832050i $$-0.687167\pi$$
0.554700 0.832050i $$-0.312833\pi$$
$$14$$ 4.00000 1.06904
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ 3.00000i 0.727607i 0.931476 + 0.363803i $$0.118522\pi$$
−0.931476 + 0.363803i $$0.881478\pi$$
$$18$$ − 2.00000i − 0.471405i
$$19$$ 3.00000 0.688247 0.344124 0.938924i $$-0.388176\pi$$
0.344124 + 0.938924i $$0.388176\pi$$
$$20$$ 0 0
$$21$$ −4.00000 −0.872872
$$22$$ − 3.00000i − 0.639602i
$$23$$ − 2.00000i − 0.417029i −0.978019 0.208514i $$-0.933137\pi$$
0.978019 0.208514i $$-0.0668628\pi$$
$$24$$ −1.00000 −0.204124
$$25$$ 0 0
$$26$$ −6.00000 −1.17670
$$27$$ 5.00000i 0.962250i
$$28$$ − 4.00000i − 0.755929i
$$29$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$30$$ 0 0
$$31$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$32$$ − 1.00000i − 0.176777i
$$33$$ 3.00000i 0.522233i
$$34$$ 3.00000 0.514496
$$35$$ 0 0
$$36$$ −2.00000 −0.333333
$$37$$ − 1.00000i − 0.164399i
$$38$$ − 3.00000i − 0.486664i
$$39$$ 6.00000 0.960769
$$40$$ 0 0
$$41$$ −3.00000 −0.468521 −0.234261 0.972174i $$-0.575267\pi$$
−0.234261 + 0.972174i $$0.575267\pi$$
$$42$$ 4.00000i 0.617213i
$$43$$ 4.00000i 0.609994i 0.952353 + 0.304997i $$0.0986555\pi$$
−0.952353 + 0.304997i $$0.901344\pi$$
$$44$$ −3.00000 −0.452267
$$45$$ 0 0
$$46$$ −2.00000 −0.294884
$$47$$ 4.00000i 0.583460i 0.956501 + 0.291730i $$0.0942309\pi$$
−0.956501 + 0.291730i $$0.905769\pi$$
$$48$$ 1.00000i 0.144338i
$$49$$ −9.00000 −1.28571
$$50$$ 0 0
$$51$$ −3.00000 −0.420084
$$52$$ 6.00000i 0.832050i
$$53$$ 2.00000i 0.274721i 0.990521 + 0.137361i $$0.0438619\pi$$
−0.990521 + 0.137361i $$0.956138\pi$$
$$54$$ 5.00000 0.680414
$$55$$ 0 0
$$56$$ −4.00000 −0.534522
$$57$$ 3.00000i 0.397360i
$$58$$ 0 0
$$59$$ 12.0000 1.56227 0.781133 0.624364i $$-0.214642\pi$$
0.781133 + 0.624364i $$0.214642\pi$$
$$60$$ 0 0
$$61$$ 12.0000 1.53644 0.768221 0.640184i $$-0.221142\pi$$
0.768221 + 0.640184i $$0.221142\pi$$
$$62$$ 0 0
$$63$$ 8.00000i 1.00791i
$$64$$ −1.00000 −0.125000
$$65$$ 0 0
$$66$$ 3.00000 0.369274
$$67$$ 9.00000i 1.09952i 0.835321 + 0.549762i $$0.185282\pi$$
−0.835321 + 0.549762i $$0.814718\pi$$
$$68$$ − 3.00000i − 0.363803i
$$69$$ 2.00000 0.240772
$$70$$ 0 0
$$71$$ −2.00000 −0.237356 −0.118678 0.992933i $$-0.537866\pi$$
−0.118678 + 0.992933i $$0.537866\pi$$
$$72$$ 2.00000i 0.235702i
$$73$$ 9.00000i 1.05337i 0.850060 + 0.526685i $$0.176565\pi$$
−0.850060 + 0.526685i $$0.823435\pi$$
$$74$$ −1.00000 −0.116248
$$75$$ 0 0
$$76$$ −3.00000 −0.344124
$$77$$ 12.0000i 1.36753i
$$78$$ − 6.00000i − 0.679366i
$$79$$ 2.00000 0.225018 0.112509 0.993651i $$-0.464111\pi$$
0.112509 + 0.993651i $$0.464111\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 3.00000i 0.331295i
$$83$$ 7.00000i 0.768350i 0.923260 + 0.384175i $$0.125514\pi$$
−0.923260 + 0.384175i $$0.874486\pi$$
$$84$$ 4.00000 0.436436
$$85$$ 0 0
$$86$$ 4.00000 0.431331
$$87$$ 0 0
$$88$$ 3.00000i 0.319801i
$$89$$ 3.00000 0.317999 0.159000 0.987279i $$-0.449173\pi$$
0.159000 + 0.987279i $$0.449173\pi$$
$$90$$ 0 0
$$91$$ 24.0000 2.51588
$$92$$ 2.00000i 0.208514i
$$93$$ 0 0
$$94$$ 4.00000 0.412568
$$95$$ 0 0
$$96$$ 1.00000 0.102062
$$97$$ 2.00000i 0.203069i 0.994832 + 0.101535i $$0.0323753\pi$$
−0.994832 + 0.101535i $$0.967625\pi$$
$$98$$ 9.00000i 0.909137i
$$99$$ 6.00000 0.603023
$$100$$ 0 0
$$101$$ −10.0000 −0.995037 −0.497519 0.867453i $$-0.665755\pi$$
−0.497519 + 0.867453i $$0.665755\pi$$
$$102$$ 3.00000i 0.297044i
$$103$$ − 14.0000i − 1.37946i −0.724066 0.689730i $$-0.757729\pi$$
0.724066 0.689730i $$-0.242271\pi$$
$$104$$ 6.00000 0.588348
$$105$$ 0 0
$$106$$ 2.00000 0.194257
$$107$$ − 1.00000i − 0.0966736i −0.998831 0.0483368i $$-0.984608\pi$$
0.998831 0.0483368i $$-0.0153921\pi$$
$$108$$ − 5.00000i − 0.481125i
$$109$$ −18.0000 −1.72409 −0.862044 0.506834i $$-0.830816\pi$$
−0.862044 + 0.506834i $$0.830816\pi$$
$$110$$ 0 0
$$111$$ 1.00000 0.0949158
$$112$$ 4.00000i 0.377964i
$$113$$ − 5.00000i − 0.470360i −0.971952 0.235180i $$-0.924432\pi$$
0.971952 0.235180i $$-0.0755680\pi$$
$$114$$ 3.00000 0.280976
$$115$$ 0 0
$$116$$ 0 0
$$117$$ − 12.0000i − 1.10940i
$$118$$ − 12.0000i − 1.10469i
$$119$$ −12.0000 −1.10004
$$120$$ 0 0
$$121$$ −2.00000 −0.181818
$$122$$ − 12.0000i − 1.08643i
$$123$$ − 3.00000i − 0.270501i
$$124$$ 0 0
$$125$$ 0 0
$$126$$ 8.00000 0.712697
$$127$$ 8.00000i 0.709885i 0.934888 + 0.354943i $$0.115500\pi$$
−0.934888 + 0.354943i $$0.884500\pi$$
$$128$$ 1.00000i 0.0883883i
$$129$$ −4.00000 −0.352180
$$130$$ 0 0
$$131$$ 12.0000 1.04844 0.524222 0.851581i $$-0.324356\pi$$
0.524222 + 0.851581i $$0.324356\pi$$
$$132$$ − 3.00000i − 0.261116i
$$133$$ 12.0000i 1.04053i
$$134$$ 9.00000 0.777482
$$135$$ 0 0
$$136$$ −3.00000 −0.257248
$$137$$ − 7.00000i − 0.598050i −0.954245 0.299025i $$-0.903339\pi$$
0.954245 0.299025i $$-0.0966615\pi$$
$$138$$ − 2.00000i − 0.170251i
$$139$$ −3.00000 −0.254457 −0.127228 0.991873i $$-0.540608\pi$$
−0.127228 + 0.991873i $$0.540608\pi$$
$$140$$ 0 0
$$141$$ −4.00000 −0.336861
$$142$$ 2.00000i 0.167836i
$$143$$ − 18.0000i − 1.50524i
$$144$$ 2.00000 0.166667
$$145$$ 0 0
$$146$$ 9.00000 0.744845
$$147$$ − 9.00000i − 0.742307i
$$148$$ 1.00000i 0.0821995i
$$149$$ 18.0000 1.47462 0.737309 0.675556i $$-0.236096\pi$$
0.737309 + 0.675556i $$0.236096\pi$$
$$150$$ 0 0
$$151$$ 22.0000 1.79033 0.895167 0.445730i $$-0.147056\pi$$
0.895167 + 0.445730i $$0.147056\pi$$
$$152$$ 3.00000i 0.243332i
$$153$$ 6.00000i 0.485071i
$$154$$ 12.0000 0.966988
$$155$$ 0 0
$$156$$ −6.00000 −0.480384
$$157$$ 24.0000i 1.91541i 0.287754 + 0.957704i $$0.407091\pi$$
−0.287754 + 0.957704i $$0.592909\pi$$
$$158$$ − 2.00000i − 0.159111i
$$159$$ −2.00000 −0.158610
$$160$$ 0 0
$$161$$ 8.00000 0.630488
$$162$$ − 1.00000i − 0.0785674i
$$163$$ 17.0000i 1.33154i 0.746156 + 0.665771i $$0.231897\pi$$
−0.746156 + 0.665771i $$0.768103\pi$$
$$164$$ 3.00000 0.234261
$$165$$ 0 0
$$166$$ 7.00000 0.543305
$$167$$ − 18.0000i − 1.39288i −0.717614 0.696441i $$-0.754766\pi$$
0.717614 0.696441i $$-0.245234\pi$$
$$168$$ − 4.00000i − 0.308607i
$$169$$ −23.0000 −1.76923
$$170$$ 0 0
$$171$$ 6.00000 0.458831
$$172$$ − 4.00000i − 0.304997i
$$173$$ − 24.0000i − 1.82469i −0.409426 0.912343i $$-0.634271\pi$$
0.409426 0.912343i $$-0.365729\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 3.00000 0.226134
$$177$$ 12.0000i 0.901975i
$$178$$ − 3.00000i − 0.224860i
$$179$$ −21.0000 −1.56961 −0.784807 0.619740i $$-0.787238\pi$$
−0.784807 + 0.619740i $$0.787238\pi$$
$$180$$ 0 0
$$181$$ −10.0000 −0.743294 −0.371647 0.928374i $$-0.621207\pi$$
−0.371647 + 0.928374i $$0.621207\pi$$
$$182$$ − 24.0000i − 1.77900i
$$183$$ 12.0000i 0.887066i
$$184$$ 2.00000 0.147442
$$185$$ 0 0
$$186$$ 0 0
$$187$$ 9.00000i 0.658145i
$$188$$ − 4.00000i − 0.291730i
$$189$$ −20.0000 −1.45479
$$190$$ 0 0
$$191$$ 6.00000 0.434145 0.217072 0.976156i $$-0.430349\pi$$
0.217072 + 0.976156i $$0.430349\pi$$
$$192$$ − 1.00000i − 0.0721688i
$$193$$ − 11.0000i − 0.791797i −0.918294 0.395899i $$-0.870433\pi$$
0.918294 0.395899i $$-0.129567\pi$$
$$194$$ 2.00000 0.143592
$$195$$ 0 0
$$196$$ 9.00000 0.642857
$$197$$ 12.0000i 0.854965i 0.904024 + 0.427482i $$0.140599\pi$$
−0.904024 + 0.427482i $$0.859401\pi$$
$$198$$ − 6.00000i − 0.426401i
$$199$$ −8.00000 −0.567105 −0.283552 0.958957i $$-0.591513\pi$$
−0.283552 + 0.958957i $$0.591513\pi$$
$$200$$ 0 0
$$201$$ −9.00000 −0.634811
$$202$$ 10.0000i 0.703598i
$$203$$ 0 0
$$204$$ 3.00000 0.210042
$$205$$ 0 0
$$206$$ −14.0000 −0.975426
$$207$$ − 4.00000i − 0.278019i
$$208$$ − 6.00000i − 0.416025i
$$209$$ 9.00000 0.622543
$$210$$ 0 0
$$211$$ 9.00000 0.619586 0.309793 0.950804i $$-0.399740\pi$$
0.309793 + 0.950804i $$0.399740\pi$$
$$212$$ − 2.00000i − 0.137361i
$$213$$ − 2.00000i − 0.137038i
$$214$$ −1.00000 −0.0683586
$$215$$ 0 0
$$216$$ −5.00000 −0.340207
$$217$$ 0 0
$$218$$ 18.0000i 1.21911i
$$219$$ −9.00000 −0.608164
$$220$$ 0 0
$$221$$ 18.0000 1.21081
$$222$$ − 1.00000i − 0.0671156i
$$223$$ − 8.00000i − 0.535720i −0.963458 0.267860i $$-0.913684\pi$$
0.963458 0.267860i $$-0.0863164\pi$$
$$224$$ 4.00000 0.267261
$$225$$ 0 0
$$226$$ −5.00000 −0.332595
$$227$$ 28.0000i 1.85843i 0.369546 + 0.929213i $$0.379513\pi$$
−0.369546 + 0.929213i $$0.620487\pi$$
$$228$$ − 3.00000i − 0.198680i
$$229$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$230$$ 0 0
$$231$$ −12.0000 −0.789542
$$232$$ 0 0
$$233$$ 26.0000i 1.70332i 0.524097 + 0.851658i $$0.324403\pi$$
−0.524097 + 0.851658i $$0.675597\pi$$
$$234$$ −12.0000 −0.784465
$$235$$ 0 0
$$236$$ −12.0000 −0.781133
$$237$$ 2.00000i 0.129914i
$$238$$ 12.0000i 0.777844i
$$239$$ 12.0000 0.776215 0.388108 0.921614i $$-0.373129\pi$$
0.388108 + 0.921614i $$0.373129\pi$$
$$240$$ 0 0
$$241$$ −21.0000 −1.35273 −0.676364 0.736567i $$-0.736446\pi$$
−0.676364 + 0.736567i $$0.736446\pi$$
$$242$$ 2.00000i 0.128565i
$$243$$ 16.0000i 1.02640i
$$244$$ −12.0000 −0.768221
$$245$$ 0 0
$$246$$ −3.00000 −0.191273
$$247$$ − 18.0000i − 1.14531i
$$248$$ 0 0
$$249$$ −7.00000 −0.443607
$$250$$ 0 0
$$251$$ −9.00000 −0.568075 −0.284037 0.958813i $$-0.591674\pi$$
−0.284037 + 0.958813i $$0.591674\pi$$
$$252$$ − 8.00000i − 0.503953i
$$253$$ − 6.00000i − 0.377217i
$$254$$ 8.00000 0.501965
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ − 22.0000i − 1.37232i −0.727450 0.686161i $$-0.759294\pi$$
0.727450 0.686161i $$-0.240706\pi$$
$$258$$ 4.00000i 0.249029i
$$259$$ 4.00000 0.248548
$$260$$ 0 0
$$261$$ 0 0
$$262$$ − 12.0000i − 0.741362i
$$263$$ − 26.0000i − 1.60323i −0.597841 0.801614i $$-0.703975\pi$$
0.597841 0.801614i $$-0.296025\pi$$
$$264$$ −3.00000 −0.184637
$$265$$ 0 0
$$266$$ 12.0000 0.735767
$$267$$ 3.00000i 0.183597i
$$268$$ − 9.00000i − 0.549762i
$$269$$ 28.0000 1.70719 0.853595 0.520937i $$-0.174417\pi$$
0.853595 + 0.520937i $$0.174417\pi$$
$$270$$ 0 0
$$271$$ −28.0000 −1.70088 −0.850439 0.526073i $$-0.823664\pi$$
−0.850439 + 0.526073i $$0.823664\pi$$
$$272$$ 3.00000i 0.181902i
$$273$$ 24.0000i 1.45255i
$$274$$ −7.00000 −0.422885
$$275$$ 0 0
$$276$$ −2.00000 −0.120386
$$277$$ 28.0000i 1.68236i 0.540758 + 0.841178i $$0.318138\pi$$
−0.540758 + 0.841178i $$0.681862\pi$$
$$278$$ 3.00000i 0.179928i
$$279$$ 0 0
$$280$$ 0 0
$$281$$ −6.00000 −0.357930 −0.178965 0.983855i $$-0.557275\pi$$
−0.178965 + 0.983855i $$0.557275\pi$$
$$282$$ 4.00000i 0.238197i
$$283$$ − 7.00000i − 0.416107i −0.978117 0.208053i $$-0.933287\pi$$
0.978117 0.208053i $$-0.0667128\pi$$
$$284$$ 2.00000 0.118678
$$285$$ 0 0
$$286$$ −18.0000 −1.06436
$$287$$ − 12.0000i − 0.708338i
$$288$$ − 2.00000i − 0.117851i
$$289$$ 8.00000 0.470588
$$290$$ 0 0
$$291$$ −2.00000 −0.117242
$$292$$ − 9.00000i − 0.526685i
$$293$$ − 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
$$296$$ 1.00000 0.0581238
$$297$$ 15.0000i 0.870388i
$$298$$ − 18.0000i − 1.04271i
$$299$$ −12.0000 −0.693978
$$300$$ 0 0
$$301$$ −16.0000 −0.922225
$$302$$ − 22.0000i − 1.26596i
$$303$$ − 10.0000i − 0.574485i
$$304$$ 3.00000 0.172062
$$305$$ 0 0
$$306$$ 6.00000 0.342997
$$307$$ − 29.0000i − 1.65512i −0.561379 0.827559i $$-0.689729\pi$$
0.561379 0.827559i $$-0.310271\pi$$
$$308$$ − 12.0000i − 0.683763i
$$309$$ 14.0000 0.796432
$$310$$ 0 0
$$311$$ 28.0000 1.58773 0.793867 0.608091i $$-0.208065\pi$$
0.793867 + 0.608091i $$0.208065\pi$$
$$312$$ 6.00000i 0.339683i
$$313$$ 10.0000i 0.565233i 0.959233 + 0.282617i $$0.0912024\pi$$
−0.959233 + 0.282617i $$0.908798\pi$$
$$314$$ 24.0000 1.35440
$$315$$ 0 0
$$316$$ −2.00000 −0.112509
$$317$$ − 18.0000i − 1.01098i −0.862832 0.505490i $$-0.831312\pi$$
0.862832 0.505490i $$-0.168688\pi$$
$$318$$ 2.00000i 0.112154i
$$319$$ 0 0
$$320$$ 0 0
$$321$$ 1.00000 0.0558146
$$322$$ − 8.00000i − 0.445823i
$$323$$ 9.00000i 0.500773i
$$324$$ −1.00000 −0.0555556
$$325$$ 0 0
$$326$$ 17.0000 0.941543
$$327$$ − 18.0000i − 0.995402i
$$328$$ − 3.00000i − 0.165647i
$$329$$ −16.0000 −0.882109
$$330$$ 0 0
$$331$$ −3.00000 −0.164895 −0.0824475 0.996595i $$-0.526274\pi$$
−0.0824475 + 0.996595i $$0.526274\pi$$
$$332$$ − 7.00000i − 0.384175i
$$333$$ − 2.00000i − 0.109599i
$$334$$ −18.0000 −0.984916
$$335$$ 0 0
$$336$$ −4.00000 −0.218218
$$337$$ − 9.00000i − 0.490261i −0.969490 0.245131i $$-0.921169\pi$$
0.969490 0.245131i $$-0.0788309\pi$$
$$338$$ 23.0000i 1.25104i
$$339$$ 5.00000 0.271563
$$340$$ 0 0
$$341$$ 0 0
$$342$$ − 6.00000i − 0.324443i
$$343$$ − 8.00000i − 0.431959i
$$344$$ −4.00000 −0.215666
$$345$$ 0 0
$$346$$ −24.0000 −1.29025
$$347$$ − 31.0000i − 1.66417i −0.554650 0.832084i $$-0.687148\pi$$
0.554650 0.832084i $$-0.312852\pi$$
$$348$$ 0 0
$$349$$ 16.0000 0.856460 0.428230 0.903670i $$-0.359137\pi$$
0.428230 + 0.903670i $$0.359137\pi$$
$$350$$ 0 0
$$351$$ 30.0000 1.60128
$$352$$ − 3.00000i − 0.159901i
$$353$$ 14.0000i 0.745145i 0.928003 + 0.372572i $$0.121524\pi$$
−0.928003 + 0.372572i $$0.878476\pi$$
$$354$$ 12.0000 0.637793
$$355$$ 0 0
$$356$$ −3.00000 −0.159000
$$357$$ − 12.0000i − 0.635107i
$$358$$ 21.0000i 1.10988i
$$359$$ −34.0000 −1.79445 −0.897226 0.441572i $$-0.854421\pi$$
−0.897226 + 0.441572i $$0.854421\pi$$
$$360$$ 0 0
$$361$$ −10.0000 −0.526316
$$362$$ 10.0000i 0.525588i
$$363$$ − 2.00000i − 0.104973i
$$364$$ −24.0000 −1.25794
$$365$$ 0 0
$$366$$ 12.0000 0.627250
$$367$$ − 2.00000i − 0.104399i −0.998637 0.0521996i $$-0.983377\pi$$
0.998637 0.0521996i $$-0.0166232\pi$$
$$368$$ − 2.00000i − 0.104257i
$$369$$ −6.00000 −0.312348
$$370$$ 0 0
$$371$$ −8.00000 −0.415339
$$372$$ 0 0
$$373$$ − 28.0000i − 1.44979i −0.688862 0.724893i $$-0.741889\pi$$
0.688862 0.724893i $$-0.258111\pi$$
$$374$$ 9.00000 0.465379
$$375$$ 0 0
$$376$$ −4.00000 −0.206284
$$377$$ 0 0
$$378$$ 20.0000i 1.02869i
$$379$$ −1.00000 −0.0513665 −0.0256833 0.999670i $$-0.508176\pi$$
−0.0256833 + 0.999670i $$0.508176\pi$$
$$380$$ 0 0
$$381$$ −8.00000 −0.409852
$$382$$ − 6.00000i − 0.306987i
$$383$$ − 24.0000i − 1.22634i −0.789950 0.613171i $$-0.789894\pi$$
0.789950 0.613171i $$-0.210106\pi$$
$$384$$ −1.00000 −0.0510310
$$385$$ 0 0
$$386$$ −11.0000 −0.559885
$$387$$ 8.00000i 0.406663i
$$388$$ − 2.00000i − 0.101535i
$$389$$ 12.0000 0.608424 0.304212 0.952604i $$-0.401607\pi$$
0.304212 + 0.952604i $$0.401607\pi$$
$$390$$ 0 0
$$391$$ 6.00000 0.303433
$$392$$ − 9.00000i − 0.454569i
$$393$$ 12.0000i 0.605320i
$$394$$ 12.0000 0.604551
$$395$$ 0 0
$$396$$ −6.00000 −0.301511
$$397$$ − 30.0000i − 1.50566i −0.658217 0.752828i $$-0.728689\pi$$
0.658217 0.752828i $$-0.271311\pi$$
$$398$$ 8.00000i 0.401004i
$$399$$ −12.0000 −0.600751
$$400$$ 0 0
$$401$$ −5.00000 −0.249688 −0.124844 0.992176i $$-0.539843\pi$$
−0.124844 + 0.992176i $$0.539843\pi$$
$$402$$ 9.00000i 0.448879i
$$403$$ 0 0
$$404$$ 10.0000 0.497519
$$405$$ 0 0
$$406$$ 0 0
$$407$$ − 3.00000i − 0.148704i
$$408$$ − 3.00000i − 0.148522i
$$409$$ −15.0000 −0.741702 −0.370851 0.928692i $$-0.620934\pi$$
−0.370851 + 0.928692i $$0.620934\pi$$
$$410$$ 0 0
$$411$$ 7.00000 0.345285
$$412$$ 14.0000i 0.689730i
$$413$$ 48.0000i 2.36193i
$$414$$ −4.00000 −0.196589
$$415$$ 0 0
$$416$$ −6.00000 −0.294174
$$417$$ − 3.00000i − 0.146911i
$$418$$ − 9.00000i − 0.440204i
$$419$$ −27.0000 −1.31904 −0.659518 0.751689i $$-0.729240\pi$$
−0.659518 + 0.751689i $$0.729240\pi$$
$$420$$ 0 0
$$421$$ −32.0000 −1.55958 −0.779792 0.626038i $$-0.784675\pi$$
−0.779792 + 0.626038i $$0.784675\pi$$
$$422$$ − 9.00000i − 0.438113i
$$423$$ 8.00000i 0.388973i
$$424$$ −2.00000 −0.0971286
$$425$$ 0 0
$$426$$ −2.00000 −0.0969003
$$427$$ 48.0000i 2.32288i
$$428$$ 1.00000i 0.0483368i
$$429$$ 18.0000 0.869048
$$430$$ 0 0
$$431$$ 2.00000 0.0963366 0.0481683 0.998839i $$-0.484662\pi$$
0.0481683 + 0.998839i $$0.484662\pi$$
$$432$$ 5.00000i 0.240563i
$$433$$ − 5.00000i − 0.240285i −0.992757 0.120142i $$-0.961665\pi$$
0.992757 0.120142i $$-0.0383351\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 18.0000 0.862044
$$437$$ − 6.00000i − 0.287019i
$$438$$ 9.00000i 0.430037i
$$439$$ −16.0000 −0.763638 −0.381819 0.924237i $$-0.624702\pi$$
−0.381819 + 0.924237i $$0.624702\pi$$
$$440$$ 0 0
$$441$$ −18.0000 −0.857143
$$442$$ − 18.0000i − 0.856173i
$$443$$ − 5.00000i − 0.237557i −0.992921 0.118779i $$-0.962102\pi$$
0.992921 0.118779i $$-0.0378979\pi$$
$$444$$ −1.00000 −0.0474579
$$445$$ 0 0
$$446$$ −8.00000 −0.378811
$$447$$ 18.0000i 0.851371i
$$448$$ − 4.00000i − 0.188982i
$$449$$ 37.0000 1.74614 0.873069 0.487597i $$-0.162126\pi$$
0.873069 + 0.487597i $$0.162126\pi$$
$$450$$ 0 0
$$451$$ −9.00000 −0.423793
$$452$$ 5.00000i 0.235180i
$$453$$ 22.0000i 1.03365i
$$454$$ 28.0000 1.31411
$$455$$ 0 0
$$456$$ −3.00000 −0.140488
$$457$$ − 33.0000i − 1.54367i −0.635820 0.771837i $$-0.719338\pi$$
0.635820 0.771837i $$-0.280662\pi$$
$$458$$ 0 0
$$459$$ −15.0000 −0.700140
$$460$$ 0 0
$$461$$ 20.0000 0.931493 0.465746 0.884918i $$-0.345786\pi$$
0.465746 + 0.884918i $$0.345786\pi$$
$$462$$ 12.0000i 0.558291i
$$463$$ − 6.00000i − 0.278844i −0.990233 0.139422i $$-0.955476\pi$$
0.990233 0.139422i $$-0.0445244\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 26.0000 1.20443
$$467$$ − 28.0000i − 1.29569i −0.761774 0.647843i $$-0.775671\pi$$
0.761774 0.647843i $$-0.224329\pi$$
$$468$$ 12.0000i 0.554700i
$$469$$ −36.0000 −1.66233
$$470$$ 0 0
$$471$$ −24.0000 −1.10586
$$472$$ 12.0000i 0.552345i
$$473$$ 12.0000i 0.551761i
$$474$$ 2.00000 0.0918630
$$475$$ 0 0
$$476$$ 12.0000 0.550019
$$477$$ 4.00000i 0.183147i
$$478$$ − 12.0000i − 0.548867i
$$479$$ −16.0000 −0.731059 −0.365529 0.930800i $$-0.619112\pi$$
−0.365529 + 0.930800i $$0.619112\pi$$
$$480$$ 0 0
$$481$$ −6.00000 −0.273576
$$482$$ 21.0000i 0.956524i
$$483$$ 8.00000i 0.364013i
$$484$$ 2.00000 0.0909091
$$485$$ 0 0
$$486$$ 16.0000 0.725775
$$487$$ 28.0000i 1.26880i 0.773004 + 0.634401i $$0.218753\pi$$
−0.773004 + 0.634401i $$0.781247\pi$$
$$488$$ 12.0000i 0.543214i
$$489$$ −17.0000 −0.768767
$$490$$ 0 0
$$491$$ −20.0000 −0.902587 −0.451294 0.892375i $$-0.649037\pi$$
−0.451294 + 0.892375i $$0.649037\pi$$
$$492$$ 3.00000i 0.135250i
$$493$$ 0 0
$$494$$ −18.0000 −0.809858
$$495$$ 0 0
$$496$$ 0 0
$$497$$ − 8.00000i − 0.358849i
$$498$$ 7.00000i 0.313678i
$$499$$ 16.0000 0.716258 0.358129 0.933672i $$-0.383415\pi$$
0.358129 + 0.933672i $$0.383415\pi$$
$$500$$ 0 0
$$501$$ 18.0000 0.804181
$$502$$ 9.00000i 0.401690i
$$503$$ 12.0000i 0.535054i 0.963550 + 0.267527i $$0.0862064\pi$$
−0.963550 + 0.267527i $$0.913794\pi$$
$$504$$ −8.00000 −0.356348
$$505$$ 0 0
$$506$$ −6.00000 −0.266733
$$507$$ − 23.0000i − 1.02147i
$$508$$ − 8.00000i − 0.354943i
$$509$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$510$$ 0 0
$$511$$ −36.0000 −1.59255
$$512$$ − 1.00000i − 0.0441942i
$$513$$ 15.0000i 0.662266i
$$514$$ −22.0000 −0.970378
$$515$$ 0 0
$$516$$ 4.00000 0.176090
$$517$$ 12.0000i 0.527759i
$$518$$ − 4.00000i − 0.175750i
$$519$$ 24.0000 1.05348
$$520$$ 0 0
$$521$$ −11.0000 −0.481919 −0.240959 0.970535i $$-0.577462\pi$$
−0.240959 + 0.970535i $$0.577462\pi$$
$$522$$ 0 0
$$523$$ − 37.0000i − 1.61790i −0.587879 0.808949i $$-0.700037\pi$$
0.587879 0.808949i $$-0.299963\pi$$
$$524$$ −12.0000 −0.524222
$$525$$ 0 0
$$526$$ −26.0000 −1.13365
$$527$$ 0 0
$$528$$ 3.00000i 0.130558i
$$529$$ 19.0000 0.826087
$$530$$ 0 0
$$531$$ 24.0000 1.04151
$$532$$ − 12.0000i − 0.520266i
$$533$$ 18.0000i 0.779667i
$$534$$ 3.00000 0.129823
$$535$$ 0 0
$$536$$ −9.00000 −0.388741
$$537$$ − 21.0000i − 0.906217i
$$538$$ − 28.0000i − 1.20717i
$$539$$ −27.0000 −1.16297
$$540$$ 0 0
$$541$$ −2.00000 −0.0859867 −0.0429934 0.999075i $$-0.513689\pi$$
−0.0429934 + 0.999075i $$0.513689\pi$$
$$542$$ 28.0000i 1.20270i
$$543$$ − 10.0000i − 0.429141i
$$544$$ 3.00000 0.128624
$$545$$ 0 0
$$546$$ 24.0000 1.02711
$$547$$ 5.00000i 0.213785i 0.994271 + 0.106892i $$0.0340900\pi$$
−0.994271 + 0.106892i $$0.965910\pi$$
$$548$$ 7.00000i 0.299025i
$$549$$ 24.0000 1.02430
$$550$$ 0 0
$$551$$ 0 0
$$552$$ 2.00000i 0.0851257i
$$553$$ 8.00000i 0.340195i
$$554$$ 28.0000 1.18961
$$555$$ 0 0
$$556$$ 3.00000 0.127228
$$557$$ 32.0000i 1.35588i 0.735116 + 0.677942i $$0.237128\pi$$
−0.735116 + 0.677942i $$0.762872\pi$$
$$558$$ 0 0
$$559$$ 24.0000 1.01509
$$560$$ 0 0
$$561$$ −9.00000 −0.379980
$$562$$ 6.00000i 0.253095i
$$563$$ − 4.00000i − 0.168580i −0.996441 0.0842900i $$-0.973138\pi$$
0.996441 0.0842900i $$-0.0268622\pi$$
$$564$$ 4.00000 0.168430
$$565$$ 0 0
$$566$$ −7.00000 −0.294232
$$567$$ 4.00000i 0.167984i
$$568$$ − 2.00000i − 0.0839181i
$$569$$ −19.0000 −0.796521 −0.398261 0.917272i $$-0.630386\pi$$
−0.398261 + 0.917272i $$0.630386\pi$$
$$570$$ 0 0
$$571$$ −4.00000 −0.167395 −0.0836974 0.996491i $$-0.526673\pi$$
−0.0836974 + 0.996491i $$0.526673\pi$$
$$572$$ 18.0000i 0.752618i
$$573$$ 6.00000i 0.250654i
$$574$$ −12.0000 −0.500870
$$575$$ 0 0
$$576$$ −2.00000 −0.0833333
$$577$$ − 15.0000i − 0.624458i −0.950007 0.312229i $$-0.898924\pi$$
0.950007 0.312229i $$-0.101076\pi$$
$$578$$ − 8.00000i − 0.332756i
$$579$$ 11.0000 0.457144
$$580$$ 0 0
$$581$$ −28.0000 −1.16164
$$582$$ 2.00000i 0.0829027i
$$583$$ 6.00000i 0.248495i
$$584$$ −9.00000 −0.372423
$$585$$ 0 0
$$586$$ −12.0000 −0.495715
$$587$$ − 13.0000i − 0.536567i −0.963340 0.268284i $$-0.913544\pi$$
0.963340 0.268284i $$-0.0864565\pi$$
$$588$$ 9.00000i 0.371154i
$$589$$ 0 0
$$590$$ 0 0
$$591$$ −12.0000 −0.493614
$$592$$ − 1.00000i − 0.0410997i
$$593$$ 9.00000i 0.369586i 0.982777 + 0.184793i $$0.0591614\pi$$
−0.982777 + 0.184793i $$0.940839\pi$$
$$594$$ 15.0000 0.615457
$$595$$ 0 0
$$596$$ −18.0000 −0.737309
$$597$$ − 8.00000i − 0.327418i
$$598$$ 12.0000i 0.490716i
$$599$$ 36.0000 1.47092 0.735460 0.677568i $$-0.236966\pi$$
0.735460 + 0.677568i $$0.236966\pi$$
$$600$$ 0 0
$$601$$ −33.0000 −1.34610 −0.673049 0.739598i $$-0.735016\pi$$
−0.673049 + 0.739598i $$0.735016\pi$$
$$602$$ 16.0000i 0.652111i
$$603$$ 18.0000i 0.733017i
$$604$$ −22.0000 −0.895167
$$605$$ 0 0
$$606$$ −10.0000 −0.406222
$$607$$ − 26.0000i − 1.05531i −0.849460 0.527654i $$-0.823072\pi$$
0.849460 0.527654i $$-0.176928\pi$$
$$608$$ − 3.00000i − 0.121666i
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 24.0000 0.970936
$$612$$ − 6.00000i − 0.242536i
$$613$$ 46.0000i 1.85792i 0.370177 + 0.928961i $$0.379297\pi$$
−0.370177 + 0.928961i $$0.620703\pi$$
$$614$$ −29.0000 −1.17034
$$615$$ 0 0
$$616$$ −12.0000 −0.483494
$$617$$ − 42.0000i − 1.69086i −0.534089 0.845428i $$-0.679345\pi$$
0.534089 0.845428i $$-0.320655\pi$$
$$618$$ − 14.0000i − 0.563163i
$$619$$ 16.0000 0.643094 0.321547 0.946894i $$-0.395797\pi$$
0.321547 + 0.946894i $$0.395797\pi$$
$$620$$ 0 0
$$621$$ 10.0000 0.401286
$$622$$ − 28.0000i − 1.12270i
$$623$$ 12.0000i 0.480770i
$$624$$ 6.00000 0.240192
$$625$$ 0 0
$$626$$ 10.0000 0.399680
$$627$$ 9.00000i 0.359425i
$$628$$ − 24.0000i − 0.957704i
$$629$$ 3.00000 0.119618
$$630$$ 0 0
$$631$$ 22.0000 0.875806 0.437903 0.899022i $$-0.355721\pi$$
0.437903 + 0.899022i $$0.355721\pi$$
$$632$$ 2.00000i 0.0795557i
$$633$$ 9.00000i 0.357718i
$$634$$ −18.0000 −0.714871
$$635$$ 0 0
$$636$$ 2.00000 0.0793052
$$637$$ 54.0000i 2.13956i
$$638$$ 0 0
$$639$$ −4.00000 −0.158238
$$640$$ 0 0
$$641$$ −6.00000 −0.236986 −0.118493 0.992955i $$-0.537806\pi$$
−0.118493 + 0.992955i $$0.537806\pi$$
$$642$$ − 1.00000i − 0.0394669i
$$643$$ − 20.0000i − 0.788723i −0.918955 0.394362i $$-0.870966\pi$$
0.918955 0.394362i $$-0.129034\pi$$
$$644$$ −8.00000 −0.315244
$$645$$ 0 0
$$646$$ 9.00000 0.354100
$$647$$ 42.0000i 1.65119i 0.564263 + 0.825595i $$0.309160\pi$$
−0.564263 + 0.825595i $$0.690840\pi$$
$$648$$ 1.00000i 0.0392837i
$$649$$ 36.0000 1.41312
$$650$$ 0 0
$$651$$ 0 0
$$652$$ − 17.0000i − 0.665771i
$$653$$ − 36.0000i − 1.40879i −0.709809 0.704394i $$-0.751219\pi$$
0.709809 0.704394i $$-0.248781\pi$$
$$654$$ −18.0000 −0.703856
$$655$$ 0 0
$$656$$ −3.00000 −0.117130
$$657$$ 18.0000i 0.702247i
$$658$$ 16.0000i 0.623745i
$$659$$ −9.00000 −0.350590 −0.175295 0.984516i $$-0.556088\pi$$
−0.175295 + 0.984516i $$0.556088\pi$$
$$660$$ 0 0
$$661$$ −16.0000 −0.622328 −0.311164 0.950356i $$-0.600719\pi$$
−0.311164 + 0.950356i $$0.600719\pi$$
$$662$$ 3.00000i 0.116598i
$$663$$ 18.0000i 0.699062i
$$664$$ −7.00000 −0.271653
$$665$$ 0 0
$$666$$ −2.00000 −0.0774984
$$667$$ 0 0
$$668$$ 18.0000i 0.696441i
$$669$$ 8.00000 0.309298
$$670$$ 0 0
$$671$$ 36.0000 1.38976
$$672$$ 4.00000i 0.154303i
$$673$$ 34.0000i 1.31060i 0.755367 + 0.655302i $$0.227459\pi$$
−0.755367 + 0.655302i $$0.772541\pi$$
$$674$$ −9.00000 −0.346667
$$675$$ 0 0
$$676$$ 23.0000 0.884615
$$677$$ − 8.00000i − 0.307465i −0.988113 0.153732i $$-0.950871\pi$$
0.988113 0.153732i $$-0.0491294\pi$$
$$678$$ − 5.00000i − 0.192024i
$$679$$ −8.00000 −0.307012
$$680$$ 0 0
$$681$$ −28.0000 −1.07296
$$682$$ 0 0
$$683$$ − 17.0000i − 0.650487i −0.945630 0.325243i $$-0.894554\pi$$
0.945630 0.325243i $$-0.105446\pi$$
$$684$$ −6.00000 −0.229416
$$685$$ 0 0
$$686$$ −8.00000 −0.305441
$$687$$ 0 0
$$688$$ 4.00000i 0.152499i
$$689$$ 12.0000 0.457164
$$690$$ 0 0
$$691$$ −1.00000 −0.0380418 −0.0190209 0.999819i $$-0.506055\pi$$
−0.0190209 + 0.999819i $$0.506055\pi$$
$$692$$ 24.0000i 0.912343i
$$693$$ 24.0000i 0.911685i
$$694$$ −31.0000 −1.17674
$$695$$ 0 0
$$696$$ 0 0
$$697$$ − 9.00000i − 0.340899i
$$698$$ − 16.0000i − 0.605609i
$$699$$ −26.0000 −0.983410
$$700$$ 0 0
$$701$$ −10.0000 −0.377695 −0.188847 0.982006i $$-0.560475\pi$$
−0.188847 + 0.982006i $$0.560475\pi$$
$$702$$ − 30.0000i − 1.13228i
$$703$$ − 3.00000i − 0.113147i
$$704$$ −3.00000 −0.113067
$$705$$ 0 0
$$706$$ 14.0000 0.526897
$$707$$ − 40.0000i − 1.50435i
$$708$$ − 12.0000i − 0.450988i
$$709$$ −32.0000 −1.20179 −0.600893 0.799330i $$-0.705188\pi$$
−0.600893 + 0.799330i $$0.705188\pi$$
$$710$$ 0 0
$$711$$ 4.00000 0.150012
$$712$$ 3.00000i 0.112430i
$$713$$ 0 0
$$714$$ −12.0000 −0.449089
$$715$$ 0 0
$$716$$ 21.0000 0.784807
$$717$$ 12.0000i 0.448148i
$$718$$ 34.0000i 1.26887i
$$719$$ 36.0000 1.34257 0.671287 0.741198i $$-0.265742\pi$$
0.671287 + 0.741198i $$0.265742\pi$$
$$720$$ 0 0
$$721$$ 56.0000 2.08555
$$722$$ 10.0000i 0.372161i
$$723$$ − 21.0000i − 0.780998i
$$724$$ 10.0000 0.371647
$$725$$ 0 0
$$726$$ −2.00000 −0.0742270
$$727$$ − 44.0000i − 1.63187i −0.578144 0.815935i $$-0.696223\pi$$
0.578144 0.815935i $$-0.303777\pi$$
$$728$$ 24.0000i 0.889499i
$$729$$ −13.0000 −0.481481
$$730$$ 0 0
$$731$$ −12.0000 −0.443836
$$732$$ − 12.0000i − 0.443533i
$$733$$ − 40.0000i − 1.47743i −0.674016 0.738717i $$-0.735432\pi$$
0.674016 0.738717i $$-0.264568\pi$$
$$734$$ −2.00000 −0.0738213
$$735$$ 0 0
$$736$$ −2.00000 −0.0737210
$$737$$ 27.0000i 0.994558i
$$738$$ 6.00000i 0.220863i
$$739$$ 12.0000 0.441427 0.220714 0.975339i $$-0.429161\pi$$
0.220714 + 0.975339i $$0.429161\pi$$
$$740$$ 0 0
$$741$$ 18.0000 0.661247
$$742$$ 8.00000i 0.293689i
$$743$$ − 6.00000i − 0.220119i −0.993925 0.110059i $$-0.964896\pi$$
0.993925 0.110059i $$-0.0351041\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ −28.0000 −1.02515
$$747$$ 14.0000i 0.512233i
$$748$$ − 9.00000i − 0.329073i
$$749$$ 4.00000 0.146157
$$750$$ 0 0
$$751$$ 20.0000 0.729810 0.364905 0.931045i $$-0.381101\pi$$
0.364905 + 0.931045i $$0.381101\pi$$
$$752$$ 4.00000i 0.145865i
$$753$$ − 9.00000i − 0.327978i
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 20.0000 0.727393
$$757$$ 16.0000i 0.581530i 0.956795 + 0.290765i $$0.0939098\pi$$
−0.956795 + 0.290765i $$0.906090\pi$$
$$758$$ 1.00000i 0.0363216i
$$759$$ 6.00000 0.217786
$$760$$ 0 0
$$761$$ 45.0000 1.63125 0.815624 0.578582i $$-0.196394\pi$$
0.815624 + 0.578582i $$0.196394\pi$$
$$762$$ 8.00000i 0.289809i
$$763$$ − 72.0000i − 2.60658i
$$764$$ −6.00000 −0.217072
$$765$$ 0 0
$$766$$ −24.0000 −0.867155
$$767$$ − 72.0000i − 2.59977i
$$768$$ 1.00000i 0.0360844i
$$769$$ −23.0000 −0.829401 −0.414701 0.909958i $$-0.636114\pi$$
−0.414701 + 0.909958i $$0.636114\pi$$
$$770$$ 0 0
$$771$$ 22.0000 0.792311
$$772$$ 11.0000i 0.395899i
$$773$$ − 46.0000i − 1.65451i −0.561830 0.827253i $$-0.689903\pi$$
0.561830 0.827253i $$-0.310097\pi$$
$$774$$ 8.00000 0.287554
$$775$$ 0 0
$$776$$ −2.00000 −0.0717958
$$777$$ 4.00000i 0.143499i
$$778$$ − 12.0000i − 0.430221i
$$779$$ −9.00000 −0.322458
$$780$$ 0 0
$$781$$ −6.00000 −0.214697
$$782$$ − 6.00000i − 0.214560i
$$783$$ 0 0
$$784$$ −9.00000 −0.321429
$$785$$ 0 0
$$786$$ 12.0000 0.428026
$$787$$ 32.0000i 1.14068i 0.821410 + 0.570338i $$0.193188\pi$$
−0.821410 + 0.570338i $$0.806812\pi$$
$$788$$ − 12.0000i − 0.427482i
$$789$$ 26.0000 0.925625
$$790$$ 0 0
$$791$$ 20.0000 0.711118
$$792$$ 6.00000i 0.213201i
$$793$$ − 72.0000i − 2.55679i
$$794$$ −30.0000 −1.06466
$$795$$ 0 0
$$796$$ 8.00000 0.283552
$$797$$ 16.0000i 0.566749i 0.959009 + 0.283375i $$0.0914540\pi$$
−0.959009 + 0.283375i $$0.908546\pi$$
$$798$$ 12.0000i 0.424795i
$$799$$ −12.0000 −0.424529
$$800$$ 0 0
$$801$$ 6.00000 0.212000
$$802$$ 5.00000i 0.176556i
$$803$$ 27.0000i 0.952809i
$$804$$ 9.00000 0.317406
$$805$$ 0 0
$$806$$ 0 0
$$807$$ 28.0000i 0.985647i
$$808$$ − 10.0000i − 0.351799i
$$809$$ 38.0000 1.33601 0.668004 0.744157i $$-0.267149\pi$$
0.668004 + 0.744157i $$0.267149\pi$$
$$810$$ 0 0
$$811$$ 20.0000 0.702295 0.351147 0.936320i $$-0.385792\pi$$
0.351147 + 0.936320i $$0.385792\pi$$
$$812$$ 0 0
$$813$$ − 28.0000i − 0.982003i
$$814$$ −3.00000 −0.105150
$$815$$ 0 0
$$816$$ −3.00000 −0.105021
$$817$$ 12.0000i 0.419827i
$$818$$ 15.0000i 0.524463i
$$819$$ 48.0000 1.67726
$$820$$ 0 0
$$821$$ −4.00000 −0.139601 −0.0698005 0.997561i $$-0.522236\pi$$
−0.0698005 + 0.997561i $$0.522236\pi$$
$$822$$ − 7.00000i − 0.244153i
$$823$$ − 42.0000i − 1.46403i −0.681290 0.732014i $$-0.738581\pi$$
0.681290 0.732014i $$-0.261419\pi$$
$$824$$ 14.0000 0.487713
$$825$$ 0 0
$$826$$ 48.0000 1.67013
$$827$$ 7.00000i 0.243414i 0.992566 + 0.121707i $$0.0388368\pi$$
−0.992566 + 0.121707i $$0.961163\pi$$
$$828$$ 4.00000i 0.139010i
$$829$$ −6.00000 −0.208389 −0.104194 0.994557i $$-0.533226\pi$$
−0.104194 + 0.994557i $$0.533226\pi$$
$$830$$ 0 0
$$831$$ −28.0000 −0.971309
$$832$$ 6.00000i 0.208013i
$$833$$ − 27.0000i − 0.935495i
$$834$$ −3.00000 −0.103882
$$835$$ 0 0
$$836$$ −9.00000 −0.311272
$$837$$ 0 0
$$838$$ 27.0000i 0.932700i
$$839$$ 42.0000 1.45000 0.725001 0.688748i $$-0.241839\pi$$
0.725001 + 0.688748i $$0.241839\pi$$
$$840$$ 0 0
$$841$$ −29.0000 −1.00000
$$842$$ 32.0000i 1.10279i
$$843$$ − 6.00000i − 0.206651i
$$844$$ −9.00000 −0.309793
$$845$$ 0 0
$$846$$ 8.00000 0.275046
$$847$$ − 8.00000i − 0.274883i
$$848$$ 2.00000i 0.0686803i
$$849$$ 7.00000 0.240239
$$850$$ 0 0
$$851$$ −2.00000 −0.0685591
$$852$$ 2.00000i 0.0685189i
$$853$$ − 40.0000i − 1.36957i −0.728743 0.684787i $$-0.759895\pi$$
0.728743 0.684787i $$-0.240105\pi$$
$$854$$ 48.0000 1.64253
$$855$$ 0 0
$$856$$ 1.00000 0.0341793
$$857$$ 3.00000i 0.102478i 0.998686 + 0.0512390i $$0.0163170\pi$$
−0.998686 + 0.0512390i $$0.983683\pi$$
$$858$$ − 18.0000i − 0.614510i
$$859$$ −9.00000 −0.307076 −0.153538 0.988143i $$-0.549067\pi$$
−0.153538 + 0.988143i $$0.549067\pi$$
$$860$$ 0 0
$$861$$ 12.0000 0.408959
$$862$$ − 2.00000i − 0.0681203i
$$863$$ − 6.00000i − 0.204242i −0.994772 0.102121i $$-0.967437\pi$$
0.994772 0.102121i $$-0.0325630\pi$$
$$864$$ 5.00000 0.170103
$$865$$ 0 0
$$866$$ −5.00000 −0.169907
$$867$$ 8.00000i 0.271694i
$$868$$ 0 0
$$869$$ 6.00000 0.203536
$$870$$ 0 0
$$871$$ 54.0000 1.82972
$$872$$ − 18.0000i − 0.609557i
$$873$$ 4.00000i 0.135379i
$$874$$ −6.00000 −0.202953
$$875$$ 0 0
$$876$$ 9.00000 0.304082
$$877$$ 26.0000i 0.877958i 0.898497 + 0.438979i $$0.144660\pi$$
−0.898497 + 0.438979i $$0.855340\pi$$
$$878$$ 16.0000i 0.539974i
$$879$$ 12.0000 0.404750
$$880$$ 0 0
$$881$$ 6.00000 0.202145 0.101073 0.994879i $$-0.467773\pi$$
0.101073 + 0.994879i $$0.467773\pi$$
$$882$$ 18.0000i 0.606092i
$$883$$ 7.00000i 0.235569i 0.993039 + 0.117784i $$0.0375792\pi$$
−0.993039 + 0.117784i $$0.962421\pi$$
$$884$$ −18.0000 −0.605406
$$885$$ 0 0
$$886$$ −5.00000 −0.167978
$$887$$ − 36.0000i − 1.20876i −0.796696 0.604381i $$-0.793421\pi$$
0.796696 0.604381i $$-0.206579\pi$$
$$888$$ 1.00000i 0.0335578i
$$889$$ −32.0000 −1.07325
$$890$$ 0 0
$$891$$ 3.00000 0.100504
$$892$$ 8.00000i 0.267860i
$$893$$ 12.0000i 0.401565i
$$894$$ 18.0000 0.602010
$$895$$ 0 0
$$896$$ −4.00000 −0.133631
$$897$$ − 12.0000i − 0.400668i
$$898$$ − 37.0000i − 1.23471i
$$899$$ 0 0
$$900$$ 0 0
$$901$$ −6.00000 −0.199889
$$902$$ 9.00000i 0.299667i
$$903$$ − 16.0000i − 0.532447i
$$904$$ 5.00000 0.166298
$$905$$ 0 0
$$906$$ 22.0000 0.730901
$$907$$ 36.0000i 1.19536i 0.801735 + 0.597680i $$0.203911\pi$$
−0.801735 + 0.597680i $$0.796089\pi$$
$$908$$ − 28.0000i − 0.929213i
$$909$$ −20.0000 −0.663358
$$910$$ 0 0
$$911$$ 18.0000 0.596367 0.298183 0.954509i $$-0.403619\pi$$
0.298183 + 0.954509i $$0.403619\pi$$
$$912$$ 3.00000i 0.0993399i
$$913$$ 21.0000i 0.694999i
$$914$$ −33.0000 −1.09154
$$915$$ 0 0
$$916$$ 0 0
$$917$$ 48.0000i 1.58510i
$$918$$ 15.0000i 0.495074i
$$919$$ 34.0000 1.12156 0.560778 0.827966i $$-0.310502\pi$$
0.560778 + 0.827966i $$0.310502\pi$$
$$920$$ 0 0
$$921$$ 29.0000 0.955582
$$922$$ − 20.0000i − 0.658665i
$$923$$ 12.0000i 0.394985i
$$924$$ 12.0000 0.394771
$$925$$ 0 0
$$926$$ −6.00000 −0.197172
$$927$$ − 28.0000i − 0.919641i
$$928$$ 0 0
$$929$$ 54.0000 1.77168 0.885841 0.463988i $$-0.153582\pi$$
0.885841 + 0.463988i $$0.153582\pi$$
$$930$$ 0 0
$$931$$ −27.0000 −0.884889
$$932$$ − 26.0000i − 0.851658i
$$933$$ 28.0000i 0.916679i
$$934$$ −28.0000 −0.916188
$$935$$ 0 0
$$936$$ 12.0000 0.392232
$$937$$ 7.00000i 0.228680i 0.993442 + 0.114340i $$0.0364753\pi$$
−0.993442 + 0.114340i $$0.963525\pi$$
$$938$$ 36.0000i 1.17544i
$$939$$ −10.0000 −0.326338
$$940$$ 0 0
$$941$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$942$$ 24.0000i 0.781962i
$$943$$ 6.00000i 0.195387i
$$944$$ 12.0000 0.390567
$$945$$ 0 0
$$946$$ 12.0000 0.390154
$$947$$ − 24.0000i − 0.779895i −0.920837 0.389948i $$-0.872493\pi$$
0.920837 0.389948i $$-0.127507\pi$$
$$948$$ − 2.00000i − 0.0649570i
$$949$$ 54.0000 1.75291
$$950$$ 0 0
$$951$$ 18.0000 0.583690
$$952$$ − 12.0000i − 0.388922i
$$953$$ − 33.0000i − 1.06897i −0.845176 0.534487i $$-0.820505\pi$$
0.845176 0.534487i $$-0.179495\pi$$
$$954$$ 4.00000 0.129505
$$955$$ 0 0
$$956$$ −12.0000 −0.388108
$$957$$ 0 0
$$958$$ 16.0000i 0.516937i
$$959$$ 28.0000 0.904167
$$960$$ 0 0
$$961$$ −31.0000 −1.00000
$$962$$ 6.00000i 0.193448i
$$963$$ − 2.00000i − 0.0644491i
$$964$$ 21.0000 0.676364
$$965$$ 0 0
$$966$$ 8.00000 0.257396
$$967$$ − 48.0000i − 1.54358i −0.635880 0.771788i $$-0.719363\pi$$
0.635880 0.771788i $$-0.280637\pi$$
$$968$$ − 2.00000i − 0.0642824i
$$969$$ −9.00000 −0.289122
$$970$$ 0 0
$$971$$ 3.00000 0.0962746 0.0481373 0.998841i $$-0.484672\pi$$
0.0481373 + 0.998841i $$0.484672\pi$$
$$972$$ − 16.0000i − 0.513200i
$$973$$ − 12.0000i − 0.384702i
$$974$$ 28.0000 0.897178
$$975$$ 0 0
$$976$$ 12.0000 0.384111
$$977$$ 23.0000i 0.735835i 0.929858 + 0.367918i $$0.119929\pi$$
−0.929858 + 0.367918i $$0.880071\pi$$
$$978$$ 17.0000i 0.543600i
$$979$$ 9.00000 0.287641
$$980$$ 0 0
$$981$$ −36.0000 −1.14939
$$982$$ 20.0000i 0.638226i
$$983$$ − 18.0000i − 0.574111i −0.957914 0.287055i $$-0.907324\pi$$
0.957914 0.287055i $$-0.0926764\pi$$
$$984$$ 3.00000 0.0956365
$$985$$ 0 0
$$986$$ 0 0
$$987$$ − 16.0000i − 0.509286i
$$988$$ 18.0000i 0.572656i
$$989$$ 8.00000 0.254385
$$990$$ 0 0
$$991$$ −12.0000 −0.381193 −0.190596 0.981669i $$-0.561042\pi$$
−0.190596 + 0.981669i $$0.561042\pi$$
$$992$$ 0 0
$$993$$ − 3.00000i − 0.0952021i
$$994$$ −8.00000 −0.253745
$$995$$ 0 0
$$996$$ 7.00000 0.221803
$$997$$ − 18.0000i − 0.570066i −0.958518 0.285033i $$-0.907995\pi$$
0.958518 0.285033i $$-0.0920045\pi$$
$$998$$ − 16.0000i − 0.506471i
$$999$$ 5.00000 0.158193
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 1850.2.b.e.149.1 2
5.2 odd 4 1850.2.a.m.1.1 yes 1
5.3 odd 4 1850.2.a.c.1.1 1
5.4 even 2 inner 1850.2.b.e.149.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
1850.2.a.c.1.1 1 5.3 odd 4
1850.2.a.m.1.1 yes 1 5.2 odd 4
1850.2.b.e.149.1 2 1.1 even 1 trivial
1850.2.b.e.149.2 2 5.4 even 2 inner