# Properties

 Label 1850.2.b.f.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.f.149.2

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-1.00000i q^{2} +2.00000i q^{3} -1.00000 q^{4} +2.00000 q^{6} -4.00000i q^{7} +1.00000i q^{8} -1.00000 q^{9} +O(q^{10})$$ $$q-1.00000i q^{2} +2.00000i q^{3} -1.00000 q^{4} +2.00000 q^{6} -4.00000i q^{7} +1.00000i q^{8} -1.00000 q^{9} -2.00000i q^{12} -2.00000i q^{13} -4.00000 q^{14} +1.00000 q^{16} +1.00000i q^{18} -5.00000 q^{19} +8.00000 q^{21} -3.00000i q^{23} -2.00000 q^{24} -2.00000 q^{26} +4.00000i q^{27} +4.00000i q^{28} -6.00000 q^{29} -4.00000 q^{31} -1.00000i q^{32} +1.00000 q^{36} +1.00000i q^{37} +5.00000i q^{38} +4.00000 q^{39} -9.00000 q^{41} -8.00000i q^{42} +7.00000i q^{43} -3.00000 q^{46} -6.00000i q^{47} +2.00000i q^{48} -9.00000 q^{49} +2.00000i q^{52} -9.00000i q^{53} +4.00000 q^{54} +4.00000 q^{56} -10.0000i q^{57} +6.00000i q^{58} -3.00000 q^{59} +2.00000 q^{61} +4.00000i q^{62} +4.00000i q^{63} -1.00000 q^{64} +2.00000i q^{67} +6.00000 q^{69} -6.00000 q^{71} -1.00000i q^{72} -11.0000i q^{73} +1.00000 q^{74} +5.00000 q^{76} -4.00000i q^{78} +1.00000 q^{79} -11.0000 q^{81} +9.00000i q^{82} -8.00000 q^{84} +7.00000 q^{86} -12.0000i q^{87} +6.00000 q^{89} -8.00000 q^{91} +3.00000i q^{92} -8.00000i q^{93} -6.00000 q^{94} +2.00000 q^{96} -4.00000i q^{97} +9.00000i q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{4} + 4 q^{6} - 2 q^{9}+O(q^{10})$$ 2 * q - 2 * q^4 + 4 * q^6 - 2 * q^9 $$2 q - 2 q^{4} + 4 q^{6} - 2 q^{9} - 8 q^{14} + 2 q^{16} - 10 q^{19} + 16 q^{21} - 4 q^{24} - 4 q^{26} - 12 q^{29} - 8 q^{31} + 2 q^{36} + 8 q^{39} - 18 q^{41} - 6 q^{46} - 18 q^{49} + 8 q^{54} + 8 q^{56} - 6 q^{59} + 4 q^{61} - 2 q^{64} + 12 q^{69} - 12 q^{71} + 2 q^{74} + 10 q^{76} + 2 q^{79} - 22 q^{81} - 16 q^{84} + 14 q^{86} + 12 q^{89} - 16 q^{91} - 12 q^{94} + 4 q^{96}+O(q^{100})$$ 2 * q - 2 * q^4 + 4 * q^6 - 2 * q^9 - 8 * q^14 + 2 * q^16 - 10 * q^19 + 16 * q^21 - 4 * q^24 - 4 * q^26 - 12 * q^29 - 8 * q^31 + 2 * q^36 + 8 * q^39 - 18 * q^41 - 6 * q^46 - 18 * q^49 + 8 * q^54 + 8 * q^56 - 6 * q^59 + 4 * q^61 - 2 * q^64 + 12 * q^69 - 12 * q^71 + 2 * q^74 + 10 * q^76 + 2 * q^79 - 22 * q^81 - 16 * q^84 + 14 * q^86 + 12 * q^89 - 16 * q^91 - 12 * q^94 + 4 * q^96

## 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
$$3$$ 2.00000i 1.15470i 0.816497 + 0.577350i $$0.195913\pi$$
−0.816497 + 0.577350i $$0.804087\pi$$
$$4$$ −1.00000 −0.500000
$$5$$ 0 0
$$6$$ 2.00000 0.816497
$$7$$ − 4.00000i − 1.51186i −0.654654 0.755929i $$-0.727186\pi$$
0.654654 0.755929i $$-0.272814\pi$$
$$8$$ 1.00000i 0.353553i
$$9$$ −1.00000 −0.333333
$$10$$ 0 0
$$11$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$12$$ − 2.00000i − 0.577350i
$$13$$ − 2.00000i − 0.554700i −0.960769 0.277350i $$-0.910544\pi$$
0.960769 0.277350i $$-0.0894562\pi$$
$$14$$ −4.00000 −1.06904
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$18$$ 1.00000i 0.235702i
$$19$$ −5.00000 −1.14708 −0.573539 0.819178i $$-0.694430\pi$$
−0.573539 + 0.819178i $$0.694430\pi$$
$$20$$ 0 0
$$21$$ 8.00000 1.74574
$$22$$ 0 0
$$23$$ − 3.00000i − 0.625543i −0.949828 0.312772i $$-0.898743\pi$$
0.949828 0.312772i $$-0.101257\pi$$
$$24$$ −2.00000 −0.408248
$$25$$ 0 0
$$26$$ −2.00000 −0.392232
$$27$$ 4.00000i 0.769800i
$$28$$ 4.00000i 0.755929i
$$29$$ −6.00000 −1.11417 −0.557086 0.830455i $$-0.688081\pi$$
−0.557086 + 0.830455i $$0.688081\pi$$
$$30$$ 0 0
$$31$$ −4.00000 −0.718421 −0.359211 0.933257i $$-0.616954\pi$$
−0.359211 + 0.933257i $$0.616954\pi$$
$$32$$ − 1.00000i − 0.176777i
$$33$$ 0 0
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 1.00000 0.166667
$$37$$ 1.00000i 0.164399i
$$38$$ 5.00000i 0.811107i
$$39$$ 4.00000 0.640513
$$40$$ 0 0
$$41$$ −9.00000 −1.40556 −0.702782 0.711405i $$-0.748059\pi$$
−0.702782 + 0.711405i $$0.748059\pi$$
$$42$$ − 8.00000i − 1.23443i
$$43$$ 7.00000i 1.06749i 0.845645 + 0.533745i $$0.179216\pi$$
−0.845645 + 0.533745i $$0.820784\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ −3.00000 −0.442326
$$47$$ − 6.00000i − 0.875190i −0.899172 0.437595i $$-0.855830\pi$$
0.899172 0.437595i $$-0.144170\pi$$
$$48$$ 2.00000i 0.288675i
$$49$$ −9.00000 −1.28571
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 2.00000i 0.277350i
$$53$$ − 9.00000i − 1.23625i −0.786082 0.618123i $$-0.787894\pi$$
0.786082 0.618123i $$-0.212106\pi$$
$$54$$ 4.00000 0.544331
$$55$$ 0 0
$$56$$ 4.00000 0.534522
$$57$$ − 10.0000i − 1.32453i
$$58$$ 6.00000i 0.787839i
$$59$$ −3.00000 −0.390567 −0.195283 0.980747i $$-0.562563\pi$$
−0.195283 + 0.980747i $$0.562563\pi$$
$$60$$ 0 0
$$61$$ 2.00000 0.256074 0.128037 0.991769i $$-0.459132\pi$$
0.128037 + 0.991769i $$0.459132\pi$$
$$62$$ 4.00000i 0.508001i
$$63$$ 4.00000i 0.503953i
$$64$$ −1.00000 −0.125000
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 2.00000i 0.244339i 0.992509 + 0.122169i $$0.0389851\pi$$
−0.992509 + 0.122169i $$0.961015\pi$$
$$68$$ 0 0
$$69$$ 6.00000 0.722315
$$70$$ 0 0
$$71$$ −6.00000 −0.712069 −0.356034 0.934473i $$-0.615871\pi$$
−0.356034 + 0.934473i $$0.615871\pi$$
$$72$$ − 1.00000i − 0.117851i
$$73$$ − 11.0000i − 1.28745i −0.765256 0.643726i $$-0.777388\pi$$
0.765256 0.643726i $$-0.222612\pi$$
$$74$$ 1.00000 0.116248
$$75$$ 0 0
$$76$$ 5.00000 0.573539
$$77$$ 0 0
$$78$$ − 4.00000i − 0.452911i
$$79$$ 1.00000 0.112509 0.0562544 0.998416i $$-0.482084\pi$$
0.0562544 + 0.998416i $$0.482084\pi$$
$$80$$ 0 0
$$81$$ −11.0000 −1.22222
$$82$$ 9.00000i 0.993884i
$$83$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$84$$ −8.00000 −0.872872
$$85$$ 0 0
$$86$$ 7.00000 0.754829
$$87$$ − 12.0000i − 1.28654i
$$88$$ 0 0
$$89$$ 6.00000 0.635999 0.317999 0.948091i $$-0.396989\pi$$
0.317999 + 0.948091i $$0.396989\pi$$
$$90$$ 0 0
$$91$$ −8.00000 −0.838628
$$92$$ 3.00000i 0.312772i
$$93$$ − 8.00000i − 0.829561i
$$94$$ −6.00000 −0.618853
$$95$$ 0 0
$$96$$ 2.00000 0.204124
$$97$$ − 4.00000i − 0.406138i −0.979164 0.203069i $$-0.934908\pi$$
0.979164 0.203069i $$-0.0650917\pi$$
$$98$$ 9.00000i 0.909137i
$$99$$ 0 0
$$100$$ 0 0
$$101$$ 3.00000 0.298511 0.149256 0.988799i $$-0.452312\pi$$
0.149256 + 0.988799i $$0.452312\pi$$
$$102$$ 0 0
$$103$$ 13.0000i 1.28093i 0.767988 + 0.640464i $$0.221258\pi$$
−0.767988 + 0.640464i $$0.778742\pi$$
$$104$$ 2.00000 0.196116
$$105$$ 0 0
$$106$$ −9.00000 −0.874157
$$107$$ − 18.0000i − 1.74013i −0.492941 0.870063i $$-0.664078\pi$$
0.492941 0.870063i $$-0.335922\pi$$
$$108$$ − 4.00000i − 0.384900i
$$109$$ 16.0000 1.53252 0.766261 0.642529i $$-0.222115\pi$$
0.766261 + 0.642529i $$0.222115\pi$$
$$110$$ 0 0
$$111$$ −2.00000 −0.189832
$$112$$ − 4.00000i − 0.377964i
$$113$$ − 18.0000i − 1.69330i −0.532152 0.846649i $$-0.678617\pi$$
0.532152 0.846649i $$-0.321383\pi$$
$$114$$ −10.0000 −0.936586
$$115$$ 0 0
$$116$$ 6.00000 0.557086
$$117$$ 2.00000i 0.184900i
$$118$$ 3.00000i 0.276172i
$$119$$ 0 0
$$120$$ 0 0
$$121$$ −11.0000 −1.00000
$$122$$ − 2.00000i − 0.181071i
$$123$$ − 18.0000i − 1.62301i
$$124$$ 4.00000 0.359211
$$125$$ 0 0
$$126$$ 4.00000 0.356348
$$127$$ − 16.0000i − 1.41977i −0.704317 0.709885i $$-0.748747\pi$$
0.704317 0.709885i $$-0.251253\pi$$
$$128$$ 1.00000i 0.0883883i
$$129$$ −14.0000 −1.23263
$$130$$ 0 0
$$131$$ −12.0000 −1.04844 −0.524222 0.851581i $$-0.675644\pi$$
−0.524222 + 0.851581i $$0.675644\pi$$
$$132$$ 0 0
$$133$$ 20.0000i 1.73422i
$$134$$ 2.00000 0.172774
$$135$$ 0 0
$$136$$ 0 0
$$137$$ 6.00000i 0.512615i 0.966595 + 0.256307i $$0.0825059\pi$$
−0.966595 + 0.256307i $$0.917494\pi$$
$$138$$ − 6.00000i − 0.510754i
$$139$$ −14.0000 −1.18746 −0.593732 0.804663i $$-0.702346\pi$$
−0.593732 + 0.804663i $$0.702346\pi$$
$$140$$ 0 0
$$141$$ 12.0000 1.01058
$$142$$ 6.00000i 0.503509i
$$143$$ 0 0
$$144$$ −1.00000 −0.0833333
$$145$$ 0 0
$$146$$ −11.0000 −0.910366
$$147$$ − 18.0000i − 1.48461i
$$148$$ − 1.00000i − 0.0821995i
$$149$$ −15.0000 −1.22885 −0.614424 0.788976i $$-0.710612\pi$$
−0.614424 + 0.788976i $$0.710612\pi$$
$$150$$ 0 0
$$151$$ −4.00000 −0.325515 −0.162758 0.986666i $$-0.552039\pi$$
−0.162758 + 0.986666i $$0.552039\pi$$
$$152$$ − 5.00000i − 0.405554i
$$153$$ 0 0
$$154$$ 0 0
$$155$$ 0 0
$$156$$ −4.00000 −0.320256
$$157$$ − 13.0000i − 1.03751i −0.854922 0.518756i $$-0.826395\pi$$
0.854922 0.518756i $$-0.173605\pi$$
$$158$$ − 1.00000i − 0.0795557i
$$159$$ 18.0000 1.42749
$$160$$ 0 0
$$161$$ −12.0000 −0.945732
$$162$$ 11.0000i 0.864242i
$$163$$ 7.00000i 0.548282i 0.961689 + 0.274141i $$0.0883936\pi$$
−0.961689 + 0.274141i $$0.911606\pi$$
$$164$$ 9.00000 0.702782
$$165$$ 0 0
$$166$$ 0 0
$$167$$ − 3.00000i − 0.232147i −0.993241 0.116073i $$-0.962969\pi$$
0.993241 0.116073i $$-0.0370308\pi$$
$$168$$ 8.00000i 0.617213i
$$169$$ 9.00000 0.692308
$$170$$ 0 0
$$171$$ 5.00000 0.382360
$$172$$ − 7.00000i − 0.533745i
$$173$$ − 6.00000i − 0.456172i −0.973641 0.228086i $$-0.926753\pi$$
0.973641 0.228086i $$-0.0732467\pi$$
$$174$$ −12.0000 −0.909718
$$175$$ 0 0
$$176$$ 0 0
$$177$$ − 6.00000i − 0.450988i
$$178$$ − 6.00000i − 0.449719i
$$179$$ 12.0000 0.896922 0.448461 0.893802i $$-0.351972\pi$$
0.448461 + 0.893802i $$0.351972\pi$$
$$180$$ 0 0
$$181$$ 5.00000 0.371647 0.185824 0.982583i $$-0.440505\pi$$
0.185824 + 0.982583i $$0.440505\pi$$
$$182$$ 8.00000i 0.592999i
$$183$$ 4.00000i 0.295689i
$$184$$ 3.00000 0.221163
$$185$$ 0 0
$$186$$ −8.00000 −0.586588
$$187$$ 0 0
$$188$$ 6.00000i 0.437595i
$$189$$ 16.0000 1.16383
$$190$$ 0 0
$$191$$ −9.00000 −0.651217 −0.325609 0.945505i $$-0.605569\pi$$
−0.325609 + 0.945505i $$0.605569\pi$$
$$192$$ − 2.00000i − 0.144338i
$$193$$ 4.00000i 0.287926i 0.989583 + 0.143963i $$0.0459847\pi$$
−0.989583 + 0.143963i $$0.954015\pi$$
$$194$$ −4.00000 −0.287183
$$195$$ 0 0
$$196$$ 9.00000 0.642857
$$197$$ 27.0000i 1.92367i 0.273629 + 0.961835i $$0.411776\pi$$
−0.273629 + 0.961835i $$0.588224\pi$$
$$198$$ 0 0
$$199$$ 13.0000 0.921546 0.460773 0.887518i $$-0.347572\pi$$
0.460773 + 0.887518i $$0.347572\pi$$
$$200$$ 0 0
$$201$$ −4.00000 −0.282138
$$202$$ − 3.00000i − 0.211079i
$$203$$ 24.0000i 1.68447i
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 13.0000 0.905753
$$207$$ 3.00000i 0.208514i
$$208$$ − 2.00000i − 0.138675i
$$209$$ 0 0
$$210$$ 0 0
$$211$$ 2.00000 0.137686 0.0688428 0.997628i $$-0.478069\pi$$
0.0688428 + 0.997628i $$0.478069\pi$$
$$212$$ 9.00000i 0.618123i
$$213$$ − 12.0000i − 0.822226i
$$214$$ −18.0000 −1.23045
$$215$$ 0 0
$$216$$ −4.00000 −0.272166
$$217$$ 16.0000i 1.08615i
$$218$$ − 16.0000i − 1.08366i
$$219$$ 22.0000 1.48662
$$220$$ 0 0
$$221$$ 0 0
$$222$$ 2.00000i 0.134231i
$$223$$ − 2.00000i − 0.133930i −0.997755 0.0669650i $$-0.978668\pi$$
0.997755 0.0669650i $$-0.0213316\pi$$
$$224$$ −4.00000 −0.267261
$$225$$ 0 0
$$226$$ −18.0000 −1.19734
$$227$$ − 9.00000i − 0.597351i −0.954355 0.298675i $$-0.903455\pi$$
0.954355 0.298675i $$-0.0965448\pi$$
$$228$$ 10.0000i 0.662266i
$$229$$ 13.0000 0.859064 0.429532 0.903052i $$-0.358679\pi$$
0.429532 + 0.903052i $$0.358679\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ − 6.00000i − 0.393919i
$$233$$ 21.0000i 1.37576i 0.725826 + 0.687878i $$0.241458\pi$$
−0.725826 + 0.687878i $$0.758542\pi$$
$$234$$ 2.00000 0.130744
$$235$$ 0 0
$$236$$ 3.00000 0.195283
$$237$$ 2.00000i 0.129914i
$$238$$ 0 0
$$239$$ −9.00000 −0.582162 −0.291081 0.956698i $$-0.594015\pi$$
−0.291081 + 0.956698i $$0.594015\pi$$
$$240$$ 0 0
$$241$$ 14.0000 0.901819 0.450910 0.892570i $$-0.351100\pi$$
0.450910 + 0.892570i $$0.351100\pi$$
$$242$$ 11.0000i 0.707107i
$$243$$ − 10.0000i − 0.641500i
$$244$$ −2.00000 −0.128037
$$245$$ 0 0
$$246$$ −18.0000 −1.14764
$$247$$ 10.0000i 0.636285i
$$248$$ − 4.00000i − 0.254000i
$$249$$ 0 0
$$250$$ 0 0
$$251$$ −27.0000 −1.70422 −0.852112 0.523359i $$-0.824679\pi$$
−0.852112 + 0.523359i $$0.824679\pi$$
$$252$$ − 4.00000i − 0.251976i
$$253$$ 0 0
$$254$$ −16.0000 −1.00393
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$258$$ 14.0000i 0.871602i
$$259$$ 4.00000 0.248548
$$260$$ 0 0
$$261$$ 6.00000 0.371391
$$262$$ 12.0000i 0.741362i
$$263$$ − 6.00000i − 0.369976i −0.982741 0.184988i $$-0.940775\pi$$
0.982741 0.184988i $$-0.0592246\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 20.0000 1.22628
$$267$$ 12.0000i 0.734388i
$$268$$ − 2.00000i − 0.122169i
$$269$$ 21.0000 1.28039 0.640196 0.768211i $$-0.278853\pi$$
0.640196 + 0.768211i $$0.278853\pi$$
$$270$$ 0 0
$$271$$ −16.0000 −0.971931 −0.485965 0.873978i $$-0.661532\pi$$
−0.485965 + 0.873978i $$0.661532\pi$$
$$272$$ 0 0
$$273$$ − 16.0000i − 0.968364i
$$274$$ 6.00000 0.362473
$$275$$ 0 0
$$276$$ −6.00000 −0.361158
$$277$$ − 22.0000i − 1.32185i −0.750451 0.660926i $$-0.770164\pi$$
0.750451 0.660926i $$-0.229836\pi$$
$$278$$ 14.0000i 0.839664i
$$279$$ 4.00000 0.239474
$$280$$ 0 0
$$281$$ 12.0000 0.715860 0.357930 0.933748i $$-0.383483\pi$$
0.357930 + 0.933748i $$0.383483\pi$$
$$282$$ − 12.0000i − 0.714590i
$$283$$ 19.0000i 1.12943i 0.825285 + 0.564716i $$0.191014\pi$$
−0.825285 + 0.564716i $$0.808986\pi$$
$$284$$ 6.00000 0.356034
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 36.0000i 2.12501i
$$288$$ 1.00000i 0.0589256i
$$289$$ 17.0000 1.00000
$$290$$ 0 0
$$291$$ 8.00000 0.468968
$$292$$ 11.0000i 0.643726i
$$293$$ 3.00000i 0.175262i 0.996153 + 0.0876309i $$0.0279296\pi$$
−0.996153 + 0.0876309i $$0.972070\pi$$
$$294$$ −18.0000 −1.04978
$$295$$ 0 0
$$296$$ −1.00000 −0.0581238
$$297$$ 0 0
$$298$$ 15.0000i 0.868927i
$$299$$ −6.00000 −0.346989
$$300$$ 0 0
$$301$$ 28.0000 1.61389
$$302$$ 4.00000i 0.230174i
$$303$$ 6.00000i 0.344691i
$$304$$ −5.00000 −0.286770
$$305$$ 0 0
$$306$$ 0 0
$$307$$ 32.0000i 1.82634i 0.407583 + 0.913168i $$0.366372\pi$$
−0.407583 + 0.913168i $$0.633628\pi$$
$$308$$ 0 0
$$309$$ −26.0000 −1.47909
$$310$$ 0 0
$$311$$ 15.0000 0.850572 0.425286 0.905059i $$-0.360174\pi$$
0.425286 + 0.905059i $$0.360174\pi$$
$$312$$ 4.00000i 0.226455i
$$313$$ 22.0000i 1.24351i 0.783210 + 0.621757i $$0.213581\pi$$
−0.783210 + 0.621757i $$0.786419\pi$$
$$314$$ −13.0000 −0.733632
$$315$$ 0 0
$$316$$ −1.00000 −0.0562544
$$317$$ 33.0000i 1.85346i 0.375722 + 0.926732i $$0.377395\pi$$
−0.375722 + 0.926732i $$0.622605\pi$$
$$318$$ − 18.0000i − 1.00939i
$$319$$ 0 0
$$320$$ 0 0
$$321$$ 36.0000 2.00932
$$322$$ 12.0000i 0.668734i
$$323$$ 0 0
$$324$$ 11.0000 0.611111
$$325$$ 0 0
$$326$$ 7.00000 0.387694
$$327$$ 32.0000i 1.76960i
$$328$$ − 9.00000i − 0.496942i
$$329$$ −24.0000 −1.32316
$$330$$ 0 0
$$331$$ 8.00000 0.439720 0.219860 0.975531i $$-0.429440\pi$$
0.219860 + 0.975531i $$0.429440\pi$$
$$332$$ 0 0
$$333$$ − 1.00000i − 0.0547997i
$$334$$ −3.00000 −0.164153
$$335$$ 0 0
$$336$$ 8.00000 0.436436
$$337$$ − 25.0000i − 1.36184i −0.732359 0.680918i $$-0.761581\pi$$
0.732359 0.680918i $$-0.238419\pi$$
$$338$$ − 9.00000i − 0.489535i
$$339$$ 36.0000 1.95525
$$340$$ 0 0
$$341$$ 0 0
$$342$$ − 5.00000i − 0.270369i
$$343$$ 8.00000i 0.431959i
$$344$$ −7.00000 −0.377415
$$345$$ 0 0
$$346$$ −6.00000 −0.322562
$$347$$ 15.0000i 0.805242i 0.915367 + 0.402621i $$0.131901\pi$$
−0.915367 + 0.402621i $$0.868099\pi$$
$$348$$ 12.0000i 0.643268i
$$349$$ 19.0000 1.01705 0.508523 0.861048i $$-0.330192\pi$$
0.508523 + 0.861048i $$0.330192\pi$$
$$350$$ 0 0
$$351$$ 8.00000 0.427008
$$352$$ 0 0
$$353$$ − 12.0000i − 0.638696i −0.947638 0.319348i $$-0.896536\pi$$
0.947638 0.319348i $$-0.103464\pi$$
$$354$$ −6.00000 −0.318896
$$355$$ 0 0
$$356$$ −6.00000 −0.317999
$$357$$ 0 0
$$358$$ − 12.0000i − 0.634220i
$$359$$ −30.0000 −1.58334 −0.791670 0.610949i $$-0.790788\pi$$
−0.791670 + 0.610949i $$0.790788\pi$$
$$360$$ 0 0
$$361$$ 6.00000 0.315789
$$362$$ − 5.00000i − 0.262794i
$$363$$ − 22.0000i − 1.15470i
$$364$$ 8.00000 0.419314
$$365$$ 0 0
$$366$$ 4.00000 0.209083
$$367$$ 2.00000i 0.104399i 0.998637 + 0.0521996i $$0.0166232\pi$$
−0.998637 + 0.0521996i $$0.983377\pi$$
$$368$$ − 3.00000i − 0.156386i
$$369$$ 9.00000 0.468521
$$370$$ 0 0
$$371$$ −36.0000 −1.86903
$$372$$ 8.00000i 0.414781i
$$373$$ − 14.0000i − 0.724893i −0.932005 0.362446i $$-0.881942\pi$$
0.932005 0.362446i $$-0.118058\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 6.00000 0.309426
$$377$$ 12.0000i 0.618031i
$$378$$ − 16.0000i − 0.822951i
$$379$$ −20.0000 −1.02733 −0.513665 0.857991i $$-0.671713\pi$$
−0.513665 + 0.857991i $$0.671713\pi$$
$$380$$ 0 0
$$381$$ 32.0000 1.63941
$$382$$ 9.00000i 0.460480i
$$383$$ 15.0000i 0.766464i 0.923652 + 0.383232i $$0.125189\pi$$
−0.923652 + 0.383232i $$0.874811\pi$$
$$384$$ −2.00000 −0.102062
$$385$$ 0 0
$$386$$ 4.00000 0.203595
$$387$$ − 7.00000i − 0.355830i
$$388$$ 4.00000i 0.203069i
$$389$$ 36.0000 1.82527 0.912636 0.408773i $$-0.134043\pi$$
0.912636 + 0.408773i $$0.134043\pi$$
$$390$$ 0 0
$$391$$ 0 0
$$392$$ − 9.00000i − 0.454569i
$$393$$ − 24.0000i − 1.21064i
$$394$$ 27.0000 1.36024
$$395$$ 0 0
$$396$$ 0 0
$$397$$ − 25.0000i − 1.25471i −0.778732 0.627357i $$-0.784137\pi$$
0.778732 0.627357i $$-0.215863\pi$$
$$398$$ − 13.0000i − 0.651631i
$$399$$ −40.0000 −2.00250
$$400$$ 0 0
$$401$$ 18.0000 0.898877 0.449439 0.893311i $$-0.351624\pi$$
0.449439 + 0.893311i $$0.351624\pi$$
$$402$$ 4.00000i 0.199502i
$$403$$ 8.00000i 0.398508i
$$404$$ −3.00000 −0.149256
$$405$$ 0 0
$$406$$ 24.0000 1.19110
$$407$$ 0 0
$$408$$ 0 0
$$409$$ −20.0000 −0.988936 −0.494468 0.869196i $$-0.664637\pi$$
−0.494468 + 0.869196i $$0.664637\pi$$
$$410$$ 0 0
$$411$$ −12.0000 −0.591916
$$412$$ − 13.0000i − 0.640464i
$$413$$ 12.0000i 0.590481i
$$414$$ 3.00000 0.147442
$$415$$ 0 0
$$416$$ −2.00000 −0.0980581
$$417$$ − 28.0000i − 1.37117i
$$418$$ 0 0
$$419$$ −12.0000 −0.586238 −0.293119 0.956076i $$-0.594693\pi$$
−0.293119 + 0.956076i $$0.594693\pi$$
$$420$$ 0 0
$$421$$ 26.0000 1.26716 0.633581 0.773676i $$-0.281584\pi$$
0.633581 + 0.773676i $$0.281584\pi$$
$$422$$ − 2.00000i − 0.0973585i
$$423$$ 6.00000i 0.291730i
$$424$$ 9.00000 0.437079
$$425$$ 0 0
$$426$$ −12.0000 −0.581402
$$427$$ − 8.00000i − 0.387147i
$$428$$ 18.0000i 0.870063i
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 15.0000 0.722525 0.361262 0.932464i $$-0.382346\pi$$
0.361262 + 0.932464i $$0.382346\pi$$
$$432$$ 4.00000i 0.192450i
$$433$$ − 26.0000i − 1.24948i −0.780833 0.624740i $$-0.785205\pi$$
0.780833 0.624740i $$-0.214795\pi$$
$$434$$ 16.0000 0.768025
$$435$$ 0 0
$$436$$ −16.0000 −0.766261
$$437$$ 15.0000i 0.717547i
$$438$$ − 22.0000i − 1.05120i
$$439$$ 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
$$442$$ 0 0
$$443$$ − 24.0000i − 1.14027i −0.821549 0.570137i $$-0.806890\pi$$
0.821549 0.570137i $$-0.193110\pi$$
$$444$$ 2.00000 0.0949158
$$445$$ 0 0
$$446$$ −2.00000 −0.0947027
$$447$$ − 30.0000i − 1.41895i
$$448$$ 4.00000i 0.188982i
$$449$$ −6.00000 −0.283158 −0.141579 0.989927i $$-0.545218\pi$$
−0.141579 + 0.989927i $$0.545218\pi$$
$$450$$ 0 0
$$451$$ 0 0
$$452$$ 18.0000i 0.846649i
$$453$$ − 8.00000i − 0.375873i
$$454$$ −9.00000 −0.422391
$$455$$ 0 0
$$456$$ 10.0000 0.468293
$$457$$ − 28.0000i − 1.30978i −0.755722 0.654892i $$-0.772714\pi$$
0.755722 0.654892i $$-0.227286\pi$$
$$458$$ − 13.0000i − 0.607450i
$$459$$ 0 0
$$460$$ 0 0
$$461$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$462$$ 0 0
$$463$$ − 32.0000i − 1.48717i −0.668644 0.743583i $$-0.733125\pi$$
0.668644 0.743583i $$-0.266875\pi$$
$$464$$ −6.00000 −0.278543
$$465$$ 0 0
$$466$$ 21.0000 0.972806
$$467$$ 12.0000i 0.555294i 0.960683 + 0.277647i $$0.0895545\pi$$
−0.960683 + 0.277647i $$0.910445\pi$$
$$468$$ − 2.00000i − 0.0924500i
$$469$$ 8.00000 0.369406
$$470$$ 0 0
$$471$$ 26.0000 1.19802
$$472$$ − 3.00000i − 0.138086i
$$473$$ 0 0
$$474$$ 2.00000 0.0918630
$$475$$ 0 0
$$476$$ 0 0
$$477$$ 9.00000i 0.412082i
$$478$$ 9.00000i 0.411650i
$$479$$ −24.0000 −1.09659 −0.548294 0.836286i $$-0.684723\pi$$
−0.548294 + 0.836286i $$0.684723\pi$$
$$480$$ 0 0
$$481$$ 2.00000 0.0911922
$$482$$ − 14.0000i − 0.637683i
$$483$$ − 24.0000i − 1.09204i
$$484$$ 11.0000 0.500000
$$485$$ 0 0
$$486$$ −10.0000 −0.453609
$$487$$ − 16.0000i − 0.725029i −0.931978 0.362515i $$-0.881918\pi$$
0.931978 0.362515i $$-0.118082\pi$$
$$488$$ 2.00000i 0.0905357i
$$489$$ −14.0000 −0.633102
$$490$$ 0 0
$$491$$ −18.0000 −0.812329 −0.406164 0.913800i $$-0.633134\pi$$
−0.406164 + 0.913800i $$0.633134\pi$$
$$492$$ 18.0000i 0.811503i
$$493$$ 0 0
$$494$$ 10.0000 0.449921
$$495$$ 0 0
$$496$$ −4.00000 −0.179605
$$497$$ 24.0000i 1.07655i
$$498$$ 0 0
$$499$$ 43.0000 1.92494 0.962472 0.271380i $$-0.0874801\pi$$
0.962472 + 0.271380i $$0.0874801\pi$$
$$500$$ 0 0
$$501$$ 6.00000 0.268060
$$502$$ 27.0000i 1.20507i
$$503$$ 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
$$506$$ 0 0
$$507$$ 18.0000i 0.799408i
$$508$$ 16.0000i 0.709885i
$$509$$ −9.00000 −0.398918 −0.199459 0.979906i $$-0.563918\pi$$
−0.199459 + 0.979906i $$0.563918\pi$$
$$510$$ 0 0
$$511$$ −44.0000 −1.94645
$$512$$ − 1.00000i − 0.0441942i
$$513$$ − 20.0000i − 0.883022i
$$514$$ 0 0
$$515$$ 0 0
$$516$$ 14.0000 0.616316
$$517$$ 0 0
$$518$$ − 4.00000i − 0.175750i
$$519$$ 12.0000 0.526742
$$520$$ 0 0
$$521$$ −33.0000 −1.44576 −0.722878 0.690976i $$-0.757181\pi$$
−0.722878 + 0.690976i $$0.757181\pi$$
$$522$$ − 6.00000i − 0.262613i
$$523$$ 28.0000i 1.22435i 0.790721 + 0.612177i $$0.209706\pi$$
−0.790721 + 0.612177i $$0.790294\pi$$
$$524$$ 12.0000 0.524222
$$525$$ 0 0
$$526$$ −6.00000 −0.261612
$$527$$ 0 0
$$528$$ 0 0
$$529$$ 14.0000 0.608696
$$530$$ 0 0
$$531$$ 3.00000 0.130189
$$532$$ − 20.0000i − 0.867110i
$$533$$ 18.0000i 0.779667i
$$534$$ 12.0000 0.519291
$$535$$ 0 0
$$536$$ −2.00000 −0.0863868
$$537$$ 24.0000i 1.03568i
$$538$$ − 21.0000i − 0.905374i
$$539$$ 0 0
$$540$$ 0 0
$$541$$ −10.0000 −0.429934 −0.214967 0.976621i $$-0.568964\pi$$
−0.214967 + 0.976621i $$0.568964\pi$$
$$542$$ 16.0000i 0.687259i
$$543$$ 10.0000i 0.429141i
$$544$$ 0 0
$$545$$ 0 0
$$546$$ −16.0000 −0.684737
$$547$$ − 28.0000i − 1.19719i −0.801050 0.598597i $$-0.795725\pi$$
0.801050 0.598597i $$-0.204275\pi$$
$$548$$ − 6.00000i − 0.256307i
$$549$$ −2.00000 −0.0853579
$$550$$ 0 0
$$551$$ 30.0000 1.27804
$$552$$ 6.00000i 0.255377i
$$553$$ − 4.00000i − 0.170097i
$$554$$ −22.0000 −0.934690
$$555$$ 0 0
$$556$$ 14.0000 0.593732
$$557$$ − 12.0000i − 0.508456i −0.967144 0.254228i $$-0.918179\pi$$
0.967144 0.254228i $$-0.0818214\pi$$
$$558$$ − 4.00000i − 0.169334i
$$559$$ 14.0000 0.592137
$$560$$ 0 0
$$561$$ 0 0
$$562$$ − 12.0000i − 0.506189i
$$563$$ − 45.0000i − 1.89652i −0.317489 0.948262i $$-0.602840\pi$$
0.317489 0.948262i $$-0.397160\pi$$
$$564$$ −12.0000 −0.505291
$$565$$ 0 0
$$566$$ 19.0000 0.798630
$$567$$ 44.0000i 1.84783i
$$568$$ − 6.00000i − 0.251754i
$$569$$ 24.0000 1.00613 0.503066 0.864248i $$-0.332205\pi$$
0.503066 + 0.864248i $$0.332205\pi$$
$$570$$ 0 0
$$571$$ 2.00000 0.0836974 0.0418487 0.999124i $$-0.486675\pi$$
0.0418487 + 0.999124i $$0.486675\pi$$
$$572$$ 0 0
$$573$$ − 18.0000i − 0.751961i
$$574$$ 36.0000 1.50261
$$575$$ 0 0
$$576$$ 1.00000 0.0416667
$$577$$ − 40.0000i − 1.66522i −0.553858 0.832611i $$-0.686845\pi$$
0.553858 0.832611i $$-0.313155\pi$$
$$578$$ − 17.0000i − 0.707107i
$$579$$ −8.00000 −0.332469
$$580$$ 0 0
$$581$$ 0 0
$$582$$ − 8.00000i − 0.331611i
$$583$$ 0 0
$$584$$ 11.0000 0.455183
$$585$$ 0 0
$$586$$ 3.00000 0.123929
$$587$$ − 3.00000i − 0.123823i −0.998082 0.0619116i $$-0.980280\pi$$
0.998082 0.0619116i $$-0.0197197\pi$$
$$588$$ 18.0000i 0.742307i
$$589$$ 20.0000 0.824086
$$590$$ 0 0
$$591$$ −54.0000 −2.22126
$$592$$ 1.00000i 0.0410997i
$$593$$ − 15.0000i − 0.615976i −0.951390 0.307988i $$-0.900344\pi$$
0.951390 0.307988i $$-0.0996557\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 15.0000 0.614424
$$597$$ 26.0000i 1.06411i
$$598$$ 6.00000i 0.245358i
$$599$$ −36.0000 −1.47092 −0.735460 0.677568i $$-0.763034\pi$$
−0.735460 + 0.677568i $$0.763034\pi$$
$$600$$ 0 0
$$601$$ −37.0000 −1.50926 −0.754631 0.656150i $$-0.772184\pi$$
−0.754631 + 0.656150i $$0.772184\pi$$
$$602$$ − 28.0000i − 1.14119i
$$603$$ − 2.00000i − 0.0814463i
$$604$$ 4.00000 0.162758
$$605$$ 0 0
$$606$$ 6.00000 0.243733
$$607$$ 32.0000i 1.29884i 0.760430 + 0.649420i $$0.224988\pi$$
−0.760430 + 0.649420i $$0.775012\pi$$
$$608$$ 5.00000i 0.202777i
$$609$$ −48.0000 −1.94506
$$610$$ 0 0
$$611$$ −12.0000 −0.485468
$$612$$ 0 0
$$613$$ 25.0000i 1.00974i 0.863195 + 0.504870i $$0.168460\pi$$
−0.863195 + 0.504870i $$0.831540\pi$$
$$614$$ 32.0000 1.29141
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 18.0000i 0.724653i 0.932051 + 0.362326i $$0.118017\pi$$
−0.932051 + 0.362326i $$0.881983\pi$$
$$618$$ 26.0000i 1.04587i
$$619$$ 46.0000 1.84890 0.924448 0.381308i $$-0.124526\pi$$
0.924448 + 0.381308i $$0.124526\pi$$
$$620$$ 0 0
$$621$$ 12.0000 0.481543
$$622$$ − 15.0000i − 0.601445i
$$623$$ − 24.0000i − 0.961540i
$$624$$ 4.00000 0.160128
$$625$$ 0 0
$$626$$ 22.0000 0.879297
$$627$$ 0 0
$$628$$ 13.0000i 0.518756i
$$629$$ 0 0
$$630$$ 0 0
$$631$$ 32.0000 1.27390 0.636950 0.770905i $$-0.280196\pi$$
0.636950 + 0.770905i $$0.280196\pi$$
$$632$$ 1.00000i 0.0397779i
$$633$$ 4.00000i 0.158986i
$$634$$ 33.0000 1.31060
$$635$$ 0 0
$$636$$ −18.0000 −0.713746
$$637$$ 18.0000i 0.713186i
$$638$$ 0 0
$$639$$ 6.00000 0.237356
$$640$$ 0 0
$$641$$ −21.0000 −0.829450 −0.414725 0.909947i $$-0.636122\pi$$
−0.414725 + 0.909947i $$0.636122\pi$$
$$642$$ − 36.0000i − 1.42081i
$$643$$ 4.00000i 0.157745i 0.996885 + 0.0788723i $$0.0251319\pi$$
−0.996885 + 0.0788723i $$0.974868\pi$$
$$644$$ 12.0000 0.472866
$$645$$ 0 0
$$646$$ 0 0
$$647$$ − 27.0000i − 1.06148i −0.847535 0.530740i $$-0.821914\pi$$
0.847535 0.530740i $$-0.178086\pi$$
$$648$$ − 11.0000i − 0.432121i
$$649$$ 0 0
$$650$$ 0 0
$$651$$ −32.0000 −1.25418
$$652$$ − 7.00000i − 0.274141i
$$653$$ 6.00000i 0.234798i 0.993085 + 0.117399i $$0.0374557\pi$$
−0.993085 + 0.117399i $$0.962544\pi$$
$$654$$ 32.0000 1.25130
$$655$$ 0 0
$$656$$ −9.00000 −0.351391
$$657$$ 11.0000i 0.429151i
$$658$$ 24.0000i 0.935617i
$$659$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$660$$ 0 0
$$661$$ 32.0000 1.24466 0.622328 0.782757i $$-0.286187\pi$$
0.622328 + 0.782757i $$0.286187\pi$$
$$662$$ − 8.00000i − 0.310929i
$$663$$ 0 0
$$664$$ 0 0
$$665$$ 0 0
$$666$$ −1.00000 −0.0387492
$$667$$ 18.0000i 0.696963i
$$668$$ 3.00000i 0.116073i
$$669$$ 4.00000 0.154649
$$670$$ 0 0
$$671$$ 0 0
$$672$$ − 8.00000i − 0.308607i
$$673$$ 37.0000i 1.42625i 0.701039 + 0.713123i $$0.252720\pi$$
−0.701039 + 0.713123i $$0.747280\pi$$
$$674$$ −25.0000 −0.962964
$$675$$ 0 0
$$676$$ −9.00000 −0.346154
$$677$$ − 39.0000i − 1.49889i −0.662066 0.749446i $$-0.730320\pi$$
0.662066 0.749446i $$-0.269680\pi$$
$$678$$ − 36.0000i − 1.38257i
$$679$$ −16.0000 −0.614024
$$680$$ 0 0
$$681$$ 18.0000 0.689761
$$682$$ 0 0
$$683$$ 48.0000i 1.83667i 0.395805 + 0.918334i $$0.370466\pi$$
−0.395805 + 0.918334i $$0.629534\pi$$
$$684$$ −5.00000 −0.191180
$$685$$ 0 0
$$686$$ 8.00000 0.305441
$$687$$ 26.0000i 0.991962i
$$688$$ 7.00000i 0.266872i
$$689$$ −18.0000 −0.685745
$$690$$ 0 0
$$691$$ −10.0000 −0.380418 −0.190209 0.981744i $$-0.560917\pi$$
−0.190209 + 0.981744i $$0.560917\pi$$
$$692$$ 6.00000i 0.228086i
$$693$$ 0 0
$$694$$ 15.0000 0.569392
$$695$$ 0 0
$$696$$ 12.0000 0.454859
$$697$$ 0 0
$$698$$ − 19.0000i − 0.719161i
$$699$$ −42.0000 −1.58859
$$700$$ 0 0
$$701$$ 18.0000 0.679851 0.339925 0.940452i $$-0.389598\pi$$
0.339925 + 0.940452i $$0.389598\pi$$
$$702$$ − 8.00000i − 0.301941i
$$703$$ − 5.00000i − 0.188579i
$$704$$ 0 0
$$705$$ 0 0
$$706$$ −12.0000 −0.451626
$$707$$ − 12.0000i − 0.451306i
$$708$$ 6.00000i 0.225494i
$$709$$ 40.0000 1.50223 0.751116 0.660171i $$-0.229516\pi$$
0.751116 + 0.660171i $$0.229516\pi$$
$$710$$ 0 0
$$711$$ −1.00000 −0.0375029
$$712$$ 6.00000i 0.224860i
$$713$$ 12.0000i 0.449404i
$$714$$ 0 0
$$715$$ 0 0
$$716$$ −12.0000 −0.448461
$$717$$ − 18.0000i − 0.672222i
$$718$$ 30.0000i 1.11959i
$$719$$ −24.0000 −0.895049 −0.447524 0.894272i $$-0.647694\pi$$
−0.447524 + 0.894272i $$0.647694\pi$$
$$720$$ 0 0
$$721$$ 52.0000 1.93658
$$722$$ − 6.00000i − 0.223297i
$$723$$ 28.0000i 1.04133i
$$724$$ −5.00000 −0.185824
$$725$$ 0 0
$$726$$ −22.0000 −0.816497
$$727$$ − 16.0000i − 0.593407i −0.954970 0.296704i $$-0.904113\pi$$
0.954970 0.296704i $$-0.0958873\pi$$
$$728$$ − 8.00000i − 0.296500i
$$729$$ −13.0000 −0.481481
$$730$$ 0 0
$$731$$ 0 0
$$732$$ − 4.00000i − 0.147844i
$$733$$ 22.0000i 0.812589i 0.913742 + 0.406294i $$0.133179\pi$$
−0.913742 + 0.406294i $$0.866821\pi$$
$$734$$ 2.00000 0.0738213
$$735$$ 0 0
$$736$$ −3.00000 −0.110581
$$737$$ 0 0
$$738$$ − 9.00000i − 0.331295i
$$739$$ 34.0000 1.25071 0.625355 0.780340i $$-0.284954\pi$$
0.625355 + 0.780340i $$0.284954\pi$$
$$740$$ 0 0
$$741$$ −20.0000 −0.734718
$$742$$ 36.0000i 1.32160i
$$743$$ 6.00000i 0.220119i 0.993925 + 0.110059i $$0.0351041\pi$$
−0.993925 + 0.110059i $$0.964896\pi$$
$$744$$ 8.00000 0.293294
$$745$$ 0 0
$$746$$ −14.0000 −0.512576
$$747$$ 0 0
$$748$$ 0 0
$$749$$ −72.0000 −2.63082
$$750$$ 0 0
$$751$$ 50.0000 1.82453 0.912263 0.409605i $$-0.134333\pi$$
0.912263 + 0.409605i $$0.134333\pi$$
$$752$$ − 6.00000i − 0.218797i
$$753$$ − 54.0000i − 1.96787i
$$754$$ 12.0000 0.437014
$$755$$ 0 0
$$756$$ −16.0000 −0.581914
$$757$$ 20.0000i 0.726912i 0.931611 + 0.363456i $$0.118403\pi$$
−0.931611 + 0.363456i $$0.881597\pi$$
$$758$$ 20.0000i 0.726433i
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 45.0000 1.63125 0.815624 0.578582i $$-0.196394\pi$$
0.815624 + 0.578582i $$0.196394\pi$$
$$762$$ − 32.0000i − 1.15924i
$$763$$ − 64.0000i − 2.31696i
$$764$$ 9.00000 0.325609
$$765$$ 0 0
$$766$$ 15.0000 0.541972
$$767$$ 6.00000i 0.216647i
$$768$$ 2.00000i 0.0721688i
$$769$$ −26.0000 −0.937584 −0.468792 0.883309i $$-0.655311\pi$$
−0.468792 + 0.883309i $$0.655311\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ − 4.00000i − 0.143963i
$$773$$ 6.00000i 0.215805i 0.994161 + 0.107903i $$0.0344134\pi$$
−0.994161 + 0.107903i $$0.965587\pi$$
$$774$$ −7.00000 −0.251610
$$775$$ 0 0
$$776$$ 4.00000 0.143592
$$777$$ 8.00000i 0.286998i
$$778$$ − 36.0000i − 1.29066i
$$779$$ 45.0000 1.61229
$$780$$ 0 0
$$781$$ 0 0
$$782$$ 0 0
$$783$$ − 24.0000i − 0.857690i
$$784$$ −9.00000 −0.321429
$$785$$ 0 0
$$786$$ −24.0000 −0.856052
$$787$$ 20.0000i 0.712923i 0.934310 + 0.356462i $$0.116017\pi$$
−0.934310 + 0.356462i $$0.883983\pi$$
$$788$$ − 27.0000i − 0.961835i
$$789$$ 12.0000 0.427211
$$790$$ 0 0
$$791$$ −72.0000 −2.56003
$$792$$ 0 0
$$793$$ − 4.00000i − 0.142044i
$$794$$ −25.0000 −0.887217
$$795$$ 0 0
$$796$$ −13.0000 −0.460773
$$797$$ − 12.0000i − 0.425062i −0.977154 0.212531i $$-0.931829\pi$$
0.977154 0.212531i $$-0.0681706\pi$$
$$798$$ 40.0000i 1.41598i
$$799$$ 0 0
$$800$$ 0 0
$$801$$ −6.00000 −0.212000
$$802$$ − 18.0000i − 0.635602i
$$803$$ 0 0
$$804$$ 4.00000 0.141069
$$805$$ 0 0
$$806$$ 8.00000 0.281788
$$807$$ 42.0000i 1.47847i
$$808$$ 3.00000i 0.105540i
$$809$$ −12.0000 −0.421898 −0.210949 0.977497i $$-0.567655\pi$$
−0.210949 + 0.977497i $$0.567655\pi$$
$$810$$ 0 0
$$811$$ −28.0000 −0.983213 −0.491606 0.870817i $$-0.663590\pi$$
−0.491606 + 0.870817i $$0.663590\pi$$
$$812$$ − 24.0000i − 0.842235i
$$813$$ − 32.0000i − 1.12229i
$$814$$ 0 0
$$815$$ 0 0
$$816$$ 0 0
$$817$$ − 35.0000i − 1.22449i
$$818$$ 20.0000i 0.699284i
$$819$$ 8.00000 0.279543
$$820$$ 0 0
$$821$$ 33.0000 1.15171 0.575854 0.817553i $$-0.304670\pi$$
0.575854 + 0.817553i $$0.304670\pi$$
$$822$$ 12.0000i 0.418548i
$$823$$ 34.0000i 1.18517i 0.805510 + 0.592583i $$0.201892\pi$$
−0.805510 + 0.592583i $$0.798108\pi$$
$$824$$ −13.0000 −0.452876
$$825$$ 0 0
$$826$$ 12.0000 0.417533
$$827$$ − 12.0000i − 0.417281i −0.977992 0.208640i $$-0.933096\pi$$
0.977992 0.208640i $$-0.0669038\pi$$
$$828$$ − 3.00000i − 0.104257i
$$829$$ −56.0000 −1.94496 −0.972480 0.232986i $$-0.925151\pi$$
−0.972480 + 0.232986i $$0.925151\pi$$
$$830$$ 0 0
$$831$$ 44.0000 1.52634
$$832$$ 2.00000i 0.0693375i
$$833$$ 0 0
$$834$$ −28.0000 −0.969561
$$835$$ 0 0
$$836$$ 0 0
$$837$$ − 16.0000i − 0.553041i
$$838$$ 12.0000i 0.414533i
$$839$$ 6.00000 0.207143 0.103572 0.994622i $$-0.466973\pi$$
0.103572 + 0.994622i $$0.466973\pi$$
$$840$$ 0 0
$$841$$ 7.00000 0.241379
$$842$$ − 26.0000i − 0.896019i
$$843$$ 24.0000i 0.826604i
$$844$$ −2.00000 −0.0688428
$$845$$ 0 0
$$846$$ 6.00000 0.206284
$$847$$ 44.0000i 1.51186i
$$848$$ − 9.00000i − 0.309061i
$$849$$ −38.0000 −1.30416
$$850$$ 0 0
$$851$$ 3.00000 0.102839
$$852$$ 12.0000i 0.411113i
$$853$$ 46.0000i 1.57501i 0.616308 + 0.787505i $$0.288628\pi$$
−0.616308 + 0.787505i $$0.711372\pi$$
$$854$$ −8.00000 −0.273754
$$855$$ 0 0
$$856$$ 18.0000 0.615227
$$857$$ − 42.0000i − 1.43469i −0.696717 0.717346i $$-0.745357\pi$$
0.696717 0.717346i $$-0.254643\pi$$
$$858$$ 0 0
$$859$$ −5.00000 −0.170598 −0.0852989 0.996355i $$-0.527185\pi$$
−0.0852989 + 0.996355i $$0.527185\pi$$
$$860$$ 0 0
$$861$$ −72.0000 −2.45375
$$862$$ − 15.0000i − 0.510902i
$$863$$ 6.00000i 0.204242i 0.994772 + 0.102121i $$0.0325630\pi$$
−0.994772 + 0.102121i $$0.967437\pi$$
$$864$$ 4.00000 0.136083
$$865$$ 0 0
$$866$$ −26.0000 −0.883516
$$867$$ 34.0000i 1.15470i
$$868$$ − 16.0000i − 0.543075i
$$869$$ 0 0
$$870$$ 0 0
$$871$$ 4.00000 0.135535
$$872$$ 16.0000i 0.541828i
$$873$$ 4.00000i 0.135379i
$$874$$ 15.0000 0.507383
$$875$$ 0 0
$$876$$ −22.0000 −0.743311
$$877$$ − 25.0000i − 0.844190i −0.906552 0.422095i $$-0.861295\pi$$
0.906552 0.422095i $$-0.138705\pi$$
$$878$$ − 7.00000i − 0.236239i
$$879$$ −6.00000 −0.202375
$$880$$ 0 0
$$881$$ −39.0000 −1.31394 −0.656972 0.753915i $$-0.728163\pi$$
−0.656972 + 0.753915i $$0.728163\pi$$
$$882$$ − 9.00000i − 0.303046i
$$883$$ − 32.0000i − 1.07689i −0.842662 0.538443i $$-0.819013\pi$$
0.842662 0.538443i $$-0.180987\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ −24.0000 −0.806296
$$887$$ − 30.0000i − 1.00730i −0.863907 0.503651i $$-0.831990\pi$$
0.863907 0.503651i $$-0.168010\pi$$
$$888$$ − 2.00000i − 0.0671156i
$$889$$ −64.0000 −2.14649
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 2.00000i 0.0669650i
$$893$$ 30.0000i 1.00391i
$$894$$ −30.0000 −1.00335
$$895$$ 0 0
$$896$$ 4.00000 0.133631
$$897$$ − 12.0000i − 0.400668i
$$898$$ 6.00000i 0.200223i
$$899$$ 24.0000 0.800445
$$900$$ 0 0
$$901$$ 0 0
$$902$$ 0 0
$$903$$ 56.0000i 1.86356i
$$904$$ 18.0000 0.598671
$$905$$ 0 0
$$906$$ −8.00000 −0.265782
$$907$$ − 37.0000i − 1.22856i −0.789086 0.614282i $$-0.789446\pi$$
0.789086 0.614282i $$-0.210554\pi$$
$$908$$ 9.00000i 0.298675i
$$909$$ −3.00000 −0.0995037
$$910$$ 0 0
$$911$$ −9.00000 −0.298183 −0.149092 0.988823i $$-0.547635\pi$$
−0.149092 + 0.988823i $$0.547635\pi$$
$$912$$ − 10.0000i − 0.331133i
$$913$$ 0 0
$$914$$ −28.0000 −0.926158
$$915$$ 0 0
$$916$$ −13.0000 −0.429532
$$917$$ 48.0000i 1.58510i
$$918$$ 0 0
$$919$$ −32.0000 −1.05558 −0.527791 0.849374i $$-0.676980\pi$$
−0.527791 + 0.849374i $$0.676980\pi$$
$$920$$ 0 0
$$921$$ −64.0000 −2.10887
$$922$$ 0 0
$$923$$ 12.0000i 0.394985i
$$924$$ 0 0
$$925$$ 0 0
$$926$$ −32.0000 −1.05159
$$927$$ − 13.0000i − 0.426976i
$$928$$ 6.00000i 0.196960i
$$929$$ −18.0000 −0.590561 −0.295280 0.955411i $$-0.595413\pi$$
−0.295280 + 0.955411i $$0.595413\pi$$
$$930$$ 0 0
$$931$$ 45.0000 1.47482
$$932$$ − 21.0000i − 0.687878i
$$933$$ 30.0000i 0.982156i
$$934$$ 12.0000 0.392652
$$935$$ 0 0
$$936$$ −2.00000 −0.0653720
$$937$$ 38.0000i 1.24141i 0.784046 + 0.620703i $$0.213153\pi$$
−0.784046 + 0.620703i $$0.786847\pi$$
$$938$$ − 8.00000i − 0.261209i
$$939$$ −44.0000 −1.43589
$$940$$ 0 0
$$941$$ 30.0000 0.977972 0.488986 0.872292i $$-0.337367\pi$$
0.488986 + 0.872292i $$0.337367\pi$$
$$942$$ − 26.0000i − 0.847126i
$$943$$ 27.0000i 0.879241i
$$944$$ −3.00000 −0.0976417
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 33.0000i 1.07236i 0.844105 + 0.536178i $$0.180132\pi$$
−0.844105 + 0.536178i $$0.819868\pi$$
$$948$$ − 2.00000i − 0.0649570i
$$949$$ −22.0000 −0.714150
$$950$$ 0 0
$$951$$ −66.0000 −2.14020
$$952$$ 0 0
$$953$$ − 9.00000i − 0.291539i −0.989319 0.145769i $$-0.953434\pi$$
0.989319 0.145769i $$-0.0465657\pi$$
$$954$$ 9.00000 0.291386
$$955$$ 0 0
$$956$$ 9.00000 0.291081
$$957$$ 0 0
$$958$$ 24.0000i 0.775405i
$$959$$ 24.0000 0.775000
$$960$$ 0 0
$$961$$ −15.0000 −0.483871
$$962$$ − 2.00000i − 0.0644826i
$$963$$ 18.0000i 0.580042i
$$964$$ −14.0000 −0.450910
$$965$$ 0 0
$$966$$ −24.0000 −0.772187
$$967$$ − 31.0000i − 0.996893i −0.866921 0.498446i $$-0.833904\pi$$
0.866921 0.498446i $$-0.166096\pi$$
$$968$$ − 11.0000i − 0.353553i
$$969$$ 0 0
$$970$$ 0 0
$$971$$ −48.0000 −1.54039 −0.770197 0.637806i $$-0.779842\pi$$
−0.770197 + 0.637806i $$0.779842\pi$$
$$972$$ 10.0000i 0.320750i
$$973$$ 56.0000i 1.79528i
$$974$$ −16.0000 −0.512673
$$975$$ 0 0
$$976$$ 2.00000 0.0640184
$$977$$ − 18.0000i − 0.575871i −0.957650 0.287936i $$-0.907031\pi$$
0.957650 0.287936i $$-0.0929689\pi$$
$$978$$ 14.0000i 0.447671i
$$979$$ 0 0
$$980$$ 0 0
$$981$$ −16.0000 −0.510841
$$982$$ 18.0000i 0.574403i
$$983$$ 54.0000i 1.72233i 0.508323 + 0.861166i $$0.330265\pi$$
−0.508323 + 0.861166i $$0.669735\pi$$
$$984$$ 18.0000 0.573819
$$985$$ 0 0
$$986$$ 0 0
$$987$$ − 48.0000i − 1.52786i
$$988$$ − 10.0000i − 0.318142i
$$989$$ 21.0000 0.667761
$$990$$ 0 0
$$991$$ −13.0000 −0.412959 −0.206479 0.978451i $$-0.566201\pi$$
−0.206479 + 0.978451i $$0.566201\pi$$
$$992$$ 4.00000i 0.127000i
$$993$$ 16.0000i 0.507745i
$$994$$ 24.0000 0.761234
$$995$$ 0 0
$$996$$ 0 0
$$997$$ − 28.0000i − 0.886769i −0.896332 0.443384i $$-0.853778\pi$$
0.896332 0.443384i $$-0.146222\pi$$
$$998$$ − 43.0000i − 1.36114i
$$999$$ −4.00000 −0.126554
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.f.149.1 2
5.2 odd 4 1850.2.a.p.1.1 yes 1
5.3 odd 4 1850.2.a.a.1.1 1
5.4 even 2 inner 1850.2.b.f.149.2 2

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