# Properties

 Label 1850.2.b.l.149.3 Level $1850$ Weight $2$ Character 1850.149 Analytic conductor $14.772$ Analytic rank $0$ Dimension $4$ 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: $$4$$ Coefficient field: $$\Q(\zeta_{12})$$ Defining polynomial: $$x^{4} - x^{2} + 1$$ x^4 - x^2 + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 370) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 149.3 Root $$-0.866025 - 0.500000i$$ of defining polynomial Character $$\chi$$ $$=$$ 1850.149 Dual form 1850.2.b.l.149.2

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+1.00000i q^{2} -2.73205i q^{3} -1.00000 q^{4} +2.73205 q^{6} +1.26795i q^{7} -1.00000i q^{8} -4.46410 q^{9} +O(q^{10})$$ $$q+1.00000i q^{2} -2.73205i q^{3} -1.00000 q^{4} +2.73205 q^{6} +1.26795i q^{7} -1.00000i q^{8} -4.46410 q^{9} +1.46410 q^{11} +2.73205i q^{12} +1.46410i q^{13} -1.26795 q^{14} +1.00000 q^{16} +1.46410i q^{17} -4.46410i q^{18} +4.19615 q^{19} +3.46410 q^{21} +1.46410i q^{22} -8.00000i q^{23} -2.73205 q^{24} -1.46410 q^{26} +4.00000i q^{27} -1.26795i q^{28} +8.92820 q^{29} -2.73205 q^{31} +1.00000i q^{32} -4.00000i q^{33} -1.46410 q^{34} +4.46410 q^{36} -1.00000i q^{37} +4.19615i q^{38} +4.00000 q^{39} -2.00000 q^{41} +3.46410i q^{42} -6.92820i q^{43} -1.46410 q^{44} +8.00000 q^{46} +1.26795i q^{47} -2.73205i q^{48} +5.39230 q^{49} +4.00000 q^{51} -1.46410i q^{52} -6.00000i q^{53} -4.00000 q^{54} +1.26795 q^{56} -11.4641i q^{57} +8.92820i q^{58} -0.196152 q^{59} +8.92820 q^{61} -2.73205i q^{62} -5.66025i q^{63} -1.00000 q^{64} +4.00000 q^{66} -13.6603i q^{67} -1.46410i q^{68} -21.8564 q^{69} -10.9282 q^{71} +4.46410i q^{72} +12.9282i q^{73} +1.00000 q^{74} -4.19615 q^{76} +1.85641i q^{77} +4.00000i q^{78} -5.26795 q^{79} -2.46410 q^{81} -2.00000i q^{82} -5.26795i q^{83} -3.46410 q^{84} +6.92820 q^{86} -24.3923i q^{87} -1.46410i q^{88} +2.00000 q^{89} -1.85641 q^{91} +8.00000i q^{92} +7.46410i q^{93} -1.26795 q^{94} +2.73205 q^{96} +2.00000i q^{97} +5.39230i q^{98} -6.53590 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 4 q^{4} + 4 q^{6} - 4 q^{9}+O(q^{10})$$ 4 * q - 4 * q^4 + 4 * q^6 - 4 * q^9 $$4 q - 4 q^{4} + 4 q^{6} - 4 q^{9} - 8 q^{11} - 12 q^{14} + 4 q^{16} - 4 q^{19} - 4 q^{24} + 8 q^{26} + 8 q^{29} - 4 q^{31} + 8 q^{34} + 4 q^{36} + 16 q^{39} - 8 q^{41} + 8 q^{44} + 32 q^{46} - 20 q^{49} + 16 q^{51} - 16 q^{54} + 12 q^{56} + 20 q^{59} + 8 q^{61} - 4 q^{64} + 16 q^{66} - 32 q^{69} - 16 q^{71} + 4 q^{74} + 4 q^{76} - 28 q^{79} + 4 q^{81} + 8 q^{89} + 48 q^{91} - 12 q^{94} + 4 q^{96} - 40 q^{99}+O(q^{100})$$ 4 * q - 4 * q^4 + 4 * q^6 - 4 * q^9 - 8 * q^11 - 12 * q^14 + 4 * q^16 - 4 * q^19 - 4 * q^24 + 8 * q^26 + 8 * q^29 - 4 * q^31 + 8 * q^34 + 4 * q^36 + 16 * q^39 - 8 * q^41 + 8 * q^44 + 32 * q^46 - 20 * q^49 + 16 * q^51 - 16 * q^54 + 12 * q^56 + 20 * q^59 + 8 * q^61 - 4 * q^64 + 16 * q^66 - 32 * q^69 - 16 * q^71 + 4 * q^74 + 4 * q^76 - 28 * q^79 + 4 * q^81 + 8 * q^89 + 48 * q^91 - 12 * q^94 + 4 * q^96 - 40 * 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
$$3$$ − 2.73205i − 1.57735i −0.614810 0.788675i $$-0.710767\pi$$
0.614810 0.788675i $$-0.289233\pi$$
$$4$$ −1.00000 −0.500000
$$5$$ 0 0
$$6$$ 2.73205 1.11536
$$7$$ 1.26795i 0.479240i 0.970867 + 0.239620i $$0.0770228\pi$$
−0.970867 + 0.239620i $$0.922977\pi$$
$$8$$ − 1.00000i − 0.353553i
$$9$$ −4.46410 −1.48803
$$10$$ 0 0
$$11$$ 1.46410 0.441443 0.220722 0.975337i $$-0.429159\pi$$
0.220722 + 0.975337i $$0.429159\pi$$
$$12$$ 2.73205i 0.788675i
$$13$$ 1.46410i 0.406069i 0.979172 + 0.203034i $$0.0650803\pi$$
−0.979172 + 0.203034i $$0.934920\pi$$
$$14$$ −1.26795 −0.338874
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ 1.46410i 0.355097i 0.984112 + 0.177548i $$0.0568166\pi$$
−0.984112 + 0.177548i $$0.943183\pi$$
$$18$$ − 4.46410i − 1.05220i
$$19$$ 4.19615 0.962663 0.481332 0.876539i $$-0.340153\pi$$
0.481332 + 0.876539i $$0.340153\pi$$
$$20$$ 0 0
$$21$$ 3.46410 0.755929
$$22$$ 1.46410i 0.312148i
$$23$$ − 8.00000i − 1.66812i −0.551677 0.834058i $$-0.686012\pi$$
0.551677 0.834058i $$-0.313988\pi$$
$$24$$ −2.73205 −0.557678
$$25$$ 0 0
$$26$$ −1.46410 −0.287134
$$27$$ 4.00000i 0.769800i
$$28$$ − 1.26795i − 0.239620i
$$29$$ 8.92820 1.65793 0.828963 0.559304i $$-0.188931\pi$$
0.828963 + 0.559304i $$0.188931\pi$$
$$30$$ 0 0
$$31$$ −2.73205 −0.490691 −0.245345 0.969436i $$-0.578901\pi$$
−0.245345 + 0.969436i $$0.578901\pi$$
$$32$$ 1.00000i 0.176777i
$$33$$ − 4.00000i − 0.696311i
$$34$$ −1.46410 −0.251091
$$35$$ 0 0
$$36$$ 4.46410 0.744017
$$37$$ − 1.00000i − 0.164399i
$$38$$ 4.19615i 0.680706i
$$39$$ 4.00000 0.640513
$$40$$ 0 0
$$41$$ −2.00000 −0.312348 −0.156174 0.987730i $$-0.549916\pi$$
−0.156174 + 0.987730i $$0.549916\pi$$
$$42$$ 3.46410i 0.534522i
$$43$$ − 6.92820i − 1.05654i −0.849076 0.528271i $$-0.822841\pi$$
0.849076 0.528271i $$-0.177159\pi$$
$$44$$ −1.46410 −0.220722
$$45$$ 0 0
$$46$$ 8.00000 1.17954
$$47$$ 1.26795i 0.184949i 0.995715 + 0.0924747i $$0.0294777\pi$$
−0.995715 + 0.0924747i $$0.970522\pi$$
$$48$$ − 2.73205i − 0.394338i
$$49$$ 5.39230 0.770329
$$50$$ 0 0
$$51$$ 4.00000 0.560112
$$52$$ − 1.46410i − 0.203034i
$$53$$ − 6.00000i − 0.824163i −0.911147 0.412082i $$-0.864802\pi$$
0.911147 0.412082i $$-0.135198\pi$$
$$54$$ −4.00000 −0.544331
$$55$$ 0 0
$$56$$ 1.26795 0.169437
$$57$$ − 11.4641i − 1.51846i
$$58$$ 8.92820i 1.17233i
$$59$$ −0.196152 −0.0255369 −0.0127684 0.999918i $$-0.504064\pi$$
−0.0127684 + 0.999918i $$0.504064\pi$$
$$60$$ 0 0
$$61$$ 8.92820 1.14314 0.571570 0.820554i $$-0.306335\pi$$
0.571570 + 0.820554i $$0.306335\pi$$
$$62$$ − 2.73205i − 0.346971i
$$63$$ − 5.66025i − 0.713125i
$$64$$ −1.00000 −0.125000
$$65$$ 0 0
$$66$$ 4.00000 0.492366
$$67$$ − 13.6603i − 1.66887i −0.551110 0.834433i $$-0.685795\pi$$
0.551110 0.834433i $$-0.314205\pi$$
$$68$$ − 1.46410i − 0.177548i
$$69$$ −21.8564 −2.63120
$$70$$ 0 0
$$71$$ −10.9282 −1.29694 −0.648470 0.761241i $$-0.724591\pi$$
−0.648470 + 0.761241i $$0.724591\pi$$
$$72$$ 4.46410i 0.526099i
$$73$$ 12.9282i 1.51313i 0.653917 + 0.756566i $$0.273124\pi$$
−0.653917 + 0.756566i $$0.726876\pi$$
$$74$$ 1.00000 0.116248
$$75$$ 0 0
$$76$$ −4.19615 −0.481332
$$77$$ 1.85641i 0.211557i
$$78$$ 4.00000i 0.452911i
$$79$$ −5.26795 −0.592691 −0.296345 0.955081i $$-0.595768\pi$$
−0.296345 + 0.955081i $$0.595768\pi$$
$$80$$ 0 0
$$81$$ −2.46410 −0.273789
$$82$$ − 2.00000i − 0.220863i
$$83$$ − 5.26795i − 0.578233i −0.957294 0.289116i $$-0.906639\pi$$
0.957294 0.289116i $$-0.0933614\pi$$
$$84$$ −3.46410 −0.377964
$$85$$ 0 0
$$86$$ 6.92820 0.747087
$$87$$ − 24.3923i − 2.61513i
$$88$$ − 1.46410i − 0.156074i
$$89$$ 2.00000 0.212000 0.106000 0.994366i $$-0.466196\pi$$
0.106000 + 0.994366i $$0.466196\pi$$
$$90$$ 0 0
$$91$$ −1.85641 −0.194604
$$92$$ 8.00000i 0.834058i
$$93$$ 7.46410i 0.773991i
$$94$$ −1.26795 −0.130779
$$95$$ 0 0
$$96$$ 2.73205 0.278839
$$97$$ 2.00000i 0.203069i 0.994832 + 0.101535i $$0.0323753\pi$$
−0.994832 + 0.101535i $$0.967625\pi$$
$$98$$ 5.39230i 0.544705i
$$99$$ −6.53590 −0.656883
$$100$$ 0 0
$$101$$ −2.53590 −0.252331 −0.126166 0.992009i $$-0.540267\pi$$
−0.126166 + 0.992009i $$0.540267\pi$$
$$102$$ 4.00000i 0.396059i
$$103$$ − 13.4641i − 1.32666i −0.748328 0.663329i $$-0.769143\pi$$
0.748328 0.663329i $$-0.230857\pi$$
$$104$$ 1.46410 0.143567
$$105$$ 0 0
$$106$$ 6.00000 0.582772
$$107$$ − 6.73205i − 0.650812i −0.945574 0.325406i $$-0.894499\pi$$
0.945574 0.325406i $$-0.105501\pi$$
$$108$$ − 4.00000i − 0.384900i
$$109$$ −2.00000 −0.191565 −0.0957826 0.995402i $$-0.530535\pi$$
−0.0957826 + 0.995402i $$0.530535\pi$$
$$110$$ 0 0
$$111$$ −2.73205 −0.259315
$$112$$ 1.26795i 0.119810i
$$113$$ − 17.4641i − 1.64288i −0.570292 0.821442i $$-0.693170\pi$$
0.570292 0.821442i $$-0.306830\pi$$
$$114$$ 11.4641 1.07371
$$115$$ 0 0
$$116$$ −8.92820 −0.828963
$$117$$ − 6.53590i − 0.604244i
$$118$$ − 0.196152i − 0.0180573i
$$119$$ −1.85641 −0.170177
$$120$$ 0 0
$$121$$ −8.85641 −0.805128
$$122$$ 8.92820i 0.808322i
$$123$$ 5.46410i 0.492681i
$$124$$ 2.73205 0.245345
$$125$$ 0 0
$$126$$ 5.66025 0.504256
$$127$$ 13.6603i 1.21215i 0.795407 + 0.606076i $$0.207257\pi$$
−0.795407 + 0.606076i $$0.792743\pi$$
$$128$$ − 1.00000i − 0.0883883i
$$129$$ −18.9282 −1.66654
$$130$$ 0 0
$$131$$ 12.5885 1.09986 0.549929 0.835211i $$-0.314655\pi$$
0.549929 + 0.835211i $$0.314655\pi$$
$$132$$ 4.00000i 0.348155i
$$133$$ 5.32051i 0.461347i
$$134$$ 13.6603 1.18007
$$135$$ 0 0
$$136$$ 1.46410 0.125546
$$137$$ − 2.00000i − 0.170872i −0.996344 0.0854358i $$-0.972772\pi$$
0.996344 0.0854358i $$-0.0272282\pi$$
$$138$$ − 21.8564i − 1.86054i
$$139$$ 6.92820 0.587643 0.293821 0.955860i $$-0.405073\pi$$
0.293821 + 0.955860i $$0.405073\pi$$
$$140$$ 0 0
$$141$$ 3.46410 0.291730
$$142$$ − 10.9282i − 0.917074i
$$143$$ 2.14359i 0.179256i
$$144$$ −4.46410 −0.372008
$$145$$ 0 0
$$146$$ −12.9282 −1.06995
$$147$$ − 14.7321i − 1.21508i
$$148$$ 1.00000i 0.0821995i
$$149$$ 16.3923 1.34291 0.671455 0.741045i $$-0.265670\pi$$
0.671455 + 0.741045i $$0.265670\pi$$
$$150$$ 0 0
$$151$$ 8.39230 0.682956 0.341478 0.939890i $$-0.389073\pi$$
0.341478 + 0.939890i $$0.389073\pi$$
$$152$$ − 4.19615i − 0.340353i
$$153$$ − 6.53590i − 0.528396i
$$154$$ −1.85641 −0.149593
$$155$$ 0 0
$$156$$ −4.00000 −0.320256
$$157$$ − 16.9282i − 1.35102i −0.737352 0.675509i $$-0.763924\pi$$
0.737352 0.675509i $$-0.236076\pi$$
$$158$$ − 5.26795i − 0.419096i
$$159$$ −16.3923 −1.29999
$$160$$ 0 0
$$161$$ 10.1436 0.799427
$$162$$ − 2.46410i − 0.193598i
$$163$$ 23.3205i 1.82660i 0.407284 + 0.913302i $$0.366476\pi$$
−0.407284 + 0.913302i $$0.633524\pi$$
$$164$$ 2.00000 0.156174
$$165$$ 0 0
$$166$$ 5.26795 0.408872
$$167$$ − 5.46410i − 0.422825i −0.977397 0.211412i $$-0.932194\pi$$
0.977397 0.211412i $$-0.0678063\pi$$
$$168$$ − 3.46410i − 0.267261i
$$169$$ 10.8564 0.835108
$$170$$ 0 0
$$171$$ −18.7321 −1.43248
$$172$$ 6.92820i 0.528271i
$$173$$ 10.0000i 0.760286i 0.924928 + 0.380143i $$0.124125\pi$$
−0.924928 + 0.380143i $$0.875875\pi$$
$$174$$ 24.3923 1.84918
$$175$$ 0 0
$$176$$ 1.46410 0.110361
$$177$$ 0.535898i 0.0402806i
$$178$$ 2.00000i 0.149906i
$$179$$ 17.6603 1.31999 0.659995 0.751270i $$-0.270559\pi$$
0.659995 + 0.751270i $$0.270559\pi$$
$$180$$ 0 0
$$181$$ −1.46410 −0.108826 −0.0544129 0.998519i $$-0.517329\pi$$
−0.0544129 + 0.998519i $$0.517329\pi$$
$$182$$ − 1.85641i − 0.137606i
$$183$$ − 24.3923i − 1.80313i
$$184$$ −8.00000 −0.589768
$$185$$ 0 0
$$186$$ −7.46410 −0.547294
$$187$$ 2.14359i 0.156755i
$$188$$ − 1.26795i − 0.0924747i
$$189$$ −5.07180 −0.368919
$$190$$ 0 0
$$191$$ −5.26795 −0.381175 −0.190588 0.981670i $$-0.561039\pi$$
−0.190588 + 0.981670i $$0.561039\pi$$
$$192$$ 2.73205i 0.197169i
$$193$$ − 11.8564i − 0.853443i −0.904383 0.426721i $$-0.859668\pi$$
0.904383 0.426721i $$-0.140332\pi$$
$$194$$ −2.00000 −0.143592
$$195$$ 0 0
$$196$$ −5.39230 −0.385165
$$197$$ − 18.7846i − 1.33835i −0.743106 0.669174i $$-0.766648\pi$$
0.743106 0.669174i $$-0.233352\pi$$
$$198$$ − 6.53590i − 0.464486i
$$199$$ −26.0526 −1.84682 −0.923408 0.383819i $$-0.874609\pi$$
−0.923408 + 0.383819i $$0.874609\pi$$
$$200$$ 0 0
$$201$$ −37.3205 −2.63239
$$202$$ − 2.53590i − 0.178425i
$$203$$ 11.3205i 0.794544i
$$204$$ −4.00000 −0.280056
$$205$$ 0 0
$$206$$ 13.4641 0.938088
$$207$$ 35.7128i 2.48221i
$$208$$ 1.46410i 0.101517i
$$209$$ 6.14359 0.424961
$$210$$ 0 0
$$211$$ 9.85641 0.678543 0.339272 0.940688i $$-0.389819\pi$$
0.339272 + 0.940688i $$0.389819\pi$$
$$212$$ 6.00000i 0.412082i
$$213$$ 29.8564i 2.04573i
$$214$$ 6.73205 0.460194
$$215$$ 0 0
$$216$$ 4.00000 0.272166
$$217$$ − 3.46410i − 0.235159i
$$218$$ − 2.00000i − 0.135457i
$$219$$ 35.3205 2.38674
$$220$$ 0 0
$$221$$ −2.14359 −0.144194
$$222$$ − 2.73205i − 0.183363i
$$223$$ − 22.0526i − 1.47675i −0.674391 0.738374i $$-0.735594\pi$$
0.674391 0.738374i $$-0.264406\pi$$
$$224$$ −1.26795 −0.0847184
$$225$$ 0 0
$$226$$ 17.4641 1.16169
$$227$$ − 3.60770i − 0.239451i −0.992807 0.119726i $$-0.961799\pi$$
0.992807 0.119726i $$-0.0382015\pi$$
$$228$$ 11.4641i 0.759229i
$$229$$ −15.8564 −1.04782 −0.523910 0.851773i $$-0.675527\pi$$
−0.523910 + 0.851773i $$0.675527\pi$$
$$230$$ 0 0
$$231$$ 5.07180 0.333700
$$232$$ − 8.92820i − 0.586165i
$$233$$ 15.0718i 0.987386i 0.869636 + 0.493693i $$0.164353\pi$$
−0.869636 + 0.493693i $$0.835647\pi$$
$$234$$ 6.53590 0.427265
$$235$$ 0 0
$$236$$ 0.196152 0.0127684
$$237$$ 14.3923i 0.934881i
$$238$$ − 1.85641i − 0.120333i
$$239$$ 17.2679 1.11697 0.558485 0.829514i $$-0.311383\pi$$
0.558485 + 0.829514i $$0.311383\pi$$
$$240$$ 0 0
$$241$$ −8.92820 −0.575116 −0.287558 0.957763i $$-0.592843\pi$$
−0.287558 + 0.957763i $$0.592843\pi$$
$$242$$ − 8.85641i − 0.569311i
$$243$$ 18.7321i 1.20166i
$$244$$ −8.92820 −0.571570
$$245$$ 0 0
$$246$$ −5.46410 −0.348378
$$247$$ 6.14359i 0.390907i
$$248$$ 2.73205i 0.173485i
$$249$$ −14.3923 −0.912075
$$250$$ 0 0
$$251$$ 22.7321 1.43483 0.717417 0.696644i $$-0.245324\pi$$
0.717417 + 0.696644i $$0.245324\pi$$
$$252$$ 5.66025i 0.356562i
$$253$$ − 11.7128i − 0.736378i
$$254$$ −13.6603 −0.857121
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ 25.4641i 1.58841i 0.607652 + 0.794204i $$0.292112\pi$$
−0.607652 + 0.794204i $$0.707888\pi$$
$$258$$ − 18.9282i − 1.17842i
$$259$$ 1.26795 0.0787865
$$260$$ 0 0
$$261$$ −39.8564 −2.46705
$$262$$ 12.5885i 0.777717i
$$263$$ 30.0526i 1.85312i 0.376147 + 0.926560i $$0.377249\pi$$
−0.376147 + 0.926560i $$0.622751\pi$$
$$264$$ −4.00000 −0.246183
$$265$$ 0 0
$$266$$ −5.32051 −0.326221
$$267$$ − 5.46410i − 0.334398i
$$268$$ 13.6603i 0.834433i
$$269$$ 0.392305 0.0239192 0.0119596 0.999928i $$-0.496193\pi$$
0.0119596 + 0.999928i $$0.496193\pi$$
$$270$$ 0 0
$$271$$ 16.7846 1.01959 0.509796 0.860295i $$-0.329721\pi$$
0.509796 + 0.860295i $$0.329721\pi$$
$$272$$ 1.46410i 0.0887742i
$$273$$ 5.07180i 0.306959i
$$274$$ 2.00000 0.120824
$$275$$ 0 0
$$276$$ 21.8564 1.31560
$$277$$ 26.2487i 1.57713i 0.614950 + 0.788566i $$0.289176\pi$$
−0.614950 + 0.788566i $$0.710824\pi$$
$$278$$ 6.92820i 0.415526i
$$279$$ 12.1962 0.730165
$$280$$ 0 0
$$281$$ 4.92820 0.293992 0.146996 0.989137i $$-0.453040\pi$$
0.146996 + 0.989137i $$0.453040\pi$$
$$282$$ 3.46410i 0.206284i
$$283$$ − 4.39230i − 0.261095i −0.991442 0.130548i $$-0.958326\pi$$
0.991442 0.130548i $$-0.0416736\pi$$
$$284$$ 10.9282 0.648470
$$285$$ 0 0
$$286$$ −2.14359 −0.126753
$$287$$ − 2.53590i − 0.149689i
$$288$$ − 4.46410i − 0.263050i
$$289$$ 14.8564 0.873906
$$290$$ 0 0
$$291$$ 5.46410 0.320311
$$292$$ − 12.9282i − 0.756566i
$$293$$ 6.00000i 0.350524i 0.984522 + 0.175262i $$0.0560772\pi$$
−0.984522 + 0.175262i $$0.943923\pi$$
$$294$$ 14.7321 0.859191
$$295$$ 0 0
$$296$$ −1.00000 −0.0581238
$$297$$ 5.85641i 0.339823i
$$298$$ 16.3923i 0.949581i
$$299$$ 11.7128 0.677369
$$300$$ 0 0
$$301$$ 8.78461 0.506336
$$302$$ 8.39230i 0.482923i
$$303$$ 6.92820i 0.398015i
$$304$$ 4.19615 0.240666
$$305$$ 0 0
$$306$$ 6.53590 0.373632
$$307$$ 12.5885i 0.718461i 0.933249 + 0.359231i $$0.116961\pi$$
−0.933249 + 0.359231i $$0.883039\pi$$
$$308$$ − 1.85641i − 0.105779i
$$309$$ −36.7846 −2.09260
$$310$$ 0 0
$$311$$ 27.1244 1.53808 0.769041 0.639200i $$-0.220734\pi$$
0.769041 + 0.639200i $$0.220734\pi$$
$$312$$ − 4.00000i − 0.226455i
$$313$$ 3.85641i 0.217977i 0.994043 + 0.108988i $$0.0347612\pi$$
−0.994043 + 0.108988i $$0.965239\pi$$
$$314$$ 16.9282 0.955314
$$315$$ 0 0
$$316$$ 5.26795 0.296345
$$317$$ 31.8564i 1.78923i 0.446834 + 0.894617i $$0.352552\pi$$
−0.446834 + 0.894617i $$0.647448\pi$$
$$318$$ − 16.3923i − 0.919235i
$$319$$ 13.0718 0.731880
$$320$$ 0 0
$$321$$ −18.3923 −1.02656
$$322$$ 10.1436i 0.565280i
$$323$$ 6.14359i 0.341839i
$$324$$ 2.46410 0.136895
$$325$$ 0 0
$$326$$ −23.3205 −1.29160
$$327$$ 5.46410i 0.302166i
$$328$$ 2.00000i 0.110432i
$$329$$ −1.60770 −0.0886351
$$330$$ 0 0
$$331$$ −8.87564 −0.487850 −0.243925 0.969794i $$-0.578435\pi$$
−0.243925 + 0.969794i $$0.578435\pi$$
$$332$$ 5.26795i 0.289116i
$$333$$ 4.46410i 0.244631i
$$334$$ 5.46410 0.298982
$$335$$ 0 0
$$336$$ 3.46410 0.188982
$$337$$ − 26.0000i − 1.41631i −0.706057 0.708155i $$-0.749528\pi$$
0.706057 0.708155i $$-0.250472\pi$$
$$338$$ 10.8564i 0.590511i
$$339$$ −47.7128 −2.59140
$$340$$ 0 0
$$341$$ −4.00000 −0.216612
$$342$$ − 18.7321i − 1.01291i
$$343$$ 15.7128i 0.848412i
$$344$$ −6.92820 −0.373544
$$345$$ 0 0
$$346$$ −10.0000 −0.537603
$$347$$ − 17.0718i − 0.916462i −0.888833 0.458231i $$-0.848483\pi$$
0.888833 0.458231i $$-0.151517\pi$$
$$348$$ 24.3923i 1.30756i
$$349$$ −19.3205 −1.03420 −0.517102 0.855924i $$-0.672989\pi$$
−0.517102 + 0.855924i $$0.672989\pi$$
$$350$$ 0 0
$$351$$ −5.85641 −0.312592
$$352$$ 1.46410i 0.0780369i
$$353$$ 15.8564i 0.843951i 0.906607 + 0.421976i $$0.138663\pi$$
−0.906607 + 0.421976i $$0.861337\pi$$
$$354$$ −0.535898 −0.0284827
$$355$$ 0 0
$$356$$ −2.00000 −0.106000
$$357$$ 5.07180i 0.268428i
$$358$$ 17.6603i 0.933373i
$$359$$ −8.39230 −0.442929 −0.221464 0.975168i $$-0.571084\pi$$
−0.221464 + 0.975168i $$0.571084\pi$$
$$360$$ 0 0
$$361$$ −1.39230 −0.0732792
$$362$$ − 1.46410i − 0.0769515i
$$363$$ 24.1962i 1.26997i
$$364$$ 1.85641 0.0973021
$$365$$ 0 0
$$366$$ 24.3923 1.27501
$$367$$ 21.6603i 1.13066i 0.824866 + 0.565328i $$0.191250\pi$$
−0.824866 + 0.565328i $$0.808750\pi$$
$$368$$ − 8.00000i − 0.417029i
$$369$$ 8.92820 0.464784
$$370$$ 0 0
$$371$$ 7.60770 0.394972
$$372$$ − 7.46410i − 0.386996i
$$373$$ 30.7846i 1.59397i 0.604001 + 0.796983i $$0.293572\pi$$
−0.604001 + 0.796983i $$0.706428\pi$$
$$374$$ −2.14359 −0.110843
$$375$$ 0 0
$$376$$ 1.26795 0.0653895
$$377$$ 13.0718i 0.673232i
$$378$$ − 5.07180i − 0.260865i
$$379$$ −11.6077 −0.596247 −0.298124 0.954527i $$-0.596361\pi$$
−0.298124 + 0.954527i $$0.596361\pi$$
$$380$$ 0 0
$$381$$ 37.3205 1.91199
$$382$$ − 5.26795i − 0.269532i
$$383$$ − 8.00000i − 0.408781i −0.978889 0.204390i $$-0.934479\pi$$
0.978889 0.204390i $$-0.0655212\pi$$
$$384$$ −2.73205 −0.139419
$$385$$ 0 0
$$386$$ 11.8564 0.603475
$$387$$ 30.9282i 1.57217i
$$388$$ − 2.00000i − 0.101535i
$$389$$ −15.8564 −0.803952 −0.401976 0.915650i $$-0.631676\pi$$
−0.401976 + 0.915650i $$0.631676\pi$$
$$390$$ 0 0
$$391$$ 11.7128 0.592342
$$392$$ − 5.39230i − 0.272353i
$$393$$ − 34.3923i − 1.73486i
$$394$$ 18.7846 0.946355
$$395$$ 0 0
$$396$$ 6.53590 0.328441
$$397$$ 22.0000i 1.10415i 0.833795 + 0.552074i $$0.186163\pi$$
−0.833795 + 0.552074i $$0.813837\pi$$
$$398$$ − 26.0526i − 1.30590i
$$399$$ 14.5359 0.727705
$$400$$ 0 0
$$401$$ −19.0718 −0.952400 −0.476200 0.879337i $$-0.657986\pi$$
−0.476200 + 0.879337i $$0.657986\pi$$
$$402$$ − 37.3205i − 1.86138i
$$403$$ − 4.00000i − 0.199254i
$$404$$ 2.53590 0.126166
$$405$$ 0 0
$$406$$ −11.3205 −0.561827
$$407$$ − 1.46410i − 0.0725728i
$$408$$ − 4.00000i − 0.198030i
$$409$$ −16.9282 −0.837046 −0.418523 0.908206i $$-0.637452\pi$$
−0.418523 + 0.908206i $$0.637452\pi$$
$$410$$ 0 0
$$411$$ −5.46410 −0.269524
$$412$$ 13.4641i 0.663329i
$$413$$ − 0.248711i − 0.0122383i
$$414$$ −35.7128 −1.75519
$$415$$ 0 0
$$416$$ −1.46410 −0.0717835
$$417$$ − 18.9282i − 0.926918i
$$418$$ 6.14359i 0.300493i
$$419$$ 34.2487 1.67316 0.836580 0.547846i $$-0.184552\pi$$
0.836580 + 0.547846i $$0.184552\pi$$
$$420$$ 0 0
$$421$$ −27.8564 −1.35764 −0.678819 0.734306i $$-0.737508\pi$$
−0.678819 + 0.734306i $$0.737508\pi$$
$$422$$ 9.85641i 0.479802i
$$423$$ − 5.66025i − 0.275211i
$$424$$ −6.00000 −0.291386
$$425$$ 0 0
$$426$$ −29.8564 −1.44655
$$427$$ 11.3205i 0.547838i
$$428$$ 6.73205i 0.325406i
$$429$$ 5.85641 0.282750
$$430$$ 0 0
$$431$$ 8.19615 0.394795 0.197397 0.980324i $$-0.436751\pi$$
0.197397 + 0.980324i $$0.436751\pi$$
$$432$$ 4.00000i 0.192450i
$$433$$ − 34.0000i − 1.63394i −0.576683 0.816968i $$-0.695653\pi$$
0.576683 0.816968i $$-0.304347\pi$$
$$434$$ 3.46410 0.166282
$$435$$ 0 0
$$436$$ 2.00000 0.0957826
$$437$$ − 33.5692i − 1.60583i
$$438$$ 35.3205i 1.68768i
$$439$$ 31.5167 1.50421 0.752104 0.659044i $$-0.229039\pi$$
0.752104 + 0.659044i $$0.229039\pi$$
$$440$$ 0 0
$$441$$ −24.0718 −1.14628
$$442$$ − 2.14359i − 0.101960i
$$443$$ − 39.1244i − 1.85885i −0.369006 0.929427i $$-0.620302\pi$$
0.369006 0.929427i $$-0.379698\pi$$
$$444$$ 2.73205 0.129657
$$445$$ 0 0
$$446$$ 22.0526 1.04422
$$447$$ − 44.7846i − 2.11824i
$$448$$ − 1.26795i − 0.0599050i
$$449$$ −33.7128 −1.59101 −0.795503 0.605950i $$-0.792793\pi$$
−0.795503 + 0.605950i $$0.792793\pi$$
$$450$$ 0 0
$$451$$ −2.92820 −0.137884
$$452$$ 17.4641i 0.821442i
$$453$$ − 22.9282i − 1.07726i
$$454$$ 3.60770 0.169318
$$455$$ 0 0
$$456$$ −11.4641 −0.536856
$$457$$ 4.14359i 0.193829i 0.995293 + 0.0969146i $$0.0308974\pi$$
−0.995293 + 0.0969146i $$0.969103\pi$$
$$458$$ − 15.8564i − 0.740921i
$$459$$ −5.85641 −0.273354
$$460$$ 0 0
$$461$$ −26.7846 −1.24748 −0.623742 0.781630i $$-0.714388\pi$$
−0.623742 + 0.781630i $$0.714388\pi$$
$$462$$ 5.07180i 0.235961i
$$463$$ 5.07180i 0.235706i 0.993031 + 0.117853i $$0.0376012\pi$$
−0.993031 + 0.117853i $$0.962399\pi$$
$$464$$ 8.92820 0.414481
$$465$$ 0 0
$$466$$ −15.0718 −0.698188
$$467$$ − 37.1769i − 1.72034i −0.510005 0.860171i $$-0.670357\pi$$
0.510005 0.860171i $$-0.329643\pi$$
$$468$$ 6.53590i 0.302122i
$$469$$ 17.3205 0.799787
$$470$$ 0 0
$$471$$ −46.2487 −2.13103
$$472$$ 0.196152i 0.00902865i
$$473$$ − 10.1436i − 0.466403i
$$474$$ −14.3923 −0.661060
$$475$$ 0 0
$$476$$ 1.85641 0.0850883
$$477$$ 26.7846i 1.22638i
$$478$$ 17.2679i 0.789818i
$$479$$ 34.0526 1.55590 0.777951 0.628325i $$-0.216259\pi$$
0.777951 + 0.628325i $$0.216259\pi$$
$$480$$ 0 0
$$481$$ 1.46410 0.0667573
$$482$$ − 8.92820i − 0.406669i
$$483$$ − 27.7128i − 1.26098i
$$484$$ 8.85641 0.402564
$$485$$ 0 0
$$486$$ −18.7321 −0.849703
$$487$$ 16.3923i 0.742806i 0.928472 + 0.371403i $$0.121123\pi$$
−0.928472 + 0.371403i $$0.878877\pi$$
$$488$$ − 8.92820i − 0.404161i
$$489$$ 63.7128 2.88119
$$490$$ 0 0
$$491$$ −1.07180 −0.0483695 −0.0241848 0.999708i $$-0.507699\pi$$
−0.0241848 + 0.999708i $$0.507699\pi$$
$$492$$ − 5.46410i − 0.246341i
$$493$$ 13.0718i 0.588724i
$$494$$ −6.14359 −0.276413
$$495$$ 0 0
$$496$$ −2.73205 −0.122673
$$497$$ − 13.8564i − 0.621545i
$$498$$ − 14.3923i − 0.644935i
$$499$$ −40.5885 −1.81699 −0.908494 0.417897i $$-0.862767\pi$$
−0.908494 + 0.417897i $$0.862767\pi$$
$$500$$ 0 0
$$501$$ −14.9282 −0.666943
$$502$$ 22.7321i 1.01458i
$$503$$ − 5.07180i − 0.226140i −0.993587 0.113070i $$-0.963932\pi$$
0.993587 0.113070i $$-0.0360685\pi$$
$$504$$ −5.66025 −0.252128
$$505$$ 0 0
$$506$$ 11.7128 0.520698
$$507$$ − 29.6603i − 1.31726i
$$508$$ − 13.6603i − 0.606076i
$$509$$ 2.00000 0.0886484 0.0443242 0.999017i $$-0.485887\pi$$
0.0443242 + 0.999017i $$0.485887\pi$$
$$510$$ 0 0
$$511$$ −16.3923 −0.725153
$$512$$ 1.00000i 0.0441942i
$$513$$ 16.7846i 0.741059i
$$514$$ −25.4641 −1.12317
$$515$$ 0 0
$$516$$ 18.9282 0.833268
$$517$$ 1.85641i 0.0816447i
$$518$$ 1.26795i 0.0557105i
$$519$$ 27.3205 1.19924
$$520$$ 0 0
$$521$$ 33.4641 1.46609 0.733044 0.680181i $$-0.238099\pi$$
0.733044 + 0.680181i $$0.238099\pi$$
$$522$$ − 39.8564i − 1.74447i
$$523$$ − 4.78461i − 0.209216i −0.994514 0.104608i $$-0.966641\pi$$
0.994514 0.104608i $$-0.0333589\pi$$
$$524$$ −12.5885 −0.549929
$$525$$ 0 0
$$526$$ −30.0526 −1.31035
$$527$$ − 4.00000i − 0.174243i
$$528$$ − 4.00000i − 0.174078i
$$529$$ −41.0000 −1.78261
$$530$$ 0 0
$$531$$ 0.875644 0.0379997
$$532$$ − 5.32051i − 0.230673i
$$533$$ − 2.92820i − 0.126835i
$$534$$ 5.46410 0.236455
$$535$$ 0 0
$$536$$ −13.6603 −0.590033
$$537$$ − 48.2487i − 2.08209i
$$538$$ 0.392305i 0.0169135i
$$539$$ 7.89488 0.340057
$$540$$ 0 0
$$541$$ −30.0000 −1.28980 −0.644900 0.764267i $$-0.723101\pi$$
−0.644900 + 0.764267i $$0.723101\pi$$
$$542$$ 16.7846i 0.720961i
$$543$$ 4.00000i 0.171656i
$$544$$ −1.46410 −0.0627728
$$545$$ 0 0
$$546$$ −5.07180 −0.217053
$$547$$ 4.00000i 0.171028i 0.996337 + 0.0855138i $$0.0272532\pi$$
−0.996337 + 0.0855138i $$0.972747\pi$$
$$548$$ 2.00000i 0.0854358i
$$549$$ −39.8564 −1.70103
$$550$$ 0 0
$$551$$ 37.4641 1.59602
$$552$$ 21.8564i 0.930270i
$$553$$ − 6.67949i − 0.284041i
$$554$$ −26.2487 −1.11520
$$555$$ 0 0
$$556$$ −6.92820 −0.293821
$$557$$ 32.1051i 1.36034i 0.733056 + 0.680169i $$0.238094\pi$$
−0.733056 + 0.680169i $$0.761906\pi$$
$$558$$ 12.1962i 0.516304i
$$559$$ 10.1436 0.429028
$$560$$ 0 0
$$561$$ 5.85641 0.247258
$$562$$ 4.92820i 0.207884i
$$563$$ 17.0718i 0.719490i 0.933051 + 0.359745i $$0.117136\pi$$
−0.933051 + 0.359745i $$0.882864\pi$$
$$564$$ −3.46410 −0.145865
$$565$$ 0 0
$$566$$ 4.39230 0.184622
$$567$$ − 3.12436i − 0.131211i
$$568$$ 10.9282i 0.458537i
$$569$$ 22.0000 0.922288 0.461144 0.887325i $$-0.347439\pi$$
0.461144 + 0.887325i $$0.347439\pi$$
$$570$$ 0 0
$$571$$ −12.0000 −0.502184 −0.251092 0.967963i $$-0.580790\pi$$
−0.251092 + 0.967963i $$0.580790\pi$$
$$572$$ − 2.14359i − 0.0896281i
$$573$$ 14.3923i 0.601247i
$$574$$ 2.53590 0.105846
$$575$$ 0 0
$$576$$ 4.46410 0.186004
$$577$$ 3.60770i 0.150190i 0.997176 + 0.0750952i $$0.0239261\pi$$
−0.997176 + 0.0750952i $$0.976074\pi$$
$$578$$ 14.8564i 0.617945i
$$579$$ −32.3923 −1.34618
$$580$$ 0 0
$$581$$ 6.67949 0.277112
$$582$$ 5.46410i 0.226494i
$$583$$ − 8.78461i − 0.363821i
$$584$$ 12.9282 0.534973
$$585$$ 0 0
$$586$$ −6.00000 −0.247858
$$587$$ − 3.21539i − 0.132713i −0.997796 0.0663567i $$-0.978862\pi$$
0.997796 0.0663567i $$-0.0211375\pi$$
$$588$$ 14.7321i 0.607540i
$$589$$ −11.4641 −0.472370
$$590$$ 0 0
$$591$$ −51.3205 −2.11104
$$592$$ − 1.00000i − 0.0410997i
$$593$$ 6.78461i 0.278611i 0.990249 + 0.139305i $$0.0444869\pi$$
−0.990249 + 0.139305i $$0.955513\pi$$
$$594$$ −5.85641 −0.240291
$$595$$ 0 0
$$596$$ −16.3923 −0.671455
$$597$$ 71.1769i 2.91308i
$$598$$ 11.7128i 0.478973i
$$599$$ −2.53590 −0.103614 −0.0518070 0.998657i $$-0.516498\pi$$
−0.0518070 + 0.998657i $$0.516498\pi$$
$$600$$ 0 0
$$601$$ 48.3923 1.97396 0.986982 0.160833i $$-0.0514180\pi$$
0.986982 + 0.160833i $$0.0514180\pi$$
$$602$$ 8.78461i 0.358034i
$$603$$ 60.9808i 2.48333i
$$604$$ −8.39230 −0.341478
$$605$$ 0 0
$$606$$ −6.92820 −0.281439
$$607$$ 40.7846i 1.65540i 0.561174 + 0.827698i $$0.310350\pi$$
−0.561174 + 0.827698i $$0.689650\pi$$
$$608$$ 4.19615i 0.170176i
$$609$$ 30.9282 1.25327
$$610$$ 0 0
$$611$$ −1.85641 −0.0751022
$$612$$ 6.53590i 0.264198i
$$613$$ 3.07180i 0.124069i 0.998074 + 0.0620344i $$0.0197588\pi$$
−0.998074 + 0.0620344i $$0.980241\pi$$
$$614$$ −12.5885 −0.508029
$$615$$ 0 0
$$616$$ 1.85641 0.0747967
$$617$$ − 12.9282i − 0.520470i −0.965545 0.260235i $$-0.916200\pi$$
0.965545 0.260235i $$-0.0838000\pi$$
$$618$$ − 36.7846i − 1.47969i
$$619$$ −9.85641 −0.396162 −0.198081 0.980186i $$-0.563471\pi$$
−0.198081 + 0.980186i $$0.563471\pi$$
$$620$$ 0 0
$$621$$ 32.0000 1.28412
$$622$$ 27.1244i 1.08759i
$$623$$ 2.53590i 0.101599i
$$624$$ 4.00000 0.160128
$$625$$ 0 0
$$626$$ −3.85641 −0.154133
$$627$$ − 16.7846i − 0.670313i
$$628$$ 16.9282i 0.675509i
$$629$$ 1.46410 0.0583776
$$630$$ 0 0
$$631$$ −20.9808 −0.835231 −0.417615 0.908624i $$-0.637134\pi$$
−0.417615 + 0.908624i $$0.637134\pi$$
$$632$$ 5.26795i 0.209548i
$$633$$ − 26.9282i − 1.07030i
$$634$$ −31.8564 −1.26518
$$635$$ 0 0
$$636$$ 16.3923 0.649997
$$637$$ 7.89488i 0.312807i
$$638$$ 13.0718i 0.517517i
$$639$$ 48.7846 1.92989
$$640$$ 0 0
$$641$$ −13.4641 −0.531800 −0.265900 0.964001i $$-0.585669\pi$$
−0.265900 + 0.964001i $$0.585669\pi$$
$$642$$ − 18.3923i − 0.725886i
$$643$$ 41.4641i 1.63518i 0.575798 + 0.817592i $$0.304692\pi$$
−0.575798 + 0.817592i $$0.695308\pi$$
$$644$$ −10.1436 −0.399714
$$645$$ 0 0
$$646$$ −6.14359 −0.241716
$$647$$ − 19.7128i − 0.774991i −0.921872 0.387495i $$-0.873340\pi$$
0.921872 0.387495i $$-0.126660\pi$$
$$648$$ 2.46410i 0.0967991i
$$649$$ −0.287187 −0.0112731
$$650$$ 0 0
$$651$$ −9.46410 −0.370927
$$652$$ − 23.3205i − 0.913302i
$$653$$ − 18.0000i − 0.704394i −0.935926 0.352197i $$-0.885435\pi$$
0.935926 0.352197i $$-0.114565\pi$$
$$654$$ −5.46410 −0.213663
$$655$$ 0 0
$$656$$ −2.00000 −0.0780869
$$657$$ − 57.7128i − 2.25159i
$$658$$ − 1.60770i − 0.0626745i
$$659$$ −6.92820 −0.269884 −0.134942 0.990853i $$-0.543085\pi$$
−0.134942 + 0.990853i $$0.543085\pi$$
$$660$$ 0 0
$$661$$ −44.9282 −1.74750 −0.873752 0.486371i $$-0.838320\pi$$
−0.873752 + 0.486371i $$0.838320\pi$$
$$662$$ − 8.87564i − 0.344962i
$$663$$ 5.85641i 0.227444i
$$664$$ −5.26795 −0.204436
$$665$$ 0 0
$$666$$ −4.46410 −0.172980
$$667$$ − 71.4256i − 2.76561i
$$668$$ 5.46410i 0.211412i
$$669$$ −60.2487 −2.32935
$$670$$ 0 0
$$671$$ 13.0718 0.504631
$$672$$ 3.46410i 0.133631i
$$673$$ − 19.0718i − 0.735164i −0.929991 0.367582i $$-0.880186\pi$$
0.929991 0.367582i $$-0.119814\pi$$
$$674$$ 26.0000 1.00148
$$675$$ 0 0
$$676$$ −10.8564 −0.417554
$$677$$ 31.8564i 1.22434i 0.790726 + 0.612171i $$0.209703\pi$$
−0.790726 + 0.612171i $$0.790297\pi$$
$$678$$ − 47.7128i − 1.83240i
$$679$$ −2.53590 −0.0973188
$$680$$ 0 0
$$681$$ −9.85641 −0.377698
$$682$$ − 4.00000i − 0.153168i
$$683$$ 36.7846i 1.40752i 0.710436 + 0.703762i $$0.248498\pi$$
−0.710436 + 0.703762i $$0.751502\pi$$
$$684$$ 18.7321 0.716238
$$685$$ 0 0
$$686$$ −15.7128 −0.599918
$$687$$ 43.3205i 1.65278i
$$688$$ − 6.92820i − 0.264135i
$$689$$ 8.78461 0.334667
$$690$$ 0 0
$$691$$ 20.3923 0.775760 0.387880 0.921710i $$-0.373208\pi$$
0.387880 + 0.921710i $$0.373208\pi$$
$$692$$ − 10.0000i − 0.380143i
$$693$$ − 8.28719i − 0.314804i
$$694$$ 17.0718 0.648037
$$695$$ 0 0
$$696$$ −24.3923 −0.924588
$$697$$ − 2.92820i − 0.110914i
$$698$$ − 19.3205i − 0.731292i
$$699$$ 41.1769 1.55745
$$700$$ 0 0
$$701$$ 15.8564 0.598888 0.299444 0.954114i $$-0.403199\pi$$
0.299444 + 0.954114i $$0.403199\pi$$
$$702$$ − 5.85641i − 0.221036i
$$703$$ − 4.19615i − 0.158261i
$$704$$ −1.46410 −0.0551804
$$705$$ 0 0
$$706$$ −15.8564 −0.596764
$$707$$ − 3.21539i − 0.120927i
$$708$$ − 0.535898i − 0.0201403i
$$709$$ −16.1436 −0.606285 −0.303143 0.952945i $$-0.598036\pi$$
−0.303143 + 0.952945i $$0.598036\pi$$
$$710$$ 0 0
$$711$$ 23.5167 0.881944
$$712$$ − 2.00000i − 0.0749532i
$$713$$ 21.8564i 0.818529i
$$714$$ −5.07180 −0.189807
$$715$$ 0 0
$$716$$ −17.6603 −0.659995
$$717$$ − 47.1769i − 1.76185i
$$718$$ − 8.39230i − 0.313198i
$$719$$ −8.39230 −0.312980 −0.156490 0.987680i $$-0.550018\pi$$
−0.156490 + 0.987680i $$0.550018\pi$$
$$720$$ 0 0
$$721$$ 17.0718 0.635787
$$722$$ − 1.39230i − 0.0518162i
$$723$$ 24.3923i 0.907160i
$$724$$ 1.46410 0.0544129
$$725$$ 0 0
$$726$$ −24.1962 −0.898003
$$727$$ 32.7846i 1.21591i 0.793970 + 0.607957i $$0.208011\pi$$
−0.793970 + 0.607957i $$0.791989\pi$$
$$728$$ 1.85641i 0.0688030i
$$729$$ 43.7846 1.62165
$$730$$ 0 0
$$731$$ 10.1436 0.375174
$$732$$ 24.3923i 0.901566i
$$733$$ 0.143594i 0.00530375i 0.999996 + 0.00265187i $$0.000844119\pi$$
−0.999996 + 0.00265187i $$0.999156\pi$$
$$734$$ −21.6603 −0.799495
$$735$$ 0 0
$$736$$ 8.00000 0.294884
$$737$$ − 20.0000i − 0.736709i
$$738$$ 8.92820i 0.328652i
$$739$$ 6.92820 0.254858 0.127429 0.991848i $$-0.459327\pi$$
0.127429 + 0.991848i $$0.459327\pi$$
$$740$$ 0 0
$$741$$ 16.7846 0.616598
$$742$$ 7.60770i 0.279287i
$$743$$ − 7.12436i − 0.261367i −0.991424 0.130684i $$-0.958283\pi$$
0.991424 0.130684i $$-0.0417172\pi$$
$$744$$ 7.46410 0.273647
$$745$$ 0 0
$$746$$ −30.7846 −1.12710
$$747$$ 23.5167i 0.860430i
$$748$$ − 2.14359i − 0.0783775i
$$749$$ 8.53590 0.311895
$$750$$ 0 0
$$751$$ 0.392305 0.0143154 0.00715770 0.999974i $$-0.497722\pi$$
0.00715770 + 0.999974i $$0.497722\pi$$
$$752$$ 1.26795i 0.0462373i
$$753$$ − 62.1051i − 2.26324i
$$754$$ −13.0718 −0.476047
$$755$$ 0 0
$$756$$ 5.07180 0.184459
$$757$$ 53.7128i 1.95223i 0.217265 + 0.976113i $$0.430286\pi$$
−0.217265 + 0.976113i $$0.569714\pi$$
$$758$$ − 11.6077i − 0.421610i
$$759$$ −32.0000 −1.16153
$$760$$ 0 0
$$761$$ 10.0000 0.362500 0.181250 0.983437i $$-0.441986\pi$$
0.181250 + 0.983437i $$0.441986\pi$$
$$762$$ 37.3205i 1.35198i
$$763$$ − 2.53590i − 0.0918057i
$$764$$ 5.26795 0.190588
$$765$$ 0 0
$$766$$ 8.00000 0.289052
$$767$$ − 0.287187i − 0.0103697i
$$768$$ − 2.73205i − 0.0985844i
$$769$$ −20.9282 −0.754690 −0.377345 0.926073i $$-0.623163\pi$$
−0.377345 + 0.926073i $$0.623163\pi$$
$$770$$ 0 0
$$771$$ 69.5692 2.50547
$$772$$ 11.8564i 0.426721i
$$773$$ − 38.7846i − 1.39499i −0.716592 0.697493i $$-0.754299\pi$$
0.716592 0.697493i $$-0.245701\pi$$
$$774$$ −30.9282 −1.11169
$$775$$ 0 0
$$776$$ 2.00000 0.0717958
$$777$$ − 3.46410i − 0.124274i
$$778$$ − 15.8564i − 0.568480i
$$779$$ −8.39230 −0.300686
$$780$$ 0 0
$$781$$ −16.0000 −0.572525
$$782$$ 11.7128i 0.418849i
$$783$$ 35.7128i 1.27627i
$$784$$ 5.39230 0.192582
$$785$$ 0 0
$$786$$ 34.3923 1.22673
$$787$$ − 11.8038i − 0.420762i −0.977620 0.210381i $$-0.932530\pi$$
0.977620 0.210381i $$-0.0674704\pi$$
$$788$$ 18.7846i 0.669174i
$$789$$ 82.1051 2.92302
$$790$$ 0 0
$$791$$ 22.1436 0.787336
$$792$$ 6.53590i 0.232243i
$$793$$ 13.0718i 0.464193i
$$794$$ −22.0000 −0.780751
$$795$$ 0 0
$$796$$ 26.0526 0.923408
$$797$$ − 17.4641i − 0.618610i −0.950963 0.309305i $$-0.899904\pi$$
0.950963 0.309305i $$-0.100096\pi$$
$$798$$ 14.5359i 0.514565i
$$799$$ −1.85641 −0.0656749
$$800$$ 0 0
$$801$$ −8.92820 −0.315463
$$802$$ − 19.0718i − 0.673449i
$$803$$ 18.9282i 0.667962i
$$804$$ 37.3205 1.31619
$$805$$ 0 0
$$806$$ 4.00000 0.140894
$$807$$ − 1.07180i − 0.0377290i
$$808$$ 2.53590i 0.0892126i
$$809$$ 39.5692 1.39118 0.695590 0.718439i $$-0.255143\pi$$
0.695590 + 0.718439i $$0.255143\pi$$
$$810$$ 0 0
$$811$$ −14.1436 −0.496649 −0.248324 0.968677i $$-0.579880\pi$$
−0.248324 + 0.968677i $$0.579880\pi$$
$$812$$ − 11.3205i − 0.397272i
$$813$$ − 45.8564i − 1.60825i
$$814$$ 1.46410 0.0513167
$$815$$ 0 0
$$816$$ 4.00000 0.140028
$$817$$ − 29.0718i − 1.01709i
$$818$$ − 16.9282i − 0.591881i
$$819$$ 8.28719 0.289578
$$820$$ 0 0
$$821$$ 30.0000 1.04701 0.523504 0.852023i $$-0.324625\pi$$
0.523504 + 0.852023i $$0.324625\pi$$
$$822$$ − 5.46410i − 0.190582i
$$823$$ − 10.4449i − 0.364085i −0.983291 0.182043i $$-0.941729\pi$$
0.983291 0.182043i $$-0.0582709\pi$$
$$824$$ −13.4641 −0.469044
$$825$$ 0 0
$$826$$ 0.248711 0.00865377
$$827$$ − 4.39230i − 0.152735i −0.997080 0.0763677i $$-0.975668\pi$$
0.997080 0.0763677i $$-0.0243323\pi$$
$$828$$ − 35.7128i − 1.24111i
$$829$$ −31.5692 −1.09644 −0.548222 0.836333i $$-0.684695\pi$$
−0.548222 + 0.836333i $$0.684695\pi$$
$$830$$ 0 0
$$831$$ 71.7128 2.48769
$$832$$ − 1.46410i − 0.0507586i
$$833$$ 7.89488i 0.273541i
$$834$$ 18.9282 0.655430
$$835$$ 0 0
$$836$$ −6.14359 −0.212481
$$837$$ − 10.9282i − 0.377734i
$$838$$ 34.2487i 1.18310i
$$839$$ −32.7846 −1.13185 −0.565925 0.824457i $$-0.691481\pi$$
−0.565925 + 0.824457i $$0.691481\pi$$
$$840$$ 0 0
$$841$$ 50.7128 1.74872
$$842$$ − 27.8564i − 0.959995i
$$843$$ − 13.4641i − 0.463728i
$$844$$ −9.85641 −0.339272
$$845$$ 0 0
$$846$$ 5.66025 0.194604
$$847$$ − 11.2295i − 0.385849i
$$848$$ − 6.00000i − 0.206041i
$$849$$ −12.0000 −0.411839
$$850$$ 0 0
$$851$$ −8.00000 −0.274236
$$852$$ − 29.8564i − 1.02286i
$$853$$ − 30.0000i − 1.02718i −0.858036 0.513590i $$-0.828315\pi$$
0.858036 0.513590i $$-0.171685\pi$$
$$854$$ −11.3205 −0.387380
$$855$$ 0 0
$$856$$ −6.73205 −0.230097
$$857$$ 4.14359i 0.141542i 0.997493 + 0.0707712i $$0.0225460\pi$$
−0.997493 + 0.0707712i $$0.977454\pi$$
$$858$$ 5.85641i 0.199934i
$$859$$ 14.4449 0.492852 0.246426 0.969162i $$-0.420744\pi$$
0.246426 + 0.969162i $$0.420744\pi$$
$$860$$ 0 0
$$861$$ −6.92820 −0.236113
$$862$$ 8.19615i 0.279162i
$$863$$ 38.4449i 1.30868i 0.756201 + 0.654339i $$0.227053\pi$$
−0.756201 + 0.654339i $$0.772947\pi$$
$$864$$ −4.00000 −0.136083
$$865$$ 0 0
$$866$$ 34.0000 1.15537
$$867$$ − 40.5885i − 1.37846i
$$868$$ 3.46410i 0.117579i
$$869$$ −7.71281 −0.261639
$$870$$ 0 0
$$871$$ 20.0000 0.677674
$$872$$ 2.00000i 0.0677285i
$$873$$ − 8.92820i − 0.302174i
$$874$$ 33.5692 1.13550
$$875$$ 0 0
$$876$$ −35.3205 −1.19337
$$877$$ 50.7846i 1.71487i 0.514589 + 0.857437i $$0.327945\pi$$
−0.514589 + 0.857437i $$0.672055\pi$$
$$878$$ 31.5167i 1.06364i
$$879$$ 16.3923 0.552899
$$880$$ 0 0
$$881$$ 47.3205 1.59427 0.797134 0.603802i $$-0.206348\pi$$
0.797134 + 0.603802i $$0.206348\pi$$
$$882$$ − 24.0718i − 0.810540i
$$883$$ − 34.2487i − 1.15256i −0.817252 0.576280i $$-0.804504\pi$$
0.817252 0.576280i $$-0.195496\pi$$
$$884$$ 2.14359 0.0720969
$$885$$ 0 0
$$886$$ 39.1244 1.31441
$$887$$ − 20.1962i − 0.678120i −0.940765 0.339060i $$-0.889891\pi$$
0.940765 0.339060i $$-0.110109\pi$$
$$888$$ 2.73205i 0.0916816i
$$889$$ −17.3205 −0.580911
$$890$$ 0 0
$$891$$ −3.60770 −0.120862
$$892$$ 22.0526i 0.738374i
$$893$$ 5.32051i 0.178044i
$$894$$ 44.7846 1.49782
$$895$$ 0 0
$$896$$ 1.26795 0.0423592
$$897$$ − 32.0000i − 1.06845i
$$898$$ − 33.7128i − 1.12501i
$$899$$ −24.3923 −0.813529
$$900$$ 0 0
$$901$$ 8.78461 0.292658
$$902$$ − 2.92820i − 0.0974985i
$$903$$ − 24.0000i − 0.798670i
$$904$$ −17.4641 −0.580847
$$905$$ 0 0
$$906$$ 22.9282 0.761739
$$907$$ − 5.75129i − 0.190968i −0.995431 0.0954842i $$-0.969560\pi$$
0.995431 0.0954842i $$-0.0304399\pi$$
$$908$$ 3.60770i 0.119726i
$$909$$ 11.3205 0.375478
$$910$$ 0 0
$$911$$ −27.9090 −0.924665 −0.462333 0.886707i $$-0.652987\pi$$
−0.462333 + 0.886707i $$0.652987\pi$$
$$912$$ − 11.4641i − 0.379614i
$$913$$ − 7.71281i − 0.255257i
$$914$$ −4.14359 −0.137058
$$915$$ 0 0
$$916$$ 15.8564 0.523910
$$917$$ 15.9615i 0.527096i
$$918$$ − 5.85641i − 0.193290i
$$919$$ 13.2679 0.437669 0.218835 0.975762i $$-0.429774\pi$$
0.218835 + 0.975762i $$0.429774\pi$$
$$920$$ 0 0
$$921$$ 34.3923 1.13326
$$922$$ − 26.7846i − 0.882104i
$$923$$ − 16.0000i − 0.526646i
$$924$$ −5.07180 −0.166850
$$925$$ 0 0
$$926$$ −5.07180 −0.166670
$$927$$ 60.1051i 1.97411i
$$928$$ 8.92820i 0.293083i
$$929$$ 15.8564 0.520232 0.260116 0.965577i $$-0.416239\pi$$
0.260116 + 0.965577i $$0.416239\pi$$
$$930$$ 0 0
$$931$$ 22.6269 0.741568
$$932$$ − 15.0718i − 0.493693i
$$933$$ − 74.1051i − 2.42609i
$$934$$ 37.1769 1.21647
$$935$$ 0 0
$$936$$ −6.53590 −0.213633
$$937$$ − 3.85641i − 0.125983i −0.998014 0.0629917i $$-0.979936\pi$$
0.998014 0.0629917i $$-0.0200642\pi$$
$$938$$ 17.3205i 0.565535i
$$939$$ 10.5359 0.343826
$$940$$ 0 0
$$941$$ 28.3923 0.925563 0.462781 0.886472i $$-0.346852\pi$$
0.462781 + 0.886472i $$0.346852\pi$$
$$942$$ − 46.2487i − 1.50686i
$$943$$ 16.0000i 0.521032i
$$944$$ −0.196152 −0.00638422
$$945$$ 0 0
$$946$$ 10.1436 0.329797
$$947$$ 45.1769i 1.46805i 0.679121 + 0.734026i $$0.262361\pi$$
−0.679121 + 0.734026i $$0.737639\pi$$
$$948$$ − 14.3923i − 0.467440i
$$949$$ −18.9282 −0.614435
$$950$$ 0 0
$$951$$ 87.0333 2.82225
$$952$$ 1.85641i 0.0601665i
$$953$$ − 11.8564i − 0.384067i −0.981388 0.192033i $$-0.938492\pi$$
0.981388 0.192033i $$-0.0615082\pi$$
$$954$$ −26.7846 −0.867184
$$955$$ 0 0
$$956$$ −17.2679 −0.558485
$$957$$ − 35.7128i − 1.15443i
$$958$$ 34.0526i 1.10019i
$$959$$ 2.53590 0.0818884
$$960$$ 0 0
$$961$$ −23.5359 −0.759223
$$962$$ 1.46410i 0.0472045i
$$963$$ 30.0526i 0.968430i
$$964$$ 8.92820 0.287558
$$965$$ 0 0
$$966$$ 27.7128 0.891645
$$967$$ 23.6077i 0.759172i 0.925156 + 0.379586i $$0.123934\pi$$
−0.925156 + 0.379586i $$0.876066\pi$$
$$968$$ 8.85641i 0.284656i
$$969$$ 16.7846 0.539199
$$970$$ 0 0
$$971$$ 14.9282 0.479069 0.239534 0.970888i $$-0.423005\pi$$
0.239534 + 0.970888i $$0.423005\pi$$
$$972$$ − 18.7321i − 0.600831i
$$973$$ 8.78461i 0.281622i
$$974$$ −16.3923 −0.525243
$$975$$ 0 0
$$976$$ 8.92820 0.285785
$$977$$ 34.0000i 1.08776i 0.839164 + 0.543878i $$0.183045\pi$$
−0.839164 + 0.543878i $$0.816955\pi$$
$$978$$ 63.7128i 2.03731i
$$979$$ 2.92820 0.0935858
$$980$$ 0 0
$$981$$ 8.92820 0.285056
$$982$$ − 1.07180i − 0.0342024i
$$983$$ − 38.4449i − 1.22620i −0.790005 0.613100i $$-0.789922\pi$$
0.790005 0.613100i $$-0.210078\pi$$
$$984$$ 5.46410 0.174189
$$985$$ 0 0
$$986$$ −13.0718 −0.416291
$$987$$ 4.39230i 0.139809i
$$988$$ − 6.14359i − 0.195454i
$$989$$ −55.4256 −1.76243
$$990$$ 0 0
$$991$$ 5.94744 0.188927 0.0944633 0.995528i $$-0.469886\pi$$
0.0944633 + 0.995528i $$0.469886\pi$$
$$992$$ − 2.73205i − 0.0867427i
$$993$$ 24.2487i 0.769510i
$$994$$ 13.8564 0.439499
$$995$$ 0 0
$$996$$ 14.3923 0.456038
$$997$$ 4.14359i 0.131229i 0.997845 + 0.0656145i $$0.0209007\pi$$
−0.997845 + 0.0656145i $$0.979099\pi$$
$$998$$ − 40.5885i − 1.28481i
$$999$$ 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.l.149.3 4
5.2 odd 4 370.2.a.e.1.1 2
5.3 odd 4 1850.2.a.x.1.2 2
5.4 even 2 inner 1850.2.b.l.149.2 4
15.2 even 4 3330.2.a.bd.1.2 2
20.7 even 4 2960.2.a.q.1.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
370.2.a.e.1.1 2 5.2 odd 4
1850.2.a.x.1.2 2 5.3 odd 4
1850.2.b.l.149.2 4 5.4 even 2 inner
1850.2.b.l.149.3 4 1.1 even 1 trivial
2960.2.a.q.1.2 2 20.7 even 4
3330.2.a.bd.1.2 2 15.2 even 4