# Properties

 Label 1450.2.b.b.349.1 Level $1450$ Weight $2$ Character 1450.349 Analytic conductor $11.578$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1450,2,Mod(349,1450)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1450, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1450.349");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1450 = 2 \cdot 5^{2} \cdot 29$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1450.b (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 Self dual: no Analytic conductor: $$11.5783082931$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(i)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 58) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 349.1 Root $$-1.00000i$$ of defining polynomial Character $$\chi$$ $$=$$ 1450.349 Dual form 1450.2.b.b.349.2

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{6} +2.00000i q^{7} +1.00000i q^{8} +2.00000 q^{9} +O(q^{10})$$ $$q-1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{6} +2.00000i q^{7} +1.00000i q^{8} +2.00000 q^{9} -3.00000 q^{11} +1.00000i q^{12} -1.00000i q^{13} +2.00000 q^{14} +1.00000 q^{16} -8.00000i q^{17} -2.00000i q^{18} +2.00000 q^{21} +3.00000i q^{22} +4.00000i q^{23} +1.00000 q^{24} -1.00000 q^{26} -5.00000i q^{27} -2.00000i q^{28} +1.00000 q^{29} -3.00000 q^{31} -1.00000i q^{32} +3.00000i q^{33} -8.00000 q^{34} -2.00000 q^{36} -8.00000i q^{37} -1.00000 q^{39} +2.00000 q^{41} -2.00000i q^{42} -11.0000i q^{43} +3.00000 q^{44} +4.00000 q^{46} -13.0000i q^{47} -1.00000i q^{48} +3.00000 q^{49} -8.00000 q^{51} +1.00000i q^{52} -11.0000i q^{53} -5.00000 q^{54} -2.00000 q^{56} -1.00000i q^{58} -8.00000 q^{61} +3.00000i q^{62} +4.00000i q^{63} -1.00000 q^{64} +3.00000 q^{66} +12.0000i q^{67} +8.00000i q^{68} +4.00000 q^{69} +2.00000 q^{71} +2.00000i q^{72} +4.00000i q^{73} -8.00000 q^{74} -6.00000i q^{77} +1.00000i q^{78} -15.0000 q^{79} +1.00000 q^{81} -2.00000i q^{82} +4.00000i q^{83} -2.00000 q^{84} -11.0000 q^{86} -1.00000i q^{87} -3.00000i q^{88} +10.0000 q^{89} +2.00000 q^{91} -4.00000i q^{92} +3.00000i q^{93} -13.0000 q^{94} -1.00000 q^{96} +2.00000i q^{97} -3.00000i q^{98} -6.00000 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{4} - 2 q^{6} + 4 q^{9}+O(q^{10})$$ 2 * q - 2 * q^4 - 2 * q^6 + 4 * q^9 $$2 q - 2 q^{4} - 2 q^{6} + 4 q^{9} - 6 q^{11} + 4 q^{14} + 2 q^{16} + 4 q^{21} + 2 q^{24} - 2 q^{26} + 2 q^{29} - 6 q^{31} - 16 q^{34} - 4 q^{36} - 2 q^{39} + 4 q^{41} + 6 q^{44} + 8 q^{46} + 6 q^{49} - 16 q^{51} - 10 q^{54} - 4 q^{56} - 16 q^{61} - 2 q^{64} + 6 q^{66} + 8 q^{69} + 4 q^{71} - 16 q^{74} - 30 q^{79} + 2 q^{81} - 4 q^{84} - 22 q^{86} + 20 q^{89} + 4 q^{91} - 26 q^{94} - 2 q^{96} - 12 q^{99}+O(q^{100})$$ 2 * q - 2 * q^4 - 2 * q^6 + 4 * q^9 - 6 * q^11 + 4 * q^14 + 2 * q^16 + 4 * q^21 + 2 * q^24 - 2 * q^26 + 2 * q^29 - 6 * q^31 - 16 * q^34 - 4 * q^36 - 2 * q^39 + 4 * q^41 + 6 * q^44 + 8 * q^46 + 6 * q^49 - 16 * q^51 - 10 * q^54 - 4 * q^56 - 16 * q^61 - 2 * q^64 + 6 * q^66 + 8 * q^69 + 4 * q^71 - 16 * q^74 - 30 * q^79 + 2 * q^81 - 4 * q^84 - 22 * q^86 + 20 * q^89 + 4 * q^91 - 26 * q^94 - 2 * q^96 - 12 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1450\mathbb{Z}\right)^\times$$.

 $$n$$ $$901$$ $$1277$$ $$\chi(n)$$ $$1$$ $$-1$$

## 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 $$-0.906785\pi$$
0.957427 0.288675i $$-0.0932147\pi$$
$$4$$ −1.00000 −0.500000
$$5$$ 0 0
$$6$$ −1.00000 −0.408248
$$7$$ 2.00000i 0.755929i 0.925820 + 0.377964i $$0.123376\pi$$
−0.925820 + 0.377964i $$0.876624\pi$$
$$8$$ 1.00000i 0.353553i
$$9$$ 2.00000 0.666667
$$10$$ 0 0
$$11$$ −3.00000 −0.904534 −0.452267 0.891883i $$-0.649385\pi$$
−0.452267 + 0.891883i $$0.649385\pi$$
$$12$$ 1.00000i 0.288675i
$$13$$ − 1.00000i − 0.277350i −0.990338 0.138675i $$-0.955716\pi$$
0.990338 0.138675i $$-0.0442844\pi$$
$$14$$ 2.00000 0.534522
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ − 8.00000i − 1.94029i −0.242536 0.970143i $$-0.577979\pi$$
0.242536 0.970143i $$-0.422021\pi$$
$$18$$ − 2.00000i − 0.471405i
$$19$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$20$$ 0 0
$$21$$ 2.00000 0.436436
$$22$$ 3.00000i 0.639602i
$$23$$ 4.00000i 0.834058i 0.908893 + 0.417029i $$0.136929\pi$$
−0.908893 + 0.417029i $$0.863071\pi$$
$$24$$ 1.00000 0.204124
$$25$$ 0 0
$$26$$ −1.00000 −0.196116
$$27$$ − 5.00000i − 0.962250i
$$28$$ − 2.00000i − 0.377964i
$$29$$ 1.00000 0.185695
$$30$$ 0 0
$$31$$ −3.00000 −0.538816 −0.269408 0.963026i $$-0.586828\pi$$
−0.269408 + 0.963026i $$0.586828\pi$$
$$32$$ − 1.00000i − 0.176777i
$$33$$ 3.00000i 0.522233i
$$34$$ −8.00000 −1.37199
$$35$$ 0 0
$$36$$ −2.00000 −0.333333
$$37$$ − 8.00000i − 1.31519i −0.753371 0.657596i $$-0.771573\pi$$
0.753371 0.657596i $$-0.228427\pi$$
$$38$$ 0 0
$$39$$ −1.00000 −0.160128
$$40$$ 0 0
$$41$$ 2.00000 0.312348 0.156174 0.987730i $$-0.450084\pi$$
0.156174 + 0.987730i $$0.450084\pi$$
$$42$$ − 2.00000i − 0.308607i
$$43$$ − 11.0000i − 1.67748i −0.544529 0.838742i $$-0.683292\pi$$
0.544529 0.838742i $$-0.316708\pi$$
$$44$$ 3.00000 0.452267
$$45$$ 0 0
$$46$$ 4.00000 0.589768
$$47$$ − 13.0000i − 1.89624i −0.317905 0.948122i $$-0.602979\pi$$
0.317905 0.948122i $$-0.397021\pi$$
$$48$$ − 1.00000i − 0.144338i
$$49$$ 3.00000 0.428571
$$50$$ 0 0
$$51$$ −8.00000 −1.12022
$$52$$ 1.00000i 0.138675i
$$53$$ − 11.0000i − 1.51097i −0.655168 0.755483i $$-0.727402\pi$$
0.655168 0.755483i $$-0.272598\pi$$
$$54$$ −5.00000 −0.680414
$$55$$ 0 0
$$56$$ −2.00000 −0.267261
$$57$$ 0 0
$$58$$ − 1.00000i − 0.131306i
$$59$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$60$$ 0 0
$$61$$ −8.00000 −1.02430 −0.512148 0.858898i $$-0.671150\pi$$
−0.512148 + 0.858898i $$0.671150\pi$$
$$62$$ 3.00000i 0.381000i
$$63$$ 4.00000i 0.503953i
$$64$$ −1.00000 −0.125000
$$65$$ 0 0
$$66$$ 3.00000 0.369274
$$67$$ 12.0000i 1.46603i 0.680211 + 0.733017i $$0.261888\pi$$
−0.680211 + 0.733017i $$0.738112\pi$$
$$68$$ 8.00000i 0.970143i
$$69$$ 4.00000 0.481543
$$70$$ 0 0
$$71$$ 2.00000 0.237356 0.118678 0.992933i $$-0.462134\pi$$
0.118678 + 0.992933i $$0.462134\pi$$
$$72$$ 2.00000i 0.235702i
$$73$$ 4.00000i 0.468165i 0.972217 + 0.234082i $$0.0752085\pi$$
−0.972217 + 0.234082i $$0.924791\pi$$
$$74$$ −8.00000 −0.929981
$$75$$ 0 0
$$76$$ 0 0
$$77$$ − 6.00000i − 0.683763i
$$78$$ 1.00000i 0.113228i
$$79$$ −15.0000 −1.68763 −0.843816 0.536633i $$-0.819696\pi$$
−0.843816 + 0.536633i $$0.819696\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ − 2.00000i − 0.220863i
$$83$$ 4.00000i 0.439057i 0.975606 + 0.219529i $$0.0704519\pi$$
−0.975606 + 0.219529i $$0.929548\pi$$
$$84$$ −2.00000 −0.218218
$$85$$ 0 0
$$86$$ −11.0000 −1.18616
$$87$$ − 1.00000i − 0.107211i
$$88$$ − 3.00000i − 0.319801i
$$89$$ 10.0000 1.06000 0.529999 0.847998i $$-0.322192\pi$$
0.529999 + 0.847998i $$0.322192\pi$$
$$90$$ 0 0
$$91$$ 2.00000 0.209657
$$92$$ − 4.00000i − 0.417029i
$$93$$ 3.00000i 0.311086i
$$94$$ −13.0000 −1.34085
$$95$$ 0 0
$$96$$ −1.00000 −0.102062
$$97$$ 2.00000i 0.203069i 0.994832 + 0.101535i $$0.0323753\pi$$
−0.994832 + 0.101535i $$0.967625\pi$$
$$98$$ − 3.00000i − 0.303046i
$$99$$ −6.00000 −0.603023
$$100$$ 0 0
$$101$$ −8.00000 −0.796030 −0.398015 0.917379i $$-0.630301\pi$$
−0.398015 + 0.917379i $$0.630301\pi$$
$$102$$ 8.00000i 0.792118i
$$103$$ 14.0000i 1.37946i 0.724066 + 0.689730i $$0.242271\pi$$
−0.724066 + 0.689730i $$0.757729\pi$$
$$104$$ 1.00000 0.0980581
$$105$$ 0 0
$$106$$ −11.0000 −1.06841
$$107$$ 2.00000i 0.193347i 0.995316 + 0.0966736i $$0.0308203\pi$$
−0.995316 + 0.0966736i $$0.969180\pi$$
$$108$$ 5.00000i 0.481125i
$$109$$ −5.00000 −0.478913 −0.239457 0.970907i $$-0.576969\pi$$
−0.239457 + 0.970907i $$0.576969\pi$$
$$110$$ 0 0
$$111$$ −8.00000 −0.759326
$$112$$ 2.00000i 0.188982i
$$113$$ − 6.00000i − 0.564433i −0.959351 0.282216i $$-0.908930\pi$$
0.959351 0.282216i $$-0.0910696\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ −1.00000 −0.0928477
$$117$$ − 2.00000i − 0.184900i
$$118$$ 0 0
$$119$$ 16.0000 1.46672
$$120$$ 0 0
$$121$$ −2.00000 −0.181818
$$122$$ 8.00000i 0.724286i
$$123$$ − 2.00000i − 0.180334i
$$124$$ 3.00000 0.269408
$$125$$ 0 0
$$126$$ 4.00000 0.356348
$$127$$ − 8.00000i − 0.709885i −0.934888 0.354943i $$-0.884500\pi$$
0.934888 0.354943i $$-0.115500\pi$$
$$128$$ 1.00000i 0.0883883i
$$129$$ −11.0000 −0.968496
$$130$$ 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
$$133$$ 0 0
$$134$$ 12.0000 1.03664
$$135$$ 0 0
$$136$$ 8.00000 0.685994
$$137$$ 12.0000i 1.02523i 0.858619 + 0.512615i $$0.171323\pi$$
−0.858619 + 0.512615i $$0.828677\pi$$
$$138$$ − 4.00000i − 0.340503i
$$139$$ 20.0000 1.69638 0.848189 0.529694i $$-0.177693\pi$$
0.848189 + 0.529694i $$0.177693\pi$$
$$140$$ 0 0
$$141$$ −13.0000 −1.09480
$$142$$ − 2.00000i − 0.167836i
$$143$$ 3.00000i 0.250873i
$$144$$ 2.00000 0.166667
$$145$$ 0 0
$$146$$ 4.00000 0.331042
$$147$$ − 3.00000i − 0.247436i
$$148$$ 8.00000i 0.657596i
$$149$$ −15.0000 −1.22885 −0.614424 0.788976i $$-0.710612\pi$$
−0.614424 + 0.788976i $$0.710612\pi$$
$$150$$ 0 0
$$151$$ 2.00000 0.162758 0.0813788 0.996683i $$-0.474068\pi$$
0.0813788 + 0.996683i $$0.474068\pi$$
$$152$$ 0 0
$$153$$ − 16.0000i − 1.29352i
$$154$$ −6.00000 −0.483494
$$155$$ 0 0
$$156$$ 1.00000 0.0800641
$$157$$ − 18.0000i − 1.43656i −0.695756 0.718278i $$-0.744931\pi$$
0.695756 0.718278i $$-0.255069\pi$$
$$158$$ 15.0000i 1.19334i
$$159$$ −11.0000 −0.872357
$$160$$ 0 0
$$161$$ −8.00000 −0.630488
$$162$$ − 1.00000i − 0.0785674i
$$163$$ 9.00000i 0.704934i 0.935824 + 0.352467i $$0.114657\pi$$
−0.935824 + 0.352467i $$0.885343\pi$$
$$164$$ −2.00000 −0.156174
$$165$$ 0 0
$$166$$ 4.00000 0.310460
$$167$$ 2.00000i 0.154765i 0.997001 + 0.0773823i $$0.0246562\pi$$
−0.997001 + 0.0773823i $$0.975344\pi$$
$$168$$ 2.00000i 0.154303i
$$169$$ 12.0000 0.923077
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 11.0000i 0.838742i
$$173$$ − 6.00000i − 0.456172i −0.973641 0.228086i $$-0.926753\pi$$
0.973641 0.228086i $$-0.0732467\pi$$
$$174$$ −1.00000 −0.0758098
$$175$$ 0 0
$$176$$ −3.00000 −0.226134
$$177$$ 0 0
$$178$$ − 10.0000i − 0.749532i
$$179$$ 10.0000 0.747435 0.373718 0.927543i $$-0.378083\pi$$
0.373718 + 0.927543i $$0.378083\pi$$
$$180$$ 0 0
$$181$$ 7.00000 0.520306 0.260153 0.965567i $$-0.416227\pi$$
0.260153 + 0.965567i $$0.416227\pi$$
$$182$$ − 2.00000i − 0.148250i
$$183$$ 8.00000i 0.591377i
$$184$$ −4.00000 −0.294884
$$185$$ 0 0
$$186$$ 3.00000 0.219971
$$187$$ 24.0000i 1.75505i
$$188$$ 13.0000i 0.948122i
$$189$$ 10.0000 0.727393
$$190$$ 0 0
$$191$$ −8.00000 −0.578860 −0.289430 0.957199i $$-0.593466\pi$$
−0.289430 + 0.957199i $$0.593466\pi$$
$$192$$ 1.00000i 0.0721688i
$$193$$ 14.0000i 1.00774i 0.863779 + 0.503871i $$0.168091\pi$$
−0.863779 + 0.503871i $$0.831909\pi$$
$$194$$ 2.00000 0.143592
$$195$$ 0 0
$$196$$ −3.00000 −0.214286
$$197$$ − 18.0000i − 1.28245i −0.767354 0.641223i $$-0.778427\pi$$
0.767354 0.641223i $$-0.221573\pi$$
$$198$$ 6.00000i 0.426401i
$$199$$ 10.0000 0.708881 0.354441 0.935079i $$-0.384671\pi$$
0.354441 + 0.935079i $$0.384671\pi$$
$$200$$ 0 0
$$201$$ 12.0000 0.846415
$$202$$ 8.00000i 0.562878i
$$203$$ 2.00000i 0.140372i
$$204$$ 8.00000 0.560112
$$205$$ 0 0
$$206$$ 14.0000 0.975426
$$207$$ 8.00000i 0.556038i
$$208$$ − 1.00000i − 0.0693375i
$$209$$ 0 0
$$210$$ 0 0
$$211$$ −3.00000 −0.206529 −0.103264 0.994654i $$-0.532929\pi$$
−0.103264 + 0.994654i $$0.532929\pi$$
$$212$$ 11.0000i 0.755483i
$$213$$ − 2.00000i − 0.137038i
$$214$$ 2.00000 0.136717
$$215$$ 0 0
$$216$$ 5.00000 0.340207
$$217$$ − 6.00000i − 0.407307i
$$218$$ 5.00000i 0.338643i
$$219$$ 4.00000 0.270295
$$220$$ 0 0
$$221$$ −8.00000 −0.538138
$$222$$ 8.00000i 0.536925i
$$223$$ − 26.0000i − 1.74109i −0.492090 0.870544i $$-0.663767\pi$$
0.492090 0.870544i $$-0.336233\pi$$
$$224$$ 2.00000 0.133631
$$225$$ 0 0
$$226$$ −6.00000 −0.399114
$$227$$ − 18.0000i − 1.19470i −0.801980 0.597351i $$-0.796220\pi$$
0.801980 0.597351i $$-0.203780\pi$$
$$228$$ 0 0
$$229$$ −10.0000 −0.660819 −0.330409 0.943838i $$-0.607187\pi$$
−0.330409 + 0.943838i $$0.607187\pi$$
$$230$$ 0 0
$$231$$ −6.00000 −0.394771
$$232$$ 1.00000i 0.0656532i
$$233$$ − 1.00000i − 0.0655122i −0.999463 0.0327561i $$-0.989572\pi$$
0.999463 0.0327561i $$-0.0104285\pi$$
$$234$$ −2.00000 −0.130744
$$235$$ 0 0
$$236$$ 0 0
$$237$$ 15.0000i 0.974355i
$$238$$ − 16.0000i − 1.03713i
$$239$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$240$$ 0 0
$$241$$ 17.0000 1.09507 0.547533 0.836784i $$-0.315567\pi$$
0.547533 + 0.836784i $$0.315567\pi$$
$$242$$ 2.00000i 0.128565i
$$243$$ − 16.0000i − 1.02640i
$$244$$ 8.00000 0.512148
$$245$$ 0 0
$$246$$ −2.00000 −0.127515
$$247$$ 0 0
$$248$$ − 3.00000i − 0.190500i
$$249$$ 4.00000 0.253490
$$250$$ 0 0
$$251$$ 27.0000 1.70422 0.852112 0.523359i $$-0.175321\pi$$
0.852112 + 0.523359i $$0.175321\pi$$
$$252$$ − 4.00000i − 0.251976i
$$253$$ − 12.0000i − 0.754434i
$$254$$ −8.00000 −0.501965
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ − 13.0000i − 0.810918i −0.914113 0.405459i $$-0.867112\pi$$
0.914113 0.405459i $$-0.132888\pi$$
$$258$$ 11.0000i 0.684830i
$$259$$ 16.0000 0.994192
$$260$$ 0 0
$$261$$ 2.00000 0.123797
$$262$$ − 12.0000i − 0.741362i
$$263$$ 9.00000i 0.554964i 0.960731 + 0.277482i $$0.0894999\pi$$
−0.960731 + 0.277482i $$0.910500\pi$$
$$264$$ −3.00000 −0.184637
$$265$$ 0 0
$$266$$ 0 0
$$267$$ − 10.0000i − 0.611990i
$$268$$ − 12.0000i − 0.733017i
$$269$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$270$$ 0 0
$$271$$ −13.0000 −0.789694 −0.394847 0.918747i $$-0.629202\pi$$
−0.394847 + 0.918747i $$0.629202\pi$$
$$272$$ − 8.00000i − 0.485071i
$$273$$ − 2.00000i − 0.121046i
$$274$$ 12.0000 0.724947
$$275$$ 0 0
$$276$$ −4.00000 −0.240772
$$277$$ 2.00000i 0.120168i 0.998193 + 0.0600842i $$0.0191369\pi$$
−0.998193 + 0.0600842i $$0.980863\pi$$
$$278$$ − 20.0000i − 1.19952i
$$279$$ −6.00000 −0.359211
$$280$$ 0 0
$$281$$ 27.0000 1.61068 0.805342 0.592810i $$-0.201981\pi$$
0.805342 + 0.592810i $$0.201981\pi$$
$$282$$ 13.0000i 0.774139i
$$283$$ 4.00000i 0.237775i 0.992908 + 0.118888i $$0.0379328\pi$$
−0.992908 + 0.118888i $$0.962067\pi$$
$$284$$ −2.00000 −0.118678
$$285$$ 0 0
$$286$$ 3.00000 0.177394
$$287$$ 4.00000i 0.236113i
$$288$$ − 2.00000i − 0.117851i
$$289$$ −47.0000 −2.76471
$$290$$ 0 0
$$291$$ 2.00000 0.117242
$$292$$ − 4.00000i − 0.234082i
$$293$$ 14.0000i 0.817889i 0.912559 + 0.408944i $$0.134103\pi$$
−0.912559 + 0.408944i $$0.865897\pi$$
$$294$$ −3.00000 −0.174964
$$295$$ 0 0
$$296$$ 8.00000 0.464991
$$297$$ 15.0000i 0.870388i
$$298$$ 15.0000i 0.868927i
$$299$$ 4.00000 0.231326
$$300$$ 0 0
$$301$$ 22.0000 1.26806
$$302$$ − 2.00000i − 0.115087i
$$303$$ 8.00000i 0.459588i
$$304$$ 0 0
$$305$$ 0 0
$$306$$ −16.0000 −0.914659
$$307$$ 7.00000i 0.399511i 0.979846 + 0.199756i $$0.0640148\pi$$
−0.979846 + 0.199756i $$0.935985\pi$$
$$308$$ 6.00000i 0.341882i
$$309$$ 14.0000 0.796432
$$310$$ 0 0
$$311$$ −8.00000 −0.453638 −0.226819 0.973937i $$-0.572833\pi$$
−0.226819 + 0.973937i $$0.572833\pi$$
$$312$$ − 1.00000i − 0.0566139i
$$313$$ 9.00000i 0.508710i 0.967111 + 0.254355i $$0.0818632\pi$$
−0.967111 + 0.254355i $$0.918137\pi$$
$$314$$ −18.0000 −1.01580
$$315$$ 0 0
$$316$$ 15.0000 0.843816
$$317$$ 12.0000i 0.673987i 0.941507 + 0.336994i $$0.109410\pi$$
−0.941507 + 0.336994i $$0.890590\pi$$
$$318$$ 11.0000i 0.616849i
$$319$$ −3.00000 −0.167968
$$320$$ 0 0
$$321$$ 2.00000 0.111629
$$322$$ 8.00000i 0.445823i
$$323$$ 0 0
$$324$$ −1.00000 −0.0555556
$$325$$ 0 0
$$326$$ 9.00000 0.498464
$$327$$ 5.00000i 0.276501i
$$328$$ 2.00000i 0.110432i
$$329$$ 26.0000 1.43343
$$330$$ 0 0
$$331$$ −23.0000 −1.26419 −0.632097 0.774889i $$-0.717806\pi$$
−0.632097 + 0.774889i $$0.717806\pi$$
$$332$$ − 4.00000i − 0.219529i
$$333$$ − 16.0000i − 0.876795i
$$334$$ 2.00000 0.109435
$$335$$ 0 0
$$336$$ 2.00000 0.109109
$$337$$ 32.0000i 1.74315i 0.490261 + 0.871576i $$0.336901\pi$$
−0.490261 + 0.871576i $$0.663099\pi$$
$$338$$ − 12.0000i − 0.652714i
$$339$$ −6.00000 −0.325875
$$340$$ 0 0
$$341$$ 9.00000 0.487377
$$342$$ 0 0
$$343$$ 20.0000i 1.07990i
$$344$$ 11.0000 0.593080
$$345$$ 0 0
$$346$$ −6.00000 −0.322562
$$347$$ 2.00000i 0.107366i 0.998558 + 0.0536828i $$0.0170960\pi$$
−0.998558 + 0.0536828i $$0.982904\pi$$
$$348$$ 1.00000i 0.0536056i
$$349$$ 15.0000 0.802932 0.401466 0.915874i $$-0.368501\pi$$
0.401466 + 0.915874i $$0.368501\pi$$
$$350$$ 0 0
$$351$$ −5.00000 −0.266880
$$352$$ 3.00000i 0.159901i
$$353$$ − 26.0000i − 1.38384i −0.721974 0.691920i $$-0.756765\pi$$
0.721974 0.691920i $$-0.243235\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ −10.0000 −0.529999
$$357$$ − 16.0000i − 0.846810i
$$358$$ − 10.0000i − 0.528516i
$$359$$ 25.0000 1.31945 0.659725 0.751507i $$-0.270673\pi$$
0.659725 + 0.751507i $$0.270673\pi$$
$$360$$ 0 0
$$361$$ −19.0000 −1.00000
$$362$$ − 7.00000i − 0.367912i
$$363$$ 2.00000i 0.104973i
$$364$$ −2.00000 −0.104828
$$365$$ 0 0
$$366$$ 8.00000 0.418167
$$367$$ 32.0000i 1.67039i 0.549957 + 0.835193i $$0.314644\pi$$
−0.549957 + 0.835193i $$0.685356\pi$$
$$368$$ 4.00000i 0.208514i
$$369$$ 4.00000 0.208232
$$370$$ 0 0
$$371$$ 22.0000 1.14218
$$372$$ − 3.00000i − 0.155543i
$$373$$ − 21.0000i − 1.08734i −0.839299 0.543669i $$-0.817035\pi$$
0.839299 0.543669i $$-0.182965\pi$$
$$374$$ 24.0000 1.24101
$$375$$ 0 0
$$376$$ 13.0000 0.670424
$$377$$ − 1.00000i − 0.0515026i
$$378$$ − 10.0000i − 0.514344i
$$379$$ −20.0000 −1.02733 −0.513665 0.857991i $$-0.671713\pi$$
−0.513665 + 0.857991i $$0.671713\pi$$
$$380$$ 0 0
$$381$$ −8.00000 −0.409852
$$382$$ 8.00000i 0.409316i
$$383$$ 14.0000i 0.715367i 0.933843 + 0.357683i $$0.116433\pi$$
−0.933843 + 0.357683i $$0.883567\pi$$
$$384$$ 1.00000 0.0510310
$$385$$ 0 0
$$386$$ 14.0000 0.712581
$$387$$ − 22.0000i − 1.11832i
$$388$$ − 2.00000i − 0.101535i
$$389$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$390$$ 0 0
$$391$$ 32.0000 1.61831
$$392$$ 3.00000i 0.151523i
$$393$$ − 12.0000i − 0.605320i
$$394$$ −18.0000 −0.906827
$$395$$ 0 0
$$396$$ 6.00000 0.301511
$$397$$ 17.0000i 0.853206i 0.904439 + 0.426603i $$0.140290\pi$$
−0.904439 + 0.426603i $$0.859710\pi$$
$$398$$ − 10.0000i − 0.501255i
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 27.0000 1.34832 0.674158 0.738587i $$-0.264507\pi$$
0.674158 + 0.738587i $$0.264507\pi$$
$$402$$ − 12.0000i − 0.598506i
$$403$$ 3.00000i 0.149441i
$$404$$ 8.00000 0.398015
$$405$$ 0 0
$$406$$ 2.00000 0.0992583
$$407$$ 24.0000i 1.18964i
$$408$$ − 8.00000i − 0.396059i
$$409$$ −30.0000 −1.48340 −0.741702 0.670729i $$-0.765981\pi$$
−0.741702 + 0.670729i $$0.765981\pi$$
$$410$$ 0 0
$$411$$ 12.0000 0.591916
$$412$$ − 14.0000i − 0.689730i
$$413$$ 0 0
$$414$$ 8.00000 0.393179
$$415$$ 0 0
$$416$$ −1.00000 −0.0490290
$$417$$ − 20.0000i − 0.979404i
$$418$$ 0 0
$$419$$ 10.0000 0.488532 0.244266 0.969708i $$-0.421453\pi$$
0.244266 + 0.969708i $$0.421453\pi$$
$$420$$ 0 0
$$421$$ 32.0000 1.55958 0.779792 0.626038i $$-0.215325\pi$$
0.779792 + 0.626038i $$0.215325\pi$$
$$422$$ 3.00000i 0.146038i
$$423$$ − 26.0000i − 1.26416i
$$424$$ 11.0000 0.534207
$$425$$ 0 0
$$426$$ −2.00000 −0.0969003
$$427$$ − 16.0000i − 0.774294i
$$428$$ − 2.00000i − 0.0966736i
$$429$$ 3.00000 0.144841
$$430$$ 0 0
$$431$$ 32.0000 1.54139 0.770693 0.637207i $$-0.219910\pi$$
0.770693 + 0.637207i $$0.219910\pi$$
$$432$$ − 5.00000i − 0.240563i
$$433$$ − 16.0000i − 0.768911i −0.923144 0.384455i $$-0.874389\pi$$
0.923144 0.384455i $$-0.125611\pi$$
$$434$$ −6.00000 −0.288009
$$435$$ 0 0
$$436$$ 5.00000 0.239457
$$437$$ 0 0
$$438$$ − 4.00000i − 0.191127i
$$439$$ −20.0000 −0.954548 −0.477274 0.878755i $$-0.658375\pi$$
−0.477274 + 0.878755i $$0.658375\pi$$
$$440$$ 0 0
$$441$$ 6.00000 0.285714
$$442$$ 8.00000i 0.380521i
$$443$$ 4.00000i 0.190046i 0.995475 + 0.0950229i $$0.0302924\pi$$
−0.995475 + 0.0950229i $$0.969708\pi$$
$$444$$ 8.00000 0.379663
$$445$$ 0 0
$$446$$ −26.0000 −1.23114
$$447$$ 15.0000i 0.709476i
$$448$$ − 2.00000i − 0.0944911i
$$449$$ 10.0000 0.471929 0.235965 0.971762i $$-0.424175\pi$$
0.235965 + 0.971762i $$0.424175\pi$$
$$450$$ 0 0
$$451$$ −6.00000 −0.282529
$$452$$ 6.00000i 0.282216i
$$453$$ − 2.00000i − 0.0939682i
$$454$$ −18.0000 −0.844782
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 2.00000i 0.0935561i 0.998905 + 0.0467780i $$0.0148953\pi$$
−0.998905 + 0.0467780i $$0.985105\pi$$
$$458$$ 10.0000i 0.467269i
$$459$$ −40.0000 −1.86704
$$460$$ 0 0
$$461$$ 2.00000 0.0931493 0.0465746 0.998915i $$-0.485169\pi$$
0.0465746 + 0.998915i $$0.485169\pi$$
$$462$$ 6.00000i 0.279145i
$$463$$ 4.00000i 0.185896i 0.995671 + 0.0929479i $$0.0296290\pi$$
−0.995671 + 0.0929479i $$0.970371\pi$$
$$464$$ 1.00000 0.0464238
$$465$$ 0 0
$$466$$ −1.00000 −0.0463241
$$467$$ 27.0000i 1.24941i 0.780860 + 0.624705i $$0.214781\pi$$
−0.780860 + 0.624705i $$0.785219\pi$$
$$468$$ 2.00000i 0.0924500i
$$469$$ −24.0000 −1.10822
$$470$$ 0 0
$$471$$ −18.0000 −0.829396
$$472$$ 0 0
$$473$$ 33.0000i 1.51734i
$$474$$ 15.0000 0.688973
$$475$$ 0 0
$$476$$ −16.0000 −0.733359
$$477$$ − 22.0000i − 1.00731i
$$478$$ 0 0
$$479$$ 5.00000 0.228456 0.114228 0.993455i $$-0.463561\pi$$
0.114228 + 0.993455i $$0.463561\pi$$
$$480$$ 0 0
$$481$$ −8.00000 −0.364769
$$482$$ − 17.0000i − 0.774329i
$$483$$ 8.00000i 0.364013i
$$484$$ 2.00000 0.0909091
$$485$$ 0 0
$$486$$ −16.0000 −0.725775
$$487$$ 22.0000i 0.996915i 0.866914 + 0.498458i $$0.166100\pi$$
−0.866914 + 0.498458i $$0.833900\pi$$
$$488$$ − 8.00000i − 0.362143i
$$489$$ 9.00000 0.406994
$$490$$ 0 0
$$491$$ −33.0000 −1.48927 −0.744635 0.667472i $$-0.767376\pi$$
−0.744635 + 0.667472i $$0.767376\pi$$
$$492$$ 2.00000i 0.0901670i
$$493$$ − 8.00000i − 0.360302i
$$494$$ 0 0
$$495$$ 0 0
$$496$$ −3.00000 −0.134704
$$497$$ 4.00000i 0.179425i
$$498$$ − 4.00000i − 0.179244i
$$499$$ 20.0000 0.895323 0.447661 0.894203i $$-0.352257\pi$$
0.447661 + 0.894203i $$0.352257\pi$$
$$500$$ 0 0
$$501$$ 2.00000 0.0893534
$$502$$ − 27.0000i − 1.20507i
$$503$$ 19.0000i 0.847168i 0.905857 + 0.423584i $$0.139228\pi$$
−0.905857 + 0.423584i $$0.860772\pi$$
$$504$$ −4.00000 −0.178174
$$505$$ 0 0
$$506$$ −12.0000 −0.533465
$$507$$ − 12.0000i − 0.532939i
$$508$$ 8.00000i 0.354943i
$$509$$ 15.0000 0.664863 0.332432 0.943127i $$-0.392131\pi$$
0.332432 + 0.943127i $$0.392131\pi$$
$$510$$ 0 0
$$511$$ −8.00000 −0.353899
$$512$$ − 1.00000i − 0.0441942i
$$513$$ 0 0
$$514$$ −13.0000 −0.573405
$$515$$ 0 0
$$516$$ 11.0000 0.484248
$$517$$ 39.0000i 1.71522i
$$518$$ − 16.0000i − 0.703000i
$$519$$ −6.00000 −0.263371
$$520$$ 0 0
$$521$$ −13.0000 −0.569540 −0.284770 0.958596i $$-0.591917\pi$$
−0.284770 + 0.958596i $$0.591917\pi$$
$$522$$ − 2.00000i − 0.0875376i
$$523$$ 24.0000i 1.04945i 0.851273 + 0.524723i $$0.175831\pi$$
−0.851273 + 0.524723i $$0.824169\pi$$
$$524$$ −12.0000 −0.524222
$$525$$ 0 0
$$526$$ 9.00000 0.392419
$$527$$ 24.0000i 1.04546i
$$528$$ 3.00000i 0.130558i
$$529$$ 7.00000 0.304348
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 0 0
$$533$$ − 2.00000i − 0.0866296i
$$534$$ −10.0000 −0.432742
$$535$$ 0 0
$$536$$ −12.0000 −0.518321
$$537$$ − 10.0000i − 0.431532i
$$538$$ 0 0
$$539$$ −9.00000 −0.387657
$$540$$ 0 0
$$541$$ −8.00000 −0.343947 −0.171973 0.985102i $$-0.555014\pi$$
−0.171973 + 0.985102i $$0.555014\pi$$
$$542$$ 13.0000i 0.558398i
$$543$$ − 7.00000i − 0.300399i
$$544$$ −8.00000 −0.342997
$$545$$ 0 0
$$546$$ −2.00000 −0.0855921
$$547$$ − 38.0000i − 1.62476i −0.583127 0.812381i $$-0.698171\pi$$
0.583127 0.812381i $$-0.301829\pi$$
$$548$$ − 12.0000i − 0.512615i
$$549$$ −16.0000 −0.682863
$$550$$ 0 0
$$551$$ 0 0
$$552$$ 4.00000i 0.170251i
$$553$$ − 30.0000i − 1.27573i
$$554$$ 2.00000 0.0849719
$$555$$ 0 0
$$556$$ −20.0000 −0.848189
$$557$$ 2.00000i 0.0847427i 0.999102 + 0.0423714i $$0.0134913\pi$$
−0.999102 + 0.0423714i $$0.986509\pi$$
$$558$$ 6.00000i 0.254000i
$$559$$ −11.0000 −0.465250
$$560$$ 0 0
$$561$$ 24.0000 1.01328
$$562$$ − 27.0000i − 1.13893i
$$563$$ − 11.0000i − 0.463595i −0.972764 0.231797i $$-0.925539\pi$$
0.972764 0.231797i $$-0.0744606\pi$$
$$564$$ 13.0000 0.547399
$$565$$ 0 0
$$566$$ 4.00000 0.168133
$$567$$ 2.00000i 0.0839921i
$$568$$ 2.00000i 0.0839181i
$$569$$ 30.0000 1.25767 0.628833 0.777541i $$-0.283533\pi$$
0.628833 + 0.777541i $$0.283533\pi$$
$$570$$ 0 0
$$571$$ −28.0000 −1.17176 −0.585882 0.810397i $$-0.699252\pi$$
−0.585882 + 0.810397i $$0.699252\pi$$
$$572$$ − 3.00000i − 0.125436i
$$573$$ 8.00000i 0.334205i
$$574$$ 4.00000 0.166957
$$575$$ 0 0
$$576$$ −2.00000 −0.0833333
$$577$$ − 8.00000i − 0.333044i −0.986038 0.166522i $$-0.946746\pi$$
0.986038 0.166522i $$-0.0532537\pi$$
$$578$$ 47.0000i 1.95494i
$$579$$ 14.0000 0.581820
$$580$$ 0 0
$$581$$ −8.00000 −0.331896
$$582$$ − 2.00000i − 0.0829027i
$$583$$ 33.0000i 1.36672i
$$584$$ −4.00000 −0.165521
$$585$$ 0 0
$$586$$ 14.0000 0.578335
$$587$$ − 28.0000i − 1.15568i −0.816149 0.577842i $$-0.803895\pi$$
0.816149 0.577842i $$-0.196105\pi$$
$$588$$ 3.00000i 0.123718i
$$589$$ 0 0
$$590$$ 0 0
$$591$$ −18.0000 −0.740421
$$592$$ − 8.00000i − 0.328798i
$$593$$ 39.0000i 1.60154i 0.598973 + 0.800769i $$0.295576\pi$$
−0.598973 + 0.800769i $$0.704424\pi$$
$$594$$ 15.0000 0.615457
$$595$$ 0 0
$$596$$ 15.0000 0.614424
$$597$$ − 10.0000i − 0.409273i
$$598$$ − 4.00000i − 0.163572i
$$599$$ 5.00000 0.204294 0.102147 0.994769i $$-0.467429\pi$$
0.102147 + 0.994769i $$0.467429\pi$$
$$600$$ 0 0
$$601$$ 2.00000 0.0815817 0.0407909 0.999168i $$-0.487012\pi$$
0.0407909 + 0.999168i $$0.487012\pi$$
$$602$$ − 22.0000i − 0.896653i
$$603$$ 24.0000i 0.977356i
$$604$$ −2.00000 −0.0813788
$$605$$ 0 0
$$606$$ 8.00000 0.324978
$$607$$ − 3.00000i − 0.121766i −0.998145 0.0608831i $$-0.980608\pi$$
0.998145 0.0608831i $$-0.0193917\pi$$
$$608$$ 0 0
$$609$$ 2.00000 0.0810441
$$610$$ 0 0
$$611$$ −13.0000 −0.525924
$$612$$ 16.0000i 0.646762i
$$613$$ − 31.0000i − 1.25208i −0.779792 0.626039i $$-0.784675\pi$$
0.779792 0.626039i $$-0.215325\pi$$
$$614$$ 7.00000 0.282497
$$615$$ 0 0
$$616$$ 6.00000 0.241747
$$617$$ 12.0000i 0.483102i 0.970388 + 0.241551i $$0.0776561\pi$$
−0.970388 + 0.241551i $$0.922344\pi$$
$$618$$ − 14.0000i − 0.563163i
$$619$$ 35.0000 1.40677 0.703384 0.710810i $$-0.251671\pi$$
0.703384 + 0.710810i $$0.251671\pi$$
$$620$$ 0 0
$$621$$ 20.0000 0.802572
$$622$$ 8.00000i 0.320771i
$$623$$ 20.0000i 0.801283i
$$624$$ −1.00000 −0.0400320
$$625$$ 0 0
$$626$$ 9.00000 0.359712
$$627$$ 0 0
$$628$$ 18.0000i 0.718278i
$$629$$ −64.0000 −2.55185
$$630$$ 0 0
$$631$$ −38.0000 −1.51276 −0.756378 0.654135i $$-0.773033\pi$$
−0.756378 + 0.654135i $$0.773033\pi$$
$$632$$ − 15.0000i − 0.596668i
$$633$$ 3.00000i 0.119239i
$$634$$ 12.0000 0.476581
$$635$$ 0 0
$$636$$ 11.0000 0.436178
$$637$$ − 3.00000i − 0.118864i
$$638$$ 3.00000i 0.118771i
$$639$$ 4.00000 0.158238
$$640$$ 0 0
$$641$$ −8.00000 −0.315981 −0.157991 0.987441i $$-0.550502\pi$$
−0.157991 + 0.987441i $$0.550502\pi$$
$$642$$ − 2.00000i − 0.0789337i
$$643$$ 34.0000i 1.34083i 0.741987 + 0.670415i $$0.233884\pi$$
−0.741987 + 0.670415i $$0.766116\pi$$
$$644$$ 8.00000 0.315244
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 32.0000i 1.25805i 0.777385 + 0.629025i $$0.216546\pi$$
−0.777385 + 0.629025i $$0.783454\pi$$
$$648$$ 1.00000i 0.0392837i
$$649$$ 0 0
$$650$$ 0 0
$$651$$ −6.00000 −0.235159
$$652$$ − 9.00000i − 0.352467i
$$653$$ − 26.0000i − 1.01746i −0.860927 0.508729i $$-0.830115\pi$$
0.860927 0.508729i $$-0.169885\pi$$
$$654$$ 5.00000 0.195515
$$655$$ 0 0
$$656$$ 2.00000 0.0780869
$$657$$ 8.00000i 0.312110i
$$658$$ − 26.0000i − 1.01359i
$$659$$ −15.0000 −0.584317 −0.292159 0.956370i $$-0.594373\pi$$
−0.292159 + 0.956370i $$0.594373\pi$$
$$660$$ 0 0
$$661$$ −18.0000 −0.700119 −0.350059 0.936727i $$-0.613839\pi$$
−0.350059 + 0.936727i $$0.613839\pi$$
$$662$$ 23.0000i 0.893920i
$$663$$ 8.00000i 0.310694i
$$664$$ −4.00000 −0.155230
$$665$$ 0 0
$$666$$ −16.0000 −0.619987
$$667$$ 4.00000i 0.154881i
$$668$$ − 2.00000i − 0.0773823i
$$669$$ −26.0000 −1.00522
$$670$$ 0 0
$$671$$ 24.0000 0.926510
$$672$$ − 2.00000i − 0.0771517i
$$673$$ 9.00000i 0.346925i 0.984841 + 0.173462i $$0.0554955\pi$$
−0.984841 + 0.173462i $$0.944505\pi$$
$$674$$ 32.0000 1.23259
$$675$$ 0 0
$$676$$ −12.0000 −0.461538
$$677$$ − 38.0000i − 1.46046i −0.683202 0.730229i $$-0.739413\pi$$
0.683202 0.730229i $$-0.260587\pi$$
$$678$$ 6.00000i 0.230429i
$$679$$ −4.00000 −0.153506
$$680$$ 0 0
$$681$$ −18.0000 −0.689761
$$682$$ − 9.00000i − 0.344628i
$$683$$ 24.0000i 0.918334i 0.888350 + 0.459167i $$0.151852\pi$$
−0.888350 + 0.459167i $$0.848148\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 20.0000 0.763604
$$687$$ 10.0000i 0.381524i
$$688$$ − 11.0000i − 0.419371i
$$689$$ −11.0000 −0.419067
$$690$$ 0 0
$$691$$ −28.0000 −1.06517 −0.532585 0.846376i $$-0.678779\pi$$
−0.532585 + 0.846376i $$0.678779\pi$$
$$692$$ 6.00000i 0.228086i
$$693$$ − 12.0000i − 0.455842i
$$694$$ 2.00000 0.0759190
$$695$$ 0 0
$$696$$ 1.00000 0.0379049
$$697$$ − 16.0000i − 0.606043i
$$698$$ − 15.0000i − 0.567758i
$$699$$ −1.00000 −0.0378235
$$700$$ 0 0
$$701$$ 27.0000 1.01978 0.509888 0.860241i $$-0.329687\pi$$
0.509888 + 0.860241i $$0.329687\pi$$
$$702$$ 5.00000i 0.188713i
$$703$$ 0 0
$$704$$ 3.00000 0.113067
$$705$$ 0 0
$$706$$ −26.0000 −0.978523
$$707$$ − 16.0000i − 0.601742i
$$708$$ 0 0
$$709$$ −15.0000 −0.563337 −0.281668 0.959512i $$-0.590888\pi$$
−0.281668 + 0.959512i $$0.590888\pi$$
$$710$$ 0 0
$$711$$ −30.0000 −1.12509
$$712$$ 10.0000i 0.374766i
$$713$$ − 12.0000i − 0.449404i
$$714$$ −16.0000 −0.598785
$$715$$ 0 0
$$716$$ −10.0000 −0.373718
$$717$$ 0 0
$$718$$ − 25.0000i − 0.932992i
$$719$$ 50.0000 1.86469 0.932343 0.361576i $$-0.117761\pi$$
0.932343 + 0.361576i $$0.117761\pi$$
$$720$$ 0 0
$$721$$ −28.0000 −1.04277
$$722$$ 19.0000i 0.707107i
$$723$$ − 17.0000i − 0.632237i
$$724$$ −7.00000 −0.260153
$$725$$ 0 0
$$726$$ 2.00000 0.0742270
$$727$$ − 8.00000i − 0.296704i −0.988935 0.148352i $$-0.952603\pi$$
0.988935 0.148352i $$-0.0473968\pi$$
$$728$$ 2.00000i 0.0741249i
$$729$$ −13.0000 −0.481481
$$730$$ 0 0
$$731$$ −88.0000 −3.25480
$$732$$ − 8.00000i − 0.295689i
$$733$$ 24.0000i 0.886460i 0.896408 + 0.443230i $$0.146168\pi$$
−0.896408 + 0.443230i $$0.853832\pi$$
$$734$$ 32.0000 1.18114
$$735$$ 0 0
$$736$$ 4.00000 0.147442
$$737$$ − 36.0000i − 1.32608i
$$738$$ − 4.00000i − 0.147242i
$$739$$ 5.00000 0.183928 0.0919640 0.995762i $$-0.470686\pi$$
0.0919640 + 0.995762i $$0.470686\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ − 22.0000i − 0.807645i
$$743$$ 44.0000i 1.61420i 0.590412 + 0.807102i $$0.298965\pi$$
−0.590412 + 0.807102i $$0.701035\pi$$
$$744$$ −3.00000 −0.109985
$$745$$ 0 0
$$746$$ −21.0000 −0.768865
$$747$$ 8.00000i 0.292705i
$$748$$ − 24.0000i − 0.877527i
$$749$$ −4.00000 −0.146157
$$750$$ 0 0
$$751$$ 32.0000 1.16770 0.583848 0.811863i $$-0.301546\pi$$
0.583848 + 0.811863i $$0.301546\pi$$
$$752$$ − 13.0000i − 0.474061i
$$753$$ − 27.0000i − 0.983935i
$$754$$ −1.00000 −0.0364179
$$755$$ 0 0
$$756$$ −10.0000 −0.363696
$$757$$ − 8.00000i − 0.290765i −0.989376 0.145382i $$-0.953559\pi$$
0.989376 0.145382i $$-0.0464413\pi$$
$$758$$ 20.0000i 0.726433i
$$759$$ −12.0000 −0.435572
$$760$$ 0 0
$$761$$ 42.0000 1.52250 0.761249 0.648459i $$-0.224586\pi$$
0.761249 + 0.648459i $$0.224586\pi$$
$$762$$ 8.00000i 0.289809i
$$763$$ − 10.0000i − 0.362024i
$$764$$ 8.00000 0.289430
$$765$$ 0 0
$$766$$ 14.0000 0.505841
$$767$$ 0 0
$$768$$ − 1.00000i − 0.0360844i
$$769$$ 20.0000 0.721218 0.360609 0.932717i $$-0.382569\pi$$
0.360609 + 0.932717i $$0.382569\pi$$
$$770$$ 0 0
$$771$$ −13.0000 −0.468184
$$772$$ − 14.0000i − 0.503871i
$$773$$ 14.0000i 0.503545i 0.967786 + 0.251773i $$0.0810135\pi$$
−0.967786 + 0.251773i $$0.918987\pi$$
$$774$$ −22.0000 −0.790774
$$775$$ 0 0
$$776$$ −2.00000 −0.0717958
$$777$$ − 16.0000i − 0.573997i
$$778$$ 0 0
$$779$$ 0 0
$$780$$ 0 0
$$781$$ −6.00000 −0.214697
$$782$$ − 32.0000i − 1.14432i
$$783$$ − 5.00000i − 0.178685i
$$784$$ 3.00000 0.107143
$$785$$ 0 0
$$786$$ −12.0000 −0.428026
$$787$$ 22.0000i 0.784215i 0.919919 + 0.392108i $$0.128254\pi$$
−0.919919 + 0.392108i $$0.871746\pi$$
$$788$$ 18.0000i 0.641223i
$$789$$ 9.00000 0.320408
$$790$$ 0 0
$$791$$ 12.0000 0.426671
$$792$$ − 6.00000i − 0.213201i
$$793$$ 8.00000i 0.284088i
$$794$$ 17.0000 0.603307
$$795$$ 0 0
$$796$$ −10.0000 −0.354441
$$797$$ 32.0000i 1.13350i 0.823890 + 0.566749i $$0.191799\pi$$
−0.823890 + 0.566749i $$0.808201\pi$$
$$798$$ 0 0
$$799$$ −104.000 −3.67926
$$800$$ 0 0
$$801$$ 20.0000 0.706665
$$802$$ − 27.0000i − 0.953403i
$$803$$ − 12.0000i − 0.423471i
$$804$$ −12.0000 −0.423207
$$805$$ 0 0
$$806$$ 3.00000 0.105670
$$807$$ 0 0
$$808$$ − 8.00000i − 0.281439i
$$809$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$810$$ 0 0
$$811$$ −18.0000 −0.632065 −0.316033 0.948748i $$-0.602351\pi$$
−0.316033 + 0.948748i $$0.602351\pi$$
$$812$$ − 2.00000i − 0.0701862i
$$813$$ 13.0000i 0.455930i
$$814$$ 24.0000 0.841200
$$815$$ 0 0
$$816$$ −8.00000 −0.280056
$$817$$ 0 0
$$818$$ 30.0000i 1.04893i
$$819$$ 4.00000 0.139771
$$820$$ 0 0
$$821$$ −33.0000 −1.15171 −0.575854 0.817553i $$-0.695330\pi$$
−0.575854 + 0.817553i $$0.695330\pi$$
$$822$$ − 12.0000i − 0.418548i
$$823$$ − 16.0000i − 0.557725i −0.960331 0.278862i $$-0.910043\pi$$
0.960331 0.278862i $$-0.0899574\pi$$
$$824$$ −14.0000 −0.487713
$$825$$ 0 0
$$826$$ 0 0
$$827$$ − 13.0000i − 0.452054i −0.974121 0.226027i $$-0.927426\pi$$
0.974121 0.226027i $$-0.0725738\pi$$
$$828$$ − 8.00000i − 0.278019i
$$829$$ −40.0000 −1.38926 −0.694629 0.719368i $$-0.744431\pi$$
−0.694629 + 0.719368i $$0.744431\pi$$
$$830$$ 0 0
$$831$$ 2.00000 0.0693792
$$832$$ 1.00000i 0.0346688i
$$833$$ − 24.0000i − 0.831551i
$$834$$ −20.0000 −0.692543
$$835$$ 0 0
$$836$$ 0 0
$$837$$ 15.0000i 0.518476i
$$838$$ − 10.0000i − 0.345444i
$$839$$ −45.0000 −1.55357 −0.776786 0.629764i $$-0.783151\pi$$
−0.776786 + 0.629764i $$0.783151\pi$$
$$840$$ 0 0
$$841$$ 1.00000 0.0344828
$$842$$ − 32.0000i − 1.10279i
$$843$$ − 27.0000i − 0.929929i
$$844$$ 3.00000 0.103264
$$845$$ 0 0
$$846$$ −26.0000 −0.893898
$$847$$ − 4.00000i − 0.137442i
$$848$$ − 11.0000i − 0.377742i
$$849$$ 4.00000 0.137280
$$850$$ 0 0
$$851$$ 32.0000 1.09695
$$852$$ 2.00000i 0.0685189i
$$853$$ 14.0000i 0.479351i 0.970853 + 0.239675i $$0.0770410\pi$$
−0.970853 + 0.239675i $$0.922959\pi$$
$$854$$ −16.0000 −0.547509
$$855$$ 0 0
$$856$$ −2.00000 −0.0683586
$$857$$ 27.0000i 0.922302i 0.887322 + 0.461151i $$0.152563\pi$$
−0.887322 + 0.461151i $$0.847437\pi$$
$$858$$ − 3.00000i − 0.102418i
$$859$$ 25.0000 0.852989 0.426494 0.904490i $$-0.359748\pi$$
0.426494 + 0.904490i $$0.359748\pi$$
$$860$$ 0 0
$$861$$ 4.00000 0.136320
$$862$$ − 32.0000i − 1.08992i
$$863$$ − 46.0000i − 1.56586i −0.622111 0.782929i $$-0.713725\pi$$
0.622111 0.782929i $$-0.286275\pi$$
$$864$$ −5.00000 −0.170103
$$865$$ 0 0
$$866$$ −16.0000 −0.543702
$$867$$ 47.0000i 1.59620i
$$868$$ 6.00000i 0.203653i
$$869$$ 45.0000 1.52652
$$870$$ 0 0
$$871$$ 12.0000 0.406604
$$872$$ − 5.00000i − 0.169321i
$$873$$ 4.00000i 0.135379i
$$874$$ 0 0
$$875$$ 0 0
$$876$$ −4.00000 −0.135147
$$877$$ − 13.0000i − 0.438979i −0.975615 0.219489i $$-0.929561\pi$$
0.975615 0.219489i $$-0.0704391\pi$$
$$878$$ 20.0000i 0.674967i
$$879$$ 14.0000 0.472208
$$880$$ 0 0
$$881$$ 42.0000 1.41502 0.707508 0.706705i $$-0.249819\pi$$
0.707508 + 0.706705i $$0.249819\pi$$
$$882$$ − 6.00000i − 0.202031i
$$883$$ − 26.0000i − 0.874970i −0.899226 0.437485i $$-0.855869\pi$$
0.899226 0.437485i $$-0.144131\pi$$
$$884$$ 8.00000 0.269069
$$885$$ 0 0
$$886$$ 4.00000 0.134383
$$887$$ − 33.0000i − 1.10803i −0.832506 0.554016i $$-0.813095\pi$$
0.832506 0.554016i $$-0.186905\pi$$
$$888$$ − 8.00000i − 0.268462i
$$889$$ 16.0000 0.536623
$$890$$ 0 0
$$891$$ −3.00000 −0.100504
$$892$$ 26.0000i 0.870544i
$$893$$ 0 0
$$894$$ 15.0000 0.501675
$$895$$ 0 0
$$896$$ −2.00000 −0.0668153
$$897$$ − 4.00000i − 0.133556i
$$898$$ − 10.0000i − 0.333704i
$$899$$ −3.00000 −0.100056
$$900$$ 0 0
$$901$$ −88.0000 −2.93171
$$902$$ 6.00000i 0.199778i
$$903$$ − 22.0000i − 0.732114i
$$904$$ 6.00000 0.199557
$$905$$ 0 0
$$906$$ −2.00000 −0.0664455
$$907$$ − 28.0000i − 0.929725i −0.885383 0.464862i $$-0.846104\pi$$
0.885383 0.464862i $$-0.153896\pi$$
$$908$$ 18.0000i 0.597351i
$$909$$ −16.0000 −0.530687
$$910$$ 0 0
$$911$$ −13.0000 −0.430709 −0.215355 0.976536i $$-0.569091\pi$$
−0.215355 + 0.976536i $$0.569091\pi$$
$$912$$ 0 0
$$913$$ − 12.0000i − 0.397142i
$$914$$ 2.00000 0.0661541
$$915$$ 0 0
$$916$$ 10.0000 0.330409
$$917$$ 24.0000i 0.792550i
$$918$$ 40.0000i 1.32020i
$$919$$ −30.0000 −0.989609 −0.494804 0.869004i $$-0.664760\pi$$
−0.494804 + 0.869004i $$0.664760\pi$$
$$920$$ 0 0
$$921$$ 7.00000 0.230658
$$922$$ − 2.00000i − 0.0658665i
$$923$$ − 2.00000i − 0.0658308i
$$924$$ 6.00000 0.197386
$$925$$ 0 0
$$926$$ 4.00000 0.131448
$$927$$ 28.0000i 0.919641i
$$928$$ − 1.00000i − 0.0328266i
$$929$$ 30.0000 0.984268 0.492134 0.870519i $$-0.336217\pi$$
0.492134 + 0.870519i $$0.336217\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ 1.00000i 0.0327561i
$$933$$ 8.00000i 0.261908i
$$934$$ 27.0000 0.883467
$$935$$ 0 0
$$936$$ 2.00000 0.0653720
$$937$$ 2.00000i 0.0653372i 0.999466 + 0.0326686i $$0.0104006\pi$$
−0.999466 + 0.0326686i $$0.989599\pi$$
$$938$$ 24.0000i 0.783628i
$$939$$ 9.00000 0.293704
$$940$$ 0 0
$$941$$ 37.0000 1.20617 0.603083 0.797679i $$-0.293939\pi$$
0.603083 + 0.797679i $$0.293939\pi$$
$$942$$ 18.0000i 0.586472i
$$943$$ 8.00000i 0.260516i
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 33.0000 1.07292
$$947$$ − 33.0000i − 1.07236i −0.844105 0.536178i $$-0.819868\pi$$
0.844105 0.536178i $$-0.180132\pi$$
$$948$$ − 15.0000i − 0.487177i
$$949$$ 4.00000 0.129845
$$950$$ 0 0
$$951$$ 12.0000 0.389127
$$952$$ 16.0000i 0.518563i
$$953$$ − 1.00000i − 0.0323932i −0.999869 0.0161966i $$-0.994844\pi$$
0.999869 0.0161966i $$-0.00515576\pi$$
$$954$$ −22.0000 −0.712276
$$955$$ 0 0
$$956$$ 0 0
$$957$$ 3.00000i 0.0969762i
$$958$$ − 5.00000i − 0.161543i
$$959$$ −24.0000 −0.775000
$$960$$ 0 0
$$961$$ −22.0000 −0.709677
$$962$$ 8.00000i 0.257930i
$$963$$ 4.00000i 0.128898i
$$964$$ −17.0000 −0.547533
$$965$$ 0 0
$$966$$ 8.00000 0.257396
$$967$$ − 13.0000i − 0.418052i −0.977910 0.209026i $$-0.932971\pi$$
0.977910 0.209026i $$-0.0670293\pi$$
$$968$$ − 2.00000i − 0.0642824i
$$969$$ 0 0
$$970$$ 0 0
$$971$$ −28.0000 −0.898563 −0.449281 0.893390i $$-0.648320\pi$$
−0.449281 + 0.893390i $$0.648320\pi$$
$$972$$ 16.0000i 0.513200i
$$973$$ 40.0000i 1.28234i
$$974$$ 22.0000 0.704925
$$975$$ 0 0
$$976$$ −8.00000 −0.256074
$$977$$ − 13.0000i − 0.415907i −0.978139 0.207953i $$-0.933320\pi$$
0.978139 0.207953i $$-0.0666802\pi$$
$$978$$ − 9.00000i − 0.287788i
$$979$$ −30.0000 −0.958804
$$980$$ 0 0
$$981$$ −10.0000 −0.319275
$$982$$ 33.0000i 1.05307i
$$983$$ 49.0000i 1.56286i 0.623995 + 0.781429i $$0.285509\pi$$
−0.623995 + 0.781429i $$0.714491\pi$$
$$984$$ 2.00000 0.0637577
$$985$$ 0 0
$$986$$ −8.00000 −0.254772
$$987$$ − 26.0000i − 0.827589i
$$988$$ 0 0
$$989$$ 44.0000 1.39912
$$990$$ 0 0
$$991$$ 22.0000 0.698853 0.349427 0.936964i $$-0.386376\pi$$
0.349427 + 0.936964i $$0.386376\pi$$
$$992$$ 3.00000i 0.0952501i
$$993$$ 23.0000i 0.729883i
$$994$$ 4.00000 0.126872
$$995$$ 0 0
$$996$$ −4.00000 −0.126745
$$997$$ − 8.00000i − 0.253363i −0.991943 0.126681i $$-0.959567\pi$$
0.991943 0.126681i $$-0.0404325\pi$$
$$998$$ − 20.0000i − 0.633089i
$$999$$ −40.0000 −1.26554
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 1450.2.b.b.349.1 2
5.2 odd 4 58.2.a.b.1.1 1
5.3 odd 4 1450.2.a.c.1.1 1
5.4 even 2 inner 1450.2.b.b.349.2 2
15.2 even 4 522.2.a.b.1.1 1
20.7 even 4 464.2.a.e.1.1 1
35.27 even 4 2842.2.a.e.1.1 1
40.27 even 4 1856.2.a.f.1.1 1
40.37 odd 4 1856.2.a.k.1.1 1
55.32 even 4 7018.2.a.a.1.1 1
60.47 odd 4 4176.2.a.n.1.1 1
65.12 odd 4 9802.2.a.a.1.1 1
145.12 even 4 1682.2.b.a.1681.2 2
145.17 even 4 1682.2.b.a.1681.1 2
145.57 odd 4 1682.2.a.d.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
58.2.a.b.1.1 1 5.2 odd 4
464.2.a.e.1.1 1 20.7 even 4
522.2.a.b.1.1 1 15.2 even 4
1450.2.a.c.1.1 1 5.3 odd 4
1450.2.b.b.349.1 2 1.1 even 1 trivial
1450.2.b.b.349.2 2 5.4 even 2 inner
1682.2.a.d.1.1 1 145.57 odd 4
1682.2.b.a.1681.1 2 145.17 even 4
1682.2.b.a.1681.2 2 145.12 even 4
1856.2.a.f.1.1 1 40.27 even 4
1856.2.a.k.1.1 1 40.37 odd 4
2842.2.a.e.1.1 1 35.27 even 4
4176.2.a.n.1.1 1 60.47 odd 4
7018.2.a.a.1.1 1 55.32 even 4
9802.2.a.a.1.1 1 65.12 odd 4