# Properties

 Label 2550.2.d.n.2449.2 Level $2550$ Weight $2$ Character 2550.2449 Analytic conductor $20.362$ 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] = [2550,2,Mod(2449,2550)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2550, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("2550.2449");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2550 = 2 \cdot 3 \cdot 5^{2} \cdot 17$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2550.d (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: $$20.3618525154$$ 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 510) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 2449.2 Root $$-1.00000i$$ of defining polynomial Character $$\chi$$ $$=$$ 2550.2449 Dual form 2550.2.d.n.2449.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} -2.00000i q^{7} -1.00000i q^{8} -1.00000 q^{9} +O(q^{10})$$ $$q+1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} -2.00000i q^{7} -1.00000i q^{8} -1.00000 q^{9} -4.00000 q^{11} +1.00000i q^{12} +4.00000i q^{13} +2.00000 q^{14} +1.00000 q^{16} -1.00000i q^{17} -1.00000i q^{18} +4.00000 q^{19} -2.00000 q^{21} -4.00000i q^{22} +8.00000i q^{23} -1.00000 q^{24} -4.00000 q^{26} +1.00000i q^{27} +2.00000i q^{28} -2.00000 q^{29} +4.00000 q^{31} +1.00000i q^{32} +4.00000i q^{33} +1.00000 q^{34} +1.00000 q^{36} -6.00000i q^{37} +4.00000i q^{38} +4.00000 q^{39} +8.00000 q^{41} -2.00000i q^{42} +6.00000i q^{43} +4.00000 q^{44} -8.00000 q^{46} -8.00000i q^{47} -1.00000i q^{48} +3.00000 q^{49} -1.00000 q^{51} -4.00000i q^{52} -2.00000i q^{53} -1.00000 q^{54} -2.00000 q^{56} -4.00000i q^{57} -2.00000i q^{58} +6.00000 q^{59} +14.0000 q^{61} +4.00000i q^{62} +2.00000i q^{63} -1.00000 q^{64} -4.00000 q^{66} +2.00000i q^{67} +1.00000i q^{68} +8.00000 q^{69} +2.00000 q^{71} +1.00000i q^{72} +4.00000i q^{73} +6.00000 q^{74} -4.00000 q^{76} +8.00000i q^{77} +4.00000i q^{78} +1.00000 q^{81} +8.00000i q^{82} -16.0000i q^{83} +2.00000 q^{84} -6.00000 q^{86} +2.00000i q^{87} +4.00000i q^{88} +2.00000 q^{89} +8.00000 q^{91} -8.00000i q^{92} -4.00000i q^{93} +8.00000 q^{94} +1.00000 q^{96} +3.00000i q^{98} +4.00000 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{4} + 2 q^{6} - 2 q^{9}+O(q^{10})$$ 2 * q - 2 * q^4 + 2 * q^6 - 2 * q^9 $$2 q - 2 q^{4} + 2 q^{6} - 2 q^{9} - 8 q^{11} + 4 q^{14} + 2 q^{16} + 8 q^{19} - 4 q^{21} - 2 q^{24} - 8 q^{26} - 4 q^{29} + 8 q^{31} + 2 q^{34} + 2 q^{36} + 8 q^{39} + 16 q^{41} + 8 q^{44} - 16 q^{46} + 6 q^{49} - 2 q^{51} - 2 q^{54} - 4 q^{56} + 12 q^{59} + 28 q^{61} - 2 q^{64} - 8 q^{66} + 16 q^{69} + 4 q^{71} + 12 q^{74} - 8 q^{76} + 2 q^{81} + 4 q^{84} - 12 q^{86} + 4 q^{89} + 16 q^{91} + 16 q^{94} + 2 q^{96} + 8 q^{99}+O(q^{100})$$ 2 * q - 2 * q^4 + 2 * q^6 - 2 * q^9 - 8 * q^11 + 4 * q^14 + 2 * q^16 + 8 * q^19 - 4 * q^21 - 2 * q^24 - 8 * q^26 - 4 * q^29 + 8 * q^31 + 2 * q^34 + 2 * q^36 + 8 * q^39 + 16 * q^41 + 8 * q^44 - 16 * q^46 + 6 * q^49 - 2 * q^51 - 2 * q^54 - 4 * q^56 + 12 * q^59 + 28 * q^61 - 2 * q^64 - 8 * q^66 + 16 * q^69 + 4 * q^71 + 12 * q^74 - 8 * q^76 + 2 * q^81 + 4 * q^84 - 12 * q^86 + 4 * q^89 + 16 * q^91 + 16 * q^94 + 2 * q^96 + 8 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/2550\mathbb{Z}\right)^\times$$.

 $$n$$ $$751$$ $$851$$ $$1327$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1$$

## Coefficient data

For each $$n$$ we display the coefficients of the $$q$$-expansion $$a_n$$, the Satake parameters $$\alpha_p$$, and the Satake angles $$\theta_p = \textrm{Arg}(\alpha_p)$$.

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 1.00000i 0.707107i
$$3$$ − 1.00000i − 0.577350i
$$4$$ −1.00000 −0.500000
$$5$$ 0 0
$$6$$ 1.00000 0.408248
$$7$$ − 2.00000i − 0.755929i −0.925820 0.377964i $$-0.876624\pi$$
0.925820 0.377964i $$-0.123376\pi$$
$$8$$ − 1.00000i − 0.353553i
$$9$$ −1.00000 −0.333333
$$10$$ 0 0
$$11$$ −4.00000 −1.20605 −0.603023 0.797724i $$-0.706037\pi$$
−0.603023 + 0.797724i $$0.706037\pi$$
$$12$$ 1.00000i 0.288675i
$$13$$ 4.00000i 1.10940i 0.832050 + 0.554700i $$0.187167\pi$$
−0.832050 + 0.554700i $$0.812833\pi$$
$$14$$ 2.00000 0.534522
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ − 1.00000i − 0.242536i
$$18$$ − 1.00000i − 0.235702i
$$19$$ 4.00000 0.917663 0.458831 0.888523i $$-0.348268\pi$$
0.458831 + 0.888523i $$0.348268\pi$$
$$20$$ 0 0
$$21$$ −2.00000 −0.436436
$$22$$ − 4.00000i − 0.852803i
$$23$$ 8.00000i 1.66812i 0.551677 + 0.834058i $$0.313988\pi$$
−0.551677 + 0.834058i $$0.686012\pi$$
$$24$$ −1.00000 −0.204124
$$25$$ 0 0
$$26$$ −4.00000 −0.784465
$$27$$ 1.00000i 0.192450i
$$28$$ 2.00000i 0.377964i
$$29$$ −2.00000 −0.371391 −0.185695 0.982607i $$-0.559454\pi$$
−0.185695 + 0.982607i $$0.559454\pi$$
$$30$$ 0 0
$$31$$ 4.00000 0.718421 0.359211 0.933257i $$-0.383046\pi$$
0.359211 + 0.933257i $$0.383046\pi$$
$$32$$ 1.00000i 0.176777i
$$33$$ 4.00000i 0.696311i
$$34$$ 1.00000 0.171499
$$35$$ 0 0
$$36$$ 1.00000 0.166667
$$37$$ − 6.00000i − 0.986394i −0.869918 0.493197i $$-0.835828\pi$$
0.869918 0.493197i $$-0.164172\pi$$
$$38$$ 4.00000i 0.648886i
$$39$$ 4.00000 0.640513
$$40$$ 0 0
$$41$$ 8.00000 1.24939 0.624695 0.780869i $$-0.285223\pi$$
0.624695 + 0.780869i $$0.285223\pi$$
$$42$$ − 2.00000i − 0.308607i
$$43$$ 6.00000i 0.914991i 0.889212 + 0.457496i $$0.151253\pi$$
−0.889212 + 0.457496i $$0.848747\pi$$
$$44$$ 4.00000 0.603023
$$45$$ 0 0
$$46$$ −8.00000 −1.17954
$$47$$ − 8.00000i − 1.16692i −0.812142 0.583460i $$-0.801699\pi$$
0.812142 0.583460i $$-0.198301\pi$$
$$48$$ − 1.00000i − 0.144338i
$$49$$ 3.00000 0.428571
$$50$$ 0 0
$$51$$ −1.00000 −0.140028
$$52$$ − 4.00000i − 0.554700i
$$53$$ − 2.00000i − 0.274721i −0.990521 0.137361i $$-0.956138\pi$$
0.990521 0.137361i $$-0.0438619\pi$$
$$54$$ −1.00000 −0.136083
$$55$$ 0 0
$$56$$ −2.00000 −0.267261
$$57$$ − 4.00000i − 0.529813i
$$58$$ − 2.00000i − 0.262613i
$$59$$ 6.00000 0.781133 0.390567 0.920575i $$-0.372279\pi$$
0.390567 + 0.920575i $$0.372279\pi$$
$$60$$ 0 0
$$61$$ 14.0000 1.79252 0.896258 0.443533i $$-0.146275\pi$$
0.896258 + 0.443533i $$0.146275\pi$$
$$62$$ 4.00000i 0.508001i
$$63$$ 2.00000i 0.251976i
$$64$$ −1.00000 −0.125000
$$65$$ 0 0
$$66$$ −4.00000 −0.492366
$$67$$ 2.00000i 0.244339i 0.992509 + 0.122169i $$0.0389851\pi$$
−0.992509 + 0.122169i $$0.961015\pi$$
$$68$$ 1.00000i 0.121268i
$$69$$ 8.00000 0.963087
$$70$$ 0 0
$$71$$ 2.00000 0.237356 0.118678 0.992933i $$-0.462134\pi$$
0.118678 + 0.992933i $$0.462134\pi$$
$$72$$ 1.00000i 0.117851i
$$73$$ 4.00000i 0.468165i 0.972217 + 0.234082i $$0.0752085\pi$$
−0.972217 + 0.234082i $$0.924791\pi$$
$$74$$ 6.00000 0.697486
$$75$$ 0 0
$$76$$ −4.00000 −0.458831
$$77$$ 8.00000i 0.911685i
$$78$$ 4.00000i 0.452911i
$$79$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 8.00000i 0.883452i
$$83$$ − 16.0000i − 1.75623i −0.478451 0.878114i $$-0.658802\pi$$
0.478451 0.878114i $$-0.341198\pi$$
$$84$$ 2.00000 0.218218
$$85$$ 0 0
$$86$$ −6.00000 −0.646997
$$87$$ 2.00000i 0.214423i
$$88$$ 4.00000i 0.426401i
$$89$$ 2.00000 0.212000 0.106000 0.994366i $$-0.466196\pi$$
0.106000 + 0.994366i $$0.466196\pi$$
$$90$$ 0 0
$$91$$ 8.00000 0.838628
$$92$$ − 8.00000i − 0.834058i
$$93$$ − 4.00000i − 0.414781i
$$94$$ 8.00000 0.825137
$$95$$ 0 0
$$96$$ 1.00000 0.102062
$$97$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$98$$ 3.00000i 0.303046i
$$99$$ 4.00000 0.402015
$$100$$ 0 0
$$101$$ 8.00000 0.796030 0.398015 0.917379i $$-0.369699\pi$$
0.398015 + 0.917379i $$0.369699\pi$$
$$102$$ − 1.00000i − 0.0990148i
$$103$$ − 12.0000i − 1.18240i −0.806527 0.591198i $$-0.798655\pi$$
0.806527 0.591198i $$-0.201345\pi$$
$$104$$ 4.00000 0.392232
$$105$$ 0 0
$$106$$ 2.00000 0.194257
$$107$$ − 12.0000i − 1.16008i −0.814587 0.580042i $$-0.803036\pi$$
0.814587 0.580042i $$-0.196964\pi$$
$$108$$ − 1.00000i − 0.0962250i
$$109$$ 14.0000 1.34096 0.670478 0.741929i $$-0.266089\pi$$
0.670478 + 0.741929i $$0.266089\pi$$
$$110$$ 0 0
$$111$$ −6.00000 −0.569495
$$112$$ − 2.00000i − 0.188982i
$$113$$ 6.00000i 0.564433i 0.959351 + 0.282216i $$0.0910696\pi$$
−0.959351 + 0.282216i $$0.908930\pi$$
$$114$$ 4.00000 0.374634
$$115$$ 0 0
$$116$$ 2.00000 0.185695
$$117$$ − 4.00000i − 0.369800i
$$118$$ 6.00000i 0.552345i
$$119$$ −2.00000 −0.183340
$$120$$ 0 0
$$121$$ 5.00000 0.454545
$$122$$ 14.0000i 1.26750i
$$123$$ − 8.00000i − 0.721336i
$$124$$ −4.00000 −0.359211
$$125$$ 0 0
$$126$$ −2.00000 −0.178174
$$127$$ 4.00000i 0.354943i 0.984126 + 0.177471i $$0.0567917\pi$$
−0.984126 + 0.177471i $$0.943208\pi$$
$$128$$ − 1.00000i − 0.0883883i
$$129$$ 6.00000 0.528271
$$130$$ 0 0
$$131$$ −16.0000 −1.39793 −0.698963 0.715158i $$-0.746355\pi$$
−0.698963 + 0.715158i $$0.746355\pi$$
$$132$$ − 4.00000i − 0.348155i
$$133$$ − 8.00000i − 0.693688i
$$134$$ −2.00000 −0.172774
$$135$$ 0 0
$$136$$ −1.00000 −0.0857493
$$137$$ 6.00000i 0.512615i 0.966595 + 0.256307i $$0.0825059\pi$$
−0.966595 + 0.256307i $$0.917494\pi$$
$$138$$ 8.00000i 0.681005i
$$139$$ 20.0000 1.69638 0.848189 0.529694i $$-0.177693\pi$$
0.848189 + 0.529694i $$0.177693\pi$$
$$140$$ 0 0
$$141$$ −8.00000 −0.673722
$$142$$ 2.00000i 0.167836i
$$143$$ − 16.0000i − 1.33799i
$$144$$ −1.00000 −0.0833333
$$145$$ 0 0
$$146$$ −4.00000 −0.331042
$$147$$ − 3.00000i − 0.247436i
$$148$$ 6.00000i 0.493197i
$$149$$ −16.0000 −1.31077 −0.655386 0.755295i $$-0.727494\pi$$
−0.655386 + 0.755295i $$0.727494\pi$$
$$150$$ 0 0
$$151$$ −8.00000 −0.651031 −0.325515 0.945537i $$-0.605538\pi$$
−0.325515 + 0.945537i $$0.605538\pi$$
$$152$$ − 4.00000i − 0.324443i
$$153$$ 1.00000i 0.0808452i
$$154$$ −8.00000 −0.644658
$$155$$ 0 0
$$156$$ −4.00000 −0.320256
$$157$$ 4.00000i 0.319235i 0.987179 + 0.159617i $$0.0510260\pi$$
−0.987179 + 0.159617i $$0.948974\pi$$
$$158$$ 0 0
$$159$$ −2.00000 −0.158610
$$160$$ 0 0
$$161$$ 16.0000 1.26098
$$162$$ 1.00000i 0.0785674i
$$163$$ − 8.00000i − 0.626608i −0.949653 0.313304i $$-0.898564\pi$$
0.949653 0.313304i $$-0.101436\pi$$
$$164$$ −8.00000 −0.624695
$$165$$ 0 0
$$166$$ 16.0000 1.24184
$$167$$ 12.0000i 0.928588i 0.885681 + 0.464294i $$0.153692\pi$$
−0.885681 + 0.464294i $$0.846308\pi$$
$$168$$ 2.00000i 0.154303i
$$169$$ −3.00000 −0.230769
$$170$$ 0 0
$$171$$ −4.00000 −0.305888
$$172$$ − 6.00000i − 0.457496i
$$173$$ 22.0000i 1.67263i 0.548250 + 0.836315i $$0.315294\pi$$
−0.548250 + 0.836315i $$0.684706\pi$$
$$174$$ −2.00000 −0.151620
$$175$$ 0 0
$$176$$ −4.00000 −0.301511
$$177$$ − 6.00000i − 0.450988i
$$178$$ 2.00000i 0.149906i
$$179$$ −6.00000 −0.448461 −0.224231 0.974536i $$-0.571987\pi$$
−0.224231 + 0.974536i $$0.571987\pi$$
$$180$$ 0 0
$$181$$ 26.0000 1.93256 0.966282 0.257485i $$-0.0828937\pi$$
0.966282 + 0.257485i $$0.0828937\pi$$
$$182$$ 8.00000i 0.592999i
$$183$$ − 14.0000i − 1.03491i
$$184$$ 8.00000 0.589768
$$185$$ 0 0
$$186$$ 4.00000 0.293294
$$187$$ 4.00000i 0.292509i
$$188$$ 8.00000i 0.583460i
$$189$$ 2.00000 0.145479
$$190$$ 0 0
$$191$$ 4.00000 0.289430 0.144715 0.989473i $$-0.453773\pi$$
0.144715 + 0.989473i $$0.453773\pi$$
$$192$$ 1.00000i 0.0721688i
$$193$$ 24.0000i 1.72756i 0.503871 + 0.863779i $$0.331909\pi$$
−0.503871 + 0.863779i $$0.668091\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ −3.00000 −0.214286
$$197$$ 26.0000i 1.85242i 0.377004 + 0.926212i $$0.376954\pi$$
−0.377004 + 0.926212i $$0.623046\pi$$
$$198$$ 4.00000i 0.284268i
$$199$$ 20.0000 1.41776 0.708881 0.705328i $$-0.249200\pi$$
0.708881 + 0.705328i $$0.249200\pi$$
$$200$$ 0 0
$$201$$ 2.00000 0.141069
$$202$$ 8.00000i 0.562878i
$$203$$ 4.00000i 0.280745i
$$204$$ 1.00000 0.0700140
$$205$$ 0 0
$$206$$ 12.0000 0.836080
$$207$$ − 8.00000i − 0.556038i
$$208$$ 4.00000i 0.277350i
$$209$$ −16.0000 −1.10674
$$210$$ 0 0
$$211$$ 4.00000 0.275371 0.137686 0.990476i $$-0.456034\pi$$
0.137686 + 0.990476i $$0.456034\pi$$
$$212$$ 2.00000i 0.137361i
$$213$$ − 2.00000i − 0.137038i
$$214$$ 12.0000 0.820303
$$215$$ 0 0
$$216$$ 1.00000 0.0680414
$$217$$ − 8.00000i − 0.543075i
$$218$$ 14.0000i 0.948200i
$$219$$ 4.00000 0.270295
$$220$$ 0 0
$$221$$ 4.00000 0.269069
$$222$$ − 6.00000i − 0.402694i
$$223$$ 28.0000i 1.87502i 0.347960 + 0.937509i $$0.386874\pi$$
−0.347960 + 0.937509i $$0.613126\pi$$
$$224$$ 2.00000 0.133631
$$225$$ 0 0
$$226$$ −6.00000 −0.399114
$$227$$ − 28.0000i − 1.85843i −0.369546 0.929213i $$-0.620487\pi$$
0.369546 0.929213i $$-0.379513\pi$$
$$228$$ 4.00000i 0.264906i
$$229$$ −6.00000 −0.396491 −0.198246 0.980152i $$-0.563524\pi$$
−0.198246 + 0.980152i $$0.563524\pi$$
$$230$$ 0 0
$$231$$ 8.00000 0.526361
$$232$$ 2.00000i 0.131306i
$$233$$ 18.0000i 1.17922i 0.807688 + 0.589610i $$0.200718\pi$$
−0.807688 + 0.589610i $$0.799282\pi$$
$$234$$ 4.00000 0.261488
$$235$$ 0 0
$$236$$ −6.00000 −0.390567
$$237$$ 0 0
$$238$$ − 2.00000i − 0.129641i
$$239$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$240$$ 0 0
$$241$$ −14.0000 −0.901819 −0.450910 0.892570i $$-0.648900\pi$$
−0.450910 + 0.892570i $$0.648900\pi$$
$$242$$ 5.00000i 0.321412i
$$243$$ − 1.00000i − 0.0641500i
$$244$$ −14.0000 −0.896258
$$245$$ 0 0
$$246$$ 8.00000 0.510061
$$247$$ 16.0000i 1.01806i
$$248$$ − 4.00000i − 0.254000i
$$249$$ −16.0000 −1.01396
$$250$$ 0 0
$$251$$ 26.0000 1.64111 0.820553 0.571571i $$-0.193666\pi$$
0.820553 + 0.571571i $$0.193666\pi$$
$$252$$ − 2.00000i − 0.125988i
$$253$$ − 32.0000i − 2.01182i
$$254$$ −4.00000 −0.250982
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ 18.0000i 1.12281i 0.827541 + 0.561405i $$0.189739\pi$$
−0.827541 + 0.561405i $$0.810261\pi$$
$$258$$ 6.00000i 0.373544i
$$259$$ −12.0000 −0.745644
$$260$$ 0 0
$$261$$ 2.00000 0.123797
$$262$$ − 16.0000i − 0.988483i
$$263$$ 8.00000i 0.493301i 0.969104 + 0.246651i $$0.0793300\pi$$
−0.969104 + 0.246651i $$0.920670\pi$$
$$264$$ 4.00000 0.246183
$$265$$ 0 0
$$266$$ 8.00000 0.490511
$$267$$ − 2.00000i − 0.122398i
$$268$$ − 2.00000i − 0.122169i
$$269$$ 30.0000 1.82913 0.914566 0.404436i $$-0.132532\pi$$
0.914566 + 0.404436i $$0.132532\pi$$
$$270$$ 0 0
$$271$$ −24.0000 −1.45790 −0.728948 0.684569i $$-0.759990\pi$$
−0.728948 + 0.684569i $$0.759990\pi$$
$$272$$ − 1.00000i − 0.0606339i
$$273$$ − 8.00000i − 0.484182i
$$274$$ −6.00000 −0.362473
$$275$$ 0 0
$$276$$ −8.00000 −0.481543
$$277$$ − 2.00000i − 0.120168i −0.998193 0.0600842i $$-0.980863\pi$$
0.998193 0.0600842i $$-0.0191369\pi$$
$$278$$ 20.0000i 1.19952i
$$279$$ −4.00000 −0.239474
$$280$$ 0 0
$$281$$ −2.00000 −0.119310 −0.0596550 0.998219i $$-0.519000\pi$$
−0.0596550 + 0.998219i $$0.519000\pi$$
$$282$$ − 8.00000i − 0.476393i
$$283$$ − 20.0000i − 1.18888i −0.804141 0.594438i $$-0.797374\pi$$
0.804141 0.594438i $$-0.202626\pi$$
$$284$$ −2.00000 −0.118678
$$285$$ 0 0
$$286$$ 16.0000 0.946100
$$287$$ − 16.0000i − 0.944450i
$$288$$ − 1.00000i − 0.0589256i
$$289$$ −1.00000 −0.0588235
$$290$$ 0 0
$$291$$ 0 0
$$292$$ − 4.00000i − 0.234082i
$$293$$ − 14.0000i − 0.817889i −0.912559 0.408944i $$-0.865897\pi$$
0.912559 0.408944i $$-0.134103\pi$$
$$294$$ 3.00000 0.174964
$$295$$ 0 0
$$296$$ −6.00000 −0.348743
$$297$$ − 4.00000i − 0.232104i
$$298$$ − 16.0000i − 0.926855i
$$299$$ −32.0000 −1.85061
$$300$$ 0 0
$$301$$ 12.0000 0.691669
$$302$$ − 8.00000i − 0.460348i
$$303$$ − 8.00000i − 0.459588i
$$304$$ 4.00000 0.229416
$$305$$ 0 0
$$306$$ −1.00000 −0.0571662
$$307$$ 10.0000i 0.570730i 0.958419 + 0.285365i $$0.0921148\pi$$
−0.958419 + 0.285365i $$0.907885\pi$$
$$308$$ − 8.00000i − 0.455842i
$$309$$ −12.0000 −0.682656
$$310$$ 0 0
$$311$$ 14.0000 0.793867 0.396934 0.917847i $$-0.370074\pi$$
0.396934 + 0.917847i $$0.370074\pi$$
$$312$$ − 4.00000i − 0.226455i
$$313$$ 16.0000i 0.904373i 0.891923 + 0.452187i $$0.149356\pi$$
−0.891923 + 0.452187i $$0.850644\pi$$
$$314$$ −4.00000 −0.225733
$$315$$ 0 0
$$316$$ 0 0
$$317$$ 2.00000i 0.112331i 0.998421 + 0.0561656i $$0.0178875\pi$$
−0.998421 + 0.0561656i $$0.982113\pi$$
$$318$$ − 2.00000i − 0.112154i
$$319$$ 8.00000 0.447914
$$320$$ 0 0
$$321$$ −12.0000 −0.669775
$$322$$ 16.0000i 0.891645i
$$323$$ − 4.00000i − 0.222566i
$$324$$ −1.00000 −0.0555556
$$325$$ 0 0
$$326$$ 8.00000 0.443079
$$327$$ − 14.0000i − 0.774202i
$$328$$ − 8.00000i − 0.441726i
$$329$$ −16.0000 −0.882109
$$330$$ 0 0
$$331$$ −8.00000 −0.439720 −0.219860 0.975531i $$-0.570560\pi$$
−0.219860 + 0.975531i $$0.570560\pi$$
$$332$$ 16.0000i 0.878114i
$$333$$ 6.00000i 0.328798i
$$334$$ −12.0000 −0.656611
$$335$$ 0 0
$$336$$ −2.00000 −0.109109
$$337$$ − 20.0000i − 1.08947i −0.838608 0.544735i $$-0.816630\pi$$
0.838608 0.544735i $$-0.183370\pi$$
$$338$$ − 3.00000i − 0.163178i
$$339$$ 6.00000 0.325875
$$340$$ 0 0
$$341$$ −16.0000 −0.866449
$$342$$ − 4.00000i − 0.216295i
$$343$$ − 20.0000i − 1.07990i
$$344$$ 6.00000 0.323498
$$345$$ 0 0
$$346$$ −22.0000 −1.18273
$$347$$ 28.0000i 1.50312i 0.659665 + 0.751559i $$0.270698\pi$$
−0.659665 + 0.751559i $$0.729302\pi$$
$$348$$ − 2.00000i − 0.107211i
$$349$$ 14.0000 0.749403 0.374701 0.927146i $$-0.377745\pi$$
0.374701 + 0.927146i $$0.377745\pi$$
$$350$$ 0 0
$$351$$ −4.00000 −0.213504
$$352$$ − 4.00000i − 0.213201i
$$353$$ − 10.0000i − 0.532246i −0.963939 0.266123i $$-0.914257\pi$$
0.963939 0.266123i $$-0.0857428\pi$$
$$354$$ 6.00000 0.318896
$$355$$ 0 0
$$356$$ −2.00000 −0.106000
$$357$$ 2.00000i 0.105851i
$$358$$ − 6.00000i − 0.317110i
$$359$$ −36.0000 −1.90001 −0.950004 0.312239i $$-0.898921\pi$$
−0.950004 + 0.312239i $$0.898921\pi$$
$$360$$ 0 0
$$361$$ −3.00000 −0.157895
$$362$$ 26.0000i 1.36653i
$$363$$ − 5.00000i − 0.262432i
$$364$$ −8.00000 −0.419314
$$365$$ 0 0
$$366$$ 14.0000 0.731792
$$367$$ − 22.0000i − 1.14839i −0.818718 0.574195i $$-0.805315\pi$$
0.818718 0.574195i $$-0.194685\pi$$
$$368$$ 8.00000i 0.417029i
$$369$$ −8.00000 −0.416463
$$370$$ 0 0
$$371$$ −4.00000 −0.207670
$$372$$ 4.00000i 0.207390i
$$373$$ − 16.0000i − 0.828449i −0.910175 0.414224i $$-0.864053\pi$$
0.910175 0.414224i $$-0.135947\pi$$
$$374$$ −4.00000 −0.206835
$$375$$ 0 0
$$376$$ −8.00000 −0.412568
$$377$$ − 8.00000i − 0.412021i
$$378$$ 2.00000i 0.102869i
$$379$$ 4.00000 0.205466 0.102733 0.994709i $$-0.467241\pi$$
0.102733 + 0.994709i $$0.467241\pi$$
$$380$$ 0 0
$$381$$ 4.00000 0.204926
$$382$$ 4.00000i 0.204658i
$$383$$ 8.00000i 0.408781i 0.978889 + 0.204390i $$0.0655212\pi$$
−0.978889 + 0.204390i $$0.934479\pi$$
$$384$$ −1.00000 −0.0510310
$$385$$ 0 0
$$386$$ −24.0000 −1.22157
$$387$$ − 6.00000i − 0.304997i
$$388$$ 0 0
$$389$$ −24.0000 −1.21685 −0.608424 0.793612i $$-0.708198\pi$$
−0.608424 + 0.793612i $$0.708198\pi$$
$$390$$ 0 0
$$391$$ 8.00000 0.404577
$$392$$ − 3.00000i − 0.151523i
$$393$$ 16.0000i 0.807093i
$$394$$ −26.0000 −1.30986
$$395$$ 0 0
$$396$$ −4.00000 −0.201008
$$397$$ − 6.00000i − 0.301131i −0.988600 0.150566i $$-0.951890\pi$$
0.988600 0.150566i $$-0.0481095\pi$$
$$398$$ 20.0000i 1.00251i
$$399$$ −8.00000 −0.400501
$$400$$ 0 0
$$401$$ −12.0000 −0.599251 −0.299626 0.954057i $$-0.596862\pi$$
−0.299626 + 0.954057i $$0.596862\pi$$
$$402$$ 2.00000i 0.0997509i
$$403$$ 16.0000i 0.797017i
$$404$$ −8.00000 −0.398015
$$405$$ 0 0
$$406$$ −4.00000 −0.198517
$$407$$ 24.0000i 1.18964i
$$408$$ 1.00000i 0.0495074i
$$409$$ 26.0000 1.28562 0.642809 0.766027i $$-0.277769\pi$$
0.642809 + 0.766027i $$0.277769\pi$$
$$410$$ 0 0
$$411$$ 6.00000 0.295958
$$412$$ 12.0000i 0.591198i
$$413$$ − 12.0000i − 0.590481i
$$414$$ 8.00000 0.393179
$$415$$ 0 0
$$416$$ −4.00000 −0.196116
$$417$$ − 20.0000i − 0.979404i
$$418$$ − 16.0000i − 0.782586i
$$419$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$420$$ 0 0
$$421$$ −18.0000 −0.877266 −0.438633 0.898666i $$-0.644537\pi$$
−0.438633 + 0.898666i $$0.644537\pi$$
$$422$$ 4.00000i 0.194717i
$$423$$ 8.00000i 0.388973i
$$424$$ −2.00000 −0.0971286
$$425$$ 0 0
$$426$$ 2.00000 0.0969003
$$427$$ − 28.0000i − 1.35501i
$$428$$ 12.0000i 0.580042i
$$429$$ −16.0000 −0.772487
$$430$$ 0 0
$$431$$ 14.0000 0.674356 0.337178 0.941441i $$-0.390528\pi$$
0.337178 + 0.941441i $$0.390528\pi$$
$$432$$ 1.00000i 0.0481125i
$$433$$ − 26.0000i − 1.24948i −0.780833 0.624740i $$-0.785205\pi$$
0.780833 0.624740i $$-0.214795\pi$$
$$434$$ 8.00000 0.384012
$$435$$ 0 0
$$436$$ −14.0000 −0.670478
$$437$$ 32.0000i 1.53077i
$$438$$ 4.00000i 0.191127i
$$439$$ 24.0000 1.14546 0.572729 0.819745i $$-0.305885\pi$$
0.572729 + 0.819745i $$0.305885\pi$$
$$440$$ 0 0
$$441$$ −3.00000 −0.142857
$$442$$ 4.00000i 0.190261i
$$443$$ 24.0000i 1.14027i 0.821549 + 0.570137i $$0.193110\pi$$
−0.821549 + 0.570137i $$0.806890\pi$$
$$444$$ 6.00000 0.284747
$$445$$ 0 0
$$446$$ −28.0000 −1.32584
$$447$$ 16.0000i 0.756774i
$$448$$ 2.00000i 0.0944911i
$$449$$ 12.0000 0.566315 0.283158 0.959073i $$-0.408618\pi$$
0.283158 + 0.959073i $$0.408618\pi$$
$$450$$ 0 0
$$451$$ −32.0000 −1.50682
$$452$$ − 6.00000i − 0.282216i
$$453$$ 8.00000i 0.375873i
$$454$$ 28.0000 1.31411
$$455$$ 0 0
$$456$$ −4.00000 −0.187317
$$457$$ 2.00000i 0.0935561i 0.998905 + 0.0467780i $$0.0148953\pi$$
−0.998905 + 0.0467780i $$0.985105\pi$$
$$458$$ − 6.00000i − 0.280362i
$$459$$ 1.00000 0.0466760
$$460$$ 0 0
$$461$$ −28.0000 −1.30409 −0.652045 0.758180i $$-0.726089\pi$$
−0.652045 + 0.758180i $$0.726089\pi$$
$$462$$ 8.00000i 0.372194i
$$463$$ 4.00000i 0.185896i 0.995671 + 0.0929479i $$0.0296290\pi$$
−0.995671 + 0.0929479i $$0.970371\pi$$
$$464$$ −2.00000 −0.0928477
$$465$$ 0 0
$$466$$ −18.0000 −0.833834
$$467$$ 12.0000i 0.555294i 0.960683 + 0.277647i $$0.0895545\pi$$
−0.960683 + 0.277647i $$0.910445\pi$$
$$468$$ 4.00000i 0.184900i
$$469$$ 4.00000 0.184703
$$470$$ 0 0
$$471$$ 4.00000 0.184310
$$472$$ − 6.00000i − 0.276172i
$$473$$ − 24.0000i − 1.10352i
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 2.00000 0.0916698
$$477$$ 2.00000i 0.0915737i
$$478$$ 0 0
$$479$$ 22.0000 1.00521 0.502603 0.864517i $$-0.332376\pi$$
0.502603 + 0.864517i $$0.332376\pi$$
$$480$$ 0 0
$$481$$ 24.0000 1.09431
$$482$$ − 14.0000i − 0.637683i
$$483$$ − 16.0000i − 0.728025i
$$484$$ −5.00000 −0.227273
$$485$$ 0 0
$$486$$ 1.00000 0.0453609
$$487$$ 22.0000i 0.996915i 0.866914 + 0.498458i $$0.166100\pi$$
−0.866914 + 0.498458i $$0.833900\pi$$
$$488$$ − 14.0000i − 0.633750i
$$489$$ −8.00000 −0.361773
$$490$$ 0 0
$$491$$ −18.0000 −0.812329 −0.406164 0.913800i $$-0.633134\pi$$
−0.406164 + 0.913800i $$0.633134\pi$$
$$492$$ 8.00000i 0.360668i
$$493$$ 2.00000i 0.0900755i
$$494$$ −16.0000 −0.719874
$$495$$ 0 0
$$496$$ 4.00000 0.179605
$$497$$ − 4.00000i − 0.179425i
$$498$$ − 16.0000i − 0.716977i
$$499$$ −4.00000 −0.179065 −0.0895323 0.995984i $$-0.528537\pi$$
−0.0895323 + 0.995984i $$0.528537\pi$$
$$500$$ 0 0
$$501$$ 12.0000 0.536120
$$502$$ 26.0000i 1.16044i
$$503$$ − 16.0000i − 0.713405i −0.934218 0.356702i $$-0.883901\pi$$
0.934218 0.356702i $$-0.116099\pi$$
$$504$$ 2.00000 0.0890871
$$505$$ 0 0
$$506$$ 32.0000 1.42257
$$507$$ 3.00000i 0.133235i
$$508$$ − 4.00000i − 0.177471i
$$509$$ −8.00000 −0.354594 −0.177297 0.984157i $$-0.556735\pi$$
−0.177297 + 0.984157i $$0.556735\pi$$
$$510$$ 0 0
$$511$$ 8.00000 0.353899
$$512$$ 1.00000i 0.0441942i
$$513$$ 4.00000i 0.176604i
$$514$$ −18.0000 −0.793946
$$515$$ 0 0
$$516$$ −6.00000 −0.264135
$$517$$ 32.0000i 1.40736i
$$518$$ − 12.0000i − 0.527250i
$$519$$ 22.0000 0.965693
$$520$$ 0 0
$$521$$ −36.0000 −1.57719 −0.788594 0.614914i $$-0.789191\pi$$
−0.788594 + 0.614914i $$0.789191\pi$$
$$522$$ 2.00000i 0.0875376i
$$523$$ − 14.0000i − 0.612177i −0.952003 0.306089i $$-0.900980\pi$$
0.952003 0.306089i $$-0.0990204\pi$$
$$524$$ 16.0000 0.698963
$$525$$ 0 0
$$526$$ −8.00000 −0.348817
$$527$$ − 4.00000i − 0.174243i
$$528$$ 4.00000i 0.174078i
$$529$$ −41.0000 −1.78261
$$530$$ 0 0
$$531$$ −6.00000 −0.260378
$$532$$ 8.00000i 0.346844i
$$533$$ 32.0000i 1.38607i
$$534$$ 2.00000 0.0865485
$$535$$ 0 0
$$536$$ 2.00000 0.0863868
$$537$$ 6.00000i 0.258919i
$$538$$ 30.0000i 1.29339i
$$539$$ −12.0000 −0.516877
$$540$$ 0 0
$$541$$ 30.0000 1.28980 0.644900 0.764267i $$-0.276899\pi$$
0.644900 + 0.764267i $$0.276899\pi$$
$$542$$ − 24.0000i − 1.03089i
$$543$$ − 26.0000i − 1.11577i
$$544$$ 1.00000 0.0428746
$$545$$ 0 0
$$546$$ 8.00000 0.342368
$$547$$ − 44.0000i − 1.88130i −0.339372 0.940652i $$-0.610215\pi$$
0.339372 0.940652i $$-0.389785\pi$$
$$548$$ − 6.00000i − 0.256307i
$$549$$ −14.0000 −0.597505
$$550$$ 0 0
$$551$$ −8.00000 −0.340811
$$552$$ − 8.00000i − 0.340503i
$$553$$ 0 0
$$554$$ 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 $$0.986509\pi$$
$$558$$ − 4.00000i − 0.169334i
$$559$$ −24.0000 −1.01509
$$560$$ 0 0
$$561$$ 4.00000 0.168880
$$562$$ − 2.00000i − 0.0843649i
$$563$$ − 24.0000i − 1.01148i −0.862686 0.505740i $$-0.831220\pi$$
0.862686 0.505740i $$-0.168780\pi$$
$$564$$ 8.00000 0.336861
$$565$$ 0 0
$$566$$ 20.0000 0.840663
$$567$$ − 2.00000i − 0.0839921i
$$568$$ − 2.00000i − 0.0839181i
$$569$$ −42.0000 −1.76073 −0.880366 0.474295i $$-0.842703\pi$$
−0.880366 + 0.474295i $$0.842703\pi$$
$$570$$ 0 0
$$571$$ 28.0000 1.17176 0.585882 0.810397i $$-0.300748\pi$$
0.585882 + 0.810397i $$0.300748\pi$$
$$572$$ 16.0000i 0.668994i
$$573$$ − 4.00000i − 0.167102i
$$574$$ 16.0000 0.667827
$$575$$ 0 0
$$576$$ 1.00000 0.0416667
$$577$$ − 6.00000i − 0.249783i −0.992170 0.124892i $$-0.960142\pi$$
0.992170 0.124892i $$-0.0398583\pi$$
$$578$$ − 1.00000i − 0.0415945i
$$579$$ 24.0000 0.997406
$$580$$ 0 0
$$581$$ −32.0000 −1.32758
$$582$$ 0 0
$$583$$ 8.00000i 0.331326i
$$584$$ 4.00000 0.165521
$$585$$ 0 0
$$586$$ 14.0000 0.578335
$$587$$ − 8.00000i − 0.330195i −0.986277 0.165098i $$-0.947206\pi$$
0.986277 0.165098i $$-0.0527939\pi$$
$$588$$ 3.00000i 0.123718i
$$589$$ 16.0000 0.659269
$$590$$ 0 0
$$591$$ 26.0000 1.06950
$$592$$ − 6.00000i − 0.246598i
$$593$$ − 18.0000i − 0.739171i −0.929197 0.369586i $$-0.879500\pi$$
0.929197 0.369586i $$-0.120500\pi$$
$$594$$ 4.00000 0.164122
$$595$$ 0 0
$$596$$ 16.0000 0.655386
$$597$$ − 20.0000i − 0.818546i
$$598$$ − 32.0000i − 1.30858i
$$599$$ −12.0000 −0.490307 −0.245153 0.969484i $$-0.578838\pi$$
−0.245153 + 0.969484i $$0.578838\pi$$
$$600$$ 0 0
$$601$$ 30.0000 1.22373 0.611863 0.790964i $$-0.290420\pi$$
0.611863 + 0.790964i $$0.290420\pi$$
$$602$$ 12.0000i 0.489083i
$$603$$ − 2.00000i − 0.0814463i
$$604$$ 8.00000 0.325515
$$605$$ 0 0
$$606$$ 8.00000 0.324978
$$607$$ − 34.0000i − 1.38002i −0.723801 0.690009i $$-0.757607\pi$$
0.723801 0.690009i $$-0.242393\pi$$
$$608$$ 4.00000i 0.162221i
$$609$$ 4.00000 0.162088
$$610$$ 0 0
$$611$$ 32.0000 1.29458
$$612$$ − 1.00000i − 0.0404226i
$$613$$ 16.0000i 0.646234i 0.946359 + 0.323117i $$0.104731\pi$$
−0.946359 + 0.323117i $$0.895269\pi$$
$$614$$ −10.0000 −0.403567
$$615$$ 0 0
$$616$$ 8.00000 0.322329
$$617$$ 6.00000i 0.241551i 0.992680 + 0.120775i $$0.0385381\pi$$
−0.992680 + 0.120775i $$0.961462\pi$$
$$618$$ − 12.0000i − 0.482711i
$$619$$ 4.00000 0.160774 0.0803868 0.996764i $$-0.474384\pi$$
0.0803868 + 0.996764i $$0.474384\pi$$
$$620$$ 0 0
$$621$$ −8.00000 −0.321029
$$622$$ 14.0000i 0.561349i
$$623$$ − 4.00000i − 0.160257i
$$624$$ 4.00000 0.160128
$$625$$ 0 0
$$626$$ −16.0000 −0.639489
$$627$$ 16.0000i 0.638978i
$$628$$ − 4.00000i − 0.159617i
$$629$$ −6.00000 −0.239236
$$630$$ 0 0
$$631$$ −16.0000 −0.636950 −0.318475 0.947931i $$-0.603171\pi$$
−0.318475 + 0.947931i $$0.603171\pi$$
$$632$$ 0 0
$$633$$ − 4.00000i − 0.158986i
$$634$$ −2.00000 −0.0794301
$$635$$ 0 0
$$636$$ 2.00000 0.0793052
$$637$$ 12.0000i 0.475457i
$$638$$ 8.00000i 0.316723i
$$639$$ −2.00000 −0.0791188
$$640$$ 0 0
$$641$$ 12.0000 0.473972 0.236986 0.971513i $$-0.423841\pi$$
0.236986 + 0.971513i $$0.423841\pi$$
$$642$$ − 12.0000i − 0.473602i
$$643$$ 16.0000i 0.630978i 0.948929 + 0.315489i $$0.102169\pi$$
−0.948929 + 0.315489i $$0.897831\pi$$
$$644$$ −16.0000 −0.630488
$$645$$ 0 0
$$646$$ 4.00000 0.157378
$$647$$ 16.0000i 0.629025i 0.949253 + 0.314512i $$0.101841\pi$$
−0.949253 + 0.314512i $$0.898159\pi$$
$$648$$ − 1.00000i − 0.0392837i
$$649$$ −24.0000 −0.942082
$$650$$ 0 0
$$651$$ −8.00000 −0.313545
$$652$$ 8.00000i 0.313304i
$$653$$ − 30.0000i − 1.17399i −0.809590 0.586995i $$-0.800311\pi$$
0.809590 0.586995i $$-0.199689\pi$$
$$654$$ 14.0000 0.547443
$$655$$ 0 0
$$656$$ 8.00000 0.312348
$$657$$ − 4.00000i − 0.156055i
$$658$$ − 16.0000i − 0.623745i
$$659$$ −14.0000 −0.545363 −0.272681 0.962104i $$-0.587910\pi$$
−0.272681 + 0.962104i $$0.587910\pi$$
$$660$$ 0 0
$$661$$ 2.00000 0.0777910 0.0388955 0.999243i $$-0.487616\pi$$
0.0388955 + 0.999243i $$0.487616\pi$$
$$662$$ − 8.00000i − 0.310929i
$$663$$ − 4.00000i − 0.155347i
$$664$$ −16.0000 −0.620920
$$665$$ 0 0
$$666$$ −6.00000 −0.232495
$$667$$ − 16.0000i − 0.619522i
$$668$$ − 12.0000i − 0.464294i
$$669$$ 28.0000 1.08254
$$670$$ 0 0
$$671$$ −56.0000 −2.16186
$$672$$ − 2.00000i − 0.0771517i
$$673$$ − 8.00000i − 0.308377i −0.988041 0.154189i $$-0.950724\pi$$
0.988041 0.154189i $$-0.0492764\pi$$
$$674$$ 20.0000 0.770371
$$675$$ 0 0
$$676$$ 3.00000 0.115385
$$677$$ − 6.00000i − 0.230599i −0.993331 0.115299i $$-0.963217\pi$$
0.993331 0.115299i $$-0.0367827\pi$$
$$678$$ 6.00000i 0.230429i
$$679$$ 0 0
$$680$$ 0 0
$$681$$ −28.0000 −1.07296
$$682$$ − 16.0000i − 0.612672i
$$683$$ 36.0000i 1.37750i 0.724998 + 0.688751i $$0.241841\pi$$
−0.724998 + 0.688751i $$0.758159\pi$$
$$684$$ 4.00000 0.152944
$$685$$ 0 0
$$686$$ 20.0000 0.763604
$$687$$ 6.00000i 0.228914i
$$688$$ 6.00000i 0.228748i
$$689$$ 8.00000 0.304776
$$690$$ 0 0
$$691$$ −20.0000 −0.760836 −0.380418 0.924815i $$-0.624220\pi$$
−0.380418 + 0.924815i $$0.624220\pi$$
$$692$$ − 22.0000i − 0.836315i
$$693$$ − 8.00000i − 0.303895i
$$694$$ −28.0000 −1.06287
$$695$$ 0 0
$$696$$ 2.00000 0.0758098
$$697$$ − 8.00000i − 0.303022i
$$698$$ 14.0000i 0.529908i
$$699$$ 18.0000 0.680823
$$700$$ 0 0
$$701$$ 36.0000 1.35970 0.679851 0.733351i $$-0.262045\pi$$
0.679851 + 0.733351i $$0.262045\pi$$
$$702$$ − 4.00000i − 0.150970i
$$703$$ − 24.0000i − 0.905177i
$$704$$ 4.00000 0.150756
$$705$$ 0 0
$$706$$ 10.0000 0.376355
$$707$$ − 16.0000i − 0.601742i
$$708$$ 6.00000i 0.225494i
$$709$$ −22.0000 −0.826227 −0.413114 0.910679i $$-0.635559\pi$$
−0.413114 + 0.910679i $$0.635559\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ − 2.00000i − 0.0749532i
$$713$$ 32.0000i 1.19841i
$$714$$ −2.00000 −0.0748481
$$715$$ 0 0
$$716$$ 6.00000 0.224231
$$717$$ 0 0
$$718$$ − 36.0000i − 1.34351i
$$719$$ −10.0000 −0.372937 −0.186469 0.982461i $$-0.559704\pi$$
−0.186469 + 0.982461i $$0.559704\pi$$
$$720$$ 0 0
$$721$$ −24.0000 −0.893807
$$722$$ − 3.00000i − 0.111648i
$$723$$ 14.0000i 0.520666i
$$724$$ −26.0000 −0.966282
$$725$$ 0 0
$$726$$ 5.00000 0.185567
$$727$$ − 44.0000i − 1.63187i −0.578144 0.815935i $$-0.696223\pi$$
0.578144 0.815935i $$-0.303777\pi$$
$$728$$ − 8.00000i − 0.296500i
$$729$$ −1.00000 −0.0370370
$$730$$ 0 0
$$731$$ 6.00000 0.221918
$$732$$ 14.0000i 0.517455i
$$733$$ 48.0000i 1.77292i 0.462805 + 0.886460i $$0.346843\pi$$
−0.462805 + 0.886460i $$0.653157\pi$$
$$734$$ 22.0000 0.812035
$$735$$ 0 0
$$736$$ −8.00000 −0.294884
$$737$$ − 8.00000i − 0.294684i
$$738$$ − 8.00000i − 0.294484i
$$739$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$740$$ 0 0
$$741$$ 16.0000 0.587775
$$742$$ − 4.00000i − 0.146845i
$$743$$ − 20.0000i − 0.733729i −0.930274 0.366864i $$-0.880431\pi$$
0.930274 0.366864i $$-0.119569\pi$$
$$744$$ −4.00000 −0.146647
$$745$$ 0 0
$$746$$ 16.0000 0.585802
$$747$$ 16.0000i 0.585409i
$$748$$ − 4.00000i − 0.146254i
$$749$$ −24.0000 −0.876941
$$750$$ 0 0
$$751$$ −24.0000 −0.875772 −0.437886 0.899030i $$-0.644273\pi$$
−0.437886 + 0.899030i $$0.644273\pi$$
$$752$$ − 8.00000i − 0.291730i
$$753$$ − 26.0000i − 0.947493i
$$754$$ 8.00000 0.291343
$$755$$ 0 0
$$756$$ −2.00000 −0.0727393
$$757$$ 28.0000i 1.01768i 0.860862 + 0.508839i $$0.169925\pi$$
−0.860862 + 0.508839i $$0.830075\pi$$
$$758$$ 4.00000i 0.145287i
$$759$$ −32.0000 −1.16153
$$760$$ 0 0
$$761$$ −6.00000 −0.217500 −0.108750 0.994069i $$-0.534685\pi$$
−0.108750 + 0.994069i $$0.534685\pi$$
$$762$$ 4.00000i 0.144905i
$$763$$ − 28.0000i − 1.01367i
$$764$$ −4.00000 −0.144715
$$765$$ 0 0
$$766$$ −8.00000 −0.289052
$$767$$ 24.0000i 0.866590i
$$768$$ − 1.00000i − 0.0360844i
$$769$$ −14.0000 −0.504853 −0.252426 0.967616i $$-0.581229\pi$$
−0.252426 + 0.967616i $$0.581229\pi$$
$$770$$ 0 0
$$771$$ 18.0000 0.648254
$$772$$ − 24.0000i − 0.863779i
$$773$$ − 42.0000i − 1.51064i −0.655359 0.755318i $$-0.727483\pi$$
0.655359 0.755318i $$-0.272517\pi$$
$$774$$ 6.00000 0.215666
$$775$$ 0 0
$$776$$ 0 0
$$777$$ 12.0000i 0.430498i
$$778$$ − 24.0000i − 0.860442i
$$779$$ 32.0000 1.14652
$$780$$ 0 0
$$781$$ −8.00000 −0.286263
$$782$$ 8.00000i 0.286079i
$$783$$ − 2.00000i − 0.0714742i
$$784$$ 3.00000 0.107143
$$785$$ 0 0
$$786$$ −16.0000 −0.570701
$$787$$ − 24.0000i − 0.855508i −0.903895 0.427754i $$-0.859305\pi$$
0.903895 0.427754i $$-0.140695\pi$$
$$788$$ − 26.0000i − 0.926212i
$$789$$ 8.00000 0.284808
$$790$$ 0 0
$$791$$ 12.0000 0.426671
$$792$$ − 4.00000i − 0.142134i
$$793$$ 56.0000i 1.98862i
$$794$$ 6.00000 0.212932
$$795$$ 0 0
$$796$$ −20.0000 −0.708881
$$797$$ 14.0000i 0.495905i 0.968772 + 0.247953i $$0.0797578\pi$$
−0.968772 + 0.247953i $$0.920242\pi$$
$$798$$ − 8.00000i − 0.283197i
$$799$$ −8.00000 −0.283020
$$800$$ 0 0
$$801$$ −2.00000 −0.0706665
$$802$$ − 12.0000i − 0.423735i
$$803$$ − 16.0000i − 0.564628i
$$804$$ −2.00000 −0.0705346
$$805$$ 0 0
$$806$$ −16.0000 −0.563576
$$807$$ − 30.0000i − 1.05605i
$$808$$ − 8.00000i − 0.281439i
$$809$$ 48.0000 1.68759 0.843795 0.536666i $$-0.180316\pi$$
0.843795 + 0.536666i $$0.180316\pi$$
$$810$$ 0 0
$$811$$ 20.0000 0.702295 0.351147 0.936320i $$-0.385792\pi$$
0.351147 + 0.936320i $$0.385792\pi$$
$$812$$ − 4.00000i − 0.140372i
$$813$$ 24.0000i 0.841717i
$$814$$ −24.0000 −0.841200
$$815$$ 0 0
$$816$$ −1.00000 −0.0350070
$$817$$ 24.0000i 0.839654i
$$818$$ 26.0000i 0.909069i
$$819$$ −8.00000 −0.279543
$$820$$ 0 0
$$821$$ −18.0000 −0.628204 −0.314102 0.949389i $$-0.601703\pi$$
−0.314102 + 0.949389i $$0.601703\pi$$
$$822$$ 6.00000i 0.209274i
$$823$$ 14.0000i 0.488009i 0.969774 + 0.244005i $$0.0784612\pi$$
−0.969774 + 0.244005i $$0.921539\pi$$
$$824$$ −12.0000 −0.418040
$$825$$ 0 0
$$826$$ 12.0000 0.417533
$$827$$ − 44.0000i − 1.53003i −0.644013 0.765015i $$-0.722732\pi$$
0.644013 0.765015i $$-0.277268\pi$$
$$828$$ 8.00000i 0.278019i
$$829$$ 46.0000 1.59765 0.798823 0.601566i $$-0.205456\pi$$
0.798823 + 0.601566i $$0.205456\pi$$
$$830$$ 0 0
$$831$$ −2.00000 −0.0693792
$$832$$ − 4.00000i − 0.138675i
$$833$$ − 3.00000i − 0.103944i
$$834$$ 20.0000 0.692543
$$835$$ 0 0
$$836$$ 16.0000 0.553372
$$837$$ 4.00000i 0.138260i
$$838$$ 0 0
$$839$$ −6.00000 −0.207143 −0.103572 0.994622i $$-0.533027\pi$$
−0.103572 + 0.994622i $$0.533027\pi$$
$$840$$ 0 0
$$841$$ −25.0000 −0.862069
$$842$$ − 18.0000i − 0.620321i
$$843$$ 2.00000i 0.0688837i
$$844$$ −4.00000 −0.137686
$$845$$ 0 0
$$846$$ −8.00000 −0.275046
$$847$$ − 10.0000i − 0.343604i
$$848$$ − 2.00000i − 0.0686803i
$$849$$ −20.0000 −0.686398
$$850$$ 0 0
$$851$$ 48.0000 1.64542
$$852$$ 2.00000i 0.0685189i
$$853$$ 2.00000i 0.0684787i 0.999414 + 0.0342393i $$0.0109009\pi$$
−0.999414 + 0.0342393i $$0.989099\pi$$
$$854$$ 28.0000 0.958140
$$855$$ 0 0
$$856$$ −12.0000 −0.410152
$$857$$ 22.0000i 0.751506i 0.926720 + 0.375753i $$0.122616\pi$$
−0.926720 + 0.375753i $$0.877384\pi$$
$$858$$ − 16.0000i − 0.546231i
$$859$$ 32.0000 1.09183 0.545913 0.837842i $$-0.316183\pi$$
0.545913 + 0.837842i $$0.316183\pi$$
$$860$$ 0 0
$$861$$ −16.0000 −0.545279
$$862$$ 14.0000i 0.476842i
$$863$$ − 16.0000i − 0.544646i −0.962206 0.272323i $$-0.912208\pi$$
0.962206 0.272323i $$-0.0877920\pi$$
$$864$$ −1.00000 −0.0340207
$$865$$ 0 0
$$866$$ 26.0000 0.883516
$$867$$ 1.00000i 0.0339618i
$$868$$ 8.00000i 0.271538i
$$869$$ 0 0
$$870$$ 0 0
$$871$$ −8.00000 −0.271070
$$872$$ − 14.0000i − 0.474100i
$$873$$ 0 0
$$874$$ −32.0000 −1.08242
$$875$$ 0 0
$$876$$ −4.00000 −0.135147
$$877$$ − 50.0000i − 1.68838i −0.536044 0.844190i $$-0.680082\pi$$
0.536044 0.844190i $$-0.319918\pi$$
$$878$$ 24.0000i 0.809961i
$$879$$ −14.0000 −0.472208
$$880$$ 0 0
$$881$$ 36.0000 1.21287 0.606435 0.795133i $$-0.292599\pi$$
0.606435 + 0.795133i $$0.292599\pi$$
$$882$$ − 3.00000i − 0.101015i
$$883$$ − 46.0000i − 1.54802i −0.633171 0.774012i $$-0.718247\pi$$
0.633171 0.774012i $$-0.281753\pi$$
$$884$$ −4.00000 −0.134535
$$885$$ 0 0
$$886$$ −24.0000 −0.806296
$$887$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$888$$ 6.00000i 0.201347i
$$889$$ 8.00000 0.268311
$$890$$ 0 0
$$891$$ −4.00000 −0.134005
$$892$$ − 28.0000i − 0.937509i
$$893$$ − 32.0000i − 1.07084i
$$894$$ −16.0000 −0.535120
$$895$$ 0 0
$$896$$ −2.00000 −0.0668153
$$897$$ 32.0000i 1.06845i
$$898$$ 12.0000i 0.400445i
$$899$$ −8.00000 −0.266815
$$900$$ 0 0
$$901$$ −2.00000 −0.0666297
$$902$$ − 32.0000i − 1.06548i
$$903$$ − 12.0000i − 0.399335i
$$904$$ 6.00000 0.199557
$$905$$ 0 0
$$906$$ −8.00000 −0.265782
$$907$$ 32.0000i 1.06254i 0.847202 + 0.531271i $$0.178286\pi$$
−0.847202 + 0.531271i $$0.821714\pi$$
$$908$$ 28.0000i 0.929213i
$$909$$ −8.00000 −0.265343
$$910$$ 0 0
$$911$$ 50.0000 1.65657 0.828287 0.560304i $$-0.189316\pi$$
0.828287 + 0.560304i $$0.189316\pi$$
$$912$$ − 4.00000i − 0.132453i
$$913$$ 64.0000i 2.11809i
$$914$$ −2.00000 −0.0661541
$$915$$ 0 0
$$916$$ 6.00000 0.198246
$$917$$ 32.0000i 1.05673i
$$918$$ 1.00000i 0.0330049i
$$919$$ 8.00000 0.263896 0.131948 0.991257i $$-0.457877\pi$$
0.131948 + 0.991257i $$0.457877\pi$$
$$920$$ 0 0
$$921$$ 10.0000 0.329511
$$922$$ − 28.0000i − 0.922131i
$$923$$ 8.00000i 0.263323i
$$924$$ −8.00000 −0.263181
$$925$$ 0 0
$$926$$ −4.00000 −0.131448
$$927$$ 12.0000i 0.394132i
$$928$$ − 2.00000i − 0.0656532i
$$929$$ 48.0000 1.57483 0.787414 0.616424i $$-0.211419\pi$$
0.787414 + 0.616424i $$0.211419\pi$$
$$930$$ 0 0
$$931$$ 12.0000 0.393284
$$932$$ − 18.0000i − 0.589610i
$$933$$ − 14.0000i − 0.458339i
$$934$$ −12.0000 −0.392652
$$935$$ 0 0
$$936$$ −4.00000 −0.130744
$$937$$ 34.0000i 1.11073i 0.831606 + 0.555366i $$0.187422\pi$$
−0.831606 + 0.555366i $$0.812578\pi$$
$$938$$ 4.00000i 0.130605i
$$939$$ 16.0000 0.522140
$$940$$ 0 0
$$941$$ −2.00000 −0.0651981 −0.0325991 0.999469i $$-0.510378\pi$$
−0.0325991 + 0.999469i $$0.510378\pi$$
$$942$$ 4.00000i 0.130327i
$$943$$ 64.0000i 2.08413i
$$944$$ 6.00000 0.195283
$$945$$ 0 0
$$946$$ 24.0000 0.780307
$$947$$ 4.00000i 0.129983i 0.997886 + 0.0649913i $$0.0207020\pi$$
−0.997886 + 0.0649913i $$0.979298\pi$$
$$948$$ 0 0
$$949$$ −16.0000 −0.519382
$$950$$ 0 0
$$951$$ 2.00000 0.0648544
$$952$$ 2.00000i 0.0648204i
$$953$$ 42.0000i 1.36051i 0.732974 + 0.680257i $$0.238132\pi$$
−0.732974 + 0.680257i $$0.761868\pi$$
$$954$$ −2.00000 −0.0647524
$$955$$ 0 0
$$956$$ 0 0
$$957$$ − 8.00000i − 0.258603i
$$958$$ 22.0000i 0.710788i
$$959$$ 12.0000 0.387500
$$960$$ 0 0
$$961$$ −15.0000 −0.483871
$$962$$ 24.0000i 0.773791i
$$963$$ 12.0000i 0.386695i
$$964$$ 14.0000 0.450910
$$965$$ 0 0
$$966$$ 16.0000 0.514792
$$967$$ − 56.0000i − 1.80084i −0.435023 0.900419i $$-0.643260\pi$$
0.435023 0.900419i $$-0.356740\pi$$
$$968$$ − 5.00000i − 0.160706i
$$969$$ −4.00000 −0.128499
$$970$$ 0 0
$$971$$ −50.0000 −1.60458 −0.802288 0.596937i $$-0.796384\pi$$
−0.802288 + 0.596937i $$0.796384\pi$$
$$972$$ 1.00000i 0.0320750i
$$973$$ − 40.0000i − 1.28234i
$$974$$ −22.0000 −0.704925
$$975$$ 0 0
$$976$$ 14.0000 0.448129
$$977$$ 42.0000i 1.34370i 0.740688 + 0.671850i $$0.234500\pi$$
−0.740688 + 0.671850i $$0.765500\pi$$
$$978$$ − 8.00000i − 0.255812i
$$979$$ −8.00000 −0.255681
$$980$$ 0 0
$$981$$ −14.0000 −0.446986
$$982$$ − 18.0000i − 0.574403i
$$983$$ − 52.0000i − 1.65854i −0.558846 0.829271i $$-0.688756\pi$$
0.558846 0.829271i $$-0.311244\pi$$
$$984$$ −8.00000 −0.255031
$$985$$ 0 0
$$986$$ −2.00000 −0.0636930
$$987$$ 16.0000i 0.509286i
$$988$$ − 16.0000i − 0.509028i
$$989$$ −48.0000 −1.52631
$$990$$ 0 0
$$991$$ 20.0000 0.635321 0.317660 0.948205i $$-0.397103\pi$$
0.317660 + 0.948205i $$0.397103\pi$$
$$992$$ 4.00000i 0.127000i
$$993$$ 8.00000i 0.253872i
$$994$$ 4.00000 0.126872
$$995$$ 0 0
$$996$$ 16.0000 0.506979
$$997$$ 50.0000i 1.58352i 0.610835 + 0.791758i $$0.290834\pi$$
−0.610835 + 0.791758i $$0.709166\pi$$
$$998$$ − 4.00000i − 0.126618i
$$999$$ 6.00000 0.189832
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 2550.2.d.n.2449.2 2
5.2 odd 4 510.2.a.a.1.1 1
5.3 odd 4 2550.2.a.bc.1.1 1
5.4 even 2 inner 2550.2.d.n.2449.1 2
15.2 even 4 1530.2.a.p.1.1 1
15.8 even 4 7650.2.a.k.1.1 1
20.7 even 4 4080.2.a.s.1.1 1
85.67 odd 4 8670.2.a.k.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
510.2.a.a.1.1 1 5.2 odd 4
1530.2.a.p.1.1 1 15.2 even 4
2550.2.a.bc.1.1 1 5.3 odd 4
2550.2.d.n.2449.1 2 5.4 even 2 inner
2550.2.d.n.2449.2 2 1.1 even 1 trivial
4080.2.a.s.1.1 1 20.7 even 4
7650.2.a.k.1.1 1 15.8 even 4
8670.2.a.k.1.1 1 85.67 odd 4