# Properties

 Label 2550.2.d.g.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 102) 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.g.2449.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{6} -2.00000i q^{7} -1.00000i q^{8} -1.00000 q^{9} +O(q^{10})$$ $$q+1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{6} -2.00000i q^{7} -1.00000i q^{8} -1.00000 q^{9} -1.00000i q^{12} +2.00000i q^{13} +2.00000 q^{14} +1.00000 q^{16} +1.00000i q^{17} -1.00000i q^{18} +4.00000 q^{19} +2.00000 q^{21} -6.00000i q^{23} +1.00000 q^{24} -2.00000 q^{26} -1.00000i q^{27} +2.00000i q^{28} -10.0000 q^{31} +1.00000i q^{32} -1.00000 q^{34} +1.00000 q^{36} -8.00000i q^{37} +4.00000i q^{38} -2.00000 q^{39} +6.00000 q^{41} +2.00000i q^{42} -4.00000i q^{43} +6.00000 q^{46} -12.0000i q^{47} +1.00000i q^{48} +3.00000 q^{49} -1.00000 q^{51} -2.00000i q^{52} +6.00000i q^{53} +1.00000 q^{54} -2.00000 q^{56} +4.00000i q^{57} +12.0000 q^{59} +8.00000 q^{61} -10.0000i q^{62} +2.00000i q^{63} -1.00000 q^{64} +4.00000i q^{67} -1.00000i q^{68} +6.00000 q^{69} +6.00000 q^{71} +1.00000i q^{72} +2.00000i q^{73} +8.00000 q^{74} -4.00000 q^{76} -2.00000i q^{78} +10.0000 q^{79} +1.00000 q^{81} +6.00000i q^{82} +12.0000i q^{83} -2.00000 q^{84} +4.00000 q^{86} +18.0000 q^{89} +4.00000 q^{91} +6.00000i q^{92} -10.0000i q^{93} +12.0000 q^{94} -1.00000 q^{96} -14.0000i q^{97} +3.00000i q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{4} - 2 q^{6} - 2 q^{9}+O(q^{10})$$ 2 * q - 2 * q^4 - 2 * q^6 - 2 * q^9 $$2 q - 2 q^{4} - 2 q^{6} - 2 q^{9} + 4 q^{14} + 2 q^{16} + 8 q^{19} + 4 q^{21} + 2 q^{24} - 4 q^{26} - 20 q^{31} - 2 q^{34} + 2 q^{36} - 4 q^{39} + 12 q^{41} + 12 q^{46} + 6 q^{49} - 2 q^{51} + 2 q^{54} - 4 q^{56} + 24 q^{59} + 16 q^{61} - 2 q^{64} + 12 q^{69} + 12 q^{71} + 16 q^{74} - 8 q^{76} + 20 q^{79} + 2 q^{81} - 4 q^{84} + 8 q^{86} + 36 q^{89} + 8 q^{91} + 24 q^{94} - 2 q^{96}+O(q^{100})$$ 2 * q - 2 * q^4 - 2 * q^6 - 2 * q^9 + 4 * q^14 + 2 * q^16 + 8 * q^19 + 4 * q^21 + 2 * q^24 - 4 * q^26 - 20 * q^31 - 2 * q^34 + 2 * q^36 - 4 * q^39 + 12 * q^41 + 12 * q^46 + 6 * q^49 - 2 * q^51 + 2 * q^54 - 4 * q^56 + 24 * q^59 + 16 * q^61 - 2 * q^64 + 12 * q^69 + 12 * q^71 + 16 * q^74 - 8 * q^76 + 20 * q^79 + 2 * q^81 - 4 * q^84 + 8 * q^86 + 36 * q^89 + 8 * q^91 + 24 * q^94 - 2 * q^96

## 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
$$11$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$12$$ − 1.00000i − 0.288675i
$$13$$ 2.00000i 0.554700i 0.960769 + 0.277350i $$0.0894562\pi$$
−0.960769 + 0.277350i $$0.910544\pi$$
$$14$$ 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
$$22$$ 0 0
$$23$$ − 6.00000i − 1.25109i −0.780189 0.625543i $$-0.784877\pi$$
0.780189 0.625543i $$-0.215123\pi$$
$$24$$ 1.00000 0.204124
$$25$$ 0 0
$$26$$ −2.00000 −0.392232
$$27$$ − 1.00000i − 0.192450i
$$28$$ 2.00000i 0.377964i
$$29$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$30$$ 0 0
$$31$$ −10.0000 −1.79605 −0.898027 0.439941i $$-0.854999\pi$$
−0.898027 + 0.439941i $$0.854999\pi$$
$$32$$ 1.00000i 0.176777i
$$33$$ 0 0
$$34$$ −1.00000 −0.171499
$$35$$ 0 0
$$36$$ 1.00000 0.166667
$$37$$ − 8.00000i − 1.31519i −0.753371 0.657596i $$-0.771573\pi$$
0.753371 0.657596i $$-0.228427\pi$$
$$38$$ 4.00000i 0.648886i
$$39$$ −2.00000 −0.320256
$$40$$ 0 0
$$41$$ 6.00000 0.937043 0.468521 0.883452i $$-0.344787\pi$$
0.468521 + 0.883452i $$0.344787\pi$$
$$42$$ 2.00000i 0.308607i
$$43$$ − 4.00000i − 0.609994i −0.952353 0.304997i $$-0.901344\pi$$
0.952353 0.304997i $$-0.0986555\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 6.00000 0.884652
$$47$$ − 12.0000i − 1.75038i −0.483779 0.875190i $$-0.660736\pi$$
0.483779 0.875190i $$-0.339264\pi$$
$$48$$ 1.00000i 0.144338i
$$49$$ 3.00000 0.428571
$$50$$ 0 0
$$51$$ −1.00000 −0.140028
$$52$$ − 2.00000i − 0.277350i
$$53$$ 6.00000i 0.824163i 0.911147 + 0.412082i $$0.135198\pi$$
−0.911147 + 0.412082i $$0.864802\pi$$
$$54$$ 1.00000 0.136083
$$55$$ 0 0
$$56$$ −2.00000 −0.267261
$$57$$ 4.00000i 0.529813i
$$58$$ 0 0
$$59$$ 12.0000 1.56227 0.781133 0.624364i $$-0.214642\pi$$
0.781133 + 0.624364i $$0.214642\pi$$
$$60$$ 0 0
$$61$$ 8.00000 1.02430 0.512148 0.858898i $$-0.328850\pi$$
0.512148 + 0.858898i $$0.328850\pi$$
$$62$$ − 10.0000i − 1.27000i
$$63$$ 2.00000i 0.251976i
$$64$$ −1.00000 −0.125000
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 4.00000i 0.488678i 0.969690 + 0.244339i $$0.0785709\pi$$
−0.969690 + 0.244339i $$0.921429\pi$$
$$68$$ − 1.00000i − 0.121268i
$$69$$ 6.00000 0.722315
$$70$$ 0 0
$$71$$ 6.00000 0.712069 0.356034 0.934473i $$-0.384129\pi$$
0.356034 + 0.934473i $$0.384129\pi$$
$$72$$ 1.00000i 0.117851i
$$73$$ 2.00000i 0.234082i 0.993127 + 0.117041i $$0.0373409\pi$$
−0.993127 + 0.117041i $$0.962659\pi$$
$$74$$ 8.00000 0.929981
$$75$$ 0 0
$$76$$ −4.00000 −0.458831
$$77$$ 0 0
$$78$$ − 2.00000i − 0.226455i
$$79$$ 10.0000 1.12509 0.562544 0.826767i $$-0.309823\pi$$
0.562544 + 0.826767i $$0.309823\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 6.00000i 0.662589i
$$83$$ 12.0000i 1.31717i 0.752506 + 0.658586i $$0.228845\pi$$
−0.752506 + 0.658586i $$0.771155\pi$$
$$84$$ −2.00000 −0.218218
$$85$$ 0 0
$$86$$ 4.00000 0.431331
$$87$$ 0 0
$$88$$ 0 0
$$89$$ 18.0000 1.90800 0.953998 0.299813i $$-0.0969242\pi$$
0.953998 + 0.299813i $$0.0969242\pi$$
$$90$$ 0 0
$$91$$ 4.00000 0.419314
$$92$$ 6.00000i 0.625543i
$$93$$ − 10.0000i − 1.03695i
$$94$$ 12.0000 1.23771
$$95$$ 0 0
$$96$$ −1.00000 −0.102062
$$97$$ − 14.0000i − 1.42148i −0.703452 0.710742i $$-0.748359\pi$$
0.703452 0.710742i $$-0.251641\pi$$
$$98$$ 3.00000i 0.303046i
$$99$$ 0 0
$$100$$ 0 0
$$101$$ 6.00000 0.597022 0.298511 0.954406i $$-0.403510\pi$$
0.298511 + 0.954406i $$0.403510\pi$$
$$102$$ − 1.00000i − 0.0990148i
$$103$$ − 4.00000i − 0.394132i −0.980390 0.197066i $$-0.936859\pi$$
0.980390 0.197066i $$-0.0631413\pi$$
$$104$$ 2.00000 0.196116
$$105$$ 0 0
$$106$$ −6.00000 −0.582772
$$107$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$108$$ 1.00000i 0.0962250i
$$109$$ −20.0000 −1.91565 −0.957826 0.287348i $$-0.907226\pi$$
−0.957826 + 0.287348i $$0.907226\pi$$
$$110$$ 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 $$-0.0910696\pi$$
$$114$$ −4.00000 −0.374634
$$115$$ 0 0
$$116$$ 0 0
$$117$$ − 2.00000i − 0.184900i
$$118$$ 12.0000i 1.10469i
$$119$$ 2.00000 0.183340
$$120$$ 0 0
$$121$$ −11.0000 −1.00000
$$122$$ 8.00000i 0.724286i
$$123$$ 6.00000i 0.541002i
$$124$$ 10.0000 0.898027
$$125$$ 0 0
$$126$$ −2.00000 −0.178174
$$127$$ − 8.00000i − 0.709885i −0.934888 0.354943i $$-0.884500\pi$$
0.934888 0.354943i $$-0.115500\pi$$
$$128$$ − 1.00000i − 0.0883883i
$$129$$ 4.00000 0.352180
$$130$$ 0 0
$$131$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$132$$ 0 0
$$133$$ − 8.00000i − 0.693688i
$$134$$ −4.00000 −0.345547
$$135$$ 0 0
$$136$$ 1.00000 0.0857493
$$137$$ 6.00000i 0.512615i 0.966595 + 0.256307i $$0.0825059\pi$$
−0.966595 + 0.256307i $$0.917494\pi$$
$$138$$ 6.00000i 0.510754i
$$139$$ −8.00000 −0.678551 −0.339276 0.940687i $$-0.610182\pi$$
−0.339276 + 0.940687i $$0.610182\pi$$
$$140$$ 0 0
$$141$$ 12.0000 1.01058
$$142$$ 6.00000i 0.503509i
$$143$$ 0 0
$$144$$ −1.00000 −0.0833333
$$145$$ 0 0
$$146$$ −2.00000 −0.165521
$$147$$ 3.00000i 0.247436i
$$148$$ 8.00000i 0.657596i
$$149$$ 6.00000 0.491539 0.245770 0.969328i $$-0.420959\pi$$
0.245770 + 0.969328i $$0.420959\pi$$
$$150$$ 0 0
$$151$$ 8.00000 0.651031 0.325515 0.945537i $$-0.394462\pi$$
0.325515 + 0.945537i $$0.394462\pi$$
$$152$$ − 4.00000i − 0.324443i
$$153$$ − 1.00000i − 0.0808452i
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 2.00000 0.160128
$$157$$ 10.0000i 0.798087i 0.916932 + 0.399043i $$0.130658\pi$$
−0.916932 + 0.399043i $$0.869342\pi$$
$$158$$ 10.0000i 0.795557i
$$159$$ −6.00000 −0.475831
$$160$$ 0 0
$$161$$ −12.0000 −0.945732
$$162$$ 1.00000i 0.0785674i
$$163$$ 20.0000i 1.56652i 0.621694 + 0.783260i $$0.286445\pi$$
−0.621694 + 0.783260i $$0.713555\pi$$
$$164$$ −6.00000 −0.468521
$$165$$ 0 0
$$166$$ −12.0000 −0.931381
$$167$$ − 18.0000i − 1.39288i −0.717614 0.696441i $$-0.754766\pi$$
0.717614 0.696441i $$-0.245234\pi$$
$$168$$ − 2.00000i − 0.154303i
$$169$$ 9.00000 0.692308
$$170$$ 0 0
$$171$$ −4.00000 −0.305888
$$172$$ 4.00000i 0.304997i
$$173$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ 12.0000i 0.901975i
$$178$$ 18.0000i 1.34916i
$$179$$ 12.0000 0.896922 0.448461 0.893802i $$-0.351972\pi$$
0.448461 + 0.893802i $$0.351972\pi$$
$$180$$ 0 0
$$181$$ 8.00000 0.594635 0.297318 0.954779i $$-0.403908\pi$$
0.297318 + 0.954779i $$0.403908\pi$$
$$182$$ 4.00000i 0.296500i
$$183$$ 8.00000i 0.591377i
$$184$$ −6.00000 −0.442326
$$185$$ 0 0
$$186$$ 10.0000 0.733236
$$187$$ 0 0
$$188$$ 12.0000i 0.875190i
$$189$$ −2.00000 −0.145479
$$190$$ 0 0
$$191$$ −12.0000 −0.868290 −0.434145 0.900843i $$-0.642949\pi$$
−0.434145 + 0.900843i $$0.642949\pi$$
$$192$$ − 1.00000i − 0.0721688i
$$193$$ 14.0000i 1.00774i 0.863779 + 0.503871i $$0.168091\pi$$
−0.863779 + 0.503871i $$0.831909\pi$$
$$194$$ 14.0000 1.00514
$$195$$ 0 0
$$196$$ −3.00000 −0.214286
$$197$$ 12.0000i 0.854965i 0.904024 + 0.427482i $$0.140599\pi$$
−0.904024 + 0.427482i $$0.859401\pi$$
$$198$$ 0 0
$$199$$ −2.00000 −0.141776 −0.0708881 0.997484i $$-0.522583\pi$$
−0.0708881 + 0.997484i $$0.522583\pi$$
$$200$$ 0 0
$$201$$ −4.00000 −0.282138
$$202$$ 6.00000i 0.422159i
$$203$$ 0 0
$$204$$ 1.00000 0.0700140
$$205$$ 0 0
$$206$$ 4.00000 0.278693
$$207$$ 6.00000i 0.417029i
$$208$$ 2.00000i 0.138675i
$$209$$ 0 0
$$210$$ 0 0
$$211$$ 8.00000 0.550743 0.275371 0.961338i $$-0.411199\pi$$
0.275371 + 0.961338i $$0.411199\pi$$
$$212$$ − 6.00000i − 0.412082i
$$213$$ 6.00000i 0.411113i
$$214$$ 0 0
$$215$$ 0 0
$$216$$ −1.00000 −0.0680414
$$217$$ 20.0000i 1.35769i
$$218$$ − 20.0000i − 1.35457i
$$219$$ −2.00000 −0.135147
$$220$$ 0 0
$$221$$ −2.00000 −0.134535
$$222$$ 8.00000i 0.536925i
$$223$$ − 28.0000i − 1.87502i −0.347960 0.937509i $$-0.613126\pi$$
0.347960 0.937509i $$-0.386874\pi$$
$$224$$ 2.00000 0.133631
$$225$$ 0 0
$$226$$ 6.00000 0.399114
$$227$$ − 12.0000i − 0.796468i −0.917284 0.398234i $$-0.869623\pi$$
0.917284 0.398234i $$-0.130377\pi$$
$$228$$ − 4.00000i − 0.264906i
$$229$$ −14.0000 −0.925146 −0.462573 0.886581i $$-0.653074\pi$$
−0.462573 + 0.886581i $$0.653074\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ − 30.0000i − 1.96537i −0.185296 0.982683i $$-0.559325\pi$$
0.185296 0.982683i $$-0.440675\pi$$
$$234$$ 2.00000 0.130744
$$235$$ 0 0
$$236$$ −12.0000 −0.781133
$$237$$ 10.0000i 0.649570i
$$238$$ 2.00000i 0.129641i
$$239$$ 12.0000 0.776215 0.388108 0.921614i $$-0.373129\pi$$
0.388108 + 0.921614i $$0.373129\pi$$
$$240$$ 0 0
$$241$$ −10.0000 −0.644157 −0.322078 0.946713i $$-0.604381\pi$$
−0.322078 + 0.946713i $$0.604381\pi$$
$$242$$ − 11.0000i − 0.707107i
$$243$$ 1.00000i 0.0641500i
$$244$$ −8.00000 −0.512148
$$245$$ 0 0
$$246$$ −6.00000 −0.382546
$$247$$ 8.00000i 0.509028i
$$248$$ 10.0000i 0.635001i
$$249$$ −12.0000 −0.760469
$$250$$ 0 0
$$251$$ 12.0000 0.757433 0.378717 0.925513i $$-0.376365\pi$$
0.378717 + 0.925513i $$0.376365\pi$$
$$252$$ − 2.00000i − 0.125988i
$$253$$ 0 0
$$254$$ 8.00000 0.501965
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ − 18.0000i − 1.12281i −0.827541 0.561405i $$-0.810261\pi$$
0.827541 0.561405i $$-0.189739\pi$$
$$258$$ 4.00000i 0.249029i
$$259$$ −16.0000 −0.994192
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 0 0
$$263$$ 12.0000i 0.739952i 0.929041 + 0.369976i $$0.120634\pi$$
−0.929041 + 0.369976i $$0.879366\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 8.00000 0.490511
$$267$$ 18.0000i 1.10158i
$$268$$ − 4.00000i − 0.244339i
$$269$$ −24.0000 −1.46331 −0.731653 0.681677i $$-0.761251\pi$$
−0.731653 + 0.681677i $$0.761251\pi$$
$$270$$ 0 0
$$271$$ −16.0000 −0.971931 −0.485965 0.873978i $$-0.661532\pi$$
−0.485965 + 0.873978i $$0.661532\pi$$
$$272$$ 1.00000i 0.0606339i
$$273$$ 4.00000i 0.242091i
$$274$$ −6.00000 −0.362473
$$275$$ 0 0
$$276$$ −6.00000 −0.361158
$$277$$ 4.00000i 0.240337i 0.992754 + 0.120168i $$0.0383434\pi$$
−0.992754 + 0.120168i $$0.961657\pi$$
$$278$$ − 8.00000i − 0.479808i
$$279$$ 10.0000 0.598684
$$280$$ 0 0
$$281$$ 30.0000 1.78965 0.894825 0.446417i $$-0.147300\pi$$
0.894825 + 0.446417i $$0.147300\pi$$
$$282$$ 12.0000i 0.714590i
$$283$$ − 16.0000i − 0.951101i −0.879688 0.475551i $$-0.842249\pi$$
0.879688 0.475551i $$-0.157751\pi$$
$$284$$ −6.00000 −0.356034
$$285$$ 0 0
$$286$$ 0 0
$$287$$ − 12.0000i − 0.708338i
$$288$$ − 1.00000i − 0.0589256i
$$289$$ −1.00000 −0.0588235
$$290$$ 0 0
$$291$$ 14.0000 0.820695
$$292$$ − 2.00000i − 0.117041i
$$293$$ 18.0000i 1.05157i 0.850617 + 0.525786i $$0.176229\pi$$
−0.850617 + 0.525786i $$0.823771\pi$$
$$294$$ −3.00000 −0.174964
$$295$$ 0 0
$$296$$ −8.00000 −0.464991
$$297$$ 0 0
$$298$$ 6.00000i 0.347571i
$$299$$ 12.0000 0.693978
$$300$$ 0 0
$$301$$ −8.00000 −0.461112
$$302$$ 8.00000i 0.460348i
$$303$$ 6.00000i 0.344691i
$$304$$ 4.00000 0.229416
$$305$$ 0 0
$$306$$ 1.00000 0.0571662
$$307$$ − 20.0000i − 1.14146i −0.821138 0.570730i $$-0.806660\pi$$
0.821138 0.570730i $$-0.193340\pi$$
$$308$$ 0 0
$$309$$ 4.00000 0.227552
$$310$$ 0 0
$$311$$ 18.0000 1.02069 0.510343 0.859971i $$-0.329518\pi$$
0.510343 + 0.859971i $$0.329518\pi$$
$$312$$ 2.00000i 0.113228i
$$313$$ 14.0000i 0.791327i 0.918396 + 0.395663i $$0.129485\pi$$
−0.918396 + 0.395663i $$0.870515\pi$$
$$314$$ −10.0000 −0.564333
$$315$$ 0 0
$$316$$ −10.0000 −0.562544
$$317$$ − 12.0000i − 0.673987i −0.941507 0.336994i $$-0.890590\pi$$
0.941507 0.336994i $$-0.109410\pi$$
$$318$$ − 6.00000i − 0.336463i
$$319$$ 0 0
$$320$$ 0 0
$$321$$ 0 0
$$322$$ − 12.0000i − 0.668734i
$$323$$ 4.00000i 0.222566i
$$324$$ −1.00000 −0.0555556
$$325$$ 0 0
$$326$$ −20.0000 −1.10770
$$327$$ − 20.0000i − 1.10600i
$$328$$ − 6.00000i − 0.331295i
$$329$$ −24.0000 −1.32316
$$330$$ 0 0
$$331$$ 20.0000 1.09930 0.549650 0.835395i $$-0.314761\pi$$
0.549650 + 0.835395i $$0.314761\pi$$
$$332$$ − 12.0000i − 0.658586i
$$333$$ 8.00000i 0.438397i
$$334$$ 18.0000 0.984916
$$335$$ 0 0
$$336$$ 2.00000 0.109109
$$337$$ 22.0000i 1.19842i 0.800593 + 0.599208i $$0.204518\pi$$
−0.800593 + 0.599208i $$0.795482\pi$$
$$338$$ 9.00000i 0.489535i
$$339$$ 6.00000 0.325875
$$340$$ 0 0
$$341$$ 0 0
$$342$$ − 4.00000i − 0.216295i
$$343$$ − 20.0000i − 1.07990i
$$344$$ −4.00000 −0.215666
$$345$$ 0 0
$$346$$ 0 0
$$347$$ 12.0000i 0.644194i 0.946707 + 0.322097i $$0.104388\pi$$
−0.946707 + 0.322097i $$0.895612\pi$$
$$348$$ 0 0
$$349$$ −26.0000 −1.39175 −0.695874 0.718164i $$-0.744983\pi$$
−0.695874 + 0.718164i $$0.744983\pi$$
$$350$$ 0 0
$$351$$ 2.00000 0.106752
$$352$$ 0 0
$$353$$ − 18.0000i − 0.958043i −0.877803 0.479022i $$-0.840992\pi$$
0.877803 0.479022i $$-0.159008\pi$$
$$354$$ −12.0000 −0.637793
$$355$$ 0 0
$$356$$ −18.0000 −0.953998
$$357$$ 2.00000i 0.105851i
$$358$$ 12.0000i 0.634220i
$$359$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$360$$ 0 0
$$361$$ −3.00000 −0.157895
$$362$$ 8.00000i 0.420471i
$$363$$ − 11.0000i − 0.577350i
$$364$$ −4.00000 −0.209657
$$365$$ 0 0
$$366$$ −8.00000 −0.418167
$$367$$ 10.0000i 0.521996i 0.965339 + 0.260998i $$0.0840516\pi$$
−0.965339 + 0.260998i $$0.915948\pi$$
$$368$$ − 6.00000i − 0.312772i
$$369$$ −6.00000 −0.312348
$$370$$ 0 0
$$371$$ 12.0000 0.623009
$$372$$ 10.0000i 0.518476i
$$373$$ − 22.0000i − 1.13912i −0.821951 0.569558i $$-0.807114\pi$$
0.821951 0.569558i $$-0.192886\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ −12.0000 −0.618853
$$377$$ 0 0
$$378$$ − 2.00000i − 0.102869i
$$379$$ 28.0000 1.43826 0.719132 0.694874i $$-0.244540\pi$$
0.719132 + 0.694874i $$0.244540\pi$$
$$380$$ 0 0
$$381$$ 8.00000 0.409852
$$382$$ − 12.0000i − 0.613973i
$$383$$ − 12.0000i − 0.613171i −0.951843 0.306586i $$-0.900813\pi$$
0.951843 0.306586i $$-0.0991866\pi$$
$$384$$ 1.00000 0.0510310
$$385$$ 0 0
$$386$$ −14.0000 −0.712581
$$387$$ 4.00000i 0.203331i
$$388$$ 14.0000i 0.710742i
$$389$$ 30.0000 1.52106 0.760530 0.649303i $$-0.224939\pi$$
0.760530 + 0.649303i $$0.224939\pi$$
$$390$$ 0 0
$$391$$ 6.00000 0.303433
$$392$$ − 3.00000i − 0.151523i
$$393$$ 0 0
$$394$$ −12.0000 −0.604551
$$395$$ 0 0
$$396$$ 0 0
$$397$$ 16.0000i 0.803017i 0.915855 + 0.401508i $$0.131514\pi$$
−0.915855 + 0.401508i $$0.868486\pi$$
$$398$$ − 2.00000i − 0.100251i
$$399$$ 8.00000 0.400501
$$400$$ 0 0
$$401$$ −18.0000 −0.898877 −0.449439 0.893311i $$-0.648376\pi$$
−0.449439 + 0.893311i $$0.648376\pi$$
$$402$$ − 4.00000i − 0.199502i
$$403$$ − 20.0000i − 0.996271i
$$404$$ −6.00000 −0.298511
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 0 0
$$408$$ 1.00000i 0.0495074i
$$409$$ −38.0000 −1.87898 −0.939490 0.342578i $$-0.888700\pi$$
−0.939490 + 0.342578i $$0.888700\pi$$
$$410$$ 0 0
$$411$$ −6.00000 −0.295958
$$412$$ 4.00000i 0.197066i
$$413$$ − 24.0000i − 1.18096i
$$414$$ −6.00000 −0.294884
$$415$$ 0 0
$$416$$ −2.00000 −0.0980581
$$417$$ − 8.00000i − 0.391762i
$$418$$ 0 0
$$419$$ 36.0000 1.75872 0.879358 0.476162i $$-0.157972\pi$$
0.879358 + 0.476162i $$0.157972\pi$$
$$420$$ 0 0
$$421$$ −22.0000 −1.07221 −0.536107 0.844150i $$-0.680106\pi$$
−0.536107 + 0.844150i $$0.680106\pi$$
$$422$$ 8.00000i 0.389434i
$$423$$ 12.0000i 0.583460i
$$424$$ 6.00000 0.291386
$$425$$ 0 0
$$426$$ −6.00000 −0.290701
$$427$$ − 16.0000i − 0.774294i
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ −30.0000 −1.44505 −0.722525 0.691345i $$-0.757018\pi$$
−0.722525 + 0.691345i $$0.757018\pi$$
$$432$$ − 1.00000i − 0.0481125i
$$433$$ 14.0000i 0.672797i 0.941720 + 0.336399i $$0.109209\pi$$
−0.941720 + 0.336399i $$0.890791\pi$$
$$434$$ −20.0000 −0.960031
$$435$$ 0 0
$$436$$ 20.0000 0.957826
$$437$$ − 24.0000i − 1.14808i
$$438$$ − 2.00000i − 0.0955637i
$$439$$ −26.0000 −1.24091 −0.620456 0.784241i $$-0.713053\pi$$
−0.620456 + 0.784241i $$0.713053\pi$$
$$440$$ 0 0
$$441$$ −3.00000 −0.142857
$$442$$ − 2.00000i − 0.0951303i
$$443$$ − 36.0000i − 1.71041i −0.518289 0.855206i $$-0.673431\pi$$
0.518289 0.855206i $$-0.326569\pi$$
$$444$$ −8.00000 −0.379663
$$445$$ 0 0
$$446$$ 28.0000 1.32584
$$447$$ 6.00000i 0.283790i
$$448$$ 2.00000i 0.0944911i
$$449$$ −6.00000 −0.283158 −0.141579 0.989927i $$-0.545218\pi$$
−0.141579 + 0.989927i $$0.545218\pi$$
$$450$$ 0 0
$$451$$ 0 0
$$452$$ 6.00000i 0.282216i
$$453$$ 8.00000i 0.375873i
$$454$$ 12.0000 0.563188
$$455$$ 0 0
$$456$$ 4.00000 0.187317
$$457$$ 10.0000i 0.467780i 0.972263 + 0.233890i $$0.0751456\pi$$
−0.972263 + 0.233890i $$0.924854\pi$$
$$458$$ − 14.0000i − 0.654177i
$$459$$ 1.00000 0.0466760
$$460$$ 0 0
$$461$$ 30.0000 1.39724 0.698620 0.715493i $$-0.253798\pi$$
0.698620 + 0.715493i $$0.253798\pi$$
$$462$$ 0 0
$$463$$ 20.0000i 0.929479i 0.885448 + 0.464739i $$0.153852\pi$$
−0.885448 + 0.464739i $$0.846148\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 30.0000 1.38972
$$467$$ − 12.0000i − 0.555294i −0.960683 0.277647i $$-0.910445\pi$$
0.960683 0.277647i $$-0.0895545\pi$$
$$468$$ 2.00000i 0.0924500i
$$469$$ 8.00000 0.369406
$$470$$ 0 0
$$471$$ −10.0000 −0.460776
$$472$$ − 12.0000i − 0.552345i
$$473$$ 0 0
$$474$$ −10.0000 −0.459315
$$475$$ 0 0
$$476$$ −2.00000 −0.0916698
$$477$$ − 6.00000i − 0.274721i
$$478$$ 12.0000i 0.548867i
$$479$$ −6.00000 −0.274147 −0.137073 0.990561i $$-0.543770\pi$$
−0.137073 + 0.990561i $$0.543770\pi$$
$$480$$ 0 0
$$481$$ 16.0000 0.729537
$$482$$ − 10.0000i − 0.455488i
$$483$$ − 12.0000i − 0.546019i
$$484$$ 11.0000 0.500000
$$485$$ 0 0
$$486$$ −1.00000 −0.0453609
$$487$$ 10.0000i 0.453143i 0.973995 + 0.226572i $$0.0727517\pi$$
−0.973995 + 0.226572i $$0.927248\pi$$
$$488$$ − 8.00000i − 0.362143i
$$489$$ −20.0000 −0.904431
$$490$$ 0 0
$$491$$ −36.0000 −1.62466 −0.812329 0.583200i $$-0.801800\pi$$
−0.812329 + 0.583200i $$0.801800\pi$$
$$492$$ − 6.00000i − 0.270501i
$$493$$ 0 0
$$494$$ −8.00000 −0.359937
$$495$$ 0 0
$$496$$ −10.0000 −0.449013
$$497$$ − 12.0000i − 0.538274i
$$498$$ − 12.0000i − 0.537733i
$$499$$ −32.0000 −1.43252 −0.716258 0.697835i $$-0.754147\pi$$
−0.716258 + 0.697835i $$0.754147\pi$$
$$500$$ 0 0
$$501$$ 18.0000 0.804181
$$502$$ 12.0000i 0.535586i
$$503$$ − 6.00000i − 0.267527i −0.991013 0.133763i $$-0.957294\pi$$
0.991013 0.133763i $$-0.0427062\pi$$
$$504$$ 2.00000 0.0890871
$$505$$ 0 0
$$506$$ 0 0
$$507$$ 9.00000i 0.399704i
$$508$$ 8.00000i 0.354943i
$$509$$ 6.00000 0.265945 0.132973 0.991120i $$-0.457548\pi$$
0.132973 + 0.991120i $$0.457548\pi$$
$$510$$ 0 0
$$511$$ 4.00000 0.176950
$$512$$ 1.00000i 0.0441942i
$$513$$ − 4.00000i − 0.176604i
$$514$$ 18.0000 0.793946
$$515$$ 0 0
$$516$$ −4.00000 −0.176090
$$517$$ 0 0
$$518$$ − 16.0000i − 0.703000i
$$519$$ 0 0
$$520$$ 0 0
$$521$$ 6.00000 0.262865 0.131432 0.991325i $$-0.458042\pi$$
0.131432 + 0.991325i $$0.458042\pi$$
$$522$$ 0 0
$$523$$ 20.0000i 0.874539i 0.899331 + 0.437269i $$0.144054\pi$$
−0.899331 + 0.437269i $$0.855946\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ −12.0000 −0.523225
$$527$$ − 10.0000i − 0.435607i
$$528$$ 0 0
$$529$$ −13.0000 −0.565217
$$530$$ 0 0
$$531$$ −12.0000 −0.520756
$$532$$ 8.00000i 0.346844i
$$533$$ 12.0000i 0.519778i
$$534$$ −18.0000 −0.778936
$$535$$ 0 0
$$536$$ 4.00000 0.172774
$$537$$ 12.0000i 0.517838i
$$538$$ − 24.0000i − 1.03471i
$$539$$ 0 0
$$540$$ 0 0
$$541$$ −16.0000 −0.687894 −0.343947 0.938989i $$-0.611764\pi$$
−0.343947 + 0.938989i $$0.611764\pi$$
$$542$$ − 16.0000i − 0.687259i
$$543$$ 8.00000i 0.343313i
$$544$$ −1.00000 −0.0428746
$$545$$ 0 0
$$546$$ −4.00000 −0.171184
$$547$$ − 8.00000i − 0.342055i −0.985266 0.171028i $$-0.945291\pi$$
0.985266 0.171028i $$-0.0547087\pi$$
$$548$$ − 6.00000i − 0.256307i
$$549$$ −8.00000 −0.341432
$$550$$ 0 0
$$551$$ 0 0
$$552$$ − 6.00000i − 0.255377i
$$553$$ − 20.0000i − 0.850487i
$$554$$ −4.00000 −0.169944
$$555$$ 0 0
$$556$$ 8.00000 0.339276
$$557$$ 42.0000i 1.77960i 0.456354 + 0.889799i $$0.349155\pi$$
−0.456354 + 0.889799i $$0.650845\pi$$
$$558$$ 10.0000i 0.423334i
$$559$$ 8.00000 0.338364
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 30.0000i 1.26547i
$$563$$ − 12.0000i − 0.505740i −0.967500 0.252870i $$-0.918626\pi$$
0.967500 0.252870i $$-0.0813744\pi$$
$$564$$ −12.0000 −0.505291
$$565$$ 0 0
$$566$$ 16.0000 0.672530
$$567$$ − 2.00000i − 0.0839921i
$$568$$ − 6.00000i − 0.251754i
$$569$$ 6.00000 0.251533 0.125767 0.992060i $$-0.459861\pi$$
0.125767 + 0.992060i $$0.459861\pi$$
$$570$$ 0 0
$$571$$ 20.0000 0.836974 0.418487 0.908223i $$-0.362561\pi$$
0.418487 + 0.908223i $$0.362561\pi$$
$$572$$ 0 0
$$573$$ − 12.0000i − 0.501307i
$$574$$ 12.0000 0.500870
$$575$$ 0 0
$$576$$ 1.00000 0.0416667
$$577$$ − 2.00000i − 0.0832611i −0.999133 0.0416305i $$-0.986745\pi$$
0.999133 0.0416305i $$-0.0132552\pi$$
$$578$$ − 1.00000i − 0.0415945i
$$579$$ −14.0000 −0.581820
$$580$$ 0 0
$$581$$ 24.0000 0.995688
$$582$$ 14.0000i 0.580319i
$$583$$ 0 0
$$584$$ 2.00000 0.0827606
$$585$$ 0 0
$$586$$ −18.0000 −0.743573
$$587$$ 12.0000i 0.495293i 0.968850 + 0.247647i $$0.0796572\pi$$
−0.968850 + 0.247647i $$0.920343\pi$$
$$588$$ − 3.00000i − 0.123718i
$$589$$ −40.0000 −1.64817
$$590$$ 0 0
$$591$$ −12.0000 −0.493614
$$592$$ − 8.00000i − 0.328798i
$$593$$ 18.0000i 0.739171i 0.929197 + 0.369586i $$0.120500\pi$$
−0.929197 + 0.369586i $$0.879500\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ −6.00000 −0.245770
$$597$$ − 2.00000i − 0.0818546i
$$598$$ 12.0000i 0.490716i
$$599$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$600$$ 0 0
$$601$$ −22.0000 −0.897399 −0.448699 0.893683i $$-0.648113\pi$$
−0.448699 + 0.893683i $$0.648113\pi$$
$$602$$ − 8.00000i − 0.326056i
$$603$$ − 4.00000i − 0.162893i
$$604$$ −8.00000 −0.325515
$$605$$ 0 0
$$606$$ −6.00000 −0.243733
$$607$$ 10.0000i 0.405887i 0.979190 + 0.202944i $$0.0650509\pi$$
−0.979190 + 0.202944i $$0.934949\pi$$
$$608$$ 4.00000i 0.162221i
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 24.0000 0.970936
$$612$$ 1.00000i 0.0404226i
$$613$$ − 34.0000i − 1.37325i −0.727013 0.686624i $$-0.759092\pi$$
0.727013 0.686624i $$-0.240908\pi$$
$$614$$ 20.0000 0.807134
$$615$$ 0 0
$$616$$ 0 0
$$617$$ − 18.0000i − 0.724653i −0.932051 0.362326i $$-0.881983\pi$$
0.932051 0.362326i $$-0.118017\pi$$
$$618$$ 4.00000i 0.160904i
$$619$$ 28.0000 1.12542 0.562708 0.826656i $$-0.309760\pi$$
0.562708 + 0.826656i $$0.309760\pi$$
$$620$$ 0 0
$$621$$ −6.00000 −0.240772
$$622$$ 18.0000i 0.721734i
$$623$$ − 36.0000i − 1.44231i
$$624$$ −2.00000 −0.0800641
$$625$$ 0 0
$$626$$ −14.0000 −0.559553
$$627$$ 0 0
$$628$$ − 10.0000i − 0.399043i
$$629$$ 8.00000 0.318981
$$630$$ 0 0
$$631$$ −4.00000 −0.159237 −0.0796187 0.996825i $$-0.525370\pi$$
−0.0796187 + 0.996825i $$0.525370\pi$$
$$632$$ − 10.0000i − 0.397779i
$$633$$ 8.00000i 0.317971i
$$634$$ 12.0000 0.476581
$$635$$ 0 0
$$636$$ 6.00000 0.237915
$$637$$ 6.00000i 0.237729i
$$638$$ 0 0
$$639$$ −6.00000 −0.237356
$$640$$ 0 0
$$641$$ 30.0000 1.18493 0.592464 0.805597i $$-0.298155\pi$$
0.592464 + 0.805597i $$0.298155\pi$$
$$642$$ 0 0
$$643$$ − 4.00000i − 0.157745i −0.996885 0.0788723i $$-0.974868\pi$$
0.996885 0.0788723i $$-0.0251319\pi$$
$$644$$ 12.0000 0.472866
$$645$$ 0 0
$$646$$ −4.00000 −0.157378
$$647$$ − 36.0000i − 1.41531i −0.706560 0.707653i $$-0.749754\pi$$
0.706560 0.707653i $$-0.250246\pi$$
$$648$$ − 1.00000i − 0.0392837i
$$649$$ 0 0
$$650$$ 0 0
$$651$$ −20.0000 −0.783862
$$652$$ − 20.0000i − 0.783260i
$$653$$ − 24.0000i − 0.939193i −0.882881 0.469596i $$-0.844399\pi$$
0.882881 0.469596i $$-0.155601\pi$$
$$654$$ 20.0000 0.782062
$$655$$ 0 0
$$656$$ 6.00000 0.234261
$$657$$ − 2.00000i − 0.0780274i
$$658$$ − 24.0000i − 0.935617i
$$659$$ −36.0000 −1.40236 −0.701180 0.712984i $$-0.747343\pi$$
−0.701180 + 0.712984i $$0.747343\pi$$
$$660$$ 0 0
$$661$$ −22.0000 −0.855701 −0.427850 0.903850i $$-0.640729\pi$$
−0.427850 + 0.903850i $$0.640729\pi$$
$$662$$ 20.0000i 0.777322i
$$663$$ − 2.00000i − 0.0776736i
$$664$$ 12.0000 0.465690
$$665$$ 0 0
$$666$$ −8.00000 −0.309994
$$667$$ 0 0
$$668$$ 18.0000i 0.696441i
$$669$$ 28.0000 1.08254
$$670$$ 0 0
$$671$$ 0 0
$$672$$ 2.00000i 0.0771517i
$$673$$ − 34.0000i − 1.31060i −0.755367 0.655302i $$-0.772541\pi$$
0.755367 0.655302i $$-0.227459\pi$$
$$674$$ −22.0000 −0.847408
$$675$$ 0 0
$$676$$ −9.00000 −0.346154
$$677$$ 12.0000i 0.461197i 0.973049 + 0.230599i $$0.0740685\pi$$
−0.973049 + 0.230599i $$0.925932\pi$$
$$678$$ 6.00000i 0.230429i
$$679$$ −28.0000 −1.07454
$$680$$ 0 0
$$681$$ 12.0000 0.459841
$$682$$ 0 0
$$683$$ − 12.0000i − 0.459167i −0.973289 0.229584i $$-0.926264\pi$$
0.973289 0.229584i $$-0.0737364\pi$$
$$684$$ 4.00000 0.152944
$$685$$ 0 0
$$686$$ 20.0000 0.763604
$$687$$ − 14.0000i − 0.534133i
$$688$$ − 4.00000i − 0.152499i
$$689$$ −12.0000 −0.457164
$$690$$ 0 0
$$691$$ 32.0000 1.21734 0.608669 0.793424i $$-0.291704\pi$$
0.608669 + 0.793424i $$0.291704\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ −12.0000 −0.455514
$$695$$ 0 0
$$696$$ 0 0
$$697$$ 6.00000i 0.227266i
$$698$$ − 26.0000i − 0.984115i
$$699$$ 30.0000 1.13470
$$700$$ 0 0
$$701$$ −42.0000 −1.58632 −0.793159 0.609015i $$-0.791565\pi$$
−0.793159 + 0.609015i $$0.791565\pi$$
$$702$$ 2.00000i 0.0754851i
$$703$$ − 32.0000i − 1.20690i
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 18.0000 0.677439
$$707$$ − 12.0000i − 0.451306i
$$708$$ − 12.0000i − 0.450988i
$$709$$ −20.0000 −0.751116 −0.375558 0.926799i $$-0.622549\pi$$
−0.375558 + 0.926799i $$0.622549\pi$$
$$710$$ 0 0
$$711$$ −10.0000 −0.375029
$$712$$ − 18.0000i − 0.674579i
$$713$$ 60.0000i 2.24702i
$$714$$ −2.00000 −0.0748481
$$715$$ 0 0
$$716$$ −12.0000 −0.448461
$$717$$ 12.0000i 0.448148i
$$718$$ 0 0
$$719$$ −6.00000 −0.223762 −0.111881 0.993722i $$-0.535688\pi$$
−0.111881 + 0.993722i $$0.535688\pi$$
$$720$$ 0 0
$$721$$ −8.00000 −0.297936
$$722$$ − 3.00000i − 0.111648i
$$723$$ − 10.0000i − 0.371904i
$$724$$ −8.00000 −0.297318
$$725$$ 0 0
$$726$$ 11.0000 0.408248
$$727$$ − 8.00000i − 0.296704i −0.988935 0.148352i $$-0.952603\pi$$
0.988935 0.148352i $$-0.0473968\pi$$
$$728$$ − 4.00000i − 0.148250i
$$729$$ −1.00000 −0.0370370
$$730$$ 0 0
$$731$$ 4.00000 0.147945
$$732$$ − 8.00000i − 0.295689i
$$733$$ 26.0000i 0.960332i 0.877178 + 0.480166i $$0.159424\pi$$
−0.877178 + 0.480166i $$0.840576\pi$$
$$734$$ −10.0000 −0.369107
$$735$$ 0 0
$$736$$ 6.00000 0.221163
$$737$$ 0 0
$$738$$ − 6.00000i − 0.220863i
$$739$$ 4.00000 0.147142 0.0735712 0.997290i $$-0.476560\pi$$
0.0735712 + 0.997290i $$0.476560\pi$$
$$740$$ 0 0
$$741$$ −8.00000 −0.293887
$$742$$ 12.0000i 0.440534i
$$743$$ 42.0000i 1.54083i 0.637542 + 0.770415i $$0.279951\pi$$
−0.637542 + 0.770415i $$0.720049\pi$$
$$744$$ −10.0000 −0.366618
$$745$$ 0 0
$$746$$ 22.0000 0.805477
$$747$$ − 12.0000i − 0.439057i
$$748$$ 0 0
$$749$$ 0 0
$$750$$ 0 0
$$751$$ 2.00000 0.0729810 0.0364905 0.999334i $$-0.488382\pi$$
0.0364905 + 0.999334i $$0.488382\pi$$
$$752$$ − 12.0000i − 0.437595i
$$753$$ 12.0000i 0.437304i
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 2.00000 0.0727393
$$757$$ 34.0000i 1.23575i 0.786276 + 0.617876i $$0.212006\pi$$
−0.786276 + 0.617876i $$0.787994\pi$$
$$758$$ 28.0000i 1.01701i
$$759$$ 0 0
$$760$$ 0 0
$$761$$ −54.0000 −1.95750 −0.978749 0.205061i $$-0.934261\pi$$
−0.978749 + 0.205061i $$0.934261\pi$$
$$762$$ 8.00000i 0.289809i
$$763$$ 40.0000i 1.44810i
$$764$$ 12.0000 0.434145
$$765$$ 0 0
$$766$$ 12.0000 0.433578
$$767$$ 24.0000i 0.866590i
$$768$$ 1.00000i 0.0360844i
$$769$$ 22.0000 0.793340 0.396670 0.917961i $$-0.370166\pi$$
0.396670 + 0.917961i $$0.370166\pi$$
$$770$$ 0 0
$$771$$ 18.0000 0.648254
$$772$$ − 14.0000i − 0.503871i
$$773$$ − 6.00000i − 0.215805i −0.994161 0.107903i $$-0.965587\pi$$
0.994161 0.107903i $$-0.0344134\pi$$
$$774$$ −4.00000 −0.143777
$$775$$ 0 0
$$776$$ −14.0000 −0.502571
$$777$$ − 16.0000i − 0.573997i
$$778$$ 30.0000i 1.07555i
$$779$$ 24.0000 0.859889
$$780$$ 0 0
$$781$$ 0 0
$$782$$ 6.00000i 0.214560i
$$783$$ 0 0
$$784$$ 3.00000 0.107143
$$785$$ 0 0
$$786$$ 0 0
$$787$$ − 32.0000i − 1.14068i −0.821410 0.570338i $$-0.806812\pi$$
0.821410 0.570338i $$-0.193188\pi$$
$$788$$ − 12.0000i − 0.427482i
$$789$$ −12.0000 −0.427211
$$790$$ 0 0
$$791$$ −12.0000 −0.426671
$$792$$ 0 0
$$793$$ 16.0000i 0.568177i
$$794$$ −16.0000 −0.567819
$$795$$ 0 0
$$796$$ 2.00000 0.0708881
$$797$$ − 6.00000i − 0.212531i −0.994338 0.106265i $$-0.966111\pi$$
0.994338 0.106265i $$-0.0338893\pi$$
$$798$$ 8.00000i 0.283197i
$$799$$ 12.0000 0.424529
$$800$$ 0 0
$$801$$ −18.0000 −0.635999
$$802$$ − 18.0000i − 0.635602i
$$803$$ 0 0
$$804$$ 4.00000 0.141069
$$805$$ 0 0
$$806$$ 20.0000 0.704470
$$807$$ − 24.0000i − 0.844840i
$$808$$ − 6.00000i − 0.211079i
$$809$$ −6.00000 −0.210949 −0.105474 0.994422i $$-0.533636\pi$$
−0.105474 + 0.994422i $$0.533636\pi$$
$$810$$ 0 0
$$811$$ −4.00000 −0.140459 −0.0702295 0.997531i $$-0.522373\pi$$
−0.0702295 + 0.997531i $$0.522373\pi$$
$$812$$ 0 0
$$813$$ − 16.0000i − 0.561144i
$$814$$ 0 0
$$815$$ 0 0
$$816$$ −1.00000 −0.0350070
$$817$$ − 16.0000i − 0.559769i
$$818$$ − 38.0000i − 1.32864i
$$819$$ −4.00000 −0.139771
$$820$$ 0 0
$$821$$ −36.0000 −1.25641 −0.628204 0.778048i $$-0.716210\pi$$
−0.628204 + 0.778048i $$0.716210\pi$$
$$822$$ − 6.00000i − 0.209274i
$$823$$ − 34.0000i − 1.18517i −0.805510 0.592583i $$-0.798108\pi$$
0.805510 0.592583i $$-0.201892\pi$$
$$824$$ −4.00000 −0.139347
$$825$$ 0 0
$$826$$ 24.0000 0.835067
$$827$$ − 48.0000i − 1.66912i −0.550914 0.834562i $$-0.685721\pi$$
0.550914 0.834562i $$-0.314279\pi$$
$$828$$ − 6.00000i − 0.208514i
$$829$$ 10.0000 0.347314 0.173657 0.984806i $$-0.444442\pi$$
0.173657 + 0.984806i $$0.444442\pi$$
$$830$$ 0 0
$$831$$ −4.00000 −0.138758
$$832$$ − 2.00000i − 0.0693375i
$$833$$ 3.00000i 0.103944i
$$834$$ 8.00000 0.277017
$$835$$ 0 0
$$836$$ 0 0
$$837$$ 10.0000i 0.345651i
$$838$$ 36.0000i 1.24360i
$$839$$ 18.0000 0.621429 0.310715 0.950503i $$-0.399432\pi$$
0.310715 + 0.950503i $$0.399432\pi$$
$$840$$ 0 0
$$841$$ −29.0000 −1.00000
$$842$$ − 22.0000i − 0.758170i
$$843$$ 30.0000i 1.03325i
$$844$$ −8.00000 −0.275371
$$845$$ 0 0
$$846$$ −12.0000 −0.412568
$$847$$ 22.0000i 0.755929i
$$848$$ 6.00000i 0.206041i
$$849$$ 16.0000 0.549119
$$850$$ 0 0
$$851$$ −48.0000 −1.64542
$$852$$ − 6.00000i − 0.205557i
$$853$$ − 28.0000i − 0.958702i −0.877623 0.479351i $$-0.840872\pi$$
0.877623 0.479351i $$-0.159128\pi$$
$$854$$ 16.0000 0.547509
$$855$$ 0 0
$$856$$ 0 0
$$857$$ − 6.00000i − 0.204956i −0.994735 0.102478i $$-0.967323\pi$$
0.994735 0.102478i $$-0.0326771\pi$$
$$858$$ 0 0
$$859$$ 28.0000 0.955348 0.477674 0.878537i $$-0.341480\pi$$
0.477674 + 0.878537i $$0.341480\pi$$
$$860$$ 0 0
$$861$$ 12.0000 0.408959
$$862$$ − 30.0000i − 1.02180i
$$863$$ 48.0000i 1.63394i 0.576681 + 0.816970i $$0.304348\pi$$
−0.576681 + 0.816970i $$0.695652\pi$$
$$864$$ 1.00000 0.0340207
$$865$$ 0 0
$$866$$ −14.0000 −0.475739
$$867$$ − 1.00000i − 0.0339618i
$$868$$ − 20.0000i − 0.678844i
$$869$$ 0 0
$$870$$ 0 0
$$871$$ −8.00000 −0.271070
$$872$$ 20.0000i 0.677285i
$$873$$ 14.0000i 0.473828i
$$874$$ 24.0000 0.811812
$$875$$ 0 0
$$876$$ 2.00000 0.0675737
$$877$$ 4.00000i 0.135070i 0.997717 + 0.0675352i $$0.0215135\pi$$
−0.997717 + 0.0675352i $$0.978487\pi$$
$$878$$ − 26.0000i − 0.877457i
$$879$$ −18.0000 −0.607125
$$880$$ 0 0
$$881$$ −54.0000 −1.81931 −0.909653 0.415369i $$-0.863653\pi$$
−0.909653 + 0.415369i $$0.863653\pi$$
$$882$$ − 3.00000i − 0.101015i
$$883$$ 20.0000i 0.673054i 0.941674 + 0.336527i $$0.109252\pi$$
−0.941674 + 0.336527i $$0.890748\pi$$
$$884$$ 2.00000 0.0672673
$$885$$ 0 0
$$886$$ 36.0000 1.20944
$$887$$ 18.0000i 0.604381i 0.953248 + 0.302190i $$0.0977178\pi$$
−0.953248 + 0.302190i $$0.902282\pi$$
$$888$$ − 8.00000i − 0.268462i
$$889$$ −16.0000 −0.536623
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 28.0000i 0.937509i
$$893$$ − 48.0000i − 1.60626i
$$894$$ −6.00000 −0.200670
$$895$$ 0 0
$$896$$ −2.00000 −0.0668153
$$897$$ 12.0000i 0.400668i
$$898$$ − 6.00000i − 0.200223i
$$899$$ 0 0
$$900$$ 0 0
$$901$$ −6.00000 −0.199889
$$902$$ 0 0
$$903$$ − 8.00000i − 0.266223i
$$904$$ −6.00000 −0.199557
$$905$$ 0 0
$$906$$ −8.00000 −0.265782
$$907$$ 40.0000i 1.32818i 0.747653 + 0.664089i $$0.231180\pi$$
−0.747653 + 0.664089i $$0.768820\pi$$
$$908$$ 12.0000i 0.398234i
$$909$$ −6.00000 −0.199007
$$910$$ 0 0
$$911$$ 6.00000 0.198789 0.0993944 0.995048i $$-0.468309\pi$$
0.0993944 + 0.995048i $$0.468309\pi$$
$$912$$ 4.00000i 0.132453i
$$913$$ 0 0
$$914$$ −10.0000 −0.330771
$$915$$ 0 0
$$916$$ 14.0000 0.462573
$$917$$ 0 0
$$918$$ 1.00000i 0.0330049i
$$919$$ 16.0000 0.527791 0.263896 0.964551i $$-0.414993\pi$$
0.263896 + 0.964551i $$0.414993\pi$$
$$920$$ 0 0
$$921$$ 20.0000 0.659022
$$922$$ 30.0000i 0.987997i
$$923$$ 12.0000i 0.394985i
$$924$$ 0 0
$$925$$ 0 0
$$926$$ −20.0000 −0.657241
$$927$$ 4.00000i 0.131377i
$$928$$ 0 0
$$929$$ 18.0000 0.590561 0.295280 0.955411i $$-0.404587\pi$$
0.295280 + 0.955411i $$0.404587\pi$$
$$930$$ 0 0
$$931$$ 12.0000 0.393284
$$932$$ 30.0000i 0.982683i
$$933$$ 18.0000i 0.589294i
$$934$$ 12.0000 0.392652
$$935$$ 0 0
$$936$$ −2.00000 −0.0653720
$$937$$ 10.0000i 0.326686i 0.986569 + 0.163343i $$0.0522277\pi$$
−0.986569 + 0.163343i $$0.947772\pi$$
$$938$$ 8.00000i 0.261209i
$$939$$ −14.0000 −0.456873
$$940$$ 0 0
$$941$$ 12.0000 0.391189 0.195594 0.980685i $$-0.437336\pi$$
0.195594 + 0.980685i $$0.437336\pi$$
$$942$$ − 10.0000i − 0.325818i
$$943$$ − 36.0000i − 1.17232i
$$944$$ 12.0000 0.390567
$$945$$ 0 0
$$946$$ 0 0
$$947$$ − 24.0000i − 0.779895i −0.920837 0.389948i $$-0.872493\pi$$
0.920837 0.389948i $$-0.127507\pi$$
$$948$$ − 10.0000i − 0.324785i
$$949$$ −4.00000 −0.129845
$$950$$ 0 0
$$951$$ 12.0000 0.389127
$$952$$ − 2.00000i − 0.0648204i
$$953$$ 54.0000i 1.74923i 0.484817 + 0.874616i $$0.338886\pi$$
−0.484817 + 0.874616i $$0.661114\pi$$
$$954$$ 6.00000 0.194257
$$955$$ 0 0
$$956$$ −12.0000 −0.388108
$$957$$ 0 0
$$958$$ − 6.00000i − 0.193851i
$$959$$ 12.0000 0.387500
$$960$$ 0 0
$$961$$ 69.0000 2.22581
$$962$$ 16.0000i 0.515861i
$$963$$ 0 0
$$964$$ 10.0000 0.322078
$$965$$ 0 0
$$966$$ 12.0000 0.386094
$$967$$ 4.00000i 0.128631i 0.997930 + 0.0643157i $$0.0204865\pi$$
−0.997930 + 0.0643157i $$0.979514\pi$$
$$968$$ 11.0000i 0.353553i
$$969$$ −4.00000 −0.128499
$$970$$ 0 0
$$971$$ 12.0000 0.385098 0.192549 0.981287i $$-0.438325\pi$$
0.192549 + 0.981287i $$0.438325\pi$$
$$972$$ − 1.00000i − 0.0320750i
$$973$$ 16.0000i 0.512936i
$$974$$ −10.0000 −0.320421
$$975$$ 0 0
$$976$$ 8.00000 0.256074
$$977$$ 30.0000i 0.959785i 0.877327 + 0.479893i $$0.159324\pi$$
−0.877327 + 0.479893i $$0.840676\pi$$
$$978$$ − 20.0000i − 0.639529i
$$979$$ 0 0
$$980$$ 0 0
$$981$$ 20.0000 0.638551
$$982$$ − 36.0000i − 1.14881i
$$983$$ 6.00000i 0.191370i 0.995412 + 0.0956851i $$0.0305042\pi$$
−0.995412 + 0.0956851i $$0.969496\pi$$
$$984$$ 6.00000 0.191273
$$985$$ 0 0
$$986$$ 0 0
$$987$$ − 24.0000i − 0.763928i
$$988$$ − 8.00000i − 0.254514i
$$989$$ −24.0000 −0.763156
$$990$$ 0 0
$$991$$ 2.00000 0.0635321 0.0317660 0.999495i $$-0.489887\pi$$
0.0317660 + 0.999495i $$0.489887\pi$$
$$992$$ − 10.0000i − 0.317500i
$$993$$ 20.0000i 0.634681i
$$994$$ 12.0000 0.380617
$$995$$ 0 0
$$996$$ 12.0000 0.380235
$$997$$ − 8.00000i − 0.253363i −0.991943 0.126681i $$-0.959567\pi$$
0.991943 0.126681i $$-0.0404325\pi$$
$$998$$ − 32.0000i − 1.01294i
$$999$$ −8.00000 −0.253109
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.g.2449.2 2
5.2 odd 4 102.2.a.b.1.1 1
5.3 odd 4 2550.2.a.u.1.1 1
5.4 even 2 inner 2550.2.d.g.2449.1 2
15.2 even 4 306.2.a.c.1.1 1
15.8 even 4 7650.2.a.j.1.1 1
20.7 even 4 816.2.a.d.1.1 1
35.27 even 4 4998.2.a.d.1.1 1
40.27 even 4 3264.2.a.w.1.1 1
40.37 odd 4 3264.2.a.i.1.1 1
60.47 odd 4 2448.2.a.i.1.1 1
85.2 odd 8 1734.2.f.b.1483.2 4
85.32 odd 8 1734.2.f.b.1483.1 4
85.42 odd 8 1734.2.f.b.829.1 4
85.47 odd 4 1734.2.b.f.577.2 2
85.67 odd 4 1734.2.a.b.1.1 1
85.72 odd 4 1734.2.b.f.577.1 2
85.77 odd 8 1734.2.f.b.829.2 4
120.77 even 4 9792.2.a.bg.1.1 1
120.107 odd 4 9792.2.a.ba.1.1 1
255.152 even 4 5202.2.a.j.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
102.2.a.b.1.1 1 5.2 odd 4
306.2.a.c.1.1 1 15.2 even 4
816.2.a.d.1.1 1 20.7 even 4
1734.2.a.b.1.1 1 85.67 odd 4
1734.2.b.f.577.1 2 85.72 odd 4
1734.2.b.f.577.2 2 85.47 odd 4
1734.2.f.b.829.1 4 85.42 odd 8
1734.2.f.b.829.2 4 85.77 odd 8
1734.2.f.b.1483.1 4 85.32 odd 8
1734.2.f.b.1483.2 4 85.2 odd 8
2448.2.a.i.1.1 1 60.47 odd 4
2550.2.a.u.1.1 1 5.3 odd 4
2550.2.d.g.2449.1 2 5.4 even 2 inner
2550.2.d.g.2449.2 2 1.1 even 1 trivial
3264.2.a.i.1.1 1 40.37 odd 4
3264.2.a.w.1.1 1 40.27 even 4
4998.2.a.d.1.1 1 35.27 even 4
5202.2.a.j.1.1 1 255.152 even 4
7650.2.a.j.1.1 1 15.8 even 4
9792.2.a.ba.1.1 1 120.107 odd 4
9792.2.a.bg.1.1 1 120.77 even 4