# Properties

 Label 2550.2.d.h.2449.1 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: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

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

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{6} +4.00000i q^{7} +1.00000i q^{8} -1.00000 q^{9} +O(q^{10})$$ $$q-1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{6} +4.00000i q^{7} +1.00000i q^{8} -1.00000 q^{9} +2.00000 q^{11} +1.00000i q^{12} -2.00000i q^{13} +4.00000 q^{14} +1.00000 q^{16} -1.00000i q^{17} +1.00000i q^{18} -8.00000 q^{19} +4.00000 q^{21} -2.00000i q^{22} -1.00000i q^{23} +1.00000 q^{24} -2.00000 q^{26} +1.00000i q^{27} -4.00000i q^{28} +4.00000 q^{29} -2.00000 q^{31} -1.00000i q^{32} -2.00000i q^{33} -1.00000 q^{34} +1.00000 q^{36} +3.00000i q^{37} +8.00000i q^{38} -2.00000 q^{39} -1.00000 q^{41} -4.00000i q^{42} +6.00000i q^{43} -2.00000 q^{44} -1.00000 q^{46} +4.00000i q^{47} -1.00000i q^{48} -9.00000 q^{49} -1.00000 q^{51} +2.00000i q^{52} +13.0000i q^{53} +1.00000 q^{54} -4.00000 q^{56} +8.00000i q^{57} -4.00000i q^{58} -15.0000 q^{59} +5.00000 q^{61} +2.00000i q^{62} -4.00000i q^{63} -1.00000 q^{64} -2.00000 q^{66} -10.0000i q^{67} +1.00000i q^{68} -1.00000 q^{69} -1.00000 q^{71} -1.00000i q^{72} +16.0000i q^{73} +3.00000 q^{74} +8.00000 q^{76} +8.00000i q^{77} +2.00000i q^{78} -12.0000 q^{79} +1.00000 q^{81} +1.00000i q^{82} +11.0000i q^{83} -4.00000 q^{84} +6.00000 q^{86} -4.00000i q^{87} +2.00000i q^{88} +2.00000 q^{89} +8.00000 q^{91} +1.00000i q^{92} +2.00000i q^{93} +4.00000 q^{94} -1.00000 q^{96} +9.00000i q^{98} -2.00000 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{4} - 2 q^{6} - 2 q^{9}+O(q^{10})$$ 2 * q - 2 * q^4 - 2 * q^6 - 2 * q^9 $$2 q - 2 q^{4} - 2 q^{6} - 2 q^{9} + 4 q^{11} + 8 q^{14} + 2 q^{16} - 16 q^{19} + 8 q^{21} + 2 q^{24} - 4 q^{26} + 8 q^{29} - 4 q^{31} - 2 q^{34} + 2 q^{36} - 4 q^{39} - 2 q^{41} - 4 q^{44} - 2 q^{46} - 18 q^{49} - 2 q^{51} + 2 q^{54} - 8 q^{56} - 30 q^{59} + 10 q^{61} - 2 q^{64} - 4 q^{66} - 2 q^{69} - 2 q^{71} + 6 q^{74} + 16 q^{76} - 24 q^{79} + 2 q^{81} - 8 q^{84} + 12 q^{86} + 4 q^{89} + 16 q^{91} + 8 q^{94} - 2 q^{96} - 4 q^{99}+O(q^{100})$$ 2 * q - 2 * q^4 - 2 * q^6 - 2 * q^9 + 4 * q^11 + 8 * q^14 + 2 * q^16 - 16 * q^19 + 8 * q^21 + 2 * q^24 - 4 * q^26 + 8 * q^29 - 4 * q^31 - 2 * q^34 + 2 * q^36 - 4 * q^39 - 2 * q^41 - 4 * q^44 - 2 * q^46 - 18 * q^49 - 2 * q^51 + 2 * q^54 - 8 * q^56 - 30 * q^59 + 10 * q^61 - 2 * q^64 - 4 * q^66 - 2 * q^69 - 2 * q^71 + 6 * q^74 + 16 * q^76 - 24 * q^79 + 2 * q^81 - 8 * q^84 + 12 * q^86 + 4 * q^89 + 16 * q^91 + 8 * q^94 - 2 * q^96 - 4 * 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
$$7$$ 4.00000i 1.51186i 0.654654 + 0.755929i $$0.272814\pi$$
−0.654654 + 0.755929i $$0.727186\pi$$
$$8$$ 1.00000i 0.353553i
$$9$$ −1.00000 −0.333333
$$10$$ 0 0
$$11$$ 2.00000 0.603023 0.301511 0.953463i $$-0.402509\pi$$
0.301511 + 0.953463i $$0.402509\pi$$
$$12$$ 1.00000i 0.288675i
$$13$$ − 2.00000i − 0.554700i −0.960769 0.277350i $$-0.910544\pi$$
0.960769 0.277350i $$-0.0894562\pi$$
$$14$$ 4.00000 1.06904
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ − 1.00000i − 0.242536i
$$18$$ 1.00000i 0.235702i
$$19$$ −8.00000 −1.83533 −0.917663 0.397360i $$-0.869927\pi$$
−0.917663 + 0.397360i $$0.869927\pi$$
$$20$$ 0 0
$$21$$ 4.00000 0.872872
$$22$$ − 2.00000i − 0.426401i
$$23$$ − 1.00000i − 0.208514i −0.994550 0.104257i $$-0.966753\pi$$
0.994550 0.104257i $$-0.0332465\pi$$
$$24$$ 1.00000 0.204124
$$25$$ 0 0
$$26$$ −2.00000 −0.392232
$$27$$ 1.00000i 0.192450i
$$28$$ − 4.00000i − 0.755929i
$$29$$ 4.00000 0.742781 0.371391 0.928477i $$-0.378881\pi$$
0.371391 + 0.928477i $$0.378881\pi$$
$$30$$ 0 0
$$31$$ −2.00000 −0.359211 −0.179605 0.983739i $$-0.557482\pi$$
−0.179605 + 0.983739i $$0.557482\pi$$
$$32$$ − 1.00000i − 0.176777i
$$33$$ − 2.00000i − 0.348155i
$$34$$ −1.00000 −0.171499
$$35$$ 0 0
$$36$$ 1.00000 0.166667
$$37$$ 3.00000i 0.493197i 0.969118 + 0.246598i $$0.0793129\pi$$
−0.969118 + 0.246598i $$0.920687\pi$$
$$38$$ 8.00000i 1.29777i
$$39$$ −2.00000 −0.320256
$$40$$ 0 0
$$41$$ −1.00000 −0.156174 −0.0780869 0.996947i $$-0.524881\pi$$
−0.0780869 + 0.996947i $$0.524881\pi$$
$$42$$ − 4.00000i − 0.617213i
$$43$$ 6.00000i 0.914991i 0.889212 + 0.457496i $$0.151253\pi$$
−0.889212 + 0.457496i $$0.848747\pi$$
$$44$$ −2.00000 −0.301511
$$45$$ 0 0
$$46$$ −1.00000 −0.147442
$$47$$ 4.00000i 0.583460i 0.956501 + 0.291730i $$0.0942309\pi$$
−0.956501 + 0.291730i $$0.905769\pi$$
$$48$$ − 1.00000i − 0.144338i
$$49$$ −9.00000 −1.28571
$$50$$ 0 0
$$51$$ −1.00000 −0.140028
$$52$$ 2.00000i 0.277350i
$$53$$ 13.0000i 1.78569i 0.450367 + 0.892844i $$0.351293\pi$$
−0.450367 + 0.892844i $$0.648707\pi$$
$$54$$ 1.00000 0.136083
$$55$$ 0 0
$$56$$ −4.00000 −0.534522
$$57$$ 8.00000i 1.05963i
$$58$$ − 4.00000i − 0.525226i
$$59$$ −15.0000 −1.95283 −0.976417 0.215894i $$-0.930733\pi$$
−0.976417 + 0.215894i $$0.930733\pi$$
$$60$$ 0 0
$$61$$ 5.00000 0.640184 0.320092 0.947386i $$-0.396286\pi$$
0.320092 + 0.947386i $$0.396286\pi$$
$$62$$ 2.00000i 0.254000i
$$63$$ − 4.00000i − 0.503953i
$$64$$ −1.00000 −0.125000
$$65$$ 0 0
$$66$$ −2.00000 −0.246183
$$67$$ − 10.0000i − 1.22169i −0.791748 0.610847i $$-0.790829\pi$$
0.791748 0.610847i $$-0.209171\pi$$
$$68$$ 1.00000i 0.121268i
$$69$$ −1.00000 −0.120386
$$70$$ 0 0
$$71$$ −1.00000 −0.118678 −0.0593391 0.998238i $$-0.518899\pi$$
−0.0593391 + 0.998238i $$0.518899\pi$$
$$72$$ − 1.00000i − 0.117851i
$$73$$ 16.0000i 1.87266i 0.351123 + 0.936329i $$0.385800\pi$$
−0.351123 + 0.936329i $$0.614200\pi$$
$$74$$ 3.00000 0.348743
$$75$$ 0 0
$$76$$ 8.00000 0.917663
$$77$$ 8.00000i 0.911685i
$$78$$ 2.00000i 0.226455i
$$79$$ −12.0000 −1.35011 −0.675053 0.737769i $$-0.735879\pi$$
−0.675053 + 0.737769i $$0.735879\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 1.00000i 0.110432i
$$83$$ 11.0000i 1.20741i 0.797209 + 0.603703i $$0.206309\pi$$
−0.797209 + 0.603703i $$0.793691\pi$$
$$84$$ −4.00000 −0.436436
$$85$$ 0 0
$$86$$ 6.00000 0.646997
$$87$$ − 4.00000i − 0.428845i
$$88$$ 2.00000i 0.213201i
$$89$$ 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
$$92$$ 1.00000i 0.104257i
$$93$$ 2.00000i 0.207390i
$$94$$ 4.00000 0.412568
$$95$$ 0 0
$$96$$ −1.00000 −0.102062
$$97$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$98$$ 9.00000i 0.909137i
$$99$$ −2.00000 −0.201008
$$100$$ 0 0
$$101$$ −10.0000 −0.995037 −0.497519 0.867453i $$-0.665755\pi$$
−0.497519 + 0.867453i $$0.665755\pi$$
$$102$$ 1.00000i 0.0990148i
$$103$$ − 3.00000i − 0.295599i −0.989017 0.147799i $$-0.952781\pi$$
0.989017 0.147799i $$-0.0472190\pi$$
$$104$$ 2.00000 0.196116
$$105$$ 0 0
$$106$$ 13.0000 1.26267
$$107$$ 12.0000i 1.16008i 0.814587 + 0.580042i $$0.196964\pi$$
−0.814587 + 0.580042i $$0.803036\pi$$
$$108$$ − 1.00000i − 0.0962250i
$$109$$ 2.00000 0.191565 0.0957826 0.995402i $$-0.469465\pi$$
0.0957826 + 0.995402i $$0.469465\pi$$
$$110$$ 0 0
$$111$$ 3.00000 0.284747
$$112$$ 4.00000i 0.377964i
$$113$$ 15.0000i 1.41108i 0.708669 + 0.705541i $$0.249296\pi$$
−0.708669 + 0.705541i $$0.750704\pi$$
$$114$$ 8.00000 0.749269
$$115$$ 0 0
$$116$$ −4.00000 −0.371391
$$117$$ 2.00000i 0.184900i
$$118$$ 15.0000i 1.38086i
$$119$$ 4.00000 0.366679
$$120$$ 0 0
$$121$$ −7.00000 −0.636364
$$122$$ − 5.00000i − 0.452679i
$$123$$ 1.00000i 0.0901670i
$$124$$ 2.00000 0.179605
$$125$$ 0 0
$$126$$ −4.00000 −0.356348
$$127$$ 16.0000i 1.41977i 0.704317 + 0.709885i $$0.251253\pi$$
−0.704317 + 0.709885i $$0.748747\pi$$
$$128$$ 1.00000i 0.0883883i
$$129$$ 6.00000 0.528271
$$130$$ 0 0
$$131$$ 8.00000 0.698963 0.349482 0.936943i $$-0.386358\pi$$
0.349482 + 0.936943i $$0.386358\pi$$
$$132$$ 2.00000i 0.174078i
$$133$$ − 32.0000i − 2.77475i
$$134$$ −10.0000 −0.863868
$$135$$ 0 0
$$136$$ 1.00000 0.0857493
$$137$$ 12.0000i 1.02523i 0.858619 + 0.512615i $$0.171323\pi$$
−0.858619 + 0.512615i $$0.828677\pi$$
$$138$$ 1.00000i 0.0851257i
$$139$$ −1.00000 −0.0848189 −0.0424094 0.999100i $$-0.513503\pi$$
−0.0424094 + 0.999100i $$0.513503\pi$$
$$140$$ 0 0
$$141$$ 4.00000 0.336861
$$142$$ 1.00000i 0.0839181i
$$143$$ − 4.00000i − 0.334497i
$$144$$ −1.00000 −0.0833333
$$145$$ 0 0
$$146$$ 16.0000 1.32417
$$147$$ 9.00000i 0.742307i
$$148$$ − 3.00000i − 0.246598i
$$149$$ −1.00000 −0.0819232 −0.0409616 0.999161i $$-0.513042\pi$$
−0.0409616 + 0.999161i $$0.513042\pi$$
$$150$$ 0 0
$$151$$ 1.00000 0.0813788 0.0406894 0.999172i $$-0.487045\pi$$
0.0406894 + 0.999172i $$0.487045\pi$$
$$152$$ − 8.00000i − 0.648886i
$$153$$ 1.00000i 0.0808452i
$$154$$ 8.00000 0.644658
$$155$$ 0 0
$$156$$ 2.00000 0.160128
$$157$$ − 14.0000i − 1.11732i −0.829396 0.558661i $$-0.811315\pi$$
0.829396 0.558661i $$-0.188685\pi$$
$$158$$ 12.0000i 0.954669i
$$159$$ 13.0000 1.03097
$$160$$ 0 0
$$161$$ 4.00000 0.315244
$$162$$ − 1.00000i − 0.0785674i
$$163$$ 7.00000i 0.548282i 0.961689 + 0.274141i $$0.0883936\pi$$
−0.961689 + 0.274141i $$0.911606\pi$$
$$164$$ 1.00000 0.0780869
$$165$$ 0 0
$$166$$ 11.0000 0.853766
$$167$$ − 24.0000i − 1.85718i −0.371113 0.928588i $$-0.621024\pi$$
0.371113 0.928588i $$-0.378976\pi$$
$$168$$ 4.00000i 0.308607i
$$169$$ 9.00000 0.692308
$$170$$ 0 0
$$171$$ 8.00000 0.611775
$$172$$ − 6.00000i − 0.457496i
$$173$$ − 8.00000i − 0.608229i −0.952636 0.304114i $$-0.901639\pi$$
0.952636 0.304114i $$-0.0983605\pi$$
$$174$$ −4.00000 −0.303239
$$175$$ 0 0
$$176$$ 2.00000 0.150756
$$177$$ 15.0000i 1.12747i
$$178$$ − 2.00000i − 0.149906i
$$179$$ −9.00000 −0.672692 −0.336346 0.941739i $$-0.609191\pi$$
−0.336346 + 0.941739i $$0.609191\pi$$
$$180$$ 0 0
$$181$$ 11.0000 0.817624 0.408812 0.912619i $$-0.365943\pi$$
0.408812 + 0.912619i $$0.365943\pi$$
$$182$$ − 8.00000i − 0.592999i
$$183$$ − 5.00000i − 0.369611i
$$184$$ 1.00000 0.0737210
$$185$$ 0 0
$$186$$ 2.00000 0.146647
$$187$$ − 2.00000i − 0.146254i
$$188$$ − 4.00000i − 0.291730i
$$189$$ −4.00000 −0.290957
$$190$$ 0 0
$$191$$ 10.0000 0.723575 0.361787 0.932261i $$-0.382167\pi$$
0.361787 + 0.932261i $$0.382167\pi$$
$$192$$ 1.00000i 0.0721688i
$$193$$ − 24.0000i − 1.72756i −0.503871 0.863779i $$-0.668091\pi$$
0.503871 0.863779i $$-0.331909\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 9.00000 0.642857
$$197$$ 2.00000i 0.142494i 0.997459 + 0.0712470i $$0.0226979\pi$$
−0.997459 + 0.0712470i $$0.977302\pi$$
$$198$$ 2.00000i 0.142134i
$$199$$ −10.0000 −0.708881 −0.354441 0.935079i $$-0.615329\pi$$
−0.354441 + 0.935079i $$0.615329\pi$$
$$200$$ 0 0
$$201$$ −10.0000 −0.705346
$$202$$ 10.0000i 0.703598i
$$203$$ 16.0000i 1.12298i
$$204$$ 1.00000 0.0700140
$$205$$ 0 0
$$206$$ −3.00000 −0.209020
$$207$$ 1.00000i 0.0695048i
$$208$$ − 2.00000i − 0.138675i
$$209$$ −16.0000 −1.10674
$$210$$ 0 0
$$211$$ 4.00000 0.275371 0.137686 0.990476i $$-0.456034\pi$$
0.137686 + 0.990476i $$0.456034\pi$$
$$212$$ − 13.0000i − 0.892844i
$$213$$ 1.00000i 0.0685189i
$$214$$ 12.0000 0.820303
$$215$$ 0 0
$$216$$ −1.00000 −0.0680414
$$217$$ − 8.00000i − 0.543075i
$$218$$ − 2.00000i − 0.135457i
$$219$$ 16.0000 1.08118
$$220$$ 0 0
$$221$$ −2.00000 −0.134535
$$222$$ − 3.00000i − 0.201347i
$$223$$ − 17.0000i − 1.13840i −0.822198 0.569202i $$-0.807252\pi$$
0.822198 0.569202i $$-0.192748\pi$$
$$224$$ 4.00000 0.267261
$$225$$ 0 0
$$226$$ 15.0000 0.997785
$$227$$ − 22.0000i − 1.46019i −0.683345 0.730096i $$-0.739475\pi$$
0.683345 0.730096i $$-0.260525\pi$$
$$228$$ − 8.00000i − 0.529813i
$$229$$ 18.0000 1.18947 0.594737 0.803921i $$-0.297256\pi$$
0.594737 + 0.803921i $$0.297256\pi$$
$$230$$ 0 0
$$231$$ 8.00000 0.526361
$$232$$ 4.00000i 0.262613i
$$233$$ − 3.00000i − 0.196537i −0.995160 0.0982683i $$-0.968670\pi$$
0.995160 0.0982683i $$-0.0313303\pi$$
$$234$$ 2.00000 0.130744
$$235$$ 0 0
$$236$$ 15.0000 0.976417
$$237$$ 12.0000i 0.779484i
$$238$$ − 4.00000i − 0.259281i
$$239$$ −6.00000 −0.388108 −0.194054 0.980991i $$-0.562164\pi$$
−0.194054 + 0.980991i $$0.562164\pi$$
$$240$$ 0 0
$$241$$ −14.0000 −0.901819 −0.450910 0.892570i $$-0.648900\pi$$
−0.450910 + 0.892570i $$0.648900\pi$$
$$242$$ 7.00000i 0.449977i
$$243$$ − 1.00000i − 0.0641500i
$$244$$ −5.00000 −0.320092
$$245$$ 0 0
$$246$$ 1.00000 0.0637577
$$247$$ 16.0000i 1.01806i
$$248$$ − 2.00000i − 0.127000i
$$249$$ 11.0000 0.697097
$$250$$ 0 0
$$251$$ −4.00000 −0.252478 −0.126239 0.992000i $$-0.540291\pi$$
−0.126239 + 0.992000i $$0.540291\pi$$
$$252$$ 4.00000i 0.251976i
$$253$$ − 2.00000i − 0.125739i
$$254$$ 16.0000 1.00393
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ 12.0000i 0.748539i 0.927320 + 0.374270i $$0.122107\pi$$
−0.927320 + 0.374270i $$0.877893\pi$$
$$258$$ − 6.00000i − 0.373544i
$$259$$ −12.0000 −0.745644
$$260$$ 0 0
$$261$$ −4.00000 −0.247594
$$262$$ − 8.00000i − 0.494242i
$$263$$ − 4.00000i − 0.246651i −0.992366 0.123325i $$-0.960644\pi$$
0.992366 0.123325i $$-0.0393559\pi$$
$$264$$ 2.00000 0.123091
$$265$$ 0 0
$$266$$ −32.0000 −1.96205
$$267$$ − 2.00000i − 0.122398i
$$268$$ 10.0000i 0.610847i
$$269$$ 6.00000 0.365826 0.182913 0.983129i $$-0.441447\pi$$
0.182913 + 0.983129i $$0.441447\pi$$
$$270$$ 0 0
$$271$$ −21.0000 −1.27566 −0.637830 0.770178i $$-0.720168\pi$$
−0.637830 + 0.770178i $$0.720168\pi$$
$$272$$ − 1.00000i − 0.0606339i
$$273$$ − 8.00000i − 0.484182i
$$274$$ 12.0000 0.724947
$$275$$ 0 0
$$276$$ 1.00000 0.0601929
$$277$$ 31.0000i 1.86261i 0.364241 + 0.931305i $$0.381328\pi$$
−0.364241 + 0.931305i $$0.618672\pi$$
$$278$$ 1.00000i 0.0599760i
$$279$$ 2.00000 0.119737
$$280$$ 0 0
$$281$$ −2.00000 −0.119310 −0.0596550 0.998219i $$-0.519000\pi$$
−0.0596550 + 0.998219i $$0.519000\pi$$
$$282$$ − 4.00000i − 0.238197i
$$283$$ 7.00000i 0.416107i 0.978117 + 0.208053i $$0.0667128\pi$$
−0.978117 + 0.208053i $$0.933287\pi$$
$$284$$ 1.00000 0.0593391
$$285$$ 0 0
$$286$$ −4.00000 −0.236525
$$287$$ − 4.00000i − 0.236113i
$$288$$ 1.00000i 0.0589256i
$$289$$ −1.00000 −0.0588235
$$290$$ 0 0
$$291$$ 0 0
$$292$$ − 16.0000i − 0.936329i
$$293$$ 31.0000i 1.81104i 0.424304 + 0.905520i $$0.360519\pi$$
−0.424304 + 0.905520i $$0.639481\pi$$
$$294$$ 9.00000 0.524891
$$295$$ 0 0
$$296$$ −3.00000 −0.174371
$$297$$ 2.00000i 0.116052i
$$298$$ 1.00000i 0.0579284i
$$299$$ −2.00000 −0.115663
$$300$$ 0 0
$$301$$ −24.0000 −1.38334
$$302$$ − 1.00000i − 0.0575435i
$$303$$ 10.0000i 0.574485i
$$304$$ −8.00000 −0.458831
$$305$$ 0 0
$$306$$ 1.00000 0.0571662
$$307$$ 22.0000i 1.25561i 0.778372 + 0.627803i $$0.216046\pi$$
−0.778372 + 0.627803i $$0.783954\pi$$
$$308$$ − 8.00000i − 0.455842i
$$309$$ −3.00000 −0.170664
$$310$$ 0 0
$$311$$ 5.00000 0.283524 0.141762 0.989901i $$-0.454723\pi$$
0.141762 + 0.989901i $$0.454723\pi$$
$$312$$ − 2.00000i − 0.113228i
$$313$$ − 20.0000i − 1.13047i −0.824931 0.565233i $$-0.808786\pi$$
0.824931 0.565233i $$-0.191214\pi$$
$$314$$ −14.0000 −0.790066
$$315$$ 0 0
$$316$$ 12.0000 0.675053
$$317$$ 26.0000i 1.46031i 0.683284 + 0.730153i $$0.260551\pi$$
−0.683284 + 0.730153i $$0.739449\pi$$
$$318$$ − 13.0000i − 0.729004i
$$319$$ 8.00000 0.447914
$$320$$ 0 0
$$321$$ 12.0000 0.669775
$$322$$ − 4.00000i − 0.222911i
$$323$$ 8.00000i 0.445132i
$$324$$ −1.00000 −0.0555556
$$325$$ 0 0
$$326$$ 7.00000 0.387694
$$327$$ − 2.00000i − 0.110600i
$$328$$ − 1.00000i − 0.0552158i
$$329$$ −16.0000 −0.882109
$$330$$ 0 0
$$331$$ 10.0000 0.549650 0.274825 0.961494i $$-0.411380\pi$$
0.274825 + 0.961494i $$0.411380\pi$$
$$332$$ − 11.0000i − 0.603703i
$$333$$ − 3.00000i − 0.164399i
$$334$$ −24.0000 −1.31322
$$335$$ 0 0
$$336$$ 4.00000 0.218218
$$337$$ − 26.0000i − 1.41631i −0.706057 0.708155i $$-0.749528\pi$$
0.706057 0.708155i $$-0.250472\pi$$
$$338$$ − 9.00000i − 0.489535i
$$339$$ 15.0000 0.814688
$$340$$ 0 0
$$341$$ −4.00000 −0.216612
$$342$$ − 8.00000i − 0.432590i
$$343$$ − 8.00000i − 0.431959i
$$344$$ −6.00000 −0.323498
$$345$$ 0 0
$$346$$ −8.00000 −0.430083
$$347$$ 16.0000i 0.858925i 0.903085 + 0.429463i $$0.141297\pi$$
−0.903085 + 0.429463i $$0.858703\pi$$
$$348$$ 4.00000i 0.214423i
$$349$$ 14.0000 0.749403 0.374701 0.927146i $$-0.377745\pi$$
0.374701 + 0.927146i $$0.377745\pi$$
$$350$$ 0 0
$$351$$ 2.00000 0.106752
$$352$$ − 2.00000i − 0.106600i
$$353$$ − 16.0000i − 0.851594i −0.904819 0.425797i $$-0.859994\pi$$
0.904819 0.425797i $$-0.140006\pi$$
$$354$$ 15.0000 0.797241
$$355$$ 0 0
$$356$$ −2.00000 −0.106000
$$357$$ − 4.00000i − 0.211702i
$$358$$ 9.00000i 0.475665i
$$359$$ −36.0000 −1.90001 −0.950004 0.312239i $$-0.898921\pi$$
−0.950004 + 0.312239i $$0.898921\pi$$
$$360$$ 0 0
$$361$$ 45.0000 2.36842
$$362$$ − 11.0000i − 0.578147i
$$363$$ 7.00000i 0.367405i
$$364$$ −8.00000 −0.419314
$$365$$ 0 0
$$366$$ −5.00000 −0.261354
$$367$$ 2.00000i 0.104399i 0.998637 + 0.0521996i $$0.0166232\pi$$
−0.998637 + 0.0521996i $$0.983377\pi$$
$$368$$ − 1.00000i − 0.0521286i
$$369$$ 1.00000 0.0520579
$$370$$ 0 0
$$371$$ −52.0000 −2.69971
$$372$$ − 2.00000i − 0.103695i
$$373$$ − 4.00000i − 0.207112i −0.994624 0.103556i $$-0.966978\pi$$
0.994624 0.103556i $$-0.0330221\pi$$
$$374$$ −2.00000 −0.103418
$$375$$ 0 0
$$376$$ −4.00000 −0.206284
$$377$$ − 8.00000i − 0.412021i
$$378$$ 4.00000i 0.205738i
$$379$$ 1.00000 0.0513665 0.0256833 0.999670i $$-0.491824\pi$$
0.0256833 + 0.999670i $$0.491824\pi$$
$$380$$ 0 0
$$381$$ 16.0000 0.819705
$$382$$ − 10.0000i − 0.511645i
$$383$$ 20.0000i 1.02195i 0.859595 + 0.510976i $$0.170716\pi$$
−0.859595 + 0.510976i $$0.829284\pi$$
$$384$$ 1.00000 0.0510310
$$385$$ 0 0
$$386$$ −24.0000 −1.22157
$$387$$ − 6.00000i − 0.304997i
$$388$$ 0 0
$$389$$ −9.00000 −0.456318 −0.228159 0.973624i $$-0.573271\pi$$
−0.228159 + 0.973624i $$0.573271\pi$$
$$390$$ 0 0
$$391$$ −1.00000 −0.0505722
$$392$$ − 9.00000i − 0.454569i
$$393$$ − 8.00000i − 0.403547i
$$394$$ 2.00000 0.100759
$$395$$ 0 0
$$396$$ 2.00000 0.100504
$$397$$ − 21.0000i − 1.05396i −0.849878 0.526980i $$-0.823324\pi$$
0.849878 0.526980i $$-0.176676\pi$$
$$398$$ 10.0000i 0.501255i
$$399$$ −32.0000 −1.60200
$$400$$ 0 0
$$401$$ 3.00000 0.149813 0.0749064 0.997191i $$-0.476134\pi$$
0.0749064 + 0.997191i $$0.476134\pi$$
$$402$$ 10.0000i 0.498755i
$$403$$ 4.00000i 0.199254i
$$404$$ 10.0000 0.497519
$$405$$ 0 0
$$406$$ 16.0000 0.794067
$$407$$ 6.00000i 0.297409i
$$408$$ − 1.00000i − 0.0495074i
$$409$$ −19.0000 −0.939490 −0.469745 0.882802i $$-0.655654\pi$$
−0.469745 + 0.882802i $$0.655654\pi$$
$$410$$ 0 0
$$411$$ 12.0000 0.591916
$$412$$ 3.00000i 0.147799i
$$413$$ − 60.0000i − 2.95241i
$$414$$ 1.00000 0.0491473
$$415$$ 0 0
$$416$$ −2.00000 −0.0980581
$$417$$ 1.00000i 0.0489702i
$$418$$ 16.0000i 0.782586i
$$419$$ −24.0000 −1.17248 −0.586238 0.810139i $$-0.699392\pi$$
−0.586238 + 0.810139i $$0.699392\pi$$
$$420$$ 0 0
$$421$$ −24.0000 −1.16969 −0.584844 0.811146i $$-0.698844\pi$$
−0.584844 + 0.811146i $$0.698844\pi$$
$$422$$ − 4.00000i − 0.194717i
$$423$$ − 4.00000i − 0.194487i
$$424$$ −13.0000 −0.631336
$$425$$ 0 0
$$426$$ 1.00000 0.0484502
$$427$$ 20.0000i 0.967868i
$$428$$ − 12.0000i − 0.580042i
$$429$$ −4.00000 −0.193122
$$430$$ 0 0
$$431$$ −28.0000 −1.34871 −0.674356 0.738406i $$-0.735579\pi$$
−0.674356 + 0.738406i $$0.735579\pi$$
$$432$$ 1.00000i 0.0481125i
$$433$$ − 14.0000i − 0.672797i −0.941720 0.336399i $$-0.890791\pi$$
0.941720 0.336399i $$-0.109209\pi$$
$$434$$ −8.00000 −0.384012
$$435$$ 0 0
$$436$$ −2.00000 −0.0957826
$$437$$ 8.00000i 0.382692i
$$438$$ − 16.0000i − 0.764510i
$$439$$ 12.0000 0.572729 0.286364 0.958121i $$-0.407553\pi$$
0.286364 + 0.958121i $$0.407553\pi$$
$$440$$ 0 0
$$441$$ 9.00000 0.428571
$$442$$ 2.00000i 0.0951303i
$$443$$ 39.0000i 1.85295i 0.376361 + 0.926473i $$0.377175\pi$$
−0.376361 + 0.926473i $$0.622825\pi$$
$$444$$ −3.00000 −0.142374
$$445$$ 0 0
$$446$$ −17.0000 −0.804973
$$447$$ 1.00000i 0.0472984i
$$448$$ − 4.00000i − 0.188982i
$$449$$ 42.0000 1.98210 0.991051 0.133482i $$-0.0426157\pi$$
0.991051 + 0.133482i $$0.0426157\pi$$
$$450$$ 0 0
$$451$$ −2.00000 −0.0941763
$$452$$ − 15.0000i − 0.705541i
$$453$$ − 1.00000i − 0.0469841i
$$454$$ −22.0000 −1.03251
$$455$$ 0 0
$$456$$ −8.00000 −0.374634
$$457$$ − 31.0000i − 1.45012i −0.688686 0.725059i $$-0.741812\pi$$
0.688686 0.725059i $$-0.258188\pi$$
$$458$$ − 18.0000i − 0.841085i
$$459$$ 1.00000 0.0466760
$$460$$ 0 0
$$461$$ −1.00000 −0.0465746 −0.0232873 0.999729i $$-0.507413\pi$$
−0.0232873 + 0.999729i $$0.507413\pi$$
$$462$$ − 8.00000i − 0.372194i
$$463$$ − 35.0000i − 1.62659i −0.581853 0.813294i $$-0.697672\pi$$
0.581853 0.813294i $$-0.302328\pi$$
$$464$$ 4.00000 0.185695
$$465$$ 0 0
$$466$$ −3.00000 −0.138972
$$467$$ 3.00000i 0.138823i 0.997588 + 0.0694117i $$0.0221122\pi$$
−0.997588 + 0.0694117i $$0.977888\pi$$
$$468$$ − 2.00000i − 0.0924500i
$$469$$ 40.0000 1.84703
$$470$$ 0 0
$$471$$ −14.0000 −0.645086
$$472$$ − 15.0000i − 0.690431i
$$473$$ 12.0000i 0.551761i
$$474$$ 12.0000 0.551178
$$475$$ 0 0
$$476$$ −4.00000 −0.183340
$$477$$ − 13.0000i − 0.595229i
$$478$$ 6.00000i 0.274434i
$$479$$ 16.0000 0.731059 0.365529 0.930800i $$-0.380888\pi$$
0.365529 + 0.930800i $$0.380888\pi$$
$$480$$ 0 0
$$481$$ 6.00000 0.273576
$$482$$ 14.0000i 0.637683i
$$483$$ − 4.00000i − 0.182006i
$$484$$ 7.00000 0.318182
$$485$$ 0 0
$$486$$ −1.00000 −0.0453609
$$487$$ − 2.00000i − 0.0906287i −0.998973 0.0453143i $$-0.985571\pi$$
0.998973 0.0453143i $$-0.0144289\pi$$
$$488$$ 5.00000i 0.226339i
$$489$$ 7.00000 0.316551
$$490$$ 0 0
$$491$$ 39.0000 1.76005 0.880023 0.474932i $$-0.157527\pi$$
0.880023 + 0.474932i $$0.157527\pi$$
$$492$$ − 1.00000i − 0.0450835i
$$493$$ − 4.00000i − 0.180151i
$$494$$ 16.0000 0.719874
$$495$$ 0 0
$$496$$ −2.00000 −0.0898027
$$497$$ − 4.00000i − 0.179425i
$$498$$ − 11.0000i − 0.492922i
$$499$$ 41.0000 1.83541 0.917706 0.397260i $$-0.130039\pi$$
0.917706 + 0.397260i $$0.130039\pi$$
$$500$$ 0 0
$$501$$ −24.0000 −1.07224
$$502$$ 4.00000i 0.178529i
$$503$$ 41.0000i 1.82810i 0.405602 + 0.914050i $$0.367062\pi$$
−0.405602 + 0.914050i $$0.632938\pi$$
$$504$$ 4.00000 0.178174
$$505$$ 0 0
$$506$$ −2.00000 −0.0889108
$$507$$ − 9.00000i − 0.399704i
$$508$$ − 16.0000i − 0.709885i
$$509$$ −2.00000 −0.0886484 −0.0443242 0.999017i $$-0.514113\pi$$
−0.0443242 + 0.999017i $$0.514113\pi$$
$$510$$ 0 0
$$511$$ −64.0000 −2.83119
$$512$$ − 1.00000i − 0.0441942i
$$513$$ − 8.00000i − 0.353209i
$$514$$ 12.0000 0.529297
$$515$$ 0 0
$$516$$ −6.00000 −0.264135
$$517$$ 8.00000i 0.351840i
$$518$$ 12.0000i 0.527250i
$$519$$ −8.00000 −0.351161
$$520$$ 0 0
$$521$$ −18.0000 −0.788594 −0.394297 0.918983i $$-0.629012\pi$$
−0.394297 + 0.918983i $$0.629012\pi$$
$$522$$ 4.00000i 0.175075i
$$523$$ 34.0000i 1.48672i 0.668894 + 0.743358i $$0.266768\pi$$
−0.668894 + 0.743358i $$0.733232\pi$$
$$524$$ −8.00000 −0.349482
$$525$$ 0 0
$$526$$ −4.00000 −0.174408
$$527$$ 2.00000i 0.0871214i
$$528$$ − 2.00000i − 0.0870388i
$$529$$ 22.0000 0.956522
$$530$$ 0 0
$$531$$ 15.0000 0.650945
$$532$$ 32.0000i 1.38738i
$$533$$ 2.00000i 0.0866296i
$$534$$ −2.00000 −0.0865485
$$535$$ 0 0
$$536$$ 10.0000 0.431934
$$537$$ 9.00000i 0.388379i
$$538$$ − 6.00000i − 0.258678i
$$539$$ −18.0000 −0.775315
$$540$$ 0 0
$$541$$ 15.0000 0.644900 0.322450 0.946586i $$-0.395494\pi$$
0.322450 + 0.946586i $$0.395494\pi$$
$$542$$ 21.0000i 0.902027i
$$543$$ − 11.0000i − 0.472055i
$$544$$ −1.00000 −0.0428746
$$545$$ 0 0
$$546$$ −8.00000 −0.342368
$$547$$ 13.0000i 0.555840i 0.960604 + 0.277920i $$0.0896450\pi$$
−0.960604 + 0.277920i $$0.910355\pi$$
$$548$$ − 12.0000i − 0.512615i
$$549$$ −5.00000 −0.213395
$$550$$ 0 0
$$551$$ −32.0000 −1.36325
$$552$$ − 1.00000i − 0.0425628i
$$553$$ − 48.0000i − 2.04117i
$$554$$ 31.0000 1.31706
$$555$$ 0 0
$$556$$ 1.00000 0.0424094
$$557$$ − 31.0000i − 1.31351i −0.754103 0.656756i $$-0.771928\pi$$
0.754103 0.656756i $$-0.228072\pi$$
$$558$$ − 2.00000i − 0.0846668i
$$559$$ 12.0000 0.507546
$$560$$ 0 0
$$561$$ −2.00000 −0.0844401
$$562$$ 2.00000i 0.0843649i
$$563$$ − 15.0000i − 0.632175i −0.948730 0.316087i $$-0.897631\pi$$
0.948730 0.316087i $$-0.102369\pi$$
$$564$$ −4.00000 −0.168430
$$565$$ 0 0
$$566$$ 7.00000 0.294232
$$567$$ 4.00000i 0.167984i
$$568$$ − 1.00000i − 0.0419591i
$$569$$ −6.00000 −0.251533 −0.125767 0.992060i $$-0.540139\pi$$
−0.125767 + 0.992060i $$0.540139\pi$$
$$570$$ 0 0
$$571$$ −41.0000 −1.71580 −0.857898 0.513820i $$-0.828230\pi$$
−0.857898 + 0.513820i $$0.828230\pi$$
$$572$$ 4.00000i 0.167248i
$$573$$ − 10.0000i − 0.417756i
$$574$$ −4.00000 −0.166957
$$575$$ 0 0
$$576$$ 1.00000 0.0416667
$$577$$ − 33.0000i − 1.37381i −0.726748 0.686904i $$-0.758969\pi$$
0.726748 0.686904i $$-0.241031\pi$$
$$578$$ 1.00000i 0.0415945i
$$579$$ −24.0000 −0.997406
$$580$$ 0 0
$$581$$ −44.0000 −1.82543
$$582$$ 0 0
$$583$$ 26.0000i 1.07681i
$$584$$ −16.0000 −0.662085
$$585$$ 0 0
$$586$$ 31.0000 1.28060
$$587$$ 25.0000i 1.03186i 0.856631 + 0.515930i $$0.172554\pi$$
−0.856631 + 0.515930i $$0.827446\pi$$
$$588$$ − 9.00000i − 0.371154i
$$589$$ 16.0000 0.659269
$$590$$ 0 0
$$591$$ 2.00000 0.0822690
$$592$$ 3.00000i 0.123299i
$$593$$ 12.0000i 0.492781i 0.969171 + 0.246390i $$0.0792446\pi$$
−0.969171 + 0.246390i $$0.920755\pi$$
$$594$$ 2.00000 0.0820610
$$595$$ 0 0
$$596$$ 1.00000 0.0409616
$$597$$ 10.0000i 0.409273i
$$598$$ 2.00000i 0.0817861i
$$599$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$600$$ 0 0
$$601$$ 6.00000 0.244745 0.122373 0.992484i $$-0.460950\pi$$
0.122373 + 0.992484i $$0.460950\pi$$
$$602$$ 24.0000i 0.978167i
$$603$$ 10.0000i 0.407231i
$$604$$ −1.00000 −0.0406894
$$605$$ 0 0
$$606$$ 10.0000 0.406222
$$607$$ − 34.0000i − 1.38002i −0.723801 0.690009i $$-0.757607\pi$$
0.723801 0.690009i $$-0.242393\pi$$
$$608$$ 8.00000i 0.324443i
$$609$$ 16.0000 0.648353
$$610$$ 0 0
$$611$$ 8.00000 0.323645
$$612$$ − 1.00000i − 0.0404226i
$$613$$ − 26.0000i − 1.05013i −0.851062 0.525065i $$-0.824041\pi$$
0.851062 0.525065i $$-0.175959\pi$$
$$614$$ 22.0000 0.887848
$$615$$ 0 0
$$616$$ −8.00000 −0.322329
$$617$$ 6.00000i 0.241551i 0.992680 + 0.120775i $$0.0385381\pi$$
−0.992680 + 0.120775i $$0.961462\pi$$
$$618$$ 3.00000i 0.120678i
$$619$$ −20.0000 −0.803868 −0.401934 0.915669i $$-0.631662\pi$$
−0.401934 + 0.915669i $$0.631662\pi$$
$$620$$ 0 0
$$621$$ 1.00000 0.0401286
$$622$$ − 5.00000i − 0.200482i
$$623$$ 8.00000i 0.320513i
$$624$$ −2.00000 −0.0800641
$$625$$ 0 0
$$626$$ −20.0000 −0.799361
$$627$$ 16.0000i 0.638978i
$$628$$ 14.0000i 0.558661i
$$629$$ 3.00000 0.119618
$$630$$ 0 0
$$631$$ −7.00000 −0.278666 −0.139333 0.990246i $$-0.544496\pi$$
−0.139333 + 0.990246i $$0.544496\pi$$
$$632$$ − 12.0000i − 0.477334i
$$633$$ − 4.00000i − 0.158986i
$$634$$ 26.0000 1.03259
$$635$$ 0 0
$$636$$ −13.0000 −0.515484
$$637$$ 18.0000i 0.713186i
$$638$$ − 8.00000i − 0.316723i
$$639$$ 1.00000 0.0395594
$$640$$ 0 0
$$641$$ −30.0000 −1.18493 −0.592464 0.805597i $$-0.701845\pi$$
−0.592464 + 0.805597i $$0.701845\pi$$
$$642$$ − 12.0000i − 0.473602i
$$643$$ − 29.0000i − 1.14365i −0.820376 0.571824i $$-0.806236\pi$$
0.820376 0.571824i $$-0.193764\pi$$
$$644$$ −4.00000 −0.157622
$$645$$ 0 0
$$646$$ 8.00000 0.314756
$$647$$ − 38.0000i − 1.49393i −0.664861 0.746967i $$-0.731509\pi$$
0.664861 0.746967i $$-0.268491\pi$$
$$648$$ 1.00000i 0.0392837i
$$649$$ −30.0000 −1.17760
$$650$$ 0 0
$$651$$ −8.00000 −0.313545
$$652$$ − 7.00000i − 0.274141i
$$653$$ 12.0000i 0.469596i 0.972044 + 0.234798i $$0.0754429\pi$$
−0.972044 + 0.234798i $$0.924557\pi$$
$$654$$ −2.00000 −0.0782062
$$655$$ 0 0
$$656$$ −1.00000 −0.0390434
$$657$$ − 16.0000i − 0.624219i
$$658$$ 16.0000i 0.623745i
$$659$$ 40.0000 1.55818 0.779089 0.626913i $$-0.215682\pi$$
0.779089 + 0.626913i $$0.215682\pi$$
$$660$$ 0 0
$$661$$ 50.0000 1.94477 0.972387 0.233373i $$-0.0749763\pi$$
0.972387 + 0.233373i $$0.0749763\pi$$
$$662$$ − 10.0000i − 0.388661i
$$663$$ 2.00000i 0.0776736i
$$664$$ −11.0000 −0.426883
$$665$$ 0 0
$$666$$ −3.00000 −0.116248
$$667$$ − 4.00000i − 0.154881i
$$668$$ 24.0000i 0.928588i
$$669$$ −17.0000 −0.657258
$$670$$ 0 0
$$671$$ 10.0000 0.386046
$$672$$ − 4.00000i − 0.154303i
$$673$$ 28.0000i 1.07932i 0.841883 + 0.539660i $$0.181447\pi$$
−0.841883 + 0.539660i $$0.818553\pi$$
$$674$$ −26.0000 −1.00148
$$675$$ 0 0
$$676$$ −9.00000 −0.346154
$$677$$ − 12.0000i − 0.461197i −0.973049 0.230599i $$-0.925932\pi$$
0.973049 0.230599i $$-0.0740685\pi$$
$$678$$ − 15.0000i − 0.576072i
$$679$$ 0 0
$$680$$ 0 0
$$681$$ −22.0000 −0.843042
$$682$$ 4.00000i 0.153168i
$$683$$ 6.00000i 0.229584i 0.993390 + 0.114792i $$0.0366201\pi$$
−0.993390 + 0.114792i $$0.963380\pi$$
$$684$$ −8.00000 −0.305888
$$685$$ 0 0
$$686$$ −8.00000 −0.305441
$$687$$ − 18.0000i − 0.686743i
$$688$$ 6.00000i 0.228748i
$$689$$ 26.0000 0.990521
$$690$$ 0 0
$$691$$ −41.0000 −1.55971 −0.779857 0.625958i $$-0.784708\pi$$
−0.779857 + 0.625958i $$0.784708\pi$$
$$692$$ 8.00000i 0.304114i
$$693$$ − 8.00000i − 0.303895i
$$694$$ 16.0000 0.607352
$$695$$ 0 0
$$696$$ 4.00000 0.151620
$$697$$ 1.00000i 0.0378777i
$$698$$ − 14.0000i − 0.529908i
$$699$$ −3.00000 −0.113470
$$700$$ 0 0
$$701$$ −3.00000 −0.113308 −0.0566542 0.998394i $$-0.518043\pi$$
−0.0566542 + 0.998394i $$0.518043\pi$$
$$702$$ − 2.00000i − 0.0754851i
$$703$$ − 24.0000i − 0.905177i
$$704$$ −2.00000 −0.0753778
$$705$$ 0 0
$$706$$ −16.0000 −0.602168
$$707$$ − 40.0000i − 1.50435i
$$708$$ − 15.0000i − 0.563735i
$$709$$ −46.0000 −1.72757 −0.863783 0.503864i $$-0.831911\pi$$
−0.863783 + 0.503864i $$0.831911\pi$$
$$710$$ 0 0
$$711$$ 12.0000 0.450035
$$712$$ 2.00000i 0.0749532i
$$713$$ 2.00000i 0.0749006i
$$714$$ −4.00000 −0.149696
$$715$$ 0 0
$$716$$ 9.00000 0.336346
$$717$$ 6.00000i 0.224074i
$$718$$ 36.0000i 1.34351i
$$719$$ 8.00000 0.298350 0.149175 0.988811i $$-0.452338\pi$$
0.149175 + 0.988811i $$0.452338\pi$$
$$720$$ 0 0
$$721$$ 12.0000 0.446903
$$722$$ − 45.0000i − 1.67473i
$$723$$ 14.0000i 0.520666i
$$724$$ −11.0000 −0.408812
$$725$$ 0 0
$$726$$ 7.00000 0.259794
$$727$$ − 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
$$732$$ 5.00000i 0.184805i
$$733$$ − 30.0000i − 1.10808i −0.832492 0.554038i $$-0.813086\pi$$
0.832492 0.554038i $$-0.186914\pi$$
$$734$$ 2.00000 0.0738213
$$735$$ 0 0
$$736$$ −1.00000 −0.0368605
$$737$$ − 20.0000i − 0.736709i
$$738$$ − 1.00000i − 0.0368105i
$$739$$ 12.0000 0.441427 0.220714 0.975339i $$-0.429161\pi$$
0.220714 + 0.975339i $$0.429161\pi$$
$$740$$ 0 0
$$741$$ 16.0000 0.587775
$$742$$ 52.0000i 1.90898i
$$743$$ − 17.0000i − 0.623670i −0.950136 0.311835i $$-0.899056\pi$$
0.950136 0.311835i $$-0.100944\pi$$
$$744$$ −2.00000 −0.0733236
$$745$$ 0 0
$$746$$ −4.00000 −0.146450
$$747$$ − 11.0000i − 0.402469i
$$748$$ 2.00000i 0.0731272i
$$749$$ −48.0000 −1.75388
$$750$$ 0 0
$$751$$ −48.0000 −1.75154 −0.875772 0.482724i $$-0.839647\pi$$
−0.875772 + 0.482724i $$0.839647\pi$$
$$752$$ 4.00000i 0.145865i
$$753$$ 4.00000i 0.145768i
$$754$$ −8.00000 −0.291343
$$755$$ 0 0
$$756$$ 4.00000 0.145479
$$757$$ 28.0000i 1.01768i 0.860862 + 0.508839i $$0.169925\pi$$
−0.860862 + 0.508839i $$0.830075\pi$$
$$758$$ − 1.00000i − 0.0363216i
$$759$$ −2.00000 −0.0725954
$$760$$ 0 0
$$761$$ 36.0000 1.30500 0.652499 0.757789i $$-0.273720\pi$$
0.652499 + 0.757789i $$0.273720\pi$$
$$762$$ − 16.0000i − 0.579619i
$$763$$ 8.00000i 0.289619i
$$764$$ −10.0000 −0.361787
$$765$$ 0 0
$$766$$ 20.0000 0.722629
$$767$$ 30.0000i 1.08324i
$$768$$ − 1.00000i − 0.0360844i
$$769$$ 37.0000 1.33425 0.667127 0.744944i $$-0.267524\pi$$
0.667127 + 0.744944i $$0.267524\pi$$
$$770$$ 0 0
$$771$$ 12.0000 0.432169
$$772$$ 24.0000i 0.863779i
$$773$$ 15.0000i 0.539513i 0.962929 + 0.269756i $$0.0869431\pi$$
−0.962929 + 0.269756i $$0.913057\pi$$
$$774$$ −6.00000 −0.215666
$$775$$ 0 0
$$776$$ 0 0
$$777$$ 12.0000i 0.430498i
$$778$$ 9.00000i 0.322666i
$$779$$ 8.00000 0.286630
$$780$$ 0 0
$$781$$ −2.00000 −0.0715656
$$782$$ 1.00000i 0.0357599i
$$783$$ 4.00000i 0.142948i
$$784$$ −9.00000 −0.321429
$$785$$ 0 0
$$786$$ −8.00000 −0.285351
$$787$$ 3.00000i 0.106938i 0.998569 + 0.0534692i $$0.0170279\pi$$
−0.998569 + 0.0534692i $$0.982972\pi$$
$$788$$ − 2.00000i − 0.0712470i
$$789$$ −4.00000 −0.142404
$$790$$ 0 0
$$791$$ −60.0000 −2.13335
$$792$$ − 2.00000i − 0.0710669i
$$793$$ − 10.0000i − 0.355110i
$$794$$ −21.0000 −0.745262
$$795$$ 0 0
$$796$$ 10.0000 0.354441
$$797$$ − 31.0000i − 1.09808i −0.835797 0.549038i $$-0.814994\pi$$
0.835797 0.549038i $$-0.185006\pi$$
$$798$$ 32.0000i 1.13279i
$$799$$ 4.00000 0.141510
$$800$$ 0 0
$$801$$ −2.00000 −0.0706665
$$802$$ − 3.00000i − 0.105934i
$$803$$ 32.0000i 1.12926i
$$804$$ 10.0000 0.352673
$$805$$ 0 0
$$806$$ 4.00000 0.140894
$$807$$ − 6.00000i − 0.211210i
$$808$$ − 10.0000i − 0.351799i
$$809$$ 18.0000 0.632846 0.316423 0.948618i $$-0.397518\pi$$
0.316423 + 0.948618i $$0.397518\pi$$
$$810$$ 0 0
$$811$$ −28.0000 −0.983213 −0.491606 0.870817i $$-0.663590\pi$$
−0.491606 + 0.870817i $$0.663590\pi$$
$$812$$ − 16.0000i − 0.561490i
$$813$$ 21.0000i 0.736502i
$$814$$ 6.00000 0.210300
$$815$$ 0 0
$$816$$ −1.00000 −0.0350070
$$817$$ − 48.0000i − 1.67931i
$$818$$ 19.0000i 0.664319i
$$819$$ −8.00000 −0.279543
$$820$$ 0 0
$$821$$ 12.0000 0.418803 0.209401 0.977830i $$-0.432848\pi$$
0.209401 + 0.977830i $$0.432848\pi$$
$$822$$ − 12.0000i − 0.418548i
$$823$$ 32.0000i 1.11545i 0.830026 + 0.557725i $$0.188326\pi$$
−0.830026 + 0.557725i $$0.811674\pi$$
$$824$$ 3.00000 0.104510
$$825$$ 0 0
$$826$$ −60.0000 −2.08767
$$827$$ − 38.0000i − 1.32139i −0.750655 0.660695i $$-0.770262\pi$$
0.750655 0.660695i $$-0.229738\pi$$
$$828$$ − 1.00000i − 0.0347524i
$$829$$ 16.0000 0.555703 0.277851 0.960624i $$-0.410378\pi$$
0.277851 + 0.960624i $$0.410378\pi$$
$$830$$ 0 0
$$831$$ 31.0000 1.07538
$$832$$ 2.00000i 0.0693375i
$$833$$ 9.00000i 0.311832i
$$834$$ 1.00000 0.0346272
$$835$$ 0 0
$$836$$ 16.0000 0.553372
$$837$$ − 2.00000i − 0.0691301i
$$838$$ 24.0000i 0.829066i
$$839$$ −27.0000 −0.932144 −0.466072 0.884747i $$-0.654331\pi$$
−0.466072 + 0.884747i $$0.654331\pi$$
$$840$$ 0 0
$$841$$ −13.0000 −0.448276
$$842$$ 24.0000i 0.827095i
$$843$$ 2.00000i 0.0688837i
$$844$$ −4.00000 −0.137686
$$845$$ 0 0
$$846$$ −4.00000 −0.137523
$$847$$ − 28.0000i − 0.962091i
$$848$$ 13.0000i 0.446422i
$$849$$ 7.00000 0.240239
$$850$$ 0 0
$$851$$ 3.00000 0.102839
$$852$$ − 1.00000i − 0.0342594i
$$853$$ 26.0000i 0.890223i 0.895475 + 0.445112i $$0.146836\pi$$
−0.895475 + 0.445112i $$0.853164\pi$$
$$854$$ 20.0000 0.684386
$$855$$ 0 0
$$856$$ −12.0000 −0.410152
$$857$$ 7.00000i 0.239115i 0.992827 + 0.119558i $$0.0381477\pi$$
−0.992827 + 0.119558i $$0.961852\pi$$
$$858$$ 4.00000i 0.136558i
$$859$$ 44.0000 1.50126 0.750630 0.660722i $$-0.229750\pi$$
0.750630 + 0.660722i $$0.229750\pi$$
$$860$$ 0 0
$$861$$ −4.00000 −0.136320
$$862$$ 28.0000i 0.953684i
$$863$$ − 40.0000i − 1.36162i −0.732462 0.680808i $$-0.761629\pi$$
0.732462 0.680808i $$-0.238371\pi$$
$$864$$ 1.00000 0.0340207
$$865$$ 0 0
$$866$$ −14.0000 −0.475739
$$867$$ 1.00000i 0.0339618i
$$868$$ 8.00000i 0.271538i
$$869$$ −24.0000 −0.814144
$$870$$ 0 0
$$871$$ −20.0000 −0.677674
$$872$$ 2.00000i 0.0677285i
$$873$$ 0 0
$$874$$ 8.00000 0.270604
$$875$$ 0 0
$$876$$ −16.0000 −0.540590
$$877$$ − 14.0000i − 0.472746i −0.971662 0.236373i $$-0.924041\pi$$
0.971662 0.236373i $$-0.0759588\pi$$
$$878$$ − 12.0000i − 0.404980i
$$879$$ 31.0000 1.04560
$$880$$ 0 0
$$881$$ 33.0000 1.11180 0.555899 0.831250i $$-0.312374\pi$$
0.555899 + 0.831250i $$0.312374\pi$$
$$882$$ − 9.00000i − 0.303046i
$$883$$ 50.0000i 1.68263i 0.540542 + 0.841317i $$0.318219\pi$$
−0.540542 + 0.841317i $$0.681781\pi$$
$$884$$ 2.00000 0.0672673
$$885$$ 0 0
$$886$$ 39.0000 1.31023
$$887$$ 15.0000i 0.503651i 0.967773 + 0.251825i $$0.0810309\pi$$
−0.967773 + 0.251825i $$0.918969\pi$$
$$888$$ 3.00000i 0.100673i
$$889$$ −64.0000 −2.14649
$$890$$ 0 0
$$891$$ 2.00000 0.0670025
$$892$$ 17.0000i 0.569202i
$$893$$ − 32.0000i − 1.07084i
$$894$$ 1.00000 0.0334450
$$895$$ 0 0
$$896$$ −4.00000 −0.133631
$$897$$ 2.00000i 0.0667781i
$$898$$ − 42.0000i − 1.40156i
$$899$$ −8.00000 −0.266815
$$900$$ 0 0
$$901$$ 13.0000 0.433093
$$902$$ 2.00000i 0.0665927i
$$903$$ 24.0000i 0.798670i
$$904$$ −15.0000 −0.498893
$$905$$ 0 0
$$906$$ −1.00000 −0.0332228
$$907$$ 23.0000i 0.763702i 0.924224 + 0.381851i $$0.124713\pi$$
−0.924224 + 0.381851i $$0.875287\pi$$
$$908$$ 22.0000i 0.730096i
$$909$$ 10.0000 0.331679
$$910$$ 0 0
$$911$$ 56.0000 1.85536 0.927681 0.373373i $$-0.121799\pi$$
0.927681 + 0.373373i $$0.121799\pi$$
$$912$$ 8.00000i 0.264906i
$$913$$ 22.0000i 0.728094i
$$914$$ −31.0000 −1.02539
$$915$$ 0 0
$$916$$ −18.0000 −0.594737
$$917$$ 32.0000i 1.05673i
$$918$$ − 1.00000i − 0.0330049i
$$919$$ 47.0000 1.55039 0.775193 0.631724i $$-0.217652\pi$$
0.775193 + 0.631724i $$0.217652\pi$$
$$920$$ 0 0
$$921$$ 22.0000 0.724925
$$922$$ 1.00000i 0.0329332i
$$923$$ 2.00000i 0.0658308i
$$924$$ −8.00000 −0.263181
$$925$$ 0 0
$$926$$ −35.0000 −1.15017
$$927$$ 3.00000i 0.0985329i
$$928$$ − 4.00000i − 0.131306i
$$929$$ 15.0000 0.492134 0.246067 0.969253i $$-0.420862\pi$$
0.246067 + 0.969253i $$0.420862\pi$$
$$930$$ 0 0
$$931$$ 72.0000 2.35970
$$932$$ 3.00000i 0.0982683i
$$933$$ − 5.00000i − 0.163693i
$$934$$ 3.00000 0.0981630
$$935$$ 0 0
$$936$$ −2.00000 −0.0653720
$$937$$ − 17.0000i − 0.555366i −0.960673 0.277683i $$-0.910434\pi$$
0.960673 0.277683i $$-0.0895665\pi$$
$$938$$ − 40.0000i − 1.30605i
$$939$$ −20.0000 −0.652675
$$940$$ 0 0
$$941$$ 22.0000 0.717180 0.358590 0.933495i $$-0.383258\pi$$
0.358590 + 0.933495i $$0.383258\pi$$
$$942$$ 14.0000i 0.456145i
$$943$$ 1.00000i 0.0325645i
$$944$$ −15.0000 −0.488208
$$945$$ 0 0
$$946$$ 12.0000 0.390154
$$947$$ 16.0000i 0.519930i 0.965618 + 0.259965i $$0.0837111\pi$$
−0.965618 + 0.259965i $$0.916289\pi$$
$$948$$ − 12.0000i − 0.389742i
$$949$$ 32.0000 1.03876
$$950$$ 0 0
$$951$$ 26.0000 0.843108
$$952$$ 4.00000i 0.129641i
$$953$$ − 42.0000i − 1.36051i −0.732974 0.680257i $$-0.761868\pi$$
0.732974 0.680257i $$-0.238132\pi$$
$$954$$ −13.0000 −0.420891
$$955$$ 0 0
$$956$$ 6.00000 0.194054
$$957$$ − 8.00000i − 0.258603i
$$958$$ − 16.0000i − 0.516937i
$$959$$ −48.0000 −1.55000
$$960$$ 0 0
$$961$$ −27.0000 −0.870968
$$962$$ − 6.00000i − 0.193448i
$$963$$ − 12.0000i − 0.386695i
$$964$$ 14.0000 0.450910
$$965$$ 0 0
$$966$$ −4.00000 −0.128698
$$967$$ 1.00000i 0.0321578i 0.999871 + 0.0160789i $$0.00511830\pi$$
−0.999871 + 0.0160789i $$0.994882\pi$$
$$968$$ − 7.00000i − 0.224989i
$$969$$ 8.00000 0.256997
$$970$$ 0 0
$$971$$ 25.0000 0.802288 0.401144 0.916015i $$-0.368613\pi$$
0.401144 + 0.916015i $$0.368613\pi$$
$$972$$ 1.00000i 0.0320750i
$$973$$ − 4.00000i − 0.128234i
$$974$$ −2.00000 −0.0640841
$$975$$ 0 0
$$976$$ 5.00000 0.160046
$$977$$ − 18.0000i − 0.575871i −0.957650 0.287936i $$-0.907031\pi$$
0.957650 0.287936i $$-0.0929689\pi$$
$$978$$ − 7.00000i − 0.223835i
$$979$$ 4.00000 0.127841
$$980$$ 0 0
$$981$$ −2.00000 −0.0638551
$$982$$ − 39.0000i − 1.24454i
$$983$$ 8.00000i 0.255160i 0.991828 + 0.127580i $$0.0407210\pi$$
−0.991828 + 0.127580i $$0.959279\pi$$
$$984$$ −1.00000 −0.0318788
$$985$$ 0 0
$$986$$ −4.00000 −0.127386
$$987$$ 16.0000i 0.509286i
$$988$$ − 16.0000i − 0.509028i
$$989$$ 6.00000 0.190789
$$990$$ 0 0
$$991$$ −4.00000 −0.127064 −0.0635321 0.997980i $$-0.520237\pi$$
−0.0635321 + 0.997980i $$0.520237\pi$$
$$992$$ 2.00000i 0.0635001i
$$993$$ − 10.0000i − 0.317340i
$$994$$ −4.00000 −0.126872
$$995$$ 0 0
$$996$$ −11.0000 −0.348548
$$997$$ 50.0000i 1.58352i 0.610835 + 0.791758i $$0.290834\pi$$
−0.610835 + 0.791758i $$0.709166\pi$$
$$998$$ − 41.0000i − 1.29783i
$$999$$ −3.00000 −0.0949158
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.h.2449.1 2
5.2 odd 4 2550.2.a.t.1.1 yes 1
5.3 odd 4 2550.2.a.q.1.1 1
5.4 even 2 inner 2550.2.d.h.2449.2 2
15.2 even 4 7650.2.a.c.1.1 1
15.8 even 4 7650.2.a.cj.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
2550.2.a.q.1.1 1 5.3 odd 4
2550.2.a.t.1.1 yes 1 5.2 odd 4
2550.2.d.h.2449.1 2 1.1 even 1 trivial
2550.2.d.h.2449.2 2 5.4 even 2 inner
7650.2.a.c.1.1 1 15.2 even 4
7650.2.a.cj.1.1 1 15.8 even 4