# Properties

 Label 2550.2.d.p.2449.1 Level $2550$ Weight $2$ Character 2550.2449 Analytic conductor $20.362$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Learn more

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.p.2449.2

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} +5.00000i q^{7} +1.00000i q^{8} -1.00000 q^{9} +O(q^{10})$$ $$q-1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} +5.00000i q^{7} +1.00000i q^{8} -1.00000 q^{9} -1.00000 q^{11} -1.00000i q^{12} +2.00000i q^{13} +5.00000 q^{14} +1.00000 q^{16} +1.00000i q^{17} +1.00000i q^{18} +7.00000 q^{19} -5.00000 q^{21} +1.00000i q^{22} -2.00000i q^{23} -1.00000 q^{24} +2.00000 q^{26} -1.00000i q^{27} -5.00000i q^{28} +10.0000 q^{29} -5.00000 q^{31} -1.00000i q^{32} -1.00000i q^{33} +1.00000 q^{34} +1.00000 q^{36} +3.00000i q^{37} -7.00000i q^{38} -2.00000 q^{39} -10.0000 q^{41} +5.00000i q^{42} +9.00000i q^{43} +1.00000 q^{44} -2.00000 q^{46} -7.00000i q^{47} +1.00000i q^{48} -18.0000 q^{49} -1.00000 q^{51} -2.00000i q^{52} +11.0000i q^{53} -1.00000 q^{54} -5.00000 q^{56} +7.00000i q^{57} -10.0000i q^{58} +2.00000 q^{61} +5.00000i q^{62} -5.00000i q^{63} -1.00000 q^{64} -1.00000 q^{66} +13.0000i q^{67} -1.00000i q^{68} +2.00000 q^{69} -10.0000 q^{71} -1.00000i q^{72} -4.00000i q^{73} +3.00000 q^{74} -7.00000 q^{76} -5.00000i q^{77} +2.00000i q^{78} -3.00000 q^{79} +1.00000 q^{81} +10.0000i q^{82} -14.0000i q^{83} +5.00000 q^{84} +9.00000 q^{86} +10.0000i q^{87} -1.00000i q^{88} -4.00000 q^{89} -10.0000 q^{91} +2.00000i q^{92} -5.00000i q^{93} -7.00000 q^{94} +1.00000 q^{96} +6.00000i q^{97} +18.0000i q^{98} +1.00000 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{4} + 2 q^{6} - 2 q^{9}+O(q^{10})$$ 2 * q - 2 * q^4 + 2 * q^6 - 2 * q^9 $$2 q - 2 q^{4} + 2 q^{6} - 2 q^{9} - 2 q^{11} + 10 q^{14} + 2 q^{16} + 14 q^{19} - 10 q^{21} - 2 q^{24} + 4 q^{26} + 20 q^{29} - 10 q^{31} + 2 q^{34} + 2 q^{36} - 4 q^{39} - 20 q^{41} + 2 q^{44} - 4 q^{46} - 36 q^{49} - 2 q^{51} - 2 q^{54} - 10 q^{56} + 4 q^{61} - 2 q^{64} - 2 q^{66} + 4 q^{69} - 20 q^{71} + 6 q^{74} - 14 q^{76} - 6 q^{79} + 2 q^{81} + 10 q^{84} + 18 q^{86} - 8 q^{89} - 20 q^{91} - 14 q^{94} + 2 q^{96} + 2 q^{99}+O(q^{100})$$ 2 * q - 2 * q^4 + 2 * q^6 - 2 * q^9 - 2 * q^11 + 10 * q^14 + 2 * q^16 + 14 * q^19 - 10 * q^21 - 2 * q^24 + 4 * q^26 + 20 * q^29 - 10 * q^31 + 2 * q^34 + 2 * q^36 - 4 * q^39 - 20 * q^41 + 2 * q^44 - 4 * q^46 - 36 * q^49 - 2 * q^51 - 2 * q^54 - 10 * q^56 + 4 * q^61 - 2 * q^64 - 2 * q^66 + 4 * q^69 - 20 * q^71 + 6 * q^74 - 14 * q^76 - 6 * q^79 + 2 * q^81 + 10 * q^84 + 18 * q^86 - 8 * q^89 - 20 * q^91 - 14 * q^94 + 2 * q^96 + 2 * 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
$$7$$ 5.00000i 1.88982i 0.327327 + 0.944911i $$0.393852\pi$$
−0.327327 + 0.944911i $$0.606148\pi$$
$$8$$ 1.00000i 0.353553i
$$9$$ −1.00000 −0.333333
$$10$$ 0 0
$$11$$ −1.00000 −0.301511 −0.150756 0.988571i $$-0.548171\pi$$
−0.150756 + 0.988571i $$0.548171\pi$$
$$12$$ − 1.00000i − 0.288675i
$$13$$ 2.00000i 0.554700i 0.960769 + 0.277350i $$0.0894562\pi$$
−0.960769 + 0.277350i $$0.910544\pi$$
$$14$$ 5.00000 1.33631
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ 1.00000i 0.242536i
$$18$$ 1.00000i 0.235702i
$$19$$ 7.00000 1.60591 0.802955 0.596040i $$-0.203260\pi$$
0.802955 + 0.596040i $$0.203260\pi$$
$$20$$ 0 0
$$21$$ −5.00000 −1.09109
$$22$$ 1.00000i 0.213201i
$$23$$ − 2.00000i − 0.417029i −0.978019 0.208514i $$-0.933137\pi$$
0.978019 0.208514i $$-0.0668628\pi$$
$$24$$ −1.00000 −0.204124
$$25$$ 0 0
$$26$$ 2.00000 0.392232
$$27$$ − 1.00000i − 0.192450i
$$28$$ − 5.00000i − 0.944911i
$$29$$ 10.0000 1.85695 0.928477 0.371391i $$-0.121119\pi$$
0.928477 + 0.371391i $$0.121119\pi$$
$$30$$ 0 0
$$31$$ −5.00000 −0.898027 −0.449013 0.893525i $$-0.648224\pi$$
−0.449013 + 0.893525i $$0.648224\pi$$
$$32$$ − 1.00000i − 0.176777i
$$33$$ − 1.00000i − 0.174078i
$$34$$ 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 $$0.920687\pi$$
$$38$$ − 7.00000i − 1.13555i
$$39$$ −2.00000 −0.320256
$$40$$ 0 0
$$41$$ −10.0000 −1.56174 −0.780869 0.624695i $$-0.785223\pi$$
−0.780869 + 0.624695i $$0.785223\pi$$
$$42$$ 5.00000i 0.771517i
$$43$$ 9.00000i 1.37249i 0.727372 + 0.686244i $$0.240742\pi$$
−0.727372 + 0.686244i $$0.759258\pi$$
$$44$$ 1.00000 0.150756
$$45$$ 0 0
$$46$$ −2.00000 −0.294884
$$47$$ − 7.00000i − 1.02105i −0.859861 0.510527i $$-0.829450\pi$$
0.859861 0.510527i $$-0.170550\pi$$
$$48$$ 1.00000i 0.144338i
$$49$$ −18.0000 −2.57143
$$50$$ 0 0
$$51$$ −1.00000 −0.140028
$$52$$ − 2.00000i − 0.277350i
$$53$$ 11.0000i 1.51097i 0.655168 + 0.755483i $$0.272598\pi$$
−0.655168 + 0.755483i $$0.727402\pi$$
$$54$$ −1.00000 −0.136083
$$55$$ 0 0
$$56$$ −5.00000 −0.668153
$$57$$ 7.00000i 0.927173i
$$58$$ − 10.0000i − 1.31306i
$$59$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$60$$ 0 0
$$61$$ 2.00000 0.256074 0.128037 0.991769i $$-0.459132\pi$$
0.128037 + 0.991769i $$0.459132\pi$$
$$62$$ 5.00000i 0.635001i
$$63$$ − 5.00000i − 0.629941i
$$64$$ −1.00000 −0.125000
$$65$$ 0 0
$$66$$ −1.00000 −0.123091
$$67$$ 13.0000i 1.58820i 0.607785 + 0.794101i $$0.292058\pi$$
−0.607785 + 0.794101i $$0.707942\pi$$
$$68$$ − 1.00000i − 0.121268i
$$69$$ 2.00000 0.240772
$$70$$ 0 0
$$71$$ −10.0000 −1.18678 −0.593391 0.804914i $$-0.702211\pi$$
−0.593391 + 0.804914i $$0.702211\pi$$
$$72$$ − 1.00000i − 0.117851i
$$73$$ − 4.00000i − 0.468165i −0.972217 0.234082i $$-0.924791\pi$$
0.972217 0.234082i $$-0.0752085\pi$$
$$74$$ 3.00000 0.348743
$$75$$ 0 0
$$76$$ −7.00000 −0.802955
$$77$$ − 5.00000i − 0.569803i
$$78$$ 2.00000i 0.226455i
$$79$$ −3.00000 −0.337526 −0.168763 0.985657i $$-0.553977\pi$$
−0.168763 + 0.985657i $$0.553977\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 10.0000i 1.10432i
$$83$$ − 14.0000i − 1.53670i −0.640030 0.768350i $$-0.721078\pi$$
0.640030 0.768350i $$-0.278922\pi$$
$$84$$ 5.00000 0.545545
$$85$$ 0 0
$$86$$ 9.00000 0.970495
$$87$$ 10.0000i 1.07211i
$$88$$ − 1.00000i − 0.106600i
$$89$$ −4.00000 −0.423999 −0.212000 0.977270i $$-0.567998\pi$$
−0.212000 + 0.977270i $$0.567998\pi$$
$$90$$ 0 0
$$91$$ −10.0000 −1.04828
$$92$$ 2.00000i 0.208514i
$$93$$ − 5.00000i − 0.518476i
$$94$$ −7.00000 −0.721995
$$95$$ 0 0
$$96$$ 1.00000 0.102062
$$97$$ 6.00000i 0.609208i 0.952479 + 0.304604i $$0.0985241\pi$$
−0.952479 + 0.304604i $$0.901476\pi$$
$$98$$ 18.0000i 1.81827i
$$99$$ 1.00000 0.100504
$$100$$ 0 0
$$101$$ −7.00000 −0.696526 −0.348263 0.937397i $$-0.613228\pi$$
−0.348263 + 0.937397i $$0.613228\pi$$
$$102$$ 1.00000i 0.0990148i
$$103$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$104$$ −2.00000 −0.196116
$$105$$ 0 0
$$106$$ 11.0000 1.06841
$$107$$ − 9.00000i − 0.870063i −0.900415 0.435031i $$-0.856737\pi$$
0.900415 0.435031i $$-0.143263\pi$$
$$108$$ 1.00000i 0.0962250i
$$109$$ −19.0000 −1.81987 −0.909935 0.414751i $$-0.863869\pi$$
−0.909935 + 0.414751i $$0.863869\pi$$
$$110$$ 0 0
$$111$$ −3.00000 −0.284747
$$112$$ 5.00000i 0.472456i
$$113$$ 3.00000i 0.282216i 0.989994 + 0.141108i $$0.0450665\pi$$
−0.989994 + 0.141108i $$0.954933\pi$$
$$114$$ 7.00000 0.655610
$$115$$ 0 0
$$116$$ −10.0000 −0.928477
$$117$$ − 2.00000i − 0.184900i
$$118$$ 0 0
$$119$$ −5.00000 −0.458349
$$120$$ 0 0
$$121$$ −10.0000 −0.909091
$$122$$ − 2.00000i − 0.181071i
$$123$$ − 10.0000i − 0.901670i
$$124$$ 5.00000 0.449013
$$125$$ 0 0
$$126$$ −5.00000 −0.445435
$$127$$ − 4.00000i − 0.354943i −0.984126 0.177471i $$-0.943208\pi$$
0.984126 0.177471i $$-0.0567917\pi$$
$$128$$ 1.00000i 0.0883883i
$$129$$ −9.00000 −0.792406
$$130$$ 0 0
$$131$$ −4.00000 −0.349482 −0.174741 0.984614i $$-0.555909\pi$$
−0.174741 + 0.984614i $$0.555909\pi$$
$$132$$ 1.00000i 0.0870388i
$$133$$ 35.0000i 3.03488i
$$134$$ 13.0000 1.12303
$$135$$ 0 0
$$136$$ −1.00000 −0.0857493
$$137$$ 12.0000i 1.02523i 0.858619 + 0.512615i $$0.171323\pi$$
−0.858619 + 0.512615i $$0.828677\pi$$
$$138$$ − 2.00000i − 0.170251i
$$139$$ 20.0000 1.69638 0.848189 0.529694i $$-0.177693\pi$$
0.848189 + 0.529694i $$0.177693\pi$$
$$140$$ 0 0
$$141$$ 7.00000 0.589506
$$142$$ 10.0000i 0.839181i
$$143$$ − 2.00000i − 0.167248i
$$144$$ −1.00000 −0.0833333
$$145$$ 0 0
$$146$$ −4.00000 −0.331042
$$147$$ − 18.0000i − 1.48461i
$$148$$ − 3.00000i − 0.246598i
$$149$$ 2.00000 0.163846 0.0819232 0.996639i $$-0.473894\pi$$
0.0819232 + 0.996639i $$0.473894\pi$$
$$150$$ 0 0
$$151$$ −8.00000 −0.651031 −0.325515 0.945537i $$-0.605538\pi$$
−0.325515 + 0.945537i $$0.605538\pi$$
$$152$$ 7.00000i 0.567775i
$$153$$ − 1.00000i − 0.0808452i
$$154$$ −5.00000 −0.402911
$$155$$ 0 0
$$156$$ 2.00000 0.160128
$$157$$ − 16.0000i − 1.27694i −0.769647 0.638470i $$-0.779568\pi$$
0.769647 0.638470i $$-0.220432\pi$$
$$158$$ 3.00000i 0.238667i
$$159$$ −11.0000 −0.872357
$$160$$ 0 0
$$161$$ 10.0000 0.788110
$$162$$ − 1.00000i − 0.0785674i
$$163$$ − 16.0000i − 1.25322i −0.779334 0.626608i $$-0.784443\pi$$
0.779334 0.626608i $$-0.215557\pi$$
$$164$$ 10.0000 0.780869
$$165$$ 0 0
$$166$$ −14.0000 −1.08661
$$167$$ − 6.00000i − 0.464294i −0.972681 0.232147i $$-0.925425\pi$$
0.972681 0.232147i $$-0.0745750\pi$$
$$168$$ − 5.00000i − 0.385758i
$$169$$ 9.00000 0.692308
$$170$$ 0 0
$$171$$ −7.00000 −0.535303
$$172$$ − 9.00000i − 0.686244i
$$173$$ 20.0000i 1.52057i 0.649589 + 0.760286i $$0.274941\pi$$
−0.649589 + 0.760286i $$0.725059\pi$$
$$174$$ 10.0000 0.758098
$$175$$ 0 0
$$176$$ −1.00000 −0.0753778
$$177$$ 0 0
$$178$$ 4.00000i 0.299813i
$$179$$ 12.0000 0.896922 0.448461 0.893802i $$-0.351972\pi$$
0.448461 + 0.893802i $$0.351972\pi$$
$$180$$ 0 0
$$181$$ 17.0000 1.26360 0.631800 0.775131i $$-0.282316\pi$$
0.631800 + 0.775131i $$0.282316\pi$$
$$182$$ 10.0000i 0.741249i
$$183$$ 2.00000i 0.147844i
$$184$$ 2.00000 0.147442
$$185$$ 0 0
$$186$$ −5.00000 −0.366618
$$187$$ − 1.00000i − 0.0731272i
$$188$$ 7.00000i 0.510527i
$$189$$ 5.00000 0.363696
$$190$$ 0 0
$$191$$ 25.0000 1.80894 0.904468 0.426541i $$-0.140268\pi$$
0.904468 + 0.426541i $$0.140268\pi$$
$$192$$ − 1.00000i − 0.0721688i
$$193$$ 24.0000i 1.72756i 0.503871 + 0.863779i $$0.331909\pi$$
−0.503871 + 0.863779i $$0.668091\pi$$
$$194$$ 6.00000 0.430775
$$195$$ 0 0
$$196$$ 18.0000 1.28571
$$197$$ − 8.00000i − 0.569976i −0.958531 0.284988i $$-0.908010\pi$$
0.958531 0.284988i $$-0.0919897\pi$$
$$198$$ − 1.00000i − 0.0710669i
$$199$$ −7.00000 −0.496217 −0.248108 0.968732i $$-0.579809\pi$$
−0.248108 + 0.968732i $$0.579809\pi$$
$$200$$ 0 0
$$201$$ −13.0000 −0.916949
$$202$$ 7.00000i 0.492518i
$$203$$ 50.0000i 3.50931i
$$204$$ 1.00000 0.0700140
$$205$$ 0 0
$$206$$ 0 0
$$207$$ 2.00000i 0.139010i
$$208$$ 2.00000i 0.138675i
$$209$$ −7.00000 −0.484200
$$210$$ 0 0
$$211$$ −14.0000 −0.963800 −0.481900 0.876226i $$-0.660053\pi$$
−0.481900 + 0.876226i $$0.660053\pi$$
$$212$$ − 11.0000i − 0.755483i
$$213$$ − 10.0000i − 0.685189i
$$214$$ −9.00000 −0.615227
$$215$$ 0 0
$$216$$ 1.00000 0.0680414
$$217$$ − 25.0000i − 1.69711i
$$218$$ 19.0000i 1.28684i
$$219$$ 4.00000 0.270295
$$220$$ 0 0
$$221$$ −2.00000 −0.134535
$$222$$ 3.00000i 0.201347i
$$223$$ − 10.0000i − 0.669650i −0.942280 0.334825i $$-0.891323\pi$$
0.942280 0.334825i $$-0.108677\pi$$
$$224$$ 5.00000 0.334077
$$225$$ 0 0
$$226$$ 3.00000 0.199557
$$227$$ 1.00000i 0.0663723i 0.999449 + 0.0331862i $$0.0105654\pi$$
−0.999449 + 0.0331862i $$0.989435\pi$$
$$228$$ − 7.00000i − 0.463586i
$$229$$ 12.0000 0.792982 0.396491 0.918039i $$-0.370228\pi$$
0.396491 + 0.918039i $$0.370228\pi$$
$$230$$ 0 0
$$231$$ 5.00000 0.328976
$$232$$ 10.0000i 0.656532i
$$233$$ 18.0000i 1.17922i 0.807688 + 0.589610i $$0.200718\pi$$
−0.807688 + 0.589610i $$0.799282\pi$$
$$234$$ −2.00000 −0.130744
$$235$$ 0 0
$$236$$ 0 0
$$237$$ − 3.00000i − 0.194871i
$$238$$ 5.00000i 0.324102i
$$239$$ −15.0000 −0.970269 −0.485135 0.874439i $$-0.661229\pi$$
−0.485135 + 0.874439i $$0.661229\pi$$
$$240$$ 0 0
$$241$$ −20.0000 −1.28831 −0.644157 0.764894i $$-0.722792\pi$$
−0.644157 + 0.764894i $$0.722792\pi$$
$$242$$ 10.0000i 0.642824i
$$243$$ 1.00000i 0.0641500i
$$244$$ −2.00000 −0.128037
$$245$$ 0 0
$$246$$ −10.0000 −0.637577
$$247$$ 14.0000i 0.890799i
$$248$$ − 5.00000i − 0.317500i
$$249$$ 14.0000 0.887214
$$250$$ 0 0
$$251$$ 2.00000 0.126239 0.0631194 0.998006i $$-0.479895\pi$$
0.0631194 + 0.998006i $$0.479895\pi$$
$$252$$ 5.00000i 0.314970i
$$253$$ 2.00000i 0.125739i
$$254$$ −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 $$0.810261\pi$$
$$258$$ 9.00000i 0.560316i
$$259$$ −15.0000 −0.932055
$$260$$ 0 0
$$261$$ −10.0000 −0.618984
$$262$$ 4.00000i 0.247121i
$$263$$ − 17.0000i − 1.04826i −0.851637 0.524132i $$-0.824390\pi$$
0.851637 0.524132i $$-0.175610\pi$$
$$264$$ 1.00000 0.0615457
$$265$$ 0 0
$$266$$ 35.0000 2.14599
$$267$$ − 4.00000i − 0.244796i
$$268$$ − 13.0000i − 0.794101i
$$269$$ −6.00000 −0.365826 −0.182913 0.983129i $$-0.558553\pi$$
−0.182913 + 0.983129i $$0.558553\pi$$
$$270$$ 0 0
$$271$$ −6.00000 −0.364474 −0.182237 0.983255i $$-0.558334\pi$$
−0.182237 + 0.983255i $$0.558334\pi$$
$$272$$ 1.00000i 0.0606339i
$$273$$ − 10.0000i − 0.605228i
$$274$$ 12.0000 0.724947
$$275$$ 0 0
$$276$$ −2.00000 −0.120386
$$277$$ 23.0000i 1.38194i 0.722885 + 0.690968i $$0.242815\pi$$
−0.722885 + 0.690968i $$0.757185\pi$$
$$278$$ − 20.0000i − 1.19952i
$$279$$ 5.00000 0.299342
$$280$$ 0 0
$$281$$ 22.0000 1.31241 0.656205 0.754583i $$-0.272161\pi$$
0.656205 + 0.754583i $$0.272161\pi$$
$$282$$ − 7.00000i − 0.416844i
$$283$$ 14.0000i 0.832214i 0.909316 + 0.416107i $$0.136606\pi$$
−0.909316 + 0.416107i $$0.863394\pi$$
$$284$$ 10.0000 0.593391
$$285$$ 0 0
$$286$$ −2.00000 −0.118262
$$287$$ − 50.0000i − 2.95141i
$$288$$ 1.00000i 0.0589256i
$$289$$ −1.00000 −0.0588235
$$290$$ 0 0
$$291$$ −6.00000 −0.351726
$$292$$ 4.00000i 0.234082i
$$293$$ 14.0000i 0.817889i 0.912559 + 0.408944i $$0.134103\pi$$
−0.912559 + 0.408944i $$0.865897\pi$$
$$294$$ −18.0000 −1.04978
$$295$$ 0 0
$$296$$ −3.00000 −0.174371
$$297$$ 1.00000i 0.0580259i
$$298$$ − 2.00000i − 0.115857i
$$299$$ 4.00000 0.231326
$$300$$ 0 0
$$301$$ −45.0000 −2.59376
$$302$$ 8.00000i 0.460348i
$$303$$ − 7.00000i − 0.402139i
$$304$$ 7.00000 0.401478
$$305$$ 0 0
$$306$$ −1.00000 −0.0571662
$$307$$ − 4.00000i − 0.228292i −0.993464 0.114146i $$-0.963587\pi$$
0.993464 0.114146i $$-0.0364132\pi$$
$$308$$ 5.00000i 0.284901i
$$309$$ 0 0
$$310$$ 0 0
$$311$$ 26.0000 1.47432 0.737162 0.675716i $$-0.236165\pi$$
0.737162 + 0.675716i $$0.236165\pi$$
$$312$$ − 2.00000i − 0.113228i
$$313$$ 2.00000i 0.113047i 0.998401 + 0.0565233i $$0.0180015\pi$$
−0.998401 + 0.0565233i $$0.981998\pi$$
$$314$$ −16.0000 −0.902932
$$315$$ 0 0
$$316$$ 3.00000 0.168763
$$317$$ − 20.0000i − 1.12331i −0.827371 0.561656i $$-0.810164\pi$$
0.827371 0.561656i $$-0.189836\pi$$
$$318$$ 11.0000i 0.616849i
$$319$$ −10.0000 −0.559893
$$320$$ 0 0
$$321$$ 9.00000 0.502331
$$322$$ − 10.0000i − 0.557278i
$$323$$ 7.00000i 0.389490i
$$324$$ −1.00000 −0.0555556
$$325$$ 0 0
$$326$$ −16.0000 −0.886158
$$327$$ − 19.0000i − 1.05070i
$$328$$ − 10.0000i − 0.552158i
$$329$$ 35.0000 1.92961
$$330$$ 0 0
$$331$$ 7.00000 0.384755 0.192377 0.981321i $$-0.438380\pi$$
0.192377 + 0.981321i $$0.438380\pi$$
$$332$$ 14.0000i 0.768350i
$$333$$ − 3.00000i − 0.164399i
$$334$$ −6.00000 −0.328305
$$335$$ 0 0
$$336$$ −5.00000 −0.272772
$$337$$ 8.00000i 0.435788i 0.975972 + 0.217894i $$0.0699187\pi$$
−0.975972 + 0.217894i $$0.930081\pi$$
$$338$$ − 9.00000i − 0.489535i
$$339$$ −3.00000 −0.162938
$$340$$ 0 0
$$341$$ 5.00000 0.270765
$$342$$ 7.00000i 0.378517i
$$343$$ − 55.0000i − 2.96972i
$$344$$ −9.00000 −0.485247
$$345$$ 0 0
$$346$$ 20.0000 1.07521
$$347$$ 5.00000i 0.268414i 0.990953 + 0.134207i $$0.0428487\pi$$
−0.990953 + 0.134207i $$0.957151\pi$$
$$348$$ − 10.0000i − 0.536056i
$$349$$ 2.00000 0.107058 0.0535288 0.998566i $$-0.482953\pi$$
0.0535288 + 0.998566i $$0.482953\pi$$
$$350$$ 0 0
$$351$$ 2.00000 0.106752
$$352$$ 1.00000i 0.0533002i
$$353$$ − 2.00000i − 0.106449i −0.998583 0.0532246i $$-0.983050\pi$$
0.998583 0.0532246i $$-0.0169499\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 4.00000 0.212000
$$357$$ − 5.00000i − 0.264628i
$$358$$ − 12.0000i − 0.634220i
$$359$$ 3.00000 0.158334 0.0791670 0.996861i $$-0.474774\pi$$
0.0791670 + 0.996861i $$0.474774\pi$$
$$360$$ 0 0
$$361$$ 30.0000 1.57895
$$362$$ − 17.0000i − 0.893500i
$$363$$ − 10.0000i − 0.524864i
$$364$$ 10.0000 0.524142
$$365$$ 0 0
$$366$$ 2.00000 0.104542
$$367$$ − 5.00000i − 0.260998i −0.991448 0.130499i $$-0.958342\pi$$
0.991448 0.130499i $$-0.0416579\pi$$
$$368$$ − 2.00000i − 0.104257i
$$369$$ 10.0000 0.520579
$$370$$ 0 0
$$371$$ −55.0000 −2.85546
$$372$$ 5.00000i 0.259238i
$$373$$ − 26.0000i − 1.34623i −0.739538 0.673114i $$-0.764956\pi$$
0.739538 0.673114i $$-0.235044\pi$$
$$374$$ −1.00000 −0.0517088
$$375$$ 0 0
$$376$$ 7.00000 0.360997
$$377$$ 20.0000i 1.03005i
$$378$$ − 5.00000i − 0.257172i
$$379$$ −20.0000 −1.02733 −0.513665 0.857991i $$-0.671713\pi$$
−0.513665 + 0.857991i $$0.671713\pi$$
$$380$$ 0 0
$$381$$ 4.00000 0.204926
$$382$$ − 25.0000i − 1.27911i
$$383$$ − 8.00000i − 0.408781i −0.978889 0.204390i $$-0.934479\pi$$
0.978889 0.204390i $$-0.0655212\pi$$
$$384$$ −1.00000 −0.0510310
$$385$$ 0 0
$$386$$ 24.0000 1.22157
$$387$$ − 9.00000i − 0.457496i
$$388$$ − 6.00000i − 0.304604i
$$389$$ 15.0000 0.760530 0.380265 0.924878i $$-0.375833\pi$$
0.380265 + 0.924878i $$0.375833\pi$$
$$390$$ 0 0
$$391$$ 2.00000 0.101144
$$392$$ − 18.0000i − 0.909137i
$$393$$ − 4.00000i − 0.201773i
$$394$$ −8.00000 −0.403034
$$395$$ 0 0
$$396$$ −1.00000 −0.0502519
$$397$$ 3.00000i 0.150566i 0.997162 + 0.0752828i $$0.0239860\pi$$
−0.997162 + 0.0752828i $$0.976014\pi$$
$$398$$ 7.00000i 0.350878i
$$399$$ −35.0000 −1.75219
$$400$$ 0 0
$$401$$ 30.0000 1.49813 0.749064 0.662497i $$-0.230503\pi$$
0.749064 + 0.662497i $$0.230503\pi$$
$$402$$ 13.0000i 0.648381i
$$403$$ − 10.0000i − 0.498135i
$$404$$ 7.00000 0.348263
$$405$$ 0 0
$$406$$ 50.0000 2.48146
$$407$$ − 3.00000i − 0.148704i
$$408$$ − 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
$$411$$ −12.0000 −0.591916
$$412$$ 0 0
$$413$$ 0 0
$$414$$ 2.00000 0.0982946
$$415$$ 0 0
$$416$$ 2.00000 0.0980581
$$417$$ 20.0000i 0.979404i
$$418$$ 7.00000i 0.342381i
$$419$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$420$$ 0 0
$$421$$ −30.0000 −1.46211 −0.731055 0.682318i $$-0.760972\pi$$
−0.731055 + 0.682318i $$0.760972\pi$$
$$422$$ 14.0000i 0.681509i
$$423$$ 7.00000i 0.340352i
$$424$$ −11.0000 −0.534207
$$425$$ 0 0
$$426$$ −10.0000 −0.484502
$$427$$ 10.0000i 0.483934i
$$428$$ 9.00000i 0.435031i
$$429$$ 2.00000 0.0965609
$$430$$ 0 0
$$431$$ −16.0000 −0.770693 −0.385346 0.922772i $$-0.625918\pi$$
−0.385346 + 0.922772i $$0.625918\pi$$
$$432$$ − 1.00000i − 0.0481125i
$$433$$ 35.0000i 1.68199i 0.541041 + 0.840996i $$0.318030\pi$$
−0.541041 + 0.840996i $$0.681970\pi$$
$$434$$ −25.0000 −1.20004
$$435$$ 0 0
$$436$$ 19.0000 0.909935
$$437$$ − 14.0000i − 0.669711i
$$438$$ − 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
$$441$$ 18.0000 0.857143
$$442$$ 2.00000i 0.0951303i
$$443$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$444$$ 3.00000 0.142374
$$445$$ 0 0
$$446$$ −10.0000 −0.473514
$$447$$ 2.00000i 0.0945968i
$$448$$ − 5.00000i − 0.236228i
$$449$$ −9.00000 −0.424736 −0.212368 0.977190i $$-0.568118\pi$$
−0.212368 + 0.977190i $$0.568118\pi$$
$$450$$ 0 0
$$451$$ 10.0000 0.470882
$$452$$ − 3.00000i − 0.141108i
$$453$$ − 8.00000i − 0.375873i
$$454$$ 1.00000 0.0469323
$$455$$ 0 0
$$456$$ −7.00000 −0.327805
$$457$$ − 17.0000i − 0.795226i −0.917553 0.397613i $$-0.869839\pi$$
0.917553 0.397613i $$-0.130161\pi$$
$$458$$ − 12.0000i − 0.560723i
$$459$$ 1.00000 0.0466760
$$460$$ 0 0
$$461$$ 35.0000 1.63011 0.815056 0.579382i $$-0.196706\pi$$
0.815056 + 0.579382i $$0.196706\pi$$
$$462$$ − 5.00000i − 0.232621i
$$463$$ − 16.0000i − 0.743583i −0.928316 0.371792i $$-0.878744\pi$$
0.928316 0.371792i $$-0.121256\pi$$
$$464$$ 10.0000 0.464238
$$465$$ 0 0
$$466$$ 18.0000 0.833834
$$467$$ 6.00000i 0.277647i 0.990317 + 0.138823i $$0.0443321\pi$$
−0.990317 + 0.138823i $$0.955668\pi$$
$$468$$ 2.00000i 0.0924500i
$$469$$ −65.0000 −3.00142
$$470$$ 0 0
$$471$$ 16.0000 0.737241
$$472$$ 0 0
$$473$$ − 9.00000i − 0.413820i
$$474$$ −3.00000 −0.137795
$$475$$ 0 0
$$476$$ 5.00000 0.229175
$$477$$ − 11.0000i − 0.503655i
$$478$$ 15.0000i 0.686084i
$$479$$ 22.0000 1.00521 0.502603 0.864517i $$-0.332376\pi$$
0.502603 + 0.864517i $$0.332376\pi$$
$$480$$ 0 0
$$481$$ −6.00000 −0.273576
$$482$$ 20.0000i 0.910975i
$$483$$ 10.0000i 0.455016i
$$484$$ 10.0000 0.454545
$$485$$ 0 0
$$486$$ 1.00000 0.0453609
$$487$$ 8.00000i 0.362515i 0.983436 + 0.181257i $$0.0580167\pi$$
−0.983436 + 0.181257i $$0.941983\pi$$
$$488$$ 2.00000i 0.0905357i
$$489$$ 16.0000 0.723545
$$490$$ 0 0
$$491$$ 12.0000 0.541552 0.270776 0.962642i $$-0.412720\pi$$
0.270776 + 0.962642i $$0.412720\pi$$
$$492$$ 10.0000i 0.450835i
$$493$$ 10.0000i 0.450377i
$$494$$ 14.0000 0.629890
$$495$$ 0 0
$$496$$ −5.00000 −0.224507
$$497$$ − 50.0000i − 2.24281i
$$498$$ − 14.0000i − 0.627355i
$$499$$ −22.0000 −0.984855 −0.492428 0.870353i $$-0.663890\pi$$
−0.492428 + 0.870353i $$0.663890\pi$$
$$500$$ 0 0
$$501$$ 6.00000 0.268060
$$502$$ − 2.00000i − 0.0892644i
$$503$$ 22.0000i 0.980932i 0.871460 + 0.490466i $$0.163173\pi$$
−0.871460 + 0.490466i $$0.836827\pi$$
$$504$$ 5.00000 0.222718
$$505$$ 0 0
$$506$$ 2.00000 0.0889108
$$507$$ 9.00000i 0.399704i
$$508$$ 4.00000i 0.177471i
$$509$$ 37.0000 1.64000 0.819998 0.572366i $$-0.193974\pi$$
0.819998 + 0.572366i $$0.193974\pi$$
$$510$$ 0 0
$$511$$ 20.0000 0.884748
$$512$$ − 1.00000i − 0.0441942i
$$513$$ − 7.00000i − 0.309058i
$$514$$ 18.0000 0.793946
$$515$$ 0 0
$$516$$ 9.00000 0.396203
$$517$$ 7.00000i 0.307860i
$$518$$ 15.0000i 0.659062i
$$519$$ −20.0000 −0.877903
$$520$$ 0 0
$$521$$ −27.0000 −1.18289 −0.591446 0.806345i $$-0.701443\pi$$
−0.591446 + 0.806345i $$0.701443\pi$$
$$522$$ 10.0000i 0.437688i
$$523$$ − 4.00000i − 0.174908i −0.996169 0.0874539i $$-0.972127\pi$$
0.996169 0.0874539i $$-0.0278730\pi$$
$$524$$ 4.00000 0.174741
$$525$$ 0 0
$$526$$ −17.0000 −0.741235
$$527$$ − 5.00000i − 0.217803i
$$528$$ − 1.00000i − 0.0435194i
$$529$$ 19.0000 0.826087
$$530$$ 0 0
$$531$$ 0 0
$$532$$ − 35.0000i − 1.51744i
$$533$$ − 20.0000i − 0.866296i
$$534$$ −4.00000 −0.173097
$$535$$ 0 0
$$536$$ −13.0000 −0.561514
$$537$$ 12.0000i 0.517838i
$$538$$ 6.00000i 0.258678i
$$539$$ 18.0000 0.775315
$$540$$ 0 0
$$541$$ −39.0000 −1.67674 −0.838370 0.545101i $$-0.816491\pi$$
−0.838370 + 0.545101i $$0.816491\pi$$
$$542$$ 6.00000i 0.257722i
$$543$$ 17.0000i 0.729540i
$$544$$ 1.00000 0.0428746
$$545$$ 0 0
$$546$$ −10.0000 −0.427960
$$547$$ 38.0000i 1.62476i 0.583127 + 0.812381i $$0.301829\pi$$
−0.583127 + 0.812381i $$0.698171\pi$$
$$548$$ − 12.0000i − 0.512615i
$$549$$ −2.00000 −0.0853579
$$550$$ 0 0
$$551$$ 70.0000 2.98210
$$552$$ 2.00000i 0.0851257i
$$553$$ − 15.0000i − 0.637865i
$$554$$ 23.0000 0.977176
$$555$$ 0 0
$$556$$ −20.0000 −0.848189
$$557$$ − 29.0000i − 1.22877i −0.789007 0.614385i $$-0.789404\pi$$
0.789007 0.614385i $$-0.210596\pi$$
$$558$$ − 5.00000i − 0.211667i
$$559$$ −18.0000 −0.761319
$$560$$ 0 0
$$561$$ 1.00000 0.0422200
$$562$$ − 22.0000i − 0.928014i
$$563$$ − 12.0000i − 0.505740i −0.967500 0.252870i $$-0.918626\pi$$
0.967500 0.252870i $$-0.0813744\pi$$
$$564$$ −7.00000 −0.294753
$$565$$ 0 0
$$566$$ 14.0000 0.588464
$$567$$ 5.00000i 0.209980i
$$568$$ − 10.0000i − 0.419591i
$$569$$ 36.0000 1.50920 0.754599 0.656186i $$-0.227831\pi$$
0.754599 + 0.656186i $$0.227831\pi$$
$$570$$ 0 0
$$571$$ −20.0000 −0.836974 −0.418487 0.908223i $$-0.637439\pi$$
−0.418487 + 0.908223i $$0.637439\pi$$
$$572$$ 2.00000i 0.0836242i
$$573$$ 25.0000i 1.04439i
$$574$$ −50.0000 −2.08696
$$575$$ 0 0
$$576$$ 1.00000 0.0416667
$$577$$ − 39.0000i − 1.62359i −0.583942 0.811796i $$-0.698490\pi$$
0.583942 0.811796i $$-0.301510\pi$$
$$578$$ 1.00000i 0.0415945i
$$579$$ −24.0000 −0.997406
$$580$$ 0 0
$$581$$ 70.0000 2.90409
$$582$$ 6.00000i 0.248708i
$$583$$ − 11.0000i − 0.455573i
$$584$$ 4.00000 0.165521
$$585$$ 0 0
$$586$$ 14.0000 0.578335
$$587$$ 26.0000i 1.07313i 0.843857 + 0.536567i $$0.180279\pi$$
−0.843857 + 0.536567i $$0.819721\pi$$
$$588$$ 18.0000i 0.742307i
$$589$$ −35.0000 −1.44215
$$590$$ 0 0
$$591$$ 8.00000 0.329076
$$592$$ 3.00000i 0.123299i
$$593$$ − 12.0000i − 0.492781i −0.969171 0.246390i $$-0.920755\pi$$
0.969171 0.246390i $$-0.0792446\pi$$
$$594$$ 1.00000 0.0410305
$$595$$ 0 0
$$596$$ −2.00000 −0.0819232
$$597$$ − 7.00000i − 0.286491i
$$598$$ − 4.00000i − 0.163572i
$$599$$ −33.0000 −1.34834 −0.674172 0.738575i $$-0.735499\pi$$
−0.674172 + 0.738575i $$0.735499\pi$$
$$600$$ 0 0
$$601$$ 18.0000 0.734235 0.367118 0.930175i $$-0.380345\pi$$
0.367118 + 0.930175i $$0.380345\pi$$
$$602$$ 45.0000i 1.83406i
$$603$$ − 13.0000i − 0.529401i
$$604$$ 8.00000 0.325515
$$605$$ 0 0
$$606$$ −7.00000 −0.284356
$$607$$ 28.0000i 1.13648i 0.822861 + 0.568242i $$0.192376\pi$$
−0.822861 + 0.568242i $$0.807624\pi$$
$$608$$ − 7.00000i − 0.283887i
$$609$$ −50.0000 −2.02610
$$610$$ 0 0
$$611$$ 14.0000 0.566379
$$612$$ 1.00000i 0.0404226i
$$613$$ 2.00000i 0.0807792i 0.999184 + 0.0403896i $$0.0128599\pi$$
−0.999184 + 0.0403896i $$0.987140\pi$$
$$614$$ −4.00000 −0.161427
$$615$$ 0 0
$$616$$ 5.00000 0.201456
$$617$$ − 9.00000i − 0.362326i −0.983453 0.181163i $$-0.942014\pi$$
0.983453 0.181163i $$-0.0579862\pi$$
$$618$$ 0 0
$$619$$ 16.0000 0.643094 0.321547 0.946894i $$-0.395797\pi$$
0.321547 + 0.946894i $$0.395797\pi$$
$$620$$ 0 0
$$621$$ −2.00000 −0.0802572
$$622$$ − 26.0000i − 1.04251i
$$623$$ − 20.0000i − 0.801283i
$$624$$ −2.00000 −0.0800641
$$625$$ 0 0
$$626$$ 2.00000 0.0799361
$$627$$ − 7.00000i − 0.279553i
$$628$$ 16.0000i 0.638470i
$$629$$ −3.00000 −0.119618
$$630$$ 0 0
$$631$$ 38.0000 1.51276 0.756378 0.654135i $$-0.226967\pi$$
0.756378 + 0.654135i $$0.226967\pi$$
$$632$$ − 3.00000i − 0.119334i
$$633$$ − 14.0000i − 0.556450i
$$634$$ −20.0000 −0.794301
$$635$$ 0 0
$$636$$ 11.0000 0.436178
$$637$$ − 36.0000i − 1.42637i
$$638$$ 10.0000i 0.395904i
$$639$$ 10.0000 0.395594
$$640$$ 0 0
$$641$$ 18.0000 0.710957 0.355479 0.934684i $$-0.384318\pi$$
0.355479 + 0.934684i $$0.384318\pi$$
$$642$$ − 9.00000i − 0.355202i
$$643$$ 14.0000i 0.552106i 0.961142 + 0.276053i $$0.0890266\pi$$
−0.961142 + 0.276053i $$0.910973\pi$$
$$644$$ −10.0000 −0.394055
$$645$$ 0 0
$$646$$ 7.00000 0.275411
$$647$$ 32.0000i 1.25805i 0.777385 + 0.629025i $$0.216546\pi$$
−0.777385 + 0.629025i $$0.783454\pi$$
$$648$$ 1.00000i 0.0392837i
$$649$$ 0 0
$$650$$ 0 0
$$651$$ 25.0000 0.979827
$$652$$ 16.0000i 0.626608i
$$653$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$654$$ −19.0000 −0.742959
$$655$$ 0 0
$$656$$ −10.0000 −0.390434
$$657$$ 4.00000i 0.156055i
$$658$$ − 35.0000i − 1.36444i
$$659$$ −44.0000 −1.71400 −0.856998 0.515319i $$-0.827673\pi$$
−0.856998 + 0.515319i $$0.827673\pi$$
$$660$$ 0 0
$$661$$ −16.0000 −0.622328 −0.311164 0.950356i $$-0.600719\pi$$
−0.311164 + 0.950356i $$0.600719\pi$$
$$662$$ − 7.00000i − 0.272063i
$$663$$ − 2.00000i − 0.0776736i
$$664$$ 14.0000 0.543305
$$665$$ 0 0
$$666$$ −3.00000 −0.116248
$$667$$ − 20.0000i − 0.774403i
$$668$$ 6.00000i 0.232147i
$$669$$ 10.0000 0.386622
$$670$$ 0 0
$$671$$ −2.00000 −0.0772091
$$672$$ 5.00000i 0.192879i
$$673$$ − 10.0000i − 0.385472i −0.981251 0.192736i $$-0.938264\pi$$
0.981251 0.192736i $$-0.0617360\pi$$
$$674$$ 8.00000 0.308148
$$675$$ 0 0
$$676$$ −9.00000 −0.346154
$$677$$ 36.0000i 1.38359i 0.722093 + 0.691796i $$0.243180\pi$$
−0.722093 + 0.691796i $$0.756820\pi$$
$$678$$ 3.00000i 0.115214i
$$679$$ −30.0000 −1.15129
$$680$$ 0 0
$$681$$ −1.00000 −0.0383201
$$682$$ − 5.00000i − 0.191460i
$$683$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$684$$ 7.00000 0.267652
$$685$$ 0 0
$$686$$ −55.0000 −2.09991
$$687$$ 12.0000i 0.457829i
$$688$$ 9.00000i 0.343122i
$$689$$ −22.0000 −0.838133
$$690$$ 0 0
$$691$$ 4.00000 0.152167 0.0760836 0.997101i $$-0.475758\pi$$
0.0760836 + 0.997101i $$0.475758\pi$$
$$692$$ − 20.0000i − 0.760286i
$$693$$ 5.00000i 0.189934i
$$694$$ 5.00000 0.189797
$$695$$ 0 0
$$696$$ −10.0000 −0.379049
$$697$$ − 10.0000i − 0.378777i
$$698$$ − 2.00000i − 0.0757011i
$$699$$ −18.0000 −0.680823
$$700$$ 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
$$703$$ 21.0000i 0.792030i
$$704$$ 1.00000 0.0376889
$$705$$ 0 0
$$706$$ −2.00000 −0.0752710
$$707$$ − 35.0000i − 1.31631i
$$708$$ 0 0
$$709$$ 17.0000 0.638448 0.319224 0.947679i $$-0.396578\pi$$
0.319224 + 0.947679i $$0.396578\pi$$
$$710$$ 0 0
$$711$$ 3.00000 0.112509
$$712$$ − 4.00000i − 0.149906i
$$713$$ 10.0000i 0.374503i
$$714$$ −5.00000 −0.187120
$$715$$ 0 0
$$716$$ −12.0000 −0.448461
$$717$$ − 15.0000i − 0.560185i
$$718$$ − 3.00000i − 0.111959i
$$719$$ 2.00000 0.0745874 0.0372937 0.999304i $$-0.488126\pi$$
0.0372937 + 0.999304i $$0.488126\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ − 30.0000i − 1.11648i
$$723$$ − 20.0000i − 0.743808i
$$724$$ −17.0000 −0.631800
$$725$$ 0 0
$$726$$ −10.0000 −0.371135
$$727$$ 26.0000i 0.964287i 0.876092 + 0.482143i $$0.160142\pi$$
−0.876092 + 0.482143i $$0.839858\pi$$
$$728$$ − 10.0000i − 0.370625i
$$729$$ −1.00000 −0.0370370
$$730$$ 0 0
$$731$$ −9.00000 −0.332877
$$732$$ − 2.00000i − 0.0739221i
$$733$$ − 36.0000i − 1.32969i −0.746981 0.664845i $$-0.768498\pi$$
0.746981 0.664845i $$-0.231502\pi$$
$$734$$ −5.00000 −0.184553
$$735$$ 0 0
$$736$$ −2.00000 −0.0737210
$$737$$ − 13.0000i − 0.478861i
$$738$$ − 10.0000i − 0.368105i
$$739$$ 51.0000 1.87607 0.938033 0.346547i $$-0.112646\pi$$
0.938033 + 0.346547i $$0.112646\pi$$
$$740$$ 0 0
$$741$$ −14.0000 −0.514303
$$742$$ 55.0000i 2.01911i
$$743$$ − 34.0000i − 1.24734i −0.781688 0.623670i $$-0.785641\pi$$
0.781688 0.623670i $$-0.214359\pi$$
$$744$$ 5.00000 0.183309
$$745$$ 0 0
$$746$$ −26.0000 −0.951928
$$747$$ 14.0000i 0.512233i
$$748$$ 1.00000i 0.0365636i
$$749$$ 45.0000 1.64426
$$750$$ 0 0
$$751$$ −24.0000 −0.875772 −0.437886 0.899030i $$-0.644273\pi$$
−0.437886 + 0.899030i $$0.644273\pi$$
$$752$$ − 7.00000i − 0.255264i
$$753$$ 2.00000i 0.0728841i
$$754$$ 20.0000 0.728357
$$755$$ 0 0
$$756$$ −5.00000 −0.181848
$$757$$ − 10.0000i − 0.363456i −0.983349 0.181728i $$-0.941831\pi$$
0.983349 0.181728i $$-0.0581691\pi$$
$$758$$ 20.0000i 0.726433i
$$759$$ −2.00000 −0.0725954
$$760$$ 0 0
$$761$$ −36.0000 −1.30500 −0.652499 0.757789i $$-0.726280\pi$$
−0.652499 + 0.757789i $$0.726280\pi$$
$$762$$ − 4.00000i − 0.144905i
$$763$$ − 95.0000i − 3.43923i
$$764$$ −25.0000 −0.904468
$$765$$ 0 0
$$766$$ −8.00000 −0.289052
$$767$$ 0 0
$$768$$ 1.00000i 0.0360844i
$$769$$ 13.0000 0.468792 0.234396 0.972141i $$-0.424689\pi$$
0.234396 + 0.972141i $$0.424689\pi$$
$$770$$ 0 0
$$771$$ −18.0000 −0.648254
$$772$$ − 24.0000i − 0.863779i
$$773$$ 6.00000i 0.215805i 0.994161 + 0.107903i $$0.0344134\pi$$
−0.994161 + 0.107903i $$0.965587\pi$$
$$774$$ −9.00000 −0.323498
$$775$$ 0 0
$$776$$ −6.00000 −0.215387
$$777$$ − 15.0000i − 0.538122i
$$778$$ − 15.0000i − 0.537776i
$$779$$ −70.0000 −2.50801
$$780$$ 0 0
$$781$$ 10.0000 0.357828
$$782$$ − 2.00000i − 0.0715199i
$$783$$ − 10.0000i − 0.357371i
$$784$$ −18.0000 −0.642857
$$785$$ 0 0
$$786$$ −4.00000 −0.142675
$$787$$ 18.0000i 0.641631i 0.947142 + 0.320815i $$0.103957\pi$$
−0.947142 + 0.320815i $$0.896043\pi$$
$$788$$ 8.00000i 0.284988i
$$789$$ 17.0000 0.605216
$$790$$ 0 0
$$791$$ −15.0000 −0.533339
$$792$$ 1.00000i 0.0355335i
$$793$$ 4.00000i 0.142044i
$$794$$ 3.00000 0.106466
$$795$$ 0 0
$$796$$ 7.00000 0.248108
$$797$$ − 41.0000i − 1.45229i −0.687539 0.726147i $$-0.741309\pi$$
0.687539 0.726147i $$-0.258691\pi$$
$$798$$ 35.0000i 1.23899i
$$799$$ 7.00000 0.247642
$$800$$ 0 0
$$801$$ 4.00000 0.141333
$$802$$ − 30.0000i − 1.05934i
$$803$$ 4.00000i 0.141157i
$$804$$ 13.0000 0.458475
$$805$$ 0 0
$$806$$ −10.0000 −0.352235
$$807$$ − 6.00000i − 0.211210i
$$808$$ − 7.00000i − 0.246259i
$$809$$ −45.0000 −1.58212 −0.791058 0.611741i $$-0.790469\pi$$
−0.791058 + 0.611741i $$0.790469\pi$$
$$810$$ 0 0
$$811$$ 8.00000 0.280918 0.140459 0.990086i $$-0.455142\pi$$
0.140459 + 0.990086i $$0.455142\pi$$
$$812$$ − 50.0000i − 1.75466i
$$813$$ − 6.00000i − 0.210429i
$$814$$ −3.00000 −0.105150
$$815$$ 0 0
$$816$$ −1.00000 −0.0350070
$$817$$ 63.0000i 2.20409i
$$818$$ − 26.0000i − 0.909069i
$$819$$ 10.0000 0.349428
$$820$$ 0 0
$$821$$ 18.0000 0.628204 0.314102 0.949389i $$-0.398297\pi$$
0.314102 + 0.949389i $$0.398297\pi$$
$$822$$ 12.0000i 0.418548i
$$823$$ 4.00000i 0.139431i 0.997567 + 0.0697156i $$0.0222092\pi$$
−0.997567 + 0.0697156i $$0.977791\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 23.0000i 0.799788i 0.916561 + 0.399894i $$0.130953\pi$$
−0.916561 + 0.399894i $$0.869047\pi$$
$$828$$ − 2.00000i − 0.0695048i
$$829$$ 22.0000 0.764092 0.382046 0.924143i $$-0.375220\pi$$
0.382046 + 0.924143i $$0.375220\pi$$
$$830$$ 0 0
$$831$$ −23.0000 −0.797861
$$832$$ − 2.00000i − 0.0693375i
$$833$$ − 18.0000i − 0.623663i
$$834$$ 20.0000 0.692543
$$835$$ 0 0
$$836$$ 7.00000 0.242100
$$837$$ 5.00000i 0.172825i
$$838$$ 0 0
$$839$$ −36.0000 −1.24286 −0.621429 0.783470i $$-0.713448\pi$$
−0.621429 + 0.783470i $$0.713448\pi$$
$$840$$ 0 0
$$841$$ 71.0000 2.44828
$$842$$ 30.0000i 1.03387i
$$843$$ 22.0000i 0.757720i
$$844$$ 14.0000 0.481900
$$845$$ 0 0
$$846$$ 7.00000 0.240665
$$847$$ − 50.0000i − 1.71802i
$$848$$ 11.0000i 0.377742i
$$849$$ −14.0000 −0.480479
$$850$$ 0 0
$$851$$ 6.00000 0.205677
$$852$$ 10.0000i 0.342594i
$$853$$ − 5.00000i − 0.171197i −0.996330 0.0855984i $$-0.972720\pi$$
0.996330 0.0855984i $$-0.0272802\pi$$
$$854$$ 10.0000 0.342193
$$855$$ 0 0
$$856$$ 9.00000 0.307614
$$857$$ 23.0000i 0.785665i 0.919610 + 0.392833i $$0.128505\pi$$
−0.919610 + 0.392833i $$0.871495\pi$$
$$858$$ − 2.00000i − 0.0682789i
$$859$$ 47.0000 1.60362 0.801810 0.597580i $$-0.203871\pi$$
0.801810 + 0.597580i $$0.203871\pi$$
$$860$$ 0 0
$$861$$ 50.0000 1.70400
$$862$$ 16.0000i 0.544962i
$$863$$ 43.0000i 1.46374i 0.681446 + 0.731869i $$0.261351\pi$$
−0.681446 + 0.731869i $$0.738649\pi$$
$$864$$ −1.00000 −0.0340207
$$865$$ 0 0
$$866$$ 35.0000 1.18935
$$867$$ − 1.00000i − 0.0339618i
$$868$$ 25.0000i 0.848555i
$$869$$ 3.00000 0.101768
$$870$$ 0 0
$$871$$ −26.0000 −0.880976
$$872$$ − 19.0000i − 0.643421i
$$873$$ − 6.00000i − 0.203069i
$$874$$ −14.0000 −0.473557
$$875$$ 0 0
$$876$$ −4.00000 −0.135147
$$877$$ 50.0000i 1.68838i 0.536044 + 0.844190i $$0.319918\pi$$
−0.536044 + 0.844190i $$0.680082\pi$$
$$878$$ − 24.0000i − 0.809961i
$$879$$ −14.0000 −0.472208
$$880$$ 0 0
$$881$$ 3.00000 0.101073 0.0505363 0.998722i $$-0.483907\pi$$
0.0505363 + 0.998722i $$0.483907\pi$$
$$882$$ − 18.0000i − 0.606092i
$$883$$ − 44.0000i − 1.48072i −0.672212 0.740359i $$-0.734656\pi$$
0.672212 0.740359i $$-0.265344\pi$$
$$884$$ 2.00000 0.0672673
$$885$$ 0 0
$$886$$ 0 0
$$887$$ − 24.0000i − 0.805841i −0.915235 0.402921i $$-0.867995\pi$$
0.915235 0.402921i $$-0.132005\pi$$
$$888$$ − 3.00000i − 0.100673i
$$889$$ 20.0000 0.670778
$$890$$ 0 0
$$891$$ −1.00000 −0.0335013
$$892$$ 10.0000i 0.334825i
$$893$$ − 49.0000i − 1.63972i
$$894$$ 2.00000 0.0668900
$$895$$ 0 0
$$896$$ −5.00000 −0.167038
$$897$$ 4.00000i 0.133556i
$$898$$ 9.00000i 0.300334i
$$899$$ −50.0000 −1.66759
$$900$$ 0 0
$$901$$ −11.0000 −0.366463
$$902$$ − 10.0000i − 0.332964i
$$903$$ − 45.0000i − 1.49751i
$$904$$ −3.00000 −0.0997785
$$905$$ 0 0
$$906$$ −8.00000 −0.265782
$$907$$ − 38.0000i − 1.26177i −0.775877 0.630885i $$-0.782692\pi$$
0.775877 0.630885i $$-0.217308\pi$$
$$908$$ − 1.00000i − 0.0331862i
$$909$$ 7.00000 0.232175
$$910$$ 0 0
$$911$$ 8.00000 0.265052 0.132526 0.991180i $$-0.457691\pi$$
0.132526 + 0.991180i $$0.457691\pi$$
$$912$$ 7.00000i 0.231793i
$$913$$ 14.0000i 0.463332i
$$914$$ −17.0000 −0.562310
$$915$$ 0 0
$$916$$ −12.0000 −0.396491
$$917$$ − 20.0000i − 0.660458i
$$918$$ − 1.00000i − 0.0330049i
$$919$$ −4.00000 −0.131948 −0.0659739 0.997821i $$-0.521015\pi$$
−0.0659739 + 0.997821i $$0.521015\pi$$
$$920$$ 0 0
$$921$$ 4.00000 0.131804
$$922$$ − 35.0000i − 1.15266i
$$923$$ − 20.0000i − 0.658308i
$$924$$ −5.00000 −0.164488
$$925$$ 0 0
$$926$$ −16.0000 −0.525793
$$927$$ 0 0
$$928$$ − 10.0000i − 0.328266i
$$929$$ 9.00000 0.295280 0.147640 0.989041i $$-0.452832\pi$$
0.147640 + 0.989041i $$0.452832\pi$$
$$930$$ 0 0
$$931$$ −126.000 −4.12948
$$932$$ − 18.0000i − 0.589610i
$$933$$ 26.0000i 0.851202i
$$934$$ 6.00000 0.196326
$$935$$ 0 0
$$936$$ 2.00000 0.0653720
$$937$$ 26.0000i 0.849383i 0.905338 + 0.424691i $$0.139617\pi$$
−0.905338 + 0.424691i $$0.860383\pi$$
$$938$$ 65.0000i 2.12233i
$$939$$ −2.00000 −0.0652675
$$940$$ 0 0
$$941$$ 10.0000 0.325991 0.162995 0.986627i $$-0.447884\pi$$
0.162995 + 0.986627i $$0.447884\pi$$
$$942$$ − 16.0000i − 0.521308i
$$943$$ 20.0000i 0.651290i
$$944$$ 0 0
$$945$$ 0 0
$$946$$ −9.00000 −0.292615
$$947$$ 41.0000i 1.33232i 0.745808 + 0.666160i $$0.232063\pi$$
−0.745808 + 0.666160i $$0.767937\pi$$
$$948$$ 3.00000i 0.0974355i
$$949$$ 8.00000 0.259691
$$950$$ 0 0
$$951$$ 20.0000 0.648544
$$952$$ − 5.00000i − 0.162051i
$$953$$ 54.0000i 1.74923i 0.484817 + 0.874616i $$0.338886\pi$$
−0.484817 + 0.874616i $$0.661114\pi$$
$$954$$ −11.0000 −0.356138
$$955$$ 0 0
$$956$$ 15.0000 0.485135
$$957$$ − 10.0000i − 0.323254i
$$958$$ − 22.0000i − 0.710788i
$$959$$ −60.0000 −1.93750
$$960$$ 0 0
$$961$$ −6.00000 −0.193548
$$962$$ 6.00000i 0.193448i
$$963$$ 9.00000i 0.290021i
$$964$$ 20.0000 0.644157
$$965$$ 0 0
$$966$$ 10.0000 0.321745
$$967$$ − 16.0000i − 0.514525i −0.966342 0.257263i $$-0.917179\pi$$
0.966342 0.257263i $$-0.0828206\pi$$
$$968$$ − 10.0000i − 0.321412i
$$969$$ −7.00000 −0.224872
$$970$$ 0 0
$$971$$ 22.0000 0.706014 0.353007 0.935621i $$-0.385159\pi$$
0.353007 + 0.935621i $$0.385159\pi$$
$$972$$ − 1.00000i − 0.0320750i
$$973$$ 100.000i 3.20585i
$$974$$ 8.00000 0.256337
$$975$$ 0 0
$$976$$ 2.00000 0.0640184
$$977$$ 18.0000i 0.575871i 0.957650 + 0.287936i $$0.0929689\pi$$
−0.957650 + 0.287936i $$0.907031\pi$$
$$978$$ − 16.0000i − 0.511624i
$$979$$ 4.00000 0.127841
$$980$$ 0 0
$$981$$ 19.0000 0.606623
$$982$$ − 12.0000i − 0.382935i
$$983$$ 10.0000i 0.318950i 0.987202 + 0.159475i $$0.0509802\pi$$
−0.987202 + 0.159475i $$0.949020\pi$$
$$984$$ 10.0000 0.318788
$$985$$ 0 0
$$986$$ 10.0000 0.318465
$$987$$ 35.0000i 1.11406i
$$988$$ − 14.0000i − 0.445399i
$$989$$ 18.0000 0.572367
$$990$$ 0 0
$$991$$ −4.00000 −0.127064 −0.0635321 0.997980i $$-0.520237\pi$$
−0.0635321 + 0.997980i $$0.520237\pi$$
$$992$$ 5.00000i 0.158750i
$$993$$ 7.00000i 0.222138i
$$994$$ −50.0000 −1.58590
$$995$$ 0 0
$$996$$ −14.0000 −0.443607
$$997$$ 43.0000i 1.36182i 0.732365 + 0.680912i $$0.238416\pi$$
−0.732365 + 0.680912i $$0.761584\pi$$
$$998$$ 22.0000i 0.696398i
$$999$$ 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.p.2449.1 2
5.2 odd 4 2550.2.a.ba.1.1 yes 1
5.3 odd 4 2550.2.a.g.1.1 1
5.4 even 2 inner 2550.2.d.p.2449.2 2
15.2 even 4 7650.2.a.b.1.1 1
15.8 even 4 7650.2.a.cp.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
2550.2.a.g.1.1 1 5.3 odd 4
2550.2.a.ba.1.1 yes 1 5.2 odd 4
2550.2.d.p.2449.1 2 1.1 even 1 trivial
2550.2.d.p.2449.2 2 5.4 even 2 inner
7650.2.a.b.1.1 1 15.2 even 4
7650.2.a.cp.1.1 1 15.8 even 4