# Properties

 Label 975.2.c.a.274.2 Level $975$ Weight $2$ Character 975.274 Analytic conductor $7.785$ 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] = [975,2,Mod(274,975)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(975, 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("975.274");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$975 = 3 \cdot 5^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 975.c (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: $$7.78541419707$$ 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: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 195) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 274.2 Root $$1.00000i$$ of defining polynomial Character $$\chi$$ $$=$$ 975.274 Dual form 975.2.c.a.274.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+2.00000i q^{2} +1.00000i q^{3} -2.00000 q^{4} -2.00000 q^{6} +3.00000i q^{7} -1.00000 q^{9} +O(q^{10})$$ $$q+2.00000i q^{2} +1.00000i q^{3} -2.00000 q^{4} -2.00000 q^{6} +3.00000i q^{7} -1.00000 q^{9} -1.00000 q^{11} -2.00000i q^{12} +1.00000i q^{13} -6.00000 q^{14} -4.00000 q^{16} -1.00000i q^{17} -2.00000i q^{18} +2.00000 q^{19} -3.00000 q^{21} -2.00000i q^{22} +3.00000i q^{23} -2.00000 q^{26} -1.00000i q^{27} -6.00000i q^{28} +2.00000 q^{29} -6.00000 q^{31} -8.00000i q^{32} -1.00000i q^{33} +2.00000 q^{34} +2.00000 q^{36} +11.0000i q^{37} +4.00000i q^{38} -1.00000 q^{39} -5.00000 q^{41} -6.00000i q^{42} -4.00000i q^{43} +2.00000 q^{44} -6.00000 q^{46} -10.0000i q^{47} -4.00000i q^{48} -2.00000 q^{49} +1.00000 q^{51} -2.00000i q^{52} -11.0000i q^{53} +2.00000 q^{54} +2.00000i q^{57} +4.00000i q^{58} -8.00000 q^{59} +13.0000 q^{61} -12.0000i q^{62} -3.00000i q^{63} +8.00000 q^{64} +2.00000 q^{66} +12.0000i q^{67} +2.00000i q^{68} -3.00000 q^{69} -5.00000 q^{71} -10.0000i q^{73} -22.0000 q^{74} -4.00000 q^{76} -3.00000i q^{77} -2.00000i q^{78} +3.00000 q^{79} +1.00000 q^{81} -10.0000i q^{82} +12.0000i q^{83} +6.00000 q^{84} +8.00000 q^{86} +2.00000i q^{87} +15.0000 q^{89} -3.00000 q^{91} -6.00000i q^{92} -6.00000i q^{93} +20.0000 q^{94} +8.00000 q^{96} +17.0000i q^{97} -4.00000i q^{98} +1.00000 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 4 q^{4} - 4 q^{6} - 2 q^{9}+O(q^{10})$$ 2 * q - 4 * q^4 - 4 * q^6 - 2 * q^9 $$2 q - 4 q^{4} - 4 q^{6} - 2 q^{9} - 2 q^{11} - 12 q^{14} - 8 q^{16} + 4 q^{19} - 6 q^{21} - 4 q^{26} + 4 q^{29} - 12 q^{31} + 4 q^{34} + 4 q^{36} - 2 q^{39} - 10 q^{41} + 4 q^{44} - 12 q^{46} - 4 q^{49} + 2 q^{51} + 4 q^{54} - 16 q^{59} + 26 q^{61} + 16 q^{64} + 4 q^{66} - 6 q^{69} - 10 q^{71} - 44 q^{74} - 8 q^{76} + 6 q^{79} + 2 q^{81} + 12 q^{84} + 16 q^{86} + 30 q^{89} - 6 q^{91} + 40 q^{94} + 16 q^{96} + 2 q^{99}+O(q^{100})$$ 2 * q - 4 * q^4 - 4 * q^6 - 2 * q^9 - 2 * q^11 - 12 * q^14 - 8 * q^16 + 4 * q^19 - 6 * q^21 - 4 * q^26 + 4 * q^29 - 12 * q^31 + 4 * q^34 + 4 * q^36 - 2 * q^39 - 10 * q^41 + 4 * q^44 - 12 * q^46 - 4 * q^49 + 2 * q^51 + 4 * q^54 - 16 * q^59 + 26 * q^61 + 16 * q^64 + 4 * q^66 - 6 * q^69 - 10 * q^71 - 44 * q^74 - 8 * q^76 + 6 * q^79 + 2 * q^81 + 12 * q^84 + 16 * q^86 + 30 * q^89 - 6 * q^91 + 40 * q^94 + 16 * q^96 + 2 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/975\mathbb{Z}\right)^\times$$.

 $$n$$ $$301$$ $$326$$ $$352$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 2.00000i 1.41421i 0.707107 + 0.707107i $$0.250000\pi$$
−0.707107 + 0.707107i $$0.750000\pi$$
$$3$$ 1.00000i 0.577350i
$$4$$ −2.00000 −1.00000
$$5$$ 0 0
$$6$$ −2.00000 −0.816497
$$7$$ 3.00000i 1.13389i 0.823754 + 0.566947i $$0.191875\pi$$
−0.823754 + 0.566947i $$0.808125\pi$$
$$8$$ 0 0
$$9$$ −1.00000 −0.333333
$$10$$ 0 0
$$11$$ −1.00000 −0.301511 −0.150756 0.988571i $$-0.548171\pi$$
−0.150756 + 0.988571i $$0.548171\pi$$
$$12$$ − 2.00000i − 0.577350i
$$13$$ 1.00000i 0.277350i
$$14$$ −6.00000 −1.60357
$$15$$ 0 0
$$16$$ −4.00000 −1.00000
$$17$$ − 1.00000i − 0.242536i −0.992620 0.121268i $$-0.961304\pi$$
0.992620 0.121268i $$-0.0386960\pi$$
$$18$$ − 2.00000i − 0.471405i
$$19$$ 2.00000 0.458831 0.229416 0.973329i $$-0.426318\pi$$
0.229416 + 0.973329i $$0.426318\pi$$
$$20$$ 0 0
$$21$$ −3.00000 −0.654654
$$22$$ − 2.00000i − 0.426401i
$$23$$ 3.00000i 0.625543i 0.949828 + 0.312772i $$0.101257\pi$$
−0.949828 + 0.312772i $$0.898743\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ −2.00000 −0.392232
$$27$$ − 1.00000i − 0.192450i
$$28$$ − 6.00000i − 1.13389i
$$29$$ 2.00000 0.371391 0.185695 0.982607i $$-0.440546\pi$$
0.185695 + 0.982607i $$0.440546\pi$$
$$30$$ 0 0
$$31$$ −6.00000 −1.07763 −0.538816 0.842424i $$-0.681128\pi$$
−0.538816 + 0.842424i $$0.681128\pi$$
$$32$$ − 8.00000i − 1.41421i
$$33$$ − 1.00000i − 0.174078i
$$34$$ 2.00000 0.342997
$$35$$ 0 0
$$36$$ 2.00000 0.333333
$$37$$ 11.0000i 1.80839i 0.427121 + 0.904194i $$0.359528\pi$$
−0.427121 + 0.904194i $$0.640472\pi$$
$$38$$ 4.00000i 0.648886i
$$39$$ −1.00000 −0.160128
$$40$$ 0 0
$$41$$ −5.00000 −0.780869 −0.390434 0.920631i $$-0.627675\pi$$
−0.390434 + 0.920631i $$0.627675\pi$$
$$42$$ − 6.00000i − 0.925820i
$$43$$ − 4.00000i − 0.609994i −0.952353 0.304997i $$-0.901344\pi$$
0.952353 0.304997i $$-0.0986555\pi$$
$$44$$ 2.00000 0.301511
$$45$$ 0 0
$$46$$ −6.00000 −0.884652
$$47$$ − 10.0000i − 1.45865i −0.684167 0.729325i $$-0.739834\pi$$
0.684167 0.729325i $$-0.260166\pi$$
$$48$$ − 4.00000i − 0.577350i
$$49$$ −2.00000 −0.285714
$$50$$ 0 0
$$51$$ 1.00000 0.140028
$$52$$ − 2.00000i − 0.277350i
$$53$$ − 11.0000i − 1.51097i −0.655168 0.755483i $$-0.727402\pi$$
0.655168 0.755483i $$-0.272598\pi$$
$$54$$ 2.00000 0.272166
$$55$$ 0 0
$$56$$ 0 0
$$57$$ 2.00000i 0.264906i
$$58$$ 4.00000i 0.525226i
$$59$$ −8.00000 −1.04151 −0.520756 0.853706i $$-0.674350\pi$$
−0.520756 + 0.853706i $$0.674350\pi$$
$$60$$ 0 0
$$61$$ 13.0000 1.66448 0.832240 0.554416i $$-0.187058\pi$$
0.832240 + 0.554416i $$0.187058\pi$$
$$62$$ − 12.0000i − 1.52400i
$$63$$ − 3.00000i − 0.377964i
$$64$$ 8.00000 1.00000
$$65$$ 0 0
$$66$$ 2.00000 0.246183
$$67$$ 12.0000i 1.46603i 0.680211 + 0.733017i $$0.261888\pi$$
−0.680211 + 0.733017i $$0.738112\pi$$
$$68$$ 2.00000i 0.242536i
$$69$$ −3.00000 −0.361158
$$70$$ 0 0
$$71$$ −5.00000 −0.593391 −0.296695 0.954972i $$-0.595885\pi$$
−0.296695 + 0.954972i $$0.595885\pi$$
$$72$$ 0 0
$$73$$ − 10.0000i − 1.17041i −0.810885 0.585206i $$-0.801014\pi$$
0.810885 0.585206i $$-0.198986\pi$$
$$74$$ −22.0000 −2.55745
$$75$$ 0 0
$$76$$ −4.00000 −0.458831
$$77$$ − 3.00000i − 0.341882i
$$78$$ − 2.00000i − 0.226455i
$$79$$ 3.00000 0.337526 0.168763 0.985657i $$-0.446023\pi$$
0.168763 + 0.985657i $$0.446023\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ − 10.0000i − 1.10432i
$$83$$ 12.0000i 1.31717i 0.752506 + 0.658586i $$0.228845\pi$$
−0.752506 + 0.658586i $$0.771155\pi$$
$$84$$ 6.00000 0.654654
$$85$$ 0 0
$$86$$ 8.00000 0.862662
$$87$$ 2.00000i 0.214423i
$$88$$ 0 0
$$89$$ 15.0000 1.59000 0.794998 0.606612i $$-0.207472\pi$$
0.794998 + 0.606612i $$0.207472\pi$$
$$90$$ 0 0
$$91$$ −3.00000 −0.314485
$$92$$ − 6.00000i − 0.625543i
$$93$$ − 6.00000i − 0.622171i
$$94$$ 20.0000 2.06284
$$95$$ 0 0
$$96$$ 8.00000 0.816497
$$97$$ 17.0000i 1.72609i 0.505128 + 0.863044i $$0.331445\pi$$
−0.505128 + 0.863044i $$0.668555\pi$$
$$98$$ − 4.00000i − 0.404061i
$$99$$ 1.00000 0.100504
$$100$$ 0 0
$$101$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$102$$ 2.00000i 0.198030i
$$103$$ 16.0000i 1.57653i 0.615338 + 0.788263i $$0.289020\pi$$
−0.615338 + 0.788263i $$0.710980\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 22.0000 2.13683
$$107$$ 9.00000i 0.870063i 0.900415 + 0.435031i $$0.143263\pi$$
−0.900415 + 0.435031i $$0.856737\pi$$
$$108$$ 2.00000i 0.192450i
$$109$$ 16.0000 1.53252 0.766261 0.642529i $$-0.222115\pi$$
0.766261 + 0.642529i $$0.222115\pi$$
$$110$$ 0 0
$$111$$ −11.0000 −1.04407
$$112$$ − 12.0000i − 1.13389i
$$113$$ − 14.0000i − 1.31701i −0.752577 0.658505i $$-0.771189\pi$$
0.752577 0.658505i $$-0.228811\pi$$
$$114$$ −4.00000 −0.374634
$$115$$ 0 0
$$116$$ −4.00000 −0.371391
$$117$$ − 1.00000i − 0.0924500i
$$118$$ − 16.0000i − 1.47292i
$$119$$ 3.00000 0.275010
$$120$$ 0 0
$$121$$ −10.0000 −0.909091
$$122$$ 26.0000i 2.35393i
$$123$$ − 5.00000i − 0.450835i
$$124$$ 12.0000 1.07763
$$125$$ 0 0
$$126$$ 6.00000 0.534522
$$127$$ 10.0000i 0.887357i 0.896186 + 0.443678i $$0.146327\pi$$
−0.896186 + 0.443678i $$0.853673\pi$$
$$128$$ 0 0
$$129$$ 4.00000 0.352180
$$130$$ 0 0
$$131$$ −6.00000 −0.524222 −0.262111 0.965038i $$-0.584419\pi$$
−0.262111 + 0.965038i $$0.584419\pi$$
$$132$$ 2.00000i 0.174078i
$$133$$ 6.00000i 0.520266i
$$134$$ −24.0000 −2.07328
$$135$$ 0 0
$$136$$ 0 0
$$137$$ 18.0000i 1.53784i 0.639343 + 0.768922i $$0.279207\pi$$
−0.639343 + 0.768922i $$0.720793\pi$$
$$138$$ − 6.00000i − 0.510754i
$$139$$ 1.00000 0.0848189 0.0424094 0.999100i $$-0.486497\pi$$
0.0424094 + 0.999100i $$0.486497\pi$$
$$140$$ 0 0
$$141$$ 10.0000 0.842152
$$142$$ − 10.0000i − 0.839181i
$$143$$ − 1.00000i − 0.0836242i
$$144$$ 4.00000 0.333333
$$145$$ 0 0
$$146$$ 20.0000 1.65521
$$147$$ − 2.00000i − 0.164957i
$$148$$ − 22.0000i − 1.80839i
$$149$$ −13.0000 −1.06500 −0.532501 0.846430i $$-0.678748\pi$$
−0.532501 + 0.846430i $$0.678748\pi$$
$$150$$ 0 0
$$151$$ 16.0000 1.30206 0.651031 0.759051i $$-0.274337\pi$$
0.651031 + 0.759051i $$0.274337\pi$$
$$152$$ 0 0
$$153$$ 1.00000i 0.0808452i
$$154$$ 6.00000 0.483494
$$155$$ 0 0
$$156$$ 2.00000 0.160128
$$157$$ − 10.0000i − 0.798087i −0.916932 0.399043i $$-0.869342\pi$$
0.916932 0.399043i $$-0.130658\pi$$
$$158$$ 6.00000i 0.477334i
$$159$$ 11.0000 0.872357
$$160$$ 0 0
$$161$$ −9.00000 −0.709299
$$162$$ 2.00000i 0.157135i
$$163$$ 13.0000i 1.01824i 0.860696 + 0.509119i $$0.170029\pi$$
−0.860696 + 0.509119i $$0.829971\pi$$
$$164$$ 10.0000 0.780869
$$165$$ 0 0
$$166$$ −24.0000 −1.86276
$$167$$ − 12.0000i − 0.928588i −0.885681 0.464294i $$-0.846308\pi$$
0.885681 0.464294i $$-0.153692\pi$$
$$168$$ 0 0
$$169$$ −1.00000 −0.0769231
$$170$$ 0 0
$$171$$ −2.00000 −0.152944
$$172$$ 8.00000i 0.609994i
$$173$$ 6.00000i 0.456172i 0.973641 + 0.228086i $$0.0732467\pi$$
−0.973641 + 0.228086i $$0.926753\pi$$
$$174$$ −4.00000 −0.303239
$$175$$ 0 0
$$176$$ 4.00000 0.301511
$$177$$ − 8.00000i − 0.601317i
$$178$$ 30.0000i 2.24860i
$$179$$ −2.00000 −0.149487 −0.0747435 0.997203i $$-0.523814\pi$$
−0.0747435 + 0.997203i $$0.523814\pi$$
$$180$$ 0 0
$$181$$ −7.00000 −0.520306 −0.260153 0.965567i $$-0.583773\pi$$
−0.260153 + 0.965567i $$0.583773\pi$$
$$182$$ − 6.00000i − 0.444750i
$$183$$ 13.0000i 0.960988i
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 12.0000 0.879883
$$187$$ 1.00000i 0.0731272i
$$188$$ 20.0000i 1.45865i
$$189$$ 3.00000 0.218218
$$190$$ 0 0
$$191$$ −8.00000 −0.578860 −0.289430 0.957199i $$-0.593466\pi$$
−0.289430 + 0.957199i $$0.593466\pi$$
$$192$$ 8.00000i 0.577350i
$$193$$ 13.0000i 0.935760i 0.883792 + 0.467880i $$0.154982\pi$$
−0.883792 + 0.467880i $$0.845018\pi$$
$$194$$ −34.0000 −2.44106
$$195$$ 0 0
$$196$$ 4.00000 0.285714
$$197$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$198$$ 2.00000i 0.142134i
$$199$$ −4.00000 −0.283552 −0.141776 0.989899i $$-0.545281\pi$$
−0.141776 + 0.989899i $$0.545281\pi$$
$$200$$ 0 0
$$201$$ −12.0000 −0.846415
$$202$$ 0 0
$$203$$ 6.00000i 0.421117i
$$204$$ −2.00000 −0.140028
$$205$$ 0 0
$$206$$ −32.0000 −2.22955
$$207$$ − 3.00000i − 0.208514i
$$208$$ − 4.00000i − 0.277350i
$$209$$ −2.00000 −0.138343
$$210$$ 0 0
$$211$$ −4.00000 −0.275371 −0.137686 0.990476i $$-0.543966\pi$$
−0.137686 + 0.990476i $$0.543966\pi$$
$$212$$ 22.0000i 1.51097i
$$213$$ − 5.00000i − 0.342594i
$$214$$ −18.0000 −1.23045
$$215$$ 0 0
$$216$$ 0 0
$$217$$ − 18.0000i − 1.22192i
$$218$$ 32.0000i 2.16731i
$$219$$ 10.0000 0.675737
$$220$$ 0 0
$$221$$ 1.00000 0.0672673
$$222$$ − 22.0000i − 1.47654i
$$223$$ − 8.00000i − 0.535720i −0.963458 0.267860i $$-0.913684\pi$$
0.963458 0.267860i $$-0.0863164\pi$$
$$224$$ 24.0000 1.60357
$$225$$ 0 0
$$226$$ 28.0000 1.86253
$$227$$ 22.0000i 1.46019i 0.683345 + 0.730096i $$0.260525\pi$$
−0.683345 + 0.730096i $$0.739475\pi$$
$$228$$ − 4.00000i − 0.264906i
$$229$$ −18.0000 −1.18947 −0.594737 0.803921i $$-0.702744\pi$$
−0.594737 + 0.803921i $$0.702744\pi$$
$$230$$ 0 0
$$231$$ 3.00000 0.197386
$$232$$ 0 0
$$233$$ 27.0000i 1.76883i 0.466702 + 0.884414i $$0.345442\pi$$
−0.466702 + 0.884414i $$0.654558\pi$$
$$234$$ 2.00000 0.130744
$$235$$ 0 0
$$236$$ 16.0000 1.04151
$$237$$ 3.00000i 0.194871i
$$238$$ 6.00000i 0.388922i
$$239$$ 13.0000 0.840900 0.420450 0.907316i $$-0.361872\pi$$
0.420450 + 0.907316i $$0.361872\pi$$
$$240$$ 0 0
$$241$$ −2.00000 −0.128831 −0.0644157 0.997923i $$-0.520518\pi$$
−0.0644157 + 0.997923i $$0.520518\pi$$
$$242$$ − 20.0000i − 1.28565i
$$243$$ 1.00000i 0.0641500i
$$244$$ −26.0000 −1.66448
$$245$$ 0 0
$$246$$ 10.0000 0.637577
$$247$$ 2.00000i 0.127257i
$$248$$ 0 0
$$249$$ −12.0000 −0.760469
$$250$$ 0 0
$$251$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$252$$ 6.00000i 0.377964i
$$253$$ − 3.00000i − 0.188608i
$$254$$ −20.0000 −1.25491
$$255$$ 0 0
$$256$$ 16.0000 1.00000
$$257$$ − 18.0000i − 1.12281i −0.827541 0.561405i $$-0.810261\pi$$
0.827541 0.561405i $$-0.189739\pi$$
$$258$$ 8.00000i 0.498058i
$$259$$ −33.0000 −2.05052
$$260$$ 0 0
$$261$$ −2.00000 −0.123797
$$262$$ − 12.0000i − 0.741362i
$$263$$ 8.00000i 0.493301i 0.969104 + 0.246651i $$0.0793300\pi$$
−0.969104 + 0.246651i $$0.920670\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ −12.0000 −0.735767
$$267$$ 15.0000i 0.917985i
$$268$$ − 24.0000i − 1.46603i
$$269$$ 4.00000 0.243884 0.121942 0.992537i $$-0.461088\pi$$
0.121942 + 0.992537i $$0.461088\pi$$
$$270$$ 0 0
$$271$$ 22.0000 1.33640 0.668202 0.743980i $$-0.267064\pi$$
0.668202 + 0.743980i $$0.267064\pi$$
$$272$$ 4.00000i 0.242536i
$$273$$ − 3.00000i − 0.181568i
$$274$$ −36.0000 −2.17484
$$275$$ 0 0
$$276$$ 6.00000 0.361158
$$277$$ − 18.0000i − 1.08152i −0.841178 0.540758i $$-0.818138\pi$$
0.841178 0.540758i $$-0.181862\pi$$
$$278$$ 2.00000i 0.119952i
$$279$$ 6.00000 0.359211
$$280$$ 0 0
$$281$$ 30.0000 1.78965 0.894825 0.446417i $$-0.147300\pi$$
0.894825 + 0.446417i $$0.147300\pi$$
$$282$$ 20.0000i 1.19098i
$$283$$ 12.0000i 0.713326i 0.934233 + 0.356663i $$0.116086\pi$$
−0.934233 + 0.356663i $$0.883914\pi$$
$$284$$ 10.0000 0.593391
$$285$$ 0 0
$$286$$ 2.00000 0.118262
$$287$$ − 15.0000i − 0.885422i
$$288$$ 8.00000i 0.471405i
$$289$$ 16.0000 0.941176
$$290$$ 0 0
$$291$$ −17.0000 −0.996558
$$292$$ 20.0000i 1.17041i
$$293$$ − 24.0000i − 1.40209i −0.713115 0.701047i $$-0.752716\pi$$
0.713115 0.701047i $$-0.247284\pi$$
$$294$$ 4.00000 0.233285
$$295$$ 0 0
$$296$$ 0 0
$$297$$ 1.00000i 0.0580259i
$$298$$ − 26.0000i − 1.50614i
$$299$$ −3.00000 −0.173494
$$300$$ 0 0
$$301$$ 12.0000 0.691669
$$302$$ 32.0000i 1.84139i
$$303$$ 0 0
$$304$$ −8.00000 −0.458831
$$305$$ 0 0
$$306$$ −2.00000 −0.114332
$$307$$ − 5.00000i − 0.285365i −0.989769 0.142683i $$-0.954427\pi$$
0.989769 0.142683i $$-0.0455728\pi$$
$$308$$ 6.00000i 0.341882i
$$309$$ −16.0000 −0.910208
$$310$$ 0 0
$$311$$ 24.0000 1.36092 0.680458 0.732787i $$-0.261781\pi$$
0.680458 + 0.732787i $$0.261781\pi$$
$$312$$ 0 0
$$313$$ 10.0000i 0.565233i 0.959233 + 0.282617i $$0.0912024\pi$$
−0.959233 + 0.282617i $$0.908798\pi$$
$$314$$ 20.0000 1.12867
$$315$$ 0 0
$$316$$ −6.00000 −0.337526
$$317$$ 28.0000i 1.57264i 0.617822 + 0.786318i $$0.288015\pi$$
−0.617822 + 0.786318i $$0.711985\pi$$
$$318$$ 22.0000i 1.23370i
$$319$$ −2.00000 −0.111979
$$320$$ 0 0
$$321$$ −9.00000 −0.502331
$$322$$ − 18.0000i − 1.00310i
$$323$$ − 2.00000i − 0.111283i
$$324$$ −2.00000 −0.111111
$$325$$ 0 0
$$326$$ −26.0000 −1.44001
$$327$$ 16.0000i 0.884802i
$$328$$ 0 0
$$329$$ 30.0000 1.65395
$$330$$ 0 0
$$331$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$332$$ − 24.0000i − 1.31717i
$$333$$ − 11.0000i − 0.602796i
$$334$$ 24.0000 1.31322
$$335$$ 0 0
$$336$$ 12.0000 0.654654
$$337$$ 4.00000i 0.217894i 0.994048 + 0.108947i $$0.0347479\pi$$
−0.994048 + 0.108947i $$0.965252\pi$$
$$338$$ − 2.00000i − 0.108786i
$$339$$ 14.0000 0.760376
$$340$$ 0 0
$$341$$ 6.00000 0.324918
$$342$$ − 4.00000i − 0.216295i
$$343$$ 15.0000i 0.809924i
$$344$$ 0 0
$$345$$ 0 0
$$346$$ −12.0000 −0.645124
$$347$$ − 19.0000i − 1.01997i −0.860182 0.509987i $$-0.829650\pi$$
0.860182 0.509987i $$-0.170350\pi$$
$$348$$ − 4.00000i − 0.214423i
$$349$$ 8.00000 0.428230 0.214115 0.976808i $$-0.431313\pi$$
0.214115 + 0.976808i $$0.431313\pi$$
$$350$$ 0 0
$$351$$ 1.00000 0.0533761
$$352$$ 8.00000i 0.426401i
$$353$$ − 6.00000i − 0.319348i −0.987170 0.159674i $$-0.948956\pi$$
0.987170 0.159674i $$-0.0510443\pi$$
$$354$$ 16.0000 0.850390
$$355$$ 0 0
$$356$$ −30.0000 −1.59000
$$357$$ 3.00000i 0.158777i
$$358$$ − 4.00000i − 0.211407i
$$359$$ −24.0000 −1.26667 −0.633336 0.773877i $$-0.718315\pi$$
−0.633336 + 0.773877i $$0.718315\pi$$
$$360$$ 0 0
$$361$$ −15.0000 −0.789474
$$362$$ − 14.0000i − 0.735824i
$$363$$ − 10.0000i − 0.524864i
$$364$$ 6.00000 0.314485
$$365$$ 0 0
$$366$$ −26.0000 −1.35904
$$367$$ 36.0000i 1.87918i 0.342296 + 0.939592i $$0.388796\pi$$
−0.342296 + 0.939592i $$0.611204\pi$$
$$368$$ − 12.0000i − 0.625543i
$$369$$ 5.00000 0.260290
$$370$$ 0 0
$$371$$ 33.0000 1.71327
$$372$$ 12.0000i 0.622171i
$$373$$ − 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
$$376$$ 0 0
$$377$$ 2.00000i 0.103005i
$$378$$ 6.00000i 0.308607i
$$379$$ 14.0000 0.719132 0.359566 0.933120i $$-0.382925\pi$$
0.359566 + 0.933120i $$0.382925\pi$$
$$380$$ 0 0
$$381$$ −10.0000 −0.512316
$$382$$ − 16.0000i − 0.818631i
$$383$$ 30.0000i 1.53293i 0.642287 + 0.766464i $$0.277986\pi$$
−0.642287 + 0.766464i $$0.722014\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ −26.0000 −1.32337
$$387$$ 4.00000i 0.203331i
$$388$$ − 34.0000i − 1.72609i
$$389$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$390$$ 0 0
$$391$$ 3.00000 0.151717
$$392$$ 0 0
$$393$$ − 6.00000i − 0.302660i
$$394$$ 0 0
$$395$$ 0 0
$$396$$ −2.00000 −0.100504
$$397$$ − 29.0000i − 1.45547i −0.685859 0.727734i $$-0.740573\pi$$
0.685859 0.727734i $$-0.259427\pi$$
$$398$$ − 8.00000i − 0.401004i
$$399$$ −6.00000 −0.300376
$$400$$ 0 0
$$401$$ 30.0000 1.49813 0.749064 0.662497i $$-0.230503\pi$$
0.749064 + 0.662497i $$0.230503\pi$$
$$402$$ − 24.0000i − 1.19701i
$$403$$ − 6.00000i − 0.298881i
$$404$$ 0 0
$$405$$ 0 0
$$406$$ −12.0000 −0.595550
$$407$$ − 11.0000i − 0.545250i
$$408$$ 0 0
$$409$$ −2.00000 −0.0988936 −0.0494468 0.998777i $$-0.515746\pi$$
−0.0494468 + 0.998777i $$0.515746\pi$$
$$410$$ 0 0
$$411$$ −18.0000 −0.887875
$$412$$ − 32.0000i − 1.57653i
$$413$$ − 24.0000i − 1.18096i
$$414$$ 6.00000 0.294884
$$415$$ 0 0
$$416$$ 8.00000 0.392232
$$417$$ 1.00000i 0.0489702i
$$418$$ − 4.00000i − 0.195646i
$$419$$ 26.0000 1.27018 0.635092 0.772437i $$-0.280962\pi$$
0.635092 + 0.772437i $$0.280962\pi$$
$$420$$ 0 0
$$421$$ 4.00000 0.194948 0.0974740 0.995238i $$-0.468924\pi$$
0.0974740 + 0.995238i $$0.468924\pi$$
$$422$$ − 8.00000i − 0.389434i
$$423$$ 10.0000i 0.486217i
$$424$$ 0 0
$$425$$ 0 0
$$426$$ 10.0000 0.484502
$$427$$ 39.0000i 1.88734i
$$428$$ − 18.0000i − 0.870063i
$$429$$ 1.00000 0.0482805
$$430$$ 0 0
$$431$$ −24.0000 −1.15604 −0.578020 0.816023i $$-0.696174\pi$$
−0.578020 + 0.816023i $$0.696174\pi$$
$$432$$ 4.00000i 0.192450i
$$433$$ − 4.00000i − 0.192228i −0.995370 0.0961139i $$-0.969359\pi$$
0.995370 0.0961139i $$-0.0306413\pi$$
$$434$$ 36.0000 1.72806
$$435$$ 0 0
$$436$$ −32.0000 −1.53252
$$437$$ 6.00000i 0.287019i
$$438$$ 20.0000i 0.955637i
$$439$$ 17.0000 0.811366 0.405683 0.914014i $$-0.367034\pi$$
0.405683 + 0.914014i $$0.367034\pi$$
$$440$$ 0 0
$$441$$ 2.00000 0.0952381
$$442$$ 2.00000i 0.0951303i
$$443$$ − 9.00000i − 0.427603i −0.976877 0.213801i $$-0.931415\pi$$
0.976877 0.213801i $$-0.0685846\pi$$
$$444$$ 22.0000 1.04407
$$445$$ 0 0
$$446$$ 16.0000 0.757622
$$447$$ − 13.0000i − 0.614879i
$$448$$ 24.0000i 1.13389i
$$449$$ 13.0000 0.613508 0.306754 0.951789i $$-0.400757\pi$$
0.306754 + 0.951789i $$0.400757\pi$$
$$450$$ 0 0
$$451$$ 5.00000 0.235441
$$452$$ 28.0000i 1.31701i
$$453$$ 16.0000i 0.751746i
$$454$$ −44.0000 −2.06502
$$455$$ 0 0
$$456$$ 0 0
$$457$$ − 11.0000i − 0.514558i −0.966337 0.257279i $$-0.917174\pi$$
0.966337 0.257279i $$-0.0828260\pi$$
$$458$$ − 36.0000i − 1.68217i
$$459$$ −1.00000 −0.0466760
$$460$$ 0 0
$$461$$ 15.0000 0.698620 0.349310 0.937007i $$-0.386416\pi$$
0.349310 + 0.937007i $$0.386416\pi$$
$$462$$ 6.00000i 0.279145i
$$463$$ − 27.0000i − 1.25480i −0.778699 0.627398i $$-0.784120\pi$$
0.778699 0.627398i $$-0.215880\pi$$
$$464$$ −8.00000 −0.371391
$$465$$ 0 0
$$466$$ −54.0000 −2.50150
$$467$$ − 23.0000i − 1.06431i −0.846646 0.532157i $$-0.821382\pi$$
0.846646 0.532157i $$-0.178618\pi$$
$$468$$ 2.00000i 0.0924500i
$$469$$ −36.0000 −1.66233
$$470$$ 0 0
$$471$$ 10.0000 0.460776
$$472$$ 0 0
$$473$$ 4.00000i 0.183920i
$$474$$ −6.00000 −0.275589
$$475$$ 0 0
$$476$$ −6.00000 −0.275010
$$477$$ 11.0000i 0.503655i
$$478$$ 26.0000i 1.18921i
$$479$$ −9.00000 −0.411220 −0.205610 0.978634i $$-0.565918\pi$$
−0.205610 + 0.978634i $$0.565918\pi$$
$$480$$ 0 0
$$481$$ −11.0000 −0.501557
$$482$$ − 4.00000i − 0.182195i
$$483$$ − 9.00000i − 0.409514i
$$484$$ 20.0000 0.909091
$$485$$ 0 0
$$486$$ −2.00000 −0.0907218
$$487$$ − 7.00000i − 0.317200i −0.987343 0.158600i $$-0.949302\pi$$
0.987343 0.158600i $$-0.0506981\pi$$
$$488$$ 0 0
$$489$$ −13.0000 −0.587880
$$490$$ 0 0
$$491$$ 28.0000 1.26362 0.631811 0.775122i $$-0.282312\pi$$
0.631811 + 0.775122i $$0.282312\pi$$
$$492$$ 10.0000i 0.450835i
$$493$$ − 2.00000i − 0.0900755i
$$494$$ −4.00000 −0.179969
$$495$$ 0 0
$$496$$ 24.0000 1.07763
$$497$$ − 15.0000i − 0.672842i
$$498$$ − 24.0000i − 1.07547i
$$499$$ 14.0000 0.626726 0.313363 0.949633i $$-0.398544\pi$$
0.313363 + 0.949633i $$0.398544\pi$$
$$500$$ 0 0
$$501$$ 12.0000 0.536120
$$502$$ 0 0
$$503$$ − 28.0000i − 1.24846i −0.781241 0.624229i $$-0.785413\pi$$
0.781241 0.624229i $$-0.214587\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 6.00000 0.266733
$$507$$ − 1.00000i − 0.0444116i
$$508$$ − 20.0000i − 0.887357i
$$509$$ 7.00000 0.310270 0.155135 0.987893i $$-0.450419\pi$$
0.155135 + 0.987893i $$0.450419\pi$$
$$510$$ 0 0
$$511$$ 30.0000 1.32712
$$512$$ 32.0000i 1.41421i
$$513$$ − 2.00000i − 0.0883022i
$$514$$ 36.0000 1.58789
$$515$$ 0 0
$$516$$ −8.00000 −0.352180
$$517$$ 10.0000i 0.439799i
$$518$$ − 66.0000i − 2.89987i
$$519$$ −6.00000 −0.263371
$$520$$ 0 0
$$521$$ −14.0000 −0.613351 −0.306676 0.951814i $$-0.599217\pi$$
−0.306676 + 0.951814i $$0.599217\pi$$
$$522$$ − 4.00000i − 0.175075i
$$523$$ − 16.0000i − 0.699631i −0.936819 0.349816i $$-0.886244\pi$$
0.936819 0.349816i $$-0.113756\pi$$
$$524$$ 12.0000 0.524222
$$525$$ 0 0
$$526$$ −16.0000 −0.697633
$$527$$ 6.00000i 0.261364i
$$528$$ 4.00000i 0.174078i
$$529$$ 14.0000 0.608696
$$530$$ 0 0
$$531$$ 8.00000 0.347170
$$532$$ − 12.0000i − 0.520266i
$$533$$ − 5.00000i − 0.216574i
$$534$$ −30.0000 −1.29823
$$535$$ 0 0
$$536$$ 0 0
$$537$$ − 2.00000i − 0.0863064i
$$538$$ 8.00000i 0.344904i
$$539$$ 2.00000 0.0861461
$$540$$ 0 0
$$541$$ −30.0000 −1.28980 −0.644900 0.764267i $$-0.723101\pi$$
−0.644900 + 0.764267i $$0.723101\pi$$
$$542$$ 44.0000i 1.88996i
$$543$$ − 7.00000i − 0.300399i
$$544$$ −8.00000 −0.342997
$$545$$ 0 0
$$546$$ 6.00000 0.256776
$$547$$ − 32.0000i − 1.36822i −0.729378 0.684111i $$-0.760191\pi$$
0.729378 0.684111i $$-0.239809\pi$$
$$548$$ − 36.0000i − 1.53784i
$$549$$ −13.0000 −0.554826
$$550$$ 0 0
$$551$$ 4.00000 0.170406
$$552$$ 0 0
$$553$$ 9.00000i 0.382719i
$$554$$ 36.0000 1.52949
$$555$$ 0 0
$$556$$ −2.00000 −0.0848189
$$557$$ − 30.0000i − 1.27114i −0.772043 0.635570i $$-0.780765\pi$$
0.772043 0.635570i $$-0.219235\pi$$
$$558$$ 12.0000i 0.508001i
$$559$$ 4.00000 0.169182
$$560$$ 0 0
$$561$$ −1.00000 −0.0422200
$$562$$ 60.0000i 2.53095i
$$563$$ − 21.0000i − 0.885044i −0.896758 0.442522i $$-0.854084\pi$$
0.896758 0.442522i $$-0.145916\pi$$
$$564$$ −20.0000 −0.842152
$$565$$ 0 0
$$566$$ −24.0000 −1.00880
$$567$$ 3.00000i 0.125988i
$$568$$ 0 0
$$569$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$570$$ 0 0
$$571$$ −31.0000 −1.29731 −0.648655 0.761083i $$-0.724668\pi$$
−0.648655 + 0.761083i $$0.724668\pi$$
$$572$$ 2.00000i 0.0836242i
$$573$$ − 8.00000i − 0.334205i
$$574$$ 30.0000 1.25218
$$575$$ 0 0
$$576$$ −8.00000 −0.333333
$$577$$ − 19.0000i − 0.790980i −0.918470 0.395490i $$-0.870575\pi$$
0.918470 0.395490i $$-0.129425\pi$$
$$578$$ 32.0000i 1.33102i
$$579$$ −13.0000 −0.540262
$$580$$ 0 0
$$581$$ −36.0000 −1.49353
$$582$$ − 34.0000i − 1.40935i
$$583$$ 11.0000i 0.455573i
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 48.0000 1.98286
$$587$$ − 18.0000i − 0.742940i −0.928445 0.371470i $$-0.878854\pi$$
0.928445 0.371470i $$-0.121146\pi$$
$$588$$ 4.00000i 0.164957i
$$589$$ −12.0000 −0.494451
$$590$$ 0 0
$$591$$ 0 0
$$592$$ − 44.0000i − 1.80839i
$$593$$ − 4.00000i − 0.164260i −0.996622 0.0821302i $$-0.973828\pi$$
0.996622 0.0821302i $$-0.0261723\pi$$
$$594$$ −2.00000 −0.0820610
$$595$$ 0 0
$$596$$ 26.0000 1.06500
$$597$$ − 4.00000i − 0.163709i
$$598$$ − 6.00000i − 0.245358i
$$599$$ −16.0000 −0.653742 −0.326871 0.945069i $$-0.605994\pi$$
−0.326871 + 0.945069i $$0.605994\pi$$
$$600$$ 0 0
$$601$$ −37.0000 −1.50926 −0.754631 0.656150i $$-0.772184\pi$$
−0.754631 + 0.656150i $$0.772184\pi$$
$$602$$ 24.0000i 0.978167i
$$603$$ − 12.0000i − 0.488678i
$$604$$ −32.0000 −1.30206
$$605$$ 0 0
$$606$$ 0 0
$$607$$ − 32.0000i − 1.29884i −0.760430 0.649420i $$-0.775012\pi$$
0.760430 0.649420i $$-0.224988\pi$$
$$608$$ − 16.0000i − 0.648886i
$$609$$ −6.00000 −0.243132
$$610$$ 0 0
$$611$$ 10.0000 0.404557
$$612$$ − 2.00000i − 0.0808452i
$$613$$ − 13.0000i − 0.525065i −0.964923 0.262533i $$-0.915442\pi$$
0.964923 0.262533i $$-0.0845577\pi$$
$$614$$ 10.0000 0.403567
$$615$$ 0 0
$$616$$ 0 0
$$617$$ − 42.0000i − 1.69086i −0.534089 0.845428i $$-0.679345\pi$$
0.534089 0.845428i $$-0.320655\pi$$
$$618$$ − 32.0000i − 1.28723i
$$619$$ 34.0000 1.36658 0.683288 0.730149i $$-0.260549\pi$$
0.683288 + 0.730149i $$0.260549\pi$$
$$620$$ 0 0
$$621$$ 3.00000 0.120386
$$622$$ 48.0000i 1.92462i
$$623$$ 45.0000i 1.80289i
$$624$$ 4.00000 0.160128
$$625$$ 0 0
$$626$$ −20.0000 −0.799361
$$627$$ − 2.00000i − 0.0798723i
$$628$$ 20.0000i 0.798087i
$$629$$ 11.0000 0.438599
$$630$$ 0 0
$$631$$ 40.0000 1.59237 0.796187 0.605050i $$-0.206847\pi$$
0.796187 + 0.605050i $$0.206847\pi$$
$$632$$ 0 0
$$633$$ − 4.00000i − 0.158986i
$$634$$ −56.0000 −2.22404
$$635$$ 0 0
$$636$$ −22.0000 −0.872357
$$637$$ − 2.00000i − 0.0792429i
$$638$$ − 4.00000i − 0.158362i
$$639$$ 5.00000 0.197797
$$640$$ 0 0
$$641$$ −12.0000 −0.473972 −0.236986 0.971513i $$-0.576159\pi$$
−0.236986 + 0.971513i $$0.576159\pi$$
$$642$$ − 18.0000i − 0.710403i
$$643$$ − 15.0000i − 0.591542i −0.955259 0.295771i $$-0.904423\pi$$
0.955259 0.295771i $$-0.0955766\pi$$
$$644$$ 18.0000 0.709299
$$645$$ 0 0
$$646$$ 4.00000 0.157378
$$647$$ 47.0000i 1.84776i 0.382682 + 0.923880i $$0.375001\pi$$
−0.382682 + 0.923880i $$0.624999\pi$$
$$648$$ 0 0
$$649$$ 8.00000 0.314027
$$650$$ 0 0
$$651$$ 18.0000 0.705476
$$652$$ − 26.0000i − 1.01824i
$$653$$ − 22.0000i − 0.860927i −0.902608 0.430463i $$-0.858350\pi$$
0.902608 0.430463i $$-0.141650\pi$$
$$654$$ −32.0000 −1.25130
$$655$$ 0 0
$$656$$ 20.0000 0.780869
$$657$$ 10.0000i 0.390137i
$$658$$ 60.0000i 2.33904i
$$659$$ −20.0000 −0.779089 −0.389545 0.921008i $$-0.627368\pi$$
−0.389545 + 0.921008i $$0.627368\pi$$
$$660$$ 0 0
$$661$$ 4.00000 0.155582 0.0777910 0.996970i $$-0.475213\pi$$
0.0777910 + 0.996970i $$0.475213\pi$$
$$662$$ 0 0
$$663$$ 1.00000i 0.0388368i
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 22.0000 0.852483
$$667$$ 6.00000i 0.232321i
$$668$$ 24.0000i 0.928588i
$$669$$ 8.00000 0.309298
$$670$$ 0 0
$$671$$ −13.0000 −0.501859
$$672$$ 24.0000i 0.925820i
$$673$$ 6.00000i 0.231283i 0.993291 + 0.115642i $$0.0368924\pi$$
−0.993291 + 0.115642i $$0.963108\pi$$
$$674$$ −8.00000 −0.308148
$$675$$ 0 0
$$676$$ 2.00000 0.0769231
$$677$$ 3.00000i 0.115299i 0.998337 + 0.0576497i $$0.0183606\pi$$
−0.998337 + 0.0576497i $$0.981639\pi$$
$$678$$ 28.0000i 1.07533i
$$679$$ −51.0000 −1.95720
$$680$$ 0 0
$$681$$ −22.0000 −0.843042
$$682$$ 12.0000i 0.459504i
$$683$$ 24.0000i 0.918334i 0.888350 + 0.459167i $$0.151852\pi$$
−0.888350 + 0.459167i $$0.848148\pi$$
$$684$$ 4.00000 0.152944
$$685$$ 0 0
$$686$$ −30.0000 −1.14541
$$687$$ − 18.0000i − 0.686743i
$$688$$ 16.0000i 0.609994i
$$689$$ 11.0000 0.419067
$$690$$ 0 0
$$691$$ 22.0000 0.836919 0.418460 0.908235i $$-0.362570\pi$$
0.418460 + 0.908235i $$0.362570\pi$$
$$692$$ − 12.0000i − 0.456172i
$$693$$ 3.00000i 0.113961i
$$694$$ 38.0000 1.44246
$$695$$ 0 0
$$696$$ 0 0
$$697$$ 5.00000i 0.189389i
$$698$$ 16.0000i 0.605609i
$$699$$ −27.0000 −1.02123
$$700$$ 0 0
$$701$$ −20.0000 −0.755390 −0.377695 0.925930i $$-0.623283\pi$$
−0.377695 + 0.925930i $$0.623283\pi$$
$$702$$ 2.00000i 0.0754851i
$$703$$ 22.0000i 0.829746i
$$704$$ −8.00000 −0.301511
$$705$$ 0 0
$$706$$ 12.0000 0.451626
$$707$$ 0 0
$$708$$ 16.0000i 0.601317i
$$709$$ −4.00000 −0.150223 −0.0751116 0.997175i $$-0.523931\pi$$
−0.0751116 + 0.997175i $$0.523931\pi$$
$$710$$ 0 0
$$711$$ −3.00000 −0.112509
$$712$$ 0 0
$$713$$ − 18.0000i − 0.674105i
$$714$$ −6.00000 −0.224544
$$715$$ 0 0
$$716$$ 4.00000 0.149487
$$717$$ 13.0000i 0.485494i
$$718$$ − 48.0000i − 1.79134i
$$719$$ 24.0000 0.895049 0.447524 0.894272i $$-0.352306\pi$$
0.447524 + 0.894272i $$0.352306\pi$$
$$720$$ 0 0
$$721$$ −48.0000 −1.78761
$$722$$ − 30.0000i − 1.11648i
$$723$$ − 2.00000i − 0.0743808i
$$724$$ 14.0000 0.520306
$$725$$ 0 0
$$726$$ 20.0000 0.742270
$$727$$ 38.0000i 1.40934i 0.709534 + 0.704671i $$0.248905\pi$$
−0.709534 + 0.704671i $$0.751095\pi$$
$$728$$ 0 0
$$729$$ −1.00000 −0.0370370
$$730$$ 0 0
$$731$$ −4.00000 −0.147945
$$732$$ − 26.0000i − 0.960988i
$$733$$ 49.0000i 1.80986i 0.425564 + 0.904928i $$0.360076\pi$$
−0.425564 + 0.904928i $$0.639924\pi$$
$$734$$ −72.0000 −2.65757
$$735$$ 0 0
$$736$$ 24.0000 0.884652
$$737$$ − 12.0000i − 0.442026i
$$738$$ 10.0000i 0.368105i
$$739$$ −10.0000 −0.367856 −0.183928 0.982940i $$-0.558881\pi$$
−0.183928 + 0.982940i $$0.558881\pi$$
$$740$$ 0 0
$$741$$ −2.00000 −0.0734718
$$742$$ 66.0000i 2.42294i
$$743$$ 34.0000i 1.24734i 0.781688 + 0.623670i $$0.214359\pi$$
−0.781688 + 0.623670i $$0.785641\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 8.00000 0.292901
$$747$$ − 12.0000i − 0.439057i
$$748$$ − 2.00000i − 0.0731272i
$$749$$ −27.0000 −0.986559
$$750$$ 0 0
$$751$$ −5.00000 −0.182453 −0.0912263 0.995830i $$-0.529079\pi$$
−0.0912263 + 0.995830i $$0.529079\pi$$
$$752$$ 40.0000i 1.45865i
$$753$$ 0 0
$$754$$ −4.00000 −0.145671
$$755$$ 0 0
$$756$$ −6.00000 −0.218218
$$757$$ − 8.00000i − 0.290765i −0.989376 0.145382i $$-0.953559\pi$$
0.989376 0.145382i $$-0.0464413\pi$$
$$758$$ 28.0000i 1.01701i
$$759$$ 3.00000 0.108893
$$760$$ 0 0
$$761$$ −6.00000 −0.217500 −0.108750 0.994069i $$-0.534685\pi$$
−0.108750 + 0.994069i $$0.534685\pi$$
$$762$$ − 20.0000i − 0.724524i
$$763$$ 48.0000i 1.73772i
$$764$$ 16.0000 0.578860
$$765$$ 0 0
$$766$$ −60.0000 −2.16789
$$767$$ − 8.00000i − 0.288863i
$$768$$ 16.0000i 0.577350i
$$769$$ 40.0000 1.44244 0.721218 0.692708i $$-0.243582\pi$$
0.721218 + 0.692708i $$0.243582\pi$$
$$770$$ 0 0
$$771$$ 18.0000 0.648254
$$772$$ − 26.0000i − 0.935760i
$$773$$ − 36.0000i − 1.29483i −0.762138 0.647415i $$-0.775850\pi$$
0.762138 0.647415i $$-0.224150\pi$$
$$774$$ −8.00000 −0.287554
$$775$$ 0 0
$$776$$ 0 0
$$777$$ − 33.0000i − 1.18387i
$$778$$ 0 0
$$779$$ −10.0000 −0.358287
$$780$$ 0 0
$$781$$ 5.00000 0.178914
$$782$$ 6.00000i 0.214560i
$$783$$ − 2.00000i − 0.0714742i
$$784$$ 8.00000 0.285714
$$785$$ 0 0
$$786$$ 12.0000 0.428026
$$787$$ 52.0000i 1.85360i 0.375555 + 0.926800i $$0.377452\pi$$
−0.375555 + 0.926800i $$0.622548\pi$$
$$788$$ 0 0
$$789$$ −8.00000 −0.284808
$$790$$ 0 0
$$791$$ 42.0000 1.49335
$$792$$ 0 0
$$793$$ 13.0000i 0.461644i
$$794$$ 58.0000 2.05834
$$795$$ 0 0
$$796$$ 8.00000 0.283552
$$797$$ − 47.0000i − 1.66483i −0.554156 0.832413i $$-0.686959\pi$$
0.554156 0.832413i $$-0.313041\pi$$
$$798$$ − 12.0000i − 0.424795i
$$799$$ −10.0000 −0.353775
$$800$$ 0 0
$$801$$ −15.0000 −0.529999
$$802$$ 60.0000i 2.11867i
$$803$$ 10.0000i 0.352892i
$$804$$ 24.0000 0.846415
$$805$$ 0 0
$$806$$ 12.0000 0.422682
$$807$$ 4.00000i 0.140807i
$$808$$ 0 0
$$809$$ 26.0000 0.914111 0.457056 0.889438i $$-0.348904\pi$$
0.457056 + 0.889438i $$0.348904\pi$$
$$810$$ 0 0
$$811$$ 36.0000 1.26413 0.632065 0.774915i $$-0.282207\pi$$
0.632065 + 0.774915i $$0.282207\pi$$
$$812$$ − 12.0000i − 0.421117i
$$813$$ 22.0000i 0.771574i
$$814$$ 22.0000 0.771100
$$815$$ 0 0
$$816$$ −4.00000 −0.140028
$$817$$ − 8.00000i − 0.279885i
$$818$$ − 4.00000i − 0.139857i
$$819$$ 3.00000 0.104828
$$820$$ 0 0
$$821$$ 27.0000 0.942306 0.471153 0.882051i $$-0.343838\pi$$
0.471153 + 0.882051i $$0.343838\pi$$
$$822$$ − 36.0000i − 1.25564i
$$823$$ − 20.0000i − 0.697156i −0.937280 0.348578i $$-0.886665\pi$$
0.937280 0.348578i $$-0.113335\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 48.0000 1.67013
$$827$$ 26.0000i 0.904109i 0.891990 + 0.452054i $$0.149309\pi$$
−0.891990 + 0.452054i $$0.850691\pi$$
$$828$$ 6.00000i 0.208514i
$$829$$ −30.0000 −1.04194 −0.520972 0.853574i $$-0.674430\pi$$
−0.520972 + 0.853574i $$0.674430\pi$$
$$830$$ 0 0
$$831$$ 18.0000 0.624413
$$832$$ 8.00000i 0.277350i
$$833$$ 2.00000i 0.0692959i
$$834$$ −2.00000 −0.0692543
$$835$$ 0 0
$$836$$ 4.00000 0.138343
$$837$$ 6.00000i 0.207390i
$$838$$ 52.0000i 1.79631i
$$839$$ 5.00000 0.172619 0.0863096 0.996268i $$-0.472493\pi$$
0.0863096 + 0.996268i $$0.472493\pi$$
$$840$$ 0 0
$$841$$ −25.0000 −0.862069
$$842$$ 8.00000i 0.275698i
$$843$$ 30.0000i 1.03325i
$$844$$ 8.00000 0.275371
$$845$$ 0 0
$$846$$ −20.0000 −0.687614
$$847$$ − 30.0000i − 1.03081i
$$848$$ 44.0000i 1.51097i
$$849$$ −12.0000 −0.411839
$$850$$ 0 0
$$851$$ −33.0000 −1.13123
$$852$$ 10.0000i 0.342594i
$$853$$ − 45.0000i − 1.54077i −0.637579 0.770385i $$-0.720064\pi$$
0.637579 0.770385i $$-0.279936\pi$$
$$854$$ −78.0000 −2.66911
$$855$$ 0 0
$$856$$ 0 0
$$857$$ 29.0000i 0.990621i 0.868716 + 0.495311i $$0.164946\pi$$
−0.868716 + 0.495311i $$0.835054\pi$$
$$858$$ 2.00000i 0.0682789i
$$859$$ 29.0000 0.989467 0.494734 0.869045i $$-0.335266\pi$$
0.494734 + 0.869045i $$0.335266\pi$$
$$860$$ 0 0
$$861$$ 15.0000 0.511199
$$862$$ − 48.0000i − 1.63489i
$$863$$ − 34.0000i − 1.15737i −0.815550 0.578687i $$-0.803565\pi$$
0.815550 0.578687i $$-0.196435\pi$$
$$864$$ −8.00000 −0.272166
$$865$$ 0 0
$$866$$ 8.00000 0.271851
$$867$$ 16.0000i 0.543388i
$$868$$ 36.0000i 1.22192i
$$869$$ −3.00000 −0.101768
$$870$$ 0 0
$$871$$ −12.0000 −0.406604
$$872$$ 0 0
$$873$$ − 17.0000i − 0.575363i
$$874$$ −12.0000 −0.405906
$$875$$ 0 0
$$876$$ −20.0000 −0.675737
$$877$$ 18.0000i 0.607817i 0.952701 + 0.303908i $$0.0982917\pi$$
−0.952701 + 0.303908i $$0.901708\pi$$
$$878$$ 34.0000i 1.14744i
$$879$$ 24.0000 0.809500
$$880$$ 0 0
$$881$$ 2.00000 0.0673817 0.0336909 0.999432i $$-0.489274\pi$$
0.0336909 + 0.999432i $$0.489274\pi$$
$$882$$ 4.00000i 0.134687i
$$883$$ 16.0000i 0.538443i 0.963078 + 0.269221i $$0.0867663\pi$$
−0.963078 + 0.269221i $$0.913234\pi$$
$$884$$ −2.00000 −0.0672673
$$885$$ 0 0
$$886$$ 18.0000 0.604722
$$887$$ − 21.0000i − 0.705111i −0.935791 0.352555i $$-0.885313\pi$$
0.935791 0.352555i $$-0.114687\pi$$
$$888$$ 0 0
$$889$$ −30.0000 −1.00617
$$890$$ 0 0
$$891$$ −1.00000 −0.0335013
$$892$$ 16.0000i 0.535720i
$$893$$ − 20.0000i − 0.669274i
$$894$$ 26.0000 0.869570
$$895$$ 0 0
$$896$$ 0 0
$$897$$ − 3.00000i − 0.100167i
$$898$$ 26.0000i 0.867631i
$$899$$ −12.0000 −0.400222
$$900$$ 0 0
$$901$$ −11.0000 −0.366463
$$902$$ 10.0000i 0.332964i
$$903$$ 12.0000i 0.399335i
$$904$$ 0 0
$$905$$ 0 0
$$906$$ −32.0000 −1.06313
$$907$$ 6.00000i 0.199227i 0.995026 + 0.0996134i $$0.0317606\pi$$
−0.995026 + 0.0996134i $$0.968239\pi$$
$$908$$ − 44.0000i − 1.46019i
$$909$$ 0 0
$$910$$ 0 0
$$911$$ 44.0000 1.45779 0.728893 0.684628i $$-0.240035\pi$$
0.728893 + 0.684628i $$0.240035\pi$$
$$912$$ − 8.00000i − 0.264906i
$$913$$ − 12.0000i − 0.397142i
$$914$$ 22.0000 0.727695
$$915$$ 0 0
$$916$$ 36.0000 1.18947
$$917$$ − 18.0000i − 0.594412i
$$918$$ − 2.00000i − 0.0660098i
$$919$$ −37.0000 −1.22052 −0.610259 0.792202i $$-0.708935\pi$$
−0.610259 + 0.792202i $$0.708935\pi$$
$$920$$ 0 0
$$921$$ 5.00000 0.164756
$$922$$ 30.0000i 0.987997i
$$923$$ − 5.00000i − 0.164577i
$$924$$ −6.00000 −0.197386
$$925$$ 0 0
$$926$$ 54.0000 1.77455
$$927$$ − 16.0000i − 0.525509i
$$928$$ − 16.0000i − 0.525226i
$$929$$ −1.00000 −0.0328089 −0.0164045 0.999865i $$-0.505222\pi$$
−0.0164045 + 0.999865i $$0.505222\pi$$
$$930$$ 0 0
$$931$$ −4.00000 −0.131095
$$932$$ − 54.0000i − 1.76883i
$$933$$ 24.0000i 0.785725i
$$934$$ 46.0000 1.50517
$$935$$ 0 0
$$936$$ 0 0
$$937$$ 30.0000i 0.980057i 0.871706 + 0.490029i $$0.163014\pi$$
−0.871706 + 0.490029i $$0.836986\pi$$
$$938$$ − 72.0000i − 2.35088i
$$939$$ −10.0000 −0.326338
$$940$$ 0 0
$$941$$ 37.0000 1.20617 0.603083 0.797679i $$-0.293939\pi$$
0.603083 + 0.797679i $$0.293939\pi$$
$$942$$ 20.0000i 0.651635i
$$943$$ − 15.0000i − 0.488467i
$$944$$ 32.0000 1.04151
$$945$$ 0 0
$$946$$ −8.00000 −0.260102
$$947$$ − 24.0000i − 0.779895i −0.920837 0.389948i $$-0.872493\pi$$
0.920837 0.389948i $$-0.127507\pi$$
$$948$$ − 6.00000i − 0.194871i
$$949$$ 10.0000 0.324614
$$950$$ 0 0
$$951$$ −28.0000 −0.907962
$$952$$ 0 0
$$953$$ 1.00000i 0.0323932i 0.999869 + 0.0161966i $$0.00515576\pi$$
−0.999869 + 0.0161966i $$0.994844\pi$$
$$954$$ −22.0000 −0.712276
$$955$$ 0 0
$$956$$ −26.0000 −0.840900
$$957$$ − 2.00000i − 0.0646508i
$$958$$ − 18.0000i − 0.581554i
$$959$$ −54.0000 −1.74375
$$960$$ 0 0
$$961$$ 5.00000 0.161290
$$962$$ − 22.0000i − 0.709308i
$$963$$ − 9.00000i − 0.290021i
$$964$$ 4.00000 0.128831
$$965$$ 0 0
$$966$$ 18.0000 0.579141
$$967$$ − 16.0000i − 0.514525i −0.966342 0.257263i $$-0.917179\pi$$
0.966342 0.257263i $$-0.0828206\pi$$
$$968$$ 0 0
$$969$$ 2.00000 0.0642493
$$970$$ 0 0
$$971$$ 12.0000 0.385098 0.192549 0.981287i $$-0.438325\pi$$
0.192549 + 0.981287i $$0.438325\pi$$
$$972$$ − 2.00000i − 0.0641500i
$$973$$ 3.00000i 0.0961756i
$$974$$ 14.0000 0.448589
$$975$$ 0 0
$$976$$ −52.0000 −1.66448
$$977$$ 32.0000i 1.02377i 0.859054 + 0.511885i $$0.171053\pi$$
−0.859054 + 0.511885i $$0.828947\pi$$
$$978$$ − 26.0000i − 0.831388i
$$979$$ −15.0000 −0.479402
$$980$$ 0 0
$$981$$ −16.0000 −0.510841
$$982$$ 56.0000i 1.78703i
$$983$$ 12.0000i 0.382741i 0.981518 + 0.191370i $$0.0612931\pi$$
−0.981518 + 0.191370i $$0.938707\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 4.00000 0.127386
$$987$$ 30.0000i 0.954911i
$$988$$ − 4.00000i − 0.127257i
$$989$$ 12.0000 0.381578
$$990$$ 0 0
$$991$$ −25.0000 −0.794151 −0.397076 0.917786i $$-0.629975\pi$$
−0.397076 + 0.917786i $$0.629975\pi$$
$$992$$ 48.0000i 1.52400i
$$993$$ 0 0
$$994$$ 30.0000 0.951542
$$995$$ 0 0
$$996$$ 24.0000 0.760469
$$997$$ − 36.0000i − 1.14013i −0.821599 0.570066i $$-0.806918\pi$$
0.821599 0.570066i $$-0.193082\pi$$
$$998$$ 28.0000i 0.886325i
$$999$$ 11.0000 0.348025
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 975.2.c.a.274.2 2
3.2 odd 2 2925.2.c.c.2224.1 2
5.2 odd 4 975.2.a.c.1.1 1
5.3 odd 4 195.2.a.b.1.1 1
5.4 even 2 inner 975.2.c.a.274.1 2
15.2 even 4 2925.2.a.q.1.1 1
15.8 even 4 585.2.a.b.1.1 1
15.14 odd 2 2925.2.c.c.2224.2 2
20.3 even 4 3120.2.a.u.1.1 1
35.13 even 4 9555.2.a.v.1.1 1
60.23 odd 4 9360.2.a.d.1.1 1
65.38 odd 4 2535.2.a.a.1.1 1
195.38 even 4 7605.2.a.u.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
195.2.a.b.1.1 1 5.3 odd 4
585.2.a.b.1.1 1 15.8 even 4
975.2.a.c.1.1 1 5.2 odd 4
975.2.c.a.274.1 2 5.4 even 2 inner
975.2.c.a.274.2 2 1.1 even 1 trivial
2535.2.a.a.1.1 1 65.38 odd 4
2925.2.a.q.1.1 1 15.2 even 4
2925.2.c.c.2224.1 2 3.2 odd 2
2925.2.c.c.2224.2 2 15.14 odd 2
3120.2.a.u.1.1 1 20.3 even 4
7605.2.a.u.1.1 1 195.38 even 4
9360.2.a.d.1.1 1 60.23 odd 4
9555.2.a.v.1.1 1 35.13 even 4