# Properties

 Label 1638.2.a.x.1.1 Level $1638$ Weight $2$ Character 1638.1 Self dual yes Analytic conductor $13.079$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1638,2,Mod(1,1638)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1638, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1638.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1638 = 2 \cdot 3^{2} \cdot 7 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1638.a (trivial)

## 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: yes Analytic conductor: $$13.0794958511$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{33})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 8$$ x^2 - x - 8 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$-2.37228$$ of defining polynomial Character $$\chi$$ $$=$$ 1638.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+1.00000 q^{2} +1.00000 q^{4} -2.37228 q^{5} -1.00000 q^{7} +1.00000 q^{8} +O(q^{10})$$ $$q+1.00000 q^{2} +1.00000 q^{4} -2.37228 q^{5} -1.00000 q^{7} +1.00000 q^{8} -2.37228 q^{10} +2.37228 q^{11} -1.00000 q^{13} -1.00000 q^{14} +1.00000 q^{16} +4.37228 q^{17} +1.62772 q^{19} -2.37228 q^{20} +2.37228 q^{22} +3.62772 q^{23} +0.627719 q^{25} -1.00000 q^{26} -1.00000 q^{28} +6.37228 q^{29} -4.74456 q^{31} +1.00000 q^{32} +4.37228 q^{34} +2.37228 q^{35} -4.37228 q^{37} +1.62772 q^{38} -2.37228 q^{40} +8.74456 q^{41} +11.1168 q^{43} +2.37228 q^{44} +3.62772 q^{46} +1.25544 q^{47} +1.00000 q^{49} +0.627719 q^{50} -1.00000 q^{52} +8.74456 q^{53} -5.62772 q^{55} -1.00000 q^{56} +6.37228 q^{58} +2.00000 q^{59} +5.11684 q^{61} -4.74456 q^{62} +1.00000 q^{64} +2.37228 q^{65} +9.48913 q^{67} +4.37228 q^{68} +2.37228 q^{70} -4.74456 q^{71} -8.37228 q^{73} -4.37228 q^{74} +1.62772 q^{76} -2.37228 q^{77} +4.74456 q^{79} -2.37228 q^{80} +8.74456 q^{82} -6.00000 q^{83} -10.3723 q^{85} +11.1168 q^{86} +2.37228 q^{88} -3.25544 q^{89} +1.00000 q^{91} +3.62772 q^{92} +1.25544 q^{94} -3.86141 q^{95} +7.48913 q^{97} +1.00000 q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{2} + 2 q^{4} + q^{5} - 2 q^{7} + 2 q^{8}+O(q^{10})$$ 2 * q + 2 * q^2 + 2 * q^4 + q^5 - 2 * q^7 + 2 * q^8 $$2 q + 2 q^{2} + 2 q^{4} + q^{5} - 2 q^{7} + 2 q^{8} + q^{10} - q^{11} - 2 q^{13} - 2 q^{14} + 2 q^{16} + 3 q^{17} + 9 q^{19} + q^{20} - q^{22} + 13 q^{23} + 7 q^{25} - 2 q^{26} - 2 q^{28} + 7 q^{29} + 2 q^{31} + 2 q^{32} + 3 q^{34} - q^{35} - 3 q^{37} + 9 q^{38} + q^{40} + 6 q^{41} + 5 q^{43} - q^{44} + 13 q^{46} + 14 q^{47} + 2 q^{49} + 7 q^{50} - 2 q^{52} + 6 q^{53} - 17 q^{55} - 2 q^{56} + 7 q^{58} + 4 q^{59} - 7 q^{61} + 2 q^{62} + 2 q^{64} - q^{65} - 4 q^{67} + 3 q^{68} - q^{70} + 2 q^{71} - 11 q^{73} - 3 q^{74} + 9 q^{76} + q^{77} - 2 q^{79} + q^{80} + 6 q^{82} - 12 q^{83} - 15 q^{85} + 5 q^{86} - q^{88} - 18 q^{89} + 2 q^{91} + 13 q^{92} + 14 q^{94} + 21 q^{95} - 8 q^{97} + 2 q^{98}+O(q^{100})$$ 2 * q + 2 * q^2 + 2 * q^4 + q^5 - 2 * q^7 + 2 * q^8 + q^10 - q^11 - 2 * q^13 - 2 * q^14 + 2 * q^16 + 3 * q^17 + 9 * q^19 + q^20 - q^22 + 13 * q^23 + 7 * q^25 - 2 * q^26 - 2 * q^28 + 7 * q^29 + 2 * q^31 + 2 * q^32 + 3 * q^34 - q^35 - 3 * q^37 + 9 * q^38 + q^40 + 6 * q^41 + 5 * q^43 - q^44 + 13 * q^46 + 14 * q^47 + 2 * q^49 + 7 * q^50 - 2 * q^52 + 6 * q^53 - 17 * q^55 - 2 * q^56 + 7 * q^58 + 4 * q^59 - 7 * q^61 + 2 * q^62 + 2 * q^64 - q^65 - 4 * q^67 + 3 * q^68 - q^70 + 2 * q^71 - 11 * q^73 - 3 * q^74 + 9 * q^76 + q^77 - 2 * q^79 + q^80 + 6 * q^82 - 12 * q^83 - 15 * q^85 + 5 * q^86 - q^88 - 18 * q^89 + 2 * q^91 + 13 * q^92 + 14 * q^94 + 21 * q^95 - 8 * q^97 + 2 * q^98

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 1.00000 0.707107
$$3$$ 0 0
$$4$$ 1.00000 0.500000
$$5$$ −2.37228 −1.06092 −0.530458 0.847711i $$-0.677980\pi$$
−0.530458 + 0.847711i $$0.677980\pi$$
$$6$$ 0 0
$$7$$ −1.00000 −0.377964
$$8$$ 1.00000 0.353553
$$9$$ 0 0
$$10$$ −2.37228 −0.750181
$$11$$ 2.37228 0.715270 0.357635 0.933862i $$-0.383583\pi$$
0.357635 + 0.933862i $$0.383583\pi$$
$$12$$ 0 0
$$13$$ −1.00000 −0.277350
$$14$$ −1.00000 −0.267261
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ 4.37228 1.06043 0.530217 0.847862i $$-0.322110\pi$$
0.530217 + 0.847862i $$0.322110\pi$$
$$18$$ 0 0
$$19$$ 1.62772 0.373424 0.186712 0.982415i $$-0.440217\pi$$
0.186712 + 0.982415i $$0.440217\pi$$
$$20$$ −2.37228 −0.530458
$$21$$ 0 0
$$22$$ 2.37228 0.505772
$$23$$ 3.62772 0.756432 0.378216 0.925717i $$-0.376538\pi$$
0.378216 + 0.925717i $$0.376538\pi$$
$$24$$ 0 0
$$25$$ 0.627719 0.125544
$$26$$ −1.00000 −0.196116
$$27$$ 0 0
$$28$$ −1.00000 −0.188982
$$29$$ 6.37228 1.18330 0.591651 0.806194i $$-0.298476\pi$$
0.591651 + 0.806194i $$0.298476\pi$$
$$30$$ 0 0
$$31$$ −4.74456 −0.852149 −0.426074 0.904688i $$-0.640104\pi$$
−0.426074 + 0.904688i $$0.640104\pi$$
$$32$$ 1.00000 0.176777
$$33$$ 0 0
$$34$$ 4.37228 0.749840
$$35$$ 2.37228 0.400989
$$36$$ 0 0
$$37$$ −4.37228 −0.718799 −0.359399 0.933184i $$-0.617018\pi$$
−0.359399 + 0.933184i $$0.617018\pi$$
$$38$$ 1.62772 0.264051
$$39$$ 0 0
$$40$$ −2.37228 −0.375091
$$41$$ 8.74456 1.36567 0.682836 0.730572i $$-0.260747\pi$$
0.682836 + 0.730572i $$0.260747\pi$$
$$42$$ 0 0
$$43$$ 11.1168 1.69530 0.847651 0.530554i $$-0.178016\pi$$
0.847651 + 0.530554i $$0.178016\pi$$
$$44$$ 2.37228 0.357635
$$45$$ 0 0
$$46$$ 3.62772 0.534878
$$47$$ 1.25544 0.183124 0.0915622 0.995799i $$-0.470814\pi$$
0.0915622 + 0.995799i $$0.470814\pi$$
$$48$$ 0 0
$$49$$ 1.00000 0.142857
$$50$$ 0.627719 0.0887728
$$51$$ 0 0
$$52$$ −1.00000 −0.138675
$$53$$ 8.74456 1.20116 0.600579 0.799565i $$-0.294937\pi$$
0.600579 + 0.799565i $$0.294937\pi$$
$$54$$ 0 0
$$55$$ −5.62772 −0.758841
$$56$$ −1.00000 −0.133631
$$57$$ 0 0
$$58$$ 6.37228 0.836722
$$59$$ 2.00000 0.260378 0.130189 0.991489i $$-0.458442\pi$$
0.130189 + 0.991489i $$0.458442\pi$$
$$60$$ 0 0
$$61$$ 5.11684 0.655145 0.327572 0.944826i $$-0.393769\pi$$
0.327572 + 0.944826i $$0.393769\pi$$
$$62$$ −4.74456 −0.602560
$$63$$ 0 0
$$64$$ 1.00000 0.125000
$$65$$ 2.37228 0.294245
$$66$$ 0 0
$$67$$ 9.48913 1.15928 0.579641 0.814872i $$-0.303193\pi$$
0.579641 + 0.814872i $$0.303193\pi$$
$$68$$ 4.37228 0.530217
$$69$$ 0 0
$$70$$ 2.37228 0.283542
$$71$$ −4.74456 −0.563076 −0.281538 0.959550i $$-0.590845\pi$$
−0.281538 + 0.959550i $$0.590845\pi$$
$$72$$ 0 0
$$73$$ −8.37228 −0.979901 −0.489951 0.871750i $$-0.662985\pi$$
−0.489951 + 0.871750i $$0.662985\pi$$
$$74$$ −4.37228 −0.508267
$$75$$ 0 0
$$76$$ 1.62772 0.186712
$$77$$ −2.37228 −0.270347
$$78$$ 0 0
$$79$$ 4.74456 0.533805 0.266903 0.963724i $$-0.414000\pi$$
0.266903 + 0.963724i $$0.414000\pi$$
$$80$$ −2.37228 −0.265229
$$81$$ 0 0
$$82$$ 8.74456 0.965675
$$83$$ −6.00000 −0.658586 −0.329293 0.944228i $$-0.606810\pi$$
−0.329293 + 0.944228i $$0.606810\pi$$
$$84$$ 0 0
$$85$$ −10.3723 −1.12503
$$86$$ 11.1168 1.19876
$$87$$ 0 0
$$88$$ 2.37228 0.252886
$$89$$ −3.25544 −0.345076 −0.172538 0.985003i $$-0.555197\pi$$
−0.172538 + 0.985003i $$0.555197\pi$$
$$90$$ 0 0
$$91$$ 1.00000 0.104828
$$92$$ 3.62772 0.378216
$$93$$ 0 0
$$94$$ 1.25544 0.129488
$$95$$ −3.86141 −0.396172
$$96$$ 0 0
$$97$$ 7.48913 0.760405 0.380203 0.924903i $$-0.375854\pi$$
0.380203 + 0.924903i $$0.375854\pi$$
$$98$$ 1.00000 0.101015
$$99$$ 0 0
$$100$$ 0.627719 0.0627719
$$101$$ −2.00000 −0.199007 −0.0995037 0.995037i $$-0.531726\pi$$
−0.0995037 + 0.995037i $$0.531726\pi$$
$$102$$ 0 0
$$103$$ −11.8614 −1.16874 −0.584370 0.811488i $$-0.698658\pi$$
−0.584370 + 0.811488i $$0.698658\pi$$
$$104$$ −1.00000 −0.0980581
$$105$$ 0 0
$$106$$ 8.74456 0.849347
$$107$$ 2.00000 0.193347 0.0966736 0.995316i $$-0.469180\pi$$
0.0966736 + 0.995316i $$0.469180\pi$$
$$108$$ 0 0
$$109$$ −3.62772 −0.347472 −0.173736 0.984792i $$-0.555584\pi$$
−0.173736 + 0.984792i $$0.555584\pi$$
$$110$$ −5.62772 −0.536582
$$111$$ 0 0
$$112$$ −1.00000 −0.0944911
$$113$$ 20.0000 1.88144 0.940721 0.339182i $$-0.110150\pi$$
0.940721 + 0.339182i $$0.110150\pi$$
$$114$$ 0 0
$$115$$ −8.60597 −0.802511
$$116$$ 6.37228 0.591651
$$117$$ 0 0
$$118$$ 2.00000 0.184115
$$119$$ −4.37228 −0.400806
$$120$$ 0 0
$$121$$ −5.37228 −0.488389
$$122$$ 5.11684 0.463257
$$123$$ 0 0
$$124$$ −4.74456 −0.426074
$$125$$ 10.3723 0.927725
$$126$$ 0 0
$$127$$ −8.00000 −0.709885 −0.354943 0.934888i $$-0.615500\pi$$
−0.354943 + 0.934888i $$0.615500\pi$$
$$128$$ 1.00000 0.0883883
$$129$$ 0 0
$$130$$ 2.37228 0.208063
$$131$$ −11.1168 −0.971283 −0.485642 0.874158i $$-0.661414\pi$$
−0.485642 + 0.874158i $$0.661414\pi$$
$$132$$ 0 0
$$133$$ −1.62772 −0.141141
$$134$$ 9.48913 0.819736
$$135$$ 0 0
$$136$$ 4.37228 0.374920
$$137$$ 1.11684 0.0954184 0.0477092 0.998861i $$-0.484808\pi$$
0.0477092 + 0.998861i $$0.484808\pi$$
$$138$$ 0 0
$$139$$ 0.744563 0.0631530 0.0315765 0.999501i $$-0.489947\pi$$
0.0315765 + 0.999501i $$0.489947\pi$$
$$140$$ 2.37228 0.200494
$$141$$ 0 0
$$142$$ −4.74456 −0.398155
$$143$$ −2.37228 −0.198380
$$144$$ 0 0
$$145$$ −15.1168 −1.25539
$$146$$ −8.37228 −0.692895
$$147$$ 0 0
$$148$$ −4.37228 −0.359399
$$149$$ −20.2337 −1.65761 −0.828804 0.559539i $$-0.810978\pi$$
−0.828804 + 0.559539i $$0.810978\pi$$
$$150$$ 0 0
$$151$$ 19.1168 1.55571 0.777853 0.628446i $$-0.216309\pi$$
0.777853 + 0.628446i $$0.216309\pi$$
$$152$$ 1.62772 0.132025
$$153$$ 0 0
$$154$$ −2.37228 −0.191164
$$155$$ 11.2554 0.904058
$$156$$ 0 0
$$157$$ −20.3723 −1.62589 −0.812943 0.582344i $$-0.802136\pi$$
−0.812943 + 0.582344i $$0.802136\pi$$
$$158$$ 4.74456 0.377457
$$159$$ 0 0
$$160$$ −2.37228 −0.187545
$$161$$ −3.62772 −0.285904
$$162$$ 0 0
$$163$$ −18.2337 −1.42817 −0.714086 0.700058i $$-0.753158\pi$$
−0.714086 + 0.700058i $$0.753158\pi$$
$$164$$ 8.74456 0.682836
$$165$$ 0 0
$$166$$ −6.00000 −0.465690
$$167$$ 13.1168 1.01501 0.507506 0.861648i $$-0.330568\pi$$
0.507506 + 0.861648i $$0.330568\pi$$
$$168$$ 0 0
$$169$$ 1.00000 0.0769231
$$170$$ −10.3723 −0.795518
$$171$$ 0 0
$$172$$ 11.1168 0.847651
$$173$$ −10.0000 −0.760286 −0.380143 0.924928i $$-0.624125\pi$$
−0.380143 + 0.924928i $$0.624125\pi$$
$$174$$ 0 0
$$175$$ −0.627719 −0.0474511
$$176$$ 2.37228 0.178817
$$177$$ 0 0
$$178$$ −3.25544 −0.244005
$$179$$ −2.74456 −0.205138 −0.102569 0.994726i $$-0.532706\pi$$
−0.102569 + 0.994726i $$0.532706\pi$$
$$180$$ 0 0
$$181$$ −23.4891 −1.74593 −0.872966 0.487780i $$-0.837807\pi$$
−0.872966 + 0.487780i $$0.837807\pi$$
$$182$$ 1.00000 0.0741249
$$183$$ 0 0
$$184$$ 3.62772 0.267439
$$185$$ 10.3723 0.762585
$$186$$ 0 0
$$187$$ 10.3723 0.758496
$$188$$ 1.25544 0.0915622
$$189$$ 0 0
$$190$$ −3.86141 −0.280136
$$191$$ 5.11684 0.370242 0.185121 0.982716i $$-0.440732\pi$$
0.185121 + 0.982716i $$0.440732\pi$$
$$192$$ 0 0
$$193$$ 2.00000 0.143963 0.0719816 0.997406i $$-0.477068\pi$$
0.0719816 + 0.997406i $$0.477068\pi$$
$$194$$ 7.48913 0.537688
$$195$$ 0 0
$$196$$ 1.00000 0.0714286
$$197$$ 6.00000 0.427482 0.213741 0.976890i $$-0.431435\pi$$
0.213741 + 0.976890i $$0.431435\pi$$
$$198$$ 0 0
$$199$$ 11.8614 0.840833 0.420416 0.907331i $$-0.361884\pi$$
0.420416 + 0.907331i $$0.361884\pi$$
$$200$$ 0.627719 0.0443864
$$201$$ 0 0
$$202$$ −2.00000 −0.140720
$$203$$ −6.37228 −0.447246
$$204$$ 0 0
$$205$$ −20.7446 −1.44886
$$206$$ −11.8614 −0.826423
$$207$$ 0 0
$$208$$ −1.00000 −0.0693375
$$209$$ 3.86141 0.267099
$$210$$ 0 0
$$211$$ −17.6277 −1.21354 −0.606771 0.794877i $$-0.707536\pi$$
−0.606771 + 0.794877i $$0.707536\pi$$
$$212$$ 8.74456 0.600579
$$213$$ 0 0
$$214$$ 2.00000 0.136717
$$215$$ −26.3723 −1.79857
$$216$$ 0 0
$$217$$ 4.74456 0.322082
$$218$$ −3.62772 −0.245700
$$219$$ 0 0
$$220$$ −5.62772 −0.379421
$$221$$ −4.37228 −0.294111
$$222$$ 0 0
$$223$$ −17.4891 −1.17116 −0.585579 0.810615i $$-0.699133\pi$$
−0.585579 + 0.810615i $$0.699133\pi$$
$$224$$ −1.00000 −0.0668153
$$225$$ 0 0
$$226$$ 20.0000 1.33038
$$227$$ 11.4891 0.762560 0.381280 0.924460i $$-0.375483\pi$$
0.381280 + 0.924460i $$0.375483\pi$$
$$228$$ 0 0
$$229$$ 6.00000 0.396491 0.198246 0.980152i $$-0.436476\pi$$
0.198246 + 0.980152i $$0.436476\pi$$
$$230$$ −8.60597 −0.567461
$$231$$ 0 0
$$232$$ 6.37228 0.418361
$$233$$ −1.48913 −0.0975558 −0.0487779 0.998810i $$-0.515533\pi$$
−0.0487779 + 0.998810i $$0.515533\pi$$
$$234$$ 0 0
$$235$$ −2.97825 −0.194280
$$236$$ 2.00000 0.130189
$$237$$ 0 0
$$238$$ −4.37228 −0.283413
$$239$$ −5.48913 −0.355062 −0.177531 0.984115i $$-0.556811\pi$$
−0.177531 + 0.984115i $$0.556811\pi$$
$$240$$ 0 0
$$241$$ −6.00000 −0.386494 −0.193247 0.981150i $$-0.561902\pi$$
−0.193247 + 0.981150i $$0.561902\pi$$
$$242$$ −5.37228 −0.345343
$$243$$ 0 0
$$244$$ 5.11684 0.327572
$$245$$ −2.37228 −0.151559
$$246$$ 0 0
$$247$$ −1.62772 −0.103569
$$248$$ −4.74456 −0.301280
$$249$$ 0 0
$$250$$ 10.3723 0.656001
$$251$$ −13.6277 −0.860174 −0.430087 0.902787i $$-0.641517\pi$$
−0.430087 + 0.902787i $$0.641517\pi$$
$$252$$ 0 0
$$253$$ 8.60597 0.541053
$$254$$ −8.00000 −0.501965
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ 11.4891 0.716672 0.358336 0.933593i $$-0.383344\pi$$
0.358336 + 0.933593i $$0.383344\pi$$
$$258$$ 0 0
$$259$$ 4.37228 0.271680
$$260$$ 2.37228 0.147123
$$261$$ 0 0
$$262$$ −11.1168 −0.686801
$$263$$ 9.25544 0.570715 0.285357 0.958421i $$-0.407888\pi$$
0.285357 + 0.958421i $$0.407888\pi$$
$$264$$ 0 0
$$265$$ −20.7446 −1.27433
$$266$$ −1.62772 −0.0998018
$$267$$ 0 0
$$268$$ 9.48913 0.579641
$$269$$ 0.510875 0.0311486 0.0155743 0.999879i $$-0.495042\pi$$
0.0155743 + 0.999879i $$0.495042\pi$$
$$270$$ 0 0
$$271$$ 30.2337 1.83657 0.918283 0.395925i $$-0.129576\pi$$
0.918283 + 0.395925i $$0.129576\pi$$
$$272$$ 4.37228 0.265108
$$273$$ 0 0
$$274$$ 1.11684 0.0674710
$$275$$ 1.48913 0.0897976
$$276$$ 0 0
$$277$$ 26.7446 1.60693 0.803463 0.595355i $$-0.202989\pi$$
0.803463 + 0.595355i $$0.202989\pi$$
$$278$$ 0.744563 0.0446559
$$279$$ 0 0
$$280$$ 2.37228 0.141771
$$281$$ 10.0000 0.596550 0.298275 0.954480i $$-0.403589\pi$$
0.298275 + 0.954480i $$0.403589\pi$$
$$282$$ 0 0
$$283$$ 14.9783 0.890365 0.445182 0.895440i $$-0.353139\pi$$
0.445182 + 0.895440i $$0.353139\pi$$
$$284$$ −4.74456 −0.281538
$$285$$ 0 0
$$286$$ −2.37228 −0.140276
$$287$$ −8.74456 −0.516175
$$288$$ 0 0
$$289$$ 2.11684 0.124520
$$290$$ −15.1168 −0.887692
$$291$$ 0 0
$$292$$ −8.37228 −0.489951
$$293$$ −28.7446 −1.67928 −0.839638 0.543147i $$-0.817233\pi$$
−0.839638 + 0.543147i $$0.817233\pi$$
$$294$$ 0 0
$$295$$ −4.74456 −0.276239
$$296$$ −4.37228 −0.254134
$$297$$ 0 0
$$298$$ −20.2337 −1.17211
$$299$$ −3.62772 −0.209796
$$300$$ 0 0
$$301$$ −11.1168 −0.640764
$$302$$ 19.1168 1.10005
$$303$$ 0 0
$$304$$ 1.62772 0.0933561
$$305$$ −12.1386 −0.695054
$$306$$ 0 0
$$307$$ 20.0000 1.14146 0.570730 0.821138i $$-0.306660\pi$$
0.570730 + 0.821138i $$0.306660\pi$$
$$308$$ −2.37228 −0.135173
$$309$$ 0 0
$$310$$ 11.2554 0.639266
$$311$$ 17.4891 0.991717 0.495859 0.868403i $$-0.334853\pi$$
0.495859 + 0.868403i $$0.334853\pi$$
$$312$$ 0 0
$$313$$ 12.9783 0.733574 0.366787 0.930305i $$-0.380458\pi$$
0.366787 + 0.930305i $$0.380458\pi$$
$$314$$ −20.3723 −1.14967
$$315$$ 0 0
$$316$$ 4.74456 0.266903
$$317$$ −28.9783 −1.62758 −0.813790 0.581159i $$-0.802600\pi$$
−0.813790 + 0.581159i $$0.802600\pi$$
$$318$$ 0 0
$$319$$ 15.1168 0.846381
$$320$$ −2.37228 −0.132615
$$321$$ 0 0
$$322$$ −3.62772 −0.202165
$$323$$ 7.11684 0.395992
$$324$$ 0 0
$$325$$ −0.627719 −0.0348196
$$326$$ −18.2337 −1.00987
$$327$$ 0 0
$$328$$ 8.74456 0.482838
$$329$$ −1.25544 −0.0692145
$$330$$ 0 0
$$331$$ 16.7446 0.920364 0.460182 0.887824i $$-0.347784\pi$$
0.460182 + 0.887824i $$0.347784\pi$$
$$332$$ −6.00000 −0.329293
$$333$$ 0 0
$$334$$ 13.1168 0.717722
$$335$$ −22.5109 −1.22990
$$336$$ 0 0
$$337$$ −1.86141 −0.101397 −0.0506986 0.998714i $$-0.516145\pi$$
−0.0506986 + 0.998714i $$0.516145\pi$$
$$338$$ 1.00000 0.0543928
$$339$$ 0 0
$$340$$ −10.3723 −0.562516
$$341$$ −11.2554 −0.609516
$$342$$ 0 0
$$343$$ −1.00000 −0.0539949
$$344$$ 11.1168 0.599380
$$345$$ 0 0
$$346$$ −10.0000 −0.537603
$$347$$ 25.7228 1.38087 0.690436 0.723393i $$-0.257418\pi$$
0.690436 + 0.723393i $$0.257418\pi$$
$$348$$ 0 0
$$349$$ −22.7446 −1.21749 −0.608744 0.793367i $$-0.708326\pi$$
−0.608744 + 0.793367i $$0.708326\pi$$
$$350$$ −0.627719 −0.0335530
$$351$$ 0 0
$$352$$ 2.37228 0.126443
$$353$$ −4.74456 −0.252528 −0.126264 0.991997i $$-0.540299\pi$$
−0.126264 + 0.991997i $$0.540299\pi$$
$$354$$ 0 0
$$355$$ 11.2554 0.597377
$$356$$ −3.25544 −0.172538
$$357$$ 0 0
$$358$$ −2.74456 −0.145055
$$359$$ −3.25544 −0.171815 −0.0859077 0.996303i $$-0.527379\pi$$
−0.0859077 + 0.996303i $$0.527379\pi$$
$$360$$ 0 0
$$361$$ −16.3505 −0.860554
$$362$$ −23.4891 −1.23456
$$363$$ 0 0
$$364$$ 1.00000 0.0524142
$$365$$ 19.8614 1.03959
$$366$$ 0 0
$$367$$ 2.51087 0.131067 0.0655333 0.997850i $$-0.479125\pi$$
0.0655333 + 0.997850i $$0.479125\pi$$
$$368$$ 3.62772 0.189108
$$369$$ 0 0
$$370$$ 10.3723 0.539229
$$371$$ −8.74456 −0.453995
$$372$$ 0 0
$$373$$ −15.4891 −0.801997 −0.400998 0.916079i $$-0.631337\pi$$
−0.400998 + 0.916079i $$0.631337\pi$$
$$374$$ 10.3723 0.536338
$$375$$ 0 0
$$376$$ 1.25544 0.0647442
$$377$$ −6.37228 −0.328189
$$378$$ 0 0
$$379$$ −28.7446 −1.47651 −0.738255 0.674522i $$-0.764350\pi$$
−0.738255 + 0.674522i $$0.764350\pi$$
$$380$$ −3.86141 −0.198086
$$381$$ 0 0
$$382$$ 5.11684 0.261801
$$383$$ 6.60597 0.337549 0.168775 0.985655i $$-0.446019\pi$$
0.168775 + 0.985655i $$0.446019\pi$$
$$384$$ 0 0
$$385$$ 5.62772 0.286815
$$386$$ 2.00000 0.101797
$$387$$ 0 0
$$388$$ 7.48913 0.380203
$$389$$ 3.25544 0.165057 0.0825286 0.996589i $$-0.473700\pi$$
0.0825286 + 0.996589i $$0.473700\pi$$
$$390$$ 0 0
$$391$$ 15.8614 0.802146
$$392$$ 1.00000 0.0505076
$$393$$ 0 0
$$394$$ 6.00000 0.302276
$$395$$ −11.2554 −0.566323
$$396$$ 0 0
$$397$$ 5.25544 0.263763 0.131881 0.991265i $$-0.457898\pi$$
0.131881 + 0.991265i $$0.457898\pi$$
$$398$$ 11.8614 0.594559
$$399$$ 0 0
$$400$$ 0.627719 0.0313859
$$401$$ −35.4891 −1.77224 −0.886121 0.463454i $$-0.846610\pi$$
−0.886121 + 0.463454i $$0.846610\pi$$
$$402$$ 0 0
$$403$$ 4.74456 0.236343
$$404$$ −2.00000 −0.0995037
$$405$$ 0 0
$$406$$ −6.37228 −0.316251
$$407$$ −10.3723 −0.514135
$$408$$ 0 0
$$409$$ 25.8614 1.27876 0.639382 0.768889i $$-0.279190\pi$$
0.639382 + 0.768889i $$0.279190\pi$$
$$410$$ −20.7446 −1.02450
$$411$$ 0 0
$$412$$ −11.8614 −0.584370
$$413$$ −2.00000 −0.0984136
$$414$$ 0 0
$$415$$ 14.2337 0.698704
$$416$$ −1.00000 −0.0490290
$$417$$ 0 0
$$418$$ 3.86141 0.188868
$$419$$ −3.86141 −0.188642 −0.0943210 0.995542i $$-0.530068\pi$$
−0.0943210 + 0.995542i $$0.530068\pi$$
$$420$$ 0 0
$$421$$ −7.48913 −0.364998 −0.182499 0.983206i $$-0.558419\pi$$
−0.182499 + 0.983206i $$0.558419\pi$$
$$422$$ −17.6277 −0.858104
$$423$$ 0 0
$$424$$ 8.74456 0.424674
$$425$$ 2.74456 0.133131
$$426$$ 0 0
$$427$$ −5.11684 −0.247621
$$428$$ 2.00000 0.0966736
$$429$$ 0 0
$$430$$ −26.3723 −1.27178
$$431$$ −32.7446 −1.57725 −0.788625 0.614874i $$-0.789207\pi$$
−0.788625 + 0.614874i $$0.789207\pi$$
$$432$$ 0 0
$$433$$ 10.0000 0.480569 0.240285 0.970702i $$-0.422759\pi$$
0.240285 + 0.970702i $$0.422759\pi$$
$$434$$ 4.74456 0.227746
$$435$$ 0 0
$$436$$ −3.62772 −0.173736
$$437$$ 5.90491 0.282470
$$438$$ 0 0
$$439$$ 9.62772 0.459506 0.229753 0.973249i $$-0.426208\pi$$
0.229753 + 0.973249i $$0.426208\pi$$
$$440$$ −5.62772 −0.268291
$$441$$ 0 0
$$442$$ −4.37228 −0.207968
$$443$$ 16.9783 0.806661 0.403331 0.915054i $$-0.367852\pi$$
0.403331 + 0.915054i $$0.367852\pi$$
$$444$$ 0 0
$$445$$ 7.72281 0.366096
$$446$$ −17.4891 −0.828134
$$447$$ 0 0
$$448$$ −1.00000 −0.0472456
$$449$$ −0.372281 −0.0175690 −0.00878452 0.999961i $$-0.502796\pi$$
−0.00878452 + 0.999961i $$0.502796\pi$$
$$450$$ 0 0
$$451$$ 20.7446 0.976823
$$452$$ 20.0000 0.940721
$$453$$ 0 0
$$454$$ 11.4891 0.539211
$$455$$ −2.37228 −0.111214
$$456$$ 0 0
$$457$$ 12.2337 0.572268 0.286134 0.958190i $$-0.407630\pi$$
0.286134 + 0.958190i $$0.407630\pi$$
$$458$$ 6.00000 0.280362
$$459$$ 0 0
$$460$$ −8.60597 −0.401255
$$461$$ −36.6060 −1.70491 −0.852455 0.522801i $$-0.824887\pi$$
−0.852455 + 0.522801i $$0.824887\pi$$
$$462$$ 0 0
$$463$$ −17.3505 −0.806348 −0.403174 0.915123i $$-0.632093\pi$$
−0.403174 + 0.915123i $$0.632093\pi$$
$$464$$ 6.37228 0.295826
$$465$$ 0 0
$$466$$ −1.48913 −0.0689824
$$467$$ −34.0951 −1.57773 −0.788866 0.614565i $$-0.789332\pi$$
−0.788866 + 0.614565i $$0.789332\pi$$
$$468$$ 0 0
$$469$$ −9.48913 −0.438167
$$470$$ −2.97825 −0.137376
$$471$$ 0 0
$$472$$ 2.00000 0.0920575
$$473$$ 26.3723 1.21260
$$474$$ 0 0
$$475$$ 1.02175 0.0468811
$$476$$ −4.37228 −0.200403
$$477$$ 0 0
$$478$$ −5.48913 −0.251067
$$479$$ 39.3505 1.79797 0.898986 0.437978i $$-0.144305\pi$$
0.898986 + 0.437978i $$0.144305\pi$$
$$480$$ 0 0
$$481$$ 4.37228 0.199359
$$482$$ −6.00000 −0.273293
$$483$$ 0 0
$$484$$ −5.37228 −0.244195
$$485$$ −17.7663 −0.806727
$$486$$ 0 0
$$487$$ −32.4674 −1.47124 −0.735619 0.677396i $$-0.763108\pi$$
−0.735619 + 0.677396i $$0.763108\pi$$
$$488$$ 5.11684 0.231629
$$489$$ 0 0
$$490$$ −2.37228 −0.107169
$$491$$ 25.2554 1.13976 0.569881 0.821727i $$-0.306989\pi$$
0.569881 + 0.821727i $$0.306989\pi$$
$$492$$ 0 0
$$493$$ 27.8614 1.25481
$$494$$ −1.62772 −0.0732345
$$495$$ 0 0
$$496$$ −4.74456 −0.213037
$$497$$ 4.74456 0.212823
$$498$$ 0 0
$$499$$ −20.0000 −0.895323 −0.447661 0.894203i $$-0.647743\pi$$
−0.447661 + 0.894203i $$0.647743\pi$$
$$500$$ 10.3723 0.463863
$$501$$ 0 0
$$502$$ −13.6277 −0.608235
$$503$$ 21.4891 0.958153 0.479076 0.877773i $$-0.340972\pi$$
0.479076 + 0.877773i $$0.340972\pi$$
$$504$$ 0 0
$$505$$ 4.74456 0.211130
$$506$$ 8.60597 0.382582
$$507$$ 0 0
$$508$$ −8.00000 −0.354943
$$509$$ −26.0951 −1.15664 −0.578322 0.815808i $$-0.696292\pi$$
−0.578322 + 0.815808i $$0.696292\pi$$
$$510$$ 0 0
$$511$$ 8.37228 0.370368
$$512$$ 1.00000 0.0441942
$$513$$ 0 0
$$514$$ 11.4891 0.506764
$$515$$ 28.1386 1.23993
$$516$$ 0 0
$$517$$ 2.97825 0.130983
$$518$$ 4.37228 0.192107
$$519$$ 0 0
$$520$$ 2.37228 0.104031
$$521$$ −28.3723 −1.24301 −0.621506 0.783409i $$-0.713479\pi$$
−0.621506 + 0.783409i $$0.713479\pi$$
$$522$$ 0 0
$$523$$ 2.23369 0.0976724 0.0488362 0.998807i $$-0.484449\pi$$
0.0488362 + 0.998807i $$0.484449\pi$$
$$524$$ −11.1168 −0.485642
$$525$$ 0 0
$$526$$ 9.25544 0.403556
$$527$$ −20.7446 −0.903647
$$528$$ 0 0
$$529$$ −9.83966 −0.427811
$$530$$ −20.7446 −0.901086
$$531$$ 0 0
$$532$$ −1.62772 −0.0705706
$$533$$ −8.74456 −0.378769
$$534$$ 0 0
$$535$$ −4.74456 −0.205125
$$536$$ 9.48913 0.409868
$$537$$ 0 0
$$538$$ 0.510875 0.0220254
$$539$$ 2.37228 0.102181
$$540$$ 0 0
$$541$$ 23.3505 1.00392 0.501959 0.864891i $$-0.332613\pi$$
0.501959 + 0.864891i $$0.332613\pi$$
$$542$$ 30.2337 1.29865
$$543$$ 0 0
$$544$$ 4.37228 0.187460
$$545$$ 8.60597 0.368639
$$546$$ 0 0
$$547$$ 28.0000 1.19719 0.598597 0.801050i $$-0.295725\pi$$
0.598597 + 0.801050i $$0.295725\pi$$
$$548$$ 1.11684 0.0477092
$$549$$ 0 0
$$550$$ 1.48913 0.0634965
$$551$$ 10.3723 0.441874
$$552$$ 0 0
$$553$$ −4.74456 −0.201759
$$554$$ 26.7446 1.13627
$$555$$ 0 0
$$556$$ 0.744563 0.0315765
$$557$$ 11.4891 0.486810 0.243405 0.969925i $$-0.421736\pi$$
0.243405 + 0.969925i $$0.421736\pi$$
$$558$$ 0 0
$$559$$ −11.1168 −0.470192
$$560$$ 2.37228 0.100247
$$561$$ 0 0
$$562$$ 10.0000 0.421825
$$563$$ −11.1168 −0.468519 −0.234260 0.972174i $$-0.575267\pi$$
−0.234260 + 0.972174i $$0.575267\pi$$
$$564$$ 0 0
$$565$$ −47.4456 −1.99605
$$566$$ 14.9783 0.629583
$$567$$ 0 0
$$568$$ −4.74456 −0.199077
$$569$$ 12.7446 0.534280 0.267140 0.963658i $$-0.413921\pi$$
0.267140 + 0.963658i $$0.413921\pi$$
$$570$$ 0 0
$$571$$ 12.0000 0.502184 0.251092 0.967963i $$-0.419210\pi$$
0.251092 + 0.967963i $$0.419210\pi$$
$$572$$ −2.37228 −0.0991901
$$573$$ 0 0
$$574$$ −8.74456 −0.364991
$$575$$ 2.27719 0.0949653
$$576$$ 0 0
$$577$$ −0.510875 −0.0212680 −0.0106340 0.999943i $$-0.503385\pi$$
−0.0106340 + 0.999943i $$0.503385\pi$$
$$578$$ 2.11684 0.0880491
$$579$$ 0 0
$$580$$ −15.1168 −0.627693
$$581$$ 6.00000 0.248922
$$582$$ 0 0
$$583$$ 20.7446 0.859152
$$584$$ −8.37228 −0.346447
$$585$$ 0 0
$$586$$ −28.7446 −1.18743
$$587$$ −27.4891 −1.13460 −0.567299 0.823512i $$-0.692012\pi$$
−0.567299 + 0.823512i $$0.692012\pi$$
$$588$$ 0 0
$$589$$ −7.72281 −0.318213
$$590$$ −4.74456 −0.195331
$$591$$ 0 0
$$592$$ −4.37228 −0.179700
$$593$$ −4.00000 −0.164260 −0.0821302 0.996622i $$-0.526172\pi$$
−0.0821302 + 0.996622i $$0.526172\pi$$
$$594$$ 0 0
$$595$$ 10.3723 0.425222
$$596$$ −20.2337 −0.828804
$$597$$ 0 0
$$598$$ −3.62772 −0.148348
$$599$$ −21.1168 −0.862811 −0.431405 0.902158i $$-0.641982\pi$$
−0.431405 + 0.902158i $$0.641982\pi$$
$$600$$ 0 0
$$601$$ −42.4674 −1.73228 −0.866140 0.499801i $$-0.833406\pi$$
−0.866140 + 0.499801i $$0.833406\pi$$
$$602$$ −11.1168 −0.453089
$$603$$ 0 0
$$604$$ 19.1168 0.777853
$$605$$ 12.7446 0.518140
$$606$$ 0 0
$$607$$ 19.1168 0.775929 0.387964 0.921674i $$-0.373178\pi$$
0.387964 + 0.921674i $$0.373178\pi$$
$$608$$ 1.62772 0.0660127
$$609$$ 0 0
$$610$$ −12.1386 −0.491477
$$611$$ −1.25544 −0.0507896
$$612$$ 0 0
$$613$$ −6.13859 −0.247935 −0.123968 0.992286i $$-0.539562\pi$$
−0.123968 + 0.992286i $$0.539562\pi$$
$$614$$ 20.0000 0.807134
$$615$$ 0 0
$$616$$ −2.37228 −0.0955819
$$617$$ 9.39403 0.378189 0.189095 0.981959i $$-0.439445\pi$$
0.189095 + 0.981959i $$0.439445\pi$$
$$618$$ 0 0
$$619$$ 11.1168 0.446824 0.223412 0.974724i $$-0.428281\pi$$
0.223412 + 0.974724i $$0.428281\pi$$
$$620$$ 11.2554 0.452029
$$621$$ 0 0
$$622$$ 17.4891 0.701250
$$623$$ 3.25544 0.130426
$$624$$ 0 0
$$625$$ −27.7446 −1.10978
$$626$$ 12.9783 0.518715
$$627$$ 0 0
$$628$$ −20.3723 −0.812943
$$629$$ −19.1168 −0.762238
$$630$$ 0 0
$$631$$ 16.6060 0.661073 0.330537 0.943793i $$-0.392770\pi$$
0.330537 + 0.943793i $$0.392770\pi$$
$$632$$ 4.74456 0.188729
$$633$$ 0 0
$$634$$ −28.9783 −1.15087
$$635$$ 18.9783 0.753129
$$636$$ 0 0
$$637$$ −1.00000 −0.0396214
$$638$$ 15.1168 0.598482
$$639$$ 0 0
$$640$$ −2.37228 −0.0937727
$$641$$ 18.2337 0.720187 0.360094 0.932916i $$-0.382745\pi$$
0.360094 + 0.932916i $$0.382745\pi$$
$$642$$ 0 0
$$643$$ −0.138593 −0.00546559 −0.00273279 0.999996i $$-0.500870\pi$$
−0.00273279 + 0.999996i $$0.500870\pi$$
$$644$$ −3.62772 −0.142952
$$645$$ 0 0
$$646$$ 7.11684 0.280008
$$647$$ 21.4891 0.844825 0.422412 0.906404i $$-0.361183\pi$$
0.422412 + 0.906404i $$0.361183\pi$$
$$648$$ 0 0
$$649$$ 4.74456 0.186240
$$650$$ −0.627719 −0.0246212
$$651$$ 0 0
$$652$$ −18.2337 −0.714086
$$653$$ 8.13859 0.318488 0.159244 0.987239i $$-0.449094\pi$$
0.159244 + 0.987239i $$0.449094\pi$$
$$654$$ 0 0
$$655$$ 26.3723 1.03045
$$656$$ 8.74456 0.341418
$$657$$ 0 0
$$658$$ −1.25544 −0.0489420
$$659$$ 9.25544 0.360541 0.180270 0.983617i $$-0.442303\pi$$
0.180270 + 0.983617i $$0.442303\pi$$
$$660$$ 0 0
$$661$$ 10.7446 0.417915 0.208958 0.977925i $$-0.432993\pi$$
0.208958 + 0.977925i $$0.432993\pi$$
$$662$$ 16.7446 0.650796
$$663$$ 0 0
$$664$$ −6.00000 −0.232845
$$665$$ 3.86141 0.149739
$$666$$ 0 0
$$667$$ 23.1168 0.895088
$$668$$ 13.1168 0.507506
$$669$$ 0 0
$$670$$ −22.5109 −0.869671
$$671$$ 12.1386 0.468605
$$672$$ 0 0
$$673$$ 16.0951 0.620420 0.310210 0.950668i $$-0.399601\pi$$
0.310210 + 0.950668i $$0.399601\pi$$
$$674$$ −1.86141 −0.0716987
$$675$$ 0 0
$$676$$ 1.00000 0.0384615
$$677$$ 35.4891 1.36396 0.681979 0.731372i $$-0.261120\pi$$
0.681979 + 0.731372i $$0.261120\pi$$
$$678$$ 0 0
$$679$$ −7.48913 −0.287406
$$680$$ −10.3723 −0.397759
$$681$$ 0 0
$$682$$ −11.2554 −0.430993
$$683$$ 11.1168 0.425374 0.212687 0.977120i $$-0.431778\pi$$
0.212687 + 0.977120i $$0.431778\pi$$
$$684$$ 0 0
$$685$$ −2.64947 −0.101231
$$686$$ −1.00000 −0.0381802
$$687$$ 0 0
$$688$$ 11.1168 0.423826
$$689$$ −8.74456 −0.333141
$$690$$ 0 0
$$691$$ 45.4891 1.73049 0.865244 0.501351i $$-0.167163\pi$$
0.865244 + 0.501351i $$0.167163\pi$$
$$692$$ −10.0000 −0.380143
$$693$$ 0 0
$$694$$ 25.7228 0.976425
$$695$$ −1.76631 −0.0670000
$$696$$ 0 0
$$697$$ 38.2337 1.44820
$$698$$ −22.7446 −0.860894
$$699$$ 0 0
$$700$$ −0.627719 −0.0237255
$$701$$ 31.7228 1.19815 0.599077 0.800691i $$-0.295534\pi$$
0.599077 + 0.800691i $$0.295534\pi$$
$$702$$ 0 0
$$703$$ −7.11684 −0.268417
$$704$$ 2.37228 0.0894087
$$705$$ 0 0
$$706$$ −4.74456 −0.178564
$$707$$ 2.00000 0.0752177
$$708$$ 0 0
$$709$$ −28.5109 −1.07075 −0.535374 0.844615i $$-0.679829\pi$$
−0.535374 + 0.844615i $$0.679829\pi$$
$$710$$ 11.2554 0.422409
$$711$$ 0 0
$$712$$ −3.25544 −0.122003
$$713$$ −17.2119 −0.644592
$$714$$ 0 0
$$715$$ 5.62772 0.210465
$$716$$ −2.74456 −0.102569
$$717$$ 0 0
$$718$$ −3.25544 −0.121492
$$719$$ −45.4891 −1.69646 −0.848229 0.529630i $$-0.822331\pi$$
−0.848229 + 0.529630i $$0.822331\pi$$
$$720$$ 0 0
$$721$$ 11.8614 0.441742
$$722$$ −16.3505 −0.608504
$$723$$ 0 0
$$724$$ −23.4891 −0.872966
$$725$$ 4.00000 0.148556
$$726$$ 0 0
$$727$$ 46.0951 1.70957 0.854786 0.518980i $$-0.173688\pi$$
0.854786 + 0.518980i $$0.173688\pi$$
$$728$$ 1.00000 0.0370625
$$729$$ 0 0
$$730$$ 19.8614 0.735104
$$731$$ 48.6060 1.79776
$$732$$ 0 0
$$733$$ −30.7446 −1.13558 −0.567788 0.823175i $$-0.692201\pi$$
−0.567788 + 0.823175i $$0.692201\pi$$
$$734$$ 2.51087 0.0926781
$$735$$ 0 0
$$736$$ 3.62772 0.133719
$$737$$ 22.5109 0.829199
$$738$$ 0 0
$$739$$ 31.7228 1.16694 0.583471 0.812134i $$-0.301694\pi$$
0.583471 + 0.812134i $$0.301694\pi$$
$$740$$ 10.3723 0.381293
$$741$$ 0 0
$$742$$ −8.74456 −0.321023
$$743$$ 44.4674 1.63135 0.815675 0.578511i $$-0.196366\pi$$
0.815675 + 0.578511i $$0.196366\pi$$
$$744$$ 0 0
$$745$$ 48.0000 1.75858
$$746$$ −15.4891 −0.567097
$$747$$ 0 0
$$748$$ 10.3723 0.379248
$$749$$ −2.00000 −0.0730784
$$750$$ 0 0
$$751$$ −10.9783 −0.400602 −0.200301 0.979734i $$-0.564192\pi$$
−0.200301 + 0.979734i $$0.564192\pi$$
$$752$$ 1.25544 0.0457811
$$753$$ 0 0
$$754$$ −6.37228 −0.232065
$$755$$ −45.3505 −1.65047
$$756$$ 0 0
$$757$$ −43.2119 −1.57056 −0.785282 0.619138i $$-0.787482\pi$$
−0.785282 + 0.619138i $$0.787482\pi$$
$$758$$ −28.7446 −1.04405
$$759$$ 0 0
$$760$$ −3.86141 −0.140068
$$761$$ −46.9783 −1.70296 −0.851480 0.524387i $$-0.824295\pi$$
−0.851480 + 0.524387i $$0.824295\pi$$
$$762$$ 0 0
$$763$$ 3.62772 0.131332
$$764$$ 5.11684 0.185121
$$765$$ 0 0
$$766$$ 6.60597 0.238683
$$767$$ −2.00000 −0.0722158
$$768$$ 0 0
$$769$$ −19.6277 −0.707794 −0.353897 0.935284i $$-0.615144\pi$$
−0.353897 + 0.935284i $$0.615144\pi$$
$$770$$ 5.62772 0.202809
$$771$$ 0 0
$$772$$ 2.00000 0.0719816
$$773$$ −34.0951 −1.22632 −0.613158 0.789961i $$-0.710101\pi$$
−0.613158 + 0.789961i $$0.710101\pi$$
$$774$$ 0 0
$$775$$ −2.97825 −0.106982
$$776$$ 7.48913 0.268844
$$777$$ 0 0
$$778$$ 3.25544 0.116713
$$779$$ 14.2337 0.509975
$$780$$ 0 0
$$781$$ −11.2554 −0.402751
$$782$$ 15.8614 0.567203
$$783$$ 0 0
$$784$$ 1.00000 0.0357143
$$785$$ 48.3288 1.72493
$$786$$ 0 0
$$787$$ 27.1168 0.966611 0.483306 0.875452i $$-0.339436\pi$$
0.483306 + 0.875452i $$0.339436\pi$$
$$788$$ 6.00000 0.213741
$$789$$ 0 0
$$790$$ −11.2554 −0.400450
$$791$$ −20.0000 −0.711118
$$792$$ 0 0
$$793$$ −5.11684 −0.181704
$$794$$ 5.25544 0.186508
$$795$$ 0 0
$$796$$ 11.8614 0.420416
$$797$$ 49.7228 1.76127 0.880636 0.473793i $$-0.157116\pi$$
0.880636 + 0.473793i $$0.157116\pi$$
$$798$$ 0 0
$$799$$ 5.48913 0.194191
$$800$$ 0.627719 0.0221932
$$801$$ 0 0
$$802$$ −35.4891 −1.25316
$$803$$ −19.8614 −0.700894
$$804$$ 0 0
$$805$$ 8.60597 0.303321
$$806$$ 4.74456 0.167120
$$807$$ 0 0
$$808$$ −2.00000 −0.0703598
$$809$$ 32.7446 1.15124 0.575619 0.817718i $$-0.304761\pi$$
0.575619 + 0.817718i $$0.304761\pi$$
$$810$$ 0 0
$$811$$ 3.11684 0.109447 0.0547236 0.998502i $$-0.482572\pi$$
0.0547236 + 0.998502i $$0.482572\pi$$
$$812$$ −6.37228 −0.223623
$$813$$ 0 0
$$814$$ −10.3723 −0.363548
$$815$$ 43.2554 1.51517
$$816$$ 0 0
$$817$$ 18.0951 0.633067
$$818$$ 25.8614 0.904223
$$819$$ 0 0
$$820$$ −20.7446 −0.724432
$$821$$ −53.7228 −1.87494 −0.937470 0.348067i $$-0.886838\pi$$
−0.937470 + 0.348067i $$0.886838\pi$$
$$822$$ 0 0
$$823$$ −31.7228 −1.10579 −0.552894 0.833252i $$-0.686477\pi$$
−0.552894 + 0.833252i $$0.686477\pi$$
$$824$$ −11.8614 −0.413212
$$825$$ 0 0
$$826$$ −2.00000 −0.0695889
$$827$$ 34.3723 1.19524 0.597621 0.801779i $$-0.296113\pi$$
0.597621 + 0.801779i $$0.296113\pi$$
$$828$$ 0 0
$$829$$ −24.3723 −0.846484 −0.423242 0.906017i $$-0.639108\pi$$
−0.423242 + 0.906017i $$0.639108\pi$$
$$830$$ 14.2337 0.494059
$$831$$ 0 0
$$832$$ −1.00000 −0.0346688
$$833$$ 4.37228 0.151491
$$834$$ 0 0
$$835$$ −31.1168 −1.07684
$$836$$ 3.86141 0.133550
$$837$$ 0 0
$$838$$ −3.86141 −0.133390
$$839$$ −1.25544 −0.0433425 −0.0216713 0.999765i $$-0.506899\pi$$
−0.0216713 + 0.999765i $$0.506899\pi$$
$$840$$ 0 0
$$841$$ 11.6060 0.400206
$$842$$ −7.48913 −0.258092
$$843$$ 0 0
$$844$$ −17.6277 −0.606771
$$845$$ −2.37228 −0.0816090
$$846$$ 0 0
$$847$$ 5.37228 0.184594
$$848$$ 8.74456 0.300290
$$849$$ 0 0
$$850$$ 2.74456 0.0941377
$$851$$ −15.8614 −0.543722
$$852$$ 0 0
$$853$$ −39.2119 −1.34259 −0.671296 0.741190i $$-0.734262\pi$$
−0.671296 + 0.741190i $$0.734262\pi$$
$$854$$ −5.11684 −0.175095
$$855$$ 0 0
$$856$$ 2.00000 0.0683586
$$857$$ −0.978251 −0.0334164 −0.0167082 0.999860i $$-0.505319\pi$$
−0.0167082 + 0.999860i $$0.505319\pi$$
$$858$$ 0 0
$$859$$ −0.744563 −0.0254041 −0.0127021 0.999919i $$-0.504043\pi$$
−0.0127021 + 0.999919i $$0.504043\pi$$
$$860$$ −26.3723 −0.899287
$$861$$ 0 0
$$862$$ −32.7446 −1.11528
$$863$$ −18.9783 −0.646027 −0.323014 0.946394i $$-0.604696\pi$$
−0.323014 + 0.946394i $$0.604696\pi$$
$$864$$ 0 0
$$865$$ 23.7228 0.806600
$$866$$ 10.0000 0.339814
$$867$$ 0 0
$$868$$ 4.74456 0.161041
$$869$$ 11.2554 0.381815
$$870$$ 0 0
$$871$$ −9.48913 −0.321527
$$872$$ −3.62772 −0.122850
$$873$$ 0 0
$$874$$ 5.90491 0.199736
$$875$$ −10.3723 −0.350647
$$876$$ 0 0
$$877$$ −52.9783 −1.78895 −0.894474 0.447120i $$-0.852450\pi$$
−0.894474 + 0.447120i $$0.852450\pi$$
$$878$$ 9.62772 0.324920
$$879$$ 0 0
$$880$$ −5.62772 −0.189710
$$881$$ −14.1386 −0.476341 −0.238171 0.971223i $$-0.576548\pi$$
−0.238171 + 0.971223i $$0.576548\pi$$
$$882$$ 0 0
$$883$$ −30.3723 −1.02211 −0.511054 0.859548i $$-0.670745\pi$$
−0.511054 + 0.859548i $$0.670745\pi$$
$$884$$ −4.37228 −0.147056
$$885$$ 0 0
$$886$$ 16.9783 0.570395
$$887$$ 53.2119 1.78668 0.893341 0.449379i $$-0.148355\pi$$
0.893341 + 0.449379i $$0.148355\pi$$
$$888$$ 0 0
$$889$$ 8.00000 0.268311
$$890$$ 7.72281 0.258869
$$891$$ 0 0
$$892$$ −17.4891 −0.585579
$$893$$ 2.04350 0.0683831
$$894$$ 0 0
$$895$$ 6.51087 0.217635
$$896$$ −1.00000 −0.0334077
$$897$$ 0 0
$$898$$ −0.372281 −0.0124232
$$899$$ −30.2337 −1.00835
$$900$$ 0 0
$$901$$ 38.2337 1.27375
$$902$$ 20.7446 0.690718
$$903$$ 0 0
$$904$$ 20.0000 0.665190
$$905$$ 55.7228 1.85229
$$906$$ 0 0
$$907$$ −57.9565 −1.92441 −0.962207 0.272319i $$-0.912209\pi$$
−0.962207 + 0.272319i $$0.912209\pi$$
$$908$$ 11.4891 0.381280
$$909$$ 0 0
$$910$$ −2.37228 −0.0786404
$$911$$ 22.8832 0.758153 0.379076 0.925365i $$-0.376242\pi$$
0.379076 + 0.925365i $$0.376242\pi$$
$$912$$ 0 0
$$913$$ −14.2337 −0.471066
$$914$$ 12.2337 0.404654
$$915$$ 0 0
$$916$$ 6.00000 0.198246
$$917$$ 11.1168 0.367111
$$918$$ 0 0
$$919$$ −49.2119 −1.62335 −0.811676 0.584108i $$-0.801444\pi$$
−0.811676 + 0.584108i $$0.801444\pi$$
$$920$$ −8.60597 −0.283730
$$921$$ 0 0
$$922$$ −36.6060 −1.20555
$$923$$ 4.74456 0.156169
$$924$$ 0 0
$$925$$ −2.74456 −0.0902407
$$926$$ −17.3505 −0.570174
$$927$$ 0 0
$$928$$ 6.37228 0.209180
$$929$$ 15.2554 0.500515 0.250257 0.968179i $$-0.419485\pi$$
0.250257 + 0.968179i $$0.419485\pi$$
$$930$$ 0 0
$$931$$ 1.62772 0.0533463
$$932$$ −1.48913 −0.0487779
$$933$$ 0 0
$$934$$ −34.0951 −1.11563
$$935$$ −24.6060 −0.804701
$$936$$ 0 0
$$937$$ 24.5109 0.800735 0.400368 0.916355i $$-0.368882\pi$$
0.400368 + 0.916355i $$0.368882\pi$$
$$938$$ −9.48913 −0.309831
$$939$$ 0 0
$$940$$ −2.97825 −0.0971398
$$941$$ 24.7446 0.806650 0.403325 0.915057i $$-0.367854\pi$$
0.403325 + 0.915057i $$0.367854\pi$$
$$942$$ 0 0
$$943$$ 31.7228 1.03304
$$944$$ 2.00000 0.0650945
$$945$$ 0 0
$$946$$ 26.3723 0.857437
$$947$$ −8.13859 −0.264469 −0.132234 0.991218i $$-0.542215\pi$$
−0.132234 + 0.991218i $$0.542215\pi$$
$$948$$ 0 0
$$949$$ 8.37228 0.271776
$$950$$ 1.02175 0.0331499
$$951$$ 0 0
$$952$$ −4.37228 −0.141706
$$953$$ −45.4891 −1.47354 −0.736769 0.676145i $$-0.763649\pi$$
−0.736769 + 0.676145i $$0.763649\pi$$
$$954$$ 0 0
$$955$$ −12.1386 −0.392796
$$956$$ −5.48913 −0.177531
$$957$$ 0 0
$$958$$ 39.3505 1.27136
$$959$$ −1.11684 −0.0360648
$$960$$ 0 0
$$961$$ −8.48913 −0.273843
$$962$$ 4.37228 0.140968
$$963$$ 0 0
$$964$$ −6.00000 −0.193247
$$965$$ −4.74456 −0.152733
$$966$$ 0 0
$$967$$ 24.8832 0.800188 0.400094 0.916474i $$-0.368977\pi$$
0.400094 + 0.916474i $$0.368977\pi$$
$$968$$ −5.37228 −0.172672
$$969$$ 0 0
$$970$$ −17.7663 −0.570442
$$971$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$972$$ 0 0
$$973$$ −0.744563 −0.0238696
$$974$$ −32.4674 −1.04032
$$975$$ 0 0
$$976$$ 5.11684 0.163786
$$977$$ −25.1168 −0.803559 −0.401780 0.915736i $$-0.631608\pi$$
−0.401780 + 0.915736i $$0.631608\pi$$
$$978$$ 0 0
$$979$$ −7.72281 −0.246822
$$980$$ −2.37228 −0.0757797
$$981$$ 0 0
$$982$$ 25.2554 0.805933
$$983$$ 35.3505 1.12751 0.563753 0.825943i $$-0.309357\pi$$
0.563753 + 0.825943i $$0.309357\pi$$
$$984$$ 0 0
$$985$$ −14.2337 −0.453523
$$986$$ 27.8614 0.887288
$$987$$ 0 0
$$988$$ −1.62772 −0.0517846
$$989$$ 40.3288 1.28238
$$990$$ 0 0
$$991$$ 14.2337 0.452148 0.226074 0.974110i $$-0.427411\pi$$
0.226074 + 0.974110i $$0.427411\pi$$
$$992$$ −4.74456 −0.150640
$$993$$ 0 0
$$994$$ 4.74456 0.150488
$$995$$ −28.1386 −0.892053
$$996$$ 0 0
$$997$$ 20.5109 0.649586 0.324793 0.945785i $$-0.394705\pi$$
0.324793 + 0.945785i $$0.394705\pi$$
$$998$$ −20.0000 −0.633089
$$999$$ 0 0
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 1638.2.a.x.1.1 yes 2
3.2 odd 2 1638.2.a.v.1.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
1638.2.a.v.1.2 2 3.2 odd 2
1638.2.a.x.1.1 yes 2 1.1 even 1 trivial