# Properties

 Label 1305.2.a.j.1.2 Level $1305$ Weight $2$ Character 1305.1 Self dual yes Analytic conductor $10.420$ 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] = [1305,2,Mod(1,1305)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1305, 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("1305.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1305 = 3^{2} \cdot 5 \cdot 29$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1305.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: $$10.4204774638$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\zeta_{10})^+$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 1$$ x^2 - x - 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 435) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$-0.618034$$ of defining polynomial Character $$\chi$$ $$=$$ 1305.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+2.23607 q^{2} +3.00000 q^{4} +1.00000 q^{5} +2.00000 q^{7} +2.23607 q^{8} +O(q^{10})$$ $$q+2.23607 q^{2} +3.00000 q^{4} +1.00000 q^{5} +2.00000 q^{7} +2.23607 q^{8} +2.23607 q^{10} +2.00000 q^{11} +2.00000 q^{13} +4.47214 q^{14} -1.00000 q^{16} -4.47214 q^{17} +2.00000 q^{19} +3.00000 q^{20} +4.47214 q^{22} +2.00000 q^{23} +1.00000 q^{25} +4.47214 q^{26} +6.00000 q^{28} +1.00000 q^{29} -2.00000 q^{31} -6.70820 q^{32} -10.0000 q^{34} +2.00000 q^{35} +8.47214 q^{37} +4.47214 q^{38} +2.23607 q^{40} -2.00000 q^{41} +4.00000 q^{43} +6.00000 q^{44} +4.47214 q^{46} -12.9443 q^{47} -3.00000 q^{49} +2.23607 q^{50} +6.00000 q^{52} -2.00000 q^{53} +2.00000 q^{55} +4.47214 q^{56} +2.23607 q^{58} -8.00000 q^{59} +6.94427 q^{61} -4.47214 q^{62} -13.0000 q^{64} +2.00000 q^{65} +2.94427 q^{67} -13.4164 q^{68} +4.47214 q^{70} -4.00000 q^{71} +3.52786 q^{73} +18.9443 q^{74} +6.00000 q^{76} +4.00000 q^{77} -2.94427 q^{79} -1.00000 q^{80} -4.47214 q^{82} +14.9443 q^{83} -4.47214 q^{85} +8.94427 q^{86} +4.47214 q^{88} +6.00000 q^{89} +4.00000 q^{91} +6.00000 q^{92} -28.9443 q^{94} +2.00000 q^{95} -17.4164 q^{97} -6.70820 q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 6 q^{4} + 2 q^{5} + 4 q^{7}+O(q^{10})$$ 2 * q + 6 * q^4 + 2 * q^5 + 4 * q^7 $$2 q + 6 q^{4} + 2 q^{5} + 4 q^{7} + 4 q^{11} + 4 q^{13} - 2 q^{16} + 4 q^{19} + 6 q^{20} + 4 q^{23} + 2 q^{25} + 12 q^{28} + 2 q^{29} - 4 q^{31} - 20 q^{34} + 4 q^{35} + 8 q^{37} - 4 q^{41} + 8 q^{43} + 12 q^{44} - 8 q^{47} - 6 q^{49} + 12 q^{52} - 4 q^{53} + 4 q^{55} - 16 q^{59} - 4 q^{61} - 26 q^{64} + 4 q^{65} - 12 q^{67} - 8 q^{71} + 16 q^{73} + 20 q^{74} + 12 q^{76} + 8 q^{77} + 12 q^{79} - 2 q^{80} + 12 q^{83} + 12 q^{89} + 8 q^{91} + 12 q^{92} - 40 q^{94} + 4 q^{95} - 8 q^{97}+O(q^{100})$$ 2 * q + 6 * q^4 + 2 * q^5 + 4 * q^7 + 4 * q^11 + 4 * q^13 - 2 * q^16 + 4 * q^19 + 6 * q^20 + 4 * q^23 + 2 * q^25 + 12 * q^28 + 2 * q^29 - 4 * q^31 - 20 * q^34 + 4 * q^35 + 8 * q^37 - 4 * q^41 + 8 * q^43 + 12 * q^44 - 8 * q^47 - 6 * q^49 + 12 * q^52 - 4 * q^53 + 4 * q^55 - 16 * q^59 - 4 * q^61 - 26 * q^64 + 4 * q^65 - 12 * q^67 - 8 * q^71 + 16 * q^73 + 20 * q^74 + 12 * q^76 + 8 * q^77 + 12 * q^79 - 2 * q^80 + 12 * q^83 + 12 * q^89 + 8 * q^91 + 12 * q^92 - 40 * q^94 + 4 * q^95 - 8 * q^97

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 2.23607 1.58114 0.790569 0.612372i $$-0.209785\pi$$
0.790569 + 0.612372i $$0.209785\pi$$
$$3$$ 0 0
$$4$$ 3.00000 1.50000
$$5$$ 1.00000 0.447214
$$6$$ 0 0
$$7$$ 2.00000 0.755929 0.377964 0.925820i $$-0.376624\pi$$
0.377964 + 0.925820i $$0.376624\pi$$
$$8$$ 2.23607 0.790569
$$9$$ 0 0
$$10$$ 2.23607 0.707107
$$11$$ 2.00000 0.603023 0.301511 0.953463i $$-0.402509\pi$$
0.301511 + 0.953463i $$0.402509\pi$$
$$12$$ 0 0
$$13$$ 2.00000 0.554700 0.277350 0.960769i $$-0.410544\pi$$
0.277350 + 0.960769i $$0.410544\pi$$
$$14$$ 4.47214 1.19523
$$15$$ 0 0
$$16$$ −1.00000 −0.250000
$$17$$ −4.47214 −1.08465 −0.542326 0.840168i $$-0.682456\pi$$
−0.542326 + 0.840168i $$0.682456\pi$$
$$18$$ 0 0
$$19$$ 2.00000 0.458831 0.229416 0.973329i $$-0.426318\pi$$
0.229416 + 0.973329i $$0.426318\pi$$
$$20$$ 3.00000 0.670820
$$21$$ 0 0
$$22$$ 4.47214 0.953463
$$23$$ 2.00000 0.417029 0.208514 0.978019i $$-0.433137\pi$$
0.208514 + 0.978019i $$0.433137\pi$$
$$24$$ 0 0
$$25$$ 1.00000 0.200000
$$26$$ 4.47214 0.877058
$$27$$ 0 0
$$28$$ 6.00000 1.13389
$$29$$ 1.00000 0.185695
$$30$$ 0 0
$$31$$ −2.00000 −0.359211 −0.179605 0.983739i $$-0.557482\pi$$
−0.179605 + 0.983739i $$0.557482\pi$$
$$32$$ −6.70820 −1.18585
$$33$$ 0 0
$$34$$ −10.0000 −1.71499
$$35$$ 2.00000 0.338062
$$36$$ 0 0
$$37$$ 8.47214 1.39281 0.696405 0.717649i $$-0.254782\pi$$
0.696405 + 0.717649i $$0.254782\pi$$
$$38$$ 4.47214 0.725476
$$39$$ 0 0
$$40$$ 2.23607 0.353553
$$41$$ −2.00000 −0.312348 −0.156174 0.987730i $$-0.549916\pi$$
−0.156174 + 0.987730i $$0.549916\pi$$
$$42$$ 0 0
$$43$$ 4.00000 0.609994 0.304997 0.952353i $$-0.401344\pi$$
0.304997 + 0.952353i $$0.401344\pi$$
$$44$$ 6.00000 0.904534
$$45$$ 0 0
$$46$$ 4.47214 0.659380
$$47$$ −12.9443 −1.88812 −0.944058 0.329779i $$-0.893026\pi$$
−0.944058 + 0.329779i $$0.893026\pi$$
$$48$$ 0 0
$$49$$ −3.00000 −0.428571
$$50$$ 2.23607 0.316228
$$51$$ 0 0
$$52$$ 6.00000 0.832050
$$53$$ −2.00000 −0.274721 −0.137361 0.990521i $$-0.543862\pi$$
−0.137361 + 0.990521i $$0.543862\pi$$
$$54$$ 0 0
$$55$$ 2.00000 0.269680
$$56$$ 4.47214 0.597614
$$57$$ 0 0
$$58$$ 2.23607 0.293610
$$59$$ −8.00000 −1.04151 −0.520756 0.853706i $$-0.674350\pi$$
−0.520756 + 0.853706i $$0.674350\pi$$
$$60$$ 0 0
$$61$$ 6.94427 0.889123 0.444561 0.895748i $$-0.353360\pi$$
0.444561 + 0.895748i $$0.353360\pi$$
$$62$$ −4.47214 −0.567962
$$63$$ 0 0
$$64$$ −13.0000 −1.62500
$$65$$ 2.00000 0.248069
$$66$$ 0 0
$$67$$ 2.94427 0.359700 0.179850 0.983694i $$-0.442439\pi$$
0.179850 + 0.983694i $$0.442439\pi$$
$$68$$ −13.4164 −1.62698
$$69$$ 0 0
$$70$$ 4.47214 0.534522
$$71$$ −4.00000 −0.474713 −0.237356 0.971423i $$-0.576281\pi$$
−0.237356 + 0.971423i $$0.576281\pi$$
$$72$$ 0 0
$$73$$ 3.52786 0.412905 0.206453 0.978457i $$-0.433808\pi$$
0.206453 + 0.978457i $$0.433808\pi$$
$$74$$ 18.9443 2.20223
$$75$$ 0 0
$$76$$ 6.00000 0.688247
$$77$$ 4.00000 0.455842
$$78$$ 0 0
$$79$$ −2.94427 −0.331256 −0.165628 0.986188i $$-0.552965\pi$$
−0.165628 + 0.986188i $$0.552965\pi$$
$$80$$ −1.00000 −0.111803
$$81$$ 0 0
$$82$$ −4.47214 −0.493865
$$83$$ 14.9443 1.64035 0.820173 0.572115i $$-0.193877\pi$$
0.820173 + 0.572115i $$0.193877\pi$$
$$84$$ 0 0
$$85$$ −4.47214 −0.485071
$$86$$ 8.94427 0.964486
$$87$$ 0 0
$$88$$ 4.47214 0.476731
$$89$$ 6.00000 0.635999 0.317999 0.948091i $$-0.396989\pi$$
0.317999 + 0.948091i $$0.396989\pi$$
$$90$$ 0 0
$$91$$ 4.00000 0.419314
$$92$$ 6.00000 0.625543
$$93$$ 0 0
$$94$$ −28.9443 −2.98537
$$95$$ 2.00000 0.205196
$$96$$ 0 0
$$97$$ −17.4164 −1.76837 −0.884184 0.467139i $$-0.845285\pi$$
−0.884184 + 0.467139i $$0.845285\pi$$
$$98$$ −6.70820 −0.677631
$$99$$ 0 0
$$100$$ 3.00000 0.300000
$$101$$ 2.94427 0.292966 0.146483 0.989213i $$-0.453205\pi$$
0.146483 + 0.989213i $$0.453205\pi$$
$$102$$ 0 0
$$103$$ −14.9443 −1.47250 −0.736251 0.676708i $$-0.763406\pi$$
−0.736251 + 0.676708i $$0.763406\pi$$
$$104$$ 4.47214 0.438529
$$105$$ 0 0
$$106$$ −4.47214 −0.434372
$$107$$ −6.94427 −0.671328 −0.335664 0.941982i $$-0.608961\pi$$
−0.335664 + 0.941982i $$0.608961\pi$$
$$108$$ 0 0
$$109$$ 2.00000 0.191565 0.0957826 0.995402i $$-0.469465\pi$$
0.0957826 + 0.995402i $$0.469465\pi$$
$$110$$ 4.47214 0.426401
$$111$$ 0 0
$$112$$ −2.00000 −0.188982
$$113$$ 4.47214 0.420703 0.210352 0.977626i $$-0.432539\pi$$
0.210352 + 0.977626i $$0.432539\pi$$
$$114$$ 0 0
$$115$$ 2.00000 0.186501
$$116$$ 3.00000 0.278543
$$117$$ 0 0
$$118$$ −17.8885 −1.64677
$$119$$ −8.94427 −0.819920
$$120$$ 0 0
$$121$$ −7.00000 −0.636364
$$122$$ 15.5279 1.40583
$$123$$ 0 0
$$124$$ −6.00000 −0.538816
$$125$$ 1.00000 0.0894427
$$126$$ 0 0
$$127$$ −8.94427 −0.793676 −0.396838 0.917889i $$-0.629892\pi$$
−0.396838 + 0.917889i $$0.629892\pi$$
$$128$$ −15.6525 −1.38350
$$129$$ 0 0
$$130$$ 4.47214 0.392232
$$131$$ 10.9443 0.956205 0.478103 0.878304i $$-0.341325\pi$$
0.478103 + 0.878304i $$0.341325\pi$$
$$132$$ 0 0
$$133$$ 4.00000 0.346844
$$134$$ 6.58359 0.568736
$$135$$ 0 0
$$136$$ −10.0000 −0.857493
$$137$$ −12.4721 −1.06557 −0.532783 0.846252i $$-0.678854\pi$$
−0.532783 + 0.846252i $$0.678854\pi$$
$$138$$ 0 0
$$139$$ −17.8885 −1.51729 −0.758643 0.651506i $$-0.774137\pi$$
−0.758643 + 0.651506i $$0.774137\pi$$
$$140$$ 6.00000 0.507093
$$141$$ 0 0
$$142$$ −8.94427 −0.750587
$$143$$ 4.00000 0.334497
$$144$$ 0 0
$$145$$ 1.00000 0.0830455
$$146$$ 7.88854 0.652861
$$147$$ 0 0
$$148$$ 25.4164 2.08922
$$149$$ −15.8885 −1.30164 −0.650820 0.759232i $$-0.725575\pi$$
−0.650820 + 0.759232i $$0.725575\pi$$
$$150$$ 0 0
$$151$$ −4.00000 −0.325515 −0.162758 0.986666i $$-0.552039\pi$$
−0.162758 + 0.986666i $$0.552039\pi$$
$$152$$ 4.47214 0.362738
$$153$$ 0 0
$$154$$ 8.94427 0.720750
$$155$$ −2.00000 −0.160644
$$156$$ 0 0
$$157$$ 7.52786 0.600789 0.300394 0.953815i $$-0.402882\pi$$
0.300394 + 0.953815i $$0.402882\pi$$
$$158$$ −6.58359 −0.523762
$$159$$ 0 0
$$160$$ −6.70820 −0.530330
$$161$$ 4.00000 0.315244
$$162$$ 0 0
$$163$$ 16.0000 1.25322 0.626608 0.779334i $$-0.284443\pi$$
0.626608 + 0.779334i $$0.284443\pi$$
$$164$$ −6.00000 −0.468521
$$165$$ 0 0
$$166$$ 33.4164 2.59362
$$167$$ 10.0000 0.773823 0.386912 0.922117i $$-0.373542\pi$$
0.386912 + 0.922117i $$0.373542\pi$$
$$168$$ 0 0
$$169$$ −9.00000 −0.692308
$$170$$ −10.0000 −0.766965
$$171$$ 0 0
$$172$$ 12.0000 0.914991
$$173$$ 2.94427 0.223849 0.111924 0.993717i $$-0.464299\pi$$
0.111924 + 0.993717i $$0.464299\pi$$
$$174$$ 0 0
$$175$$ 2.00000 0.151186
$$176$$ −2.00000 −0.150756
$$177$$ 0 0
$$178$$ 13.4164 1.00560
$$179$$ −13.8885 −1.03808 −0.519039 0.854750i $$-0.673710\pi$$
−0.519039 + 0.854750i $$0.673710\pi$$
$$180$$ 0 0
$$181$$ −19.8885 −1.47830 −0.739152 0.673539i $$-0.764773\pi$$
−0.739152 + 0.673539i $$0.764773\pi$$
$$182$$ 8.94427 0.662994
$$183$$ 0 0
$$184$$ 4.47214 0.329690
$$185$$ 8.47214 0.622884
$$186$$ 0 0
$$187$$ −8.94427 −0.654070
$$188$$ −38.8328 −2.83217
$$189$$ 0 0
$$190$$ 4.47214 0.324443
$$191$$ 10.0000 0.723575 0.361787 0.932261i $$-0.382167\pi$$
0.361787 + 0.932261i $$0.382167\pi$$
$$192$$ 0 0
$$193$$ −0.472136 −0.0339851 −0.0169925 0.999856i $$-0.505409\pi$$
−0.0169925 + 0.999856i $$0.505409\pi$$
$$194$$ −38.9443 −2.79604
$$195$$ 0 0
$$196$$ −9.00000 −0.642857
$$197$$ −14.9443 −1.06474 −0.532368 0.846513i $$-0.678698\pi$$
−0.532368 + 0.846513i $$0.678698\pi$$
$$198$$ 0 0
$$199$$ 21.8885 1.55164 0.775819 0.630956i $$-0.217337\pi$$
0.775819 + 0.630956i $$0.217337\pi$$
$$200$$ 2.23607 0.158114
$$201$$ 0 0
$$202$$ 6.58359 0.463220
$$203$$ 2.00000 0.140372
$$204$$ 0 0
$$205$$ −2.00000 −0.139686
$$206$$ −33.4164 −2.32823
$$207$$ 0 0
$$208$$ −2.00000 −0.138675
$$209$$ 4.00000 0.276686
$$210$$ 0 0
$$211$$ −1.05573 −0.0726793 −0.0363397 0.999339i $$-0.511570\pi$$
−0.0363397 + 0.999339i $$0.511570\pi$$
$$212$$ −6.00000 −0.412082
$$213$$ 0 0
$$214$$ −15.5279 −1.06146
$$215$$ 4.00000 0.272798
$$216$$ 0 0
$$217$$ −4.00000 −0.271538
$$218$$ 4.47214 0.302891
$$219$$ 0 0
$$220$$ 6.00000 0.404520
$$221$$ −8.94427 −0.601657
$$222$$ 0 0
$$223$$ −22.9443 −1.53646 −0.768231 0.640173i $$-0.778863\pi$$
−0.768231 + 0.640173i $$0.778863\pi$$
$$224$$ −13.4164 −0.896421
$$225$$ 0 0
$$226$$ 10.0000 0.665190
$$227$$ 23.8885 1.58554 0.792769 0.609522i $$-0.208639\pi$$
0.792769 + 0.609522i $$0.208639\pi$$
$$228$$ 0 0
$$229$$ 5.05573 0.334092 0.167046 0.985949i $$-0.446577\pi$$
0.167046 + 0.985949i $$0.446577\pi$$
$$230$$ 4.47214 0.294884
$$231$$ 0 0
$$232$$ 2.23607 0.146805
$$233$$ 18.9443 1.24108 0.620540 0.784175i $$-0.286913\pi$$
0.620540 + 0.784175i $$0.286913\pi$$
$$234$$ 0 0
$$235$$ −12.9443 −0.844391
$$236$$ −24.0000 −1.56227
$$237$$ 0 0
$$238$$ −20.0000 −1.29641
$$239$$ 21.8885 1.41585 0.707926 0.706287i $$-0.249631\pi$$
0.707926 + 0.706287i $$0.249631\pi$$
$$240$$ 0 0
$$241$$ 15.8885 1.02347 0.511736 0.859143i $$-0.329003\pi$$
0.511736 + 0.859143i $$0.329003\pi$$
$$242$$ −15.6525 −1.00618
$$243$$ 0 0
$$244$$ 20.8328 1.33368
$$245$$ −3.00000 −0.191663
$$246$$ 0 0
$$247$$ 4.00000 0.254514
$$248$$ −4.47214 −0.283981
$$249$$ 0 0
$$250$$ 2.23607 0.141421
$$251$$ 18.0000 1.13615 0.568075 0.822977i $$-0.307688\pi$$
0.568075 + 0.822977i $$0.307688\pi$$
$$252$$ 0 0
$$253$$ 4.00000 0.251478
$$254$$ −20.0000 −1.25491
$$255$$ 0 0
$$256$$ −9.00000 −0.562500
$$257$$ −18.0000 −1.12281 −0.561405 0.827541i $$-0.689739\pi$$
−0.561405 + 0.827541i $$0.689739\pi$$
$$258$$ 0 0
$$259$$ 16.9443 1.05287
$$260$$ 6.00000 0.372104
$$261$$ 0 0
$$262$$ 24.4721 1.51189
$$263$$ 30.8328 1.90123 0.950616 0.310368i $$-0.100452\pi$$
0.950616 + 0.310368i $$0.100452\pi$$
$$264$$ 0 0
$$265$$ −2.00000 −0.122859
$$266$$ 8.94427 0.548408
$$267$$ 0 0
$$268$$ 8.83282 0.539550
$$269$$ −30.9443 −1.88671 −0.943353 0.331791i $$-0.892347\pi$$
−0.943353 + 0.331791i $$0.892347\pi$$
$$270$$ 0 0
$$271$$ −9.05573 −0.550096 −0.275048 0.961430i $$-0.588694\pi$$
−0.275048 + 0.961430i $$0.588694\pi$$
$$272$$ 4.47214 0.271163
$$273$$ 0 0
$$274$$ −27.8885 −1.68481
$$275$$ 2.00000 0.120605
$$276$$ 0 0
$$277$$ −10.9443 −0.657578 −0.328789 0.944403i $$-0.606640\pi$$
−0.328789 + 0.944403i $$0.606640\pi$$
$$278$$ −40.0000 −2.39904
$$279$$ 0 0
$$280$$ 4.47214 0.267261
$$281$$ 10.0000 0.596550 0.298275 0.954480i $$-0.403589\pi$$
0.298275 + 0.954480i $$0.403589\pi$$
$$282$$ 0 0
$$283$$ −1.05573 −0.0627565 −0.0313783 0.999508i $$-0.509990\pi$$
−0.0313783 + 0.999508i $$0.509990\pi$$
$$284$$ −12.0000 −0.712069
$$285$$ 0 0
$$286$$ 8.94427 0.528886
$$287$$ −4.00000 −0.236113
$$288$$ 0 0
$$289$$ 3.00000 0.176471
$$290$$ 2.23607 0.131306
$$291$$ 0 0
$$292$$ 10.5836 0.619358
$$293$$ −15.5279 −0.907148 −0.453574 0.891219i $$-0.649851\pi$$
−0.453574 + 0.891219i $$0.649851\pi$$
$$294$$ 0 0
$$295$$ −8.00000 −0.465778
$$296$$ 18.9443 1.10111
$$297$$ 0 0
$$298$$ −35.5279 −2.05807
$$299$$ 4.00000 0.231326
$$300$$ 0 0
$$301$$ 8.00000 0.461112
$$302$$ −8.94427 −0.514685
$$303$$ 0 0
$$304$$ −2.00000 −0.114708
$$305$$ 6.94427 0.397628
$$306$$ 0 0
$$307$$ 8.00000 0.456584 0.228292 0.973593i $$-0.426686\pi$$
0.228292 + 0.973593i $$0.426686\pi$$
$$308$$ 12.0000 0.683763
$$309$$ 0 0
$$310$$ −4.47214 −0.254000
$$311$$ −20.8328 −1.18132 −0.590660 0.806920i $$-0.701133\pi$$
−0.590660 + 0.806920i $$0.701133\pi$$
$$312$$ 0 0
$$313$$ −2.94427 −0.166420 −0.0832100 0.996532i $$-0.526517\pi$$
−0.0832100 + 0.996532i $$0.526517\pi$$
$$314$$ 16.8328 0.949931
$$315$$ 0 0
$$316$$ −8.83282 −0.496885
$$317$$ 9.41641 0.528878 0.264439 0.964402i $$-0.414813\pi$$
0.264439 + 0.964402i $$0.414813\pi$$
$$318$$ 0 0
$$319$$ 2.00000 0.111979
$$320$$ −13.0000 −0.726722
$$321$$ 0 0
$$322$$ 8.94427 0.498445
$$323$$ −8.94427 −0.497673
$$324$$ 0 0
$$325$$ 2.00000 0.110940
$$326$$ 35.7771 1.98151
$$327$$ 0 0
$$328$$ −4.47214 −0.246932
$$329$$ −25.8885 −1.42728
$$330$$ 0 0
$$331$$ 19.8885 1.09317 0.546587 0.837403i $$-0.315927\pi$$
0.546587 + 0.837403i $$0.315927\pi$$
$$332$$ 44.8328 2.46052
$$333$$ 0 0
$$334$$ 22.3607 1.22352
$$335$$ 2.94427 0.160863
$$336$$ 0 0
$$337$$ 34.3607 1.87175 0.935873 0.352338i $$-0.114613\pi$$
0.935873 + 0.352338i $$0.114613\pi$$
$$338$$ −20.1246 −1.09463
$$339$$ 0 0
$$340$$ −13.4164 −0.727607
$$341$$ −4.00000 −0.216612
$$342$$ 0 0
$$343$$ −20.0000 −1.07990
$$344$$ 8.94427 0.482243
$$345$$ 0 0
$$346$$ 6.58359 0.353936
$$347$$ 30.9443 1.66118 0.830588 0.556888i $$-0.188005\pi$$
0.830588 + 0.556888i $$0.188005\pi$$
$$348$$ 0 0
$$349$$ −7.88854 −0.422264 −0.211132 0.977458i $$-0.567715\pi$$
−0.211132 + 0.977458i $$0.567715\pi$$
$$350$$ 4.47214 0.239046
$$351$$ 0 0
$$352$$ −13.4164 −0.715097
$$353$$ −30.9443 −1.64700 −0.823499 0.567318i $$-0.807981\pi$$
−0.823499 + 0.567318i $$0.807981\pi$$
$$354$$ 0 0
$$355$$ −4.00000 −0.212298
$$356$$ 18.0000 0.953998
$$357$$ 0 0
$$358$$ −31.0557 −1.64135
$$359$$ −6.94427 −0.366505 −0.183252 0.983066i $$-0.558663\pi$$
−0.183252 + 0.983066i $$0.558663\pi$$
$$360$$ 0 0
$$361$$ −15.0000 −0.789474
$$362$$ −44.4721 −2.33740
$$363$$ 0 0
$$364$$ 12.0000 0.628971
$$365$$ 3.52786 0.184657
$$366$$ 0 0
$$367$$ 7.05573 0.368306 0.184153 0.982898i $$-0.441046\pi$$
0.184153 + 0.982898i $$0.441046\pi$$
$$368$$ −2.00000 −0.104257
$$369$$ 0 0
$$370$$ 18.9443 0.984866
$$371$$ −4.00000 −0.207670
$$372$$ 0 0
$$373$$ 19.8885 1.02979 0.514895 0.857253i $$-0.327831\pi$$
0.514895 + 0.857253i $$0.327831\pi$$
$$374$$ −20.0000 −1.03418
$$375$$ 0 0
$$376$$ −28.9443 −1.49269
$$377$$ 2.00000 0.103005
$$378$$ 0 0
$$379$$ −1.05573 −0.0542291 −0.0271146 0.999632i $$-0.508632\pi$$
−0.0271146 + 0.999632i $$0.508632\pi$$
$$380$$ 6.00000 0.307794
$$381$$ 0 0
$$382$$ 22.3607 1.14407
$$383$$ 17.0557 0.871507 0.435753 0.900066i $$-0.356482\pi$$
0.435753 + 0.900066i $$0.356482\pi$$
$$384$$ 0 0
$$385$$ 4.00000 0.203859
$$386$$ −1.05573 −0.0537351
$$387$$ 0 0
$$388$$ −52.2492 −2.65255
$$389$$ 28.8328 1.46188 0.730941 0.682441i $$-0.239081\pi$$
0.730941 + 0.682441i $$0.239081\pi$$
$$390$$ 0 0
$$391$$ −8.94427 −0.452331
$$392$$ −6.70820 −0.338815
$$393$$ 0 0
$$394$$ −33.4164 −1.68349
$$395$$ −2.94427 −0.148142
$$396$$ 0 0
$$397$$ 19.8885 0.998177 0.499089 0.866551i $$-0.333668\pi$$
0.499089 + 0.866551i $$0.333668\pi$$
$$398$$ 48.9443 2.45335
$$399$$ 0 0
$$400$$ −1.00000 −0.0500000
$$401$$ 23.8885 1.19294 0.596468 0.802637i $$-0.296570\pi$$
0.596468 + 0.802637i $$0.296570\pi$$
$$402$$ 0 0
$$403$$ −4.00000 −0.199254
$$404$$ 8.83282 0.439449
$$405$$ 0 0
$$406$$ 4.47214 0.221948
$$407$$ 16.9443 0.839896
$$408$$ 0 0
$$409$$ −30.0000 −1.48340 −0.741702 0.670729i $$-0.765981\pi$$
−0.741702 + 0.670729i $$0.765981\pi$$
$$410$$ −4.47214 −0.220863
$$411$$ 0 0
$$412$$ −44.8328 −2.20875
$$413$$ −16.0000 −0.787309
$$414$$ 0 0
$$415$$ 14.9443 0.733585
$$416$$ −13.4164 −0.657794
$$417$$ 0 0
$$418$$ 8.94427 0.437479
$$419$$ 4.00000 0.195413 0.0977064 0.995215i $$-0.468849\pi$$
0.0977064 + 0.995215i $$0.468849\pi$$
$$420$$ 0 0
$$421$$ 24.8328 1.21028 0.605139 0.796120i $$-0.293118\pi$$
0.605139 + 0.796120i $$0.293118\pi$$
$$422$$ −2.36068 −0.114916
$$423$$ 0 0
$$424$$ −4.47214 −0.217186
$$425$$ −4.47214 −0.216930
$$426$$ 0 0
$$427$$ 13.8885 0.672114
$$428$$ −20.8328 −1.00699
$$429$$ 0 0
$$430$$ 8.94427 0.431331
$$431$$ 8.00000 0.385346 0.192673 0.981263i $$-0.438284\pi$$
0.192673 + 0.981263i $$0.438284\pi$$
$$432$$ 0 0
$$433$$ 4.47214 0.214917 0.107459 0.994210i $$-0.465729\pi$$
0.107459 + 0.994210i $$0.465729\pi$$
$$434$$ −8.94427 −0.429339
$$435$$ 0 0
$$436$$ 6.00000 0.287348
$$437$$ 4.00000 0.191346
$$438$$ 0 0
$$439$$ 8.00000 0.381819 0.190910 0.981608i $$-0.438856\pi$$
0.190910 + 0.981608i $$0.438856\pi$$
$$440$$ 4.47214 0.213201
$$441$$ 0 0
$$442$$ −20.0000 −0.951303
$$443$$ 24.0000 1.14027 0.570137 0.821549i $$-0.306890\pi$$
0.570137 + 0.821549i $$0.306890\pi$$
$$444$$ 0 0
$$445$$ 6.00000 0.284427
$$446$$ −51.3050 −2.42936
$$447$$ 0 0
$$448$$ −26.0000 −1.22838
$$449$$ 9.05573 0.427366 0.213683 0.976903i $$-0.431454\pi$$
0.213683 + 0.976903i $$0.431454\pi$$
$$450$$ 0 0
$$451$$ −4.00000 −0.188353
$$452$$ 13.4164 0.631055
$$453$$ 0 0
$$454$$ 53.4164 2.50696
$$455$$ 4.00000 0.187523
$$456$$ 0 0
$$457$$ 11.8885 0.556123 0.278061 0.960563i $$-0.410308\pi$$
0.278061 + 0.960563i $$0.410308\pi$$
$$458$$ 11.3050 0.528246
$$459$$ 0 0
$$460$$ 6.00000 0.279751
$$461$$ 10.9443 0.509726 0.254863 0.966977i $$-0.417970\pi$$
0.254863 + 0.966977i $$0.417970\pi$$
$$462$$ 0 0
$$463$$ 32.8328 1.52587 0.762935 0.646475i $$-0.223758\pi$$
0.762935 + 0.646475i $$0.223758\pi$$
$$464$$ −1.00000 −0.0464238
$$465$$ 0 0
$$466$$ 42.3607 1.96232
$$467$$ 6.11146 0.282804 0.141402 0.989952i $$-0.454839\pi$$
0.141402 + 0.989952i $$0.454839\pi$$
$$468$$ 0 0
$$469$$ 5.88854 0.271908
$$470$$ −28.9443 −1.33510
$$471$$ 0 0
$$472$$ −17.8885 −0.823387
$$473$$ 8.00000 0.367840
$$474$$ 0 0
$$475$$ 2.00000 0.0917663
$$476$$ −26.8328 −1.22988
$$477$$ 0 0
$$478$$ 48.9443 2.23866
$$479$$ 26.0000 1.18797 0.593985 0.804476i $$-0.297554\pi$$
0.593985 + 0.804476i $$0.297554\pi$$
$$480$$ 0 0
$$481$$ 16.9443 0.772592
$$482$$ 35.5279 1.61825
$$483$$ 0 0
$$484$$ −21.0000 −0.954545
$$485$$ −17.4164 −0.790838
$$486$$ 0 0
$$487$$ 27.8885 1.26375 0.631875 0.775070i $$-0.282285\pi$$
0.631875 + 0.775070i $$0.282285\pi$$
$$488$$ 15.5279 0.702913
$$489$$ 0 0
$$490$$ −6.70820 −0.303046
$$491$$ 10.9443 0.493908 0.246954 0.969027i $$-0.420570\pi$$
0.246954 + 0.969027i $$0.420570\pi$$
$$492$$ 0 0
$$493$$ −4.47214 −0.201415
$$494$$ 8.94427 0.402422
$$495$$ 0 0
$$496$$ 2.00000 0.0898027
$$497$$ −8.00000 −0.358849
$$498$$ 0 0
$$499$$ 20.0000 0.895323 0.447661 0.894203i $$-0.352257\pi$$
0.447661 + 0.894203i $$0.352257\pi$$
$$500$$ 3.00000 0.134164
$$501$$ 0 0
$$502$$ 40.2492 1.79641
$$503$$ −18.8328 −0.839714 −0.419857 0.907590i $$-0.637920\pi$$
−0.419857 + 0.907590i $$0.637920\pi$$
$$504$$ 0 0
$$505$$ 2.94427 0.131018
$$506$$ 8.94427 0.397621
$$507$$ 0 0
$$508$$ −26.8328 −1.19051
$$509$$ −34.0000 −1.50702 −0.753512 0.657434i $$-0.771642\pi$$
−0.753512 + 0.657434i $$0.771642\pi$$
$$510$$ 0 0
$$511$$ 7.05573 0.312127
$$512$$ 11.1803 0.494106
$$513$$ 0 0
$$514$$ −40.2492 −1.77532
$$515$$ −14.9443 −0.658523
$$516$$ 0 0
$$517$$ −25.8885 −1.13858
$$518$$ 37.8885 1.66473
$$519$$ 0 0
$$520$$ 4.47214 0.196116
$$521$$ −7.88854 −0.345603 −0.172802 0.984957i $$-0.555282\pi$$
−0.172802 + 0.984957i $$0.555282\pi$$
$$522$$ 0 0
$$523$$ −38.9443 −1.70291 −0.851457 0.524424i $$-0.824281\pi$$
−0.851457 + 0.524424i $$0.824281\pi$$
$$524$$ 32.8328 1.43431
$$525$$ 0 0
$$526$$ 68.9443 3.00611
$$527$$ 8.94427 0.389619
$$528$$ 0 0
$$529$$ −19.0000 −0.826087
$$530$$ −4.47214 −0.194257
$$531$$ 0 0
$$532$$ 12.0000 0.520266
$$533$$ −4.00000 −0.173259
$$534$$ 0 0
$$535$$ −6.94427 −0.300227
$$536$$ 6.58359 0.284368
$$537$$ 0 0
$$538$$ −69.1935 −2.98314
$$539$$ −6.00000 −0.258438
$$540$$ 0 0
$$541$$ 0.111456 0.00479188 0.00239594 0.999997i $$-0.499237\pi$$
0.00239594 + 0.999997i $$0.499237\pi$$
$$542$$ −20.2492 −0.869779
$$543$$ 0 0
$$544$$ 30.0000 1.28624
$$545$$ 2.00000 0.0856706
$$546$$ 0 0
$$547$$ −6.00000 −0.256541 −0.128271 0.991739i $$-0.540943\pi$$
−0.128271 + 0.991739i $$0.540943\pi$$
$$548$$ −37.4164 −1.59835
$$549$$ 0 0
$$550$$ 4.47214 0.190693
$$551$$ 2.00000 0.0852029
$$552$$ 0 0
$$553$$ −5.88854 −0.250406
$$554$$ −24.4721 −1.03972
$$555$$ 0 0
$$556$$ −53.6656 −2.27593
$$557$$ −35.8885 −1.52065 −0.760323 0.649545i $$-0.774959\pi$$
−0.760323 + 0.649545i $$0.774959\pi$$
$$558$$ 0 0
$$559$$ 8.00000 0.338364
$$560$$ −2.00000 −0.0845154
$$561$$ 0 0
$$562$$ 22.3607 0.943228
$$563$$ 35.7771 1.50782 0.753912 0.656975i $$-0.228164\pi$$
0.753912 + 0.656975i $$0.228164\pi$$
$$564$$ 0 0
$$565$$ 4.47214 0.188144
$$566$$ −2.36068 −0.0992268
$$567$$ 0 0
$$568$$ −8.94427 −0.375293
$$569$$ 34.9443 1.46494 0.732470 0.680799i $$-0.238367\pi$$
0.732470 + 0.680799i $$0.238367\pi$$
$$570$$ 0 0
$$571$$ −45.8885 −1.92038 −0.960188 0.279355i $$-0.909879\pi$$
−0.960188 + 0.279355i $$0.909879\pi$$
$$572$$ 12.0000 0.501745
$$573$$ 0 0
$$574$$ −8.94427 −0.373327
$$575$$ 2.00000 0.0834058
$$576$$ 0 0
$$577$$ −16.4721 −0.685744 −0.342872 0.939382i $$-0.611400\pi$$
−0.342872 + 0.939382i $$0.611400\pi$$
$$578$$ 6.70820 0.279024
$$579$$ 0 0
$$580$$ 3.00000 0.124568
$$581$$ 29.8885 1.23999
$$582$$ 0 0
$$583$$ −4.00000 −0.165663
$$584$$ 7.88854 0.326430
$$585$$ 0 0
$$586$$ −34.7214 −1.43433
$$587$$ −0.111456 −0.00460029 −0.00230014 0.999997i $$-0.500732\pi$$
−0.00230014 + 0.999997i $$0.500732\pi$$
$$588$$ 0 0
$$589$$ −4.00000 −0.164817
$$590$$ −17.8885 −0.736460
$$591$$ 0 0
$$592$$ −8.47214 −0.348203
$$593$$ 26.9443 1.10647 0.553234 0.833026i $$-0.313393\pi$$
0.553234 + 0.833026i $$0.313393\pi$$
$$594$$ 0 0
$$595$$ −8.94427 −0.366679
$$596$$ −47.6656 −1.95246
$$597$$ 0 0
$$598$$ 8.94427 0.365758
$$599$$ −0.111456 −0.00455398 −0.00227699 0.999997i $$-0.500725\pi$$
−0.00227699 + 0.999997i $$0.500725\pi$$
$$600$$ 0 0
$$601$$ −18.9443 −0.772753 −0.386376 0.922341i $$-0.626273\pi$$
−0.386376 + 0.922341i $$0.626273\pi$$
$$602$$ 17.8885 0.729083
$$603$$ 0 0
$$604$$ −12.0000 −0.488273
$$605$$ −7.00000 −0.284590
$$606$$ 0 0
$$607$$ 36.9443 1.49952 0.749761 0.661709i $$-0.230169\pi$$
0.749761 + 0.661709i $$0.230169\pi$$
$$608$$ −13.4164 −0.544107
$$609$$ 0 0
$$610$$ 15.5279 0.628705
$$611$$ −25.8885 −1.04734
$$612$$ 0 0
$$613$$ 30.9443 1.24983 0.624914 0.780694i $$-0.285134\pi$$
0.624914 + 0.780694i $$0.285134\pi$$
$$614$$ 17.8885 0.721923
$$615$$ 0 0
$$616$$ 8.94427 0.360375
$$617$$ 15.5279 0.625128 0.312564 0.949897i $$-0.398812\pi$$
0.312564 + 0.949897i $$0.398812\pi$$
$$618$$ 0 0
$$619$$ 1.05573 0.0424333 0.0212166 0.999775i $$-0.493246\pi$$
0.0212166 + 0.999775i $$0.493246\pi$$
$$620$$ −6.00000 −0.240966
$$621$$ 0 0
$$622$$ −46.5836 −1.86783
$$623$$ 12.0000 0.480770
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ −6.58359 −0.263133
$$627$$ 0 0
$$628$$ 22.5836 0.901183
$$629$$ −37.8885 −1.51072
$$630$$ 0 0
$$631$$ −47.7771 −1.90198 −0.950988 0.309228i $$-0.899929\pi$$
−0.950988 + 0.309228i $$0.899929\pi$$
$$632$$ −6.58359 −0.261881
$$633$$ 0 0
$$634$$ 21.0557 0.836230
$$635$$ −8.94427 −0.354943
$$636$$ 0 0
$$637$$ −6.00000 −0.237729
$$638$$ 4.47214 0.177054
$$639$$ 0 0
$$640$$ −15.6525 −0.618718
$$641$$ −35.8885 −1.41751 −0.708756 0.705454i $$-0.750743\pi$$
−0.708756 + 0.705454i $$0.750743\pi$$
$$642$$ 0 0
$$643$$ 30.9443 1.22032 0.610161 0.792277i $$-0.291105\pi$$
0.610161 + 0.792277i $$0.291105\pi$$
$$644$$ 12.0000 0.472866
$$645$$ 0 0
$$646$$ −20.0000 −0.786889
$$647$$ −9.05573 −0.356017 −0.178009 0.984029i $$-0.556966\pi$$
−0.178009 + 0.984029i $$0.556966\pi$$
$$648$$ 0 0
$$649$$ −16.0000 −0.628055
$$650$$ 4.47214 0.175412
$$651$$ 0 0
$$652$$ 48.0000 1.87983
$$653$$ 23.3050 0.911993 0.455997 0.889982i $$-0.349283\pi$$
0.455997 + 0.889982i $$0.349283\pi$$
$$654$$ 0 0
$$655$$ 10.9443 0.427628
$$656$$ 2.00000 0.0780869
$$657$$ 0 0
$$658$$ −57.8885 −2.25673
$$659$$ −8.83282 −0.344078 −0.172039 0.985090i $$-0.555035\pi$$
−0.172039 + 0.985090i $$0.555035\pi$$
$$660$$ 0 0
$$661$$ 15.8885 0.617993 0.308996 0.951063i $$-0.400007\pi$$
0.308996 + 0.951063i $$0.400007\pi$$
$$662$$ 44.4721 1.72846
$$663$$ 0 0
$$664$$ 33.4164 1.29681
$$665$$ 4.00000 0.155113
$$666$$ 0 0
$$667$$ 2.00000 0.0774403
$$668$$ 30.0000 1.16073
$$669$$ 0 0
$$670$$ 6.58359 0.254346
$$671$$ 13.8885 0.536161
$$672$$ 0 0
$$673$$ 46.9443 1.80957 0.904784 0.425870i $$-0.140032\pi$$
0.904784 + 0.425870i $$0.140032\pi$$
$$674$$ 76.8328 2.95949
$$675$$ 0 0
$$676$$ −27.0000 −1.03846
$$677$$ −31.5279 −1.21171 −0.605857 0.795573i $$-0.707170\pi$$
−0.605857 + 0.795573i $$0.707170\pi$$
$$678$$ 0 0
$$679$$ −34.8328 −1.33676
$$680$$ −10.0000 −0.383482
$$681$$ 0 0
$$682$$ −8.94427 −0.342494
$$683$$ 45.7771 1.75161 0.875806 0.482664i $$-0.160331\pi$$
0.875806 + 0.482664i $$0.160331\pi$$
$$684$$ 0 0
$$685$$ −12.4721 −0.476536
$$686$$ −44.7214 −1.70747
$$687$$ 0 0
$$688$$ −4.00000 −0.152499
$$689$$ −4.00000 −0.152388
$$690$$ 0 0
$$691$$ −2.11146 −0.0803236 −0.0401618 0.999193i $$-0.512787\pi$$
−0.0401618 + 0.999193i $$0.512787\pi$$
$$692$$ 8.83282 0.335773
$$693$$ 0 0
$$694$$ 69.1935 2.62655
$$695$$ −17.8885 −0.678551
$$696$$ 0 0
$$697$$ 8.94427 0.338788
$$698$$ −17.6393 −0.667658
$$699$$ 0 0
$$700$$ 6.00000 0.226779
$$701$$ 3.88854 0.146868 0.0734341 0.997300i $$-0.476604\pi$$
0.0734341 + 0.997300i $$0.476604\pi$$
$$702$$ 0 0
$$703$$ 16.9443 0.639065
$$704$$ −26.0000 −0.979912
$$705$$ 0 0
$$706$$ −69.1935 −2.60413
$$707$$ 5.88854 0.221461
$$708$$ 0 0
$$709$$ −41.7771 −1.56897 −0.784486 0.620147i $$-0.787073\pi$$
−0.784486 + 0.620147i $$0.787073\pi$$
$$710$$ −8.94427 −0.335673
$$711$$ 0 0
$$712$$ 13.4164 0.502801
$$713$$ −4.00000 −0.149801
$$714$$ 0 0
$$715$$ 4.00000 0.149592
$$716$$ −41.6656 −1.55712
$$717$$ 0 0
$$718$$ −15.5279 −0.579495
$$719$$ −39.7771 −1.48344 −0.741718 0.670712i $$-0.765988\pi$$
−0.741718 + 0.670712i $$0.765988\pi$$
$$720$$ 0 0
$$721$$ −29.8885 −1.11311
$$722$$ −33.5410 −1.24827
$$723$$ 0 0
$$724$$ −59.6656 −2.21746
$$725$$ 1.00000 0.0371391
$$726$$ 0 0
$$727$$ 50.8328 1.88528 0.942642 0.333804i $$-0.108332\pi$$
0.942642 + 0.333804i $$0.108332\pi$$
$$728$$ 8.94427 0.331497
$$729$$ 0 0
$$730$$ 7.88854 0.291968
$$731$$ −17.8885 −0.661632
$$732$$ 0 0
$$733$$ 49.1935 1.81700 0.908502 0.417881i $$-0.137227\pi$$
0.908502 + 0.417881i $$0.137227\pi$$
$$734$$ 15.7771 0.582343
$$735$$ 0 0
$$736$$ −13.4164 −0.494535
$$737$$ 5.88854 0.216907
$$738$$ 0 0
$$739$$ −23.8885 −0.878754 −0.439377 0.898303i $$-0.644801\pi$$
−0.439377 + 0.898303i $$0.644801\pi$$
$$740$$ 25.4164 0.934326
$$741$$ 0 0
$$742$$ −8.94427 −0.328355
$$743$$ 0.944272 0.0346420 0.0173210 0.999850i $$-0.494486\pi$$
0.0173210 + 0.999850i $$0.494486\pi$$
$$744$$ 0 0
$$745$$ −15.8885 −0.582111
$$746$$ 44.4721 1.62824
$$747$$ 0 0
$$748$$ −26.8328 −0.981105
$$749$$ −13.8885 −0.507476
$$750$$ 0 0
$$751$$ −6.00000 −0.218943 −0.109472 0.993990i $$-0.534916\pi$$
−0.109472 + 0.993990i $$0.534916\pi$$
$$752$$ 12.9443 0.472029
$$753$$ 0 0
$$754$$ 4.47214 0.162866
$$755$$ −4.00000 −0.145575
$$756$$ 0 0
$$757$$ −21.4164 −0.778393 −0.389196 0.921155i $$-0.627247\pi$$
−0.389196 + 0.921155i $$0.627247\pi$$
$$758$$ −2.36068 −0.0857438
$$759$$ 0 0
$$760$$ 4.47214 0.162221
$$761$$ 29.7771 1.07942 0.539709 0.841851i $$-0.318534\pi$$
0.539709 + 0.841851i $$0.318534\pi$$
$$762$$ 0 0
$$763$$ 4.00000 0.144810
$$764$$ 30.0000 1.08536
$$765$$ 0 0
$$766$$ 38.1378 1.37797
$$767$$ −16.0000 −0.577727
$$768$$ 0 0
$$769$$ 5.05573 0.182314 0.0911571 0.995837i $$-0.470943\pi$$
0.0911571 + 0.995837i $$0.470943\pi$$
$$770$$ 8.94427 0.322329
$$771$$ 0 0
$$772$$ −1.41641 −0.0509776
$$773$$ −50.3607 −1.81135 −0.905674 0.423975i $$-0.860634\pi$$
−0.905674 + 0.423975i $$0.860634\pi$$
$$774$$ 0 0
$$775$$ −2.00000 −0.0718421
$$776$$ −38.9443 −1.39802
$$777$$ 0 0
$$778$$ 64.4721 2.31144
$$779$$ −4.00000 −0.143315
$$780$$ 0 0
$$781$$ −8.00000 −0.286263
$$782$$ −20.0000 −0.715199
$$783$$ 0 0
$$784$$ 3.00000 0.107143
$$785$$ 7.52786 0.268681
$$786$$ 0 0
$$787$$ −11.8885 −0.423781 −0.211890 0.977293i $$-0.567962\pi$$
−0.211890 + 0.977293i $$0.567962\pi$$
$$788$$ −44.8328 −1.59710
$$789$$ 0 0
$$790$$ −6.58359 −0.234234
$$791$$ 8.94427 0.318022
$$792$$ 0 0
$$793$$ 13.8885 0.493197
$$794$$ 44.4721 1.57826
$$795$$ 0 0
$$796$$ 65.6656 2.32746
$$797$$ 13.4164 0.475234 0.237617 0.971359i $$-0.423634\pi$$
0.237617 + 0.971359i $$0.423634\pi$$
$$798$$ 0 0
$$799$$ 57.8885 2.04795
$$800$$ −6.70820 −0.237171
$$801$$ 0 0
$$802$$ 53.4164 1.88620
$$803$$ 7.05573 0.248991
$$804$$ 0 0
$$805$$ 4.00000 0.140981
$$806$$ −8.94427 −0.315049
$$807$$ 0 0
$$808$$ 6.58359 0.231610
$$809$$ −0.111456 −0.00391859 −0.00195930 0.999998i $$-0.500624\pi$$
−0.00195930 + 0.999998i $$0.500624\pi$$
$$810$$ 0 0
$$811$$ −41.8885 −1.47091 −0.735453 0.677576i $$-0.763031\pi$$
−0.735453 + 0.677576i $$0.763031\pi$$
$$812$$ 6.00000 0.210559
$$813$$ 0 0
$$814$$ 37.8885 1.32799
$$815$$ 16.0000 0.560456
$$816$$ 0 0
$$817$$ 8.00000 0.279885
$$818$$ −67.0820 −2.34547
$$819$$ 0 0
$$820$$ −6.00000 −0.209529
$$821$$ −14.0000 −0.488603 −0.244302 0.969699i $$-0.578559\pi$$
−0.244302 + 0.969699i $$0.578559\pi$$
$$822$$ 0 0
$$823$$ 28.9443 1.00893 0.504467 0.863431i $$-0.331689\pi$$
0.504467 + 0.863431i $$0.331689\pi$$
$$824$$ −33.4164 −1.16412
$$825$$ 0 0
$$826$$ −35.7771 −1.24484
$$827$$ −29.8885 −1.03933 −0.519663 0.854371i $$-0.673943\pi$$
−0.519663 + 0.854371i $$0.673943\pi$$
$$828$$ 0 0
$$829$$ 29.7771 1.03420 0.517101 0.855925i $$-0.327011\pi$$
0.517101 + 0.855925i $$0.327011\pi$$
$$830$$ 33.4164 1.15990
$$831$$ 0 0
$$832$$ −26.0000 −0.901388
$$833$$ 13.4164 0.464851
$$834$$ 0 0
$$835$$ 10.0000 0.346064
$$836$$ 12.0000 0.415029
$$837$$ 0 0
$$838$$ 8.94427 0.308975
$$839$$ −12.1115 −0.418134 −0.209067 0.977901i $$-0.567043\pi$$
−0.209067 + 0.977901i $$0.567043\pi$$
$$840$$ 0 0
$$841$$ 1.00000 0.0344828
$$842$$ 55.5279 1.91362
$$843$$ 0 0
$$844$$ −3.16718 −0.109019
$$845$$ −9.00000 −0.309609
$$846$$ 0 0
$$847$$ −14.0000 −0.481046
$$848$$ 2.00000 0.0686803
$$849$$ 0 0
$$850$$ −10.0000 −0.342997
$$851$$ 16.9443 0.580842
$$852$$ 0 0
$$853$$ 27.5279 0.942536 0.471268 0.881990i $$-0.343796\pi$$
0.471268 + 0.881990i $$0.343796\pi$$
$$854$$ 31.0557 1.06271
$$855$$ 0 0
$$856$$ −15.5279 −0.530731
$$857$$ −26.0000 −0.888143 −0.444072 0.895991i $$-0.646466\pi$$
−0.444072 + 0.895991i $$0.646466\pi$$
$$858$$ 0 0
$$859$$ −38.9443 −1.32876 −0.664381 0.747394i $$-0.731305\pi$$
−0.664381 + 0.747394i $$0.731305\pi$$
$$860$$ 12.0000 0.409197
$$861$$ 0 0
$$862$$ 17.8885 0.609286
$$863$$ 34.9443 1.18952 0.594758 0.803904i $$-0.297248\pi$$
0.594758 + 0.803904i $$0.297248\pi$$
$$864$$ 0 0
$$865$$ 2.94427 0.100108
$$866$$ 10.0000 0.339814
$$867$$ 0 0
$$868$$ −12.0000 −0.407307
$$869$$ −5.88854 −0.199755
$$870$$ 0 0
$$871$$ 5.88854 0.199526
$$872$$ 4.47214 0.151446
$$873$$ 0 0
$$874$$ 8.94427 0.302545
$$875$$ 2.00000 0.0676123
$$876$$ 0 0
$$877$$ −9.05573 −0.305790 −0.152895 0.988242i $$-0.548860\pi$$
−0.152895 + 0.988242i $$0.548860\pi$$
$$878$$ 17.8885 0.603709
$$879$$ 0 0
$$880$$ −2.00000 −0.0674200
$$881$$ −35.8885 −1.20912 −0.604558 0.796561i $$-0.706650\pi$$
−0.604558 + 0.796561i $$0.706650\pi$$
$$882$$ 0 0
$$883$$ 43.8885 1.47697 0.738484 0.674271i $$-0.235542\pi$$
0.738484 + 0.674271i $$0.235542\pi$$
$$884$$ −26.8328 −0.902485
$$885$$ 0 0
$$886$$ 53.6656 1.80293
$$887$$ 16.9443 0.568933 0.284466 0.958686i $$-0.408184\pi$$
0.284466 + 0.958686i $$0.408184\pi$$
$$888$$ 0 0
$$889$$ −17.8885 −0.599963
$$890$$ 13.4164 0.449719
$$891$$ 0 0
$$892$$ −68.8328 −2.30469
$$893$$ −25.8885 −0.866327
$$894$$ 0 0
$$895$$ −13.8885 −0.464243
$$896$$ −31.3050 −1.04583
$$897$$ 0 0
$$898$$ 20.2492 0.675725
$$899$$ −2.00000 −0.0667037
$$900$$ 0 0
$$901$$ 8.94427 0.297977
$$902$$ −8.94427 −0.297812
$$903$$ 0 0
$$904$$ 10.0000 0.332595
$$905$$ −19.8885 −0.661118
$$906$$ 0 0
$$907$$ 31.7771 1.05514 0.527570 0.849511i $$-0.323103\pi$$
0.527570 + 0.849511i $$0.323103\pi$$
$$908$$ 71.6656 2.37831
$$909$$ 0 0
$$910$$ 8.94427 0.296500
$$911$$ 28.8328 0.955274 0.477637 0.878557i $$-0.341493\pi$$
0.477637 + 0.878557i $$0.341493\pi$$
$$912$$ 0 0
$$913$$ 29.8885 0.989166
$$914$$ 26.5836 0.879307
$$915$$ 0 0
$$916$$ 15.1672 0.501138
$$917$$ 21.8885 0.722823
$$918$$ 0 0
$$919$$ 5.88854 0.194245 0.0971226 0.995272i $$-0.469036\pi$$
0.0971226 + 0.995272i $$0.469036\pi$$
$$920$$ 4.47214 0.147442
$$921$$ 0 0
$$922$$ 24.4721 0.805947
$$923$$ −8.00000 −0.263323
$$924$$ 0 0
$$925$$ 8.47214 0.278562
$$926$$ 73.4164 2.41261
$$927$$ 0 0
$$928$$ −6.70820 −0.220208
$$929$$ −34.0000 −1.11550 −0.557752 0.830008i $$-0.688336\pi$$
−0.557752 + 0.830008i $$0.688336\pi$$
$$930$$ 0 0
$$931$$ −6.00000 −0.196642
$$932$$ 56.8328 1.86162
$$933$$ 0 0
$$934$$ 13.6656 0.447153
$$935$$ −8.94427 −0.292509
$$936$$ 0 0
$$937$$ −57.7771 −1.88750 −0.943748 0.330667i $$-0.892726\pi$$
−0.943748 + 0.330667i $$0.892726\pi$$
$$938$$ 13.1672 0.429924
$$939$$ 0 0
$$940$$ −38.8328 −1.26659
$$941$$ −31.8885 −1.03954 −0.519768 0.854307i $$-0.673982\pi$$
−0.519768 + 0.854307i $$0.673982\pi$$
$$942$$ 0 0
$$943$$ −4.00000 −0.130258
$$944$$ 8.00000 0.260378
$$945$$ 0 0
$$946$$ 17.8885 0.581607
$$947$$ 40.0000 1.29983 0.649913 0.760009i $$-0.274805\pi$$
0.649913 + 0.760009i $$0.274805\pi$$
$$948$$ 0 0
$$949$$ 7.05573 0.229039
$$950$$ 4.47214 0.145095
$$951$$ 0 0
$$952$$ −20.0000 −0.648204
$$953$$ −14.9443 −0.484092 −0.242046 0.970265i $$-0.577819\pi$$
−0.242046 + 0.970265i $$0.577819\pi$$
$$954$$ 0 0
$$955$$ 10.0000 0.323592
$$956$$ 65.6656 2.12378
$$957$$ 0 0
$$958$$ 58.1378 1.87835
$$959$$ −24.9443 −0.805493
$$960$$ 0 0
$$961$$ −27.0000 −0.870968
$$962$$ 37.8885 1.22158
$$963$$ 0 0
$$964$$ 47.6656 1.53521
$$965$$ −0.472136 −0.0151986
$$966$$ 0 0
$$967$$ −48.7214 −1.56677 −0.783387 0.621535i $$-0.786509\pi$$
−0.783387 + 0.621535i $$0.786509\pi$$
$$968$$ −15.6525 −0.503090
$$969$$ 0 0
$$970$$ −38.9443 −1.25043
$$971$$ 48.8328 1.56712 0.783560 0.621316i $$-0.213402\pi$$
0.783560 + 0.621316i $$0.213402\pi$$
$$972$$ 0 0
$$973$$ −35.7771 −1.14696
$$974$$ 62.3607 1.99817
$$975$$ 0 0
$$976$$ −6.94427 −0.222281
$$977$$ 15.8885 0.508320 0.254160 0.967162i $$-0.418201\pi$$
0.254160 + 0.967162i $$0.418201\pi$$
$$978$$ 0 0
$$979$$ 12.0000 0.383522
$$980$$ −9.00000 −0.287494
$$981$$ 0 0
$$982$$ 24.4721 0.780937
$$983$$ 31.0557 0.990524 0.495262 0.868744i $$-0.335072\pi$$
0.495262 + 0.868744i $$0.335072\pi$$
$$984$$ 0 0
$$985$$ −14.9443 −0.476164
$$986$$ −10.0000 −0.318465
$$987$$ 0 0
$$988$$ 12.0000 0.381771
$$989$$ 8.00000 0.254385
$$990$$ 0 0
$$991$$ 37.8885 1.20357 0.601785 0.798658i $$-0.294457\pi$$
0.601785 + 0.798658i $$0.294457\pi$$
$$992$$ 13.4164 0.425971
$$993$$ 0 0
$$994$$ −17.8885 −0.567390
$$995$$ 21.8885 0.693913
$$996$$ 0 0
$$997$$ −27.5279 −0.871816 −0.435908 0.899991i $$-0.643573\pi$$
−0.435908 + 0.899991i $$0.643573\pi$$
$$998$$ 44.7214 1.41563
$$999$$ 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 1305.2.a.j.1.2 2
3.2 odd 2 435.2.a.g.1.1 2
5.4 even 2 6525.2.a.y.1.1 2
12.11 even 2 6960.2.a.bp.1.2 2
15.2 even 4 2175.2.c.g.349.1 4
15.8 even 4 2175.2.c.g.349.4 4
15.14 odd 2 2175.2.a.o.1.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
435.2.a.g.1.1 2 3.2 odd 2
1305.2.a.j.1.2 2 1.1 even 1 trivial
2175.2.a.o.1.2 2 15.14 odd 2
2175.2.c.g.349.1 4 15.2 even 4
2175.2.c.g.349.4 4 15.8 even 4
6525.2.a.y.1.1 2 5.4 even 2
6960.2.a.bp.1.2 2 12.11 even 2