# Properties

 Label 1520.2.a.t.1.2 Level $1520$ Weight $2$ Character 1520.1 Self dual yes Analytic conductor $12.137$ Analytic rank $0$ Dimension $4$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1520,2,Mod(1,1520)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1520, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("1520.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1520 = 2^{4} \cdot 5 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1520.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: $$12.1372611072$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: 4.4.11344.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - 2x^{3} - 4x^{2} + 4x + 3$$ x^4 - 2*x^3 - 4*x^2 + 4*x + 3 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 95) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$-1.51658$$ of defining polynomial Character $$\chi$$ $$=$$ 1520.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-1.53844 q^{3} -1.00000 q^{5} -5.03316 q^{7} -0.633188 q^{9} +O(q^{10})$$ $$q-1.53844 q^{3} -1.00000 q^{5} -5.03316 q^{7} -0.633188 q^{9} +3.03316 q^{11} -4.57160 q^{13} +1.53844 q^{15} -1.07689 q^{17} -1.00000 q^{19} +7.74324 q^{21} -4.11005 q^{23} +1.00000 q^{25} +5.58946 q^{27} -1.07689 q^{29} -5.58946 q^{31} -4.66635 q^{33} +5.03316 q^{35} +0.0947438 q^{37} +7.03316 q^{39} +10.6663 q^{41} -5.03316 q^{43} +0.633188 q^{45} +12.2995 q^{47} +18.3327 q^{49} +1.65673 q^{51} -4.09474 q^{53} -3.03316 q^{55} +1.53844 q^{57} +1.39997 q^{59} -5.69951 q^{61} +3.18694 q^{63} +4.57160 q^{65} -5.28168 q^{67} +6.32308 q^{69} +5.67692 q^{71} +9.07689 q^{73} -1.53844 q^{75} -15.2664 q^{77} +5.39997 q^{79} -6.69951 q^{81} -1.95627 q^{83} +1.07689 q^{85} +1.65673 q^{87} -2.18949 q^{89} +23.0096 q^{91} +8.59907 q^{93} +1.00000 q^{95} -2.16106 q^{97} -1.92056 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 2 q^{3} - 4 q^{5} - 4 q^{7} + 8 q^{9}+O(q^{10})$$ 4 * q - 2 * q^3 - 4 * q^5 - 4 * q^7 + 8 * q^9 $$4 q - 2 q^{3} - 4 q^{5} - 4 q^{7} + 8 q^{9} - 4 q^{11} + 2 q^{13} + 2 q^{15} + 4 q^{17} - 4 q^{19} - 4 q^{21} + 8 q^{23} + 4 q^{25} + 4 q^{27} + 4 q^{29} - 4 q^{31} + 8 q^{33} + 4 q^{35} - 6 q^{37} + 12 q^{39} + 16 q^{41} - 4 q^{43} - 8 q^{45} + 12 q^{47} + 20 q^{49} + 36 q^{51} - 10 q^{53} + 4 q^{55} + 2 q^{57} + 20 q^{61} - 20 q^{63} - 2 q^{65} + 18 q^{67} + 28 q^{69} + 20 q^{71} + 28 q^{73} - 2 q^{75} - 40 q^{77} + 16 q^{79} + 16 q^{81} - 4 q^{85} + 36 q^{87} + 4 q^{89} + 36 q^{91} - 40 q^{93} + 4 q^{95} + 30 q^{97} + 4 q^{99}+O(q^{100})$$ 4 * q - 2 * q^3 - 4 * q^5 - 4 * q^7 + 8 * q^9 - 4 * q^11 + 2 * q^13 + 2 * q^15 + 4 * q^17 - 4 * q^19 - 4 * q^21 + 8 * q^23 + 4 * q^25 + 4 * q^27 + 4 * q^29 - 4 * q^31 + 8 * q^33 + 4 * q^35 - 6 * q^37 + 12 * q^39 + 16 * q^41 - 4 * q^43 - 8 * q^45 + 12 * q^47 + 20 * q^49 + 36 * q^51 - 10 * q^53 + 4 * q^55 + 2 * q^57 + 20 * q^61 - 20 * q^63 - 2 * q^65 + 18 * q^67 + 28 * q^69 + 20 * q^71 + 28 * q^73 - 2 * q^75 - 40 * q^77 + 16 * q^79 + 16 * q^81 - 4 * q^85 + 36 * q^87 + 4 * q^89 + 36 * q^91 - 40 * q^93 + 4 * q^95 + 30 * q^97 + 4 * q^99

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0 0
$$3$$ −1.53844 −0.888221 −0.444111 0.895972i $$-0.646480\pi$$
−0.444111 + 0.895972i $$0.646480\pi$$
$$4$$ 0 0
$$5$$ −1.00000 −0.447214
$$6$$ 0 0
$$7$$ −5.03316 −1.90236 −0.951178 0.308644i $$-0.900125\pi$$
−0.951178 + 0.308644i $$0.900125\pi$$
$$8$$ 0 0
$$9$$ −0.633188 −0.211063
$$10$$ 0 0
$$11$$ 3.03316 0.914532 0.457266 0.889330i $$-0.348829\pi$$
0.457266 + 0.889330i $$0.348829\pi$$
$$12$$ 0 0
$$13$$ −4.57160 −1.26793 −0.633967 0.773360i $$-0.718575\pi$$
−0.633967 + 0.773360i $$0.718575\pi$$
$$14$$ 0 0
$$15$$ 1.53844 0.397225
$$16$$ 0 0
$$17$$ −1.07689 −0.261184 −0.130592 0.991436i $$-0.541688\pi$$
−0.130592 + 0.991436i $$0.541688\pi$$
$$18$$ 0 0
$$19$$ −1.00000 −0.229416
$$20$$ 0 0
$$21$$ 7.74324 1.68971
$$22$$ 0 0
$$23$$ −4.11005 −0.857004 −0.428502 0.903541i $$-0.640959\pi$$
−0.428502 + 0.903541i $$0.640959\pi$$
$$24$$ 0 0
$$25$$ 1.00000 0.200000
$$26$$ 0 0
$$27$$ 5.58946 1.07569
$$28$$ 0 0
$$29$$ −1.07689 −0.199973 −0.0999866 0.994989i $$-0.531880\pi$$
−0.0999866 + 0.994989i $$0.531880\pi$$
$$30$$ 0 0
$$31$$ −5.58946 −1.00390 −0.501948 0.864898i $$-0.667383\pi$$
−0.501948 + 0.864898i $$0.667383\pi$$
$$32$$ 0 0
$$33$$ −4.66635 −0.812307
$$34$$ 0 0
$$35$$ 5.03316 0.850759
$$36$$ 0 0
$$37$$ 0.0947438 0.0155758 0.00778789 0.999970i $$-0.497521\pi$$
0.00778789 + 0.999970i $$0.497521\pi$$
$$38$$ 0 0
$$39$$ 7.03316 1.12621
$$40$$ 0 0
$$41$$ 10.6663 1.66580 0.832902 0.553421i $$-0.186678\pi$$
0.832902 + 0.553421i $$0.186678\pi$$
$$42$$ 0 0
$$43$$ −5.03316 −0.767550 −0.383775 0.923427i $$-0.625376\pi$$
−0.383775 + 0.923427i $$0.625376\pi$$
$$44$$ 0 0
$$45$$ 0.633188 0.0943901
$$46$$ 0 0
$$47$$ 12.2995 1.79407 0.897036 0.441958i $$-0.145716\pi$$
0.897036 + 0.441958i $$0.145716\pi$$
$$48$$ 0 0
$$49$$ 18.3327 2.61896
$$50$$ 0 0
$$51$$ 1.65673 0.231989
$$52$$ 0 0
$$53$$ −4.09474 −0.562456 −0.281228 0.959641i $$-0.590742\pi$$
−0.281228 + 0.959641i $$0.590742\pi$$
$$54$$ 0 0
$$55$$ −3.03316 −0.408991
$$56$$ 0 0
$$57$$ 1.53844 0.203772
$$58$$ 0 0
$$59$$ 1.39997 0.182261 0.0911304 0.995839i $$-0.470952\pi$$
0.0911304 + 0.995839i $$0.470952\pi$$
$$60$$ 0 0
$$61$$ −5.69951 −0.729747 −0.364874 0.931057i $$-0.618888\pi$$
−0.364874 + 0.931057i $$0.618888\pi$$
$$62$$ 0 0
$$63$$ 3.18694 0.401516
$$64$$ 0 0
$$65$$ 4.57160 0.567038
$$66$$ 0 0
$$67$$ −5.28168 −0.645260 −0.322630 0.946525i $$-0.604567\pi$$
−0.322630 + 0.946525i $$0.604567\pi$$
$$68$$ 0 0
$$69$$ 6.32308 0.761210
$$70$$ 0 0
$$71$$ 5.67692 0.673726 0.336863 0.941554i $$-0.390634\pi$$
0.336863 + 0.941554i $$0.390634\pi$$
$$72$$ 0 0
$$73$$ 9.07689 1.06237 0.531185 0.847256i $$-0.321747\pi$$
0.531185 + 0.847256i $$0.321747\pi$$
$$74$$ 0 0
$$75$$ −1.53844 −0.177644
$$76$$ 0 0
$$77$$ −15.2664 −1.73977
$$78$$ 0 0
$$79$$ 5.39997 0.607544 0.303772 0.952745i $$-0.401754\pi$$
0.303772 + 0.952745i $$0.401754\pi$$
$$80$$ 0 0
$$81$$ −6.69951 −0.744390
$$82$$ 0 0
$$83$$ −1.95627 −0.214729 −0.107364 0.994220i $$-0.534241\pi$$
−0.107364 + 0.994220i $$0.534241\pi$$
$$84$$ 0 0
$$85$$ 1.07689 0.116805
$$86$$ 0 0
$$87$$ 1.65673 0.177621
$$88$$ 0 0
$$89$$ −2.18949 −0.232085 −0.116043 0.993244i $$-0.537021\pi$$
−0.116043 + 0.993244i $$0.537021\pi$$
$$90$$ 0 0
$$91$$ 23.0096 2.41206
$$92$$ 0 0
$$93$$ 8.59907 0.891682
$$94$$ 0 0
$$95$$ 1.00000 0.102598
$$96$$ 0 0
$$97$$ −2.16106 −0.219423 −0.109711 0.993963i $$-0.534993\pi$$
−0.109711 + 0.993963i $$0.534993\pi$$
$$98$$ 0 0
$$99$$ −1.92056 −0.193024
$$100$$ 0 0
$$101$$ 12.5869 1.25244 0.626222 0.779645i $$-0.284600\pi$$
0.626222 + 0.779645i $$0.284600\pi$$
$$102$$ 0 0
$$103$$ 6.20479 0.611376 0.305688 0.952132i $$-0.401113\pi$$
0.305688 + 0.952132i $$0.401113\pi$$
$$104$$ 0 0
$$105$$ −7.74324 −0.755663
$$106$$ 0 0
$$107$$ 12.5481 1.21307 0.606533 0.795058i $$-0.292560\pi$$
0.606533 + 0.795058i $$0.292560\pi$$
$$108$$ 0 0
$$109$$ −15.8096 −1.51428 −0.757140 0.653252i $$-0.773404\pi$$
−0.757140 + 0.653252i $$0.773404\pi$$
$$110$$ 0 0
$$111$$ −0.145758 −0.0138347
$$112$$ 0 0
$$113$$ −5.49472 −0.516899 −0.258450 0.966025i $$-0.583212\pi$$
−0.258450 + 0.966025i $$0.583212\pi$$
$$114$$ 0 0
$$115$$ 4.11005 0.383264
$$116$$ 0 0
$$117$$ 2.89469 0.267614
$$118$$ 0 0
$$119$$ 5.42015 0.496865
$$120$$ 0 0
$$121$$ −1.79994 −0.163631
$$122$$ 0 0
$$123$$ −16.4096 −1.47960
$$124$$ 0 0
$$125$$ −1.00000 −0.0894427
$$126$$ 0 0
$$127$$ 8.61533 0.764487 0.382244 0.924062i $$-0.375152\pi$$
0.382244 + 0.924062i $$0.375152\pi$$
$$128$$ 0 0
$$129$$ 7.74324 0.681754
$$130$$ 0 0
$$131$$ 2.15378 0.188176 0.0940882 0.995564i $$-0.470006\pi$$
0.0940882 + 0.995564i $$0.470006\pi$$
$$132$$ 0 0
$$133$$ 5.03316 0.436430
$$134$$ 0 0
$$135$$ −5.58946 −0.481064
$$136$$ 0 0
$$137$$ 2.18949 0.187061 0.0935303 0.995616i $$-0.470185\pi$$
0.0935303 + 0.995616i $$0.470185\pi$$
$$138$$ 0 0
$$139$$ −22.5196 −1.91009 −0.955045 0.296460i $$-0.904194\pi$$
−0.955045 + 0.296460i $$0.904194\pi$$
$$140$$ 0 0
$$141$$ −18.9222 −1.59353
$$142$$ 0 0
$$143$$ −13.8664 −1.15957
$$144$$ 0 0
$$145$$ 1.07689 0.0894308
$$146$$ 0 0
$$147$$ −28.2038 −2.32621
$$148$$ 0 0
$$149$$ 9.78697 0.801780 0.400890 0.916126i $$-0.368701\pi$$
0.400890 + 0.916126i $$0.368701\pi$$
$$150$$ 0 0
$$151$$ −5.87683 −0.478250 −0.239125 0.970989i $$-0.576861\pi$$
−0.239125 + 0.970989i $$0.576861\pi$$
$$152$$ 0 0
$$153$$ 0.681874 0.0551262
$$154$$ 0 0
$$155$$ 5.58946 0.448956
$$156$$ 0 0
$$157$$ 10.1895 0.813210 0.406605 0.913604i $$-0.366713\pi$$
0.406605 + 0.913604i $$0.366713\pi$$
$$158$$ 0 0
$$159$$ 6.29954 0.499586
$$160$$ 0 0
$$161$$ 20.6865 1.63033
$$162$$ 0 0
$$163$$ 10.3659 0.811916 0.405958 0.913892i $$-0.366938\pi$$
0.405958 + 0.913892i $$0.366938\pi$$
$$164$$ 0 0
$$165$$ 4.66635 0.363275
$$166$$ 0 0
$$167$$ 15.6048 1.20753 0.603766 0.797161i $$-0.293666\pi$$
0.603766 + 0.797161i $$0.293666\pi$$
$$168$$ 0 0
$$169$$ 7.89956 0.607659
$$170$$ 0 0
$$171$$ 0.633188 0.0484211
$$172$$ 0 0
$$173$$ 0.571604 0.0434583 0.0217291 0.999764i $$-0.493083\pi$$
0.0217291 + 0.999764i $$0.493083\pi$$
$$174$$ 0 0
$$175$$ −5.03316 −0.380471
$$176$$ 0 0
$$177$$ −2.15378 −0.161888
$$178$$ 0 0
$$179$$ −24.8864 −1.86010 −0.930050 0.367433i $$-0.880237\pi$$
−0.930050 + 0.367433i $$0.880237\pi$$
$$180$$ 0 0
$$181$$ 6.95372 0.516866 0.258433 0.966029i $$-0.416794\pi$$
0.258433 + 0.966029i $$0.416794\pi$$
$$182$$ 0 0
$$183$$ 8.76838 0.648177
$$184$$ 0 0
$$185$$ −0.0947438 −0.00696570
$$186$$ 0 0
$$187$$ −3.26638 −0.238861
$$188$$ 0 0
$$189$$ −28.1326 −2.04635
$$190$$ 0 0
$$191$$ 3.91254 0.283102 0.141551 0.989931i $$-0.454791\pi$$
0.141551 + 0.989931i $$0.454791\pi$$
$$192$$ 0 0
$$193$$ −9.20734 −0.662759 −0.331379 0.943498i $$-0.607514\pi$$
−0.331379 + 0.943498i $$0.607514\pi$$
$$194$$ 0 0
$$195$$ −7.03316 −0.503655
$$196$$ 0 0
$$197$$ −3.84622 −0.274032 −0.137016 0.990569i $$-0.543751\pi$$
−0.137016 + 0.990569i $$0.543751\pi$$
$$198$$ 0 0
$$199$$ 18.2864 1.29629 0.648145 0.761517i $$-0.275545\pi$$
0.648145 + 0.761517i $$0.275545\pi$$
$$200$$ 0 0
$$201$$ 8.12557 0.573134
$$202$$ 0 0
$$203$$ 5.42015 0.380420
$$204$$ 0 0
$$205$$ −10.6663 −0.744970
$$206$$ 0 0
$$207$$ 2.60243 0.180882
$$208$$ 0 0
$$209$$ −3.03316 −0.209808
$$210$$ 0 0
$$211$$ 19.2970 1.32846 0.664230 0.747529i $$-0.268760\pi$$
0.664230 + 0.747529i $$0.268760\pi$$
$$212$$ 0 0
$$213$$ −8.73362 −0.598418
$$214$$ 0 0
$$215$$ 5.03316 0.343259
$$216$$ 0 0
$$217$$ 28.1326 1.90977
$$218$$ 0 0
$$219$$ −13.9643 −0.943619
$$220$$ 0 0
$$221$$ 4.92311 0.331164
$$222$$ 0 0
$$223$$ −0.615334 −0.0412058 −0.0206029 0.999788i $$-0.506559\pi$$
−0.0206029 + 0.999788i $$0.506559\pi$$
$$224$$ 0 0
$$225$$ −0.633188 −0.0422126
$$226$$ 0 0
$$227$$ −17.4712 −1.15960 −0.579801 0.814758i $$-0.696870\pi$$
−0.579801 + 0.814758i $$0.696870\pi$$
$$228$$ 0 0
$$229$$ 9.69951 0.640962 0.320481 0.947255i $$-0.396156\pi$$
0.320481 + 0.947255i $$0.396156\pi$$
$$230$$ 0 0
$$231$$ 23.4865 1.54530
$$232$$ 0 0
$$233$$ −8.15378 −0.534172 −0.267086 0.963673i $$-0.586061\pi$$
−0.267086 + 0.963673i $$0.586061\pi$$
$$234$$ 0 0
$$235$$ −12.2995 −0.802333
$$236$$ 0 0
$$237$$ −8.30756 −0.539634
$$238$$ 0 0
$$239$$ −24.5991 −1.59118 −0.795591 0.605834i $$-0.792839\pi$$
−0.795591 + 0.605834i $$0.792839\pi$$
$$240$$ 0 0
$$241$$ 25.7738 1.66024 0.830120 0.557585i $$-0.188272\pi$$
0.830120 + 0.557585i $$0.188272\pi$$
$$242$$ 0 0
$$243$$ −6.46156 −0.414509
$$244$$ 0 0
$$245$$ −18.3327 −1.17123
$$246$$ 0 0
$$247$$ 4.57160 0.290884
$$248$$ 0 0
$$249$$ 3.00961 0.190727
$$250$$ 0 0
$$251$$ −12.5991 −0.795246 −0.397623 0.917549i $$-0.630165\pi$$
−0.397623 + 0.917549i $$0.630165\pi$$
$$252$$ 0 0
$$253$$ −12.4664 −0.783758
$$254$$ 0 0
$$255$$ −1.65673 −0.103749
$$256$$ 0 0
$$257$$ 0.182203 0.0113655 0.00568275 0.999984i $$-0.498191\pi$$
0.00568275 + 0.999984i $$0.498191\pi$$
$$258$$ 0 0
$$259$$ −0.476860 −0.0296307
$$260$$ 0 0
$$261$$ 0.681874 0.0422069
$$262$$ 0 0
$$263$$ −7.37643 −0.454850 −0.227425 0.973796i $$-0.573031\pi$$
−0.227425 + 0.973796i $$0.573031\pi$$
$$264$$ 0 0
$$265$$ 4.09474 0.251538
$$266$$ 0 0
$$267$$ 3.36841 0.206143
$$268$$ 0 0
$$269$$ 4.70206 0.286689 0.143345 0.989673i $$-0.454214\pi$$
0.143345 + 0.989673i $$0.454214\pi$$
$$270$$ 0 0
$$271$$ −9.92056 −0.602631 −0.301316 0.953524i $$-0.597426\pi$$
−0.301316 + 0.953524i $$0.597426\pi$$
$$272$$ 0 0
$$273$$ −35.3990 −2.14245
$$274$$ 0 0
$$275$$ 3.03316 0.182906
$$276$$ 0 0
$$277$$ −22.6297 −1.35969 −0.679843 0.733358i $$-0.737952\pi$$
−0.679843 + 0.733358i $$0.737952\pi$$
$$278$$ 0 0
$$279$$ 3.53918 0.211885
$$280$$ 0 0
$$281$$ −3.95386 −0.235868 −0.117934 0.993021i $$-0.537627\pi$$
−0.117934 + 0.993021i $$0.537627\pi$$
$$282$$ 0 0
$$283$$ 13.0638 0.776561 0.388280 0.921541i $$-0.373069\pi$$
0.388280 + 0.921541i $$0.373069\pi$$
$$284$$ 0 0
$$285$$ −1.53844 −0.0911296
$$286$$ 0 0
$$287$$ −53.6854 −3.16895
$$288$$ 0 0
$$289$$ −15.8403 −0.931783
$$290$$ 0 0
$$291$$ 3.32468 0.194896
$$292$$ 0 0
$$293$$ −9.69463 −0.566366 −0.283183 0.959066i $$-0.591390\pi$$
−0.283183 + 0.959066i $$0.591390\pi$$
$$294$$ 0 0
$$295$$ −1.39997 −0.0815095
$$296$$ 0 0
$$297$$ 16.9537 0.983755
$$298$$ 0 0
$$299$$ 18.7895 1.08663
$$300$$ 0 0
$$301$$ 25.3327 1.46015
$$302$$ 0 0
$$303$$ −19.3643 −1.11245
$$304$$ 0 0
$$305$$ 5.69951 0.326353
$$306$$ 0 0
$$307$$ 28.5481 1.62932 0.814662 0.579936i $$-0.196923\pi$$
0.814662 + 0.579936i $$0.196923\pi$$
$$308$$ 0 0
$$309$$ −9.54573 −0.543038
$$310$$ 0 0
$$311$$ −25.4070 −1.44070 −0.720350 0.693610i $$-0.756019\pi$$
−0.720350 + 0.693610i $$0.756019\pi$$
$$312$$ 0 0
$$313$$ 16.7999 0.949589 0.474794 0.880097i $$-0.342522\pi$$
0.474794 + 0.880097i $$0.342522\pi$$
$$314$$ 0 0
$$315$$ −3.18694 −0.179564
$$316$$ 0 0
$$317$$ −31.7505 −1.78329 −0.891643 0.452738i $$-0.850447\pi$$
−0.891643 + 0.452738i $$0.850447\pi$$
$$318$$ 0 0
$$319$$ −3.26638 −0.182882
$$320$$ 0 0
$$321$$ −19.3045 −1.07747
$$322$$ 0 0
$$323$$ 1.07689 0.0599197
$$324$$ 0 0
$$325$$ −4.57160 −0.253587
$$326$$ 0 0
$$327$$ 24.3221 1.34502
$$328$$ 0 0
$$329$$ −61.9055 −3.41296
$$330$$ 0 0
$$331$$ 9.96429 0.547687 0.273843 0.961774i $$-0.411705\pi$$
0.273843 + 0.961774i $$0.411705\pi$$
$$332$$ 0 0
$$333$$ −0.0599906 −0.00328747
$$334$$ 0 0
$$335$$ 5.28168 0.288569
$$336$$ 0 0
$$337$$ 23.1918 1.26334 0.631669 0.775238i $$-0.282370\pi$$
0.631669 + 0.775238i $$0.282370\pi$$
$$338$$ 0 0
$$339$$ 8.45331 0.459121
$$340$$ 0 0
$$341$$ −16.9537 −0.918095
$$342$$ 0 0
$$343$$ −57.0393 −3.07983
$$344$$ 0 0
$$345$$ −6.32308 −0.340423
$$346$$ 0 0
$$347$$ −21.1352 −1.13460 −0.567298 0.823512i $$-0.692011\pi$$
−0.567298 + 0.823512i $$0.692011\pi$$
$$348$$ 0 0
$$349$$ −5.04628 −0.270121 −0.135061 0.990837i $$-0.543123\pi$$
−0.135061 + 0.990837i $$0.543123\pi$$
$$350$$ 0 0
$$351$$ −25.5528 −1.36391
$$352$$ 0 0
$$353$$ 12.5634 0.668680 0.334340 0.942452i $$-0.391487\pi$$
0.334340 + 0.942452i $$0.391487\pi$$
$$354$$ 0 0
$$355$$ −5.67692 −0.301300
$$356$$ 0 0
$$357$$ −8.33861 −0.441326
$$358$$ 0 0
$$359$$ 17.6534 0.931709 0.465855 0.884861i $$-0.345747\pi$$
0.465855 + 0.884861i $$0.345747\pi$$
$$360$$ 0 0
$$361$$ 1.00000 0.0526316
$$362$$ 0 0
$$363$$ 2.76911 0.145341
$$364$$ 0 0
$$365$$ −9.07689 −0.475106
$$366$$ 0 0
$$367$$ −4.43409 −0.231457 −0.115729 0.993281i $$-0.536920\pi$$
−0.115729 + 0.993281i $$0.536920\pi$$
$$368$$ 0 0
$$369$$ −6.75381 −0.351589
$$370$$ 0 0
$$371$$ 20.6095 1.06999
$$372$$ 0 0
$$373$$ −20.8735 −1.08079 −0.540396 0.841411i $$-0.681725\pi$$
−0.540396 + 0.841411i $$0.681725\pi$$
$$374$$ 0 0
$$375$$ 1.53844 0.0794449
$$376$$ 0 0
$$377$$ 4.92311 0.253553
$$378$$ 0 0
$$379$$ 6.79994 0.349290 0.174645 0.984632i $$-0.444122\pi$$
0.174645 + 0.984632i $$0.444122\pi$$
$$380$$ 0 0
$$381$$ −13.2542 −0.679034
$$382$$ 0 0
$$383$$ 10.9078 0.557363 0.278681 0.960384i $$-0.410103\pi$$
0.278681 + 0.960384i $$0.410103\pi$$
$$384$$ 0 0
$$385$$ 15.2664 0.778047
$$386$$ 0 0
$$387$$ 3.18694 0.162001
$$388$$ 0 0
$$389$$ 20.9326 1.06132 0.530662 0.847584i $$-0.321943\pi$$
0.530662 + 0.847584i $$0.321943\pi$$
$$390$$ 0 0
$$391$$ 4.42607 0.223836
$$392$$ 0 0
$$393$$ −3.31347 −0.167142
$$394$$ 0 0
$$395$$ −5.39997 −0.271702
$$396$$ 0 0
$$397$$ 13.1221 0.658578 0.329289 0.944229i $$-0.393191\pi$$
0.329289 + 0.944229i $$0.393191\pi$$
$$398$$ 0 0
$$399$$ −7.74324 −0.387647
$$400$$ 0 0
$$401$$ −11.5528 −0.576919 −0.288459 0.957492i $$-0.593143\pi$$
−0.288459 + 0.957492i $$0.593143\pi$$
$$402$$ 0 0
$$403$$ 25.5528 1.27288
$$404$$ 0 0
$$405$$ 6.69951 0.332901
$$406$$ 0 0
$$407$$ 0.287373 0.0142445
$$408$$ 0 0
$$409$$ −18.9433 −0.936686 −0.468343 0.883547i $$-0.655149\pi$$
−0.468343 + 0.883547i $$0.655149\pi$$
$$410$$ 0 0
$$411$$ −3.36841 −0.166151
$$412$$ 0 0
$$413$$ −7.04628 −0.346725
$$414$$ 0 0
$$415$$ 1.95627 0.0960295
$$416$$ 0 0
$$417$$ 34.6452 1.69658
$$418$$ 0 0
$$419$$ −27.2664 −1.33205 −0.666025 0.745930i $$-0.732006\pi$$
−0.666025 + 0.745930i $$0.732006\pi$$
$$420$$ 0 0
$$421$$ 6.66635 0.324898 0.162449 0.986717i $$-0.448061\pi$$
0.162449 + 0.986717i $$0.448061\pi$$
$$422$$ 0 0
$$423$$ −7.78792 −0.378662
$$424$$ 0 0
$$425$$ −1.07689 −0.0522368
$$426$$ 0 0
$$427$$ 28.6865 1.38824
$$428$$ 0 0
$$429$$ 21.3327 1.02995
$$430$$ 0 0
$$431$$ 32.4548 1.56329 0.781646 0.623723i $$-0.214381\pi$$
0.781646 + 0.623723i $$0.214381\pi$$
$$432$$ 0 0
$$433$$ 33.8168 1.62513 0.812567 0.582868i $$-0.198070\pi$$
0.812567 + 0.582868i $$0.198070\pi$$
$$434$$ 0 0
$$435$$ −1.65673 −0.0794343
$$436$$ 0 0
$$437$$ 4.11005 0.196610
$$438$$ 0 0
$$439$$ 27.8453 1.32898 0.664491 0.747296i $$-0.268648\pi$$
0.664491 + 0.747296i $$0.268648\pi$$
$$440$$ 0 0
$$441$$ −11.6080 −0.552764
$$442$$ 0 0
$$443$$ −22.7754 −1.08209 −0.541047 0.840992i $$-0.681972\pi$$
−0.541047 + 0.840992i $$0.681972\pi$$
$$444$$ 0 0
$$445$$ 2.18949 0.103792
$$446$$ 0 0
$$447$$ −15.0567 −0.712158
$$448$$ 0 0
$$449$$ 24.8507 1.17278 0.586389 0.810029i $$-0.300549\pi$$
0.586389 + 0.810029i $$0.300549\pi$$
$$450$$ 0 0
$$451$$ 32.3527 1.52343
$$452$$ 0 0
$$453$$ 9.04118 0.424792
$$454$$ 0 0
$$455$$ −23.0096 −1.07871
$$456$$ 0 0
$$457$$ −7.33270 −0.343009 −0.171505 0.985183i $$-0.554863\pi$$
−0.171505 + 0.985183i $$0.554863\pi$$
$$458$$ 0 0
$$459$$ −6.01923 −0.280953
$$460$$ 0 0
$$461$$ 10.3076 0.480071 0.240035 0.970764i $$-0.422841\pi$$
0.240035 + 0.970764i $$0.422841\pi$$
$$462$$ 0 0
$$463$$ −7.80249 −0.362613 −0.181306 0.983427i $$-0.558033\pi$$
−0.181306 + 0.983427i $$0.558033\pi$$
$$464$$ 0 0
$$465$$ −8.59907 −0.398772
$$466$$ 0 0
$$467$$ 7.37643 0.341340 0.170670 0.985328i $$-0.445407\pi$$
0.170670 + 0.985328i $$0.445407\pi$$
$$468$$ 0 0
$$469$$ 26.5835 1.22751
$$470$$ 0 0
$$471$$ −15.6760 −0.722310
$$472$$ 0 0
$$473$$ −15.2664 −0.701949
$$474$$ 0 0
$$475$$ −1.00000 −0.0458831
$$476$$ 0 0
$$477$$ 2.59274 0.118714
$$478$$ 0 0
$$479$$ 16.1458 0.737719 0.368859 0.929485i $$-0.379748\pi$$
0.368859 + 0.929485i $$0.379748\pi$$
$$480$$ 0 0
$$481$$ −0.433131 −0.0197491
$$482$$ 0 0
$$483$$ −31.8251 −1.44809
$$484$$ 0 0
$$485$$ 2.16106 0.0981288
$$486$$ 0 0
$$487$$ 8.40566 0.380897 0.190448 0.981697i $$-0.439006\pi$$
0.190448 + 0.981697i $$0.439006\pi$$
$$488$$ 0 0
$$489$$ −15.9473 −0.721162
$$490$$ 0 0
$$491$$ 36.0855 1.62852 0.814259 0.580502i $$-0.197144\pi$$
0.814259 + 0.580502i $$0.197144\pi$$
$$492$$ 0 0
$$493$$ 1.15969 0.0522298
$$494$$ 0 0
$$495$$ 1.92056 0.0863228
$$496$$ 0 0
$$497$$ −28.5728 −1.28167
$$498$$ 0 0
$$499$$ 3.30035 0.147744 0.0738720 0.997268i $$-0.476464\pi$$
0.0738720 + 0.997268i $$0.476464\pi$$
$$500$$ 0 0
$$501$$ −24.0071 −1.07256
$$502$$ 0 0
$$503$$ −3.53530 −0.157631 −0.0788157 0.996889i $$-0.525114\pi$$
−0.0788157 + 0.996889i $$0.525114\pi$$
$$504$$ 0 0
$$505$$ −12.5869 −0.560110
$$506$$ 0 0
$$507$$ −12.1530 −0.539736
$$508$$ 0 0
$$509$$ 22.4096 0.993287 0.496644 0.867955i $$-0.334566\pi$$
0.496644 + 0.867955i $$0.334566\pi$$
$$510$$ 0 0
$$511$$ −45.6854 −2.02100
$$512$$ 0 0
$$513$$ −5.58946 −0.246781
$$514$$ 0 0
$$515$$ −6.20479 −0.273416
$$516$$ 0 0
$$517$$ 37.3065 1.64074
$$518$$ 0 0
$$519$$ −0.879381 −0.0386006
$$520$$ 0 0
$$521$$ 34.4402 1.50885 0.754426 0.656385i $$-0.227915\pi$$
0.754426 + 0.656385i $$0.227915\pi$$
$$522$$ 0 0
$$523$$ 30.4451 1.33127 0.665635 0.746277i $$-0.268161\pi$$
0.665635 + 0.746277i $$0.268161\pi$$
$$524$$ 0 0
$$525$$ 7.74324 0.337943
$$526$$ 0 0
$$527$$ 6.01923 0.262202
$$528$$ 0 0
$$529$$ −6.10750 −0.265543
$$530$$ 0 0
$$531$$ −0.886445 −0.0384685
$$532$$ 0 0
$$533$$ −48.7623 −2.11213
$$534$$ 0 0
$$535$$ −12.5481 −0.542500
$$536$$ 0 0
$$537$$ 38.2864 1.65218
$$538$$ 0 0
$$539$$ 55.6060 2.39512
$$540$$ 0 0
$$541$$ −12.2794 −0.527931 −0.263965 0.964532i $$-0.585030\pi$$
−0.263965 + 0.964532i $$0.585030\pi$$
$$542$$ 0 0
$$543$$ −10.6979 −0.459091
$$544$$ 0 0
$$545$$ 15.8096 0.677207
$$546$$ 0 0
$$547$$ −0.250928 −0.0107289 −0.00536446 0.999986i $$-0.501708\pi$$
−0.00536446 + 0.999986i $$0.501708\pi$$
$$548$$ 0 0
$$549$$ 3.60886 0.154022
$$550$$ 0 0
$$551$$ 1.07689 0.0458770
$$552$$ 0 0
$$553$$ −27.1789 −1.15577
$$554$$ 0 0
$$555$$ 0.145758 0.00618708
$$556$$ 0 0
$$557$$ 1.67596 0.0710128 0.0355064 0.999369i $$-0.488696\pi$$
0.0355064 + 0.999369i $$0.488696\pi$$
$$558$$ 0 0
$$559$$ 23.0096 0.973203
$$560$$ 0 0
$$561$$ 5.02514 0.212162
$$562$$ 0 0
$$563$$ −20.6605 −0.870737 −0.435368 0.900252i $$-0.643382\pi$$
−0.435368 + 0.900252i $$0.643382\pi$$
$$564$$ 0 0
$$565$$ 5.49472 0.231164
$$566$$ 0 0
$$567$$ 33.7197 1.41609
$$568$$ 0 0
$$569$$ 40.9673 1.71744 0.858720 0.512445i $$-0.171260\pi$$
0.858720 + 0.512445i $$0.171260\pi$$
$$570$$ 0 0
$$571$$ −24.2070 −1.01303 −0.506515 0.862231i $$-0.669067\pi$$
−0.506515 + 0.862231i $$0.669067\pi$$
$$572$$ 0 0
$$573$$ −6.01923 −0.251457
$$574$$ 0 0
$$575$$ −4.11005 −0.171401
$$576$$ 0 0
$$577$$ 40.2864 1.67715 0.838573 0.544790i $$-0.183391\pi$$
0.838573 + 0.544790i $$0.183391\pi$$
$$578$$ 0 0
$$579$$ 14.1650 0.588677
$$580$$ 0 0
$$581$$ 9.84622 0.408490
$$582$$ 0 0
$$583$$ −12.4200 −0.514384
$$584$$ 0 0
$$585$$ −2.89469 −0.119681
$$586$$ 0 0
$$587$$ 34.7754 1.43534 0.717668 0.696385i $$-0.245210\pi$$
0.717668 + 0.696385i $$0.245210\pi$$
$$588$$ 0 0
$$589$$ 5.58946 0.230310
$$590$$ 0 0
$$591$$ 5.91720 0.243401
$$592$$ 0 0
$$593$$ −35.0864 −1.44082 −0.720412 0.693546i $$-0.756047\pi$$
−0.720412 + 0.693546i $$0.756047\pi$$
$$594$$ 0 0
$$595$$ −5.42015 −0.222205
$$596$$ 0 0
$$597$$ −28.1326 −1.15139
$$598$$ 0 0
$$599$$ −30.8201 −1.25928 −0.629638 0.776889i $$-0.716797\pi$$
−0.629638 + 0.776889i $$0.716797\pi$$
$$600$$ 0 0
$$601$$ −7.10654 −0.289882 −0.144941 0.989440i $$-0.546299\pi$$
−0.144941 + 0.989440i $$0.546299\pi$$
$$602$$ 0 0
$$603$$ 3.34430 0.136190
$$604$$ 0 0
$$605$$ 1.79994 0.0731781
$$606$$ 0 0
$$607$$ −2.01531 −0.0817987 −0.0408994 0.999163i $$-0.513022\pi$$
−0.0408994 + 0.999163i $$0.513022\pi$$
$$608$$ 0 0
$$609$$ −8.33861 −0.337897
$$610$$ 0 0
$$611$$ −56.2286 −2.27477
$$612$$ 0 0
$$613$$ −11.6096 −0.468909 −0.234455 0.972127i $$-0.575330\pi$$
−0.234455 + 0.972127i $$0.575330\pi$$
$$614$$ 0 0
$$615$$ 16.4096 0.661698
$$616$$ 0 0
$$617$$ 12.4307 0.500442 0.250221 0.968189i $$-0.419497\pi$$
0.250221 + 0.968189i $$0.419497\pi$$
$$618$$ 0 0
$$619$$ −3.85424 −0.154915 −0.0774575 0.996996i $$-0.524680\pi$$
−0.0774575 + 0.996996i $$0.524680\pi$$
$$620$$ 0 0
$$621$$ −22.9729 −0.921873
$$622$$ 0 0
$$623$$ 11.0200 0.441509
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ 0 0
$$627$$ 4.66635 0.186356
$$628$$ 0 0
$$629$$ −0.102029 −0.00406814
$$630$$ 0 0
$$631$$ 16.8794 0.671958 0.335979 0.941870i $$-0.390933\pi$$
0.335979 + 0.941870i $$0.390933\pi$$
$$632$$ 0 0
$$633$$ −29.6873 −1.17997
$$634$$ 0 0
$$635$$ −8.61533 −0.341889
$$636$$ 0 0
$$637$$ −83.8098 −3.32067
$$638$$ 0 0
$$639$$ −3.59456 −0.142199
$$640$$ 0 0
$$641$$ 43.4855 1.71757 0.858787 0.512332i $$-0.171218\pi$$
0.858787 + 0.512332i $$0.171218\pi$$
$$642$$ 0 0
$$643$$ −11.0689 −0.436514 −0.218257 0.975891i $$-0.570037\pi$$
−0.218257 + 0.975891i $$0.570037\pi$$
$$644$$ 0 0
$$645$$ −7.74324 −0.304890
$$646$$ 0 0
$$647$$ −5.61766 −0.220853 −0.110427 0.993884i $$-0.535222\pi$$
−0.110427 + 0.993884i $$0.535222\pi$$
$$648$$ 0 0
$$649$$ 4.24634 0.166683
$$650$$ 0 0
$$651$$ −43.2805 −1.69630
$$652$$ 0 0
$$653$$ 44.5211 1.74224 0.871122 0.491066i $$-0.163393\pi$$
0.871122 + 0.491066i $$0.163393\pi$$
$$654$$ 0 0
$$655$$ −2.15378 −0.0841551
$$656$$ 0 0
$$657$$ −5.74738 −0.224227
$$658$$ 0 0
$$659$$ 6.89154 0.268456 0.134228 0.990950i $$-0.457144\pi$$
0.134228 + 0.990950i $$0.457144\pi$$
$$660$$ 0 0
$$661$$ −44.3989 −1.72692 −0.863458 0.504421i $$-0.831706\pi$$
−0.863458 + 0.504421i $$0.831706\pi$$
$$662$$ 0 0
$$663$$ −7.57393 −0.294147
$$664$$ 0 0
$$665$$ −5.03316 −0.195178
$$666$$ 0 0
$$667$$ 4.42607 0.171378
$$668$$ 0 0
$$669$$ 0.946657 0.0365999
$$670$$ 0 0
$$671$$ −17.2875 −0.667377
$$672$$ 0 0
$$673$$ 38.6214 1.48875 0.744374 0.667763i $$-0.232748\pi$$
0.744374 + 0.667763i $$0.232748\pi$$
$$674$$ 0 0
$$675$$ 5.58946 0.215138
$$676$$ 0 0
$$677$$ 43.3917 1.66768 0.833840 0.552006i $$-0.186138\pi$$
0.833840 + 0.552006i $$0.186138\pi$$
$$678$$ 0 0
$$679$$ 10.8770 0.417420
$$680$$ 0 0
$$681$$ 26.8784 1.02998
$$682$$ 0 0
$$683$$ 34.7123 1.32823 0.664114 0.747631i $$-0.268809\pi$$
0.664114 + 0.747631i $$0.268809\pi$$
$$684$$ 0 0
$$685$$ −2.18949 −0.0836560
$$686$$ 0 0
$$687$$ −14.9222 −0.569316
$$688$$ 0 0
$$689$$ 18.7195 0.713158
$$690$$ 0 0
$$691$$ −2.94570 −0.112060 −0.0560299 0.998429i $$-0.517844\pi$$
−0.0560299 + 0.998429i $$0.517844\pi$$
$$692$$ 0 0
$$693$$ 9.66649 0.367200
$$694$$ 0 0
$$695$$ 22.5196 0.854218
$$696$$ 0 0
$$697$$ −11.4865 −0.435081
$$698$$ 0 0
$$699$$ 12.5441 0.474463
$$700$$ 0 0
$$701$$ 18.5920 0.702210 0.351105 0.936336i $$-0.385806\pi$$
0.351105 + 0.936336i $$0.385806\pi$$
$$702$$ 0 0
$$703$$ −0.0947438 −0.00357333
$$704$$ 0 0
$$705$$ 18.9222 0.712650
$$706$$ 0 0
$$707$$ −63.3519 −2.38259
$$708$$ 0 0
$$709$$ 39.8443 1.49638 0.748192 0.663482i $$-0.230922\pi$$
0.748192 + 0.663482i $$0.230922\pi$$
$$710$$ 0 0
$$711$$ −3.41920 −0.128230
$$712$$ 0 0
$$713$$ 22.9729 0.860344
$$714$$ 0 0
$$715$$ 13.8664 0.518574
$$716$$ 0 0
$$717$$ 37.8443 1.41332
$$718$$ 0 0
$$719$$ −21.2321 −0.791824 −0.395912 0.918288i $$-0.629572\pi$$
−0.395912 + 0.918288i $$0.629572\pi$$
$$720$$ 0 0
$$721$$ −31.2297 −1.16306
$$722$$ 0 0
$$723$$ −39.6516 −1.47466
$$724$$ 0 0
$$725$$ −1.07689 −0.0399947
$$726$$ 0 0
$$727$$ 49.1658 1.82346 0.911729 0.410792i $$-0.134748\pi$$
0.911729 + 0.410792i $$0.134748\pi$$
$$728$$ 0 0
$$729$$ 30.0393 1.11257
$$730$$ 0 0
$$731$$ 5.42015 0.200472
$$732$$ 0 0
$$733$$ 1.60498 0.0592815 0.0296407 0.999561i $$-0.490564\pi$$
0.0296407 + 0.999561i $$0.490564\pi$$
$$734$$ 0 0
$$735$$ 28.2038 1.04031
$$736$$ 0 0
$$737$$ −16.0202 −0.590111
$$738$$ 0 0
$$739$$ −8.13264 −0.299164 −0.149582 0.988749i $$-0.547793\pi$$
−0.149582 + 0.988749i $$0.547793\pi$$
$$740$$ 0 0
$$741$$ −7.03316 −0.258370
$$742$$ 0 0
$$743$$ 42.9261 1.57481 0.787403 0.616439i $$-0.211425\pi$$
0.787403 + 0.616439i $$0.211425\pi$$
$$744$$ 0 0
$$745$$ −9.78697 −0.358567
$$746$$ 0 0
$$747$$ 1.23869 0.0453212
$$748$$ 0 0
$$749$$ −63.1564 −2.30768
$$750$$ 0 0
$$751$$ −4.05685 −0.148037 −0.0740183 0.997257i $$-0.523582\pi$$
−0.0740183 + 0.997257i $$0.523582\pi$$
$$752$$ 0 0
$$753$$ 19.3830 0.706355
$$754$$ 0 0
$$755$$ 5.87683 0.213880
$$756$$ 0 0
$$757$$ 22.6151 0.821960 0.410980 0.911644i $$-0.365187\pi$$
0.410980 + 0.911644i $$0.365187\pi$$
$$758$$ 0 0
$$759$$ 19.1789 0.696151
$$760$$ 0 0
$$761$$ −41.3609 −1.49933 −0.749666 0.661817i $$-0.769786\pi$$
−0.749666 + 0.661817i $$0.769786\pi$$
$$762$$ 0 0
$$763$$ 79.5720 2.88070
$$764$$ 0 0
$$765$$ −0.681874 −0.0246532
$$766$$ 0 0
$$767$$ −6.40011 −0.231095
$$768$$ 0 0
$$769$$ 17.9196 0.646197 0.323099 0.946365i $$-0.395275\pi$$
0.323099 + 0.946365i $$0.395275\pi$$
$$770$$ 0 0
$$771$$ −0.280309 −0.0100951
$$772$$ 0 0
$$773$$ −21.9043 −0.787843 −0.393921 0.919144i $$-0.628882\pi$$
−0.393921 + 0.919144i $$0.628882\pi$$
$$774$$ 0 0
$$775$$ −5.58946 −0.200779
$$776$$ 0 0
$$777$$ 0.733623 0.0263186
$$778$$ 0 0
$$779$$ −10.6663 −0.382162
$$780$$ 0 0
$$781$$ 17.2190 0.616144
$$782$$ 0 0
$$783$$ −6.01923 −0.215110
$$784$$ 0 0
$$785$$ −10.1895 −0.363678
$$786$$ 0 0
$$787$$ 31.6354 1.12768 0.563840 0.825884i $$-0.309324\pi$$
0.563840 + 0.825884i $$0.309324\pi$$
$$788$$ 0 0
$$789$$ 11.3482 0.404007
$$790$$ 0 0
$$791$$ 27.6558 0.983326
$$792$$ 0 0
$$793$$ 26.0559 0.925272
$$794$$ 0 0
$$795$$ −6.29954 −0.223422
$$796$$ 0 0
$$797$$ −15.4486 −0.547217 −0.273608 0.961841i $$-0.588217\pi$$
−0.273608 + 0.961841i $$0.588217\pi$$
$$798$$ 0 0
$$799$$ −13.2452 −0.468583
$$800$$ 0 0
$$801$$ 1.38636 0.0489845
$$802$$ 0 0
$$803$$ 27.5317 0.971571
$$804$$ 0 0
$$805$$ −20.6865 −0.729104
$$806$$ 0 0
$$807$$ −7.23385 −0.254644
$$808$$ 0 0
$$809$$ −32.9326 −1.15785 −0.578924 0.815382i $$-0.696527\pi$$
−0.578924 + 0.815382i $$0.696527\pi$$
$$810$$ 0 0
$$811$$ 25.5895 0.898567 0.449284 0.893389i $$-0.351679\pi$$
0.449284 + 0.893389i $$0.351679\pi$$
$$812$$ 0 0
$$813$$ 15.2622 0.535270
$$814$$ 0 0
$$815$$ −10.3659 −0.363100
$$816$$ 0 0
$$817$$ 5.03316 0.176088
$$818$$ 0 0
$$819$$ −14.5694 −0.509097
$$820$$ 0 0
$$821$$ 17.8674 0.623575 0.311788 0.950152i $$-0.399072\pi$$
0.311788 + 0.950152i $$0.399072\pi$$
$$822$$ 0 0
$$823$$ 18.4533 0.643242 0.321621 0.946868i $$-0.395772\pi$$
0.321621 + 0.946868i $$0.395772\pi$$
$$824$$ 0 0
$$825$$ −4.66635 −0.162461
$$826$$ 0 0
$$827$$ −47.5460 −1.65334 −0.826668 0.562690i $$-0.809767\pi$$
−0.826668 + 0.562690i $$0.809767\pi$$
$$828$$ 0 0
$$829$$ 19.0308 0.660965 0.330483 0.943812i $$-0.392788\pi$$
0.330483 + 0.943812i $$0.392788\pi$$
$$830$$ 0 0
$$831$$ 34.8145 1.20770
$$832$$ 0 0
$$833$$ −19.7423 −0.684029
$$834$$ 0 0
$$835$$ −15.6048 −0.540025
$$836$$ 0 0
$$837$$ −31.2420 −1.07988
$$838$$ 0 0
$$839$$ −3.04022 −0.104960 −0.0524801 0.998622i $$-0.516713\pi$$
−0.0524801 + 0.998622i $$0.516713\pi$$
$$840$$ 0 0
$$841$$ −27.8403 −0.960011
$$842$$ 0 0
$$843$$ 6.08280 0.209503
$$844$$ 0 0
$$845$$ −7.89956 −0.271753
$$846$$ 0 0
$$847$$ 9.05940 0.311285
$$848$$ 0 0
$$849$$ −20.0979 −0.689758
$$850$$ 0 0
$$851$$ −0.389401 −0.0133485
$$852$$ 0 0
$$853$$ −18.2201 −0.623844 −0.311922 0.950108i $$-0.600973\pi$$
−0.311922 + 0.950108i $$0.600973\pi$$
$$854$$ 0 0
$$855$$ −0.633188 −0.0216546
$$856$$ 0 0
$$857$$ 19.3611 0.661363 0.330682 0.943742i $$-0.392721\pi$$
0.330682 + 0.943742i $$0.392721\pi$$
$$858$$ 0 0
$$859$$ −3.44129 −0.117415 −0.0587077 0.998275i $$-0.518698\pi$$
−0.0587077 + 0.998275i $$0.518698\pi$$
$$860$$ 0 0
$$861$$ 82.5921 2.81473
$$862$$ 0 0
$$863$$ 26.5471 0.903674 0.451837 0.892101i $$-0.350769\pi$$
0.451837 + 0.892101i $$0.350769\pi$$
$$864$$ 0 0
$$865$$ −0.571604 −0.0194351
$$866$$ 0 0
$$867$$ 24.3694 0.827630
$$868$$ 0 0
$$869$$ 16.3790 0.555619
$$870$$ 0 0
$$871$$ 24.1458 0.818148
$$872$$ 0 0
$$873$$ 1.36836 0.0463120
$$874$$ 0 0
$$875$$ 5.03316 0.170152
$$876$$ 0 0
$$877$$ 20.5495 0.693908 0.346954 0.937882i $$-0.387216\pi$$
0.346954 + 0.937882i $$0.387216\pi$$
$$878$$ 0 0
$$879$$ 14.9146 0.503059
$$880$$ 0 0
$$881$$ −31.1911 −1.05085 −0.525427 0.850839i $$-0.676094\pi$$
−0.525427 + 0.850839i $$0.676094\pi$$
$$882$$ 0 0
$$883$$ −14.1861 −0.477401 −0.238701 0.971093i $$-0.576721\pi$$
−0.238701 + 0.971093i $$0.576721\pi$$
$$884$$ 0 0
$$885$$ 2.15378 0.0723985
$$886$$ 0 0
$$887$$ −46.6046 −1.56483 −0.782415 0.622757i $$-0.786012\pi$$
−0.782415 + 0.622757i $$0.786012\pi$$
$$888$$ 0 0
$$889$$ −43.3623 −1.45433
$$890$$ 0 0
$$891$$ −20.3207 −0.680768
$$892$$ 0 0
$$893$$ −12.2995 −0.411588
$$894$$ 0 0
$$895$$ 24.8864 0.831862
$$896$$ 0 0
$$897$$ −28.9066 −0.965164
$$898$$ 0 0
$$899$$ 6.01923 0.200752
$$900$$ 0 0
$$901$$ 4.40959 0.146905
$$902$$ 0 0
$$903$$ −38.9729 −1.29694
$$904$$ 0 0
$$905$$ −6.95372 −0.231150
$$906$$ 0 0
$$907$$ 0.754028 0.0250371 0.0125185 0.999922i $$-0.496015\pi$$
0.0125185 + 0.999922i $$0.496015\pi$$
$$908$$ 0 0
$$909$$ −7.96988 −0.264344
$$910$$ 0 0
$$911$$ 53.1413 1.76065 0.880325 0.474371i $$-0.157325\pi$$
0.880325 + 0.474371i $$0.157325\pi$$
$$912$$ 0 0
$$913$$ −5.93368 −0.196376
$$914$$ 0 0
$$915$$ −8.76838 −0.289874
$$916$$ 0 0
$$917$$ −10.8403 −0.357979
$$918$$ 0 0
$$919$$ −37.4202 −1.23438 −0.617189 0.786815i $$-0.711728\pi$$
−0.617189 + 0.786815i $$0.711728\pi$$
$$920$$ 0 0
$$921$$ −43.9196 −1.44720
$$922$$ 0 0
$$923$$ −25.9526 −0.854241
$$924$$ 0 0
$$925$$ 0.0947438 0.00311516
$$926$$ 0 0
$$927$$ −3.92880 −0.129039
$$928$$ 0 0
$$929$$ −19.1981 −0.629871 −0.314935 0.949113i $$-0.601983\pi$$
−0.314935 + 0.949113i $$0.601983\pi$$
$$930$$ 0 0
$$931$$ −18.3327 −0.600830
$$932$$ 0 0
$$933$$ 39.0873 1.27966
$$934$$ 0 0
$$935$$ 3.26638 0.106822
$$936$$ 0 0
$$937$$ −20.0095 −0.653681 −0.326840 0.945080i $$-0.605984\pi$$
−0.326840 + 0.945080i $$0.605984\pi$$
$$938$$ 0 0
$$939$$ −25.8458 −0.843445
$$940$$ 0 0
$$941$$ 15.3327 0.499832 0.249916 0.968268i $$-0.419597\pi$$
0.249916 + 0.968268i $$0.419597\pi$$
$$942$$ 0 0
$$943$$ −43.8392 −1.42760
$$944$$ 0 0
$$945$$ 28.1326 0.915155
$$946$$ 0 0
$$947$$ 29.8217 0.969076 0.484538 0.874770i $$-0.338988\pi$$
0.484538 + 0.874770i $$0.338988\pi$$
$$948$$ 0 0
$$949$$ −41.4959 −1.34702
$$950$$ 0 0
$$951$$ 48.8464 1.58395
$$952$$ 0 0
$$953$$ 31.4938 1.02018 0.510091 0.860120i $$-0.329612\pi$$
0.510091 + 0.860120i $$0.329612\pi$$
$$954$$ 0 0
$$955$$ −3.91254 −0.126607
$$956$$ 0 0
$$957$$ 5.02514 0.162440
$$958$$ 0 0
$$959$$ −11.0200 −0.355856
$$960$$ 0 0
$$961$$ 0.242050 0.00780806
$$962$$ 0 0
$$963$$ −7.94528 −0.256033
$$964$$ 0 0
$$965$$ 9.20734 0.296395
$$966$$ 0 0
$$967$$ −1.16143 −0.0373490 −0.0186745 0.999826i $$-0.505945\pi$$
−0.0186745 + 0.999826i $$0.505945\pi$$
$$968$$ 0 0
$$969$$ −1.65673 −0.0532220
$$970$$ 0 0
$$971$$ −18.4557 −0.592272 −0.296136 0.955146i $$-0.595698\pi$$
−0.296136 + 0.955146i $$0.595698\pi$$
$$972$$ 0 0
$$973$$ 113.345 3.63367
$$974$$ 0 0
$$975$$ 7.03316 0.225241
$$976$$ 0 0
$$977$$ 5.62735 0.180035 0.0900175 0.995940i $$-0.471308\pi$$
0.0900175 + 0.995940i $$0.471308\pi$$
$$978$$ 0 0
$$979$$ −6.64107 −0.212249
$$980$$ 0 0
$$981$$ 10.0104 0.319608
$$982$$ 0 0
$$983$$ 25.1228 0.801293 0.400647 0.916233i $$-0.368786\pi$$
0.400647 + 0.916233i $$0.368786\pi$$
$$984$$ 0 0
$$985$$ 3.84622 0.122551
$$986$$ 0 0
$$987$$ 95.2382 3.03147
$$988$$ 0 0
$$989$$ 20.6865 0.657793
$$990$$ 0 0
$$991$$ 41.8749 1.33020 0.665100 0.746754i $$-0.268389\pi$$
0.665100 + 0.746754i $$0.268389\pi$$
$$992$$ 0 0
$$993$$ −15.3295 −0.486467
$$994$$ 0 0
$$995$$ −18.2864 −0.579718
$$996$$ 0 0
$$997$$ −37.0346 −1.17290 −0.586449 0.809986i $$-0.699475\pi$$
−0.586449 + 0.809986i $$0.699475\pi$$
$$998$$ 0 0
$$999$$ 0.529566 0.0167547
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 1520.2.a.t.1.2 4
4.3 odd 2 95.2.a.b.1.3 4
5.4 even 2 7600.2.a.cf.1.3 4
8.3 odd 2 6080.2.a.cc.1.2 4
8.5 even 2 6080.2.a.ch.1.3 4
12.11 even 2 855.2.a.m.1.2 4
20.3 even 4 475.2.b.e.324.4 8
20.7 even 4 475.2.b.e.324.5 8
20.19 odd 2 475.2.a.i.1.2 4
28.27 even 2 4655.2.a.y.1.3 4
60.59 even 2 4275.2.a.bo.1.3 4
76.75 even 2 1805.2.a.p.1.2 4
380.379 even 2 9025.2.a.bf.1.3 4

By twisted newform
Twist Min Dim Char Parity Ord Type
95.2.a.b.1.3 4 4.3 odd 2
475.2.a.i.1.2 4 20.19 odd 2
475.2.b.e.324.4 8 20.3 even 4
475.2.b.e.324.5 8 20.7 even 4
855.2.a.m.1.2 4 12.11 even 2
1520.2.a.t.1.2 4 1.1 even 1 trivial
1805.2.a.p.1.2 4 76.75 even 2
4275.2.a.bo.1.3 4 60.59 even 2
4655.2.a.y.1.3 4 28.27 even 2
6080.2.a.cc.1.2 4 8.3 odd 2
6080.2.a.ch.1.3 4 8.5 even 2
7600.2.a.cf.1.3 4 5.4 even 2
9025.2.a.bf.1.3 4 380.379 even 2