# Properties

 Label 4680.2.l.e.2809.6 Level $4680$ Weight $2$ Character 4680.2809 Analytic conductor $37.370$ Analytic rank $0$ Dimension $6$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4680,2,Mod(2809,4680)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4680, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 1, 0]))

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

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

chi := DirichletCharacter("4680.2809");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$4680 = 2^{3} \cdot 3^{2} \cdot 5 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4680.l (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 Self dual: no Analytic conductor: $$37.3699881460$$ Analytic rank: $$0$$ Dimension: $$6$$ Coefficient field: 6.0.5161984.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{6} - 4x^{3} + 25x^{2} - 20x + 8$$ x^6 - 4*x^3 + 25*x^2 - 20*x + 8 Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 1560) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 2809.6 Root $$-1.75233 - 1.75233i$$ of defining polynomial Character $$\chi$$ $$=$$ 4680.2809 Dual form 4680.2.l.e.2809.5

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+(1.75233 + 1.38900i) q^{5} +O(q^{10})$$ $$q+(1.75233 + 1.38900i) q^{5} -1.50466 q^{11} +1.00000i q^{13} -2.72666i q^{17} -0.726656 q^{19} +4.72666i q^{23} +(1.14134 + 4.86799i) q^{25} -7.55602 q^{29} -3.00933 q^{31} +5.00933i q^{37} -5.78734 q^{41} +2.72666i q^{43} -10.2313i q^{47} +7.00000 q^{49} +7.55602i q^{53} +(-2.63667 - 2.08998i) q^{55} -12.5140 q^{59} +6.28267 q^{61} +(-1.38900 + 1.75233i) q^{65} +12.5653i q^{67} -4.77801 q^{71} +12.0187i q^{73} -5.27334 q^{79} -7.78734i q^{83} +(3.78734 - 4.77801i) q^{85} +1.78734 q^{89} +(-1.27334 - 1.00933i) q^{95} +6.00000i q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q+O(q^{10})$$ 6 * q $$6 q + 12 q^{11} + 4 q^{19} - 10 q^{25} - 20 q^{29} + 24 q^{31} + 20 q^{41} + 42 q^{49} - 20 q^{55} - 12 q^{59} + 4 q^{61} - 2 q^{65} - 16 q^{71} - 40 q^{79} - 32 q^{85} - 44 q^{89} - 16 q^{95}+O(q^{100})$$ 6 * q + 12 * q^11 + 4 * q^19 - 10 * q^25 - 20 * q^29 + 24 * q^31 + 20 * q^41 + 42 * q^49 - 20 * q^55 - 12 * q^59 + 4 * q^61 - 2 * q^65 - 16 * q^71 - 40 * q^79 - 32 * q^85 - 44 * q^89 - 16 * q^95

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/4680\mathbb{Z}\right)^\times$$.

 $$n$$ $$937$$ $$1081$$ $$2081$$ $$2341$$ $$3511$$ $$\chi(n)$$ $$-1$$ $$1$$ $$1$$ $$1$$ $$1$$

## Coefficient data

For each $$n$$ we display the coefficients of the $$q$$-expansion $$a_n$$, the Satake parameters $$\alpha_p$$, and the Satake angles $$\theta_p = \textrm{Arg}(\alpha_p)$$.

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0 0
$$3$$ 0 0
$$4$$ 0 0
$$5$$ 1.75233 + 1.38900i 0.783667 + 0.621181i
$$6$$ 0 0
$$7$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$8$$ 0 0
$$9$$ 0 0
$$10$$ 0 0
$$11$$ −1.50466 −0.453673 −0.226837 0.973933i $$-0.572838\pi$$
−0.226837 + 0.973933i $$0.572838\pi$$
$$12$$ 0 0
$$13$$ 1.00000i 0.277350i
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ 2.72666i 0.661311i −0.943751 0.330656i $$-0.892730\pi$$
0.943751 0.330656i $$-0.107270\pi$$
$$18$$ 0 0
$$19$$ −0.726656 −0.166706 −0.0833532 0.996520i $$-0.526563\pi$$
−0.0833532 + 0.996520i $$0.526563\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ 4.72666i 0.985576i 0.870149 + 0.492788i $$0.164022\pi$$
−0.870149 + 0.492788i $$0.835978\pi$$
$$24$$ 0 0
$$25$$ 1.14134 + 4.86799i 0.228267 + 0.973599i
$$26$$ 0 0
$$27$$ 0 0
$$28$$ 0 0
$$29$$ −7.55602 −1.40312 −0.701558 0.712612i $$-0.747512\pi$$
−0.701558 + 0.712612i $$0.747512\pi$$
$$30$$ 0 0
$$31$$ −3.00933 −0.540491 −0.270246 0.962791i $$-0.587105\pi$$
−0.270246 + 0.962791i $$0.587105\pi$$
$$32$$ 0 0
$$33$$ 0 0
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 5.00933i 0.823529i 0.911290 + 0.411764i $$0.135087\pi$$
−0.911290 + 0.411764i $$0.864913\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 0 0
$$41$$ −5.78734 −0.903830 −0.451915 0.892061i $$-0.649259\pi$$
−0.451915 + 0.892061i $$0.649259\pi$$
$$42$$ 0 0
$$43$$ 2.72666i 0.415811i 0.978149 + 0.207906i $$0.0666647\pi$$
−0.978149 + 0.207906i $$0.933335\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ 10.2313i 1.49239i −0.665727 0.746196i $$-0.731878\pi$$
0.665727 0.746196i $$-0.268122\pi$$
$$48$$ 0 0
$$49$$ 7.00000 1.00000
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 0 0
$$53$$ 7.55602i 1.03790i 0.854805 + 0.518949i $$0.173677\pi$$
−0.854805 + 0.518949i $$0.826323\pi$$
$$54$$ 0 0
$$55$$ −2.63667 2.08998i −0.355529 0.281813i
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 0 0
$$59$$ −12.5140 −1.62918 −0.814592 0.580035i $$-0.803039\pi$$
−0.814592 + 0.580035i $$0.803039\pi$$
$$60$$ 0 0
$$61$$ 6.28267 0.804414 0.402207 0.915549i $$-0.368243\pi$$
0.402207 + 0.915549i $$0.368243\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ −1.38900 + 1.75233i −0.172285 + 0.217350i
$$66$$ 0 0
$$67$$ 12.5653i 1.53510i 0.640988 + 0.767551i $$0.278525\pi$$
−0.640988 + 0.767551i $$0.721475\pi$$
$$68$$ 0 0
$$69$$ 0 0
$$70$$ 0 0
$$71$$ −4.77801 −0.567045 −0.283523 0.958966i $$-0.591503\pi$$
−0.283523 + 0.958966i $$0.591503\pi$$
$$72$$ 0 0
$$73$$ 12.0187i 1.40668i 0.710855 + 0.703339i $$0.248308\pi$$
−0.710855 + 0.703339i $$0.751692\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ 0 0
$$78$$ 0 0
$$79$$ −5.27334 −0.593297 −0.296649 0.954987i $$-0.595869\pi$$
−0.296649 + 0.954987i $$0.595869\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ 0 0
$$83$$ 7.78734i 0.854771i −0.904069 0.427386i $$-0.859435\pi$$
0.904069 0.427386i $$-0.140565\pi$$
$$84$$ 0 0
$$85$$ 3.78734 4.77801i 0.410794 0.518248i
$$86$$ 0 0
$$87$$ 0 0
$$88$$ 0 0
$$89$$ 1.78734 0.189457 0.0947286 0.995503i $$-0.469802\pi$$
0.0947286 + 0.995503i $$0.469802\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 0 0
$$93$$ 0 0
$$94$$ 0 0
$$95$$ −1.27334 1.00933i −0.130642 0.103555i
$$96$$ 0 0
$$97$$ 6.00000i 0.609208i 0.952479 + 0.304604i $$0.0985241\pi$$
−0.952479 + 0.304604i $$0.901476\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 0 0
$$101$$ 2.99067 0.297583 0.148791 0.988869i $$-0.452462\pi$$
0.148791 + 0.988869i $$0.452462\pi$$
$$102$$ 0 0
$$103$$ 0.443984i 0.0437471i −0.999761 0.0218735i $$-0.993037\pi$$
0.999761 0.0218735i $$-0.00696312\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 7.00933i 0.677617i 0.940855 + 0.338809i $$0.110024\pi$$
−0.940855 + 0.338809i $$0.889976\pi$$
$$108$$ 0 0
$$109$$ 13.8387 1.32551 0.662753 0.748838i $$-0.269388\pi$$
0.662753 + 0.748838i $$0.269388\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 0 0
$$113$$ 4.28267i 0.402880i 0.979501 + 0.201440i $$0.0645621\pi$$
−0.979501 + 0.201440i $$0.935438\pi$$
$$114$$ 0 0
$$115$$ −6.56534 + 8.28267i −0.612222 + 0.772363i
$$116$$ 0 0
$$117$$ 0 0
$$118$$ 0 0
$$119$$ 0 0
$$120$$ 0 0
$$121$$ −8.73599 −0.794180
$$122$$ 0 0
$$123$$ 0 0
$$124$$ 0 0
$$125$$ −4.76166 + 10.1157i −0.425896 + 0.904772i
$$126$$ 0 0
$$127$$ 5.71733i 0.507331i 0.967292 + 0.253665i $$0.0816362\pi$$
−0.967292 + 0.253665i $$0.918364\pi$$
$$128$$ 0 0
$$129$$ 0 0
$$130$$ 0 0
$$131$$ −5.55602 −0.485431 −0.242716 0.970097i $$-0.578038\pi$$
−0.242716 + 0.970097i $$0.578038\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ 6.67531i 0.570310i −0.958481 0.285155i $$-0.907955\pi$$
0.958481 0.285155i $$-0.0920450\pi$$
$$138$$ 0 0
$$139$$ −19.4720 −1.65159 −0.825795 0.563970i $$-0.809273\pi$$
−0.825795 + 0.563970i $$0.809273\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 0 0
$$143$$ 1.50466i 0.125826i
$$144$$ 0 0
$$145$$ −13.2406 10.4953i −1.09958 0.871590i
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ −14.5140 −1.18903 −0.594516 0.804084i $$-0.702656\pi$$
−0.594516 + 0.804084i $$0.702656\pi$$
$$150$$ 0 0
$$151$$ −4.46264 −0.363165 −0.181582 0.983376i $$-0.558122\pi$$
−0.181582 + 0.983376i $$0.558122\pi$$
$$152$$ 0 0
$$153$$ 0 0
$$154$$ 0 0
$$155$$ −5.27334 4.17997i −0.423565 0.335743i
$$156$$ 0 0
$$157$$ 8.30133i 0.662518i −0.943540 0.331259i $$-0.892527\pi$$
0.943540 0.331259i $$-0.107473\pi$$
$$158$$ 0 0
$$159$$ 0 0
$$160$$ 0 0
$$161$$ 0 0
$$162$$ 0 0
$$163$$ 11.0093i 0.862317i −0.902276 0.431159i $$-0.858105\pi$$
0.902276 0.431159i $$-0.141895\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 5.76868i 0.446394i −0.974773 0.223197i $$-0.928351\pi$$
0.974773 0.223197i $$-0.0716493\pi$$
$$168$$ 0 0
$$169$$ −1.00000 −0.0769231
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 0 0
$$173$$ 4.90663i 0.373044i −0.982451 0.186522i $$-0.940278\pi$$
0.982451 0.186522i $$-0.0597215\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ 0 0
$$178$$ 0 0
$$179$$ 9.45331 0.706574 0.353287 0.935515i $$-0.385064\pi$$
0.353287 + 0.935515i $$0.385064\pi$$
$$180$$ 0 0
$$181$$ 17.4720 1.29868 0.649341 0.760498i $$-0.275045\pi$$
0.649341 + 0.760498i $$0.275045\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 0 0
$$185$$ −6.95798 + 8.77801i −0.511561 + 0.645372i
$$186$$ 0 0
$$187$$ 4.10270i 0.300019i
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 11.1120 0.804038 0.402019 0.915631i $$-0.368309\pi$$
0.402019 + 0.915631i $$0.368309\pi$$
$$192$$ 0 0
$$193$$ 6.10270i 0.439282i 0.975581 + 0.219641i $$0.0704886\pi$$
−0.975581 + 0.219641i $$0.929511\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 21.3620i 1.52198i 0.648764 + 0.760990i $$0.275286\pi$$
−0.648764 + 0.760990i $$0.724714\pi$$
$$198$$ 0 0
$$199$$ −8.38538 −0.594423 −0.297212 0.954812i $$-0.596057\pi$$
−0.297212 + 0.954812i $$0.596057\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 0 0
$$203$$ 0 0
$$204$$ 0 0
$$205$$ −10.1413 8.03863i −0.708302 0.561443i
$$206$$ 0 0
$$207$$ 0 0
$$208$$ 0 0
$$209$$ 1.09337 0.0756303
$$210$$ 0 0
$$211$$ −1.27334 −0.0876606 −0.0438303 0.999039i $$-0.513956\pi$$
−0.0438303 + 0.999039i $$0.513956\pi$$
$$212$$ 0 0
$$213$$ 0 0
$$214$$ 0 0
$$215$$ −3.78734 + 4.77801i −0.258294 + 0.325857i
$$216$$ 0 0
$$217$$ 0 0
$$218$$ 0 0
$$219$$ 0 0
$$220$$ 0 0
$$221$$ 2.72666 0.183415
$$222$$ 0 0
$$223$$ 16.4626i 1.10242i −0.834367 0.551210i $$-0.814166\pi$$
0.834367 0.551210i $$-0.185834\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ 12.2500i 0.813060i −0.913638 0.406530i $$-0.866739\pi$$
0.913638 0.406530i $$-0.133261\pi$$
$$228$$ 0 0
$$229$$ 1.27334 0.0841449 0.0420725 0.999115i $$-0.486604\pi$$
0.0420725 + 0.999115i $$0.486604\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ 18.3013i 1.19896i 0.800390 + 0.599480i $$0.204626\pi$$
−0.800390 + 0.599480i $$0.795374\pi$$
$$234$$ 0 0
$$235$$ 14.2113 17.9287i 0.927046 1.16954i
$$236$$ 0 0
$$237$$ 0 0
$$238$$ 0 0
$$239$$ 6.23132 0.403071 0.201535 0.979481i $$-0.435407\pi$$
0.201535 + 0.979481i $$0.435407\pi$$
$$240$$ 0 0
$$241$$ 19.5560 1.25971 0.629857 0.776711i $$-0.283114\pi$$
0.629857 + 0.776711i $$0.283114\pi$$
$$242$$ 0 0
$$243$$ 0 0
$$244$$ 0 0
$$245$$ 12.2663 + 9.72303i 0.783667 + 0.621181i
$$246$$ 0 0
$$247$$ 0.726656i 0.0462360i
$$248$$ 0 0
$$249$$ 0 0
$$250$$ 0 0
$$251$$ −26.4813 −1.67148 −0.835742 0.549122i $$-0.814962\pi$$
−0.835742 + 0.549122i $$0.814962\pi$$
$$252$$ 0 0
$$253$$ 7.11203i 0.447130i
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ 14.8294i 0.925030i −0.886611 0.462515i $$-0.846947\pi$$
0.886611 0.462515i $$-0.153053\pi$$
$$258$$ 0 0
$$259$$ 0 0
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 0 0
$$263$$ 22.9507i 1.41520i 0.706612 + 0.707601i $$0.250223\pi$$
−0.706612 + 0.707601i $$0.749777\pi$$
$$264$$ 0 0
$$265$$ −10.4953 + 13.2406i −0.644723 + 0.813367i
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 0 0
$$269$$ −27.9160 −1.70207 −0.851033 0.525112i $$-0.824023\pi$$
−0.851033 + 0.525112i $$0.824023\pi$$
$$270$$ 0 0
$$271$$ −5.65872 −0.343743 −0.171871 0.985119i $$-0.554981\pi$$
−0.171871 + 0.985119i $$0.554981\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ −1.71733 7.32469i −0.103559 0.441696i
$$276$$ 0 0
$$277$$ 7.73599i 0.464810i 0.972619 + 0.232405i $$0.0746595\pi$$
−0.972619 + 0.232405i $$0.925340\pi$$
$$278$$ 0 0
$$279$$ 0 0
$$280$$ 0 0
$$281$$ −5.11929 −0.305391 −0.152696 0.988273i $$-0.548795\pi$$
−0.152696 + 0.988273i $$0.548795\pi$$
$$282$$ 0 0
$$283$$ 6.90663i 0.410556i −0.978704 0.205278i $$-0.934190\pi$$
0.978704 0.205278i $$-0.0658099\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 0 0
$$288$$ 0 0
$$289$$ 9.56534 0.562667
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 0 0
$$293$$ 26.3527i 1.53954i 0.638321 + 0.769770i $$0.279629\pi$$
−0.638321 + 0.769770i $$0.720371\pi$$
$$294$$ 0 0
$$295$$ −21.9287 17.3820i −1.27674 1.01202i
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ −4.72666 −0.273350
$$300$$ 0 0
$$301$$ 0 0
$$302$$ 0 0
$$303$$ 0 0
$$304$$ 0 0
$$305$$ 11.0093 + 8.72666i 0.630392 + 0.499687i
$$306$$ 0 0
$$307$$ 20.0000i 1.14146i 0.821138 + 0.570730i $$0.193340\pi$$
−0.821138 + 0.570730i $$0.806660\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 0 0
$$311$$ 11.8973 0.674634 0.337317 0.941391i $$-0.390481\pi$$
0.337317 + 0.941391i $$0.390481\pi$$
$$312$$ 0 0
$$313$$ 17.7360i 1.00250i −0.865303 0.501249i $$-0.832874\pi$$
0.865303 0.501249i $$-0.167126\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ 1.32469i 0.0744023i 0.999308 + 0.0372011i $$0.0118442\pi$$
−0.999308 + 0.0372011i $$0.988156\pi$$
$$318$$ 0 0
$$319$$ 11.3693 0.636557
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 0 0
$$323$$ 1.98134i 0.110245i
$$324$$ 0 0
$$325$$ −4.86799 + 1.14134i −0.270028 + 0.0633099i
$$326$$ 0 0
$$327$$ 0 0
$$328$$ 0 0
$$329$$ 0 0
$$330$$ 0 0
$$331$$ −3.73599 −0.205348 −0.102674 0.994715i $$-0.532740\pi$$
−0.102674 + 0.994715i $$0.532740\pi$$
$$332$$ 0 0
$$333$$ 0 0
$$334$$ 0 0
$$335$$ −17.4533 + 22.0187i −0.953576 + 1.20301i
$$336$$ 0 0
$$337$$ 23.3947i 1.27439i 0.770702 + 0.637195i $$0.219906\pi$$
−0.770702 + 0.637195i $$0.780094\pi$$
$$338$$ 0 0
$$339$$ 0 0
$$340$$ 0 0
$$341$$ 4.52803 0.245207
$$342$$ 0 0
$$343$$ 0 0
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ 30.5840i 1.64184i 0.571047 + 0.820918i $$0.306538\pi$$
−0.571047 + 0.820918i $$0.693462\pi$$
$$348$$ 0 0
$$349$$ −1.37605 −0.0736581 −0.0368290 0.999322i $$-0.511726\pi$$
−0.0368290 + 0.999322i $$0.511726\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 0 0
$$353$$ 19.9087i 1.05963i −0.848112 0.529817i $$-0.822261\pi$$
0.848112 0.529817i $$-0.177739\pi$$
$$354$$ 0 0
$$355$$ −8.37266 6.63667i −0.444374 0.352238i
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ 26.6940 1.40885 0.704427 0.709777i $$-0.251204\pi$$
0.704427 + 0.709777i $$0.251204\pi$$
$$360$$ 0 0
$$361$$ −18.4720 −0.972209
$$362$$ 0 0
$$363$$ 0 0
$$364$$ 0 0
$$365$$ −16.6940 + 21.0607i −0.873802 + 1.10237i
$$366$$ 0 0
$$367$$ 8.12136i 0.423932i −0.977277 0.211966i $$-0.932013\pi$$
0.977277 0.211966i $$-0.0679865\pi$$
$$368$$ 0 0
$$369$$ 0 0
$$370$$ 0 0
$$371$$ 0 0
$$372$$ 0 0
$$373$$ 9.47197i 0.490440i 0.969467 + 0.245220i $$0.0788602\pi$$
−0.969467 + 0.245220i $$0.921140\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 7.55602i 0.389155i
$$378$$ 0 0
$$379$$ −16.7267 −0.859191 −0.429595 0.903022i $$-0.641344\pi$$
−0.429595 + 0.903022i $$0.641344\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 0 0
$$383$$ 19.6846i 1.00584i −0.864334 0.502919i $$-0.832259\pi$$
0.864334 0.502919i $$-0.167741\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ 0 0
$$388$$ 0 0
$$389$$ −14.0000 −0.709828 −0.354914 0.934899i $$-0.615490\pi$$
−0.354914 + 0.934899i $$0.615490\pi$$
$$390$$ 0 0
$$391$$ 12.8880 0.651773
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 0 0
$$395$$ −9.24065 7.32469i −0.464948 0.368545i
$$396$$ 0 0
$$397$$ 3.35061i 0.168162i 0.996459 + 0.0840812i $$0.0267955\pi$$
−0.996459 + 0.0840812i $$0.973205\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 36.3713 1.81630 0.908149 0.418647i $$-0.137496\pi$$
0.908149 + 0.418647i $$0.137496\pi$$
$$402$$ 0 0
$$403$$ 3.00933i 0.149905i
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 7.53736i 0.373613i
$$408$$ 0 0
$$409$$ −35.5933 −1.75998 −0.879988 0.474995i $$-0.842450\pi$$
−0.879988 + 0.474995i $$0.842450\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 0 0
$$413$$ 0 0
$$414$$ 0 0
$$415$$ 10.8166 13.6460i 0.530968 0.669856i
$$416$$ 0 0
$$417$$ 0 0
$$418$$ 0 0
$$419$$ −13.9160 −0.679839 −0.339919 0.940455i $$-0.610400\pi$$
−0.339919 + 0.940455i $$0.610400\pi$$
$$420$$ 0 0
$$421$$ 4.70800 0.229454 0.114727 0.993397i $$-0.463401\pi$$
0.114727 + 0.993397i $$0.463401\pi$$
$$422$$ 0 0
$$423$$ 0 0
$$424$$ 0 0
$$425$$ 13.2733 3.11203i 0.643852 0.150956i
$$426$$ 0 0
$$427$$ 0 0
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ −7.89004 −0.380050 −0.190025 0.981779i $$-0.560857\pi$$
−0.190025 + 0.981779i $$0.560857\pi$$
$$432$$ 0 0
$$433$$ 2.30133i 0.110595i −0.998470 0.0552974i $$-0.982389\pi$$
0.998470 0.0552974i $$-0.0176107\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ 3.43466i 0.164302i
$$438$$ 0 0
$$439$$ −41.1307 −1.96306 −0.981530 0.191307i $$-0.938728\pi$$
−0.981530 + 0.191307i $$0.938728\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ 29.5933i 1.40602i 0.711179 + 0.703011i $$0.248161\pi$$
−0.711179 + 0.703011i $$0.751839\pi$$
$$444$$ 0 0
$$445$$ 3.13201 + 2.48262i 0.148471 + 0.117687i
$$446$$ 0 0
$$447$$ 0 0
$$448$$ 0 0
$$449$$ −11.4461 −0.540173 −0.270086 0.962836i $$-0.587052\pi$$
−0.270086 + 0.962836i $$0.587052\pi$$
$$450$$ 0 0
$$451$$ 8.70800 0.410044
$$452$$ 0 0
$$453$$ 0 0
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 16.5467i 0.774021i 0.922075 + 0.387011i $$0.126492\pi$$
−0.922075 + 0.387011i $$0.873508\pi$$
$$458$$ 0 0
$$459$$ 0 0
$$460$$ 0 0
$$461$$ −3.50466 −0.163228 −0.0816142 0.996664i $$-0.526008\pi$$
−0.0816142 + 0.996664i $$0.526008\pi$$
$$462$$ 0 0
$$463$$ 16.0000i 0.743583i −0.928316 0.371792i $$-0.878744\pi$$
0.928316 0.371792i $$-0.121256\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ 6.44398i 0.298192i −0.988823 0.149096i $$-0.952364\pi$$
0.988823 0.149096i $$-0.0476363\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 0 0
$$473$$ 4.10270i 0.188642i
$$474$$ 0 0
$$475$$ −0.829359 3.53736i −0.0380536 0.162305i
$$476$$ 0 0
$$477$$ 0 0
$$478$$ 0 0
$$479$$ 22.2313 1.01577 0.507887 0.861423i $$-0.330427\pi$$
0.507887 + 0.861423i $$0.330427\pi$$
$$480$$ 0 0
$$481$$ −5.00933 −0.228406
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ −8.33402 + 10.5140i −0.378429 + 0.477416i
$$486$$ 0 0
$$487$$ 13.9160i 0.630592i −0.948993 0.315296i $$-0.897896\pi$$
0.948993 0.315296i $$-0.102104\pi$$
$$488$$ 0 0
$$489$$ 0 0
$$490$$ 0 0
$$491$$ 28.1400 1.26994 0.634971 0.772536i $$-0.281012\pi$$
0.634971 + 0.772536i $$0.281012\pi$$
$$492$$ 0 0
$$493$$ 20.6027i 0.927897i
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 0 0
$$498$$ 0 0
$$499$$ −38.8480 −1.73908 −0.869538 0.493866i $$-0.835583\pi$$
−0.869538 + 0.493866i $$0.835583\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 0 0
$$503$$ 8.19863i 0.365559i 0.983154 + 0.182779i $$0.0585094\pi$$
−0.983154 + 0.182779i $$0.941491\pi$$
$$504$$ 0 0
$$505$$ 5.24065 + 4.15405i 0.233206 + 0.184853i
$$506$$ 0 0
$$507$$ 0 0
$$508$$ 0 0
$$509$$ 16.0700 0.712291 0.356145 0.934431i $$-0.384091\pi$$
0.356145 + 0.934431i $$0.384091\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ 0 0
$$513$$ 0 0
$$514$$ 0 0
$$515$$ 0.616696 0.778008i 0.0271749 0.0342831i
$$516$$ 0 0
$$517$$ 15.3947i 0.677058i
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 0 0
$$521$$ 23.0280 1.00887 0.504437 0.863448i $$-0.331700\pi$$
0.504437 + 0.863448i $$0.331700\pi$$
$$522$$ 0 0
$$523$$ 5.47875i 0.239569i −0.992800 0.119784i $$-0.961780\pi$$
0.992800 0.119784i $$-0.0382204\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 8.20541i 0.357433i
$$528$$ 0 0
$$529$$ 0.658719 0.0286399
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 0 0
$$533$$ 5.78734i 0.250677i
$$534$$ 0 0
$$535$$ −9.73599 + 12.2827i −0.420923 + 0.531026i
$$536$$ 0 0
$$537$$ 0 0
$$538$$ 0 0
$$539$$ −10.5327 −0.453673
$$540$$ 0 0
$$541$$ 17.8387 0.766945 0.383473 0.923552i $$-0.374728\pi$$
0.383473 + 0.923552i $$0.374728\pi$$
$$542$$ 0 0
$$543$$ 0 0
$$544$$ 0 0
$$545$$ 24.2500 + 19.2220i 1.03875 + 0.823380i
$$546$$ 0 0
$$547$$ 29.1053i 1.24445i 0.782838 + 0.622225i $$0.213771\pi$$
−0.782838 + 0.622225i $$0.786229\pi$$
$$548$$ 0 0
$$549$$ 0 0
$$550$$ 0 0
$$551$$ 5.49063 0.233909
$$552$$ 0 0
$$553$$ 0 0
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ 44.0114i 1.86482i −0.361399 0.932411i $$-0.617701\pi$$
0.361399 0.932411i $$-0.382299\pi$$
$$558$$ 0 0
$$559$$ −2.72666 −0.115325
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ 7.43466i 0.313333i 0.987652 + 0.156667i $$0.0500748\pi$$
−0.987652 + 0.156667i $$0.949925\pi$$
$$564$$ 0 0
$$565$$ −5.94865 + 7.50466i −0.250262 + 0.315724i
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ 8.48130 0.355555 0.177777 0.984071i $$-0.443109\pi$$
0.177777 + 0.984071i $$0.443109\pi$$
$$570$$ 0 0
$$571$$ 40.7826 1.70670 0.853350 0.521339i $$-0.174567\pi$$
0.853350 + 0.521339i $$0.174567\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 0 0
$$575$$ −23.0093 + 5.39470i −0.959555 + 0.224975i
$$576$$ 0 0
$$577$$ 1.57467i 0.0655545i 0.999463 + 0.0327773i $$0.0104352\pi$$
−0.999463 + 0.0327773i $$0.989565\pi$$
$$578$$ 0 0
$$579$$ 0 0
$$580$$ 0 0
$$581$$ 0 0
$$582$$ 0 0
$$583$$ 11.3693i 0.470867i
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ 27.8247i 1.14845i −0.818699 0.574223i $$-0.805304\pi$$
0.818699 0.574223i $$-0.194696\pi$$
$$588$$ 0 0
$$589$$ 2.18675 0.0901034
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 0 0
$$593$$ 32.6940i 1.34258i 0.741195 + 0.671290i $$0.234260\pi$$
−0.741195 + 0.671290i $$0.765740\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ 0 0
$$598$$ 0 0
$$599$$ −32.1400 −1.31321 −0.656603 0.754237i $$-0.728007\pi$$
−0.656603 + 0.754237i $$0.728007\pi$$
$$600$$ 0 0
$$601$$ 40.8667 1.66699 0.833493 0.552530i $$-0.186337\pi$$
0.833493 + 0.552530i $$0.186337\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ 0 0
$$605$$ −15.3083 12.1343i −0.622373 0.493330i
$$606$$ 0 0
$$607$$ 14.9907i 0.608453i 0.952600 + 0.304226i $$0.0983979\pi$$
−0.952600 + 0.304226i $$0.901602\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 10.2313 0.413915
$$612$$ 0 0
$$613$$ 13.5747i 0.548276i −0.961690 0.274138i $$-0.911608\pi$$
0.961690 0.274138i $$-0.0883925\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 4.69396i 0.188972i 0.995526 + 0.0944859i $$0.0301207\pi$$
−0.995526 + 0.0944859i $$0.969879\pi$$
$$618$$ 0 0
$$619$$ 20.8667 0.838702 0.419351 0.907824i $$-0.362258\pi$$
0.419351 + 0.907824i $$0.362258\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 0 0
$$623$$ 0 0
$$624$$ 0 0
$$625$$ −22.3947 + 11.1120i −0.895788 + 0.444481i
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ 13.6587 0.544609
$$630$$ 0 0
$$631$$ 45.9533 1.82937 0.914685 0.404167i $$-0.132438\pi$$
0.914685 + 0.404167i $$0.132438\pi$$
$$632$$ 0 0
$$633$$ 0 0
$$634$$ 0 0
$$635$$ −7.94139 + 10.0187i −0.315144 + 0.397578i
$$636$$ 0 0
$$637$$ 7.00000i 0.277350i
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 0 0
$$641$$ −20.0187 −0.790689 −0.395345 0.918533i $$-0.629375\pi$$
−0.395345 + 0.918533i $$0.629375\pi$$
$$642$$ 0 0
$$643$$ 26.5840i 1.04837i 0.851604 + 0.524185i $$0.175630\pi$$
−0.851604 + 0.524185i $$0.824370\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 9.39470i 0.369344i −0.982800 0.184672i $$-0.940878\pi$$
0.982800 0.184672i $$-0.0591223\pi$$
$$648$$ 0 0
$$649$$ 18.8294 0.739117
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ 2.00000i 0.0782660i 0.999234 + 0.0391330i $$0.0124596\pi$$
−0.999234 + 0.0391330i $$0.987540\pi$$
$$654$$ 0 0
$$655$$ −9.73599 7.71733i −0.380416 0.301541i
$$656$$ 0 0
$$657$$ 0 0
$$658$$ 0 0
$$659$$ 47.6774 1.85725 0.928623 0.371024i $$-0.120993\pi$$
0.928623 + 0.371024i $$0.120993\pi$$
$$660$$ 0 0
$$661$$ −22.0959 −0.859432 −0.429716 0.902964i $$-0.641386\pi$$
−0.429716 + 0.902964i $$0.641386\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 35.7147i 1.38288i
$$668$$ 0 0
$$669$$ 0 0
$$670$$ 0 0
$$671$$ −9.45331 −0.364941
$$672$$ 0 0
$$673$$ 48.6027i 1.87349i 0.350006 + 0.936747i $$0.386180\pi$$
−0.350006 + 0.936747i $$0.613820\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ 10.4253i 0.400678i −0.979727 0.200339i $$-0.935796\pi$$
0.979727 0.200339i $$-0.0642043\pi$$
$$678$$ 0 0
$$679$$ 0 0
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 0 0
$$683$$ 9.13795i 0.349654i −0.984599 0.174827i $$-0.944063\pi$$
0.984599 0.174827i $$-0.0559366\pi$$
$$684$$ 0 0
$$685$$ 9.27203 11.6974i 0.354266 0.446933i
$$686$$ 0 0
$$687$$ 0 0
$$688$$ 0 0
$$689$$ −7.55602 −0.287861
$$690$$ 0 0
$$691$$ 33.7173 1.28267 0.641334 0.767262i $$-0.278381\pi$$
0.641334 + 0.767262i $$0.278381\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ −34.1214 27.0466i −1.29430 1.02594i
$$696$$ 0 0
$$697$$ 15.7801i 0.597713i
$$698$$ 0 0
$$699$$ 0 0
$$700$$ 0 0
$$701$$ 19.9813 0.754685 0.377342 0.926074i $$-0.376838\pi$$
0.377342 + 0.926074i $$0.376838\pi$$
$$702$$ 0 0
$$703$$ 3.64006i 0.137288i
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ 0 0
$$708$$ 0 0
$$709$$ 43.3293 1.62727 0.813633 0.581378i $$-0.197486\pi$$
0.813633 + 0.581378i $$0.197486\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 0 0
$$713$$ 14.2241i 0.532695i
$$714$$ 0 0
$$715$$ 2.08998 2.63667i 0.0781610 0.0986059i
$$716$$ 0 0
$$717$$ 0 0
$$718$$ 0 0
$$719$$ −34.6867 −1.29360 −0.646798 0.762661i $$-0.723892\pi$$
−0.646798 + 0.762661i $$0.723892\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ 0 0
$$723$$ 0 0
$$724$$ 0 0
$$725$$ −8.62395 36.7826i −0.320286 1.36607i
$$726$$ 0 0
$$727$$ 5.39470i 0.200078i −0.994983 0.100039i $$-0.968103\pi$$
0.994983 0.100039i $$-0.0318968\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ 0 0
$$731$$ 7.43466 0.274981
$$732$$ 0 0
$$733$$ 9.11203i 0.336561i 0.985739 + 0.168280i $$0.0538214\pi$$
−0.985739 + 0.168280i $$0.946179\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 18.9066i 0.696435i
$$738$$ 0 0
$$739$$ 4.82936 0.177651 0.0888254 0.996047i $$-0.471689\pi$$
0.0888254 + 0.996047i $$0.471689\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ 50.4087i 1.84931i −0.380801 0.924657i $$-0.624352\pi$$
0.380801 0.924657i $$-0.375648\pi$$
$$744$$ 0 0
$$745$$ −25.4333 20.1600i −0.931805 0.738605i
$$746$$ 0 0
$$747$$ 0 0
$$748$$ 0 0
$$749$$ 0 0
$$750$$ 0 0
$$751$$ 4.88797 0.178365 0.0891823 0.996015i $$-0.471575\pi$$
0.0891823 + 0.996015i $$0.471575\pi$$
$$752$$ 0 0
$$753$$ 0 0
$$754$$ 0 0
$$755$$ −7.82003 6.19863i −0.284600 0.225591i
$$756$$ 0 0
$$757$$ 32.2241i 1.17120i −0.810599 0.585602i $$-0.800858\pi$$
0.810599 0.585602i $$-0.199142\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ −36.4740 −1.32218 −0.661091 0.750305i $$-0.729906\pi$$
−0.661091 + 0.750305i $$0.729906\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ 12.5140i 0.451854i
$$768$$ 0 0
$$769$$ −13.1120 −0.472832 −0.236416 0.971652i $$-0.575973\pi$$
−0.236416 + 0.971652i $$0.575973\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 0 0
$$773$$ 12.5913i 0.452876i 0.974026 + 0.226438i $$0.0727081\pi$$
−0.974026 + 0.226438i $$0.927292\pi$$
$$774$$ 0 0
$$775$$ −3.43466 14.6494i −0.123376 0.526222i
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ 4.20541 0.150674
$$780$$ 0 0
$$781$$ 7.18930 0.257253
$$782$$ 0 0
$$783$$ 0 0
$$784$$ 0 0
$$785$$ 11.5306 14.5467i 0.411544 0.519194i
$$786$$ 0 0
$$787$$ 36.9253i 1.31624i −0.752911 0.658122i $$-0.771351\pi$$
0.752911 0.658122i $$-0.228649\pi$$
$$788$$ 0 0
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 0 0
$$792$$ 0 0
$$793$$ 6.28267i 0.223104i
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ 25.0466i 0.887198i −0.896225 0.443599i $$-0.853702\pi$$
0.896225 0.443599i $$-0.146298\pi$$
$$798$$ 0 0
$$799$$ −27.8973 −0.986935
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 0