Properties

 Label 441.2.a.i.1.2 Level $441$ Weight $2$ Character 441.1 Self dual yes Analytic conductor $3.521$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [441,2,Mod(1,441)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(441, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("441.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$441 = 3^{2} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 441.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: $$3.52140272914$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\zeta_{8})^+$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - 2$$ x^2 - 2 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 147) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

 Embedding label 1.2 Root $$1.41421$$ of defining polynomial Character $$\chi$$ $$=$$ 441.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+2.41421 q^{2} +3.82843 q^{4} -0.585786 q^{5} +4.41421 q^{8} +O(q^{10})$$ $$q+2.41421 q^{2} +3.82843 q^{4} -0.585786 q^{5} +4.41421 q^{8} -1.41421 q^{10} +2.00000 q^{11} +5.41421 q^{13} +3.00000 q^{16} -6.24264 q^{17} +2.82843 q^{19} -2.24264 q^{20} +4.82843 q^{22} -3.65685 q^{23} -4.65685 q^{25} +13.0711 q^{26} +1.17157 q^{29} -6.82843 q^{31} -1.58579 q^{32} -15.0711 q^{34} -4.00000 q^{37} +6.82843 q^{38} -2.58579 q^{40} +2.24264 q^{41} -5.65685 q^{43} +7.65685 q^{44} -8.82843 q^{46} -2.82843 q^{47} -11.2426 q^{50} +20.7279 q^{52} +2.00000 q^{53} -1.17157 q^{55} +2.82843 q^{58} +6.82843 q^{59} +3.75736 q^{61} -16.4853 q^{62} -9.82843 q^{64} -3.17157 q^{65} +5.65685 q^{67} -23.8995 q^{68} +13.3137 q^{71} -5.89949 q^{73} -9.65685 q^{74} +10.8284 q^{76} +2.34315 q^{79} -1.75736 q^{80} +5.41421 q^{82} +15.3137 q^{83} +3.65685 q^{85} -13.6569 q^{86} +8.82843 q^{88} +5.75736 q^{89} -14.0000 q^{92} -6.82843 q^{94} -1.65685 q^{95} +5.41421 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{2} + 2 q^{4} - 4 q^{5} + 6 q^{8}+O(q^{10})$$ 2 * q + 2 * q^2 + 2 * q^4 - 4 * q^5 + 6 * q^8 $$2 q + 2 q^{2} + 2 q^{4} - 4 q^{5} + 6 q^{8} + 4 q^{11} + 8 q^{13} + 6 q^{16} - 4 q^{17} + 4 q^{20} + 4 q^{22} + 4 q^{23} + 2 q^{25} + 12 q^{26} + 8 q^{29} - 8 q^{31} - 6 q^{32} - 16 q^{34} - 8 q^{37} + 8 q^{38} - 8 q^{40} - 4 q^{41} + 4 q^{44} - 12 q^{46} - 14 q^{50} + 16 q^{52} + 4 q^{53} - 8 q^{55} + 8 q^{59} + 16 q^{61} - 16 q^{62} - 14 q^{64} - 12 q^{65} - 28 q^{68} + 4 q^{71} + 8 q^{73} - 8 q^{74} + 16 q^{76} + 16 q^{79} - 12 q^{80} + 8 q^{82} + 8 q^{83} - 4 q^{85} - 16 q^{86} + 12 q^{88} + 20 q^{89} - 28 q^{92} - 8 q^{94} + 8 q^{95} + 8 q^{97}+O(q^{100})$$ 2 * q + 2 * q^2 + 2 * q^4 - 4 * q^5 + 6 * q^8 + 4 * q^11 + 8 * q^13 + 6 * q^16 - 4 * q^17 + 4 * q^20 + 4 * q^22 + 4 * q^23 + 2 * q^25 + 12 * q^26 + 8 * q^29 - 8 * q^31 - 6 * q^32 - 16 * q^34 - 8 * q^37 + 8 * q^38 - 8 * q^40 - 4 * q^41 + 4 * q^44 - 12 * q^46 - 14 * q^50 + 16 * q^52 + 4 * q^53 - 8 * q^55 + 8 * q^59 + 16 * q^61 - 16 * q^62 - 14 * q^64 - 12 * q^65 - 28 * q^68 + 4 * q^71 + 8 * q^73 - 8 * q^74 + 16 * q^76 + 16 * q^79 - 12 * q^80 + 8 * q^82 + 8 * q^83 - 4 * q^85 - 16 * q^86 + 12 * q^88 + 20 * q^89 - 28 * q^92 - 8 * q^94 + 8 * 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 2.41421 1.70711 0.853553 0.521005i $$-0.174443\pi$$
0.853553 + 0.521005i $$0.174443\pi$$
$$3$$ 0 0
$$4$$ 3.82843 1.91421
$$5$$ −0.585786 −0.261972 −0.130986 0.991384i $$-0.541814\pi$$
−0.130986 + 0.991384i $$0.541814\pi$$
$$6$$ 0 0
$$7$$ 0 0
$$8$$ 4.41421 1.56066
$$9$$ 0 0
$$10$$ −1.41421 −0.447214
$$11$$ 2.00000 0.603023 0.301511 0.953463i $$-0.402509\pi$$
0.301511 + 0.953463i $$0.402509\pi$$
$$12$$ 0 0
$$13$$ 5.41421 1.50163 0.750816 0.660511i $$-0.229660\pi$$
0.750816 + 0.660511i $$0.229660\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 3.00000 0.750000
$$17$$ −6.24264 −1.51406 −0.757031 0.653379i $$-0.773351\pi$$
−0.757031 + 0.653379i $$0.773351\pi$$
$$18$$ 0 0
$$19$$ 2.82843 0.648886 0.324443 0.945905i $$-0.394823\pi$$
0.324443 + 0.945905i $$0.394823\pi$$
$$20$$ −2.24264 −0.501470
$$21$$ 0 0
$$22$$ 4.82843 1.02942
$$23$$ −3.65685 −0.762507 −0.381253 0.924471i $$-0.624507\pi$$
−0.381253 + 0.924471i $$0.624507\pi$$
$$24$$ 0 0
$$25$$ −4.65685 −0.931371
$$26$$ 13.0711 2.56345
$$27$$ 0 0
$$28$$ 0 0
$$29$$ 1.17157 0.217556 0.108778 0.994066i $$-0.465306\pi$$
0.108778 + 0.994066i $$0.465306\pi$$
$$30$$ 0 0
$$31$$ −6.82843 −1.22642 −0.613211 0.789919i $$-0.710122\pi$$
−0.613211 + 0.789919i $$0.710122\pi$$
$$32$$ −1.58579 −0.280330
$$33$$ 0 0
$$34$$ −15.0711 −2.58467
$$35$$ 0 0
$$36$$ 0 0
$$37$$ −4.00000 −0.657596 −0.328798 0.944400i $$-0.606644\pi$$
−0.328798 + 0.944400i $$0.606644\pi$$
$$38$$ 6.82843 1.10772
$$39$$ 0 0
$$40$$ −2.58579 −0.408849
$$41$$ 2.24264 0.350242 0.175121 0.984547i $$-0.443968\pi$$
0.175121 + 0.984547i $$0.443968\pi$$
$$42$$ 0 0
$$43$$ −5.65685 −0.862662 −0.431331 0.902194i $$-0.641956\pi$$
−0.431331 + 0.902194i $$0.641956\pi$$
$$44$$ 7.65685 1.15431
$$45$$ 0 0
$$46$$ −8.82843 −1.30168
$$47$$ −2.82843 −0.412568 −0.206284 0.978492i $$-0.566137\pi$$
−0.206284 + 0.978492i $$0.566137\pi$$
$$48$$ 0 0
$$49$$ 0 0
$$50$$ −11.2426 −1.58995
$$51$$ 0 0
$$52$$ 20.7279 2.87445
$$53$$ 2.00000 0.274721 0.137361 0.990521i $$-0.456138\pi$$
0.137361 + 0.990521i $$0.456138\pi$$
$$54$$ 0 0
$$55$$ −1.17157 −0.157975
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 2.82843 0.371391
$$59$$ 6.82843 0.888985 0.444493 0.895782i $$-0.353384\pi$$
0.444493 + 0.895782i $$0.353384\pi$$
$$60$$ 0 0
$$61$$ 3.75736 0.481081 0.240540 0.970639i $$-0.422675\pi$$
0.240540 + 0.970639i $$0.422675\pi$$
$$62$$ −16.4853 −2.09363
$$63$$ 0 0
$$64$$ −9.82843 −1.22855
$$65$$ −3.17157 −0.393385
$$66$$ 0 0
$$67$$ 5.65685 0.691095 0.345547 0.938401i $$-0.387693\pi$$
0.345547 + 0.938401i $$0.387693\pi$$
$$68$$ −23.8995 −2.89824
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 13.3137 1.58005 0.790023 0.613077i $$-0.210068\pi$$
0.790023 + 0.613077i $$0.210068\pi$$
$$72$$ 0 0
$$73$$ −5.89949 −0.690484 −0.345242 0.938514i $$-0.612203\pi$$
−0.345242 + 0.938514i $$0.612203\pi$$
$$74$$ −9.65685 −1.12259
$$75$$ 0 0
$$76$$ 10.8284 1.24211
$$77$$ 0 0
$$78$$ 0 0
$$79$$ 2.34315 0.263624 0.131812 0.991275i $$-0.457920\pi$$
0.131812 + 0.991275i $$0.457920\pi$$
$$80$$ −1.75736 −0.196479
$$81$$ 0 0
$$82$$ 5.41421 0.597900
$$83$$ 15.3137 1.68090 0.840449 0.541891i $$-0.182291\pi$$
0.840449 + 0.541891i $$0.182291\pi$$
$$84$$ 0 0
$$85$$ 3.65685 0.396642
$$86$$ −13.6569 −1.47266
$$87$$ 0 0
$$88$$ 8.82843 0.941113
$$89$$ 5.75736 0.610279 0.305139 0.952308i $$-0.401297\pi$$
0.305139 + 0.952308i $$0.401297\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ −14.0000 −1.45960
$$93$$ 0 0
$$94$$ −6.82843 −0.704298
$$95$$ −1.65685 −0.169990
$$96$$ 0 0
$$97$$ 5.41421 0.549730 0.274865 0.961483i $$-0.411367\pi$$
0.274865 + 0.961483i $$0.411367\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ −17.8284 −1.78284
$$101$$ −17.0711 −1.69863 −0.849317 0.527883i $$-0.822986\pi$$
−0.849317 + 0.527883i $$0.822986\pi$$
$$102$$ 0 0
$$103$$ −12.4853 −1.23021 −0.615106 0.788445i $$-0.710887\pi$$
−0.615106 + 0.788445i $$0.710887\pi$$
$$104$$ 23.8995 2.34354
$$105$$ 0 0
$$106$$ 4.82843 0.468978
$$107$$ 11.6569 1.12691 0.563455 0.826147i $$-0.309472\pi$$
0.563455 + 0.826147i $$0.309472\pi$$
$$108$$ 0 0
$$109$$ 5.65685 0.541828 0.270914 0.962604i $$-0.412674\pi$$
0.270914 + 0.962604i $$0.412674\pi$$
$$110$$ −2.82843 −0.269680
$$111$$ 0 0
$$112$$ 0 0
$$113$$ −17.3137 −1.62874 −0.814368 0.580348i $$-0.802916\pi$$
−0.814368 + 0.580348i $$0.802916\pi$$
$$114$$ 0 0
$$115$$ 2.14214 0.199755
$$116$$ 4.48528 0.416448
$$117$$ 0 0
$$118$$ 16.4853 1.51759
$$119$$ 0 0
$$120$$ 0 0
$$121$$ −7.00000 −0.636364
$$122$$ 9.07107 0.821256
$$123$$ 0 0
$$124$$ −26.1421 −2.34763
$$125$$ 5.65685 0.505964
$$126$$ 0 0
$$127$$ 9.65685 0.856907 0.428454 0.903564i $$-0.359059\pi$$
0.428454 + 0.903564i $$0.359059\pi$$
$$128$$ −20.5563 −1.81694
$$129$$ 0 0
$$130$$ −7.65685 −0.671551
$$131$$ −7.31371 −0.639002 −0.319501 0.947586i $$-0.603515\pi$$
−0.319501 + 0.947586i $$0.603515\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ 13.6569 1.17977
$$135$$ 0 0
$$136$$ −27.5563 −2.36294
$$137$$ 14.1421 1.20824 0.604122 0.796892i $$-0.293524\pi$$
0.604122 + 0.796892i $$0.293524\pi$$
$$138$$ 0 0
$$139$$ −6.34315 −0.538019 −0.269009 0.963138i $$-0.586696\pi$$
−0.269009 + 0.963138i $$0.586696\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 32.1421 2.69731
$$143$$ 10.8284 0.905519
$$144$$ 0 0
$$145$$ −0.686292 −0.0569934
$$146$$ −14.2426 −1.17873
$$147$$ 0 0
$$148$$ −15.3137 −1.25878
$$149$$ 5.31371 0.435316 0.217658 0.976025i $$-0.430158\pi$$
0.217658 + 0.976025i $$0.430158\pi$$
$$150$$ 0 0
$$151$$ 12.0000 0.976546 0.488273 0.872691i $$-0.337627\pi$$
0.488273 + 0.872691i $$0.337627\pi$$
$$152$$ 12.4853 1.01269
$$153$$ 0 0
$$154$$ 0 0
$$155$$ 4.00000 0.321288
$$156$$ 0 0
$$157$$ 20.2426 1.61554 0.807769 0.589499i $$-0.200675\pi$$
0.807769 + 0.589499i $$0.200675\pi$$
$$158$$ 5.65685 0.450035
$$159$$ 0 0
$$160$$ 0.928932 0.0734385
$$161$$ 0 0
$$162$$ 0 0
$$163$$ 11.3137 0.886158 0.443079 0.896483i $$-0.353886\pi$$
0.443079 + 0.896483i $$0.353886\pi$$
$$164$$ 8.58579 0.670437
$$165$$ 0 0
$$166$$ 36.9706 2.86947
$$167$$ −19.7990 −1.53209 −0.766046 0.642786i $$-0.777779\pi$$
−0.766046 + 0.642786i $$0.777779\pi$$
$$168$$ 0 0
$$169$$ 16.3137 1.25490
$$170$$ 8.82843 0.677109
$$171$$ 0 0
$$172$$ −21.6569 −1.65132
$$173$$ −6.92893 −0.526797 −0.263398 0.964687i $$-0.584843\pi$$
−0.263398 + 0.964687i $$0.584843\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 6.00000 0.452267
$$177$$ 0 0
$$178$$ 13.8995 1.04181
$$179$$ 8.34315 0.623596 0.311798 0.950148i $$-0.399069\pi$$
0.311798 + 0.950148i $$0.399069\pi$$
$$180$$ 0 0
$$181$$ −5.41421 −0.402435 −0.201218 0.979547i $$-0.564490\pi$$
−0.201218 + 0.979547i $$0.564490\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ −16.1421 −1.19001
$$185$$ 2.34315 0.172272
$$186$$ 0 0
$$187$$ −12.4853 −0.913014
$$188$$ −10.8284 −0.789744
$$189$$ 0 0
$$190$$ −4.00000 −0.290191
$$191$$ 18.0000 1.30243 0.651217 0.758891i $$-0.274259\pi$$
0.651217 + 0.758891i $$0.274259\pi$$
$$192$$ 0 0
$$193$$ −17.3137 −1.24627 −0.623134 0.782115i $$-0.714141\pi$$
−0.623134 + 0.782115i $$0.714141\pi$$
$$194$$ 13.0711 0.938448
$$195$$ 0 0
$$196$$ 0 0
$$197$$ −2.00000 −0.142494 −0.0712470 0.997459i $$-0.522698\pi$$
−0.0712470 + 0.997459i $$0.522698\pi$$
$$198$$ 0 0
$$199$$ 10.3431 0.733206 0.366603 0.930377i $$-0.380521\pi$$
0.366603 + 0.930377i $$0.380521\pi$$
$$200$$ −20.5563 −1.45355
$$201$$ 0 0
$$202$$ −41.2132 −2.89975
$$203$$ 0 0
$$204$$ 0 0
$$205$$ −1.31371 −0.0917534
$$206$$ −30.1421 −2.10010
$$207$$ 0 0
$$208$$ 16.2426 1.12622
$$209$$ 5.65685 0.391293
$$210$$ 0 0
$$211$$ −20.9706 −1.44367 −0.721837 0.692064i $$-0.756702\pi$$
−0.721837 + 0.692064i $$0.756702\pi$$
$$212$$ 7.65685 0.525875
$$213$$ 0 0
$$214$$ 28.1421 1.92376
$$215$$ 3.31371 0.225993
$$216$$ 0 0
$$217$$ 0 0
$$218$$ 13.6569 0.924959
$$219$$ 0 0
$$220$$ −4.48528 −0.302398
$$221$$ −33.7990 −2.27357
$$222$$ 0 0
$$223$$ −8.97056 −0.600713 −0.300357 0.953827i $$-0.597106\pi$$
−0.300357 + 0.953827i $$0.597106\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ −41.7990 −2.78043
$$227$$ 15.7990 1.04862 0.524308 0.851529i $$-0.324324\pi$$
0.524308 + 0.851529i $$0.324324\pi$$
$$228$$ 0 0
$$229$$ 8.24264 0.544689 0.272345 0.962200i $$-0.412201\pi$$
0.272345 + 0.962200i $$0.412201\pi$$
$$230$$ 5.17157 0.341003
$$231$$ 0 0
$$232$$ 5.17157 0.339530
$$233$$ −22.1421 −1.45058 −0.725290 0.688444i $$-0.758294\pi$$
−0.725290 + 0.688444i $$0.758294\pi$$
$$234$$ 0 0
$$235$$ 1.65685 0.108081
$$236$$ 26.1421 1.70171
$$237$$ 0 0
$$238$$ 0 0
$$239$$ 4.34315 0.280935 0.140467 0.990085i $$-0.455139\pi$$
0.140467 + 0.990085i $$0.455139\pi$$
$$240$$ 0 0
$$241$$ −7.75736 −0.499695 −0.249848 0.968285i $$-0.580381\pi$$
−0.249848 + 0.968285i $$0.580381\pi$$
$$242$$ −16.8995 −1.08634
$$243$$ 0 0
$$244$$ 14.3848 0.920891
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 15.3137 0.974388
$$248$$ −30.1421 −1.91403
$$249$$ 0 0
$$250$$ 13.6569 0.863735
$$251$$ −4.48528 −0.283108 −0.141554 0.989931i $$-0.545210\pi$$
−0.141554 + 0.989931i $$0.545210\pi$$
$$252$$ 0 0
$$253$$ −7.31371 −0.459809
$$254$$ 23.3137 1.46283
$$255$$ 0 0
$$256$$ −29.9706 −1.87316
$$257$$ 19.2132 1.19849 0.599243 0.800567i $$-0.295468\pi$$
0.599243 + 0.800567i $$0.295468\pi$$
$$258$$ 0 0
$$259$$ 0 0
$$260$$ −12.1421 −0.753023
$$261$$ 0 0
$$262$$ −17.6569 −1.09084
$$263$$ 17.3137 1.06761 0.533805 0.845608i $$-0.320762\pi$$
0.533805 + 0.845608i $$0.320762\pi$$
$$264$$ 0 0
$$265$$ −1.17157 −0.0719691
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 21.6569 1.32290
$$269$$ −10.7279 −0.654093 −0.327046 0.945008i $$-0.606053\pi$$
−0.327046 + 0.945008i $$0.606053\pi$$
$$270$$ 0 0
$$271$$ 18.1421 1.10206 0.551028 0.834487i $$-0.314236\pi$$
0.551028 + 0.834487i $$0.314236\pi$$
$$272$$ −18.7279 −1.13555
$$273$$ 0 0
$$274$$ 34.1421 2.06260
$$275$$ −9.31371 −0.561638
$$276$$ 0 0
$$277$$ 13.3137 0.799943 0.399972 0.916528i $$-0.369020\pi$$
0.399972 + 0.916528i $$0.369020\pi$$
$$278$$ −15.3137 −0.918455
$$279$$ 0 0
$$280$$ 0 0
$$281$$ 16.4853 0.983429 0.491715 0.870756i $$-0.336370\pi$$
0.491715 + 0.870756i $$0.336370\pi$$
$$282$$ 0 0
$$283$$ 8.48528 0.504398 0.252199 0.967675i $$-0.418846\pi$$
0.252199 + 0.967675i $$0.418846\pi$$
$$284$$ 50.9706 3.02455
$$285$$ 0 0
$$286$$ 26.1421 1.54582
$$287$$ 0 0
$$288$$ 0 0
$$289$$ 21.9706 1.29239
$$290$$ −1.65685 −0.0972938
$$291$$ 0 0
$$292$$ −22.5858 −1.32173
$$293$$ −19.4142 −1.13419 −0.567095 0.823652i $$-0.691933\pi$$
−0.567095 + 0.823652i $$0.691933\pi$$
$$294$$ 0 0
$$295$$ −4.00000 −0.232889
$$296$$ −17.6569 −1.02628
$$297$$ 0 0
$$298$$ 12.8284 0.743131
$$299$$ −19.7990 −1.14501
$$300$$ 0 0
$$301$$ 0 0
$$302$$ 28.9706 1.66707
$$303$$ 0 0
$$304$$ 8.48528 0.486664
$$305$$ −2.20101 −0.126029
$$306$$ 0 0
$$307$$ 1.85786 0.106034 0.0530170 0.998594i $$-0.483116\pi$$
0.0530170 + 0.998594i $$0.483116\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 9.65685 0.548472
$$311$$ 22.1421 1.25557 0.627783 0.778389i $$-0.283963\pi$$
0.627783 + 0.778389i $$0.283963\pi$$
$$312$$ 0 0
$$313$$ −17.8995 −1.01174 −0.505870 0.862610i $$-0.668828\pi$$
−0.505870 + 0.862610i $$0.668828\pi$$
$$314$$ 48.8701 2.75790
$$315$$ 0 0
$$316$$ 8.97056 0.504634
$$317$$ −10.0000 −0.561656 −0.280828 0.959758i $$-0.590609\pi$$
−0.280828 + 0.959758i $$0.590609\pi$$
$$318$$ 0 0
$$319$$ 2.34315 0.131191
$$320$$ 5.75736 0.321846
$$321$$ 0 0
$$322$$ 0 0
$$323$$ −17.6569 −0.982454
$$324$$ 0 0
$$325$$ −25.2132 −1.39858
$$326$$ 27.3137 1.51277
$$327$$ 0 0
$$328$$ 9.89949 0.546608
$$329$$ 0 0
$$330$$ 0 0
$$331$$ −4.00000 −0.219860 −0.109930 0.993939i $$-0.535063\pi$$
−0.109930 + 0.993939i $$0.535063\pi$$
$$332$$ 58.6274 3.21760
$$333$$ 0 0
$$334$$ −47.7990 −2.61544
$$335$$ −3.31371 −0.181047
$$336$$ 0 0
$$337$$ −18.3431 −0.999215 −0.499607 0.866252i $$-0.666522\pi$$
−0.499607 + 0.866252i $$0.666522\pi$$
$$338$$ 39.3848 2.14225
$$339$$ 0 0
$$340$$ 14.0000 0.759257
$$341$$ −13.6569 −0.739560
$$342$$ 0 0
$$343$$ 0 0
$$344$$ −24.9706 −1.34632
$$345$$ 0 0
$$346$$ −16.7279 −0.899299
$$347$$ −10.6863 −0.573670 −0.286835 0.957980i $$-0.592603\pi$$
−0.286835 + 0.957980i $$0.592603\pi$$
$$348$$ 0 0
$$349$$ 9.89949 0.529908 0.264954 0.964261i $$-0.414643\pi$$
0.264954 + 0.964261i $$0.414643\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ −3.17157 −0.169045
$$353$$ −10.7279 −0.570990 −0.285495 0.958380i $$-0.592158\pi$$
−0.285495 + 0.958380i $$0.592158\pi$$
$$354$$ 0 0
$$355$$ −7.79899 −0.413927
$$356$$ 22.0416 1.16820
$$357$$ 0 0
$$358$$ 20.1421 1.06454
$$359$$ 11.6569 0.615225 0.307613 0.951512i $$-0.400470\pi$$
0.307613 + 0.951512i $$0.400470\pi$$
$$360$$ 0 0
$$361$$ −11.0000 −0.578947
$$362$$ −13.0711 −0.687000
$$363$$ 0 0
$$364$$ 0 0
$$365$$ 3.45584 0.180887
$$366$$ 0 0
$$367$$ 19.3137 1.00817 0.504084 0.863655i $$-0.331830\pi$$
0.504084 + 0.863655i $$0.331830\pi$$
$$368$$ −10.9706 −0.571880
$$369$$ 0 0
$$370$$ 5.65685 0.294086
$$371$$ 0 0
$$372$$ 0 0
$$373$$ −33.3137 −1.72492 −0.862459 0.506127i $$-0.831077\pi$$
−0.862459 + 0.506127i $$0.831077\pi$$
$$374$$ −30.1421 −1.55861
$$375$$ 0 0
$$376$$ −12.4853 −0.643879
$$377$$ 6.34315 0.326689
$$378$$ 0 0
$$379$$ 31.3137 1.60848 0.804239 0.594307i $$-0.202573\pi$$
0.804239 + 0.594307i $$0.202573\pi$$
$$380$$ −6.34315 −0.325397
$$381$$ 0 0
$$382$$ 43.4558 2.22339
$$383$$ 29.6569 1.51539 0.757697 0.652606i $$-0.226324\pi$$
0.757697 + 0.652606i $$0.226324\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ −41.7990 −2.12751
$$387$$ 0 0
$$388$$ 20.7279 1.05230
$$389$$ −10.1421 −0.514227 −0.257113 0.966381i $$-0.582771\pi$$
−0.257113 + 0.966381i $$0.582771\pi$$
$$390$$ 0 0
$$391$$ 22.8284 1.15448
$$392$$ 0 0
$$393$$ 0 0
$$394$$ −4.82843 −0.243253
$$395$$ −1.37258 −0.0690621
$$396$$ 0 0
$$397$$ 34.3848 1.72572 0.862861 0.505441i $$-0.168670\pi$$
0.862861 + 0.505441i $$0.168670\pi$$
$$398$$ 24.9706 1.25166
$$399$$ 0 0
$$400$$ −13.9706 −0.698528
$$401$$ −22.1421 −1.10573 −0.552863 0.833272i $$-0.686465\pi$$
−0.552863 + 0.833272i $$0.686465\pi$$
$$402$$ 0 0
$$403$$ −36.9706 −1.84163
$$404$$ −65.3553 −3.25155
$$405$$ 0 0
$$406$$ 0 0
$$407$$ −8.00000 −0.396545
$$408$$ 0 0
$$409$$ −18.5858 −0.919008 −0.459504 0.888176i $$-0.651973\pi$$
−0.459504 + 0.888176i $$0.651973\pi$$
$$410$$ −3.17157 −0.156633
$$411$$ 0 0
$$412$$ −47.7990 −2.35489
$$413$$ 0 0
$$414$$ 0 0
$$415$$ −8.97056 −0.440348
$$416$$ −8.58579 −0.420953
$$417$$ 0 0
$$418$$ 13.6569 0.667979
$$419$$ −38.8284 −1.89689 −0.948446 0.316938i $$-0.897345\pi$$
−0.948446 + 0.316938i $$0.897345\pi$$
$$420$$ 0 0
$$421$$ −28.6274 −1.39521 −0.697607 0.716480i $$-0.745752\pi$$
−0.697607 + 0.716480i $$0.745752\pi$$
$$422$$ −50.6274 −2.46450
$$423$$ 0 0
$$424$$ 8.82843 0.428746
$$425$$ 29.0711 1.41015
$$426$$ 0 0
$$427$$ 0 0
$$428$$ 44.6274 2.15715
$$429$$ 0 0
$$430$$ 8.00000 0.385794
$$431$$ −6.97056 −0.335760 −0.167880 0.985807i $$-0.553692\pi$$
−0.167880 + 0.985807i $$0.553692\pi$$
$$432$$ 0 0
$$433$$ −11.7574 −0.565023 −0.282511 0.959264i $$-0.591167\pi$$
−0.282511 + 0.959264i $$0.591167\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 21.6569 1.03718
$$437$$ −10.3431 −0.494780
$$438$$ 0 0
$$439$$ 35.3137 1.68543 0.842716 0.538359i $$-0.180956\pi$$
0.842716 + 0.538359i $$0.180956\pi$$
$$440$$ −5.17157 −0.246545
$$441$$ 0 0
$$442$$ −81.5980 −3.88122
$$443$$ −1.02944 −0.0489100 −0.0244550 0.999701i $$-0.507785\pi$$
−0.0244550 + 0.999701i $$0.507785\pi$$
$$444$$ 0 0
$$445$$ −3.37258 −0.159876
$$446$$ −21.6569 −1.02548
$$447$$ 0 0
$$448$$ 0 0
$$449$$ −17.3137 −0.817084 −0.408542 0.912739i $$-0.633963\pi$$
−0.408542 + 0.912739i $$0.633963\pi$$
$$450$$ 0 0
$$451$$ 4.48528 0.211204
$$452$$ −66.2843 −3.11775
$$453$$ 0 0
$$454$$ 38.1421 1.79010
$$455$$ 0 0
$$456$$ 0 0
$$457$$ −18.0000 −0.842004 −0.421002 0.907060i $$-0.638322\pi$$
−0.421002 + 0.907060i $$0.638322\pi$$
$$458$$ 19.8995 0.929842
$$459$$ 0 0
$$460$$ 8.20101 0.382374
$$461$$ 19.4142 0.904210 0.452105 0.891965i $$-0.350673\pi$$
0.452105 + 0.891965i $$0.350673\pi$$
$$462$$ 0 0
$$463$$ 18.6274 0.865689 0.432845 0.901468i $$-0.357510\pi$$
0.432845 + 0.901468i $$0.357510\pi$$
$$464$$ 3.51472 0.163167
$$465$$ 0 0
$$466$$ −53.4558 −2.47629
$$467$$ −39.7990 −1.84168 −0.920839 0.389943i $$-0.872495\pi$$
−0.920839 + 0.389943i $$0.872495\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ 4.00000 0.184506
$$471$$ 0 0
$$472$$ 30.1421 1.38740
$$473$$ −11.3137 −0.520205
$$474$$ 0 0
$$475$$ −13.1716 −0.604353
$$476$$ 0 0
$$477$$ 0 0
$$478$$ 10.4853 0.479586
$$479$$ −30.1421 −1.37723 −0.688615 0.725127i $$-0.741781\pi$$
−0.688615 + 0.725127i $$0.741781\pi$$
$$480$$ 0 0
$$481$$ −21.6569 −0.987468
$$482$$ −18.7279 −0.853033
$$483$$ 0 0
$$484$$ −26.7990 −1.21814
$$485$$ −3.17157 −0.144014
$$486$$ 0 0
$$487$$ −18.6274 −0.844089 −0.422044 0.906575i $$-0.638687\pi$$
−0.422044 + 0.906575i $$0.638687\pi$$
$$488$$ 16.5858 0.750803
$$489$$ 0 0
$$490$$ 0 0
$$491$$ −38.9706 −1.75872 −0.879358 0.476160i $$-0.842028\pi$$
−0.879358 + 0.476160i $$0.842028\pi$$
$$492$$ 0 0
$$493$$ −7.31371 −0.329393
$$494$$ 36.9706 1.66338
$$495$$ 0 0
$$496$$ −20.4853 −0.919816
$$497$$ 0 0
$$498$$ 0 0
$$499$$ −19.3137 −0.864600 −0.432300 0.901730i $$-0.642298\pi$$
−0.432300 + 0.901730i $$0.642298\pi$$
$$500$$ 21.6569 0.968524
$$501$$ 0 0
$$502$$ −10.8284 −0.483296
$$503$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$504$$ 0 0
$$505$$ 10.0000 0.444994
$$506$$ −17.6569 −0.784943
$$507$$ 0 0
$$508$$ 36.9706 1.64030
$$509$$ 25.5563 1.13277 0.566383 0.824142i $$-0.308342\pi$$
0.566383 + 0.824142i $$0.308342\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ −31.2426 −1.38074
$$513$$ 0 0
$$514$$ 46.3848 2.04594
$$515$$ 7.31371 0.322281
$$516$$ 0 0
$$517$$ −5.65685 −0.248788
$$518$$ 0 0
$$519$$ 0 0
$$520$$ −14.0000 −0.613941
$$521$$ 32.5858 1.42761 0.713805 0.700345i $$-0.246970\pi$$
0.713805 + 0.700345i $$0.246970\pi$$
$$522$$ 0 0
$$523$$ −14.3431 −0.627182 −0.313591 0.949558i $$-0.601532\pi$$
−0.313591 + 0.949558i $$0.601532\pi$$
$$524$$ −28.0000 −1.22319
$$525$$ 0 0
$$526$$ 41.7990 1.82252
$$527$$ 42.6274 1.85688
$$528$$ 0 0
$$529$$ −9.62742 −0.418583
$$530$$ −2.82843 −0.122859
$$531$$ 0 0
$$532$$ 0 0
$$533$$ 12.1421 0.525934
$$534$$ 0 0
$$535$$ −6.82843 −0.295219
$$536$$ 24.9706 1.07856
$$537$$ 0 0
$$538$$ −25.8995 −1.11661
$$539$$ 0 0
$$540$$ 0 0
$$541$$ −5.31371 −0.228454 −0.114227 0.993455i $$-0.536439\pi$$
−0.114227 + 0.993455i $$0.536439\pi$$
$$542$$ 43.7990 1.88133
$$543$$ 0 0
$$544$$ 9.89949 0.424437
$$545$$ −3.31371 −0.141944
$$546$$ 0 0
$$547$$ −3.02944 −0.129529 −0.0647647 0.997901i $$-0.520630\pi$$
−0.0647647 + 0.997901i $$0.520630\pi$$
$$548$$ 54.1421 2.31284
$$549$$ 0 0
$$550$$ −22.4853 −0.958776
$$551$$ 3.31371 0.141169
$$552$$ 0 0
$$553$$ 0 0
$$554$$ 32.1421 1.36559
$$555$$ 0 0
$$556$$ −24.2843 −1.02988
$$557$$ −26.0000 −1.10166 −0.550828 0.834619i $$-0.685688\pi$$
−0.550828 + 0.834619i $$0.685688\pi$$
$$558$$ 0 0
$$559$$ −30.6274 −1.29540
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 39.7990 1.67882
$$563$$ 6.82843 0.287784 0.143892 0.989593i $$-0.454038\pi$$
0.143892 + 0.989593i $$0.454038\pi$$
$$564$$ 0 0
$$565$$ 10.1421 0.426683
$$566$$ 20.4853 0.861061
$$567$$ 0 0
$$568$$ 58.7696 2.46592
$$569$$ −0.485281 −0.0203441 −0.0101720 0.999948i $$-0.503238\pi$$
−0.0101720 + 0.999948i $$0.503238\pi$$
$$570$$ 0 0
$$571$$ 33.6569 1.40850 0.704248 0.709954i $$-0.251284\pi$$
0.704248 + 0.709954i $$0.251284\pi$$
$$572$$ 41.4558 1.73336
$$573$$ 0 0
$$574$$ 0 0
$$575$$ 17.0294 0.710177
$$576$$ 0 0
$$577$$ −14.1005 −0.587012 −0.293506 0.955957i $$-0.594822\pi$$
−0.293506 + 0.955957i $$0.594822\pi$$
$$578$$ 53.0416 2.20624
$$579$$ 0 0
$$580$$ −2.62742 −0.109098
$$581$$ 0 0
$$582$$ 0 0
$$583$$ 4.00000 0.165663
$$584$$ −26.0416 −1.07761
$$585$$ 0 0
$$586$$ −46.8701 −1.93618
$$587$$ 17.1716 0.708747 0.354373 0.935104i $$-0.384694\pi$$
0.354373 + 0.935104i $$0.384694\pi$$
$$588$$ 0 0
$$589$$ −19.3137 −0.795807
$$590$$ −9.65685 −0.397566
$$591$$ 0 0
$$592$$ −12.0000 −0.493197
$$593$$ 21.0711 0.865285 0.432643 0.901566i $$-0.357581\pi$$
0.432643 + 0.901566i $$0.357581\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 20.3431 0.833288
$$597$$ 0 0
$$598$$ −47.7990 −1.95465
$$599$$ 2.00000 0.0817178 0.0408589 0.999165i $$-0.486991\pi$$
0.0408589 + 0.999165i $$0.486991\pi$$
$$600$$ 0 0
$$601$$ −0.928932 −0.0378919 −0.0189460 0.999821i $$-0.506031\pi$$
−0.0189460 + 0.999821i $$0.506031\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ 45.9411 1.86932
$$605$$ 4.10051 0.166709
$$606$$ 0 0
$$607$$ −29.6569 −1.20373 −0.601867 0.798596i $$-0.705576\pi$$
−0.601867 + 0.798596i $$0.705576\pi$$
$$608$$ −4.48528 −0.181902
$$609$$ 0 0
$$610$$ −5.31371 −0.215146
$$611$$ −15.3137 −0.619526
$$612$$ 0 0
$$613$$ 27.3137 1.10319 0.551595 0.834112i $$-0.314019\pi$$
0.551595 + 0.834112i $$0.314019\pi$$
$$614$$ 4.48528 0.181011
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 7.51472 0.302531 0.151266 0.988493i $$-0.451665\pi$$
0.151266 + 0.988493i $$0.451665\pi$$
$$618$$ 0 0
$$619$$ −4.97056 −0.199784 −0.0998919 0.994998i $$-0.531850\pi$$
−0.0998919 + 0.994998i $$0.531850\pi$$
$$620$$ 15.3137 0.615013
$$621$$ 0 0
$$622$$ 53.4558 2.14338
$$623$$ 0 0
$$624$$ 0 0
$$625$$ 19.9706 0.798823
$$626$$ −43.2132 −1.72715
$$627$$ 0 0
$$628$$ 77.4975 3.09249
$$629$$ 24.9706 0.995642
$$630$$ 0 0
$$631$$ 0.686292 0.0273208 0.0136604 0.999907i $$-0.495652\pi$$
0.0136604 + 0.999907i $$0.495652\pi$$
$$632$$ 10.3431 0.411428
$$633$$ 0 0
$$634$$ −24.1421 −0.958807
$$635$$ −5.65685 −0.224485
$$636$$ 0 0
$$637$$ 0 0
$$638$$ 5.65685 0.223957
$$639$$ 0 0
$$640$$ 12.0416 0.475987
$$641$$ 5.17157 0.204265 0.102132 0.994771i $$-0.467433\pi$$
0.102132 + 0.994771i $$0.467433\pi$$
$$642$$ 0 0
$$643$$ 50.4264 1.98862 0.994312 0.106510i $$-0.0339675\pi$$
0.994312 + 0.106510i $$0.0339675\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ −42.6274 −1.67715
$$647$$ −21.1716 −0.832340 −0.416170 0.909287i $$-0.636628\pi$$
−0.416170 + 0.909287i $$0.636628\pi$$
$$648$$ 0 0
$$649$$ 13.6569 0.536078
$$650$$ −60.8701 −2.38752
$$651$$ 0 0
$$652$$ 43.3137 1.69630
$$653$$ −19.5147 −0.763670 −0.381835 0.924231i $$-0.624708\pi$$
−0.381835 + 0.924231i $$0.624708\pi$$
$$654$$ 0 0
$$655$$ 4.28427 0.167400
$$656$$ 6.72792 0.262681
$$657$$ 0 0
$$658$$ 0 0
$$659$$ 13.3137 0.518628 0.259314 0.965793i $$-0.416503\pi$$
0.259314 + 0.965793i $$0.416503\pi$$
$$660$$ 0 0
$$661$$ 7.55635 0.293908 0.146954 0.989143i $$-0.453053\pi$$
0.146954 + 0.989143i $$0.453053\pi$$
$$662$$ −9.65685 −0.375324
$$663$$ 0 0
$$664$$ 67.5980 2.62331
$$665$$ 0 0
$$666$$ 0 0
$$667$$ −4.28427 −0.165888
$$668$$ −75.7990 −2.93275
$$669$$ 0 0
$$670$$ −8.00000 −0.309067
$$671$$ 7.51472 0.290102
$$672$$ 0 0
$$673$$ 0.686292 0.0264546 0.0132273 0.999913i $$-0.495789\pi$$
0.0132273 + 0.999913i $$0.495789\pi$$
$$674$$ −44.2843 −1.70577
$$675$$ 0 0
$$676$$ 62.4558 2.40215
$$677$$ −28.5858 −1.09864 −0.549321 0.835612i $$-0.685113\pi$$
−0.549321 + 0.835612i $$0.685113\pi$$
$$678$$ 0 0
$$679$$ 0 0
$$680$$ 16.1421 0.619023
$$681$$ 0 0
$$682$$ −32.9706 −1.26251
$$683$$ 8.34315 0.319242 0.159621 0.987178i $$-0.448973\pi$$
0.159621 + 0.987178i $$0.448973\pi$$
$$684$$ 0 0
$$685$$ −8.28427 −0.316526
$$686$$ 0 0
$$687$$ 0 0
$$688$$ −16.9706 −0.646997
$$689$$ 10.8284 0.412530
$$690$$ 0 0
$$691$$ −23.3137 −0.886895 −0.443448 0.896300i $$-0.646245\pi$$
−0.443448 + 0.896300i $$0.646245\pi$$
$$692$$ −26.5269 −1.00840
$$693$$ 0 0
$$694$$ −25.7990 −0.979316
$$695$$ 3.71573 0.140946
$$696$$ 0 0
$$697$$ −14.0000 −0.530288
$$698$$ 23.8995 0.904609
$$699$$ 0 0
$$700$$ 0 0
$$701$$ 22.8284 0.862218 0.431109 0.902300i $$-0.358122\pi$$
0.431109 + 0.902300i $$0.358122\pi$$
$$702$$ 0 0
$$703$$ −11.3137 −0.426705
$$704$$ −19.6569 −0.740846
$$705$$ 0 0
$$706$$ −25.8995 −0.974740
$$707$$ 0 0
$$708$$ 0 0
$$709$$ 20.2843 0.761792 0.380896 0.924618i $$-0.375616\pi$$
0.380896 + 0.924618i $$0.375616\pi$$
$$710$$ −18.8284 −0.706618
$$711$$ 0 0
$$712$$ 25.4142 0.952438
$$713$$ 24.9706 0.935155
$$714$$ 0 0
$$715$$ −6.34315 −0.237220
$$716$$ 31.9411 1.19370
$$717$$ 0 0
$$718$$ 28.1421 1.05026
$$719$$ 25.9411 0.967441 0.483720 0.875223i $$-0.339285\pi$$
0.483720 + 0.875223i $$0.339285\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ −26.5563 −0.988325
$$723$$ 0 0
$$724$$ −20.7279 −0.770347
$$725$$ −5.45584 −0.202625
$$726$$ 0 0
$$727$$ −4.48528 −0.166350 −0.0831749 0.996535i $$-0.526506\pi$$
−0.0831749 + 0.996535i $$0.526506\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ 8.34315 0.308794
$$731$$ 35.3137 1.30612
$$732$$ 0 0
$$733$$ −9.69848 −0.358222 −0.179111 0.983829i $$-0.557322\pi$$
−0.179111 + 0.983829i $$0.557322\pi$$
$$734$$ 46.6274 1.72105
$$735$$ 0 0
$$736$$ 5.79899 0.213754
$$737$$ 11.3137 0.416746
$$738$$ 0 0
$$739$$ 27.3137 1.00475 0.502376 0.864650i $$-0.332460\pi$$
0.502376 + 0.864650i $$0.332460\pi$$
$$740$$ 8.97056 0.329764
$$741$$ 0 0
$$742$$ 0 0
$$743$$ 17.0294 0.624749 0.312375 0.949959i $$-0.398876\pi$$
0.312375 + 0.949959i $$0.398876\pi$$
$$744$$ 0 0
$$745$$ −3.11270 −0.114040
$$746$$ −80.4264 −2.94462
$$747$$ 0 0
$$748$$ −47.7990 −1.74770
$$749$$ 0 0
$$750$$ 0 0
$$751$$ 2.34315 0.0855026 0.0427513 0.999086i $$-0.486388\pi$$
0.0427513 + 0.999086i $$0.486388\pi$$
$$752$$ −8.48528 −0.309426
$$753$$ 0 0
$$754$$ 15.3137 0.557692
$$755$$ −7.02944 −0.255827
$$756$$ 0 0
$$757$$ 37.6569 1.36866 0.684331 0.729172i $$-0.260094\pi$$
0.684331 + 0.729172i $$0.260094\pi$$
$$758$$ 75.5980 2.74584
$$759$$ 0 0
$$760$$ −7.31371 −0.265296
$$761$$ −46.5269 −1.68660 −0.843300 0.537444i $$-0.819390\pi$$
−0.843300 + 0.537444i $$0.819390\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ 68.9117 2.49314
$$765$$ 0 0
$$766$$ 71.5980 2.58694
$$767$$ 36.9706 1.33493
$$768$$ 0 0
$$769$$ −29.6985 −1.07095 −0.535477 0.844550i $$-0.679868\pi$$
−0.535477 + 0.844550i $$0.679868\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ −66.2843 −2.38562
$$773$$ −21.5563 −0.775328 −0.387664 0.921801i $$-0.626718\pi$$
−0.387664 + 0.921801i $$0.626718\pi$$
$$774$$ 0 0
$$775$$ 31.7990 1.14225
$$776$$ 23.8995 0.857942
$$777$$ 0 0
$$778$$ −24.4853 −0.877840
$$779$$ 6.34315 0.227267
$$780$$ 0 0
$$781$$ 26.6274 0.952804
$$782$$ 55.1127 1.97083
$$783$$ 0 0
$$784$$ 0 0
$$785$$ −11.8579 −0.423225
$$786$$ 0 0
$$787$$ 47.3137 1.68655 0.843276 0.537481i $$-0.180624\pi$$
0.843276 + 0.537481i $$0.180624\pi$$
$$788$$ −7.65685 −0.272764
$$789$$ 0 0
$$790$$ −3.31371 −0.117896
$$791$$ 0 0
$$792$$ 0 0
$$793$$ 20.3431 0.722406
$$794$$ 83.0122 2.94599
$$795$$ 0 0
$$796$$ 39.5980 1.40351
$$797$$ −28.3848 −1.00544 −0.502720 0.864449i $$-0.667667\pi$$
−0.502720 + 0.864449i $$0.667667\pi$$
$$798$$ 0 0
$$799$$ 17.6569 0.624655
$$800$$ 7.38478 0.261091
$$801$$ 0 0
$$802$$ −53.4558 −1.88759
$$803$$ −11.7990 −0.416377
$$804$$ 0 0
$$805$$ 0 0
$$806$$ −89.2548 −3.14387
$$807$$ 0 0
$$808$$ −75.3553 −2.65099
$$809$$ 47.9411 1.68552 0.842760 0.538289i $$-0.180929\pi$$
0.842760 + 0.538289i $$0.180929\pi$$
$$810$$ 0 0
$$811$$ −6.34315 −0.222738 −0.111369 0.993779i $$-0.535524\pi$$
−0.111369 + 0.993779i $$0.535524\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ −19.3137 −0.676945
$$815$$ −6.62742 −0.232148
$$816$$ 0 0
$$817$$ −16.0000 −0.559769
$$818$$ −44.8701 −1.56884
$$819$$ 0 0
$$820$$ −5.02944 −0.175636
$$821$$ 33.3137 1.16266 0.581328 0.813669i $$-0.302533\pi$$
0.581328 + 0.813669i $$0.302533\pi$$
$$822$$ 0 0
$$823$$ −24.9706 −0.870419 −0.435210 0.900329i $$-0.643326\pi$$
−0.435210 + 0.900329i $$0.643326\pi$$
$$824$$ −55.1127 −1.91994
$$825$$ 0 0
$$826$$ 0 0
$$827$$ −36.3431 −1.26378 −0.631888 0.775060i $$-0.717720\pi$$
−0.631888 + 0.775060i $$0.717720\pi$$
$$828$$ 0 0
$$829$$ 24.7279 0.858836 0.429418 0.903106i $$-0.358719\pi$$
0.429418 + 0.903106i $$0.358719\pi$$
$$830$$ −21.6569 −0.751720
$$831$$ 0 0
$$832$$ −53.2132 −1.84484
$$833$$ 0 0
$$834$$ 0 0
$$835$$ 11.5980 0.401365
$$836$$ 21.6569 0.749018
$$837$$ 0 0
$$838$$ −93.7401 −3.23820
$$839$$ −45.1716 −1.55950 −0.779748 0.626094i $$-0.784653\pi$$
−0.779748 + 0.626094i $$0.784653\pi$$
$$840$$ 0 0
$$841$$ −27.6274 −0.952670
$$842$$ −69.1127 −2.38178
$$843$$ 0 0
$$844$$ −80.2843 −2.76350
$$845$$ −9.55635 −0.328748
$$846$$ 0 0
$$847$$ 0 0
$$848$$ 6.00000 0.206041
$$849$$ 0 0
$$850$$ 70.1838 2.40728
$$851$$ 14.6274 0.501421
$$852$$ 0 0
$$853$$ 49.4975 1.69476 0.847381 0.530986i $$-0.178178\pi$$
0.847381 + 0.530986i $$0.178178\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 51.4558 1.75872
$$857$$ −12.5858 −0.429922 −0.214961 0.976623i $$-0.568962\pi$$
−0.214961 + 0.976623i $$0.568962\pi$$
$$858$$ 0 0
$$859$$ 6.54416 0.223284 0.111642 0.993749i $$-0.464389\pi$$
0.111642 + 0.993749i $$0.464389\pi$$
$$860$$ 12.6863 0.432599
$$861$$ 0 0
$$862$$ −16.8284 −0.573179
$$863$$ 5.31371 0.180881 0.0904404 0.995902i $$-0.471173\pi$$
0.0904404 + 0.995902i $$0.471173\pi$$
$$864$$ 0 0
$$865$$ 4.05887 0.138006
$$866$$ −28.3848 −0.964554
$$867$$ 0 0
$$868$$ 0 0
$$869$$ 4.68629 0.158972
$$870$$ 0 0
$$871$$ 30.6274 1.03777
$$872$$ 24.9706 0.845610
$$873$$ 0 0
$$874$$ −24.9706 −0.844642
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 11.3137 0.382037 0.191018 0.981586i $$-0.438821\pi$$
0.191018 + 0.981586i $$0.438821\pi$$
$$878$$ 85.2548 2.87721
$$879$$ 0 0
$$880$$ −3.51472 −0.118481
$$881$$ −30.2426 −1.01890 −0.509450 0.860500i $$-0.670151\pi$$
−0.509450 + 0.860500i $$0.670151\pi$$
$$882$$ 0 0
$$883$$ −27.3137 −0.919179 −0.459590 0.888131i $$-0.652004\pi$$
−0.459590 + 0.888131i $$0.652004\pi$$
$$884$$ −129.397 −4.35209
$$885$$ 0 0
$$886$$ −2.48528 −0.0834947
$$887$$ −2.82843 −0.0949693 −0.0474846 0.998872i $$-0.515121\pi$$
−0.0474846 + 0.998872i $$0.515121\pi$$
$$888$$ 0 0
$$889$$ 0 0
$$890$$ −8.14214 −0.272925
$$891$$ 0 0
$$892$$ −34.3431 −1.14989
$$893$$ −8.00000 −0.267710
$$894$$ 0 0
$$895$$ −4.88730 −0.163364
$$896$$ 0 0
$$897$$ 0 0
$$898$$ −41.7990 −1.39485
$$899$$ −8.00000 −0.266815
$$900$$ 0 0
$$901$$ −12.4853 −0.415945
$$902$$ 10.8284 0.360547
$$903$$ 0 0
$$904$$ −76.4264 −2.54190
$$905$$ 3.17157 0.105427
$$906$$ 0 0
$$907$$ −16.0000 −0.531271 −0.265636 0.964073i $$-0.585582\pi$$
−0.265636 + 0.964073i $$0.585582\pi$$
$$908$$ 60.4853 2.00727
$$909$$ 0 0
$$910$$ 0 0
$$911$$ 34.9706 1.15863 0.579313 0.815105i $$-0.303321\pi$$
0.579313 + 0.815105i $$0.303321\pi$$
$$912$$ 0 0
$$913$$ 30.6274 1.01362
$$914$$ −43.4558 −1.43739
$$915$$ 0 0
$$916$$ 31.5563 1.04265
$$917$$ 0 0
$$918$$ 0 0
$$919$$ 48.2843 1.59275 0.796376 0.604802i $$-0.206748\pi$$
0.796376 + 0.604802i $$0.206748\pi$$
$$920$$ 9.45584 0.311750
$$921$$ 0 0
$$922$$ 46.8701 1.54358
$$923$$ 72.0833 2.37265
$$924$$ 0 0
$$925$$ 18.6274 0.612466
$$926$$ 44.9706 1.47782
$$927$$ 0 0
$$928$$ −1.85786 −0.0609874
$$929$$ −3.21320 −0.105422 −0.0527109 0.998610i $$-0.516786\pi$$
−0.0527109 + 0.998610i $$0.516786\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ −84.7696 −2.77672
$$933$$ 0 0
$$934$$ −96.0833 −3.14394
$$935$$ 7.31371 0.239184
$$936$$ 0 0
$$937$$ 33.4142 1.09159 0.545797 0.837917i $$-0.316227\pi$$
0.545797 + 0.837917i $$0.316227\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ 6.34315 0.206891
$$941$$ 7.21320 0.235144 0.117572 0.993064i $$-0.462489\pi$$
0.117572 + 0.993064i $$0.462489\pi$$
$$942$$ 0 0
$$943$$ −8.20101 −0.267062
$$944$$ 20.4853 0.666739
$$945$$ 0 0
$$946$$ −27.3137 −0.888045
$$947$$ −53.3137 −1.73246 −0.866231 0.499643i $$-0.833465\pi$$
−0.866231 + 0.499643i $$0.833465\pi$$
$$948$$ 0 0
$$949$$ −31.9411 −1.03685
$$950$$ −31.7990 −1.03170
$$951$$ 0 0
$$952$$ 0 0
$$953$$ −2.00000 −0.0647864 −0.0323932 0.999475i $$-0.510313\pi$$
−0.0323932 + 0.999475i $$0.510313\pi$$
$$954$$ 0 0
$$955$$ −10.5442 −0.341201
$$956$$ 16.6274 0.537769
$$957$$ 0 0
$$958$$ −72.7696 −2.35108
$$959$$ 0 0
$$960$$ 0 0
$$961$$ 15.6274 0.504110
$$962$$ −52.2843 −1.68571
$$963$$ 0 0
$$964$$ −29.6985 −0.956524
$$965$$ 10.1421 0.326487
$$966$$ 0 0
$$967$$ 22.3431 0.718507 0.359254 0.933240i $$-0.383031\pi$$
0.359254 + 0.933240i $$0.383031\pi$$
$$968$$ −30.8995 −0.993147
$$969$$ 0 0
$$970$$ −7.65685 −0.245847
$$971$$ 5.37258 0.172414 0.0862072 0.996277i $$-0.472525\pi$$
0.0862072 + 0.996277i $$0.472525\pi$$
$$972$$ 0 0
$$973$$ 0 0
$$974$$ −44.9706 −1.44095
$$975$$ 0 0
$$976$$ 11.2721 0.360810
$$977$$ 26.8284 0.858317 0.429159 0.903229i $$-0.358810\pi$$
0.429159 + 0.903229i $$0.358810\pi$$
$$978$$ 0 0
$$979$$ 11.5147 0.368012
$$980$$ 0 0
$$981$$ 0 0
$$982$$ −94.0833 −3.00232
$$983$$ −37.2548 −1.18824 −0.594122 0.804375i $$-0.702501\pi$$
−0.594122 + 0.804375i $$0.702501\pi$$
$$984$$ 0 0
$$985$$ 1.17157 0.0373294
$$986$$ −17.6569 −0.562309
$$987$$ 0 0
$$988$$ 58.6274 1.86519
$$989$$ 20.6863 0.657786
$$990$$ 0 0
$$991$$ 20.9706 0.666152 0.333076 0.942900i $$-0.391913\pi$$
0.333076 + 0.942900i $$0.391913\pi$$
$$992$$ 10.8284 0.343803
$$993$$ 0 0
$$994$$ 0 0
$$995$$ −6.05887 −0.192079
$$996$$ 0 0
$$997$$ −10.3848 −0.328889 −0.164445 0.986386i $$-0.552583\pi$$
−0.164445 + 0.986386i $$0.552583\pi$$
$$998$$ −46.6274 −1.47597
$$999$$ 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 441.2.a.i.1.2 2
3.2 odd 2 147.2.a.e.1.1 yes 2
4.3 odd 2 7056.2.a.cf.1.2 2
7.2 even 3 441.2.e.g.361.1 4
7.3 odd 6 441.2.e.f.226.1 4
7.4 even 3 441.2.e.g.226.1 4
7.5 odd 6 441.2.e.f.361.1 4
7.6 odd 2 441.2.a.j.1.2 2
12.11 even 2 2352.2.a.bc.1.1 2
15.14 odd 2 3675.2.a.bd.1.2 2
21.2 odd 6 147.2.e.d.67.2 4
21.5 even 6 147.2.e.e.67.2 4
21.11 odd 6 147.2.e.d.79.2 4
21.17 even 6 147.2.e.e.79.2 4
21.20 even 2 147.2.a.d.1.1 2
24.5 odd 2 9408.2.a.di.1.2 2
24.11 even 2 9408.2.a.dt.1.2 2
28.27 even 2 7056.2.a.cv.1.1 2
84.11 even 6 2352.2.q.bd.961.2 4
84.23 even 6 2352.2.q.bd.1537.2 4
84.47 odd 6 2352.2.q.bb.1537.1 4
84.59 odd 6 2352.2.q.bb.961.1 4
84.83 odd 2 2352.2.a.be.1.2 2
105.104 even 2 3675.2.a.bf.1.2 2
168.83 odd 2 9408.2.a.dq.1.1 2
168.125 even 2 9408.2.a.ef.1.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
147.2.a.d.1.1 2 21.20 even 2
147.2.a.e.1.1 yes 2 3.2 odd 2
147.2.e.d.67.2 4 21.2 odd 6
147.2.e.d.79.2 4 21.11 odd 6
147.2.e.e.67.2 4 21.5 even 6
147.2.e.e.79.2 4 21.17 even 6
441.2.a.i.1.2 2 1.1 even 1 trivial
441.2.a.j.1.2 2 7.6 odd 2
441.2.e.f.226.1 4 7.3 odd 6
441.2.e.f.361.1 4 7.5 odd 6
441.2.e.g.226.1 4 7.4 even 3
441.2.e.g.361.1 4 7.2 even 3
2352.2.a.bc.1.1 2 12.11 even 2
2352.2.a.be.1.2 2 84.83 odd 2
2352.2.q.bb.961.1 4 84.59 odd 6
2352.2.q.bb.1537.1 4 84.47 odd 6
2352.2.q.bd.961.2 4 84.11 even 6
2352.2.q.bd.1537.2 4 84.23 even 6
3675.2.a.bd.1.2 2 15.14 odd 2
3675.2.a.bf.1.2 2 105.104 even 2
7056.2.a.cf.1.2 2 4.3 odd 2
7056.2.a.cv.1.1 2 28.27 even 2
9408.2.a.di.1.2 2 24.5 odd 2
9408.2.a.dq.1.1 2 168.83 odd 2
9408.2.a.dt.1.2 2 24.11 even 2
9408.2.a.ef.1.1 2 168.125 even 2