# Properties

 Label 1089.4.a.t.1.2 Level $1089$ Weight $4$ Character 1089.1 Self dual yes Analytic conductor $64.253$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1089,4,Mod(1,1089)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1089, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("1089.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1089 = 3^{2} \cdot 11^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 1089.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: $$64.2530799963$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{33})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 8$$ x^2 - x - 8 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 33) Fricke sign: $$+1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$3.37228$$ of defining polynomial Character $$\chi$$ $$=$$ 1089.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+3.37228 q^{2} +3.37228 q^{4} +3.48913 q^{5} +4.74456 q^{7} -15.6060 q^{8} +O(q^{10})$$ $$q+3.37228 q^{2} +3.37228 q^{4} +3.48913 q^{5} +4.74456 q^{7} -15.6060 q^{8} +11.7663 q^{10} +15.0217 q^{13} +16.0000 q^{14} -79.6060 q^{16} +73.1684 q^{17} +78.7011 q^{19} +11.7663 q^{20} -112.000 q^{23} -112.826 q^{25} +50.6576 q^{26} +16.0000 q^{28} +243.125 q^{29} +278.717 q^{31} -143.606 q^{32} +246.745 q^{34} +16.5544 q^{35} +102.380 q^{37} +265.402 q^{38} -54.4512 q^{40} -241.255 q^{41} +280.016 q^{43} -377.696 q^{46} +169.870 q^{47} -320.489 q^{49} -380.481 q^{50} +50.6576 q^{52} +409.652 q^{53} -74.0435 q^{56} +819.886 q^{58} -196.000 q^{59} +701.359 q^{61} +939.913 q^{62} +152.568 q^{64} +52.4128 q^{65} +900.587 q^{67} +246.745 q^{68} +55.8260 q^{70} -756.500 q^{71} +1019.81 q^{73} +345.255 q^{74} +265.402 q^{76} +327.549 q^{79} -277.755 q^{80} -813.581 q^{82} -756.619 q^{83} +255.294 q^{85} +944.293 q^{86} -508.978 q^{89} +71.2716 q^{91} -377.696 q^{92} +572.848 q^{94} +274.598 q^{95} +614.358 q^{97} -1080.78 q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + q^{2} + q^{4} - 16 q^{5} - 2 q^{7} + 9 q^{8}+O(q^{10})$$ 2 * q + q^2 + q^4 - 16 * q^5 - 2 * q^7 + 9 * q^8 $$2 q + q^{2} + q^{4} - 16 q^{5} - 2 q^{7} + 9 q^{8} + 58 q^{10} + 76 q^{13} + 32 q^{14} - 119 q^{16} - 26 q^{17} + 54 q^{19} + 58 q^{20} - 224 q^{23} + 142 q^{25} - 94 q^{26} + 32 q^{28} + 222 q^{29} - 40 q^{31} - 247 q^{32} + 482 q^{34} + 148 q^{35} - 48 q^{37} + 324 q^{38} - 534 q^{40} - 494 q^{41} + 66 q^{43} - 112 q^{46} + 64 q^{47} - 618 q^{49} - 985 q^{50} - 94 q^{52} + 84 q^{53} - 240 q^{56} + 870 q^{58} - 392 q^{59} + 1104 q^{61} + 1696 q^{62} + 713 q^{64} - 1136 q^{65} + 928 q^{67} + 482 q^{68} - 256 q^{70} - 456 q^{71} + 592 q^{73} + 702 q^{74} + 324 q^{76} + 230 q^{79} + 490 q^{80} - 214 q^{82} + 348 q^{83} + 2188 q^{85} + 1452 q^{86} - 972 q^{89} - 340 q^{91} - 112 q^{92} + 824 q^{94} + 756 q^{95} - 1184 q^{97} - 375 q^{98}+O(q^{100})$$ 2 * q + q^2 + q^4 - 16 * q^5 - 2 * q^7 + 9 * q^8 + 58 * q^10 + 76 * q^13 + 32 * q^14 - 119 * q^16 - 26 * q^17 + 54 * q^19 + 58 * q^20 - 224 * q^23 + 142 * q^25 - 94 * q^26 + 32 * q^28 + 222 * q^29 - 40 * q^31 - 247 * q^32 + 482 * q^34 + 148 * q^35 - 48 * q^37 + 324 * q^38 - 534 * q^40 - 494 * q^41 + 66 * q^43 - 112 * q^46 + 64 * q^47 - 618 * q^49 - 985 * q^50 - 94 * q^52 + 84 * q^53 - 240 * q^56 + 870 * q^58 - 392 * q^59 + 1104 * q^61 + 1696 * q^62 + 713 * q^64 - 1136 * q^65 + 928 * q^67 + 482 * q^68 - 256 * q^70 - 456 * q^71 + 592 * q^73 + 702 * q^74 + 324 * q^76 + 230 * q^79 + 490 * q^80 - 214 * q^82 + 348 * q^83 + 2188 * q^85 + 1452 * q^86 - 972 * q^89 - 340 * q^91 - 112 * q^92 + 824 * q^94 + 756 * q^95 - 1184 * q^97 - 375 * q^98

## Coefficient data

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

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 3.37228 1.19228 0.596141 0.802880i $$-0.296700\pi$$
0.596141 + 0.802880i $$0.296700\pi$$
$$3$$ 0 0
$$4$$ 3.37228 0.421535
$$5$$ 3.48913 0.312077 0.156038 0.987751i $$-0.450128\pi$$
0.156038 + 0.987751i $$0.450128\pi$$
$$6$$ 0 0
$$7$$ 4.74456 0.256182 0.128091 0.991762i $$-0.459115\pi$$
0.128091 + 0.991762i $$0.459115\pi$$
$$8$$ −15.6060 −0.689693
$$9$$ 0 0
$$10$$ 11.7663 0.372083
$$11$$ 0 0
$$12$$ 0 0
$$13$$ 15.0217 0.320483 0.160242 0.987078i $$-0.448773\pi$$
0.160242 + 0.987078i $$0.448773\pi$$
$$14$$ 16.0000 0.305441
$$15$$ 0 0
$$16$$ −79.6060 −1.24384
$$17$$ 73.1684 1.04388 0.521940 0.852982i $$-0.325209\pi$$
0.521940 + 0.852982i $$0.325209\pi$$
$$18$$ 0 0
$$19$$ 78.7011 0.950277 0.475138 0.879911i $$-0.342398\pi$$
0.475138 + 0.879911i $$0.342398\pi$$
$$20$$ 11.7663 0.131551
$$21$$ 0 0
$$22$$ 0 0
$$23$$ −112.000 −1.01537 −0.507687 0.861541i $$-0.669499\pi$$
−0.507687 + 0.861541i $$0.669499\pi$$
$$24$$ 0 0
$$25$$ −112.826 −0.902608
$$26$$ 50.6576 0.382106
$$27$$ 0 0
$$28$$ 16.0000 0.107990
$$29$$ 243.125 1.55680 0.778399 0.627769i $$-0.216032\pi$$
0.778399 + 0.627769i $$0.216032\pi$$
$$30$$ 0 0
$$31$$ 278.717 1.61481 0.807405 0.589998i $$-0.200871\pi$$
0.807405 + 0.589998i $$0.200871\pi$$
$$32$$ −143.606 −0.793318
$$33$$ 0 0
$$34$$ 246.745 1.24460
$$35$$ 16.5544 0.0799486
$$36$$ 0 0
$$37$$ 102.380 0.454898 0.227449 0.973790i $$-0.426961\pi$$
0.227449 + 0.973790i $$0.426961\pi$$
$$38$$ 265.402 1.13300
$$39$$ 0 0
$$40$$ −54.4512 −0.215237
$$41$$ −241.255 −0.918970 −0.459485 0.888186i $$-0.651966\pi$$
−0.459485 + 0.888186i $$0.651966\pi$$
$$42$$ 0 0
$$43$$ 280.016 0.993071 0.496536 0.868016i $$-0.334605\pi$$
0.496536 + 0.868016i $$0.334605\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ −377.696 −1.21061
$$47$$ 169.870 0.527192 0.263596 0.964633i $$-0.415091\pi$$
0.263596 + 0.964633i $$0.415091\pi$$
$$48$$ 0 0
$$49$$ −320.489 −0.934371
$$50$$ −380.481 −1.07616
$$51$$ 0 0
$$52$$ 50.6576 0.135095
$$53$$ 409.652 1.06170 0.530849 0.847466i $$-0.321873\pi$$
0.530849 + 0.847466i $$0.321873\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ −74.0435 −0.176687
$$57$$ 0 0
$$58$$ 819.886 1.85614
$$59$$ −196.000 −0.432492 −0.216246 0.976339i $$-0.569381\pi$$
−0.216246 + 0.976339i $$0.569381\pi$$
$$60$$ 0 0
$$61$$ 701.359 1.47213 0.736064 0.676912i $$-0.236682\pi$$
0.736064 + 0.676912i $$0.236682\pi$$
$$62$$ 939.913 1.92531
$$63$$ 0 0
$$64$$ 152.568 0.297984
$$65$$ 52.4128 0.100015
$$66$$ 0 0
$$67$$ 900.587 1.64215 0.821076 0.570819i $$-0.193374\pi$$
0.821076 + 0.570819i $$0.193374\pi$$
$$68$$ 246.745 0.440032
$$69$$ 0 0
$$70$$ 55.8260 0.0953212
$$71$$ −756.500 −1.26451 −0.632254 0.774762i $$-0.717870\pi$$
−0.632254 + 0.774762i $$0.717870\pi$$
$$72$$ 0 0
$$73$$ 1019.81 1.63507 0.817536 0.575877i $$-0.195339\pi$$
0.817536 + 0.575877i $$0.195339\pi$$
$$74$$ 345.255 0.542367
$$75$$ 0 0
$$76$$ 265.402 0.400575
$$77$$ 0 0
$$78$$ 0 0
$$79$$ 327.549 0.466483 0.233241 0.972419i $$-0.425067\pi$$
0.233241 + 0.972419i $$0.425067\pi$$
$$80$$ −277.755 −0.388175
$$81$$ 0 0
$$82$$ −813.581 −1.09567
$$83$$ −756.619 −1.00060 −0.500300 0.865852i $$-0.666777\pi$$
−0.500300 + 0.865852i $$0.666777\pi$$
$$84$$ 0 0
$$85$$ 255.294 0.325771
$$86$$ 944.293 1.18402
$$87$$ 0 0
$$88$$ 0 0
$$89$$ −508.978 −0.606198 −0.303099 0.952959i $$-0.598021\pi$$
−0.303099 + 0.952959i $$0.598021\pi$$
$$90$$ 0 0
$$91$$ 71.2716 0.0821022
$$92$$ −377.696 −0.428016
$$93$$ 0 0
$$94$$ 572.848 0.628561
$$95$$ 274.598 0.296559
$$96$$ 0 0
$$97$$ 614.358 0.643079 0.321539 0.946896i $$-0.395800\pi$$
0.321539 + 0.946896i $$0.395800\pi$$
$$98$$ −1080.78 −1.11403
$$99$$ 0 0
$$100$$ −380.481 −0.380481
$$101$$ −1015.92 −1.00087 −0.500434 0.865775i $$-0.666826\pi$$
−0.500434 + 0.865775i $$0.666826\pi$$
$$102$$ 0 0
$$103$$ 1102.16 1.05436 0.527181 0.849753i $$-0.323249\pi$$
0.527181 + 0.849753i $$0.323249\pi$$
$$104$$ −234.429 −0.221035
$$105$$ 0 0
$$106$$ 1381.46 1.26584
$$107$$ 1377.58 1.24463 0.622315 0.782767i $$-0.286192\pi$$
0.622315 + 0.782767i $$0.286192\pi$$
$$108$$ 0 0
$$109$$ −320.217 −0.281388 −0.140694 0.990053i $$-0.544933\pi$$
−0.140694 + 0.990053i $$0.544933\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ −377.696 −0.318651
$$113$$ 1629.45 1.35651 0.678254 0.734828i $$-0.262737\pi$$
0.678254 + 0.734828i $$0.262737\pi$$
$$114$$ 0 0
$$115$$ −390.782 −0.316875
$$116$$ 819.886 0.656245
$$117$$ 0 0
$$118$$ −660.967 −0.515652
$$119$$ 347.152 0.267423
$$120$$ 0 0
$$121$$ 0 0
$$122$$ 2365.18 1.75519
$$123$$ 0 0
$$124$$ 939.913 0.680699
$$125$$ −829.805 −0.593760
$$126$$ 0 0
$$127$$ −2291.26 −1.60091 −0.800457 0.599390i $$-0.795410\pi$$
−0.800457 + 0.599390i $$0.795410\pi$$
$$128$$ 1663.35 1.14860
$$129$$ 0 0
$$130$$ 176.751 0.119247
$$131$$ −1147.41 −0.765267 −0.382633 0.923900i $$-0.624983\pi$$
−0.382633 + 0.923900i $$0.624983\pi$$
$$132$$ 0 0
$$133$$ 373.402 0.243444
$$134$$ 3037.03 1.95791
$$135$$ 0 0
$$136$$ −1141.86 −0.719956
$$137$$ −1268.60 −0.791121 −0.395561 0.918440i $$-0.629450\pi$$
−0.395561 + 0.918440i $$0.629450\pi$$
$$138$$ 0 0
$$139$$ 486.288 0.296737 0.148368 0.988932i $$-0.452598\pi$$
0.148368 + 0.988932i $$0.452598\pi$$
$$140$$ 55.8260 0.0337011
$$141$$ 0 0
$$142$$ −2551.13 −1.50765
$$143$$ 0 0
$$144$$ 0 0
$$145$$ 848.293 0.485841
$$146$$ 3439.10 1.94947
$$147$$ 0 0
$$148$$ 345.255 0.191756
$$149$$ 2354.11 1.29434 0.647169 0.762346i $$-0.275953\pi$$
0.647169 + 0.762346i $$0.275953\pi$$
$$150$$ 0 0
$$151$$ 570.070 0.307229 0.153615 0.988131i $$-0.450909\pi$$
0.153615 + 0.988131i $$0.450909\pi$$
$$152$$ −1228.21 −0.655399
$$153$$ 0 0
$$154$$ 0 0
$$155$$ 972.479 0.503945
$$156$$ 0 0
$$157$$ −2072.67 −1.05361 −0.526807 0.849985i $$-0.676611\pi$$
−0.526807 + 0.849985i $$0.676611\pi$$
$$158$$ 1104.59 0.556179
$$159$$ 0 0
$$160$$ −501.059 −0.247576
$$161$$ −531.391 −0.260121
$$162$$ 0 0
$$163$$ 2676.51 1.28614 0.643069 0.765808i $$-0.277661\pi$$
0.643069 + 0.765808i $$0.277661\pi$$
$$164$$ −813.581 −0.387378
$$165$$ 0 0
$$166$$ −2551.53 −1.19300
$$167$$ −1188.12 −0.550536 −0.275268 0.961368i $$-0.588767\pi$$
−0.275268 + 0.961368i $$0.588767\pi$$
$$168$$ 0 0
$$169$$ −1971.35 −0.897290
$$170$$ 860.923 0.388410
$$171$$ 0 0
$$172$$ 944.293 0.418615
$$173$$ 807.147 0.354718 0.177359 0.984146i $$-0.443245\pi$$
0.177359 + 0.984146i $$0.443245\pi$$
$$174$$ 0 0
$$175$$ −535.310 −0.231232
$$176$$ 0 0
$$177$$ 0 0
$$178$$ −1716.42 −0.722758
$$179$$ 1950.39 0.814408 0.407204 0.913337i $$-0.366504\pi$$
0.407204 + 0.913337i $$0.366504\pi$$
$$180$$ 0 0
$$181$$ 1061.61 0.435959 0.217980 0.975953i $$-0.430053\pi$$
0.217980 + 0.975953i $$0.430053\pi$$
$$182$$ 240.348 0.0978889
$$183$$ 0 0
$$184$$ 1747.87 0.700297
$$185$$ 357.218 0.141963
$$186$$ 0 0
$$187$$ 0 0
$$188$$ 572.848 0.222230
$$189$$ 0 0
$$190$$ 926.021 0.353582
$$191$$ −2136.41 −0.809348 −0.404674 0.914461i $$-0.632615\pi$$
−0.404674 + 0.914461i $$0.632615\pi$$
$$192$$ 0 0
$$193$$ −3947.76 −1.47236 −0.736181 0.676784i $$-0.763373\pi$$
−0.736181 + 0.676784i $$0.763373\pi$$
$$194$$ 2071.79 0.766731
$$195$$ 0 0
$$196$$ −1080.78 −0.393870
$$197$$ 923.886 0.334133 0.167066 0.985946i $$-0.446571\pi$$
0.167066 + 0.985946i $$0.446571\pi$$
$$198$$ 0 0
$$199$$ −476.152 −0.169616 −0.0848078 0.996397i $$-0.527028\pi$$
−0.0848078 + 0.996397i $$0.527028\pi$$
$$200$$ 1760.76 0.622522
$$201$$ 0 0
$$202$$ −3425.96 −1.19332
$$203$$ 1153.52 0.398824
$$204$$ 0 0
$$205$$ −841.770 −0.286789
$$206$$ 3716.80 1.25710
$$207$$ 0 0
$$208$$ −1195.82 −0.398631
$$209$$ 0 0
$$210$$ 0 0
$$211$$ 4918.24 1.60467 0.802336 0.596872i $$-0.203590\pi$$
0.802336 + 0.596872i $$0.203590\pi$$
$$212$$ 1381.46 0.447543
$$213$$ 0 0
$$214$$ 4645.57 1.48395
$$215$$ 977.012 0.309915
$$216$$ 0 0
$$217$$ 1322.39 0.413686
$$218$$ −1079.86 −0.335494
$$219$$ 0 0
$$220$$ 0 0
$$221$$ 1099.12 0.334546
$$222$$ 0 0
$$223$$ 2100.29 0.630700 0.315350 0.948975i $$-0.397878\pi$$
0.315350 + 0.948975i $$0.397878\pi$$
$$224$$ −681.348 −0.203234
$$225$$ 0 0
$$226$$ 5494.95 1.61734
$$227$$ −2257.16 −0.659970 −0.329985 0.943986i $$-0.607044\pi$$
−0.329985 + 0.943986i $$0.607044\pi$$
$$228$$ 0 0
$$229$$ −5311.07 −1.53260 −0.766301 0.642482i $$-0.777905\pi$$
−0.766301 + 0.642482i $$0.777905\pi$$
$$230$$ −1317.83 −0.377804
$$231$$ 0 0
$$232$$ −3794.20 −1.07371
$$233$$ 2466.27 0.693435 0.346718 0.937970i $$-0.387296\pi$$
0.346718 + 0.937970i $$0.387296\pi$$
$$234$$ 0 0
$$235$$ 592.696 0.164524
$$236$$ −660.967 −0.182311
$$237$$ 0 0
$$238$$ 1170.70 0.318844
$$239$$ 1429.40 0.386863 0.193432 0.981114i $$-0.438038\pi$$
0.193432 + 0.981114i $$0.438038\pi$$
$$240$$ 0 0
$$241$$ 978.989 0.261669 0.130835 0.991404i $$-0.458234\pi$$
0.130835 + 0.991404i $$0.458234\pi$$
$$242$$ 0 0
$$243$$ 0 0
$$244$$ 2365.18 0.620553
$$245$$ −1118.23 −0.291595
$$246$$ 0 0
$$247$$ 1182.23 0.304548
$$248$$ −4349.65 −1.11372
$$249$$ 0 0
$$250$$ −2798.33 −0.707929
$$251$$ 6530.63 1.64227 0.821135 0.570734i $$-0.193341\pi$$
0.821135 + 0.570734i $$0.193341\pi$$
$$252$$ 0 0
$$253$$ 0 0
$$254$$ −7726.76 −1.90874
$$255$$ 0 0
$$256$$ 4388.74 1.07147
$$257$$ −8130.26 −1.97335 −0.986676 0.162696i $$-0.947981\pi$$
−0.986676 + 0.162696i $$0.947981\pi$$
$$258$$ 0 0
$$259$$ 485.750 0.116537
$$260$$ 176.751 0.0421600
$$261$$ 0 0
$$262$$ −3869.40 −0.912414
$$263$$ −4549.42 −1.06665 −0.533326 0.845910i $$-0.679058\pi$$
−0.533326 + 0.845910i $$0.679058\pi$$
$$264$$ 0 0
$$265$$ 1429.33 0.331332
$$266$$ 1259.22 0.290254
$$267$$ 0 0
$$268$$ 3037.03 0.692225
$$269$$ 29.1522 0.00660760 0.00330380 0.999995i $$-0.498948\pi$$
0.00330380 + 0.999995i $$0.498948\pi$$
$$270$$ 0 0
$$271$$ −7711.22 −1.72850 −0.864250 0.503063i $$-0.832206\pi$$
−0.864250 + 0.503063i $$0.832206\pi$$
$$272$$ −5824.64 −1.29842
$$273$$ 0 0
$$274$$ −4278.07 −0.943239
$$275$$ 0 0
$$276$$ 0 0
$$277$$ −1127.52 −0.244571 −0.122286 0.992495i $$-0.539022\pi$$
−0.122286 + 0.992495i $$0.539022\pi$$
$$278$$ 1639.90 0.353794
$$279$$ 0 0
$$280$$ −258.347 −0.0551400
$$281$$ −1872.47 −0.397517 −0.198758 0.980049i $$-0.563691\pi$$
−0.198758 + 0.980049i $$0.563691\pi$$
$$282$$ 0 0
$$283$$ −2124.48 −0.446245 −0.223123 0.974790i $$-0.571625\pi$$
−0.223123 + 0.974790i $$0.571625\pi$$
$$284$$ −2551.13 −0.533034
$$285$$ 0 0
$$286$$ 0 0
$$287$$ −1144.65 −0.235424
$$288$$ 0 0
$$289$$ 440.621 0.0896846
$$290$$ 2860.68 0.579259
$$291$$ 0 0
$$292$$ 3439.10 0.689241
$$293$$ −3324.19 −0.662802 −0.331401 0.943490i $$-0.607521\pi$$
−0.331401 + 0.943490i $$0.607521\pi$$
$$294$$ 0 0
$$295$$ −683.869 −0.134971
$$296$$ −1597.75 −0.313740
$$297$$ 0 0
$$298$$ 7938.73 1.54322
$$299$$ −1682.44 −0.325411
$$300$$ 0 0
$$301$$ 1328.55 0.254407
$$302$$ 1922.44 0.366304
$$303$$ 0 0
$$304$$ −6265.07 −1.18200
$$305$$ 2447.13 0.459417
$$306$$ 0 0
$$307$$ 1698.94 0.315843 0.157921 0.987452i $$-0.449521\pi$$
0.157921 + 0.987452i $$0.449521\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 3279.47 0.600844
$$311$$ −6928.83 −1.26334 −0.631668 0.775239i $$-0.717630\pi$$
−0.631668 + 0.775239i $$0.717630\pi$$
$$312$$ 0 0
$$313$$ −3560.75 −0.643020 −0.321510 0.946906i $$-0.604190\pi$$
−0.321510 + 0.946906i $$0.604190\pi$$
$$314$$ −6989.64 −1.25620
$$315$$ 0 0
$$316$$ 1104.59 0.196639
$$317$$ −332.750 −0.0589561 −0.0294780 0.999565i $$-0.509385\pi$$
−0.0294780 + 0.999565i $$0.509385\pi$$
$$318$$ 0 0
$$319$$ 0 0
$$320$$ 532.329 0.0929940
$$321$$ 0 0
$$322$$ −1792.00 −0.310137
$$323$$ 5758.43 0.991975
$$324$$ 0 0
$$325$$ −1694.84 −0.289271
$$326$$ 9025.94 1.53344
$$327$$ 0 0
$$328$$ 3765.02 0.633807
$$329$$ 805.957 0.135057
$$330$$ 0 0
$$331$$ −541.445 −0.0899108 −0.0449554 0.998989i $$-0.514315\pi$$
−0.0449554 + 0.998989i $$0.514315\pi$$
$$332$$ −2551.53 −0.421788
$$333$$ 0 0
$$334$$ −4006.67 −0.656393
$$335$$ 3142.26 0.512478
$$336$$ 0 0
$$337$$ −816.531 −0.131986 −0.0659930 0.997820i $$-0.521022\pi$$
−0.0659930 + 0.997820i $$0.521022\pi$$
$$338$$ −6647.94 −1.06982
$$339$$ 0 0
$$340$$ 860.923 0.137324
$$341$$ 0 0
$$342$$ 0 0
$$343$$ −3147.97 −0.495552
$$344$$ −4369.92 −0.684914
$$345$$ 0 0
$$346$$ 2721.93 0.422924
$$347$$ 6260.53 0.968539 0.484269 0.874919i $$-0.339086\pi$$
0.484269 + 0.874919i $$0.339086\pi$$
$$348$$ 0 0
$$349$$ 12768.5 1.95840 0.979198 0.202906i $$-0.0650386\pi$$
0.979198 + 0.202906i $$0.0650386\pi$$
$$350$$ −1805.22 −0.275694
$$351$$ 0 0
$$352$$ 0 0
$$353$$ 2649.28 0.399453 0.199727 0.979852i $$-0.435995\pi$$
0.199727 + 0.979852i $$0.435995\pi$$
$$354$$ 0 0
$$355$$ −2639.52 −0.394623
$$356$$ −1716.42 −0.255534
$$357$$ 0 0
$$358$$ 6577.27 0.971004
$$359$$ −3203.91 −0.471020 −0.235510 0.971872i $$-0.575676\pi$$
−0.235510 + 0.971872i $$0.575676\pi$$
$$360$$ 0 0
$$361$$ −665.143 −0.0969737
$$362$$ 3580.04 0.519786
$$363$$ 0 0
$$364$$ 240.348 0.0346089
$$365$$ 3558.26 0.510268
$$366$$ 0 0
$$367$$ −8429.40 −1.19894 −0.599470 0.800397i $$-0.704622\pi$$
−0.599470 + 0.800397i $$0.704622\pi$$
$$368$$ 8915.87 1.26297
$$369$$ 0 0
$$370$$ 1204.64 0.169260
$$371$$ 1943.62 0.271988
$$372$$ 0 0
$$373$$ 9388.53 1.30327 0.651635 0.758533i $$-0.274083\pi$$
0.651635 + 0.758533i $$0.274083\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ −2650.98 −0.363600
$$377$$ 3652.16 0.498928
$$378$$ 0 0
$$379$$ −14264.5 −1.93329 −0.966647 0.256112i $$-0.917558\pi$$
−0.966647 + 0.256112i $$0.917558\pi$$
$$380$$ 926.021 0.125010
$$381$$ 0 0
$$382$$ −7204.58 −0.964970
$$383$$ −13462.2 −1.79605 −0.898026 0.439942i $$-0.854999\pi$$
−0.898026 + 0.439942i $$0.854999\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ −13313.0 −1.75547
$$387$$ 0 0
$$388$$ 2071.79 0.271080
$$389$$ 941.881 0.122764 0.0613821 0.998114i $$-0.480449\pi$$
0.0613821 + 0.998114i $$0.480449\pi$$
$$390$$ 0 0
$$391$$ −8194.87 −1.05993
$$392$$ 5001.54 0.644429
$$393$$ 0 0
$$394$$ 3115.60 0.398380
$$395$$ 1142.86 0.145578
$$396$$ 0 0
$$397$$ −847.839 −0.107183 −0.0535917 0.998563i $$-0.517067\pi$$
−0.0535917 + 0.998563i $$0.517067\pi$$
$$398$$ −1605.72 −0.202230
$$399$$ 0 0
$$400$$ 8981.62 1.12270
$$401$$ −12203.6 −1.51975 −0.759875 0.650069i $$-0.774740\pi$$
−0.759875 + 0.650069i $$0.774740\pi$$
$$402$$ 0 0
$$403$$ 4186.82 0.517520
$$404$$ −3425.96 −0.421901
$$405$$ 0 0
$$406$$ 3890.00 0.475511
$$407$$ 0 0
$$408$$ 0 0
$$409$$ −8759.53 −1.05900 −0.529500 0.848310i $$-0.677620\pi$$
−0.529500 + 0.848310i $$0.677620\pi$$
$$410$$ −2838.69 −0.341934
$$411$$ 0 0
$$412$$ 3716.80 0.444451
$$413$$ −929.934 −0.110797
$$414$$ 0 0
$$415$$ −2639.94 −0.312264
$$416$$ −2157.21 −0.254245
$$417$$ 0 0
$$418$$ 0 0
$$419$$ 11188.4 1.30451 0.652256 0.757999i $$-0.273823\pi$$
0.652256 + 0.757999i $$0.273823\pi$$
$$420$$ 0 0
$$421$$ −14082.3 −1.63023 −0.815116 0.579298i $$-0.803327\pi$$
−0.815116 + 0.579298i $$0.803327\pi$$
$$422$$ 16585.7 1.91322
$$423$$ 0 0
$$424$$ −6393.02 −0.732246
$$425$$ −8255.30 −0.942214
$$426$$ 0 0
$$427$$ 3327.64 0.377133
$$428$$ 4645.57 0.524655
$$429$$ 0 0
$$430$$ 3294.76 0.369505
$$431$$ −5616.05 −0.627647 −0.313823 0.949481i $$-0.601610\pi$$
−0.313823 + 0.949481i $$0.601610\pi$$
$$432$$ 0 0
$$433$$ 7195.75 0.798627 0.399314 0.916814i $$-0.369248\pi$$
0.399314 + 0.916814i $$0.369248\pi$$
$$434$$ 4459.48 0.493230
$$435$$ 0 0
$$436$$ −1079.86 −0.118615
$$437$$ −8814.52 −0.964887
$$438$$ 0 0
$$439$$ −101.959 −0.0110848 −0.00554240 0.999985i $$-0.501764\pi$$
−0.00554240 + 0.999985i $$0.501764\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 3706.53 0.398873
$$443$$ −4953.74 −0.531285 −0.265642 0.964072i $$-0.585584\pi$$
−0.265642 + 0.964072i $$0.585584\pi$$
$$444$$ 0 0
$$445$$ −1775.89 −0.189180
$$446$$ 7082.78 0.751972
$$447$$ 0 0
$$448$$ 723.869 0.0763383
$$449$$ 11602.0 1.21945 0.609723 0.792615i $$-0.291281\pi$$
0.609723 + 0.792615i $$0.291281\pi$$
$$450$$ 0 0
$$451$$ 0 0
$$452$$ 5494.95 0.571816
$$453$$ 0 0
$$454$$ −7611.79 −0.786870
$$455$$ 248.676 0.0256222
$$456$$ 0 0
$$457$$ 3530.68 0.361397 0.180698 0.983539i $$-0.442164\pi$$
0.180698 + 0.983539i $$0.442164\pi$$
$$458$$ −17910.4 −1.82729
$$459$$ 0 0
$$460$$ −1317.83 −0.133574
$$461$$ 11566.3 1.16854 0.584271 0.811559i $$-0.301381\pi$$
0.584271 + 0.811559i $$0.301381\pi$$
$$462$$ 0 0
$$463$$ 10888.5 1.09294 0.546470 0.837479i $$-0.315971\pi$$
0.546470 + 0.837479i $$0.315971\pi$$
$$464$$ −19354.2 −1.93641
$$465$$ 0 0
$$466$$ 8316.94 0.826770
$$467$$ −10688.0 −1.05906 −0.529529 0.848292i $$-0.677631\pi$$
−0.529529 + 0.848292i $$0.677631\pi$$
$$468$$ 0 0
$$469$$ 4272.89 0.420690
$$470$$ 1998.74 0.196159
$$471$$ 0 0
$$472$$ 3058.77 0.298287
$$473$$ 0 0
$$474$$ 0 0
$$475$$ −8879.53 −0.857728
$$476$$ 1170.70 0.112728
$$477$$ 0 0
$$478$$ 4820.35 0.461250
$$479$$ 2341.90 0.223391 0.111696 0.993742i $$-0.464372\pi$$
0.111696 + 0.993742i $$0.464372\pi$$
$$480$$ 0 0
$$481$$ 1537.93 0.145787
$$482$$ 3301.43 0.311983
$$483$$ 0 0
$$484$$ 0 0
$$485$$ 2143.57 0.200690
$$486$$ 0 0
$$487$$ 6748.91 0.627972 0.313986 0.949428i $$-0.398335\pi$$
0.313986 + 0.949428i $$0.398335\pi$$
$$488$$ −10945.4 −1.01532
$$489$$ 0 0
$$490$$ −3770.98 −0.347664
$$491$$ 7361.40 0.676609 0.338305 0.941037i $$-0.390147\pi$$
0.338305 + 0.941037i $$0.390147\pi$$
$$492$$ 0 0
$$493$$ 17789.1 1.62511
$$494$$ 3986.80 0.363107
$$495$$ 0 0
$$496$$ −22187.6 −2.00857
$$497$$ −3589.26 −0.323944
$$498$$ 0 0
$$499$$ 10381.7 0.931359 0.465680 0.884953i $$-0.345810\pi$$
0.465680 + 0.884953i $$0.345810\pi$$
$$500$$ −2798.33 −0.250291
$$501$$ 0 0
$$502$$ 22023.1 1.95805
$$503$$ 19149.0 1.69744 0.848721 0.528840i $$-0.177373\pi$$
0.848721 + 0.528840i $$0.177373\pi$$
$$504$$ 0 0
$$505$$ −3544.67 −0.312348
$$506$$ 0 0
$$507$$ 0 0
$$508$$ −7726.76 −0.674841
$$509$$ −16073.2 −1.39967 −0.699836 0.714303i $$-0.746744\pi$$
−0.699836 + 0.714303i $$0.746744\pi$$
$$510$$ 0 0
$$511$$ 4838.58 0.418877
$$512$$ 1493.27 0.128894
$$513$$ 0 0
$$514$$ −27417.5 −2.35279
$$515$$ 3845.58 0.329042
$$516$$ 0 0
$$517$$ 0 0
$$518$$ 1638.09 0.138945
$$519$$ 0 0
$$520$$ −817.952 −0.0689799
$$521$$ 18955.3 1.59395 0.796975 0.604012i $$-0.206432\pi$$
0.796975 + 0.604012i $$0.206432\pi$$
$$522$$ 0 0
$$523$$ 4442.19 0.371402 0.185701 0.982606i $$-0.440544\pi$$
0.185701 + 0.982606i $$0.440544\pi$$
$$524$$ −3869.40 −0.322587
$$525$$ 0 0
$$526$$ −15341.9 −1.27175
$$527$$ 20393.3 1.68567
$$528$$ 0 0
$$529$$ 377.000 0.0309855
$$530$$ 4820.09 0.395041
$$531$$ 0 0
$$532$$ 1259.22 0.102620
$$533$$ −3624.08 −0.294515
$$534$$ 0 0
$$535$$ 4806.54 0.388420
$$536$$ −14054.5 −1.13258
$$537$$ 0 0
$$538$$ 98.3096 0.00787812
$$539$$ 0 0
$$540$$ 0 0
$$541$$ −2180.90 −0.173316 −0.0866580 0.996238i $$-0.527619\pi$$
−0.0866580 + 0.996238i $$0.527619\pi$$
$$542$$ −26004.4 −2.06086
$$543$$ 0 0
$$544$$ −10507.4 −0.828129
$$545$$ −1117.28 −0.0878146
$$546$$ 0 0
$$547$$ −8225.04 −0.642920 −0.321460 0.946923i $$-0.604174\pi$$
−0.321460 + 0.946923i $$0.604174\pi$$
$$548$$ −4278.07 −0.333485
$$549$$ 0 0
$$550$$ 0 0
$$551$$ 19134.2 1.47939
$$552$$ 0 0
$$553$$ 1554.08 0.119505
$$554$$ −3802.32 −0.291598
$$555$$ 0 0
$$556$$ 1639.90 0.125085
$$557$$ −25181.9 −1.91561 −0.957804 0.287423i $$-0.907201\pi$$
−0.957804 + 0.287423i $$0.907201\pi$$
$$558$$ 0 0
$$559$$ 4206.33 0.318263
$$560$$ −1317.83 −0.0994435
$$561$$ 0 0
$$562$$ −6314.50 −0.473952
$$563$$ −4504.50 −0.337197 −0.168599 0.985685i $$-0.553924\pi$$
−0.168599 + 0.985685i $$0.553924\pi$$
$$564$$ 0 0
$$565$$ 5685.34 0.423335
$$566$$ −7164.36 −0.532050
$$567$$ 0 0
$$568$$ 11805.9 0.872122
$$569$$ −13447.0 −0.990732 −0.495366 0.868684i $$-0.664966\pi$$
−0.495366 + 0.868684i $$0.664966\pi$$
$$570$$ 0 0
$$571$$ 2605.52 0.190959 0.0954795 0.995431i $$-0.469562\pi$$
0.0954795 + 0.995431i $$0.469562\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ −3860.09 −0.280691
$$575$$ 12636.5 0.916485
$$576$$ 0 0
$$577$$ 6339.65 0.457406 0.228703 0.973496i $$-0.426552\pi$$
0.228703 + 0.973496i $$0.426552\pi$$
$$578$$ 1485.90 0.106929
$$579$$ 0 0
$$580$$ 2860.68 0.204799
$$581$$ −3589.83 −0.256336
$$582$$ 0 0
$$583$$ 0 0
$$584$$ −15915.2 −1.12770
$$585$$ 0 0
$$586$$ −11210.1 −0.790247
$$587$$ 13370.6 0.940140 0.470070 0.882629i $$-0.344229\pi$$
0.470070 + 0.882629i $$0.344229\pi$$
$$588$$ 0 0
$$589$$ 21935.3 1.53452
$$590$$ −2306.20 −0.160923
$$591$$ 0 0
$$592$$ −8150.09 −0.565822
$$593$$ 14319.3 0.991608 0.495804 0.868434i $$-0.334873\pi$$
0.495804 + 0.868434i $$0.334873\pi$$
$$594$$ 0 0
$$595$$ 1211.26 0.0834567
$$596$$ 7938.73 0.545609
$$597$$ 0 0
$$598$$ −5673.65 −0.387981
$$599$$ 5788.63 0.394853 0.197427 0.980318i $$-0.436742\pi$$
0.197427 + 0.980318i $$0.436742\pi$$
$$600$$ 0 0
$$601$$ −23968.1 −1.62675 −0.813375 0.581739i $$-0.802372\pi$$
−0.813375 + 0.581739i $$0.802372\pi$$
$$602$$ 4480.26 0.303325
$$603$$ 0 0
$$604$$ 1922.44 0.129508
$$605$$ 0 0
$$606$$ 0 0
$$607$$ 23526.6 1.57317 0.786585 0.617482i $$-0.211847\pi$$
0.786585 + 0.617482i $$0.211847\pi$$
$$608$$ −11301.9 −0.753872
$$609$$ 0 0
$$610$$ 8252.40 0.547754
$$611$$ 2551.74 0.168956
$$612$$ 0 0
$$613$$ −1228.07 −0.0809159 −0.0404579 0.999181i $$-0.512882\pi$$
−0.0404579 + 0.999181i $$0.512882\pi$$
$$614$$ 5729.31 0.376573
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 9844.90 0.642368 0.321184 0.947017i $$-0.395919\pi$$
0.321184 + 0.947017i $$0.395919\pi$$
$$618$$ 0 0
$$619$$ −6551.68 −0.425419 −0.212709 0.977115i $$-0.568229\pi$$
−0.212709 + 0.977115i $$0.568229\pi$$
$$620$$ 3279.47 0.212430
$$621$$ 0 0
$$622$$ −23365.9 −1.50625
$$623$$ −2414.88 −0.155297
$$624$$ 0 0
$$625$$ 11208.0 0.717309
$$626$$ −12007.8 −0.766661
$$627$$ 0 0
$$628$$ −6989.64 −0.444135
$$629$$ 7491.01 0.474859
$$630$$ 0 0
$$631$$ −26440.5 −1.66812 −0.834058 0.551677i $$-0.813988\pi$$
−0.834058 + 0.551677i $$0.813988\pi$$
$$632$$ −5111.72 −0.321730
$$633$$ 0 0
$$634$$ −1122.13 −0.0702923
$$635$$ −7994.48 −0.499608
$$636$$ 0 0
$$637$$ −4814.31 −0.299450
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 5803.64 0.358451
$$641$$ 27927.2 1.72084 0.860421 0.509584i $$-0.170201\pi$$
0.860421 + 0.509584i $$0.170201\pi$$
$$642$$ 0 0
$$643$$ −16737.7 −1.02655 −0.513274 0.858225i $$-0.671568\pi$$
−0.513274 + 0.858225i $$0.671568\pi$$
$$644$$ −1792.00 −0.109650
$$645$$ 0 0
$$646$$ 19419.1 1.18271
$$647$$ −7818.70 −0.475092 −0.237546 0.971376i $$-0.576343\pi$$
−0.237546 + 0.971376i $$0.576343\pi$$
$$648$$ 0 0
$$649$$ 0 0
$$650$$ −5715.49 −0.344892
$$651$$ 0 0
$$652$$ 9025.94 0.542152
$$653$$ −19747.6 −1.18344 −0.591719 0.806144i $$-0.701550\pi$$
−0.591719 + 0.806144i $$0.701550\pi$$
$$654$$ 0 0
$$655$$ −4003.47 −0.238822
$$656$$ 19205.4 1.14305
$$657$$ 0 0
$$658$$ 2717.91 0.161026
$$659$$ 7867.72 0.465072 0.232536 0.972588i $$-0.425298\pi$$
0.232536 + 0.972588i $$0.425298\pi$$
$$660$$ 0 0
$$661$$ 4227.41 0.248755 0.124378 0.992235i $$-0.460307\pi$$
0.124378 + 0.992235i $$0.460307\pi$$
$$662$$ −1825.90 −0.107199
$$663$$ 0 0
$$664$$ 11807.8 0.690106
$$665$$ 1302.85 0.0759733
$$666$$ 0 0
$$667$$ −27230.0 −1.58073
$$668$$ −4006.67 −0.232070
$$669$$ 0 0
$$670$$ 10596.6 0.611018
$$671$$ 0 0
$$672$$ 0 0
$$673$$ −29397.6 −1.68379 −0.841897 0.539638i $$-0.818561\pi$$
−0.841897 + 0.539638i $$0.818561\pi$$
$$674$$ −2753.57 −0.157364
$$675$$ 0 0
$$676$$ −6647.94 −0.378239
$$677$$ 5737.14 0.325696 0.162848 0.986651i $$-0.447932\pi$$
0.162848 + 0.986651i $$0.447932\pi$$
$$678$$ 0 0
$$679$$ 2914.86 0.164745
$$680$$ −3984.11 −0.224682
$$681$$ 0 0
$$682$$ 0 0
$$683$$ −32097.6 −1.79821 −0.899107 0.437729i $$-0.855783\pi$$
−0.899107 + 0.437729i $$0.855783\pi$$
$$684$$ 0 0
$$685$$ −4426.30 −0.246891
$$686$$ −10615.8 −0.590837
$$687$$ 0 0
$$688$$ −22291.0 −1.23523
$$689$$ 6153.69 0.340257
$$690$$ 0 0
$$691$$ −16456.2 −0.905965 −0.452983 0.891519i $$-0.649640\pi$$
−0.452983 + 0.891519i $$0.649640\pi$$
$$692$$ 2721.93 0.149526
$$693$$ 0 0
$$694$$ 21112.3 1.15477
$$695$$ 1696.72 0.0926047
$$696$$ 0 0
$$697$$ −17652.3 −0.959294
$$698$$ 43058.9 2.33496
$$699$$ 0 0
$$700$$ −1805.22 −0.0974725
$$701$$ 27238.1 1.46758 0.733788 0.679379i $$-0.237751\pi$$
0.733788 + 0.679379i $$0.237751\pi$$
$$702$$ 0 0
$$703$$ 8057.44 0.432279
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 8934.12 0.476261
$$707$$ −4820.09 −0.256405
$$708$$ 0 0
$$709$$ 28761.4 1.52349 0.761747 0.647875i $$-0.224342\pi$$
0.761747 + 0.647875i $$0.224342\pi$$
$$710$$ −8901.21 −0.470502
$$711$$ 0 0
$$712$$ 7943.10 0.418090
$$713$$ −31216.3 −1.63964
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 6577.27 0.343302
$$717$$ 0 0
$$718$$ −10804.5 −0.561588
$$719$$ 27272.0 1.41456 0.707282 0.706931i $$-0.249921\pi$$
0.707282 + 0.706931i $$0.249921\pi$$
$$720$$ 0 0
$$721$$ 5229.28 0.270109
$$722$$ −2243.05 −0.115620
$$723$$ 0 0
$$724$$ 3580.04 0.183772
$$725$$ −27430.8 −1.40518
$$726$$ 0 0
$$727$$ 3979.75 0.203027 0.101514 0.994834i $$-0.467631\pi$$
0.101514 + 0.994834i $$0.467631\pi$$
$$728$$ −1112.26 −0.0566253
$$729$$ 0 0
$$730$$ 11999.5 0.608383
$$731$$ 20488.3 1.03665
$$732$$ 0 0
$$733$$ −9342.48 −0.470767 −0.235384 0.971903i $$-0.575635\pi$$
−0.235384 + 0.971903i $$0.575635\pi$$
$$734$$ −28426.3 −1.42947
$$735$$ 0 0
$$736$$ 16083.9 0.805515
$$737$$ 0 0
$$738$$ 0 0
$$739$$ 28928.0 1.43997 0.719983 0.693992i $$-0.244150\pi$$
0.719983 + 0.693992i $$0.244150\pi$$
$$740$$ 1204.64 0.0598425
$$741$$ 0 0
$$742$$ 6554.43 0.324287
$$743$$ 4857.04 0.239822 0.119911 0.992785i $$-0.461739\pi$$
0.119911 + 0.992785i $$0.461739\pi$$
$$744$$ 0 0
$$745$$ 8213.80 0.403933
$$746$$ 31660.8 1.55386
$$747$$ 0 0
$$748$$ 0 0
$$749$$ 6536.00 0.318852
$$750$$ 0 0
$$751$$ 14355.4 0.697517 0.348759 0.937213i $$-0.386603\pi$$
0.348759 + 0.937213i $$0.386603\pi$$
$$752$$ −13522.6 −0.655744
$$753$$ 0 0
$$754$$ 12316.1 0.594863
$$755$$ 1989.05 0.0958792
$$756$$ 0 0
$$757$$ −17714.9 −0.850538 −0.425269 0.905067i $$-0.639821\pi$$
−0.425269 + 0.905067i $$0.639821\pi$$
$$758$$ −48103.9 −2.30503
$$759$$ 0 0
$$760$$ −4285.37 −0.204535
$$761$$ −7945.82 −0.378497 −0.189248 0.981929i $$-0.560605\pi$$
−0.189248 + 0.981929i $$0.560605\pi$$
$$762$$ 0 0
$$763$$ −1519.29 −0.0720866
$$764$$ −7204.58 −0.341168
$$765$$ 0 0
$$766$$ −45398.4 −2.14140
$$767$$ −2944.26 −0.138606
$$768$$ 0 0
$$769$$ −27308.1 −1.28057 −0.640284 0.768139i $$-0.721183\pi$$
−0.640284 + 0.768139i $$0.721183\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ −13313.0 −0.620653
$$773$$ 18872.6 0.878136 0.439068 0.898454i $$-0.355309\pi$$
0.439068 + 0.898454i $$0.355309\pi$$
$$774$$ 0 0
$$775$$ −31446.6 −1.45754
$$776$$ −9587.65 −0.443527
$$777$$ 0 0
$$778$$ 3176.29 0.146369
$$779$$ −18987.1 −0.873276
$$780$$ 0 0
$$781$$ 0 0
$$782$$ −27635.4 −1.26373
$$783$$ 0 0
$$784$$ 25512.8 1.16221
$$785$$ −7231.82 −0.328808
$$786$$ 0 0
$$787$$ 14512.1 0.657307 0.328654 0.944450i $$-0.393405\pi$$
0.328654 + 0.944450i $$0.393405\pi$$
$$788$$ 3115.60 0.140849
$$789$$ 0 0
$$790$$ 3854.04 0.173570
$$791$$ 7731.01 0.347513
$$792$$ 0 0
$$793$$ 10535.6 0.471792
$$794$$ −2859.15 −0.127793
$$795$$ 0 0
$$796$$ −1605.72 −0.0714989
$$797$$ −29108.9 −1.29371 −0.646856 0.762612i $$-0.723917\pi$$
−0.646856 + 0.762612i $$0.723917\pi$$
$$798$$ 0 0
$$799$$ 12429.1 0.550325
$$800$$ 16202.5 0.716056
$$801$$ 0 0
$$802$$ −41154.0 −1.81197
$$803$$ 0 0
$$804$$ 0 0
$$805$$ −1854.09 −0.0811777
$$806$$ 14119.1 0.617029
$$807$$ 0 0
$$808$$ 15854.4 0.690291
$$809$$ −3000.83 −0.130413 −0.0652063 0.997872i $$-0.520771\pi$$
−0.0652063 + 0.997872i $$0.520771\pi$$
$$810$$ 0 0
$$811$$ −6239.39 −0.270154 −0.135077 0.990835i $$-0.543128\pi$$
−0.135077 + 0.990835i $$0.543128\pi$$
$$812$$ 3890.00 0.168118
$$813$$ 0 0
$$814$$ 0 0
$$815$$ 9338.68 0.401374
$$816$$ 0 0
$$817$$ 22037.6 0.943693
$$818$$ −29539.6 −1.26263
$$819$$ 0 0
$$820$$ −2838.69 −0.120892
$$821$$ 14922.4 0.634342 0.317171 0.948368i $$-0.397267\pi$$
0.317171 + 0.948368i $$0.397267\pi$$
$$822$$ 0 0
$$823$$ −25737.8 −1.09011 −0.545057 0.838399i $$-0.683492\pi$$
−0.545057 + 0.838399i $$0.683492\pi$$
$$824$$ −17200.3 −0.727186
$$825$$ 0 0
$$826$$ −3136.00 −0.132101
$$827$$ 27043.4 1.13711 0.568555 0.822645i $$-0.307503\pi$$
0.568555 + 0.822645i $$0.307503\pi$$
$$828$$ 0 0
$$829$$ −9795.41 −0.410384 −0.205192 0.978722i $$-0.565782\pi$$
−0.205192 + 0.978722i $$0.565782\pi$$
$$830$$ −8902.62 −0.372306
$$831$$ 0 0
$$832$$ 2291.84 0.0954990
$$833$$ −23449.7 −0.975370
$$834$$ 0 0
$$835$$ −4145.50 −0.171809
$$836$$ 0 0
$$837$$ 0 0
$$838$$ 37730.5 1.55535
$$839$$ −28875.5 −1.18819 −0.594095 0.804395i $$-0.702490\pi$$
−0.594095 + 0.804395i $$0.702490\pi$$
$$840$$ 0 0
$$841$$ 34720.7 1.42362
$$842$$ −47489.3 −1.94369
$$843$$ 0 0
$$844$$ 16585.7 0.676426
$$845$$ −6878.28 −0.280024
$$846$$ 0 0
$$847$$ 0 0
$$848$$ −32610.7 −1.32059
$$849$$ 0 0
$$850$$ −27839.2 −1.12338
$$851$$ −11466.6 −0.461892
$$852$$ 0 0
$$853$$ 47157.1 1.89288 0.946441 0.322878i $$-0.104650\pi$$
0.946441 + 0.322878i $$0.104650\pi$$
$$854$$ 11221.7 0.449649
$$855$$ 0 0
$$856$$ −21498.4 −0.858412
$$857$$ 5021.31 0.200145 0.100073 0.994980i $$-0.468092\pi$$
0.100073 + 0.994980i $$0.468092\pi$$
$$858$$ 0 0
$$859$$ −22921.1 −0.910428 −0.455214 0.890382i $$-0.650437\pi$$
−0.455214 + 0.890382i $$0.650437\pi$$
$$860$$ 3294.76 0.130640
$$861$$ 0 0
$$862$$ −18938.9 −0.748332
$$863$$ 19488.1 0.768693 0.384347 0.923189i $$-0.374427\pi$$
0.384347 + 0.923189i $$0.374427\pi$$
$$864$$ 0 0
$$865$$ 2816.24 0.110699
$$866$$ 24266.1 0.952188
$$867$$ 0 0
$$868$$ 4459.48 0.174383
$$869$$ 0 0
$$870$$ 0 0
$$871$$ 13528.4 0.526282
$$872$$ 4997.30 0.194071
$$873$$ 0 0
$$874$$ −29725.0 −1.15042
$$875$$ −3937.06 −0.152111
$$876$$ 0 0
$$877$$ 8455.67 0.325573 0.162787 0.986661i $$-0.447952\pi$$
0.162787 + 0.986661i $$0.447952\pi$$
$$878$$ −343.834 −0.0132162
$$879$$ 0 0
$$880$$ 0 0
$$881$$ 11291.2 0.431794 0.215897 0.976416i $$-0.430732\pi$$
0.215897 + 0.976416i $$0.430732\pi$$
$$882$$ 0 0
$$883$$ 31818.1 1.21264 0.606322 0.795219i $$-0.292644\pi$$
0.606322 + 0.795219i $$0.292644\pi$$
$$884$$ 3706.53 0.141023
$$885$$ 0 0
$$886$$ −16705.4 −0.633441
$$887$$ 17481.1 0.661732 0.330866 0.943678i $$-0.392659\pi$$
0.330866 + 0.943678i $$0.392659\pi$$
$$888$$ 0 0
$$889$$ −10871.0 −0.410126
$$890$$ −5988.80 −0.225556
$$891$$ 0 0
$$892$$ 7082.78 0.265862
$$893$$ 13368.9 0.500978
$$894$$ 0 0
$$895$$ 6805.16 0.254158
$$896$$ 7891.87 0.294251
$$897$$ 0 0
$$898$$ 39125.1 1.45392
$$899$$ 67763.1 2.51393
$$900$$ 0 0
$$901$$ 29973.6 1.10829
$$902$$ 0 0
$$903$$ 0 0
$$904$$ −25429.1 −0.935574
$$905$$ 3704.08 0.136053
$$906$$ 0 0
$$907$$ 10607.4 0.388326 0.194163 0.980969i $$-0.437801\pi$$
0.194163 + 0.980969i $$0.437801\pi$$
$$908$$ −7611.79 −0.278201
$$909$$ 0 0
$$910$$ 838.604 0.0305489
$$911$$ 41249.2 1.50016 0.750080 0.661347i $$-0.230015\pi$$
0.750080 + 0.661347i $$0.230015\pi$$
$$912$$ 0 0
$$913$$ 0 0
$$914$$ 11906.5 0.430887
$$915$$ 0 0
$$916$$ −17910.4 −0.646045
$$917$$ −5443.97 −0.196048
$$918$$ 0 0
$$919$$ 13858.1 0.497429 0.248714 0.968577i $$-0.419992\pi$$
0.248714 + 0.968577i $$0.419992\pi$$
$$920$$ 6098.53 0.218546
$$921$$ 0 0
$$922$$ 39004.9 1.39323
$$923$$ −11363.9 −0.405253
$$924$$ 0 0
$$925$$ −11551.2 −0.410595
$$926$$ 36719.0 1.30309
$$927$$ 0 0
$$928$$ −34914.2 −1.23504
$$929$$ −20893.7 −0.737890 −0.368945 0.929451i $$-0.620281\pi$$
−0.368945 + 0.929451i $$0.620281\pi$$
$$930$$ 0 0
$$931$$ −25222.8 −0.887911
$$932$$ 8316.94 0.292307
$$933$$ 0 0
$$934$$ −36042.9 −1.26270
$$935$$ 0 0
$$936$$ 0 0
$$937$$ −3203.52 −0.111691 −0.0558454 0.998439i $$-0.517785\pi$$
−0.0558454 + 0.998439i $$0.517785\pi$$
$$938$$ 14409.4 0.501581
$$939$$ 0 0
$$940$$ 1998.74 0.0693528
$$941$$ 19951.6 0.691182 0.345591 0.938385i $$-0.387678\pi$$
0.345591 + 0.938385i $$0.387678\pi$$
$$942$$ 0 0
$$943$$ 27020.6 0.933099
$$944$$ 15602.8 0.537952
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 38216.7 1.31138 0.655689 0.755031i $$-0.272378\pi$$
0.655689 + 0.755031i $$0.272378\pi$$
$$948$$ 0 0
$$949$$ 15319.4 0.524014
$$950$$ −29944.3 −1.02265
$$951$$ 0 0
$$952$$ −5417.65 −0.184440
$$953$$ −47661.4 −1.62004 −0.810022 0.586399i $$-0.800545\pi$$
−0.810022 + 0.586399i $$0.800545\pi$$
$$954$$ 0 0
$$955$$ −7454.21 −0.252579
$$956$$ 4820.35 0.163077
$$957$$ 0 0
$$958$$ 7897.56 0.266345
$$959$$ −6018.94 −0.202671
$$960$$ 0 0
$$961$$ 47892.3 1.60761
$$962$$ 5186.34 0.173819
$$963$$ 0 0
$$964$$ 3301.43 0.110303
$$965$$ −13774.2 −0.459490
$$966$$ 0 0
$$967$$ −18933.2 −0.629628 −0.314814 0.949153i $$-0.601942\pi$$
−0.314814 + 0.949153i $$0.601942\pi$$
$$968$$ 0 0
$$969$$ 0 0
$$970$$ 7228.73 0.239279
$$971$$ 40660.3 1.34382 0.671911 0.740632i $$-0.265474\pi$$
0.671911 + 0.740632i $$0.265474\pi$$
$$972$$ 0 0
$$973$$ 2307.23 0.0760188
$$974$$ 22759.2 0.748720
$$975$$ 0 0
$$976$$ −55832.3 −1.83110
$$977$$ 22502.8 0.736876 0.368438 0.929652i $$-0.379893\pi$$
0.368438 + 0.929652i $$0.379893\pi$$
$$978$$ 0 0
$$979$$ 0 0
$$980$$ −3770.98 −0.122918
$$981$$ 0 0
$$982$$ 24824.7 0.806709
$$983$$ 4435.20 0.143907 0.0719536 0.997408i $$-0.477077\pi$$
0.0719536 + 0.997408i $$0.477077\pi$$
$$984$$ 0 0
$$985$$ 3223.55 0.104275
$$986$$ 59989.8 1.93759
$$987$$ 0 0
$$988$$ 3986.80 0.128378
$$989$$ −31361.8 −1.00834
$$990$$ 0 0
$$991$$ 7362.76 0.236010 0.118005 0.993013i $$-0.462350\pi$$
0.118005 + 0.993013i $$0.462350\pi$$
$$992$$ −40025.5 −1.28106
$$993$$ 0 0
$$994$$ −12104.0 −0.386233
$$995$$ −1661.35 −0.0529331
$$996$$ 0 0
$$997$$ 53480.1 1.69883 0.849413 0.527728i $$-0.176956\pi$$
0.849413 + 0.527728i $$0.176956\pi$$
$$998$$ 35010.0 1.11044
$$999$$ 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 1089.4.a.t.1.2 2
3.2 odd 2 363.4.a.j.1.1 2
11.10 odd 2 99.4.a.e.1.1 2
33.32 even 2 33.4.a.d.1.2 2
44.43 even 2 1584.4.a.x.1.2 2
55.54 odd 2 2475.4.a.o.1.2 2
132.131 odd 2 528.4.a.o.1.1 2
165.32 odd 4 825.4.c.i.199.4 4
165.98 odd 4 825.4.c.i.199.1 4
165.164 even 2 825.4.a.k.1.1 2
231.230 odd 2 1617.4.a.j.1.2 2
264.131 odd 2 2112.4.a.bh.1.2 2
264.197 even 2 2112.4.a.ba.1.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
33.4.a.d.1.2 2 33.32 even 2
99.4.a.e.1.1 2 11.10 odd 2
363.4.a.j.1.1 2 3.2 odd 2
528.4.a.o.1.1 2 132.131 odd 2
825.4.a.k.1.1 2 165.164 even 2
825.4.c.i.199.1 4 165.98 odd 4
825.4.c.i.199.4 4 165.32 odd 4
1089.4.a.t.1.2 2 1.1 even 1 trivial
1584.4.a.x.1.2 2 44.43 even 2
1617.4.a.j.1.2 2 231.230 odd 2
2112.4.a.ba.1.2 2 264.197 even 2
2112.4.a.bh.1.2 2 264.131 odd 2
2475.4.a.o.1.2 2 55.54 odd 2