# Properties

 Label 99.4.a.e.1.1 Level $99$ Weight $4$ Character 99.1 Self dual yes Analytic conductor $5.841$ Analytic rank $1$ 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] = [99,4,Mod(1,99)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(99, 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("99.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$99 = 3^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 99.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: $$5.84118909057$$ Analytic rank: $$1$$ 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 1.1 Root $$3.37228$$ of defining polynomial Character $$\chi$$ $$=$$ 99.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-3.37228 q^{2} +3.37228 q^{4} +3.48913 q^{5} -4.74456 q^{7} +15.6060 q^{8} +O(q^{10})$$ $$q-3.37228 q^{2} +3.37228 q^{4} +3.48913 q^{5} -4.74456 q^{7} +15.6060 q^{8} -11.7663 q^{10} -11.0000 q^{11} -15.0217 q^{13} +16.0000 q^{14} -79.6060 q^{16} -73.1684 q^{17} -78.7011 q^{19} +11.7663 q^{20} +37.0951 q^{22} -112.000 q^{23} -112.826 q^{25} +50.6576 q^{26} -16.0000 q^{28} -243.125 q^{29} +278.717 q^{31} +143.606 q^{32} +246.745 q^{34} -16.5544 q^{35} +102.380 q^{37} +265.402 q^{38} +54.4512 q^{40} +241.255 q^{41} -280.016 q^{43} -37.0951 q^{44} +377.696 q^{46} +169.870 q^{47} -320.489 q^{49} +380.481 q^{50} -50.6576 q^{52} +409.652 q^{53} -38.3804 q^{55} -74.0435 q^{56} +819.886 q^{58} -196.000 q^{59} -701.359 q^{61} -939.913 q^{62} +152.568 q^{64} -52.4128 q^{65} +900.587 q^{67} -246.745 q^{68} +55.8260 q^{70} -756.500 q^{71} -1019.81 q^{73} -345.255 q^{74} -265.402 q^{76} +52.1902 q^{77} -327.549 q^{79} -277.755 q^{80} -813.581 q^{82} +756.619 q^{83} -255.294 q^{85} +944.293 q^{86} -171.666 q^{88} -508.978 q^{89} +71.2716 q^{91} -377.696 q^{92} -572.848 q^{94} -274.598 q^{95} +614.358 q^{97} +1080.78 q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - q^{2} + q^{4} - 16 q^{5} + 2 q^{7} - 9 q^{8}+O(q^{10})$$ 2 * q - q^2 + q^4 - 16 * q^5 + 2 * q^7 - 9 * q^8 $$2 q - q^{2} + q^{4} - 16 q^{5} + 2 q^{7} - 9 q^{8} - 58 q^{10} - 22 q^{11} - 76 q^{13} + 32 q^{14} - 119 q^{16} + 26 q^{17} - 54 q^{19} + 58 q^{20} + 11 q^{22} - 224 q^{23} + 142 q^{25} - 94 q^{26} - 32 q^{28} - 222 q^{29} - 40 q^{31} + 247 q^{32} + 482 q^{34} - 148 q^{35} - 48 q^{37} + 324 q^{38} + 534 q^{40} + 494 q^{41} - 66 q^{43} - 11 q^{44} + 112 q^{46} + 64 q^{47} - 618 q^{49} + 985 q^{50} + 94 q^{52} + 84 q^{53} + 176 q^{55} - 240 q^{56} + 870 q^{58} - 392 q^{59} - 1104 q^{61} - 1696 q^{62} + 713 q^{64} + 1136 q^{65} + 928 q^{67} - 482 q^{68} - 256 q^{70} - 456 q^{71} - 592 q^{73} - 702 q^{74} - 324 q^{76} - 22 q^{77} - 230 q^{79} + 490 q^{80} - 214 q^{82} - 348 q^{83} - 2188 q^{85} + 1452 q^{86} + 99 q^{88} - 972 q^{89} - 340 q^{91} - 112 q^{92} - 824 q^{94} - 756 q^{95} - 1184 q^{97} + 375 q^{98}+O(q^{100})$$ 2 * q - q^2 + q^4 - 16 * q^5 + 2 * q^7 - 9 * q^8 - 58 * q^10 - 22 * q^11 - 76 * q^13 + 32 * q^14 - 119 * q^16 + 26 * q^17 - 54 * q^19 + 58 * q^20 + 11 * q^22 - 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 - 11 * q^44 + 112 * q^46 + 64 * q^47 - 618 * q^49 + 985 * q^50 + 94 * q^52 + 84 * q^53 + 176 * q^55 - 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 - 22 * q^77 - 230 * q^79 + 490 * q^80 - 214 * q^82 - 348 * q^83 - 2188 * q^85 + 1452 * q^86 + 99 * q^88 - 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 $$-0.703300\pi$$
−0.596141 + 0.802880i $$0.703300\pi$$
$$3$$ 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 $$-0.540885\pi$$
−0.128091 + 0.991762i $$0.540885\pi$$
$$8$$ 15.6060 0.689693
$$9$$ 0 0
$$10$$ −11.7663 −0.372083
$$11$$ −11.0000 −0.301511
$$12$$ 0 0
$$13$$ −15.0217 −0.320483 −0.160242 0.987078i $$-0.551227\pi$$
−0.160242 + 0.987078i $$0.551227\pi$$
$$14$$ 16.0000 0.305441
$$15$$ 0 0
$$16$$ −79.6060 −1.24384
$$17$$ −73.1684 −1.04388 −0.521940 0.852982i $$-0.674791\pi$$
−0.521940 + 0.852982i $$0.674791\pi$$
$$18$$ 0 0
$$19$$ −78.7011 −0.950277 −0.475138 0.879911i $$-0.657602\pi$$
−0.475138 + 0.879911i $$0.657602\pi$$
$$20$$ 11.7663 0.131551
$$21$$ 0 0
$$22$$ 37.0951 0.359486
$$23$$ −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 $$-0.783968\pi$$
−0.778399 + 0.627769i $$0.783968\pi$$
$$30$$ 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 $$-0.348034\pi$$
0.459485 + 0.888186i $$0.348034\pi$$
$$42$$ 0 0
$$43$$ −280.016 −0.993071 −0.496536 0.868016i $$-0.665395\pi$$
−0.496536 + 0.868016i $$0.665395\pi$$
$$44$$ −37.0951 −0.127098
$$45$$ 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
$$55$$ −38.3804 −0.0940947
$$56$$ −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 $$-0.763318\pi$$
−0.736064 + 0.676912i $$0.763318\pi$$
$$62$$ −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 $$-0.804661\pi$$
−0.817536 + 0.575877i $$0.804661\pi$$
$$74$$ −345.255 −0.542367
$$75$$ 0 0
$$76$$ −265.402 −0.400575
$$77$$ 52.1902 0.0772419
$$78$$ 0 0
$$79$$ −327.549 −0.466483 −0.233241 0.972419i $$-0.574933\pi$$
−0.233241 + 0.972419i $$0.574933\pi$$
$$80$$ −277.755 −0.388175
$$81$$ 0 0
$$82$$ −813.581 −1.09567
$$83$$ 756.619 1.00060 0.500300 0.865852i $$-0.333223\pi$$
0.500300 + 0.865852i $$0.333223\pi$$
$$84$$ 0 0
$$85$$ −255.294 −0.325771
$$86$$ 944.293 1.18402
$$87$$ 0 0
$$88$$ −171.666 −0.207950
$$89$$ −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 $$-0.333174\pi$$
0.500434 + 0.865775i $$0.333174\pi$$
$$102$$ 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 $$-0.713808\pi$$
−0.622315 + 0.782767i $$0.713808\pi$$
$$108$$ 0 0
$$109$$ 320.217 0.281388 0.140694 0.990053i $$-0.455067\pi$$
0.140694 + 0.990053i $$0.455067\pi$$
$$110$$ 129.429 0.112187
$$111$$ 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
$$121$$ 121.000 0.0909091
$$122$$ 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 $$-0.204590\pi$$
0.800457 + 0.599390i $$0.204590\pi$$
$$128$$ −1663.35 −1.14860
$$129$$ 0 0
$$130$$ 176.751 0.119247
$$131$$ 1147.41 0.765267 0.382633 0.923900i $$-0.375017\pi$$
0.382633 + 0.923900i $$0.375017\pi$$
$$132$$ 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 $$-0.547402\pi$$
−0.148368 + 0.988932i $$0.547402\pi$$
$$140$$ −55.8260 −0.0337011
$$141$$ 0 0
$$142$$ 2551.13 1.50765
$$143$$ 165.239 0.0966294
$$144$$ 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 $$-0.724047\pi$$
−0.647169 + 0.762346i $$0.724047\pi$$
$$150$$ 0 0
$$151$$ −570.070 −0.307229 −0.153615 0.988131i $$-0.549091\pi$$
−0.153615 + 0.988131i $$0.549091\pi$$
$$152$$ −1228.21 −0.655399
$$153$$ 0 0
$$154$$ −176.000 −0.0920941
$$155$$ 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 $$-0.411233\pi$$
0.275268 + 0.961368i $$0.411233\pi$$
$$168$$ 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 $$-0.556755\pi$$
−0.177359 + 0.984146i $$0.556755\pi$$
$$174$$ 0 0
$$175$$ 535.310 0.231232
$$176$$ 875.666 0.375033
$$177$$ 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
$$187$$ 804.853 0.314742
$$188$$ 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 $$-0.236627\pi$$
0.736181 + 0.676784i $$0.236627\pi$$
$$194$$ −2071.79 −0.766731
$$195$$ 0 0
$$196$$ −1080.78 −0.393870
$$197$$ −923.886 −0.334133 −0.167066 0.985946i $$-0.553429\pi$$
−0.167066 + 0.985946i $$0.553429\pi$$
$$198$$ 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
$$209$$ 865.712 0.286519
$$210$$ 0 0
$$211$$ −4918.24 −1.60467 −0.802336 0.596872i $$-0.796410\pi$$
−0.802336 + 0.596872i $$0.796410\pi$$
$$212$$ 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
$$220$$ −129.429 −0.0396642
$$221$$ 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 $$-0.392956\pi$$
0.329985 + 0.943986i $$0.392956\pi$$
$$228$$ 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 $$-0.612704\pi$$
−0.346718 + 0.937970i $$0.612704\pi$$
$$234$$ 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 $$-0.561962\pi$$
−0.193432 + 0.981114i $$0.561962\pi$$
$$240$$ 0 0
$$241$$ −978.989 −0.261669 −0.130835 0.991404i $$-0.541766\pi$$
−0.130835 + 0.991404i $$0.541766\pi$$
$$242$$ −408.046 −0.108389
$$243$$ 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
$$253$$ 1232.00 0.306147
$$254$$ −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 $$-0.320942\pi$$
0.533326 + 0.845910i $$0.320942\pi$$
$$264$$ 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 $$-0.167794\pi$$
0.864250 + 0.503063i $$0.167794\pi$$
$$272$$ 5824.64 1.29842
$$273$$ 0 0
$$274$$ 4278.07 0.943239
$$275$$ 1241.09 0.272147
$$276$$ 0 0
$$277$$ 1127.52 0.244571 0.122286 0.992495i $$-0.460978\pi$$
0.122286 + 0.992495i $$0.460978\pi$$
$$278$$ 1639.90 0.353794
$$279$$ 0 0
$$280$$ −258.347 −0.0551400
$$281$$ 1872.47 0.397517 0.198758 0.980049i $$-0.436309\pi$$
0.198758 + 0.980049i $$0.436309\pi$$
$$282$$ 0 0
$$283$$ 2124.48 0.446245 0.223123 0.974790i $$-0.428375\pi$$
0.223123 + 0.974790i $$0.428375\pi$$
$$284$$ −2551.13 −0.533034
$$285$$ 0 0
$$286$$ −557.233 −0.115209
$$287$$ −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 $$-0.392479\pi$$
0.331401 + 0.943490i $$0.392479\pi$$
$$294$$ 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 $$-0.550479\pi$$
−0.157921 + 0.987452i $$0.550479\pi$$
$$308$$ 176.000 0.0325602
$$309$$ 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
$$319$$ 2674.37 0.469393
$$320$$ 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 $$-0.478978\pi$$
0.0659930 + 0.997820i $$0.478978\pi$$
$$338$$ 6647.94 1.06982
$$339$$ 0 0
$$340$$ −860.923 −0.137324
$$341$$ −3065.89 −0.486883
$$342$$ 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 $$-0.660914\pi$$
−0.484269 + 0.874919i $$0.660914\pi$$
$$348$$ 0 0
$$349$$ −12768.5 −1.95840 −0.979198 0.202906i $$-0.934961\pi$$
−0.979198 + 0.202906i $$0.934961\pi$$
$$350$$ −1805.22 −0.275694
$$351$$ 0 0
$$352$$ −1579.67 −0.239194
$$353$$ 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 $$-0.424324\pi$$
0.235510 + 0.971872i $$0.424324\pi$$
$$360$$ 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 $$-0.725917\pi$$
−0.651635 + 0.758533i $$0.725917\pi$$
$$374$$ −2714.19 −0.375261
$$375$$ 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
$$385$$ 182.098 0.0241054
$$386$$ −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
$$407$$ −1126.18 −0.137157
$$408$$ 0 0
$$409$$ 8759.53 1.05900 0.529500 0.848310i $$-0.322380\pi$$
0.529500 + 0.848310i $$0.322380\pi$$
$$410$$ −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
$$418$$ −2919.42 −0.341612
$$419$$ 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 $$-0.398390\pi$$
0.313823 + 0.949481i $$0.398390\pi$$
$$432$$ 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 $$-0.498236\pi$$
0.00554240 + 0.999985i $$0.498236\pi$$
$$440$$ −598.963 −0.0648965
$$441$$ 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
$$451$$ −2653.81 −0.277080
$$452$$ 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 $$-0.557836\pi$$
−0.180698 + 0.983539i $$0.557836\pi$$
$$458$$ 17910.4 1.82729
$$459$$ 0 0
$$460$$ −1317.83 −0.133574
$$461$$ −11566.3 −1.16854 −0.584271 0.811559i $$-0.698619\pi$$
−0.584271 + 0.811559i $$0.698619\pi$$
$$462$$ 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
$$473$$ 3080.18 0.299422
$$474$$ 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 $$-0.535628\pi$$
−0.111696 + 0.993742i $$0.535628\pi$$
$$480$$ 0 0
$$481$$ −1537.93 −0.145787
$$482$$ 3301.43 0.311983
$$483$$ 0 0
$$484$$ 408.046 0.0383214
$$485$$ 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 $$-0.609853\pi$$
−0.338305 + 0.941037i $$0.609853\pi$$
$$492$$ 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 $$-0.822627\pi$$
−0.848721 + 0.528840i $$0.822627\pi$$
$$504$$ 0 0
$$505$$ 3544.67 0.312348
$$506$$ −4154.65 −0.365013
$$507$$ 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
$$517$$ −1868.56 −0.158954
$$518$$ 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 $$-0.559456\pi$$
−0.185701 + 0.982606i $$0.559456\pi$$
$$524$$ 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
$$539$$ 3525.38 0.281723
$$540$$ 0 0
$$541$$ 2180.90 0.173316 0.0866580 0.996238i $$-0.472381\pi$$
0.0866580 + 0.996238i $$0.472381\pi$$
$$542$$ −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 $$-0.395826\pi$$
0.321460 + 0.946923i $$0.395826\pi$$
$$548$$ −4278.07 −0.333485
$$549$$ 0 0
$$550$$ −4185.29 −0.324475
$$551$$ 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 $$-0.0927986\pi$$
0.957804 + 0.287423i $$0.0927986\pi$$
$$558$$ 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 $$-0.446076\pi$$
0.168599 + 0.985685i $$0.446076\pi$$
$$564$$ 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 $$-0.335034\pi$$
0.495366 + 0.868684i $$0.335034\pi$$
$$570$$ 0 0
$$571$$ −2605.52 −0.190959 −0.0954795 0.995431i $$-0.530438\pi$$
−0.0954795 + 0.995431i $$0.530438\pi$$
$$572$$ 557.233 0.0407327
$$573$$ 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
$$583$$ −4506.17 −0.320114
$$584$$ −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 $$-0.665127\pi$$
−0.495804 + 0.868434i $$0.665127\pi$$
$$594$$ 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 $$-0.197628\pi$$
0.813375 + 0.581739i $$0.197628\pi$$
$$602$$ −4480.26 −0.303325
$$603$$ 0 0
$$604$$ −1922.44 −0.129508
$$605$$ 422.184 0.0283706
$$606$$ 0 0
$$607$$ −23526.6 −1.57317 −0.786585 0.617482i $$-0.788153\pi$$
−0.786585 + 0.617482i $$0.788153\pi$$
$$608$$ −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 $$-0.487118\pi$$
0.0404579 + 0.999181i $$0.487118\pi$$
$$614$$ 5729.31 0.376573
$$615$$ 0 0
$$616$$ 814.478 0.0532732
$$617$$ 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
$$638$$ −9018.74 −0.559648
$$639$$ 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
$$649$$ 2156.00 0.130401
$$650$$ −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 $$-0.574702\pi$$
−0.232536 + 0.972588i $$0.574702\pi$$
$$660$$ 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
$$671$$ 7714.94 0.443863
$$672$$ 0 0
$$673$$ 29397.6 1.68379 0.841897 0.539638i $$-0.181439\pi$$
0.841897 + 0.539638i $$0.181439\pi$$
$$674$$ −2753.57 −0.157364
$$675$$ 0 0
$$676$$ −6647.94 −0.378239
$$677$$ −5737.14 −0.325696 −0.162848 0.986651i $$-0.552068\pi$$
−0.162848 + 0.986651i $$0.552068\pi$$
$$678$$ 0 0
$$679$$ −2914.86 −0.164745
$$680$$ −3984.11 −0.224682
$$681$$ 0 0
$$682$$ 10339.0 0.580502
$$683$$ −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 $$-0.762249\pi$$
−0.733788 + 0.679379i $$0.762249\pi$$
$$702$$ 0 0
$$703$$ −8057.44 −0.432279
$$704$$ −1678.25 −0.0898457
$$705$$ 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
$$715$$ 576.540 0.0301558
$$716$$ 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 $$-0.424365\pi$$
0.235384 + 0.971903i $$0.424365\pi$$
$$734$$ 28426.3 1.42947
$$735$$ 0 0
$$736$$ −16083.9 −0.805515
$$737$$ −9906.45 −0.495127
$$738$$ 0 0
$$739$$ −28928.0 −1.43997 −0.719983 0.693992i $$-0.755850\pi$$
−0.719983 + 0.693992i $$0.755850\pi$$
$$740$$ 1204.64 0.0598425
$$741$$ 0 0
$$742$$ 6554.43 0.324287
$$743$$ −4857.04 −0.239822 −0.119911 0.992785i $$-0.538261\pi$$
−0.119911 + 0.992785i $$0.538261\pi$$
$$744$$ 0 0
$$745$$ −8213.80 −0.403933
$$746$$ 31660.8 1.55386
$$747$$ 0 0
$$748$$ 2714.19 0.132675
$$749$$ 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 $$-0.439395\pi$$
0.189248 + 0.981929i $$0.439395\pi$$
$$762$$ 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 $$-0.278817\pi$$
0.640284 + 0.768139i $$0.278817\pi$$
$$770$$ −614.086 −0.0287404
$$771$$ 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
$$781$$ 8321.50 0.381263
$$782$$ −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 $$-0.606595\pi$$
−0.328654 + 0.944450i $$0.606595\pi$$
$$788$$ −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
$$803$$ 11218.0 0.492993
$$804$$ 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 $$-0.479229\pi$$
0.0652063 + 0.997872i $$0.479229\pi$$
$$810$$ 0 0
$$811$$ 6239.39 0.270154 0.135077 0.990835i $$-0.456872\pi$$
0.135077 + 0.990835i $$0.456872\pi$$
$$812$$ 3890.00 0.168118
$$813$$ 0 0
$$814$$ 3797.81 0.163530
$$815$$ 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 $$-0.602733\pi$$
−0.317171 + 0.948368i $$0.602733\pi$$
$$822$$ 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 $$-0.692497\pi$$
−0.568555 + 0.822645i $$0.692497\pi$$
$$828$$ 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
$$836$$ 2919.42 0.120778
$$837$$ 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
$$847$$ −574.092 −0.0232893
$$848$$ −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 $$-0.895350\pi$$
−0.946441 + 0.322878i $$0.895350\pi$$
$$854$$ −11221.7 −0.449649
$$855$$ 0 0
$$856$$ −21498.4 −0.858412
$$857$$ −5021.31 −0.200145 −0.100073 0.994980i $$-0.531908\pi$$
−0.100073 + 0.994980i $$0.531908\pi$$
$$858$$ 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
$$869$$ 3603.04 0.140650
$$870$$ 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 $$-0.552048\pi$$
−0.162787 + 0.986661i $$0.552048\pi$$
$$878$$ −343.834 −0.0132162
$$879$$ 0 0
$$880$$ 3055.31 0.117039
$$881$$ 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 $$-0.607341\pi$$
−0.330866 + 0.943678i $$0.607341\pi$$
$$888$$ 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
$$902$$ 8949.39 0.330357
$$903$$ 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
$$913$$ −8322.81 −0.301692
$$914$$ 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 $$-0.580008\pi$$
−0.248714 + 0.968577i $$0.580008\pi$$
$$920$$ −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
$$935$$ 2808.23 0.0982236
$$936$$ 0 0
$$937$$ 3203.52 0.111691 0.0558454 0.998439i $$-0.482215\pi$$
0.0558454 + 0.998439i $$0.482215\pi$$
$$938$$ 14409.4 0.501581
$$939$$ 0 0
$$940$$ 1998.74 0.0693528
$$941$$ −19951.6 −0.691182 −0.345591 0.938385i $$-0.612322\pi$$
−0.345591 + 0.938385i $$0.612322\pi$$
$$942$$ 0 0
$$943$$ −27020.6 −0.933099
$$944$$ 15602.8 0.537952
$$945$$ 0 0
$$946$$ −10387.2 −0.356996
$$947$$ 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 $$-0.199455\pi$$
0.810022 + 0.586399i $$0.199455\pi$$
$$954$$ 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 $$-0.398058\pi$$
0.314814 + 0.949153i $$0.398058\pi$$
$$968$$ 1888.32 0.0626994
$$969$$ 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
$$979$$ 5598.76 0.182775
$$980$$ −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 $$-0.823044\pi$$
−0.849413 + 0.527728i $$0.823044\pi$$
$$998$$ −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 99.4.a.e.1.1 2
3.2 odd 2 33.4.a.d.1.2 2
4.3 odd 2 1584.4.a.x.1.2 2
5.4 even 2 2475.4.a.o.1.2 2
11.10 odd 2 1089.4.a.t.1.2 2
12.11 even 2 528.4.a.o.1.1 2
15.2 even 4 825.4.c.i.199.4 4
15.8 even 4 825.4.c.i.199.1 4
15.14 odd 2 825.4.a.k.1.1 2
21.20 even 2 1617.4.a.j.1.2 2
24.5 odd 2 2112.4.a.ba.1.2 2
24.11 even 2 2112.4.a.bh.1.2 2
33.32 even 2 363.4.a.j.1.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
33.4.a.d.1.2 2 3.2 odd 2
99.4.a.e.1.1 2 1.1 even 1 trivial
363.4.a.j.1.1 2 33.32 even 2
528.4.a.o.1.1 2 12.11 even 2
825.4.a.k.1.1 2 15.14 odd 2
825.4.c.i.199.1 4 15.8 even 4
825.4.c.i.199.4 4 15.2 even 4
1089.4.a.t.1.2 2 11.10 odd 2
1584.4.a.x.1.2 2 4.3 odd 2
1617.4.a.j.1.2 2 21.20 even 2
2112.4.a.ba.1.2 2 24.5 odd 2
2112.4.a.bh.1.2 2 24.11 even 2
2475.4.a.o.1.2 2 5.4 even 2