# Properties

 Label 363.4.a.j.1.1 Level $363$ Weight $4$ Character 363.1 Self dual yes Analytic conductor $21.418$ 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] = [363,4,Mod(1,363)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(363, 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("363.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$363 = 3 \cdot 11^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 363.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: $$21.4176933321$$ 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 1.1 Root $$3.37228$$ of defining polynomial Character $$\chi$$ $$=$$ 363.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-3.37228 q^{2} +3.00000 q^{3} +3.37228 q^{4} -3.48913 q^{5} -10.1168 q^{6} +4.74456 q^{7} +15.6060 q^{8} +9.00000 q^{9} +O(q^{10})$$ $$q-3.37228 q^{2} +3.00000 q^{3} +3.37228 q^{4} -3.48913 q^{5} -10.1168 q^{6} +4.74456 q^{7} +15.6060 q^{8} +9.00000 q^{9} +11.7663 q^{10} +10.1168 q^{12} +15.0217 q^{13} -16.0000 q^{14} -10.4674 q^{15} -79.6060 q^{16} -73.1684 q^{17} -30.3505 q^{18} +78.7011 q^{19} -11.7663 q^{20} +14.2337 q^{21} +112.000 q^{23} +46.8179 q^{24} -112.826 q^{25} -50.6576 q^{26} +27.0000 q^{27} +16.0000 q^{28} -243.125 q^{29} +35.2989 q^{30} +278.717 q^{31} +143.606 q^{32} +246.745 q^{34} -16.5544 q^{35} +30.3505 q^{36} +102.380 q^{37} -265.402 q^{38} +45.0652 q^{39} -54.4512 q^{40} +241.255 q^{41} -48.0000 q^{42} +280.016 q^{43} -31.4021 q^{45} -377.696 q^{46} -169.870 q^{47} -238.818 q^{48} -320.489 q^{49} +380.481 q^{50} -219.505 q^{51} +50.6576 q^{52} -409.652 q^{53} -91.0516 q^{54} +74.0435 q^{56} +236.103 q^{57} +819.886 q^{58} +196.000 q^{59} -35.2989 q^{60} +701.359 q^{61} -939.913 q^{62} +42.7011 q^{63} +152.568 q^{64} -52.4128 q^{65} +900.587 q^{67} -246.745 q^{68} +336.000 q^{69} +55.8260 q^{70} +756.500 q^{71} +140.454 q^{72} +1019.81 q^{73} -345.255 q^{74} -338.478 q^{75} +265.402 q^{76} -151.973 q^{78} +327.549 q^{79} +277.755 q^{80} +81.0000 q^{81} -813.581 q^{82} +756.619 q^{83} +48.0000 q^{84} +255.294 q^{85} -944.293 q^{86} -729.375 q^{87} +508.978 q^{89} +105.897 q^{90} +71.2716 q^{91} +377.696 q^{92} +836.152 q^{93} +572.848 q^{94} -274.598 q^{95} +430.818 q^{96} +614.358 q^{97} +1080.78 q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - q^{2} + 6 q^{3} + q^{4} + 16 q^{5} - 3 q^{6} - 2 q^{7} - 9 q^{8} + 18 q^{9}+O(q^{10})$$ 2 * q - q^2 + 6 * q^3 + q^4 + 16 * q^5 - 3 * q^6 - 2 * q^7 - 9 * q^8 + 18 * q^9 $$2 q - q^{2} + 6 q^{3} + q^{4} + 16 q^{5} - 3 q^{6} - 2 q^{7} - 9 q^{8} + 18 q^{9} + 58 q^{10} + 3 q^{12} + 76 q^{13} - 32 q^{14} + 48 q^{15} - 119 q^{16} + 26 q^{17} - 9 q^{18} + 54 q^{19} - 58 q^{20} - 6 q^{21} + 224 q^{23} - 27 q^{24} + 142 q^{25} + 94 q^{26} + 54 q^{27} + 32 q^{28} - 222 q^{29} + 174 q^{30} - 40 q^{31} + 247 q^{32} + 482 q^{34} - 148 q^{35} + 9 q^{36} - 48 q^{37} - 324 q^{38} + 228 q^{39} - 534 q^{40} + 494 q^{41} - 96 q^{42} + 66 q^{43} + 144 q^{45} - 112 q^{46} - 64 q^{47} - 357 q^{48} - 618 q^{49} + 985 q^{50} + 78 q^{51} - 94 q^{52} - 84 q^{53} - 27 q^{54} + 240 q^{56} + 162 q^{57} + 870 q^{58} + 392 q^{59} - 174 q^{60} + 1104 q^{61} - 1696 q^{62} - 18 q^{63} + 713 q^{64} + 1136 q^{65} + 928 q^{67} - 482 q^{68} + 672 q^{69} - 256 q^{70} + 456 q^{71} - 81 q^{72} + 592 q^{73} - 702 q^{74} + 426 q^{75} + 324 q^{76} + 282 q^{78} + 230 q^{79} - 490 q^{80} + 162 q^{81} - 214 q^{82} - 348 q^{83} + 96 q^{84} + 2188 q^{85} - 1452 q^{86} - 666 q^{87} + 972 q^{89} + 522 q^{90} - 340 q^{91} + 112 q^{92} - 120 q^{93} + 824 q^{94} - 756 q^{95} + 741 q^{96} - 1184 q^{97} + 375 q^{98}+O(q^{100})$$ 2 * q - q^2 + 6 * q^3 + q^4 + 16 * q^5 - 3 * q^6 - 2 * q^7 - 9 * q^8 + 18 * q^9 + 58 * q^10 + 3 * q^12 + 76 * q^13 - 32 * q^14 + 48 * q^15 - 119 * q^16 + 26 * q^17 - 9 * q^18 + 54 * q^19 - 58 * q^20 - 6 * q^21 + 224 * q^23 - 27 * q^24 + 142 * q^25 + 94 * q^26 + 54 * q^27 + 32 * q^28 - 222 * q^29 + 174 * q^30 - 40 * q^31 + 247 * q^32 + 482 * q^34 - 148 * q^35 + 9 * q^36 - 48 * q^37 - 324 * q^38 + 228 * q^39 - 534 * q^40 + 494 * q^41 - 96 * q^42 + 66 * q^43 + 144 * q^45 - 112 * q^46 - 64 * q^47 - 357 * q^48 - 618 * q^49 + 985 * q^50 + 78 * q^51 - 94 * q^52 - 84 * q^53 - 27 * q^54 + 240 * q^56 + 162 * q^57 + 870 * q^58 + 392 * q^59 - 174 * q^60 + 1104 * q^61 - 1696 * q^62 - 18 * q^63 + 713 * q^64 + 1136 * q^65 + 928 * q^67 - 482 * q^68 + 672 * q^69 - 256 * q^70 + 456 * q^71 - 81 * q^72 + 592 * q^73 - 702 * q^74 + 426 * q^75 + 324 * q^76 + 282 * q^78 + 230 * q^79 - 490 * q^80 + 162 * q^81 - 214 * q^82 - 348 * q^83 + 96 * q^84 + 2188 * q^85 - 1452 * q^86 - 666 * q^87 + 972 * q^89 + 522 * q^90 - 340 * q^91 + 112 * q^92 - 120 * q^93 + 824 * q^94 - 756 * q^95 + 741 * q^96 - 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 $$0.703300\pi$$
$$3$$ 3.00000 0.577350
$$4$$ 3.37228 0.421535
$$5$$ −3.48913 −0.312077 −0.156038 0.987751i $$-0.549872\pi$$
−0.156038 + 0.987751i $$0.549872\pi$$
$$6$$ −10.1168 −0.688364
$$7$$ 4.74456 0.256182 0.128091 0.991762i $$-0.459115\pi$$
0.128091 + 0.991762i $$0.459115\pi$$
$$8$$ 15.6060 0.689693
$$9$$ 9.00000 0.333333
$$10$$ 11.7663 0.372083
$$11$$ 0 0
$$12$$ 10.1168 0.243373
$$13$$ 15.0217 0.320483 0.160242 0.987078i $$-0.448773\pi$$
0.160242 + 0.987078i $$0.448773\pi$$
$$14$$ −16.0000 −0.305441
$$15$$ −10.4674 −0.180178
$$16$$ −79.6060 −1.24384
$$17$$ −73.1684 −1.04388 −0.521940 0.852982i $$-0.674791\pi$$
−0.521940 + 0.852982i $$0.674791\pi$$
$$18$$ −30.3505 −0.397427
$$19$$ 78.7011 0.950277 0.475138 0.879911i $$-0.342398\pi$$
0.475138 + 0.879911i $$0.342398\pi$$
$$20$$ −11.7663 −0.131551
$$21$$ 14.2337 0.147907
$$22$$ 0 0
$$23$$ 112.000 1.01537 0.507687 0.861541i $$-0.330501\pi$$
0.507687 + 0.861541i $$0.330501\pi$$
$$24$$ 46.8179 0.398194
$$25$$ −112.826 −0.902608
$$26$$ −50.6576 −0.382106
$$27$$ 27.0000 0.192450
$$28$$ 16.0000 0.107990
$$29$$ −243.125 −1.55680 −0.778399 0.627769i $$-0.783968\pi$$
−0.778399 + 0.627769i $$0.783968\pi$$
$$30$$ 35.2989 0.214822
$$31$$ 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
$$36$$ 30.3505 0.140512
$$37$$ 102.380 0.454898 0.227449 0.973790i $$-0.426961\pi$$
0.227449 + 0.973790i $$0.426961\pi$$
$$38$$ −265.402 −1.13300
$$39$$ 45.0652 0.185031
$$40$$ −54.4512 −0.215237
$$41$$ 241.255 0.918970 0.459485 0.888186i $$-0.348034\pi$$
0.459485 + 0.888186i $$0.348034\pi$$
$$42$$ −48.0000 −0.176347
$$43$$ 280.016 0.993071 0.496536 0.868016i $$-0.334605\pi$$
0.496536 + 0.868016i $$0.334605\pi$$
$$44$$ 0 0
$$45$$ −31.4021 −0.104026
$$46$$ −377.696 −1.21061
$$47$$ −169.870 −0.527192 −0.263596 0.964633i $$-0.584909\pi$$
−0.263596 + 0.964633i $$0.584909\pi$$
$$48$$ −238.818 −0.718133
$$49$$ −320.489 −0.934371
$$50$$ 380.481 1.07616
$$51$$ −219.505 −0.602684
$$52$$ 50.6576 0.135095
$$53$$ −409.652 −1.06170 −0.530849 0.847466i $$-0.678127\pi$$
−0.530849 + 0.847466i $$0.678127\pi$$
$$54$$ −91.0516 −0.229455
$$55$$ 0 0
$$56$$ 74.0435 0.176687
$$57$$ 236.103 0.548643
$$58$$ 819.886 1.85614
$$59$$ 196.000 0.432492 0.216246 0.976339i $$-0.430619\pi$$
0.216246 + 0.976339i $$0.430619\pi$$
$$60$$ −35.2989 −0.0759512
$$61$$ 701.359 1.47213 0.736064 0.676912i $$-0.236682\pi$$
0.736064 + 0.676912i $$0.236682\pi$$
$$62$$ −939.913 −1.92531
$$63$$ 42.7011 0.0853941
$$64$$ 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
$$69$$ 336.000 0.586227
$$70$$ 55.8260 0.0953212
$$71$$ 756.500 1.26451 0.632254 0.774762i $$-0.282130\pi$$
0.632254 + 0.774762i $$0.282130\pi$$
$$72$$ 140.454 0.229898
$$73$$ 1019.81 1.63507 0.817536 0.575877i $$-0.195339\pi$$
0.817536 + 0.575877i $$0.195339\pi$$
$$74$$ −345.255 −0.542367
$$75$$ −338.478 −0.521121
$$76$$ 265.402 0.400575
$$77$$ 0 0
$$78$$ −151.973 −0.220609
$$79$$ 327.549 0.466483 0.233241 0.972419i $$-0.425067\pi$$
0.233241 + 0.972419i $$0.425067\pi$$
$$80$$ 277.755 0.388175
$$81$$ 81.0000 0.111111
$$82$$ −813.581 −1.09567
$$83$$ 756.619 1.00060 0.500300 0.865852i $$-0.333223\pi$$
0.500300 + 0.865852i $$0.333223\pi$$
$$84$$ 48.0000 0.0623480
$$85$$ 255.294 0.325771
$$86$$ −944.293 −1.18402
$$87$$ −729.375 −0.898818
$$88$$ 0 0
$$89$$ 508.978 0.606198 0.303099 0.952959i $$-0.401979\pi$$
0.303099 + 0.952959i $$0.401979\pi$$
$$90$$ 105.897 0.124028
$$91$$ 71.2716 0.0821022
$$92$$ 377.696 0.428016
$$93$$ 836.152 0.932311
$$94$$ 572.848 0.628561
$$95$$ −274.598 −0.296559
$$96$$ 430.818 0.458023
$$97$$ 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 $$0.333174\pi$$
$$102$$ 740.234 0.718569
$$103$$ 1102.16 1.05436 0.527181 0.849753i $$-0.323249\pi$$
0.527181 + 0.849753i $$0.323249\pi$$
$$104$$ 234.429 0.221035
$$105$$ −49.6631 −0.0461583
$$106$$ 1381.46 1.26584
$$107$$ −1377.58 −1.24463 −0.622315 0.782767i $$-0.713808\pi$$
−0.622315 + 0.782767i $$0.713808\pi$$
$$108$$ 91.0516 0.0811245
$$109$$ −320.217 −0.281388 −0.140694 0.990053i $$-0.544933\pi$$
−0.140694 + 0.990053i $$0.544933\pi$$
$$110$$ 0 0
$$111$$ 307.141 0.262636
$$112$$ −377.696 −0.318651
$$113$$ −1629.45 −1.35651 −0.678254 0.734828i $$-0.737263\pi$$
−0.678254 + 0.734828i $$0.737263\pi$$
$$114$$ −796.206 −0.654136
$$115$$ −390.782 −0.316875
$$116$$ −819.886 −0.656245
$$117$$ 135.196 0.106828
$$118$$ −660.967 −0.515652
$$119$$ −347.152 −0.267423
$$120$$ −163.354 −0.124267
$$121$$ 0 0
$$122$$ −2365.18 −1.75519
$$123$$ 723.766 0.530568
$$124$$ 939.913 0.680699
$$125$$ 829.805 0.593760
$$126$$ −144.000 −0.101814
$$127$$ −2291.26 −1.60091 −0.800457 0.599390i $$-0.795410\pi$$
−0.800457 + 0.599390i $$0.795410\pi$$
$$128$$ −1663.35 −1.14860
$$129$$ 840.049 0.573350
$$130$$ 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
$$135$$ −94.2064 −0.0600592
$$136$$ −1141.86 −0.719956
$$137$$ 1268.60 0.791121 0.395561 0.918440i $$-0.370550\pi$$
0.395561 + 0.918440i $$0.370550\pi$$
$$138$$ −1133.09 −0.698947
$$139$$ 486.288 0.296737 0.148368 0.988932i $$-0.452598\pi$$
0.148368 + 0.988932i $$0.452598\pi$$
$$140$$ −55.8260 −0.0337011
$$141$$ −509.609 −0.304374
$$142$$ −2551.13 −1.50765
$$143$$ 0 0
$$144$$ −716.454 −0.414614
$$145$$ 848.293 0.485841
$$146$$ −3439.10 −1.94947
$$147$$ −961.467 −0.539459
$$148$$ 345.255 0.191756
$$149$$ −2354.11 −1.29434 −0.647169 0.762346i $$-0.724047\pi$$
−0.647169 + 0.762346i $$0.724047\pi$$
$$150$$ 1141.44 0.621323
$$151$$ 570.070 0.307229 0.153615 0.988131i $$-0.450909\pi$$
0.153615 + 0.988131i $$0.450909\pi$$
$$152$$ 1228.21 0.655399
$$153$$ −658.516 −0.347960
$$154$$ 0 0
$$155$$ −972.479 −0.503945
$$156$$ 151.973 0.0779971
$$157$$ −2072.67 −1.05361 −0.526807 0.849985i $$-0.676611\pi$$
−0.526807 + 0.849985i $$0.676611\pi$$
$$158$$ −1104.59 −0.556179
$$159$$ −1228.96 −0.612972
$$160$$ −501.059 −0.247576
$$161$$ 531.391 0.260121
$$162$$ −273.155 −0.132476
$$163$$ 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 $$0.411233\pi$$
$$168$$ 222.130 0.102010
$$169$$ −1971.35 −0.897290
$$170$$ −860.923 −0.388410
$$171$$ 708.310 0.316759
$$172$$ 944.293 0.418615
$$173$$ −807.147 −0.354718 −0.177359 0.984146i $$-0.556755\pi$$
−0.177359 + 0.984146i $$0.556755\pi$$
$$174$$ 2459.66 1.07164
$$175$$ −535.310 −0.231232
$$176$$ 0 0
$$177$$ 588.000 0.249699
$$178$$ −1716.42 −0.722758
$$179$$ −1950.39 −0.814408 −0.407204 0.913337i $$-0.633496\pi$$
−0.407204 + 0.913337i $$0.633496\pi$$
$$180$$ −105.897 −0.0438505
$$181$$ 1061.61 0.435959 0.217980 0.975953i $$-0.430053\pi$$
0.217980 + 0.975953i $$0.430053\pi$$
$$182$$ −240.348 −0.0978889
$$183$$ 2104.08 0.849933
$$184$$ 1747.87 0.700297
$$185$$ −357.218 −0.141963
$$186$$ −2819.74 −1.11158
$$187$$ 0 0
$$188$$ −572.848 −0.222230
$$189$$ 128.103 0.0493023
$$190$$ 926.021 0.353582
$$191$$ 2136.41 0.809348 0.404674 0.914461i $$-0.367385\pi$$
0.404674 + 0.914461i $$0.367385\pi$$
$$192$$ 457.704 0.172041
$$193$$ −3947.76 −1.47236 −0.736181 0.676784i $$-0.763373\pi$$
−0.736181 + 0.676784i $$0.763373\pi$$
$$194$$ −2071.79 −0.766731
$$195$$ −157.238 −0.0577439
$$196$$ −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
$$201$$ 2701.76 0.948097
$$202$$ −3425.96 −1.19332
$$203$$ −1153.52 −0.398824
$$204$$ −740.234 −0.254053
$$205$$ −841.770 −0.286789
$$206$$ −3716.80 −1.25710
$$207$$ 1008.00 0.338458
$$208$$ −1195.82 −0.398631
$$209$$ 0 0
$$210$$ 167.478 0.0550337
$$211$$ 4918.24 1.60467 0.802336 0.596872i $$-0.203590\pi$$
0.802336 + 0.596872i $$0.203590\pi$$
$$212$$ −1381.46 −0.447543
$$213$$ 2269.50 0.730064
$$214$$ 4645.57 1.48395
$$215$$ −977.012 −0.309915
$$216$$ 421.361 0.132731
$$217$$ 1322.39 0.413686
$$218$$ 1079.86 0.335494
$$219$$ 3059.44 0.944010
$$220$$ 0 0
$$221$$ −1099.12 −0.334546
$$222$$ −1035.77 −0.313136
$$223$$ 2100.29 0.630700 0.315350 0.948975i $$-0.397878\pi$$
0.315350 + 0.948975i $$0.397878\pi$$
$$224$$ 681.348 0.203234
$$225$$ −1015.43 −0.300869
$$226$$ 5494.95 1.61734
$$227$$ 2257.16 0.659970 0.329985 0.943986i $$-0.392956\pi$$
0.329985 + 0.943986i $$0.392956\pi$$
$$228$$ 796.206 0.231272
$$229$$ −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 $$0.612704\pi$$
$$234$$ −455.918 −0.127369
$$235$$ 592.696 0.164524
$$236$$ 660.967 0.182311
$$237$$ 982.646 0.269324
$$238$$ 1170.70 0.318844
$$239$$ −1429.40 −0.386863 −0.193432 0.981114i $$-0.561962\pi$$
−0.193432 + 0.981114i $$0.561962\pi$$
$$240$$ 833.266 0.224113
$$241$$ 978.989 0.261669 0.130835 0.991404i $$-0.458234\pi$$
0.130835 + 0.991404i $$0.458234\pi$$
$$242$$ 0 0
$$243$$ 243.000 0.0641500
$$244$$ 2365.18 0.620553
$$245$$ 1118.23 0.291595
$$246$$ −2440.74 −0.632586
$$247$$ 1182.23 0.304548
$$248$$ 4349.65 1.11372
$$249$$ 2269.86 0.577696
$$250$$ −2798.33 −0.707929
$$251$$ −6530.63 −1.64227 −0.821135 0.570734i $$-0.806659\pi$$
−0.821135 + 0.570734i $$0.806659\pi$$
$$252$$ 144.000 0.0359966
$$253$$ 0 0
$$254$$ 7726.76 1.90874
$$255$$ 765.882 0.188084
$$256$$ 4388.74 1.07147
$$257$$ 8130.26 1.97335 0.986676 0.162696i $$-0.0520188\pi$$
0.986676 + 0.162696i $$0.0520188\pi$$
$$258$$ −2832.88 −0.683595
$$259$$ 485.750 0.116537
$$260$$ −176.751 −0.0421600
$$261$$ −2188.12 −0.518933
$$262$$ −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
$$267$$ 1526.93 0.349988
$$268$$ 3037.03 0.692225
$$269$$ −29.1522 −0.00660760 −0.00330380 0.999995i $$-0.501052\pi$$
−0.00330380 + 0.999995i $$0.501052\pi$$
$$270$$ 317.690 0.0716075
$$271$$ −7711.22 −1.72850 −0.864250 0.503063i $$-0.832206\pi$$
−0.864250 + 0.503063i $$0.832206\pi$$
$$272$$ 5824.64 1.29842
$$273$$ 213.815 0.0474017
$$274$$ −4278.07 −0.943239
$$275$$ 0 0
$$276$$ 1133.09 0.247115
$$277$$ −1127.52 −0.244571 −0.122286 0.992495i $$-0.539022\pi$$
−0.122286 + 0.992495i $$0.539022\pi$$
$$278$$ −1639.90 −0.353794
$$279$$ 2508.46 0.538270
$$280$$ −258.347 −0.0551400
$$281$$ 1872.47 0.397517 0.198758 0.980049i $$-0.436309\pi$$
0.198758 + 0.980049i $$0.436309\pi$$
$$282$$ 1718.54 0.362900
$$283$$ −2124.48 −0.446245 −0.223123 0.974790i $$-0.571625\pi$$
−0.223123 + 0.974790i $$0.571625\pi$$
$$284$$ 2551.13 0.533034
$$285$$ −823.794 −0.171219
$$286$$ 0 0
$$287$$ 1144.65 0.235424
$$288$$ 1292.45 0.264439
$$289$$ 440.621 0.0896846
$$290$$ −2860.68 −0.579259
$$291$$ 1843.07 0.371282
$$292$$ 3439.10 0.689241
$$293$$ 3324.19 0.662802 0.331401 0.943490i $$-0.392479\pi$$
0.331401 + 0.943490i $$0.392479\pi$$
$$294$$ 3242.34 0.643187
$$295$$ −683.869 −0.134971
$$296$$ 1597.75 0.313740
$$297$$ 0 0
$$298$$ 7938.73 1.54322
$$299$$ 1682.44 0.325411
$$300$$ −1141.44 −0.219671
$$301$$ 1328.55 0.254407
$$302$$ −1922.44 −0.366304
$$303$$ 3047.75 0.577851
$$304$$ −6265.07 −1.18200
$$305$$ −2447.13 −0.459417
$$306$$ 2220.70 0.414866
$$307$$ 1698.94 0.315843 0.157921 0.987452i $$-0.449521\pi$$
0.157921 + 0.987452i $$0.449521\pi$$
$$308$$ 0 0
$$309$$ 3306.49 0.608736
$$310$$ 3279.47 0.600844
$$311$$ 6928.83 1.26334 0.631668 0.775239i $$-0.282370\pi$$
0.631668 + 0.775239i $$0.282370\pi$$
$$312$$ 703.287 0.127615
$$313$$ −3560.75 −0.643020 −0.321510 0.946906i $$-0.604190\pi$$
−0.321510 + 0.946906i $$0.604190\pi$$
$$314$$ 6989.64 1.25620
$$315$$ −148.989 −0.0266495
$$316$$ 1104.59 0.196639
$$317$$ 332.750 0.0589561 0.0294780 0.999565i $$-0.490615\pi$$
0.0294780 + 0.999565i $$0.490615\pi$$
$$318$$ 4144.39 0.730835
$$319$$ 0 0
$$320$$ −532.329 −0.0929940
$$321$$ −4132.73 −0.718587
$$322$$ −1792.00 −0.310137
$$323$$ −5758.43 −0.991975
$$324$$ 273.155 0.0468372
$$325$$ −1694.84 −0.289271
$$326$$ −9025.94 −1.53344
$$327$$ −960.652 −0.162459
$$328$$ 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
$$333$$ 921.423 0.151633
$$334$$ −4006.67 −0.656393
$$335$$ −3142.26 −0.512478
$$336$$ −1133.09 −0.183973
$$337$$ −816.531 −0.131986 −0.0659930 0.997820i $$-0.521022\pi$$
−0.0659930 + 0.997820i $$0.521022\pi$$
$$338$$ 6647.94 1.06982
$$339$$ −4888.34 −0.783180
$$340$$ 860.923 0.137324
$$341$$ 0 0
$$342$$ −2388.62 −0.377666
$$343$$ −3147.97 −0.495552
$$344$$ 4369.92 0.684914
$$345$$ −1172.35 −0.182948
$$346$$ 2721.93 0.422924
$$347$$ −6260.53 −0.968539 −0.484269 0.874919i $$-0.660914\pi$$
−0.484269 + 0.874919i $$0.660914\pi$$
$$348$$ −2459.66 −0.378884
$$349$$ 12768.5 1.95840 0.979198 0.202906i $$-0.0650386\pi$$
0.979198 + 0.202906i $$0.0650386\pi$$
$$350$$ 1805.22 0.275694
$$351$$ 405.587 0.0616771
$$352$$ 0 0
$$353$$ −2649.28 −0.399453 −0.199727 0.979852i $$-0.564005\pi$$
−0.199727 + 0.979852i $$0.564005\pi$$
$$354$$ −1982.90 −0.297712
$$355$$ −2639.52 −0.394623
$$356$$ 1716.42 0.255534
$$357$$ −1041.46 −0.154397
$$358$$ 6577.27 0.971004
$$359$$ 3203.91 0.471020 0.235510 0.971872i $$-0.424324\pi$$
0.235510 + 0.971872i $$0.424324\pi$$
$$360$$ −490.061 −0.0717457
$$361$$ −665.143 −0.0969737
$$362$$ −3580.04 −0.519786
$$363$$ 0 0
$$364$$ 240.348 0.0346089
$$365$$ −3558.26 −0.510268
$$366$$ −7095.54 −1.01336
$$367$$ −8429.40 −1.19894 −0.599470 0.800397i $$-0.704622\pi$$
−0.599470 + 0.800397i $$0.704622\pi$$
$$368$$ −8915.87 −1.26297
$$369$$ 2171.30 0.306323
$$370$$ 1204.64 0.169260
$$371$$ −1943.62 −0.271988
$$372$$ 2819.74 0.393002
$$373$$ 9388.53 1.30327 0.651635 0.758533i $$-0.274083\pi$$
0.651635 + 0.758533i $$0.274083\pi$$
$$374$$ 0 0
$$375$$ 2489.41 0.342807
$$376$$ −2650.98 −0.363600
$$377$$ −3652.16 −0.498928
$$378$$ −432.000 −0.0587822
$$379$$ −14264.5 −1.93329 −0.966647 0.256112i $$-0.917558\pi$$
−0.966647 + 0.256112i $$0.917558\pi$$
$$380$$ −926.021 −0.125010
$$381$$ −6873.77 −0.924288
$$382$$ −7204.58 −0.964970
$$383$$ 13462.2 1.79605 0.898026 0.439942i $$-0.145001\pi$$
0.898026 + 0.439942i $$0.145001\pi$$
$$384$$ −4990.05 −0.663144
$$385$$ 0 0
$$386$$ 13313.0 1.75547
$$387$$ 2520.15 0.331024
$$388$$ 2071.79 0.271080
$$389$$ −941.881 −0.122764 −0.0613821 0.998114i $$-0.519551\pi$$
−0.0613821 + 0.998114i $$0.519551\pi$$
$$390$$ 530.252 0.0688470
$$391$$ −8194.87 −1.05993
$$392$$ −5001.54 −0.644429
$$393$$ 3442.24 0.441827
$$394$$ 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
$$399$$ 1120.21 0.140553
$$400$$ 8981.62 1.12270
$$401$$ 12203.6 1.51975 0.759875 0.650069i $$-0.225260\pi$$
0.759875 + 0.650069i $$0.225260\pi$$
$$402$$ −9111.10 −1.13040
$$403$$ 4186.82 0.517520
$$404$$ 3425.96 0.421901
$$405$$ −282.619 −0.0346752
$$406$$ 3890.00 0.475511
$$407$$ 0 0
$$408$$ −3425.59 −0.415667
$$409$$ −8759.53 −1.05900 −0.529500 0.848310i $$-0.677620\pi$$
−0.529500 + 0.848310i $$0.677620\pi$$
$$410$$ 2838.69 0.341934
$$411$$ 3805.79 0.456754
$$412$$ 3716.80 0.444451
$$413$$ 929.934 0.110797
$$414$$ −3399.26 −0.403537
$$415$$ −2639.94 −0.312264
$$416$$ 2157.21 0.254245
$$417$$ 1458.86 0.171321
$$418$$ 0 0
$$419$$ −11188.4 −1.30451 −0.652256 0.757999i $$-0.726177\pi$$
−0.652256 + 0.757999i $$0.726177\pi$$
$$420$$ −167.478 −0.0194574
$$421$$ −14082.3 −1.63023 −0.815116 0.579298i $$-0.803327\pi$$
−0.815116 + 0.579298i $$0.803327\pi$$
$$422$$ −16585.7 −1.91322
$$423$$ −1528.83 −0.175731
$$424$$ −6393.02 −0.732246
$$425$$ 8255.30 0.942214
$$426$$ −7653.39 −0.870441
$$427$$ 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 $$0.398390\pi$$
$$432$$ −2149.36 −0.239378
$$433$$ 7195.75 0.798627 0.399314 0.916814i $$-0.369248\pi$$
0.399314 + 0.916814i $$0.369248\pi$$
$$434$$ −4459.48 −0.493230
$$435$$ 2544.88 0.280500
$$436$$ −1079.86 −0.118615
$$437$$ 8814.52 0.964887
$$438$$ −10317.3 −1.12553
$$439$$ −101.959 −0.0110848 −0.00554240 0.999985i $$-0.501764\pi$$
−0.00554240 + 0.999985i $$0.501764\pi$$
$$440$$ 0 0
$$441$$ −2884.40 −0.311457
$$442$$ 3706.53 0.398873
$$443$$ 4953.74 0.531285 0.265642 0.964072i $$-0.414416\pi$$
0.265642 + 0.964072i $$0.414416\pi$$
$$444$$ 1035.77 0.110710
$$445$$ −1775.89 −0.189180
$$446$$ −7082.78 −0.751972
$$447$$ −7062.34 −0.747287
$$448$$ 723.869 0.0763383
$$449$$ −11602.0 −1.21945 −0.609723 0.792615i $$-0.708719\pi$$
−0.609723 + 0.792615i $$0.708719\pi$$
$$450$$ 3424.33 0.358721
$$451$$ 0 0
$$452$$ −5494.95 −0.571816
$$453$$ 1710.21 0.177379
$$454$$ −7611.79 −0.786870
$$455$$ −248.676 −0.0256222
$$456$$ 3684.62 0.378395
$$457$$ 3530.68 0.361397 0.180698 0.983539i $$-0.442164\pi$$
0.180698 + 0.983539i $$0.442164\pi$$
$$458$$ 17910.4 1.82729
$$459$$ −1975.55 −0.200895
$$460$$ −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
$$465$$ −2917.44 −0.290953
$$466$$ 8316.94 0.826770
$$467$$ 10688.0 1.05906 0.529529 0.848292i $$-0.322369\pi$$
0.529529 + 0.848292i $$0.322369\pi$$
$$468$$ 455.918 0.0450317
$$469$$ 4272.89 0.420690
$$470$$ −1998.74 −0.196159
$$471$$ −6218.02 −0.608304
$$472$$ 3058.77 0.298287
$$473$$ 0 0
$$474$$ −3313.76 −0.321110
$$475$$ −8879.53 −0.857728
$$476$$ −1170.70 −0.112728
$$477$$ −3686.87 −0.353900
$$478$$ 4820.35 0.461250
$$479$$ −2341.90 −0.223391 −0.111696 0.993742i $$-0.535628\pi$$
−0.111696 + 0.993742i $$0.535628\pi$$
$$480$$ −1503.18 −0.142938
$$481$$ 1537.93 0.145787
$$482$$ −3301.43 −0.311983
$$483$$ 1594.17 0.150181
$$484$$ 0 0
$$485$$ −2143.57 −0.200690
$$486$$ −819.464 −0.0764849
$$487$$ 6748.91 0.627972 0.313986 0.949428i $$-0.398335\pi$$
0.313986 + 0.949428i $$0.398335\pi$$
$$488$$ 10945.4 1.01532
$$489$$ 8029.53 0.742552
$$490$$ −3770.98 −0.347664
$$491$$ −7361.40 −0.676609 −0.338305 0.941037i $$-0.609853\pi$$
−0.338305 + 0.941037i $$0.609853\pi$$
$$492$$ 2440.74 0.223653
$$493$$ 17789.1 1.62511
$$494$$ −3986.80 −0.363107
$$495$$ 0 0
$$496$$ −22187.6 −2.00857
$$497$$ 3589.26 0.323944
$$498$$ −7654.60 −0.688777
$$499$$ 10381.7 0.931359 0.465680 0.884953i $$-0.345810\pi$$
0.465680 + 0.884953i $$0.345810\pi$$
$$500$$ 2798.33 0.250291
$$501$$ 3564.36 0.317852
$$502$$ 22023.1 1.95805
$$503$$ −19149.0 −1.69744 −0.848721 0.528840i $$-0.822627\pi$$
−0.848721 + 0.528840i $$0.822627\pi$$
$$504$$ 666.391 0.0588957
$$505$$ −3544.67 −0.312348
$$506$$ 0 0
$$507$$ −5914.04 −0.518051
$$508$$ −7726.76 −0.674841
$$509$$ 16073.2 1.39967 0.699836 0.714303i $$-0.253256\pi$$
0.699836 + 0.714303i $$0.253256\pi$$
$$510$$ −2582.77 −0.224249
$$511$$ 4838.58 0.418877
$$512$$ −1493.27 −0.128894
$$513$$ 2124.93 0.182881
$$514$$ −27417.5 −2.35279
$$515$$ −3845.58 −0.329042
$$516$$ 2832.88 0.241687
$$517$$ 0 0
$$518$$ −1638.09 −0.138945
$$519$$ −2421.44 −0.204797
$$520$$ −817.952 −0.0689799
$$521$$ −18955.3 −1.59395 −0.796975 0.604012i $$-0.793568\pi$$
−0.796975 + 0.604012i $$0.793568\pi$$
$$522$$ 7378.97 0.618714
$$523$$ 4442.19 0.371402 0.185701 0.982606i $$-0.440544\pi$$
0.185701 + 0.982606i $$0.440544\pi$$
$$524$$ 3869.40 0.322587
$$525$$ −1605.93 −0.133502
$$526$$ −15341.9 −1.27175
$$527$$ −20393.3 −1.68567
$$528$$ 0 0
$$529$$ 377.000 0.0309855
$$530$$ −4820.09 −0.395041
$$531$$ 1764.00 0.144164
$$532$$ 1259.22 0.102620
$$533$$ 3624.08 0.294515
$$534$$ −5149.25 −0.417285
$$535$$ 4806.54 0.388420
$$536$$ 14054.5 1.13258
$$537$$ −5851.17 −0.470199
$$538$$ 98.3096 0.00787812
$$539$$ 0 0
$$540$$ −317.690 −0.0253171
$$541$$ −2180.90 −0.173316 −0.0866580 0.996238i $$-0.527619\pi$$
−0.0866580 + 0.996238i $$0.527619\pi$$
$$542$$ 26004.4 2.06086
$$543$$ 3184.82 0.251701
$$544$$ −10507.4 −0.828129
$$545$$ 1117.28 0.0878146
$$546$$ −721.044 −0.0565162
$$547$$ −8225.04 −0.642920 −0.321460 0.946923i $$-0.604174\pi$$
−0.321460 + 0.946923i $$0.604174\pi$$
$$548$$ 4278.07 0.333485
$$549$$ 6312.23 0.490709
$$550$$ 0 0
$$551$$ −19134.2 −1.47939
$$552$$ 5243.61 0.404316
$$553$$ 1554.08 0.119505
$$554$$ 3802.32 0.291598
$$555$$ −1071.65 −0.0819625
$$556$$ 1639.90 0.125085
$$557$$ 25181.9 1.91561 0.957804 0.287423i $$-0.0927986\pi$$
0.957804 + 0.287423i $$0.0927986\pi$$
$$558$$ −8459.22 −0.641769
$$559$$ 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 $$0.446076\pi$$
$$564$$ −1718.54 −0.128304
$$565$$ 5685.34 0.423335
$$566$$ 7164.36 0.532050
$$567$$ 384.310 0.0284647
$$568$$ 11805.9 0.872122
$$569$$ 13447.0 0.990732 0.495366 0.868684i $$-0.335034\pi$$
0.495366 + 0.868684i $$0.335034\pi$$
$$570$$ 2778.06 0.204141
$$571$$ 2605.52 0.190959 0.0954795 0.995431i $$-0.469562\pi$$
0.0954795 + 0.995431i $$0.469562\pi$$
$$572$$ 0 0
$$573$$ 6409.24 0.467277
$$574$$ −3860.09 −0.280691
$$575$$ −12636.5 −0.916485
$$576$$ 1373.11 0.0993281
$$577$$ 6339.65 0.457406 0.228703 0.973496i $$-0.426552\pi$$
0.228703 + 0.973496i $$0.426552\pi$$
$$578$$ −1485.90 −0.106929
$$579$$ −11843.3 −0.850069
$$580$$ 2860.68 0.204799
$$581$$ 3589.83 0.256336
$$582$$ −6215.37 −0.442672
$$583$$ 0 0
$$584$$ 15915.2 1.12770
$$585$$ −471.715 −0.0333385
$$586$$ −11210.1 −0.790247
$$587$$ −13370.6 −0.940140 −0.470070 0.882629i $$-0.655771\pi$$
−0.470070 + 0.882629i $$0.655771\pi$$
$$588$$ −3242.34 −0.227401
$$589$$ 21935.3 1.53452
$$590$$ 2306.20 0.160923
$$591$$ −2771.66 −0.192912
$$592$$ −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
$$597$$ −1428.46 −0.0979276
$$598$$ −5673.65 −0.387981
$$599$$ −5788.63 −0.394853 −0.197427 0.980318i $$-0.563258\pi$$
−0.197427 + 0.980318i $$0.563258\pi$$
$$600$$ −5282.28 −0.359413
$$601$$ −23968.1 −1.62675 −0.813375 0.581739i $$-0.802372\pi$$
−0.813375 + 0.581739i $$0.802372\pi$$
$$602$$ −4480.26 −0.303325
$$603$$ 8105.28 0.547384
$$604$$ 1922.44 0.129508
$$605$$ 0 0
$$606$$ −10277.9 −0.688961
$$607$$ 23526.6 1.57317 0.786585 0.617482i $$-0.211847\pi$$
0.786585 + 0.617482i $$0.211847\pi$$
$$608$$ 11301.9 0.753872
$$609$$ −3460.56 −0.230261
$$610$$ 8252.40 0.547754
$$611$$ −2551.74 −0.168956
$$612$$ −2220.70 −0.146677
$$613$$ −1228.07 −0.0809159 −0.0404579 0.999181i $$-0.512882\pi$$
−0.0404579 + 0.999181i $$0.512882\pi$$
$$614$$ −5729.31 −0.376573
$$615$$ −2525.31 −0.165578
$$616$$ 0 0
$$617$$ −9844.90 −0.642368 −0.321184 0.947017i $$-0.604081\pi$$
−0.321184 + 0.947017i $$0.604081\pi$$
$$618$$ −11150.4 −0.725785
$$619$$ −6551.68 −0.425419 −0.212709 0.977115i $$-0.568229\pi$$
−0.212709 + 0.977115i $$0.568229\pi$$
$$620$$ −3279.47 −0.212430
$$621$$ 3024.00 0.195409
$$622$$ −23365.9 −1.50625
$$623$$ 2414.88 0.155297
$$624$$ −3587.46 −0.230150
$$625$$ 11208.0 0.717309
$$626$$ 12007.8 0.766661
$$627$$ 0 0
$$628$$ −6989.64 −0.444135
$$629$$ −7491.01 −0.474859
$$630$$ 502.434 0.0317737
$$631$$ −26440.5 −1.66812 −0.834058 0.551677i $$-0.813988\pi$$
−0.834058 + 0.551677i $$0.813988\pi$$
$$632$$ 5111.72 0.321730
$$633$$ 14754.7 0.926458
$$634$$ −1122.13 −0.0702923
$$635$$ 7994.48 0.499608
$$636$$ −4144.39 −0.258389
$$637$$ −4814.31 −0.299450
$$638$$ 0 0
$$639$$ 6808.50 0.421502
$$640$$ 5803.64 0.358451
$$641$$ −27927.2 −1.72084 −0.860421 0.509584i $$-0.829799\pi$$
−0.860421 + 0.509584i $$0.829799\pi$$
$$642$$ 13936.7 0.856758
$$643$$ −16737.7 −1.02655 −0.513274 0.858225i $$-0.671568\pi$$
−0.513274 + 0.858225i $$0.671568\pi$$
$$644$$ 1792.00 0.109650
$$645$$ −2931.03 −0.178929
$$646$$ 19419.1 1.18271
$$647$$ 7818.70 0.475092 0.237546 0.971376i $$-0.423657\pi$$
0.237546 + 0.971376i $$0.423657\pi$$
$$648$$ 1264.08 0.0766325
$$649$$ 0 0
$$650$$ 5715.49 0.344892
$$651$$ 3967.17 0.238842
$$652$$ 9025.94 0.542152
$$653$$ 19747.6 1.18344 0.591719 0.806144i $$-0.298450\pi$$
0.591719 + 0.806144i $$0.298450\pi$$
$$654$$ 3239.59 0.193697
$$655$$ −4003.47 −0.238822
$$656$$ −19205.4 −1.14305
$$657$$ 9178.33 0.545024
$$658$$ 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
$$663$$ −3297.35 −0.193150
$$664$$ 11807.8 0.690106
$$665$$ −1302.85 −0.0759733
$$666$$ −3107.30 −0.180789
$$667$$ −27230.0 −1.58073
$$668$$ 4006.67 0.232070
$$669$$ 6300.88 0.364135
$$670$$ 10596.6 0.611018
$$671$$ 0 0
$$672$$ 2044.04 0.117337
$$673$$ −29397.6 −1.68379 −0.841897 0.539638i $$-0.818561\pi$$
−0.841897 + 0.539638i $$0.818561\pi$$
$$674$$ 2753.57 0.157364
$$675$$ −3046.30 −0.173707
$$676$$ −6647.94 −0.378239
$$677$$ −5737.14 −0.325696 −0.162848 0.986651i $$-0.552068\pi$$
−0.162848 + 0.986651i $$0.552068\pi$$
$$678$$ 16484.8 0.933771
$$679$$ 2914.86 0.164745
$$680$$ 3984.11 0.224682
$$681$$ 6771.49 0.381034
$$682$$ 0 0
$$683$$ 32097.6 1.79821 0.899107 0.437729i $$-0.144217\pi$$
0.899107 + 0.437729i $$0.144217\pi$$
$$684$$ 2388.62 0.133525
$$685$$ −4426.30 −0.246891
$$686$$ 10615.8 0.590837
$$687$$ −15933.2 −0.884848
$$688$$ −22291.0 −1.23523
$$689$$ −6153.69 −0.340257
$$690$$ 3953.48 0.218125
$$691$$ −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
$$696$$ −11382.6 −0.619909
$$697$$ −17652.3 −0.959294
$$698$$ −43058.9 −2.33496
$$699$$ −7398.80 −0.400355
$$700$$ −1805.22 −0.0974725
$$701$$ −27238.1 −1.46758 −0.733788 0.679379i $$-0.762249\pi$$
−0.733788 + 0.679379i $$0.762249\pi$$
$$702$$ −1367.75 −0.0735364
$$703$$ 8057.44 0.432279
$$704$$ 0 0
$$705$$ 1778.09 0.0949882
$$706$$ 8934.12 0.476261
$$707$$ 4820.09 0.256405
$$708$$ 1982.90 0.105257
$$709$$ 28761.4 1.52349 0.761747 0.647875i $$-0.224342\pi$$
0.761747 + 0.647875i $$0.224342\pi$$
$$710$$ 8901.21 0.470502
$$711$$ 2947.94 0.155494
$$712$$ 7943.10 0.418090
$$713$$ 31216.3 1.63964
$$714$$ 3512.09 0.184085
$$715$$ 0 0
$$716$$ −6577.27 −0.343302
$$717$$ −4288.21 −0.223356
$$718$$ −10804.5 −0.561588
$$719$$ −27272.0 −1.41456 −0.707282 0.706931i $$-0.750079\pi$$
−0.707282 + 0.706931i $$0.750079\pi$$
$$720$$ 2499.80 0.129392
$$721$$ 5229.28 0.270109
$$722$$ 2243.05 0.115620
$$723$$ 2936.97 0.151075
$$724$$ 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
$$729$$ 729.000 0.0370370
$$730$$ 11999.5 0.608383
$$731$$ −20488.3 −1.03665
$$732$$ 7095.54 0.358277
$$733$$ −9342.48 −0.470767 −0.235384 0.971903i $$-0.575635\pi$$
−0.235384 + 0.971903i $$0.575635\pi$$
$$734$$ 28426.3 1.42947
$$735$$ 3354.68 0.168353
$$736$$ 16083.9 0.805515
$$737$$ 0 0
$$738$$ −7322.23 −0.365224
$$739$$ 28928.0 1.43997 0.719983 0.693992i $$-0.244150\pi$$
0.719983 + 0.693992i $$0.244150\pi$$
$$740$$ −1204.64 −0.0598425
$$741$$ 3546.68 0.175831
$$742$$ 6554.43 0.324287
$$743$$ −4857.04 −0.239822 −0.119911 0.992785i $$-0.538261\pi$$
−0.119911 + 0.992785i $$0.538261\pi$$
$$744$$ 13049.0 0.643008
$$745$$ 8213.80 0.403933
$$746$$ −31660.8 −1.55386
$$747$$ 6809.57 0.333533
$$748$$ 0 0
$$749$$ −6536.00 −0.318852
$$750$$ −8395.00 −0.408723
$$751$$ 14355.4 0.697517 0.348759 0.937213i $$-0.386603\pi$$
0.348759 + 0.937213i $$0.386603\pi$$
$$752$$ 13522.6 0.655744
$$753$$ −19591.9 −0.948165
$$754$$ 12316.1 0.594863
$$755$$ −1989.05 −0.0958792
$$756$$ 432.000 0.0207827
$$757$$ −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 $$0.439395\pi$$
$$762$$ 23180.3 1.10201
$$763$$ −1519.29 −0.0720866
$$764$$ 7204.58 0.341168
$$765$$ 2297.64 0.108590
$$766$$ −45398.4 −2.14140
$$767$$ 2944.26 0.138606
$$768$$ 13166.2 0.618613
$$769$$ −27308.1 −1.28057 −0.640284 0.768139i $$-0.721183\pi$$
−0.640284 + 0.768139i $$0.721183\pi$$
$$770$$ 0 0
$$771$$ 24390.8 1.13932
$$772$$ −13313.0 −0.620653
$$773$$ −18872.6 −0.878136 −0.439068 0.898454i $$-0.644691\pi$$
−0.439068 + 0.898454i $$0.644691\pi$$
$$774$$ −8498.64 −0.394674
$$775$$ −31446.6 −1.45754
$$776$$ 9587.65 0.443527
$$777$$ 1457.25 0.0672826
$$778$$ 3176.29 0.146369
$$779$$ 18987.1 0.873276
$$780$$ −530.252 −0.0243411
$$781$$ 0 0
$$782$$ 27635.4 1.26373
$$783$$ −6564.37 −0.299606
$$784$$ 25512.8 1.16221
$$785$$ 7231.82 0.328808
$$786$$ −11608.2 −0.526782
$$787$$ 14512.1 0.657307 0.328654 0.944450i $$-0.393405\pi$$
0.328654 + 0.944450i $$0.393405\pi$$
$$788$$ −3115.60 −0.140849
$$789$$ 13648.3 0.615832
$$790$$ 3854.04 0.173570
$$791$$ −7731.01 −0.347513
$$792$$ 0 0
$$793$$ 10535.6 0.471792
$$794$$ 2859.15 0.127793
$$795$$ 4287.98 0.191294
$$796$$ −1605.72 −0.0714989
$$797$$ 29108.9 1.29371 0.646856 0.762612i $$-0.276083\pi$$
0.646856 + 0.762612i $$0.276083\pi$$
$$798$$ −3777.65 −0.167578
$$799$$ 12429.1 0.550325
$$800$$ −16202.5 −0.716056
$$801$$ 4580.80 0.202066
$$802$$ −41154.0 −1.81197
$$803$$ 0 0
$$804$$ 9111.10 0.399656
$$805$$ −1854.09 −0.0811777
$$806$$ −14119.1 −0.617029
$$807$$ −87.4567 −0.00381490
$$808$$ 15854.4 0.690291
$$809$$ 3000.83 0.130413 0.0652063 0.997872i $$-0.479229\pi$$
0.0652063 + 0.997872i $$0.479229\pi$$
$$810$$ 953.071 0.0413426
$$811$$ −6239.39 −0.270154 −0.135077 0.990835i $$-0.543128\pi$$
−0.135077 + 0.990835i $$0.543128\pi$$
$$812$$ −3890.00 −0.168118
$$813$$ −23133.7 −0.997950
$$814$$ 0 0
$$815$$ −9338.68 −0.401374
$$816$$ 17473.9 0.749645
$$817$$ 22037.6 0.943693
$$818$$ 29539.6 1.26263
$$819$$ 641.445 0.0273674
$$820$$ −2838.69 −0.120892
$$821$$ −14922.4 −0.634342 −0.317171 0.948368i $$-0.602733\pi$$
−0.317171 + 0.948368i $$0.602733\pi$$
$$822$$ −12834.2 −0.544580
$$823$$ −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 $$0.692497\pi$$
$$828$$ 3399.26 0.142672
$$829$$ −9795.41 −0.410384 −0.205192 0.978722i $$-0.565782\pi$$
−0.205192 + 0.978722i $$0.565782\pi$$
$$830$$ 8902.62 0.372306
$$831$$ −3382.56 −0.141203
$$832$$ 2291.84 0.0954990
$$833$$ 23449.7 0.975370
$$834$$ −4919.70 −0.204263
$$835$$ −4145.50 −0.171809
$$836$$ 0 0
$$837$$ 7525.37 0.310770
$$838$$ 37730.5 1.55535
$$839$$ 28875.5 1.18819 0.594095 0.804395i $$-0.297510\pi$$
0.594095 + 0.804395i $$0.297510\pi$$
$$840$$ −775.041 −0.0318351
$$841$$ 34720.7 1.42362
$$842$$ 47489.3 1.94369
$$843$$ 5617.41 0.229506
$$844$$ 16585.7 0.676426
$$845$$ 6878.28 0.280024
$$846$$ 5155.63 0.209520
$$847$$ 0 0
$$848$$ 32610.7 1.32059
$$849$$ −6373.45 −0.257640
$$850$$ −27839.2 −1.12338
$$851$$ 11466.6 0.461892
$$852$$ 7653.39 0.307747
$$853$$ 47157.1 1.89288 0.946441 0.322878i $$-0.104650\pi$$
0.946441 + 0.322878i $$0.104650\pi$$
$$854$$ −11221.7 −0.449649
$$855$$ −2471.38 −0.0988531
$$856$$ −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
$$861$$ 3433.95 0.135922
$$862$$ −18938.9 −0.748332
$$863$$ −19488.1 −0.768693 −0.384347 0.923189i $$-0.625573\pi$$
−0.384347 + 0.923189i $$0.625573\pi$$
$$864$$ 3877.36 0.152674
$$865$$ 2816.24 0.110699
$$866$$ −24266.1 −0.952188
$$867$$ 1321.86 0.0517794
$$868$$ 4459.48 0.174383
$$869$$ 0 0
$$870$$ −8582.05 −0.334435
$$871$$ 13528.4 0.526282
$$872$$ −4997.30 −0.194071
$$873$$ 5529.22 0.214360
$$874$$ −29725.0 −1.15042
$$875$$ 3937.06 0.152111
$$876$$ 10317.3 0.397933
$$877$$ 8455.67 0.325573 0.162787 0.986661i $$-0.447952\pi$$
0.162787 + 0.986661i $$0.447952\pi$$
$$878$$ 343.834 0.0132162
$$879$$ 9972.56 0.382669
$$880$$ 0 0
$$881$$ −11291.2 −0.431794 −0.215897 0.976416i $$-0.569268\pi$$
−0.215897 + 0.976416i $$0.569268\pi$$
$$882$$ 9727.02 0.371344
$$883$$ 31818.1 1.21264 0.606322 0.795219i $$-0.292644\pi$$
0.606322 + 0.795219i $$0.292644\pi$$
$$884$$ −3706.53 −0.141023
$$885$$ −2051.61 −0.0779254
$$886$$ −16705.4 −0.633441
$$887$$ −17481.1 −0.661732 −0.330866 0.943678i $$-0.607341\pi$$
−0.330866 + 0.943678i $$0.607341\pi$$
$$888$$ 4793.24 0.181138
$$889$$ −10871.0 −0.410126
$$890$$ 5988.80 0.225556
$$891$$ 0 0
$$892$$ 7082.78 0.265862
$$893$$ −13368.9 −0.500978
$$894$$ 23816.2 0.890976
$$895$$ 6805.16 0.254158
$$896$$ −7891.87 −0.294251
$$897$$ 5047.31 0.187876
$$898$$ 39125.1 1.45392
$$899$$ −67763.1 −2.51393
$$900$$ −3424.33 −0.126827
$$901$$ 29973.6 1.10829
$$902$$ 0 0
$$903$$ 3985.66 0.146882
$$904$$ −25429.1 −0.935574
$$905$$ −3704.08 −0.136053
$$906$$ −5767.31 −0.211486
$$907$$ 10607.4 0.388326 0.194163 0.980969i $$-0.437801\pi$$
0.194163 + 0.980969i $$0.437801\pi$$
$$908$$ 7611.79 0.278201
$$909$$ 9143.26 0.333623
$$910$$ 838.604 0.0305489
$$911$$ −41249.2 −1.50016 −0.750080 0.661347i $$-0.769985\pi$$
−0.750080 + 0.661347i $$0.769985\pi$$
$$912$$ −18795.2 −0.682425
$$913$$ 0 0
$$914$$ −11906.5 −0.430887
$$915$$ −7341.38 −0.265244
$$916$$ −17910.4 −0.646045
$$917$$ 5443.97 0.196048
$$918$$ 6662.10 0.239523
$$919$$ 13858.1 0.497429 0.248714 0.968577i $$-0.419992\pi$$
0.248714 + 0.968577i $$0.419992\pi$$
$$920$$ −6098.53 −0.218546
$$921$$ 5096.83 0.182352
$$922$$ 39004.9 1.39323
$$923$$ 11363.9 0.405253
$$924$$ 0 0
$$925$$ −11551.2 −0.410595
$$926$$ −36719.0 −1.30309
$$927$$ 9919.47 0.351454
$$928$$ −34914.2 −1.23504
$$929$$ 20893.7 0.737890 0.368945 0.929451i $$-0.379719\pi$$
0.368945 + 0.929451i $$0.379719\pi$$
$$930$$ 9838.42 0.346897
$$931$$ −25222.8 −0.887911
$$932$$ −8316.94 −0.292307
$$933$$ 20786.5 0.729388
$$934$$ −36042.9 −1.26270
$$935$$ 0 0
$$936$$ 2109.86 0.0736784
$$937$$ −3203.52 −0.111691 −0.0558454 0.998439i $$-0.517785\pi$$
−0.0558454 + 0.998439i $$0.517785\pi$$
$$938$$ −14409.4 −0.501581
$$939$$ −10682.2 −0.371248
$$940$$ 1998.74 0.0693528
$$941$$ −19951.6 −0.691182 −0.345591 0.938385i $$-0.612322\pi$$
−0.345591 + 0.938385i $$0.612322\pi$$
$$942$$ 20968.9 0.725270
$$943$$ 27020.6 0.933099
$$944$$ −15602.8 −0.537952
$$945$$ −446.968 −0.0153861
$$946$$ 0 0
$$947$$ −38216.7 −1.31138 −0.655689 0.755031i $$-0.727622\pi$$
−0.655689 + 0.755031i $$0.727622\pi$$
$$948$$ 3313.76 0.113529
$$949$$ 15319.4 0.524014
$$950$$ 29944.3 1.02265
$$951$$ 998.249 0.0340383
$$952$$ −5417.65 −0.184440
$$953$$ 47661.4 1.62004 0.810022 0.586399i $$-0.199455\pi$$
0.810022 + 0.586399i $$0.199455\pi$$
$$954$$ 12433.2 0.421948
$$955$$ −7454.21 −0.252579
$$956$$ −4820.35 −0.163077
$$957$$ 0 0
$$958$$ 7897.56 0.266345
$$959$$ 6018.94 0.202671
$$960$$ −1596.99 −0.0536901
$$961$$ 47892.3 1.60761
$$962$$ −5186.34 −0.173819
$$963$$ −12398.2 −0.414876
$$964$$ 3301.43 0.110303
$$965$$ 13774.2 0.459490
$$966$$ −5376.00 −0.179058
$$967$$ −18933.2 −0.629628 −0.314814 0.949153i $$-0.601942\pi$$
−0.314814 + 0.949153i $$0.601942\pi$$
$$968$$ 0 0
$$969$$ −17275.3 −0.572717
$$970$$ 7228.73 0.239279
$$971$$ −40660.3 −1.34382 −0.671911 0.740632i $$-0.734526\pi$$
−0.671911 + 0.740632i $$0.734526\pi$$
$$972$$ 819.464 0.0270415
$$973$$ 2307.23 0.0760188
$$974$$ −22759.2 −0.748720
$$975$$ −5084.53 −0.167011
$$976$$ −55832.3 −1.83110
$$977$$ −22502.8 −0.736876 −0.368438 0.929652i $$-0.620107\pi$$
−0.368438 + 0.929652i $$0.620107\pi$$
$$978$$ −27077.8 −0.885331
$$979$$ 0 0
$$980$$ 3770.98 0.122918
$$981$$ −2881.96 −0.0937959
$$982$$ 24824.7 0.806709
$$983$$ −4435.20 −0.143907 −0.0719536 0.997408i $$-0.522923\pi$$
−0.0719536 + 0.997408i $$0.522923\pi$$
$$984$$ 11295.1 0.365929
$$985$$ 3223.55 0.104275
$$986$$ −59989.8 −1.93759
$$987$$ −2417.87 −0.0779753
$$988$$ 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
$$993$$ −1624.33 −0.0519101
$$994$$ −12104.0 −0.386233
$$995$$ 1661.35 0.0529331
$$996$$ 7654.60 0.243519
$$997$$ 53480.1 1.69883 0.849413 0.527728i $$-0.176956\pi$$
0.849413 + 0.527728i $$0.176956\pi$$
$$998$$ −35010.0 −1.11044
$$999$$ 2764.27 0.0875452
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 363.4.a.j.1.1 2
3.2 odd 2 1089.4.a.t.1.2 2
11.10 odd 2 33.4.a.d.1.2 2
33.32 even 2 99.4.a.e.1.1 2
44.43 even 2 528.4.a.o.1.1 2
55.32 even 4 825.4.c.i.199.4 4
55.43 even 4 825.4.c.i.199.1 4
55.54 odd 2 825.4.a.k.1.1 2
77.76 even 2 1617.4.a.j.1.2 2
88.21 odd 2 2112.4.a.ba.1.2 2
88.43 even 2 2112.4.a.bh.1.2 2
132.131 odd 2 1584.4.a.x.1.2 2
165.164 even 2 2475.4.a.o.1.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
33.4.a.d.1.2 2 11.10 odd 2
99.4.a.e.1.1 2 33.32 even 2
363.4.a.j.1.1 2 1.1 even 1 trivial
528.4.a.o.1.1 2 44.43 even 2
825.4.a.k.1.1 2 55.54 odd 2
825.4.c.i.199.1 4 55.43 even 4
825.4.c.i.199.4 4 55.32 even 4
1089.4.a.t.1.2 2 3.2 odd 2
1584.4.a.x.1.2 2 132.131 odd 2
1617.4.a.j.1.2 2 77.76 even 2
2112.4.a.ba.1.2 2 88.21 odd 2
2112.4.a.bh.1.2 2 88.43 even 2
2475.4.a.o.1.2 2 165.164 even 2