# Properties

 Label 1617.4.a.j.1.2 Level $1617$ Weight $4$ Character 1617.1 Self dual yes Analytic conductor $95.406$ 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] = [1617,4,Mod(1,1617)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1617, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("1617.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1617 = 3 \cdot 7^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 1617.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: $$95.4060884793$$ 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 $$\chi$$ $$=$$ 1617.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+3.37228 q^{2} -3.00000 q^{3} +3.37228 q^{4} +3.48913 q^{5} -10.1168 q^{6} -15.6060 q^{8} +9.00000 q^{9} +O(q^{10})$$ $$q+3.37228 q^{2} -3.00000 q^{3} +3.37228 q^{4} +3.48913 q^{5} -10.1168 q^{6} -15.6060 q^{8} +9.00000 q^{9} +11.7663 q^{10} +11.0000 q^{11} -10.1168 q^{12} +15.0217 q^{13} -10.4674 q^{15} -79.6060 q^{16} -73.1684 q^{17} +30.3505 q^{18} +78.7011 q^{19} +11.7663 q^{20} +37.0951 q^{22} +112.000 q^{23} +46.8179 q^{24} -112.826 q^{25} +50.6576 q^{26} -27.0000 q^{27} +243.125 q^{29} -35.2989 q^{30} -278.717 q^{31} -143.606 q^{32} -33.0000 q^{33} -246.745 q^{34} +30.3505 q^{36} +102.380 q^{37} +265.402 q^{38} -45.0652 q^{39} -54.4512 q^{40} +241.255 q^{41} -280.016 q^{43} +37.0951 q^{44} +31.4021 q^{45} +377.696 q^{46} +169.870 q^{47} +238.818 q^{48} -380.481 q^{50} +219.505 q^{51} +50.6576 q^{52} -409.652 q^{53} -91.0516 q^{54} +38.3804 q^{55} -236.103 q^{57} +819.886 q^{58} -196.000 q^{59} -35.2989 q^{60} +701.359 q^{61} -939.913 q^{62} +152.568 q^{64} +52.4128 q^{65} -111.285 q^{66} +900.587 q^{67} -246.745 q^{68} -336.000 q^{69} +756.500 q^{71} -140.454 q^{72} +1019.81 q^{73} +345.255 q^{74} +338.478 q^{75} +265.402 q^{76} -151.973 q^{78} -327.549 q^{79} -277.755 q^{80} +81.0000 q^{81} +813.581 q^{82} +756.619 q^{83} -255.294 q^{85} -944.293 q^{86} -729.375 q^{87} -171.666 q^{88} -508.978 q^{89} +105.897 q^{90} +377.696 q^{92} +836.152 q^{93} +572.848 q^{94} +274.598 q^{95} +430.818 q^{96} -614.358 q^{97} +99.0000 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + q^{2} - 6 q^{3} + q^{4} - 16 q^{5} - 3 q^{6} + 9 q^{8} + 18 q^{9}+O(q^{10})$$ 2 * q + q^2 - 6 * q^3 + q^4 - 16 * q^5 - 3 * q^6 + 9 * q^8 + 18 * q^9 $$2 q + q^{2} - 6 q^{3} + q^{4} - 16 q^{5} - 3 q^{6} + 9 q^{8} + 18 q^{9} + 58 q^{10} + 22 q^{11} - 3 q^{12} + 76 q^{13} + 48 q^{15} - 119 q^{16} + 26 q^{17} + 9 q^{18} + 54 q^{19} + 58 q^{20} + 11 q^{22} + 224 q^{23} - 27 q^{24} + 142 q^{25} - 94 q^{26} - 54 q^{27} + 222 q^{29} - 174 q^{30} + 40 q^{31} - 247 q^{32} - 66 q^{33} - 482 q^{34} + 9 q^{36} - 48 q^{37} + 324 q^{38} - 228 q^{39} - 534 q^{40} + 494 q^{41} - 66 q^{43} + 11 q^{44} - 144 q^{45} + 112 q^{46} + 64 q^{47} + 357 q^{48} - 985 q^{50} - 78 q^{51} - 94 q^{52} - 84 q^{53} - 27 q^{54} - 176 q^{55} - 162 q^{57} + 870 q^{58} - 392 q^{59} - 174 q^{60} + 1104 q^{61} - 1696 q^{62} + 713 q^{64} - 1136 q^{65} - 33 q^{66} + 928 q^{67} - 482 q^{68} - 672 q^{69} + 456 q^{71} + 81 q^{72} + 592 q^{73} + 702 q^{74} - 426 q^{75} + 324 q^{76} + 282 q^{78} - 230 q^{79} + 490 q^{80} + 162 q^{81} + 214 q^{82} - 348 q^{83} - 2188 q^{85} - 1452 q^{86} - 666 q^{87} + 99 q^{88} - 972 q^{89} + 522 q^{90} + 112 q^{92} - 120 q^{93} + 824 q^{94} + 756 q^{95} + 741 q^{96} + 1184 q^{97} + 198 q^{99}+O(q^{100})$$ 2 * q + q^2 - 6 * q^3 + q^4 - 16 * q^5 - 3 * q^6 + 9 * q^8 + 18 * q^9 + 58 * q^10 + 22 * q^11 - 3 * q^12 + 76 * q^13 + 48 * q^15 - 119 * q^16 + 26 * q^17 + 9 * q^18 + 54 * q^19 + 58 * q^20 + 11 * q^22 + 224 * q^23 - 27 * q^24 + 142 * q^25 - 94 * q^26 - 54 * q^27 + 222 * q^29 - 174 * q^30 + 40 * q^31 - 247 * q^32 - 66 * q^33 - 482 * q^34 + 9 * q^36 - 48 * q^37 + 324 * q^38 - 228 * q^39 - 534 * q^40 + 494 * q^41 - 66 * q^43 + 11 * q^44 - 144 * q^45 + 112 * q^46 + 64 * q^47 + 357 * q^48 - 985 * q^50 - 78 * q^51 - 94 * q^52 - 84 * q^53 - 27 * q^54 - 176 * q^55 - 162 * q^57 + 870 * q^58 - 392 * q^59 - 174 * q^60 + 1104 * q^61 - 1696 * q^62 + 713 * q^64 - 1136 * q^65 - 33 * q^66 + 928 * q^67 - 482 * q^68 - 672 * q^69 + 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 - 2188 * q^85 - 1452 * q^86 - 666 * q^87 + 99 * q^88 - 972 * q^89 + 522 * q^90 + 112 * q^92 - 120 * q^93 + 824 * q^94 + 756 * q^95 + 741 * q^96 + 1184 * q^97 + 198 * q^99

## 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 $$0.296700\pi$$
$$3$$ −3.00000 −0.577350
$$4$$ 3.37228 0.421535
$$5$$ 3.48913 0.312077 0.156038 0.987751i $$-0.450128\pi$$
0.156038 + 0.987751i $$0.450128\pi$$
$$6$$ −10.1168 −0.688364
$$7$$ 0 0
$$8$$ −15.6060 −0.689693
$$9$$ 9.00000 0.333333
$$10$$ 11.7663 0.372083
$$11$$ 11.0000 0.301511
$$12$$ −10.1168 −0.243373
$$13$$ 15.0217 0.320483 0.160242 0.987078i $$-0.448773\pi$$
0.160242 + 0.987078i $$0.448773\pi$$
$$14$$ 0 0
$$15$$ −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
$$21$$ 0 0
$$22$$ 37.0951 0.359486
$$23$$ 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
$$28$$ 0 0
$$29$$ 243.125 1.55680 0.778399 0.627769i $$-0.216032\pi$$
0.778399 + 0.627769i $$0.216032\pi$$
$$30$$ −35.2989 −0.214822
$$31$$ −278.717 −1.61481 −0.807405 0.589998i $$-0.799129\pi$$
−0.807405 + 0.589998i $$0.799129\pi$$
$$32$$ −143.606 −0.793318
$$33$$ −33.0000 −0.174078
$$34$$ −246.745 −1.24460
$$35$$ 0 0
$$36$$ 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 $$0.348034\pi$$
$$42$$ 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
$$45$$ 31.4021 0.104026
$$46$$ 377.696 1.21061
$$47$$ 169.870 0.527192 0.263596 0.964633i $$-0.415091\pi$$
0.263596 + 0.964633i $$0.415091\pi$$
$$48$$ 238.818 0.718133
$$49$$ 0 0
$$50$$ −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
$$55$$ 38.3804 0.0940947
$$56$$ 0 0
$$57$$ −236.103 −0.548643
$$58$$ 819.886 1.85614
$$59$$ −196.000 −0.432492 −0.216246 0.976339i $$-0.569381\pi$$
−0.216246 + 0.976339i $$0.569381\pi$$
$$60$$ −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
$$63$$ 0 0
$$64$$ 152.568 0.297984
$$65$$ 52.4128 0.100015
$$66$$ −111.285 −0.207550
$$67$$ 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
$$70$$ 0 0
$$71$$ 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 $$-0.574933\pi$$
−0.233241 + 0.972419i $$0.574933\pi$$
$$80$$ −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 $$0.333223\pi$$
$$84$$ 0 0
$$85$$ −255.294 −0.325771
$$86$$ −944.293 −1.18402
$$87$$ −729.375 −0.898818
$$88$$ −171.666 −0.207950
$$89$$ −508.978 −0.606198 −0.303099 0.952959i $$-0.598021\pi$$
−0.303099 + 0.952959i $$0.598021\pi$$
$$90$$ 105.897 0.124028
$$91$$ 0 0
$$92$$ 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 $$-0.604200\pi$$
−0.321539 + 0.946896i $$0.604200\pi$$
$$98$$ 0 0
$$99$$ 99.0000 0.100504
$$100$$ −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 $$-0.676751\pi$$
−0.527181 + 0.849753i $$0.676751\pi$$
$$104$$ −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 $$0.286192\pi$$
$$108$$ −91.0516 −0.0811245
$$109$$ 320.217 0.281388 0.140694 0.990053i $$-0.455067\pi$$
0.140694 + 0.990053i $$0.455067\pi$$
$$110$$ 129.429 0.112187
$$111$$ −307.141 −0.262636
$$112$$ 0 0
$$113$$ −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
$$119$$ 0 0
$$120$$ 163.354 0.124267
$$121$$ 121.000 0.0909091
$$122$$ 2365.18 1.75519
$$123$$ −723.766 −0.530568
$$124$$ −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
$$129$$ 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 $$0.375017\pi$$
$$132$$ −111.285 −0.0733799
$$133$$ 0 0
$$134$$ 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 $$0.452598\pi$$
$$140$$ 0 0
$$141$$ −509.609 −0.304374
$$142$$ 2551.13 1.50765
$$143$$ 165.239 0.0966294
$$144$$ −716.454 −0.414614
$$145$$ 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 $$0.275953\pi$$
$$150$$ 1141.44 0.621323
$$151$$ −570.070 −0.307229 −0.153615 0.988131i $$-0.549091\pi$$
−0.153615 + 0.988131i $$0.549091\pi$$
$$152$$ −1228.21 −0.655399
$$153$$ −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 $$-0.323389\pi$$
0.526807 + 0.849985i $$0.323389\pi$$
$$158$$ −1104.59 −0.556179
$$159$$ 1228.96 0.612972
$$160$$ −501.059 −0.247576
$$161$$ 0 0
$$162$$ 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
$$165$$ −115.141 −0.0543256
$$166$$ 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
$$171$$ 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
$$175$$ 0 0
$$176$$ −875.666 −0.375033
$$177$$ 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 $$-0.569947\pi$$
−0.217980 + 0.975953i $$0.569947\pi$$
$$182$$ 0 0
$$183$$ −2104.08 −0.849933
$$184$$ −1747.87 −0.700297
$$185$$ 357.218 0.141963
$$186$$ 2819.74 1.11158
$$187$$ −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 $$-0.367385\pi$$
0.404674 + 0.914461i $$0.367385\pi$$
$$192$$ −457.704 −0.172041
$$193$$ 3947.76 1.47236 0.736181 0.676784i $$-0.236627\pi$$
0.736181 + 0.676784i $$0.236627\pi$$
$$194$$ −2071.79 −0.766731
$$195$$ −157.238 −0.0577439
$$196$$ 0 0
$$197$$ 923.886 0.334133 0.167066 0.985946i $$-0.446571\pi$$
0.167066 + 0.985946i $$0.446571\pi$$
$$198$$ 333.856 0.119829
$$199$$ 476.152 0.169616 0.0848078 0.996397i $$-0.472972\pi$$
0.0848078 + 0.996397i $$0.472972\pi$$
$$200$$ 1760.76 0.622522
$$201$$ −2701.76 −0.948097
$$202$$ 3425.96 1.19332
$$203$$ 0 0
$$204$$ 740.234 0.254053
$$205$$ 841.770 0.286789
$$206$$ −3716.80 −1.25710
$$207$$ 1008.00 0.338458
$$208$$ −1195.82 −0.398631
$$209$$ 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
$$213$$ −2269.50 −0.730064
$$214$$ 4645.57 1.48395
$$215$$ −977.012 −0.309915
$$216$$ 421.361 0.132731
$$217$$ 0 0
$$218$$ 1079.86 0.335494
$$219$$ −3059.44 −0.944010
$$220$$ 129.429 0.0396642
$$221$$ −1099.12 −0.334546
$$222$$ −1035.77 −0.313136
$$223$$ −2100.29 −0.630700 −0.315350 0.948975i $$-0.602122\pi$$
−0.315350 + 0.948975i $$0.602122\pi$$
$$224$$ 0 0
$$225$$ −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 $$-0.222095\pi$$
0.766301 + 0.642482i $$0.222095\pi$$
$$230$$ 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 $$0.387296\pi$$
$$234$$ 455.918 0.127369
$$235$$ 592.696 0.164524
$$236$$ −660.967 −0.182311
$$237$$ 982.646 0.269324
$$238$$ 0 0
$$239$$ 1429.40 0.386863 0.193432 0.981114i $$-0.438038\pi$$
0.193432 + 0.981114i $$0.438038\pi$$
$$240$$ 833.266 0.224113
$$241$$ 978.989 0.261669 0.130835 0.991404i $$-0.458234\pi$$
0.130835 + 0.991404i $$0.458234\pi$$
$$242$$ 408.046 0.108389
$$243$$ −243.000 −0.0641500
$$244$$ 2365.18 0.620553
$$245$$ 0 0
$$246$$ −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 $$-0.193341\pi$$
0.821135 + 0.570734i $$0.193341\pi$$
$$252$$ 0 0
$$253$$ 1232.00 0.306147
$$254$$ 7726.76 1.90874
$$255$$ 765.882 0.188084
$$256$$ 4388.74 1.07147
$$257$$ −8130.26 −1.97335 −0.986676 0.162696i $$-0.947981\pi$$
−0.986676 + 0.162696i $$0.947981\pi$$
$$258$$ 2832.88 0.683595
$$259$$ 0 0
$$260$$ 176.751 0.0421600
$$261$$ 2188.12 0.518933
$$262$$ 3869.40 0.912414
$$263$$ −4549.42 −1.06665 −0.533326 0.845910i $$-0.679058\pi$$
−0.533326 + 0.845910i $$0.679058\pi$$
$$264$$ 514.997 0.120060
$$265$$ −1429.33 −0.331332
$$266$$ 0 0
$$267$$ 1526.93 0.349988
$$268$$ 3037.03 0.692225
$$269$$ 29.1522 0.00660760 0.00330380 0.999995i $$-0.498948\pi$$
0.00330380 + 0.999995i $$0.498948\pi$$
$$270$$ −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
$$273$$ 0 0
$$274$$ 4278.07 0.943239
$$275$$ −1241.09 −0.272147
$$276$$ −1133.09 −0.247115
$$277$$ 1127.52 0.244571 0.122286 0.992495i $$-0.460978\pi$$
0.122286 + 0.992495i $$0.460978\pi$$
$$278$$ 1639.90 0.353794
$$279$$ −2508.46 −0.538270
$$280$$ 0 0
$$281$$ −1872.47 −0.397517 −0.198758 0.980049i $$-0.563691\pi$$
−0.198758 + 0.980049i $$0.563691\pi$$
$$282$$ −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
$$286$$ 557.233 0.115209
$$287$$ 0 0
$$288$$ −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 $$0.392479\pi$$
$$294$$ 0 0
$$295$$ −683.869 −0.134971
$$296$$ −1597.75 −0.313740
$$297$$ −297.000 −0.0580259
$$298$$ 7938.73 1.54322
$$299$$ 1682.44 0.325411
$$300$$ 1141.44 0.219671
$$301$$ 0 0
$$302$$ −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 $$-0.717630\pi$$
−0.631668 + 0.775239i $$0.717630\pi$$
$$312$$ 703.287 0.127615
$$313$$ 3560.75 0.643020 0.321510 0.946906i $$-0.395810\pi$$
0.321510 + 0.946906i $$0.395810\pi$$
$$314$$ 6989.64 1.25620
$$315$$ 0 0
$$316$$ −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
$$319$$ 2674.37 0.469393
$$320$$ 532.329 0.0929940
$$321$$ −4132.73 −0.718587
$$322$$ 0 0
$$323$$ −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
$$329$$ 0 0
$$330$$ −388.288 −0.0647714
$$331$$ −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
$$336$$ 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
$$339$$ 4888.34 0.783180
$$340$$ −860.923 −0.137324
$$341$$ −3065.89 −0.486883
$$342$$ 2388.62 0.377666
$$343$$ 0 0
$$344$$ 4369.92 0.684914
$$345$$ −1172.35 −0.182948
$$346$$ −2721.93 −0.422924
$$347$$ 6260.53 0.968539 0.484269 0.874919i $$-0.339086\pi$$
0.484269 + 0.874919i $$0.339086\pi$$
$$348$$ −2459.66 −0.378884
$$349$$ 12768.5 1.95840 0.979198 0.202906i $$-0.0650386\pi$$
0.979198 + 0.202906i $$0.0650386\pi$$
$$350$$ 0 0
$$351$$ −405.587 −0.0616771
$$352$$ −1579.67 −0.239194
$$353$$ 2649.28 0.399453 0.199727 0.979852i $$-0.435995\pi$$
0.199727 + 0.979852i $$0.435995\pi$$
$$354$$ 1982.90 0.297712
$$355$$ 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 $$0.575676\pi$$
$$360$$ −490.061 −0.0717457
$$361$$ −665.143 −0.0969737
$$362$$ −3580.04 −0.519786
$$363$$ −363.000 −0.0524864
$$364$$ 0 0
$$365$$ 3558.26 0.510268
$$366$$ −7095.54 −1.01336
$$367$$ 8429.40 1.19894 0.599470 0.800397i $$-0.295378\pi$$
0.599470 + 0.800397i $$0.295378\pi$$
$$368$$ −8915.87 −1.26297
$$369$$ 2171.30 0.306323
$$370$$ 1204.64 0.169260
$$371$$ 0 0
$$372$$ 2819.74 0.393002
$$373$$ −9388.53 −1.30327 −0.651635 0.758533i $$-0.725917\pi$$
−0.651635 + 0.758533i $$0.725917\pi$$
$$374$$ −2714.19 −0.375261
$$375$$ 2489.41 0.342807
$$376$$ −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
$$381$$ −6873.77 −0.924288
$$382$$ 7204.58 0.964970
$$383$$ −13462.2 −1.79605 −0.898026 0.439942i $$-0.854999\pi$$
−0.898026 + 0.439942i $$0.854999\pi$$
$$384$$ −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
$$392$$ 0 0
$$393$$ −3442.24 −0.441827
$$394$$ 3115.60 0.398380
$$395$$ −1142.86 −0.145578
$$396$$ 333.856 0.0423659
$$397$$ 847.839 0.107183 0.0535917 0.998563i $$-0.482933\pi$$
0.0535917 + 0.998563i $$0.482933\pi$$
$$398$$ 1605.72 0.202230
$$399$$ 0 0
$$400$$ 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
$$406$$ 0 0
$$407$$ 1126.18 0.137157
$$408$$ −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
$$413$$ 0 0
$$414$$ 3399.26 0.403537
$$415$$ 2639.94 0.312264
$$416$$ −2157.21 −0.254245
$$417$$ −1458.86 −0.171321
$$418$$ 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
$$423$$ 1528.83 0.175731
$$424$$ 6393.02 0.732246
$$425$$ 8255.30 0.942214
$$426$$ −7653.39 −0.870441
$$427$$ 0 0
$$428$$ 4645.57 0.524655
$$429$$ −495.718 −0.0557890
$$430$$ −3294.76 −0.369505
$$431$$ −5616.05 −0.627647 −0.313823 0.949481i $$-0.601610\pi$$
−0.313823 + 0.949481i $$0.601610\pi$$
$$432$$ 2149.36 0.239378
$$433$$ −7195.75 −0.798627 −0.399314 0.916814i $$-0.630752\pi$$
−0.399314 + 0.916814i $$0.630752\pi$$
$$434$$ 0 0
$$435$$ −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 $$0.501764\pi$$
$$440$$ −598.963 −0.0648965
$$441$$ 0 0
$$442$$ −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
$$448$$ 0 0
$$449$$ −11602.0 −1.21945 −0.609723 0.792615i $$-0.708719\pi$$
−0.609723 + 0.792615i $$0.708719\pi$$
$$450$$ −3424.33 −0.358721
$$451$$ 2653.81 0.277080
$$452$$ −5494.95 −0.571816
$$453$$ 1710.21 0.177379
$$454$$ 7611.79 0.786870
$$455$$ 0 0
$$456$$ 3684.62 0.378395
$$457$$ −3530.68 −0.361397 −0.180698 0.983539i $$-0.557836\pi$$
−0.180698 + 0.983539i $$0.557836\pi$$
$$458$$ 17910.4 1.82729
$$459$$ 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 $$-0.677631\pi$$
−0.529529 + 0.848292i $$0.677631\pi$$
$$468$$ 455.918 0.0450317
$$469$$ 0 0
$$470$$ 1998.74 0.196159
$$471$$ −6218.02 −0.608304
$$472$$ 3058.77 0.298287
$$473$$ −3080.18 −0.299422
$$474$$ 3313.76 0.321110
$$475$$ −8879.53 −0.857728
$$476$$ 0 0
$$477$$ −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
$$483$$ 0 0
$$484$$ 408.046 0.0383214
$$485$$ −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
$$490$$ 0 0
$$491$$ 7361.40 0.676609 0.338305 0.941037i $$-0.390147\pi$$
0.338305 + 0.941037i $$0.390147\pi$$
$$492$$ −2440.74 −0.223653
$$493$$ −17789.1 −1.62511
$$494$$ 3986.80 0.363107
$$495$$ 345.423 0.0313649
$$496$$ 22187.6 2.00857
$$497$$ 0 0
$$498$$ −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 $$0.822627\pi$$
$$504$$ 0 0
$$505$$ 3544.67 0.312348
$$506$$ 4154.65 0.365013
$$507$$ 5914.04 0.518051
$$508$$ 7726.76 0.674841
$$509$$ −16073.2 −1.39967 −0.699836 0.714303i $$-0.746744\pi$$
−0.699836 + 0.714303i $$0.746744\pi$$
$$510$$ 2582.77 0.224249
$$511$$ 0 0
$$512$$ 1493.27 0.128894
$$513$$ −2124.93 −0.182881
$$514$$ −27417.5 −2.35279
$$515$$ −3845.58 −0.329042
$$516$$ 2832.88 0.241687
$$517$$ 1868.56 0.158954
$$518$$ 0 0
$$519$$ 2421.44 0.204797
$$520$$ −817.952 −0.0689799
$$521$$ 18955.3 1.59395 0.796975 0.604012i $$-0.206432\pi$$
0.796975 + 0.604012i $$0.206432\pi$$
$$522$$ 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
$$525$$ 0 0
$$526$$ −15341.9 −1.27175
$$527$$ 20393.3 1.68567
$$528$$ 2627.00 0.216525
$$529$$ 377.000 0.0309855
$$530$$ −4820.09 −0.395041
$$531$$ −1764.00 −0.144164
$$532$$ 0 0
$$533$$ 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 $$-0.472381\pi$$
0.0866580 + 0.996238i $$0.472381\pi$$
$$542$$ −26004.4 −2.06086
$$543$$ 3184.82 0.251701
$$544$$ 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
$$549$$ 6312.23 0.490709
$$550$$ −4185.29 −0.324475
$$551$$ 19134.2 1.47939
$$552$$ 5243.61 0.404316
$$553$$ 0 0
$$554$$ 3802.32 0.291598
$$555$$ −1071.65 −0.0819625
$$556$$ 1639.90 0.125085
$$557$$ −25181.9 −1.91561 −0.957804 0.287423i $$-0.907201\pi$$
−0.957804 + 0.287423i $$0.907201\pi$$
$$558$$ −8459.22 −0.641769
$$559$$ −4206.33 −0.318263
$$560$$ 0 0
$$561$$ 2414.56 0.181716
$$562$$ −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
$$567$$ 0 0
$$568$$ −11805.9 −0.872122
$$569$$ −13447.0 −0.990732 −0.495366 0.868684i $$-0.664966\pi$$
−0.495366 + 0.868684i $$0.664966\pi$$
$$570$$ −2778.06 −0.204141
$$571$$ −2605.52 −0.190959 −0.0954795 0.995431i $$-0.530438\pi$$
−0.0954795 + 0.995431i $$0.530438\pi$$
$$572$$ 557.233 0.0407327
$$573$$ −6409.24 −0.467277
$$574$$ 0 0
$$575$$ −12636.5 −0.916485
$$576$$ 1373.11 0.0993281
$$577$$ −6339.65 −0.457406 −0.228703 0.973496i $$-0.573448\pi$$
−0.228703 + 0.973496i $$0.573448\pi$$
$$578$$ 1485.90 0.106929
$$579$$ −11843.3 −0.850069
$$580$$ 2860.68 0.204799
$$581$$ 0 0
$$582$$ 6215.37 0.442672
$$583$$ −4506.17 −0.320114
$$584$$ −15915.2 −1.12770
$$585$$ 471.715 0.0333385
$$586$$ 11210.1 0.790247
$$587$$ 13370.6 0.940140 0.470070 0.882629i $$-0.344229\pi$$
0.470070 + 0.882629i $$0.344229\pi$$
$$588$$ 0 0
$$589$$ −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 $$0.665127\pi$$
$$594$$ −1001.57 −0.0691832
$$595$$ 0 0
$$596$$ 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 $$0.802372\pi$$
$$602$$ 0 0
$$603$$ 8105.28 0.547384
$$604$$ −1922.44 −0.129508
$$605$$ 422.184 0.0283706
$$606$$ −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
$$609$$ 0 0
$$610$$ 8252.40 0.547754
$$611$$ 2551.74 0.168956
$$612$$ −2220.70 −0.146677
$$613$$ 1228.07 0.0809159 0.0404579 0.999181i $$-0.487118\pi$$
0.0404579 + 0.999181i $$0.487118\pi$$
$$614$$ 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 $$-0.431771\pi$$
0.212709 + 0.977115i $$0.431771\pi$$
$$620$$ −3279.47 −0.212430
$$621$$ −3024.00 −0.195409
$$622$$ −23365.9 −1.50625
$$623$$ 0 0
$$624$$ 3587.46 0.230150
$$625$$ 11208.0 0.717309
$$626$$ 12007.8 0.766661
$$627$$ −2597.14 −0.165422
$$628$$ 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
$$633$$ 14754.7 0.926458
$$634$$ 1122.13 0.0702923
$$635$$ 7994.48 0.499608
$$636$$ 4144.39 0.258389
$$637$$ 0 0
$$638$$ 9018.74 0.559648
$$639$$ 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 $$-0.328432\pi$$
0.513274 + 0.858225i $$0.328432\pi$$
$$644$$ 0 0
$$645$$ 2931.03 0.178929
$$646$$ −19419.1 −1.18271
$$647$$ −7818.70 −0.475092 −0.237546 0.971376i $$-0.576343\pi$$
−0.237546 + 0.971376i $$0.576343\pi$$
$$648$$ −1264.08 −0.0766325
$$649$$ −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 $$-0.298450\pi$$
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
$$658$$ 0 0
$$659$$ 7867.72 0.465072 0.232536 0.972588i $$-0.425298\pi$$
0.232536 + 0.972588i $$0.425298\pi$$
$$660$$ −388.288 −0.0229002
$$661$$ −4227.41 −0.248755 −0.124378 0.992235i $$-0.539693\pi$$
−0.124378 + 0.992235i $$0.539693\pi$$
$$662$$ −1825.90 −0.107199
$$663$$ 3297.35 0.193150
$$664$$ −11807.8 −0.690106
$$665$$ 0 0
$$666$$ 3107.30 0.180789
$$667$$ 27230.0 1.58073
$$668$$ 4006.67 0.232070
$$669$$ 6300.88 0.364135
$$670$$ 10596.6 0.611018
$$671$$ 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
$$675$$ 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
$$679$$ 0 0
$$680$$ 3984.11 0.224682
$$681$$ −6771.49 −0.381034
$$682$$ −10339.0 −0.580502
$$683$$ 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
$$686$$ 0 0
$$687$$ −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 $$-0.350360\pi$$
0.452983 + 0.891519i $$0.350360\pi$$
$$692$$ −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
$$700$$ 0 0
$$701$$ 27238.1 1.46758 0.733788 0.679379i $$-0.237751\pi$$
0.733788 + 0.679379i $$0.237751\pi$$
$$702$$ −1367.75 −0.0735364
$$703$$ 8057.44 0.432279
$$704$$ 1678.25 0.0898457
$$705$$ −1778.09 −0.0949882
$$706$$ 8934.12 0.476261
$$707$$ 0 0
$$708$$ 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
$$714$$ 0 0
$$715$$ 576.540 0.0301558
$$716$$ −6577.27 −0.343302
$$717$$ −4288.21 −0.223356
$$718$$ −10804.5 −0.561588
$$719$$ 27272.0 1.41456 0.707282 0.706931i $$-0.249921\pi$$
0.707282 + 0.706931i $$0.249921\pi$$
$$720$$ −2499.80 −0.129392
$$721$$ 0 0
$$722$$ −2243.05 −0.115620
$$723$$ −2936.97 −0.151075
$$724$$ −3580.04 −0.183772
$$725$$ −27430.8 −1.40518
$$726$$ −1224.14 −0.0625785
$$727$$ −3979.75 −0.203027 −0.101514 0.994834i $$-0.532369\pi$$
−0.101514 + 0.994834i $$0.532369\pi$$
$$728$$ 0 0
$$729$$ 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
$$735$$ 0 0
$$736$$ −16083.9 −0.805515
$$737$$ 9906.45 0.495127
$$738$$ 7322.23 0.365224
$$739$$ −28928.0 −1.43997 −0.719983 0.693992i $$-0.755850\pi$$
−0.719983 + 0.693992i $$0.755850\pi$$
$$740$$ 1204.64 0.0598425
$$741$$ −3546.68 −0.175831
$$742$$ 0 0
$$743$$ 4857.04 0.239822 0.119911 0.992785i $$-0.461739\pi$$
0.119911 + 0.992785i $$0.461739\pi$$
$$744$$ −13049.0 −0.643008
$$745$$ 8213.80 0.403933
$$746$$ −31660.8 −1.55386
$$747$$ 6809.57 0.333533
$$748$$ −2714.19 −0.132675
$$749$$ 0 0
$$750$$ 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
$$756$$ 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
$$759$$ −3696.00 −0.176754
$$760$$ −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
$$763$$ 0 0
$$764$$ 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 $$-0.355309\pi$$
0.439068 + 0.898454i $$0.355309\pi$$
$$774$$ −8498.64 −0.394674
$$775$$ 31446.6 1.45754
$$776$$ 9587.65 0.443527
$$777$$ 0 0
$$778$$ −3176.29 −0.146369
$$779$$ 18987.1 0.873276
$$780$$ −530.252 −0.0243411
$$781$$ 8321.50 0.381263
$$782$$ −27635.4 −1.26373
$$783$$ −6564.37 −0.299606
$$784$$ 0 0
$$785$$ 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
$$791$$ 0 0
$$792$$ −1544.99 −0.0693167
$$793$$ 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 $$-0.723917\pi$$
−0.646856 + 0.762612i $$0.723917\pi$$
$$798$$ 0 0
$$799$$ −12429.1 −0.550325
$$800$$ 16202.5 0.716056
$$801$$ −4580.80 −0.202066
$$802$$ 41154.0 1.81197
$$803$$ 11218.0 0.492993
$$804$$ −9111.10 −0.399656
$$805$$ 0 0
$$806$$ −14119.1 −0.617029
$$807$$ −87.4567 −0.00381490
$$808$$ −15854.4 −0.690291
$$809$$ −3000.83 −0.130413 −0.0652063 0.997872i $$-0.520771\pi$$
−0.0652063 + 0.997872i $$0.520771\pi$$
$$810$$ 953.071 0.0413426
$$811$$ −6239.39 −0.270154 −0.135077 0.990835i $$-0.543128\pi$$
−0.135077 + 0.990835i $$0.543128\pi$$
$$812$$ 0 0
$$813$$ 23133.7 0.997950
$$814$$ 3797.81 0.163530
$$815$$ 9338.68 0.401374
$$816$$ −17473.9 −0.749645
$$817$$ −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 $$0.397267\pi$$
$$822$$ −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
$$825$$ 3723.26 0.157124
$$826$$ 0 0
$$827$$ 27043.4 1.13711 0.568555 0.822645i $$-0.307503\pi$$
0.568555 + 0.822645i $$0.307503\pi$$
$$828$$ 3399.26 0.142672
$$829$$ 9795.41 0.410384 0.205192 0.978722i $$-0.434218\pi$$
0.205192 + 0.978722i $$0.434218\pi$$
$$830$$ 8902.62 0.372306
$$831$$ −3382.56 −0.141203
$$832$$ 2291.84 0.0954990
$$833$$ 0 0
$$834$$ −4919.70 −0.204263
$$835$$ 4145.50 0.171809
$$836$$ 2919.42 0.120778
$$837$$ 7525.37 0.310770
$$838$$ 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
$$843$$ 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 $$0.104650\pi$$
$$854$$ 0 0
$$855$$ 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 $$0.531908\pi$$
$$858$$ −1671.70 −0.0665162
$$859$$ 22921.1 0.910428 0.455214 0.890382i $$-0.349563\pi$$
0.455214 + 0.890382i $$0.349563\pi$$
$$860$$ −3294.76 −0.130640
$$861$$ 0 0
$$862$$ −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
$$868$$ 0 0
$$869$$ −3603.04 −0.140650
$$870$$ −8582.05 −0.334435
$$871$$ 13528.4 0.526282
$$872$$ −4997.30 −0.194071
$$873$$ −5529.22 −0.214360
$$874$$ 29725.0 1.15042
$$875$$ 0 0
$$876$$ −10317.3 −0.397933
$$877$$ −8455.67 −0.325573 −0.162787 0.986661i $$-0.552048\pi$$
−0.162787 + 0.986661i $$0.552048\pi$$
$$878$$ −343.834 −0.0132162
$$879$$ −9972.56 −0.382669
$$880$$ −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
$$885$$ 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
$$889$$ 0 0
$$890$$ −5988.80 −0.225556
$$891$$ 891.000 0.0335013
$$892$$ −7082.78 −0.265862
$$893$$ 13368.9 0.500978
$$894$$ −23816.2 −0.890976
$$895$$ −6805.16 −0.254158
$$896$$ 0 0
$$897$$ −5047.31 −0.187876
$$898$$ −39125.1 −1.45392
$$899$$ −67763.1 −2.51393
$$900$$ −3424.33 −0.126827
$$901$$ 29973.6 1.10829
$$902$$ 8949.39 0.330357
$$903$$ 0 0
$$904$$ 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
$$910$$ 0 0
$$911$$ −41249.2 −1.50016 −0.750080 0.661347i $$-0.769985\pi$$
−0.750080 + 0.661347i $$0.769985\pi$$
$$912$$ 18795.2 0.682425
$$913$$ 8322.81 0.301692
$$914$$ −11906.5 −0.430887
$$915$$ −7341.38 −0.265244
$$916$$ 17910.4 0.646045
$$917$$ 0 0
$$918$$ 6662.10 0.239523
$$919$$ −13858.1 −0.497429 −0.248714 0.968577i $$-0.580008\pi$$
−0.248714 + 0.968577i $$0.580008\pi$$
$$920$$ −6098.53 −0.218546
$$921$$ −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 $$-0.620281\pi$$
−0.368945 + 0.929451i $$0.620281\pi$$
$$930$$ 9838.42 0.346897
$$931$$ 0 0
$$932$$ 8316.94 0.292307
$$933$$ 20786.5 0.729388
$$934$$ −36042.9 −1.26270
$$935$$ −2808.23 −0.0982236
$$936$$ −2109.86 −0.0736784
$$937$$ −3203.52 −0.111691 −0.0558454 0.998439i $$-0.517785\pi$$
−0.0558454 + 0.998439i $$0.517785\pi$$
$$938$$ 0 0
$$939$$ −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
$$945$$ 0 0
$$946$$ −10387.2 −0.356996
$$947$$ −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
$$952$$ 0 0
$$953$$ −47661.4 −1.62004 −0.810022 0.586399i $$-0.800545\pi$$
−0.810022 + 0.586399i $$0.800545\pi$$
$$954$$ −12433.2 −0.421948
$$955$$ 7454.21 0.252579
$$956$$ 4820.35 0.163077
$$957$$ −8023.12 −0.271004
$$958$$ −7897.56 −0.266345
$$959$$ 0 0
$$960$$ −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
$$966$$ 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
$$969$$ 17275.3 0.572717
$$970$$ −7228.73 −0.239279
$$971$$ 40660.3 1.34382 0.671911 0.740632i $$-0.265474\pi$$
0.671911 + 0.740632i $$0.265474\pi$$
$$972$$ −819.464 −0.0270415
$$973$$ 0 0
$$974$$ 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
$$979$$ −5598.76 −0.182775
$$980$$ 0 0
$$981$$ 2881.96 0.0937959
$$982$$ 24824.7 0.806709
$$983$$ 4435.20 0.143907 0.0719536 0.997408i $$-0.477077\pi$$
0.0719536 + 0.997408i $$0.477077\pi$$
$$984$$ 11295.1 0.365929
$$985$$ 3223.55 0.104275
$$986$$ −59989.8 −1.93759
$$987$$ 0 0
$$988$$ 3986.80 0.128378
$$989$$ −31361.8 −1.00834
$$990$$ 1164.86 0.0373958
$$991$$ 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
$$994$$ 0 0
$$995$$ 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 1617.4.a.j.1.2 2
7.6 odd 2 33.4.a.d.1.2 2
21.20 even 2 99.4.a.e.1.1 2
28.27 even 2 528.4.a.o.1.1 2
35.13 even 4 825.4.c.i.199.1 4
35.27 even 4 825.4.c.i.199.4 4
35.34 odd 2 825.4.a.k.1.1 2
56.13 odd 2 2112.4.a.ba.1.2 2
56.27 even 2 2112.4.a.bh.1.2 2
77.76 even 2 363.4.a.j.1.1 2
84.83 odd 2 1584.4.a.x.1.2 2
105.104 even 2 2475.4.a.o.1.2 2
231.230 odd 2 1089.4.a.t.1.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
33.4.a.d.1.2 2 7.6 odd 2
99.4.a.e.1.1 2 21.20 even 2
363.4.a.j.1.1 2 77.76 even 2
528.4.a.o.1.1 2 28.27 even 2
825.4.a.k.1.1 2 35.34 odd 2
825.4.c.i.199.1 4 35.13 even 4
825.4.c.i.199.4 4 35.27 even 4
1089.4.a.t.1.2 2 231.230 odd 2
1584.4.a.x.1.2 2 84.83 odd 2
1617.4.a.j.1.2 2 1.1 even 1 trivial
2112.4.a.ba.1.2 2 56.13 odd 2
2112.4.a.bh.1.2 2 56.27 even 2
2475.4.a.o.1.2 2 105.104 even 2