# Properties

 Label 528.4.a.o.1.1 Level $528$ Weight $4$ Character 528.1 Self dual yes Analytic conductor $31.153$ 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] = [528,4,Mod(1,528)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(528, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("528.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$528 = 2^{4} \cdot 3 \cdot 11$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 528.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: $$31.1530084830$$ 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: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 33) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$-2.37228$$ of defining polynomial Character $$\chi$$ $$=$$ 528.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-3.00000 q^{3} -3.48913 q^{5} +4.74456 q^{7} +9.00000 q^{9} +O(q^{10})$$ $$q-3.00000 q^{3} -3.48913 q^{5} +4.74456 q^{7} +9.00000 q^{9} -11.0000 q^{11} -15.0217 q^{13} +10.4674 q^{15} +73.1684 q^{17} +78.7011 q^{19} -14.2337 q^{21} -112.000 q^{23} -112.826 q^{25} -27.0000 q^{27} +243.125 q^{29} -278.717 q^{31} +33.0000 q^{33} -16.5544 q^{35} +102.380 q^{37} +45.0652 q^{39} -241.255 q^{41} +280.016 q^{43} -31.4021 q^{45} +169.870 q^{47} -320.489 q^{49} -219.505 q^{51} -409.652 q^{53} +38.3804 q^{55} -236.103 q^{57} -196.000 q^{59} -701.359 q^{61} +42.7011 q^{63} +52.4128 q^{65} -900.587 q^{67} +336.000 q^{69} -756.500 q^{71} -1019.81 q^{73} +338.478 q^{75} -52.1902 q^{77} +327.549 q^{79} +81.0000 q^{81} +756.619 q^{83} -255.294 q^{85} -729.375 q^{87} +508.978 q^{89} -71.2716 q^{91} +836.152 q^{93} -274.598 q^{95} +614.358 q^{97} -99.0000 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 6 q^{3} + 16 q^{5} - 2 q^{7} + 18 q^{9}+O(q^{10})$$ 2 * q - 6 * q^3 + 16 * q^5 - 2 * q^7 + 18 * q^9 $$2 q - 6 q^{3} + 16 q^{5} - 2 q^{7} + 18 q^{9} - 22 q^{11} - 76 q^{13} - 48 q^{15} - 26 q^{17} + 54 q^{19} + 6 q^{21} - 224 q^{23} + 142 q^{25} - 54 q^{27} + 222 q^{29} + 40 q^{31} + 66 q^{33} - 148 q^{35} - 48 q^{37} + 228 q^{39} - 494 q^{41} + 66 q^{43} + 144 q^{45} + 64 q^{47} - 618 q^{49} + 78 q^{51} - 84 q^{53} - 176 q^{55} - 162 q^{57} - 392 q^{59} - 1104 q^{61} - 18 q^{63} - 1136 q^{65} - 928 q^{67} + 672 q^{69} - 456 q^{71} - 592 q^{73} - 426 q^{75} + 22 q^{77} + 230 q^{79} + 162 q^{81} - 348 q^{83} - 2188 q^{85} - 666 q^{87} + 972 q^{89} + 340 q^{91} - 120 q^{93} - 756 q^{95} - 1184 q^{97} - 198 q^{99}+O(q^{100})$$ 2 * q - 6 * q^3 + 16 * q^5 - 2 * q^7 + 18 * q^9 - 22 * q^11 - 76 * q^13 - 48 * q^15 - 26 * q^17 + 54 * q^19 + 6 * q^21 - 224 * q^23 + 142 * q^25 - 54 * q^27 + 222 * q^29 + 40 * q^31 + 66 * q^33 - 148 * q^35 - 48 * q^37 + 228 * q^39 - 494 * q^41 + 66 * q^43 + 144 * q^45 + 64 * q^47 - 618 * q^49 + 78 * q^51 - 84 * q^53 - 176 * q^55 - 162 * q^57 - 392 * q^59 - 1104 * q^61 - 18 * q^63 - 1136 * q^65 - 928 * q^67 + 672 * q^69 - 456 * q^71 - 592 * q^73 - 426 * q^75 + 22 * q^77 + 230 * q^79 + 162 * q^81 - 348 * q^83 - 2188 * q^85 - 666 * q^87 + 972 * q^89 + 340 * q^91 - 120 * q^93 - 756 * q^95 - 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0 0
$$3$$ −3.00000 −0.577350
$$4$$ 0 0
$$5$$ −3.48913 −0.312077 −0.156038 0.987751i $$-0.549872\pi$$
−0.156038 + 0.987751i $$0.549872\pi$$
$$6$$ 0 0
$$7$$ 4.74456 0.256182 0.128091 0.991762i $$-0.459115\pi$$
0.128091 + 0.991762i $$0.459115\pi$$
$$8$$ 0 0
$$9$$ 9.00000 0.333333
$$10$$ 0 0
$$11$$ −11.0000 −0.301511
$$12$$ 0 0
$$13$$ −15.0217 −0.320483 −0.160242 0.987078i $$-0.551227\pi$$
−0.160242 + 0.987078i $$0.551227\pi$$
$$14$$ 0 0
$$15$$ 10.4674 0.180178
$$16$$ 0 0
$$17$$ 73.1684 1.04388 0.521940 0.852982i $$-0.325209\pi$$
0.521940 + 0.852982i $$0.325209\pi$$
$$18$$ 0 0
$$19$$ 78.7011 0.950277 0.475138 0.879911i $$-0.342398\pi$$
0.475138 + 0.879911i $$0.342398\pi$$
$$20$$ 0 0
$$21$$ −14.2337 −0.147907
$$22$$ 0 0
$$23$$ −112.000 −1.01537 −0.507687 0.861541i $$-0.669499\pi$$
−0.507687 + 0.861541i $$0.669499\pi$$
$$24$$ 0 0
$$25$$ −112.826 −0.902608
$$26$$ 0 0
$$27$$ −27.0000 −0.192450
$$28$$ 0 0
$$29$$ 243.125 1.55680 0.778399 0.627769i $$-0.216032\pi$$
0.778399 + 0.627769i $$0.216032\pi$$
$$30$$ 0 0
$$31$$ −278.717 −1.61481 −0.807405 0.589998i $$-0.799129\pi$$
−0.807405 + 0.589998i $$0.799129\pi$$
$$32$$ 0 0
$$33$$ 33.0000 0.174078
$$34$$ 0 0
$$35$$ −16.5544 −0.0799486
$$36$$ 0 0
$$37$$ 102.380 0.454898 0.227449 0.973790i $$-0.426961\pi$$
0.227449 + 0.973790i $$0.426961\pi$$
$$38$$ 0 0
$$39$$ 45.0652 0.185031
$$40$$ 0 0
$$41$$ −241.255 −0.918970 −0.459485 0.888186i $$-0.651966\pi$$
−0.459485 + 0.888186i $$0.651966\pi$$
$$42$$ 0 0
$$43$$ 280.016 0.993071 0.496536 0.868016i $$-0.334605\pi$$
0.496536 + 0.868016i $$0.334605\pi$$
$$44$$ 0 0
$$45$$ −31.4021 −0.104026
$$46$$ 0 0
$$47$$ 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
$$50$$ 0 0
$$51$$ −219.505 −0.602684
$$52$$ 0 0
$$53$$ −409.652 −1.06170 −0.530849 0.847466i $$-0.678127\pi$$
−0.530849 + 0.847466i $$0.678127\pi$$
$$54$$ 0 0
$$55$$ 38.3804 0.0940947
$$56$$ 0 0
$$57$$ −236.103 −0.548643
$$58$$ 0 0
$$59$$ −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 $$0.763318\pi$$
$$62$$ 0 0
$$63$$ 42.7011 0.0853941
$$64$$ 0 0
$$65$$ 52.4128 0.100015
$$66$$ 0 0
$$67$$ −900.587 −1.64215 −0.821076 0.570819i $$-0.806626\pi$$
−0.821076 + 0.570819i $$0.806626\pi$$
$$68$$ 0 0
$$69$$ 336.000 0.586227
$$70$$ 0 0
$$71$$ −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 $$0.804661\pi$$
$$74$$ 0 0
$$75$$ 338.478 0.521121
$$76$$ 0 0
$$77$$ −52.1902 −0.0772419
$$78$$ 0 0
$$79$$ 327.549 0.466483 0.233241 0.972419i $$-0.425067\pi$$
0.233241 + 0.972419i $$0.425067\pi$$
$$80$$ 0 0
$$81$$ 81.0000 0.111111
$$82$$ 0 0
$$83$$ 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
$$86$$ 0 0
$$87$$ −729.375 −0.898818
$$88$$ 0 0
$$89$$ 508.978 0.606198 0.303099 0.952959i $$-0.401979\pi$$
0.303099 + 0.952959i $$0.401979\pi$$
$$90$$ 0 0
$$91$$ −71.2716 −0.0821022
$$92$$ 0 0
$$93$$ 836.152 0.932311
$$94$$ 0 0
$$95$$ −274.598 −0.296559
$$96$$ 0 0
$$97$$ 614.358 0.643079 0.321539 0.946896i $$-0.395800\pi$$
0.321539 + 0.946896i $$0.395800\pi$$
$$98$$ 0 0
$$99$$ −99.0000 −0.100504
$$100$$ 0 0
$$101$$ −1015.92 −1.00087 −0.500434 0.865775i $$-0.666826\pi$$
−0.500434 + 0.865775i $$0.666826\pi$$
$$102$$ 0 0
$$103$$ −1102.16 −1.05436 −0.527181 0.849753i $$-0.676751\pi$$
−0.527181 + 0.849753i $$0.676751\pi$$
$$104$$ 0 0
$$105$$ 49.6631 0.0461583
$$106$$ 0 0
$$107$$ −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 $$0.455067\pi$$
$$110$$ 0 0
$$111$$ −307.141 −0.262636
$$112$$ 0 0
$$113$$ −1629.45 −1.35651 −0.678254 0.734828i $$-0.737263\pi$$
−0.678254 + 0.734828i $$0.737263\pi$$
$$114$$ 0 0
$$115$$ 390.782 0.316875
$$116$$ 0 0
$$117$$ −135.196 −0.106828
$$118$$ 0 0
$$119$$ 347.152 0.267423
$$120$$ 0 0
$$121$$ 121.000 0.0909091
$$122$$ 0 0
$$123$$ 723.766 0.530568
$$124$$ 0 0
$$125$$ 829.805 0.593760
$$126$$ 0 0
$$127$$ −2291.26 −1.60091 −0.800457 0.599390i $$-0.795410\pi$$
−0.800457 + 0.599390i $$0.795410\pi$$
$$128$$ 0 0
$$129$$ −840.049 −0.573350
$$130$$ 0 0
$$131$$ 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
$$134$$ 0 0
$$135$$ 94.2064 0.0600592
$$136$$ 0 0
$$137$$ 1268.60 0.791121 0.395561 0.918440i $$-0.370550\pi$$
0.395561 + 0.918440i $$0.370550\pi$$
$$138$$ 0 0
$$139$$ 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
$$142$$ 0 0
$$143$$ 165.239 0.0966294
$$144$$ 0 0
$$145$$ −848.293 −0.485841
$$146$$ 0 0
$$147$$ 961.467 0.539459
$$148$$ 0 0
$$149$$ 2354.11 1.29434 0.647169 0.762346i $$-0.275953\pi$$
0.647169 + 0.762346i $$0.275953\pi$$
$$150$$ 0 0
$$151$$ 570.070 0.307229 0.153615 0.988131i $$-0.450909\pi$$
0.153615 + 0.988131i $$0.450909\pi$$
$$152$$ 0 0
$$153$$ 658.516 0.347960
$$154$$ 0 0
$$155$$ 972.479 0.503945
$$156$$ 0 0
$$157$$ −2072.67 −1.05361 −0.526807 0.849985i $$-0.676611\pi$$
−0.526807 + 0.849985i $$0.676611\pi$$
$$158$$ 0 0
$$159$$ 1228.96 0.612972
$$160$$ 0 0
$$161$$ −531.391 −0.260121
$$162$$ 0 0
$$163$$ −2676.51 −1.28614 −0.643069 0.765808i $$-0.722339\pi$$
−0.643069 + 0.765808i $$0.722339\pi$$
$$164$$ 0 0
$$165$$ −115.141 −0.0543256
$$166$$ 0 0
$$167$$ 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
$$170$$ 0 0
$$171$$ 708.310 0.316759
$$172$$ 0 0
$$173$$ 807.147 0.354718 0.177359 0.984146i $$-0.443245\pi$$
0.177359 + 0.984146i $$0.443245\pi$$
$$174$$ 0 0
$$175$$ −535.310 −0.231232
$$176$$ 0 0
$$177$$ 588.000 0.249699
$$178$$ 0 0
$$179$$ 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 $$0.430053\pi$$
$$182$$ 0 0
$$183$$ 2104.08 0.849933
$$184$$ 0 0
$$185$$ −357.218 −0.141963
$$186$$ 0 0
$$187$$ −804.853 −0.314742
$$188$$ 0 0
$$189$$ −128.103 −0.0493023
$$190$$ 0 0
$$191$$ −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 $$0.236627\pi$$
$$194$$ 0 0
$$195$$ −157.238 −0.0577439
$$196$$ 0 0
$$197$$ 923.886 0.334133 0.167066 0.985946i $$-0.446571\pi$$
0.167066 + 0.985946i $$0.446571\pi$$
$$198$$ 0 0
$$199$$ 476.152 0.169616 0.0848078 0.996397i $$-0.472972\pi$$
0.0848078 + 0.996397i $$0.472972\pi$$
$$200$$ 0 0
$$201$$ 2701.76 0.948097
$$202$$ 0 0
$$203$$ 1153.52 0.398824
$$204$$ 0 0
$$205$$ 841.770 0.286789
$$206$$ 0 0
$$207$$ −1008.00 −0.338458
$$208$$ 0 0
$$209$$ −865.712 −0.286519
$$210$$ 0 0
$$211$$ 4918.24 1.60467 0.802336 0.596872i $$-0.203590\pi$$
0.802336 + 0.596872i $$0.203590\pi$$
$$212$$ 0 0
$$213$$ 2269.50 0.730064
$$214$$ 0 0
$$215$$ −977.012 −0.309915
$$216$$ 0 0
$$217$$ −1322.39 −0.413686
$$218$$ 0 0
$$219$$ 3059.44 0.944010
$$220$$ 0 0
$$221$$ −1099.12 −0.334546
$$222$$ 0 0
$$223$$ −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
$$226$$ 0 0
$$227$$ 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 $$0.777905\pi$$
$$230$$ 0 0
$$231$$ 156.571 0.0445956
$$232$$ 0 0
$$233$$ 2466.27 0.693435 0.346718 0.937970i $$-0.387296\pi$$
0.346718 + 0.937970i $$0.387296\pi$$
$$234$$ 0 0
$$235$$ −592.696 −0.164524
$$236$$ 0 0
$$237$$ −982.646 −0.269324
$$238$$ 0 0
$$239$$ −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 $$0.541766\pi$$
$$242$$ 0 0
$$243$$ −243.000 −0.0641500
$$244$$ 0 0
$$245$$ 1118.23 0.291595
$$246$$ 0 0
$$247$$ −1182.23 −0.304548
$$248$$ 0 0
$$249$$ −2269.86 −0.577696
$$250$$ 0 0
$$251$$ 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
$$254$$ 0 0
$$255$$ 765.882 0.188084
$$256$$ 0 0
$$257$$ 8130.26 1.97335 0.986676 0.162696i $$-0.0520188\pi$$
0.986676 + 0.162696i $$0.0520188\pi$$
$$258$$ 0 0
$$259$$ 485.750 0.116537
$$260$$ 0 0
$$261$$ 2188.12 0.518933
$$262$$ 0 0
$$263$$ 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
$$266$$ 0 0
$$267$$ −1526.93 −0.349988
$$268$$ 0 0
$$269$$ −29.1522 −0.00660760 −0.00330380 0.999995i $$-0.501052\pi$$
−0.00330380 + 0.999995i $$0.501052\pi$$
$$270$$ 0 0
$$271$$ −7711.22 −1.72850 −0.864250 0.503063i $$-0.832206\pi$$
−0.864250 + 0.503063i $$0.832206\pi$$
$$272$$ 0 0
$$273$$ 213.815 0.0474017
$$274$$ 0 0
$$275$$ 1241.09 0.272147
$$276$$ 0 0
$$277$$ 1127.52 0.244571 0.122286 0.992495i $$-0.460978\pi$$
0.122286 + 0.992495i $$0.460978\pi$$
$$278$$ 0 0
$$279$$ −2508.46 −0.538270
$$280$$ 0 0
$$281$$ −1872.47 −0.397517 −0.198758 0.980049i $$-0.563691\pi$$
−0.198758 + 0.980049i $$0.563691\pi$$
$$282$$ 0 0
$$283$$ −2124.48 −0.446245 −0.223123 0.974790i $$-0.571625\pi$$
−0.223123 + 0.974790i $$0.571625\pi$$
$$284$$ 0 0
$$285$$ 823.794 0.171219
$$286$$ 0 0
$$287$$ −1144.65 −0.235424
$$288$$ 0 0
$$289$$ 440.621 0.0896846
$$290$$ 0 0
$$291$$ −1843.07 −0.371282
$$292$$ 0 0
$$293$$ −3324.19 −0.662802 −0.331401 0.943490i $$-0.607521\pi$$
−0.331401 + 0.943490i $$0.607521\pi$$
$$294$$ 0 0
$$295$$ 683.869 0.134971
$$296$$ 0 0
$$297$$ 297.000 0.0580259
$$298$$ 0 0
$$299$$ 1682.44 0.325411
$$300$$ 0 0
$$301$$ 1328.55 0.254407
$$302$$ 0 0
$$303$$ 3047.75 0.577851
$$304$$ 0 0
$$305$$ 2447.13 0.459417
$$306$$ 0 0
$$307$$ 1698.94 0.315843 0.157921 0.987452i $$-0.449521\pi$$
0.157921 + 0.987452i $$0.449521\pi$$
$$308$$ 0 0
$$309$$ 3306.49 0.608736
$$310$$ 0 0
$$311$$ −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 $$0.604190\pi$$
$$314$$ 0 0
$$315$$ −148.989 −0.0266495
$$316$$ 0 0
$$317$$ 332.750 0.0589561 0.0294780 0.999565i $$-0.490615\pi$$
0.0294780 + 0.999565i $$0.490615\pi$$
$$318$$ 0 0
$$319$$ −2674.37 −0.469393
$$320$$ 0 0
$$321$$ 4132.73 0.718587
$$322$$ 0 0
$$323$$ 5758.43 0.991975
$$324$$ 0 0
$$325$$ 1694.84 0.289271
$$326$$ 0 0
$$327$$ −960.652 −0.162459
$$328$$ 0 0
$$329$$ 805.957 0.135057
$$330$$ 0 0
$$331$$ 541.445 0.0899108 0.0449554 0.998989i $$-0.485685\pi$$
0.0449554 + 0.998989i $$0.485685\pi$$
$$332$$ 0 0
$$333$$ 921.423 0.151633
$$334$$ 0 0
$$335$$ 3142.26 0.512478
$$336$$ 0 0
$$337$$ 816.531 0.131986 0.0659930 0.997820i $$-0.478978\pi$$
0.0659930 + 0.997820i $$0.478978\pi$$
$$338$$ 0 0
$$339$$ 4888.34 0.783180
$$340$$ 0 0
$$341$$ 3065.89 0.486883
$$342$$ 0 0
$$343$$ −3147.97 −0.495552
$$344$$ 0 0
$$345$$ −1172.35 −0.182948
$$346$$ 0 0
$$347$$ −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 $$0.934961\pi$$
$$350$$ 0 0
$$351$$ 405.587 0.0616771
$$352$$ 0 0
$$353$$ −2649.28 −0.399453 −0.199727 0.979852i $$-0.564005\pi$$
−0.199727 + 0.979852i $$0.564005\pi$$
$$354$$ 0 0
$$355$$ 2639.52 0.394623
$$356$$ 0 0
$$357$$ −1041.46 −0.154397
$$358$$ 0 0
$$359$$ 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
$$362$$ 0 0
$$363$$ −363.000 −0.0524864
$$364$$ 0 0
$$365$$ 3558.26 0.510268
$$366$$ 0 0
$$367$$ 8429.40 1.19894 0.599470 0.800397i $$-0.295378\pi$$
0.599470 + 0.800397i $$0.295378\pi$$
$$368$$ 0 0
$$369$$ −2171.30 −0.306323
$$370$$ 0 0
$$371$$ −1943.62 −0.271988
$$372$$ 0 0
$$373$$ −9388.53 −1.30327 −0.651635 0.758533i $$-0.725917\pi$$
−0.651635 + 0.758533i $$0.725917\pi$$
$$374$$ 0 0
$$375$$ −2489.41 −0.342807
$$376$$ 0 0
$$377$$ −3652.16 −0.498928
$$378$$ 0 0
$$379$$ 14264.5 1.93329 0.966647 0.256112i $$-0.0824415\pi$$
0.966647 + 0.256112i $$0.0824415\pi$$
$$380$$ 0 0
$$381$$ 6873.77 0.924288
$$382$$ 0 0
$$383$$ −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
$$386$$ 0 0
$$387$$ 2520.15 0.331024
$$388$$ 0 0
$$389$$ −941.881 −0.122764 −0.0613821 0.998114i $$-0.519551\pi$$
−0.0613821 + 0.998114i $$0.519551\pi$$
$$390$$ 0 0
$$391$$ −8194.87 −1.05993
$$392$$ 0 0
$$393$$ −3442.24 −0.441827
$$394$$ 0 0
$$395$$ −1142.86 −0.145578
$$396$$ 0 0
$$397$$ −847.839 −0.107183 −0.0535917 0.998563i $$-0.517067\pi$$
−0.0535917 + 0.998563i $$0.517067\pi$$
$$398$$ 0 0
$$399$$ −1120.21 −0.140553
$$400$$ 0 0
$$401$$ 12203.6 1.51975 0.759875 0.650069i $$-0.225260\pi$$
0.759875 + 0.650069i $$0.225260\pi$$
$$402$$ 0 0
$$403$$ 4186.82 0.517520
$$404$$ 0 0
$$405$$ −282.619 −0.0346752
$$406$$ 0 0
$$407$$ −1126.18 −0.137157
$$408$$ 0 0
$$409$$ 8759.53 1.05900 0.529500 0.848310i $$-0.322380\pi$$
0.529500 + 0.848310i $$0.322380\pi$$
$$410$$ 0 0
$$411$$ −3805.79 −0.456754
$$412$$ 0 0
$$413$$ −929.934 −0.110797
$$414$$ 0 0
$$415$$ −2639.94 −0.312264
$$416$$ 0 0
$$417$$ −1458.86 −0.171321
$$418$$ 0 0
$$419$$ 11188.4 1.30451 0.652256 0.757999i $$-0.273823\pi$$
0.652256 + 0.757999i $$0.273823\pi$$
$$420$$ 0 0
$$421$$ −14082.3 −1.63023 −0.815116 0.579298i $$-0.803327\pi$$
−0.815116 + 0.579298i $$0.803327\pi$$
$$422$$ 0 0
$$423$$ 1528.83 0.175731
$$424$$ 0 0
$$425$$ −8255.30 −0.942214
$$426$$ 0 0
$$427$$ −3327.64 −0.377133
$$428$$ 0 0
$$429$$ −495.718 −0.0557890
$$430$$ 0 0
$$431$$ 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 $$0.369248\pi$$
$$434$$ 0 0
$$435$$ 2544.88 0.280500
$$436$$ 0 0
$$437$$ −8814.52 −0.964887
$$438$$ 0 0
$$439$$ −101.959 −0.0110848 −0.00554240 0.999985i $$-0.501764\pi$$
−0.00554240 + 0.999985i $$0.501764\pi$$
$$440$$ 0 0
$$441$$ −2884.40 −0.311457
$$442$$ 0 0
$$443$$ −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
$$446$$ 0 0
$$447$$ −7062.34 −0.747287
$$448$$ 0 0
$$449$$ −11602.0 −1.21945 −0.609723 0.792615i $$-0.708719\pi$$
−0.609723 + 0.792615i $$0.708719\pi$$
$$450$$ 0 0
$$451$$ 2653.81 0.277080
$$452$$ 0 0
$$453$$ −1710.21 −0.177379
$$454$$ 0 0
$$455$$ 248.676 0.0256222
$$456$$ 0 0
$$457$$ −3530.68 −0.361397 −0.180698 0.983539i $$-0.557836\pi$$
−0.180698 + 0.983539i $$0.557836\pi$$
$$458$$ 0 0
$$459$$ −1975.55 −0.200895
$$460$$ 0 0
$$461$$ 11566.3 1.16854 0.584271 0.811559i $$-0.301381\pi$$
0.584271 + 0.811559i $$0.301381\pi$$
$$462$$ 0 0
$$463$$ −10888.5 −1.09294 −0.546470 0.837479i $$-0.684029\pi$$
−0.546470 + 0.837479i $$0.684029\pi$$
$$464$$ 0 0
$$465$$ −2917.44 −0.290953
$$466$$ 0 0
$$467$$ −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
$$470$$ 0 0
$$471$$ 6218.02 0.608304
$$472$$ 0 0
$$473$$ −3080.18 −0.299422
$$474$$ 0 0
$$475$$ −8879.53 −0.857728
$$476$$ 0 0
$$477$$ −3686.87 −0.353900
$$478$$ 0 0
$$479$$ −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
$$482$$ 0 0
$$483$$ 1594.17 0.150181
$$484$$ 0 0
$$485$$ −2143.57 −0.200690
$$486$$ 0 0
$$487$$ −6748.91 −0.627972 −0.313986 0.949428i $$-0.601665\pi$$
−0.313986 + 0.949428i $$0.601665\pi$$
$$488$$ 0 0
$$489$$ 8029.53 0.742552
$$490$$ 0 0
$$491$$ −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
$$494$$ 0 0
$$495$$ 345.423 0.0313649
$$496$$ 0 0
$$497$$ −3589.26 −0.323944
$$498$$ 0 0
$$499$$ −10381.7 −0.931359 −0.465680 0.884953i $$-0.654190\pi$$
−0.465680 + 0.884953i $$0.654190\pi$$
$$500$$ 0 0
$$501$$ −3564.36 −0.317852
$$502$$ 0 0
$$503$$ −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
$$506$$ 0 0
$$507$$ 5914.04 0.518051
$$508$$ 0 0
$$509$$ 16073.2 1.39967 0.699836 0.714303i $$-0.253256\pi$$
0.699836 + 0.714303i $$0.253256\pi$$
$$510$$ 0 0
$$511$$ −4838.58 −0.418877
$$512$$ 0 0
$$513$$ −2124.93 −0.182881
$$514$$ 0 0
$$515$$ 3845.58 0.329042
$$516$$ 0 0
$$517$$ −1868.56 −0.158954
$$518$$ 0 0
$$519$$ −2421.44 −0.204797
$$520$$ 0 0
$$521$$ −18955.3 −1.59395 −0.796975 0.604012i $$-0.793568\pi$$
−0.796975 + 0.604012i $$0.793568\pi$$
$$522$$ 0 0
$$523$$ 4442.19 0.371402 0.185701 0.982606i $$-0.440544\pi$$
0.185701 + 0.982606i $$0.440544\pi$$
$$524$$ 0 0
$$525$$ 1605.93 0.133502
$$526$$ 0 0
$$527$$ −20393.3 −1.68567
$$528$$ 0 0
$$529$$ 377.000 0.0309855
$$530$$ 0 0
$$531$$ −1764.00 −0.144164
$$532$$ 0 0
$$533$$ 3624.08 0.294515
$$534$$ 0 0
$$535$$ 4806.54 0.388420
$$536$$ 0 0
$$537$$ −5851.17 −0.470199
$$538$$ 0 0
$$539$$ 3525.38 0.281723
$$540$$ 0 0
$$541$$ 2180.90 0.173316 0.0866580 0.996238i $$-0.472381\pi$$
0.0866580 + 0.996238i $$0.472381\pi$$
$$542$$ 0 0
$$543$$ −3184.82 −0.251701
$$544$$ 0 0
$$545$$ −1117.28 −0.0878146
$$546$$ 0 0
$$547$$ −8225.04 −0.642920 −0.321460 0.946923i $$-0.604174\pi$$
−0.321460 + 0.946923i $$0.604174\pi$$
$$548$$ 0 0
$$549$$ −6312.23 −0.490709
$$550$$ 0 0
$$551$$ 19134.2 1.47939
$$552$$ 0 0
$$553$$ 1554.08 0.119505
$$554$$ 0 0
$$555$$ 1071.65 0.0819625
$$556$$ 0 0
$$557$$ −25181.9 −1.91561 −0.957804 0.287423i $$-0.907201\pi$$
−0.957804 + 0.287423i $$0.907201\pi$$
$$558$$ 0 0
$$559$$ −4206.33 −0.318263
$$560$$ 0 0
$$561$$ 2414.56 0.181716
$$562$$ 0 0
$$563$$ 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
$$566$$ 0 0
$$567$$ 384.310 0.0284647
$$568$$ 0 0
$$569$$ −13447.0 −0.990732 −0.495366 0.868684i $$-0.664966\pi$$
−0.495366 + 0.868684i $$0.664966\pi$$
$$570$$ 0 0
$$571$$ 2605.52 0.190959 0.0954795 0.995431i $$-0.469562\pi$$
0.0954795 + 0.995431i $$0.469562\pi$$
$$572$$ 0 0
$$573$$ 6409.24 0.467277
$$574$$ 0 0
$$575$$ 12636.5 0.916485
$$576$$ 0 0
$$577$$ 6339.65 0.457406 0.228703 0.973496i $$-0.426552\pi$$
0.228703 + 0.973496i $$0.426552\pi$$
$$578$$ 0 0
$$579$$ −11843.3 −0.850069
$$580$$ 0 0
$$581$$ 3589.83 0.256336
$$582$$ 0 0
$$583$$ 4506.17 0.320114
$$584$$ 0 0
$$585$$ 471.715 0.0333385
$$586$$ 0 0
$$587$$ 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
$$590$$ 0 0
$$591$$ −2771.66 −0.192912
$$592$$ 0 0
$$593$$ 14319.3 0.991608 0.495804 0.868434i $$-0.334873\pi$$
0.495804 + 0.868434i $$0.334873\pi$$
$$594$$ 0 0
$$595$$ −1211.26 −0.0834567
$$596$$ 0 0
$$597$$ −1428.46 −0.0979276
$$598$$ 0 0
$$599$$ 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 $$0.197628\pi$$
$$602$$ 0 0
$$603$$ −8105.28 −0.547384
$$604$$ 0 0
$$605$$ −422.184 −0.0283706
$$606$$ 0 0
$$607$$ 23526.6 1.57317 0.786585 0.617482i $$-0.211847\pi$$
0.786585 + 0.617482i $$0.211847\pi$$
$$608$$ 0 0
$$609$$ −3460.56 −0.230261
$$610$$ 0 0
$$611$$ −2551.74 −0.168956
$$612$$ 0 0
$$613$$ 1228.07 0.0809159 0.0404579 0.999181i $$-0.487118\pi$$
0.0404579 + 0.999181i $$0.487118\pi$$
$$614$$ 0 0
$$615$$ −2525.31 −0.165578
$$616$$ 0 0
$$617$$ −9844.90 −0.642368 −0.321184 0.947017i $$-0.604081\pi$$
−0.321184 + 0.947017i $$0.604081\pi$$
$$618$$ 0 0
$$619$$ 6551.68 0.425419 0.212709 0.977115i $$-0.431771\pi$$
0.212709 + 0.977115i $$0.431771\pi$$
$$620$$ 0 0
$$621$$ 3024.00 0.195409
$$622$$ 0 0
$$623$$ 2414.88 0.155297
$$624$$ 0 0
$$625$$ 11208.0 0.717309
$$626$$ 0 0
$$627$$ 2597.14 0.165422
$$628$$ 0 0
$$629$$ 7491.01 0.474859
$$630$$ 0 0
$$631$$ 26440.5 1.66812 0.834058 0.551677i $$-0.186012\pi$$
0.834058 + 0.551677i $$0.186012\pi$$
$$632$$ 0 0
$$633$$ −14754.7 −0.926458
$$634$$ 0 0
$$635$$ 7994.48 0.499608
$$636$$ 0 0
$$637$$ 4814.31 0.299450
$$638$$ 0 0
$$639$$ −6808.50 −0.421502
$$640$$ 0 0
$$641$$ −27927.2 −1.72084 −0.860421 0.509584i $$-0.829799\pi$$
−0.860421 + 0.509584i $$0.829799\pi$$
$$642$$ 0 0
$$643$$ 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
$$646$$ 0 0
$$647$$ −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
$$650$$ 0 0
$$651$$ 3967.17 0.238842
$$652$$ 0 0
$$653$$ 19747.6 1.18344 0.591719 0.806144i $$-0.298450\pi$$
0.591719 + 0.806144i $$0.298450\pi$$
$$654$$ 0 0
$$655$$ −4003.47 −0.238822
$$656$$ 0 0
$$657$$ −9178.33 −0.545024
$$658$$ 0 0
$$659$$ −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 $$0.460307\pi$$
$$662$$ 0 0
$$663$$ 3297.35 0.193150
$$664$$ 0 0
$$665$$ −1302.85 −0.0759733
$$666$$ 0 0
$$667$$ −27230.0 −1.58073
$$668$$ 0 0
$$669$$ 6300.88 0.364135
$$670$$ 0 0
$$671$$ 7714.94 0.443863
$$672$$ 0 0
$$673$$ 29397.6 1.68379 0.841897 0.539638i $$-0.181439\pi$$
0.841897 + 0.539638i $$0.181439\pi$$
$$674$$ 0 0
$$675$$ 3046.30 0.173707
$$676$$ 0 0
$$677$$ 5737.14 0.325696 0.162848 0.986651i $$-0.447932\pi$$
0.162848 + 0.986651i $$0.447932\pi$$
$$678$$ 0 0
$$679$$ 2914.86 0.164745
$$680$$ 0 0
$$681$$ −6771.49 −0.381034
$$682$$ 0 0
$$683$$ −32097.6 −1.79821 −0.899107 0.437729i $$-0.855783\pi$$
−0.899107 + 0.437729i $$0.855783\pi$$
$$684$$ 0 0
$$685$$ −4426.30 −0.246891
$$686$$ 0 0
$$687$$ 15933.2 0.884848
$$688$$ 0 0
$$689$$ 6153.69 0.340257
$$690$$ 0 0
$$691$$ 16456.2 0.905965 0.452983 0.891519i $$-0.350360\pi$$
0.452983 + 0.891519i $$0.350360\pi$$
$$692$$ 0 0
$$693$$ −469.712 −0.0257473
$$694$$ 0 0
$$695$$ −1696.72 −0.0926047
$$696$$ 0 0
$$697$$ −17652.3 −0.959294
$$698$$ 0 0
$$699$$ −7398.80 −0.400355
$$700$$ 0 0
$$701$$ 27238.1 1.46758 0.733788 0.679379i $$-0.237751\pi$$
0.733788 + 0.679379i $$0.237751\pi$$
$$702$$ 0 0
$$703$$ 8057.44 0.432279
$$704$$ 0 0
$$705$$ 1778.09 0.0949882
$$706$$ 0 0
$$707$$ −4820.09 −0.256405
$$708$$ 0 0
$$709$$ 28761.4 1.52349 0.761747 0.647875i $$-0.224342\pi$$
0.761747 + 0.647875i $$0.224342\pi$$
$$710$$ 0 0
$$711$$ 2947.94 0.155494
$$712$$ 0 0
$$713$$ 31216.3 1.63964
$$714$$ 0 0
$$715$$ −576.540 −0.0301558
$$716$$ 0 0
$$717$$ 4288.21 0.223356
$$718$$ 0 0
$$719$$ 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
$$722$$ 0 0
$$723$$ 2936.97 0.151075
$$724$$ 0 0
$$725$$ −27430.8 −1.40518
$$726$$ 0 0
$$727$$ −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
$$730$$ 0 0
$$731$$ 20488.3 1.03665
$$732$$ 0 0
$$733$$ 9342.48 0.470767 0.235384 0.971903i $$-0.424365\pi$$
0.235384 + 0.971903i $$0.424365\pi$$
$$734$$ 0 0
$$735$$ −3354.68 −0.168353
$$736$$ 0 0
$$737$$ 9906.45 0.495127
$$738$$ 0 0
$$739$$ 28928.0 1.43997 0.719983 0.693992i $$-0.244150\pi$$
0.719983 + 0.693992i $$0.244150\pi$$
$$740$$ 0 0
$$741$$ 3546.68 0.175831
$$742$$ 0 0
$$743$$ −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
$$746$$ 0 0
$$747$$ 6809.57 0.333533
$$748$$ 0 0
$$749$$ −6536.00 −0.318852
$$750$$ 0 0
$$751$$ −14355.4 −0.697517 −0.348759 0.937213i $$-0.613397\pi$$
−0.348759 + 0.937213i $$0.613397\pi$$
$$752$$ 0 0
$$753$$ −19591.9 −0.948165
$$754$$ 0 0
$$755$$ −1989.05 −0.0958792
$$756$$ 0 0
$$757$$ −17714.9 −0.850538 −0.425269 0.905067i $$-0.639821\pi$$
−0.425269 + 0.905067i $$0.639821\pi$$
$$758$$ 0 0
$$759$$ −3696.00 −0.176754
$$760$$ 0 0
$$761$$ −7945.82 −0.378497 −0.189248 0.981929i $$-0.560605\pi$$
−0.189248 + 0.981929i $$0.560605\pi$$
$$762$$ 0 0
$$763$$ 1519.29 0.0720866
$$764$$ 0 0
$$765$$ −2297.64 −0.108590
$$766$$ 0 0
$$767$$ 2944.26 0.138606
$$768$$ 0 0
$$769$$ 27308.1 1.28057 0.640284 0.768139i $$-0.278817\pi$$
0.640284 + 0.768139i $$0.278817\pi$$
$$770$$ 0 0
$$771$$ −24390.8 −1.13932
$$772$$ 0 0
$$773$$ −18872.6 −0.878136 −0.439068 0.898454i $$-0.644691\pi$$
−0.439068 + 0.898454i $$0.644691\pi$$
$$774$$ 0 0
$$775$$ 31446.6 1.45754
$$776$$ 0 0
$$777$$ −1457.25 −0.0672826
$$778$$ 0 0
$$779$$ −18987.1 −0.873276
$$780$$ 0 0
$$781$$ 8321.50 0.381263
$$782$$ 0 0
$$783$$ −6564.37 −0.299606
$$784$$ 0 0
$$785$$ 7231.82 0.328808
$$786$$ 0 0
$$787$$ 14512.1 0.657307 0.328654 0.944450i $$-0.393405\pi$$
0.328654 + 0.944450i $$0.393405\pi$$
$$788$$ 0 0
$$789$$ −13648.3 −0.615832
$$790$$ 0 0
$$791$$ −7731.01 −0.347513
$$792$$ 0 0
$$793$$ 10535.6 0.471792
$$794$$ 0 0
$$795$$ −4287.98 −0.191294
$$796$$ 0 0
$$797$$ 29108.9 1.29371 0.646856 0.762612i $$-0.276083\pi$$
0.646856 + 0.762612i $$0.276083\pi$$
$$798$$ 0 0
$$799$$ 12429.1 0.550325
$$800$$ 0 0
$$801$$ 4580.80 0.202066
$$802$$ 0 0
$$803$$ 11218.0 0.492993
$$804$$ 0 0
$$805$$ 1854.09 0.0811777
$$806$$ 0 0
$$807$$ 87.4567 0.00381490
$$808$$ 0 0
$$809$$ −3000.83 −0.130413 −0.0652063 0.997872i $$-0.520771\pi$$
−0.0652063 + 0.997872i $$0.520771\pi$$
$$810$$ 0 0
$$811$$ −6239.39 −0.270154 −0.135077 0.990835i $$-0.543128\pi$$
−0.135077 + 0.990835i $$0.543128\pi$$
$$812$$ 0 0
$$813$$ 23133.7 0.997950
$$814$$ 0 0
$$815$$ 9338.68 0.401374
$$816$$ 0 0
$$817$$ 22037.6 0.943693
$$818$$ 0 0
$$819$$ −641.445 −0.0273674
$$820$$ 0 0
$$821$$ 14922.4 0.634342 0.317171 0.948368i $$-0.397267\pi$$
0.317171 + 0.948368i $$0.397267\pi$$
$$822$$ 0 0
$$823$$ 25737.8 1.09011 0.545057 0.838399i $$-0.316508\pi$$
0.545057 + 0.838399i $$0.316508\pi$$
$$824$$ 0 0
$$825$$ −3723.26 −0.157124
$$826$$ 0 0
$$827$$ −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 $$0.565782\pi$$
$$830$$ 0 0
$$831$$ −3382.56 −0.141203
$$832$$ 0 0
$$833$$ −23449.7 −0.975370
$$834$$ 0 0
$$835$$ −4145.50 −0.171809
$$836$$ 0 0
$$837$$ 7525.37 0.310770
$$838$$ 0 0
$$839$$ −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
$$842$$ 0 0
$$843$$ 5617.41 0.229506
$$844$$ 0 0
$$845$$ 6878.28 0.280024
$$846$$ 0 0
$$847$$ 574.092 0.0232893
$$848$$ 0 0
$$849$$ 6373.45 0.257640
$$850$$ 0 0
$$851$$ −11466.6 −0.461892
$$852$$ 0 0
$$853$$ −47157.1 −1.89288 −0.946441 0.322878i $$-0.895350\pi$$
−0.946441 + 0.322878i $$0.895350\pi$$
$$854$$ 0 0
$$855$$ −2471.38 −0.0988531
$$856$$ 0 0
$$857$$ 5021.31 0.200145 0.100073 0.994980i $$-0.468092\pi$$
0.100073 + 0.994980i $$0.468092\pi$$
$$858$$ 0 0
$$859$$ 22921.1 0.910428 0.455214 0.890382i $$-0.349563\pi$$
0.455214 + 0.890382i $$0.349563\pi$$
$$860$$ 0 0
$$861$$ 3433.95 0.135922
$$862$$ 0 0
$$863$$ 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
$$866$$ 0 0
$$867$$ −1321.86 −0.0517794
$$868$$ 0 0
$$869$$ −3603.04 −0.140650
$$870$$ 0 0
$$871$$ 13528.4 0.526282
$$872$$ 0 0
$$873$$ 5529.22 0.214360
$$874$$ 0 0
$$875$$ 3937.06 0.152111
$$876$$ 0 0
$$877$$ −8455.67 −0.325573 −0.162787 0.986661i $$-0.552048\pi$$
−0.162787 + 0.986661i $$0.552048\pi$$
$$878$$ 0 0
$$879$$ 9972.56 0.382669
$$880$$ 0 0
$$881$$ −11291.2 −0.431794 −0.215897 0.976416i $$-0.569268\pi$$
−0.215897 + 0.976416i $$0.569268\pi$$
$$882$$ 0 0
$$883$$ −31818.1 −1.21264 −0.606322 0.795219i $$-0.707356\pi$$
−0.606322 + 0.795219i $$0.707356\pi$$
$$884$$ 0 0
$$885$$ −2051.61 −0.0779254
$$886$$ 0 0
$$887$$ −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
$$890$$ 0 0
$$891$$ −891.000 −0.0335013
$$892$$ 0 0
$$893$$ 13368.9 0.500978
$$894$$ 0 0
$$895$$ −6805.16 −0.254158
$$896$$ 0 0
$$897$$ −5047.31 −0.187876
$$898$$ 0 0
$$899$$ −67763.1 −2.51393
$$900$$ 0 0
$$901$$ −29973.6 −1.10829
$$902$$ 0 0
$$903$$ −3985.66 −0.146882
$$904$$ 0 0
$$905$$ −3704.08 −0.136053
$$906$$ 0 0
$$907$$ −10607.4 −0.388326 −0.194163 0.980969i $$-0.562199\pi$$
−0.194163 + 0.980969i $$0.562199\pi$$
$$908$$ 0 0
$$909$$ −9143.26 −0.333623
$$910$$ 0 0
$$911$$ 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
$$914$$ 0 0
$$915$$ −7341.38 −0.265244
$$916$$ 0 0
$$917$$ 5443.97 0.196048
$$918$$ 0 0
$$919$$ 13858.1 0.497429 0.248714 0.968577i $$-0.419992\pi$$
0.248714 + 0.968577i $$0.419992\pi$$
$$920$$ 0 0
$$921$$ −5096.83 −0.182352
$$922$$ 0 0
$$923$$ 11363.9 0.405253
$$924$$ 0 0
$$925$$ −11551.2 −0.410595
$$926$$ 0 0
$$927$$ −9919.47 −0.351454
$$928$$ 0 0
$$929$$ 20893.7 0.737890 0.368945 0.929451i $$-0.379719\pi$$
0.368945 + 0.929451i $$0.379719\pi$$
$$930$$ 0 0
$$931$$ −25222.8 −0.887911
$$932$$ 0 0
$$933$$ 20786.5 0.729388
$$934$$ 0 0
$$935$$ 2808.23 0.0982236
$$936$$ 0 0
$$937$$ 3203.52 0.111691 0.0558454 0.998439i $$-0.482215\pi$$
0.0558454 + 0.998439i $$0.482215\pi$$
$$938$$ 0 0
$$939$$ 10682.2 0.371248
$$940$$ 0 0
$$941$$ 19951.6 0.691182 0.345591 0.938385i $$-0.387678\pi$$
0.345591 + 0.938385i $$0.387678\pi$$
$$942$$ 0 0
$$943$$ 27020.6 0.933099
$$944$$ 0 0
$$945$$ 446.968 0.0153861
$$946$$ 0 0
$$947$$ 38216.7 1.31138 0.655689 0.755031i $$-0.272378\pi$$
0.655689 + 0.755031i $$0.272378\pi$$
$$948$$ 0 0
$$949$$ 15319.4 0.524014
$$950$$ 0 0
$$951$$ −998.249 −0.0340383
$$952$$ 0 0
$$953$$ −47661.4 −1.62004 −0.810022 0.586399i $$-0.800545\pi$$
−0.810022 + 0.586399i $$0.800545\pi$$
$$954$$ 0 0
$$955$$ 7454.21 0.252579
$$956$$ 0 0
$$957$$ 8023.12 0.271004
$$958$$ 0 0
$$959$$ 6018.94 0.202671
$$960$$ 0 0
$$961$$ 47892.3 1.60761
$$962$$ 0 0
$$963$$ −12398.2 −0.414876
$$964$$ 0 0
$$965$$ −13774.2 −0.459490
$$966$$ 0 0
$$967$$ −18933.2 −0.629628 −0.314814 0.949153i $$-0.601942\pi$$
−0.314814 + 0.949153i $$0.601942\pi$$
$$968$$ 0 0
$$969$$ −17275.3 −0.572717
$$970$$ 0 0
$$971$$ 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
$$974$$ 0 0
$$975$$ −5084.53 −0.167011
$$976$$ 0 0
$$977$$ −22502.8 −0.736876 −0.368438 0.929652i $$-0.620107\pi$$
−0.368438 + 0.929652i $$0.620107\pi$$
$$978$$ 0 0
$$979$$ −5598.76 −0.182775
$$980$$ 0 0
$$981$$ 2881.96 0.0937959
$$982$$ 0 0
$$983$$ 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
$$986$$ 0 0
$$987$$ −2417.87 −0.0779753
$$988$$ 0 0
$$989$$ −31361.8 −1.00834
$$990$$ 0 0
$$991$$ −7362.76 −0.236010 −0.118005 0.993013i $$-0.537650\pi$$
−0.118005 + 0.993013i $$0.537650\pi$$
$$992$$ 0 0
$$993$$ −1624.33 −0.0519101
$$994$$ 0 0
$$995$$ −1661.35 −0.0529331
$$996$$ 0 0
$$997$$ −53480.1 −1.69883 −0.849413 0.527728i $$-0.823044\pi$$
−0.849413 + 0.527728i $$0.823044\pi$$
$$998$$ 0 0
$$999$$ −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 528.4.a.o.1.1 2
3.2 odd 2 1584.4.a.x.1.2 2
4.3 odd 2 33.4.a.d.1.2 2
8.3 odd 2 2112.4.a.ba.1.2 2
8.5 even 2 2112.4.a.bh.1.2 2
12.11 even 2 99.4.a.e.1.1 2
20.3 even 4 825.4.c.i.199.1 4
20.7 even 4 825.4.c.i.199.4 4
20.19 odd 2 825.4.a.k.1.1 2
28.27 even 2 1617.4.a.j.1.2 2
44.43 even 2 363.4.a.j.1.1 2
60.59 even 2 2475.4.a.o.1.2 2
132.131 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 4.3 odd 2
99.4.a.e.1.1 2 12.11 even 2
363.4.a.j.1.1 2 44.43 even 2
528.4.a.o.1.1 2 1.1 even 1 trivial
825.4.a.k.1.1 2 20.19 odd 2
825.4.c.i.199.1 4 20.3 even 4
825.4.c.i.199.4 4 20.7 even 4
1089.4.a.t.1.2 2 132.131 odd 2
1584.4.a.x.1.2 2 3.2 odd 2
1617.4.a.j.1.2 2 28.27 even 2
2112.4.a.ba.1.2 2 8.3 odd 2
2112.4.a.bh.1.2 2 8.5 even 2
2475.4.a.o.1.2 2 60.59 even 2