Properties

 Label 1452.4.a.t.1.4 Level $1452$ Weight $4$ Character 1452.1 Self dual yes Analytic conductor $85.671$ Analytic rank $1$ Dimension $6$ CM no Inner twists $1$

Related objects

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1452,4,Mod(1,1452)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1452, 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("1452.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1452 = 2^{2} \cdot 3 \cdot 11^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 1452.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: $$85.6707733283$$ Analytic rank: $$1$$ Dimension: $$6$$ Coefficient field: $$\mathbb{Q}[x]/(x^{6} - \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{6} - x^{5} - 174x^{4} + 63x^{3} + 7614x^{2} + 1579x - 12611$$ x^6 - x^5 - 174*x^4 + 63*x^3 + 7614*x^2 + 1579*x - 12611 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{2}\cdot 11$$ Twist minimal: no (minimal twist has level 132) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

 Embedding label 1.4 Root $$-8.43670$$ of defining polynomial Character $$\chi$$ $$=$$ 1452.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+3.00000 q^{3} +3.10899 q^{5} -29.8524 q^{7} +9.00000 q^{9} +O(q^{10})$$ $$q+3.00000 q^{3} +3.10899 q^{5} -29.8524 q^{7} +9.00000 q^{9} +32.2030 q^{13} +9.32697 q^{15} +70.3884 q^{17} -47.1971 q^{19} -89.5572 q^{21} +56.3381 q^{23} -115.334 q^{25} +27.0000 q^{27} -194.765 q^{29} +146.250 q^{31} -92.8108 q^{35} +310.527 q^{37} +96.6091 q^{39} -332.509 q^{41} +87.1866 q^{43} +27.9809 q^{45} -101.101 q^{47} +548.166 q^{49} +211.165 q^{51} +365.931 q^{53} -141.591 q^{57} -641.892 q^{59} -758.230 q^{61} -268.672 q^{63} +100.119 q^{65} +123.262 q^{67} +169.014 q^{69} -1066.33 q^{71} +525.946 q^{73} -346.003 q^{75} -1155.63 q^{79} +81.0000 q^{81} -330.379 q^{83} +218.837 q^{85} -584.295 q^{87} -300.630 q^{89} -961.338 q^{91} +438.749 q^{93} -146.735 q^{95} -1548.21 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q + 18 q^{3} + 13 q^{5} - 23 q^{7} + 54 q^{9}+O(q^{10})$$ 6 * q + 18 * q^3 + 13 * q^5 - 23 * q^7 + 54 * q^9 $$6 q + 18 q^{3} + 13 q^{5} - 23 q^{7} + 54 q^{9} - 66 q^{13} + 39 q^{15} - 44 q^{17} - 270 q^{19} - 69 q^{21} - 124 q^{23} + 131 q^{25} + 162 q^{27} - 141 q^{29} - 253 q^{31} - 884 q^{35} + 288 q^{37} - 198 q^{39} - 428 q^{41} - 1006 q^{43} + 117 q^{45} - 674 q^{47} + 181 q^{49} - 132 q^{51} - 773 q^{53} - 810 q^{57} - 17 q^{59} - 1016 q^{61} - 207 q^{63} - 1220 q^{65} + 1836 q^{67} - 372 q^{69} - 208 q^{71} - 1521 q^{73} + 393 q^{75} - 1425 q^{79} + 486 q^{81} - 3065 q^{83} - 1304 q^{85} - 423 q^{87} + 1444 q^{89} + 1328 q^{91} - 759 q^{93} - 1760 q^{95} - 3887 q^{97}+O(q^{100})$$ 6 * q + 18 * q^3 + 13 * q^5 - 23 * q^7 + 54 * q^9 - 66 * q^13 + 39 * q^15 - 44 * q^17 - 270 * q^19 - 69 * q^21 - 124 * q^23 + 131 * q^25 + 162 * q^27 - 141 * q^29 - 253 * q^31 - 884 * q^35 + 288 * q^37 - 198 * q^39 - 428 * q^41 - 1006 * q^43 + 117 * q^45 - 674 * q^47 + 181 * q^49 - 132 * q^51 - 773 * q^53 - 810 * q^57 - 17 * q^59 - 1016 * q^61 - 207 * q^63 - 1220 * q^65 + 1836 * q^67 - 372 * q^69 - 208 * q^71 - 1521 * q^73 + 393 * q^75 - 1425 * q^79 + 486 * q^81 - 3065 * q^83 - 1304 * q^85 - 423 * q^87 + 1444 * q^89 + 1328 * q^91 - 759 * q^93 - 1760 * q^95 - 3887 * q^97

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
$$5$$ 3.10899 0.278076 0.139038 0.990287i $$-0.455599\pi$$
0.139038 + 0.990287i $$0.455599\pi$$
$$6$$ 0 0
$$7$$ −29.8524 −1.61188 −0.805939 0.591999i $$-0.798339\pi$$
−0.805939 + 0.591999i $$0.798339\pi$$
$$8$$ 0 0
$$9$$ 9.00000 0.333333
$$10$$ 0 0
$$11$$ 0 0
$$12$$ 0 0
$$13$$ 32.2030 0.687040 0.343520 0.939145i $$-0.388381\pi$$
0.343520 + 0.939145i $$0.388381\pi$$
$$14$$ 0 0
$$15$$ 9.32697 0.160548
$$16$$ 0 0
$$17$$ 70.3884 1.00422 0.502109 0.864805i $$-0.332558\pi$$
0.502109 + 0.864805i $$0.332558\pi$$
$$18$$ 0 0
$$19$$ −47.1971 −0.569882 −0.284941 0.958545i $$-0.591974\pi$$
−0.284941 + 0.958545i $$0.591974\pi$$
$$20$$ 0 0
$$21$$ −89.5572 −0.930618
$$22$$ 0 0
$$23$$ 56.3381 0.510753 0.255376 0.966842i $$-0.417801\pi$$
0.255376 + 0.966842i $$0.417801\pi$$
$$24$$ 0 0
$$25$$ −115.334 −0.922673
$$26$$ 0 0
$$27$$ 27.0000 0.192450
$$28$$ 0 0
$$29$$ −194.765 −1.24714 −0.623568 0.781769i $$-0.714318\pi$$
−0.623568 + 0.781769i $$0.714318\pi$$
$$30$$ 0 0
$$31$$ 146.250 0.847330 0.423665 0.905819i $$-0.360743\pi$$
0.423665 + 0.905819i $$0.360743\pi$$
$$32$$ 0 0
$$33$$ 0 0
$$34$$ 0 0
$$35$$ −92.8108 −0.448225
$$36$$ 0 0
$$37$$ 310.527 1.37974 0.689869 0.723935i $$-0.257668\pi$$
0.689869 + 0.723935i $$0.257668\pi$$
$$38$$ 0 0
$$39$$ 96.6091 0.396663
$$40$$ 0 0
$$41$$ −332.509 −1.26657 −0.633284 0.773920i $$-0.718293\pi$$
−0.633284 + 0.773920i $$0.718293\pi$$
$$42$$ 0 0
$$43$$ 87.1866 0.309205 0.154603 0.987977i $$-0.450590\pi$$
0.154603 + 0.987977i $$0.450590\pi$$
$$44$$ 0 0
$$45$$ 27.9809 0.0926922
$$46$$ 0 0
$$47$$ −101.101 −0.313769 −0.156885 0.987617i $$-0.550145\pi$$
−0.156885 + 0.987617i $$0.550145\pi$$
$$48$$ 0 0
$$49$$ 548.166 1.59815
$$50$$ 0 0
$$51$$ 211.165 0.579785
$$52$$ 0 0
$$53$$ 365.931 0.948388 0.474194 0.880420i $$-0.342740\pi$$
0.474194 + 0.880420i $$0.342740\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ −141.591 −0.329021
$$58$$ 0 0
$$59$$ −641.892 −1.41639 −0.708197 0.706015i $$-0.750491\pi$$
−0.708197 + 0.706015i $$0.750491\pi$$
$$60$$ 0 0
$$61$$ −758.230 −1.59150 −0.795750 0.605626i $$-0.792923\pi$$
−0.795750 + 0.605626i $$0.792923\pi$$
$$62$$ 0 0
$$63$$ −268.672 −0.537293
$$64$$ 0 0
$$65$$ 100.119 0.191050
$$66$$ 0 0
$$67$$ 123.262 0.224759 0.112380 0.993665i $$-0.464153\pi$$
0.112380 + 0.993665i $$0.464153\pi$$
$$68$$ 0 0
$$69$$ 169.014 0.294883
$$70$$ 0 0
$$71$$ −1066.33 −1.78240 −0.891200 0.453611i $$-0.850135\pi$$
−0.891200 + 0.453611i $$0.850135\pi$$
$$72$$ 0 0
$$73$$ 525.946 0.843251 0.421625 0.906770i $$-0.361460\pi$$
0.421625 + 0.906770i $$0.361460\pi$$
$$74$$ 0 0
$$75$$ −346.003 −0.532706
$$76$$ 0 0
$$77$$ 0 0
$$78$$ 0 0
$$79$$ −1155.63 −1.64580 −0.822899 0.568187i $$-0.807645\pi$$
−0.822899 + 0.568187i $$0.807645\pi$$
$$80$$ 0 0
$$81$$ 81.0000 0.111111
$$82$$ 0 0
$$83$$ −330.379 −0.436913 −0.218456 0.975847i $$-0.570102\pi$$
−0.218456 + 0.975847i $$0.570102\pi$$
$$84$$ 0 0
$$85$$ 218.837 0.279249
$$86$$ 0 0
$$87$$ −584.295 −0.720034
$$88$$ 0 0
$$89$$ −300.630 −0.358053 −0.179027 0.983844i $$-0.557295\pi$$
−0.179027 + 0.983844i $$0.557295\pi$$
$$90$$ 0 0
$$91$$ −961.338 −1.10742
$$92$$ 0 0
$$93$$ 438.749 0.489206
$$94$$ 0 0
$$95$$ −146.735 −0.158471
$$96$$ 0 0
$$97$$ −1548.21 −1.62058 −0.810292 0.586027i $$-0.800691\pi$$
−0.810292 + 0.586027i $$0.800691\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 0 0
$$101$$ 715.641 0.705039 0.352520 0.935804i $$-0.385325\pi$$
0.352520 + 0.935804i $$0.385325\pi$$
$$102$$ 0 0
$$103$$ −245.027 −0.234400 −0.117200 0.993108i $$-0.537392\pi$$
−0.117200 + 0.993108i $$0.537392\pi$$
$$104$$ 0 0
$$105$$ −278.432 −0.258783
$$106$$ 0 0
$$107$$ −1511.96 −1.36604 −0.683020 0.730400i $$-0.739334\pi$$
−0.683020 + 0.730400i $$0.739334\pi$$
$$108$$ 0 0
$$109$$ −1213.16 −1.06605 −0.533024 0.846100i $$-0.678945\pi$$
−0.533024 + 0.846100i $$0.678945\pi$$
$$110$$ 0 0
$$111$$ 931.580 0.796592
$$112$$ 0 0
$$113$$ 370.312 0.308284 0.154142 0.988049i $$-0.450739\pi$$
0.154142 + 0.988049i $$0.450739\pi$$
$$114$$ 0 0
$$115$$ 175.155 0.142028
$$116$$ 0 0
$$117$$ 289.827 0.229013
$$118$$ 0 0
$$119$$ −2101.26 −1.61868
$$120$$ 0 0
$$121$$ 0 0
$$122$$ 0 0
$$123$$ −997.528 −0.731253
$$124$$ 0 0
$$125$$ −747.196 −0.534650
$$126$$ 0 0
$$127$$ 179.811 0.125635 0.0628175 0.998025i $$-0.479991\pi$$
0.0628175 + 0.998025i $$0.479991\pi$$
$$128$$ 0 0
$$129$$ 261.560 0.178520
$$130$$ 0 0
$$131$$ 1146.39 0.764583 0.382291 0.924042i $$-0.375135\pi$$
0.382291 + 0.924042i $$0.375135\pi$$
$$132$$ 0 0
$$133$$ 1408.95 0.918580
$$134$$ 0 0
$$135$$ 83.9427 0.0535158
$$136$$ 0 0
$$137$$ 2053.05 1.28032 0.640160 0.768242i $$-0.278868\pi$$
0.640160 + 0.768242i $$0.278868\pi$$
$$138$$ 0 0
$$139$$ 215.546 0.131528 0.0657641 0.997835i $$-0.479052\pi$$
0.0657641 + 0.997835i $$0.479052\pi$$
$$140$$ 0 0
$$141$$ −303.304 −0.181155
$$142$$ 0 0
$$143$$ 0 0
$$144$$ 0 0
$$145$$ −605.522 −0.346799
$$146$$ 0 0
$$147$$ 1644.50 0.922693
$$148$$ 0 0
$$149$$ −1159.51 −0.637522 −0.318761 0.947835i $$-0.603267\pi$$
−0.318761 + 0.947835i $$0.603267\pi$$
$$150$$ 0 0
$$151$$ −2723.16 −1.46760 −0.733799 0.679367i $$-0.762255\pi$$
−0.733799 + 0.679367i $$0.762255\pi$$
$$152$$ 0 0
$$153$$ 633.496 0.334739
$$154$$ 0 0
$$155$$ 454.689 0.235622
$$156$$ 0 0
$$157$$ 2513.02 1.27746 0.638728 0.769433i $$-0.279461\pi$$
0.638728 + 0.769433i $$0.279461\pi$$
$$158$$ 0 0
$$159$$ 1097.79 0.547552
$$160$$ 0 0
$$161$$ −1681.83 −0.823271
$$162$$ 0 0
$$163$$ 976.748 0.469355 0.234677 0.972073i $$-0.424597\pi$$
0.234677 + 0.972073i $$0.424597\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ −1117.72 −0.517915 −0.258957 0.965889i $$-0.583379\pi$$
−0.258957 + 0.965889i $$0.583379\pi$$
$$168$$ 0 0
$$169$$ −1159.96 −0.527976
$$170$$ 0 0
$$171$$ −424.774 −0.189961
$$172$$ 0 0
$$173$$ −4055.13 −1.78211 −0.891057 0.453891i $$-0.850036\pi$$
−0.891057 + 0.453891i $$0.850036\pi$$
$$174$$ 0 0
$$175$$ 3443.00 1.48724
$$176$$ 0 0
$$177$$ −1925.68 −0.817755
$$178$$ 0 0
$$179$$ −3518.15 −1.46904 −0.734522 0.678585i $$-0.762593\pi$$
−0.734522 + 0.678585i $$0.762593\pi$$
$$180$$ 0 0
$$181$$ 1824.02 0.749051 0.374526 0.927217i $$-0.377806\pi$$
0.374526 + 0.927217i $$0.377806\pi$$
$$182$$ 0 0
$$183$$ −2274.69 −0.918853
$$184$$ 0 0
$$185$$ 965.424 0.383672
$$186$$ 0 0
$$187$$ 0 0
$$188$$ 0 0
$$189$$ −806.015 −0.310206
$$190$$ 0 0
$$191$$ −1270.69 −0.481382 −0.240691 0.970602i $$-0.577374\pi$$
−0.240691 + 0.970602i $$0.577374\pi$$
$$192$$ 0 0
$$193$$ 1028.31 0.383520 0.191760 0.981442i $$-0.438581\pi$$
0.191760 + 0.981442i $$0.438581\pi$$
$$194$$ 0 0
$$195$$ 300.357 0.110303
$$196$$ 0 0
$$197$$ 3541.77 1.28092 0.640459 0.767992i $$-0.278744\pi$$
0.640459 + 0.767992i $$0.278744\pi$$
$$198$$ 0 0
$$199$$ 2616.56 0.932074 0.466037 0.884765i $$-0.345681\pi$$
0.466037 + 0.884765i $$0.345681\pi$$
$$200$$ 0 0
$$201$$ 369.786 0.129765
$$202$$ 0 0
$$203$$ 5814.20 2.01023
$$204$$ 0 0
$$205$$ −1033.77 −0.352203
$$206$$ 0 0
$$207$$ 507.043 0.170251
$$208$$ 0 0
$$209$$ 0 0
$$210$$ 0 0
$$211$$ −3030.15 −0.988646 −0.494323 0.869278i $$-0.664584\pi$$
−0.494323 + 0.869278i $$0.664584\pi$$
$$212$$ 0 0
$$213$$ −3199.00 −1.02907
$$214$$ 0 0
$$215$$ 271.062 0.0859828
$$216$$ 0 0
$$217$$ −4365.90 −1.36579
$$218$$ 0 0
$$219$$ 1577.84 0.486851
$$220$$ 0 0
$$221$$ 2266.72 0.689937
$$222$$ 0 0
$$223$$ 446.011 0.133933 0.0669666 0.997755i $$-0.478668\pi$$
0.0669666 + 0.997755i $$0.478668\pi$$
$$224$$ 0 0
$$225$$ −1038.01 −0.307558
$$226$$ 0 0
$$227$$ 3770.73 1.10252 0.551260 0.834334i $$-0.314147\pi$$
0.551260 + 0.834334i $$0.314147\pi$$
$$228$$ 0 0
$$229$$ −5645.24 −1.62903 −0.814514 0.580143i $$-0.802997\pi$$
−0.814514 + 0.580143i $$0.802997\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ −684.619 −0.192493 −0.0962466 0.995358i $$-0.530684\pi$$
−0.0962466 + 0.995358i $$0.530684\pi$$
$$234$$ 0 0
$$235$$ −314.323 −0.0872519
$$236$$ 0 0
$$237$$ −3466.88 −0.950202
$$238$$ 0 0
$$239$$ 2955.66 0.799941 0.399971 0.916528i $$-0.369020\pi$$
0.399971 + 0.916528i $$0.369020\pi$$
$$240$$ 0 0
$$241$$ 481.604 0.128725 0.0643627 0.997927i $$-0.479499\pi$$
0.0643627 + 0.997927i $$0.479499\pi$$
$$242$$ 0 0
$$243$$ 243.000 0.0641500
$$244$$ 0 0
$$245$$ 1704.24 0.444408
$$246$$ 0 0
$$247$$ −1519.89 −0.391531
$$248$$ 0 0
$$249$$ −991.136 −0.252252
$$250$$ 0 0
$$251$$ −7561.16 −1.90142 −0.950710 0.310083i $$-0.899643\pi$$
−0.950710 + 0.310083i $$0.899643\pi$$
$$252$$ 0 0
$$253$$ 0 0
$$254$$ 0 0
$$255$$ 656.510 0.161225
$$256$$ 0 0
$$257$$ −2562.02 −0.621845 −0.310922 0.950435i $$-0.600638\pi$$
−0.310922 + 0.950435i $$0.600638\pi$$
$$258$$ 0 0
$$259$$ −9269.97 −2.22397
$$260$$ 0 0
$$261$$ −1752.88 −0.415712
$$262$$ 0 0
$$263$$ −7518.10 −1.76268 −0.881342 0.472480i $$-0.843359\pi$$
−0.881342 + 0.472480i $$0.843359\pi$$
$$264$$ 0 0
$$265$$ 1137.68 0.263724
$$266$$ 0 0
$$267$$ −901.890 −0.206722
$$268$$ 0 0
$$269$$ 8339.96 1.89032 0.945160 0.326608i $$-0.105906\pi$$
0.945160 + 0.326608i $$0.105906\pi$$
$$270$$ 0 0
$$271$$ −2005.14 −0.449460 −0.224730 0.974421i $$-0.572150\pi$$
−0.224730 + 0.974421i $$0.572150\pi$$
$$272$$ 0 0
$$273$$ −2884.01 −0.639372
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 0 0
$$277$$ 1146.94 0.248784 0.124392 0.992233i $$-0.460302\pi$$
0.124392 + 0.992233i $$0.460302\pi$$
$$278$$ 0 0
$$279$$ 1316.25 0.282443
$$280$$ 0 0
$$281$$ 3845.96 0.816480 0.408240 0.912875i $$-0.366143\pi$$
0.408240 + 0.912875i $$0.366143\pi$$
$$282$$ 0 0
$$283$$ −1078.38 −0.226513 −0.113256 0.993566i $$-0.536128\pi$$
−0.113256 + 0.993566i $$0.536128\pi$$
$$284$$ 0 0
$$285$$ −440.206 −0.0914931
$$286$$ 0 0
$$287$$ 9926.21 2.04155
$$288$$ 0 0
$$289$$ 41.5253 0.00845214
$$290$$ 0 0
$$291$$ −4644.62 −0.935644
$$292$$ 0 0
$$293$$ 3978.34 0.793233 0.396616 0.917985i $$-0.370184\pi$$
0.396616 + 0.917985i $$0.370184\pi$$
$$294$$ 0 0
$$295$$ −1995.64 −0.393866
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ 1814.26 0.350907
$$300$$ 0 0
$$301$$ −2602.73 −0.498401
$$302$$ 0 0
$$303$$ 2146.92 0.407055
$$304$$ 0 0
$$305$$ −2357.33 −0.442559
$$306$$ 0 0
$$307$$ 2166.24 0.402716 0.201358 0.979518i $$-0.435465\pi$$
0.201358 + 0.979518i $$0.435465\pi$$
$$308$$ 0 0
$$309$$ −735.081 −0.135331
$$310$$ 0 0
$$311$$ 1978.32 0.360709 0.180355 0.983602i $$-0.442276\pi$$
0.180355 + 0.983602i $$0.442276\pi$$
$$312$$ 0 0
$$313$$ −4766.98 −0.860848 −0.430424 0.902627i $$-0.641636\pi$$
−0.430424 + 0.902627i $$0.641636\pi$$
$$314$$ 0 0
$$315$$ −835.297 −0.149408
$$316$$ 0 0
$$317$$ −3540.69 −0.627333 −0.313667 0.949533i $$-0.601557\pi$$
−0.313667 + 0.949533i $$0.601557\pi$$
$$318$$ 0 0
$$319$$ 0 0
$$320$$ 0 0
$$321$$ −4535.87 −0.788683
$$322$$ 0 0
$$323$$ −3322.13 −0.572285
$$324$$ 0 0
$$325$$ −3714.11 −0.633913
$$326$$ 0 0
$$327$$ −3639.47 −0.615483
$$328$$ 0 0
$$329$$ 3018.12 0.505758
$$330$$ 0 0
$$331$$ −10927.7 −1.81463 −0.907315 0.420451i $$-0.861872\pi$$
−0.907315 + 0.420451i $$0.861872\pi$$
$$332$$ 0 0
$$333$$ 2794.74 0.459912
$$334$$ 0 0
$$335$$ 383.220 0.0625002
$$336$$ 0 0
$$337$$ −9153.38 −1.47957 −0.739787 0.672842i $$-0.765074\pi$$
−0.739787 + 0.672842i $$0.765074\pi$$
$$338$$ 0 0
$$339$$ 1110.94 0.177988
$$340$$ 0 0
$$341$$ 0 0
$$342$$ 0 0
$$343$$ −6124.69 −0.964146
$$344$$ 0 0
$$345$$ 525.464 0.0820001
$$346$$ 0 0
$$347$$ −7626.15 −1.17981 −0.589903 0.807474i $$-0.700834\pi$$
−0.589903 + 0.807474i $$0.700834\pi$$
$$348$$ 0 0
$$349$$ 3708.63 0.568821 0.284410 0.958703i $$-0.408202\pi$$
0.284410 + 0.958703i $$0.408202\pi$$
$$350$$ 0 0
$$351$$ 869.482 0.132221
$$352$$ 0 0
$$353$$ 7111.77 1.07230 0.536149 0.844123i $$-0.319878\pi$$
0.536149 + 0.844123i $$0.319878\pi$$
$$354$$ 0 0
$$355$$ −3315.22 −0.495643
$$356$$ 0 0
$$357$$ −6303.79 −0.934543
$$358$$ 0 0
$$359$$ 12035.1 1.76933 0.884665 0.466228i $$-0.154387\pi$$
0.884665 + 0.466228i $$0.154387\pi$$
$$360$$ 0 0
$$361$$ −4631.43 −0.675235
$$362$$ 0 0
$$363$$ 0 0
$$364$$ 0 0
$$365$$ 1635.16 0.234488
$$366$$ 0 0
$$367$$ 2187.73 0.311167 0.155584 0.987823i $$-0.450274\pi$$
0.155584 + 0.987823i $$0.450274\pi$$
$$368$$ 0 0
$$369$$ −2992.59 −0.422189
$$370$$ 0 0
$$371$$ −10923.9 −1.52869
$$372$$ 0 0
$$373$$ 2599.10 0.360794 0.180397 0.983594i $$-0.442262\pi$$
0.180397 + 0.983594i $$0.442262\pi$$
$$374$$ 0 0
$$375$$ −2241.59 −0.308680
$$376$$ 0 0
$$377$$ −6272.02 −0.856832
$$378$$ 0 0
$$379$$ 9772.05 1.32442 0.662212 0.749317i $$-0.269618\pi$$
0.662212 + 0.749317i $$0.269618\pi$$
$$380$$ 0 0
$$381$$ 539.433 0.0725354
$$382$$ 0 0
$$383$$ 10067.7 1.34317 0.671586 0.740927i $$-0.265614\pi$$
0.671586 + 0.740927i $$0.265614\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ 784.680 0.103068
$$388$$ 0 0
$$389$$ −1025.63 −0.133680 −0.0668401 0.997764i $$-0.521292\pi$$
−0.0668401 + 0.997764i $$0.521292\pi$$
$$390$$ 0 0
$$391$$ 3965.55 0.512907
$$392$$ 0 0
$$393$$ 3439.16 0.441432
$$394$$ 0 0
$$395$$ −3592.83 −0.457658
$$396$$ 0 0
$$397$$ −6069.83 −0.767346 −0.383673 0.923469i $$-0.625341\pi$$
−0.383673 + 0.923469i $$0.625341\pi$$
$$398$$ 0 0
$$399$$ 4226.84 0.530342
$$400$$ 0 0
$$401$$ 14193.3 1.76753 0.883767 0.467927i $$-0.154999\pi$$
0.883767 + 0.467927i $$0.154999\pi$$
$$402$$ 0 0
$$403$$ 4709.69 0.582149
$$404$$ 0 0
$$405$$ 251.828 0.0308974
$$406$$ 0 0
$$407$$ 0 0
$$408$$ 0 0
$$409$$ −11547.7 −1.39608 −0.698039 0.716060i $$-0.745944\pi$$
−0.698039 + 0.716060i $$0.745944\pi$$
$$410$$ 0 0
$$411$$ 6159.14 0.739193
$$412$$ 0 0
$$413$$ 19162.0 2.28305
$$414$$ 0 0
$$415$$ −1027.14 −0.121495
$$416$$ 0 0
$$417$$ 646.639 0.0759378
$$418$$ 0 0
$$419$$ −10748.2 −1.25318 −0.626590 0.779349i $$-0.715550\pi$$
−0.626590 + 0.779349i $$0.715550\pi$$
$$420$$ 0 0
$$421$$ −3124.08 −0.361658 −0.180829 0.983515i $$-0.557878\pi$$
−0.180829 + 0.983515i $$0.557878\pi$$
$$422$$ 0 0
$$423$$ −909.913 −0.104590
$$424$$ 0 0
$$425$$ −8118.19 −0.926565
$$426$$ 0 0
$$427$$ 22635.0 2.56530
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 7448.50 0.832440 0.416220 0.909264i $$-0.363355\pi$$
0.416220 + 0.909264i $$0.363355\pi$$
$$432$$ 0 0
$$433$$ 9124.75 1.01272 0.506360 0.862322i $$-0.330991\pi$$
0.506360 + 0.862322i $$0.330991\pi$$
$$434$$ 0 0
$$435$$ −1816.57 −0.200225
$$436$$ 0 0
$$437$$ −2659.00 −0.291069
$$438$$ 0 0
$$439$$ −14044.5 −1.52689 −0.763447 0.645870i $$-0.776495\pi$$
−0.763447 + 0.645870i $$0.776495\pi$$
$$440$$ 0 0
$$441$$ 4933.49 0.532717
$$442$$ 0 0
$$443$$ −7537.00 −0.808338 −0.404169 0.914684i $$-0.632439\pi$$
−0.404169 + 0.914684i $$0.632439\pi$$
$$444$$ 0 0
$$445$$ −934.656 −0.0995661
$$446$$ 0 0
$$447$$ −3478.53 −0.368073
$$448$$ 0 0
$$449$$ 7767.34 0.816400 0.408200 0.912893i $$-0.366157\pi$$
0.408200 + 0.912893i $$0.366157\pi$$
$$450$$ 0 0
$$451$$ 0 0
$$452$$ 0 0
$$453$$ −8169.47 −0.847318
$$454$$ 0 0
$$455$$ −2988.79 −0.307949
$$456$$ 0 0
$$457$$ 169.378 0.0173373 0.00866866 0.999962i $$-0.497241\pi$$
0.00866866 + 0.999962i $$0.497241\pi$$
$$458$$ 0 0
$$459$$ 1900.49 0.193262
$$460$$ 0 0
$$461$$ 5648.85 0.570701 0.285351 0.958423i $$-0.407890\pi$$
0.285351 + 0.958423i $$0.407890\pi$$
$$462$$ 0 0
$$463$$ 16305.5 1.63668 0.818338 0.574738i $$-0.194896\pi$$
0.818338 + 0.574738i $$0.194896\pi$$
$$464$$ 0 0
$$465$$ 1364.07 0.136037
$$466$$ 0 0
$$467$$ −15725.6 −1.55823 −0.779117 0.626879i $$-0.784332\pi$$
−0.779117 + 0.626879i $$0.784332\pi$$
$$468$$ 0 0
$$469$$ −3679.67 −0.362284
$$470$$ 0 0
$$471$$ 7539.05 0.737539
$$472$$ 0 0
$$473$$ 0 0
$$474$$ 0 0
$$475$$ 5443.44 0.525815
$$476$$ 0 0
$$477$$ 3293.38 0.316129
$$478$$ 0 0
$$479$$ 13243.8 1.26331 0.631656 0.775249i $$-0.282376\pi$$
0.631656 + 0.775249i $$0.282376\pi$$
$$480$$ 0 0
$$481$$ 9999.90 0.947934
$$482$$ 0 0
$$483$$ −5045.48 −0.475316
$$484$$ 0 0
$$485$$ −4813.36 −0.450646
$$486$$ 0 0
$$487$$ 6558.26 0.610232 0.305116 0.952315i $$-0.401305\pi$$
0.305116 + 0.952315i $$0.401305\pi$$
$$488$$ 0 0
$$489$$ 2930.24 0.270982
$$490$$ 0 0
$$491$$ 6925.61 0.636554 0.318277 0.947998i $$-0.396896\pi$$
0.318277 + 0.947998i $$0.396896\pi$$
$$492$$ 0 0
$$493$$ −13709.2 −1.25240
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 31832.6 2.87301
$$498$$ 0 0
$$499$$ 15286.3 1.37136 0.685680 0.727903i $$-0.259505\pi$$
0.685680 + 0.727903i $$0.259505\pi$$
$$500$$ 0 0
$$501$$ −3353.16 −0.299018
$$502$$ 0 0
$$503$$ 3998.72 0.354461 0.177231 0.984169i $$-0.443286\pi$$
0.177231 + 0.984169i $$0.443286\pi$$
$$504$$ 0 0
$$505$$ 2224.92 0.196055
$$506$$ 0 0
$$507$$ −3479.89 −0.304827
$$508$$ 0 0
$$509$$ −18157.1 −1.58114 −0.790570 0.612371i $$-0.790216\pi$$
−0.790570 + 0.612371i $$0.790216\pi$$
$$510$$ 0 0
$$511$$ −15700.7 −1.35922
$$512$$ 0 0
$$513$$ −1274.32 −0.109674
$$514$$ 0 0
$$515$$ −761.786 −0.0651812
$$516$$ 0 0
$$517$$ 0 0
$$518$$ 0 0
$$519$$ −12165.4 −1.02890
$$520$$ 0 0
$$521$$ −8815.08 −0.741258 −0.370629 0.928781i $$-0.620858\pi$$
−0.370629 + 0.928781i $$0.620858\pi$$
$$522$$ 0 0
$$523$$ 5258.60 0.439660 0.219830 0.975538i $$-0.429450\pi$$
0.219830 + 0.975538i $$0.429450\pi$$
$$524$$ 0 0
$$525$$ 10329.0 0.858657
$$526$$ 0 0
$$527$$ 10294.3 0.850903
$$528$$ 0 0
$$529$$ −8993.02 −0.739132
$$530$$ 0 0
$$531$$ −5777.03 −0.472131
$$532$$ 0 0
$$533$$ −10707.8 −0.870182
$$534$$ 0 0
$$535$$ −4700.65 −0.379864
$$536$$ 0 0
$$537$$ −10554.4 −0.848153
$$538$$ 0 0
$$539$$ 0 0
$$540$$ 0 0
$$541$$ 2737.07 0.217515 0.108758 0.994068i $$-0.465313\pi$$
0.108758 + 0.994068i $$0.465313\pi$$
$$542$$ 0 0
$$543$$ 5472.06 0.432465
$$544$$ 0 0
$$545$$ −3771.69 −0.296443
$$546$$ 0 0
$$547$$ −10663.8 −0.833549 −0.416775 0.909010i $$-0.636840\pi$$
−0.416775 + 0.909010i $$0.636840\pi$$
$$548$$ 0 0
$$549$$ −6824.07 −0.530500
$$550$$ 0 0
$$551$$ 9192.34 0.710720
$$552$$ 0 0
$$553$$ 34498.2 2.65283
$$554$$ 0 0
$$555$$ 2896.27 0.221513
$$556$$ 0 0
$$557$$ −5987.45 −0.455469 −0.227735 0.973723i $$-0.573132\pi$$
−0.227735 + 0.973723i $$0.573132\pi$$
$$558$$ 0 0
$$559$$ 2807.67 0.212436
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ 18636.9 1.39512 0.697559 0.716528i $$-0.254270\pi$$
0.697559 + 0.716528i $$0.254270\pi$$
$$564$$ 0 0
$$565$$ 1151.30 0.0857264
$$566$$ 0 0
$$567$$ −2418.04 −0.179098
$$568$$ 0 0
$$569$$ 18099.5 1.33352 0.666759 0.745273i $$-0.267681\pi$$
0.666759 + 0.745273i $$0.267681\pi$$
$$570$$ 0 0
$$571$$ −21590.0 −1.58234 −0.791168 0.611599i $$-0.790526\pi$$
−0.791168 + 0.611599i $$0.790526\pi$$
$$572$$ 0 0
$$573$$ −3812.08 −0.277926
$$574$$ 0 0
$$575$$ −6497.71 −0.471258
$$576$$ 0 0
$$577$$ −22439.3 −1.61900 −0.809499 0.587121i $$-0.800261\pi$$
−0.809499 + 0.587121i $$0.800261\pi$$
$$578$$ 0 0
$$579$$ 3084.93 0.221425
$$580$$ 0 0
$$581$$ 9862.60 0.704250
$$582$$ 0 0
$$583$$ 0 0
$$584$$ 0 0
$$585$$ 901.070 0.0636832
$$586$$ 0 0
$$587$$ −24440.1 −1.71849 −0.859244 0.511566i $$-0.829066\pi$$
−0.859244 + 0.511566i $$0.829066\pi$$
$$588$$ 0 0
$$589$$ −6902.56 −0.482878
$$590$$ 0 0
$$591$$ 10625.3 0.739539
$$592$$ 0 0
$$593$$ −2703.41 −0.187211 −0.0936053 0.995609i $$-0.529839\pi$$
−0.0936053 + 0.995609i $$0.529839\pi$$
$$594$$ 0 0
$$595$$ −6532.80 −0.450116
$$596$$ 0 0
$$597$$ 7849.67 0.538133
$$598$$ 0 0
$$599$$ 14555.5 0.992856 0.496428 0.868078i $$-0.334645\pi$$
0.496428 + 0.868078i $$0.334645\pi$$
$$600$$ 0 0
$$601$$ 25383.3 1.72281 0.861403 0.507922i $$-0.169586\pi$$
0.861403 + 0.507922i $$0.169586\pi$$
$$602$$ 0 0
$$603$$ 1109.36 0.0749197
$$604$$ 0 0
$$605$$ 0 0
$$606$$ 0 0
$$607$$ −17684.1 −1.18250 −0.591249 0.806489i $$-0.701365\pi$$
−0.591249 + 0.806489i $$0.701365\pi$$
$$608$$ 0 0
$$609$$ 17442.6 1.16061
$$610$$ 0 0
$$611$$ −3255.77 −0.215572
$$612$$ 0 0
$$613$$ 20806.2 1.37089 0.685444 0.728125i $$-0.259608\pi$$
0.685444 + 0.728125i $$0.259608\pi$$
$$614$$ 0 0
$$615$$ −3101.31 −0.203344
$$616$$ 0 0
$$617$$ −25625.0 −1.67200 −0.835999 0.548731i $$-0.815111\pi$$
−0.835999 + 0.548731i $$0.815111\pi$$
$$618$$ 0 0
$$619$$ −14375.1 −0.933417 −0.466709 0.884411i $$-0.654560\pi$$
−0.466709 + 0.884411i $$0.654560\pi$$
$$620$$ 0 0
$$621$$ 1521.13 0.0982944
$$622$$ 0 0
$$623$$ 8974.53 0.577138
$$624$$ 0 0
$$625$$ 12093.7 0.774000
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ 21857.5 1.38556
$$630$$ 0 0
$$631$$ 6824.66 0.430563 0.215282 0.976552i $$-0.430933\pi$$
0.215282 + 0.976552i $$0.430933\pi$$
$$632$$ 0 0
$$633$$ −9090.46 −0.570795
$$634$$ 0 0
$$635$$ 559.030 0.0349361
$$636$$ 0 0
$$637$$ 17652.6 1.09799
$$638$$ 0 0
$$639$$ −9596.99 −0.594133
$$640$$ 0 0
$$641$$ 24879.1 1.53302 0.766509 0.642233i $$-0.221992\pi$$
0.766509 + 0.642233i $$0.221992\pi$$
$$642$$ 0 0
$$643$$ −20537.3 −1.25959 −0.629793 0.776763i $$-0.716860\pi$$
−0.629793 + 0.776763i $$0.716860\pi$$
$$644$$ 0 0
$$645$$ 813.187 0.0496422
$$646$$ 0 0
$$647$$ 6296.05 0.382571 0.191285 0.981534i $$-0.438734\pi$$
0.191285 + 0.981534i $$0.438734\pi$$
$$648$$ 0 0
$$649$$ 0 0
$$650$$ 0 0
$$651$$ −13097.7 −0.788541
$$652$$ 0 0
$$653$$ −16220.2 −0.972047 −0.486024 0.873946i $$-0.661553\pi$$
−0.486024 + 0.873946i $$0.661553\pi$$
$$654$$ 0 0
$$655$$ 3564.11 0.212613
$$656$$ 0 0
$$657$$ 4733.51 0.281084
$$658$$ 0 0
$$659$$ −7204.55 −0.425872 −0.212936 0.977066i $$-0.568303\pi$$
−0.212936 + 0.977066i $$0.568303\pi$$
$$660$$ 0 0
$$661$$ 10969.0 0.645456 0.322728 0.946492i $$-0.395400\pi$$
0.322728 + 0.946492i $$0.395400\pi$$
$$662$$ 0 0
$$663$$ 6800.16 0.398335
$$664$$ 0 0
$$665$$ 4380.40 0.255436
$$666$$ 0 0
$$667$$ −10972.7 −0.636978
$$668$$ 0 0
$$669$$ 1338.03 0.0773264
$$670$$ 0 0
$$671$$ 0 0
$$672$$ 0 0
$$673$$ 11005.3 0.630345 0.315172 0.949034i $$-0.397938\pi$$
0.315172 + 0.949034i $$0.397938\pi$$
$$674$$ 0 0
$$675$$ −3114.02 −0.177569
$$676$$ 0 0
$$677$$ 24640.3 1.39882 0.699411 0.714720i $$-0.253446\pi$$
0.699411 + 0.714720i $$0.253446\pi$$
$$678$$ 0 0
$$679$$ 46217.7 2.61218
$$680$$ 0 0
$$681$$ 11312.2 0.636540
$$682$$ 0 0
$$683$$ −1240.24 −0.0694825 −0.0347413 0.999396i $$-0.511061\pi$$
−0.0347413 + 0.999396i $$0.511061\pi$$
$$684$$ 0 0
$$685$$ 6382.90 0.356027
$$686$$ 0 0
$$687$$ −16935.7 −0.940520
$$688$$ 0 0
$$689$$ 11784.1 0.651580
$$690$$ 0 0
$$691$$ −13126.4 −0.722648 −0.361324 0.932440i $$-0.617675\pi$$
−0.361324 + 0.932440i $$0.617675\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ 670.132 0.0365749
$$696$$ 0 0
$$697$$ −23404.8 −1.27191
$$698$$ 0 0
$$699$$ −2053.86 −0.111136
$$700$$ 0 0
$$701$$ 15373.2 0.828299 0.414149 0.910209i $$-0.364079\pi$$
0.414149 + 0.910209i $$0.364079\pi$$
$$702$$ 0 0
$$703$$ −14656.0 −0.786287
$$704$$ 0 0
$$705$$ −942.970 −0.0503749
$$706$$ 0 0
$$707$$ −21363.6 −1.13644
$$708$$ 0 0
$$709$$ 20921.9 1.10823 0.554117 0.832439i $$-0.313056\pi$$
0.554117 + 0.832439i $$0.313056\pi$$
$$710$$ 0 0
$$711$$ −10400.6 −0.548600
$$712$$ 0 0
$$713$$ 8239.43 0.432776
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ 8866.99 0.461846
$$718$$ 0 0
$$719$$ 11994.3 0.622131 0.311065 0.950389i $$-0.399314\pi$$
0.311065 + 0.950389i $$0.399314\pi$$
$$720$$ 0 0
$$721$$ 7314.64 0.377824
$$722$$ 0 0
$$723$$ 1444.81 0.0743197
$$724$$ 0 0
$$725$$ 22463.1 1.15070
$$726$$ 0 0
$$727$$ −35755.6 −1.82408 −0.912038 0.410106i $$-0.865492\pi$$
−0.912038 + 0.410106i $$0.865492\pi$$
$$728$$ 0 0
$$729$$ 729.000 0.0370370
$$730$$ 0 0
$$731$$ 6136.93 0.310509
$$732$$ 0 0
$$733$$ −13984.4 −0.704672 −0.352336 0.935873i $$-0.614613\pi$$
−0.352336 + 0.935873i $$0.614613\pi$$
$$734$$ 0 0
$$735$$ 5112.72 0.256579
$$736$$ 0 0
$$737$$ 0 0
$$738$$ 0 0
$$739$$ 13255.7 0.659837 0.329919 0.944009i $$-0.392979\pi$$
0.329919 + 0.944009i $$0.392979\pi$$
$$740$$ 0 0
$$741$$ −4559.67 −0.226051
$$742$$ 0 0
$$743$$ 27444.5 1.35510 0.677552 0.735475i $$-0.263041\pi$$
0.677552 + 0.735475i $$0.263041\pi$$
$$744$$ 0 0
$$745$$ −3604.90 −0.177280
$$746$$ 0 0
$$747$$ −2973.41 −0.145638
$$748$$ 0 0
$$749$$ 45135.5 2.20189
$$750$$ 0 0
$$751$$ −10600.1 −0.515053 −0.257527 0.966271i $$-0.582907\pi$$
−0.257527 + 0.966271i $$0.582907\pi$$
$$752$$ 0 0
$$753$$ −22683.5 −1.09778
$$754$$ 0 0
$$755$$ −8466.26 −0.408104
$$756$$ 0 0
$$757$$ 22837.7 1.09650 0.548250 0.836314i $$-0.315294\pi$$
0.548250 + 0.836314i $$0.315294\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 35381.9 1.68540 0.842701 0.538381i $$-0.180964\pi$$
0.842701 + 0.538381i $$0.180964\pi$$
$$762$$ 0 0
$$763$$ 36215.6 1.71834
$$764$$ 0 0
$$765$$ 1969.53 0.0930831
$$766$$ 0 0
$$767$$ −20670.9 −0.973119
$$768$$ 0 0
$$769$$ 21448.1 1.00577 0.502885 0.864353i $$-0.332272\pi$$
0.502885 + 0.864353i $$0.332272\pi$$
$$770$$ 0 0
$$771$$ −7686.05 −0.359022
$$772$$ 0 0
$$773$$ 16900.6 0.786380 0.393190 0.919457i $$-0.371372\pi$$
0.393190 + 0.919457i $$0.371372\pi$$
$$774$$ 0 0
$$775$$ −16867.6 −0.781809
$$776$$ 0 0
$$777$$ −27809.9 −1.28401
$$778$$ 0 0
$$779$$ 15693.5 0.721794
$$780$$ 0 0
$$781$$ 0 0
$$782$$ 0 0
$$783$$ −5258.65 −0.240011
$$784$$ 0 0
$$785$$ 7812.94 0.355230
$$786$$ 0 0
$$787$$ 7633.16 0.345734 0.172867 0.984945i $$-0.444697\pi$$
0.172867 + 0.984945i $$0.444697\pi$$
$$788$$ 0 0
$$789$$ −22554.3 −1.01769
$$790$$ 0 0
$$791$$ −11054.7 −0.496916
$$792$$ 0 0
$$793$$ −24417.3 −1.09342
$$794$$ 0 0
$$795$$ 3413.03 0.152261
$$796$$ 0 0
$$797$$ 488.565 0.0217138 0.0108569 0.999941i $$-0.496544\pi$$
0.0108569 + 0.999941i $$0.496544\pi$$
$$798$$ 0 0
$$799$$ −7116.37 −0.315093
$$800$$ 0 0
$$801$$ −2705.67 −0.119351
$$802$$ 0 0
$$803$$ 0 0
$$804$$ 0 0
$$805$$ −5228.79 −0.228932
$$806$$ 0 0
$$807$$ 25019.9 1.09138
$$808$$ 0 0
$$809$$ 27235.7 1.18363 0.591814 0.806075i $$-0.298412\pi$$
0.591814 + 0.806075i $$0.298412\pi$$
$$810$$ 0 0
$$811$$ −7346.61 −0.318094 −0.159047 0.987271i $$-0.550842\pi$$
−0.159047 + 0.987271i $$0.550842\pi$$
$$812$$ 0 0
$$813$$ −6015.42 −0.259496
$$814$$ 0 0
$$815$$ 3036.70 0.130516
$$816$$ 0 0
$$817$$ −4114.95 −0.176211
$$818$$ 0 0
$$819$$ −8652.04 −0.369141
$$820$$ 0 0
$$821$$ 30190.3 1.28337 0.641686 0.766967i $$-0.278235\pi$$
0.641686 + 0.766967i $$0.278235\pi$$
$$822$$ 0 0
$$823$$ 21592.1 0.914524 0.457262 0.889332i $$-0.348830\pi$$
0.457262 + 0.889332i $$0.348830\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 20421.1 0.858659 0.429330 0.903148i $$-0.358750\pi$$
0.429330 + 0.903148i $$0.358750\pi$$
$$828$$ 0 0
$$829$$ 28570.2 1.19696 0.598482 0.801136i $$-0.295771\pi$$
0.598482 + 0.801136i $$0.295771\pi$$
$$830$$ 0 0
$$831$$ 3440.83 0.143635
$$832$$ 0 0
$$833$$ 38584.5 1.60489
$$834$$ 0 0
$$835$$ −3474.98 −0.144020
$$836$$ 0 0
$$837$$ 3948.74 0.163069
$$838$$ 0 0
$$839$$ −23875.5 −0.982447 −0.491224 0.871034i $$-0.663450\pi$$
−0.491224 + 0.871034i $$0.663450\pi$$
$$840$$ 0 0
$$841$$ 13544.4 0.555348
$$842$$ 0 0
$$843$$ 11537.9 0.471395
$$844$$ 0 0
$$845$$ −3606.32 −0.146818
$$846$$ 0 0
$$847$$ 0 0
$$848$$ 0 0
$$849$$ −3235.14 −0.130777
$$850$$ 0 0
$$851$$ 17494.5 0.704704
$$852$$ 0 0
$$853$$ −28777.2 −1.15511 −0.577557 0.816351i $$-0.695994\pi$$
−0.577557 + 0.816351i $$0.695994\pi$$
$$854$$ 0 0
$$855$$ −1320.62 −0.0528236
$$856$$ 0 0
$$857$$ −25020.3 −0.997291 −0.498646 0.866806i $$-0.666169\pi$$
−0.498646 + 0.866806i $$0.666169\pi$$
$$858$$ 0 0
$$859$$ 40749.7 1.61858 0.809290 0.587409i $$-0.199852\pi$$
0.809290 + 0.587409i $$0.199852\pi$$
$$860$$ 0 0
$$861$$ 29778.6 1.17869
$$862$$ 0 0
$$863$$ 43635.0 1.72115 0.860575 0.509324i $$-0.170104\pi$$
0.860575 + 0.509324i $$0.170104\pi$$
$$864$$ 0 0
$$865$$ −12607.4 −0.495564
$$866$$ 0 0
$$867$$ 124.576 0.00487984
$$868$$ 0 0
$$869$$ 0 0
$$870$$ 0 0
$$871$$ 3969.41 0.154418
$$872$$ 0 0
$$873$$ −13933.9 −0.540195
$$874$$ 0 0
$$875$$ 22305.6 0.861791
$$876$$ 0 0
$$877$$ 35002.0 1.34770 0.673850 0.738868i $$-0.264639\pi$$
0.673850 + 0.738868i $$0.264639\pi$$
$$878$$ 0 0
$$879$$ 11935.0 0.457973
$$880$$ 0 0
$$881$$ 5760.33 0.220284 0.110142 0.993916i $$-0.464869\pi$$
0.110142 + 0.993916i $$0.464869\pi$$
$$882$$ 0 0
$$883$$ −14286.6 −0.544487 −0.272244 0.962228i $$-0.587766\pi$$
−0.272244 + 0.962228i $$0.587766\pi$$
$$884$$ 0 0
$$885$$ −5986.91 −0.227398
$$886$$ 0 0
$$887$$ 9440.24 0.357353 0.178677 0.983908i $$-0.442818\pi$$
0.178677 + 0.983908i $$0.442818\pi$$
$$888$$ 0 0
$$889$$ −5367.79 −0.202508
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 0 0
$$893$$ 4771.69 0.178811
$$894$$ 0 0
$$895$$ −10937.9 −0.408506
$$896$$ 0 0
$$897$$ 5442.78 0.202596
$$898$$ 0 0
$$899$$ −28484.3 −1.05674
$$900$$ 0 0
$$901$$ 25757.3 0.952387
$$902$$ 0 0
$$903$$ −7808.19 −0.287752
$$904$$ 0 0
$$905$$ 5670.86 0.208294
$$906$$ 0 0
$$907$$ −49416.3 −1.80909 −0.904543 0.426383i $$-0.859788\pi$$
−0.904543 + 0.426383i $$0.859788\pi$$
$$908$$ 0 0
$$909$$ 6440.77 0.235013
$$910$$ 0 0
$$911$$ −5531.63 −0.201176 −0.100588 0.994928i $$-0.532072\pi$$
−0.100588 + 0.994928i $$0.532072\pi$$
$$912$$ 0 0
$$913$$ 0 0
$$914$$ 0 0
$$915$$ −7071.99 −0.255511
$$916$$ 0 0
$$917$$ −34222.4 −1.23241
$$918$$ 0 0
$$919$$ −23529.6 −0.844582 −0.422291 0.906460i $$-0.638774\pi$$
−0.422291 + 0.906460i $$0.638774\pi$$
$$920$$ 0 0
$$921$$ 6498.72 0.232508
$$922$$ 0 0
$$923$$ −34339.1 −1.22458
$$924$$ 0 0
$$925$$ −35814.3 −1.27305
$$926$$ 0 0
$$927$$ −2205.24 −0.0781334
$$928$$ 0 0
$$929$$ 32996.3 1.16531 0.582656 0.812719i $$-0.302014\pi$$
0.582656 + 0.812719i $$0.302014\pi$$
$$930$$ 0 0
$$931$$ −25871.8 −0.910757
$$932$$ 0 0
$$933$$ 5934.97 0.208255
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ −28998.0 −1.01102 −0.505508 0.862822i $$-0.668695\pi$$
−0.505508 + 0.862822i $$0.668695\pi$$
$$938$$ 0 0
$$939$$ −14300.9 −0.497011
$$940$$ 0 0
$$941$$ −7317.71 −0.253507 −0.126754 0.991934i $$-0.540456\pi$$
−0.126754 + 0.991934i $$0.540456\pi$$
$$942$$ 0 0
$$943$$ −18733.0 −0.646903
$$944$$ 0 0
$$945$$ −2505.89 −0.0862610
$$946$$ 0 0
$$947$$ −11124.6 −0.381732 −0.190866 0.981616i $$-0.561130\pi$$
−0.190866 + 0.981616i $$0.561130\pi$$
$$948$$ 0 0
$$949$$ 16937.1 0.579347
$$950$$ 0 0
$$951$$ −10622.1 −0.362191
$$952$$ 0 0
$$953$$ 43980.0 1.49491 0.747457 0.664310i $$-0.231275\pi$$
0.747457 + 0.664310i $$0.231275\pi$$
$$954$$ 0 0
$$955$$ −3950.57 −0.133861
$$956$$ 0 0
$$957$$ 0 0
$$958$$ 0 0
$$959$$ −61288.4 −2.06372
$$960$$ 0 0
$$961$$ −8402.02 −0.282032
$$962$$ 0 0
$$963$$ −13607.6 −0.455347
$$964$$ 0 0
$$965$$ 3197.00 0.106648
$$966$$ 0 0
$$967$$ 15009.5 0.499143 0.249572 0.968356i $$-0.419710\pi$$
0.249572 + 0.968356i $$0.419710\pi$$
$$968$$ 0 0
$$969$$ −9966.38 −0.330409
$$970$$ 0 0
$$971$$ −3141.24 −0.103818 −0.0519089 0.998652i $$-0.516531\pi$$
−0.0519089 + 0.998652i $$0.516531\pi$$
$$972$$ 0 0
$$973$$ −6434.58 −0.212007
$$974$$ 0 0
$$975$$ −11142.3 −0.365990
$$976$$ 0 0
$$977$$ −46209.5 −1.51318 −0.756588 0.653892i $$-0.773135\pi$$
−0.756588 + 0.653892i $$0.773135\pi$$
$$978$$ 0 0
$$979$$ 0 0
$$980$$ 0 0
$$981$$ −10918.4 −0.355349
$$982$$ 0 0
$$983$$ 11471.1 0.372198 0.186099 0.982531i $$-0.440415\pi$$
0.186099 + 0.982531i $$0.440415\pi$$
$$984$$ 0 0
$$985$$ 11011.3 0.356193
$$986$$ 0 0
$$987$$ 9054.36 0.292000
$$988$$ 0 0
$$989$$ 4911.93 0.157928
$$990$$ 0 0
$$991$$ −39758.4 −1.27444 −0.637218 0.770684i $$-0.719915\pi$$
−0.637218 + 0.770684i $$0.719915\pi$$
$$992$$ 0 0
$$993$$ −32783.2 −1.04768
$$994$$ 0 0
$$995$$ 8134.85 0.259188
$$996$$ 0 0
$$997$$ 15019.4 0.477099 0.238550 0.971130i $$-0.423328\pi$$
0.238550 + 0.971130i $$0.423328\pi$$
$$998$$ 0 0
$$999$$ 8384.22 0.265531
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 1452.4.a.t.1.4 6
11.2 odd 10 132.4.i.c.37.2 yes 12
11.6 odd 10 132.4.i.c.25.2 12
11.10 odd 2 1452.4.a.u.1.4 6
33.2 even 10 396.4.j.c.37.2 12
33.17 even 10 396.4.j.c.289.2 12

By twisted newform
Twist Min Dim Char Parity Ord Type
132.4.i.c.25.2 12 11.6 odd 10
132.4.i.c.37.2 yes 12 11.2 odd 10
396.4.j.c.37.2 12 33.2 even 10
396.4.j.c.289.2 12 33.17 even 10
1452.4.a.t.1.4 6 1.1 even 1 trivial
1452.4.a.u.1.4 6 11.10 odd 2