# Properties

 Label 950.4.b.i.799.3 Level $950$ Weight $4$ Character 950.799 Analytic conductor $56.052$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [950,4,Mod(799,950)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(950, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("950.799");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$950 = 2 \cdot 5^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 950.b (of order $$2$$, degree $$1$$, not minimal)

## 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: no Analytic conductor: $$56.0518145055$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(i, \sqrt{73})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} + 37x^{2} + 324$$ x^4 + 37*x^2 + 324 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 38) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 799.3 Root $$-3.77200i$$ of defining polynomial Character $$\chi$$ $$=$$ 950.799 Dual form 950.4.b.i.799.2

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+2.00000i q^{2} -8.77200i q^{3} -4.00000 q^{4} +17.5440 q^{6} -26.0880i q^{7} -8.00000i q^{8} -49.9480 q^{9} +O(q^{10})$$ $$q+2.00000i q^{2} -8.77200i q^{3} -4.00000 q^{4} +17.5440 q^{6} -26.0880i q^{7} -8.00000i q^{8} -49.9480 q^{9} -4.22800 q^{11} +35.0880i q^{12} -64.0360i q^{13} +52.1760 q^{14} +16.0000 q^{16} -48.5440i q^{17} -99.8960i q^{18} -19.0000 q^{19} -228.844 q^{21} -8.45600i q^{22} -92.0360i q^{23} -70.1760 q^{24} +128.072 q^{26} +201.300i q^{27} +104.352i q^{28} +88.2120 q^{29} -81.9681 q^{31} +32.0000i q^{32} +37.0880i q^{33} +97.0880 q^{34} +199.792 q^{36} -23.6161i q^{37} -38.0000i q^{38} -561.724 q^{39} +17.7200 q^{41} -457.688i q^{42} -368.404i q^{43} +16.9120 q^{44} +184.072 q^{46} -497.812i q^{47} -140.352i q^{48} -337.584 q^{49} -425.828 q^{51} +256.144i q^{52} +536.876i q^{53} -402.600 q^{54} -208.704 q^{56} +166.668i q^{57} +176.424i q^{58} +36.7000 q^{59} +630.692 q^{61} -163.936i q^{62} +1303.04i q^{63} -64.0000 q^{64} -74.1760 q^{66} +282.556i q^{67} +194.176i q^{68} -807.340 q^{69} +595.552 q^{71} +399.584i q^{72} +597.048i q^{73} +47.2321 q^{74} +76.0000 q^{76} +110.300i q^{77} -1123.45i q^{78} -427.224 q^{79} +417.208 q^{81} +35.4400i q^{82} -493.768i q^{83} +915.376 q^{84} +736.808 q^{86} -773.796i q^{87} +33.8240i q^{88} +921.136 q^{89} -1670.57 q^{91} +368.144i q^{92} +719.024i q^{93} +995.624 q^{94} +280.704 q^{96} +1082.74i q^{97} -675.168i q^{98} +211.180 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 16 q^{4} + 36 q^{6} - 46 q^{9}+O(q^{10})$$ 4 * q - 16 * q^4 + 36 * q^6 - 46 * q^9 $$4 q - 16 q^{4} + 36 q^{6} - 46 q^{9} - 34 q^{11} + 72 q^{14} + 64 q^{16} - 76 q^{19} - 454 q^{21} - 144 q^{24} + 68 q^{26} - 6 q^{29} + 424 q^{31} + 320 q^{34} + 184 q^{36} - 1102 q^{39} - 100 q^{41} + 136 q^{44} + 292 q^{46} - 120 q^{49} - 866 q^{51} - 756 q^{54} - 288 q^{56} + 574 q^{59} + 626 q^{61} - 256 q^{64} - 160 q^{66} - 1606 q^{69} + 400 q^{71} - 768 q^{74} + 304 q^{76} - 2700 q^{79} + 2284 q^{81} + 1816 q^{84} + 2708 q^{86} + 472 q^{89} - 4102 q^{91} + 1556 q^{94} + 576 q^{96} - 266 q^{99}+O(q^{100})$$ 4 * q - 16 * q^4 + 36 * q^6 - 46 * q^9 - 34 * q^11 + 72 * q^14 + 64 * q^16 - 76 * q^19 - 454 * q^21 - 144 * q^24 + 68 * q^26 - 6 * q^29 + 424 * q^31 + 320 * q^34 + 184 * q^36 - 1102 * q^39 - 100 * q^41 + 136 * q^44 + 292 * q^46 - 120 * q^49 - 866 * q^51 - 756 * q^54 - 288 * q^56 + 574 * q^59 + 626 * q^61 - 256 * q^64 - 160 * q^66 - 1606 * q^69 + 400 * q^71 - 768 * q^74 + 304 * q^76 - 2700 * q^79 + 2284 * q^81 + 1816 * q^84 + 2708 * q^86 + 472 * q^89 - 4102 * q^91 + 1556 * q^94 + 576 * q^96 - 266 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/950\mathbb{Z}\right)^\times$$.

 $$n$$ $$77$$ $$401$$ $$\chi(n)$$ $$-1$$ $$1$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 2.00000i 0.707107i
$$3$$ − 8.77200i − 1.68817i −0.536207 0.844086i $$-0.680144\pi$$
0.536207 0.844086i $$-0.319856\pi$$
$$4$$ −4.00000 −0.500000
$$5$$ 0 0
$$6$$ 17.5440 1.19372
$$7$$ − 26.0880i − 1.40862i −0.709893 0.704310i $$-0.751257\pi$$
0.709893 0.704310i $$-0.248743\pi$$
$$8$$ − 8.00000i − 0.353553i
$$9$$ −49.9480 −1.84993
$$10$$ 0 0
$$11$$ −4.22800 −0.115890 −0.0579450 0.998320i $$-0.518455\pi$$
−0.0579450 + 0.998320i $$0.518455\pi$$
$$12$$ 35.0880i 0.844086i
$$13$$ − 64.0360i − 1.36618i −0.730332 0.683092i $$-0.760635\pi$$
0.730332 0.683092i $$-0.239365\pi$$
$$14$$ 52.1760 0.996045
$$15$$ 0 0
$$16$$ 16.0000 0.250000
$$17$$ − 48.5440i − 0.692568i −0.938130 0.346284i $$-0.887443\pi$$
0.938130 0.346284i $$-0.112557\pi$$
$$18$$ − 99.8960i − 1.30810i
$$19$$ −19.0000 −0.229416
$$20$$ 0 0
$$21$$ −228.844 −2.37799
$$22$$ − 8.45600i − 0.0819466i
$$23$$ − 92.0360i − 0.834384i −0.908818 0.417192i $$-0.863014\pi$$
0.908818 0.417192i $$-0.136986\pi$$
$$24$$ −70.1760 −0.596859
$$25$$ 0 0
$$26$$ 128.072 0.966038
$$27$$ 201.300i 1.43482i
$$28$$ 104.352i 0.704310i
$$29$$ 88.2120 0.564847 0.282424 0.959290i $$-0.408862\pi$$
0.282424 + 0.959290i $$0.408862\pi$$
$$30$$ 0 0
$$31$$ −81.9681 −0.474900 −0.237450 0.971400i $$-0.576312\pi$$
−0.237450 + 0.971400i $$0.576312\pi$$
$$32$$ 32.0000i 0.176777i
$$33$$ 37.0880i 0.195642i
$$34$$ 97.0880 0.489719
$$35$$ 0 0
$$36$$ 199.792 0.924963
$$37$$ − 23.6161i − 0.104931i −0.998623 0.0524656i $$-0.983292\pi$$
0.998623 0.0524656i $$-0.0167080\pi$$
$$38$$ − 38.0000i − 0.162221i
$$39$$ −561.724 −2.30636
$$40$$ 0 0
$$41$$ 17.7200 0.0674976 0.0337488 0.999430i $$-0.489255\pi$$
0.0337488 + 0.999430i $$0.489255\pi$$
$$42$$ − 457.688i − 1.68150i
$$43$$ − 368.404i − 1.30654i −0.757126 0.653268i $$-0.773397\pi$$
0.757126 0.653268i $$-0.226603\pi$$
$$44$$ 16.9120 0.0579450
$$45$$ 0 0
$$46$$ 184.072 0.589999
$$47$$ − 497.812i − 1.54497i −0.635036 0.772483i $$-0.719015\pi$$
0.635036 0.772483i $$-0.280985\pi$$
$$48$$ − 140.352i − 0.422043i
$$49$$ −337.584 −0.984210
$$50$$ 0 0
$$51$$ −425.828 −1.16917
$$52$$ 256.144i 0.683092i
$$53$$ 536.876i 1.39143i 0.718320 + 0.695713i $$0.244911\pi$$
−0.718320 + 0.695713i $$0.755089\pi$$
$$54$$ −402.600 −1.01457
$$55$$ 0 0
$$56$$ −208.704 −0.498022
$$57$$ 166.668i 0.387293i
$$58$$ 176.424i 0.399407i
$$59$$ 36.7000 0.0809818 0.0404909 0.999180i $$-0.487108\pi$$
0.0404909 + 0.999180i $$0.487108\pi$$
$$60$$ 0 0
$$61$$ 630.692 1.32380 0.661901 0.749592i $$-0.269750\pi$$
0.661901 + 0.749592i $$0.269750\pi$$
$$62$$ − 163.936i − 0.335805i
$$63$$ 1303.04i 2.60584i
$$64$$ −64.0000 −0.125000
$$65$$ 0 0
$$66$$ −74.1760 −0.138340
$$67$$ 282.556i 0.515219i 0.966249 + 0.257610i $$0.0829349\pi$$
−0.966249 + 0.257610i $$0.917065\pi$$
$$68$$ 194.176i 0.346284i
$$69$$ −807.340 −1.40858
$$70$$ 0 0
$$71$$ 595.552 0.995480 0.497740 0.867326i $$-0.334163\pi$$
0.497740 + 0.867326i $$0.334163\pi$$
$$72$$ 399.584i 0.654048i
$$73$$ 597.048i 0.957250i 0.878020 + 0.478625i $$0.158865\pi$$
−0.878020 + 0.478625i $$0.841135\pi$$
$$74$$ 47.2321 0.0741976
$$75$$ 0 0
$$76$$ 76.0000 0.114708
$$77$$ 110.300i 0.163245i
$$78$$ − 1123.45i − 1.63084i
$$79$$ −427.224 −0.608436 −0.304218 0.952602i $$-0.598395\pi$$
−0.304218 + 0.952602i $$0.598395\pi$$
$$80$$ 0 0
$$81$$ 417.208 0.572302
$$82$$ 35.4400i 0.0477280i
$$83$$ − 493.768i − 0.652989i −0.945199 0.326495i $$-0.894132\pi$$
0.945199 0.326495i $$-0.105868\pi$$
$$84$$ 915.376 1.18900
$$85$$ 0 0
$$86$$ 736.808 0.923861
$$87$$ − 773.796i − 0.953559i
$$88$$ 33.8240i 0.0409733i
$$89$$ 921.136 1.09708 0.548541 0.836124i $$-0.315184\pi$$
0.548541 + 0.836124i $$0.315184\pi$$
$$90$$ 0 0
$$91$$ −1670.57 −1.92443
$$92$$ 368.144i 0.417192i
$$93$$ 719.024i 0.801713i
$$94$$ 995.624 1.09246
$$95$$ 0 0
$$96$$ 280.704 0.298430
$$97$$ 1082.74i 1.13336i 0.823938 + 0.566680i $$0.191773\pi$$
−0.823938 + 0.566680i $$0.808227\pi$$
$$98$$ − 675.168i − 0.695942i
$$99$$ 211.180 0.214388
$$100$$ 0 0
$$101$$ −712.448 −0.701893 −0.350947 0.936395i $$-0.614140\pi$$
−0.350947 + 0.936395i $$0.614140\pi$$
$$102$$ − 851.656i − 0.826731i
$$103$$ 26.4797i 0.0253313i 0.999920 + 0.0126656i $$0.00403171\pi$$
−0.999920 + 0.0126656i $$0.995968\pi$$
$$104$$ −512.288 −0.483019
$$105$$ 0 0
$$106$$ −1073.75 −0.983887
$$107$$ − 740.996i − 0.669484i −0.942310 0.334742i $$-0.891351\pi$$
0.942310 0.334742i $$-0.108649\pi$$
$$108$$ − 805.200i − 0.717411i
$$109$$ 1983.08 1.74261 0.871304 0.490744i $$-0.163275\pi$$
0.871304 + 0.490744i $$0.163275\pi$$
$$110$$ 0 0
$$111$$ −207.160 −0.177142
$$112$$ − 417.408i − 0.352155i
$$113$$ 718.720i 0.598332i 0.954201 + 0.299166i $$0.0967085\pi$$
−0.954201 + 0.299166i $$0.903292\pi$$
$$114$$ −333.336 −0.273858
$$115$$ 0 0
$$116$$ −352.848 −0.282424
$$117$$ 3198.47i 2.52734i
$$118$$ 73.3999i 0.0572628i
$$119$$ −1266.42 −0.975565
$$120$$ 0 0
$$121$$ −1313.12 −0.986570
$$122$$ 1261.38i 0.936069i
$$123$$ − 155.440i − 0.113948i
$$124$$ 327.872 0.237450
$$125$$ 0 0
$$126$$ −2606.09 −1.84261
$$127$$ 2610.72i 1.82413i 0.410050 + 0.912063i $$0.365511\pi$$
−0.410050 + 0.912063i $$0.634489\pi$$
$$128$$ − 128.000i − 0.0883883i
$$129$$ −3231.64 −2.20566
$$130$$ 0 0
$$131$$ −1216.69 −0.811472 −0.405736 0.913990i $$-0.632985\pi$$
−0.405736 + 0.913990i $$0.632985\pi$$
$$132$$ − 148.352i − 0.0978211i
$$133$$ 495.672i 0.323160i
$$134$$ −565.112 −0.364315
$$135$$ 0 0
$$136$$ −388.352 −0.244860
$$137$$ 1170.67i 0.730053i 0.930997 + 0.365026i $$0.118940\pi$$
−0.930997 + 0.365026i $$0.881060\pi$$
$$138$$ − 1614.68i − 0.996020i
$$139$$ 271.083 0.165417 0.0827086 0.996574i $$-0.473643\pi$$
0.0827086 + 0.996574i $$0.473643\pi$$
$$140$$ 0 0
$$141$$ −4366.81 −2.60817
$$142$$ 1191.10i 0.703910i
$$143$$ 270.744i 0.158327i
$$144$$ −799.168 −0.462482
$$145$$ 0 0
$$146$$ −1194.10 −0.676878
$$147$$ 2961.29i 1.66152i
$$148$$ 94.4642i 0.0524656i
$$149$$ −1841.19 −1.01232 −0.506161 0.862439i $$-0.668936\pi$$
−0.506161 + 0.862439i $$0.668936\pi$$
$$150$$ 0 0
$$151$$ 3322.32 1.79051 0.895254 0.445557i $$-0.146994\pi$$
0.895254 + 0.445557i $$0.146994\pi$$
$$152$$ 152.000i 0.0811107i
$$153$$ 2424.68i 1.28120i
$$154$$ −220.600 −0.115432
$$155$$ 0 0
$$156$$ 2246.90 1.15318
$$157$$ 243.616i 0.123839i 0.998081 + 0.0619194i $$0.0197222\pi$$
−0.998081 + 0.0619194i $$0.980278\pi$$
$$158$$ − 854.448i − 0.430229i
$$159$$ 4709.48 2.34897
$$160$$ 0 0
$$161$$ −2401.04 −1.17533
$$162$$ 834.416i 0.404678i
$$163$$ 2598.11i 1.24847i 0.781238 + 0.624233i $$0.214588\pi$$
−0.781238 + 0.624233i $$0.785412\pi$$
$$164$$ −70.8801 −0.0337488
$$165$$ 0 0
$$166$$ 987.537 0.461733
$$167$$ − 491.064i − 0.227543i −0.993507 0.113772i $$-0.963707\pi$$
0.993507 0.113772i $$-0.0362932\pi$$
$$168$$ 1830.75i 0.840748i
$$169$$ −1903.61 −0.866460
$$170$$ 0 0
$$171$$ 949.012 0.424402
$$172$$ 1473.62i 0.653268i
$$173$$ − 1648.56i − 0.724496i −0.932082 0.362248i $$-0.882009\pi$$
0.932082 0.362248i $$-0.117991\pi$$
$$174$$ 1547.59 0.674268
$$175$$ 0 0
$$176$$ −67.6480 −0.0289725
$$177$$ − 321.932i − 0.136711i
$$178$$ 1842.27i 0.775754i
$$179$$ −2326.81 −0.971586 −0.485793 0.874074i $$-0.661469\pi$$
−0.485793 + 0.874074i $$0.661469\pi$$
$$180$$ 0 0
$$181$$ −4637.46 −1.90442 −0.952208 0.305449i $$-0.901193\pi$$
−0.952208 + 0.305449i $$0.901193\pi$$
$$182$$ − 3341.14i − 1.36078i
$$183$$ − 5532.43i − 2.23480i
$$184$$ −736.288 −0.294999
$$185$$ 0 0
$$186$$ −1438.05 −0.566897
$$187$$ 205.244i 0.0802616i
$$188$$ 1991.25i 0.772483i
$$189$$ 5251.52 2.02112
$$190$$ 0 0
$$191$$ 5260.38 1.99281 0.996407 0.0846903i $$-0.0269901\pi$$
0.996407 + 0.0846903i $$0.0269901\pi$$
$$192$$ 561.408i 0.211022i
$$193$$ − 16.1833i − 0.00603575i −0.999995 0.00301787i $$-0.999039\pi$$
0.999995 0.00301787i $$-0.000960620\pi$$
$$194$$ −2165.49 −0.801407
$$195$$ 0 0
$$196$$ 1350.34 0.492105
$$197$$ 3784.71i 1.36878i 0.729116 + 0.684390i $$0.239931\pi$$
−0.729116 + 0.684390i $$0.760069\pi$$
$$198$$ 422.360i 0.151595i
$$199$$ −73.2079 −0.0260783 −0.0130391 0.999915i $$-0.504151\pi$$
−0.0130391 + 0.999915i $$0.504151\pi$$
$$200$$ 0 0
$$201$$ 2478.58 0.869779
$$202$$ − 1424.90i − 0.496313i
$$203$$ − 2301.28i − 0.795655i
$$204$$ 1703.31 0.584587
$$205$$ 0 0
$$206$$ −52.9594 −0.0179119
$$207$$ 4597.02i 1.54355i
$$208$$ − 1024.58i − 0.341546i
$$209$$ 80.3320 0.0265870
$$210$$ 0 0
$$211$$ −2945.44 −0.961006 −0.480503 0.876993i $$-0.659546\pi$$
−0.480503 + 0.876993i $$0.659546\pi$$
$$212$$ − 2147.50i − 0.695713i
$$213$$ − 5224.19i − 1.68054i
$$214$$ 1481.99 0.473397
$$215$$ 0 0
$$216$$ 1610.40 0.507286
$$217$$ 2138.38i 0.668954i
$$218$$ 3966.15i 1.23221i
$$219$$ 5237.31 1.61600
$$220$$ 0 0
$$221$$ −3108.57 −0.946175
$$222$$ − 414.320i − 0.125258i
$$223$$ − 3125.30i − 0.938499i −0.883066 0.469250i $$-0.844524\pi$$
0.883066 0.469250i $$-0.155476\pi$$
$$224$$ 834.816 0.249011
$$225$$ 0 0
$$226$$ −1437.44 −0.423085
$$227$$ − 3577.80i − 1.04611i −0.852299 0.523055i $$-0.824792\pi$$
0.852299 0.523055i $$-0.175208\pi$$
$$228$$ − 666.672i − 0.193647i
$$229$$ 4802.00 1.38570 0.692850 0.721082i $$-0.256355\pi$$
0.692850 + 0.721082i $$0.256355\pi$$
$$230$$ 0 0
$$231$$ 967.552 0.275586
$$232$$ − 705.696i − 0.199704i
$$233$$ − 5829.49i − 1.63907i −0.573031 0.819534i $$-0.694232\pi$$
0.573031 0.819534i $$-0.305768\pi$$
$$234$$ −6396.94 −1.78710
$$235$$ 0 0
$$236$$ −146.800 −0.0404909
$$237$$ 3747.61i 1.02714i
$$238$$ − 2532.83i − 0.689828i
$$239$$ −1364.33 −0.369251 −0.184625 0.982809i $$-0.559107\pi$$
−0.184625 + 0.982809i $$0.559107\pi$$
$$240$$ 0 0
$$241$$ −2647.22 −0.707563 −0.353782 0.935328i $$-0.615104\pi$$
−0.353782 + 0.935328i $$0.615104\pi$$
$$242$$ − 2626.25i − 0.697610i
$$243$$ 1775.35i 0.468679i
$$244$$ −2522.77 −0.661901
$$245$$ 0 0
$$246$$ 310.880 0.0805731
$$247$$ 1216.68i 0.313424i
$$248$$ 655.745i 0.167903i
$$249$$ −4331.34 −1.10236
$$250$$ 0 0
$$251$$ 1970.73 0.495582 0.247791 0.968814i $$-0.420295\pi$$
0.247791 + 0.968814i $$0.420295\pi$$
$$252$$ − 5212.18i − 1.30292i
$$253$$ 389.128i 0.0966967i
$$254$$ −5221.44 −1.28985
$$255$$ 0 0
$$256$$ 256.000 0.0625000
$$257$$ − 7915.82i − 1.92131i −0.277752 0.960653i $$-0.589589\pi$$
0.277752 0.960653i $$-0.410411\pi$$
$$258$$ − 6463.28i − 1.55964i
$$259$$ −616.096 −0.147808
$$260$$ 0 0
$$261$$ −4406.02 −1.04493
$$262$$ − 2433.38i − 0.573798i
$$263$$ − 3287.96i − 0.770892i −0.922730 0.385446i $$-0.874048\pi$$
0.922730 0.385446i $$-0.125952\pi$$
$$264$$ 296.704 0.0691700
$$265$$ 0 0
$$266$$ −991.344 −0.228508
$$267$$ − 8080.21i − 1.85206i
$$268$$ − 1130.22i − 0.257610i
$$269$$ 4749.61 1.07654 0.538269 0.842773i $$-0.319078\pi$$
0.538269 + 0.842773i $$0.319078\pi$$
$$270$$ 0 0
$$271$$ 242.661 0.0543933 0.0271967 0.999630i $$-0.491342\pi$$
0.0271967 + 0.999630i $$0.491342\pi$$
$$272$$ − 776.704i − 0.173142i
$$273$$ 14654.3i 3.24878i
$$274$$ −2341.34 −0.516225
$$275$$ 0 0
$$276$$ 3229.36 0.704292
$$277$$ − 4131.13i − 0.896086i −0.894012 0.448043i $$-0.852121\pi$$
0.894012 0.448043i $$-0.147879\pi$$
$$278$$ 542.167i 0.116968i
$$279$$ 4094.14 0.878530
$$280$$ 0 0
$$281$$ 1007.19 0.213822 0.106911 0.994269i $$-0.465904\pi$$
0.106911 + 0.994269i $$0.465904\pi$$
$$282$$ − 8733.62i − 1.84425i
$$283$$ − 2333.63i − 0.490176i −0.969501 0.245088i $$-0.921183\pi$$
0.969501 0.245088i $$-0.0788169\pi$$
$$284$$ −2382.21 −0.497740
$$285$$ 0 0
$$286$$ −541.488 −0.111954
$$287$$ − 462.280i − 0.0950785i
$$288$$ − 1598.34i − 0.327024i
$$289$$ 2556.48 0.520350
$$290$$ 0 0
$$291$$ 9497.83 1.91331
$$292$$ − 2388.19i − 0.478625i
$$293$$ 1588.68i 0.316763i 0.987378 + 0.158381i $$0.0506275\pi$$
−0.987378 + 0.158381i $$0.949372\pi$$
$$294$$ −5922.58 −1.17487
$$295$$ 0 0
$$296$$ −188.928 −0.0370988
$$297$$ − 851.096i − 0.166282i
$$298$$ − 3682.38i − 0.715820i
$$299$$ −5893.62 −1.13992
$$300$$ 0 0
$$301$$ −9610.93 −1.84041
$$302$$ 6644.64i 1.26608i
$$303$$ 6249.59i 1.18492i
$$304$$ −304.000 −0.0573539
$$305$$ 0 0
$$306$$ −4849.35 −0.905945
$$307$$ 4057.46i 0.754304i 0.926151 + 0.377152i $$0.123097\pi$$
−0.926151 + 0.377152i $$0.876903\pi$$
$$308$$ − 441.200i − 0.0816224i
$$309$$ 232.280 0.0427636
$$310$$ 0 0
$$311$$ 2871.92 0.523638 0.261819 0.965117i $$-0.415678\pi$$
0.261819 + 0.965117i $$0.415678\pi$$
$$312$$ 4493.79i 0.815420i
$$313$$ − 4322.67i − 0.780612i −0.920685 0.390306i $$-0.872369\pi$$
0.920685 0.390306i $$-0.127631\pi$$
$$314$$ −487.232 −0.0875672
$$315$$ 0 0
$$316$$ 1708.90 0.304218
$$317$$ 2513.56i 0.445349i 0.974893 + 0.222674i $$0.0714786\pi$$
−0.974893 + 0.222674i $$0.928521\pi$$
$$318$$ 9418.95i 1.66097i
$$319$$ −372.960 −0.0654601
$$320$$ 0 0
$$321$$ −6500.02 −1.13021
$$322$$ − 4802.07i − 0.831084i
$$323$$ 922.336i 0.158886i
$$324$$ −1668.83 −0.286151
$$325$$ 0 0
$$326$$ −5196.22 −0.882798
$$327$$ − 17395.6i − 2.94182i
$$328$$ − 141.760i − 0.0238640i
$$329$$ −12986.9 −2.17627
$$330$$ 0 0
$$331$$ −4573.78 −0.759509 −0.379754 0.925087i $$-0.623992\pi$$
−0.379754 + 0.925087i $$0.623992\pi$$
$$332$$ 1975.07i 0.326495i
$$333$$ 1179.57i 0.194115i
$$334$$ 982.129 0.160897
$$335$$ 0 0
$$336$$ −3661.50 −0.594498
$$337$$ 9001.71i 1.45506i 0.686077 + 0.727529i $$0.259331\pi$$
−0.686077 + 0.727529i $$0.740669\pi$$
$$338$$ − 3807.22i − 0.612680i
$$339$$ 6304.62 1.01009
$$340$$ 0 0
$$341$$ 346.561 0.0550361
$$342$$ 1898.02i 0.300098i
$$343$$ − 141.289i − 0.0222417i
$$344$$ −2947.23 −0.461931
$$345$$ 0 0
$$346$$ 3297.12 0.512296
$$347$$ 9358.68i 1.44784i 0.689884 + 0.723920i $$0.257661\pi$$
−0.689884 + 0.723920i $$0.742339\pi$$
$$348$$ 3095.18i 0.476780i
$$349$$ −5787.76 −0.887712 −0.443856 0.896098i $$-0.646390\pi$$
−0.443856 + 0.896098i $$0.646390\pi$$
$$350$$ 0 0
$$351$$ 12890.5 1.96023
$$352$$ − 135.296i − 0.0204866i
$$353$$ − 5784.59i − 0.872188i −0.899901 0.436094i $$-0.856361\pi$$
0.899901 0.436094i $$-0.143639\pi$$
$$354$$ 643.864 0.0966695
$$355$$ 0 0
$$356$$ −3684.55 −0.548541
$$357$$ 11109.0i 1.64692i
$$358$$ − 4653.62i − 0.687015i
$$359$$ 10132.3 1.48959 0.744796 0.667292i $$-0.232547\pi$$
0.744796 + 0.667292i $$0.232547\pi$$
$$360$$ 0 0
$$361$$ 361.000 0.0526316
$$362$$ − 9274.91i − 1.34663i
$$363$$ 11518.7i 1.66550i
$$364$$ 6682.29 0.962217
$$365$$ 0 0
$$366$$ 11064.9 1.58025
$$367$$ − 6993.81i − 0.994752i −0.867535 0.497376i $$-0.834297\pi$$
0.867535 0.497376i $$-0.165703\pi$$
$$368$$ − 1472.58i − 0.208596i
$$369$$ −885.080 −0.124866
$$370$$ 0 0
$$371$$ 14006.0 1.95999
$$372$$ − 2876.10i − 0.400857i
$$373$$ − 6523.15i − 0.905512i −0.891634 0.452756i $$-0.850441\pi$$
0.891634 0.452756i $$-0.149559\pi$$
$$374$$ −410.488 −0.0567535
$$375$$ 0 0
$$376$$ −3982.50 −0.546228
$$377$$ − 5648.75i − 0.771685i
$$378$$ 10503.0i 1.42915i
$$379$$ 9782.00 1.32577 0.662886 0.748720i $$-0.269331\pi$$
0.662886 + 0.748720i $$0.269331\pi$$
$$380$$ 0 0
$$381$$ 22901.2 3.07944
$$382$$ 10520.8i 1.40913i
$$383$$ − 9878.11i − 1.31788i −0.752196 0.658940i $$-0.771005\pi$$
0.752196 0.658940i $$-0.228995\pi$$
$$384$$ −1122.82 −0.149215
$$385$$ 0 0
$$386$$ 32.3666 0.00426792
$$387$$ 18401.0i 2.41700i
$$388$$ − 4330.98i − 0.566680i
$$389$$ 7891.25 1.02854 0.514270 0.857628i $$-0.328063\pi$$
0.514270 + 0.857628i $$0.328063\pi$$
$$390$$ 0 0
$$391$$ −4467.80 −0.577868
$$392$$ 2700.67i 0.347971i
$$393$$ 10672.8i 1.36991i
$$394$$ −7569.43 −0.967874
$$395$$ 0 0
$$396$$ −844.720 −0.107194
$$397$$ − 2787.84i − 0.352437i −0.984351 0.176219i $$-0.943613\pi$$
0.984351 0.176219i $$-0.0563865\pi$$
$$398$$ − 146.416i − 0.0184401i
$$399$$ 4348.04 0.545549
$$400$$ 0 0
$$401$$ 1264.42 0.157461 0.0787306 0.996896i $$-0.474913\pi$$
0.0787306 + 0.996896i $$0.474913\pi$$
$$402$$ 4957.16i 0.615027i
$$403$$ 5248.91i 0.648801i
$$404$$ 2849.79 0.350947
$$405$$ 0 0
$$406$$ 4602.55 0.562613
$$407$$ 99.8486i 0.0121605i
$$408$$ 3406.62i 0.413365i
$$409$$ 8140.55 0.984166 0.492083 0.870548i $$-0.336236\pi$$
0.492083 + 0.870548i $$0.336236\pi$$
$$410$$ 0 0
$$411$$ 10269.1 1.23246
$$412$$ − 105.919i − 0.0126656i
$$413$$ − 957.429i − 0.114073i
$$414$$ −9194.03 −1.09145
$$415$$ 0 0
$$416$$ 2049.15 0.241510
$$417$$ − 2377.94i − 0.279253i
$$418$$ 160.664i 0.0187998i
$$419$$ 9601.15 1.11944 0.559722 0.828680i $$-0.310908\pi$$
0.559722 + 0.828680i $$0.310908\pi$$
$$420$$ 0 0
$$421$$ 5702.48 0.660147 0.330074 0.943955i $$-0.392926\pi$$
0.330074 + 0.943955i $$0.392926\pi$$
$$422$$ − 5890.87i − 0.679534i
$$423$$ 24864.7i 2.85807i
$$424$$ 4295.01 0.491943
$$425$$ 0 0
$$426$$ 10448.4 1.18832
$$427$$ − 16453.5i − 1.86473i
$$428$$ 2963.99i 0.334742i
$$429$$ 2374.97 0.267283
$$430$$ 0 0
$$431$$ −4025.72 −0.449912 −0.224956 0.974369i $$-0.572224\pi$$
−0.224956 + 0.974369i $$0.572224\pi$$
$$432$$ 3220.80i 0.358706i
$$433$$ 1347.10i 0.149510i 0.997202 + 0.0747548i $$0.0238174\pi$$
−0.997202 + 0.0747548i $$0.976183\pi$$
$$434$$ −4276.77 −0.473022
$$435$$ 0 0
$$436$$ −7932.31 −0.871304
$$437$$ 1748.68i 0.191421i
$$438$$ 10474.6i 1.14269i
$$439$$ −4109.36 −0.446763 −0.223381 0.974731i $$-0.571710\pi$$
−0.223381 + 0.974731i $$0.571710\pi$$
$$440$$ 0 0
$$441$$ 16861.7 1.82072
$$442$$ − 6217.13i − 0.669047i
$$443$$ − 6964.84i − 0.746974i −0.927635 0.373487i $$-0.878162\pi$$
0.927635 0.373487i $$-0.121838\pi$$
$$444$$ 828.640 0.0885710
$$445$$ 0 0
$$446$$ 6250.59 0.663619
$$447$$ 16150.9i 1.70897i
$$448$$ 1669.63i 0.176078i
$$449$$ −3041.21 −0.319652 −0.159826 0.987145i $$-0.551093\pi$$
−0.159826 + 0.987145i $$0.551093\pi$$
$$450$$ 0 0
$$451$$ −74.9202 −0.00782229
$$452$$ − 2874.88i − 0.299166i
$$453$$ − 29143.4i − 3.02269i
$$454$$ 7155.60 0.739711
$$455$$ 0 0
$$456$$ 1333.34 0.136929
$$457$$ 11984.3i 1.22670i 0.789810 + 0.613352i $$0.210179\pi$$
−0.789810 + 0.613352i $$0.789821\pi$$
$$458$$ 9604.01i 0.979838i
$$459$$ 9771.91 0.993712
$$460$$ 0 0
$$461$$ −12126.7 −1.22515 −0.612577 0.790411i $$-0.709867\pi$$
−0.612577 + 0.790411i $$0.709867\pi$$
$$462$$ 1935.10i 0.194868i
$$463$$ 6399.19i 0.642323i 0.947024 + 0.321162i $$0.104073\pi$$
−0.947024 + 0.321162i $$0.895927\pi$$
$$464$$ 1411.39 0.141212
$$465$$ 0 0
$$466$$ 11659.0 1.15900
$$467$$ 993.366i 0.0984315i 0.998788 + 0.0492157i $$0.0156722\pi$$
−0.998788 + 0.0492157i $$0.984328\pi$$
$$468$$ − 12793.9i − 1.26367i
$$469$$ 7371.32 0.725748
$$470$$ 0 0
$$471$$ 2137.00 0.209061
$$472$$ − 293.600i − 0.0286314i
$$473$$ 1557.61i 0.151414i
$$474$$ −7495.22 −0.726301
$$475$$ 0 0
$$476$$ 5065.67 0.487782
$$477$$ − 26815.9i − 2.57404i
$$478$$ − 2728.65i − 0.261100i
$$479$$ −6639.36 −0.633320 −0.316660 0.948539i $$-0.602561\pi$$
−0.316660 + 0.948539i $$0.602561\pi$$
$$480$$ 0 0
$$481$$ −1512.28 −0.143355
$$482$$ − 5294.45i − 0.500323i
$$483$$ 21061.9i 1.98416i
$$484$$ 5252.50 0.493285
$$485$$ 0 0
$$486$$ −3550.70 −0.331406
$$487$$ − 11088.8i − 1.03179i −0.856652 0.515894i $$-0.827460\pi$$
0.856652 0.515894i $$-0.172540\pi$$
$$488$$ − 5045.54i − 0.468034i
$$489$$ 22790.6 2.10762
$$490$$ 0 0
$$491$$ −13215.2 −1.21465 −0.607324 0.794454i $$-0.707757\pi$$
−0.607324 + 0.794454i $$0.707757\pi$$
$$492$$ 621.760i 0.0569738i
$$493$$ − 4282.17i − 0.391195i
$$494$$ −2433.37 −0.221624
$$495$$ 0 0
$$496$$ −1311.49 −0.118725
$$497$$ − 15536.8i − 1.40225i
$$498$$ − 8662.67i − 0.779485i
$$499$$ −410.640 −0.0368393 −0.0184196 0.999830i $$-0.505863\pi$$
−0.0184196 + 0.999830i $$0.505863\pi$$
$$500$$ 0 0
$$501$$ −4307.62 −0.384132
$$502$$ 3941.45i 0.350429i
$$503$$ 9407.88i 0.833950i 0.908918 + 0.416975i $$0.136910\pi$$
−0.908918 + 0.416975i $$0.863090\pi$$
$$504$$ 10424.4 0.921305
$$505$$ 0 0
$$506$$ −778.256 −0.0683749
$$507$$ 16698.5i 1.46273i
$$508$$ − 10442.9i − 0.912063i
$$509$$ −10482.2 −0.912803 −0.456402 0.889774i $$-0.650862\pi$$
−0.456402 + 0.889774i $$0.650862\pi$$
$$510$$ 0 0
$$511$$ 15575.8 1.34840
$$512$$ 512.000i 0.0441942i
$$513$$ − 3824.70i − 0.329171i
$$514$$ 15831.6 1.35857
$$515$$ 0 0
$$516$$ 12926.6 1.10283
$$517$$ 2104.75i 0.179046i
$$518$$ − 1232.19i − 0.104516i
$$519$$ −14461.2 −1.22307
$$520$$ 0 0
$$521$$ −3181.02 −0.267492 −0.133746 0.991016i $$-0.542701\pi$$
−0.133746 + 0.991016i $$0.542701\pi$$
$$522$$ − 8812.03i − 0.738874i
$$523$$ 4360.12i 0.364541i 0.983248 + 0.182270i $$0.0583446\pi$$
−0.983248 + 0.182270i $$0.941655\pi$$
$$524$$ 4866.77 0.405736
$$525$$ 0 0
$$526$$ 6575.93 0.545103
$$527$$ 3979.06i 0.328900i
$$528$$ 593.408i 0.0489106i
$$529$$ 3696.37 0.303803
$$530$$ 0 0
$$531$$ −1833.09 −0.149810
$$532$$ − 1982.69i − 0.161580i
$$533$$ − 1134.72i − 0.0922142i
$$534$$ 16160.4 1.30961
$$535$$ 0 0
$$536$$ 2260.45 0.182158
$$537$$ 20410.8i 1.64020i
$$538$$ 9499.22i 0.761227i
$$539$$ 1427.31 0.114060
$$540$$ 0 0
$$541$$ −23681.2 −1.88195 −0.940973 0.338481i $$-0.890087\pi$$
−0.940973 + 0.338481i $$0.890087\pi$$
$$542$$ 485.322i 0.0384619i
$$543$$ 40679.8i 3.21498i
$$544$$ 1553.41 0.122430
$$545$$ 0 0
$$546$$ −29308.5 −2.29723
$$547$$ − 7373.25i − 0.576339i −0.957579 0.288169i $$-0.906953\pi$$
0.957579 0.288169i $$-0.0930466\pi$$
$$548$$ − 4682.69i − 0.365026i
$$549$$ −31501.8 −2.44893
$$550$$ 0 0
$$551$$ −1676.03 −0.129585
$$552$$ 6458.72i 0.498010i
$$553$$ 11145.4i 0.857055i
$$554$$ 8262.27 0.633628
$$555$$ 0 0
$$556$$ −1084.33 −0.0827086
$$557$$ 4772.14i 0.363020i 0.983389 + 0.181510i $$0.0580984\pi$$
−0.983389 + 0.181510i $$0.941902\pi$$
$$558$$ 8188.29i 0.621215i
$$559$$ −23591.1 −1.78497
$$560$$ 0 0
$$561$$ 1800.40 0.135495
$$562$$ 2014.38i 0.151195i
$$563$$ − 7276.49i − 0.544702i −0.962198 0.272351i $$-0.912199\pi$$
0.962198 0.272351i $$-0.0878012\pi$$
$$564$$ 17467.2 1.30408
$$565$$ 0 0
$$566$$ 4667.26 0.346607
$$567$$ − 10884.1i − 0.806156i
$$568$$ − 4764.42i − 0.351955i
$$569$$ 10685.1 0.787245 0.393622 0.919272i $$-0.371222\pi$$
0.393622 + 0.919272i $$0.371222\pi$$
$$570$$ 0 0
$$571$$ 14856.1 1.08881 0.544404 0.838823i $$-0.316756\pi$$
0.544404 + 0.838823i $$0.316756\pi$$
$$572$$ − 1082.98i − 0.0791635i
$$573$$ − 46144.0i − 3.36421i
$$574$$ 924.560 0.0672306
$$575$$ 0 0
$$576$$ 3196.67 0.231241
$$577$$ 3212.67i 0.231794i 0.993261 + 0.115897i $$0.0369742\pi$$
−0.993261 + 0.115897i $$0.963026\pi$$
$$578$$ 5112.96i 0.367943i
$$579$$ −141.960 −0.0101894
$$580$$ 0 0
$$581$$ −12881.4 −0.919814
$$582$$ 18995.7i 1.35291i
$$583$$ − 2269.91i − 0.161252i
$$584$$ 4776.39 0.338439
$$585$$ 0 0
$$586$$ −3177.35 −0.223985
$$587$$ − 22321.1i − 1.56949i −0.619818 0.784745i $$-0.712794\pi$$
0.619818 0.784745i $$-0.287206\pi$$
$$588$$ − 11845.2i − 0.830758i
$$589$$ 1557.39 0.108950
$$590$$ 0 0
$$591$$ 33199.5 2.31074
$$592$$ − 377.857i − 0.0262328i
$$593$$ 8202.50i 0.568021i 0.958821 + 0.284010i $$0.0916650\pi$$
−0.958821 + 0.284010i $$0.908335\pi$$
$$594$$ 1702.19 0.117579
$$595$$ 0 0
$$596$$ 7364.75 0.506161
$$597$$ 642.180i 0.0440246i
$$598$$ − 11787.2i − 0.806047i
$$599$$ −10583.3 −0.721906 −0.360953 0.932584i $$-0.617548\pi$$
−0.360953 + 0.932584i $$0.617548\pi$$
$$600$$ 0 0
$$601$$ −9051.94 −0.614370 −0.307185 0.951650i $$-0.599387\pi$$
−0.307185 + 0.951650i $$0.599387\pi$$
$$602$$ − 19221.9i − 1.30137i
$$603$$ − 14113.1i − 0.953118i
$$604$$ −13289.3 −0.895254
$$605$$ 0 0
$$606$$ −12499.2 −0.837863
$$607$$ 8123.48i 0.543199i 0.962410 + 0.271599i $$0.0875526\pi$$
−0.962410 + 0.271599i $$0.912447\pi$$
$$608$$ − 608.000i − 0.0405554i
$$609$$ −20186.8 −1.34320
$$610$$ 0 0
$$611$$ −31877.9 −2.11071
$$612$$ − 9698.71i − 0.640600i
$$613$$ − 22384.7i − 1.47490i −0.675404 0.737448i $$-0.736031\pi$$
0.675404 0.737448i $$-0.263969\pi$$
$$614$$ −8114.91 −0.533373
$$615$$ 0 0
$$616$$ 882.400 0.0577158
$$617$$ 11349.1i 0.740517i 0.928929 + 0.370259i $$0.120731\pi$$
−0.928929 + 0.370259i $$0.879269\pi$$
$$618$$ 464.560i 0.0302384i
$$619$$ 9106.25 0.591294 0.295647 0.955297i $$-0.404465\pi$$
0.295647 + 0.955297i $$0.404465\pi$$
$$620$$ 0 0
$$621$$ 18526.9 1.19719
$$622$$ 5743.84i 0.370268i
$$623$$ − 24030.6i − 1.54537i
$$624$$ −8987.59 −0.576589
$$625$$ 0 0
$$626$$ 8645.34 0.551976
$$627$$ − 704.672i − 0.0448834i
$$628$$ − 974.464i − 0.0619194i
$$629$$ −1146.42 −0.0726720
$$630$$ 0 0
$$631$$ −27784.2 −1.75289 −0.876444 0.481505i $$-0.840090\pi$$
−0.876444 + 0.481505i $$0.840090\pi$$
$$632$$ 3417.79i 0.215115i
$$633$$ 25837.4i 1.62234i
$$634$$ −5027.12 −0.314909
$$635$$ 0 0
$$636$$ −18837.9 −1.17448
$$637$$ 21617.5i 1.34461i
$$638$$ − 745.921i − 0.0462873i
$$639$$ −29746.7 −1.84156
$$640$$ 0 0
$$641$$ −16958.3 −1.04495 −0.522476 0.852654i $$-0.674992\pi$$
−0.522476 + 0.852654i $$0.674992\pi$$
$$642$$ − 13000.0i − 0.799176i
$$643$$ − 4754.37i − 0.291592i −0.989315 0.145796i $$-0.953426\pi$$
0.989315 0.145796i $$-0.0465744\pi$$
$$644$$ 9604.15 0.587665
$$645$$ 0 0
$$646$$ −1844.67 −0.112349
$$647$$ − 11254.0i − 0.683831i −0.939731 0.341916i $$-0.888924\pi$$
0.939731 0.341916i $$-0.111076\pi$$
$$648$$ − 3337.66i − 0.202339i
$$649$$ −155.167 −0.00938498
$$650$$ 0 0
$$651$$ 18757.9 1.12931
$$652$$ − 10392.4i − 0.624233i
$$653$$ 15515.1i 0.929793i 0.885365 + 0.464896i $$0.153908\pi$$
−0.885365 + 0.464896i $$0.846092\pi$$
$$654$$ 34791.1 2.08018
$$655$$ 0 0
$$656$$ 283.520 0.0168744
$$657$$ − 29821.4i − 1.77084i
$$658$$ − 25973.9i − 1.53885i
$$659$$ −17203.2 −1.01691 −0.508453 0.861090i $$-0.669783\pi$$
−0.508453 + 0.861090i $$0.669783\pi$$
$$660$$ 0 0
$$661$$ 2305.65 0.135672 0.0678361 0.997696i $$-0.478390\pi$$
0.0678361 + 0.997696i $$0.478390\pi$$
$$662$$ − 9147.55i − 0.537054i
$$663$$ 27268.3i 1.59731i
$$664$$ −3950.15 −0.230867
$$665$$ 0 0
$$666$$ −2359.15 −0.137260
$$667$$ − 8118.69i − 0.471299i
$$668$$ 1964.26i 0.113772i
$$669$$ −27415.1 −1.58435
$$670$$ 0 0
$$671$$ −2666.57 −0.153415
$$672$$ − 7323.01i − 0.420374i
$$673$$ 14242.8i 0.815782i 0.913031 + 0.407891i $$0.133736\pi$$
−0.913031 + 0.407891i $$0.866264\pi$$
$$674$$ −18003.4 −1.02888
$$675$$ 0 0
$$676$$ 7614.45 0.433230
$$677$$ − 13480.0i − 0.765256i −0.923902 0.382628i $$-0.875019\pi$$
0.923902 0.382628i $$-0.124981\pi$$
$$678$$ 12609.2i 0.714240i
$$679$$ 28246.6 1.59647
$$680$$ 0 0
$$681$$ −31384.5 −1.76601
$$682$$ 693.122i 0.0389164i
$$683$$ − 27626.1i − 1.54771i −0.633365 0.773854i $$-0.718327\pi$$
0.633365 0.773854i $$-0.281673\pi$$
$$684$$ −3796.05 −0.212201
$$685$$ 0 0
$$686$$ 282.578 0.0157272
$$687$$ − 42123.2i − 2.33930i
$$688$$ − 5894.46i − 0.326634i
$$689$$ 34379.4 1.90094
$$690$$ 0 0
$$691$$ 17419.7 0.959009 0.479505 0.877539i $$-0.340816\pi$$
0.479505 + 0.877539i $$0.340816\pi$$
$$692$$ 6594.24i 0.362248i
$$693$$ − 5509.27i − 0.301991i
$$694$$ −18717.4 −1.02378
$$695$$ 0 0
$$696$$ −6190.37 −0.337134
$$697$$ − 860.201i − 0.0467467i
$$698$$ − 11575.5i − 0.627707i
$$699$$ −51136.3 −2.76703
$$700$$ 0 0
$$701$$ 5069.39 0.273136 0.136568 0.990631i $$-0.456393\pi$$
0.136568 + 0.990631i $$0.456393\pi$$
$$702$$ 25780.9i 1.38609i
$$703$$ 448.705i 0.0240729i
$$704$$ 270.592 0.0144862
$$705$$ 0 0
$$706$$ 11569.2 0.616730
$$707$$ 18586.3i 0.988701i
$$708$$ 1287.73i 0.0683556i
$$709$$ 16758.9 0.887719 0.443860 0.896096i $$-0.353609\pi$$
0.443860 + 0.896096i $$0.353609\pi$$
$$710$$ 0 0
$$711$$ 21339.0 1.12556
$$712$$ − 7369.09i − 0.387877i
$$713$$ 7544.02i 0.396249i
$$714$$ −22218.0 −1.16455
$$715$$ 0 0
$$716$$ 9307.24 0.485793
$$717$$ 11967.9i 0.623359i
$$718$$ 20264.6i 1.05330i
$$719$$ 3885.84 0.201554 0.100777 0.994909i $$-0.467867\pi$$
0.100777 + 0.994909i $$0.467867\pi$$
$$720$$ 0 0
$$721$$ 690.803 0.0356822
$$722$$ 722.000i 0.0372161i
$$723$$ 23221.5i 1.19449i
$$724$$ 18549.8 0.952208
$$725$$ 0 0
$$726$$ −23037.5 −1.17769
$$727$$ 6468.37i 0.329984i 0.986295 + 0.164992i $$0.0527598\pi$$
−0.986295 + 0.164992i $$0.947240\pi$$
$$728$$ 13364.6i 0.680390i
$$729$$ 26838.0 1.36351
$$730$$ 0 0
$$731$$ −17883.8 −0.904865
$$732$$ 22129.7i 1.11740i
$$733$$ − 25245.5i − 1.27212i −0.771640 0.636059i $$-0.780563\pi$$
0.771640 0.636059i $$-0.219437\pi$$
$$734$$ 13987.6 0.703396
$$735$$ 0 0
$$736$$ 2945.15 0.147500
$$737$$ − 1194.65i − 0.0597087i
$$738$$ − 1770.16i − 0.0882933i
$$739$$ −3229.28 −0.160746 −0.0803728 0.996765i $$-0.525611\pi$$
−0.0803728 + 0.996765i $$0.525611\pi$$
$$740$$ 0 0
$$741$$ 10672.8 0.529114
$$742$$ 28012.0i 1.38592i
$$743$$ − 18876.2i − 0.932033i −0.884776 0.466016i $$-0.845689\pi$$
0.884776 0.466016i $$-0.154311\pi$$
$$744$$ 5752.19 0.283448
$$745$$ 0 0
$$746$$ 13046.3 0.640294
$$747$$ 24662.8i 1.20798i
$$748$$ − 820.976i − 0.0401308i
$$749$$ −19331.1 −0.943049
$$750$$ 0 0
$$751$$ 24895.8 1.20967 0.604833 0.796352i $$-0.293240\pi$$
0.604833 + 0.796352i $$0.293240\pi$$
$$752$$ − 7964.99i − 0.386241i
$$753$$ − 17287.2i − 0.836628i
$$754$$ 11297.5 0.545664
$$755$$ 0 0
$$756$$ −21006.1 −1.01056
$$757$$ − 36203.2i − 1.73821i −0.494624 0.869107i $$-0.664694\pi$$
0.494624 0.869107i $$-0.335306\pi$$
$$758$$ 19564.0i 0.937462i
$$759$$ 3413.43 0.163241
$$760$$ 0 0
$$761$$ 11417.5 0.543868 0.271934 0.962316i $$-0.412337\pi$$
0.271934 + 0.962316i $$0.412337\pi$$
$$762$$ 45802.5i 2.17749i
$$763$$ − 51734.5i − 2.45467i
$$764$$ −21041.5 −0.996407
$$765$$ 0 0
$$766$$ 19756.2 0.931881
$$767$$ − 2350.12i − 0.110636i
$$768$$ − 2245.63i − 0.105511i
$$769$$ −39414.5 −1.84828 −0.924138 0.382058i $$-0.875215\pi$$
−0.924138 + 0.382058i $$0.875215\pi$$
$$770$$ 0 0
$$771$$ −69437.6 −3.24350
$$772$$ 64.7332i 0.00301787i
$$773$$ 14268.5i 0.663910i 0.943295 + 0.331955i $$0.107708\pi$$
−0.943295 + 0.331955i $$0.892292\pi$$
$$774$$ −36802.1 −1.70908
$$775$$ 0 0
$$776$$ 8661.95 0.400704
$$777$$ 5404.39i 0.249526i
$$778$$ 15782.5i 0.727288i
$$779$$ −336.680 −0.0154850
$$780$$ 0 0
$$781$$ −2517.99 −0.115366
$$782$$ − 8935.59i − 0.408614i
$$783$$ 17757.1i 0.810455i
$$784$$ −5401.35 −0.246053
$$785$$ 0 0
$$786$$ −21345.7 −0.968669
$$787$$ − 2922.28i − 0.132361i −0.997808 0.0661804i $$-0.978919\pi$$
0.997808 0.0661804i $$-0.0210813\pi$$
$$788$$ − 15138.9i − 0.684390i
$$789$$ −28842.0 −1.30140
$$790$$ 0 0
$$791$$ 18750.0 0.842823
$$792$$ − 1689.44i − 0.0757976i
$$793$$ − 40387.0i − 1.80856i
$$794$$ 5575.67 0.249211
$$795$$ 0 0
$$796$$ 292.832 0.0130391
$$797$$ − 7724.25i − 0.343296i −0.985158 0.171648i $$-0.945091\pi$$
0.985158 0.171648i $$-0.0549092\pi$$
$$798$$ 8696.07i 0.385762i
$$799$$ −24165.8 −1.06999
$$800$$ 0 0
$$801$$ −46008.9 −2.02952
$$802$$ 2528.83i 0.111342i
$$803$$ − 2524.32i − 0.110936i
$$804$$ −9914.32 −0.434890
$$805$$ 0 0
$$806$$ −10497.8 −0.458772
$$807$$ − 41663.6i − 1.81738i
$$808$$ 5699.58i 0.248157i
$$809$$ −42980.8 −1.86789 −0.933947 0.357412i $$-0.883659\pi$$
−0.933947 + 0.357412i $$0.883659\pi$$
$$810$$ 0 0
$$811$$ 28749.5 1.24480 0.622398 0.782701i $$-0.286158\pi$$
0.622398 + 0.782701i $$0.286158\pi$$
$$812$$ 9205.11i 0.397827i
$$813$$ − 2128.62i − 0.0918253i
$$814$$ −199.697 −0.00859875
$$815$$ 0 0
$$816$$ −6813.25 −0.292293
$$817$$ 6999.68i 0.299740i
$$818$$ 16281.1i 0.695911i
$$819$$ 83441.8 3.56006
$$820$$ 0 0
$$821$$ −30274.8 −1.28696 −0.643482 0.765461i $$-0.722511\pi$$
−0.643482 + 0.765461i $$0.722511\pi$$
$$822$$ 20538.3i 0.871477i
$$823$$ − 17296.1i − 0.732568i −0.930503 0.366284i $$-0.880630\pi$$
0.930503 0.366284i $$-0.119370\pi$$
$$824$$ 211.838 0.00895596
$$825$$ 0 0
$$826$$ 1914.86 0.0806615
$$827$$ − 2022.80i − 0.0850541i −0.999095 0.0425271i $$-0.986459\pi$$
0.999095 0.0425271i $$-0.0135409\pi$$
$$828$$ − 18388.1i − 0.771775i
$$829$$ 43239.0 1.81152 0.905762 0.423786i $$-0.139299\pi$$
0.905762 + 0.423786i $$0.139299\pi$$
$$830$$ 0 0
$$831$$ −36238.3 −1.51275
$$832$$ 4098.31i 0.170773i
$$833$$ 16387.7i 0.681632i
$$834$$ 4755.89 0.197462
$$835$$ 0 0
$$836$$ −321.328 −0.0132935
$$837$$ − 16500.2i − 0.681397i
$$838$$ 19202.3i 0.791567i
$$839$$ −27435.9 −1.12895 −0.564477 0.825449i $$-0.690922\pi$$
−0.564477 + 0.825449i $$0.690922\pi$$
$$840$$ 0 0
$$841$$ −16607.6 −0.680948
$$842$$ 11405.0i 0.466795i
$$843$$ − 8835.09i − 0.360969i
$$844$$ 11781.7 0.480503
$$845$$ 0 0
$$846$$ −49729.5 −2.02096
$$847$$ 34256.8i 1.38970i
$$848$$ 8590.02i 0.347857i
$$849$$ −20470.6 −0.827502
$$850$$ 0 0
$$851$$ −2173.53 −0.0875530
$$852$$ 20896.7i 0.840271i
$$853$$ 20978.4i 0.842071i 0.907044 + 0.421035i $$0.138333\pi$$
−0.907044 + 0.421035i $$0.861667\pi$$
$$854$$ 32907.0 1.31857
$$855$$ 0 0
$$856$$ −5927.97 −0.236698
$$857$$ − 30822.4i − 1.22856i −0.789089 0.614279i $$-0.789447\pi$$
0.789089 0.614279i $$-0.210553\pi$$
$$858$$ 4749.94i 0.188998i
$$859$$ 39267.6 1.55971 0.779856 0.625959i $$-0.215292\pi$$
0.779856 + 0.625959i $$0.215292\pi$$
$$860$$ 0 0
$$861$$ −4055.12 −0.160509
$$862$$ − 8051.45i − 0.318136i
$$863$$ 24131.3i 0.951842i 0.879488 + 0.475921i $$0.157885\pi$$
−0.879488 + 0.475921i $$0.842115\pi$$
$$864$$ −6441.60 −0.253643
$$865$$ 0 0
$$866$$ −2694.21 −0.105719
$$867$$ − 22425.4i − 0.878441i
$$868$$ − 8553.54i − 0.334477i
$$869$$ 1806.30 0.0705116
$$870$$ 0 0
$$871$$ 18093.8 0.703885
$$872$$ − 15864.6i − 0.616105i
$$873$$ − 54080.9i − 2.09663i
$$874$$ −3497.37 −0.135355
$$875$$ 0 0
$$876$$ −20949.2 −0.808001
$$877$$ 39380.6i 1.51629i 0.652084 + 0.758147i $$0.273895\pi$$
−0.652084 + 0.758147i $$0.726105\pi$$
$$878$$ − 8218.71i − 0.315909i
$$879$$ 13935.9 0.534750
$$880$$ 0 0
$$881$$ 30887.5 1.18119 0.590595 0.806968i $$-0.298893\pi$$
0.590595 + 0.806968i $$0.298893\pi$$
$$882$$ 33723.3i 1.28744i
$$883$$ − 28191.9i − 1.07444i −0.843441 0.537221i $$-0.819474\pi$$
0.843441 0.537221i $$-0.180526\pi$$
$$884$$ 12434.3 0.473088
$$885$$ 0 0
$$886$$ 13929.7 0.528190
$$887$$ 2760.58i 0.104500i 0.998634 + 0.0522498i $$0.0166392\pi$$
−0.998634 + 0.0522498i $$0.983361\pi$$
$$888$$ 1657.28i 0.0626292i
$$889$$ 68108.5 2.56950
$$890$$ 0 0
$$891$$ −1763.95 −0.0663240
$$892$$ 12501.2i 0.469250i
$$893$$ 9458.43i 0.354439i
$$894$$ −32301.8 −1.20843
$$895$$ 0 0
$$896$$ −3339.26 −0.124506
$$897$$ 51698.9i 1.92439i
$$898$$ − 6082.42i − 0.226028i
$$899$$ −7230.57 −0.268246
$$900$$ 0 0
$$901$$ 26062.1 0.963657
$$902$$ − 149.840i − 0.00553120i
$$903$$ 84307.1i 3.10694i
$$904$$ 5749.76 0.211542
$$905$$ 0 0
$$906$$ 58286.8 2.13736
$$907$$ − 18969.1i − 0.694443i −0.937783 0.347222i $$-0.887125\pi$$
0.937783 0.347222i $$-0.112875\pi$$
$$908$$ 14311.2i 0.523055i
$$909$$ 35585.4 1.29845
$$910$$ 0 0
$$911$$ −48732.9 −1.77233 −0.886164 0.463371i $$-0.846640\pi$$
−0.886164 + 0.463371i $$0.846640\pi$$
$$912$$ 2666.69i 0.0968233i
$$913$$ 2087.65i 0.0756749i
$$914$$ −23968.7 −0.867411
$$915$$ 0 0
$$916$$ −19208.0 −0.692850
$$917$$ 31741.1i 1.14306i
$$918$$ 19543.8i 0.702660i
$$919$$ 35850.4 1.28683 0.643414 0.765518i $$-0.277517\pi$$
0.643414 + 0.765518i $$0.277517\pi$$
$$920$$ 0 0
$$921$$ 35592.0 1.27340
$$922$$ − 24253.4i − 0.866315i
$$923$$ − 38136.8i − 1.36001i
$$924$$ −3870.21 −0.137793
$$925$$ 0 0
$$926$$ −12798.4 −0.454191
$$927$$ − 1322.61i − 0.0468610i
$$928$$ 2822.79i 0.0998518i
$$929$$ −22936.8 −0.810044 −0.405022 0.914307i $$-0.632736\pi$$
−0.405022 + 0.914307i $$0.632736\pi$$
$$930$$ 0 0
$$931$$ 6414.10 0.225793
$$932$$ 23318.0i 0.819534i
$$933$$ − 25192.5i − 0.883992i
$$934$$ −1986.73 −0.0696016
$$935$$ 0 0
$$936$$ 25587.8 0.893550
$$937$$ 47925.4i 1.67092i 0.549548 + 0.835462i $$0.314800\pi$$
−0.549548 + 0.835462i $$0.685200\pi$$
$$938$$ 14742.6i 0.513181i
$$939$$ −37918.5 −1.31781
$$940$$ 0 0
$$941$$ −25842.2 −0.895251 −0.447626 0.894221i $$-0.647730\pi$$
−0.447626 + 0.894221i $$0.647730\pi$$
$$942$$ 4274.00i 0.147829i
$$943$$ − 1630.88i − 0.0563189i
$$944$$ 587.199 0.0202455
$$945$$ 0 0
$$946$$ −3115.22 −0.107066
$$947$$ 36562.8i 1.25463i 0.778766 + 0.627314i $$0.215846\pi$$
−0.778766 + 0.627314i $$0.784154\pi$$
$$948$$ − 14990.4i − 0.513572i
$$949$$ 38232.6 1.30778
$$950$$ 0 0
$$951$$ 22048.9 0.751825
$$952$$ 10131.3i 0.344914i
$$953$$ − 29813.1i − 1.01337i −0.862132 0.506684i $$-0.830871\pi$$
0.862132 0.506684i $$-0.169129\pi$$
$$954$$ 53631.8 1.82012
$$955$$ 0 0
$$956$$ 5457.31 0.184625
$$957$$ 3271.61i 0.110508i
$$958$$ − 13278.7i − 0.447825i
$$959$$ 30540.5 1.02837
$$960$$ 0 0
$$961$$ −23072.2 −0.774470
$$962$$ − 3024.56i − 0.101368i
$$963$$ 37011.3i 1.23850i
$$964$$ 10588.9 0.353782
$$965$$ 0 0
$$966$$ −42123.8 −1.40301
$$967$$ 30315.5i 1.00815i 0.863660 + 0.504075i $$0.168167\pi$$
−0.863660 + 0.504075i $$0.831833\pi$$
$$968$$ 10505.0i 0.348805i
$$969$$ 8090.73 0.268227
$$970$$ 0 0
$$971$$ 26455.6 0.874357 0.437178 0.899375i $$-0.355978\pi$$
0.437178 + 0.899375i $$0.355978\pi$$
$$972$$ − 7101.41i − 0.234339i
$$973$$ − 7072.03i − 0.233010i
$$974$$ 22177.6 0.729584
$$975$$ 0 0
$$976$$ 10091.1 0.330950
$$977$$ 30207.7i 0.989183i 0.869126 + 0.494591i $$0.164682\pi$$
−0.869126 + 0.494591i $$0.835318\pi$$
$$978$$ 45581.3i 1.49032i
$$979$$ −3894.56 −0.127141
$$980$$ 0 0
$$981$$ −99050.7 −3.22370
$$982$$ − 26430.3i − 0.858886i
$$983$$ 5878.48i 0.190737i 0.995442 + 0.0953685i $$0.0304029\pi$$
−0.995442 + 0.0953685i $$0.969597\pi$$
$$984$$ −1243.52 −0.0402866
$$985$$ 0 0
$$986$$ 8564.33 0.276616
$$987$$ 113921.i 3.67392i
$$988$$ − 4866.74i − 0.156712i
$$989$$ −33906.4 −1.09015
$$990$$ 0 0
$$991$$ −42532.3 −1.36335 −0.681676 0.731654i $$-0.738749\pi$$
−0.681676 + 0.731654i $$0.738749\pi$$
$$992$$ − 2622.98i − 0.0839513i
$$993$$ 40121.2i 1.28218i
$$994$$ 31073.5 0.991542
$$995$$ 0 0
$$996$$ 17325.3 0.551179
$$997$$ 6320.28i 0.200767i 0.994949 + 0.100384i $$0.0320070\pi$$
−0.994949 + 0.100384i $$0.967993\pi$$
$$998$$ − 821.281i − 0.0260493i
$$999$$ 4753.91 0.150558
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 950.4.b.i.799.3 4
5.2 odd 4 950.4.a.e.1.1 2
5.3 odd 4 38.4.a.c.1.2 2
5.4 even 2 inner 950.4.b.i.799.2 4
15.8 even 4 342.4.a.h.1.2 2
20.3 even 4 304.4.a.c.1.1 2
35.13 even 4 1862.4.a.e.1.1 2
40.3 even 4 1216.4.a.p.1.2 2
40.13 odd 4 1216.4.a.g.1.1 2
95.18 even 4 722.4.a.f.1.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
38.4.a.c.1.2 2 5.3 odd 4
304.4.a.c.1.1 2 20.3 even 4
342.4.a.h.1.2 2 15.8 even 4
722.4.a.f.1.1 2 95.18 even 4
950.4.a.e.1.1 2 5.2 odd 4
950.4.b.i.799.2 4 5.4 even 2 inner
950.4.b.i.799.3 4 1.1 even 1 trivial
1216.4.a.g.1.1 2 40.13 odd 4
1216.4.a.p.1.2 2 40.3 even 4
1862.4.a.e.1.1 2 35.13 even 4