# Properties

 Label 950.4.a.e.1.1 Level $950$ Weight $4$ Character 950.1 Self dual yes Analytic conductor $56.052$ 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] = [950,4,Mod(1,950)]

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([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("950.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$950 = 2 \cdot 5^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 950.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: $$56.0518145055$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{73})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 18$$ x^2 - x - 18 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 38) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$4.77200$$ of defining polynomial Character $$\chi$$ $$=$$ 950.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-2.00000 q^{2} -8.77200 q^{3} +4.00000 q^{4} +17.5440 q^{6} +26.0880 q^{7} -8.00000 q^{8} +49.9480 q^{9} +O(q^{10})$$ $$q-2.00000 q^{2} -8.77200 q^{3} +4.00000 q^{4} +17.5440 q^{6} +26.0880 q^{7} -8.00000 q^{8} +49.9480 q^{9} -4.22800 q^{11} -35.0880 q^{12} -64.0360 q^{13} -52.1760 q^{14} +16.0000 q^{16} +48.5440 q^{17} -99.8960 q^{18} +19.0000 q^{19} -228.844 q^{21} +8.45600 q^{22} -92.0360 q^{23} +70.1760 q^{24} +128.072 q^{26} -201.300 q^{27} +104.352 q^{28} -88.2120 q^{29} -81.9681 q^{31} -32.0000 q^{32} +37.0880 q^{33} -97.0880 q^{34} +199.792 q^{36} +23.6161 q^{37} -38.0000 q^{38} +561.724 q^{39} +17.7200 q^{41} +457.688 q^{42} -368.404 q^{43} -16.9120 q^{44} +184.072 q^{46} +497.812 q^{47} -140.352 q^{48} +337.584 q^{49} -425.828 q^{51} -256.144 q^{52} +536.876 q^{53} +402.600 q^{54} -208.704 q^{56} -166.668 q^{57} +176.424 q^{58} -36.7000 q^{59} +630.692 q^{61} +163.936 q^{62} +1303.04 q^{63} +64.0000 q^{64} -74.1760 q^{66} -282.556 q^{67} +194.176 q^{68} +807.340 q^{69} +595.552 q^{71} -399.584 q^{72} +597.048 q^{73} -47.2321 q^{74} +76.0000 q^{76} -110.300 q^{77} -1123.45 q^{78} +427.224 q^{79} +417.208 q^{81} -35.4400 q^{82} -493.768 q^{83} -915.376 q^{84} +736.808 q^{86} +773.796 q^{87} +33.8240 q^{88} -921.136 q^{89} -1670.57 q^{91} -368.144 q^{92} +719.024 q^{93} -995.624 q^{94} +280.704 q^{96} -1082.74 q^{97} -675.168 q^{98} -211.180 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 4 q^{2} - 9 q^{3} + 8 q^{4} + 18 q^{6} + 18 q^{7} - 16 q^{8} + 23 q^{9}+O(q^{10})$$ 2 * q - 4 * q^2 - 9 * q^3 + 8 * q^4 + 18 * q^6 + 18 * q^7 - 16 * q^8 + 23 * q^9 $$2 q - 4 q^{2} - 9 q^{3} + 8 q^{4} + 18 q^{6} + 18 q^{7} - 16 q^{8} + 23 q^{9} - 17 q^{11} - 36 q^{12} - 17 q^{13} - 36 q^{14} + 32 q^{16} + 80 q^{17} - 46 q^{18} + 38 q^{19} - 227 q^{21} + 34 q^{22} - 73 q^{23} + 72 q^{24} + 34 q^{26} - 189 q^{27} + 72 q^{28} + 3 q^{29} + 212 q^{31} - 64 q^{32} + 40 q^{33} - 160 q^{34} + 92 q^{36} - 192 q^{37} - 76 q^{38} + 551 q^{39} - 50 q^{41} + 454 q^{42} - 677 q^{43} - 68 q^{44} + 146 q^{46} + 389 q^{47} - 144 q^{48} + 60 q^{49} - 433 q^{51} - 68 q^{52} + 1219 q^{53} + 378 q^{54} - 144 q^{56} - 171 q^{57} - 6 q^{58} - 287 q^{59} + 313 q^{61} - 424 q^{62} + 1521 q^{63} + 128 q^{64} - 80 q^{66} - 1223 q^{67} + 320 q^{68} + 803 q^{69} + 200 q^{71} - 184 q^{72} - 378 q^{73} + 384 q^{74} + 152 q^{76} - 7 q^{77} - 1102 q^{78} + 1350 q^{79} + 1142 q^{81} + 100 q^{82} + 670 q^{83} - 908 q^{84} + 1354 q^{86} + 753 q^{87} + 136 q^{88} - 236 q^{89} - 2051 q^{91} - 292 q^{92} + 652 q^{93} - 778 q^{94} + 288 q^{96} - 1294 q^{97} - 120 q^{98} + 133 q^{99}+O(q^{100})$$ 2 * q - 4 * q^2 - 9 * q^3 + 8 * q^4 + 18 * q^6 + 18 * q^7 - 16 * q^8 + 23 * q^9 - 17 * q^11 - 36 * q^12 - 17 * q^13 - 36 * q^14 + 32 * q^16 + 80 * q^17 - 46 * q^18 + 38 * q^19 - 227 * q^21 + 34 * q^22 - 73 * q^23 + 72 * q^24 + 34 * q^26 - 189 * q^27 + 72 * q^28 + 3 * q^29 + 212 * q^31 - 64 * q^32 + 40 * q^33 - 160 * q^34 + 92 * q^36 - 192 * q^37 - 76 * q^38 + 551 * q^39 - 50 * q^41 + 454 * q^42 - 677 * q^43 - 68 * q^44 + 146 * q^46 + 389 * q^47 - 144 * q^48 + 60 * q^49 - 433 * q^51 - 68 * q^52 + 1219 * q^53 + 378 * q^54 - 144 * q^56 - 171 * q^57 - 6 * q^58 - 287 * q^59 + 313 * q^61 - 424 * q^62 + 1521 * q^63 + 128 * q^64 - 80 * q^66 - 1223 * q^67 + 320 * q^68 + 803 * q^69 + 200 * q^71 - 184 * q^72 - 378 * q^73 + 384 * q^74 + 152 * q^76 - 7 * q^77 - 1102 * q^78 + 1350 * q^79 + 1142 * q^81 + 100 * q^82 + 670 * q^83 - 908 * q^84 + 1354 * q^86 + 753 * q^87 + 136 * q^88 - 236 * q^89 - 2051 * q^91 - 292 * q^92 + 652 * q^93 - 778 * q^94 + 288 * q^96 - 1294 * q^97 - 120 * q^98 + 133 * 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −2.00000 −0.707107
$$3$$ −8.77200 −1.68817 −0.844086 0.536207i $$-0.819856\pi$$
−0.844086 + 0.536207i $$0.819856\pi$$
$$4$$ 4.00000 0.500000
$$5$$ 0 0
$$6$$ 17.5440 1.19372
$$7$$ 26.0880 1.40862 0.704310 0.709893i $$-0.251257\pi$$
0.704310 + 0.709893i $$0.251257\pi$$
$$8$$ −8.00000 −0.353553
$$9$$ 49.9480 1.84993
$$10$$ 0 0
$$11$$ −4.22800 −0.115890 −0.0579450 0.998320i $$-0.518455\pi$$
−0.0579450 + 0.998320i $$0.518455\pi$$
$$12$$ −35.0880 −0.844086
$$13$$ −64.0360 −1.36618 −0.683092 0.730332i $$-0.739365\pi$$
−0.683092 + 0.730332i $$0.739365\pi$$
$$14$$ −52.1760 −0.996045
$$15$$ 0 0
$$16$$ 16.0000 0.250000
$$17$$ 48.5440 0.692568 0.346284 0.938130i $$-0.387443\pi$$
0.346284 + 0.938130i $$0.387443\pi$$
$$18$$ −99.8960 −1.30810
$$19$$ 19.0000 0.229416
$$20$$ 0 0
$$21$$ −228.844 −2.37799
$$22$$ 8.45600 0.0819466
$$23$$ −92.0360 −0.834384 −0.417192 0.908818i $$-0.636986\pi$$
−0.417192 + 0.908818i $$0.636986\pi$$
$$24$$ 70.1760 0.596859
$$25$$ 0 0
$$26$$ 128.072 0.966038
$$27$$ −201.300 −1.43482
$$28$$ 104.352 0.704310
$$29$$ −88.2120 −0.564847 −0.282424 0.959290i $$-0.591138\pi$$
−0.282424 + 0.959290i $$0.591138\pi$$
$$30$$ 0 0
$$31$$ −81.9681 −0.474900 −0.237450 0.971400i $$-0.576312\pi$$
−0.237450 + 0.971400i $$0.576312\pi$$
$$32$$ −32.0000 −0.176777
$$33$$ 37.0880 0.195642
$$34$$ −97.0880 −0.489719
$$35$$ 0 0
$$36$$ 199.792 0.924963
$$37$$ 23.6161 0.104931 0.0524656 0.998623i $$-0.483292\pi$$
0.0524656 + 0.998623i $$0.483292\pi$$
$$38$$ −38.0000 −0.162221
$$39$$ 561.724 2.30636
$$40$$ 0 0
$$41$$ 17.7200 0.0674976 0.0337488 0.999430i $$-0.489255\pi$$
0.0337488 + 0.999430i $$0.489255\pi$$
$$42$$ 457.688 1.68150
$$43$$ −368.404 −1.30654 −0.653268 0.757126i $$-0.726603\pi$$
−0.653268 + 0.757126i $$0.726603\pi$$
$$44$$ −16.9120 −0.0579450
$$45$$ 0 0
$$46$$ 184.072 0.589999
$$47$$ 497.812 1.54497 0.772483 0.635036i $$-0.219015\pi$$
0.772483 + 0.635036i $$0.219015\pi$$
$$48$$ −140.352 −0.422043
$$49$$ 337.584 0.984210
$$50$$ 0 0
$$51$$ −425.828 −1.16917
$$52$$ −256.144 −0.683092
$$53$$ 536.876 1.39143 0.695713 0.718320i $$-0.255089\pi$$
0.695713 + 0.718320i $$0.255089\pi$$
$$54$$ 402.600 1.01457
$$55$$ 0 0
$$56$$ −208.704 −0.498022
$$57$$ −166.668 −0.387293
$$58$$ 176.424 0.399407
$$59$$ −36.7000 −0.0809818 −0.0404909 0.999180i $$-0.512892\pi$$
−0.0404909 + 0.999180i $$0.512892\pi$$
$$60$$ 0 0
$$61$$ 630.692 1.32380 0.661901 0.749592i $$-0.269750\pi$$
0.661901 + 0.749592i $$0.269750\pi$$
$$62$$ 163.936 0.335805
$$63$$ 1303.04 2.60584
$$64$$ 64.0000 0.125000
$$65$$ 0 0
$$66$$ −74.1760 −0.138340
$$67$$ −282.556 −0.515219 −0.257610 0.966249i $$-0.582935\pi$$
−0.257610 + 0.966249i $$0.582935\pi$$
$$68$$ 194.176 0.346284
$$69$$ 807.340 1.40858
$$70$$ 0 0
$$71$$ 595.552 0.995480 0.497740 0.867326i $$-0.334163\pi$$
0.497740 + 0.867326i $$0.334163\pi$$
$$72$$ −399.584 −0.654048
$$73$$ 597.048 0.957250 0.478625 0.878020i $$-0.341135\pi$$
0.478625 + 0.878020i $$0.341135\pi$$
$$74$$ −47.2321 −0.0741976
$$75$$ 0 0
$$76$$ 76.0000 0.114708
$$77$$ −110.300 −0.163245
$$78$$ −1123.45 −1.63084
$$79$$ 427.224 0.608436 0.304218 0.952602i $$-0.401605\pi$$
0.304218 + 0.952602i $$0.401605\pi$$
$$80$$ 0 0
$$81$$ 417.208 0.572302
$$82$$ −35.4400 −0.0477280
$$83$$ −493.768 −0.652989 −0.326495 0.945199i $$-0.605868\pi$$
−0.326495 + 0.945199i $$0.605868\pi$$
$$84$$ −915.376 −1.18900
$$85$$ 0 0
$$86$$ 736.808 0.923861
$$87$$ 773.796 0.953559
$$88$$ 33.8240 0.0409733
$$89$$ −921.136 −1.09708 −0.548541 0.836124i $$-0.684816\pi$$
−0.548541 + 0.836124i $$0.684816\pi$$
$$90$$ 0 0
$$91$$ −1670.57 −1.92443
$$92$$ −368.144 −0.417192
$$93$$ 719.024 0.801713
$$94$$ −995.624 −1.09246
$$95$$ 0 0
$$96$$ 280.704 0.298430
$$97$$ −1082.74 −1.13336 −0.566680 0.823938i $$-0.691773\pi$$
−0.566680 + 0.823938i $$0.691773\pi$$
$$98$$ −675.168 −0.695942
$$99$$ −211.180 −0.214388
$$100$$ 0 0
$$101$$ −712.448 −0.701893 −0.350947 0.936395i $$-0.614140\pi$$
−0.350947 + 0.936395i $$0.614140\pi$$
$$102$$ 851.656 0.826731
$$103$$ 26.4797 0.0253313 0.0126656 0.999920i $$-0.495968\pi$$
0.0126656 + 0.999920i $$0.495968\pi$$
$$104$$ 512.288 0.483019
$$105$$ 0 0
$$106$$ −1073.75 −0.983887
$$107$$ 740.996 0.669484 0.334742 0.942310i $$-0.391351\pi$$
0.334742 + 0.942310i $$0.391351\pi$$
$$108$$ −805.200 −0.717411
$$109$$ −1983.08 −1.74261 −0.871304 0.490744i $$-0.836725\pi$$
−0.871304 + 0.490744i $$0.836725\pi$$
$$110$$ 0 0
$$111$$ −207.160 −0.177142
$$112$$ 417.408 0.352155
$$113$$ 718.720 0.598332 0.299166 0.954201i $$-0.403292\pi$$
0.299166 + 0.954201i $$0.403292\pi$$
$$114$$ 333.336 0.273858
$$115$$ 0 0
$$116$$ −352.848 −0.282424
$$117$$ −3198.47 −2.52734
$$118$$ 73.3999 0.0572628
$$119$$ 1266.42 0.975565
$$120$$ 0 0
$$121$$ −1313.12 −0.986570
$$122$$ −1261.38 −0.936069
$$123$$ −155.440 −0.113948
$$124$$ −327.872 −0.237450
$$125$$ 0 0
$$126$$ −2606.09 −1.84261
$$127$$ −2610.72 −1.82413 −0.912063 0.410050i $$-0.865511\pi$$
−0.912063 + 0.410050i $$0.865511\pi$$
$$128$$ −128.000 −0.0883883
$$129$$ 3231.64 2.20566
$$130$$ 0 0
$$131$$ −1216.69 −0.811472 −0.405736 0.913990i $$-0.632985\pi$$
−0.405736 + 0.913990i $$0.632985\pi$$
$$132$$ 148.352 0.0978211
$$133$$ 495.672 0.323160
$$134$$ 565.112 0.364315
$$135$$ 0 0
$$136$$ −388.352 −0.244860
$$137$$ −1170.67 −0.730053 −0.365026 0.930997i $$-0.618940\pi$$
−0.365026 + 0.930997i $$0.618940\pi$$
$$138$$ −1614.68 −0.996020
$$139$$ −271.083 −0.165417 −0.0827086 0.996574i $$-0.526357\pi$$
−0.0827086 + 0.996574i $$0.526357\pi$$
$$140$$ 0 0
$$141$$ −4366.81 −2.60817
$$142$$ −1191.10 −0.703910
$$143$$ 270.744 0.158327
$$144$$ 799.168 0.462482
$$145$$ 0 0
$$146$$ −1194.10 −0.676878
$$147$$ −2961.29 −1.66152
$$148$$ 94.4642 0.0524656
$$149$$ 1841.19 1.01232 0.506161 0.862439i $$-0.331064\pi$$
0.506161 + 0.862439i $$0.331064\pi$$
$$150$$ 0 0
$$151$$ 3322.32 1.79051 0.895254 0.445557i $$-0.146994\pi$$
0.895254 + 0.445557i $$0.146994\pi$$
$$152$$ −152.000 −0.0811107
$$153$$ 2424.68 1.28120
$$154$$ 220.600 0.115432
$$155$$ 0 0
$$156$$ 2246.90 1.15318
$$157$$ −243.616 −0.123839 −0.0619194 0.998081i $$-0.519722\pi$$
−0.0619194 + 0.998081i $$0.519722\pi$$
$$158$$ −854.448 −0.430229
$$159$$ −4709.48 −2.34897
$$160$$ 0 0
$$161$$ −2401.04 −1.17533
$$162$$ −834.416 −0.404678
$$163$$ 2598.11 1.24847 0.624233 0.781238i $$-0.285412\pi$$
0.624233 + 0.781238i $$0.285412\pi$$
$$164$$ 70.8801 0.0337488
$$165$$ 0 0
$$166$$ 987.537 0.461733
$$167$$ 491.064 0.227543 0.113772 0.993507i $$-0.463707\pi$$
0.113772 + 0.993507i $$0.463707\pi$$
$$168$$ 1830.75 0.840748
$$169$$ 1903.61 0.866460
$$170$$ 0 0
$$171$$ 949.012 0.424402
$$172$$ −1473.62 −0.653268
$$173$$ −1648.56 −0.724496 −0.362248 0.932082i $$-0.617991\pi$$
−0.362248 + 0.932082i $$0.617991\pi$$
$$174$$ −1547.59 −0.674268
$$175$$ 0 0
$$176$$ −67.6480 −0.0289725
$$177$$ 321.932 0.136711
$$178$$ 1842.27 0.775754
$$179$$ 2326.81 0.971586 0.485793 0.874074i $$-0.338531\pi$$
0.485793 + 0.874074i $$0.338531\pi$$
$$180$$ 0 0
$$181$$ −4637.46 −1.90442 −0.952208 0.305449i $$-0.901193\pi$$
−0.952208 + 0.305449i $$0.901193\pi$$
$$182$$ 3341.14 1.36078
$$183$$ −5532.43 −2.23480
$$184$$ 736.288 0.294999
$$185$$ 0 0
$$186$$ −1438.05 −0.566897
$$187$$ −205.244 −0.0802616
$$188$$ 1991.25 0.772483
$$189$$ −5251.52 −2.02112
$$190$$ 0 0
$$191$$ 5260.38 1.99281 0.996407 0.0846903i $$-0.0269901\pi$$
0.996407 + 0.0846903i $$0.0269901\pi$$
$$192$$ −561.408 −0.211022
$$193$$ −16.1833 −0.00603575 −0.00301787 0.999995i $$-0.500961\pi$$
−0.00301787 + 0.999995i $$0.500961\pi$$
$$194$$ 2165.49 0.801407
$$195$$ 0 0
$$196$$ 1350.34 0.492105
$$197$$ −3784.71 −1.36878 −0.684390 0.729116i $$-0.739931\pi$$
−0.684390 + 0.729116i $$0.739931\pi$$
$$198$$ 422.360 0.151595
$$199$$ 73.2079 0.0260783 0.0130391 0.999915i $$-0.495849\pi$$
0.0130391 + 0.999915i $$0.495849\pi$$
$$200$$ 0 0
$$201$$ 2478.58 0.869779
$$202$$ 1424.90 0.496313
$$203$$ −2301.28 −0.795655
$$204$$ −1703.31 −0.584587
$$205$$ 0 0
$$206$$ −52.9594 −0.0179119
$$207$$ −4597.02 −1.54355
$$208$$ −1024.58 −0.341546
$$209$$ −80.3320 −0.0265870
$$210$$ 0 0
$$211$$ −2945.44 −0.961006 −0.480503 0.876993i $$-0.659546\pi$$
−0.480503 + 0.876993i $$0.659546\pi$$
$$212$$ 2147.50 0.695713
$$213$$ −5224.19 −1.68054
$$214$$ −1481.99 −0.473397
$$215$$ 0 0
$$216$$ 1610.40 0.507286
$$217$$ −2138.38 −0.668954
$$218$$ 3966.15 1.23221
$$219$$ −5237.31 −1.61600
$$220$$ 0 0
$$221$$ −3108.57 −0.946175
$$222$$ 414.320 0.125258
$$223$$ −3125.30 −0.938499 −0.469250 0.883066i $$-0.655476\pi$$
−0.469250 + 0.883066i $$0.655476\pi$$
$$224$$ −834.816 −0.249011
$$225$$ 0 0
$$226$$ −1437.44 −0.423085
$$227$$ 3577.80 1.04611 0.523055 0.852299i $$-0.324792\pi$$
0.523055 + 0.852299i $$0.324792\pi$$
$$228$$ −666.672 −0.193647
$$229$$ −4802.00 −1.38570 −0.692850 0.721082i $$-0.743645\pi$$
−0.692850 + 0.721082i $$0.743645\pi$$
$$230$$ 0 0
$$231$$ 967.552 0.275586
$$232$$ 705.696 0.199704
$$233$$ −5829.49 −1.63907 −0.819534 0.573031i $$-0.805768\pi$$
−0.819534 + 0.573031i $$0.805768\pi$$
$$234$$ 6396.94 1.78710
$$235$$ 0 0
$$236$$ −146.800 −0.0404909
$$237$$ −3747.61 −1.02714
$$238$$ −2532.83 −0.689828
$$239$$ 1364.33 0.369251 0.184625 0.982809i $$-0.440893\pi$$
0.184625 + 0.982809i $$0.440893\pi$$
$$240$$ 0 0
$$241$$ −2647.22 −0.707563 −0.353782 0.935328i $$-0.615104\pi$$
−0.353782 + 0.935328i $$0.615104\pi$$
$$242$$ 2626.25 0.697610
$$243$$ 1775.35 0.468679
$$244$$ 2522.77 0.661901
$$245$$ 0 0
$$246$$ 310.880 0.0805731
$$247$$ −1216.68 −0.313424
$$248$$ 655.745 0.167903
$$249$$ 4331.34 1.10236
$$250$$ 0 0
$$251$$ 1970.73 0.495582 0.247791 0.968814i $$-0.420295\pi$$
0.247791 + 0.968814i $$0.420295\pi$$
$$252$$ 5212.18 1.30292
$$253$$ 389.128 0.0966967
$$254$$ 5221.44 1.28985
$$255$$ 0 0
$$256$$ 256.000 0.0625000
$$257$$ 7915.82 1.92131 0.960653 0.277752i $$-0.0895892\pi$$
0.960653 + 0.277752i $$0.0895892\pi$$
$$258$$ −6463.28 −1.55964
$$259$$ 616.096 0.147808
$$260$$ 0 0
$$261$$ −4406.02 −1.04493
$$262$$ 2433.38 0.573798
$$263$$ −3287.96 −0.770892 −0.385446 0.922730i $$-0.625952\pi$$
−0.385446 + 0.922730i $$0.625952\pi$$
$$264$$ −296.704 −0.0691700
$$265$$ 0 0
$$266$$ −991.344 −0.228508
$$267$$ 8080.21 1.85206
$$268$$ −1130.22 −0.257610
$$269$$ −4749.61 −1.07654 −0.538269 0.842773i $$-0.680922\pi$$
−0.538269 + 0.842773i $$0.680922\pi$$
$$270$$ 0 0
$$271$$ 242.661 0.0543933 0.0271967 0.999630i $$-0.491342\pi$$
0.0271967 + 0.999630i $$0.491342\pi$$
$$272$$ 776.704 0.173142
$$273$$ 14654.3 3.24878
$$274$$ 2341.34 0.516225
$$275$$ 0 0
$$276$$ 3229.36 0.704292
$$277$$ 4131.13 0.896086 0.448043 0.894012i $$-0.352121\pi$$
0.448043 + 0.894012i $$0.352121\pi$$
$$278$$ 542.167 0.116968
$$279$$ −4094.14 −0.878530
$$280$$ 0 0
$$281$$ 1007.19 0.213822 0.106911 0.994269i $$-0.465904\pi$$
0.106911 + 0.994269i $$0.465904\pi$$
$$282$$ 8733.62 1.84425
$$283$$ −2333.63 −0.490176 −0.245088 0.969501i $$-0.578817\pi$$
−0.245088 + 0.969501i $$0.578817\pi$$
$$284$$ 2382.21 0.497740
$$285$$ 0 0
$$286$$ −541.488 −0.111954
$$287$$ 462.280 0.0950785
$$288$$ −1598.34 −0.327024
$$289$$ −2556.48 −0.520350
$$290$$ 0 0
$$291$$ 9497.83 1.91331
$$292$$ 2388.19 0.478625
$$293$$ 1588.68 0.316763 0.158381 0.987378i $$-0.449372\pi$$
0.158381 + 0.987378i $$0.449372\pi$$
$$294$$ 5922.58 1.17487
$$295$$ 0 0
$$296$$ −188.928 −0.0370988
$$297$$ 851.096 0.166282
$$298$$ −3682.38 −0.715820
$$299$$ 5893.62 1.13992
$$300$$ 0 0
$$301$$ −9610.93 −1.84041
$$302$$ −6644.64 −1.26608
$$303$$ 6249.59 1.18492
$$304$$ 304.000 0.0573539
$$305$$ 0 0
$$306$$ −4849.35 −0.905945
$$307$$ −4057.46 −0.754304 −0.377152 0.926151i $$-0.623097\pi$$
−0.377152 + 0.926151i $$0.623097\pi$$
$$308$$ −441.200 −0.0816224
$$309$$ −232.280 −0.0427636
$$310$$ 0 0
$$311$$ 2871.92 0.523638 0.261819 0.965117i $$-0.415678\pi$$
0.261819 + 0.965117i $$0.415678\pi$$
$$312$$ −4493.79 −0.815420
$$313$$ −4322.67 −0.780612 −0.390306 0.920685i $$-0.627631\pi$$
−0.390306 + 0.920685i $$0.627631\pi$$
$$314$$ 487.232 0.0875672
$$315$$ 0 0
$$316$$ 1708.90 0.304218
$$317$$ −2513.56 −0.445349 −0.222674 0.974893i $$-0.571479\pi$$
−0.222674 + 0.974893i $$0.571479\pi$$
$$318$$ 9418.95 1.66097
$$319$$ 372.960 0.0654601
$$320$$ 0 0
$$321$$ −6500.02 −1.13021
$$322$$ 4802.07 0.831084
$$323$$ 922.336 0.158886
$$324$$ 1668.83 0.286151
$$325$$ 0 0
$$326$$ −5196.22 −0.882798
$$327$$ 17395.6 2.94182
$$328$$ −141.760 −0.0238640
$$329$$ 12986.9 2.17627
$$330$$ 0 0
$$331$$ −4573.78 −0.759509 −0.379754 0.925087i $$-0.623992\pi$$
−0.379754 + 0.925087i $$0.623992\pi$$
$$332$$ −1975.07 −0.326495
$$333$$ 1179.57 0.194115
$$334$$ −982.129 −0.160897
$$335$$ 0 0
$$336$$ −3661.50 −0.594498
$$337$$ −9001.71 −1.45506 −0.727529 0.686077i $$-0.759331\pi$$
−0.727529 + 0.686077i $$0.759331\pi$$
$$338$$ −3807.22 −0.612680
$$339$$ −6304.62 −1.01009
$$340$$ 0 0
$$341$$ 346.561 0.0550361
$$342$$ −1898.02 −0.300098
$$343$$ −141.289 −0.0222417
$$344$$ 2947.23 0.461931
$$345$$ 0 0
$$346$$ 3297.12 0.512296
$$347$$ −9358.68 −1.44784 −0.723920 0.689884i $$-0.757661\pi$$
−0.723920 + 0.689884i $$0.757661\pi$$
$$348$$ 3095.18 0.476780
$$349$$ 5787.76 0.887712 0.443856 0.896098i $$-0.353610\pi$$
0.443856 + 0.896098i $$0.353610\pi$$
$$350$$ 0 0
$$351$$ 12890.5 1.96023
$$352$$ 135.296 0.0204866
$$353$$ −5784.59 −0.872188 −0.436094 0.899901i $$-0.643639\pi$$
−0.436094 + 0.899901i $$0.643639\pi$$
$$354$$ −643.864 −0.0966695
$$355$$ 0 0
$$356$$ −3684.55 −0.548541
$$357$$ −11109.0 −1.64692
$$358$$ −4653.62 −0.687015
$$359$$ −10132.3 −1.48959 −0.744796 0.667292i $$-0.767453\pi$$
−0.744796 + 0.667292i $$0.767453\pi$$
$$360$$ 0 0
$$361$$ 361.000 0.0526316
$$362$$ 9274.91 1.34663
$$363$$ 11518.7 1.66550
$$364$$ −6682.29 −0.962217
$$365$$ 0 0
$$366$$ 11064.9 1.58025
$$367$$ 6993.81 0.994752 0.497376 0.867535i $$-0.334297\pi$$
0.497376 + 0.867535i $$0.334297\pi$$
$$368$$ −1472.58 −0.208596
$$369$$ 885.080 0.124866
$$370$$ 0 0
$$371$$ 14006.0 1.95999
$$372$$ 2876.10 0.400857
$$373$$ −6523.15 −0.905512 −0.452756 0.891634i $$-0.649559\pi$$
−0.452756 + 0.891634i $$0.649559\pi$$
$$374$$ 410.488 0.0567535
$$375$$ 0 0
$$376$$ −3982.50 −0.546228
$$377$$ 5648.75 0.771685
$$378$$ 10503.0 1.42915
$$379$$ −9782.00 −1.32577 −0.662886 0.748720i $$-0.730669\pi$$
−0.662886 + 0.748720i $$0.730669\pi$$
$$380$$ 0 0
$$381$$ 22901.2 3.07944
$$382$$ −10520.8 −1.40913
$$383$$ −9878.11 −1.31788 −0.658940 0.752196i $$-0.728995\pi$$
−0.658940 + 0.752196i $$0.728995\pi$$
$$384$$ 1122.82 0.149215
$$385$$ 0 0
$$386$$ 32.3666 0.00426792
$$387$$ −18401.0 −2.41700
$$388$$ −4330.98 −0.566680
$$389$$ −7891.25 −1.02854 −0.514270 0.857628i $$-0.671937\pi$$
−0.514270 + 0.857628i $$0.671937\pi$$
$$390$$ 0 0
$$391$$ −4467.80 −0.577868
$$392$$ −2700.67 −0.347971
$$393$$ 10672.8 1.36991
$$394$$ 7569.43 0.967874
$$395$$ 0 0
$$396$$ −844.720 −0.107194
$$397$$ 2787.84 0.352437 0.176219 0.984351i $$-0.443613\pi$$
0.176219 + 0.984351i $$0.443613\pi$$
$$398$$ −146.416 −0.0184401
$$399$$ −4348.04 −0.545549
$$400$$ 0 0
$$401$$ 1264.42 0.157461 0.0787306 0.996896i $$-0.474913\pi$$
0.0787306 + 0.996896i $$0.474913\pi$$
$$402$$ −4957.16 −0.615027
$$403$$ 5248.91 0.648801
$$404$$ −2849.79 −0.350947
$$405$$ 0 0
$$406$$ 4602.55 0.562613
$$407$$ −99.8486 −0.0121605
$$408$$ 3406.62 0.413365
$$409$$ −8140.55 −0.984166 −0.492083 0.870548i $$-0.663764\pi$$
−0.492083 + 0.870548i $$0.663764\pi$$
$$410$$ 0 0
$$411$$ 10269.1 1.23246
$$412$$ 105.919 0.0126656
$$413$$ −957.429 −0.114073
$$414$$ 9194.03 1.09145
$$415$$ 0 0
$$416$$ 2049.15 0.241510
$$417$$ 2377.94 0.279253
$$418$$ 160.664 0.0187998
$$419$$ −9601.15 −1.11944 −0.559722 0.828680i $$-0.689092\pi$$
−0.559722 + 0.828680i $$0.689092\pi$$
$$420$$ 0 0
$$421$$ 5702.48 0.660147 0.330074 0.943955i $$-0.392926\pi$$
0.330074 + 0.943955i $$0.392926\pi$$
$$422$$ 5890.87 0.679534
$$423$$ 24864.7 2.85807
$$424$$ −4295.01 −0.491943
$$425$$ 0 0
$$426$$ 10448.4 1.18832
$$427$$ 16453.5 1.86473
$$428$$ 2963.99 0.334742
$$429$$ −2374.97 −0.267283
$$430$$ 0 0
$$431$$ −4025.72 −0.449912 −0.224956 0.974369i $$-0.572224\pi$$
−0.224956 + 0.974369i $$0.572224\pi$$
$$432$$ −3220.80 −0.358706
$$433$$ 1347.10 0.149510 0.0747548 0.997202i $$-0.476183\pi$$
0.0747548 + 0.997202i $$0.476183\pi$$
$$434$$ 4276.77 0.473022
$$435$$ 0 0
$$436$$ −7932.31 −0.871304
$$437$$ −1748.68 −0.191421
$$438$$ 10474.6 1.14269
$$439$$ 4109.36 0.446763 0.223381 0.974731i $$-0.428290\pi$$
0.223381 + 0.974731i $$0.428290\pi$$
$$440$$ 0 0
$$441$$ 16861.7 1.82072
$$442$$ 6217.13 0.669047
$$443$$ −6964.84 −0.746974 −0.373487 0.927635i $$-0.621838\pi$$
−0.373487 + 0.927635i $$0.621838\pi$$
$$444$$ −828.640 −0.0885710
$$445$$ 0 0
$$446$$ 6250.59 0.663619
$$447$$ −16150.9 −1.70897
$$448$$ 1669.63 0.176078
$$449$$ 3041.21 0.319652 0.159826 0.987145i $$-0.448907\pi$$
0.159826 + 0.987145i $$0.448907\pi$$
$$450$$ 0 0
$$451$$ −74.9202 −0.00782229
$$452$$ 2874.88 0.299166
$$453$$ −29143.4 −3.02269
$$454$$ −7155.60 −0.739711
$$455$$ 0 0
$$456$$ 1333.34 0.136929
$$457$$ −11984.3 −1.22670 −0.613352 0.789810i $$-0.710179\pi$$
−0.613352 + 0.789810i $$0.710179\pi$$
$$458$$ 9604.01 0.979838
$$459$$ −9771.91 −0.993712
$$460$$ 0 0
$$461$$ −12126.7 −1.22515 −0.612577 0.790411i $$-0.709867\pi$$
−0.612577 + 0.790411i $$0.709867\pi$$
$$462$$ −1935.10 −0.194868
$$463$$ 6399.19 0.642323 0.321162 0.947024i $$-0.395927\pi$$
0.321162 + 0.947024i $$0.395927\pi$$
$$464$$ −1411.39 −0.141212
$$465$$ 0 0
$$466$$ 11659.0 1.15900
$$467$$ −993.366 −0.0984315 −0.0492157 0.998788i $$-0.515672\pi$$
−0.0492157 + 0.998788i $$0.515672\pi$$
$$468$$ −12793.9 −1.26367
$$469$$ −7371.32 −0.725748
$$470$$ 0 0
$$471$$ 2137.00 0.209061
$$472$$ 293.600 0.0286314
$$473$$ 1557.61 0.151414
$$474$$ 7495.22 0.726301
$$475$$ 0 0
$$476$$ 5065.67 0.487782
$$477$$ 26815.9 2.57404
$$478$$ −2728.65 −0.261100
$$479$$ 6639.36 0.633320 0.316660 0.948539i $$-0.397439\pi$$
0.316660 + 0.948539i $$0.397439\pi$$
$$480$$ 0 0
$$481$$ −1512.28 −0.143355
$$482$$ 5294.45 0.500323
$$483$$ 21061.9 1.98416
$$484$$ −5252.50 −0.493285
$$485$$ 0 0
$$486$$ −3550.70 −0.331406
$$487$$ 11088.8 1.03179 0.515894 0.856652i $$-0.327460\pi$$
0.515894 + 0.856652i $$0.327460\pi$$
$$488$$ −5045.54 −0.468034
$$489$$ −22790.6 −2.10762
$$490$$ 0 0
$$491$$ −13215.2 −1.21465 −0.607324 0.794454i $$-0.707757\pi$$
−0.607324 + 0.794454i $$0.707757\pi$$
$$492$$ −621.760 −0.0569738
$$493$$ −4282.17 −0.391195
$$494$$ 2433.37 0.221624
$$495$$ 0 0
$$496$$ −1311.49 −0.118725
$$497$$ 15536.8 1.40225
$$498$$ −8662.67 −0.779485
$$499$$ 410.640 0.0368393 0.0184196 0.999830i $$-0.494137\pi$$
0.0184196 + 0.999830i $$0.494137\pi$$
$$500$$ 0 0
$$501$$ −4307.62 −0.384132
$$502$$ −3941.45 −0.350429
$$503$$ 9407.88 0.833950 0.416975 0.908918i $$-0.363090\pi$$
0.416975 + 0.908918i $$0.363090\pi$$
$$504$$ −10424.4 −0.921305
$$505$$ 0 0
$$506$$ −778.256 −0.0683749
$$507$$ −16698.5 −1.46273
$$508$$ −10442.9 −0.912063
$$509$$ 10482.2 0.912803 0.456402 0.889774i $$-0.349138\pi$$
0.456402 + 0.889774i $$0.349138\pi$$
$$510$$ 0 0
$$511$$ 15575.8 1.34840
$$512$$ −512.000 −0.0441942
$$513$$ −3824.70 −0.329171
$$514$$ −15831.6 −1.35857
$$515$$ 0 0
$$516$$ 12926.6 1.10283
$$517$$ −2104.75 −0.179046
$$518$$ −1232.19 −0.104516
$$519$$ 14461.2 1.22307
$$520$$ 0 0
$$521$$ −3181.02 −0.267492 −0.133746 0.991016i $$-0.542701\pi$$
−0.133746 + 0.991016i $$0.542701\pi$$
$$522$$ 8812.03 0.738874
$$523$$ 4360.12 0.364541 0.182270 0.983248i $$-0.441655\pi$$
0.182270 + 0.983248i $$0.441655\pi$$
$$524$$ −4866.77 −0.405736
$$525$$ 0 0
$$526$$ 6575.93 0.545103
$$527$$ −3979.06 −0.328900
$$528$$ 593.408 0.0489106
$$529$$ −3696.37 −0.303803
$$530$$ 0 0
$$531$$ −1833.09 −0.149810
$$532$$ 1982.69 0.161580
$$533$$ −1134.72 −0.0922142
$$534$$ −16160.4 −1.30961
$$535$$ 0 0
$$536$$ 2260.45 0.182158
$$537$$ −20410.8 −1.64020
$$538$$ 9499.22 0.761227
$$539$$ −1427.31 −0.114060
$$540$$ 0 0
$$541$$ −23681.2 −1.88195 −0.940973 0.338481i $$-0.890087\pi$$
−0.940973 + 0.338481i $$0.890087\pi$$
$$542$$ −485.322 −0.0384619
$$543$$ 40679.8 3.21498
$$544$$ −1553.41 −0.122430
$$545$$ 0 0
$$546$$ −29308.5 −2.29723
$$547$$ 7373.25 0.576339 0.288169 0.957579i $$-0.406953\pi$$
0.288169 + 0.957579i $$0.406953\pi$$
$$548$$ −4682.69 −0.365026
$$549$$ 31501.8 2.44893
$$550$$ 0 0
$$551$$ −1676.03 −0.129585
$$552$$ −6458.72 −0.498010
$$553$$ 11145.4 0.857055
$$554$$ −8262.27 −0.633628
$$555$$ 0 0
$$556$$ −1084.33 −0.0827086
$$557$$ −4772.14 −0.363020 −0.181510 0.983389i $$-0.558098\pi$$
−0.181510 + 0.983389i $$0.558098\pi$$
$$558$$ 8188.29 0.621215
$$559$$ 23591.1 1.78497
$$560$$ 0 0
$$561$$ 1800.40 0.135495
$$562$$ −2014.38 −0.151195
$$563$$ −7276.49 −0.544702 −0.272351 0.962198i $$-0.587801\pi$$
−0.272351 + 0.962198i $$0.587801\pi$$
$$564$$ −17467.2 −1.30408
$$565$$ 0 0
$$566$$ 4667.26 0.346607
$$567$$ 10884.1 0.806156
$$568$$ −4764.42 −0.351955
$$569$$ −10685.1 −0.787245 −0.393622 0.919272i $$-0.628778\pi$$
−0.393622 + 0.919272i $$0.628778\pi$$
$$570$$ 0 0
$$571$$ 14856.1 1.08881 0.544404 0.838823i $$-0.316756\pi$$
0.544404 + 0.838823i $$0.316756\pi$$
$$572$$ 1082.98 0.0791635
$$573$$ −46144.0 −3.36421
$$574$$ −924.560 −0.0672306
$$575$$ 0 0
$$576$$ 3196.67 0.231241
$$577$$ −3212.67 −0.231794 −0.115897 0.993261i $$-0.536974\pi$$
−0.115897 + 0.993261i $$0.536974\pi$$
$$578$$ 5112.96 0.367943
$$579$$ 141.960 0.0101894
$$580$$ 0 0
$$581$$ −12881.4 −0.919814
$$582$$ −18995.7 −1.35291
$$583$$ −2269.91 −0.161252
$$584$$ −4776.39 −0.338439
$$585$$ 0 0
$$586$$ −3177.35 −0.223985
$$587$$ 22321.1 1.56949 0.784745 0.619818i $$-0.212794\pi$$
0.784745 + 0.619818i $$0.212794\pi$$
$$588$$ −11845.2 −0.830758
$$589$$ −1557.39 −0.108950
$$590$$ 0 0
$$591$$ 33199.5 2.31074
$$592$$ 377.857 0.0262328
$$593$$ 8202.50 0.568021 0.284010 0.958821i $$-0.408335\pi$$
0.284010 + 0.958821i $$0.408335\pi$$
$$594$$ −1702.19 −0.117579
$$595$$ 0 0
$$596$$ 7364.75 0.506161
$$597$$ −642.180 −0.0440246
$$598$$ −11787.2 −0.806047
$$599$$ 10583.3 0.721906 0.360953 0.932584i $$-0.382452\pi$$
0.360953 + 0.932584i $$0.382452\pi$$
$$600$$ 0 0
$$601$$ −9051.94 −0.614370 −0.307185 0.951650i $$-0.599387\pi$$
−0.307185 + 0.951650i $$0.599387\pi$$
$$602$$ 19221.9 1.30137
$$603$$ −14113.1 −0.953118
$$604$$ 13289.3 0.895254
$$605$$ 0 0
$$606$$ −12499.2 −0.837863
$$607$$ −8123.48 −0.543199 −0.271599 0.962410i $$-0.587553\pi$$
−0.271599 + 0.962410i $$0.587553\pi$$
$$608$$ −608.000 −0.0405554
$$609$$ 20186.8 1.34320
$$610$$ 0 0
$$611$$ −31877.9 −2.11071
$$612$$ 9698.71 0.640600
$$613$$ −22384.7 −1.47490 −0.737448 0.675404i $$-0.763969\pi$$
−0.737448 + 0.675404i $$0.763969\pi$$
$$614$$ 8114.91 0.533373
$$615$$ 0 0
$$616$$ 882.400 0.0577158
$$617$$ −11349.1 −0.740517 −0.370259 0.928929i $$-0.620731\pi$$
−0.370259 + 0.928929i $$0.620731\pi$$
$$618$$ 464.560 0.0302384
$$619$$ −9106.25 −0.591294 −0.295647 0.955297i $$-0.595535\pi$$
−0.295647 + 0.955297i $$0.595535\pi$$
$$620$$ 0 0
$$621$$ 18526.9 1.19719
$$622$$ −5743.84 −0.370268
$$623$$ −24030.6 −1.54537
$$624$$ 8987.59 0.576589
$$625$$ 0 0
$$626$$ 8645.34 0.551976
$$627$$ 704.672 0.0448834
$$628$$ −974.464 −0.0619194
$$629$$ 1146.42 0.0726720
$$630$$ 0 0
$$631$$ −27784.2 −1.75289 −0.876444 0.481505i $$-0.840090\pi$$
−0.876444 + 0.481505i $$0.840090\pi$$
$$632$$ −3417.79 −0.215115
$$633$$ 25837.4 1.62234
$$634$$ 5027.12 0.314909
$$635$$ 0 0
$$636$$ −18837.9 −1.17448
$$637$$ −21617.5 −1.34461
$$638$$ −745.921 −0.0462873
$$639$$ 29746.7 1.84156
$$640$$ 0 0
$$641$$ −16958.3 −1.04495 −0.522476 0.852654i $$-0.674992\pi$$
−0.522476 + 0.852654i $$0.674992\pi$$
$$642$$ 13000.0 0.799176
$$643$$ −4754.37 −0.291592 −0.145796 0.989315i $$-0.546574\pi$$
−0.145796 + 0.989315i $$0.546574\pi$$
$$644$$ −9604.15 −0.587665
$$645$$ 0 0
$$646$$ −1844.67 −0.112349
$$647$$ 11254.0 0.683831 0.341916 0.939731i $$-0.388924\pi$$
0.341916 + 0.939731i $$0.388924\pi$$
$$648$$ −3337.66 −0.202339
$$649$$ 155.167 0.00938498
$$650$$ 0 0
$$651$$ 18757.9 1.12931
$$652$$ 10392.4 0.624233
$$653$$ 15515.1 0.929793 0.464896 0.885365i $$-0.346092\pi$$
0.464896 + 0.885365i $$0.346092\pi$$
$$654$$ −34791.1 −2.08018
$$655$$ 0 0
$$656$$ 283.520 0.0168744
$$657$$ 29821.4 1.77084
$$658$$ −25973.9 −1.53885
$$659$$ 17203.2 1.01691 0.508453 0.861090i $$-0.330217\pi$$
0.508453 + 0.861090i $$0.330217\pi$$
$$660$$ 0 0
$$661$$ 2305.65 0.135672 0.0678361 0.997696i $$-0.478390\pi$$
0.0678361 + 0.997696i $$0.478390\pi$$
$$662$$ 9147.55 0.537054
$$663$$ 27268.3 1.59731
$$664$$ 3950.15 0.230867
$$665$$ 0 0
$$666$$ −2359.15 −0.137260
$$667$$ 8118.69 0.471299
$$668$$ 1964.26 0.113772
$$669$$ 27415.1 1.58435
$$670$$ 0 0
$$671$$ −2666.57 −0.153415
$$672$$ 7323.01 0.420374
$$673$$ 14242.8 0.815782 0.407891 0.913031i $$-0.366264\pi$$
0.407891 + 0.913031i $$0.366264\pi$$
$$674$$ 18003.4 1.02888
$$675$$ 0 0
$$676$$ 7614.45 0.433230
$$677$$ 13480.0 0.765256 0.382628 0.923902i $$-0.375019\pi$$
0.382628 + 0.923902i $$0.375019\pi$$
$$678$$ 12609.2 0.714240
$$679$$ −28246.6 −1.59647
$$680$$ 0 0
$$681$$ −31384.5 −1.76601
$$682$$ −693.122 −0.0389164
$$683$$ −27626.1 −1.54771 −0.773854 0.633365i $$-0.781673\pi$$
−0.773854 + 0.633365i $$0.781673\pi$$
$$684$$ 3796.05 0.212201
$$685$$ 0 0
$$686$$ 282.578 0.0157272
$$687$$ 42123.2 2.33930
$$688$$ −5894.46 −0.326634
$$689$$ −34379.4 −1.90094
$$690$$ 0 0
$$691$$ 17419.7 0.959009 0.479505 0.877539i $$-0.340816\pi$$
0.479505 + 0.877539i $$0.340816\pi$$
$$692$$ −6594.24 −0.362248
$$693$$ −5509.27 −0.301991
$$694$$ 18717.4 1.02378
$$695$$ 0 0
$$696$$ −6190.37 −0.337134
$$697$$ 860.201 0.0467467
$$698$$ −11575.5 −0.627707
$$699$$ 51136.3 2.76703
$$700$$ 0 0
$$701$$ 5069.39 0.273136 0.136568 0.990631i $$-0.456393\pi$$
0.136568 + 0.990631i $$0.456393\pi$$
$$702$$ −25780.9 −1.38609
$$703$$ 448.705 0.0240729
$$704$$ −270.592 −0.0144862
$$705$$ 0 0
$$706$$ 11569.2 0.616730
$$707$$ −18586.3 −0.988701
$$708$$ 1287.73 0.0683556
$$709$$ −16758.9 −0.887719 −0.443860 0.896096i $$-0.646391\pi$$
−0.443860 + 0.896096i $$0.646391\pi$$
$$710$$ 0 0
$$711$$ 21339.0 1.12556
$$712$$ 7369.09 0.387877
$$713$$ 7544.02 0.396249
$$714$$ 22218.0 1.16455
$$715$$ 0 0
$$716$$ 9307.24 0.485793
$$717$$ −11967.9 −0.623359
$$718$$ 20264.6 1.05330
$$719$$ −3885.84 −0.201554 −0.100777 0.994909i $$-0.532133\pi$$
−0.100777 + 0.994909i $$0.532133\pi$$
$$720$$ 0 0
$$721$$ 690.803 0.0356822
$$722$$ −722.000 −0.0372161
$$723$$ 23221.5 1.19449
$$724$$ −18549.8 −0.952208
$$725$$ 0 0
$$726$$ −23037.5 −1.17769
$$727$$ −6468.37 −0.329984 −0.164992 0.986295i $$-0.552760\pi$$
−0.164992 + 0.986295i $$0.552760\pi$$
$$728$$ 13364.6 0.680390
$$729$$ −26838.0 −1.36351
$$730$$ 0 0
$$731$$ −17883.8 −0.904865
$$732$$ −22129.7 −1.11740
$$733$$ −25245.5 −1.27212 −0.636059 0.771640i $$-0.719437\pi$$
−0.636059 + 0.771640i $$0.719437\pi$$
$$734$$ −13987.6 −0.703396
$$735$$ 0 0
$$736$$ 2945.15 0.147500
$$737$$ 1194.65 0.0597087
$$738$$ −1770.16 −0.0882933
$$739$$ 3229.28 0.160746 0.0803728 0.996765i $$-0.474389\pi$$
0.0803728 + 0.996765i $$0.474389\pi$$
$$740$$ 0 0
$$741$$ 10672.8 0.529114
$$742$$ −28012.0 −1.38592
$$743$$ −18876.2 −0.932033 −0.466016 0.884776i $$-0.654311\pi$$
−0.466016 + 0.884776i $$0.654311\pi$$
$$744$$ −5752.19 −0.283448
$$745$$ 0 0
$$746$$ 13046.3 0.640294
$$747$$ −24662.8 −1.20798
$$748$$ −820.976 −0.0401308
$$749$$ 19331.1 0.943049
$$750$$ 0 0
$$751$$ 24895.8 1.20967 0.604833 0.796352i $$-0.293240\pi$$
0.604833 + 0.796352i $$0.293240\pi$$
$$752$$ 7964.99 0.386241
$$753$$ −17287.2 −0.836628
$$754$$ −11297.5 −0.545664
$$755$$ 0 0
$$756$$ −21006.1 −1.01056
$$757$$ 36203.2 1.73821 0.869107 0.494624i $$-0.164694\pi$$
0.869107 + 0.494624i $$0.164694\pi$$
$$758$$ 19564.0 0.937462
$$759$$ −3413.43 −0.163241
$$760$$ 0 0
$$761$$ 11417.5 0.543868 0.271934 0.962316i $$-0.412337\pi$$
0.271934 + 0.962316i $$0.412337\pi$$
$$762$$ −45802.5 −2.17749
$$763$$ −51734.5 −2.45467
$$764$$ 21041.5 0.996407
$$765$$ 0 0
$$766$$ 19756.2 0.931881
$$767$$ 2350.12 0.110636
$$768$$ −2245.63 −0.105511
$$769$$ 39414.5 1.84828 0.924138 0.382058i $$-0.124785\pi$$
0.924138 + 0.382058i $$0.124785\pi$$
$$770$$ 0 0
$$771$$ −69437.6 −3.24350
$$772$$ −64.7332 −0.00301787
$$773$$ 14268.5 0.663910 0.331955 0.943295i $$-0.392292\pi$$
0.331955 + 0.943295i $$0.392292\pi$$
$$774$$ 36802.1 1.70908
$$775$$ 0 0
$$776$$ 8661.95 0.400704
$$777$$ −5404.39 −0.249526
$$778$$ 15782.5 0.727288
$$779$$ 336.680 0.0154850
$$780$$ 0 0
$$781$$ −2517.99 −0.115366
$$782$$ 8935.59 0.408614
$$783$$ 17757.1 0.810455
$$784$$ 5401.35 0.246053
$$785$$ 0 0
$$786$$ −21345.7 −0.968669
$$787$$ 2922.28 0.132361 0.0661804 0.997808i $$-0.478919\pi$$
0.0661804 + 0.997808i $$0.478919\pi$$
$$788$$ −15138.9 −0.684390
$$789$$ 28842.0 1.30140
$$790$$ 0 0
$$791$$ 18750.0 0.842823
$$792$$ 1689.44 0.0757976
$$793$$ −40387.0 −1.80856
$$794$$ −5575.67 −0.249211
$$795$$ 0 0
$$796$$ 292.832 0.0130391
$$797$$ 7724.25 0.343296 0.171648 0.985158i $$-0.445091\pi$$
0.171648 + 0.985158i $$0.445091\pi$$
$$798$$ 8696.07 0.385762
$$799$$ 24165.8 1.06999
$$800$$ 0 0
$$801$$ −46008.9 −2.02952
$$802$$ −2528.83 −0.111342
$$803$$ −2524.32 −0.110936
$$804$$ 9914.32 0.434890
$$805$$ 0 0
$$806$$ −10497.8 −0.458772
$$807$$ 41663.6 1.81738
$$808$$ 5699.58 0.248157
$$809$$ 42980.8 1.86789 0.933947 0.357412i $$-0.116341\pi$$
0.933947 + 0.357412i $$0.116341\pi$$
$$810$$ 0 0
$$811$$ 28749.5 1.24480 0.622398 0.782701i $$-0.286158\pi$$
0.622398 + 0.782701i $$0.286158\pi$$
$$812$$ −9205.11 −0.397827
$$813$$ −2128.62 −0.0918253
$$814$$ 199.697 0.00859875
$$815$$ 0 0
$$816$$ −6813.25 −0.292293
$$817$$ −6999.68 −0.299740
$$818$$ 16281.1 0.695911
$$819$$ −83441.8 −3.56006
$$820$$ 0 0
$$821$$ −30274.8 −1.28696 −0.643482 0.765461i $$-0.722511\pi$$
−0.643482 + 0.765461i $$0.722511\pi$$
$$822$$ −20538.3 −0.871477
$$823$$ −17296.1 −0.732568 −0.366284 0.930503i $$-0.619370\pi$$
−0.366284 + 0.930503i $$0.619370\pi$$
$$824$$ −211.838 −0.00895596
$$825$$ 0 0
$$826$$ 1914.86 0.0806615
$$827$$ 2022.80 0.0850541 0.0425271 0.999095i $$-0.486459\pi$$
0.0425271 + 0.999095i $$0.486459\pi$$
$$828$$ −18388.1 −0.771775
$$829$$ −43239.0 −1.81152 −0.905762 0.423786i $$-0.860701\pi$$
−0.905762 + 0.423786i $$0.860701\pi$$
$$830$$ 0 0
$$831$$ −36238.3 −1.51275
$$832$$ −4098.31 −0.170773
$$833$$ 16387.7 0.681632
$$834$$ −4755.89 −0.197462
$$835$$ 0 0
$$836$$ −321.328 −0.0132935
$$837$$ 16500.2 0.681397
$$838$$ 19202.3 0.791567
$$839$$ 27435.9 1.12895 0.564477 0.825449i $$-0.309078\pi$$
0.564477 + 0.825449i $$0.309078\pi$$
$$840$$ 0 0
$$841$$ −16607.6 −0.680948
$$842$$ −11405.0 −0.466795
$$843$$ −8835.09 −0.360969
$$844$$ −11781.7 −0.480503
$$845$$ 0 0
$$846$$ −49729.5 −2.02096
$$847$$ −34256.8 −1.38970
$$848$$ 8590.02 0.347857
$$849$$ 20470.6 0.827502
$$850$$ 0 0
$$851$$ −2173.53 −0.0875530
$$852$$ −20896.7 −0.840271
$$853$$ 20978.4 0.842071 0.421035 0.907044i $$-0.361667\pi$$
0.421035 + 0.907044i $$0.361667\pi$$
$$854$$ −32907.0 −1.31857
$$855$$ 0 0
$$856$$ −5927.97 −0.236698
$$857$$ 30822.4 1.22856 0.614279 0.789089i $$-0.289447\pi$$
0.614279 + 0.789089i $$0.289447\pi$$
$$858$$ 4749.94 0.188998
$$859$$ −39267.6 −1.55971 −0.779856 0.625959i $$-0.784708\pi$$
−0.779856 + 0.625959i $$0.784708\pi$$
$$860$$ 0 0
$$861$$ −4055.12 −0.160509
$$862$$ 8051.45 0.318136
$$863$$ 24131.3 0.951842 0.475921 0.879488i $$-0.342115\pi$$
0.475921 + 0.879488i $$0.342115\pi$$
$$864$$ 6441.60 0.253643
$$865$$ 0 0
$$866$$ −2694.21 −0.105719
$$867$$ 22425.4 0.878441
$$868$$ −8553.54 −0.334477
$$869$$ −1806.30 −0.0705116
$$870$$ 0 0
$$871$$ 18093.8 0.703885
$$872$$ 15864.6 0.616105
$$873$$ −54080.9 −2.09663
$$874$$ 3497.37 0.135355
$$875$$ 0 0
$$876$$ −20949.2 −0.808001
$$877$$ −39380.6 −1.51629 −0.758147 0.652084i $$-0.773895\pi$$
−0.758147 + 0.652084i $$0.773895\pi$$
$$878$$ −8218.71 −0.315909
$$879$$ −13935.9 −0.534750
$$880$$ 0 0
$$881$$ 30887.5 1.18119 0.590595 0.806968i $$-0.298893\pi$$
0.590595 + 0.806968i $$0.298893\pi$$
$$882$$ −33723.3 −1.28744
$$883$$ −28191.9 −1.07444 −0.537221 0.843441i $$-0.680526\pi$$
−0.537221 + 0.843441i $$0.680526\pi$$
$$884$$ −12434.3 −0.473088
$$885$$ 0 0
$$886$$ 13929.7 0.528190
$$887$$ −2760.58 −0.104500 −0.0522498 0.998634i $$-0.516639\pi$$
−0.0522498 + 0.998634i $$0.516639\pi$$
$$888$$ 1657.28 0.0626292
$$889$$ −68108.5 −2.56950
$$890$$ 0 0
$$891$$ −1763.95 −0.0663240
$$892$$ −12501.2 −0.469250
$$893$$ 9458.43 0.354439
$$894$$ 32301.8 1.20843
$$895$$ 0 0
$$896$$ −3339.26 −0.124506
$$897$$ −51698.9 −1.92439
$$898$$ −6082.42 −0.226028
$$899$$ 7230.57 0.268246
$$900$$ 0 0
$$901$$ 26062.1 0.963657
$$902$$ 149.840 0.00553120
$$903$$ 84307.1 3.10694
$$904$$ −5749.76 −0.211542
$$905$$ 0 0
$$906$$ 58286.8 2.13736
$$907$$ 18969.1 0.694443 0.347222 0.937783i $$-0.387125\pi$$
0.347222 + 0.937783i $$0.387125\pi$$
$$908$$ 14311.2 0.523055
$$909$$ −35585.4 −1.29845
$$910$$ 0 0
$$911$$ −48732.9 −1.77233 −0.886164 0.463371i $$-0.846640\pi$$
−0.886164 + 0.463371i $$0.846640\pi$$
$$912$$ −2666.69 −0.0968233
$$913$$ 2087.65 0.0756749
$$914$$ 23968.7 0.867411
$$915$$ 0 0
$$916$$ −19208.0 −0.692850
$$917$$ −31741.1 −1.14306
$$918$$ 19543.8 0.702660
$$919$$ −35850.4 −1.28683 −0.643414 0.765518i $$-0.722483\pi$$
−0.643414 + 0.765518i $$0.722483\pi$$
$$920$$ 0 0
$$921$$ 35592.0 1.27340
$$922$$ 24253.4 0.866315
$$923$$ −38136.8 −1.36001
$$924$$ 3870.21 0.137793
$$925$$ 0 0
$$926$$ −12798.4 −0.454191
$$927$$ 1322.61 0.0468610
$$928$$ 2822.79 0.0998518
$$929$$ 22936.8 0.810044 0.405022 0.914307i $$-0.367264\pi$$
0.405022 + 0.914307i $$0.367264\pi$$
$$930$$ 0 0
$$931$$ 6414.10 0.225793
$$932$$ −23318.0 −0.819534
$$933$$ −25192.5 −0.883992
$$934$$ 1986.73 0.0696016
$$935$$ 0 0
$$936$$ 25587.8 0.893550
$$937$$ −47925.4 −1.67092 −0.835462 0.549548i $$-0.814800\pi$$
−0.835462 + 0.549548i $$0.814800\pi$$
$$938$$ 14742.6 0.513181
$$939$$ 37918.5 1.31781
$$940$$ 0 0
$$941$$ −25842.2 −0.895251 −0.447626 0.894221i $$-0.647730\pi$$
−0.447626 + 0.894221i $$0.647730\pi$$
$$942$$ −4274.00 −0.147829
$$943$$ −1630.88 −0.0563189
$$944$$ −587.199 −0.0202455
$$945$$ 0 0
$$946$$ −3115.22 −0.107066
$$947$$ −36562.8 −1.25463 −0.627314 0.778766i $$-0.715846\pi$$
−0.627314 + 0.778766i $$0.715846\pi$$
$$948$$ −14990.4 −0.513572
$$949$$ −38232.6 −1.30778
$$950$$ 0 0
$$951$$ 22048.9 0.751825
$$952$$ −10131.3 −0.344914
$$953$$ −29813.1 −1.01337 −0.506684 0.862132i $$-0.669129\pi$$
−0.506684 + 0.862132i $$0.669129\pi$$
$$954$$ −53631.8 −1.82012
$$955$$ 0 0
$$956$$ 5457.31 0.184625
$$957$$ −3271.61 −0.110508
$$958$$ −13278.7 −0.447825
$$959$$ −30540.5 −1.02837
$$960$$ 0 0
$$961$$ −23072.2 −0.774470
$$962$$ 3024.56 0.101368
$$963$$ 37011.3 1.23850
$$964$$ −10588.9 −0.353782
$$965$$ 0 0
$$966$$ −42123.8 −1.40301
$$967$$ −30315.5 −1.00815 −0.504075 0.863660i $$-0.668167\pi$$
−0.504075 + 0.863660i $$0.668167\pi$$
$$968$$ 10505.0 0.348805
$$969$$ −8090.73 −0.268227
$$970$$ 0 0
$$971$$ 26455.6 0.874357 0.437178 0.899375i $$-0.355978\pi$$
0.437178 + 0.899375i $$0.355978\pi$$
$$972$$ 7101.41 0.234339
$$973$$ −7072.03 −0.233010
$$974$$ −22177.6 −0.729584
$$975$$ 0 0
$$976$$ 10091.1 0.330950
$$977$$ −30207.7 −0.989183 −0.494591 0.869126i $$-0.664682\pi$$
−0.494591 + 0.869126i $$0.664682\pi$$
$$978$$ 45581.3 1.49032
$$979$$ 3894.56 0.127141
$$980$$ 0 0
$$981$$ −99050.7 −3.22370
$$982$$ 26430.3 0.858886
$$983$$ 5878.48 0.190737 0.0953685 0.995442i $$-0.469597\pi$$
0.0953685 + 0.995442i $$0.469597\pi$$
$$984$$ 1243.52 0.0402866
$$985$$ 0 0
$$986$$ 8564.33 0.276616
$$987$$ −113921. −3.67392
$$988$$ −4866.74 −0.156712
$$989$$ 33906.4 1.09015
$$990$$ 0 0
$$991$$ −42532.3 −1.36335 −0.681676 0.731654i $$-0.738749\pi$$
−0.681676 + 0.731654i $$0.738749\pi$$
$$992$$ 2622.98 0.0839513
$$993$$ 40121.2 1.28218
$$994$$ −31073.5 −0.991542
$$995$$ 0 0
$$996$$ 17325.3 0.551179
$$997$$ −6320.28 −0.200767 −0.100384 0.994949i $$-0.532007\pi$$
−0.100384 + 0.994949i $$0.532007\pi$$
$$998$$ −821.281 −0.0260493
$$999$$ −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.a.e.1.1 2
5.2 odd 4 950.4.b.i.799.2 4
5.3 odd 4 950.4.b.i.799.3 4
5.4 even 2 38.4.a.c.1.2 2
15.14 odd 2 342.4.a.h.1.2 2
20.19 odd 2 304.4.a.c.1.1 2
35.34 odd 2 1862.4.a.e.1.1 2
40.19 odd 2 1216.4.a.p.1.2 2
40.29 even 2 1216.4.a.g.1.1 2
95.94 odd 2 722.4.a.f.1.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
38.4.a.c.1.2 2 5.4 even 2
304.4.a.c.1.1 2 20.19 odd 2
342.4.a.h.1.2 2 15.14 odd 2
722.4.a.f.1.1 2 95.94 odd 2
950.4.a.e.1.1 2 1.1 even 1 trivial
950.4.b.i.799.2 4 5.2 odd 4
950.4.b.i.799.3 4 5.3 odd 4
1216.4.a.g.1.1 2 40.29 even 2
1216.4.a.p.1.2 2 40.19 odd 2
1862.4.a.e.1.1 2 35.34 odd 2