# Properties

 Label 1216.4.a.g.1.1 Level $1216$ Weight $4$ Character 1216.1 Self dual yes Analytic conductor $71.746$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1216,4,Mod(1,1216)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1216, 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("1216.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1216 = 2^{6} \cdot 19$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 1216.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: $$71.7463225670$$ Analytic rank: $$0$$ 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 $$\chi$$ $$=$$ 1216.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-8.77200 q^{3} +17.3160 q^{5} -26.0880 q^{7} +49.9480 q^{9} +O(q^{10})$$ $$q-8.77200 q^{3} +17.3160 q^{5} -26.0880 q^{7} +49.9480 q^{9} +4.22800 q^{11} -64.0360 q^{13} -151.896 q^{15} -48.5440 q^{17} -19.0000 q^{19} +228.844 q^{21} +92.0360 q^{23} +174.844 q^{25} -201.300 q^{27} +88.2120 q^{29} -81.9681 q^{31} -37.0880 q^{33} -451.740 q^{35} +23.6161 q^{37} +561.724 q^{39} +17.7200 q^{41} -368.404 q^{43} +864.900 q^{45} -497.812 q^{47} +337.584 q^{49} +425.828 q^{51} +536.876 q^{53} +73.2120 q^{55} +166.668 q^{57} +36.7000 q^{59} -630.692 q^{61} -1303.04 q^{63} -1108.85 q^{65} -282.556 q^{67} -807.340 q^{69} +595.552 q^{71} -597.048 q^{73} -1533.73 q^{75} -110.300 q^{77} +427.224 q^{79} +417.208 q^{81} -493.768 q^{83} -840.588 q^{85} -773.796 q^{87} -921.136 q^{89} +1670.57 q^{91} +719.024 q^{93} -329.004 q^{95} +1082.74 q^{97} +211.180 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 9 q^{3} + 9 q^{5} - 18 q^{7} + 23 q^{9}+O(q^{10})$$ 2 * q - 9 * q^3 + 9 * q^5 - 18 * q^7 + 23 * q^9 $$2 q - 9 q^{3} + 9 q^{5} - 18 q^{7} + 23 q^{9} + 17 q^{11} - 17 q^{13} - 150 q^{15} - 80 q^{17} - 38 q^{19} + 227 q^{21} + 73 q^{23} + 119 q^{25} - 189 q^{27} - 3 q^{29} + 212 q^{31} - 40 q^{33} - 519 q^{35} - 192 q^{37} + 551 q^{39} - 50 q^{41} - 677 q^{43} + 1089 q^{45} - 389 q^{47} + 60 q^{49} + 433 q^{51} + 1219 q^{53} - 33 q^{55} + 171 q^{57} + 287 q^{59} - 313 q^{61} - 1521 q^{63} - 1500 q^{65} - 1223 q^{67} - 803 q^{69} + 200 q^{71} + 378 q^{73} - 1521 q^{75} - 7 q^{77} + 1350 q^{79} + 1142 q^{81} + 670 q^{83} - 579 q^{85} - 753 q^{87} - 236 q^{89} + 2051 q^{91} + 652 q^{93} - 171 q^{95} + 1294 q^{97} - 133 q^{99}+O(q^{100})$$ 2 * q - 9 * q^3 + 9 * q^5 - 18 * q^7 + 23 * q^9 + 17 * q^11 - 17 * q^13 - 150 * q^15 - 80 * q^17 - 38 * q^19 + 227 * q^21 + 73 * q^23 + 119 * q^25 - 189 * q^27 - 3 * q^29 + 212 * q^31 - 40 * q^33 - 519 * q^35 - 192 * q^37 + 551 * q^39 - 50 * q^41 - 677 * q^43 + 1089 * q^45 - 389 * q^47 + 60 * q^49 + 433 * q^51 + 1219 * q^53 - 33 * q^55 + 171 * q^57 + 287 * q^59 - 313 * q^61 - 1521 * q^63 - 1500 * q^65 - 1223 * q^67 - 803 * q^69 + 200 * q^71 + 378 * q^73 - 1521 * q^75 - 7 * q^77 + 1350 * q^79 + 1142 * q^81 + 670 * q^83 - 579 * q^85 - 753 * q^87 - 236 * q^89 + 2051 * q^91 + 652 * q^93 - 171 * q^95 + 1294 * q^97 - 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0 0
$$3$$ −8.77200 −1.68817 −0.844086 0.536207i $$-0.819856\pi$$
−0.844086 + 0.536207i $$0.819856\pi$$
$$4$$ 0 0
$$5$$ 17.3160 1.54879 0.774395 0.632702i $$-0.218054\pi$$
0.774395 + 0.632702i $$0.218054\pi$$
$$6$$ 0 0
$$7$$ −26.0880 −1.40862 −0.704310 0.709893i $$-0.748743\pi$$
−0.704310 + 0.709893i $$0.748743\pi$$
$$8$$ 0 0
$$9$$ 49.9480 1.84993
$$10$$ 0 0
$$11$$ 4.22800 0.115890 0.0579450 0.998320i $$-0.481545\pi$$
0.0579450 + 0.998320i $$0.481545\pi$$
$$12$$ 0 0
$$13$$ −64.0360 −1.36618 −0.683092 0.730332i $$-0.739365\pi$$
−0.683092 + 0.730332i $$0.739365\pi$$
$$14$$ 0 0
$$15$$ −151.896 −2.61463
$$16$$ 0 0
$$17$$ −48.5440 −0.692568 −0.346284 0.938130i $$-0.612557\pi$$
−0.346284 + 0.938130i $$0.612557\pi$$
$$18$$ 0 0
$$19$$ −19.0000 −0.229416
$$20$$ 0 0
$$21$$ 228.844 2.37799
$$22$$ 0 0
$$23$$ 92.0360 0.834384 0.417192 0.908818i $$-0.363014\pi$$
0.417192 + 0.908818i $$0.363014\pi$$
$$24$$ 0 0
$$25$$ 174.844 1.39875
$$26$$ 0 0
$$27$$ −201.300 −1.43482
$$28$$ 0 0
$$29$$ 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 $$0.576312\pi$$
$$32$$ 0 0
$$33$$ −37.0880 −0.195642
$$34$$ 0 0
$$35$$ −451.740 −2.18166
$$36$$ 0 0
$$37$$ 23.6161 0.104931 0.0524656 0.998623i $$-0.483292\pi$$
0.0524656 + 0.998623i $$0.483292\pi$$
$$38$$ 0 0
$$39$$ 561.724 2.30636
$$40$$ 0 0
$$41$$ 17.7200 0.0674976 0.0337488 0.999430i $$-0.489255\pi$$
0.0337488 + 0.999430i $$0.489255\pi$$
$$42$$ 0 0
$$43$$ −368.404 −1.30654 −0.653268 0.757126i $$-0.726603\pi$$
−0.653268 + 0.757126i $$0.726603\pi$$
$$44$$ 0 0
$$45$$ 864.900 2.86515
$$46$$ 0 0
$$47$$ −497.812 −1.54497 −0.772483 0.635036i $$-0.780985\pi$$
−0.772483 + 0.635036i $$0.780985\pi$$
$$48$$ 0 0
$$49$$ 337.584 0.984210
$$50$$ 0 0
$$51$$ 425.828 1.16917
$$52$$ 0 0
$$53$$ 536.876 1.39143 0.695713 0.718320i $$-0.255089\pi$$
0.695713 + 0.718320i $$0.255089\pi$$
$$54$$ 0 0
$$55$$ 73.2120 0.179489
$$56$$ 0 0
$$57$$ 166.668 0.387293
$$58$$ 0 0
$$59$$ 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 $$-0.730250\pi$$
−0.661901 + 0.749592i $$0.730250\pi$$
$$62$$ 0 0
$$63$$ −1303.04 −2.60584
$$64$$ 0 0
$$65$$ −1108.85 −2.11593
$$66$$ 0 0
$$67$$ −282.556 −0.515219 −0.257610 0.966249i $$-0.582935\pi$$
−0.257610 + 0.966249i $$0.582935\pi$$
$$68$$ 0 0
$$69$$ −807.340 −1.40858
$$70$$ 0 0
$$71$$ 595.552 0.995480 0.497740 0.867326i $$-0.334163\pi$$
0.497740 + 0.867326i $$0.334163\pi$$
$$72$$ 0 0
$$73$$ −597.048 −0.957250 −0.478625 0.878020i $$-0.658865\pi$$
−0.478625 + 0.878020i $$0.658865\pi$$
$$74$$ 0 0
$$75$$ −1533.73 −2.36134
$$76$$ 0 0
$$77$$ −110.300 −0.163245
$$78$$ 0 0
$$79$$ 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
$$82$$ 0 0
$$83$$ −493.768 −0.652989 −0.326495 0.945199i $$-0.605868\pi$$
−0.326495 + 0.945199i $$0.605868\pi$$
$$84$$ 0 0
$$85$$ −840.588 −1.07264
$$86$$ 0 0
$$87$$ −773.796 −0.953559
$$88$$ 0 0
$$89$$ −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
$$92$$ 0 0
$$93$$ 719.024 0.801713
$$94$$ 0 0
$$95$$ −329.004 −0.355317
$$96$$ 0 0
$$97$$ 1082.74 1.13336 0.566680 0.823938i $$-0.308227\pi$$
0.566680 + 0.823938i $$0.308227\pi$$
$$98$$ 0 0
$$99$$ 211.180 0.214388
$$100$$ 0 0
$$101$$ 712.448 0.701893 0.350947 0.936395i $$-0.385860\pi$$
0.350947 + 0.936395i $$0.385860\pi$$
$$102$$ 0 0
$$103$$ −26.4797 −0.0253313 −0.0126656 0.999920i $$-0.504032\pi$$
−0.0126656 + 0.999920i $$0.504032\pi$$
$$104$$ 0 0
$$105$$ 3962.66 3.68301
$$106$$ 0 0
$$107$$ 740.996 0.669484 0.334742 0.942310i $$-0.391351\pi$$
0.334742 + 0.942310i $$0.391351\pi$$
$$108$$ 0 0
$$109$$ 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
$$112$$ 0 0
$$113$$ −718.720 −0.598332 −0.299166 0.954201i $$-0.596708\pi$$
−0.299166 + 0.954201i $$0.596708\pi$$
$$114$$ 0 0
$$115$$ 1593.70 1.29229
$$116$$ 0 0
$$117$$ −3198.47 −2.52734
$$118$$ 0 0
$$119$$ 1266.42 0.975565
$$120$$ 0 0
$$121$$ −1313.12 −0.986570
$$122$$ 0 0
$$123$$ −155.440 −0.113948
$$124$$ 0 0
$$125$$ 863.100 0.617584
$$126$$ 0 0
$$127$$ 2610.72 1.82413 0.912063 0.410050i $$-0.134489\pi$$
0.912063 + 0.410050i $$0.134489\pi$$
$$128$$ 0 0
$$129$$ 3231.64 2.20566
$$130$$ 0 0
$$131$$ 1216.69 0.811472 0.405736 0.913990i $$-0.367015\pi$$
0.405736 + 0.913990i $$0.367015\pi$$
$$132$$ 0 0
$$133$$ 495.672 0.323160
$$134$$ 0 0
$$135$$ −3485.71 −2.22224
$$136$$ 0 0
$$137$$ 1170.67 0.730053 0.365026 0.930997i $$-0.381060\pi$$
0.365026 + 0.930997i $$0.381060\pi$$
$$138$$ 0 0
$$139$$ 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
$$142$$ 0 0
$$143$$ −270.744 −0.158327
$$144$$ 0 0
$$145$$ 1527.48 0.874830
$$146$$ 0 0
$$147$$ −2961.29 −1.66152
$$148$$ 0 0
$$149$$ −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 $$0.146994\pi$$
$$152$$ 0 0
$$153$$ −2424.68 −1.28120
$$154$$ 0 0
$$155$$ −1419.36 −0.735521
$$156$$ 0 0
$$157$$ −243.616 −0.123839 −0.0619194 0.998081i $$-0.519722\pi$$
−0.0619194 + 0.998081i $$0.519722\pi$$
$$158$$ 0 0
$$159$$ −4709.48 −2.34897
$$160$$ 0 0
$$161$$ −2401.04 −1.17533
$$162$$ 0 0
$$163$$ 2598.11 1.24847 0.624233 0.781238i $$-0.285412\pi$$
0.624233 + 0.781238i $$0.285412\pi$$
$$164$$ 0 0
$$165$$ −642.216 −0.303009
$$166$$ 0 0
$$167$$ −491.064 −0.227543 −0.113772 0.993507i $$-0.536293\pi$$
−0.113772 + 0.993507i $$0.536293\pi$$
$$168$$ 0 0
$$169$$ 1903.61 0.866460
$$170$$ 0 0
$$171$$ −949.012 −0.424402
$$172$$ 0 0
$$173$$ −1648.56 −0.724496 −0.362248 0.932082i $$-0.617991\pi$$
−0.362248 + 0.932082i $$0.617991\pi$$
$$174$$ 0 0
$$175$$ −4561.33 −1.97031
$$176$$ 0 0
$$177$$ −321.932 −0.136711
$$178$$ 0 0
$$179$$ −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 $$-0.0988066\pi$$
0.952208 + 0.305449i $$0.0988066\pi$$
$$182$$ 0 0
$$183$$ 5532.43 2.23480
$$184$$ 0 0
$$185$$ 408.936 0.162516
$$186$$ 0 0
$$187$$ −205.244 −0.0802616
$$188$$ 0 0
$$189$$ 5251.52 2.02112
$$190$$ 0 0
$$191$$ 5260.38 1.99281 0.996407 0.0846903i $$-0.0269901\pi$$
0.996407 + 0.0846903i $$0.0269901\pi$$
$$192$$ 0 0
$$193$$ 16.1833 0.00603575 0.00301787 0.999995i $$-0.499039\pi$$
0.00301787 + 0.999995i $$0.499039\pi$$
$$194$$ 0 0
$$195$$ 9726.82 3.57206
$$196$$ 0 0
$$197$$ −3784.71 −1.36878 −0.684390 0.729116i $$-0.739931\pi$$
−0.684390 + 0.729116i $$0.739931\pi$$
$$198$$ 0 0
$$199$$ 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
$$202$$ 0 0
$$203$$ −2301.28 −0.795655
$$204$$ 0 0
$$205$$ 306.840 0.104540
$$206$$ 0 0
$$207$$ 4597.02 1.54355
$$208$$ 0 0
$$209$$ −80.3320 −0.0265870
$$210$$ 0 0
$$211$$ 2945.44 0.961006 0.480503 0.876993i $$-0.340454\pi$$
0.480503 + 0.876993i $$0.340454\pi$$
$$212$$ 0 0
$$213$$ −5224.19 −1.68054
$$214$$ 0 0
$$215$$ −6379.29 −2.02355
$$216$$ 0 0
$$217$$ 2138.38 0.668954
$$218$$ 0 0
$$219$$ 5237.31 1.61600
$$220$$ 0 0
$$221$$ 3108.57 0.946175
$$222$$ 0 0
$$223$$ 3125.30 0.938499 0.469250 0.883066i $$-0.344524\pi$$
0.469250 + 0.883066i $$0.344524\pi$$
$$224$$ 0 0
$$225$$ 8733.11 2.58759
$$226$$ 0 0
$$227$$ 3577.80 1.04611 0.523055 0.852299i $$-0.324792\pi$$
0.523055 + 0.852299i $$0.324792\pi$$
$$228$$ 0 0
$$229$$ 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
$$232$$ 0 0
$$233$$ 5829.49 1.63907 0.819534 0.573031i $$-0.194232\pi$$
0.819534 + 0.573031i $$0.194232\pi$$
$$234$$ 0 0
$$235$$ −8620.12 −2.39283
$$236$$ 0 0
$$237$$ −3747.61 −1.02714
$$238$$ 0 0
$$239$$ 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 $$0.615104\pi$$
$$242$$ 0 0
$$243$$ 1775.35 0.468679
$$244$$ 0 0
$$245$$ 5845.61 1.52434
$$246$$ 0 0
$$247$$ 1216.68 0.313424
$$248$$ 0 0
$$249$$ 4331.34 1.10236
$$250$$ 0 0
$$251$$ −1970.73 −0.495582 −0.247791 0.968814i $$-0.579705\pi$$
−0.247791 + 0.968814i $$0.579705\pi$$
$$252$$ 0 0
$$253$$ 389.128 0.0966967
$$254$$ 0 0
$$255$$ 7373.64 1.81081
$$256$$ 0 0
$$257$$ −7915.82 −1.92131 −0.960653 0.277752i $$-0.910411\pi$$
−0.960653 + 0.277752i $$0.910411\pi$$
$$258$$ 0 0
$$259$$ −616.096 −0.147808
$$260$$ 0 0
$$261$$ 4406.02 1.04493
$$262$$ 0 0
$$263$$ 3287.96 0.770892 0.385446 0.922730i $$-0.374048\pi$$
0.385446 + 0.922730i $$0.374048\pi$$
$$264$$ 0 0
$$265$$ 9296.55 2.15503
$$266$$ 0 0
$$267$$ 8080.21 1.85206
$$268$$ 0 0
$$269$$ 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 $$0.491342\pi$$
$$272$$ 0 0
$$273$$ −14654.3 −3.24878
$$274$$ 0 0
$$275$$ 739.240 0.162101
$$276$$ 0 0
$$277$$ 4131.13 0.896086 0.448043 0.894012i $$-0.352121\pi$$
0.448043 + 0.894012i $$0.352121\pi$$
$$278$$ 0 0
$$279$$ −4094.14 −0.878530
$$280$$ 0 0
$$281$$ 1007.19 0.213822 0.106911 0.994269i $$-0.465904\pi$$
0.106911 + 0.994269i $$0.465904\pi$$
$$282$$ 0 0
$$283$$ −2333.63 −0.490176 −0.245088 0.969501i $$-0.578817\pi$$
−0.245088 + 0.969501i $$0.578817\pi$$
$$284$$ 0 0
$$285$$ 2886.02 0.599836
$$286$$ 0 0
$$287$$ −462.280 −0.0950785
$$288$$ 0 0
$$289$$ −2556.48 −0.520350
$$290$$ 0 0
$$291$$ −9497.83 −1.91331
$$292$$ 0 0
$$293$$ 1588.68 0.316763 0.158381 0.987378i $$-0.449372\pi$$
0.158381 + 0.987378i $$0.449372\pi$$
$$294$$ 0 0
$$295$$ 635.497 0.125424
$$296$$ 0 0
$$297$$ −851.096 −0.166282
$$298$$ 0 0
$$299$$ −5893.62 −1.13992
$$300$$ 0 0
$$301$$ 9610.93 1.84041
$$302$$ 0 0
$$303$$ −6249.59 −1.18492
$$304$$ 0 0
$$305$$ −10921.1 −2.05029
$$306$$ 0 0
$$307$$ −4057.46 −0.754304 −0.377152 0.926151i $$-0.623097\pi$$
−0.377152 + 0.926151i $$0.623097\pi$$
$$308$$ 0 0
$$309$$ 232.280 0.0427636
$$310$$ 0 0
$$311$$ 2871.92 0.523638 0.261819 0.965117i $$-0.415678\pi$$
0.261819 + 0.965117i $$0.415678\pi$$
$$312$$ 0 0
$$313$$ 4322.67 0.780612 0.390306 0.920685i $$-0.372369\pi$$
0.390306 + 0.920685i $$0.372369\pi$$
$$314$$ 0 0
$$315$$ −22563.5 −4.03591
$$316$$ 0 0
$$317$$ −2513.56 −0.445349 −0.222674 0.974893i $$-0.571479\pi$$
−0.222674 + 0.974893i $$0.571479\pi$$
$$318$$ 0 0
$$319$$ 372.960 0.0654601
$$320$$ 0 0
$$321$$ −6500.02 −1.13021
$$322$$ 0 0
$$323$$ 922.336 0.158886
$$324$$ 0 0
$$325$$ −11196.3 −1.91095
$$326$$ 0 0
$$327$$ −17395.6 −2.94182
$$328$$ 0 0
$$329$$ 12986.9 2.17627
$$330$$ 0 0
$$331$$ 4573.78 0.759509 0.379754 0.925087i $$-0.376008\pi$$
0.379754 + 0.925087i $$0.376008\pi$$
$$332$$ 0 0
$$333$$ 1179.57 0.194115
$$334$$ 0 0
$$335$$ −4892.74 −0.797967
$$336$$ 0 0
$$337$$ 9001.71 1.45506 0.727529 0.686077i $$-0.240669\pi$$
0.727529 + 0.686077i $$0.240669\pi$$
$$338$$ 0 0
$$339$$ 6304.62 1.01009
$$340$$ 0 0
$$341$$ −346.561 −0.0550361
$$342$$ 0 0
$$343$$ 141.289 0.0222417
$$344$$ 0 0
$$345$$ −13979.9 −2.18160
$$346$$ 0 0
$$347$$ −9358.68 −1.44784 −0.723920 0.689884i $$-0.757661\pi$$
−0.723920 + 0.689884i $$0.757661\pi$$
$$348$$ 0 0
$$349$$ −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
$$352$$ 0 0
$$353$$ 5784.59 0.872188 0.436094 0.899901i $$-0.356361\pi$$
0.436094 + 0.899901i $$0.356361\pi$$
$$354$$ 0 0
$$355$$ 10312.6 1.54179
$$356$$ 0 0
$$357$$ −11109.0 −1.64692
$$358$$ 0 0
$$359$$ −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
$$362$$ 0 0
$$363$$ 11518.7 1.66550
$$364$$ 0 0
$$365$$ −10338.5 −1.48258
$$366$$ 0 0
$$367$$ −6993.81 −0.994752 −0.497376 0.867535i $$-0.665703\pi$$
−0.497376 + 0.867535i $$0.665703\pi$$
$$368$$ 0 0
$$369$$ 885.080 0.124866
$$370$$ 0 0
$$371$$ −14006.0 −1.95999
$$372$$ 0 0
$$373$$ −6523.15 −0.905512 −0.452756 0.891634i $$-0.649559\pi$$
−0.452756 + 0.891634i $$0.649559\pi$$
$$374$$ 0 0
$$375$$ −7571.11 −1.04259
$$376$$ 0 0
$$377$$ −5648.75 −0.771685
$$378$$ 0 0
$$379$$ 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
$$382$$ 0 0
$$383$$ 9878.11 1.31788 0.658940 0.752196i $$-0.271005\pi$$
0.658940 + 0.752196i $$0.271005\pi$$
$$384$$ 0 0
$$385$$ −1909.96 −0.252832
$$386$$ 0 0
$$387$$ −18401.0 −2.41700
$$388$$ 0 0
$$389$$ 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
$$392$$ 0 0
$$393$$ −10672.8 −1.36991
$$394$$ 0 0
$$395$$ 7397.81 0.942340
$$396$$ 0 0
$$397$$ 2787.84 0.352437 0.176219 0.984351i $$-0.443613\pi$$
0.176219 + 0.984351i $$0.443613\pi$$
$$398$$ 0 0
$$399$$ −4348.04 −0.545549
$$400$$ 0 0
$$401$$ 1264.42 0.157461 0.0787306 0.996896i $$-0.474913\pi$$
0.0787306 + 0.996896i $$0.474913\pi$$
$$402$$ 0 0
$$403$$ 5248.91 0.648801
$$404$$ 0 0
$$405$$ 7224.37 0.886375
$$406$$ 0 0
$$407$$ 99.8486 0.0121605
$$408$$ 0 0
$$409$$ −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
$$412$$ 0 0
$$413$$ −957.429 −0.114073
$$414$$ 0 0
$$415$$ −8550.10 −1.01134
$$416$$ 0 0
$$417$$ −2377.94 −0.279253
$$418$$ 0 0
$$419$$ 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 $$-0.607074\pi$$
−0.330074 + 0.943955i $$0.607074\pi$$
$$422$$ 0 0
$$423$$ −24864.7 −2.85807
$$424$$ 0 0
$$425$$ −8487.63 −0.968731
$$426$$ 0 0
$$427$$ 16453.5 1.86473
$$428$$ 0 0
$$429$$ 2374.97 0.267283
$$430$$ 0 0
$$431$$ −4025.72 −0.449912 −0.224956 0.974369i $$-0.572224\pi$$
−0.224956 + 0.974369i $$0.572224\pi$$
$$432$$ 0 0
$$433$$ −1347.10 −0.149510 −0.0747548 0.997202i $$-0.523817\pi$$
−0.0747548 + 0.997202i $$0.523817\pi$$
$$434$$ 0 0
$$435$$ −13399.1 −1.47686
$$436$$ 0 0
$$437$$ −1748.68 −0.191421
$$438$$ 0 0
$$439$$ 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
$$442$$ 0 0
$$443$$ −6964.84 −0.746974 −0.373487 0.927635i $$-0.621838\pi$$
−0.373487 + 0.927635i $$0.621838\pi$$
$$444$$ 0 0
$$445$$ −15950.4 −1.69915
$$446$$ 0 0
$$447$$ 16150.9 1.70897
$$448$$ 0 0
$$449$$ 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
$$452$$ 0 0
$$453$$ −29143.4 −3.02269
$$454$$ 0 0
$$455$$ 28927.6 2.98055
$$456$$ 0 0
$$457$$ 11984.3 1.22670 0.613352 0.789810i $$-0.289821\pi$$
0.613352 + 0.789810i $$0.289821\pi$$
$$458$$ 0 0
$$459$$ 9771.91 0.993712
$$460$$ 0 0
$$461$$ 12126.7 1.22515 0.612577 0.790411i $$-0.290133\pi$$
0.612577 + 0.790411i $$0.290133\pi$$
$$462$$ 0 0
$$463$$ −6399.19 −0.642323 −0.321162 0.947024i $$-0.604073\pi$$
−0.321162 + 0.947024i $$0.604073\pi$$
$$464$$ 0 0
$$465$$ 12450.6 1.24169
$$466$$ 0 0
$$467$$ −993.366 −0.0984315 −0.0492157 0.998788i $$-0.515672\pi$$
−0.0492157 + 0.998788i $$0.515672\pi$$
$$468$$ 0 0
$$469$$ 7371.32 0.725748
$$470$$ 0 0
$$471$$ 2137.00 0.209061
$$472$$ 0 0
$$473$$ −1557.61 −0.151414
$$474$$ 0 0
$$475$$ −3322.04 −0.320896
$$476$$ 0 0
$$477$$ 26815.9 2.57404
$$478$$ 0 0
$$479$$ 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
$$482$$ 0 0
$$483$$ 21061.9 1.98416
$$484$$ 0 0
$$485$$ 18748.8 1.75534
$$486$$ 0 0
$$487$$ −11088.8 −1.03179 −0.515894 0.856652i $$-0.672540\pi$$
−0.515894 + 0.856652i $$0.672540\pi$$
$$488$$ 0 0
$$489$$ −22790.6 −2.10762
$$490$$ 0 0
$$491$$ 13215.2 1.21465 0.607324 0.794454i $$-0.292243\pi$$
0.607324 + 0.794454i $$0.292243\pi$$
$$492$$ 0 0
$$493$$ −4282.17 −0.391195
$$494$$ 0 0
$$495$$ 3656.80 0.332042
$$496$$ 0 0
$$497$$ −15536.8 −1.40225
$$498$$ 0 0
$$499$$ −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
$$502$$ 0 0
$$503$$ −9407.88 −0.833950 −0.416975 0.908918i $$-0.636910\pi$$
−0.416975 + 0.908918i $$0.636910\pi$$
$$504$$ 0 0
$$505$$ 12336.7 1.08709
$$506$$ 0 0
$$507$$ −16698.5 −1.46273
$$508$$ 0 0
$$509$$ −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
$$512$$ 0 0
$$513$$ 3824.70 0.329171
$$514$$ 0 0
$$515$$ −458.523 −0.0392329
$$516$$ 0 0
$$517$$ −2104.75 −0.179046
$$518$$ 0 0
$$519$$ 14461.2 1.22307
$$520$$ 0 0
$$521$$ −3181.02 −0.267492 −0.133746 0.991016i $$-0.542701\pi$$
−0.133746 + 0.991016i $$0.542701\pi$$
$$522$$ 0 0
$$523$$ 4360.12 0.364541 0.182270 0.983248i $$-0.441655\pi$$
0.182270 + 0.983248i $$0.441655\pi$$
$$524$$ 0 0
$$525$$ 40012.0 3.32622
$$526$$ 0 0
$$527$$ 3979.06 0.328900
$$528$$ 0 0
$$529$$ −3696.37 −0.303803
$$530$$ 0 0
$$531$$ 1833.09 0.149810
$$532$$ 0 0
$$533$$ −1134.72 −0.0922142
$$534$$ 0 0
$$535$$ 12831.1 1.03689
$$536$$ 0 0
$$537$$ 20410.8 1.64020
$$538$$ 0 0
$$539$$ 1427.31 0.114060
$$540$$ 0 0
$$541$$ 23681.2 1.88195 0.940973 0.338481i $$-0.109913\pi$$
0.940973 + 0.338481i $$0.109913\pi$$
$$542$$ 0 0
$$543$$ −40679.8 −3.21498
$$544$$ 0 0
$$545$$ 34339.0 2.69894
$$546$$ 0 0
$$547$$ 7373.25 0.576339 0.288169 0.957579i $$-0.406953\pi$$
0.288169 + 0.957579i $$0.406953\pi$$
$$548$$ 0 0
$$549$$ −31501.8 −2.44893
$$550$$ 0 0
$$551$$ −1676.03 −0.129585
$$552$$ 0 0
$$553$$ −11145.4 −0.857055
$$554$$ 0 0
$$555$$ −3587.18 −0.274356
$$556$$ 0 0
$$557$$ −4772.14 −0.363020 −0.181510 0.983389i $$-0.558098\pi$$
−0.181510 + 0.983389i $$0.558098\pi$$
$$558$$ 0 0
$$559$$ 23591.1 1.78497
$$560$$ 0 0
$$561$$ 1800.40 0.135495
$$562$$ 0 0
$$563$$ −7276.49 −0.544702 −0.272351 0.962198i $$-0.587801\pi$$
−0.272351 + 0.962198i $$0.587801\pi$$
$$564$$ 0 0
$$565$$ −12445.4 −0.926691
$$566$$ 0 0
$$567$$ −10884.1 −0.806156
$$568$$ 0 0
$$569$$ −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 $$-0.683244\pi$$
−0.544404 + 0.838823i $$0.683244\pi$$
$$572$$ 0 0
$$573$$ −46144.0 −3.36421
$$574$$ 0 0
$$575$$ 16092.0 1.16710
$$576$$ 0 0
$$577$$ 3212.67 0.231794 0.115897 0.993261i $$-0.463026\pi$$
0.115897 + 0.993261i $$0.463026\pi$$
$$578$$ 0 0
$$579$$ −141.960 −0.0101894
$$580$$ 0 0
$$581$$ 12881.4 0.919814
$$582$$ 0 0
$$583$$ 2269.91 0.161252
$$584$$ 0 0
$$585$$ −55384.8 −3.91432
$$586$$ 0 0
$$587$$ 22321.1 1.56949 0.784745 0.619818i $$-0.212794\pi$$
0.784745 + 0.619818i $$0.212794\pi$$
$$588$$ 0 0
$$589$$ 1557.39 0.108950
$$590$$ 0 0
$$591$$ 33199.5 2.31074
$$592$$ 0 0
$$593$$ −8202.50 −0.568021 −0.284010 0.958821i $$-0.591665\pi$$
−0.284010 + 0.958821i $$0.591665\pi$$
$$594$$ 0 0
$$595$$ 21929.3 1.51095
$$596$$ 0 0
$$597$$ −642.180 −0.0440246
$$598$$ 0 0
$$599$$ 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 $$0.599387\pi$$
$$602$$ 0 0
$$603$$ −14113.1 −0.953118
$$604$$ 0 0
$$605$$ −22738.1 −1.52799
$$606$$ 0 0
$$607$$ 8123.48 0.543199 0.271599 0.962410i $$-0.412447\pi$$
0.271599 + 0.962410i $$0.412447\pi$$
$$608$$ 0 0
$$609$$ 20186.8 1.34320
$$610$$ 0 0
$$611$$ 31877.9 2.11071
$$612$$ 0 0
$$613$$ −22384.7 −1.47490 −0.737448 0.675404i $$-0.763969\pi$$
−0.737448 + 0.675404i $$0.763969\pi$$
$$614$$ 0 0
$$615$$ −2691.60 −0.176481
$$616$$ 0 0
$$617$$ 11349.1 0.740517 0.370259 0.928929i $$-0.379269\pi$$
0.370259 + 0.928929i $$0.379269\pi$$
$$618$$ 0 0
$$619$$ 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
$$622$$ 0 0
$$623$$ 24030.6 1.54537
$$624$$ 0 0
$$625$$ −6910.06 −0.442244
$$626$$ 0 0
$$627$$ 704.672 0.0448834
$$628$$ 0 0
$$629$$ −1146.42 −0.0726720
$$630$$ 0 0
$$631$$ −27784.2 −1.75289 −0.876444 0.481505i $$-0.840090\pi$$
−0.876444 + 0.481505i $$0.840090\pi$$
$$632$$ 0 0
$$633$$ −25837.4 −1.62234
$$634$$ 0 0
$$635$$ 45207.3 2.82519
$$636$$ 0 0
$$637$$ −21617.5 −1.34461
$$638$$ 0 0
$$639$$ 29746.7 1.84156
$$640$$ 0 0
$$641$$ −16958.3 −1.04495 −0.522476 0.852654i $$-0.674992\pi$$
−0.522476 + 0.852654i $$0.674992\pi$$
$$642$$ 0 0
$$643$$ −4754.37 −0.291592 −0.145796 0.989315i $$-0.546574\pi$$
−0.145796 + 0.989315i $$0.546574\pi$$
$$644$$ 0 0
$$645$$ 55959.1 3.41611
$$646$$ 0 0
$$647$$ −11254.0 −0.683831 −0.341916 0.939731i $$-0.611076\pi$$
−0.341916 + 0.939731i $$0.611076\pi$$
$$648$$ 0 0
$$649$$ 155.167 0.00938498
$$650$$ 0 0
$$651$$ −18757.9 −1.12931
$$652$$ 0 0
$$653$$ 15515.1 0.929793 0.464896 0.885365i $$-0.346092\pi$$
0.464896 + 0.885365i $$0.346092\pi$$
$$654$$ 0 0
$$655$$ 21068.2 1.25680
$$656$$ 0 0
$$657$$ −29821.4 −1.77084
$$658$$ 0 0
$$659$$ −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 $$-0.521610\pi$$
−0.0678361 + 0.997696i $$0.521610\pi$$
$$662$$ 0 0
$$663$$ −27268.3 −1.59731
$$664$$ 0 0
$$665$$ 8583.06 0.500507
$$666$$ 0 0
$$667$$ 8118.69 0.471299
$$668$$ 0 0
$$669$$ −27415.1 −1.58435
$$670$$ 0 0
$$671$$ −2666.57 −0.153415
$$672$$ 0 0
$$673$$ −14242.8 −0.815782 −0.407891 0.913031i $$-0.633736\pi$$
−0.407891 + 0.913031i $$0.633736\pi$$
$$674$$ 0 0
$$675$$ −35196.1 −2.00696
$$676$$ 0 0
$$677$$ 13480.0 0.765256 0.382628 0.923902i $$-0.375019\pi$$
0.382628 + 0.923902i $$0.375019\pi$$
$$678$$ 0 0
$$679$$ −28246.6 −1.59647
$$680$$ 0 0
$$681$$ −31384.5 −1.76601
$$682$$ 0 0
$$683$$ −27626.1 −1.54771 −0.773854 0.633365i $$-0.781673\pi$$
−0.773854 + 0.633365i $$0.781673\pi$$
$$684$$ 0 0
$$685$$ 20271.4 1.13070
$$686$$ 0 0
$$687$$ −42123.2 −2.33930
$$688$$ 0 0
$$689$$ −34379.4 −1.90094
$$690$$ 0 0
$$691$$ −17419.7 −0.959009 −0.479505 0.877539i $$-0.659184\pi$$
−0.479505 + 0.877539i $$0.659184\pi$$
$$692$$ 0 0
$$693$$ −5509.27 −0.301991
$$694$$ 0 0
$$695$$ 4694.08 0.256197
$$696$$ 0 0
$$697$$ −860.201 −0.0467467
$$698$$ 0 0
$$699$$ −51136.3 −2.76703
$$700$$ 0 0
$$701$$ −5069.39 −0.273136 −0.136568 0.990631i $$-0.543607\pi$$
−0.136568 + 0.990631i $$0.543607\pi$$
$$702$$ 0 0
$$703$$ −448.705 −0.0240729
$$704$$ 0 0
$$705$$ 75615.7 4.03951
$$706$$ 0 0
$$707$$ −18586.3 −0.988701
$$708$$ 0 0
$$709$$ 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
$$712$$ 0 0
$$713$$ −7544.02 −0.396249
$$714$$ 0 0
$$715$$ −4688.21 −0.245215
$$716$$ 0 0
$$717$$ −11967.9 −0.623359
$$718$$ 0 0
$$719$$ −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
$$722$$ 0 0
$$723$$ 23221.5 1.19449
$$724$$ 0 0
$$725$$ 15423.4 0.790081
$$726$$ 0 0
$$727$$ 6468.37 0.329984 0.164992 0.986295i $$-0.447240\pi$$
0.164992 + 0.986295i $$0.447240\pi$$
$$728$$ 0 0
$$729$$ −26838.0 −1.36351
$$730$$ 0 0
$$731$$ 17883.8 0.904865
$$732$$ 0 0
$$733$$ −25245.5 −1.27212 −0.636059 0.771640i $$-0.719437\pi$$
−0.636059 + 0.771640i $$0.719437\pi$$
$$734$$ 0 0
$$735$$ −51277.7 −2.57334
$$736$$ 0 0
$$737$$ −1194.65 −0.0597087
$$738$$ 0 0
$$739$$ −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
$$742$$ 0 0
$$743$$ 18876.2 0.932033 0.466016 0.884776i $$-0.345689\pi$$
0.466016 + 0.884776i $$0.345689\pi$$
$$744$$ 0 0
$$745$$ −31882.0 −1.56788
$$746$$ 0 0
$$747$$ −24662.8 −1.20798
$$748$$ 0 0
$$749$$ −19331.1 −0.943049
$$750$$ 0 0
$$751$$ 24895.8 1.20967 0.604833 0.796352i $$-0.293240\pi$$
0.604833 + 0.796352i $$0.293240\pi$$
$$752$$ 0 0
$$753$$ 17287.2 0.836628
$$754$$ 0 0
$$755$$ 57529.3 2.77312
$$756$$ 0 0
$$757$$ 36203.2 1.73821 0.869107 0.494624i $$-0.164694\pi$$
0.869107 + 0.494624i $$0.164694\pi$$
$$758$$ 0 0
$$759$$ −3413.43 −0.163241
$$760$$ 0 0
$$761$$ 11417.5 0.543868 0.271934 0.962316i $$-0.412337\pi$$
0.271934 + 0.962316i $$0.412337\pi$$
$$762$$ 0 0
$$763$$ −51734.5 −2.45467
$$764$$ 0 0
$$765$$ −41985.7 −1.98431
$$766$$ 0 0
$$767$$ −2350.12 −0.110636
$$768$$ 0 0
$$769$$ 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
$$772$$ 0 0
$$773$$ 14268.5 0.663910 0.331955 0.943295i $$-0.392292\pi$$
0.331955 + 0.943295i $$0.392292\pi$$
$$774$$ 0 0
$$775$$ −14331.6 −0.664268
$$776$$ 0 0
$$777$$ 5404.39 0.249526
$$778$$ 0 0
$$779$$ −336.680 −0.0154850
$$780$$ 0 0
$$781$$ 2517.99 0.115366
$$782$$ 0 0
$$783$$ −17757.1 −0.810455
$$784$$ 0 0
$$785$$ −4218.46 −0.191800
$$786$$ 0 0
$$787$$ 2922.28 0.132361 0.0661804 0.997808i $$-0.478919\pi$$
0.0661804 + 0.997808i $$0.478919\pi$$
$$788$$ 0 0
$$789$$ −28842.0 −1.30140
$$790$$ 0 0
$$791$$ 18750.0 0.842823
$$792$$ 0 0
$$793$$ 40387.0 1.80856
$$794$$ 0 0
$$795$$ −81549.3 −3.63806
$$796$$ 0 0
$$797$$ 7724.25 0.343296 0.171648 0.985158i $$-0.445091\pi$$
0.171648 + 0.985158i $$0.445091\pi$$
$$798$$ 0 0
$$799$$ 24165.8 1.06999
$$800$$ 0 0
$$801$$ −46008.9 −2.02952
$$802$$ 0 0
$$803$$ −2524.32 −0.110936
$$804$$ 0 0
$$805$$ −41576.4 −1.82034
$$806$$ 0 0
$$807$$ −41663.6 −1.81738
$$808$$ 0 0
$$809$$ 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 $$-0.713842\pi$$
−0.622398 + 0.782701i $$0.713842\pi$$
$$812$$ 0 0
$$813$$ −2128.62 −0.0918253
$$814$$ 0 0
$$815$$ 44988.9 1.93361
$$816$$ 0 0
$$817$$ 6999.68 0.299740
$$818$$ 0 0
$$819$$ 83441.8 3.56006
$$820$$ 0 0
$$821$$ 30274.8 1.28696 0.643482 0.765461i $$-0.277489\pi$$
0.643482 + 0.765461i $$0.277489\pi$$
$$822$$ 0 0
$$823$$ 17296.1 0.732568 0.366284 0.930503i $$-0.380630\pi$$
0.366284 + 0.930503i $$0.380630\pi$$
$$824$$ 0 0
$$825$$ −6484.62 −0.273655
$$826$$ 0 0
$$827$$ 2022.80 0.0850541 0.0425271 0.999095i $$-0.486459\pi$$
0.0425271 + 0.999095i $$0.486459\pi$$
$$828$$ 0 0
$$829$$ 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
$$832$$ 0 0
$$833$$ −16387.7 −0.681632
$$834$$ 0 0
$$835$$ −8503.27 −0.352417
$$836$$ 0 0
$$837$$ 16500.2 0.681397
$$838$$ 0 0
$$839$$ 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
$$842$$ 0 0
$$843$$ −8835.09 −0.360969
$$844$$ 0 0
$$845$$ 32963.0 1.34196
$$846$$ 0 0
$$847$$ 34256.8 1.38970
$$848$$ 0 0
$$849$$ 20470.6 0.827502
$$850$$ 0 0
$$851$$ 2173.53 0.0875530
$$852$$ 0 0
$$853$$ 20978.4 0.842071 0.421035 0.907044i $$-0.361667\pi$$
0.421035 + 0.907044i $$0.361667\pi$$
$$854$$ 0 0
$$855$$ −16433.1 −0.657310
$$856$$ 0 0
$$857$$ −30822.4 −1.22856 −0.614279 0.789089i $$-0.710553\pi$$
−0.614279 + 0.789089i $$0.710553\pi$$
$$858$$ 0 0
$$859$$ 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
$$862$$ 0 0
$$863$$ −24131.3 −0.951842 −0.475921 0.879488i $$-0.657885\pi$$
−0.475921 + 0.879488i $$0.657885\pi$$
$$864$$ 0 0
$$865$$ −28546.5 −1.12209
$$866$$ 0 0
$$867$$ 22425.4 0.878441
$$868$$ 0 0
$$869$$ 1806.30 0.0705116
$$870$$ 0 0
$$871$$ 18093.8 0.703885
$$872$$ 0 0
$$873$$ 54080.9 2.09663
$$874$$ 0 0
$$875$$ −22516.6 −0.869941
$$876$$ 0 0
$$877$$ −39380.6 −1.51629 −0.758147 0.652084i $$-0.773895\pi$$
−0.758147 + 0.652084i $$0.773895\pi$$
$$878$$ 0 0
$$879$$ −13935.9 −0.534750
$$880$$ 0 0
$$881$$ 30887.5 1.18119 0.590595 0.806968i $$-0.298893\pi$$
0.590595 + 0.806968i $$0.298893\pi$$
$$882$$ 0 0
$$883$$ −28191.9 −1.07444 −0.537221 0.843441i $$-0.680526\pi$$
−0.537221 + 0.843441i $$0.680526\pi$$
$$884$$ 0 0
$$885$$ −5574.58 −0.211737
$$886$$ 0 0
$$887$$ 2760.58 0.104500 0.0522498 0.998634i $$-0.483361\pi$$
0.0522498 + 0.998634i $$0.483361\pi$$
$$888$$ 0 0
$$889$$ −68108.5 −2.56950
$$890$$ 0 0
$$891$$ 1763.95 0.0663240
$$892$$ 0 0
$$893$$ 9458.43 0.354439
$$894$$ 0 0
$$895$$ −40291.0 −1.50478
$$896$$ 0 0
$$897$$ 51698.9 1.92439
$$898$$ 0 0
$$899$$ −7230.57 −0.268246
$$900$$ 0 0
$$901$$ −26062.1 −0.963657
$$902$$ 0 0
$$903$$ −84307.1 −3.10694
$$904$$ 0 0
$$905$$ 80302.2 2.94954
$$906$$ 0 0
$$907$$ 18969.1 0.694443 0.347222 0.937783i $$-0.387125\pi$$
0.347222 + 0.937783i $$0.387125\pi$$
$$908$$ 0 0
$$909$$ 35585.4 1.29845
$$910$$ 0 0
$$911$$ −48732.9 −1.77233 −0.886164 0.463371i $$-0.846640\pi$$
−0.886164 + 0.463371i $$0.846640\pi$$
$$912$$ 0 0
$$913$$ −2087.65 −0.0756749
$$914$$ 0 0
$$915$$ 95799.6 3.46124
$$916$$ 0 0
$$917$$ −31741.1 −1.14306
$$918$$ 0 0
$$919$$ −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
$$922$$ 0 0
$$923$$ −38136.8 −1.36001
$$924$$ 0 0
$$925$$ 4129.13 0.146773
$$926$$ 0 0
$$927$$ −1322.61 −0.0468610
$$928$$ 0 0
$$929$$ 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
$$932$$ 0 0
$$933$$ −25192.5 −0.883992
$$934$$ 0 0
$$935$$ −3554.01 −0.124308
$$936$$ 0 0
$$937$$ 47925.4 1.67092 0.835462 0.549548i $$-0.185200\pi$$
0.835462 + 0.549548i $$0.185200\pi$$
$$938$$ 0 0
$$939$$ −37918.5 −1.31781
$$940$$ 0 0
$$941$$ 25842.2 0.895251 0.447626 0.894221i $$-0.352270\pi$$
0.447626 + 0.894221i $$0.352270\pi$$
$$942$$ 0 0
$$943$$ 1630.88 0.0563189
$$944$$ 0 0
$$945$$ 90935.3 3.13029
$$946$$ 0 0
$$947$$ −36562.8 −1.25463 −0.627314 0.778766i $$-0.715846\pi$$
−0.627314 + 0.778766i $$0.715846\pi$$
$$948$$ 0 0
$$949$$ 38232.6 1.30778
$$950$$ 0 0
$$951$$ 22048.9 0.751825
$$952$$ 0 0
$$953$$ 29813.1 1.01337 0.506684 0.862132i $$-0.330871\pi$$
0.506684 + 0.862132i $$0.330871\pi$$
$$954$$ 0 0
$$955$$ 91088.7 3.08645
$$956$$ 0 0
$$957$$ −3271.61 −0.110508
$$958$$ 0 0
$$959$$ −30540.5 −1.02837
$$960$$ 0 0
$$961$$ −23072.2 −0.774470
$$962$$ 0 0
$$963$$ 37011.3 1.23850
$$964$$ 0 0
$$965$$ 280.230 0.00934811
$$966$$ 0 0
$$967$$ 30315.5 1.00815 0.504075 0.863660i $$-0.331833\pi$$
0.504075 + 0.863660i $$0.331833\pi$$
$$968$$ 0 0
$$969$$ −8090.73 −0.268227
$$970$$ 0 0
$$971$$ −26455.6 −0.874357 −0.437178 0.899375i $$-0.644022\pi$$
−0.437178 + 0.899375i $$0.644022\pi$$
$$972$$ 0 0
$$973$$ −7072.03 −0.233010
$$974$$ 0 0
$$975$$ 98214.1 3.22602
$$976$$ 0 0
$$977$$ 30207.7 0.989183 0.494591 0.869126i $$-0.335318\pi$$
0.494591 + 0.869126i $$0.335318\pi$$
$$978$$ 0 0
$$979$$ −3894.56 −0.127141
$$980$$ 0 0
$$981$$ 99050.7 3.22370
$$982$$ 0 0
$$983$$ −5878.48 −0.190737 −0.0953685 0.995442i $$-0.530403\pi$$
−0.0953685 + 0.995442i $$0.530403\pi$$
$$984$$ 0 0
$$985$$ −65536.1 −2.11995
$$986$$ 0 0
$$987$$ −113921. −3.67392
$$988$$ 0 0
$$989$$ −33906.4 −1.09015
$$990$$ 0 0
$$991$$ −42532.3 −1.36335 −0.681676 0.731654i $$-0.738749\pi$$
−0.681676 + 0.731654i $$0.738749\pi$$
$$992$$ 0 0
$$993$$ −40121.2 −1.28218
$$994$$ 0 0
$$995$$ 1267.67 0.0403898
$$996$$ 0 0
$$997$$ −6320.28 −0.200767 −0.100384 0.994949i $$-0.532007\pi$$
−0.100384 + 0.994949i $$0.532007\pi$$
$$998$$ 0 0
$$999$$ −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 1216.4.a.g.1.1 2
4.3 odd 2 1216.4.a.p.1.2 2
8.3 odd 2 304.4.a.c.1.1 2
8.5 even 2 38.4.a.c.1.2 2
24.5 odd 2 342.4.a.h.1.2 2
40.13 odd 4 950.4.b.i.799.2 4
40.29 even 2 950.4.a.e.1.1 2
40.37 odd 4 950.4.b.i.799.3 4
56.13 odd 2 1862.4.a.e.1.1 2
152.37 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 8.5 even 2
304.4.a.c.1.1 2 8.3 odd 2
342.4.a.h.1.2 2 24.5 odd 2
722.4.a.f.1.1 2 152.37 odd 2
950.4.a.e.1.1 2 40.29 even 2
950.4.b.i.799.2 4 40.13 odd 4
950.4.b.i.799.3 4 40.37 odd 4
1216.4.a.g.1.1 2 1.1 even 1 trivial
1216.4.a.p.1.2 2 4.3 odd 2
1862.4.a.e.1.1 2 56.13 odd 2