# Properties

 Label 304.4.a.c.1.1 Level $304$ Weight $4$ Character 304.1 Self dual yes Analytic conductor $17.937$ 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] = [304,4,Mod(1,304)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(304, 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("304.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$304 = 2^{4} \cdot 19$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 304.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: $$17.9365806417$$ 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 $$\chi$$ $$=$$ 304.1

## $q$-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 $$-0.781946\pi$$
−0.774395 + 0.632702i $$0.781946\pi$$
$$6$$ 0 0
$$7$$ 26.0880 1.40862 0.704310 0.709893i $$-0.251257\pi$$
0.704310 + 0.709893i $$0.251257\pi$$
$$8$$ 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 $$-0.260635\pi$$
0.683092 + 0.730332i $$0.260635\pi$$
$$14$$ 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 $$-0.636986\pi$$
−0.417192 + 0.908818i $$0.636986\pi$$
$$24$$ 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 $$-0.591138\pi$$
−0.282424 + 0.959290i $$0.591138\pi$$
$$30$$ 0 0
$$31$$ 81.9681 0.474900 0.237450 0.971400i $$-0.423688\pi$$
0.237450 + 0.971400i $$0.423688\pi$$
$$32$$ 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 $$-0.516708\pi$$
−0.0524656 + 0.998623i $$0.516708\pi$$
$$38$$ 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 $$-0.219015\pi$$
0.772483 + 0.635036i $$0.219015\pi$$
$$48$$ 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 $$-0.744911\pi$$
−0.695713 + 0.718320i $$0.744911\pi$$
$$54$$ 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 $$-0.269750\pi$$
0.661901 + 0.749592i $$0.269750\pi$$
$$62$$ 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 $$-0.665837\pi$$
−0.497740 + 0.867326i $$0.665837\pi$$
$$72$$ 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 $$-0.598395\pi$$
−0.304218 + 0.952602i $$0.598395\pi$$
$$80$$ 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 $$-0.614140\pi$$
−0.350947 + 0.936395i $$0.614140\pi$$
$$102$$ 0 0
$$103$$ 26.4797 0.0253313 0.0126656 0.999920i $$-0.495968\pi$$
0.0126656 + 0.999920i $$0.495968\pi$$
$$104$$ 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 $$-0.836725\pi$$
−0.871304 + 0.490744i $$0.836725\pi$$
$$110$$ 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 $$-0.865511\pi$$
−0.912063 + 0.410050i $$0.865511\pi$$
$$128$$ 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 $$-0.331064\pi$$
0.506161 + 0.862439i $$0.331064\pi$$
$$150$$ 0 0
$$151$$ −3322.32 −1.79051 −0.895254 0.445557i $$-0.853006\pi$$
−0.895254 + 0.445557i $$0.853006\pi$$
$$152$$ 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 $$-0.480278\pi$$
0.0619194 + 0.998081i $$0.480278\pi$$
$$158$$ 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 $$-0.463707\pi$$
0.113772 + 0.993507i $$0.463707\pi$$
$$168$$ 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 $$-0.382009\pi$$
0.362248 + 0.932082i $$0.382009\pi$$
$$174$$ 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 $$-0.901193\pi$$
−0.952208 + 0.305449i $$0.901193\pi$$
$$182$$ 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 $$-0.973010\pi$$
−0.996407 + 0.0846903i $$0.973010\pi$$
$$192$$ 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 $$-0.260069\pi$$
0.684390 + 0.729116i $$0.260069\pi$$
$$198$$ 0 0
$$199$$ −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
$$202$$ 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 $$-0.655476\pi$$
−0.469250 + 0.883066i $$0.655476\pi$$
$$224$$ 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 $$-0.743645\pi$$
−0.692850 + 0.721082i $$0.743645\pi$$
$$230$$ 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 $$-0.559107\pi$$
−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 $$0.615104\pi$$
$$242$$ 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 $$-0.625952\pi$$
−0.385446 + 0.922730i $$0.625952\pi$$
$$264$$ 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 $$-0.680922\pi$$
−0.538269 + 0.842773i $$0.680922\pi$$
$$270$$ 0 0
$$271$$ −242.661 −0.0543933 −0.0271967 0.999630i $$-0.508658\pi$$
−0.0271967 + 0.999630i $$0.508658\pi$$
$$272$$ 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 $$-0.647879\pi$$
−0.448043 + 0.894012i $$0.647879\pi$$
$$278$$ 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 $$-0.550628\pi$$
−0.158381 + 0.987378i $$0.550628\pi$$
$$294$$ 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 $$-0.584322\pi$$
−0.261819 + 0.965117i $$0.584322\pi$$
$$312$$ 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 $$-0.428521\pi$$
0.222674 + 0.974893i $$0.428521\pi$$
$$318$$ 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 $$-0.353610\pi$$
0.443856 + 0.896098i $$0.353610\pi$$
$$350$$ 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 $$-0.232547\pi$$
0.744796 + 0.667292i $$0.232547\pi$$
$$360$$ 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 $$-0.334297\pi$$
0.497376 + 0.867535i $$0.334297\pi$$
$$368$$ 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 $$-0.350441\pi$$
0.452756 + 0.891634i $$0.350441\pi$$
$$374$$ 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 $$-0.728995\pi$$
−0.658940 + 0.752196i $$0.728995\pi$$
$$384$$ 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 $$-0.671937\pi$$
−0.514270 + 0.857628i $$0.671937\pi$$
$$390$$ 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 $$-0.556387\pi$$
−0.176219 + 0.984351i $$0.556387\pi$$
$$398$$ 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 $$-0.392926\pi$$
0.330074 + 0.943955i $$0.392926\pi$$
$$422$$ 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 $$-0.427776\pi$$
0.224956 + 0.974369i $$0.427776\pi$$
$$432$$ 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 $$-0.571710\pi$$
−0.223381 + 0.974731i $$0.571710\pi$$
$$440$$ 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 $$-0.709867\pi$$
−0.612577 + 0.790411i $$0.709867\pi$$
$$462$$ 0 0
$$463$$ 6399.19 0.642323 0.321162 0.947024i $$-0.395927\pi$$
0.321162 + 0.947024i $$0.395927\pi$$
$$464$$ 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 $$-0.602561\pi$$
−0.316660 + 0.948539i $$0.602561\pi$$
$$480$$ 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 $$-0.327460\pi$$
0.515894 + 0.856652i $$0.327460\pi$$
$$488$$ 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 $$-0.363090\pi$$
0.416975 + 0.908918i $$0.363090\pi$$
$$504$$ 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 $$-0.349138\pi$$
0.456402 + 0.889774i $$0.349138\pi$$
$$510$$ 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 $$-0.890087\pi$$
−0.940973 + 0.338481i $$0.890087\pi$$
$$542$$ 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 $$-0.441902\pi$$
0.181510 + 0.983389i $$0.441902\pi$$
$$558$$ 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 $$-0.617548\pi$$
−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 $$0.599387\pi$$
$$602$$ 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 $$-0.587553\pi$$
−0.271599 + 0.962410i $$0.587553\pi$$
$$608$$ 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 $$-0.236031\pi$$
0.737448 + 0.675404i $$0.236031\pi$$
$$614$$ 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 $$-0.159910\pi$$
0.876444 + 0.481505i $$0.159910\pi$$
$$632$$ 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 $$-0.388924\pi$$
0.341916 + 0.939731i $$0.388924\pi$$
$$648$$ 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 $$-0.653908\pi$$
−0.464896 + 0.885365i $$0.653908\pi$$
$$654$$ 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 $$-0.478390\pi$$
0.0678361 + 0.997696i $$0.478390\pi$$
$$662$$ 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 $$-0.624981\pi$$
−0.382628 + 0.923902i $$0.624981\pi$$
$$678$$ 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 $$-0.456393\pi$$
0.136568 + 0.990631i $$0.456393\pi$$
$$702$$ 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 $$-0.646391\pi$$
−0.443860 + 0.896096i $$0.646391\pi$$
$$710$$ 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 $$-0.467867\pi$$
0.100777 + 0.994909i $$0.467867\pi$$
$$720$$ 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 $$-0.552760\pi$$
−0.164992 + 0.986295i $$0.552760\pi$$
$$728$$ 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 $$-0.280563\pi$$
0.636059 + 0.771640i $$0.280563\pi$$
$$734$$ 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 $$-0.654311\pi$$
−0.466016 + 0.884776i $$0.654311\pi$$
$$744$$ 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 $$-0.706760\pi$$
−0.604833 + 0.796352i $$0.706760\pi$$
$$752$$ 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 $$-0.835306\pi$$
−0.869107 + 0.494624i $$0.835306\pi$$
$$758$$ 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 $$-0.607708\pi$$
−0.331955 + 0.943295i $$0.607708\pi$$
$$774$$ 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 $$-0.554909\pi$$
−0.171648 + 0.985158i $$0.554909\pi$$
$$798$$ 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 $$-0.722511\pi$$
−0.643482 + 0.765461i $$0.722511\pi$$
$$822$$ 0 0
$$823$$ −17296.1 −0.732568 −0.366284 0.930503i $$-0.619370\pi$$
−0.366284 + 0.930503i $$0.619370\pi$$
$$824$$ 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 $$-0.860701\pi$$
−0.905762 + 0.423786i $$0.860701\pi$$
$$830$$ 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 $$-0.690922\pi$$
−0.564477 + 0.825449i $$0.690922\pi$$
$$840$$ 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 $$-0.638333\pi$$
−0.421035 + 0.907044i $$0.638333\pi$$
$$854$$ 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 $$-0.342115\pi$$
0.475921 + 0.879488i $$0.342115\pi$$
$$864$$ 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 $$-0.226105\pi$$
0.758147 + 0.652084i $$0.226105\pi$$
$$878$$ 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 $$-0.516639\pi$$
−0.0522498 + 0.998634i $$0.516639\pi$$
$$888$$ 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 $$-0.153360\pi$$
0.886164 + 0.463371i $$0.153360\pi$$
$$912$$ 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 $$-0.277517\pi$$
0.643414 + 0.765518i $$0.277517\pi$$
$$920$$ 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 $$-0.647730\pi$$
−0.447626 + 0.894221i $$0.647730\pi$$
$$942$$ 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 $$-0.668167\pi$$
−0.504075 + 0.863660i $$0.668167\pi$$
$$968$$ 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 $$-0.469597\pi$$
0.0953685 + 0.995442i $$0.469597\pi$$
$$984$$ 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 $$-0.261251\pi$$
0.681676 + 0.731654i $$0.261251\pi$$
$$992$$ 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 $$-0.467993\pi$$
0.100384 + 0.994949i $$0.467993\pi$$
$$998$$ 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 304.4.a.c.1.1 2
4.3 odd 2 38.4.a.c.1.2 2
8.3 odd 2 1216.4.a.g.1.1 2
8.5 even 2 1216.4.a.p.1.2 2
12.11 even 2 342.4.a.h.1.2 2
20.3 even 4 950.4.b.i.799.2 4
20.7 even 4 950.4.b.i.799.3 4
20.19 odd 2 950.4.a.e.1.1 2
28.27 even 2 1862.4.a.e.1.1 2
76.75 even 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 4.3 odd 2
304.4.a.c.1.1 2 1.1 even 1 trivial
342.4.a.h.1.2 2 12.11 even 2
722.4.a.f.1.1 2 76.75 even 2
950.4.a.e.1.1 2 20.19 odd 2
950.4.b.i.799.2 4 20.3 even 4
950.4.b.i.799.3 4 20.7 even 4
1216.4.a.g.1.1 2 8.3 odd 2
1216.4.a.p.1.2 2 8.5 even 2
1862.4.a.e.1.1 2 28.27 even 2