# Properties

 Label 608.4.a.j.1.2 Level $608$ Weight $4$ Character 608.1 Self dual yes Analytic conductor $35.873$ Analytic rank $1$ Dimension $7$ CM no Inner twists $1$

# Learn more

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [608,4,Mod(1,608)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(608, 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("608.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$608 = 2^{5} \cdot 19$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 608.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: $$35.8731612835$$ Analytic rank: $$1$$ Dimension: $$7$$ Coefficient field: $$\mathbb{Q}[x]/(x^{7} - \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{7} - 3x^{6} - 121x^{5} + 402x^{4} + 4234x^{3} - 14542x^{2} - 40996x + 141664$$ x^7 - 3*x^6 - 121*x^5 + 402*x^4 + 4234*x^3 - 14542*x^2 - 40996*x + 141664 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{6}$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$7.37394$$ of defining polynomial Character $$\chi$$ $$=$$ 608.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-6.88542 q^{3} +4.69330 q^{5} -3.62949 q^{7} +20.4091 q^{9} +O(q^{10})$$ $$q-6.88542 q^{3} +4.69330 q^{5} -3.62949 q^{7} +20.4091 q^{9} -43.4975 q^{11} +75.0368 q^{13} -32.3154 q^{15} -13.1508 q^{17} +19.0000 q^{19} +24.9906 q^{21} +96.5591 q^{23} -102.973 q^{25} +45.3813 q^{27} +40.5580 q^{29} -138.828 q^{31} +299.499 q^{33} -17.0343 q^{35} +348.686 q^{37} -516.660 q^{39} -44.3243 q^{41} -214.385 q^{43} +95.7860 q^{45} +334.307 q^{47} -329.827 q^{49} +90.5487 q^{51} +311.122 q^{53} -204.147 q^{55} -130.823 q^{57} -502.875 q^{59} -54.7193 q^{61} -74.0746 q^{63} +352.170 q^{65} -312.959 q^{67} -664.850 q^{69} -642.657 q^{71} -837.348 q^{73} +709.012 q^{75} +157.874 q^{77} -349.595 q^{79} -863.515 q^{81} -521.399 q^{83} -61.7206 q^{85} -279.259 q^{87} +63.4094 q^{89} -272.345 q^{91} +955.887 q^{93} +89.1728 q^{95} -1229.92 q^{97} -887.745 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$7 q - 3 q^{3} - 17 q^{5} - 42 q^{7} + 86 q^{9}+O(q^{10})$$ 7 * q - 3 * q^3 - 17 * q^5 - 42 * q^7 + 86 * q^9 $$7 q - 3 q^{3} - 17 q^{5} - 42 q^{7} + 86 q^{9} + 33 q^{11} - 35 q^{13} - 120 q^{15} + 66 q^{17} + 133 q^{19} + 33 q^{21} - 389 q^{23} + 44 q^{25} + 39 q^{27} + 233 q^{29} - 158 q^{31} - 206 q^{33} - 123 q^{35} - 436 q^{37} - 807 q^{39} - 94 q^{41} - 645 q^{43} + 103 q^{45} - 1451 q^{47} + 93 q^{49} - 1741 q^{51} + 3 q^{53} - 1971 q^{55} - 57 q^{57} - 297 q^{59} + 93 q^{61} - 2999 q^{63} - 788 q^{65} - 1641 q^{67} + 945 q^{69} - 2392 q^{71} + 324 q^{73} - 1909 q^{75} - 711 q^{77} - 2492 q^{79} + 143 q^{81} - 310 q^{83} + 2353 q^{85} - 4795 q^{87} - 440 q^{89} + 107 q^{91} + 900 q^{93} - 323 q^{95} - 532 q^{97} + 1591 q^{99}+O(q^{100})$$ 7 * q - 3 * q^3 - 17 * q^5 - 42 * q^7 + 86 * q^9 + 33 * q^11 - 35 * q^13 - 120 * q^15 + 66 * q^17 + 133 * q^19 + 33 * q^21 - 389 * q^23 + 44 * q^25 + 39 * q^27 + 233 * q^29 - 158 * q^31 - 206 * q^33 - 123 * q^35 - 436 * q^37 - 807 * q^39 - 94 * q^41 - 645 * q^43 + 103 * q^45 - 1451 * q^47 + 93 * q^49 - 1741 * q^51 + 3 * q^53 - 1971 * q^55 - 57 * q^57 - 297 * q^59 + 93 * q^61 - 2999 * q^63 - 788 * q^65 - 1641 * q^67 + 945 * q^69 - 2392 * q^71 + 324 * q^73 - 1909 * q^75 - 711 * q^77 - 2492 * q^79 + 143 * q^81 - 310 * q^83 + 2353 * q^85 - 4795 * q^87 - 440 * q^89 + 107 * q^91 + 900 * q^93 - 323 * q^95 - 532 * q^97 + 1591 * 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
$$3$$ −6.88542 −1.32510 −0.662550 0.749017i $$-0.730526\pi$$
−0.662550 + 0.749017i $$0.730526\pi$$
$$4$$ 0 0
$$5$$ 4.69330 0.419782 0.209891 0.977725i $$-0.432689\pi$$
0.209891 + 0.977725i $$0.432689\pi$$
$$6$$ 0 0
$$7$$ −3.62949 −0.195974 −0.0979871 0.995188i $$-0.531240\pi$$
−0.0979871 + 0.995188i $$0.531240\pi$$
$$8$$ 0 0
$$9$$ 20.4091 0.755892
$$10$$ 0 0
$$11$$ −43.4975 −1.19227 −0.596137 0.802883i $$-0.703298\pi$$
−0.596137 + 0.802883i $$0.703298\pi$$
$$12$$ 0 0
$$13$$ 75.0368 1.60088 0.800441 0.599412i $$-0.204599\pi$$
0.800441 + 0.599412i $$0.204599\pi$$
$$14$$ 0 0
$$15$$ −32.3154 −0.556253
$$16$$ 0 0
$$17$$ −13.1508 −0.187620 −0.0938098 0.995590i $$-0.529905\pi$$
−0.0938098 + 0.995590i $$0.529905\pi$$
$$18$$ 0 0
$$19$$ 19.0000 0.229416
$$20$$ 0 0
$$21$$ 24.9906 0.259686
$$22$$ 0 0
$$23$$ 96.5591 0.875390 0.437695 0.899124i $$-0.355795\pi$$
0.437695 + 0.899124i $$0.355795\pi$$
$$24$$ 0 0
$$25$$ −102.973 −0.823783
$$26$$ 0 0
$$27$$ 45.3813 0.323468
$$28$$ 0 0
$$29$$ 40.5580 0.259704 0.129852 0.991533i $$-0.458550\pi$$
0.129852 + 0.991533i $$0.458550\pi$$
$$30$$ 0 0
$$31$$ −138.828 −0.804328 −0.402164 0.915568i $$-0.631742\pi$$
−0.402164 + 0.915568i $$0.631742\pi$$
$$32$$ 0 0
$$33$$ 299.499 1.57988
$$34$$ 0 0
$$35$$ −17.0343 −0.0822664
$$36$$ 0 0
$$37$$ 348.686 1.54929 0.774644 0.632398i $$-0.217929\pi$$
0.774644 + 0.632398i $$0.217929\pi$$
$$38$$ 0 0
$$39$$ −516.660 −2.12133
$$40$$ 0 0
$$41$$ −44.3243 −0.168836 −0.0844182 0.996430i $$-0.526903\pi$$
−0.0844182 + 0.996430i $$0.526903\pi$$
$$42$$ 0 0
$$43$$ −214.385 −0.760313 −0.380157 0.924922i $$-0.624130\pi$$
−0.380157 + 0.924922i $$0.624130\pi$$
$$44$$ 0 0
$$45$$ 95.7860 0.317310
$$46$$ 0 0
$$47$$ 334.307 1.03752 0.518762 0.854919i $$-0.326393\pi$$
0.518762 + 0.854919i $$0.326393\pi$$
$$48$$ 0 0
$$49$$ −329.827 −0.961594
$$50$$ 0 0
$$51$$ 90.5487 0.248615
$$52$$ 0 0
$$53$$ 311.122 0.806338 0.403169 0.915126i $$-0.367909\pi$$
0.403169 + 0.915126i $$0.367909\pi$$
$$54$$ 0 0
$$55$$ −204.147 −0.500495
$$56$$ 0 0
$$57$$ −130.823 −0.303999
$$58$$ 0 0
$$59$$ −502.875 −1.10964 −0.554820 0.831971i $$-0.687213\pi$$
−0.554820 + 0.831971i $$0.687213\pi$$
$$60$$ 0 0
$$61$$ −54.7193 −0.114854 −0.0574270 0.998350i $$-0.518290\pi$$
−0.0574270 + 0.998350i $$0.518290\pi$$
$$62$$ 0 0
$$63$$ −74.0746 −0.148135
$$64$$ 0 0
$$65$$ 352.170 0.672021
$$66$$ 0 0
$$67$$ −312.959 −0.570658 −0.285329 0.958430i $$-0.592103\pi$$
−0.285329 + 0.958430i $$0.592103\pi$$
$$68$$ 0 0
$$69$$ −664.850 −1.15998
$$70$$ 0 0
$$71$$ −642.657 −1.07422 −0.537108 0.843513i $$-0.680483\pi$$
−0.537108 + 0.843513i $$0.680483\pi$$
$$72$$ 0 0
$$73$$ −837.348 −1.34252 −0.671261 0.741221i $$-0.734247\pi$$
−0.671261 + 0.741221i $$0.734247\pi$$
$$74$$ 0 0
$$75$$ 709.012 1.09160
$$76$$ 0 0
$$77$$ 157.874 0.233655
$$78$$ 0 0
$$79$$ −349.595 −0.497880 −0.248940 0.968519i $$-0.580082\pi$$
−0.248940 + 0.968519i $$0.580082\pi$$
$$80$$ 0 0
$$81$$ −863.515 −1.18452
$$82$$ 0 0
$$83$$ −521.399 −0.689530 −0.344765 0.938689i $$-0.612041\pi$$
−0.344765 + 0.938689i $$0.612041\pi$$
$$84$$ 0 0
$$85$$ −61.7206 −0.0787593
$$86$$ 0 0
$$87$$ −279.259 −0.344135
$$88$$ 0 0
$$89$$ 63.4094 0.0755211 0.0377606 0.999287i $$-0.487978\pi$$
0.0377606 + 0.999287i $$0.487978\pi$$
$$90$$ 0 0
$$91$$ −272.345 −0.313731
$$92$$ 0 0
$$93$$ 955.887 1.06582
$$94$$ 0 0
$$95$$ 89.1728 0.0963046
$$96$$ 0 0
$$97$$ −1229.92 −1.28742 −0.643711 0.765269i $$-0.722606\pi$$
−0.643711 + 0.765269i $$0.722606\pi$$
$$98$$ 0 0
$$99$$ −887.745 −0.901229
$$100$$ 0 0
$$101$$ −1077.89 −1.06192 −0.530962 0.847396i $$-0.678169\pi$$
−0.530962 + 0.847396i $$0.678169\pi$$
$$102$$ 0 0
$$103$$ −1896.77 −1.81451 −0.907255 0.420580i $$-0.861826\pi$$
−0.907255 + 0.420580i $$0.861826\pi$$
$$104$$ 0 0
$$105$$ 117.288 0.109011
$$106$$ 0 0
$$107$$ 480.564 0.434186 0.217093 0.976151i $$-0.430343\pi$$
0.217093 + 0.976151i $$0.430343\pi$$
$$108$$ 0 0
$$109$$ 124.669 0.109552 0.0547758 0.998499i $$-0.482556\pi$$
0.0547758 + 0.998499i $$0.482556\pi$$
$$110$$ 0 0
$$111$$ −2400.85 −2.05296
$$112$$ 0 0
$$113$$ −1778.75 −1.48080 −0.740400 0.672166i $$-0.765364\pi$$
−0.740400 + 0.672166i $$0.765364\pi$$
$$114$$ 0 0
$$115$$ 453.181 0.367473
$$116$$ 0 0
$$117$$ 1531.43 1.21009
$$118$$ 0 0
$$119$$ 47.7307 0.0367686
$$120$$ 0 0
$$121$$ 561.036 0.421515
$$122$$ 0 0
$$123$$ 305.191 0.223725
$$124$$ 0 0
$$125$$ −1069.95 −0.765591
$$126$$ 0 0
$$127$$ −1077.80 −0.753068 −0.376534 0.926403i $$-0.622884\pi$$
−0.376534 + 0.926403i $$0.622884\pi$$
$$128$$ 0 0
$$129$$ 1476.13 1.00749
$$130$$ 0 0
$$131$$ 1092.79 0.728835 0.364417 0.931236i $$-0.381268\pi$$
0.364417 + 0.931236i $$0.381268\pi$$
$$132$$ 0 0
$$133$$ −68.9604 −0.0449596
$$134$$ 0 0
$$135$$ 212.988 0.135786
$$136$$ 0 0
$$137$$ 3052.21 1.90341 0.951706 0.307010i $$-0.0993286\pi$$
0.951706 + 0.307010i $$0.0993286\pi$$
$$138$$ 0 0
$$139$$ 1428.73 0.871821 0.435911 0.899990i $$-0.356426\pi$$
0.435911 + 0.899990i $$0.356426\pi$$
$$140$$ 0 0
$$141$$ −2301.84 −1.37482
$$142$$ 0 0
$$143$$ −3263.92 −1.90869
$$144$$ 0 0
$$145$$ 190.351 0.109019
$$146$$ 0 0
$$147$$ 2271.00 1.27421
$$148$$ 0 0
$$149$$ 571.022 0.313959 0.156980 0.987602i $$-0.449824\pi$$
0.156980 + 0.987602i $$0.449824\pi$$
$$150$$ 0 0
$$151$$ 997.782 0.537737 0.268869 0.963177i $$-0.413350\pi$$
0.268869 + 0.963177i $$0.413350\pi$$
$$152$$ 0 0
$$153$$ −268.395 −0.141820
$$154$$ 0 0
$$155$$ −651.560 −0.337642
$$156$$ 0 0
$$157$$ −2171.23 −1.10371 −0.551857 0.833939i $$-0.686080\pi$$
−0.551857 + 0.833939i $$0.686080\pi$$
$$158$$ 0 0
$$159$$ −2142.21 −1.06848
$$160$$ 0 0
$$161$$ −350.461 −0.171554
$$162$$ 0 0
$$163$$ −305.488 −0.146795 −0.0733977 0.997303i $$-0.523384\pi$$
−0.0733977 + 0.997303i $$0.523384\pi$$
$$164$$ 0 0
$$165$$ 1405.64 0.663206
$$166$$ 0 0
$$167$$ 136.673 0.0633300 0.0316650 0.999499i $$-0.489919\pi$$
0.0316650 + 0.999499i $$0.489919\pi$$
$$168$$ 0 0
$$169$$ 3433.52 1.56282
$$170$$ 0 0
$$171$$ 387.772 0.173413
$$172$$ 0 0
$$173$$ −1508.96 −0.663144 −0.331572 0.943430i $$-0.607579\pi$$
−0.331572 + 0.943430i $$0.607579\pi$$
$$174$$ 0 0
$$175$$ 373.739 0.161440
$$176$$ 0 0
$$177$$ 3462.51 1.47038
$$178$$ 0 0
$$179$$ −770.548 −0.321751 −0.160875 0.986975i $$-0.551432\pi$$
−0.160875 + 0.986975i $$0.551432\pi$$
$$180$$ 0 0
$$181$$ 2292.62 0.941486 0.470743 0.882270i $$-0.343986\pi$$
0.470743 + 0.882270i $$0.343986\pi$$
$$182$$ 0 0
$$183$$ 376.766 0.152193
$$184$$ 0 0
$$185$$ 1636.49 0.650363
$$186$$ 0 0
$$187$$ 572.027 0.223694
$$188$$ 0 0
$$189$$ −164.711 −0.0633914
$$190$$ 0 0
$$191$$ 1670.39 0.632803 0.316402 0.948625i $$-0.397525\pi$$
0.316402 + 0.948625i $$0.397525\pi$$
$$192$$ 0 0
$$193$$ −663.849 −0.247590 −0.123795 0.992308i $$-0.539507\pi$$
−0.123795 + 0.992308i $$0.539507\pi$$
$$194$$ 0 0
$$195$$ −2424.84 −0.890495
$$196$$ 0 0
$$197$$ −2470.86 −0.893611 −0.446806 0.894631i $$-0.647438\pi$$
−0.446806 + 0.894631i $$0.647438\pi$$
$$198$$ 0 0
$$199$$ 4348.26 1.54894 0.774472 0.632608i $$-0.218016\pi$$
0.774472 + 0.632608i $$0.218016\pi$$
$$200$$ 0 0
$$201$$ 2154.86 0.756179
$$202$$ 0 0
$$203$$ −147.205 −0.0508954
$$204$$ 0 0
$$205$$ −208.027 −0.0708744
$$206$$ 0 0
$$207$$ 1970.68 0.661700
$$208$$ 0 0
$$209$$ −826.453 −0.273526
$$210$$ 0 0
$$211$$ 3910.79 1.27597 0.637986 0.770048i $$-0.279768\pi$$
0.637986 + 0.770048i $$0.279768\pi$$
$$212$$ 0 0
$$213$$ 4424.97 1.42344
$$214$$ 0 0
$$215$$ −1006.18 −0.319166
$$216$$ 0 0
$$217$$ 503.874 0.157628
$$218$$ 0 0
$$219$$ 5765.50 1.77898
$$220$$ 0 0
$$221$$ −986.792 −0.300357
$$222$$ 0 0
$$223$$ 70.6739 0.0212228 0.0106114 0.999944i $$-0.496622\pi$$
0.0106114 + 0.999944i $$0.496622\pi$$
$$224$$ 0 0
$$225$$ −2101.58 −0.622691
$$226$$ 0 0
$$227$$ 1132.37 0.331091 0.165546 0.986202i $$-0.447061\pi$$
0.165546 + 0.986202i $$0.447061\pi$$
$$228$$ 0 0
$$229$$ 2922.00 0.843194 0.421597 0.906783i $$-0.361470\pi$$
0.421597 + 0.906783i $$0.361470\pi$$
$$230$$ 0 0
$$231$$ −1087.03 −0.309616
$$232$$ 0 0
$$233$$ −926.052 −0.260376 −0.130188 0.991489i $$-0.541558\pi$$
−0.130188 + 0.991489i $$0.541558\pi$$
$$234$$ 0 0
$$235$$ 1569.00 0.435534
$$236$$ 0 0
$$237$$ 2407.11 0.659740
$$238$$ 0 0
$$239$$ −4376.11 −1.18438 −0.592190 0.805798i $$-0.701737\pi$$
−0.592190 + 0.805798i $$0.701737\pi$$
$$240$$ 0 0
$$241$$ −3964.69 −1.05970 −0.529850 0.848091i $$-0.677752\pi$$
−0.529850 + 0.848091i $$0.677752\pi$$
$$242$$ 0 0
$$243$$ 4720.37 1.24614
$$244$$ 0 0
$$245$$ −1547.98 −0.403660
$$246$$ 0 0
$$247$$ 1425.70 0.367267
$$248$$ 0 0
$$249$$ 3590.05 0.913697
$$250$$ 0 0
$$251$$ 219.854 0.0552870 0.0276435 0.999618i $$-0.491200\pi$$
0.0276435 + 0.999618i $$0.491200\pi$$
$$252$$ 0 0
$$253$$ −4200.08 −1.04370
$$254$$ 0 0
$$255$$ 424.973 0.104364
$$256$$ 0 0
$$257$$ −5582.44 −1.35495 −0.677477 0.735544i $$-0.736926\pi$$
−0.677477 + 0.735544i $$0.736926\pi$$
$$258$$ 0 0
$$259$$ −1265.55 −0.303620
$$260$$ 0 0
$$261$$ 827.751 0.196308
$$262$$ 0 0
$$263$$ 2971.76 0.696755 0.348378 0.937354i $$-0.386733\pi$$
0.348378 + 0.937354i $$0.386733\pi$$
$$264$$ 0 0
$$265$$ 1460.19 0.338486
$$266$$ 0 0
$$267$$ −436.601 −0.100073
$$268$$ 0 0
$$269$$ −5969.51 −1.35304 −0.676519 0.736425i $$-0.736513\pi$$
−0.676519 + 0.736425i $$0.736513\pi$$
$$270$$ 0 0
$$271$$ −6355.51 −1.42461 −0.712306 0.701869i $$-0.752349\pi$$
−0.712306 + 0.701869i $$0.752349\pi$$
$$272$$ 0 0
$$273$$ 1875.21 0.415726
$$274$$ 0 0
$$275$$ 4479.07 0.982174
$$276$$ 0 0
$$277$$ −8038.80 −1.74370 −0.871849 0.489774i $$-0.837079\pi$$
−0.871849 + 0.489774i $$0.837079\pi$$
$$278$$ 0 0
$$279$$ −2833.34 −0.607985
$$280$$ 0 0
$$281$$ −4478.81 −0.950831 −0.475416 0.879761i $$-0.657702\pi$$
−0.475416 + 0.879761i $$0.657702\pi$$
$$282$$ 0 0
$$283$$ −8812.45 −1.85105 −0.925523 0.378691i $$-0.876374\pi$$
−0.925523 + 0.378691i $$0.876374\pi$$
$$284$$ 0 0
$$285$$ −613.992 −0.127613
$$286$$ 0 0
$$287$$ 160.875 0.0330876
$$288$$ 0 0
$$289$$ −4740.06 −0.964799
$$290$$ 0 0
$$291$$ 8468.55 1.70596
$$292$$ 0 0
$$293$$ 942.016 0.187826 0.0939132 0.995580i $$-0.470062\pi$$
0.0939132 + 0.995580i $$0.470062\pi$$
$$294$$ 0 0
$$295$$ −2360.14 −0.465807
$$296$$ 0 0
$$297$$ −1973.98 −0.385662
$$298$$ 0 0
$$299$$ 7245.48 1.40140
$$300$$ 0 0
$$301$$ 778.110 0.149002
$$302$$ 0 0
$$303$$ 7421.74 1.40715
$$304$$ 0 0
$$305$$ −256.815 −0.0482136
$$306$$ 0 0
$$307$$ 5371.60 0.998610 0.499305 0.866426i $$-0.333589\pi$$
0.499305 + 0.866426i $$0.333589\pi$$
$$308$$ 0 0
$$309$$ 13060.1 2.40441
$$310$$ 0 0
$$311$$ 8180.21 1.49150 0.745751 0.666225i $$-0.232091\pi$$
0.745751 + 0.666225i $$0.232091\pi$$
$$312$$ 0 0
$$313$$ −621.565 −0.112246 −0.0561229 0.998424i $$-0.517874\pi$$
−0.0561229 + 0.998424i $$0.517874\pi$$
$$314$$ 0 0
$$315$$ −347.655 −0.0621845
$$316$$ 0 0
$$317$$ −2641.47 −0.468012 −0.234006 0.972235i $$-0.575184\pi$$
−0.234006 + 0.972235i $$0.575184\pi$$
$$318$$ 0 0
$$319$$ −1764.17 −0.309639
$$320$$ 0 0
$$321$$ −3308.89 −0.575340
$$322$$ 0 0
$$323$$ −249.865 −0.0430429
$$324$$ 0 0
$$325$$ −7726.76 −1.31878
$$326$$ 0 0
$$327$$ −858.398 −0.145167
$$328$$ 0 0
$$329$$ −1213.36 −0.203328
$$330$$ 0 0
$$331$$ −7450.56 −1.23722 −0.618610 0.785698i $$-0.712304\pi$$
−0.618610 + 0.785698i $$0.712304\pi$$
$$332$$ 0 0
$$333$$ 7116.36 1.17109
$$334$$ 0 0
$$335$$ −1468.81 −0.239552
$$336$$ 0 0
$$337$$ 121.084 0.0195722 0.00978612 0.999952i $$-0.496885\pi$$
0.00978612 + 0.999952i $$0.496885\pi$$
$$338$$ 0 0
$$339$$ 12247.4 1.96221
$$340$$ 0 0
$$341$$ 6038.66 0.958978
$$342$$ 0 0
$$343$$ 2442.02 0.384422
$$344$$ 0 0
$$345$$ −3120.35 −0.486938
$$346$$ 0 0
$$347$$ 9376.43 1.45059 0.725293 0.688441i $$-0.241704\pi$$
0.725293 + 0.688441i $$0.241704\pi$$
$$348$$ 0 0
$$349$$ −4225.67 −0.648124 −0.324062 0.946036i $$-0.605049\pi$$
−0.324062 + 0.946036i $$0.605049\pi$$
$$350$$ 0 0
$$351$$ 3405.27 0.517834
$$352$$ 0 0
$$353$$ 10833.3 1.63343 0.816714 0.577043i $$-0.195794\pi$$
0.816714 + 0.577043i $$0.195794\pi$$
$$354$$ 0 0
$$355$$ −3016.19 −0.450937
$$356$$ 0 0
$$357$$ −328.646 −0.0487221
$$358$$ 0 0
$$359$$ 1002.24 0.147343 0.0736715 0.997283i $$-0.476528\pi$$
0.0736715 + 0.997283i $$0.476528\pi$$
$$360$$ 0 0
$$361$$ 361.000 0.0526316
$$362$$ 0 0
$$363$$ −3862.97 −0.558550
$$364$$ 0 0
$$365$$ −3929.93 −0.563567
$$366$$ 0 0
$$367$$ 4868.42 0.692451 0.346225 0.938151i $$-0.387463\pi$$
0.346225 + 0.938151i $$0.387463\pi$$
$$368$$ 0 0
$$369$$ −904.617 −0.127622
$$370$$ 0 0
$$371$$ −1129.22 −0.158021
$$372$$ 0 0
$$373$$ 10466.0 1.45284 0.726421 0.687250i $$-0.241182\pi$$
0.726421 + 0.687250i $$0.241182\pi$$
$$374$$ 0 0
$$375$$ 7367.03 1.01449
$$376$$ 0 0
$$377$$ 3043.34 0.415756
$$378$$ 0 0
$$379$$ −11355.1 −1.53898 −0.769490 0.638659i $$-0.779490\pi$$
−0.769490 + 0.638659i $$0.779490\pi$$
$$380$$ 0 0
$$381$$ 7421.14 0.997891
$$382$$ 0 0
$$383$$ 8723.90 1.16389 0.581946 0.813228i $$-0.302292\pi$$
0.581946 + 0.813228i $$0.302292\pi$$
$$384$$ 0 0
$$385$$ 740.951 0.0980840
$$386$$ 0 0
$$387$$ −4375.41 −0.574714
$$388$$ 0 0
$$389$$ 2155.33 0.280924 0.140462 0.990086i $$-0.455141\pi$$
0.140462 + 0.990086i $$0.455141\pi$$
$$390$$ 0 0
$$391$$ −1269.83 −0.164240
$$392$$ 0 0
$$393$$ −7524.31 −0.965780
$$394$$ 0 0
$$395$$ −1640.75 −0.209001
$$396$$ 0 0
$$397$$ −4708.96 −0.595305 −0.297652 0.954674i $$-0.596204\pi$$
−0.297652 + 0.954674i $$0.596204\pi$$
$$398$$ 0 0
$$399$$ 474.821 0.0595759
$$400$$ 0 0
$$401$$ −1051.63 −0.130962 −0.0654812 0.997854i $$-0.520858\pi$$
−0.0654812 + 0.997854i $$0.520858\pi$$
$$402$$ 0 0
$$403$$ −10417.2 −1.28763
$$404$$ 0 0
$$405$$ −4052.74 −0.497240
$$406$$ 0 0
$$407$$ −15167.0 −1.84717
$$408$$ 0 0
$$409$$ 3346.05 0.404527 0.202263 0.979331i $$-0.435170\pi$$
0.202263 + 0.979331i $$0.435170\pi$$
$$410$$ 0 0
$$411$$ −21015.7 −2.52221
$$412$$ 0 0
$$413$$ 1825.18 0.217461
$$414$$ 0 0
$$415$$ −2447.08 −0.289452
$$416$$ 0 0
$$417$$ −9837.40 −1.15525
$$418$$ 0 0
$$419$$ 15743.9 1.83565 0.917827 0.396982i $$-0.129942\pi$$
0.917827 + 0.396982i $$0.129942\pi$$
$$420$$ 0 0
$$421$$ 3755.30 0.434732 0.217366 0.976090i $$-0.430254\pi$$
0.217366 + 0.976090i $$0.430254\pi$$
$$422$$ 0 0
$$423$$ 6822.89 0.784255
$$424$$ 0 0
$$425$$ 1354.17 0.154558
$$426$$ 0 0
$$427$$ 198.603 0.0225084
$$428$$ 0 0
$$429$$ 22473.4 2.52920
$$430$$ 0 0
$$431$$ −9670.09 −1.08072 −0.540362 0.841433i $$-0.681713\pi$$
−0.540362 + 0.841433i $$0.681713\pi$$
$$432$$ 0 0
$$433$$ −10656.7 −1.18274 −0.591370 0.806401i $$-0.701413\pi$$
−0.591370 + 0.806401i $$0.701413\pi$$
$$434$$ 0 0
$$435$$ −1310.65 −0.144461
$$436$$ 0 0
$$437$$ 1834.62 0.200828
$$438$$ 0 0
$$439$$ −5113.47 −0.555929 −0.277964 0.960591i $$-0.589660\pi$$
−0.277964 + 0.960591i $$0.589660\pi$$
$$440$$ 0 0
$$441$$ −6731.46 −0.726861
$$442$$ 0 0
$$443$$ −2552.54 −0.273758 −0.136879 0.990588i $$-0.543707\pi$$
−0.136879 + 0.990588i $$0.543707\pi$$
$$444$$ 0 0
$$445$$ 297.600 0.0317024
$$446$$ 0 0
$$447$$ −3931.73 −0.416028
$$448$$ 0 0
$$449$$ 4504.62 0.473466 0.236733 0.971575i $$-0.423923\pi$$
0.236733 + 0.971575i $$0.423923\pi$$
$$450$$ 0 0
$$451$$ 1928.00 0.201299
$$452$$ 0 0
$$453$$ −6870.15 −0.712556
$$454$$ 0 0
$$455$$ −1278.20 −0.131699
$$456$$ 0 0
$$457$$ 9278.63 0.949751 0.474875 0.880053i $$-0.342493\pi$$
0.474875 + 0.880053i $$0.342493\pi$$
$$458$$ 0 0
$$459$$ −596.800 −0.0606890
$$460$$ 0 0
$$461$$ 3861.47 0.390123 0.195062 0.980791i $$-0.437509\pi$$
0.195062 + 0.980791i $$0.437509\pi$$
$$462$$ 0 0
$$463$$ 154.016 0.0154595 0.00772975 0.999970i $$-0.497540\pi$$
0.00772975 + 0.999970i $$0.497540\pi$$
$$464$$ 0 0
$$465$$ 4486.27 0.447410
$$466$$ 0 0
$$467$$ −16010.0 −1.58641 −0.793205 0.608954i $$-0.791589\pi$$
−0.793205 + 0.608954i $$0.791589\pi$$
$$468$$ 0 0
$$469$$ 1135.88 0.111834
$$470$$ 0 0
$$471$$ 14949.8 1.46253
$$472$$ 0 0
$$473$$ 9325.24 0.906501
$$474$$ 0 0
$$475$$ −1956.49 −0.188989
$$476$$ 0 0
$$477$$ 6349.71 0.609504
$$478$$ 0 0
$$479$$ −10467.1 −0.998441 −0.499220 0.866475i $$-0.666380\pi$$
−0.499220 + 0.866475i $$0.666380\pi$$
$$480$$ 0 0
$$481$$ 26164.3 2.48023
$$482$$ 0 0
$$483$$ 2413.07 0.227326
$$484$$ 0 0
$$485$$ −5772.41 −0.540436
$$486$$ 0 0
$$487$$ −5868.52 −0.546054 −0.273027 0.962006i $$-0.588025\pi$$
−0.273027 + 0.962006i $$0.588025\pi$$
$$488$$ 0 0
$$489$$ 2103.41 0.194519
$$490$$ 0 0
$$491$$ 20259.1 1.86208 0.931041 0.364915i $$-0.118902\pi$$
0.931041 + 0.364915i $$0.118902\pi$$
$$492$$ 0 0
$$493$$ −533.369 −0.0487257
$$494$$ 0 0
$$495$$ −4166.46 −0.378320
$$496$$ 0 0
$$497$$ 2332.52 0.210519
$$498$$ 0 0
$$499$$ −6557.19 −0.588256 −0.294128 0.955766i $$-0.595029\pi$$
−0.294128 + 0.955766i $$0.595029\pi$$
$$500$$ 0 0
$$501$$ −941.054 −0.0839186
$$502$$ 0 0
$$503$$ 20542.5 1.82097 0.910484 0.413545i $$-0.135709\pi$$
0.910484 + 0.413545i $$0.135709\pi$$
$$504$$ 0 0
$$505$$ −5058.87 −0.445776
$$506$$ 0 0
$$507$$ −23641.2 −2.07090
$$508$$ 0 0
$$509$$ −5464.15 −0.475823 −0.237912 0.971287i $$-0.576463\pi$$
−0.237912 + 0.971287i $$0.576463\pi$$
$$510$$ 0 0
$$511$$ 3039.15 0.263100
$$512$$ 0 0
$$513$$ 862.245 0.0742087
$$514$$ 0 0
$$515$$ −8902.13 −0.761699
$$516$$ 0 0
$$517$$ −14541.5 −1.23701
$$518$$ 0 0
$$519$$ 10389.8 0.878733
$$520$$ 0 0
$$521$$ −12113.9 −1.01865 −0.509327 0.860573i $$-0.670106\pi$$
−0.509327 + 0.860573i $$0.670106\pi$$
$$522$$ 0 0
$$523$$ −12706.6 −1.06237 −0.531185 0.847256i $$-0.678253\pi$$
−0.531185 + 0.847256i $$0.678253\pi$$
$$524$$ 0 0
$$525$$ −2573.35 −0.213925
$$526$$ 0 0
$$527$$ 1825.69 0.150908
$$528$$ 0 0
$$529$$ −2843.34 −0.233693
$$530$$ 0 0
$$531$$ −10263.2 −0.838767
$$532$$ 0 0
$$533$$ −3325.95 −0.270287
$$534$$ 0 0
$$535$$ 2255.43 0.182263
$$536$$ 0 0
$$537$$ 5305.55 0.426352
$$538$$ 0 0
$$539$$ 14346.7 1.14648
$$540$$ 0 0
$$541$$ −20011.9 −1.59035 −0.795173 0.606383i $$-0.792620\pi$$
−0.795173 + 0.606383i $$0.792620\pi$$
$$542$$ 0 0
$$543$$ −15785.6 −1.24756
$$544$$ 0 0
$$545$$ 585.109 0.0459877
$$546$$ 0 0
$$547$$ −14732.5 −1.15158 −0.575790 0.817597i $$-0.695306\pi$$
−0.575790 + 0.817597i $$0.695306\pi$$
$$548$$ 0 0
$$549$$ −1116.77 −0.0868172
$$550$$ 0 0
$$551$$ 770.602 0.0595803
$$552$$ 0 0
$$553$$ 1268.85 0.0975715
$$554$$ 0 0
$$555$$ −11267.9 −0.861796
$$556$$ 0 0
$$557$$ −2202.91 −0.167577 −0.0837883 0.996484i $$-0.526702\pi$$
−0.0837883 + 0.996484i $$0.526702\pi$$
$$558$$ 0 0
$$559$$ −16086.8 −1.21717
$$560$$ 0 0
$$561$$ −3938.65 −0.296417
$$562$$ 0 0
$$563$$ −14661.2 −1.09751 −0.548755 0.835983i $$-0.684898\pi$$
−0.548755 + 0.835983i $$0.684898\pi$$
$$564$$ 0 0
$$565$$ −8348.20 −0.621613
$$566$$ 0 0
$$567$$ 3134.12 0.232135
$$568$$ 0 0
$$569$$ 2252.50 0.165957 0.0829785 0.996551i $$-0.473557\pi$$
0.0829785 + 0.996551i $$0.473557\pi$$
$$570$$ 0 0
$$571$$ −13749.4 −1.00770 −0.503848 0.863793i $$-0.668083\pi$$
−0.503848 + 0.863793i $$0.668083\pi$$
$$572$$ 0 0
$$573$$ −11501.4 −0.838528
$$574$$ 0 0
$$575$$ −9942.97 −0.721131
$$576$$ 0 0
$$577$$ −16355.8 −1.18007 −0.590037 0.807376i $$-0.700887\pi$$
−0.590037 + 0.807376i $$0.700887\pi$$
$$578$$ 0 0
$$579$$ 4570.88 0.328082
$$580$$ 0 0
$$581$$ 1892.41 0.135130
$$582$$ 0 0
$$583$$ −13533.0 −0.961375
$$584$$ 0 0
$$585$$ 7187.47 0.507975
$$586$$ 0 0
$$587$$ −12704.1 −0.893279 −0.446640 0.894714i $$-0.647379\pi$$
−0.446640 + 0.894714i $$0.647379\pi$$
$$588$$ 0 0
$$589$$ −2637.72 −0.184525
$$590$$ 0 0
$$591$$ 17012.9 1.18412
$$592$$ 0 0
$$593$$ 7442.67 0.515403 0.257701 0.966225i $$-0.417035\pi$$
0.257701 + 0.966225i $$0.417035\pi$$
$$594$$ 0 0
$$595$$ 224.015 0.0154348
$$596$$ 0 0
$$597$$ −29939.6 −2.05251
$$598$$ 0 0
$$599$$ 17522.7 1.19525 0.597627 0.801774i $$-0.296110\pi$$
0.597627 + 0.801774i $$0.296110\pi$$
$$600$$ 0 0
$$601$$ 14983.9 1.01698 0.508490 0.861068i $$-0.330204\pi$$
0.508490 + 0.861068i $$0.330204\pi$$
$$602$$ 0 0
$$603$$ −6387.21 −0.431355
$$604$$ 0 0
$$605$$ 2633.11 0.176944
$$606$$ 0 0
$$607$$ 18937.9 1.26634 0.633169 0.774013i $$-0.281754\pi$$
0.633169 + 0.774013i $$0.281754\pi$$
$$608$$ 0 0
$$609$$ 1013.57 0.0674415
$$610$$ 0 0
$$611$$ 25085.3 1.66095
$$612$$ 0 0
$$613$$ 7209.14 0.474999 0.237500 0.971388i $$-0.423672\pi$$
0.237500 + 0.971388i $$0.423672\pi$$
$$614$$ 0 0
$$615$$ 1432.36 0.0939158
$$616$$ 0 0
$$617$$ −25907.8 −1.69045 −0.845226 0.534408i $$-0.820534\pi$$
−0.845226 + 0.534408i $$0.820534\pi$$
$$618$$ 0 0
$$619$$ −26596.0 −1.72695 −0.863475 0.504391i $$-0.831717\pi$$
−0.863475 + 0.504391i $$0.831717\pi$$
$$620$$ 0 0
$$621$$ 4381.98 0.283161
$$622$$ 0 0
$$623$$ −230.144 −0.0148002
$$624$$ 0 0
$$625$$ 7850.03 0.502402
$$626$$ 0 0
$$627$$ 5690.48 0.362450
$$628$$ 0 0
$$629$$ −4585.50 −0.290677
$$630$$ 0 0
$$631$$ 231.639 0.0146140 0.00730699 0.999973i $$-0.497674\pi$$
0.00730699 + 0.999973i $$0.497674\pi$$
$$632$$ 0 0
$$633$$ −26927.5 −1.69079
$$634$$ 0 0
$$635$$ −5058.46 −0.316124
$$636$$ 0 0
$$637$$ −24749.1 −1.53940
$$638$$ 0 0
$$639$$ −13116.0 −0.811991
$$640$$ 0 0
$$641$$ 9194.84 0.566574 0.283287 0.959035i $$-0.408575\pi$$
0.283287 + 0.959035i $$0.408575\pi$$
$$642$$ 0 0
$$643$$ 12339.1 0.756775 0.378387 0.925647i $$-0.376479\pi$$
0.378387 + 0.925647i $$0.376479\pi$$
$$644$$ 0 0
$$645$$ 6927.95 0.422927
$$646$$ 0 0
$$647$$ 10852.4 0.659428 0.329714 0.944081i $$-0.393048\pi$$
0.329714 + 0.944081i $$0.393048\pi$$
$$648$$ 0 0
$$649$$ 21873.8 1.32299
$$650$$ 0 0
$$651$$ −3469.38 −0.208872
$$652$$ 0 0
$$653$$ 23310.7 1.39696 0.698482 0.715627i $$-0.253859\pi$$
0.698482 + 0.715627i $$0.253859\pi$$
$$654$$ 0 0
$$655$$ 5128.79 0.305952
$$656$$ 0 0
$$657$$ −17089.5 −1.01480
$$658$$ 0 0
$$659$$ −21318.0 −1.26014 −0.630069 0.776539i $$-0.716973\pi$$
−0.630069 + 0.776539i $$0.716973\pi$$
$$660$$ 0 0
$$661$$ 26645.4 1.56791 0.783954 0.620818i $$-0.213200\pi$$
0.783954 + 0.620818i $$0.213200\pi$$
$$662$$ 0 0
$$663$$ 6794.49 0.398003
$$664$$ 0 0
$$665$$ −323.652 −0.0188732
$$666$$ 0 0
$$667$$ 3916.24 0.227343
$$668$$ 0 0
$$669$$ −486.620 −0.0281223
$$670$$ 0 0
$$671$$ 2380.16 0.136937
$$672$$ 0 0
$$673$$ 12426.1 0.711728 0.355864 0.934538i $$-0.384187\pi$$
0.355864 + 0.934538i $$0.384187\pi$$
$$674$$ 0 0
$$675$$ −4673.05 −0.266468
$$676$$ 0 0
$$677$$ −28485.7 −1.61712 −0.808562 0.588411i $$-0.799754\pi$$
−0.808562 + 0.588411i $$0.799754\pi$$
$$678$$ 0 0
$$679$$ 4464.00 0.252301
$$680$$ 0 0
$$681$$ −7796.81 −0.438729
$$682$$ 0 0
$$683$$ 1899.14 0.106396 0.0531981 0.998584i $$-0.483059\pi$$
0.0531981 + 0.998584i $$0.483059\pi$$
$$684$$ 0 0
$$685$$ 14324.9 0.799018
$$686$$ 0 0
$$687$$ −20119.2 −1.11732
$$688$$ 0 0
$$689$$ 23345.6 1.29085
$$690$$ 0 0
$$691$$ 29568.6 1.62785 0.813923 0.580973i $$-0.197328\pi$$
0.813923 + 0.580973i $$0.197328\pi$$
$$692$$ 0 0
$$693$$ 3222.06 0.176618
$$694$$ 0 0
$$695$$ 6705.46 0.365975
$$696$$ 0 0
$$697$$ 582.899 0.0316770
$$698$$ 0 0
$$699$$ 6376.26 0.345025
$$700$$ 0 0
$$701$$ −6906.35 −0.372110 −0.186055 0.982539i $$-0.559570\pi$$
−0.186055 + 0.982539i $$0.559570\pi$$
$$702$$ 0 0
$$703$$ 6625.04 0.355431
$$704$$ 0 0
$$705$$ −10803.2 −0.577126
$$706$$ 0 0
$$707$$ 3912.20 0.208110
$$708$$ 0 0
$$709$$ 14410.0 0.763301 0.381650 0.924307i $$-0.375356\pi$$
0.381650 + 0.924307i $$0.375356\pi$$
$$710$$ 0 0
$$711$$ −7134.90 −0.376343
$$712$$ 0 0
$$713$$ −13405.1 −0.704100
$$714$$ 0 0
$$715$$ −15318.6 −0.801233
$$716$$ 0 0
$$717$$ 30131.4 1.56942
$$718$$ 0 0
$$719$$ 11170.9 0.579419 0.289710 0.957115i $$-0.406441\pi$$
0.289710 + 0.957115i $$0.406441\pi$$
$$720$$ 0 0
$$721$$ 6884.32 0.355597
$$722$$ 0 0
$$723$$ 27298.6 1.40421
$$724$$ 0 0
$$725$$ −4176.37 −0.213940
$$726$$ 0 0
$$727$$ −29676.9 −1.51397 −0.756985 0.653433i $$-0.773328\pi$$
−0.756985 + 0.653433i $$0.773328\pi$$
$$728$$ 0 0
$$729$$ −9186.85 −0.466740
$$730$$ 0 0
$$731$$ 2819.34 0.142650
$$732$$ 0 0
$$733$$ −252.315 −0.0127142 −0.00635708 0.999980i $$-0.502024\pi$$
−0.00635708 + 0.999980i $$0.502024\pi$$
$$734$$ 0 0
$$735$$ 10658.5 0.534890
$$736$$ 0 0
$$737$$ 13613.0 0.680380
$$738$$ 0 0
$$739$$ −1702.08 −0.0847252 −0.0423626 0.999102i $$-0.513488\pi$$
−0.0423626 + 0.999102i $$0.513488\pi$$
$$740$$ 0 0
$$741$$ −9816.54 −0.486666
$$742$$ 0 0
$$743$$ 11210.7 0.553539 0.276769 0.960936i $$-0.410736\pi$$
0.276769 + 0.960936i $$0.410736\pi$$
$$744$$ 0 0
$$745$$ 2679.98 0.131794
$$746$$ 0 0
$$747$$ −10641.3 −0.521210
$$748$$ 0 0
$$749$$ −1744.20 −0.0850892
$$750$$ 0 0
$$751$$ −37318.2 −1.81326 −0.906631 0.421925i $$-0.861354\pi$$
−0.906631 + 0.421925i $$0.861354\pi$$
$$752$$ 0 0
$$753$$ −1513.79 −0.0732609
$$754$$ 0 0
$$755$$ 4682.90 0.225732
$$756$$ 0 0
$$757$$ −13356.3 −0.641274 −0.320637 0.947202i $$-0.603897\pi$$
−0.320637 + 0.947202i $$0.603897\pi$$
$$758$$ 0 0
$$759$$ 28919.4 1.38301
$$760$$ 0 0
$$761$$ 37538.5 1.78814 0.894068 0.447932i $$-0.147839\pi$$
0.894068 + 0.447932i $$0.147839\pi$$
$$762$$ 0 0
$$763$$ −452.485 −0.0214693
$$764$$ 0 0
$$765$$ −1259.66 −0.0595335
$$766$$ 0 0
$$767$$ −37734.1 −1.77640
$$768$$ 0 0
$$769$$ 422.829 0.0198278 0.00991391 0.999951i $$-0.496844\pi$$
0.00991391 + 0.999951i $$0.496844\pi$$
$$770$$ 0 0
$$771$$ 38437.5 1.79545
$$772$$ 0 0
$$773$$ 23033.6 1.07175 0.535873 0.844298i $$-0.319982\pi$$
0.535873 + 0.844298i $$0.319982\pi$$
$$774$$ 0 0
$$775$$ 14295.5 0.662592
$$776$$ 0 0
$$777$$ 8713.88 0.402328
$$778$$ 0 0
$$779$$ −842.161 −0.0387337
$$780$$ 0 0
$$781$$ 27954.0 1.28076
$$782$$ 0 0
$$783$$ 1840.58 0.0840061
$$784$$ 0 0
$$785$$ −10190.2 −0.463319
$$786$$ 0 0
$$787$$ −37994.2 −1.72090 −0.860448 0.509538i $$-0.829816\pi$$
−0.860448 + 0.509538i $$0.829816\pi$$
$$788$$ 0 0
$$789$$ −20461.8 −0.923271
$$790$$ 0 0
$$791$$ 6455.95 0.290199
$$792$$ 0 0
$$793$$ −4105.96 −0.183868
$$794$$ 0 0
$$795$$ −10054.0 −0.448528
$$796$$ 0 0
$$797$$ −23765.7 −1.05624 −0.528122 0.849169i $$-0.677103\pi$$
−0.528122 + 0.849169i $$0.677103\pi$$
$$798$$ 0 0
$$799$$ −4396.39 −0.194660
$$800$$ 0 0
$$801$$ 1294.13 0.0570858
$$802$$ 0 0
$$803$$ 36422.6 1.60065
$$804$$ 0 0
$$805$$ −1644.82 −0.0720152
$$806$$ 0 0
$$807$$ 41102.6 1.79291
$$808$$ 0 0
$$809$$ −7019.58 −0.305062 −0.152531 0.988299i $$-0.548742\pi$$
−0.152531 + 0.988299i $$0.548742\pi$$
$$810$$ 0 0
$$811$$ −35061.1 −1.51808 −0.759039 0.651045i $$-0.774331\pi$$
−0.759039 + 0.651045i $$0.774331\pi$$
$$812$$ 0 0
$$813$$ 43760.4 1.88775
$$814$$ 0 0
$$815$$ −1433.75 −0.0616220
$$816$$ 0 0
$$817$$ −4073.32 −0.174428
$$818$$ 0 0
$$819$$ −5558.32 −0.237147
$$820$$ 0 0
$$821$$ −306.783 −0.0130412 −0.00652059 0.999979i $$-0.502076\pi$$
−0.00652059 + 0.999979i $$0.502076\pi$$
$$822$$ 0 0
$$823$$ 23239.1 0.984281 0.492140 0.870516i $$-0.336215\pi$$
0.492140 + 0.870516i $$0.336215\pi$$
$$824$$ 0 0
$$825$$ −30840.3 −1.30148
$$826$$ 0 0
$$827$$ 24374.9 1.02491 0.512455 0.858714i $$-0.328736\pi$$
0.512455 + 0.858714i $$0.328736\pi$$
$$828$$ 0 0
$$829$$ 27129.6 1.13661 0.568306 0.822817i $$-0.307599\pi$$
0.568306 + 0.822817i $$0.307599\pi$$
$$830$$ 0 0
$$831$$ 55350.5 2.31058
$$832$$ 0 0
$$833$$ 4337.48 0.180414
$$834$$ 0 0
$$835$$ 641.450 0.0265848
$$836$$ 0 0
$$837$$ −6300.18 −0.260174
$$838$$ 0 0
$$839$$ 4376.25 0.180077 0.0900386 0.995938i $$-0.471301\pi$$
0.0900386 + 0.995938i $$0.471301\pi$$
$$840$$ 0 0
$$841$$ −22744.0 −0.932554
$$842$$ 0 0
$$843$$ 30838.5 1.25995
$$844$$ 0 0
$$845$$ 16114.6 0.656044
$$846$$ 0 0
$$847$$ −2036.28 −0.0826061
$$848$$ 0 0
$$849$$ 60677.5 2.45282
$$850$$ 0 0
$$851$$ 33668.8 1.35623
$$852$$ 0 0
$$853$$ 46278.7 1.85762 0.928812 0.370550i $$-0.120831\pi$$
0.928812 + 0.370550i $$0.120831\pi$$
$$854$$ 0 0
$$855$$ 1819.93 0.0727958
$$856$$ 0 0
$$857$$ −5108.69 −0.203628 −0.101814 0.994803i $$-0.532465\pi$$
−0.101814 + 0.994803i $$0.532465\pi$$
$$858$$ 0 0
$$859$$ −12440.5 −0.494137 −0.247069 0.968998i $$-0.579467\pi$$
−0.247069 + 0.968998i $$0.579467\pi$$
$$860$$ 0 0
$$861$$ −1107.69 −0.0438444
$$862$$ 0 0
$$863$$ −33694.7 −1.32906 −0.664531 0.747260i $$-0.731369\pi$$
−0.664531 + 0.747260i $$0.731369\pi$$
$$864$$ 0 0
$$865$$ −7082.00 −0.278376
$$866$$ 0 0
$$867$$ 32637.3 1.27846
$$868$$ 0 0
$$869$$ 15206.5 0.593608
$$870$$ 0 0
$$871$$ −23483.5 −0.913555
$$872$$ 0 0
$$873$$ −25101.6 −0.973151
$$874$$ 0 0
$$875$$ 3883.36 0.150036
$$876$$ 0 0
$$877$$ 6266.75 0.241292 0.120646 0.992696i $$-0.461503\pi$$
0.120646 + 0.992696i $$0.461503\pi$$
$$878$$ 0 0
$$879$$ −6486.18 −0.248889
$$880$$ 0 0
$$881$$ −18875.2 −0.721818 −0.360909 0.932601i $$-0.617534\pi$$
−0.360909 + 0.932601i $$0.617534\pi$$
$$882$$ 0 0
$$883$$ 24369.5 0.928765 0.464383 0.885635i $$-0.346276\pi$$
0.464383 + 0.885635i $$0.346276\pi$$
$$884$$ 0 0
$$885$$ 16250.6 0.617241
$$886$$ 0 0
$$887$$ 46836.8 1.77297 0.886486 0.462754i $$-0.153139\pi$$
0.886486 + 0.462754i $$0.153139\pi$$
$$888$$ 0 0
$$889$$ 3911.88 0.147582
$$890$$ 0 0
$$891$$ 37560.8 1.41227
$$892$$ 0 0
$$893$$ 6351.82 0.238024
$$894$$ 0 0
$$895$$ −3616.41 −0.135065
$$896$$ 0 0
$$897$$ −49888.2 −1.85699
$$898$$ 0 0
$$899$$ −5630.57 −0.208888
$$900$$ 0 0
$$901$$ −4091.50 −0.151285
$$902$$ 0 0
$$903$$ −5357.62 −0.197442
$$904$$ 0 0
$$905$$ 10760.0 0.395219
$$906$$ 0 0
$$907$$ 19015.8 0.696150 0.348075 0.937467i $$-0.386836\pi$$
0.348075 + 0.937467i $$0.386836\pi$$
$$908$$ 0 0
$$909$$ −21998.8 −0.802699
$$910$$ 0 0
$$911$$ 30625.3 1.11379 0.556894 0.830583i $$-0.311993\pi$$
0.556894 + 0.830583i $$0.311993\pi$$
$$912$$ 0 0
$$913$$ 22679.6 0.822108
$$914$$ 0 0
$$915$$ 1768.28 0.0638879
$$916$$ 0 0
$$917$$ −3966.27 −0.142833
$$918$$ 0 0
$$919$$ −48642.7 −1.74600 −0.873001 0.487719i $$-0.837829\pi$$
−0.873001 + 0.487719i $$0.837829\pi$$
$$920$$ 0 0
$$921$$ −36985.7 −1.32326
$$922$$ 0 0
$$923$$ −48222.9 −1.71969
$$924$$ 0 0
$$925$$ −35905.2 −1.27628
$$926$$ 0 0
$$927$$ −38711.4 −1.37157
$$928$$ 0 0
$$929$$ 42194.6 1.49016 0.745080 0.666975i $$-0.232411\pi$$
0.745080 + 0.666975i $$0.232411\pi$$
$$930$$ 0 0
$$931$$ −6266.71 −0.220605
$$932$$ 0 0
$$933$$ −56324.2 −1.97639
$$934$$ 0 0
$$935$$ 2684.70 0.0939026
$$936$$ 0 0
$$937$$ −28438.1 −0.991498 −0.495749 0.868466i $$-0.665106\pi$$
−0.495749 + 0.868466i $$0.665106\pi$$
$$938$$ 0 0
$$939$$ 4279.74 0.148737
$$940$$ 0 0
$$941$$ 29883.2 1.03524 0.517622 0.855610i $$-0.326818\pi$$
0.517622 + 0.855610i $$0.326818\pi$$
$$942$$ 0 0
$$943$$ −4279.91 −0.147798
$$944$$ 0 0
$$945$$ −773.040 −0.0266106
$$946$$ 0 0
$$947$$ 46897.4 1.60925 0.804626 0.593782i $$-0.202366\pi$$
0.804626 + 0.593782i $$0.202366\pi$$
$$948$$ 0 0
$$949$$ −62831.9 −2.14922
$$950$$ 0 0
$$951$$ 18187.7 0.620163
$$952$$ 0 0
$$953$$ −20401.1 −0.693449 −0.346724 0.937967i $$-0.612706\pi$$
−0.346724 + 0.937967i $$0.612706\pi$$
$$954$$ 0 0
$$955$$ 7839.66 0.265639
$$956$$ 0 0
$$957$$ 12147.1 0.410302
$$958$$ 0 0
$$959$$ −11078.0 −0.373020
$$960$$ 0 0
$$961$$ −10517.9 −0.353057
$$962$$ 0 0
$$963$$ 9807.87 0.328197
$$964$$ 0 0
$$965$$ −3115.65 −0.103934
$$966$$ 0 0
$$967$$ 23598.5 0.784775 0.392388 0.919800i $$-0.371649\pi$$
0.392388 + 0.919800i $$0.371649\pi$$
$$968$$ 0 0
$$969$$ 1720.43 0.0570362
$$970$$ 0 0
$$971$$ 17149.1 0.566776 0.283388 0.959005i $$-0.408542\pi$$
0.283388 + 0.959005i $$0.408542\pi$$
$$972$$ 0 0
$$973$$ −5185.56 −0.170854
$$974$$ 0 0
$$975$$ 53202.0 1.74752
$$976$$ 0 0
$$977$$ −56653.2 −1.85517 −0.927583 0.373618i $$-0.878117\pi$$
−0.927583 + 0.373618i $$0.878117\pi$$
$$978$$ 0 0
$$979$$ −2758.15 −0.0900418
$$980$$ 0 0
$$981$$ 2544.38 0.0828091
$$982$$ 0 0
$$983$$ 7895.00 0.256166 0.128083 0.991763i $$-0.459118\pi$$
0.128083 + 0.991763i $$0.459118\pi$$
$$984$$ 0 0
$$985$$ −11596.5 −0.375122
$$986$$ 0 0
$$987$$ 8354.52 0.269430
$$988$$ 0 0
$$989$$ −20700.9 −0.665570
$$990$$ 0 0
$$991$$ −48458.9 −1.55333 −0.776664 0.629915i $$-0.783090\pi$$
−0.776664 + 0.629915i $$0.783090\pi$$
$$992$$ 0 0
$$993$$ 51300.3 1.63944
$$994$$ 0 0
$$995$$ 20407.7 0.650219
$$996$$ 0 0
$$997$$ 40325.2 1.28095 0.640477 0.767977i $$-0.278737\pi$$
0.640477 + 0.767977i $$0.278737\pi$$
$$998$$ 0 0
$$999$$ 15823.8 0.501145
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 608.4.a.j.1.2 7
4.3 odd 2 608.4.a.k.1.6 yes 7
8.3 odd 2 1216.4.a.bf.1.2 7
8.5 even 2 1216.4.a.bg.1.6 7

By twisted newform
Twist Min Dim Char Parity Ord Type
608.4.a.j.1.2 7 1.1 even 1 trivial
608.4.a.k.1.6 yes 7 4.3 odd 2
1216.4.a.bf.1.2 7 8.3 odd 2
1216.4.a.bg.1.6 7 8.5 even 2