# Properties

 Label 847.4.a.b.1.1 Level $847$ Weight $4$ Character 847.1 Self dual yes Analytic conductor $49.975$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [847,4,Mod(1,847)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(847, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("847.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$847 = 7 \cdot 11^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 847.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: $$49.9746177749$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 7) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 847.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+1.00000 q^{2} -2.00000 q^{3} -7.00000 q^{4} +16.0000 q^{5} -2.00000 q^{6} +7.00000 q^{7} -15.0000 q^{8} -23.0000 q^{9} +O(q^{10})$$ $$q+1.00000 q^{2} -2.00000 q^{3} -7.00000 q^{4} +16.0000 q^{5} -2.00000 q^{6} +7.00000 q^{7} -15.0000 q^{8} -23.0000 q^{9} +16.0000 q^{10} +14.0000 q^{12} -28.0000 q^{13} +7.00000 q^{14} -32.0000 q^{15} +41.0000 q^{16} -54.0000 q^{17} -23.0000 q^{18} +110.000 q^{19} -112.000 q^{20} -14.0000 q^{21} +48.0000 q^{23} +30.0000 q^{24} +131.000 q^{25} -28.0000 q^{26} +100.000 q^{27} -49.0000 q^{28} +110.000 q^{29} -32.0000 q^{30} +12.0000 q^{31} +161.000 q^{32} -54.0000 q^{34} +112.000 q^{35} +161.000 q^{36} -246.000 q^{37} +110.000 q^{38} +56.0000 q^{39} -240.000 q^{40} -182.000 q^{41} -14.0000 q^{42} -128.000 q^{43} -368.000 q^{45} +48.0000 q^{46} +324.000 q^{47} -82.0000 q^{48} +49.0000 q^{49} +131.000 q^{50} +108.000 q^{51} +196.000 q^{52} -162.000 q^{53} +100.000 q^{54} -105.000 q^{56} -220.000 q^{57} +110.000 q^{58} +810.000 q^{59} +224.000 q^{60} +488.000 q^{61} +12.0000 q^{62} -161.000 q^{63} -167.000 q^{64} -448.000 q^{65} +244.000 q^{67} +378.000 q^{68} -96.0000 q^{69} +112.000 q^{70} -768.000 q^{71} +345.000 q^{72} +702.000 q^{73} -246.000 q^{74} -262.000 q^{75} -770.000 q^{76} +56.0000 q^{78} -440.000 q^{79} +656.000 q^{80} +421.000 q^{81} -182.000 q^{82} +1302.00 q^{83} +98.0000 q^{84} -864.000 q^{85} -128.000 q^{86} -220.000 q^{87} +730.000 q^{89} -368.000 q^{90} -196.000 q^{91} -336.000 q^{92} -24.0000 q^{93} +324.000 q^{94} +1760.00 q^{95} -322.000 q^{96} +294.000 q^{97} +49.0000 q^{98} +O(q^{100})$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 1.00000 0.353553 0.176777 0.984251i $$-0.443433\pi$$
0.176777 + 0.984251i $$0.443433\pi$$
$$3$$ −2.00000 −0.384900 −0.192450 0.981307i $$-0.561643\pi$$
−0.192450 + 0.981307i $$0.561643\pi$$
$$4$$ −7.00000 −0.875000
$$5$$ 16.0000 1.43108 0.715542 0.698570i $$-0.246180\pi$$
0.715542 + 0.698570i $$0.246180\pi$$
$$6$$ −2.00000 −0.136083
$$7$$ 7.00000 0.377964
$$8$$ −15.0000 −0.662913
$$9$$ −23.0000 −0.851852
$$10$$ 16.0000 0.505964
$$11$$ 0 0
$$12$$ 14.0000 0.336788
$$13$$ −28.0000 −0.597369 −0.298685 0.954352i $$-0.596548\pi$$
−0.298685 + 0.954352i $$0.596548\pi$$
$$14$$ 7.00000 0.133631
$$15$$ −32.0000 −0.550824
$$16$$ 41.0000 0.640625
$$17$$ −54.0000 −0.770407 −0.385204 0.922832i $$-0.625869\pi$$
−0.385204 + 0.922832i $$0.625869\pi$$
$$18$$ −23.0000 −0.301175
$$19$$ 110.000 1.32820 0.664098 0.747645i $$-0.268816\pi$$
0.664098 + 0.747645i $$0.268816\pi$$
$$20$$ −112.000 −1.25220
$$21$$ −14.0000 −0.145479
$$22$$ 0 0
$$23$$ 48.0000 0.435161 0.217580 0.976042i $$-0.430184\pi$$
0.217580 + 0.976042i $$0.430184\pi$$
$$24$$ 30.0000 0.255155
$$25$$ 131.000 1.04800
$$26$$ −28.0000 −0.211202
$$27$$ 100.000 0.712778
$$28$$ −49.0000 −0.330719
$$29$$ 110.000 0.704362 0.352181 0.935932i $$-0.385440\pi$$
0.352181 + 0.935932i $$0.385440\pi$$
$$30$$ −32.0000 −0.194746
$$31$$ 12.0000 0.0695246 0.0347623 0.999396i $$-0.488933\pi$$
0.0347623 + 0.999396i $$0.488933\pi$$
$$32$$ 161.000 0.889408
$$33$$ 0 0
$$34$$ −54.0000 −0.272380
$$35$$ 112.000 0.540899
$$36$$ 161.000 0.745370
$$37$$ −246.000 −1.09303 −0.546516 0.837449i $$-0.684046\pi$$
−0.546516 + 0.837449i $$0.684046\pi$$
$$38$$ 110.000 0.469588
$$39$$ 56.0000 0.229928
$$40$$ −240.000 −0.948683
$$41$$ −182.000 −0.693259 −0.346630 0.938002i $$-0.612674\pi$$
−0.346630 + 0.938002i $$0.612674\pi$$
$$42$$ −14.0000 −0.0514344
$$43$$ −128.000 −0.453949 −0.226975 0.973901i $$-0.572883\pi$$
−0.226975 + 0.973901i $$0.572883\pi$$
$$44$$ 0 0
$$45$$ −368.000 −1.21907
$$46$$ 48.0000 0.153852
$$47$$ 324.000 1.00554 0.502769 0.864421i $$-0.332315\pi$$
0.502769 + 0.864421i $$0.332315\pi$$
$$48$$ −82.0000 −0.246577
$$49$$ 49.0000 0.142857
$$50$$ 131.000 0.370524
$$51$$ 108.000 0.296530
$$52$$ 196.000 0.522698
$$53$$ −162.000 −0.419857 −0.209928 0.977717i $$-0.567323\pi$$
−0.209928 + 0.977717i $$0.567323\pi$$
$$54$$ 100.000 0.252005
$$55$$ 0 0
$$56$$ −105.000 −0.250557
$$57$$ −220.000 −0.511223
$$58$$ 110.000 0.249029
$$59$$ 810.000 1.78734 0.893670 0.448725i $$-0.148122\pi$$
0.893670 + 0.448725i $$0.148122\pi$$
$$60$$ 224.000 0.481971
$$61$$ 488.000 1.02430 0.512148 0.858898i $$-0.328850\pi$$
0.512148 + 0.858898i $$0.328850\pi$$
$$62$$ 12.0000 0.0245807
$$63$$ −161.000 −0.321970
$$64$$ −167.000 −0.326172
$$65$$ −448.000 −0.854886
$$66$$ 0 0
$$67$$ 244.000 0.444916 0.222458 0.974942i $$-0.428592\pi$$
0.222458 + 0.974942i $$0.428592\pi$$
$$68$$ 378.000 0.674106
$$69$$ −96.0000 −0.167493
$$70$$ 112.000 0.191237
$$71$$ −768.000 −1.28373 −0.641865 0.766818i $$-0.721839\pi$$
−0.641865 + 0.766818i $$0.721839\pi$$
$$72$$ 345.000 0.564703
$$73$$ 702.000 1.12552 0.562759 0.826621i $$-0.309740\pi$$
0.562759 + 0.826621i $$0.309740\pi$$
$$74$$ −246.000 −0.386445
$$75$$ −262.000 −0.403375
$$76$$ −770.000 −1.16217
$$77$$ 0 0
$$78$$ 56.0000 0.0812917
$$79$$ −440.000 −0.626631 −0.313316 0.949649i $$-0.601440\pi$$
−0.313316 + 0.949649i $$0.601440\pi$$
$$80$$ 656.000 0.916788
$$81$$ 421.000 0.577503
$$82$$ −182.000 −0.245104
$$83$$ 1302.00 1.72184 0.860922 0.508737i $$-0.169887\pi$$
0.860922 + 0.508737i $$0.169887\pi$$
$$84$$ 98.0000 0.127294
$$85$$ −864.000 −1.10252
$$86$$ −128.000 −0.160495
$$87$$ −220.000 −0.271109
$$88$$ 0 0
$$89$$ 730.000 0.869436 0.434718 0.900567i $$-0.356848\pi$$
0.434718 + 0.900567i $$0.356848\pi$$
$$90$$ −368.000 −0.431007
$$91$$ −196.000 −0.225784
$$92$$ −336.000 −0.380765
$$93$$ −24.0000 −0.0267600
$$94$$ 324.000 0.355511
$$95$$ 1760.00 1.90076
$$96$$ −322.000 −0.342333
$$97$$ 294.000 0.307744 0.153872 0.988091i $$-0.450826\pi$$
0.153872 + 0.988091i $$0.450826\pi$$
$$98$$ 49.0000 0.0505076
$$99$$ 0 0
$$100$$ −917.000 −0.917000
$$101$$ 688.000 0.677808 0.338904 0.940821i $$-0.389944\pi$$
0.338904 + 0.940821i $$0.389944\pi$$
$$102$$ 108.000 0.104839
$$103$$ 1388.00 1.32780 0.663901 0.747820i $$-0.268899\pi$$
0.663901 + 0.747820i $$0.268899\pi$$
$$104$$ 420.000 0.396004
$$105$$ −224.000 −0.208192
$$106$$ −162.000 −0.148442
$$107$$ −244.000 −0.220452 −0.110226 0.993907i $$-0.535157\pi$$
−0.110226 + 0.993907i $$0.535157\pi$$
$$108$$ −700.000 −0.623681
$$109$$ −90.0000 −0.0790866 −0.0395433 0.999218i $$-0.512590\pi$$
−0.0395433 + 0.999218i $$0.512590\pi$$
$$110$$ 0 0
$$111$$ 492.000 0.420708
$$112$$ 287.000 0.242133
$$113$$ 1318.00 1.09723 0.548615 0.836075i $$-0.315155\pi$$
0.548615 + 0.836075i $$0.315155\pi$$
$$114$$ −220.000 −0.180745
$$115$$ 768.000 0.622751
$$116$$ −770.000 −0.616316
$$117$$ 644.000 0.508870
$$118$$ 810.000 0.631920
$$119$$ −378.000 −0.291187
$$120$$ 480.000 0.365148
$$121$$ 0 0
$$122$$ 488.000 0.362143
$$123$$ 364.000 0.266836
$$124$$ −84.0000 −0.0608341
$$125$$ 96.0000 0.0686920
$$126$$ −161.000 −0.113833
$$127$$ 1776.00 1.24090 0.620451 0.784245i $$-0.286950\pi$$
0.620451 + 0.784245i $$0.286950\pi$$
$$128$$ −1455.00 −1.00473
$$129$$ 256.000 0.174725
$$130$$ −448.000 −0.302248
$$131$$ 1118.00 0.745650 0.372825 0.927902i $$-0.378389\pi$$
0.372825 + 0.927902i $$0.378389\pi$$
$$132$$ 0 0
$$133$$ 770.000 0.502011
$$134$$ 244.000 0.157301
$$135$$ 1600.00 1.02004
$$136$$ 810.000 0.510713
$$137$$ 2274.00 1.41811 0.709054 0.705154i $$-0.249122\pi$$
0.709054 + 0.705154i $$0.249122\pi$$
$$138$$ −96.0000 −0.0592178
$$139$$ 210.000 0.128144 0.0640718 0.997945i $$-0.479591\pi$$
0.0640718 + 0.997945i $$0.479591\pi$$
$$140$$ −784.000 −0.473286
$$141$$ −648.000 −0.387032
$$142$$ −768.000 −0.453867
$$143$$ 0 0
$$144$$ −943.000 −0.545718
$$145$$ 1760.00 1.00800
$$146$$ 702.000 0.397931
$$147$$ −98.0000 −0.0549857
$$148$$ 1722.00 0.956402
$$149$$ 2010.00 1.10514 0.552569 0.833467i $$-0.313648\pi$$
0.552569 + 0.833467i $$0.313648\pi$$
$$150$$ −262.000 −0.142615
$$151$$ −1112.00 −0.599293 −0.299647 0.954050i $$-0.596869\pi$$
−0.299647 + 0.954050i $$0.596869\pi$$
$$152$$ −1650.00 −0.880478
$$153$$ 1242.00 0.656273
$$154$$ 0 0
$$155$$ 192.000 0.0994956
$$156$$ −392.000 −0.201187
$$157$$ 124.000 0.0630336 0.0315168 0.999503i $$-0.489966\pi$$
0.0315168 + 0.999503i $$0.489966\pi$$
$$158$$ −440.000 −0.221548
$$159$$ 324.000 0.161603
$$160$$ 2576.00 1.27282
$$161$$ 336.000 0.164475
$$162$$ 421.000 0.204178
$$163$$ 2008.00 0.964900 0.482450 0.875924i $$-0.339747\pi$$
0.482450 + 0.875924i $$0.339747\pi$$
$$164$$ 1274.00 0.606602
$$165$$ 0 0
$$166$$ 1302.00 0.608764
$$167$$ −2884.00 −1.33635 −0.668176 0.744004i $$-0.732924\pi$$
−0.668176 + 0.744004i $$0.732924\pi$$
$$168$$ 210.000 0.0964396
$$169$$ −1413.00 −0.643150
$$170$$ −864.000 −0.389799
$$171$$ −2530.00 −1.13143
$$172$$ 896.000 0.397206
$$173$$ −2228.00 −0.979143 −0.489571 0.871963i $$-0.662847\pi$$
−0.489571 + 0.871963i $$0.662847\pi$$
$$174$$ −220.000 −0.0958515
$$175$$ 917.000 0.396107
$$176$$ 0 0
$$177$$ −1620.00 −0.687947
$$178$$ 730.000 0.307392
$$179$$ −820.000 −0.342400 −0.171200 0.985236i $$-0.554764\pi$$
−0.171200 + 0.985236i $$0.554764\pi$$
$$180$$ 2576.00 1.06669
$$181$$ 3892.00 1.59829 0.799144 0.601140i $$-0.205287\pi$$
0.799144 + 0.601140i $$0.205287\pi$$
$$182$$ −196.000 −0.0798268
$$183$$ −976.000 −0.394251
$$184$$ −720.000 −0.288473
$$185$$ −3936.00 −1.56422
$$186$$ −24.0000 −0.00946110
$$187$$ 0 0
$$188$$ −2268.00 −0.879845
$$189$$ 700.000 0.269405
$$190$$ 1760.00 0.672020
$$191$$ −5048.00 −1.91236 −0.956179 0.292782i $$-0.905419\pi$$
−0.956179 + 0.292782i $$0.905419\pi$$
$$192$$ 334.000 0.125544
$$193$$ 2962.00 1.10471 0.552356 0.833608i $$-0.313729\pi$$
0.552356 + 0.833608i $$0.313729\pi$$
$$194$$ 294.000 0.108804
$$195$$ 896.000 0.329046
$$196$$ −343.000 −0.125000
$$197$$ −3334.00 −1.20577 −0.602887 0.797826i $$-0.705983\pi$$
−0.602887 + 0.797826i $$0.705983\pi$$
$$198$$ 0 0
$$199$$ 1860.00 0.662572 0.331286 0.943530i $$-0.392517\pi$$
0.331286 + 0.943530i $$0.392517\pi$$
$$200$$ −1965.00 −0.694732
$$201$$ −488.000 −0.171248
$$202$$ 688.000 0.239641
$$203$$ 770.000 0.266224
$$204$$ −756.000 −0.259464
$$205$$ −2912.00 −0.992112
$$206$$ 1388.00 0.469449
$$207$$ −1104.00 −0.370692
$$208$$ −1148.00 −0.382690
$$209$$ 0 0
$$210$$ −224.000 −0.0736070
$$211$$ 4268.00 1.39252 0.696259 0.717791i $$-0.254847\pi$$
0.696259 + 0.717791i $$0.254847\pi$$
$$212$$ 1134.00 0.367375
$$213$$ 1536.00 0.494108
$$214$$ −244.000 −0.0779416
$$215$$ −2048.00 −0.649639
$$216$$ −1500.00 −0.472510
$$217$$ 84.0000 0.0262778
$$218$$ −90.0000 −0.0279613
$$219$$ −1404.00 −0.433212
$$220$$ 0 0
$$221$$ 1512.00 0.460218
$$222$$ 492.000 0.148743
$$223$$ −5432.00 −1.63118 −0.815591 0.578629i $$-0.803588\pi$$
−0.815591 + 0.578629i $$0.803588\pi$$
$$224$$ 1127.00 0.336165
$$225$$ −3013.00 −0.892741
$$226$$ 1318.00 0.387929
$$227$$ 2046.00 0.598228 0.299114 0.954217i $$-0.403309\pi$$
0.299114 + 0.954217i $$0.403309\pi$$
$$228$$ 1540.00 0.447320
$$229$$ −2980.00 −0.859930 −0.429965 0.902846i $$-0.641474\pi$$
−0.429965 + 0.902846i $$0.641474\pi$$
$$230$$ 768.000 0.220176
$$231$$ 0 0
$$232$$ −1650.00 −0.466930
$$233$$ −4458.00 −1.25345 −0.626724 0.779241i $$-0.715605\pi$$
−0.626724 + 0.779241i $$0.715605\pi$$
$$234$$ 644.000 0.179913
$$235$$ 5184.00 1.43901
$$236$$ −5670.00 −1.56392
$$237$$ 880.000 0.241190
$$238$$ −378.000 −0.102950
$$239$$ −4440.00 −1.20167 −0.600836 0.799372i $$-0.705166\pi$$
−0.600836 + 0.799372i $$0.705166\pi$$
$$240$$ −1312.00 −0.352872
$$241$$ −3302.00 −0.882575 −0.441287 0.897366i $$-0.645478\pi$$
−0.441287 + 0.897366i $$0.645478\pi$$
$$242$$ 0 0
$$243$$ −3542.00 −0.935059
$$244$$ −3416.00 −0.896258
$$245$$ 784.000 0.204441
$$246$$ 364.000 0.0943406
$$247$$ −3080.00 −0.793424
$$248$$ −180.000 −0.0460888
$$249$$ −2604.00 −0.662738
$$250$$ 96.0000 0.0242863
$$251$$ 1582.00 0.397829 0.198914 0.980017i $$-0.436258\pi$$
0.198914 + 0.980017i $$0.436258\pi$$
$$252$$ 1127.00 0.281724
$$253$$ 0 0
$$254$$ 1776.00 0.438725
$$255$$ 1728.00 0.424359
$$256$$ −119.000 −0.0290527
$$257$$ 2354.00 0.571356 0.285678 0.958326i $$-0.407781\pi$$
0.285678 + 0.958326i $$0.407781\pi$$
$$258$$ 256.000 0.0617747
$$259$$ −1722.00 −0.413127
$$260$$ 3136.00 0.748025
$$261$$ −2530.00 −0.600012
$$262$$ 1118.00 0.263627
$$263$$ 3872.00 0.907824 0.453912 0.891046i $$-0.350028\pi$$
0.453912 + 0.891046i $$0.350028\pi$$
$$264$$ 0 0
$$265$$ −2592.00 −0.600850
$$266$$ 770.000 0.177488
$$267$$ −1460.00 −0.334646
$$268$$ −1708.00 −0.389301
$$269$$ 180.000 0.0407985 0.0203992 0.999792i $$-0.493506\pi$$
0.0203992 + 0.999792i $$0.493506\pi$$
$$270$$ 1600.00 0.360640
$$271$$ −2032.00 −0.455480 −0.227740 0.973722i $$-0.573134\pi$$
−0.227740 + 0.973722i $$0.573134\pi$$
$$272$$ −2214.00 −0.493542
$$273$$ 392.000 0.0869045
$$274$$ 2274.00 0.501377
$$275$$ 0 0
$$276$$ 672.000 0.146557
$$277$$ 5426.00 1.17696 0.588478 0.808513i $$-0.299727\pi$$
0.588478 + 0.808513i $$0.299727\pi$$
$$278$$ 210.000 0.0453056
$$279$$ −276.000 −0.0592247
$$280$$ −1680.00 −0.358569
$$281$$ −842.000 −0.178753 −0.0893764 0.995998i $$-0.528487\pi$$
−0.0893764 + 0.995998i $$0.528487\pi$$
$$282$$ −648.000 −0.136836
$$283$$ 3782.00 0.794405 0.397202 0.917731i $$-0.369981\pi$$
0.397202 + 0.917731i $$0.369981\pi$$
$$284$$ 5376.00 1.12326
$$285$$ −3520.00 −0.731603
$$286$$ 0 0
$$287$$ −1274.00 −0.262027
$$288$$ −3703.00 −0.757644
$$289$$ −1997.00 −0.406473
$$290$$ 1760.00 0.356382
$$291$$ −588.000 −0.118451
$$292$$ −4914.00 −0.984829
$$293$$ 4312.00 0.859760 0.429880 0.902886i $$-0.358556\pi$$
0.429880 + 0.902886i $$0.358556\pi$$
$$294$$ −98.0000 −0.0194404
$$295$$ 12960.0 2.55783
$$296$$ 3690.00 0.724584
$$297$$ 0 0
$$298$$ 2010.00 0.390725
$$299$$ −1344.00 −0.259952
$$300$$ 1834.00 0.352953
$$301$$ −896.000 −0.171577
$$302$$ −1112.00 −0.211882
$$303$$ −1376.00 −0.260888
$$304$$ 4510.00 0.850876
$$305$$ 7808.00 1.46585
$$306$$ 1242.00 0.232027
$$307$$ −2674.00 −0.497112 −0.248556 0.968618i $$-0.579956\pi$$
−0.248556 + 0.968618i $$0.579956\pi$$
$$308$$ 0 0
$$309$$ −2776.00 −0.511072
$$310$$ 192.000 0.0351770
$$311$$ −3768.00 −0.687021 −0.343511 0.939149i $$-0.611616\pi$$
−0.343511 + 0.939149i $$0.611616\pi$$
$$312$$ −840.000 −0.152422
$$313$$ 2438.00 0.440268 0.220134 0.975470i $$-0.429351\pi$$
0.220134 + 0.975470i $$0.429351\pi$$
$$314$$ 124.000 0.0222857
$$315$$ −2576.00 −0.460766
$$316$$ 3080.00 0.548302
$$317$$ −3186.00 −0.564491 −0.282245 0.959342i $$-0.591079\pi$$
−0.282245 + 0.959342i $$0.591079\pi$$
$$318$$ 324.000 0.0571353
$$319$$ 0 0
$$320$$ −2672.00 −0.466779
$$321$$ 488.000 0.0848520
$$322$$ 336.000 0.0581508
$$323$$ −5940.00 −1.02325
$$324$$ −2947.00 −0.505316
$$325$$ −3668.00 −0.626043
$$326$$ 2008.00 0.341144
$$327$$ 180.000 0.0304404
$$328$$ 2730.00 0.459570
$$329$$ 2268.00 0.380057
$$330$$ 0 0
$$331$$ 8672.00 1.44005 0.720025 0.693949i $$-0.244131\pi$$
0.720025 + 0.693949i $$0.244131\pi$$
$$332$$ −9114.00 −1.50661
$$333$$ 5658.00 0.931101
$$334$$ −2884.00 −0.472471
$$335$$ 3904.00 0.636711
$$336$$ −574.000 −0.0931972
$$337$$ −814.000 −0.131577 −0.0657884 0.997834i $$-0.520956\pi$$
−0.0657884 + 0.997834i $$0.520956\pi$$
$$338$$ −1413.00 −0.227388
$$339$$ −2636.00 −0.422324
$$340$$ 6048.00 0.964703
$$341$$ 0 0
$$342$$ −2530.00 −0.400020
$$343$$ 343.000 0.0539949
$$344$$ 1920.00 0.300929
$$345$$ −1536.00 −0.239697
$$346$$ −2228.00 −0.346179
$$347$$ −9344.00 −1.44557 −0.722784 0.691074i $$-0.757138\pi$$
−0.722784 + 0.691074i $$0.757138\pi$$
$$348$$ 1540.00 0.237220
$$349$$ 5180.00 0.794496 0.397248 0.917711i $$-0.369965\pi$$
0.397248 + 0.917711i $$0.369965\pi$$
$$350$$ 917.000 0.140045
$$351$$ −2800.00 −0.425792
$$352$$ 0 0
$$353$$ 12178.0 1.83617 0.918087 0.396379i $$-0.129733\pi$$
0.918087 + 0.396379i $$0.129733\pi$$
$$354$$ −1620.00 −0.243226
$$355$$ −12288.0 −1.83712
$$356$$ −5110.00 −0.760757
$$357$$ 756.000 0.112078
$$358$$ −820.000 −0.121057
$$359$$ −440.000 −0.0646861 −0.0323431 0.999477i $$-0.510297\pi$$
−0.0323431 + 0.999477i $$0.510297\pi$$
$$360$$ 5520.00 0.808138
$$361$$ 5241.00 0.764106
$$362$$ 3892.00 0.565080
$$363$$ 0 0
$$364$$ 1372.00 0.197561
$$365$$ 11232.0 1.61071
$$366$$ −976.000 −0.139389
$$367$$ −9816.00 −1.39616 −0.698080 0.716019i $$-0.745962\pi$$
−0.698080 + 0.716019i $$0.745962\pi$$
$$368$$ 1968.00 0.278775
$$369$$ 4186.00 0.590554
$$370$$ −3936.00 −0.553035
$$371$$ −1134.00 −0.158691
$$372$$ 168.000 0.0234150
$$373$$ 442.000 0.0613563 0.0306781 0.999529i $$-0.490233\pi$$
0.0306781 + 0.999529i $$0.490233\pi$$
$$374$$ 0 0
$$375$$ −192.000 −0.0264396
$$376$$ −4860.00 −0.666583
$$377$$ −3080.00 −0.420764
$$378$$ 700.000 0.0952490
$$379$$ −3960.00 −0.536706 −0.268353 0.963321i $$-0.586479\pi$$
−0.268353 + 0.963321i $$0.586479\pi$$
$$380$$ −12320.0 −1.66316
$$381$$ −3552.00 −0.477623
$$382$$ −5048.00 −0.676121
$$383$$ 6708.00 0.894942 0.447471 0.894298i $$-0.352325\pi$$
0.447471 + 0.894298i $$0.352325\pi$$
$$384$$ 2910.00 0.386720
$$385$$ 0 0
$$386$$ 2962.00 0.390575
$$387$$ 2944.00 0.386697
$$388$$ −2058.00 −0.269276
$$389$$ −13350.0 −1.74003 −0.870015 0.493025i $$-0.835891\pi$$
−0.870015 + 0.493025i $$0.835891\pi$$
$$390$$ 896.000 0.116335
$$391$$ −2592.00 −0.335251
$$392$$ −735.000 −0.0947018
$$393$$ −2236.00 −0.287001
$$394$$ −3334.00 −0.426306
$$395$$ −7040.00 −0.896762
$$396$$ 0 0
$$397$$ −1356.00 −0.171425 −0.0857125 0.996320i $$-0.527317\pi$$
−0.0857125 + 0.996320i $$0.527317\pi$$
$$398$$ 1860.00 0.234255
$$399$$ −1540.00 −0.193224
$$400$$ 5371.00 0.671375
$$401$$ 6222.00 0.774843 0.387421 0.921903i $$-0.373366\pi$$
0.387421 + 0.921903i $$0.373366\pi$$
$$402$$ −488.000 −0.0605453
$$403$$ −336.000 −0.0415319
$$404$$ −4816.00 −0.593082
$$405$$ 6736.00 0.826456
$$406$$ 770.000 0.0941243
$$407$$ 0 0
$$408$$ −1620.00 −0.196573
$$409$$ −5150.00 −0.622619 −0.311309 0.950309i $$-0.600768\pi$$
−0.311309 + 0.950309i $$0.600768\pi$$
$$410$$ −2912.00 −0.350764
$$411$$ −4548.00 −0.545830
$$412$$ −9716.00 −1.16183
$$413$$ 5670.00 0.675551
$$414$$ −1104.00 −0.131060
$$415$$ 20832.0 2.46410
$$416$$ −4508.00 −0.531305
$$417$$ −420.000 −0.0493225
$$418$$ 0 0
$$419$$ 2310.00 0.269334 0.134667 0.990891i $$-0.457004\pi$$
0.134667 + 0.990891i $$0.457004\pi$$
$$420$$ 1568.00 0.182168
$$421$$ 1262.00 0.146095 0.0730476 0.997328i $$-0.476727\pi$$
0.0730476 + 0.997328i $$0.476727\pi$$
$$422$$ 4268.00 0.492329
$$423$$ −7452.00 −0.856569
$$424$$ 2430.00 0.278328
$$425$$ −7074.00 −0.807387
$$426$$ 1536.00 0.174694
$$427$$ 3416.00 0.387147
$$428$$ 1708.00 0.192896
$$429$$ 0 0
$$430$$ −2048.00 −0.229682
$$431$$ 4488.00 0.501576 0.250788 0.968042i $$-0.419310\pi$$
0.250788 + 0.968042i $$0.419310\pi$$
$$432$$ 4100.00 0.456623
$$433$$ 17038.0 1.89098 0.945490 0.325652i $$-0.105584\pi$$
0.945490 + 0.325652i $$0.105584\pi$$
$$434$$ 84.0000 0.00929062
$$435$$ −3520.00 −0.387979
$$436$$ 630.000 0.0692008
$$437$$ 5280.00 0.577979
$$438$$ −1404.00 −0.153164
$$439$$ −16200.0 −1.76124 −0.880619 0.473824i $$-0.842873\pi$$
−0.880619 + 0.473824i $$0.842873\pi$$
$$440$$ 0 0
$$441$$ −1127.00 −0.121693
$$442$$ 1512.00 0.162712
$$443$$ −8772.00 −0.940791 −0.470395 0.882456i $$-0.655889\pi$$
−0.470395 + 0.882456i $$0.655889\pi$$
$$444$$ −3444.00 −0.368119
$$445$$ 11680.0 1.24424
$$446$$ −5432.00 −0.576710
$$447$$ −4020.00 −0.425368
$$448$$ −1169.00 −0.123281
$$449$$ 2130.00 0.223877 0.111939 0.993715i $$-0.464294\pi$$
0.111939 + 0.993715i $$0.464294\pi$$
$$450$$ −3013.00 −0.315632
$$451$$ 0 0
$$452$$ −9226.00 −0.960076
$$453$$ 2224.00 0.230668
$$454$$ 2046.00 0.211506
$$455$$ −3136.00 −0.323116
$$456$$ 3300.00 0.338896
$$457$$ −10534.0 −1.07825 −0.539124 0.842226i $$-0.681245\pi$$
−0.539124 + 0.842226i $$0.681245\pi$$
$$458$$ −2980.00 −0.304031
$$459$$ −5400.00 −0.549129
$$460$$ −5376.00 −0.544907
$$461$$ 9268.00 0.936342 0.468171 0.883638i $$-0.344913\pi$$
0.468171 + 0.883638i $$0.344913\pi$$
$$462$$ 0 0
$$463$$ −9392.00 −0.942728 −0.471364 0.881939i $$-0.656238\pi$$
−0.471364 + 0.881939i $$0.656238\pi$$
$$464$$ 4510.00 0.451232
$$465$$ −384.000 −0.0382959
$$466$$ −4458.00 −0.443161
$$467$$ −10806.0 −1.07075 −0.535377 0.844613i $$-0.679830\pi$$
−0.535377 + 0.844613i $$0.679830\pi$$
$$468$$ −4508.00 −0.445261
$$469$$ 1708.00 0.168162
$$470$$ 5184.00 0.508766
$$471$$ −248.000 −0.0242616
$$472$$ −12150.0 −1.18485
$$473$$ 0 0
$$474$$ 880.000 0.0852737
$$475$$ 14410.0 1.39195
$$476$$ 2646.00 0.254788
$$477$$ 3726.00 0.357656
$$478$$ −4440.00 −0.424855
$$479$$ −4940.00 −0.471220 −0.235610 0.971848i $$-0.575709\pi$$
−0.235610 + 0.971848i $$0.575709\pi$$
$$480$$ −5152.00 −0.489907
$$481$$ 6888.00 0.652943
$$482$$ −3302.00 −0.312037
$$483$$ −672.000 −0.0633065
$$484$$ 0 0
$$485$$ 4704.00 0.440407
$$486$$ −3542.00 −0.330593
$$487$$ −5216.00 −0.485338 −0.242669 0.970109i $$-0.578023\pi$$
−0.242669 + 0.970109i $$0.578023\pi$$
$$488$$ −7320.00 −0.679018
$$489$$ −4016.00 −0.371390
$$490$$ 784.000 0.0722806
$$491$$ −4412.00 −0.405521 −0.202760 0.979228i $$-0.564991\pi$$
−0.202760 + 0.979228i $$0.564991\pi$$
$$492$$ −2548.00 −0.233481
$$493$$ −5940.00 −0.542645
$$494$$ −3080.00 −0.280518
$$495$$ 0 0
$$496$$ 492.000 0.0445392
$$497$$ −5376.00 −0.485204
$$498$$ −2604.00 −0.234313
$$499$$ 19060.0 1.70991 0.854953 0.518706i $$-0.173586\pi$$
0.854953 + 0.518706i $$0.173586\pi$$
$$500$$ −672.000 −0.0601055
$$501$$ 5768.00 0.514362
$$502$$ 1582.00 0.140654
$$503$$ −12768.0 −1.13180 −0.565902 0.824473i $$-0.691472\pi$$
−0.565902 + 0.824473i $$0.691472\pi$$
$$504$$ 2415.00 0.213438
$$505$$ 11008.0 0.969999
$$506$$ 0 0
$$507$$ 2826.00 0.247548
$$508$$ −12432.0 −1.08579
$$509$$ −5500.00 −0.478945 −0.239473 0.970903i $$-0.576975\pi$$
−0.239473 + 0.970903i $$0.576975\pi$$
$$510$$ 1728.00 0.150034
$$511$$ 4914.00 0.425406
$$512$$ 11521.0 0.994455
$$513$$ 11000.0 0.946709
$$514$$ 2354.00 0.202005
$$515$$ 22208.0 1.90020
$$516$$ −1792.00 −0.152884
$$517$$ 0 0
$$518$$ −1722.00 −0.146062
$$519$$ 4456.00 0.376872
$$520$$ 6720.00 0.566714
$$521$$ −7338.00 −0.617051 −0.308526 0.951216i $$-0.599836\pi$$
−0.308526 + 0.951216i $$0.599836\pi$$
$$522$$ −2530.00 −0.212136
$$523$$ 17582.0 1.46999 0.734997 0.678070i $$-0.237183\pi$$
0.734997 + 0.678070i $$0.237183\pi$$
$$524$$ −7826.00 −0.652444
$$525$$ −1834.00 −0.152462
$$526$$ 3872.00 0.320964
$$527$$ −648.000 −0.0535623
$$528$$ 0 0
$$529$$ −9863.00 −0.810635
$$530$$ −2592.00 −0.212433
$$531$$ −18630.0 −1.52255
$$532$$ −5390.00 −0.439260
$$533$$ 5096.00 0.414132
$$534$$ −1460.00 −0.118315
$$535$$ −3904.00 −0.315485
$$536$$ −3660.00 −0.294940
$$537$$ 1640.00 0.131790
$$538$$ 180.000 0.0144244
$$539$$ 0 0
$$540$$ −11200.0 −0.892539
$$541$$ 1618.00 0.128583 0.0642914 0.997931i $$-0.479521\pi$$
0.0642914 + 0.997931i $$0.479521\pi$$
$$542$$ −2032.00 −0.161037
$$543$$ −7784.00 −0.615181
$$544$$ −8694.00 −0.685206
$$545$$ −1440.00 −0.113179
$$546$$ 392.000 0.0307254
$$547$$ −16144.0 −1.26192 −0.630958 0.775817i $$-0.717338\pi$$
−0.630958 + 0.775817i $$0.717338\pi$$
$$548$$ −15918.0 −1.24085
$$549$$ −11224.0 −0.872548
$$550$$ 0 0
$$551$$ 12100.0 0.935531
$$552$$ 1440.00 0.111033
$$553$$ −3080.00 −0.236844
$$554$$ 5426.00 0.416117
$$555$$ 7872.00 0.602068
$$556$$ −1470.00 −0.112126
$$557$$ −4654.00 −0.354033 −0.177016 0.984208i $$-0.556645\pi$$
−0.177016 + 0.984208i $$0.556645\pi$$
$$558$$ −276.000 −0.0209391
$$559$$ 3584.00 0.271175
$$560$$ 4592.00 0.346513
$$561$$ 0 0
$$562$$ −842.000 −0.0631986
$$563$$ −10078.0 −0.754418 −0.377209 0.926128i $$-0.623116\pi$$
−0.377209 + 0.926128i $$0.623116\pi$$
$$564$$ 4536.00 0.338653
$$565$$ 21088.0 1.57023
$$566$$ 3782.00 0.280865
$$567$$ 2947.00 0.218276
$$568$$ 11520.0 0.851001
$$569$$ 5930.00 0.436904 0.218452 0.975848i $$-0.429899\pi$$
0.218452 + 0.975848i $$0.429899\pi$$
$$570$$ −3520.00 −0.258661
$$571$$ 19048.0 1.39603 0.698016 0.716082i $$-0.254067\pi$$
0.698016 + 0.716082i $$0.254067\pi$$
$$572$$ 0 0
$$573$$ 10096.0 0.736067
$$574$$ −1274.00 −0.0926406
$$575$$ 6288.00 0.456048
$$576$$ 3841.00 0.277850
$$577$$ −14366.0 −1.03651 −0.518253 0.855227i $$-0.673418\pi$$
−0.518253 + 0.855227i $$0.673418\pi$$
$$578$$ −1997.00 −0.143710
$$579$$ −5924.00 −0.425204
$$580$$ −12320.0 −0.882000
$$581$$ 9114.00 0.650796
$$582$$ −588.000 −0.0418787
$$583$$ 0 0
$$584$$ −10530.0 −0.746121
$$585$$ 10304.0 0.728236
$$586$$ 4312.00 0.303971
$$587$$ −3626.00 −0.254959 −0.127480 0.991841i $$-0.540689\pi$$
−0.127480 + 0.991841i $$0.540689\pi$$
$$588$$ 686.000 0.0481125
$$589$$ 1320.00 0.0923424
$$590$$ 12960.0 0.904330
$$591$$ 6668.00 0.464103
$$592$$ −10086.0 −0.700223
$$593$$ 1062.00 0.0735432 0.0367716 0.999324i $$-0.488293\pi$$
0.0367716 + 0.999324i $$0.488293\pi$$
$$594$$ 0 0
$$595$$ −6048.00 −0.416712
$$596$$ −14070.0 −0.966996
$$597$$ −3720.00 −0.255024
$$598$$ −1344.00 −0.0919068
$$599$$ −10200.0 −0.695761 −0.347880 0.937539i $$-0.613098\pi$$
−0.347880 + 0.937539i $$0.613098\pi$$
$$600$$ 3930.00 0.267403
$$601$$ 25158.0 1.70751 0.853757 0.520671i $$-0.174318\pi$$
0.853757 + 0.520671i $$0.174318\pi$$
$$602$$ −896.000 −0.0606615
$$603$$ −5612.00 −0.379002
$$604$$ 7784.00 0.524382
$$605$$ 0 0
$$606$$ −1376.00 −0.0922379
$$607$$ −25664.0 −1.71609 −0.858047 0.513570i $$-0.828323\pi$$
−0.858047 + 0.513570i $$0.828323\pi$$
$$608$$ 17710.0 1.18131
$$609$$ −1540.00 −0.102470
$$610$$ 7808.00 0.518257
$$611$$ −9072.00 −0.600677
$$612$$ −8694.00 −0.574239
$$613$$ −19018.0 −1.25307 −0.626533 0.779395i $$-0.715527\pi$$
−0.626533 + 0.779395i $$0.715527\pi$$
$$614$$ −2674.00 −0.175755
$$615$$ 5824.00 0.381864
$$616$$ 0 0
$$617$$ 17334.0 1.13102 0.565511 0.824741i $$-0.308679\pi$$
0.565511 + 0.824741i $$0.308679\pi$$
$$618$$ −2776.00 −0.180691
$$619$$ 18730.0 1.21619 0.608096 0.793864i $$-0.291934\pi$$
0.608096 + 0.793864i $$0.291934\pi$$
$$620$$ −1344.00 −0.0870586
$$621$$ 4800.00 0.310173
$$622$$ −3768.00 −0.242899
$$623$$ 5110.00 0.328616
$$624$$ 2296.00 0.147297
$$625$$ −14839.0 −0.949696
$$626$$ 2438.00 0.155658
$$627$$ 0 0
$$628$$ −868.000 −0.0551544
$$629$$ 13284.0 0.842079
$$630$$ −2576.00 −0.162905
$$631$$ −6928.00 −0.437083 −0.218541 0.975828i $$-0.570130\pi$$
−0.218541 + 0.975828i $$0.570130\pi$$
$$632$$ 6600.00 0.415402
$$633$$ −8536.00 −0.535980
$$634$$ −3186.00 −0.199578
$$635$$ 28416.0 1.77583
$$636$$ −2268.00 −0.141403
$$637$$ −1372.00 −0.0853385
$$638$$ 0 0
$$639$$ 17664.0 1.09355
$$640$$ −23280.0 −1.43785
$$641$$ 16302.0 1.00451 0.502255 0.864720i $$-0.332504\pi$$
0.502255 + 0.864720i $$0.332504\pi$$
$$642$$ 488.000 0.0299997
$$643$$ 4718.00 0.289362 0.144681 0.989478i $$-0.453784\pi$$
0.144681 + 0.989478i $$0.453784\pi$$
$$644$$ −2352.00 −0.143916
$$645$$ 4096.00 0.250046
$$646$$ −5940.00 −0.361774
$$647$$ −21436.0 −1.30253 −0.651264 0.758851i $$-0.725761\pi$$
−0.651264 + 0.758851i $$0.725761\pi$$
$$648$$ −6315.00 −0.382834
$$649$$ 0 0
$$650$$ −3668.00 −0.221340
$$651$$ −168.000 −0.0101143
$$652$$ −14056.0 −0.844287
$$653$$ 4458.00 0.267159 0.133580 0.991038i $$-0.457353\pi$$
0.133580 + 0.991038i $$0.457353\pi$$
$$654$$ 180.000 0.0107623
$$655$$ 17888.0 1.06709
$$656$$ −7462.00 −0.444119
$$657$$ −16146.0 −0.958775
$$658$$ 2268.00 0.134371
$$659$$ 26640.0 1.57473 0.787365 0.616487i $$-0.211445\pi$$
0.787365 + 0.616487i $$0.211445\pi$$
$$660$$ 0 0
$$661$$ 7432.00 0.437324 0.218662 0.975801i $$-0.429831\pi$$
0.218662 + 0.975801i $$0.429831\pi$$
$$662$$ 8672.00 0.509134
$$663$$ −3024.00 −0.177138
$$664$$ −19530.0 −1.14143
$$665$$ 12320.0 0.718420
$$666$$ 5658.00 0.329194
$$667$$ 5280.00 0.306510
$$668$$ 20188.0 1.16931
$$669$$ 10864.0 0.627842
$$670$$ 3904.00 0.225111
$$671$$ 0 0
$$672$$ −2254.00 −0.129390
$$673$$ −58.0000 −0.00332204 −0.00166102 0.999999i $$-0.500529\pi$$
−0.00166102 + 0.999999i $$0.500529\pi$$
$$674$$ −814.000 −0.0465194
$$675$$ 13100.0 0.746991
$$676$$ 9891.00 0.562756
$$677$$ 21516.0 1.22146 0.610729 0.791840i $$-0.290876\pi$$
0.610729 + 0.791840i $$0.290876\pi$$
$$678$$ −2636.00 −0.149314
$$679$$ 2058.00 0.116316
$$680$$ 12960.0 0.730873
$$681$$ −4092.00 −0.230258
$$682$$ 0 0
$$683$$ 18108.0 1.01447 0.507235 0.861808i $$-0.330668\pi$$
0.507235 + 0.861808i $$0.330668\pi$$
$$684$$ 17710.0 0.989998
$$685$$ 36384.0 2.02943
$$686$$ 343.000 0.0190901
$$687$$ 5960.00 0.330987
$$688$$ −5248.00 −0.290811
$$689$$ 4536.00 0.250810
$$690$$ −1536.00 −0.0847457
$$691$$ −10078.0 −0.554827 −0.277413 0.960751i $$-0.589477\pi$$
−0.277413 + 0.960751i $$0.589477\pi$$
$$692$$ 15596.0 0.856750
$$693$$ 0 0
$$694$$ −9344.00 −0.511086
$$695$$ 3360.00 0.183384
$$696$$ 3300.00 0.179722
$$697$$ 9828.00 0.534092
$$698$$ 5180.00 0.280897
$$699$$ 8916.00 0.482452
$$700$$ −6419.00 −0.346593
$$701$$ −18762.0 −1.01089 −0.505443 0.862860i $$-0.668671\pi$$
−0.505443 + 0.862860i $$0.668671\pi$$
$$702$$ −2800.00 −0.150540
$$703$$ −27060.0 −1.45176
$$704$$ 0 0
$$705$$ −10368.0 −0.553874
$$706$$ 12178.0 0.649186
$$707$$ 4816.00 0.256187
$$708$$ 11340.0 0.601954
$$709$$ 6810.00 0.360726 0.180363 0.983600i $$-0.442273\pi$$
0.180363 + 0.983600i $$0.442273\pi$$
$$710$$ −12288.0 −0.649522
$$711$$ 10120.0 0.533797
$$712$$ −10950.0 −0.576360
$$713$$ 576.000 0.0302544
$$714$$ 756.000 0.0396255
$$715$$ 0 0
$$716$$ 5740.00 0.299600
$$717$$ 8880.00 0.462524
$$718$$ −440.000 −0.0228700
$$719$$ 4860.00 0.252083 0.126041 0.992025i $$-0.459773\pi$$
0.126041 + 0.992025i $$0.459773\pi$$
$$720$$ −15088.0 −0.780967
$$721$$ 9716.00 0.501862
$$722$$ 5241.00 0.270152
$$723$$ 6604.00 0.339703
$$724$$ −27244.0 −1.39850
$$725$$ 14410.0 0.738171
$$726$$ 0 0
$$727$$ −13636.0 −0.695641 −0.347821 0.937561i $$-0.613078\pi$$
−0.347821 + 0.937561i $$0.613078\pi$$
$$728$$ 2940.00 0.149675
$$729$$ −4283.00 −0.217599
$$730$$ 11232.0 0.569473
$$731$$ 6912.00 0.349726
$$732$$ 6832.00 0.344970
$$733$$ −2088.00 −0.105214 −0.0526071 0.998615i $$-0.516753\pi$$
−0.0526071 + 0.998615i $$0.516753\pi$$
$$734$$ −9816.00 −0.493617
$$735$$ −1568.00 −0.0786892
$$736$$ 7728.00 0.387035
$$737$$ 0 0
$$738$$ 4186.00 0.208792
$$739$$ 5160.00 0.256852 0.128426 0.991719i $$-0.459008\pi$$
0.128426 + 0.991719i $$0.459008\pi$$
$$740$$ 27552.0 1.36869
$$741$$ 6160.00 0.305389
$$742$$ −1134.00 −0.0561057
$$743$$ 28152.0 1.39004 0.695018 0.718992i $$-0.255396\pi$$
0.695018 + 0.718992i $$0.255396\pi$$
$$744$$ 360.000 0.0177396
$$745$$ 32160.0 1.58155
$$746$$ 442.000 0.0216927
$$747$$ −29946.0 −1.46676
$$748$$ 0 0
$$749$$ −1708.00 −0.0833230
$$750$$ −192.000 −0.00934780
$$751$$ −16808.0 −0.816688 −0.408344 0.912828i $$-0.633894\pi$$
−0.408344 + 0.912828i $$0.633894\pi$$
$$752$$ 13284.0 0.644172
$$753$$ −3164.00 −0.153124
$$754$$ −3080.00 −0.148763
$$755$$ −17792.0 −0.857639
$$756$$ −4900.00 −0.235729
$$757$$ 21674.0 1.04063 0.520314 0.853975i $$-0.325815\pi$$
0.520314 + 0.853975i $$0.325815\pi$$
$$758$$ −3960.00 −0.189754
$$759$$ 0 0
$$760$$ −26400.0 −1.26004
$$761$$ −7422.00 −0.353544 −0.176772 0.984252i $$-0.556566\pi$$
−0.176772 + 0.984252i $$0.556566\pi$$
$$762$$ −3552.00 −0.168865
$$763$$ −630.000 −0.0298919
$$764$$ 35336.0 1.67331
$$765$$ 19872.0 0.939181
$$766$$ 6708.00 0.316410
$$767$$ −22680.0 −1.06770
$$768$$ 238.000 0.0111824
$$769$$ −13790.0 −0.646658 −0.323329 0.946287i $$-0.604802\pi$$
−0.323329 + 0.946287i $$0.604802\pi$$
$$770$$ 0 0
$$771$$ −4708.00 −0.219915
$$772$$ −20734.0 −0.966623
$$773$$ −6232.00 −0.289973 −0.144987 0.989434i $$-0.546314\pi$$
−0.144987 + 0.989434i $$0.546314\pi$$
$$774$$ 2944.00 0.136718
$$775$$ 1572.00 0.0728618
$$776$$ −4410.00 −0.204007
$$777$$ 3444.00 0.159013
$$778$$ −13350.0 −0.615194
$$779$$ −20020.0 −0.920784
$$780$$ −6272.00 −0.287915
$$781$$ 0 0
$$782$$ −2592.00 −0.118529
$$783$$ 11000.0 0.502054
$$784$$ 2009.00 0.0915179
$$785$$ 1984.00 0.0902064
$$786$$ −2236.00 −0.101470
$$787$$ 1766.00 0.0799887 0.0399943 0.999200i $$-0.487266\pi$$
0.0399943 + 0.999200i $$0.487266\pi$$
$$788$$ 23338.0 1.05505
$$789$$ −7744.00 −0.349422
$$790$$ −7040.00 −0.317053
$$791$$ 9226.00 0.414714
$$792$$ 0 0
$$793$$ −13664.0 −0.611883
$$794$$ −1356.00 −0.0606079
$$795$$ 5184.00 0.231267
$$796$$ −13020.0 −0.579751
$$797$$ 1204.00 0.0535105 0.0267552 0.999642i $$-0.491483\pi$$
0.0267552 + 0.999642i $$0.491483\pi$$
$$798$$ −1540.00 −0.0683150
$$799$$ −17496.0 −0.774673
$$800$$ 21091.0 0.932099
$$801$$ −16790.0 −0.740631
$$802$$ 6222.00 0.273948
$$803$$ 0 0
$$804$$ 3416.00 0.149842
$$805$$ 5376.00 0.235378
$$806$$ −336.000 −0.0146837
$$807$$ −360.000 −0.0157033
$$808$$ −10320.0 −0.449327
$$809$$ 7050.00 0.306384 0.153192 0.988196i $$-0.451045\pi$$
0.153192 + 0.988196i $$0.451045\pi$$
$$810$$ 6736.00 0.292196
$$811$$ −23282.0 −1.00807 −0.504033 0.863684i $$-0.668151\pi$$
−0.504033 + 0.863684i $$0.668151\pi$$
$$812$$ −5390.00 −0.232946
$$813$$ 4064.00 0.175315
$$814$$ 0 0
$$815$$ 32128.0 1.38085
$$816$$ 4428.00 0.189964
$$817$$ −14080.0 −0.602934
$$818$$ −5150.00 −0.220129
$$819$$ 4508.00 0.192335
$$820$$ 20384.0 0.868098
$$821$$ −10142.0 −0.431131 −0.215565 0.976489i $$-0.569159\pi$$
−0.215565 + 0.976489i $$0.569159\pi$$
$$822$$ −4548.00 −0.192980
$$823$$ −9192.00 −0.389323 −0.194662 0.980870i $$-0.562361\pi$$
−0.194662 + 0.980870i $$0.562361\pi$$
$$824$$ −20820.0 −0.880217
$$825$$ 0 0
$$826$$ 5670.00 0.238843
$$827$$ 46716.0 1.96430 0.982149 0.188104i $$-0.0602344\pi$$
0.982149 + 0.188104i $$0.0602344\pi$$
$$828$$ 7728.00 0.324356
$$829$$ 11240.0 0.470906 0.235453 0.971886i $$-0.424343\pi$$
0.235453 + 0.971886i $$0.424343\pi$$
$$830$$ 20832.0 0.871192
$$831$$ −10852.0 −0.453010
$$832$$ 4676.00 0.194845
$$833$$ −2646.00 −0.110058
$$834$$ −420.000 −0.0174381
$$835$$ −46144.0 −1.91243
$$836$$ 0 0
$$837$$ 1200.00 0.0495556
$$838$$ 2310.00 0.0952239
$$839$$ 700.000 0.0288042 0.0144021 0.999896i $$-0.495416\pi$$
0.0144021 + 0.999896i $$0.495416\pi$$
$$840$$ 3360.00 0.138013
$$841$$ −12289.0 −0.503875
$$842$$ 1262.00 0.0516525
$$843$$ 1684.00 0.0688019
$$844$$ −29876.0 −1.21845
$$845$$ −22608.0 −0.920401
$$846$$ −7452.00 −0.302843
$$847$$ 0 0
$$848$$ −6642.00 −0.268971
$$849$$ −7564.00 −0.305767
$$850$$ −7074.00 −0.285454
$$851$$ −11808.0 −0.475644
$$852$$ −10752.0 −0.432344
$$853$$ 37492.0 1.50493 0.752463 0.658635i $$-0.228866\pi$$
0.752463 + 0.658635i $$0.228866\pi$$
$$854$$ 3416.00 0.136877
$$855$$ −40480.0 −1.61917
$$856$$ 3660.00 0.146140
$$857$$ −28894.0 −1.15169 −0.575846 0.817558i $$-0.695327\pi$$
−0.575846 + 0.817558i $$0.695327\pi$$
$$858$$ 0 0
$$859$$ −2770.00 −0.110025 −0.0550123 0.998486i $$-0.517520\pi$$
−0.0550123 + 0.998486i $$0.517520\pi$$
$$860$$ 14336.0 0.568434
$$861$$ 2548.00 0.100854
$$862$$ 4488.00 0.177334
$$863$$ 17688.0 0.697690 0.348845 0.937180i $$-0.386574\pi$$
0.348845 + 0.937180i $$0.386574\pi$$
$$864$$ 16100.0 0.633950
$$865$$ −35648.0 −1.40124
$$866$$ 17038.0 0.668562
$$867$$ 3994.00 0.156451
$$868$$ −588.000 −0.0229931
$$869$$ 0 0
$$870$$ −3520.00 −0.137171
$$871$$ −6832.00 −0.265779
$$872$$ 1350.00 0.0524275
$$873$$ −6762.00 −0.262152
$$874$$ 5280.00 0.204346
$$875$$ 672.000 0.0259631
$$876$$ 9828.00 0.379061
$$877$$ 33566.0 1.29241 0.646205 0.763164i $$-0.276355\pi$$
0.646205 + 0.763164i $$0.276355\pi$$
$$878$$ −16200.0 −0.622692
$$879$$ −8624.00 −0.330922
$$880$$ 0 0
$$881$$ −16758.0 −0.640853 −0.320426 0.947273i $$-0.603826\pi$$
−0.320426 + 0.947273i $$0.603826\pi$$
$$882$$ −1127.00 −0.0430250
$$883$$ 11468.0 0.437066 0.218533 0.975830i $$-0.429873\pi$$
0.218533 + 0.975830i $$0.429873\pi$$
$$884$$ −10584.0 −0.402691
$$885$$ −25920.0 −0.984510
$$886$$ −8772.00 −0.332620
$$887$$ 50356.0 1.90619 0.953094 0.302674i $$-0.0978793\pi$$
0.953094 + 0.302674i $$0.0978793\pi$$
$$888$$ −7380.00 −0.278893
$$889$$ 12432.0 0.469017
$$890$$ 11680.0 0.439904
$$891$$ 0 0
$$892$$ 38024.0 1.42728
$$893$$ 35640.0 1.33555
$$894$$ −4020.00 −0.150390
$$895$$ −13120.0 −0.490004
$$896$$ −10185.0 −0.379751
$$897$$ 2688.00 0.100055
$$898$$ 2130.00 0.0791526
$$899$$ 1320.00 0.0489705
$$900$$ 21091.0 0.781148
$$901$$ 8748.00 0.323461
$$902$$ 0 0
$$903$$ 1792.00 0.0660399
$$904$$ −19770.0 −0.727368
$$905$$ 62272.0 2.28728
$$906$$ 2224.00 0.0815535
$$907$$ −8716.00 −0.319085 −0.159542 0.987191i $$-0.551002\pi$$
−0.159542 + 0.987191i $$0.551002\pi$$
$$908$$ −14322.0 −0.523450
$$909$$ −15824.0 −0.577392
$$910$$ −3136.00 −0.114239
$$911$$ 7632.00 0.277563 0.138781 0.990323i $$-0.455682\pi$$
0.138781 + 0.990323i $$0.455682\pi$$
$$912$$ −9020.00 −0.327502
$$913$$ 0 0
$$914$$ −10534.0 −0.381219
$$915$$ −15616.0 −0.564207
$$916$$ 20860.0 0.752439
$$917$$ 7826.00 0.281829
$$918$$ −5400.00 −0.194147
$$919$$ 23080.0 0.828443 0.414221 0.910176i $$-0.364054\pi$$
0.414221 + 0.910176i $$0.364054\pi$$
$$920$$ −11520.0 −0.412830
$$921$$ 5348.00 0.191338
$$922$$ 9268.00 0.331047
$$923$$ 21504.0 0.766861
$$924$$ 0 0
$$925$$ −32226.0 −1.14550
$$926$$ −9392.00 −0.333305
$$927$$ −31924.0 −1.13109
$$928$$ 17710.0 0.626465
$$929$$ 45110.0 1.59312 0.796561 0.604558i $$-0.206650\pi$$
0.796561 + 0.604558i $$0.206650\pi$$
$$930$$ −384.000 −0.0135396
$$931$$ 5390.00 0.189742
$$932$$ 31206.0 1.09677
$$933$$ 7536.00 0.264435
$$934$$ −10806.0 −0.378569
$$935$$ 0 0
$$936$$ −9660.00 −0.337337
$$937$$ −16674.0 −0.581340 −0.290670 0.956823i $$-0.593878\pi$$
−0.290670 + 0.956823i $$0.593878\pi$$
$$938$$ 1708.00 0.0594543
$$939$$ −4876.00 −0.169459
$$940$$ −36288.0 −1.25913
$$941$$ −43832.0 −1.51847 −0.759236 0.650815i $$-0.774427\pi$$
−0.759236 + 0.650815i $$0.774427\pi$$
$$942$$ −248.000 −0.00857779
$$943$$ −8736.00 −0.301679
$$944$$ 33210.0 1.14501
$$945$$ 11200.0 0.385541
$$946$$ 0 0
$$947$$ −736.000 −0.0252553 −0.0126277 0.999920i $$-0.504020\pi$$
−0.0126277 + 0.999920i $$0.504020\pi$$
$$948$$ −6160.00 −0.211042
$$949$$ −19656.0 −0.672351
$$950$$ 14410.0 0.492129
$$951$$ 6372.00 0.217273
$$952$$ 5670.00 0.193031
$$953$$ −38138.0 −1.29634 −0.648169 0.761496i $$-0.724465\pi$$
−0.648169 + 0.761496i $$0.724465\pi$$
$$954$$ 3726.00 0.126450
$$955$$ −80768.0 −2.73674
$$956$$ 31080.0 1.05146
$$957$$ 0 0
$$958$$ −4940.00 −0.166601
$$959$$ 15918.0 0.535995
$$960$$ 5344.00 0.179663
$$961$$ −29647.0 −0.995166
$$962$$ 6888.00 0.230850
$$963$$ 5612.00 0.187792
$$964$$ 23114.0 0.772253
$$965$$ 47392.0 1.58094
$$966$$ −672.000 −0.0223822
$$967$$ −26224.0 −0.872086 −0.436043 0.899926i $$-0.643620\pi$$
−0.436043 + 0.899926i $$0.643620\pi$$
$$968$$ 0 0
$$969$$ 11880.0 0.393850
$$970$$ 4704.00 0.155708
$$971$$ 18762.0 0.620084 0.310042 0.950723i $$-0.399657\pi$$
0.310042 + 0.950723i $$0.399657\pi$$
$$972$$ 24794.0 0.818177
$$973$$ 1470.00 0.0484337
$$974$$ −5216.00 −0.171593
$$975$$ 7336.00 0.240964
$$976$$ 20008.0 0.656189
$$977$$ 38394.0 1.25725 0.628625 0.777709i $$-0.283618\pi$$
0.628625 + 0.777709i $$0.283618\pi$$
$$978$$ −4016.00 −0.131306
$$979$$ 0 0
$$980$$ −5488.00 −0.178885
$$981$$ 2070.00 0.0673700
$$982$$ −4412.00 −0.143373
$$983$$ 5388.00 0.174822 0.0874112 0.996172i $$-0.472141\pi$$
0.0874112 + 0.996172i $$0.472141\pi$$
$$984$$ −5460.00 −0.176889
$$985$$ −53344.0 −1.72556
$$986$$ −5940.00 −0.191854
$$987$$ −4536.00 −0.146284
$$988$$ 21560.0 0.694246
$$989$$ −6144.00 −0.197541
$$990$$ 0 0
$$991$$ 25472.0 0.816493 0.408247 0.912872i $$-0.366140\pi$$
0.408247 + 0.912872i $$0.366140\pi$$
$$992$$ 1932.00 0.0618357
$$993$$ −17344.0 −0.554275
$$994$$ −5376.00 −0.171546
$$995$$ 29760.0 0.948196
$$996$$ 18228.0 0.579896
$$997$$ 17096.0 0.543065 0.271532 0.962429i $$-0.412470\pi$$
0.271532 + 0.962429i $$0.412470\pi$$
$$998$$ 19060.0 0.604543
$$999$$ −24600.0 −0.779089
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 847.4.a.b.1.1 1
11.10 odd 2 7.4.a.a.1.1 1
33.32 even 2 63.4.a.b.1.1 1
44.43 even 2 112.4.a.f.1.1 1
55.32 even 4 175.4.b.b.99.1 2
55.43 even 4 175.4.b.b.99.2 2
55.54 odd 2 175.4.a.b.1.1 1
77.10 even 6 49.4.c.b.30.1 2
77.32 odd 6 49.4.c.c.30.1 2
77.54 even 6 49.4.c.b.18.1 2
77.65 odd 6 49.4.c.c.18.1 2
77.76 even 2 49.4.a.b.1.1 1
88.21 odd 2 448.4.a.i.1.1 1
88.43 even 2 448.4.a.e.1.1 1
132.131 odd 2 1008.4.a.c.1.1 1
143.142 odd 2 1183.4.a.b.1.1 1
165.164 even 2 1575.4.a.e.1.1 1
187.186 odd 2 2023.4.a.a.1.1 1
231.32 even 6 441.4.e.h.226.1 2
231.65 even 6 441.4.e.h.361.1 2
231.131 odd 6 441.4.e.e.361.1 2
231.164 odd 6 441.4.e.e.226.1 2
231.230 odd 2 441.4.a.i.1.1 1
308.307 odd 2 784.4.a.g.1.1 1
385.384 even 2 1225.4.a.j.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
7.4.a.a.1.1 1 11.10 odd 2
49.4.a.b.1.1 1 77.76 even 2
49.4.c.b.18.1 2 77.54 even 6
49.4.c.b.30.1 2 77.10 even 6
49.4.c.c.18.1 2 77.65 odd 6
49.4.c.c.30.1 2 77.32 odd 6
63.4.a.b.1.1 1 33.32 even 2
112.4.a.f.1.1 1 44.43 even 2
175.4.a.b.1.1 1 55.54 odd 2
175.4.b.b.99.1 2 55.32 even 4
175.4.b.b.99.2 2 55.43 even 4
441.4.a.i.1.1 1 231.230 odd 2
441.4.e.e.226.1 2 231.164 odd 6
441.4.e.e.361.1 2 231.131 odd 6
441.4.e.h.226.1 2 231.32 even 6
441.4.e.h.361.1 2 231.65 even 6
448.4.a.e.1.1 1 88.43 even 2
448.4.a.i.1.1 1 88.21 odd 2
784.4.a.g.1.1 1 308.307 odd 2
847.4.a.b.1.1 1 1.1 even 1 trivial
1008.4.a.c.1.1 1 132.131 odd 2
1183.4.a.b.1.1 1 143.142 odd 2
1225.4.a.j.1.1 1 385.384 even 2
1575.4.a.e.1.1 1 165.164 even 2
2023.4.a.a.1.1 1 187.186 odd 2