# Properties

 Label 5.4.a.a.1.1 Level $5$ Weight $4$ Character 5.1 Self dual yes Analytic conductor $0.295$ 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] = [5,4,Mod(1,5)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("5.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$5$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 5.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: $$0.295009550029$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 5.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-4.00000 q^{2} +2.00000 q^{3} +8.00000 q^{4} -5.00000 q^{5} -8.00000 q^{6} +6.00000 q^{7} -23.0000 q^{9} +O(q^{10})$$ $$q-4.00000 q^{2} +2.00000 q^{3} +8.00000 q^{4} -5.00000 q^{5} -8.00000 q^{6} +6.00000 q^{7} -23.0000 q^{9} +20.0000 q^{10} +32.0000 q^{11} +16.0000 q^{12} -38.0000 q^{13} -24.0000 q^{14} -10.0000 q^{15} -64.0000 q^{16} +26.0000 q^{17} +92.0000 q^{18} +100.000 q^{19} -40.0000 q^{20} +12.0000 q^{21} -128.000 q^{22} -78.0000 q^{23} +25.0000 q^{25} +152.000 q^{26} -100.000 q^{27} +48.0000 q^{28} -50.0000 q^{29} +40.0000 q^{30} -108.000 q^{31} +256.000 q^{32} +64.0000 q^{33} -104.000 q^{34} -30.0000 q^{35} -184.000 q^{36} +266.000 q^{37} -400.000 q^{38} -76.0000 q^{39} +22.0000 q^{41} -48.0000 q^{42} +442.000 q^{43} +256.000 q^{44} +115.000 q^{45} +312.000 q^{46} -514.000 q^{47} -128.000 q^{48} -307.000 q^{49} -100.000 q^{50} +52.0000 q^{51} -304.000 q^{52} +2.00000 q^{53} +400.000 q^{54} -160.000 q^{55} +200.000 q^{57} +200.000 q^{58} +500.000 q^{59} -80.0000 q^{60} -518.000 q^{61} +432.000 q^{62} -138.000 q^{63} -512.000 q^{64} +190.000 q^{65} -256.000 q^{66} +126.000 q^{67} +208.000 q^{68} -156.000 q^{69} +120.000 q^{70} +412.000 q^{71} -878.000 q^{73} -1064.00 q^{74} +50.0000 q^{75} +800.000 q^{76} +192.000 q^{77} +304.000 q^{78} +600.000 q^{79} +320.000 q^{80} +421.000 q^{81} -88.0000 q^{82} +282.000 q^{83} +96.0000 q^{84} -130.000 q^{85} -1768.00 q^{86} -100.000 q^{87} -150.000 q^{89} -460.000 q^{90} -228.000 q^{91} -624.000 q^{92} -216.000 q^{93} +2056.00 q^{94} -500.000 q^{95} +512.000 q^{96} +386.000 q^{97} +1228.00 q^{98} -736.000 q^{99} +O(q^{100})$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −4.00000 −1.41421 −0.707107 0.707107i $$-0.750000\pi$$
−0.707107 + 0.707107i $$0.750000\pi$$
$$3$$ 2.00000 0.384900 0.192450 0.981307i $$-0.438357\pi$$
0.192450 + 0.981307i $$0.438357\pi$$
$$4$$ 8.00000 1.00000
$$5$$ −5.00000 −0.447214
$$6$$ −8.00000 −0.544331
$$7$$ 6.00000 0.323970 0.161985 0.986793i $$-0.448210\pi$$
0.161985 + 0.986793i $$0.448210\pi$$
$$8$$ 0 0
$$9$$ −23.0000 −0.851852
$$10$$ 20.0000 0.632456
$$11$$ 32.0000 0.877124 0.438562 0.898701i $$-0.355488\pi$$
0.438562 + 0.898701i $$0.355488\pi$$
$$12$$ 16.0000 0.384900
$$13$$ −38.0000 −0.810716 −0.405358 0.914158i $$-0.632853\pi$$
−0.405358 + 0.914158i $$0.632853\pi$$
$$14$$ −24.0000 −0.458162
$$15$$ −10.0000 −0.172133
$$16$$ −64.0000 −1.00000
$$17$$ 26.0000 0.370937 0.185468 0.982650i $$-0.440620\pi$$
0.185468 + 0.982650i $$0.440620\pi$$
$$18$$ 92.0000 1.20470
$$19$$ 100.000 1.20745 0.603726 0.797192i $$-0.293682\pi$$
0.603726 + 0.797192i $$0.293682\pi$$
$$20$$ −40.0000 −0.447214
$$21$$ 12.0000 0.124696
$$22$$ −128.000 −1.24044
$$23$$ −78.0000 −0.707136 −0.353568 0.935409i $$-0.615032\pi$$
−0.353568 + 0.935409i $$0.615032\pi$$
$$24$$ 0 0
$$25$$ 25.0000 0.200000
$$26$$ 152.000 1.14653
$$27$$ −100.000 −0.712778
$$28$$ 48.0000 0.323970
$$29$$ −50.0000 −0.320164 −0.160082 0.987104i $$-0.551176\pi$$
−0.160082 + 0.987104i $$0.551176\pi$$
$$30$$ 40.0000 0.243432
$$31$$ −108.000 −0.625722 −0.312861 0.949799i $$-0.601287\pi$$
−0.312861 + 0.949799i $$0.601287\pi$$
$$32$$ 256.000 1.41421
$$33$$ 64.0000 0.337605
$$34$$ −104.000 −0.524584
$$35$$ −30.0000 −0.144884
$$36$$ −184.000 −0.851852
$$37$$ 266.000 1.18190 0.590948 0.806710i $$-0.298754\pi$$
0.590948 + 0.806710i $$0.298754\pi$$
$$38$$ −400.000 −1.70759
$$39$$ −76.0000 −0.312045
$$40$$ 0 0
$$41$$ 22.0000 0.0838006 0.0419003 0.999122i $$-0.486659\pi$$
0.0419003 + 0.999122i $$0.486659\pi$$
$$42$$ −48.0000 −0.176347
$$43$$ 442.000 1.56754 0.783772 0.621049i $$-0.213293\pi$$
0.783772 + 0.621049i $$0.213293\pi$$
$$44$$ 256.000 0.877124
$$45$$ 115.000 0.380960
$$46$$ 312.000 1.00004
$$47$$ −514.000 −1.59520 −0.797602 0.603184i $$-0.793899\pi$$
−0.797602 + 0.603184i $$0.793899\pi$$
$$48$$ −128.000 −0.384900
$$49$$ −307.000 −0.895044
$$50$$ −100.000 −0.282843
$$51$$ 52.0000 0.142774
$$52$$ −304.000 −0.810716
$$53$$ 2.00000 0.00518342 0.00259171 0.999997i $$-0.499175\pi$$
0.00259171 + 0.999997i $$0.499175\pi$$
$$54$$ 400.000 1.00802
$$55$$ −160.000 −0.392262
$$56$$ 0 0
$$57$$ 200.000 0.464748
$$58$$ 200.000 0.452781
$$59$$ 500.000 1.10330 0.551648 0.834077i $$-0.313999\pi$$
0.551648 + 0.834077i $$0.313999\pi$$
$$60$$ −80.0000 −0.172133
$$61$$ −518.000 −1.08726 −0.543632 0.839324i $$-0.682951\pi$$
−0.543632 + 0.839324i $$0.682951\pi$$
$$62$$ 432.000 0.884904
$$63$$ −138.000 −0.275974
$$64$$ −512.000 −1.00000
$$65$$ 190.000 0.362563
$$66$$ −256.000 −0.477446
$$67$$ 126.000 0.229751 0.114876 0.993380i $$-0.463353\pi$$
0.114876 + 0.993380i $$0.463353\pi$$
$$68$$ 208.000 0.370937
$$69$$ −156.000 −0.272177
$$70$$ 120.000 0.204896
$$71$$ 412.000 0.688668 0.344334 0.938847i $$-0.388105\pi$$
0.344334 + 0.938847i $$0.388105\pi$$
$$72$$ 0 0
$$73$$ −878.000 −1.40770 −0.703850 0.710348i $$-0.748537\pi$$
−0.703850 + 0.710348i $$0.748537\pi$$
$$74$$ −1064.00 −1.67145
$$75$$ 50.0000 0.0769800
$$76$$ 800.000 1.20745
$$77$$ 192.000 0.284161
$$78$$ 304.000 0.441298
$$79$$ 600.000 0.854497 0.427249 0.904134i $$-0.359483\pi$$
0.427249 + 0.904134i $$0.359483\pi$$
$$80$$ 320.000 0.447214
$$81$$ 421.000 0.577503
$$82$$ −88.0000 −0.118512
$$83$$ 282.000 0.372934 0.186467 0.982461i $$-0.440296\pi$$
0.186467 + 0.982461i $$0.440296\pi$$
$$84$$ 96.0000 0.124696
$$85$$ −130.000 −0.165888
$$86$$ −1768.00 −2.21684
$$87$$ −100.000 −0.123231
$$88$$ 0 0
$$89$$ −150.000 −0.178651 −0.0893257 0.996002i $$-0.528471\pi$$
−0.0893257 + 0.996002i $$0.528471\pi$$
$$90$$ −460.000 −0.538758
$$91$$ −228.000 −0.262647
$$92$$ −624.000 −0.707136
$$93$$ −216.000 −0.240840
$$94$$ 2056.00 2.25596
$$95$$ −500.000 −0.539989
$$96$$ 512.000 0.544331
$$97$$ 386.000 0.404045 0.202022 0.979381i $$-0.435249\pi$$
0.202022 + 0.979381i $$0.435249\pi$$
$$98$$ 1228.00 1.26578
$$99$$ −736.000 −0.747180
$$100$$ 200.000 0.200000
$$101$$ 702.000 0.691600 0.345800 0.938308i $$-0.387608\pi$$
0.345800 + 0.938308i $$0.387608\pi$$
$$102$$ −208.000 −0.201912
$$103$$ −598.000 −0.572065 −0.286032 0.958220i $$-0.592337\pi$$
−0.286032 + 0.958220i $$0.592337\pi$$
$$104$$ 0 0
$$105$$ −60.0000 −0.0557657
$$106$$ −8.00000 −0.00733046
$$107$$ −1194.00 −1.07877 −0.539385 0.842059i $$-0.681343\pi$$
−0.539385 + 0.842059i $$0.681343\pi$$
$$108$$ −800.000 −0.712778
$$109$$ −550.000 −0.483307 −0.241653 0.970363i $$-0.577690\pi$$
−0.241653 + 0.970363i $$0.577690\pi$$
$$110$$ 640.000 0.554742
$$111$$ 532.000 0.454912
$$112$$ −384.000 −0.323970
$$113$$ 1562.00 1.30036 0.650180 0.759781i $$-0.274694\pi$$
0.650180 + 0.759781i $$0.274694\pi$$
$$114$$ −800.000 −0.657253
$$115$$ 390.000 0.316241
$$116$$ −400.000 −0.320164
$$117$$ 874.000 0.690610
$$118$$ −2000.00 −1.56030
$$119$$ 156.000 0.120172
$$120$$ 0 0
$$121$$ −307.000 −0.230654
$$122$$ 2072.00 1.53762
$$123$$ 44.0000 0.0322548
$$124$$ −864.000 −0.625722
$$125$$ −125.000 −0.0894427
$$126$$ 552.000 0.390286
$$127$$ 1846.00 1.28981 0.644906 0.764262i $$-0.276897\pi$$
0.644906 + 0.764262i $$0.276897\pi$$
$$128$$ 0 0
$$129$$ 884.000 0.603348
$$130$$ −760.000 −0.512742
$$131$$ −2208.00 −1.47262 −0.736312 0.676642i $$-0.763435\pi$$
−0.736312 + 0.676642i $$0.763435\pi$$
$$132$$ 512.000 0.337605
$$133$$ 600.000 0.391177
$$134$$ −504.000 −0.324918
$$135$$ 500.000 0.318764
$$136$$ 0 0
$$137$$ −2334.00 −1.45553 −0.727763 0.685829i $$-0.759440\pi$$
−0.727763 + 0.685829i $$0.759440\pi$$
$$138$$ 624.000 0.384916
$$139$$ −700.000 −0.427146 −0.213573 0.976927i $$-0.568510\pi$$
−0.213573 + 0.976927i $$0.568510\pi$$
$$140$$ −240.000 −0.144884
$$141$$ −1028.00 −0.613994
$$142$$ −1648.00 −0.973923
$$143$$ −1216.00 −0.711098
$$144$$ 1472.00 0.851852
$$145$$ 250.000 0.143182
$$146$$ 3512.00 1.99079
$$147$$ −614.000 −0.344502
$$148$$ 2128.00 1.18190
$$149$$ 2050.00 1.12713 0.563566 0.826071i $$-0.309429\pi$$
0.563566 + 0.826071i $$0.309429\pi$$
$$150$$ −200.000 −0.108866
$$151$$ 1852.00 0.998103 0.499052 0.866572i $$-0.333682\pi$$
0.499052 + 0.866572i $$0.333682\pi$$
$$152$$ 0 0
$$153$$ −598.000 −0.315983
$$154$$ −768.000 −0.401865
$$155$$ 540.000 0.279831
$$156$$ −608.000 −0.312045
$$157$$ −2494.00 −1.26779 −0.633894 0.773420i $$-0.718545\pi$$
−0.633894 + 0.773420i $$0.718545\pi$$
$$158$$ −2400.00 −1.20844
$$159$$ 4.00000 0.00199510
$$160$$ −1280.00 −0.632456
$$161$$ −468.000 −0.229090
$$162$$ −1684.00 −0.816713
$$163$$ 2762.00 1.32722 0.663609 0.748080i $$-0.269024\pi$$
0.663609 + 0.748080i $$0.269024\pi$$
$$164$$ 176.000 0.0838006
$$165$$ −320.000 −0.150982
$$166$$ −1128.00 −0.527408
$$167$$ 3126.00 1.44849 0.724243 0.689545i $$-0.242189\pi$$
0.724243 + 0.689545i $$0.242189\pi$$
$$168$$ 0 0
$$169$$ −753.000 −0.342740
$$170$$ 520.000 0.234601
$$171$$ −2300.00 −1.02857
$$172$$ 3536.00 1.56754
$$173$$ −78.0000 −0.0342788 −0.0171394 0.999853i $$-0.505456\pi$$
−0.0171394 + 0.999853i $$0.505456\pi$$
$$174$$ 400.000 0.174275
$$175$$ 150.000 0.0647939
$$176$$ −2048.00 −0.877124
$$177$$ 1000.00 0.424659
$$178$$ 600.000 0.252651
$$179$$ −1300.00 −0.542830 −0.271415 0.962462i $$-0.587492\pi$$
−0.271415 + 0.962462i $$0.587492\pi$$
$$180$$ 920.000 0.380960
$$181$$ 1742.00 0.715369 0.357685 0.933842i $$-0.383566\pi$$
0.357685 + 0.933842i $$0.383566\pi$$
$$182$$ 912.000 0.371439
$$183$$ −1036.00 −0.418488
$$184$$ 0 0
$$185$$ −1330.00 −0.528560
$$186$$ 864.000 0.340600
$$187$$ 832.000 0.325358
$$188$$ −4112.00 −1.59520
$$189$$ −600.000 −0.230918
$$190$$ 2000.00 0.763659
$$191$$ 3772.00 1.42897 0.714483 0.699653i $$-0.246662\pi$$
0.714483 + 0.699653i $$0.246662\pi$$
$$192$$ −1024.00 −0.384900
$$193$$ −358.000 −0.133520 −0.0667601 0.997769i $$-0.521266\pi$$
−0.0667601 + 0.997769i $$0.521266\pi$$
$$194$$ −1544.00 −0.571406
$$195$$ 380.000 0.139551
$$196$$ −2456.00 −0.895044
$$197$$ −2214.00 −0.800716 −0.400358 0.916359i $$-0.631114\pi$$
−0.400358 + 0.916359i $$0.631114\pi$$
$$198$$ 2944.00 1.05667
$$199$$ −2600.00 −0.926176 −0.463088 0.886312i $$-0.653259\pi$$
−0.463088 + 0.886312i $$0.653259\pi$$
$$200$$ 0 0
$$201$$ 252.000 0.0884314
$$202$$ −2808.00 −0.978070
$$203$$ −300.000 −0.103724
$$204$$ 416.000 0.142774
$$205$$ −110.000 −0.0374767
$$206$$ 2392.00 0.809022
$$207$$ 1794.00 0.602375
$$208$$ 2432.00 0.810716
$$209$$ 3200.00 1.05908
$$210$$ 240.000 0.0788646
$$211$$ −1168.00 −0.381083 −0.190541 0.981679i $$-0.561024\pi$$
−0.190541 + 0.981679i $$0.561024\pi$$
$$212$$ 16.0000 0.00518342
$$213$$ 824.000 0.265068
$$214$$ 4776.00 1.52561
$$215$$ −2210.00 −0.701027
$$216$$ 0 0
$$217$$ −648.000 −0.202715
$$218$$ 2200.00 0.683499
$$219$$ −1756.00 −0.541824
$$220$$ −1280.00 −0.392262
$$221$$ −988.000 −0.300724
$$222$$ −2128.00 −0.643342
$$223$$ −6478.00 −1.94529 −0.972643 0.232303i $$-0.925374\pi$$
−0.972643 + 0.232303i $$0.925374\pi$$
$$224$$ 1536.00 0.458162
$$225$$ −575.000 −0.170370
$$226$$ −6248.00 −1.83899
$$227$$ 646.000 0.188883 0.0944417 0.995530i $$-0.469893\pi$$
0.0944417 + 0.995530i $$0.469893\pi$$
$$228$$ 1600.00 0.464748
$$229$$ 3750.00 1.08213 0.541063 0.840982i $$-0.318022\pi$$
0.541063 + 0.840982i $$0.318022\pi$$
$$230$$ −1560.00 −0.447232
$$231$$ 384.000 0.109374
$$232$$ 0 0
$$233$$ 1482.00 0.416691 0.208346 0.978055i $$-0.433192\pi$$
0.208346 + 0.978055i $$0.433192\pi$$
$$234$$ −3496.00 −0.976670
$$235$$ 2570.00 0.713397
$$236$$ 4000.00 1.10330
$$237$$ 1200.00 0.328896
$$238$$ −624.000 −0.169949
$$239$$ 1400.00 0.378906 0.189453 0.981890i $$-0.439329\pi$$
0.189453 + 0.981890i $$0.439329\pi$$
$$240$$ 640.000 0.172133
$$241$$ 3022.00 0.807735 0.403867 0.914817i $$-0.367666\pi$$
0.403867 + 0.914817i $$0.367666\pi$$
$$242$$ 1228.00 0.326194
$$243$$ 3542.00 0.935059
$$244$$ −4144.00 −1.08726
$$245$$ 1535.00 0.400276
$$246$$ −176.000 −0.0456152
$$247$$ −3800.00 −0.978900
$$248$$ 0 0
$$249$$ 564.000 0.143542
$$250$$ 500.000 0.126491
$$251$$ −1248.00 −0.313837 −0.156918 0.987612i $$-0.550156\pi$$
−0.156918 + 0.987612i $$0.550156\pi$$
$$252$$ −1104.00 −0.275974
$$253$$ −2496.00 −0.620246
$$254$$ −7384.00 −1.82407
$$255$$ −260.000 −0.0638503
$$256$$ 4096.00 1.00000
$$257$$ 2106.00 0.511162 0.255581 0.966788i $$-0.417733\pi$$
0.255581 + 0.966788i $$0.417733\pi$$
$$258$$ −3536.00 −0.853263
$$259$$ 1596.00 0.382898
$$260$$ 1520.00 0.362563
$$261$$ 1150.00 0.272733
$$262$$ 8832.00 2.08261
$$263$$ −3638.00 −0.852961 −0.426480 0.904497i $$-0.640247\pi$$
−0.426480 + 0.904497i $$0.640247\pi$$
$$264$$ 0 0
$$265$$ −10.0000 −0.00231809
$$266$$ −2400.00 −0.553208
$$267$$ −300.000 −0.0687629
$$268$$ 1008.00 0.229751
$$269$$ −6550.00 −1.48461 −0.742306 0.670061i $$-0.766268\pi$$
−0.742306 + 0.670061i $$0.766268\pi$$
$$270$$ −2000.00 −0.450800
$$271$$ −4388.00 −0.983587 −0.491793 0.870712i $$-0.663658\pi$$
−0.491793 + 0.870712i $$0.663658\pi$$
$$272$$ −1664.00 −0.370937
$$273$$ −456.000 −0.101093
$$274$$ 9336.00 2.05842
$$275$$ 800.000 0.175425
$$276$$ −1248.00 −0.272177
$$277$$ 546.000 0.118433 0.0592165 0.998245i $$-0.481140\pi$$
0.0592165 + 0.998245i $$0.481140\pi$$
$$278$$ 2800.00 0.604075
$$279$$ 2484.00 0.533022
$$280$$ 0 0
$$281$$ −6858.00 −1.45592 −0.727961 0.685619i $$-0.759532\pi$$
−0.727961 + 0.685619i $$0.759532\pi$$
$$282$$ 4112.00 0.868319
$$283$$ 9282.00 1.94967 0.974837 0.222920i $$-0.0715588\pi$$
0.974837 + 0.222920i $$0.0715588\pi$$
$$284$$ 3296.00 0.688668
$$285$$ −1000.00 −0.207842
$$286$$ 4864.00 1.00564
$$287$$ 132.000 0.0271488
$$288$$ −5888.00 −1.20470
$$289$$ −4237.00 −0.862406
$$290$$ −1000.00 −0.202490
$$291$$ 772.000 0.155517
$$292$$ −7024.00 −1.40770
$$293$$ 4842.00 0.965436 0.482718 0.875776i $$-0.339650\pi$$
0.482718 + 0.875776i $$0.339650\pi$$
$$294$$ 2456.00 0.487200
$$295$$ −2500.00 −0.493409
$$296$$ 0 0
$$297$$ −3200.00 −0.625195
$$298$$ −8200.00 −1.59400
$$299$$ 2964.00 0.573286
$$300$$ 400.000 0.0769800
$$301$$ 2652.00 0.507836
$$302$$ −7408.00 −1.41153
$$303$$ 1404.00 0.266197
$$304$$ −6400.00 −1.20745
$$305$$ 2590.00 0.486239
$$306$$ 2392.00 0.446868
$$307$$ −2594.00 −0.482239 −0.241120 0.970495i $$-0.577515\pi$$
−0.241120 + 0.970495i $$0.577515\pi$$
$$308$$ 1536.00 0.284161
$$309$$ −1196.00 −0.220188
$$310$$ −2160.00 −0.395741
$$311$$ 7332.00 1.33685 0.668424 0.743781i $$-0.266969\pi$$
0.668424 + 0.743781i $$0.266969\pi$$
$$312$$ 0 0
$$313$$ 1562.00 0.282075 0.141037 0.990004i $$-0.454956\pi$$
0.141037 + 0.990004i $$0.454956\pi$$
$$314$$ 9976.00 1.79292
$$315$$ 690.000 0.123419
$$316$$ 4800.00 0.854497
$$317$$ 1426.00 0.252657 0.126328 0.991988i $$-0.459681\pi$$
0.126328 + 0.991988i $$0.459681\pi$$
$$318$$ −16.0000 −0.00282150
$$319$$ −1600.00 −0.280824
$$320$$ 2560.00 0.447214
$$321$$ −2388.00 −0.415219
$$322$$ 1872.00 0.323983
$$323$$ 2600.00 0.447888
$$324$$ 3368.00 0.577503
$$325$$ −950.000 −0.162143
$$326$$ −11048.0 −1.87697
$$327$$ −1100.00 −0.186025
$$328$$ 0 0
$$329$$ −3084.00 −0.516798
$$330$$ 1280.00 0.213520
$$331$$ −4008.00 −0.665558 −0.332779 0.943005i $$-0.607986\pi$$
−0.332779 + 0.943005i $$0.607986\pi$$
$$332$$ 2256.00 0.372934
$$333$$ −6118.00 −1.00680
$$334$$ −12504.0 −2.04847
$$335$$ −630.000 −0.102748
$$336$$ −768.000 −0.124696
$$337$$ 8866.00 1.43312 0.716561 0.697525i $$-0.245715\pi$$
0.716561 + 0.697525i $$0.245715\pi$$
$$338$$ 3012.00 0.484708
$$339$$ 3124.00 0.500509
$$340$$ −1040.00 −0.165888
$$341$$ −3456.00 −0.548835
$$342$$ 9200.00 1.45462
$$343$$ −3900.00 −0.613936
$$344$$ 0 0
$$345$$ 780.000 0.121721
$$346$$ 312.000 0.0484775
$$347$$ −1714.00 −0.265165 −0.132583 0.991172i $$-0.542327\pi$$
−0.132583 + 0.991172i $$0.542327\pi$$
$$348$$ −800.000 −0.123231
$$349$$ 1150.00 0.176384 0.0881921 0.996103i $$-0.471891\pi$$
0.0881921 + 0.996103i $$0.471891\pi$$
$$350$$ −600.000 −0.0916324
$$351$$ 3800.00 0.577860
$$352$$ 8192.00 1.24044
$$353$$ −4398.00 −0.663122 −0.331561 0.943434i $$-0.607575\pi$$
−0.331561 + 0.943434i $$0.607575\pi$$
$$354$$ −4000.00 −0.600558
$$355$$ −2060.00 −0.307982
$$356$$ −1200.00 −0.178651
$$357$$ 312.000 0.0462543
$$358$$ 5200.00 0.767677
$$359$$ 1800.00 0.264625 0.132312 0.991208i $$-0.457760\pi$$
0.132312 + 0.991208i $$0.457760\pi$$
$$360$$ 0 0
$$361$$ 3141.00 0.457938
$$362$$ −6968.00 −1.01168
$$363$$ −614.000 −0.0887786
$$364$$ −1824.00 −0.262647
$$365$$ 4390.00 0.629543
$$366$$ 4144.00 0.591832
$$367$$ −5874.00 −0.835478 −0.417739 0.908567i $$-0.637177\pi$$
−0.417739 + 0.908567i $$0.637177\pi$$
$$368$$ 4992.00 0.707136
$$369$$ −506.000 −0.0713857
$$370$$ 5320.00 0.747496
$$371$$ 12.0000 0.00167927
$$372$$ −1728.00 −0.240840
$$373$$ −2078.00 −0.288458 −0.144229 0.989544i $$-0.546070\pi$$
−0.144229 + 0.989544i $$0.546070\pi$$
$$374$$ −3328.00 −0.460125
$$375$$ −250.000 −0.0344265
$$376$$ 0 0
$$377$$ 1900.00 0.259562
$$378$$ 2400.00 0.326568
$$379$$ 7900.00 1.07070 0.535351 0.844630i $$-0.320179\pi$$
0.535351 + 0.844630i $$0.320179\pi$$
$$380$$ −4000.00 −0.539989
$$381$$ 3692.00 0.496449
$$382$$ −15088.0 −2.02086
$$383$$ −7518.00 −1.00301 −0.501504 0.865155i $$-0.667220\pi$$
−0.501504 + 0.865155i $$0.667220\pi$$
$$384$$ 0 0
$$385$$ −960.000 −0.127081
$$386$$ 1432.00 0.188826
$$387$$ −10166.0 −1.33531
$$388$$ 3088.00 0.404045
$$389$$ −1950.00 −0.254162 −0.127081 0.991892i $$-0.540561\pi$$
−0.127081 + 0.991892i $$0.540561\pi$$
$$390$$ −1520.00 −0.197354
$$391$$ −2028.00 −0.262303
$$392$$ 0 0
$$393$$ −4416.00 −0.566814
$$394$$ 8856.00 1.13238
$$395$$ −3000.00 −0.382143
$$396$$ −5888.00 −0.747180
$$397$$ 13786.0 1.74282 0.871410 0.490555i $$-0.163206\pi$$
0.871410 + 0.490555i $$0.163206\pi$$
$$398$$ 10400.0 1.30981
$$399$$ 1200.00 0.150564
$$400$$ −1600.00 −0.200000
$$401$$ 6402.00 0.797258 0.398629 0.917112i $$-0.369486\pi$$
0.398629 + 0.917112i $$0.369486\pi$$
$$402$$ −1008.00 −0.125061
$$403$$ 4104.00 0.507282
$$404$$ 5616.00 0.691600
$$405$$ −2105.00 −0.258267
$$406$$ 1200.00 0.146687
$$407$$ 8512.00 1.03667
$$408$$ 0 0
$$409$$ 11150.0 1.34800 0.674000 0.738731i $$-0.264575\pi$$
0.674000 + 0.738731i $$0.264575\pi$$
$$410$$ 440.000 0.0530001
$$411$$ −4668.00 −0.560232
$$412$$ −4784.00 −0.572065
$$413$$ 3000.00 0.357434
$$414$$ −7176.00 −0.851887
$$415$$ −1410.00 −0.166781
$$416$$ −9728.00 −1.14653
$$417$$ −1400.00 −0.164408
$$418$$ −12800.0 −1.49777
$$419$$ −13700.0 −1.59735 −0.798674 0.601764i $$-0.794465\pi$$
−0.798674 + 0.601764i $$0.794465\pi$$
$$420$$ −480.000 −0.0557657
$$421$$ −5438.00 −0.629529 −0.314765 0.949170i $$-0.601926\pi$$
−0.314765 + 0.949170i $$0.601926\pi$$
$$422$$ 4672.00 0.538932
$$423$$ 11822.0 1.35888
$$424$$ 0 0
$$425$$ 650.000 0.0741874
$$426$$ −3296.00 −0.374863
$$427$$ −3108.00 −0.352240
$$428$$ −9552.00 −1.07877
$$429$$ −2432.00 −0.273702
$$430$$ 8840.00 0.991402
$$431$$ 7692.00 0.859653 0.429827 0.902911i $$-0.358575\pi$$
0.429827 + 0.902911i $$0.358575\pi$$
$$432$$ 6400.00 0.712778
$$433$$ −1118.00 −0.124082 −0.0620412 0.998074i $$-0.519761\pi$$
−0.0620412 + 0.998074i $$0.519761\pi$$
$$434$$ 2592.00 0.286682
$$435$$ 500.000 0.0551107
$$436$$ −4400.00 −0.483307
$$437$$ −7800.00 −0.853832
$$438$$ 7024.00 0.766255
$$439$$ −2600.00 −0.282668 −0.141334 0.989962i $$-0.545139\pi$$
−0.141334 + 0.989962i $$0.545139\pi$$
$$440$$ 0 0
$$441$$ 7061.00 0.762445
$$442$$ 3952.00 0.425288
$$443$$ −11958.0 −1.28249 −0.641243 0.767337i $$-0.721581\pi$$
−0.641243 + 0.767337i $$0.721581\pi$$
$$444$$ 4256.00 0.454912
$$445$$ 750.000 0.0798953
$$446$$ 25912.0 2.75105
$$447$$ 4100.00 0.433833
$$448$$ −3072.00 −0.323970
$$449$$ −17050.0 −1.79207 −0.896035 0.443984i $$-0.853565\pi$$
−0.896035 + 0.443984i $$0.853565\pi$$
$$450$$ 2300.00 0.240940
$$451$$ 704.000 0.0735035
$$452$$ 12496.0 1.30036
$$453$$ 3704.00 0.384170
$$454$$ −2584.00 −0.267121
$$455$$ 1140.00 0.117459
$$456$$ 0 0
$$457$$ −9494.00 −0.971796 −0.485898 0.874016i $$-0.661507\pi$$
−0.485898 + 0.874016i $$0.661507\pi$$
$$458$$ −15000.0 −1.53036
$$459$$ −2600.00 −0.264396
$$460$$ 3120.00 0.316241
$$461$$ −11418.0 −1.15356 −0.576778 0.816901i $$-0.695690\pi$$
−0.576778 + 0.816901i $$0.695690\pi$$
$$462$$ −1536.00 −0.154678
$$463$$ 7962.00 0.799191 0.399596 0.916692i $$-0.369151\pi$$
0.399596 + 0.916692i $$0.369151\pi$$
$$464$$ 3200.00 0.320164
$$465$$ 1080.00 0.107707
$$466$$ −5928.00 −0.589290
$$467$$ 6526.00 0.646654 0.323327 0.946287i $$-0.395199\pi$$
0.323327 + 0.946287i $$0.395199\pi$$
$$468$$ 6992.00 0.690610
$$469$$ 756.000 0.0744325
$$470$$ −10280.0 −1.00890
$$471$$ −4988.00 −0.487972
$$472$$ 0 0
$$473$$ 14144.0 1.37493
$$474$$ −4800.00 −0.465129
$$475$$ 2500.00 0.241490
$$476$$ 1248.00 0.120172
$$477$$ −46.0000 −0.00441550
$$478$$ −5600.00 −0.535854
$$479$$ 17400.0 1.65976 0.829881 0.557940i $$-0.188408\pi$$
0.829881 + 0.557940i $$0.188408\pi$$
$$480$$ −2560.00 −0.243432
$$481$$ −10108.0 −0.958181
$$482$$ −12088.0 −1.14231
$$483$$ −936.000 −0.0881770
$$484$$ −2456.00 −0.230654
$$485$$ −1930.00 −0.180694
$$486$$ −14168.0 −1.32237
$$487$$ 1166.00 0.108494 0.0542469 0.998528i $$-0.482724\pi$$
0.0542469 + 0.998528i $$0.482724\pi$$
$$488$$ 0 0
$$489$$ 5524.00 0.510846
$$490$$ −6140.00 −0.566075
$$491$$ 7072.00 0.650010 0.325005 0.945712i $$-0.394634\pi$$
0.325005 + 0.945712i $$0.394634\pi$$
$$492$$ 352.000 0.0322548
$$493$$ −1300.00 −0.118761
$$494$$ 15200.0 1.38437
$$495$$ 3680.00 0.334149
$$496$$ 6912.00 0.625722
$$497$$ 2472.00 0.223107
$$498$$ −2256.00 −0.203000
$$499$$ 100.000 0.00897117 0.00448559 0.999990i $$-0.498572\pi$$
0.00448559 + 0.999990i $$0.498572\pi$$
$$500$$ −1000.00 −0.0894427
$$501$$ 6252.00 0.557522
$$502$$ 4992.00 0.443832
$$503$$ 2602.00 0.230651 0.115325 0.993328i $$-0.463209\pi$$
0.115325 + 0.993328i $$0.463209\pi$$
$$504$$ 0 0
$$505$$ −3510.00 −0.309293
$$506$$ 9984.00 0.877160
$$507$$ −1506.00 −0.131921
$$508$$ 14768.0 1.28981
$$509$$ 11150.0 0.970953 0.485476 0.874250i $$-0.338646\pi$$
0.485476 + 0.874250i $$0.338646\pi$$
$$510$$ 1040.00 0.0902980
$$511$$ −5268.00 −0.456052
$$512$$ −16384.0 −1.41421
$$513$$ −10000.0 −0.860645
$$514$$ −8424.00 −0.722892
$$515$$ 2990.00 0.255835
$$516$$ 7072.00 0.603348
$$517$$ −16448.0 −1.39919
$$518$$ −6384.00 −0.541500
$$519$$ −156.000 −0.0131939
$$520$$ 0 0
$$521$$ −3638.00 −0.305919 −0.152959 0.988232i $$-0.548880\pi$$
−0.152959 + 0.988232i $$0.548880\pi$$
$$522$$ −4600.00 −0.385702
$$523$$ −2078.00 −0.173737 −0.0868686 0.996220i $$-0.527686\pi$$
−0.0868686 + 0.996220i $$0.527686\pi$$
$$524$$ −17664.0 −1.47262
$$525$$ 300.000 0.0249392
$$526$$ 14552.0 1.20627
$$527$$ −2808.00 −0.232103
$$528$$ −4096.00 −0.337605
$$529$$ −6083.00 −0.499959
$$530$$ 40.0000 0.00327828
$$531$$ −11500.0 −0.939845
$$532$$ 4800.00 0.391177
$$533$$ −836.000 −0.0679384
$$534$$ 1200.00 0.0972455
$$535$$ 5970.00 0.482440
$$536$$ 0 0
$$537$$ −2600.00 −0.208935
$$538$$ 26200.0 2.09956
$$539$$ −9824.00 −0.785064
$$540$$ 4000.00 0.318764
$$541$$ 5622.00 0.446781 0.223391 0.974729i $$-0.428287\pi$$
0.223391 + 0.974729i $$0.428287\pi$$
$$542$$ 17552.0 1.39100
$$543$$ 3484.00 0.275346
$$544$$ 6656.00 0.524584
$$545$$ 2750.00 0.216141
$$546$$ 1824.00 0.142967
$$547$$ 16486.0 1.28865 0.644324 0.764753i $$-0.277139\pi$$
0.644324 + 0.764753i $$0.277139\pi$$
$$548$$ −18672.0 −1.45553
$$549$$ 11914.0 0.926188
$$550$$ −3200.00 −0.248088
$$551$$ −5000.00 −0.386583
$$552$$ 0 0
$$553$$ 3600.00 0.276831
$$554$$ −2184.00 −0.167490
$$555$$ −2660.00 −0.203443
$$556$$ −5600.00 −0.427146
$$557$$ 11706.0 0.890483 0.445242 0.895410i $$-0.353118\pi$$
0.445242 + 0.895410i $$0.353118\pi$$
$$558$$ −9936.00 −0.753807
$$559$$ −16796.0 −1.27083
$$560$$ 1920.00 0.144884
$$561$$ 1664.00 0.125230
$$562$$ 27432.0 2.05898
$$563$$ −25038.0 −1.87429 −0.937146 0.348939i $$-0.886542\pi$$
−0.937146 + 0.348939i $$0.886542\pi$$
$$564$$ −8224.00 −0.613994
$$565$$ −7810.00 −0.581538
$$566$$ −37128.0 −2.75725
$$567$$ 2526.00 0.187094
$$568$$ 0 0
$$569$$ 17550.0 1.29303 0.646515 0.762901i $$-0.276226\pi$$
0.646515 + 0.762901i $$0.276226\pi$$
$$570$$ 4000.00 0.293933
$$571$$ 10712.0 0.785084 0.392542 0.919734i $$-0.371596\pi$$
0.392542 + 0.919734i $$0.371596\pi$$
$$572$$ −9728.00 −0.711098
$$573$$ 7544.00 0.550009
$$574$$ −528.000 −0.0383942
$$575$$ −1950.00 −0.141427
$$576$$ 11776.0 0.851852
$$577$$ −13654.0 −0.985136 −0.492568 0.870274i $$-0.663942\pi$$
−0.492568 + 0.870274i $$0.663942\pi$$
$$578$$ 16948.0 1.21963
$$579$$ −716.000 −0.0513920
$$580$$ 2000.00 0.143182
$$581$$ 1692.00 0.120819
$$582$$ −3088.00 −0.219934
$$583$$ 64.0000 0.00454650
$$584$$ 0 0
$$585$$ −4370.00 −0.308850
$$586$$ −19368.0 −1.36533
$$587$$ 14166.0 0.996071 0.498035 0.867157i $$-0.334055\pi$$
0.498035 + 0.867157i $$0.334055\pi$$
$$588$$ −4912.00 −0.344502
$$589$$ −10800.0 −0.755528
$$590$$ 10000.0 0.697786
$$591$$ −4428.00 −0.308196
$$592$$ −17024.0 −1.18190
$$593$$ 17842.0 1.23555 0.617777 0.786354i $$-0.288034\pi$$
0.617777 + 0.786354i $$0.288034\pi$$
$$594$$ 12800.0 0.884159
$$595$$ −780.000 −0.0537427
$$596$$ 16400.0 1.12713
$$597$$ −5200.00 −0.356485
$$598$$ −11856.0 −0.810749
$$599$$ −17600.0 −1.20053 −0.600264 0.799802i $$-0.704938\pi$$
−0.600264 + 0.799802i $$0.704938\pi$$
$$600$$ 0 0
$$601$$ 27302.0 1.85303 0.926516 0.376256i $$-0.122789\pi$$
0.926516 + 0.376256i $$0.122789\pi$$
$$602$$ −10608.0 −0.718189
$$603$$ −2898.00 −0.195714
$$604$$ 14816.0 0.998103
$$605$$ 1535.00 0.103151
$$606$$ −5616.00 −0.376459
$$607$$ −3794.00 −0.253696 −0.126848 0.991922i $$-0.540486\pi$$
−0.126848 + 0.991922i $$0.540486\pi$$
$$608$$ 25600.0 1.70759
$$609$$ −600.000 −0.0399232
$$610$$ −10360.0 −0.687646
$$611$$ 19532.0 1.29326
$$612$$ −4784.00 −0.315983
$$613$$ −13238.0 −0.872231 −0.436116 0.899891i $$-0.643646\pi$$
−0.436116 + 0.899891i $$0.643646\pi$$
$$614$$ 10376.0 0.681989
$$615$$ −220.000 −0.0144248
$$616$$ 0 0
$$617$$ −11574.0 −0.755189 −0.377595 0.925971i $$-0.623249\pi$$
−0.377595 + 0.925971i $$0.623249\pi$$
$$618$$ 4784.00 0.311393
$$619$$ 8300.00 0.538942 0.269471 0.963008i $$-0.413151\pi$$
0.269471 + 0.963008i $$0.413151\pi$$
$$620$$ 4320.00 0.279831
$$621$$ 7800.00 0.504031
$$622$$ −29328.0 −1.89059
$$623$$ −900.000 −0.0578776
$$624$$ 4864.00 0.312045
$$625$$ 625.000 0.0400000
$$626$$ −6248.00 −0.398914
$$627$$ 6400.00 0.407642
$$628$$ −19952.0 −1.26779
$$629$$ 6916.00 0.438409
$$630$$ −2760.00 −0.174541
$$631$$ −7508.00 −0.473675 −0.236837 0.971549i $$-0.576111\pi$$
−0.236837 + 0.971549i $$0.576111\pi$$
$$632$$ 0 0
$$633$$ −2336.00 −0.146679
$$634$$ −5704.00 −0.357310
$$635$$ −9230.00 −0.576821
$$636$$ 32.0000 0.00199510
$$637$$ 11666.0 0.725626
$$638$$ 6400.00 0.397145
$$639$$ −9476.00 −0.586643
$$640$$ 0 0
$$641$$ −27378.0 −1.68700 −0.843499 0.537130i $$-0.819508\pi$$
−0.843499 + 0.537130i $$0.819508\pi$$
$$642$$ 9552.00 0.587208
$$643$$ 1842.00 0.112973 0.0564863 0.998403i $$-0.482010\pi$$
0.0564863 + 0.998403i $$0.482010\pi$$
$$644$$ −3744.00 −0.229090
$$645$$ −4420.00 −0.269825
$$646$$ −10400.0 −0.633409
$$647$$ −10114.0 −0.614563 −0.307282 0.951619i $$-0.599419\pi$$
−0.307282 + 0.951619i $$0.599419\pi$$
$$648$$ 0 0
$$649$$ 16000.0 0.967727
$$650$$ 3800.00 0.229305
$$651$$ −1296.00 −0.0780250
$$652$$ 22096.0 1.32722
$$653$$ 10402.0 0.623372 0.311686 0.950185i $$-0.399106\pi$$
0.311686 + 0.950185i $$0.399106\pi$$
$$654$$ 4400.00 0.263079
$$655$$ 11040.0 0.658578
$$656$$ −1408.00 −0.0838006
$$657$$ 20194.0 1.19915
$$658$$ 12336.0 0.730862
$$659$$ 7100.00 0.419692 0.209846 0.977734i $$-0.432704\pi$$
0.209846 + 0.977734i $$0.432704\pi$$
$$660$$ −2560.00 −0.150982
$$661$$ −7118.00 −0.418847 −0.209424 0.977825i $$-0.567159\pi$$
−0.209424 + 0.977825i $$0.567159\pi$$
$$662$$ 16032.0 0.941241
$$663$$ −1976.00 −0.115749
$$664$$ 0 0
$$665$$ −3000.00 −0.174940
$$666$$ 24472.0 1.42383
$$667$$ 3900.00 0.226400
$$668$$ 25008.0 1.44849
$$669$$ −12956.0 −0.748741
$$670$$ 2520.00 0.145308
$$671$$ −16576.0 −0.953665
$$672$$ 3072.00 0.176347
$$673$$ −31278.0 −1.79150 −0.895749 0.444560i $$-0.853360\pi$$
−0.895749 + 0.444560i $$0.853360\pi$$
$$674$$ −35464.0 −2.02674
$$675$$ −2500.00 −0.142556
$$676$$ −6024.00 −0.342740
$$677$$ −30054.0 −1.70616 −0.853079 0.521782i $$-0.825268\pi$$
−0.853079 + 0.521782i $$0.825268\pi$$
$$678$$ −12496.0 −0.707826
$$679$$ 2316.00 0.130898
$$680$$ 0 0
$$681$$ 1292.00 0.0727012
$$682$$ 13824.0 0.776171
$$683$$ −4518.00 −0.253113 −0.126557 0.991959i $$-0.540393\pi$$
−0.126557 + 0.991959i $$0.540393\pi$$
$$684$$ −18400.0 −1.02857
$$685$$ 11670.0 0.650931
$$686$$ 15600.0 0.868237
$$687$$ 7500.00 0.416511
$$688$$ −28288.0 −1.56754
$$689$$ −76.0000 −0.00420228
$$690$$ −3120.00 −0.172140
$$691$$ 29272.0 1.61152 0.805759 0.592243i $$-0.201758\pi$$
0.805759 + 0.592243i $$0.201758\pi$$
$$692$$ −624.000 −0.0342788
$$693$$ −4416.00 −0.242063
$$694$$ 6856.00 0.375000
$$695$$ 3500.00 0.191025
$$696$$ 0 0
$$697$$ 572.000 0.0310847
$$698$$ −4600.00 −0.249445
$$699$$ 2964.00 0.160385
$$700$$ 1200.00 0.0647939
$$701$$ −5798.00 −0.312393 −0.156196 0.987726i $$-0.549923\pi$$
−0.156196 + 0.987726i $$0.549923\pi$$
$$702$$ −15200.0 −0.817218
$$703$$ 26600.0 1.42708
$$704$$ −16384.0 −0.877124
$$705$$ 5140.00 0.274587
$$706$$ 17592.0 0.937796
$$707$$ 4212.00 0.224057
$$708$$ 8000.00 0.424659
$$709$$ 8950.00 0.474082 0.237041 0.971500i $$-0.423822\pi$$
0.237041 + 0.971500i $$0.423822\pi$$
$$710$$ 8240.00 0.435552
$$711$$ −13800.0 −0.727905
$$712$$ 0 0
$$713$$ 8424.00 0.442470
$$714$$ −1248.00 −0.0654135
$$715$$ 6080.00 0.318013
$$716$$ −10400.0 −0.542830
$$717$$ 2800.00 0.145841
$$718$$ −7200.00 −0.374236
$$719$$ 7800.00 0.404577 0.202289 0.979326i $$-0.435162\pi$$
0.202289 + 0.979326i $$0.435162\pi$$
$$720$$ −7360.00 −0.380960
$$721$$ −3588.00 −0.185332
$$722$$ −12564.0 −0.647623
$$723$$ 6044.00 0.310897
$$724$$ 13936.0 0.715369
$$725$$ −1250.00 −0.0640329
$$726$$ 2456.00 0.125552
$$727$$ −8554.00 −0.436383 −0.218191 0.975906i $$-0.570016\pi$$
−0.218191 + 0.975906i $$0.570016\pi$$
$$728$$ 0 0
$$729$$ −4283.00 −0.217599
$$730$$ −17560.0 −0.890308
$$731$$ 11492.0 0.581460
$$732$$ −8288.00 −0.418488
$$733$$ 2882.00 0.145224 0.0726119 0.997360i $$-0.476867\pi$$
0.0726119 + 0.997360i $$0.476867\pi$$
$$734$$ 23496.0 1.18154
$$735$$ 3070.00 0.154066
$$736$$ −19968.0 −1.00004
$$737$$ 4032.00 0.201521
$$738$$ 2024.00 0.100955
$$739$$ 18700.0 0.930840 0.465420 0.885090i $$-0.345903\pi$$
0.465420 + 0.885090i $$0.345903\pi$$
$$740$$ −10640.0 −0.528560
$$741$$ −7600.00 −0.376779
$$742$$ −48.0000 −0.00237485
$$743$$ 12242.0 0.604462 0.302231 0.953235i $$-0.402269\pi$$
0.302231 + 0.953235i $$0.402269\pi$$
$$744$$ 0 0
$$745$$ −10250.0 −0.504068
$$746$$ 8312.00 0.407941
$$747$$ −6486.00 −0.317685
$$748$$ 6656.00 0.325358
$$749$$ −7164.00 −0.349488
$$750$$ 1000.00 0.0486864
$$751$$ −31148.0 −1.51346 −0.756729 0.653729i $$-0.773204\pi$$
−0.756729 + 0.653729i $$0.773204\pi$$
$$752$$ 32896.0 1.59520
$$753$$ −2496.00 −0.120796
$$754$$ −7600.00 −0.367076
$$755$$ −9260.00 −0.446365
$$756$$ −4800.00 −0.230918
$$757$$ −7694.00 −0.369410 −0.184705 0.982794i $$-0.559133\pi$$
−0.184705 + 0.982794i $$0.559133\pi$$
$$758$$ −31600.0 −1.51420
$$759$$ −4992.00 −0.238733
$$760$$ 0 0
$$761$$ −4518.00 −0.215213 −0.107607 0.994194i $$-0.534319\pi$$
−0.107607 + 0.994194i $$0.534319\pi$$
$$762$$ −14768.0 −0.702084
$$763$$ −3300.00 −0.156577
$$764$$ 30176.0 1.42897
$$765$$ 2990.00 0.141312
$$766$$ 30072.0 1.41847
$$767$$ −19000.0 −0.894459
$$768$$ 8192.00 0.384900
$$769$$ −39550.0 −1.85463 −0.927314 0.374283i $$-0.877889\pi$$
−0.927314 + 0.374283i $$0.877889\pi$$
$$770$$ 3840.00 0.179719
$$771$$ 4212.00 0.196746
$$772$$ −2864.00 −0.133520
$$773$$ 22122.0 1.02933 0.514666 0.857391i $$-0.327916\pi$$
0.514666 + 0.857391i $$0.327916\pi$$
$$774$$ 40664.0 1.88842
$$775$$ −2700.00 −0.125144
$$776$$ 0 0
$$777$$ 3192.00 0.147378
$$778$$ 7800.00 0.359439
$$779$$ 2200.00 0.101185
$$780$$ 3040.00 0.139551
$$781$$ 13184.0 0.604047
$$782$$ 8112.00 0.370952
$$783$$ 5000.00 0.228206
$$784$$ 19648.0 0.895044
$$785$$ 12470.0 0.566972
$$786$$ 17664.0 0.801595
$$787$$ −16634.0 −0.753416 −0.376708 0.926332i $$-0.622944\pi$$
−0.376708 + 0.926332i $$0.622944\pi$$
$$788$$ −17712.0 −0.800716
$$789$$ −7276.00 −0.328305
$$790$$ 12000.0 0.540431
$$791$$ 9372.00 0.421277
$$792$$ 0 0
$$793$$ 19684.0 0.881462
$$794$$ −55144.0 −2.46472
$$795$$ −20.0000 −0.000892235 0
$$796$$ −20800.0 −0.926176
$$797$$ 27586.0 1.22603 0.613015 0.790071i $$-0.289956\pi$$
0.613015 + 0.790071i $$0.289956\pi$$
$$798$$ −4800.00 −0.212930
$$799$$ −13364.0 −0.591720
$$800$$ 6400.00 0.282843
$$801$$ 3450.00 0.152184
$$802$$ −25608.0 −1.12749
$$803$$ −28096.0 −1.23473
$$804$$ 2016.00 0.0884314
$$805$$ 2340.00 0.102452
$$806$$ −16416.0 −0.717406
$$807$$ −13100.0 −0.571427
$$808$$ 0 0
$$809$$ 3850.00 0.167316 0.0836581 0.996495i $$-0.473340\pi$$
0.0836581 + 0.996495i $$0.473340\pi$$
$$810$$ 8420.00 0.365245
$$811$$ 10032.0 0.434366 0.217183 0.976131i $$-0.430313\pi$$
0.217183 + 0.976131i $$0.430313\pi$$
$$812$$ −2400.00 −0.103724
$$813$$ −8776.00 −0.378583
$$814$$ −34048.0 −1.46607
$$815$$ −13810.0 −0.593550
$$816$$ −3328.00 −0.142774
$$817$$ 44200.0 1.89273
$$818$$ −44600.0 −1.90636
$$819$$ 5244.00 0.223736
$$820$$ −880.000 −0.0374767
$$821$$ 20562.0 0.874079 0.437039 0.899442i $$-0.356027\pi$$
0.437039 + 0.899442i $$0.356027\pi$$
$$822$$ 18672.0 0.792288
$$823$$ 10322.0 0.437184 0.218592 0.975816i $$-0.429854\pi$$
0.218592 + 0.975816i $$0.429854\pi$$
$$824$$ 0 0
$$825$$ 1600.00 0.0675210
$$826$$ −12000.0 −0.505488
$$827$$ 8846.00 0.371954 0.185977 0.982554i $$-0.440455\pi$$
0.185977 + 0.982554i $$0.440455\pi$$
$$828$$ 14352.0 0.602375
$$829$$ −25350.0 −1.06205 −0.531026 0.847355i $$-0.678194\pi$$
−0.531026 + 0.847355i $$0.678194\pi$$
$$830$$ 5640.00 0.235864
$$831$$ 1092.00 0.0455849
$$832$$ 19456.0 0.810716
$$833$$ −7982.00 −0.332005
$$834$$ 5600.00 0.232509
$$835$$ −15630.0 −0.647783
$$836$$ 25600.0 1.05908
$$837$$ 10800.0 0.446001
$$838$$ 54800.0 2.25899
$$839$$ 46000.0 1.89284 0.946422 0.322932i $$-0.104669\pi$$
0.946422 + 0.322932i $$0.104669\pi$$
$$840$$ 0 0
$$841$$ −21889.0 −0.897495
$$842$$ 21752.0 0.890289
$$843$$ −13716.0 −0.560385
$$844$$ −9344.00 −0.381083
$$845$$ 3765.00 0.153278
$$846$$ −47288.0 −1.92174
$$847$$ −1842.00 −0.0747248
$$848$$ −128.000 −0.00518342
$$849$$ 18564.0 0.750430
$$850$$ −2600.00 −0.104917
$$851$$ −20748.0 −0.835761
$$852$$ 6592.00 0.265068
$$853$$ −16998.0 −0.682298 −0.341149 0.940009i $$-0.610816\pi$$
−0.341149 + 0.940009i $$0.610816\pi$$
$$854$$ 12432.0 0.498143
$$855$$ 11500.0 0.459990
$$856$$ 0 0
$$857$$ −26494.0 −1.05603 −0.528015 0.849235i $$-0.677064\pi$$
−0.528015 + 0.849235i $$0.677064\pi$$
$$858$$ 9728.00 0.387073
$$859$$ −21500.0 −0.853982 −0.426991 0.904256i $$-0.640426\pi$$
−0.426991 + 0.904256i $$0.640426\pi$$
$$860$$ −17680.0 −0.701027
$$861$$ 264.000 0.0104496
$$862$$ −30768.0 −1.21573
$$863$$ 25762.0 1.01616 0.508082 0.861309i $$-0.330355\pi$$
0.508082 + 0.861309i $$0.330355\pi$$
$$864$$ −25600.0 −1.00802
$$865$$ 390.000 0.0153299
$$866$$ 4472.00 0.175479
$$867$$ −8474.00 −0.331940
$$868$$ −5184.00 −0.202715
$$869$$ 19200.0 0.749500
$$870$$ −2000.00 −0.0779383
$$871$$ −4788.00 −0.186263
$$872$$ 0 0
$$873$$ −8878.00 −0.344186
$$874$$ 31200.0 1.20750
$$875$$ −750.000 −0.0289767
$$876$$ −14048.0 −0.541824
$$877$$ 30546.0 1.17613 0.588064 0.808814i $$-0.299890\pi$$
0.588064 + 0.808814i $$0.299890\pi$$
$$878$$ 10400.0 0.399753
$$879$$ 9684.00 0.371596
$$880$$ 10240.0 0.392262
$$881$$ 32942.0 1.25976 0.629878 0.776694i $$-0.283105\pi$$
0.629878 + 0.776694i $$0.283105\pi$$
$$882$$ −28244.0 −1.07826
$$883$$ −27118.0 −1.03351 −0.516757 0.856132i $$-0.672861\pi$$
−0.516757 + 0.856132i $$0.672861\pi$$
$$884$$ −7904.00 −0.300724
$$885$$ −5000.00 −0.189913
$$886$$ 47832.0 1.81371
$$887$$ −38634.0 −1.46246 −0.731230 0.682131i $$-0.761054\pi$$
−0.731230 + 0.682131i $$0.761054\pi$$
$$888$$ 0 0
$$889$$ 11076.0 0.417860
$$890$$ −3000.00 −0.112989
$$891$$ 13472.0 0.506542
$$892$$ −51824.0 −1.94529
$$893$$ −51400.0 −1.92613
$$894$$ −16400.0 −0.613532
$$895$$ 6500.00 0.242761
$$896$$ 0 0
$$897$$ 5928.00 0.220658
$$898$$ 68200.0 2.53437
$$899$$ 5400.00 0.200334
$$900$$ −4600.00 −0.170370
$$901$$ 52.0000 0.00192272
$$902$$ −2816.00 −0.103950
$$903$$ 5304.00 0.195466
$$904$$ 0 0
$$905$$ −8710.00 −0.319923
$$906$$ −14816.0 −0.543299
$$907$$ −1794.00 −0.0656767 −0.0328384 0.999461i $$-0.510455\pi$$
−0.0328384 + 0.999461i $$0.510455\pi$$
$$908$$ 5168.00 0.188883
$$909$$ −16146.0 −0.589141
$$910$$ −4560.00 −0.166113
$$911$$ 41732.0 1.51772 0.758860 0.651254i $$-0.225757\pi$$
0.758860 + 0.651254i $$0.225757\pi$$
$$912$$ −12800.0 −0.464748
$$913$$ 9024.00 0.327109
$$914$$ 37976.0 1.37433
$$915$$ 5180.00 0.187154
$$916$$ 30000.0 1.08213
$$917$$ −13248.0 −0.477086
$$918$$ 10400.0 0.373912
$$919$$ 29200.0 1.04812 0.524058 0.851682i $$-0.324417\pi$$
0.524058 + 0.851682i $$0.324417\pi$$
$$920$$ 0 0
$$921$$ −5188.00 −0.185614
$$922$$ 45672.0 1.63137
$$923$$ −15656.0 −0.558314
$$924$$ 3072.00 0.109374
$$925$$ 6650.00 0.236379
$$926$$ −31848.0 −1.13023
$$927$$ 13754.0 0.487315
$$928$$ −12800.0 −0.452781
$$929$$ −48650.0 −1.71814 −0.859071 0.511856i $$-0.828958\pi$$
−0.859071 + 0.511856i $$0.828958\pi$$
$$930$$ −4320.00 −0.152321
$$931$$ −30700.0 −1.08072
$$932$$ 11856.0 0.416691
$$933$$ 14664.0 0.514553
$$934$$ −26104.0 −0.914506
$$935$$ −4160.00 −0.145504
$$936$$ 0 0
$$937$$ −11334.0 −0.395161 −0.197580 0.980287i $$-0.563308\pi$$
−0.197580 + 0.980287i $$0.563308\pi$$
$$938$$ −3024.00 −0.105263
$$939$$ 3124.00 0.108571
$$940$$ 20560.0 0.713397
$$941$$ −31178.0 −1.08010 −0.540050 0.841633i $$-0.681595\pi$$
−0.540050 + 0.841633i $$0.681595\pi$$
$$942$$ 19952.0 0.690097
$$943$$ −1716.00 −0.0592584
$$944$$ −32000.0 −1.10330
$$945$$ 3000.00 0.103270
$$946$$ −56576.0 −1.94444
$$947$$ 4686.00 0.160797 0.0803984 0.996763i $$-0.474381\pi$$
0.0803984 + 0.996763i $$0.474381\pi$$
$$948$$ 9600.00 0.328896
$$949$$ 33364.0 1.14124
$$950$$ −10000.0 −0.341519
$$951$$ 2852.00 0.0972476
$$952$$ 0 0
$$953$$ −598.000 −0.0203265 −0.0101632 0.999948i $$-0.503235\pi$$
−0.0101632 + 0.999948i $$0.503235\pi$$
$$954$$ 184.000 0.00624447
$$955$$ −18860.0 −0.639053
$$956$$ 11200.0 0.378906
$$957$$ −3200.00 −0.108089
$$958$$ −69600.0 −2.34726
$$959$$ −14004.0 −0.471546
$$960$$ 5120.00 0.172133
$$961$$ −18127.0 −0.608472
$$962$$ 40432.0 1.35507
$$963$$ 27462.0 0.918952
$$964$$ 24176.0 0.807735
$$965$$ 1790.00 0.0597121
$$966$$ 3744.00 0.124701
$$967$$ 41726.0 1.38761 0.693804 0.720163i $$-0.255933\pi$$
0.693804 + 0.720163i $$0.255933\pi$$
$$968$$ 0 0
$$969$$ 5200.00 0.172392
$$970$$ 7720.00 0.255540
$$971$$ 24312.0 0.803511 0.401756 0.915747i $$-0.368400\pi$$
0.401756 + 0.915747i $$0.368400\pi$$
$$972$$ 28336.0 0.935059
$$973$$ −4200.00 −0.138382
$$974$$ −4664.00 −0.153433
$$975$$ −1900.00 −0.0624089
$$976$$ 33152.0 1.08726
$$977$$ 40946.0 1.34082 0.670409 0.741992i $$-0.266119\pi$$
0.670409 + 0.741992i $$0.266119\pi$$
$$978$$ −22096.0 −0.722446
$$979$$ −4800.00 −0.156699
$$980$$ 12280.0 0.400276
$$981$$ 12650.0 0.411706
$$982$$ −28288.0 −0.919253
$$983$$ 42282.0 1.37191 0.685954 0.727645i $$-0.259385\pi$$
0.685954 + 0.727645i $$0.259385\pi$$
$$984$$ 0 0
$$985$$ 11070.0 0.358091
$$986$$ 5200.00 0.167953
$$987$$ −6168.00 −0.198916
$$988$$ −30400.0 −0.978900
$$989$$ −34476.0 −1.10847
$$990$$ −14720.0 −0.472558
$$991$$ 1172.00 0.0375679 0.0187840 0.999824i $$-0.494021\pi$$
0.0187840 + 0.999824i $$0.494021\pi$$
$$992$$ −27648.0 −0.884904
$$993$$ −8016.00 −0.256173
$$994$$ −9888.00 −0.315521
$$995$$ 13000.0 0.414199
$$996$$ 4512.00 0.143542
$$997$$ −31614.0 −1.00424 −0.502119 0.864798i $$-0.667446\pi$$
−0.502119 + 0.864798i $$0.667446\pi$$
$$998$$ −400.000 −0.0126872
$$999$$ −26600.0 −0.842429
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 5.4.a.a.1.1 1
3.2 odd 2 45.4.a.d.1.1 1
4.3 odd 2 80.4.a.d.1.1 1
5.2 odd 4 25.4.b.a.24.1 2
5.3 odd 4 25.4.b.a.24.2 2
5.4 even 2 25.4.a.c.1.1 1
7.2 even 3 245.4.e.f.116.1 2
7.3 odd 6 245.4.e.g.226.1 2
7.4 even 3 245.4.e.f.226.1 2
7.5 odd 6 245.4.e.g.116.1 2
7.6 odd 2 245.4.a.a.1.1 1
8.3 odd 2 320.4.a.h.1.1 1
8.5 even 2 320.4.a.g.1.1 1
9.2 odd 6 405.4.e.c.271.1 2
9.4 even 3 405.4.e.l.136.1 2
9.5 odd 6 405.4.e.c.136.1 2
9.7 even 3 405.4.e.l.271.1 2
11.10 odd 2 605.4.a.d.1.1 1
12.11 even 2 720.4.a.u.1.1 1
13.12 even 2 845.4.a.b.1.1 1
15.2 even 4 225.4.b.c.199.2 2
15.8 even 4 225.4.b.c.199.1 2
15.14 odd 2 225.4.a.b.1.1 1
16.3 odd 4 1280.4.d.l.641.1 2
16.5 even 4 1280.4.d.e.641.1 2
16.11 odd 4 1280.4.d.l.641.2 2
16.13 even 4 1280.4.d.e.641.2 2
17.16 even 2 1445.4.a.a.1.1 1
19.18 odd 2 1805.4.a.h.1.1 1
20.3 even 4 400.4.c.k.49.1 2
20.7 even 4 400.4.c.k.49.2 2
20.19 odd 2 400.4.a.m.1.1 1
21.20 even 2 2205.4.a.q.1.1 1
35.34 odd 2 1225.4.a.k.1.1 1
40.19 odd 2 1600.4.a.s.1.1 1
40.29 even 2 1600.4.a.bi.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
5.4.a.a.1.1 1 1.1 even 1 trivial
25.4.a.c.1.1 1 5.4 even 2
25.4.b.a.24.1 2 5.2 odd 4
25.4.b.a.24.2 2 5.3 odd 4
45.4.a.d.1.1 1 3.2 odd 2
80.4.a.d.1.1 1 4.3 odd 2
225.4.a.b.1.1 1 15.14 odd 2
225.4.b.c.199.1 2 15.8 even 4
225.4.b.c.199.2 2 15.2 even 4
245.4.a.a.1.1 1 7.6 odd 2
245.4.e.f.116.1 2 7.2 even 3
245.4.e.f.226.1 2 7.4 even 3
245.4.e.g.116.1 2 7.5 odd 6
245.4.e.g.226.1 2 7.3 odd 6
320.4.a.g.1.1 1 8.5 even 2
320.4.a.h.1.1 1 8.3 odd 2
400.4.a.m.1.1 1 20.19 odd 2
400.4.c.k.49.1 2 20.3 even 4
400.4.c.k.49.2 2 20.7 even 4
405.4.e.c.136.1 2 9.5 odd 6
405.4.e.c.271.1 2 9.2 odd 6
405.4.e.l.136.1 2 9.4 even 3
405.4.e.l.271.1 2 9.7 even 3
605.4.a.d.1.1 1 11.10 odd 2
720.4.a.u.1.1 1 12.11 even 2
845.4.a.b.1.1 1 13.12 even 2
1225.4.a.k.1.1 1 35.34 odd 2
1280.4.d.e.641.1 2 16.5 even 4
1280.4.d.e.641.2 2 16.13 even 4
1280.4.d.l.641.1 2 16.3 odd 4
1280.4.d.l.641.2 2 16.11 odd 4
1445.4.a.a.1.1 1 17.16 even 2
1600.4.a.s.1.1 1 40.19 odd 2
1600.4.a.bi.1.1 1 40.29 even 2
1805.4.a.h.1.1 1 19.18 odd 2
2205.4.a.q.1.1 1 21.20 even 2