Properties

 Label 245.4.a.a.1.1 Level $245$ Weight $4$ Character 245.1 Self dual yes Analytic conductor $14.455$ 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] = [245,4,Mod(1,245)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(245, 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("245.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$245 = 5 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 245.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: $$14.4554679514$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 5) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 245.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

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

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 $$-0.561643\pi$$
−0.192450 + 0.981307i $$0.561643\pi$$
$$4$$ 8.00000 1.00000
$$5$$ 5.00000 0.447214
$$6$$ 8.00000 0.544331
$$7$$ 0 0
$$8$$ 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 $$-0.367147\pi$$
0.405358 + 0.914158i $$0.367147\pi$$
$$14$$ 0 0
$$15$$ −10.0000 −0.172133
$$16$$ −64.0000 −1.00000
$$17$$ −26.0000 −0.370937 −0.185468 0.982650i $$-0.559380\pi$$
−0.185468 + 0.982650i $$0.559380\pi$$
$$18$$ 92.0000 1.20470
$$19$$ −100.000 −1.20745 −0.603726 0.797192i $$-0.706318\pi$$
−0.603726 + 0.797192i $$0.706318\pi$$
$$20$$ 40.0000 0.447214
$$21$$ 0 0
$$22$$ −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
$$28$$ 0 0
$$29$$ −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 $$-0.398713\pi$$
0.312861 + 0.949799i $$0.398713\pi$$
$$32$$ 256.000 1.41421
$$33$$ −64.0000 −0.337605
$$34$$ 104.000 0.524584
$$35$$ 0 0
$$36$$ −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 $$-0.513341\pi$$
−0.0419003 + 0.999122i $$0.513341\pi$$
$$42$$ 0 0
$$43$$ 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 $$-0.206101\pi$$
0.797602 + 0.603184i $$0.206101\pi$$
$$48$$ 128.000 0.384900
$$49$$ 0 0
$$50$$ −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 $$-0.686001\pi$$
−0.551648 + 0.834077i $$0.686001\pi$$
$$60$$ −80.0000 −0.172133
$$61$$ 518.000 1.08726 0.543632 0.839324i $$-0.317049\pi$$
0.543632 + 0.839324i $$0.317049\pi$$
$$62$$ −432.000 −0.884904
$$63$$ 0 0
$$64$$ −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
$$70$$ 0 0
$$71$$ 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 $$-0.251463\pi$$
0.703850 + 0.710348i $$0.251463\pi$$
$$74$$ −1064.00 −1.67145
$$75$$ −50.0000 −0.0769800
$$76$$ −800.000 −1.20745
$$77$$ 0 0
$$78$$ 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 $$-0.559704\pi$$
−0.186467 + 0.982461i $$0.559704\pi$$
$$84$$ 0 0
$$85$$ −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 $$-0.471529\pi$$
0.0893257 + 0.996002i $$0.471529\pi$$
$$90$$ 460.000 0.538758
$$91$$ 0 0
$$92$$ −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 $$-0.564751\pi$$
−0.202022 + 0.979381i $$0.564751\pi$$
$$98$$ 0 0
$$99$$ −736.000 −0.747180
$$100$$ 200.000 0.200000
$$101$$ −702.000 −0.691600 −0.345800 0.938308i $$-0.612392\pi$$
−0.345800 + 0.938308i $$0.612392\pi$$
$$102$$ −208.000 −0.201912
$$103$$ 598.000 0.572065 0.286032 0.958220i $$-0.407663\pi$$
0.286032 + 0.958220i $$0.407663\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ −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
$$112$$ 0 0
$$113$$ 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
$$119$$ 0 0
$$120$$ 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
$$126$$ 0 0
$$127$$ 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 $$-0.236565\pi$$
0.736312 + 0.676642i $$0.236565\pi$$
$$132$$ −512.000 −0.337605
$$133$$ 0 0
$$134$$ −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 $$-0.431490\pi$$
0.213573 + 0.976927i $$0.431490\pi$$
$$140$$ 0 0
$$141$$ −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
$$147$$ 0 0
$$148$$ 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
$$154$$ 0 0
$$155$$ 540.000 0.279831
$$156$$ −608.000 −0.312045
$$157$$ 2494.00 1.26779 0.633894 0.773420i $$-0.281455\pi$$
0.633894 + 0.773420i $$0.281455\pi$$
$$158$$ −2400.00 −1.20844
$$159$$ −4.00000 −0.00199510
$$160$$ 1280.00 0.632456
$$161$$ 0 0
$$162$$ −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 $$-0.757811\pi$$
−0.724243 + 0.689545i $$0.757811\pi$$
$$168$$ 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 $$-0.494544\pi$$
0.0171394 + 0.999853i $$0.494544\pi$$
$$174$$ −400.000 −0.174275
$$175$$ 0 0
$$176$$ −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 $$-0.616434\pi$$
−0.357685 + 0.933842i $$0.616434\pi$$
$$182$$ 0 0
$$183$$ −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
$$189$$ 0 0
$$190$$ 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
$$196$$ 0 0
$$197$$ −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 $$-0.346741\pi$$
0.463088 + 0.886312i $$0.346741\pi$$
$$200$$ 0 0
$$201$$ −252.000 −0.0884314
$$202$$ 2808.00 0.978070
$$203$$ 0 0
$$204$$ 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
$$210$$ 0 0
$$211$$ −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
$$217$$ 0 0
$$218$$ 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 $$-0.0746262\pi$$
0.972643 + 0.232303i $$0.0746262\pi$$
$$224$$ 0 0
$$225$$ −575.000 −0.170370
$$226$$ −6248.00 −1.83899
$$227$$ −646.000 −0.188883 −0.0944417 0.995530i $$-0.530107\pi$$
−0.0944417 + 0.995530i $$0.530107\pi$$
$$228$$ 1600.00 0.464748
$$229$$ −3750.00 −1.08213 −0.541063 0.840982i $$-0.681978\pi$$
−0.541063 + 0.840982i $$0.681978\pi$$
$$230$$ 1560.00 0.447232
$$231$$ 0 0
$$232$$ 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
$$238$$ 0 0
$$239$$ 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 $$-0.632334\pi$$
−0.403867 + 0.914817i $$0.632334\pi$$
$$242$$ 1228.00 0.326194
$$243$$ −3542.00 −0.935059
$$244$$ 4144.00 1.08726
$$245$$ 0 0
$$246$$ −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 $$-0.449844\pi$$
0.156918 + 0.987612i $$0.449844\pi$$
$$252$$ 0 0
$$253$$ −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 $$-0.582267\pi$$
−0.255581 + 0.966788i $$0.582267\pi$$
$$258$$ 3536.00 0.853263
$$259$$ 0 0
$$260$$ 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
$$266$$ 0 0
$$267$$ −300.000 −0.0687629
$$268$$ 1008.00 0.229751
$$269$$ 6550.00 1.48461 0.742306 0.670061i $$-0.233732\pi$$
0.742306 + 0.670061i $$0.233732\pi$$
$$270$$ −2000.00 −0.450800
$$271$$ 4388.00 0.983587 0.491793 0.870712i $$-0.336342\pi$$
0.491793 + 0.870712i $$0.336342\pi$$
$$272$$ 1664.00 0.370937
$$273$$ 0 0
$$274$$ 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 $$-0.928441\pi$$
−0.974837 + 0.222920i $$0.928441\pi$$
$$284$$ 3296.00 0.688668
$$285$$ 1000.00 0.207842
$$286$$ −4864.00 −1.00564
$$287$$ 0 0
$$288$$ −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 $$-0.660350\pi$$
−0.482718 + 0.875776i $$0.660350\pi$$
$$294$$ 0 0
$$295$$ −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
$$301$$ 0 0
$$302$$ −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 $$-0.422485\pi$$
0.241120 + 0.970495i $$0.422485\pi$$
$$308$$ 0 0
$$309$$ −1196.00 −0.220188
$$310$$ −2160.00 −0.395741
$$311$$ −7332.00 −1.33685 −0.668424 0.743781i $$-0.733031\pi$$
−0.668424 + 0.743781i $$0.733031\pi$$
$$312$$ 0 0
$$313$$ −1562.00 −0.282075 −0.141037 0.990004i $$-0.545044\pi$$
−0.141037 + 0.990004i $$0.545044\pi$$
$$314$$ −9976.00 −1.79292
$$315$$ 0 0
$$316$$ 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
$$322$$ 0 0
$$323$$ 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
$$329$$ 0 0
$$330$$ 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
$$336$$ 0 0
$$337$$ 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
$$343$$ 0 0
$$344$$ 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 $$-0.528109\pi$$
−0.0881921 + 0.996103i $$0.528109\pi$$
$$350$$ 0 0
$$351$$ 3800.00 0.577860
$$352$$ 8192.00 1.24044
$$353$$ 4398.00 0.663122 0.331561 0.943434i $$-0.392425\pi$$
0.331561 + 0.943434i $$0.392425\pi$$
$$354$$ −4000.00 −0.600558
$$355$$ 2060.00 0.307982
$$356$$ 1200.00 0.178651
$$357$$ 0 0
$$358$$ 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
$$364$$ 0 0
$$365$$ 4390.00 0.629543
$$366$$ 4144.00 0.591832
$$367$$ 5874.00 0.835478 0.417739 0.908567i $$-0.362823\pi$$
0.417739 + 0.908567i $$0.362823\pi$$
$$368$$ 4992.00 0.707136
$$369$$ 506.000 0.0713857
$$370$$ −5320.00 −0.747496
$$371$$ 0 0
$$372$$ −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
$$378$$ 0 0
$$379$$ 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 $$-0.332780\pi$$
0.501504 + 0.865155i $$0.332780\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 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 $$-0.836794\pi$$
−0.871410 + 0.490555i $$0.836794\pi$$
$$398$$ −10400.0 −1.30981
$$399$$ 0 0
$$400$$ −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
$$406$$ 0 0
$$407$$ 8512.00 1.03667
$$408$$ 0 0
$$409$$ −11150.0 −1.34800 −0.674000 0.738731i $$-0.735425\pi$$
−0.674000 + 0.738731i $$0.735425\pi$$
$$410$$ 440.000 0.0530001
$$411$$ 4668.00 0.560232
$$412$$ 4784.00 0.572065
$$413$$ 0 0
$$414$$ −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 $$-0.205535\pi$$
0.798674 + 0.601764i $$0.205535\pi$$
$$420$$ 0 0
$$421$$ −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
$$427$$ 0 0
$$428$$ −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 $$-0.480239\pi$$
0.0620412 + 0.998074i $$0.480239\pi$$
$$434$$ 0 0
$$435$$ 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 $$-0.454861\pi$$
0.141334 + 0.989962i $$0.454861\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 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
$$448$$ 0 0
$$449$$ −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
$$455$$ 0 0
$$456$$ 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 $$-0.304310\pi$$
0.576778 + 0.816901i $$0.304310\pi$$
$$462$$ 0 0
$$463$$ 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 $$-0.604801\pi$$
−0.323327 + 0.946287i $$0.604801\pi$$
$$468$$ −6992.00 −0.690610
$$469$$ 0 0
$$470$$ −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
$$476$$ 0 0
$$477$$ −46.0000 −0.00441550
$$478$$ −5600.00 −0.535854
$$479$$ −17400.0 −1.65976 −0.829881 0.557940i $$-0.811592\pi$$
−0.829881 + 0.557940i $$0.811592\pi$$
$$480$$ −2560.00 −0.243432
$$481$$ 10108.0 0.958181
$$482$$ 12088.0 1.14231
$$483$$ 0 0
$$484$$ −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
$$490$$ 0 0
$$491$$ 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
$$497$$ 0 0
$$498$$ −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 $$-0.536791\pi$$
−0.115325 + 0.993328i $$0.536791\pi$$
$$504$$ 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 $$-0.661354\pi$$
−0.485476 + 0.874250i $$0.661354\pi$$
$$510$$ −1040.00 −0.0902980
$$511$$ 0 0
$$512$$ −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
$$518$$ 0 0
$$519$$ −156.000 −0.0131939
$$520$$ 0 0
$$521$$ 3638.00 0.305919 0.152959 0.988232i $$-0.451120\pi$$
0.152959 + 0.988232i $$0.451120\pi$$
$$522$$ −4600.00 −0.385702
$$523$$ 2078.00 0.173737 0.0868686 0.996220i $$-0.472314\pi$$
0.0868686 + 0.996220i $$0.472314\pi$$
$$524$$ 17664.0 1.47262
$$525$$ 0 0
$$526$$ 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
$$532$$ 0 0
$$533$$ −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
$$539$$ 0 0
$$540$$ 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
$$546$$ 0 0
$$547$$ 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
$$553$$ 0 0
$$554$$ −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
$$560$$ 0 0
$$561$$ 1664.00 0.125230
$$562$$ 27432.0 2.05898
$$563$$ 25038.0 1.87429 0.937146 0.348939i $$-0.113458\pi$$
0.937146 + 0.348939i $$0.113458\pi$$
$$564$$ −8224.00 −0.613994
$$565$$ 7810.00 0.581538
$$566$$ 37128.0 2.75725
$$567$$ 0 0
$$568$$ 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
$$574$$ 0 0
$$575$$ −1950.00 −0.141427
$$576$$ 11776.0 0.851852
$$577$$ 13654.0 0.985136 0.492568 0.870274i $$-0.336058\pi$$
0.492568 + 0.870274i $$0.336058\pi$$
$$578$$ 16948.0 1.21963
$$579$$ 716.000 0.0513920
$$580$$ −2000.00 −0.143182
$$581$$ 0 0
$$582$$ −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 $$-0.665945\pi$$
−0.498035 + 0.867157i $$0.665945\pi$$
$$588$$ 0 0
$$589$$ −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 $$-0.711966\pi$$
−0.617777 + 0.786354i $$0.711966\pi$$
$$594$$ −12800.0 −0.884159
$$595$$ 0 0
$$596$$ 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 $$-0.877211\pi$$
−0.926516 + 0.376256i $$0.877211\pi$$
$$602$$ 0 0
$$603$$ −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 $$-0.459514\pi$$
0.126848 + 0.991922i $$0.459514\pi$$
$$608$$ −25600.0 −1.70759
$$609$$ 0 0
$$610$$ −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 $$-0.586849\pi$$
−0.269471 + 0.963008i $$0.586849\pi$$
$$620$$ 4320.00 0.279831
$$621$$ −7800.00 −0.504031
$$622$$ 29328.0 1.89059
$$623$$ 0 0
$$624$$ 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
$$630$$ 0 0
$$631$$ −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
$$637$$ 0 0
$$638$$ 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 $$-0.517990\pi$$
−0.0564863 + 0.998403i $$0.517990\pi$$
$$644$$ 0 0
$$645$$ −4420.00 −0.269825
$$646$$ −10400.0 −0.633409
$$647$$ 10114.0 0.614563 0.307282 0.951619i $$-0.400581\pi$$
0.307282 + 0.951619i $$0.400581\pi$$
$$648$$ 0 0
$$649$$ −16000.0 −0.967727
$$650$$ −3800.00 −0.229305
$$651$$ 0 0
$$652$$ 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
$$658$$ 0 0
$$659$$ 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 $$-0.432841\pi$$
0.209424 + 0.977825i $$0.432841\pi$$
$$662$$ 16032.0 0.941241
$$663$$ 1976.00 0.115749
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 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
$$672$$ 0 0
$$673$$ −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 $$-0.174732\pi$$
0.853079 + 0.521782i $$0.174732\pi$$
$$678$$ 12496.0 0.707826
$$679$$ 0 0
$$680$$ 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
$$686$$ 0 0
$$687$$ 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 $$-0.798242\pi$$
−0.805759 + 0.592243i $$0.798242\pi$$
$$692$$ 624.000 0.0342788
$$693$$ 0 0
$$694$$ 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
$$700$$ 0 0
$$701$$ −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
$$707$$ 0 0
$$708$$ 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
$$714$$ 0 0
$$715$$ 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 $$-0.564838\pi$$
−0.202289 + 0.979326i $$0.564838\pi$$
$$720$$ 7360.00 0.380960
$$721$$ 0 0
$$722$$ −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 $$-0.429984\pi$$
0.218191 + 0.975906i $$0.429984\pi$$
$$728$$ 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 $$-0.523133\pi$$
−0.0726119 + 0.997360i $$0.523133\pi$$
$$734$$ −23496.0 −1.18154
$$735$$ 0 0
$$736$$ −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
$$742$$ 0 0
$$743$$ 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
$$749$$ 0 0
$$750$$ 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
$$756$$ 0 0
$$757$$ −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 $$-0.465681\pi$$
0.107607 + 0.994194i $$0.465681\pi$$
$$762$$ 14768.0 0.702084
$$763$$ 0 0
$$764$$ 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 $$-0.122111\pi$$
0.927314 + 0.374283i $$0.122111\pi$$
$$770$$ 0 0
$$771$$ 4212.00 0.196746
$$772$$ −2864.00 −0.133520
$$773$$ −22122.0 −1.02933 −0.514666 0.857391i $$-0.672084\pi$$
−0.514666 + 0.857391i $$0.672084\pi$$
$$774$$ 40664.0 1.88842
$$775$$ 2700.00 0.125144
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 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
$$784$$ 0 0
$$785$$ 12470.0 0.566972
$$786$$ 17664.0 0.801595
$$787$$ 16634.0 0.753416 0.376708 0.926332i $$-0.377056\pi$$
0.376708 + 0.926332i $$0.377056\pi$$
$$788$$ −17712.0 −0.800716
$$789$$ 7276.00 0.328305
$$790$$ −12000.0 −0.540431
$$791$$ 0 0
$$792$$ 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 $$-0.710044\pi$$
−0.613015 + 0.790071i $$0.710044\pi$$
$$798$$ 0 0
$$799$$ −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
$$805$$ 0 0
$$806$$ −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 $$-0.569687\pi$$
−0.217183 + 0.976131i $$0.569687\pi$$
$$812$$ 0 0
$$813$$ −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
$$819$$ 0 0
$$820$$ −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
$$826$$ 0 0
$$827$$ 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 $$-0.321806\pi$$
0.531026 + 0.847355i $$0.321806\pi$$
$$830$$ 5640.00 0.235864
$$831$$ −1092.00 −0.0455849
$$832$$ −19456.0 −0.810716
$$833$$ 0 0
$$834$$ 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 $$-0.895331\pi$$
−0.946422 + 0.322932i $$0.895331\pi$$
$$840$$ 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
$$847$$ 0 0
$$848$$ −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 $$-0.389184\pi$$
0.341149 + 0.940009i $$0.389184\pi$$
$$854$$ 0 0
$$855$$ 11500.0 0.459990
$$856$$ 0 0
$$857$$ 26494.0 1.05603 0.528015 0.849235i $$-0.322936\pi$$
0.528015 + 0.849235i $$0.322936\pi$$
$$858$$ 9728.00 0.387073
$$859$$ 21500.0 0.853982 0.426991 0.904256i $$-0.359574\pi$$
0.426991 + 0.904256i $$0.359574\pi$$
$$860$$ 17680.0 0.701027
$$861$$ 0 0
$$862$$ −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
$$868$$ 0 0
$$869$$ 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
$$875$$ 0 0
$$876$$ −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 $$-0.716895\pi$$
−0.629878 + 0.776694i $$0.716895\pi$$
$$882$$ 0 0
$$883$$ −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 $$-0.238946\pi$$
0.731230 + 0.682131i $$0.238946\pi$$
$$888$$ 0 0
$$889$$ 0 0
$$890$$ −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
$$903$$ 0 0
$$904$$ 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
$$910$$ 0 0
$$911$$ 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
$$917$$ 0 0
$$918$$ 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
$$924$$ 0 0
$$925$$ 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 $$-0.171042\pi$$
0.859071 + 0.511856i $$0.171042\pi$$
$$930$$ 4320.00 0.152321
$$931$$ 0 0
$$932$$ 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 $$-0.436692\pi$$
0.197580 + 0.980287i $$0.436692\pi$$
$$938$$ 0 0
$$939$$ 3124.00 0.108571
$$940$$ 20560.0 0.713397
$$941$$ 31178.0 1.08010 0.540050 0.841633i $$-0.318405\pi$$
0.540050 + 0.841633i $$0.318405\pi$$
$$942$$ 19952.0 0.690097
$$943$$ 1716.00 0.0592584
$$944$$ 32000.0 1.10330
$$945$$ 0 0
$$946$$ −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
$$959$$ 0 0
$$960$$ 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
$$966$$ 0 0
$$967$$ 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 $$-0.631600\pi$$
−0.401756 + 0.915747i $$0.631600\pi$$
$$972$$ −28336.0 −0.935059
$$973$$ 0 0
$$974$$ −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
$$980$$ 0 0
$$981$$ 12650.0 0.411706
$$982$$ −28288.0 −0.919253
$$983$$ −42282.0 −1.37191 −0.685954 0.727645i $$-0.740615\pi$$
−0.685954 + 0.727645i $$0.740615\pi$$
$$984$$ 0 0
$$985$$ −11070.0 −0.358091
$$986$$ −5200.00 −0.167953
$$987$$ 0 0
$$988$$ −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
$$994$$ 0 0
$$995$$ 13000.0 0.414199
$$996$$ 4512.00 0.143542
$$997$$ 31614.0 1.00424 0.502119 0.864798i $$-0.332554\pi$$
0.502119 + 0.864798i $$0.332554\pi$$
$$998$$ −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 245.4.a.a.1.1 1
3.2 odd 2 2205.4.a.q.1.1 1
5.4 even 2 1225.4.a.k.1.1 1
7.2 even 3 245.4.e.g.116.1 2
7.3 odd 6 245.4.e.f.226.1 2
7.4 even 3 245.4.e.g.226.1 2
7.5 odd 6 245.4.e.f.116.1 2
7.6 odd 2 5.4.a.a.1.1 1
21.20 even 2 45.4.a.d.1.1 1
28.27 even 2 80.4.a.d.1.1 1
35.13 even 4 25.4.b.a.24.2 2
35.27 even 4 25.4.b.a.24.1 2
35.34 odd 2 25.4.a.c.1.1 1
56.13 odd 2 320.4.a.g.1.1 1
56.27 even 2 320.4.a.h.1.1 1
63.13 odd 6 405.4.e.l.136.1 2
63.20 even 6 405.4.e.c.271.1 2
63.34 odd 6 405.4.e.l.271.1 2
63.41 even 6 405.4.e.c.136.1 2
77.76 even 2 605.4.a.d.1.1 1
84.83 odd 2 720.4.a.u.1.1 1
91.90 odd 2 845.4.a.b.1.1 1
105.62 odd 4 225.4.b.c.199.2 2
105.83 odd 4 225.4.b.c.199.1 2
105.104 even 2 225.4.a.b.1.1 1
112.13 odd 4 1280.4.d.e.641.2 2
112.27 even 4 1280.4.d.l.641.2 2
112.69 odd 4 1280.4.d.e.641.1 2
112.83 even 4 1280.4.d.l.641.1 2
119.118 odd 2 1445.4.a.a.1.1 1
133.132 even 2 1805.4.a.h.1.1 1
140.27 odd 4 400.4.c.k.49.2 2
140.83 odd 4 400.4.c.k.49.1 2
140.139 even 2 400.4.a.m.1.1 1
280.69 odd 2 1600.4.a.bi.1.1 1
280.139 even 2 1600.4.a.s.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
5.4.a.a.1.1 1 7.6 odd 2
25.4.a.c.1.1 1 35.34 odd 2
25.4.b.a.24.1 2 35.27 even 4
25.4.b.a.24.2 2 35.13 even 4
45.4.a.d.1.1 1 21.20 even 2
80.4.a.d.1.1 1 28.27 even 2
225.4.a.b.1.1 1 105.104 even 2
225.4.b.c.199.1 2 105.83 odd 4
225.4.b.c.199.2 2 105.62 odd 4
245.4.a.a.1.1 1 1.1 even 1 trivial
245.4.e.f.116.1 2 7.5 odd 6
245.4.e.f.226.1 2 7.3 odd 6
245.4.e.g.116.1 2 7.2 even 3
245.4.e.g.226.1 2 7.4 even 3
320.4.a.g.1.1 1 56.13 odd 2
320.4.a.h.1.1 1 56.27 even 2
400.4.a.m.1.1 1 140.139 even 2
400.4.c.k.49.1 2 140.83 odd 4
400.4.c.k.49.2 2 140.27 odd 4
405.4.e.c.136.1 2 63.41 even 6
405.4.e.c.271.1 2 63.20 even 6
405.4.e.l.136.1 2 63.13 odd 6
405.4.e.l.271.1 2 63.34 odd 6
605.4.a.d.1.1 1 77.76 even 2
720.4.a.u.1.1 1 84.83 odd 2
845.4.a.b.1.1 1 91.90 odd 2
1225.4.a.k.1.1 1 5.4 even 2
1280.4.d.e.641.1 2 112.69 odd 4
1280.4.d.e.641.2 2 112.13 odd 4
1280.4.d.l.641.1 2 112.83 even 4
1280.4.d.l.641.2 2 112.27 even 4
1445.4.a.a.1.1 1 119.118 odd 2
1600.4.a.s.1.1 1 280.139 even 2
1600.4.a.bi.1.1 1 280.69 odd 2
1805.4.a.h.1.1 1 133.132 even 2
2205.4.a.q.1.1 1 3.2 odd 2