# Properties

 Label 15.4.a.a.1.1 Level $15$ Weight $4$ Character 15.1 Self dual yes Analytic conductor $0.885$ 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] = [15,4,Mod(1,15)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(15, 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("15.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$15 = 3 \cdot 5$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 15.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: $$0.885028650086$$ 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 $$\chi$$ $$=$$ 15.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+1.00000 q^{2} +3.00000 q^{3} -7.00000 q^{4} +5.00000 q^{5} +3.00000 q^{6} -24.0000 q^{7} -15.0000 q^{8} +9.00000 q^{9} +O(q^{10})$$ $$q+1.00000 q^{2} +3.00000 q^{3} -7.00000 q^{4} +5.00000 q^{5} +3.00000 q^{6} -24.0000 q^{7} -15.0000 q^{8} +9.00000 q^{9} +5.00000 q^{10} +52.0000 q^{11} -21.0000 q^{12} +22.0000 q^{13} -24.0000 q^{14} +15.0000 q^{15} +41.0000 q^{16} -14.0000 q^{17} +9.00000 q^{18} -20.0000 q^{19} -35.0000 q^{20} -72.0000 q^{21} +52.0000 q^{22} -168.000 q^{23} -45.0000 q^{24} +25.0000 q^{25} +22.0000 q^{26} +27.0000 q^{27} +168.000 q^{28} +230.000 q^{29} +15.0000 q^{30} -288.000 q^{31} +161.000 q^{32} +156.000 q^{33} -14.0000 q^{34} -120.000 q^{35} -63.0000 q^{36} -34.0000 q^{37} -20.0000 q^{38} +66.0000 q^{39} -75.0000 q^{40} +122.000 q^{41} -72.0000 q^{42} -188.000 q^{43} -364.000 q^{44} +45.0000 q^{45} -168.000 q^{46} +256.000 q^{47} +123.000 q^{48} +233.000 q^{49} +25.0000 q^{50} -42.0000 q^{51} -154.000 q^{52} -338.000 q^{53} +27.0000 q^{54} +260.000 q^{55} +360.000 q^{56} -60.0000 q^{57} +230.000 q^{58} +100.000 q^{59} -105.000 q^{60} +742.000 q^{61} -288.000 q^{62} -216.000 q^{63} -167.000 q^{64} +110.000 q^{65} +156.000 q^{66} -84.0000 q^{67} +98.0000 q^{68} -504.000 q^{69} -120.000 q^{70} -328.000 q^{71} -135.000 q^{72} -38.0000 q^{73} -34.0000 q^{74} +75.0000 q^{75} +140.000 q^{76} -1248.00 q^{77} +66.0000 q^{78} -240.000 q^{79} +205.000 q^{80} +81.0000 q^{81} +122.000 q^{82} +1212.00 q^{83} +504.000 q^{84} -70.0000 q^{85} -188.000 q^{86} +690.000 q^{87} -780.000 q^{88} +330.000 q^{89} +45.0000 q^{90} -528.000 q^{91} +1176.00 q^{92} -864.000 q^{93} +256.000 q^{94} -100.000 q^{95} +483.000 q^{96} +866.000 q^{97} +233.000 q^{98} +468.000 q^{99} +O(q^{100})$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 1.00000 0.353553 0.176777 0.984251i $$-0.443433\pi$$
0.176777 + 0.984251i $$0.443433\pi$$
$$3$$ 3.00000 0.577350
$$4$$ −7.00000 −0.875000
$$5$$ 5.00000 0.447214
$$6$$ 3.00000 0.204124
$$7$$ −24.0000 −1.29588 −0.647939 0.761692i $$-0.724369\pi$$
−0.647939 + 0.761692i $$0.724369\pi$$
$$8$$ −15.0000 −0.662913
$$9$$ 9.00000 0.333333
$$10$$ 5.00000 0.158114
$$11$$ 52.0000 1.42533 0.712663 0.701506i $$-0.247489\pi$$
0.712663 + 0.701506i $$0.247489\pi$$
$$12$$ −21.0000 −0.505181
$$13$$ 22.0000 0.469362 0.234681 0.972072i $$-0.424595\pi$$
0.234681 + 0.972072i $$0.424595\pi$$
$$14$$ −24.0000 −0.458162
$$15$$ 15.0000 0.258199
$$16$$ 41.0000 0.640625
$$17$$ −14.0000 −0.199735 −0.0998676 0.995001i $$-0.531842\pi$$
−0.0998676 + 0.995001i $$0.531842\pi$$
$$18$$ 9.00000 0.117851
$$19$$ −20.0000 −0.241490 −0.120745 0.992684i $$-0.538528\pi$$
−0.120745 + 0.992684i $$0.538528\pi$$
$$20$$ −35.0000 −0.391312
$$21$$ −72.0000 −0.748176
$$22$$ 52.0000 0.503929
$$23$$ −168.000 −1.52306 −0.761531 0.648129i $$-0.775552\pi$$
−0.761531 + 0.648129i $$0.775552\pi$$
$$24$$ −45.0000 −0.382733
$$25$$ 25.0000 0.200000
$$26$$ 22.0000 0.165944
$$27$$ 27.0000 0.192450
$$28$$ 168.000 1.13389
$$29$$ 230.000 1.47276 0.736378 0.676570i $$-0.236535\pi$$
0.736378 + 0.676570i $$0.236535\pi$$
$$30$$ 15.0000 0.0912871
$$31$$ −288.000 −1.66859 −0.834296 0.551317i $$-0.814125\pi$$
−0.834296 + 0.551317i $$0.814125\pi$$
$$32$$ 161.000 0.889408
$$33$$ 156.000 0.822913
$$34$$ −14.0000 −0.0706171
$$35$$ −120.000 −0.579534
$$36$$ −63.0000 −0.291667
$$37$$ −34.0000 −0.151069 −0.0755347 0.997143i $$-0.524066\pi$$
−0.0755347 + 0.997143i $$0.524066\pi$$
$$38$$ −20.0000 −0.0853797
$$39$$ 66.0000 0.270986
$$40$$ −75.0000 −0.296464
$$41$$ 122.000 0.464712 0.232356 0.972631i $$-0.425357\pi$$
0.232356 + 0.972631i $$0.425357\pi$$
$$42$$ −72.0000 −0.264520
$$43$$ −188.000 −0.666738 −0.333369 0.942796i $$-0.608185\pi$$
−0.333369 + 0.942796i $$0.608185\pi$$
$$44$$ −364.000 −1.24716
$$45$$ 45.0000 0.149071
$$46$$ −168.000 −0.538484
$$47$$ 256.000 0.794499 0.397249 0.917711i $$-0.369965\pi$$
0.397249 + 0.917711i $$0.369965\pi$$
$$48$$ 123.000 0.369865
$$49$$ 233.000 0.679300
$$50$$ 25.0000 0.0707107
$$51$$ −42.0000 −0.115317
$$52$$ −154.000 −0.410691
$$53$$ −338.000 −0.875998 −0.437999 0.898976i $$-0.644313\pi$$
−0.437999 + 0.898976i $$0.644313\pi$$
$$54$$ 27.0000 0.0680414
$$55$$ 260.000 0.637425
$$56$$ 360.000 0.859054
$$57$$ −60.0000 −0.139424
$$58$$ 230.000 0.520698
$$59$$ 100.000 0.220659 0.110330 0.993895i $$-0.464809\pi$$
0.110330 + 0.993895i $$0.464809\pi$$
$$60$$ −105.000 −0.225924
$$61$$ 742.000 1.55743 0.778716 0.627376i $$-0.215871\pi$$
0.778716 + 0.627376i $$0.215871\pi$$
$$62$$ −288.000 −0.589936
$$63$$ −216.000 −0.431959
$$64$$ −167.000 −0.326172
$$65$$ 110.000 0.209905
$$66$$ 156.000 0.290944
$$67$$ −84.0000 −0.153168 −0.0765838 0.997063i $$-0.524401\pi$$
−0.0765838 + 0.997063i $$0.524401\pi$$
$$68$$ 98.0000 0.174768
$$69$$ −504.000 −0.879340
$$70$$ −120.000 −0.204896
$$71$$ −328.000 −0.548260 −0.274130 0.961693i $$-0.588390\pi$$
−0.274130 + 0.961693i $$0.588390\pi$$
$$72$$ −135.000 −0.220971
$$73$$ −38.0000 −0.0609255 −0.0304628 0.999536i $$-0.509698\pi$$
−0.0304628 + 0.999536i $$0.509698\pi$$
$$74$$ −34.0000 −0.0534111
$$75$$ 75.0000 0.115470
$$76$$ 140.000 0.211304
$$77$$ −1248.00 −1.84705
$$78$$ 66.0000 0.0958081
$$79$$ −240.000 −0.341799 −0.170899 0.985288i $$-0.554667\pi$$
−0.170899 + 0.985288i $$0.554667\pi$$
$$80$$ 205.000 0.286496
$$81$$ 81.0000 0.111111
$$82$$ 122.000 0.164301
$$83$$ 1212.00 1.60282 0.801411 0.598114i $$-0.204083\pi$$
0.801411 + 0.598114i $$0.204083\pi$$
$$84$$ 504.000 0.654654
$$85$$ −70.0000 −0.0893243
$$86$$ −188.000 −0.235727
$$87$$ 690.000 0.850296
$$88$$ −780.000 −0.944867
$$89$$ 330.000 0.393033 0.196516 0.980501i $$-0.437037\pi$$
0.196516 + 0.980501i $$0.437037\pi$$
$$90$$ 45.0000 0.0527046
$$91$$ −528.000 −0.608236
$$92$$ 1176.00 1.33268
$$93$$ −864.000 −0.963362
$$94$$ 256.000 0.280898
$$95$$ −100.000 −0.107998
$$96$$ 483.000 0.513500
$$97$$ 866.000 0.906484 0.453242 0.891387i $$-0.350267\pi$$
0.453242 + 0.891387i $$0.350267\pi$$
$$98$$ 233.000 0.240169
$$99$$ 468.000 0.475109
$$100$$ −175.000 −0.175000
$$101$$ −1218.00 −1.19996 −0.599978 0.800017i $$-0.704824\pi$$
−0.599978 + 0.800017i $$0.704824\pi$$
$$102$$ −42.0000 −0.0407708
$$103$$ −88.0000 −0.0841835 −0.0420917 0.999114i $$-0.513402\pi$$
−0.0420917 + 0.999114i $$0.513402\pi$$
$$104$$ −330.000 −0.311146
$$105$$ −360.000 −0.334594
$$106$$ −338.000 −0.309712
$$107$$ 36.0000 0.0325257 0.0162629 0.999868i $$-0.494823\pi$$
0.0162629 + 0.999868i $$0.494823\pi$$
$$108$$ −189.000 −0.168394
$$109$$ −970.000 −0.852378 −0.426189 0.904634i $$-0.640144\pi$$
−0.426189 + 0.904634i $$0.640144\pi$$
$$110$$ 260.000 0.225364
$$111$$ −102.000 −0.0872199
$$112$$ −984.000 −0.830172
$$113$$ 1042.00 0.867461 0.433731 0.901043i $$-0.357197\pi$$
0.433731 + 0.901043i $$0.357197\pi$$
$$114$$ −60.0000 −0.0492940
$$115$$ −840.000 −0.681134
$$116$$ −1610.00 −1.28866
$$117$$ 198.000 0.156454
$$118$$ 100.000 0.0780148
$$119$$ 336.000 0.258833
$$120$$ −225.000 −0.171163
$$121$$ 1373.00 1.03156
$$122$$ 742.000 0.550635
$$123$$ 366.000 0.268302
$$124$$ 2016.00 1.46002
$$125$$ 125.000 0.0894427
$$126$$ −216.000 −0.152721
$$127$$ 1936.00 1.35269 0.676347 0.736583i $$-0.263562\pi$$
0.676347 + 0.736583i $$0.263562\pi$$
$$128$$ −1455.00 −1.00473
$$129$$ −564.000 −0.384941
$$130$$ 110.000 0.0742126
$$131$$ 732.000 0.488207 0.244104 0.969749i $$-0.421506\pi$$
0.244104 + 0.969749i $$0.421506\pi$$
$$132$$ −1092.00 −0.720048
$$133$$ 480.000 0.312942
$$134$$ −84.0000 −0.0541529
$$135$$ 135.000 0.0860663
$$136$$ 210.000 0.132407
$$137$$ −2214.00 −1.38069 −0.690346 0.723479i $$-0.742542\pi$$
−0.690346 + 0.723479i $$0.742542\pi$$
$$138$$ −504.000 −0.310894
$$139$$ 20.0000 0.0122042 0.00610208 0.999981i $$-0.498058\pi$$
0.00610208 + 0.999981i $$0.498058\pi$$
$$140$$ 840.000 0.507093
$$141$$ 768.000 0.458704
$$142$$ −328.000 −0.193839
$$143$$ 1144.00 0.668994
$$144$$ 369.000 0.213542
$$145$$ 1150.00 0.658637
$$146$$ −38.0000 −0.0215404
$$147$$ 699.000 0.392194
$$148$$ 238.000 0.132186
$$149$$ −1330.00 −0.731261 −0.365630 0.930760i $$-0.619147\pi$$
−0.365630 + 0.930760i $$0.619147\pi$$
$$150$$ 75.0000 0.0408248
$$151$$ −1208.00 −0.651031 −0.325515 0.945537i $$-0.605538\pi$$
−0.325515 + 0.945537i $$0.605538\pi$$
$$152$$ 300.000 0.160087
$$153$$ −126.000 −0.0665784
$$154$$ −1248.00 −0.653031
$$155$$ −1440.00 −0.746217
$$156$$ −462.000 −0.237113
$$157$$ −3514.00 −1.78629 −0.893146 0.449768i $$-0.851507\pi$$
−0.893146 + 0.449768i $$0.851507\pi$$
$$158$$ −240.000 −0.120844
$$159$$ −1014.00 −0.505757
$$160$$ 805.000 0.397755
$$161$$ 4032.00 1.97370
$$162$$ 81.0000 0.0392837
$$163$$ −2068.00 −0.993732 −0.496866 0.867827i $$-0.665516\pi$$
−0.496866 + 0.867827i $$0.665516\pi$$
$$164$$ −854.000 −0.406623
$$165$$ 780.000 0.368018
$$166$$ 1212.00 0.566683
$$167$$ −24.0000 −0.0111208 −0.00556041 0.999985i $$-0.501770\pi$$
−0.00556041 + 0.999985i $$0.501770\pi$$
$$168$$ 1080.00 0.495975
$$169$$ −1713.00 −0.779700
$$170$$ −70.0000 −0.0315809
$$171$$ −180.000 −0.0804967
$$172$$ 1316.00 0.583396
$$173$$ −618.000 −0.271593 −0.135797 0.990737i $$-0.543359\pi$$
−0.135797 + 0.990737i $$0.543359\pi$$
$$174$$ 690.000 0.300625
$$175$$ −600.000 −0.259176
$$176$$ 2132.00 0.913100
$$177$$ 300.000 0.127398
$$178$$ 330.000 0.138958
$$179$$ 3340.00 1.39466 0.697328 0.716752i $$-0.254372\pi$$
0.697328 + 0.716752i $$0.254372\pi$$
$$180$$ −315.000 −0.130437
$$181$$ −178.000 −0.0730974 −0.0365487 0.999332i $$-0.511636\pi$$
−0.0365487 + 0.999332i $$0.511636\pi$$
$$182$$ −528.000 −0.215044
$$183$$ 2226.00 0.899184
$$184$$ 2520.00 1.00966
$$185$$ −170.000 −0.0675603
$$186$$ −864.000 −0.340600
$$187$$ −728.000 −0.284688
$$188$$ −1792.00 −0.695186
$$189$$ −648.000 −0.249392
$$190$$ −100.000 −0.0381830
$$191$$ −1888.00 −0.715240 −0.357620 0.933867i $$-0.616412\pi$$
−0.357620 + 0.933867i $$0.616412\pi$$
$$192$$ −501.000 −0.188315
$$193$$ 1922.00 0.716832 0.358416 0.933562i $$-0.383317\pi$$
0.358416 + 0.933562i $$0.383317\pi$$
$$194$$ 866.000 0.320491
$$195$$ 330.000 0.121189
$$196$$ −1631.00 −0.594388
$$197$$ 2526.00 0.913554 0.456777 0.889581i $$-0.349004\pi$$
0.456777 + 0.889581i $$0.349004\pi$$
$$198$$ 468.000 0.167976
$$199$$ −1160.00 −0.413217 −0.206609 0.978424i $$-0.566243\pi$$
−0.206609 + 0.978424i $$0.566243\pi$$
$$200$$ −375.000 −0.132583
$$201$$ −252.000 −0.0884314
$$202$$ −1218.00 −0.424248
$$203$$ −5520.00 −1.90851
$$204$$ 294.000 0.100903
$$205$$ 610.000 0.207826
$$206$$ −88.0000 −0.0297634
$$207$$ −1512.00 −0.507687
$$208$$ 902.000 0.300685
$$209$$ −1040.00 −0.344202
$$210$$ −360.000 −0.118297
$$211$$ −4468.00 −1.45777 −0.728886 0.684635i $$-0.759961\pi$$
−0.728886 + 0.684635i $$0.759961\pi$$
$$212$$ 2366.00 0.766498
$$213$$ −984.000 −0.316538
$$214$$ 36.0000 0.0114996
$$215$$ −940.000 −0.298174
$$216$$ −405.000 −0.127578
$$217$$ 6912.00 2.16229
$$218$$ −970.000 −0.301361
$$219$$ −114.000 −0.0351754
$$220$$ −1820.00 −0.557747
$$221$$ −308.000 −0.0937481
$$222$$ −102.000 −0.0308369
$$223$$ 6032.00 1.81136 0.905678 0.423965i $$-0.139362\pi$$
0.905678 + 0.423965i $$0.139362\pi$$
$$224$$ −3864.00 −1.15256
$$225$$ 225.000 0.0666667
$$226$$ 1042.00 0.306694
$$227$$ 2636.00 0.770738 0.385369 0.922763i $$-0.374074\pi$$
0.385369 + 0.922763i $$0.374074\pi$$
$$228$$ 420.000 0.121996
$$229$$ 4830.00 1.39378 0.696889 0.717179i $$-0.254567\pi$$
0.696889 + 0.717179i $$0.254567\pi$$
$$230$$ −840.000 −0.240817
$$231$$ −3744.00 −1.06639
$$232$$ −3450.00 −0.976309
$$233$$ 2682.00 0.754093 0.377046 0.926194i $$-0.376940\pi$$
0.377046 + 0.926194i $$0.376940\pi$$
$$234$$ 198.000 0.0553148
$$235$$ 1280.00 0.355311
$$236$$ −700.000 −0.193077
$$237$$ −720.000 −0.197338
$$238$$ 336.000 0.0915111
$$239$$ 2320.00 0.627901 0.313950 0.949439i $$-0.398347\pi$$
0.313950 + 0.949439i $$0.398347\pi$$
$$240$$ 615.000 0.165409
$$241$$ 2002.00 0.535104 0.267552 0.963543i $$-0.413785\pi$$
0.267552 + 0.963543i $$0.413785\pi$$
$$242$$ 1373.00 0.364710
$$243$$ 243.000 0.0641500
$$244$$ −5194.00 −1.36275
$$245$$ 1165.00 0.303792
$$246$$ 366.000 0.0948590
$$247$$ −440.000 −0.113346
$$248$$ 4320.00 1.10613
$$249$$ 3636.00 0.925390
$$250$$ 125.000 0.0316228
$$251$$ 132.000 0.0331943 0.0165971 0.999862i $$-0.494717\pi$$
0.0165971 + 0.999862i $$0.494717\pi$$
$$252$$ 1512.00 0.377964
$$253$$ −8736.00 −2.17086
$$254$$ 1936.00 0.478250
$$255$$ −210.000 −0.0515714
$$256$$ −119.000 −0.0290527
$$257$$ −7614.00 −1.84805 −0.924024 0.382335i $$-0.875120\pi$$
−0.924024 + 0.382335i $$0.875120\pi$$
$$258$$ −564.000 −0.136097
$$259$$ 816.000 0.195767
$$260$$ −770.000 −0.183667
$$261$$ 2070.00 0.490919
$$262$$ 732.000 0.172607
$$263$$ −4888.00 −1.14603 −0.573017 0.819543i $$-0.694227\pi$$
−0.573017 + 0.819543i $$0.694227\pi$$
$$264$$ −2340.00 −0.545519
$$265$$ −1690.00 −0.391758
$$266$$ 480.000 0.110642
$$267$$ 990.000 0.226918
$$268$$ 588.000 0.134022
$$269$$ 1270.00 0.287856 0.143928 0.989588i $$-0.454027\pi$$
0.143928 + 0.989588i $$0.454027\pi$$
$$270$$ 135.000 0.0304290
$$271$$ 1072.00 0.240293 0.120146 0.992756i $$-0.461664\pi$$
0.120146 + 0.992756i $$0.461664\pi$$
$$272$$ −574.000 −0.127955
$$273$$ −1584.00 −0.351165
$$274$$ −2214.00 −0.488148
$$275$$ 1300.00 0.285065
$$276$$ 3528.00 0.769423
$$277$$ −5394.00 −1.17001 −0.585007 0.811028i $$-0.698908\pi$$
−0.585007 + 0.811028i $$0.698908\pi$$
$$278$$ 20.0000 0.00431482
$$279$$ −2592.00 −0.556197
$$280$$ 1800.00 0.384181
$$281$$ 2442.00 0.518425 0.259213 0.965820i $$-0.416537\pi$$
0.259213 + 0.965820i $$0.416537\pi$$
$$282$$ 768.000 0.162176
$$283$$ 2772.00 0.582255 0.291128 0.956684i $$-0.405970\pi$$
0.291128 + 0.956684i $$0.405970\pi$$
$$284$$ 2296.00 0.479727
$$285$$ −300.000 −0.0623525
$$286$$ 1144.00 0.236525
$$287$$ −2928.00 −0.602210
$$288$$ 1449.00 0.296469
$$289$$ −4717.00 −0.960106
$$290$$ 1150.00 0.232863
$$291$$ 2598.00 0.523359
$$292$$ 266.000 0.0533098
$$293$$ 4542.00 0.905619 0.452810 0.891607i $$-0.350422\pi$$
0.452810 + 0.891607i $$0.350422\pi$$
$$294$$ 699.000 0.138662
$$295$$ 500.000 0.0986818
$$296$$ 510.000 0.100146
$$297$$ 1404.00 0.274304
$$298$$ −1330.00 −0.258540
$$299$$ −3696.00 −0.714867
$$300$$ −525.000 −0.101036
$$301$$ 4512.00 0.864011
$$302$$ −1208.00 −0.230174
$$303$$ −3654.00 −0.692795
$$304$$ −820.000 −0.154705
$$305$$ 3710.00 0.696505
$$306$$ −126.000 −0.0235390
$$307$$ 5116.00 0.951093 0.475546 0.879691i $$-0.342250\pi$$
0.475546 + 0.879691i $$0.342250\pi$$
$$308$$ 8736.00 1.61617
$$309$$ −264.000 −0.0486034
$$310$$ −1440.00 −0.263827
$$311$$ −2808.00 −0.511984 −0.255992 0.966679i $$-0.582402\pi$$
−0.255992 + 0.966679i $$0.582402\pi$$
$$312$$ −990.000 −0.179640
$$313$$ −7318.00 −1.32153 −0.660763 0.750594i $$-0.729767\pi$$
−0.660763 + 0.750594i $$0.729767\pi$$
$$314$$ −3514.00 −0.631549
$$315$$ −1080.00 −0.193178
$$316$$ 1680.00 0.299074
$$317$$ 2246.00 0.397943 0.198971 0.980005i $$-0.436240\pi$$
0.198971 + 0.980005i $$0.436240\pi$$
$$318$$ −1014.00 −0.178812
$$319$$ 11960.0 2.09916
$$320$$ −835.000 −0.145868
$$321$$ 108.000 0.0187787
$$322$$ 4032.00 0.697809
$$323$$ 280.000 0.0482341
$$324$$ −567.000 −0.0972222
$$325$$ 550.000 0.0938723
$$326$$ −2068.00 −0.351337
$$327$$ −2910.00 −0.492120
$$328$$ −1830.00 −0.308064
$$329$$ −6144.00 −1.02957
$$330$$ 780.000 0.130114
$$331$$ 1332.00 0.221188 0.110594 0.993866i $$-0.464725\pi$$
0.110594 + 0.993866i $$0.464725\pi$$
$$332$$ −8484.00 −1.40247
$$333$$ −306.000 −0.0503564
$$334$$ −24.0000 −0.00393180
$$335$$ −420.000 −0.0684987
$$336$$ −2952.00 −0.479300
$$337$$ −11534.0 −1.86438 −0.932191 0.361966i $$-0.882106\pi$$
−0.932191 + 0.361966i $$0.882106\pi$$
$$338$$ −1713.00 −0.275665
$$339$$ 3126.00 0.500829
$$340$$ 490.000 0.0781588
$$341$$ −14976.0 −2.37829
$$342$$ −180.000 −0.0284599
$$343$$ 2640.00 0.415588
$$344$$ 2820.00 0.441989
$$345$$ −2520.00 −0.393253
$$346$$ −618.000 −0.0960228
$$347$$ 11956.0 1.84966 0.924830 0.380382i $$-0.124207\pi$$
0.924830 + 0.380382i $$0.124207\pi$$
$$348$$ −4830.00 −0.744009
$$349$$ 4870.00 0.746949 0.373474 0.927640i $$-0.378166\pi$$
0.373474 + 0.927640i $$0.378166\pi$$
$$350$$ −600.000 −0.0916324
$$351$$ 594.000 0.0903287
$$352$$ 8372.00 1.26770
$$353$$ 10722.0 1.61664 0.808321 0.588742i $$-0.200377\pi$$
0.808321 + 0.588742i $$0.200377\pi$$
$$354$$ 300.000 0.0450419
$$355$$ −1640.00 −0.245189
$$356$$ −2310.00 −0.343904
$$357$$ 1008.00 0.149437
$$358$$ 3340.00 0.493085
$$359$$ 120.000 0.0176417 0.00882083 0.999961i $$-0.497192\pi$$
0.00882083 + 0.999961i $$0.497192\pi$$
$$360$$ −675.000 −0.0988212
$$361$$ −6459.00 −0.941682
$$362$$ −178.000 −0.0258438
$$363$$ 4119.00 0.595569
$$364$$ 3696.00 0.532206
$$365$$ −190.000 −0.0272467
$$366$$ 2226.00 0.317910
$$367$$ 3936.00 0.559830 0.279915 0.960025i $$-0.409694\pi$$
0.279915 + 0.960025i $$0.409694\pi$$
$$368$$ −6888.00 −0.975711
$$369$$ 1098.00 0.154904
$$370$$ −170.000 −0.0238862
$$371$$ 8112.00 1.13519
$$372$$ 6048.00 0.842941
$$373$$ 3022.00 0.419499 0.209750 0.977755i $$-0.432735\pi$$
0.209750 + 0.977755i $$0.432735\pi$$
$$374$$ −728.000 −0.100652
$$375$$ 375.000 0.0516398
$$376$$ −3840.00 −0.526683
$$377$$ 5060.00 0.691255
$$378$$ −648.000 −0.0881733
$$379$$ −13340.0 −1.80799 −0.903997 0.427539i $$-0.859381\pi$$
−0.903997 + 0.427539i $$0.859381\pi$$
$$380$$ 700.000 0.0944980
$$381$$ 5808.00 0.780979
$$382$$ −1888.00 −0.252876
$$383$$ −1008.00 −0.134481 −0.0672407 0.997737i $$-0.521420\pi$$
−0.0672407 + 0.997737i $$0.521420\pi$$
$$384$$ −4365.00 −0.580079
$$385$$ −6240.00 −0.826026
$$386$$ 1922.00 0.253438
$$387$$ −1692.00 −0.222246
$$388$$ −6062.00 −0.793174
$$389$$ 9630.00 1.25517 0.627584 0.778549i $$-0.284044\pi$$
0.627584 + 0.778549i $$0.284044\pi$$
$$390$$ 330.000 0.0428467
$$391$$ 2352.00 0.304209
$$392$$ −3495.00 −0.450317
$$393$$ 2196.00 0.281867
$$394$$ 2526.00 0.322990
$$395$$ −1200.00 −0.152857
$$396$$ −3276.00 −0.415720
$$397$$ 7126.00 0.900866 0.450433 0.892810i $$-0.351270\pi$$
0.450433 + 0.892810i $$0.351270\pi$$
$$398$$ −1160.00 −0.146094
$$399$$ 1440.00 0.180677
$$400$$ 1025.00 0.128125
$$401$$ −8718.00 −1.08568 −0.542838 0.839837i $$-0.682650\pi$$
−0.542838 + 0.839837i $$0.682650\pi$$
$$402$$ −252.000 −0.0312652
$$403$$ −6336.00 −0.783173
$$404$$ 8526.00 1.04996
$$405$$ 405.000 0.0496904
$$406$$ −5520.00 −0.674761
$$407$$ −1768.00 −0.215323
$$408$$ 630.000 0.0764452
$$409$$ −10870.0 −1.31415 −0.657074 0.753826i $$-0.728206\pi$$
−0.657074 + 0.753826i $$0.728206\pi$$
$$410$$ 610.000 0.0734774
$$411$$ −6642.00 −0.797143
$$412$$ 616.000 0.0736605
$$413$$ −2400.00 −0.285947
$$414$$ −1512.00 −0.179495
$$415$$ 6060.00 0.716804
$$416$$ 3542.00 0.417454
$$417$$ 60.0000 0.00704607
$$418$$ −1040.00 −0.121694
$$419$$ −9700.00 −1.13097 −0.565484 0.824759i $$-0.691311\pi$$
−0.565484 + 0.824759i $$0.691311\pi$$
$$420$$ 2520.00 0.292770
$$421$$ 862.000 0.0997893 0.0498947 0.998754i $$-0.484111\pi$$
0.0498947 + 0.998754i $$0.484111\pi$$
$$422$$ −4468.00 −0.515400
$$423$$ 2304.00 0.264833
$$424$$ 5070.00 0.580710
$$425$$ −350.000 −0.0399470
$$426$$ −984.000 −0.111913
$$427$$ −17808.0 −2.01824
$$428$$ −252.000 −0.0284600
$$429$$ 3432.00 0.386244
$$430$$ −940.000 −0.105421
$$431$$ 15792.0 1.76490 0.882452 0.470402i $$-0.155891\pi$$
0.882452 + 0.470402i $$0.155891\pi$$
$$432$$ 1107.00 0.123288
$$433$$ 11602.0 1.28766 0.643830 0.765169i $$-0.277345\pi$$
0.643830 + 0.765169i $$0.277345\pi$$
$$434$$ 6912.00 0.764485
$$435$$ 3450.00 0.380264
$$436$$ 6790.00 0.745830
$$437$$ 3360.00 0.367805
$$438$$ −114.000 −0.0124364
$$439$$ −440.000 −0.0478361 −0.0239181 0.999714i $$-0.507614\pi$$
−0.0239181 + 0.999714i $$0.507614\pi$$
$$440$$ −3900.00 −0.422557
$$441$$ 2097.00 0.226433
$$442$$ −308.000 −0.0331449
$$443$$ −10188.0 −1.09266 −0.546328 0.837571i $$-0.683975\pi$$
−0.546328 + 0.837571i $$0.683975\pi$$
$$444$$ 714.000 0.0763174
$$445$$ 1650.00 0.175770
$$446$$ 6032.00 0.640411
$$447$$ −3990.00 −0.422194
$$448$$ 4008.00 0.422679
$$449$$ −13310.0 −1.39897 −0.699485 0.714647i $$-0.746587\pi$$
−0.699485 + 0.714647i $$0.746587\pi$$
$$450$$ 225.000 0.0235702
$$451$$ 6344.00 0.662367
$$452$$ −7294.00 −0.759029
$$453$$ −3624.00 −0.375873
$$454$$ 2636.00 0.272497
$$455$$ −2640.00 −0.272011
$$456$$ 900.000 0.0924262
$$457$$ 3226.00 0.330210 0.165105 0.986276i $$-0.447204\pi$$
0.165105 + 0.986276i $$0.447204\pi$$
$$458$$ 4830.00 0.492775
$$459$$ −378.000 −0.0384391
$$460$$ 5880.00 0.595992
$$461$$ 6582.00 0.664977 0.332488 0.943107i $$-0.392112\pi$$
0.332488 + 0.943107i $$0.392112\pi$$
$$462$$ −3744.00 −0.377027
$$463$$ 15072.0 1.51286 0.756431 0.654073i $$-0.226941\pi$$
0.756431 + 0.654073i $$0.226941\pi$$
$$464$$ 9430.00 0.943484
$$465$$ −4320.00 −0.430828
$$466$$ 2682.00 0.266612
$$467$$ 476.000 0.0471663 0.0235831 0.999722i $$-0.492493\pi$$
0.0235831 + 0.999722i $$0.492493\pi$$
$$468$$ −1386.00 −0.136897
$$469$$ 2016.00 0.198487
$$470$$ 1280.00 0.125621
$$471$$ −10542.0 −1.03132
$$472$$ −1500.00 −0.146278
$$473$$ −9776.00 −0.950319
$$474$$ −720.000 −0.0697694
$$475$$ −500.000 −0.0482980
$$476$$ −2352.00 −0.226478
$$477$$ −3042.00 −0.291999
$$478$$ 2320.00 0.221997
$$479$$ −19680.0 −1.87725 −0.938624 0.344941i $$-0.887899\pi$$
−0.938624 + 0.344941i $$0.887899\pi$$
$$480$$ 2415.00 0.229644
$$481$$ −748.000 −0.0709062
$$482$$ 2002.00 0.189188
$$483$$ 12096.0 1.13952
$$484$$ −9611.00 −0.902611
$$485$$ 4330.00 0.405392
$$486$$ 243.000 0.0226805
$$487$$ −5944.00 −0.553077 −0.276538 0.961003i $$-0.589187\pi$$
−0.276538 + 0.961003i $$0.589187\pi$$
$$488$$ −11130.0 −1.03244
$$489$$ −6204.00 −0.573731
$$490$$ 1165.00 0.107407
$$491$$ 10772.0 0.990089 0.495044 0.868868i $$-0.335152\pi$$
0.495044 + 0.868868i $$0.335152\pi$$
$$492$$ −2562.00 −0.234764
$$493$$ −3220.00 −0.294161
$$494$$ −440.000 −0.0400740
$$495$$ 2340.00 0.212475
$$496$$ −11808.0 −1.06894
$$497$$ 7872.00 0.710478
$$498$$ 3636.00 0.327175
$$499$$ 8140.00 0.730253 0.365127 0.930958i $$-0.381026\pi$$
0.365127 + 0.930958i $$0.381026\pi$$
$$500$$ −875.000 −0.0782624
$$501$$ −72.0000 −0.00642060
$$502$$ 132.000 0.0117360
$$503$$ −13768.0 −1.22045 −0.610223 0.792229i $$-0.708920\pi$$
−0.610223 + 0.792229i $$0.708920\pi$$
$$504$$ 3240.00 0.286351
$$505$$ −6090.00 −0.536637
$$506$$ −8736.00 −0.767515
$$507$$ −5139.00 −0.450160
$$508$$ −13552.0 −1.18361
$$509$$ 22150.0 1.92884 0.964422 0.264368i $$-0.0851633\pi$$
0.964422 + 0.264368i $$0.0851633\pi$$
$$510$$ −210.000 −0.0182332
$$511$$ 912.000 0.0789521
$$512$$ 11521.0 0.994455
$$513$$ −540.000 −0.0464748
$$514$$ −7614.00 −0.653384
$$515$$ −440.000 −0.0376480
$$516$$ 3948.00 0.336824
$$517$$ 13312.0 1.13242
$$518$$ 816.000 0.0692143
$$519$$ −1854.00 −0.156805
$$520$$ −1650.00 −0.139149
$$521$$ 1562.00 0.131348 0.0656741 0.997841i $$-0.479080\pi$$
0.0656741 + 0.997841i $$0.479080\pi$$
$$522$$ 2070.00 0.173566
$$523$$ −668.000 −0.0558501 −0.0279250 0.999610i $$-0.508890\pi$$
−0.0279250 + 0.999610i $$0.508890\pi$$
$$524$$ −5124.00 −0.427181
$$525$$ −1800.00 −0.149635
$$526$$ −4888.00 −0.405184
$$527$$ 4032.00 0.333276
$$528$$ 6396.00 0.527178
$$529$$ 16057.0 1.31972
$$530$$ −1690.00 −0.138507
$$531$$ 900.000 0.0735531
$$532$$ −3360.00 −0.273824
$$533$$ 2684.00 0.218118
$$534$$ 990.000 0.0802275
$$535$$ 180.000 0.0145459
$$536$$ 1260.00 0.101537
$$537$$ 10020.0 0.805205
$$538$$ 1270.00 0.101772
$$539$$ 12116.0 0.968225
$$540$$ −945.000 −0.0753080
$$541$$ −6138.00 −0.487788 −0.243894 0.969802i $$-0.578425\pi$$
−0.243894 + 0.969802i $$0.578425\pi$$
$$542$$ 1072.00 0.0849564
$$543$$ −534.000 −0.0422028
$$544$$ −2254.00 −0.177646
$$545$$ −4850.00 −0.381195
$$546$$ −1584.00 −0.124156
$$547$$ −10484.0 −0.819494 −0.409747 0.912199i $$-0.634383\pi$$
−0.409747 + 0.912199i $$0.634383\pi$$
$$548$$ 15498.0 1.20811
$$549$$ 6678.00 0.519144
$$550$$ 1300.00 0.100786
$$551$$ −4600.00 −0.355656
$$552$$ 7560.00 0.582926
$$553$$ 5760.00 0.442930
$$554$$ −5394.00 −0.413663
$$555$$ −510.000 −0.0390059
$$556$$ −140.000 −0.0106786
$$557$$ 3606.00 0.274311 0.137155 0.990550i $$-0.456204\pi$$
0.137155 + 0.990550i $$0.456204\pi$$
$$558$$ −2592.00 −0.196645
$$559$$ −4136.00 −0.312941
$$560$$ −4920.00 −0.371264
$$561$$ −2184.00 −0.164365
$$562$$ 2442.00 0.183291
$$563$$ 12252.0 0.917159 0.458579 0.888654i $$-0.348359\pi$$
0.458579 + 0.888654i $$0.348359\pi$$
$$564$$ −5376.00 −0.401366
$$565$$ 5210.00 0.387940
$$566$$ 2772.00 0.205858
$$567$$ −1944.00 −0.143986
$$568$$ 4920.00 0.363448
$$569$$ −14550.0 −1.07200 −0.536000 0.844218i $$-0.680065\pi$$
−0.536000 + 0.844218i $$0.680065\pi$$
$$570$$ −300.000 −0.0220449
$$571$$ −25468.0 −1.86655 −0.933277 0.359157i $$-0.883064\pi$$
−0.933277 + 0.359157i $$0.883064\pi$$
$$572$$ −8008.00 −0.585369
$$573$$ −5664.00 −0.412944
$$574$$ −2928.00 −0.212914
$$575$$ −4200.00 −0.304612
$$576$$ −1503.00 −0.108724
$$577$$ 12866.0 0.928282 0.464141 0.885761i $$-0.346363\pi$$
0.464141 + 0.885761i $$0.346363\pi$$
$$578$$ −4717.00 −0.339449
$$579$$ 5766.00 0.413863
$$580$$ −8050.00 −0.576307
$$581$$ −29088.0 −2.07706
$$582$$ 2598.00 0.185035
$$583$$ −17576.0 −1.24858
$$584$$ 570.000 0.0403883
$$585$$ 990.000 0.0699683
$$586$$ 4542.00 0.320185
$$587$$ −14844.0 −1.04374 −0.521872 0.853024i $$-0.674766\pi$$
−0.521872 + 0.853024i $$0.674766\pi$$
$$588$$ −4893.00 −0.343170
$$589$$ 5760.00 0.402948
$$590$$ 500.000 0.0348893
$$591$$ 7578.00 0.527440
$$592$$ −1394.00 −0.0967788
$$593$$ 20402.0 1.41283 0.706416 0.707797i $$-0.250311\pi$$
0.706416 + 0.707797i $$0.250311\pi$$
$$594$$ 1404.00 0.0969812
$$595$$ 1680.00 0.115753
$$596$$ 9310.00 0.639853
$$597$$ −3480.00 −0.238571
$$598$$ −3696.00 −0.252744
$$599$$ 10760.0 0.733959 0.366980 0.930229i $$-0.380392\pi$$
0.366980 + 0.930229i $$0.380392\pi$$
$$600$$ −1125.00 −0.0765466
$$601$$ 14282.0 0.969343 0.484671 0.874696i $$-0.338939\pi$$
0.484671 + 0.874696i $$0.338939\pi$$
$$602$$ 4512.00 0.305474
$$603$$ −756.000 −0.0510559
$$604$$ 8456.00 0.569652
$$605$$ 6865.00 0.461326
$$606$$ −3654.00 −0.244940
$$607$$ 11056.0 0.739290 0.369645 0.929173i $$-0.379479\pi$$
0.369645 + 0.929173i $$0.379479\pi$$
$$608$$ −3220.00 −0.214783
$$609$$ −16560.0 −1.10188
$$610$$ 3710.00 0.246252
$$611$$ 5632.00 0.372907
$$612$$ 882.000 0.0582561
$$613$$ −16418.0 −1.08176 −0.540878 0.841101i $$-0.681908\pi$$
−0.540878 + 0.841101i $$0.681908\pi$$
$$614$$ 5116.00 0.336262
$$615$$ 1830.00 0.119988
$$616$$ 18720.0 1.22443
$$617$$ −10374.0 −0.676891 −0.338445 0.940986i $$-0.609901\pi$$
−0.338445 + 0.940986i $$0.609901\pi$$
$$618$$ −264.000 −0.0171839
$$619$$ −5260.00 −0.341546 −0.170773 0.985310i $$-0.554627\pi$$
−0.170773 + 0.985310i $$0.554627\pi$$
$$620$$ 10080.0 0.652940
$$621$$ −4536.00 −0.293113
$$622$$ −2808.00 −0.181014
$$623$$ −7920.00 −0.509323
$$624$$ 2706.00 0.173600
$$625$$ 625.000 0.0400000
$$626$$ −7318.00 −0.467230
$$627$$ −3120.00 −0.198725
$$628$$ 24598.0 1.56300
$$629$$ 476.000 0.0301739
$$630$$ −1080.00 −0.0682988
$$631$$ 21352.0 1.34708 0.673542 0.739149i $$-0.264772\pi$$
0.673542 + 0.739149i $$0.264772\pi$$
$$632$$ 3600.00 0.226583
$$633$$ −13404.0 −0.841645
$$634$$ 2246.00 0.140694
$$635$$ 9680.00 0.604943
$$636$$ 7098.00 0.442538
$$637$$ 5126.00 0.318838
$$638$$ 11960.0 0.742164
$$639$$ −2952.00 −0.182753
$$640$$ −7275.00 −0.449328
$$641$$ −29118.0 −1.79422 −0.897108 0.441812i $$-0.854336\pi$$
−0.897108 + 0.441812i $$0.854336\pi$$
$$642$$ 108.000 0.00663928
$$643$$ 5772.00 0.354005 0.177003 0.984210i $$-0.443360\pi$$
0.177003 + 0.984210i $$0.443360\pi$$
$$644$$ −28224.0 −1.72699
$$645$$ −2820.00 −0.172151
$$646$$ 280.000 0.0170533
$$647$$ −14264.0 −0.866732 −0.433366 0.901218i $$-0.642674\pi$$
−0.433366 + 0.901218i $$0.642674\pi$$
$$648$$ −1215.00 −0.0736570
$$649$$ 5200.00 0.314511
$$650$$ 550.000 0.0331889
$$651$$ 20736.0 1.24840
$$652$$ 14476.0 0.869515
$$653$$ 6902.00 0.413623 0.206812 0.978381i $$-0.433691\pi$$
0.206812 + 0.978381i $$0.433691\pi$$
$$654$$ −2910.00 −0.173991
$$655$$ 3660.00 0.218333
$$656$$ 5002.00 0.297706
$$657$$ −342.000 −0.0203085
$$658$$ −6144.00 −0.364009
$$659$$ 20140.0 1.19051 0.595253 0.803539i $$-0.297052\pi$$
0.595253 + 0.803539i $$0.297052\pi$$
$$660$$ −5460.00 −0.322015
$$661$$ −3218.00 −0.189358 −0.0946790 0.995508i $$-0.530182\pi$$
−0.0946790 + 0.995508i $$0.530182\pi$$
$$662$$ 1332.00 0.0782019
$$663$$ −924.000 −0.0541255
$$664$$ −18180.0 −1.06253
$$665$$ 2400.00 0.139952
$$666$$ −306.000 −0.0178037
$$667$$ −38640.0 −2.24310
$$668$$ 168.000 0.00973071
$$669$$ 18096.0 1.04579
$$670$$ −420.000 −0.0242179
$$671$$ 38584.0 2.21985
$$672$$ −11592.0 −0.665433
$$673$$ −7518.00 −0.430606 −0.215303 0.976547i $$-0.569074\pi$$
−0.215303 + 0.976547i $$0.569074\pi$$
$$674$$ −11534.0 −0.659159
$$675$$ 675.000 0.0384900
$$676$$ 11991.0 0.682237
$$677$$ −18114.0 −1.02833 −0.514164 0.857692i $$-0.671898\pi$$
−0.514164 + 0.857692i $$0.671898\pi$$
$$678$$ 3126.00 0.177070
$$679$$ −20784.0 −1.17469
$$680$$ 1050.00 0.0592142
$$681$$ 7908.00 0.444986
$$682$$ −14976.0 −0.840851
$$683$$ −23868.0 −1.33716 −0.668582 0.743638i $$-0.733099\pi$$
−0.668582 + 0.743638i $$0.733099\pi$$
$$684$$ 1260.00 0.0704347
$$685$$ −11070.0 −0.617464
$$686$$ 2640.00 0.146932
$$687$$ 14490.0 0.804699
$$688$$ −7708.00 −0.427129
$$689$$ −7436.00 −0.411160
$$690$$ −2520.00 −0.139036
$$691$$ 172.000 0.00946916 0.00473458 0.999989i $$-0.498493\pi$$
0.00473458 + 0.999989i $$0.498493\pi$$
$$692$$ 4326.00 0.237644
$$693$$ −11232.0 −0.615683
$$694$$ 11956.0 0.653953
$$695$$ 100.000 0.00545787
$$696$$ −10350.0 −0.563672
$$697$$ −1708.00 −0.0928194
$$698$$ 4870.00 0.264086
$$699$$ 8046.00 0.435376
$$700$$ 4200.00 0.226779
$$701$$ −22138.0 −1.19278 −0.596391 0.802694i $$-0.703399\pi$$
−0.596391 + 0.802694i $$0.703399\pi$$
$$702$$ 594.000 0.0319360
$$703$$ 680.000 0.0364818
$$704$$ −8684.00 −0.464901
$$705$$ 3840.00 0.205139
$$706$$ 10722.0 0.571569
$$707$$ 29232.0 1.55500
$$708$$ −2100.00 −0.111473
$$709$$ 3070.00 0.162618 0.0813091 0.996689i $$-0.474090\pi$$
0.0813091 + 0.996689i $$0.474090\pi$$
$$710$$ −1640.00 −0.0866875
$$711$$ −2160.00 −0.113933
$$712$$ −4950.00 −0.260546
$$713$$ 48384.0 2.54137
$$714$$ 1008.00 0.0528340
$$715$$ 5720.00 0.299183
$$716$$ −23380.0 −1.22032
$$717$$ 6960.00 0.362519
$$718$$ 120.000 0.00623727
$$719$$ 15600.0 0.809154 0.404577 0.914504i $$-0.367419\pi$$
0.404577 + 0.914504i $$0.367419\pi$$
$$720$$ 1845.00 0.0954987
$$721$$ 2112.00 0.109092
$$722$$ −6459.00 −0.332935
$$723$$ 6006.00 0.308943
$$724$$ 1246.00 0.0639603
$$725$$ 5750.00 0.294551
$$726$$ 4119.00 0.210565
$$727$$ 20696.0 1.05581 0.527904 0.849304i $$-0.322978\pi$$
0.527904 + 0.849304i $$0.322978\pi$$
$$728$$ 7920.00 0.403207
$$729$$ 729.000 0.0370370
$$730$$ −190.000 −0.00963317
$$731$$ 2632.00 0.133171
$$732$$ −15582.0 −0.786786
$$733$$ −30778.0 −1.55090 −0.775451 0.631408i $$-0.782478\pi$$
−0.775451 + 0.631408i $$0.782478\pi$$
$$734$$ 3936.00 0.197930
$$735$$ 3495.00 0.175395
$$736$$ −27048.0 −1.35462
$$737$$ −4368.00 −0.218314
$$738$$ 1098.00 0.0547669
$$739$$ 11740.0 0.584388 0.292194 0.956359i $$-0.405615\pi$$
0.292194 + 0.956359i $$0.405615\pi$$
$$740$$ 1190.00 0.0591152
$$741$$ −1320.00 −0.0654405
$$742$$ 8112.00 0.401349
$$743$$ 2632.00 0.129958 0.0649789 0.997887i $$-0.479302\pi$$
0.0649789 + 0.997887i $$0.479302\pi$$
$$744$$ 12960.0 0.638625
$$745$$ −6650.00 −0.327030
$$746$$ 3022.00 0.148315
$$747$$ 10908.0 0.534274
$$748$$ 5096.00 0.249102
$$749$$ −864.000 −0.0421494
$$750$$ 375.000 0.0182574
$$751$$ −20528.0 −0.997440 −0.498720 0.866763i $$-0.666196\pi$$
−0.498720 + 0.866763i $$0.666196\pi$$
$$752$$ 10496.0 0.508976
$$753$$ 396.000 0.0191647
$$754$$ 5060.00 0.244396
$$755$$ −6040.00 −0.291150
$$756$$ 4536.00 0.218218
$$757$$ 21646.0 1.03928 0.519642 0.854384i $$-0.326066\pi$$
0.519642 + 0.854384i $$0.326066\pi$$
$$758$$ −13340.0 −0.639222
$$759$$ −26208.0 −1.25335
$$760$$ 1500.00 0.0715931
$$761$$ 18282.0 0.870857 0.435428 0.900223i $$-0.356597\pi$$
0.435428 + 0.900223i $$0.356597\pi$$
$$762$$ 5808.00 0.276118
$$763$$ 23280.0 1.10458
$$764$$ 13216.0 0.625835
$$765$$ −630.000 −0.0297748
$$766$$ −1008.00 −0.0475464
$$767$$ 2200.00 0.103569
$$768$$ −357.000 −0.0167736
$$769$$ −24190.0 −1.13435 −0.567174 0.823598i $$-0.691963\pi$$
−0.567174 + 0.823598i $$0.691963\pi$$
$$770$$ −6240.00 −0.292044
$$771$$ −22842.0 −1.06697
$$772$$ −13454.0 −0.627228
$$773$$ −25698.0 −1.19572 −0.597861 0.801600i $$-0.703982\pi$$
−0.597861 + 0.801600i $$0.703982\pi$$
$$774$$ −1692.00 −0.0785758
$$775$$ −7200.00 −0.333718
$$776$$ −12990.0 −0.600920
$$777$$ 2448.00 0.113026
$$778$$ 9630.00 0.443769
$$779$$ −2440.00 −0.112223
$$780$$ −2310.00 −0.106040
$$781$$ −17056.0 −0.781449
$$782$$ 2352.00 0.107554
$$783$$ 6210.00 0.283432
$$784$$ 9553.00 0.435177
$$785$$ −17570.0 −0.798854
$$786$$ 2196.00 0.0996549
$$787$$ 33436.0 1.51444 0.757220 0.653160i $$-0.226557\pi$$
0.757220 + 0.653160i $$0.226557\pi$$
$$788$$ −17682.0 −0.799359
$$789$$ −14664.0 −0.661663
$$790$$ −1200.00 −0.0540431
$$791$$ −25008.0 −1.12412
$$792$$ −7020.00 −0.314956
$$793$$ 16324.0 0.730999
$$794$$ 7126.00 0.318504
$$795$$ −5070.00 −0.226182
$$796$$ 8120.00 0.361565
$$797$$ −37594.0 −1.67083 −0.835413 0.549623i $$-0.814771\pi$$
−0.835413 + 0.549623i $$0.814771\pi$$
$$798$$ 1440.00 0.0638790
$$799$$ −3584.00 −0.158689
$$800$$ 4025.00 0.177882
$$801$$ 2970.00 0.131011
$$802$$ −8718.00 −0.383844
$$803$$ −1976.00 −0.0868388
$$804$$ 1764.00 0.0773775
$$805$$ 20160.0 0.882667
$$806$$ −6336.00 −0.276893
$$807$$ 3810.00 0.166194
$$808$$ 18270.0 0.795466
$$809$$ 4730.00 0.205560 0.102780 0.994704i $$-0.467226\pi$$
0.102780 + 0.994704i $$0.467226\pi$$
$$810$$ 405.000 0.0175682
$$811$$ −8748.00 −0.378772 −0.189386 0.981903i $$-0.560650\pi$$
−0.189386 + 0.981903i $$0.560650\pi$$
$$812$$ 38640.0 1.66995
$$813$$ 3216.00 0.138733
$$814$$ −1768.00 −0.0761282
$$815$$ −10340.0 −0.444410
$$816$$ −1722.00 −0.0738751
$$817$$ 3760.00 0.161011
$$818$$ −10870.0 −0.464622
$$819$$ −4752.00 −0.202745
$$820$$ −4270.00 −0.181847
$$821$$ 44142.0 1.87645 0.938226 0.346024i $$-0.112468\pi$$
0.938226 + 0.346024i $$0.112468\pi$$
$$822$$ −6642.00 −0.281833
$$823$$ 3992.00 0.169079 0.0845397 0.996420i $$-0.473058\pi$$
0.0845397 + 0.996420i $$0.473058\pi$$
$$824$$ 1320.00 0.0558063
$$825$$ 3900.00 0.164583
$$826$$ −2400.00 −0.101098
$$827$$ −14444.0 −0.607336 −0.303668 0.952778i $$-0.598211\pi$$
−0.303668 + 0.952778i $$0.598211\pi$$
$$828$$ 10584.0 0.444226
$$829$$ 42150.0 1.76590 0.882949 0.469468i $$-0.155554\pi$$
0.882949 + 0.469468i $$0.155554\pi$$
$$830$$ 6060.00 0.253429
$$831$$ −16182.0 −0.675508
$$832$$ −3674.00 −0.153093
$$833$$ −3262.00 −0.135680
$$834$$ 60.0000 0.00249116
$$835$$ −120.000 −0.00497338
$$836$$ 7280.00 0.301177
$$837$$ −7776.00 −0.321121
$$838$$ −9700.00 −0.399858
$$839$$ 13400.0 0.551394 0.275697 0.961245i $$-0.411091\pi$$
0.275697 + 0.961245i $$0.411091\pi$$
$$840$$ 5400.00 0.221807
$$841$$ 28511.0 1.16901
$$842$$ 862.000 0.0352809
$$843$$ 7326.00 0.299313
$$844$$ 31276.0 1.27555
$$845$$ −8565.00 −0.348692
$$846$$ 2304.00 0.0936326
$$847$$ −32952.0 −1.33677
$$848$$ −13858.0 −0.561186
$$849$$ 8316.00 0.336165
$$850$$ −350.000 −0.0141234
$$851$$ 5712.00 0.230088
$$852$$ 6888.00 0.276971
$$853$$ −8658.00 −0.347531 −0.173766 0.984787i $$-0.555594\pi$$
−0.173766 + 0.984787i $$0.555594\pi$$
$$854$$ −17808.0 −0.713556
$$855$$ −900.000 −0.0359992
$$856$$ −540.000 −0.0215617
$$857$$ 42826.0 1.70701 0.853505 0.521084i $$-0.174472\pi$$
0.853505 + 0.521084i $$0.174472\pi$$
$$858$$ 3432.00 0.136558
$$859$$ −35900.0 −1.42595 −0.712976 0.701189i $$-0.752653\pi$$
−0.712976 + 0.701189i $$0.752653\pi$$
$$860$$ 6580.00 0.260902
$$861$$ −8784.00 −0.347686
$$862$$ 15792.0 0.623988
$$863$$ −3088.00 −0.121804 −0.0609019 0.998144i $$-0.519398\pi$$
−0.0609019 + 0.998144i $$0.519398\pi$$
$$864$$ 4347.00 0.171167
$$865$$ −3090.00 −0.121460
$$866$$ 11602.0 0.455256
$$867$$ −14151.0 −0.554317
$$868$$ −48384.0 −1.89200
$$869$$ −12480.0 −0.487175
$$870$$ 3450.00 0.134444
$$871$$ −1848.00 −0.0718910
$$872$$ 14550.0 0.565052
$$873$$ 7794.00 0.302161
$$874$$ 3360.00 0.130039
$$875$$ −3000.00 −0.115907
$$876$$ 798.000 0.0307784
$$877$$ −35274.0 −1.35817 −0.679087 0.734058i $$-0.737624\pi$$
−0.679087 + 0.734058i $$0.737624\pi$$
$$878$$ −440.000 −0.0169126
$$879$$ 13626.0 0.522860
$$880$$ 10660.0 0.408351
$$881$$ 25042.0 0.957646 0.478823 0.877911i $$-0.341064\pi$$
0.478823 + 0.877911i $$0.341064\pi$$
$$882$$ 2097.00 0.0800563
$$883$$ 12572.0 0.479141 0.239570 0.970879i $$-0.422993\pi$$
0.239570 + 0.970879i $$0.422993\pi$$
$$884$$ 2156.00 0.0820296
$$885$$ 1500.00 0.0569740
$$886$$ −10188.0 −0.386312
$$887$$ −21864.0 −0.827645 −0.413823 0.910358i $$-0.635807\pi$$
−0.413823 + 0.910358i $$0.635807\pi$$
$$888$$ 1530.00 0.0578192
$$889$$ −46464.0 −1.75293
$$890$$ 1650.00 0.0621440
$$891$$ 4212.00 0.158370
$$892$$ −42224.0 −1.58494
$$893$$ −5120.00 −0.191864
$$894$$ −3990.00 −0.149268
$$895$$ 16700.0 0.623709
$$896$$ 34920.0 1.30200
$$897$$ −11088.0 −0.412729
$$898$$ −13310.0 −0.494611
$$899$$ −66240.0 −2.45743
$$900$$ −1575.00 −0.0583333
$$901$$ 4732.00 0.174968
$$902$$ 6344.00 0.234182
$$903$$ 13536.0 0.498837
$$904$$ −15630.0 −0.575051
$$905$$ −890.000 −0.0326902
$$906$$ −3624.00 −0.132891
$$907$$ 31236.0 1.14352 0.571761 0.820420i $$-0.306260\pi$$
0.571761 + 0.820420i $$0.306260\pi$$
$$908$$ −18452.0 −0.674396
$$909$$ −10962.0 −0.399985
$$910$$ −2640.00 −0.0961705
$$911$$ 8272.00 0.300838 0.150419 0.988622i $$-0.451938\pi$$
0.150419 + 0.988622i $$0.451938\pi$$
$$912$$ −2460.00 −0.0893188
$$913$$ 63024.0 2.28455
$$914$$ 3226.00 0.116747
$$915$$ 11130.0 0.402127
$$916$$ −33810.0 −1.21956
$$917$$ −17568.0 −0.632657
$$918$$ −378.000 −0.0135903
$$919$$ 20200.0 0.725067 0.362533 0.931971i $$-0.381912\pi$$
0.362533 + 0.931971i $$0.381912\pi$$
$$920$$ 12600.0 0.451532
$$921$$ 15348.0 0.549114
$$922$$ 6582.00 0.235105
$$923$$ −7216.00 −0.257332
$$924$$ 26208.0 0.933095
$$925$$ −850.000 −0.0302139
$$926$$ 15072.0 0.534878
$$927$$ −792.000 −0.0280612
$$928$$ 37030.0 1.30988
$$929$$ 31010.0 1.09516 0.547581 0.836753i $$-0.315549\pi$$
0.547581 + 0.836753i $$0.315549\pi$$
$$930$$ −4320.00 −0.152321
$$931$$ −4660.00 −0.164044
$$932$$ −18774.0 −0.659831
$$933$$ −8424.00 −0.295594
$$934$$ 476.000 0.0166758
$$935$$ −3640.00 −0.127316
$$936$$ −2970.00 −0.103715
$$937$$ −39174.0 −1.36580 −0.682902 0.730510i $$-0.739283\pi$$
−0.682902 + 0.730510i $$0.739283\pi$$
$$938$$ 2016.00 0.0701756
$$939$$ −21954.0 −0.762984
$$940$$ −8960.00 −0.310897
$$941$$ −4138.00 −0.143353 −0.0716764 0.997428i $$-0.522835\pi$$
−0.0716764 + 0.997428i $$0.522835\pi$$
$$942$$ −10542.0 −0.364625
$$943$$ −20496.0 −0.707785
$$944$$ 4100.00 0.141360
$$945$$ −3240.00 −0.111531
$$946$$ −9776.00 −0.335989
$$947$$ 23676.0 0.812425 0.406213 0.913779i $$-0.366849\pi$$
0.406213 + 0.913779i $$0.366849\pi$$
$$948$$ 5040.00 0.172670
$$949$$ −836.000 −0.0285961
$$950$$ −500.000 −0.0170759
$$951$$ 6738.00 0.229752
$$952$$ −5040.00 −0.171583
$$953$$ 18922.0 0.643173 0.321586 0.946880i $$-0.395784\pi$$
0.321586 + 0.946880i $$0.395784\pi$$
$$954$$ −3042.00 −0.103237
$$955$$ −9440.00 −0.319865
$$956$$ −16240.0 −0.549413
$$957$$ 35880.0 1.21195
$$958$$ −19680.0 −0.663708
$$959$$ 53136.0 1.78921
$$960$$ −2505.00 −0.0842172
$$961$$ 53153.0 1.78420
$$962$$ −748.000 −0.0250691
$$963$$ 324.000 0.0108419
$$964$$ −14014.0 −0.468216
$$965$$ 9610.00 0.320577
$$966$$ 12096.0 0.402880
$$967$$ 39656.0 1.31877 0.659385 0.751805i $$-0.270817\pi$$
0.659385 + 0.751805i $$0.270817\pi$$
$$968$$ −20595.0 −0.683831
$$969$$ 840.000 0.0278480
$$970$$ 4330.00 0.143328
$$971$$ −33228.0 −1.09818 −0.549092 0.835762i $$-0.685026\pi$$
−0.549092 + 0.835762i $$0.685026\pi$$
$$972$$ −1701.00 −0.0561313
$$973$$ −480.000 −0.0158151
$$974$$ −5944.00 −0.195542
$$975$$ 1650.00 0.0541972
$$976$$ 30422.0 0.997730
$$977$$ −974.000 −0.0318946 −0.0159473 0.999873i $$-0.505076\pi$$
−0.0159473 + 0.999873i $$0.505076\pi$$
$$978$$ −6204.00 −0.202845
$$979$$ 17160.0 0.560200
$$980$$ −8155.00 −0.265818
$$981$$ −8730.00 −0.284126
$$982$$ 10772.0 0.350049
$$983$$ −13608.0 −0.441534 −0.220767 0.975327i $$-0.570856\pi$$
−0.220767 + 0.975327i $$0.570856\pi$$
$$984$$ −5490.00 −0.177861
$$985$$ 12630.0 0.408554
$$986$$ −3220.00 −0.104002
$$987$$ −18432.0 −0.594425
$$988$$ 3080.00 0.0991780
$$989$$ 31584.0 1.01548
$$990$$ 2340.00 0.0751213
$$991$$ 13472.0 0.431839 0.215919 0.976411i $$-0.430725\pi$$
0.215919 + 0.976411i $$0.430725\pi$$
$$992$$ −46368.0 −1.48406
$$993$$ 3996.00 0.127703
$$994$$ 7872.00 0.251192
$$995$$ −5800.00 −0.184796
$$996$$ −25452.0 −0.809716
$$997$$ −3234.00 −0.102730 −0.0513650 0.998680i $$-0.516357\pi$$
−0.0513650 + 0.998680i $$0.516357\pi$$
$$998$$ 8140.00 0.258184
$$999$$ −918.000 −0.0290733
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 15.4.a.a.1.1 1
3.2 odd 2 45.4.a.c.1.1 1
4.3 odd 2 240.4.a.e.1.1 1
5.2 odd 4 75.4.b.b.49.2 2
5.3 odd 4 75.4.b.b.49.1 2
5.4 even 2 75.4.a.b.1.1 1
7.6 odd 2 735.4.a.e.1.1 1
8.3 odd 2 960.4.a.ba.1.1 1
8.5 even 2 960.4.a.b.1.1 1
9.2 odd 6 405.4.e.i.271.1 2
9.4 even 3 405.4.e.g.136.1 2
9.5 odd 6 405.4.e.i.136.1 2
9.7 even 3 405.4.e.g.271.1 2
11.10 odd 2 1815.4.a.e.1.1 1
12.11 even 2 720.4.a.n.1.1 1
15.2 even 4 225.4.b.e.199.1 2
15.8 even 4 225.4.b.e.199.2 2
15.14 odd 2 225.4.a.f.1.1 1
20.3 even 4 1200.4.f.b.49.1 2
20.7 even 4 1200.4.f.b.49.2 2
20.19 odd 2 1200.4.a.t.1.1 1
21.20 even 2 2205.4.a.l.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
15.4.a.a.1.1 1 1.1 even 1 trivial
45.4.a.c.1.1 1 3.2 odd 2
75.4.a.b.1.1 1 5.4 even 2
75.4.b.b.49.1 2 5.3 odd 4
75.4.b.b.49.2 2 5.2 odd 4
225.4.a.f.1.1 1 15.14 odd 2
225.4.b.e.199.1 2 15.2 even 4
225.4.b.e.199.2 2 15.8 even 4
240.4.a.e.1.1 1 4.3 odd 2
405.4.e.g.136.1 2 9.4 even 3
405.4.e.g.271.1 2 9.7 even 3
405.4.e.i.136.1 2 9.5 odd 6
405.4.e.i.271.1 2 9.2 odd 6
720.4.a.n.1.1 1 12.11 even 2
735.4.a.e.1.1 1 7.6 odd 2
960.4.a.b.1.1 1 8.5 even 2
960.4.a.ba.1.1 1 8.3 odd 2
1200.4.a.t.1.1 1 20.19 odd 2
1200.4.f.b.49.1 2 20.3 even 4
1200.4.f.b.49.2 2 20.7 even 4
1815.4.a.e.1.1 1 11.10 odd 2
2205.4.a.l.1.1 1 21.20 even 2