# Properties

 Label 45.4.a.c.1.1 Level $45$ Weight $4$ Character 45.1 Self dual yes Analytic conductor $2.655$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

# Learn more

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [45,4,Mod(1,45)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(45, 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("45.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$45 = 3^{2} \cdot 5$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 45.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: $$2.65508595026$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 15) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 45.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-1.00000 q^{2} -7.00000 q^{4} -5.00000 q^{5} -24.0000 q^{7} +15.0000 q^{8} +O(q^{10})$$ $$q-1.00000 q^{2} -7.00000 q^{4} -5.00000 q^{5} -24.0000 q^{7} +15.0000 q^{8} +5.00000 q^{10} -52.0000 q^{11} +22.0000 q^{13} +24.0000 q^{14} +41.0000 q^{16} +14.0000 q^{17} -20.0000 q^{19} +35.0000 q^{20} +52.0000 q^{22} +168.000 q^{23} +25.0000 q^{25} -22.0000 q^{26} +168.000 q^{28} -230.000 q^{29} -288.000 q^{31} -161.000 q^{32} -14.0000 q^{34} +120.000 q^{35} -34.0000 q^{37} +20.0000 q^{38} -75.0000 q^{40} -122.000 q^{41} -188.000 q^{43} +364.000 q^{44} -168.000 q^{46} -256.000 q^{47} +233.000 q^{49} -25.0000 q^{50} -154.000 q^{52} +338.000 q^{53} +260.000 q^{55} -360.000 q^{56} +230.000 q^{58} -100.000 q^{59} +742.000 q^{61} +288.000 q^{62} -167.000 q^{64} -110.000 q^{65} -84.0000 q^{67} -98.0000 q^{68} -120.000 q^{70} +328.000 q^{71} -38.0000 q^{73} +34.0000 q^{74} +140.000 q^{76} +1248.00 q^{77} -240.000 q^{79} -205.000 q^{80} +122.000 q^{82} -1212.00 q^{83} -70.0000 q^{85} +188.000 q^{86} -780.000 q^{88} -330.000 q^{89} -528.000 q^{91} -1176.00 q^{92} +256.000 q^{94} +100.000 q^{95} +866.000 q^{97} -233.000 q^{98} +O(q^{100})$$

## Coefficient data

For each $$n$$ we display the coefficients of the $$q$$-expansion $$a_n$$, the Satake parameters $$\alpha_p$$, and the Satake angles $$\theta_p = \textrm{Arg}(\alpha_p)$$.

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −1.00000 −0.353553 −0.176777 0.984251i $$-0.556567\pi$$
−0.176777 + 0.984251i $$0.556567\pi$$
$$3$$ 0 0
$$4$$ −7.00000 −0.875000
$$5$$ −5.00000 −0.447214
$$6$$ 0 0
$$7$$ −24.0000 −1.29588 −0.647939 0.761692i $$-0.724369\pi$$
−0.647939 + 0.761692i $$0.724369\pi$$
$$8$$ 15.0000 0.662913
$$9$$ 0 0
$$10$$ 5.00000 0.158114
$$11$$ −52.0000 −1.42533 −0.712663 0.701506i $$-0.752511\pi$$
−0.712663 + 0.701506i $$0.752511\pi$$
$$12$$ 0 0
$$13$$ 22.0000 0.469362 0.234681 0.972072i $$-0.424595\pi$$
0.234681 + 0.972072i $$0.424595\pi$$
$$14$$ 24.0000 0.458162
$$15$$ 0 0
$$16$$ 41.0000 0.640625
$$17$$ 14.0000 0.199735 0.0998676 0.995001i $$-0.468158\pi$$
0.0998676 + 0.995001i $$0.468158\pi$$
$$18$$ 0 0
$$19$$ −20.0000 −0.241490 −0.120745 0.992684i $$-0.538528\pi$$
−0.120745 + 0.992684i $$0.538528\pi$$
$$20$$ 35.0000 0.391312
$$21$$ 0 0
$$22$$ 52.0000 0.503929
$$23$$ 168.000 1.52306 0.761531 0.648129i $$-0.224448\pi$$
0.761531 + 0.648129i $$0.224448\pi$$
$$24$$ 0 0
$$25$$ 25.0000 0.200000
$$26$$ −22.0000 −0.165944
$$27$$ 0 0
$$28$$ 168.000 1.13389
$$29$$ −230.000 −1.47276 −0.736378 0.676570i $$-0.763465\pi$$
−0.736378 + 0.676570i $$0.763465\pi$$
$$30$$ 0 0
$$31$$ −288.000 −1.66859 −0.834296 0.551317i $$-0.814125\pi$$
−0.834296 + 0.551317i $$0.814125\pi$$
$$32$$ −161.000 −0.889408
$$33$$ 0 0
$$34$$ −14.0000 −0.0706171
$$35$$ 120.000 0.579534
$$36$$ 0 0
$$37$$ −34.0000 −0.151069 −0.0755347 0.997143i $$-0.524066\pi$$
−0.0755347 + 0.997143i $$0.524066\pi$$
$$38$$ 20.0000 0.0853797
$$39$$ 0 0
$$40$$ −75.0000 −0.296464
$$41$$ −122.000 −0.464712 −0.232356 0.972631i $$-0.574643\pi$$
−0.232356 + 0.972631i $$0.574643\pi$$
$$42$$ 0 0
$$43$$ −188.000 −0.666738 −0.333369 0.942796i $$-0.608185\pi$$
−0.333369 + 0.942796i $$0.608185\pi$$
$$44$$ 364.000 1.24716
$$45$$ 0 0
$$46$$ −168.000 −0.538484
$$47$$ −256.000 −0.794499 −0.397249 0.917711i $$-0.630035\pi$$
−0.397249 + 0.917711i $$0.630035\pi$$
$$48$$ 0 0
$$49$$ 233.000 0.679300
$$50$$ −25.0000 −0.0707107
$$51$$ 0 0
$$52$$ −154.000 −0.410691
$$53$$ 338.000 0.875998 0.437999 0.898976i $$-0.355687\pi$$
0.437999 + 0.898976i $$0.355687\pi$$
$$54$$ 0 0
$$55$$ 260.000 0.637425
$$56$$ −360.000 −0.859054
$$57$$ 0 0
$$58$$ 230.000 0.520698
$$59$$ −100.000 −0.220659 −0.110330 0.993895i $$-0.535191\pi$$
−0.110330 + 0.993895i $$0.535191\pi$$
$$60$$ 0 0
$$61$$ 742.000 1.55743 0.778716 0.627376i $$-0.215871\pi$$
0.778716 + 0.627376i $$0.215871\pi$$
$$62$$ 288.000 0.589936
$$63$$ 0 0
$$64$$ −167.000 −0.326172
$$65$$ −110.000 −0.209905
$$66$$ 0 0
$$67$$ −84.0000 −0.153168 −0.0765838 0.997063i $$-0.524401\pi$$
−0.0765838 + 0.997063i $$0.524401\pi$$
$$68$$ −98.0000 −0.174768
$$69$$ 0 0
$$70$$ −120.000 −0.204896
$$71$$ 328.000 0.548260 0.274130 0.961693i $$-0.411610\pi$$
0.274130 + 0.961693i $$0.411610\pi$$
$$72$$ 0 0
$$73$$ −38.0000 −0.0609255 −0.0304628 0.999536i $$-0.509698\pi$$
−0.0304628 + 0.999536i $$0.509698\pi$$
$$74$$ 34.0000 0.0534111
$$75$$ 0 0
$$76$$ 140.000 0.211304
$$77$$ 1248.00 1.84705
$$78$$ 0 0
$$79$$ −240.000 −0.341799 −0.170899 0.985288i $$-0.554667\pi$$
−0.170899 + 0.985288i $$0.554667\pi$$
$$80$$ −205.000 −0.286496
$$81$$ 0 0
$$82$$ 122.000 0.164301
$$83$$ −1212.00 −1.60282 −0.801411 0.598114i $$-0.795917\pi$$
−0.801411 + 0.598114i $$0.795917\pi$$
$$84$$ 0 0
$$85$$ −70.0000 −0.0893243
$$86$$ 188.000 0.235727
$$87$$ 0 0
$$88$$ −780.000 −0.944867
$$89$$ −330.000 −0.393033 −0.196516 0.980501i $$-0.562963\pi$$
−0.196516 + 0.980501i $$0.562963\pi$$
$$90$$ 0 0
$$91$$ −528.000 −0.608236
$$92$$ −1176.00 −1.33268
$$93$$ 0 0
$$94$$ 256.000 0.280898
$$95$$ 100.000 0.107998
$$96$$ 0 0
$$97$$ 866.000 0.906484 0.453242 0.891387i $$-0.350267\pi$$
0.453242 + 0.891387i $$0.350267\pi$$
$$98$$ −233.000 −0.240169
$$99$$ 0 0
$$100$$ −175.000 −0.175000
$$101$$ 1218.00 1.19996 0.599978 0.800017i $$-0.295176\pi$$
0.599978 + 0.800017i $$0.295176\pi$$
$$102$$ 0 0
$$103$$ −88.0000 −0.0841835 −0.0420917 0.999114i $$-0.513402\pi$$
−0.0420917 + 0.999114i $$0.513402\pi$$
$$104$$ 330.000 0.311146
$$105$$ 0 0
$$106$$ −338.000 −0.309712
$$107$$ −36.0000 −0.0325257 −0.0162629 0.999868i $$-0.505177\pi$$
−0.0162629 + 0.999868i $$0.505177\pi$$
$$108$$ 0 0
$$109$$ −970.000 −0.852378 −0.426189 0.904634i $$-0.640144\pi$$
−0.426189 + 0.904634i $$0.640144\pi$$
$$110$$ −260.000 −0.225364
$$111$$ 0 0
$$112$$ −984.000 −0.830172
$$113$$ −1042.00 −0.867461 −0.433731 0.901043i $$-0.642803\pi$$
−0.433731 + 0.901043i $$0.642803\pi$$
$$114$$ 0 0
$$115$$ −840.000 −0.681134
$$116$$ 1610.00 1.28866
$$117$$ 0 0
$$118$$ 100.000 0.0780148
$$119$$ −336.000 −0.258833
$$120$$ 0 0
$$121$$ 1373.00 1.03156
$$122$$ −742.000 −0.550635
$$123$$ 0 0
$$124$$ 2016.00 1.46002
$$125$$ −125.000 −0.0894427
$$126$$ 0 0
$$127$$ 1936.00 1.35269 0.676347 0.736583i $$-0.263562\pi$$
0.676347 + 0.736583i $$0.263562\pi$$
$$128$$ 1455.00 1.00473
$$129$$ 0 0
$$130$$ 110.000 0.0742126
$$131$$ −732.000 −0.488207 −0.244104 0.969749i $$-0.578494\pi$$
−0.244104 + 0.969749i $$0.578494\pi$$
$$132$$ 0 0
$$133$$ 480.000 0.312942
$$134$$ 84.0000 0.0541529
$$135$$ 0 0
$$136$$ 210.000 0.132407
$$137$$ 2214.00 1.38069 0.690346 0.723479i $$-0.257458\pi$$
0.690346 + 0.723479i $$0.257458\pi$$
$$138$$ 0 0
$$139$$ 20.0000 0.0122042 0.00610208 0.999981i $$-0.498058\pi$$
0.00610208 + 0.999981i $$0.498058\pi$$
$$140$$ −840.000 −0.507093
$$141$$ 0 0
$$142$$ −328.000 −0.193839
$$143$$ −1144.00 −0.668994
$$144$$ 0 0
$$145$$ 1150.00 0.658637
$$146$$ 38.0000 0.0215404
$$147$$ 0 0
$$148$$ 238.000 0.132186
$$149$$ 1330.00 0.731261 0.365630 0.930760i $$-0.380853\pi$$
0.365630 + 0.930760i $$0.380853\pi$$
$$150$$ 0 0
$$151$$ −1208.00 −0.651031 −0.325515 0.945537i $$-0.605538\pi$$
−0.325515 + 0.945537i $$0.605538\pi$$
$$152$$ −300.000 −0.160087
$$153$$ 0 0
$$154$$ −1248.00 −0.653031
$$155$$ 1440.00 0.746217
$$156$$ 0 0
$$157$$ −3514.00 −1.78629 −0.893146 0.449768i $$-0.851507\pi$$
−0.893146 + 0.449768i $$0.851507\pi$$
$$158$$ 240.000 0.120844
$$159$$ 0 0
$$160$$ 805.000 0.397755
$$161$$ −4032.00 −1.97370
$$162$$ 0 0
$$163$$ −2068.00 −0.993732 −0.496866 0.867827i $$-0.665516\pi$$
−0.496866 + 0.867827i $$0.665516\pi$$
$$164$$ 854.000 0.406623
$$165$$ 0 0
$$166$$ 1212.00 0.566683
$$167$$ 24.0000 0.0111208 0.00556041 0.999985i $$-0.498230\pi$$
0.00556041 + 0.999985i $$0.498230\pi$$
$$168$$ 0 0
$$169$$ −1713.00 −0.779700
$$170$$ 70.0000 0.0315809
$$171$$ 0 0
$$172$$ 1316.00 0.583396
$$173$$ 618.000 0.271593 0.135797 0.990737i $$-0.456641\pi$$
0.135797 + 0.990737i $$0.456641\pi$$
$$174$$ 0 0
$$175$$ −600.000 −0.259176
$$176$$ −2132.00 −0.913100
$$177$$ 0 0
$$178$$ 330.000 0.138958
$$179$$ −3340.00 −1.39466 −0.697328 0.716752i $$-0.745628\pi$$
−0.697328 + 0.716752i $$0.745628\pi$$
$$180$$ 0 0
$$181$$ −178.000 −0.0730974 −0.0365487 0.999332i $$-0.511636\pi$$
−0.0365487 + 0.999332i $$0.511636\pi$$
$$182$$ 528.000 0.215044
$$183$$ 0 0
$$184$$ 2520.00 1.00966
$$185$$ 170.000 0.0675603
$$186$$ 0 0
$$187$$ −728.000 −0.284688
$$188$$ 1792.00 0.695186
$$189$$ 0 0
$$190$$ −100.000 −0.0381830
$$191$$ 1888.00 0.715240 0.357620 0.933867i $$-0.383588\pi$$
0.357620 + 0.933867i $$0.383588\pi$$
$$192$$ 0 0
$$193$$ 1922.00 0.716832 0.358416 0.933562i $$-0.383317\pi$$
0.358416 + 0.933562i $$0.383317\pi$$
$$194$$ −866.000 −0.320491
$$195$$ 0 0
$$196$$ −1631.00 −0.594388
$$197$$ −2526.00 −0.913554 −0.456777 0.889581i $$-0.650996\pi$$
−0.456777 + 0.889581i $$0.650996\pi$$
$$198$$ 0 0
$$199$$ −1160.00 −0.413217 −0.206609 0.978424i $$-0.566243\pi$$
−0.206609 + 0.978424i $$0.566243\pi$$
$$200$$ 375.000 0.132583
$$201$$ 0 0
$$202$$ −1218.00 −0.424248
$$203$$ 5520.00 1.90851
$$204$$ 0 0
$$205$$ 610.000 0.207826
$$206$$ 88.0000 0.0297634
$$207$$ 0 0
$$208$$ 902.000 0.300685
$$209$$ 1040.00 0.344202
$$210$$ 0 0
$$211$$ −4468.00 −1.45777 −0.728886 0.684635i $$-0.759961\pi$$
−0.728886 + 0.684635i $$0.759961\pi$$
$$212$$ −2366.00 −0.766498
$$213$$ 0 0
$$214$$ 36.0000 0.0114996
$$215$$ 940.000 0.298174
$$216$$ 0 0
$$217$$ 6912.00 2.16229
$$218$$ 970.000 0.301361
$$219$$ 0 0
$$220$$ −1820.00 −0.557747
$$221$$ 308.000 0.0937481
$$222$$ 0 0
$$223$$ 6032.00 1.81136 0.905678 0.423965i $$-0.139362\pi$$
0.905678 + 0.423965i $$0.139362\pi$$
$$224$$ 3864.00 1.15256
$$225$$ 0 0
$$226$$ 1042.00 0.306694
$$227$$ −2636.00 −0.770738 −0.385369 0.922763i $$-0.625926\pi$$
−0.385369 + 0.922763i $$0.625926\pi$$
$$228$$ 0 0
$$229$$ 4830.00 1.39378 0.696889 0.717179i $$-0.254567\pi$$
0.696889 + 0.717179i $$0.254567\pi$$
$$230$$ 840.000 0.240817
$$231$$ 0 0
$$232$$ −3450.00 −0.976309
$$233$$ −2682.00 −0.754093 −0.377046 0.926194i $$-0.623060\pi$$
−0.377046 + 0.926194i $$0.623060\pi$$
$$234$$ 0 0
$$235$$ 1280.00 0.355311
$$236$$ 700.000 0.193077
$$237$$ 0 0
$$238$$ 336.000 0.0915111
$$239$$ −2320.00 −0.627901 −0.313950 0.949439i $$-0.601653\pi$$
−0.313950 + 0.949439i $$0.601653\pi$$
$$240$$ 0 0
$$241$$ 2002.00 0.535104 0.267552 0.963543i $$-0.413785\pi$$
0.267552 + 0.963543i $$0.413785\pi$$
$$242$$ −1373.00 −0.364710
$$243$$ 0 0
$$244$$ −5194.00 −1.36275
$$245$$ −1165.00 −0.303792
$$246$$ 0 0
$$247$$ −440.000 −0.113346
$$248$$ −4320.00 −1.10613
$$249$$ 0 0
$$250$$ 125.000 0.0316228
$$251$$ −132.000 −0.0331943 −0.0165971 0.999862i $$-0.505283\pi$$
−0.0165971 + 0.999862i $$0.505283\pi$$
$$252$$ 0 0
$$253$$ −8736.00 −2.17086
$$254$$ −1936.00 −0.478250
$$255$$ 0 0
$$256$$ −119.000 −0.0290527
$$257$$ 7614.00 1.84805 0.924024 0.382335i $$-0.124880\pi$$
0.924024 + 0.382335i $$0.124880\pi$$
$$258$$ 0 0
$$259$$ 816.000 0.195767
$$260$$ 770.000 0.183667
$$261$$ 0 0
$$262$$ 732.000 0.172607
$$263$$ 4888.00 1.14603 0.573017 0.819543i $$-0.305773\pi$$
0.573017 + 0.819543i $$0.305773\pi$$
$$264$$ 0 0
$$265$$ −1690.00 −0.391758
$$266$$ −480.000 −0.110642
$$267$$ 0 0
$$268$$ 588.000 0.134022
$$269$$ −1270.00 −0.287856 −0.143928 0.989588i $$-0.545973\pi$$
−0.143928 + 0.989588i $$0.545973\pi$$
$$270$$ 0 0
$$271$$ 1072.00 0.240293 0.120146 0.992756i $$-0.461664\pi$$
0.120146 + 0.992756i $$0.461664\pi$$
$$272$$ 574.000 0.127955
$$273$$ 0 0
$$274$$ −2214.00 −0.488148
$$275$$ −1300.00 −0.285065
$$276$$ 0 0
$$277$$ −5394.00 −1.17001 −0.585007 0.811028i $$-0.698908\pi$$
−0.585007 + 0.811028i $$0.698908\pi$$
$$278$$ −20.0000 −0.00431482
$$279$$ 0 0
$$280$$ 1800.00 0.384181
$$281$$ −2442.00 −0.518425 −0.259213 0.965820i $$-0.583463\pi$$
−0.259213 + 0.965820i $$0.583463\pi$$
$$282$$ 0 0
$$283$$ 2772.00 0.582255 0.291128 0.956684i $$-0.405970\pi$$
0.291128 + 0.956684i $$0.405970\pi$$
$$284$$ −2296.00 −0.479727
$$285$$ 0 0
$$286$$ 1144.00 0.236525
$$287$$ 2928.00 0.602210
$$288$$ 0 0
$$289$$ −4717.00 −0.960106
$$290$$ −1150.00 −0.232863
$$291$$ 0 0
$$292$$ 266.000 0.0533098
$$293$$ −4542.00 −0.905619 −0.452810 0.891607i $$-0.649578\pi$$
−0.452810 + 0.891607i $$0.649578\pi$$
$$294$$ 0 0
$$295$$ 500.000 0.0986818
$$296$$ −510.000 −0.100146
$$297$$ 0 0
$$298$$ −1330.00 −0.258540
$$299$$ 3696.00 0.714867
$$300$$ 0 0
$$301$$ 4512.00 0.864011
$$302$$ 1208.00 0.230174
$$303$$ 0 0
$$304$$ −820.000 −0.154705
$$305$$ −3710.00 −0.696505
$$306$$ 0 0
$$307$$ 5116.00 0.951093 0.475546 0.879691i $$-0.342250\pi$$
0.475546 + 0.879691i $$0.342250\pi$$
$$308$$ −8736.00 −1.61617
$$309$$ 0 0
$$310$$ −1440.00 −0.263827
$$311$$ 2808.00 0.511984 0.255992 0.966679i $$-0.417598\pi$$
0.255992 + 0.966679i $$0.417598\pi$$
$$312$$ 0 0
$$313$$ −7318.00 −1.32153 −0.660763 0.750594i $$-0.729767\pi$$
−0.660763 + 0.750594i $$0.729767\pi$$
$$314$$ 3514.00 0.631549
$$315$$ 0 0
$$316$$ 1680.00 0.299074
$$317$$ −2246.00 −0.397943 −0.198971 0.980005i $$-0.563760\pi$$
−0.198971 + 0.980005i $$0.563760\pi$$
$$318$$ 0 0
$$319$$ 11960.0 2.09916
$$320$$ 835.000 0.145868
$$321$$ 0 0
$$322$$ 4032.00 0.697809
$$323$$ −280.000 −0.0482341
$$324$$ 0 0
$$325$$ 550.000 0.0938723
$$326$$ 2068.00 0.351337
$$327$$ 0 0
$$328$$ −1830.00 −0.308064
$$329$$ 6144.00 1.02957
$$330$$ 0 0
$$331$$ 1332.00 0.221188 0.110594 0.993866i $$-0.464725\pi$$
0.110594 + 0.993866i $$0.464725\pi$$
$$332$$ 8484.00 1.40247
$$333$$ 0 0
$$334$$ −24.0000 −0.00393180
$$335$$ 420.000 0.0684987
$$336$$ 0 0
$$337$$ −11534.0 −1.86438 −0.932191 0.361966i $$-0.882106\pi$$
−0.932191 + 0.361966i $$0.882106\pi$$
$$338$$ 1713.00 0.275665
$$339$$ 0 0
$$340$$ 490.000 0.0781588
$$341$$ 14976.0 2.37829
$$342$$ 0 0
$$343$$ 2640.00 0.415588
$$344$$ −2820.00 −0.441989
$$345$$ 0 0
$$346$$ −618.000 −0.0960228
$$347$$ −11956.0 −1.84966 −0.924830 0.380382i $$-0.875793\pi$$
−0.924830 + 0.380382i $$0.875793\pi$$
$$348$$ 0 0
$$349$$ 4870.00 0.746949 0.373474 0.927640i $$-0.378166\pi$$
0.373474 + 0.927640i $$0.378166\pi$$
$$350$$ 600.000 0.0916324
$$351$$ 0 0
$$352$$ 8372.00 1.26770
$$353$$ −10722.0 −1.61664 −0.808321 0.588742i $$-0.799623\pi$$
−0.808321 + 0.588742i $$0.799623\pi$$
$$354$$ 0 0
$$355$$ −1640.00 −0.245189
$$356$$ 2310.00 0.343904
$$357$$ 0 0
$$358$$ 3340.00 0.493085
$$359$$ −120.000 −0.0176417 −0.00882083 0.999961i $$-0.502808\pi$$
−0.00882083 + 0.999961i $$0.502808\pi$$
$$360$$ 0 0
$$361$$ −6459.00 −0.941682
$$362$$ 178.000 0.0258438
$$363$$ 0 0
$$364$$ 3696.00 0.532206
$$365$$ 190.000 0.0272467
$$366$$ 0 0
$$367$$ 3936.00 0.559830 0.279915 0.960025i $$-0.409694\pi$$
0.279915 + 0.960025i $$0.409694\pi$$
$$368$$ 6888.00 0.975711
$$369$$ 0 0
$$370$$ −170.000 −0.0238862
$$371$$ −8112.00 −1.13519
$$372$$ 0 0
$$373$$ 3022.00 0.419499 0.209750 0.977755i $$-0.432735\pi$$
0.209750 + 0.977755i $$0.432735\pi$$
$$374$$ 728.000 0.100652
$$375$$ 0 0
$$376$$ −3840.00 −0.526683
$$377$$ −5060.00 −0.691255
$$378$$ 0 0
$$379$$ −13340.0 −1.80799 −0.903997 0.427539i $$-0.859381\pi$$
−0.903997 + 0.427539i $$0.859381\pi$$
$$380$$ −700.000 −0.0944980
$$381$$ 0 0
$$382$$ −1888.00 −0.252876
$$383$$ 1008.00 0.134481 0.0672407 0.997737i $$-0.478580\pi$$
0.0672407 + 0.997737i $$0.478580\pi$$
$$384$$ 0 0
$$385$$ −6240.00 −0.826026
$$386$$ −1922.00 −0.253438
$$387$$ 0 0
$$388$$ −6062.00 −0.793174
$$389$$ −9630.00 −1.25517 −0.627584 0.778549i $$-0.715956\pi$$
−0.627584 + 0.778549i $$0.715956\pi$$
$$390$$ 0 0
$$391$$ 2352.00 0.304209
$$392$$ 3495.00 0.450317
$$393$$ 0 0
$$394$$ 2526.00 0.322990
$$395$$ 1200.00 0.152857
$$396$$ 0 0
$$397$$ 7126.00 0.900866 0.450433 0.892810i $$-0.351270\pi$$
0.450433 + 0.892810i $$0.351270\pi$$
$$398$$ 1160.00 0.146094
$$399$$ 0 0
$$400$$ 1025.00 0.128125
$$401$$ 8718.00 1.08568 0.542838 0.839837i $$-0.317350\pi$$
0.542838 + 0.839837i $$0.317350\pi$$
$$402$$ 0 0
$$403$$ −6336.00 −0.783173
$$404$$ −8526.00 −1.04996
$$405$$ 0 0
$$406$$ −5520.00 −0.674761
$$407$$ 1768.00 0.215323
$$408$$ 0 0
$$409$$ −10870.0 −1.31415 −0.657074 0.753826i $$-0.728206\pi$$
−0.657074 + 0.753826i $$0.728206\pi$$
$$410$$ −610.000 −0.0734774
$$411$$ 0 0
$$412$$ 616.000 0.0736605
$$413$$ 2400.00 0.285947
$$414$$ 0 0
$$415$$ 6060.00 0.716804
$$416$$ −3542.00 −0.417454
$$417$$ 0 0
$$418$$ −1040.00 −0.121694
$$419$$ 9700.00 1.13097 0.565484 0.824759i $$-0.308689\pi$$
0.565484 + 0.824759i $$0.308689\pi$$
$$420$$ 0 0
$$421$$ 862.000 0.0997893 0.0498947 0.998754i $$-0.484111\pi$$
0.0498947 + 0.998754i $$0.484111\pi$$
$$422$$ 4468.00 0.515400
$$423$$ 0 0
$$424$$ 5070.00 0.580710
$$425$$ 350.000 0.0399470
$$426$$ 0 0
$$427$$ −17808.0 −2.01824
$$428$$ 252.000 0.0284600
$$429$$ 0 0
$$430$$ −940.000 −0.105421
$$431$$ −15792.0 −1.76490 −0.882452 0.470402i $$-0.844109\pi$$
−0.882452 + 0.470402i $$0.844109\pi$$
$$432$$ 0 0
$$433$$ 11602.0 1.28766 0.643830 0.765169i $$-0.277345\pi$$
0.643830 + 0.765169i $$0.277345\pi$$
$$434$$ −6912.00 −0.764485
$$435$$ 0 0
$$436$$ 6790.00 0.745830
$$437$$ −3360.00 −0.367805
$$438$$ 0 0
$$439$$ −440.000 −0.0478361 −0.0239181 0.999714i $$-0.507614\pi$$
−0.0239181 + 0.999714i $$0.507614\pi$$
$$440$$ 3900.00 0.422557
$$441$$ 0 0
$$442$$ −308.000 −0.0331449
$$443$$ 10188.0 1.09266 0.546328 0.837571i $$-0.316025\pi$$
0.546328 + 0.837571i $$0.316025\pi$$
$$444$$ 0 0
$$445$$ 1650.00 0.175770
$$446$$ −6032.00 −0.640411
$$447$$ 0 0
$$448$$ 4008.00 0.422679
$$449$$ 13310.0 1.39897 0.699485 0.714647i $$-0.253413\pi$$
0.699485 + 0.714647i $$0.253413\pi$$
$$450$$ 0 0
$$451$$ 6344.00 0.662367
$$452$$ 7294.00 0.759029
$$453$$ 0 0
$$454$$ 2636.00 0.272497
$$455$$ 2640.00 0.272011
$$456$$ 0 0
$$457$$ 3226.00 0.330210 0.165105 0.986276i $$-0.447204\pi$$
0.165105 + 0.986276i $$0.447204\pi$$
$$458$$ −4830.00 −0.492775
$$459$$ 0 0
$$460$$ 5880.00 0.595992
$$461$$ −6582.00 −0.664977 −0.332488 0.943107i $$-0.607888\pi$$
−0.332488 + 0.943107i $$0.607888\pi$$
$$462$$ 0 0
$$463$$ 15072.0 1.51286 0.756431 0.654073i $$-0.226941\pi$$
0.756431 + 0.654073i $$0.226941\pi$$
$$464$$ −9430.00 −0.943484
$$465$$ 0 0
$$466$$ 2682.00 0.266612
$$467$$ −476.000 −0.0471663 −0.0235831 0.999722i $$-0.507507\pi$$
−0.0235831 + 0.999722i $$0.507507\pi$$
$$468$$ 0 0
$$469$$ 2016.00 0.198487
$$470$$ −1280.00 −0.125621
$$471$$ 0 0
$$472$$ −1500.00 −0.146278
$$473$$ 9776.00 0.950319
$$474$$ 0 0
$$475$$ −500.000 −0.0482980
$$476$$ 2352.00 0.226478
$$477$$ 0 0
$$478$$ 2320.00 0.221997
$$479$$ 19680.0 1.87725 0.938624 0.344941i $$-0.112101\pi$$
0.938624 + 0.344941i $$0.112101\pi$$
$$480$$ 0 0
$$481$$ −748.000 −0.0709062
$$482$$ −2002.00 −0.189188
$$483$$ 0 0
$$484$$ −9611.00 −0.902611
$$485$$ −4330.00 −0.405392
$$486$$ 0 0
$$487$$ −5944.00 −0.553077 −0.276538 0.961003i $$-0.589187\pi$$
−0.276538 + 0.961003i $$0.589187\pi$$
$$488$$ 11130.0 1.03244
$$489$$ 0 0
$$490$$ 1165.00 0.107407
$$491$$ −10772.0 −0.990089 −0.495044 0.868868i $$-0.664848\pi$$
−0.495044 + 0.868868i $$0.664848\pi$$
$$492$$ 0 0
$$493$$ −3220.00 −0.294161
$$494$$ 440.000 0.0400740
$$495$$ 0 0
$$496$$ −11808.0 −1.06894
$$497$$ −7872.00 −0.710478
$$498$$ 0 0
$$499$$ 8140.00 0.730253 0.365127 0.930958i $$-0.381026\pi$$
0.365127 + 0.930958i $$0.381026\pi$$
$$500$$ 875.000 0.0782624
$$501$$ 0 0
$$502$$ 132.000 0.0117360
$$503$$ 13768.0 1.22045 0.610223 0.792229i $$-0.291080\pi$$
0.610223 + 0.792229i $$0.291080\pi$$
$$504$$ 0 0
$$505$$ −6090.00 −0.536637
$$506$$ 8736.00 0.767515
$$507$$ 0 0
$$508$$ −13552.0 −1.18361
$$509$$ −22150.0 −1.92884 −0.964422 0.264368i $$-0.914837\pi$$
−0.964422 + 0.264368i $$0.914837\pi$$
$$510$$ 0 0
$$511$$ 912.000 0.0789521
$$512$$ −11521.0 −0.994455
$$513$$ 0 0
$$514$$ −7614.00 −0.653384
$$515$$ 440.000 0.0376480
$$516$$ 0 0
$$517$$ 13312.0 1.13242
$$518$$ −816.000 −0.0692143
$$519$$ 0 0
$$520$$ −1650.00 −0.139149
$$521$$ −1562.00 −0.131348 −0.0656741 0.997841i $$-0.520920\pi$$
−0.0656741 + 0.997841i $$0.520920\pi$$
$$522$$ 0 0
$$523$$ −668.000 −0.0558501 −0.0279250 0.999610i $$-0.508890\pi$$
−0.0279250 + 0.999610i $$0.508890\pi$$
$$524$$ 5124.00 0.427181
$$525$$ 0 0
$$526$$ −4888.00 −0.405184
$$527$$ −4032.00 −0.333276
$$528$$ 0 0
$$529$$ 16057.0 1.31972
$$530$$ 1690.00 0.138507
$$531$$ 0 0
$$532$$ −3360.00 −0.273824
$$533$$ −2684.00 −0.218118
$$534$$ 0 0
$$535$$ 180.000 0.0145459
$$536$$ −1260.00 −0.101537
$$537$$ 0 0
$$538$$ 1270.00 0.101772
$$539$$ −12116.0 −0.968225
$$540$$ 0 0
$$541$$ −6138.00 −0.487788 −0.243894 0.969802i $$-0.578425\pi$$
−0.243894 + 0.969802i $$0.578425\pi$$
$$542$$ −1072.00 −0.0849564
$$543$$ 0 0
$$544$$ −2254.00 −0.177646
$$545$$ 4850.00 0.381195
$$546$$ 0 0
$$547$$ −10484.0 −0.819494 −0.409747 0.912199i $$-0.634383\pi$$
−0.409747 + 0.912199i $$0.634383\pi$$
$$548$$ −15498.0 −1.20811
$$549$$ 0 0
$$550$$ 1300.00 0.100786
$$551$$ 4600.00 0.355656
$$552$$ 0 0
$$553$$ 5760.00 0.442930
$$554$$ 5394.00 0.413663
$$555$$ 0 0
$$556$$ −140.000 −0.0106786
$$557$$ −3606.00 −0.274311 −0.137155 0.990550i $$-0.543796\pi$$
−0.137155 + 0.990550i $$0.543796\pi$$
$$558$$ 0 0
$$559$$ −4136.00 −0.312941
$$560$$ 4920.00 0.371264
$$561$$ 0 0
$$562$$ 2442.00 0.183291
$$563$$ −12252.0 −0.917159 −0.458579 0.888654i $$-0.651641\pi$$
−0.458579 + 0.888654i $$0.651641\pi$$
$$564$$ 0 0
$$565$$ 5210.00 0.387940
$$566$$ −2772.00 −0.205858
$$567$$ 0 0
$$568$$ 4920.00 0.363448
$$569$$ 14550.0 1.07200 0.536000 0.844218i $$-0.319935\pi$$
0.536000 + 0.844218i $$0.319935\pi$$
$$570$$ 0 0
$$571$$ −25468.0 −1.86655 −0.933277 0.359157i $$-0.883064\pi$$
−0.933277 + 0.359157i $$0.883064\pi$$
$$572$$ 8008.00 0.585369
$$573$$ 0 0
$$574$$ −2928.00 −0.212914
$$575$$ 4200.00 0.304612
$$576$$ 0 0
$$577$$ 12866.0 0.928282 0.464141 0.885761i $$-0.346363\pi$$
0.464141 + 0.885761i $$0.346363\pi$$
$$578$$ 4717.00 0.339449
$$579$$ 0 0
$$580$$ −8050.00 −0.576307
$$581$$ 29088.0 2.07706
$$582$$ 0 0
$$583$$ −17576.0 −1.24858
$$584$$ −570.000 −0.0403883
$$585$$ 0 0
$$586$$ 4542.00 0.320185
$$587$$ 14844.0 1.04374 0.521872 0.853024i $$-0.325234\pi$$
0.521872 + 0.853024i $$0.325234\pi$$
$$588$$ 0 0
$$589$$ 5760.00 0.402948
$$590$$ −500.000 −0.0348893
$$591$$ 0 0
$$592$$ −1394.00 −0.0967788
$$593$$ −20402.0 −1.41283 −0.706416 0.707797i $$-0.749689\pi$$
−0.706416 + 0.707797i $$0.749689\pi$$
$$594$$ 0 0
$$595$$ 1680.00 0.115753
$$596$$ −9310.00 −0.639853
$$597$$ 0 0
$$598$$ −3696.00 −0.252744
$$599$$ −10760.0 −0.733959 −0.366980 0.930229i $$-0.619608\pi$$
−0.366980 + 0.930229i $$0.619608\pi$$
$$600$$ 0 0
$$601$$ 14282.0 0.969343 0.484671 0.874696i $$-0.338939\pi$$
0.484671 + 0.874696i $$0.338939\pi$$
$$602$$ −4512.00 −0.305474
$$603$$ 0 0
$$604$$ 8456.00 0.569652
$$605$$ −6865.00 −0.461326
$$606$$ 0 0
$$607$$ 11056.0 0.739290 0.369645 0.929173i $$-0.379479\pi$$
0.369645 + 0.929173i $$0.379479\pi$$
$$608$$ 3220.00 0.214783
$$609$$ 0 0
$$610$$ 3710.00 0.246252
$$611$$ −5632.00 −0.372907
$$612$$ 0 0
$$613$$ −16418.0 −1.08176 −0.540878 0.841101i $$-0.681908\pi$$
−0.540878 + 0.841101i $$0.681908\pi$$
$$614$$ −5116.00 −0.336262
$$615$$ 0 0
$$616$$ 18720.0 1.22443
$$617$$ 10374.0 0.676891 0.338445 0.940986i $$-0.390099\pi$$
0.338445 + 0.940986i $$0.390099\pi$$
$$618$$ 0 0
$$619$$ −5260.00 −0.341546 −0.170773 0.985310i $$-0.554627\pi$$
−0.170773 + 0.985310i $$0.554627\pi$$
$$620$$ −10080.0 −0.652940
$$621$$ 0 0
$$622$$ −2808.00 −0.181014
$$623$$ 7920.00 0.509323
$$624$$ 0 0
$$625$$ 625.000 0.0400000
$$626$$ 7318.00 0.467230
$$627$$ 0 0
$$628$$ 24598.0 1.56300
$$629$$ −476.000 −0.0301739
$$630$$ 0 0
$$631$$ 21352.0 1.34708 0.673542 0.739149i $$-0.264772\pi$$
0.673542 + 0.739149i $$0.264772\pi$$
$$632$$ −3600.00 −0.226583
$$633$$ 0 0
$$634$$ 2246.00 0.140694
$$635$$ −9680.00 −0.604943
$$636$$ 0 0
$$637$$ 5126.00 0.318838
$$638$$ −11960.0 −0.742164
$$639$$ 0 0
$$640$$ −7275.00 −0.449328
$$641$$ 29118.0 1.79422 0.897108 0.441812i $$-0.145664\pi$$
0.897108 + 0.441812i $$0.145664\pi$$
$$642$$ 0 0
$$643$$ 5772.00 0.354005 0.177003 0.984210i $$-0.443360\pi$$
0.177003 + 0.984210i $$0.443360\pi$$
$$644$$ 28224.0 1.72699
$$645$$ 0 0
$$646$$ 280.000 0.0170533
$$647$$ 14264.0 0.866732 0.433366 0.901218i $$-0.357326\pi$$
0.433366 + 0.901218i $$0.357326\pi$$
$$648$$ 0 0
$$649$$ 5200.00 0.314511
$$650$$ −550.000 −0.0331889
$$651$$ 0 0
$$652$$ 14476.0 0.869515
$$653$$ −6902.00 −0.413623 −0.206812 0.978381i $$-0.566309\pi$$
−0.206812 + 0.978381i $$0.566309\pi$$
$$654$$ 0 0
$$655$$ 3660.00 0.218333
$$656$$ −5002.00 −0.297706
$$657$$ 0 0
$$658$$ −6144.00 −0.364009
$$659$$ −20140.0 −1.19051 −0.595253 0.803539i $$-0.702948\pi$$
−0.595253 + 0.803539i $$0.702948\pi$$
$$660$$ 0 0
$$661$$ −3218.00 −0.189358 −0.0946790 0.995508i $$-0.530182\pi$$
−0.0946790 + 0.995508i $$0.530182\pi$$
$$662$$ −1332.00 −0.0782019
$$663$$ 0 0
$$664$$ −18180.0 −1.06253
$$665$$ −2400.00 −0.139952
$$666$$ 0 0
$$667$$ −38640.0 −2.24310
$$668$$ −168.000 −0.00973071
$$669$$ 0 0
$$670$$ −420.000 −0.0242179
$$671$$ −38584.0 −2.21985
$$672$$ 0 0
$$673$$ −7518.00 −0.430606 −0.215303 0.976547i $$-0.569074\pi$$
−0.215303 + 0.976547i $$0.569074\pi$$
$$674$$ 11534.0 0.659159
$$675$$ 0 0
$$676$$ 11991.0 0.682237
$$677$$ 18114.0 1.02833 0.514164 0.857692i $$-0.328102\pi$$
0.514164 + 0.857692i $$0.328102\pi$$
$$678$$ 0 0
$$679$$ −20784.0 −1.17469
$$680$$ −1050.00 −0.0592142
$$681$$ 0 0
$$682$$ −14976.0 −0.840851
$$683$$ 23868.0 1.33716 0.668582 0.743638i $$-0.266901\pi$$
0.668582 + 0.743638i $$0.266901\pi$$
$$684$$ 0 0
$$685$$ −11070.0 −0.617464
$$686$$ −2640.00 −0.146932
$$687$$ 0 0
$$688$$ −7708.00 −0.427129
$$689$$ 7436.00 0.411160
$$690$$ 0 0
$$691$$ 172.000 0.00946916 0.00473458 0.999989i $$-0.498493\pi$$
0.00473458 + 0.999989i $$0.498493\pi$$
$$692$$ −4326.00 −0.237644
$$693$$ 0 0
$$694$$ 11956.0 0.653953
$$695$$ −100.000 −0.00545787
$$696$$ 0 0
$$697$$ −1708.00 −0.0928194
$$698$$ −4870.00 −0.264086
$$699$$ 0 0
$$700$$ 4200.00 0.226779
$$701$$ 22138.0 1.19278 0.596391 0.802694i $$-0.296601\pi$$
0.596391 + 0.802694i $$0.296601\pi$$
$$702$$ 0 0
$$703$$ 680.000 0.0364818
$$704$$ 8684.00 0.464901
$$705$$ 0 0
$$706$$ 10722.0 0.571569
$$707$$ −29232.0 −1.55500
$$708$$ 0 0
$$709$$ 3070.00 0.162618 0.0813091 0.996689i $$-0.474090\pi$$
0.0813091 + 0.996689i $$0.474090\pi$$
$$710$$ 1640.00 0.0866875
$$711$$ 0 0
$$712$$ −4950.00 −0.260546
$$713$$ −48384.0 −2.54137
$$714$$ 0 0
$$715$$ 5720.00 0.299183
$$716$$ 23380.0 1.22032
$$717$$ 0 0
$$718$$ 120.000 0.00623727
$$719$$ −15600.0 −0.809154 −0.404577 0.914504i $$-0.632581\pi$$
−0.404577 + 0.914504i $$0.632581\pi$$
$$720$$ 0 0
$$721$$ 2112.00 0.109092
$$722$$ 6459.00 0.332935
$$723$$ 0 0
$$724$$ 1246.00 0.0639603
$$725$$ −5750.00 −0.294551
$$726$$ 0 0
$$727$$ 20696.0 1.05581 0.527904 0.849304i $$-0.322978\pi$$
0.527904 + 0.849304i $$0.322978\pi$$
$$728$$ −7920.00 −0.403207
$$729$$ 0 0
$$730$$ −190.000 −0.00963317
$$731$$ −2632.00 −0.133171
$$732$$ 0 0
$$733$$ −30778.0 −1.55090 −0.775451 0.631408i $$-0.782478\pi$$
−0.775451 + 0.631408i $$0.782478\pi$$
$$734$$ −3936.00 −0.197930
$$735$$ 0 0
$$736$$ −27048.0 −1.35462
$$737$$ 4368.00 0.218314
$$738$$ 0 0
$$739$$ 11740.0 0.584388 0.292194 0.956359i $$-0.405615\pi$$
0.292194 + 0.956359i $$0.405615\pi$$
$$740$$ −1190.00 −0.0591152
$$741$$ 0 0
$$742$$ 8112.00 0.401349
$$743$$ −2632.00 −0.129958 −0.0649789 0.997887i $$-0.520698\pi$$
−0.0649789 + 0.997887i $$0.520698\pi$$
$$744$$ 0 0
$$745$$ −6650.00 −0.327030
$$746$$ −3022.00 −0.148315
$$747$$ 0 0
$$748$$ 5096.00 0.249102
$$749$$ 864.000 0.0421494
$$750$$ 0 0
$$751$$ −20528.0 −0.997440 −0.498720 0.866763i $$-0.666196\pi$$
−0.498720 + 0.866763i $$0.666196\pi$$
$$752$$ −10496.0 −0.508976
$$753$$ 0 0
$$754$$ 5060.00 0.244396
$$755$$ 6040.00 0.291150
$$756$$ 0 0
$$757$$ 21646.0 1.03928 0.519642 0.854384i $$-0.326066\pi$$
0.519642 + 0.854384i $$0.326066\pi$$
$$758$$ 13340.0 0.639222
$$759$$ 0 0
$$760$$ 1500.00 0.0715931
$$761$$ −18282.0 −0.870857 −0.435428 0.900223i $$-0.643403\pi$$
−0.435428 + 0.900223i $$0.643403\pi$$
$$762$$ 0 0
$$763$$ 23280.0 1.10458
$$764$$ −13216.0 −0.625835
$$765$$ 0 0
$$766$$ −1008.00 −0.0475464
$$767$$ −2200.00 −0.103569
$$768$$ 0 0
$$769$$ −24190.0 −1.13435 −0.567174 0.823598i $$-0.691963\pi$$
−0.567174 + 0.823598i $$0.691963\pi$$
$$770$$ 6240.00 0.292044
$$771$$ 0 0
$$772$$ −13454.0 −0.627228
$$773$$ 25698.0 1.19572 0.597861 0.801600i $$-0.296018\pi$$
0.597861 + 0.801600i $$0.296018\pi$$
$$774$$ 0 0
$$775$$ −7200.00 −0.333718
$$776$$ 12990.0 0.600920
$$777$$ 0 0
$$778$$ 9630.00 0.443769
$$779$$ 2440.00 0.112223
$$780$$ 0 0
$$781$$ −17056.0 −0.781449
$$782$$ −2352.00 −0.107554
$$783$$ 0 0
$$784$$ 9553.00 0.435177
$$785$$ 17570.0 0.798854
$$786$$ 0 0
$$787$$ 33436.0 1.51444 0.757220 0.653160i $$-0.226557\pi$$
0.757220 + 0.653160i $$0.226557\pi$$
$$788$$ 17682.0 0.799359
$$789$$ 0 0
$$790$$ −1200.00 −0.0540431
$$791$$ 25008.0 1.12412
$$792$$ 0 0
$$793$$ 16324.0 0.730999
$$794$$ −7126.00 −0.318504
$$795$$ 0 0
$$796$$ 8120.00 0.361565
$$797$$ 37594.0 1.67083 0.835413 0.549623i $$-0.185229\pi$$
0.835413 + 0.549623i $$0.185229\pi$$
$$798$$ 0 0
$$799$$ −3584.00 −0.158689
$$800$$ −4025.00 −0.177882
$$801$$ 0 0
$$802$$ −8718.00 −0.383844
$$803$$ 1976.00 0.0868388
$$804$$ 0 0
$$805$$ 20160.0 0.882667
$$806$$ 6336.00 0.276893
$$807$$ 0 0
$$808$$ 18270.0 0.795466
$$809$$ −4730.00 −0.205560 −0.102780 0.994704i $$-0.532774\pi$$
−0.102780 + 0.994704i $$0.532774\pi$$
$$810$$ 0 0
$$811$$ −8748.00 −0.378772 −0.189386 0.981903i $$-0.560650\pi$$
−0.189386 + 0.981903i $$0.560650\pi$$
$$812$$ −38640.0 −1.66995
$$813$$ 0 0
$$814$$ −1768.00 −0.0761282
$$815$$ 10340.0 0.444410
$$816$$ 0 0
$$817$$ 3760.00 0.161011
$$818$$ 10870.0 0.464622
$$819$$ 0 0
$$820$$ −4270.00 −0.181847
$$821$$ −44142.0 −1.87645 −0.938226 0.346024i $$-0.887532\pi$$
−0.938226 + 0.346024i $$0.887532\pi$$
$$822$$ 0 0
$$823$$ 3992.00 0.169079 0.0845397 0.996420i $$-0.473058\pi$$
0.0845397 + 0.996420i $$0.473058\pi$$
$$824$$ −1320.00 −0.0558063
$$825$$ 0 0
$$826$$ −2400.00 −0.101098
$$827$$ 14444.0 0.607336 0.303668 0.952778i $$-0.401789\pi$$
0.303668 + 0.952778i $$0.401789\pi$$
$$828$$ 0 0
$$829$$ 42150.0 1.76590 0.882949 0.469468i $$-0.155554\pi$$
0.882949 + 0.469468i $$0.155554\pi$$
$$830$$ −6060.00 −0.253429
$$831$$ 0 0
$$832$$ −3674.00 −0.153093
$$833$$ 3262.00 0.135680
$$834$$ 0 0
$$835$$ −120.000 −0.00497338
$$836$$ −7280.00 −0.301177
$$837$$ 0 0
$$838$$ −9700.00 −0.399858
$$839$$ −13400.0 −0.551394 −0.275697 0.961245i $$-0.588909\pi$$
−0.275697 + 0.961245i $$0.588909\pi$$
$$840$$ 0 0
$$841$$ 28511.0 1.16901
$$842$$ −862.000 −0.0352809
$$843$$ 0 0
$$844$$ 31276.0 1.27555
$$845$$ 8565.00 0.348692
$$846$$ 0 0
$$847$$ −32952.0 −1.33677
$$848$$ 13858.0 0.561186
$$849$$ 0 0
$$850$$ −350.000 −0.0141234
$$851$$ −5712.00 −0.230088
$$852$$ 0 0
$$853$$ −8658.00 −0.347531 −0.173766 0.984787i $$-0.555594\pi$$
−0.173766 + 0.984787i $$0.555594\pi$$
$$854$$ 17808.0 0.713556
$$855$$ 0 0
$$856$$ −540.000 −0.0215617
$$857$$ −42826.0 −1.70701 −0.853505 0.521084i $$-0.825528\pi$$
−0.853505 + 0.521084i $$0.825528\pi$$
$$858$$ 0 0
$$859$$ −35900.0 −1.42595 −0.712976 0.701189i $$-0.752653\pi$$
−0.712976 + 0.701189i $$0.752653\pi$$
$$860$$ −6580.00 −0.260902
$$861$$ 0 0
$$862$$ 15792.0 0.623988
$$863$$ 3088.00 0.121804 0.0609019 0.998144i $$-0.480602\pi$$
0.0609019 + 0.998144i $$0.480602\pi$$
$$864$$ 0 0
$$865$$ −3090.00 −0.121460
$$866$$ −11602.0 −0.455256
$$867$$ 0 0
$$868$$ −48384.0 −1.89200
$$869$$ 12480.0 0.487175
$$870$$ 0 0
$$871$$ −1848.00 −0.0718910
$$872$$ −14550.0 −0.565052
$$873$$ 0 0
$$874$$ 3360.00 0.130039
$$875$$ 3000.00 0.115907
$$876$$ 0 0
$$877$$ −35274.0 −1.35817 −0.679087 0.734058i $$-0.737624\pi$$
−0.679087 + 0.734058i $$0.737624\pi$$
$$878$$ 440.000 0.0169126
$$879$$ 0 0
$$880$$ 10660.0 0.408351
$$881$$ −25042.0 −0.957646 −0.478823 0.877911i $$-0.658936\pi$$
−0.478823 + 0.877911i $$0.658936\pi$$
$$882$$ 0 0
$$883$$ 12572.0 0.479141 0.239570 0.970879i $$-0.422993\pi$$
0.239570 + 0.970879i $$0.422993\pi$$
$$884$$ −2156.00 −0.0820296
$$885$$ 0 0
$$886$$ −10188.0 −0.386312
$$887$$ 21864.0 0.827645 0.413823 0.910358i $$-0.364193\pi$$
0.413823 + 0.910358i $$0.364193\pi$$
$$888$$ 0 0
$$889$$ −46464.0 −1.75293
$$890$$ −1650.00 −0.0621440
$$891$$ 0 0
$$892$$ −42224.0 −1.58494
$$893$$ 5120.00 0.191864
$$894$$ 0 0
$$895$$ 16700.0 0.623709
$$896$$ −34920.0 −1.30200
$$897$$ 0 0
$$898$$ −13310.0 −0.494611
$$899$$ 66240.0 2.45743
$$900$$ 0 0
$$901$$ 4732.00 0.174968
$$902$$ −6344.00 −0.234182
$$903$$ 0 0
$$904$$ −15630.0 −0.575051
$$905$$ 890.000 0.0326902
$$906$$ 0 0
$$907$$ 31236.0 1.14352 0.571761 0.820420i $$-0.306260\pi$$
0.571761 + 0.820420i $$0.306260\pi$$
$$908$$ 18452.0 0.674396
$$909$$ 0 0
$$910$$ −2640.00 −0.0961705
$$911$$ −8272.00 −0.300838 −0.150419 0.988622i $$-0.548062\pi$$
−0.150419 + 0.988622i $$0.548062\pi$$
$$912$$ 0 0
$$913$$ 63024.0 2.28455
$$914$$ −3226.00 −0.116747
$$915$$ 0 0
$$916$$ −33810.0 −1.21956
$$917$$ 17568.0 0.632657
$$918$$ 0 0
$$919$$ 20200.0 0.725067 0.362533 0.931971i $$-0.381912\pi$$
0.362533 + 0.931971i $$0.381912\pi$$
$$920$$ −12600.0 −0.451532
$$921$$ 0 0
$$922$$ 6582.00 0.235105
$$923$$ 7216.00 0.257332
$$924$$ 0 0
$$925$$ −850.000 −0.0302139
$$926$$ −15072.0 −0.534878
$$927$$ 0 0
$$928$$ 37030.0 1.30988
$$929$$ −31010.0 −1.09516 −0.547581 0.836753i $$-0.684451\pi$$
−0.547581 + 0.836753i $$0.684451\pi$$
$$930$$ 0 0
$$931$$ −4660.00 −0.164044
$$932$$ 18774.0 0.659831
$$933$$ 0 0
$$934$$ 476.000 0.0166758
$$935$$ 3640.00 0.127316
$$936$$ 0 0
$$937$$ −39174.0 −1.36580 −0.682902 0.730510i $$-0.739283\pi$$
−0.682902 + 0.730510i $$0.739283\pi$$
$$938$$ −2016.00 −0.0701756
$$939$$ 0 0
$$940$$ −8960.00 −0.310897
$$941$$ 4138.00 0.143353 0.0716764 0.997428i $$-0.477165\pi$$
0.0716764 + 0.997428i $$0.477165\pi$$
$$942$$ 0 0
$$943$$ −20496.0 −0.707785
$$944$$ −4100.00 −0.141360
$$945$$ 0 0
$$946$$ −9776.00 −0.335989
$$947$$ −23676.0 −0.812425 −0.406213 0.913779i $$-0.633151\pi$$
−0.406213 + 0.913779i $$0.633151\pi$$
$$948$$ 0 0
$$949$$ −836.000 −0.0285961
$$950$$ 500.000 0.0170759
$$951$$ 0 0
$$952$$ −5040.00 −0.171583
$$953$$ −18922.0 −0.643173 −0.321586 0.946880i $$-0.604216\pi$$
−0.321586 + 0.946880i $$0.604216\pi$$
$$954$$ 0 0
$$955$$ −9440.00 −0.319865
$$956$$ 16240.0 0.549413
$$957$$ 0 0
$$958$$ −19680.0 −0.663708
$$959$$ −53136.0 −1.78921
$$960$$ 0 0
$$961$$ 53153.0 1.78420
$$962$$ 748.000 0.0250691
$$963$$ 0 0
$$964$$ −14014.0 −0.468216
$$965$$ −9610.00 −0.320577
$$966$$ 0 0
$$967$$ 39656.0 1.31877 0.659385 0.751805i $$-0.270817\pi$$
0.659385 + 0.751805i $$0.270817\pi$$
$$968$$ 20595.0 0.683831
$$969$$ 0 0
$$970$$ 4330.00 0.143328
$$971$$ 33228.0 1.09818 0.549092 0.835762i $$-0.314974\pi$$
0.549092 + 0.835762i $$0.314974\pi$$
$$972$$ 0 0
$$973$$ −480.000 −0.0158151
$$974$$ 5944.00 0.195542
$$975$$ 0 0
$$976$$ 30422.0 0.997730
$$977$$ 974.000 0.0318946 0.0159473 0.999873i $$-0.494924\pi$$
0.0159473 + 0.999873i $$0.494924\pi$$
$$978$$ 0 0
$$979$$ 17160.0 0.560200
$$980$$ 8155.00 0.265818
$$981$$ 0 0
$$982$$ 10772.0 0.350049
$$983$$ 13608.0 0.441534 0.220767 0.975327i $$-0.429144\pi$$
0.220767 + 0.975327i $$0.429144\pi$$
$$984$$ 0 0
$$985$$ 12630.0 0.408554
$$986$$ 3220.00 0.104002
$$987$$ 0 0
$$988$$ 3080.00 0.0991780
$$989$$ −31584.0 −1.01548
$$990$$ 0 0
$$991$$ 13472.0 0.431839 0.215919 0.976411i $$-0.430725\pi$$
0.215919 + 0.976411i $$0.430725\pi$$
$$992$$ 46368.0 1.48406
$$993$$ 0 0
$$994$$ 7872.00 0.251192
$$995$$ 5800.00 0.184796
$$996$$ 0 0
$$997$$ −3234.00 −0.102730 −0.0513650 0.998680i $$-0.516357\pi$$
−0.0513650 + 0.998680i $$0.516357\pi$$
$$998$$ −8140.00 −0.258184
$$999$$ 0 0
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 45.4.a.c.1.1 1
3.2 odd 2 15.4.a.a.1.1 1
4.3 odd 2 720.4.a.n.1.1 1
5.2 odd 4 225.4.b.e.199.1 2
5.3 odd 4 225.4.b.e.199.2 2
5.4 even 2 225.4.a.f.1.1 1
7.6 odd 2 2205.4.a.l.1.1 1
9.2 odd 6 405.4.e.g.271.1 2
9.4 even 3 405.4.e.i.136.1 2
9.5 odd 6 405.4.e.g.136.1 2
9.7 even 3 405.4.e.i.271.1 2
12.11 even 2 240.4.a.e.1.1 1
15.2 even 4 75.4.b.b.49.2 2
15.8 even 4 75.4.b.b.49.1 2
15.14 odd 2 75.4.a.b.1.1 1
21.20 even 2 735.4.a.e.1.1 1
24.5 odd 2 960.4.a.b.1.1 1
24.11 even 2 960.4.a.ba.1.1 1
33.32 even 2 1815.4.a.e.1.1 1
60.23 odd 4 1200.4.f.b.49.1 2
60.47 odd 4 1200.4.f.b.49.2 2
60.59 even 2 1200.4.a.t.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
15.4.a.a.1.1 1 3.2 odd 2
45.4.a.c.1.1 1 1.1 even 1 trivial
75.4.a.b.1.1 1 15.14 odd 2
75.4.b.b.49.1 2 15.8 even 4
75.4.b.b.49.2 2 15.2 even 4
225.4.a.f.1.1 1 5.4 even 2
225.4.b.e.199.1 2 5.2 odd 4
225.4.b.e.199.2 2 5.3 odd 4
240.4.a.e.1.1 1 12.11 even 2
405.4.e.g.136.1 2 9.5 odd 6
405.4.e.g.271.1 2 9.2 odd 6
405.4.e.i.136.1 2 9.4 even 3
405.4.e.i.271.1 2 9.7 even 3
720.4.a.n.1.1 1 4.3 odd 2
735.4.a.e.1.1 1 21.20 even 2
960.4.a.b.1.1 1 24.5 odd 2
960.4.a.ba.1.1 1 24.11 even 2
1200.4.a.t.1.1 1 60.59 even 2
1200.4.f.b.49.1 2 60.23 odd 4
1200.4.f.b.49.2 2 60.47 odd 4
1815.4.a.e.1.1 1 33.32 even 2
2205.4.a.l.1.1 1 7.6 odd 2