# Properties

 Label 338.4.a.b.1.1 Level $338$ Weight $4$ Character 338.1 Self dual yes Analytic conductor $19.943$ 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] = [338,4,Mod(1,338)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(338, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("338.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$338 = 2 \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 338.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: $$19.9426455819$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 26) Fricke sign: $$+1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 338.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-2.00000 q^{2} -1.00000 q^{3} +4.00000 q^{4} -17.0000 q^{5} +2.00000 q^{6} +35.0000 q^{7} -8.00000 q^{8} -26.0000 q^{9} +O(q^{10})$$ $$q-2.00000 q^{2} -1.00000 q^{3} +4.00000 q^{4} -17.0000 q^{5} +2.00000 q^{6} +35.0000 q^{7} -8.00000 q^{8} -26.0000 q^{9} +34.0000 q^{10} -2.00000 q^{11} -4.00000 q^{12} -70.0000 q^{14} +17.0000 q^{15} +16.0000 q^{16} -19.0000 q^{17} +52.0000 q^{18} -94.0000 q^{19} -68.0000 q^{20} -35.0000 q^{21} +4.00000 q^{22} -72.0000 q^{23} +8.00000 q^{24} +164.000 q^{25} +53.0000 q^{27} +140.000 q^{28} +246.000 q^{29} -34.0000 q^{30} +100.000 q^{31} -32.0000 q^{32} +2.00000 q^{33} +38.0000 q^{34} -595.000 q^{35} -104.000 q^{36} +11.0000 q^{37} +188.000 q^{38} +136.000 q^{40} +280.000 q^{41} +70.0000 q^{42} +241.000 q^{43} -8.00000 q^{44} +442.000 q^{45} +144.000 q^{46} -137.000 q^{47} -16.0000 q^{48} +882.000 q^{49} -328.000 q^{50} +19.0000 q^{51} -232.000 q^{53} -106.000 q^{54} +34.0000 q^{55} -280.000 q^{56} +94.0000 q^{57} -492.000 q^{58} +386.000 q^{59} +68.0000 q^{60} +64.0000 q^{61} -200.000 q^{62} -910.000 q^{63} +64.0000 q^{64} -4.00000 q^{66} +670.000 q^{67} -76.0000 q^{68} +72.0000 q^{69} +1190.00 q^{70} -55.0000 q^{71} +208.000 q^{72} +838.000 q^{73} -22.0000 q^{74} -164.000 q^{75} -376.000 q^{76} -70.0000 q^{77} +1016.00 q^{79} -272.000 q^{80} +649.000 q^{81} -560.000 q^{82} -420.000 q^{83} -140.000 q^{84} +323.000 q^{85} -482.000 q^{86} -246.000 q^{87} +16.0000 q^{88} +934.000 q^{89} -884.000 q^{90} -288.000 q^{92} -100.000 q^{93} +274.000 q^{94} +1598.00 q^{95} +32.0000 q^{96} +1154.00 q^{97} -1764.00 q^{98} +52.0000 q^{99} +O(q^{100})$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −2.00000 −0.707107
$$3$$ −1.00000 −0.192450 −0.0962250 0.995360i $$-0.530677\pi$$
−0.0962250 + 0.995360i $$0.530677\pi$$
$$4$$ 4.00000 0.500000
$$5$$ −17.0000 −1.52053 −0.760263 0.649615i $$-0.774930\pi$$
−0.760263 + 0.649615i $$0.774930\pi$$
$$6$$ 2.00000 0.136083
$$7$$ 35.0000 1.88982 0.944911 0.327327i $$-0.106148\pi$$
0.944911 + 0.327327i $$0.106148\pi$$
$$8$$ −8.00000 −0.353553
$$9$$ −26.0000 −0.962963
$$10$$ 34.0000 1.07517
$$11$$ −2.00000 −0.0548202 −0.0274101 0.999624i $$-0.508726\pi$$
−0.0274101 + 0.999624i $$0.508726\pi$$
$$12$$ −4.00000 −0.0962250
$$13$$ 0 0
$$14$$ −70.0000 −1.33631
$$15$$ 17.0000 0.292625
$$16$$ 16.0000 0.250000
$$17$$ −19.0000 −0.271069 −0.135535 0.990773i $$-0.543275\pi$$
−0.135535 + 0.990773i $$0.543275\pi$$
$$18$$ 52.0000 0.680918
$$19$$ −94.0000 −1.13500 −0.567502 0.823372i $$-0.692090\pi$$
−0.567502 + 0.823372i $$0.692090\pi$$
$$20$$ −68.0000 −0.760263
$$21$$ −35.0000 −0.363696
$$22$$ 4.00000 0.0387638
$$23$$ −72.0000 −0.652741 −0.326370 0.945242i $$-0.605826\pi$$
−0.326370 + 0.945242i $$0.605826\pi$$
$$24$$ 8.00000 0.0680414
$$25$$ 164.000 1.31200
$$26$$ 0 0
$$27$$ 53.0000 0.377772
$$28$$ 140.000 0.944911
$$29$$ 246.000 1.57521 0.787604 0.616181i $$-0.211321\pi$$
0.787604 + 0.616181i $$0.211321\pi$$
$$30$$ −34.0000 −0.206917
$$31$$ 100.000 0.579372 0.289686 0.957122i $$-0.406449\pi$$
0.289686 + 0.957122i $$0.406449\pi$$
$$32$$ −32.0000 −0.176777
$$33$$ 2.00000 0.0105502
$$34$$ 38.0000 0.191675
$$35$$ −595.000 −2.87352
$$36$$ −104.000 −0.481481
$$37$$ 11.0000 0.0488754 0.0244377 0.999701i $$-0.492220\pi$$
0.0244377 + 0.999701i $$0.492220\pi$$
$$38$$ 188.000 0.802569
$$39$$ 0 0
$$40$$ 136.000 0.537587
$$41$$ 280.000 1.06655 0.533276 0.845941i $$-0.320961\pi$$
0.533276 + 0.845941i $$0.320961\pi$$
$$42$$ 70.0000 0.257172
$$43$$ 241.000 0.854701 0.427351 0.904086i $$-0.359447\pi$$
0.427351 + 0.904086i $$0.359447\pi$$
$$44$$ −8.00000 −0.0274101
$$45$$ 442.000 1.46421
$$46$$ 144.000 0.461557
$$47$$ −137.000 −0.425181 −0.212590 0.977141i $$-0.568190\pi$$
−0.212590 + 0.977141i $$0.568190\pi$$
$$48$$ −16.0000 −0.0481125
$$49$$ 882.000 2.57143
$$50$$ −328.000 −0.927724
$$51$$ 19.0000 0.0521673
$$52$$ 0 0
$$53$$ −232.000 −0.601276 −0.300638 0.953738i $$-0.597200\pi$$
−0.300638 + 0.953738i $$0.597200\pi$$
$$54$$ −106.000 −0.267125
$$55$$ 34.0000 0.0833556
$$56$$ −280.000 −0.668153
$$57$$ 94.0000 0.218432
$$58$$ −492.000 −1.11384
$$59$$ 386.000 0.851744 0.425872 0.904783i $$-0.359967\pi$$
0.425872 + 0.904783i $$0.359967\pi$$
$$60$$ 68.0000 0.146313
$$61$$ 64.0000 0.134334 0.0671669 0.997742i $$-0.478604\pi$$
0.0671669 + 0.997742i $$0.478604\pi$$
$$62$$ −200.000 −0.409678
$$63$$ −910.000 −1.81983
$$64$$ 64.0000 0.125000
$$65$$ 0 0
$$66$$ −4.00000 −0.00746009
$$67$$ 670.000 1.22169 0.610847 0.791748i $$-0.290829\pi$$
0.610847 + 0.791748i $$0.290829\pi$$
$$68$$ −76.0000 −0.135535
$$69$$ 72.0000 0.125620
$$70$$ 1190.00 2.03189
$$71$$ −55.0000 −0.0919338 −0.0459669 0.998943i $$-0.514637\pi$$
−0.0459669 + 0.998943i $$0.514637\pi$$
$$72$$ 208.000 0.340459
$$73$$ 838.000 1.34357 0.671784 0.740747i $$-0.265528\pi$$
0.671784 + 0.740747i $$0.265528\pi$$
$$74$$ −22.0000 −0.0345601
$$75$$ −164.000 −0.252495
$$76$$ −376.000 −0.567502
$$77$$ −70.0000 −0.103601
$$78$$ 0 0
$$79$$ 1016.00 1.44695 0.723474 0.690351i $$-0.242544\pi$$
0.723474 + 0.690351i $$0.242544\pi$$
$$80$$ −272.000 −0.380132
$$81$$ 649.000 0.890261
$$82$$ −560.000 −0.754167
$$83$$ −420.000 −0.555434 −0.277717 0.960663i $$-0.589578\pi$$
−0.277717 + 0.960663i $$0.589578\pi$$
$$84$$ −140.000 −0.181848
$$85$$ 323.000 0.412168
$$86$$ −482.000 −0.604365
$$87$$ −246.000 −0.303149
$$88$$ 16.0000 0.0193819
$$89$$ 934.000 1.11240 0.556201 0.831048i $$-0.312258\pi$$
0.556201 + 0.831048i $$0.312258\pi$$
$$90$$ −884.000 −1.03535
$$91$$ 0 0
$$92$$ −288.000 −0.326370
$$93$$ −100.000 −0.111500
$$94$$ 274.000 0.300648
$$95$$ 1598.00 1.72580
$$96$$ 32.0000 0.0340207
$$97$$ 1154.00 1.20795 0.603974 0.797004i $$-0.293583\pi$$
0.603974 + 0.797004i $$0.293583\pi$$
$$98$$ −1764.00 −1.81827
$$99$$ 52.0000 0.0527899
$$100$$ 656.000 0.656000
$$101$$ 292.000 0.287674 0.143837 0.989601i $$-0.454056\pi$$
0.143837 + 0.989601i $$0.454056\pi$$
$$102$$ −38.0000 −0.0368878
$$103$$ 52.0000 0.0497448 0.0248724 0.999691i $$-0.492082\pi$$
0.0248724 + 0.999691i $$0.492082\pi$$
$$104$$ 0 0
$$105$$ 595.000 0.553010
$$106$$ 464.000 0.425167
$$107$$ 12.0000 0.0108419 0.00542095 0.999985i $$-0.498274\pi$$
0.00542095 + 0.999985i $$0.498274\pi$$
$$108$$ 212.000 0.188886
$$109$$ −1213.00 −1.06591 −0.532956 0.846143i $$-0.678919\pi$$
−0.532956 + 0.846143i $$0.678919\pi$$
$$110$$ −68.0000 −0.0589413
$$111$$ −11.0000 −0.00940607
$$112$$ 560.000 0.472456
$$113$$ −1158.00 −0.964031 −0.482015 0.876163i $$-0.660095\pi$$
−0.482015 + 0.876163i $$0.660095\pi$$
$$114$$ −188.000 −0.154455
$$115$$ 1224.00 0.992509
$$116$$ 984.000 0.787604
$$117$$ 0 0
$$118$$ −772.000 −0.602274
$$119$$ −665.000 −0.512273
$$120$$ −136.000 −0.103459
$$121$$ −1327.00 −0.996995
$$122$$ −128.000 −0.0949883
$$123$$ −280.000 −0.205258
$$124$$ 400.000 0.289686
$$125$$ −663.000 −0.474404
$$126$$ 1820.00 1.28681
$$127$$ −540.000 −0.377301 −0.188651 0.982044i $$-0.560411\pi$$
−0.188651 + 0.982044i $$0.560411\pi$$
$$128$$ −128.000 −0.0883883
$$129$$ −241.000 −0.164487
$$130$$ 0 0
$$131$$ −1571.00 −1.04778 −0.523889 0.851787i $$-0.675519\pi$$
−0.523889 + 0.851787i $$0.675519\pi$$
$$132$$ 8.00000 0.00527508
$$133$$ −3290.00 −2.14496
$$134$$ −1340.00 −0.863868
$$135$$ −901.000 −0.574413
$$136$$ 152.000 0.0958374
$$137$$ 568.000 0.354215 0.177108 0.984191i $$-0.443326\pi$$
0.177108 + 0.984191i $$0.443326\pi$$
$$138$$ −144.000 −0.0888268
$$139$$ 1845.00 1.12583 0.562917 0.826514i $$-0.309679\pi$$
0.562917 + 0.826514i $$0.309679\pi$$
$$140$$ −2380.00 −1.43676
$$141$$ 137.000 0.0818261
$$142$$ 110.000 0.0650070
$$143$$ 0 0
$$144$$ −416.000 −0.240741
$$145$$ −4182.00 −2.39515
$$146$$ −1676.00 −0.950046
$$147$$ −882.000 −0.494872
$$148$$ 44.0000 0.0244377
$$149$$ −2850.00 −1.56699 −0.783494 0.621400i $$-0.786564\pi$$
−0.783494 + 0.621400i $$0.786564\pi$$
$$150$$ 328.000 0.178541
$$151$$ −2763.00 −1.48907 −0.744536 0.667583i $$-0.767329\pi$$
−0.744536 + 0.667583i $$0.767329\pi$$
$$152$$ 752.000 0.401285
$$153$$ 494.000 0.261030
$$154$$ 140.000 0.0732566
$$155$$ −1700.00 −0.880950
$$156$$ 0 0
$$157$$ −466.000 −0.236884 −0.118442 0.992961i $$-0.537790\pi$$
−0.118442 + 0.992961i $$0.537790\pi$$
$$158$$ −2032.00 −1.02315
$$159$$ 232.000 0.115716
$$160$$ 544.000 0.268794
$$161$$ −2520.00 −1.23356
$$162$$ −1298.00 −0.629509
$$163$$ −2552.00 −1.22631 −0.613154 0.789964i $$-0.710099\pi$$
−0.613154 + 0.789964i $$0.710099\pi$$
$$164$$ 1120.00 0.533276
$$165$$ −34.0000 −0.0160418
$$166$$ 840.000 0.392751
$$167$$ 3768.00 1.74597 0.872984 0.487749i $$-0.162182\pi$$
0.872984 + 0.487749i $$0.162182\pi$$
$$168$$ 280.000 0.128586
$$169$$ 0 0
$$170$$ −646.000 −0.291447
$$171$$ 2444.00 1.09297
$$172$$ 964.000 0.427351
$$173$$ 3296.00 1.44850 0.724249 0.689538i $$-0.242187\pi$$
0.724249 + 0.689538i $$0.242187\pi$$
$$174$$ 492.000 0.214359
$$175$$ 5740.00 2.47945
$$176$$ −32.0000 −0.0137051
$$177$$ −386.000 −0.163918
$$178$$ −1868.00 −0.786587
$$179$$ 45.0000 0.0187903 0.00939513 0.999956i $$-0.497009\pi$$
0.00939513 + 0.999956i $$0.497009\pi$$
$$180$$ 1768.00 0.732105
$$181$$ −364.000 −0.149480 −0.0747401 0.997203i $$-0.523813\pi$$
−0.0747401 + 0.997203i $$0.523813\pi$$
$$182$$ 0 0
$$183$$ −64.0000 −0.0258525
$$184$$ 576.000 0.230779
$$185$$ −187.000 −0.0743163
$$186$$ 200.000 0.0788425
$$187$$ 38.0000 0.0148601
$$188$$ −548.000 −0.212590
$$189$$ 1855.00 0.713923
$$190$$ −3196.00 −1.22033
$$191$$ 2482.00 0.940268 0.470134 0.882595i $$-0.344206\pi$$
0.470134 + 0.882595i $$0.344206\pi$$
$$192$$ −64.0000 −0.0240563
$$193$$ 1220.00 0.455013 0.227507 0.973777i $$-0.426943\pi$$
0.227507 + 0.973777i $$0.426943\pi$$
$$194$$ −2308.00 −0.854148
$$195$$ 0 0
$$196$$ 3528.00 1.28571
$$197$$ 1593.00 0.576125 0.288062 0.957612i $$-0.406989\pi$$
0.288062 + 0.957612i $$0.406989\pi$$
$$198$$ −104.000 −0.0373281
$$199$$ 1606.00 0.572092 0.286046 0.958216i $$-0.407659\pi$$
0.286046 + 0.958216i $$0.407659\pi$$
$$200$$ −1312.00 −0.463862
$$201$$ −670.000 −0.235115
$$202$$ −584.000 −0.203416
$$203$$ 8610.00 2.97686
$$204$$ 76.0000 0.0260836
$$205$$ −4760.00 −1.62172
$$206$$ −104.000 −0.0351749
$$207$$ 1872.00 0.628565
$$208$$ 0 0
$$209$$ 188.000 0.0622212
$$210$$ −1190.00 −0.391037
$$211$$ 2469.00 0.805559 0.402780 0.915297i $$-0.368044\pi$$
0.402780 + 0.915297i $$0.368044\pi$$
$$212$$ −928.000 −0.300638
$$213$$ 55.0000 0.0176927
$$214$$ −24.0000 −0.00766638
$$215$$ −4097.00 −1.29960
$$216$$ −424.000 −0.133563
$$217$$ 3500.00 1.09491
$$218$$ 2426.00 0.753713
$$219$$ −838.000 −0.258570
$$220$$ 136.000 0.0416778
$$221$$ 0 0
$$222$$ 22.0000 0.00665110
$$223$$ −1943.00 −0.583466 −0.291733 0.956500i $$-0.594232\pi$$
−0.291733 + 0.956500i $$0.594232\pi$$
$$224$$ −1120.00 −0.334077
$$225$$ −4264.00 −1.26341
$$226$$ 2316.00 0.681673
$$227$$ −3032.00 −0.886524 −0.443262 0.896392i $$-0.646179\pi$$
−0.443262 + 0.896392i $$0.646179\pi$$
$$228$$ 376.000 0.109216
$$229$$ −5311.00 −1.53258 −0.766290 0.642495i $$-0.777899\pi$$
−0.766290 + 0.642495i $$0.777899\pi$$
$$230$$ −2448.00 −0.701810
$$231$$ 70.0000 0.0199379
$$232$$ −1968.00 −0.556920
$$233$$ −507.000 −0.142552 −0.0712761 0.997457i $$-0.522707\pi$$
−0.0712761 + 0.997457i $$0.522707\pi$$
$$234$$ 0 0
$$235$$ 2329.00 0.646499
$$236$$ 1544.00 0.425872
$$237$$ −1016.00 −0.278465
$$238$$ 1330.00 0.362231
$$239$$ 6795.00 1.83905 0.919523 0.393036i $$-0.128575\pi$$
0.919523 + 0.393036i $$0.128575\pi$$
$$240$$ 272.000 0.0731564
$$241$$ 4442.00 1.18728 0.593640 0.804731i $$-0.297690\pi$$
0.593640 + 0.804731i $$0.297690\pi$$
$$242$$ 2654.00 0.704982
$$243$$ −2080.00 −0.549103
$$244$$ 256.000 0.0671669
$$245$$ −14994.0 −3.90992
$$246$$ 560.000 0.145139
$$247$$ 0 0
$$248$$ −800.000 −0.204839
$$249$$ 420.000 0.106893
$$250$$ 1326.00 0.335454
$$251$$ −5024.00 −1.26339 −0.631697 0.775215i $$-0.717641\pi$$
−0.631697 + 0.775215i $$0.717641\pi$$
$$252$$ −3640.00 −0.909914
$$253$$ 144.000 0.0357834
$$254$$ 1080.00 0.266792
$$255$$ −323.000 −0.0793217
$$256$$ 256.000 0.0625000
$$257$$ 7065.00 1.71480 0.857398 0.514654i $$-0.172080\pi$$
0.857398 + 0.514654i $$0.172080\pi$$
$$258$$ 482.000 0.116310
$$259$$ 385.000 0.0923658
$$260$$ 0 0
$$261$$ −6396.00 −1.51687
$$262$$ 3142.00 0.740891
$$263$$ 1364.00 0.319802 0.159901 0.987133i $$-0.448883\pi$$
0.159901 + 0.987133i $$0.448883\pi$$
$$264$$ −16.0000 −0.00373005
$$265$$ 3944.00 0.914257
$$266$$ 6580.00 1.51671
$$267$$ −934.000 −0.214082
$$268$$ 2680.00 0.610847
$$269$$ −4304.00 −0.975537 −0.487769 0.872973i $$-0.662189\pi$$
−0.487769 + 0.872973i $$0.662189\pi$$
$$270$$ 1802.00 0.406171
$$271$$ 1519.00 0.340490 0.170245 0.985402i $$-0.445544\pi$$
0.170245 + 0.985402i $$0.445544\pi$$
$$272$$ −304.000 −0.0677673
$$273$$ 0 0
$$274$$ −1136.00 −0.250468
$$275$$ −328.000 −0.0719242
$$276$$ 288.000 0.0628100
$$277$$ 1996.00 0.432953 0.216477 0.976288i $$-0.430544\pi$$
0.216477 + 0.976288i $$0.430544\pi$$
$$278$$ −3690.00 −0.796085
$$279$$ −2600.00 −0.557914
$$280$$ 4760.00 1.01594
$$281$$ 2918.00 0.619478 0.309739 0.950822i $$-0.399758\pi$$
0.309739 + 0.950822i $$0.399758\pi$$
$$282$$ −274.000 −0.0578598
$$283$$ 812.000 0.170560 0.0852798 0.996357i $$-0.472822\pi$$
0.0852798 + 0.996357i $$0.472822\pi$$
$$284$$ −220.000 −0.0459669
$$285$$ −1598.00 −0.332131
$$286$$ 0 0
$$287$$ 9800.00 2.01559
$$288$$ 832.000 0.170229
$$289$$ −4552.00 −0.926521
$$290$$ 8364.00 1.69362
$$291$$ −1154.00 −0.232470
$$292$$ 3352.00 0.671784
$$293$$ 1855.00 0.369864 0.184932 0.982751i $$-0.440793\pi$$
0.184932 + 0.982751i $$0.440793\pi$$
$$294$$ 1764.00 0.349927
$$295$$ −6562.00 −1.29510
$$296$$ −88.0000 −0.0172801
$$297$$ −106.000 −0.0207096
$$298$$ 5700.00 1.10803
$$299$$ 0 0
$$300$$ −656.000 −0.126247
$$301$$ 8435.00 1.61523
$$302$$ 5526.00 1.05293
$$303$$ −292.000 −0.0553629
$$304$$ −1504.00 −0.283751
$$305$$ −1088.00 −0.204258
$$306$$ −988.000 −0.184576
$$307$$ −5350.00 −0.994595 −0.497297 0.867580i $$-0.665674\pi$$
−0.497297 + 0.867580i $$0.665674\pi$$
$$308$$ −280.000 −0.0518003
$$309$$ −52.0000 −0.00957339
$$310$$ 3400.00 0.622926
$$311$$ 2262.00 0.412432 0.206216 0.978507i $$-0.433885\pi$$
0.206216 + 0.978507i $$0.433885\pi$$
$$312$$ 0 0
$$313$$ −1857.00 −0.335348 −0.167674 0.985843i $$-0.553626\pi$$
−0.167674 + 0.985843i $$0.553626\pi$$
$$314$$ 932.000 0.167503
$$315$$ 15470.0 2.76710
$$316$$ 4064.00 0.723474
$$317$$ 3870.00 0.685681 0.342840 0.939394i $$-0.388611\pi$$
0.342840 + 0.939394i $$0.388611\pi$$
$$318$$ −464.000 −0.0818234
$$319$$ −492.000 −0.0863533
$$320$$ −1088.00 −0.190066
$$321$$ −12.0000 −0.00208653
$$322$$ 5040.00 0.872262
$$323$$ 1786.00 0.307665
$$324$$ 2596.00 0.445130
$$325$$ 0 0
$$326$$ 5104.00 0.867130
$$327$$ 1213.00 0.205135
$$328$$ −2240.00 −0.377083
$$329$$ −4795.00 −0.803516
$$330$$ 68.0000 0.0113433
$$331$$ −10520.0 −1.74692 −0.873461 0.486893i $$-0.838130\pi$$
−0.873461 + 0.486893i $$0.838130\pi$$
$$332$$ −1680.00 −0.277717
$$333$$ −286.000 −0.0470652
$$334$$ −7536.00 −1.23459
$$335$$ −11390.0 −1.85762
$$336$$ −560.000 −0.0909241
$$337$$ 7839.00 1.26711 0.633557 0.773696i $$-0.281594\pi$$
0.633557 + 0.773696i $$0.281594\pi$$
$$338$$ 0 0
$$339$$ 1158.00 0.185528
$$340$$ 1292.00 0.206084
$$341$$ −200.000 −0.0317613
$$342$$ −4888.00 −0.772844
$$343$$ 18865.0 2.96972
$$344$$ −1928.00 −0.302183
$$345$$ −1224.00 −0.191009
$$346$$ −6592.00 −1.02424
$$347$$ −1275.00 −0.197250 −0.0986248 0.995125i $$-0.531444\pi$$
−0.0986248 + 0.995125i $$0.531444\pi$$
$$348$$ −984.000 −0.151575
$$349$$ 375.000 0.0575166 0.0287583 0.999586i $$-0.490845\pi$$
0.0287583 + 0.999586i $$0.490845\pi$$
$$350$$ −11480.0 −1.75323
$$351$$ 0 0
$$352$$ 64.0000 0.00969094
$$353$$ −1592.00 −0.240039 −0.120019 0.992772i $$-0.538296\pi$$
−0.120019 + 0.992772i $$0.538296\pi$$
$$354$$ 772.000 0.115908
$$355$$ 935.000 0.139788
$$356$$ 3736.00 0.556201
$$357$$ 665.000 0.0985869
$$358$$ −90.0000 −0.0132867
$$359$$ 2424.00 0.356362 0.178181 0.983998i $$-0.442979\pi$$
0.178181 + 0.983998i $$0.442979\pi$$
$$360$$ −3536.00 −0.517677
$$361$$ 1977.00 0.288234
$$362$$ 728.000 0.105698
$$363$$ 1327.00 0.191872
$$364$$ 0 0
$$365$$ −14246.0 −2.04293
$$366$$ 128.000 0.0182805
$$367$$ −7970.00 −1.13360 −0.566799 0.823856i $$-0.691818\pi$$
−0.566799 + 0.823856i $$0.691818\pi$$
$$368$$ −1152.00 −0.163185
$$369$$ −7280.00 −1.02705
$$370$$ 374.000 0.0525496
$$371$$ −8120.00 −1.13631
$$372$$ −400.000 −0.0557501
$$373$$ 12492.0 1.73408 0.867039 0.498240i $$-0.166020\pi$$
0.867039 + 0.498240i $$0.166020\pi$$
$$374$$ −76.0000 −0.0105077
$$375$$ 663.000 0.0912991
$$376$$ 1096.00 0.150324
$$377$$ 0 0
$$378$$ −3710.00 −0.504820
$$379$$ 4284.00 0.580618 0.290309 0.956933i $$-0.406242\pi$$
0.290309 + 0.956933i $$0.406242\pi$$
$$380$$ 6392.00 0.862902
$$381$$ 540.000 0.0726116
$$382$$ −4964.00 −0.664870
$$383$$ 2337.00 0.311789 0.155894 0.987774i $$-0.450174\pi$$
0.155894 + 0.987774i $$0.450174\pi$$
$$384$$ 128.000 0.0170103
$$385$$ 1190.00 0.157527
$$386$$ −2440.00 −0.321743
$$387$$ −6266.00 −0.823046
$$388$$ 4616.00 0.603974
$$389$$ 3562.00 0.464269 0.232134 0.972684i $$-0.425429\pi$$
0.232134 + 0.972684i $$0.425429\pi$$
$$390$$ 0 0
$$391$$ 1368.00 0.176938
$$392$$ −7056.00 −0.909137
$$393$$ 1571.00 0.201645
$$394$$ −3186.00 −0.407382
$$395$$ −17272.0 −2.20012
$$396$$ 208.000 0.0263949
$$397$$ 5274.00 0.666737 0.333368 0.942797i $$-0.391815\pi$$
0.333368 + 0.942797i $$0.391815\pi$$
$$398$$ −3212.00 −0.404530
$$399$$ 3290.00 0.412797
$$400$$ 2624.00 0.328000
$$401$$ −4812.00 −0.599251 −0.299626 0.954057i $$-0.596862\pi$$
−0.299626 + 0.954057i $$0.596862\pi$$
$$402$$ 1340.00 0.166252
$$403$$ 0 0
$$404$$ 1168.00 0.143837
$$405$$ −11033.0 −1.35366
$$406$$ −17220.0 −2.10496
$$407$$ −22.0000 −0.00267936
$$408$$ −152.000 −0.0184439
$$409$$ 9448.00 1.14223 0.571117 0.820869i $$-0.306510\pi$$
0.571117 + 0.820869i $$0.306510\pi$$
$$410$$ 9520.00 1.14673
$$411$$ −568.000 −0.0681688
$$412$$ 208.000 0.0248724
$$413$$ 13510.0 1.60965
$$414$$ −3744.00 −0.444463
$$415$$ 7140.00 0.844551
$$416$$ 0 0
$$417$$ −1845.00 −0.216667
$$418$$ −376.000 −0.0439970
$$419$$ −521.000 −0.0607459 −0.0303729 0.999539i $$-0.509669\pi$$
−0.0303729 + 0.999539i $$0.509669\pi$$
$$420$$ 2380.00 0.276505
$$421$$ −7141.00 −0.826677 −0.413339 0.910577i $$-0.635637\pi$$
−0.413339 + 0.910577i $$0.635637\pi$$
$$422$$ −4938.00 −0.569616
$$423$$ 3562.00 0.409433
$$424$$ 1856.00 0.212583
$$425$$ −3116.00 −0.355643
$$426$$ −110.000 −0.0125106
$$427$$ 2240.00 0.253867
$$428$$ 48.0000 0.00542095
$$429$$ 0 0
$$430$$ 8194.00 0.918953
$$431$$ 6299.00 0.703973 0.351986 0.936005i $$-0.385506\pi$$
0.351986 + 0.936005i $$0.385506\pi$$
$$432$$ 848.000 0.0944431
$$433$$ 3231.00 0.358596 0.179298 0.983795i $$-0.442617\pi$$
0.179298 + 0.983795i $$0.442617\pi$$
$$434$$ −7000.00 −0.774218
$$435$$ 4182.00 0.460946
$$436$$ −4852.00 −0.532956
$$437$$ 6768.00 0.740863
$$438$$ 1676.00 0.182836
$$439$$ −410.000 −0.0445746 −0.0222873 0.999752i $$-0.507095\pi$$
−0.0222873 + 0.999752i $$0.507095\pi$$
$$440$$ −272.000 −0.0294707
$$441$$ −22932.0 −2.47619
$$442$$ 0 0
$$443$$ −9397.00 −1.00782 −0.503911 0.863756i $$-0.668106\pi$$
−0.503911 + 0.863756i $$0.668106\pi$$
$$444$$ −44.0000 −0.00470304
$$445$$ −15878.0 −1.69144
$$446$$ 3886.00 0.412573
$$447$$ 2850.00 0.301567
$$448$$ 2240.00 0.236228
$$449$$ −654.000 −0.0687398 −0.0343699 0.999409i $$-0.510942\pi$$
−0.0343699 + 0.999409i $$0.510942\pi$$
$$450$$ 8528.00 0.893364
$$451$$ −560.000 −0.0584687
$$452$$ −4632.00 −0.482015
$$453$$ 2763.00 0.286572
$$454$$ 6064.00 0.626867
$$455$$ 0 0
$$456$$ −752.000 −0.0772273
$$457$$ −14982.0 −1.53354 −0.766771 0.641921i $$-0.778138\pi$$
−0.766771 + 0.641921i $$0.778138\pi$$
$$458$$ 10622.0 1.08370
$$459$$ −1007.00 −0.102402
$$460$$ 4896.00 0.496255
$$461$$ −10845.0 −1.09567 −0.547833 0.836588i $$-0.684547\pi$$
−0.547833 + 0.836588i $$0.684547\pi$$
$$462$$ −140.000 −0.0140982
$$463$$ −13072.0 −1.31211 −0.656055 0.754713i $$-0.727776\pi$$
−0.656055 + 0.754713i $$0.727776\pi$$
$$464$$ 3936.00 0.393802
$$465$$ 1700.00 0.169539
$$466$$ 1014.00 0.100800
$$467$$ −15340.0 −1.52002 −0.760011 0.649910i $$-0.774807\pi$$
−0.760011 + 0.649910i $$0.774807\pi$$
$$468$$ 0 0
$$469$$ 23450.0 2.30879
$$470$$ −4658.00 −0.457144
$$471$$ 466.000 0.0455884
$$472$$ −3088.00 −0.301137
$$473$$ −482.000 −0.0468549
$$474$$ 2032.00 0.196905
$$475$$ −15416.0 −1.48913
$$476$$ −2660.00 −0.256136
$$477$$ 6032.00 0.579007
$$478$$ −13590.0 −1.30040
$$479$$ 19615.0 1.87105 0.935524 0.353262i $$-0.114928\pi$$
0.935524 + 0.353262i $$0.114928\pi$$
$$480$$ −544.000 −0.0517294
$$481$$ 0 0
$$482$$ −8884.00 −0.839533
$$483$$ 2520.00 0.237400
$$484$$ −5308.00 −0.498497
$$485$$ −19618.0 −1.83672
$$486$$ 4160.00 0.388275
$$487$$ 10904.0 1.01459 0.507297 0.861771i $$-0.330645\pi$$
0.507297 + 0.861771i $$0.330645\pi$$
$$488$$ −512.000 −0.0474942
$$489$$ 2552.00 0.236003
$$490$$ 29988.0 2.76473
$$491$$ −18519.0 −1.70214 −0.851070 0.525052i $$-0.824046\pi$$
−0.851070 + 0.525052i $$0.824046\pi$$
$$492$$ −1120.00 −0.102629
$$493$$ −4674.00 −0.426991
$$494$$ 0 0
$$495$$ −884.000 −0.0802684
$$496$$ 1600.00 0.144843
$$497$$ −1925.00 −0.173739
$$498$$ −840.000 −0.0755849
$$499$$ 14400.0 1.29185 0.645924 0.763401i $$-0.276472\pi$$
0.645924 + 0.763401i $$0.276472\pi$$
$$500$$ −2652.00 −0.237202
$$501$$ −3768.00 −0.336012
$$502$$ 10048.0 0.893355
$$503$$ 20422.0 1.81028 0.905141 0.425111i $$-0.139765\pi$$
0.905141 + 0.425111i $$0.139765\pi$$
$$504$$ 7280.00 0.643407
$$505$$ −4964.00 −0.437416
$$506$$ −288.000 −0.0253027
$$507$$ 0 0
$$508$$ −2160.00 −0.188651
$$509$$ −15606.0 −1.35899 −0.679493 0.733682i $$-0.737800\pi$$
−0.679493 + 0.733682i $$0.737800\pi$$
$$510$$ 646.000 0.0560889
$$511$$ 29330.0 2.53911
$$512$$ −512.000 −0.0441942
$$513$$ −4982.00 −0.428773
$$514$$ −14130.0 −1.21254
$$515$$ −884.000 −0.0756382
$$516$$ −964.000 −0.0822437
$$517$$ 274.000 0.0233085
$$518$$ −770.000 −0.0653125
$$519$$ −3296.00 −0.278764
$$520$$ 0 0
$$521$$ 1783.00 0.149932 0.0749661 0.997186i $$-0.476115\pi$$
0.0749661 + 0.997186i $$0.476115\pi$$
$$522$$ 12792.0 1.07259
$$523$$ 11140.0 0.931392 0.465696 0.884945i $$-0.345804\pi$$
0.465696 + 0.884945i $$0.345804\pi$$
$$524$$ −6284.00 −0.523889
$$525$$ −5740.00 −0.477170
$$526$$ −2728.00 −0.226134
$$527$$ −1900.00 −0.157050
$$528$$ 32.0000 0.00263754
$$529$$ −6983.00 −0.573929
$$530$$ −7888.00 −0.646477
$$531$$ −10036.0 −0.820198
$$532$$ −13160.0 −1.07248
$$533$$ 0 0
$$534$$ 1868.00 0.151379
$$535$$ −204.000 −0.0164854
$$536$$ −5360.00 −0.431934
$$537$$ −45.0000 −0.00361619
$$538$$ 8608.00 0.689809
$$539$$ −1764.00 −0.140966
$$540$$ −3604.00 −0.287206
$$541$$ −839.000 −0.0666755 −0.0333377 0.999444i $$-0.510614\pi$$
−0.0333377 + 0.999444i $$0.510614\pi$$
$$542$$ −3038.00 −0.240762
$$543$$ 364.000 0.0287675
$$544$$ 608.000 0.0479187
$$545$$ 20621.0 1.62075
$$546$$ 0 0
$$547$$ 17279.0 1.35063 0.675317 0.737528i $$-0.264007\pi$$
0.675317 + 0.737528i $$0.264007\pi$$
$$548$$ 2272.00 0.177108
$$549$$ −1664.00 −0.129358
$$550$$ 656.000 0.0508581
$$551$$ −23124.0 −1.78787
$$552$$ −576.000 −0.0444134
$$553$$ 35560.0 2.73448
$$554$$ −3992.00 −0.306144
$$555$$ 187.000 0.0143022
$$556$$ 7380.00 0.562917
$$557$$ 20577.0 1.56531 0.782653 0.622458i $$-0.213866\pi$$
0.782653 + 0.622458i $$0.213866\pi$$
$$558$$ 5200.00 0.394505
$$559$$ 0 0
$$560$$ −9520.00 −0.718381
$$561$$ −38.0000 −0.00285982
$$562$$ −5836.00 −0.438037
$$563$$ 11785.0 0.882200 0.441100 0.897458i $$-0.354588\pi$$
0.441100 + 0.897458i $$0.354588\pi$$
$$564$$ 548.000 0.0409131
$$565$$ 19686.0 1.46583
$$566$$ −1624.00 −0.120604
$$567$$ 22715.0 1.68243
$$568$$ 440.000 0.0325035
$$569$$ 10887.0 0.802121 0.401060 0.916052i $$-0.368642\pi$$
0.401060 + 0.916052i $$0.368642\pi$$
$$570$$ 3196.00 0.234852
$$571$$ −11453.0 −0.839393 −0.419696 0.907665i $$-0.637863\pi$$
−0.419696 + 0.907665i $$0.637863\pi$$
$$572$$ 0 0
$$573$$ −2482.00 −0.180955
$$574$$ −19600.0 −1.42524
$$575$$ −11808.0 −0.856396
$$576$$ −1664.00 −0.120370
$$577$$ −14382.0 −1.03766 −0.518831 0.854877i $$-0.673632\pi$$
−0.518831 + 0.854877i $$0.673632\pi$$
$$578$$ 9104.00 0.655150
$$579$$ −1220.00 −0.0875673
$$580$$ −16728.0 −1.19757
$$581$$ −14700.0 −1.04967
$$582$$ 2308.00 0.164381
$$583$$ 464.000 0.0329621
$$584$$ −6704.00 −0.475023
$$585$$ 0 0
$$586$$ −3710.00 −0.261534
$$587$$ 10160.0 0.714392 0.357196 0.934029i $$-0.383733\pi$$
0.357196 + 0.934029i $$0.383733\pi$$
$$588$$ −3528.00 −0.247436
$$589$$ −9400.00 −0.657590
$$590$$ 13124.0 0.915774
$$591$$ −1593.00 −0.110875
$$592$$ 176.000 0.0122188
$$593$$ −16842.0 −1.16630 −0.583152 0.812363i $$-0.698181\pi$$
−0.583152 + 0.812363i $$0.698181\pi$$
$$594$$ 212.000 0.0146439
$$595$$ 11305.0 0.778924
$$596$$ −11400.0 −0.783494
$$597$$ −1606.00 −0.110099
$$598$$ 0 0
$$599$$ −13830.0 −0.943370 −0.471685 0.881767i $$-0.656354\pi$$
−0.471685 + 0.881767i $$0.656354\pi$$
$$600$$ 1312.00 0.0892703
$$601$$ −15835.0 −1.07475 −0.537374 0.843344i $$-0.680583\pi$$
−0.537374 + 0.843344i $$0.680583\pi$$
$$602$$ −16870.0 −1.14214
$$603$$ −17420.0 −1.17645
$$604$$ −11052.0 −0.744536
$$605$$ 22559.0 1.51596
$$606$$ 584.000 0.0391475
$$607$$ −22894.0 −1.53087 −0.765436 0.643513i $$-0.777476\pi$$
−0.765436 + 0.643513i $$0.777476\pi$$
$$608$$ 3008.00 0.200642
$$609$$ −8610.00 −0.572898
$$610$$ 2176.00 0.144432
$$611$$ 0 0
$$612$$ 1976.00 0.130515
$$613$$ −9886.00 −0.651373 −0.325687 0.945478i $$-0.605595\pi$$
−0.325687 + 0.945478i $$0.605595\pi$$
$$614$$ 10700.0 0.703285
$$615$$ 4760.00 0.312100
$$616$$ 560.000 0.0366283
$$617$$ −8856.00 −0.577843 −0.288922 0.957353i $$-0.593297\pi$$
−0.288922 + 0.957353i $$0.593297\pi$$
$$618$$ 104.000 0.00676941
$$619$$ −3764.00 −0.244407 −0.122204 0.992505i $$-0.538996\pi$$
−0.122204 + 0.992505i $$0.538996\pi$$
$$620$$ −6800.00 −0.440475
$$621$$ −3816.00 −0.246587
$$622$$ −4524.00 −0.291633
$$623$$ 32690.0 2.10224
$$624$$ 0 0
$$625$$ −9229.00 −0.590656
$$626$$ 3714.00 0.237127
$$627$$ −188.000 −0.0119745
$$628$$ −1864.00 −0.118442
$$629$$ −209.000 −0.0132486
$$630$$ −30940.0 −1.95663
$$631$$ −17775.0 −1.12141 −0.560706 0.828015i $$-0.689470\pi$$
−0.560706 + 0.828015i $$0.689470\pi$$
$$632$$ −8128.00 −0.511574
$$633$$ −2469.00 −0.155030
$$634$$ −7740.00 −0.484850
$$635$$ 9180.00 0.573696
$$636$$ 928.000 0.0578579
$$637$$ 0 0
$$638$$ 984.000 0.0610610
$$639$$ 1430.00 0.0885288
$$640$$ 2176.00 0.134397
$$641$$ 13358.0 0.823103 0.411552 0.911386i $$-0.364987\pi$$
0.411552 + 0.911386i $$0.364987\pi$$
$$642$$ 24.0000 0.00147540
$$643$$ 26062.0 1.59842 0.799211 0.601051i $$-0.205251\pi$$
0.799211 + 0.601051i $$0.205251\pi$$
$$644$$ −10080.0 −0.616782
$$645$$ 4097.00 0.250107
$$646$$ −3572.00 −0.217552
$$647$$ 20598.0 1.25161 0.625804 0.779980i $$-0.284771\pi$$
0.625804 + 0.779980i $$0.284771\pi$$
$$648$$ −5192.00 −0.314755
$$649$$ −772.000 −0.0466928
$$650$$ 0 0
$$651$$ −3500.00 −0.210716
$$652$$ −10208.0 −0.613154
$$653$$ 27104.0 1.62429 0.812145 0.583456i $$-0.198300\pi$$
0.812145 + 0.583456i $$0.198300\pi$$
$$654$$ −2426.00 −0.145052
$$655$$ 26707.0 1.59317
$$656$$ 4480.00 0.266638
$$657$$ −21788.0 −1.29381
$$658$$ 9590.00 0.568172
$$659$$ 7500.00 0.443336 0.221668 0.975122i $$-0.428850\pi$$
0.221668 + 0.975122i $$0.428850\pi$$
$$660$$ −136.000 −0.00802090
$$661$$ −10402.0 −0.612089 −0.306045 0.952017i $$-0.599006\pi$$
−0.306045 + 0.952017i $$0.599006\pi$$
$$662$$ 21040.0 1.23526
$$663$$ 0 0
$$664$$ 3360.00 0.196375
$$665$$ 55930.0 3.26146
$$666$$ 572.000 0.0332801
$$667$$ −17712.0 −1.02820
$$668$$ 15072.0 0.872984
$$669$$ 1943.00 0.112288
$$670$$ 22780.0 1.31353
$$671$$ −128.000 −0.00736421
$$672$$ 1120.00 0.0642931
$$673$$ −19715.0 −1.12921 −0.564604 0.825362i $$-0.690971\pi$$
−0.564604 + 0.825362i $$0.690971\pi$$
$$674$$ −15678.0 −0.895985
$$675$$ 8692.00 0.495637
$$676$$ 0 0
$$677$$ −1168.00 −0.0663071 −0.0331535 0.999450i $$-0.510555\pi$$
−0.0331535 + 0.999450i $$0.510555\pi$$
$$678$$ −2316.00 −0.131188
$$679$$ 40390.0 2.28281
$$680$$ −2584.00 −0.145723
$$681$$ 3032.00 0.170612
$$682$$ 400.000 0.0224586
$$683$$ −3800.00 −0.212889 −0.106444 0.994319i $$-0.533947\pi$$
−0.106444 + 0.994319i $$0.533947\pi$$
$$684$$ 9776.00 0.546483
$$685$$ −9656.00 −0.538594
$$686$$ −37730.0 −2.09991
$$687$$ 5311.00 0.294945
$$688$$ 3856.00 0.213675
$$689$$ 0 0
$$690$$ 2448.00 0.135063
$$691$$ 17336.0 0.954403 0.477202 0.878794i $$-0.341651\pi$$
0.477202 + 0.878794i $$0.341651\pi$$
$$692$$ 13184.0 0.724249
$$693$$ 1820.00 0.0997635
$$694$$ 2550.00 0.139476
$$695$$ −31365.0 −1.71186
$$696$$ 1968.00 0.107179
$$697$$ −5320.00 −0.289110
$$698$$ −750.000 −0.0406704
$$699$$ 507.000 0.0274342
$$700$$ 22960.0 1.23972
$$701$$ −10088.0 −0.543536 −0.271768 0.962363i $$-0.587608\pi$$
−0.271768 + 0.962363i $$0.587608\pi$$
$$702$$ 0 0
$$703$$ −1034.00 −0.0554738
$$704$$ −128.000 −0.00685253
$$705$$ −2329.00 −0.124419
$$706$$ 3184.00 0.169733
$$707$$ 10220.0 0.543653
$$708$$ −1544.00 −0.0819591
$$709$$ −11730.0 −0.621339 −0.310670 0.950518i $$-0.600553\pi$$
−0.310670 + 0.950518i $$0.600553\pi$$
$$710$$ −1870.00 −0.0988449
$$711$$ −26416.0 −1.39336
$$712$$ −7472.00 −0.393294
$$713$$ −7200.00 −0.378180
$$714$$ −1330.00 −0.0697115
$$715$$ 0 0
$$716$$ 180.000 0.00939513
$$717$$ −6795.00 −0.353925
$$718$$ −4848.00 −0.251986
$$719$$ −2190.00 −0.113593 −0.0567964 0.998386i $$-0.518089\pi$$
−0.0567964 + 0.998386i $$0.518089\pi$$
$$720$$ 7072.00 0.366053
$$721$$ 1820.00 0.0940088
$$722$$ −3954.00 −0.203813
$$723$$ −4442.00 −0.228492
$$724$$ −1456.00 −0.0747401
$$725$$ 40344.0 2.06667
$$726$$ −2654.00 −0.135674
$$727$$ −2470.00 −0.126007 −0.0630036 0.998013i $$-0.520068\pi$$
−0.0630036 + 0.998013i $$0.520068\pi$$
$$728$$ 0 0
$$729$$ −15443.0 −0.784586
$$730$$ 28492.0 1.44457
$$731$$ −4579.00 −0.231683
$$732$$ −256.000 −0.0129263
$$733$$ 8693.00 0.438040 0.219020 0.975720i $$-0.429714\pi$$
0.219020 + 0.975720i $$0.429714\pi$$
$$734$$ 15940.0 0.801575
$$735$$ 14994.0 0.752465
$$736$$ 2304.00 0.115389
$$737$$ −1340.00 −0.0669736
$$738$$ 14560.0 0.726234
$$739$$ 14636.0 0.728544 0.364272 0.931293i $$-0.381318\pi$$
0.364272 + 0.931293i $$0.381318\pi$$
$$740$$ −748.000 −0.0371581
$$741$$ 0 0
$$742$$ 16240.0 0.803489
$$743$$ −27501.0 −1.35789 −0.678946 0.734188i $$-0.737563\pi$$
−0.678946 + 0.734188i $$0.737563\pi$$
$$744$$ 800.000 0.0394213
$$745$$ 48450.0 2.38265
$$746$$ −24984.0 −1.22618
$$747$$ 10920.0 0.534862
$$748$$ 152.000 0.00743004
$$749$$ 420.000 0.0204893
$$750$$ −1326.00 −0.0645582
$$751$$ 11888.0 0.577629 0.288814 0.957385i $$-0.406739\pi$$
0.288814 + 0.957385i $$0.406739\pi$$
$$752$$ −2192.00 −0.106295
$$753$$ 5024.00 0.243140
$$754$$ 0 0
$$755$$ 46971.0 2.26417
$$756$$ 7420.00 0.356961
$$757$$ 10776.0 0.517385 0.258692 0.965960i $$-0.416708\pi$$
0.258692 + 0.965960i $$0.416708\pi$$
$$758$$ −8568.00 −0.410559
$$759$$ −144.000 −0.00688652
$$760$$ −12784.0 −0.610164
$$761$$ 21758.0 1.03643 0.518217 0.855249i $$-0.326596\pi$$
0.518217 + 0.855249i $$0.326596\pi$$
$$762$$ −1080.00 −0.0513442
$$763$$ −42455.0 −2.01438
$$764$$ 9928.00 0.470134
$$765$$ −8398.00 −0.396902
$$766$$ −4674.00 −0.220468
$$767$$ 0 0
$$768$$ −256.000 −0.0120281
$$769$$ 8080.00 0.378898 0.189449 0.981891i $$-0.439330\pi$$
0.189449 + 0.981891i $$0.439330\pi$$
$$770$$ −2380.00 −0.111389
$$771$$ −7065.00 −0.330013
$$772$$ 4880.00 0.227507
$$773$$ −3981.00 −0.185235 −0.0926175 0.995702i $$-0.529523\pi$$
−0.0926175 + 0.995702i $$0.529523\pi$$
$$774$$ 12532.0 0.581981
$$775$$ 16400.0 0.760136
$$776$$ −9232.00 −0.427074
$$777$$ −385.000 −0.0177758
$$778$$ −7124.00 −0.328288
$$779$$ −26320.0 −1.21054
$$780$$ 0 0
$$781$$ 110.000 0.00503983
$$782$$ −2736.00 −0.125114
$$783$$ 13038.0 0.595070
$$784$$ 14112.0 0.642857
$$785$$ 7922.00 0.360189
$$786$$ −3142.00 −0.142585
$$787$$ 26048.0 1.17981 0.589905 0.807472i $$-0.299165\pi$$
0.589905 + 0.807472i $$0.299165\pi$$
$$788$$ 6372.00 0.288062
$$789$$ −1364.00 −0.0615459
$$790$$ 34544.0 1.55572
$$791$$ −40530.0 −1.82185
$$792$$ −416.000 −0.0186640
$$793$$ 0 0
$$794$$ −10548.0 −0.471454
$$795$$ −3944.00 −0.175949
$$796$$ 6424.00 0.286046
$$797$$ −12434.0 −0.552616 −0.276308 0.961069i $$-0.589111\pi$$
−0.276308 + 0.961069i $$0.589111\pi$$
$$798$$ −6580.00 −0.291892
$$799$$ 2603.00 0.115253
$$800$$ −5248.00 −0.231931
$$801$$ −24284.0 −1.07120
$$802$$ 9624.00 0.423735
$$803$$ −1676.00 −0.0736547
$$804$$ −2680.00 −0.117558
$$805$$ 42840.0 1.87567
$$806$$ 0 0
$$807$$ 4304.00 0.187742
$$808$$ −2336.00 −0.101708
$$809$$ −8113.00 −0.352581 −0.176290 0.984338i $$-0.556410\pi$$
−0.176290 + 0.984338i $$0.556410\pi$$
$$810$$ 22066.0 0.957185
$$811$$ −732.000 −0.0316942 −0.0158471 0.999874i $$-0.505044\pi$$
−0.0158471 + 0.999874i $$0.505044\pi$$
$$812$$ 34440.0 1.48843
$$813$$ −1519.00 −0.0655273
$$814$$ 44.0000 0.00189459
$$815$$ 43384.0 1.86463
$$816$$ 304.000 0.0130418
$$817$$ −22654.0 −0.970090
$$818$$ −18896.0 −0.807681
$$819$$ 0 0
$$820$$ −19040.0 −0.810861
$$821$$ 32559.0 1.38406 0.692032 0.721867i $$-0.256716\pi$$
0.692032 + 0.721867i $$0.256716\pi$$
$$822$$ 1136.00 0.0482026
$$823$$ −43582.0 −1.84590 −0.922948 0.384924i $$-0.874228\pi$$
−0.922948 + 0.384924i $$0.874228\pi$$
$$824$$ −416.000 −0.0175874
$$825$$ 328.000 0.0138418
$$826$$ −27020.0 −1.13819
$$827$$ 23042.0 0.968862 0.484431 0.874829i $$-0.339027\pi$$
0.484431 + 0.874829i $$0.339027\pi$$
$$828$$ 7488.00 0.314283
$$829$$ −20690.0 −0.866820 −0.433410 0.901197i $$-0.642690\pi$$
−0.433410 + 0.901197i $$0.642690\pi$$
$$830$$ −14280.0 −0.597188
$$831$$ −1996.00 −0.0833219
$$832$$ 0 0
$$833$$ −16758.0 −0.697035
$$834$$ 3690.00 0.153207
$$835$$ −64056.0 −2.65479
$$836$$ 752.000 0.0311106
$$837$$ 5300.00 0.218871
$$838$$ 1042.00 0.0429538
$$839$$ 43000.0 1.76940 0.884699 0.466163i $$-0.154364\pi$$
0.884699 + 0.466163i $$0.154364\pi$$
$$840$$ −4760.00 −0.195519
$$841$$ 36127.0 1.48128
$$842$$ 14282.0 0.584549
$$843$$ −2918.00 −0.119219
$$844$$ 9876.00 0.402780
$$845$$ 0 0
$$846$$ −7124.00 −0.289513
$$847$$ −46445.0 −1.88414
$$848$$ −3712.00 −0.150319
$$849$$ −812.000 −0.0328242
$$850$$ 6232.00 0.251477
$$851$$ −792.000 −0.0319029
$$852$$ 220.000 0.00884633
$$853$$ 33065.0 1.32723 0.663613 0.748076i $$-0.269022\pi$$
0.663613 + 0.748076i $$0.269022\pi$$
$$854$$ −4480.00 −0.179511
$$855$$ −41548.0 −1.66188
$$856$$ −96.0000 −0.00383319
$$857$$ −28938.0 −1.15345 −0.576723 0.816940i $$-0.695669\pi$$
−0.576723 + 0.816940i $$0.695669\pi$$
$$858$$ 0 0
$$859$$ 12508.0 0.496819 0.248409 0.968655i $$-0.420092\pi$$
0.248409 + 0.968655i $$0.420092\pi$$
$$860$$ −16388.0 −0.649798
$$861$$ −9800.00 −0.387901
$$862$$ −12598.0 −0.497784
$$863$$ 19535.0 0.770544 0.385272 0.922803i $$-0.374108\pi$$
0.385272 + 0.922803i $$0.374108\pi$$
$$864$$ −1696.00 −0.0667814
$$865$$ −56032.0 −2.20248
$$866$$ −6462.00 −0.253565
$$867$$ 4552.00 0.178309
$$868$$ 14000.0 0.547455
$$869$$ −2032.00 −0.0793221
$$870$$ −8364.00 −0.325938
$$871$$ 0 0
$$872$$ 9704.00 0.376857
$$873$$ −30004.0 −1.16321
$$874$$ −13536.0 −0.523870
$$875$$ −23205.0 −0.896540
$$876$$ −3352.00 −0.129285
$$877$$ −18775.0 −0.722904 −0.361452 0.932391i $$-0.617719\pi$$
−0.361452 + 0.932391i $$0.617719\pi$$
$$878$$ 820.000 0.0315190
$$879$$ −1855.00 −0.0711804
$$880$$ 544.000 0.0208389
$$881$$ −43531.0 −1.66470 −0.832348 0.554254i $$-0.813004\pi$$
−0.832348 + 0.554254i $$0.813004\pi$$
$$882$$ 45864.0 1.75093
$$883$$ 6955.00 0.265067 0.132534 0.991179i $$-0.457689\pi$$
0.132534 + 0.991179i $$0.457689\pi$$
$$884$$ 0 0
$$885$$ 6562.00 0.249242
$$886$$ 18794.0 0.712637
$$887$$ 40728.0 1.54173 0.770864 0.637000i $$-0.219825\pi$$
0.770864 + 0.637000i $$0.219825\pi$$
$$888$$ 88.0000 0.00332555
$$889$$ −18900.0 −0.713032
$$890$$ 31756.0 1.19603
$$891$$ −1298.00 −0.0488043
$$892$$ −7772.00 −0.291733
$$893$$ 12878.0 0.482582
$$894$$ −5700.00 −0.213240
$$895$$ −765.000 −0.0285711
$$896$$ −4480.00 −0.167038
$$897$$ 0 0
$$898$$ 1308.00 0.0486064
$$899$$ 24600.0 0.912632
$$900$$ −17056.0 −0.631704
$$901$$ 4408.00 0.162988
$$902$$ 1120.00 0.0413436
$$903$$ −8435.00 −0.310852
$$904$$ 9264.00 0.340836
$$905$$ 6188.00 0.227288
$$906$$ −5526.00 −0.202637
$$907$$ 23653.0 0.865915 0.432958 0.901414i $$-0.357470\pi$$
0.432958 + 0.901414i $$0.357470\pi$$
$$908$$ −12128.0 −0.443262
$$909$$ −7592.00 −0.277020
$$910$$ 0 0
$$911$$ −206.000 −0.00749186 −0.00374593 0.999993i $$-0.501192\pi$$
−0.00374593 + 0.999993i $$0.501192\pi$$
$$912$$ 1504.00 0.0546079
$$913$$ 840.000 0.0304490
$$914$$ 29964.0 1.08438
$$915$$ 1088.00 0.0393095
$$916$$ −21244.0 −0.766290
$$917$$ −54985.0 −1.98011
$$918$$ 2014.00 0.0724095
$$919$$ 28352.0 1.01768 0.508839 0.860862i $$-0.330075\pi$$
0.508839 + 0.860862i $$0.330075\pi$$
$$920$$ −9792.00 −0.350905
$$921$$ 5350.00 0.191410
$$922$$ 21690.0 0.774753
$$923$$ 0 0
$$924$$ 280.000 0.00996897
$$925$$ 1804.00 0.0641245
$$926$$ 26144.0 0.927803
$$927$$ −1352.00 −0.0479024
$$928$$ −7872.00 −0.278460
$$929$$ 17612.0 0.621992 0.310996 0.950411i $$-0.399337\pi$$
0.310996 + 0.950411i $$0.399337\pi$$
$$930$$ −3400.00 −0.119882
$$931$$ −82908.0 −2.91858
$$932$$ −2028.00 −0.0712761
$$933$$ −2262.00 −0.0793725
$$934$$ 30680.0 1.07482
$$935$$ −646.000 −0.0225951
$$936$$ 0 0
$$937$$ 12014.0 0.418869 0.209435 0.977823i $$-0.432838\pi$$
0.209435 + 0.977823i $$0.432838\pi$$
$$938$$ −46900.0 −1.63256
$$939$$ 1857.00 0.0645377
$$940$$ 9316.00 0.323249
$$941$$ 26737.0 0.926250 0.463125 0.886293i $$-0.346728\pi$$
0.463125 + 0.886293i $$0.346728\pi$$
$$942$$ −932.000 −0.0322359
$$943$$ −20160.0 −0.696182
$$944$$ 6176.00 0.212936
$$945$$ −31535.0 −1.08554
$$946$$ 964.000 0.0331314
$$947$$ 35566.0 1.22042 0.610211 0.792239i $$-0.291085\pi$$
0.610211 + 0.792239i $$0.291085\pi$$
$$948$$ −4064.00 −0.139233
$$949$$ 0 0
$$950$$ 30832.0 1.05297
$$951$$ −3870.00 −0.131959
$$952$$ 5320.00 0.181116
$$953$$ 12279.0 0.417372 0.208686 0.977983i $$-0.433081\pi$$
0.208686 + 0.977983i $$0.433081\pi$$
$$954$$ −12064.0 −0.409420
$$955$$ −42194.0 −1.42970
$$956$$ 27180.0 0.919523
$$957$$ 492.000 0.0166187
$$958$$ −39230.0 −1.32303
$$959$$ 19880.0 0.669404
$$960$$ 1088.00 0.0365782
$$961$$ −19791.0 −0.664328
$$962$$ 0 0
$$963$$ −312.000 −0.0104404
$$964$$ 17768.0 0.593640
$$965$$ −20740.0 −0.691859
$$966$$ −5040.00 −0.167867
$$967$$ −51523.0 −1.71341 −0.856705 0.515806i $$-0.827492\pi$$
−0.856705 + 0.515806i $$0.827492\pi$$
$$968$$ 10616.0 0.352491
$$969$$ −1786.00 −0.0592101
$$970$$ 39236.0 1.29875
$$971$$ 41027.0 1.35594 0.677971 0.735089i $$-0.262860\pi$$
0.677971 + 0.735089i $$0.262860\pi$$
$$972$$ −8320.00 −0.274552
$$973$$ 64575.0 2.12763
$$974$$ −21808.0 −0.717426
$$975$$ 0 0
$$976$$ 1024.00 0.0335834
$$977$$ −23562.0 −0.771561 −0.385781 0.922591i $$-0.626068\pi$$
−0.385781 + 0.922591i $$0.626068\pi$$
$$978$$ −5104.00 −0.166879
$$979$$ −1868.00 −0.0609822
$$980$$ −59976.0 −1.95496
$$981$$ 31538.0 1.02643
$$982$$ 37038.0 1.20359
$$983$$ 41307.0 1.34027 0.670137 0.742238i $$-0.266235\pi$$
0.670137 + 0.742238i $$0.266235\pi$$
$$984$$ 2240.00 0.0725697
$$985$$ −27081.0 −0.876013
$$986$$ 9348.00 0.301928
$$987$$ 4795.00 0.154637
$$988$$ 0 0
$$989$$ −17352.0 −0.557898
$$990$$ 1768.00 0.0567583
$$991$$ −26586.0 −0.852202 −0.426101 0.904676i $$-0.640113\pi$$
−0.426101 + 0.904676i $$0.640113\pi$$
$$992$$ −3200.00 −0.102419
$$993$$ 10520.0 0.336195
$$994$$ 3850.00 0.122852
$$995$$ −27302.0 −0.869881
$$996$$ 1680.00 0.0534466
$$997$$ 39298.0 1.24833 0.624163 0.781295i $$-0.285440\pi$$
0.624163 + 0.781295i $$0.285440\pi$$
$$998$$ −28800.0 −0.913475
$$999$$ 583.000 0.0184638
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 338.4.a.b.1.1 1
13.2 odd 12 338.4.e.c.147.1 4
13.3 even 3 338.4.c.g.191.1 2
13.4 even 6 338.4.c.c.315.1 2
13.5 odd 4 338.4.b.b.337.2 2
13.6 odd 12 338.4.e.c.23.2 4
13.7 odd 12 338.4.e.c.23.1 4
13.8 odd 4 338.4.b.b.337.1 2
13.9 even 3 338.4.c.g.315.1 2
13.10 even 6 338.4.c.c.191.1 2
13.11 odd 12 338.4.e.c.147.2 4
13.12 even 2 26.4.a.b.1.1 1
39.38 odd 2 234.4.a.a.1.1 1
52.51 odd 2 208.4.a.e.1.1 1
65.12 odd 4 650.4.b.d.599.2 2
65.38 odd 4 650.4.b.d.599.1 2
65.64 even 2 650.4.a.c.1.1 1
91.90 odd 2 1274.4.a.f.1.1 1
104.51 odd 2 832.4.a.g.1.1 1
104.77 even 2 832.4.a.j.1.1 1
156.155 even 2 1872.4.a.b.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
26.4.a.b.1.1 1 13.12 even 2
208.4.a.e.1.1 1 52.51 odd 2
234.4.a.a.1.1 1 39.38 odd 2
338.4.a.b.1.1 1 1.1 even 1 trivial
338.4.b.b.337.1 2 13.8 odd 4
338.4.b.b.337.2 2 13.5 odd 4
338.4.c.c.191.1 2 13.10 even 6
338.4.c.c.315.1 2 13.4 even 6
338.4.c.g.191.1 2 13.3 even 3
338.4.c.g.315.1 2 13.9 even 3
338.4.e.c.23.1 4 13.7 odd 12
338.4.e.c.23.2 4 13.6 odd 12
338.4.e.c.147.1 4 13.2 odd 12
338.4.e.c.147.2 4 13.11 odd 12
650.4.a.c.1.1 1 65.64 even 2
650.4.b.d.599.1 2 65.38 odd 4
650.4.b.d.599.2 2 65.12 odd 4
832.4.a.g.1.1 1 104.51 odd 2
832.4.a.j.1.1 1 104.77 even 2
1274.4.a.f.1.1 1 91.90 odd 2
1872.4.a.b.1.1 1 156.155 even 2