# Properties

 Label 363.4.a.d.1.1 Level $363$ Weight $4$ Character 363.1 Self dual yes Analytic conductor $21.418$ 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] = [363,4,Mod(1,363)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(363, 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("363.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$363 = 3 \cdot 11^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 363.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: $$21.4176933321$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 33) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 363.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+1.00000 q^{2} -3.00000 q^{3} -7.00000 q^{4} -4.00000 q^{5} -3.00000 q^{6} +26.0000 q^{7} -15.0000 q^{8} +9.00000 q^{9} +O(q^{10})$$ $$q+1.00000 q^{2} -3.00000 q^{3} -7.00000 q^{4} -4.00000 q^{5} -3.00000 q^{6} +26.0000 q^{7} -15.0000 q^{8} +9.00000 q^{9} -4.00000 q^{10} +21.0000 q^{12} +32.0000 q^{13} +26.0000 q^{14} +12.0000 q^{15} +41.0000 q^{16} -74.0000 q^{17} +9.00000 q^{18} +60.0000 q^{19} +28.0000 q^{20} -78.0000 q^{21} -182.000 q^{23} +45.0000 q^{24} -109.000 q^{25} +32.0000 q^{26} -27.0000 q^{27} -182.000 q^{28} +90.0000 q^{29} +12.0000 q^{30} -8.00000 q^{31} +161.000 q^{32} -74.0000 q^{34} -104.000 q^{35} -63.0000 q^{36} -66.0000 q^{37} +60.0000 q^{38} -96.0000 q^{39} +60.0000 q^{40} -422.000 q^{41} -78.0000 q^{42} -408.000 q^{43} -36.0000 q^{45} -182.000 q^{46} -506.000 q^{47} -123.000 q^{48} +333.000 q^{49} -109.000 q^{50} +222.000 q^{51} -224.000 q^{52} +348.000 q^{53} -27.0000 q^{54} -390.000 q^{56} -180.000 q^{57} +90.0000 q^{58} -200.000 q^{59} -84.0000 q^{60} -132.000 q^{61} -8.00000 q^{62} +234.000 q^{63} -167.000 q^{64} -128.000 q^{65} -1036.00 q^{67} +518.000 q^{68} +546.000 q^{69} -104.000 q^{70} +762.000 q^{71} -135.000 q^{72} +542.000 q^{73} -66.0000 q^{74} +327.000 q^{75} -420.000 q^{76} -96.0000 q^{78} +550.000 q^{79} -164.000 q^{80} +81.0000 q^{81} -422.000 q^{82} +132.000 q^{83} +546.000 q^{84} +296.000 q^{85} -408.000 q^{86} -270.000 q^{87} +570.000 q^{89} -36.0000 q^{90} +832.000 q^{91} +1274.00 q^{92} +24.0000 q^{93} -506.000 q^{94} -240.000 q^{95} -483.000 q^{96} +14.0000 q^{97} +333.000 q^{98} +O(q^{100})$$

## 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 $$-0.443433\pi$$
0.176777 + 0.984251i $$0.443433\pi$$
$$3$$ −3.00000 −0.577350
$$4$$ −7.00000 −0.875000
$$5$$ −4.00000 −0.357771 −0.178885 0.983870i $$-0.557249\pi$$
−0.178885 + 0.983870i $$0.557249\pi$$
$$6$$ −3.00000 −0.204124
$$7$$ 26.0000 1.40387 0.701934 0.712242i $$-0.252320\pi$$
0.701934 + 0.712242i $$0.252320\pi$$
$$8$$ −15.0000 −0.662913
$$9$$ 9.00000 0.333333
$$10$$ −4.00000 −0.126491
$$11$$ 0 0
$$12$$ 21.0000 0.505181
$$13$$ 32.0000 0.682708 0.341354 0.939935i $$-0.389115\pi$$
0.341354 + 0.939935i $$0.389115\pi$$
$$14$$ 26.0000 0.496342
$$15$$ 12.0000 0.206559
$$16$$ 41.0000 0.640625
$$17$$ −74.0000 −1.05574 −0.527872 0.849324i $$-0.677010\pi$$
−0.527872 + 0.849324i $$0.677010\pi$$
$$18$$ 9.00000 0.117851
$$19$$ 60.0000 0.724471 0.362235 0.932087i $$-0.382014\pi$$
0.362235 + 0.932087i $$0.382014\pi$$
$$20$$ 28.0000 0.313050
$$21$$ −78.0000 −0.810524
$$22$$ 0 0
$$23$$ −182.000 −1.64998 −0.824992 0.565145i $$-0.808820\pi$$
−0.824992 + 0.565145i $$0.808820\pi$$
$$24$$ 45.0000 0.382733
$$25$$ −109.000 −0.872000
$$26$$ 32.0000 0.241374
$$27$$ −27.0000 −0.192450
$$28$$ −182.000 −1.22838
$$29$$ 90.0000 0.576296 0.288148 0.957586i $$-0.406961\pi$$
0.288148 + 0.957586i $$0.406961\pi$$
$$30$$ 12.0000 0.0730297
$$31$$ −8.00000 −0.0463498 −0.0231749 0.999731i $$-0.507377\pi$$
−0.0231749 + 0.999731i $$0.507377\pi$$
$$32$$ 161.000 0.889408
$$33$$ 0 0
$$34$$ −74.0000 −0.373262
$$35$$ −104.000 −0.502263
$$36$$ −63.0000 −0.291667
$$37$$ −66.0000 −0.293252 −0.146626 0.989192i $$-0.546841\pi$$
−0.146626 + 0.989192i $$0.546841\pi$$
$$38$$ 60.0000 0.256139
$$39$$ −96.0000 −0.394162
$$40$$ 60.0000 0.237171
$$41$$ −422.000 −1.60745 −0.803724 0.595003i $$-0.797151\pi$$
−0.803724 + 0.595003i $$0.797151\pi$$
$$42$$ −78.0000 −0.286563
$$43$$ −408.000 −1.44696 −0.723482 0.690344i $$-0.757459\pi$$
−0.723482 + 0.690344i $$0.757459\pi$$
$$44$$ 0 0
$$45$$ −36.0000 −0.119257
$$46$$ −182.000 −0.583357
$$47$$ −506.000 −1.57038 −0.785188 0.619257i $$-0.787434\pi$$
−0.785188 + 0.619257i $$0.787434\pi$$
$$48$$ −123.000 −0.369865
$$49$$ 333.000 0.970845
$$50$$ −109.000 −0.308299
$$51$$ 222.000 0.609534
$$52$$ −224.000 −0.597369
$$53$$ 348.000 0.901915 0.450957 0.892546i $$-0.351083\pi$$
0.450957 + 0.892546i $$0.351083\pi$$
$$54$$ −27.0000 −0.0680414
$$55$$ 0 0
$$56$$ −390.000 −0.930642
$$57$$ −180.000 −0.418273
$$58$$ 90.0000 0.203751
$$59$$ −200.000 −0.441318 −0.220659 0.975351i $$-0.570821\pi$$
−0.220659 + 0.975351i $$0.570821\pi$$
$$60$$ −84.0000 −0.180739
$$61$$ −132.000 −0.277063 −0.138532 0.990358i $$-0.544238\pi$$
−0.138532 + 0.990358i $$0.544238\pi$$
$$62$$ −8.00000 −0.0163871
$$63$$ 234.000 0.467956
$$64$$ −167.000 −0.326172
$$65$$ −128.000 −0.244253
$$66$$ 0 0
$$67$$ −1036.00 −1.88907 −0.944534 0.328414i $$-0.893486\pi$$
−0.944534 + 0.328414i $$0.893486\pi$$
$$68$$ 518.000 0.923775
$$69$$ 546.000 0.952618
$$70$$ −104.000 −0.177577
$$71$$ 762.000 1.27370 0.636850 0.770987i $$-0.280237\pi$$
0.636850 + 0.770987i $$0.280237\pi$$
$$72$$ −135.000 −0.220971
$$73$$ 542.000 0.868990 0.434495 0.900674i $$-0.356927\pi$$
0.434495 + 0.900674i $$0.356927\pi$$
$$74$$ −66.0000 −0.103680
$$75$$ 327.000 0.503449
$$76$$ −420.000 −0.633912
$$77$$ 0 0
$$78$$ −96.0000 −0.139357
$$79$$ 550.000 0.783289 0.391645 0.920117i $$-0.371906\pi$$
0.391645 + 0.920117i $$0.371906\pi$$
$$80$$ −164.000 −0.229197
$$81$$ 81.0000 0.111111
$$82$$ −422.000 −0.568318
$$83$$ 132.000 0.174565 0.0872824 0.996184i $$-0.472182\pi$$
0.0872824 + 0.996184i $$0.472182\pi$$
$$84$$ 546.000 0.709208
$$85$$ 296.000 0.377714
$$86$$ −408.000 −0.511579
$$87$$ −270.000 −0.332725
$$88$$ 0 0
$$89$$ 570.000 0.678875 0.339438 0.940629i $$-0.389763\pi$$
0.339438 + 0.940629i $$0.389763\pi$$
$$90$$ −36.0000 −0.0421637
$$91$$ 832.000 0.958432
$$92$$ 1274.00 1.44374
$$93$$ 24.0000 0.0267600
$$94$$ −506.000 −0.555212
$$95$$ −240.000 −0.259195
$$96$$ −483.000 −0.513500
$$97$$ 14.0000 0.0146545 0.00732724 0.999973i $$-0.497668\pi$$
0.00732724 + 0.999973i $$0.497668\pi$$
$$98$$ 333.000 0.343246
$$99$$ 0 0
$$100$$ 763.000 0.763000
$$101$$ −1702.00 −1.67679 −0.838393 0.545067i $$-0.816504\pi$$
−0.838393 + 0.545067i $$0.816504\pi$$
$$102$$ 222.000 0.215503
$$103$$ −1132.00 −1.08291 −0.541453 0.840731i $$-0.682126\pi$$
−0.541453 + 0.840731i $$0.682126\pi$$
$$104$$ −480.000 −0.452576
$$105$$ 312.000 0.289982
$$106$$ 348.000 0.318875
$$107$$ −564.000 −0.509570 −0.254785 0.966998i $$-0.582005\pi$$
−0.254785 + 0.966998i $$0.582005\pi$$
$$108$$ 189.000 0.168394
$$109$$ 320.000 0.281197 0.140598 0.990067i $$-0.455097\pi$$
0.140598 + 0.990067i $$0.455097\pi$$
$$110$$ 0 0
$$111$$ 198.000 0.169309
$$112$$ 1066.00 0.899353
$$113$$ −2142.00 −1.78321 −0.891604 0.452817i $$-0.850419\pi$$
−0.891604 + 0.452817i $$0.850419\pi$$
$$114$$ −180.000 −0.147882
$$115$$ 728.000 0.590316
$$116$$ −630.000 −0.504259
$$117$$ 288.000 0.227569
$$118$$ −200.000 −0.156030
$$119$$ −1924.00 −1.48212
$$120$$ −180.000 −0.136931
$$121$$ 0 0
$$122$$ −132.000 −0.0979567
$$123$$ 1266.00 0.928060
$$124$$ 56.0000 0.0405560
$$125$$ 936.000 0.669747
$$126$$ 234.000 0.165447
$$127$$ 1606.00 1.12212 0.561061 0.827775i $$-0.310393\pi$$
0.561061 + 0.827775i $$0.310393\pi$$
$$128$$ −1455.00 −1.00473
$$129$$ 1224.00 0.835405
$$130$$ −128.000 −0.0863565
$$131$$ 1908.00 1.27254 0.636270 0.771466i $$-0.280476\pi$$
0.636270 + 0.771466i $$0.280476\pi$$
$$132$$ 0 0
$$133$$ 1560.00 1.01706
$$134$$ −1036.00 −0.667886
$$135$$ 108.000 0.0688530
$$136$$ 1110.00 0.699866
$$137$$ −2186.00 −1.36323 −0.681615 0.731711i $$-0.738722\pi$$
−0.681615 + 0.731711i $$0.738722\pi$$
$$138$$ 546.000 0.336801
$$139$$ −2740.00 −1.67197 −0.835985 0.548753i $$-0.815103\pi$$
−0.835985 + 0.548753i $$0.815103\pi$$
$$140$$ 728.000 0.439480
$$141$$ 1518.00 0.906657
$$142$$ 762.000 0.450321
$$143$$ 0 0
$$144$$ 369.000 0.213542
$$145$$ −360.000 −0.206182
$$146$$ 542.000 0.307235
$$147$$ −999.000 −0.560518
$$148$$ 462.000 0.256596
$$149$$ 1310.00 0.720264 0.360132 0.932901i $$-0.382732\pi$$
0.360132 + 0.932901i $$0.382732\pi$$
$$150$$ 327.000 0.177996
$$151$$ 1198.00 0.645641 0.322821 0.946460i $$-0.395369\pi$$
0.322821 + 0.946460i $$0.395369\pi$$
$$152$$ −900.000 −0.480261
$$153$$ −666.000 −0.351914
$$154$$ 0 0
$$155$$ 32.0000 0.0165826
$$156$$ 672.000 0.344891
$$157$$ 2114.00 1.07462 0.537311 0.843384i $$-0.319440\pi$$
0.537311 + 0.843384i $$0.319440\pi$$
$$158$$ 550.000 0.276934
$$159$$ −1044.00 −0.520721
$$160$$ −644.000 −0.318204
$$161$$ −4732.00 −2.31636
$$162$$ 81.0000 0.0392837
$$163$$ 3868.00 1.85868 0.929341 0.369223i $$-0.120376\pi$$
0.929341 + 0.369223i $$0.120376\pi$$
$$164$$ 2954.00 1.40652
$$165$$ 0 0
$$166$$ 132.000 0.0617180
$$167$$ −2004.00 −0.928588 −0.464294 0.885681i $$-0.653692\pi$$
−0.464294 + 0.885681i $$0.653692\pi$$
$$168$$ 1170.00 0.537306
$$169$$ −1173.00 −0.533910
$$170$$ 296.000 0.133542
$$171$$ 540.000 0.241490
$$172$$ 2856.00 1.26609
$$173$$ −678.000 −0.297962 −0.148981 0.988840i $$-0.547599\pi$$
−0.148981 + 0.988840i $$0.547599\pi$$
$$174$$ −270.000 −0.117636
$$175$$ −2834.00 −1.22417
$$176$$ 0 0
$$177$$ 600.000 0.254795
$$178$$ 570.000 0.240019
$$179$$ −1680.00 −0.701503 −0.350752 0.936469i $$-0.614074\pi$$
−0.350752 + 0.936469i $$0.614074\pi$$
$$180$$ 252.000 0.104350
$$181$$ −4358.00 −1.78966 −0.894828 0.446412i $$-0.852702\pi$$
−0.894828 + 0.446412i $$0.852702\pi$$
$$182$$ 832.000 0.338857
$$183$$ 396.000 0.159963
$$184$$ 2730.00 1.09379
$$185$$ 264.000 0.104917
$$186$$ 24.0000 0.00946110
$$187$$ 0 0
$$188$$ 3542.00 1.37408
$$189$$ −702.000 −0.270175
$$190$$ −240.000 −0.0916391
$$191$$ −1778.00 −0.673568 −0.336784 0.941582i $$-0.609339\pi$$
−0.336784 + 0.941582i $$0.609339\pi$$
$$192$$ 501.000 0.188315
$$193$$ 3962.00 1.47767 0.738837 0.673884i $$-0.235375\pi$$
0.738837 + 0.673884i $$0.235375\pi$$
$$194$$ 14.0000 0.00518114
$$195$$ 384.000 0.141020
$$196$$ −2331.00 −0.849490
$$197$$ −374.000 −0.135261 −0.0676304 0.997710i $$-0.521544\pi$$
−0.0676304 + 0.997710i $$0.521544\pi$$
$$198$$ 0 0
$$199$$ 2100.00 0.748066 0.374033 0.927415i $$-0.377975\pi$$
0.374033 + 0.927415i $$0.377975\pi$$
$$200$$ 1635.00 0.578060
$$201$$ 3108.00 1.09065
$$202$$ −1702.00 −0.592833
$$203$$ 2340.00 0.809043
$$204$$ −1554.00 −0.533342
$$205$$ 1688.00 0.575098
$$206$$ −1132.00 −0.382865
$$207$$ −1638.00 −0.549995
$$208$$ 1312.00 0.437360
$$209$$ 0 0
$$210$$ 312.000 0.102524
$$211$$ −2232.00 −0.728233 −0.364117 0.931353i $$-0.618629\pi$$
−0.364117 + 0.931353i $$0.618629\pi$$
$$212$$ −2436.00 −0.789175
$$213$$ −2286.00 −0.735372
$$214$$ −564.000 −0.180160
$$215$$ 1632.00 0.517681
$$216$$ 405.000 0.127578
$$217$$ −208.000 −0.0650689
$$218$$ 320.000 0.0994180
$$219$$ −1626.00 −0.501712
$$220$$ 0 0
$$221$$ −2368.00 −0.720764
$$222$$ 198.000 0.0598599
$$223$$ 2128.00 0.639020 0.319510 0.947583i $$-0.396482\pi$$
0.319510 + 0.947583i $$0.396482\pi$$
$$224$$ 4186.00 1.24861
$$225$$ −981.000 −0.290667
$$226$$ −2142.00 −0.630459
$$227$$ −2964.00 −0.866641 −0.433321 0.901240i $$-0.642658\pi$$
−0.433321 + 0.901240i $$0.642658\pi$$
$$228$$ 1260.00 0.365989
$$229$$ −2550.00 −0.735846 −0.367923 0.929856i $$-0.619931\pi$$
−0.367923 + 0.929856i $$0.619931\pi$$
$$230$$ 728.000 0.208708
$$231$$ 0 0
$$232$$ −1350.00 −0.382034
$$233$$ 3042.00 0.855314 0.427657 0.903941i $$-0.359339\pi$$
0.427657 + 0.903941i $$0.359339\pi$$
$$234$$ 288.000 0.0804579
$$235$$ 2024.00 0.561835
$$236$$ 1400.00 0.386154
$$237$$ −1650.00 −0.452232
$$238$$ −1924.00 −0.524010
$$239$$ −2700.00 −0.730747 −0.365373 0.930861i $$-0.619059\pi$$
−0.365373 + 0.930861i $$0.619059\pi$$
$$240$$ 492.000 0.132327
$$241$$ 578.000 0.154491 0.0772453 0.997012i $$-0.475388\pi$$
0.0772453 + 0.997012i $$0.475388\pi$$
$$242$$ 0 0
$$243$$ −243.000 −0.0641500
$$244$$ 924.000 0.242430
$$245$$ −1332.00 −0.347340
$$246$$ 1266.00 0.328119
$$247$$ 1920.00 0.494602
$$248$$ 120.000 0.0307258
$$249$$ −396.000 −0.100785
$$250$$ 936.000 0.236791
$$251$$ 3752.00 0.943522 0.471761 0.881726i $$-0.343618\pi$$
0.471761 + 0.881726i $$0.343618\pi$$
$$252$$ −1638.00 −0.409462
$$253$$ 0 0
$$254$$ 1606.00 0.396730
$$255$$ −888.000 −0.218073
$$256$$ −119.000 −0.0290527
$$257$$ 674.000 0.163591 0.0817957 0.996649i $$-0.473935\pi$$
0.0817957 + 0.996649i $$0.473935\pi$$
$$258$$ 1224.00 0.295360
$$259$$ −1716.00 −0.411687
$$260$$ 896.000 0.213721
$$261$$ 810.000 0.192099
$$262$$ 1908.00 0.449911
$$263$$ 4352.00 1.02036 0.510182 0.860066i $$-0.329578\pi$$
0.510182 + 0.860066i $$0.329578\pi$$
$$264$$ 0 0
$$265$$ −1392.00 −0.322679
$$266$$ 1560.00 0.359585
$$267$$ −1710.00 −0.391949
$$268$$ 7252.00 1.65293
$$269$$ 500.000 0.113329 0.0566646 0.998393i $$-0.481953\pi$$
0.0566646 + 0.998393i $$0.481953\pi$$
$$270$$ 108.000 0.0243432
$$271$$ 6538.00 1.46552 0.732759 0.680489i $$-0.238232\pi$$
0.732759 + 0.680489i $$0.238232\pi$$
$$272$$ −3034.00 −0.676336
$$273$$ −2496.00 −0.553351
$$274$$ −2186.00 −0.481975
$$275$$ 0 0
$$276$$ −3822.00 −0.833541
$$277$$ −124.000 −0.0268969 −0.0134484 0.999910i $$-0.504281\pi$$
−0.0134484 + 0.999910i $$0.504281\pi$$
$$278$$ −2740.00 −0.591131
$$279$$ −72.0000 −0.0154499
$$280$$ 1560.00 0.332957
$$281$$ −3642.00 −0.773180 −0.386590 0.922252i $$-0.626347\pi$$
−0.386590 + 0.922252i $$0.626347\pi$$
$$282$$ 1518.00 0.320552
$$283$$ −4648.00 −0.976307 −0.488154 0.872758i $$-0.662329\pi$$
−0.488154 + 0.872758i $$0.662329\pi$$
$$284$$ −5334.00 −1.11449
$$285$$ 720.000 0.149646
$$286$$ 0 0
$$287$$ −10972.0 −2.25664
$$288$$ 1449.00 0.296469
$$289$$ 563.000 0.114594
$$290$$ −360.000 −0.0728963
$$291$$ −42.0000 −0.00846077
$$292$$ −3794.00 −0.760367
$$293$$ 3102.00 0.618501 0.309250 0.950981i $$-0.399922\pi$$
0.309250 + 0.950981i $$0.399922\pi$$
$$294$$ −999.000 −0.198173
$$295$$ 800.000 0.157891
$$296$$ 990.000 0.194401
$$297$$ 0 0
$$298$$ 1310.00 0.254652
$$299$$ −5824.00 −1.12646
$$300$$ −2289.00 −0.440518
$$301$$ −10608.0 −2.03135
$$302$$ 1198.00 0.228269
$$303$$ 5106.00 0.968093
$$304$$ 2460.00 0.464114
$$305$$ 528.000 0.0991252
$$306$$ −666.000 −0.124421
$$307$$ −1244.00 −0.231267 −0.115633 0.993292i $$-0.536890\pi$$
−0.115633 + 0.993292i $$0.536890\pi$$
$$308$$ 0 0
$$309$$ 3396.00 0.625216
$$310$$ 32.0000 0.00586283
$$311$$ 2082.00 0.379612 0.189806 0.981822i $$-0.439214\pi$$
0.189806 + 0.981822i $$0.439214\pi$$
$$312$$ 1440.00 0.261295
$$313$$ 2378.00 0.429433 0.214716 0.976676i $$-0.431117\pi$$
0.214716 + 0.976676i $$0.431117\pi$$
$$314$$ 2114.00 0.379936
$$315$$ −936.000 −0.167421
$$316$$ −3850.00 −0.685378
$$317$$ −496.000 −0.0878806 −0.0439403 0.999034i $$-0.513991\pi$$
−0.0439403 + 0.999034i $$0.513991\pi$$
$$318$$ −1044.00 −0.184103
$$319$$ 0 0
$$320$$ 668.000 0.116695
$$321$$ 1692.00 0.294200
$$322$$ −4732.00 −0.818957
$$323$$ −4440.00 −0.764855
$$324$$ −567.000 −0.0972222
$$325$$ −3488.00 −0.595321
$$326$$ 3868.00 0.657143
$$327$$ −960.000 −0.162349
$$328$$ 6330.00 1.06560
$$329$$ −13156.0 −2.20460
$$330$$ 0 0
$$331$$ −2708.00 −0.449683 −0.224842 0.974395i $$-0.572186\pi$$
−0.224842 + 0.974395i $$0.572186\pi$$
$$332$$ −924.000 −0.152744
$$333$$ −594.000 −0.0977507
$$334$$ −2004.00 −0.328305
$$335$$ 4144.00 0.675853
$$336$$ −3198.00 −0.519242
$$337$$ −4034.00 −0.652065 −0.326033 0.945359i $$-0.605712\pi$$
−0.326033 + 0.945359i $$0.605712\pi$$
$$338$$ −1173.00 −0.188766
$$339$$ 6426.00 1.02954
$$340$$ −2072.00 −0.330500
$$341$$ 0 0
$$342$$ 540.000 0.0853797
$$343$$ −260.000 −0.0409291
$$344$$ 6120.00 0.959210
$$345$$ −2184.00 −0.340819
$$346$$ −678.000 −0.105345
$$347$$ −11084.0 −1.71476 −0.857378 0.514687i $$-0.827908\pi$$
−0.857378 + 0.514687i $$0.827908\pi$$
$$348$$ 1890.00 0.291134
$$349$$ 3120.00 0.478538 0.239269 0.970953i $$-0.423092\pi$$
0.239269 + 0.970953i $$0.423092\pi$$
$$350$$ −2834.00 −0.432810
$$351$$ −864.000 −0.131387
$$352$$ 0 0
$$353$$ −5622.00 −0.847674 −0.423837 0.905739i $$-0.639317\pi$$
−0.423837 + 0.905739i $$0.639317\pi$$
$$354$$ 600.000 0.0900837
$$355$$ −3048.00 −0.455693
$$356$$ −3990.00 −0.594016
$$357$$ 5772.00 0.855705
$$358$$ −1680.00 −0.248019
$$359$$ 8500.00 1.24962 0.624809 0.780778i $$-0.285177\pi$$
0.624809 + 0.780778i $$0.285177\pi$$
$$360$$ 540.000 0.0790569
$$361$$ −3259.00 −0.475142
$$362$$ −4358.00 −0.632739
$$363$$ 0 0
$$364$$ −5824.00 −0.838628
$$365$$ −2168.00 −0.310899
$$366$$ 396.000 0.0565553
$$367$$ 7144.00 1.01611 0.508057 0.861324i $$-0.330364\pi$$
0.508057 + 0.861324i $$0.330364\pi$$
$$368$$ −7462.00 −1.05702
$$369$$ −3798.00 −0.535816
$$370$$ 264.000 0.0370938
$$371$$ 9048.00 1.26617
$$372$$ −168.000 −0.0234150
$$373$$ 632.000 0.0877312 0.0438656 0.999037i $$-0.486033\pi$$
0.0438656 + 0.999037i $$0.486033\pi$$
$$374$$ 0 0
$$375$$ −2808.00 −0.386679
$$376$$ 7590.00 1.04102
$$377$$ 2880.00 0.393442
$$378$$ −702.000 −0.0955211
$$379$$ −4220.00 −0.571944 −0.285972 0.958238i $$-0.592316\pi$$
−0.285972 + 0.958238i $$0.592316\pi$$
$$380$$ 1680.00 0.226795
$$381$$ −4818.00 −0.647857
$$382$$ −1778.00 −0.238142
$$383$$ 8458.00 1.12842 0.564208 0.825632i $$-0.309181\pi$$
0.564208 + 0.825632i $$0.309181\pi$$
$$384$$ 4365.00 0.580079
$$385$$ 0 0
$$386$$ 3962.00 0.522437
$$387$$ −3672.00 −0.482321
$$388$$ −98.0000 −0.0128227
$$389$$ 1740.00 0.226790 0.113395 0.993550i $$-0.463827\pi$$
0.113395 + 0.993550i $$0.463827\pi$$
$$390$$ 384.000 0.0498579
$$391$$ 13468.0 1.74196
$$392$$ −4995.00 −0.643586
$$393$$ −5724.00 −0.734701
$$394$$ −374.000 −0.0478219
$$395$$ −2200.00 −0.280238
$$396$$ 0 0
$$397$$ −5126.00 −0.648027 −0.324013 0.946053i $$-0.605032\pi$$
−0.324013 + 0.946053i $$0.605032\pi$$
$$398$$ 2100.00 0.264481
$$399$$ −4680.00 −0.587201
$$400$$ −4469.00 −0.558625
$$401$$ −3098.00 −0.385802 −0.192901 0.981218i $$-0.561790\pi$$
−0.192901 + 0.981218i $$0.561790\pi$$
$$402$$ 3108.00 0.385604
$$403$$ −256.000 −0.0316433
$$404$$ 11914.0 1.46719
$$405$$ −324.000 −0.0397523
$$406$$ 2340.00 0.286040
$$407$$ 0 0
$$408$$ −3330.00 −0.404068
$$409$$ −6390.00 −0.772531 −0.386265 0.922388i $$-0.626235\pi$$
−0.386265 + 0.922388i $$0.626235\pi$$
$$410$$ 1688.00 0.203328
$$411$$ 6558.00 0.787062
$$412$$ 7924.00 0.947542
$$413$$ −5200.00 −0.619553
$$414$$ −1638.00 −0.194452
$$415$$ −528.000 −0.0624542
$$416$$ 5152.00 0.607206
$$417$$ 8220.00 0.965312
$$418$$ 0 0
$$419$$ 9760.00 1.13796 0.568982 0.822350i $$-0.307337\pi$$
0.568982 + 0.822350i $$0.307337\pi$$
$$420$$ −2184.00 −0.253734
$$421$$ −5138.00 −0.594800 −0.297400 0.954753i $$-0.596119\pi$$
−0.297400 + 0.954753i $$0.596119\pi$$
$$422$$ −2232.00 −0.257469
$$423$$ −4554.00 −0.523459
$$424$$ −5220.00 −0.597891
$$425$$ 8066.00 0.920608
$$426$$ −2286.00 −0.259993
$$427$$ −3432.00 −0.388960
$$428$$ 3948.00 0.445873
$$429$$ 0 0
$$430$$ 1632.00 0.183028
$$431$$ 7008.00 0.783210 0.391605 0.920133i $$-0.371920\pi$$
0.391605 + 0.920133i $$0.371920\pi$$
$$432$$ −1107.00 −0.123288
$$433$$ 5578.00 0.619080 0.309540 0.950886i $$-0.399825\pi$$
0.309540 + 0.950886i $$0.399825\pi$$
$$434$$ −208.000 −0.0230053
$$435$$ 1080.00 0.119039
$$436$$ −2240.00 −0.246047
$$437$$ −10920.0 −1.19536
$$438$$ −1626.00 −0.177382
$$439$$ 10430.0 1.13393 0.566967 0.823741i $$-0.308117\pi$$
0.566967 + 0.823741i $$0.308117\pi$$
$$440$$ 0 0
$$441$$ 2997.00 0.323615
$$442$$ −2368.00 −0.254829
$$443$$ −4432.00 −0.475329 −0.237664 0.971347i $$-0.576382\pi$$
−0.237664 + 0.971347i $$0.576382\pi$$
$$444$$ −1386.00 −0.148146
$$445$$ −2280.00 −0.242882
$$446$$ 2128.00 0.225928
$$447$$ −3930.00 −0.415845
$$448$$ −4342.00 −0.457902
$$449$$ −6290.00 −0.661121 −0.330561 0.943785i $$-0.607238\pi$$
−0.330561 + 0.943785i $$0.607238\pi$$
$$450$$ −981.000 −0.102766
$$451$$ 0 0
$$452$$ 14994.0 1.56031
$$453$$ −3594.00 −0.372761
$$454$$ −2964.00 −0.306404
$$455$$ −3328.00 −0.342899
$$456$$ 2700.00 0.277279
$$457$$ −3054.00 −0.312604 −0.156302 0.987709i $$-0.549957\pi$$
−0.156302 + 0.987709i $$0.549957\pi$$
$$458$$ −2550.00 −0.260161
$$459$$ 1998.00 0.203178
$$460$$ −5096.00 −0.516527
$$461$$ −12882.0 −1.30146 −0.650732 0.759308i $$-0.725538\pi$$
−0.650732 + 0.759308i $$0.725538\pi$$
$$462$$ 0 0
$$463$$ 6148.00 0.617110 0.308555 0.951207i $$-0.400155\pi$$
0.308555 + 0.951207i $$0.400155\pi$$
$$464$$ 3690.00 0.369190
$$465$$ −96.0000 −0.00957396
$$466$$ 3042.00 0.302399
$$467$$ 5124.00 0.507731 0.253866 0.967240i $$-0.418298\pi$$
0.253866 + 0.967240i $$0.418298\pi$$
$$468$$ −2016.00 −0.199123
$$469$$ −26936.0 −2.65200
$$470$$ 2024.00 0.198639
$$471$$ −6342.00 −0.620433
$$472$$ 3000.00 0.292555
$$473$$ 0 0
$$474$$ −1650.00 −0.159888
$$475$$ −6540.00 −0.631738
$$476$$ 13468.0 1.29686
$$477$$ 3132.00 0.300638
$$478$$ −2700.00 −0.258358
$$479$$ 16520.0 1.57582 0.787910 0.615790i $$-0.211163\pi$$
0.787910 + 0.615790i $$0.211163\pi$$
$$480$$ 1932.00 0.183715
$$481$$ −2112.00 −0.200206
$$482$$ 578.000 0.0546207
$$483$$ 14196.0 1.33735
$$484$$ 0 0
$$485$$ −56.0000 −0.00524295
$$486$$ −243.000 −0.0226805
$$487$$ 524.000 0.0487571 0.0243785 0.999703i $$-0.492239\pi$$
0.0243785 + 0.999703i $$0.492239\pi$$
$$488$$ 1980.00 0.183669
$$489$$ −11604.0 −1.07311
$$490$$ −1332.00 −0.122803
$$491$$ 15028.0 1.38127 0.690636 0.723203i $$-0.257331\pi$$
0.690636 + 0.723203i $$0.257331\pi$$
$$492$$ −8862.00 −0.812052
$$493$$ −6660.00 −0.608421
$$494$$ 1920.00 0.174868
$$495$$ 0 0
$$496$$ −328.000 −0.0296928
$$497$$ 19812.0 1.78811
$$498$$ −396.000 −0.0356329
$$499$$ 9020.00 0.809200 0.404600 0.914494i $$-0.367411\pi$$
0.404600 + 0.914494i $$0.367411\pi$$
$$500$$ −6552.00 −0.586029
$$501$$ 6012.00 0.536120
$$502$$ 3752.00 0.333586
$$503$$ 14812.0 1.31299 0.656495 0.754330i $$-0.272038\pi$$
0.656495 + 0.754330i $$0.272038\pi$$
$$504$$ −3510.00 −0.310214
$$505$$ 6808.00 0.599905
$$506$$ 0 0
$$507$$ 3519.00 0.308253
$$508$$ −11242.0 −0.981856
$$509$$ 12660.0 1.10245 0.551223 0.834358i $$-0.314161\pi$$
0.551223 + 0.834358i $$0.314161\pi$$
$$510$$ −888.000 −0.0771006
$$511$$ 14092.0 1.21995
$$512$$ 11521.0 0.994455
$$513$$ −1620.00 −0.139424
$$514$$ 674.000 0.0578383
$$515$$ 4528.00 0.387432
$$516$$ −8568.00 −0.730979
$$517$$ 0 0
$$518$$ −1716.00 −0.145553
$$519$$ 2034.00 0.172028
$$520$$ 1920.00 0.161918
$$521$$ −3738.00 −0.314328 −0.157164 0.987573i $$-0.550235\pi$$
−0.157164 + 0.987573i $$0.550235\pi$$
$$522$$ 810.000 0.0679171
$$523$$ 6352.00 0.531078 0.265539 0.964100i $$-0.414450\pi$$
0.265539 + 0.964100i $$0.414450\pi$$
$$524$$ −13356.0 −1.11347
$$525$$ 8502.00 0.706777
$$526$$ 4352.00 0.360753
$$527$$ 592.000 0.0489334
$$528$$ 0 0
$$529$$ 20957.0 1.72245
$$530$$ −1392.00 −0.114084
$$531$$ −1800.00 −0.147106
$$532$$ −10920.0 −0.889929
$$533$$ −13504.0 −1.09742
$$534$$ −1710.00 −0.138575
$$535$$ 2256.00 0.182309
$$536$$ 15540.0 1.25229
$$537$$ 5040.00 0.405013
$$538$$ 500.000 0.0400679
$$539$$ 0 0
$$540$$ −756.000 −0.0602464
$$541$$ 24728.0 1.96514 0.982569 0.185898i $$-0.0595193\pi$$
0.982569 + 0.185898i $$0.0595193\pi$$
$$542$$ 6538.00 0.518139
$$543$$ 13074.0 1.03326
$$544$$ −11914.0 −0.938986
$$545$$ −1280.00 −0.100604
$$546$$ −2496.00 −0.195639
$$547$$ 22756.0 1.77875 0.889375 0.457178i $$-0.151140\pi$$
0.889375 + 0.457178i $$0.151140\pi$$
$$548$$ 15302.0 1.19283
$$549$$ −1188.00 −0.0923545
$$550$$ 0 0
$$551$$ 5400.00 0.417509
$$552$$ −8190.00 −0.631503
$$553$$ 14300.0 1.09963
$$554$$ −124.000 −0.00950949
$$555$$ −792.000 −0.0605739
$$556$$ 19180.0 1.46297
$$557$$ 9526.00 0.724649 0.362325 0.932052i $$-0.381983\pi$$
0.362325 + 0.932052i $$0.381983\pi$$
$$558$$ −72.0000 −0.00546237
$$559$$ −13056.0 −0.987853
$$560$$ −4264.00 −0.321762
$$561$$ 0 0
$$562$$ −3642.00 −0.273360
$$563$$ −12068.0 −0.903385 −0.451692 0.892174i $$-0.649180\pi$$
−0.451692 + 0.892174i $$0.649180\pi$$
$$564$$ −10626.0 −0.793325
$$565$$ 8568.00 0.637980
$$566$$ −4648.00 −0.345177
$$567$$ 2106.00 0.155985
$$568$$ −11430.0 −0.844352
$$569$$ −15090.0 −1.11179 −0.555893 0.831254i $$-0.687623\pi$$
−0.555893 + 0.831254i $$0.687623\pi$$
$$570$$ 720.000 0.0529079
$$571$$ −4412.00 −0.323356 −0.161678 0.986844i $$-0.551691\pi$$
−0.161678 + 0.986844i $$0.551691\pi$$
$$572$$ 0 0
$$573$$ 5334.00 0.388885
$$574$$ −10972.0 −0.797844
$$575$$ 19838.0 1.43879
$$576$$ −1503.00 −0.108724
$$577$$ −3906.00 −0.281818 −0.140909 0.990023i $$-0.545002\pi$$
−0.140909 + 0.990023i $$0.545002\pi$$
$$578$$ 563.000 0.0405151
$$579$$ −11886.0 −0.853135
$$580$$ 2520.00 0.180409
$$581$$ 3432.00 0.245066
$$582$$ −42.0000 −0.00299133
$$583$$ 0 0
$$584$$ −8130.00 −0.576065
$$585$$ −1152.00 −0.0814177
$$586$$ 3102.00 0.218673
$$587$$ −12016.0 −0.844895 −0.422448 0.906387i $$-0.638829\pi$$
−0.422448 + 0.906387i $$0.638829\pi$$
$$588$$ 6993.00 0.490453
$$589$$ −480.000 −0.0335790
$$590$$ 800.000 0.0558228
$$591$$ 1122.00 0.0780929
$$592$$ −2706.00 −0.187865
$$593$$ 11342.0 0.785430 0.392715 0.919660i $$-0.371536\pi$$
0.392715 + 0.919660i $$0.371536\pi$$
$$594$$ 0 0
$$595$$ 7696.00 0.530261
$$596$$ −9170.00 −0.630231
$$597$$ −6300.00 −0.431896
$$598$$ −5824.00 −0.398263
$$599$$ 20690.0 1.41130 0.705651 0.708559i $$-0.250654\pi$$
0.705651 + 0.708559i $$0.250654\pi$$
$$600$$ −4905.00 −0.333743
$$601$$ 598.000 0.0405872 0.0202936 0.999794i $$-0.493540\pi$$
0.0202936 + 0.999794i $$0.493540\pi$$
$$602$$ −10608.0 −0.718189
$$603$$ −9324.00 −0.629689
$$604$$ −8386.00 −0.564936
$$605$$ 0 0
$$606$$ 5106.00 0.342272
$$607$$ 166.000 0.0111001 0.00555003 0.999985i $$-0.498233\pi$$
0.00555003 + 0.999985i $$0.498233\pi$$
$$608$$ 9660.00 0.644350
$$609$$ −7020.00 −0.467101
$$610$$ 528.000 0.0350461
$$611$$ −16192.0 −1.07211
$$612$$ 4662.00 0.307925
$$613$$ −20108.0 −1.32488 −0.662442 0.749113i $$-0.730480\pi$$
−0.662442 + 0.749113i $$0.730480\pi$$
$$614$$ −1244.00 −0.0817651
$$615$$ −5064.00 −0.332033
$$616$$ 0 0
$$617$$ −2286.00 −0.149159 −0.0745793 0.997215i $$-0.523761\pi$$
−0.0745793 + 0.997215i $$0.523761\pi$$
$$618$$ 3396.00 0.221047
$$619$$ −25660.0 −1.66618 −0.833088 0.553141i $$-0.813429\pi$$
−0.833088 + 0.553141i $$0.813429\pi$$
$$620$$ −224.000 −0.0145098
$$621$$ 4914.00 0.317539
$$622$$ 2082.00 0.134213
$$623$$ 14820.0 0.953051
$$624$$ −3936.00 −0.252510
$$625$$ 9881.00 0.632384
$$626$$ 2378.00 0.151827
$$627$$ 0 0
$$628$$ −14798.0 −0.940294
$$629$$ 4884.00 0.309599
$$630$$ −936.000 −0.0591923
$$631$$ −11408.0 −0.719723 −0.359862 0.933006i $$-0.617176\pi$$
−0.359862 + 0.933006i $$0.617176\pi$$
$$632$$ −8250.00 −0.519252
$$633$$ 6696.00 0.420446
$$634$$ −496.000 −0.0310705
$$635$$ −6424.00 −0.401462
$$636$$ 7308.00 0.455631
$$637$$ 10656.0 0.662804
$$638$$ 0 0
$$639$$ 6858.00 0.424567
$$640$$ 5820.00 0.359462
$$641$$ −3378.00 −0.208148 −0.104074 0.994570i $$-0.533188\pi$$
−0.104074 + 0.994570i $$0.533188\pi$$
$$642$$ 1692.00 0.104015
$$643$$ −11212.0 −0.687649 −0.343824 0.939034i $$-0.611722\pi$$
−0.343824 + 0.939034i $$0.611722\pi$$
$$644$$ 33124.0 2.02681
$$645$$ −4896.00 −0.298883
$$646$$ −4440.00 −0.270417
$$647$$ −86.0000 −0.00522567 −0.00261284 0.999997i $$-0.500832\pi$$
−0.00261284 + 0.999997i $$0.500832\pi$$
$$648$$ −1215.00 −0.0736570
$$649$$ 0 0
$$650$$ −3488.00 −0.210478
$$651$$ 624.000 0.0375676
$$652$$ −27076.0 −1.62635
$$653$$ −4432.00 −0.265601 −0.132801 0.991143i $$-0.542397\pi$$
−0.132801 + 0.991143i $$0.542397\pi$$
$$654$$ −960.000 −0.0573990
$$655$$ −7632.00 −0.455278
$$656$$ −17302.0 −1.02977
$$657$$ 4878.00 0.289663
$$658$$ −13156.0 −0.779444
$$659$$ −4580.00 −0.270731 −0.135365 0.990796i $$-0.543221\pi$$
−0.135365 + 0.990796i $$0.543221\pi$$
$$660$$ 0 0
$$661$$ 4282.00 0.251967 0.125984 0.992032i $$-0.459791\pi$$
0.125984 + 0.992032i $$0.459791\pi$$
$$662$$ −2708.00 −0.158987
$$663$$ 7104.00 0.416133
$$664$$ −1980.00 −0.115721
$$665$$ −6240.00 −0.363875
$$666$$ −594.000 −0.0345601
$$667$$ −16380.0 −0.950879
$$668$$ 14028.0 0.812514
$$669$$ −6384.00 −0.368938
$$670$$ 4144.00 0.238950
$$671$$ 0 0
$$672$$ −12558.0 −0.720886
$$673$$ −8438.00 −0.483300 −0.241650 0.970363i $$-0.577689\pi$$
−0.241650 + 0.970363i $$0.577689\pi$$
$$674$$ −4034.00 −0.230540
$$675$$ 2943.00 0.167816
$$676$$ 8211.00 0.467171
$$677$$ −34494.0 −1.95822 −0.979108 0.203341i $$-0.934820\pi$$
−0.979108 + 0.203341i $$0.934820\pi$$
$$678$$ 6426.00 0.363996
$$679$$ 364.000 0.0205730
$$680$$ −4440.00 −0.250392
$$681$$ 8892.00 0.500356
$$682$$ 0 0
$$683$$ −13712.0 −0.768192 −0.384096 0.923293i $$-0.625487\pi$$
−0.384096 + 0.923293i $$0.625487\pi$$
$$684$$ −3780.00 −0.211304
$$685$$ 8744.00 0.487724
$$686$$ −260.000 −0.0144706
$$687$$ 7650.00 0.424841
$$688$$ −16728.0 −0.926961
$$689$$ 11136.0 0.615744
$$690$$ −2184.00 −0.120498
$$691$$ 11372.0 0.626066 0.313033 0.949742i $$-0.398655\pi$$
0.313033 + 0.949742i $$0.398655\pi$$
$$692$$ 4746.00 0.260717
$$693$$ 0 0
$$694$$ −11084.0 −0.606258
$$695$$ 10960.0 0.598182
$$696$$ 4050.00 0.220567
$$697$$ 31228.0 1.69705
$$698$$ 3120.00 0.169189
$$699$$ −9126.00 −0.493815
$$700$$ 19838.0 1.07115
$$701$$ 6398.00 0.344721 0.172360 0.985034i $$-0.444861\pi$$
0.172360 + 0.985034i $$0.444861\pi$$
$$702$$ −864.000 −0.0464524
$$703$$ −3960.00 −0.212453
$$704$$ 0 0
$$705$$ −6072.00 −0.324376
$$706$$ −5622.00 −0.299698
$$707$$ −44252.0 −2.35399
$$708$$ −4200.00 −0.222946
$$709$$ −5830.00 −0.308816 −0.154408 0.988007i $$-0.549347\pi$$
−0.154408 + 0.988007i $$0.549347\pi$$
$$710$$ −3048.00 −0.161112
$$711$$ 4950.00 0.261096
$$712$$ −8550.00 −0.450035
$$713$$ 1456.00 0.0764763
$$714$$ 5772.00 0.302537
$$715$$ 0 0
$$716$$ 11760.0 0.613815
$$717$$ 8100.00 0.421897
$$718$$ 8500.00 0.441807
$$719$$ 34530.0 1.79103 0.895516 0.445030i $$-0.146807\pi$$
0.895516 + 0.445030i $$0.146807\pi$$
$$720$$ −1476.00 −0.0763990
$$721$$ −29432.0 −1.52026
$$722$$ −3259.00 −0.167988
$$723$$ −1734.00 −0.0891952
$$724$$ 30506.0 1.56595
$$725$$ −9810.00 −0.502530
$$726$$ 0 0
$$727$$ −17316.0 −0.883377 −0.441688 0.897169i $$-0.645620\pi$$
−0.441688 + 0.897169i $$0.645620\pi$$
$$728$$ −12480.0 −0.635357
$$729$$ 729.000 0.0370370
$$730$$ −2168.00 −0.109920
$$731$$ 30192.0 1.52762
$$732$$ −2772.00 −0.139967
$$733$$ 27072.0 1.36416 0.682079 0.731279i $$-0.261076\pi$$
0.682079 + 0.731279i $$0.261076\pi$$
$$734$$ 7144.00 0.359250
$$735$$ 3996.00 0.200537
$$736$$ −29302.0 −1.46751
$$737$$ 0 0
$$738$$ −3798.00 −0.189439
$$739$$ 17320.0 0.862147 0.431073 0.902317i $$-0.358135\pi$$
0.431073 + 0.902317i $$0.358135\pi$$
$$740$$ −1848.00 −0.0918025
$$741$$ −5760.00 −0.285559
$$742$$ 9048.00 0.447658
$$743$$ −14588.0 −0.720299 −0.360149 0.932895i $$-0.617274\pi$$
−0.360149 + 0.932895i $$0.617274\pi$$
$$744$$ −360.000 −0.0177396
$$745$$ −5240.00 −0.257690
$$746$$ 632.000 0.0310176
$$747$$ 1188.00 0.0581883
$$748$$ 0 0
$$749$$ −14664.0 −0.715368
$$750$$ −2808.00 −0.136712
$$751$$ 26152.0 1.27071 0.635353 0.772222i $$-0.280855\pi$$
0.635353 + 0.772222i $$0.280855\pi$$
$$752$$ −20746.0 −1.00602
$$753$$ −11256.0 −0.544743
$$754$$ 2880.00 0.139103
$$755$$ −4792.00 −0.230992
$$756$$ 4914.00 0.236403
$$757$$ −1066.00 −0.0511815 −0.0255908 0.999673i $$-0.508147\pi$$
−0.0255908 + 0.999673i $$0.508147\pi$$
$$758$$ −4220.00 −0.202213
$$759$$ 0 0
$$760$$ 3600.00 0.171823
$$761$$ 37518.0 1.78716 0.893578 0.448907i $$-0.148187\pi$$
0.893578 + 0.448907i $$0.148187\pi$$
$$762$$ −4818.00 −0.229052
$$763$$ 8320.00 0.394763
$$764$$ 12446.0 0.589372
$$765$$ 2664.00 0.125905
$$766$$ 8458.00 0.398956
$$767$$ −6400.00 −0.301292
$$768$$ 357.000 0.0167736
$$769$$ 17290.0 0.810785 0.405392 0.914143i $$-0.367135\pi$$
0.405392 + 0.914143i $$0.367135\pi$$
$$770$$ 0 0
$$771$$ −2022.00 −0.0944495
$$772$$ −27734.0 −1.29296
$$773$$ −17172.0 −0.799009 −0.399504 0.916731i $$-0.630818\pi$$
−0.399504 + 0.916731i $$0.630818\pi$$
$$774$$ −3672.00 −0.170526
$$775$$ 872.000 0.0404170
$$776$$ −210.000 −0.00971464
$$777$$ 5148.00 0.237688
$$778$$ 1740.00 0.0801825
$$779$$ −25320.0 −1.16455
$$780$$ −2688.00 −0.123392
$$781$$ 0 0
$$782$$ 13468.0 0.615876
$$783$$ −2430.00 −0.110908
$$784$$ 13653.0 0.621948
$$785$$ −8456.00 −0.384468
$$786$$ −5724.00 −0.259756
$$787$$ 9536.00 0.431921 0.215960 0.976402i $$-0.430712\pi$$
0.215960 + 0.976402i $$0.430712\pi$$
$$788$$ 2618.00 0.118353
$$789$$ −13056.0 −0.589108
$$790$$ −2200.00 −0.0990791
$$791$$ −55692.0 −2.50339
$$792$$ 0 0
$$793$$ −4224.00 −0.189153
$$794$$ −5126.00 −0.229112
$$795$$ 4176.00 0.186299
$$796$$ −14700.0 −0.654557
$$797$$ −20516.0 −0.911812 −0.455906 0.890028i $$-0.650685\pi$$
−0.455906 + 0.890028i $$0.650685\pi$$
$$798$$ −4680.00 −0.207607
$$799$$ 37444.0 1.65791
$$800$$ −17549.0 −0.775564
$$801$$ 5130.00 0.226292
$$802$$ −3098.00 −0.136402
$$803$$ 0 0
$$804$$ −21756.0 −0.954322
$$805$$ 18928.0 0.828726
$$806$$ −256.000 −0.0111876
$$807$$ −1500.00 −0.0654306
$$808$$ 25530.0 1.11156
$$809$$ −22470.0 −0.976518 −0.488259 0.872699i $$-0.662368\pi$$
−0.488259 + 0.872699i $$0.662368\pi$$
$$810$$ −324.000 −0.0140546
$$811$$ 3368.00 0.145828 0.0729140 0.997338i $$-0.476770\pi$$
0.0729140 + 0.997338i $$0.476770\pi$$
$$812$$ −16380.0 −0.707913
$$813$$ −19614.0 −0.846117
$$814$$ 0 0
$$815$$ −15472.0 −0.664982
$$816$$ 9102.00 0.390483
$$817$$ −24480.0 −1.04828
$$818$$ −6390.00 −0.273131
$$819$$ 7488.00 0.319477
$$820$$ −11816.0 −0.503211
$$821$$ 10738.0 0.456466 0.228233 0.973607i $$-0.426705\pi$$
0.228233 + 0.973607i $$0.426705\pi$$
$$822$$ 6558.00 0.278268
$$823$$ −15912.0 −0.673946 −0.336973 0.941514i $$-0.609403\pi$$
−0.336973 + 0.941514i $$0.609403\pi$$
$$824$$ 16980.0 0.717872
$$825$$ 0 0
$$826$$ −5200.00 −0.219045
$$827$$ −22924.0 −0.963900 −0.481950 0.876199i $$-0.660071\pi$$
−0.481950 + 0.876199i $$0.660071\pi$$
$$828$$ 11466.0 0.481245
$$829$$ −41690.0 −1.74663 −0.873313 0.487159i $$-0.838033\pi$$
−0.873313 + 0.487159i $$0.838033\pi$$
$$830$$ −528.000 −0.0220809
$$831$$ 372.000 0.0155289
$$832$$ −5344.00 −0.222680
$$833$$ −24642.0 −1.02496
$$834$$ 8220.00 0.341289
$$835$$ 8016.00 0.332222
$$836$$ 0 0
$$837$$ 216.000 0.00892001
$$838$$ 9760.00 0.402331
$$839$$ −16450.0 −0.676898 −0.338449 0.940985i $$-0.609902\pi$$
−0.338449 + 0.940985i $$0.609902\pi$$
$$840$$ −4680.00 −0.192233
$$841$$ −16289.0 −0.667883
$$842$$ −5138.00 −0.210294
$$843$$ 10926.0 0.446396
$$844$$ 15624.0 0.637204
$$845$$ 4692.00 0.191017
$$846$$ −4554.00 −0.185071
$$847$$ 0 0
$$848$$ 14268.0 0.577789
$$849$$ 13944.0 0.563671
$$850$$ 8066.00 0.325484
$$851$$ 12012.0 0.483861
$$852$$ 16002.0 0.643450
$$853$$ 30892.0 1.24000 0.620001 0.784601i $$-0.287132\pi$$
0.620001 + 0.784601i $$0.287132\pi$$
$$854$$ −3432.00 −0.137518
$$855$$ −2160.00 −0.0863982
$$856$$ 8460.00 0.337800
$$857$$ 38906.0 1.55076 0.775381 0.631493i $$-0.217558\pi$$
0.775381 + 0.631493i $$0.217558\pi$$
$$858$$ 0 0
$$859$$ −1020.00 −0.0405145 −0.0202572 0.999795i $$-0.506449\pi$$
−0.0202572 + 0.999795i $$0.506449\pi$$
$$860$$ −11424.0 −0.452971
$$861$$ 32916.0 1.30287
$$862$$ 7008.00 0.276907
$$863$$ 15078.0 0.594741 0.297370 0.954762i $$-0.403890\pi$$
0.297370 + 0.954762i $$0.403890\pi$$
$$864$$ −4347.00 −0.171167
$$865$$ 2712.00 0.106602
$$866$$ 5578.00 0.218878
$$867$$ −1689.00 −0.0661608
$$868$$ 1456.00 0.0569353
$$869$$ 0 0
$$870$$ 1080.00 0.0420867
$$871$$ −33152.0 −1.28968
$$872$$ −4800.00 −0.186409
$$873$$ 126.000 0.00488483
$$874$$ −10920.0 −0.422625
$$875$$ 24336.0 0.940237
$$876$$ 11382.0 0.438998
$$877$$ −22704.0 −0.874184 −0.437092 0.899417i $$-0.643992\pi$$
−0.437092 + 0.899417i $$0.643992\pi$$
$$878$$ 10430.0 0.400906
$$879$$ −9306.00 −0.357092
$$880$$ 0 0
$$881$$ −19358.0 −0.740281 −0.370141 0.928976i $$-0.620690\pi$$
−0.370141 + 0.928976i $$0.620690\pi$$
$$882$$ 2997.00 0.114415
$$883$$ −11252.0 −0.428833 −0.214417 0.976742i $$-0.568785\pi$$
−0.214417 + 0.976742i $$0.568785\pi$$
$$884$$ 16576.0 0.630669
$$885$$ −2400.00 −0.0911583
$$886$$ −4432.00 −0.168054
$$887$$ −43684.0 −1.65362 −0.826812 0.562478i $$-0.809848\pi$$
−0.826812 + 0.562478i $$0.809848\pi$$
$$888$$ −2970.00 −0.112237
$$889$$ 41756.0 1.57531
$$890$$ −2280.00 −0.0858717
$$891$$ 0 0
$$892$$ −14896.0 −0.559142
$$893$$ −30360.0 −1.13769
$$894$$ −3930.00 −0.147023
$$895$$ 6720.00 0.250977
$$896$$ −37830.0 −1.41050
$$897$$ 17472.0 0.650360
$$898$$ −6290.00 −0.233742
$$899$$ −720.000 −0.0267112
$$900$$ 6867.00 0.254333
$$901$$ −25752.0 −0.952190
$$902$$ 0 0
$$903$$ 31824.0 1.17280
$$904$$ 32130.0 1.18211
$$905$$ 17432.0 0.640287
$$906$$ −3594.00 −0.131791
$$907$$ 45804.0 1.67684 0.838422 0.545022i $$-0.183479\pi$$
0.838422 + 0.545022i $$0.183479\pi$$
$$908$$ 20748.0 0.758311
$$909$$ −15318.0 −0.558928
$$910$$ −3328.00 −0.121233
$$911$$ −15318.0 −0.557089 −0.278544 0.960423i $$-0.589852\pi$$
−0.278544 + 0.960423i $$0.589852\pi$$
$$912$$ −7380.00 −0.267956
$$913$$ 0 0
$$914$$ −3054.00 −0.110522
$$915$$ −1584.00 −0.0572300
$$916$$ 17850.0 0.643865
$$917$$ 49608.0 1.78648
$$918$$ 1998.00 0.0718342
$$919$$ −11350.0 −0.407401 −0.203701 0.979033i $$-0.565297\pi$$
−0.203701 + 0.979033i $$0.565297\pi$$
$$920$$ −10920.0 −0.391328
$$921$$ 3732.00 0.133522
$$922$$ −12882.0 −0.460137
$$923$$ 24384.0 0.869566
$$924$$ 0 0
$$925$$ 7194.00 0.255716
$$926$$ 6148.00 0.218181
$$927$$ −10188.0 −0.360969
$$928$$ 14490.0 0.512562
$$929$$ 33030.0 1.16650 0.583250 0.812292i $$-0.301781\pi$$
0.583250 + 0.812292i $$0.301781\pi$$
$$930$$ −96.0000 −0.00338491
$$931$$ 19980.0 0.703349
$$932$$ −21294.0 −0.748399
$$933$$ −6246.00 −0.219169
$$934$$ 5124.00 0.179510
$$935$$ 0 0
$$936$$ −4320.00 −0.150859
$$937$$ 10006.0 0.348860 0.174430 0.984670i $$-0.444192\pi$$
0.174430 + 0.984670i $$0.444192\pi$$
$$938$$ −26936.0 −0.937624
$$939$$ −7134.00 −0.247933
$$940$$ −14168.0 −0.491606
$$941$$ −2622.00 −0.0908340 −0.0454170 0.998968i $$-0.514462\pi$$
−0.0454170 + 0.998968i $$0.514462\pi$$
$$942$$ −6342.00 −0.219356
$$943$$ 76804.0 2.65226
$$944$$ −8200.00 −0.282720
$$945$$ 2808.00 0.0966606
$$946$$ 0 0
$$947$$ −39876.0 −1.36832 −0.684158 0.729334i $$-0.739830\pi$$
−0.684158 + 0.729334i $$0.739830\pi$$
$$948$$ 11550.0 0.395703
$$949$$ 17344.0 0.593267
$$950$$ −6540.00 −0.223353
$$951$$ 1488.00 0.0507379
$$952$$ 28860.0 0.982519
$$953$$ −38918.0 −1.32285 −0.661426 0.750011i $$-0.730048\pi$$
−0.661426 + 0.750011i $$0.730048\pi$$
$$954$$ 3132.00 0.106292
$$955$$ 7112.00 0.240983
$$956$$ 18900.0 0.639403
$$957$$ 0 0
$$958$$ 16520.0 0.557137
$$959$$ −56836.0 −1.91380
$$960$$ −2004.00 −0.0673738
$$961$$ −29727.0 −0.997852
$$962$$ −2112.00 −0.0707834
$$963$$ −5076.00 −0.169857
$$964$$ −4046.00 −0.135179
$$965$$ −15848.0 −0.528669
$$966$$ 14196.0 0.472825
$$967$$ −1114.00 −0.0370464 −0.0185232 0.999828i $$-0.505896\pi$$
−0.0185232 + 0.999828i $$0.505896\pi$$
$$968$$ 0 0
$$969$$ 13320.0 0.441589
$$970$$ −56.0000 −0.00185366
$$971$$ −1688.00 −0.0557884 −0.0278942 0.999611i $$-0.508880\pi$$
−0.0278942 + 0.999611i $$0.508880\pi$$
$$972$$ 1701.00 0.0561313
$$973$$ −71240.0 −2.34722
$$974$$ 524.000 0.0172382
$$975$$ 10464.0 0.343709
$$976$$ −5412.00 −0.177494
$$977$$ −41826.0 −1.36963 −0.684817 0.728715i $$-0.740118\pi$$
−0.684817 + 0.728715i $$0.740118\pi$$
$$978$$ −11604.0 −0.379402
$$979$$ 0 0
$$980$$ 9324.00 0.303923
$$981$$ 2880.00 0.0937322
$$982$$ 15028.0 0.488353
$$983$$ 978.000 0.0317328 0.0158664 0.999874i $$-0.494949\pi$$
0.0158664 + 0.999874i $$0.494949\pi$$
$$984$$ −18990.0 −0.615223
$$985$$ 1496.00 0.0483924
$$986$$ −6660.00 −0.215109
$$987$$ 39468.0 1.27283
$$988$$ −13440.0 −0.432777
$$989$$ 74256.0 2.38747
$$990$$ 0 0
$$991$$ 47272.0 1.51528 0.757641 0.652671i $$-0.226352\pi$$
0.757641 + 0.652671i $$0.226352\pi$$
$$992$$ −1288.00 −0.0412238
$$993$$ 8124.00 0.259625
$$994$$ 19812.0 0.632192
$$995$$ −8400.00 −0.267636
$$996$$ 2772.00 0.0881869
$$997$$ −51104.0 −1.62335 −0.811675 0.584109i $$-0.801444\pi$$
−0.811675 + 0.584109i $$0.801444\pi$$
$$998$$ 9020.00 0.286095
$$999$$ 1782.00 0.0564364
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 363.4.a.d.1.1 1
3.2 odd 2 1089.4.a.e.1.1 1
11.10 odd 2 33.4.a.b.1.1 1
33.32 even 2 99.4.a.a.1.1 1
44.43 even 2 528.4.a.h.1.1 1
55.32 even 4 825.4.c.f.199.1 2
55.43 even 4 825.4.c.f.199.2 2
55.54 odd 2 825.4.a.f.1.1 1
77.76 even 2 1617.4.a.d.1.1 1
88.21 odd 2 2112.4.a.u.1.1 1
88.43 even 2 2112.4.a.h.1.1 1
132.131 odd 2 1584.4.a.l.1.1 1
165.164 even 2 2475.4.a.e.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
33.4.a.b.1.1 1 11.10 odd 2
99.4.a.a.1.1 1 33.32 even 2
363.4.a.d.1.1 1 1.1 even 1 trivial
528.4.a.h.1.1 1 44.43 even 2
825.4.a.f.1.1 1 55.54 odd 2
825.4.c.f.199.1 2 55.32 even 4
825.4.c.f.199.2 2 55.43 even 4
1089.4.a.e.1.1 1 3.2 odd 2
1584.4.a.l.1.1 1 132.131 odd 2
1617.4.a.d.1.1 1 77.76 even 2
2112.4.a.h.1.1 1 88.43 even 2
2112.4.a.u.1.1 1 88.21 odd 2
2475.4.a.e.1.1 1 165.164 even 2