# Properties

 Label 33.4.a.b.1.1 Level $33$ Weight $4$ Character 33.1 Self dual yes Analytic conductor $1.947$ Analytic rank $1$ 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] = [33,4,Mod(1,33)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(33, 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("33.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$33 = 3 \cdot 11$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 33.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: $$1.94706303019$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 33.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-1.00000 q^{2} -3.00000 q^{3} -7.00000 q^{4} -4.00000 q^{5} +3.00000 q^{6} -26.0000 q^{7} +15.0000 q^{8} +9.00000 q^{9} +O(q^{10})$$ $$q-1.00000 q^{2} -3.00000 q^{3} -7.00000 q^{4} -4.00000 q^{5} +3.00000 q^{6} -26.0000 q^{7} +15.0000 q^{8} +9.00000 q^{9} +4.00000 q^{10} +11.0000 q^{11} +21.0000 q^{12} -32.0000 q^{13} +26.0000 q^{14} +12.0000 q^{15} +41.0000 q^{16} +74.0000 q^{17} -9.00000 q^{18} -60.0000 q^{19} +28.0000 q^{20} +78.0000 q^{21} -11.0000 q^{22} -182.000 q^{23} -45.0000 q^{24} -109.000 q^{25} +32.0000 q^{26} -27.0000 q^{27} +182.000 q^{28} -90.0000 q^{29} -12.0000 q^{30} -8.00000 q^{31} -161.000 q^{32} -33.0000 q^{33} -74.0000 q^{34} +104.000 q^{35} -63.0000 q^{36} -66.0000 q^{37} +60.0000 q^{38} +96.0000 q^{39} -60.0000 q^{40} +422.000 q^{41} -78.0000 q^{42} +408.000 q^{43} -77.0000 q^{44} -36.0000 q^{45} +182.000 q^{46} -506.000 q^{47} -123.000 q^{48} +333.000 q^{49} +109.000 q^{50} -222.000 q^{51} +224.000 q^{52} +348.000 q^{53} +27.0000 q^{54} -44.0000 q^{55} -390.000 q^{56} +180.000 q^{57} +90.0000 q^{58} -200.000 q^{59} -84.0000 q^{60} +132.000 q^{61} +8.00000 q^{62} -234.000 q^{63} -167.000 q^{64} +128.000 q^{65} +33.0000 q^{66} -1036.00 q^{67} -518.000 q^{68} +546.000 q^{69} -104.000 q^{70} +762.000 q^{71} +135.000 q^{72} -542.000 q^{73} +66.0000 q^{74} +327.000 q^{75} +420.000 q^{76} -286.000 q^{77} -96.0000 q^{78} -550.000 q^{79} -164.000 q^{80} +81.0000 q^{81} -422.000 q^{82} -132.000 q^{83} -546.000 q^{84} -296.000 q^{85} -408.000 q^{86} +270.000 q^{87} +165.000 q^{88} +570.000 q^{89} +36.0000 q^{90} +832.000 q^{91} +1274.00 q^{92} +24.0000 q^{93} +506.000 q^{94} +240.000 q^{95} +483.000 q^{96} +14.0000 q^{97} -333.000 q^{98} +99.0000 q^{99} +O(q^{100})$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −1.00000 −0.353553 −0.176777 0.984251i $$-0.556567\pi$$
−0.176777 + 0.984251i $$0.556567\pi$$
$$3$$ −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 $$-0.747680\pi$$
−0.701934 + 0.712242i $$0.747680\pi$$
$$8$$ 15.0000 0.662913
$$9$$ 9.00000 0.333333
$$10$$ 4.00000 0.126491
$$11$$ 11.0000 0.301511
$$12$$ 21.0000 0.505181
$$13$$ −32.0000 −0.682708 −0.341354 0.939935i $$-0.610885\pi$$
−0.341354 + 0.939935i $$0.610885\pi$$
$$14$$ 26.0000 0.496342
$$15$$ 12.0000 0.206559
$$16$$ 41.0000 0.640625
$$17$$ 74.0000 1.05574 0.527872 0.849324i $$-0.322990\pi$$
0.527872 + 0.849324i $$0.322990\pi$$
$$18$$ −9.00000 −0.117851
$$19$$ −60.0000 −0.724471 −0.362235 0.932087i $$-0.617986\pi$$
−0.362235 + 0.932087i $$0.617986\pi$$
$$20$$ 28.0000 0.313050
$$21$$ 78.0000 0.810524
$$22$$ −11.0000 −0.106600
$$23$$ −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 $$-0.593039\pi$$
−0.288148 + 0.957586i $$0.593039\pi$$
$$30$$ −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
$$33$$ −33.0000 −0.174078
$$34$$ −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 $$-0.202849\pi$$
0.803724 + 0.595003i $$0.202849\pi$$
$$42$$ −78.0000 −0.286563
$$43$$ 408.000 1.44696 0.723482 0.690344i $$-0.242541\pi$$
0.723482 + 0.690344i $$0.242541\pi$$
$$44$$ −77.0000 −0.263822
$$45$$ −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
$$55$$ −44.0000 −0.107872
$$56$$ −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 $$-0.455762\pi$$
0.138532 + 0.990358i $$0.455762\pi$$
$$62$$ 8.00000 0.0163871
$$63$$ −234.000 −0.467956
$$64$$ −167.000 −0.326172
$$65$$ 128.000 0.244253
$$66$$ 33.0000 0.0615457
$$67$$ −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 $$-0.643073\pi$$
−0.434495 + 0.900674i $$0.643073\pi$$
$$74$$ 66.0000 0.103680
$$75$$ 327.000 0.503449
$$76$$ 420.000 0.633912
$$77$$ −286.000 −0.423282
$$78$$ −96.0000 −0.139357
$$79$$ −550.000 −0.783289 −0.391645 0.920117i $$-0.628094\pi$$
−0.391645 + 0.920117i $$0.628094\pi$$
$$80$$ −164.000 −0.229197
$$81$$ 81.0000 0.111111
$$82$$ −422.000 −0.568318
$$83$$ −132.000 −0.174565 −0.0872824 0.996184i $$-0.527818\pi$$
−0.0872824 + 0.996184i $$0.527818\pi$$
$$84$$ −546.000 −0.709208
$$85$$ −296.000 −0.377714
$$86$$ −408.000 −0.511579
$$87$$ 270.000 0.332725
$$88$$ 165.000 0.199876
$$89$$ 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
$$99$$ 99.0000 0.100504
$$100$$ 763.000 0.763000
$$101$$ 1702.00 1.67679 0.838393 0.545067i $$-0.183496\pi$$
0.838393 + 0.545067i $$0.183496\pi$$
$$102$$ 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 $$-0.417995\pi$$
0.254785 + 0.966998i $$0.417995\pi$$
$$108$$ 189.000 0.168394
$$109$$ −320.000 −0.281197 −0.140598 0.990067i $$-0.544903\pi$$
−0.140598 + 0.990067i $$0.544903\pi$$
$$110$$ 44.0000 0.0381385
$$111$$ 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
$$121$$ 121.000 0.0909091
$$122$$ −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 $$-0.689607\pi$$
−0.561061 + 0.827775i $$0.689607\pi$$
$$128$$ 1455.00 1.00473
$$129$$ −1224.00 −0.835405
$$130$$ −128.000 −0.0863565
$$131$$ −1908.00 −1.27254 −0.636270 0.771466i $$-0.719524\pi$$
−0.636270 + 0.771466i $$0.719524\pi$$
$$132$$ 231.000 0.152318
$$133$$ 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 $$-0.184897\pi$$
0.835985 + 0.548753i $$0.184897\pi$$
$$140$$ −728.000 −0.439480
$$141$$ 1518.00 0.906657
$$142$$ −762.000 −0.450321
$$143$$ −352.000 −0.205844
$$144$$ 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 $$-0.617268\pi$$
−0.360132 + 0.932901i $$0.617268\pi$$
$$150$$ −327.000 −0.177996
$$151$$ −1198.00 −0.645641 −0.322821 0.946460i $$-0.604631\pi$$
−0.322821 + 0.946460i $$0.604631\pi$$
$$152$$ −900.000 −0.480261
$$153$$ 666.000 0.351914
$$154$$ 286.000 0.149653
$$155$$ 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
$$165$$ 132.000 0.0622799
$$166$$ 132.000 0.0617180
$$167$$ 2004.00 0.928588 0.464294 0.885681i $$-0.346308\pi$$
0.464294 + 0.885681i $$0.346308\pi$$
$$168$$ 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 $$-0.452401\pi$$
0.148981 + 0.988840i $$0.452401\pi$$
$$174$$ −270.000 −0.117636
$$175$$ 2834.00 1.22417
$$176$$ 451.000 0.193156
$$177$$ 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
$$187$$ 814.000 0.318319
$$188$$ 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 $$-0.764625\pi$$
−0.738837 + 0.673884i $$0.764625\pi$$
$$194$$ −14.0000 −0.00518114
$$195$$ −384.000 −0.141020
$$196$$ −2331.00 −0.849490
$$197$$ 374.000 0.135261 0.0676304 0.997710i $$-0.478456\pi$$
0.0676304 + 0.997710i $$0.478456\pi$$
$$198$$ −99.0000 −0.0355335
$$199$$ 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
$$209$$ −660.000 −0.218436
$$210$$ 312.000 0.102524
$$211$$ 2232.00 0.728233 0.364117 0.931353i $$-0.381371\pi$$
0.364117 + 0.931353i $$0.381371\pi$$
$$212$$ −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
$$220$$ 308.000 0.0943880
$$221$$ −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 $$-0.357342\pi$$
0.433321 + 0.901240i $$0.357342\pi$$
$$228$$ −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
$$231$$ 858.000 0.244382
$$232$$ −1350.00 −0.382034
$$233$$ −3042.00 −0.855314 −0.427657 0.903941i $$-0.640661\pi$$
−0.427657 + 0.903941i $$0.640661\pi$$
$$234$$ 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 $$-0.380941\pi$$
0.365373 + 0.930861i $$0.380941\pi$$
$$240$$ 492.000 0.132327
$$241$$ −578.000 −0.154491 −0.0772453 0.997012i $$-0.524612\pi$$
−0.0772453 + 0.997012i $$0.524612\pi$$
$$242$$ −121.000 −0.0321412
$$243$$ −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
$$253$$ −2002.00 −0.497489
$$254$$ 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 $$-0.670422\pi$$
−0.510182 + 0.860066i $$0.670422\pi$$
$$264$$ −495.000 −0.115398
$$265$$ −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 $$-0.761768\pi$$
−0.732759 + 0.680489i $$0.761768\pi$$
$$272$$ 3034.00 0.676336
$$273$$ −2496.00 −0.553351
$$274$$ 2186.00 0.481975
$$275$$ −1199.00 −0.262918
$$276$$ −3822.00 −0.833541
$$277$$ 124.000 0.0268969 0.0134484 0.999910i $$-0.495719\pi$$
0.0134484 + 0.999910i $$0.495719\pi$$
$$278$$ −2740.00 −0.591131
$$279$$ −72.0000 −0.0154499
$$280$$ 1560.00 0.332957
$$281$$ 3642.00 0.773180 0.386590 0.922252i $$-0.373653\pi$$
0.386590 + 0.922252i $$0.373653\pi$$
$$282$$ −1518.00 −0.320552
$$283$$ 4648.00 0.976307 0.488154 0.872758i $$-0.337671\pi$$
0.488154 + 0.872758i $$0.337671\pi$$
$$284$$ −5334.00 −1.11449
$$285$$ −720.000 −0.149646
$$286$$ 352.000 0.0727769
$$287$$ −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 $$-0.600078\pi$$
−0.309250 + 0.950981i $$0.600078\pi$$
$$294$$ 999.000 0.198173
$$295$$ 800.000 0.157891
$$296$$ −990.000 −0.194401
$$297$$ −297.000 −0.0580259
$$298$$ 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 $$-0.463110\pi$$
0.115633 + 0.993292i $$0.463110\pi$$
$$308$$ 2002.00 0.370372
$$309$$ 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
$$319$$ −990.000 −0.173760
$$320$$ 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
$$330$$ −132.000 −0.0220193
$$331$$ −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 $$-0.394288\pi$$
0.326033 + 0.945359i $$0.394288\pi$$
$$338$$ 1173.00 0.188766
$$339$$ 6426.00 1.02954
$$340$$ 2072.00 0.330500
$$341$$ −88.0000 −0.0139750
$$342$$ 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 $$-0.172092\pi$$
0.857378 + 0.514687i $$0.172092\pi$$
$$348$$ −1890.00 −0.291134
$$349$$ −3120.00 −0.478538 −0.239269 0.970953i $$-0.576908\pi$$
−0.239269 + 0.970953i $$0.576908\pi$$
$$350$$ −2834.00 −0.432810
$$351$$ 864.000 0.131387
$$352$$ −1771.00 −0.268167
$$353$$ −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 $$-0.714823\pi$$
−0.624809 + 0.780778i $$0.714823\pi$$
$$360$$ −540.000 −0.0790569
$$361$$ −3259.00 −0.475142
$$362$$ 4358.00 0.632739
$$363$$ −363.000 −0.0524864
$$364$$ −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 $$-0.513967\pi$$
−0.0438656 + 0.999037i $$0.513967\pi$$
$$374$$ −814.000 −0.112543
$$375$$ −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
$$385$$ 1144.00 0.151438
$$386$$ 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
$$396$$ −693.000 −0.0879408
$$397$$ −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
$$407$$ −726.000 −0.0884189
$$408$$ −3330.00 −0.404068
$$409$$ 6390.00 0.772531 0.386265 0.922388i $$-0.373765\pi$$
0.386265 + 0.922388i $$0.373765\pi$$
$$410$$ 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
$$418$$ 660.000 0.0772288
$$419$$ 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
$$429$$ 1056.00 0.118844
$$430$$ 1632.00 0.183028
$$431$$ −7008.00 −0.783210 −0.391605 0.920133i $$-0.628080\pi$$
−0.391605 + 0.920133i $$0.628080\pi$$
$$432$$ −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 $$-0.691883\pi$$
−0.566967 + 0.823741i $$0.691883\pi$$
$$440$$ −660.000 −0.0715097
$$441$$ 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
$$451$$ 4642.00 0.484664
$$452$$ 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 $$-0.450043\pi$$
0.156302 + 0.987709i $$0.450043\pi$$
$$458$$ 2550.00 0.260161
$$459$$ −1998.00 −0.203178
$$460$$ −5096.00 −0.516527
$$461$$ 12882.0 1.30146 0.650732 0.759308i $$-0.274462\pi$$
0.650732 + 0.759308i $$0.274462\pi$$
$$462$$ −858.000 −0.0864021
$$463$$ 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
$$473$$ 4488.00 0.436276
$$474$$ −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 $$-0.788837\pi$$
−0.787910 + 0.615790i $$0.788837\pi$$
$$480$$ −1932.00 −0.183715
$$481$$ 2112.00 0.200206
$$482$$ 578.000 0.0546207
$$483$$ −14196.0 −1.33735
$$484$$ −847.000 −0.0795455
$$485$$ −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 $$-0.742669\pi$$
−0.690636 + 0.723203i $$0.742669\pi$$
$$492$$ 8862.00 0.812052
$$493$$ −6660.00 −0.608421
$$494$$ −1920.00 −0.174868
$$495$$ −396.000 −0.0359573
$$496$$ −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 $$-0.727962\pi$$
−0.656495 + 0.754330i $$0.727962\pi$$
$$504$$ −3510.00 −0.310214
$$505$$ −6808.00 −0.599905
$$506$$ 2002.00 0.175889
$$507$$ 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
$$517$$ −5566.00 −0.473486
$$518$$ −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 $$-0.585550\pi$$
−0.265539 + 0.964100i $$0.585550\pi$$
$$524$$ 13356.0 1.11347
$$525$$ −8502.00 −0.706777
$$526$$ 4352.00 0.360753
$$527$$ −592.000 −0.0489334
$$528$$ −1353.00 −0.111518
$$529$$ 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
$$539$$ 3663.00 0.292721
$$540$$ −756.000 −0.0602464
$$541$$ −24728.0 −1.96514 −0.982569 0.185898i $$-0.940481\pi$$
−0.982569 + 0.185898i $$0.940481\pi$$
$$542$$ 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 $$-0.848860\pi$$
−0.889375 + 0.457178i $$0.848860\pi$$
$$548$$ 15302.0 1.19283
$$549$$ 1188.00 0.0923545
$$550$$ 1199.00 0.0929555
$$551$$ 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 $$-0.618017\pi$$
−0.362325 + 0.932052i $$0.618017\pi$$
$$558$$ 72.0000 0.00546237
$$559$$ −13056.0 −0.987853
$$560$$ 4264.00 0.321762
$$561$$ −2442.00 −0.183781
$$562$$ −3642.00 −0.273360
$$563$$ 12068.0 0.903385 0.451692 0.892174i $$-0.350820\pi$$
0.451692 + 0.892174i $$0.350820\pi$$
$$564$$ −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 $$-0.312377\pi$$
0.555893 + 0.831254i $$0.312377\pi$$
$$570$$ 720.000 0.0529079
$$571$$ 4412.00 0.323356 0.161678 0.986844i $$-0.448309\pi$$
0.161678 + 0.986844i $$0.448309\pi$$
$$572$$ 2464.00 0.180114
$$573$$ 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
$$583$$ 3828.00 0.271937
$$584$$ −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 $$-0.628464\pi$$
−0.392715 + 0.919660i $$0.628464\pi$$
$$594$$ 297.000 0.0205152
$$595$$ 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 $$-0.506460\pi$$
−0.0202936 + 0.999794i $$0.506460\pi$$
$$602$$ 10608.0 0.718189
$$603$$ −9324.00 −0.629689
$$604$$ 8386.00 0.564936
$$605$$ −484.000 −0.0325246
$$606$$ 5106.00 0.342272
$$607$$ −166.000 −0.0111001 −0.00555003 0.999985i $$-0.501767\pi$$
−0.00555003 + 0.999985i $$0.501767\pi$$
$$608$$ 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 $$-0.269520\pi$$
0.662442 + 0.749113i $$0.269520\pi$$
$$614$$ −1244.00 −0.0817651
$$615$$ 5064.00 0.332033
$$616$$ −4290.00 −0.280599
$$617$$ −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
$$627$$ 1980.00 0.126114
$$628$$ −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
$$638$$ 990.000 0.0614333
$$639$$ 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
$$649$$ −2200.00 −0.133062
$$650$$ −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 $$-0.456779\pi$$
0.135365 + 0.990796i $$0.456779\pi$$
$$660$$ −924.000 −0.0544949
$$661$$ 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
$$671$$ 1452.00 0.0835378
$$672$$ −12558.0 −0.720886
$$673$$ 8438.00 0.483300 0.241650 0.970363i $$-0.422311\pi$$
0.241650 + 0.970363i $$0.422311\pi$$
$$674$$ −4034.00 −0.230540
$$675$$ 2943.00 0.167816
$$676$$ 8211.00 0.467171
$$677$$ 34494.0 1.95822 0.979108 0.203341i $$-0.0651801\pi$$
0.979108 + 0.203341i $$0.0651801\pi$$
$$678$$ −6426.00 −0.363996
$$679$$ −364.000 −0.0205730
$$680$$ −4440.00 −0.250392
$$681$$ −8892.00 −0.500356
$$682$$ 88.0000 0.00494090
$$683$$ −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
$$693$$ −2574.00 −0.141094
$$694$$ −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 $$-0.555139\pi$$
−0.172360 + 0.985034i $$0.555139\pi$$
$$702$$ −864.000 −0.0464524
$$703$$ 3960.00 0.212453
$$704$$ −1837.00 −0.0983445
$$705$$ −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
$$715$$ 1408.00 0.0736451
$$716$$ 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
$$726$$ 363.000 0.0185567
$$727$$ −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 $$-0.738924\pi$$
−0.682079 + 0.731279i $$0.738924\pi$$
$$734$$ −7144.00 −0.359250
$$735$$ 3996.00 0.200537
$$736$$ 29302.0 1.46751
$$737$$ −11396.0 −0.569575
$$738$$ −3798.00 −0.189439
$$739$$ −17320.0 −0.862147 −0.431073 0.902317i $$-0.641865\pi$$
−0.431073 + 0.902317i $$0.641865\pi$$
$$740$$ −1848.00 −0.0918025
$$741$$ −5760.00 −0.285559
$$742$$ 9048.00 0.447658
$$743$$ 14588.0 0.720299 0.360149 0.932895i $$-0.382726\pi$$
0.360149 + 0.932895i $$0.382726\pi$$
$$744$$ 360.000 0.0177396
$$745$$ 5240.00 0.257690
$$746$$ 632.000 0.0310176
$$747$$ −1188.00 −0.0581883
$$748$$ −5698.00 −0.278529
$$749$$ −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
$$759$$ 6006.00 0.287225
$$760$$ 3600.00 0.171823
$$761$$ −37518.0 −1.78716 −0.893578 0.448907i $$-0.851813\pi$$
−0.893578 + 0.448907i $$0.851813\pi$$
$$762$$ −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 $$-0.632865\pi$$
−0.405392 + 0.914143i $$0.632865\pi$$
$$770$$ −1144.00 −0.0535414
$$771$$ −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
$$781$$ 8382.00 0.384035
$$782$$ 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 $$-0.569288\pi$$
−0.215960 + 0.976402i $$0.569288\pi$$
$$788$$ −2618.00 −0.118353
$$789$$ 13056.0 0.589108
$$790$$ −2200.00 −0.0990791
$$791$$ 55692.0 2.50339
$$792$$ 1485.00 0.0666252
$$793$$ −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
$$803$$ −5962.00 −0.262010
$$804$$ −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 $$-0.337632\pi$$
0.488259 + 0.872699i $$0.337632\pi$$
$$810$$ 324.000 0.0140546
$$811$$ −3368.00 −0.145828 −0.0729140 0.997338i $$-0.523230\pi$$
−0.0729140 + 0.997338i $$0.523230\pi$$
$$812$$ −16380.0 −0.707913
$$813$$ 19614.0 0.846117
$$814$$ 726.000 0.0312608
$$815$$ −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 $$-0.573295\pi$$
−0.228233 + 0.973607i $$0.573295\pi$$
$$822$$ −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
$$825$$ 3597.00 0.151796
$$826$$ −5200.00 −0.219045
$$827$$ 22924.0 0.963900 0.481950 0.876199i $$-0.339929\pi$$
0.481950 + 0.876199i $$0.339929\pi$$
$$828$$ 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
$$836$$ 4620.00 0.191132
$$837$$ 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
$$847$$ −3146.00 −0.127624
$$848$$ 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 $$-0.712868\pi$$
−0.620001 + 0.784601i $$0.712868\pi$$
$$854$$ 3432.00 0.137518
$$855$$ 2160.00 0.0863982
$$856$$ 8460.00 0.337800
$$857$$ −38906.0 −1.55076 −0.775381 0.631493i $$-0.782442\pi$$
−0.775381 + 0.631493i $$0.782442\pi$$
$$858$$ −1056.00 −0.0420178
$$859$$ −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
$$869$$ −6050.00 −0.236171
$$870$$ 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 $$-0.356008\pi$$
0.437092 + 0.899417i $$0.356008\pi$$
$$878$$ 10430.0 0.400906
$$879$$ 9306.00 0.357092
$$880$$ −1804.00 −0.0691055
$$881$$ −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 $$-0.190152\pi$$
0.826812 + 0.562478i $$0.190152\pi$$
$$888$$ 2970.00 0.112237
$$889$$ 41756.0 1.57531
$$890$$ 2280.00 0.0858717
$$891$$ 891.000 0.0335013
$$892$$ −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
$$902$$ −4642.00 −0.171354
$$903$$ 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
$$913$$ −1452.00 −0.0526333
$$914$$ −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 $$-0.434703\pi$$
0.203701 + 0.979033i $$0.434703\pi$$
$$920$$ 10920.0 0.391328
$$921$$ −3732.00 −0.133522
$$922$$ −12882.0 −0.460137
$$923$$ −24384.0 −0.869566
$$924$$ −6006.00 −0.213834
$$925$$ 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
$$935$$ −3256.00 −0.113885
$$936$$ −4320.00 −0.150859
$$937$$ −10006.0 −0.348860 −0.174430 0.984670i $$-0.555808\pi$$
−0.174430 + 0.984670i $$0.555808\pi$$
$$938$$ −26936.0 −0.937624
$$939$$ −7134.00 −0.247933
$$940$$ −14168.0 −0.491606
$$941$$ 2622.00 0.0908340 0.0454170 0.998968i $$-0.485538\pi$$
0.0454170 + 0.998968i $$0.485538\pi$$
$$942$$ 6342.00 0.219356
$$943$$ −76804.0 −2.65226
$$944$$ −8200.00 −0.282720
$$945$$ −2808.00 −0.0966606
$$946$$ −4488.00 −0.154247
$$947$$ −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 $$-0.269952\pi$$
0.661426 + 0.750011i $$0.269952\pi$$
$$954$$ −3132.00 −0.106292
$$955$$ 7112.00 0.240983
$$956$$ −18900.0 −0.639403
$$957$$ 2970.00 0.100320
$$958$$ 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 $$-0.494104\pi$$
0.0185232 + 0.999828i $$0.494104\pi$$
$$968$$ 1815.00 0.0602648
$$969$$ 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
$$979$$ 6270.00 0.204689
$$980$$ 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
$$990$$ 396.000 0.0127128
$$991$$ 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 $$-0.198556\pi$$
0.811675 + 0.584109i $$0.198556\pi$$
$$998$$ −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 33.4.a.b.1.1 1
3.2 odd 2 99.4.a.a.1.1 1
4.3 odd 2 528.4.a.h.1.1 1
5.2 odd 4 825.4.c.f.199.1 2
5.3 odd 4 825.4.c.f.199.2 2
5.4 even 2 825.4.a.f.1.1 1
7.6 odd 2 1617.4.a.d.1.1 1
8.3 odd 2 2112.4.a.h.1.1 1
8.5 even 2 2112.4.a.u.1.1 1
11.10 odd 2 363.4.a.d.1.1 1
12.11 even 2 1584.4.a.l.1.1 1
15.14 odd 2 2475.4.a.e.1.1 1
33.32 even 2 1089.4.a.e.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
33.4.a.b.1.1 1 1.1 even 1 trivial
99.4.a.a.1.1 1 3.2 odd 2
363.4.a.d.1.1 1 11.10 odd 2
528.4.a.h.1.1 1 4.3 odd 2
825.4.a.f.1.1 1 5.4 even 2
825.4.c.f.199.1 2 5.2 odd 4
825.4.c.f.199.2 2 5.3 odd 4
1089.4.a.e.1.1 1 33.32 even 2
1584.4.a.l.1.1 1 12.11 even 2
1617.4.a.d.1.1 1 7.6 odd 2
2112.4.a.h.1.1 1 8.3 odd 2
2112.4.a.u.1.1 1 8.5 even 2
2475.4.a.e.1.1 1 15.14 odd 2