# Properties

 Label 33.4.a.a.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)

 $$f(q)$$ $$=$$ $$q-5.00000 q^{2} +3.00000 q^{3} +17.0000 q^{4} -14.0000 q^{5} -15.0000 q^{6} -32.0000 q^{7} -45.0000 q^{8} +9.00000 q^{9} +O(q^{10})$$ $$q-5.00000 q^{2} +3.00000 q^{3} +17.0000 q^{4} -14.0000 q^{5} -15.0000 q^{6} -32.0000 q^{7} -45.0000 q^{8} +9.00000 q^{9} +70.0000 q^{10} -11.0000 q^{11} +51.0000 q^{12} -38.0000 q^{13} +160.000 q^{14} -42.0000 q^{15} +89.0000 q^{16} -2.00000 q^{17} -45.0000 q^{18} +72.0000 q^{19} -238.000 q^{20} -96.0000 q^{21} +55.0000 q^{22} +68.0000 q^{23} -135.000 q^{24} +71.0000 q^{25} +190.000 q^{26} +27.0000 q^{27} -544.000 q^{28} -54.0000 q^{29} +210.000 q^{30} -152.000 q^{31} -85.0000 q^{32} -33.0000 q^{33} +10.0000 q^{34} +448.000 q^{35} +153.000 q^{36} +174.000 q^{37} -360.000 q^{38} -114.000 q^{39} +630.000 q^{40} +94.0000 q^{41} +480.000 q^{42} -528.000 q^{43} -187.000 q^{44} -126.000 q^{45} -340.000 q^{46} -340.000 q^{47} +267.000 q^{48} +681.000 q^{49} -355.000 q^{50} -6.00000 q^{51} -646.000 q^{52} -438.000 q^{53} -135.000 q^{54} +154.000 q^{55} +1440.00 q^{56} +216.000 q^{57} +270.000 q^{58} +20.0000 q^{59} -714.000 q^{60} +570.000 q^{61} +760.000 q^{62} -288.000 q^{63} -287.000 q^{64} +532.000 q^{65} +165.000 q^{66} -460.000 q^{67} -34.0000 q^{68} +204.000 q^{69} -2240.00 q^{70} -1092.00 q^{71} -405.000 q^{72} +562.000 q^{73} -870.000 q^{74} +213.000 q^{75} +1224.00 q^{76} +352.000 q^{77} +570.000 q^{78} -16.0000 q^{79} -1246.00 q^{80} +81.0000 q^{81} -470.000 q^{82} +372.000 q^{83} -1632.00 q^{84} +28.0000 q^{85} +2640.00 q^{86} -162.000 q^{87} +495.000 q^{88} -966.000 q^{89} +630.000 q^{90} +1216.00 q^{91} +1156.00 q^{92} -456.000 q^{93} +1700.00 q^{94} -1008.00 q^{95} -255.000 q^{96} -526.000 q^{97} -3405.00 q^{98} -99.0000 q^{99} +O(q^{100})$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −5.00000 −1.76777 −0.883883 0.467707i $$-0.845080\pi$$
−0.883883 + 0.467707i $$0.845080\pi$$
$$3$$ 3.00000 0.577350
$$4$$ 17.0000 2.12500
$$5$$ −14.0000 −1.25220 −0.626099 0.779744i $$-0.715349\pi$$
−0.626099 + 0.779744i $$0.715349\pi$$
$$6$$ −15.0000 −1.02062
$$7$$ −32.0000 −1.72784 −0.863919 0.503631i $$-0.831997\pi$$
−0.863919 + 0.503631i $$0.831997\pi$$
$$8$$ −45.0000 −1.98874
$$9$$ 9.00000 0.333333
$$10$$ 70.0000 2.21359
$$11$$ −11.0000 −0.301511
$$12$$ 51.0000 1.22687
$$13$$ −38.0000 −0.810716 −0.405358 0.914158i $$-0.632853\pi$$
−0.405358 + 0.914158i $$0.632853\pi$$
$$14$$ 160.000 3.05441
$$15$$ −42.0000 −0.722957
$$16$$ 89.0000 1.39062
$$17$$ −2.00000 −0.0285336 −0.0142668 0.999898i $$-0.504541\pi$$
−0.0142668 + 0.999898i $$0.504541\pi$$
$$18$$ −45.0000 −0.589256
$$19$$ 72.0000 0.869365 0.434682 0.900584i $$-0.356861\pi$$
0.434682 + 0.900584i $$0.356861\pi$$
$$20$$ −238.000 −2.66092
$$21$$ −96.0000 −0.997567
$$22$$ 55.0000 0.533002
$$23$$ 68.0000 0.616477 0.308239 0.951309i $$-0.400260\pi$$
0.308239 + 0.951309i $$0.400260\pi$$
$$24$$ −135.000 −1.14820
$$25$$ 71.0000 0.568000
$$26$$ 190.000 1.43316
$$27$$ 27.0000 0.192450
$$28$$ −544.000 −3.67165
$$29$$ −54.0000 −0.345778 −0.172889 0.984941i $$-0.555310\pi$$
−0.172889 + 0.984941i $$0.555310\pi$$
$$30$$ 210.000 1.27802
$$31$$ −152.000 −0.880645 −0.440323 0.897840i $$-0.645136\pi$$
−0.440323 + 0.897840i $$0.645136\pi$$
$$32$$ −85.0000 −0.469563
$$33$$ −33.0000 −0.174078
$$34$$ 10.0000 0.0504408
$$35$$ 448.000 2.16359
$$36$$ 153.000 0.708333
$$37$$ 174.000 0.773120 0.386560 0.922264i $$-0.373663\pi$$
0.386560 + 0.922264i $$0.373663\pi$$
$$38$$ −360.000 −1.53683
$$39$$ −114.000 −0.468067
$$40$$ 630.000 2.49029
$$41$$ 94.0000 0.358057 0.179028 0.983844i $$-0.442705\pi$$
0.179028 + 0.983844i $$0.442705\pi$$
$$42$$ 480.000 1.76347
$$43$$ −528.000 −1.87254 −0.936270 0.351280i $$-0.885746\pi$$
−0.936270 + 0.351280i $$0.885746\pi$$
$$44$$ −187.000 −0.640712
$$45$$ −126.000 −0.417399
$$46$$ −340.000 −1.08979
$$47$$ −340.000 −1.05519 −0.527597 0.849495i $$-0.676907\pi$$
−0.527597 + 0.849495i $$0.676907\pi$$
$$48$$ 267.000 0.802878
$$49$$ 681.000 1.98542
$$50$$ −355.000 −1.00409
$$51$$ −6.00000 −0.0164739
$$52$$ −646.000 −1.72277
$$53$$ −438.000 −1.13517 −0.567584 0.823315i $$-0.692122\pi$$
−0.567584 + 0.823315i $$0.692122\pi$$
$$54$$ −135.000 −0.340207
$$55$$ 154.000 0.377552
$$56$$ 1440.00 3.43622
$$57$$ 216.000 0.501928
$$58$$ 270.000 0.611254
$$59$$ 20.0000 0.0441318 0.0220659 0.999757i $$-0.492976\pi$$
0.0220659 + 0.999757i $$0.492976\pi$$
$$60$$ −714.000 −1.53628
$$61$$ 570.000 1.19641 0.598205 0.801343i $$-0.295881\pi$$
0.598205 + 0.801343i $$0.295881\pi$$
$$62$$ 760.000 1.55678
$$63$$ −288.000 −0.575946
$$64$$ −287.000 −0.560547
$$65$$ 532.000 1.01518
$$66$$ 165.000 0.307729
$$67$$ −460.000 −0.838775 −0.419388 0.907807i $$-0.637755\pi$$
−0.419388 + 0.907807i $$0.637755\pi$$
$$68$$ −34.0000 −0.0606339
$$69$$ 204.000 0.355923
$$70$$ −2240.00 −3.82473
$$71$$ −1092.00 −1.82530 −0.912652 0.408738i $$-0.865969\pi$$
−0.912652 + 0.408738i $$0.865969\pi$$
$$72$$ −405.000 −0.662913
$$73$$ 562.000 0.901057 0.450528 0.892762i $$-0.351236\pi$$
0.450528 + 0.892762i $$0.351236\pi$$
$$74$$ −870.000 −1.36670
$$75$$ 213.000 0.327935
$$76$$ 1224.00 1.84740
$$77$$ 352.000 0.520963
$$78$$ 570.000 0.827433
$$79$$ −16.0000 −0.0227866 −0.0113933 0.999935i $$-0.503627\pi$$
−0.0113933 + 0.999935i $$0.503627\pi$$
$$80$$ −1246.00 −1.74134
$$81$$ 81.0000 0.111111
$$82$$ −470.000 −0.632961
$$83$$ 372.000 0.491955 0.245978 0.969275i $$-0.420891\pi$$
0.245978 + 0.969275i $$0.420891\pi$$
$$84$$ −1632.00 −2.11983
$$85$$ 28.0000 0.0357297
$$86$$ 2640.00 3.31022
$$87$$ −162.000 −0.199635
$$88$$ 495.000 0.599627
$$89$$ −966.000 −1.15051 −0.575257 0.817973i $$-0.695098\pi$$
−0.575257 + 0.817973i $$0.695098\pi$$
$$90$$ 630.000 0.737865
$$91$$ 1216.00 1.40079
$$92$$ 1156.00 1.31001
$$93$$ −456.000 −0.508441
$$94$$ 1700.00 1.86534
$$95$$ −1008.00 −1.08862
$$96$$ −255.000 −0.271102
$$97$$ −526.000 −0.550590 −0.275295 0.961360i $$-0.588775\pi$$
−0.275295 + 0.961360i $$0.588775\pi$$
$$98$$ −3405.00 −3.50976
$$99$$ −99.0000 −0.100504
$$100$$ 1207.00 1.20700
$$101$$ 50.0000 0.0492593 0.0246296 0.999697i $$-0.492159\pi$$
0.0246296 + 0.999697i $$0.492159\pi$$
$$102$$ 30.0000 0.0291220
$$103$$ 944.000 0.903059 0.451530 0.892256i $$-0.350879\pi$$
0.451530 + 0.892256i $$0.350879\pi$$
$$104$$ 1710.00 1.61230
$$105$$ 1344.00 1.24915
$$106$$ 2190.00 2.00671
$$107$$ 468.000 0.422834 0.211417 0.977396i $$-0.432192\pi$$
0.211417 + 0.977396i $$0.432192\pi$$
$$108$$ 459.000 0.408956
$$109$$ 154.000 0.135326 0.0676630 0.997708i $$-0.478446\pi$$
0.0676630 + 0.997708i $$0.478446\pi$$
$$110$$ −770.000 −0.667424
$$111$$ 522.000 0.446361
$$112$$ −2848.00 −2.40277
$$113$$ −54.0000 −0.0449548 −0.0224774 0.999747i $$-0.507155\pi$$
−0.0224774 + 0.999747i $$0.507155\pi$$
$$114$$ −1080.00 −0.887292
$$115$$ −952.000 −0.771952
$$116$$ −918.000 −0.734777
$$117$$ −342.000 −0.270239
$$118$$ −100.000 −0.0780148
$$119$$ 64.0000 0.0493014
$$120$$ 1890.00 1.43777
$$121$$ 121.000 0.0909091
$$122$$ −2850.00 −2.11497
$$123$$ 282.000 0.206724
$$124$$ −2584.00 −1.87137
$$125$$ 756.000 0.540950
$$126$$ 1440.00 1.01814
$$127$$ −2224.00 −1.55392 −0.776961 0.629549i $$-0.783240\pi$$
−0.776961 + 0.629549i $$0.783240\pi$$
$$128$$ 2115.00 1.46048
$$129$$ −1584.00 −1.08111
$$130$$ −2660.00 −1.79460
$$131$$ −2772.00 −1.84878 −0.924392 0.381443i $$-0.875427\pi$$
−0.924392 + 0.381443i $$0.875427\pi$$
$$132$$ −561.000 −0.369915
$$133$$ −2304.00 −1.50212
$$134$$ 2300.00 1.48276
$$135$$ −378.000 −0.240986
$$136$$ 90.0000 0.0567459
$$137$$ 1130.00 0.704689 0.352345 0.935870i $$-0.385385\pi$$
0.352345 + 0.935870i $$0.385385\pi$$
$$138$$ −1020.00 −0.629190
$$139$$ −1616.00 −0.986096 −0.493048 0.870002i $$-0.664117\pi$$
−0.493048 + 0.870002i $$0.664117\pi$$
$$140$$ 7616.00 4.59764
$$141$$ −1020.00 −0.609216
$$142$$ 5460.00 3.22671
$$143$$ 418.000 0.244440
$$144$$ 801.000 0.463542
$$145$$ 756.000 0.432982
$$146$$ −2810.00 −1.59286
$$147$$ 2043.00 1.14628
$$148$$ 2958.00 1.64288
$$149$$ 2066.00 1.13593 0.567964 0.823053i $$-0.307731\pi$$
0.567964 + 0.823053i $$0.307731\pi$$
$$150$$ −1065.00 −0.579713
$$151$$ 248.000 0.133655 0.0668277 0.997765i $$-0.478712\pi$$
0.0668277 + 0.997765i $$0.478712\pi$$
$$152$$ −3240.00 −1.72894
$$153$$ −18.0000 −0.00951120
$$154$$ −1760.00 −0.920941
$$155$$ 2128.00 1.10274
$$156$$ −1938.00 −0.994642
$$157$$ 2366.00 1.20272 0.601361 0.798977i $$-0.294625\pi$$
0.601361 + 0.798977i $$0.294625\pi$$
$$158$$ 80.0000 0.0402814
$$159$$ −1314.00 −0.655390
$$160$$ 1190.00 0.587986
$$161$$ −2176.00 −1.06517
$$162$$ −405.000 −0.196419
$$163$$ −284.000 −0.136470 −0.0682350 0.997669i $$-0.521737\pi$$
−0.0682350 + 0.997669i $$0.521737\pi$$
$$164$$ 1598.00 0.760871
$$165$$ 462.000 0.217980
$$166$$ −1860.00 −0.869663
$$167$$ 600.000 0.278020 0.139010 0.990291i $$-0.455608\pi$$
0.139010 + 0.990291i $$0.455608\pi$$
$$168$$ 4320.00 1.98390
$$169$$ −753.000 −0.342740
$$170$$ −140.000 −0.0631618
$$171$$ 648.000 0.289788
$$172$$ −8976.00 −3.97915
$$173$$ 138.000 0.0606471 0.0303235 0.999540i $$-0.490346\pi$$
0.0303235 + 0.999540i $$0.490346\pi$$
$$174$$ 810.000 0.352908
$$175$$ −2272.00 −0.981412
$$176$$ −979.000 −0.419289
$$177$$ 60.0000 0.0254795
$$178$$ 4830.00 2.03384
$$179$$ 3972.00 1.65855 0.829277 0.558838i $$-0.188752\pi$$
0.829277 + 0.558838i $$0.188752\pi$$
$$180$$ −2142.00 −0.886974
$$181$$ 2230.00 0.915771 0.457886 0.889011i $$-0.348607\pi$$
0.457886 + 0.889011i $$0.348607\pi$$
$$182$$ −6080.00 −2.47626
$$183$$ 1710.00 0.690748
$$184$$ −3060.00 −1.22601
$$185$$ −2436.00 −0.968099
$$186$$ 2280.00 0.898805
$$187$$ 22.0000 0.00860320
$$188$$ −5780.00 −2.24229
$$189$$ −864.000 −0.332522
$$190$$ 5040.00 1.92442
$$191$$ −772.000 −0.292461 −0.146230 0.989251i $$-0.546714\pi$$
−0.146230 + 0.989251i $$0.546714\pi$$
$$192$$ −861.000 −0.323632
$$193$$ 394.000 0.146947 0.0734734 0.997297i $$-0.476592\pi$$
0.0734734 + 0.997297i $$0.476592\pi$$
$$194$$ 2630.00 0.973314
$$195$$ 1596.00 0.586112
$$196$$ 11577.0 4.21902
$$197$$ 3058.00 1.10596 0.552978 0.833196i $$-0.313491\pi$$
0.552978 + 0.833196i $$0.313491\pi$$
$$198$$ 495.000 0.177667
$$199$$ 2664.00 0.948975 0.474487 0.880262i $$-0.342633\pi$$
0.474487 + 0.880262i $$0.342633\pi$$
$$200$$ −3195.00 −1.12960
$$201$$ −1380.00 −0.484267
$$202$$ −250.000 −0.0870789
$$203$$ 1728.00 0.597447
$$204$$ −102.000 −0.0350070
$$205$$ −1316.00 −0.448358
$$206$$ −4720.00 −1.59640
$$207$$ 612.000 0.205492
$$208$$ −3382.00 −1.12740
$$209$$ −792.000 −0.262123
$$210$$ −6720.00 −2.20821
$$211$$ −6000.00 −1.95762 −0.978808 0.204779i $$-0.934352\pi$$
−0.978808 + 0.204779i $$0.934352\pi$$
$$212$$ −7446.00 −2.41223
$$213$$ −3276.00 −1.05384
$$214$$ −2340.00 −0.747472
$$215$$ 7392.00 2.34479
$$216$$ −1215.00 −0.382733
$$217$$ 4864.00 1.52161
$$218$$ −770.000 −0.239225
$$219$$ 1686.00 0.520225
$$220$$ 2618.00 0.802298
$$221$$ 76.0000 0.0231326
$$222$$ −2610.00 −0.789062
$$223$$ −560.000 −0.168163 −0.0840816 0.996459i $$-0.526796\pi$$
−0.0840816 + 0.996459i $$0.526796\pi$$
$$224$$ 2720.00 0.811329
$$225$$ 639.000 0.189333
$$226$$ 270.000 0.0794696
$$227$$ 5292.00 1.54732 0.773662 0.633599i $$-0.218423\pi$$
0.773662 + 0.633599i $$0.218423\pi$$
$$228$$ 3672.00 1.06660
$$229$$ −5322.00 −1.53575 −0.767877 0.640597i $$-0.778687\pi$$
−0.767877 + 0.640597i $$0.778687\pi$$
$$230$$ 4760.00 1.36463
$$231$$ 1056.00 0.300778
$$232$$ 2430.00 0.687661
$$233$$ −3954.00 −1.11174 −0.555869 0.831270i $$-0.687615\pi$$
−0.555869 + 0.831270i $$0.687615\pi$$
$$234$$ 1710.00 0.477719
$$235$$ 4760.00 1.32131
$$236$$ 340.000 0.0937801
$$237$$ −48.0000 −0.0131558
$$238$$ −320.000 −0.0871534
$$239$$ −3360.00 −0.909374 −0.454687 0.890651i $$-0.650249\pi$$
−0.454687 + 0.890651i $$0.650249\pi$$
$$240$$ −3738.00 −1.00536
$$241$$ −3278.00 −0.876160 −0.438080 0.898936i $$-0.644341\pi$$
−0.438080 + 0.898936i $$0.644341\pi$$
$$242$$ −605.000 −0.160706
$$243$$ 243.000 0.0641500
$$244$$ 9690.00 2.54237
$$245$$ −9534.00 −2.48614
$$246$$ −1410.00 −0.365440
$$247$$ −2736.00 −0.704808
$$248$$ 6840.00 1.75137
$$249$$ 1116.00 0.284031
$$250$$ −3780.00 −0.956273
$$251$$ 2092.00 0.526079 0.263040 0.964785i $$-0.415275\pi$$
0.263040 + 0.964785i $$0.415275\pi$$
$$252$$ −4896.00 −1.22388
$$253$$ −748.000 −0.185875
$$254$$ 11120.0 2.74697
$$255$$ 84.0000 0.0206286
$$256$$ −8279.00 −2.02124
$$257$$ 658.000 0.159708 0.0798539 0.996807i $$-0.474555\pi$$
0.0798539 + 0.996807i $$0.474555\pi$$
$$258$$ 7920.00 1.91115
$$259$$ −5568.00 −1.33583
$$260$$ 9044.00 2.15725
$$261$$ −486.000 −0.115259
$$262$$ 13860.0 3.26822
$$263$$ −5104.00 −1.19668 −0.598339 0.801243i $$-0.704172\pi$$
−0.598339 + 0.801243i $$0.704172\pi$$
$$264$$ 1485.00 0.346195
$$265$$ 6132.00 1.42146
$$266$$ 11520.0 2.65540
$$267$$ −2898.00 −0.664250
$$268$$ −7820.00 −1.78240
$$269$$ −4238.00 −0.960578 −0.480289 0.877110i $$-0.659468\pi$$
−0.480289 + 0.877110i $$0.659468\pi$$
$$270$$ 1890.00 0.426006
$$271$$ −3376.00 −0.756743 −0.378372 0.925654i $$-0.623516\pi$$
−0.378372 + 0.925654i $$0.623516\pi$$
$$272$$ −178.000 −0.0396795
$$273$$ 3648.00 0.808744
$$274$$ −5650.00 −1.24573
$$275$$ −781.000 −0.171258
$$276$$ 3468.00 0.756337
$$277$$ 2074.00 0.449872 0.224936 0.974374i $$-0.427783\pi$$
0.224936 + 0.974374i $$0.427783\pi$$
$$278$$ 8080.00 1.74319
$$279$$ −1368.00 −0.293548
$$280$$ −20160.0 −4.30282
$$281$$ 702.000 0.149031 0.0745157 0.997220i $$-0.476259\pi$$
0.0745157 + 0.997220i $$0.476259\pi$$
$$282$$ 5100.00 1.07695
$$283$$ 4912.00 1.03176 0.515880 0.856661i $$-0.327465\pi$$
0.515880 + 0.856661i $$0.327465\pi$$
$$284$$ −18564.0 −3.87877
$$285$$ −3024.00 −0.628513
$$286$$ −2090.00 −0.432113
$$287$$ −3008.00 −0.618664
$$288$$ −765.000 −0.156521
$$289$$ −4909.00 −0.999186
$$290$$ −3780.00 −0.765411
$$291$$ −1578.00 −0.317883
$$292$$ 9554.00 1.91475
$$293$$ −3486.00 −0.695066 −0.347533 0.937668i $$-0.612981\pi$$
−0.347533 + 0.937668i $$0.612981\pi$$
$$294$$ −10215.0 −2.02636
$$295$$ −280.000 −0.0552618
$$296$$ −7830.00 −1.53753
$$297$$ −297.000 −0.0580259
$$298$$ −10330.0 −2.00806
$$299$$ −2584.00 −0.499788
$$300$$ 3621.00 0.696862
$$301$$ 16896.0 3.23545
$$302$$ −1240.00 −0.236271
$$303$$ 150.000 0.0284399
$$304$$ 6408.00 1.20896
$$305$$ −7980.00 −1.49814
$$306$$ 90.0000 0.0168136
$$307$$ 8360.00 1.55417 0.777085 0.629395i $$-0.216697\pi$$
0.777085 + 0.629395i $$0.216697\pi$$
$$308$$ 5984.00 1.10705
$$309$$ 2832.00 0.521381
$$310$$ −10640.0 −1.94939
$$311$$ −5532.00 −1.00865 −0.504326 0.863513i $$-0.668259\pi$$
−0.504326 + 0.863513i $$0.668259\pi$$
$$312$$ 5130.00 0.930862
$$313$$ 4826.00 0.871507 0.435753 0.900066i $$-0.356482\pi$$
0.435753 + 0.900066i $$0.356482\pi$$
$$314$$ −11830.0 −2.12613
$$315$$ 4032.00 0.721198
$$316$$ −272.000 −0.0484215
$$317$$ 7570.00 1.34124 0.670621 0.741800i $$-0.266028\pi$$
0.670621 + 0.741800i $$0.266028\pi$$
$$318$$ 6570.00 1.15858
$$319$$ 594.000 0.104256
$$320$$ 4018.00 0.701916
$$321$$ 1404.00 0.244123
$$322$$ 10880.0 1.88298
$$323$$ −144.000 −0.0248061
$$324$$ 1377.00 0.236111
$$325$$ −2698.00 −0.460487
$$326$$ 1420.00 0.241247
$$327$$ 462.000 0.0781305
$$328$$ −4230.00 −0.712081
$$329$$ 10880.0 1.82320
$$330$$ −2310.00 −0.385337
$$331$$ 3676.00 0.610427 0.305213 0.952284i $$-0.401272\pi$$
0.305213 + 0.952284i $$0.401272\pi$$
$$332$$ 6324.00 1.04541
$$333$$ 1566.00 0.257707
$$334$$ −3000.00 −0.491475
$$335$$ 6440.00 1.05031
$$336$$ −8544.00 −1.38724
$$337$$ −5686.00 −0.919098 −0.459549 0.888152i $$-0.651989\pi$$
−0.459549 + 0.888152i $$0.651989\pi$$
$$338$$ 3765.00 0.605885
$$339$$ −162.000 −0.0259547
$$340$$ 476.000 0.0759257
$$341$$ 1672.00 0.265525
$$342$$ −3240.00 −0.512278
$$343$$ −10816.0 −1.70265
$$344$$ 23760.0 3.72399
$$345$$ −2856.00 −0.445687
$$346$$ −690.000 −0.107210
$$347$$ −1652.00 −0.255574 −0.127787 0.991802i $$-0.540787\pi$$
−0.127787 + 0.991802i $$0.540787\pi$$
$$348$$ −2754.00 −0.424224
$$349$$ −6990.00 −1.07211 −0.536055 0.844183i $$-0.680086\pi$$
−0.536055 + 0.844183i $$0.680086\pi$$
$$350$$ 11360.0 1.73491
$$351$$ −1026.00 −0.156022
$$352$$ 935.000 0.141579
$$353$$ −8094.00 −1.22040 −0.610199 0.792249i $$-0.708910\pi$$
−0.610199 + 0.792249i $$0.708910\pi$$
$$354$$ −300.000 −0.0450419
$$355$$ 15288.0 2.28564
$$356$$ −16422.0 −2.44484
$$357$$ 192.000 0.0284642
$$358$$ −19860.0 −2.93194
$$359$$ 1024.00 0.150542 0.0752711 0.997163i $$-0.476018\pi$$
0.0752711 + 0.997163i $$0.476018\pi$$
$$360$$ 5670.00 0.830098
$$361$$ −1675.00 −0.244205
$$362$$ −11150.0 −1.61887
$$363$$ 363.000 0.0524864
$$364$$ 20672.0 2.97667
$$365$$ −7868.00 −1.12830
$$366$$ −8550.00 −1.22108
$$367$$ −13664.0 −1.94347 −0.971737 0.236066i $$-0.924142\pi$$
−0.971737 + 0.236066i $$0.924142\pi$$
$$368$$ 6052.00 0.857289
$$369$$ 846.000 0.119352
$$370$$ 12180.0 1.71137
$$371$$ 14016.0 1.96139
$$372$$ −7752.00 −1.08044
$$373$$ −1958.00 −0.271800 −0.135900 0.990723i $$-0.543393\pi$$
−0.135900 + 0.990723i $$0.543393\pi$$
$$374$$ −110.000 −0.0152085
$$375$$ 2268.00 0.312317
$$376$$ 15300.0 2.09850
$$377$$ 2052.00 0.280327
$$378$$ 4320.00 0.587822
$$379$$ 6124.00 0.829997 0.414998 0.909822i $$-0.363782\pi$$
0.414998 + 0.909822i $$0.363782\pi$$
$$380$$ −17136.0 −2.31331
$$381$$ −6672.00 −0.897157
$$382$$ 3860.00 0.517002
$$383$$ 5612.00 0.748720 0.374360 0.927283i $$-0.377862\pi$$
0.374360 + 0.927283i $$0.377862\pi$$
$$384$$ 6345.00 0.843208
$$385$$ −4928.00 −0.652348
$$386$$ −1970.00 −0.259768
$$387$$ −4752.00 −0.624180
$$388$$ −8942.00 −1.17000
$$389$$ 12450.0 1.62273 0.811363 0.584543i $$-0.198726\pi$$
0.811363 + 0.584543i $$0.198726\pi$$
$$390$$ −7980.00 −1.03611
$$391$$ −136.000 −0.0175903
$$392$$ −30645.0 −3.94849
$$393$$ −8316.00 −1.06740
$$394$$ −15290.0 −1.95507
$$395$$ 224.000 0.0285333
$$396$$ −1683.00 −0.213571
$$397$$ 14830.0 1.87480 0.937401 0.348252i $$-0.113225\pi$$
0.937401 + 0.348252i $$0.113225\pi$$
$$398$$ −13320.0 −1.67757
$$399$$ −6912.00 −0.867250
$$400$$ 6319.00 0.789875
$$401$$ −3358.00 −0.418181 −0.209090 0.977896i $$-0.567050\pi$$
−0.209090 + 0.977896i $$0.567050\pi$$
$$402$$ 6900.00 0.856071
$$403$$ 5776.00 0.713953
$$404$$ 850.000 0.104676
$$405$$ −1134.00 −0.139133
$$406$$ −8640.00 −1.05615
$$407$$ −1914.00 −0.233104
$$408$$ 270.000 0.0327622
$$409$$ 10698.0 1.29335 0.646677 0.762764i $$-0.276158\pi$$
0.646677 + 0.762764i $$0.276158\pi$$
$$410$$ 6580.00 0.792593
$$411$$ 3390.00 0.406852
$$412$$ 16048.0 1.91900
$$413$$ −640.000 −0.0762526
$$414$$ −3060.00 −0.363263
$$415$$ −5208.00 −0.616026
$$416$$ 3230.00 0.380682
$$417$$ −4848.00 −0.569323
$$418$$ 3960.00 0.463373
$$419$$ −2044.00 −0.238320 −0.119160 0.992875i $$-0.538020\pi$$
−0.119160 + 0.992875i $$0.538020\pi$$
$$420$$ 22848.0 2.65445
$$421$$ 3070.00 0.355398 0.177699 0.984085i $$-0.443135\pi$$
0.177699 + 0.984085i $$0.443135\pi$$
$$422$$ 30000.0 3.46061
$$423$$ −3060.00 −0.351731
$$424$$ 19710.0 2.25755
$$425$$ −142.000 −0.0162071
$$426$$ 16380.0 1.86294
$$427$$ −18240.0 −2.06720
$$428$$ 7956.00 0.898523
$$429$$ 1254.00 0.141127
$$430$$ −36960.0 −4.14505
$$431$$ −12600.0 −1.40817 −0.704084 0.710116i $$-0.748642\pi$$
−0.704084 + 0.710116i $$0.748642\pi$$
$$432$$ 2403.00 0.267626
$$433$$ −9902.00 −1.09898 −0.549492 0.835499i $$-0.685179\pi$$
−0.549492 + 0.835499i $$0.685179\pi$$
$$434$$ −24320.0 −2.68986
$$435$$ 2268.00 0.249982
$$436$$ 2618.00 0.287568
$$437$$ 4896.00 0.535944
$$438$$ −8430.00 −0.919637
$$439$$ 11440.0 1.24374 0.621869 0.783121i $$-0.286373\pi$$
0.621869 + 0.783121i $$0.286373\pi$$
$$440$$ −6930.00 −0.750852
$$441$$ 6129.00 0.661808
$$442$$ −380.000 −0.0408931
$$443$$ −5180.00 −0.555551 −0.277776 0.960646i $$-0.589597\pi$$
−0.277776 + 0.960646i $$0.589597\pi$$
$$444$$ 8874.00 0.948517
$$445$$ 13524.0 1.44067
$$446$$ 2800.00 0.297273
$$447$$ 6198.00 0.655829
$$448$$ 9184.00 0.968534
$$449$$ 10826.0 1.13789 0.568943 0.822377i $$-0.307353\pi$$
0.568943 + 0.822377i $$0.307353\pi$$
$$450$$ −3195.00 −0.334697
$$451$$ −1034.00 −0.107958
$$452$$ −918.000 −0.0955290
$$453$$ 744.000 0.0771659
$$454$$ −26460.0 −2.73531
$$455$$ −17024.0 −1.75406
$$456$$ −9720.00 −0.998203
$$457$$ −15798.0 −1.61707 −0.808533 0.588451i $$-0.799738\pi$$
−0.808533 + 0.588451i $$0.799738\pi$$
$$458$$ 26610.0 2.71486
$$459$$ −54.0000 −0.00549129
$$460$$ −16184.0 −1.64040
$$461$$ −3894.00 −0.393409 −0.196705 0.980463i $$-0.563024\pi$$
−0.196705 + 0.980463i $$0.563024\pi$$
$$462$$ −5280.00 −0.531705
$$463$$ −15992.0 −1.60521 −0.802604 0.596512i $$-0.796553\pi$$
−0.802604 + 0.596512i $$0.796553\pi$$
$$464$$ −4806.00 −0.480847
$$465$$ 6384.00 0.636669
$$466$$ 19770.0 1.96530
$$467$$ 11844.0 1.17361 0.586804 0.809729i $$-0.300386\pi$$
0.586804 + 0.809729i $$0.300386\pi$$
$$468$$ −5814.00 −0.574257
$$469$$ 14720.0 1.44927
$$470$$ −23800.0 −2.33577
$$471$$ 7098.00 0.694392
$$472$$ −900.000 −0.0877666
$$473$$ 5808.00 0.564592
$$474$$ 240.000 0.0232565
$$475$$ 5112.00 0.493799
$$476$$ 1088.00 0.104766
$$477$$ −3942.00 −0.378389
$$478$$ 16800.0 1.60756
$$479$$ 14936.0 1.42472 0.712362 0.701812i $$-0.247625\pi$$
0.712362 + 0.701812i $$0.247625\pi$$
$$480$$ 3570.00 0.339474
$$481$$ −6612.00 −0.626780
$$482$$ 16390.0 1.54885
$$483$$ −6528.00 −0.614978
$$484$$ 2057.00 0.193182
$$485$$ 7364.00 0.689447
$$486$$ −1215.00 −0.113402
$$487$$ −2056.00 −0.191306 −0.0956532 0.995415i $$-0.530494\pi$$
−0.0956532 + 0.995415i $$0.530494\pi$$
$$488$$ −25650.0 −2.37935
$$489$$ −852.000 −0.0787909
$$490$$ 47670.0 4.39492
$$491$$ −17852.0 −1.64083 −0.820417 0.571766i $$-0.806259\pi$$
−0.820417 + 0.571766i $$0.806259\pi$$
$$492$$ 4794.00 0.439289
$$493$$ 108.000 0.00986628
$$494$$ 13680.0 1.24594
$$495$$ 1386.00 0.125851
$$496$$ −13528.0 −1.22465
$$497$$ 34944.0 3.15383
$$498$$ −5580.00 −0.502100
$$499$$ 4508.00 0.404420 0.202210 0.979342i $$-0.435188\pi$$
0.202210 + 0.979342i $$0.435188\pi$$
$$500$$ 12852.0 1.14952
$$501$$ 1800.00 0.160515
$$502$$ −10460.0 −0.929985
$$503$$ −5912.00 −0.524062 −0.262031 0.965059i $$-0.584392\pi$$
−0.262031 + 0.965059i $$0.584392\pi$$
$$504$$ 12960.0 1.14541
$$505$$ −700.000 −0.0616824
$$506$$ 3740.00 0.328584
$$507$$ −2259.00 −0.197881
$$508$$ −37808.0 −3.30208
$$509$$ −11406.0 −0.993246 −0.496623 0.867966i $$-0.665427\pi$$
−0.496623 + 0.867966i $$0.665427\pi$$
$$510$$ −420.000 −0.0364665
$$511$$ −17984.0 −1.55688
$$512$$ 24475.0 2.11260
$$513$$ 1944.00 0.167309
$$514$$ −3290.00 −0.282326
$$515$$ −13216.0 −1.13081
$$516$$ −26928.0 −2.29736
$$517$$ 3740.00 0.318153
$$518$$ 27840.0 2.36143
$$519$$ 414.000 0.0350146
$$520$$ −23940.0 −2.01892
$$521$$ −1542.00 −0.129667 −0.0648333 0.997896i $$-0.520652\pi$$
−0.0648333 + 0.997896i $$0.520652\pi$$
$$522$$ 2430.00 0.203751
$$523$$ −7504.00 −0.627394 −0.313697 0.949523i $$-0.601568\pi$$
−0.313697 + 0.949523i $$0.601568\pi$$
$$524$$ −47124.0 −3.92867
$$525$$ −6816.00 −0.566618
$$526$$ 25520.0 2.11545
$$527$$ 304.000 0.0251280
$$528$$ −2937.00 −0.242077
$$529$$ −7543.00 −0.619956
$$530$$ −30660.0 −2.51280
$$531$$ 180.000 0.0147106
$$532$$ −39168.0 −3.19201
$$533$$ −3572.00 −0.290282
$$534$$ 14490.0 1.17424
$$535$$ −6552.00 −0.529472
$$536$$ 20700.0 1.66810
$$537$$ 11916.0 0.957567
$$538$$ 21190.0 1.69808
$$539$$ −7491.00 −0.598627
$$540$$ −6426.00 −0.512094
$$541$$ 1018.00 0.0809006 0.0404503 0.999182i $$-0.487121\pi$$
0.0404503 + 0.999182i $$0.487121\pi$$
$$542$$ 16880.0 1.33775
$$543$$ 6690.00 0.528721
$$544$$ 170.000 0.0133983
$$545$$ −2156.00 −0.169455
$$546$$ −18240.0 −1.42967
$$547$$ 7904.00 0.617826 0.308913 0.951090i $$-0.400035\pi$$
0.308913 + 0.951090i $$0.400035\pi$$
$$548$$ 19210.0 1.49746
$$549$$ 5130.00 0.398803
$$550$$ 3905.00 0.302745
$$551$$ −3888.00 −0.300607
$$552$$ −9180.00 −0.707838
$$553$$ 512.000 0.0393715
$$554$$ −10370.0 −0.795269
$$555$$ −7308.00 −0.558932
$$556$$ −27472.0 −2.09545
$$557$$ −22934.0 −1.74460 −0.872302 0.488967i $$-0.837374\pi$$
−0.872302 + 0.488967i $$0.837374\pi$$
$$558$$ 6840.00 0.518925
$$559$$ 20064.0 1.51810
$$560$$ 39872.0 3.00875
$$561$$ 66.0000 0.00496706
$$562$$ −3510.00 −0.263453
$$563$$ 14020.0 1.04951 0.524754 0.851254i $$-0.324157\pi$$
0.524754 + 0.851254i $$0.324157\pi$$
$$564$$ −17340.0 −1.29458
$$565$$ 756.000 0.0562923
$$566$$ −24560.0 −1.82391
$$567$$ −2592.00 −0.191982
$$568$$ 49140.0 3.63005
$$569$$ 4230.00 0.311653 0.155827 0.987784i $$-0.450196\pi$$
0.155827 + 0.987784i $$0.450196\pi$$
$$570$$ 15120.0 1.11107
$$571$$ −8536.00 −0.625605 −0.312803 0.949818i $$-0.601268\pi$$
−0.312803 + 0.949818i $$0.601268\pi$$
$$572$$ 7106.00 0.519435
$$573$$ −2316.00 −0.168852
$$574$$ 15040.0 1.09365
$$575$$ 4828.00 0.350159
$$576$$ −2583.00 −0.186849
$$577$$ −11982.0 −0.864501 −0.432251 0.901754i $$-0.642280\pi$$
−0.432251 + 0.901754i $$0.642280\pi$$
$$578$$ 24545.0 1.76633
$$579$$ 1182.00 0.0848398
$$580$$ 12852.0 0.920087
$$581$$ −11904.0 −0.850019
$$582$$ 7890.00 0.561943
$$583$$ 4818.00 0.342266
$$584$$ −25290.0 −1.79197
$$585$$ 4788.00 0.338392
$$586$$ 17430.0 1.22871
$$587$$ −20396.0 −1.43413 −0.717064 0.697007i $$-0.754514\pi$$
−0.717064 + 0.697007i $$0.754514\pi$$
$$588$$ 34731.0 2.43585
$$589$$ −10944.0 −0.765602
$$590$$ 1400.00 0.0976900
$$591$$ 9174.00 0.638524
$$592$$ 15486.0 1.07512
$$593$$ 12518.0 0.866868 0.433434 0.901185i $$-0.357302\pi$$
0.433434 + 0.901185i $$0.357302\pi$$
$$594$$ 1485.00 0.102576
$$595$$ −896.000 −0.0617352
$$596$$ 35122.0 2.41385
$$597$$ 7992.00 0.547891
$$598$$ 12920.0 0.883509
$$599$$ −25292.0 −1.72521 −0.862607 0.505875i $$-0.831170\pi$$
−0.862607 + 0.505875i $$0.831170\pi$$
$$600$$ −9585.00 −0.652177
$$601$$ 15962.0 1.08337 0.541683 0.840583i $$-0.317787\pi$$
0.541683 + 0.840583i $$0.317787\pi$$
$$602$$ −84480.0 −5.71951
$$603$$ −4140.00 −0.279592
$$604$$ 4216.00 0.284018
$$605$$ −1694.00 −0.113836
$$606$$ −750.000 −0.0502750
$$607$$ −1600.00 −0.106988 −0.0534942 0.998568i $$-0.517036\pi$$
−0.0534942 + 0.998568i $$0.517036\pi$$
$$608$$ −6120.00 −0.408222
$$609$$ 5184.00 0.344936
$$610$$ 39900.0 2.64837
$$611$$ 12920.0 0.855462
$$612$$ −306.000 −0.0202113
$$613$$ 2162.00 0.142451 0.0712254 0.997460i $$-0.477309\pi$$
0.0712254 + 0.997460i $$0.477309\pi$$
$$614$$ −41800.0 −2.74741
$$615$$ −3948.00 −0.258860
$$616$$ −15840.0 −1.03606
$$617$$ −18126.0 −1.18270 −0.591350 0.806415i $$-0.701405\pi$$
−0.591350 + 0.806415i $$0.701405\pi$$
$$618$$ −14160.0 −0.921681
$$619$$ 17348.0 1.12645 0.563227 0.826302i $$-0.309560\pi$$
0.563227 + 0.826302i $$0.309560\pi$$
$$620$$ 36176.0 2.34333
$$621$$ 1836.00 0.118641
$$622$$ 27660.0 1.78306
$$623$$ 30912.0 1.98790
$$624$$ −10146.0 −0.650906
$$625$$ −19459.0 −1.24538
$$626$$ −24130.0 −1.54062
$$627$$ −2376.00 −0.151337
$$628$$ 40222.0 2.55578
$$629$$ −348.000 −0.0220599
$$630$$ −20160.0 −1.27491
$$631$$ 10096.0 0.636950 0.318475 0.947931i $$-0.396829\pi$$
0.318475 + 0.947931i $$0.396829\pi$$
$$632$$ 720.000 0.0453166
$$633$$ −18000.0 −1.13023
$$634$$ −37850.0 −2.37100
$$635$$ 31136.0 1.94582
$$636$$ −22338.0 −1.39270
$$637$$ −25878.0 −1.60961
$$638$$ −2970.00 −0.184300
$$639$$ −9828.00 −0.608435
$$640$$ −29610.0 −1.82881
$$641$$ 8922.00 0.549763 0.274881 0.961478i $$-0.411361\pi$$
0.274881 + 0.961478i $$0.411361\pi$$
$$642$$ −7020.00 −0.431553
$$643$$ −14644.0 −0.898138 −0.449069 0.893497i $$-0.648244\pi$$
−0.449069 + 0.893497i $$0.648244\pi$$
$$644$$ −36992.0 −2.26349
$$645$$ 22176.0 1.35377
$$646$$ 720.000 0.0438514
$$647$$ 6932.00 0.421213 0.210607 0.977571i $$-0.432456\pi$$
0.210607 + 0.977571i $$0.432456\pi$$
$$648$$ −3645.00 −0.220971
$$649$$ −220.000 −0.0133062
$$650$$ 13490.0 0.814033
$$651$$ 14592.0 0.878503
$$652$$ −4828.00 −0.289999
$$653$$ −5942.00 −0.356093 −0.178046 0.984022i $$-0.556978\pi$$
−0.178046 + 0.984022i $$0.556978\pi$$
$$654$$ −2310.00 −0.138116
$$655$$ 38808.0 2.31504
$$656$$ 8366.00 0.497923
$$657$$ 5058.00 0.300352
$$658$$ −54400.0 −3.22300
$$659$$ 484.000 0.0286100 0.0143050 0.999898i $$-0.495446\pi$$
0.0143050 + 0.999898i $$0.495446\pi$$
$$660$$ 7854.00 0.463207
$$661$$ −17114.0 −1.00705 −0.503523 0.863982i $$-0.667963\pi$$
−0.503523 + 0.863982i $$0.667963\pi$$
$$662$$ −18380.0 −1.07909
$$663$$ 228.000 0.0133556
$$664$$ −16740.0 −0.978370
$$665$$ 32256.0 1.88095
$$666$$ −7830.00 −0.455565
$$667$$ −3672.00 −0.213164
$$668$$ 10200.0 0.590793
$$669$$ −1680.00 −0.0970890
$$670$$ −32200.0 −1.85671
$$671$$ −6270.00 −0.360731
$$672$$ 8160.00 0.468421
$$673$$ 16154.0 0.925247 0.462623 0.886555i $$-0.346908\pi$$
0.462623 + 0.886555i $$0.346908\pi$$
$$674$$ 28430.0 1.62475
$$675$$ 1917.00 0.109312
$$676$$ −12801.0 −0.728323
$$677$$ −3390.00 −0.192449 −0.0962247 0.995360i $$-0.530677\pi$$
−0.0962247 + 0.995360i $$0.530677\pi$$
$$678$$ 810.000 0.0458818
$$679$$ 16832.0 0.951330
$$680$$ −1260.00 −0.0710571
$$681$$ 15876.0 0.893347
$$682$$ −8360.00 −0.469386
$$683$$ −25540.0 −1.43084 −0.715418 0.698697i $$-0.753764\pi$$
−0.715418 + 0.698697i $$0.753764\pi$$
$$684$$ 11016.0 0.615800
$$685$$ −15820.0 −0.882410
$$686$$ 54080.0 3.00989
$$687$$ −15966.0 −0.886668
$$688$$ −46992.0 −2.60400
$$689$$ 16644.0 0.920299
$$690$$ 14280.0 0.787870
$$691$$ 12476.0 0.686844 0.343422 0.939181i $$-0.388414\pi$$
0.343422 + 0.939181i $$0.388414\pi$$
$$692$$ 2346.00 0.128875
$$693$$ 3168.00 0.173654
$$694$$ 8260.00 0.451794
$$695$$ 22624.0 1.23479
$$696$$ 7290.00 0.397021
$$697$$ −188.000 −0.0102167
$$698$$ 34950.0 1.89524
$$699$$ −11862.0 −0.641863
$$700$$ −38624.0 −2.08550
$$701$$ −20806.0 −1.12102 −0.560508 0.828149i $$-0.689394\pi$$
−0.560508 + 0.828149i $$0.689394\pi$$
$$702$$ 5130.00 0.275811
$$703$$ 12528.0 0.672123
$$704$$ 3157.00 0.169011
$$705$$ 14280.0 0.762859
$$706$$ 40470.0 2.15738
$$707$$ −1600.00 −0.0851120
$$708$$ 1020.00 0.0541440
$$709$$ 14198.0 0.752069 0.376035 0.926606i $$-0.377287\pi$$
0.376035 + 0.926606i $$0.377287\pi$$
$$710$$ −76440.0 −4.04048
$$711$$ −144.000 −0.00759553
$$712$$ 43470.0 2.28807
$$713$$ −10336.0 −0.542898
$$714$$ −960.000 −0.0503181
$$715$$ −5852.00 −0.306087
$$716$$ 67524.0 3.52443
$$717$$ −10080.0 −0.525027
$$718$$ −5120.00 −0.266124
$$719$$ 4596.00 0.238389 0.119195 0.992871i $$-0.461969\pi$$
0.119195 + 0.992871i $$0.461969\pi$$
$$720$$ −11214.0 −0.580446
$$721$$ −30208.0 −1.56034
$$722$$ 8375.00 0.431697
$$723$$ −9834.00 −0.505851
$$724$$ 37910.0 1.94601
$$725$$ −3834.00 −0.196402
$$726$$ −1815.00 −0.0927837
$$727$$ 19560.0 0.997855 0.498927 0.866644i $$-0.333727\pi$$
0.498927 + 0.866644i $$0.333727\pi$$
$$728$$ −54720.0 −2.78579
$$729$$ 729.000 0.0370370
$$730$$ 39340.0 1.99457
$$731$$ 1056.00 0.0534303
$$732$$ 29070.0 1.46784
$$733$$ −1638.00 −0.0825388 −0.0412694 0.999148i $$-0.513140\pi$$
−0.0412694 + 0.999148i $$0.513140\pi$$
$$734$$ 68320.0 3.43561
$$735$$ −28602.0 −1.43538
$$736$$ −5780.00 −0.289475
$$737$$ 5060.00 0.252900
$$738$$ −4230.00 −0.210987
$$739$$ −15592.0 −0.776131 −0.388066 0.921632i $$-0.626857\pi$$
−0.388066 + 0.921632i $$0.626857\pi$$
$$740$$ −41412.0 −2.05721
$$741$$ −8208.00 −0.406921
$$742$$ −70080.0 −3.46727
$$743$$ 592.000 0.0292307 0.0146153 0.999893i $$-0.495348\pi$$
0.0146153 + 0.999893i $$0.495348\pi$$
$$744$$ 20520.0 1.01116
$$745$$ −28924.0 −1.42241
$$746$$ 9790.00 0.480479
$$747$$ 3348.00 0.163985
$$748$$ 374.000 0.0182818
$$749$$ −14976.0 −0.730589
$$750$$ −11340.0 −0.552104
$$751$$ 39832.0 1.93541 0.967703 0.252092i $$-0.0811186\pi$$
0.967703 + 0.252092i $$0.0811186\pi$$
$$752$$ −30260.0 −1.46738
$$753$$ 6276.00 0.303732
$$754$$ −10260.0 −0.495553
$$755$$ −3472.00 −0.167363
$$756$$ −14688.0 −0.706610
$$757$$ 10958.0 0.526123 0.263062 0.964779i $$-0.415268\pi$$
0.263062 + 0.964779i $$0.415268\pi$$
$$758$$ −30620.0 −1.46724
$$759$$ −2244.00 −0.107315
$$760$$ 45360.0 2.16497
$$761$$ −8970.00 −0.427283 −0.213641 0.976912i $$-0.568532\pi$$
−0.213641 + 0.976912i $$0.568532\pi$$
$$762$$ 33360.0 1.58596
$$763$$ −4928.00 −0.233821
$$764$$ −13124.0 −0.621479
$$765$$ 252.000 0.0119099
$$766$$ −28060.0 −1.32356
$$767$$ −760.000 −0.0357784
$$768$$ −24837.0 −1.16696
$$769$$ −10054.0 −0.471465 −0.235732 0.971818i $$-0.575749\pi$$
−0.235732 + 0.971818i $$0.575749\pi$$
$$770$$ 24640.0 1.15320
$$771$$ 1974.00 0.0922074
$$772$$ 6698.00 0.312262
$$773$$ 26346.0 1.22587 0.612936 0.790132i $$-0.289988\pi$$
0.612936 + 0.790132i $$0.289988\pi$$
$$774$$ 23760.0 1.10341
$$775$$ −10792.0 −0.500207
$$776$$ 23670.0 1.09498
$$777$$ −16704.0 −0.771239
$$778$$ −62250.0 −2.86860
$$779$$ 6768.00 0.311282
$$780$$ 27132.0 1.24549
$$781$$ 12012.0 0.550350
$$782$$ 680.000 0.0310956
$$783$$ −1458.00 −0.0665449
$$784$$ 60609.0 2.76098
$$785$$ −33124.0 −1.50605
$$786$$ 41580.0 1.88691
$$787$$ −16040.0 −0.726511 −0.363256 0.931690i $$-0.618335\pi$$
−0.363256 + 0.931690i $$0.618335\pi$$
$$788$$ 51986.0 2.35016
$$789$$ −15312.0 −0.690902
$$790$$ −1120.00 −0.0504403
$$791$$ 1728.00 0.0776746
$$792$$ 4455.00 0.199876
$$793$$ −21660.0 −0.969948
$$794$$ −74150.0 −3.31421
$$795$$ 18396.0 0.820678
$$796$$ 45288.0 2.01657
$$797$$ 32810.0 1.45821 0.729103 0.684404i $$-0.239938\pi$$
0.729103 + 0.684404i $$0.239938\pi$$
$$798$$ 34560.0 1.53310
$$799$$ 680.000 0.0301085
$$800$$ −6035.00 −0.266712
$$801$$ −8694.00 −0.383505
$$802$$ 16790.0 0.739246
$$803$$ −6182.00 −0.271679
$$804$$ −23460.0 −1.02907
$$805$$ 30464.0 1.33381
$$806$$ −28880.0 −1.26210
$$807$$ −12714.0 −0.554590
$$808$$ −2250.00 −0.0979638
$$809$$ 18918.0 0.822153 0.411076 0.911601i $$-0.365153\pi$$
0.411076 + 0.911601i $$0.365153\pi$$
$$810$$ 5670.00 0.245955
$$811$$ −8552.00 −0.370285 −0.185143 0.982712i $$-0.559275\pi$$
−0.185143 + 0.982712i $$0.559275\pi$$
$$812$$ 29376.0 1.26958
$$813$$ −10128.0 −0.436906
$$814$$ 9570.00 0.412074
$$815$$ 3976.00 0.170887
$$816$$ −534.000 −0.0229090
$$817$$ −38016.0 −1.62792
$$818$$ −53490.0 −2.28635
$$819$$ 10944.0 0.466928
$$820$$ −22372.0 −0.952761
$$821$$ −46430.0 −1.97371 −0.986856 0.161600i $$-0.948335\pi$$
−0.986856 + 0.161600i $$0.948335\pi$$
$$822$$ −16950.0 −0.719220
$$823$$ 16392.0 0.694276 0.347138 0.937814i $$-0.387154\pi$$
0.347138 + 0.937814i $$0.387154\pi$$
$$824$$ −42480.0 −1.79595
$$825$$ −2343.00 −0.0988761
$$826$$ 3200.00 0.134797
$$827$$ −13876.0 −0.583453 −0.291727 0.956502i $$-0.594230\pi$$
−0.291727 + 0.956502i $$0.594230\pi$$
$$828$$ 10404.0 0.436671
$$829$$ −24554.0 −1.02870 −0.514352 0.857579i $$-0.671968\pi$$
−0.514352 + 0.857579i $$0.671968\pi$$
$$830$$ 26040.0 1.08899
$$831$$ 6222.00 0.259734
$$832$$ 10906.0 0.454444
$$833$$ −1362.00 −0.0566513
$$834$$ 24240.0 1.00643
$$835$$ −8400.00 −0.348137
$$836$$ −13464.0 −0.557012
$$837$$ −4104.00 −0.169480
$$838$$ 10220.0 0.421294
$$839$$ 19900.0 0.818861 0.409430 0.912341i $$-0.365727\pi$$
0.409430 + 0.912341i $$0.365727\pi$$
$$840$$ −60480.0 −2.48424
$$841$$ −21473.0 −0.880438
$$842$$ −15350.0 −0.628261
$$843$$ 2106.00 0.0860433
$$844$$ −102000. −4.15993
$$845$$ 10542.0 0.429178
$$846$$ 15300.0 0.621779
$$847$$ −3872.00 −0.157076
$$848$$ −38982.0 −1.57859
$$849$$ 14736.0 0.595687
$$850$$ 710.000 0.0286504
$$851$$ 11832.0 0.476611
$$852$$ −55692.0 −2.23941
$$853$$ 41138.0 1.65128 0.825638 0.564200i $$-0.190815\pi$$
0.825638 + 0.564200i $$0.190815\pi$$
$$854$$ 91200.0 3.65433
$$855$$ −9072.00 −0.362872
$$856$$ −21060.0 −0.840907
$$857$$ 19910.0 0.793597 0.396799 0.917906i $$-0.370121\pi$$
0.396799 + 0.917906i $$0.370121\pi$$
$$858$$ −6270.00 −0.249481
$$859$$ 42924.0 1.70495 0.852473 0.522772i $$-0.175102\pi$$
0.852473 + 0.522772i $$0.175102\pi$$
$$860$$ 125664. 4.98268
$$861$$ −9024.00 −0.357186
$$862$$ 63000.0 2.48931
$$863$$ −46236.0 −1.82374 −0.911872 0.410474i $$-0.865363\pi$$
−0.911872 + 0.410474i $$0.865363\pi$$
$$864$$ −2295.00 −0.0903675
$$865$$ −1932.00 −0.0759422
$$866$$ 49510.0 1.94275
$$867$$ −14727.0 −0.576880
$$868$$ 82688.0 3.23343
$$869$$ 176.000 0.00687042
$$870$$ −11340.0 −0.441910
$$871$$ 17480.0 0.680008
$$872$$ −6930.00 −0.269128
$$873$$ −4734.00 −0.183530
$$874$$ −24480.0 −0.947424
$$875$$ −24192.0 −0.934673
$$876$$ 28662.0 1.10548
$$877$$ 25746.0 0.991312 0.495656 0.868519i $$-0.334928\pi$$
0.495656 + 0.868519i $$0.334928\pi$$
$$878$$ −57200.0 −2.19864
$$879$$ −10458.0 −0.401296
$$880$$ 13706.0 0.525033
$$881$$ −24550.0 −0.938831 −0.469416 0.882977i $$-0.655535\pi$$
−0.469416 + 0.882977i $$0.655535\pi$$
$$882$$ −30645.0 −1.16992
$$883$$ −19436.0 −0.740740 −0.370370 0.928884i $$-0.620769\pi$$
−0.370370 + 0.928884i $$0.620769\pi$$
$$884$$ 1292.00 0.0491569
$$885$$ −840.000 −0.0319054
$$886$$ 25900.0 0.982085
$$887$$ −22912.0 −0.867316 −0.433658 0.901077i $$-0.642777\pi$$
−0.433658 + 0.901077i $$0.642777\pi$$
$$888$$ −23490.0 −0.887695
$$889$$ 71168.0 2.68492
$$890$$ −67620.0 −2.54677
$$891$$ −891.000 −0.0335013
$$892$$ −9520.00 −0.357347
$$893$$ −24480.0 −0.917348
$$894$$ −30990.0 −1.15935
$$895$$ −55608.0 −2.07684
$$896$$ −67680.0 −2.52347
$$897$$ −7752.00 −0.288553
$$898$$ −54130.0 −2.01152
$$899$$ 8208.00 0.304507
$$900$$ 10863.0 0.402333
$$901$$ 876.000 0.0323904
$$902$$ 5170.00 0.190845
$$903$$ 50688.0 1.86799
$$904$$ 2430.00 0.0894033
$$905$$ −31220.0 −1.14673
$$906$$ −3720.00 −0.136411
$$907$$ −39900.0 −1.46070 −0.730352 0.683071i $$-0.760644\pi$$
−0.730352 + 0.683071i $$0.760644\pi$$
$$908$$ 89964.0 3.28806
$$909$$ 450.000 0.0164198
$$910$$ 85120.0 3.10077
$$911$$ 29460.0 1.07141 0.535704 0.844406i $$-0.320046\pi$$
0.535704 + 0.844406i $$0.320046\pi$$
$$912$$ 19224.0 0.697994
$$913$$ −4092.00 −0.148330
$$914$$ 78990.0 2.85860
$$915$$ −23940.0 −0.864953
$$916$$ −90474.0 −3.26348
$$917$$ 88704.0 3.19440
$$918$$ 270.000 0.00970733
$$919$$ 29368.0 1.05415 0.527073 0.849820i $$-0.323289\pi$$
0.527073 + 0.849820i $$0.323289\pi$$
$$920$$ 42840.0 1.53521
$$921$$ 25080.0 0.897301
$$922$$ 19470.0 0.695456
$$923$$ 41496.0 1.47980
$$924$$ 17952.0 0.639153
$$925$$ 12354.0 0.439132
$$926$$ 79960.0 2.83763
$$927$$ 8496.00 0.301020
$$928$$ 4590.00 0.162364
$$929$$ 33954.0 1.19913 0.599567 0.800325i $$-0.295340\pi$$
0.599567 + 0.800325i $$0.295340\pi$$
$$930$$ −31920.0 −1.12548
$$931$$ 49032.0 1.72606
$$932$$ −67218.0 −2.36245
$$933$$ −16596.0 −0.582346
$$934$$ −59220.0 −2.07467
$$935$$ −308.000 −0.0107729
$$936$$ 15390.0 0.537434
$$937$$ −2854.00 −0.0995049 −0.0497525 0.998762i $$-0.515843\pi$$
−0.0497525 + 0.998762i $$0.515843\pi$$
$$938$$ −73600.0 −2.56197
$$939$$ 14478.0 0.503165
$$940$$ 80920.0 2.80779
$$941$$ −6294.00 −0.218043 −0.109022 0.994039i $$-0.534772\pi$$
−0.109022 + 0.994039i $$0.534772\pi$$
$$942$$ −35490.0 −1.22752
$$943$$ 6392.00 0.220734
$$944$$ 1780.00 0.0613708
$$945$$ 12096.0 0.416384
$$946$$ −29040.0 −0.998067
$$947$$ 2268.00 0.0778248 0.0389124 0.999243i $$-0.487611\pi$$
0.0389124 + 0.999243i $$0.487611\pi$$
$$948$$ −816.000 −0.0279562
$$949$$ −21356.0 −0.730501
$$950$$ −25560.0 −0.872922
$$951$$ 22710.0 0.774366
$$952$$ −2880.00 −0.0980476
$$953$$ 26566.0 0.902998 0.451499 0.892272i $$-0.350889\pi$$
0.451499 + 0.892272i $$0.350889\pi$$
$$954$$ 19710.0 0.668904
$$955$$ 10808.0 0.366218
$$956$$ −57120.0 −1.93242
$$957$$ 1782.00 0.0601921
$$958$$ −74680.0 −2.51858
$$959$$ −36160.0 −1.21759
$$960$$ 12054.0 0.405251
$$961$$ −6687.00 −0.224464
$$962$$ 33060.0 1.10800
$$963$$ 4212.00 0.140945
$$964$$ −55726.0 −1.86184
$$965$$ −5516.00 −0.184007
$$966$$ 32640.0 1.08714
$$967$$ 11176.0 0.371661 0.185830 0.982582i $$-0.440503\pi$$
0.185830 + 0.982582i $$0.440503\pi$$
$$968$$ −5445.00 −0.180794
$$969$$ −432.000 −0.0143218
$$970$$ −36820.0 −1.21878
$$971$$ −42316.0 −1.39854 −0.699271 0.714856i $$-0.746492\pi$$
−0.699271 + 0.714856i $$0.746492\pi$$
$$972$$ 4131.00 0.136319
$$973$$ 51712.0 1.70381
$$974$$ 10280.0 0.338185
$$975$$ −8094.00 −0.265862
$$976$$ 50730.0 1.66376
$$977$$ −45054.0 −1.47534 −0.737669 0.675163i $$-0.764073\pi$$
−0.737669 + 0.675163i $$0.764073\pi$$
$$978$$ 4260.00 0.139284
$$979$$ 10626.0 0.346893
$$980$$ −162078. −5.28305
$$981$$ 1386.00 0.0451086
$$982$$ 89260.0 2.90061
$$983$$ −12300.0 −0.399094 −0.199547 0.979888i $$-0.563947\pi$$
−0.199547 + 0.979888i $$0.563947\pi$$
$$984$$ −12690.0 −0.411120
$$985$$ −42812.0 −1.38488
$$986$$ −540.000 −0.0174413
$$987$$ 32640.0 1.05263
$$988$$ −46512.0 −1.49772
$$989$$ −35904.0 −1.15438
$$990$$ −6930.00 −0.222475
$$991$$ 36280.0 1.16294 0.581469 0.813568i $$-0.302478\pi$$
0.581469 + 0.813568i $$0.302478\pi$$
$$992$$ 12920.0 0.413519
$$993$$ 11028.0 0.352430
$$994$$ −174720. −5.57523
$$995$$ −37296.0 −1.18830
$$996$$ 18972.0 0.603565
$$997$$ 3290.00 0.104509 0.0522544 0.998634i $$-0.483359\pi$$
0.0522544 + 0.998634i $$0.483359\pi$$
$$998$$ −22540.0 −0.714921
$$999$$ 4698.00 0.148787
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.a.1.1 1
3.2 odd 2 99.4.a.b.1.1 1
4.3 odd 2 528.4.a.a.1.1 1
5.2 odd 4 825.4.c.a.199.1 2
5.3 odd 4 825.4.c.a.199.2 2
5.4 even 2 825.4.a.i.1.1 1
7.6 odd 2 1617.4.a.a.1.1 1
8.3 odd 2 2112.4.a.y.1.1 1
8.5 even 2 2112.4.a.l.1.1 1
11.10 odd 2 363.4.a.h.1.1 1
12.11 even 2 1584.4.a.t.1.1 1
15.14 odd 2 2475.4.a.b.1.1 1
33.32 even 2 1089.4.a.a.1.1 1

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