# Properties

 Label 176.4.a.e.1.1 Level $176$ Weight $4$ Character 176.1 Self dual yes Analytic conductor $10.384$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [176,4,Mod(1,176)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(176, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("176.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$176 = 2^{4} \cdot 11$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 176.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: $$10.3843361610$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 44) Fricke sign: $$+1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 176.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+5.00000 q^{3} -7.00000 q^{5} +26.0000 q^{7} -2.00000 q^{9} +O(q^{10})$$ $$q+5.00000 q^{3} -7.00000 q^{5} +26.0000 q^{7} -2.00000 q^{9} +11.0000 q^{11} +52.0000 q^{13} -35.0000 q^{15} +46.0000 q^{17} +96.0000 q^{19} +130.000 q^{21} -27.0000 q^{23} -76.0000 q^{25} -145.000 q^{27} +16.0000 q^{29} +293.000 q^{31} +55.0000 q^{33} -182.000 q^{35} -29.0000 q^{37} +260.000 q^{39} -472.000 q^{41} +110.000 q^{43} +14.0000 q^{45} +224.000 q^{47} +333.000 q^{49} +230.000 q^{51} +754.000 q^{53} -77.0000 q^{55} +480.000 q^{57} -825.000 q^{59} -548.000 q^{61} -52.0000 q^{63} -364.000 q^{65} +123.000 q^{67} -135.000 q^{69} -1001.00 q^{71} -1020.00 q^{73} -380.000 q^{75} +286.000 q^{77} -526.000 q^{79} -671.000 q^{81} +158.000 q^{83} -322.000 q^{85} +80.0000 q^{87} -1217.00 q^{89} +1352.00 q^{91} +1465.00 q^{93} -672.000 q^{95} -263.000 q^{97} -22.0000 q^{99} +O(q^{100})$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0 0
$$3$$ 5.00000 0.962250 0.481125 0.876652i $$-0.340228\pi$$
0.481125 + 0.876652i $$0.340228\pi$$
$$4$$ 0 0
$$5$$ −7.00000 −0.626099 −0.313050 0.949737i $$-0.601351\pi$$
−0.313050 + 0.949737i $$0.601351\pi$$
$$6$$ 0 0
$$7$$ 26.0000 1.40387 0.701934 0.712242i $$-0.252320\pi$$
0.701934 + 0.712242i $$0.252320\pi$$
$$8$$ 0 0
$$9$$ −2.00000 −0.0740741
$$10$$ 0 0
$$11$$ 11.0000 0.301511
$$12$$ 0 0
$$13$$ 52.0000 1.10940 0.554700 0.832050i $$-0.312833\pi$$
0.554700 + 0.832050i $$0.312833\pi$$
$$14$$ 0 0
$$15$$ −35.0000 −0.602464
$$16$$ 0 0
$$17$$ 46.0000 0.656273 0.328136 0.944630i $$-0.393579\pi$$
0.328136 + 0.944630i $$0.393579\pi$$
$$18$$ 0 0
$$19$$ 96.0000 1.15915 0.579577 0.814918i $$-0.303218\pi$$
0.579577 + 0.814918i $$0.303218\pi$$
$$20$$ 0 0
$$21$$ 130.000 1.35087
$$22$$ 0 0
$$23$$ −27.0000 −0.244778 −0.122389 0.992482i $$-0.539056\pi$$
−0.122389 + 0.992482i $$0.539056\pi$$
$$24$$ 0 0
$$25$$ −76.0000 −0.608000
$$26$$ 0 0
$$27$$ −145.000 −1.03353
$$28$$ 0 0
$$29$$ 16.0000 0.102453 0.0512263 0.998687i $$-0.483687\pi$$
0.0512263 + 0.998687i $$0.483687\pi$$
$$30$$ 0 0
$$31$$ 293.000 1.69756 0.848780 0.528746i $$-0.177338\pi$$
0.848780 + 0.528746i $$0.177338\pi$$
$$32$$ 0 0
$$33$$ 55.0000 0.290129
$$34$$ 0 0
$$35$$ −182.000 −0.878960
$$36$$ 0 0
$$37$$ −29.0000 −0.128853 −0.0644266 0.997922i $$-0.520522\pi$$
−0.0644266 + 0.997922i $$0.520522\pi$$
$$38$$ 0 0
$$39$$ 260.000 1.06752
$$40$$ 0 0
$$41$$ −472.000 −1.79790 −0.898951 0.438048i $$-0.855670\pi$$
−0.898951 + 0.438048i $$0.855670\pi$$
$$42$$ 0 0
$$43$$ 110.000 0.390113 0.195056 0.980792i $$-0.437511\pi$$
0.195056 + 0.980792i $$0.437511\pi$$
$$44$$ 0 0
$$45$$ 14.0000 0.0463777
$$46$$ 0 0
$$47$$ 224.000 0.695186 0.347593 0.937645i $$-0.386999\pi$$
0.347593 + 0.937645i $$0.386999\pi$$
$$48$$ 0 0
$$49$$ 333.000 0.970845
$$50$$ 0 0
$$51$$ 230.000 0.631499
$$52$$ 0 0
$$53$$ 754.000 1.95415 0.977074 0.212899i $$-0.0682905\pi$$
0.977074 + 0.212899i $$0.0682905\pi$$
$$54$$ 0 0
$$55$$ −77.0000 −0.188776
$$56$$ 0 0
$$57$$ 480.000 1.11540
$$58$$ 0 0
$$59$$ −825.000 −1.82044 −0.910219 0.414127i $$-0.864087\pi$$
−0.910219 + 0.414127i $$0.864087\pi$$
$$60$$ 0 0
$$61$$ −548.000 −1.15023 −0.575116 0.818072i $$-0.695043\pi$$
−0.575116 + 0.818072i $$0.695043\pi$$
$$62$$ 0 0
$$63$$ −52.0000 −0.103990
$$64$$ 0 0
$$65$$ −364.000 −0.694595
$$66$$ 0 0
$$67$$ 123.000 0.224281 0.112141 0.993692i $$-0.464229\pi$$
0.112141 + 0.993692i $$0.464229\pi$$
$$68$$ 0 0
$$69$$ −135.000 −0.235538
$$70$$ 0 0
$$71$$ −1001.00 −1.67319 −0.836597 0.547818i $$-0.815459\pi$$
−0.836597 + 0.547818i $$0.815459\pi$$
$$72$$ 0 0
$$73$$ −1020.00 −1.63537 −0.817685 0.575666i $$-0.804743\pi$$
−0.817685 + 0.575666i $$0.804743\pi$$
$$74$$ 0 0
$$75$$ −380.000 −0.585048
$$76$$ 0 0
$$77$$ 286.000 0.423282
$$78$$ 0 0
$$79$$ −526.000 −0.749109 −0.374555 0.927205i $$-0.622204\pi$$
−0.374555 + 0.927205i $$0.622204\pi$$
$$80$$ 0 0
$$81$$ −671.000 −0.920439
$$82$$ 0 0
$$83$$ 158.000 0.208949 0.104474 0.994528i $$-0.466684\pi$$
0.104474 + 0.994528i $$0.466684\pi$$
$$84$$ 0 0
$$85$$ −322.000 −0.410892
$$86$$ 0 0
$$87$$ 80.0000 0.0985851
$$88$$ 0 0
$$89$$ −1217.00 −1.44946 −0.724729 0.689034i $$-0.758035\pi$$
−0.724729 + 0.689034i $$0.758035\pi$$
$$90$$ 0 0
$$91$$ 1352.00 1.55745
$$92$$ 0 0
$$93$$ 1465.00 1.63348
$$94$$ 0 0
$$95$$ −672.000 −0.725745
$$96$$ 0 0
$$97$$ −263.000 −0.275295 −0.137647 0.990481i $$-0.543954\pi$$
−0.137647 + 0.990481i $$0.543954\pi$$
$$98$$ 0 0
$$99$$ −22.0000 −0.0223342
$$100$$ 0 0
$$101$$ −814.000 −0.801941 −0.400970 0.916091i $$-0.631327\pi$$
−0.400970 + 0.916091i $$0.631327\pi$$
$$102$$ 0 0
$$103$$ −376.000 −0.359693 −0.179847 0.983695i $$-0.557560\pi$$
−0.179847 + 0.983695i $$0.557560\pi$$
$$104$$ 0 0
$$105$$ −910.000 −0.845780
$$106$$ 0 0
$$107$$ 1494.00 1.34982 0.674909 0.737901i $$-0.264183\pi$$
0.674909 + 0.737901i $$0.264183\pi$$
$$108$$ 0 0
$$109$$ 842.000 0.739899 0.369949 0.929052i $$-0.379375\pi$$
0.369949 + 0.929052i $$0.379375\pi$$
$$110$$ 0 0
$$111$$ −145.000 −0.123989
$$112$$ 0 0
$$113$$ 1281.00 1.06643 0.533214 0.845980i $$-0.320984\pi$$
0.533214 + 0.845980i $$0.320984\pi$$
$$114$$ 0 0
$$115$$ 189.000 0.153255
$$116$$ 0 0
$$117$$ −104.000 −0.0821778
$$118$$ 0 0
$$119$$ 1196.00 0.921321
$$120$$ 0 0
$$121$$ 121.000 0.0909091
$$122$$ 0 0
$$123$$ −2360.00 −1.73003
$$124$$ 0 0
$$125$$ 1407.00 1.00677
$$126$$ 0 0
$$127$$ −1736.00 −1.21295 −0.606477 0.795101i $$-0.707418\pi$$
−0.606477 + 0.795101i $$0.707418\pi$$
$$128$$ 0 0
$$129$$ 550.000 0.375386
$$130$$ 0 0
$$131$$ −6.00000 −0.00400170 −0.00200085 0.999998i $$-0.500637\pi$$
−0.00200085 + 0.999998i $$0.500637\pi$$
$$132$$ 0 0
$$133$$ 2496.00 1.62730
$$134$$ 0 0
$$135$$ 1015.00 0.647091
$$136$$ 0 0
$$137$$ −1439.00 −0.897387 −0.448694 0.893686i $$-0.648111\pi$$
−0.448694 + 0.893686i $$0.648111\pi$$
$$138$$ 0 0
$$139$$ 318.000 0.194046 0.0970231 0.995282i $$-0.469068\pi$$
0.0970231 + 0.995282i $$0.469068\pi$$
$$140$$ 0 0
$$141$$ 1120.00 0.668943
$$142$$ 0 0
$$143$$ 572.000 0.334497
$$144$$ 0 0
$$145$$ −112.000 −0.0641455
$$146$$ 0 0
$$147$$ 1665.00 0.934196
$$148$$ 0 0
$$149$$ −922.000 −0.506934 −0.253467 0.967344i $$-0.581571\pi$$
−0.253467 + 0.967344i $$0.581571\pi$$
$$150$$ 0 0
$$151$$ 1030.00 0.555101 0.277550 0.960711i $$-0.410477\pi$$
0.277550 + 0.960711i $$0.410477\pi$$
$$152$$ 0 0
$$153$$ −92.0000 −0.0486128
$$154$$ 0 0
$$155$$ −2051.00 −1.06284
$$156$$ 0 0
$$157$$ 1017.00 0.516977 0.258489 0.966014i $$-0.416776\pi$$
0.258489 + 0.966014i $$0.416776\pi$$
$$158$$ 0 0
$$159$$ 3770.00 1.88038
$$160$$ 0 0
$$161$$ −702.000 −0.343636
$$162$$ 0 0
$$163$$ 2444.00 1.17441 0.587205 0.809438i $$-0.300228\pi$$
0.587205 + 0.809438i $$0.300228\pi$$
$$164$$ 0 0
$$165$$ −385.000 −0.181650
$$166$$ 0 0
$$167$$ 1452.00 0.672809 0.336405 0.941718i $$-0.390789\pi$$
0.336405 + 0.941718i $$0.390789\pi$$
$$168$$ 0 0
$$169$$ 507.000 0.230769
$$170$$ 0 0
$$171$$ −192.000 −0.0858632
$$172$$ 0 0
$$173$$ 1914.00 0.841149 0.420574 0.907258i $$-0.361829\pi$$
0.420574 + 0.907258i $$0.361829\pi$$
$$174$$ 0 0
$$175$$ −1976.00 −0.853552
$$176$$ 0 0
$$177$$ −4125.00 −1.75172
$$178$$ 0 0
$$179$$ −1293.00 −0.539907 −0.269954 0.962873i $$-0.587008\pi$$
−0.269954 + 0.962873i $$0.587008\pi$$
$$180$$ 0 0
$$181$$ 455.000 0.186850 0.0934251 0.995626i $$-0.470218\pi$$
0.0934251 + 0.995626i $$0.470218\pi$$
$$182$$ 0 0
$$183$$ −2740.00 −1.10681
$$184$$ 0 0
$$185$$ 203.000 0.0806749
$$186$$ 0 0
$$187$$ 506.000 0.197874
$$188$$ 0 0
$$189$$ −3770.00 −1.45094
$$190$$ 0 0
$$191$$ 1115.00 0.422401 0.211200 0.977443i $$-0.432263\pi$$
0.211200 + 0.977443i $$0.432263\pi$$
$$192$$ 0 0
$$193$$ 5012.00 1.86928 0.934642 0.355591i $$-0.115720\pi$$
0.934642 + 0.355591i $$0.115720\pi$$
$$194$$ 0 0
$$195$$ −1820.00 −0.668374
$$196$$ 0 0
$$197$$ −4146.00 −1.49944 −0.749721 0.661753i $$-0.769813\pi$$
−0.749721 + 0.661753i $$0.769813\pi$$
$$198$$ 0 0
$$199$$ 1240.00 0.441715 0.220857 0.975306i $$-0.429114\pi$$
0.220857 + 0.975306i $$0.429114\pi$$
$$200$$ 0 0
$$201$$ 615.000 0.215815
$$202$$ 0 0
$$203$$ 416.000 0.143830
$$204$$ 0 0
$$205$$ 3304.00 1.12567
$$206$$ 0 0
$$207$$ 54.0000 0.0181317
$$208$$ 0 0
$$209$$ 1056.00 0.349498
$$210$$ 0 0
$$211$$ 2820.00 0.920080 0.460040 0.887898i $$-0.347835\pi$$
0.460040 + 0.887898i $$0.347835\pi$$
$$212$$ 0 0
$$213$$ −5005.00 −1.61003
$$214$$ 0 0
$$215$$ −770.000 −0.244249
$$216$$ 0 0
$$217$$ 7618.00 2.38315
$$218$$ 0 0
$$219$$ −5100.00 −1.57363
$$220$$ 0 0
$$221$$ 2392.00 0.728069
$$222$$ 0 0
$$223$$ −3695.00 −1.10958 −0.554788 0.831992i $$-0.687201\pi$$
−0.554788 + 0.831992i $$0.687201\pi$$
$$224$$ 0 0
$$225$$ 152.000 0.0450370
$$226$$ 0 0
$$227$$ 486.000 0.142101 0.0710506 0.997473i $$-0.477365\pi$$
0.0710506 + 0.997473i $$0.477365\pi$$
$$228$$ 0 0
$$229$$ 4231.00 1.22093 0.610464 0.792044i $$-0.290983\pi$$
0.610464 + 0.792044i $$0.290983\pi$$
$$230$$ 0 0
$$231$$ 1430.00 0.407303
$$232$$ 0 0
$$233$$ −3336.00 −0.937977 −0.468988 0.883204i $$-0.655381\pi$$
−0.468988 + 0.883204i $$0.655381\pi$$
$$234$$ 0 0
$$235$$ −1568.00 −0.435255
$$236$$ 0 0
$$237$$ −2630.00 −0.720831
$$238$$ 0 0
$$239$$ −3610.00 −0.977036 −0.488518 0.872554i $$-0.662462\pi$$
−0.488518 + 0.872554i $$0.662462\pi$$
$$240$$ 0 0
$$241$$ −6408.00 −1.71276 −0.856381 0.516345i $$-0.827292\pi$$
−0.856381 + 0.516345i $$0.827292\pi$$
$$242$$ 0 0
$$243$$ 560.000 0.147835
$$244$$ 0 0
$$245$$ −2331.00 −0.607845
$$246$$ 0 0
$$247$$ 4992.00 1.28596
$$248$$ 0 0
$$249$$ 790.000 0.201061
$$250$$ 0 0
$$251$$ 1787.00 0.449380 0.224690 0.974430i $$-0.427863\pi$$
0.224690 + 0.974430i $$0.427863\pi$$
$$252$$ 0 0
$$253$$ −297.000 −0.0738033
$$254$$ 0 0
$$255$$ −1610.00 −0.395381
$$256$$ 0 0
$$257$$ −354.000 −0.0859218 −0.0429609 0.999077i $$-0.513679\pi$$
−0.0429609 + 0.999077i $$0.513679\pi$$
$$258$$ 0 0
$$259$$ −754.000 −0.180893
$$260$$ 0 0
$$261$$ −32.0000 −0.00758908
$$262$$ 0 0
$$263$$ 2026.00 0.475013 0.237507 0.971386i $$-0.423670\pi$$
0.237507 + 0.971386i $$0.423670\pi$$
$$264$$ 0 0
$$265$$ −5278.00 −1.22349
$$266$$ 0 0
$$267$$ −6085.00 −1.39474
$$268$$ 0 0
$$269$$ −8750.00 −1.98326 −0.991630 0.129112i $$-0.958787\pi$$
−0.991630 + 0.129112i $$0.958787\pi$$
$$270$$ 0 0
$$271$$ 5036.00 1.12884 0.564419 0.825488i $$-0.309100\pi$$
0.564419 + 0.825488i $$0.309100\pi$$
$$272$$ 0 0
$$273$$ 6760.00 1.49866
$$274$$ 0 0
$$275$$ −836.000 −0.183319
$$276$$ 0 0
$$277$$ −4306.00 −0.934016 −0.467008 0.884253i $$-0.654668\pi$$
−0.467008 + 0.884253i $$0.654668\pi$$
$$278$$ 0 0
$$279$$ −586.000 −0.125745
$$280$$ 0 0
$$281$$ −1202.00 −0.255179 −0.127590 0.991827i $$-0.540724\pi$$
−0.127590 + 0.991827i $$0.540724\pi$$
$$282$$ 0 0
$$283$$ 6620.00 1.39052 0.695262 0.718757i $$-0.255288\pi$$
0.695262 + 0.718757i $$0.255288\pi$$
$$284$$ 0 0
$$285$$ −3360.00 −0.698348
$$286$$ 0 0
$$287$$ −12272.0 −2.52402
$$288$$ 0 0
$$289$$ −2797.00 −0.569306
$$290$$ 0 0
$$291$$ −1315.00 −0.264903
$$292$$ 0 0
$$293$$ 6968.00 1.38933 0.694667 0.719331i $$-0.255552\pi$$
0.694667 + 0.719331i $$0.255552\pi$$
$$294$$ 0 0
$$295$$ 5775.00 1.13977
$$296$$ 0 0
$$297$$ −1595.00 −0.311620
$$298$$ 0 0
$$299$$ −1404.00 −0.271557
$$300$$ 0 0
$$301$$ 2860.00 0.547667
$$302$$ 0 0
$$303$$ −4070.00 −0.771668
$$304$$ 0 0
$$305$$ 3836.00 0.720160
$$306$$ 0 0
$$307$$ −7640.00 −1.42032 −0.710159 0.704041i $$-0.751377\pi$$
−0.710159 + 0.704041i $$0.751377\pi$$
$$308$$ 0 0
$$309$$ −1880.00 −0.346115
$$310$$ 0 0
$$311$$ 652.000 0.118880 0.0594398 0.998232i $$-0.481069\pi$$
0.0594398 + 0.998232i $$0.481069\pi$$
$$312$$ 0 0
$$313$$ 8055.00 1.45462 0.727309 0.686310i $$-0.240771\pi$$
0.727309 + 0.686310i $$0.240771\pi$$
$$314$$ 0 0
$$315$$ 364.000 0.0651082
$$316$$ 0 0
$$317$$ −5675.00 −1.00549 −0.502744 0.864435i $$-0.667676\pi$$
−0.502744 + 0.864435i $$0.667676\pi$$
$$318$$ 0 0
$$319$$ 176.000 0.0308906
$$320$$ 0 0
$$321$$ 7470.00 1.29886
$$322$$ 0 0
$$323$$ 4416.00 0.760721
$$324$$ 0 0
$$325$$ −3952.00 −0.674515
$$326$$ 0 0
$$327$$ 4210.00 0.711968
$$328$$ 0 0
$$329$$ 5824.00 0.975950
$$330$$ 0 0
$$331$$ 2245.00 0.372799 0.186399 0.982474i $$-0.440318\pi$$
0.186399 + 0.982474i $$0.440318\pi$$
$$332$$ 0 0
$$333$$ 58.0000 0.00954469
$$334$$ 0 0
$$335$$ −861.000 −0.140422
$$336$$ 0 0
$$337$$ −694.000 −0.112180 −0.0560899 0.998426i $$-0.517863\pi$$
−0.0560899 + 0.998426i $$0.517863\pi$$
$$338$$ 0 0
$$339$$ 6405.00 1.02617
$$340$$ 0 0
$$341$$ 3223.00 0.511834
$$342$$ 0 0
$$343$$ −260.000 −0.0409291
$$344$$ 0 0
$$345$$ 945.000 0.147470
$$346$$ 0 0
$$347$$ −4092.00 −0.633055 −0.316527 0.948583i $$-0.602517\pi$$
−0.316527 + 0.948583i $$0.602517\pi$$
$$348$$ 0 0
$$349$$ 334.000 0.0512281 0.0256141 0.999672i $$-0.491846\pi$$
0.0256141 + 0.999672i $$0.491846\pi$$
$$350$$ 0 0
$$351$$ −7540.00 −1.14660
$$352$$ 0 0
$$353$$ 891.000 0.134343 0.0671716 0.997741i $$-0.478603\pi$$
0.0671716 + 0.997741i $$0.478603\pi$$
$$354$$ 0 0
$$355$$ 7007.00 1.04759
$$356$$ 0 0
$$357$$ 5980.00 0.886541
$$358$$ 0 0
$$359$$ −2476.00 −0.364006 −0.182003 0.983298i $$-0.558258\pi$$
−0.182003 + 0.983298i $$0.558258\pi$$
$$360$$ 0 0
$$361$$ 2357.00 0.343636
$$362$$ 0 0
$$363$$ 605.000 0.0874773
$$364$$ 0 0
$$365$$ 7140.00 1.02390
$$366$$ 0 0
$$367$$ −1379.00 −0.196140 −0.0980698 0.995180i $$-0.531267\pi$$
−0.0980698 + 0.995180i $$0.531267\pi$$
$$368$$ 0 0
$$369$$ 944.000 0.133178
$$370$$ 0 0
$$371$$ 19604.0 2.74337
$$372$$ 0 0
$$373$$ −6266.00 −0.869816 −0.434908 0.900475i $$-0.643219\pi$$
−0.434908 + 0.900475i $$0.643219\pi$$
$$374$$ 0 0
$$375$$ 7035.00 0.968762
$$376$$ 0 0
$$377$$ 832.000 0.113661
$$378$$ 0 0
$$379$$ −151.000 −0.0204653 −0.0102327 0.999948i $$-0.503257\pi$$
−0.0102327 + 0.999948i $$0.503257\pi$$
$$380$$ 0 0
$$381$$ −8680.00 −1.16717
$$382$$ 0 0
$$383$$ 1989.00 0.265361 0.132680 0.991159i $$-0.457642\pi$$
0.132680 + 0.991159i $$0.457642\pi$$
$$384$$ 0 0
$$385$$ −2002.00 −0.265017
$$386$$ 0 0
$$387$$ −220.000 −0.0288972
$$388$$ 0 0
$$389$$ 6817.00 0.888523 0.444262 0.895897i $$-0.353466\pi$$
0.444262 + 0.895897i $$0.353466\pi$$
$$390$$ 0 0
$$391$$ −1242.00 −0.160641
$$392$$ 0 0
$$393$$ −30.0000 −0.00385064
$$394$$ 0 0
$$395$$ 3682.00 0.469017
$$396$$ 0 0
$$397$$ −4170.00 −0.527170 −0.263585 0.964636i $$-0.584905\pi$$
−0.263585 + 0.964636i $$0.584905\pi$$
$$398$$ 0 0
$$399$$ 12480.0 1.56587
$$400$$ 0 0
$$401$$ 10914.0 1.35915 0.679575 0.733606i $$-0.262164\pi$$
0.679575 + 0.733606i $$0.262164\pi$$
$$402$$ 0 0
$$403$$ 15236.0 1.88327
$$404$$ 0 0
$$405$$ 4697.00 0.576286
$$406$$ 0 0
$$407$$ −319.000 −0.0388507
$$408$$ 0 0
$$409$$ −1102.00 −0.133228 −0.0666142 0.997779i $$-0.521220\pi$$
−0.0666142 + 0.997779i $$0.521220\pi$$
$$410$$ 0 0
$$411$$ −7195.00 −0.863511
$$412$$ 0 0
$$413$$ −21450.0 −2.55565
$$414$$ 0 0
$$415$$ −1106.00 −0.130823
$$416$$ 0 0
$$417$$ 1590.00 0.186721
$$418$$ 0 0
$$419$$ −11028.0 −1.28581 −0.642903 0.765947i $$-0.722270\pi$$
−0.642903 + 0.765947i $$0.722270\pi$$
$$420$$ 0 0
$$421$$ 2622.00 0.303536 0.151768 0.988416i $$-0.451503\pi$$
0.151768 + 0.988416i $$0.451503\pi$$
$$422$$ 0 0
$$423$$ −448.000 −0.0514953
$$424$$ 0 0
$$425$$ −3496.00 −0.399014
$$426$$ 0 0
$$427$$ −14248.0 −1.61478
$$428$$ 0 0
$$429$$ 2860.00 0.321870
$$430$$ 0 0
$$431$$ −16598.0 −1.85498 −0.927491 0.373845i $$-0.878039\pi$$
−0.927491 + 0.373845i $$0.878039\pi$$
$$432$$ 0 0
$$433$$ −5763.00 −0.639612 −0.319806 0.947483i $$-0.603618\pi$$
−0.319806 + 0.947483i $$0.603618\pi$$
$$434$$ 0 0
$$435$$ −560.000 −0.0617240
$$436$$ 0 0
$$437$$ −2592.00 −0.283735
$$438$$ 0 0
$$439$$ 3128.00 0.340071 0.170036 0.985438i $$-0.445612\pi$$
0.170036 + 0.985438i $$0.445612\pi$$
$$440$$ 0 0
$$441$$ −666.000 −0.0719145
$$442$$ 0 0
$$443$$ −6369.00 −0.683071 −0.341535 0.939869i $$-0.610947\pi$$
−0.341535 + 0.939869i $$0.610947\pi$$
$$444$$ 0 0
$$445$$ 8519.00 0.907504
$$446$$ 0 0
$$447$$ −4610.00 −0.487798
$$448$$ 0 0
$$449$$ 8691.00 0.913483 0.456741 0.889600i $$-0.349017\pi$$
0.456741 + 0.889600i $$0.349017\pi$$
$$450$$ 0 0
$$451$$ −5192.00 −0.542088
$$452$$ 0 0
$$453$$ 5150.00 0.534146
$$454$$ 0 0
$$455$$ −9464.00 −0.975119
$$456$$ 0 0
$$457$$ 2260.00 0.231331 0.115666 0.993288i $$-0.463100\pi$$
0.115666 + 0.993288i $$0.463100\pi$$
$$458$$ 0 0
$$459$$ −6670.00 −0.678277
$$460$$ 0 0
$$461$$ −12756.0 −1.28873 −0.644367 0.764717i $$-0.722879\pi$$
−0.644367 + 0.764717i $$0.722879\pi$$
$$462$$ 0 0
$$463$$ 6887.00 0.691287 0.345644 0.938366i $$-0.387661\pi$$
0.345644 + 0.938366i $$0.387661\pi$$
$$464$$ 0 0
$$465$$ −10255.0 −1.02272
$$466$$ 0 0
$$467$$ 1535.00 0.152101 0.0760507 0.997104i $$-0.475769\pi$$
0.0760507 + 0.997104i $$0.475769\pi$$
$$468$$ 0 0
$$469$$ 3198.00 0.314861
$$470$$ 0 0
$$471$$ 5085.00 0.497462
$$472$$ 0 0
$$473$$ 1210.00 0.117623
$$474$$ 0 0
$$475$$ −7296.00 −0.704765
$$476$$ 0 0
$$477$$ −1508.00 −0.144752
$$478$$ 0 0
$$479$$ 17564.0 1.67541 0.837703 0.546126i $$-0.183898\pi$$
0.837703 + 0.546126i $$0.183898\pi$$
$$480$$ 0 0
$$481$$ −1508.00 −0.142950
$$482$$ 0 0
$$483$$ −3510.00 −0.330664
$$484$$ 0 0
$$485$$ 1841.00 0.172362
$$486$$ 0 0
$$487$$ 7541.00 0.701674 0.350837 0.936437i $$-0.385897\pi$$
0.350837 + 0.936437i $$0.385897\pi$$
$$488$$ 0 0
$$489$$ 12220.0 1.13008
$$490$$ 0 0
$$491$$ −12552.0 −1.15369 −0.576847 0.816852i $$-0.695717\pi$$
−0.576847 + 0.816852i $$0.695717\pi$$
$$492$$ 0 0
$$493$$ 736.000 0.0672369
$$494$$ 0 0
$$495$$ 154.000 0.0139834
$$496$$ 0 0
$$497$$ −26026.0 −2.34894
$$498$$ 0 0
$$499$$ 8396.00 0.753220 0.376610 0.926372i $$-0.377090\pi$$
0.376610 + 0.926372i $$0.377090\pi$$
$$500$$ 0 0
$$501$$ 7260.00 0.647411
$$502$$ 0 0
$$503$$ 12194.0 1.08092 0.540461 0.841369i $$-0.318250\pi$$
0.540461 + 0.841369i $$0.318250\pi$$
$$504$$ 0 0
$$505$$ 5698.00 0.502094
$$506$$ 0 0
$$507$$ 2535.00 0.222058
$$508$$ 0 0
$$509$$ 18295.0 1.59315 0.796573 0.604542i $$-0.206644\pi$$
0.796573 + 0.604542i $$0.206644\pi$$
$$510$$ 0 0
$$511$$ −26520.0 −2.29584
$$512$$ 0 0
$$513$$ −13920.0 −1.19802
$$514$$ 0 0
$$515$$ 2632.00 0.225203
$$516$$ 0 0
$$517$$ 2464.00 0.209607
$$518$$ 0 0
$$519$$ 9570.00 0.809396
$$520$$ 0 0
$$521$$ 7101.00 0.597122 0.298561 0.954391i $$-0.403493\pi$$
0.298561 + 0.954391i $$0.403493\pi$$
$$522$$ 0 0
$$523$$ 4912.00 0.410682 0.205341 0.978690i $$-0.434170\pi$$
0.205341 + 0.978690i $$0.434170\pi$$
$$524$$ 0 0
$$525$$ −9880.00 −0.821331
$$526$$ 0 0
$$527$$ 13478.0 1.11406
$$528$$ 0 0
$$529$$ −11438.0 −0.940084
$$530$$ 0 0
$$531$$ 1650.00 0.134847
$$532$$ 0 0
$$533$$ −24544.0 −1.99459
$$534$$ 0 0
$$535$$ −10458.0 −0.845119
$$536$$ 0 0
$$537$$ −6465.00 −0.519526
$$538$$ 0 0
$$539$$ 3663.00 0.292721
$$540$$ 0 0
$$541$$ 11496.0 0.913589 0.456794 0.889572i $$-0.348997\pi$$
0.456794 + 0.889572i $$0.348997\pi$$
$$542$$ 0 0
$$543$$ 2275.00 0.179797
$$544$$ 0 0
$$545$$ −5894.00 −0.463250
$$546$$ 0 0
$$547$$ −19048.0 −1.48891 −0.744455 0.667673i $$-0.767291\pi$$
−0.744455 + 0.667673i $$0.767291\pi$$
$$548$$ 0 0
$$549$$ 1096.00 0.0852024
$$550$$ 0 0
$$551$$ 1536.00 0.118758
$$552$$ 0 0
$$553$$ −13676.0 −1.05165
$$554$$ 0 0
$$555$$ 1015.00 0.0776295
$$556$$ 0 0
$$557$$ 10718.0 0.815325 0.407663 0.913133i $$-0.366344\pi$$
0.407663 + 0.913133i $$0.366344\pi$$
$$558$$ 0 0
$$559$$ 5720.00 0.432791
$$560$$ 0 0
$$561$$ 2530.00 0.190404
$$562$$ 0 0
$$563$$ −6660.00 −0.498553 −0.249277 0.968432i $$-0.580193\pi$$
−0.249277 + 0.968432i $$0.580193\pi$$
$$564$$ 0 0
$$565$$ −8967.00 −0.667689
$$566$$ 0 0
$$567$$ −17446.0 −1.29217
$$568$$ 0 0
$$569$$ −10960.0 −0.807499 −0.403750 0.914870i $$-0.632293\pi$$
−0.403750 + 0.914870i $$0.632293\pi$$
$$570$$ 0 0
$$571$$ −18596.0 −1.36290 −0.681452 0.731863i $$-0.738651\pi$$
−0.681452 + 0.731863i $$0.738651\pi$$
$$572$$ 0 0
$$573$$ 5575.00 0.406455
$$574$$ 0 0
$$575$$ 2052.00 0.148825
$$576$$ 0 0
$$577$$ −21119.0 −1.52374 −0.761868 0.647733i $$-0.775717\pi$$
−0.761868 + 0.647733i $$0.775717\pi$$
$$578$$ 0 0
$$579$$ 25060.0 1.79872
$$580$$ 0 0
$$581$$ 4108.00 0.293337
$$582$$ 0 0
$$583$$ 8294.00 0.589198
$$584$$ 0 0
$$585$$ 728.000 0.0514514
$$586$$ 0 0
$$587$$ 2836.00 0.199411 0.0997055 0.995017i $$-0.468210\pi$$
0.0997055 + 0.995017i $$0.468210\pi$$
$$588$$ 0 0
$$589$$ 28128.0 1.96773
$$590$$ 0 0
$$591$$ −20730.0 −1.44284
$$592$$ 0 0
$$593$$ 18044.0 1.24954 0.624771 0.780808i $$-0.285192\pi$$
0.624771 + 0.780808i $$0.285192\pi$$
$$594$$ 0 0
$$595$$ −8372.00 −0.576838
$$596$$ 0 0
$$597$$ 6200.00 0.425040
$$598$$ 0 0
$$599$$ 9264.00 0.631914 0.315957 0.948773i $$-0.397674\pi$$
0.315957 + 0.948773i $$0.397674\pi$$
$$600$$ 0 0
$$601$$ −19326.0 −1.31169 −0.655844 0.754897i $$-0.727687\pi$$
−0.655844 + 0.754897i $$0.727687\pi$$
$$602$$ 0 0
$$603$$ −246.000 −0.0166134
$$604$$ 0 0
$$605$$ −847.000 −0.0569181
$$606$$ 0 0
$$607$$ −10082.0 −0.674161 −0.337081 0.941476i $$-0.609439\pi$$
−0.337081 + 0.941476i $$0.609439\pi$$
$$608$$ 0 0
$$609$$ 2080.00 0.138400
$$610$$ 0 0
$$611$$ 11648.0 0.771240
$$612$$ 0 0
$$613$$ 13088.0 0.862348 0.431174 0.902269i $$-0.358100\pi$$
0.431174 + 0.902269i $$0.358100\pi$$
$$614$$ 0 0
$$615$$ 16520.0 1.08317
$$616$$ 0 0
$$617$$ 23426.0 1.52852 0.764259 0.644910i $$-0.223105\pi$$
0.764259 + 0.644910i $$0.223105\pi$$
$$618$$ 0 0
$$619$$ −23587.0 −1.53157 −0.765785 0.643097i $$-0.777649\pi$$
−0.765785 + 0.643097i $$0.777649\pi$$
$$620$$ 0 0
$$621$$ 3915.00 0.252985
$$622$$ 0 0
$$623$$ −31642.0 −2.03485
$$624$$ 0 0
$$625$$ −349.000 −0.0223360
$$626$$ 0 0
$$627$$ 5280.00 0.336304
$$628$$ 0 0
$$629$$ −1334.00 −0.0845629
$$630$$ 0 0
$$631$$ −19683.0 −1.24179 −0.620894 0.783895i $$-0.713230\pi$$
−0.620894 + 0.783895i $$0.713230\pi$$
$$632$$ 0 0
$$633$$ 14100.0 0.885347
$$634$$ 0 0
$$635$$ 12152.0 0.759429
$$636$$ 0 0
$$637$$ 17316.0 1.07706
$$638$$ 0 0
$$639$$ 2002.00 0.123940
$$640$$ 0 0
$$641$$ 375.000 0.0231070 0.0115535 0.999933i $$-0.496322\pi$$
0.0115535 + 0.999933i $$0.496322\pi$$
$$642$$ 0 0
$$643$$ 21055.0 1.29133 0.645667 0.763619i $$-0.276579\pi$$
0.645667 + 0.763619i $$0.276579\pi$$
$$644$$ 0 0
$$645$$ −3850.00 −0.235029
$$646$$ 0 0
$$647$$ 6427.00 0.390528 0.195264 0.980751i $$-0.437444\pi$$
0.195264 + 0.980751i $$0.437444\pi$$
$$648$$ 0 0
$$649$$ −9075.00 −0.548883
$$650$$ 0 0
$$651$$ 38090.0 2.29319
$$652$$ 0 0
$$653$$ −7617.00 −0.456472 −0.228236 0.973606i $$-0.573296\pi$$
−0.228236 + 0.973606i $$0.573296\pi$$
$$654$$ 0 0
$$655$$ 42.0000 0.00250546
$$656$$ 0 0
$$657$$ 2040.00 0.121138
$$658$$ 0 0
$$659$$ 17630.0 1.04214 0.521068 0.853515i $$-0.325534\pi$$
0.521068 + 0.853515i $$0.325534\pi$$
$$660$$ 0 0
$$661$$ 4605.00 0.270974 0.135487 0.990779i $$-0.456740\pi$$
0.135487 + 0.990779i $$0.456740\pi$$
$$662$$ 0 0
$$663$$ 11960.0 0.700585
$$664$$ 0 0
$$665$$ −17472.0 −1.01885
$$666$$ 0 0
$$667$$ −432.000 −0.0250781
$$668$$ 0 0
$$669$$ −18475.0 −1.06769
$$670$$ 0 0
$$671$$ −6028.00 −0.346808
$$672$$ 0 0
$$673$$ −2818.00 −0.161406 −0.0807028 0.996738i $$-0.525716\pi$$
−0.0807028 + 0.996738i $$0.525716\pi$$
$$674$$ 0 0
$$675$$ 11020.0 0.628385
$$676$$ 0 0
$$677$$ 8438.00 0.479023 0.239512 0.970894i $$-0.423013\pi$$
0.239512 + 0.970894i $$0.423013\pi$$
$$678$$ 0 0
$$679$$ −6838.00 −0.386478
$$680$$ 0 0
$$681$$ 2430.00 0.136737
$$682$$ 0 0
$$683$$ 17344.0 0.971669 0.485834 0.874051i $$-0.338516\pi$$
0.485834 + 0.874051i $$0.338516\pi$$
$$684$$ 0 0
$$685$$ 10073.0 0.561853
$$686$$ 0 0
$$687$$ 21155.0 1.17484
$$688$$ 0 0
$$689$$ 39208.0 2.16793
$$690$$ 0 0
$$691$$ 3947.00 0.217295 0.108648 0.994080i $$-0.465348\pi$$
0.108648 + 0.994080i $$0.465348\pi$$
$$692$$ 0 0
$$693$$ −572.000 −0.0313542
$$694$$ 0 0
$$695$$ −2226.00 −0.121492
$$696$$ 0 0
$$697$$ −21712.0 −1.17991
$$698$$ 0 0
$$699$$ −16680.0 −0.902569
$$700$$ 0 0
$$701$$ −7998.00 −0.430928 −0.215464 0.976512i $$-0.569126\pi$$
−0.215464 + 0.976512i $$0.569126\pi$$
$$702$$ 0 0
$$703$$ −2784.00 −0.149361
$$704$$ 0 0
$$705$$ −7840.00 −0.418825
$$706$$ 0 0
$$707$$ −21164.0 −1.12582
$$708$$ 0 0
$$709$$ −881.000 −0.0466666 −0.0233333 0.999728i $$-0.507428\pi$$
−0.0233333 + 0.999728i $$0.507428\pi$$
$$710$$ 0 0
$$711$$ 1052.00 0.0554896
$$712$$ 0 0
$$713$$ −7911.00 −0.415525
$$714$$ 0 0
$$715$$ −4004.00 −0.209428
$$716$$ 0 0
$$717$$ −18050.0 −0.940153
$$718$$ 0 0
$$719$$ 26093.0 1.35341 0.676707 0.736252i $$-0.263406\pi$$
0.676707 + 0.736252i $$0.263406\pi$$
$$720$$ 0 0
$$721$$ −9776.00 −0.504962
$$722$$ 0 0
$$723$$ −32040.0 −1.64811
$$724$$ 0 0
$$725$$ −1216.00 −0.0622912
$$726$$ 0 0
$$727$$ 7481.00 0.381644 0.190822 0.981625i $$-0.438885\pi$$
0.190822 + 0.981625i $$0.438885\pi$$
$$728$$ 0 0
$$729$$ 20917.0 1.06269
$$730$$ 0 0
$$731$$ 5060.00 0.256020
$$732$$ 0 0
$$733$$ 17788.0 0.896337 0.448168 0.893949i $$-0.352077\pi$$
0.448168 + 0.893949i $$0.352077\pi$$
$$734$$ 0 0
$$735$$ −11655.0 −0.584900
$$736$$ 0 0
$$737$$ 1353.00 0.0676233
$$738$$ 0 0
$$739$$ 32182.0 1.60194 0.800970 0.598704i $$-0.204317\pi$$
0.800970 + 0.598704i $$0.204317\pi$$
$$740$$ 0 0
$$741$$ 24960.0 1.23742
$$742$$ 0 0
$$743$$ 32044.0 1.58221 0.791104 0.611682i $$-0.209507\pi$$
0.791104 + 0.611682i $$0.209507\pi$$
$$744$$ 0 0
$$745$$ 6454.00 0.317391
$$746$$ 0 0
$$747$$ −316.000 −0.0154777
$$748$$ 0 0
$$749$$ 38844.0 1.89497
$$750$$ 0 0
$$751$$ 33779.0 1.64130 0.820648 0.571434i $$-0.193613\pi$$
0.820648 + 0.571434i $$0.193613\pi$$
$$752$$ 0 0
$$753$$ 8935.00 0.432416
$$754$$ 0 0
$$755$$ −7210.00 −0.347548
$$756$$ 0 0
$$757$$ −15630.0 −0.750439 −0.375219 0.926936i $$-0.622433\pi$$
−0.375219 + 0.926936i $$0.622433\pi$$
$$758$$ 0 0
$$759$$ −1485.00 −0.0710172
$$760$$ 0 0
$$761$$ 1948.00 0.0927923 0.0463962 0.998923i $$-0.485226\pi$$
0.0463962 + 0.998923i $$0.485226\pi$$
$$762$$ 0 0
$$763$$ 21892.0 1.03872
$$764$$ 0 0
$$765$$ 644.000 0.0304364
$$766$$ 0 0
$$767$$ −42900.0 −2.01959
$$768$$ 0 0
$$769$$ −17420.0 −0.816881 −0.408440 0.912785i $$-0.633927\pi$$
−0.408440 + 0.912785i $$0.633927\pi$$
$$770$$ 0 0
$$771$$ −1770.00 −0.0826783
$$772$$ 0 0
$$773$$ 11122.0 0.517504 0.258752 0.965944i $$-0.416689\pi$$
0.258752 + 0.965944i $$0.416689\pi$$
$$774$$ 0 0
$$775$$ −22268.0 −1.03212
$$776$$ 0 0
$$777$$ −3770.00 −0.174064
$$778$$ 0 0
$$779$$ −45312.0 −2.08404
$$780$$ 0 0
$$781$$ −11011.0 −0.504487
$$782$$ 0 0
$$783$$ −2320.00 −0.105888
$$784$$ 0 0
$$785$$ −7119.00 −0.323679
$$786$$ 0 0
$$787$$ −29648.0 −1.34287 −0.671434 0.741064i $$-0.734321\pi$$
−0.671434 + 0.741064i $$0.734321\pi$$
$$788$$ 0 0
$$789$$ 10130.0 0.457082
$$790$$ 0 0
$$791$$ 33306.0 1.49712
$$792$$ 0 0
$$793$$ −28496.0 −1.27607
$$794$$ 0 0
$$795$$ −26390.0 −1.17730
$$796$$ 0 0
$$797$$ −23219.0 −1.03194 −0.515972 0.856606i $$-0.672569\pi$$
−0.515972 + 0.856606i $$0.672569\pi$$
$$798$$ 0 0
$$799$$ 10304.0 0.456232
$$800$$ 0 0
$$801$$ 2434.00 0.107367
$$802$$ 0 0
$$803$$ −11220.0 −0.493082
$$804$$ 0 0
$$805$$ 4914.00 0.215150
$$806$$ 0 0
$$807$$ −43750.0 −1.90839
$$808$$ 0 0
$$809$$ 9232.00 0.401211 0.200606 0.979672i $$-0.435709\pi$$
0.200606 + 0.979672i $$0.435709\pi$$
$$810$$ 0 0
$$811$$ 32286.0 1.39792 0.698961 0.715160i $$-0.253646\pi$$
0.698961 + 0.715160i $$0.253646\pi$$
$$812$$ 0 0
$$813$$ 25180.0 1.08623
$$814$$ 0 0
$$815$$ −17108.0 −0.735297
$$816$$ 0 0
$$817$$ 10560.0 0.452200
$$818$$ 0 0
$$819$$ −2704.00 −0.115367
$$820$$ 0 0
$$821$$ −31706.0 −1.34780 −0.673902 0.738821i $$-0.735383\pi$$
−0.673902 + 0.738821i $$0.735383\pi$$
$$822$$ 0 0
$$823$$ −41139.0 −1.74242 −0.871212 0.490906i $$-0.836666\pi$$
−0.871212 + 0.490906i $$0.836666\pi$$
$$824$$ 0 0
$$825$$ −4180.00 −0.176399
$$826$$ 0 0
$$827$$ 15252.0 0.641311 0.320655 0.947196i $$-0.396097\pi$$
0.320655 + 0.947196i $$0.396097\pi$$
$$828$$ 0 0
$$829$$ 369.000 0.0154595 0.00772973 0.999970i $$-0.497540\pi$$
0.00772973 + 0.999970i $$0.497540\pi$$
$$830$$ 0 0
$$831$$ −21530.0 −0.898757
$$832$$ 0 0
$$833$$ 15318.0 0.637140
$$834$$ 0 0
$$835$$ −10164.0 −0.421245
$$836$$ 0 0
$$837$$ −42485.0 −1.75448
$$838$$ 0 0
$$839$$ 10257.0 0.422063 0.211032 0.977479i $$-0.432318\pi$$
0.211032 + 0.977479i $$0.432318\pi$$
$$840$$ 0 0
$$841$$ −24133.0 −0.989503
$$842$$ 0 0
$$843$$ −6010.00 −0.245546
$$844$$ 0 0
$$845$$ −3549.00 −0.144484
$$846$$ 0 0
$$847$$ 3146.00 0.127624
$$848$$ 0 0
$$849$$ 33100.0 1.33803
$$850$$ 0 0
$$851$$ 783.000 0.0315404
$$852$$ 0 0
$$853$$ −34386.0 −1.38025 −0.690126 0.723690i $$-0.742445\pi$$
−0.690126 + 0.723690i $$0.742445\pi$$
$$854$$ 0 0
$$855$$ 1344.00 0.0537589
$$856$$ 0 0
$$857$$ −23464.0 −0.935257 −0.467628 0.883925i $$-0.654891\pi$$
−0.467628 + 0.883925i $$0.654891\pi$$
$$858$$ 0 0
$$859$$ 22475.0 0.892709 0.446355 0.894856i $$-0.352722\pi$$
0.446355 + 0.894856i $$0.352722\pi$$
$$860$$ 0 0
$$861$$ −61360.0 −2.42874
$$862$$ 0 0
$$863$$ −2880.00 −0.113599 −0.0567997 0.998386i $$-0.518090\pi$$
−0.0567997 + 0.998386i $$0.518090\pi$$
$$864$$ 0 0
$$865$$ −13398.0 −0.526642
$$866$$ 0 0
$$867$$ −13985.0 −0.547815
$$868$$ 0 0
$$869$$ −5786.00 −0.225865
$$870$$ 0 0
$$871$$ 6396.00 0.248818
$$872$$ 0 0
$$873$$ 526.000 0.0203922
$$874$$ 0 0
$$875$$ 36582.0 1.41337
$$876$$ 0 0
$$877$$ −11084.0 −0.426773 −0.213387 0.976968i $$-0.568449\pi$$
−0.213387 + 0.976968i $$0.568449\pi$$
$$878$$ 0 0
$$879$$ 34840.0 1.33689
$$880$$ 0 0
$$881$$ 41797.0 1.59838 0.799192 0.601076i $$-0.205261\pi$$
0.799192 + 0.601076i $$0.205261\pi$$
$$882$$ 0 0
$$883$$ −23780.0 −0.906298 −0.453149 0.891435i $$-0.649699\pi$$
−0.453149 + 0.891435i $$0.649699\pi$$
$$884$$ 0 0
$$885$$ 28875.0 1.09675
$$886$$ 0 0
$$887$$ 14334.0 0.542603 0.271301 0.962494i $$-0.412546\pi$$
0.271301 + 0.962494i $$0.412546\pi$$
$$888$$ 0 0
$$889$$ −45136.0 −1.70283
$$890$$ 0 0
$$891$$ −7381.00 −0.277523
$$892$$ 0 0
$$893$$ 21504.0 0.805827
$$894$$ 0 0
$$895$$ 9051.00 0.338035
$$896$$ 0 0
$$897$$ −7020.00 −0.261305
$$898$$ 0 0
$$899$$ 4688.00 0.173919
$$900$$ 0 0
$$901$$ 34684.0 1.28245
$$902$$ 0 0
$$903$$ 14300.0 0.526992
$$904$$ 0 0
$$905$$ −3185.00 −0.116987
$$906$$ 0 0
$$907$$ −25716.0 −0.941440 −0.470720 0.882283i $$-0.656006\pi$$
−0.470720 + 0.882283i $$0.656006\pi$$
$$908$$ 0 0
$$909$$ 1628.00 0.0594030
$$910$$ 0 0
$$911$$ 28300.0 1.02922 0.514611 0.857424i $$-0.327936\pi$$
0.514611 + 0.857424i $$0.327936\pi$$
$$912$$ 0 0
$$913$$ 1738.00 0.0630004
$$914$$ 0 0
$$915$$ 19180.0 0.692974
$$916$$ 0 0
$$917$$ −156.000 −0.00561786
$$918$$ 0 0
$$919$$ −21410.0 −0.768499 −0.384250 0.923229i $$-0.625540\pi$$
−0.384250 + 0.923229i $$0.625540\pi$$
$$920$$ 0 0
$$921$$ −38200.0 −1.36670
$$922$$ 0 0
$$923$$ −52052.0 −1.85624
$$924$$ 0 0
$$925$$ 2204.00 0.0783428
$$926$$ 0 0
$$927$$ 752.000 0.0266439
$$928$$ 0 0
$$929$$ 19938.0 0.704138 0.352069 0.935974i $$-0.385478\pi$$
0.352069 + 0.935974i $$0.385478\pi$$
$$930$$ 0 0
$$931$$ 31968.0 1.12536
$$932$$ 0 0
$$933$$ 3260.00 0.114392
$$934$$ 0 0
$$935$$ −3542.00 −0.123889
$$936$$ 0 0
$$937$$ 26728.0 0.931874 0.465937 0.884818i $$-0.345717\pi$$
0.465937 + 0.884818i $$0.345717\pi$$
$$938$$ 0 0
$$939$$ 40275.0 1.39971
$$940$$ 0 0
$$941$$ 8634.00 0.299108 0.149554 0.988754i $$-0.452216\pi$$
0.149554 + 0.988754i $$0.452216\pi$$
$$942$$ 0 0
$$943$$ 12744.0 0.440087
$$944$$ 0 0
$$945$$ 26390.0 0.908430
$$946$$ 0 0
$$947$$ −24841.0 −0.852401 −0.426201 0.904629i $$-0.640148\pi$$
−0.426201 + 0.904629i $$0.640148\pi$$
$$948$$ 0 0
$$949$$ −53040.0 −1.81428
$$950$$ 0 0
$$951$$ −28375.0 −0.967531
$$952$$ 0 0
$$953$$ 51234.0 1.74148 0.870741 0.491742i $$-0.163640\pi$$
0.870741 + 0.491742i $$0.163640\pi$$
$$954$$ 0 0
$$955$$ −7805.00 −0.264465
$$956$$ 0 0
$$957$$ 880.000 0.0297245
$$958$$ 0 0
$$959$$ −37414.0 −1.25981
$$960$$ 0 0
$$961$$ 56058.0 1.88171
$$962$$ 0 0
$$963$$ −2988.00 −0.0999865
$$964$$ 0 0
$$965$$ −35084.0 −1.17036
$$966$$ 0 0
$$967$$ 26752.0 0.889645 0.444822 0.895619i $$-0.353267\pi$$
0.444822 + 0.895619i $$0.353267\pi$$
$$968$$ 0 0
$$969$$ 22080.0 0.732004
$$970$$ 0 0
$$971$$ −21155.0 −0.699172 −0.349586 0.936904i $$-0.613678\pi$$
−0.349586 + 0.936904i $$0.613678\pi$$
$$972$$ 0 0
$$973$$ 8268.00 0.272415
$$974$$ 0 0
$$975$$ −19760.0 −0.649053
$$976$$ 0 0
$$977$$ −56323.0 −1.84435 −0.922176 0.386770i $$-0.873591\pi$$
−0.922176 + 0.386770i $$0.873591\pi$$
$$978$$ 0 0
$$979$$ −13387.0 −0.437028
$$980$$ 0 0
$$981$$ −1684.00 −0.0548073
$$982$$ 0 0
$$983$$ −24683.0 −0.800880 −0.400440 0.916323i $$-0.631143\pi$$
−0.400440 + 0.916323i $$0.631143\pi$$
$$984$$ 0 0
$$985$$ 29022.0 0.938800
$$986$$ 0 0
$$987$$ 29120.0 0.939108
$$988$$ 0 0
$$989$$ −2970.00 −0.0954909
$$990$$ 0 0
$$991$$ 26816.0 0.859574 0.429787 0.902930i $$-0.358589\pi$$
0.429787 + 0.902930i $$0.358589\pi$$
$$992$$ 0 0
$$993$$ 11225.0 0.358726
$$994$$ 0 0
$$995$$ −8680.00 −0.276557
$$996$$ 0 0
$$997$$ 18614.0 0.591285 0.295643 0.955299i $$-0.404466\pi$$
0.295643 + 0.955299i $$0.404466\pi$$
$$998$$ 0 0
$$999$$ 4205.00 0.133173
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 176.4.a.e.1.1 1
3.2 odd 2 1584.4.a.p.1.1 1
4.3 odd 2 44.4.a.a.1.1 1
8.3 odd 2 704.4.a.j.1.1 1
8.5 even 2 704.4.a.c.1.1 1
11.10 odd 2 1936.4.a.m.1.1 1
12.11 even 2 396.4.a.e.1.1 1
20.3 even 4 1100.4.b.c.749.1 2
20.7 even 4 1100.4.b.c.749.2 2
20.19 odd 2 1100.4.a.d.1.1 1
28.27 even 2 2156.4.a.b.1.1 1
44.43 even 2 484.4.a.a.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
44.4.a.a.1.1 1 4.3 odd 2
176.4.a.e.1.1 1 1.1 even 1 trivial
396.4.a.e.1.1 1 12.11 even 2
484.4.a.a.1.1 1 44.43 even 2
704.4.a.c.1.1 1 8.5 even 2
704.4.a.j.1.1 1 8.3 odd 2
1100.4.a.d.1.1 1 20.19 odd 2
1100.4.b.c.749.1 2 20.3 even 4
1100.4.b.c.749.2 2 20.7 even 4
1584.4.a.p.1.1 1 3.2 odd 2
1936.4.a.m.1.1 1 11.10 odd 2
2156.4.a.b.1.1 1 28.27 even 2