# Properties

 Label 624.4.a.a.1.1 Level $624$ Weight $4$ Character 624.1 Self dual yes Analytic conductor $36.817$ 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] = [624,4,Mod(1,624)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(624, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("624.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$624 = 2^{4} \cdot 3 \cdot 13$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 624.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: $$36.8171918436$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 78) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 624.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-3.00000 q^{3} -16.0000 q^{5} +8.00000 q^{7} +9.00000 q^{9} +O(q^{10})$$ $$q-3.00000 q^{3} -16.0000 q^{5} +8.00000 q^{7} +9.00000 q^{9} +38.0000 q^{11} -13.0000 q^{13} +48.0000 q^{15} -78.0000 q^{17} +72.0000 q^{19} -24.0000 q^{21} +52.0000 q^{23} +131.000 q^{25} -27.0000 q^{27} +242.000 q^{29} -76.0000 q^{31} -114.000 q^{33} -128.000 q^{35} +342.000 q^{37} +39.0000 q^{39} -336.000 q^{41} -76.0000 q^{43} -144.000 q^{45} -94.0000 q^{47} -279.000 q^{49} +234.000 q^{51} -450.000 q^{53} -608.000 q^{55} -216.000 q^{57} -854.000 q^{59} -110.000 q^{61} +72.0000 q^{63} +208.000 q^{65} +908.000 q^{67} -156.000 q^{69} -838.000 q^{71} -970.000 q^{73} -393.000 q^{75} +304.000 q^{77} +352.000 q^{79} +81.0000 q^{81} -474.000 q^{83} +1248.00 q^{85} -726.000 q^{87} -1452.00 q^{89} -104.000 q^{91} +228.000 q^{93} -1152.00 q^{95} -562.000 q^{97} +342.000 q^{99} +O(q^{100})$$

## 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
$$3$$ −3.00000 −0.577350
$$4$$ 0 0
$$5$$ −16.0000 −1.43108 −0.715542 0.698570i $$-0.753820\pi$$
−0.715542 + 0.698570i $$0.753820\pi$$
$$6$$ 0 0
$$7$$ 8.00000 0.431959 0.215980 0.976398i $$-0.430705\pi$$
0.215980 + 0.976398i $$0.430705\pi$$
$$8$$ 0 0
$$9$$ 9.00000 0.333333
$$10$$ 0 0
$$11$$ 38.0000 1.04158 0.520792 0.853683i $$-0.325637\pi$$
0.520792 + 0.853683i $$0.325637\pi$$
$$12$$ 0 0
$$13$$ −13.0000 −0.277350
$$14$$ 0 0
$$15$$ 48.0000 0.826236
$$16$$ 0 0
$$17$$ −78.0000 −1.11281 −0.556405 0.830911i $$-0.687820\pi$$
−0.556405 + 0.830911i $$0.687820\pi$$
$$18$$ 0 0
$$19$$ 72.0000 0.869365 0.434682 0.900584i $$-0.356861\pi$$
0.434682 + 0.900584i $$0.356861\pi$$
$$20$$ 0 0
$$21$$ −24.0000 −0.249392
$$22$$ 0 0
$$23$$ 52.0000 0.471424 0.235712 0.971823i $$-0.424258\pi$$
0.235712 + 0.971823i $$0.424258\pi$$
$$24$$ 0 0
$$25$$ 131.000 1.04800
$$26$$ 0 0
$$27$$ −27.0000 −0.192450
$$28$$ 0 0
$$29$$ 242.000 1.54960 0.774798 0.632209i $$-0.217852\pi$$
0.774798 + 0.632209i $$0.217852\pi$$
$$30$$ 0 0
$$31$$ −76.0000 −0.440323 −0.220161 0.975463i $$-0.570658\pi$$
−0.220161 + 0.975463i $$0.570658\pi$$
$$32$$ 0 0
$$33$$ −114.000 −0.601359
$$34$$ 0 0
$$35$$ −128.000 −0.618170
$$36$$ 0 0
$$37$$ 342.000 1.51958 0.759790 0.650169i $$-0.225302\pi$$
0.759790 + 0.650169i $$0.225302\pi$$
$$38$$ 0 0
$$39$$ 39.0000 0.160128
$$40$$ 0 0
$$41$$ −336.000 −1.27986 −0.639932 0.768432i $$-0.721037\pi$$
−0.639932 + 0.768432i $$0.721037\pi$$
$$42$$ 0 0
$$43$$ −76.0000 −0.269532 −0.134766 0.990877i $$-0.543028\pi$$
−0.134766 + 0.990877i $$0.543028\pi$$
$$44$$ 0 0
$$45$$ −144.000 −0.477028
$$46$$ 0 0
$$47$$ −94.0000 −0.291730 −0.145865 0.989305i $$-0.546597\pi$$
−0.145865 + 0.989305i $$0.546597\pi$$
$$48$$ 0 0
$$49$$ −279.000 −0.813411
$$50$$ 0 0
$$51$$ 234.000 0.642481
$$52$$ 0 0
$$53$$ −450.000 −1.16627 −0.583134 0.812376i $$-0.698174\pi$$
−0.583134 + 0.812376i $$0.698174\pi$$
$$54$$ 0 0
$$55$$ −608.000 −1.49059
$$56$$ 0 0
$$57$$ −216.000 −0.501928
$$58$$ 0 0
$$59$$ −854.000 −1.88443 −0.942215 0.335010i $$-0.891260\pi$$
−0.942215 + 0.335010i $$0.891260\pi$$
$$60$$ 0 0
$$61$$ −110.000 −0.230886 −0.115443 0.993314i $$-0.536829\pi$$
−0.115443 + 0.993314i $$0.536829\pi$$
$$62$$ 0 0
$$63$$ 72.0000 0.143986
$$64$$ 0 0
$$65$$ 208.000 0.396911
$$66$$ 0 0
$$67$$ 908.000 1.65567 0.827835 0.560972i $$-0.189572\pi$$
0.827835 + 0.560972i $$0.189572\pi$$
$$68$$ 0 0
$$69$$ −156.000 −0.272177
$$70$$ 0 0
$$71$$ −838.000 −1.40074 −0.700368 0.713782i $$-0.746981\pi$$
−0.700368 + 0.713782i $$0.746981\pi$$
$$72$$ 0 0
$$73$$ −970.000 −1.55520 −0.777602 0.628757i $$-0.783564\pi$$
−0.777602 + 0.628757i $$0.783564\pi$$
$$74$$ 0 0
$$75$$ −393.000 −0.605063
$$76$$ 0 0
$$77$$ 304.000 0.449922
$$78$$ 0 0
$$79$$ 352.000 0.501305 0.250652 0.968077i $$-0.419355\pi$$
0.250652 + 0.968077i $$0.419355\pi$$
$$80$$ 0 0
$$81$$ 81.0000 0.111111
$$82$$ 0 0
$$83$$ −474.000 −0.626846 −0.313423 0.949614i $$-0.601476\pi$$
−0.313423 + 0.949614i $$0.601476\pi$$
$$84$$ 0 0
$$85$$ 1248.00 1.59252
$$86$$ 0 0
$$87$$ −726.000 −0.894659
$$88$$ 0 0
$$89$$ −1452.00 −1.72934 −0.864672 0.502336i $$-0.832474\pi$$
−0.864672 + 0.502336i $$0.832474\pi$$
$$90$$ 0 0
$$91$$ −104.000 −0.119804
$$92$$ 0 0
$$93$$ 228.000 0.254220
$$94$$ 0 0
$$95$$ −1152.00 −1.24413
$$96$$ 0 0
$$97$$ −562.000 −0.588273 −0.294136 0.955763i $$-0.595032\pi$$
−0.294136 + 0.955763i $$0.595032\pi$$
$$98$$ 0 0
$$99$$ 342.000 0.347195
$$100$$ 0 0
$$101$$ 466.000 0.459096 0.229548 0.973297i $$-0.426275\pi$$
0.229548 + 0.973297i $$0.426275\pi$$
$$102$$ 0 0
$$103$$ 1448.00 1.38520 0.692600 0.721321i $$-0.256465\pi$$
0.692600 + 0.721321i $$0.256465\pi$$
$$104$$ 0 0
$$105$$ 384.000 0.356901
$$106$$ 0 0
$$107$$ 424.000 0.383081 0.191540 0.981485i $$-0.438652\pi$$
0.191540 + 0.981485i $$0.438652\pi$$
$$108$$ 0 0
$$109$$ −782.000 −0.687174 −0.343587 0.939121i $$-0.611642\pi$$
−0.343587 + 0.939121i $$0.611642\pi$$
$$110$$ 0 0
$$111$$ −1026.00 −0.877330
$$112$$ 0 0
$$113$$ −634.000 −0.527803 −0.263901 0.964550i $$-0.585009\pi$$
−0.263901 + 0.964550i $$0.585009\pi$$
$$114$$ 0 0
$$115$$ −832.000 −0.674647
$$116$$ 0 0
$$117$$ −117.000 −0.0924500
$$118$$ 0 0
$$119$$ −624.000 −0.480689
$$120$$ 0 0
$$121$$ 113.000 0.0848986
$$122$$ 0 0
$$123$$ 1008.00 0.738929
$$124$$ 0 0
$$125$$ −96.0000 −0.0686920
$$126$$ 0 0
$$127$$ −256.000 −0.178869 −0.0894344 0.995993i $$-0.528506\pi$$
−0.0894344 + 0.995993i $$0.528506\pi$$
$$128$$ 0 0
$$129$$ 228.000 0.155615
$$130$$ 0 0
$$131$$ 1360.00 0.907052 0.453526 0.891243i $$-0.350166\pi$$
0.453526 + 0.891243i $$0.350166\pi$$
$$132$$ 0 0
$$133$$ 576.000 0.375530
$$134$$ 0 0
$$135$$ 432.000 0.275412
$$136$$ 0 0
$$137$$ 2976.00 1.85589 0.927945 0.372718i $$-0.121574\pi$$
0.927945 + 0.372718i $$0.121574\pi$$
$$138$$ 0 0
$$139$$ −2764.00 −1.68661 −0.843307 0.537432i $$-0.819395\pi$$
−0.843307 + 0.537432i $$0.819395\pi$$
$$140$$ 0 0
$$141$$ 282.000 0.168430
$$142$$ 0 0
$$143$$ −494.000 −0.288884
$$144$$ 0 0
$$145$$ −3872.00 −2.21760
$$146$$ 0 0
$$147$$ 837.000 0.469623
$$148$$ 0 0
$$149$$ −2940.00 −1.61647 −0.808236 0.588859i $$-0.799577\pi$$
−0.808236 + 0.588859i $$0.799577\pi$$
$$150$$ 0 0
$$151$$ −1188.00 −0.640252 −0.320126 0.947375i $$-0.603725\pi$$
−0.320126 + 0.947375i $$0.603725\pi$$
$$152$$ 0 0
$$153$$ −702.000 −0.370937
$$154$$ 0 0
$$155$$ 1216.00 0.630139
$$156$$ 0 0
$$157$$ −2410.00 −1.22509 −0.612544 0.790436i $$-0.709854\pi$$
−0.612544 + 0.790436i $$0.709854\pi$$
$$158$$ 0 0
$$159$$ 1350.00 0.673346
$$160$$ 0 0
$$161$$ 416.000 0.203636
$$162$$ 0 0
$$163$$ 2248.00 1.08023 0.540113 0.841592i $$-0.318381\pi$$
0.540113 + 0.841592i $$0.318381\pi$$
$$164$$ 0 0
$$165$$ 1824.00 0.860595
$$166$$ 0 0
$$167$$ 1530.00 0.708952 0.354476 0.935065i $$-0.384659\pi$$
0.354476 + 0.935065i $$0.384659\pi$$
$$168$$ 0 0
$$169$$ 169.000 0.0769231
$$170$$ 0 0
$$171$$ 648.000 0.289788
$$172$$ 0 0
$$173$$ −1030.00 −0.452656 −0.226328 0.974051i $$-0.572672\pi$$
−0.226328 + 0.974051i $$0.572672\pi$$
$$174$$ 0 0
$$175$$ 1048.00 0.452693
$$176$$ 0 0
$$177$$ 2562.00 1.08798
$$178$$ 0 0
$$179$$ −1380.00 −0.576235 −0.288117 0.957595i $$-0.593029\pi$$
−0.288117 + 0.957595i $$0.593029\pi$$
$$180$$ 0 0
$$181$$ 2286.00 0.938768 0.469384 0.882994i $$-0.344476\pi$$
0.469384 + 0.882994i $$0.344476\pi$$
$$182$$ 0 0
$$183$$ 330.000 0.133302
$$184$$ 0 0
$$185$$ −5472.00 −2.17465
$$186$$ 0 0
$$187$$ −2964.00 −1.15909
$$188$$ 0 0
$$189$$ −216.000 −0.0831306
$$190$$ 0 0
$$191$$ −4720.00 −1.78810 −0.894050 0.447967i $$-0.852148\pi$$
−0.894050 + 0.447967i $$0.852148\pi$$
$$192$$ 0 0
$$193$$ 2042.00 0.761587 0.380794 0.924660i $$-0.375651\pi$$
0.380794 + 0.924660i $$0.375651\pi$$
$$194$$ 0 0
$$195$$ −624.000 −0.229157
$$196$$ 0 0
$$197$$ −1512.00 −0.546830 −0.273415 0.961896i $$-0.588153\pi$$
−0.273415 + 0.961896i $$0.588153\pi$$
$$198$$ 0 0
$$199$$ 2224.00 0.792237 0.396119 0.918199i $$-0.370357\pi$$
0.396119 + 0.918199i $$0.370357\pi$$
$$200$$ 0 0
$$201$$ −2724.00 −0.955901
$$202$$ 0 0
$$203$$ 1936.00 0.669362
$$204$$ 0 0
$$205$$ 5376.00 1.83159
$$206$$ 0 0
$$207$$ 468.000 0.157141
$$208$$ 0 0
$$209$$ 2736.00 0.905517
$$210$$ 0 0
$$211$$ −4652.00 −1.51781 −0.758903 0.651204i $$-0.774264\pi$$
−0.758903 + 0.651204i $$0.774264\pi$$
$$212$$ 0 0
$$213$$ 2514.00 0.808716
$$214$$ 0 0
$$215$$ 1216.00 0.385723
$$216$$ 0 0
$$217$$ −608.000 −0.190202
$$218$$ 0 0
$$219$$ 2910.00 0.897898
$$220$$ 0 0
$$221$$ 1014.00 0.308638
$$222$$ 0 0
$$223$$ 1812.00 0.544128 0.272064 0.962279i $$-0.412294\pi$$
0.272064 + 0.962279i $$0.412294\pi$$
$$224$$ 0 0
$$225$$ 1179.00 0.349333
$$226$$ 0 0
$$227$$ −126.000 −0.0368410 −0.0184205 0.999830i $$-0.505864\pi$$
−0.0184205 + 0.999830i $$0.505864\pi$$
$$228$$ 0 0
$$229$$ 3186.00 0.919375 0.459687 0.888081i $$-0.347961\pi$$
0.459687 + 0.888081i $$0.347961\pi$$
$$230$$ 0 0
$$231$$ −912.000 −0.259763
$$232$$ 0 0
$$233$$ −2378.00 −0.668618 −0.334309 0.942464i $$-0.608503\pi$$
−0.334309 + 0.942464i $$0.608503\pi$$
$$234$$ 0 0
$$235$$ 1504.00 0.417490
$$236$$ 0 0
$$237$$ −1056.00 −0.289429
$$238$$ 0 0
$$239$$ 1338.00 0.362126 0.181063 0.983472i $$-0.442046\pi$$
0.181063 + 0.983472i $$0.442046\pi$$
$$240$$ 0 0
$$241$$ 6870.00 1.83625 0.918124 0.396294i $$-0.129704\pi$$
0.918124 + 0.396294i $$0.129704\pi$$
$$242$$ 0 0
$$243$$ −243.000 −0.0641500
$$244$$ 0 0
$$245$$ 4464.00 1.16406
$$246$$ 0 0
$$247$$ −936.000 −0.241118
$$248$$ 0 0
$$249$$ 1422.00 0.361910
$$250$$ 0 0
$$251$$ −6768.00 −1.70196 −0.850981 0.525197i $$-0.823992\pi$$
−0.850981 + 0.525197i $$0.823992\pi$$
$$252$$ 0 0
$$253$$ 1976.00 0.491028
$$254$$ 0 0
$$255$$ −3744.00 −0.919445
$$256$$ 0 0
$$257$$ −3546.00 −0.860675 −0.430337 0.902668i $$-0.641605\pi$$
−0.430337 + 0.902668i $$0.641605\pi$$
$$258$$ 0 0
$$259$$ 2736.00 0.656397
$$260$$ 0 0
$$261$$ 2178.00 0.516532
$$262$$ 0 0
$$263$$ 5340.00 1.25201 0.626005 0.779819i $$-0.284689\pi$$
0.626005 + 0.779819i $$0.284689\pi$$
$$264$$ 0 0
$$265$$ 7200.00 1.66903
$$266$$ 0 0
$$267$$ 4356.00 0.998438
$$268$$ 0 0
$$269$$ −3486.00 −0.790131 −0.395065 0.918653i $$-0.629278\pi$$
−0.395065 + 0.918653i $$0.629278\pi$$
$$270$$ 0 0
$$271$$ −256.000 −0.0573834 −0.0286917 0.999588i $$-0.509134\pi$$
−0.0286917 + 0.999588i $$0.509134\pi$$
$$272$$ 0 0
$$273$$ 312.000 0.0691689
$$274$$ 0 0
$$275$$ 4978.00 1.09158
$$276$$ 0 0
$$277$$ −3354.00 −0.727517 −0.363759 0.931493i $$-0.618507\pi$$
−0.363759 + 0.931493i $$0.618507\pi$$
$$278$$ 0 0
$$279$$ −684.000 −0.146774
$$280$$ 0 0
$$281$$ −6608.00 −1.40285 −0.701424 0.712744i $$-0.747452\pi$$
−0.701424 + 0.712744i $$0.747452\pi$$
$$282$$ 0 0
$$283$$ 1148.00 0.241136 0.120568 0.992705i $$-0.461528\pi$$
0.120568 + 0.992705i $$0.461528\pi$$
$$284$$ 0 0
$$285$$ 3456.00 0.718301
$$286$$ 0 0
$$287$$ −2688.00 −0.552849
$$288$$ 0 0
$$289$$ 1171.00 0.238347
$$290$$ 0 0
$$291$$ 1686.00 0.339639
$$292$$ 0 0
$$293$$ 1972.00 0.393193 0.196596 0.980485i $$-0.437011\pi$$
0.196596 + 0.980485i $$0.437011\pi$$
$$294$$ 0 0
$$295$$ 13664.0 2.69678
$$296$$ 0 0
$$297$$ −1026.00 −0.200453
$$298$$ 0 0
$$299$$ −676.000 −0.130749
$$300$$ 0 0
$$301$$ −608.000 −0.116427
$$302$$ 0 0
$$303$$ −1398.00 −0.265059
$$304$$ 0 0
$$305$$ 1760.00 0.330417
$$306$$ 0 0
$$307$$ 7876.00 1.46419 0.732096 0.681201i $$-0.238542\pi$$
0.732096 + 0.681201i $$0.238542\pi$$
$$308$$ 0 0
$$309$$ −4344.00 −0.799746
$$310$$ 0 0
$$311$$ 6852.00 1.24933 0.624664 0.780893i $$-0.285236\pi$$
0.624664 + 0.780893i $$0.285236\pi$$
$$312$$ 0 0
$$313$$ −4714.00 −0.851281 −0.425641 0.904892i $$-0.639951\pi$$
−0.425641 + 0.904892i $$0.639951\pi$$
$$314$$ 0 0
$$315$$ −1152.00 −0.206057
$$316$$ 0 0
$$317$$ −480.000 −0.0850457 −0.0425228 0.999095i $$-0.513540\pi$$
−0.0425228 + 0.999095i $$0.513540\pi$$
$$318$$ 0 0
$$319$$ 9196.00 1.61403
$$320$$ 0 0
$$321$$ −1272.00 −0.221172
$$322$$ 0 0
$$323$$ −5616.00 −0.967438
$$324$$ 0 0
$$325$$ −1703.00 −0.290663
$$326$$ 0 0
$$327$$ 2346.00 0.396740
$$328$$ 0 0
$$329$$ −752.000 −0.126016
$$330$$ 0 0
$$331$$ −7628.00 −1.26669 −0.633343 0.773872i $$-0.718318\pi$$
−0.633343 + 0.773872i $$0.718318\pi$$
$$332$$ 0 0
$$333$$ 3078.00 0.506527
$$334$$ 0 0
$$335$$ −14528.0 −2.36940
$$336$$ 0 0
$$337$$ −9346.00 −1.51071 −0.755355 0.655316i $$-0.772535\pi$$
−0.755355 + 0.655316i $$0.772535\pi$$
$$338$$ 0 0
$$339$$ 1902.00 0.304727
$$340$$ 0 0
$$341$$ −2888.00 −0.458633
$$342$$ 0 0
$$343$$ −4976.00 −0.783320
$$344$$ 0 0
$$345$$ 2496.00 0.389508
$$346$$ 0 0
$$347$$ 492.000 0.0761151 0.0380576 0.999276i $$-0.487883\pi$$
0.0380576 + 0.999276i $$0.487883\pi$$
$$348$$ 0 0
$$349$$ 358.000 0.0549092 0.0274546 0.999623i $$-0.491260\pi$$
0.0274546 + 0.999623i $$0.491260\pi$$
$$350$$ 0 0
$$351$$ 351.000 0.0533761
$$352$$ 0 0
$$353$$ 1648.00 0.248482 0.124241 0.992252i $$-0.460350\pi$$
0.124241 + 0.992252i $$0.460350\pi$$
$$354$$ 0 0
$$355$$ 13408.0 2.00457
$$356$$ 0 0
$$357$$ 1872.00 0.277526
$$358$$ 0 0
$$359$$ −9750.00 −1.43339 −0.716693 0.697389i $$-0.754345\pi$$
−0.716693 + 0.697389i $$0.754345\pi$$
$$360$$ 0 0
$$361$$ −1675.00 −0.244205
$$362$$ 0 0
$$363$$ −339.000 −0.0490162
$$364$$ 0 0
$$365$$ 15520.0 2.22563
$$366$$ 0 0
$$367$$ −10856.0 −1.54408 −0.772042 0.635572i $$-0.780764\pi$$
−0.772042 + 0.635572i $$0.780764\pi$$
$$368$$ 0 0
$$369$$ −3024.00 −0.426621
$$370$$ 0 0
$$371$$ −3600.00 −0.503781
$$372$$ 0 0
$$373$$ 1826.00 0.253476 0.126738 0.991936i $$-0.459549\pi$$
0.126738 + 0.991936i $$0.459549\pi$$
$$374$$ 0 0
$$375$$ 288.000 0.0396593
$$376$$ 0 0
$$377$$ −3146.00 −0.429780
$$378$$ 0 0
$$379$$ 896.000 0.121436 0.0607182 0.998155i $$-0.480661\pi$$
0.0607182 + 0.998155i $$0.480661\pi$$
$$380$$ 0 0
$$381$$ 768.000 0.103270
$$382$$ 0 0
$$383$$ −2826.00 −0.377028 −0.188514 0.982070i $$-0.560367\pi$$
−0.188514 + 0.982070i $$0.560367\pi$$
$$384$$ 0 0
$$385$$ −4864.00 −0.643876
$$386$$ 0 0
$$387$$ −684.000 −0.0898441
$$388$$ 0 0
$$389$$ 9846.00 1.28332 0.641661 0.766989i $$-0.278246\pi$$
0.641661 + 0.766989i $$0.278246\pi$$
$$390$$ 0 0
$$391$$ −4056.00 −0.524605
$$392$$ 0 0
$$393$$ −4080.00 −0.523686
$$394$$ 0 0
$$395$$ −5632.00 −0.717409
$$396$$ 0 0
$$397$$ 8678.00 1.09707 0.548534 0.836128i $$-0.315186\pi$$
0.548534 + 0.836128i $$0.315186\pi$$
$$398$$ 0 0
$$399$$ −1728.00 −0.216813
$$400$$ 0 0
$$401$$ −9948.00 −1.23885 −0.619426 0.785055i $$-0.712634\pi$$
−0.619426 + 0.785055i $$0.712634\pi$$
$$402$$ 0 0
$$403$$ 988.000 0.122124
$$404$$ 0 0
$$405$$ −1296.00 −0.159009
$$406$$ 0 0
$$407$$ 12996.0 1.58277
$$408$$ 0 0
$$409$$ −98.0000 −0.0118479 −0.00592395 0.999982i $$-0.501886\pi$$
−0.00592395 + 0.999982i $$0.501886\pi$$
$$410$$ 0 0
$$411$$ −8928.00 −1.07150
$$412$$ 0 0
$$413$$ −6832.00 −0.813997
$$414$$ 0 0
$$415$$ 7584.00 0.897070
$$416$$ 0 0
$$417$$ 8292.00 0.973767
$$418$$ 0 0
$$419$$ 3216.00 0.374969 0.187484 0.982268i $$-0.439967\pi$$
0.187484 + 0.982268i $$0.439967\pi$$
$$420$$ 0 0
$$421$$ 4738.00 0.548494 0.274247 0.961659i $$-0.411571\pi$$
0.274247 + 0.961659i $$0.411571\pi$$
$$422$$ 0 0
$$423$$ −846.000 −0.0972433
$$424$$ 0 0
$$425$$ −10218.0 −1.16623
$$426$$ 0 0
$$427$$ −880.000 −0.0997335
$$428$$ 0 0
$$429$$ 1482.00 0.166787
$$430$$ 0 0
$$431$$ 2598.00 0.290351 0.145175 0.989406i $$-0.453625\pi$$
0.145175 + 0.989406i $$0.453625\pi$$
$$432$$ 0 0
$$433$$ −7490.00 −0.831285 −0.415643 0.909528i $$-0.636443\pi$$
−0.415643 + 0.909528i $$0.636443\pi$$
$$434$$ 0 0
$$435$$ 11616.0 1.28033
$$436$$ 0 0
$$437$$ 3744.00 0.409839
$$438$$ 0 0
$$439$$ 17632.0 1.91692 0.958462 0.285221i $$-0.0920670\pi$$
0.958462 + 0.285221i $$0.0920670\pi$$
$$440$$ 0 0
$$441$$ −2511.00 −0.271137
$$442$$ 0 0
$$443$$ −9696.00 −1.03989 −0.519945 0.854200i $$-0.674047\pi$$
−0.519945 + 0.854200i $$0.674047\pi$$
$$444$$ 0 0
$$445$$ 23232.0 2.47484
$$446$$ 0 0
$$447$$ 8820.00 0.933270
$$448$$ 0 0
$$449$$ −4436.00 −0.466253 −0.233127 0.972446i $$-0.574896\pi$$
−0.233127 + 0.972446i $$0.574896\pi$$
$$450$$ 0 0
$$451$$ −12768.0 −1.33309
$$452$$ 0 0
$$453$$ 3564.00 0.369650
$$454$$ 0 0
$$455$$ 1664.00 0.171450
$$456$$ 0 0
$$457$$ 12862.0 1.31654 0.658270 0.752782i $$-0.271288\pi$$
0.658270 + 0.752782i $$0.271288\pi$$
$$458$$ 0 0
$$459$$ 2106.00 0.214160
$$460$$ 0 0
$$461$$ 9816.00 0.991707 0.495853 0.868406i $$-0.334855\pi$$
0.495853 + 0.868406i $$0.334855\pi$$
$$462$$ 0 0
$$463$$ −10408.0 −1.04471 −0.522355 0.852728i $$-0.674946\pi$$
−0.522355 + 0.852728i $$0.674946\pi$$
$$464$$ 0 0
$$465$$ −3648.00 −0.363811
$$466$$ 0 0
$$467$$ −10472.0 −1.03766 −0.518829 0.854878i $$-0.673632\pi$$
−0.518829 + 0.854878i $$0.673632\pi$$
$$468$$ 0 0
$$469$$ 7264.00 0.715182
$$470$$ 0 0
$$471$$ 7230.00 0.707305
$$472$$ 0 0
$$473$$ −2888.00 −0.280741
$$474$$ 0 0
$$475$$ 9432.00 0.911094
$$476$$ 0 0
$$477$$ −4050.00 −0.388756
$$478$$ 0 0
$$479$$ 13398.0 1.27802 0.639009 0.769200i $$-0.279345\pi$$
0.639009 + 0.769200i $$0.279345\pi$$
$$480$$ 0 0
$$481$$ −4446.00 −0.421456
$$482$$ 0 0
$$483$$ −1248.00 −0.117569
$$484$$ 0 0
$$485$$ 8992.00 0.841867
$$486$$ 0 0
$$487$$ −14780.0 −1.37525 −0.687624 0.726067i $$-0.741346\pi$$
−0.687624 + 0.726067i $$0.741346\pi$$
$$488$$ 0 0
$$489$$ −6744.00 −0.623669
$$490$$ 0 0
$$491$$ 12632.0 1.16105 0.580524 0.814243i $$-0.302848\pi$$
0.580524 + 0.814243i $$0.302848\pi$$
$$492$$ 0 0
$$493$$ −18876.0 −1.72441
$$494$$ 0 0
$$495$$ −5472.00 −0.496865
$$496$$ 0 0
$$497$$ −6704.00 −0.605061
$$498$$ 0 0
$$499$$ −17260.0 −1.54842 −0.774212 0.632926i $$-0.781854\pi$$
−0.774212 + 0.632926i $$0.781854\pi$$
$$500$$ 0 0
$$501$$ −4590.00 −0.409314
$$502$$ 0 0
$$503$$ −76.0000 −0.00673692 −0.00336846 0.999994i $$-0.501072\pi$$
−0.00336846 + 0.999994i $$0.501072\pi$$
$$504$$ 0 0
$$505$$ −7456.00 −0.657005
$$506$$ 0 0
$$507$$ −507.000 −0.0444116
$$508$$ 0 0
$$509$$ −11144.0 −0.970430 −0.485215 0.874395i $$-0.661259\pi$$
−0.485215 + 0.874395i $$0.661259\pi$$
$$510$$ 0 0
$$511$$ −7760.00 −0.671785
$$512$$ 0 0
$$513$$ −1944.00 −0.167309
$$514$$ 0 0
$$515$$ −23168.0 −1.98234
$$516$$ 0 0
$$517$$ −3572.00 −0.303861
$$518$$ 0 0
$$519$$ 3090.00 0.261341
$$520$$ 0 0
$$521$$ −4242.00 −0.356709 −0.178355 0.983966i $$-0.557077\pi$$
−0.178355 + 0.983966i $$0.557077\pi$$
$$522$$ 0 0
$$523$$ 9564.00 0.799626 0.399813 0.916597i $$-0.369075\pi$$
0.399813 + 0.916597i $$0.369075\pi$$
$$524$$ 0 0
$$525$$ −3144.00 −0.261363
$$526$$ 0 0
$$527$$ 5928.00 0.489996
$$528$$ 0 0
$$529$$ −9463.00 −0.777760
$$530$$ 0 0
$$531$$ −7686.00 −0.628143
$$532$$ 0 0
$$533$$ 4368.00 0.354970
$$534$$ 0 0
$$535$$ −6784.00 −0.548220
$$536$$ 0 0
$$537$$ 4140.00 0.332689
$$538$$ 0 0
$$539$$ −10602.0 −0.847236
$$540$$ 0 0
$$541$$ −16078.0 −1.27772 −0.638861 0.769322i $$-0.720594\pi$$
−0.638861 + 0.769322i $$0.720594\pi$$
$$542$$ 0 0
$$543$$ −6858.00 −0.541998
$$544$$ 0 0
$$545$$ 12512.0 0.983404
$$546$$ 0 0
$$547$$ −6292.00 −0.491822 −0.245911 0.969292i $$-0.579087\pi$$
−0.245911 + 0.969292i $$0.579087\pi$$
$$548$$ 0 0
$$549$$ −990.000 −0.0769621
$$550$$ 0 0
$$551$$ 17424.0 1.34716
$$552$$ 0 0
$$553$$ 2816.00 0.216543
$$554$$ 0 0
$$555$$ 16416.0 1.25553
$$556$$ 0 0
$$557$$ −3588.00 −0.272942 −0.136471 0.990644i $$-0.543576\pi$$
−0.136471 + 0.990644i $$0.543576\pi$$
$$558$$ 0 0
$$559$$ 988.000 0.0747548
$$560$$ 0 0
$$561$$ 8892.00 0.669199
$$562$$ 0 0
$$563$$ 5932.00 0.444057 0.222028 0.975040i $$-0.428732\pi$$
0.222028 + 0.975040i $$0.428732\pi$$
$$564$$ 0 0
$$565$$ 10144.0 0.755330
$$566$$ 0 0
$$567$$ 648.000 0.0479955
$$568$$ 0 0
$$569$$ −1178.00 −0.0867914 −0.0433957 0.999058i $$-0.513818\pi$$
−0.0433957 + 0.999058i $$0.513818\pi$$
$$570$$ 0 0
$$571$$ −18444.0 −1.35176 −0.675882 0.737010i $$-0.736237\pi$$
−0.675882 + 0.737010i $$0.736237\pi$$
$$572$$ 0 0
$$573$$ 14160.0 1.03236
$$574$$ 0 0
$$575$$ 6812.00 0.494052
$$576$$ 0 0
$$577$$ −2382.00 −0.171861 −0.0859306 0.996301i $$-0.527386\pi$$
−0.0859306 + 0.996301i $$0.527386\pi$$
$$578$$ 0 0
$$579$$ −6126.00 −0.439703
$$580$$ 0 0
$$581$$ −3792.00 −0.270772
$$582$$ 0 0
$$583$$ −17100.0 −1.21477
$$584$$ 0 0
$$585$$ 1872.00 0.132304
$$586$$ 0 0
$$587$$ −15698.0 −1.10379 −0.551896 0.833913i $$-0.686095\pi$$
−0.551896 + 0.833913i $$0.686095\pi$$
$$588$$ 0 0
$$589$$ −5472.00 −0.382801
$$590$$ 0 0
$$591$$ 4536.00 0.315713
$$592$$ 0 0
$$593$$ −8452.00 −0.585299 −0.292649 0.956220i $$-0.594537\pi$$
−0.292649 + 0.956220i $$0.594537\pi$$
$$594$$ 0 0
$$595$$ 9984.00 0.687906
$$596$$ 0 0
$$597$$ −6672.00 −0.457398
$$598$$ 0 0
$$599$$ −5836.00 −0.398084 −0.199042 0.979991i $$-0.563783\pi$$
−0.199042 + 0.979991i $$0.563783\pi$$
$$600$$ 0 0
$$601$$ −25850.0 −1.75448 −0.877241 0.480051i $$-0.840618\pi$$
−0.877241 + 0.480051i $$0.840618\pi$$
$$602$$ 0 0
$$603$$ 8172.00 0.551890
$$604$$ 0 0
$$605$$ −1808.00 −0.121497
$$606$$ 0 0
$$607$$ −21624.0 −1.44595 −0.722975 0.690875i $$-0.757226\pi$$
−0.722975 + 0.690875i $$0.757226\pi$$
$$608$$ 0 0
$$609$$ −5808.00 −0.386457
$$610$$ 0 0
$$611$$ 1222.00 0.0809113
$$612$$ 0 0
$$613$$ 3902.00 0.257097 0.128548 0.991703i $$-0.458968\pi$$
0.128548 + 0.991703i $$0.458968\pi$$
$$614$$ 0 0
$$615$$ −16128.0 −1.05747
$$616$$ 0 0
$$617$$ 16888.0 1.10192 0.550961 0.834531i $$-0.314261\pi$$
0.550961 + 0.834531i $$0.314261\pi$$
$$618$$ 0 0
$$619$$ 27452.0 1.78253 0.891267 0.453478i $$-0.149817\pi$$
0.891267 + 0.453478i $$0.149817\pi$$
$$620$$ 0 0
$$621$$ −1404.00 −0.0907256
$$622$$ 0 0
$$623$$ −11616.0 −0.747007
$$624$$ 0 0
$$625$$ −14839.0 −0.949696
$$626$$ 0 0
$$627$$ −8208.00 −0.522801
$$628$$ 0 0
$$629$$ −26676.0 −1.69100
$$630$$ 0 0
$$631$$ −5548.00 −0.350020 −0.175010 0.984567i $$-0.555996\pi$$
−0.175010 + 0.984567i $$0.555996\pi$$
$$632$$ 0 0
$$633$$ 13956.0 0.876305
$$634$$ 0 0
$$635$$ 4096.00 0.255976
$$636$$ 0 0
$$637$$ 3627.00 0.225600
$$638$$ 0 0
$$639$$ −7542.00 −0.466912
$$640$$ 0 0
$$641$$ 1618.00 0.0996992 0.0498496 0.998757i $$-0.484126\pi$$
0.0498496 + 0.998757i $$0.484126\pi$$
$$642$$ 0 0
$$643$$ 19900.0 1.22050 0.610248 0.792210i $$-0.291070\pi$$
0.610248 + 0.792210i $$0.291070\pi$$
$$644$$ 0 0
$$645$$ −3648.00 −0.222697
$$646$$ 0 0
$$647$$ −18832.0 −1.14430 −0.572150 0.820149i $$-0.693891\pi$$
−0.572150 + 0.820149i $$0.693891\pi$$
$$648$$ 0 0
$$649$$ −32452.0 −1.96279
$$650$$ 0 0
$$651$$ 1824.00 0.109813
$$652$$ 0 0
$$653$$ −4542.00 −0.272193 −0.136097 0.990696i $$-0.543456\pi$$
−0.136097 + 0.990696i $$0.543456\pi$$
$$654$$ 0 0
$$655$$ −21760.0 −1.29807
$$656$$ 0 0
$$657$$ −8730.00 −0.518401
$$658$$ 0 0
$$659$$ −8820.00 −0.521363 −0.260682 0.965425i $$-0.583947\pi$$
−0.260682 + 0.965425i $$0.583947\pi$$
$$660$$ 0 0
$$661$$ 21014.0 1.23654 0.618268 0.785968i $$-0.287835\pi$$
0.618268 + 0.785968i $$0.287835\pi$$
$$662$$ 0 0
$$663$$ −3042.00 −0.178192
$$664$$ 0 0
$$665$$ −9216.00 −0.537415
$$666$$ 0 0
$$667$$ 12584.0 0.730516
$$668$$ 0 0
$$669$$ −5436.00 −0.314152
$$670$$ 0 0
$$671$$ −4180.00 −0.240487
$$672$$ 0 0
$$673$$ 1714.00 0.0981721 0.0490861 0.998795i $$-0.484369\pi$$
0.0490861 + 0.998795i $$0.484369\pi$$
$$674$$ 0 0
$$675$$ −3537.00 −0.201688
$$676$$ 0 0
$$677$$ −15114.0 −0.858018 −0.429009 0.903300i $$-0.641137\pi$$
−0.429009 + 0.903300i $$0.641137\pi$$
$$678$$ 0 0
$$679$$ −4496.00 −0.254110
$$680$$ 0 0
$$681$$ 378.000 0.0212702
$$682$$ 0 0
$$683$$ 20486.0 1.14769 0.573847 0.818963i $$-0.305450\pi$$
0.573847 + 0.818963i $$0.305450\pi$$
$$684$$ 0 0
$$685$$ −47616.0 −2.65593
$$686$$ 0 0
$$687$$ −9558.00 −0.530801
$$688$$ 0 0
$$689$$ 5850.00 0.323465
$$690$$ 0 0
$$691$$ 8948.00 0.492616 0.246308 0.969192i $$-0.420782\pi$$
0.246308 + 0.969192i $$0.420782\pi$$
$$692$$ 0 0
$$693$$ 2736.00 0.149974
$$694$$ 0 0
$$695$$ 44224.0 2.41369
$$696$$ 0 0
$$697$$ 26208.0 1.42425
$$698$$ 0 0
$$699$$ 7134.00 0.386027
$$700$$ 0 0
$$701$$ 1350.00 0.0727372 0.0363686 0.999338i $$-0.488421\pi$$
0.0363686 + 0.999338i $$0.488421\pi$$
$$702$$ 0 0
$$703$$ 24624.0 1.32107
$$704$$ 0 0
$$705$$ −4512.00 −0.241038
$$706$$ 0 0
$$707$$ 3728.00 0.198311
$$708$$ 0 0
$$709$$ 19802.0 1.04891 0.524457 0.851437i $$-0.324268\pi$$
0.524457 + 0.851437i $$0.324268\pi$$
$$710$$ 0 0
$$711$$ 3168.00 0.167102
$$712$$ 0 0
$$713$$ −3952.00 −0.207579
$$714$$ 0 0
$$715$$ 7904.00 0.413417
$$716$$ 0 0
$$717$$ −4014.00 −0.209073
$$718$$ 0 0
$$719$$ 28204.0 1.46291 0.731455 0.681890i $$-0.238842\pi$$
0.731455 + 0.681890i $$0.238842\pi$$
$$720$$ 0 0
$$721$$ 11584.0 0.598350
$$722$$ 0 0
$$723$$ −20610.0 −1.06016
$$724$$ 0 0
$$725$$ 31702.0 1.62398
$$726$$ 0 0
$$727$$ −20992.0 −1.07091 −0.535454 0.844564i $$-0.679859\pi$$
−0.535454 + 0.844564i $$0.679859\pi$$
$$728$$ 0 0
$$729$$ 729.000 0.0370370
$$730$$ 0 0
$$731$$ 5928.00 0.299938
$$732$$ 0 0
$$733$$ −19894.0 −1.00246 −0.501229 0.865315i $$-0.667119\pi$$
−0.501229 + 0.865315i $$0.667119\pi$$
$$734$$ 0 0
$$735$$ −13392.0 −0.672070
$$736$$ 0 0
$$737$$ 34504.0 1.72452
$$738$$ 0 0
$$739$$ −6252.00 −0.311209 −0.155605 0.987819i $$-0.549733\pi$$
−0.155605 + 0.987819i $$0.549733\pi$$
$$740$$ 0 0
$$741$$ 2808.00 0.139210
$$742$$ 0 0
$$743$$ 30938.0 1.52760 0.763799 0.645454i $$-0.223332\pi$$
0.763799 + 0.645454i $$0.223332\pi$$
$$744$$ 0 0
$$745$$ 47040.0 2.31331
$$746$$ 0 0
$$747$$ −4266.00 −0.208949
$$748$$ 0 0
$$749$$ 3392.00 0.165475
$$750$$ 0 0
$$751$$ −11328.0 −0.550419 −0.275209 0.961384i $$-0.588747\pi$$
−0.275209 + 0.961384i $$0.588747\pi$$
$$752$$ 0 0
$$753$$ 20304.0 0.982628
$$754$$ 0 0
$$755$$ 19008.0 0.916254
$$756$$ 0 0
$$757$$ 32754.0 1.57261 0.786304 0.617840i $$-0.211992\pi$$
0.786304 + 0.617840i $$0.211992\pi$$
$$758$$ 0 0
$$759$$ −5928.00 −0.283495
$$760$$ 0 0
$$761$$ −19776.0 −0.942023 −0.471011 0.882127i $$-0.656111\pi$$
−0.471011 + 0.882127i $$0.656111\pi$$
$$762$$ 0 0
$$763$$ −6256.00 −0.296831
$$764$$ 0 0
$$765$$ 11232.0 0.530842
$$766$$ 0 0
$$767$$ 11102.0 0.522647
$$768$$ 0 0
$$769$$ 28362.0 1.32999 0.664993 0.746849i $$-0.268434\pi$$
0.664993 + 0.746849i $$0.268434\pi$$
$$770$$ 0 0
$$771$$ 10638.0 0.496911
$$772$$ 0 0
$$773$$ 17108.0 0.796031 0.398016 0.917379i $$-0.369699\pi$$
0.398016 + 0.917379i $$0.369699\pi$$
$$774$$ 0 0
$$775$$ −9956.00 −0.461458
$$776$$ 0 0
$$777$$ −8208.00 −0.378971
$$778$$ 0 0
$$779$$ −24192.0 −1.11267
$$780$$ 0 0
$$781$$ −31844.0 −1.45899
$$782$$ 0 0
$$783$$ −6534.00 −0.298220
$$784$$ 0 0
$$785$$ 38560.0 1.75320
$$786$$ 0 0
$$787$$ 21364.0 0.967655 0.483827 0.875163i $$-0.339246\pi$$
0.483827 + 0.875163i $$0.339246\pi$$
$$788$$ 0 0
$$789$$ −16020.0 −0.722848
$$790$$ 0 0
$$791$$ −5072.00 −0.227989
$$792$$ 0 0
$$793$$ 1430.00 0.0640363
$$794$$ 0 0
$$795$$ −21600.0 −0.963614
$$796$$ 0 0
$$797$$ 28542.0 1.26852 0.634259 0.773120i $$-0.281305\pi$$
0.634259 + 0.773120i $$0.281305\pi$$
$$798$$ 0 0
$$799$$ 7332.00 0.324640
$$800$$ 0 0
$$801$$ −13068.0 −0.576448
$$802$$ 0 0
$$803$$ −36860.0 −1.61988
$$804$$ 0 0
$$805$$ −6656.00 −0.291420
$$806$$ 0 0
$$807$$ 10458.0 0.456182
$$808$$ 0 0
$$809$$ 26046.0 1.13193 0.565963 0.824430i $$-0.308504\pi$$
0.565963 + 0.824430i $$0.308504\pi$$
$$810$$ 0 0
$$811$$ 15352.0 0.664712 0.332356 0.943154i $$-0.392156\pi$$
0.332356 + 0.943154i $$0.392156\pi$$
$$812$$ 0 0
$$813$$ 768.000 0.0331303
$$814$$ 0 0
$$815$$ −35968.0 −1.54589
$$816$$ 0 0
$$817$$ −5472.00 −0.234322
$$818$$ 0 0
$$819$$ −936.000 −0.0399347
$$820$$ 0 0
$$821$$ −31972.0 −1.35911 −0.679556 0.733624i $$-0.737827\pi$$
−0.679556 + 0.733624i $$0.737827\pi$$
$$822$$ 0 0
$$823$$ 32208.0 1.36416 0.682078 0.731279i $$-0.261076\pi$$
0.682078 + 0.731279i $$0.261076\pi$$
$$824$$ 0 0
$$825$$ −14934.0 −0.630224
$$826$$ 0 0
$$827$$ 27006.0 1.13554 0.567769 0.823188i $$-0.307807\pi$$
0.567769 + 0.823188i $$0.307807\pi$$
$$828$$ 0 0
$$829$$ 5818.00 0.243748 0.121874 0.992546i $$-0.461110\pi$$
0.121874 + 0.992546i $$0.461110\pi$$
$$830$$ 0 0
$$831$$ 10062.0 0.420032
$$832$$ 0 0
$$833$$ 21762.0 0.905172
$$834$$ 0 0
$$835$$ −24480.0 −1.01457
$$836$$ 0 0
$$837$$ 2052.00 0.0847401
$$838$$ 0 0
$$839$$ 5926.00 0.243848 0.121924 0.992539i $$-0.461094\pi$$
0.121924 + 0.992539i $$0.461094\pi$$
$$840$$ 0 0
$$841$$ 34175.0 1.40125
$$842$$ 0 0
$$843$$ 19824.0 0.809935
$$844$$ 0 0
$$845$$ −2704.00 −0.110083
$$846$$ 0 0
$$847$$ 904.000 0.0366727
$$848$$ 0 0
$$849$$ −3444.00 −0.139220
$$850$$ 0 0
$$851$$ 17784.0 0.716366
$$852$$ 0 0
$$853$$ −40874.0 −1.64068 −0.820339 0.571877i $$-0.806215\pi$$
−0.820339 + 0.571877i $$0.806215\pi$$
$$854$$ 0 0
$$855$$ −10368.0 −0.414711
$$856$$ 0 0
$$857$$ −3530.00 −0.140703 −0.0703515 0.997522i $$-0.522412\pi$$
−0.0703515 + 0.997522i $$0.522412\pi$$
$$858$$ 0 0
$$859$$ 34756.0 1.38051 0.690256 0.723565i $$-0.257498\pi$$
0.690256 + 0.723565i $$0.257498\pi$$
$$860$$ 0 0
$$861$$ 8064.00 0.319187
$$862$$ 0 0
$$863$$ 9878.00 0.389630 0.194815 0.980840i $$-0.437589\pi$$
0.194815 + 0.980840i $$0.437589\pi$$
$$864$$ 0 0
$$865$$ 16480.0 0.647788
$$866$$ 0 0
$$867$$ −3513.00 −0.137610
$$868$$ 0 0
$$869$$ 13376.0 0.522152
$$870$$ 0 0
$$871$$ −11804.0 −0.459200
$$872$$ 0 0
$$873$$ −5058.00 −0.196091
$$874$$ 0 0
$$875$$ −768.000 −0.0296722
$$876$$ 0 0
$$877$$ 150.000 0.00577553 0.00288777 0.999996i $$-0.499081\pi$$
0.00288777 + 0.999996i $$0.499081\pi$$
$$878$$ 0 0
$$879$$ −5916.00 −0.227010
$$880$$ 0 0
$$881$$ −3666.00 −0.140194 −0.0700969 0.997540i $$-0.522331\pi$$
−0.0700969 + 0.997540i $$0.522331\pi$$
$$882$$ 0 0
$$883$$ −24316.0 −0.926725 −0.463363 0.886169i $$-0.653357\pi$$
−0.463363 + 0.886169i $$0.653357\pi$$
$$884$$ 0 0
$$885$$ −40992.0 −1.55698
$$886$$ 0 0
$$887$$ −2992.00 −0.113260 −0.0566299 0.998395i $$-0.518036\pi$$
−0.0566299 + 0.998395i $$0.518036\pi$$
$$888$$ 0 0
$$889$$ −2048.00 −0.0772640
$$890$$ 0 0
$$891$$ 3078.00 0.115732
$$892$$ 0 0
$$893$$ −6768.00 −0.253620
$$894$$ 0 0
$$895$$ 22080.0 0.824640
$$896$$ 0 0
$$897$$ 2028.00 0.0754882
$$898$$ 0 0
$$899$$ −18392.0 −0.682322
$$900$$ 0 0
$$901$$ 35100.0 1.29784
$$902$$ 0 0
$$903$$ 1824.00 0.0672192
$$904$$ 0 0
$$905$$ −36576.0 −1.34346
$$906$$ 0 0
$$907$$ 1956.00 0.0716074 0.0358037 0.999359i $$-0.488601\pi$$
0.0358037 + 0.999359i $$0.488601\pi$$
$$908$$ 0 0
$$909$$ 4194.00 0.153032
$$910$$ 0 0
$$911$$ 38832.0 1.41225 0.706126 0.708086i $$-0.250441\pi$$
0.706126 + 0.708086i $$0.250441\pi$$
$$912$$ 0 0
$$913$$ −18012.0 −0.652914
$$914$$ 0 0
$$915$$ −5280.00 −0.190767
$$916$$ 0 0
$$917$$ 10880.0 0.391809
$$918$$ 0 0
$$919$$ −504.000 −0.0180908 −0.00904539 0.999959i $$-0.502879\pi$$
−0.00904539 + 0.999959i $$0.502879\pi$$
$$920$$ 0 0
$$921$$ −23628.0 −0.845352
$$922$$ 0 0
$$923$$ 10894.0 0.388494
$$924$$ 0 0
$$925$$ 44802.0 1.59252
$$926$$ 0 0
$$927$$ 13032.0 0.461734
$$928$$ 0 0
$$929$$ −2976.00 −0.105102 −0.0525508 0.998618i $$-0.516735\pi$$
−0.0525508 + 0.998618i $$0.516735\pi$$
$$930$$ 0 0
$$931$$ −20088.0 −0.707151
$$932$$ 0 0
$$933$$ −20556.0 −0.721300
$$934$$ 0 0
$$935$$ 47424.0 1.65875
$$936$$ 0 0
$$937$$ −14082.0 −0.490970 −0.245485 0.969400i $$-0.578947\pi$$
−0.245485 + 0.969400i $$0.578947\pi$$
$$938$$ 0 0
$$939$$ 14142.0 0.491487
$$940$$ 0 0
$$941$$ −3260.00 −0.112936 −0.0564681 0.998404i $$-0.517984\pi$$
−0.0564681 + 0.998404i $$0.517984\pi$$
$$942$$ 0 0
$$943$$ −17472.0 −0.603358
$$944$$ 0 0
$$945$$ 3456.00 0.118967
$$946$$ 0 0
$$947$$ −5886.00 −0.201974 −0.100987 0.994888i $$-0.532200\pi$$
−0.100987 + 0.994888i $$0.532200\pi$$
$$948$$ 0 0
$$949$$ 12610.0 0.431336
$$950$$ 0 0
$$951$$ 1440.00 0.0491012
$$952$$ 0 0
$$953$$ 22574.0 0.767307 0.383654 0.923477i $$-0.374666\pi$$
0.383654 + 0.923477i $$0.374666\pi$$
$$954$$ 0 0
$$955$$ 75520.0 2.55892
$$956$$ 0 0
$$957$$ −27588.0 −0.931864
$$958$$ 0 0
$$959$$ 23808.0 0.801669
$$960$$ 0 0
$$961$$ −24015.0 −0.806116
$$962$$ 0 0
$$963$$ 3816.00 0.127694
$$964$$ 0 0
$$965$$ −32672.0 −1.08990
$$966$$ 0 0
$$967$$ −32996.0 −1.09729 −0.548645 0.836055i $$-0.684856\pi$$
−0.548645 + 0.836055i $$0.684856\pi$$
$$968$$ 0 0
$$969$$ 16848.0 0.558551
$$970$$ 0 0
$$971$$ −14292.0 −0.472350 −0.236175 0.971711i $$-0.575894\pi$$
−0.236175 + 0.971711i $$0.575894\pi$$
$$972$$ 0 0
$$973$$ −22112.0 −0.728549
$$974$$ 0 0
$$975$$ 5109.00 0.167814
$$976$$ 0 0
$$977$$ 48756.0 1.59656 0.798282 0.602284i $$-0.205743\pi$$
0.798282 + 0.602284i $$0.205743\pi$$
$$978$$ 0 0
$$979$$ −55176.0 −1.80126
$$980$$ 0 0
$$981$$ −7038.00 −0.229058
$$982$$ 0 0
$$983$$ 42022.0 1.36347 0.681736 0.731598i $$-0.261225\pi$$
0.681736 + 0.731598i $$0.261225\pi$$
$$984$$ 0 0
$$985$$ 24192.0 0.782560
$$986$$ 0 0
$$987$$ 2256.00 0.0727551
$$988$$ 0 0
$$989$$ −3952.00 −0.127064
$$990$$ 0 0
$$991$$ −46752.0 −1.49861 −0.749307 0.662223i $$-0.769613\pi$$
−0.749307 + 0.662223i $$0.769613\pi$$
$$992$$ 0 0
$$993$$ 22884.0 0.731321
$$994$$ 0 0
$$995$$ −35584.0 −1.13376
$$996$$ 0 0
$$997$$ −37666.0 −1.19648 −0.598242 0.801316i $$-0.704134\pi$$
−0.598242 + 0.801316i $$0.704134\pi$$
$$998$$ 0 0
$$999$$ −9234.00 −0.292443
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 624.4.a.a.1.1 1
3.2 odd 2 1872.4.a.p.1.1 1
4.3 odd 2 78.4.a.b.1.1 1
8.3 odd 2 2496.4.a.h.1.1 1
8.5 even 2 2496.4.a.p.1.1 1
12.11 even 2 234.4.a.j.1.1 1
20.19 odd 2 1950.4.a.k.1.1 1
52.31 even 4 1014.4.b.f.337.2 2
52.47 even 4 1014.4.b.f.337.1 2
52.51 odd 2 1014.4.a.k.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
78.4.a.b.1.1 1 4.3 odd 2
234.4.a.j.1.1 1 12.11 even 2
624.4.a.a.1.1 1 1.1 even 1 trivial
1014.4.a.k.1.1 1 52.51 odd 2
1014.4.b.f.337.1 2 52.47 even 4
1014.4.b.f.337.2 2 52.31 even 4
1872.4.a.p.1.1 1 3.2 odd 2
1950.4.a.k.1.1 1 20.19 odd 2
2496.4.a.h.1.1 1 8.3 odd 2
2496.4.a.p.1.1 1 8.5 even 2