# Properties

 Label 480.4.a.i.1.1 Level $480$ Weight $4$ Character 480.1 Self dual yes Analytic conductor $28.321$ 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] = [480,4,Mod(1,480)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(480, 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("480.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$480 = 2^{5} \cdot 3 \cdot 5$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 480.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: $$28.3209168028$$ 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 $$\chi$$ $$=$$ 480.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+3.00000 q^{3} +5.00000 q^{5} -16.0000 q^{7} +9.00000 q^{9} +O(q^{10})$$ $$q+3.00000 q^{3} +5.00000 q^{5} -16.0000 q^{7} +9.00000 q^{9} -24.0000 q^{11} -14.0000 q^{13} +15.0000 q^{15} -18.0000 q^{17} -36.0000 q^{19} -48.0000 q^{21} -104.000 q^{23} +25.0000 q^{25} +27.0000 q^{27} -250.000 q^{29} +28.0000 q^{31} -72.0000 q^{33} -80.0000 q^{35} -54.0000 q^{37} -42.0000 q^{39} +354.000 q^{41} -228.000 q^{43} +45.0000 q^{45} -408.000 q^{47} -87.0000 q^{49} -54.0000 q^{51} +262.000 q^{53} -120.000 q^{55} -108.000 q^{57} +64.0000 q^{59} +374.000 q^{61} -144.000 q^{63} -70.0000 q^{65} -300.000 q^{67} -312.000 q^{69} -1016.00 q^{71} +274.000 q^{73} +75.0000 q^{75} +384.000 q^{77} -788.000 q^{79} +81.0000 q^{81} +396.000 q^{83} -90.0000 q^{85} -750.000 q^{87} +786.000 q^{89} +224.000 q^{91} +84.0000 q^{93} -180.000 q^{95} -1086.00 q^{97} -216.000 q^{99} +O(q^{100})$$

## 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
$$5$$ 5.00000 0.447214
$$6$$ 0 0
$$7$$ −16.0000 −0.863919 −0.431959 0.901893i $$-0.642178\pi$$
−0.431959 + 0.901893i $$0.642178\pi$$
$$8$$ 0 0
$$9$$ 9.00000 0.333333
$$10$$ 0 0
$$11$$ −24.0000 −0.657843 −0.328921 0.944357i $$-0.606685\pi$$
−0.328921 + 0.944357i $$0.606685\pi$$
$$12$$ 0 0
$$13$$ −14.0000 −0.298685 −0.149342 0.988786i $$-0.547716\pi$$
−0.149342 + 0.988786i $$0.547716\pi$$
$$14$$ 0 0
$$15$$ 15.0000 0.258199
$$16$$ 0 0
$$17$$ −18.0000 −0.256802 −0.128401 0.991722i $$-0.540985\pi$$
−0.128401 + 0.991722i $$0.540985\pi$$
$$18$$ 0 0
$$19$$ −36.0000 −0.434682 −0.217341 0.976096i $$-0.569738\pi$$
−0.217341 + 0.976096i $$0.569738\pi$$
$$20$$ 0 0
$$21$$ −48.0000 −0.498784
$$22$$ 0 0
$$23$$ −104.000 −0.942848 −0.471424 0.881907i $$-0.656260\pi$$
−0.471424 + 0.881907i $$0.656260\pi$$
$$24$$ 0 0
$$25$$ 25.0000 0.200000
$$26$$ 0 0
$$27$$ 27.0000 0.192450
$$28$$ 0 0
$$29$$ −250.000 −1.60082 −0.800411 0.599452i $$-0.795385\pi$$
−0.800411 + 0.599452i $$0.795385\pi$$
$$30$$ 0 0
$$31$$ 28.0000 0.162224 0.0811121 0.996705i $$-0.474153\pi$$
0.0811121 + 0.996705i $$0.474153\pi$$
$$32$$ 0 0
$$33$$ −72.0000 −0.379806
$$34$$ 0 0
$$35$$ −80.0000 −0.386356
$$36$$ 0 0
$$37$$ −54.0000 −0.239934 −0.119967 0.992778i $$-0.538279\pi$$
−0.119967 + 0.992778i $$0.538279\pi$$
$$38$$ 0 0
$$39$$ −42.0000 −0.172446
$$40$$ 0 0
$$41$$ 354.000 1.34843 0.674214 0.738536i $$-0.264483\pi$$
0.674214 + 0.738536i $$0.264483\pi$$
$$42$$ 0 0
$$43$$ −228.000 −0.808597 −0.404299 0.914627i $$-0.632484\pi$$
−0.404299 + 0.914627i $$0.632484\pi$$
$$44$$ 0 0
$$45$$ 45.0000 0.149071
$$46$$ 0 0
$$47$$ −408.000 −1.26623 −0.633116 0.774057i $$-0.718224\pi$$
−0.633116 + 0.774057i $$0.718224\pi$$
$$48$$ 0 0
$$49$$ −87.0000 −0.253644
$$50$$ 0 0
$$51$$ −54.0000 −0.148265
$$52$$ 0 0
$$53$$ 262.000 0.679028 0.339514 0.940601i $$-0.389737\pi$$
0.339514 + 0.940601i $$0.389737\pi$$
$$54$$ 0 0
$$55$$ −120.000 −0.294196
$$56$$ 0 0
$$57$$ −108.000 −0.250964
$$58$$ 0 0
$$59$$ 64.0000 0.141222 0.0706109 0.997504i $$-0.477505\pi$$
0.0706109 + 0.997504i $$0.477505\pi$$
$$60$$ 0 0
$$61$$ 374.000 0.785013 0.392507 0.919749i $$-0.371608\pi$$
0.392507 + 0.919749i $$0.371608\pi$$
$$62$$ 0 0
$$63$$ −144.000 −0.287973
$$64$$ 0 0
$$65$$ −70.0000 −0.133576
$$66$$ 0 0
$$67$$ −300.000 −0.547027 −0.273514 0.961868i $$-0.588186\pi$$
−0.273514 + 0.961868i $$0.588186\pi$$
$$68$$ 0 0
$$69$$ −312.000 −0.544353
$$70$$ 0 0
$$71$$ −1016.00 −1.69827 −0.849134 0.528178i $$-0.822876\pi$$
−0.849134 + 0.528178i $$0.822876\pi$$
$$72$$ 0 0
$$73$$ 274.000 0.439305 0.219653 0.975578i $$-0.429508\pi$$
0.219653 + 0.975578i $$0.429508\pi$$
$$74$$ 0 0
$$75$$ 75.0000 0.115470
$$76$$ 0 0
$$77$$ 384.000 0.568323
$$78$$ 0 0
$$79$$ −788.000 −1.12224 −0.561120 0.827735i $$-0.689629\pi$$
−0.561120 + 0.827735i $$0.689629\pi$$
$$80$$ 0 0
$$81$$ 81.0000 0.111111
$$82$$ 0 0
$$83$$ 396.000 0.523695 0.261847 0.965109i $$-0.415668\pi$$
0.261847 + 0.965109i $$0.415668\pi$$
$$84$$ 0 0
$$85$$ −90.0000 −0.114846
$$86$$ 0 0
$$87$$ −750.000 −0.924235
$$88$$ 0 0
$$89$$ 786.000 0.936133 0.468066 0.883693i $$-0.344951\pi$$
0.468066 + 0.883693i $$0.344951\pi$$
$$90$$ 0 0
$$91$$ 224.000 0.258039
$$92$$ 0 0
$$93$$ 84.0000 0.0936602
$$94$$ 0 0
$$95$$ −180.000 −0.194396
$$96$$ 0 0
$$97$$ −1086.00 −1.13677 −0.568385 0.822763i $$-0.692431\pi$$
−0.568385 + 0.822763i $$0.692431\pi$$
$$98$$ 0 0
$$99$$ −216.000 −0.219281
$$100$$ 0 0
$$101$$ 78.0000 0.0768445 0.0384222 0.999262i $$-0.487767\pi$$
0.0384222 + 0.999262i $$0.487767\pi$$
$$102$$ 0 0
$$103$$ −1208.00 −1.15561 −0.577805 0.816175i $$-0.696090\pi$$
−0.577805 + 0.816175i $$0.696090\pi$$
$$104$$ 0 0
$$105$$ −240.000 −0.223063
$$106$$ 0 0
$$107$$ 44.0000 0.0397537 0.0198768 0.999802i $$-0.493673\pi$$
0.0198768 + 0.999802i $$0.493673\pi$$
$$108$$ 0 0
$$109$$ −1122.00 −0.985946 −0.492973 0.870045i $$-0.664090\pi$$
−0.492973 + 0.870045i $$0.664090\pi$$
$$110$$ 0 0
$$111$$ −162.000 −0.138526
$$112$$ 0 0
$$113$$ 606.000 0.504493 0.252246 0.967663i $$-0.418831\pi$$
0.252246 + 0.967663i $$0.418831\pi$$
$$114$$ 0 0
$$115$$ −520.000 −0.421654
$$116$$ 0 0
$$117$$ −126.000 −0.0995616
$$118$$ 0 0
$$119$$ 288.000 0.221856
$$120$$ 0 0
$$121$$ −755.000 −0.567243
$$122$$ 0 0
$$123$$ 1062.00 0.778515
$$124$$ 0 0
$$125$$ 125.000 0.0894427
$$126$$ 0 0
$$127$$ −1744.00 −1.21854 −0.609272 0.792962i $$-0.708538\pi$$
−0.609272 + 0.792962i $$0.708538\pi$$
$$128$$ 0 0
$$129$$ −684.000 −0.466844
$$130$$ 0 0
$$131$$ 480.000 0.320136 0.160068 0.987106i $$-0.448829\pi$$
0.160068 + 0.987106i $$0.448829\pi$$
$$132$$ 0 0
$$133$$ 576.000 0.375530
$$134$$ 0 0
$$135$$ 135.000 0.0860663
$$136$$ 0 0
$$137$$ 1598.00 0.996543 0.498271 0.867021i $$-0.333968\pi$$
0.498271 + 0.867021i $$0.333968\pi$$
$$138$$ 0 0
$$139$$ 2964.00 1.80866 0.904328 0.426838i $$-0.140373\pi$$
0.904328 + 0.426838i $$0.140373\pi$$
$$140$$ 0 0
$$141$$ −1224.00 −0.731060
$$142$$ 0 0
$$143$$ 336.000 0.196488
$$144$$ 0 0
$$145$$ −1250.00 −0.715909
$$146$$ 0 0
$$147$$ −261.000 −0.146442
$$148$$ 0 0
$$149$$ 334.000 0.183640 0.0918200 0.995776i $$-0.470732\pi$$
0.0918200 + 0.995776i $$0.470732\pi$$
$$150$$ 0 0
$$151$$ 1148.00 0.618695 0.309347 0.950949i $$-0.399889\pi$$
0.309347 + 0.950949i $$0.399889\pi$$
$$152$$ 0 0
$$153$$ −162.000 −0.0856008
$$154$$ 0 0
$$155$$ 140.000 0.0725488
$$156$$ 0 0
$$157$$ 906.000 0.460552 0.230276 0.973125i $$-0.426037\pi$$
0.230276 + 0.973125i $$0.426037\pi$$
$$158$$ 0 0
$$159$$ 786.000 0.392037
$$160$$ 0 0
$$161$$ 1664.00 0.814544
$$162$$ 0 0
$$163$$ 1916.00 0.920691 0.460346 0.887740i $$-0.347725\pi$$
0.460346 + 0.887740i $$0.347725\pi$$
$$164$$ 0 0
$$165$$ −360.000 −0.169854
$$166$$ 0 0
$$167$$ −1152.00 −0.533799 −0.266900 0.963724i $$-0.585999\pi$$
−0.266900 + 0.963724i $$0.585999\pi$$
$$168$$ 0 0
$$169$$ −2001.00 −0.910787
$$170$$ 0 0
$$171$$ −324.000 −0.144894
$$172$$ 0 0
$$173$$ 3142.00 1.38082 0.690410 0.723418i $$-0.257430\pi$$
0.690410 + 0.723418i $$0.257430\pi$$
$$174$$ 0 0
$$175$$ −400.000 −0.172784
$$176$$ 0 0
$$177$$ 192.000 0.0815345
$$178$$ 0 0
$$179$$ 1032.00 0.430923 0.215462 0.976512i $$-0.430874\pi$$
0.215462 + 0.976512i $$0.430874\pi$$
$$180$$ 0 0
$$181$$ −1562.00 −0.641451 −0.320725 0.947172i $$-0.603927\pi$$
−0.320725 + 0.947172i $$0.603927\pi$$
$$182$$ 0 0
$$183$$ 1122.00 0.453227
$$184$$ 0 0
$$185$$ −270.000 −0.107302
$$186$$ 0 0
$$187$$ 432.000 0.168936
$$188$$ 0 0
$$189$$ −432.000 −0.166261
$$190$$ 0 0
$$191$$ −1960.00 −0.742516 −0.371258 0.928530i $$-0.621074\pi$$
−0.371258 + 0.928530i $$0.621074\pi$$
$$192$$ 0 0
$$193$$ −4006.00 −1.49408 −0.747042 0.664777i $$-0.768527\pi$$
−0.747042 + 0.664777i $$0.768527\pi$$
$$194$$ 0 0
$$195$$ −210.000 −0.0771201
$$196$$ 0 0
$$197$$ 2118.00 0.765996 0.382998 0.923749i $$-0.374892\pi$$
0.382998 + 0.923749i $$0.374892\pi$$
$$198$$ 0 0
$$199$$ 3748.00 1.33512 0.667559 0.744556i $$-0.267339\pi$$
0.667559 + 0.744556i $$0.267339\pi$$
$$200$$ 0 0
$$201$$ −900.000 −0.315826
$$202$$ 0 0
$$203$$ 4000.00 1.38298
$$204$$ 0 0
$$205$$ 1770.00 0.603035
$$206$$ 0 0
$$207$$ −936.000 −0.314283
$$208$$ 0 0
$$209$$ 864.000 0.285953
$$210$$ 0 0
$$211$$ 4796.00 1.56479 0.782394 0.622784i $$-0.213998\pi$$
0.782394 + 0.622784i $$0.213998\pi$$
$$212$$ 0 0
$$213$$ −3048.00 −0.980495
$$214$$ 0 0
$$215$$ −1140.00 −0.361616
$$216$$ 0 0
$$217$$ −448.000 −0.140148
$$218$$ 0 0
$$219$$ 822.000 0.253633
$$220$$ 0 0
$$221$$ 252.000 0.0767030
$$222$$ 0 0
$$223$$ −2560.00 −0.768746 −0.384373 0.923178i $$-0.625582\pi$$
−0.384373 + 0.923178i $$0.625582\pi$$
$$224$$ 0 0
$$225$$ 225.000 0.0666667
$$226$$ 0 0
$$227$$ −3500.00 −1.02336 −0.511681 0.859176i $$-0.670977\pi$$
−0.511681 + 0.859176i $$0.670977\pi$$
$$228$$ 0 0
$$229$$ 1966.00 0.567323 0.283661 0.958924i $$-0.408451\pi$$
0.283661 + 0.958924i $$0.408451\pi$$
$$230$$ 0 0
$$231$$ 1152.00 0.328121
$$232$$ 0 0
$$233$$ 3246.00 0.912672 0.456336 0.889808i $$-0.349162\pi$$
0.456336 + 0.889808i $$0.349162\pi$$
$$234$$ 0 0
$$235$$ −2040.00 −0.566276
$$236$$ 0 0
$$237$$ −2364.00 −0.647925
$$238$$ 0 0
$$239$$ 7320.00 1.98114 0.990568 0.137023i $$-0.0437534\pi$$
0.990568 + 0.137023i $$0.0437534\pi$$
$$240$$ 0 0
$$241$$ 3490.00 0.932824 0.466412 0.884568i $$-0.345546\pi$$
0.466412 + 0.884568i $$0.345546\pi$$
$$242$$ 0 0
$$243$$ 243.000 0.0641500
$$244$$ 0 0
$$245$$ −435.000 −0.113433
$$246$$ 0 0
$$247$$ 504.000 0.129833
$$248$$ 0 0
$$249$$ 1188.00 0.302355
$$250$$ 0 0
$$251$$ 7456.00 1.87497 0.937487 0.348020i $$-0.113146\pi$$
0.937487 + 0.348020i $$0.113146\pi$$
$$252$$ 0 0
$$253$$ 2496.00 0.620246
$$254$$ 0 0
$$255$$ −270.000 −0.0663061
$$256$$ 0 0
$$257$$ 4558.00 1.10630 0.553152 0.833080i $$-0.313425\pi$$
0.553152 + 0.833080i $$0.313425\pi$$
$$258$$ 0 0
$$259$$ 864.000 0.207283
$$260$$ 0 0
$$261$$ −2250.00 −0.533607
$$262$$ 0 0
$$263$$ 2848.00 0.667738 0.333869 0.942619i $$-0.391646\pi$$
0.333869 + 0.942619i $$0.391646\pi$$
$$264$$ 0 0
$$265$$ 1310.00 0.303670
$$266$$ 0 0
$$267$$ 2358.00 0.540477
$$268$$ 0 0
$$269$$ 3110.00 0.704907 0.352454 0.935829i $$-0.385347\pi$$
0.352454 + 0.935829i $$0.385347\pi$$
$$270$$ 0 0
$$271$$ −1700.00 −0.381061 −0.190531 0.981681i $$-0.561021\pi$$
−0.190531 + 0.981681i $$0.561021\pi$$
$$272$$ 0 0
$$273$$ 672.000 0.148979
$$274$$ 0 0
$$275$$ −600.000 −0.131569
$$276$$ 0 0
$$277$$ −6494.00 −1.40862 −0.704308 0.709895i $$-0.748743\pi$$
−0.704308 + 0.709895i $$0.748743\pi$$
$$278$$ 0 0
$$279$$ 252.000 0.0540747
$$280$$ 0 0
$$281$$ 2498.00 0.530314 0.265157 0.964205i $$-0.414576\pi$$
0.265157 + 0.964205i $$0.414576\pi$$
$$282$$ 0 0
$$283$$ 5324.00 1.11830 0.559150 0.829066i $$-0.311128\pi$$
0.559150 + 0.829066i $$0.311128\pi$$
$$284$$ 0 0
$$285$$ −540.000 −0.112235
$$286$$ 0 0
$$287$$ −5664.00 −1.16493
$$288$$ 0 0
$$289$$ −4589.00 −0.934053
$$290$$ 0 0
$$291$$ −3258.00 −0.656314
$$292$$ 0 0
$$293$$ −522.000 −0.104080 −0.0520402 0.998645i $$-0.516572\pi$$
−0.0520402 + 0.998645i $$0.516572\pi$$
$$294$$ 0 0
$$295$$ 320.000 0.0631563
$$296$$ 0 0
$$297$$ −648.000 −0.126602
$$298$$ 0 0
$$299$$ 1456.00 0.281614
$$300$$ 0 0
$$301$$ 3648.00 0.698562
$$302$$ 0 0
$$303$$ 234.000 0.0443662
$$304$$ 0 0
$$305$$ 1870.00 0.351068
$$306$$ 0 0
$$307$$ −7844.00 −1.45824 −0.729122 0.684384i $$-0.760071\pi$$
−0.729122 + 0.684384i $$0.760071\pi$$
$$308$$ 0 0
$$309$$ −3624.00 −0.667191
$$310$$ 0 0
$$311$$ 3248.00 0.592210 0.296105 0.955155i $$-0.404312\pi$$
0.296105 + 0.955155i $$0.404312\pi$$
$$312$$ 0 0
$$313$$ −5374.00 −0.970468 −0.485234 0.874384i $$-0.661266\pi$$
−0.485234 + 0.874384i $$0.661266\pi$$
$$314$$ 0 0
$$315$$ −720.000 −0.128785
$$316$$ 0 0
$$317$$ −6786.00 −1.20233 −0.601167 0.799124i $$-0.705297\pi$$
−0.601167 + 0.799124i $$0.705297\pi$$
$$318$$ 0 0
$$319$$ 6000.00 1.05309
$$320$$ 0 0
$$321$$ 132.000 0.0229518
$$322$$ 0 0
$$323$$ 648.000 0.111628
$$324$$ 0 0
$$325$$ −350.000 −0.0597369
$$326$$ 0 0
$$327$$ −3366.00 −0.569236
$$328$$ 0 0
$$329$$ 6528.00 1.09392
$$330$$ 0 0
$$331$$ −6596.00 −1.09531 −0.547657 0.836703i $$-0.684480\pi$$
−0.547657 + 0.836703i $$0.684480\pi$$
$$332$$ 0 0
$$333$$ −486.000 −0.0799779
$$334$$ 0 0
$$335$$ −1500.00 −0.244638
$$336$$ 0 0
$$337$$ −5830.00 −0.942375 −0.471187 0.882033i $$-0.656174\pi$$
−0.471187 + 0.882033i $$0.656174\pi$$
$$338$$ 0 0
$$339$$ 1818.00 0.291269
$$340$$ 0 0
$$341$$ −672.000 −0.106718
$$342$$ 0 0
$$343$$ 6880.00 1.08305
$$344$$ 0 0
$$345$$ −1560.00 −0.243442
$$346$$ 0 0
$$347$$ −11732.0 −1.81501 −0.907503 0.420047i $$-0.862014\pi$$
−0.907503 + 0.420047i $$0.862014\pi$$
$$348$$ 0 0
$$349$$ 1014.00 0.155525 0.0777624 0.996972i $$-0.475222\pi$$
0.0777624 + 0.996972i $$0.475222\pi$$
$$350$$ 0 0
$$351$$ −378.000 −0.0574819
$$352$$ 0 0
$$353$$ −8202.00 −1.23668 −0.618341 0.785910i $$-0.712195\pi$$
−0.618341 + 0.785910i $$0.712195\pi$$
$$354$$ 0 0
$$355$$ −5080.00 −0.759488
$$356$$ 0 0
$$357$$ 864.000 0.128089
$$358$$ 0 0
$$359$$ 8160.00 1.19963 0.599817 0.800138i $$-0.295240\pi$$
0.599817 + 0.800138i $$0.295240\pi$$
$$360$$ 0 0
$$361$$ −5563.00 −0.811051
$$362$$ 0 0
$$363$$ −2265.00 −0.327498
$$364$$ 0 0
$$365$$ 1370.00 0.196463
$$366$$ 0 0
$$367$$ −12360.0 −1.75800 −0.879001 0.476820i $$-0.841789\pi$$
−0.879001 + 0.476820i $$0.841789\pi$$
$$368$$ 0 0
$$369$$ 3186.00 0.449476
$$370$$ 0 0
$$371$$ −4192.00 −0.586625
$$372$$ 0 0
$$373$$ 930.000 0.129098 0.0645490 0.997915i $$-0.479439\pi$$
0.0645490 + 0.997915i $$0.479439\pi$$
$$374$$ 0 0
$$375$$ 375.000 0.0516398
$$376$$ 0 0
$$377$$ 3500.00 0.478141
$$378$$ 0 0
$$379$$ −4228.00 −0.573028 −0.286514 0.958076i $$-0.592497\pi$$
−0.286514 + 0.958076i $$0.592497\pi$$
$$380$$ 0 0
$$381$$ −5232.00 −0.703526
$$382$$ 0 0
$$383$$ −8384.00 −1.11854 −0.559272 0.828984i $$-0.688919\pi$$
−0.559272 + 0.828984i $$0.688919\pi$$
$$384$$ 0 0
$$385$$ 1920.00 0.254162
$$386$$ 0 0
$$387$$ −2052.00 −0.269532
$$388$$ 0 0
$$389$$ 5534.00 0.721298 0.360649 0.932702i $$-0.382555\pi$$
0.360649 + 0.932702i $$0.382555\pi$$
$$390$$ 0 0
$$391$$ 1872.00 0.242126
$$392$$ 0 0
$$393$$ 1440.00 0.184831
$$394$$ 0 0
$$395$$ −3940.00 −0.501881
$$396$$ 0 0
$$397$$ 5426.00 0.685952 0.342976 0.939344i $$-0.388565\pi$$
0.342976 + 0.939344i $$0.388565\pi$$
$$398$$ 0 0
$$399$$ 1728.00 0.216813
$$400$$ 0 0
$$401$$ −78.0000 −0.00971355 −0.00485678 0.999988i $$-0.501546\pi$$
−0.00485678 + 0.999988i $$0.501546\pi$$
$$402$$ 0 0
$$403$$ −392.000 −0.0484539
$$404$$ 0 0
$$405$$ 405.000 0.0496904
$$406$$ 0 0
$$407$$ 1296.00 0.157839
$$408$$ 0 0
$$409$$ −454.000 −0.0548872 −0.0274436 0.999623i $$-0.508737\pi$$
−0.0274436 + 0.999623i $$0.508737\pi$$
$$410$$ 0 0
$$411$$ 4794.00 0.575354
$$412$$ 0 0
$$413$$ −1024.00 −0.122004
$$414$$ 0 0
$$415$$ 1980.00 0.234203
$$416$$ 0 0
$$417$$ 8892.00 1.04423
$$418$$ 0 0
$$419$$ −12296.0 −1.43365 −0.716824 0.697254i $$-0.754405\pi$$
−0.716824 + 0.697254i $$0.754405\pi$$
$$420$$ 0 0
$$421$$ 12798.0 1.48156 0.740780 0.671748i $$-0.234456\pi$$
0.740780 + 0.671748i $$0.234456\pi$$
$$422$$ 0 0
$$423$$ −3672.00 −0.422077
$$424$$ 0 0
$$425$$ −450.000 −0.0513605
$$426$$ 0 0
$$427$$ −5984.00 −0.678187
$$428$$ 0 0
$$429$$ 1008.00 0.113442
$$430$$ 0 0
$$431$$ −9912.00 −1.10776 −0.553880 0.832597i $$-0.686853\pi$$
−0.553880 + 0.832597i $$0.686853\pi$$
$$432$$ 0 0
$$433$$ −6774.00 −0.751819 −0.375910 0.926656i $$-0.622670\pi$$
−0.375910 + 0.926656i $$0.622670\pi$$
$$434$$ 0 0
$$435$$ −3750.00 −0.413330
$$436$$ 0 0
$$437$$ 3744.00 0.409839
$$438$$ 0 0
$$439$$ 16628.0 1.80777 0.903885 0.427775i $$-0.140703\pi$$
0.903885 + 0.427775i $$0.140703\pi$$
$$440$$ 0 0
$$441$$ −783.000 −0.0845481
$$442$$ 0 0
$$443$$ 940.000 0.100814 0.0504072 0.998729i $$-0.483948\pi$$
0.0504072 + 0.998729i $$0.483948\pi$$
$$444$$ 0 0
$$445$$ 3930.00 0.418651
$$446$$ 0 0
$$447$$ 1002.00 0.106025
$$448$$ 0 0
$$449$$ −1662.00 −0.174687 −0.0873437 0.996178i $$-0.527838\pi$$
−0.0873437 + 0.996178i $$0.527838\pi$$
$$450$$ 0 0
$$451$$ −8496.00 −0.887053
$$452$$ 0 0
$$453$$ 3444.00 0.357204
$$454$$ 0 0
$$455$$ 1120.00 0.115399
$$456$$ 0 0
$$457$$ −13942.0 −1.42709 −0.713544 0.700610i $$-0.752911\pi$$
−0.713544 + 0.700610i $$0.752911\pi$$
$$458$$ 0 0
$$459$$ −486.000 −0.0494217
$$460$$ 0 0
$$461$$ −16170.0 −1.63365 −0.816824 0.576887i $$-0.804267\pi$$
−0.816824 + 0.576887i $$0.804267\pi$$
$$462$$ 0 0
$$463$$ −1048.00 −0.105194 −0.0525969 0.998616i $$-0.516750\pi$$
−0.0525969 + 0.998616i $$0.516750\pi$$
$$464$$ 0 0
$$465$$ 420.000 0.0418861
$$466$$ 0 0
$$467$$ −13716.0 −1.35910 −0.679551 0.733628i $$-0.737825\pi$$
−0.679551 + 0.733628i $$0.737825\pi$$
$$468$$ 0 0
$$469$$ 4800.00 0.472587
$$470$$ 0 0
$$471$$ 2718.00 0.265900
$$472$$ 0 0
$$473$$ 5472.00 0.531930
$$474$$ 0 0
$$475$$ −900.000 −0.0869365
$$476$$ 0 0
$$477$$ 2358.00 0.226343
$$478$$ 0 0
$$479$$ 8832.00 0.842473 0.421236 0.906951i $$-0.361596\pi$$
0.421236 + 0.906951i $$0.361596\pi$$
$$480$$ 0 0
$$481$$ 756.000 0.0716645
$$482$$ 0 0
$$483$$ 4992.00 0.470277
$$484$$ 0 0
$$485$$ −5430.00 −0.508379
$$486$$ 0 0
$$487$$ −10120.0 −0.941645 −0.470822 0.882228i $$-0.656043\pi$$
−0.470822 + 0.882228i $$0.656043\pi$$
$$488$$ 0 0
$$489$$ 5748.00 0.531561
$$490$$ 0 0
$$491$$ 4376.00 0.402212 0.201106 0.979569i $$-0.435546\pi$$
0.201106 + 0.979569i $$0.435546\pi$$
$$492$$ 0 0
$$493$$ 4500.00 0.411095
$$494$$ 0 0
$$495$$ −1080.00 −0.0980654
$$496$$ 0 0
$$497$$ 16256.0 1.46717
$$498$$ 0 0
$$499$$ −12364.0 −1.10920 −0.554598 0.832119i $$-0.687128\pi$$
−0.554598 + 0.832119i $$0.687128\pi$$
$$500$$ 0 0
$$501$$ −3456.00 −0.308189
$$502$$ 0 0
$$503$$ 1248.00 0.110627 0.0553137 0.998469i $$-0.482384\pi$$
0.0553137 + 0.998469i $$0.482384\pi$$
$$504$$ 0 0
$$505$$ 390.000 0.0343659
$$506$$ 0 0
$$507$$ −6003.00 −0.525843
$$508$$ 0 0
$$509$$ −12730.0 −1.10854 −0.554270 0.832337i $$-0.687003\pi$$
−0.554270 + 0.832337i $$0.687003\pi$$
$$510$$ 0 0
$$511$$ −4384.00 −0.379524
$$512$$ 0 0
$$513$$ −972.000 −0.0836547
$$514$$ 0 0
$$515$$ −6040.00 −0.516804
$$516$$ 0 0
$$517$$ 9792.00 0.832982
$$518$$ 0 0
$$519$$ 9426.00 0.797217
$$520$$ 0 0
$$521$$ −13286.0 −1.11722 −0.558609 0.829431i $$-0.688665\pi$$
−0.558609 + 0.829431i $$0.688665\pi$$
$$522$$ 0 0
$$523$$ 15892.0 1.32870 0.664349 0.747423i $$-0.268709\pi$$
0.664349 + 0.747423i $$0.268709\pi$$
$$524$$ 0 0
$$525$$ −1200.00 −0.0997567
$$526$$ 0 0
$$527$$ −504.000 −0.0416596
$$528$$ 0 0
$$529$$ −1351.00 −0.111038
$$530$$ 0 0
$$531$$ 576.000 0.0470740
$$532$$ 0 0
$$533$$ −4956.00 −0.402755
$$534$$ 0 0
$$535$$ 220.000 0.0177784
$$536$$ 0 0
$$537$$ 3096.00 0.248794
$$538$$ 0 0
$$539$$ 2088.00 0.166858
$$540$$ 0 0
$$541$$ 9662.00 0.767841 0.383920 0.923366i $$-0.374574\pi$$
0.383920 + 0.923366i $$0.374574\pi$$
$$542$$ 0 0
$$543$$ −4686.00 −0.370342
$$544$$ 0 0
$$545$$ −5610.00 −0.440928
$$546$$ 0 0
$$547$$ −9596.00 −0.750083 −0.375041 0.927008i $$-0.622372\pi$$
−0.375041 + 0.927008i $$0.622372\pi$$
$$548$$ 0 0
$$549$$ 3366.00 0.261671
$$550$$ 0 0
$$551$$ 9000.00 0.695849
$$552$$ 0 0
$$553$$ 12608.0 0.969524
$$554$$ 0 0
$$555$$ −810.000 −0.0619506
$$556$$ 0 0
$$557$$ −4458.00 −0.339123 −0.169562 0.985520i $$-0.554235\pi$$
−0.169562 + 0.985520i $$0.554235\pi$$
$$558$$ 0 0
$$559$$ 3192.00 0.241516
$$560$$ 0 0
$$561$$ 1296.00 0.0975350
$$562$$ 0 0
$$563$$ 4708.00 0.352431 0.176215 0.984352i $$-0.443614\pi$$
0.176215 + 0.984352i $$0.443614\pi$$
$$564$$ 0 0
$$565$$ 3030.00 0.225616
$$566$$ 0 0
$$567$$ −1296.00 −0.0959910
$$568$$ 0 0
$$569$$ −12358.0 −0.910500 −0.455250 0.890364i $$-0.650450\pi$$
−0.455250 + 0.890364i $$0.650450\pi$$
$$570$$ 0 0
$$571$$ 7532.00 0.552022 0.276011 0.961155i $$-0.410987\pi$$
0.276011 + 0.961155i $$0.410987\pi$$
$$572$$ 0 0
$$573$$ −5880.00 −0.428692
$$574$$ 0 0
$$575$$ −2600.00 −0.188570
$$576$$ 0 0
$$577$$ −18878.0 −1.36205 −0.681024 0.732261i $$-0.738465\pi$$
−0.681024 + 0.732261i $$0.738465\pi$$
$$578$$ 0 0
$$579$$ −12018.0 −0.862610
$$580$$ 0 0
$$581$$ −6336.00 −0.452430
$$582$$ 0 0
$$583$$ −6288.00 −0.446694
$$584$$ 0 0
$$585$$ −630.000 −0.0445253
$$586$$ 0 0
$$587$$ 22380.0 1.57363 0.786816 0.617188i $$-0.211728\pi$$
0.786816 + 0.617188i $$0.211728\pi$$
$$588$$ 0 0
$$589$$ −1008.00 −0.0705160
$$590$$ 0 0
$$591$$ 6354.00 0.442248
$$592$$ 0 0
$$593$$ 7726.00 0.535023 0.267512 0.963555i $$-0.413799\pi$$
0.267512 + 0.963555i $$0.413799\pi$$
$$594$$ 0 0
$$595$$ 1440.00 0.0992172
$$596$$ 0 0
$$597$$ 11244.0 0.770831
$$598$$ 0 0
$$599$$ 21232.0 1.44827 0.724137 0.689656i $$-0.242238\pi$$
0.724137 + 0.689656i $$0.242238\pi$$
$$600$$ 0 0
$$601$$ 18954.0 1.28644 0.643219 0.765682i $$-0.277598\pi$$
0.643219 + 0.765682i $$0.277598\pi$$
$$602$$ 0 0
$$603$$ −2700.00 −0.182342
$$604$$ 0 0
$$605$$ −3775.00 −0.253679
$$606$$ 0 0
$$607$$ −1896.00 −0.126781 −0.0633907 0.997989i $$-0.520191\pi$$
−0.0633907 + 0.997989i $$0.520191\pi$$
$$608$$ 0 0
$$609$$ 12000.0 0.798464
$$610$$ 0 0
$$611$$ 5712.00 0.378204
$$612$$ 0 0
$$613$$ −9862.00 −0.649792 −0.324896 0.945750i $$-0.605329\pi$$
−0.324896 + 0.945750i $$0.605329\pi$$
$$614$$ 0 0
$$615$$ 5310.00 0.348162
$$616$$ 0 0
$$617$$ −20434.0 −1.33329 −0.666647 0.745374i $$-0.732271\pi$$
−0.666647 + 0.745374i $$0.732271\pi$$
$$618$$ 0 0
$$619$$ 12644.0 0.821010 0.410505 0.911858i $$-0.365352\pi$$
0.410505 + 0.911858i $$0.365352\pi$$
$$620$$ 0 0
$$621$$ −2808.00 −0.181451
$$622$$ 0 0
$$623$$ −12576.0 −0.808743
$$624$$ 0 0
$$625$$ 625.000 0.0400000
$$626$$ 0 0
$$627$$ 2592.00 0.165095
$$628$$ 0 0
$$629$$ 972.000 0.0616155
$$630$$ 0 0
$$631$$ 4660.00 0.293996 0.146998 0.989137i $$-0.453039\pi$$
0.146998 + 0.989137i $$0.453039\pi$$
$$632$$ 0 0
$$633$$ 14388.0 0.903431
$$634$$ 0 0
$$635$$ −8720.00 −0.544949
$$636$$ 0 0
$$637$$ 1218.00 0.0757597
$$638$$ 0 0
$$639$$ −9144.00 −0.566089
$$640$$ 0 0
$$641$$ −8598.00 −0.529798 −0.264899 0.964276i $$-0.585339\pi$$
−0.264899 + 0.964276i $$0.585339\pi$$
$$642$$ 0 0
$$643$$ 1836.00 0.112605 0.0563023 0.998414i $$-0.482069\pi$$
0.0563023 + 0.998414i $$0.482069\pi$$
$$644$$ 0 0
$$645$$ −3420.00 −0.208779
$$646$$ 0 0
$$647$$ −1696.00 −0.103055 −0.0515275 0.998672i $$-0.516409\pi$$
−0.0515275 + 0.998672i $$0.516409\pi$$
$$648$$ 0 0
$$649$$ −1536.00 −0.0929018
$$650$$ 0 0
$$651$$ −1344.00 −0.0809148
$$652$$ 0 0
$$653$$ −24730.0 −1.48202 −0.741010 0.671493i $$-0.765653\pi$$
−0.741010 + 0.671493i $$0.765653\pi$$
$$654$$ 0 0
$$655$$ 2400.00 0.143169
$$656$$ 0 0
$$657$$ 2466.00 0.146435
$$658$$ 0 0
$$659$$ −4800.00 −0.283735 −0.141868 0.989886i $$-0.545311\pi$$
−0.141868 + 0.989886i $$0.545311\pi$$
$$660$$ 0 0
$$661$$ 32174.0 1.89323 0.946614 0.322370i $$-0.104479\pi$$
0.946614 + 0.322370i $$0.104479\pi$$
$$662$$ 0 0
$$663$$ 756.000 0.0442845
$$664$$ 0 0
$$665$$ 2880.00 0.167942
$$666$$ 0 0
$$667$$ 26000.0 1.50933
$$668$$ 0 0
$$669$$ −7680.00 −0.443836
$$670$$ 0 0
$$671$$ −8976.00 −0.516415
$$672$$ 0 0
$$673$$ 7114.00 0.407466 0.203733 0.979026i $$-0.434693\pi$$
0.203733 + 0.979026i $$0.434693\pi$$
$$674$$ 0 0
$$675$$ 675.000 0.0384900
$$676$$ 0 0
$$677$$ −20466.0 −1.16185 −0.580925 0.813957i $$-0.697309\pi$$
−0.580925 + 0.813957i $$0.697309\pi$$
$$678$$ 0 0
$$679$$ 17376.0 0.982076
$$680$$ 0 0
$$681$$ −10500.0 −0.590838
$$682$$ 0 0
$$683$$ 34068.0 1.90860 0.954301 0.298846i $$-0.0966016\pi$$
0.954301 + 0.298846i $$0.0966016\pi$$
$$684$$ 0 0
$$685$$ 7990.00 0.445667
$$686$$ 0 0
$$687$$ 5898.00 0.327544
$$688$$ 0 0
$$689$$ −3668.00 −0.202815
$$690$$ 0 0
$$691$$ 21340.0 1.17484 0.587418 0.809284i $$-0.300144\pi$$
0.587418 + 0.809284i $$0.300144\pi$$
$$692$$ 0 0
$$693$$ 3456.00 0.189441
$$694$$ 0 0
$$695$$ 14820.0 0.808856
$$696$$ 0 0
$$697$$ −6372.00 −0.346279
$$698$$ 0 0
$$699$$ 9738.00 0.526931
$$700$$ 0 0
$$701$$ −5370.00 −0.289333 −0.144666 0.989481i $$-0.546211\pi$$
−0.144666 + 0.989481i $$0.546211\pi$$
$$702$$ 0 0
$$703$$ 1944.00 0.104295
$$704$$ 0 0
$$705$$ −6120.00 −0.326940
$$706$$ 0 0
$$707$$ −1248.00 −0.0663874
$$708$$ 0 0
$$709$$ −18690.0 −0.990011 −0.495005 0.868890i $$-0.664834\pi$$
−0.495005 + 0.868890i $$0.664834\pi$$
$$710$$ 0 0
$$711$$ −7092.00 −0.374080
$$712$$ 0 0
$$713$$ −2912.00 −0.152953
$$714$$ 0 0
$$715$$ 1680.00 0.0878719
$$716$$ 0 0
$$717$$ 21960.0 1.14381
$$718$$ 0 0
$$719$$ 14328.0 0.743177 0.371588 0.928398i $$-0.378813\pi$$
0.371588 + 0.928398i $$0.378813\pi$$
$$720$$ 0 0
$$721$$ 19328.0 0.998353
$$722$$ 0 0
$$723$$ 10470.0 0.538566
$$724$$ 0 0
$$725$$ −6250.00 −0.320164
$$726$$ 0 0
$$727$$ 14488.0 0.739106 0.369553 0.929210i $$-0.379511\pi$$
0.369553 + 0.929210i $$0.379511\pi$$
$$728$$ 0 0
$$729$$ 729.000 0.0370370
$$730$$ 0 0
$$731$$ 4104.00 0.207650
$$732$$ 0 0
$$733$$ 25354.0 1.27759 0.638794 0.769378i $$-0.279434\pi$$
0.638794 + 0.769378i $$0.279434\pi$$
$$734$$ 0 0
$$735$$ −1305.00 −0.0654907
$$736$$ 0 0
$$737$$ 7200.00 0.359858
$$738$$ 0 0
$$739$$ −33100.0 −1.64764 −0.823818 0.566854i $$-0.808160\pi$$
−0.823818 + 0.566854i $$0.808160\pi$$
$$740$$ 0 0
$$741$$ 1512.00 0.0749591
$$742$$ 0 0
$$743$$ 4456.00 0.220020 0.110010 0.993930i $$-0.464912\pi$$
0.110010 + 0.993930i $$0.464912\pi$$
$$744$$ 0 0
$$745$$ 1670.00 0.0821263
$$746$$ 0 0
$$747$$ 3564.00 0.174565
$$748$$ 0 0
$$749$$ −704.000 −0.0343439
$$750$$ 0 0
$$751$$ −23268.0 −1.13057 −0.565287 0.824894i $$-0.691235\pi$$
−0.565287 + 0.824894i $$0.691235\pi$$
$$752$$ 0 0
$$753$$ 22368.0 1.08252
$$754$$ 0 0
$$755$$ 5740.00 0.276689
$$756$$ 0 0
$$757$$ −35726.0 −1.71530 −0.857651 0.514232i $$-0.828077\pi$$
−0.857651 + 0.514232i $$0.828077\pi$$
$$758$$ 0 0
$$759$$ 7488.00 0.358099
$$760$$ 0 0
$$761$$ −12278.0 −0.584858 −0.292429 0.956287i $$-0.594464\pi$$
−0.292429 + 0.956287i $$0.594464\pi$$
$$762$$ 0 0
$$763$$ 17952.0 0.851777
$$764$$ 0 0
$$765$$ −810.000 −0.0382818
$$766$$ 0 0
$$767$$ −896.000 −0.0421808
$$768$$ 0 0
$$769$$ −26542.0 −1.24464 −0.622321 0.782763i $$-0.713810\pi$$
−0.622321 + 0.782763i $$0.713810\pi$$
$$770$$ 0 0
$$771$$ 13674.0 0.638725
$$772$$ 0 0
$$773$$ 9942.00 0.462599 0.231299 0.972883i $$-0.425702\pi$$
0.231299 + 0.972883i $$0.425702\pi$$
$$774$$ 0 0
$$775$$ 700.000 0.0324448
$$776$$ 0 0
$$777$$ 2592.00 0.119675
$$778$$ 0 0
$$779$$ −12744.0 −0.586138
$$780$$ 0 0
$$781$$ 24384.0 1.11719
$$782$$ 0 0
$$783$$ −6750.00 −0.308078
$$784$$ 0 0
$$785$$ 4530.00 0.205965
$$786$$ 0 0
$$787$$ 11132.0 0.504210 0.252105 0.967700i $$-0.418877\pi$$
0.252105 + 0.967700i $$0.418877\pi$$
$$788$$ 0 0
$$789$$ 8544.00 0.385519
$$790$$ 0 0
$$791$$ −9696.00 −0.435841
$$792$$ 0 0
$$793$$ −5236.00 −0.234471
$$794$$ 0 0
$$795$$ 3930.00 0.175324
$$796$$ 0 0
$$797$$ 23910.0 1.06265 0.531327 0.847167i $$-0.321693\pi$$
0.531327 + 0.847167i $$0.321693\pi$$
$$798$$ 0 0
$$799$$ 7344.00 0.325172
$$800$$ 0 0
$$801$$ 7074.00 0.312044
$$802$$ 0 0
$$803$$ −6576.00 −0.288994
$$804$$ 0 0
$$805$$ 8320.00 0.364275
$$806$$ 0 0
$$807$$ 9330.00 0.406978
$$808$$ 0 0
$$809$$ −15934.0 −0.692472 −0.346236 0.938148i $$-0.612540\pi$$
−0.346236 + 0.938148i $$0.612540\pi$$
$$810$$ 0 0
$$811$$ 23756.0 1.02859 0.514295 0.857614i $$-0.328054\pi$$
0.514295 + 0.857614i $$0.328054\pi$$
$$812$$ 0 0
$$813$$ −5100.00 −0.220006
$$814$$ 0 0
$$815$$ 9580.00 0.411746
$$816$$ 0 0
$$817$$ 8208.00 0.351483
$$818$$ 0 0
$$819$$ 2016.00 0.0860131
$$820$$ 0 0
$$821$$ −114.000 −0.00484607 −0.00242304 0.999997i $$-0.500771\pi$$
−0.00242304 + 0.999997i $$0.500771\pi$$
$$822$$ 0 0
$$823$$ −43784.0 −1.85445 −0.927226 0.374502i $$-0.877814\pi$$
−0.927226 + 0.374502i $$0.877814\pi$$
$$824$$ 0 0
$$825$$ −1800.00 −0.0759612
$$826$$ 0 0
$$827$$ 17044.0 0.716660 0.358330 0.933595i $$-0.383346\pi$$
0.358330 + 0.933595i $$0.383346\pi$$
$$828$$ 0 0
$$829$$ −21682.0 −0.908380 −0.454190 0.890905i $$-0.650071\pi$$
−0.454190 + 0.890905i $$0.650071\pi$$
$$830$$ 0 0
$$831$$ −19482.0 −0.813265
$$832$$ 0 0
$$833$$ 1566.00 0.0651365
$$834$$ 0 0
$$835$$ −5760.00 −0.238722
$$836$$ 0 0
$$837$$ 756.000 0.0312201
$$838$$ 0 0
$$839$$ −39488.0 −1.62488 −0.812442 0.583042i $$-0.801862\pi$$
−0.812442 + 0.583042i $$0.801862\pi$$
$$840$$ 0 0
$$841$$ 38111.0 1.56263
$$842$$ 0 0
$$843$$ 7494.00 0.306177
$$844$$ 0 0
$$845$$ −10005.0 −0.407317
$$846$$ 0 0
$$847$$ 12080.0 0.490052
$$848$$ 0 0
$$849$$ 15972.0 0.645651
$$850$$ 0 0
$$851$$ 5616.00 0.226221
$$852$$ 0 0
$$853$$ −14182.0 −0.569264 −0.284632 0.958637i $$-0.591871\pi$$
−0.284632 + 0.958637i $$0.591871\pi$$
$$854$$ 0 0
$$855$$ −1620.00 −0.0647986
$$856$$ 0 0
$$857$$ 27094.0 1.07995 0.539973 0.841682i $$-0.318435\pi$$
0.539973 + 0.841682i $$0.318435\pi$$
$$858$$ 0 0
$$859$$ −26692.0 −1.06021 −0.530104 0.847932i $$-0.677847\pi$$
−0.530104 + 0.847932i $$0.677847\pi$$
$$860$$ 0 0
$$861$$ −16992.0 −0.672574
$$862$$ 0 0
$$863$$ −38872.0 −1.53328 −0.766639 0.642079i $$-0.778072\pi$$
−0.766639 + 0.642079i $$0.778072\pi$$
$$864$$ 0 0
$$865$$ 15710.0 0.617521
$$866$$ 0 0
$$867$$ −13767.0 −0.539275
$$868$$ 0 0
$$869$$ 18912.0 0.738257
$$870$$ 0 0
$$871$$ 4200.00 0.163389
$$872$$ 0 0
$$873$$ −9774.00 −0.378923
$$874$$ 0 0
$$875$$ −2000.00 −0.0772712
$$876$$ 0 0
$$877$$ 6490.00 0.249888 0.124944 0.992164i $$-0.460125\pi$$
0.124944 + 0.992164i $$0.460125\pi$$
$$878$$ 0 0
$$879$$ −1566.00 −0.0600909
$$880$$ 0 0
$$881$$ −35766.0 −1.36775 −0.683875 0.729600i $$-0.739706\pi$$
−0.683875 + 0.729600i $$0.739706\pi$$
$$882$$ 0 0
$$883$$ −1316.00 −0.0501551 −0.0250775 0.999686i $$-0.507983\pi$$
−0.0250775 + 0.999686i $$0.507983\pi$$
$$884$$ 0 0
$$885$$ 960.000 0.0364633
$$886$$ 0 0
$$887$$ −6656.00 −0.251958 −0.125979 0.992033i $$-0.540207\pi$$
−0.125979 + 0.992033i $$0.540207\pi$$
$$888$$ 0 0
$$889$$ 27904.0 1.05272
$$890$$ 0 0
$$891$$ −1944.00 −0.0730937
$$892$$ 0 0
$$893$$ 14688.0 0.550409
$$894$$ 0 0
$$895$$ 5160.00 0.192715
$$896$$ 0 0
$$897$$ 4368.00 0.162590
$$898$$ 0 0
$$899$$ −7000.00 −0.259692
$$900$$ 0 0
$$901$$ −4716.00 −0.174376
$$902$$ 0 0
$$903$$ 10944.0 0.403315
$$904$$ 0 0
$$905$$ −7810.00 −0.286865
$$906$$ 0 0
$$907$$ 15772.0 0.577399 0.288699 0.957420i $$-0.406777\pi$$
0.288699 + 0.957420i $$0.406777\pi$$
$$908$$ 0 0
$$909$$ 702.000 0.0256148
$$910$$ 0 0
$$911$$ −15168.0 −0.551634 −0.275817 0.961210i $$-0.588948\pi$$
−0.275817 + 0.961210i $$0.588948\pi$$
$$912$$ 0 0
$$913$$ −9504.00 −0.344509
$$914$$ 0 0
$$915$$ 5610.00 0.202689
$$916$$ 0 0
$$917$$ −7680.00 −0.276571
$$918$$ 0 0
$$919$$ −7148.00 −0.256573 −0.128287 0.991737i $$-0.540948\pi$$
−0.128287 + 0.991737i $$0.540948\pi$$
$$920$$ 0 0
$$921$$ −23532.0 −0.841917
$$922$$ 0 0
$$923$$ 14224.0 0.507247
$$924$$ 0 0
$$925$$ −1350.00 −0.0479867
$$926$$ 0 0
$$927$$ −10872.0 −0.385203
$$928$$ 0 0
$$929$$ −8206.00 −0.289806 −0.144903 0.989446i $$-0.546287\pi$$
−0.144903 + 0.989446i $$0.546287\pi$$
$$930$$ 0 0
$$931$$ 3132.00 0.110255
$$932$$ 0 0
$$933$$ 9744.00 0.341912
$$934$$ 0 0
$$935$$ 2160.00 0.0755503
$$936$$ 0 0
$$937$$ −55574.0 −1.93759 −0.968796 0.247860i $$-0.920273\pi$$
−0.968796 + 0.247860i $$0.920273\pi$$
$$938$$ 0 0
$$939$$ −16122.0 −0.560300
$$940$$ 0 0
$$941$$ −3690.00 −0.127833 −0.0639163 0.997955i $$-0.520359\pi$$
−0.0639163 + 0.997955i $$0.520359\pi$$
$$942$$ 0 0
$$943$$ −36816.0 −1.27136
$$944$$ 0 0
$$945$$ −2160.00 −0.0743543
$$946$$ 0 0
$$947$$ 46700.0 1.60248 0.801239 0.598345i $$-0.204175\pi$$
0.801239 + 0.598345i $$0.204175\pi$$
$$948$$ 0 0
$$949$$ −3836.00 −0.131214
$$950$$ 0 0
$$951$$ −20358.0 −0.694168
$$952$$ 0 0
$$953$$ −40018.0 −1.36024 −0.680121 0.733100i $$-0.738073\pi$$
−0.680121 + 0.733100i $$0.738073\pi$$
$$954$$ 0 0
$$955$$ −9800.00 −0.332063
$$956$$ 0 0
$$957$$ 18000.0 0.608001
$$958$$ 0 0
$$959$$ −25568.0 −0.860932
$$960$$ 0 0
$$961$$ −29007.0 −0.973683
$$962$$ 0 0
$$963$$ 396.000 0.0132512
$$964$$ 0 0
$$965$$ −20030.0 −0.668175
$$966$$ 0 0
$$967$$ 1064.00 0.0353836 0.0176918 0.999843i $$-0.494368\pi$$
0.0176918 + 0.999843i $$0.494368\pi$$
$$968$$ 0 0
$$969$$ 1944.00 0.0644482
$$970$$ 0 0
$$971$$ −5664.00 −0.187195 −0.0935975 0.995610i $$-0.529837\pi$$
−0.0935975 + 0.995610i $$0.529837\pi$$
$$972$$ 0 0
$$973$$ −47424.0 −1.56253
$$974$$ 0 0
$$975$$ −1050.00 −0.0344891
$$976$$ 0 0
$$977$$ 33870.0 1.10911 0.554553 0.832148i $$-0.312889\pi$$
0.554553 + 0.832148i $$0.312889\pi$$
$$978$$ 0 0
$$979$$ −18864.0 −0.615828
$$980$$ 0 0
$$981$$ −10098.0 −0.328649
$$982$$ 0 0
$$983$$ −19976.0 −0.648154 −0.324077 0.946031i $$-0.605054\pi$$
−0.324077 + 0.946031i $$0.605054\pi$$
$$984$$ 0 0
$$985$$ 10590.0 0.342564
$$986$$ 0 0
$$987$$ 19584.0 0.631576
$$988$$ 0 0
$$989$$ 23712.0 0.762384
$$990$$ 0 0
$$991$$ −28748.0 −0.921504 −0.460752 0.887529i $$-0.652420\pi$$
−0.460752 + 0.887529i $$0.652420\pi$$
$$992$$ 0 0
$$993$$ −19788.0 −0.632380
$$994$$ 0 0
$$995$$ 18740.0 0.597083
$$996$$ 0 0
$$997$$ −16830.0 −0.534615 −0.267308 0.963611i $$-0.586134\pi$$
−0.267308 + 0.963611i $$0.586134\pi$$
$$998$$ 0 0
$$999$$ −1458.00 −0.0461753
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 480.4.a.i.1.1 yes 1
3.2 odd 2 1440.4.a.c.1.1 1
4.3 odd 2 480.4.a.f.1.1 1
5.4 even 2 2400.4.a.h.1.1 1
8.3 odd 2 960.4.a.z.1.1 1
8.5 even 2 960.4.a.c.1.1 1
12.11 even 2 1440.4.a.h.1.1 1
20.19 odd 2 2400.4.a.o.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
480.4.a.f.1.1 1 4.3 odd 2
480.4.a.i.1.1 yes 1 1.1 even 1 trivial
960.4.a.c.1.1 1 8.5 even 2
960.4.a.z.1.1 1 8.3 odd 2
1440.4.a.c.1.1 1 3.2 odd 2
1440.4.a.h.1.1 1 12.11 even 2
2400.4.a.h.1.1 1 5.4 even 2
2400.4.a.o.1.1 1 20.19 odd 2