# Properties

 Label 120.4.a.a.1.1 Level $120$ Weight $4$ Character 120.1 Self dual yes Analytic conductor $7.080$ 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] = [120,4,Mod(1,120)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(120, 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("120.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$120 = 2^{3} \cdot 3 \cdot 5$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 120.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: $$7.08022920069$$ Analytic rank: $$0$$ 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 $$\chi$$ $$=$$ 120.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-3.00000 q^{3} -5.00000 q^{5} +4.00000 q^{7} +9.00000 q^{9} +O(q^{10})$$ $$q-3.00000 q^{3} -5.00000 q^{5} +4.00000 q^{7} +9.00000 q^{9} +72.0000 q^{11} -6.00000 q^{13} +15.0000 q^{15} +38.0000 q^{17} +52.0000 q^{19} -12.0000 q^{21} +152.000 q^{23} +25.0000 q^{25} -27.0000 q^{27} -78.0000 q^{29} +120.000 q^{31} -216.000 q^{33} -20.0000 q^{35} -150.000 q^{37} +18.0000 q^{39} +362.000 q^{41} -484.000 q^{43} -45.0000 q^{45} +280.000 q^{47} -327.000 q^{49} -114.000 q^{51} -670.000 q^{53} -360.000 q^{55} -156.000 q^{57} +696.000 q^{59} +222.000 q^{61} +36.0000 q^{63} +30.0000 q^{65} -4.00000 q^{67} -456.000 q^{69} +96.0000 q^{71} +178.000 q^{73} -75.0000 q^{75} +288.000 q^{77} -632.000 q^{79} +81.0000 q^{81} -612.000 q^{83} -190.000 q^{85} +234.000 q^{87} +994.000 q^{89} -24.0000 q^{91} -360.000 q^{93} -260.000 q^{95} +1634.00 q^{97} +648.000 q^{99} +O(q^{100})$$

## 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
$$7$$ 4.00000 0.215980 0.107990 0.994152i $$-0.465559\pi$$
0.107990 + 0.994152i $$0.465559\pi$$
$$8$$ 0 0
$$9$$ 9.00000 0.333333
$$10$$ 0 0
$$11$$ 72.0000 1.97353 0.986764 0.162160i $$-0.0518462\pi$$
0.986764 + 0.162160i $$0.0518462\pi$$
$$12$$ 0 0
$$13$$ −6.00000 −0.128008 −0.0640039 0.997950i $$-0.520387\pi$$
−0.0640039 + 0.997950i $$0.520387\pi$$
$$14$$ 0 0
$$15$$ 15.0000 0.258199
$$16$$ 0 0
$$17$$ 38.0000 0.542138 0.271069 0.962560i $$-0.412623\pi$$
0.271069 + 0.962560i $$0.412623\pi$$
$$18$$ 0 0
$$19$$ 52.0000 0.627875 0.313937 0.949444i $$-0.398352\pi$$
0.313937 + 0.949444i $$0.398352\pi$$
$$20$$ 0 0
$$21$$ −12.0000 −0.124696
$$22$$ 0 0
$$23$$ 152.000 1.37801 0.689004 0.724757i $$-0.258048\pi$$
0.689004 + 0.724757i $$0.258048\pi$$
$$24$$ 0 0
$$25$$ 25.0000 0.200000
$$26$$ 0 0
$$27$$ −27.0000 −0.192450
$$28$$ 0 0
$$29$$ −78.0000 −0.499456 −0.249728 0.968316i $$-0.580341\pi$$
−0.249728 + 0.968316i $$0.580341\pi$$
$$30$$ 0 0
$$31$$ 120.000 0.695246 0.347623 0.937634i $$-0.386989\pi$$
0.347623 + 0.937634i $$0.386989\pi$$
$$32$$ 0 0
$$33$$ −216.000 −1.13942
$$34$$ 0 0
$$35$$ −20.0000 −0.0965891
$$36$$ 0 0
$$37$$ −150.000 −0.666482 −0.333241 0.942842i $$-0.608142\pi$$
−0.333241 + 0.942842i $$0.608142\pi$$
$$38$$ 0 0
$$39$$ 18.0000 0.0739053
$$40$$ 0 0
$$41$$ 362.000 1.37890 0.689450 0.724333i $$-0.257852\pi$$
0.689450 + 0.724333i $$0.257852\pi$$
$$42$$ 0 0
$$43$$ −484.000 −1.71650 −0.858248 0.513236i $$-0.828447\pi$$
−0.858248 + 0.513236i $$0.828447\pi$$
$$44$$ 0 0
$$45$$ −45.0000 −0.149071
$$46$$ 0 0
$$47$$ 280.000 0.868983 0.434491 0.900676i $$-0.356928\pi$$
0.434491 + 0.900676i $$0.356928\pi$$
$$48$$ 0 0
$$49$$ −327.000 −0.953353
$$50$$ 0 0
$$51$$ −114.000 −0.313004
$$52$$ 0 0
$$53$$ −670.000 −1.73644 −0.868222 0.496175i $$-0.834737\pi$$
−0.868222 + 0.496175i $$0.834737\pi$$
$$54$$ 0 0
$$55$$ −360.000 −0.882589
$$56$$ 0 0
$$57$$ −156.000 −0.362504
$$58$$ 0 0
$$59$$ 696.000 1.53579 0.767894 0.640577i $$-0.221305\pi$$
0.767894 + 0.640577i $$0.221305\pi$$
$$60$$ 0 0
$$61$$ 222.000 0.465970 0.232985 0.972480i $$-0.425151\pi$$
0.232985 + 0.972480i $$0.425151\pi$$
$$62$$ 0 0
$$63$$ 36.0000 0.0719932
$$64$$ 0 0
$$65$$ 30.0000 0.0572468
$$66$$ 0 0
$$67$$ −4.00000 −0.00729370 −0.00364685 0.999993i $$-0.501161\pi$$
−0.00364685 + 0.999993i $$0.501161\pi$$
$$68$$ 0 0
$$69$$ −456.000 −0.795593
$$70$$ 0 0
$$71$$ 96.0000 0.160466 0.0802331 0.996776i $$-0.474434\pi$$
0.0802331 + 0.996776i $$0.474434\pi$$
$$72$$ 0 0
$$73$$ 178.000 0.285388 0.142694 0.989767i $$-0.454424\pi$$
0.142694 + 0.989767i $$0.454424\pi$$
$$74$$ 0 0
$$75$$ −75.0000 −0.115470
$$76$$ 0 0
$$77$$ 288.000 0.426242
$$78$$ 0 0
$$79$$ −632.000 −0.900070 −0.450035 0.893011i $$-0.648589\pi$$
−0.450035 + 0.893011i $$0.648589\pi$$
$$80$$ 0 0
$$81$$ 81.0000 0.111111
$$82$$ 0 0
$$83$$ −612.000 −0.809346 −0.404673 0.914461i $$-0.632615\pi$$
−0.404673 + 0.914461i $$0.632615\pi$$
$$84$$ 0 0
$$85$$ −190.000 −0.242452
$$86$$ 0 0
$$87$$ 234.000 0.288361
$$88$$ 0 0
$$89$$ 994.000 1.18386 0.591931 0.805988i $$-0.298366\pi$$
0.591931 + 0.805988i $$0.298366\pi$$
$$90$$ 0 0
$$91$$ −24.0000 −0.0276471
$$92$$ 0 0
$$93$$ −360.000 −0.401401
$$94$$ 0 0
$$95$$ −260.000 −0.280794
$$96$$ 0 0
$$97$$ 1634.00 1.71039 0.855194 0.518309i $$-0.173438\pi$$
0.855194 + 0.518309i $$0.173438\pi$$
$$98$$ 0 0
$$99$$ 648.000 0.657843
$$100$$ 0 0
$$101$$ 890.000 0.876815 0.438407 0.898776i $$-0.355543\pi$$
0.438407 + 0.898776i $$0.355543\pi$$
$$102$$ 0 0
$$103$$ −524.000 −0.501274 −0.250637 0.968081i $$-0.580640\pi$$
−0.250637 + 0.968081i $$0.580640\pi$$
$$104$$ 0 0
$$105$$ 60.0000 0.0557657
$$106$$ 0 0
$$107$$ 932.000 0.842055 0.421027 0.907048i $$-0.361670\pi$$
0.421027 + 0.907048i $$0.361670\pi$$
$$108$$ 0 0
$$109$$ 446.000 0.391918 0.195959 0.980612i $$-0.437218\pi$$
0.195959 + 0.980612i $$0.437218\pi$$
$$110$$ 0 0
$$111$$ 450.000 0.384794
$$112$$ 0 0
$$113$$ −786.000 −0.654342 −0.327171 0.944965i $$-0.606095\pi$$
−0.327171 + 0.944965i $$0.606095\pi$$
$$114$$ 0 0
$$115$$ −760.000 −0.616264
$$116$$ 0 0
$$117$$ −54.0000 −0.0426692
$$118$$ 0 0
$$119$$ 152.000 0.117091
$$120$$ 0 0
$$121$$ 3853.00 2.89482
$$122$$ 0 0
$$123$$ −1086.00 −0.796108
$$124$$ 0 0
$$125$$ −125.000 −0.0894427
$$126$$ 0 0
$$127$$ 716.000 0.500273 0.250137 0.968211i $$-0.419524\pi$$
0.250137 + 0.968211i $$0.419524\pi$$
$$128$$ 0 0
$$129$$ 1452.00 0.991019
$$130$$ 0 0
$$131$$ −808.000 −0.538895 −0.269448 0.963015i $$-0.586841\pi$$
−0.269448 + 0.963015i $$0.586841\pi$$
$$132$$ 0 0
$$133$$ 208.000 0.135608
$$134$$ 0 0
$$135$$ 135.000 0.0860663
$$136$$ 0 0
$$137$$ −1770.00 −1.10381 −0.551903 0.833909i $$-0.686098\pi$$
−0.551903 + 0.833909i $$0.686098\pi$$
$$138$$ 0 0
$$139$$ −924.000 −0.563832 −0.281916 0.959439i $$-0.590970\pi$$
−0.281916 + 0.959439i $$0.590970\pi$$
$$140$$ 0 0
$$141$$ −840.000 −0.501708
$$142$$ 0 0
$$143$$ −432.000 −0.252627
$$144$$ 0 0
$$145$$ 390.000 0.223364
$$146$$ 0 0
$$147$$ 981.000 0.550418
$$148$$ 0 0
$$149$$ −3198.00 −1.75832 −0.879162 0.476522i $$-0.841897\pi$$
−0.879162 + 0.476522i $$0.841897\pi$$
$$150$$ 0 0
$$151$$ −3384.00 −1.82375 −0.911874 0.410470i $$-0.865365\pi$$
−0.911874 + 0.410470i $$0.865365\pi$$
$$152$$ 0 0
$$153$$ 342.000 0.180713
$$154$$ 0 0
$$155$$ −600.000 −0.310924
$$156$$ 0 0
$$157$$ −3302.00 −1.67852 −0.839262 0.543727i $$-0.817013\pi$$
−0.839262 + 0.543727i $$0.817013\pi$$
$$158$$ 0 0
$$159$$ 2010.00 1.00254
$$160$$ 0 0
$$161$$ 608.000 0.297622
$$162$$ 0 0
$$163$$ 2252.00 1.08215 0.541074 0.840975i $$-0.318018\pi$$
0.541074 + 0.840975i $$0.318018\pi$$
$$164$$ 0 0
$$165$$ 1080.00 0.509563
$$166$$ 0 0
$$167$$ −184.000 −0.0852596 −0.0426298 0.999091i $$-0.513574\pi$$
−0.0426298 + 0.999091i $$0.513574\pi$$
$$168$$ 0 0
$$169$$ −2161.00 −0.983614
$$170$$ 0 0
$$171$$ 468.000 0.209292
$$172$$ 0 0
$$173$$ −2646.00 −1.16284 −0.581421 0.813603i $$-0.697503\pi$$
−0.581421 + 0.813603i $$0.697503\pi$$
$$174$$ 0 0
$$175$$ 100.000 0.0431959
$$176$$ 0 0
$$177$$ −2088.00 −0.886688
$$178$$ 0 0
$$179$$ −608.000 −0.253877 −0.126939 0.991911i $$-0.540515\pi$$
−0.126939 + 0.991911i $$0.540515\pi$$
$$180$$ 0 0
$$181$$ 2246.00 0.922342 0.461171 0.887311i $$-0.347430\pi$$
0.461171 + 0.887311i $$0.347430\pi$$
$$182$$ 0 0
$$183$$ −666.000 −0.269028
$$184$$ 0 0
$$185$$ 750.000 0.298060
$$186$$ 0 0
$$187$$ 2736.00 1.06993
$$188$$ 0 0
$$189$$ −108.000 −0.0415653
$$190$$ 0 0
$$191$$ −3848.00 −1.45776 −0.728878 0.684643i $$-0.759958\pi$$
−0.728878 + 0.684643i $$0.759958\pi$$
$$192$$ 0 0
$$193$$ 2058.00 0.767555 0.383777 0.923426i $$-0.374623\pi$$
0.383777 + 0.923426i $$0.374623\pi$$
$$194$$ 0 0
$$195$$ −90.0000 −0.0330515
$$196$$ 0 0
$$197$$ −3838.00 −1.38805 −0.694026 0.719950i $$-0.744165\pi$$
−0.694026 + 0.719950i $$0.744165\pi$$
$$198$$ 0 0
$$199$$ −1992.00 −0.709594 −0.354797 0.934943i $$-0.615450\pi$$
−0.354797 + 0.934943i $$0.615450\pi$$
$$200$$ 0 0
$$201$$ 12.0000 0.00421102
$$202$$ 0 0
$$203$$ −312.000 −0.107872
$$204$$ 0 0
$$205$$ −1810.00 −0.616663
$$206$$ 0 0
$$207$$ 1368.00 0.459336
$$208$$ 0 0
$$209$$ 3744.00 1.23913
$$210$$ 0 0
$$211$$ 4764.00 1.55435 0.777174 0.629286i $$-0.216653\pi$$
0.777174 + 0.629286i $$0.216653\pi$$
$$212$$ 0 0
$$213$$ −288.000 −0.0926452
$$214$$ 0 0
$$215$$ 2420.00 0.767640
$$216$$ 0 0
$$217$$ 480.000 0.150159
$$218$$ 0 0
$$219$$ −534.000 −0.164769
$$220$$ 0 0
$$221$$ −228.000 −0.0693979
$$222$$ 0 0
$$223$$ 4092.00 1.22879 0.614396 0.788998i $$-0.289400\pi$$
0.614396 + 0.788998i $$0.289400\pi$$
$$224$$ 0 0
$$225$$ 225.000 0.0666667
$$226$$ 0 0
$$227$$ 468.000 0.136838 0.0684191 0.997657i $$-0.478205\pi$$
0.0684191 + 0.997657i $$0.478205\pi$$
$$228$$ 0 0
$$229$$ −5586.00 −1.61194 −0.805968 0.591959i $$-0.798355\pi$$
−0.805968 + 0.591959i $$0.798355\pi$$
$$230$$ 0 0
$$231$$ −864.000 −0.246091
$$232$$ 0 0
$$233$$ −1058.00 −0.297476 −0.148738 0.988877i $$-0.547521\pi$$
−0.148738 + 0.988877i $$0.547521\pi$$
$$234$$ 0 0
$$235$$ −1400.00 −0.388621
$$236$$ 0 0
$$237$$ 1896.00 0.519656
$$238$$ 0 0
$$239$$ 6840.00 1.85123 0.925613 0.378472i $$-0.123550\pi$$
0.925613 + 0.378472i $$0.123550\pi$$
$$240$$ 0 0
$$241$$ −6430.00 −1.71864 −0.859321 0.511437i $$-0.829113\pi$$
−0.859321 + 0.511437i $$0.829113\pi$$
$$242$$ 0 0
$$243$$ −243.000 −0.0641500
$$244$$ 0 0
$$245$$ 1635.00 0.426352
$$246$$ 0 0
$$247$$ −312.000 −0.0803728
$$248$$ 0 0
$$249$$ 1836.00 0.467276
$$250$$ 0 0
$$251$$ −6352.00 −1.59735 −0.798675 0.601763i $$-0.794465\pi$$
−0.798675 + 0.601763i $$0.794465\pi$$
$$252$$ 0 0
$$253$$ 10944.0 2.71954
$$254$$ 0 0
$$255$$ 570.000 0.139980
$$256$$ 0 0
$$257$$ 1422.00 0.345144 0.172572 0.984997i $$-0.444792\pi$$
0.172572 + 0.984997i $$0.444792\pi$$
$$258$$ 0 0
$$259$$ −600.000 −0.143947
$$260$$ 0 0
$$261$$ −702.000 −0.166485
$$262$$ 0 0
$$263$$ 7224.00 1.69373 0.846865 0.531808i $$-0.178487\pi$$
0.846865 + 0.531808i $$0.178487\pi$$
$$264$$ 0 0
$$265$$ 3350.00 0.776562
$$266$$ 0 0
$$267$$ −2982.00 −0.683504
$$268$$ 0 0
$$269$$ 3186.00 0.722133 0.361067 0.932540i $$-0.382413\pi$$
0.361067 + 0.932540i $$0.382413\pi$$
$$270$$ 0 0
$$271$$ −256.000 −0.0573834 −0.0286917 0.999588i $$-0.509134\pi$$
−0.0286917 + 0.999588i $$0.509134\pi$$
$$272$$ 0 0
$$273$$ 72.0000 0.0159620
$$274$$ 0 0
$$275$$ 1800.00 0.394706
$$276$$ 0 0
$$277$$ −5942.00 −1.28888 −0.644441 0.764654i $$-0.722910\pi$$
−0.644441 + 0.764654i $$0.722910\pi$$
$$278$$ 0 0
$$279$$ 1080.00 0.231749
$$280$$ 0 0
$$281$$ 3202.00 0.679770 0.339885 0.940467i $$-0.389612\pi$$
0.339885 + 0.940467i $$0.389612\pi$$
$$282$$ 0 0
$$283$$ 3940.00 0.827593 0.413796 0.910370i $$-0.364203\pi$$
0.413796 + 0.910370i $$0.364203\pi$$
$$284$$ 0 0
$$285$$ 780.000 0.162117
$$286$$ 0 0
$$287$$ 1448.00 0.297814
$$288$$ 0 0
$$289$$ −3469.00 −0.706086
$$290$$ 0 0
$$291$$ −4902.00 −0.987493
$$292$$ 0 0
$$293$$ 1826.00 0.364082 0.182041 0.983291i $$-0.441730\pi$$
0.182041 + 0.983291i $$0.441730\pi$$
$$294$$ 0 0
$$295$$ −3480.00 −0.686825
$$296$$ 0 0
$$297$$ −1944.00 −0.379806
$$298$$ 0 0
$$299$$ −912.000 −0.176396
$$300$$ 0 0
$$301$$ −1936.00 −0.370728
$$302$$ 0 0
$$303$$ −2670.00 −0.506229
$$304$$ 0 0
$$305$$ −1110.00 −0.208388
$$306$$ 0 0
$$307$$ 6580.00 1.22326 0.611629 0.791144i $$-0.290514\pi$$
0.611629 + 0.791144i $$0.290514\pi$$
$$308$$ 0 0
$$309$$ 1572.00 0.289411
$$310$$ 0 0
$$311$$ −5728.00 −1.04439 −0.522195 0.852826i $$-0.674887\pi$$
−0.522195 + 0.852826i $$0.674887\pi$$
$$312$$ 0 0
$$313$$ −1742.00 −0.314580 −0.157290 0.987552i $$-0.550276\pi$$
−0.157290 + 0.987552i $$0.550276\pi$$
$$314$$ 0 0
$$315$$ −180.000 −0.0321964
$$316$$ 0 0
$$317$$ 8746.00 1.54960 0.774802 0.632204i $$-0.217850\pi$$
0.774802 + 0.632204i $$0.217850\pi$$
$$318$$ 0 0
$$319$$ −5616.00 −0.985692
$$320$$ 0 0
$$321$$ −2796.00 −0.486160
$$322$$ 0 0
$$323$$ 1976.00 0.340395
$$324$$ 0 0
$$325$$ −150.000 −0.0256015
$$326$$ 0 0
$$327$$ −1338.00 −0.226274
$$328$$ 0 0
$$329$$ 1120.00 0.187683
$$330$$ 0 0
$$331$$ −2564.00 −0.425771 −0.212885 0.977077i $$-0.568286\pi$$
−0.212885 + 0.977077i $$0.568286\pi$$
$$332$$ 0 0
$$333$$ −1350.00 −0.222161
$$334$$ 0 0
$$335$$ 20.0000 0.00326184
$$336$$ 0 0
$$337$$ −4166.00 −0.673402 −0.336701 0.941612i $$-0.609311\pi$$
−0.336701 + 0.941612i $$0.609311\pi$$
$$338$$ 0 0
$$339$$ 2358.00 0.377785
$$340$$ 0 0
$$341$$ 8640.00 1.37209
$$342$$ 0 0
$$343$$ −2680.00 −0.421885
$$344$$ 0 0
$$345$$ 2280.00 0.355800
$$346$$ 0 0
$$347$$ −9444.00 −1.46104 −0.730519 0.682892i $$-0.760722\pi$$
−0.730519 + 0.682892i $$0.760722\pi$$
$$348$$ 0 0
$$349$$ −9218.00 −1.41383 −0.706917 0.707296i $$-0.749915\pi$$
−0.706917 + 0.707296i $$0.749915\pi$$
$$350$$ 0 0
$$351$$ 162.000 0.0246351
$$352$$ 0 0
$$353$$ −4698.00 −0.708355 −0.354177 0.935178i $$-0.615239\pi$$
−0.354177 + 0.935178i $$0.615239\pi$$
$$354$$ 0 0
$$355$$ −480.000 −0.0717627
$$356$$ 0 0
$$357$$ −456.000 −0.0676025
$$358$$ 0 0
$$359$$ −6056.00 −0.890316 −0.445158 0.895452i $$-0.646852\pi$$
−0.445158 + 0.895452i $$0.646852\pi$$
$$360$$ 0 0
$$361$$ −4155.00 −0.605773
$$362$$ 0 0
$$363$$ −11559.0 −1.67132
$$364$$ 0 0
$$365$$ −890.000 −0.127629
$$366$$ 0 0
$$367$$ −8228.00 −1.17029 −0.585147 0.810927i $$-0.698963\pi$$
−0.585147 + 0.810927i $$0.698963\pi$$
$$368$$ 0 0
$$369$$ 3258.00 0.459633
$$370$$ 0 0
$$371$$ −2680.00 −0.375037
$$372$$ 0 0
$$373$$ 5954.00 0.826505 0.413253 0.910616i $$-0.364393\pi$$
0.413253 + 0.910616i $$0.364393\pi$$
$$374$$ 0 0
$$375$$ 375.000 0.0516398
$$376$$ 0 0
$$377$$ 468.000 0.0639343
$$378$$ 0 0
$$379$$ 5284.00 0.716150 0.358075 0.933693i $$-0.383433\pi$$
0.358075 + 0.933693i $$0.383433\pi$$
$$380$$ 0 0
$$381$$ −2148.00 −0.288833
$$382$$ 0 0
$$383$$ 9832.00 1.31173 0.655864 0.754879i $$-0.272305\pi$$
0.655864 + 0.754879i $$0.272305\pi$$
$$384$$ 0 0
$$385$$ −1440.00 −0.190621
$$386$$ 0 0
$$387$$ −4356.00 −0.572165
$$388$$ 0 0
$$389$$ −222.000 −0.0289353 −0.0144677 0.999895i $$-0.504605\pi$$
−0.0144677 + 0.999895i $$0.504605\pi$$
$$390$$ 0 0
$$391$$ 5776.00 0.747071
$$392$$ 0 0
$$393$$ 2424.00 0.311131
$$394$$ 0 0
$$395$$ 3160.00 0.402524
$$396$$ 0 0
$$397$$ 12098.0 1.52942 0.764712 0.644372i $$-0.222881\pi$$
0.764712 + 0.644372i $$0.222881\pi$$
$$398$$ 0 0
$$399$$ −624.000 −0.0782934
$$400$$ 0 0
$$401$$ −5958.00 −0.741966 −0.370983 0.928640i $$-0.620979\pi$$
−0.370983 + 0.928640i $$0.620979\pi$$
$$402$$ 0 0
$$403$$ −720.000 −0.0889969
$$404$$ 0 0
$$405$$ −405.000 −0.0496904
$$406$$ 0 0
$$407$$ −10800.0 −1.31532
$$408$$ 0 0
$$409$$ 1930.00 0.233331 0.116665 0.993171i $$-0.462779\pi$$
0.116665 + 0.993171i $$0.462779\pi$$
$$410$$ 0 0
$$411$$ 5310.00 0.637282
$$412$$ 0 0
$$413$$ 2784.00 0.331699
$$414$$ 0 0
$$415$$ 3060.00 0.361951
$$416$$ 0 0
$$417$$ 2772.00 0.325529
$$418$$ 0 0
$$419$$ 4744.00 0.553125 0.276563 0.960996i $$-0.410805\pi$$
0.276563 + 0.960996i $$0.410805\pi$$
$$420$$ 0 0
$$421$$ 1614.00 0.186845 0.0934223 0.995627i $$-0.470219\pi$$
0.0934223 + 0.995627i $$0.470219\pi$$
$$422$$ 0 0
$$423$$ 2520.00 0.289661
$$424$$ 0 0
$$425$$ 950.000 0.108428
$$426$$ 0 0
$$427$$ 888.000 0.100640
$$428$$ 0 0
$$429$$ 1296.00 0.145854
$$430$$ 0 0
$$431$$ 9296.00 1.03892 0.519458 0.854496i $$-0.326134\pi$$
0.519458 + 0.854496i $$0.326134\pi$$
$$432$$ 0 0
$$433$$ −3494.00 −0.387785 −0.193893 0.981023i $$-0.562111\pi$$
−0.193893 + 0.981023i $$0.562111\pi$$
$$434$$ 0 0
$$435$$ −1170.00 −0.128959
$$436$$ 0 0
$$437$$ 7904.00 0.865216
$$438$$ 0 0
$$439$$ −12584.0 −1.36811 −0.684056 0.729429i $$-0.739786\pi$$
−0.684056 + 0.729429i $$0.739786\pi$$
$$440$$ 0 0
$$441$$ −2943.00 −0.317784
$$442$$ 0 0
$$443$$ −12852.0 −1.37837 −0.689184 0.724586i $$-0.742031\pi$$
−0.689184 + 0.724586i $$0.742031\pi$$
$$444$$ 0 0
$$445$$ −4970.00 −0.529440
$$446$$ 0 0
$$447$$ 9594.00 1.01517
$$448$$ 0 0
$$449$$ 14458.0 1.51963 0.759816 0.650138i $$-0.225289\pi$$
0.759816 + 0.650138i $$0.225289\pi$$
$$450$$ 0 0
$$451$$ 26064.0 2.72130
$$452$$ 0 0
$$453$$ 10152.0 1.05294
$$454$$ 0 0
$$455$$ 120.000 0.0123641
$$456$$ 0 0
$$457$$ −4310.00 −0.441167 −0.220583 0.975368i $$-0.570796\pi$$
−0.220583 + 0.975368i $$0.570796\pi$$
$$458$$ 0 0
$$459$$ −1026.00 −0.104335
$$460$$ 0 0
$$461$$ 5338.00 0.539296 0.269648 0.962959i $$-0.413093\pi$$
0.269648 + 0.962959i $$0.413093\pi$$
$$462$$ 0 0
$$463$$ 1156.00 0.116034 0.0580171 0.998316i $$-0.481522\pi$$
0.0580171 + 0.998316i $$0.481522\pi$$
$$464$$ 0 0
$$465$$ 1800.00 0.179512
$$466$$ 0 0
$$467$$ 5948.00 0.589380 0.294690 0.955593i $$-0.404784\pi$$
0.294690 + 0.955593i $$0.404784\pi$$
$$468$$ 0 0
$$469$$ −16.0000 −0.00157529
$$470$$ 0 0
$$471$$ 9906.00 0.969096
$$472$$ 0 0
$$473$$ −34848.0 −3.38755
$$474$$ 0 0
$$475$$ 1300.00 0.125575
$$476$$ 0 0
$$477$$ −6030.00 −0.578815
$$478$$ 0 0
$$479$$ 6888.00 0.657037 0.328519 0.944498i $$-0.393451\pi$$
0.328519 + 0.944498i $$0.393451\pi$$
$$480$$ 0 0
$$481$$ 900.000 0.0853149
$$482$$ 0 0
$$483$$ −1824.00 −0.171832
$$484$$ 0 0
$$485$$ −8170.00 −0.764908
$$486$$ 0 0
$$487$$ 2892.00 0.269095 0.134547 0.990907i $$-0.457042\pi$$
0.134547 + 0.990907i $$0.457042\pi$$
$$488$$ 0 0
$$489$$ −6756.00 −0.624779
$$490$$ 0 0
$$491$$ 4096.00 0.376476 0.188238 0.982123i $$-0.439722\pi$$
0.188238 + 0.982123i $$0.439722\pi$$
$$492$$ 0 0
$$493$$ −2964.00 −0.270775
$$494$$ 0 0
$$495$$ −3240.00 −0.294196
$$496$$ 0 0
$$497$$ 384.000 0.0346575
$$498$$ 0 0
$$499$$ 11060.0 0.992212 0.496106 0.868262i $$-0.334763\pi$$
0.496106 + 0.868262i $$0.334763\pi$$
$$500$$ 0 0
$$501$$ 552.000 0.0492246
$$502$$ 0 0
$$503$$ −9648.00 −0.855235 −0.427617 0.903960i $$-0.640647\pi$$
−0.427617 + 0.903960i $$0.640647\pi$$
$$504$$ 0 0
$$505$$ −4450.00 −0.392124
$$506$$ 0 0
$$507$$ 6483.00 0.567890
$$508$$ 0 0
$$509$$ −10062.0 −0.876209 −0.438104 0.898924i $$-0.644350\pi$$
−0.438104 + 0.898924i $$0.644350\pi$$
$$510$$ 0 0
$$511$$ 712.000 0.0616380
$$512$$ 0 0
$$513$$ −1404.00 −0.120835
$$514$$ 0 0
$$515$$ 2620.00 0.224177
$$516$$ 0 0
$$517$$ 20160.0 1.71496
$$518$$ 0 0
$$519$$ 7938.00 0.671367
$$520$$ 0 0
$$521$$ −7966.00 −0.669859 −0.334930 0.942243i $$-0.608713\pi$$
−0.334930 + 0.942243i $$0.608713\pi$$
$$522$$ 0 0
$$523$$ 7668.00 0.641106 0.320553 0.947231i $$-0.396131\pi$$
0.320553 + 0.947231i $$0.396131\pi$$
$$524$$ 0 0
$$525$$ −300.000 −0.0249392
$$526$$ 0 0
$$527$$ 4560.00 0.376920
$$528$$ 0 0
$$529$$ 10937.0 0.898907
$$530$$ 0 0
$$531$$ 6264.00 0.511929
$$532$$ 0 0
$$533$$ −2172.00 −0.176510
$$534$$ 0 0
$$535$$ −4660.00 −0.376578
$$536$$ 0 0
$$537$$ 1824.00 0.146576
$$538$$ 0 0
$$539$$ −23544.0 −1.88147
$$540$$ 0 0
$$541$$ 6590.00 0.523708 0.261854 0.965107i $$-0.415666\pi$$
0.261854 + 0.965107i $$0.415666\pi$$
$$542$$ 0 0
$$543$$ −6738.00 −0.532514
$$544$$ 0 0
$$545$$ −2230.00 −0.175271
$$546$$ 0 0
$$547$$ −4700.00 −0.367381 −0.183691 0.982984i $$-0.558804\pi$$
−0.183691 + 0.982984i $$0.558804\pi$$
$$548$$ 0 0
$$549$$ 1998.00 0.155323
$$550$$ 0 0
$$551$$ −4056.00 −0.313596
$$552$$ 0 0
$$553$$ −2528.00 −0.194397
$$554$$ 0 0
$$555$$ −2250.00 −0.172085
$$556$$ 0 0
$$557$$ −15766.0 −1.19933 −0.599665 0.800251i $$-0.704700\pi$$
−0.599665 + 0.800251i $$0.704700\pi$$
$$558$$ 0 0
$$559$$ 2904.00 0.219725
$$560$$ 0 0
$$561$$ −8208.00 −0.617722
$$562$$ 0 0
$$563$$ 22788.0 1.70586 0.852930 0.522025i $$-0.174823\pi$$
0.852930 + 0.522025i $$0.174823\pi$$
$$564$$ 0 0
$$565$$ 3930.00 0.292631
$$566$$ 0 0
$$567$$ 324.000 0.0239977
$$568$$ 0 0
$$569$$ −3358.00 −0.247407 −0.123704 0.992319i $$-0.539477\pi$$
−0.123704 + 0.992319i $$0.539477\pi$$
$$570$$ 0 0
$$571$$ −11444.0 −0.838733 −0.419366 0.907817i $$-0.637748\pi$$
−0.419366 + 0.907817i $$0.637748\pi$$
$$572$$ 0 0
$$573$$ 11544.0 0.841636
$$574$$ 0 0
$$575$$ 3800.00 0.275602
$$576$$ 0 0
$$577$$ −10622.0 −0.766377 −0.383189 0.923670i $$-0.625174\pi$$
−0.383189 + 0.923670i $$0.625174\pi$$
$$578$$ 0 0
$$579$$ −6174.00 −0.443148
$$580$$ 0 0
$$581$$ −2448.00 −0.174802
$$582$$ 0 0
$$583$$ −48240.0 −3.42692
$$584$$ 0 0
$$585$$ 270.000 0.0190823
$$586$$ 0 0
$$587$$ −6588.00 −0.463230 −0.231615 0.972808i $$-0.574401\pi$$
−0.231615 + 0.972808i $$0.574401\pi$$
$$588$$ 0 0
$$589$$ 6240.00 0.436528
$$590$$ 0 0
$$591$$ 11514.0 0.801392
$$592$$ 0 0
$$593$$ −11362.0 −0.786815 −0.393408 0.919364i $$-0.628704\pi$$
−0.393408 + 0.919364i $$0.628704\pi$$
$$594$$ 0 0
$$595$$ −760.000 −0.0523646
$$596$$ 0 0
$$597$$ 5976.00 0.409684
$$598$$ 0 0
$$599$$ 1624.00 0.110776 0.0553880 0.998465i $$-0.482360\pi$$
0.0553880 + 0.998465i $$0.482360\pi$$
$$600$$ 0 0
$$601$$ −14950.0 −1.01468 −0.507340 0.861746i $$-0.669371\pi$$
−0.507340 + 0.861746i $$0.669371\pi$$
$$602$$ 0 0
$$603$$ −36.0000 −0.00243123
$$604$$ 0 0
$$605$$ −19265.0 −1.29460
$$606$$ 0 0
$$607$$ 8244.00 0.551258 0.275629 0.961264i $$-0.411114\pi$$
0.275629 + 0.961264i $$0.411114\pi$$
$$608$$ 0 0
$$609$$ 936.000 0.0622802
$$610$$ 0 0
$$611$$ −1680.00 −0.111237
$$612$$ 0 0
$$613$$ 6698.00 0.441321 0.220660 0.975351i $$-0.429179\pi$$
0.220660 + 0.975351i $$0.429179\pi$$
$$614$$ 0 0
$$615$$ 5430.00 0.356030
$$616$$ 0 0
$$617$$ 22670.0 1.47919 0.739595 0.673053i $$-0.235017\pi$$
0.739595 + 0.673053i $$0.235017\pi$$
$$618$$ 0 0
$$619$$ −10060.0 −0.653224 −0.326612 0.945159i $$-0.605907\pi$$
−0.326612 + 0.945159i $$0.605907\pi$$
$$620$$ 0 0
$$621$$ −4104.00 −0.265198
$$622$$ 0 0
$$623$$ 3976.00 0.255690
$$624$$ 0 0
$$625$$ 625.000 0.0400000
$$626$$ 0 0
$$627$$ −11232.0 −0.715411
$$628$$ 0 0
$$629$$ −5700.00 −0.361326
$$630$$ 0 0
$$631$$ 10240.0 0.646035 0.323017 0.946393i $$-0.395303\pi$$
0.323017 + 0.946393i $$0.395303\pi$$
$$632$$ 0 0
$$633$$ −14292.0 −0.897403
$$634$$ 0 0
$$635$$ −3580.00 −0.223729
$$636$$ 0 0
$$637$$ 1962.00 0.122037
$$638$$ 0 0
$$639$$ 864.000 0.0534888
$$640$$ 0 0
$$641$$ 13218.0 0.814477 0.407238 0.913322i $$-0.366492\pi$$
0.407238 + 0.913322i $$0.366492\pi$$
$$642$$ 0 0
$$643$$ −23412.0 −1.43589 −0.717946 0.696098i $$-0.754918\pi$$
−0.717946 + 0.696098i $$0.754918\pi$$
$$644$$ 0 0
$$645$$ −7260.00 −0.443197
$$646$$ 0 0
$$647$$ −15264.0 −0.927496 −0.463748 0.885967i $$-0.653496\pi$$
−0.463748 + 0.885967i $$0.653496\pi$$
$$648$$ 0 0
$$649$$ 50112.0 3.03092
$$650$$ 0 0
$$651$$ −1440.00 −0.0866944
$$652$$ 0 0
$$653$$ 1482.00 0.0888134 0.0444067 0.999014i $$-0.485860\pi$$
0.0444067 + 0.999014i $$0.485860\pi$$
$$654$$ 0 0
$$655$$ 4040.00 0.241001
$$656$$ 0 0
$$657$$ 1602.00 0.0951293
$$658$$ 0 0
$$659$$ −18920.0 −1.11839 −0.559195 0.829036i $$-0.688890\pi$$
−0.559195 + 0.829036i $$0.688890\pi$$
$$660$$ 0 0
$$661$$ −24218.0 −1.42507 −0.712535 0.701637i $$-0.752453\pi$$
−0.712535 + 0.701637i $$0.752453\pi$$
$$662$$ 0 0
$$663$$ 684.000 0.0400669
$$664$$ 0 0
$$665$$ −1040.00 −0.0606458
$$666$$ 0 0
$$667$$ −11856.0 −0.688255
$$668$$ 0 0
$$669$$ −12276.0 −0.709443
$$670$$ 0 0
$$671$$ 15984.0 0.919606
$$672$$ 0 0
$$673$$ 890.000 0.0509762 0.0254881 0.999675i $$-0.491886\pi$$
0.0254881 + 0.999675i $$0.491886\pi$$
$$674$$ 0 0
$$675$$ −675.000 −0.0384900
$$676$$ 0 0
$$677$$ 29250.0 1.66052 0.830258 0.557380i $$-0.188193\pi$$
0.830258 + 0.557380i $$0.188193\pi$$
$$678$$ 0 0
$$679$$ 6536.00 0.369409
$$680$$ 0 0
$$681$$ −1404.00 −0.0790035
$$682$$ 0 0
$$683$$ 14580.0 0.816820 0.408410 0.912799i $$-0.366083\pi$$
0.408410 + 0.912799i $$0.366083\pi$$
$$684$$ 0 0
$$685$$ 8850.00 0.493637
$$686$$ 0 0
$$687$$ 16758.0 0.930651
$$688$$ 0 0
$$689$$ 4020.00 0.222278
$$690$$ 0 0
$$691$$ 23668.0 1.30300 0.651500 0.758649i $$-0.274140\pi$$
0.651500 + 0.758649i $$0.274140\pi$$
$$692$$ 0 0
$$693$$ 2592.00 0.142081
$$694$$ 0 0
$$695$$ 4620.00 0.252153
$$696$$ 0 0
$$697$$ 13756.0 0.747555
$$698$$ 0 0
$$699$$ 3174.00 0.171748
$$700$$ 0 0
$$701$$ 32402.0 1.74580 0.872901 0.487898i $$-0.162236\pi$$
0.872901 + 0.487898i $$0.162236\pi$$
$$702$$ 0 0
$$703$$ −7800.00 −0.418467
$$704$$ 0 0
$$705$$ 4200.00 0.224370
$$706$$ 0 0
$$707$$ 3560.00 0.189374
$$708$$ 0 0
$$709$$ −30626.0 −1.62226 −0.811131 0.584865i $$-0.801148\pi$$
−0.811131 + 0.584865i $$0.801148\pi$$
$$710$$ 0 0
$$711$$ −5688.00 −0.300023
$$712$$ 0 0
$$713$$ 18240.0 0.958055
$$714$$ 0 0
$$715$$ 2160.00 0.112978
$$716$$ 0 0
$$717$$ −20520.0 −1.06881
$$718$$ 0 0
$$719$$ 13440.0 0.697117 0.348559 0.937287i $$-0.386671\pi$$
0.348559 + 0.937287i $$0.386671\pi$$
$$720$$ 0 0
$$721$$ −2096.00 −0.108265
$$722$$ 0 0
$$723$$ 19290.0 0.992258
$$724$$ 0 0
$$725$$ −1950.00 −0.0998913
$$726$$ 0 0
$$727$$ −24820.0 −1.26619 −0.633097 0.774073i $$-0.718217\pi$$
−0.633097 + 0.774073i $$0.718217\pi$$
$$728$$ 0 0
$$729$$ 729.000 0.0370370
$$730$$ 0 0
$$731$$ −18392.0 −0.930578
$$732$$ 0 0
$$733$$ 21986.0 1.10787 0.553937 0.832559i $$-0.313125\pi$$
0.553937 + 0.832559i $$0.313125\pi$$
$$734$$ 0 0
$$735$$ −4905.00 −0.246155
$$736$$ 0 0
$$737$$ −288.000 −0.0143943
$$738$$ 0 0
$$739$$ 4420.00 0.220017 0.110008 0.993931i $$-0.464912\pi$$
0.110008 + 0.993931i $$0.464912\pi$$
$$740$$ 0 0
$$741$$ 936.000 0.0464033
$$742$$ 0 0
$$743$$ 34560.0 1.70644 0.853219 0.521553i $$-0.174647\pi$$
0.853219 + 0.521553i $$0.174647\pi$$
$$744$$ 0 0
$$745$$ 15990.0 0.786347
$$746$$ 0 0
$$747$$ −5508.00 −0.269782
$$748$$ 0 0
$$749$$ 3728.00 0.181867
$$750$$ 0 0
$$751$$ −24792.0 −1.20462 −0.602312 0.798261i $$-0.705754\pi$$
−0.602312 + 0.798261i $$0.705754\pi$$
$$752$$ 0 0
$$753$$ 19056.0 0.922230
$$754$$ 0 0
$$755$$ 16920.0 0.815605
$$756$$ 0 0
$$757$$ −2166.00 −0.103996 −0.0519978 0.998647i $$-0.516559\pi$$
−0.0519978 + 0.998647i $$0.516559\pi$$
$$758$$ 0 0
$$759$$ −32832.0 −1.57013
$$760$$ 0 0
$$761$$ −10622.0 −0.505975 −0.252988 0.967470i $$-0.581413\pi$$
−0.252988 + 0.967470i $$0.581413\pi$$
$$762$$ 0 0
$$763$$ 1784.00 0.0846463
$$764$$ 0 0
$$765$$ −1710.00 −0.0808172
$$766$$ 0 0
$$767$$ −4176.00 −0.196593
$$768$$ 0 0
$$769$$ 29826.0 1.39864 0.699319 0.714809i $$-0.253487\pi$$
0.699319 + 0.714809i $$0.253487\pi$$
$$770$$ 0 0
$$771$$ −4266.00 −0.199269
$$772$$ 0 0
$$773$$ 6386.00 0.297139 0.148570 0.988902i $$-0.452533\pi$$
0.148570 + 0.988902i $$0.452533\pi$$
$$774$$ 0 0
$$775$$ 3000.00 0.139049
$$776$$ 0 0
$$777$$ 1800.00 0.0831076
$$778$$ 0 0
$$779$$ 18824.0 0.865776
$$780$$ 0 0
$$781$$ 6912.00 0.316685
$$782$$ 0 0
$$783$$ 2106.00 0.0961204
$$784$$ 0 0
$$785$$ 16510.0 0.750659
$$786$$ 0 0
$$787$$ 3516.00 0.159253 0.0796263 0.996825i $$-0.474627\pi$$
0.0796263 + 0.996825i $$0.474627\pi$$
$$788$$ 0 0
$$789$$ −21672.0 −0.977875
$$790$$ 0 0
$$791$$ −3144.00 −0.141325
$$792$$ 0 0
$$793$$ −1332.00 −0.0596478
$$794$$ 0 0
$$795$$ −10050.0 −0.448348
$$796$$ 0 0
$$797$$ −25030.0 −1.11243 −0.556216 0.831038i $$-0.687747\pi$$
−0.556216 + 0.831038i $$0.687747\pi$$
$$798$$ 0 0
$$799$$ 10640.0 0.471109
$$800$$ 0 0
$$801$$ 8946.00 0.394621
$$802$$ 0 0
$$803$$ 12816.0 0.563221
$$804$$ 0 0
$$805$$ −3040.00 −0.133101
$$806$$ 0 0
$$807$$ −9558.00 −0.416924
$$808$$ 0 0
$$809$$ 7962.00 0.346019 0.173009 0.984920i $$-0.444651\pi$$
0.173009 + 0.984920i $$0.444651\pi$$
$$810$$ 0 0
$$811$$ −34668.0 −1.50106 −0.750529 0.660837i $$-0.770201\pi$$
−0.750529 + 0.660837i $$0.770201\pi$$
$$812$$ 0 0
$$813$$ 768.000 0.0331303
$$814$$ 0 0
$$815$$ −11260.0 −0.483952
$$816$$ 0 0
$$817$$ −25168.0 −1.07774
$$818$$ 0 0
$$819$$ −216.000 −0.00921569
$$820$$ 0 0
$$821$$ 250.000 0.0106274 0.00531368 0.999986i $$-0.498309\pi$$
0.00531368 + 0.999986i $$0.498309\pi$$
$$822$$ 0 0
$$823$$ −6388.00 −0.270561 −0.135280 0.990807i $$-0.543194\pi$$
−0.135280 + 0.990807i $$0.543194\pi$$
$$824$$ 0 0
$$825$$ −5400.00 −0.227883
$$826$$ 0 0
$$827$$ 3932.00 0.165331 0.0826657 0.996577i $$-0.473657\pi$$
0.0826657 + 0.996577i $$0.473657\pi$$
$$828$$ 0 0
$$829$$ −25906.0 −1.08535 −0.542673 0.839944i $$-0.682588\pi$$
−0.542673 + 0.839944i $$0.682588\pi$$
$$830$$ 0 0
$$831$$ 17826.0 0.744136
$$832$$ 0 0
$$833$$ −12426.0 −0.516849
$$834$$ 0 0
$$835$$ 920.000 0.0381292
$$836$$ 0 0
$$837$$ −3240.00 −0.133800
$$838$$ 0 0
$$839$$ −9944.00 −0.409184 −0.204592 0.978847i $$-0.565587\pi$$
−0.204592 + 0.978847i $$0.565587\pi$$
$$840$$ 0 0
$$841$$ −18305.0 −0.750543
$$842$$ 0 0
$$843$$ −9606.00 −0.392465
$$844$$ 0 0
$$845$$ 10805.0 0.439886
$$846$$ 0 0
$$847$$ 15412.0 0.625221
$$848$$ 0 0
$$849$$ −11820.0 −0.477811
$$850$$ 0 0
$$851$$ −22800.0 −0.918418
$$852$$ 0 0
$$853$$ −14630.0 −0.587247 −0.293623 0.955921i $$-0.594861\pi$$
−0.293623 + 0.955921i $$0.594861\pi$$
$$854$$ 0 0
$$855$$ −2340.00 −0.0935980
$$856$$ 0 0
$$857$$ 478.000 0.0190527 0.00952635 0.999955i $$-0.496968\pi$$
0.00952635 + 0.999955i $$0.496968\pi$$
$$858$$ 0 0
$$859$$ 24132.0 0.958525 0.479263 0.877672i $$-0.340904\pi$$
0.479263 + 0.877672i $$0.340904\pi$$
$$860$$ 0 0
$$861$$ −4344.00 −0.171943
$$862$$ 0 0
$$863$$ 15776.0 0.622273 0.311136 0.950365i $$-0.399290\pi$$
0.311136 + 0.950365i $$0.399290\pi$$
$$864$$ 0 0
$$865$$ 13230.0 0.520039
$$866$$ 0 0
$$867$$ 10407.0 0.407659
$$868$$ 0 0
$$869$$ −45504.0 −1.77631
$$870$$ 0 0
$$871$$ 24.0000 0.000933650 0
$$872$$ 0 0
$$873$$ 14706.0 0.570129
$$874$$ 0 0
$$875$$ −500.000 −0.0193178
$$876$$ 0 0
$$877$$ −33542.0 −1.29149 −0.645743 0.763555i $$-0.723452\pi$$
−0.645743 + 0.763555i $$0.723452\pi$$
$$878$$ 0 0
$$879$$ −5478.00 −0.210203
$$880$$ 0 0
$$881$$ 22858.0 0.874127 0.437063 0.899431i $$-0.356019\pi$$
0.437063 + 0.899431i $$0.356019\pi$$
$$882$$ 0 0
$$883$$ 2764.00 0.105341 0.0526704 0.998612i $$-0.483227\pi$$
0.0526704 + 0.998612i $$0.483227\pi$$
$$884$$ 0 0
$$885$$ 10440.0 0.396539
$$886$$ 0 0
$$887$$ 6216.00 0.235302 0.117651 0.993055i $$-0.462464\pi$$
0.117651 + 0.993055i $$0.462464\pi$$
$$888$$ 0 0
$$889$$ 2864.00 0.108049
$$890$$ 0 0
$$891$$ 5832.00 0.219281
$$892$$ 0 0
$$893$$ 14560.0 0.545612
$$894$$ 0 0
$$895$$ 3040.00 0.113537
$$896$$ 0 0
$$897$$ 2736.00 0.101842
$$898$$ 0 0
$$899$$ −9360.00 −0.347245
$$900$$ 0 0
$$901$$ −25460.0 −0.941394
$$902$$ 0 0
$$903$$ 5808.00 0.214040
$$904$$ 0 0
$$905$$ −11230.0 −0.412484
$$906$$ 0 0
$$907$$ −18884.0 −0.691326 −0.345663 0.938359i $$-0.612346\pi$$
−0.345663 + 0.938359i $$0.612346\pi$$
$$908$$ 0 0
$$909$$ 8010.00 0.292272
$$910$$ 0 0
$$911$$ −15232.0 −0.553961 −0.276981 0.960876i $$-0.589334\pi$$
−0.276981 + 0.960876i $$0.589334\pi$$
$$912$$ 0 0
$$913$$ −44064.0 −1.59727
$$914$$ 0 0
$$915$$ 3330.00 0.120313
$$916$$ 0 0
$$917$$ −3232.00 −0.116390
$$918$$ 0 0
$$919$$ 7744.00 0.277966 0.138983 0.990295i $$-0.455617\pi$$
0.138983 + 0.990295i $$0.455617\pi$$
$$920$$ 0 0
$$921$$ −19740.0 −0.706249
$$922$$ 0 0
$$923$$ −576.000 −0.0205409
$$924$$ 0 0
$$925$$ −3750.00 −0.133296
$$926$$ 0 0
$$927$$ −4716.00 −0.167091
$$928$$ 0 0
$$929$$ 22266.0 0.786355 0.393177 0.919463i $$-0.371376\pi$$
0.393177 + 0.919463i $$0.371376\pi$$
$$930$$ 0 0
$$931$$ −17004.0 −0.598586
$$932$$ 0 0
$$933$$ 17184.0 0.602978
$$934$$ 0 0
$$935$$ −13680.0 −0.478485
$$936$$ 0 0
$$937$$ 16202.0 0.564884 0.282442 0.959284i $$-0.408856\pi$$
0.282442 + 0.959284i $$0.408856\pi$$
$$938$$ 0 0
$$939$$ 5226.00 0.181623
$$940$$ 0 0
$$941$$ −53494.0 −1.85319 −0.926596 0.376057i $$-0.877280\pi$$
−0.926596 + 0.376057i $$0.877280\pi$$
$$942$$ 0 0
$$943$$ 55024.0 1.90014
$$944$$ 0 0
$$945$$ 540.000 0.0185886
$$946$$ 0 0
$$947$$ −2332.00 −0.0800209 −0.0400105 0.999199i $$-0.512739\pi$$
−0.0400105 + 0.999199i $$0.512739\pi$$
$$948$$ 0 0
$$949$$ −1068.00 −0.0365319
$$950$$ 0 0
$$951$$ −26238.0 −0.894664
$$952$$ 0 0
$$953$$ 15414.0 0.523933 0.261967 0.965077i $$-0.415629\pi$$
0.261967 + 0.965077i $$0.415629\pi$$
$$954$$ 0 0
$$955$$ 19240.0 0.651929
$$956$$ 0 0
$$957$$ 16848.0 0.569089
$$958$$ 0 0
$$959$$ −7080.00 −0.238400
$$960$$ 0 0
$$961$$ −15391.0 −0.516633
$$962$$ 0 0
$$963$$ 8388.00 0.280685
$$964$$ 0 0
$$965$$ −10290.0 −0.343261
$$966$$ 0 0
$$967$$ 35012.0 1.16433 0.582167 0.813070i $$-0.302205\pi$$
0.582167 + 0.813070i $$0.302205\pi$$
$$968$$ 0 0
$$969$$ −5928.00 −0.196527
$$970$$ 0 0
$$971$$ 11360.0 0.375448 0.187724 0.982222i $$-0.439889\pi$$
0.187724 + 0.982222i $$0.439889\pi$$
$$972$$ 0 0
$$973$$ −3696.00 −0.121776
$$974$$ 0 0
$$975$$ 450.000 0.0147811
$$976$$ 0 0
$$977$$ −24586.0 −0.805093 −0.402546 0.915400i $$-0.631875\pi$$
−0.402546 + 0.915400i $$0.631875\pi$$
$$978$$ 0 0
$$979$$ 71568.0 2.33639
$$980$$ 0 0
$$981$$ 4014.00 0.130639
$$982$$ 0 0
$$983$$ 8832.00 0.286569 0.143284 0.989682i $$-0.454234\pi$$
0.143284 + 0.989682i $$0.454234\pi$$
$$984$$ 0 0
$$985$$ 19190.0 0.620756
$$986$$ 0 0
$$987$$ −3360.00 −0.108359
$$988$$ 0 0
$$989$$ −73568.0 −2.36535
$$990$$ 0 0
$$991$$ −22912.0 −0.734434 −0.367217 0.930135i $$-0.619689\pi$$
−0.367217 + 0.930135i $$0.619689\pi$$
$$992$$ 0 0
$$993$$ 7692.00 0.245819
$$994$$ 0 0
$$995$$ 9960.00 0.317340
$$996$$ 0 0
$$997$$ −10974.0 −0.348596 −0.174298 0.984693i $$-0.555766\pi$$
−0.174298 + 0.984693i $$0.555766\pi$$
$$998$$ 0 0
$$999$$ 4050.00 0.128265
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 120.4.a.a.1.1 1
3.2 odd 2 360.4.a.l.1.1 1
4.3 odd 2 240.4.a.h.1.1 1
5.2 odd 4 600.4.f.i.49.2 2
5.3 odd 4 600.4.f.i.49.1 2
5.4 even 2 600.4.a.l.1.1 1
8.3 odd 2 960.4.a.o.1.1 1
8.5 even 2 960.4.a.bf.1.1 1
12.11 even 2 720.4.a.v.1.1 1
15.2 even 4 1800.4.f.a.649.2 2
15.8 even 4 1800.4.f.a.649.1 2
15.14 odd 2 1800.4.a.n.1.1 1
20.3 even 4 1200.4.f.a.49.2 2
20.7 even 4 1200.4.f.a.49.1 2
20.19 odd 2 1200.4.a.k.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
120.4.a.a.1.1 1 1.1 even 1 trivial
240.4.a.h.1.1 1 4.3 odd 2
360.4.a.l.1.1 1 3.2 odd 2
600.4.a.l.1.1 1 5.4 even 2
600.4.f.i.49.1 2 5.3 odd 4
600.4.f.i.49.2 2 5.2 odd 4
720.4.a.v.1.1 1 12.11 even 2
960.4.a.o.1.1 1 8.3 odd 2
960.4.a.bf.1.1 1 8.5 even 2
1200.4.a.k.1.1 1 20.19 odd 2
1200.4.f.a.49.1 2 20.7 even 4
1200.4.f.a.49.2 2 20.3 even 4
1800.4.a.n.1.1 1 15.14 odd 2
1800.4.f.a.649.1 2 15.8 even 4
1800.4.f.a.649.2 2 15.2 even 4