# Properties

 Label 240.4.a.h.1.1 Level $240$ Weight $4$ Character 240.1 Self dual yes Analytic conductor $14.160$ 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] = [240,4,Mod(1,240)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(240, 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("240.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$240 = 2^{4} \cdot 3 \cdot 5$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 240.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: $$14.1604584014$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 120) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 240.1

## $q$-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 $$-0.534441\pi$$
−0.107990 + 0.994152i $$0.534441\pi$$
$$8$$ 0 0
$$9$$ 9.00000 0.333333
$$10$$ 0 0
$$11$$ −72.0000 −1.97353 −0.986764 0.162160i $$-0.948154\pi$$
−0.986764 + 0.162160i $$0.948154\pi$$
$$12$$ 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 $$-0.601648\pi$$
−0.313937 + 0.949444i $$0.601648\pi$$
$$20$$ 0 0
$$21$$ −12.0000 −0.124696
$$22$$ 0 0
$$23$$ −152.000 −1.37801 −0.689004 0.724757i $$-0.741952\pi$$
−0.689004 + 0.724757i $$0.741952\pi$$
$$24$$ 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 $$-0.613011\pi$$
−0.347623 + 0.937634i $$0.613011\pi$$
$$32$$ 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 $$-0.171553\pi$$
0.858248 + 0.513236i $$0.171553\pi$$
$$44$$ 0 0
$$45$$ −45.0000 −0.149071
$$46$$ 0 0
$$47$$ −280.000 −0.868983 −0.434491 0.900676i $$-0.643072\pi$$
−0.434491 + 0.900676i $$0.643072\pi$$
$$48$$ 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 $$-0.778695\pi$$
−0.767894 + 0.640577i $$0.778695\pi$$
$$60$$ 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 $$-0.498839\pi$$
0.00364685 + 0.999993i $$0.498839\pi$$
$$68$$ 0 0
$$69$$ −456.000 −0.795593
$$70$$ 0 0
$$71$$ −96.0000 −0.160466 −0.0802331 0.996776i $$-0.525566\pi$$
−0.0802331 + 0.996776i $$0.525566\pi$$
$$72$$ 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 $$-0.351411\pi$$
0.450035 + 0.893011i $$0.351411\pi$$
$$80$$ 0 0
$$81$$ 81.0000 0.111111
$$82$$ 0 0
$$83$$ 612.000 0.809346 0.404673 0.914461i $$-0.367385\pi$$
0.404673 + 0.914461i $$0.367385\pi$$
$$84$$ 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 $$-0.419360\pi$$
0.250637 + 0.968081i $$0.419360\pi$$
$$104$$ 0 0
$$105$$ 60.0000 0.0557657
$$106$$ 0 0
$$107$$ −932.000 −0.842055 −0.421027 0.907048i $$-0.638330\pi$$
−0.421027 + 0.907048i $$0.638330\pi$$
$$108$$ 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 $$-0.580476\pi$$
−0.250137 + 0.968211i $$0.580476\pi$$
$$128$$ 0 0
$$129$$ 1452.00 0.991019
$$130$$ 0 0
$$131$$ 808.000 0.538895 0.269448 0.963015i $$-0.413159\pi$$
0.269448 + 0.963015i $$0.413159\pi$$
$$132$$ 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 $$-0.409030\pi$$
0.281916 + 0.959439i $$0.409030\pi$$
$$140$$ 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 $$-0.134635\pi$$
0.911874 + 0.410470i $$0.134635\pi$$
$$152$$ 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 $$-0.681982\pi$$
−0.541074 + 0.840975i $$0.681982\pi$$
$$164$$ 0 0
$$165$$ 1080.00 0.509563
$$166$$ 0 0
$$167$$ 184.000 0.0852596 0.0426298 0.999091i $$-0.486426\pi$$
0.0426298 + 0.999091i $$0.486426\pi$$
$$168$$ 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 $$-0.459485\pi$$
0.126939 + 0.991911i $$0.459485\pi$$
$$180$$ 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 $$-0.240042\pi$$
0.728878 + 0.684643i $$0.240042\pi$$
$$192$$ 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 $$-0.384550\pi$$
0.354797 + 0.934943i $$0.384550\pi$$
$$200$$ 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 $$-0.783347\pi$$
−0.777174 + 0.629286i $$0.783347\pi$$
$$212$$ 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 $$-0.710600\pi$$
−0.614396 + 0.788998i $$0.710600\pi$$
$$224$$ 0 0
$$225$$ 225.000 0.0666667
$$226$$ 0 0
$$227$$ −468.000 −0.136838 −0.0684191 0.997657i $$-0.521795\pi$$
−0.0684191 + 0.997657i $$0.521795\pi$$
$$228$$ 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 $$-0.876450\pi$$
−0.925613 + 0.378472i $$0.876450\pi$$
$$240$$ 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 $$-0.205535\pi$$
0.798675 + 0.601763i $$0.205535\pi$$
$$252$$ 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 $$-0.821513\pi$$
−0.846865 + 0.531808i $$0.821513\pi$$
$$264$$ 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 $$-0.490866\pi$$
0.0286917 + 0.999588i $$0.490866\pi$$
$$272$$ 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 $$-0.635797\pi$$
−0.413796 + 0.910370i $$0.635797\pi$$
$$284$$ 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 $$-0.709486\pi$$
−0.611629 + 0.791144i $$0.709486\pi$$
$$308$$ 0 0
$$309$$ 1572.00 0.289411
$$310$$ 0 0
$$311$$ 5728.00 1.04439 0.522195 0.852826i $$-0.325113\pi$$
0.522195 + 0.852826i $$0.325113\pi$$
$$312$$ 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 $$-0.431714\pi$$
0.212885 + 0.977077i $$0.431714\pi$$
$$332$$ 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 $$-0.239278\pi$$
0.730519 + 0.682892i $$0.239278\pi$$
$$348$$ 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 $$-0.353148\pi$$
0.445158 + 0.895452i $$0.353148\pi$$
$$360$$ 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 $$-0.301037\pi$$
0.585147 + 0.810927i $$0.301037\pi$$
$$368$$ 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 $$-0.616567\pi$$
−0.358075 + 0.933693i $$0.616567\pi$$
$$380$$ 0 0
$$381$$ −2148.00 −0.288833
$$382$$ 0 0
$$383$$ −9832.00 −1.31173 −0.655864 0.754879i $$-0.727695\pi$$
−0.655864 + 0.754879i $$0.727695\pi$$
$$384$$ 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 $$-0.589195\pi$$
−0.276563 + 0.960996i $$0.589195\pi$$
$$420$$ 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 $$-0.673866\pi$$
−0.519458 + 0.854496i $$0.673866\pi$$
$$432$$ 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 $$-0.260214\pi$$
0.684056 + 0.729429i $$0.260214\pi$$
$$440$$ 0 0
$$441$$ −2943.00 −0.317784
$$442$$ 0 0
$$443$$ 12852.0 1.37837 0.689184 0.724586i $$-0.257969\pi$$
0.689184 + 0.724586i $$0.257969\pi$$
$$444$$ 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 $$-0.518478\pi$$
−0.0580171 + 0.998316i $$0.518478\pi$$
$$464$$ 0 0
$$465$$ 1800.00 0.179512
$$466$$ 0 0
$$467$$ −5948.00 −0.589380 −0.294690 0.955593i $$-0.595216\pi$$
−0.294690 + 0.955593i $$0.595216\pi$$
$$468$$ 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 $$-0.606549\pi$$
−0.328519 + 0.944498i $$0.606549\pi$$
$$480$$ 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 $$-0.542958\pi$$
−0.134547 + 0.990907i $$0.542958\pi$$
$$488$$ 0 0
$$489$$ −6756.00 −0.624779
$$490$$ 0 0
$$491$$ −4096.00 −0.376476 −0.188238 0.982123i $$-0.560278\pi$$
−0.188238 + 0.982123i $$0.560278\pi$$
$$492$$ 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 $$-0.665237\pi$$
−0.496106 + 0.868262i $$0.665237\pi$$
$$500$$ 0 0
$$501$$ 552.000 0.0492246
$$502$$ 0 0
$$503$$ 9648.00 0.855235 0.427617 0.903960i $$-0.359353\pi$$
0.427617 + 0.903960i $$0.359353\pi$$
$$504$$ 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 $$-0.603869\pi$$
−0.320553 + 0.947231i $$0.603869\pi$$
$$524$$ 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 $$-0.441196\pi$$
0.183691 + 0.982984i $$0.441196\pi$$
$$548$$ 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 $$-0.825177\pi$$
−0.852930 + 0.522025i $$0.825177\pi$$
$$564$$ 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 $$-0.362252\pi$$
0.419366 + 0.907817i $$0.362252\pi$$
$$572$$ 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 $$-0.425599\pi$$
0.231615 + 0.972808i $$0.425599\pi$$
$$588$$ 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 $$-0.517640\pi$$
−0.0553880 + 0.998465i $$0.517640\pi$$
$$600$$ 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 $$-0.588886\pi$$
−0.275629 + 0.961264i $$0.588886\pi$$
$$608$$ 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 $$-0.394093\pi$$
0.326612 + 0.945159i $$0.394093\pi$$
$$620$$ 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 $$-0.604697\pi$$
−0.323017 + 0.946393i $$0.604697\pi$$
$$632$$ 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 $$-0.245082\pi$$
0.717946 + 0.696098i $$0.245082\pi$$
$$644$$ 0 0
$$645$$ −7260.00 −0.443197
$$646$$ 0 0
$$647$$ 15264.0 0.927496 0.463748 0.885967i $$-0.346504\pi$$
0.463748 + 0.885967i $$0.346504\pi$$
$$648$$ 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 $$-0.311110\pi$$
0.559195 + 0.829036i $$0.311110\pi$$
$$660$$ 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 $$-0.633917\pi$$
−0.408410 + 0.912799i $$0.633917\pi$$
$$684$$ 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 $$-0.725860\pi$$
−0.651500 + 0.758649i $$0.725860\pi$$
$$692$$ 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 $$-0.613329\pi$$
−0.348559 + 0.937287i $$0.613329\pi$$
$$720$$ 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 $$-0.281783\pi$$
0.633097 + 0.774073i $$0.281783\pi$$
$$728$$ 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 $$-0.535088\pi$$
−0.110008 + 0.993931i $$0.535088\pi$$
$$740$$ 0 0
$$741$$ 936.000 0.0464033
$$742$$ 0 0
$$743$$ −34560.0 −1.70644 −0.853219 0.521553i $$-0.825353\pi$$
−0.853219 + 0.521553i $$0.825353\pi$$
$$744$$ 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 $$-0.294246\pi$$
0.602312 + 0.798261i $$0.294246\pi$$
$$752$$ 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 $$-0.525373\pi$$
−0.0796263 + 0.996825i $$0.525373\pi$$
$$788$$ 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 $$-0.229799\pi$$
0.750529 + 0.660837i $$0.229799\pi$$
$$812$$ 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 $$-0.456806\pi$$
0.135280 + 0.990807i $$0.456806\pi$$
$$824$$ 0 0
$$825$$ −5400.00 −0.227883
$$826$$ 0 0
$$827$$ −3932.00 −0.165331 −0.0826657 0.996577i $$-0.526343\pi$$
−0.0826657 + 0.996577i $$0.526343\pi$$
$$828$$ 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 $$-0.434413\pi$$
0.204592 + 0.978847i $$0.434413\pi$$
$$840$$ 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 $$-0.659096\pi$$
−0.479263 + 0.877672i $$0.659096\pi$$
$$860$$ 0 0
$$861$$ −4344.00 −0.171943
$$862$$ 0 0
$$863$$ −15776.0 −0.622273 −0.311136 0.950365i $$-0.600710\pi$$
−0.311136 + 0.950365i $$0.600710\pi$$
$$864$$ 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 $$-0.516773\pi$$
−0.0526704 + 0.998612i $$0.516773\pi$$
$$884$$ 0 0
$$885$$ 10440.0 0.396539
$$886$$ 0 0
$$887$$ −6216.00 −0.235302 −0.117651 0.993055i $$-0.537536\pi$$
−0.117651 + 0.993055i $$0.537536\pi$$
$$888$$ 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 $$-0.387654\pi$$
0.345663 + 0.938359i $$0.387654\pi$$
$$908$$ 0 0
$$909$$ 8010.00 0.292272
$$910$$ 0 0
$$911$$ 15232.0 0.553961 0.276981 0.960876i $$-0.410666\pi$$
0.276981 + 0.960876i $$0.410666\pi$$
$$912$$ 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 $$-0.544383\pi$$
−0.138983 + 0.990295i $$0.544383\pi$$
$$920$$ 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 $$-0.487261\pi$$
0.0400105 + 0.999199i $$0.487261\pi$$
$$948$$ 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 $$-0.697795\pi$$
−0.582167 + 0.813070i $$0.697795\pi$$
$$968$$ 0 0
$$969$$ −5928.00 −0.196527
$$970$$ 0 0
$$971$$ −11360.0 −0.375448 −0.187724 0.982222i $$-0.560111\pi$$
−0.187724 + 0.982222i $$0.560111\pi$$
$$972$$ 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 $$-0.545766\pi$$
−0.143284 + 0.989682i $$0.545766\pi$$
$$984$$ 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 $$-0.380311\pi$$
0.367217 + 0.930135i $$0.380311\pi$$
$$992$$ 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 240.4.a.h.1.1 1
3.2 odd 2 720.4.a.v.1.1 1
4.3 odd 2 120.4.a.a.1.1 1
5.2 odd 4 1200.4.f.a.49.1 2
5.3 odd 4 1200.4.f.a.49.2 2
5.4 even 2 1200.4.a.k.1.1 1
8.3 odd 2 960.4.a.bf.1.1 1
8.5 even 2 960.4.a.o.1.1 1
12.11 even 2 360.4.a.l.1.1 1
20.3 even 4 600.4.f.i.49.1 2
20.7 even 4 600.4.f.i.49.2 2
20.19 odd 2 600.4.a.l.1.1 1
60.23 odd 4 1800.4.f.a.649.1 2
60.47 odd 4 1800.4.f.a.649.2 2
60.59 even 2 1800.4.a.n.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
120.4.a.a.1.1 1 4.3 odd 2
240.4.a.h.1.1 1 1.1 even 1 trivial
360.4.a.l.1.1 1 12.11 even 2
600.4.a.l.1.1 1 20.19 odd 2
600.4.f.i.49.1 2 20.3 even 4
600.4.f.i.49.2 2 20.7 even 4
720.4.a.v.1.1 1 3.2 odd 2
960.4.a.o.1.1 1 8.5 even 2
960.4.a.bf.1.1 1 8.3 odd 2
1200.4.a.k.1.1 1 5.4 even 2
1200.4.f.a.49.1 2 5.2 odd 4
1200.4.f.a.49.2 2 5.3 odd 4
1800.4.a.n.1.1 1 60.59 even 2
1800.4.f.a.649.1 2 60.23 odd 4
1800.4.f.a.649.2 2 60.47 odd 4