# Properties

 Label 600.4.f.i.49.2 Level $600$ Weight $4$ Character 600.49 Analytic conductor $35.401$ Analytic rank $0$ Dimension $2$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [600,4,Mod(49,600)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(600, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 1]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("600.49");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$600 = 2^{3} \cdot 3 \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 600.f (of order $$2$$, degree $$1$$, not minimal)

## 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: no Analytic conductor: $$35.4011460034$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(i)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 120) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 49.2 Root $$1.00000i$$ of defining polynomial Character $$\chi$$ $$=$$ 600.49 Dual form 600.4.f.i.49.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+3.00000i q^{3} +4.00000i q^{7} -9.00000 q^{9} +O(q^{10})$$ $$q+3.00000i q^{3} +4.00000i q^{7} -9.00000 q^{9} +72.0000 q^{11} +6.00000i q^{13} +38.0000i q^{17} -52.0000 q^{19} -12.0000 q^{21} -152.000i q^{23} -27.0000i q^{27} +78.0000 q^{29} +120.000 q^{31} +216.000i q^{33} -150.000i q^{37} -18.0000 q^{39} +362.000 q^{41} +484.000i q^{43} +280.000i q^{47} +327.000 q^{49} -114.000 q^{51} +670.000i q^{53} -156.000i q^{57} -696.000 q^{59} +222.000 q^{61} -36.0000i q^{63} -4.00000i q^{67} +456.000 q^{69} +96.0000 q^{71} -178.000i q^{73} +288.000i q^{77} +632.000 q^{79} +81.0000 q^{81} +612.000i q^{83} +234.000i q^{87} -994.000 q^{89} -24.0000 q^{91} +360.000i q^{93} +1634.00i q^{97} -648.000 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 18 q^{9}+O(q^{10})$$ 2 * q - 18 * q^9 $$2 q - 18 q^{9} + 144 q^{11} - 104 q^{19} - 24 q^{21} + 156 q^{29} + 240 q^{31} - 36 q^{39} + 724 q^{41} + 654 q^{49} - 228 q^{51} - 1392 q^{59} + 444 q^{61} + 912 q^{69} + 192 q^{71} + 1264 q^{79} + 162 q^{81} - 1988 q^{89} - 48 q^{91} - 1296 q^{99}+O(q^{100})$$ 2 * q - 18 * q^9 + 144 * q^11 - 104 * q^19 - 24 * q^21 + 156 * q^29 + 240 * q^31 - 36 * q^39 + 724 * q^41 + 654 * q^49 - 228 * q^51 - 1392 * q^59 + 444 * q^61 + 912 * q^69 + 192 * q^71 + 1264 * q^79 + 162 * q^81 - 1988 * q^89 - 48 * q^91 - 1296 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/600\mathbb{Z}\right)^\times$$.

 $$n$$ $$151$$ $$301$$ $$401$$ $$577$$ $$\chi(n)$$ $$1$$ $$1$$ $$1$$ $$-1$$

## 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
$$3$$ 3.00000i 0.577350i
$$4$$ 0 0
$$5$$ 0 0
$$6$$ 0 0
$$7$$ 4.00000i 0.215980i 0.994152 + 0.107990i $$0.0344414\pi$$
−0.994152 + 0.107990i $$0.965559\pi$$
$$8$$ 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
$$13$$ 6.00000i 0.128008i 0.997950 + 0.0640039i $$0.0203870\pi$$
−0.997950 + 0.0640039i $$0.979613\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ 38.0000i 0.542138i 0.962560 + 0.271069i $$0.0873772\pi$$
−0.962560 + 0.271069i $$0.912623\pi$$
$$18$$ 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
$$23$$ − 152.000i − 1.37801i −0.724757 0.689004i $$-0.758048\pi$$
0.724757 0.689004i $$-0.241952\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ 0 0
$$27$$ − 27.0000i − 0.192450i
$$28$$ 0 0
$$29$$ 78.0000 0.499456 0.249728 0.968316i $$-0.419659\pi$$
0.249728 + 0.968316i $$0.419659\pi$$
$$30$$ 0 0
$$31$$ 120.000 0.695246 0.347623 0.937634i $$-0.386989\pi$$
0.347623 + 0.937634i $$0.386989\pi$$
$$32$$ 0 0
$$33$$ 216.000i 1.13942i
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ − 150.000i − 0.666482i −0.942842 0.333241i $$-0.891858\pi$$
0.942842 0.333241i $$-0.108142\pi$$
$$38$$ 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
$$43$$ 484.000i 1.71650i 0.513236 + 0.858248i $$0.328447\pi$$
−0.513236 + 0.858248i $$0.671553\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ 280.000i 0.868983i 0.900676 + 0.434491i $$0.143072\pi$$
−0.900676 + 0.434491i $$0.856928\pi$$
$$48$$ 0 0
$$49$$ 327.000 0.953353
$$50$$ 0 0
$$51$$ −114.000 −0.313004
$$52$$ 0 0
$$53$$ 670.000i 1.73644i 0.496175 + 0.868222i $$0.334737\pi$$
−0.496175 + 0.868222i $$0.665263\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ − 156.000i − 0.362504i
$$58$$ 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
$$63$$ − 36.0000i − 0.0719932i
$$64$$ 0 0
$$65$$ 0 0
$$66$$ 0 0
$$67$$ − 4.00000i − 0.00729370i −0.999993 0.00364685i $$-0.998839\pi$$
0.999993 0.00364685i $$-0.00116083\pi$$
$$68$$ 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
$$73$$ − 178.000i − 0.285388i −0.989767 0.142694i $$-0.954424\pi$$
0.989767 0.142694i $$-0.0455765\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ 288.000i 0.426242i
$$78$$ 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
$$83$$ 612.000i 0.809346i 0.914461 + 0.404673i $$0.132615\pi$$
−0.914461 + 0.404673i $$0.867385\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 0 0
$$87$$ 234.000i 0.288361i
$$88$$ 0 0
$$89$$ −994.000 −1.18386 −0.591931 0.805988i $$-0.701634\pi$$
−0.591931 + 0.805988i $$0.701634\pi$$
$$90$$ 0 0
$$91$$ −24.0000 −0.0276471
$$92$$ 0 0
$$93$$ 360.000i 0.401401i
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 0 0
$$97$$ 1634.00i 1.71039i 0.518309 + 0.855194i $$0.326562\pi$$
−0.518309 + 0.855194i $$0.673438\pi$$
$$98$$ 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
$$103$$ 524.000i 0.501274i 0.968081 + 0.250637i $$0.0806401\pi$$
−0.968081 + 0.250637i $$0.919360\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 932.000i 0.842055i 0.907048 + 0.421027i $$0.138330\pi$$
−0.907048 + 0.421027i $$0.861670\pi$$
$$108$$ 0 0
$$109$$ −446.000 −0.391918 −0.195959 0.980612i $$-0.562782\pi$$
−0.195959 + 0.980612i $$0.562782\pi$$
$$110$$ 0 0
$$111$$ 450.000 0.384794
$$112$$ 0 0
$$113$$ 786.000i 0.654342i 0.944965 + 0.327171i $$0.106095\pi$$
−0.944965 + 0.327171i $$0.893905\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 0 0
$$117$$ − 54.0000i − 0.0426692i
$$118$$ 0 0
$$119$$ −152.000 −0.117091
$$120$$ 0 0
$$121$$ 3853.00 2.89482
$$122$$ 0 0
$$123$$ 1086.00i 0.796108i
$$124$$ 0 0
$$125$$ 0 0
$$126$$ 0 0
$$127$$ 716.000i 0.500273i 0.968211 + 0.250137i $$0.0804756\pi$$
−0.968211 + 0.250137i $$0.919524\pi$$
$$128$$ 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
$$133$$ − 208.000i − 0.135608i
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ − 1770.00i − 1.10381i −0.833909 0.551903i $$-0.813902\pi$$
0.833909 0.551903i $$-0.186098\pi$$
$$138$$ 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
$$143$$ 432.000i 0.252627i
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 0 0
$$147$$ 981.000i 0.550418i
$$148$$ 0 0
$$149$$ 3198.00 1.75832 0.879162 0.476522i $$-0.158103\pi$$
0.879162 + 0.476522i $$0.158103\pi$$
$$150$$ 0 0
$$151$$ −3384.00 −1.82375 −0.911874 0.410470i $$-0.865365\pi$$
−0.911874 + 0.410470i $$0.865365\pi$$
$$152$$ 0 0
$$153$$ − 342.000i − 0.180713i
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ − 3302.00i − 1.67852i −0.543727 0.839262i $$-0.682987\pi$$
0.543727 0.839262i $$-0.317013\pi$$
$$158$$ 0 0
$$159$$ −2010.00 −1.00254
$$160$$ 0 0
$$161$$ 608.000 0.297622
$$162$$ 0 0
$$163$$ − 2252.00i − 1.08215i −0.840975 0.541074i $$-0.818018\pi$$
0.840975 0.541074i $$-0.181982\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ − 184.000i − 0.0852596i −0.999091 0.0426298i $$-0.986426\pi$$
0.999091 0.0426298i $$-0.0135736\pi$$
$$168$$ 0 0
$$169$$ 2161.00 0.983614
$$170$$ 0 0
$$171$$ 468.000 0.209292
$$172$$ 0 0
$$173$$ 2646.00i 1.16284i 0.813603 + 0.581421i $$0.197503\pi$$
−0.813603 + 0.581421i $$0.802497\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ − 2088.00i − 0.886688i
$$178$$ 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
$$183$$ 666.000i 0.269028i
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 0 0
$$187$$ 2736.00i 1.06993i
$$188$$ 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
$$193$$ − 2058.00i − 0.767555i −0.923426 0.383777i $$-0.874623\pi$$
0.923426 0.383777i $$-0.125377\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ − 3838.00i − 1.38805i −0.719950 0.694026i $$-0.755835\pi$$
0.719950 0.694026i $$-0.244165\pi$$
$$198$$ 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
$$203$$ 312.000i 0.107872i
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 0 0
$$207$$ 1368.00i 0.459336i
$$208$$ 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
$$213$$ 288.000i 0.0926452i
$$214$$ 0 0
$$215$$ 0 0
$$216$$ 0 0
$$217$$ 480.000i 0.150159i
$$218$$ 0 0
$$219$$ 534.000 0.164769
$$220$$ 0 0
$$221$$ −228.000 −0.0693979
$$222$$ 0 0
$$223$$ − 4092.00i − 1.22879i −0.788998 0.614396i $$-0.789400\pi$$
0.788998 0.614396i $$-0.210600\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ 468.000i 0.136838i 0.997657 + 0.0684191i $$0.0217955\pi$$
−0.997657 + 0.0684191i $$0.978205\pi$$
$$228$$ 0 0
$$229$$ 5586.00 1.61194 0.805968 0.591959i $$-0.201645\pi$$
0.805968 + 0.591959i $$0.201645\pi$$
$$230$$ 0 0
$$231$$ −864.000 −0.246091
$$232$$ 0 0
$$233$$ 1058.00i 0.297476i 0.988877 + 0.148738i $$0.0475211\pi$$
−0.988877 + 0.148738i $$0.952479\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 0 0
$$237$$ 1896.00i 0.519656i
$$238$$ 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
$$243$$ 243.000i 0.0641500i
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ − 312.000i − 0.0803728i
$$248$$ 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
$$253$$ − 10944.0i − 2.71954i
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ 1422.00i 0.345144i 0.984997 + 0.172572i $$0.0552077\pi$$
−0.984997 + 0.172572i $$0.944792\pi$$
$$258$$ 0 0
$$259$$ 600.000 0.143947
$$260$$ 0 0
$$261$$ −702.000 −0.166485
$$262$$ 0 0
$$263$$ − 7224.00i − 1.69373i −0.531808 0.846865i $$-0.678487\pi$$
0.531808 0.846865i $$-0.321513\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 0 0
$$267$$ − 2982.00i − 0.683504i
$$268$$ 0 0
$$269$$ −3186.00 −0.722133 −0.361067 0.932540i $$-0.617587\pi$$
−0.361067 + 0.932540i $$0.617587\pi$$
$$270$$ 0 0
$$271$$ −256.000 −0.0573834 −0.0286917 0.999588i $$-0.509134\pi$$
−0.0286917 + 0.999588i $$0.509134\pi$$
$$272$$ 0 0
$$273$$ − 72.0000i − 0.0159620i
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 0 0
$$277$$ − 5942.00i − 1.28888i −0.764654 0.644441i $$-0.777090\pi$$
0.764654 0.644441i $$-0.222910\pi$$
$$278$$ 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
$$283$$ − 3940.00i − 0.827593i −0.910370 0.413796i $$-0.864203\pi$$
0.910370 0.413796i $$-0.135797\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 1448.00i 0.297814i
$$288$$ 0 0
$$289$$ 3469.00 0.706086
$$290$$ 0 0
$$291$$ −4902.00 −0.987493
$$292$$ 0 0
$$293$$ − 1826.00i − 0.364082i −0.983291 0.182041i $$-0.941730\pi$$
0.983291 0.182041i $$-0.0582704\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 0 0
$$297$$ − 1944.00i − 0.379806i
$$298$$ 0 0
$$299$$ 912.000 0.176396
$$300$$ 0 0
$$301$$ −1936.00 −0.370728
$$302$$ 0 0
$$303$$ 2670.00i 0.506229i
$$304$$ 0 0
$$305$$ 0 0
$$306$$ 0 0
$$307$$ 6580.00i 1.22326i 0.791144 + 0.611629i $$0.209486\pi$$
−0.791144 + 0.611629i $$0.790514\pi$$
$$308$$ 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
$$313$$ 1742.00i 0.314580i 0.987552 + 0.157290i $$0.0502758\pi$$
−0.987552 + 0.157290i $$0.949724\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ 8746.00i 1.54960i 0.632204 + 0.774802i $$0.282150\pi$$
−0.632204 + 0.774802i $$0.717850\pi$$
$$318$$ 0 0
$$319$$ 5616.00 0.985692
$$320$$ 0 0
$$321$$ −2796.00 −0.486160
$$322$$ 0 0
$$323$$ − 1976.00i − 0.340395i
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 0 0
$$327$$ − 1338.00i − 0.226274i
$$328$$ 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
$$333$$ 1350.00i 0.222161i
$$334$$ 0 0
$$335$$ 0 0
$$336$$ 0 0
$$337$$ − 4166.00i − 0.673402i −0.941612 0.336701i $$-0.890689\pi$$
0.941612 0.336701i $$-0.109311\pi$$
$$338$$ 0 0
$$339$$ −2358.00 −0.377785
$$340$$ 0 0
$$341$$ 8640.00 1.37209
$$342$$ 0 0
$$343$$ 2680.00i 0.421885i
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ − 9444.00i − 1.46104i −0.682892 0.730519i $$-0.739278\pi$$
0.682892 0.730519i $$-0.260722\pi$$
$$348$$ 0 0
$$349$$ 9218.00 1.41383 0.706917 0.707296i $$-0.250085\pi$$
0.706917 + 0.707296i $$0.250085\pi$$
$$350$$ 0 0
$$351$$ 162.000 0.0246351
$$352$$ 0 0
$$353$$ 4698.00i 0.708355i 0.935178 + 0.354177i $$0.115239\pi$$
−0.935178 + 0.354177i $$0.884761\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0 0
$$357$$ − 456.000i − 0.0676025i
$$358$$ 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
$$363$$ 11559.0i 1.67132i
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 0 0
$$367$$ − 8228.00i − 1.17029i −0.810927 0.585147i $$-0.801037\pi$$
0.810927 0.585147i $$-0.198963\pi$$
$$368$$ 0 0
$$369$$ −3258.00 −0.459633
$$370$$ 0 0
$$371$$ −2680.00 −0.375037
$$372$$ 0 0
$$373$$ − 5954.00i − 0.826505i −0.910616 0.413253i $$-0.864393\pi$$
0.910616 0.413253i $$-0.135607\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 468.000i 0.0639343i
$$378$$ 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
$$383$$ − 9832.00i − 1.31173i −0.754879 0.655864i $$-0.772305\pi$$
0.754879 0.655864i $$-0.227695\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ − 4356.00i − 0.572165i
$$388$$ 0 0
$$389$$ 222.000 0.0289353 0.0144677 0.999895i $$-0.495395\pi$$
0.0144677 + 0.999895i $$0.495395\pi$$
$$390$$ 0 0
$$391$$ 5776.00 0.747071
$$392$$ 0 0
$$393$$ − 2424.00i − 0.311131i
$$394$$ 0 0
$$395$$ 0 0
$$396$$ 0 0
$$397$$ 12098.0i 1.52942i 0.644372 + 0.764712i $$0.277119\pi$$
−0.644372 + 0.764712i $$0.722881\pi$$
$$398$$ 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
$$403$$ 720.000i 0.0889969i
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ − 10800.0i − 1.31532i
$$408$$ 0 0
$$409$$ −1930.00 −0.233331 −0.116665 0.993171i $$-0.537221\pi$$
−0.116665 + 0.993171i $$0.537221\pi$$
$$410$$ 0 0
$$411$$ 5310.00 0.637282
$$412$$ 0 0
$$413$$ − 2784.00i − 0.331699i
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 0 0
$$417$$ 2772.00i 0.325529i
$$418$$ 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
$$423$$ − 2520.00i − 0.289661i
$$424$$ 0 0
$$425$$ 0 0
$$426$$ 0 0
$$427$$ 888.000i 0.100640i
$$428$$ 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
$$433$$ 3494.00i 0.387785i 0.981023 + 0.193893i $$0.0621113\pi$$
−0.981023 + 0.193893i $$0.937889\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ 7904.00i 0.865216i
$$438$$ 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
$$443$$ 12852.0i 1.37837i 0.724586 + 0.689184i $$0.242031\pi$$
−0.724586 + 0.689184i $$0.757969\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 0 0
$$447$$ 9594.00i 1.01517i
$$448$$ 0 0
$$449$$ −14458.0 −1.51963 −0.759816 0.650138i $$-0.774711\pi$$
−0.759816 + 0.650138i $$0.774711\pi$$
$$450$$ 0 0
$$451$$ 26064.0 2.72130
$$452$$ 0 0
$$453$$ − 10152.0i − 1.05294i
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ − 4310.00i − 0.441167i −0.975368 0.220583i $$-0.929204\pi$$
0.975368 0.220583i $$-0.0707961\pi$$
$$458$$ 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
$$463$$ − 1156.00i − 0.116034i −0.998316 0.0580171i $$-0.981522\pi$$
0.998316 0.0580171i $$-0.0184778\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ 5948.00i 0.589380i 0.955593 + 0.294690i $$0.0952164\pi$$
−0.955593 + 0.294690i $$0.904784\pi$$
$$468$$ 0 0
$$469$$ 16.0000 0.00157529
$$470$$ 0 0
$$471$$ 9906.00 0.969096
$$472$$ 0 0
$$473$$ 34848.0i 3.38755i
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 0 0
$$477$$ − 6030.00i − 0.578815i
$$478$$ 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
$$483$$ 1824.00i 0.171832i
$$484$$ 0 0
$$485$$ 0 0
$$486$$ 0 0
$$487$$ 2892.00i 0.269095i 0.990907 + 0.134547i $$0.0429580\pi$$
−0.990907 + 0.134547i $$0.957042\pi$$
$$488$$ 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
$$493$$ 2964.00i 0.270775i
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 384.000i 0.0346575i
$$498$$ 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
$$503$$ 9648.00i 0.855235i 0.903960 + 0.427617i $$0.140647\pi$$
−0.903960 + 0.427617i $$0.859353\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 0 0
$$507$$ 6483.00i 0.567890i
$$508$$ 0 0
$$509$$ 10062.0 0.876209 0.438104 0.898924i $$-0.355650\pi$$
0.438104 + 0.898924i $$0.355650\pi$$
$$510$$ 0 0
$$511$$ 712.000 0.0616380
$$512$$ 0 0
$$513$$ 1404.00i 0.120835i
$$514$$ 0 0
$$515$$ 0 0
$$516$$ 0 0
$$517$$ 20160.0i 1.71496i
$$518$$ 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
$$523$$ − 7668.00i − 0.641106i −0.947231 0.320553i $$-0.896131\pi$$
0.947231 0.320553i $$-0.103869\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 4560.00i 0.376920i
$$528$$ 0 0
$$529$$ −10937.0 −0.898907
$$530$$ 0 0
$$531$$ 6264.00 0.511929
$$532$$ 0 0
$$533$$ 2172.00i 0.176510i
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 0 0
$$537$$ 1824.00i 0.146576i
$$538$$ 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
$$543$$ 6738.00i 0.532514i
$$544$$ 0 0
$$545$$ 0 0
$$546$$ 0 0
$$547$$ − 4700.00i − 0.367381i −0.982984 0.183691i $$-0.941196\pi$$
0.982984 0.183691i $$-0.0588044\pi$$
$$548$$ 0 0
$$549$$ −1998.00 −0.155323
$$550$$ 0 0
$$551$$ −4056.00 −0.313596
$$552$$ 0 0
$$553$$ 2528.00i 0.194397i
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ − 15766.0i − 1.19933i −0.800251 0.599665i $$-0.795300\pi$$
0.800251 0.599665i $$-0.204700\pi$$
$$558$$ 0 0
$$559$$ −2904.00 −0.219725
$$560$$ 0 0
$$561$$ −8208.00 −0.617722
$$562$$ 0 0
$$563$$ − 22788.0i − 1.70586i −0.522025 0.852930i $$-0.674823\pi$$
0.522025 0.852930i $$-0.325177\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 0 0
$$567$$ 324.000i 0.0239977i
$$568$$ 0 0
$$569$$ 3358.00 0.247407 0.123704 0.992319i $$-0.460523\pi$$
0.123704 + 0.992319i $$0.460523\pi$$
$$570$$ 0 0
$$571$$ −11444.0 −0.838733 −0.419366 0.907817i $$-0.637748\pi$$
−0.419366 + 0.907817i $$0.637748\pi$$
$$572$$ 0 0
$$573$$ − 11544.0i − 0.841636i
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 0 0
$$577$$ − 10622.0i − 0.766377i −0.923670 0.383189i $$-0.874826\pi$$
0.923670 0.383189i $$-0.125174\pi$$
$$578$$ 0 0
$$579$$ 6174.00 0.443148
$$580$$ 0 0
$$581$$ −2448.00 −0.174802
$$582$$ 0 0
$$583$$ 48240.0i 3.42692i
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ − 6588.00i − 0.463230i −0.972808 0.231615i $$-0.925599\pi$$
0.972808 0.231615i $$-0.0744009\pi$$
$$588$$ 0 0
$$589$$ −6240.00 −0.436528
$$590$$ 0 0
$$591$$ 11514.0 0.801392
$$592$$ 0 0
$$593$$ 11362.0i 0.786815i 0.919364 + 0.393408i $$0.128704\pi$$
−0.919364 + 0.393408i $$0.871296\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ 5976.00i 0.409684i
$$598$$ 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
$$603$$ 36.0000i 0.00243123i
$$604$$ 0 0
$$605$$ 0 0
$$606$$ 0 0
$$607$$ 8244.00i 0.551258i 0.961264 + 0.275629i $$0.0888861\pi$$
−0.961264 + 0.275629i $$0.911114\pi$$
$$608$$ 0 0
$$609$$ −936.000 −0.0622802
$$610$$ 0 0
$$611$$ −1680.00 −0.111237
$$612$$ 0 0
$$613$$ − 6698.00i − 0.441321i −0.975351 0.220660i $$-0.929179\pi$$
0.975351 0.220660i $$-0.0708213\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 22670.0i 1.47919i 0.673053 + 0.739595i $$0.264983\pi$$
−0.673053 + 0.739595i $$0.735017\pi$$
$$618$$ 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
$$623$$ − 3976.00i − 0.255690i
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 0 0
$$627$$ − 11232.0i − 0.715411i
$$628$$ 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
$$633$$ 14292.0i 0.897403i
$$634$$ 0 0
$$635$$ 0 0
$$636$$ 0 0
$$637$$ 1962.00i 0.122037i
$$638$$ 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
$$643$$ 23412.0i 1.43589i 0.696098 + 0.717946i $$0.254918\pi$$
−0.696098 + 0.717946i $$0.745082\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ − 15264.0i − 0.927496i −0.885967 0.463748i $$-0.846504\pi$$
0.885967 0.463748i $$-0.153496\pi$$
$$648$$ 0 0
$$649$$ −50112.0 −3.03092
$$650$$ 0 0
$$651$$ −1440.00 −0.0866944
$$652$$ 0 0
$$653$$ − 1482.00i − 0.0888134i −0.999014 0.0444067i $$-0.985860\pi$$
0.999014 0.0444067i $$-0.0141397\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 0 0
$$657$$ 1602.00i 0.0951293i
$$658$$ 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
$$663$$ − 684.000i − 0.0400669i
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ − 11856.0i − 0.688255i
$$668$$ 0 0
$$669$$ 12276.0 0.709443
$$670$$ 0 0
$$671$$ 15984.0 0.919606
$$672$$ 0 0
$$673$$ − 890.000i − 0.0509762i −0.999675 0.0254881i $$-0.991886\pi$$
0.999675 0.0254881i $$-0.00811399\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ 29250.0i 1.66052i 0.557380 + 0.830258i $$0.311807\pi$$
−0.557380 + 0.830258i $$0.688193\pi$$
$$678$$ 0 0
$$679$$ −6536.00 −0.369409
$$680$$ 0 0
$$681$$ −1404.00 −0.0790035
$$682$$ 0 0
$$683$$ − 14580.0i − 0.816820i −0.912799 0.408410i $$-0.866083\pi$$
0.912799 0.408410i $$-0.133917\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 0 0
$$687$$ 16758.0i 0.930651i
$$688$$ 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
$$693$$ − 2592.00i − 0.142081i
$$694$$ 0 0
$$695$$ 0 0
$$696$$ 0 0
$$697$$ 13756.0i 0.747555i
$$698$$ 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
$$703$$ 7800.00i 0.418467i
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ 3560.00i 0.189374i
$$708$$ 0 0
$$709$$ 30626.0 1.62226 0.811131 0.584865i $$-0.198852\pi$$
0.811131 + 0.584865i $$0.198852\pi$$
$$710$$ 0 0
$$711$$ −5688.00 −0.300023
$$712$$ 0 0
$$713$$ − 18240.0i − 0.958055i
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ − 20520.0i − 1.06881i
$$718$$ 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
$$723$$ − 19290.0i − 0.992258i
$$724$$ 0 0
$$725$$ 0 0
$$726$$ 0 0
$$727$$ − 24820.0i − 1.26619i −0.774073 0.633097i $$-0.781783\pi$$
0.774073 0.633097i $$-0.218217\pi$$
$$728$$ 0 0
$$729$$ −729.000 −0.0370370
$$730$$ 0 0
$$731$$ −18392.0 −0.930578
$$732$$ 0 0
$$733$$ − 21986.0i − 1.10787i −0.832559 0.553937i $$-0.813125\pi$$
0.832559 0.553937i $$-0.186875\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ − 288.000i − 0.0143943i
$$738$$ 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
$$743$$ − 34560.0i − 1.70644i −0.521553 0.853219i $$-0.674647\pi$$
0.521553 0.853219i $$-0.325353\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 0 0
$$747$$ − 5508.00i − 0.269782i
$$748$$ 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
$$753$$ − 19056.0i − 0.922230i
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 0 0
$$757$$ − 2166.00i − 0.103996i −0.998647 0.0519978i $$-0.983441\pi$$
0.998647 0.0519978i $$-0.0165589\pi$$
$$758$$ 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
$$763$$ − 1784.00i − 0.0846463i
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ − 4176.00i − 0.196593i
$$768$$ 0 0
$$769$$ −29826.0 −1.39864 −0.699319 0.714809i $$-0.746513\pi$$
−0.699319 + 0.714809i $$0.746513\pi$$
$$770$$ 0 0
$$771$$ −4266.00 −0.199269
$$772$$ 0 0
$$773$$ − 6386.00i − 0.297139i −0.988902 0.148570i $$-0.952533\pi$$
0.988902 0.148570i $$-0.0474669\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 0 0
$$777$$ 1800.00i 0.0831076i
$$778$$ 0 0
$$779$$ −18824.0 −0.865776
$$780$$ 0 0
$$781$$ 6912.00 0.316685
$$782$$ 0 0
$$783$$ − 2106.00i − 0.0961204i
$$784$$ 0 0
$$785$$ 0 0
$$786$$ 0 0
$$787$$ 3516.00i 0.159253i 0.996825 + 0.0796263i $$0.0253727\pi$$
−0.996825 + 0.0796263i $$0.974627\pi$$
$$788$$ 0 0
$$789$$ 21672.0 0.977875
$$790$$ 0 0
$$791$$ −3144.00 −0.141325
$$792$$ 0 0
$$793$$ 1332.00i 0.0596478i
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ − 25030.0i − 1.11243i −0.831038 0.556216i $$-0.812253\pi$$
0.831038 0.556216i $$-0.187747\pi$$
$$798$$ 0 0
$$799$$ −10640.0 −0.471109
$$800$$ 0 0
$$801$$ 8946.00 0.394621
$$802$$ 0 0
$$803$$ − 12816.0i − 0.563221i
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ − 9558.00i − 0.416924i
$$808$$ 0 0
$$809$$ −7962.00 −0.346019 −0.173009 0.984920i $$-0.555349\pi$$
−0.173009 + 0.984920i $$0.555349\pi$$
$$810$$ 0 0
$$811$$ −34668.0 −1.50106 −0.750529 0.660837i $$-0.770201\pi$$
−0.750529 + 0.660837i $$0.770201\pi$$
$$812$$ 0 0
$$813$$ − 768.000i − 0.0331303i
$$814$$ 0 0
$$815$$ 0 0
$$816$$ 0 0
$$817$$ − 25168.0i − 1.07774i
$$818$$ 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
$$823$$ 6388.00i 0.270561i 0.990807 + 0.135280i $$0.0431936\pi$$
−0.990807 + 0.135280i $$0.956806\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 3932.00i 0.165331i 0.996577 + 0.0826657i $$0.0263434\pi$$
−0.996577 + 0.0826657i $$0.973657\pi$$
$$828$$ 0 0
$$829$$ 25906.0 1.08535 0.542673 0.839944i $$-0.317412\pi$$
0.542673 + 0.839944i $$0.317412\pi$$
$$830$$ 0 0
$$831$$ 17826.0 0.744136
$$832$$ 0 0
$$833$$ 12426.0i 0.516849i
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 0 0
$$837$$ − 3240.00i − 0.133800i
$$838$$ 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
$$843$$ 9606.00i 0.392465i
$$844$$ 0 0
$$845$$ 0 0
$$846$$ 0 0
$$847$$ 15412.0i 0.625221i
$$848$$ 0 0
$$849$$ 11820.0 0.477811
$$850$$ 0 0
$$851$$ −22800.0 −0.918418
$$852$$ 0 0
$$853$$ 14630.0i 0.587247i 0.955921 + 0.293623i $$0.0948612\pi$$
−0.955921 + 0.293623i $$0.905139\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ 478.000i 0.0190527i 0.999955 + 0.00952635i $$0.00303238\pi$$
−0.999955 + 0.00952635i $$0.996968\pi$$
$$858$$ 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
$$863$$ − 15776.0i − 0.622273i −0.950365 0.311136i $$-0.899290\pi$$
0.950365 0.311136i $$-0.100710\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 0 0
$$867$$ 10407.0i 0.407659i
$$868$$ 0 0
$$869$$ 45504.0 1.77631
$$870$$ 0 0
$$871$$ 24.0000 0.000933650 0
$$872$$ 0 0
$$873$$ − 14706.0i − 0.570129i
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ − 33542.0i − 1.29149i −0.763555 0.645743i $$-0.776548\pi$$
0.763555 0.645743i $$-0.223452\pi$$
$$878$$ 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
$$883$$ − 2764.00i − 0.105341i −0.998612 0.0526704i $$-0.983227\pi$$
0.998612 0.0526704i $$-0.0167733\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ 6216.00i 0.235302i 0.993055 + 0.117651i $$0.0375364\pi$$
−0.993055 + 0.117651i $$0.962464\pi$$
$$888$$ 0 0
$$889$$ −2864.00 −0.108049
$$890$$ 0 0
$$891$$ 5832.00 0.219281
$$892$$ 0 0
$$893$$ − 14560.0i − 0.545612i
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 0 0
$$897$$ 2736.00i 0.101842i
$$898$$ 0 0
$$899$$ 9360.00 0.347245
$$900$$ 0 0
$$901$$ −25460.0 −0.941394
$$902$$ 0 0
$$903$$ − 5808.00i − 0.214040i
$$904$$ 0 0
$$905$$ 0 0
$$906$$ 0 0
$$907$$ − 18884.0i − 0.691326i −0.938359 0.345663i $$-0.887654\pi$$
0.938359 0.345663i $$-0.112346\pi$$
$$908$$ 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
$$913$$ 44064.0i 1.59727i
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ − 3232.00i − 0.116390i
$$918$$ 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
$$923$$ 576.000i 0.0205409i
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 0 0
$$927$$ − 4716.00i − 0.167091i
$$928$$ 0 0
$$929$$ −22266.0 −0.786355 −0.393177 0.919463i $$-0.628624\pi$$
−0.393177 + 0.919463i $$0.628624\pi$$
$$930$$ 0 0
$$931$$ −17004.0 −0.598586
$$932$$ 0 0
$$933$$ − 17184.0i − 0.602978i
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ 16202.0i 0.564884i 0.959284 + 0.282442i $$0.0911445\pi$$
−0.959284 + 0.282442i $$0.908856\pi$$
$$938$$ 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
$$943$$ − 55024.0i − 1.90014i
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ − 2332.00i − 0.0800209i −0.999199 0.0400105i $$-0.987261\pi$$
0.999199 0.0400105i $$-0.0127391\pi$$
$$948$$ 0 0
$$949$$ 1068.00 0.0365319
$$950$$ 0 0
$$951$$ −26238.0 −0.894664
$$952$$ 0 0
$$953$$ − 15414.0i − 0.523933i −0.965077 0.261967i $$-0.915629\pi$$
0.965077 0.261967i $$-0.0843710\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 0 0
$$957$$ 16848.0i 0.569089i
$$958$$ 0 0
$$959$$ 7080.00 0.238400
$$960$$ 0 0
$$961$$ −15391.0 −0.516633
$$962$$ 0 0
$$963$$ − 8388.00i − 0.280685i
$$964$$ 0 0
$$965$$ 0 0
$$966$$ 0 0
$$967$$ 35012.0i 1.16433i 0.813070 + 0.582167i $$0.197795\pi$$
−0.813070 + 0.582167i $$0.802205\pi$$
$$968$$ 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
$$973$$ 3696.00i 0.121776i
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ − 24586.0i − 0.805093i −0.915400 0.402546i $$-0.868125\pi$$
0.915400 0.402546i $$-0.131875\pi$$
$$978$$ 0 0
$$979$$ −71568.0 −2.33639
$$980$$ 0 0
$$981$$ 4014.00 0.130639
$$982$$ 0 0
$$983$$ − 8832.00i − 0.286569i −0.989682 0.143284i $$-0.954234\pi$$
0.989682 0.143284i $$-0.0457663\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 0 0
$$987$$ − 3360.00i − 0.108359i
$$988$$ 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
$$993$$ − 7692.00i − 0.245819i
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 0 0
$$997$$ − 10974.0i − 0.348596i −0.984693 0.174298i $$-0.944234\pi$$
0.984693 0.174298i $$-0.0557656\pi$$
$$998$$ 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 600.4.f.i.49.2 2
3.2 odd 2 1800.4.f.a.649.2 2
4.3 odd 2 1200.4.f.a.49.1 2
5.2 odd 4 600.4.a.l.1.1 1
5.3 odd 4 120.4.a.a.1.1 1
5.4 even 2 inner 600.4.f.i.49.1 2
15.2 even 4 1800.4.a.n.1.1 1
15.8 even 4 360.4.a.l.1.1 1
15.14 odd 2 1800.4.f.a.649.1 2
20.3 even 4 240.4.a.h.1.1 1
20.7 even 4 1200.4.a.k.1.1 1
20.19 odd 2 1200.4.f.a.49.2 2
40.3 even 4 960.4.a.o.1.1 1
40.13 odd 4 960.4.a.bf.1.1 1
60.23 odd 4 720.4.a.v.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
120.4.a.a.1.1 1 5.3 odd 4
240.4.a.h.1.1 1 20.3 even 4
360.4.a.l.1.1 1 15.8 even 4
600.4.a.l.1.1 1 5.2 odd 4
600.4.f.i.49.1 2 5.4 even 2 inner
600.4.f.i.49.2 2 1.1 even 1 trivial
720.4.a.v.1.1 1 60.23 odd 4
960.4.a.o.1.1 1 40.3 even 4
960.4.a.bf.1.1 1 40.13 odd 4
1200.4.a.k.1.1 1 20.7 even 4
1200.4.f.a.49.1 2 4.3 odd 2
1200.4.f.a.49.2 2 20.19 odd 2
1800.4.a.n.1.1 1 15.2 even 4
1800.4.f.a.649.1 2 15.14 odd 2
1800.4.f.a.649.2 2 3.2 odd 2