# Properties

 Label 1200.4.f.a.49.2 Level $1200$ Weight $4$ Character 1200.49 Analytic conductor $70.802$ 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] = [1200,4,Mod(49,1200)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1200, 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("1200.49");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1200 = 2^{4} \cdot 3 \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 1200.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: $$70.8022920069$$ 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 $$\chi$$ $$=$$ 1200.49 Dual form 1200.4.f.a.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 $$\left(\mathbb{Z}/1200\mathbb{Z}\right)^\times$$.

 $$n$$ $$401$$ $$577$$ $$751$$ $$901$$ $$\chi(n)$$ $$1$$ $$-1$$ $$1$$ $$1$$

## 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 $$-0.948154\pi$$
−0.986764 + 0.162160i $$0.948154\pi$$
$$12$$ 0 0
$$13$$ − 6.00000i − 0.128008i −0.997950 0.0640039i $$-0.979613\pi$$
0.997950 0.0640039i $$-0.0203870\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ − 38.0000i − 0.542138i −0.962560 0.271069i $$-0.912623\pi$$
0.962560 0.271069i $$-0.0873772\pi$$
$$18$$ 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
$$23$$ − 152.000i − 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 $$-0.613011\pi$$
−0.347623 + 0.937634i $$0.613011\pi$$
$$32$$ 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 $$0.108142\pi$$
−0.942842 + 0.333241i $$0.891858\pi$$
$$38$$ 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 $$-0.665263\pi$$
0.496175 0.868222i $$-0.334737\pi$$
$$54$$ 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 $$-0.221305\pi$$
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
$$63$$ − 36.0000i − 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 $$-0.525566\pi$$
−0.0802331 + 0.996776i $$0.525566\pi$$
$$72$$ 0 0
$$73$$ 178.000i 0.285388i 0.989767 + 0.142694i $$0.0455765\pi$$
−0.989767 + 0.142694i $$0.954424\pi$$
$$74$$ 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 $$-0.648589\pi$$
−0.450035 + 0.893011i $$0.648589\pi$$
$$80$$ 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 $$-0.673438\pi$$
0.518309 0.855194i $$-0.326562\pi$$
$$98$$ 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 $$-0.893905\pi$$
0.944965 0.327171i $$-0.106095\pi$$
$$114$$ 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 $$-0.413159\pi$$
0.269448 + 0.963015i $$0.413159\pi$$
$$132$$ 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 $$0.186098\pi$$
−0.833909 + 0.551903i $$0.813902\pi$$
$$138$$ 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
$$143$$ 432.000i 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 $$-0.134635\pi$$
0.911874 + 0.410470i $$0.134635\pi$$
$$152$$ 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 $$0.317013\pi$$
−0.543727 + 0.839262i $$0.682987\pi$$
$$158$$ 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 $$-0.802497\pi$$
0.813603 0.581421i $$-0.197503\pi$$
$$174$$ 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 $$-0.540515\pi$$
−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
$$183$$ 666.000i 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 $$-0.240042\pi$$
0.728878 + 0.684643i $$0.240042\pi$$
$$192$$ 0 0
$$193$$ 2058.00i 0.767555i 0.923426 + 0.383777i $$0.125377\pi$$
−0.923426 + 0.383777i $$0.874623\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 3838.00i 1.38805i 0.719950 + 0.694026i $$0.244165\pi$$
−0.719950 + 0.694026i $$0.755835\pi$$
$$198$$ 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
$$203$$ 312.000i 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 $$-0.783347\pi$$
−0.777174 + 0.629286i $$0.783347\pi$$
$$212$$ 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 $$-0.952479\pi$$
0.988877 0.148738i $$-0.0475211\pi$$
$$234$$ 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 $$-0.123550\pi$$
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
$$243$$ 243.000i 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 $$-0.205535\pi$$
0.798675 + 0.601763i $$0.205535\pi$$
$$252$$ 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 $$-0.944792\pi$$
0.984997 0.172572i $$-0.0552077\pi$$
$$258$$ 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 $$-0.490866\pi$$
0.0286917 + 0.999588i $$0.490866\pi$$
$$272$$ 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 $$0.222910\pi$$
−0.764654 + 0.644441i $$0.777090\pi$$
$$278$$ 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 $$0.0582704\pi$$
−0.983291 + 0.182041i $$0.941730\pi$$
$$294$$ 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 $$-0.325113\pi$$
0.522195 + 0.852826i $$0.325113\pi$$
$$312$$ 0 0
$$313$$ − 1742.00i − 0.314580i −0.987552 0.157290i $$-0.949724\pi$$
0.987552 0.157290i $$-0.0502758\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ − 8746.00i − 1.54960i −0.632204 0.774802i $$-0.717850\pi$$
0.632204 0.774802i $$-0.282150\pi$$
$$318$$ 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 $$-0.431714\pi$$
0.212885 + 0.977077i $$0.431714\pi$$
$$332$$ 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 $$0.109311\pi$$
−0.941612 + 0.336701i $$0.890689\pi$$
$$338$$ 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 $$-0.884761\pi$$
0.935178 0.354177i $$-0.115239\pi$$
$$354$$ 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 $$-0.646852\pi$$
−0.445158 + 0.895452i $$0.646852\pi$$
$$360$$ 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 $$0.135607\pi$$
−0.910616 + 0.413253i $$0.864393\pi$$
$$374$$ 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 $$-0.383433\pi$$
0.358075 + 0.933693i $$0.383433\pi$$
$$380$$ 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 $$-0.722881\pi$$
0.644372 0.764712i $$-0.277119\pi$$
$$398$$ 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 $$-0.410805\pi$$
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
$$423$$ − 2520.00i − 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 $$-0.673866\pi$$
−0.519458 + 0.854496i $$0.673866\pi$$
$$432$$ 0 0
$$433$$ − 3494.00i − 0.387785i −0.981023 0.193893i $$-0.937889\pi$$
0.981023 0.193893i $$-0.0621113\pi$$
$$434$$ 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 $$-0.739786\pi$$
−0.684056 + 0.729429i $$0.739786\pi$$
$$440$$ 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 $$0.0707961\pi$$
−0.975368 + 0.220583i $$0.929204\pi$$
$$458$$ 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 $$-0.393451\pi$$
0.328519 + 0.944498i $$0.393451\pi$$
$$480$$ 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 $$-0.560278\pi$$
−0.188238 + 0.982123i $$0.560278\pi$$
$$492$$ 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 $$-0.334763\pi$$
0.496106 + 0.868262i $$0.334763\pi$$
$$500$$ 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 $$0.204700\pi$$
−0.800251 + 0.599665i $$0.795300\pi$$
$$558$$ 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 $$-0.362252\pi$$
0.419366 + 0.907817i $$0.362252\pi$$
$$572$$ 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 $$0.125174\pi$$
−0.923670 + 0.383189i $$0.874826\pi$$
$$578$$ 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 $$-0.871296\pi$$
0.919364 0.393408i $$-0.128704\pi$$
$$594$$ 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 $$-0.482360\pi$$
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
$$603$$ 36.0000i 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 $$0.0708213\pi$$
−0.975351 + 0.220660i $$0.929179\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ − 22670.0i − 1.47919i −0.673053 0.739595i $$-0.735017\pi$$
0.673053 0.739595i $$-0.264983\pi$$
$$618$$ 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
$$623$$ − 3976.00i − 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 $$-0.604697\pi$$
−0.323017 + 0.946393i $$0.604697\pi$$
$$632$$ 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 $$0.0141397\pi$$
−0.999014 + 0.0444067i $$0.985860\pi$$
$$654$$ 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 $$-0.688890\pi$$
−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
$$663$$ − 684.000i − 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 $$0.00811399\pi$$
−0.999675 + 0.0254881i $$0.991886\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ − 29250.0i − 1.66052i −0.557380 0.830258i $$-0.688193\pi$$
0.557380 0.830258i $$-0.311807\pi$$
$$678$$ 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 $$-0.725860\pi$$
−0.651500 + 0.758649i $$0.725860\pi$$
$$692$$ 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 $$-0.386671\pi$$
0.348559 + 0.937287i $$0.386671\pi$$
$$720$$ 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 $$0.186875\pi$$
−0.832559 + 0.553937i $$0.813125\pi$$
$$734$$ 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 $$-0.464912\pi$$
0.110008 + 0.993931i $$0.464912\pi$$
$$740$$ 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 $$-0.294246\pi$$
0.602312 + 0.798261i $$0.294246\pi$$
$$752$$ 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 $$0.0165589\pi$$
−0.998647 + 0.0519978i $$0.983441\pi$$
$$758$$ 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 $$0.0474669\pi$$
−0.988902 + 0.148570i $$0.952533\pi$$
$$774$$ 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 $$0.187747\pi$$
−0.831038 + 0.556216i $$0.812253\pi$$
$$798$$ 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 $$-0.229799\pi$$
0.750529 + 0.660837i $$0.229799\pi$$
$$812$$ 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 $$-0.565587\pi$$
−0.204592 + 0.978847i $$0.565587\pi$$
$$840$$ 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 $$-0.905139\pi$$
0.955921 0.293623i $$-0.0948612\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ − 478.000i − 0.0190527i −0.999955 0.00952635i $$-0.996968\pi$$
0.999955 0.00952635i $$-0.00303238\pi$$
$$858$$ 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
$$863$$ − 15776.0i − 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 $$0.223452\pi$$
−0.763555 + 0.645743i $$0.776548\pi$$
$$878$$ 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 $$-0.410666\pi$$
0.276981 + 0.960876i $$0.410666\pi$$
$$912$$ 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 $$-0.455617\pi$$
0.138983 + 0.990295i $$0.455617\pi$$
$$920$$ 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 $$-0.908856\pi$$
0.959284 0.282442i $$-0.0911445\pi$$
$$938$$ 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 $$0.0843710\pi$$
−0.965077 + 0.261967i $$0.915629\pi$$
$$954$$ 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 $$-0.560111\pi$$
−0.187724 + 0.982222i $$0.560111\pi$$
$$972$$ 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 $$0.131875\pi$$
−0.915400 + 0.402546i $$0.868125\pi$$
$$978$$ 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 $$-0.380311\pi$$
0.367217 + 0.930135i $$0.380311\pi$$
$$992$$ 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 $$0.0557656\pi$$
−0.984693 + 0.174298i $$0.944234\pi$$
$$998$$ 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 1200.4.f.a.49.2 2
4.3 odd 2 600.4.f.i.49.1 2
5.2 odd 4 240.4.a.h.1.1 1
5.3 odd 4 1200.4.a.k.1.1 1
5.4 even 2 inner 1200.4.f.a.49.1 2
12.11 even 2 1800.4.f.a.649.1 2
15.2 even 4 720.4.a.v.1.1 1
20.3 even 4 600.4.a.l.1.1 1
20.7 even 4 120.4.a.a.1.1 1
20.19 odd 2 600.4.f.i.49.2 2
40.27 even 4 960.4.a.bf.1.1 1
40.37 odd 4 960.4.a.o.1.1 1
60.23 odd 4 1800.4.a.n.1.1 1
60.47 odd 4 360.4.a.l.1.1 1
60.59 even 2 1800.4.f.a.649.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
120.4.a.a.1.1 1 20.7 even 4
240.4.a.h.1.1 1 5.2 odd 4
360.4.a.l.1.1 1 60.47 odd 4
600.4.a.l.1.1 1 20.3 even 4
600.4.f.i.49.1 2 4.3 odd 2
600.4.f.i.49.2 2 20.19 odd 2
720.4.a.v.1.1 1 15.2 even 4
960.4.a.o.1.1 1 40.37 odd 4
960.4.a.bf.1.1 1 40.27 even 4
1200.4.a.k.1.1 1 5.3 odd 4
1200.4.f.a.49.1 2 5.4 even 2 inner
1200.4.f.a.49.2 2 1.1 even 1 trivial
1800.4.a.n.1.1 1 60.23 odd 4
1800.4.f.a.649.1 2 12.11 even 2
1800.4.f.a.649.2 2 60.59 even 2