# Properties

 Label 9075.2.a.r.1.1 Level $9075$ Weight $2$ Character 9075.1 Self dual yes Analytic conductor $72.464$ 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] = [9075,2,Mod(1,9075)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9075, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("9075.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$9075 = 3 \cdot 5^{2} \cdot 11^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 9075.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: $$72.4642398343$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 1815) Fricke sign: $$+1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 9075.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+1.00000 q^{2} +1.00000 q^{3} -1.00000 q^{4} +1.00000 q^{6} +4.00000 q^{7} -3.00000 q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q+1.00000 q^{2} +1.00000 q^{3} -1.00000 q^{4} +1.00000 q^{6} +4.00000 q^{7} -3.00000 q^{8} +1.00000 q^{9} -1.00000 q^{12} +2.00000 q^{13} +4.00000 q^{14} -1.00000 q^{16} +2.00000 q^{17} +1.00000 q^{18} -8.00000 q^{19} +4.00000 q^{21} -4.00000 q^{23} -3.00000 q^{24} +2.00000 q^{26} +1.00000 q^{27} -4.00000 q^{28} -4.00000 q^{29} -8.00000 q^{31} +5.00000 q^{32} +2.00000 q^{34} -1.00000 q^{36} -8.00000 q^{37} -8.00000 q^{38} +2.00000 q^{39} -12.0000 q^{41} +4.00000 q^{42} -8.00000 q^{43} -4.00000 q^{46} +4.00000 q^{47} -1.00000 q^{48} +9.00000 q^{49} +2.00000 q^{51} -2.00000 q^{52} +4.00000 q^{53} +1.00000 q^{54} -12.0000 q^{56} -8.00000 q^{57} -4.00000 q^{58} -8.00000 q^{59} -8.00000 q^{62} +4.00000 q^{63} +7.00000 q^{64} +4.00000 q^{67} -2.00000 q^{68} -4.00000 q^{69} -12.0000 q^{71} -3.00000 q^{72} +2.00000 q^{73} -8.00000 q^{74} +8.00000 q^{76} +2.00000 q^{78} -8.00000 q^{79} +1.00000 q^{81} -12.0000 q^{82} +4.00000 q^{83} -4.00000 q^{84} -8.00000 q^{86} -4.00000 q^{87} +6.00000 q^{89} +8.00000 q^{91} +4.00000 q^{92} -8.00000 q^{93} +4.00000 q^{94} +5.00000 q^{96} +8.00000 q^{97} +9.00000 q^{98} +O(q^{100})$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 1.00000 0.707107 0.353553 0.935414i $$-0.384973\pi$$
0.353553 + 0.935414i $$0.384973\pi$$
$$3$$ 1.00000 0.577350
$$4$$ −1.00000 −0.500000
$$5$$ 0 0
$$6$$ 1.00000 0.408248
$$7$$ 4.00000 1.51186 0.755929 0.654654i $$-0.227186\pi$$
0.755929 + 0.654654i $$0.227186\pi$$
$$8$$ −3.00000 −1.06066
$$9$$ 1.00000 0.333333
$$10$$ 0 0
$$11$$ 0 0
$$12$$ −1.00000 −0.288675
$$13$$ 2.00000 0.554700 0.277350 0.960769i $$-0.410544\pi$$
0.277350 + 0.960769i $$0.410544\pi$$
$$14$$ 4.00000 1.06904
$$15$$ 0 0
$$16$$ −1.00000 −0.250000
$$17$$ 2.00000 0.485071 0.242536 0.970143i $$-0.422021\pi$$
0.242536 + 0.970143i $$0.422021\pi$$
$$18$$ 1.00000 0.235702
$$19$$ −8.00000 −1.83533 −0.917663 0.397360i $$-0.869927\pi$$
−0.917663 + 0.397360i $$0.869927\pi$$
$$20$$ 0 0
$$21$$ 4.00000 0.872872
$$22$$ 0 0
$$23$$ −4.00000 −0.834058 −0.417029 0.908893i $$-0.636929\pi$$
−0.417029 + 0.908893i $$0.636929\pi$$
$$24$$ −3.00000 −0.612372
$$25$$ 0 0
$$26$$ 2.00000 0.392232
$$27$$ 1.00000 0.192450
$$28$$ −4.00000 −0.755929
$$29$$ −4.00000 −0.742781 −0.371391 0.928477i $$-0.621119\pi$$
−0.371391 + 0.928477i $$0.621119\pi$$
$$30$$ 0 0
$$31$$ −8.00000 −1.43684 −0.718421 0.695608i $$-0.755135\pi$$
−0.718421 + 0.695608i $$0.755135\pi$$
$$32$$ 5.00000 0.883883
$$33$$ 0 0
$$34$$ 2.00000 0.342997
$$35$$ 0 0
$$36$$ −1.00000 −0.166667
$$37$$ −8.00000 −1.31519 −0.657596 0.753371i $$-0.728427\pi$$
−0.657596 + 0.753371i $$0.728427\pi$$
$$38$$ −8.00000 −1.29777
$$39$$ 2.00000 0.320256
$$40$$ 0 0
$$41$$ −12.0000 −1.87409 −0.937043 0.349215i $$-0.886448\pi$$
−0.937043 + 0.349215i $$0.886448\pi$$
$$42$$ 4.00000 0.617213
$$43$$ −8.00000 −1.21999 −0.609994 0.792406i $$-0.708828\pi$$
−0.609994 + 0.792406i $$0.708828\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ −4.00000 −0.589768
$$47$$ 4.00000 0.583460 0.291730 0.956501i $$-0.405769\pi$$
0.291730 + 0.956501i $$0.405769\pi$$
$$48$$ −1.00000 −0.144338
$$49$$ 9.00000 1.28571
$$50$$ 0 0
$$51$$ 2.00000 0.280056
$$52$$ −2.00000 −0.277350
$$53$$ 4.00000 0.549442 0.274721 0.961524i $$-0.411414\pi$$
0.274721 + 0.961524i $$0.411414\pi$$
$$54$$ 1.00000 0.136083
$$55$$ 0 0
$$56$$ −12.0000 −1.60357
$$57$$ −8.00000 −1.05963
$$58$$ −4.00000 −0.525226
$$59$$ −8.00000 −1.04151 −0.520756 0.853706i $$-0.674350\pi$$
−0.520756 + 0.853706i $$0.674350\pi$$
$$60$$ 0 0
$$61$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$62$$ −8.00000 −1.01600
$$63$$ 4.00000 0.503953
$$64$$ 7.00000 0.875000
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 4.00000 0.488678 0.244339 0.969690i $$-0.421429\pi$$
0.244339 + 0.969690i $$0.421429\pi$$
$$68$$ −2.00000 −0.242536
$$69$$ −4.00000 −0.481543
$$70$$ 0 0
$$71$$ −12.0000 −1.42414 −0.712069 0.702109i $$-0.752242\pi$$
−0.712069 + 0.702109i $$0.752242\pi$$
$$72$$ −3.00000 −0.353553
$$73$$ 2.00000 0.234082 0.117041 0.993127i $$-0.462659\pi$$
0.117041 + 0.993127i $$0.462659\pi$$
$$74$$ −8.00000 −0.929981
$$75$$ 0 0
$$76$$ 8.00000 0.917663
$$77$$ 0 0
$$78$$ 2.00000 0.226455
$$79$$ −8.00000 −0.900070 −0.450035 0.893011i $$-0.648589\pi$$
−0.450035 + 0.893011i $$0.648589\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ −12.0000 −1.32518
$$83$$ 4.00000 0.439057 0.219529 0.975606i $$-0.429548\pi$$
0.219529 + 0.975606i $$0.429548\pi$$
$$84$$ −4.00000 −0.436436
$$85$$ 0 0
$$86$$ −8.00000 −0.862662
$$87$$ −4.00000 −0.428845
$$88$$ 0 0
$$89$$ 6.00000 0.635999 0.317999 0.948091i $$-0.396989\pi$$
0.317999 + 0.948091i $$0.396989\pi$$
$$90$$ 0 0
$$91$$ 8.00000 0.838628
$$92$$ 4.00000 0.417029
$$93$$ −8.00000 −0.829561
$$94$$ 4.00000 0.412568
$$95$$ 0 0
$$96$$ 5.00000 0.510310
$$97$$ 8.00000 0.812277 0.406138 0.913812i $$-0.366875\pi$$
0.406138 + 0.913812i $$0.366875\pi$$
$$98$$ 9.00000 0.909137
$$99$$ 0 0
$$100$$ 0 0
$$101$$ 4.00000 0.398015 0.199007 0.979998i $$-0.436228\pi$$
0.199007 + 0.979998i $$0.436228\pi$$
$$102$$ 2.00000 0.198030
$$103$$ 12.0000 1.18240 0.591198 0.806527i $$-0.298655\pi$$
0.591198 + 0.806527i $$0.298655\pi$$
$$104$$ −6.00000 −0.588348
$$105$$ 0 0
$$106$$ 4.00000 0.388514
$$107$$ 12.0000 1.16008 0.580042 0.814587i $$-0.303036\pi$$
0.580042 + 0.814587i $$0.303036\pi$$
$$108$$ −1.00000 −0.0962250
$$109$$ 8.00000 0.766261 0.383131 0.923694i $$-0.374846\pi$$
0.383131 + 0.923694i $$0.374846\pi$$
$$110$$ 0 0
$$111$$ −8.00000 −0.759326
$$112$$ −4.00000 −0.377964
$$113$$ −12.0000 −1.12887 −0.564433 0.825479i $$-0.690905\pi$$
−0.564433 + 0.825479i $$0.690905\pi$$
$$114$$ −8.00000 −0.749269
$$115$$ 0 0
$$116$$ 4.00000 0.371391
$$117$$ 2.00000 0.184900
$$118$$ −8.00000 −0.736460
$$119$$ 8.00000 0.733359
$$120$$ 0 0
$$121$$ 0 0
$$122$$ 0 0
$$123$$ −12.0000 −1.08200
$$124$$ 8.00000 0.718421
$$125$$ 0 0
$$126$$ 4.00000 0.356348
$$127$$ −4.00000 −0.354943 −0.177471 0.984126i $$-0.556792\pi$$
−0.177471 + 0.984126i $$0.556792\pi$$
$$128$$ −3.00000 −0.265165
$$129$$ −8.00000 −0.704361
$$130$$ 0 0
$$131$$ −8.00000 −0.698963 −0.349482 0.936943i $$-0.613642\pi$$
−0.349482 + 0.936943i $$0.613642\pi$$
$$132$$ 0 0
$$133$$ −32.0000 −2.77475
$$134$$ 4.00000 0.345547
$$135$$ 0 0
$$136$$ −6.00000 −0.514496
$$137$$ 4.00000 0.341743 0.170872 0.985293i $$-0.445342\pi$$
0.170872 + 0.985293i $$0.445342\pi$$
$$138$$ −4.00000 −0.340503
$$139$$ −16.0000 −1.35710 −0.678551 0.734553i $$-0.737392\pi$$
−0.678551 + 0.734553i $$0.737392\pi$$
$$140$$ 0 0
$$141$$ 4.00000 0.336861
$$142$$ −12.0000 −1.00702
$$143$$ 0 0
$$144$$ −1.00000 −0.0833333
$$145$$ 0 0
$$146$$ 2.00000 0.165521
$$147$$ 9.00000 0.742307
$$148$$ 8.00000 0.657596
$$149$$ −20.0000 −1.63846 −0.819232 0.573462i $$-0.805600\pi$$
−0.819232 + 0.573462i $$0.805600\pi$$
$$150$$ 0 0
$$151$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$152$$ 24.0000 1.94666
$$153$$ 2.00000 0.161690
$$154$$ 0 0
$$155$$ 0 0
$$156$$ −2.00000 −0.160128
$$157$$ −8.00000 −0.638470 −0.319235 0.947676i $$-0.603426\pi$$
−0.319235 + 0.947676i $$0.603426\pi$$
$$158$$ −8.00000 −0.636446
$$159$$ 4.00000 0.317221
$$160$$ 0 0
$$161$$ −16.0000 −1.26098
$$162$$ 1.00000 0.0785674
$$163$$ 20.0000 1.56652 0.783260 0.621694i $$-0.213555\pi$$
0.783260 + 0.621694i $$0.213555\pi$$
$$164$$ 12.0000 0.937043
$$165$$ 0 0
$$166$$ 4.00000 0.310460
$$167$$ 24.0000 1.85718 0.928588 0.371113i $$-0.121024\pi$$
0.928588 + 0.371113i $$0.121024\pi$$
$$168$$ −12.0000 −0.925820
$$169$$ −9.00000 −0.692308
$$170$$ 0 0
$$171$$ −8.00000 −0.611775
$$172$$ 8.00000 0.609994
$$173$$ 14.0000 1.06440 0.532200 0.846619i $$-0.321365\pi$$
0.532200 + 0.846619i $$0.321365\pi$$
$$174$$ −4.00000 −0.303239
$$175$$ 0 0
$$176$$ 0 0
$$177$$ −8.00000 −0.601317
$$178$$ 6.00000 0.449719
$$179$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$180$$ 0 0
$$181$$ −10.0000 −0.743294 −0.371647 0.928374i $$-0.621207\pi$$
−0.371647 + 0.928374i $$0.621207\pi$$
$$182$$ 8.00000 0.592999
$$183$$ 0 0
$$184$$ 12.0000 0.884652
$$185$$ 0 0
$$186$$ −8.00000 −0.586588
$$187$$ 0 0
$$188$$ −4.00000 −0.291730
$$189$$ 4.00000 0.290957
$$190$$ 0 0
$$191$$ 12.0000 0.868290 0.434145 0.900843i $$-0.357051\pi$$
0.434145 + 0.900843i $$0.357051\pi$$
$$192$$ 7.00000 0.505181
$$193$$ −6.00000 −0.431889 −0.215945 0.976406i $$-0.569283\pi$$
−0.215945 + 0.976406i $$0.569283\pi$$
$$194$$ 8.00000 0.574367
$$195$$ 0 0
$$196$$ −9.00000 −0.642857
$$197$$ 6.00000 0.427482 0.213741 0.976890i $$-0.431435\pi$$
0.213741 + 0.976890i $$0.431435\pi$$
$$198$$ 0 0
$$199$$ −16.0000 −1.13421 −0.567105 0.823646i $$-0.691937\pi$$
−0.567105 + 0.823646i $$0.691937\pi$$
$$200$$ 0 0
$$201$$ 4.00000 0.282138
$$202$$ 4.00000 0.281439
$$203$$ −16.0000 −1.12298
$$204$$ −2.00000 −0.140028
$$205$$ 0 0
$$206$$ 12.0000 0.836080
$$207$$ −4.00000 −0.278019
$$208$$ −2.00000 −0.138675
$$209$$ 0 0
$$210$$ 0 0
$$211$$ 16.0000 1.10149 0.550743 0.834675i $$-0.314345\pi$$
0.550743 + 0.834675i $$0.314345\pi$$
$$212$$ −4.00000 −0.274721
$$213$$ −12.0000 −0.822226
$$214$$ 12.0000 0.820303
$$215$$ 0 0
$$216$$ −3.00000 −0.204124
$$217$$ −32.0000 −2.17230
$$218$$ 8.00000 0.541828
$$219$$ 2.00000 0.135147
$$220$$ 0 0
$$221$$ 4.00000 0.269069
$$222$$ −8.00000 −0.536925
$$223$$ 4.00000 0.267860 0.133930 0.990991i $$-0.457240\pi$$
0.133930 + 0.990991i $$0.457240\pi$$
$$224$$ 20.0000 1.33631
$$225$$ 0 0
$$226$$ −12.0000 −0.798228
$$227$$ −12.0000 −0.796468 −0.398234 0.917284i $$-0.630377\pi$$
−0.398234 + 0.917284i $$0.630377\pi$$
$$228$$ 8.00000 0.529813
$$229$$ −22.0000 −1.45380 −0.726900 0.686743i $$-0.759040\pi$$
−0.726900 + 0.686743i $$0.759040\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 12.0000 0.787839
$$233$$ 6.00000 0.393073 0.196537 0.980497i $$-0.437031\pi$$
0.196537 + 0.980497i $$0.437031\pi$$
$$234$$ 2.00000 0.130744
$$235$$ 0 0
$$236$$ 8.00000 0.520756
$$237$$ −8.00000 −0.519656
$$238$$ 8.00000 0.518563
$$239$$ 24.0000 1.55243 0.776215 0.630468i $$-0.217137\pi$$
0.776215 + 0.630468i $$0.217137\pi$$
$$240$$ 0 0
$$241$$ −8.00000 −0.515325 −0.257663 0.966235i $$-0.582952\pi$$
−0.257663 + 0.966235i $$0.582952\pi$$
$$242$$ 0 0
$$243$$ 1.00000 0.0641500
$$244$$ 0 0
$$245$$ 0 0
$$246$$ −12.0000 −0.765092
$$247$$ −16.0000 −1.01806
$$248$$ 24.0000 1.52400
$$249$$ 4.00000 0.253490
$$250$$ 0 0
$$251$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$252$$ −4.00000 −0.251976
$$253$$ 0 0
$$254$$ −4.00000 −0.250982
$$255$$ 0 0
$$256$$ −17.0000 −1.06250
$$257$$ −4.00000 −0.249513 −0.124757 0.992187i $$-0.539815\pi$$
−0.124757 + 0.992187i $$0.539815\pi$$
$$258$$ −8.00000 −0.498058
$$259$$ −32.0000 −1.98838
$$260$$ 0 0
$$261$$ −4.00000 −0.247594
$$262$$ −8.00000 −0.494242
$$263$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ −32.0000 −1.96205
$$267$$ 6.00000 0.367194
$$268$$ −4.00000 −0.244339
$$269$$ −26.0000 −1.58525 −0.792624 0.609711i $$-0.791286\pi$$
−0.792624 + 0.609711i $$0.791286\pi$$
$$270$$ 0 0
$$271$$ −16.0000 −0.971931 −0.485965 0.873978i $$-0.661532\pi$$
−0.485965 + 0.873978i $$0.661532\pi$$
$$272$$ −2.00000 −0.121268
$$273$$ 8.00000 0.484182
$$274$$ 4.00000 0.241649
$$275$$ 0 0
$$276$$ 4.00000 0.240772
$$277$$ 10.0000 0.600842 0.300421 0.953807i $$-0.402873\pi$$
0.300421 + 0.953807i $$0.402873\pi$$
$$278$$ −16.0000 −0.959616
$$279$$ −8.00000 −0.478947
$$280$$ 0 0
$$281$$ −12.0000 −0.715860 −0.357930 0.933748i $$-0.616517\pi$$
−0.357930 + 0.933748i $$0.616517\pi$$
$$282$$ 4.00000 0.238197
$$283$$ −16.0000 −0.951101 −0.475551 0.879688i $$-0.657751\pi$$
−0.475551 + 0.879688i $$0.657751\pi$$
$$284$$ 12.0000 0.712069
$$285$$ 0 0
$$286$$ 0 0
$$287$$ −48.0000 −2.83335
$$288$$ 5.00000 0.294628
$$289$$ −13.0000 −0.764706
$$290$$ 0 0
$$291$$ 8.00000 0.468968
$$292$$ −2.00000 −0.117041
$$293$$ −6.00000 −0.350524 −0.175262 0.984522i $$-0.556077\pi$$
−0.175262 + 0.984522i $$0.556077\pi$$
$$294$$ 9.00000 0.524891
$$295$$ 0 0
$$296$$ 24.0000 1.39497
$$297$$ 0 0
$$298$$ −20.0000 −1.15857
$$299$$ −8.00000 −0.462652
$$300$$ 0 0
$$301$$ −32.0000 −1.84445
$$302$$ 0 0
$$303$$ 4.00000 0.229794
$$304$$ 8.00000 0.458831
$$305$$ 0 0
$$306$$ 2.00000 0.114332
$$307$$ 24.0000 1.36975 0.684876 0.728659i $$-0.259856\pi$$
0.684876 + 0.728659i $$0.259856\pi$$
$$308$$ 0 0
$$309$$ 12.0000 0.682656
$$310$$ 0 0
$$311$$ 28.0000 1.58773 0.793867 0.608091i $$-0.208065\pi$$
0.793867 + 0.608091i $$0.208065\pi$$
$$312$$ −6.00000 −0.339683
$$313$$ −16.0000 −0.904373 −0.452187 0.891923i $$-0.649356\pi$$
−0.452187 + 0.891923i $$0.649356\pi$$
$$314$$ −8.00000 −0.451466
$$315$$ 0 0
$$316$$ 8.00000 0.450035
$$317$$ −20.0000 −1.12331 −0.561656 0.827371i $$-0.689836\pi$$
−0.561656 + 0.827371i $$0.689836\pi$$
$$318$$ 4.00000 0.224309
$$319$$ 0 0
$$320$$ 0 0
$$321$$ 12.0000 0.669775
$$322$$ −16.0000 −0.891645
$$323$$ −16.0000 −0.890264
$$324$$ −1.00000 −0.0555556
$$325$$ 0 0
$$326$$ 20.0000 1.10770
$$327$$ 8.00000 0.442401
$$328$$ 36.0000 1.98777
$$329$$ 16.0000 0.882109
$$330$$ 0 0
$$331$$ 4.00000 0.219860 0.109930 0.993939i $$-0.464937\pi$$
0.109930 + 0.993939i $$0.464937\pi$$
$$332$$ −4.00000 −0.219529
$$333$$ −8.00000 −0.438397
$$334$$ 24.0000 1.31322
$$335$$ 0 0
$$336$$ −4.00000 −0.218218
$$337$$ 26.0000 1.41631 0.708155 0.706057i $$-0.249528\pi$$
0.708155 + 0.706057i $$0.249528\pi$$
$$338$$ −9.00000 −0.489535
$$339$$ −12.0000 −0.651751
$$340$$ 0 0
$$341$$ 0 0
$$342$$ −8.00000 −0.432590
$$343$$ 8.00000 0.431959
$$344$$ 24.0000 1.29399
$$345$$ 0 0
$$346$$ 14.0000 0.752645
$$347$$ 12.0000 0.644194 0.322097 0.946707i $$-0.395612\pi$$
0.322097 + 0.946707i $$0.395612\pi$$
$$348$$ 4.00000 0.214423
$$349$$ −16.0000 −0.856460 −0.428230 0.903670i $$-0.640863\pi$$
−0.428230 + 0.903670i $$0.640863\pi$$
$$350$$ 0 0
$$351$$ 2.00000 0.106752
$$352$$ 0 0
$$353$$ −12.0000 −0.638696 −0.319348 0.947638i $$-0.603464\pi$$
−0.319348 + 0.947638i $$0.603464\pi$$
$$354$$ −8.00000 −0.425195
$$355$$ 0 0
$$356$$ −6.00000 −0.317999
$$357$$ 8.00000 0.423405
$$358$$ 0 0
$$359$$ −24.0000 −1.26667 −0.633336 0.773877i $$-0.718315\pi$$
−0.633336 + 0.773877i $$0.718315\pi$$
$$360$$ 0 0
$$361$$ 45.0000 2.36842
$$362$$ −10.0000 −0.525588
$$363$$ 0 0
$$364$$ −8.00000 −0.419314
$$365$$ 0 0
$$366$$ 0 0
$$367$$ −36.0000 −1.87918 −0.939592 0.342296i $$-0.888796\pi$$
−0.939592 + 0.342296i $$0.888796\pi$$
$$368$$ 4.00000 0.208514
$$369$$ −12.0000 −0.624695
$$370$$ 0 0
$$371$$ 16.0000 0.830679
$$372$$ 8.00000 0.414781
$$373$$ 22.0000 1.13912 0.569558 0.821951i $$-0.307114\pi$$
0.569558 + 0.821951i $$0.307114\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ −12.0000 −0.618853
$$377$$ −8.00000 −0.412021
$$378$$ 4.00000 0.205738
$$379$$ −4.00000 −0.205466 −0.102733 0.994709i $$-0.532759\pi$$
−0.102733 + 0.994709i $$0.532759\pi$$
$$380$$ 0 0
$$381$$ −4.00000 −0.204926
$$382$$ 12.0000 0.613973
$$383$$ 36.0000 1.83951 0.919757 0.392488i $$-0.128386\pi$$
0.919757 + 0.392488i $$0.128386\pi$$
$$384$$ −3.00000 −0.153093
$$385$$ 0 0
$$386$$ −6.00000 −0.305392
$$387$$ −8.00000 −0.406663
$$388$$ −8.00000 −0.406138
$$389$$ 18.0000 0.912636 0.456318 0.889817i $$-0.349168\pi$$
0.456318 + 0.889817i $$0.349168\pi$$
$$390$$ 0 0
$$391$$ −8.00000 −0.404577
$$392$$ −27.0000 −1.36371
$$393$$ −8.00000 −0.403547
$$394$$ 6.00000 0.302276
$$395$$ 0 0
$$396$$ 0 0
$$397$$ −8.00000 −0.401508 −0.200754 0.979642i $$-0.564339\pi$$
−0.200754 + 0.979642i $$0.564339\pi$$
$$398$$ −16.0000 −0.802008
$$399$$ −32.0000 −1.60200
$$400$$ 0 0
$$401$$ −2.00000 −0.0998752 −0.0499376 0.998752i $$-0.515902\pi$$
−0.0499376 + 0.998752i $$0.515902\pi$$
$$402$$ 4.00000 0.199502
$$403$$ −16.0000 −0.797017
$$404$$ −4.00000 −0.199007
$$405$$ 0 0
$$406$$ −16.0000 −0.794067
$$407$$ 0 0
$$408$$ −6.00000 −0.297044
$$409$$ 40.0000 1.97787 0.988936 0.148340i $$-0.0473931\pi$$
0.988936 + 0.148340i $$0.0473931\pi$$
$$410$$ 0 0
$$411$$ 4.00000 0.197305
$$412$$ −12.0000 −0.591198
$$413$$ −32.0000 −1.57462
$$414$$ −4.00000 −0.196589
$$415$$ 0 0
$$416$$ 10.0000 0.490290
$$417$$ −16.0000 −0.783523
$$418$$ 0 0
$$419$$ 8.00000 0.390826 0.195413 0.980721i $$-0.437395\pi$$
0.195413 + 0.980721i $$0.437395\pi$$
$$420$$ 0 0
$$421$$ −10.0000 −0.487370 −0.243685 0.969854i $$-0.578356\pi$$
−0.243685 + 0.969854i $$0.578356\pi$$
$$422$$ 16.0000 0.778868
$$423$$ 4.00000 0.194487
$$424$$ −12.0000 −0.582772
$$425$$ 0 0
$$426$$ −12.0000 −0.581402
$$427$$ 0 0
$$428$$ −12.0000 −0.580042
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 24.0000 1.15604 0.578020 0.816023i $$-0.303826\pi$$
0.578020 + 0.816023i $$0.303826\pi$$
$$432$$ −1.00000 −0.0481125
$$433$$ −16.0000 −0.768911 −0.384455 0.923144i $$-0.625611\pi$$
−0.384455 + 0.923144i $$0.625611\pi$$
$$434$$ −32.0000 −1.53605
$$435$$ 0 0
$$436$$ −8.00000 −0.383131
$$437$$ 32.0000 1.53077
$$438$$ 2.00000 0.0955637
$$439$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$440$$ 0 0
$$441$$ 9.00000 0.428571
$$442$$ 4.00000 0.190261
$$443$$ −12.0000 −0.570137 −0.285069 0.958507i $$-0.592016\pi$$
−0.285069 + 0.958507i $$0.592016\pi$$
$$444$$ 8.00000 0.379663
$$445$$ 0 0
$$446$$ 4.00000 0.189405
$$447$$ −20.0000 −0.945968
$$448$$ 28.0000 1.32288
$$449$$ −30.0000 −1.41579 −0.707894 0.706319i $$-0.750354\pi$$
−0.707894 + 0.706319i $$0.750354\pi$$
$$450$$ 0 0
$$451$$ 0 0
$$452$$ 12.0000 0.564433
$$453$$ 0 0
$$454$$ −12.0000 −0.563188
$$455$$ 0 0
$$456$$ 24.0000 1.12390
$$457$$ −2.00000 −0.0935561 −0.0467780 0.998905i $$-0.514895\pi$$
−0.0467780 + 0.998905i $$0.514895\pi$$
$$458$$ −22.0000 −1.02799
$$459$$ 2.00000 0.0933520
$$460$$ 0 0
$$461$$ 12.0000 0.558896 0.279448 0.960161i $$-0.409849\pi$$
0.279448 + 0.960161i $$0.409849\pi$$
$$462$$ 0 0
$$463$$ −28.0000 −1.30127 −0.650635 0.759390i $$-0.725497\pi$$
−0.650635 + 0.759390i $$0.725497\pi$$
$$464$$ 4.00000 0.185695
$$465$$ 0 0
$$466$$ 6.00000 0.277945
$$467$$ −12.0000 −0.555294 −0.277647 0.960683i $$-0.589555\pi$$
−0.277647 + 0.960683i $$0.589555\pi$$
$$468$$ −2.00000 −0.0924500
$$469$$ 16.0000 0.738811
$$470$$ 0 0
$$471$$ −8.00000 −0.368621
$$472$$ 24.0000 1.10469
$$473$$ 0 0
$$474$$ −8.00000 −0.367452
$$475$$ 0 0
$$476$$ −8.00000 −0.366679
$$477$$ 4.00000 0.183147
$$478$$ 24.0000 1.09773
$$479$$ 24.0000 1.09659 0.548294 0.836286i $$-0.315277\pi$$
0.548294 + 0.836286i $$0.315277\pi$$
$$480$$ 0 0
$$481$$ −16.0000 −0.729537
$$482$$ −8.00000 −0.364390
$$483$$ −16.0000 −0.728025
$$484$$ 0 0
$$485$$ 0 0
$$486$$ 1.00000 0.0453609
$$487$$ −20.0000 −0.906287 −0.453143 0.891438i $$-0.649697\pi$$
−0.453143 + 0.891438i $$0.649697\pi$$
$$488$$ 0 0
$$489$$ 20.0000 0.904431
$$490$$ 0 0
$$491$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$492$$ 12.0000 0.541002
$$493$$ −8.00000 −0.360302
$$494$$ −16.0000 −0.719874
$$495$$ 0 0
$$496$$ 8.00000 0.359211
$$497$$ −48.0000 −2.15309
$$498$$ 4.00000 0.179244
$$499$$ −12.0000 −0.537194 −0.268597 0.963253i $$-0.586560\pi$$
−0.268597 + 0.963253i $$0.586560\pi$$
$$500$$ 0 0
$$501$$ 24.0000 1.07224
$$502$$ 0 0
$$503$$ 24.0000 1.07011 0.535054 0.844818i $$-0.320291\pi$$
0.535054 + 0.844818i $$0.320291\pi$$
$$504$$ −12.0000 −0.534522
$$505$$ 0 0
$$506$$ 0 0
$$507$$ −9.00000 −0.399704
$$508$$ 4.00000 0.177471
$$509$$ 26.0000 1.15243 0.576215 0.817298i $$-0.304529\pi$$
0.576215 + 0.817298i $$0.304529\pi$$
$$510$$ 0 0
$$511$$ 8.00000 0.353899
$$512$$ −11.0000 −0.486136
$$513$$ −8.00000 −0.353209
$$514$$ −4.00000 −0.176432
$$515$$ 0 0
$$516$$ 8.00000 0.352180
$$517$$ 0 0
$$518$$ −32.0000 −1.40600
$$519$$ 14.0000 0.614532
$$520$$ 0 0
$$521$$ 22.0000 0.963837 0.481919 0.876216i $$-0.339940\pi$$
0.481919 + 0.876216i $$0.339940\pi$$
$$522$$ −4.00000 −0.175075
$$523$$ 24.0000 1.04945 0.524723 0.851273i $$-0.324169\pi$$
0.524723 + 0.851273i $$0.324169\pi$$
$$524$$ 8.00000 0.349482
$$525$$ 0 0
$$526$$ 0 0
$$527$$ −16.0000 −0.696971
$$528$$ 0 0
$$529$$ −7.00000 −0.304348
$$530$$ 0 0
$$531$$ −8.00000 −0.347170
$$532$$ 32.0000 1.38738
$$533$$ −24.0000 −1.03956
$$534$$ 6.00000 0.259645
$$535$$ 0 0
$$536$$ −12.0000 −0.518321
$$537$$ 0 0
$$538$$ −26.0000 −1.12094
$$539$$ 0 0
$$540$$ 0 0
$$541$$ −8.00000 −0.343947 −0.171973 0.985102i $$-0.555014\pi$$
−0.171973 + 0.985102i $$0.555014\pi$$
$$542$$ −16.0000 −0.687259
$$543$$ −10.0000 −0.429141
$$544$$ 10.0000 0.428746
$$545$$ 0 0
$$546$$ 8.00000 0.342368
$$547$$ −8.00000 −0.342055 −0.171028 0.985266i $$-0.554709\pi$$
−0.171028 + 0.985266i $$0.554709\pi$$
$$548$$ −4.00000 −0.170872
$$549$$ 0 0
$$550$$ 0 0
$$551$$ 32.0000 1.36325
$$552$$ 12.0000 0.510754
$$553$$ −32.0000 −1.36078
$$554$$ 10.0000 0.424859
$$555$$ 0 0
$$556$$ 16.0000 0.678551
$$557$$ 14.0000 0.593199 0.296600 0.955002i $$-0.404147\pi$$
0.296600 + 0.955002i $$0.404147\pi$$
$$558$$ −8.00000 −0.338667
$$559$$ −16.0000 −0.676728
$$560$$ 0 0
$$561$$ 0 0
$$562$$ −12.0000 −0.506189
$$563$$ 36.0000 1.51722 0.758610 0.651546i $$-0.225879\pi$$
0.758610 + 0.651546i $$0.225879\pi$$
$$564$$ −4.00000 −0.168430
$$565$$ 0 0
$$566$$ −16.0000 −0.672530
$$567$$ 4.00000 0.167984
$$568$$ 36.0000 1.51053
$$569$$ −44.0000 −1.84458 −0.922288 0.386503i $$-0.873683\pi$$
−0.922288 + 0.386503i $$0.873683\pi$$
$$570$$ 0 0
$$571$$ 40.0000 1.67395 0.836974 0.547243i $$-0.184323\pi$$
0.836974 + 0.547243i $$0.184323\pi$$
$$572$$ 0 0
$$573$$ 12.0000 0.501307
$$574$$ −48.0000 −2.00348
$$575$$ 0 0
$$576$$ 7.00000 0.291667
$$577$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$578$$ −13.0000 −0.540729
$$579$$ −6.00000 −0.249351
$$580$$ 0 0
$$581$$ 16.0000 0.663792
$$582$$ 8.00000 0.331611
$$583$$ 0 0
$$584$$ −6.00000 −0.248282
$$585$$ 0 0
$$586$$ −6.00000 −0.247858
$$587$$ −12.0000 −0.495293 −0.247647 0.968850i $$-0.579657\pi$$
−0.247647 + 0.968850i $$0.579657\pi$$
$$588$$ −9.00000 −0.371154
$$589$$ 64.0000 2.63707
$$590$$ 0 0
$$591$$ 6.00000 0.246807
$$592$$ 8.00000 0.328798
$$593$$ −14.0000 −0.574911 −0.287456 0.957794i $$-0.592809\pi$$
−0.287456 + 0.957794i $$0.592809\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 20.0000 0.819232
$$597$$ −16.0000 −0.654836
$$598$$ −8.00000 −0.327144
$$599$$ −12.0000 −0.490307 −0.245153 0.969484i $$-0.578838\pi$$
−0.245153 + 0.969484i $$0.578838\pi$$
$$600$$ 0 0
$$601$$ 32.0000 1.30531 0.652654 0.757656i $$-0.273656\pi$$
0.652654 + 0.757656i $$0.273656\pi$$
$$602$$ −32.0000 −1.30422
$$603$$ 4.00000 0.162893
$$604$$ 0 0
$$605$$ 0 0
$$606$$ 4.00000 0.162489
$$607$$ 20.0000 0.811775 0.405887 0.913923i $$-0.366962\pi$$
0.405887 + 0.913923i $$0.366962\pi$$
$$608$$ −40.0000 −1.62221
$$609$$ −16.0000 −0.648353
$$610$$ 0 0
$$611$$ 8.00000 0.323645
$$612$$ −2.00000 −0.0808452
$$613$$ −42.0000 −1.69636 −0.848182 0.529705i $$-0.822303\pi$$
−0.848182 + 0.529705i $$0.822303\pi$$
$$614$$ 24.0000 0.968561
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 28.0000 1.12724 0.563619 0.826035i $$-0.309409\pi$$
0.563619 + 0.826035i $$0.309409\pi$$
$$618$$ 12.0000 0.482711
$$619$$ −36.0000 −1.44696 −0.723481 0.690344i $$-0.757459\pi$$
−0.723481 + 0.690344i $$0.757459\pi$$
$$620$$ 0 0
$$621$$ −4.00000 −0.160514
$$622$$ 28.0000 1.12270
$$623$$ 24.0000 0.961540
$$624$$ −2.00000 −0.0800641
$$625$$ 0 0
$$626$$ −16.0000 −0.639489
$$627$$ 0 0
$$628$$ 8.00000 0.319235
$$629$$ −16.0000 −0.637962
$$630$$ 0 0
$$631$$ 8.00000 0.318475 0.159237 0.987240i $$-0.449096\pi$$
0.159237 + 0.987240i $$0.449096\pi$$
$$632$$ 24.0000 0.954669
$$633$$ 16.0000 0.635943
$$634$$ −20.0000 −0.794301
$$635$$ 0 0
$$636$$ −4.00000 −0.158610
$$637$$ 18.0000 0.713186
$$638$$ 0 0
$$639$$ −12.0000 −0.474713
$$640$$ 0 0
$$641$$ −2.00000 −0.0789953 −0.0394976 0.999220i $$-0.512576\pi$$
−0.0394976 + 0.999220i $$0.512576\pi$$
$$642$$ 12.0000 0.473602
$$643$$ −12.0000 −0.473234 −0.236617 0.971603i $$-0.576039\pi$$
−0.236617 + 0.971603i $$0.576039\pi$$
$$644$$ 16.0000 0.630488
$$645$$ 0 0
$$646$$ −16.0000 −0.629512
$$647$$ 28.0000 1.10079 0.550397 0.834903i $$-0.314476\pi$$
0.550397 + 0.834903i $$0.314476\pi$$
$$648$$ −3.00000 −0.117851
$$649$$ 0 0
$$650$$ 0 0
$$651$$ −32.0000 −1.25418
$$652$$ −20.0000 −0.783260
$$653$$ −28.0000 −1.09572 −0.547862 0.836569i $$-0.684558\pi$$
−0.547862 + 0.836569i $$0.684558\pi$$
$$654$$ 8.00000 0.312825
$$655$$ 0 0
$$656$$ 12.0000 0.468521
$$657$$ 2.00000 0.0780274
$$658$$ 16.0000 0.623745
$$659$$ −24.0000 −0.934907 −0.467454 0.884018i $$-0.654829\pi$$
−0.467454 + 0.884018i $$0.654829\pi$$
$$660$$ 0 0
$$661$$ 6.00000 0.233373 0.116686 0.993169i $$-0.462773\pi$$
0.116686 + 0.993169i $$0.462773\pi$$
$$662$$ 4.00000 0.155464
$$663$$ 4.00000 0.155347
$$664$$ −12.0000 −0.465690
$$665$$ 0 0
$$666$$ −8.00000 −0.309994
$$667$$ 16.0000 0.619522
$$668$$ −24.0000 −0.928588
$$669$$ 4.00000 0.154649
$$670$$ 0 0
$$671$$ 0 0
$$672$$ 20.0000 0.771517
$$673$$ −38.0000 −1.46479 −0.732396 0.680879i $$-0.761598\pi$$
−0.732396 + 0.680879i $$0.761598\pi$$
$$674$$ 26.0000 1.00148
$$675$$ 0 0
$$676$$ 9.00000 0.346154
$$677$$ −38.0000 −1.46046 −0.730229 0.683202i $$-0.760587\pi$$
−0.730229 + 0.683202i $$0.760587\pi$$
$$678$$ −12.0000 −0.460857
$$679$$ 32.0000 1.22805
$$680$$ 0 0
$$681$$ −12.0000 −0.459841
$$682$$ 0 0
$$683$$ −4.00000 −0.153056 −0.0765279 0.997067i $$-0.524383\pi$$
−0.0765279 + 0.997067i $$0.524383\pi$$
$$684$$ 8.00000 0.305888
$$685$$ 0 0
$$686$$ 8.00000 0.305441
$$687$$ −22.0000 −0.839352
$$688$$ 8.00000 0.304997
$$689$$ 8.00000 0.304776
$$690$$ 0 0
$$691$$ −20.0000 −0.760836 −0.380418 0.924815i $$-0.624220\pi$$
−0.380418 + 0.924815i $$0.624220\pi$$
$$692$$ −14.0000 −0.532200
$$693$$ 0 0
$$694$$ 12.0000 0.455514
$$695$$ 0 0
$$696$$ 12.0000 0.454859
$$697$$ −24.0000 −0.909065
$$698$$ −16.0000 −0.605609
$$699$$ 6.00000 0.226941
$$700$$ 0 0
$$701$$ −12.0000 −0.453234 −0.226617 0.973984i $$-0.572767\pi$$
−0.226617 + 0.973984i $$0.572767\pi$$
$$702$$ 2.00000 0.0754851
$$703$$ 64.0000 2.41381
$$704$$ 0 0
$$705$$ 0 0
$$706$$ −12.0000 −0.451626
$$707$$ 16.0000 0.601742
$$708$$ 8.00000 0.300658
$$709$$ 26.0000 0.976450 0.488225 0.872718i $$-0.337644\pi$$
0.488225 + 0.872718i $$0.337644\pi$$
$$710$$ 0 0
$$711$$ −8.00000 −0.300023
$$712$$ −18.0000 −0.674579
$$713$$ 32.0000 1.19841
$$714$$ 8.00000 0.299392
$$715$$ 0 0
$$716$$ 0 0
$$717$$ 24.0000 0.896296
$$718$$ −24.0000 −0.895672
$$719$$ 36.0000 1.34257 0.671287 0.741198i $$-0.265742\pi$$
0.671287 + 0.741198i $$0.265742\pi$$
$$720$$ 0 0
$$721$$ 48.0000 1.78761
$$722$$ 45.0000 1.67473
$$723$$ −8.00000 −0.297523
$$724$$ 10.0000 0.371647
$$725$$ 0 0
$$726$$ 0 0
$$727$$ −28.0000 −1.03846 −0.519231 0.854634i $$-0.673782\pi$$
−0.519231 + 0.854634i $$0.673782\pi$$
$$728$$ −24.0000 −0.889499
$$729$$ 1.00000 0.0370370
$$730$$ 0 0
$$731$$ −16.0000 −0.591781
$$732$$ 0 0
$$733$$ 14.0000 0.517102 0.258551 0.965998i $$-0.416755\pi$$
0.258551 + 0.965998i $$0.416755\pi$$
$$734$$ −36.0000 −1.32878
$$735$$ 0 0
$$736$$ −20.0000 −0.737210
$$737$$ 0 0
$$738$$ −12.0000 −0.441726
$$739$$ −16.0000 −0.588570 −0.294285 0.955718i $$-0.595081\pi$$
−0.294285 + 0.955718i $$0.595081\pi$$
$$740$$ 0 0
$$741$$ −16.0000 −0.587775
$$742$$ 16.0000 0.587378
$$743$$ −24.0000 −0.880475 −0.440237 0.897881i $$-0.645106\pi$$
−0.440237 + 0.897881i $$0.645106\pi$$
$$744$$ 24.0000 0.879883
$$745$$ 0 0
$$746$$ 22.0000 0.805477
$$747$$ 4.00000 0.146352
$$748$$ 0 0
$$749$$ 48.0000 1.75388
$$750$$ 0 0
$$751$$ −16.0000 −0.583848 −0.291924 0.956441i $$-0.594295\pi$$
−0.291924 + 0.956441i $$0.594295\pi$$
$$752$$ −4.00000 −0.145865
$$753$$ 0 0
$$754$$ −8.00000 −0.291343
$$755$$ 0 0
$$756$$ −4.00000 −0.145479
$$757$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$758$$ −4.00000 −0.145287
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 36.0000 1.30500 0.652499 0.757789i $$-0.273720\pi$$
0.652499 + 0.757789i $$0.273720\pi$$
$$762$$ −4.00000 −0.144905
$$763$$ 32.0000 1.15848
$$764$$ −12.0000 −0.434145
$$765$$ 0 0
$$766$$ 36.0000 1.30073
$$767$$ −16.0000 −0.577727
$$768$$ −17.0000 −0.613435
$$769$$ −16.0000 −0.576975 −0.288487 0.957484i $$-0.593152\pi$$
−0.288487 + 0.957484i $$0.593152\pi$$
$$770$$ 0 0
$$771$$ −4.00000 −0.144056
$$772$$ 6.00000 0.215945
$$773$$ 36.0000 1.29483 0.647415 0.762138i $$-0.275850\pi$$
0.647415 + 0.762138i $$0.275850\pi$$
$$774$$ −8.00000 −0.287554
$$775$$ 0 0
$$776$$ −24.0000 −0.861550
$$777$$ −32.0000 −1.14799
$$778$$ 18.0000 0.645331
$$779$$ 96.0000 3.43956
$$780$$ 0 0
$$781$$ 0 0
$$782$$ −8.00000 −0.286079
$$783$$ −4.00000 −0.142948
$$784$$ −9.00000 −0.321429
$$785$$ 0 0
$$786$$ −8.00000 −0.285351
$$787$$ 32.0000 1.14068 0.570338 0.821410i $$-0.306812\pi$$
0.570338 + 0.821410i $$0.306812\pi$$
$$788$$ −6.00000 −0.213741
$$789$$ 0 0
$$790$$ 0 0
$$791$$ −48.0000 −1.70668
$$792$$ 0 0
$$793$$ 0 0
$$794$$ −8.00000 −0.283909
$$795$$ 0 0
$$796$$ 16.0000 0.567105
$$797$$ −12.0000 −0.425062 −0.212531 0.977154i $$-0.568171\pi$$
−0.212531 + 0.977154i $$0.568171\pi$$
$$798$$ −32.0000 −1.13279
$$799$$ 8.00000 0.283020
$$800$$ 0 0
$$801$$ 6.00000 0.212000
$$802$$ −2.00000 −0.0706225
$$803$$ 0 0
$$804$$ −4.00000 −0.141069
$$805$$ 0 0
$$806$$ −16.0000 −0.563576
$$807$$ −26.0000 −0.915243
$$808$$ −12.0000 −0.422159
$$809$$ 44.0000 1.54696 0.773479 0.633822i $$-0.218515\pi$$
0.773479 + 0.633822i $$0.218515\pi$$
$$810$$ 0 0
$$811$$ 8.00000 0.280918 0.140459 0.990086i $$-0.455142\pi$$
0.140459 + 0.990086i $$0.455142\pi$$
$$812$$ 16.0000 0.561490
$$813$$ −16.0000 −0.561144
$$814$$ 0 0
$$815$$ 0 0
$$816$$ −2.00000 −0.0700140
$$817$$ 64.0000 2.23908
$$818$$ 40.0000 1.39857
$$819$$ 8.00000 0.279543
$$820$$ 0 0
$$821$$ −20.0000 −0.698005 −0.349002 0.937122i $$-0.613479\pi$$
−0.349002 + 0.937122i $$0.613479\pi$$
$$822$$ 4.00000 0.139516
$$823$$ −20.0000 −0.697156 −0.348578 0.937280i $$-0.613335\pi$$
−0.348578 + 0.937280i $$0.613335\pi$$
$$824$$ −36.0000 −1.25412
$$825$$ 0 0
$$826$$ −32.0000 −1.11342
$$827$$ −20.0000 −0.695468 −0.347734 0.937593i $$-0.613049\pi$$
−0.347734 + 0.937593i $$0.613049\pi$$
$$828$$ 4.00000 0.139010
$$829$$ −14.0000 −0.486240 −0.243120 0.969996i $$-0.578171\pi$$
−0.243120 + 0.969996i $$0.578171\pi$$
$$830$$ 0 0
$$831$$ 10.0000 0.346896
$$832$$ 14.0000 0.485363
$$833$$ 18.0000 0.623663
$$834$$ −16.0000 −0.554035
$$835$$ 0 0
$$836$$ 0 0
$$837$$ −8.00000 −0.276520
$$838$$ 8.00000 0.276355
$$839$$ 20.0000 0.690477 0.345238 0.938515i $$-0.387798\pi$$
0.345238 + 0.938515i $$0.387798\pi$$
$$840$$ 0 0
$$841$$ −13.0000 −0.448276
$$842$$ −10.0000 −0.344623
$$843$$ −12.0000 −0.413302
$$844$$ −16.0000 −0.550743
$$845$$ 0 0
$$846$$ 4.00000 0.137523
$$847$$ 0 0
$$848$$ −4.00000 −0.137361
$$849$$ −16.0000 −0.549119
$$850$$ 0 0
$$851$$ 32.0000 1.09695
$$852$$ 12.0000 0.411113
$$853$$ 6.00000 0.205436 0.102718 0.994711i $$-0.467246\pi$$
0.102718 + 0.994711i $$0.467246\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ −36.0000 −1.23045
$$857$$ −38.0000 −1.29806 −0.649028 0.760765i $$-0.724824\pi$$
−0.649028 + 0.760765i $$0.724824\pi$$
$$858$$ 0 0
$$859$$ 4.00000 0.136478 0.0682391 0.997669i $$-0.478262\pi$$
0.0682391 + 0.997669i $$0.478262\pi$$
$$860$$ 0 0
$$861$$ −48.0000 −1.63584
$$862$$ 24.0000 0.817443
$$863$$ 36.0000 1.22545 0.612727 0.790295i $$-0.290072\pi$$
0.612727 + 0.790295i $$0.290072\pi$$
$$864$$ 5.00000 0.170103
$$865$$ 0 0
$$866$$ −16.0000 −0.543702
$$867$$ −13.0000 −0.441503
$$868$$ 32.0000 1.08615
$$869$$ 0 0
$$870$$ 0 0
$$871$$ 8.00000 0.271070
$$872$$ −24.0000 −0.812743
$$873$$ 8.00000 0.270759
$$874$$ 32.0000 1.08242
$$875$$ 0 0
$$876$$ −2.00000 −0.0675737
$$877$$ −2.00000 −0.0675352 −0.0337676 0.999430i $$-0.510751\pi$$
−0.0337676 + 0.999430i $$0.510751\pi$$
$$878$$ 0 0
$$879$$ −6.00000 −0.202375
$$880$$ 0 0
$$881$$ −30.0000 −1.01073 −0.505363 0.862907i $$-0.668641\pi$$
−0.505363 + 0.862907i $$0.668641\pi$$
$$882$$ 9.00000 0.303046
$$883$$ 4.00000 0.134611 0.0673054 0.997732i $$-0.478560\pi$$
0.0673054 + 0.997732i $$0.478560\pi$$
$$884$$ −4.00000 −0.134535
$$885$$ 0 0
$$886$$ −12.0000 −0.403148
$$887$$ −16.0000 −0.537227 −0.268614 0.963248i $$-0.586566\pi$$
−0.268614 + 0.963248i $$0.586566\pi$$
$$888$$ 24.0000 0.805387
$$889$$ −16.0000 −0.536623
$$890$$ 0 0
$$891$$ 0 0
$$892$$ −4.00000 −0.133930
$$893$$ −32.0000 −1.07084
$$894$$ −20.0000 −0.668900
$$895$$ 0 0
$$896$$ −12.0000 −0.400892
$$897$$ −8.00000 −0.267112
$$898$$ −30.0000 −1.00111
$$899$$ 32.0000 1.06726
$$900$$ 0 0
$$901$$ 8.00000 0.266519
$$902$$ 0 0
$$903$$ −32.0000 −1.06489
$$904$$ 36.0000 1.19734
$$905$$ 0 0
$$906$$ 0 0
$$907$$ 52.0000 1.72663 0.863316 0.504664i $$-0.168384\pi$$
0.863316 + 0.504664i $$0.168384\pi$$
$$908$$ 12.0000 0.398234
$$909$$ 4.00000 0.132672
$$910$$ 0 0
$$911$$ −36.0000 −1.19273 −0.596367 0.802712i $$-0.703390\pi$$
−0.596367 + 0.802712i $$0.703390\pi$$
$$912$$ 8.00000 0.264906
$$913$$ 0 0
$$914$$ −2.00000 −0.0661541
$$915$$ 0 0
$$916$$ 22.0000 0.726900
$$917$$ −32.0000 −1.05673
$$918$$ 2.00000 0.0660098
$$919$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$920$$ 0 0
$$921$$ 24.0000 0.790827
$$922$$ 12.0000 0.395199
$$923$$ −24.0000 −0.789970
$$924$$ 0 0
$$925$$ 0 0
$$926$$ −28.0000 −0.920137
$$927$$ 12.0000 0.394132
$$928$$ −20.0000 −0.656532
$$929$$ −18.0000 −0.590561 −0.295280 0.955411i $$-0.595413\pi$$
−0.295280 + 0.955411i $$0.595413\pi$$
$$930$$ 0 0
$$931$$ −72.0000 −2.35970
$$932$$ −6.00000 −0.196537
$$933$$ 28.0000 0.916679
$$934$$ −12.0000 −0.392652
$$935$$ 0 0
$$936$$ −6.00000 −0.196116
$$937$$ 14.0000 0.457360 0.228680 0.973502i $$-0.426559\pi$$
0.228680 + 0.973502i $$0.426559\pi$$
$$938$$ 16.0000 0.522419
$$939$$ −16.0000 −0.522140
$$940$$ 0 0
$$941$$ 4.00000 0.130396 0.0651981 0.997872i $$-0.479232\pi$$
0.0651981 + 0.997872i $$0.479232\pi$$
$$942$$ −8.00000 −0.260654
$$943$$ 48.0000 1.56310
$$944$$ 8.00000 0.260378
$$945$$ 0 0
$$946$$ 0 0
$$947$$ −28.0000 −0.909878 −0.454939 0.890523i $$-0.650339\pi$$
−0.454939 + 0.890523i $$0.650339\pi$$
$$948$$ 8.00000 0.259828
$$949$$ 4.00000 0.129845
$$950$$ 0 0
$$951$$ −20.0000 −0.648544
$$952$$ −24.0000 −0.777844
$$953$$ 54.0000 1.74923 0.874616 0.484817i $$-0.161114\pi$$
0.874616 + 0.484817i $$0.161114\pi$$
$$954$$ 4.00000 0.129505
$$955$$ 0 0
$$956$$ −24.0000 −0.776215
$$957$$ 0 0
$$958$$ 24.0000 0.775405
$$959$$ 16.0000 0.516667
$$960$$ 0 0
$$961$$ 33.0000 1.06452
$$962$$ −16.0000 −0.515861
$$963$$ 12.0000 0.386695
$$964$$ 8.00000 0.257663
$$965$$ 0 0
$$966$$ −16.0000 −0.514792
$$967$$ 4.00000 0.128631 0.0643157 0.997930i $$-0.479514\pi$$
0.0643157 + 0.997930i $$0.479514\pi$$
$$968$$ 0 0
$$969$$ −16.0000 −0.513994
$$970$$ 0 0
$$971$$ 16.0000 0.513464 0.256732 0.966483i $$-0.417354\pi$$
0.256732 + 0.966483i $$0.417354\pi$$
$$972$$ −1.00000 −0.0320750
$$973$$ −64.0000 −2.05175
$$974$$ −20.0000 −0.640841
$$975$$ 0 0
$$976$$ 0 0
$$977$$ −36.0000 −1.15174 −0.575871 0.817541i $$-0.695337\pi$$
−0.575871 + 0.817541i $$0.695337\pi$$
$$978$$ 20.0000 0.639529
$$979$$ 0 0
$$980$$ 0 0
$$981$$ 8.00000 0.255420
$$982$$ 0 0
$$983$$ 4.00000 0.127580 0.0637901 0.997963i $$-0.479681\pi$$
0.0637901 + 0.997963i $$0.479681\pi$$
$$984$$ 36.0000 1.14764
$$985$$ 0 0
$$986$$ −8.00000 −0.254772
$$987$$ 16.0000 0.509286
$$988$$ 16.0000 0.509028
$$989$$ 32.0000 1.01754
$$990$$ 0 0
$$991$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$992$$ −40.0000 −1.27000
$$993$$ 4.00000 0.126936
$$994$$ −48.0000 −1.52247
$$995$$ 0 0
$$996$$ −4.00000 −0.126745
$$997$$ −42.0000 −1.33015 −0.665077 0.746775i $$-0.731601\pi$$
−0.665077 + 0.746775i $$0.731601\pi$$
$$998$$ −12.0000 −0.379853
$$999$$ −8.00000 −0.253109
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 9075.2.a.r.1.1 1
5.2 odd 4 1815.2.c.b.364.2 yes 2
5.3 odd 4 1815.2.c.b.364.1 yes 2
5.4 even 2 9075.2.a.c.1.1 1
11.10 odd 2 9075.2.a.f.1.1 1
55.32 even 4 1815.2.c.a.364.1 2
55.43 even 4 1815.2.c.a.364.2 yes 2
55.54 odd 2 9075.2.a.o.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
1815.2.c.a.364.1 2 55.32 even 4
1815.2.c.a.364.2 yes 2 55.43 even 4
1815.2.c.b.364.1 yes 2 5.3 odd 4
1815.2.c.b.364.2 yes 2 5.2 odd 4
9075.2.a.c.1.1 1 5.4 even 2
9075.2.a.f.1.1 1 11.10 odd 2
9075.2.a.o.1.1 1 55.54 odd 2
9075.2.a.r.1.1 1 1.1 even 1 trivial