# Properties

 Label 405.2.a.b.1.1 Level $405$ Weight $2$ Character 405.1 Self dual yes Analytic conductor $3.234$ 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] = [405,2,Mod(1,405)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(405, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("405.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$405 = 3^{4} \cdot 5$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 405.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: $$3.23394128186$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 45) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 405.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-1.00000 q^{2} -1.00000 q^{4} +1.00000 q^{5} -3.00000 q^{7} +3.00000 q^{8} +O(q^{10})$$ $$q-1.00000 q^{2} -1.00000 q^{4} +1.00000 q^{5} -3.00000 q^{7} +3.00000 q^{8} -1.00000 q^{10} +2.00000 q^{11} -2.00000 q^{13} +3.00000 q^{14} -1.00000 q^{16} -4.00000 q^{17} -8.00000 q^{19} -1.00000 q^{20} -2.00000 q^{22} -3.00000 q^{23} +1.00000 q^{25} +2.00000 q^{26} +3.00000 q^{28} +1.00000 q^{29} -5.00000 q^{32} +4.00000 q^{34} -3.00000 q^{35} -4.00000 q^{37} +8.00000 q^{38} +3.00000 q^{40} -5.00000 q^{41} -8.00000 q^{43} -2.00000 q^{44} +3.00000 q^{46} -7.00000 q^{47} +2.00000 q^{49} -1.00000 q^{50} +2.00000 q^{52} +2.00000 q^{53} +2.00000 q^{55} -9.00000 q^{56} -1.00000 q^{58} +14.0000 q^{59} +7.00000 q^{61} +7.00000 q^{64} -2.00000 q^{65} -3.00000 q^{67} +4.00000 q^{68} +3.00000 q^{70} -2.00000 q^{71} +4.00000 q^{73} +4.00000 q^{74} +8.00000 q^{76} -6.00000 q^{77} -6.00000 q^{79} -1.00000 q^{80} +5.00000 q^{82} -9.00000 q^{83} -4.00000 q^{85} +8.00000 q^{86} +6.00000 q^{88} +15.0000 q^{89} +6.00000 q^{91} +3.00000 q^{92} +7.00000 q^{94} -8.00000 q^{95} +2.00000 q^{97} -2.00000 q^{98} +O(q^{100})$$

## 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 $$-0.615027\pi$$
−0.353553 + 0.935414i $$0.615027\pi$$
$$3$$ 0 0
$$4$$ −1.00000 −0.500000
$$5$$ 1.00000 0.447214
$$6$$ 0 0
$$7$$ −3.00000 −1.13389 −0.566947 0.823754i $$-0.691875\pi$$
−0.566947 + 0.823754i $$0.691875\pi$$
$$8$$ 3.00000 1.06066
$$9$$ 0 0
$$10$$ −1.00000 −0.316228
$$11$$ 2.00000 0.603023 0.301511 0.953463i $$-0.402509\pi$$
0.301511 + 0.953463i $$0.402509\pi$$
$$12$$ 0 0
$$13$$ −2.00000 −0.554700 −0.277350 0.960769i $$-0.589456\pi$$
−0.277350 + 0.960769i $$0.589456\pi$$
$$14$$ 3.00000 0.801784
$$15$$ 0 0
$$16$$ −1.00000 −0.250000
$$17$$ −4.00000 −0.970143 −0.485071 0.874475i $$-0.661206\pi$$
−0.485071 + 0.874475i $$0.661206\pi$$
$$18$$ 0 0
$$19$$ −8.00000 −1.83533 −0.917663 0.397360i $$-0.869927\pi$$
−0.917663 + 0.397360i $$0.869927\pi$$
$$20$$ −1.00000 −0.223607
$$21$$ 0 0
$$22$$ −2.00000 −0.426401
$$23$$ −3.00000 −0.625543 −0.312772 0.949828i $$-0.601257\pi$$
−0.312772 + 0.949828i $$0.601257\pi$$
$$24$$ 0 0
$$25$$ 1.00000 0.200000
$$26$$ 2.00000 0.392232
$$27$$ 0 0
$$28$$ 3.00000 0.566947
$$29$$ 1.00000 0.185695 0.0928477 0.995680i $$-0.470403\pi$$
0.0928477 + 0.995680i $$0.470403\pi$$
$$30$$ 0 0
$$31$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$32$$ −5.00000 −0.883883
$$33$$ 0 0
$$34$$ 4.00000 0.685994
$$35$$ −3.00000 −0.507093
$$36$$ 0 0
$$37$$ −4.00000 −0.657596 −0.328798 0.944400i $$-0.606644\pi$$
−0.328798 + 0.944400i $$0.606644\pi$$
$$38$$ 8.00000 1.29777
$$39$$ 0 0
$$40$$ 3.00000 0.474342
$$41$$ −5.00000 −0.780869 −0.390434 0.920631i $$-0.627675\pi$$
−0.390434 + 0.920631i $$0.627675\pi$$
$$42$$ 0 0
$$43$$ −8.00000 −1.21999 −0.609994 0.792406i $$-0.708828\pi$$
−0.609994 + 0.792406i $$0.708828\pi$$
$$44$$ −2.00000 −0.301511
$$45$$ 0 0
$$46$$ 3.00000 0.442326
$$47$$ −7.00000 −1.02105 −0.510527 0.859861i $$-0.670550\pi$$
−0.510527 + 0.859861i $$0.670550\pi$$
$$48$$ 0 0
$$49$$ 2.00000 0.285714
$$50$$ −1.00000 −0.141421
$$51$$ 0 0
$$52$$ 2.00000 0.277350
$$53$$ 2.00000 0.274721 0.137361 0.990521i $$-0.456138\pi$$
0.137361 + 0.990521i $$0.456138\pi$$
$$54$$ 0 0
$$55$$ 2.00000 0.269680
$$56$$ −9.00000 −1.20268
$$57$$ 0 0
$$58$$ −1.00000 −0.131306
$$59$$ 14.0000 1.82264 0.911322 0.411693i $$-0.135063\pi$$
0.911322 + 0.411693i $$0.135063\pi$$
$$60$$ 0 0
$$61$$ 7.00000 0.896258 0.448129 0.893969i $$-0.352090\pi$$
0.448129 + 0.893969i $$0.352090\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 7.00000 0.875000
$$65$$ −2.00000 −0.248069
$$66$$ 0 0
$$67$$ −3.00000 −0.366508 −0.183254 0.983066i $$-0.558663\pi$$
−0.183254 + 0.983066i $$0.558663\pi$$
$$68$$ 4.00000 0.485071
$$69$$ 0 0
$$70$$ 3.00000 0.358569
$$71$$ −2.00000 −0.237356 −0.118678 0.992933i $$-0.537866\pi$$
−0.118678 + 0.992933i $$0.537866\pi$$
$$72$$ 0 0
$$73$$ 4.00000 0.468165 0.234082 0.972217i $$-0.424791\pi$$
0.234082 + 0.972217i $$0.424791\pi$$
$$74$$ 4.00000 0.464991
$$75$$ 0 0
$$76$$ 8.00000 0.917663
$$77$$ −6.00000 −0.683763
$$78$$ 0 0
$$79$$ −6.00000 −0.675053 −0.337526 0.941316i $$-0.609590\pi$$
−0.337526 + 0.941316i $$0.609590\pi$$
$$80$$ −1.00000 −0.111803
$$81$$ 0 0
$$82$$ 5.00000 0.552158
$$83$$ −9.00000 −0.987878 −0.493939 0.869496i $$-0.664443\pi$$
−0.493939 + 0.869496i $$0.664443\pi$$
$$84$$ 0 0
$$85$$ −4.00000 −0.433861
$$86$$ 8.00000 0.862662
$$87$$ 0 0
$$88$$ 6.00000 0.639602
$$89$$ 15.0000 1.59000 0.794998 0.606612i $$-0.207472\pi$$
0.794998 + 0.606612i $$0.207472\pi$$
$$90$$ 0 0
$$91$$ 6.00000 0.628971
$$92$$ 3.00000 0.312772
$$93$$ 0 0
$$94$$ 7.00000 0.721995
$$95$$ −8.00000 −0.820783
$$96$$ 0 0
$$97$$ 2.00000 0.203069 0.101535 0.994832i $$-0.467625\pi$$
0.101535 + 0.994832i $$0.467625\pi$$
$$98$$ −2.00000 −0.202031
$$99$$ 0 0
$$100$$ −1.00000 −0.100000
$$101$$ 18.0000 1.79107 0.895533 0.444994i $$-0.146794\pi$$
0.895533 + 0.444994i $$0.146794\pi$$
$$102$$ 0 0
$$103$$ 8.00000 0.788263 0.394132 0.919054i $$-0.371045\pi$$
0.394132 + 0.919054i $$0.371045\pi$$
$$104$$ −6.00000 −0.588348
$$105$$ 0 0
$$106$$ −2.00000 −0.194257
$$107$$ −3.00000 −0.290021 −0.145010 0.989430i $$-0.546322\pi$$
−0.145010 + 0.989430i $$0.546322\pi$$
$$108$$ 0 0
$$109$$ 5.00000 0.478913 0.239457 0.970907i $$-0.423031\pi$$
0.239457 + 0.970907i $$0.423031\pi$$
$$110$$ −2.00000 −0.190693
$$111$$ 0 0
$$112$$ 3.00000 0.283473
$$113$$ 8.00000 0.752577 0.376288 0.926503i $$-0.377200\pi$$
0.376288 + 0.926503i $$0.377200\pi$$
$$114$$ 0 0
$$115$$ −3.00000 −0.279751
$$116$$ −1.00000 −0.0928477
$$117$$ 0 0
$$118$$ −14.0000 −1.28880
$$119$$ 12.0000 1.10004
$$120$$ 0 0
$$121$$ −7.00000 −0.636364
$$122$$ −7.00000 −0.633750
$$123$$ 0 0
$$124$$ 0 0
$$125$$ 1.00000 0.0894427
$$126$$ 0 0
$$127$$ −5.00000 −0.443678 −0.221839 0.975083i $$-0.571206\pi$$
−0.221839 + 0.975083i $$0.571206\pi$$
$$128$$ 3.00000 0.265165
$$129$$ 0 0
$$130$$ 2.00000 0.175412
$$131$$ 6.00000 0.524222 0.262111 0.965038i $$-0.415581\pi$$
0.262111 + 0.965038i $$0.415581\pi$$
$$132$$ 0 0
$$133$$ 24.0000 2.08106
$$134$$ 3.00000 0.259161
$$135$$ 0 0
$$136$$ −12.0000 −1.02899
$$137$$ −12.0000 −1.02523 −0.512615 0.858619i $$-0.671323\pi$$
−0.512615 + 0.858619i $$0.671323\pi$$
$$138$$ 0 0
$$139$$ −16.0000 −1.35710 −0.678551 0.734553i $$-0.737392\pi$$
−0.678551 + 0.734553i $$0.737392\pi$$
$$140$$ 3.00000 0.253546
$$141$$ 0 0
$$142$$ 2.00000 0.167836
$$143$$ −4.00000 −0.334497
$$144$$ 0 0
$$145$$ 1.00000 0.0830455
$$146$$ −4.00000 −0.331042
$$147$$ 0 0
$$148$$ 4.00000 0.328798
$$149$$ −17.0000 −1.39269 −0.696347 0.717705i $$-0.745193\pi$$
−0.696347 + 0.717705i $$0.745193\pi$$
$$150$$ 0 0
$$151$$ −2.00000 −0.162758 −0.0813788 0.996683i $$-0.525932\pi$$
−0.0813788 + 0.996683i $$0.525932\pi$$
$$152$$ −24.0000 −1.94666
$$153$$ 0 0
$$154$$ 6.00000 0.483494
$$155$$ 0 0
$$156$$ 0 0
$$157$$ 14.0000 1.11732 0.558661 0.829396i $$-0.311315\pi$$
0.558661 + 0.829396i $$0.311315\pi$$
$$158$$ 6.00000 0.477334
$$159$$ 0 0
$$160$$ −5.00000 −0.395285
$$161$$ 9.00000 0.709299
$$162$$ 0 0
$$163$$ −4.00000 −0.313304 −0.156652 0.987654i $$-0.550070\pi$$
−0.156652 + 0.987654i $$0.550070\pi$$
$$164$$ 5.00000 0.390434
$$165$$ 0 0
$$166$$ 9.00000 0.698535
$$167$$ 9.00000 0.696441 0.348220 0.937413i $$-0.386786\pi$$
0.348220 + 0.937413i $$0.386786\pi$$
$$168$$ 0 0
$$169$$ −9.00000 −0.692308
$$170$$ 4.00000 0.306786
$$171$$ 0 0
$$172$$ 8.00000 0.609994
$$173$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$174$$ 0 0
$$175$$ −3.00000 −0.226779
$$176$$ −2.00000 −0.150756
$$177$$ 0 0
$$178$$ −15.0000 −1.12430
$$179$$ 2.00000 0.149487 0.0747435 0.997203i $$-0.476186\pi$$
0.0747435 + 0.997203i $$0.476186\pi$$
$$180$$ 0 0
$$181$$ −7.00000 −0.520306 −0.260153 0.965567i $$-0.583773\pi$$
−0.260153 + 0.965567i $$0.583773\pi$$
$$182$$ −6.00000 −0.444750
$$183$$ 0 0
$$184$$ −9.00000 −0.663489
$$185$$ −4.00000 −0.294086
$$186$$ 0 0
$$187$$ −8.00000 −0.585018
$$188$$ 7.00000 0.510527
$$189$$ 0 0
$$190$$ 8.00000 0.580381
$$191$$ −8.00000 −0.578860 −0.289430 0.957199i $$-0.593466\pi$$
−0.289430 + 0.957199i $$0.593466\pi$$
$$192$$ 0 0
$$193$$ −10.0000 −0.719816 −0.359908 0.932988i $$-0.617192\pi$$
−0.359908 + 0.932988i $$0.617192\pi$$
$$194$$ −2.00000 −0.143592
$$195$$ 0 0
$$196$$ −2.00000 −0.142857
$$197$$ 12.0000 0.854965 0.427482 0.904024i $$-0.359401\pi$$
0.427482 + 0.904024i $$0.359401\pi$$
$$198$$ 0 0
$$199$$ 4.00000 0.283552 0.141776 0.989899i $$-0.454719\pi$$
0.141776 + 0.989899i $$0.454719\pi$$
$$200$$ 3.00000 0.212132
$$201$$ 0 0
$$202$$ −18.0000 −1.26648
$$203$$ −3.00000 −0.210559
$$204$$ 0 0
$$205$$ −5.00000 −0.349215
$$206$$ −8.00000 −0.557386
$$207$$ 0 0
$$208$$ 2.00000 0.138675
$$209$$ −16.0000 −1.10674
$$210$$ 0 0
$$211$$ −22.0000 −1.51454 −0.757271 0.653101i $$-0.773468\pi$$
−0.757271 + 0.653101i $$0.773468\pi$$
$$212$$ −2.00000 −0.137361
$$213$$ 0 0
$$214$$ 3.00000 0.205076
$$215$$ −8.00000 −0.545595
$$216$$ 0 0
$$217$$ 0 0
$$218$$ −5.00000 −0.338643
$$219$$ 0 0
$$220$$ −2.00000 −0.134840
$$221$$ 8.00000 0.538138
$$222$$ 0 0
$$223$$ −19.0000 −1.27233 −0.636167 0.771551i $$-0.719481\pi$$
−0.636167 + 0.771551i $$0.719481\pi$$
$$224$$ 15.0000 1.00223
$$225$$ 0 0
$$226$$ −8.00000 −0.532152
$$227$$ 4.00000 0.265489 0.132745 0.991150i $$-0.457621\pi$$
0.132745 + 0.991150i $$0.457621\pi$$
$$228$$ 0 0
$$229$$ 15.0000 0.991228 0.495614 0.868543i $$-0.334943\pi$$
0.495614 + 0.868543i $$0.334943\pi$$
$$230$$ 3.00000 0.197814
$$231$$ 0 0
$$232$$ 3.00000 0.196960
$$233$$ −24.0000 −1.57229 −0.786146 0.618041i $$-0.787927\pi$$
−0.786146 + 0.618041i $$0.787927\pi$$
$$234$$ 0 0
$$235$$ −7.00000 −0.456630
$$236$$ −14.0000 −0.911322
$$237$$ 0 0
$$238$$ −12.0000 −0.777844
$$239$$ 8.00000 0.517477 0.258738 0.965947i $$-0.416693\pi$$
0.258738 + 0.965947i $$0.416693\pi$$
$$240$$ 0 0
$$241$$ −11.0000 −0.708572 −0.354286 0.935137i $$-0.615276\pi$$
−0.354286 + 0.935137i $$0.615276\pi$$
$$242$$ 7.00000 0.449977
$$243$$ 0 0
$$244$$ −7.00000 −0.448129
$$245$$ 2.00000 0.127775
$$246$$ 0 0
$$247$$ 16.0000 1.01806
$$248$$ 0 0
$$249$$ 0 0
$$250$$ −1.00000 −0.0632456
$$251$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$252$$ 0 0
$$253$$ −6.00000 −0.377217
$$254$$ 5.00000 0.313728
$$255$$ 0 0
$$256$$ −17.0000 −1.06250
$$257$$ 6.00000 0.374270 0.187135 0.982334i $$-0.440080\pi$$
0.187135 + 0.982334i $$0.440080\pi$$
$$258$$ 0 0
$$259$$ 12.0000 0.745644
$$260$$ 2.00000 0.124035
$$261$$ 0 0
$$262$$ −6.00000 −0.370681
$$263$$ 16.0000 0.986602 0.493301 0.869859i $$-0.335790\pi$$
0.493301 + 0.869859i $$0.335790\pi$$
$$264$$ 0 0
$$265$$ 2.00000 0.122859
$$266$$ −24.0000 −1.47153
$$267$$ 0 0
$$268$$ 3.00000 0.183254
$$269$$ −25.0000 −1.52428 −0.762138 0.647414i $$-0.775850\pi$$
−0.762138 + 0.647414i $$0.775850\pi$$
$$270$$ 0 0
$$271$$ −8.00000 −0.485965 −0.242983 0.970031i $$-0.578126\pi$$
−0.242983 + 0.970031i $$0.578126\pi$$
$$272$$ 4.00000 0.242536
$$273$$ 0 0
$$274$$ 12.0000 0.724947
$$275$$ 2.00000 0.120605
$$276$$ 0 0
$$277$$ 12.0000 0.721010 0.360505 0.932757i $$-0.382604\pi$$
0.360505 + 0.932757i $$0.382604\pi$$
$$278$$ 16.0000 0.959616
$$279$$ 0 0
$$280$$ −9.00000 −0.537853
$$281$$ 15.0000 0.894825 0.447412 0.894328i $$-0.352346\pi$$
0.447412 + 0.894328i $$0.352346\pi$$
$$282$$ 0 0
$$283$$ 21.0000 1.24832 0.624160 0.781296i $$-0.285441\pi$$
0.624160 + 0.781296i $$0.285441\pi$$
$$284$$ 2.00000 0.118678
$$285$$ 0 0
$$286$$ 4.00000 0.236525
$$287$$ 15.0000 0.885422
$$288$$ 0 0
$$289$$ −1.00000 −0.0588235
$$290$$ −1.00000 −0.0587220
$$291$$ 0 0
$$292$$ −4.00000 −0.234082
$$293$$ −12.0000 −0.701047 −0.350524 0.936554i $$-0.613996\pi$$
−0.350524 + 0.936554i $$0.613996\pi$$
$$294$$ 0 0
$$295$$ 14.0000 0.815112
$$296$$ −12.0000 −0.697486
$$297$$ 0 0
$$298$$ 17.0000 0.984784
$$299$$ 6.00000 0.346989
$$300$$ 0 0
$$301$$ 24.0000 1.38334
$$302$$ 2.00000 0.115087
$$303$$ 0 0
$$304$$ 8.00000 0.458831
$$305$$ 7.00000 0.400819
$$306$$ 0 0
$$307$$ 7.00000 0.399511 0.199756 0.979846i $$-0.435985\pi$$
0.199756 + 0.979846i $$0.435985\pi$$
$$308$$ 6.00000 0.341882
$$309$$ 0 0
$$310$$ 0 0
$$311$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$312$$ 0 0
$$313$$ 14.0000 0.791327 0.395663 0.918396i $$-0.370515\pi$$
0.395663 + 0.918396i $$0.370515\pi$$
$$314$$ −14.0000 −0.790066
$$315$$ 0 0
$$316$$ 6.00000 0.337526
$$317$$ 34.0000 1.90963 0.954815 0.297200i $$-0.0960529\pi$$
0.954815 + 0.297200i $$0.0960529\pi$$
$$318$$ 0 0
$$319$$ 2.00000 0.111979
$$320$$ 7.00000 0.391312
$$321$$ 0 0
$$322$$ −9.00000 −0.501550
$$323$$ 32.0000 1.78053
$$324$$ 0 0
$$325$$ −2.00000 −0.110940
$$326$$ 4.00000 0.221540
$$327$$ 0 0
$$328$$ −15.0000 −0.828236
$$329$$ 21.0000 1.15777
$$330$$ 0 0
$$331$$ −6.00000 −0.329790 −0.164895 0.986311i $$-0.552728\pi$$
−0.164895 + 0.986311i $$0.552728\pi$$
$$332$$ 9.00000 0.493939
$$333$$ 0 0
$$334$$ −9.00000 −0.492458
$$335$$ −3.00000 −0.163908
$$336$$ 0 0
$$337$$ −8.00000 −0.435788 −0.217894 0.975972i $$-0.569919\pi$$
−0.217894 + 0.975972i $$0.569919\pi$$
$$338$$ 9.00000 0.489535
$$339$$ 0 0
$$340$$ 4.00000 0.216930
$$341$$ 0 0
$$342$$ 0 0
$$343$$ 15.0000 0.809924
$$344$$ −24.0000 −1.29399
$$345$$ 0 0
$$346$$ 0 0
$$347$$ −4.00000 −0.214731 −0.107366 0.994220i $$-0.534242\pi$$
−0.107366 + 0.994220i $$0.534242\pi$$
$$348$$ 0 0
$$349$$ −5.00000 −0.267644 −0.133822 0.991005i $$-0.542725\pi$$
−0.133822 + 0.991005i $$0.542725\pi$$
$$350$$ 3.00000 0.160357
$$351$$ 0 0
$$352$$ −10.0000 −0.533002
$$353$$ −24.0000 −1.27739 −0.638696 0.769460i $$-0.720526\pi$$
−0.638696 + 0.769460i $$0.720526\pi$$
$$354$$ 0 0
$$355$$ −2.00000 −0.106149
$$356$$ −15.0000 −0.794998
$$357$$ 0 0
$$358$$ −2.00000 −0.105703
$$359$$ −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
$$362$$ 7.00000 0.367912
$$363$$ 0 0
$$364$$ −6.00000 −0.314485
$$365$$ 4.00000 0.209370
$$366$$ 0 0
$$367$$ 24.0000 1.25279 0.626395 0.779506i $$-0.284530\pi$$
0.626395 + 0.779506i $$0.284530\pi$$
$$368$$ 3.00000 0.156386
$$369$$ 0 0
$$370$$ 4.00000 0.207950
$$371$$ −6.00000 −0.311504
$$372$$ 0 0
$$373$$ 10.0000 0.517780 0.258890 0.965907i $$-0.416643\pi$$
0.258890 + 0.965907i $$0.416643\pi$$
$$374$$ 8.00000 0.413670
$$375$$ 0 0
$$376$$ −21.0000 −1.08299
$$377$$ −2.00000 −0.103005
$$378$$ 0 0
$$379$$ −26.0000 −1.33553 −0.667765 0.744372i $$-0.732749\pi$$
−0.667765 + 0.744372i $$0.732749\pi$$
$$380$$ 8.00000 0.410391
$$381$$ 0 0
$$382$$ 8.00000 0.409316
$$383$$ −36.0000 −1.83951 −0.919757 0.392488i $$-0.871614\pi$$
−0.919757 + 0.392488i $$0.871614\pi$$
$$384$$ 0 0
$$385$$ −6.00000 −0.305788
$$386$$ 10.0000 0.508987
$$387$$ 0 0
$$388$$ −2.00000 −0.101535
$$389$$ −33.0000 −1.67317 −0.836583 0.547840i $$-0.815450\pi$$
−0.836583 + 0.547840i $$0.815450\pi$$
$$390$$ 0 0
$$391$$ 12.0000 0.606866
$$392$$ 6.00000 0.303046
$$393$$ 0 0
$$394$$ −12.0000 −0.604551
$$395$$ −6.00000 −0.301893
$$396$$ 0 0
$$397$$ 34.0000 1.70641 0.853206 0.521575i $$-0.174655\pi$$
0.853206 + 0.521575i $$0.174655\pi$$
$$398$$ −4.00000 −0.200502
$$399$$ 0 0
$$400$$ −1.00000 −0.0500000
$$401$$ 18.0000 0.898877 0.449439 0.893311i $$-0.351624\pi$$
0.449439 + 0.893311i $$0.351624\pi$$
$$402$$ 0 0
$$403$$ 0 0
$$404$$ −18.0000 −0.895533
$$405$$ 0 0
$$406$$ 3.00000 0.148888
$$407$$ −8.00000 −0.396545
$$408$$ 0 0
$$409$$ 14.0000 0.692255 0.346128 0.938187i $$-0.387496\pi$$
0.346128 + 0.938187i $$0.387496\pi$$
$$410$$ 5.00000 0.246932
$$411$$ 0 0
$$412$$ −8.00000 −0.394132
$$413$$ −42.0000 −2.06668
$$414$$ 0 0
$$415$$ −9.00000 −0.441793
$$416$$ 10.0000 0.490290
$$417$$ 0 0
$$418$$ 16.0000 0.782586
$$419$$ −26.0000 −1.27018 −0.635092 0.772437i $$-0.719038\pi$$
−0.635092 + 0.772437i $$0.719038\pi$$
$$420$$ 0 0
$$421$$ 34.0000 1.65706 0.828529 0.559946i $$-0.189178\pi$$
0.828529 + 0.559946i $$0.189178\pi$$
$$422$$ 22.0000 1.07094
$$423$$ 0 0
$$424$$ 6.00000 0.291386
$$425$$ −4.00000 −0.194029
$$426$$ 0 0
$$427$$ −21.0000 −1.01626
$$428$$ 3.00000 0.145010
$$429$$ 0 0
$$430$$ 8.00000 0.385794
$$431$$ 30.0000 1.44505 0.722525 0.691345i $$-0.242982\pi$$
0.722525 + 0.691345i $$0.242982\pi$$
$$432$$ 0 0
$$433$$ 28.0000 1.34559 0.672797 0.739827i $$-0.265093\pi$$
0.672797 + 0.739827i $$0.265093\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ −5.00000 −0.239457
$$437$$ 24.0000 1.14808
$$438$$ 0 0
$$439$$ 28.0000 1.33637 0.668184 0.743996i $$-0.267072\pi$$
0.668184 + 0.743996i $$0.267072\pi$$
$$440$$ 6.00000 0.286039
$$441$$ 0 0
$$442$$ −8.00000 −0.380521
$$443$$ −15.0000 −0.712672 −0.356336 0.934358i $$-0.615974\pi$$
−0.356336 + 0.934358i $$0.615974\pi$$
$$444$$ 0 0
$$445$$ 15.0000 0.711068
$$446$$ 19.0000 0.899676
$$447$$ 0 0
$$448$$ −21.0000 −0.992157
$$449$$ 26.0000 1.22702 0.613508 0.789689i $$-0.289758\pi$$
0.613508 + 0.789689i $$0.289758\pi$$
$$450$$ 0 0
$$451$$ −10.0000 −0.470882
$$452$$ −8.00000 −0.376288
$$453$$ 0 0
$$454$$ −4.00000 −0.187729
$$455$$ 6.00000 0.281284
$$456$$ 0 0
$$457$$ −20.0000 −0.935561 −0.467780 0.883845i $$-0.654946\pi$$
−0.467780 + 0.883845i $$0.654946\pi$$
$$458$$ −15.0000 −0.700904
$$459$$ 0 0
$$460$$ 3.00000 0.139876
$$461$$ −9.00000 −0.419172 −0.209586 0.977790i $$-0.567212\pi$$
−0.209586 + 0.977790i $$0.567212\pi$$
$$462$$ 0 0
$$463$$ −36.0000 −1.67306 −0.836531 0.547920i $$-0.815420\pi$$
−0.836531 + 0.547920i $$0.815420\pi$$
$$464$$ −1.00000 −0.0464238
$$465$$ 0 0
$$466$$ 24.0000 1.11178
$$467$$ −20.0000 −0.925490 −0.462745 0.886492i $$-0.653135\pi$$
−0.462745 + 0.886492i $$0.653135\pi$$
$$468$$ 0 0
$$469$$ 9.00000 0.415581
$$470$$ 7.00000 0.322886
$$471$$ 0 0
$$472$$ 42.0000 1.93321
$$473$$ −16.0000 −0.735681
$$474$$ 0 0
$$475$$ −8.00000 −0.367065
$$476$$ −12.0000 −0.550019
$$477$$ 0 0
$$478$$ −8.00000 −0.365911
$$479$$ 18.0000 0.822441 0.411220 0.911536i $$-0.365103\pi$$
0.411220 + 0.911536i $$0.365103\pi$$
$$480$$ 0 0
$$481$$ 8.00000 0.364769
$$482$$ 11.0000 0.501036
$$483$$ 0 0
$$484$$ 7.00000 0.318182
$$485$$ 2.00000 0.0908153
$$486$$ 0 0
$$487$$ −16.0000 −0.725029 −0.362515 0.931978i $$-0.618082\pi$$
−0.362515 + 0.931978i $$0.618082\pi$$
$$488$$ 21.0000 0.950625
$$489$$ 0 0
$$490$$ −2.00000 −0.0903508
$$491$$ −20.0000 −0.902587 −0.451294 0.892375i $$-0.649037\pi$$
−0.451294 + 0.892375i $$0.649037\pi$$
$$492$$ 0 0
$$493$$ −4.00000 −0.180151
$$494$$ −16.0000 −0.719874
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 6.00000 0.269137
$$498$$ 0 0
$$499$$ −32.0000 −1.43252 −0.716258 0.697835i $$-0.754147\pi$$
−0.716258 + 0.697835i $$0.754147\pi$$
$$500$$ −1.00000 −0.0447214
$$501$$ 0 0
$$502$$ 0 0
$$503$$ 7.00000 0.312115 0.156057 0.987748i $$-0.450122\pi$$
0.156057 + 0.987748i $$0.450122\pi$$
$$504$$ 0 0
$$505$$ 18.0000 0.800989
$$506$$ 6.00000 0.266733
$$507$$ 0 0
$$508$$ 5.00000 0.221839
$$509$$ −43.0000 −1.90594 −0.952971 0.303062i $$-0.901991\pi$$
−0.952971 + 0.303062i $$0.901991\pi$$
$$510$$ 0 0
$$511$$ −12.0000 −0.530849
$$512$$ 11.0000 0.486136
$$513$$ 0 0
$$514$$ −6.00000 −0.264649
$$515$$ 8.00000 0.352522
$$516$$ 0 0
$$517$$ −14.0000 −0.615719
$$518$$ −12.0000 −0.527250
$$519$$ 0 0
$$520$$ −6.00000 −0.263117
$$521$$ −11.0000 −0.481919 −0.240959 0.970535i $$-0.577462\pi$$
−0.240959 + 0.970535i $$0.577462\pi$$
$$522$$ 0 0
$$523$$ −29.0000 −1.26808 −0.634041 0.773300i $$-0.718605\pi$$
−0.634041 + 0.773300i $$0.718605\pi$$
$$524$$ −6.00000 −0.262111
$$525$$ 0 0
$$526$$ −16.0000 −0.697633
$$527$$ 0 0
$$528$$ 0 0
$$529$$ −14.0000 −0.608696
$$530$$ −2.00000 −0.0868744
$$531$$ 0 0
$$532$$ −24.0000 −1.04053
$$533$$ 10.0000 0.433148
$$534$$ 0 0
$$535$$ −3.00000 −0.129701
$$536$$ −9.00000 −0.388741
$$537$$ 0 0
$$538$$ 25.0000 1.07783
$$539$$ 4.00000 0.172292
$$540$$ 0 0
$$541$$ −39.0000 −1.67674 −0.838370 0.545101i $$-0.816491\pi$$
−0.838370 + 0.545101i $$0.816491\pi$$
$$542$$ 8.00000 0.343629
$$543$$ 0 0
$$544$$ 20.0000 0.857493
$$545$$ 5.00000 0.214176
$$546$$ 0 0
$$547$$ −29.0000 −1.23995 −0.619975 0.784621i $$-0.712857\pi$$
−0.619975 + 0.784621i $$0.712857\pi$$
$$548$$ 12.0000 0.512615
$$549$$ 0 0
$$550$$ −2.00000 −0.0852803
$$551$$ −8.00000 −0.340811
$$552$$ 0 0
$$553$$ 18.0000 0.765438
$$554$$ −12.0000 −0.509831
$$555$$ 0 0
$$556$$ 16.0000 0.678551
$$557$$ 30.0000 1.27114 0.635570 0.772043i $$-0.280765\pi$$
0.635570 + 0.772043i $$0.280765\pi$$
$$558$$ 0 0
$$559$$ 16.0000 0.676728
$$560$$ 3.00000 0.126773
$$561$$ 0 0
$$562$$ −15.0000 −0.632737
$$563$$ 21.0000 0.885044 0.442522 0.896758i $$-0.354084\pi$$
0.442522 + 0.896758i $$0.354084\pi$$
$$564$$ 0 0
$$565$$ 8.00000 0.336563
$$566$$ −21.0000 −0.882696
$$567$$ 0 0
$$568$$ −6.00000 −0.251754
$$569$$ 6.00000 0.251533 0.125767 0.992060i $$-0.459861\pi$$
0.125767 + 0.992060i $$0.459861\pi$$
$$570$$ 0 0
$$571$$ 32.0000 1.33916 0.669579 0.742741i $$-0.266474\pi$$
0.669579 + 0.742741i $$0.266474\pi$$
$$572$$ 4.00000 0.167248
$$573$$ 0 0
$$574$$ −15.0000 −0.626088
$$575$$ −3.00000 −0.125109
$$576$$ 0 0
$$577$$ −10.0000 −0.416305 −0.208153 0.978096i $$-0.566745\pi$$
−0.208153 + 0.978096i $$0.566745\pi$$
$$578$$ 1.00000 0.0415945
$$579$$ 0 0
$$580$$ −1.00000 −0.0415227
$$581$$ 27.0000 1.12015
$$582$$ 0 0
$$583$$ 4.00000 0.165663
$$584$$ 12.0000 0.496564
$$585$$ 0 0
$$586$$ 12.0000 0.495715
$$587$$ 33.0000 1.36206 0.681028 0.732257i $$-0.261533\pi$$
0.681028 + 0.732257i $$0.261533\pi$$
$$588$$ 0 0
$$589$$ 0 0
$$590$$ −14.0000 −0.576371
$$591$$ 0 0
$$592$$ 4.00000 0.164399
$$593$$ −20.0000 −0.821302 −0.410651 0.911793i $$-0.634698\pi$$
−0.410651 + 0.911793i $$0.634698\pi$$
$$594$$ 0 0
$$595$$ 12.0000 0.491952
$$596$$ 17.0000 0.696347
$$597$$ 0 0
$$598$$ −6.00000 −0.245358
$$599$$ 10.0000 0.408589 0.204294 0.978909i $$-0.434510\pi$$
0.204294 + 0.978909i $$0.434510\pi$$
$$600$$ 0 0
$$601$$ 2.00000 0.0815817 0.0407909 0.999168i $$-0.487012\pi$$
0.0407909 + 0.999168i $$0.487012\pi$$
$$602$$ −24.0000 −0.978167
$$603$$ 0 0
$$604$$ 2.00000 0.0813788
$$605$$ −7.00000 −0.284590
$$606$$ 0 0
$$607$$ −41.0000 −1.66414 −0.832069 0.554672i $$-0.812844\pi$$
−0.832069 + 0.554672i $$0.812844\pi$$
$$608$$ 40.0000 1.62221
$$609$$ 0 0
$$610$$ −7.00000 −0.283422
$$611$$ 14.0000 0.566379
$$612$$ 0 0
$$613$$ −44.0000 −1.77714 −0.888572 0.458738i $$-0.848302\pi$$
−0.888572 + 0.458738i $$0.848302\pi$$
$$614$$ −7.00000 −0.282497
$$615$$ 0 0
$$616$$ −18.0000 −0.725241
$$617$$ 36.0000 1.44931 0.724653 0.689114i $$-0.242000\pi$$
0.724653 + 0.689114i $$0.242000\pi$$
$$618$$ 0 0
$$619$$ −4.00000 −0.160774 −0.0803868 0.996764i $$-0.525616\pi$$
−0.0803868 + 0.996764i $$0.525616\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 0 0
$$623$$ −45.0000 −1.80289
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ −14.0000 −0.559553
$$627$$ 0 0
$$628$$ −14.0000 −0.558661
$$629$$ 16.0000 0.637962
$$630$$ 0 0
$$631$$ 16.0000 0.636950 0.318475 0.947931i $$-0.396829\pi$$
0.318475 + 0.947931i $$0.396829\pi$$
$$632$$ −18.0000 −0.716002
$$633$$ 0 0
$$634$$ −34.0000 −1.35031
$$635$$ −5.00000 −0.198419
$$636$$ 0 0
$$637$$ −4.00000 −0.158486
$$638$$ −2.00000 −0.0791808
$$639$$ 0 0
$$640$$ 3.00000 0.118585
$$641$$ −33.0000 −1.30342 −0.651711 0.758468i $$-0.725948\pi$$
−0.651711 + 0.758468i $$0.725948\pi$$
$$642$$ 0 0
$$643$$ −9.00000 −0.354925 −0.177463 0.984128i $$-0.556789\pi$$
−0.177463 + 0.984128i $$0.556789\pi$$
$$644$$ −9.00000 −0.354650
$$645$$ 0 0
$$646$$ −32.0000 −1.25902
$$647$$ 17.0000 0.668339 0.334169 0.942513i $$-0.391544\pi$$
0.334169 + 0.942513i $$0.391544\pi$$
$$648$$ 0 0
$$649$$ 28.0000 1.09910
$$650$$ 2.00000 0.0784465
$$651$$ 0 0
$$652$$ 4.00000 0.156652
$$653$$ 4.00000 0.156532 0.0782660 0.996933i $$-0.475062\pi$$
0.0782660 + 0.996933i $$0.475062\pi$$
$$654$$ 0 0
$$655$$ 6.00000 0.234439
$$656$$ 5.00000 0.195217
$$657$$ 0 0
$$658$$ −21.0000 −0.818665
$$659$$ 8.00000 0.311636 0.155818 0.987786i $$-0.450199\pi$$
0.155818 + 0.987786i $$0.450199\pi$$
$$660$$ 0 0
$$661$$ −14.0000 −0.544537 −0.272268 0.962221i $$-0.587774\pi$$
−0.272268 + 0.962221i $$0.587774\pi$$
$$662$$ 6.00000 0.233197
$$663$$ 0 0
$$664$$ −27.0000 −1.04780
$$665$$ 24.0000 0.930680
$$666$$ 0 0
$$667$$ −3.00000 −0.116160
$$668$$ −9.00000 −0.348220
$$669$$ 0 0
$$670$$ 3.00000 0.115900
$$671$$ 14.0000 0.540464
$$672$$ 0 0
$$673$$ 6.00000 0.231283 0.115642 0.993291i $$-0.463108\pi$$
0.115642 + 0.993291i $$0.463108\pi$$
$$674$$ 8.00000 0.308148
$$675$$ 0 0
$$676$$ 9.00000 0.346154
$$677$$ 42.0000 1.61419 0.807096 0.590421i $$-0.201038\pi$$
0.807096 + 0.590421i $$0.201038\pi$$
$$678$$ 0 0
$$679$$ −6.00000 −0.230259
$$680$$ −12.0000 −0.460179
$$681$$ 0 0
$$682$$ 0 0
$$683$$ −12.0000 −0.459167 −0.229584 0.973289i $$-0.573736\pi$$
−0.229584 + 0.973289i $$0.573736\pi$$
$$684$$ 0 0
$$685$$ −12.0000 −0.458496
$$686$$ −15.0000 −0.572703
$$687$$ 0 0
$$688$$ 8.00000 0.304997
$$689$$ −4.00000 −0.152388
$$690$$ 0 0
$$691$$ −14.0000 −0.532585 −0.266293 0.963892i $$-0.585799\pi$$
−0.266293 + 0.963892i $$0.585799\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 4.00000 0.151838
$$695$$ −16.0000 −0.606915
$$696$$ 0 0
$$697$$ 20.0000 0.757554
$$698$$ 5.00000 0.189253
$$699$$ 0 0
$$700$$ 3.00000 0.113389
$$701$$ −23.0000 −0.868698 −0.434349 0.900745i $$-0.643022\pi$$
−0.434349 + 0.900745i $$0.643022\pi$$
$$702$$ 0 0
$$703$$ 32.0000 1.20690
$$704$$ 14.0000 0.527645
$$705$$ 0 0
$$706$$ 24.0000 0.903252
$$707$$ −54.0000 −2.03088
$$708$$ 0 0
$$709$$ −41.0000 −1.53979 −0.769894 0.638172i $$-0.779691\pi$$
−0.769894 + 0.638172i $$0.779691\pi$$
$$710$$ 2.00000 0.0750587
$$711$$ 0 0
$$712$$ 45.0000 1.68645
$$713$$ 0 0
$$714$$ 0 0
$$715$$ −4.00000 −0.149592
$$716$$ −2.00000 −0.0747435
$$717$$ 0 0
$$718$$ 24.0000 0.895672
$$719$$ −6.00000 −0.223762 −0.111881 0.993722i $$-0.535688\pi$$
−0.111881 + 0.993722i $$0.535688\pi$$
$$720$$ 0 0
$$721$$ −24.0000 −0.893807
$$722$$ −45.0000 −1.67473
$$723$$ 0 0
$$724$$ 7.00000 0.260153
$$725$$ 1.00000 0.0371391
$$726$$ 0 0
$$727$$ 23.0000 0.853023 0.426511 0.904482i $$-0.359742\pi$$
0.426511 + 0.904482i $$0.359742\pi$$
$$728$$ 18.0000 0.667124
$$729$$ 0 0
$$730$$ −4.00000 −0.148047
$$731$$ 32.0000 1.18356
$$732$$ 0 0
$$733$$ −34.0000 −1.25582 −0.627909 0.778287i $$-0.716089\pi$$
−0.627909 + 0.778287i $$0.716089\pi$$
$$734$$ −24.0000 −0.885856
$$735$$ 0 0
$$736$$ 15.0000 0.552907
$$737$$ −6.00000 −0.221013
$$738$$ 0 0
$$739$$ −2.00000 −0.0735712 −0.0367856 0.999323i $$-0.511712\pi$$
−0.0367856 + 0.999323i $$0.511712\pi$$
$$740$$ 4.00000 0.147043
$$741$$ 0 0
$$742$$ 6.00000 0.220267
$$743$$ 29.0000 1.06391 0.531953 0.846774i $$-0.321458\pi$$
0.531953 + 0.846774i $$0.321458\pi$$
$$744$$ 0 0
$$745$$ −17.0000 −0.622832
$$746$$ −10.0000 −0.366126
$$747$$ 0 0
$$748$$ 8.00000 0.292509
$$749$$ 9.00000 0.328853
$$750$$ 0 0
$$751$$ 10.0000 0.364905 0.182453 0.983215i $$-0.441596\pi$$
0.182453 + 0.983215i $$0.441596\pi$$
$$752$$ 7.00000 0.255264
$$753$$ 0 0
$$754$$ 2.00000 0.0728357
$$755$$ −2.00000 −0.0727875
$$756$$ 0 0
$$757$$ −26.0000 −0.944986 −0.472493 0.881334i $$-0.656646\pi$$
−0.472493 + 0.881334i $$0.656646\pi$$
$$758$$ 26.0000 0.944363
$$759$$ 0 0
$$760$$ −24.0000 −0.870572
$$761$$ −15.0000 −0.543750 −0.271875 0.962333i $$-0.587644\pi$$
−0.271875 + 0.962333i $$0.587644\pi$$
$$762$$ 0 0
$$763$$ −15.0000 −0.543036
$$764$$ 8.00000 0.289430
$$765$$ 0 0
$$766$$ 36.0000 1.30073
$$767$$ −28.0000 −1.01102
$$768$$ 0 0
$$769$$ 5.00000 0.180305 0.0901523 0.995928i $$-0.471265\pi$$
0.0901523 + 0.995928i $$0.471265\pi$$
$$770$$ 6.00000 0.216225
$$771$$ 0 0
$$772$$ 10.0000 0.359908
$$773$$ 24.0000 0.863220 0.431610 0.902060i $$-0.357946\pi$$
0.431610 + 0.902060i $$0.357946\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 6.00000 0.215387
$$777$$ 0 0
$$778$$ 33.0000 1.18311
$$779$$ 40.0000 1.43315
$$780$$ 0 0
$$781$$ −4.00000 −0.143131
$$782$$ −12.0000 −0.429119
$$783$$ 0 0
$$784$$ −2.00000 −0.0714286
$$785$$ 14.0000 0.499681
$$786$$ 0 0
$$787$$ 28.0000 0.998092 0.499046 0.866575i $$-0.333684\pi$$
0.499046 + 0.866575i $$0.333684\pi$$
$$788$$ −12.0000 −0.427482
$$789$$ 0 0
$$790$$ 6.00000 0.213470
$$791$$ −24.0000 −0.853342
$$792$$ 0 0
$$793$$ −14.0000 −0.497155
$$794$$ −34.0000 −1.20661
$$795$$ 0 0
$$796$$ −4.00000 −0.141776
$$797$$ −26.0000 −0.920967 −0.460484 0.887668i $$-0.652324\pi$$
−0.460484 + 0.887668i $$0.652324\pi$$
$$798$$ 0 0
$$799$$ 28.0000 0.990569
$$800$$ −5.00000 −0.176777
$$801$$ 0 0
$$802$$ −18.0000 −0.635602
$$803$$ 8.00000 0.282314
$$804$$ 0 0
$$805$$ 9.00000 0.317208
$$806$$ 0 0
$$807$$ 0 0
$$808$$ 54.0000 1.89971
$$809$$ −26.0000 −0.914111 −0.457056 0.889438i $$-0.651096\pi$$
−0.457056 + 0.889438i $$0.651096\pi$$
$$810$$ 0 0
$$811$$ −42.0000 −1.47482 −0.737410 0.675446i $$-0.763951\pi$$
−0.737410 + 0.675446i $$0.763951\pi$$
$$812$$ 3.00000 0.105279
$$813$$ 0 0
$$814$$ 8.00000 0.280400
$$815$$ −4.00000 −0.140114
$$816$$ 0 0
$$817$$ 64.0000 2.23908
$$818$$ −14.0000 −0.489499
$$819$$ 0 0
$$820$$ 5.00000 0.174608
$$821$$ −15.0000 −0.523504 −0.261752 0.965135i $$-0.584300\pi$$
−0.261752 + 0.965135i $$0.584300\pi$$
$$822$$ 0 0
$$823$$ 53.0000 1.84746 0.923732 0.383040i $$-0.125123\pi$$
0.923732 + 0.383040i $$0.125123\pi$$
$$824$$ 24.0000 0.836080
$$825$$ 0 0
$$826$$ 42.0000 1.46137
$$827$$ −37.0000 −1.28662 −0.643308 0.765607i $$-0.722439\pi$$
−0.643308 + 0.765607i $$0.722439\pi$$
$$828$$ 0 0
$$829$$ −3.00000 −0.104194 −0.0520972 0.998642i $$-0.516591\pi$$
−0.0520972 + 0.998642i $$0.516591\pi$$
$$830$$ 9.00000 0.312395
$$831$$ 0 0
$$832$$ −14.0000 −0.485363
$$833$$ −8.00000 −0.277184
$$834$$ 0 0
$$835$$ 9.00000 0.311458
$$836$$ 16.0000 0.553372
$$837$$ 0 0
$$838$$ 26.0000 0.898155
$$839$$ 40.0000 1.38095 0.690477 0.723355i $$-0.257401\pi$$
0.690477 + 0.723355i $$0.257401\pi$$
$$840$$ 0 0
$$841$$ −28.0000 −0.965517
$$842$$ −34.0000 −1.17172
$$843$$ 0 0
$$844$$ 22.0000 0.757271
$$845$$ −9.00000 −0.309609
$$846$$ 0 0
$$847$$ 21.0000 0.721569
$$848$$ −2.00000 −0.0686803
$$849$$ 0 0
$$850$$ 4.00000 0.137199
$$851$$ 12.0000 0.411355
$$852$$ 0 0
$$853$$ −54.0000 −1.84892 −0.924462 0.381273i $$-0.875486\pi$$
−0.924462 + 0.381273i $$0.875486\pi$$
$$854$$ 21.0000 0.718605
$$855$$ 0 0
$$856$$ −9.00000 −0.307614
$$857$$ −10.0000 −0.341593 −0.170797 0.985306i $$-0.554634\pi$$
−0.170797 + 0.985306i $$0.554634\pi$$
$$858$$ 0 0
$$859$$ 22.0000 0.750630 0.375315 0.926897i $$-0.377534\pi$$
0.375315 + 0.926897i $$0.377534\pi$$
$$860$$ 8.00000 0.272798
$$861$$ 0 0
$$862$$ −30.0000 −1.02180
$$863$$ −17.0000 −0.578687 −0.289343 0.957225i $$-0.593437\pi$$
−0.289343 + 0.957225i $$0.593437\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ −28.0000 −0.951479
$$867$$ 0 0
$$868$$ 0 0
$$869$$ −12.0000 −0.407072
$$870$$ 0 0
$$871$$ 6.00000 0.203302
$$872$$ 15.0000 0.507964
$$873$$ 0 0
$$874$$ −24.0000 −0.811812
$$875$$ −3.00000 −0.101419
$$876$$ 0 0
$$877$$ −18.0000 −0.607817 −0.303908 0.952701i $$-0.598292\pi$$
−0.303908 + 0.952701i $$0.598292\pi$$
$$878$$ −28.0000 −0.944954
$$879$$ 0 0
$$880$$ −2.00000 −0.0674200
$$881$$ 35.0000 1.17918 0.589590 0.807703i $$-0.299289\pi$$
0.589590 + 0.807703i $$0.299289\pi$$
$$882$$ 0 0
$$883$$ 23.0000 0.774012 0.387006 0.922077i $$-0.373509\pi$$
0.387006 + 0.922077i $$0.373509\pi$$
$$884$$ −8.00000 −0.269069
$$885$$ 0 0
$$886$$ 15.0000 0.503935
$$887$$ −36.0000 −1.20876 −0.604381 0.796696i $$-0.706579\pi$$
−0.604381 + 0.796696i $$0.706579\pi$$
$$888$$ 0 0
$$889$$ 15.0000 0.503084
$$890$$ −15.0000 −0.502801
$$891$$ 0 0
$$892$$ 19.0000 0.636167
$$893$$ 56.0000 1.87397
$$894$$ 0 0
$$895$$ 2.00000 0.0668526
$$896$$ −9.00000 −0.300669
$$897$$ 0 0
$$898$$ −26.0000 −0.867631
$$899$$ 0 0
$$900$$ 0 0
$$901$$ −8.00000 −0.266519
$$902$$ 10.0000 0.332964
$$903$$ 0 0
$$904$$ 24.0000 0.798228
$$905$$ −7.00000 −0.232688
$$906$$ 0 0
$$907$$ −51.0000 −1.69343 −0.846714 0.532049i $$-0.821422\pi$$
−0.846714 + 0.532049i $$0.821422\pi$$
$$908$$ −4.00000 −0.132745
$$909$$ 0 0
$$910$$ −6.00000 −0.198898
$$911$$ 50.0000 1.65657 0.828287 0.560304i $$-0.189316\pi$$
0.828287 + 0.560304i $$0.189316\pi$$
$$912$$ 0 0
$$913$$ −18.0000 −0.595713
$$914$$ 20.0000 0.661541
$$915$$ 0 0
$$916$$ −15.0000 −0.495614
$$917$$ −18.0000 −0.594412
$$918$$ 0 0
$$919$$ 10.0000 0.329870 0.164935 0.986304i $$-0.447259\pi$$
0.164935 + 0.986304i $$0.447259\pi$$
$$920$$ −9.00000 −0.296721
$$921$$ 0 0
$$922$$ 9.00000 0.296399
$$923$$ 4.00000 0.131662
$$924$$ 0 0
$$925$$ −4.00000 −0.131519
$$926$$ 36.0000 1.18303
$$927$$ 0 0
$$928$$ −5.00000 −0.164133
$$929$$ −14.0000 −0.459325 −0.229663 0.973270i $$-0.573762\pi$$
−0.229663 + 0.973270i $$0.573762\pi$$
$$930$$ 0 0
$$931$$ −16.0000 −0.524379
$$932$$ 24.0000 0.786146
$$933$$ 0 0
$$934$$ 20.0000 0.654420
$$935$$ −8.00000 −0.261628
$$936$$ 0 0
$$937$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$938$$ −9.00000 −0.293860
$$939$$ 0 0
$$940$$ 7.00000 0.228315
$$941$$ 7.00000 0.228193 0.114097 0.993470i $$-0.463603\pi$$
0.114097 + 0.993470i $$0.463603\pi$$
$$942$$ 0 0
$$943$$ 15.0000 0.488467
$$944$$ −14.0000 −0.455661
$$945$$ 0 0
$$946$$ 16.0000 0.520205
$$947$$ −57.0000 −1.85225 −0.926126 0.377215i $$-0.876882\pi$$
−0.926126 + 0.377215i $$0.876882\pi$$
$$948$$ 0 0
$$949$$ −8.00000 −0.259691
$$950$$ 8.00000 0.259554
$$951$$ 0 0
$$952$$ 36.0000 1.16677
$$953$$ 26.0000 0.842223 0.421111 0.907009i $$-0.361640\pi$$
0.421111 + 0.907009i $$0.361640\pi$$
$$954$$ 0 0
$$955$$ −8.00000 −0.258874
$$956$$ −8.00000 −0.258738
$$957$$ 0 0
$$958$$ −18.0000 −0.581554
$$959$$ 36.0000 1.16250
$$960$$ 0 0
$$961$$ −31.0000 −1.00000
$$962$$ −8.00000 −0.257930
$$963$$ 0 0
$$964$$ 11.0000 0.354286
$$965$$ −10.0000 −0.321911
$$966$$ 0 0
$$967$$ 41.0000 1.31847 0.659236 0.751936i $$-0.270880\pi$$
0.659236 + 0.751936i $$0.270880\pi$$
$$968$$ −21.0000 −0.674966
$$969$$ 0 0
$$970$$ −2.00000 −0.0642161
$$971$$ 36.0000 1.15529 0.577647 0.816286i $$-0.303971\pi$$
0.577647 + 0.816286i $$0.303971\pi$$
$$972$$ 0 0
$$973$$ 48.0000 1.53881
$$974$$ 16.0000 0.512673
$$975$$ 0 0
$$976$$ −7.00000 −0.224065
$$977$$ 38.0000 1.21573 0.607864 0.794041i $$-0.292027\pi$$
0.607864 + 0.794041i $$0.292027\pi$$
$$978$$ 0 0
$$979$$ 30.0000 0.958804
$$980$$ −2.00000 −0.0638877
$$981$$ 0 0
$$982$$ 20.0000 0.638226
$$983$$ −3.00000 −0.0956851 −0.0478426 0.998855i $$-0.515235\pi$$
−0.0478426 + 0.998855i $$0.515235\pi$$
$$984$$ 0 0
$$985$$ 12.0000 0.382352
$$986$$ 4.00000 0.127386
$$987$$ 0 0
$$988$$ −16.0000 −0.509028
$$989$$ 24.0000 0.763156
$$990$$ 0 0
$$991$$ 26.0000 0.825917 0.412959 0.910750i $$-0.364495\pi$$
0.412959 + 0.910750i $$0.364495\pi$$
$$992$$ 0 0
$$993$$ 0 0
$$994$$ −6.00000 −0.190308
$$995$$ 4.00000 0.126809
$$996$$ 0 0
$$997$$ −18.0000 −0.570066 −0.285033 0.958518i $$-0.592005\pi$$
−0.285033 + 0.958518i $$0.592005\pi$$
$$998$$ 32.0000 1.01294
$$999$$ 0 0
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 405.2.a.b.1.1 1
3.2 odd 2 405.2.a.e.1.1 1
4.3 odd 2 6480.2.a.x.1.1 1
5.2 odd 4 2025.2.b.d.649.1 2
5.3 odd 4 2025.2.b.d.649.2 2
5.4 even 2 2025.2.a.e.1.1 1
9.2 odd 6 45.2.e.a.31.1 yes 2
9.4 even 3 135.2.e.a.46.1 2
9.5 odd 6 45.2.e.a.16.1 2
9.7 even 3 135.2.e.a.91.1 2
12.11 even 2 6480.2.a.k.1.1 1
15.2 even 4 2025.2.b.c.649.2 2
15.8 even 4 2025.2.b.c.649.1 2
15.14 odd 2 2025.2.a.b.1.1 1
36.7 odd 6 2160.2.q.a.1441.1 2
36.11 even 6 720.2.q.d.481.1 2
36.23 even 6 720.2.q.d.241.1 2
36.31 odd 6 2160.2.q.a.721.1 2
45.2 even 12 225.2.k.a.49.2 4
45.4 even 6 675.2.e.a.451.1 2
45.7 odd 12 675.2.k.a.199.1 4
45.13 odd 12 675.2.k.a.424.1 4
45.14 odd 6 225.2.e.a.151.1 2
45.22 odd 12 675.2.k.a.424.2 4
45.23 even 12 225.2.k.a.124.2 4
45.29 odd 6 225.2.e.a.76.1 2
45.32 even 12 225.2.k.a.124.1 4
45.34 even 6 675.2.e.a.226.1 2
45.38 even 12 225.2.k.a.49.1 4
45.43 odd 12 675.2.k.a.199.2 4

By twisted newform
Twist Min Dim Char Parity Ord Type
45.2.e.a.16.1 2 9.5 odd 6
45.2.e.a.31.1 yes 2 9.2 odd 6
135.2.e.a.46.1 2 9.4 even 3
135.2.e.a.91.1 2 9.7 even 3
225.2.e.a.76.1 2 45.29 odd 6
225.2.e.a.151.1 2 45.14 odd 6
225.2.k.a.49.1 4 45.38 even 12
225.2.k.a.49.2 4 45.2 even 12
225.2.k.a.124.1 4 45.32 even 12
225.2.k.a.124.2 4 45.23 even 12
405.2.a.b.1.1 1 1.1 even 1 trivial
405.2.a.e.1.1 1 3.2 odd 2
675.2.e.a.226.1 2 45.34 even 6
675.2.e.a.451.1 2 45.4 even 6
675.2.k.a.199.1 4 45.7 odd 12
675.2.k.a.199.2 4 45.43 odd 12
675.2.k.a.424.1 4 45.13 odd 12
675.2.k.a.424.2 4 45.22 odd 12
720.2.q.d.241.1 2 36.23 even 6
720.2.q.d.481.1 2 36.11 even 6
2025.2.a.b.1.1 1 15.14 odd 2
2025.2.a.e.1.1 1 5.4 even 2
2025.2.b.c.649.1 2 15.8 even 4
2025.2.b.c.649.2 2 15.2 even 4
2025.2.b.d.649.1 2 5.2 odd 4
2025.2.b.d.649.2 2 5.3 odd 4
2160.2.q.a.721.1 2 36.31 odd 6
2160.2.q.a.1441.1 2 36.7 odd 6
6480.2.a.k.1.1 1 12.11 even 2
6480.2.a.x.1.1 1 4.3 odd 2