# Properties

 Label 1225.2.a.b.1.1 Level $1225$ Weight $2$ Character 1225.1 Self dual yes Analytic conductor $9.782$ 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] = [1225,2,Mod(1,1225)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1225, 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("1225.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1225 = 5^{2} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1225.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: $$9.78167424761$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 35) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 1225.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-1.00000 q^{2} -1.00000 q^{3} -1.00000 q^{4} +1.00000 q^{6} +3.00000 q^{8} -2.00000 q^{9} +O(q^{10})$$ $$q-1.00000 q^{2} -1.00000 q^{3} -1.00000 q^{4} +1.00000 q^{6} +3.00000 q^{8} -2.00000 q^{9} +1.00000 q^{12} +2.00000 q^{13} -1.00000 q^{16} +2.00000 q^{17} +2.00000 q^{18} -6.00000 q^{19} +3.00000 q^{23} -3.00000 q^{24} -2.00000 q^{26} +5.00000 q^{27} +7.00000 q^{29} -2.00000 q^{31} -5.00000 q^{32} -2.00000 q^{34} +2.00000 q^{36} +8.00000 q^{37} +6.00000 q^{38} -2.00000 q^{39} -5.00000 q^{41} -7.00000 q^{43} -3.00000 q^{46} +1.00000 q^{48} -2.00000 q^{51} -2.00000 q^{52} -6.00000 q^{53} -5.00000 q^{54} +6.00000 q^{57} -7.00000 q^{58} -10.0000 q^{59} -7.00000 q^{61} +2.00000 q^{62} +7.00000 q^{64} +5.00000 q^{67} -2.00000 q^{68} -3.00000 q^{69} -2.00000 q^{71} -6.00000 q^{72} -6.00000 q^{73} -8.00000 q^{74} +6.00000 q^{76} +2.00000 q^{78} -2.00000 q^{79} +1.00000 q^{81} +5.00000 q^{82} -11.0000 q^{83} +7.00000 q^{86} -7.00000 q^{87} -9.00000 q^{89} -3.00000 q^{92} +2.00000 q^{93} +5.00000 q^{96} -16.0000 q^{97} +O(q^{100})$$

## 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 $$0.615027\pi$$
$$3$$ −1.00000 −0.577350 −0.288675 0.957427i $$-0.593215\pi$$
−0.288675 + 0.957427i $$0.593215\pi$$
$$4$$ −1.00000 −0.500000
$$5$$ 0 0
$$6$$ 1.00000 0.408248
$$7$$ 0 0
$$8$$ 3.00000 1.06066
$$9$$ −2.00000 −0.666667
$$10$$ 0 0
$$11$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$12$$ 1.00000 0.288675
$$13$$ 2.00000 0.554700 0.277350 0.960769i $$-0.410544\pi$$
0.277350 + 0.960769i $$0.410544\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ −1.00000 −0.250000
$$17$$ 2.00000 0.485071 0.242536 0.970143i $$-0.422021\pi$$
0.242536 + 0.970143i $$0.422021\pi$$
$$18$$ 2.00000 0.471405
$$19$$ −6.00000 −1.37649 −0.688247 0.725476i $$-0.741620\pi$$
−0.688247 + 0.725476i $$0.741620\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ 3.00000 0.625543 0.312772 0.949828i $$-0.398743\pi$$
0.312772 + 0.949828i $$0.398743\pi$$
$$24$$ −3.00000 −0.612372
$$25$$ 0 0
$$26$$ −2.00000 −0.392232
$$27$$ 5.00000 0.962250
$$28$$ 0 0
$$29$$ 7.00000 1.29987 0.649934 0.759991i $$-0.274797\pi$$
0.649934 + 0.759991i $$0.274797\pi$$
$$30$$ 0 0
$$31$$ −2.00000 −0.359211 −0.179605 0.983739i $$-0.557482\pi$$
−0.179605 + 0.983739i $$0.557482\pi$$
$$32$$ −5.00000 −0.883883
$$33$$ 0 0
$$34$$ −2.00000 −0.342997
$$35$$ 0 0
$$36$$ 2.00000 0.333333
$$37$$ 8.00000 1.31519 0.657596 0.753371i $$-0.271573\pi$$
0.657596 + 0.753371i $$0.271573\pi$$
$$38$$ 6.00000 0.973329
$$39$$ −2.00000 −0.320256
$$40$$ 0 0
$$41$$ −5.00000 −0.780869 −0.390434 0.920631i $$-0.627675\pi$$
−0.390434 + 0.920631i $$0.627675\pi$$
$$42$$ 0 0
$$43$$ −7.00000 −1.06749 −0.533745 0.845645i $$-0.679216\pi$$
−0.533745 + 0.845645i $$0.679216\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ −3.00000 −0.442326
$$47$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$48$$ 1.00000 0.144338
$$49$$ 0 0
$$50$$ 0 0
$$51$$ −2.00000 −0.280056
$$52$$ −2.00000 −0.277350
$$53$$ −6.00000 −0.824163 −0.412082 0.911147i $$-0.635198\pi$$
−0.412082 + 0.911147i $$0.635198\pi$$
$$54$$ −5.00000 −0.680414
$$55$$ 0 0
$$56$$ 0 0
$$57$$ 6.00000 0.794719
$$58$$ −7.00000 −0.919145
$$59$$ −10.0000 −1.30189 −0.650945 0.759125i $$-0.725627\pi$$
−0.650945 + 0.759125i $$0.725627\pi$$
$$60$$ 0 0
$$61$$ −7.00000 −0.896258 −0.448129 0.893969i $$-0.647910\pi$$
−0.448129 + 0.893969i $$0.647910\pi$$
$$62$$ 2.00000 0.254000
$$63$$ 0 0
$$64$$ 7.00000 0.875000
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 5.00000 0.610847 0.305424 0.952217i $$-0.401202\pi$$
0.305424 + 0.952217i $$0.401202\pi$$
$$68$$ −2.00000 −0.242536
$$69$$ −3.00000 −0.361158
$$70$$ 0 0
$$71$$ −2.00000 −0.237356 −0.118678 0.992933i $$-0.537866\pi$$
−0.118678 + 0.992933i $$0.537866\pi$$
$$72$$ −6.00000 −0.707107
$$73$$ −6.00000 −0.702247 −0.351123 0.936329i $$-0.614200\pi$$
−0.351123 + 0.936329i $$0.614200\pi$$
$$74$$ −8.00000 −0.929981
$$75$$ 0 0
$$76$$ 6.00000 0.688247
$$77$$ 0 0
$$78$$ 2.00000 0.226455
$$79$$ −2.00000 −0.225018 −0.112509 0.993651i $$-0.535889\pi$$
−0.112509 + 0.993651i $$0.535889\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 5.00000 0.552158
$$83$$ −11.0000 −1.20741 −0.603703 0.797209i $$-0.706309\pi$$
−0.603703 + 0.797209i $$0.706309\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 7.00000 0.754829
$$87$$ −7.00000 −0.750479
$$88$$ 0 0
$$89$$ −9.00000 −0.953998 −0.476999 0.878904i $$-0.658275\pi$$
−0.476999 + 0.878904i $$0.658275\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ −3.00000 −0.312772
$$93$$ 2.00000 0.207390
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 5.00000 0.510310
$$97$$ −16.0000 −1.62455 −0.812277 0.583272i $$-0.801772\pi$$
−0.812277 + 0.583272i $$0.801772\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 0 0
$$101$$ −9.00000 −0.895533 −0.447767 0.894150i $$-0.647781\pi$$
−0.447767 + 0.894150i $$0.647781\pi$$
$$102$$ 2.00000 0.198030
$$103$$ 7.00000 0.689730 0.344865 0.938652i $$-0.387925\pi$$
0.344865 + 0.938652i $$0.387925\pi$$
$$104$$ 6.00000 0.588348
$$105$$ 0 0
$$106$$ 6.00000 0.582772
$$107$$ −11.0000 −1.06341 −0.531705 0.846930i $$-0.678449\pi$$
−0.531705 + 0.846930i $$0.678449\pi$$
$$108$$ −5.00000 −0.481125
$$109$$ 5.00000 0.478913 0.239457 0.970907i $$-0.423031\pi$$
0.239457 + 0.970907i $$0.423031\pi$$
$$110$$ 0 0
$$111$$ −8.00000 −0.759326
$$112$$ 0 0
$$113$$ 6.00000 0.564433 0.282216 0.959351i $$-0.408930\pi$$
0.282216 + 0.959351i $$0.408930\pi$$
$$114$$ −6.00000 −0.561951
$$115$$ 0 0
$$116$$ −7.00000 −0.649934
$$117$$ −4.00000 −0.369800
$$118$$ 10.0000 0.920575
$$119$$ 0 0
$$120$$ 0 0
$$121$$ −11.0000 −1.00000
$$122$$ 7.00000 0.633750
$$123$$ 5.00000 0.450835
$$124$$ 2.00000 0.179605
$$125$$ 0 0
$$126$$ 0 0
$$127$$ −16.0000 −1.41977 −0.709885 0.704317i $$-0.751253\pi$$
−0.709885 + 0.704317i $$0.751253\pi$$
$$128$$ 3.00000 0.265165
$$129$$ 7.00000 0.616316
$$130$$ 0 0
$$131$$ −4.00000 −0.349482 −0.174741 0.984614i $$-0.555909\pi$$
−0.174741 + 0.984614i $$0.555909\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ −5.00000 −0.431934
$$135$$ 0 0
$$136$$ 6.00000 0.514496
$$137$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$138$$ 3.00000 0.255377
$$139$$ −8.00000 −0.678551 −0.339276 0.940687i $$-0.610182\pi$$
−0.339276 + 0.940687i $$0.610182\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 2.00000 0.167836
$$143$$ 0 0
$$144$$ 2.00000 0.166667
$$145$$ 0 0
$$146$$ 6.00000 0.496564
$$147$$ 0 0
$$148$$ −8.00000 −0.657596
$$149$$ 1.00000 0.0819232 0.0409616 0.999161i $$-0.486958\pi$$
0.0409616 + 0.999161i $$0.486958\pi$$
$$150$$ 0 0
$$151$$ 14.0000 1.13930 0.569652 0.821886i $$-0.307078\pi$$
0.569652 + 0.821886i $$0.307078\pi$$
$$152$$ −18.0000 −1.45999
$$153$$ −4.00000 −0.323381
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 2.00000 0.160128
$$157$$ 12.0000 0.957704 0.478852 0.877896i $$-0.341053\pi$$
0.478852 + 0.877896i $$0.341053\pi$$
$$158$$ 2.00000 0.159111
$$159$$ 6.00000 0.475831
$$160$$ 0 0
$$161$$ 0 0
$$162$$ −1.00000 −0.0785674
$$163$$ 4.00000 0.313304 0.156652 0.987654i $$-0.449930\pi$$
0.156652 + 0.987654i $$0.449930\pi$$
$$164$$ 5.00000 0.390434
$$165$$ 0 0
$$166$$ 11.0000 0.853766
$$167$$ −3.00000 −0.232147 −0.116073 0.993241i $$-0.537031\pi$$
−0.116073 + 0.993241i $$0.537031\pi$$
$$168$$ 0 0
$$169$$ −9.00000 −0.692308
$$170$$ 0 0
$$171$$ 12.0000 0.917663
$$172$$ 7.00000 0.533745
$$173$$ −12.0000 −0.912343 −0.456172 0.889892i $$-0.650780\pi$$
−0.456172 + 0.889892i $$0.650780\pi$$
$$174$$ 7.00000 0.530669
$$175$$ 0 0
$$176$$ 0 0
$$177$$ 10.0000 0.751646
$$178$$ 9.00000 0.674579
$$179$$ −2.00000 −0.149487 −0.0747435 0.997203i $$-0.523814\pi$$
−0.0747435 + 0.997203i $$0.523814\pi$$
$$180$$ 0 0
$$181$$ 3.00000 0.222988 0.111494 0.993765i $$-0.464436\pi$$
0.111494 + 0.993765i $$0.464436\pi$$
$$182$$ 0 0
$$183$$ 7.00000 0.517455
$$184$$ 9.00000 0.663489
$$185$$ 0 0
$$186$$ −2.00000 −0.146647
$$187$$ 0 0
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 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
$$193$$ 4.00000 0.287926 0.143963 0.989583i $$-0.454015\pi$$
0.143963 + 0.989583i $$0.454015\pi$$
$$194$$ 16.0000 1.14873
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 8.00000 0.569976 0.284988 0.958531i $$-0.408010\pi$$
0.284988 + 0.958531i $$0.408010\pi$$
$$198$$ 0 0
$$199$$ 4.00000 0.283552 0.141776 0.989899i $$-0.454719\pi$$
0.141776 + 0.989899i $$0.454719\pi$$
$$200$$ 0 0
$$201$$ −5.00000 −0.352673
$$202$$ 9.00000 0.633238
$$203$$ 0 0
$$204$$ 2.00000 0.140028
$$205$$ 0 0
$$206$$ −7.00000 −0.487713
$$207$$ −6.00000 −0.417029
$$208$$ −2.00000 −0.138675
$$209$$ 0 0
$$210$$ 0 0
$$211$$ −10.0000 −0.688428 −0.344214 0.938891i $$-0.611855\pi$$
−0.344214 + 0.938891i $$0.611855\pi$$
$$212$$ 6.00000 0.412082
$$213$$ 2.00000 0.137038
$$214$$ 11.0000 0.751945
$$215$$ 0 0
$$216$$ 15.0000 1.02062
$$217$$ 0 0
$$218$$ −5.00000 −0.338643
$$219$$ 6.00000 0.405442
$$220$$ 0 0
$$221$$ 4.00000 0.269069
$$222$$ 8.00000 0.536925
$$223$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ −6.00000 −0.399114
$$227$$ −20.0000 −1.32745 −0.663723 0.747978i $$-0.731025\pi$$
−0.663723 + 0.747978i $$0.731025\pi$$
$$228$$ −6.00000 −0.397360
$$229$$ 22.0000 1.45380 0.726900 0.686743i $$-0.240960\pi$$
0.726900 + 0.686743i $$0.240960\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 21.0000 1.37872
$$233$$ −14.0000 −0.917170 −0.458585 0.888650i $$-0.651644\pi$$
−0.458585 + 0.888650i $$0.651644\pi$$
$$234$$ 4.00000 0.261488
$$235$$ 0 0
$$236$$ 10.0000 0.650945
$$237$$ 2.00000 0.129914
$$238$$ 0 0
$$239$$ −18.0000 −1.16432 −0.582162 0.813073i $$-0.697793\pi$$
−0.582162 + 0.813073i $$0.697793\pi$$
$$240$$ 0 0
$$241$$ 14.0000 0.901819 0.450910 0.892570i $$-0.351100\pi$$
0.450910 + 0.892570i $$0.351100\pi$$
$$242$$ 11.0000 0.707107
$$243$$ −16.0000 −1.02640
$$244$$ 7.00000 0.448129
$$245$$ 0 0
$$246$$ −5.00000 −0.318788
$$247$$ −12.0000 −0.763542
$$248$$ −6.00000 −0.381000
$$249$$ 11.0000 0.697097
$$250$$ 0 0
$$251$$ −30.0000 −1.89358 −0.946792 0.321847i $$-0.895696\pi$$
−0.946792 + 0.321847i $$0.895696\pi$$
$$252$$ 0 0
$$253$$ 0 0
$$254$$ 16.0000 1.00393
$$255$$ 0 0
$$256$$ −17.0000 −1.06250
$$257$$ 20.0000 1.24757 0.623783 0.781598i $$-0.285595\pi$$
0.623783 + 0.781598i $$0.285595\pi$$
$$258$$ −7.00000 −0.435801
$$259$$ 0 0
$$260$$ 0 0
$$261$$ −14.0000 −0.866578
$$262$$ 4.00000 0.247121
$$263$$ 9.00000 0.554964 0.277482 0.960731i $$-0.410500\pi$$
0.277482 + 0.960731i $$0.410500\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 0 0
$$267$$ 9.00000 0.550791
$$268$$ −5.00000 −0.305424
$$269$$ 11.0000 0.670682 0.335341 0.942097i $$-0.391148\pi$$
0.335341 + 0.942097i $$0.391148\pi$$
$$270$$ 0 0
$$271$$ 2.00000 0.121491 0.0607457 0.998153i $$-0.480652\pi$$
0.0607457 + 0.998153i $$0.480652\pi$$
$$272$$ −2.00000 −0.121268
$$273$$ 0 0
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 3.00000 0.180579
$$277$$ −10.0000 −0.600842 −0.300421 0.953807i $$-0.597127\pi$$
−0.300421 + 0.953807i $$0.597127\pi$$
$$278$$ 8.00000 0.479808
$$279$$ 4.00000 0.239474
$$280$$ 0 0
$$281$$ −14.0000 −0.835170 −0.417585 0.908638i $$-0.637123\pi$$
−0.417585 + 0.908638i $$0.637123\pi$$
$$282$$ 0 0
$$283$$ 16.0000 0.951101 0.475551 0.879688i $$-0.342249\pi$$
0.475551 + 0.879688i $$0.342249\pi$$
$$284$$ 2.00000 0.118678
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 0 0
$$288$$ 10.0000 0.589256
$$289$$ −13.0000 −0.764706
$$290$$ 0 0
$$291$$ 16.0000 0.937937
$$292$$ 6.00000 0.351123
$$293$$ 24.0000 1.40209 0.701047 0.713115i $$-0.252716\pi$$
0.701047 + 0.713115i $$0.252716\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 24.0000 1.39497
$$297$$ 0 0
$$298$$ −1.00000 −0.0579284
$$299$$ 6.00000 0.346989
$$300$$ 0 0
$$301$$ 0 0
$$302$$ −14.0000 −0.805609
$$303$$ 9.00000 0.517036
$$304$$ 6.00000 0.344124
$$305$$ 0 0
$$306$$ 4.00000 0.228665
$$307$$ 23.0000 1.31268 0.656340 0.754466i $$-0.272104\pi$$
0.656340 + 0.754466i $$0.272104\pi$$
$$308$$ 0 0
$$309$$ −7.00000 −0.398216
$$310$$ 0 0
$$311$$ −30.0000 −1.70114 −0.850572 0.525859i $$-0.823744\pi$$
−0.850572 + 0.525859i $$0.823744\pi$$
$$312$$ −6.00000 −0.339683
$$313$$ 24.0000 1.35656 0.678280 0.734803i $$-0.262726\pi$$
0.678280 + 0.734803i $$0.262726\pi$$
$$314$$ −12.0000 −0.677199
$$315$$ 0 0
$$316$$ 2.00000 0.112509
$$317$$ 6.00000 0.336994 0.168497 0.985702i $$-0.446109\pi$$
0.168497 + 0.985702i $$0.446109\pi$$
$$318$$ −6.00000 −0.336463
$$319$$ 0 0
$$320$$ 0 0
$$321$$ 11.0000 0.613960
$$322$$ 0 0
$$323$$ −12.0000 −0.667698
$$324$$ −1.00000 −0.0555556
$$325$$ 0 0
$$326$$ −4.00000 −0.221540
$$327$$ −5.00000 −0.276501
$$328$$ −15.0000 −0.828236
$$329$$ 0 0
$$330$$ 0 0
$$331$$ 6.00000 0.329790 0.164895 0.986311i $$-0.447272\pi$$
0.164895 + 0.986311i $$0.447272\pi$$
$$332$$ 11.0000 0.603703
$$333$$ −16.0000 −0.876795
$$334$$ 3.00000 0.164153
$$335$$ 0 0
$$336$$ 0 0
$$337$$ −18.0000 −0.980522 −0.490261 0.871576i $$-0.663099\pi$$
−0.490261 + 0.871576i $$0.663099\pi$$
$$338$$ 9.00000 0.489535
$$339$$ −6.00000 −0.325875
$$340$$ 0 0
$$341$$ 0 0
$$342$$ −12.0000 −0.648886
$$343$$ 0 0
$$344$$ −21.0000 −1.13224
$$345$$ 0 0
$$346$$ 12.0000 0.645124
$$347$$ −15.0000 −0.805242 −0.402621 0.915367i $$-0.631901\pi$$
−0.402621 + 0.915367i $$0.631901\pi$$
$$348$$ 7.00000 0.375239
$$349$$ 17.0000 0.909989 0.454995 0.890494i $$-0.349641\pi$$
0.454995 + 0.890494i $$0.349641\pi$$
$$350$$ 0 0
$$351$$ 10.0000 0.533761
$$352$$ 0 0
$$353$$ −26.0000 −1.38384 −0.691920 0.721974i $$-0.743235\pi$$
−0.691920 + 0.721974i $$0.743235\pi$$
$$354$$ −10.0000 −0.531494
$$355$$ 0 0
$$356$$ 9.00000 0.476999
$$357$$ 0 0
$$358$$ 2.00000 0.105703
$$359$$ 10.0000 0.527780 0.263890 0.964553i $$-0.414994\pi$$
0.263890 + 0.964553i $$0.414994\pi$$
$$360$$ 0 0
$$361$$ 17.0000 0.894737
$$362$$ −3.00000 −0.157676
$$363$$ 11.0000 0.577350
$$364$$ 0 0
$$365$$ 0 0
$$366$$ −7.00000 −0.365896
$$367$$ 27.0000 1.40939 0.704694 0.709511i $$-0.251084\pi$$
0.704694 + 0.709511i $$0.251084\pi$$
$$368$$ −3.00000 −0.156386
$$369$$ 10.0000 0.520579
$$370$$ 0 0
$$371$$ 0 0
$$372$$ −2.00000 −0.103695
$$373$$ 24.0000 1.24267 0.621336 0.783544i $$-0.286590\pi$$
0.621336 + 0.783544i $$0.286590\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 14.0000 0.721037
$$378$$ 0 0
$$379$$ −10.0000 −0.513665 −0.256833 0.966456i $$-0.582679\pi$$
−0.256833 + 0.966456i $$0.582679\pi$$
$$380$$ 0 0
$$381$$ 16.0000 0.819705
$$382$$ −12.0000 −0.613973
$$383$$ 5.00000 0.255488 0.127744 0.991807i $$-0.459226\pi$$
0.127744 + 0.991807i $$0.459226\pi$$
$$384$$ −3.00000 −0.153093
$$385$$ 0 0
$$386$$ −4.00000 −0.203595
$$387$$ 14.0000 0.711660
$$388$$ 16.0000 0.812277
$$389$$ 14.0000 0.709828 0.354914 0.934899i $$-0.384510\pi$$
0.354914 + 0.934899i $$0.384510\pi$$
$$390$$ 0 0
$$391$$ 6.00000 0.303433
$$392$$ 0 0
$$393$$ 4.00000 0.201773
$$394$$ −8.00000 −0.403034
$$395$$ 0 0
$$396$$ 0 0
$$397$$ −16.0000 −0.803017 −0.401508 0.915855i $$-0.631514\pi$$
−0.401508 + 0.915855i $$0.631514\pi$$
$$398$$ −4.00000 −0.200502
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 3.00000 0.149813 0.0749064 0.997191i $$-0.476134\pi$$
0.0749064 + 0.997191i $$0.476134\pi$$
$$402$$ 5.00000 0.249377
$$403$$ −4.00000 −0.199254
$$404$$ 9.00000 0.447767
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 0 0
$$408$$ −6.00000 −0.297044
$$409$$ 25.0000 1.23617 0.618085 0.786111i $$-0.287909\pi$$
0.618085 + 0.786111i $$0.287909\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ −7.00000 −0.344865
$$413$$ 0 0
$$414$$ 6.00000 0.294884
$$415$$ 0 0
$$416$$ −10.0000 −0.490290
$$417$$ 8.00000 0.391762
$$418$$ 0 0
$$419$$ −24.0000 −1.17248 −0.586238 0.810139i $$-0.699392\pi$$
−0.586238 + 0.810139i $$0.699392\pi$$
$$420$$ 0 0
$$421$$ 15.0000 0.731055 0.365528 0.930800i $$-0.380889\pi$$
0.365528 + 0.930800i $$0.380889\pi$$
$$422$$ 10.0000 0.486792
$$423$$ 0 0
$$424$$ −18.0000 −0.874157
$$425$$ 0 0
$$426$$ −2.00000 −0.0969003
$$427$$ 0 0
$$428$$ 11.0000 0.531705
$$429$$ 0 0
$$430$$ 0 0
$$431$$ −32.0000 −1.54139 −0.770693 0.637207i $$-0.780090\pi$$
−0.770693 + 0.637207i $$0.780090\pi$$
$$432$$ −5.00000 −0.240563
$$433$$ −2.00000 −0.0961139 −0.0480569 0.998845i $$-0.515303\pi$$
−0.0480569 + 0.998845i $$0.515303\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ −5.00000 −0.239457
$$437$$ −18.0000 −0.861057
$$438$$ −6.00000 −0.286691
$$439$$ −24.0000 −1.14546 −0.572729 0.819745i $$-0.694115\pi$$
−0.572729 + 0.819745i $$0.694115\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ −4.00000 −0.190261
$$443$$ −31.0000 −1.47285 −0.736427 0.676517i $$-0.763489\pi$$
−0.736427 + 0.676517i $$0.763489\pi$$
$$444$$ 8.00000 0.379663
$$445$$ 0 0
$$446$$ 0 0
$$447$$ −1.00000 −0.0472984
$$448$$ 0 0
$$449$$ 31.0000 1.46298 0.731490 0.681852i $$-0.238825\pi$$
0.731490 + 0.681852i $$0.238825\pi$$
$$450$$ 0 0
$$451$$ 0 0
$$452$$ −6.00000 −0.282216
$$453$$ −14.0000 −0.657777
$$454$$ 20.0000 0.938647
$$455$$ 0 0
$$456$$ 18.0000 0.842927
$$457$$ −32.0000 −1.49690 −0.748448 0.663193i $$-0.769201\pi$$
−0.748448 + 0.663193i $$0.769201\pi$$
$$458$$ −22.0000 −1.02799
$$459$$ 10.0000 0.466760
$$460$$ 0 0
$$461$$ −18.0000 −0.838344 −0.419172 0.907907i $$-0.637680\pi$$
−0.419172 + 0.907907i $$0.637680\pi$$
$$462$$ 0 0
$$463$$ −3.00000 −0.139422 −0.0697109 0.997567i $$-0.522208\pi$$
−0.0697109 + 0.997567i $$0.522208\pi$$
$$464$$ −7.00000 −0.324967
$$465$$ 0 0
$$466$$ 14.0000 0.648537
$$467$$ 3.00000 0.138823 0.0694117 0.997588i $$-0.477888\pi$$
0.0694117 + 0.997588i $$0.477888\pi$$
$$468$$ 4.00000 0.184900
$$469$$ 0 0
$$470$$ 0 0
$$471$$ −12.0000 −0.552931
$$472$$ −30.0000 −1.38086
$$473$$ 0 0
$$474$$ −2.00000 −0.0918630
$$475$$ 0 0
$$476$$ 0 0
$$477$$ 12.0000 0.549442
$$478$$ 18.0000 0.823301
$$479$$ −42.0000 −1.91903 −0.959514 0.281659i $$-0.909115\pi$$
−0.959514 + 0.281659i $$0.909115\pi$$
$$480$$ 0 0
$$481$$ 16.0000 0.729537
$$482$$ −14.0000 −0.637683
$$483$$ 0 0
$$484$$ 11.0000 0.500000
$$485$$ 0 0
$$486$$ 16.0000 0.725775
$$487$$ 12.0000 0.543772 0.271886 0.962329i $$-0.412353\pi$$
0.271886 + 0.962329i $$0.412353\pi$$
$$488$$ −21.0000 −0.950625
$$489$$ −4.00000 −0.180886
$$490$$ 0 0
$$491$$ −26.0000 −1.17336 −0.586682 0.809818i $$-0.699566\pi$$
−0.586682 + 0.809818i $$0.699566\pi$$
$$492$$ −5.00000 −0.225417
$$493$$ 14.0000 0.630528
$$494$$ 12.0000 0.539906
$$495$$ 0 0
$$496$$ 2.00000 0.0898027
$$497$$ 0 0
$$498$$ −11.0000 −0.492922
$$499$$ 16.0000 0.716258 0.358129 0.933672i $$-0.383415\pi$$
0.358129 + 0.933672i $$0.383415\pi$$
$$500$$ 0 0
$$501$$ 3.00000 0.134030
$$502$$ 30.0000 1.33897
$$503$$ −15.0000 −0.668817 −0.334408 0.942428i $$-0.608537\pi$$
−0.334408 + 0.942428i $$0.608537\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 0 0
$$507$$ 9.00000 0.399704
$$508$$ 16.0000 0.709885
$$509$$ −27.0000 −1.19675 −0.598377 0.801215i $$-0.704187\pi$$
−0.598377 + 0.801215i $$0.704187\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ 11.0000 0.486136
$$513$$ −30.0000 −1.32453
$$514$$ −20.0000 −0.882162
$$515$$ 0 0
$$516$$ −7.00000 −0.308158
$$517$$ 0 0
$$518$$ 0 0
$$519$$ 12.0000 0.526742
$$520$$ 0 0
$$521$$ −14.0000 −0.613351 −0.306676 0.951814i $$-0.599217\pi$$
−0.306676 + 0.951814i $$0.599217\pi$$
$$522$$ 14.0000 0.612763
$$523$$ 4.00000 0.174908 0.0874539 0.996169i $$-0.472127\pi$$
0.0874539 + 0.996169i $$0.472127\pi$$
$$524$$ 4.00000 0.174741
$$525$$ 0 0
$$526$$ −9.00000 −0.392419
$$527$$ −4.00000 −0.174243
$$528$$ 0 0
$$529$$ −14.0000 −0.608696
$$530$$ 0 0
$$531$$ 20.0000 0.867926
$$532$$ 0 0
$$533$$ −10.0000 −0.433148
$$534$$ −9.00000 −0.389468
$$535$$ 0 0
$$536$$ 15.0000 0.647901
$$537$$ 2.00000 0.0863064
$$538$$ −11.0000 −0.474244
$$539$$ 0 0
$$540$$ 0 0
$$541$$ −3.00000 −0.128980 −0.0644900 0.997918i $$-0.520542\pi$$
−0.0644900 + 0.997918i $$0.520542\pi$$
$$542$$ −2.00000 −0.0859074
$$543$$ −3.00000 −0.128742
$$544$$ −10.0000 −0.428746
$$545$$ 0 0
$$546$$ 0 0
$$547$$ 17.0000 0.726868 0.363434 0.931620i $$-0.381604\pi$$
0.363434 + 0.931620i $$0.381604\pi$$
$$548$$ 0 0
$$549$$ 14.0000 0.597505
$$550$$ 0 0
$$551$$ −42.0000 −1.78926
$$552$$ −9.00000 −0.383065
$$553$$ 0 0
$$554$$ 10.0000 0.424859
$$555$$ 0 0
$$556$$ 8.00000 0.339276
$$557$$ 44.0000 1.86434 0.932170 0.362021i $$-0.117913\pi$$
0.932170 + 0.362021i $$0.117913\pi$$
$$558$$ −4.00000 −0.169334
$$559$$ −14.0000 −0.592137
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 14.0000 0.590554
$$563$$ 33.0000 1.39078 0.695392 0.718631i $$-0.255231\pi$$
0.695392 + 0.718631i $$0.255231\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ −16.0000 −0.672530
$$567$$ 0 0
$$568$$ −6.00000 −0.251754
$$569$$ −18.0000 −0.754599 −0.377300 0.926091i $$-0.623147\pi$$
−0.377300 + 0.926091i $$0.623147\pi$$
$$570$$ 0 0
$$571$$ −18.0000 −0.753277 −0.376638 0.926360i $$-0.622920\pi$$
−0.376638 + 0.926360i $$0.622920\pi$$
$$572$$ 0 0
$$573$$ −12.0000 −0.501307
$$574$$ 0 0
$$575$$ 0 0
$$576$$ −14.0000 −0.583333
$$577$$ −32.0000 −1.33218 −0.666089 0.745873i $$-0.732033\pi$$
−0.666089 + 0.745873i $$0.732033\pi$$
$$578$$ 13.0000 0.540729
$$579$$ −4.00000 −0.166234
$$580$$ 0 0
$$581$$ 0 0
$$582$$ −16.0000 −0.663221
$$583$$ 0 0
$$584$$ −18.0000 −0.744845
$$585$$ 0 0
$$586$$ −24.0000 −0.991431
$$587$$ −36.0000 −1.48588 −0.742940 0.669359i $$-0.766569\pi$$
−0.742940 + 0.669359i $$0.766569\pi$$
$$588$$ 0 0
$$589$$ 12.0000 0.494451
$$590$$ 0 0
$$591$$ −8.00000 −0.329076
$$592$$ −8.00000 −0.328798
$$593$$ 22.0000 0.903432 0.451716 0.892162i $$-0.350812\pi$$
0.451716 + 0.892162i $$0.350812\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ −1.00000 −0.0409616
$$597$$ −4.00000 −0.163709
$$598$$ −6.00000 −0.245358
$$599$$ −44.0000 −1.79779 −0.898896 0.438163i $$-0.855629\pi$$
−0.898896 + 0.438163i $$0.855629\pi$$
$$600$$ 0 0
$$601$$ −10.0000 −0.407909 −0.203954 0.978980i $$-0.565379\pi$$
−0.203954 + 0.978980i $$0.565379\pi$$
$$602$$ 0 0
$$603$$ −10.0000 −0.407231
$$604$$ −14.0000 −0.569652
$$605$$ 0 0
$$606$$ −9.00000 −0.365600
$$607$$ −33.0000 −1.33943 −0.669714 0.742619i $$-0.733583\pi$$
−0.669714 + 0.742619i $$0.733583\pi$$
$$608$$ 30.0000 1.21666
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 0 0
$$612$$ 4.00000 0.161690
$$613$$ 34.0000 1.37325 0.686624 0.727013i $$-0.259092\pi$$
0.686624 + 0.727013i $$0.259092\pi$$
$$614$$ −23.0000 −0.928204
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 4.00000 0.161034 0.0805170 0.996753i $$-0.474343\pi$$
0.0805170 + 0.996753i $$0.474343\pi$$
$$618$$ 7.00000 0.281581
$$619$$ −28.0000 −1.12542 −0.562708 0.826656i $$-0.690240\pi$$
−0.562708 + 0.826656i $$0.690240\pi$$
$$620$$ 0 0
$$621$$ 15.0000 0.601929
$$622$$ 30.0000 1.20289
$$623$$ 0 0
$$624$$ 2.00000 0.0800641
$$625$$ 0 0
$$626$$ −24.0000 −0.959233
$$627$$ 0 0
$$628$$ −12.0000 −0.478852
$$629$$ 16.0000 0.637962
$$630$$ 0 0
$$631$$ 40.0000 1.59237 0.796187 0.605050i $$-0.206847\pi$$
0.796187 + 0.605050i $$0.206847\pi$$
$$632$$ −6.00000 −0.238667
$$633$$ 10.0000 0.397464
$$634$$ −6.00000 −0.238290
$$635$$ 0 0
$$636$$ −6.00000 −0.237915
$$637$$ 0 0
$$638$$ 0 0
$$639$$ 4.00000 0.158238
$$640$$ 0 0
$$641$$ −11.0000 −0.434474 −0.217237 0.976119i $$-0.569704\pi$$
−0.217237 + 0.976119i $$0.569704\pi$$
$$642$$ −11.0000 −0.434135
$$643$$ −4.00000 −0.157745 −0.0788723 0.996885i $$-0.525132\pi$$
−0.0788723 + 0.996885i $$0.525132\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 12.0000 0.472134
$$647$$ 47.0000 1.84776 0.923880 0.382682i $$-0.124999\pi$$
0.923880 + 0.382682i $$0.124999\pi$$
$$648$$ 3.00000 0.117851
$$649$$ 0 0
$$650$$ 0 0
$$651$$ 0 0
$$652$$ −4.00000 −0.156652
$$653$$ 26.0000 1.01746 0.508729 0.860927i $$-0.330115\pi$$
0.508729 + 0.860927i $$0.330115\pi$$
$$654$$ 5.00000 0.195515
$$655$$ 0 0
$$656$$ 5.00000 0.195217
$$657$$ 12.0000 0.468165
$$658$$ 0 0
$$659$$ 6.00000 0.233727 0.116863 0.993148i $$-0.462716\pi$$
0.116863 + 0.993148i $$0.462716\pi$$
$$660$$ 0 0
$$661$$ 9.00000 0.350059 0.175030 0.984563i $$-0.443998\pi$$
0.175030 + 0.984563i $$0.443998\pi$$
$$662$$ −6.00000 −0.233197
$$663$$ −4.00000 −0.155347
$$664$$ −33.0000 −1.28065
$$665$$ 0 0
$$666$$ 16.0000 0.619987
$$667$$ 21.0000 0.813123
$$668$$ 3.00000 0.116073
$$669$$ 0 0
$$670$$ 0 0
$$671$$ 0 0
$$672$$ 0 0
$$673$$ 36.0000 1.38770 0.693849 0.720121i $$-0.255914\pi$$
0.693849 + 0.720121i $$0.255914\pi$$
$$674$$ 18.0000 0.693334
$$675$$ 0 0
$$676$$ 9.00000 0.346154
$$677$$ −14.0000 −0.538064 −0.269032 0.963131i $$-0.586704\pi$$
−0.269032 + 0.963131i $$0.586704\pi$$
$$678$$ 6.00000 0.230429
$$679$$ 0 0
$$680$$ 0 0
$$681$$ 20.0000 0.766402
$$682$$ 0 0
$$683$$ −35.0000 −1.33924 −0.669619 0.742705i $$-0.733543\pi$$
−0.669619 + 0.742705i $$0.733543\pi$$
$$684$$ −12.0000 −0.458831
$$685$$ 0 0
$$686$$ 0 0
$$687$$ −22.0000 −0.839352
$$688$$ 7.00000 0.266872
$$689$$ −12.0000 −0.457164
$$690$$ 0 0
$$691$$ 20.0000 0.760836 0.380418 0.924815i $$-0.375780\pi$$
0.380418 + 0.924815i $$0.375780\pi$$
$$692$$ 12.0000 0.456172
$$693$$ 0 0
$$694$$ 15.0000 0.569392
$$695$$ 0 0
$$696$$ −21.0000 −0.796003
$$697$$ −10.0000 −0.378777
$$698$$ −17.0000 −0.643459
$$699$$ 14.0000 0.529529
$$700$$ 0 0
$$701$$ −1.00000 −0.0377695 −0.0188847 0.999822i $$-0.506012\pi$$
−0.0188847 + 0.999822i $$0.506012\pi$$
$$702$$ −10.0000 −0.377426
$$703$$ −48.0000 −1.81035
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 26.0000 0.978523
$$707$$ 0 0
$$708$$ −10.0000 −0.375823
$$709$$ −49.0000 −1.84023 −0.920117 0.391644i $$-0.871906\pi$$
−0.920117 + 0.391644i $$0.871906\pi$$
$$710$$ 0 0
$$711$$ 4.00000 0.150012
$$712$$ −27.0000 −1.01187
$$713$$ −6.00000 −0.224702
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 2.00000 0.0747435
$$717$$ 18.0000 0.672222
$$718$$ −10.0000 −0.373197
$$719$$ −36.0000 −1.34257 −0.671287 0.741198i $$-0.734258\pi$$
−0.671287 + 0.741198i $$0.734258\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ −17.0000 −0.632674
$$723$$ −14.0000 −0.520666
$$724$$ −3.00000 −0.111494
$$725$$ 0 0
$$726$$ −11.0000 −0.408248
$$727$$ −47.0000 −1.74313 −0.871567 0.490277i $$-0.836896\pi$$
−0.871567 + 0.490277i $$0.836896\pi$$
$$728$$ 0 0
$$729$$ 13.0000 0.481481
$$730$$ 0 0
$$731$$ −14.0000 −0.517809
$$732$$ −7.00000 −0.258727
$$733$$ −16.0000 −0.590973 −0.295487 0.955347i $$-0.595482\pi$$
−0.295487 + 0.955347i $$0.595482\pi$$
$$734$$ −27.0000 −0.996588
$$735$$ 0 0
$$736$$ −15.0000 −0.552907
$$737$$ 0 0
$$738$$ −10.0000 −0.368105
$$739$$ 16.0000 0.588570 0.294285 0.955718i $$-0.404919\pi$$
0.294285 + 0.955718i $$0.404919\pi$$
$$740$$ 0 0
$$741$$ 12.0000 0.440831
$$742$$ 0 0
$$743$$ 49.0000 1.79764 0.898818 0.438322i $$-0.144427\pi$$
0.898818 + 0.438322i $$0.144427\pi$$
$$744$$ 6.00000 0.219971
$$745$$ 0 0
$$746$$ −24.0000 −0.878702
$$747$$ 22.0000 0.804938
$$748$$ 0 0
$$749$$ 0 0
$$750$$ 0 0
$$751$$ −12.0000 −0.437886 −0.218943 0.975738i $$-0.570261\pi$$
−0.218943 + 0.975738i $$0.570261\pi$$
$$752$$ 0 0
$$753$$ 30.0000 1.09326
$$754$$ −14.0000 −0.509850
$$755$$ 0 0
$$756$$ 0 0
$$757$$ −22.0000 −0.799604 −0.399802 0.916602i $$-0.630921\pi$$
−0.399802 + 0.916602i $$0.630921\pi$$
$$758$$ 10.0000 0.363216
$$759$$ 0 0
$$760$$ 0 0
$$761$$ −14.0000 −0.507500 −0.253750 0.967270i $$-0.581664\pi$$
−0.253750 + 0.967270i $$0.581664\pi$$
$$762$$ −16.0000 −0.579619
$$763$$ 0 0
$$764$$ −12.0000 −0.434145
$$765$$ 0 0
$$766$$ −5.00000 −0.180657
$$767$$ −20.0000 −0.722158
$$768$$ 17.0000 0.613435
$$769$$ 30.0000 1.08183 0.540914 0.841078i $$-0.318079\pi$$
0.540914 + 0.841078i $$0.318079\pi$$
$$770$$ 0 0
$$771$$ −20.0000 −0.720282
$$772$$ −4.00000 −0.143963
$$773$$ 18.0000 0.647415 0.323708 0.946157i $$-0.395071\pi$$
0.323708 + 0.946157i $$0.395071\pi$$
$$774$$ −14.0000 −0.503220
$$775$$ 0 0
$$776$$ −48.0000 −1.72310
$$777$$ 0 0
$$778$$ −14.0000 −0.501924
$$779$$ 30.0000 1.07486
$$780$$ 0 0
$$781$$ 0 0
$$782$$ −6.00000 −0.214560
$$783$$ 35.0000 1.25080
$$784$$ 0 0
$$785$$ 0 0
$$786$$ −4.00000 −0.142675
$$787$$ −17.0000 −0.605985 −0.302992 0.952993i $$-0.597986\pi$$
−0.302992 + 0.952993i $$0.597986\pi$$
$$788$$ −8.00000 −0.284988
$$789$$ −9.00000 −0.320408
$$790$$ 0 0
$$791$$ 0 0
$$792$$ 0 0
$$793$$ −14.0000 −0.497155
$$794$$ 16.0000 0.567819
$$795$$ 0 0
$$796$$ −4.00000 −0.141776
$$797$$ 40.0000 1.41687 0.708436 0.705775i $$-0.249401\pi$$
0.708436 + 0.705775i $$0.249401\pi$$
$$798$$ 0 0
$$799$$ 0 0
$$800$$ 0 0
$$801$$ 18.0000 0.635999
$$802$$ −3.00000 −0.105934
$$803$$ 0 0
$$804$$ 5.00000 0.176336
$$805$$ 0 0
$$806$$ 4.00000 0.140894
$$807$$ −11.0000 −0.387218
$$808$$ −27.0000 −0.949857
$$809$$ 9.00000 0.316423 0.158212 0.987405i $$-0.449427\pi$$
0.158212 + 0.987405i $$0.449427\pi$$
$$810$$ 0 0
$$811$$ 2.00000 0.0702295 0.0351147 0.999383i $$-0.488820\pi$$
0.0351147 + 0.999383i $$0.488820\pi$$
$$812$$ 0 0
$$813$$ −2.00000 −0.0701431
$$814$$ 0 0
$$815$$ 0 0
$$816$$ 2.00000 0.0700140
$$817$$ 42.0000 1.46939
$$818$$ −25.0000 −0.874105
$$819$$ 0 0
$$820$$ 0 0
$$821$$ 22.0000 0.767805 0.383903 0.923374i $$-0.374580\pi$$
0.383903 + 0.923374i $$0.374580\pi$$
$$822$$ 0 0
$$823$$ −25.0000 −0.871445 −0.435723 0.900081i $$-0.643507\pi$$
−0.435723 + 0.900081i $$0.643507\pi$$
$$824$$ 21.0000 0.731570
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 39.0000 1.35616 0.678081 0.734987i $$-0.262812\pi$$
0.678081 + 0.734987i $$0.262812\pi$$
$$828$$ 6.00000 0.208514
$$829$$ 30.0000 1.04194 0.520972 0.853574i $$-0.325570\pi$$
0.520972 + 0.853574i $$0.325570\pi$$
$$830$$ 0 0
$$831$$ 10.0000 0.346896
$$832$$ 14.0000 0.485363
$$833$$ 0 0
$$834$$ −8.00000 −0.277017
$$835$$ 0 0
$$836$$ 0 0
$$837$$ −10.0000 −0.345651
$$838$$ 24.0000 0.829066
$$839$$ −48.0000 −1.65714 −0.828572 0.559883i $$-0.810846\pi$$
−0.828572 + 0.559883i $$0.810846\pi$$
$$840$$ 0 0
$$841$$ 20.0000 0.689655
$$842$$ −15.0000 −0.516934
$$843$$ 14.0000 0.482186
$$844$$ 10.0000 0.344214
$$845$$ 0 0
$$846$$ 0 0
$$847$$ 0 0
$$848$$ 6.00000 0.206041
$$849$$ −16.0000 −0.549119
$$850$$ 0 0
$$851$$ 24.0000 0.822709
$$852$$ −2.00000 −0.0685189
$$853$$ −6.00000 −0.205436 −0.102718 0.994711i $$-0.532754\pi$$
−0.102718 + 0.994711i $$0.532754\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ −33.0000 −1.12792
$$857$$ −12.0000 −0.409912 −0.204956 0.978771i $$-0.565705\pi$$
−0.204956 + 0.978771i $$0.565705\pi$$
$$858$$ 0 0
$$859$$ 32.0000 1.09183 0.545913 0.837842i $$-0.316183\pi$$
0.545913 + 0.837842i $$0.316183\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 32.0000 1.08992
$$863$$ 9.00000 0.306364 0.153182 0.988198i $$-0.451048\pi$$
0.153182 + 0.988198i $$0.451048\pi$$
$$864$$ −25.0000 −0.850517
$$865$$ 0 0
$$866$$ 2.00000 0.0679628
$$867$$ 13.0000 0.441503
$$868$$ 0 0
$$869$$ 0 0
$$870$$ 0 0
$$871$$ 10.0000 0.338837
$$872$$ 15.0000 0.507964
$$873$$ 32.0000 1.08304
$$874$$ 18.0000 0.608859
$$875$$ 0 0
$$876$$ −6.00000 −0.202721
$$877$$ 2.00000 0.0675352 0.0337676 0.999430i $$-0.489249\pi$$
0.0337676 + 0.999430i $$0.489249\pi$$
$$878$$ 24.0000 0.809961
$$879$$ −24.0000 −0.809500
$$880$$ 0 0
$$881$$ 43.0000 1.44871 0.724353 0.689429i $$-0.242138\pi$$
0.724353 + 0.689429i $$0.242138\pi$$
$$882$$ 0 0
$$883$$ −36.0000 −1.21150 −0.605748 0.795656i $$-0.707126\pi$$
−0.605748 + 0.795656i $$0.707126\pi$$
$$884$$ −4.00000 −0.134535
$$885$$ 0 0
$$886$$ 31.0000 1.04147
$$887$$ 43.0000 1.44380 0.721899 0.691998i $$-0.243269\pi$$
0.721899 + 0.691998i $$0.243269\pi$$
$$888$$ −24.0000 −0.805387
$$889$$ 0 0
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 0 0
$$893$$ 0 0
$$894$$ 1.00000 0.0334450
$$895$$ 0 0
$$896$$ 0 0
$$897$$ −6.00000 −0.200334
$$898$$ −31.0000 −1.03448
$$899$$ −14.0000 −0.466926
$$900$$ 0 0
$$901$$ −12.0000 −0.399778
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 18.0000 0.598671
$$905$$ 0 0
$$906$$ 14.0000 0.465119
$$907$$ 19.0000 0.630885 0.315442 0.948945i $$-0.397847\pi$$
0.315442 + 0.948945i $$0.397847\pi$$
$$908$$ 20.0000 0.663723
$$909$$ 18.0000 0.597022
$$910$$ 0 0
$$911$$ 24.0000 0.795155 0.397578 0.917568i $$-0.369851\pi$$
0.397578 + 0.917568i $$0.369851\pi$$
$$912$$ −6.00000 −0.198680
$$913$$ 0 0
$$914$$ 32.0000 1.05847
$$915$$ 0 0
$$916$$ −22.0000 −0.726900
$$917$$ 0 0
$$918$$ −10.0000 −0.330049
$$919$$ 34.0000 1.12156 0.560778 0.827966i $$-0.310502\pi$$
0.560778 + 0.827966i $$0.310502\pi$$
$$920$$ 0 0
$$921$$ −23.0000 −0.757876
$$922$$ 18.0000 0.592798
$$923$$ −4.00000 −0.131662
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 3.00000 0.0985861
$$927$$ −14.0000 −0.459820
$$928$$ −35.0000 −1.14893
$$929$$ 29.0000 0.951459 0.475730 0.879592i $$-0.342184\pi$$
0.475730 + 0.879592i $$0.342184\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ 14.0000 0.458585
$$933$$ 30.0000 0.982156
$$934$$ −3.00000 −0.0981630
$$935$$ 0 0
$$936$$ −12.0000 −0.392232
$$937$$ −8.00000 −0.261349 −0.130674 0.991425i $$-0.541714\pi$$
−0.130674 + 0.991425i $$0.541714\pi$$
$$938$$ 0 0
$$939$$ −24.0000 −0.783210
$$940$$ 0 0
$$941$$ 18.0000 0.586783 0.293392 0.955992i $$-0.405216\pi$$
0.293392 + 0.955992i $$0.405216\pi$$
$$942$$ 12.0000 0.390981
$$943$$ −15.0000 −0.488467
$$944$$ 10.0000 0.325472
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 57.0000 1.85225 0.926126 0.377215i $$-0.123118\pi$$
0.926126 + 0.377215i $$0.123118\pi$$
$$948$$ −2.00000 −0.0649570
$$949$$ −12.0000 −0.389536
$$950$$ 0 0
$$951$$ −6.00000 −0.194563
$$952$$ 0 0
$$953$$ −60.0000 −1.94359 −0.971795 0.235826i $$-0.924220\pi$$
−0.971795 + 0.235826i $$0.924220\pi$$
$$954$$ −12.0000 −0.388514
$$955$$ 0 0
$$956$$ 18.0000 0.582162
$$957$$ 0 0
$$958$$ 42.0000 1.35696
$$959$$ 0 0
$$960$$ 0 0
$$961$$ −27.0000 −0.870968
$$962$$ −16.0000 −0.515861
$$963$$ 22.0000 0.708940
$$964$$ −14.0000 −0.450910
$$965$$ 0 0
$$966$$ 0 0
$$967$$ 13.0000 0.418052 0.209026 0.977910i $$-0.432971\pi$$
0.209026 + 0.977910i $$0.432971\pi$$
$$968$$ −33.0000 −1.06066
$$969$$ 12.0000 0.385496
$$970$$ 0 0
$$971$$ 32.0000 1.02693 0.513464 0.858111i $$-0.328362\pi$$
0.513464 + 0.858111i $$0.328362\pi$$
$$972$$ 16.0000 0.513200
$$973$$ 0 0
$$974$$ −12.0000 −0.384505
$$975$$ 0 0
$$976$$ 7.00000 0.224065
$$977$$ −60.0000 −1.91957 −0.959785 0.280736i $$-0.909421\pi$$
−0.959785 + 0.280736i $$0.909421\pi$$
$$978$$ 4.00000 0.127906
$$979$$ 0 0
$$980$$ 0 0
$$981$$ −10.0000 −0.319275
$$982$$ 26.0000 0.829693
$$983$$ 3.00000 0.0956851 0.0478426 0.998855i $$-0.484765\pi$$
0.0478426 + 0.998855i $$0.484765\pi$$
$$984$$ 15.0000 0.478183
$$985$$ 0 0
$$986$$ −14.0000 −0.445851
$$987$$ 0 0
$$988$$ 12.0000 0.381771
$$989$$ −21.0000 −0.667761
$$990$$ 0 0
$$991$$ −50.0000 −1.58830 −0.794151 0.607720i $$-0.792084\pi$$
−0.794151 + 0.607720i $$0.792084\pi$$
$$992$$ 10.0000 0.317500
$$993$$ −6.00000 −0.190404
$$994$$ 0 0
$$995$$ 0 0
$$996$$ −11.0000 −0.348548
$$997$$ −2.00000 −0.0633406 −0.0316703 0.999498i $$-0.510083\pi$$
−0.0316703 + 0.999498i $$0.510083\pi$$
$$998$$ −16.0000 −0.506471
$$999$$ 40.0000 1.26554
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 1225.2.a.b.1.1 1
5.2 odd 4 245.2.b.b.99.1 2
5.3 odd 4 245.2.b.b.99.2 2
5.4 even 2 1225.2.a.g.1.1 1
7.3 odd 6 175.2.e.b.51.1 2
7.5 odd 6 175.2.e.b.151.1 2
7.6 odd 2 1225.2.a.d.1.1 1
15.2 even 4 2205.2.d.e.1324.2 2
15.8 even 4 2205.2.d.e.1324.1 2
35.2 odd 12 245.2.j.c.214.1 4
35.3 even 12 35.2.j.a.9.1 yes 4
35.12 even 12 35.2.j.a.4.1 4
35.13 even 4 245.2.b.c.99.2 2
35.17 even 12 35.2.j.a.9.2 yes 4
35.18 odd 12 245.2.j.c.79.1 4
35.19 odd 6 175.2.e.a.151.1 2
35.23 odd 12 245.2.j.c.214.2 4
35.24 odd 6 175.2.e.a.51.1 2
35.27 even 4 245.2.b.c.99.1 2
35.32 odd 12 245.2.j.c.79.2 4
35.33 even 12 35.2.j.a.4.2 yes 4
35.34 odd 2 1225.2.a.f.1.1 1
105.17 odd 12 315.2.bf.a.289.1 4
105.38 odd 12 315.2.bf.a.289.2 4
105.47 odd 12 315.2.bf.a.109.2 4
105.62 odd 4 2205.2.d.d.1324.2 2
105.68 odd 12 315.2.bf.a.109.1 4
105.83 odd 4 2205.2.d.d.1324.1 2
140.3 odd 12 560.2.bw.b.289.1 4
140.47 odd 12 560.2.bw.b.529.1 4
140.87 odd 12 560.2.bw.b.289.2 4
140.103 odd 12 560.2.bw.b.529.2 4

By twisted newform
Twist Min Dim Char Parity Ord Type
35.2.j.a.4.1 4 35.12 even 12
35.2.j.a.4.2 yes 4 35.33 even 12
35.2.j.a.9.1 yes 4 35.3 even 12
35.2.j.a.9.2 yes 4 35.17 even 12
175.2.e.a.51.1 2 35.24 odd 6
175.2.e.a.151.1 2 35.19 odd 6
175.2.e.b.51.1 2 7.3 odd 6
175.2.e.b.151.1 2 7.5 odd 6
245.2.b.b.99.1 2 5.2 odd 4
245.2.b.b.99.2 2 5.3 odd 4
245.2.b.c.99.1 2 35.27 even 4
245.2.b.c.99.2 2 35.13 even 4
245.2.j.c.79.1 4 35.18 odd 12
245.2.j.c.79.2 4 35.32 odd 12
245.2.j.c.214.1 4 35.2 odd 12
245.2.j.c.214.2 4 35.23 odd 12
315.2.bf.a.109.1 4 105.68 odd 12
315.2.bf.a.109.2 4 105.47 odd 12
315.2.bf.a.289.1 4 105.17 odd 12
315.2.bf.a.289.2 4 105.38 odd 12
560.2.bw.b.289.1 4 140.3 odd 12
560.2.bw.b.289.2 4 140.87 odd 12
560.2.bw.b.529.1 4 140.47 odd 12
560.2.bw.b.529.2 4 140.103 odd 12
1225.2.a.b.1.1 1 1.1 even 1 trivial
1225.2.a.d.1.1 1 7.6 odd 2
1225.2.a.f.1.1 1 35.34 odd 2
1225.2.a.g.1.1 1 5.4 even 2
2205.2.d.d.1324.1 2 105.83 odd 4
2205.2.d.d.1324.2 2 105.62 odd 4
2205.2.d.e.1324.1 2 15.8 even 4
2205.2.d.e.1324.2 2 15.2 even 4