# Properties

 Label 390.2.a.a.1.1 Level $390$ Weight $2$ Character 390.1 Self dual yes Analytic conductor $3.114$ 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] = [390,2,Mod(1,390)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(390, 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("390.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$390 = 2 \cdot 3 \cdot 5 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 390.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: $$3.11416567883$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 390.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{10} -1.00000 q^{12} -1.00000 q^{13} +1.00000 q^{15} +1.00000 q^{16} -6.00000 q^{17} -1.00000 q^{18} -1.00000 q^{20} -4.00000 q^{23} +1.00000 q^{24} +1.00000 q^{25} +1.00000 q^{26} -1.00000 q^{27} -10.0000 q^{29} -1.00000 q^{30} -1.00000 q^{32} +6.00000 q^{34} +1.00000 q^{36} -6.00000 q^{37} +1.00000 q^{39} +1.00000 q^{40} +2.00000 q^{41} -4.00000 q^{43} -1.00000 q^{45} +4.00000 q^{46} -1.00000 q^{48} -7.00000 q^{49} -1.00000 q^{50} +6.00000 q^{51} -1.00000 q^{52} -6.00000 q^{53} +1.00000 q^{54} +10.0000 q^{58} +1.00000 q^{60} +6.00000 q^{61} +1.00000 q^{64} +1.00000 q^{65} +4.00000 q^{67} -6.00000 q^{68} +4.00000 q^{69} +16.0000 q^{71} -1.00000 q^{72} -2.00000 q^{73} +6.00000 q^{74} -1.00000 q^{75} -1.00000 q^{78} -1.00000 q^{80} +1.00000 q^{81} -2.00000 q^{82} +4.00000 q^{83} +6.00000 q^{85} +4.00000 q^{86} +10.0000 q^{87} -6.00000 q^{89} +1.00000 q^{90} -4.00000 q^{92} +1.00000 q^{96} +14.0000 q^{97} +7.00000 q^{98} +O(q^{100})$$

## 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
$$3$$ −1.00000 −0.577350
$$4$$ 1.00000 0.500000
$$5$$ −1.00000 −0.447214
$$6$$ 1.00000 0.408248
$$7$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$8$$ −1.00000 −0.353553
$$9$$ 1.00000 0.333333
$$10$$ 1.00000 0.316228
$$11$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$12$$ −1.00000 −0.288675
$$13$$ −1.00000 −0.277350
$$14$$ 0 0
$$15$$ 1.00000 0.258199
$$16$$ 1.00000 0.250000
$$17$$ −6.00000 −1.45521 −0.727607 0.685994i $$-0.759367\pi$$
−0.727607 + 0.685994i $$0.759367\pi$$
$$18$$ −1.00000 −0.235702
$$19$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$20$$ −1.00000 −0.223607
$$21$$ 0 0
$$22$$ 0 0
$$23$$ −4.00000 −0.834058 −0.417029 0.908893i $$-0.636929\pi$$
−0.417029 + 0.908893i $$0.636929\pi$$
$$24$$ 1.00000 0.204124
$$25$$ 1.00000 0.200000
$$26$$ 1.00000 0.196116
$$27$$ −1.00000 −0.192450
$$28$$ 0 0
$$29$$ −10.0000 −1.85695 −0.928477 0.371391i $$-0.878881\pi$$
−0.928477 + 0.371391i $$0.878881\pi$$
$$30$$ −1.00000 −0.182574
$$31$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$32$$ −1.00000 −0.176777
$$33$$ 0 0
$$34$$ 6.00000 1.02899
$$35$$ 0 0
$$36$$ 1.00000 0.166667
$$37$$ −6.00000 −0.986394 −0.493197 0.869918i $$-0.664172\pi$$
−0.493197 + 0.869918i $$0.664172\pi$$
$$38$$ 0 0
$$39$$ 1.00000 0.160128
$$40$$ 1.00000 0.158114
$$41$$ 2.00000 0.312348 0.156174 0.987730i $$-0.450084\pi$$
0.156174 + 0.987730i $$0.450084\pi$$
$$42$$ 0 0
$$43$$ −4.00000 −0.609994 −0.304997 0.952353i $$-0.598656\pi$$
−0.304997 + 0.952353i $$0.598656\pi$$
$$44$$ 0 0
$$45$$ −1.00000 −0.149071
$$46$$ 4.00000 0.589768
$$47$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$48$$ −1.00000 −0.144338
$$49$$ −7.00000 −1.00000
$$50$$ −1.00000 −0.141421
$$51$$ 6.00000 0.840168
$$52$$ −1.00000 −0.138675
$$53$$ −6.00000 −0.824163 −0.412082 0.911147i $$-0.635198\pi$$
−0.412082 + 0.911147i $$0.635198\pi$$
$$54$$ 1.00000 0.136083
$$55$$ 0 0
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 10.0000 1.31306
$$59$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$60$$ 1.00000 0.129099
$$61$$ 6.00000 0.768221 0.384111 0.923287i $$-0.374508\pi$$
0.384111 + 0.923287i $$0.374508\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 1.00000 0.125000
$$65$$ 1.00000 0.124035
$$66$$ 0 0
$$67$$ 4.00000 0.488678 0.244339 0.969690i $$-0.421429\pi$$
0.244339 + 0.969690i $$0.421429\pi$$
$$68$$ −6.00000 −0.727607
$$69$$ 4.00000 0.481543
$$70$$ 0 0
$$71$$ 16.0000 1.89885 0.949425 0.313993i $$-0.101667\pi$$
0.949425 + 0.313993i $$0.101667\pi$$
$$72$$ −1.00000 −0.117851
$$73$$ −2.00000 −0.234082 −0.117041 0.993127i $$-0.537341\pi$$
−0.117041 + 0.993127i $$0.537341\pi$$
$$74$$ 6.00000 0.697486
$$75$$ −1.00000 −0.115470
$$76$$ 0 0
$$77$$ 0 0
$$78$$ −1.00000 −0.113228
$$79$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$80$$ −1.00000 −0.111803
$$81$$ 1.00000 0.111111
$$82$$ −2.00000 −0.220863
$$83$$ 4.00000 0.439057 0.219529 0.975606i $$-0.429548\pi$$
0.219529 + 0.975606i $$0.429548\pi$$
$$84$$ 0 0
$$85$$ 6.00000 0.650791
$$86$$ 4.00000 0.431331
$$87$$ 10.0000 1.07211
$$88$$ 0 0
$$89$$ −6.00000 −0.635999 −0.317999 0.948091i $$-0.603011\pi$$
−0.317999 + 0.948091i $$0.603011\pi$$
$$90$$ 1.00000 0.105409
$$91$$ 0 0
$$92$$ −4.00000 −0.417029
$$93$$ 0 0
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 1.00000 0.102062
$$97$$ 14.0000 1.42148 0.710742 0.703452i $$-0.248359\pi$$
0.710742 + 0.703452i $$0.248359\pi$$
$$98$$ 7.00000 0.707107
$$99$$ 0 0
$$100$$ 1.00000 0.100000
$$101$$ 6.00000 0.597022 0.298511 0.954406i $$-0.403510\pi$$
0.298511 + 0.954406i $$0.403510\pi$$
$$102$$ −6.00000 −0.594089
$$103$$ 12.0000 1.18240 0.591198 0.806527i $$-0.298655\pi$$
0.591198 + 0.806527i $$0.298655\pi$$
$$104$$ 1.00000 0.0980581
$$105$$ 0 0
$$106$$ 6.00000 0.582772
$$107$$ −4.00000 −0.386695 −0.193347 0.981130i $$-0.561934\pi$$
−0.193347 + 0.981130i $$0.561934\pi$$
$$108$$ −1.00000 −0.0962250
$$109$$ −14.0000 −1.34096 −0.670478 0.741929i $$-0.733911\pi$$
−0.670478 + 0.741929i $$0.733911\pi$$
$$110$$ 0 0
$$111$$ 6.00000 0.569495
$$112$$ 0 0
$$113$$ 10.0000 0.940721 0.470360 0.882474i $$-0.344124\pi$$
0.470360 + 0.882474i $$0.344124\pi$$
$$114$$ 0 0
$$115$$ 4.00000 0.373002
$$116$$ −10.0000 −0.928477
$$117$$ −1.00000 −0.0924500
$$118$$ 0 0
$$119$$ 0 0
$$120$$ −1.00000 −0.0912871
$$121$$ −11.0000 −1.00000
$$122$$ −6.00000 −0.543214
$$123$$ −2.00000 −0.180334
$$124$$ 0 0
$$125$$ −1.00000 −0.0894427
$$126$$ 0 0
$$127$$ −12.0000 −1.06483 −0.532414 0.846484i $$-0.678715\pi$$
−0.532414 + 0.846484i $$0.678715\pi$$
$$128$$ −1.00000 −0.0883883
$$129$$ 4.00000 0.352180
$$130$$ −1.00000 −0.0877058
$$131$$ 12.0000 1.04844 0.524222 0.851581i $$-0.324356\pi$$
0.524222 + 0.851581i $$0.324356\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ −4.00000 −0.345547
$$135$$ 1.00000 0.0860663
$$136$$ 6.00000 0.514496
$$137$$ 6.00000 0.512615 0.256307 0.966595i $$-0.417494\pi$$
0.256307 + 0.966595i $$0.417494\pi$$
$$138$$ −4.00000 −0.340503
$$139$$ 4.00000 0.339276 0.169638 0.985506i $$-0.445740\pi$$
0.169638 + 0.985506i $$0.445740\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ −16.0000 −1.34269
$$143$$ 0 0
$$144$$ 1.00000 0.0833333
$$145$$ 10.0000 0.830455
$$146$$ 2.00000 0.165521
$$147$$ 7.00000 0.577350
$$148$$ −6.00000 −0.493197
$$149$$ −14.0000 −1.14692 −0.573462 0.819232i $$-0.694400\pi$$
−0.573462 + 0.819232i $$0.694400\pi$$
$$150$$ 1.00000 0.0816497
$$151$$ −16.0000 −1.30206 −0.651031 0.759051i $$-0.725663\pi$$
−0.651031 + 0.759051i $$0.725663\pi$$
$$152$$ 0 0
$$153$$ −6.00000 −0.485071
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 1.00000 0.0800641
$$157$$ −14.0000 −1.11732 −0.558661 0.829396i $$-0.688685\pi$$
−0.558661 + 0.829396i $$0.688685\pi$$
$$158$$ 0 0
$$159$$ 6.00000 0.475831
$$160$$ 1.00000 0.0790569
$$161$$ 0 0
$$162$$ −1.00000 −0.0785674
$$163$$ −4.00000 −0.313304 −0.156652 0.987654i $$-0.550070\pi$$
−0.156652 + 0.987654i $$0.550070\pi$$
$$164$$ 2.00000 0.156174
$$165$$ 0 0
$$166$$ −4.00000 −0.310460
$$167$$ 24.0000 1.85718 0.928588 0.371113i $$-0.121024\pi$$
0.928588 + 0.371113i $$0.121024\pi$$
$$168$$ 0 0
$$169$$ 1.00000 0.0769231
$$170$$ −6.00000 −0.460179
$$171$$ 0 0
$$172$$ −4.00000 −0.304997
$$173$$ −14.0000 −1.06440 −0.532200 0.846619i $$-0.678635\pi$$
−0.532200 + 0.846619i $$0.678635\pi$$
$$174$$ −10.0000 −0.758098
$$175$$ 0 0
$$176$$ 0 0
$$177$$ 0 0
$$178$$ 6.00000 0.449719
$$179$$ −20.0000 −1.49487 −0.747435 0.664335i $$-0.768715\pi$$
−0.747435 + 0.664335i $$0.768715\pi$$
$$180$$ −1.00000 −0.0745356
$$181$$ −10.0000 −0.743294 −0.371647 0.928374i $$-0.621207\pi$$
−0.371647 + 0.928374i $$0.621207\pi$$
$$182$$ 0 0
$$183$$ −6.00000 −0.443533
$$184$$ 4.00000 0.294884
$$185$$ 6.00000 0.441129
$$186$$ 0 0
$$187$$ 0 0
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 24.0000 1.73658 0.868290 0.496058i $$-0.165220\pi$$
0.868290 + 0.496058i $$0.165220\pi$$
$$192$$ −1.00000 −0.0721688
$$193$$ 14.0000 1.00774 0.503871 0.863779i $$-0.331909\pi$$
0.503871 + 0.863779i $$0.331909\pi$$
$$194$$ −14.0000 −1.00514
$$195$$ −1.00000 −0.0716115
$$196$$ −7.00000 −0.500000
$$197$$ −22.0000 −1.56744 −0.783718 0.621117i $$-0.786679\pi$$
−0.783718 + 0.621117i $$0.786679\pi$$
$$198$$ 0 0
$$199$$ −16.0000 −1.13421 −0.567105 0.823646i $$-0.691937\pi$$
−0.567105 + 0.823646i $$0.691937\pi$$
$$200$$ −1.00000 −0.0707107
$$201$$ −4.00000 −0.282138
$$202$$ −6.00000 −0.422159
$$203$$ 0 0
$$204$$ 6.00000 0.420084
$$205$$ −2.00000 −0.139686
$$206$$ −12.0000 −0.836080
$$207$$ −4.00000 −0.278019
$$208$$ −1.00000 −0.0693375
$$209$$ 0 0
$$210$$ 0 0
$$211$$ −12.0000 −0.826114 −0.413057 0.910705i $$-0.635539\pi$$
−0.413057 + 0.910705i $$0.635539\pi$$
$$212$$ −6.00000 −0.412082
$$213$$ −16.0000 −1.09630
$$214$$ 4.00000 0.273434
$$215$$ 4.00000 0.272798
$$216$$ 1.00000 0.0680414
$$217$$ 0 0
$$218$$ 14.0000 0.948200
$$219$$ 2.00000 0.135147
$$220$$ 0 0
$$221$$ 6.00000 0.403604
$$222$$ −6.00000 −0.402694
$$223$$ 16.0000 1.07144 0.535720 0.844396i $$-0.320040\pi$$
0.535720 + 0.844396i $$0.320040\pi$$
$$224$$ 0 0
$$225$$ 1.00000 0.0666667
$$226$$ −10.0000 −0.665190
$$227$$ 20.0000 1.32745 0.663723 0.747978i $$-0.268975\pi$$
0.663723 + 0.747978i $$0.268975\pi$$
$$228$$ 0 0
$$229$$ 2.00000 0.132164 0.0660819 0.997814i $$-0.478950\pi$$
0.0660819 + 0.997814i $$0.478950\pi$$
$$230$$ −4.00000 −0.263752
$$231$$ 0 0
$$232$$ 10.0000 0.656532
$$233$$ 18.0000 1.17922 0.589610 0.807688i $$-0.299282\pi$$
0.589610 + 0.807688i $$0.299282\pi$$
$$234$$ 1.00000 0.0653720
$$235$$ 0 0
$$236$$ 0 0
$$237$$ 0 0
$$238$$ 0 0
$$239$$ 16.0000 1.03495 0.517477 0.855697i $$-0.326871\pi$$
0.517477 + 0.855697i $$0.326871\pi$$
$$240$$ 1.00000 0.0645497
$$241$$ −14.0000 −0.901819 −0.450910 0.892570i $$-0.648900\pi$$
−0.450910 + 0.892570i $$0.648900\pi$$
$$242$$ 11.0000 0.707107
$$243$$ −1.00000 −0.0641500
$$244$$ 6.00000 0.384111
$$245$$ 7.00000 0.447214
$$246$$ 2.00000 0.127515
$$247$$ 0 0
$$248$$ 0 0
$$249$$ −4.00000 −0.253490
$$250$$ 1.00000 0.0632456
$$251$$ −4.00000 −0.252478 −0.126239 0.992000i $$-0.540291\pi$$
−0.126239 + 0.992000i $$0.540291\pi$$
$$252$$ 0 0
$$253$$ 0 0
$$254$$ 12.0000 0.752947
$$255$$ −6.00000 −0.375735
$$256$$ 1.00000 0.0625000
$$257$$ −6.00000 −0.374270 −0.187135 0.982334i $$-0.559920\pi$$
−0.187135 + 0.982334i $$0.559920\pi$$
$$258$$ −4.00000 −0.249029
$$259$$ 0 0
$$260$$ 1.00000 0.0620174
$$261$$ −10.0000 −0.618984
$$262$$ −12.0000 −0.741362
$$263$$ −28.0000 −1.72655 −0.863277 0.504730i $$-0.831592\pi$$
−0.863277 + 0.504730i $$0.831592\pi$$
$$264$$ 0 0
$$265$$ 6.00000 0.368577
$$266$$ 0 0
$$267$$ 6.00000 0.367194
$$268$$ 4.00000 0.244339
$$269$$ 14.0000 0.853595 0.426798 0.904347i $$-0.359642\pi$$
0.426798 + 0.904347i $$0.359642\pi$$
$$270$$ −1.00000 −0.0608581
$$271$$ −24.0000 −1.45790 −0.728948 0.684569i $$-0.759990\pi$$
−0.728948 + 0.684569i $$0.759990\pi$$
$$272$$ −6.00000 −0.363803
$$273$$ 0 0
$$274$$ −6.00000 −0.362473
$$275$$ 0 0
$$276$$ 4.00000 0.240772
$$277$$ 2.00000 0.120168 0.0600842 0.998193i $$-0.480863\pi$$
0.0600842 + 0.998193i $$0.480863\pi$$
$$278$$ −4.00000 −0.239904
$$279$$ 0 0
$$280$$ 0 0
$$281$$ −6.00000 −0.357930 −0.178965 0.983855i $$-0.557275\pi$$
−0.178965 + 0.983855i $$0.557275\pi$$
$$282$$ 0 0
$$283$$ −12.0000 −0.713326 −0.356663 0.934233i $$-0.616086\pi$$
−0.356663 + 0.934233i $$0.616086\pi$$
$$284$$ 16.0000 0.949425
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 0 0
$$288$$ −1.00000 −0.0589256
$$289$$ 19.0000 1.11765
$$290$$ −10.0000 −0.587220
$$291$$ −14.0000 −0.820695
$$292$$ −2.00000 −0.117041
$$293$$ 26.0000 1.51894 0.759468 0.650545i $$-0.225459\pi$$
0.759468 + 0.650545i $$0.225459\pi$$
$$294$$ −7.00000 −0.408248
$$295$$ 0 0
$$296$$ 6.00000 0.348743
$$297$$ 0 0
$$298$$ 14.0000 0.810998
$$299$$ 4.00000 0.231326
$$300$$ −1.00000 −0.0577350
$$301$$ 0 0
$$302$$ 16.0000 0.920697
$$303$$ −6.00000 −0.344691
$$304$$ 0 0
$$305$$ −6.00000 −0.343559
$$306$$ 6.00000 0.342997
$$307$$ −4.00000 −0.228292 −0.114146 0.993464i $$-0.536413\pi$$
−0.114146 + 0.993464i $$0.536413\pi$$
$$308$$ 0 0
$$309$$ −12.0000 −0.682656
$$310$$ 0 0
$$311$$ −16.0000 −0.907277 −0.453638 0.891186i $$-0.649874\pi$$
−0.453638 + 0.891186i $$0.649874\pi$$
$$312$$ −1.00000 −0.0566139
$$313$$ 26.0000 1.46961 0.734803 0.678280i $$-0.237274\pi$$
0.734803 + 0.678280i $$0.237274\pi$$
$$314$$ 14.0000 0.790066
$$315$$ 0 0
$$316$$ 0 0
$$317$$ 2.00000 0.112331 0.0561656 0.998421i $$-0.482113\pi$$
0.0561656 + 0.998421i $$0.482113\pi$$
$$318$$ −6.00000 −0.336463
$$319$$ 0 0
$$320$$ −1.00000 −0.0559017
$$321$$ 4.00000 0.223258
$$322$$ 0 0
$$323$$ 0 0
$$324$$ 1.00000 0.0555556
$$325$$ −1.00000 −0.0554700
$$326$$ 4.00000 0.221540
$$327$$ 14.0000 0.774202
$$328$$ −2.00000 −0.110432
$$329$$ 0 0
$$330$$ 0 0
$$331$$ 8.00000 0.439720 0.219860 0.975531i $$-0.429440\pi$$
0.219860 + 0.975531i $$0.429440\pi$$
$$332$$ 4.00000 0.219529
$$333$$ −6.00000 −0.328798
$$334$$ −24.0000 −1.31322
$$335$$ −4.00000 −0.218543
$$336$$ 0 0
$$337$$ 18.0000 0.980522 0.490261 0.871576i $$-0.336901\pi$$
0.490261 + 0.871576i $$0.336901\pi$$
$$338$$ −1.00000 −0.0543928
$$339$$ −10.0000 −0.543125
$$340$$ 6.00000 0.325396
$$341$$ 0 0
$$342$$ 0 0
$$343$$ 0 0
$$344$$ 4.00000 0.215666
$$345$$ −4.00000 −0.215353
$$346$$ 14.0000 0.752645
$$347$$ 12.0000 0.644194 0.322097 0.946707i $$-0.395612\pi$$
0.322097 + 0.946707i $$0.395612\pi$$
$$348$$ 10.0000 0.536056
$$349$$ 34.0000 1.81998 0.909989 0.414632i $$-0.136090\pi$$
0.909989 + 0.414632i $$0.136090\pi$$
$$350$$ 0 0
$$351$$ 1.00000 0.0533761
$$352$$ 0 0
$$353$$ −26.0000 −1.38384 −0.691920 0.721974i $$-0.743235\pi$$
−0.691920 + 0.721974i $$0.743235\pi$$
$$354$$ 0 0
$$355$$ −16.0000 −0.849192
$$356$$ −6.00000 −0.317999
$$357$$ 0 0
$$358$$ 20.0000 1.05703
$$359$$ 24.0000 1.26667 0.633336 0.773877i $$-0.281685\pi$$
0.633336 + 0.773877i $$0.281685\pi$$
$$360$$ 1.00000 0.0527046
$$361$$ −19.0000 −1.00000
$$362$$ 10.0000 0.525588
$$363$$ 11.0000 0.577350
$$364$$ 0 0
$$365$$ 2.00000 0.104685
$$366$$ 6.00000 0.313625
$$367$$ 20.0000 1.04399 0.521996 0.852948i $$-0.325188\pi$$
0.521996 + 0.852948i $$0.325188\pi$$
$$368$$ −4.00000 −0.208514
$$369$$ 2.00000 0.104116
$$370$$ −6.00000 −0.311925
$$371$$ 0 0
$$372$$ 0 0
$$373$$ 10.0000 0.517780 0.258890 0.965907i $$-0.416643\pi$$
0.258890 + 0.965907i $$0.416643\pi$$
$$374$$ 0 0
$$375$$ 1.00000 0.0516398
$$376$$ 0 0
$$377$$ 10.0000 0.515026
$$378$$ 0 0
$$379$$ 24.0000 1.23280 0.616399 0.787434i $$-0.288591\pi$$
0.616399 + 0.787434i $$0.288591\pi$$
$$380$$ 0 0
$$381$$ 12.0000 0.614779
$$382$$ −24.0000 −1.22795
$$383$$ −8.00000 −0.408781 −0.204390 0.978889i $$-0.565521\pi$$
−0.204390 + 0.978889i $$0.565521\pi$$
$$384$$ 1.00000 0.0510310
$$385$$ 0 0
$$386$$ −14.0000 −0.712581
$$387$$ −4.00000 −0.203331
$$388$$ 14.0000 0.710742
$$389$$ −18.0000 −0.912636 −0.456318 0.889817i $$-0.650832\pi$$
−0.456318 + 0.889817i $$0.650832\pi$$
$$390$$ 1.00000 0.0506370
$$391$$ 24.0000 1.21373
$$392$$ 7.00000 0.353553
$$393$$ −12.0000 −0.605320
$$394$$ 22.0000 1.10834
$$395$$ 0 0
$$396$$ 0 0
$$397$$ −14.0000 −0.702640 −0.351320 0.936255i $$-0.614267\pi$$
−0.351320 + 0.936255i $$0.614267\pi$$
$$398$$ 16.0000 0.802008
$$399$$ 0 0
$$400$$ 1.00000 0.0500000
$$401$$ 18.0000 0.898877 0.449439 0.893311i $$-0.351624\pi$$
0.449439 + 0.893311i $$0.351624\pi$$
$$402$$ 4.00000 0.199502
$$403$$ 0 0
$$404$$ 6.00000 0.298511
$$405$$ −1.00000 −0.0496904
$$406$$ 0 0
$$407$$ 0 0
$$408$$ −6.00000 −0.297044
$$409$$ 2.00000 0.0988936 0.0494468 0.998777i $$-0.484254\pi$$
0.0494468 + 0.998777i $$0.484254\pi$$
$$410$$ 2.00000 0.0987730
$$411$$ −6.00000 −0.295958
$$412$$ 12.0000 0.591198
$$413$$ 0 0
$$414$$ 4.00000 0.196589
$$415$$ −4.00000 −0.196352
$$416$$ 1.00000 0.0490290
$$417$$ −4.00000 −0.195881
$$418$$ 0 0
$$419$$ 12.0000 0.586238 0.293119 0.956076i $$-0.405307\pi$$
0.293119 + 0.956076i $$0.405307\pi$$
$$420$$ 0 0
$$421$$ −22.0000 −1.07221 −0.536107 0.844150i $$-0.680106\pi$$
−0.536107 + 0.844150i $$0.680106\pi$$
$$422$$ 12.0000 0.584151
$$423$$ 0 0
$$424$$ 6.00000 0.291386
$$425$$ −6.00000 −0.291043
$$426$$ 16.0000 0.775203
$$427$$ 0 0
$$428$$ −4.00000 −0.193347
$$429$$ 0 0
$$430$$ −4.00000 −0.192897
$$431$$ −24.0000 −1.15604 −0.578020 0.816023i $$-0.696174\pi$$
−0.578020 + 0.816023i $$0.696174\pi$$
$$432$$ −1.00000 −0.0481125
$$433$$ 2.00000 0.0961139 0.0480569 0.998845i $$-0.484697\pi$$
0.0480569 + 0.998845i $$0.484697\pi$$
$$434$$ 0 0
$$435$$ −10.0000 −0.479463
$$436$$ −14.0000 −0.670478
$$437$$ 0 0
$$438$$ −2.00000 −0.0955637
$$439$$ −24.0000 −1.14546 −0.572729 0.819745i $$-0.694115\pi$$
−0.572729 + 0.819745i $$0.694115\pi$$
$$440$$ 0 0
$$441$$ −7.00000 −0.333333
$$442$$ −6.00000 −0.285391
$$443$$ −4.00000 −0.190046 −0.0950229 0.995475i $$-0.530292\pi$$
−0.0950229 + 0.995475i $$0.530292\pi$$
$$444$$ 6.00000 0.284747
$$445$$ 6.00000 0.284427
$$446$$ −16.0000 −0.757622
$$447$$ 14.0000 0.662177
$$448$$ 0 0
$$449$$ −14.0000 −0.660701 −0.330350 0.943858i $$-0.607167\pi$$
−0.330350 + 0.943858i $$0.607167\pi$$
$$450$$ −1.00000 −0.0471405
$$451$$ 0 0
$$452$$ 10.0000 0.470360
$$453$$ 16.0000 0.751746
$$454$$ −20.0000 −0.938647
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 6.00000 0.280668 0.140334 0.990104i $$-0.455182\pi$$
0.140334 + 0.990104i $$0.455182\pi$$
$$458$$ −2.00000 −0.0934539
$$459$$ 6.00000 0.280056
$$460$$ 4.00000 0.186501
$$461$$ −14.0000 −0.652045 −0.326023 0.945362i $$-0.605709\pi$$
−0.326023 + 0.945362i $$0.605709\pi$$
$$462$$ 0 0
$$463$$ −24.0000 −1.11537 −0.557687 0.830051i $$-0.688311\pi$$
−0.557687 + 0.830051i $$0.688311\pi$$
$$464$$ −10.0000 −0.464238
$$465$$ 0 0
$$466$$ −18.0000 −0.833834
$$467$$ −28.0000 −1.29569 −0.647843 0.761774i $$-0.724329\pi$$
−0.647843 + 0.761774i $$0.724329\pi$$
$$468$$ −1.00000 −0.0462250
$$469$$ 0 0
$$470$$ 0 0
$$471$$ 14.0000 0.645086
$$472$$ 0 0
$$473$$ 0 0
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 0 0
$$477$$ −6.00000 −0.274721
$$478$$ −16.0000 −0.731823
$$479$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$480$$ −1.00000 −0.0456435
$$481$$ 6.00000 0.273576
$$482$$ 14.0000 0.637683
$$483$$ 0 0
$$484$$ −11.0000 −0.500000
$$485$$ −14.0000 −0.635707
$$486$$ 1.00000 0.0453609
$$487$$ 8.00000 0.362515 0.181257 0.983436i $$-0.441983\pi$$
0.181257 + 0.983436i $$0.441983\pi$$
$$488$$ −6.00000 −0.271607
$$489$$ 4.00000 0.180886
$$490$$ −7.00000 −0.316228
$$491$$ 20.0000 0.902587 0.451294 0.892375i $$-0.350963\pi$$
0.451294 + 0.892375i $$0.350963\pi$$
$$492$$ −2.00000 −0.0901670
$$493$$ 60.0000 2.70226
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 0 0
$$498$$ 4.00000 0.179244
$$499$$ −40.0000 −1.79065 −0.895323 0.445418i $$-0.853055\pi$$
−0.895323 + 0.445418i $$0.853055\pi$$
$$500$$ −1.00000 −0.0447214
$$501$$ −24.0000 −1.07224
$$502$$ 4.00000 0.178529
$$503$$ −36.0000 −1.60516 −0.802580 0.596544i $$-0.796540\pi$$
−0.802580 + 0.596544i $$0.796540\pi$$
$$504$$ 0 0
$$505$$ −6.00000 −0.266996
$$506$$ 0 0
$$507$$ −1.00000 −0.0444116
$$508$$ −12.0000 −0.532414
$$509$$ −38.0000 −1.68432 −0.842160 0.539227i $$-0.818716\pi$$
−0.842160 + 0.539227i $$0.818716\pi$$
$$510$$ 6.00000 0.265684
$$511$$ 0 0
$$512$$ −1.00000 −0.0441942
$$513$$ 0 0
$$514$$ 6.00000 0.264649
$$515$$ −12.0000 −0.528783
$$516$$ 4.00000 0.176090
$$517$$ 0 0
$$518$$ 0 0
$$519$$ 14.0000 0.614532
$$520$$ −1.00000 −0.0438529
$$521$$ 10.0000 0.438108 0.219054 0.975713i $$-0.429703\pi$$
0.219054 + 0.975713i $$0.429703\pi$$
$$522$$ 10.0000 0.437688
$$523$$ −20.0000 −0.874539 −0.437269 0.899331i $$-0.644054\pi$$
−0.437269 + 0.899331i $$0.644054\pi$$
$$524$$ 12.0000 0.524222
$$525$$ 0 0
$$526$$ 28.0000 1.22086
$$527$$ 0 0
$$528$$ 0 0
$$529$$ −7.00000 −0.304348
$$530$$ −6.00000 −0.260623
$$531$$ 0 0
$$532$$ 0 0
$$533$$ −2.00000 −0.0866296
$$534$$ −6.00000 −0.259645
$$535$$ 4.00000 0.172935
$$536$$ −4.00000 −0.172774
$$537$$ 20.0000 0.863064
$$538$$ −14.0000 −0.603583
$$539$$ 0 0
$$540$$ 1.00000 0.0430331
$$541$$ −22.0000 −0.945854 −0.472927 0.881102i $$-0.656803\pi$$
−0.472927 + 0.881102i $$0.656803\pi$$
$$542$$ 24.0000 1.03089
$$543$$ 10.0000 0.429141
$$544$$ 6.00000 0.257248
$$545$$ 14.0000 0.599694
$$546$$ 0 0
$$547$$ 28.0000 1.19719 0.598597 0.801050i $$-0.295725\pi$$
0.598597 + 0.801050i $$0.295725\pi$$
$$548$$ 6.00000 0.256307
$$549$$ 6.00000 0.256074
$$550$$ 0 0
$$551$$ 0 0
$$552$$ −4.00000 −0.170251
$$553$$ 0 0
$$554$$ −2.00000 −0.0849719
$$555$$ −6.00000 −0.254686
$$556$$ 4.00000 0.169638
$$557$$ −30.0000 −1.27114 −0.635570 0.772043i $$-0.719235\pi$$
−0.635570 + 0.772043i $$0.719235\pi$$
$$558$$ 0 0
$$559$$ 4.00000 0.169182
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 6.00000 0.253095
$$563$$ −28.0000 −1.18006 −0.590030 0.807382i $$-0.700884\pi$$
−0.590030 + 0.807382i $$0.700884\pi$$
$$564$$ 0 0
$$565$$ −10.0000 −0.420703
$$566$$ 12.0000 0.504398
$$567$$ 0 0
$$568$$ −16.0000 −0.671345
$$569$$ −6.00000 −0.251533 −0.125767 0.992060i $$-0.540139\pi$$
−0.125767 + 0.992060i $$0.540139\pi$$
$$570$$ 0 0
$$571$$ −4.00000 −0.167395 −0.0836974 0.996491i $$-0.526673\pi$$
−0.0836974 + 0.996491i $$0.526673\pi$$
$$572$$ 0 0
$$573$$ −24.0000 −1.00261
$$574$$ 0 0
$$575$$ −4.00000 −0.166812
$$576$$ 1.00000 0.0416667
$$577$$ 22.0000 0.915872 0.457936 0.888985i $$-0.348589\pi$$
0.457936 + 0.888985i $$0.348589\pi$$
$$578$$ −19.0000 −0.790296
$$579$$ −14.0000 −0.581820
$$580$$ 10.0000 0.415227
$$581$$ 0 0
$$582$$ 14.0000 0.580319
$$583$$ 0 0
$$584$$ 2.00000 0.0827606
$$585$$ 1.00000 0.0413449
$$586$$ −26.0000 −1.07405
$$587$$ −12.0000 −0.495293 −0.247647 0.968850i $$-0.579657\pi$$
−0.247647 + 0.968850i $$0.579657\pi$$
$$588$$ 7.00000 0.288675
$$589$$ 0 0
$$590$$ 0 0
$$591$$ 22.0000 0.904959
$$592$$ −6.00000 −0.246598
$$593$$ 6.00000 0.246390 0.123195 0.992382i $$-0.460686\pi$$
0.123195 + 0.992382i $$0.460686\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ −14.0000 −0.573462
$$597$$ 16.0000 0.654836
$$598$$ −4.00000 −0.163572
$$599$$ 24.0000 0.980613 0.490307 0.871550i $$-0.336885\pi$$
0.490307 + 0.871550i $$0.336885\pi$$
$$600$$ 1.00000 0.0408248
$$601$$ 42.0000 1.71322 0.856608 0.515968i $$-0.172568\pi$$
0.856608 + 0.515968i $$0.172568\pi$$
$$602$$ 0 0
$$603$$ 4.00000 0.162893
$$604$$ −16.0000 −0.651031
$$605$$ 11.0000 0.447214
$$606$$ 6.00000 0.243733
$$607$$ −36.0000 −1.46119 −0.730597 0.682808i $$-0.760758\pi$$
−0.730597 + 0.682808i $$0.760758\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 6.00000 0.242933
$$611$$ 0 0
$$612$$ −6.00000 −0.242536
$$613$$ −22.0000 −0.888572 −0.444286 0.895885i $$-0.646543\pi$$
−0.444286 + 0.895885i $$0.646543\pi$$
$$614$$ 4.00000 0.161427
$$615$$ 2.00000 0.0806478
$$616$$ 0 0
$$617$$ 30.0000 1.20775 0.603877 0.797077i $$-0.293622\pi$$
0.603877 + 0.797077i $$0.293622\pi$$
$$618$$ 12.0000 0.482711
$$619$$ 40.0000 1.60774 0.803868 0.594808i $$-0.202772\pi$$
0.803868 + 0.594808i $$0.202772\pi$$
$$620$$ 0 0
$$621$$ 4.00000 0.160514
$$622$$ 16.0000 0.641542
$$623$$ 0 0
$$624$$ 1.00000 0.0400320
$$625$$ 1.00000 0.0400000
$$626$$ −26.0000 −1.03917
$$627$$ 0 0
$$628$$ −14.0000 −0.558661
$$629$$ 36.0000 1.43541
$$630$$ 0 0
$$631$$ −48.0000 −1.91085 −0.955425 0.295234i $$-0.904602\pi$$
−0.955425 + 0.295234i $$0.904602\pi$$
$$632$$ 0 0
$$633$$ 12.0000 0.476957
$$634$$ −2.00000 −0.0794301
$$635$$ 12.0000 0.476205
$$636$$ 6.00000 0.237915
$$637$$ 7.00000 0.277350
$$638$$ 0 0
$$639$$ 16.0000 0.632950
$$640$$ 1.00000 0.0395285
$$641$$ 2.00000 0.0789953 0.0394976 0.999220i $$-0.487424\pi$$
0.0394976 + 0.999220i $$0.487424\pi$$
$$642$$ −4.00000 −0.157867
$$643$$ 4.00000 0.157745 0.0788723 0.996885i $$-0.474868\pi$$
0.0788723 + 0.996885i $$0.474868\pi$$
$$644$$ 0 0
$$645$$ −4.00000 −0.157500
$$646$$ 0 0
$$647$$ −36.0000 −1.41531 −0.707653 0.706560i $$-0.750246\pi$$
−0.707653 + 0.706560i $$0.750246\pi$$
$$648$$ −1.00000 −0.0392837
$$649$$ 0 0
$$650$$ 1.00000 0.0392232
$$651$$ 0 0
$$652$$ −4.00000 −0.156652
$$653$$ −6.00000 −0.234798 −0.117399 0.993085i $$-0.537456\pi$$
−0.117399 + 0.993085i $$0.537456\pi$$
$$654$$ −14.0000 −0.547443
$$655$$ −12.0000 −0.468879
$$656$$ 2.00000 0.0780869
$$657$$ −2.00000 −0.0780274
$$658$$ 0 0
$$659$$ −20.0000 −0.779089 −0.389545 0.921008i $$-0.627368\pi$$
−0.389545 + 0.921008i $$0.627368\pi$$
$$660$$ 0 0
$$661$$ −30.0000 −1.16686 −0.583432 0.812162i $$-0.698291\pi$$
−0.583432 + 0.812162i $$0.698291\pi$$
$$662$$ −8.00000 −0.310929
$$663$$ −6.00000 −0.233021
$$664$$ −4.00000 −0.155230
$$665$$ 0 0
$$666$$ 6.00000 0.232495
$$667$$ 40.0000 1.54881
$$668$$ 24.0000 0.928588
$$669$$ −16.0000 −0.618596
$$670$$ 4.00000 0.154533
$$671$$ 0 0
$$672$$ 0 0
$$673$$ 42.0000 1.61898 0.809491 0.587133i $$-0.199743\pi$$
0.809491 + 0.587133i $$0.199743\pi$$
$$674$$ −18.0000 −0.693334
$$675$$ −1.00000 −0.0384900
$$676$$ 1.00000 0.0384615
$$677$$ −30.0000 −1.15299 −0.576497 0.817099i $$-0.695581\pi$$
−0.576497 + 0.817099i $$0.695581\pi$$
$$678$$ 10.0000 0.384048
$$679$$ 0 0
$$680$$ −6.00000 −0.230089
$$681$$ −20.0000 −0.766402
$$682$$ 0 0
$$683$$ 12.0000 0.459167 0.229584 0.973289i $$-0.426264\pi$$
0.229584 + 0.973289i $$0.426264\pi$$
$$684$$ 0 0
$$685$$ −6.00000 −0.229248
$$686$$ 0 0
$$687$$ −2.00000 −0.0763048
$$688$$ −4.00000 −0.152499
$$689$$ 6.00000 0.228582
$$690$$ 4.00000 0.152277
$$691$$ 32.0000 1.21734 0.608669 0.793424i $$-0.291704\pi$$
0.608669 + 0.793424i $$0.291704\pi$$
$$692$$ −14.0000 −0.532200
$$693$$ 0 0
$$694$$ −12.0000 −0.455514
$$695$$ −4.00000 −0.151729
$$696$$ −10.0000 −0.379049
$$697$$ −12.0000 −0.454532
$$698$$ −34.0000 −1.28692
$$699$$ −18.0000 −0.680823
$$700$$ 0 0
$$701$$ −18.0000 −0.679851 −0.339925 0.940452i $$-0.610402\pi$$
−0.339925 + 0.940452i $$0.610402\pi$$
$$702$$ −1.00000 −0.0377426
$$703$$ 0 0
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 26.0000 0.978523
$$707$$ 0 0
$$708$$ 0 0
$$709$$ 2.00000 0.0751116 0.0375558 0.999295i $$-0.488043\pi$$
0.0375558 + 0.999295i $$0.488043\pi$$
$$710$$ 16.0000 0.600469
$$711$$ 0 0
$$712$$ 6.00000 0.224860
$$713$$ 0 0
$$714$$ 0 0
$$715$$ 0 0
$$716$$ −20.0000 −0.747435
$$717$$ −16.0000 −0.597531
$$718$$ −24.0000 −0.895672
$$719$$ 24.0000 0.895049 0.447524 0.894272i $$-0.352306\pi$$
0.447524 + 0.894272i $$0.352306\pi$$
$$720$$ −1.00000 −0.0372678
$$721$$ 0 0
$$722$$ 19.0000 0.707107
$$723$$ 14.0000 0.520666
$$724$$ −10.0000 −0.371647
$$725$$ −10.0000 −0.371391
$$726$$ −11.0000 −0.408248
$$727$$ 12.0000 0.445055 0.222528 0.974926i $$-0.428569\pi$$
0.222528 + 0.974926i $$0.428569\pi$$
$$728$$ 0 0
$$729$$ 1.00000 0.0370370
$$730$$ −2.00000 −0.0740233
$$731$$ 24.0000 0.887672
$$732$$ −6.00000 −0.221766
$$733$$ 50.0000 1.84679 0.923396 0.383849i $$-0.125402\pi$$
0.923396 + 0.383849i $$0.125402\pi$$
$$734$$ −20.0000 −0.738213
$$735$$ −7.00000 −0.258199
$$736$$ 4.00000 0.147442
$$737$$ 0 0
$$738$$ −2.00000 −0.0736210
$$739$$ −40.0000 −1.47142 −0.735712 0.677295i $$-0.763152\pi$$
−0.735712 + 0.677295i $$0.763152\pi$$
$$740$$ 6.00000 0.220564
$$741$$ 0 0
$$742$$ 0 0
$$743$$ −48.0000 −1.76095 −0.880475 0.474093i $$-0.842776\pi$$
−0.880475 + 0.474093i $$0.842776\pi$$
$$744$$ 0 0
$$745$$ 14.0000 0.512920
$$746$$ −10.0000 −0.366126
$$747$$ 4.00000 0.146352
$$748$$ 0 0
$$749$$ 0 0
$$750$$ −1.00000 −0.0365148
$$751$$ 24.0000 0.875772 0.437886 0.899030i $$-0.355727\pi$$
0.437886 + 0.899030i $$0.355727\pi$$
$$752$$ 0 0
$$753$$ 4.00000 0.145768
$$754$$ −10.0000 −0.364179
$$755$$ 16.0000 0.582300
$$756$$ 0 0
$$757$$ 2.00000 0.0726912 0.0363456 0.999339i $$-0.488428\pi$$
0.0363456 + 0.999339i $$0.488428\pi$$
$$758$$ −24.0000 −0.871719
$$759$$ 0 0
$$760$$ 0 0
$$761$$ −22.0000 −0.797499 −0.398750 0.917060i $$-0.630556\pi$$
−0.398750 + 0.917060i $$0.630556\pi$$
$$762$$ −12.0000 −0.434714
$$763$$ 0 0
$$764$$ 24.0000 0.868290
$$765$$ 6.00000 0.216930
$$766$$ 8.00000 0.289052
$$767$$ 0 0
$$768$$ −1.00000 −0.0360844
$$769$$ 34.0000 1.22607 0.613036 0.790055i $$-0.289948\pi$$
0.613036 + 0.790055i $$0.289948\pi$$
$$770$$ 0 0
$$771$$ 6.00000 0.216085
$$772$$ 14.0000 0.503871
$$773$$ 26.0000 0.935155 0.467578 0.883952i $$-0.345127\pi$$
0.467578 + 0.883952i $$0.345127\pi$$
$$774$$ 4.00000 0.143777
$$775$$ 0 0
$$776$$ −14.0000 −0.502571
$$777$$ 0 0
$$778$$ 18.0000 0.645331
$$779$$ 0 0
$$780$$ −1.00000 −0.0358057
$$781$$ 0 0
$$782$$ −24.0000 −0.858238
$$783$$ 10.0000 0.357371
$$784$$ −7.00000 −0.250000
$$785$$ 14.0000 0.499681
$$786$$ 12.0000 0.428026
$$787$$ −36.0000 −1.28326 −0.641631 0.767014i $$-0.721742\pi$$
−0.641631 + 0.767014i $$0.721742\pi$$
$$788$$ −22.0000 −0.783718
$$789$$ 28.0000 0.996826
$$790$$ 0 0
$$791$$ 0 0
$$792$$ 0 0
$$793$$ −6.00000 −0.213066
$$794$$ 14.0000 0.496841
$$795$$ −6.00000 −0.212798
$$796$$ −16.0000 −0.567105
$$797$$ 18.0000 0.637593 0.318796 0.947823i $$-0.396721\pi$$
0.318796 + 0.947823i $$0.396721\pi$$
$$798$$ 0 0
$$799$$ 0 0
$$800$$ −1.00000 −0.0353553
$$801$$ −6.00000 −0.212000
$$802$$ −18.0000 −0.635602
$$803$$ 0 0
$$804$$ −4.00000 −0.141069
$$805$$ 0 0
$$806$$ 0 0
$$807$$ −14.0000 −0.492823
$$808$$ −6.00000 −0.211079
$$809$$ 26.0000 0.914111 0.457056 0.889438i $$-0.348904\pi$$
0.457056 + 0.889438i $$0.348904\pi$$
$$810$$ 1.00000 0.0351364
$$811$$ 32.0000 1.12367 0.561836 0.827249i $$-0.310095\pi$$
0.561836 + 0.827249i $$0.310095\pi$$
$$812$$ 0 0
$$813$$ 24.0000 0.841717
$$814$$ 0 0
$$815$$ 4.00000 0.140114
$$816$$ 6.00000 0.210042
$$817$$ 0 0
$$818$$ −2.00000 −0.0699284
$$819$$ 0 0
$$820$$ −2.00000 −0.0698430
$$821$$ −22.0000 −0.767805 −0.383903 0.923374i $$-0.625420\pi$$
−0.383903 + 0.923374i $$0.625420\pi$$
$$822$$ 6.00000 0.209274
$$823$$ −44.0000 −1.53374 −0.766872 0.641800i $$-0.778188\pi$$
−0.766872 + 0.641800i $$0.778188\pi$$
$$824$$ −12.0000 −0.418040
$$825$$ 0 0
$$826$$ 0 0
$$827$$ −12.0000 −0.417281 −0.208640 0.977992i $$-0.566904\pi$$
−0.208640 + 0.977992i $$0.566904\pi$$
$$828$$ −4.00000 −0.139010
$$829$$ 54.0000 1.87550 0.937749 0.347314i $$-0.112906\pi$$
0.937749 + 0.347314i $$0.112906\pi$$
$$830$$ 4.00000 0.138842
$$831$$ −2.00000 −0.0693792
$$832$$ −1.00000 −0.0346688
$$833$$ 42.0000 1.45521
$$834$$ 4.00000 0.138509
$$835$$ −24.0000 −0.830554
$$836$$ 0 0
$$837$$ 0 0
$$838$$ −12.0000 −0.414533
$$839$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$840$$ 0 0
$$841$$ 71.0000 2.44828
$$842$$ 22.0000 0.758170
$$843$$ 6.00000 0.206651
$$844$$ −12.0000 −0.413057
$$845$$ −1.00000 −0.0344010
$$846$$ 0 0
$$847$$ 0 0
$$848$$ −6.00000 −0.206041
$$849$$ 12.0000 0.411839
$$850$$ 6.00000 0.205798
$$851$$ 24.0000 0.822709
$$852$$ −16.0000 −0.548151
$$853$$ −38.0000 −1.30110 −0.650548 0.759465i $$-0.725461\pi$$
−0.650548 + 0.759465i $$0.725461\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 4.00000 0.136717
$$857$$ −38.0000 −1.29806 −0.649028 0.760765i $$-0.724824\pi$$
−0.649028 + 0.760765i $$0.724824\pi$$
$$858$$ 0 0
$$859$$ −20.0000 −0.682391 −0.341196 0.939992i $$-0.610832\pi$$
−0.341196 + 0.939992i $$0.610832\pi$$
$$860$$ 4.00000 0.136399
$$861$$ 0 0
$$862$$ 24.0000 0.817443
$$863$$ 48.0000 1.63394 0.816970 0.576681i $$-0.195652\pi$$
0.816970 + 0.576681i $$0.195652\pi$$
$$864$$ 1.00000 0.0340207
$$865$$ 14.0000 0.476014
$$866$$ −2.00000 −0.0679628
$$867$$ −19.0000 −0.645274
$$868$$ 0 0
$$869$$ 0 0
$$870$$ 10.0000 0.339032
$$871$$ −4.00000 −0.135535
$$872$$ 14.0000 0.474100
$$873$$ 14.0000 0.473828
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 2.00000 0.0675737
$$877$$ −14.0000 −0.472746 −0.236373 0.971662i $$-0.575959\pi$$
−0.236373 + 0.971662i $$0.575959\pi$$
$$878$$ 24.0000 0.809961
$$879$$ −26.0000 −0.876958
$$880$$ 0 0
$$881$$ 34.0000 1.14549 0.572745 0.819734i $$-0.305879\pi$$
0.572745 + 0.819734i $$0.305879\pi$$
$$882$$ 7.00000 0.235702
$$883$$ 52.0000 1.74994 0.874970 0.484178i $$-0.160881\pi$$
0.874970 + 0.484178i $$0.160881\pi$$
$$884$$ 6.00000 0.201802
$$885$$ 0 0
$$886$$ 4.00000 0.134383
$$887$$ −36.0000 −1.20876 −0.604381 0.796696i $$-0.706579\pi$$
−0.604381 + 0.796696i $$0.706579\pi$$
$$888$$ −6.00000 −0.201347
$$889$$ 0 0
$$890$$ −6.00000 −0.201120
$$891$$ 0 0
$$892$$ 16.0000 0.535720
$$893$$ 0 0
$$894$$ −14.0000 −0.468230
$$895$$ 20.0000 0.668526
$$896$$ 0 0
$$897$$ −4.00000 −0.133556
$$898$$ 14.0000 0.467186
$$899$$ 0 0
$$900$$ 1.00000 0.0333333
$$901$$ 36.0000 1.19933
$$902$$ 0 0
$$903$$ 0 0
$$904$$ −10.0000 −0.332595
$$905$$ 10.0000 0.332411
$$906$$ −16.0000 −0.531564
$$907$$ 28.0000 0.929725 0.464862 0.885383i $$-0.346104\pi$$
0.464862 + 0.885383i $$0.346104\pi$$
$$908$$ 20.0000 0.663723
$$909$$ 6.00000 0.199007
$$910$$ 0 0
$$911$$ 40.0000 1.32526 0.662630 0.748947i $$-0.269440\pi$$
0.662630 + 0.748947i $$0.269440\pi$$
$$912$$ 0 0
$$913$$ 0 0
$$914$$ −6.00000 −0.198462
$$915$$ 6.00000 0.198354
$$916$$ 2.00000 0.0660819
$$917$$ 0 0
$$918$$ −6.00000 −0.198030
$$919$$ −16.0000 −0.527791 −0.263896 0.964551i $$-0.585007\pi$$
−0.263896 + 0.964551i $$0.585007\pi$$
$$920$$ −4.00000 −0.131876
$$921$$ 4.00000 0.131804
$$922$$ 14.0000 0.461065
$$923$$ −16.0000 −0.526646
$$924$$ 0 0
$$925$$ −6.00000 −0.197279
$$926$$ 24.0000 0.788689
$$927$$ 12.0000 0.394132
$$928$$ 10.0000 0.328266
$$929$$ −46.0000 −1.50921 −0.754606 0.656179i $$-0.772172\pi$$
−0.754606 + 0.656179i $$0.772172\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ 18.0000 0.589610
$$933$$ 16.0000 0.523816
$$934$$ 28.0000 0.916188
$$935$$ 0 0
$$936$$ 1.00000 0.0326860
$$937$$ −22.0000 −0.718709 −0.359354 0.933201i $$-0.617003\pi$$
−0.359354 + 0.933201i $$0.617003\pi$$
$$938$$ 0 0
$$939$$ −26.0000 −0.848478
$$940$$ 0 0
$$941$$ 10.0000 0.325991 0.162995 0.986627i $$-0.447884\pi$$
0.162995 + 0.986627i $$0.447884\pi$$
$$942$$ −14.0000 −0.456145
$$943$$ −8.00000 −0.260516
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ −36.0000 −1.16984 −0.584921 0.811090i $$-0.698875\pi$$
−0.584921 + 0.811090i $$0.698875\pi$$
$$948$$ 0 0
$$949$$ 2.00000 0.0649227
$$950$$ 0 0
$$951$$ −2.00000 −0.0648544
$$952$$ 0 0
$$953$$ 10.0000 0.323932 0.161966 0.986796i $$-0.448217\pi$$
0.161966 + 0.986796i $$0.448217\pi$$
$$954$$ 6.00000 0.194257
$$955$$ −24.0000 −0.776622
$$956$$ 16.0000 0.517477
$$957$$ 0 0
$$958$$ 0 0
$$959$$ 0 0
$$960$$ 1.00000 0.0322749
$$961$$ −31.0000 −1.00000
$$962$$ −6.00000 −0.193448
$$963$$ −4.00000 −0.128898
$$964$$ −14.0000 −0.450910
$$965$$ −14.0000 −0.450676
$$966$$ 0 0
$$967$$ −32.0000 −1.02905 −0.514525 0.857475i $$-0.672032\pi$$
−0.514525 + 0.857475i $$0.672032\pi$$
$$968$$ 11.0000 0.353553
$$969$$ 0 0
$$970$$ 14.0000 0.449513
$$971$$ 12.0000 0.385098 0.192549 0.981287i $$-0.438325\pi$$
0.192549 + 0.981287i $$0.438325\pi$$
$$972$$ −1.00000 −0.0320750
$$973$$ 0 0
$$974$$ −8.00000 −0.256337
$$975$$ 1.00000 0.0320256
$$976$$ 6.00000 0.192055
$$977$$ −2.00000 −0.0639857 −0.0319928 0.999488i $$-0.510185\pi$$
−0.0319928 + 0.999488i $$0.510185\pi$$
$$978$$ −4.00000 −0.127906
$$979$$ 0 0
$$980$$ 7.00000 0.223607
$$981$$ −14.0000 −0.446986
$$982$$ −20.0000 −0.638226
$$983$$ 16.0000 0.510321 0.255160 0.966899i $$-0.417872\pi$$
0.255160 + 0.966899i $$0.417872\pi$$
$$984$$ 2.00000 0.0637577
$$985$$ 22.0000 0.700978
$$986$$ −60.0000 −1.91079
$$987$$ 0 0
$$988$$ 0 0
$$989$$ 16.0000 0.508770
$$990$$ 0 0
$$991$$ −8.00000 −0.254128 −0.127064 0.991894i $$-0.540555\pi$$
−0.127064 + 0.991894i $$0.540555\pi$$
$$992$$ 0 0
$$993$$ −8.00000 −0.253872
$$994$$ 0 0
$$995$$ 16.0000 0.507234
$$996$$ −4.00000 −0.126745
$$997$$ −22.0000 −0.696747 −0.348373 0.937356i $$-0.613266\pi$$
−0.348373 + 0.937356i $$0.613266\pi$$
$$998$$ 40.0000 1.26618
$$999$$ 6.00000 0.189832
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 390.2.a.a.1.1 1
3.2 odd 2 1170.2.a.m.1.1 1
4.3 odd 2 3120.2.a.q.1.1 1
5.2 odd 4 1950.2.e.l.1249.1 2
5.3 odd 4 1950.2.e.l.1249.2 2
5.4 even 2 1950.2.a.y.1.1 1
12.11 even 2 9360.2.a.bn.1.1 1
13.5 odd 4 5070.2.b.c.1351.2 2
13.8 odd 4 5070.2.b.c.1351.1 2
13.12 even 2 5070.2.a.s.1.1 1
15.2 even 4 5850.2.e.p.5149.2 2
15.8 even 4 5850.2.e.p.5149.1 2
15.14 odd 2 5850.2.a.m.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
390.2.a.a.1.1 1 1.1 even 1 trivial
1170.2.a.m.1.1 1 3.2 odd 2
1950.2.a.y.1.1 1 5.4 even 2
1950.2.e.l.1249.1 2 5.2 odd 4
1950.2.e.l.1249.2 2 5.3 odd 4
3120.2.a.q.1.1 1 4.3 odd 2
5070.2.a.s.1.1 1 13.12 even 2
5070.2.b.c.1351.1 2 13.8 odd 4
5070.2.b.c.1351.2 2 13.5 odd 4
5850.2.a.m.1.1 1 15.14 odd 2
5850.2.e.p.5149.1 2 15.8 even 4
5850.2.e.p.5149.2 2 15.2 even 4
9360.2.a.bn.1.1 1 12.11 even 2