# Properties

 Label 3234.2.a.e.1.1 Level $3234$ Weight $2$ Character 3234.1 Self dual yes Analytic conductor $25.824$ 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] = [3234,2,Mod(1,3234)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3234, 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("3234.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$3234 = 2 \cdot 3 \cdot 7^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3234.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: $$25.8236200137$$ 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 $$\chi$$ $$=$$ 3234.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{11} -1.00000 q^{12} -4.00000 q^{13} +1.00000 q^{16} +4.00000 q^{17} -1.00000 q^{18} -1.00000 q^{22} +1.00000 q^{24} -5.00000 q^{25} +4.00000 q^{26} -1.00000 q^{27} +2.00000 q^{29} -4.00000 q^{31} -1.00000 q^{32} -1.00000 q^{33} -4.00000 q^{34} +1.00000 q^{36} +10.0000 q^{37} +4.00000 q^{39} -12.0000 q^{41} +4.00000 q^{43} +1.00000 q^{44} +4.00000 q^{47} -1.00000 q^{48} +5.00000 q^{50} -4.00000 q^{51} -4.00000 q^{52} -10.0000 q^{53} +1.00000 q^{54} -2.00000 q^{58} -4.00000 q^{59} +4.00000 q^{61} +4.00000 q^{62} +1.00000 q^{64} +1.00000 q^{66} +4.00000 q^{67} +4.00000 q^{68} -8.00000 q^{71} -1.00000 q^{72} +4.00000 q^{73} -10.0000 q^{74} +5.00000 q^{75} -4.00000 q^{78} +8.00000 q^{79} +1.00000 q^{81} +12.0000 q^{82} -16.0000 q^{83} -4.00000 q^{86} -2.00000 q^{87} -1.00000 q^{88} +8.00000 q^{89} +4.00000 q^{93} -4.00000 q^{94} +1.00000 q^{96} -8.00000 q^{97} +1.00000 q^{99} +O(q^{100})$$

## 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
$$5$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$6$$ 1.00000 0.408248
$$7$$ 0 0
$$8$$ −1.00000 −0.353553
$$9$$ 1.00000 0.333333
$$10$$ 0 0
$$11$$ 1.00000 0.301511
$$12$$ −1.00000 −0.288675
$$13$$ −4.00000 −1.10940 −0.554700 0.832050i $$-0.687167\pi$$
−0.554700 + 0.832050i $$0.687167\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ 4.00000 0.970143 0.485071 0.874475i $$-0.338794\pi$$
0.485071 + 0.874475i $$0.338794\pi$$
$$18$$ −1.00000 −0.235702
$$19$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ −1.00000 −0.213201
$$23$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$24$$ 1.00000 0.204124
$$25$$ −5.00000 −1.00000
$$26$$ 4.00000 0.784465
$$27$$ −1.00000 −0.192450
$$28$$ 0 0
$$29$$ 2.00000 0.371391 0.185695 0.982607i $$-0.440546\pi$$
0.185695 + 0.982607i $$0.440546\pi$$
$$30$$ 0 0
$$31$$ −4.00000 −0.718421 −0.359211 0.933257i $$-0.616954\pi$$
−0.359211 + 0.933257i $$0.616954\pi$$
$$32$$ −1.00000 −0.176777
$$33$$ −1.00000 −0.174078
$$34$$ −4.00000 −0.685994
$$35$$ 0 0
$$36$$ 1.00000 0.166667
$$37$$ 10.0000 1.64399 0.821995 0.569495i $$-0.192861\pi$$
0.821995 + 0.569495i $$0.192861\pi$$
$$38$$ 0 0
$$39$$ 4.00000 0.640513
$$40$$ 0 0
$$41$$ −12.0000 −1.87409 −0.937043 0.349215i $$-0.886448\pi$$
−0.937043 + 0.349215i $$0.886448\pi$$
$$42$$ 0 0
$$43$$ 4.00000 0.609994 0.304997 0.952353i $$-0.401344\pi$$
0.304997 + 0.952353i $$0.401344\pi$$
$$44$$ 1.00000 0.150756
$$45$$ 0 0
$$46$$ 0 0
$$47$$ 4.00000 0.583460 0.291730 0.956501i $$-0.405769\pi$$
0.291730 + 0.956501i $$0.405769\pi$$
$$48$$ −1.00000 −0.144338
$$49$$ 0 0
$$50$$ 5.00000 0.707107
$$51$$ −4.00000 −0.560112
$$52$$ −4.00000 −0.554700
$$53$$ −10.0000 −1.37361 −0.686803 0.726844i $$-0.740986\pi$$
−0.686803 + 0.726844i $$0.740986\pi$$
$$54$$ 1.00000 0.136083
$$55$$ 0 0
$$56$$ 0 0
$$57$$ 0 0
$$58$$ −2.00000 −0.262613
$$59$$ −4.00000 −0.520756 −0.260378 0.965507i $$-0.583847\pi$$
−0.260378 + 0.965507i $$0.583847\pi$$
$$60$$ 0 0
$$61$$ 4.00000 0.512148 0.256074 0.966657i $$-0.417571\pi$$
0.256074 + 0.966657i $$0.417571\pi$$
$$62$$ 4.00000 0.508001
$$63$$ 0 0
$$64$$ 1.00000 0.125000
$$65$$ 0 0
$$66$$ 1.00000 0.123091
$$67$$ 4.00000 0.488678 0.244339 0.969690i $$-0.421429\pi$$
0.244339 + 0.969690i $$0.421429\pi$$
$$68$$ 4.00000 0.485071
$$69$$ 0 0
$$70$$ 0 0
$$71$$ −8.00000 −0.949425 −0.474713 0.880141i $$-0.657448\pi$$
−0.474713 + 0.880141i $$0.657448\pi$$
$$72$$ −1.00000 −0.117851
$$73$$ 4.00000 0.468165 0.234082 0.972217i $$-0.424791\pi$$
0.234082 + 0.972217i $$0.424791\pi$$
$$74$$ −10.0000 −1.16248
$$75$$ 5.00000 0.577350
$$76$$ 0 0
$$77$$ 0 0
$$78$$ −4.00000 −0.452911
$$79$$ 8.00000 0.900070 0.450035 0.893011i $$-0.351411\pi$$
0.450035 + 0.893011i $$0.351411\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 12.0000 1.32518
$$83$$ −16.0000 −1.75623 −0.878114 0.478451i $$-0.841198\pi$$
−0.878114 + 0.478451i $$0.841198\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ −4.00000 −0.431331
$$87$$ −2.00000 −0.214423
$$88$$ −1.00000 −0.106600
$$89$$ 8.00000 0.847998 0.423999 0.905663i $$-0.360626\pi$$
0.423999 + 0.905663i $$0.360626\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 0 0
$$93$$ 4.00000 0.414781
$$94$$ −4.00000 −0.412568
$$95$$ 0 0
$$96$$ 1.00000 0.102062
$$97$$ −8.00000 −0.812277 −0.406138 0.913812i $$-0.633125\pi$$
−0.406138 + 0.913812i $$0.633125\pi$$
$$98$$ 0 0
$$99$$ 1.00000 0.100504
$$100$$ −5.00000 −0.500000
$$101$$ −12.0000 −1.19404 −0.597022 0.802225i $$-0.703650\pi$$
−0.597022 + 0.802225i $$0.703650\pi$$
$$102$$ 4.00000 0.396059
$$103$$ 4.00000 0.394132 0.197066 0.980390i $$-0.436859\pi$$
0.197066 + 0.980390i $$0.436859\pi$$
$$104$$ 4.00000 0.392232
$$105$$ 0 0
$$106$$ 10.0000 0.971286
$$107$$ −12.0000 −1.16008 −0.580042 0.814587i $$-0.696964\pi$$
−0.580042 + 0.814587i $$0.696964\pi$$
$$108$$ −1.00000 −0.0962250
$$109$$ 18.0000 1.72409 0.862044 0.506834i $$-0.169184\pi$$
0.862044 + 0.506834i $$0.169184\pi$$
$$110$$ 0 0
$$111$$ −10.0000 −0.949158
$$112$$ 0 0
$$113$$ 2.00000 0.188144 0.0940721 0.995565i $$-0.470012\pi$$
0.0940721 + 0.995565i $$0.470012\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 2.00000 0.185695
$$117$$ −4.00000 −0.369800
$$118$$ 4.00000 0.368230
$$119$$ 0 0
$$120$$ 0 0
$$121$$ 1.00000 0.0909091
$$122$$ −4.00000 −0.362143
$$123$$ 12.0000 1.08200
$$124$$ −4.00000 −0.359211
$$125$$ 0 0
$$126$$ 0 0
$$127$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$128$$ −1.00000 −0.0883883
$$129$$ −4.00000 −0.352180
$$130$$ 0 0
$$131$$ −8.00000 −0.698963 −0.349482 0.936943i $$-0.613642\pi$$
−0.349482 + 0.936943i $$0.613642\pi$$
$$132$$ −1.00000 −0.0870388
$$133$$ 0 0
$$134$$ −4.00000 −0.345547
$$135$$ 0 0
$$136$$ −4.00000 −0.342997
$$137$$ 6.00000 0.512615 0.256307 0.966595i $$-0.417494\pi$$
0.256307 + 0.966595i $$0.417494\pi$$
$$138$$ 0 0
$$139$$ −16.0000 −1.35710 −0.678551 0.734553i $$-0.737392\pi$$
−0.678551 + 0.734553i $$0.737392\pi$$
$$140$$ 0 0
$$141$$ −4.00000 −0.336861
$$142$$ 8.00000 0.671345
$$143$$ −4.00000 −0.334497
$$144$$ 1.00000 0.0833333
$$145$$ 0 0
$$146$$ −4.00000 −0.331042
$$147$$ 0 0
$$148$$ 10.0000 0.821995
$$149$$ −10.0000 −0.819232 −0.409616 0.912258i $$-0.634337\pi$$
−0.409616 + 0.912258i $$0.634337\pi$$
$$150$$ −5.00000 −0.408248
$$151$$ −8.00000 −0.651031 −0.325515 0.945537i $$-0.605538\pi$$
−0.325515 + 0.945537i $$0.605538\pi$$
$$152$$ 0 0
$$153$$ 4.00000 0.323381
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 4.00000 0.320256
$$157$$ 8.00000 0.638470 0.319235 0.947676i $$-0.396574\pi$$
0.319235 + 0.947676i $$0.396574\pi$$
$$158$$ −8.00000 −0.636446
$$159$$ 10.0000 0.793052
$$160$$ 0 0
$$161$$ 0 0
$$162$$ −1.00000 −0.0785674
$$163$$ 12.0000 0.939913 0.469956 0.882690i $$-0.344270\pi$$
0.469956 + 0.882690i $$0.344270\pi$$
$$164$$ −12.0000 −0.937043
$$165$$ 0 0
$$166$$ 16.0000 1.24184
$$167$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$168$$ 0 0
$$169$$ 3.00000 0.230769
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 4.00000 0.304997
$$173$$ 4.00000 0.304114 0.152057 0.988372i $$-0.451410\pi$$
0.152057 + 0.988372i $$0.451410\pi$$
$$174$$ 2.00000 0.151620
$$175$$ 0 0
$$176$$ 1.00000 0.0753778
$$177$$ 4.00000 0.300658
$$178$$ −8.00000 −0.599625
$$179$$ −4.00000 −0.298974 −0.149487 0.988764i $$-0.547762\pi$$
−0.149487 + 0.988764i $$0.547762\pi$$
$$180$$ 0 0
$$181$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$182$$ 0 0
$$183$$ −4.00000 −0.295689
$$184$$ 0 0
$$185$$ 0 0
$$186$$ −4.00000 −0.293294
$$187$$ 4.00000 0.292509
$$188$$ 4.00000 0.291730
$$189$$ 0 0
$$190$$ 0 0
$$191$$ −16.0000 −1.15772 −0.578860 0.815427i $$-0.696502\pi$$
−0.578860 + 0.815427i $$0.696502\pi$$
$$192$$ −1.00000 −0.0721688
$$193$$ −14.0000 −1.00774 −0.503871 0.863779i $$-0.668091\pi$$
−0.503871 + 0.863779i $$0.668091\pi$$
$$194$$ 8.00000 0.574367
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 6.00000 0.427482 0.213741 0.976890i $$-0.431435\pi$$
0.213741 + 0.976890i $$0.431435\pi$$
$$198$$ −1.00000 −0.0710669
$$199$$ −12.0000 −0.850657 −0.425329 0.905039i $$-0.639842\pi$$
−0.425329 + 0.905039i $$0.639842\pi$$
$$200$$ 5.00000 0.353553
$$201$$ −4.00000 −0.282138
$$202$$ 12.0000 0.844317
$$203$$ 0 0
$$204$$ −4.00000 −0.280056
$$205$$ 0 0
$$206$$ −4.00000 −0.278693
$$207$$ 0 0
$$208$$ −4.00000 −0.277350
$$209$$ 0 0
$$210$$ 0 0
$$211$$ −12.0000 −0.826114 −0.413057 0.910705i $$-0.635539\pi$$
−0.413057 + 0.910705i $$0.635539\pi$$
$$212$$ −10.0000 −0.686803
$$213$$ 8.00000 0.548151
$$214$$ 12.0000 0.820303
$$215$$ 0 0
$$216$$ 1.00000 0.0680414
$$217$$ 0 0
$$218$$ −18.0000 −1.21911
$$219$$ −4.00000 −0.270295
$$220$$ 0 0
$$221$$ −16.0000 −1.07628
$$222$$ 10.0000 0.671156
$$223$$ −12.0000 −0.803579 −0.401790 0.915732i $$-0.631612\pi$$
−0.401790 + 0.915732i $$0.631612\pi$$
$$224$$ 0 0
$$225$$ −5.00000 −0.333333
$$226$$ −2.00000 −0.133038
$$227$$ −8.00000 −0.530979 −0.265489 0.964114i $$-0.585534\pi$$
−0.265489 + 0.964114i $$0.585534\pi$$
$$228$$ 0 0
$$229$$ −16.0000 −1.05731 −0.528655 0.848837i $$-0.677303\pi$$
−0.528655 + 0.848837i $$0.677303\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ −2.00000 −0.131306
$$233$$ 6.00000 0.393073 0.196537 0.980497i $$-0.437031\pi$$
0.196537 + 0.980497i $$0.437031\pi$$
$$234$$ 4.00000 0.261488
$$235$$ 0 0
$$236$$ −4.00000 −0.260378
$$237$$ −8.00000 −0.519656
$$238$$ 0 0
$$239$$ 8.00000 0.517477 0.258738 0.965947i $$-0.416693\pi$$
0.258738 + 0.965947i $$0.416693\pi$$
$$240$$ 0 0
$$241$$ −20.0000 −1.28831 −0.644157 0.764894i $$-0.722792\pi$$
−0.644157 + 0.764894i $$0.722792\pi$$
$$242$$ −1.00000 −0.0642824
$$243$$ −1.00000 −0.0641500
$$244$$ 4.00000 0.256074
$$245$$ 0 0
$$246$$ −12.0000 −0.765092
$$247$$ 0 0
$$248$$ 4.00000 0.254000
$$249$$ 16.0000 1.01396
$$250$$ 0 0
$$251$$ 4.00000 0.252478 0.126239 0.992000i $$-0.459709\pi$$
0.126239 + 0.992000i $$0.459709\pi$$
$$252$$ 0 0
$$253$$ 0 0
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ −24.0000 −1.49708 −0.748539 0.663090i $$-0.769245\pi$$
−0.748539 + 0.663090i $$0.769245\pi$$
$$258$$ 4.00000 0.249029
$$259$$ 0 0
$$260$$ 0 0
$$261$$ 2.00000 0.123797
$$262$$ 8.00000 0.494242
$$263$$ −8.00000 −0.493301 −0.246651 0.969104i $$-0.579330\pi$$
−0.246651 + 0.969104i $$0.579330\pi$$
$$264$$ 1.00000 0.0615457
$$265$$ 0 0
$$266$$ 0 0
$$267$$ −8.00000 −0.489592
$$268$$ 4.00000 0.244339
$$269$$ −8.00000 −0.487769 −0.243884 0.969804i $$-0.578422\pi$$
−0.243884 + 0.969804i $$0.578422\pi$$
$$270$$ 0 0
$$271$$ −24.0000 −1.45790 −0.728948 0.684569i $$-0.759990\pi$$
−0.728948 + 0.684569i $$0.759990\pi$$
$$272$$ 4.00000 0.242536
$$273$$ 0 0
$$274$$ −6.00000 −0.362473
$$275$$ −5.00000 −0.301511
$$276$$ 0 0
$$277$$ −10.0000 −0.600842 −0.300421 0.953807i $$-0.597127\pi$$
−0.300421 + 0.953807i $$0.597127\pi$$
$$278$$ 16.0000 0.959616
$$279$$ −4.00000 −0.239474
$$280$$ 0 0
$$281$$ −10.0000 −0.596550 −0.298275 0.954480i $$-0.596411\pi$$
−0.298275 + 0.954480i $$0.596411\pi$$
$$282$$ 4.00000 0.238197
$$283$$ −24.0000 −1.42665 −0.713326 0.700832i $$-0.752812\pi$$
−0.713326 + 0.700832i $$0.752812\pi$$
$$284$$ −8.00000 −0.474713
$$285$$ 0 0
$$286$$ 4.00000 0.236525
$$287$$ 0 0
$$288$$ −1.00000 −0.0589256
$$289$$ −1.00000 −0.0588235
$$290$$ 0 0
$$291$$ 8.00000 0.468968
$$292$$ 4.00000 0.234082
$$293$$ −4.00000 −0.233682 −0.116841 0.993151i $$-0.537277\pi$$
−0.116841 + 0.993151i $$0.537277\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ −10.0000 −0.581238
$$297$$ −1.00000 −0.0580259
$$298$$ 10.0000 0.579284
$$299$$ 0 0
$$300$$ 5.00000 0.288675
$$301$$ 0 0
$$302$$ 8.00000 0.460348
$$303$$ 12.0000 0.689382
$$304$$ 0 0
$$305$$ 0 0
$$306$$ −4.00000 −0.228665
$$307$$ −8.00000 −0.456584 −0.228292 0.973593i $$-0.573314\pi$$
−0.228292 + 0.973593i $$0.573314\pi$$
$$308$$ 0 0
$$309$$ −4.00000 −0.227552
$$310$$ 0 0
$$311$$ 4.00000 0.226819 0.113410 0.993548i $$-0.463823\pi$$
0.113410 + 0.993548i $$0.463823\pi$$
$$312$$ −4.00000 −0.226455
$$313$$ 32.0000 1.80875 0.904373 0.426742i $$-0.140339\pi$$
0.904373 + 0.426742i $$0.140339\pi$$
$$314$$ −8.00000 −0.451466
$$315$$ 0 0
$$316$$ 8.00000 0.450035
$$317$$ −18.0000 −1.01098 −0.505490 0.862832i $$-0.668688\pi$$
−0.505490 + 0.862832i $$0.668688\pi$$
$$318$$ −10.0000 −0.560772
$$319$$ 2.00000 0.111979
$$320$$ 0 0
$$321$$ 12.0000 0.669775
$$322$$ 0 0
$$323$$ 0 0
$$324$$ 1.00000 0.0555556
$$325$$ 20.0000 1.10940
$$326$$ −12.0000 −0.664619
$$327$$ −18.0000 −0.995402
$$328$$ 12.0000 0.662589
$$329$$ 0 0
$$330$$ 0 0
$$331$$ 20.0000 1.09930 0.549650 0.835395i $$-0.314761\pi$$
0.549650 + 0.835395i $$0.314761\pi$$
$$332$$ −16.0000 −0.878114
$$333$$ 10.0000 0.547997
$$334$$ 0 0
$$335$$ 0 0
$$336$$ 0 0
$$337$$ −2.00000 −0.108947 −0.0544735 0.998515i $$-0.517348\pi$$
−0.0544735 + 0.998515i $$0.517348\pi$$
$$338$$ −3.00000 −0.163178
$$339$$ −2.00000 −0.108625
$$340$$ 0 0
$$341$$ −4.00000 −0.216612
$$342$$ 0 0
$$343$$ 0 0
$$344$$ −4.00000 −0.215666
$$345$$ 0 0
$$346$$ −4.00000 −0.215041
$$347$$ 12.0000 0.644194 0.322097 0.946707i $$-0.395612\pi$$
0.322097 + 0.946707i $$0.395612\pi$$
$$348$$ −2.00000 −0.107211
$$349$$ −28.0000 −1.49881 −0.749403 0.662114i $$-0.769659\pi$$
−0.749403 + 0.662114i $$0.769659\pi$$
$$350$$ 0 0
$$351$$ 4.00000 0.213504
$$352$$ −1.00000 −0.0533002
$$353$$ 24.0000 1.27739 0.638696 0.769460i $$-0.279474\pi$$
0.638696 + 0.769460i $$0.279474\pi$$
$$354$$ −4.00000 −0.212598
$$355$$ 0 0
$$356$$ 8.00000 0.423999
$$357$$ 0 0
$$358$$ 4.00000 0.211407
$$359$$ −32.0000 −1.68890 −0.844448 0.535638i $$-0.820071\pi$$
−0.844448 + 0.535638i $$0.820071\pi$$
$$360$$ 0 0
$$361$$ −19.0000 −1.00000
$$362$$ 0 0
$$363$$ −1.00000 −0.0524864
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 4.00000 0.209083
$$367$$ −12.0000 −0.626395 −0.313197 0.949688i $$-0.601400\pi$$
−0.313197 + 0.949688i $$0.601400\pi$$
$$368$$ 0 0
$$369$$ −12.0000 −0.624695
$$370$$ 0 0
$$371$$ 0 0
$$372$$ 4.00000 0.207390
$$373$$ 22.0000 1.13912 0.569558 0.821951i $$-0.307114\pi$$
0.569558 + 0.821951i $$0.307114\pi$$
$$374$$ −4.00000 −0.206835
$$375$$ 0 0
$$376$$ −4.00000 −0.206284
$$377$$ −8.00000 −0.412021
$$378$$ 0 0
$$379$$ 4.00000 0.205466 0.102733 0.994709i $$-0.467241\pi$$
0.102733 + 0.994709i $$0.467241\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 16.0000 0.818631
$$383$$ −4.00000 −0.204390 −0.102195 0.994764i $$-0.532587\pi$$
−0.102195 + 0.994764i $$0.532587\pi$$
$$384$$ 1.00000 0.0510310
$$385$$ 0 0
$$386$$ 14.0000 0.712581
$$387$$ 4.00000 0.203331
$$388$$ −8.00000 −0.406138
$$389$$ 10.0000 0.507020 0.253510 0.967333i $$-0.418415\pi$$
0.253510 + 0.967333i $$0.418415\pi$$
$$390$$ 0 0
$$391$$ 0 0
$$392$$ 0 0
$$393$$ 8.00000 0.403547
$$394$$ −6.00000 −0.302276
$$395$$ 0 0
$$396$$ 1.00000 0.0502519
$$397$$ −8.00000 −0.401508 −0.200754 0.979642i $$-0.564339\pi$$
−0.200754 + 0.979642i $$0.564339\pi$$
$$398$$ 12.0000 0.601506
$$399$$ 0 0
$$400$$ −5.00000 −0.250000
$$401$$ −34.0000 −1.69788 −0.848939 0.528490i $$-0.822758\pi$$
−0.848939 + 0.528490i $$0.822758\pi$$
$$402$$ 4.00000 0.199502
$$403$$ 16.0000 0.797017
$$404$$ −12.0000 −0.597022
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 10.0000 0.495682
$$408$$ 4.00000 0.198030
$$409$$ 20.0000 0.988936 0.494468 0.869196i $$-0.335363\pi$$
0.494468 + 0.869196i $$0.335363\pi$$
$$410$$ 0 0
$$411$$ −6.00000 −0.295958
$$412$$ 4.00000 0.197066
$$413$$ 0 0
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 4.00000 0.196116
$$417$$ 16.0000 0.783523
$$418$$ 0 0
$$419$$ −36.0000 −1.75872 −0.879358 0.476162i $$-0.842028\pi$$
−0.879358 + 0.476162i $$0.842028\pi$$
$$420$$ 0 0
$$421$$ 22.0000 1.07221 0.536107 0.844150i $$-0.319894\pi$$
0.536107 + 0.844150i $$0.319894\pi$$
$$422$$ 12.0000 0.584151
$$423$$ 4.00000 0.194487
$$424$$ 10.0000 0.485643
$$425$$ −20.0000 −0.970143
$$426$$ −8.00000 −0.387601
$$427$$ 0 0
$$428$$ −12.0000 −0.580042
$$429$$ 4.00000 0.193122
$$430$$ 0 0
$$431$$ −40.0000 −1.92673 −0.963366 0.268190i $$-0.913575\pi$$
−0.963366 + 0.268190i $$0.913575\pi$$
$$432$$ −1.00000 −0.0481125
$$433$$ 40.0000 1.92228 0.961139 0.276066i $$-0.0890309\pi$$
0.961139 + 0.276066i $$0.0890309\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 18.0000 0.862044
$$437$$ 0 0
$$438$$ 4.00000 0.191127
$$439$$ −40.0000 −1.90910 −0.954548 0.298057i $$-0.903661\pi$$
−0.954548 + 0.298057i $$0.903661\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 16.0000 0.761042
$$443$$ 4.00000 0.190046 0.0950229 0.995475i $$-0.469708\pi$$
0.0950229 + 0.995475i $$0.469708\pi$$
$$444$$ −10.0000 −0.474579
$$445$$ 0 0
$$446$$ 12.0000 0.568216
$$447$$ 10.0000 0.472984
$$448$$ 0 0
$$449$$ 18.0000 0.849473 0.424736 0.905317i $$-0.360367\pi$$
0.424736 + 0.905317i $$0.360367\pi$$
$$450$$ 5.00000 0.235702
$$451$$ −12.0000 −0.565058
$$452$$ 2.00000 0.0940721
$$453$$ 8.00000 0.375873
$$454$$ 8.00000 0.375459
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 26.0000 1.21623 0.608114 0.793849i $$-0.291926\pi$$
0.608114 + 0.793849i $$0.291926\pi$$
$$458$$ 16.0000 0.747631
$$459$$ −4.00000 −0.186704
$$460$$ 0 0
$$461$$ 28.0000 1.30409 0.652045 0.758180i $$-0.273911\pi$$
0.652045 + 0.758180i $$0.273911\pi$$
$$462$$ 0 0
$$463$$ −40.0000 −1.85896 −0.929479 0.368875i $$-0.879743\pi$$
−0.929479 + 0.368875i $$0.879743\pi$$
$$464$$ 2.00000 0.0928477
$$465$$ 0 0
$$466$$ −6.00000 −0.277945
$$467$$ 36.0000 1.66588 0.832941 0.553362i $$-0.186655\pi$$
0.832941 + 0.553362i $$0.186655\pi$$
$$468$$ −4.00000 −0.184900
$$469$$ 0 0
$$470$$ 0 0
$$471$$ −8.00000 −0.368621
$$472$$ 4.00000 0.184115
$$473$$ 4.00000 0.183920
$$474$$ 8.00000 0.367452
$$475$$ 0 0
$$476$$ 0 0
$$477$$ −10.0000 −0.457869
$$478$$ −8.00000 −0.365911
$$479$$ 24.0000 1.09659 0.548294 0.836286i $$-0.315277\pi$$
0.548294 + 0.836286i $$0.315277\pi$$
$$480$$ 0 0
$$481$$ −40.0000 −1.82384
$$482$$ 20.0000 0.910975
$$483$$ 0 0
$$484$$ 1.00000 0.0454545
$$485$$ 0 0
$$486$$ 1.00000 0.0453609
$$487$$ −8.00000 −0.362515 −0.181257 0.983436i $$-0.558017\pi$$
−0.181257 + 0.983436i $$0.558017\pi$$
$$488$$ −4.00000 −0.181071
$$489$$ −12.0000 −0.542659
$$490$$ 0 0
$$491$$ 20.0000 0.902587 0.451294 0.892375i $$-0.350963\pi$$
0.451294 + 0.892375i $$0.350963\pi$$
$$492$$ 12.0000 0.541002
$$493$$ 8.00000 0.360302
$$494$$ 0 0
$$495$$ 0 0
$$496$$ −4.00000 −0.179605
$$497$$ 0 0
$$498$$ −16.0000 −0.716977
$$499$$ 20.0000 0.895323 0.447661 0.894203i $$-0.352257\pi$$
0.447661 + 0.894203i $$0.352257\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ −4.00000 −0.178529
$$503$$ −24.0000 −1.07011 −0.535054 0.844818i $$-0.679709\pi$$
−0.535054 + 0.844818i $$0.679709\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 0 0
$$507$$ −3.00000 −0.133235
$$508$$ 0 0
$$509$$ −24.0000 −1.06378 −0.531891 0.846813i $$-0.678518\pi$$
−0.531891 + 0.846813i $$0.678518\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ −1.00000 −0.0441942
$$513$$ 0 0
$$514$$ 24.0000 1.05859
$$515$$ 0 0
$$516$$ −4.00000 −0.176090
$$517$$ 4.00000 0.175920
$$518$$ 0 0
$$519$$ −4.00000 −0.175581
$$520$$ 0 0
$$521$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$522$$ −2.00000 −0.0875376
$$523$$ 8.00000 0.349816 0.174908 0.984585i $$-0.444037\pi$$
0.174908 + 0.984585i $$0.444037\pi$$
$$524$$ −8.00000 −0.349482
$$525$$ 0 0
$$526$$ 8.00000 0.348817
$$527$$ −16.0000 −0.696971
$$528$$ −1.00000 −0.0435194
$$529$$ −23.0000 −1.00000
$$530$$ 0 0
$$531$$ −4.00000 −0.173585
$$532$$ 0 0
$$533$$ 48.0000 2.07911
$$534$$ 8.00000 0.346194
$$535$$ 0 0
$$536$$ −4.00000 −0.172774
$$537$$ 4.00000 0.172613
$$538$$ 8.00000 0.344904
$$539$$ 0 0
$$540$$ 0 0
$$541$$ −34.0000 −1.46177 −0.730887 0.682498i $$-0.760893\pi$$
−0.730887 + 0.682498i $$0.760893\pi$$
$$542$$ 24.0000 1.03089
$$543$$ 0 0
$$544$$ −4.00000 −0.171499
$$545$$ 0 0
$$546$$ 0 0
$$547$$ −36.0000 −1.53925 −0.769624 0.638497i $$-0.779557\pi$$
−0.769624 + 0.638497i $$0.779557\pi$$
$$548$$ 6.00000 0.256307
$$549$$ 4.00000 0.170716
$$550$$ 5.00000 0.213201
$$551$$ 0 0
$$552$$ 0 0
$$553$$ 0 0
$$554$$ 10.0000 0.424859
$$555$$ 0 0
$$556$$ −16.0000 −0.678551
$$557$$ 14.0000 0.593199 0.296600 0.955002i $$-0.404147\pi$$
0.296600 + 0.955002i $$0.404147\pi$$
$$558$$ 4.00000 0.169334
$$559$$ −16.0000 −0.676728
$$560$$ 0 0
$$561$$ −4.00000 −0.168880
$$562$$ 10.0000 0.421825
$$563$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$564$$ −4.00000 −0.168430
$$565$$ 0 0
$$566$$ 24.0000 1.00880
$$567$$ 0 0
$$568$$ 8.00000 0.335673
$$569$$ 6.00000 0.251533 0.125767 0.992060i $$-0.459861\pi$$
0.125767 + 0.992060i $$0.459861\pi$$
$$570$$ 0 0
$$571$$ −12.0000 −0.502184 −0.251092 0.967963i $$-0.580790\pi$$
−0.251092 + 0.967963i $$0.580790\pi$$
$$572$$ −4.00000 −0.167248
$$573$$ 16.0000 0.668410
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 1.00000 0.0416667
$$577$$ 40.0000 1.66522 0.832611 0.553858i $$-0.186845\pi$$
0.832611 + 0.553858i $$0.186845\pi$$
$$578$$ 1.00000 0.0415945
$$579$$ 14.0000 0.581820
$$580$$ 0 0
$$581$$ 0 0
$$582$$ −8.00000 −0.331611
$$583$$ −10.0000 −0.414158
$$584$$ −4.00000 −0.165521
$$585$$ 0 0
$$586$$ 4.00000 0.165238
$$587$$ 36.0000 1.48588 0.742940 0.669359i $$-0.233431\pi$$
0.742940 + 0.669359i $$0.233431\pi$$
$$588$$ 0 0
$$589$$ 0 0
$$590$$ 0 0
$$591$$ −6.00000 −0.246807
$$592$$ 10.0000 0.410997
$$593$$ −36.0000 −1.47834 −0.739171 0.673517i $$-0.764783\pi$$
−0.739171 + 0.673517i $$0.764783\pi$$
$$594$$ 1.00000 0.0410305
$$595$$ 0 0
$$596$$ −10.0000 −0.409616
$$597$$ 12.0000 0.491127
$$598$$ 0 0
$$599$$ −24.0000 −0.980613 −0.490307 0.871550i $$-0.663115\pi$$
−0.490307 + 0.871550i $$0.663115\pi$$
$$600$$ −5.00000 −0.204124
$$601$$ 28.0000 1.14214 0.571072 0.820900i $$-0.306528\pi$$
0.571072 + 0.820900i $$0.306528\pi$$
$$602$$ 0 0
$$603$$ 4.00000 0.162893
$$604$$ −8.00000 −0.325515
$$605$$ 0 0
$$606$$ −12.0000 −0.487467
$$607$$ 16.0000 0.649420 0.324710 0.945814i $$-0.394733\pi$$
0.324710 + 0.945814i $$0.394733\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ −16.0000 −0.647291
$$612$$ 4.00000 0.161690
$$613$$ 26.0000 1.05013 0.525065 0.851062i $$-0.324041\pi$$
0.525065 + 0.851062i $$0.324041\pi$$
$$614$$ 8.00000 0.322854
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 22.0000 0.885687 0.442843 0.896599i $$-0.353970\pi$$
0.442843 + 0.896599i $$0.353970\pi$$
$$618$$ 4.00000 0.160904
$$619$$ 36.0000 1.44696 0.723481 0.690344i $$-0.242541\pi$$
0.723481 + 0.690344i $$0.242541\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ −4.00000 −0.160385
$$623$$ 0 0
$$624$$ 4.00000 0.160128
$$625$$ 25.0000 1.00000
$$626$$ −32.0000 −1.27898
$$627$$ 0 0
$$628$$ 8.00000 0.319235
$$629$$ 40.0000 1.59490
$$630$$ 0 0
$$631$$ −16.0000 −0.636950 −0.318475 0.947931i $$-0.603171\pi$$
−0.318475 + 0.947931i $$0.603171\pi$$
$$632$$ −8.00000 −0.318223
$$633$$ 12.0000 0.476957
$$634$$ 18.0000 0.714871
$$635$$ 0 0
$$636$$ 10.0000 0.396526
$$637$$ 0 0
$$638$$ −2.00000 −0.0791808
$$639$$ −8.00000 −0.316475
$$640$$ 0 0
$$641$$ −2.00000 −0.0789953 −0.0394976 0.999220i $$-0.512576\pi$$
−0.0394976 + 0.999220i $$0.512576\pi$$
$$642$$ −12.0000 −0.473602
$$643$$ 20.0000 0.788723 0.394362 0.918955i $$-0.370966\pi$$
0.394362 + 0.918955i $$0.370966\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ −20.0000 −0.786281 −0.393141 0.919478i $$-0.628611\pi$$
−0.393141 + 0.919478i $$0.628611\pi$$
$$648$$ −1.00000 −0.0392837
$$649$$ −4.00000 −0.157014
$$650$$ −20.0000 −0.784465
$$651$$ 0 0
$$652$$ 12.0000 0.469956
$$653$$ −30.0000 −1.17399 −0.586995 0.809590i $$-0.699689\pi$$
−0.586995 + 0.809590i $$0.699689\pi$$
$$654$$ 18.0000 0.703856
$$655$$ 0 0
$$656$$ −12.0000 −0.468521
$$657$$ 4.00000 0.156055
$$658$$ 0 0
$$659$$ 4.00000 0.155818 0.0779089 0.996960i $$-0.475176\pi$$
0.0779089 + 0.996960i $$0.475176\pi$$
$$660$$ 0 0
$$661$$ 40.0000 1.55582 0.777910 0.628376i $$-0.216280\pi$$
0.777910 + 0.628376i $$0.216280\pi$$
$$662$$ −20.0000 −0.777322
$$663$$ 16.0000 0.621389
$$664$$ 16.0000 0.620920
$$665$$ 0 0
$$666$$ −10.0000 −0.387492
$$667$$ 0 0
$$668$$ 0 0
$$669$$ 12.0000 0.463947
$$670$$ 0 0
$$671$$ 4.00000 0.154418
$$672$$ 0 0
$$673$$ −2.00000 −0.0770943 −0.0385472 0.999257i $$-0.512273\pi$$
−0.0385472 + 0.999257i $$0.512273\pi$$
$$674$$ 2.00000 0.0770371
$$675$$ 5.00000 0.192450
$$676$$ 3.00000 0.115385
$$677$$ 4.00000 0.153732 0.0768662 0.997041i $$-0.475509\pi$$
0.0768662 + 0.997041i $$0.475509\pi$$
$$678$$ 2.00000 0.0768095
$$679$$ 0 0
$$680$$ 0 0
$$681$$ 8.00000 0.306561
$$682$$ 4.00000 0.153168
$$683$$ 4.00000 0.153056 0.0765279 0.997067i $$-0.475617\pi$$
0.0765279 + 0.997067i $$0.475617\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 0 0
$$687$$ 16.0000 0.610438
$$688$$ 4.00000 0.152499
$$689$$ 40.0000 1.52388
$$690$$ 0 0
$$691$$ 28.0000 1.06517 0.532585 0.846376i $$-0.321221\pi$$
0.532585 + 0.846376i $$0.321221\pi$$
$$692$$ 4.00000 0.152057
$$693$$ 0 0
$$694$$ −12.0000 −0.455514
$$695$$ 0 0
$$696$$ 2.00000 0.0758098
$$697$$ −48.0000 −1.81813
$$698$$ 28.0000 1.05982
$$699$$ −6.00000 −0.226941
$$700$$ 0 0
$$701$$ 2.00000 0.0755390 0.0377695 0.999286i $$-0.487975\pi$$
0.0377695 + 0.999286i $$0.487975\pi$$
$$702$$ −4.00000 −0.150970
$$703$$ 0 0
$$704$$ 1.00000 0.0376889
$$705$$ 0 0
$$706$$ −24.0000 −0.903252
$$707$$ 0 0
$$708$$ 4.00000 0.150329
$$709$$ 26.0000 0.976450 0.488225 0.872718i $$-0.337644\pi$$
0.488225 + 0.872718i $$0.337644\pi$$
$$710$$ 0 0
$$711$$ 8.00000 0.300023
$$712$$ −8.00000 −0.299813
$$713$$ 0 0
$$714$$ 0 0
$$715$$ 0 0
$$716$$ −4.00000 −0.149487
$$717$$ −8.00000 −0.298765
$$718$$ 32.0000 1.19423
$$719$$ −36.0000 −1.34257 −0.671287 0.741198i $$-0.734258\pi$$
−0.671287 + 0.741198i $$0.734258\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ 19.0000 0.707107
$$723$$ 20.0000 0.743808
$$724$$ 0 0
$$725$$ −10.0000 −0.371391
$$726$$ 1.00000 0.0371135
$$727$$ 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
$$730$$ 0 0
$$731$$ 16.0000 0.591781
$$732$$ −4.00000 −0.147844
$$733$$ 4.00000 0.147743 0.0738717 0.997268i $$-0.476464\pi$$
0.0738717 + 0.997268i $$0.476464\pi$$
$$734$$ 12.0000 0.442928
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 4.00000 0.147342
$$738$$ 12.0000 0.441726
$$739$$ 28.0000 1.03000 0.514998 0.857191i $$-0.327793\pi$$
0.514998 + 0.857191i $$0.327793\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ 32.0000 1.17397 0.586983 0.809599i $$-0.300316\pi$$
0.586983 + 0.809599i $$0.300316\pi$$
$$744$$ −4.00000 −0.146647
$$745$$ 0 0
$$746$$ −22.0000 −0.805477
$$747$$ −16.0000 −0.585409
$$748$$ 4.00000 0.146254
$$749$$ 0 0
$$750$$ 0 0
$$751$$ 32.0000 1.16770 0.583848 0.811863i $$-0.301546\pi$$
0.583848 + 0.811863i $$0.301546\pi$$
$$752$$ 4.00000 0.145865
$$753$$ −4.00000 −0.145768
$$754$$ 8.00000 0.291343
$$755$$ 0 0
$$756$$ 0 0
$$757$$ −22.0000 −0.799604 −0.399802 0.916602i $$-0.630921\pi$$
−0.399802 + 0.916602i $$0.630921\pi$$
$$758$$ −4.00000 −0.145287
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 44.0000 1.59500 0.797499 0.603320i $$-0.206156\pi$$
0.797499 + 0.603320i $$0.206156\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ −16.0000 −0.578860
$$765$$ 0 0
$$766$$ 4.00000 0.144526
$$767$$ 16.0000 0.577727
$$768$$ −1.00000 −0.0360844
$$769$$ 20.0000 0.721218 0.360609 0.932717i $$-0.382569\pi$$
0.360609 + 0.932717i $$0.382569\pi$$
$$770$$ 0 0
$$771$$ 24.0000 0.864339
$$772$$ −14.0000 −0.503871
$$773$$ 40.0000 1.43870 0.719350 0.694648i $$-0.244440\pi$$
0.719350 + 0.694648i $$0.244440\pi$$
$$774$$ −4.00000 −0.143777
$$775$$ 20.0000 0.718421
$$776$$ 8.00000 0.287183
$$777$$ 0 0
$$778$$ −10.0000 −0.358517
$$779$$ 0 0
$$780$$ 0 0
$$781$$ −8.00000 −0.286263
$$782$$ 0 0
$$783$$ −2.00000 −0.0714742
$$784$$ 0 0
$$785$$ 0 0
$$786$$ −8.00000 −0.285351
$$787$$ 32.0000 1.14068 0.570338 0.821410i $$-0.306812\pi$$
0.570338 + 0.821410i $$0.306812\pi$$
$$788$$ 6.00000 0.213741
$$789$$ 8.00000 0.284808
$$790$$ 0 0
$$791$$ 0 0
$$792$$ −1.00000 −0.0355335
$$793$$ −16.0000 −0.568177
$$794$$ 8.00000 0.283909
$$795$$ 0 0
$$796$$ −12.0000 −0.425329
$$797$$ 16.0000 0.566749 0.283375 0.959009i $$-0.408546\pi$$
0.283375 + 0.959009i $$0.408546\pi$$
$$798$$ 0 0
$$799$$ 16.0000 0.566039
$$800$$ 5.00000 0.176777
$$801$$ 8.00000 0.282666
$$802$$ 34.0000 1.20058
$$803$$ 4.00000 0.141157
$$804$$ −4.00000 −0.141069
$$805$$ 0 0
$$806$$ −16.0000 −0.563576
$$807$$ 8.00000 0.281613
$$808$$ 12.0000 0.422159
$$809$$ 10.0000 0.351581 0.175791 0.984428i $$-0.443752\pi$$
0.175791 + 0.984428i $$0.443752\pi$$
$$810$$ 0 0
$$811$$ −8.00000 −0.280918 −0.140459 0.990086i $$-0.544858\pi$$
−0.140459 + 0.990086i $$0.544858\pi$$
$$812$$ 0 0
$$813$$ 24.0000 0.841717
$$814$$ −10.0000 −0.350500
$$815$$ 0 0
$$816$$ −4.00000 −0.140028
$$817$$ 0 0
$$818$$ −20.0000 −0.699284
$$819$$ 0 0
$$820$$ 0 0
$$821$$ 54.0000 1.88461 0.942306 0.334751i $$-0.108652\pi$$
0.942306 + 0.334751i $$0.108652\pi$$
$$822$$ 6.00000 0.209274
$$823$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$824$$ −4.00000 −0.139347
$$825$$ 5.00000 0.174078
$$826$$ 0 0
$$827$$ −28.0000 −0.973655 −0.486828 0.873498i $$-0.661846\pi$$
−0.486828 + 0.873498i $$0.661846\pi$$
$$828$$ 0 0
$$829$$ −24.0000 −0.833554 −0.416777 0.909009i $$-0.636840\pi$$
−0.416777 + 0.909009i $$0.636840\pi$$
$$830$$ 0 0
$$831$$ 10.0000 0.346896
$$832$$ −4.00000 −0.138675
$$833$$ 0 0
$$834$$ −16.0000 −0.554035
$$835$$ 0 0
$$836$$ 0 0
$$837$$ 4.00000 0.138260
$$838$$ 36.0000 1.24360
$$839$$ −36.0000 −1.24286 −0.621429 0.783470i $$-0.713448\pi$$
−0.621429 + 0.783470i $$0.713448\pi$$
$$840$$ 0 0
$$841$$ −25.0000 −0.862069
$$842$$ −22.0000 −0.758170
$$843$$ 10.0000 0.344418
$$844$$ −12.0000 −0.413057
$$845$$ 0 0
$$846$$ −4.00000 −0.137523
$$847$$ 0 0
$$848$$ −10.0000 −0.343401
$$849$$ 24.0000 0.823678
$$850$$ 20.0000 0.685994
$$851$$ 0 0
$$852$$ 8.00000 0.274075
$$853$$ −4.00000 −0.136957 −0.0684787 0.997653i $$-0.521815\pi$$
−0.0684787 + 0.997653i $$0.521815\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 12.0000 0.410152
$$857$$ −4.00000 −0.136637 −0.0683187 0.997664i $$-0.521763\pi$$
−0.0683187 + 0.997664i $$0.521763\pi$$
$$858$$ −4.00000 −0.136558
$$859$$ 52.0000 1.77422 0.887109 0.461561i $$-0.152710\pi$$
0.887109 + 0.461561i $$0.152710\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 40.0000 1.36241
$$863$$ −32.0000 −1.08929 −0.544646 0.838666i $$-0.683336\pi$$
−0.544646 + 0.838666i $$0.683336\pi$$
$$864$$ 1.00000 0.0340207
$$865$$ 0 0
$$866$$ −40.0000 −1.35926
$$867$$ 1.00000 0.0339618
$$868$$ 0 0
$$869$$ 8.00000 0.271381
$$870$$ 0 0
$$871$$ −16.0000 −0.542139
$$872$$ −18.0000 −0.609557
$$873$$ −8.00000 −0.270759
$$874$$ 0 0
$$875$$ 0 0
$$876$$ −4.00000 −0.135147
$$877$$ −46.0000 −1.55331 −0.776655 0.629926i $$-0.783085\pi$$
−0.776655 + 0.629926i $$0.783085\pi$$
$$878$$ 40.0000 1.34993
$$879$$ 4.00000 0.134917
$$880$$ 0 0
$$881$$ 16.0000 0.539054 0.269527 0.962993i $$-0.413133\pi$$
0.269527 + 0.962993i $$0.413133\pi$$
$$882$$ 0 0
$$883$$ 4.00000 0.134611 0.0673054 0.997732i $$-0.478560\pi$$
0.0673054 + 0.997732i $$0.478560\pi$$
$$884$$ −16.0000 −0.538138
$$885$$ 0 0
$$886$$ −4.00000 −0.134383
$$887$$ −8.00000 −0.268614 −0.134307 0.990940i $$-0.542881\pi$$
−0.134307 + 0.990940i $$0.542881\pi$$
$$888$$ 10.0000 0.335578
$$889$$ 0 0
$$890$$ 0 0
$$891$$ 1.00000 0.0335013
$$892$$ −12.0000 −0.401790
$$893$$ 0 0
$$894$$ −10.0000 −0.334450
$$895$$ 0 0
$$896$$ 0 0
$$897$$ 0 0
$$898$$ −18.0000 −0.600668
$$899$$ −8.00000 −0.266815
$$900$$ −5.00000 −0.166667
$$901$$ −40.0000 −1.33259
$$902$$ 12.0000 0.399556
$$903$$ 0 0
$$904$$ −2.00000 −0.0665190
$$905$$ 0 0
$$906$$ −8.00000 −0.265782
$$907$$ 12.0000 0.398453 0.199227 0.979953i $$-0.436157\pi$$
0.199227 + 0.979953i $$0.436157\pi$$
$$908$$ −8.00000 −0.265489
$$909$$ −12.0000 −0.398015
$$910$$ 0 0
$$911$$ −40.0000 −1.32526 −0.662630 0.748947i $$-0.730560\pi$$
−0.662630 + 0.748947i $$0.730560\pi$$
$$912$$ 0 0
$$913$$ −16.0000 −0.529523
$$914$$ −26.0000 −0.860004
$$915$$ 0 0
$$916$$ −16.0000 −0.528655
$$917$$ 0 0
$$918$$ 4.00000 0.132020
$$919$$ −16.0000 −0.527791 −0.263896 0.964551i $$-0.585007\pi$$
−0.263896 + 0.964551i $$0.585007\pi$$
$$920$$ 0 0
$$921$$ 8.00000 0.263609
$$922$$ −28.0000 −0.922131
$$923$$ 32.0000 1.05329
$$924$$ 0 0
$$925$$ −50.0000 −1.64399
$$926$$ 40.0000 1.31448
$$927$$ 4.00000 0.131377
$$928$$ −2.00000 −0.0656532
$$929$$ 48.0000 1.57483 0.787414 0.616424i $$-0.211419\pi$$
0.787414 + 0.616424i $$0.211419\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ 6.00000 0.196537
$$933$$ −4.00000 −0.130954
$$934$$ −36.0000 −1.17796
$$935$$ 0 0
$$936$$ 4.00000 0.130744
$$937$$ 36.0000 1.17607 0.588034 0.808836i $$-0.299902\pi$$
0.588034 + 0.808836i $$0.299902\pi$$
$$938$$ 0 0
$$939$$ −32.0000 −1.04428
$$940$$ 0 0
$$941$$ 28.0000 0.912774 0.456387 0.889781i $$-0.349143\pi$$
0.456387 + 0.889781i $$0.349143\pi$$
$$942$$ 8.00000 0.260654
$$943$$ 0 0
$$944$$ −4.00000 −0.130189
$$945$$ 0 0
$$946$$ −4.00000 −0.130051
$$947$$ −12.0000 −0.389948 −0.194974 0.980808i $$-0.562462\pi$$
−0.194974 + 0.980808i $$0.562462\pi$$
$$948$$ −8.00000 −0.259828
$$949$$ −16.0000 −0.519382
$$950$$ 0 0
$$951$$ 18.0000 0.583690
$$952$$ 0 0
$$953$$ −54.0000 −1.74923 −0.874616 0.484817i $$-0.838886\pi$$
−0.874616 + 0.484817i $$0.838886\pi$$
$$954$$ 10.0000 0.323762
$$955$$ 0 0
$$956$$ 8.00000 0.258738
$$957$$ −2.00000 −0.0646508
$$958$$ −24.0000 −0.775405
$$959$$ 0 0
$$960$$ 0 0
$$961$$ −15.0000 −0.483871
$$962$$ 40.0000 1.28965
$$963$$ −12.0000 −0.386695
$$964$$ −20.0000 −0.644157
$$965$$ 0 0
$$966$$ 0 0
$$967$$ −8.00000 −0.257263 −0.128631 0.991692i $$-0.541058\pi$$
−0.128631 + 0.991692i $$0.541058\pi$$
$$968$$ −1.00000 −0.0321412
$$969$$ 0 0
$$970$$ 0 0
$$971$$ 20.0000 0.641831 0.320915 0.947108i $$-0.396010\pi$$
0.320915 + 0.947108i $$0.396010\pi$$
$$972$$ −1.00000 −0.0320750
$$973$$ 0 0
$$974$$ 8.00000 0.256337
$$975$$ −20.0000 −0.640513
$$976$$ 4.00000 0.128037
$$977$$ −50.0000 −1.59964 −0.799821 0.600239i $$-0.795072\pi$$
−0.799821 + 0.600239i $$0.795072\pi$$
$$978$$ 12.0000 0.383718
$$979$$ 8.00000 0.255681
$$980$$ 0 0
$$981$$ 18.0000 0.574696
$$982$$ −20.0000 −0.638226
$$983$$ −60.0000 −1.91370 −0.956851 0.290578i $$-0.906153\pi$$
−0.956851 + 0.290578i $$0.906153\pi$$
$$984$$ −12.0000 −0.382546
$$985$$ 0 0
$$986$$ −8.00000 −0.254772
$$987$$ 0 0
$$988$$ 0 0
$$989$$ 0 0
$$990$$ 0 0
$$991$$ 56.0000 1.77890 0.889449 0.457034i $$-0.151088\pi$$
0.889449 + 0.457034i $$0.151088\pi$$
$$992$$ 4.00000 0.127000
$$993$$ −20.0000 −0.634681
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 16.0000 0.506979
$$997$$ 28.0000 0.886769 0.443384 0.896332i $$-0.353778\pi$$
0.443384 + 0.896332i $$0.353778\pi$$
$$998$$ −20.0000 −0.633089
$$999$$ −10.0000 −0.316386
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 3234.2.a.e.1.1 1
3.2 odd 2 9702.2.a.bk.1.1 1
7.6 odd 2 3234.2.a.m.1.1 yes 1
21.20 even 2 9702.2.a.bo.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
3234.2.a.e.1.1 1 1.1 even 1 trivial
3234.2.a.m.1.1 yes 1 7.6 odd 2
9702.2.a.bk.1.1 1 3.2 odd 2
9702.2.a.bo.1.1 1 21.20 even 2