# Properties

 Label 3822.2.a.e.1.1 Level $3822$ Weight $2$ Character 3822.1 Self dual yes Analytic conductor $30.519$ 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] = [3822,2,Mod(1,3822)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3822, 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("3822.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$3822 = 2 \cdot 3 \cdot 7^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3822.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: $$30.5188236525$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 546) Fricke sign: $$+1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 3822.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{10} +3.00000 q^{11} -1.00000 q^{12} +1.00000 q^{13} +1.00000 q^{15} +1.00000 q^{16} -5.00000 q^{17} -1.00000 q^{18} -1.00000 q^{19} -1.00000 q^{20} -3.00000 q^{22} +3.00000 q^{23} +1.00000 q^{24} -4.00000 q^{25} -1.00000 q^{26} -1.00000 q^{27} +5.00000 q^{29} -1.00000 q^{30} -4.00000 q^{31} -1.00000 q^{32} -3.00000 q^{33} +5.00000 q^{34} +1.00000 q^{36} -5.00000 q^{37} +1.00000 q^{38} -1.00000 q^{39} +1.00000 q^{40} +8.00000 q^{41} -1.00000 q^{43} +3.00000 q^{44} -1.00000 q^{45} -3.00000 q^{46} -8.00000 q^{47} -1.00000 q^{48} +4.00000 q^{50} +5.00000 q^{51} +1.00000 q^{52} +6.00000 q^{53} +1.00000 q^{54} -3.00000 q^{55} +1.00000 q^{57} -5.00000 q^{58} +1.00000 q^{60} -13.0000 q^{61} +4.00000 q^{62} +1.00000 q^{64} -1.00000 q^{65} +3.00000 q^{66} -10.0000 q^{67} -5.00000 q^{68} -3.00000 q^{69} +8.00000 q^{71} -1.00000 q^{72} +15.0000 q^{73} +5.00000 q^{74} +4.00000 q^{75} -1.00000 q^{76} +1.00000 q^{78} +6.00000 q^{79} -1.00000 q^{80} +1.00000 q^{81} -8.00000 q^{82} +2.00000 q^{83} +5.00000 q^{85} +1.00000 q^{86} -5.00000 q^{87} -3.00000 q^{88} +2.00000 q^{89} +1.00000 q^{90} +3.00000 q^{92} +4.00000 q^{93} +8.00000 q^{94} +1.00000 q^{95} +1.00000 q^{96} +2.00000 q^{97} +3.00000 q^{99} +O(q^{100})$$

## 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
$$5$$ −1.00000 −0.447214 −0.223607 0.974679i $$-0.571783\pi$$
−0.223607 + 0.974679i $$0.571783\pi$$
$$6$$ 1.00000 0.408248
$$7$$ 0 0
$$8$$ −1.00000 −0.353553
$$9$$ 1.00000 0.333333
$$10$$ 1.00000 0.316228
$$11$$ 3.00000 0.904534 0.452267 0.891883i $$-0.350615\pi$$
0.452267 + 0.891883i $$0.350615\pi$$
$$12$$ −1.00000 −0.288675
$$13$$ 1.00000 0.277350
$$14$$ 0 0
$$15$$ 1.00000 0.258199
$$16$$ 1.00000 0.250000
$$17$$ −5.00000 −1.21268 −0.606339 0.795206i $$-0.707363\pi$$
−0.606339 + 0.795206i $$0.707363\pi$$
$$18$$ −1.00000 −0.235702
$$19$$ −1.00000 −0.229416 −0.114708 0.993399i $$-0.536593\pi$$
−0.114708 + 0.993399i $$0.536593\pi$$
$$20$$ −1.00000 −0.223607
$$21$$ 0 0
$$22$$ −3.00000 −0.639602
$$23$$ 3.00000 0.625543 0.312772 0.949828i $$-0.398743\pi$$
0.312772 + 0.949828i $$0.398743\pi$$
$$24$$ 1.00000 0.204124
$$25$$ −4.00000 −0.800000
$$26$$ −1.00000 −0.196116
$$27$$ −1.00000 −0.192450
$$28$$ 0 0
$$29$$ 5.00000 0.928477 0.464238 0.885710i $$-0.346328\pi$$
0.464238 + 0.885710i $$0.346328\pi$$
$$30$$ −1.00000 −0.182574
$$31$$ −4.00000 −0.718421 −0.359211 0.933257i $$-0.616954\pi$$
−0.359211 + 0.933257i $$0.616954\pi$$
$$32$$ −1.00000 −0.176777
$$33$$ −3.00000 −0.522233
$$34$$ 5.00000 0.857493
$$35$$ 0 0
$$36$$ 1.00000 0.166667
$$37$$ −5.00000 −0.821995 −0.410997 0.911636i $$-0.634819\pi$$
−0.410997 + 0.911636i $$0.634819\pi$$
$$38$$ 1.00000 0.162221
$$39$$ −1.00000 −0.160128
$$40$$ 1.00000 0.158114
$$41$$ 8.00000 1.24939 0.624695 0.780869i $$-0.285223\pi$$
0.624695 + 0.780869i $$0.285223\pi$$
$$42$$ 0 0
$$43$$ −1.00000 −0.152499 −0.0762493 0.997089i $$-0.524294\pi$$
−0.0762493 + 0.997089i $$0.524294\pi$$
$$44$$ 3.00000 0.452267
$$45$$ −1.00000 −0.149071
$$46$$ −3.00000 −0.442326
$$47$$ −8.00000 −1.16692 −0.583460 0.812142i $$-0.698301\pi$$
−0.583460 + 0.812142i $$0.698301\pi$$
$$48$$ −1.00000 −0.144338
$$49$$ 0 0
$$50$$ 4.00000 0.565685
$$51$$ 5.00000 0.700140
$$52$$ 1.00000 0.138675
$$53$$ 6.00000 0.824163 0.412082 0.911147i $$-0.364802\pi$$
0.412082 + 0.911147i $$0.364802\pi$$
$$54$$ 1.00000 0.136083
$$55$$ −3.00000 −0.404520
$$56$$ 0 0
$$57$$ 1.00000 0.132453
$$58$$ −5.00000 −0.656532
$$59$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$60$$ 1.00000 0.129099
$$61$$ −13.0000 −1.66448 −0.832240 0.554416i $$-0.812942\pi$$
−0.832240 + 0.554416i $$0.812942\pi$$
$$62$$ 4.00000 0.508001
$$63$$ 0 0
$$64$$ 1.00000 0.125000
$$65$$ −1.00000 −0.124035
$$66$$ 3.00000 0.369274
$$67$$ −10.0000 −1.22169 −0.610847 0.791748i $$-0.709171\pi$$
−0.610847 + 0.791748i $$0.709171\pi$$
$$68$$ −5.00000 −0.606339
$$69$$ −3.00000 −0.361158
$$70$$ 0 0
$$71$$ 8.00000 0.949425 0.474713 0.880141i $$-0.342552\pi$$
0.474713 + 0.880141i $$0.342552\pi$$
$$72$$ −1.00000 −0.117851
$$73$$ 15.0000 1.75562 0.877809 0.479012i $$-0.159005\pi$$
0.877809 + 0.479012i $$0.159005\pi$$
$$74$$ 5.00000 0.581238
$$75$$ 4.00000 0.461880
$$76$$ −1.00000 −0.114708
$$77$$ 0 0
$$78$$ 1.00000 0.113228
$$79$$ 6.00000 0.675053 0.337526 0.941316i $$-0.390410\pi$$
0.337526 + 0.941316i $$0.390410\pi$$
$$80$$ −1.00000 −0.111803
$$81$$ 1.00000 0.111111
$$82$$ −8.00000 −0.883452
$$83$$ 2.00000 0.219529 0.109764 0.993958i $$-0.464990\pi$$
0.109764 + 0.993958i $$0.464990\pi$$
$$84$$ 0 0
$$85$$ 5.00000 0.542326
$$86$$ 1.00000 0.107833
$$87$$ −5.00000 −0.536056
$$88$$ −3.00000 −0.319801
$$89$$ 2.00000 0.212000 0.106000 0.994366i $$-0.466196\pi$$
0.106000 + 0.994366i $$0.466196\pi$$
$$90$$ 1.00000 0.105409
$$91$$ 0 0
$$92$$ 3.00000 0.312772
$$93$$ 4.00000 0.414781
$$94$$ 8.00000 0.825137
$$95$$ 1.00000 0.102598
$$96$$ 1.00000 0.102062
$$97$$ 2.00000 0.203069 0.101535 0.994832i $$-0.467625\pi$$
0.101535 + 0.994832i $$0.467625\pi$$
$$98$$ 0 0
$$99$$ 3.00000 0.301511
$$100$$ −4.00000 −0.400000
$$101$$ 16.0000 1.59206 0.796030 0.605257i $$-0.206930\pi$$
0.796030 + 0.605257i $$0.206930\pi$$
$$102$$ −5.00000 −0.495074
$$103$$ −1.00000 −0.0985329 −0.0492665 0.998786i $$-0.515688\pi$$
−0.0492665 + 0.998786i $$0.515688\pi$$
$$104$$ −1.00000 −0.0980581
$$105$$ 0 0
$$106$$ −6.00000 −0.582772
$$107$$ −6.00000 −0.580042 −0.290021 0.957020i $$-0.593662\pi$$
−0.290021 + 0.957020i $$0.593662\pi$$
$$108$$ −1.00000 −0.0962250
$$109$$ −7.00000 −0.670478 −0.335239 0.942133i $$-0.608817\pi$$
−0.335239 + 0.942133i $$0.608817\pi$$
$$110$$ 3.00000 0.286039
$$111$$ 5.00000 0.474579
$$112$$ 0 0
$$113$$ −6.00000 −0.564433 −0.282216 0.959351i $$-0.591070\pi$$
−0.282216 + 0.959351i $$0.591070\pi$$
$$114$$ −1.00000 −0.0936586
$$115$$ −3.00000 −0.279751
$$116$$ 5.00000 0.464238
$$117$$ 1.00000 0.0924500
$$118$$ 0 0
$$119$$ 0 0
$$120$$ −1.00000 −0.0912871
$$121$$ −2.00000 −0.181818
$$122$$ 13.0000 1.17696
$$123$$ −8.00000 −0.721336
$$124$$ −4.00000 −0.359211
$$125$$ 9.00000 0.804984
$$126$$ 0 0
$$127$$ −18.0000 −1.59724 −0.798621 0.601834i $$-0.794437\pi$$
−0.798621 + 0.601834i $$0.794437\pi$$
$$128$$ −1.00000 −0.0883883
$$129$$ 1.00000 0.0880451
$$130$$ 1.00000 0.0877058
$$131$$ −17.0000 −1.48530 −0.742648 0.669681i $$-0.766431\pi$$
−0.742648 + 0.669681i $$0.766431\pi$$
$$132$$ −3.00000 −0.261116
$$133$$ 0 0
$$134$$ 10.0000 0.863868
$$135$$ 1.00000 0.0860663
$$136$$ 5.00000 0.428746
$$137$$ −15.0000 −1.28154 −0.640768 0.767734i $$-0.721384\pi$$
−0.640768 + 0.767734i $$0.721384\pi$$
$$138$$ 3.00000 0.255377
$$139$$ 16.0000 1.35710 0.678551 0.734553i $$-0.262608\pi$$
0.678551 + 0.734553i $$0.262608\pi$$
$$140$$ 0 0
$$141$$ 8.00000 0.673722
$$142$$ −8.00000 −0.671345
$$143$$ 3.00000 0.250873
$$144$$ 1.00000 0.0833333
$$145$$ −5.00000 −0.415227
$$146$$ −15.0000 −1.24141
$$147$$ 0 0
$$148$$ −5.00000 −0.410997
$$149$$ −16.0000 −1.31077 −0.655386 0.755295i $$-0.727494\pi$$
−0.655386 + 0.755295i $$0.727494\pi$$
$$150$$ −4.00000 −0.326599
$$151$$ −5.00000 −0.406894 −0.203447 0.979086i $$-0.565214\pi$$
−0.203447 + 0.979086i $$0.565214\pi$$
$$152$$ 1.00000 0.0811107
$$153$$ −5.00000 −0.404226
$$154$$ 0 0
$$155$$ 4.00000 0.321288
$$156$$ −1.00000 −0.0800641
$$157$$ −9.00000 −0.718278 −0.359139 0.933284i $$-0.616930\pi$$
−0.359139 + 0.933284i $$0.616930\pi$$
$$158$$ −6.00000 −0.477334
$$159$$ −6.00000 −0.475831
$$160$$ 1.00000 0.0790569
$$161$$ 0 0
$$162$$ −1.00000 −0.0785674
$$163$$ 10.0000 0.783260 0.391630 0.920123i $$-0.371911\pi$$
0.391630 + 0.920123i $$0.371911\pi$$
$$164$$ 8.00000 0.624695
$$165$$ 3.00000 0.233550
$$166$$ −2.00000 −0.155230
$$167$$ 15.0000 1.16073 0.580367 0.814355i $$-0.302909\pi$$
0.580367 + 0.814355i $$0.302909\pi$$
$$168$$ 0 0
$$169$$ 1.00000 0.0769231
$$170$$ −5.00000 −0.383482
$$171$$ −1.00000 −0.0764719
$$172$$ −1.00000 −0.0762493
$$173$$ −6.00000 −0.456172 −0.228086 0.973641i $$-0.573247\pi$$
−0.228086 + 0.973641i $$0.573247\pi$$
$$174$$ 5.00000 0.379049
$$175$$ 0 0
$$176$$ 3.00000 0.226134
$$177$$ 0 0
$$178$$ −2.00000 −0.149906
$$179$$ −16.0000 −1.19590 −0.597948 0.801535i $$-0.704017\pi$$
−0.597948 + 0.801535i $$0.704017\pi$$
$$180$$ −1.00000 −0.0745356
$$181$$ −6.00000 −0.445976 −0.222988 0.974821i $$-0.571581\pi$$
−0.222988 + 0.974821i $$0.571581\pi$$
$$182$$ 0 0
$$183$$ 13.0000 0.960988
$$184$$ −3.00000 −0.221163
$$185$$ 5.00000 0.367607
$$186$$ −4.00000 −0.293294
$$187$$ −15.0000 −1.09691
$$188$$ −8.00000 −0.583460
$$189$$ 0 0
$$190$$ −1.00000 −0.0725476
$$191$$ 11.0000 0.795932 0.397966 0.917400i $$-0.369716\pi$$
0.397966 + 0.917400i $$0.369716\pi$$
$$192$$ −1.00000 −0.0721688
$$193$$ −20.0000 −1.43963 −0.719816 0.694165i $$-0.755774\pi$$
−0.719816 + 0.694165i $$0.755774\pi$$
$$194$$ −2.00000 −0.143592
$$195$$ 1.00000 0.0716115
$$196$$ 0 0
$$197$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$198$$ −3.00000 −0.213201
$$199$$ 5.00000 0.354441 0.177220 0.984171i $$-0.443289\pi$$
0.177220 + 0.984171i $$0.443289\pi$$
$$200$$ 4.00000 0.282843
$$201$$ 10.0000 0.705346
$$202$$ −16.0000 −1.12576
$$203$$ 0 0
$$204$$ 5.00000 0.350070
$$205$$ −8.00000 −0.558744
$$206$$ 1.00000 0.0696733
$$207$$ 3.00000 0.208514
$$208$$ 1.00000 0.0693375
$$209$$ −3.00000 −0.207514
$$210$$ 0 0
$$211$$ 5.00000 0.344214 0.172107 0.985078i $$-0.444942\pi$$
0.172107 + 0.985078i $$0.444942\pi$$
$$212$$ 6.00000 0.412082
$$213$$ −8.00000 −0.548151
$$214$$ 6.00000 0.410152
$$215$$ 1.00000 0.0681994
$$216$$ 1.00000 0.0680414
$$217$$ 0 0
$$218$$ 7.00000 0.474100
$$219$$ −15.0000 −1.01361
$$220$$ −3.00000 −0.202260
$$221$$ −5.00000 −0.336336
$$222$$ −5.00000 −0.335578
$$223$$ −26.0000 −1.74109 −0.870544 0.492090i $$-0.836233\pi$$
−0.870544 + 0.492090i $$0.836233\pi$$
$$224$$ 0 0
$$225$$ −4.00000 −0.266667
$$226$$ 6.00000 0.399114
$$227$$ −2.00000 −0.132745 −0.0663723 0.997795i $$-0.521143\pi$$
−0.0663723 + 0.997795i $$0.521143\pi$$
$$228$$ 1.00000 0.0662266
$$229$$ 6.00000 0.396491 0.198246 0.980152i $$-0.436476\pi$$
0.198246 + 0.980152i $$0.436476\pi$$
$$230$$ 3.00000 0.197814
$$231$$ 0 0
$$232$$ −5.00000 −0.328266
$$233$$ −4.00000 −0.262049 −0.131024 0.991379i $$-0.541827\pi$$
−0.131024 + 0.991379i $$0.541827\pi$$
$$234$$ −1.00000 −0.0653720
$$235$$ 8.00000 0.521862
$$236$$ 0 0
$$237$$ −6.00000 −0.389742
$$238$$ 0 0
$$239$$ −12.0000 −0.776215 −0.388108 0.921614i $$-0.626871\pi$$
−0.388108 + 0.921614i $$0.626871\pi$$
$$240$$ 1.00000 0.0645497
$$241$$ −22.0000 −1.41714 −0.708572 0.705638i $$-0.750660\pi$$
−0.708572 + 0.705638i $$0.750660\pi$$
$$242$$ 2.00000 0.128565
$$243$$ −1.00000 −0.0641500
$$244$$ −13.0000 −0.832240
$$245$$ 0 0
$$246$$ 8.00000 0.510061
$$247$$ −1.00000 −0.0636285
$$248$$ 4.00000 0.254000
$$249$$ −2.00000 −0.126745
$$250$$ −9.00000 −0.569210
$$251$$ −21.0000 −1.32551 −0.662754 0.748837i $$-0.730613\pi$$
−0.662754 + 0.748837i $$0.730613\pi$$
$$252$$ 0 0
$$253$$ 9.00000 0.565825
$$254$$ 18.0000 1.12942
$$255$$ −5.00000 −0.313112
$$256$$ 1.00000 0.0625000
$$257$$ 18.0000 1.12281 0.561405 0.827541i $$-0.310261\pi$$
0.561405 + 0.827541i $$0.310261\pi$$
$$258$$ −1.00000 −0.0622573
$$259$$ 0 0
$$260$$ −1.00000 −0.0620174
$$261$$ 5.00000 0.309492
$$262$$ 17.0000 1.05026
$$263$$ 12.0000 0.739952 0.369976 0.929041i $$-0.379366\pi$$
0.369976 + 0.929041i $$0.379366\pi$$
$$264$$ 3.00000 0.184637
$$265$$ −6.00000 −0.368577
$$266$$ 0 0
$$267$$ −2.00000 −0.122398
$$268$$ −10.0000 −0.610847
$$269$$ −6.00000 −0.365826 −0.182913 0.983129i $$-0.558553\pi$$
−0.182913 + 0.983129i $$0.558553\pi$$
$$270$$ −1.00000 −0.0608581
$$271$$ −2.00000 −0.121491 −0.0607457 0.998153i $$-0.519348\pi$$
−0.0607457 + 0.998153i $$0.519348\pi$$
$$272$$ −5.00000 −0.303170
$$273$$ 0 0
$$274$$ 15.0000 0.906183
$$275$$ −12.0000 −0.723627
$$276$$ −3.00000 −0.180579
$$277$$ −18.0000 −1.08152 −0.540758 0.841178i $$-0.681862\pi$$
−0.540758 + 0.841178i $$0.681862\pi$$
$$278$$ −16.0000 −0.959616
$$279$$ −4.00000 −0.239474
$$280$$ 0 0
$$281$$ 10.0000 0.596550 0.298275 0.954480i $$-0.403589\pi$$
0.298275 + 0.954480i $$0.403589\pi$$
$$282$$ −8.00000 −0.476393
$$283$$ 8.00000 0.475551 0.237775 0.971320i $$-0.423582\pi$$
0.237775 + 0.971320i $$0.423582\pi$$
$$284$$ 8.00000 0.474713
$$285$$ −1.00000 −0.0592349
$$286$$ −3.00000 −0.177394
$$287$$ 0 0
$$288$$ −1.00000 −0.0589256
$$289$$ 8.00000 0.470588
$$290$$ 5.00000 0.293610
$$291$$ −2.00000 −0.117242
$$292$$ 15.0000 0.877809
$$293$$ 6.00000 0.350524 0.175262 0.984522i $$-0.443923\pi$$
0.175262 + 0.984522i $$0.443923\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 5.00000 0.290619
$$297$$ −3.00000 −0.174078
$$298$$ 16.0000 0.926855
$$299$$ 3.00000 0.173494
$$300$$ 4.00000 0.230940
$$301$$ 0 0
$$302$$ 5.00000 0.287718
$$303$$ −16.0000 −0.919176
$$304$$ −1.00000 −0.0573539
$$305$$ 13.0000 0.744378
$$306$$ 5.00000 0.285831
$$307$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$308$$ 0 0
$$309$$ 1.00000 0.0568880
$$310$$ −4.00000 −0.227185
$$311$$ −26.0000 −1.47432 −0.737162 0.675716i $$-0.763835\pi$$
−0.737162 + 0.675716i $$0.763835\pi$$
$$312$$ 1.00000 0.0566139
$$313$$ −28.0000 −1.58265 −0.791327 0.611393i $$-0.790609\pi$$
−0.791327 + 0.611393i $$0.790609\pi$$
$$314$$ 9.00000 0.507899
$$315$$ 0 0
$$316$$ 6.00000 0.337526
$$317$$ −24.0000 −1.34797 −0.673987 0.738743i $$-0.735420\pi$$
−0.673987 + 0.738743i $$0.735420\pi$$
$$318$$ 6.00000 0.336463
$$319$$ 15.0000 0.839839
$$320$$ −1.00000 −0.0559017
$$321$$ 6.00000 0.334887
$$322$$ 0 0
$$323$$ 5.00000 0.278207
$$324$$ 1.00000 0.0555556
$$325$$ −4.00000 −0.221880
$$326$$ −10.0000 −0.553849
$$327$$ 7.00000 0.387101
$$328$$ −8.00000 −0.441726
$$329$$ 0 0
$$330$$ −3.00000 −0.165145
$$331$$ 34.0000 1.86881 0.934405 0.356214i $$-0.115932\pi$$
0.934405 + 0.356214i $$0.115932\pi$$
$$332$$ 2.00000 0.109764
$$333$$ −5.00000 −0.273998
$$334$$ −15.0000 −0.820763
$$335$$ 10.0000 0.546358
$$336$$ 0 0
$$337$$ −29.0000 −1.57973 −0.789865 0.613280i $$-0.789850\pi$$
−0.789865 + 0.613280i $$0.789850\pi$$
$$338$$ −1.00000 −0.0543928
$$339$$ 6.00000 0.325875
$$340$$ 5.00000 0.271163
$$341$$ −12.0000 −0.649836
$$342$$ 1.00000 0.0540738
$$343$$ 0 0
$$344$$ 1.00000 0.0539164
$$345$$ 3.00000 0.161515
$$346$$ 6.00000 0.322562
$$347$$ 32.0000 1.71785 0.858925 0.512101i $$-0.171133\pi$$
0.858925 + 0.512101i $$0.171133\pi$$
$$348$$ −5.00000 −0.268028
$$349$$ −6.00000 −0.321173 −0.160586 0.987022i $$-0.551338\pi$$
−0.160586 + 0.987022i $$0.551338\pi$$
$$350$$ 0 0
$$351$$ −1.00000 −0.0533761
$$352$$ −3.00000 −0.159901
$$353$$ −2.00000 −0.106449 −0.0532246 0.998583i $$-0.516950\pi$$
−0.0532246 + 0.998583i $$0.516950\pi$$
$$354$$ 0 0
$$355$$ −8.00000 −0.424596
$$356$$ 2.00000 0.106000
$$357$$ 0 0
$$358$$ 16.0000 0.845626
$$359$$ 24.0000 1.26667 0.633336 0.773877i $$-0.281685\pi$$
0.633336 + 0.773877i $$0.281685\pi$$
$$360$$ 1.00000 0.0527046
$$361$$ −18.0000 −0.947368
$$362$$ 6.00000 0.315353
$$363$$ 2.00000 0.104973
$$364$$ 0 0
$$365$$ −15.0000 −0.785136
$$366$$ −13.0000 −0.679521
$$367$$ −8.00000 −0.417597 −0.208798 0.977959i $$-0.566955\pi$$
−0.208798 + 0.977959i $$0.566955\pi$$
$$368$$ 3.00000 0.156386
$$369$$ 8.00000 0.416463
$$370$$ −5.00000 −0.259938
$$371$$ 0 0
$$372$$ 4.00000 0.207390
$$373$$ −32.0000 −1.65690 −0.828449 0.560065i $$-0.810776\pi$$
−0.828449 + 0.560065i $$0.810776\pi$$
$$374$$ 15.0000 0.775632
$$375$$ −9.00000 −0.464758
$$376$$ 8.00000 0.412568
$$377$$ 5.00000 0.257513
$$378$$ 0 0
$$379$$ −8.00000 −0.410932 −0.205466 0.978664i $$-0.565871\pi$$
−0.205466 + 0.978664i $$0.565871\pi$$
$$380$$ 1.00000 0.0512989
$$381$$ 18.0000 0.922168
$$382$$ −11.0000 −0.562809
$$383$$ −1.00000 −0.0510976 −0.0255488 0.999674i $$-0.508133\pi$$
−0.0255488 + 0.999674i $$0.508133\pi$$
$$384$$ 1.00000 0.0510310
$$385$$ 0 0
$$386$$ 20.0000 1.01797
$$387$$ −1.00000 −0.0508329
$$388$$ 2.00000 0.101535
$$389$$ −18.0000 −0.912636 −0.456318 0.889817i $$-0.650832\pi$$
−0.456318 + 0.889817i $$0.650832\pi$$
$$390$$ −1.00000 −0.0506370
$$391$$ −15.0000 −0.758583
$$392$$ 0 0
$$393$$ 17.0000 0.857537
$$394$$ 0 0
$$395$$ −6.00000 −0.301893
$$396$$ 3.00000 0.150756
$$397$$ −30.0000 −1.50566 −0.752828 0.658217i $$-0.771311\pi$$
−0.752828 + 0.658217i $$0.771311\pi$$
$$398$$ −5.00000 −0.250627
$$399$$ 0 0
$$400$$ −4.00000 −0.200000
$$401$$ −30.0000 −1.49813 −0.749064 0.662497i $$-0.769497\pi$$
−0.749064 + 0.662497i $$0.769497\pi$$
$$402$$ −10.0000 −0.498755
$$403$$ −4.00000 −0.199254
$$404$$ 16.0000 0.796030
$$405$$ −1.00000 −0.0496904
$$406$$ 0 0
$$407$$ −15.0000 −0.743522
$$408$$ −5.00000 −0.247537
$$409$$ −3.00000 −0.148340 −0.0741702 0.997246i $$-0.523631\pi$$
−0.0741702 + 0.997246i $$0.523631\pi$$
$$410$$ 8.00000 0.395092
$$411$$ 15.0000 0.739895
$$412$$ −1.00000 −0.0492665
$$413$$ 0 0
$$414$$ −3.00000 −0.147442
$$415$$ −2.00000 −0.0981761
$$416$$ −1.00000 −0.0490290
$$417$$ −16.0000 −0.783523
$$418$$ 3.00000 0.146735
$$419$$ 5.00000 0.244266 0.122133 0.992514i $$-0.461027\pi$$
0.122133 + 0.992514i $$0.461027\pi$$
$$420$$ 0 0
$$421$$ −22.0000 −1.07221 −0.536107 0.844150i $$-0.680106\pi$$
−0.536107 + 0.844150i $$0.680106\pi$$
$$422$$ −5.00000 −0.243396
$$423$$ −8.00000 −0.388973
$$424$$ −6.00000 −0.291386
$$425$$ 20.0000 0.970143
$$426$$ 8.00000 0.387601
$$427$$ 0 0
$$428$$ −6.00000 −0.290021
$$429$$ −3.00000 −0.144841
$$430$$ −1.00000 −0.0482243
$$431$$ 10.0000 0.481683 0.240842 0.970564i $$-0.422577\pi$$
0.240842 + 0.970564i $$0.422577\pi$$
$$432$$ −1.00000 −0.0481125
$$433$$ −20.0000 −0.961139 −0.480569 0.876957i $$-0.659570\pi$$
−0.480569 + 0.876957i $$0.659570\pi$$
$$434$$ 0 0
$$435$$ 5.00000 0.239732
$$436$$ −7.00000 −0.335239
$$437$$ −3.00000 −0.143509
$$438$$ 15.0000 0.716728
$$439$$ 11.0000 0.525001 0.262501 0.964932i $$-0.415453\pi$$
0.262501 + 0.964932i $$0.415453\pi$$
$$440$$ 3.00000 0.143019
$$441$$ 0 0
$$442$$ 5.00000 0.237826
$$443$$ −18.0000 −0.855206 −0.427603 0.903967i $$-0.640642\pi$$
−0.427603 + 0.903967i $$0.640642\pi$$
$$444$$ 5.00000 0.237289
$$445$$ −2.00000 −0.0948091
$$446$$ 26.0000 1.23114
$$447$$ 16.0000 0.756774
$$448$$ 0 0
$$449$$ 13.0000 0.613508 0.306754 0.951789i $$-0.400757\pi$$
0.306754 + 0.951789i $$0.400757\pi$$
$$450$$ 4.00000 0.188562
$$451$$ 24.0000 1.13012
$$452$$ −6.00000 −0.282216
$$453$$ 5.00000 0.234920
$$454$$ 2.00000 0.0938647
$$455$$ 0 0
$$456$$ −1.00000 −0.0468293
$$457$$ 8.00000 0.374224 0.187112 0.982339i $$-0.440087\pi$$
0.187112 + 0.982339i $$0.440087\pi$$
$$458$$ −6.00000 −0.280362
$$459$$ 5.00000 0.233380
$$460$$ −3.00000 −0.139876
$$461$$ 13.0000 0.605470 0.302735 0.953075i $$-0.402100\pi$$
0.302735 + 0.953075i $$0.402100\pi$$
$$462$$ 0 0
$$463$$ −13.0000 −0.604161 −0.302081 0.953282i $$-0.597681\pi$$
−0.302081 + 0.953282i $$0.597681\pi$$
$$464$$ 5.00000 0.232119
$$465$$ −4.00000 −0.185496
$$466$$ 4.00000 0.185296
$$467$$ −21.0000 −0.971764 −0.485882 0.874024i $$-0.661502\pi$$
−0.485882 + 0.874024i $$0.661502\pi$$
$$468$$ 1.00000 0.0462250
$$469$$ 0 0
$$470$$ −8.00000 −0.369012
$$471$$ 9.00000 0.414698
$$472$$ 0 0
$$473$$ −3.00000 −0.137940
$$474$$ 6.00000 0.275589
$$475$$ 4.00000 0.183533
$$476$$ 0 0
$$477$$ 6.00000 0.274721
$$478$$ 12.0000 0.548867
$$479$$ −27.0000 −1.23366 −0.616831 0.787096i $$-0.711584\pi$$
−0.616831 + 0.787096i $$0.711584\pi$$
$$480$$ −1.00000 −0.0456435
$$481$$ −5.00000 −0.227980
$$482$$ 22.0000 1.00207
$$483$$ 0 0
$$484$$ −2.00000 −0.0909091
$$485$$ −2.00000 −0.0908153
$$486$$ 1.00000 0.0453609
$$487$$ 8.00000 0.362515 0.181257 0.983436i $$-0.441983\pi$$
0.181257 + 0.983436i $$0.441983\pi$$
$$488$$ 13.0000 0.588482
$$489$$ −10.0000 −0.452216
$$490$$ 0 0
$$491$$ −38.0000 −1.71492 −0.857458 0.514554i $$-0.827958\pi$$
−0.857458 + 0.514554i $$0.827958\pi$$
$$492$$ −8.00000 −0.360668
$$493$$ −25.0000 −1.12594
$$494$$ 1.00000 0.0449921
$$495$$ −3.00000 −0.134840
$$496$$ −4.00000 −0.179605
$$497$$ 0 0
$$498$$ 2.00000 0.0896221
$$499$$ 8.00000 0.358129 0.179065 0.983837i $$-0.442693\pi$$
0.179065 + 0.983837i $$0.442693\pi$$
$$500$$ 9.00000 0.402492
$$501$$ −15.0000 −0.670151
$$502$$ 21.0000 0.937276
$$503$$ 6.00000 0.267527 0.133763 0.991013i $$-0.457294\pi$$
0.133763 + 0.991013i $$0.457294\pi$$
$$504$$ 0 0
$$505$$ −16.0000 −0.711991
$$506$$ −9.00000 −0.400099
$$507$$ −1.00000 −0.0444116
$$508$$ −18.0000 −0.798621
$$509$$ −11.0000 −0.487566 −0.243783 0.969830i $$-0.578389\pi$$
−0.243783 + 0.969830i $$0.578389\pi$$
$$510$$ 5.00000 0.221404
$$511$$ 0 0
$$512$$ −1.00000 −0.0441942
$$513$$ 1.00000 0.0441511
$$514$$ −18.0000 −0.793946
$$515$$ 1.00000 0.0440653
$$516$$ 1.00000 0.0440225
$$517$$ −24.0000 −1.05552
$$518$$ 0 0
$$519$$ 6.00000 0.263371
$$520$$ 1.00000 0.0438529
$$521$$ −39.0000 −1.70862 −0.854311 0.519763i $$-0.826020\pi$$
−0.854311 + 0.519763i $$0.826020\pi$$
$$522$$ −5.00000 −0.218844
$$523$$ 24.0000 1.04945 0.524723 0.851273i $$-0.324169\pi$$
0.524723 + 0.851273i $$0.324169\pi$$
$$524$$ −17.0000 −0.742648
$$525$$ 0 0
$$526$$ −12.0000 −0.523225
$$527$$ 20.0000 0.871214
$$528$$ −3.00000 −0.130558
$$529$$ −14.0000 −0.608696
$$530$$ 6.00000 0.260623
$$531$$ 0 0
$$532$$ 0 0
$$533$$ 8.00000 0.346518
$$534$$ 2.00000 0.0865485
$$535$$ 6.00000 0.259403
$$536$$ 10.0000 0.431934
$$537$$ 16.0000 0.690451
$$538$$ 6.00000 0.258678
$$539$$ 0 0
$$540$$ 1.00000 0.0430331
$$541$$ 41.0000 1.76273 0.881364 0.472438i $$-0.156626\pi$$
0.881364 + 0.472438i $$0.156626\pi$$
$$542$$ 2.00000 0.0859074
$$543$$ 6.00000 0.257485
$$544$$ 5.00000 0.214373
$$545$$ 7.00000 0.299847
$$546$$ 0 0
$$547$$ −44.0000 −1.88130 −0.940652 0.339372i $$-0.889785\pi$$
−0.940652 + 0.339372i $$0.889785\pi$$
$$548$$ −15.0000 −0.640768
$$549$$ −13.0000 −0.554826
$$550$$ 12.0000 0.511682
$$551$$ −5.00000 −0.213007
$$552$$ 3.00000 0.127688
$$553$$ 0 0
$$554$$ 18.0000 0.764747
$$555$$ −5.00000 −0.212238
$$556$$ 16.0000 0.678551
$$557$$ −24.0000 −1.01691 −0.508456 0.861088i $$-0.669784\pi$$
−0.508456 + 0.861088i $$0.669784\pi$$
$$558$$ 4.00000 0.169334
$$559$$ −1.00000 −0.0422955
$$560$$ 0 0
$$561$$ 15.0000 0.633300
$$562$$ −10.0000 −0.421825
$$563$$ −9.00000 −0.379305 −0.189652 0.981851i $$-0.560736\pi$$
−0.189652 + 0.981851i $$0.560736\pi$$
$$564$$ 8.00000 0.336861
$$565$$ 6.00000 0.252422
$$566$$ −8.00000 −0.336265
$$567$$ 0 0
$$568$$ −8.00000 −0.335673
$$569$$ 34.0000 1.42535 0.712677 0.701492i $$-0.247483\pi$$
0.712677 + 0.701492i $$0.247483\pi$$
$$570$$ 1.00000 0.0418854
$$571$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$572$$ 3.00000 0.125436
$$573$$ −11.0000 −0.459532
$$574$$ 0 0
$$575$$ −12.0000 −0.500435
$$576$$ 1.00000 0.0416667
$$577$$ 34.0000 1.41544 0.707719 0.706494i $$-0.249724\pi$$
0.707719 + 0.706494i $$0.249724\pi$$
$$578$$ −8.00000 −0.332756
$$579$$ 20.0000 0.831172
$$580$$ −5.00000 −0.207614
$$581$$ 0 0
$$582$$ 2.00000 0.0829027
$$583$$ 18.0000 0.745484
$$584$$ −15.0000 −0.620704
$$585$$ −1.00000 −0.0413449
$$586$$ −6.00000 −0.247858
$$587$$ 26.0000 1.07313 0.536567 0.843857i $$-0.319721\pi$$
0.536567 + 0.843857i $$0.319721\pi$$
$$588$$ 0 0
$$589$$ 4.00000 0.164817
$$590$$ 0 0
$$591$$ 0 0
$$592$$ −5.00000 −0.205499
$$593$$ −38.0000 −1.56047 −0.780236 0.625485i $$-0.784901\pi$$
−0.780236 + 0.625485i $$0.784901\pi$$
$$594$$ 3.00000 0.123091
$$595$$ 0 0
$$596$$ −16.0000 −0.655386
$$597$$ −5.00000 −0.204636
$$598$$ −3.00000 −0.122679
$$599$$ 37.0000 1.51178 0.755890 0.654699i $$-0.227205\pi$$
0.755890 + 0.654699i $$0.227205\pi$$
$$600$$ −4.00000 −0.163299
$$601$$ −20.0000 −0.815817 −0.407909 0.913023i $$-0.633742\pi$$
−0.407909 + 0.913023i $$0.633742\pi$$
$$602$$ 0 0
$$603$$ −10.0000 −0.407231
$$604$$ −5.00000 −0.203447
$$605$$ 2.00000 0.0813116
$$606$$ 16.0000 0.649956
$$607$$ 27.0000 1.09590 0.547948 0.836512i $$-0.315409\pi$$
0.547948 + 0.836512i $$0.315409\pi$$
$$608$$ 1.00000 0.0405554
$$609$$ 0 0
$$610$$ −13.0000 −0.526355
$$611$$ −8.00000 −0.323645
$$612$$ −5.00000 −0.202113
$$613$$ 41.0000 1.65597 0.827987 0.560747i $$-0.189486\pi$$
0.827987 + 0.560747i $$0.189486\pi$$
$$614$$ 0 0
$$615$$ 8.00000 0.322591
$$616$$ 0 0
$$617$$ 43.0000 1.73111 0.865557 0.500810i $$-0.166964\pi$$
0.865557 + 0.500810i $$0.166964\pi$$
$$618$$ −1.00000 −0.0402259
$$619$$ 31.0000 1.24600 0.622998 0.782224i $$-0.285915\pi$$
0.622998 + 0.782224i $$0.285915\pi$$
$$620$$ 4.00000 0.160644
$$621$$ −3.00000 −0.120386
$$622$$ 26.0000 1.04251
$$623$$ 0 0
$$624$$ −1.00000 −0.0400320
$$625$$ 11.0000 0.440000
$$626$$ 28.0000 1.11911
$$627$$ 3.00000 0.119808
$$628$$ −9.00000 −0.359139
$$629$$ 25.0000 0.996815
$$630$$ 0 0
$$631$$ −9.00000 −0.358284 −0.179142 0.983823i $$-0.557332\pi$$
−0.179142 + 0.983823i $$0.557332\pi$$
$$632$$ −6.00000 −0.238667
$$633$$ −5.00000 −0.198732
$$634$$ 24.0000 0.953162
$$635$$ 18.0000 0.714308
$$636$$ −6.00000 −0.237915
$$637$$ 0 0
$$638$$ −15.0000 −0.593856
$$639$$ 8.00000 0.316475
$$640$$ 1.00000 0.0395285
$$641$$ 30.0000 1.18493 0.592464 0.805597i $$-0.298155\pi$$
0.592464 + 0.805597i $$0.298155\pi$$
$$642$$ −6.00000 −0.236801
$$643$$ 1.00000 0.0394362 0.0197181 0.999806i $$-0.493723\pi$$
0.0197181 + 0.999806i $$0.493723\pi$$
$$644$$ 0 0
$$645$$ −1.00000 −0.0393750
$$646$$ −5.00000 −0.196722
$$647$$ −28.0000 −1.10079 −0.550397 0.834903i $$-0.685524\pi$$
−0.550397 + 0.834903i $$0.685524\pi$$
$$648$$ −1.00000 −0.0392837
$$649$$ 0 0
$$650$$ 4.00000 0.156893
$$651$$ 0 0
$$652$$ 10.0000 0.391630
$$653$$ 3.00000 0.117399 0.0586995 0.998276i $$-0.481305\pi$$
0.0586995 + 0.998276i $$0.481305\pi$$
$$654$$ −7.00000 −0.273722
$$655$$ 17.0000 0.664245
$$656$$ 8.00000 0.312348
$$657$$ 15.0000 0.585206
$$658$$ 0 0
$$659$$ 6.00000 0.233727 0.116863 0.993148i $$-0.462716\pi$$
0.116863 + 0.993148i $$0.462716\pi$$
$$660$$ 3.00000 0.116775
$$661$$ −10.0000 −0.388955 −0.194477 0.980907i $$-0.562301\pi$$
−0.194477 + 0.980907i $$0.562301\pi$$
$$662$$ −34.0000 −1.32145
$$663$$ 5.00000 0.194184
$$664$$ −2.00000 −0.0776151
$$665$$ 0 0
$$666$$ 5.00000 0.193746
$$667$$ 15.0000 0.580802
$$668$$ 15.0000 0.580367
$$669$$ 26.0000 1.00522
$$670$$ −10.0000 −0.386334
$$671$$ −39.0000 −1.50558
$$672$$ 0 0
$$673$$ −1.00000 −0.0385472 −0.0192736 0.999814i $$-0.506135\pi$$
−0.0192736 + 0.999814i $$0.506135\pi$$
$$674$$ 29.0000 1.11704
$$675$$ 4.00000 0.153960
$$676$$ 1.00000 0.0384615
$$677$$ 18.0000 0.691796 0.345898 0.938272i $$-0.387574\pi$$
0.345898 + 0.938272i $$0.387574\pi$$
$$678$$ −6.00000 −0.230429
$$679$$ 0 0
$$680$$ −5.00000 −0.191741
$$681$$ 2.00000 0.0766402
$$682$$ 12.0000 0.459504
$$683$$ 33.0000 1.26271 0.631355 0.775494i $$-0.282499\pi$$
0.631355 + 0.775494i $$0.282499\pi$$
$$684$$ −1.00000 −0.0382360
$$685$$ 15.0000 0.573121
$$686$$ 0 0
$$687$$ −6.00000 −0.228914
$$688$$ −1.00000 −0.0381246
$$689$$ 6.00000 0.228582
$$690$$ −3.00000 −0.114208
$$691$$ 20.0000 0.760836 0.380418 0.924815i $$-0.375780\pi$$
0.380418 + 0.924815i $$0.375780\pi$$
$$692$$ −6.00000 −0.228086
$$693$$ 0 0
$$694$$ −32.0000 −1.21470
$$695$$ −16.0000 −0.606915
$$696$$ 5.00000 0.189525
$$697$$ −40.0000 −1.51511
$$698$$ 6.00000 0.227103
$$699$$ 4.00000 0.151294
$$700$$ 0 0
$$701$$ −14.0000 −0.528773 −0.264386 0.964417i $$-0.585169\pi$$
−0.264386 + 0.964417i $$0.585169\pi$$
$$702$$ 1.00000 0.0377426
$$703$$ 5.00000 0.188579
$$704$$ 3.00000 0.113067
$$705$$ −8.00000 −0.301297
$$706$$ 2.00000 0.0752710
$$707$$ 0 0
$$708$$ 0 0
$$709$$ 50.0000 1.87779 0.938895 0.344204i $$-0.111851\pi$$
0.938895 + 0.344204i $$0.111851\pi$$
$$710$$ 8.00000 0.300235
$$711$$ 6.00000 0.225018
$$712$$ −2.00000 −0.0749532
$$713$$ −12.0000 −0.449404
$$714$$ 0 0
$$715$$ −3.00000 −0.112194
$$716$$ −16.0000 −0.597948
$$717$$ 12.0000 0.448148
$$718$$ −24.0000 −0.895672
$$719$$ −30.0000 −1.11881 −0.559406 0.828894i $$-0.688971\pi$$
−0.559406 + 0.828894i $$0.688971\pi$$
$$720$$ −1.00000 −0.0372678
$$721$$ 0 0
$$722$$ 18.0000 0.669891
$$723$$ 22.0000 0.818189
$$724$$ −6.00000 −0.222988
$$725$$ −20.0000 −0.742781
$$726$$ −2.00000 −0.0742270
$$727$$ 27.0000 1.00137 0.500687 0.865628i $$-0.333081\pi$$
0.500687 + 0.865628i $$0.333081\pi$$
$$728$$ 0 0
$$729$$ 1.00000 0.0370370
$$730$$ 15.0000 0.555175
$$731$$ 5.00000 0.184932
$$732$$ 13.0000 0.480494
$$733$$ −20.0000 −0.738717 −0.369358 0.929287i $$-0.620423\pi$$
−0.369358 + 0.929287i $$0.620423\pi$$
$$734$$ 8.00000 0.295285
$$735$$ 0 0
$$736$$ −3.00000 −0.110581
$$737$$ −30.0000 −1.10506
$$738$$ −8.00000 −0.294484
$$739$$ −26.0000 −0.956425 −0.478213 0.878244i $$-0.658715\pi$$
−0.478213 + 0.878244i $$0.658715\pi$$
$$740$$ 5.00000 0.183804
$$741$$ 1.00000 0.0367359
$$742$$ 0 0
$$743$$ −42.0000 −1.54083 −0.770415 0.637542i $$-0.779951\pi$$
−0.770415 + 0.637542i $$0.779951\pi$$
$$744$$ −4.00000 −0.146647
$$745$$ 16.0000 0.586195
$$746$$ 32.0000 1.17160
$$747$$ 2.00000 0.0731762
$$748$$ −15.0000 −0.548454
$$749$$ 0 0
$$750$$ 9.00000 0.328634
$$751$$ 24.0000 0.875772 0.437886 0.899030i $$-0.355727\pi$$
0.437886 + 0.899030i $$0.355727\pi$$
$$752$$ −8.00000 −0.291730
$$753$$ 21.0000 0.765283
$$754$$ −5.00000 −0.182089
$$755$$ 5.00000 0.181969
$$756$$ 0 0
$$757$$ −34.0000 −1.23575 −0.617876 0.786276i $$-0.712006\pi$$
−0.617876 + 0.786276i $$0.712006\pi$$
$$758$$ 8.00000 0.290573
$$759$$ −9.00000 −0.326679
$$760$$ −1.00000 −0.0362738
$$761$$ 24.0000 0.869999 0.435000 0.900431i $$-0.356748\pi$$
0.435000 + 0.900431i $$0.356748\pi$$
$$762$$ −18.0000 −0.652071
$$763$$ 0 0
$$764$$ 11.0000 0.397966
$$765$$ 5.00000 0.180775
$$766$$ 1.00000 0.0361315
$$767$$ 0 0
$$768$$ −1.00000 −0.0360844
$$769$$ −43.0000 −1.55062 −0.775310 0.631581i $$-0.782406\pi$$
−0.775310 + 0.631581i $$0.782406\pi$$
$$770$$ 0 0
$$771$$ −18.0000 −0.648254
$$772$$ −20.0000 −0.719816
$$773$$ −11.0000 −0.395643 −0.197821 0.980238i $$-0.563387\pi$$
−0.197821 + 0.980238i $$0.563387\pi$$
$$774$$ 1.00000 0.0359443
$$775$$ 16.0000 0.574737
$$776$$ −2.00000 −0.0717958
$$777$$ 0 0
$$778$$ 18.0000 0.645331
$$779$$ −8.00000 −0.286630
$$780$$ 1.00000 0.0358057
$$781$$ 24.0000 0.858788
$$782$$ 15.0000 0.536399
$$783$$ −5.00000 −0.178685
$$784$$ 0 0
$$785$$ 9.00000 0.321224
$$786$$ −17.0000 −0.606370
$$787$$ 47.0000 1.67537 0.837685 0.546154i $$-0.183909\pi$$
0.837685 + 0.546154i $$0.183909\pi$$
$$788$$ 0 0
$$789$$ −12.0000 −0.427211
$$790$$ 6.00000 0.213470
$$791$$ 0 0
$$792$$ −3.00000 −0.106600
$$793$$ −13.0000 −0.461644
$$794$$ 30.0000 1.06466
$$795$$ 6.00000 0.212798
$$796$$ 5.00000 0.177220
$$797$$ 28.0000 0.991811 0.495905 0.868377i $$-0.334836\pi$$
0.495905 + 0.868377i $$0.334836\pi$$
$$798$$ 0 0
$$799$$ 40.0000 1.41510
$$800$$ 4.00000 0.141421
$$801$$ 2.00000 0.0706665
$$802$$ 30.0000 1.05934
$$803$$ 45.0000 1.58802
$$804$$ 10.0000 0.352673
$$805$$ 0 0
$$806$$ 4.00000 0.140894
$$807$$ 6.00000 0.211210
$$808$$ −16.0000 −0.562878
$$809$$ 10.0000 0.351581 0.175791 0.984428i $$-0.443752\pi$$
0.175791 + 0.984428i $$0.443752\pi$$
$$810$$ 1.00000 0.0351364
$$811$$ 19.0000 0.667180 0.333590 0.942718i $$-0.391740\pi$$
0.333590 + 0.942718i $$0.391740\pi$$
$$812$$ 0 0
$$813$$ 2.00000 0.0701431
$$814$$ 15.0000 0.525750
$$815$$ −10.0000 −0.350285
$$816$$ 5.00000 0.175035
$$817$$ 1.00000 0.0349856
$$818$$ 3.00000 0.104893
$$819$$ 0 0
$$820$$ −8.00000 −0.279372
$$821$$ 24.0000 0.837606 0.418803 0.908077i $$-0.362450\pi$$
0.418803 + 0.908077i $$0.362450\pi$$
$$822$$ −15.0000 −0.523185
$$823$$ 24.0000 0.836587 0.418294 0.908312i $$-0.362628\pi$$
0.418294 + 0.908312i $$0.362628\pi$$
$$824$$ 1.00000 0.0348367
$$825$$ 12.0000 0.417786
$$826$$ 0 0
$$827$$ 43.0000 1.49526 0.747628 0.664117i $$-0.231193\pi$$
0.747628 + 0.664117i $$0.231193\pi$$
$$828$$ 3.00000 0.104257
$$829$$ 19.0000 0.659897 0.329949 0.943999i $$-0.392969\pi$$
0.329949 + 0.943999i $$0.392969\pi$$
$$830$$ 2.00000 0.0694210
$$831$$ 18.0000 0.624413
$$832$$ 1.00000 0.0346688
$$833$$ 0 0
$$834$$ 16.0000 0.554035
$$835$$ −15.0000 −0.519096
$$836$$ −3.00000 −0.103757
$$837$$ 4.00000 0.138260
$$838$$ −5.00000 −0.172722
$$839$$ 12.0000 0.414286 0.207143 0.978311i $$-0.433583\pi$$
0.207143 + 0.978311i $$0.433583\pi$$
$$840$$ 0 0
$$841$$ −4.00000 −0.137931
$$842$$ 22.0000 0.758170
$$843$$ −10.0000 −0.344418
$$844$$ 5.00000 0.172107
$$845$$ −1.00000 −0.0344010
$$846$$ 8.00000 0.275046
$$847$$ 0 0
$$848$$ 6.00000 0.206041
$$849$$ −8.00000 −0.274559
$$850$$ −20.0000 −0.685994
$$851$$ −15.0000 −0.514193
$$852$$ −8.00000 −0.274075
$$853$$ 28.0000 0.958702 0.479351 0.877623i $$-0.340872\pi$$
0.479351 + 0.877623i $$0.340872\pi$$
$$854$$ 0 0
$$855$$ 1.00000 0.0341993
$$856$$ 6.00000 0.205076
$$857$$ −42.0000 −1.43469 −0.717346 0.696717i $$-0.754643\pi$$
−0.717346 + 0.696717i $$0.754643\pi$$
$$858$$ 3.00000 0.102418
$$859$$ 28.0000 0.955348 0.477674 0.878537i $$-0.341480\pi$$
0.477674 + 0.878537i $$0.341480\pi$$
$$860$$ 1.00000 0.0340997
$$861$$ 0 0
$$862$$ −10.0000 −0.340601
$$863$$ −2.00000 −0.0680808 −0.0340404 0.999420i $$-0.510837\pi$$
−0.0340404 + 0.999420i $$0.510837\pi$$
$$864$$ 1.00000 0.0340207
$$865$$ 6.00000 0.204006
$$866$$ 20.0000 0.679628
$$867$$ −8.00000 −0.271694
$$868$$ 0 0
$$869$$ 18.0000 0.610608
$$870$$ −5.00000 −0.169516
$$871$$ −10.0000 −0.338837
$$872$$ 7.00000 0.237050
$$873$$ 2.00000 0.0676897
$$874$$ 3.00000 0.101477
$$875$$ 0 0
$$876$$ −15.0000 −0.506803
$$877$$ 34.0000 1.14810 0.574049 0.818821i $$-0.305372\pi$$
0.574049 + 0.818821i $$0.305372\pi$$
$$878$$ −11.0000 −0.371232
$$879$$ −6.00000 −0.202375
$$880$$ −3.00000 −0.101130
$$881$$ −37.0000 −1.24656 −0.623281 0.781998i $$-0.714201\pi$$
−0.623281 + 0.781998i $$0.714201\pi$$
$$882$$ 0 0
$$883$$ −41.0000 −1.37976 −0.689880 0.723924i $$-0.742337\pi$$
−0.689880 + 0.723924i $$0.742337\pi$$
$$884$$ −5.00000 −0.168168
$$885$$ 0 0
$$886$$ 18.0000 0.604722
$$887$$ 20.0000 0.671534 0.335767 0.941945i $$-0.391004\pi$$
0.335767 + 0.941945i $$0.391004\pi$$
$$888$$ −5.00000 −0.167789
$$889$$ 0 0
$$890$$ 2.00000 0.0670402
$$891$$ 3.00000 0.100504
$$892$$ −26.0000 −0.870544
$$893$$ 8.00000 0.267710
$$894$$ −16.0000 −0.535120
$$895$$ 16.0000 0.534821
$$896$$ 0 0
$$897$$ −3.00000 −0.100167
$$898$$ −13.0000 −0.433816
$$899$$ −20.0000 −0.667037
$$900$$ −4.00000 −0.133333
$$901$$ −30.0000 −0.999445
$$902$$ −24.0000 −0.799113
$$903$$ 0 0
$$904$$ 6.00000 0.199557
$$905$$ 6.00000 0.199447
$$906$$ −5.00000 −0.166114
$$907$$ −20.0000 −0.664089 −0.332045 0.943264i $$-0.607738\pi$$
−0.332045 + 0.943264i $$0.607738\pi$$
$$908$$ −2.00000 −0.0663723
$$909$$ 16.0000 0.530687
$$910$$ 0 0
$$911$$ 29.0000 0.960813 0.480406 0.877046i $$-0.340489\pi$$
0.480406 + 0.877046i $$0.340489\pi$$
$$912$$ 1.00000 0.0331133
$$913$$ 6.00000 0.198571
$$914$$ −8.00000 −0.264616
$$915$$ −13.0000 −0.429767
$$916$$ 6.00000 0.198246
$$917$$ 0 0
$$918$$ −5.00000 −0.165025
$$919$$ −50.0000 −1.64935 −0.824674 0.565608i $$-0.808641\pi$$
−0.824674 + 0.565608i $$0.808641\pi$$
$$920$$ 3.00000 0.0989071
$$921$$ 0 0
$$922$$ −13.0000 −0.428132
$$923$$ 8.00000 0.263323
$$924$$ 0 0
$$925$$ 20.0000 0.657596
$$926$$ 13.0000 0.427207
$$927$$ −1.00000 −0.0328443
$$928$$ −5.00000 −0.164133
$$929$$ 4.00000 0.131236 0.0656179 0.997845i $$-0.479098\pi$$
0.0656179 + 0.997845i $$0.479098\pi$$
$$930$$ 4.00000 0.131165
$$931$$ 0 0
$$932$$ −4.00000 −0.131024
$$933$$ 26.0000 0.851202
$$934$$ 21.0000 0.687141
$$935$$ 15.0000 0.490552
$$936$$ −1.00000 −0.0326860
$$937$$ 14.0000 0.457360 0.228680 0.973502i $$-0.426559\pi$$
0.228680 + 0.973502i $$0.426559\pi$$
$$938$$ 0 0
$$939$$ 28.0000 0.913745
$$940$$ 8.00000 0.260931
$$941$$ 18.0000 0.586783 0.293392 0.955992i $$-0.405216\pi$$
0.293392 + 0.955992i $$0.405216\pi$$
$$942$$ −9.00000 −0.293236
$$943$$ 24.0000 0.781548
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 3.00000 0.0975384
$$947$$ 47.0000 1.52729 0.763647 0.645634i $$-0.223407\pi$$
0.763647 + 0.645634i $$0.223407\pi$$
$$948$$ −6.00000 −0.194871
$$949$$ 15.0000 0.486921
$$950$$ −4.00000 −0.129777
$$951$$ 24.0000 0.778253
$$952$$ 0 0
$$953$$ −44.0000 −1.42530 −0.712650 0.701520i $$-0.752505\pi$$
−0.712650 + 0.701520i $$0.752505\pi$$
$$954$$ −6.00000 −0.194257
$$955$$ −11.0000 −0.355952
$$956$$ −12.0000 −0.388108
$$957$$ −15.0000 −0.484881
$$958$$ 27.0000 0.872330
$$959$$ 0 0
$$960$$ 1.00000 0.0322749
$$961$$ −15.0000 −0.483871
$$962$$ 5.00000 0.161206
$$963$$ −6.00000 −0.193347
$$964$$ −22.0000 −0.708572
$$965$$ 20.0000 0.643823
$$966$$ 0 0
$$967$$ −31.0000 −0.996893 −0.498446 0.866921i $$-0.666096\pi$$
−0.498446 + 0.866921i $$0.666096\pi$$
$$968$$ 2.00000 0.0642824
$$969$$ −5.00000 −0.160623
$$970$$ 2.00000 0.0642161
$$971$$ 32.0000 1.02693 0.513464 0.858111i $$-0.328362\pi$$
0.513464 + 0.858111i $$0.328362\pi$$
$$972$$ −1.00000 −0.0320750
$$973$$ 0 0
$$974$$ −8.00000 −0.256337
$$975$$ 4.00000 0.128103
$$976$$ −13.0000 −0.416120
$$977$$ −25.0000 −0.799821 −0.399910 0.916554i $$-0.630959\pi$$
−0.399910 + 0.916554i $$0.630959\pi$$
$$978$$ 10.0000 0.319765
$$979$$ 6.00000 0.191761
$$980$$ 0 0
$$981$$ −7.00000 −0.223493
$$982$$ 38.0000 1.21263
$$983$$ 5.00000 0.159475 0.0797376 0.996816i $$-0.474592\pi$$
0.0797376 + 0.996816i $$0.474592\pi$$
$$984$$ 8.00000 0.255031
$$985$$ 0 0
$$986$$ 25.0000 0.796162
$$987$$ 0 0
$$988$$ −1.00000 −0.0318142
$$989$$ −3.00000 −0.0953945
$$990$$ 3.00000 0.0953463
$$991$$ −30.0000 −0.952981 −0.476491 0.879180i $$-0.658091\pi$$
−0.476491 + 0.879180i $$0.658091\pi$$
$$992$$ 4.00000 0.127000
$$993$$ −34.0000 −1.07896
$$994$$ 0 0
$$995$$ −5.00000 −0.158511
$$996$$ −2.00000 −0.0633724
$$997$$ 50.0000 1.58352 0.791758 0.610835i $$-0.209166\pi$$
0.791758 + 0.610835i $$0.209166\pi$$
$$998$$ −8.00000 −0.253236
$$999$$ 5.00000 0.158193
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 3822.2.a.e.1.1 1
7.6 odd 2 546.2.a.c.1.1 1
21.20 even 2 1638.2.a.o.1.1 1
28.27 even 2 4368.2.a.h.1.1 1
91.90 odd 2 7098.2.a.z.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
546.2.a.c.1.1 1 7.6 odd 2
1638.2.a.o.1.1 1 21.20 even 2
3822.2.a.e.1.1 1 1.1 even 1 trivial
4368.2.a.h.1.1 1 28.27 even 2
7098.2.a.z.1.1 1 91.90 odd 2