# Properties

 Label 1638.2.a.o.1.1 Level $1638$ Weight $2$ Character 1638.1 Self dual yes Analytic conductor $13.079$ 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] = [1638,2,Mod(1,1638)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1638, 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("1638.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1638 = 2 \cdot 3^{2} \cdot 7 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1638.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: $$13.0794958511$$ 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 $$\chi$$ $$=$$ 1638.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{7} +1.00000 q^{8} +O(q^{10})$$ $$q+1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{7} +1.00000 q^{8} -1.00000 q^{10} -3.00000 q^{11} -1.00000 q^{13} -1.00000 q^{14} +1.00000 q^{16} -5.00000 q^{17} +1.00000 q^{19} -1.00000 q^{20} -3.00000 q^{22} -3.00000 q^{23} -4.00000 q^{25} -1.00000 q^{26} -1.00000 q^{28} -5.00000 q^{29} +4.00000 q^{31} +1.00000 q^{32} -5.00000 q^{34} +1.00000 q^{35} -5.00000 q^{37} +1.00000 q^{38} -1.00000 q^{40} +8.00000 q^{41} -1.00000 q^{43} -3.00000 q^{44} -3.00000 q^{46} -8.00000 q^{47} +1.00000 q^{49} -4.00000 q^{50} -1.00000 q^{52} -6.00000 q^{53} +3.00000 q^{55} -1.00000 q^{56} -5.00000 q^{58} +13.0000 q^{61} +4.00000 q^{62} +1.00000 q^{64} +1.00000 q^{65} -10.0000 q^{67} -5.00000 q^{68} +1.00000 q^{70} -8.00000 q^{71} -15.0000 q^{73} -5.00000 q^{74} +1.00000 q^{76} +3.00000 q^{77} +6.00000 q^{79} -1.00000 q^{80} +8.00000 q^{82} +2.00000 q^{83} +5.00000 q^{85} -1.00000 q^{86} -3.00000 q^{88} +2.00000 q^{89} +1.00000 q^{91} -3.00000 q^{92} -8.00000 q^{94} -1.00000 q^{95} -2.00000 q^{97} +1.00000 q^{98} +O(q^{100})$$

## 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
$$3$$ 0 0
$$4$$ 1.00000 0.500000
$$5$$ −1.00000 −0.447214 −0.223607 0.974679i $$-0.571783\pi$$
−0.223607 + 0.974679i $$0.571783\pi$$
$$6$$ 0 0
$$7$$ −1.00000 −0.377964
$$8$$ 1.00000 0.353553
$$9$$ 0 0
$$10$$ −1.00000 −0.316228
$$11$$ −3.00000 −0.904534 −0.452267 0.891883i $$-0.649385\pi$$
−0.452267 + 0.891883i $$0.649385\pi$$
$$12$$ 0 0
$$13$$ −1.00000 −0.277350
$$14$$ −1.00000 −0.267261
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ −5.00000 −1.21268 −0.606339 0.795206i $$-0.707363\pi$$
−0.606339 + 0.795206i $$0.707363\pi$$
$$18$$ 0 0
$$19$$ 1.00000 0.229416 0.114708 0.993399i $$-0.463407\pi$$
0.114708 + 0.993399i $$0.463407\pi$$
$$20$$ −1.00000 −0.223607
$$21$$ 0 0
$$22$$ −3.00000 −0.639602
$$23$$ −3.00000 −0.625543 −0.312772 0.949828i $$-0.601257\pi$$
−0.312772 + 0.949828i $$0.601257\pi$$
$$24$$ 0 0
$$25$$ −4.00000 −0.800000
$$26$$ −1.00000 −0.196116
$$27$$ 0 0
$$28$$ −1.00000 −0.188982
$$29$$ −5.00000 −0.928477 −0.464238 0.885710i $$-0.653672\pi$$
−0.464238 + 0.885710i $$0.653672\pi$$
$$30$$ 0 0
$$31$$ 4.00000 0.718421 0.359211 0.933257i $$-0.383046\pi$$
0.359211 + 0.933257i $$0.383046\pi$$
$$32$$ 1.00000 0.176777
$$33$$ 0 0
$$34$$ −5.00000 −0.857493
$$35$$ 1.00000 0.169031
$$36$$ 0 0
$$37$$ −5.00000 −0.821995 −0.410997 0.911636i $$-0.634819\pi$$
−0.410997 + 0.911636i $$0.634819\pi$$
$$38$$ 1.00000 0.162221
$$39$$ 0 0
$$40$$ −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
$$45$$ 0 0
$$46$$ −3.00000 −0.442326
$$47$$ −8.00000 −1.16692 −0.583460 0.812142i $$-0.698301\pi$$
−0.583460 + 0.812142i $$0.698301\pi$$
$$48$$ 0 0
$$49$$ 1.00000 0.142857
$$50$$ −4.00000 −0.565685
$$51$$ 0 0
$$52$$ −1.00000 −0.138675
$$53$$ −6.00000 −0.824163 −0.412082 0.911147i $$-0.635198\pi$$
−0.412082 + 0.911147i $$0.635198\pi$$
$$54$$ 0 0
$$55$$ 3.00000 0.404520
$$56$$ −1.00000 −0.133631
$$57$$ 0 0
$$58$$ −5.00000 −0.656532
$$59$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$60$$ 0 0
$$61$$ 13.0000 1.66448 0.832240 0.554416i $$-0.187058\pi$$
0.832240 + 0.554416i $$0.187058\pi$$
$$62$$ 4.00000 0.508001
$$63$$ 0 0
$$64$$ 1.00000 0.125000
$$65$$ 1.00000 0.124035
$$66$$ 0 0
$$67$$ −10.0000 −1.22169 −0.610847 0.791748i $$-0.709171\pi$$
−0.610847 + 0.791748i $$0.709171\pi$$
$$68$$ −5.00000 −0.606339
$$69$$ 0 0
$$70$$ 1.00000 0.119523
$$71$$ −8.00000 −0.949425 −0.474713 0.880141i $$-0.657448\pi$$
−0.474713 + 0.880141i $$0.657448\pi$$
$$72$$ 0 0
$$73$$ −15.0000 −1.75562 −0.877809 0.479012i $$-0.840995\pi$$
−0.877809 + 0.479012i $$0.840995\pi$$
$$74$$ −5.00000 −0.581238
$$75$$ 0 0
$$76$$ 1.00000 0.114708
$$77$$ 3.00000 0.341882
$$78$$ 0 0
$$79$$ 6.00000 0.675053 0.337526 0.941316i $$-0.390410\pi$$
0.337526 + 0.941316i $$0.390410\pi$$
$$80$$ −1.00000 −0.111803
$$81$$ 0 0
$$82$$ 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
$$87$$ 0 0
$$88$$ −3.00000 −0.319801
$$89$$ 2.00000 0.212000 0.106000 0.994366i $$-0.466196\pi$$
0.106000 + 0.994366i $$0.466196\pi$$
$$90$$ 0 0
$$91$$ 1.00000 0.104828
$$92$$ −3.00000 −0.312772
$$93$$ 0 0
$$94$$ −8.00000 −0.825137
$$95$$ −1.00000 −0.102598
$$96$$ 0 0
$$97$$ −2.00000 −0.203069 −0.101535 0.994832i $$-0.532375\pi$$
−0.101535 + 0.994832i $$0.532375\pi$$
$$98$$ 1.00000 0.101015
$$99$$ 0 0
$$100$$ −4.00000 −0.400000
$$101$$ 16.0000 1.59206 0.796030 0.605257i $$-0.206930\pi$$
0.796030 + 0.605257i $$0.206930\pi$$
$$102$$ 0 0
$$103$$ 1.00000 0.0985329 0.0492665 0.998786i $$-0.484312\pi$$
0.0492665 + 0.998786i $$0.484312\pi$$
$$104$$ −1.00000 −0.0980581
$$105$$ 0 0
$$106$$ −6.00000 −0.582772
$$107$$ 6.00000 0.580042 0.290021 0.957020i $$-0.406338\pi$$
0.290021 + 0.957020i $$0.406338\pi$$
$$108$$ 0 0
$$109$$ −7.00000 −0.670478 −0.335239 0.942133i $$-0.608817\pi$$
−0.335239 + 0.942133i $$0.608817\pi$$
$$110$$ 3.00000 0.286039
$$111$$ 0 0
$$112$$ −1.00000 −0.0944911
$$113$$ 6.00000 0.564433 0.282216 0.959351i $$-0.408930\pi$$
0.282216 + 0.959351i $$0.408930\pi$$
$$114$$ 0 0
$$115$$ 3.00000 0.279751
$$116$$ −5.00000 −0.464238
$$117$$ 0 0
$$118$$ 0 0
$$119$$ 5.00000 0.458349
$$120$$ 0 0
$$121$$ −2.00000 −0.181818
$$122$$ 13.0000 1.17696
$$123$$ 0 0
$$124$$ 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
$$129$$ 0 0
$$130$$ 1.00000 0.0877058
$$131$$ −17.0000 −1.48530 −0.742648 0.669681i $$-0.766431\pi$$
−0.742648 + 0.669681i $$0.766431\pi$$
$$132$$ 0 0
$$133$$ −1.00000 −0.0867110
$$134$$ −10.0000 −0.863868
$$135$$ 0 0
$$136$$ −5.00000 −0.428746
$$137$$ 15.0000 1.28154 0.640768 0.767734i $$-0.278616\pi$$
0.640768 + 0.767734i $$0.278616\pi$$
$$138$$ 0 0
$$139$$ −16.0000 −1.35710 −0.678551 0.734553i $$-0.737392\pi$$
−0.678551 + 0.734553i $$0.737392\pi$$
$$140$$ 1.00000 0.0845154
$$141$$ 0 0
$$142$$ −8.00000 −0.671345
$$143$$ 3.00000 0.250873
$$144$$ 0 0
$$145$$ 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 $$-0.272506\pi$$
0.655386 + 0.755295i $$0.272506\pi$$
$$150$$ 0 0
$$151$$ −5.00000 −0.406894 −0.203447 0.979086i $$-0.565214\pi$$
−0.203447 + 0.979086i $$0.565214\pi$$
$$152$$ 1.00000 0.0811107
$$153$$ 0 0
$$154$$ 3.00000 0.241747
$$155$$ −4.00000 −0.321288
$$156$$ 0 0
$$157$$ 9.00000 0.718278 0.359139 0.933284i $$-0.383070\pi$$
0.359139 + 0.933284i $$0.383070\pi$$
$$158$$ 6.00000 0.477334
$$159$$ 0 0
$$160$$ −1.00000 −0.0790569
$$161$$ 3.00000 0.236433
$$162$$ 0 0
$$163$$ 10.0000 0.783260 0.391630 0.920123i $$-0.371911\pi$$
0.391630 + 0.920123i $$0.371911\pi$$
$$164$$ 8.00000 0.624695
$$165$$ 0 0
$$166$$ 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
$$171$$ 0 0
$$172$$ −1.00000 −0.0762493
$$173$$ −6.00000 −0.456172 −0.228086 0.973641i $$-0.573247\pi$$
−0.228086 + 0.973641i $$0.573247\pi$$
$$174$$ 0 0
$$175$$ 4.00000 0.302372
$$176$$ −3.00000 −0.226134
$$177$$ 0 0
$$178$$ 2.00000 0.149906
$$179$$ 16.0000 1.19590 0.597948 0.801535i $$-0.295983\pi$$
0.597948 + 0.801535i $$0.295983\pi$$
$$180$$ 0 0
$$181$$ 6.00000 0.445976 0.222988 0.974821i $$-0.428419\pi$$
0.222988 + 0.974821i $$0.428419\pi$$
$$182$$ 1.00000 0.0741249
$$183$$ 0 0
$$184$$ −3.00000 −0.221163
$$185$$ 5.00000 0.367607
$$186$$ 0 0
$$187$$ 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 $$-0.630284\pi$$
−0.397966 + 0.917400i $$0.630284\pi$$
$$192$$ 0 0
$$193$$ −20.0000 −1.43963 −0.719816 0.694165i $$-0.755774\pi$$
−0.719816 + 0.694165i $$0.755774\pi$$
$$194$$ −2.00000 −0.143592
$$195$$ 0 0
$$196$$ 1.00000 0.0714286
$$197$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$198$$ 0 0
$$199$$ −5.00000 −0.354441 −0.177220 0.984171i $$-0.556711\pi$$
−0.177220 + 0.984171i $$0.556711\pi$$
$$200$$ −4.00000 −0.282843
$$201$$ 0 0
$$202$$ 16.0000 1.12576
$$203$$ 5.00000 0.350931
$$204$$ 0 0
$$205$$ −8.00000 −0.558744
$$206$$ 1.00000 0.0696733
$$207$$ 0 0
$$208$$ −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
$$213$$ 0 0
$$214$$ 6.00000 0.410152
$$215$$ 1.00000 0.0681994
$$216$$ 0 0
$$217$$ −4.00000 −0.271538
$$218$$ −7.00000 −0.474100
$$219$$ 0 0
$$220$$ 3.00000 0.202260
$$221$$ 5.00000 0.336336
$$222$$ 0 0
$$223$$ 26.0000 1.74109 0.870544 0.492090i $$-0.163767\pi$$
0.870544 + 0.492090i $$0.163767\pi$$
$$224$$ −1.00000 −0.0668153
$$225$$ 0 0
$$226$$ 6.00000 0.399114
$$227$$ −2.00000 −0.132745 −0.0663723 0.997795i $$-0.521143\pi$$
−0.0663723 + 0.997795i $$0.521143\pi$$
$$228$$ 0 0
$$229$$ −6.00000 −0.396491 −0.198246 0.980152i $$-0.563524\pi$$
−0.198246 + 0.980152i $$0.563524\pi$$
$$230$$ 3.00000 0.197814
$$231$$ 0 0
$$232$$ −5.00000 −0.328266
$$233$$ 4.00000 0.262049 0.131024 0.991379i $$-0.458173\pi$$
0.131024 + 0.991379i $$0.458173\pi$$
$$234$$ 0 0
$$235$$ 8.00000 0.521862
$$236$$ 0 0
$$237$$ 0 0
$$238$$ 5.00000 0.324102
$$239$$ 12.0000 0.776215 0.388108 0.921614i $$-0.373129\pi$$
0.388108 + 0.921614i $$0.373129\pi$$
$$240$$ 0 0
$$241$$ 22.0000 1.41714 0.708572 0.705638i $$-0.249340\pi$$
0.708572 + 0.705638i $$0.249340\pi$$
$$242$$ −2.00000 −0.128565
$$243$$ 0 0
$$244$$ 13.0000 0.832240
$$245$$ −1.00000 −0.0638877
$$246$$ 0 0
$$247$$ −1.00000 −0.0636285
$$248$$ 4.00000 0.254000
$$249$$ 0 0
$$250$$ 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
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ 18.0000 1.12281 0.561405 0.827541i $$-0.310261\pi$$
0.561405 + 0.827541i $$0.310261\pi$$
$$258$$ 0 0
$$259$$ 5.00000 0.310685
$$260$$ 1.00000 0.0620174
$$261$$ 0 0
$$262$$ −17.0000 −1.05026
$$263$$ −12.0000 −0.739952 −0.369976 0.929041i $$-0.620634\pi$$
−0.369976 + 0.929041i $$0.620634\pi$$
$$264$$ 0 0
$$265$$ 6.00000 0.368577
$$266$$ −1.00000 −0.0613139
$$267$$ 0 0
$$268$$ −10.0000 −0.610847
$$269$$ −6.00000 −0.365826 −0.182913 0.983129i $$-0.558553\pi$$
−0.182913 + 0.983129i $$0.558553\pi$$
$$270$$ 0 0
$$271$$ 2.00000 0.121491 0.0607457 0.998153i $$-0.480652\pi$$
0.0607457 + 0.998153i $$0.480652\pi$$
$$272$$ −5.00000 −0.303170
$$273$$ 0 0
$$274$$ 15.0000 0.906183
$$275$$ 12.0000 0.723627
$$276$$ 0 0
$$277$$ −18.0000 −1.08152 −0.540758 0.841178i $$-0.681862\pi$$
−0.540758 + 0.841178i $$0.681862\pi$$
$$278$$ −16.0000 −0.959616
$$279$$ 0 0
$$280$$ 1.00000 0.0597614
$$281$$ −10.0000 −0.596550 −0.298275 0.954480i $$-0.596411\pi$$
−0.298275 + 0.954480i $$0.596411\pi$$
$$282$$ 0 0
$$283$$ −8.00000 −0.475551 −0.237775 0.971320i $$-0.576418\pi$$
−0.237775 + 0.971320i $$0.576418\pi$$
$$284$$ −8.00000 −0.474713
$$285$$ 0 0
$$286$$ 3.00000 0.177394
$$287$$ −8.00000 −0.472225
$$288$$ 0 0
$$289$$ 8.00000 0.470588
$$290$$ 5.00000 0.293610
$$291$$ 0 0
$$292$$ −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
$$297$$ 0 0
$$298$$ 16.0000 0.926855
$$299$$ 3.00000 0.173494
$$300$$ 0 0
$$301$$ 1.00000 0.0576390
$$302$$ −5.00000 −0.287718
$$303$$ 0 0
$$304$$ 1.00000 0.0573539
$$305$$ −13.0000 −0.744378
$$306$$ 0 0
$$307$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$308$$ 3.00000 0.170941
$$309$$ 0 0
$$310$$ −4.00000 −0.227185
$$311$$ −26.0000 −1.47432 −0.737162 0.675716i $$-0.763835\pi$$
−0.737162 + 0.675716i $$0.763835\pi$$
$$312$$ 0 0
$$313$$ 28.0000 1.58265 0.791327 0.611393i $$-0.209391\pi$$
0.791327 + 0.611393i $$0.209391\pi$$
$$314$$ 9.00000 0.507899
$$315$$ 0 0
$$316$$ 6.00000 0.337526
$$317$$ 24.0000 1.34797 0.673987 0.738743i $$-0.264580\pi$$
0.673987 + 0.738743i $$0.264580\pi$$
$$318$$ 0 0
$$319$$ 15.0000 0.839839
$$320$$ −1.00000 −0.0559017
$$321$$ 0 0
$$322$$ 3.00000 0.167183
$$323$$ −5.00000 −0.278207
$$324$$ 0 0
$$325$$ 4.00000 0.221880
$$326$$ 10.0000 0.553849
$$327$$ 0 0
$$328$$ 8.00000 0.441726
$$329$$ 8.00000 0.441054
$$330$$ 0 0
$$331$$ 34.0000 1.86881 0.934405 0.356214i $$-0.115932\pi$$
0.934405 + 0.356214i $$0.115932\pi$$
$$332$$ 2.00000 0.109764
$$333$$ 0 0
$$334$$ 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
$$339$$ 0 0
$$340$$ 5.00000 0.271163
$$341$$ −12.0000 −0.649836
$$342$$ 0 0
$$343$$ −1.00000 −0.0539949
$$344$$ −1.00000 −0.0539164
$$345$$ 0 0
$$346$$ −6.00000 −0.322562
$$347$$ −32.0000 −1.71785 −0.858925 0.512101i $$-0.828867\pi$$
−0.858925 + 0.512101i $$0.828867\pi$$
$$348$$ 0 0
$$349$$ 6.00000 0.321173 0.160586 0.987022i $$-0.448662\pi$$
0.160586 + 0.987022i $$0.448662\pi$$
$$350$$ 4.00000 0.213809
$$351$$ 0 0
$$352$$ −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 $$-0.718315\pi$$
−0.633336 + 0.773877i $$0.718315\pi$$
$$360$$ 0 0
$$361$$ −18.0000 −0.947368
$$362$$ 6.00000 0.315353
$$363$$ 0 0
$$364$$ 1.00000 0.0524142
$$365$$ 15.0000 0.785136
$$366$$ 0 0
$$367$$ 8.00000 0.417597 0.208798 0.977959i $$-0.433045\pi$$
0.208798 + 0.977959i $$0.433045\pi$$
$$368$$ −3.00000 −0.156386
$$369$$ 0 0
$$370$$ 5.00000 0.259938
$$371$$ 6.00000 0.311504
$$372$$ 0 0
$$373$$ −32.0000 −1.65690 −0.828449 0.560065i $$-0.810776\pi$$
−0.828449 + 0.560065i $$0.810776\pi$$
$$374$$ 15.0000 0.775632
$$375$$ 0 0
$$376$$ −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
$$381$$ 0 0
$$382$$ −11.0000 −0.562809
$$383$$ −1.00000 −0.0510976 −0.0255488 0.999674i $$-0.508133\pi$$
−0.0255488 + 0.999674i $$0.508133\pi$$
$$384$$ 0 0
$$385$$ −3.00000 −0.152894
$$386$$ −20.0000 −1.01797
$$387$$ 0 0
$$388$$ −2.00000 −0.101535
$$389$$ 18.0000 0.912636 0.456318 0.889817i $$-0.349168\pi$$
0.456318 + 0.889817i $$0.349168\pi$$
$$390$$ 0 0
$$391$$ 15.0000 0.758583
$$392$$ 1.00000 0.0505076
$$393$$ 0 0
$$394$$ 0 0
$$395$$ −6.00000 −0.301893
$$396$$ 0 0
$$397$$ 30.0000 1.50566 0.752828 0.658217i $$-0.228689\pi$$
0.752828 + 0.658217i $$0.228689\pi$$
$$398$$ −5.00000 −0.250627
$$399$$ 0 0
$$400$$ −4.00000 −0.200000
$$401$$ 30.0000 1.49813 0.749064 0.662497i $$-0.230503\pi$$
0.749064 + 0.662497i $$0.230503\pi$$
$$402$$ 0 0
$$403$$ −4.00000 −0.199254
$$404$$ 16.0000 0.796030
$$405$$ 0 0
$$406$$ 5.00000 0.248146
$$407$$ 15.0000 0.743522
$$408$$ 0 0
$$409$$ 3.00000 0.148340 0.0741702 0.997246i $$-0.476369\pi$$
0.0741702 + 0.997246i $$0.476369\pi$$
$$410$$ −8.00000 −0.395092
$$411$$ 0 0
$$412$$ 1.00000 0.0492665
$$413$$ 0 0
$$414$$ 0 0
$$415$$ −2.00000 −0.0981761
$$416$$ −1.00000 −0.0490290
$$417$$ 0 0
$$418$$ −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
$$423$$ 0 0
$$424$$ −6.00000 −0.291386
$$425$$ 20.0000 0.970143
$$426$$ 0 0
$$427$$ −13.0000 −0.629114
$$428$$ 6.00000 0.290021
$$429$$ 0 0
$$430$$ 1.00000 0.0482243
$$431$$ −10.0000 −0.481683 −0.240842 0.970564i $$-0.577423\pi$$
−0.240842 + 0.970564i $$0.577423\pi$$
$$432$$ 0 0
$$433$$ 20.0000 0.961139 0.480569 0.876957i $$-0.340430\pi$$
0.480569 + 0.876957i $$0.340430\pi$$
$$434$$ −4.00000 −0.192006
$$435$$ 0 0
$$436$$ −7.00000 −0.335239
$$437$$ −3.00000 −0.143509
$$438$$ 0 0
$$439$$ −11.0000 −0.525001 −0.262501 0.964932i $$-0.584547\pi$$
−0.262501 + 0.964932i $$0.584547\pi$$
$$440$$ 3.00000 0.143019
$$441$$ 0 0
$$442$$ 5.00000 0.237826
$$443$$ 18.0000 0.855206 0.427603 0.903967i $$-0.359358\pi$$
0.427603 + 0.903967i $$0.359358\pi$$
$$444$$ 0 0
$$445$$ −2.00000 −0.0948091
$$446$$ 26.0000 1.23114
$$447$$ 0 0
$$448$$ −1.00000 −0.0472456
$$449$$ −13.0000 −0.613508 −0.306754 0.951789i $$-0.599243\pi$$
−0.306754 + 0.951789i $$0.599243\pi$$
$$450$$ 0 0
$$451$$ −24.0000 −1.13012
$$452$$ 6.00000 0.282216
$$453$$ 0 0
$$454$$ −2.00000 −0.0938647
$$455$$ −1.00000 −0.0468807
$$456$$ 0 0
$$457$$ 8.00000 0.374224 0.187112 0.982339i $$-0.440087\pi$$
0.187112 + 0.982339i $$0.440087\pi$$
$$458$$ −6.00000 −0.280362
$$459$$ 0 0
$$460$$ 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
$$465$$ 0 0
$$466$$ 4.00000 0.185296
$$467$$ −21.0000 −0.971764 −0.485882 0.874024i $$-0.661502\pi$$
−0.485882 + 0.874024i $$0.661502\pi$$
$$468$$ 0 0
$$469$$ 10.0000 0.461757
$$470$$ 8.00000 0.369012
$$471$$ 0 0
$$472$$ 0 0
$$473$$ 3.00000 0.137940
$$474$$ 0 0
$$475$$ −4.00000 −0.183533
$$476$$ 5.00000 0.229175
$$477$$ 0 0
$$478$$ 12.0000 0.548867
$$479$$ −27.0000 −1.23366 −0.616831 0.787096i $$-0.711584\pi$$
−0.616831 + 0.787096i $$0.711584\pi$$
$$480$$ 0 0
$$481$$ 5.00000 0.227980
$$482$$ 22.0000 1.00207
$$483$$ 0 0
$$484$$ −2.00000 −0.0909091
$$485$$ 2.00000 0.0908153
$$486$$ 0 0
$$487$$ 8.00000 0.362515 0.181257 0.983436i $$-0.441983\pi$$
0.181257 + 0.983436i $$0.441983\pi$$
$$488$$ 13.0000 0.588482
$$489$$ 0 0
$$490$$ −1.00000 −0.0451754
$$491$$ 38.0000 1.71492 0.857458 0.514554i $$-0.172042\pi$$
0.857458 + 0.514554i $$0.172042\pi$$
$$492$$ 0 0
$$493$$ 25.0000 1.12594
$$494$$ −1.00000 −0.0449921
$$495$$ 0 0
$$496$$ 4.00000 0.179605
$$497$$ 8.00000 0.358849
$$498$$ 0 0
$$499$$ 8.00000 0.358129 0.179065 0.983837i $$-0.442693\pi$$
0.179065 + 0.983837i $$0.442693\pi$$
$$500$$ 9.00000 0.402492
$$501$$ 0 0
$$502$$ −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
$$507$$ 0 0
$$508$$ −18.0000 −0.798621
$$509$$ −11.0000 −0.487566 −0.243783 0.969830i $$-0.578389\pi$$
−0.243783 + 0.969830i $$0.578389\pi$$
$$510$$ 0 0
$$511$$ 15.0000 0.663561
$$512$$ 1.00000 0.0441942
$$513$$ 0 0
$$514$$ 18.0000 0.793946
$$515$$ −1.00000 −0.0440653
$$516$$ 0 0
$$517$$ 24.0000 1.05552
$$518$$ 5.00000 0.219687
$$519$$ 0 0
$$520$$ 1.00000 0.0438529
$$521$$ −39.0000 −1.70862 −0.854311 0.519763i $$-0.826020\pi$$
−0.854311 + 0.519763i $$0.826020\pi$$
$$522$$ 0 0
$$523$$ −24.0000 −1.04945 −0.524723 0.851273i $$-0.675831\pi$$
−0.524723 + 0.851273i $$0.675831\pi$$
$$524$$ −17.0000 −0.742648
$$525$$ 0 0
$$526$$ −12.0000 −0.523225
$$527$$ −20.0000 −0.871214
$$528$$ 0 0
$$529$$ −14.0000 −0.608696
$$530$$ 6.00000 0.260623
$$531$$ 0 0
$$532$$ −1.00000 −0.0433555
$$533$$ −8.00000 −0.346518
$$534$$ 0 0
$$535$$ −6.00000 −0.259403
$$536$$ −10.0000 −0.431934
$$537$$ 0 0
$$538$$ −6.00000 −0.258678
$$539$$ −3.00000 −0.129219
$$540$$ 0 0
$$541$$ 41.0000 1.76273 0.881364 0.472438i $$-0.156626\pi$$
0.881364 + 0.472438i $$0.156626\pi$$
$$542$$ 2.00000 0.0859074
$$543$$ 0 0
$$544$$ −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
$$549$$ 0 0
$$550$$ 12.0000 0.511682
$$551$$ −5.00000 −0.213007
$$552$$ 0 0
$$553$$ −6.00000 −0.255146
$$554$$ −18.0000 −0.764747
$$555$$ 0 0
$$556$$ −16.0000 −0.678551
$$557$$ 24.0000 1.01691 0.508456 0.861088i $$-0.330216\pi$$
0.508456 + 0.861088i $$0.330216\pi$$
$$558$$ 0 0
$$559$$ 1.00000 0.0422955
$$560$$ 1.00000 0.0422577
$$561$$ 0 0
$$562$$ −10.0000 −0.421825
$$563$$ −9.00000 −0.379305 −0.189652 0.981851i $$-0.560736\pi$$
−0.189652 + 0.981851i $$0.560736\pi$$
$$564$$ 0 0
$$565$$ −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 $$-0.752517\pi$$
−0.712677 + 0.701492i $$0.752517\pi$$
$$570$$ 0 0
$$571$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$572$$ 3.00000 0.125436
$$573$$ 0 0
$$574$$ −8.00000 −0.333914
$$575$$ 12.0000 0.500435
$$576$$ 0 0
$$577$$ −34.0000 −1.41544 −0.707719 0.706494i $$-0.750276\pi$$
−0.707719 + 0.706494i $$0.750276\pi$$
$$578$$ 8.00000 0.332756
$$579$$ 0 0
$$580$$ 5.00000 0.207614
$$581$$ −2.00000 −0.0829740
$$582$$ 0 0
$$583$$ 18.0000 0.745484
$$584$$ −15.0000 −0.620704
$$585$$ 0 0
$$586$$ 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 $$0.784901\pi$$
$$594$$ 0 0
$$595$$ −5.00000 −0.204980
$$596$$ 16.0000 0.655386
$$597$$ 0 0
$$598$$ 3.00000 0.122679
$$599$$ −37.0000 −1.51178 −0.755890 0.654699i $$-0.772795\pi$$
−0.755890 + 0.654699i $$0.772795\pi$$
$$600$$ 0 0
$$601$$ 20.0000 0.815817 0.407909 0.913023i $$-0.366258\pi$$
0.407909 + 0.913023i $$0.366258\pi$$
$$602$$ 1.00000 0.0407570
$$603$$ 0 0
$$604$$ −5.00000 −0.203447
$$605$$ 2.00000 0.0813116
$$606$$ 0 0
$$607$$ −27.0000 −1.09590 −0.547948 0.836512i $$-0.684591\pi$$
−0.547948 + 0.836512i $$0.684591\pi$$
$$608$$ 1.00000 0.0405554
$$609$$ 0 0
$$610$$ −13.0000 −0.526355
$$611$$ 8.00000 0.323645
$$612$$ 0 0
$$613$$ 41.0000 1.65597 0.827987 0.560747i $$-0.189486\pi$$
0.827987 + 0.560747i $$0.189486\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 3.00000 0.120873
$$617$$ −43.0000 −1.73111 −0.865557 0.500810i $$-0.833036\pi$$
−0.865557 + 0.500810i $$0.833036\pi$$
$$618$$ 0 0
$$619$$ −31.0000 −1.24600 −0.622998 0.782224i $$-0.714085\pi$$
−0.622998 + 0.782224i $$0.714085\pi$$
$$620$$ −4.00000 −0.160644
$$621$$ 0 0
$$622$$ −26.0000 −1.04251
$$623$$ −2.00000 −0.0801283
$$624$$ 0 0
$$625$$ 11.0000 0.440000
$$626$$ 28.0000 1.11911
$$627$$ 0 0
$$628$$ 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
$$633$$ 0 0
$$634$$ 24.0000 0.953162
$$635$$ 18.0000 0.714308
$$636$$ 0 0
$$637$$ −1.00000 −0.0396214
$$638$$ 15.0000 0.593856
$$639$$ 0 0
$$640$$ −1.00000 −0.0395285
$$641$$ −30.0000 −1.18493 −0.592464 0.805597i $$-0.701845\pi$$
−0.592464 + 0.805597i $$0.701845\pi$$
$$642$$ 0 0
$$643$$ −1.00000 −0.0394362 −0.0197181 0.999806i $$-0.506277\pi$$
−0.0197181 + 0.999806i $$0.506277\pi$$
$$644$$ 3.00000 0.118217
$$645$$ 0 0
$$646$$ −5.00000 −0.196722
$$647$$ −28.0000 −1.10079 −0.550397 0.834903i $$-0.685524\pi$$
−0.550397 + 0.834903i $$0.685524\pi$$
$$648$$ 0 0
$$649$$ 0 0
$$650$$ 4.00000 0.156893
$$651$$ 0 0
$$652$$ 10.0000 0.391630
$$653$$ −3.00000 −0.117399 −0.0586995 0.998276i $$-0.518695\pi$$
−0.0586995 + 0.998276i $$0.518695\pi$$
$$654$$ 0 0
$$655$$ 17.0000 0.664245
$$656$$ 8.00000 0.312348
$$657$$ 0 0
$$658$$ 8.00000 0.311872
$$659$$ −6.00000 −0.233727 −0.116863 0.993148i $$-0.537284\pi$$
−0.116863 + 0.993148i $$0.537284\pi$$
$$660$$ 0 0
$$661$$ 10.0000 0.388955 0.194477 0.980907i $$-0.437699\pi$$
0.194477 + 0.980907i $$0.437699\pi$$
$$662$$ 34.0000 1.32145
$$663$$ 0 0
$$664$$ 2.00000 0.0776151
$$665$$ 1.00000 0.0387783
$$666$$ 0 0
$$667$$ 15.0000 0.580802
$$668$$ 15.0000 0.580367
$$669$$ 0 0
$$670$$ 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
$$675$$ 0 0
$$676$$ 1.00000 0.0384615
$$677$$ 18.0000 0.691796 0.345898 0.938272i $$-0.387574\pi$$
0.345898 + 0.938272i $$0.387574\pi$$
$$678$$ 0 0
$$679$$ 2.00000 0.0767530
$$680$$ 5.00000 0.191741
$$681$$ 0 0
$$682$$ −12.0000 −0.459504
$$683$$ −33.0000 −1.26271 −0.631355 0.775494i $$-0.717501\pi$$
−0.631355 + 0.775494i $$0.717501\pi$$
$$684$$ 0 0
$$685$$ −15.0000 −0.573121
$$686$$ −1.00000 −0.0381802
$$687$$ 0 0
$$688$$ −1.00000 −0.0381246
$$689$$ 6.00000 0.228582
$$690$$ 0 0
$$691$$ −20.0000 −0.760836 −0.380418 0.924815i $$-0.624220\pi$$
−0.380418 + 0.924815i $$0.624220\pi$$
$$692$$ −6.00000 −0.228086
$$693$$ 0 0
$$694$$ −32.0000 −1.21470
$$695$$ 16.0000 0.606915
$$696$$ 0 0
$$697$$ −40.0000 −1.51511
$$698$$ 6.00000 0.227103
$$699$$ 0 0
$$700$$ 4.00000 0.151186
$$701$$ 14.0000 0.528773 0.264386 0.964417i $$-0.414831\pi$$
0.264386 + 0.964417i $$0.414831\pi$$
$$702$$ 0 0
$$703$$ −5.00000 −0.188579
$$704$$ −3.00000 −0.113067
$$705$$ 0 0
$$706$$ −2.00000 −0.0752710
$$707$$ −16.0000 −0.601742
$$708$$ 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
$$711$$ 0 0
$$712$$ 2.00000 0.0749532
$$713$$ −12.0000 −0.449404
$$714$$ 0 0
$$715$$ −3.00000 −0.112194
$$716$$ 16.0000 0.597948
$$717$$ 0 0
$$718$$ −24.0000 −0.895672
$$719$$ −30.0000 −1.11881 −0.559406 0.828894i $$-0.688971\pi$$
−0.559406 + 0.828894i $$0.688971\pi$$
$$720$$ 0 0
$$721$$ −1.00000 −0.0372419
$$722$$ −18.0000 −0.669891
$$723$$ 0 0
$$724$$ 6.00000 0.222988
$$725$$ 20.0000 0.742781
$$726$$ 0 0
$$727$$ −27.0000 −1.00137 −0.500687 0.865628i $$-0.666919\pi$$
−0.500687 + 0.865628i $$0.666919\pi$$
$$728$$ 1.00000 0.0370625
$$729$$ 0 0
$$730$$ 15.0000 0.555175
$$731$$ 5.00000 0.184932
$$732$$ 0 0
$$733$$ 20.0000 0.738717 0.369358 0.929287i $$-0.379577\pi$$
0.369358 + 0.929287i $$0.379577\pi$$
$$734$$ 8.00000 0.295285
$$735$$ 0 0
$$736$$ −3.00000 −0.110581
$$737$$ 30.0000 1.10506
$$738$$ 0 0
$$739$$ −26.0000 −0.956425 −0.478213 0.878244i $$-0.658715\pi$$
−0.478213 + 0.878244i $$0.658715\pi$$
$$740$$ 5.00000 0.183804
$$741$$ 0 0
$$742$$ 6.00000 0.220267
$$743$$ 42.0000 1.54083 0.770415 0.637542i $$-0.220049\pi$$
0.770415 + 0.637542i $$0.220049\pi$$
$$744$$ 0 0
$$745$$ −16.0000 −0.586195
$$746$$ −32.0000 −1.17160
$$747$$ 0 0
$$748$$ 15.0000 0.548454
$$749$$ −6.00000 −0.219235
$$750$$ 0 0
$$751$$ 24.0000 0.875772 0.437886 0.899030i $$-0.355727\pi$$
0.437886 + 0.899030i $$0.355727\pi$$
$$752$$ −8.00000 −0.291730
$$753$$ 0 0
$$754$$ 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
$$759$$ 0 0
$$760$$ −1.00000 −0.0362738
$$761$$ 24.0000 0.869999 0.435000 0.900431i $$-0.356748\pi$$
0.435000 + 0.900431i $$0.356748\pi$$
$$762$$ 0 0
$$763$$ 7.00000 0.253417
$$764$$ −11.0000 −0.397966
$$765$$ 0 0
$$766$$ −1.00000 −0.0361315
$$767$$ 0 0
$$768$$ 0 0
$$769$$ 43.0000 1.55062 0.775310 0.631581i $$-0.217594\pi$$
0.775310 + 0.631581i $$0.217594\pi$$
$$770$$ −3.00000 −0.108112
$$771$$ 0 0
$$772$$ −20.0000 −0.719816
$$773$$ −11.0000 −0.395643 −0.197821 0.980238i $$-0.563387\pi$$
−0.197821 + 0.980238i $$0.563387\pi$$
$$774$$ 0 0
$$775$$ −16.0000 −0.574737
$$776$$ −2.00000 −0.0717958
$$777$$ 0 0
$$778$$ 18.0000 0.645331
$$779$$ 8.00000 0.286630
$$780$$ 0 0
$$781$$ 24.0000 0.858788
$$782$$ 15.0000 0.536399
$$783$$ 0 0
$$784$$ 1.00000 0.0357143
$$785$$ −9.00000 −0.321224
$$786$$ 0 0
$$787$$ −47.0000 −1.67537 −0.837685 0.546154i $$-0.816091\pi$$
−0.837685 + 0.546154i $$0.816091\pi$$
$$788$$ 0 0
$$789$$ 0 0
$$790$$ −6.00000 −0.213470
$$791$$ −6.00000 −0.213335
$$792$$ 0 0
$$793$$ −13.0000 −0.461644
$$794$$ 30.0000 1.06466
$$795$$ 0 0
$$796$$ −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
$$801$$ 0 0
$$802$$ 30.0000 1.05934
$$803$$ 45.0000 1.58802
$$804$$ 0 0
$$805$$ −3.00000 −0.105736
$$806$$ −4.00000 −0.140894
$$807$$ 0 0
$$808$$ 16.0000 0.562878
$$809$$ −10.0000 −0.351581 −0.175791 0.984428i $$-0.556248\pi$$
−0.175791 + 0.984428i $$0.556248\pi$$
$$810$$ 0 0
$$811$$ −19.0000 −0.667180 −0.333590 0.942718i $$-0.608260\pi$$
−0.333590 + 0.942718i $$0.608260\pi$$
$$812$$ 5.00000 0.175466
$$813$$ 0 0
$$814$$ 15.0000 0.525750
$$815$$ −10.0000 −0.350285
$$816$$ 0 0
$$817$$ −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 $$-0.637550\pi$$
−0.418803 + 0.908077i $$0.637550\pi$$
$$822$$ 0 0
$$823$$ 24.0000 0.836587 0.418294 0.908312i $$-0.362628\pi$$
0.418294 + 0.908312i $$0.362628\pi$$
$$824$$ 1.00000 0.0348367
$$825$$ 0 0
$$826$$ 0 0
$$827$$ −43.0000 −1.49526 −0.747628 0.664117i $$-0.768807\pi$$
−0.747628 + 0.664117i $$0.768807\pi$$
$$828$$ 0 0
$$829$$ −19.0000 −0.659897 −0.329949 0.943999i $$-0.607031\pi$$
−0.329949 + 0.943999i $$0.607031\pi$$
$$830$$ −2.00000 −0.0694210
$$831$$ 0 0
$$832$$ −1.00000 −0.0346688
$$833$$ −5.00000 −0.173240
$$834$$ 0 0
$$835$$ −15.0000 −0.519096
$$836$$ −3.00000 −0.103757
$$837$$ 0 0
$$838$$ 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
$$843$$ 0 0
$$844$$ 5.00000 0.172107
$$845$$ −1.00000 −0.0344010
$$846$$ 0 0
$$847$$ 2.00000 0.0687208
$$848$$ −6.00000 −0.206041
$$849$$ 0 0
$$850$$ 20.0000 0.685994
$$851$$ 15.0000 0.514193
$$852$$ 0 0
$$853$$ −28.0000 −0.958702 −0.479351 0.877623i $$-0.659128\pi$$
−0.479351 + 0.877623i $$0.659128\pi$$
$$854$$ −13.0000 −0.444851
$$855$$ 0 0
$$856$$ 6.00000 0.205076
$$857$$ −42.0000 −1.43469 −0.717346 0.696717i $$-0.754643\pi$$
−0.717346 + 0.696717i $$0.754643\pi$$
$$858$$ 0 0
$$859$$ −28.0000 −0.955348 −0.477674 0.878537i $$-0.658520\pi$$
−0.477674 + 0.878537i $$0.658520\pi$$
$$860$$ 1.00000 0.0340997
$$861$$ 0 0
$$862$$ −10.0000 −0.340601
$$863$$ 2.00000 0.0680808 0.0340404 0.999420i $$-0.489163\pi$$
0.0340404 + 0.999420i $$0.489163\pi$$
$$864$$ 0 0
$$865$$ 6.00000 0.204006
$$866$$ 20.0000 0.679628
$$867$$ 0 0
$$868$$ −4.00000 −0.135769
$$869$$ −18.0000 −0.610608
$$870$$ 0 0
$$871$$ 10.0000 0.338837
$$872$$ −7.00000 −0.237050
$$873$$ 0 0
$$874$$ −3.00000 −0.101477
$$875$$ −9.00000 −0.304256
$$876$$ 0 0
$$877$$ 34.0000 1.14810 0.574049 0.818821i $$-0.305372\pi$$
0.574049 + 0.818821i $$0.305372\pi$$
$$878$$ −11.0000 −0.371232
$$879$$ 0 0
$$880$$ 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 $$0.391004\pi$$
$$888$$ 0 0
$$889$$ 18.0000 0.603701
$$890$$ −2.00000 −0.0670402
$$891$$ 0 0
$$892$$ 26.0000 0.870544
$$893$$ −8.00000 −0.267710
$$894$$ 0 0
$$895$$ −16.0000 −0.534821
$$896$$ −1.00000 −0.0334077
$$897$$ 0 0
$$898$$ −13.0000 −0.433816
$$899$$ −20.0000 −0.667037
$$900$$ 0 0
$$901$$ 30.0000 0.999445
$$902$$ −24.0000 −0.799113
$$903$$ 0 0
$$904$$ 6.00000 0.199557
$$905$$ −6.00000 −0.199447
$$906$$ 0 0
$$907$$ −20.0000 −0.664089 −0.332045 0.943264i $$-0.607738\pi$$
−0.332045 + 0.943264i $$0.607738\pi$$
$$908$$ −2.00000 −0.0663723
$$909$$ 0 0
$$910$$ −1.00000 −0.0331497
$$911$$ −29.0000 −0.960813 −0.480406 0.877046i $$-0.659511\pi$$
−0.480406 + 0.877046i $$0.659511\pi$$
$$912$$ 0 0
$$913$$ −6.00000 −0.198571
$$914$$ 8.00000 0.264616
$$915$$ 0 0
$$916$$ −6.00000 −0.198246
$$917$$ 17.0000 0.561389
$$918$$ 0 0
$$919$$ −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
$$927$$ 0 0
$$928$$ −5.00000 −0.164133
$$929$$ 4.00000 0.131236 0.0656179 0.997845i $$-0.479098\pi$$
0.0656179 + 0.997845i $$0.479098\pi$$
$$930$$ 0 0
$$931$$ 1.00000 0.0327737
$$932$$ 4.00000 0.131024
$$933$$ 0 0
$$934$$ −21.0000 −0.687141
$$935$$ −15.0000 −0.490552
$$936$$ 0 0
$$937$$ −14.0000 −0.457360 −0.228680 0.973502i $$-0.573441\pi$$
−0.228680 + 0.973502i $$0.573441\pi$$
$$938$$ 10.0000 0.326512
$$939$$ 0 0
$$940$$ 8.00000 0.260931
$$941$$ 18.0000 0.586783 0.293392 0.955992i $$-0.405216\pi$$
0.293392 + 0.955992i $$0.405216\pi$$
$$942$$ 0 0
$$943$$ −24.0000 −0.781548
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 3.00000 0.0975384
$$947$$ −47.0000 −1.52729 −0.763647 0.645634i $$-0.776593\pi$$
−0.763647 + 0.645634i $$0.776593\pi$$
$$948$$ 0 0
$$949$$ 15.0000 0.486921
$$950$$ −4.00000 −0.129777
$$951$$ 0 0
$$952$$ 5.00000 0.162051
$$953$$ 44.0000 1.42530 0.712650 0.701520i $$-0.247495\pi$$
0.712650 + 0.701520i $$0.247495\pi$$
$$954$$ 0 0
$$955$$ 11.0000 0.355952
$$956$$ 12.0000 0.388108
$$957$$ 0 0
$$958$$ −27.0000 −0.872330
$$959$$ −15.0000 −0.484375
$$960$$ 0 0
$$961$$ −15.0000 −0.483871
$$962$$ 5.00000 0.161206
$$963$$ 0 0
$$964$$ 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
$$969$$ 0 0
$$970$$ 2.00000 0.0642161
$$971$$ 32.0000 1.02693 0.513464 0.858111i $$-0.328362\pi$$
0.513464 + 0.858111i $$0.328362\pi$$
$$972$$ 0 0
$$973$$ 16.0000 0.512936
$$974$$ 8.00000 0.256337
$$975$$ 0 0
$$976$$ 13.0000 0.416120
$$977$$ 25.0000 0.799821 0.399910 0.916554i $$-0.369041\pi$$
0.399910 + 0.916554i $$0.369041\pi$$
$$978$$ 0 0
$$979$$ −6.00000 −0.191761
$$980$$ −1.00000 −0.0319438
$$981$$ 0 0
$$982$$ 38.0000 1.21263
$$983$$ 5.00000 0.159475 0.0797376 0.996816i $$-0.474592\pi$$
0.0797376 + 0.996816i $$0.474592\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 25.0000 0.796162
$$987$$ 0 0
$$988$$ −1.00000 −0.0318142
$$989$$ 3.00000 0.0953945
$$990$$ 0 0
$$991$$ −30.0000 −0.952981 −0.476491 0.879180i $$-0.658091\pi$$
−0.476491 + 0.879180i $$0.658091\pi$$
$$992$$ 4.00000 0.127000
$$993$$ 0 0
$$994$$ 8.00000 0.253745
$$995$$ 5.00000 0.158511
$$996$$ 0 0
$$997$$ −50.0000 −1.58352 −0.791758 0.610835i $$-0.790834\pi$$
−0.791758 + 0.610835i $$0.790834\pi$$
$$998$$ 8.00000 0.253236
$$999$$ 0 0
Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000

## Twists

By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 1638.2.a.o.1.1 1
3.2 odd 2 546.2.a.c.1.1 1
12.11 even 2 4368.2.a.h.1.1 1
21.20 even 2 3822.2.a.e.1.1 1
39.38 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 3.2 odd 2
1638.2.a.o.1.1 1 1.1 even 1 trivial
3822.2.a.e.1.1 1 21.20 even 2
4368.2.a.h.1.1 1 12.11 even 2
7098.2.a.z.1.1 1 39.38 odd 2