# Properties

 Label 3120.2.a.n.1.1 Level $3120$ Weight $2$ Character 3120.1 Self dual yes Analytic conductor $24.913$ Analytic rank $0$ 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] = [3120,2,Mod(1,3120)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3120, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 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("3120.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$3120 = 2^{4} \cdot 3 \cdot 5 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3120.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: $$24.9133254306$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 195) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 3120.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-1.00000 q^{3} +1.00000 q^{5} +3.00000 q^{7} +1.00000 q^{9} +O(q^{10})$$ $$q-1.00000 q^{3} +1.00000 q^{5} +3.00000 q^{7} +1.00000 q^{9} +5.00000 q^{11} +1.00000 q^{13} -1.00000 q^{15} +5.00000 q^{17} -2.00000 q^{19} -3.00000 q^{21} +1.00000 q^{23} +1.00000 q^{25} -1.00000 q^{27} +10.0000 q^{29} +2.00000 q^{31} -5.00000 q^{33} +3.00000 q^{35} -3.00000 q^{37} -1.00000 q^{39} -9.00000 q^{41} +4.00000 q^{43} +1.00000 q^{45} -10.0000 q^{47} +2.00000 q^{49} -5.00000 q^{51} +9.00000 q^{53} +5.00000 q^{55} +2.00000 q^{57} -11.0000 q^{61} +3.00000 q^{63} +1.00000 q^{65} +4.00000 q^{67} -1.00000 q^{69} -15.0000 q^{71} +6.00000 q^{73} -1.00000 q^{75} +15.0000 q^{77} +11.0000 q^{79} +1.00000 q^{81} -8.00000 q^{83} +5.00000 q^{85} -10.0000 q^{87} -11.0000 q^{89} +3.00000 q^{91} -2.00000 q^{93} -2.00000 q^{95} -9.00000 q^{97} +5.00000 q^{99} +O(q^{100})$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0 0
$$3$$ −1.00000 −0.577350
$$4$$ 0 0
$$5$$ 1.00000 0.447214
$$6$$ 0 0
$$7$$ 3.00000 1.13389 0.566947 0.823754i $$-0.308125\pi$$
0.566947 + 0.823754i $$0.308125\pi$$
$$8$$ 0 0
$$9$$ 1.00000 0.333333
$$10$$ 0 0
$$11$$ 5.00000 1.50756 0.753778 0.657129i $$-0.228229\pi$$
0.753778 + 0.657129i $$0.228229\pi$$
$$12$$ 0 0
$$13$$ 1.00000 0.277350
$$14$$ 0 0
$$15$$ −1.00000 −0.258199
$$16$$ 0 0
$$17$$ 5.00000 1.21268 0.606339 0.795206i $$-0.292637\pi$$
0.606339 + 0.795206i $$0.292637\pi$$
$$18$$ 0 0
$$19$$ −2.00000 −0.458831 −0.229416 0.973329i $$-0.573682\pi$$
−0.229416 + 0.973329i $$0.573682\pi$$
$$20$$ 0 0
$$21$$ −3.00000 −0.654654
$$22$$ 0 0
$$23$$ 1.00000 0.208514 0.104257 0.994550i $$-0.466753\pi$$
0.104257 + 0.994550i $$0.466753\pi$$
$$24$$ 0 0
$$25$$ 1.00000 0.200000
$$26$$ 0 0
$$27$$ −1.00000 −0.192450
$$28$$ 0 0
$$29$$ 10.0000 1.85695 0.928477 0.371391i $$-0.121119\pi$$
0.928477 + 0.371391i $$0.121119\pi$$
$$30$$ 0 0
$$31$$ 2.00000 0.359211 0.179605 0.983739i $$-0.442518\pi$$
0.179605 + 0.983739i $$0.442518\pi$$
$$32$$ 0 0
$$33$$ −5.00000 −0.870388
$$34$$ 0 0
$$35$$ 3.00000 0.507093
$$36$$ 0 0
$$37$$ −3.00000 −0.493197 −0.246598 0.969118i $$-0.579313\pi$$
−0.246598 + 0.969118i $$0.579313\pi$$
$$38$$ 0 0
$$39$$ −1.00000 −0.160128
$$40$$ 0 0
$$41$$ −9.00000 −1.40556 −0.702782 0.711405i $$-0.748059\pi$$
−0.702782 + 0.711405i $$0.748059\pi$$
$$42$$ 0 0
$$43$$ 4.00000 0.609994 0.304997 0.952353i $$-0.401344\pi$$
0.304997 + 0.952353i $$0.401344\pi$$
$$44$$ 0 0
$$45$$ 1.00000 0.149071
$$46$$ 0 0
$$47$$ −10.0000 −1.45865 −0.729325 0.684167i $$-0.760166\pi$$
−0.729325 + 0.684167i $$0.760166\pi$$
$$48$$ 0 0
$$49$$ 2.00000 0.285714
$$50$$ 0 0
$$51$$ −5.00000 −0.700140
$$52$$ 0 0
$$53$$ 9.00000 1.23625 0.618123 0.786082i $$-0.287894\pi$$
0.618123 + 0.786082i $$0.287894\pi$$
$$54$$ 0 0
$$55$$ 5.00000 0.674200
$$56$$ 0 0
$$57$$ 2.00000 0.264906
$$58$$ 0 0
$$59$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$60$$ 0 0
$$61$$ −11.0000 −1.40841 −0.704203 0.709999i $$-0.748695\pi$$
−0.704203 + 0.709999i $$0.748695\pi$$
$$62$$ 0 0
$$63$$ 3.00000 0.377964
$$64$$ 0 0
$$65$$ 1.00000 0.124035
$$66$$ 0 0
$$67$$ 4.00000 0.488678 0.244339 0.969690i $$-0.421429\pi$$
0.244339 + 0.969690i $$0.421429\pi$$
$$68$$ 0 0
$$69$$ −1.00000 −0.120386
$$70$$ 0 0
$$71$$ −15.0000 −1.78017 −0.890086 0.455792i $$-0.849356\pi$$
−0.890086 + 0.455792i $$0.849356\pi$$
$$72$$ 0 0
$$73$$ 6.00000 0.702247 0.351123 0.936329i $$-0.385800\pi$$
0.351123 + 0.936329i $$0.385800\pi$$
$$74$$ 0 0
$$75$$ −1.00000 −0.115470
$$76$$ 0 0
$$77$$ 15.0000 1.70941
$$78$$ 0 0
$$79$$ 11.0000 1.23760 0.618798 0.785550i $$-0.287620\pi$$
0.618798 + 0.785550i $$0.287620\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 0 0
$$83$$ −8.00000 −0.878114 −0.439057 0.898459i $$-0.644687\pi$$
−0.439057 + 0.898459i $$0.644687\pi$$
$$84$$ 0 0
$$85$$ 5.00000 0.542326
$$86$$ 0 0
$$87$$ −10.0000 −1.07211
$$88$$ 0 0
$$89$$ −11.0000 −1.16600 −0.582999 0.812473i $$-0.698121\pi$$
−0.582999 + 0.812473i $$0.698121\pi$$
$$90$$ 0 0
$$91$$ 3.00000 0.314485
$$92$$ 0 0
$$93$$ −2.00000 −0.207390
$$94$$ 0 0
$$95$$ −2.00000 −0.205196
$$96$$ 0 0
$$97$$ −9.00000 −0.913812 −0.456906 0.889515i $$-0.651042\pi$$
−0.456906 + 0.889515i $$0.651042\pi$$
$$98$$ 0 0
$$99$$ 5.00000 0.502519
$$100$$ 0 0
$$101$$ −12.0000 −1.19404 −0.597022 0.802225i $$-0.703650\pi$$
−0.597022 + 0.802225i $$0.703650\pi$$
$$102$$ 0 0
$$103$$ −4.00000 −0.394132 −0.197066 0.980390i $$-0.563141\pi$$
−0.197066 + 0.980390i $$0.563141\pi$$
$$104$$ 0 0
$$105$$ −3.00000 −0.292770
$$106$$ 0 0
$$107$$ −3.00000 −0.290021 −0.145010 0.989430i $$-0.546322\pi$$
−0.145010 + 0.989430i $$0.546322\pi$$
$$108$$ 0 0
$$109$$ 16.0000 1.53252 0.766261 0.642529i $$-0.222115\pi$$
0.766261 + 0.642529i $$0.222115\pi$$
$$110$$ 0 0
$$111$$ 3.00000 0.284747
$$112$$ 0 0
$$113$$ 2.00000 0.188144 0.0940721 0.995565i $$-0.470012\pi$$
0.0940721 + 0.995565i $$0.470012\pi$$
$$114$$ 0 0
$$115$$ 1.00000 0.0932505
$$116$$ 0 0
$$117$$ 1.00000 0.0924500
$$118$$ 0 0
$$119$$ 15.0000 1.37505
$$120$$ 0 0
$$121$$ 14.0000 1.27273
$$122$$ 0 0
$$123$$ 9.00000 0.811503
$$124$$ 0 0
$$125$$ 1.00000 0.0894427
$$126$$ 0 0
$$127$$ −14.0000 −1.24230 −0.621150 0.783692i $$-0.713334\pi$$
−0.621150 + 0.783692i $$0.713334\pi$$
$$128$$ 0 0
$$129$$ −4.00000 −0.352180
$$130$$ 0 0
$$131$$ −6.00000 −0.524222 −0.262111 0.965038i $$-0.584419\pi$$
−0.262111 + 0.965038i $$0.584419\pi$$
$$132$$ 0 0
$$133$$ −6.00000 −0.520266
$$134$$ 0 0
$$135$$ −1.00000 −0.0860663
$$136$$ 0 0
$$137$$ 6.00000 0.512615 0.256307 0.966595i $$-0.417494\pi$$
0.256307 + 0.966595i $$0.417494\pi$$
$$138$$ 0 0
$$139$$ 17.0000 1.44192 0.720961 0.692976i $$-0.243701\pi$$
0.720961 + 0.692976i $$0.243701\pi$$
$$140$$ 0 0
$$141$$ 10.0000 0.842152
$$142$$ 0 0
$$143$$ 5.00000 0.418121
$$144$$ 0 0
$$145$$ 10.0000 0.830455
$$146$$ 0 0
$$147$$ −2.00000 −0.164957
$$148$$ 0 0
$$149$$ −7.00000 −0.573462 −0.286731 0.958011i $$-0.592569\pi$$
−0.286731 + 0.958011i $$0.592569\pi$$
$$150$$ 0 0
$$151$$ 12.0000 0.976546 0.488273 0.872691i $$-0.337627\pi$$
0.488273 + 0.872691i $$0.337627\pi$$
$$152$$ 0 0
$$153$$ 5.00000 0.404226
$$154$$ 0 0
$$155$$ 2.00000 0.160644
$$156$$ 0 0
$$157$$ 22.0000 1.75579 0.877896 0.478852i $$-0.158947\pi$$
0.877896 + 0.478852i $$0.158947\pi$$
$$158$$ 0 0
$$159$$ −9.00000 −0.713746
$$160$$ 0 0
$$161$$ 3.00000 0.236433
$$162$$ 0 0
$$163$$ 11.0000 0.861586 0.430793 0.902451i $$-0.358234\pi$$
0.430793 + 0.902451i $$0.358234\pi$$
$$164$$ 0 0
$$165$$ −5.00000 −0.389249
$$166$$ 0 0
$$167$$ 8.00000 0.619059 0.309529 0.950890i $$-0.399829\pi$$
0.309529 + 0.950890i $$0.399829\pi$$
$$168$$ 0 0
$$169$$ 1.00000 0.0769231
$$170$$ 0 0
$$171$$ −2.00000 −0.152944
$$172$$ 0 0
$$173$$ −2.00000 −0.152057 −0.0760286 0.997106i $$-0.524224\pi$$
−0.0760286 + 0.997106i $$0.524224\pi$$
$$174$$ 0 0
$$175$$ 3.00000 0.226779
$$176$$ 0 0
$$177$$ 0 0
$$178$$ 0 0
$$179$$ 6.00000 0.448461 0.224231 0.974536i $$-0.428013\pi$$
0.224231 + 0.974536i $$0.428013\pi$$
$$180$$ 0 0
$$181$$ −23.0000 −1.70958 −0.854788 0.518977i $$-0.826313\pi$$
−0.854788 + 0.518977i $$0.826313\pi$$
$$182$$ 0 0
$$183$$ 11.0000 0.813143
$$184$$ 0 0
$$185$$ −3.00000 −0.220564
$$186$$ 0 0
$$187$$ 25.0000 1.82818
$$188$$ 0 0
$$189$$ −3.00000 −0.218218
$$190$$ 0 0
$$191$$ 20.0000 1.44715 0.723575 0.690246i $$-0.242498\pi$$
0.723575 + 0.690246i $$0.242498\pi$$
$$192$$ 0 0
$$193$$ 13.0000 0.935760 0.467880 0.883792i $$-0.345018\pi$$
0.467880 + 0.883792i $$0.345018\pi$$
$$194$$ 0 0
$$195$$ −1.00000 −0.0716115
$$196$$ 0 0
$$197$$ −12.0000 −0.854965 −0.427482 0.904024i $$-0.640599\pi$$
−0.427482 + 0.904024i $$0.640599\pi$$
$$198$$ 0 0
$$199$$ −4.00000 −0.283552 −0.141776 0.989899i $$-0.545281\pi$$
−0.141776 + 0.989899i $$0.545281\pi$$
$$200$$ 0 0
$$201$$ −4.00000 −0.282138
$$202$$ 0 0
$$203$$ 30.0000 2.10559
$$204$$ 0 0
$$205$$ −9.00000 −0.628587
$$206$$ 0 0
$$207$$ 1.00000 0.0695048
$$208$$ 0 0
$$209$$ −10.0000 −0.691714
$$210$$ 0 0
$$211$$ 4.00000 0.275371 0.137686 0.990476i $$-0.456034\pi$$
0.137686 + 0.990476i $$0.456034\pi$$
$$212$$ 0 0
$$213$$ 15.0000 1.02778
$$214$$ 0 0
$$215$$ 4.00000 0.272798
$$216$$ 0 0
$$217$$ 6.00000 0.407307
$$218$$ 0 0
$$219$$ −6.00000 −0.405442
$$220$$ 0 0
$$221$$ 5.00000 0.336336
$$222$$ 0 0
$$223$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$224$$ 0 0
$$225$$ 1.00000 0.0666667
$$226$$ 0 0
$$227$$ 18.0000 1.19470 0.597351 0.801980i $$-0.296220\pi$$
0.597351 + 0.801980i $$0.296220\pi$$
$$228$$ 0 0
$$229$$ −14.0000 −0.925146 −0.462573 0.886581i $$-0.653074\pi$$
−0.462573 + 0.886581i $$0.653074\pi$$
$$230$$ 0 0
$$231$$ −15.0000 −0.986928
$$232$$ 0 0
$$233$$ −25.0000 −1.63780 −0.818902 0.573933i $$-0.805417\pi$$
−0.818902 + 0.573933i $$0.805417\pi$$
$$234$$ 0 0
$$235$$ −10.0000 −0.652328
$$236$$ 0 0
$$237$$ −11.0000 −0.714527
$$238$$ 0 0
$$239$$ −15.0000 −0.970269 −0.485135 0.874439i $$-0.661229\pi$$
−0.485135 + 0.874439i $$0.661229\pi$$
$$240$$ 0 0
$$241$$ −14.0000 −0.901819 −0.450910 0.892570i $$-0.648900\pi$$
−0.450910 + 0.892570i $$0.648900\pi$$
$$242$$ 0 0
$$243$$ −1.00000 −0.0641500
$$244$$ 0 0
$$245$$ 2.00000 0.127775
$$246$$ 0 0
$$247$$ −2.00000 −0.127257
$$248$$ 0 0
$$249$$ 8.00000 0.506979
$$250$$ 0 0
$$251$$ −20.0000 −1.26239 −0.631194 0.775625i $$-0.717435\pi$$
−0.631194 + 0.775625i $$0.717435\pi$$
$$252$$ 0 0
$$253$$ 5.00000 0.314347
$$254$$ 0 0
$$255$$ −5.00000 −0.313112
$$256$$ 0 0
$$257$$ 18.0000 1.12281 0.561405 0.827541i $$-0.310261\pi$$
0.561405 + 0.827541i $$0.310261\pi$$
$$258$$ 0 0
$$259$$ −9.00000 −0.559233
$$260$$ 0 0
$$261$$ 10.0000 0.618984
$$262$$ 0 0
$$263$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$264$$ 0 0
$$265$$ 9.00000 0.552866
$$266$$ 0 0
$$267$$ 11.0000 0.673189
$$268$$ 0 0
$$269$$ 32.0000 1.95107 0.975537 0.219834i $$-0.0705517\pi$$
0.975537 + 0.219834i $$0.0705517\pi$$
$$270$$ 0 0
$$271$$ 2.00000 0.121491 0.0607457 0.998153i $$-0.480652\pi$$
0.0607457 + 0.998153i $$0.480652\pi$$
$$272$$ 0 0
$$273$$ −3.00000 −0.181568
$$274$$ 0 0
$$275$$ 5.00000 0.301511
$$276$$ 0 0
$$277$$ 26.0000 1.56219 0.781094 0.624413i $$-0.214662\pi$$
0.781094 + 0.624413i $$0.214662\pi$$
$$278$$ 0 0
$$279$$ 2.00000 0.119737
$$280$$ 0 0
$$281$$ −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 $$-0.423582\pi$$
0.237775 + 0.971320i $$0.423582\pi$$
$$284$$ 0 0
$$285$$ 2.00000 0.118470
$$286$$ 0 0
$$287$$ −27.0000 −1.59376
$$288$$ 0 0
$$289$$ 8.00000 0.470588
$$290$$ 0 0
$$291$$ 9.00000 0.527589
$$292$$ 0 0
$$293$$ 24.0000 1.40209 0.701047 0.713115i $$-0.252716\pi$$
0.701047 + 0.713115i $$0.252716\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 0 0
$$297$$ −5.00000 −0.290129
$$298$$ 0 0
$$299$$ 1.00000 0.0578315
$$300$$ 0 0
$$301$$ 12.0000 0.691669
$$302$$ 0 0
$$303$$ 12.0000 0.689382
$$304$$ 0 0
$$305$$ −11.0000 −0.629858
$$306$$ 0 0
$$307$$ 19.0000 1.08439 0.542194 0.840254i $$-0.317594\pi$$
0.542194 + 0.840254i $$0.317594\pi$$
$$308$$ 0 0
$$309$$ 4.00000 0.227552
$$310$$ 0 0
$$311$$ −24.0000 −1.36092 −0.680458 0.732787i $$-0.738219\pi$$
−0.680458 + 0.732787i $$0.738219\pi$$
$$312$$ 0 0
$$313$$ −10.0000 −0.565233 −0.282617 0.959233i $$-0.591202\pi$$
−0.282617 + 0.959233i $$0.591202\pi$$
$$314$$ 0 0
$$315$$ 3.00000 0.169031
$$316$$ 0 0
$$317$$ 12.0000 0.673987 0.336994 0.941507i $$-0.390590\pi$$
0.336994 + 0.941507i $$0.390590\pi$$
$$318$$ 0 0
$$319$$ 50.0000 2.79946
$$320$$ 0 0
$$321$$ 3.00000 0.167444
$$322$$ 0 0
$$323$$ −10.0000 −0.556415
$$324$$ 0 0
$$325$$ 1.00000 0.0554700
$$326$$ 0 0
$$327$$ −16.0000 −0.884802
$$328$$ 0 0
$$329$$ −30.0000 −1.65395
$$330$$ 0 0
$$331$$ −32.0000 −1.75888 −0.879440 0.476011i $$-0.842082\pi$$
−0.879440 + 0.476011i $$0.842082\pi$$
$$332$$ 0 0
$$333$$ −3.00000 −0.164399
$$334$$ 0 0
$$335$$ 4.00000 0.218543
$$336$$ 0 0
$$337$$ 4.00000 0.217894 0.108947 0.994048i $$-0.465252\pi$$
0.108947 + 0.994048i $$0.465252\pi$$
$$338$$ 0 0
$$339$$ −2.00000 −0.108625
$$340$$ 0 0
$$341$$ 10.0000 0.541530
$$342$$ 0 0
$$343$$ −15.0000 −0.809924
$$344$$ 0 0
$$345$$ −1.00000 −0.0538382
$$346$$ 0 0
$$347$$ 1.00000 0.0536828 0.0268414 0.999640i $$-0.491455\pi$$
0.0268414 + 0.999640i $$0.491455\pi$$
$$348$$ 0 0
$$349$$ 20.0000 1.07058 0.535288 0.844670i $$-0.320203\pi$$
0.535288 + 0.844670i $$0.320203\pi$$
$$350$$ 0 0
$$351$$ −1.00000 −0.0533761
$$352$$ 0 0
$$353$$ −14.0000 −0.745145 −0.372572 0.928003i $$-0.621524\pi$$
−0.372572 + 0.928003i $$0.621524\pi$$
$$354$$ 0 0
$$355$$ −15.0000 −0.796117
$$356$$ 0 0
$$357$$ −15.0000 −0.793884
$$358$$ 0 0
$$359$$ −16.0000 −0.844448 −0.422224 0.906492i $$-0.638750\pi$$
−0.422224 + 0.906492i $$0.638750\pi$$
$$360$$ 0 0
$$361$$ −15.0000 −0.789474
$$362$$ 0 0
$$363$$ −14.0000 −0.734809
$$364$$ 0 0
$$365$$ 6.00000 0.314054
$$366$$ 0 0
$$367$$ −8.00000 −0.417597 −0.208798 0.977959i $$-0.566955\pi$$
−0.208798 + 0.977959i $$0.566955\pi$$
$$368$$ 0 0
$$369$$ −9.00000 −0.468521
$$370$$ 0 0
$$371$$ 27.0000 1.40177
$$372$$ 0 0
$$373$$ 16.0000 0.828449 0.414224 0.910175i $$-0.364053\pi$$
0.414224 + 0.910175i $$0.364053\pi$$
$$374$$ 0 0
$$375$$ −1.00000 −0.0516398
$$376$$ 0 0
$$377$$ 10.0000 0.515026
$$378$$ 0 0
$$379$$ −6.00000 −0.308199 −0.154100 0.988055i $$-0.549248\pi$$
−0.154100 + 0.988055i $$0.549248\pi$$
$$380$$ 0 0
$$381$$ 14.0000 0.717242
$$382$$ 0 0
$$383$$ 18.0000 0.919757 0.459879 0.887982i $$-0.347893\pi$$
0.459879 + 0.887982i $$0.347893\pi$$
$$384$$ 0 0
$$385$$ 15.0000 0.764471
$$386$$ 0 0
$$387$$ 4.00000 0.203331
$$388$$ 0 0
$$389$$ −24.0000 −1.21685 −0.608424 0.793612i $$-0.708198\pi$$
−0.608424 + 0.793612i $$0.708198\pi$$
$$390$$ 0 0
$$391$$ 5.00000 0.252861
$$392$$ 0 0
$$393$$ 6.00000 0.302660
$$394$$ 0 0
$$395$$ 11.0000 0.553470
$$396$$ 0 0
$$397$$ −19.0000 −0.953583 −0.476791 0.879017i $$-0.658200\pi$$
−0.476791 + 0.879017i $$0.658200\pi$$
$$398$$ 0 0
$$399$$ 6.00000 0.300376
$$400$$ 0 0
$$401$$ −18.0000 −0.898877 −0.449439 0.893311i $$-0.648376\pi$$
−0.449439 + 0.893311i $$0.648376\pi$$
$$402$$ 0 0
$$403$$ 2.00000 0.0996271
$$404$$ 0 0
$$405$$ 1.00000 0.0496904
$$406$$ 0 0
$$407$$ −15.0000 −0.743522
$$408$$ 0 0
$$409$$ −26.0000 −1.28562 −0.642809 0.766027i $$-0.722231\pi$$
−0.642809 + 0.766027i $$0.722231\pi$$
$$410$$ 0 0
$$411$$ −6.00000 −0.295958
$$412$$ 0 0
$$413$$ 0 0
$$414$$ 0 0
$$415$$ −8.00000 −0.392705
$$416$$ 0 0
$$417$$ −17.0000 −0.832494
$$418$$ 0 0
$$419$$ −26.0000 −1.27018 −0.635092 0.772437i $$-0.719038\pi$$
−0.635092 + 0.772437i $$0.719038\pi$$
$$420$$ 0 0
$$421$$ 32.0000 1.55958 0.779792 0.626038i $$-0.215325\pi$$
0.779792 + 0.626038i $$0.215325\pi$$
$$422$$ 0 0
$$423$$ −10.0000 −0.486217
$$424$$ 0 0
$$425$$ 5.00000 0.242536
$$426$$ 0 0
$$427$$ −33.0000 −1.59698
$$428$$ 0 0
$$429$$ −5.00000 −0.241402
$$430$$ 0 0
$$431$$ 16.0000 0.770693 0.385346 0.922772i $$-0.374082\pi$$
0.385346 + 0.922772i $$0.374082\pi$$
$$432$$ 0 0
$$433$$ 24.0000 1.15337 0.576683 0.816968i $$-0.304347\pi$$
0.576683 + 0.816968i $$0.304347\pi$$
$$434$$ 0 0
$$435$$ −10.0000 −0.479463
$$436$$ 0 0
$$437$$ −2.00000 −0.0956730
$$438$$ 0 0
$$439$$ 33.0000 1.57500 0.787502 0.616312i $$-0.211374\pi$$
0.787502 + 0.616312i $$0.211374\pi$$
$$440$$ 0 0
$$441$$ 2.00000 0.0952381
$$442$$ 0 0
$$443$$ −35.0000 −1.66290 −0.831450 0.555599i $$-0.812489\pi$$
−0.831450 + 0.555599i $$0.812489\pi$$
$$444$$ 0 0
$$445$$ −11.0000 −0.521450
$$446$$ 0 0
$$447$$ 7.00000 0.331089
$$448$$ 0 0
$$449$$ 15.0000 0.707894 0.353947 0.935266i $$-0.384839\pi$$
0.353947 + 0.935266i $$0.384839\pi$$
$$450$$ 0 0
$$451$$ −45.0000 −2.11897
$$452$$ 0 0
$$453$$ −12.0000 −0.563809
$$454$$ 0 0
$$455$$ 3.00000 0.140642
$$456$$ 0 0
$$457$$ −13.0000 −0.608114 −0.304057 0.952654i $$-0.598341\pi$$
−0.304057 + 0.952654i $$0.598341\pi$$
$$458$$ 0 0
$$459$$ −5.00000 −0.233380
$$460$$ 0 0
$$461$$ 3.00000 0.139724 0.0698620 0.997557i $$-0.477744\pi$$
0.0698620 + 0.997557i $$0.477744\pi$$
$$462$$ 0 0
$$463$$ −5.00000 −0.232370 −0.116185 0.993228i $$-0.537067\pi$$
−0.116185 + 0.993228i $$0.537067\pi$$
$$464$$ 0 0
$$465$$ −2.00000 −0.0927478
$$466$$ 0 0
$$467$$ 29.0000 1.34196 0.670980 0.741475i $$-0.265874\pi$$
0.670980 + 0.741475i $$0.265874\pi$$
$$468$$ 0 0
$$469$$ 12.0000 0.554109
$$470$$ 0 0
$$471$$ −22.0000 −1.01371
$$472$$ 0 0
$$473$$ 20.0000 0.919601
$$474$$ 0 0
$$475$$ −2.00000 −0.0917663
$$476$$ 0 0
$$477$$ 9.00000 0.412082
$$478$$ 0 0
$$479$$ −5.00000 −0.228456 −0.114228 0.993455i $$-0.536439\pi$$
−0.114228 + 0.993455i $$0.536439\pi$$
$$480$$ 0 0
$$481$$ −3.00000 −0.136788
$$482$$ 0 0
$$483$$ −3.00000 −0.136505
$$484$$ 0 0
$$485$$ −9.00000 −0.408669
$$486$$ 0 0
$$487$$ −7.00000 −0.317200 −0.158600 0.987343i $$-0.550698\pi$$
−0.158600 + 0.987343i $$0.550698\pi$$
$$488$$ 0 0
$$489$$ −11.0000 −0.497437
$$490$$ 0 0
$$491$$ −16.0000 −0.722070 −0.361035 0.932552i $$-0.617576\pi$$
−0.361035 + 0.932552i $$0.617576\pi$$
$$492$$ 0 0
$$493$$ 50.0000 2.25189
$$494$$ 0 0
$$495$$ 5.00000 0.224733
$$496$$ 0 0
$$497$$ −45.0000 −2.01853
$$498$$ 0 0
$$499$$ −34.0000 −1.52205 −0.761025 0.648723i $$-0.775303\pi$$
−0.761025 + 0.648723i $$0.775303\pi$$
$$500$$ 0 0
$$501$$ −8.00000 −0.357414
$$502$$ 0 0
$$503$$ −12.0000 −0.535054 −0.267527 0.963550i $$-0.586206\pi$$
−0.267527 + 0.963550i $$0.586206\pi$$
$$504$$ 0 0
$$505$$ −12.0000 −0.533993
$$506$$ 0 0
$$507$$ −1.00000 −0.0444116
$$508$$ 0 0
$$509$$ 21.0000 0.930809 0.465404 0.885098i $$-0.345909\pi$$
0.465404 + 0.885098i $$0.345909\pi$$
$$510$$ 0 0
$$511$$ 18.0000 0.796273
$$512$$ 0 0
$$513$$ 2.00000 0.0883022
$$514$$ 0 0
$$515$$ −4.00000 −0.176261
$$516$$ 0 0
$$517$$ −50.0000 −2.19900
$$518$$ 0 0
$$519$$ 2.00000 0.0877903
$$520$$ 0 0
$$521$$ −22.0000 −0.963837 −0.481919 0.876216i $$-0.660060\pi$$
−0.481919 + 0.876216i $$0.660060\pi$$
$$522$$ 0 0
$$523$$ 20.0000 0.874539 0.437269 0.899331i $$-0.355946\pi$$
0.437269 + 0.899331i $$0.355946\pi$$
$$524$$ 0 0
$$525$$ −3.00000 −0.130931
$$526$$ 0 0
$$527$$ 10.0000 0.435607
$$528$$ 0 0
$$529$$ −22.0000 −0.956522
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 0 0
$$533$$ −9.00000 −0.389833
$$534$$ 0 0
$$535$$ −3.00000 −0.129701
$$536$$ 0 0
$$537$$ −6.00000 −0.258919
$$538$$ 0 0
$$539$$ 10.0000 0.430730
$$540$$ 0 0
$$541$$ 38.0000 1.63375 0.816874 0.576816i $$-0.195705\pi$$
0.816874 + 0.576816i $$0.195705\pi$$
$$542$$ 0 0
$$543$$ 23.0000 0.987024
$$544$$ 0 0
$$545$$ 16.0000 0.685365
$$546$$ 0 0
$$547$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$548$$ 0 0
$$549$$ −11.0000 −0.469469
$$550$$ 0 0
$$551$$ −20.0000 −0.852029
$$552$$ 0 0
$$553$$ 33.0000 1.40330
$$554$$ 0 0
$$555$$ 3.00000 0.127343
$$556$$ 0 0
$$557$$ −30.0000 −1.27114 −0.635570 0.772043i $$-0.719235\pi$$
−0.635570 + 0.772043i $$0.719235\pi$$
$$558$$ 0 0
$$559$$ 4.00000 0.169182
$$560$$ 0 0
$$561$$ −25.0000 −1.05550
$$562$$ 0 0
$$563$$ 41.0000 1.72794 0.863972 0.503540i $$-0.167969\pi$$
0.863972 + 0.503540i $$0.167969\pi$$
$$564$$ 0 0
$$565$$ 2.00000 0.0841406
$$566$$ 0 0
$$567$$ 3.00000 0.125988
$$568$$ 0 0
$$569$$ 16.0000 0.670755 0.335377 0.942084i $$-0.391136\pi$$
0.335377 + 0.942084i $$0.391136\pi$$
$$570$$ 0 0
$$571$$ −17.0000 −0.711428 −0.355714 0.934595i $$-0.615762\pi$$
−0.355714 + 0.934595i $$0.615762\pi$$
$$572$$ 0 0
$$573$$ −20.0000 −0.835512
$$574$$ 0 0
$$575$$ 1.00000 0.0417029
$$576$$ 0 0
$$577$$ −21.0000 −0.874241 −0.437121 0.899403i $$-0.644002\pi$$
−0.437121 + 0.899403i $$0.644002\pi$$
$$578$$ 0 0
$$579$$ −13.0000 −0.540262
$$580$$ 0 0
$$581$$ −24.0000 −0.995688
$$582$$ 0 0
$$583$$ 45.0000 1.86371
$$584$$ 0 0
$$585$$ 1.00000 0.0413449
$$586$$ 0 0
$$587$$ −42.0000 −1.73353 −0.866763 0.498721i $$-0.833803\pi$$
−0.866763 + 0.498721i $$0.833803\pi$$
$$588$$ 0 0
$$589$$ −4.00000 −0.164817
$$590$$ 0 0
$$591$$ 12.0000 0.493614
$$592$$ 0 0
$$593$$ −36.0000 −1.47834 −0.739171 0.673517i $$-0.764783\pi$$
−0.739171 + 0.673517i $$0.764783\pi$$
$$594$$ 0 0
$$595$$ 15.0000 0.614940
$$596$$ 0 0
$$597$$ 4.00000 0.163709
$$598$$ 0 0
$$599$$ −4.00000 −0.163436 −0.0817178 0.996656i $$-0.526041\pi$$
−0.0817178 + 0.996656i $$0.526041\pi$$
$$600$$ 0 0
$$601$$ −5.00000 −0.203954 −0.101977 0.994787i $$-0.532517\pi$$
−0.101977 + 0.994787i $$0.532517\pi$$
$$602$$ 0 0
$$603$$ 4.00000 0.162893
$$604$$ 0 0
$$605$$ 14.0000 0.569181
$$606$$ 0 0
$$607$$ 40.0000 1.62355 0.811775 0.583970i $$-0.198502\pi$$
0.811775 + 0.583970i $$0.198502\pi$$
$$608$$ 0 0
$$609$$ −30.0000 −1.21566
$$610$$ 0 0
$$611$$ −10.0000 −0.404557
$$612$$ 0 0
$$613$$ 3.00000 0.121169 0.0605844 0.998163i $$-0.480704\pi$$
0.0605844 + 0.998163i $$0.480704\pi$$
$$614$$ 0 0
$$615$$ 9.00000 0.362915
$$616$$ 0 0
$$617$$ 22.0000 0.885687 0.442843 0.896599i $$-0.353970\pi$$
0.442843 + 0.896599i $$0.353970\pi$$
$$618$$ 0 0
$$619$$ 2.00000 0.0803868 0.0401934 0.999192i $$-0.487203\pi$$
0.0401934 + 0.999192i $$0.487203\pi$$
$$620$$ 0 0
$$621$$ −1.00000 −0.0401286
$$622$$ 0 0
$$623$$ −33.0000 −1.32212
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ 0 0
$$627$$ 10.0000 0.399362
$$628$$ 0 0
$$629$$ −15.0000 −0.598089
$$630$$ 0 0
$$631$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$632$$ 0 0
$$633$$ −4.00000 −0.158986
$$634$$ 0 0
$$635$$ −14.0000 −0.555573
$$636$$ 0 0
$$637$$ 2.00000 0.0792429
$$638$$ 0 0
$$639$$ −15.0000 −0.593391
$$640$$ 0 0
$$641$$ −36.0000 −1.42191 −0.710957 0.703235i $$-0.751738\pi$$
−0.710957 + 0.703235i $$0.751738\pi$$
$$642$$ 0 0
$$643$$ −1.00000 −0.0394362 −0.0197181 0.999806i $$-0.506277\pi$$
−0.0197181 + 0.999806i $$0.506277\pi$$
$$644$$ 0 0
$$645$$ −4.00000 −0.157500
$$646$$ 0 0
$$647$$ −21.0000 −0.825595 −0.412798 0.910823i $$-0.635448\pi$$
−0.412798 + 0.910823i $$0.635448\pi$$
$$648$$ 0 0
$$649$$ 0 0
$$650$$ 0 0
$$651$$ −6.00000 −0.235159
$$652$$ 0 0
$$653$$ −30.0000 −1.17399 −0.586995 0.809590i $$-0.699689\pi$$
−0.586995 + 0.809590i $$0.699689\pi$$
$$654$$ 0 0
$$655$$ −6.00000 −0.234439
$$656$$ 0 0
$$657$$ 6.00000 0.234082
$$658$$ 0 0
$$659$$ −12.0000 −0.467454 −0.233727 0.972302i $$-0.575092\pi$$
−0.233727 + 0.972302i $$0.575092\pi$$
$$660$$ 0 0
$$661$$ −16.0000 −0.622328 −0.311164 0.950356i $$-0.600719\pi$$
−0.311164 + 0.950356i $$0.600719\pi$$
$$662$$ 0 0
$$663$$ −5.00000 −0.194184
$$664$$ 0 0
$$665$$ −6.00000 −0.232670
$$666$$ 0 0
$$667$$ 10.0000 0.387202
$$668$$ 0 0
$$669$$ 0 0
$$670$$ 0 0
$$671$$ −55.0000 −2.12325
$$672$$ 0 0
$$673$$ 22.0000 0.848038 0.424019 0.905653i $$-0.360619\pi$$
0.424019 + 0.905653i $$0.360619\pi$$
$$674$$ 0 0
$$675$$ −1.00000 −0.0384900
$$676$$ 0 0
$$677$$ −7.00000 −0.269032 −0.134516 0.990911i $$-0.542948\pi$$
−0.134516 + 0.990911i $$0.542948\pi$$
$$678$$ 0 0
$$679$$ −27.0000 −1.03616
$$680$$ 0 0
$$681$$ −18.0000 −0.689761
$$682$$ 0 0
$$683$$ −4.00000 −0.153056 −0.0765279 0.997067i $$-0.524383\pi$$
−0.0765279 + 0.997067i $$0.524383\pi$$
$$684$$ 0 0
$$685$$ 6.00000 0.229248
$$686$$ 0 0
$$687$$ 14.0000 0.534133
$$688$$ 0 0
$$689$$ 9.00000 0.342873
$$690$$ 0 0
$$691$$ −6.00000 −0.228251 −0.114125 0.993466i $$-0.536407\pi$$
−0.114125 + 0.993466i $$0.536407\pi$$
$$692$$ 0 0
$$693$$ 15.0000 0.569803
$$694$$ 0 0
$$695$$ 17.0000 0.644847
$$696$$ 0 0
$$697$$ −45.0000 −1.70450
$$698$$ 0 0
$$699$$ 25.0000 0.945587
$$700$$ 0 0
$$701$$ −4.00000 −0.151078 −0.0755390 0.997143i $$-0.524068\pi$$
−0.0755390 + 0.997143i $$0.524068\pi$$
$$702$$ 0 0
$$703$$ 6.00000 0.226294
$$704$$ 0 0
$$705$$ 10.0000 0.376622
$$706$$ 0 0
$$707$$ −36.0000 −1.35392
$$708$$ 0 0
$$709$$ 20.0000 0.751116 0.375558 0.926799i $$-0.377451\pi$$
0.375558 + 0.926799i $$0.377451\pi$$
$$710$$ 0 0
$$711$$ 11.0000 0.412532
$$712$$ 0 0
$$713$$ 2.00000 0.0749006
$$714$$ 0 0
$$715$$ 5.00000 0.186989
$$716$$ 0 0
$$717$$ 15.0000 0.560185
$$718$$ 0 0
$$719$$ −12.0000 −0.447524 −0.223762 0.974644i $$-0.571834\pi$$
−0.223762 + 0.974644i $$0.571834\pi$$
$$720$$ 0 0
$$721$$ −12.0000 −0.446903
$$722$$ 0 0
$$723$$ 14.0000 0.520666
$$724$$ 0 0
$$725$$ 10.0000 0.371391
$$726$$ 0 0
$$727$$ −6.00000 −0.222528 −0.111264 0.993791i $$-0.535490\pi$$
−0.111264 + 0.993791i $$0.535490\pi$$
$$728$$ 0 0
$$729$$ 1.00000 0.0370370
$$730$$ 0 0
$$731$$ 20.0000 0.739727
$$732$$ 0 0
$$733$$ −15.0000 −0.554038 −0.277019 0.960864i $$-0.589346\pi$$
−0.277019 + 0.960864i $$0.589346\pi$$
$$734$$ 0 0
$$735$$ −2.00000 −0.0737711
$$736$$ 0 0
$$737$$ 20.0000 0.736709
$$738$$ 0 0
$$739$$ −38.0000 −1.39785 −0.698926 0.715194i $$-0.746338\pi$$
−0.698926 + 0.715194i $$0.746338\pi$$
$$740$$ 0 0
$$741$$ 2.00000 0.0734718
$$742$$ 0 0
$$743$$ −6.00000 −0.220119 −0.110059 0.993925i $$-0.535104\pi$$
−0.110059 + 0.993925i $$0.535104\pi$$
$$744$$ 0 0
$$745$$ −7.00000 −0.256460
$$746$$ 0 0
$$747$$ −8.00000 −0.292705
$$748$$ 0 0
$$749$$ −9.00000 −0.328853
$$750$$ 0 0
$$751$$ 45.0000 1.64207 0.821037 0.570875i $$-0.193396\pi$$
0.821037 + 0.570875i $$0.193396\pi$$
$$752$$ 0 0
$$753$$ 20.0000 0.728841
$$754$$ 0 0
$$755$$ 12.0000 0.436725
$$756$$ 0 0
$$757$$ 36.0000 1.30844 0.654221 0.756303i $$-0.272997\pi$$
0.654221 + 0.756303i $$0.272997\pi$$
$$758$$ 0 0
$$759$$ −5.00000 −0.181489
$$760$$ 0 0
$$761$$ −14.0000 −0.507500 −0.253750 0.967270i $$-0.581664\pi$$
−0.253750 + 0.967270i $$0.581664\pi$$
$$762$$ 0 0
$$763$$ 48.0000 1.73772
$$764$$ 0 0
$$765$$ 5.00000 0.180775
$$766$$ 0 0
$$767$$ 0 0
$$768$$ 0 0
$$769$$ −12.0000 −0.432731 −0.216366 0.976312i $$-0.569420\pi$$
−0.216366 + 0.976312i $$0.569420\pi$$
$$770$$ 0 0
$$771$$ −18.0000 −0.648254
$$772$$ 0 0
$$773$$ 24.0000 0.863220 0.431610 0.902060i $$-0.357946\pi$$
0.431610 + 0.902060i $$0.357946\pi$$
$$774$$ 0 0
$$775$$ 2.00000 0.0718421
$$776$$ 0 0
$$777$$ 9.00000 0.322873
$$778$$ 0 0
$$779$$ 18.0000 0.644917
$$780$$ 0 0
$$781$$ −75.0000 −2.68371
$$782$$ 0 0
$$783$$ −10.0000 −0.357371
$$784$$ 0 0
$$785$$ 22.0000 0.785214
$$786$$ 0 0
$$787$$ −44.0000 −1.56843 −0.784215 0.620489i $$-0.786934\pi$$
−0.784215 + 0.620489i $$0.786934\pi$$
$$788$$ 0 0
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 6.00000 0.213335
$$792$$ 0 0
$$793$$ −11.0000 −0.390621
$$794$$ 0 0
$$795$$ −9.00000 −0.319197
$$796$$ 0 0
$$797$$ −5.00000 −0.177109 −0.0885545 0.996071i $$-0.528225\pi$$
−0.0885545 + 0.996071i $$0.528225\pi$$
$$798$$ 0 0
$$799$$ −50.0000 −1.76887
$$800$$ 0 0
$$801$$ −11.0000 −0.388666
$$802$$ 0 0
$$803$$ 30.0000 1.05868
$$804$$ 0 0
$$805$$ 3.00000 0.105736
$$806$$ 0 0
$$807$$ −32.0000 −1.12645
$$808$$ 0 0
$$809$$ −6.00000 −0.210949 −0.105474 0.994422i $$-0.533636\pi$$
−0.105474 + 0.994422i $$0.533636\pi$$
$$810$$ 0 0
$$811$$ 4.00000 0.140459 0.0702295 0.997531i $$-0.477627\pi$$
0.0702295 + 0.997531i $$0.477627\pi$$
$$812$$ 0 0
$$813$$ −2.00000 −0.0701431
$$814$$ 0 0
$$815$$ 11.0000 0.385313
$$816$$ 0 0
$$817$$ −8.00000 −0.279885
$$818$$ 0 0
$$819$$ 3.00000 0.104828
$$820$$ 0 0
$$821$$ −41.0000 −1.43091 −0.715455 0.698659i $$-0.753781\pi$$
−0.715455 + 0.698659i $$0.753781\pi$$
$$822$$ 0 0
$$823$$ 48.0000 1.67317 0.836587 0.547833i $$-0.184547\pi$$
0.836587 + 0.547833i $$0.184547\pi$$
$$824$$ 0 0
$$825$$ −5.00000 −0.174078
$$826$$ 0 0
$$827$$ 42.0000 1.46048 0.730242 0.683189i $$-0.239408\pi$$
0.730242 + 0.683189i $$0.239408\pi$$
$$828$$ 0 0
$$829$$ 14.0000 0.486240 0.243120 0.969996i $$-0.421829\pi$$
0.243120 + 0.969996i $$0.421829\pi$$
$$830$$ 0 0
$$831$$ −26.0000 −0.901930
$$832$$ 0 0
$$833$$ 10.0000 0.346479
$$834$$ 0 0
$$835$$ 8.00000 0.276851
$$836$$ 0 0
$$837$$ −2.00000 −0.0691301
$$838$$ 0 0
$$839$$ −7.00000 −0.241667 −0.120833 0.992673i $$-0.538557\pi$$
−0.120833 + 0.992673i $$0.538557\pi$$
$$840$$ 0 0
$$841$$ 71.0000 2.44828
$$842$$ 0 0
$$843$$ 10.0000 0.344418
$$844$$ 0 0
$$845$$ 1.00000 0.0344010
$$846$$ 0 0
$$847$$ 42.0000 1.44314
$$848$$ 0 0
$$849$$ −8.00000 −0.274559
$$850$$ 0 0
$$851$$ −3.00000 −0.102839
$$852$$ 0 0
$$853$$ 51.0000 1.74621 0.873103 0.487535i $$-0.162104\pi$$
0.873103 + 0.487535i $$0.162104\pi$$
$$854$$ 0 0
$$855$$ −2.00000 −0.0683986
$$856$$ 0 0
$$857$$ −17.0000 −0.580709 −0.290354 0.956919i $$-0.593773\pi$$
−0.290354 + 0.956919i $$0.593773\pi$$
$$858$$ 0 0
$$859$$ −35.0000 −1.19418 −0.597092 0.802173i $$-0.703677\pi$$
−0.597092 + 0.802173i $$0.703677\pi$$
$$860$$ 0 0
$$861$$ 27.0000 0.920158
$$862$$ 0 0
$$863$$ 22.0000 0.748889 0.374444 0.927249i $$-0.377833\pi$$
0.374444 + 0.927249i $$0.377833\pi$$
$$864$$ 0 0
$$865$$ −2.00000 −0.0680020
$$866$$ 0 0
$$867$$ −8.00000 −0.271694
$$868$$ 0 0
$$869$$ 55.0000 1.86575
$$870$$ 0 0
$$871$$ 4.00000 0.135535
$$872$$ 0 0
$$873$$ −9.00000 −0.304604
$$874$$ 0 0
$$875$$ 3.00000 0.101419
$$876$$ 0 0
$$877$$ 22.0000 0.742887 0.371444 0.928456i $$-0.378863\pi$$
0.371444 + 0.928456i $$0.378863\pi$$
$$878$$ 0 0
$$879$$ −24.0000 −0.809500
$$880$$ 0 0
$$881$$ −14.0000 −0.471672 −0.235836 0.971793i $$-0.575783\pi$$
−0.235836 + 0.971793i $$0.575783\pi$$
$$882$$ 0 0
$$883$$ −12.0000 −0.403832 −0.201916 0.979403i $$-0.564717\pi$$
−0.201916 + 0.979403i $$0.564717\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ 15.0000 0.503651 0.251825 0.967773i $$-0.418969\pi$$
0.251825 + 0.967773i $$0.418969\pi$$
$$888$$ 0 0
$$889$$ −42.0000 −1.40863
$$890$$ 0 0
$$891$$ 5.00000 0.167506
$$892$$ 0 0
$$893$$ 20.0000 0.669274
$$894$$ 0 0
$$895$$ 6.00000 0.200558
$$896$$ 0 0
$$897$$ −1.00000 −0.0333890
$$898$$ 0 0
$$899$$ 20.0000 0.667037
$$900$$ 0 0
$$901$$ 45.0000 1.49917
$$902$$ 0 0
$$903$$ −12.0000 −0.399335
$$904$$ 0 0
$$905$$ −23.0000 −0.764546
$$906$$ 0 0
$$907$$ −2.00000 −0.0664089 −0.0332045 0.999449i $$-0.510571\pi$$
−0.0332045 + 0.999449i $$0.510571\pi$$
$$908$$ 0 0
$$909$$ −12.0000 −0.398015
$$910$$ 0 0
$$911$$ −8.00000 −0.265052 −0.132526 0.991180i $$-0.542309\pi$$
−0.132526 + 0.991180i $$0.542309\pi$$
$$912$$ 0 0
$$913$$ −40.0000 −1.32381
$$914$$ 0 0
$$915$$ 11.0000 0.363649
$$916$$ 0 0
$$917$$ −18.0000 −0.594412
$$918$$ 0 0
$$919$$ −29.0000 −0.956622 −0.478311 0.878191i $$-0.658751\pi$$
−0.478311 + 0.878191i $$0.658751\pi$$
$$920$$ 0 0
$$921$$ −19.0000 −0.626071
$$922$$ 0 0
$$923$$ −15.0000 −0.493731
$$924$$ 0 0
$$925$$ −3.00000 −0.0986394
$$926$$ 0 0
$$927$$ −4.00000 −0.131377
$$928$$ 0 0
$$929$$ 21.0000 0.688988 0.344494 0.938789i $$-0.388051\pi$$
0.344494 + 0.938789i $$0.388051\pi$$
$$930$$ 0 0
$$931$$ −4.00000 −0.131095
$$932$$ 0 0
$$933$$ 24.0000 0.785725
$$934$$ 0 0
$$935$$ 25.0000 0.817587
$$936$$ 0 0
$$937$$ 2.00000 0.0653372 0.0326686 0.999466i $$-0.489599\pi$$
0.0326686 + 0.999466i $$0.489599\pi$$
$$938$$ 0 0
$$939$$ 10.0000 0.326338
$$940$$ 0 0
$$941$$ −23.0000 −0.749779 −0.374889 0.927070i $$-0.622319\pi$$
−0.374889 + 0.927070i $$0.622319\pi$$
$$942$$ 0 0
$$943$$ −9.00000 −0.293080
$$944$$ 0 0
$$945$$ −3.00000 −0.0975900
$$946$$ 0 0
$$947$$ −36.0000 −1.16984 −0.584921 0.811090i $$-0.698875\pi$$
−0.584921 + 0.811090i $$0.698875\pi$$
$$948$$ 0 0
$$949$$ 6.00000 0.194768
$$950$$ 0 0
$$951$$ −12.0000 −0.389127
$$952$$ 0 0
$$953$$ −11.0000 −0.356325 −0.178162 0.984001i $$-0.557015\pi$$
−0.178162 + 0.984001i $$0.557015\pi$$
$$954$$ 0 0
$$955$$ 20.0000 0.647185
$$956$$ 0 0
$$957$$ −50.0000 −1.61627
$$958$$ 0 0
$$959$$ 18.0000 0.581250
$$960$$ 0 0
$$961$$ −27.0000 −0.870968
$$962$$ 0 0
$$963$$ −3.00000 −0.0966736
$$964$$ 0 0
$$965$$ 13.0000 0.418485
$$966$$ 0 0
$$967$$ 40.0000 1.28631 0.643157 0.765735i $$-0.277624\pi$$
0.643157 + 0.765735i $$0.277624\pi$$
$$968$$ 0 0
$$969$$ 10.0000 0.321246
$$970$$ 0 0
$$971$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$972$$ 0 0
$$973$$ 51.0000 1.63498
$$974$$ 0 0
$$975$$ −1.00000 −0.0320256
$$976$$ 0 0
$$977$$ 12.0000 0.383914 0.191957 0.981403i $$-0.438517\pi$$
0.191957 + 0.981403i $$0.438517\pi$$
$$978$$ 0 0
$$979$$ −55.0000 −1.75781
$$980$$ 0 0
$$981$$ 16.0000 0.510841
$$982$$ 0 0
$$983$$ −36.0000 −1.14822 −0.574111 0.818778i $$-0.694652\pi$$
−0.574111 + 0.818778i $$0.694652\pi$$
$$984$$ 0 0
$$985$$ −12.0000 −0.382352
$$986$$ 0 0
$$987$$ 30.0000 0.954911
$$988$$ 0 0
$$989$$ 4.00000 0.127193
$$990$$ 0 0
$$991$$ −39.0000 −1.23888 −0.619438 0.785046i $$-0.712639\pi$$
−0.619438 + 0.785046i $$0.712639\pi$$
$$992$$ 0 0
$$993$$ 32.0000 1.01549
$$994$$ 0 0
$$995$$ −4.00000 −0.126809
$$996$$ 0 0
$$997$$ −16.0000 −0.506725 −0.253363 0.967371i $$-0.581537\pi$$
−0.253363 + 0.967371i $$0.581537\pi$$
$$998$$ 0 0
$$999$$ 3.00000 0.0949158
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 3120.2.a.n.1.1 1
3.2 odd 2 9360.2.a.w.1.1 1
4.3 odd 2 195.2.a.d.1.1 1
12.11 even 2 585.2.a.a.1.1 1
20.3 even 4 975.2.c.b.274.1 2
20.7 even 4 975.2.c.b.274.2 2
20.19 odd 2 975.2.a.b.1.1 1
28.27 even 2 9555.2.a.t.1.1 1
52.51 odd 2 2535.2.a.b.1.1 1
60.23 odd 4 2925.2.c.d.2224.2 2
60.47 odd 4 2925.2.c.d.2224.1 2
60.59 even 2 2925.2.a.t.1.1 1
156.155 even 2 7605.2.a.v.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
195.2.a.d.1.1 1 4.3 odd 2
585.2.a.a.1.1 1 12.11 even 2
975.2.a.b.1.1 1 20.19 odd 2
975.2.c.b.274.1 2 20.3 even 4
975.2.c.b.274.2 2 20.7 even 4
2535.2.a.b.1.1 1 52.51 odd 2
2925.2.a.t.1.1 1 60.59 even 2
2925.2.c.d.2224.1 2 60.47 odd 4
2925.2.c.d.2224.2 2 60.23 odd 4
3120.2.a.n.1.1 1 1.1 even 1 trivial
7605.2.a.v.1.1 1 156.155 even 2
9360.2.a.w.1.1 1 3.2 odd 2
9555.2.a.t.1.1 1 28.27 even 2