# Properties

 Label 3330.2.a.p.1.1 Level $3330$ Weight $2$ Character 3330.1 Self dual yes Analytic conductor $26.590$ 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] = [3330,2,Mod(1,3330)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3330, 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("3330.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$3330 = 2 \cdot 3^{2} \cdot 5 \cdot 37$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3330.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: $$26.5901838731$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 370) Fricke sign: $$+1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 3330.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{7} +1.00000 q^{8} +O(q^{10})$$ $$q+1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{7} +1.00000 q^{8} -1.00000 q^{10} -3.00000 q^{11} +1.00000 q^{14} +1.00000 q^{16} -3.00000 q^{17} -6.00000 q^{19} -1.00000 q^{20} -3.00000 q^{22} -2.00000 q^{23} +1.00000 q^{25} +1.00000 q^{28} +3.00000 q^{29} +3.00000 q^{31} +1.00000 q^{32} -3.00000 q^{34} -1.00000 q^{35} -1.00000 q^{37} -6.00000 q^{38} -1.00000 q^{40} -3.00000 q^{41} -1.00000 q^{43} -3.00000 q^{44} -2.00000 q^{46} -4.00000 q^{47} -6.00000 q^{49} +1.00000 q^{50} -13.0000 q^{53} +3.00000 q^{55} +1.00000 q^{56} +3.00000 q^{58} -15.0000 q^{61} +3.00000 q^{62} +1.00000 q^{64} -3.00000 q^{68} -1.00000 q^{70} +2.00000 q^{71} -1.00000 q^{74} -6.00000 q^{76} -3.00000 q^{77} -8.00000 q^{79} -1.00000 q^{80} -3.00000 q^{82} +4.00000 q^{83} +3.00000 q^{85} -1.00000 q^{86} -3.00000 q^{88} +18.0000 q^{89} -2.00000 q^{92} -4.00000 q^{94} +6.00000 q^{95} -7.00000 q^{97} -6.00000 q^{98} +O(q^{100})$$

## 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
$$6$$ 0 0
$$7$$ 1.00000 0.377964 0.188982 0.981981i $$-0.439481\pi$$
0.188982 + 0.981981i $$0.439481\pi$$
$$8$$ 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
$$13$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$14$$ 1.00000 0.267261
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ −3.00000 −0.727607 −0.363803 0.931476i $$-0.618522\pi$$
−0.363803 + 0.931476i $$0.618522\pi$$
$$18$$ 0 0
$$19$$ −6.00000 −1.37649 −0.688247 0.725476i $$-0.741620\pi$$
−0.688247 + 0.725476i $$0.741620\pi$$
$$20$$ −1.00000 −0.223607
$$21$$ 0 0
$$22$$ −3.00000 −0.639602
$$23$$ −2.00000 −0.417029 −0.208514 0.978019i $$-0.566863\pi$$
−0.208514 + 0.978019i $$0.566863\pi$$
$$24$$ 0 0
$$25$$ 1.00000 0.200000
$$26$$ 0 0
$$27$$ 0 0
$$28$$ 1.00000 0.188982
$$29$$ 3.00000 0.557086 0.278543 0.960424i $$-0.410149\pi$$
0.278543 + 0.960424i $$0.410149\pi$$
$$30$$ 0 0
$$31$$ 3.00000 0.538816 0.269408 0.963026i $$-0.413172\pi$$
0.269408 + 0.963026i $$0.413172\pi$$
$$32$$ 1.00000 0.176777
$$33$$ 0 0
$$34$$ −3.00000 −0.514496
$$35$$ −1.00000 −0.169031
$$36$$ 0 0
$$37$$ −1.00000 −0.164399
$$38$$ −6.00000 −0.973329
$$39$$ 0 0
$$40$$ −1.00000 −0.158114
$$41$$ −3.00000 −0.468521 −0.234261 0.972174i $$-0.575267\pi$$
−0.234261 + 0.972174i $$0.575267\pi$$
$$42$$ 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
$$46$$ −2.00000 −0.294884
$$47$$ −4.00000 −0.583460 −0.291730 0.956501i $$-0.594231\pi$$
−0.291730 + 0.956501i $$0.594231\pi$$
$$48$$ 0 0
$$49$$ −6.00000 −0.857143
$$50$$ 1.00000 0.141421
$$51$$ 0 0
$$52$$ 0 0
$$53$$ −13.0000 −1.78569 −0.892844 0.450367i $$-0.851293\pi$$
−0.892844 + 0.450367i $$0.851293\pi$$
$$54$$ 0 0
$$55$$ 3.00000 0.404520
$$56$$ 1.00000 0.133631
$$57$$ 0 0
$$58$$ 3.00000 0.393919
$$59$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$60$$ 0 0
$$61$$ −15.0000 −1.92055 −0.960277 0.279050i $$-0.909981\pi$$
−0.960277 + 0.279050i $$0.909981\pi$$
$$62$$ 3.00000 0.381000
$$63$$ 0 0
$$64$$ 1.00000 0.125000
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$68$$ −3.00000 −0.363803
$$69$$ 0 0
$$70$$ −1.00000 −0.119523
$$71$$ 2.00000 0.237356 0.118678 0.992933i $$-0.462134\pi$$
0.118678 + 0.992933i $$0.462134\pi$$
$$72$$ 0 0
$$73$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$74$$ −1.00000 −0.116248
$$75$$ 0 0
$$76$$ −6.00000 −0.688247
$$77$$ −3.00000 −0.341882
$$78$$ 0 0
$$79$$ −8.00000 −0.900070 −0.450035 0.893011i $$-0.648589\pi$$
−0.450035 + 0.893011i $$0.648589\pi$$
$$80$$ −1.00000 −0.111803
$$81$$ 0 0
$$82$$ −3.00000 −0.331295
$$83$$ 4.00000 0.439057 0.219529 0.975606i $$-0.429548\pi$$
0.219529 + 0.975606i $$0.429548\pi$$
$$84$$ 0 0
$$85$$ 3.00000 0.325396
$$86$$ −1.00000 −0.107833
$$87$$ 0 0
$$88$$ −3.00000 −0.319801
$$89$$ 18.0000 1.90800 0.953998 0.299813i $$-0.0969242\pi$$
0.953998 + 0.299813i $$0.0969242\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ −2.00000 −0.208514
$$93$$ 0 0
$$94$$ −4.00000 −0.412568
$$95$$ 6.00000 0.615587
$$96$$ 0 0
$$97$$ −7.00000 −0.710742 −0.355371 0.934725i $$-0.615646\pi$$
−0.355371 + 0.934725i $$0.615646\pi$$
$$98$$ −6.00000 −0.606092
$$99$$ 0 0
$$100$$ 1.00000 0.100000
$$101$$ 10.0000 0.995037 0.497519 0.867453i $$-0.334245\pi$$
0.497519 + 0.867453i $$0.334245\pi$$
$$102$$ 0 0
$$103$$ 8.00000 0.788263 0.394132 0.919054i $$-0.371045\pi$$
0.394132 + 0.919054i $$0.371045\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ −13.0000 −1.26267
$$107$$ −2.00000 −0.193347 −0.0966736 0.995316i $$-0.530820\pi$$
−0.0966736 + 0.995316i $$0.530820\pi$$
$$108$$ 0 0
$$109$$ −3.00000 −0.287348 −0.143674 0.989625i $$-0.545892\pi$$
−0.143674 + 0.989625i $$0.545892\pi$$
$$110$$ 3.00000 0.286039
$$111$$ 0 0
$$112$$ 1.00000 0.0944911
$$113$$ 7.00000 0.658505 0.329252 0.944242i $$-0.393203\pi$$
0.329252 + 0.944242i $$0.393203\pi$$
$$114$$ 0 0
$$115$$ 2.00000 0.186501
$$116$$ 3.00000 0.278543
$$117$$ 0 0
$$118$$ 0 0
$$119$$ −3.00000 −0.275010
$$120$$ 0 0
$$121$$ −2.00000 −0.181818
$$122$$ −15.0000 −1.35804
$$123$$ 0 0
$$124$$ 3.00000 0.269408
$$125$$ −1.00000 −0.0894427
$$126$$ 0 0
$$127$$ −4.00000 −0.354943 −0.177471 0.984126i $$-0.556792\pi$$
−0.177471 + 0.984126i $$0.556792\pi$$
$$128$$ 1.00000 0.0883883
$$129$$ 0 0
$$130$$ 0 0
$$131$$ 12.0000 1.04844 0.524222 0.851581i $$-0.324356\pi$$
0.524222 + 0.851581i $$0.324356\pi$$
$$132$$ 0 0
$$133$$ −6.00000 −0.520266
$$134$$ 0 0
$$135$$ 0 0
$$136$$ −3.00000 −0.257248
$$137$$ −8.00000 −0.683486 −0.341743 0.939793i $$-0.611017\pi$$
−0.341743 + 0.939793i $$0.611017\pi$$
$$138$$ 0 0
$$139$$ 3.00000 0.254457 0.127228 0.991873i $$-0.459392\pi$$
0.127228 + 0.991873i $$0.459392\pi$$
$$140$$ −1.00000 −0.0845154
$$141$$ 0 0
$$142$$ 2.00000 0.167836
$$143$$ 0 0
$$144$$ 0 0
$$145$$ −3.00000 −0.249136
$$146$$ 0 0
$$147$$ 0 0
$$148$$ −1.00000 −0.0821995
$$149$$ −18.0000 −1.47462 −0.737309 0.675556i $$-0.763904\pi$$
−0.737309 + 0.675556i $$0.763904\pi$$
$$150$$ 0 0
$$151$$ 16.0000 1.30206 0.651031 0.759051i $$-0.274337\pi$$
0.651031 + 0.759051i $$0.274337\pi$$
$$152$$ −6.00000 −0.486664
$$153$$ 0 0
$$154$$ −3.00000 −0.241747
$$155$$ −3.00000 −0.240966
$$156$$ 0 0
$$157$$ −3.00000 −0.239426 −0.119713 0.992809i $$-0.538197\pi$$
−0.119713 + 0.992809i $$0.538197\pi$$
$$158$$ −8.00000 −0.636446
$$159$$ 0 0
$$160$$ −1.00000 −0.0790569
$$161$$ −2.00000 −0.157622
$$162$$ 0 0
$$163$$ −5.00000 −0.391630 −0.195815 0.980641i $$-0.562735\pi$$
−0.195815 + 0.980641i $$0.562735\pi$$
$$164$$ −3.00000 −0.234261
$$165$$ 0 0
$$166$$ 4.00000 0.310460
$$167$$ −12.0000 −0.928588 −0.464294 0.885681i $$-0.653692\pi$$
−0.464294 + 0.885681i $$0.653692\pi$$
$$168$$ 0 0
$$169$$ −13.0000 −1.00000
$$170$$ 3.00000 0.230089
$$171$$ 0 0
$$172$$ −1.00000 −0.0762493
$$173$$ −9.00000 −0.684257 −0.342129 0.939653i $$-0.611148\pi$$
−0.342129 + 0.939653i $$0.611148\pi$$
$$174$$ 0 0
$$175$$ 1.00000 0.0755929
$$176$$ −3.00000 −0.226134
$$177$$ 0 0
$$178$$ 18.0000 1.34916
$$179$$ −24.0000 −1.79384 −0.896922 0.442189i $$-0.854202\pi$$
−0.896922 + 0.442189i $$0.854202\pi$$
$$180$$ 0 0
$$181$$ 2.00000 0.148659 0.0743294 0.997234i $$-0.476318\pi$$
0.0743294 + 0.997234i $$0.476318\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ −2.00000 −0.147442
$$185$$ 1.00000 0.0735215
$$186$$ 0 0
$$187$$ 9.00000 0.658145
$$188$$ −4.00000 −0.291730
$$189$$ 0 0
$$190$$ 6.00000 0.435286
$$191$$ −21.0000 −1.51951 −0.759753 0.650211i $$-0.774680\pi$$
−0.759753 + 0.650211i $$0.774680\pi$$
$$192$$ 0 0
$$193$$ −10.0000 −0.719816 −0.359908 0.932988i $$-0.617192\pi$$
−0.359908 + 0.932988i $$0.617192\pi$$
$$194$$ −7.00000 −0.502571
$$195$$ 0 0
$$196$$ −6.00000 −0.428571
$$197$$ 18.0000 1.28245 0.641223 0.767354i $$-0.278427\pi$$
0.641223 + 0.767354i $$0.278427\pi$$
$$198$$ 0 0
$$199$$ 8.00000 0.567105 0.283552 0.958957i $$-0.408487\pi$$
0.283552 + 0.958957i $$0.408487\pi$$
$$200$$ 1.00000 0.0707107
$$201$$ 0 0
$$202$$ 10.0000 0.703598
$$203$$ 3.00000 0.210559
$$204$$ 0 0
$$205$$ 3.00000 0.209529
$$206$$ 8.00000 0.557386
$$207$$ 0 0
$$208$$ 0 0
$$209$$ 18.0000 1.24509
$$210$$ 0 0
$$211$$ −3.00000 −0.206529 −0.103264 0.994654i $$-0.532929\pi$$
−0.103264 + 0.994654i $$0.532929\pi$$
$$212$$ −13.0000 −0.892844
$$213$$ 0 0
$$214$$ −2.00000 −0.136717
$$215$$ 1.00000 0.0681994
$$216$$ 0 0
$$217$$ 3.00000 0.203653
$$218$$ −3.00000 −0.203186
$$219$$ 0 0
$$220$$ 3.00000 0.202260
$$221$$ 0 0
$$222$$ 0 0
$$223$$ 23.0000 1.54019 0.770097 0.637927i $$-0.220208\pi$$
0.770097 + 0.637927i $$0.220208\pi$$
$$224$$ 1.00000 0.0668153
$$225$$ 0 0
$$226$$ 7.00000 0.465633
$$227$$ −13.0000 −0.862840 −0.431420 0.902151i $$-0.641987\pi$$
−0.431420 + 0.902151i $$0.641987\pi$$
$$228$$ 0 0
$$229$$ −6.00000 −0.396491 −0.198246 0.980152i $$-0.563524\pi$$
−0.198246 + 0.980152i $$0.563524\pi$$
$$230$$ 2.00000 0.131876
$$231$$ 0 0
$$232$$ 3.00000 0.196960
$$233$$ −4.00000 −0.262049 −0.131024 0.991379i $$-0.541827\pi$$
−0.131024 + 0.991379i $$0.541827\pi$$
$$234$$ 0 0
$$235$$ 4.00000 0.260931
$$236$$ 0 0
$$237$$ 0 0
$$238$$ −3.00000 −0.194461
$$239$$ −9.00000 −0.582162 −0.291081 0.956698i $$-0.594015\pi$$
−0.291081 + 0.956698i $$0.594015\pi$$
$$240$$ 0 0
$$241$$ 24.0000 1.54598 0.772988 0.634421i $$-0.218761\pi$$
0.772988 + 0.634421i $$0.218761\pi$$
$$242$$ −2.00000 −0.128565
$$243$$ 0 0
$$244$$ −15.0000 −0.960277
$$245$$ 6.00000 0.383326
$$246$$ 0 0
$$247$$ 0 0
$$248$$ 3.00000 0.190500
$$249$$ 0 0
$$250$$ −1.00000 −0.0632456
$$251$$ −6.00000 −0.378717 −0.189358 0.981908i $$-0.560641\pi$$
−0.189358 + 0.981908i $$0.560641\pi$$
$$252$$ 0 0
$$253$$ 6.00000 0.377217
$$254$$ −4.00000 −0.250982
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ 22.0000 1.37232 0.686161 0.727450i $$-0.259294\pi$$
0.686161 + 0.727450i $$0.259294\pi$$
$$258$$ 0 0
$$259$$ −1.00000 −0.0621370
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 12.0000 0.741362
$$263$$ 1.00000 0.0616626 0.0308313 0.999525i $$-0.490185\pi$$
0.0308313 + 0.999525i $$0.490185\pi$$
$$264$$ 0 0
$$265$$ 13.0000 0.798584
$$266$$ −6.00000 −0.367884
$$267$$ 0 0
$$268$$ 0 0
$$269$$ 10.0000 0.609711 0.304855 0.952399i $$-0.401392\pi$$
0.304855 + 0.952399i $$0.401392\pi$$
$$270$$ 0 0
$$271$$ −10.0000 −0.607457 −0.303728 0.952759i $$-0.598232\pi$$
−0.303728 + 0.952759i $$0.598232\pi$$
$$272$$ −3.00000 −0.181902
$$273$$ 0 0
$$274$$ −8.00000 −0.483298
$$275$$ −3.00000 −0.180907
$$276$$ 0 0
$$277$$ −20.0000 −1.20168 −0.600842 0.799368i $$-0.705168\pi$$
−0.600842 + 0.799368i $$0.705168\pi$$
$$278$$ 3.00000 0.179928
$$279$$ 0 0
$$280$$ −1.00000 −0.0597614
$$281$$ 24.0000 1.43172 0.715860 0.698244i $$-0.246035\pi$$
0.715860 + 0.698244i $$0.246035\pi$$
$$282$$ 0 0
$$283$$ −20.0000 −1.18888 −0.594438 0.804141i $$-0.702626\pi$$
−0.594438 + 0.804141i $$0.702626\pi$$
$$284$$ 2.00000 0.118678
$$285$$ 0 0
$$286$$ 0 0
$$287$$ −3.00000 −0.177084
$$288$$ 0 0
$$289$$ −8.00000 −0.470588
$$290$$ −3.00000 −0.176166
$$291$$ 0 0
$$292$$ 0 0
$$293$$ −3.00000 −0.175262 −0.0876309 0.996153i $$-0.527930\pi$$
−0.0876309 + 0.996153i $$0.527930\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ −1.00000 −0.0581238
$$297$$ 0 0
$$298$$ −18.0000 −1.04271
$$299$$ 0 0
$$300$$ 0 0
$$301$$ −1.00000 −0.0576390
$$302$$ 16.0000 0.920697
$$303$$ 0 0
$$304$$ −6.00000 −0.344124
$$305$$ 15.0000 0.858898
$$306$$ 0 0
$$307$$ 34.0000 1.94048 0.970241 0.242140i $$-0.0778494\pi$$
0.970241 + 0.242140i $$0.0778494\pi$$
$$308$$ −3.00000 −0.170941
$$309$$ 0 0
$$310$$ −3.00000 −0.170389
$$311$$ −25.0000 −1.41762 −0.708810 0.705399i $$-0.750768\pi$$
−0.708810 + 0.705399i $$0.750768\pi$$
$$312$$ 0 0
$$313$$ 26.0000 1.46961 0.734803 0.678280i $$-0.237274\pi$$
0.734803 + 0.678280i $$0.237274\pi$$
$$314$$ −3.00000 −0.169300
$$315$$ 0 0
$$316$$ −8.00000 −0.450035
$$317$$ −21.0000 −1.17948 −0.589739 0.807594i $$-0.700769\pi$$
−0.589739 + 0.807594i $$0.700769\pi$$
$$318$$ 0 0
$$319$$ −9.00000 −0.503903
$$320$$ −1.00000 −0.0559017
$$321$$ 0 0
$$322$$ −2.00000 −0.111456
$$323$$ 18.0000 1.00155
$$324$$ 0 0
$$325$$ 0 0
$$326$$ −5.00000 −0.276924
$$327$$ 0 0
$$328$$ −3.00000 −0.165647
$$329$$ −4.00000 −0.220527
$$330$$ 0 0
$$331$$ 30.0000 1.64895 0.824475 0.565899i $$-0.191471\pi$$
0.824475 + 0.565899i $$0.191471\pi$$
$$332$$ 4.00000 0.219529
$$333$$ 0 0
$$334$$ −12.0000 −0.656611
$$335$$ 0 0
$$336$$ 0 0
$$337$$ 12.0000 0.653682 0.326841 0.945079i $$-0.394016\pi$$
0.326841 + 0.945079i $$0.394016\pi$$
$$338$$ −13.0000 −0.707107
$$339$$ 0 0
$$340$$ 3.00000 0.162698
$$341$$ −9.00000 −0.487377
$$342$$ 0 0
$$343$$ −13.0000 −0.701934
$$344$$ −1.00000 −0.0539164
$$345$$ 0 0
$$346$$ −9.00000 −0.483843
$$347$$ 28.0000 1.50312 0.751559 0.659665i $$-0.229302\pi$$
0.751559 + 0.659665i $$0.229302\pi$$
$$348$$ 0 0
$$349$$ −22.0000 −1.17763 −0.588817 0.808267i $$-0.700406\pi$$
−0.588817 + 0.808267i $$0.700406\pi$$
$$350$$ 1.00000 0.0534522
$$351$$ 0 0
$$352$$ −3.00000 −0.159901
$$353$$ −19.0000 −1.01127 −0.505634 0.862748i $$-0.668741\pi$$
−0.505634 + 0.862748i $$0.668741\pi$$
$$354$$ 0 0
$$355$$ −2.00000 −0.106149
$$356$$ 18.0000 0.953998
$$357$$ 0 0
$$358$$ −24.0000 −1.26844
$$359$$ −16.0000 −0.844448 −0.422224 0.906492i $$-0.638750\pi$$
−0.422224 + 0.906492i $$0.638750\pi$$
$$360$$ 0 0
$$361$$ 17.0000 0.894737
$$362$$ 2.00000 0.105118
$$363$$ 0 0
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 0 0
$$367$$ 13.0000 0.678594 0.339297 0.940679i $$-0.389811\pi$$
0.339297 + 0.940679i $$0.389811\pi$$
$$368$$ −2.00000 −0.104257
$$369$$ 0 0
$$370$$ 1.00000 0.0519875
$$371$$ −13.0000 −0.674926
$$372$$ 0 0
$$373$$ 22.0000 1.13912 0.569558 0.821951i $$-0.307114\pi$$
0.569558 + 0.821951i $$0.307114\pi$$
$$374$$ 9.00000 0.465379
$$375$$ 0 0
$$376$$ −4.00000 −0.206284
$$377$$ 0 0
$$378$$ 0 0
$$379$$ 16.0000 0.821865 0.410932 0.911666i $$-0.365203\pi$$
0.410932 + 0.911666i $$0.365203\pi$$
$$380$$ 6.00000 0.307794
$$381$$ 0 0
$$382$$ −21.0000 −1.07445
$$383$$ 12.0000 0.613171 0.306586 0.951843i $$-0.400813\pi$$
0.306586 + 0.951843i $$0.400813\pi$$
$$384$$ 0 0
$$385$$ 3.00000 0.152894
$$386$$ −10.0000 −0.508987
$$387$$ 0 0
$$388$$ −7.00000 −0.355371
$$389$$ −15.0000 −0.760530 −0.380265 0.924878i $$-0.624167\pi$$
−0.380265 + 0.924878i $$0.624167\pi$$
$$390$$ 0 0
$$391$$ 6.00000 0.303433
$$392$$ −6.00000 −0.303046
$$393$$ 0 0
$$394$$ 18.0000 0.906827
$$395$$ 8.00000 0.402524
$$396$$ 0 0
$$397$$ 18.0000 0.903394 0.451697 0.892171i $$-0.350819\pi$$
0.451697 + 0.892171i $$0.350819\pi$$
$$398$$ 8.00000 0.401004
$$399$$ 0 0
$$400$$ 1.00000 0.0500000
$$401$$ −34.0000 −1.69788 −0.848939 0.528490i $$-0.822758\pi$$
−0.848939 + 0.528490i $$0.822758\pi$$
$$402$$ 0 0
$$403$$ 0 0
$$404$$ 10.0000 0.497519
$$405$$ 0 0
$$406$$ 3.00000 0.148888
$$407$$ 3.00000 0.148704
$$408$$ 0 0
$$409$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$410$$ 3.00000 0.148159
$$411$$ 0 0
$$412$$ 8.00000 0.394132
$$413$$ 0 0
$$414$$ 0 0
$$415$$ −4.00000 −0.196352
$$416$$ 0 0
$$417$$ 0 0
$$418$$ 18.0000 0.880409
$$419$$ −12.0000 −0.586238 −0.293119 0.956076i $$-0.594693\pi$$
−0.293119 + 0.956076i $$0.594693\pi$$
$$420$$ 0 0
$$421$$ 34.0000 1.65706 0.828529 0.559946i $$-0.189178\pi$$
0.828529 + 0.559946i $$0.189178\pi$$
$$422$$ −3.00000 −0.146038
$$423$$ 0 0
$$424$$ −13.0000 −0.631336
$$425$$ −3.00000 −0.145521
$$426$$ 0 0
$$427$$ −15.0000 −0.725901
$$428$$ −2.00000 −0.0966736
$$429$$ 0 0
$$430$$ 1.00000 0.0482243
$$431$$ 31.0000 1.49322 0.746609 0.665263i $$-0.231681\pi$$
0.746609 + 0.665263i $$0.231681\pi$$
$$432$$ 0 0
$$433$$ −22.0000 −1.05725 −0.528626 0.848855i $$-0.677293\pi$$
−0.528626 + 0.848855i $$0.677293\pi$$
$$434$$ 3.00000 0.144005
$$435$$ 0 0
$$436$$ −3.00000 −0.143674
$$437$$ 12.0000 0.574038
$$438$$ 0 0
$$439$$ 37.0000 1.76591 0.882957 0.469454i $$-0.155549\pi$$
0.882957 + 0.469454i $$0.155549\pi$$
$$440$$ 3.00000 0.143019
$$441$$ 0 0
$$442$$ 0 0
$$443$$ 34.0000 1.61539 0.807694 0.589601i $$-0.200715\pi$$
0.807694 + 0.589601i $$0.200715\pi$$
$$444$$ 0 0
$$445$$ −18.0000 −0.853282
$$446$$ 23.0000 1.08908
$$447$$ 0 0
$$448$$ 1.00000 0.0472456
$$449$$ 10.0000 0.471929 0.235965 0.971762i $$-0.424175\pi$$
0.235965 + 0.971762i $$0.424175\pi$$
$$450$$ 0 0
$$451$$ 9.00000 0.423793
$$452$$ 7.00000 0.329252
$$453$$ 0 0
$$454$$ −13.0000 −0.610120
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 9.00000 0.421002 0.210501 0.977594i $$-0.432490\pi$$
0.210501 + 0.977594i $$0.432490\pi$$
$$458$$ −6.00000 −0.280362
$$459$$ 0 0
$$460$$ 2.00000 0.0932505
$$461$$ 1.00000 0.0465746 0.0232873 0.999729i $$-0.492587\pi$$
0.0232873 + 0.999729i $$0.492587\pi$$
$$462$$ 0 0
$$463$$ 12.0000 0.557687 0.278844 0.960337i $$-0.410049\pi$$
0.278844 + 0.960337i $$0.410049\pi$$
$$464$$ 3.00000 0.139272
$$465$$ 0 0
$$466$$ −4.00000 −0.185296
$$467$$ 37.0000 1.71216 0.856078 0.516847i $$-0.172894\pi$$
0.856078 + 0.516847i $$0.172894\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ 4.00000 0.184506
$$471$$ 0 0
$$472$$ 0 0
$$473$$ 3.00000 0.137940
$$474$$ 0 0
$$475$$ −6.00000 −0.275299
$$476$$ −3.00000 −0.137505
$$477$$ 0 0
$$478$$ −9.00000 −0.411650
$$479$$ 8.00000 0.365529 0.182765 0.983157i $$-0.441495\pi$$
0.182765 + 0.983157i $$0.441495\pi$$
$$480$$ 0 0
$$481$$ 0 0
$$482$$ 24.0000 1.09317
$$483$$ 0 0
$$484$$ −2.00000 −0.0909091
$$485$$ 7.00000 0.317854
$$486$$ 0 0
$$487$$ 34.0000 1.54069 0.770344 0.637629i $$-0.220085\pi$$
0.770344 + 0.637629i $$0.220085\pi$$
$$488$$ −15.0000 −0.679018
$$489$$ 0 0
$$490$$ 6.00000 0.271052
$$491$$ 20.0000 0.902587 0.451294 0.892375i $$-0.350963\pi$$
0.451294 + 0.892375i $$0.350963\pi$$
$$492$$ 0 0
$$493$$ −9.00000 −0.405340
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 3.00000 0.134704
$$497$$ 2.00000 0.0897123
$$498$$ 0 0
$$499$$ −22.0000 −0.984855 −0.492428 0.870353i $$-0.663890\pi$$
−0.492428 + 0.870353i $$0.663890\pi$$
$$500$$ −1.00000 −0.0447214
$$501$$ 0 0
$$502$$ −6.00000 −0.267793
$$503$$ 24.0000 1.07011 0.535054 0.844818i $$-0.320291\pi$$
0.535054 + 0.844818i $$0.320291\pi$$
$$504$$ 0 0
$$505$$ −10.0000 −0.444994
$$506$$ 6.00000 0.266733
$$507$$ 0 0
$$508$$ −4.00000 −0.177471
$$509$$ −36.0000 −1.59567 −0.797836 0.602875i $$-0.794022\pi$$
−0.797836 + 0.602875i $$0.794022\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ 1.00000 0.0441942
$$513$$ 0 0
$$514$$ 22.0000 0.970378
$$515$$ −8.00000 −0.352522
$$516$$ 0 0
$$517$$ 12.0000 0.527759
$$518$$ −1.00000 −0.0439375
$$519$$ 0 0
$$520$$ 0 0
$$521$$ 23.0000 1.00765 0.503824 0.863806i $$-0.331926\pi$$
0.503824 + 0.863806i $$0.331926\pi$$
$$522$$ 0 0
$$523$$ −44.0000 −1.92399 −0.961993 0.273075i $$-0.911959\pi$$
−0.961993 + 0.273075i $$0.911959\pi$$
$$524$$ 12.0000 0.524222
$$525$$ 0 0
$$526$$ 1.00000 0.0436021
$$527$$ −9.00000 −0.392046
$$528$$ 0 0
$$529$$ −19.0000 −0.826087
$$530$$ 13.0000 0.564684
$$531$$ 0 0
$$532$$ −6.00000 −0.260133
$$533$$ 0 0
$$534$$ 0 0
$$535$$ 2.00000 0.0864675
$$536$$ 0 0
$$537$$ 0 0
$$538$$ 10.0000 0.431131
$$539$$ 18.0000 0.775315
$$540$$ 0 0
$$541$$ −2.00000 −0.0859867 −0.0429934 0.999075i $$-0.513689\pi$$
−0.0429934 + 0.999075i $$0.513689\pi$$
$$542$$ −10.0000 −0.429537
$$543$$ 0 0
$$544$$ −3.00000 −0.128624
$$545$$ 3.00000 0.128506
$$546$$ 0 0
$$547$$ 23.0000 0.983409 0.491704 0.870762i $$-0.336374\pi$$
0.491704 + 0.870762i $$0.336374\pi$$
$$548$$ −8.00000 −0.341743
$$549$$ 0 0
$$550$$ −3.00000 −0.127920
$$551$$ −18.0000 −0.766826
$$552$$ 0 0
$$553$$ −8.00000 −0.340195
$$554$$ −20.0000 −0.849719
$$555$$ 0 0
$$556$$ 3.00000 0.127228
$$557$$ 40.0000 1.69485 0.847427 0.530912i $$-0.178150\pi$$
0.847427 + 0.530912i $$0.178150\pi$$
$$558$$ 0 0
$$559$$ 0 0
$$560$$ −1.00000 −0.0422577
$$561$$ 0 0
$$562$$ 24.0000 1.01238
$$563$$ 11.0000 0.463595 0.231797 0.972764i $$-0.425539\pi$$
0.231797 + 0.972764i $$0.425539\pi$$
$$564$$ 0 0
$$565$$ −7.00000 −0.294492
$$566$$ −20.0000 −0.840663
$$567$$ 0 0
$$568$$ 2.00000 0.0839181
$$569$$ −10.0000 −0.419222 −0.209611 0.977785i $$-0.567220\pi$$
−0.209611 + 0.977785i $$0.567220\pi$$
$$570$$ 0 0
$$571$$ −31.0000 −1.29731 −0.648655 0.761083i $$-0.724668\pi$$
−0.648655 + 0.761083i $$0.724668\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ −3.00000 −0.125218
$$575$$ −2.00000 −0.0834058
$$576$$ 0 0
$$577$$ 30.0000 1.24892 0.624458 0.781058i $$-0.285320\pi$$
0.624458 + 0.781058i $$0.285320\pi$$
$$578$$ −8.00000 −0.332756
$$579$$ 0 0
$$580$$ −3.00000 −0.124568
$$581$$ 4.00000 0.165948
$$582$$ 0 0
$$583$$ 39.0000 1.61521
$$584$$ 0 0
$$585$$ 0 0
$$586$$ −3.00000 −0.123929
$$587$$ −35.0000 −1.44460 −0.722302 0.691577i $$-0.756916\pi$$
−0.722302 + 0.691577i $$0.756916\pi$$
$$588$$ 0 0
$$589$$ −18.0000 −0.741677
$$590$$ 0 0
$$591$$ 0 0
$$592$$ −1.00000 −0.0410997
$$593$$ −12.0000 −0.492781 −0.246390 0.969171i $$-0.579245\pi$$
−0.246390 + 0.969171i $$0.579245\pi$$
$$594$$ 0 0
$$595$$ 3.00000 0.122988
$$596$$ −18.0000 −0.737309
$$597$$ 0 0
$$598$$ 0 0
$$599$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$600$$ 0 0
$$601$$ 21.0000 0.856608 0.428304 0.903635i $$-0.359111\pi$$
0.428304 + 0.903635i $$0.359111\pi$$
$$602$$ −1.00000 −0.0407570
$$603$$ 0 0
$$604$$ 16.0000 0.651031
$$605$$ 2.00000 0.0813116
$$606$$ 0 0
$$607$$ 22.0000 0.892952 0.446476 0.894795i $$-0.352679\pi$$
0.446476 + 0.894795i $$0.352679\pi$$
$$608$$ −6.00000 −0.243332
$$609$$ 0 0
$$610$$ 15.0000 0.607332
$$611$$ 0 0
$$612$$ 0 0
$$613$$ 11.0000 0.444286 0.222143 0.975014i $$-0.428695\pi$$
0.222143 + 0.975014i $$0.428695\pi$$
$$614$$ 34.0000 1.37213
$$615$$ 0 0
$$616$$ −3.00000 −0.120873
$$617$$ −36.0000 −1.44931 −0.724653 0.689114i $$-0.758000\pi$$
−0.724653 + 0.689114i $$0.758000\pi$$
$$618$$ 0 0
$$619$$ −37.0000 −1.48716 −0.743578 0.668649i $$-0.766873\pi$$
−0.743578 + 0.668649i $$0.766873\pi$$
$$620$$ −3.00000 −0.120483
$$621$$ 0 0
$$622$$ −25.0000 −1.00241
$$623$$ 18.0000 0.721155
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ 26.0000 1.03917
$$627$$ 0 0
$$628$$ −3.00000 −0.119713
$$629$$ 3.00000 0.119618
$$630$$ 0 0
$$631$$ −41.0000 −1.63218 −0.816092 0.577922i $$-0.803864\pi$$
−0.816092 + 0.577922i $$0.803864\pi$$
$$632$$ −8.00000 −0.318223
$$633$$ 0 0
$$634$$ −21.0000 −0.834017
$$635$$ 4.00000 0.158735
$$636$$ 0 0
$$637$$ 0 0
$$638$$ −9.00000 −0.356313
$$639$$ 0 0
$$640$$ −1.00000 −0.0395285
$$641$$ 33.0000 1.30342 0.651711 0.758468i $$-0.274052\pi$$
0.651711 + 0.758468i $$0.274052\pi$$
$$642$$ 0 0
$$643$$ 11.0000 0.433798 0.216899 0.976194i $$-0.430406\pi$$
0.216899 + 0.976194i $$0.430406\pi$$
$$644$$ −2.00000 −0.0788110
$$645$$ 0 0
$$646$$ 18.0000 0.708201
$$647$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$648$$ 0 0
$$649$$ 0 0
$$650$$ 0 0
$$651$$ 0 0
$$652$$ −5.00000 −0.195815
$$653$$ −30.0000 −1.17399 −0.586995 0.809590i $$-0.699689\pi$$
−0.586995 + 0.809590i $$0.699689\pi$$
$$654$$ 0 0
$$655$$ −12.0000 −0.468879
$$656$$ −3.00000 −0.117130
$$657$$ 0 0
$$658$$ −4.00000 −0.155936
$$659$$ −36.0000 −1.40236 −0.701180 0.712984i $$-0.747343\pi$$
−0.701180 + 0.712984i $$0.747343\pi$$
$$660$$ 0 0
$$661$$ −13.0000 −0.505641 −0.252821 0.967513i $$-0.581358\pi$$
−0.252821 + 0.967513i $$0.581358\pi$$
$$662$$ 30.0000 1.16598
$$663$$ 0 0
$$664$$ 4.00000 0.155230
$$665$$ 6.00000 0.232670
$$666$$ 0 0
$$667$$ −6.00000 −0.232321
$$668$$ −12.0000 −0.464294
$$669$$ 0 0
$$670$$ 0 0
$$671$$ 45.0000 1.73721
$$672$$ 0 0
$$673$$ −34.0000 −1.31060 −0.655302 0.755367i $$-0.727459\pi$$
−0.655302 + 0.755367i $$0.727459\pi$$
$$674$$ 12.0000 0.462223
$$675$$ 0 0
$$676$$ −13.0000 −0.500000
$$677$$ −34.0000 −1.30673 −0.653363 0.757045i $$-0.726642\pi$$
−0.653363 + 0.757045i $$0.726642\pi$$
$$678$$ 0 0
$$679$$ −7.00000 −0.268635
$$680$$ 3.00000 0.115045
$$681$$ 0 0
$$682$$ −9.00000 −0.344628
$$683$$ 31.0000 1.18618 0.593091 0.805135i $$-0.297907\pi$$
0.593091 + 0.805135i $$0.297907\pi$$
$$684$$ 0 0
$$685$$ 8.00000 0.305664
$$686$$ −13.0000 −0.496342
$$687$$ 0 0
$$688$$ −1.00000 −0.0381246
$$689$$ 0 0
$$690$$ 0 0
$$691$$ 5.00000 0.190209 0.0951045 0.995467i $$-0.469681\pi$$
0.0951045 + 0.995467i $$0.469681\pi$$
$$692$$ −9.00000 −0.342129
$$693$$ 0 0
$$694$$ 28.0000 1.06287
$$695$$ −3.00000 −0.113796
$$696$$ 0 0
$$697$$ 9.00000 0.340899
$$698$$ −22.0000 −0.832712
$$699$$ 0 0
$$700$$ 1.00000 0.0377964
$$701$$ 22.0000 0.830929 0.415464 0.909610i $$-0.363619\pi$$
0.415464 + 0.909610i $$0.363619\pi$$
$$702$$ 0 0
$$703$$ 6.00000 0.226294
$$704$$ −3.00000 −0.113067
$$705$$ 0 0
$$706$$ −19.0000 −0.715074
$$707$$ 10.0000 0.376089
$$708$$ 0 0
$$709$$ −19.0000 −0.713560 −0.356780 0.934188i $$-0.616125\pi$$
−0.356780 + 0.934188i $$0.616125\pi$$
$$710$$ −2.00000 −0.0750587
$$711$$ 0 0
$$712$$ 18.0000 0.674579
$$713$$ −6.00000 −0.224702
$$714$$ 0 0
$$715$$ 0 0
$$716$$ −24.0000 −0.896922
$$717$$ 0 0
$$718$$ −16.0000 −0.597115
$$719$$ 18.0000 0.671287 0.335643 0.941989i $$-0.391046\pi$$
0.335643 + 0.941989i $$0.391046\pi$$
$$720$$ 0 0
$$721$$ 8.00000 0.297936
$$722$$ 17.0000 0.632674
$$723$$ 0 0
$$724$$ 2.00000 0.0743294
$$725$$ 3.00000 0.111417
$$726$$ 0 0
$$727$$ 52.0000 1.92857 0.964287 0.264861i $$-0.0853260\pi$$
0.964287 + 0.264861i $$0.0853260\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ 0 0
$$731$$ 3.00000 0.110959
$$732$$ 0 0
$$733$$ −41.0000 −1.51437 −0.757185 0.653201i $$-0.773426\pi$$
−0.757185 + 0.653201i $$0.773426\pi$$
$$734$$ 13.0000 0.479839
$$735$$ 0 0
$$736$$ −2.00000 −0.0737210
$$737$$ 0 0
$$738$$ 0 0
$$739$$ 27.0000 0.993211 0.496606 0.867976i $$-0.334580\pi$$
0.496606 + 0.867976i $$0.334580\pi$$
$$740$$ 1.00000 0.0367607
$$741$$ 0 0
$$742$$ −13.0000 −0.477245
$$743$$ −3.00000 −0.110059 −0.0550297 0.998485i $$-0.517525\pi$$
−0.0550297 + 0.998485i $$0.517525\pi$$
$$744$$ 0 0
$$745$$ 18.0000 0.659469
$$746$$ 22.0000 0.805477
$$747$$ 0 0
$$748$$ 9.00000 0.329073
$$749$$ −2.00000 −0.0730784
$$750$$ 0 0
$$751$$ −22.0000 −0.802791 −0.401396 0.915905i $$-0.631475\pi$$
−0.401396 + 0.915905i $$0.631475\pi$$
$$752$$ −4.00000 −0.145865
$$753$$ 0 0
$$754$$ 0 0
$$755$$ −16.0000 −0.582300
$$756$$ 0 0
$$757$$ −50.0000 −1.81728 −0.908640 0.417579i $$-0.862879\pi$$
−0.908640 + 0.417579i $$0.862879\pi$$
$$758$$ 16.0000 0.581146
$$759$$ 0 0
$$760$$ 6.00000 0.217643
$$761$$ 21.0000 0.761249 0.380625 0.924730i $$-0.375709\pi$$
0.380625 + 0.924730i $$0.375709\pi$$
$$762$$ 0 0
$$763$$ −3.00000 −0.108607
$$764$$ −21.0000 −0.759753
$$765$$ 0 0
$$766$$ 12.0000 0.433578
$$767$$ 0 0
$$768$$ 0 0
$$769$$ 8.00000 0.288487 0.144244 0.989542i $$-0.453925\pi$$
0.144244 + 0.989542i $$0.453925\pi$$
$$770$$ 3.00000 0.108112
$$771$$ 0 0
$$772$$ −10.0000 −0.359908
$$773$$ 41.0000 1.47467 0.737334 0.675529i $$-0.236085\pi$$
0.737334 + 0.675529i $$0.236085\pi$$
$$774$$ 0 0
$$775$$ 3.00000 0.107763
$$776$$ −7.00000 −0.251285
$$777$$ 0 0
$$778$$ −15.0000 −0.537776
$$779$$ 18.0000 0.644917
$$780$$ 0 0
$$781$$ −6.00000 −0.214697
$$782$$ 6.00000 0.214560
$$783$$ 0 0
$$784$$ −6.00000 −0.214286
$$785$$ 3.00000 0.107075
$$786$$ 0 0
$$787$$ −10.0000 −0.356462 −0.178231 0.983989i $$-0.557037\pi$$
−0.178231 + 0.983989i $$0.557037\pi$$
$$788$$ 18.0000 0.641223
$$789$$ 0 0
$$790$$ 8.00000 0.284627
$$791$$ 7.00000 0.248891
$$792$$ 0 0
$$793$$ 0 0
$$794$$ 18.0000 0.638796
$$795$$ 0 0
$$796$$ 8.00000 0.283552
$$797$$ 26.0000 0.920967 0.460484 0.887668i $$-0.347676\pi$$
0.460484 + 0.887668i $$0.347676\pi$$
$$798$$ 0 0
$$799$$ 12.0000 0.424529
$$800$$ 1.00000 0.0353553
$$801$$ 0 0
$$802$$ −34.0000 −1.20058
$$803$$ 0 0
$$804$$ 0 0
$$805$$ 2.00000 0.0704907
$$806$$ 0 0
$$807$$ 0 0
$$808$$ 10.0000 0.351799
$$809$$ −34.0000 −1.19538 −0.597688 0.801729i $$-0.703914\pi$$
−0.597688 + 0.801729i $$0.703914\pi$$
$$810$$ 0 0
$$811$$ −16.0000 −0.561836 −0.280918 0.959732i $$-0.590639\pi$$
−0.280918 + 0.959732i $$0.590639\pi$$
$$812$$ 3.00000 0.105279
$$813$$ 0 0
$$814$$ 3.00000 0.105150
$$815$$ 5.00000 0.175142
$$816$$ 0 0
$$817$$ 6.00000 0.209913
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 3.00000 0.104765
$$821$$ −8.00000 −0.279202 −0.139601 0.990208i $$-0.544582\pi$$
−0.139601 + 0.990208i $$0.544582\pi$$
$$822$$ 0 0
$$823$$ 24.0000 0.836587 0.418294 0.908312i $$-0.362628\pi$$
0.418294 + 0.908312i $$0.362628\pi$$
$$824$$ 8.00000 0.278693
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 11.0000 0.382507 0.191254 0.981541i $$-0.438745\pi$$
0.191254 + 0.981541i $$0.438745\pi$$
$$828$$ 0 0
$$829$$ −39.0000 −1.35453 −0.677263 0.735741i $$-0.736834\pi$$
−0.677263 + 0.735741i $$0.736834\pi$$
$$830$$ −4.00000 −0.138842
$$831$$ 0 0
$$832$$ 0 0
$$833$$ 18.0000 0.623663
$$834$$ 0 0
$$835$$ 12.0000 0.415277
$$836$$ 18.0000 0.622543
$$837$$ 0 0
$$838$$ −12.0000 −0.414533
$$839$$ −36.0000 −1.24286 −0.621429 0.783470i $$-0.713448\pi$$
−0.621429 + 0.783470i $$0.713448\pi$$
$$840$$ 0 0
$$841$$ −20.0000 −0.689655
$$842$$ 34.0000 1.17172
$$843$$ 0 0
$$844$$ −3.00000 −0.103264
$$845$$ 13.0000 0.447214
$$846$$ 0 0
$$847$$ −2.00000 −0.0687208
$$848$$ −13.0000 −0.446422
$$849$$ 0 0
$$850$$ −3.00000 −0.102899
$$851$$ 2.00000 0.0685591
$$852$$ 0 0
$$853$$ 22.0000 0.753266 0.376633 0.926363i $$-0.377082\pi$$
0.376633 + 0.926363i $$0.377082\pi$$
$$854$$ −15.0000 −0.513289
$$855$$ 0 0
$$856$$ −2.00000 −0.0683586
$$857$$ −33.0000 −1.12726 −0.563629 0.826028i $$-0.690595\pi$$
−0.563629 + 0.826028i $$0.690595\pi$$
$$858$$ 0 0
$$859$$ −36.0000 −1.22830 −0.614152 0.789188i $$-0.710502\pi$$
−0.614152 + 0.789188i $$0.710502\pi$$
$$860$$ 1.00000 0.0340997
$$861$$ 0 0
$$862$$ 31.0000 1.05586
$$863$$ −33.0000 −1.12333 −0.561667 0.827364i $$-0.689840\pi$$
−0.561667 + 0.827364i $$0.689840\pi$$
$$864$$ 0 0
$$865$$ 9.00000 0.306009
$$866$$ −22.0000 −0.747590
$$867$$ 0 0
$$868$$ 3.00000 0.101827
$$869$$ 24.0000 0.814144
$$870$$ 0 0
$$871$$ 0 0
$$872$$ −3.00000 −0.101593
$$873$$ 0 0
$$874$$ 12.0000 0.405906
$$875$$ −1.00000 −0.0338062
$$876$$ 0 0
$$877$$ −43.0000 −1.45201 −0.726003 0.687691i $$-0.758624\pi$$
−0.726003 + 0.687691i $$0.758624\pi$$
$$878$$ 37.0000 1.24869
$$879$$ 0 0
$$880$$ 3.00000 0.101130
$$881$$ 9.00000 0.303218 0.151609 0.988441i $$-0.451555\pi$$
0.151609 + 0.988441i $$0.451555\pi$$
$$882$$ 0 0
$$883$$ 29.0000 0.975928 0.487964 0.872864i $$-0.337740\pi$$
0.487964 + 0.872864i $$0.337740\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 34.0000 1.14225
$$887$$ 33.0000 1.10803 0.554016 0.832506i $$-0.313095\pi$$
0.554016 + 0.832506i $$0.313095\pi$$
$$888$$ 0 0
$$889$$ −4.00000 −0.134156
$$890$$ −18.0000 −0.603361
$$891$$ 0 0
$$892$$ 23.0000 0.770097
$$893$$ 24.0000 0.803129
$$894$$ 0 0
$$895$$ 24.0000 0.802232
$$896$$ 1.00000 0.0334077
$$897$$ 0 0
$$898$$ 10.0000 0.333704
$$899$$ 9.00000 0.300167
$$900$$ 0 0
$$901$$ 39.0000 1.29928
$$902$$ 9.00000 0.299667
$$903$$ 0 0
$$904$$ 7.00000 0.232817
$$905$$ −2.00000 −0.0664822
$$906$$ 0 0
$$907$$ −12.0000 −0.398453 −0.199227 0.979953i $$-0.563843\pi$$
−0.199227 + 0.979953i $$0.563843\pi$$
$$908$$ −13.0000 −0.431420
$$909$$ 0 0
$$910$$ 0 0
$$911$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$912$$ 0 0
$$913$$ −12.0000 −0.397142
$$914$$ 9.00000 0.297694
$$915$$ 0 0
$$916$$ −6.00000 −0.198246
$$917$$ 12.0000 0.396275
$$918$$ 0 0
$$919$$ 32.0000 1.05558 0.527791 0.849374i $$-0.323020\pi$$
0.527791 + 0.849374i $$0.323020\pi$$
$$920$$ 2.00000 0.0659380
$$921$$ 0 0
$$922$$ 1.00000 0.0329332
$$923$$ 0 0
$$924$$ 0 0
$$925$$ −1.00000 −0.0328798
$$926$$ 12.0000 0.394344
$$927$$ 0 0
$$928$$ 3.00000 0.0984798
$$929$$ 3.00000 0.0984268 0.0492134 0.998788i $$-0.484329\pi$$
0.0492134 + 0.998788i $$0.484329\pi$$
$$930$$ 0 0
$$931$$ 36.0000 1.17985
$$932$$ −4.00000 −0.131024
$$933$$ 0 0
$$934$$ 37.0000 1.21068
$$935$$ −9.00000 −0.294331
$$936$$ 0 0
$$937$$ −14.0000 −0.457360 −0.228680 0.973502i $$-0.573441\pi$$
−0.228680 + 0.973502i $$0.573441\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ 4.00000 0.130466
$$941$$ 6.00000 0.195594 0.0977972 0.995206i $$-0.468820\pi$$
0.0977972 + 0.995206i $$0.468820\pi$$
$$942$$ 0 0
$$943$$ 6.00000 0.195387
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 3.00000 0.0975384
$$947$$ −9.00000 −0.292461 −0.146230 0.989251i $$-0.546714\pi$$
−0.146230 + 0.989251i $$0.546714\pi$$
$$948$$ 0 0
$$949$$ 0 0
$$950$$ −6.00000 −0.194666
$$951$$ 0 0
$$952$$ −3.00000 −0.0972306
$$953$$ −36.0000 −1.16615 −0.583077 0.812417i $$-0.698151\pi$$
−0.583077 + 0.812417i $$0.698151\pi$$
$$954$$ 0 0
$$955$$ 21.0000 0.679544
$$956$$ −9.00000 −0.291081
$$957$$ 0 0
$$958$$ 8.00000 0.258468
$$959$$ −8.00000 −0.258333
$$960$$ 0 0
$$961$$ −22.0000 −0.709677
$$962$$ 0 0
$$963$$ 0 0
$$964$$ 24.0000 0.772988
$$965$$ 10.0000 0.321911
$$966$$ 0 0
$$967$$ 48.0000 1.54358 0.771788 0.635880i $$-0.219363\pi$$
0.771788 + 0.635880i $$0.219363\pi$$
$$968$$ −2.00000 −0.0642824
$$969$$ 0 0
$$970$$ 7.00000 0.224756
$$971$$ −9.00000 −0.288824 −0.144412 0.989518i $$-0.546129\pi$$
−0.144412 + 0.989518i $$0.546129\pi$$
$$972$$ 0 0
$$973$$ 3.00000 0.0961756
$$974$$ 34.0000 1.08943
$$975$$ 0 0
$$976$$ −15.0000 −0.480138
$$977$$ −35.0000 −1.11975 −0.559875 0.828577i $$-0.689151\pi$$
−0.559875 + 0.828577i $$0.689151\pi$$
$$978$$ 0 0
$$979$$ −54.0000 −1.72585
$$980$$ 6.00000 0.191663
$$981$$ 0 0
$$982$$ 20.0000 0.638226
$$983$$ 27.0000 0.861166 0.430583 0.902551i $$-0.358308\pi$$
0.430583 + 0.902551i $$0.358308\pi$$
$$984$$ 0 0
$$985$$ −18.0000 −0.573528
$$986$$ −9.00000 −0.286618
$$987$$ 0 0
$$988$$ 0 0
$$989$$ 2.00000 0.0635963
$$990$$ 0 0
$$991$$ 33.0000 1.04828 0.524140 0.851632i $$-0.324387\pi$$
0.524140 + 0.851632i $$0.324387\pi$$
$$992$$ 3.00000 0.0952501
$$993$$ 0 0
$$994$$ 2.00000 0.0634361
$$995$$ −8.00000 −0.253617
$$996$$ 0 0
$$997$$ −12.0000 −0.380044 −0.190022 0.981780i $$-0.560856\pi$$
−0.190022 + 0.981780i $$0.560856\pi$$
$$998$$ −22.0000 −0.696398
$$999$$ 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 3330.2.a.p.1.1 1
3.2 odd 2 370.2.a.c.1.1 1
12.11 even 2 2960.2.a.c.1.1 1
15.2 even 4 1850.2.b.c.149.1 2
15.8 even 4 1850.2.b.c.149.2 2
15.14 odd 2 1850.2.a.i.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
370.2.a.c.1.1 1 3.2 odd 2
1850.2.a.i.1.1 1 15.14 odd 2
1850.2.b.c.149.1 2 15.2 even 4
1850.2.b.c.149.2 2 15.8 even 4
2960.2.a.c.1.1 1 12.11 even 2
3330.2.a.p.1.1 1 1.1 even 1 trivial