# Properties

 Label 370.2.a.c.1.1 Level $370$ Weight $2$ Character 370.1 Self dual yes Analytic conductor $2.954$ 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] = [370,2,Mod(1,370)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(370, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("370.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$370 = 2 \cdot 5 \cdot 37$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 370.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: $$2.95446487479$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 370.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-1.00000 q^{2} +2.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} -2.00000 q^{6} +1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q-1.00000 q^{2} +2.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} -2.00000 q^{6} +1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} +3.00000 q^{11} +2.00000 q^{12} -1.00000 q^{14} +2.00000 q^{15} +1.00000 q^{16} +3.00000 q^{17} -1.00000 q^{18} -6.00000 q^{19} +1.00000 q^{20} +2.00000 q^{21} -3.00000 q^{22} +2.00000 q^{23} -2.00000 q^{24} +1.00000 q^{25} -4.00000 q^{27} +1.00000 q^{28} -3.00000 q^{29} -2.00000 q^{30} +3.00000 q^{31} -1.00000 q^{32} +6.00000 q^{33} -3.00000 q^{34} +1.00000 q^{35} +1.00000 q^{36} -1.00000 q^{37} +6.00000 q^{38} -1.00000 q^{40} +3.00000 q^{41} -2.00000 q^{42} -1.00000 q^{43} +3.00000 q^{44} +1.00000 q^{45} -2.00000 q^{46} +4.00000 q^{47} +2.00000 q^{48} -6.00000 q^{49} -1.00000 q^{50} +6.00000 q^{51} +13.0000 q^{53} +4.00000 q^{54} +3.00000 q^{55} -1.00000 q^{56} -12.0000 q^{57} +3.00000 q^{58} +2.00000 q^{60} -15.0000 q^{61} -3.00000 q^{62} +1.00000 q^{63} +1.00000 q^{64} -6.00000 q^{66} +3.00000 q^{68} +4.00000 q^{69} -1.00000 q^{70} -2.00000 q^{71} -1.00000 q^{72} +1.00000 q^{74} +2.00000 q^{75} -6.00000 q^{76} +3.00000 q^{77} -8.00000 q^{79} +1.00000 q^{80} -11.0000 q^{81} -3.00000 q^{82} -4.00000 q^{83} +2.00000 q^{84} +3.00000 q^{85} +1.00000 q^{86} -6.00000 q^{87} -3.00000 q^{88} -18.0000 q^{89} -1.00000 q^{90} +2.00000 q^{92} +6.00000 q^{93} -4.00000 q^{94} -6.00000 q^{95} -2.00000 q^{96} -7.00000 q^{97} +6.00000 q^{98} +3.00000 q^{99} +O(q^{100})$$

## Coefficient data

For each $$n$$ we display the coefficients of the $$q$$-expansion $$a_n$$, the Satake parameters $$\alpha_p$$, and the Satake angles $$\theta_p = \textrm{Arg}(\alpha_p)$$.

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −1.00000 −0.707107
$$3$$ 2.00000 1.15470 0.577350 0.816497i $$-0.304087\pi$$
0.577350 + 0.816497i $$0.304087\pi$$
$$4$$ 1.00000 0.500000
$$5$$ 1.00000 0.447214
$$6$$ −2.00000 −0.816497
$$7$$ 1.00000 0.377964 0.188982 0.981981i $$-0.439481\pi$$
0.188982 + 0.981981i $$0.439481\pi$$
$$8$$ −1.00000 −0.353553
$$9$$ 1.00000 0.333333
$$10$$ −1.00000 −0.316228
$$11$$ 3.00000 0.904534 0.452267 0.891883i $$-0.350615\pi$$
0.452267 + 0.891883i $$0.350615\pi$$
$$12$$ 2.00000 0.577350
$$13$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$14$$ −1.00000 −0.267261
$$15$$ 2.00000 0.516398
$$16$$ 1.00000 0.250000
$$17$$ 3.00000 0.727607 0.363803 0.931476i $$-0.381478\pi$$
0.363803 + 0.931476i $$0.381478\pi$$
$$18$$ −1.00000 −0.235702
$$19$$ −6.00000 −1.37649 −0.688247 0.725476i $$-0.741620\pi$$
−0.688247 + 0.725476i $$0.741620\pi$$
$$20$$ 1.00000 0.223607
$$21$$ 2.00000 0.436436
$$22$$ −3.00000 −0.639602
$$23$$ 2.00000 0.417029 0.208514 0.978019i $$-0.433137\pi$$
0.208514 + 0.978019i $$0.433137\pi$$
$$24$$ −2.00000 −0.408248
$$25$$ 1.00000 0.200000
$$26$$ 0 0
$$27$$ −4.00000 −0.769800
$$28$$ 1.00000 0.188982
$$29$$ −3.00000 −0.557086 −0.278543 0.960424i $$-0.589851\pi$$
−0.278543 + 0.960424i $$0.589851\pi$$
$$30$$ −2.00000 −0.365148
$$31$$ 3.00000 0.538816 0.269408 0.963026i $$-0.413172\pi$$
0.269408 + 0.963026i $$0.413172\pi$$
$$32$$ −1.00000 −0.176777
$$33$$ 6.00000 1.04447
$$34$$ −3.00000 −0.514496
$$35$$ 1.00000 0.169031
$$36$$ 1.00000 0.166667
$$37$$ −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 $$-0.424733\pi$$
0.234261 + 0.972174i $$0.424733\pi$$
$$42$$ −2.00000 −0.308607
$$43$$ −1.00000 −0.152499 −0.0762493 0.997089i $$-0.524294\pi$$
−0.0762493 + 0.997089i $$0.524294\pi$$
$$44$$ 3.00000 0.452267
$$45$$ 1.00000 0.149071
$$46$$ −2.00000 −0.294884
$$47$$ 4.00000 0.583460 0.291730 0.956501i $$-0.405769\pi$$
0.291730 + 0.956501i $$0.405769\pi$$
$$48$$ 2.00000 0.288675
$$49$$ −6.00000 −0.857143
$$50$$ −1.00000 −0.141421
$$51$$ 6.00000 0.840168
$$52$$ 0 0
$$53$$ 13.0000 1.78569 0.892844 0.450367i $$-0.148707\pi$$
0.892844 + 0.450367i $$0.148707\pi$$
$$54$$ 4.00000 0.544331
$$55$$ 3.00000 0.404520
$$56$$ −1.00000 −0.133631
$$57$$ −12.0000 −1.58944
$$58$$ 3.00000 0.393919
$$59$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$60$$ 2.00000 0.258199
$$61$$ −15.0000 −1.92055 −0.960277 0.279050i $$-0.909981\pi$$
−0.960277 + 0.279050i $$0.909981\pi$$
$$62$$ −3.00000 −0.381000
$$63$$ 1.00000 0.125988
$$64$$ 1.00000 0.125000
$$65$$ 0 0
$$66$$ −6.00000 −0.738549
$$67$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$68$$ 3.00000 0.363803
$$69$$ 4.00000 0.481543
$$70$$ −1.00000 −0.119523
$$71$$ −2.00000 −0.237356 −0.118678 0.992933i $$-0.537866\pi$$
−0.118678 + 0.992933i $$0.537866\pi$$
$$72$$ −1.00000 −0.117851
$$73$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$74$$ 1.00000 0.116248
$$75$$ 2.00000 0.230940
$$76$$ −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
$$81$$ −11.0000 −1.22222
$$82$$ −3.00000 −0.331295
$$83$$ −4.00000 −0.439057 −0.219529 0.975606i $$-0.570452\pi$$
−0.219529 + 0.975606i $$0.570452\pi$$
$$84$$ 2.00000 0.218218
$$85$$ 3.00000 0.325396
$$86$$ 1.00000 0.107833
$$87$$ −6.00000 −0.643268
$$88$$ −3.00000 −0.319801
$$89$$ −18.0000 −1.90800 −0.953998 0.299813i $$-0.903076\pi$$
−0.953998 + 0.299813i $$0.903076\pi$$
$$90$$ −1.00000 −0.105409
$$91$$ 0 0
$$92$$ 2.00000 0.208514
$$93$$ 6.00000 0.622171
$$94$$ −4.00000 −0.412568
$$95$$ −6.00000 −0.615587
$$96$$ −2.00000 −0.204124
$$97$$ −7.00000 −0.710742 −0.355371 0.934725i $$-0.615646\pi$$
−0.355371 + 0.934725i $$0.615646\pi$$
$$98$$ 6.00000 0.606092
$$99$$ 3.00000 0.301511
$$100$$ 1.00000 0.100000
$$101$$ −10.0000 −0.995037 −0.497519 0.867453i $$-0.665755\pi$$
−0.497519 + 0.867453i $$0.665755\pi$$
$$102$$ −6.00000 −0.594089
$$103$$ 8.00000 0.788263 0.394132 0.919054i $$-0.371045\pi$$
0.394132 + 0.919054i $$0.371045\pi$$
$$104$$ 0 0
$$105$$ 2.00000 0.195180
$$106$$ −13.0000 −1.26267
$$107$$ 2.00000 0.193347 0.0966736 0.995316i $$-0.469180\pi$$
0.0966736 + 0.995316i $$0.469180\pi$$
$$108$$ −4.00000 −0.384900
$$109$$ −3.00000 −0.287348 −0.143674 0.989625i $$-0.545892\pi$$
−0.143674 + 0.989625i $$0.545892\pi$$
$$110$$ −3.00000 −0.286039
$$111$$ −2.00000 −0.189832
$$112$$ 1.00000 0.0944911
$$113$$ −7.00000 −0.658505 −0.329252 0.944242i $$-0.606797\pi$$
−0.329252 + 0.944242i $$0.606797\pi$$
$$114$$ 12.0000 1.12390
$$115$$ 2.00000 0.186501
$$116$$ −3.00000 −0.278543
$$117$$ 0 0
$$118$$ 0 0
$$119$$ 3.00000 0.275010
$$120$$ −2.00000 −0.182574
$$121$$ −2.00000 −0.181818
$$122$$ 15.0000 1.35804
$$123$$ 6.00000 0.541002
$$124$$ 3.00000 0.269408
$$125$$ 1.00000 0.0894427
$$126$$ −1.00000 −0.0890871
$$127$$ −4.00000 −0.354943 −0.177471 0.984126i $$-0.556792\pi$$
−0.177471 + 0.984126i $$0.556792\pi$$
$$128$$ −1.00000 −0.0883883
$$129$$ −2.00000 −0.176090
$$130$$ 0 0
$$131$$ −12.0000 −1.04844 −0.524222 0.851581i $$-0.675644\pi$$
−0.524222 + 0.851581i $$0.675644\pi$$
$$132$$ 6.00000 0.522233
$$133$$ −6.00000 −0.520266
$$134$$ 0 0
$$135$$ −4.00000 −0.344265
$$136$$ −3.00000 −0.257248
$$137$$ 8.00000 0.683486 0.341743 0.939793i $$-0.388983\pi$$
0.341743 + 0.939793i $$0.388983\pi$$
$$138$$ −4.00000 −0.340503
$$139$$ 3.00000 0.254457 0.127228 0.991873i $$-0.459392\pi$$
0.127228 + 0.991873i $$0.459392\pi$$
$$140$$ 1.00000 0.0845154
$$141$$ 8.00000 0.673722
$$142$$ 2.00000 0.167836
$$143$$ 0 0
$$144$$ 1.00000 0.0833333
$$145$$ −3.00000 −0.249136
$$146$$ 0 0
$$147$$ −12.0000 −0.989743
$$148$$ −1.00000 −0.0821995
$$149$$ 18.0000 1.47462 0.737309 0.675556i $$-0.236096\pi$$
0.737309 + 0.675556i $$0.236096\pi$$
$$150$$ −2.00000 −0.163299
$$151$$ 16.0000 1.30206 0.651031 0.759051i $$-0.274337\pi$$
0.651031 + 0.759051i $$0.274337\pi$$
$$152$$ 6.00000 0.486664
$$153$$ 3.00000 0.242536
$$154$$ −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
$$159$$ 26.0000 2.06193
$$160$$ −1.00000 −0.0790569
$$161$$ 2.00000 0.157622
$$162$$ 11.0000 0.864242
$$163$$ −5.00000 −0.391630 −0.195815 0.980641i $$-0.562735\pi$$
−0.195815 + 0.980641i $$0.562735\pi$$
$$164$$ 3.00000 0.234261
$$165$$ 6.00000 0.467099
$$166$$ 4.00000 0.310460
$$167$$ 12.0000 0.928588 0.464294 0.885681i $$-0.346308\pi$$
0.464294 + 0.885681i $$0.346308\pi$$
$$168$$ −2.00000 −0.154303
$$169$$ −13.0000 −1.00000
$$170$$ −3.00000 −0.230089
$$171$$ −6.00000 −0.458831
$$172$$ −1.00000 −0.0762493
$$173$$ 9.00000 0.684257 0.342129 0.939653i $$-0.388852\pi$$
0.342129 + 0.939653i $$0.388852\pi$$
$$174$$ 6.00000 0.454859
$$175$$ 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 $$-0.145798\pi$$
0.896922 + 0.442189i $$0.145798\pi$$
$$180$$ 1.00000 0.0745356
$$181$$ 2.00000 0.148659 0.0743294 0.997234i $$-0.476318\pi$$
0.0743294 + 0.997234i $$0.476318\pi$$
$$182$$ 0 0
$$183$$ −30.0000 −2.21766
$$184$$ −2.00000 −0.147442
$$185$$ −1.00000 −0.0735215
$$186$$ −6.00000 −0.439941
$$187$$ 9.00000 0.658145
$$188$$ 4.00000 0.291730
$$189$$ −4.00000 −0.290957
$$190$$ 6.00000 0.435286
$$191$$ 21.0000 1.51951 0.759753 0.650211i $$-0.225320\pi$$
0.759753 + 0.650211i $$0.225320\pi$$
$$192$$ 2.00000 0.144338
$$193$$ −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 $$-0.721573\pi$$
−0.641223 + 0.767354i $$0.721573\pi$$
$$198$$ −3.00000 −0.213201
$$199$$ 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
$$204$$ 6.00000 0.420084
$$205$$ 3.00000 0.209529
$$206$$ −8.00000 −0.557386
$$207$$ 2.00000 0.139010
$$208$$ 0 0
$$209$$ −18.0000 −1.24509
$$210$$ −2.00000 −0.138013
$$211$$ −3.00000 −0.206529 −0.103264 0.994654i $$-0.532929\pi$$
−0.103264 + 0.994654i $$0.532929\pi$$
$$212$$ 13.0000 0.892844
$$213$$ −4.00000 −0.274075
$$214$$ −2.00000 −0.136717
$$215$$ −1.00000 −0.0681994
$$216$$ 4.00000 0.272166
$$217$$ 3.00000 0.203653
$$218$$ 3.00000 0.203186
$$219$$ 0 0
$$220$$ 3.00000 0.202260
$$221$$ 0 0
$$222$$ 2.00000 0.134231
$$223$$ 23.0000 1.54019 0.770097 0.637927i $$-0.220208\pi$$
0.770097 + 0.637927i $$0.220208\pi$$
$$224$$ −1.00000 −0.0668153
$$225$$ 1.00000 0.0666667
$$226$$ 7.00000 0.465633
$$227$$ 13.0000 0.862840 0.431420 0.902151i $$-0.358013\pi$$
0.431420 + 0.902151i $$0.358013\pi$$
$$228$$ −12.0000 −0.794719
$$229$$ −6.00000 −0.396491 −0.198246 0.980152i $$-0.563524\pi$$
−0.198246 + 0.980152i $$0.563524\pi$$
$$230$$ −2.00000 −0.131876
$$231$$ 6.00000 0.394771
$$232$$ 3.00000 0.196960
$$233$$ 4.00000 0.262049 0.131024 0.991379i $$-0.458173\pi$$
0.131024 + 0.991379i $$0.458173\pi$$
$$234$$ 0 0
$$235$$ 4.00000 0.260931
$$236$$ 0 0
$$237$$ −16.0000 −1.03931
$$238$$ −3.00000 −0.194461
$$239$$ 9.00000 0.582162 0.291081 0.956698i $$-0.405985\pi$$
0.291081 + 0.956698i $$0.405985\pi$$
$$240$$ 2.00000 0.129099
$$241$$ 24.0000 1.54598 0.772988 0.634421i $$-0.218761\pi$$
0.772988 + 0.634421i $$0.218761\pi$$
$$242$$ 2.00000 0.128565
$$243$$ −10.0000 −0.641500
$$244$$ −15.0000 −0.960277
$$245$$ −6.00000 −0.383326
$$246$$ −6.00000 −0.382546
$$247$$ 0 0
$$248$$ −3.00000 −0.190500
$$249$$ −8.00000 −0.506979
$$250$$ −1.00000 −0.0632456
$$251$$ 6.00000 0.378717 0.189358 0.981908i $$-0.439359\pi$$
0.189358 + 0.981908i $$0.439359\pi$$
$$252$$ 1.00000 0.0629941
$$253$$ 6.00000 0.377217
$$254$$ 4.00000 0.250982
$$255$$ 6.00000 0.375735
$$256$$ 1.00000 0.0625000
$$257$$ −22.0000 −1.37232 −0.686161 0.727450i $$-0.740706\pi$$
−0.686161 + 0.727450i $$0.740706\pi$$
$$258$$ 2.00000 0.124515
$$259$$ −1.00000 −0.0621370
$$260$$ 0 0
$$261$$ −3.00000 −0.185695
$$262$$ 12.0000 0.741362
$$263$$ −1.00000 −0.0616626 −0.0308313 0.999525i $$-0.509815\pi$$
−0.0308313 + 0.999525i $$0.509815\pi$$
$$264$$ −6.00000 −0.369274
$$265$$ 13.0000 0.798584
$$266$$ 6.00000 0.367884
$$267$$ −36.0000 −2.20316
$$268$$ 0 0
$$269$$ −10.0000 −0.609711 −0.304855 0.952399i $$-0.598608\pi$$
−0.304855 + 0.952399i $$0.598608\pi$$
$$270$$ 4.00000 0.243432
$$271$$ −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
$$276$$ 4.00000 0.240772
$$277$$ −20.0000 −1.20168 −0.600842 0.799368i $$-0.705168\pi$$
−0.600842 + 0.799368i $$0.705168\pi$$
$$278$$ −3.00000 −0.179928
$$279$$ 3.00000 0.179605
$$280$$ −1.00000 −0.0597614
$$281$$ −24.0000 −1.43172 −0.715860 0.698244i $$-0.753965\pi$$
−0.715860 + 0.698244i $$0.753965\pi$$
$$282$$ −8.00000 −0.476393
$$283$$ −20.0000 −1.18888 −0.594438 0.804141i $$-0.702626\pi$$
−0.594438 + 0.804141i $$0.702626\pi$$
$$284$$ −2.00000 −0.118678
$$285$$ −12.0000 −0.710819
$$286$$ 0 0
$$287$$ 3.00000 0.177084
$$288$$ −1.00000 −0.0589256
$$289$$ −8.00000 −0.470588
$$290$$ 3.00000 0.176166
$$291$$ −14.0000 −0.820695
$$292$$ 0 0
$$293$$ 3.00000 0.175262 0.0876309 0.996153i $$-0.472070\pi$$
0.0876309 + 0.996153i $$0.472070\pi$$
$$294$$ 12.0000 0.699854
$$295$$ 0 0
$$296$$ 1.00000 0.0581238
$$297$$ −12.0000 −0.696311
$$298$$ −18.0000 −1.04271
$$299$$ 0 0
$$300$$ 2.00000 0.115470
$$301$$ −1.00000 −0.0576390
$$302$$ −16.0000 −0.920697
$$303$$ −20.0000 −1.14897
$$304$$ −6.00000 −0.344124
$$305$$ −15.0000 −0.858898
$$306$$ −3.00000 −0.171499
$$307$$ 34.0000 1.94048 0.970241 0.242140i $$-0.0778494\pi$$
0.970241 + 0.242140i $$0.0778494\pi$$
$$308$$ 3.00000 0.170941
$$309$$ 16.0000 0.910208
$$310$$ −3.00000 −0.170389
$$311$$ 25.0000 1.41762 0.708810 0.705399i $$-0.249232\pi$$
0.708810 + 0.705399i $$0.249232\pi$$
$$312$$ 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
$$315$$ 1.00000 0.0563436
$$316$$ −8.00000 −0.450035
$$317$$ 21.0000 1.17948 0.589739 0.807594i $$-0.299231\pi$$
0.589739 + 0.807594i $$0.299231\pi$$
$$318$$ −26.0000 −1.45801
$$319$$ −9.00000 −0.503903
$$320$$ 1.00000 0.0559017
$$321$$ 4.00000 0.223258
$$322$$ −2.00000 −0.111456
$$323$$ −18.0000 −1.00155
$$324$$ −11.0000 −0.611111
$$325$$ 0 0
$$326$$ 5.00000 0.276924
$$327$$ −6.00000 −0.331801
$$328$$ −3.00000 −0.165647
$$329$$ 4.00000 0.220527
$$330$$ −6.00000 −0.330289
$$331$$ 30.0000 1.64895 0.824475 0.565899i $$-0.191471\pi$$
0.824475 + 0.565899i $$0.191471\pi$$
$$332$$ −4.00000 −0.219529
$$333$$ −1.00000 −0.0547997
$$334$$ −12.0000 −0.656611
$$335$$ 0 0
$$336$$ 2.00000 0.109109
$$337$$ 12.0000 0.653682 0.326841 0.945079i $$-0.394016\pi$$
0.326841 + 0.945079i $$0.394016\pi$$
$$338$$ 13.0000 0.707107
$$339$$ −14.0000 −0.760376
$$340$$ 3.00000 0.162698
$$341$$ 9.00000 0.487377
$$342$$ 6.00000 0.324443
$$343$$ −13.0000 −0.701934
$$344$$ 1.00000 0.0539164
$$345$$ 4.00000 0.215353
$$346$$ −9.00000 −0.483843
$$347$$ −28.0000 −1.50312 −0.751559 0.659665i $$-0.770698\pi$$
−0.751559 + 0.659665i $$0.770698\pi$$
$$348$$ −6.00000 −0.321634
$$349$$ −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 $$-0.331259\pi$$
0.505634 + 0.862748i $$0.331259\pi$$
$$354$$ 0 0
$$355$$ −2.00000 −0.106149
$$356$$ −18.0000 −0.953998
$$357$$ 6.00000 0.317554
$$358$$ −24.0000 −1.26844
$$359$$ 16.0000 0.844448 0.422224 0.906492i $$-0.361250\pi$$
0.422224 + 0.906492i $$0.361250\pi$$
$$360$$ −1.00000 −0.0527046
$$361$$ 17.0000 0.894737
$$362$$ −2.00000 −0.105118
$$363$$ −4.00000 −0.209946
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 30.0000 1.56813
$$367$$ 13.0000 0.678594 0.339297 0.940679i $$-0.389811\pi$$
0.339297 + 0.940679i $$0.389811\pi$$
$$368$$ 2.00000 0.104257
$$369$$ 3.00000 0.156174
$$370$$ 1.00000 0.0519875
$$371$$ 13.0000 0.674926
$$372$$ 6.00000 0.311086
$$373$$ 22.0000 1.13912 0.569558 0.821951i $$-0.307114\pi$$
0.569558 + 0.821951i $$0.307114\pi$$
$$374$$ −9.00000 −0.465379
$$375$$ 2.00000 0.103280
$$376$$ −4.00000 −0.206284
$$377$$ 0 0
$$378$$ 4.00000 0.205738
$$379$$ 16.0000 0.821865 0.410932 0.911666i $$-0.365203\pi$$
0.410932 + 0.911666i $$0.365203\pi$$
$$380$$ −6.00000 −0.307794
$$381$$ −8.00000 −0.409852
$$382$$ −21.0000 −1.07445
$$383$$ −12.0000 −0.613171 −0.306586 0.951843i $$-0.599187\pi$$
−0.306586 + 0.951843i $$0.599187\pi$$
$$384$$ −2.00000 −0.102062
$$385$$ 3.00000 0.152894
$$386$$ 10.0000 0.508987
$$387$$ −1.00000 −0.0508329
$$388$$ −7.00000 −0.355371
$$389$$ 15.0000 0.760530 0.380265 0.924878i $$-0.375833\pi$$
0.380265 + 0.924878i $$0.375833\pi$$
$$390$$ 0 0
$$391$$ 6.00000 0.303433
$$392$$ 6.00000 0.303046
$$393$$ −24.0000 −1.21064
$$394$$ 18.0000 0.906827
$$395$$ −8.00000 −0.402524
$$396$$ 3.00000 0.150756
$$397$$ 18.0000 0.903394 0.451697 0.892171i $$-0.350819\pi$$
0.451697 + 0.892171i $$0.350819\pi$$
$$398$$ −8.00000 −0.401004
$$399$$ −12.0000 −0.600751
$$400$$ 1.00000 0.0500000
$$401$$ 34.0000 1.69788 0.848939 0.528490i $$-0.177242\pi$$
0.848939 + 0.528490i $$0.177242\pi$$
$$402$$ 0 0
$$403$$ 0 0
$$404$$ −10.0000 −0.497519
$$405$$ −11.0000 −0.546594
$$406$$ 3.00000 0.148888
$$407$$ −3.00000 −0.148704
$$408$$ −6.00000 −0.297044
$$409$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$410$$ −3.00000 −0.148159
$$411$$ 16.0000 0.789222
$$412$$ 8.00000 0.394132
$$413$$ 0 0
$$414$$ −2.00000 −0.0982946
$$415$$ −4.00000 −0.196352
$$416$$ 0 0
$$417$$ 6.00000 0.293821
$$418$$ 18.0000 0.880409
$$419$$ 12.0000 0.586238 0.293119 0.956076i $$-0.405307\pi$$
0.293119 + 0.956076i $$0.405307\pi$$
$$420$$ 2.00000 0.0975900
$$421$$ 34.0000 1.65706 0.828529 0.559946i $$-0.189178\pi$$
0.828529 + 0.559946i $$0.189178\pi$$
$$422$$ 3.00000 0.146038
$$423$$ 4.00000 0.194487
$$424$$ −13.0000 −0.631336
$$425$$ 3.00000 0.145521
$$426$$ 4.00000 0.193801
$$427$$ −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 $$-0.768319\pi$$
−0.746609 + 0.665263i $$0.768319\pi$$
$$432$$ −4.00000 −0.192450
$$433$$ −22.0000 −1.05725 −0.528626 0.848855i $$-0.677293\pi$$
−0.528626 + 0.848855i $$0.677293\pi$$
$$434$$ −3.00000 −0.144005
$$435$$ −6.00000 −0.287678
$$436$$ −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
$$441$$ −6.00000 −0.285714
$$442$$ 0 0
$$443$$ −34.0000 −1.61539 −0.807694 0.589601i $$-0.799285\pi$$
−0.807694 + 0.589601i $$0.799285\pi$$
$$444$$ −2.00000 −0.0949158
$$445$$ −18.0000 −0.853282
$$446$$ −23.0000 −1.08908
$$447$$ 36.0000 1.70274
$$448$$ 1.00000 0.0472456
$$449$$ −10.0000 −0.471929 −0.235965 0.971762i $$-0.575825\pi$$
−0.235965 + 0.971762i $$0.575825\pi$$
$$450$$ −1.00000 −0.0471405
$$451$$ 9.00000 0.423793
$$452$$ −7.00000 −0.329252
$$453$$ 32.0000 1.50349
$$454$$ −13.0000 −0.610120
$$455$$ 0 0
$$456$$ 12.0000 0.561951
$$457$$ 9.00000 0.421002 0.210501 0.977594i $$-0.432490\pi$$
0.210501 + 0.977594i $$0.432490\pi$$
$$458$$ 6.00000 0.280362
$$459$$ −12.0000 −0.560112
$$460$$ 2.00000 0.0932505
$$461$$ −1.00000 −0.0465746 −0.0232873 0.999729i $$-0.507413\pi$$
−0.0232873 + 0.999729i $$0.507413\pi$$
$$462$$ −6.00000 −0.279145
$$463$$ 12.0000 0.557687 0.278844 0.960337i $$-0.410049\pi$$
0.278844 + 0.960337i $$0.410049\pi$$
$$464$$ −3.00000 −0.139272
$$465$$ 6.00000 0.278243
$$466$$ −4.00000 −0.185296
$$467$$ −37.0000 −1.71216 −0.856078 0.516847i $$-0.827106\pi$$
−0.856078 + 0.516847i $$0.827106\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ −4.00000 −0.184506
$$471$$ −6.00000 −0.276465
$$472$$ 0 0
$$473$$ −3.00000 −0.137940
$$474$$ 16.0000 0.734904
$$475$$ −6.00000 −0.275299
$$476$$ 3.00000 0.137505
$$477$$ 13.0000 0.595229
$$478$$ −9.00000 −0.411650
$$479$$ −8.00000 −0.365529 −0.182765 0.983157i $$-0.558505\pi$$
−0.182765 + 0.983157i $$0.558505\pi$$
$$480$$ −2.00000 −0.0912871
$$481$$ 0 0
$$482$$ −24.0000 −1.09317
$$483$$ 4.00000 0.182006
$$484$$ −2.00000 −0.0909091
$$485$$ −7.00000 −0.317854
$$486$$ 10.0000 0.453609
$$487$$ 34.0000 1.54069 0.770344 0.637629i $$-0.220085\pi$$
0.770344 + 0.637629i $$0.220085\pi$$
$$488$$ 15.0000 0.679018
$$489$$ −10.0000 −0.452216
$$490$$ 6.00000 0.271052
$$491$$ −20.0000 −0.902587 −0.451294 0.892375i $$-0.649037\pi$$
−0.451294 + 0.892375i $$0.649037\pi$$
$$492$$ 6.00000 0.270501
$$493$$ −9.00000 −0.405340
$$494$$ 0 0
$$495$$ 3.00000 0.134840
$$496$$ 3.00000 0.134704
$$497$$ −2.00000 −0.0897123
$$498$$ 8.00000 0.358489
$$499$$ −22.0000 −0.984855 −0.492428 0.870353i $$-0.663890\pi$$
−0.492428 + 0.870353i $$0.663890\pi$$
$$500$$ 1.00000 0.0447214
$$501$$ 24.0000 1.07224
$$502$$ −6.00000 −0.267793
$$503$$ −24.0000 −1.07011 −0.535054 0.844818i $$-0.679709\pi$$
−0.535054 + 0.844818i $$0.679709\pi$$
$$504$$ −1.00000 −0.0445435
$$505$$ −10.0000 −0.444994
$$506$$ −6.00000 −0.266733
$$507$$ −26.0000 −1.15470
$$508$$ −4.00000 −0.177471
$$509$$ 36.0000 1.59567 0.797836 0.602875i $$-0.205978\pi$$
0.797836 + 0.602875i $$0.205978\pi$$
$$510$$ −6.00000 −0.265684
$$511$$ 0 0
$$512$$ −1.00000 −0.0441942
$$513$$ 24.0000 1.05963
$$514$$ 22.0000 0.970378
$$515$$ 8.00000 0.352522
$$516$$ −2.00000 −0.0880451
$$517$$ 12.0000 0.527759
$$518$$ 1.00000 0.0439375
$$519$$ 18.0000 0.790112
$$520$$ 0 0
$$521$$ −23.0000 −1.00765 −0.503824 0.863806i $$-0.668074\pi$$
−0.503824 + 0.863806i $$0.668074\pi$$
$$522$$ 3.00000 0.131306
$$523$$ −44.0000 −1.92399 −0.961993 0.273075i $$-0.911959\pi$$
−0.961993 + 0.273075i $$0.911959\pi$$
$$524$$ −12.0000 −0.524222
$$525$$ 2.00000 0.0872872
$$526$$ 1.00000 0.0436021
$$527$$ 9.00000 0.392046
$$528$$ 6.00000 0.261116
$$529$$ −19.0000 −0.826087
$$530$$ −13.0000 −0.564684
$$531$$ 0 0
$$532$$ −6.00000 −0.260133
$$533$$ 0 0
$$534$$ 36.0000 1.55787
$$535$$ 2.00000 0.0864675
$$536$$ 0 0
$$537$$ 48.0000 2.07135
$$538$$ 10.0000 0.431131
$$539$$ −18.0000 −0.775315
$$540$$ −4.00000 −0.172133
$$541$$ −2.00000 −0.0859867 −0.0429934 0.999075i $$-0.513689\pi$$
−0.0429934 + 0.999075i $$0.513689\pi$$
$$542$$ 10.0000 0.429537
$$543$$ 4.00000 0.171656
$$544$$ −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
$$549$$ −15.0000 −0.640184
$$550$$ −3.00000 −0.127920
$$551$$ 18.0000 0.766826
$$552$$ −4.00000 −0.170251
$$553$$ −8.00000 −0.340195
$$554$$ 20.0000 0.849719
$$555$$ −2.00000 −0.0848953
$$556$$ 3.00000 0.127228
$$557$$ −40.0000 −1.69485 −0.847427 0.530912i $$-0.821850\pi$$
−0.847427 + 0.530912i $$0.821850\pi$$
$$558$$ −3.00000 −0.127000
$$559$$ 0 0
$$560$$ 1.00000 0.0422577
$$561$$ 18.0000 0.759961
$$562$$ 24.0000 1.01238
$$563$$ −11.0000 −0.463595 −0.231797 0.972764i $$-0.574461\pi$$
−0.231797 + 0.972764i $$0.574461\pi$$
$$564$$ 8.00000 0.336861
$$565$$ −7.00000 −0.294492
$$566$$ 20.0000 0.840663
$$567$$ −11.0000 −0.461957
$$568$$ 2.00000 0.0839181
$$569$$ 10.0000 0.419222 0.209611 0.977785i $$-0.432780\pi$$
0.209611 + 0.977785i $$0.432780\pi$$
$$570$$ 12.0000 0.502625
$$571$$ −31.0000 −1.29731 −0.648655 0.761083i $$-0.724668\pi$$
−0.648655 + 0.761083i $$0.724668\pi$$
$$572$$ 0 0
$$573$$ 42.0000 1.75458
$$574$$ −3.00000 −0.125218
$$575$$ 2.00000 0.0834058
$$576$$ 1.00000 0.0416667
$$577$$ 30.0000 1.24892 0.624458 0.781058i $$-0.285320\pi$$
0.624458 + 0.781058i $$0.285320\pi$$
$$578$$ 8.00000 0.332756
$$579$$ −20.0000 −0.831172
$$580$$ −3.00000 −0.124568
$$581$$ −4.00000 −0.165948
$$582$$ 14.0000 0.580319
$$583$$ 39.0000 1.61521
$$584$$ 0 0
$$585$$ 0 0
$$586$$ −3.00000 −0.123929
$$587$$ 35.0000 1.44460 0.722302 0.691577i $$-0.243084\pi$$
0.722302 + 0.691577i $$0.243084\pi$$
$$588$$ −12.0000 −0.494872
$$589$$ −18.0000 −0.741677
$$590$$ 0 0
$$591$$ −36.0000 −1.48084
$$592$$ −1.00000 −0.0410997
$$593$$ 12.0000 0.492781 0.246390 0.969171i $$-0.420755\pi$$
0.246390 + 0.969171i $$0.420755\pi$$
$$594$$ 12.0000 0.492366
$$595$$ 3.00000 0.122988
$$596$$ 18.0000 0.737309
$$597$$ 16.0000 0.654836
$$598$$ 0 0
$$599$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$600$$ −2.00000 −0.0816497
$$601$$ 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
$$606$$ 20.0000 0.812444
$$607$$ 22.0000 0.892952 0.446476 0.894795i $$-0.352679\pi$$
0.446476 + 0.894795i $$0.352679\pi$$
$$608$$ 6.00000 0.243332
$$609$$ −6.00000 −0.243132
$$610$$ 15.0000 0.607332
$$611$$ 0 0
$$612$$ 3.00000 0.121268
$$613$$ 11.0000 0.444286 0.222143 0.975014i $$-0.428695\pi$$
0.222143 + 0.975014i $$0.428695\pi$$
$$614$$ −34.0000 −1.37213
$$615$$ 6.00000 0.241943
$$616$$ −3.00000 −0.120873
$$617$$ 36.0000 1.44931 0.724653 0.689114i $$-0.242000\pi$$
0.724653 + 0.689114i $$0.242000\pi$$
$$618$$ −16.0000 −0.643614
$$619$$ −37.0000 −1.48716 −0.743578 0.668649i $$-0.766873\pi$$
−0.743578 + 0.668649i $$0.766873\pi$$
$$620$$ 3.00000 0.120483
$$621$$ −8.00000 −0.321029
$$622$$ −25.0000 −1.00241
$$623$$ −18.0000 −0.721155
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ −26.0000 −1.03917
$$627$$ −36.0000 −1.43770
$$628$$ −3.00000 −0.119713
$$629$$ −3.00000 −0.119618
$$630$$ −1.00000 −0.0398410
$$631$$ −41.0000 −1.63218 −0.816092 0.577922i $$-0.803864\pi$$
−0.816092 + 0.577922i $$0.803864\pi$$
$$632$$ 8.00000 0.318223
$$633$$ −6.00000 −0.238479
$$634$$ −21.0000 −0.834017
$$635$$ −4.00000 −0.158735
$$636$$ 26.0000 1.03097
$$637$$ 0 0
$$638$$ 9.00000 0.356313
$$639$$ −2.00000 −0.0791188
$$640$$ −1.00000 −0.0395285
$$641$$ −33.0000 −1.30342 −0.651711 0.758468i $$-0.725948\pi$$
−0.651711 + 0.758468i $$0.725948\pi$$
$$642$$ −4.00000 −0.157867
$$643$$ 11.0000 0.433798 0.216899 0.976194i $$-0.430406\pi$$
0.216899 + 0.976194i $$0.430406\pi$$
$$644$$ 2.00000 0.0788110
$$645$$ −2.00000 −0.0787499
$$646$$ 18.0000 0.708201
$$647$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$648$$ 11.0000 0.432121
$$649$$ 0 0
$$650$$ 0 0
$$651$$ 6.00000 0.235159
$$652$$ −5.00000 −0.195815
$$653$$ 30.0000 1.17399 0.586995 0.809590i $$-0.300311\pi$$
0.586995 + 0.809590i $$0.300311\pi$$
$$654$$ 6.00000 0.234619
$$655$$ −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 $$-0.252657\pi$$
0.701180 + 0.712984i $$0.252657\pi$$
$$660$$ 6.00000 0.233550
$$661$$ −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
$$666$$ 1.00000 0.0387492
$$667$$ −6.00000 −0.232321
$$668$$ 12.0000 0.464294
$$669$$ 46.0000 1.77846
$$670$$ 0 0
$$671$$ −45.0000 −1.73721
$$672$$ −2.00000 −0.0771517
$$673$$ −34.0000 −1.31060 −0.655302 0.755367i $$-0.727459\pi$$
−0.655302 + 0.755367i $$0.727459\pi$$
$$674$$ −12.0000 −0.462223
$$675$$ −4.00000 −0.153960
$$676$$ −13.0000 −0.500000
$$677$$ 34.0000 1.30673 0.653363 0.757045i $$-0.273358\pi$$
0.653363 + 0.757045i $$0.273358\pi$$
$$678$$ 14.0000 0.537667
$$679$$ −7.00000 −0.268635
$$680$$ −3.00000 −0.115045
$$681$$ 26.0000 0.996322
$$682$$ −9.00000 −0.344628
$$683$$ −31.0000 −1.18618 −0.593091 0.805135i $$-0.702093\pi$$
−0.593091 + 0.805135i $$0.702093\pi$$
$$684$$ −6.00000 −0.229416
$$685$$ 8.00000 0.305664
$$686$$ 13.0000 0.496342
$$687$$ −12.0000 −0.457829
$$688$$ −1.00000 −0.0381246
$$689$$ 0 0
$$690$$ −4.00000 −0.152277
$$691$$ 5.00000 0.190209 0.0951045 0.995467i $$-0.469681\pi$$
0.0951045 + 0.995467i $$0.469681\pi$$
$$692$$ 9.00000 0.342129
$$693$$ 3.00000 0.113961
$$694$$ 28.0000 1.06287
$$695$$ 3.00000 0.113796
$$696$$ 6.00000 0.227429
$$697$$ 9.00000 0.340899
$$698$$ 22.0000 0.832712
$$699$$ 8.00000 0.302588
$$700$$ 1.00000 0.0377964
$$701$$ −22.0000 −0.830929 −0.415464 0.909610i $$-0.636381\pi$$
−0.415464 + 0.909610i $$0.636381\pi$$
$$702$$ 0 0
$$703$$ 6.00000 0.226294
$$704$$ 3.00000 0.113067
$$705$$ 8.00000 0.301297
$$706$$ −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
$$711$$ −8.00000 −0.300023
$$712$$ 18.0000 0.674579
$$713$$ 6.00000 0.224702
$$714$$ −6.00000 −0.224544
$$715$$ 0 0
$$716$$ 24.0000 0.896922
$$717$$ 18.0000 0.672222
$$718$$ −16.0000 −0.597115
$$719$$ −18.0000 −0.671287 −0.335643 0.941989i $$-0.608954\pi$$
−0.335643 + 0.941989i $$0.608954\pi$$
$$720$$ 1.00000 0.0372678
$$721$$ 8.00000 0.297936
$$722$$ −17.0000 −0.632674
$$723$$ 48.0000 1.78514
$$724$$ 2.00000 0.0743294
$$725$$ −3.00000 −0.111417
$$726$$ 4.00000 0.148454
$$727$$ 52.0000 1.92857 0.964287 0.264861i $$-0.0853260\pi$$
0.964287 + 0.264861i $$0.0853260\pi$$
$$728$$ 0 0
$$729$$ 13.0000 0.481481
$$730$$ 0 0
$$731$$ −3.00000 −0.110959
$$732$$ −30.0000 −1.10883
$$733$$ −41.0000 −1.51437 −0.757185 0.653201i $$-0.773426\pi$$
−0.757185 + 0.653201i $$0.773426\pi$$
$$734$$ −13.0000 −0.479839
$$735$$ −12.0000 −0.442627
$$736$$ −2.00000 −0.0737210
$$737$$ 0 0
$$738$$ −3.00000 −0.110432
$$739$$ 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 $$-0.482475\pi$$
0.0550297 + 0.998485i $$0.482475\pi$$
$$744$$ −6.00000 −0.219971
$$745$$ 18.0000 0.659469
$$746$$ −22.0000 −0.805477
$$747$$ −4.00000 −0.146352
$$748$$ 9.00000 0.329073
$$749$$ 2.00000 0.0730784
$$750$$ −2.00000 −0.0730297
$$751$$ −22.0000 −0.802791 −0.401396 0.915905i $$-0.631475\pi$$
−0.401396 + 0.915905i $$0.631475\pi$$
$$752$$ 4.00000 0.145865
$$753$$ 12.0000 0.437304
$$754$$ 0 0
$$755$$ 16.0000 0.582300
$$756$$ −4.00000 −0.145479
$$757$$ −50.0000 −1.81728 −0.908640 0.417579i $$-0.862879\pi$$
−0.908640 + 0.417579i $$0.862879\pi$$
$$758$$ −16.0000 −0.581146
$$759$$ 12.0000 0.435572
$$760$$ 6.00000 0.217643
$$761$$ −21.0000 −0.761249 −0.380625 0.924730i $$-0.624291\pi$$
−0.380625 + 0.924730i $$0.624291\pi$$
$$762$$ 8.00000 0.289809
$$763$$ −3.00000 −0.108607
$$764$$ 21.0000 0.759753
$$765$$ 3.00000 0.108465
$$766$$ 12.0000 0.433578
$$767$$ 0 0
$$768$$ 2.00000 0.0721688
$$769$$ 8.00000 0.288487 0.144244 0.989542i $$-0.453925\pi$$
0.144244 + 0.989542i $$0.453925\pi$$
$$770$$ −3.00000 −0.108112
$$771$$ −44.0000 −1.58462
$$772$$ −10.0000 −0.359908
$$773$$ −41.0000 −1.47467 −0.737334 0.675529i $$-0.763915\pi$$
−0.737334 + 0.675529i $$0.763915\pi$$
$$774$$ 1.00000 0.0359443
$$775$$ 3.00000 0.107763
$$776$$ 7.00000 0.251285
$$777$$ −2.00000 −0.0717496
$$778$$ −15.0000 −0.537776
$$779$$ −18.0000 −0.644917
$$780$$ 0 0
$$781$$ −6.00000 −0.214697
$$782$$ −6.00000 −0.214560
$$783$$ 12.0000 0.428845
$$784$$ −6.00000 −0.214286
$$785$$ −3.00000 −0.107075
$$786$$ 24.0000 0.856052
$$787$$ −10.0000 −0.356462 −0.178231 0.983989i $$-0.557037\pi$$
−0.178231 + 0.983989i $$0.557037\pi$$
$$788$$ −18.0000 −0.641223
$$789$$ −2.00000 −0.0712019
$$790$$ 8.00000 0.284627
$$791$$ −7.00000 −0.248891
$$792$$ −3.00000 −0.106600
$$793$$ 0 0
$$794$$ −18.0000 −0.638796
$$795$$ 26.0000 0.922125
$$796$$ 8.00000 0.283552
$$797$$ −26.0000 −0.920967 −0.460484 0.887668i $$-0.652324\pi$$
−0.460484 + 0.887668i $$0.652324\pi$$
$$798$$ 12.0000 0.424795
$$799$$ 12.0000 0.424529
$$800$$ −1.00000 −0.0353553
$$801$$ −18.0000 −0.635999
$$802$$ −34.0000 −1.20058
$$803$$ 0 0
$$804$$ 0 0
$$805$$ 2.00000 0.0704907
$$806$$ 0 0
$$807$$ −20.0000 −0.704033
$$808$$ 10.0000 0.351799
$$809$$ 34.0000 1.19538 0.597688 0.801729i $$-0.296086\pi$$
0.597688 + 0.801729i $$0.296086\pi$$
$$810$$ 11.0000 0.386501
$$811$$ −16.0000 −0.561836 −0.280918 0.959732i $$-0.590639\pi$$
−0.280918 + 0.959732i $$0.590639\pi$$
$$812$$ −3.00000 −0.105279
$$813$$ −20.0000 −0.701431
$$814$$ 3.00000 0.105150
$$815$$ −5.00000 −0.175142
$$816$$ 6.00000 0.210042
$$817$$ 6.00000 0.209913
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 3.00000 0.104765
$$821$$ 8.00000 0.279202 0.139601 0.990208i $$-0.455418\pi$$
0.139601 + 0.990208i $$0.455418\pi$$
$$822$$ −16.0000 −0.558064
$$823$$ 24.0000 0.836587 0.418294 0.908312i $$-0.362628\pi$$
0.418294 + 0.908312i $$0.362628\pi$$
$$824$$ −8.00000 −0.278693
$$825$$ 6.00000 0.208893
$$826$$ 0 0
$$827$$ −11.0000 −0.382507 −0.191254 0.981541i $$-0.561255\pi$$
−0.191254 + 0.981541i $$0.561255\pi$$
$$828$$ 2.00000 0.0695048
$$829$$ −39.0000 −1.35453 −0.677263 0.735741i $$-0.736834\pi$$
−0.677263 + 0.735741i $$0.736834\pi$$
$$830$$ 4.00000 0.138842
$$831$$ −40.0000 −1.38758
$$832$$ 0 0
$$833$$ −18.0000 −0.623663
$$834$$ −6.00000 −0.207763
$$835$$ 12.0000 0.415277
$$836$$ −18.0000 −0.622543
$$837$$ −12.0000 −0.414781
$$838$$ −12.0000 −0.414533
$$839$$ 36.0000 1.24286 0.621429 0.783470i $$-0.286552\pi$$
0.621429 + 0.783470i $$0.286552\pi$$
$$840$$ −2.00000 −0.0690066
$$841$$ −20.0000 −0.689655
$$842$$ −34.0000 −1.17172
$$843$$ −48.0000 −1.65321
$$844$$ −3.00000 −0.103264
$$845$$ −13.0000 −0.447214
$$846$$ −4.00000 −0.137523
$$847$$ −2.00000 −0.0687208
$$848$$ 13.0000 0.446422
$$849$$ −40.0000 −1.37280
$$850$$ −3.00000 −0.102899
$$851$$ −2.00000 −0.0685591
$$852$$ −4.00000 −0.137038
$$853$$ 22.0000 0.753266 0.376633 0.926363i $$-0.377082\pi$$
0.376633 + 0.926363i $$0.377082\pi$$
$$854$$ 15.0000 0.513289
$$855$$ −6.00000 −0.205196
$$856$$ −2.00000 −0.0683586
$$857$$ 33.0000 1.12726 0.563629 0.826028i $$-0.309405\pi$$
0.563629 + 0.826028i $$0.309405\pi$$
$$858$$ 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
$$861$$ 6.00000 0.204479
$$862$$ 31.0000 1.05586
$$863$$ 33.0000 1.12333 0.561667 0.827364i $$-0.310160\pi$$
0.561667 + 0.827364i $$0.310160\pi$$
$$864$$ 4.00000 0.136083
$$865$$ 9.00000 0.306009
$$866$$ 22.0000 0.747590
$$867$$ −16.0000 −0.543388
$$868$$ 3.00000 0.101827
$$869$$ −24.0000 −0.814144
$$870$$ 6.00000 0.203419
$$871$$ 0 0
$$872$$ 3.00000 0.101593
$$873$$ −7.00000 −0.236914
$$874$$ 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
$$879$$ 6.00000 0.202375
$$880$$ 3.00000 0.101130
$$881$$ −9.00000 −0.303218 −0.151609 0.988441i $$-0.548445\pi$$
−0.151609 + 0.988441i $$0.548445\pi$$
$$882$$ 6.00000 0.202031
$$883$$ 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 $$-0.686905\pi$$
−0.554016 + 0.832506i $$0.686905\pi$$
$$888$$ 2.00000 0.0671156
$$889$$ −4.00000 −0.134156
$$890$$ 18.0000 0.603361
$$891$$ −33.0000 −1.10554
$$892$$ 23.0000 0.770097
$$893$$ −24.0000 −0.803129
$$894$$ −36.0000 −1.20402
$$895$$ 24.0000 0.802232
$$896$$ −1.00000 −0.0334077
$$897$$ 0 0
$$898$$ 10.0000 0.333704
$$899$$ −9.00000 −0.300167
$$900$$ 1.00000 0.0333333
$$901$$ 39.0000 1.29928
$$902$$ −9.00000 −0.299667
$$903$$ −2.00000 −0.0665558
$$904$$ 7.00000 0.232817
$$905$$ 2.00000 0.0664822
$$906$$ −32.0000 −1.06313
$$907$$ −12.0000 −0.398453 −0.199227 0.979953i $$-0.563843\pi$$
−0.199227 + 0.979953i $$0.563843\pi$$
$$908$$ 13.0000 0.431420
$$909$$ −10.0000 −0.331679
$$910$$ 0 0
$$911$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$912$$ −12.0000 −0.397360
$$913$$ −12.0000 −0.397142
$$914$$ −9.00000 −0.297694
$$915$$ −30.0000 −0.991769
$$916$$ −6.00000 −0.198246
$$917$$ −12.0000 −0.396275
$$918$$ 12.0000 0.396059
$$919$$ 32.0000 1.05558 0.527791 0.849374i $$-0.323020\pi$$
0.527791 + 0.849374i $$0.323020\pi$$
$$920$$ −2.00000 −0.0659380
$$921$$ 68.0000 2.24068
$$922$$ 1.00000 0.0329332
$$923$$ 0 0
$$924$$ 6.00000 0.197386
$$925$$ −1.00000 −0.0328798
$$926$$ −12.0000 −0.394344
$$927$$ 8.00000 0.262754
$$928$$ 3.00000 0.0984798
$$929$$ −3.00000 −0.0984268 −0.0492134 0.998788i $$-0.515671\pi$$
−0.0492134 + 0.998788i $$0.515671\pi$$
$$930$$ −6.00000 −0.196748
$$931$$ 36.0000 1.17985
$$932$$ 4.00000 0.131024
$$933$$ 50.0000 1.63693
$$934$$ 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
$$939$$ 52.0000 1.69696
$$940$$ 4.00000 0.130466
$$941$$ −6.00000 −0.195594 −0.0977972 0.995206i $$-0.531180\pi$$
−0.0977972 + 0.995206i $$0.531180\pi$$
$$942$$ 6.00000 0.195491
$$943$$ 6.00000 0.195387
$$944$$ 0 0
$$945$$ −4.00000 −0.130120
$$946$$ 3.00000 0.0975384
$$947$$ 9.00000 0.292461 0.146230 0.989251i $$-0.453286\pi$$
0.146230 + 0.989251i $$0.453286\pi$$
$$948$$ −16.0000 −0.519656
$$949$$ 0 0
$$950$$ 6.00000 0.194666
$$951$$ 42.0000 1.36194
$$952$$ −3.00000 −0.0972306
$$953$$ 36.0000 1.16615 0.583077 0.812417i $$-0.301849\pi$$
0.583077 + 0.812417i $$0.301849\pi$$
$$954$$ −13.0000 −0.420891
$$955$$ 21.0000 0.679544
$$956$$ 9.00000 0.291081
$$957$$ −18.0000 −0.581857
$$958$$ 8.00000 0.258468
$$959$$ 8.00000 0.258333
$$960$$ 2.00000 0.0645497
$$961$$ −22.0000 −0.709677
$$962$$ 0 0
$$963$$ 2.00000 0.0644491
$$964$$ 24.0000 0.772988
$$965$$ −10.0000 −0.321911
$$966$$ −4.00000 −0.128698
$$967$$ 48.0000 1.54358 0.771788 0.635880i $$-0.219363\pi$$
0.771788 + 0.635880i $$0.219363\pi$$
$$968$$ 2.00000 0.0642824
$$969$$ −36.0000 −1.15649
$$970$$ 7.00000 0.224756
$$971$$ 9.00000 0.288824 0.144412 0.989518i $$-0.453871\pi$$
0.144412 + 0.989518i $$0.453871\pi$$
$$972$$ −10.0000 −0.320750
$$973$$ 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 $$-0.310849\pi$$
0.559875 + 0.828577i $$0.310849\pi$$
$$978$$ 10.0000 0.319765
$$979$$ −54.0000 −1.72585
$$980$$ −6.00000 −0.191663
$$981$$ −3.00000 −0.0957826
$$982$$ 20.0000 0.638226
$$983$$ −27.0000 −0.861166 −0.430583 0.902551i $$-0.641692\pi$$
−0.430583 + 0.902551i $$0.641692\pi$$
$$984$$ −6.00000 −0.191273
$$985$$ −18.0000 −0.573528
$$986$$ 9.00000 0.286618
$$987$$ 8.00000 0.254643
$$988$$ 0 0
$$989$$ −2.00000 −0.0635963
$$990$$ −3.00000 −0.0953463
$$991$$ 33.0000 1.04828 0.524140 0.851632i $$-0.324387\pi$$
0.524140 + 0.851632i $$0.324387\pi$$
$$992$$ −3.00000 −0.0952501
$$993$$ 60.0000 1.90404
$$994$$ 2.00000 0.0634361
$$995$$ 8.00000 0.253617
$$996$$ −8.00000 −0.253490
$$997$$ −12.0000 −0.380044 −0.190022 0.981780i $$-0.560856\pi$$
−0.190022 + 0.981780i $$0.560856\pi$$
$$998$$ 22.0000 0.696398
$$999$$ 4.00000 0.126554
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 370.2.a.c.1.1 1
3.2 odd 2 3330.2.a.p.1.1 1
4.3 odd 2 2960.2.a.c.1.1 1
5.2 odd 4 1850.2.b.c.149.1 2
5.3 odd 4 1850.2.b.c.149.2 2
5.4 even 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 1.1 even 1 trivial
1850.2.a.i.1.1 1 5.4 even 2
1850.2.b.c.149.1 2 5.2 odd 4
1850.2.b.c.149.2 2 5.3 odd 4
2960.2.a.c.1.1 1 4.3 odd 2
3330.2.a.p.1.1 1 3.2 odd 2