# Properties

 Label 175.2.a.c.1.1 Level $175$ Weight $2$ Character 175.1 Self dual yes Analytic conductor $1.397$ 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] = [175,2,Mod(1,175)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(175, 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("175.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$175 = 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 175.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: $$1.39738203537$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 35) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 175.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+2.00000 q^{2} +1.00000 q^{3} +2.00000 q^{4} +2.00000 q^{6} -1.00000 q^{7} -2.00000 q^{9} +O(q^{10})$$ $$q+2.00000 q^{2} +1.00000 q^{3} +2.00000 q^{4} +2.00000 q^{6} -1.00000 q^{7} -2.00000 q^{9} -3.00000 q^{11} +2.00000 q^{12} +1.00000 q^{13} -2.00000 q^{14} -4.00000 q^{16} +7.00000 q^{17} -4.00000 q^{18} -1.00000 q^{21} -6.00000 q^{22} +6.00000 q^{23} +2.00000 q^{26} -5.00000 q^{27} -2.00000 q^{28} -5.00000 q^{29} +2.00000 q^{31} -8.00000 q^{32} -3.00000 q^{33} +14.0000 q^{34} -4.00000 q^{36} +2.00000 q^{37} +1.00000 q^{39} +2.00000 q^{41} -2.00000 q^{42} -4.00000 q^{43} -6.00000 q^{44} +12.0000 q^{46} -3.00000 q^{47} -4.00000 q^{48} +1.00000 q^{49} +7.00000 q^{51} +2.00000 q^{52} +6.00000 q^{53} -10.0000 q^{54} -10.0000 q^{58} +10.0000 q^{59} -8.00000 q^{61} +4.00000 q^{62} +2.00000 q^{63} -8.00000 q^{64} -6.00000 q^{66} +2.00000 q^{67} +14.0000 q^{68} +6.00000 q^{69} -8.00000 q^{71} +6.00000 q^{73} +4.00000 q^{74} +3.00000 q^{77} +2.00000 q^{78} -5.00000 q^{79} +1.00000 q^{81} +4.00000 q^{82} -4.00000 q^{83} -2.00000 q^{84} -8.00000 q^{86} -5.00000 q^{87} -1.00000 q^{91} +12.0000 q^{92} +2.00000 q^{93} -6.00000 q^{94} -8.00000 q^{96} +7.00000 q^{97} +2.00000 q^{98} +6.00000 q^{99} +O(q^{100})$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 2.00000 1.41421 0.707107 0.707107i $$-0.250000\pi$$
0.707107 + 0.707107i $$0.250000\pi$$
$$3$$ 1.00000 0.577350 0.288675 0.957427i $$-0.406785\pi$$
0.288675 + 0.957427i $$0.406785\pi$$
$$4$$ 2.00000 1.00000
$$5$$ 0 0
$$6$$ 2.00000 0.816497
$$7$$ −1.00000 −0.377964
$$8$$ 0 0
$$9$$ −2.00000 −0.666667
$$10$$ 0 0
$$11$$ −3.00000 −0.904534 −0.452267 0.891883i $$-0.649385\pi$$
−0.452267 + 0.891883i $$0.649385\pi$$
$$12$$ 2.00000 0.577350
$$13$$ 1.00000 0.277350 0.138675 0.990338i $$-0.455716\pi$$
0.138675 + 0.990338i $$0.455716\pi$$
$$14$$ −2.00000 −0.534522
$$15$$ 0 0
$$16$$ −4.00000 −1.00000
$$17$$ 7.00000 1.69775 0.848875 0.528594i $$-0.177281\pi$$
0.848875 + 0.528594i $$0.177281\pi$$
$$18$$ −4.00000 −0.942809
$$19$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$20$$ 0 0
$$21$$ −1.00000 −0.218218
$$22$$ −6.00000 −1.27920
$$23$$ 6.00000 1.25109 0.625543 0.780189i $$-0.284877\pi$$
0.625543 + 0.780189i $$0.284877\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ 2.00000 0.392232
$$27$$ −5.00000 −0.962250
$$28$$ −2.00000 −0.377964
$$29$$ −5.00000 −0.928477 −0.464238 0.885710i $$-0.653672\pi$$
−0.464238 + 0.885710i $$0.653672\pi$$
$$30$$ 0 0
$$31$$ 2.00000 0.359211 0.179605 0.983739i $$-0.442518\pi$$
0.179605 + 0.983739i $$0.442518\pi$$
$$32$$ −8.00000 −1.41421
$$33$$ −3.00000 −0.522233
$$34$$ 14.0000 2.40098
$$35$$ 0 0
$$36$$ −4.00000 −0.666667
$$37$$ 2.00000 0.328798 0.164399 0.986394i $$-0.447432\pi$$
0.164399 + 0.986394i $$0.447432\pi$$
$$38$$ 0 0
$$39$$ 1.00000 0.160128
$$40$$ 0 0
$$41$$ 2.00000 0.312348 0.156174 0.987730i $$-0.450084\pi$$
0.156174 + 0.987730i $$0.450084\pi$$
$$42$$ −2.00000 −0.308607
$$43$$ −4.00000 −0.609994 −0.304997 0.952353i $$-0.598656\pi$$
−0.304997 + 0.952353i $$0.598656\pi$$
$$44$$ −6.00000 −0.904534
$$45$$ 0 0
$$46$$ 12.0000 1.76930
$$47$$ −3.00000 −0.437595 −0.218797 0.975770i $$-0.570213\pi$$
−0.218797 + 0.975770i $$0.570213\pi$$
$$48$$ −4.00000 −0.577350
$$49$$ 1.00000 0.142857
$$50$$ 0 0
$$51$$ 7.00000 0.980196
$$52$$ 2.00000 0.277350
$$53$$ 6.00000 0.824163 0.412082 0.911147i $$-0.364802\pi$$
0.412082 + 0.911147i $$0.364802\pi$$
$$54$$ −10.0000 −1.36083
$$55$$ 0 0
$$56$$ 0 0
$$57$$ 0 0
$$58$$ −10.0000 −1.31306
$$59$$ 10.0000 1.30189 0.650945 0.759125i $$-0.274373\pi$$
0.650945 + 0.759125i $$0.274373\pi$$
$$60$$ 0 0
$$61$$ −8.00000 −1.02430 −0.512148 0.858898i $$-0.671150\pi$$
−0.512148 + 0.858898i $$0.671150\pi$$
$$62$$ 4.00000 0.508001
$$63$$ 2.00000 0.251976
$$64$$ −8.00000 −1.00000
$$65$$ 0 0
$$66$$ −6.00000 −0.738549
$$67$$ 2.00000 0.244339 0.122169 0.992509i $$-0.461015\pi$$
0.122169 + 0.992509i $$0.461015\pi$$
$$68$$ 14.0000 1.69775
$$69$$ 6.00000 0.722315
$$70$$ 0 0
$$71$$ −8.00000 −0.949425 −0.474713 0.880141i $$-0.657448\pi$$
−0.474713 + 0.880141i $$0.657448\pi$$
$$72$$ 0 0
$$73$$ 6.00000 0.702247 0.351123 0.936329i $$-0.385800\pi$$
0.351123 + 0.936329i $$0.385800\pi$$
$$74$$ 4.00000 0.464991
$$75$$ 0 0
$$76$$ 0 0
$$77$$ 3.00000 0.341882
$$78$$ 2.00000 0.226455
$$79$$ −5.00000 −0.562544 −0.281272 0.959628i $$-0.590756\pi$$
−0.281272 + 0.959628i $$0.590756\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 4.00000 0.441726
$$83$$ −4.00000 −0.439057 −0.219529 0.975606i $$-0.570452\pi$$
−0.219529 + 0.975606i $$0.570452\pi$$
$$84$$ −2.00000 −0.218218
$$85$$ 0 0
$$86$$ −8.00000 −0.862662
$$87$$ −5.00000 −0.536056
$$88$$ 0 0
$$89$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$90$$ 0 0
$$91$$ −1.00000 −0.104828
$$92$$ 12.0000 1.25109
$$93$$ 2.00000 0.207390
$$94$$ −6.00000 −0.618853
$$95$$ 0 0
$$96$$ −8.00000 −0.816497
$$97$$ 7.00000 0.710742 0.355371 0.934725i $$-0.384354\pi$$
0.355371 + 0.934725i $$0.384354\pi$$
$$98$$ 2.00000 0.202031
$$99$$ 6.00000 0.603023
$$100$$ 0 0
$$101$$ 12.0000 1.19404 0.597022 0.802225i $$-0.296350\pi$$
0.597022 + 0.802225i $$0.296350\pi$$
$$102$$ 14.0000 1.38621
$$103$$ −19.0000 −1.87213 −0.936063 0.351833i $$-0.885559\pi$$
−0.936063 + 0.351833i $$0.885559\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 12.0000 1.16554
$$107$$ −8.00000 −0.773389 −0.386695 0.922208i $$-0.626383\pi$$
−0.386695 + 0.922208i $$0.626383\pi$$
$$108$$ −10.0000 −0.962250
$$109$$ 5.00000 0.478913 0.239457 0.970907i $$-0.423031\pi$$
0.239457 + 0.970907i $$0.423031\pi$$
$$110$$ 0 0
$$111$$ 2.00000 0.189832
$$112$$ 4.00000 0.377964
$$113$$ 6.00000 0.564433 0.282216 0.959351i $$-0.408930\pi$$
0.282216 + 0.959351i $$0.408930\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ −10.0000 −0.928477
$$117$$ −2.00000 −0.184900
$$118$$ 20.0000 1.84115
$$119$$ −7.00000 −0.641689
$$120$$ 0 0
$$121$$ −2.00000 −0.181818
$$122$$ −16.0000 −1.44857
$$123$$ 2.00000 0.180334
$$124$$ 4.00000 0.359211
$$125$$ 0 0
$$126$$ 4.00000 0.356348
$$127$$ 2.00000 0.177471 0.0887357 0.996055i $$-0.471717\pi$$
0.0887357 + 0.996055i $$0.471717\pi$$
$$128$$ 0 0
$$129$$ −4.00000 −0.352180
$$130$$ 0 0
$$131$$ 22.0000 1.92215 0.961074 0.276289i $$-0.0891049\pi$$
0.961074 + 0.276289i $$0.0891049\pi$$
$$132$$ −6.00000 −0.522233
$$133$$ 0 0
$$134$$ 4.00000 0.345547
$$135$$ 0 0
$$136$$ 0 0
$$137$$ 12.0000 1.02523 0.512615 0.858619i $$-0.328677\pi$$
0.512615 + 0.858619i $$0.328677\pi$$
$$138$$ 12.0000 1.02151
$$139$$ −10.0000 −0.848189 −0.424094 0.905618i $$-0.639408\pi$$
−0.424094 + 0.905618i $$0.639408\pi$$
$$140$$ 0 0
$$141$$ −3.00000 −0.252646
$$142$$ −16.0000 −1.34269
$$143$$ −3.00000 −0.250873
$$144$$ 8.00000 0.666667
$$145$$ 0 0
$$146$$ 12.0000 0.993127
$$147$$ 1.00000 0.0824786
$$148$$ 4.00000 0.328798
$$149$$ 10.0000 0.819232 0.409616 0.912258i $$-0.365663\pi$$
0.409616 + 0.912258i $$0.365663\pi$$
$$150$$ 0 0
$$151$$ −13.0000 −1.05792 −0.528962 0.848645i $$-0.677419\pi$$
−0.528962 + 0.848645i $$0.677419\pi$$
$$152$$ 0 0
$$153$$ −14.0000 −1.13183
$$154$$ 6.00000 0.483494
$$155$$ 0 0
$$156$$ 2.00000 0.160128
$$157$$ −18.0000 −1.43656 −0.718278 0.695756i $$-0.755069\pi$$
−0.718278 + 0.695756i $$0.755069\pi$$
$$158$$ −10.0000 −0.795557
$$159$$ 6.00000 0.475831
$$160$$ 0 0
$$161$$ −6.00000 −0.472866
$$162$$ 2.00000 0.157135
$$163$$ −14.0000 −1.09656 −0.548282 0.836293i $$-0.684718\pi$$
−0.548282 + 0.836293i $$0.684718\pi$$
$$164$$ 4.00000 0.312348
$$165$$ 0 0
$$166$$ −8.00000 −0.620920
$$167$$ −3.00000 −0.232147 −0.116073 0.993241i $$-0.537031\pi$$
−0.116073 + 0.993241i $$0.537031\pi$$
$$168$$ 0 0
$$169$$ −12.0000 −0.923077
$$170$$ 0 0
$$171$$ 0 0
$$172$$ −8.00000 −0.609994
$$173$$ −9.00000 −0.684257 −0.342129 0.939653i $$-0.611148\pi$$
−0.342129 + 0.939653i $$0.611148\pi$$
$$174$$ −10.0000 −0.758098
$$175$$ 0 0
$$176$$ 12.0000 0.904534
$$177$$ 10.0000 0.751646
$$178$$ 0 0
$$179$$ −20.0000 −1.49487 −0.747435 0.664335i $$-0.768715\pi$$
−0.747435 + 0.664335i $$0.768715\pi$$
$$180$$ 0 0
$$181$$ −18.0000 −1.33793 −0.668965 0.743294i $$-0.733262\pi$$
−0.668965 + 0.743294i $$0.733262\pi$$
$$182$$ −2.00000 −0.148250
$$183$$ −8.00000 −0.591377
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 4.00000 0.293294
$$187$$ −21.0000 −1.53567
$$188$$ −6.00000 −0.437595
$$189$$ 5.00000 0.363696
$$190$$ 0 0
$$191$$ −3.00000 −0.217072 −0.108536 0.994092i $$-0.534616\pi$$
−0.108536 + 0.994092i $$0.534616\pi$$
$$192$$ −8.00000 −0.577350
$$193$$ 16.0000 1.15171 0.575853 0.817554i $$-0.304670\pi$$
0.575853 + 0.817554i $$0.304670\pi$$
$$194$$ 14.0000 1.00514
$$195$$ 0 0
$$196$$ 2.00000 0.142857
$$197$$ 2.00000 0.142494 0.0712470 0.997459i $$-0.477302\pi$$
0.0712470 + 0.997459i $$0.477302\pi$$
$$198$$ 12.0000 0.852803
$$199$$ −10.0000 −0.708881 −0.354441 0.935079i $$-0.615329\pi$$
−0.354441 + 0.935079i $$0.615329\pi$$
$$200$$ 0 0
$$201$$ 2.00000 0.141069
$$202$$ 24.0000 1.68863
$$203$$ 5.00000 0.350931
$$204$$ 14.0000 0.980196
$$205$$ 0 0
$$206$$ −38.0000 −2.64759
$$207$$ −12.0000 −0.834058
$$208$$ −4.00000 −0.277350
$$209$$ 0 0
$$210$$ 0 0
$$211$$ −13.0000 −0.894957 −0.447478 0.894295i $$-0.647678\pi$$
−0.447478 + 0.894295i $$0.647678\pi$$
$$212$$ 12.0000 0.824163
$$213$$ −8.00000 −0.548151
$$214$$ −16.0000 −1.09374
$$215$$ 0 0
$$216$$ 0 0
$$217$$ −2.00000 −0.135769
$$218$$ 10.0000 0.677285
$$219$$ 6.00000 0.405442
$$220$$ 0 0
$$221$$ 7.00000 0.470871
$$222$$ 4.00000 0.268462
$$223$$ 21.0000 1.40626 0.703132 0.711059i $$-0.251784\pi$$
0.703132 + 0.711059i $$0.251784\pi$$
$$224$$ 8.00000 0.534522
$$225$$ 0 0
$$226$$ 12.0000 0.798228
$$227$$ 17.0000 1.12833 0.564165 0.825662i $$-0.309198\pi$$
0.564165 + 0.825662i $$0.309198\pi$$
$$228$$ 0 0
$$229$$ −10.0000 −0.660819 −0.330409 0.943838i $$-0.607187\pi$$
−0.330409 + 0.943838i $$0.607187\pi$$
$$230$$ 0 0
$$231$$ 3.00000 0.197386
$$232$$ 0 0
$$233$$ 16.0000 1.04819 0.524097 0.851658i $$-0.324403\pi$$
0.524097 + 0.851658i $$0.324403\pi$$
$$234$$ −4.00000 −0.261488
$$235$$ 0 0
$$236$$ 20.0000 1.30189
$$237$$ −5.00000 −0.324785
$$238$$ −14.0000 −0.907485
$$239$$ 15.0000 0.970269 0.485135 0.874439i $$-0.338771\pi$$
0.485135 + 0.874439i $$0.338771\pi$$
$$240$$ 0 0
$$241$$ 22.0000 1.41714 0.708572 0.705638i $$-0.249340\pi$$
0.708572 + 0.705638i $$0.249340\pi$$
$$242$$ −4.00000 −0.257130
$$243$$ 16.0000 1.02640
$$244$$ −16.0000 −1.02430
$$245$$ 0 0
$$246$$ 4.00000 0.255031
$$247$$ 0 0
$$248$$ 0 0
$$249$$ −4.00000 −0.253490
$$250$$ 0 0
$$251$$ −18.0000 −1.13615 −0.568075 0.822977i $$-0.692312\pi$$
−0.568075 + 0.822977i $$0.692312\pi$$
$$252$$ 4.00000 0.251976
$$253$$ −18.0000 −1.13165
$$254$$ 4.00000 0.250982
$$255$$ 0 0
$$256$$ 16.0000 1.00000
$$257$$ 22.0000 1.37232 0.686161 0.727450i $$-0.259294\pi$$
0.686161 + 0.727450i $$0.259294\pi$$
$$258$$ −8.00000 −0.498058
$$259$$ −2.00000 −0.124274
$$260$$ 0 0
$$261$$ 10.0000 0.618984
$$262$$ 44.0000 2.71833
$$263$$ −24.0000 −1.47990 −0.739952 0.672660i $$-0.765152\pi$$
−0.739952 + 0.672660i $$0.765152\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 4.00000 0.244339
$$269$$ 10.0000 0.609711 0.304855 0.952399i $$-0.401392\pi$$
0.304855 + 0.952399i $$0.401392\pi$$
$$270$$ 0 0
$$271$$ −8.00000 −0.485965 −0.242983 0.970031i $$-0.578126\pi$$
−0.242983 + 0.970031i $$0.578126\pi$$
$$272$$ −28.0000 −1.69775
$$273$$ −1.00000 −0.0605228
$$274$$ 24.0000 1.44989
$$275$$ 0 0
$$276$$ 12.0000 0.722315
$$277$$ 2.00000 0.120168 0.0600842 0.998193i $$-0.480863\pi$$
0.0600842 + 0.998193i $$0.480863\pi$$
$$278$$ −20.0000 −1.19952
$$279$$ −4.00000 −0.239474
$$280$$ 0 0
$$281$$ 7.00000 0.417585 0.208792 0.977960i $$-0.433047\pi$$
0.208792 + 0.977960i $$0.433047\pi$$
$$282$$ −6.00000 −0.357295
$$283$$ 11.0000 0.653882 0.326941 0.945045i $$-0.393982\pi$$
0.326941 + 0.945045i $$0.393982\pi$$
$$284$$ −16.0000 −0.949425
$$285$$ 0 0
$$286$$ −6.00000 −0.354787
$$287$$ −2.00000 −0.118056
$$288$$ 16.0000 0.942809
$$289$$ 32.0000 1.88235
$$290$$ 0 0
$$291$$ 7.00000 0.410347
$$292$$ 12.0000 0.702247
$$293$$ −9.00000 −0.525786 −0.262893 0.964825i $$-0.584677\pi$$
−0.262893 + 0.964825i $$0.584677\pi$$
$$294$$ 2.00000 0.116642
$$295$$ 0 0
$$296$$ 0 0
$$297$$ 15.0000 0.870388
$$298$$ 20.0000 1.15857
$$299$$ 6.00000 0.346989
$$300$$ 0 0
$$301$$ 4.00000 0.230556
$$302$$ −26.0000 −1.49613
$$303$$ 12.0000 0.689382
$$304$$ 0 0
$$305$$ 0 0
$$306$$ −28.0000 −1.60065
$$307$$ 7.00000 0.399511 0.199756 0.979846i $$-0.435985\pi$$
0.199756 + 0.979846i $$0.435985\pi$$
$$308$$ 6.00000 0.341882
$$309$$ −19.0000 −1.08087
$$310$$ 0 0
$$311$$ 12.0000 0.680458 0.340229 0.940343i $$-0.389495\pi$$
0.340229 + 0.940343i $$0.389495\pi$$
$$312$$ 0 0
$$313$$ 21.0000 1.18699 0.593495 0.804838i $$-0.297748\pi$$
0.593495 + 0.804838i $$0.297748\pi$$
$$314$$ −36.0000 −2.03160
$$315$$ 0 0
$$316$$ −10.0000 −0.562544
$$317$$ 12.0000 0.673987 0.336994 0.941507i $$-0.390590\pi$$
0.336994 + 0.941507i $$0.390590\pi$$
$$318$$ 12.0000 0.672927
$$319$$ 15.0000 0.839839
$$320$$ 0 0
$$321$$ −8.00000 −0.446516
$$322$$ −12.0000 −0.668734
$$323$$ 0 0
$$324$$ 2.00000 0.111111
$$325$$ 0 0
$$326$$ −28.0000 −1.55078
$$327$$ 5.00000 0.276501
$$328$$ 0 0
$$329$$ 3.00000 0.165395
$$330$$ 0 0
$$331$$ 12.0000 0.659580 0.329790 0.944054i $$-0.393022\pi$$
0.329790 + 0.944054i $$0.393022\pi$$
$$332$$ −8.00000 −0.439057
$$333$$ −4.00000 −0.219199
$$334$$ −6.00000 −0.328305
$$335$$ 0 0
$$336$$ 4.00000 0.218218
$$337$$ −18.0000 −0.980522 −0.490261 0.871576i $$-0.663099\pi$$
−0.490261 + 0.871576i $$0.663099\pi$$
$$338$$ −24.0000 −1.30543
$$339$$ 6.00000 0.325875
$$340$$ 0 0
$$341$$ −6.00000 −0.324918
$$342$$ 0 0
$$343$$ −1.00000 −0.0539949
$$344$$ 0 0
$$345$$ 0 0
$$346$$ −18.0000 −0.967686
$$347$$ −18.0000 −0.966291 −0.483145 0.875540i $$-0.660506\pi$$
−0.483145 + 0.875540i $$0.660506\pi$$
$$348$$ −10.0000 −0.536056
$$349$$ −20.0000 −1.07058 −0.535288 0.844670i $$-0.679797\pi$$
−0.535288 + 0.844670i $$0.679797\pi$$
$$350$$ 0 0
$$351$$ −5.00000 −0.266880
$$352$$ 24.0000 1.27920
$$353$$ 11.0000 0.585471 0.292735 0.956193i $$-0.405434\pi$$
0.292735 + 0.956193i $$0.405434\pi$$
$$354$$ 20.0000 1.06299
$$355$$ 0 0
$$356$$ 0 0
$$357$$ −7.00000 −0.370479
$$358$$ −40.0000 −2.11407
$$359$$ −20.0000 −1.05556 −0.527780 0.849381i $$-0.676975\pi$$
−0.527780 + 0.849381i $$0.676975\pi$$
$$360$$ 0 0
$$361$$ −19.0000 −1.00000
$$362$$ −36.0000 −1.89212
$$363$$ −2.00000 −0.104973
$$364$$ −2.00000 −0.104828
$$365$$ 0 0
$$366$$ −16.0000 −0.836333
$$367$$ −3.00000 −0.156599 −0.0782994 0.996930i $$-0.524949\pi$$
−0.0782994 + 0.996930i $$0.524949\pi$$
$$368$$ −24.0000 −1.25109
$$369$$ −4.00000 −0.208232
$$370$$ 0 0
$$371$$ −6.00000 −0.311504
$$372$$ 4.00000 0.207390
$$373$$ −24.0000 −1.24267 −0.621336 0.783544i $$-0.713410\pi$$
−0.621336 + 0.783544i $$0.713410\pi$$
$$374$$ −42.0000 −2.17177
$$375$$ 0 0
$$376$$ 0 0
$$377$$ −5.00000 −0.257513
$$378$$ 10.0000 0.514344
$$379$$ 20.0000 1.02733 0.513665 0.857991i $$-0.328287\pi$$
0.513665 + 0.857991i $$0.328287\pi$$
$$380$$ 0 0
$$381$$ 2.00000 0.102463
$$382$$ −6.00000 −0.306987
$$383$$ 16.0000 0.817562 0.408781 0.912633i $$-0.365954\pi$$
0.408781 + 0.912633i $$0.365954\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 32.0000 1.62876
$$387$$ 8.00000 0.406663
$$388$$ 14.0000 0.710742
$$389$$ 5.00000 0.253510 0.126755 0.991934i $$-0.459544\pi$$
0.126755 + 0.991934i $$0.459544\pi$$
$$390$$ 0 0
$$391$$ 42.0000 2.12403
$$392$$ 0 0
$$393$$ 22.0000 1.10975
$$394$$ 4.00000 0.201517
$$395$$ 0 0
$$396$$ 12.0000 0.603023
$$397$$ 7.00000 0.351320 0.175660 0.984451i $$-0.443794\pi$$
0.175660 + 0.984451i $$0.443794\pi$$
$$398$$ −20.0000 −1.00251
$$399$$ 0 0
$$400$$ 0 0
$$401$$ −3.00000 −0.149813 −0.0749064 0.997191i $$-0.523866\pi$$
−0.0749064 + 0.997191i $$0.523866\pi$$
$$402$$ 4.00000 0.199502
$$403$$ 2.00000 0.0996271
$$404$$ 24.0000 1.19404
$$405$$ 0 0
$$406$$ 10.0000 0.496292
$$407$$ −6.00000 −0.297409
$$408$$ 0 0
$$409$$ 20.0000 0.988936 0.494468 0.869196i $$-0.335363\pi$$
0.494468 + 0.869196i $$0.335363\pi$$
$$410$$ 0 0
$$411$$ 12.0000 0.591916
$$412$$ −38.0000 −1.87213
$$413$$ −10.0000 −0.492068
$$414$$ −24.0000 −1.17954
$$415$$ 0 0
$$416$$ −8.00000 −0.392232
$$417$$ −10.0000 −0.489702
$$418$$ 0 0
$$419$$ 30.0000 1.46560 0.732798 0.680446i $$-0.238214\pi$$
0.732798 + 0.680446i $$0.238214\pi$$
$$420$$ 0 0
$$421$$ −3.00000 −0.146211 −0.0731055 0.997324i $$-0.523291\pi$$
−0.0731055 + 0.997324i $$0.523291\pi$$
$$422$$ −26.0000 −1.26566
$$423$$ 6.00000 0.291730
$$424$$ 0 0
$$425$$ 0 0
$$426$$ −16.0000 −0.775203
$$427$$ 8.00000 0.387147
$$428$$ −16.0000 −0.773389
$$429$$ −3.00000 −0.144841
$$430$$ 0 0
$$431$$ −23.0000 −1.10787 −0.553936 0.832560i $$-0.686875\pi$$
−0.553936 + 0.832560i $$0.686875\pi$$
$$432$$ 20.0000 0.962250
$$433$$ 26.0000 1.24948 0.624740 0.780833i $$-0.285205\pi$$
0.624740 + 0.780833i $$0.285205\pi$$
$$434$$ −4.00000 −0.192006
$$435$$ 0 0
$$436$$ 10.0000 0.478913
$$437$$ 0 0
$$438$$ 12.0000 0.573382
$$439$$ 30.0000 1.43182 0.715911 0.698192i $$-0.246012\pi$$
0.715911 + 0.698192i $$0.246012\pi$$
$$440$$ 0 0
$$441$$ −2.00000 −0.0952381
$$442$$ 14.0000 0.665912
$$443$$ −4.00000 −0.190046 −0.0950229 0.995475i $$-0.530292\pi$$
−0.0950229 + 0.995475i $$0.530292\pi$$
$$444$$ 4.00000 0.189832
$$445$$ 0 0
$$446$$ 42.0000 1.98876
$$447$$ 10.0000 0.472984
$$448$$ 8.00000 0.377964
$$449$$ −5.00000 −0.235965 −0.117982 0.993016i $$-0.537643\pi$$
−0.117982 + 0.993016i $$0.537643\pi$$
$$450$$ 0 0
$$451$$ −6.00000 −0.282529
$$452$$ 12.0000 0.564433
$$453$$ −13.0000 −0.610793
$$454$$ 34.0000 1.59570
$$455$$ 0 0
$$456$$ 0 0
$$457$$ −38.0000 −1.77757 −0.888783 0.458329i $$-0.848448\pi$$
−0.888783 + 0.458329i $$0.848448\pi$$
$$458$$ −20.0000 −0.934539
$$459$$ −35.0000 −1.63366
$$460$$ 0 0
$$461$$ 12.0000 0.558896 0.279448 0.960161i $$-0.409849\pi$$
0.279448 + 0.960161i $$0.409849\pi$$
$$462$$ 6.00000 0.279145
$$463$$ 36.0000 1.67306 0.836531 0.547920i $$-0.184580\pi$$
0.836531 + 0.547920i $$0.184580\pi$$
$$464$$ 20.0000 0.928477
$$465$$ 0 0
$$466$$ 32.0000 1.48237
$$467$$ 27.0000 1.24941 0.624705 0.780860i $$-0.285219\pi$$
0.624705 + 0.780860i $$0.285219\pi$$
$$468$$ −4.00000 −0.184900
$$469$$ −2.00000 −0.0923514
$$470$$ 0 0
$$471$$ −18.0000 −0.829396
$$472$$ 0 0
$$473$$ 12.0000 0.551761
$$474$$ −10.0000 −0.459315
$$475$$ 0 0
$$476$$ −14.0000 −0.641689
$$477$$ −12.0000 −0.549442
$$478$$ 30.0000 1.37217
$$479$$ −30.0000 −1.37073 −0.685367 0.728197i $$-0.740358\pi$$
−0.685367 + 0.728197i $$0.740358\pi$$
$$480$$ 0 0
$$481$$ 2.00000 0.0911922
$$482$$ 44.0000 2.00415
$$483$$ −6.00000 −0.273009
$$484$$ −4.00000 −0.181818
$$485$$ 0 0
$$486$$ 32.0000 1.45155
$$487$$ 42.0000 1.90320 0.951601 0.307337i $$-0.0994378\pi$$
0.951601 + 0.307337i $$0.0994378\pi$$
$$488$$ 0 0
$$489$$ −14.0000 −0.633102
$$490$$ 0 0
$$491$$ 7.00000 0.315906 0.157953 0.987447i $$-0.449511\pi$$
0.157953 + 0.987447i $$0.449511\pi$$
$$492$$ 4.00000 0.180334
$$493$$ −35.0000 −1.57632
$$494$$ 0 0
$$495$$ 0 0
$$496$$ −8.00000 −0.359211
$$497$$ 8.00000 0.358849
$$498$$ −8.00000 −0.358489
$$499$$ −35.0000 −1.56682 −0.783408 0.621508i $$-0.786520\pi$$
−0.783408 + 0.621508i $$0.786520\pi$$
$$500$$ 0 0
$$501$$ −3.00000 −0.134030
$$502$$ −36.0000 −1.60676
$$503$$ −9.00000 −0.401290 −0.200645 0.979664i $$-0.564304\pi$$
−0.200645 + 0.979664i $$0.564304\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ −36.0000 −1.60040
$$507$$ −12.0000 −0.532939
$$508$$ 4.00000 0.177471
$$509$$ −30.0000 −1.32973 −0.664863 0.746965i $$-0.731510\pi$$
−0.664863 + 0.746965i $$0.731510\pi$$
$$510$$ 0 0
$$511$$ −6.00000 −0.265424
$$512$$ 32.0000 1.41421
$$513$$ 0 0
$$514$$ 44.0000 1.94076
$$515$$ 0 0
$$516$$ −8.00000 −0.352180
$$517$$ 9.00000 0.395820
$$518$$ −4.00000 −0.175750
$$519$$ −9.00000 −0.395056
$$520$$ 0 0
$$521$$ −28.0000 −1.22670 −0.613351 0.789810i $$-0.710179\pi$$
−0.613351 + 0.789810i $$0.710179\pi$$
$$522$$ 20.0000 0.875376
$$523$$ −4.00000 −0.174908 −0.0874539 0.996169i $$-0.527873\pi$$
−0.0874539 + 0.996169i $$0.527873\pi$$
$$524$$ 44.0000 1.92215
$$525$$ 0 0
$$526$$ −48.0000 −2.09290
$$527$$ 14.0000 0.609850
$$528$$ 12.0000 0.522233
$$529$$ 13.0000 0.565217
$$530$$ 0 0
$$531$$ −20.0000 −0.867926
$$532$$ 0 0
$$533$$ 2.00000 0.0866296
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 0 0
$$537$$ −20.0000 −0.863064
$$538$$ 20.0000 0.862261
$$539$$ −3.00000 −0.129219
$$540$$ 0 0
$$541$$ 27.0000 1.16082 0.580410 0.814324i $$-0.302892\pi$$
0.580410 + 0.814324i $$0.302892\pi$$
$$542$$ −16.0000 −0.687259
$$543$$ −18.0000 −0.772454
$$544$$ −56.0000 −2.40098
$$545$$ 0 0
$$546$$ −2.00000 −0.0855921
$$547$$ 32.0000 1.36822 0.684111 0.729378i $$-0.260191\pi$$
0.684111 + 0.729378i $$0.260191\pi$$
$$548$$ 24.0000 1.02523
$$549$$ 16.0000 0.682863
$$550$$ 0 0
$$551$$ 0 0
$$552$$ 0 0
$$553$$ 5.00000 0.212622
$$554$$ 4.00000 0.169944
$$555$$ 0 0
$$556$$ −20.0000 −0.848189
$$557$$ −28.0000 −1.18640 −0.593199 0.805056i $$-0.702135\pi$$
−0.593199 + 0.805056i $$0.702135\pi$$
$$558$$ −8.00000 −0.338667
$$559$$ −4.00000 −0.169182
$$560$$ 0 0
$$561$$ −21.0000 −0.886621
$$562$$ 14.0000 0.590554
$$563$$ −24.0000 −1.01148 −0.505740 0.862686i $$-0.668780\pi$$
−0.505740 + 0.862686i $$0.668780\pi$$
$$564$$ −6.00000 −0.252646
$$565$$ 0 0
$$566$$ 22.0000 0.924729
$$567$$ −1.00000 −0.0419961
$$568$$ 0 0
$$569$$ −30.0000 −1.25767 −0.628833 0.777541i $$-0.716467\pi$$
−0.628833 + 0.777541i $$0.716467\pi$$
$$570$$ 0 0
$$571$$ 12.0000 0.502184 0.251092 0.967963i $$-0.419210\pi$$
0.251092 + 0.967963i $$0.419210\pi$$
$$572$$ −6.00000 −0.250873
$$573$$ −3.00000 −0.125327
$$574$$ −4.00000 −0.166957
$$575$$ 0 0
$$576$$ 16.0000 0.666667
$$577$$ −43.0000 −1.79011 −0.895057 0.445952i $$-0.852865\pi$$
−0.895057 + 0.445952i $$0.852865\pi$$
$$578$$ 64.0000 2.66205
$$579$$ 16.0000 0.664937
$$580$$ 0 0
$$581$$ 4.00000 0.165948
$$582$$ 14.0000 0.580319
$$583$$ −18.0000 −0.745484
$$584$$ 0 0
$$585$$ 0 0
$$586$$ −18.0000 −0.743573
$$587$$ 12.0000 0.495293 0.247647 0.968850i $$-0.420343\pi$$
0.247647 + 0.968850i $$0.420343\pi$$
$$588$$ 2.00000 0.0824786
$$589$$ 0 0
$$590$$ 0 0
$$591$$ 2.00000 0.0822690
$$592$$ −8.00000 −0.328798
$$593$$ 41.0000 1.68367 0.841834 0.539736i $$-0.181476\pi$$
0.841834 + 0.539736i $$0.181476\pi$$
$$594$$ 30.0000 1.23091
$$595$$ 0 0
$$596$$ 20.0000 0.819232
$$597$$ −10.0000 −0.409273
$$598$$ 12.0000 0.490716
$$599$$ 25.0000 1.02147 0.510736 0.859738i $$-0.329373\pi$$
0.510736 + 0.859738i $$0.329373\pi$$
$$600$$ 0 0
$$601$$ −8.00000 −0.326327 −0.163163 0.986599i $$-0.552170\pi$$
−0.163163 + 0.986599i $$0.552170\pi$$
$$602$$ 8.00000 0.326056
$$603$$ −4.00000 −0.162893
$$604$$ −26.0000 −1.05792
$$605$$ 0 0
$$606$$ 24.0000 0.974933
$$607$$ 27.0000 1.09590 0.547948 0.836512i $$-0.315409\pi$$
0.547948 + 0.836512i $$0.315409\pi$$
$$608$$ 0 0
$$609$$ 5.00000 0.202610
$$610$$ 0 0
$$611$$ −3.00000 −0.121367
$$612$$ −28.0000 −1.13183
$$613$$ −44.0000 −1.77714 −0.888572 0.458738i $$-0.848302\pi$$
−0.888572 + 0.458738i $$0.848302\pi$$
$$614$$ 14.0000 0.564994
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 22.0000 0.885687 0.442843 0.896599i $$-0.353970\pi$$
0.442843 + 0.896599i $$0.353970\pi$$
$$618$$ −38.0000 −1.52858
$$619$$ 10.0000 0.401934 0.200967 0.979598i $$-0.435592\pi$$
0.200967 + 0.979598i $$0.435592\pi$$
$$620$$ 0 0
$$621$$ −30.0000 −1.20386
$$622$$ 24.0000 0.962312
$$623$$ 0 0
$$624$$ −4.00000 −0.160128
$$625$$ 0 0
$$626$$ 42.0000 1.67866
$$627$$ 0 0
$$628$$ −36.0000 −1.43656
$$629$$ 14.0000 0.558217
$$630$$ 0 0
$$631$$ 37.0000 1.47295 0.736473 0.676467i $$-0.236490\pi$$
0.736473 + 0.676467i $$0.236490\pi$$
$$632$$ 0 0
$$633$$ −13.0000 −0.516704
$$634$$ 24.0000 0.953162
$$635$$ 0 0
$$636$$ 12.0000 0.475831
$$637$$ 1.00000 0.0396214
$$638$$ 30.0000 1.18771
$$639$$ 16.0000 0.632950
$$640$$ 0 0
$$641$$ 22.0000 0.868948 0.434474 0.900684i $$-0.356934\pi$$
0.434474 + 0.900684i $$0.356934\pi$$
$$642$$ −16.0000 −0.631470
$$643$$ 1.00000 0.0394362 0.0197181 0.999806i $$-0.493723\pi$$
0.0197181 + 0.999806i $$0.493723\pi$$
$$644$$ −12.0000 −0.472866
$$645$$ 0 0
$$646$$ 0 0
$$647$$ −8.00000 −0.314512 −0.157256 0.987558i $$-0.550265\pi$$
−0.157256 + 0.987558i $$0.550265\pi$$
$$648$$ 0 0
$$649$$ −30.0000 −1.17760
$$650$$ 0 0
$$651$$ −2.00000 −0.0783862
$$652$$ −28.0000 −1.09656
$$653$$ −4.00000 −0.156532 −0.0782660 0.996933i $$-0.524938\pi$$
−0.0782660 + 0.996933i $$0.524938\pi$$
$$654$$ 10.0000 0.391031
$$655$$ 0 0
$$656$$ −8.00000 −0.312348
$$657$$ −12.0000 −0.468165
$$658$$ 6.00000 0.233904
$$659$$ −15.0000 −0.584317 −0.292159 0.956370i $$-0.594373\pi$$
−0.292159 + 0.956370i $$0.594373\pi$$
$$660$$ 0 0
$$661$$ 12.0000 0.466746 0.233373 0.972387i $$-0.425024\pi$$
0.233373 + 0.972387i $$0.425024\pi$$
$$662$$ 24.0000 0.932786
$$663$$ 7.00000 0.271857
$$664$$ 0 0
$$665$$ 0 0
$$666$$ −8.00000 −0.309994
$$667$$ −30.0000 −1.16160
$$668$$ −6.00000 −0.232147
$$669$$ 21.0000 0.811907
$$670$$ 0 0
$$671$$ 24.0000 0.926510
$$672$$ 8.00000 0.308607
$$673$$ −24.0000 −0.925132 −0.462566 0.886585i $$-0.653071\pi$$
−0.462566 + 0.886585i $$0.653071\pi$$
$$674$$ −36.0000 −1.38667
$$675$$ 0 0
$$676$$ −24.0000 −0.923077
$$677$$ −43.0000 −1.65262 −0.826312 0.563212i $$-0.809565\pi$$
−0.826312 + 0.563212i $$0.809565\pi$$
$$678$$ 12.0000 0.460857
$$679$$ −7.00000 −0.268635
$$680$$ 0 0
$$681$$ 17.0000 0.651441
$$682$$ −12.0000 −0.459504
$$683$$ 16.0000 0.612223 0.306111 0.951996i $$-0.400972\pi$$
0.306111 + 0.951996i $$0.400972\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ −2.00000 −0.0763604
$$687$$ −10.0000 −0.381524
$$688$$ 16.0000 0.609994
$$689$$ 6.00000 0.228582
$$690$$ 0 0
$$691$$ −8.00000 −0.304334 −0.152167 0.988355i $$-0.548625\pi$$
−0.152167 + 0.988355i $$0.548625\pi$$
$$692$$ −18.0000 −0.684257
$$693$$ −6.00000 −0.227921
$$694$$ −36.0000 −1.36654
$$695$$ 0 0
$$696$$ 0 0
$$697$$ 14.0000 0.530288
$$698$$ −40.0000 −1.51402
$$699$$ 16.0000 0.605176
$$700$$ 0 0
$$701$$ −13.0000 −0.491003 −0.245502 0.969396i $$-0.578953\pi$$
−0.245502 + 0.969396i $$0.578953\pi$$
$$702$$ −10.0000 −0.377426
$$703$$ 0 0
$$704$$ 24.0000 0.904534
$$705$$ 0 0
$$706$$ 22.0000 0.827981
$$707$$ −12.0000 −0.451306
$$708$$ 20.0000 0.751646
$$709$$ 35.0000 1.31445 0.657226 0.753693i $$-0.271730\pi$$
0.657226 + 0.753693i $$0.271730\pi$$
$$710$$ 0 0
$$711$$ 10.0000 0.375029
$$712$$ 0 0
$$713$$ 12.0000 0.449404
$$714$$ −14.0000 −0.523937
$$715$$ 0 0
$$716$$ −40.0000 −1.49487
$$717$$ 15.0000 0.560185
$$718$$ −40.0000 −1.49279
$$719$$ 30.0000 1.11881 0.559406 0.828894i $$-0.311029\pi$$
0.559406 + 0.828894i $$0.311029\pi$$
$$720$$ 0 0
$$721$$ 19.0000 0.707597
$$722$$ −38.0000 −1.41421
$$723$$ 22.0000 0.818189
$$724$$ −36.0000 −1.33793
$$725$$ 0 0
$$726$$ −4.00000 −0.148454
$$727$$ −28.0000 −1.03846 −0.519231 0.854634i $$-0.673782\pi$$
−0.519231 + 0.854634i $$0.673782\pi$$
$$728$$ 0 0
$$729$$ 13.0000 0.481481
$$730$$ 0 0
$$731$$ −28.0000 −1.03562
$$732$$ −16.0000 −0.591377
$$733$$ 1.00000 0.0369358 0.0184679 0.999829i $$-0.494121\pi$$
0.0184679 + 0.999829i $$0.494121\pi$$
$$734$$ −6.00000 −0.221464
$$735$$ 0 0
$$736$$ −48.0000 −1.76930
$$737$$ −6.00000 −0.221013
$$738$$ −8.00000 −0.294484
$$739$$ −35.0000 −1.28750 −0.643748 0.765238i $$-0.722621\pi$$
−0.643748 + 0.765238i $$0.722621\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ −12.0000 −0.440534
$$743$$ −14.0000 −0.513610 −0.256805 0.966463i $$-0.582670\pi$$
−0.256805 + 0.966463i $$0.582670\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ −48.0000 −1.75740
$$747$$ 8.00000 0.292705
$$748$$ −42.0000 −1.53567
$$749$$ 8.00000 0.292314
$$750$$ 0 0
$$751$$ −33.0000 −1.20419 −0.602094 0.798426i $$-0.705667\pi$$
−0.602094 + 0.798426i $$0.705667\pi$$
$$752$$ 12.0000 0.437595
$$753$$ −18.0000 −0.655956
$$754$$ −10.0000 −0.364179
$$755$$ 0 0
$$756$$ 10.0000 0.363696
$$757$$ 32.0000 1.16306 0.581530 0.813525i $$-0.302454\pi$$
0.581530 + 0.813525i $$0.302454\pi$$
$$758$$ 40.0000 1.45287
$$759$$ −18.0000 −0.653359
$$760$$ 0 0
$$761$$ 2.00000 0.0724999 0.0362500 0.999343i $$-0.488459\pi$$
0.0362500 + 0.999343i $$0.488459\pi$$
$$762$$ 4.00000 0.144905
$$763$$ −5.00000 −0.181012
$$764$$ −6.00000 −0.217072
$$765$$ 0 0
$$766$$ 32.0000 1.15621
$$767$$ 10.0000 0.361079
$$768$$ 16.0000 0.577350
$$769$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$770$$ 0 0
$$771$$ 22.0000 0.792311
$$772$$ 32.0000 1.15171
$$773$$ 21.0000 0.755318 0.377659 0.925945i $$-0.376729\pi$$
0.377659 + 0.925945i $$0.376729\pi$$
$$774$$ 16.0000 0.575108
$$775$$ 0 0
$$776$$ 0 0
$$777$$ −2.00000 −0.0717496
$$778$$ 10.0000 0.358517
$$779$$ 0 0
$$780$$ 0 0
$$781$$ 24.0000 0.858788
$$782$$ 84.0000 3.00383
$$783$$ 25.0000 0.893427
$$784$$ −4.00000 −0.142857
$$785$$ 0 0
$$786$$ 44.0000 1.56943
$$787$$ 17.0000 0.605985 0.302992 0.952993i $$-0.402014\pi$$
0.302992 + 0.952993i $$0.402014\pi$$
$$788$$ 4.00000 0.142494
$$789$$ −24.0000 −0.854423
$$790$$ 0 0
$$791$$ −6.00000 −0.213335
$$792$$ 0 0
$$793$$ −8.00000 −0.284088
$$794$$ 14.0000 0.496841
$$795$$ 0 0
$$796$$ −20.0000 −0.708881
$$797$$ −13.0000 −0.460484 −0.230242 0.973133i $$-0.573952\pi$$
−0.230242 + 0.973133i $$0.573952\pi$$
$$798$$ 0 0
$$799$$ −21.0000 −0.742927
$$800$$ 0 0
$$801$$ 0 0
$$802$$ −6.00000 −0.211867
$$803$$ −18.0000 −0.635206
$$804$$ 4.00000 0.141069
$$805$$ 0 0
$$806$$ 4.00000 0.140894
$$807$$ 10.0000 0.352017
$$808$$ 0 0
$$809$$ 45.0000 1.58212 0.791058 0.611741i $$-0.209531\pi$$
0.791058 + 0.611741i $$0.209531\pi$$
$$810$$ 0 0
$$811$$ −38.0000 −1.33436 −0.667180 0.744896i $$-0.732499\pi$$
−0.667180 + 0.744896i $$0.732499\pi$$
$$812$$ 10.0000 0.350931
$$813$$ −8.00000 −0.280572
$$814$$ −12.0000 −0.420600
$$815$$ 0 0
$$816$$ −28.0000 −0.980196
$$817$$ 0 0
$$818$$ 40.0000 1.39857
$$819$$ 2.00000 0.0698857
$$820$$ 0 0
$$821$$ −23.0000 −0.802706 −0.401353 0.915924i $$-0.631460\pi$$
−0.401353 + 0.915924i $$0.631460\pi$$
$$822$$ 24.0000 0.837096
$$823$$ 26.0000 0.906303 0.453152 0.891434i $$-0.350300\pi$$
0.453152 + 0.891434i $$0.350300\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ −20.0000 −0.695889
$$827$$ −18.0000 −0.625921 −0.312961 0.949766i $$-0.601321\pi$$
−0.312961 + 0.949766i $$0.601321\pi$$
$$828$$ −24.0000 −0.834058
$$829$$ −30.0000 −1.04194 −0.520972 0.853574i $$-0.674430\pi$$
−0.520972 + 0.853574i $$0.674430\pi$$
$$830$$ 0 0
$$831$$ 2.00000 0.0693792
$$832$$ −8.00000 −0.277350
$$833$$ 7.00000 0.242536
$$834$$ −20.0000 −0.692543
$$835$$ 0 0
$$836$$ 0 0
$$837$$ −10.0000 −0.345651
$$838$$ 60.0000 2.07267
$$839$$ 30.0000 1.03572 0.517858 0.855467i $$-0.326730\pi$$
0.517858 + 0.855467i $$0.326730\pi$$
$$840$$ 0 0
$$841$$ −4.00000 −0.137931
$$842$$ −6.00000 −0.206774
$$843$$ 7.00000 0.241093
$$844$$ −26.0000 −0.894957
$$845$$ 0 0
$$846$$ 12.0000 0.412568
$$847$$ 2.00000 0.0687208
$$848$$ −24.0000 −0.824163
$$849$$ 11.0000 0.377519
$$850$$ 0 0
$$851$$ 12.0000 0.411355
$$852$$ −16.0000 −0.548151
$$853$$ −54.0000 −1.84892 −0.924462 0.381273i $$-0.875486\pi$$
−0.924462 + 0.381273i $$0.875486\pi$$
$$854$$ 16.0000 0.547509
$$855$$ 0 0
$$856$$ 0 0
$$857$$ 42.0000 1.43469 0.717346 0.696717i $$-0.245357\pi$$
0.717346 + 0.696717i $$0.245357\pi$$
$$858$$ −6.00000 −0.204837
$$859$$ 40.0000 1.36478 0.682391 0.730987i $$-0.260940\pi$$
0.682391 + 0.730987i $$0.260940\pi$$
$$860$$ 0 0
$$861$$ −2.00000 −0.0681598
$$862$$ −46.0000 −1.56677
$$863$$ −54.0000 −1.83818 −0.919091 0.394046i $$-0.871075\pi$$
−0.919091 + 0.394046i $$0.871075\pi$$
$$864$$ 40.0000 1.36083
$$865$$ 0 0
$$866$$ 52.0000 1.76703
$$867$$ 32.0000 1.08678
$$868$$ −4.00000 −0.135769
$$869$$ 15.0000 0.508840
$$870$$ 0 0
$$871$$ 2.00000 0.0677674
$$872$$ 0 0
$$873$$ −14.0000 −0.473828
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 12.0000 0.405442
$$877$$ 32.0000 1.08056 0.540282 0.841484i $$-0.318318\pi$$
0.540282 + 0.841484i $$0.318318\pi$$
$$878$$ 60.0000 2.02490
$$879$$ −9.00000 −0.303562
$$880$$ 0 0
$$881$$ 32.0000 1.07811 0.539054 0.842271i $$-0.318782\pi$$
0.539054 + 0.842271i $$0.318782\pi$$
$$882$$ −4.00000 −0.134687
$$883$$ 36.0000 1.21150 0.605748 0.795656i $$-0.292874\pi$$
0.605748 + 0.795656i $$0.292874\pi$$
$$884$$ 14.0000 0.470871
$$885$$ 0 0
$$886$$ −8.00000 −0.268765
$$887$$ −28.0000 −0.940148 −0.470074 0.882627i $$-0.655773\pi$$
−0.470074 + 0.882627i $$0.655773\pi$$
$$888$$ 0 0
$$889$$ −2.00000 −0.0670778
$$890$$ 0 0
$$891$$ −3.00000 −0.100504
$$892$$ 42.0000 1.40626
$$893$$ 0 0
$$894$$ 20.0000 0.668900
$$895$$ 0 0
$$896$$ 0 0
$$897$$ 6.00000 0.200334
$$898$$ −10.0000 −0.333704
$$899$$ −10.0000 −0.333519
$$900$$ 0 0
$$901$$ 42.0000 1.39922
$$902$$ −12.0000 −0.399556
$$903$$ 4.00000 0.133112
$$904$$ 0 0
$$905$$ 0 0
$$906$$ −26.0000 −0.863792
$$907$$ −38.0000 −1.26177 −0.630885 0.775877i $$-0.717308\pi$$
−0.630885 + 0.775877i $$0.717308\pi$$
$$908$$ 34.0000 1.12833
$$909$$ −24.0000 −0.796030
$$910$$ 0 0
$$911$$ 12.0000 0.397578 0.198789 0.980042i $$-0.436299\pi$$
0.198789 + 0.980042i $$0.436299\pi$$
$$912$$ 0 0
$$913$$ 12.0000 0.397142
$$914$$ −76.0000 −2.51386
$$915$$ 0 0
$$916$$ −20.0000 −0.660819
$$917$$ −22.0000 −0.726504
$$918$$ −70.0000 −2.31034
$$919$$ −5.00000 −0.164935 −0.0824674 0.996594i $$-0.526280\pi$$
−0.0824674 + 0.996594i $$0.526280\pi$$
$$920$$ 0 0
$$921$$ 7.00000 0.230658
$$922$$ 24.0000 0.790398
$$923$$ −8.00000 −0.263323
$$924$$ 6.00000 0.197386
$$925$$ 0 0
$$926$$ 72.0000 2.36607
$$927$$ 38.0000 1.24808
$$928$$ 40.0000 1.31306
$$929$$ −50.0000 −1.64045 −0.820223 0.572043i $$-0.806151\pi$$
−0.820223 + 0.572043i $$0.806151\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ 32.0000 1.04819
$$933$$ 12.0000 0.392862
$$934$$ 54.0000 1.76693
$$935$$ 0 0
$$936$$ 0 0
$$937$$ −13.0000 −0.424691 −0.212346 0.977195i $$-0.568110\pi$$
−0.212346 + 0.977195i $$0.568110\pi$$
$$938$$ −4.00000 −0.130605
$$939$$ 21.0000 0.685309
$$940$$ 0 0
$$941$$ −48.0000 −1.56476 −0.782378 0.622804i $$-0.785993\pi$$
−0.782378 + 0.622804i $$0.785993\pi$$
$$942$$ −36.0000 −1.17294
$$943$$ 12.0000 0.390774
$$944$$ −40.0000 −1.30189
$$945$$ 0 0
$$946$$ 24.0000 0.780307
$$947$$ 42.0000 1.36482 0.682408 0.730971i $$-0.260933\pi$$
0.682408 + 0.730971i $$0.260933\pi$$
$$948$$ −10.0000 −0.324785
$$949$$ 6.00000 0.194768
$$950$$ 0 0
$$951$$ 12.0000 0.389127
$$952$$ 0 0
$$953$$ 6.00000 0.194359 0.0971795 0.995267i $$-0.469018\pi$$
0.0971795 + 0.995267i $$0.469018\pi$$
$$954$$ −24.0000 −0.777029
$$955$$ 0 0
$$956$$ 30.0000 0.970269
$$957$$ 15.0000 0.484881
$$958$$ −60.0000 −1.93851
$$959$$ −12.0000 −0.387500
$$960$$ 0 0
$$961$$ −27.0000 −0.870968
$$962$$ 4.00000 0.128965
$$963$$ 16.0000 0.515593
$$964$$ 44.0000 1.41714
$$965$$ 0 0
$$966$$ −12.0000 −0.386094
$$967$$ −8.00000 −0.257263 −0.128631 0.991692i $$-0.541058\pi$$
−0.128631 + 0.991692i $$0.541058\pi$$
$$968$$ 0 0
$$969$$ 0 0
$$970$$ 0 0
$$971$$ 22.0000 0.706014 0.353007 0.935621i $$-0.385159\pi$$
0.353007 + 0.935621i $$0.385159\pi$$
$$972$$ 32.0000 1.02640
$$973$$ 10.0000 0.320585
$$974$$ 84.0000 2.69153
$$975$$ 0 0
$$976$$ 32.0000 1.02430
$$977$$ 12.0000 0.383914 0.191957 0.981403i $$-0.438517\pi$$
0.191957 + 0.981403i $$0.438517\pi$$
$$978$$ −28.0000 −0.895341
$$979$$ 0 0
$$980$$ 0 0
$$981$$ −10.0000 −0.319275
$$982$$ 14.0000 0.446758
$$983$$ 51.0000 1.62665 0.813324 0.581811i $$-0.197656\pi$$
0.813324 + 0.581811i $$0.197656\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ −70.0000 −2.22925
$$987$$ 3.00000 0.0954911
$$988$$ 0 0
$$989$$ −24.0000 −0.763156
$$990$$ 0 0
$$991$$ −8.00000 −0.254128 −0.127064 0.991894i $$-0.540555\pi$$
−0.127064 + 0.991894i $$0.540555\pi$$
$$992$$ −16.0000 −0.508001
$$993$$ 12.0000 0.380808
$$994$$ 16.0000 0.507489
$$995$$ 0 0
$$996$$ −8.00000 −0.253490
$$997$$ −13.0000 −0.411714 −0.205857 0.978582i $$-0.565998\pi$$
−0.205857 + 0.978582i $$0.565998\pi$$
$$998$$ −70.0000 −2.21581
$$999$$ −10.0000 −0.316386
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 175.2.a.c.1.1 1
3.2 odd 2 1575.2.a.a.1.1 1
4.3 odd 2 2800.2.a.l.1.1 1
5.2 odd 4 35.2.b.a.29.2 yes 2
5.3 odd 4 35.2.b.a.29.1 2
5.4 even 2 175.2.a.a.1.1 1
7.6 odd 2 1225.2.a.i.1.1 1
15.2 even 4 315.2.d.a.64.1 2
15.8 even 4 315.2.d.a.64.2 2
15.14 odd 2 1575.2.a.k.1.1 1
20.3 even 4 560.2.g.b.449.1 2
20.7 even 4 560.2.g.b.449.2 2
20.19 odd 2 2800.2.a.w.1.1 1
35.2 odd 12 245.2.j.e.214.2 4
35.3 even 12 245.2.j.d.79.2 4
35.12 even 12 245.2.j.d.214.2 4
35.13 even 4 245.2.b.a.99.1 2
35.17 even 12 245.2.j.d.79.1 4
35.18 odd 12 245.2.j.e.79.2 4
35.23 odd 12 245.2.j.e.214.1 4
35.27 even 4 245.2.b.a.99.2 2
35.32 odd 12 245.2.j.e.79.1 4
35.33 even 12 245.2.j.d.214.1 4
35.34 odd 2 1225.2.a.a.1.1 1
40.3 even 4 2240.2.g.g.449.2 2
40.13 odd 4 2240.2.g.h.449.1 2
40.27 even 4 2240.2.g.g.449.1 2
40.37 odd 4 2240.2.g.h.449.2 2
60.23 odd 4 5040.2.t.p.1009.1 2
60.47 odd 4 5040.2.t.p.1009.2 2
105.62 odd 4 2205.2.d.b.1324.1 2
105.83 odd 4 2205.2.d.b.1324.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
35.2.b.a.29.1 2 5.3 odd 4
35.2.b.a.29.2 yes 2 5.2 odd 4
175.2.a.a.1.1 1 5.4 even 2
175.2.a.c.1.1 1 1.1 even 1 trivial
245.2.b.a.99.1 2 35.13 even 4
245.2.b.a.99.2 2 35.27 even 4
245.2.j.d.79.1 4 35.17 even 12
245.2.j.d.79.2 4 35.3 even 12
245.2.j.d.214.1 4 35.33 even 12
245.2.j.d.214.2 4 35.12 even 12
245.2.j.e.79.1 4 35.32 odd 12
245.2.j.e.79.2 4 35.18 odd 12
245.2.j.e.214.1 4 35.23 odd 12
245.2.j.e.214.2 4 35.2 odd 12
315.2.d.a.64.1 2 15.2 even 4
315.2.d.a.64.2 2 15.8 even 4
560.2.g.b.449.1 2 20.3 even 4
560.2.g.b.449.2 2 20.7 even 4
1225.2.a.a.1.1 1 35.34 odd 2
1225.2.a.i.1.1 1 7.6 odd 2
1575.2.a.a.1.1 1 3.2 odd 2
1575.2.a.k.1.1 1 15.14 odd 2
2205.2.d.b.1324.1 2 105.62 odd 4
2205.2.d.b.1324.2 2 105.83 odd 4
2240.2.g.g.449.1 2 40.27 even 4
2240.2.g.g.449.2 2 40.3 even 4
2240.2.g.h.449.1 2 40.13 odd 4
2240.2.g.h.449.2 2 40.37 odd 4
2800.2.a.l.1.1 1 4.3 odd 2
2800.2.a.w.1.1 1 20.19 odd 2
5040.2.t.p.1009.1 2 60.23 odd 4
5040.2.t.p.1009.2 2 60.47 odd 4