# Properties

 Label 1960.2.a.o.1.1 Level $1960$ Weight $2$ Character 1960.1 Self dual yes Analytic conductor $15.651$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Learn more

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1960,2,Mod(1,1960)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1960, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1960.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1960 = 2^{3} \cdot 5 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1960.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: $$15.6506787962$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 280) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 1960.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+3.00000 q^{3} -1.00000 q^{5} +6.00000 q^{9} +O(q^{10})$$ $$q+3.00000 q^{3} -1.00000 q^{5} +6.00000 q^{9} -5.00000 q^{11} +5.00000 q^{13} -3.00000 q^{15} +7.00000 q^{17} +2.00000 q^{19} -2.00000 q^{23} +1.00000 q^{25} +9.00000 q^{27} +7.00000 q^{29} -4.00000 q^{31} -15.0000 q^{33} -6.00000 q^{37} +15.0000 q^{39} +12.0000 q^{41} -2.00000 q^{43} -6.00000 q^{45} -1.00000 q^{47} +21.0000 q^{51} +5.00000 q^{55} +6.00000 q^{57} +4.00000 q^{59} -4.00000 q^{61} -5.00000 q^{65} +8.00000 q^{67} -6.00000 q^{69} -6.00000 q^{73} +3.00000 q^{75} -3.00000 q^{79} +9.00000 q^{81} +4.00000 q^{83} -7.00000 q^{85} +21.0000 q^{87} -12.0000 q^{93} -2.00000 q^{95} -13.0000 q^{97} -30.0000 q^{99} +O(q^{100})$$

## Coefficient data

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

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0 0
$$3$$ 3.00000 1.73205 0.866025 0.500000i $$-0.166667\pi$$
0.866025 + 0.500000i $$0.166667\pi$$
$$4$$ 0 0
$$5$$ −1.00000 −0.447214
$$6$$ 0 0
$$7$$ 0 0
$$8$$ 0 0
$$9$$ 6.00000 2.00000
$$10$$ 0 0
$$11$$ −5.00000 −1.50756 −0.753778 0.657129i $$-0.771771\pi$$
−0.753778 + 0.657129i $$0.771771\pi$$
$$12$$ 0 0
$$13$$ 5.00000 1.38675 0.693375 0.720577i $$-0.256123\pi$$
0.693375 + 0.720577i $$0.256123\pi$$
$$14$$ 0 0
$$15$$ −3.00000 −0.774597
$$16$$ 0 0
$$17$$ 7.00000 1.69775 0.848875 0.528594i $$-0.177281\pi$$
0.848875 + 0.528594i $$0.177281\pi$$
$$18$$ 0 0
$$19$$ 2.00000 0.458831 0.229416 0.973329i $$-0.426318\pi$$
0.229416 + 0.973329i $$0.426318\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ −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
$$27$$ 9.00000 1.73205
$$28$$ 0 0
$$29$$ 7.00000 1.29987 0.649934 0.759991i $$-0.274797\pi$$
0.649934 + 0.759991i $$0.274797\pi$$
$$30$$ 0 0
$$31$$ −4.00000 −0.718421 −0.359211 0.933257i $$-0.616954\pi$$
−0.359211 + 0.933257i $$0.616954\pi$$
$$32$$ 0 0
$$33$$ −15.0000 −2.61116
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ −6.00000 −0.986394 −0.493197 0.869918i $$-0.664172\pi$$
−0.493197 + 0.869918i $$0.664172\pi$$
$$38$$ 0 0
$$39$$ 15.0000 2.40192
$$40$$ 0 0
$$41$$ 12.0000 1.87409 0.937043 0.349215i $$-0.113552\pi$$
0.937043 + 0.349215i $$0.113552\pi$$
$$42$$ 0 0
$$43$$ −2.00000 −0.304997 −0.152499 0.988304i $$-0.548732\pi$$
−0.152499 + 0.988304i $$0.548732\pi$$
$$44$$ 0 0
$$45$$ −6.00000 −0.894427
$$46$$ 0 0
$$47$$ −1.00000 −0.145865 −0.0729325 0.997337i $$-0.523236\pi$$
−0.0729325 + 0.997337i $$0.523236\pi$$
$$48$$ 0 0
$$49$$ 0 0
$$50$$ 0 0
$$51$$ 21.0000 2.94059
$$52$$ 0 0
$$53$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$54$$ 0 0
$$55$$ 5.00000 0.674200
$$56$$ 0 0
$$57$$ 6.00000 0.794719
$$58$$ 0 0
$$59$$ 4.00000 0.520756 0.260378 0.965507i $$-0.416153\pi$$
0.260378 + 0.965507i $$0.416153\pi$$
$$60$$ 0 0
$$61$$ −4.00000 −0.512148 −0.256074 0.966657i $$-0.582429\pi$$
−0.256074 + 0.966657i $$0.582429\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ −5.00000 −0.620174
$$66$$ 0 0
$$67$$ 8.00000 0.977356 0.488678 0.872464i $$-0.337479\pi$$
0.488678 + 0.872464i $$0.337479\pi$$
$$68$$ 0 0
$$69$$ −6.00000 −0.722315
$$70$$ 0 0
$$71$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$72$$ 0 0
$$73$$ −6.00000 −0.702247 −0.351123 0.936329i $$-0.614200\pi$$
−0.351123 + 0.936329i $$0.614200\pi$$
$$74$$ 0 0
$$75$$ 3.00000 0.346410
$$76$$ 0 0
$$77$$ 0 0
$$78$$ 0 0
$$79$$ −3.00000 −0.337526 −0.168763 0.985657i $$-0.553977\pi$$
−0.168763 + 0.985657i $$0.553977\pi$$
$$80$$ 0 0
$$81$$ 9.00000 1.00000
$$82$$ 0 0
$$83$$ 4.00000 0.439057 0.219529 0.975606i $$-0.429548\pi$$
0.219529 + 0.975606i $$0.429548\pi$$
$$84$$ 0 0
$$85$$ −7.00000 −0.759257
$$86$$ 0 0
$$87$$ 21.0000 2.25144
$$88$$ 0 0
$$89$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 0 0
$$93$$ −12.0000 −1.24434
$$94$$ 0 0
$$95$$ −2.00000 −0.205196
$$96$$ 0 0
$$97$$ −13.0000 −1.31995 −0.659975 0.751288i $$-0.729433\pi$$
−0.659975 + 0.751288i $$0.729433\pi$$
$$98$$ 0 0
$$99$$ −30.0000 −3.01511
$$100$$ 0 0
$$101$$ 18.0000 1.79107 0.895533 0.444994i $$-0.146794\pi$$
0.895533 + 0.444994i $$0.146794\pi$$
$$102$$ 0 0
$$103$$ −13.0000 −1.28093 −0.640464 0.767988i $$-0.721258\pi$$
−0.640464 + 0.767988i $$0.721258\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 18.0000 1.74013 0.870063 0.492941i $$-0.164078\pi$$
0.870063 + 0.492941i $$0.164078\pi$$
$$108$$ 0 0
$$109$$ 5.00000 0.478913 0.239457 0.970907i $$-0.423031\pi$$
0.239457 + 0.970907i $$0.423031\pi$$
$$110$$ 0 0
$$111$$ −18.0000 −1.70848
$$112$$ 0 0
$$113$$ 6.00000 0.564433 0.282216 0.959351i $$-0.408930\pi$$
0.282216 + 0.959351i $$0.408930\pi$$
$$114$$ 0 0
$$115$$ 2.00000 0.186501
$$116$$ 0 0
$$117$$ 30.0000 2.77350
$$118$$ 0 0
$$119$$ 0 0
$$120$$ 0 0
$$121$$ 14.0000 1.27273
$$122$$ 0 0
$$123$$ 36.0000 3.24601
$$124$$ 0 0
$$125$$ −1.00000 −0.0894427
$$126$$ 0 0
$$127$$ −12.0000 −1.06483 −0.532414 0.846484i $$-0.678715\pi$$
−0.532414 + 0.846484i $$0.678715\pi$$
$$128$$ 0 0
$$129$$ −6.00000 −0.528271
$$130$$ 0 0
$$131$$ 6.00000 0.524222 0.262111 0.965038i $$-0.415581\pi$$
0.262111 + 0.965038i $$0.415581\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ 0 0
$$135$$ −9.00000 −0.774597
$$136$$ 0 0
$$137$$ −8.00000 −0.683486 −0.341743 0.939793i $$-0.611017\pi$$
−0.341743 + 0.939793i $$0.611017\pi$$
$$138$$ 0 0
$$139$$ −18.0000 −1.52674 −0.763370 0.645961i $$-0.776457\pi$$
−0.763370 + 0.645961i $$0.776457\pi$$
$$140$$ 0 0
$$141$$ −3.00000 −0.252646
$$142$$ 0 0
$$143$$ −25.0000 −2.09061
$$144$$ 0 0
$$145$$ −7.00000 −0.581318
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ −22.0000 −1.80231 −0.901155 0.433497i $$-0.857280\pi$$
−0.901155 + 0.433497i $$0.857280\pi$$
$$150$$ 0 0
$$151$$ −19.0000 −1.54620 −0.773099 0.634285i $$-0.781294\pi$$
−0.773099 + 0.634285i $$0.781294\pi$$
$$152$$ 0 0
$$153$$ 42.0000 3.39550
$$154$$ 0 0
$$155$$ 4.00000 0.321288
$$156$$ 0 0
$$157$$ −10.0000 −0.798087 −0.399043 0.916932i $$-0.630658\pi$$
−0.399043 + 0.916932i $$0.630658\pi$$
$$158$$ 0 0
$$159$$ 0 0
$$160$$ 0 0
$$161$$ 0 0
$$162$$ 0 0
$$163$$ −14.0000 −1.09656 −0.548282 0.836293i $$-0.684718\pi$$
−0.548282 + 0.836293i $$0.684718\pi$$
$$164$$ 0 0
$$165$$ 15.0000 1.16775
$$166$$ 0 0
$$167$$ 3.00000 0.232147 0.116073 0.993241i $$-0.462969\pi$$
0.116073 + 0.993241i $$0.462969\pi$$
$$168$$ 0 0
$$169$$ 12.0000 0.923077
$$170$$ 0 0
$$171$$ 12.0000 0.917663
$$172$$ 0 0
$$173$$ 7.00000 0.532200 0.266100 0.963945i $$-0.414265\pi$$
0.266100 + 0.963945i $$0.414265\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ 12.0000 0.901975
$$178$$ 0 0
$$179$$ −4.00000 −0.298974 −0.149487 0.988764i $$-0.547762\pi$$
−0.149487 + 0.988764i $$0.547762\pi$$
$$180$$ 0 0
$$181$$ −8.00000 −0.594635 −0.297318 0.954779i $$-0.596092\pi$$
−0.297318 + 0.954779i $$0.596092\pi$$
$$182$$ 0 0
$$183$$ −12.0000 −0.887066
$$184$$ 0 0
$$185$$ 6.00000 0.441129
$$186$$ 0 0
$$187$$ −35.0000 −2.55945
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ −13.0000 −0.940647 −0.470323 0.882494i $$-0.655863\pi$$
−0.470323 + 0.882494i $$0.655863\pi$$
$$192$$ 0 0
$$193$$ −8.00000 −0.575853 −0.287926 0.957653i $$-0.592966\pi$$
−0.287926 + 0.957653i $$0.592966\pi$$
$$194$$ 0 0
$$195$$ −15.0000 −1.07417
$$196$$ 0 0
$$197$$ −8.00000 −0.569976 −0.284988 0.958531i $$-0.591990\pi$$
−0.284988 + 0.958531i $$0.591990\pi$$
$$198$$ 0 0
$$199$$ 4.00000 0.283552 0.141776 0.989899i $$-0.454719\pi$$
0.141776 + 0.989899i $$0.454719\pi$$
$$200$$ 0 0
$$201$$ 24.0000 1.69283
$$202$$ 0 0
$$203$$ 0 0
$$204$$ 0 0
$$205$$ −12.0000 −0.838116
$$206$$ 0 0
$$207$$ −12.0000 −0.834058
$$208$$ 0 0
$$209$$ −10.0000 −0.691714
$$210$$ 0 0
$$211$$ 5.00000 0.344214 0.172107 0.985078i $$-0.444942\pi$$
0.172107 + 0.985078i $$0.444942\pi$$
$$212$$ 0 0
$$213$$ 0 0
$$214$$ 0 0
$$215$$ 2.00000 0.136399
$$216$$ 0 0
$$217$$ 0 0
$$218$$ 0 0
$$219$$ −18.0000 −1.21633
$$220$$ 0 0
$$221$$ 35.0000 2.35435
$$222$$ 0 0
$$223$$ 19.0000 1.27233 0.636167 0.771551i $$-0.280519\pi$$
0.636167 + 0.771551i $$0.280519\pi$$
$$224$$ 0 0
$$225$$ 6.00000 0.400000
$$226$$ 0 0
$$227$$ −9.00000 −0.597351 −0.298675 0.954355i $$-0.596545\pi$$
−0.298675 + 0.954355i $$0.596545\pi$$
$$228$$ 0 0
$$229$$ −4.00000 −0.264327 −0.132164 0.991228i $$-0.542192\pi$$
−0.132164 + 0.991228i $$0.542192\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ −24.0000 −1.57229 −0.786146 0.618041i $$-0.787927\pi$$
−0.786146 + 0.618041i $$0.787927\pi$$
$$234$$ 0 0
$$235$$ 1.00000 0.0652328
$$236$$ 0 0
$$237$$ −9.00000 −0.584613
$$238$$ 0 0
$$239$$ 9.00000 0.582162 0.291081 0.956698i $$-0.405985\pi$$
0.291081 + 0.956698i $$0.405985\pi$$
$$240$$ 0 0
$$241$$ 22.0000 1.41714 0.708572 0.705638i $$-0.249340\pi$$
0.708572 + 0.705638i $$0.249340\pi$$
$$242$$ 0 0
$$243$$ 0 0
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 10.0000 0.636285
$$248$$ 0 0
$$249$$ 12.0000 0.760469
$$250$$ 0 0
$$251$$ −6.00000 −0.378717 −0.189358 0.981908i $$-0.560641\pi$$
−0.189358 + 0.981908i $$0.560641\pi$$
$$252$$ 0 0
$$253$$ 10.0000 0.628695
$$254$$ 0 0
$$255$$ −21.0000 −1.31507
$$256$$ 0 0
$$257$$ −18.0000 −1.12281 −0.561405 0.827541i $$-0.689739\pi$$
−0.561405 + 0.827541i $$0.689739\pi$$
$$258$$ 0 0
$$259$$ 0 0
$$260$$ 0 0
$$261$$ 42.0000 2.59973
$$262$$ 0 0
$$263$$ −30.0000 −1.84988 −0.924940 0.380114i $$-0.875885\pi$$
−0.924940 + 0.380114i $$0.875885\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 0 0
$$269$$ −26.0000 −1.58525 −0.792624 0.609711i $$-0.791286\pi$$
−0.792624 + 0.609711i $$0.791286\pi$$
$$270$$ 0 0
$$271$$ −12.0000 −0.728948 −0.364474 0.931214i $$-0.618751\pi$$
−0.364474 + 0.931214i $$0.618751\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ −5.00000 −0.301511
$$276$$ 0 0
$$277$$ 2.00000 0.120168 0.0600842 0.998193i $$-0.480863\pi$$
0.0600842 + 0.998193i $$0.480863\pi$$
$$278$$ 0 0
$$279$$ −24.0000 −1.43684
$$280$$ 0 0
$$281$$ 19.0000 1.13344 0.566722 0.823909i $$-0.308211\pi$$
0.566722 + 0.823909i $$0.308211\pi$$
$$282$$ 0 0
$$283$$ 25.0000 1.48610 0.743048 0.669238i $$-0.233379\pi$$
0.743048 + 0.669238i $$0.233379\pi$$
$$284$$ 0 0
$$285$$ −6.00000 −0.355409
$$286$$ 0 0
$$287$$ 0 0
$$288$$ 0 0
$$289$$ 32.0000 1.88235
$$290$$ 0 0
$$291$$ −39.0000 −2.28622
$$292$$ 0 0
$$293$$ −13.0000 −0.759468 −0.379734 0.925096i $$-0.623985\pi$$
−0.379734 + 0.925096i $$0.623985\pi$$
$$294$$ 0 0
$$295$$ −4.00000 −0.232889
$$296$$ 0 0
$$297$$ −45.0000 −2.61116
$$298$$ 0 0
$$299$$ −10.0000 −0.578315
$$300$$ 0 0
$$301$$ 0 0
$$302$$ 0 0
$$303$$ 54.0000 3.10222
$$304$$ 0 0
$$305$$ 4.00000 0.229039
$$306$$ 0 0
$$307$$ 17.0000 0.970241 0.485121 0.874447i $$-0.338776\pi$$
0.485121 + 0.874447i $$0.338776\pi$$
$$308$$ 0 0
$$309$$ −39.0000 −2.21863
$$310$$ 0 0
$$311$$ 34.0000 1.92796 0.963982 0.265969i $$-0.0856919\pi$$
0.963982 + 0.265969i $$0.0856919\pi$$
$$312$$ 0 0
$$313$$ 1.00000 0.0565233 0.0282617 0.999601i $$-0.491003\pi$$
0.0282617 + 0.999601i $$0.491003\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ 6.00000 0.336994 0.168497 0.985702i $$-0.446109\pi$$
0.168497 + 0.985702i $$0.446109\pi$$
$$318$$ 0 0
$$319$$ −35.0000 −1.95962
$$320$$ 0 0
$$321$$ 54.0000 3.01399
$$322$$ 0 0
$$323$$ 14.0000 0.778981
$$324$$ 0 0
$$325$$ 5.00000 0.277350
$$326$$ 0 0
$$327$$ 15.0000 0.829502
$$328$$ 0 0
$$329$$ 0 0
$$330$$ 0 0
$$331$$ −12.0000 −0.659580 −0.329790 0.944054i $$-0.606978\pi$$
−0.329790 + 0.944054i $$0.606978\pi$$
$$332$$ 0 0
$$333$$ −36.0000 −1.97279
$$334$$ 0 0
$$335$$ −8.00000 −0.437087
$$336$$ 0 0
$$337$$ 34.0000 1.85210 0.926049 0.377403i $$-0.123183\pi$$
0.926049 + 0.377403i $$0.123183\pi$$
$$338$$ 0 0
$$339$$ 18.0000 0.977626
$$340$$ 0 0
$$341$$ 20.0000 1.08306
$$342$$ 0 0
$$343$$ 0 0
$$344$$ 0 0
$$345$$ 6.00000 0.323029
$$346$$ 0 0
$$347$$ 26.0000 1.39575 0.697877 0.716218i $$-0.254128\pi$$
0.697877 + 0.716218i $$0.254128\pi$$
$$348$$ 0 0
$$349$$ −18.0000 −0.963518 −0.481759 0.876304i $$-0.660002\pi$$
−0.481759 + 0.876304i $$0.660002\pi$$
$$350$$ 0 0
$$351$$ 45.0000 2.40192
$$352$$ 0 0
$$353$$ −5.00000 −0.266123 −0.133062 0.991108i $$-0.542481\pi$$
−0.133062 + 0.991108i $$0.542481\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$360$$ 0 0
$$361$$ −15.0000 −0.789474
$$362$$ 0 0
$$363$$ 42.0000 2.20443
$$364$$ 0 0
$$365$$ 6.00000 0.314054
$$366$$ 0 0
$$367$$ 7.00000 0.365397 0.182699 0.983169i $$-0.441517\pi$$
0.182699 + 0.983169i $$0.441517\pi$$
$$368$$ 0 0
$$369$$ 72.0000 3.74817
$$370$$ 0 0
$$371$$ 0 0
$$372$$ 0 0
$$373$$ 16.0000 0.828449 0.414224 0.910175i $$-0.364053\pi$$
0.414224 + 0.910175i $$0.364053\pi$$
$$374$$ 0 0
$$375$$ −3.00000 −0.154919
$$376$$ 0 0
$$377$$ 35.0000 1.80259
$$378$$ 0 0
$$379$$ −20.0000 −1.02733 −0.513665 0.857991i $$-0.671713\pi$$
−0.513665 + 0.857991i $$0.671713\pi$$
$$380$$ 0 0
$$381$$ −36.0000 −1.84434
$$382$$ 0 0
$$383$$ −12.0000 −0.613171 −0.306586 0.951843i $$-0.599187\pi$$
−0.306586 + 0.951843i $$0.599187\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ −12.0000 −0.609994
$$388$$ 0 0
$$389$$ −15.0000 −0.760530 −0.380265 0.924878i $$-0.624167\pi$$
−0.380265 + 0.924878i $$0.624167\pi$$
$$390$$ 0 0
$$391$$ −14.0000 −0.708010
$$392$$ 0 0
$$393$$ 18.0000 0.907980
$$394$$ 0 0
$$395$$ 3.00000 0.150946
$$396$$ 0 0
$$397$$ 23.0000 1.15434 0.577168 0.816625i $$-0.304158\pi$$
0.577168 + 0.816625i $$0.304158\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ −15.0000 −0.749064 −0.374532 0.927214i $$-0.622197\pi$$
−0.374532 + 0.927214i $$0.622197\pi$$
$$402$$ 0 0
$$403$$ −20.0000 −0.996271
$$404$$ 0 0
$$405$$ −9.00000 −0.447214
$$406$$ 0 0
$$407$$ 30.0000 1.48704
$$408$$ 0 0
$$409$$ 14.0000 0.692255 0.346128 0.938187i $$-0.387496\pi$$
0.346128 + 0.938187i $$0.387496\pi$$
$$410$$ 0 0
$$411$$ −24.0000 −1.18383
$$412$$ 0 0
$$413$$ 0 0
$$414$$ 0 0
$$415$$ −4.00000 −0.196352
$$416$$ 0 0
$$417$$ −54.0000 −2.64439
$$418$$ 0 0
$$419$$ 24.0000 1.17248 0.586238 0.810139i $$-0.300608\pi$$
0.586238 + 0.810139i $$0.300608\pi$$
$$420$$ 0 0
$$421$$ −3.00000 −0.146211 −0.0731055 0.997324i $$-0.523291\pi$$
−0.0731055 + 0.997324i $$0.523291\pi$$
$$422$$ 0 0
$$423$$ −6.00000 −0.291730
$$424$$ 0 0
$$425$$ 7.00000 0.339550
$$426$$ 0 0
$$427$$ 0 0
$$428$$ 0 0
$$429$$ −75.0000 −3.62103
$$430$$ 0 0
$$431$$ −25.0000 −1.20421 −0.602104 0.798418i $$-0.705671\pi$$
−0.602104 + 0.798418i $$0.705671\pi$$
$$432$$ 0 0
$$433$$ −14.0000 −0.672797 −0.336399 0.941720i $$-0.609209\pi$$
−0.336399 + 0.941720i $$0.609209\pi$$
$$434$$ 0 0
$$435$$ −21.0000 −1.00687
$$436$$ 0 0
$$437$$ −4.00000 −0.191346
$$438$$ 0 0
$$439$$ 26.0000 1.24091 0.620456 0.784241i $$-0.286947\pi$$
0.620456 + 0.784241i $$0.286947\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ 6.00000 0.285069 0.142534 0.989790i $$-0.454475\pi$$
0.142534 + 0.989790i $$0.454475\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 0 0
$$447$$ −66.0000 −3.12169
$$448$$ 0 0
$$449$$ −9.00000 −0.424736 −0.212368 0.977190i $$-0.568118\pi$$
−0.212368 + 0.977190i $$0.568118\pi$$
$$450$$ 0 0
$$451$$ −60.0000 −2.82529
$$452$$ 0 0
$$453$$ −57.0000 −2.67809
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ −8.00000 −0.374224 −0.187112 0.982339i $$-0.559913\pi$$
−0.187112 + 0.982339i $$0.559913\pi$$
$$458$$ 0 0
$$459$$ 63.0000 2.94059
$$460$$ 0 0
$$461$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$462$$ 0 0
$$463$$ −36.0000 −1.67306 −0.836531 0.547920i $$-0.815420\pi$$
−0.836531 + 0.547920i $$0.815420\pi$$
$$464$$ 0 0
$$465$$ 12.0000 0.556487
$$466$$ 0 0
$$467$$ −11.0000 −0.509019 −0.254510 0.967070i $$-0.581914\pi$$
−0.254510 + 0.967070i $$0.581914\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ 0 0
$$471$$ −30.0000 −1.38233
$$472$$ 0 0
$$473$$ 10.0000 0.459800
$$474$$ 0 0
$$475$$ 2.00000 0.0917663
$$476$$ 0 0
$$477$$ 0 0
$$478$$ 0 0
$$479$$ −34.0000 −1.55350 −0.776750 0.629809i $$-0.783133\pi$$
−0.776750 + 0.629809i $$0.783133\pi$$
$$480$$ 0 0
$$481$$ −30.0000 −1.36788
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ 13.0000 0.590300
$$486$$ 0 0
$$487$$ 22.0000 0.996915 0.498458 0.866914i $$-0.333900\pi$$
0.498458 + 0.866914i $$0.333900\pi$$
$$488$$ 0 0
$$489$$ −42.0000 −1.89931
$$490$$ 0 0
$$491$$ 9.00000 0.406164 0.203082 0.979162i $$-0.434904\pi$$
0.203082 + 0.979162i $$0.434904\pi$$
$$492$$ 0 0
$$493$$ 49.0000 2.20685
$$494$$ 0 0
$$495$$ 30.0000 1.34840
$$496$$ 0 0
$$497$$ 0 0
$$498$$ 0 0
$$499$$ −1.00000 −0.0447661 −0.0223831 0.999749i $$-0.507125\pi$$
−0.0223831 + 0.999749i $$0.507125\pi$$
$$500$$ 0 0
$$501$$ 9.00000 0.402090
$$502$$ 0 0
$$503$$ −3.00000 −0.133763 −0.0668817 0.997761i $$-0.521305\pi$$
−0.0668817 + 0.997761i $$0.521305\pi$$
$$504$$ 0 0
$$505$$ −18.0000 −0.800989
$$506$$ 0 0
$$507$$ 36.0000 1.59882
$$508$$ 0 0
$$509$$ 14.0000 0.620539 0.310270 0.950649i $$-0.399581\pi$$
0.310270 + 0.950649i $$0.399581\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ 0 0
$$513$$ 18.0000 0.794719
$$514$$ 0 0
$$515$$ 13.0000 0.572848
$$516$$ 0 0
$$517$$ 5.00000 0.219900
$$518$$ 0 0
$$519$$ 21.0000 0.921798
$$520$$ 0 0
$$521$$ −22.0000 −0.963837 −0.481919 0.876216i $$-0.660060\pi$$
−0.481919 + 0.876216i $$0.660060\pi$$
$$522$$ 0 0
$$523$$ 44.0000 1.92399 0.961993 0.273075i $$-0.0880406\pi$$
0.961993 + 0.273075i $$0.0880406\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ −28.0000 −1.21970
$$528$$ 0 0
$$529$$ −19.0000 −0.826087
$$530$$ 0 0
$$531$$ 24.0000 1.04151
$$532$$ 0 0
$$533$$ 60.0000 2.59889
$$534$$ 0 0
$$535$$ −18.0000 −0.778208
$$536$$ 0 0
$$537$$ −12.0000 −0.517838
$$538$$ 0 0
$$539$$ 0 0
$$540$$ 0 0
$$541$$ 7.00000 0.300954 0.150477 0.988614i $$-0.451919\pi$$
0.150477 + 0.988614i $$0.451919\pi$$
$$542$$ 0 0
$$543$$ −24.0000 −1.02994
$$544$$ 0 0
$$545$$ −5.00000 −0.214176
$$546$$ 0 0
$$547$$ −28.0000 −1.19719 −0.598597 0.801050i $$-0.704275\pi$$
−0.598597 + 0.801050i $$0.704275\pi$$
$$548$$ 0 0
$$549$$ −24.0000 −1.02430
$$550$$ 0 0
$$551$$ 14.0000 0.596420
$$552$$ 0 0
$$553$$ 0 0
$$554$$ 0 0
$$555$$ 18.0000 0.764057
$$556$$ 0 0
$$557$$ −16.0000 −0.677942 −0.338971 0.940797i $$-0.610079\pi$$
−0.338971 + 0.940797i $$0.610079\pi$$
$$558$$ 0 0
$$559$$ −10.0000 −0.422955
$$560$$ 0 0
$$561$$ −105.000 −4.43310
$$562$$ 0 0
$$563$$ −4.00000 −0.168580 −0.0842900 0.996441i $$-0.526862\pi$$
−0.0842900 + 0.996441i $$0.526862\pi$$
$$564$$ 0 0
$$565$$ −6.00000 −0.252422
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ −22.0000 −0.922288 −0.461144 0.887325i $$-0.652561\pi$$
−0.461144 + 0.887325i $$0.652561\pi$$
$$570$$ 0 0
$$571$$ 4.00000 0.167395 0.0836974 0.996491i $$-0.473327\pi$$
0.0836974 + 0.996491i $$0.473327\pi$$
$$572$$ 0 0
$$573$$ −39.0000 −1.62925
$$574$$ 0 0
$$575$$ −2.00000 −0.0834058
$$576$$ 0 0
$$577$$ 37.0000 1.54033 0.770165 0.637845i $$-0.220174\pi$$
0.770165 + 0.637845i $$0.220174\pi$$
$$578$$ 0 0
$$579$$ −24.0000 −0.997406
$$580$$ 0 0
$$581$$ 0 0
$$582$$ 0 0
$$583$$ 0 0
$$584$$ 0 0
$$585$$ −30.0000 −1.24035
$$586$$ 0 0
$$587$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$588$$ 0 0
$$589$$ −8.00000 −0.329634
$$590$$ 0 0
$$591$$ −24.0000 −0.987228
$$592$$ 0 0
$$593$$ −27.0000 −1.10876 −0.554379 0.832265i $$-0.687044\pi$$
−0.554379 + 0.832265i $$0.687044\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ 12.0000 0.491127
$$598$$ 0 0
$$599$$ 15.0000 0.612883 0.306442 0.951889i $$-0.400862\pi$$
0.306442 + 0.951889i $$0.400862\pi$$
$$600$$ 0 0
$$601$$ −10.0000 −0.407909 −0.203954 0.978980i $$-0.565379\pi$$
−0.203954 + 0.978980i $$0.565379\pi$$
$$602$$ 0 0
$$603$$ 48.0000 1.95471
$$604$$ 0 0
$$605$$ −14.0000 −0.569181
$$606$$ 0 0
$$607$$ −3.00000 −0.121766 −0.0608831 0.998145i $$-0.519392\pi$$
−0.0608831 + 0.998145i $$0.519392\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ −5.00000 −0.202278
$$612$$ 0 0
$$613$$ −30.0000 −1.21169 −0.605844 0.795583i $$-0.707165\pi$$
−0.605844 + 0.795583i $$0.707165\pi$$
$$614$$ 0 0
$$615$$ −36.0000 −1.45166
$$616$$ 0 0
$$617$$ 6.00000 0.241551 0.120775 0.992680i $$-0.461462\pi$$
0.120775 + 0.992680i $$0.461462\pi$$
$$618$$ 0 0
$$619$$ −26.0000 −1.04503 −0.522514 0.852631i $$-0.675006\pi$$
−0.522514 + 0.852631i $$0.675006\pi$$
$$620$$ 0 0
$$621$$ −18.0000 −0.722315
$$622$$ 0 0
$$623$$ 0 0
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ 0 0
$$627$$ −30.0000 −1.19808
$$628$$ 0 0
$$629$$ −42.0000 −1.67465
$$630$$ 0 0
$$631$$ 7.00000 0.278666 0.139333 0.990246i $$-0.455504\pi$$
0.139333 + 0.990246i $$0.455504\pi$$
$$632$$ 0 0
$$633$$ 15.0000 0.596196
$$634$$ 0 0
$$635$$ 12.0000 0.476205
$$636$$ 0 0
$$637$$ 0 0
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 0 0
$$641$$ 18.0000 0.710957 0.355479 0.934684i $$-0.384318\pi$$
0.355479 + 0.934684i $$0.384318\pi$$
$$642$$ 0 0
$$643$$ −21.0000 −0.828159 −0.414080 0.910241i $$-0.635896\pi$$
−0.414080 + 0.910241i $$0.635896\pi$$
$$644$$ 0 0
$$645$$ 6.00000 0.236250
$$646$$ 0 0
$$647$$ −24.0000 −0.943537 −0.471769 0.881722i $$-0.656384\pi$$
−0.471769 + 0.881722i $$0.656384\pi$$
$$648$$ 0 0
$$649$$ −20.0000 −0.785069
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ 38.0000 1.48705 0.743527 0.668705i $$-0.233151\pi$$
0.743527 + 0.668705i $$0.233151\pi$$
$$654$$ 0 0
$$655$$ −6.00000 −0.234439
$$656$$ 0 0
$$657$$ −36.0000 −1.40449
$$658$$ 0 0
$$659$$ 15.0000 0.584317 0.292159 0.956370i $$-0.405627\pi$$
0.292159 + 0.956370i $$0.405627\pi$$
$$660$$ 0 0
$$661$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$662$$ 0 0
$$663$$ 105.000 4.07786
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ −14.0000 −0.542082
$$668$$ 0 0
$$669$$ 57.0000 2.20375
$$670$$ 0 0
$$671$$ 20.0000 0.772091
$$672$$ 0 0
$$673$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$674$$ 0 0
$$675$$ 9.00000 0.346410
$$676$$ 0 0
$$677$$ 29.0000 1.11456 0.557280 0.830324i $$-0.311845\pi$$
0.557280 + 0.830324i $$0.311845\pi$$
$$678$$ 0 0
$$679$$ 0 0
$$680$$ 0 0
$$681$$ −27.0000 −1.03464
$$682$$ 0 0
$$683$$ 12.0000 0.459167 0.229584 0.973289i $$-0.426264\pi$$
0.229584 + 0.973289i $$0.426264\pi$$
$$684$$ 0 0
$$685$$ 8.00000 0.305664
$$686$$ 0 0
$$687$$ −12.0000 −0.457829
$$688$$ 0 0
$$689$$ 0 0
$$690$$ 0 0
$$691$$ −44.0000 −1.67384 −0.836919 0.547326i $$-0.815646\pi$$
−0.836919 + 0.547326i $$0.815646\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ 18.0000 0.682779
$$696$$ 0 0
$$697$$ 84.0000 3.18173
$$698$$ 0 0
$$699$$ −72.0000 −2.72329
$$700$$ 0 0
$$701$$ 35.0000 1.32193 0.660966 0.750416i $$-0.270147\pi$$
0.660966 + 0.750416i $$0.270147\pi$$
$$702$$ 0 0
$$703$$ −12.0000 −0.452589
$$704$$ 0 0
$$705$$ 3.00000 0.112987
$$706$$ 0 0
$$707$$ 0 0
$$708$$ 0 0
$$709$$ −25.0000 −0.938895 −0.469447 0.882960i $$-0.655547\pi$$
−0.469447 + 0.882960i $$0.655547\pi$$
$$710$$ 0 0
$$711$$ −18.0000 −0.675053
$$712$$ 0 0
$$713$$ 8.00000 0.299602
$$714$$ 0 0
$$715$$ 25.0000 0.934947
$$716$$ 0 0
$$717$$ 27.0000 1.00833
$$718$$ 0 0
$$719$$ 14.0000 0.522112 0.261056 0.965324i $$-0.415929\pi$$
0.261056 + 0.965324i $$0.415929\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ 0 0
$$723$$ 66.0000 2.45457
$$724$$ 0 0
$$725$$ 7.00000 0.259973
$$726$$ 0 0
$$727$$ 40.0000 1.48352 0.741759 0.670667i $$-0.233992\pi$$
0.741759 + 0.670667i $$0.233992\pi$$
$$728$$ 0 0
$$729$$ −27.0000 −1.00000
$$730$$ 0 0
$$731$$ −14.0000 −0.517809
$$732$$ 0 0
$$733$$ 33.0000 1.21888 0.609441 0.792831i $$-0.291394\pi$$
0.609441 + 0.792831i $$0.291394\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ −40.0000 −1.47342
$$738$$ 0 0
$$739$$ 51.0000 1.87607 0.938033 0.346547i $$-0.112646\pi$$
0.938033 + 0.346547i $$0.112646\pi$$
$$740$$ 0 0
$$741$$ 30.0000 1.10208
$$742$$ 0 0
$$743$$ 24.0000 0.880475 0.440237 0.897881i $$-0.354894\pi$$
0.440237 + 0.897881i $$0.354894\pi$$
$$744$$ 0 0
$$745$$ 22.0000 0.806018
$$746$$ 0 0
$$747$$ 24.0000 0.878114
$$748$$ 0 0
$$749$$ 0 0
$$750$$ 0 0
$$751$$ −3.00000 −0.109472 −0.0547358 0.998501i $$-0.517432\pi$$
−0.0547358 + 0.998501i $$0.517432\pi$$
$$752$$ 0 0
$$753$$ −18.0000 −0.655956
$$754$$ 0 0
$$755$$ 19.0000 0.691481
$$756$$ 0 0
$$757$$ −40.0000 −1.45382 −0.726912 0.686730i $$-0.759045\pi$$
−0.726912 + 0.686730i $$0.759045\pi$$
$$758$$ 0 0
$$759$$ 30.0000 1.08893
$$760$$ 0 0
$$761$$ −18.0000 −0.652499 −0.326250 0.945284i $$-0.605785\pi$$
−0.326250 + 0.945284i $$0.605785\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ 0 0
$$765$$ −42.0000 −1.51851
$$766$$ 0 0
$$767$$ 20.0000 0.722158
$$768$$ 0 0
$$769$$ −34.0000 −1.22607 −0.613036 0.790055i $$-0.710052\pi$$
−0.613036 + 0.790055i $$0.710052\pi$$
$$770$$ 0 0
$$771$$ −54.0000 −1.94476
$$772$$ 0 0
$$773$$ 21.0000 0.755318 0.377659 0.925945i $$-0.376729\pi$$
0.377659 + 0.925945i $$0.376729\pi$$
$$774$$ 0 0
$$775$$ −4.00000 −0.143684
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ 24.0000 0.859889
$$780$$ 0 0
$$781$$ 0 0
$$782$$ 0 0
$$783$$ 63.0000 2.25144
$$784$$ 0 0
$$785$$ 10.0000 0.356915
$$786$$ 0 0
$$787$$ −17.0000 −0.605985 −0.302992 0.952993i $$-0.597986\pi$$
−0.302992 + 0.952993i $$0.597986\pi$$
$$788$$ 0 0
$$789$$ −90.0000 −3.20408
$$790$$ 0 0
$$791$$ 0 0
$$792$$ 0 0
$$793$$ −20.0000 −0.710221
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ 39.0000 1.38145 0.690725 0.723117i $$-0.257291\pi$$
0.690725 + 0.723117i $$0.257291\pi$$
$$798$$ 0 0
$$799$$ −7.00000 −0.247642
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 0 0
$$803$$ 30.0000 1.05868
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ −78.0000 −2.74573
$$808$$ 0 0
$$809$$ −39.0000 −1.37117 −0.685583 0.727994i $$-0.740453\pi$$
−0.685583 + 0.727994i $$0.740453\pi$$
$$810$$ 0 0
$$811$$ 10.0000 0.351147 0.175574 0.984466i $$-0.443822\pi$$
0.175574 + 0.984466i $$0.443822\pi$$
$$812$$ 0 0
$$813$$ −36.0000 −1.26258
$$814$$ 0 0
$$815$$ 14.0000 0.490399
$$816$$ 0 0
$$817$$ −4.00000 −0.139942
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ 1.00000 0.0349002 0.0174501 0.999848i $$-0.494445\pi$$
0.0174501 + 0.999848i $$0.494445\pi$$
$$822$$ 0 0
$$823$$ 28.0000 0.976019 0.488009 0.872838i $$-0.337723\pi$$
0.488009 + 0.872838i $$0.337723\pi$$
$$824$$ 0 0
$$825$$ −15.0000 −0.522233
$$826$$ 0 0
$$827$$ 6.00000 0.208640 0.104320 0.994544i $$-0.466733\pi$$
0.104320 + 0.994544i $$0.466733\pi$$
$$828$$ 0 0
$$829$$ 20.0000 0.694629 0.347314 0.937749i $$-0.387094\pi$$
0.347314 + 0.937749i $$0.387094\pi$$
$$830$$ 0 0
$$831$$ 6.00000 0.208138
$$832$$ 0 0
$$833$$ 0 0
$$834$$ 0 0
$$835$$ −3.00000 −0.103819
$$836$$ 0 0
$$837$$ −36.0000 −1.24434
$$838$$ 0 0
$$839$$ 34.0000 1.17381 0.586905 0.809656i $$-0.300346\pi$$
0.586905 + 0.809656i $$0.300346\pi$$
$$840$$ 0 0
$$841$$ 20.0000 0.689655
$$842$$ 0 0
$$843$$ 57.0000 1.96318
$$844$$ 0 0
$$845$$ −12.0000 −0.412813
$$846$$ 0 0
$$847$$ 0 0
$$848$$ 0 0
$$849$$ 75.0000 2.57399
$$850$$ 0 0
$$851$$ 12.0000 0.411355
$$852$$ 0 0
$$853$$ 22.0000 0.753266 0.376633 0.926363i $$-0.377082\pi$$
0.376633 + 0.926363i $$0.377082\pi$$
$$854$$ 0 0
$$855$$ −12.0000 −0.410391
$$856$$ 0 0
$$857$$ −6.00000 −0.204956 −0.102478 0.994735i $$-0.532677\pi$$
−0.102478 + 0.994735i $$0.532677\pi$$
$$858$$ 0 0
$$859$$ 4.00000 0.136478 0.0682391 0.997669i $$-0.478262\pi$$
0.0682391 + 0.997669i $$0.478262\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ 48.0000 1.63394 0.816970 0.576681i $$-0.195652\pi$$
0.816970 + 0.576681i $$0.195652\pi$$
$$864$$ 0 0
$$865$$ −7.00000 −0.238007
$$866$$ 0 0
$$867$$ 96.0000 3.26033
$$868$$ 0 0
$$869$$ 15.0000 0.508840
$$870$$ 0 0
$$871$$ 40.0000 1.35535
$$872$$ 0 0
$$873$$ −78.0000 −2.63990
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 22.0000 0.742887 0.371444 0.928456i $$-0.378863\pi$$
0.371444 + 0.928456i $$0.378863\pi$$
$$878$$ 0 0
$$879$$ −39.0000 −1.31544
$$880$$ 0 0
$$881$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$882$$ 0 0
$$883$$ −40.0000 −1.34611 −0.673054 0.739594i $$-0.735018\pi$$
−0.673054 + 0.739594i $$0.735018\pi$$
$$884$$ 0 0
$$885$$ −12.0000 −0.403376
$$886$$ 0 0
$$887$$ −16.0000 −0.537227 −0.268614 0.963248i $$-0.586566\pi$$
−0.268614 + 0.963248i $$0.586566\pi$$
$$888$$ 0 0
$$889$$ 0 0
$$890$$ 0 0
$$891$$ −45.0000 −1.50756
$$892$$ 0 0
$$893$$ −2.00000 −0.0669274
$$894$$ 0 0
$$895$$ 4.00000 0.133705
$$896$$ 0 0
$$897$$ −30.0000 −1.00167
$$898$$ 0 0
$$899$$ −28.0000 −0.933852
$$900$$ 0 0
$$901$$ 0 0
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ 8.00000 0.265929
$$906$$ 0 0
$$907$$ −18.0000 −0.597680 −0.298840 0.954303i $$-0.596600\pi$$
−0.298840 + 0.954303i $$0.596600\pi$$
$$908$$ 0 0
$$909$$ 108.000 3.58213
$$910$$ 0 0
$$911$$ −8.00000 −0.265052 −0.132526 0.991180i $$-0.542309\pi$$
−0.132526 + 0.991180i $$0.542309\pi$$
$$912$$ 0 0
$$913$$ −20.0000 −0.661903
$$914$$ 0 0
$$915$$ 12.0000 0.396708
$$916$$ 0 0
$$917$$ 0 0
$$918$$ 0 0
$$919$$ −55.0000 −1.81428 −0.907141 0.420826i $$-0.861740\pi$$
−0.907141 + 0.420826i $$0.861740\pi$$
$$920$$ 0 0
$$921$$ 51.0000 1.68051
$$922$$ 0 0
$$923$$ 0 0
$$924$$ 0 0
$$925$$ −6.00000 −0.197279
$$926$$ 0 0
$$927$$ −78.0000 −2.56186
$$928$$ 0 0
$$929$$ 32.0000 1.04989 0.524943 0.851137i $$-0.324087\pi$$
0.524943 + 0.851137i $$0.324087\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ 0 0
$$933$$ 102.000 3.33933
$$934$$ 0 0
$$935$$ 35.0000 1.14462
$$936$$ 0 0
$$937$$ −13.0000 −0.424691 −0.212346 0.977195i $$-0.568110\pi$$
−0.212346 + 0.977195i $$0.568110\pi$$
$$938$$ 0 0
$$939$$ 3.00000 0.0979013
$$940$$ 0 0
$$941$$ −28.0000 −0.912774 −0.456387 0.889781i $$-0.650857\pi$$
−0.456387 + 0.889781i $$0.650857\pi$$
$$942$$ 0 0
$$943$$ −24.0000 −0.781548
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 40.0000 1.29983 0.649913 0.760009i $$-0.274805\pi$$
0.649913 + 0.760009i $$0.274805\pi$$
$$948$$ 0 0
$$949$$ −30.0000 −0.973841
$$950$$ 0 0
$$951$$ 18.0000 0.583690
$$952$$ 0 0
$$953$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$954$$ 0 0
$$955$$ 13.0000 0.420670
$$956$$ 0 0
$$957$$ −105.000 −3.39417
$$958$$ 0 0
$$959$$ 0 0
$$960$$ 0 0
$$961$$ −15.0000 −0.483871
$$962$$ 0 0
$$963$$ 108.000 3.48025
$$964$$ 0 0
$$965$$ 8.00000 0.257529
$$966$$ 0 0
$$967$$ 6.00000 0.192947 0.0964735 0.995336i $$-0.469244\pi$$
0.0964735 + 0.995336i $$0.469244\pi$$
$$968$$ 0 0
$$969$$ 42.0000 1.34923
$$970$$ 0 0
$$971$$ 16.0000 0.513464 0.256732 0.966483i $$-0.417354\pi$$
0.256732 + 0.966483i $$0.417354\pi$$
$$972$$ 0 0
$$973$$ 0 0
$$974$$ 0 0
$$975$$ 15.0000 0.480384
$$976$$ 0 0
$$977$$ 26.0000 0.831814 0.415907 0.909407i $$-0.363464\pi$$
0.415907 + 0.909407i $$0.363464\pi$$
$$978$$ 0 0
$$979$$ 0 0
$$980$$ 0 0
$$981$$ 30.0000 0.957826
$$982$$ 0 0
$$983$$ −3.00000 −0.0956851 −0.0478426 0.998855i $$-0.515235\pi$$
−0.0478426 + 0.998855i $$0.515235\pi$$
$$984$$ 0 0
$$985$$ 8.00000 0.254901
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ 4.00000 0.127193
$$990$$ 0 0
$$991$$ 32.0000 1.01651 0.508257 0.861206i $$-0.330290\pi$$
0.508257 + 0.861206i $$0.330290\pi$$
$$992$$ 0 0
$$993$$ −36.0000 −1.14243
$$994$$ 0 0
$$995$$ −4.00000 −0.126809
$$996$$ 0 0
$$997$$ 3.00000 0.0950110 0.0475055 0.998871i $$-0.484873\pi$$
0.0475055 + 0.998871i $$0.484873\pi$$
$$998$$ 0 0
$$999$$ −54.0000 −1.70848
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 1960.2.a.o.1.1 1
4.3 odd 2 3920.2.a.c.1.1 1
5.4 even 2 9800.2.a.a.1.1 1
7.2 even 3 1960.2.q.a.361.1 2
7.3 odd 6 1960.2.q.o.961.1 2
7.4 even 3 1960.2.q.a.961.1 2
7.5 odd 6 1960.2.q.o.361.1 2
7.6 odd 2 280.2.a.a.1.1 1
21.20 even 2 2520.2.a.i.1.1 1
28.27 even 2 560.2.a.f.1.1 1
35.13 even 4 1400.2.g.a.449.1 2
35.27 even 4 1400.2.g.a.449.2 2
35.34 odd 2 1400.2.a.n.1.1 1
56.13 odd 2 2240.2.a.z.1.1 1
56.27 even 2 2240.2.a.a.1.1 1
84.83 odd 2 5040.2.a.a.1.1 1
140.27 odd 4 2800.2.g.b.449.1 2
140.83 odd 4 2800.2.g.b.449.2 2
140.139 even 2 2800.2.a.c.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
280.2.a.a.1.1 1 7.6 odd 2
560.2.a.f.1.1 1 28.27 even 2
1400.2.a.n.1.1 1 35.34 odd 2
1400.2.g.a.449.1 2 35.13 even 4
1400.2.g.a.449.2 2 35.27 even 4
1960.2.a.o.1.1 1 1.1 even 1 trivial
1960.2.q.a.361.1 2 7.2 even 3
1960.2.q.a.961.1 2 7.4 even 3
1960.2.q.o.361.1 2 7.5 odd 6
1960.2.q.o.961.1 2 7.3 odd 6
2240.2.a.a.1.1 1 56.27 even 2
2240.2.a.z.1.1 1 56.13 odd 2
2520.2.a.i.1.1 1 21.20 even 2
2800.2.a.c.1.1 1 140.139 even 2
2800.2.g.b.449.1 2 140.27 odd 4
2800.2.g.b.449.2 2 140.83 odd 4
3920.2.a.c.1.1 1 4.3 odd 2
5040.2.a.a.1.1 1 84.83 odd 2
9800.2.a.a.1.1 1 5.4 even 2