# Properties

 Label 1840.2.a.f.1.1 Level $1840$ Weight $2$ Character 1840.1 Self dual yes Analytic conductor $14.692$ 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] = [1840,2,Mod(1,1840)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1840, 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("1840.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1840 = 2^{4} \cdot 5 \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1840.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: $$14.6924739719$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 920) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 1840.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+1.00000 q^{5} -1.00000 q^{7} -3.00000 q^{9} +O(q^{10})$$ $$q+1.00000 q^{5} -1.00000 q^{7} -3.00000 q^{9} +6.00000 q^{11} -2.00000 q^{13} -3.00000 q^{17} +6.00000 q^{19} -1.00000 q^{23} +1.00000 q^{25} +3.00000 q^{29} +3.00000 q^{31} -1.00000 q^{35} +1.00000 q^{37} +9.00000 q^{41} +8.00000 q^{43} -3.00000 q^{45} -4.00000 q^{47} -6.00000 q^{49} +1.00000 q^{53} +6.00000 q^{55} -1.00000 q^{59} +8.00000 q^{61} +3.00000 q^{63} -2.00000 q^{65} +7.00000 q^{67} +5.00000 q^{71} -6.00000 q^{73} -6.00000 q^{77} +9.00000 q^{81} +11.0000 q^{83} -3.00000 q^{85} +4.00000 q^{89} +2.00000 q^{91} +6.00000 q^{95} +6.00000 q^{97} -18.0000 q^{99} +O(q^{100})$$

## 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
$$3$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$4$$ 0 0
$$5$$ 1.00000 0.447214
$$6$$ 0 0
$$7$$ −1.00000 −0.377964 −0.188982 0.981981i $$-0.560519\pi$$
−0.188982 + 0.981981i $$0.560519\pi$$
$$8$$ 0 0
$$9$$ −3.00000 −1.00000
$$10$$ 0 0
$$11$$ 6.00000 1.80907 0.904534 0.426401i $$-0.140219\pi$$
0.904534 + 0.426401i $$0.140219\pi$$
$$12$$ 0 0
$$13$$ −2.00000 −0.554700 −0.277350 0.960769i $$-0.589456\pi$$
−0.277350 + 0.960769i $$0.589456\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ −3.00000 −0.727607 −0.363803 0.931476i $$-0.618522\pi$$
−0.363803 + 0.931476i $$0.618522\pi$$
$$18$$ 0 0
$$19$$ 6.00000 1.37649 0.688247 0.725476i $$-0.258380\pi$$
0.688247 + 0.725476i $$0.258380\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ −1.00000 −0.208514
$$24$$ 0 0
$$25$$ 1.00000 0.200000
$$26$$ 0 0
$$27$$ 0 0
$$28$$ 0 0
$$29$$ 3.00000 0.557086 0.278543 0.960424i $$-0.410149\pi$$
0.278543 + 0.960424i $$0.410149\pi$$
$$30$$ 0 0
$$31$$ 3.00000 0.538816 0.269408 0.963026i $$-0.413172\pi$$
0.269408 + 0.963026i $$0.413172\pi$$
$$32$$ 0 0
$$33$$ 0 0
$$34$$ 0 0
$$35$$ −1.00000 −0.169031
$$36$$ 0 0
$$37$$ 1.00000 0.164399 0.0821995 0.996616i $$-0.473806\pi$$
0.0821995 + 0.996616i $$0.473806\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 0 0
$$41$$ 9.00000 1.40556 0.702782 0.711405i $$-0.251941\pi$$
0.702782 + 0.711405i $$0.251941\pi$$
$$42$$ 0 0
$$43$$ 8.00000 1.21999 0.609994 0.792406i $$-0.291172\pi$$
0.609994 + 0.792406i $$0.291172\pi$$
$$44$$ 0 0
$$45$$ −3.00000 −0.447214
$$46$$ 0 0
$$47$$ −4.00000 −0.583460 −0.291730 0.956501i $$-0.594231\pi$$
−0.291730 + 0.956501i $$0.594231\pi$$
$$48$$ 0 0
$$49$$ −6.00000 −0.857143
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 0 0
$$53$$ 1.00000 0.137361 0.0686803 0.997639i $$-0.478121\pi$$
0.0686803 + 0.997639i $$0.478121\pi$$
$$54$$ 0 0
$$55$$ 6.00000 0.809040
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 0 0
$$59$$ −1.00000 −0.130189 −0.0650945 0.997879i $$-0.520735\pi$$
−0.0650945 + 0.997879i $$0.520735\pi$$
$$60$$ 0 0
$$61$$ 8.00000 1.02430 0.512148 0.858898i $$-0.328850\pi$$
0.512148 + 0.858898i $$0.328850\pi$$
$$62$$ 0 0
$$63$$ 3.00000 0.377964
$$64$$ 0 0
$$65$$ −2.00000 −0.248069
$$66$$ 0 0
$$67$$ 7.00000 0.855186 0.427593 0.903971i $$-0.359362\pi$$
0.427593 + 0.903971i $$0.359362\pi$$
$$68$$ 0 0
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 5.00000 0.593391 0.296695 0.954972i $$-0.404115\pi$$
0.296695 + 0.954972i $$0.404115\pi$$
$$72$$ 0 0
$$73$$ −6.00000 −0.702247 −0.351123 0.936329i $$-0.614200\pi$$
−0.351123 + 0.936329i $$0.614200\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ −6.00000 −0.683763
$$78$$ 0 0
$$79$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$80$$ 0 0
$$81$$ 9.00000 1.00000
$$82$$ 0 0
$$83$$ 11.0000 1.20741 0.603703 0.797209i $$-0.293691\pi$$
0.603703 + 0.797209i $$0.293691\pi$$
$$84$$ 0 0
$$85$$ −3.00000 −0.325396
$$86$$ 0 0
$$87$$ 0 0
$$88$$ 0 0
$$89$$ 4.00000 0.423999 0.212000 0.977270i $$-0.432002\pi$$
0.212000 + 0.977270i $$0.432002\pi$$
$$90$$ 0 0
$$91$$ 2.00000 0.209657
$$92$$ 0 0
$$93$$ 0 0
$$94$$ 0 0
$$95$$ 6.00000 0.615587
$$96$$ 0 0
$$97$$ 6.00000 0.609208 0.304604 0.952479i $$-0.401476\pi$$
0.304604 + 0.952479i $$0.401476\pi$$
$$98$$ 0 0
$$99$$ −18.0000 −1.80907
$$100$$ 0 0
$$101$$ −9.00000 −0.895533 −0.447767 0.894150i $$-0.647781\pi$$
−0.447767 + 0.894150i $$0.647781\pi$$
$$102$$ 0 0
$$103$$ 16.0000 1.57653 0.788263 0.615338i $$-0.210980\pi$$
0.788263 + 0.615338i $$0.210980\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ −5.00000 −0.483368 −0.241684 0.970355i $$-0.577700\pi$$
−0.241684 + 0.970355i $$0.577700\pi$$
$$108$$ 0 0
$$109$$ −6.00000 −0.574696 −0.287348 0.957826i $$-0.592774\pi$$
−0.287348 + 0.957826i $$0.592774\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 0 0
$$113$$ −7.00000 −0.658505 −0.329252 0.944242i $$-0.606797\pi$$
−0.329252 + 0.944242i $$0.606797\pi$$
$$114$$ 0 0
$$115$$ −1.00000 −0.0932505
$$116$$ 0 0
$$117$$ 6.00000 0.554700
$$118$$ 0 0
$$119$$ 3.00000 0.275010
$$120$$ 0 0
$$121$$ 25.0000 2.27273
$$122$$ 0 0
$$123$$ 0 0
$$124$$ 0 0
$$125$$ 1.00000 0.0894427
$$126$$ 0 0
$$127$$ −8.00000 −0.709885 −0.354943 0.934888i $$-0.615500\pi$$
−0.354943 + 0.934888i $$0.615500\pi$$
$$128$$ 0 0
$$129$$ 0 0
$$130$$ 0 0
$$131$$ −12.0000 −1.04844 −0.524222 0.851581i $$-0.675644\pi$$
−0.524222 + 0.851581i $$0.675644\pi$$
$$132$$ 0 0
$$133$$ −6.00000 −0.520266
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ 10.0000 0.854358 0.427179 0.904167i $$-0.359507\pi$$
0.427179 + 0.904167i $$0.359507\pi$$
$$138$$ 0 0
$$139$$ −7.00000 −0.593732 −0.296866 0.954919i $$-0.595942\pi$$
−0.296866 + 0.954919i $$0.595942\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 0 0
$$143$$ −12.0000 −1.00349
$$144$$ 0 0
$$145$$ 3.00000 0.249136
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ 20.0000 1.63846 0.819232 0.573462i $$-0.194400\pi$$
0.819232 + 0.573462i $$0.194400\pi$$
$$150$$ 0 0
$$151$$ −4.00000 −0.325515 −0.162758 0.986666i $$-0.552039\pi$$
−0.162758 + 0.986666i $$0.552039\pi$$
$$152$$ 0 0
$$153$$ 9.00000 0.727607
$$154$$ 0 0
$$155$$ 3.00000 0.240966
$$156$$ 0 0
$$157$$ 3.00000 0.239426 0.119713 0.992809i $$-0.461803\pi$$
0.119713 + 0.992809i $$0.461803\pi$$
$$158$$ 0 0
$$159$$ 0 0
$$160$$ 0 0
$$161$$ 1.00000 0.0788110
$$162$$ 0 0
$$163$$ 14.0000 1.09656 0.548282 0.836293i $$-0.315282\pi$$
0.548282 + 0.836293i $$0.315282\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 8.00000 0.619059 0.309529 0.950890i $$-0.399829\pi$$
0.309529 + 0.950890i $$0.399829\pi$$
$$168$$ 0 0
$$169$$ −9.00000 −0.692308
$$170$$ 0 0
$$171$$ −18.0000 −1.37649
$$172$$ 0 0
$$173$$ −16.0000 −1.21646 −0.608229 0.793762i $$-0.708120\pi$$
−0.608229 + 0.793762i $$0.708120\pi$$
$$174$$ 0 0
$$175$$ −1.00000 −0.0755929
$$176$$ 0 0
$$177$$ 0 0
$$178$$ 0 0
$$179$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$180$$ 0 0
$$181$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 0 0
$$185$$ 1.00000 0.0735215
$$186$$ 0 0
$$187$$ −18.0000 −1.31629
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ −8.00000 −0.578860 −0.289430 0.957199i $$-0.593466\pi$$
−0.289430 + 0.957199i $$0.593466\pi$$
$$192$$ 0 0
$$193$$ −4.00000 −0.287926 −0.143963 0.989583i $$-0.545985\pi$$
−0.143963 + 0.989583i $$0.545985\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ −18.0000 −1.28245 −0.641223 0.767354i $$-0.721573\pi$$
−0.641223 + 0.767354i $$0.721573\pi$$
$$198$$ 0 0
$$199$$ 22.0000 1.55954 0.779769 0.626067i $$-0.215336\pi$$
0.779769 + 0.626067i $$0.215336\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 0 0
$$203$$ −3.00000 −0.210559
$$204$$ 0 0
$$205$$ 9.00000 0.628587
$$206$$ 0 0
$$207$$ 3.00000 0.208514
$$208$$ 0 0
$$209$$ 36.0000 2.49017
$$210$$ 0 0
$$211$$ 19.0000 1.30801 0.654007 0.756489i $$-0.273087\pi$$
0.654007 + 0.756489i $$0.273087\pi$$
$$212$$ 0 0
$$213$$ 0 0
$$214$$ 0 0
$$215$$ 8.00000 0.545595
$$216$$ 0 0
$$217$$ −3.00000 −0.203653
$$218$$ 0 0
$$219$$ 0 0
$$220$$ 0 0
$$221$$ 6.00000 0.403604
$$222$$ 0 0
$$223$$ −26.0000 −1.74109 −0.870544 0.492090i $$-0.836233\pi$$
−0.870544 + 0.492090i $$0.836233\pi$$
$$224$$ 0 0
$$225$$ −3.00000 −0.200000
$$226$$ 0 0
$$227$$ 4.00000 0.265489 0.132745 0.991150i $$-0.457621\pi$$
0.132745 + 0.991150i $$0.457621\pi$$
$$228$$ 0 0
$$229$$ 4.00000 0.264327 0.132164 0.991228i $$-0.457808\pi$$
0.132164 + 0.991228i $$0.457808\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ 24.0000 1.57229 0.786146 0.618041i $$-0.212073\pi$$
0.786146 + 0.618041i $$0.212073\pi$$
$$234$$ 0 0
$$235$$ −4.00000 −0.260931
$$236$$ 0 0
$$237$$ 0 0
$$238$$ 0 0
$$239$$ −5.00000 −0.323423 −0.161712 0.986838i $$-0.551701\pi$$
−0.161712 + 0.986838i $$0.551701\pi$$
$$240$$ 0 0
$$241$$ 2.00000 0.128831 0.0644157 0.997923i $$-0.479482\pi$$
0.0644157 + 0.997923i $$0.479482\pi$$
$$242$$ 0 0
$$243$$ 0 0
$$244$$ 0 0
$$245$$ −6.00000 −0.383326
$$246$$ 0 0
$$247$$ −12.0000 −0.763542
$$248$$ 0 0
$$249$$ 0 0
$$250$$ 0 0
$$251$$ 2.00000 0.126239 0.0631194 0.998006i $$-0.479895\pi$$
0.0631194 + 0.998006i $$0.479895\pi$$
$$252$$ 0 0
$$253$$ −6.00000 −0.377217
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ 20.0000 1.24757 0.623783 0.781598i $$-0.285595\pi$$
0.623783 + 0.781598i $$0.285595\pi$$
$$258$$ 0 0
$$259$$ −1.00000 −0.0621370
$$260$$ 0 0
$$261$$ −9.00000 −0.557086
$$262$$ 0 0
$$263$$ −3.00000 −0.184988 −0.0924940 0.995713i $$-0.529484\pi$$
−0.0924940 + 0.995713i $$0.529484\pi$$
$$264$$ 0 0
$$265$$ 1.00000 0.0614295
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 0 0
$$269$$ −31.0000 −1.89010 −0.945052 0.326921i $$-0.893989\pi$$
−0.945052 + 0.326921i $$0.893989\pi$$
$$270$$ 0 0
$$271$$ −19.0000 −1.15417 −0.577084 0.816685i $$-0.695809\pi$$
−0.577084 + 0.816685i $$0.695809\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ 6.00000 0.361814
$$276$$ 0 0
$$277$$ 10.0000 0.600842 0.300421 0.953807i $$-0.402873\pi$$
0.300421 + 0.953807i $$0.402873\pi$$
$$278$$ 0 0
$$279$$ −9.00000 −0.538816
$$280$$ 0 0
$$281$$ 30.0000 1.78965 0.894825 0.446417i $$-0.147300\pi$$
0.894825 + 0.446417i $$0.147300\pi$$
$$282$$ 0 0
$$283$$ 17.0000 1.01055 0.505273 0.862960i $$-0.331392\pi$$
0.505273 + 0.862960i $$0.331392\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ −9.00000 −0.531253
$$288$$ 0 0
$$289$$ −8.00000 −0.470588
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 0 0
$$293$$ −21.0000 −1.22683 −0.613417 0.789760i $$-0.710205\pi$$
−0.613417 + 0.789760i $$0.710205\pi$$
$$294$$ 0 0
$$295$$ −1.00000 −0.0582223
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ 2.00000 0.115663
$$300$$ 0 0
$$301$$ −8.00000 −0.461112
$$302$$ 0 0
$$303$$ 0 0
$$304$$ 0 0
$$305$$ 8.00000 0.458079
$$306$$ 0 0
$$307$$ −22.0000 −1.25561 −0.627803 0.778372i $$-0.716046\pi$$
−0.627803 + 0.778372i $$0.716046\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 0 0
$$311$$ 8.00000 0.453638 0.226819 0.973937i $$-0.427167\pi$$
0.226819 + 0.973937i $$0.427167\pi$$
$$312$$ 0 0
$$313$$ −29.0000 −1.63918 −0.819588 0.572953i $$-0.805798\pi$$
−0.819588 + 0.572953i $$0.805798\pi$$
$$314$$ 0 0
$$315$$ 3.00000 0.169031
$$316$$ 0 0
$$317$$ −8.00000 −0.449325 −0.224662 0.974437i $$-0.572128\pi$$
−0.224662 + 0.974437i $$0.572128\pi$$
$$318$$ 0 0
$$319$$ 18.0000 1.00781
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 0 0
$$323$$ −18.0000 −1.00155
$$324$$ 0 0
$$325$$ −2.00000 −0.110940
$$326$$ 0 0
$$327$$ 0 0
$$328$$ 0 0
$$329$$ 4.00000 0.220527
$$330$$ 0 0
$$331$$ 7.00000 0.384755 0.192377 0.981321i $$-0.438380\pi$$
0.192377 + 0.981321i $$0.438380\pi$$
$$332$$ 0 0
$$333$$ −3.00000 −0.164399
$$334$$ 0 0
$$335$$ 7.00000 0.382451
$$336$$ 0 0
$$337$$ −34.0000 −1.85210 −0.926049 0.377403i $$-0.876817\pi$$
−0.926049 + 0.377403i $$0.876817\pi$$
$$338$$ 0 0
$$339$$ 0 0
$$340$$ 0 0
$$341$$ 18.0000 0.974755
$$342$$ 0 0
$$343$$ 13.0000 0.701934
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ −24.0000 −1.28839 −0.644194 0.764862i $$-0.722807\pi$$
−0.644194 + 0.764862i $$0.722807\pi$$
$$348$$ 0 0
$$349$$ −25.0000 −1.33822 −0.669110 0.743164i $$-0.733324\pi$$
−0.669110 + 0.743164i $$0.733324\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 0 0
$$353$$ −8.00000 −0.425797 −0.212899 0.977074i $$-0.568290\pi$$
−0.212899 + 0.977074i $$0.568290\pi$$
$$354$$ 0 0
$$355$$ 5.00000 0.265372
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ −36.0000 −1.90001 −0.950004 0.312239i $$-0.898921\pi$$
−0.950004 + 0.312239i $$0.898921\pi$$
$$360$$ 0 0
$$361$$ 17.0000 0.894737
$$362$$ 0 0
$$363$$ 0 0
$$364$$ 0 0
$$365$$ −6.00000 −0.314054
$$366$$ 0 0
$$367$$ −35.0000 −1.82699 −0.913493 0.406855i $$-0.866625\pi$$
−0.913493 + 0.406855i $$0.866625\pi$$
$$368$$ 0 0
$$369$$ −27.0000 −1.40556
$$370$$ 0 0
$$371$$ −1.00000 −0.0519174
$$372$$ 0 0
$$373$$ −38.0000 −1.96757 −0.983783 0.179364i $$-0.942596\pi$$
−0.983783 + 0.179364i $$0.942596\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ −6.00000 −0.309016
$$378$$ 0 0
$$379$$ −20.0000 −1.02733 −0.513665 0.857991i $$-0.671713\pi$$
−0.513665 + 0.857991i $$0.671713\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 0 0
$$383$$ 21.0000 1.07305 0.536525 0.843884i $$-0.319737\pi$$
0.536525 + 0.843884i $$0.319737\pi$$
$$384$$ 0 0
$$385$$ −6.00000 −0.305788
$$386$$ 0 0
$$387$$ −24.0000 −1.21999
$$388$$ 0 0
$$389$$ 26.0000 1.31825 0.659126 0.752032i $$-0.270926\pi$$
0.659126 + 0.752032i $$0.270926\pi$$
$$390$$ 0 0
$$391$$ 3.00000 0.151717
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 0 0
$$395$$ 0 0
$$396$$ 0 0
$$397$$ −32.0000 −1.60603 −0.803017 0.595956i $$-0.796773\pi$$
−0.803017 + 0.595956i $$0.796773\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 18.0000 0.898877 0.449439 0.893311i $$-0.351624\pi$$
0.449439 + 0.893311i $$0.351624\pi$$
$$402$$ 0 0
$$403$$ −6.00000 −0.298881
$$404$$ 0 0
$$405$$ 9.00000 0.447214
$$406$$ 0 0
$$407$$ 6.00000 0.297409
$$408$$ 0 0
$$409$$ −35.0000 −1.73064 −0.865319 0.501221i $$-0.832884\pi$$
−0.865319 + 0.501221i $$0.832884\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 0 0
$$413$$ 1.00000 0.0492068
$$414$$ 0 0
$$415$$ 11.0000 0.539969
$$416$$ 0 0
$$417$$ 0 0
$$418$$ 0 0
$$419$$ 14.0000 0.683945 0.341972 0.939710i $$-0.388905\pi$$
0.341972 + 0.939710i $$0.388905\pi$$
$$420$$ 0 0
$$421$$ −34.0000 −1.65706 −0.828529 0.559946i $$-0.810822\pi$$
−0.828529 + 0.559946i $$0.810822\pi$$
$$422$$ 0 0
$$423$$ 12.0000 0.583460
$$424$$ 0 0
$$425$$ −3.00000 −0.145521
$$426$$ 0 0
$$427$$ −8.00000 −0.387147
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$432$$ 0 0
$$433$$ 17.0000 0.816968 0.408484 0.912766i $$-0.366058\pi$$
0.408484 + 0.912766i $$0.366058\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ −6.00000 −0.287019
$$438$$ 0 0
$$439$$ −32.0000 −1.52728 −0.763638 0.645644i $$-0.776589\pi$$
−0.763638 + 0.645644i $$0.776589\pi$$
$$440$$ 0 0
$$441$$ 18.0000 0.857143
$$442$$ 0 0
$$443$$ 36.0000 1.71041 0.855206 0.518289i $$-0.173431\pi$$
0.855206 + 0.518289i $$0.173431\pi$$
$$444$$ 0 0
$$445$$ 4.00000 0.189618
$$446$$ 0 0
$$447$$ 0 0
$$448$$ 0 0
$$449$$ 3.00000 0.141579 0.0707894 0.997491i $$-0.477448\pi$$
0.0707894 + 0.997491i $$0.477448\pi$$
$$450$$ 0 0
$$451$$ 54.0000 2.54276
$$452$$ 0 0
$$453$$ 0 0
$$454$$ 0 0
$$455$$ 2.00000 0.0937614
$$456$$ 0 0
$$457$$ −25.0000 −1.16945 −0.584725 0.811231i $$-0.698798\pi$$
−0.584725 + 0.811231i $$0.698798\pi$$
$$458$$ 0 0
$$459$$ 0 0
$$460$$ 0 0
$$461$$ 22.0000 1.02464 0.512321 0.858794i $$-0.328786\pi$$
0.512321 + 0.858794i $$0.328786\pi$$
$$462$$ 0 0
$$463$$ −22.0000 −1.02243 −0.511213 0.859454i $$-0.670804\pi$$
−0.511213 + 0.859454i $$0.670804\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ 27.0000 1.24941 0.624705 0.780860i $$-0.285219\pi$$
0.624705 + 0.780860i $$0.285219\pi$$
$$468$$ 0 0
$$469$$ −7.00000 −0.323230
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 0 0
$$473$$ 48.0000 2.20704
$$474$$ 0 0
$$475$$ 6.00000 0.275299
$$476$$ 0 0
$$477$$ −3.00000 −0.137361
$$478$$ 0 0
$$479$$ 4.00000 0.182765 0.0913823 0.995816i $$-0.470871\pi$$
0.0913823 + 0.995816i $$0.470871\pi$$
$$480$$ 0 0
$$481$$ −2.00000 −0.0911922
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ 6.00000 0.272446
$$486$$ 0 0
$$487$$ 40.0000 1.81257 0.906287 0.422664i $$-0.138905\pi$$
0.906287 + 0.422664i $$0.138905\pi$$
$$488$$ 0 0
$$489$$ 0 0
$$490$$ 0 0
$$491$$ −11.0000 −0.496423 −0.248212 0.968706i $$-0.579843\pi$$
−0.248212 + 0.968706i $$0.579843\pi$$
$$492$$ 0 0
$$493$$ −9.00000 −0.405340
$$494$$ 0 0
$$495$$ −18.0000 −0.809040
$$496$$ 0 0
$$497$$ −5.00000 −0.224281
$$498$$ 0 0
$$499$$ 13.0000 0.581960 0.290980 0.956729i $$-0.406019\pi$$
0.290980 + 0.956729i $$0.406019\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 0 0
$$503$$ 17.0000 0.757993 0.378996 0.925398i $$-0.376269\pi$$
0.378996 + 0.925398i $$0.376269\pi$$
$$504$$ 0 0
$$505$$ −9.00000 −0.400495
$$506$$ 0 0
$$507$$ 0 0
$$508$$ 0 0
$$509$$ 18.0000 0.797836 0.398918 0.916987i $$-0.369386\pi$$
0.398918 + 0.916987i $$0.369386\pi$$
$$510$$ 0 0
$$511$$ 6.00000 0.265424
$$512$$ 0 0
$$513$$ 0 0
$$514$$ 0 0
$$515$$ 16.0000 0.705044
$$516$$ 0 0
$$517$$ −24.0000 −1.05552
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 0 0
$$521$$ −10.0000 −0.438108 −0.219054 0.975713i $$-0.570297\pi$$
−0.219054 + 0.975713i $$0.570297\pi$$
$$522$$ 0 0
$$523$$ −12.0000 −0.524723 −0.262362 0.964970i $$-0.584501\pi$$
−0.262362 + 0.964970i $$0.584501\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ −9.00000 −0.392046
$$528$$ 0 0
$$529$$ 1.00000 0.0434783
$$530$$ 0 0
$$531$$ 3.00000 0.130189
$$532$$ 0 0
$$533$$ −18.0000 −0.779667
$$534$$ 0 0
$$535$$ −5.00000 −0.216169
$$536$$ 0 0
$$537$$ 0 0
$$538$$ 0 0
$$539$$ −36.0000 −1.55063
$$540$$ 0 0
$$541$$ 38.0000 1.63375 0.816874 0.576816i $$-0.195705\pi$$
0.816874 + 0.576816i $$0.195705\pi$$
$$542$$ 0 0
$$543$$ 0 0
$$544$$ 0 0
$$545$$ −6.00000 −0.257012
$$546$$ 0 0
$$547$$ −8.00000 −0.342055 −0.171028 0.985266i $$-0.554709\pi$$
−0.171028 + 0.985266i $$0.554709\pi$$
$$548$$ 0 0
$$549$$ −24.0000 −1.02430
$$550$$ 0 0
$$551$$ 18.0000 0.766826
$$552$$ 0 0
$$553$$ 0 0
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ −31.0000 −1.31351 −0.656756 0.754103i $$-0.728072\pi$$
−0.656756 + 0.754103i $$0.728072\pi$$
$$558$$ 0 0
$$559$$ −16.0000 −0.676728
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ −11.0000 −0.463595 −0.231797 0.972764i $$-0.574461\pi$$
−0.231797 + 0.972764i $$0.574461\pi$$
$$564$$ 0 0
$$565$$ −7.00000 −0.294492
$$566$$ 0 0
$$567$$ −9.00000 −0.377964
$$568$$ 0 0
$$569$$ −42.0000 −1.76073 −0.880366 0.474295i $$-0.842703\pi$$
−0.880366 + 0.474295i $$0.842703\pi$$
$$570$$ 0 0
$$571$$ 16.0000 0.669579 0.334790 0.942293i $$-0.391335\pi$$
0.334790 + 0.942293i $$0.391335\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 0 0
$$575$$ −1.00000 −0.0417029
$$576$$ 0 0
$$577$$ 12.0000 0.499567 0.249783 0.968302i $$-0.419641\pi$$
0.249783 + 0.968302i $$0.419641\pi$$
$$578$$ 0 0
$$579$$ 0 0
$$580$$ 0 0
$$581$$ −11.0000 −0.456357
$$582$$ 0 0
$$583$$ 6.00000 0.248495
$$584$$ 0 0
$$585$$ 6.00000 0.248069
$$586$$ 0 0
$$587$$ 18.0000 0.742940 0.371470 0.928445i $$-0.378854\pi$$
0.371470 + 0.928445i $$0.378854\pi$$
$$588$$ 0 0
$$589$$ 18.0000 0.741677
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 0 0
$$593$$ 28.0000 1.14982 0.574911 0.818216i $$-0.305037\pi$$
0.574911 + 0.818216i $$0.305037\pi$$
$$594$$ 0 0
$$595$$ 3.00000 0.122988
$$596$$ 0 0
$$597$$ 0 0
$$598$$ 0 0
$$599$$ 24.0000 0.980613 0.490307 0.871550i $$-0.336885\pi$$
0.490307 + 0.871550i $$0.336885\pi$$
$$600$$ 0 0
$$601$$ −5.00000 −0.203954 −0.101977 0.994787i $$-0.532517\pi$$
−0.101977 + 0.994787i $$0.532517\pi$$
$$602$$ 0 0
$$603$$ −21.0000 −0.855186
$$604$$ 0 0
$$605$$ 25.0000 1.01639
$$606$$ 0 0
$$607$$ 32.0000 1.29884 0.649420 0.760430i $$-0.275012\pi$$
0.649420 + 0.760430i $$0.275012\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 8.00000 0.323645
$$612$$ 0 0
$$613$$ −22.0000 −0.888572 −0.444286 0.895885i $$-0.646543\pi$$
−0.444286 + 0.895885i $$0.646543\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ −21.0000 −0.845428 −0.422714 0.906263i $$-0.638923\pi$$
−0.422714 + 0.906263i $$0.638923\pi$$
$$618$$ 0 0
$$619$$ 4.00000 0.160774 0.0803868 0.996764i $$-0.474384\pi$$
0.0803868 + 0.996764i $$0.474384\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 0 0
$$623$$ −4.00000 −0.160257
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ −3.00000 −0.119618
$$630$$ 0 0
$$631$$ 4.00000 0.159237 0.0796187 0.996825i $$-0.474630\pi$$
0.0796187 + 0.996825i $$0.474630\pi$$
$$632$$ 0 0
$$633$$ 0 0
$$634$$ 0 0
$$635$$ −8.00000 −0.317470
$$636$$ 0 0
$$637$$ 12.0000 0.475457
$$638$$ 0 0
$$639$$ −15.0000 −0.593391
$$640$$ 0 0
$$641$$ −22.0000 −0.868948 −0.434474 0.900684i $$-0.643066\pi$$
−0.434474 + 0.900684i $$0.643066\pi$$
$$642$$ 0 0
$$643$$ −37.0000 −1.45914 −0.729569 0.683907i $$-0.760279\pi$$
−0.729569 + 0.683907i $$0.760279\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ −42.0000 −1.65119 −0.825595 0.564263i $$-0.809160\pi$$
−0.825595 + 0.564263i $$0.809160\pi$$
$$648$$ 0 0
$$649$$ −6.00000 −0.235521
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ 6.00000 0.234798 0.117399 0.993085i $$-0.462544\pi$$
0.117399 + 0.993085i $$0.462544\pi$$
$$654$$ 0 0
$$655$$ −12.0000 −0.468879
$$656$$ 0 0
$$657$$ 18.0000 0.702247
$$658$$ 0 0
$$659$$ 22.0000 0.856998 0.428499 0.903542i $$-0.359042\pi$$
0.428499 + 0.903542i $$0.359042\pi$$
$$660$$ 0 0
$$661$$ −42.0000 −1.63361 −0.816805 0.576913i $$-0.804257\pi$$
−0.816805 + 0.576913i $$0.804257\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ −6.00000 −0.232670
$$666$$ 0 0
$$667$$ −3.00000 −0.116160
$$668$$ 0 0
$$669$$ 0 0
$$670$$ 0 0
$$671$$ 48.0000 1.85302
$$672$$ 0 0
$$673$$ 28.0000 1.07932 0.539660 0.841883i $$-0.318553\pi$$
0.539660 + 0.841883i $$0.318553\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ 47.0000 1.80636 0.903178 0.429265i $$-0.141228\pi$$
0.903178 + 0.429265i $$0.141228\pi$$
$$678$$ 0 0
$$679$$ −6.00000 −0.230259
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 0 0
$$683$$ −44.0000 −1.68361 −0.841807 0.539779i $$-0.818508\pi$$
−0.841807 + 0.539779i $$0.818508\pi$$
$$684$$ 0 0
$$685$$ 10.0000 0.382080
$$686$$ 0 0
$$687$$ 0 0
$$688$$ 0 0
$$689$$ −2.00000 −0.0761939
$$690$$ 0 0
$$691$$ −4.00000 −0.152167 −0.0760836 0.997101i $$-0.524242\pi$$
−0.0760836 + 0.997101i $$0.524242\pi$$
$$692$$ 0 0
$$693$$ 18.0000 0.683763
$$694$$ 0 0
$$695$$ −7.00000 −0.265525
$$696$$ 0 0
$$697$$ −27.0000 −1.02270
$$698$$ 0 0
$$699$$ 0 0
$$700$$ 0 0
$$701$$ −24.0000 −0.906467 −0.453234 0.891392i $$-0.649730\pi$$
−0.453234 + 0.891392i $$0.649730\pi$$
$$702$$ 0 0
$$703$$ 6.00000 0.226294
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ 9.00000 0.338480
$$708$$ 0 0
$$709$$ 8.00000 0.300446 0.150223 0.988652i $$-0.452001\pi$$
0.150223 + 0.988652i $$0.452001\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 0 0
$$713$$ −3.00000 −0.112351
$$714$$ 0 0
$$715$$ −12.0000 −0.448775
$$716$$ 0 0
$$717$$ 0 0
$$718$$ 0 0
$$719$$ −13.0000 −0.484818 −0.242409 0.970174i $$-0.577938\pi$$
−0.242409 + 0.970174i $$0.577938\pi$$
$$720$$ 0 0
$$721$$ −16.0000 −0.595871
$$722$$ 0 0
$$723$$ 0 0
$$724$$ 0 0
$$725$$ 3.00000 0.111417
$$726$$ 0 0
$$727$$ 21.0000 0.778847 0.389423 0.921059i $$-0.372674\pi$$
0.389423 + 0.921059i $$0.372674\pi$$
$$728$$ 0 0
$$729$$ −27.0000 −1.00000
$$730$$ 0 0
$$731$$ −24.0000 −0.887672
$$732$$ 0 0
$$733$$ 25.0000 0.923396 0.461698 0.887037i $$-0.347240\pi$$
0.461698 + 0.887037i $$0.347240\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 42.0000 1.54709
$$738$$ 0 0
$$739$$ 19.0000 0.698926 0.349463 0.936950i $$-0.386364\pi$$
0.349463 + 0.936950i $$0.386364\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ 24.0000 0.880475 0.440237 0.897881i $$-0.354894\pi$$
0.440237 + 0.897881i $$0.354894\pi$$
$$744$$ 0 0
$$745$$ 20.0000 0.732743
$$746$$ 0 0
$$747$$ −33.0000 −1.20741
$$748$$ 0 0
$$749$$ 5.00000 0.182696
$$750$$ 0 0
$$751$$ 14.0000 0.510867 0.255434 0.966827i $$-0.417782\pi$$
0.255434 + 0.966827i $$0.417782\pi$$
$$752$$ 0 0
$$753$$ 0 0
$$754$$ 0 0
$$755$$ −4.00000 −0.145575
$$756$$ 0 0
$$757$$ 15.0000 0.545184 0.272592 0.962130i $$-0.412119\pi$$
0.272592 + 0.962130i $$0.412119\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 29.0000 1.05125 0.525625 0.850717i $$-0.323832\pi$$
0.525625 + 0.850717i $$0.323832\pi$$
$$762$$ 0 0
$$763$$ 6.00000 0.217215
$$764$$ 0 0
$$765$$ 9.00000 0.325396
$$766$$ 0 0
$$767$$ 2.00000 0.0722158
$$768$$ 0 0
$$769$$ 44.0000 1.58668 0.793340 0.608778i $$-0.208340\pi$$
0.793340 + 0.608778i $$0.208340\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 0 0
$$773$$ −22.0000 −0.791285 −0.395643 0.918405i $$-0.629478\pi$$
−0.395643 + 0.918405i $$0.629478\pi$$
$$774$$ 0 0
$$775$$ 3.00000 0.107763
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ 54.0000 1.93475
$$780$$ 0 0
$$781$$ 30.0000 1.07348
$$782$$ 0 0
$$783$$ 0 0
$$784$$ 0 0
$$785$$ 3.00000 0.107075
$$786$$ 0 0
$$787$$ −13.0000 −0.463400 −0.231700 0.972787i $$-0.574429\pi$$
−0.231700 + 0.972787i $$0.574429\pi$$
$$788$$ 0 0
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 7.00000 0.248891
$$792$$ 0 0
$$793$$ −16.0000 −0.568177
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ −33.0000 −1.16892 −0.584460 0.811423i $$-0.698694\pi$$
−0.584460 + 0.811423i $$0.698694\pi$$
$$798$$ 0 0
$$799$$ 12.0000 0.424529
$$800$$ 0 0
$$801$$ −12.0000 −0.423999
$$802$$ 0 0
$$803$$ −36.0000 −1.27041
$$804$$ 0 0
$$805$$ 1.00000 0.0352454
$$806$$ 0 0
$$807$$ 0 0
$$808$$ 0 0
$$809$$ 47.0000 1.65243 0.826216 0.563353i $$-0.190489\pi$$
0.826216 + 0.563353i $$0.190489\pi$$
$$810$$ 0 0
$$811$$ −37.0000 −1.29925 −0.649623 0.760257i $$-0.725073\pi$$
−0.649623 + 0.760257i $$0.725073\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ 0 0
$$815$$ 14.0000 0.490399
$$816$$ 0 0
$$817$$ 48.0000 1.67931
$$818$$ 0 0
$$819$$ −6.00000 −0.209657
$$820$$ 0 0
$$821$$ 42.0000 1.46581 0.732905 0.680331i $$-0.238164\pi$$
0.732905 + 0.680331i $$0.238164\pi$$
$$822$$ 0 0
$$823$$ −24.0000 −0.836587 −0.418294 0.908312i $$-0.637372\pi$$
−0.418294 + 0.908312i $$0.637372\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 57.0000 1.98208 0.991042 0.133550i $$-0.0426376\pi$$
0.991042 + 0.133550i $$0.0426376\pi$$
$$828$$ 0 0
$$829$$ 33.0000 1.14614 0.573069 0.819507i $$-0.305753\pi$$
0.573069 + 0.819507i $$0.305753\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 0 0
$$833$$ 18.0000 0.623663
$$834$$ 0 0
$$835$$ 8.00000 0.276851
$$836$$ 0 0
$$837$$ 0 0
$$838$$ 0 0
$$839$$ −30.0000 −1.03572 −0.517858 0.855467i $$-0.673270\pi$$
−0.517858 + 0.855467i $$0.673270\pi$$
$$840$$ 0 0
$$841$$ −20.0000 −0.689655
$$842$$ 0 0
$$843$$ 0 0
$$844$$ 0 0
$$845$$ −9.00000 −0.309609
$$846$$ 0 0
$$847$$ −25.0000 −0.859010
$$848$$ 0 0
$$849$$ 0 0
$$850$$ 0 0
$$851$$ −1.00000 −0.0342796
$$852$$ 0 0
$$853$$ −2.00000 −0.0684787 −0.0342393 0.999414i $$-0.510901\pi$$
−0.0342393 + 0.999414i $$0.510901\pi$$
$$854$$ 0 0
$$855$$ −18.0000 −0.615587
$$856$$ 0 0
$$857$$ 14.0000 0.478231 0.239115 0.970991i $$-0.423143\pi$$
0.239115 + 0.970991i $$0.423143\pi$$
$$858$$ 0 0
$$859$$ −1.00000 −0.0341196 −0.0170598 0.999854i $$-0.505431\pi$$
−0.0170598 + 0.999854i $$0.505431\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ −22.0000 −0.748889 −0.374444 0.927249i $$-0.622167\pi$$
−0.374444 + 0.927249i $$0.622167\pi$$
$$864$$ 0 0
$$865$$ −16.0000 −0.544016
$$866$$ 0 0
$$867$$ 0 0
$$868$$ 0 0
$$869$$ 0 0
$$870$$ 0 0
$$871$$ −14.0000 −0.474372
$$872$$ 0 0
$$873$$ −18.0000 −0.609208
$$874$$ 0 0
$$875$$ −1.00000 −0.0338062
$$876$$ 0 0
$$877$$ 6.00000 0.202606 0.101303 0.994856i $$-0.467699\pi$$
0.101303 + 0.994856i $$0.467699\pi$$
$$878$$ 0 0
$$879$$ 0 0
$$880$$ 0 0
$$881$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$882$$ 0 0
$$883$$ 36.0000 1.21150 0.605748 0.795656i $$-0.292874\pi$$
0.605748 + 0.795656i $$0.292874\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ 30.0000 1.00730 0.503651 0.863907i $$-0.331990\pi$$
0.503651 + 0.863907i $$0.331990\pi$$
$$888$$ 0 0
$$889$$ 8.00000 0.268311
$$890$$ 0 0
$$891$$ 54.0000 1.80907
$$892$$ 0 0
$$893$$ −24.0000 −0.803129
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 0 0
$$899$$ 9.00000 0.300167
$$900$$ 0 0
$$901$$ −3.00000 −0.0999445
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ 0 0
$$906$$ 0 0
$$907$$ 7.00000 0.232431 0.116216 0.993224i $$-0.462924\pi$$
0.116216 + 0.993224i $$0.462924\pi$$
$$908$$ 0 0
$$909$$ 27.0000 0.895533
$$910$$ 0 0
$$911$$ 20.0000 0.662630 0.331315 0.943520i $$-0.392508\pi$$
0.331315 + 0.943520i $$0.392508\pi$$
$$912$$ 0 0
$$913$$ 66.0000 2.18428
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ 12.0000 0.396275
$$918$$ 0 0
$$919$$ 32.0000 1.05558 0.527791 0.849374i $$-0.323020\pi$$
0.527791 + 0.849374i $$0.323020\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 0 0
$$923$$ −10.0000 −0.329154
$$924$$ 0 0
$$925$$ 1.00000 0.0328798
$$926$$ 0 0
$$927$$ −48.0000 −1.57653
$$928$$ 0 0
$$929$$ −29.0000 −0.951459 −0.475730 0.879592i $$-0.657816\pi$$
−0.475730 + 0.879592i $$0.657816\pi$$
$$930$$ 0 0
$$931$$ −36.0000 −1.17985
$$932$$ 0 0
$$933$$ 0 0
$$934$$ 0 0
$$935$$ −18.0000 −0.588663
$$936$$ 0 0
$$937$$ 22.0000 0.718709 0.359354 0.933201i $$-0.382997\pi$$
0.359354 + 0.933201i $$0.382997\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ 0 0
$$941$$ −24.0000 −0.782378 −0.391189 0.920310i $$-0.627936\pi$$
−0.391189 + 0.920310i $$0.627936\pi$$
$$942$$ 0 0
$$943$$ −9.00000 −0.293080
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 8.00000 0.259965 0.129983 0.991516i $$-0.458508\pi$$
0.129983 + 0.991516i $$0.458508\pi$$
$$948$$ 0 0
$$949$$ 12.0000 0.389536
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 0 0
$$953$$ 54.0000 1.74923 0.874616 0.484817i $$-0.161114\pi$$
0.874616 + 0.484817i $$0.161114\pi$$
$$954$$ 0 0
$$955$$ −8.00000 −0.258874
$$956$$ 0 0
$$957$$ 0 0
$$958$$ 0 0
$$959$$ −10.0000 −0.322917
$$960$$ 0 0
$$961$$ −22.0000 −0.709677
$$962$$ 0 0
$$963$$ 15.0000 0.483368
$$964$$ 0 0
$$965$$ −4.00000 −0.128765
$$966$$ 0 0
$$967$$ 56.0000 1.80084 0.900419 0.435023i $$-0.143260\pi$$
0.900419 + 0.435023i $$0.143260\pi$$
$$968$$ 0 0
$$969$$ 0 0
$$970$$ 0 0
$$971$$ −40.0000 −1.28366 −0.641831 0.766846i $$-0.721825\pi$$
−0.641831 + 0.766846i $$0.721825\pi$$
$$972$$ 0 0
$$973$$ 7.00000 0.224410
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ −33.0000 −1.05576 −0.527882 0.849318i $$-0.677014\pi$$
−0.527882 + 0.849318i $$0.677014\pi$$
$$978$$ 0 0
$$979$$ 24.0000 0.767043
$$980$$ 0 0
$$981$$ 18.0000 0.574696
$$982$$ 0 0
$$983$$ −1.00000 −0.0318950 −0.0159475 0.999873i $$-0.505076\pi$$
−0.0159475 + 0.999873i $$0.505076\pi$$
$$984$$ 0 0
$$985$$ −18.0000 −0.573528
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ −8.00000 −0.254385
$$990$$ 0 0
$$991$$ 15.0000 0.476491 0.238245 0.971205i $$-0.423428\pi$$
0.238245 + 0.971205i $$0.423428\pi$$
$$992$$ 0 0
$$993$$ 0 0
$$994$$ 0 0
$$995$$ 22.0000 0.697447
$$996$$ 0 0
$$997$$ −32.0000 −1.01345 −0.506725 0.862108i $$-0.669144\pi$$
−0.506725 + 0.862108i $$0.669144\pi$$
$$998$$ 0 0
$$999$$ 0 0
Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000

## Twists

By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 1840.2.a.f.1.1 1
4.3 odd 2 920.2.a.c.1.1 1
5.4 even 2 9200.2.a.w.1.1 1
8.3 odd 2 7360.2.a.l.1.1 1
8.5 even 2 7360.2.a.k.1.1 1
12.11 even 2 8280.2.a.j.1.1 1
20.3 even 4 4600.2.e.j.4049.1 2
20.7 even 4 4600.2.e.j.4049.2 2
20.19 odd 2 4600.2.a.h.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
920.2.a.c.1.1 1 4.3 odd 2
1840.2.a.f.1.1 1 1.1 even 1 trivial
4600.2.a.h.1.1 1 20.19 odd 2
4600.2.e.j.4049.1 2 20.3 even 4
4600.2.e.j.4049.2 2 20.7 even 4
7360.2.a.k.1.1 1 8.5 even 2
7360.2.a.l.1.1 1 8.3 odd 2
8280.2.a.j.1.1 1 12.11 even 2
9200.2.a.w.1.1 1 5.4 even 2