# Properties

 Label 4140.2.a.e.1.1 Level $4140$ Weight $2$ Character 4140.1 Self dual yes Analytic conductor $33.058$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4140,2,Mod(1,4140)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4140, 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("4140.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$4140 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4140.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: $$33.0580664368$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 1380) Fricke sign: $$+1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 4140.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+1.00000 q^{5} -5.00000 q^{7} +O(q^{10})$$ $$q+1.00000 q^{5} -5.00000 q^{7} +4.00000 q^{13} +3.00000 q^{17} -4.00000 q^{19} +1.00000 q^{23} +1.00000 q^{25} -1.00000 q^{29} +1.00000 q^{31} -5.00000 q^{35} -1.00000 q^{37} -11.0000 q^{41} +4.00000 q^{43} -6.00000 q^{47} +18.0000 q^{49} -1.00000 q^{53} +1.00000 q^{59} +6.00000 q^{61} +4.00000 q^{65} -9.00000 q^{67} -13.0000 q^{71} +16.0000 q^{79} -9.00000 q^{83} +3.00000 q^{85} -8.00000 q^{89} -20.0000 q^{91} -4.00000 q^{95} +2.00000 q^{97} +O(q^{100})$$

## 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
$$4$$ 0 0
$$5$$ 1.00000 0.447214
$$6$$ 0 0
$$7$$ −5.00000 −1.88982 −0.944911 0.327327i $$-0.893852\pi$$
−0.944911 + 0.327327i $$0.893852\pi$$
$$8$$ 0 0
$$9$$ 0 0
$$10$$ 0 0
$$11$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$12$$ 0 0
$$13$$ 4.00000 1.10940 0.554700 0.832050i $$-0.312833\pi$$
0.554700 + 0.832050i $$0.312833\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ 3.00000 0.727607 0.363803 0.931476i $$-0.381478\pi$$
0.363803 + 0.931476i $$0.381478\pi$$
$$18$$ 0 0
$$19$$ −4.00000 −0.917663 −0.458831 0.888523i $$-0.651732\pi$$
−0.458831 + 0.888523i $$0.651732\pi$$
$$20$$ 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
$$29$$ −1.00000 −0.185695 −0.0928477 0.995680i $$-0.529597\pi$$
−0.0928477 + 0.995680i $$0.529597\pi$$
$$30$$ 0 0
$$31$$ 1.00000 0.179605 0.0898027 0.995960i $$-0.471376\pi$$
0.0898027 + 0.995960i $$0.471376\pi$$
$$32$$ 0 0
$$33$$ 0 0
$$34$$ 0 0
$$35$$ −5.00000 −0.845154
$$36$$ 0 0
$$37$$ −1.00000 −0.164399 −0.0821995 0.996616i $$-0.526194\pi$$
−0.0821995 + 0.996616i $$0.526194\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 0 0
$$41$$ −11.0000 −1.71791 −0.858956 0.512050i $$-0.828886\pi$$
−0.858956 + 0.512050i $$0.828886\pi$$
$$42$$ 0 0
$$43$$ 4.00000 0.609994 0.304997 0.952353i $$-0.401344\pi$$
0.304997 + 0.952353i $$0.401344\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ −6.00000 −0.875190 −0.437595 0.899172i $$-0.644170\pi$$
−0.437595 + 0.899172i $$0.644170\pi$$
$$48$$ 0 0
$$49$$ 18.0000 2.57143
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 0 0
$$53$$ −1.00000 −0.137361 −0.0686803 0.997639i $$-0.521879\pi$$
−0.0686803 + 0.997639i $$0.521879\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 0 0
$$59$$ 1.00000 0.130189 0.0650945 0.997879i $$-0.479265\pi$$
0.0650945 + 0.997879i $$0.479265\pi$$
$$60$$ 0 0
$$61$$ 6.00000 0.768221 0.384111 0.923287i $$-0.374508\pi$$
0.384111 + 0.923287i $$0.374508\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ 4.00000 0.496139
$$66$$ 0 0
$$67$$ −9.00000 −1.09952 −0.549762 0.835321i $$-0.685282\pi$$
−0.549762 + 0.835321i $$0.685282\pi$$
$$68$$ 0 0
$$69$$ 0 0
$$70$$ 0 0
$$71$$ −13.0000 −1.54282 −0.771408 0.636341i $$-0.780447\pi$$
−0.771408 + 0.636341i $$0.780447\pi$$
$$72$$ 0 0
$$73$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ 0 0
$$78$$ 0 0
$$79$$ 16.0000 1.80014 0.900070 0.435745i $$-0.143515\pi$$
0.900070 + 0.435745i $$0.143515\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ 0 0
$$83$$ −9.00000 −0.987878 −0.493939 0.869496i $$-0.664443\pi$$
−0.493939 + 0.869496i $$0.664443\pi$$
$$84$$ 0 0
$$85$$ 3.00000 0.325396
$$86$$ 0 0
$$87$$ 0 0
$$88$$ 0 0
$$89$$ −8.00000 −0.847998 −0.423999 0.905663i $$-0.639374\pi$$
−0.423999 + 0.905663i $$0.639374\pi$$
$$90$$ 0 0
$$91$$ −20.0000 −2.09657
$$92$$ 0 0
$$93$$ 0 0
$$94$$ 0 0
$$95$$ −4.00000 −0.410391
$$96$$ 0 0
$$97$$ 2.00000 0.203069 0.101535 0.994832i $$-0.467625\pi$$
0.101535 + 0.994832i $$0.467625\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 0 0
$$101$$ 7.00000 0.696526 0.348263 0.937397i $$-0.386772\pi$$
0.348263 + 0.937397i $$0.386772\pi$$
$$102$$ 0 0
$$103$$ −16.0000 −1.57653 −0.788263 0.615338i $$-0.789020\pi$$
−0.788263 + 0.615338i $$0.789020\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 3.00000 0.290021 0.145010 0.989430i $$-0.453678\pi$$
0.145010 + 0.989430i $$0.453678\pi$$
$$108$$ 0 0
$$109$$ −12.0000 −1.14939 −0.574696 0.818367i $$-0.694880\pi$$
−0.574696 + 0.818367i $$0.694880\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 0 0
$$113$$ −1.00000 −0.0940721 −0.0470360 0.998893i $$-0.514978\pi$$
−0.0470360 + 0.998893i $$0.514978\pi$$
$$114$$ 0 0
$$115$$ 1.00000 0.0932505
$$116$$ 0 0
$$117$$ 0 0
$$118$$ 0 0
$$119$$ −15.0000 −1.37505
$$120$$ 0 0
$$121$$ −11.0000 −1.00000
$$122$$ 0 0
$$123$$ 0 0
$$124$$ 0 0
$$125$$ 1.00000 0.0894427
$$126$$ 0 0
$$127$$ −18.0000 −1.59724 −0.798621 0.601834i $$-0.794437\pi$$
−0.798621 + 0.601834i $$0.794437\pi$$
$$128$$ 0 0
$$129$$ 0 0
$$130$$ 0 0
$$131$$ −20.0000 −1.74741 −0.873704 0.486458i $$-0.838289\pi$$
−0.873704 + 0.486458i $$0.838289\pi$$
$$132$$ 0 0
$$133$$ 20.0000 1.73422
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ −2.00000 −0.170872 −0.0854358 0.996344i $$-0.527228\pi$$
−0.0854358 + 0.996344i $$0.527228\pi$$
$$138$$ 0 0
$$139$$ 11.0000 0.933008 0.466504 0.884519i $$-0.345513\pi$$
0.466504 + 0.884519i $$0.345513\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 0 0
$$143$$ 0 0
$$144$$ 0 0
$$145$$ −1.00000 −0.0830455
$$146$$ 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
$$151$$ −16.0000 −1.30206 −0.651031 0.759051i $$-0.725663\pi$$
−0.651031 + 0.759051i $$0.725663\pi$$
$$152$$ 0 0
$$153$$ 0 0
$$154$$ 0 0
$$155$$ 1.00000 0.0803219
$$156$$ 0 0
$$157$$ 5.00000 0.399043 0.199522 0.979893i $$-0.436061\pi$$
0.199522 + 0.979893i $$0.436061\pi$$
$$158$$ 0 0
$$159$$ 0 0
$$160$$ 0 0
$$161$$ −5.00000 −0.394055
$$162$$ 0 0
$$163$$ 18.0000 1.40987 0.704934 0.709273i $$-0.250976\pi$$
0.704934 + 0.709273i $$0.250976\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 6.00000 0.464294 0.232147 0.972681i $$-0.425425\pi$$
0.232147 + 0.972681i $$0.425425\pi$$
$$168$$ 0 0
$$169$$ 3.00000 0.230769
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 0 0
$$173$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$174$$ 0 0
$$175$$ −5.00000 −0.377964
$$176$$ 0 0
$$177$$ 0 0
$$178$$ 0 0
$$179$$ −24.0000 −1.79384 −0.896922 0.442189i $$-0.854202\pi$$
−0.896922 + 0.442189i $$0.854202\pi$$
$$180$$ 0 0
$$181$$ −8.00000 −0.594635 −0.297318 0.954779i $$-0.596092\pi$$
−0.297318 + 0.954779i $$0.596092\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 0 0
$$185$$ −1.00000 −0.0735215
$$186$$ 0 0
$$187$$ 0 0
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 10.0000 0.723575 0.361787 0.932261i $$-0.382167\pi$$
0.361787 + 0.932261i $$0.382167\pi$$
$$192$$ 0 0
$$193$$ 12.0000 0.863779 0.431889 0.901927i $$-0.357847\pi$$
0.431889 + 0.901927i $$0.357847\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ −2.00000 −0.142494 −0.0712470 0.997459i $$-0.522698\pi$$
−0.0712470 + 0.997459i $$0.522698\pi$$
$$198$$ 0 0
$$199$$ 2.00000 0.141776 0.0708881 0.997484i $$-0.477417\pi$$
0.0708881 + 0.997484i $$0.477417\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 0 0
$$203$$ 5.00000 0.350931
$$204$$ 0 0
$$205$$ −11.0000 −0.768273
$$206$$ 0 0
$$207$$ 0 0
$$208$$ 0 0
$$209$$ 0 0
$$210$$ 0 0
$$211$$ −23.0000 −1.58339 −0.791693 0.610920i $$-0.790800\pi$$
−0.791693 + 0.610920i $$0.790800\pi$$
$$212$$ 0 0
$$213$$ 0 0
$$214$$ 0 0
$$215$$ 4.00000 0.272798
$$216$$ 0 0
$$217$$ −5.00000 −0.339422
$$218$$ 0 0
$$219$$ 0 0
$$220$$ 0 0
$$221$$ 12.0000 0.807207
$$222$$ 0 0
$$223$$ 14.0000 0.937509 0.468755 0.883328i $$-0.344703\pi$$
0.468755 + 0.883328i $$0.344703\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ −28.0000 −1.85843 −0.929213 0.369546i $$-0.879513\pi$$
−0.929213 + 0.369546i $$0.879513\pi$$
$$228$$ 0 0
$$229$$ −20.0000 −1.32164 −0.660819 0.750546i $$-0.729791\pi$$
−0.660819 + 0.750546i $$0.729791\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ −8.00000 −0.524097 −0.262049 0.965055i $$-0.584398\pi$$
−0.262049 + 0.965055i $$0.584398\pi$$
$$234$$ 0 0
$$235$$ −6.00000 −0.391397
$$236$$ 0 0
$$237$$ 0 0
$$238$$ 0 0
$$239$$ 21.0000 1.35838 0.679189 0.733964i $$-0.262332\pi$$
0.679189 + 0.733964i $$0.262332\pi$$
$$240$$ 0 0
$$241$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$242$$ 0 0
$$243$$ 0 0
$$244$$ 0 0
$$245$$ 18.0000 1.14998
$$246$$ 0 0
$$247$$ −16.0000 −1.01806
$$248$$ 0 0
$$249$$ 0 0
$$250$$ 0 0
$$251$$ −6.00000 −0.378717 −0.189358 0.981908i $$-0.560641\pi$$
−0.189358 + 0.981908i $$0.560641\pi$$
$$252$$ 0 0
$$253$$ 0 0
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ 18.0000 1.12281 0.561405 0.827541i $$-0.310261\pi$$
0.561405 + 0.827541i $$0.310261\pi$$
$$258$$ 0 0
$$259$$ 5.00000 0.310685
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 0 0
$$263$$ 29.0000 1.78822 0.894108 0.447851i $$-0.147810\pi$$
0.894108 + 0.447851i $$0.147810\pi$$
$$264$$ 0 0
$$265$$ −1.00000 −0.0614295
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 0 0
$$269$$ −15.0000 −0.914566 −0.457283 0.889321i $$-0.651177\pi$$
−0.457283 + 0.889321i $$0.651177\pi$$
$$270$$ 0 0
$$271$$ −13.0000 −0.789694 −0.394847 0.918747i $$-0.629202\pi$$
−0.394847 + 0.918747i $$0.629202\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 0 0
$$277$$ 10.0000 0.600842 0.300421 0.953807i $$-0.402873\pi$$
0.300421 + 0.953807i $$0.402873\pi$$
$$278$$ 0 0
$$279$$ 0 0
$$280$$ 0 0
$$281$$ −20.0000 −1.19310 −0.596550 0.802576i $$-0.703462\pi$$
−0.596550 + 0.802576i $$0.703462\pi$$
$$282$$ 0 0
$$283$$ 1.00000 0.0594438 0.0297219 0.999558i $$-0.490538\pi$$
0.0297219 + 0.999558i $$0.490538\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 55.0000 3.24655
$$288$$ 0 0
$$289$$ −8.00000 −0.470588
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 0 0
$$293$$ −7.00000 −0.408944 −0.204472 0.978872i $$-0.565548\pi$$
−0.204472 + 0.978872i $$0.565548\pi$$
$$294$$ 0 0
$$295$$ 1.00000 0.0582223
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ 4.00000 0.231326
$$300$$ 0 0
$$301$$ −20.0000 −1.15278
$$302$$ 0 0
$$303$$ 0 0
$$304$$ 0 0
$$305$$ 6.00000 0.343559
$$306$$ 0 0
$$307$$ 12.0000 0.684876 0.342438 0.939540i $$-0.388747\pi$$
0.342438 + 0.939540i $$0.388747\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 0 0
$$311$$ 32.0000 1.81455 0.907277 0.420534i $$-0.138157\pi$$
0.907277 + 0.420534i $$0.138157\pi$$
$$312$$ 0 0
$$313$$ 21.0000 1.18699 0.593495 0.804838i $$-0.297748\pi$$
0.593495 + 0.804838i $$0.297748\pi$$
$$314$$ 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
$$319$$ 0 0
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 0 0
$$323$$ −12.0000 −0.667698
$$324$$ 0 0
$$325$$ 4.00000 0.221880
$$326$$ 0 0
$$327$$ 0 0
$$328$$ 0 0
$$329$$ 30.0000 1.65395
$$330$$ 0 0
$$331$$ 25.0000 1.37412 0.687062 0.726599i $$-0.258900\pi$$
0.687062 + 0.726599i $$0.258900\pi$$
$$332$$ 0 0
$$333$$ 0 0
$$334$$ 0 0
$$335$$ −9.00000 −0.491723
$$336$$ 0 0
$$337$$ 10.0000 0.544735 0.272367 0.962193i $$-0.412193\pi$$
0.272367 + 0.962193i $$0.412193\pi$$
$$338$$ 0 0
$$339$$ 0 0
$$340$$ 0 0
$$341$$ 0 0
$$342$$ 0 0
$$343$$ −55.0000 −2.96972
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ 8.00000 0.429463 0.214731 0.976673i $$-0.431112\pi$$
0.214731 + 0.976673i $$0.431112\pi$$
$$348$$ 0 0
$$349$$ 19.0000 1.01705 0.508523 0.861048i $$-0.330192\pi$$
0.508523 + 0.861048i $$0.330192\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 0 0
$$353$$ 18.0000 0.958043 0.479022 0.877803i $$-0.340992\pi$$
0.479022 + 0.877803i $$0.340992\pi$$
$$354$$ 0 0
$$355$$ −13.0000 −0.689968
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ 10.0000 0.527780 0.263890 0.964553i $$-0.414994\pi$$
0.263890 + 0.964553i $$0.414994\pi$$
$$360$$ 0 0
$$361$$ −3.00000 −0.157895
$$362$$ 0 0
$$363$$ 0 0
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 0 0
$$367$$ −7.00000 −0.365397 −0.182699 0.983169i $$-0.558483\pi$$
−0.182699 + 0.983169i $$0.558483\pi$$
$$368$$ 0 0
$$369$$ 0 0
$$370$$ 0 0
$$371$$ 5.00000 0.259587
$$372$$ 0 0
$$373$$ −22.0000 −1.13912 −0.569558 0.821951i $$-0.692886\pi$$
−0.569558 + 0.821951i $$0.692886\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ −4.00000 −0.206010
$$378$$ 0 0
$$379$$ −24.0000 −1.23280 −0.616399 0.787434i $$-0.711409\pi$$
−0.616399 + 0.787434i $$0.711409\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 0 0
$$383$$ −27.0000 −1.37964 −0.689818 0.723983i $$-0.742309\pi$$
−0.689818 + 0.723983i $$0.742309\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ 0 0
$$388$$ 0 0
$$389$$ −10.0000 −0.507020 −0.253510 0.967333i $$-0.581585\pi$$
−0.253510 + 0.967333i $$0.581585\pi$$
$$390$$ 0 0
$$391$$ 3.00000 0.151717
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 0 0
$$395$$ 16.0000 0.805047
$$396$$ 0 0
$$397$$ 32.0000 1.60603 0.803017 0.595956i $$-0.203227\pi$$
0.803017 + 0.595956i $$0.203227\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ −38.0000 −1.89763 −0.948815 0.315833i $$-0.897716\pi$$
−0.948815 + 0.315833i $$0.897716\pi$$
$$402$$ 0 0
$$403$$ 4.00000 0.199254
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 0 0
$$408$$ 0 0
$$409$$ −7.00000 −0.346128 −0.173064 0.984911i $$-0.555367\pi$$
−0.173064 + 0.984911i $$0.555367\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 0 0
$$413$$ −5.00000 −0.246034
$$414$$ 0 0
$$415$$ −9.00000 −0.441793
$$416$$ 0 0
$$417$$ 0 0
$$418$$ 0 0
$$419$$ −20.0000 −0.977064 −0.488532 0.872546i $$-0.662467\pi$$
−0.488532 + 0.872546i $$0.662467\pi$$
$$420$$ 0 0
$$421$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$422$$ 0 0
$$423$$ 0 0
$$424$$ 0 0
$$425$$ 3.00000 0.145521
$$426$$ 0 0
$$427$$ −30.0000 −1.45180
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ −12.0000 −0.578020 −0.289010 0.957326i $$-0.593326\pi$$
−0.289010 + 0.957326i $$0.593326\pi$$
$$432$$ 0 0
$$433$$ −17.0000 −0.816968 −0.408484 0.912766i $$-0.633942\pi$$
−0.408484 + 0.912766i $$0.633942\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ −4.00000 −0.191346
$$438$$ 0 0
$$439$$ 24.0000 1.14546 0.572729 0.819745i $$-0.305885\pi$$
0.572729 + 0.819745i $$0.305885\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ 2.00000 0.0950229 0.0475114 0.998871i $$-0.484871\pi$$
0.0475114 + 0.998871i $$0.484871\pi$$
$$444$$ 0 0
$$445$$ −8.00000 −0.379236
$$446$$ 0 0
$$447$$ 0 0
$$448$$ 0 0
$$449$$ −1.00000 −0.0471929 −0.0235965 0.999722i $$-0.507512\pi$$
−0.0235965 + 0.999722i $$0.507512\pi$$
$$450$$ 0 0
$$451$$ 0 0
$$452$$ 0 0
$$453$$ 0 0
$$454$$ 0 0
$$455$$ −20.0000 −0.937614
$$456$$ 0 0
$$457$$ 9.00000 0.421002 0.210501 0.977594i $$-0.432490\pi$$
0.210501 + 0.977594i $$0.432490\pi$$
$$458$$ 0 0
$$459$$ 0 0
$$460$$ 0 0
$$461$$ −34.0000 −1.58354 −0.791769 0.610821i $$-0.790840\pi$$
−0.791769 + 0.610821i $$0.790840\pi$$
$$462$$ 0 0
$$463$$ 32.0000 1.48717 0.743583 0.668644i $$-0.233125\pi$$
0.743583 + 0.668644i $$0.233125\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ 3.00000 0.138823 0.0694117 0.997588i $$-0.477888\pi$$
0.0694117 + 0.997588i $$0.477888\pi$$
$$468$$ 0 0
$$469$$ 45.0000 2.07791
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 0 0
$$473$$ 0 0
$$474$$ 0 0
$$475$$ −4.00000 −0.183533
$$476$$ 0 0
$$477$$ 0 0
$$478$$ 0 0
$$479$$ −38.0000 −1.73626 −0.868132 0.496333i $$-0.834679\pi$$
−0.868132 + 0.496333i $$0.834679\pi$$
$$480$$ 0 0
$$481$$ −4.00000 −0.182384
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ 2.00000 0.0908153
$$486$$ 0 0
$$487$$ −14.0000 −0.634401 −0.317200 0.948359i $$-0.602743\pi$$
−0.317200 + 0.948359i $$0.602743\pi$$
$$488$$ 0 0
$$489$$ 0 0
$$490$$ 0 0
$$491$$ −1.00000 −0.0451294 −0.0225647 0.999745i $$-0.507183\pi$$
−0.0225647 + 0.999745i $$0.507183\pi$$
$$492$$ 0 0
$$493$$ −3.00000 −0.135113
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 65.0000 2.91565
$$498$$ 0 0
$$499$$ 3.00000 0.134298 0.0671492 0.997743i $$-0.478610\pi$$
0.0671492 + 0.997743i $$0.478610\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 0 0
$$503$$ −31.0000 −1.38222 −0.691111 0.722749i $$-0.742878\pi$$
−0.691111 + 0.722749i $$0.742878\pi$$
$$504$$ 0 0
$$505$$ 7.00000 0.311496
$$506$$ 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
$$511$$ 0 0
$$512$$ 0 0
$$513$$ 0 0
$$514$$ 0 0
$$515$$ −16.0000 −0.705044
$$516$$ 0 0
$$517$$ 0 0
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 0 0
$$521$$ −32.0000 −1.40195 −0.700973 0.713188i $$-0.747251\pi$$
−0.700973 + 0.713188i $$0.747251\pi$$
$$522$$ 0 0
$$523$$ 28.0000 1.22435 0.612177 0.790721i $$-0.290294\pi$$
0.612177 + 0.790721i $$0.290294\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 3.00000 0.130682
$$528$$ 0 0
$$529$$ 1.00000 0.0434783
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 0 0
$$533$$ −44.0000 −1.90585
$$534$$ 0 0
$$535$$ 3.00000 0.129701
$$536$$ 0 0
$$537$$ 0 0
$$538$$ 0 0
$$539$$ 0 0
$$540$$ 0 0
$$541$$ −18.0000 −0.773880 −0.386940 0.922105i $$-0.626468\pi$$
−0.386940 + 0.922105i $$0.626468\pi$$
$$542$$ 0 0
$$543$$ 0 0
$$544$$ 0 0
$$545$$ −12.0000 −0.514024
$$546$$ 0 0
$$547$$ −12.0000 −0.513083 −0.256541 0.966533i $$-0.582583\pi$$
−0.256541 + 0.966533i $$0.582583\pi$$
$$548$$ 0 0
$$549$$ 0 0
$$550$$ 0 0
$$551$$ 4.00000 0.170406
$$552$$ 0 0
$$553$$ −80.0000 −3.40195
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ 23.0000 0.974541 0.487271 0.873251i $$-0.337993\pi$$
0.487271 + 0.873251i $$0.337993\pi$$
$$558$$ 0 0
$$559$$ 16.0000 0.676728
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ −27.0000 −1.13791 −0.568957 0.822367i $$-0.692653\pi$$
−0.568957 + 0.822367i $$0.692653\pi$$
$$564$$ 0 0
$$565$$ −1.00000 −0.0420703
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ 30.0000 1.25767 0.628833 0.777541i $$-0.283533\pi$$
0.628833 + 0.777541i $$0.283533\pi$$
$$570$$ 0 0
$$571$$ 10.0000 0.418487 0.209243 0.977864i $$-0.432900\pi$$
0.209243 + 0.977864i $$0.432900\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 0 0
$$575$$ 1.00000 0.0417029
$$576$$ 0 0
$$577$$ 4.00000 0.166522 0.0832611 0.996528i $$-0.473466\pi$$
0.0832611 + 0.996528i $$0.473466\pi$$
$$578$$ 0 0
$$579$$ 0 0
$$580$$ 0 0
$$581$$ 45.0000 1.86691
$$582$$ 0 0
$$583$$ 0 0
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ 6.00000 0.247647 0.123823 0.992304i $$-0.460484\pi$$
0.123823 + 0.992304i $$0.460484\pi$$
$$588$$ 0 0
$$589$$ −4.00000 −0.164817
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 0 0
$$593$$ −6.00000 −0.246390 −0.123195 0.992382i $$-0.539314\pi$$
−0.123195 + 0.992382i $$0.539314\pi$$
$$594$$ 0 0
$$595$$ −15.0000 −0.614940
$$596$$ 0 0
$$597$$ 0 0
$$598$$ 0 0
$$599$$ 44.0000 1.79779 0.898896 0.438163i $$-0.144371\pi$$
0.898896 + 0.438163i $$0.144371\pi$$
$$600$$ 0 0
$$601$$ 7.00000 0.285536 0.142768 0.989756i $$-0.454400\pi$$
0.142768 + 0.989756i $$0.454400\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ 0 0
$$605$$ −11.0000 −0.447214
$$606$$ 0 0
$$607$$ 22.0000 0.892952 0.446476 0.894795i $$-0.352679\pi$$
0.446476 + 0.894795i $$0.352679\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ −24.0000 −0.970936
$$612$$ 0 0
$$613$$ 14.0000 0.565455 0.282727 0.959200i $$-0.408761\pi$$
0.282727 + 0.959200i $$0.408761\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 1.00000 0.0402585 0.0201292 0.999797i $$-0.493592\pi$$
0.0201292 + 0.999797i $$0.493592\pi$$
$$618$$ 0 0
$$619$$ −20.0000 −0.803868 −0.401934 0.915669i $$-0.631662\pi$$
−0.401934 + 0.915669i $$0.631662\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 0 0
$$623$$ 40.0000 1.60257
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ −3.00000 −0.119618
$$630$$ 0 0
$$631$$ 38.0000 1.51276 0.756378 0.654135i $$-0.226967\pi$$
0.756378 + 0.654135i $$0.226967\pi$$
$$632$$ 0 0
$$633$$ 0 0
$$634$$ 0 0
$$635$$ −18.0000 −0.714308
$$636$$ 0 0
$$637$$ 72.0000 2.85274
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 0 0
$$641$$ −24.0000 −0.947943 −0.473972 0.880540i $$-0.657180\pi$$
−0.473972 + 0.880540i $$0.657180\pi$$
$$642$$ 0 0
$$643$$ −1.00000 −0.0394362 −0.0197181 0.999806i $$-0.506277\pi$$
−0.0197181 + 0.999806i $$0.506277\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 48.0000 1.88707 0.943537 0.331266i $$-0.107476\pi$$
0.943537 + 0.331266i $$0.107476\pi$$
$$648$$ 0 0
$$649$$ 0 0
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ 44.0000 1.72185 0.860927 0.508729i $$-0.169885\pi$$
0.860927 + 0.508729i $$0.169885\pi$$
$$654$$ 0 0
$$655$$ −20.0000 −0.781465
$$656$$ 0 0
$$657$$ 0 0
$$658$$ 0 0
$$659$$ −24.0000 −0.934907 −0.467454 0.884018i $$-0.654829\pi$$
−0.467454 + 0.884018i $$0.654829\pi$$
$$660$$ 0 0
$$661$$ −14.0000 −0.544537 −0.272268 0.962221i $$-0.587774\pi$$
−0.272268 + 0.962221i $$0.587774\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ 20.0000 0.775567
$$666$$ 0 0
$$667$$ −1.00000 −0.0387202
$$668$$ 0 0
$$669$$ 0 0
$$670$$ 0 0
$$671$$ 0 0
$$672$$ 0 0
$$673$$ −6.00000 −0.231283 −0.115642 0.993291i $$-0.536892\pi$$
−0.115642 + 0.993291i $$0.536892\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ −39.0000 −1.49889 −0.749446 0.662066i $$-0.769680\pi$$
−0.749446 + 0.662066i $$0.769680\pi$$
$$678$$ 0 0
$$679$$ −10.0000 −0.383765
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 0 0
$$683$$ −34.0000 −1.30097 −0.650487 0.759517i $$-0.725435\pi$$
−0.650487 + 0.759517i $$0.725435\pi$$
$$684$$ 0 0
$$685$$ −2.00000 −0.0764161
$$686$$ 0 0
$$687$$ 0 0
$$688$$ 0 0
$$689$$ −4.00000 −0.152388
$$690$$ 0 0
$$691$$ 8.00000 0.304334 0.152167 0.988355i $$-0.451375\pi$$
0.152167 + 0.988355i $$0.451375\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ 11.0000 0.417254
$$696$$ 0 0
$$697$$ −33.0000 −1.24996
$$698$$ 0 0
$$699$$ 0 0
$$700$$ 0 0
$$701$$ 18.0000 0.679851 0.339925 0.940452i $$-0.389598\pi$$
0.339925 + 0.940452i $$0.389598\pi$$
$$702$$ 0 0
$$703$$ 4.00000 0.150863
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ −35.0000 −1.31631
$$708$$ 0 0
$$709$$ 18.0000 0.676004 0.338002 0.941145i $$-0.390249\pi$$
0.338002 + 0.941145i $$0.390249\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 0 0
$$713$$ 1.00000 0.0374503
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ 0 0
$$718$$ 0 0
$$719$$ 25.0000 0.932343 0.466171 0.884694i $$-0.345633\pi$$
0.466171 + 0.884694i $$0.345633\pi$$
$$720$$ 0 0
$$721$$ 80.0000 2.97936
$$722$$ 0 0
$$723$$ 0 0
$$724$$ 0 0
$$725$$ −1.00000 −0.0371391
$$726$$ 0 0
$$727$$ 13.0000 0.482143 0.241072 0.970507i $$-0.422501\pi$$
0.241072 + 0.970507i $$0.422501\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ 0 0
$$731$$ 12.0000 0.443836
$$732$$ 0 0
$$733$$ −21.0000 −0.775653 −0.387826 0.921732i $$-0.626774\pi$$
−0.387826 + 0.921732i $$0.626774\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 0 0
$$738$$ 0 0
$$739$$ 45.0000 1.65535 0.827676 0.561206i $$-0.189663\pi$$
0.827676 + 0.561206i $$0.189663\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ −16.0000 −0.586983 −0.293492 0.955962i $$-0.594817\pi$$
−0.293492 + 0.955962i $$0.594817\pi$$
$$744$$ 0 0
$$745$$ −22.0000 −0.806018
$$746$$ 0 0
$$747$$ 0 0
$$748$$ 0 0
$$749$$ −15.0000 −0.548088
$$750$$ 0 0
$$751$$ 20.0000 0.729810 0.364905 0.931045i $$-0.381101\pi$$
0.364905 + 0.931045i $$0.381101\pi$$
$$752$$ 0 0
$$753$$ 0 0
$$754$$ 0 0
$$755$$ −16.0000 −0.582300
$$756$$ 0 0
$$757$$ 29.0000 1.05402 0.527011 0.849858i $$-0.323312\pi$$
0.527011 + 0.849858i $$0.323312\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 45.0000 1.63125 0.815624 0.578582i $$-0.196394\pi$$
0.815624 + 0.578582i $$0.196394\pi$$
$$762$$ 0 0
$$763$$ 60.0000 2.17215
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ 4.00000 0.144432
$$768$$ 0 0
$$769$$ 10.0000 0.360609 0.180305 0.983611i $$-0.442292\pi$$
0.180305 + 0.983611i $$0.442292\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 0 0
$$773$$ 6.00000 0.215805 0.107903 0.994161i $$-0.465587\pi$$
0.107903 + 0.994161i $$0.465587\pi$$
$$774$$ 0 0
$$775$$ 1.00000 0.0359211
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ 44.0000 1.57646
$$780$$ 0 0
$$781$$ 0 0
$$782$$ 0 0
$$783$$ 0 0
$$784$$ 0 0
$$785$$ 5.00000 0.178458
$$786$$ 0 0
$$787$$ 23.0000 0.819861 0.409931 0.912117i $$-0.365553\pi$$
0.409931 + 0.912117i $$0.365553\pi$$
$$788$$ 0 0
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 5.00000 0.177780
$$792$$ 0 0
$$793$$ 24.0000 0.852265
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ −15.0000 −0.531327 −0.265664 0.964066i $$-0.585591\pi$$
−0.265664 + 0.964066i $$0.585591\pi$$
$$798$$ 0 0
$$799$$ −18.0000 −0.636794
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 0 0
$$803$$ 0 0
$$804$$ 0 0
$$805$$ −5.00000 −0.176227
$$806$$ 0 0
$$807$$ 0 0
$$808$$ 0 0
$$809$$ −13.0000 −0.457056 −0.228528 0.973537i $$-0.573391\pi$$
−0.228528 + 0.973537i $$0.573391\pi$$
$$810$$ 0 0
$$811$$ −39.0000 −1.36948 −0.684738 0.728790i $$-0.740083\pi$$
−0.684738 + 0.728790i $$0.740083\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ 0 0
$$815$$ 18.0000 0.630512
$$816$$ 0 0
$$817$$ −16.0000 −0.559769
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ 42.0000 1.46581 0.732905 0.680331i $$-0.238164\pi$$
0.732905 + 0.680331i $$0.238164\pi$$
$$822$$ 0 0
$$823$$ 40.0000 1.39431 0.697156 0.716919i $$-0.254448\pi$$
0.697156 + 0.716919i $$0.254448\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ −27.0000 −0.938882 −0.469441 0.882964i $$-0.655545\pi$$
−0.469441 + 0.882964i $$0.655545\pi$$
$$828$$ 0 0
$$829$$ 17.0000 0.590434 0.295217 0.955430i $$-0.404608\pi$$
0.295217 + 0.955430i $$0.404608\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 0 0
$$833$$ 54.0000 1.87099
$$834$$ 0 0
$$835$$ 6.00000 0.207639
$$836$$ 0 0
$$837$$ 0 0
$$838$$ 0 0
$$839$$ 26.0000 0.897620 0.448810 0.893627i $$-0.351848\pi$$
0.448810 + 0.893627i $$0.351848\pi$$
$$840$$ 0 0
$$841$$ −28.0000 −0.965517
$$842$$ 0 0
$$843$$ 0 0
$$844$$ 0 0
$$845$$ 3.00000 0.103203
$$846$$ 0 0
$$847$$ 55.0000 1.88982
$$848$$ 0 0
$$849$$ 0 0
$$850$$ 0 0
$$851$$ −1.00000 −0.0342796
$$852$$ 0 0
$$853$$ −54.0000 −1.84892 −0.924462 0.381273i $$-0.875486\pi$$
−0.924462 + 0.381273i $$0.875486\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ 2.00000 0.0683187 0.0341593 0.999416i $$-0.489125\pi$$
0.0341593 + 0.999416i $$0.489125\pi$$
$$858$$ 0 0
$$859$$ −27.0000 −0.921228 −0.460614 0.887601i $$-0.652371\pi$$
−0.460614 + 0.887601i $$0.652371\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ −42.0000 −1.42970 −0.714848 0.699280i $$-0.753504\pi$$
−0.714848 + 0.699280i $$0.753504\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 0 0
$$867$$ 0 0
$$868$$ 0 0
$$869$$ 0 0
$$870$$ 0 0
$$871$$ −36.0000 −1.21981
$$872$$ 0 0
$$873$$ 0 0
$$874$$ 0 0
$$875$$ −5.00000 −0.169031
$$876$$ 0 0
$$877$$ 42.0000 1.41824 0.709120 0.705088i $$-0.249093\pi$$
0.709120 + 0.705088i $$0.249093\pi$$
$$878$$ 0 0
$$879$$ 0 0
$$880$$ 0 0
$$881$$ 6.00000 0.202145 0.101073 0.994879i $$-0.467773\pi$$
0.101073 + 0.994879i $$0.467773\pi$$
$$882$$ 0 0
$$883$$ 46.0000 1.54802 0.774012 0.633171i $$-0.218247\pi$$
0.774012 + 0.633171i $$0.218247\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ −38.0000 −1.27592 −0.637958 0.770072i $$-0.720220\pi$$
−0.637958 + 0.770072i $$0.720220\pi$$
$$888$$ 0 0
$$889$$ 90.0000 3.01850
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 0 0
$$893$$ 24.0000 0.803129
$$894$$ 0 0
$$895$$ −24.0000 −0.802232
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 0 0
$$899$$ −1.00000 −0.0333519
$$900$$ 0 0
$$901$$ −3.00000 −0.0999445
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ −8.00000 −0.265929
$$906$$ 0 0
$$907$$ −17.0000 −0.564476 −0.282238 0.959344i $$-0.591077\pi$$
−0.282238 + 0.959344i $$0.591077\pi$$
$$908$$ 0 0
$$909$$ 0 0
$$910$$ 0 0
$$911$$ 24.0000 0.795155 0.397578 0.917568i $$-0.369851\pi$$
0.397578 + 0.917568i $$0.369851\pi$$
$$912$$ 0 0
$$913$$ 0 0
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ 100.000 3.30229
$$918$$ 0 0
$$919$$ 48.0000 1.58337 0.791687 0.610927i $$-0.209203\pi$$
0.791687 + 0.610927i $$0.209203\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 0 0
$$923$$ −52.0000 −1.71160
$$924$$ 0 0
$$925$$ −1.00000 −0.0328798
$$926$$ 0 0
$$927$$ 0 0
$$928$$ 0 0
$$929$$ −5.00000 −0.164045 −0.0820223 0.996630i $$-0.526138\pi$$
−0.0820223 + 0.996630i $$0.526138\pi$$
$$930$$ 0 0
$$931$$ −72.0000 −2.35970
$$932$$ 0 0
$$933$$ 0 0
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ 2.00000 0.0653372 0.0326686 0.999466i $$-0.489599\pi$$
0.0326686 + 0.999466i $$0.489599\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ 0 0
$$941$$ −6.00000 −0.195594 −0.0977972 0.995206i $$-0.531180\pi$$
−0.0977972 + 0.995206i $$0.531180\pi$$
$$942$$ 0 0
$$943$$ −11.0000 −0.358209
$$944$$ 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
$$949$$ 0 0
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 0 0
$$953$$ 6.00000 0.194359 0.0971795 0.995267i $$-0.469018\pi$$
0.0971795 + 0.995267i $$0.469018\pi$$
$$954$$ 0 0
$$955$$ 10.0000 0.323592
$$956$$ 0 0
$$957$$ 0 0
$$958$$ 0 0
$$959$$ 10.0000 0.322917
$$960$$ 0 0
$$961$$ −30.0000 −0.967742
$$962$$ 0 0
$$963$$ 0 0
$$964$$ 0 0
$$965$$ 12.0000 0.386294
$$966$$ 0 0
$$967$$ 2.00000 0.0643157 0.0321578 0.999483i $$-0.489762\pi$$
0.0321578 + 0.999483i $$0.489762\pi$$
$$968$$ 0 0
$$969$$ 0 0
$$970$$ 0 0
$$971$$ −20.0000 −0.641831 −0.320915 0.947108i $$-0.603990\pi$$
−0.320915 + 0.947108i $$0.603990\pi$$
$$972$$ 0 0
$$973$$ −55.0000 −1.76322
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ −31.0000 −0.991778 −0.495889 0.868386i $$-0.665158\pi$$
−0.495889 + 0.868386i $$0.665158\pi$$
$$978$$ 0 0
$$979$$ 0 0
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 0 0
$$983$$ 11.0000 0.350846 0.175423 0.984493i $$-0.443871\pi$$
0.175423 + 0.984493i $$0.443871\pi$$
$$984$$ 0 0
$$985$$ −2.00000 −0.0637253
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ 4.00000 0.127193
$$990$$ 0 0
$$991$$ −35.0000 −1.11181 −0.555906 0.831245i $$-0.687628\pi$$
−0.555906 + 0.831245i $$0.687628\pi$$
$$992$$ 0 0
$$993$$ 0 0
$$994$$ 0 0
$$995$$ 2.00000 0.0634043
$$996$$ 0 0
$$997$$ 40.0000 1.26681 0.633406 0.773819i $$-0.281656\pi$$
0.633406 + 0.773819i $$0.281656\pi$$
$$998$$ 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 4140.2.a.e.1.1 1
3.2 odd 2 1380.2.a.a.1.1 1
12.11 even 2 5520.2.a.ba.1.1 1
15.2 even 4 6900.2.f.f.6349.2 2
15.8 even 4 6900.2.f.f.6349.1 2
15.14 odd 2 6900.2.a.i.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
1380.2.a.a.1.1 1 3.2 odd 2
4140.2.a.e.1.1 1 1.1 even 1 trivial
5520.2.a.ba.1.1 1 12.11 even 2
6900.2.a.i.1.1 1 15.14 odd 2
6900.2.f.f.6349.1 2 15.8 even 4
6900.2.f.f.6349.2 2 15.2 even 4