# Properties

 Label 9680.2.a.e.1.1 Level $9680$ Weight $2$ Character 9680.1 Self dual yes Analytic conductor $77.295$ 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] = [9680,2,Mod(1,9680)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9680, 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("9680.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$9680 = 2^{4} \cdot 5 \cdot 11^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 9680.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: $$77.2951891566$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 4840) Fricke sign: $$+1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 9680.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-1.00000 q^{3} -1.00000 q^{5} -3.00000 q^{7} -2.00000 q^{9} +O(q^{10})$$ $$q-1.00000 q^{3} -1.00000 q^{5} -3.00000 q^{7} -2.00000 q^{9} -4.00000 q^{13} +1.00000 q^{15} +3.00000 q^{17} +3.00000 q^{19} +3.00000 q^{21} -4.00000 q^{23} +1.00000 q^{25} +5.00000 q^{27} -1.00000 q^{29} +5.00000 q^{31} +3.00000 q^{35} -5.00000 q^{37} +4.00000 q^{39} -2.00000 q^{41} +8.00000 q^{43} +2.00000 q^{45} +10.0000 q^{47} +2.00000 q^{49} -3.00000 q^{51} +11.0000 q^{53} -3.00000 q^{57} +1.00000 q^{61} +6.00000 q^{63} +4.00000 q^{65} -4.00000 q^{67} +4.00000 q^{69} -5.00000 q^{71} +10.0000 q^{73} -1.00000 q^{75} +8.00000 q^{79} +1.00000 q^{81} -3.00000 q^{85} +1.00000 q^{87} -15.0000 q^{89} +12.0000 q^{91} -5.00000 q^{93} -3.00000 q^{95} -6.00000 q^{97} +O(q^{100})$$

## 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
$$3$$ −1.00000 −0.577350 −0.288675 0.957427i $$-0.593215\pi$$
−0.288675 + 0.957427i $$0.593215\pi$$
$$4$$ 0 0
$$5$$ −1.00000 −0.447214
$$6$$ 0 0
$$7$$ −3.00000 −1.13389 −0.566947 0.823754i $$-0.691875\pi$$
−0.566947 + 0.823754i $$0.691875\pi$$
$$8$$ 0 0
$$9$$ −2.00000 −0.666667
$$10$$ 0 0
$$11$$ 0 0
$$12$$ 0 0
$$13$$ −4.00000 −1.10940 −0.554700 0.832050i $$-0.687167\pi$$
−0.554700 + 0.832050i $$0.687167\pi$$
$$14$$ 0 0
$$15$$ 1.00000 0.258199
$$16$$ 0 0
$$17$$ 3.00000 0.727607 0.363803 0.931476i $$-0.381478\pi$$
0.363803 + 0.931476i $$0.381478\pi$$
$$18$$ 0 0
$$19$$ 3.00000 0.688247 0.344124 0.938924i $$-0.388176\pi$$
0.344124 + 0.938924i $$0.388176\pi$$
$$20$$ 0 0
$$21$$ 3.00000 0.654654
$$22$$ 0 0
$$23$$ −4.00000 −0.834058 −0.417029 0.908893i $$-0.636929\pi$$
−0.417029 + 0.908893i $$0.636929\pi$$
$$24$$ 0 0
$$25$$ 1.00000 0.200000
$$26$$ 0 0
$$27$$ 5.00000 0.962250
$$28$$ 0 0
$$29$$ −1.00000 −0.185695 −0.0928477 0.995680i $$-0.529597\pi$$
−0.0928477 + 0.995680i $$0.529597\pi$$
$$30$$ 0 0
$$31$$ 5.00000 0.898027 0.449013 0.893525i $$-0.351776\pi$$
0.449013 + 0.893525i $$0.351776\pi$$
$$32$$ 0 0
$$33$$ 0 0
$$34$$ 0 0
$$35$$ 3.00000 0.507093
$$36$$ 0 0
$$37$$ −5.00000 −0.821995 −0.410997 0.911636i $$-0.634819\pi$$
−0.410997 + 0.911636i $$0.634819\pi$$
$$38$$ 0 0
$$39$$ 4.00000 0.640513
$$40$$ 0 0
$$41$$ −2.00000 −0.312348 −0.156174 0.987730i $$-0.549916\pi$$
−0.156174 + 0.987730i $$0.549916\pi$$
$$42$$ 0 0
$$43$$ 8.00000 1.21999 0.609994 0.792406i $$-0.291172\pi$$
0.609994 + 0.792406i $$0.291172\pi$$
$$44$$ 0 0
$$45$$ 2.00000 0.298142
$$46$$ 0 0
$$47$$ 10.0000 1.45865 0.729325 0.684167i $$-0.239834\pi$$
0.729325 + 0.684167i $$0.239834\pi$$
$$48$$ 0 0
$$49$$ 2.00000 0.285714
$$50$$ 0 0
$$51$$ −3.00000 −0.420084
$$52$$ 0 0
$$53$$ 11.0000 1.51097 0.755483 0.655168i $$-0.227402\pi$$
0.755483 + 0.655168i $$0.227402\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ −3.00000 −0.397360
$$58$$ 0 0
$$59$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$60$$ 0 0
$$61$$ 1.00000 0.128037 0.0640184 0.997949i $$-0.479608\pi$$
0.0640184 + 0.997949i $$0.479608\pi$$
$$62$$ 0 0
$$63$$ 6.00000 0.755929
$$64$$ 0 0
$$65$$ 4.00000 0.496139
$$66$$ 0 0
$$67$$ −4.00000 −0.488678 −0.244339 0.969690i $$-0.578571\pi$$
−0.244339 + 0.969690i $$0.578571\pi$$
$$68$$ 0 0
$$69$$ 4.00000 0.481543
$$70$$ 0 0
$$71$$ −5.00000 −0.593391 −0.296695 0.954972i $$-0.595885\pi$$
−0.296695 + 0.954972i $$0.595885\pi$$
$$72$$ 0 0
$$73$$ 10.0000 1.17041 0.585206 0.810885i $$-0.301014\pi$$
0.585206 + 0.810885i $$0.301014\pi$$
$$74$$ 0 0
$$75$$ −1.00000 −0.115470
$$76$$ 0 0
$$77$$ 0 0
$$78$$ 0 0
$$79$$ 8.00000 0.900070 0.450035 0.893011i $$-0.351411\pi$$
0.450035 + 0.893011i $$0.351411\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 0 0
$$83$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$84$$ 0 0
$$85$$ −3.00000 −0.325396
$$86$$ 0 0
$$87$$ 1.00000 0.107211
$$88$$ 0 0
$$89$$ −15.0000 −1.59000 −0.794998 0.606612i $$-0.792528\pi$$
−0.794998 + 0.606612i $$0.792528\pi$$
$$90$$ 0 0
$$91$$ 12.0000 1.25794
$$92$$ 0 0
$$93$$ −5.00000 −0.518476
$$94$$ 0 0
$$95$$ −3.00000 −0.307794
$$96$$ 0 0
$$97$$ −6.00000 −0.609208 −0.304604 0.952479i $$-0.598524\pi$$
−0.304604 + 0.952479i $$0.598524\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 0 0
$$101$$ −18.0000 −1.79107 −0.895533 0.444994i $$-0.853206\pi$$
−0.895533 + 0.444994i $$0.853206\pi$$
$$102$$ 0 0
$$103$$ −10.0000 −0.985329 −0.492665 0.870219i $$-0.663977\pi$$
−0.492665 + 0.870219i $$0.663977\pi$$
$$104$$ 0 0
$$105$$ −3.00000 −0.292770
$$106$$ 0 0
$$107$$ 18.0000 1.74013 0.870063 0.492941i $$-0.164078\pi$$
0.870063 + 0.492941i $$0.164078\pi$$
$$108$$ 0 0
$$109$$ −14.0000 −1.34096 −0.670478 0.741929i $$-0.733911\pi$$
−0.670478 + 0.741929i $$0.733911\pi$$
$$110$$ 0 0
$$111$$ 5.00000 0.474579
$$112$$ 0 0
$$113$$ 16.0000 1.50515 0.752577 0.658505i $$-0.228811\pi$$
0.752577 + 0.658505i $$0.228811\pi$$
$$114$$ 0 0
$$115$$ 4.00000 0.373002
$$116$$ 0 0
$$117$$ 8.00000 0.739600
$$118$$ 0 0
$$119$$ −9.00000 −0.825029
$$120$$ 0 0
$$121$$ 0 0
$$122$$ 0 0
$$123$$ 2.00000 0.180334
$$124$$ 0 0
$$125$$ −1.00000 −0.0894427
$$126$$ 0 0
$$127$$ 16.0000 1.41977 0.709885 0.704317i $$-0.248747\pi$$
0.709885 + 0.704317i $$0.248747\pi$$
$$128$$ 0 0
$$129$$ −8.00000 −0.704361
$$130$$ 0 0
$$131$$ 15.0000 1.31056 0.655278 0.755388i $$-0.272551\pi$$
0.655278 + 0.755388i $$0.272551\pi$$
$$132$$ 0 0
$$133$$ −9.00000 −0.780399
$$134$$ 0 0
$$135$$ −5.00000 −0.430331
$$136$$ 0 0
$$137$$ 4.00000 0.341743 0.170872 0.985293i $$-0.445342\pi$$
0.170872 + 0.985293i $$0.445342\pi$$
$$138$$ 0 0
$$139$$ 4.00000 0.339276 0.169638 0.985506i $$-0.445740\pi$$
0.169638 + 0.985506i $$0.445740\pi$$
$$140$$ 0 0
$$141$$ −10.0000 −0.842152
$$142$$ 0 0
$$143$$ 0 0
$$144$$ 0 0
$$145$$ 1.00000 0.0830455
$$146$$ 0 0
$$147$$ −2.00000 −0.164957
$$148$$ 0 0
$$149$$ −19.0000 −1.55654 −0.778270 0.627929i $$-0.783903\pi$$
−0.778270 + 0.627929i $$0.783903\pi$$
$$150$$ 0 0
$$151$$ −12.0000 −0.976546 −0.488273 0.872691i $$-0.662373\pi$$
−0.488273 + 0.872691i $$0.662373\pi$$
$$152$$ 0 0
$$153$$ −6.00000 −0.485071
$$154$$ 0 0
$$155$$ −5.00000 −0.401610
$$156$$ 0 0
$$157$$ 11.0000 0.877896 0.438948 0.898513i $$-0.355351\pi$$
0.438948 + 0.898513i $$0.355351\pi$$
$$158$$ 0 0
$$159$$ −11.0000 −0.872357
$$160$$ 0 0
$$161$$ 12.0000 0.945732
$$162$$ 0 0
$$163$$ −5.00000 −0.391630 −0.195815 0.980641i $$-0.562735\pi$$
−0.195815 + 0.980641i $$0.562735\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 1.00000 0.0773823 0.0386912 0.999251i $$-0.487681\pi$$
0.0386912 + 0.999251i $$0.487681\pi$$
$$168$$ 0 0
$$169$$ 3.00000 0.230769
$$170$$ 0 0
$$171$$ −6.00000 −0.458831
$$172$$ 0 0
$$173$$ 14.0000 1.06440 0.532200 0.846619i $$-0.321365\pi$$
0.532200 + 0.846619i $$0.321365\pi$$
$$174$$ 0 0
$$175$$ −3.00000 −0.226779
$$176$$ 0 0
$$177$$ 0 0
$$178$$ 0 0
$$179$$ 2.00000 0.149487 0.0747435 0.997203i $$-0.476186\pi$$
0.0747435 + 0.997203i $$0.476186\pi$$
$$180$$ 0 0
$$181$$ 16.0000 1.18927 0.594635 0.803996i $$-0.297296\pi$$
0.594635 + 0.803996i $$0.297296\pi$$
$$182$$ 0 0
$$183$$ −1.00000 −0.0739221
$$184$$ 0 0
$$185$$ 5.00000 0.367607
$$186$$ 0 0
$$187$$ 0 0
$$188$$ 0 0
$$189$$ −15.0000 −1.09109
$$190$$ 0 0
$$191$$ 4.00000 0.289430 0.144715 0.989473i $$-0.453773\pi$$
0.144715 + 0.989473i $$0.453773\pi$$
$$192$$ 0 0
$$193$$ −5.00000 −0.359908 −0.179954 0.983675i $$-0.557595\pi$$
−0.179954 + 0.983675i $$0.557595\pi$$
$$194$$ 0 0
$$195$$ −4.00000 −0.286446
$$196$$ 0 0
$$197$$ −26.0000 −1.85242 −0.926212 0.377004i $$-0.876954\pi$$
−0.926212 + 0.377004i $$0.876954\pi$$
$$198$$ 0 0
$$199$$ −17.0000 −1.20510 −0.602549 0.798082i $$-0.705848\pi$$
−0.602549 + 0.798082i $$0.705848\pi$$
$$200$$ 0 0
$$201$$ 4.00000 0.282138
$$202$$ 0 0
$$203$$ 3.00000 0.210559
$$204$$ 0 0
$$205$$ 2.00000 0.139686
$$206$$ 0 0
$$207$$ 8.00000 0.556038
$$208$$ 0 0
$$209$$ 0 0
$$210$$ 0 0
$$211$$ 23.0000 1.58339 0.791693 0.610920i $$-0.209200\pi$$
0.791693 + 0.610920i $$0.209200\pi$$
$$212$$ 0 0
$$213$$ 5.00000 0.342594
$$214$$ 0 0
$$215$$ −8.00000 −0.545595
$$216$$ 0 0
$$217$$ −15.0000 −1.01827
$$218$$ 0 0
$$219$$ −10.0000 −0.675737
$$220$$ 0 0
$$221$$ −12.0000 −0.807207
$$222$$ 0 0
$$223$$ −2.00000 −0.133930 −0.0669650 0.997755i $$-0.521332\pi$$
−0.0669650 + 0.997755i $$0.521332\pi$$
$$224$$ 0 0
$$225$$ −2.00000 −0.133333
$$226$$ 0 0
$$227$$ −14.0000 −0.929213 −0.464606 0.885517i $$-0.653804\pi$$
−0.464606 + 0.885517i $$0.653804\pi$$
$$228$$ 0 0
$$229$$ −16.0000 −1.05731 −0.528655 0.848837i $$-0.677303\pi$$
−0.528655 + 0.848837i $$0.677303\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ −13.0000 −0.851658 −0.425829 0.904804i $$-0.640018\pi$$
−0.425829 + 0.904804i $$0.640018\pi$$
$$234$$ 0 0
$$235$$ −10.0000 −0.652328
$$236$$ 0 0
$$237$$ −8.00000 −0.519656
$$238$$ 0 0
$$239$$ −14.0000 −0.905585 −0.452792 0.891616i $$-0.649572\pi$$
−0.452792 + 0.891616i $$0.649572\pi$$
$$240$$ 0 0
$$241$$ 22.0000 1.41714 0.708572 0.705638i $$-0.249340\pi$$
0.708572 + 0.705638i $$0.249340\pi$$
$$242$$ 0 0
$$243$$ −16.0000 −1.02640
$$244$$ 0 0
$$245$$ −2.00000 −0.127775
$$246$$ 0 0
$$247$$ −12.0000 −0.763542
$$248$$ 0 0
$$249$$ 0 0
$$250$$ 0 0
$$251$$ −16.0000 −1.00991 −0.504956 0.863145i $$-0.668491\pi$$
−0.504956 + 0.863145i $$0.668491\pi$$
$$252$$ 0 0
$$253$$ 0 0
$$254$$ 0 0
$$255$$ 3.00000 0.187867
$$256$$ 0 0
$$257$$ −12.0000 −0.748539 −0.374270 0.927320i $$-0.622107\pi$$
−0.374270 + 0.927320i $$0.622107\pi$$
$$258$$ 0 0
$$259$$ 15.0000 0.932055
$$260$$ 0 0
$$261$$ 2.00000 0.123797
$$262$$ 0 0
$$263$$ 11.0000 0.678289 0.339145 0.940734i $$-0.389862\pi$$
0.339145 + 0.940734i $$0.389862\pi$$
$$264$$ 0 0
$$265$$ −11.0000 −0.675725
$$266$$ 0 0
$$267$$ 15.0000 0.917985
$$268$$ 0 0
$$269$$ −4.00000 −0.243884 −0.121942 0.992537i $$-0.538912\pi$$
−0.121942 + 0.992537i $$0.538912\pi$$
$$270$$ 0 0
$$271$$ −32.0000 −1.94386 −0.971931 0.235267i $$-0.924404\pi$$
−0.971931 + 0.235267i $$0.924404\pi$$
$$272$$ 0 0
$$273$$ −12.0000 −0.726273
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 0 0
$$277$$ 22.0000 1.32185 0.660926 0.750451i $$-0.270164\pi$$
0.660926 + 0.750451i $$0.270164\pi$$
$$278$$ 0 0
$$279$$ −10.0000 −0.598684
$$280$$ 0 0
$$281$$ −18.0000 −1.07379 −0.536895 0.843649i $$-0.680403\pi$$
−0.536895 + 0.843649i $$0.680403\pi$$
$$282$$ 0 0
$$283$$ −4.00000 −0.237775 −0.118888 0.992908i $$-0.537933\pi$$
−0.118888 + 0.992908i $$0.537933\pi$$
$$284$$ 0 0
$$285$$ 3.00000 0.177705
$$286$$ 0 0
$$287$$ 6.00000 0.354169
$$288$$ 0 0
$$289$$ −8.00000 −0.470588
$$290$$ 0 0
$$291$$ 6.00000 0.351726
$$292$$ 0 0
$$293$$ −10.0000 −0.584206 −0.292103 0.956387i $$-0.594355\pi$$
−0.292103 + 0.956387i $$0.594355\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ 16.0000 0.925304
$$300$$ 0 0
$$301$$ −24.0000 −1.38334
$$302$$ 0 0
$$303$$ 18.0000 1.03407
$$304$$ 0 0
$$305$$ −1.00000 −0.0572598
$$306$$ 0 0
$$307$$ −2.00000 −0.114146 −0.0570730 0.998370i $$-0.518177\pi$$
−0.0570730 + 0.998370i $$0.518177\pi$$
$$308$$ 0 0
$$309$$ 10.0000 0.568880
$$310$$ 0 0
$$311$$ 21.0000 1.19080 0.595400 0.803429i $$-0.296993\pi$$
0.595400 + 0.803429i $$0.296993\pi$$
$$312$$ 0 0
$$313$$ −34.0000 −1.92179 −0.960897 0.276907i $$-0.910691\pi$$
−0.960897 + 0.276907i $$0.910691\pi$$
$$314$$ 0 0
$$315$$ −6.00000 −0.338062
$$316$$ 0 0
$$317$$ 17.0000 0.954815 0.477408 0.878682i $$-0.341577\pi$$
0.477408 + 0.878682i $$0.341577\pi$$
$$318$$ 0 0
$$319$$ 0 0
$$320$$ 0 0
$$321$$ −18.0000 −1.00466
$$322$$ 0 0
$$323$$ 9.00000 0.500773
$$324$$ 0 0
$$325$$ −4.00000 −0.221880
$$326$$ 0 0
$$327$$ 14.0000 0.774202
$$328$$ 0 0
$$329$$ −30.0000 −1.65395
$$330$$ 0 0
$$331$$ −20.0000 −1.09930 −0.549650 0.835395i $$-0.685239\pi$$
−0.549650 + 0.835395i $$0.685239\pi$$
$$332$$ 0 0
$$333$$ 10.0000 0.547997
$$334$$ 0 0
$$335$$ 4.00000 0.218543
$$336$$ 0 0
$$337$$ 29.0000 1.57973 0.789865 0.613280i $$-0.210150\pi$$
0.789865 + 0.613280i $$0.210150\pi$$
$$338$$ 0 0
$$339$$ −16.0000 −0.869001
$$340$$ 0 0
$$341$$ 0 0
$$342$$ 0 0
$$343$$ 15.0000 0.809924
$$344$$ 0 0
$$345$$ −4.00000 −0.215353
$$346$$ 0 0
$$347$$ 18.0000 0.966291 0.483145 0.875540i $$-0.339494\pi$$
0.483145 + 0.875540i $$0.339494\pi$$
$$348$$ 0 0
$$349$$ 10.0000 0.535288 0.267644 0.963518i $$-0.413755\pi$$
0.267644 + 0.963518i $$0.413755\pi$$
$$350$$ 0 0
$$351$$ −20.0000 −1.06752
$$352$$ 0 0
$$353$$ −12.0000 −0.638696 −0.319348 0.947638i $$-0.603464\pi$$
−0.319348 + 0.947638i $$0.603464\pi$$
$$354$$ 0 0
$$355$$ 5.00000 0.265372
$$356$$ 0 0
$$357$$ 9.00000 0.476331
$$358$$ 0 0
$$359$$ −2.00000 −0.105556 −0.0527780 0.998606i $$-0.516808\pi$$
−0.0527780 + 0.998606i $$0.516808\pi$$
$$360$$ 0 0
$$361$$ −10.0000 −0.526316
$$362$$ 0 0
$$363$$ 0 0
$$364$$ 0 0
$$365$$ −10.0000 −0.523424
$$366$$ 0 0
$$367$$ −28.0000 −1.46159 −0.730794 0.682598i $$-0.760850\pi$$
−0.730794 + 0.682598i $$0.760850\pi$$
$$368$$ 0 0
$$369$$ 4.00000 0.208232
$$370$$ 0 0
$$371$$ −33.0000 −1.71327
$$372$$ 0 0
$$373$$ −6.00000 −0.310668 −0.155334 0.987862i $$-0.549645\pi$$
−0.155334 + 0.987862i $$0.549645\pi$$
$$374$$ 0 0
$$375$$ 1.00000 0.0516398
$$376$$ 0 0
$$377$$ 4.00000 0.206010
$$378$$ 0 0
$$379$$ 32.0000 1.64373 0.821865 0.569683i $$-0.192934\pi$$
0.821865 + 0.569683i $$0.192934\pi$$
$$380$$ 0 0
$$381$$ −16.0000 −0.819705
$$382$$ 0 0
$$383$$ 2.00000 0.102195 0.0510976 0.998694i $$-0.483728\pi$$
0.0510976 + 0.998694i $$0.483728\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ −16.0000 −0.813326
$$388$$ 0 0
$$389$$ −24.0000 −1.21685 −0.608424 0.793612i $$-0.708198\pi$$
−0.608424 + 0.793612i $$0.708198\pi$$
$$390$$ 0 0
$$391$$ −12.0000 −0.606866
$$392$$ 0 0
$$393$$ −15.0000 −0.756650
$$394$$ 0 0
$$395$$ −8.00000 −0.402524
$$396$$ 0 0
$$397$$ −10.0000 −0.501886 −0.250943 0.968002i $$-0.580741\pi$$
−0.250943 + 0.968002i $$0.580741\pi$$
$$398$$ 0 0
$$399$$ 9.00000 0.450564
$$400$$ 0 0
$$401$$ −25.0000 −1.24844 −0.624220 0.781248i $$-0.714583\pi$$
−0.624220 + 0.781248i $$0.714583\pi$$
$$402$$ 0 0
$$403$$ −20.0000 −0.996271
$$404$$ 0 0
$$405$$ −1.00000 −0.0496904
$$406$$ 0 0
$$407$$ 0 0
$$408$$ 0 0
$$409$$ 38.0000 1.87898 0.939490 0.342578i $$-0.111300\pi$$
0.939490 + 0.342578i $$0.111300\pi$$
$$410$$ 0 0
$$411$$ −4.00000 −0.197305
$$412$$ 0 0
$$413$$ 0 0
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 0 0
$$417$$ −4.00000 −0.195881
$$418$$ 0 0
$$419$$ −10.0000 −0.488532 −0.244266 0.969708i $$-0.578547\pi$$
−0.244266 + 0.969708i $$0.578547\pi$$
$$420$$ 0 0
$$421$$ 26.0000 1.26716 0.633581 0.773676i $$-0.281584\pi$$
0.633581 + 0.773676i $$0.281584\pi$$
$$422$$ 0 0
$$423$$ −20.0000 −0.972433
$$424$$ 0 0
$$425$$ 3.00000 0.145521
$$426$$ 0 0
$$427$$ −3.00000 −0.145180
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 12.0000 0.578020 0.289010 0.957326i $$-0.406674\pi$$
0.289010 + 0.957326i $$0.406674\pi$$
$$432$$ 0 0
$$433$$ −14.0000 −0.672797 −0.336399 0.941720i $$-0.609209\pi$$
−0.336399 + 0.941720i $$0.609209\pi$$
$$434$$ 0 0
$$435$$ −1.00000 −0.0479463
$$436$$ 0 0
$$437$$ −12.0000 −0.574038
$$438$$ 0 0
$$439$$ 38.0000 1.81364 0.906821 0.421517i $$-0.138502\pi$$
0.906821 + 0.421517i $$0.138502\pi$$
$$440$$ 0 0
$$441$$ −4.00000 −0.190476
$$442$$ 0 0
$$443$$ 24.0000 1.14027 0.570137 0.821549i $$-0.306890\pi$$
0.570137 + 0.821549i $$0.306890\pi$$
$$444$$ 0 0
$$445$$ 15.0000 0.711068
$$446$$ 0 0
$$447$$ 19.0000 0.898669
$$448$$ 0 0
$$449$$ −30.0000 −1.41579 −0.707894 0.706319i $$-0.750354\pi$$
−0.707894 + 0.706319i $$0.750354\pi$$
$$450$$ 0 0
$$451$$ 0 0
$$452$$ 0 0
$$453$$ 12.0000 0.563809
$$454$$ 0 0
$$455$$ −12.0000 −0.562569
$$456$$ 0 0
$$457$$ −3.00000 −0.140334 −0.0701670 0.997535i $$-0.522353\pi$$
−0.0701670 + 0.997535i $$0.522353\pi$$
$$458$$ 0 0
$$459$$ 15.0000 0.700140
$$460$$ 0 0
$$461$$ 29.0000 1.35066 0.675332 0.737514i $$-0.264000\pi$$
0.675332 + 0.737514i $$0.264000\pi$$
$$462$$ 0 0
$$463$$ −8.00000 −0.371792 −0.185896 0.982569i $$-0.559519\pi$$
−0.185896 + 0.982569i $$0.559519\pi$$
$$464$$ 0 0
$$465$$ 5.00000 0.231869
$$466$$ 0 0
$$467$$ 15.0000 0.694117 0.347059 0.937843i $$-0.387180\pi$$
0.347059 + 0.937843i $$0.387180\pi$$
$$468$$ 0 0
$$469$$ 12.0000 0.554109
$$470$$ 0 0
$$471$$ −11.0000 −0.506853
$$472$$ 0 0
$$473$$ 0 0
$$474$$ 0 0
$$475$$ 3.00000 0.137649
$$476$$ 0 0
$$477$$ −22.0000 −1.00731
$$478$$ 0 0
$$479$$ 6.00000 0.274147 0.137073 0.990561i $$-0.456230\pi$$
0.137073 + 0.990561i $$0.456230\pi$$
$$480$$ 0 0
$$481$$ 20.0000 0.911922
$$482$$ 0 0
$$483$$ −12.0000 −0.546019
$$484$$ 0 0
$$485$$ 6.00000 0.272446
$$486$$ 0 0
$$487$$ −32.0000 −1.45006 −0.725029 0.688718i $$-0.758174\pi$$
−0.725029 + 0.688718i $$0.758174\pi$$
$$488$$ 0 0
$$489$$ 5.00000 0.226108
$$490$$ 0 0
$$491$$ 11.0000 0.496423 0.248212 0.968706i $$-0.420157\pi$$
0.248212 + 0.968706i $$0.420157\pi$$
$$492$$ 0 0
$$493$$ −3.00000 −0.135113
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 15.0000 0.672842
$$498$$ 0 0
$$499$$ 10.0000 0.447661 0.223831 0.974628i $$-0.428144\pi$$
0.223831 + 0.974628i $$0.428144\pi$$
$$500$$ 0 0
$$501$$ −1.00000 −0.0446767
$$502$$ 0 0
$$503$$ 4.00000 0.178351 0.0891756 0.996016i $$-0.471577\pi$$
0.0891756 + 0.996016i $$0.471577\pi$$
$$504$$ 0 0
$$505$$ 18.0000 0.800989
$$506$$ 0 0
$$507$$ −3.00000 −0.133235
$$508$$ 0 0
$$509$$ −6.00000 −0.265945 −0.132973 0.991120i $$-0.542452\pi$$
−0.132973 + 0.991120i $$0.542452\pi$$
$$510$$ 0 0
$$511$$ −30.0000 −1.32712
$$512$$ 0 0
$$513$$ 15.0000 0.662266
$$514$$ 0 0
$$515$$ 10.0000 0.440653
$$516$$ 0 0
$$517$$ 0 0
$$518$$ 0 0
$$519$$ −14.0000 −0.614532
$$520$$ 0 0
$$521$$ 10.0000 0.438108 0.219054 0.975713i $$-0.429703\pi$$
0.219054 + 0.975713i $$0.429703\pi$$
$$522$$ 0 0
$$523$$ 18.0000 0.787085 0.393543 0.919306i $$-0.371249\pi$$
0.393543 + 0.919306i $$0.371249\pi$$
$$524$$ 0 0
$$525$$ 3.00000 0.130931
$$526$$ 0 0
$$527$$ 15.0000 0.653410
$$528$$ 0 0
$$529$$ −7.00000 −0.304348
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 0 0
$$533$$ 8.00000 0.346518
$$534$$ 0 0
$$535$$ −18.0000 −0.778208
$$536$$ 0 0
$$537$$ −2.00000 −0.0863064
$$538$$ 0 0
$$539$$ 0 0
$$540$$ 0 0
$$541$$ −9.00000 −0.386940 −0.193470 0.981106i $$-0.561974\pi$$
−0.193470 + 0.981106i $$0.561974\pi$$
$$542$$ 0 0
$$543$$ −16.0000 −0.686626
$$544$$ 0 0
$$545$$ 14.0000 0.599694
$$546$$ 0 0
$$547$$ 26.0000 1.11168 0.555840 0.831289i $$-0.312397\pi$$
0.555840 + 0.831289i $$0.312397\pi$$
$$548$$ 0 0
$$549$$ −2.00000 −0.0853579
$$550$$ 0 0
$$551$$ −3.00000 −0.127804
$$552$$ 0 0
$$553$$ −24.0000 −1.02058
$$554$$ 0 0
$$555$$ −5.00000 −0.212238
$$556$$ 0 0
$$557$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$558$$ 0 0
$$559$$ −32.0000 −1.35346
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ −18.0000 −0.758610 −0.379305 0.925272i $$-0.623837\pi$$
−0.379305 + 0.925272i $$0.623837\pi$$
$$564$$ 0 0
$$565$$ −16.0000 −0.673125
$$566$$ 0 0
$$567$$ −3.00000 −0.125988
$$568$$ 0 0
$$569$$ 10.0000 0.419222 0.209611 0.977785i $$-0.432780\pi$$
0.209611 + 0.977785i $$0.432780\pi$$
$$570$$ 0 0
$$571$$ −23.0000 −0.962520 −0.481260 0.876578i $$-0.659821\pi$$
−0.481260 + 0.876578i $$0.659821\pi$$
$$572$$ 0 0
$$573$$ −4.00000 −0.167102
$$574$$ 0 0
$$575$$ −4.00000 −0.166812
$$576$$ 0 0
$$577$$ 24.0000 0.999133 0.499567 0.866276i $$-0.333493\pi$$
0.499567 + 0.866276i $$0.333493\pi$$
$$578$$ 0 0
$$579$$ 5.00000 0.207793
$$580$$ 0 0
$$581$$ 0 0
$$582$$ 0 0
$$583$$ 0 0
$$584$$ 0 0
$$585$$ −8.00000 −0.330759
$$586$$ 0 0
$$587$$ −39.0000 −1.60970 −0.804851 0.593477i $$-0.797755\pi$$
−0.804851 + 0.593477i $$0.797755\pi$$
$$588$$ 0 0
$$589$$ 15.0000 0.618064
$$590$$ 0 0
$$591$$ 26.0000 1.06950
$$592$$ 0 0
$$593$$ −14.0000 −0.574911 −0.287456 0.957794i $$-0.592809\pi$$
−0.287456 + 0.957794i $$0.592809\pi$$
$$594$$ 0 0
$$595$$ 9.00000 0.368964
$$596$$ 0 0
$$597$$ 17.0000 0.695764
$$598$$ 0 0
$$599$$ −19.0000 −0.776319 −0.388159 0.921592i $$-0.626889\pi$$
−0.388159 + 0.921592i $$0.626889\pi$$
$$600$$ 0 0
$$601$$ 2.00000 0.0815817 0.0407909 0.999168i $$-0.487012\pi$$
0.0407909 + 0.999168i $$0.487012\pi$$
$$602$$ 0 0
$$603$$ 8.00000 0.325785
$$604$$ 0 0
$$605$$ 0 0
$$606$$ 0 0
$$607$$ −29.0000 −1.17707 −0.588537 0.808470i $$-0.700296\pi$$
−0.588537 + 0.808470i $$0.700296\pi$$
$$608$$ 0 0
$$609$$ −3.00000 −0.121566
$$610$$ 0 0
$$611$$ −40.0000 −1.61823
$$612$$ 0 0
$$613$$ −16.0000 −0.646234 −0.323117 0.946359i $$-0.604731\pi$$
−0.323117 + 0.946359i $$0.604731\pi$$
$$614$$ 0 0
$$615$$ −2.00000 −0.0806478
$$616$$ 0 0
$$617$$ 6.00000 0.241551 0.120775 0.992680i $$-0.461462\pi$$
0.120775 + 0.992680i $$0.461462\pi$$
$$618$$ 0 0
$$619$$ −18.0000 −0.723481 −0.361741 0.932279i $$-0.617817\pi$$
−0.361741 + 0.932279i $$0.617817\pi$$
$$620$$ 0 0
$$621$$ −20.0000 −0.802572
$$622$$ 0 0
$$623$$ 45.0000 1.80289
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ −15.0000 −0.598089
$$630$$ 0 0
$$631$$ −49.0000 −1.95066 −0.975330 0.220754i $$-0.929148\pi$$
−0.975330 + 0.220754i $$0.929148\pi$$
$$632$$ 0 0
$$633$$ −23.0000 −0.914168
$$634$$ 0 0
$$635$$ −16.0000 −0.634941
$$636$$ 0 0
$$637$$ −8.00000 −0.316972
$$638$$ 0 0
$$639$$ 10.0000 0.395594
$$640$$ 0 0
$$641$$ 19.0000 0.750455 0.375227 0.926933i $$-0.377565\pi$$
0.375227 + 0.926933i $$0.377565\pi$$
$$642$$ 0 0
$$643$$ −33.0000 −1.30139 −0.650696 0.759338i $$-0.725523\pi$$
−0.650696 + 0.759338i $$0.725523\pi$$
$$644$$ 0 0
$$645$$ 8.00000 0.315000
$$646$$ 0 0
$$647$$ 12.0000 0.471769 0.235884 0.971781i $$-0.424201\pi$$
0.235884 + 0.971781i $$0.424201\pi$$
$$648$$ 0 0
$$649$$ 0 0
$$650$$ 0 0
$$651$$ 15.0000 0.587896
$$652$$ 0 0
$$653$$ −3.00000 −0.117399 −0.0586995 0.998276i $$-0.518695\pi$$
−0.0586995 + 0.998276i $$0.518695\pi$$
$$654$$ 0 0
$$655$$ −15.0000 −0.586098
$$656$$ 0 0
$$657$$ −20.0000 −0.780274
$$658$$ 0 0
$$659$$ 33.0000 1.28550 0.642749 0.766077i $$-0.277794\pi$$
0.642749 + 0.766077i $$0.277794\pi$$
$$660$$ 0 0
$$661$$ 22.0000 0.855701 0.427850 0.903850i $$-0.359271\pi$$
0.427850 + 0.903850i $$0.359271\pi$$
$$662$$ 0 0
$$663$$ 12.0000 0.466041
$$664$$ 0 0
$$665$$ 9.00000 0.349005
$$666$$ 0 0
$$667$$ 4.00000 0.154881
$$668$$ 0 0
$$669$$ 2.00000 0.0773245
$$670$$ 0 0
$$671$$ 0 0
$$672$$ 0 0
$$673$$ −5.00000 −0.192736 −0.0963679 0.995346i $$-0.530723\pi$$
−0.0963679 + 0.995346i $$0.530723\pi$$
$$674$$ 0 0
$$675$$ 5.00000 0.192450
$$676$$ 0 0
$$677$$ 2.00000 0.0768662 0.0384331 0.999261i $$-0.487763\pi$$
0.0384331 + 0.999261i $$0.487763\pi$$
$$678$$ 0 0
$$679$$ 18.0000 0.690777
$$680$$ 0 0
$$681$$ 14.0000 0.536481
$$682$$ 0 0
$$683$$ 1.00000 0.0382639 0.0191320 0.999817i $$-0.493910\pi$$
0.0191320 + 0.999817i $$0.493910\pi$$
$$684$$ 0 0
$$685$$ −4.00000 −0.152832
$$686$$ 0 0
$$687$$ 16.0000 0.610438
$$688$$ 0 0
$$689$$ −44.0000 −1.67627
$$690$$ 0 0
$$691$$ 10.0000 0.380418 0.190209 0.981744i $$-0.439083\pi$$
0.190209 + 0.981744i $$0.439083\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ −4.00000 −0.151729
$$696$$ 0 0
$$697$$ −6.00000 −0.227266
$$698$$ 0 0
$$699$$ 13.0000 0.491705
$$700$$ 0 0
$$701$$ 45.0000 1.69963 0.849813 0.527084i $$-0.176715\pi$$
0.849813 + 0.527084i $$0.176715\pi$$
$$702$$ 0 0
$$703$$ −15.0000 −0.565736
$$704$$ 0 0
$$705$$ 10.0000 0.376622
$$706$$ 0 0
$$707$$ 54.0000 2.03088
$$708$$ 0 0
$$709$$ −18.0000 −0.676004 −0.338002 0.941145i $$-0.609751\pi$$
−0.338002 + 0.941145i $$0.609751\pi$$
$$710$$ 0 0
$$711$$ −16.0000 −0.600047
$$712$$ 0 0
$$713$$ −20.0000 −0.749006
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ 14.0000 0.522840
$$718$$ 0 0
$$719$$ −31.0000 −1.15610 −0.578052 0.816000i $$-0.696187\pi$$
−0.578052 + 0.816000i $$0.696187\pi$$
$$720$$ 0 0
$$721$$ 30.0000 1.11726
$$722$$ 0 0
$$723$$ −22.0000 −0.818189
$$724$$ 0 0
$$725$$ −1.00000 −0.0371391
$$726$$ 0 0
$$727$$ 44.0000 1.63187 0.815935 0.578144i $$-0.196223\pi$$
0.815935 + 0.578144i $$0.196223\pi$$
$$728$$ 0 0
$$729$$ 13.0000 0.481481
$$730$$ 0 0
$$731$$ 24.0000 0.887672
$$732$$ 0 0
$$733$$ −32.0000 −1.18195 −0.590973 0.806691i $$-0.701256\pi$$
−0.590973 + 0.806691i $$0.701256\pi$$
$$734$$ 0 0
$$735$$ 2.00000 0.0737711
$$736$$ 0 0
$$737$$ 0 0
$$738$$ 0 0
$$739$$ −48.0000 −1.76571 −0.882854 0.469647i $$-0.844381\pi$$
−0.882854 + 0.469647i $$0.844381\pi$$
$$740$$ 0 0
$$741$$ 12.0000 0.440831
$$742$$ 0 0
$$743$$ −23.0000 −0.843788 −0.421894 0.906645i $$-0.638635\pi$$
−0.421894 + 0.906645i $$0.638635\pi$$
$$744$$ 0 0
$$745$$ 19.0000 0.696106
$$746$$ 0 0
$$747$$ 0 0
$$748$$ 0 0
$$749$$ −54.0000 −1.97312
$$750$$ 0 0
$$751$$ −23.0000 −0.839282 −0.419641 0.907690i $$-0.637844\pi$$
−0.419641 + 0.907690i $$0.637844\pi$$
$$752$$ 0 0
$$753$$ 16.0000 0.583072
$$754$$ 0 0
$$755$$ 12.0000 0.436725
$$756$$ 0 0
$$757$$ 38.0000 1.38113 0.690567 0.723269i $$-0.257361\pi$$
0.690567 + 0.723269i $$0.257361\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ −54.0000 −1.95750 −0.978749 0.205061i $$-0.934261\pi$$
−0.978749 + 0.205061i $$0.934261\pi$$
$$762$$ 0 0
$$763$$ 42.0000 1.52050
$$764$$ 0 0
$$765$$ 6.00000 0.216930
$$766$$ 0 0
$$767$$ 0 0
$$768$$ 0 0
$$769$$ 38.0000 1.37032 0.685158 0.728395i $$-0.259733\pi$$
0.685158 + 0.728395i $$0.259733\pi$$
$$770$$ 0 0
$$771$$ 12.0000 0.432169
$$772$$ 0 0
$$773$$ 15.0000 0.539513 0.269756 0.962929i $$-0.413057\pi$$
0.269756 + 0.962929i $$0.413057\pi$$
$$774$$ 0 0
$$775$$ 5.00000 0.179605
$$776$$ 0 0
$$777$$ −15.0000 −0.538122
$$778$$ 0 0
$$779$$ −6.00000 −0.214972
$$780$$ 0 0
$$781$$ 0 0
$$782$$ 0 0
$$783$$ −5.00000 −0.178685
$$784$$ 0 0
$$785$$ −11.0000 −0.392607
$$786$$ 0 0
$$787$$ −6.00000 −0.213877 −0.106938 0.994266i $$-0.534105\pi$$
−0.106938 + 0.994266i $$0.534105\pi$$
$$788$$ 0 0
$$789$$ −11.0000 −0.391610
$$790$$ 0 0
$$791$$ −48.0000 −1.70668
$$792$$ 0 0
$$793$$ −4.00000 −0.142044
$$794$$ 0 0
$$795$$ 11.0000 0.390130
$$796$$ 0 0
$$797$$ −26.0000 −0.920967 −0.460484 0.887668i $$-0.652324\pi$$
−0.460484 + 0.887668i $$0.652324\pi$$
$$798$$ 0 0
$$799$$ 30.0000 1.06132
$$800$$ 0 0
$$801$$ 30.0000 1.06000
$$802$$ 0 0
$$803$$ 0 0
$$804$$ 0 0
$$805$$ −12.0000 −0.422944
$$806$$ 0 0
$$807$$ 4.00000 0.140807
$$808$$ 0 0
$$809$$ 24.0000 0.843795 0.421898 0.906644i $$-0.361364\pi$$
0.421898 + 0.906644i $$0.361364\pi$$
$$810$$ 0 0
$$811$$ −31.0000 −1.08856 −0.544279 0.838905i $$-0.683197\pi$$
−0.544279 + 0.838905i $$0.683197\pi$$
$$812$$ 0 0
$$813$$ 32.0000 1.12229
$$814$$ 0 0
$$815$$ 5.00000 0.175142
$$816$$ 0 0
$$817$$ 24.0000 0.839654
$$818$$ 0 0
$$819$$ −24.0000 −0.838628
$$820$$ 0 0
$$821$$ 6.00000 0.209401 0.104701 0.994504i $$-0.466612\pi$$
0.104701 + 0.994504i $$0.466612\pi$$
$$822$$ 0 0
$$823$$ −46.0000 −1.60346 −0.801730 0.597687i $$-0.796087\pi$$
−0.801730 + 0.597687i $$0.796087\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ −30.0000 −1.04320 −0.521601 0.853189i $$-0.674665\pi$$
−0.521601 + 0.853189i $$0.674665\pi$$
$$828$$ 0 0
$$829$$ 4.00000 0.138926 0.0694629 0.997585i $$-0.477871\pi$$
0.0694629 + 0.997585i $$0.477871\pi$$
$$830$$ 0 0
$$831$$ −22.0000 −0.763172
$$832$$ 0 0
$$833$$ 6.00000 0.207888
$$834$$ 0 0
$$835$$ −1.00000 −0.0346064
$$836$$ 0 0
$$837$$ 25.0000 0.864126
$$838$$ 0 0
$$839$$ −44.0000 −1.51905 −0.759524 0.650479i $$-0.774568\pi$$
−0.759524 + 0.650479i $$0.774568\pi$$
$$840$$ 0 0
$$841$$ −28.0000 −0.965517
$$842$$ 0 0
$$843$$ 18.0000 0.619953
$$844$$ 0 0
$$845$$ −3.00000 −0.103203
$$846$$ 0 0
$$847$$ 0 0
$$848$$ 0 0
$$849$$ 4.00000 0.137280
$$850$$ 0 0
$$851$$ 20.0000 0.685591
$$852$$ 0 0
$$853$$ 40.0000 1.36957 0.684787 0.728743i $$-0.259895\pi$$
0.684787 + 0.728743i $$0.259895\pi$$
$$854$$ 0 0
$$855$$ 6.00000 0.205196
$$856$$ 0 0
$$857$$ −25.0000 −0.853984 −0.426992 0.904255i $$-0.640427\pi$$
−0.426992 + 0.904255i $$0.640427\pi$$
$$858$$ 0 0
$$859$$ −52.0000 −1.77422 −0.887109 0.461561i $$-0.847290\pi$$
−0.887109 + 0.461561i $$0.847290\pi$$
$$860$$ 0 0
$$861$$ −6.00000 −0.204479
$$862$$ 0 0
$$863$$ −50.0000 −1.70202 −0.851010 0.525150i $$-0.824009\pi$$
−0.851010 + 0.525150i $$0.824009\pi$$
$$864$$ 0 0
$$865$$ −14.0000 −0.476014
$$866$$ 0 0
$$867$$ 8.00000 0.271694
$$868$$ 0 0
$$869$$ 0 0
$$870$$ 0 0
$$871$$ 16.0000 0.542139
$$872$$ 0 0
$$873$$ 12.0000 0.406138
$$874$$ 0 0
$$875$$ 3.00000 0.101419
$$876$$ 0 0
$$877$$ −42.0000 −1.41824 −0.709120 0.705088i $$-0.750907\pi$$
−0.709120 + 0.705088i $$0.750907\pi$$
$$878$$ 0 0
$$879$$ 10.0000 0.337292
$$880$$ 0 0
$$881$$ −26.0000 −0.875962 −0.437981 0.898984i $$-0.644306\pi$$
−0.437981 + 0.898984i $$0.644306\pi$$
$$882$$ 0 0
$$883$$ 57.0000 1.91820 0.959101 0.283063i $$-0.0913505\pi$$
0.959101 + 0.283063i $$0.0913505\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ 48.0000 1.61168 0.805841 0.592132i $$-0.201714\pi$$
0.805841 + 0.592132i $$0.201714\pi$$
$$888$$ 0 0
$$889$$ −48.0000 −1.60987
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 0 0
$$893$$ 30.0000 1.00391
$$894$$ 0 0
$$895$$ −2.00000 −0.0668526
$$896$$ 0 0
$$897$$ −16.0000 −0.534224
$$898$$ 0 0
$$899$$ −5.00000 −0.166759
$$900$$ 0 0
$$901$$ 33.0000 1.09939
$$902$$ 0 0
$$903$$ 24.0000 0.798670
$$904$$ 0 0
$$905$$ −16.0000 −0.531858
$$906$$ 0 0
$$907$$ 35.0000 1.16216 0.581078 0.813848i $$-0.302631\pi$$
0.581078 + 0.813848i $$0.302631\pi$$
$$908$$ 0 0
$$909$$ 36.0000 1.19404
$$910$$ 0 0
$$911$$ −9.00000 −0.298183 −0.149092 0.988823i $$-0.547635\pi$$
−0.149092 + 0.988823i $$0.547635\pi$$
$$912$$ 0 0
$$913$$ 0 0
$$914$$ 0 0
$$915$$ 1.00000 0.0330590
$$916$$ 0 0
$$917$$ −45.0000 −1.48603
$$918$$ 0 0
$$919$$ −32.0000 −1.05558 −0.527791 0.849374i $$-0.676980\pi$$
−0.527791 + 0.849374i $$0.676980\pi$$
$$920$$ 0 0
$$921$$ 2.00000 0.0659022
$$922$$ 0 0
$$923$$ 20.0000 0.658308
$$924$$ 0 0
$$925$$ −5.00000 −0.164399
$$926$$ 0 0
$$927$$ 20.0000 0.656886
$$928$$ 0 0
$$929$$ −21.0000 −0.688988 −0.344494 0.938789i $$-0.611949\pi$$
−0.344494 + 0.938789i $$0.611949\pi$$
$$930$$ 0 0
$$931$$ 6.00000 0.196642
$$932$$ 0 0
$$933$$ −21.0000 −0.687509
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ 42.0000 1.37208 0.686040 0.727564i $$-0.259347\pi$$
0.686040 + 0.727564i $$0.259347\pi$$
$$938$$ 0 0
$$939$$ 34.0000 1.10955
$$940$$ 0 0
$$941$$ 39.0000 1.27136 0.635682 0.771951i $$-0.280719\pi$$
0.635682 + 0.771951i $$0.280719\pi$$
$$942$$ 0 0
$$943$$ 8.00000 0.260516
$$944$$ 0 0
$$945$$ 15.0000 0.487950
$$946$$ 0 0
$$947$$ −59.0000 −1.91724 −0.958621 0.284685i $$-0.908111\pi$$
−0.958621 + 0.284685i $$0.908111\pi$$
$$948$$ 0 0
$$949$$ −40.0000 −1.29845
$$950$$ 0 0
$$951$$ −17.0000 −0.551263
$$952$$ 0 0
$$953$$ −43.0000 −1.39291 −0.696453 0.717602i $$-0.745240\pi$$
−0.696453 + 0.717602i $$0.745240\pi$$
$$954$$ 0 0
$$955$$ −4.00000 −0.129437
$$956$$ 0 0
$$957$$ 0 0
$$958$$ 0 0
$$959$$ −12.0000 −0.387500
$$960$$ 0 0
$$961$$ −6.00000 −0.193548
$$962$$ 0 0
$$963$$ −36.0000 −1.16008
$$964$$ 0 0
$$965$$ 5.00000 0.160956
$$966$$ 0 0
$$967$$ 55.0000 1.76868 0.884340 0.466843i $$-0.154609\pi$$
0.884340 + 0.466843i $$0.154609\pi$$
$$968$$ 0 0
$$969$$ −9.00000 −0.289122
$$970$$ 0 0
$$971$$ −22.0000 −0.706014 −0.353007 0.935621i $$-0.614841\pi$$
−0.353007 + 0.935621i $$0.614841\pi$$
$$972$$ 0 0
$$973$$ −12.0000 −0.384702
$$974$$ 0 0
$$975$$ 4.00000 0.128103
$$976$$ 0 0
$$977$$ 6.00000 0.191957 0.0959785 0.995383i $$-0.469402\pi$$
0.0959785 + 0.995383i $$0.469402\pi$$
$$978$$ 0 0
$$979$$ 0 0
$$980$$ 0 0
$$981$$ 28.0000 0.893971
$$982$$ 0 0
$$983$$ −4.00000 −0.127580 −0.0637901 0.997963i $$-0.520319\pi$$
−0.0637901 + 0.997963i $$0.520319\pi$$
$$984$$ 0 0
$$985$$ 26.0000 0.828429
$$986$$ 0 0
$$987$$ 30.0000 0.954911
$$988$$ 0 0
$$989$$ −32.0000 −1.01754
$$990$$ 0 0
$$991$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$992$$ 0 0
$$993$$ 20.0000 0.634681
$$994$$ 0 0
$$995$$ 17.0000 0.538936
$$996$$ 0 0
$$997$$ −48.0000 −1.52018 −0.760088 0.649821i $$-0.774844\pi$$
−0.760088 + 0.649821i $$0.774844\pi$$
$$998$$ 0 0
$$999$$ −25.0000 −0.790965
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 9680.2.a.e.1.1 1
4.3 odd 2 4840.2.a.h.1.1 yes 1
11.10 odd 2 9680.2.a.i.1.1 1
44.43 even 2 4840.2.a.g.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
4840.2.a.g.1.1 1 44.43 even 2
4840.2.a.h.1.1 yes 1 4.3 odd 2
9680.2.a.e.1.1 1 1.1 even 1 trivial
9680.2.a.i.1.1 1 11.10 odd 2