# Properties

 Label 200.4.a.f.1.1 Level $200$ Weight $4$ Character 200.1 Self dual yes Analytic conductor $11.800$ 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] = [200,4,Mod(1,200)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(200, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("200.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$200 = 2^{3} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 200.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: $$11.8003820011$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 200.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+1.00000 q^{3} -6.00000 q^{7} -26.0000 q^{9} +O(q^{10})$$ $$q+1.00000 q^{3} -6.00000 q^{7} -26.0000 q^{9} -19.0000 q^{11} +12.0000 q^{13} -75.0000 q^{17} -91.0000 q^{19} -6.00000 q^{21} +174.000 q^{23} -53.0000 q^{27} -272.000 q^{29} -230.000 q^{31} -19.0000 q^{33} -182.000 q^{37} +12.0000 q^{39} +117.000 q^{41} +372.000 q^{43} -52.0000 q^{47} -307.000 q^{49} -75.0000 q^{51} -402.000 q^{53} -91.0000 q^{57} +312.000 q^{59} +170.000 q^{61} +156.000 q^{63} +763.000 q^{67} +174.000 q^{69} -52.0000 q^{71} -981.000 q^{73} +114.000 q^{77} +1054.00 q^{79} +649.000 q^{81} +351.000 q^{83} -272.000 q^{87} +799.000 q^{89} -72.0000 q^{91} -230.000 q^{93} +962.000 q^{97} +494.000 q^{99} +O(q^{100})$$

## 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
$$3$$ 1.00000 0.192450 0.0962250 0.995360i $$-0.469323\pi$$
0.0962250 + 0.995360i $$0.469323\pi$$
$$4$$ 0 0
$$5$$ 0 0
$$6$$ 0 0
$$7$$ −6.00000 −0.323970 −0.161985 0.986793i $$-0.551790\pi$$
−0.161985 + 0.986793i $$0.551790\pi$$
$$8$$ 0 0
$$9$$ −26.0000 −0.962963
$$10$$ 0 0
$$11$$ −19.0000 −0.520792 −0.260396 0.965502i $$-0.583853\pi$$
−0.260396 + 0.965502i $$0.583853\pi$$
$$12$$ 0 0
$$13$$ 12.0000 0.256015 0.128008 0.991773i $$-0.459142\pi$$
0.128008 + 0.991773i $$0.459142\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ −75.0000 −1.07001 −0.535005 0.844849i $$-0.679690\pi$$
−0.535005 + 0.844849i $$0.679690\pi$$
$$18$$ 0 0
$$19$$ −91.0000 −1.09878 −0.549390 0.835566i $$-0.685140\pi$$
−0.549390 + 0.835566i $$0.685140\pi$$
$$20$$ 0 0
$$21$$ −6.00000 −0.0623480
$$22$$ 0 0
$$23$$ 174.000 1.57746 0.788728 0.614742i $$-0.210740\pi$$
0.788728 + 0.614742i $$0.210740\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ 0 0
$$27$$ −53.0000 −0.377772
$$28$$ 0 0
$$29$$ −272.000 −1.74169 −0.870847 0.491554i $$-0.836429\pi$$
−0.870847 + 0.491554i $$0.836429\pi$$
$$30$$ 0 0
$$31$$ −230.000 −1.33256 −0.666278 0.745704i $$-0.732113\pi$$
−0.666278 + 0.745704i $$0.732113\pi$$
$$32$$ 0 0
$$33$$ −19.0000 −0.100227
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ −182.000 −0.808665 −0.404333 0.914612i $$-0.632496\pi$$
−0.404333 + 0.914612i $$0.632496\pi$$
$$38$$ 0 0
$$39$$ 12.0000 0.0492702
$$40$$ 0 0
$$41$$ 117.000 0.445667 0.222833 0.974857i $$-0.428469\pi$$
0.222833 + 0.974857i $$0.428469\pi$$
$$42$$ 0 0
$$43$$ 372.000 1.31929 0.659645 0.751577i $$-0.270707\pi$$
0.659645 + 0.751577i $$0.270707\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ −52.0000 −0.161383 −0.0806913 0.996739i $$-0.525713\pi$$
−0.0806913 + 0.996739i $$0.525713\pi$$
$$48$$ 0 0
$$49$$ −307.000 −0.895044
$$50$$ 0 0
$$51$$ −75.0000 −0.205924
$$52$$ 0 0
$$53$$ −402.000 −1.04187 −0.520933 0.853597i $$-0.674416\pi$$
−0.520933 + 0.853597i $$0.674416\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ −91.0000 −0.211460
$$58$$ 0 0
$$59$$ 312.000 0.688457 0.344228 0.938886i $$-0.388141\pi$$
0.344228 + 0.938886i $$0.388141\pi$$
$$60$$ 0 0
$$61$$ 170.000 0.356824 0.178412 0.983956i $$-0.442904\pi$$
0.178412 + 0.983956i $$0.442904\pi$$
$$62$$ 0 0
$$63$$ 156.000 0.311971
$$64$$ 0 0
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 763.000 1.39127 0.695636 0.718394i $$-0.255122\pi$$
0.695636 + 0.718394i $$0.255122\pi$$
$$68$$ 0 0
$$69$$ 174.000 0.303582
$$70$$ 0 0
$$71$$ −52.0000 −0.0869192 −0.0434596 0.999055i $$-0.513838\pi$$
−0.0434596 + 0.999055i $$0.513838\pi$$
$$72$$ 0 0
$$73$$ −981.000 −1.57284 −0.786420 0.617692i $$-0.788068\pi$$
−0.786420 + 0.617692i $$0.788068\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ 114.000 0.168721
$$78$$ 0 0
$$79$$ 1054.00 1.50107 0.750533 0.660833i $$-0.229797\pi$$
0.750533 + 0.660833i $$0.229797\pi$$
$$80$$ 0 0
$$81$$ 649.000 0.890261
$$82$$ 0 0
$$83$$ 351.000 0.464184 0.232092 0.972694i $$-0.425443\pi$$
0.232092 + 0.972694i $$0.425443\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 0 0
$$87$$ −272.000 −0.335189
$$88$$ 0 0
$$89$$ 799.000 0.951616 0.475808 0.879549i $$-0.342156\pi$$
0.475808 + 0.879549i $$0.342156\pi$$
$$90$$ 0 0
$$91$$ −72.0000 −0.0829412
$$92$$ 0 0
$$93$$ −230.000 −0.256450
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 0 0
$$97$$ 962.000 1.00697 0.503486 0.864003i $$-0.332051\pi$$
0.503486 + 0.864003i $$0.332051\pi$$
$$98$$ 0 0
$$99$$ 494.000 0.501504
$$100$$ 0 0
$$101$$ 486.000 0.478800 0.239400 0.970921i $$-0.423049\pi$$
0.239400 + 0.970921i $$0.423049\pi$$
$$102$$ 0 0
$$103$$ −1188.00 −1.13648 −0.568238 0.822864i $$-0.692375\pi$$
−0.568238 + 0.822864i $$0.692375\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 1325.00 1.19713 0.598563 0.801075i $$-0.295738\pi$$
0.598563 + 0.801075i $$0.295738\pi$$
$$108$$ 0 0
$$109$$ 126.000 0.110721 0.0553606 0.998466i $$-0.482369\pi$$
0.0553606 + 0.998466i $$0.482369\pi$$
$$110$$ 0 0
$$111$$ −182.000 −0.155628
$$112$$ 0 0
$$113$$ 183.000 0.152347 0.0761734 0.997095i $$-0.475730\pi$$
0.0761734 + 0.997095i $$0.475730\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 0 0
$$117$$ −312.000 −0.246533
$$118$$ 0 0
$$119$$ 450.000 0.346651
$$120$$ 0 0
$$121$$ −970.000 −0.728775
$$122$$ 0 0
$$123$$ 117.000 0.0857686
$$124$$ 0 0
$$125$$ 0 0
$$126$$ 0 0
$$127$$ −902.000 −0.630233 −0.315116 0.949053i $$-0.602044\pi$$
−0.315116 + 0.949053i $$0.602044\pi$$
$$128$$ 0 0
$$129$$ 372.000 0.253897
$$130$$ 0 0
$$131$$ 2068.00 1.37925 0.689626 0.724166i $$-0.257775\pi$$
0.689626 + 0.724166i $$0.257775\pi$$
$$132$$ 0 0
$$133$$ 546.000 0.355971
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ −1339.00 −0.835025 −0.417513 0.908671i $$-0.637098\pi$$
−0.417513 + 0.908671i $$0.637098\pi$$
$$138$$ 0 0
$$139$$ −2939.00 −1.79340 −0.896700 0.442638i $$-0.854043\pi$$
−0.896700 + 0.442638i $$0.854043\pi$$
$$140$$ 0 0
$$141$$ −52.0000 −0.0310581
$$142$$ 0 0
$$143$$ −228.000 −0.133331
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 0 0
$$147$$ −307.000 −0.172251
$$148$$ 0 0
$$149$$ −208.000 −0.114363 −0.0571813 0.998364i $$-0.518211\pi$$
−0.0571813 + 0.998364i $$0.518211\pi$$
$$150$$ 0 0
$$151$$ −2678.00 −1.44326 −0.721631 0.692278i $$-0.756607\pi$$
−0.721631 + 0.692278i $$0.756607\pi$$
$$152$$ 0 0
$$153$$ 1950.00 1.03038
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ 1482.00 0.753353 0.376677 0.926345i $$-0.377067\pi$$
0.376677 + 0.926345i $$0.377067\pi$$
$$158$$ 0 0
$$159$$ −402.000 −0.200507
$$160$$ 0 0
$$161$$ −1044.00 −0.511048
$$162$$ 0 0
$$163$$ −1469.00 −0.705895 −0.352948 0.935643i $$-0.614821\pi$$
−0.352948 + 0.935643i $$0.614821\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ −4004.00 −1.85532 −0.927661 0.373423i $$-0.878184\pi$$
−0.927661 + 0.373423i $$0.878184\pi$$
$$168$$ 0 0
$$169$$ −2053.00 −0.934456
$$170$$ 0 0
$$171$$ 2366.00 1.05809
$$172$$ 0 0
$$173$$ 3224.00 1.41686 0.708428 0.705783i $$-0.249405\pi$$
0.708428 + 0.705783i $$0.249405\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ 312.000 0.132494
$$178$$ 0 0
$$179$$ −4191.00 −1.75000 −0.875000 0.484123i $$-0.839139\pi$$
−0.875000 + 0.484123i $$0.839139\pi$$
$$180$$ 0 0
$$181$$ −3718.00 −1.52683 −0.763416 0.645907i $$-0.776480\pi$$
−0.763416 + 0.645907i $$0.776480\pi$$
$$182$$ 0 0
$$183$$ 170.000 0.0686708
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 0 0
$$187$$ 1425.00 0.557253
$$188$$ 0 0
$$189$$ 318.000 0.122387
$$190$$ 0 0
$$191$$ 870.000 0.329586 0.164793 0.986328i $$-0.447304\pi$$
0.164793 + 0.986328i $$0.447304\pi$$
$$192$$ 0 0
$$193$$ 2197.00 0.819396 0.409698 0.912221i $$-0.365634\pi$$
0.409698 + 0.912221i $$0.365634\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ −2314.00 −0.836882 −0.418441 0.908244i $$-0.637423\pi$$
−0.418441 + 0.908244i $$0.637423\pi$$
$$198$$ 0 0
$$199$$ −252.000 −0.0897679 −0.0448839 0.998992i $$-0.514292\pi$$
−0.0448839 + 0.998992i $$0.514292\pi$$
$$200$$ 0 0
$$201$$ 763.000 0.267751
$$202$$ 0 0
$$203$$ 1632.00 0.564256
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 0 0
$$207$$ −4524.00 −1.51903
$$208$$ 0 0
$$209$$ 1729.00 0.572237
$$210$$ 0 0
$$211$$ −741.000 −0.241766 −0.120883 0.992667i $$-0.538573\pi$$
−0.120883 + 0.992667i $$0.538573\pi$$
$$212$$ 0 0
$$213$$ −52.0000 −0.0167276
$$214$$ 0 0
$$215$$ 0 0
$$216$$ 0 0
$$217$$ 1380.00 0.431707
$$218$$ 0 0
$$219$$ −981.000 −0.302693
$$220$$ 0 0
$$221$$ −900.000 −0.273939
$$222$$ 0 0
$$223$$ −5092.00 −1.52908 −0.764542 0.644574i $$-0.777035\pi$$
−0.764542 + 0.644574i $$0.777035\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ 5876.00 1.71808 0.859039 0.511910i $$-0.171062\pi$$
0.859039 + 0.511910i $$0.171062\pi$$
$$228$$ 0 0
$$229$$ −604.000 −0.174295 −0.0871473 0.996195i $$-0.527775\pi$$
−0.0871473 + 0.996195i $$0.527775\pi$$
$$230$$ 0 0
$$231$$ 114.000 0.0324703
$$232$$ 0 0
$$233$$ 278.000 0.0781647 0.0390824 0.999236i $$-0.487557\pi$$
0.0390824 + 0.999236i $$0.487557\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 0 0
$$237$$ 1054.00 0.288880
$$238$$ 0 0
$$239$$ 2496.00 0.675535 0.337767 0.941230i $$-0.390328\pi$$
0.337767 + 0.941230i $$0.390328\pi$$
$$240$$ 0 0
$$241$$ −2567.00 −0.686120 −0.343060 0.939313i $$-0.611463\pi$$
−0.343060 + 0.939313i $$0.611463\pi$$
$$242$$ 0 0
$$243$$ 2080.00 0.549103
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ −1092.00 −0.281305
$$248$$ 0 0
$$249$$ 351.000 0.0893322
$$250$$ 0 0
$$251$$ 5395.00 1.35669 0.678345 0.734743i $$-0.262697\pi$$
0.678345 + 0.734743i $$0.262697\pi$$
$$252$$ 0 0
$$253$$ −3306.00 −0.821527
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ −1490.00 −0.361648 −0.180824 0.983515i $$-0.557877\pi$$
−0.180824 + 0.983515i $$0.557877\pi$$
$$258$$ 0 0
$$259$$ 1092.00 0.261983
$$260$$ 0 0
$$261$$ 7072.00 1.67719
$$262$$ 0 0
$$263$$ −3330.00 −0.780748 −0.390374 0.920656i $$-0.627654\pi$$
−0.390374 + 0.920656i $$0.627654\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 0 0
$$267$$ 799.000 0.183139
$$268$$ 0 0
$$269$$ 6096.00 1.38171 0.690854 0.722994i $$-0.257235\pi$$
0.690854 + 0.722994i $$0.257235\pi$$
$$270$$ 0 0
$$271$$ −6006.00 −1.34627 −0.673134 0.739521i $$-0.735052\pi$$
−0.673134 + 0.739521i $$0.735052\pi$$
$$272$$ 0 0
$$273$$ −72.0000 −0.0159620
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 0 0
$$277$$ 6976.00 1.51317 0.756583 0.653897i $$-0.226867\pi$$
0.756583 + 0.653897i $$0.226867\pi$$
$$278$$ 0 0
$$279$$ 5980.00 1.28320
$$280$$ 0 0
$$281$$ 3998.00 0.848757 0.424378 0.905485i $$-0.360493\pi$$
0.424378 + 0.905485i $$0.360493\pi$$
$$282$$ 0 0
$$283$$ −13.0000 −0.00273064 −0.00136532 0.999999i $$-0.500435\pi$$
−0.00136532 + 0.999999i $$0.500435\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ −702.000 −0.144382
$$288$$ 0 0
$$289$$ 712.000 0.144922
$$290$$ 0 0
$$291$$ 962.000 0.193792
$$292$$ 0 0
$$293$$ −4466.00 −0.890466 −0.445233 0.895415i $$-0.646879\pi$$
−0.445233 + 0.895415i $$0.646879\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 0 0
$$297$$ 1007.00 0.196741
$$298$$ 0 0
$$299$$ 2088.00 0.403853
$$300$$ 0 0
$$301$$ −2232.00 −0.427410
$$302$$ 0 0
$$303$$ 486.000 0.0921451
$$304$$ 0 0
$$305$$ 0 0
$$306$$ 0 0
$$307$$ 9041.00 1.68077 0.840386 0.541988i $$-0.182328\pi$$
0.840386 + 0.541988i $$0.182328\pi$$
$$308$$ 0 0
$$309$$ −1188.00 −0.218715
$$310$$ 0 0
$$311$$ −346.000 −0.0630864 −0.0315432 0.999502i $$-0.510042\pi$$
−0.0315432 + 0.999502i $$0.510042\pi$$
$$312$$ 0 0
$$313$$ −10646.0 −1.92252 −0.961258 0.275650i $$-0.911107\pi$$
−0.961258 + 0.275650i $$0.911107\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ −8116.00 −1.43798 −0.718990 0.695020i $$-0.755396\pi$$
−0.718990 + 0.695020i $$0.755396\pi$$
$$318$$ 0 0
$$319$$ 5168.00 0.907061
$$320$$ 0 0
$$321$$ 1325.00 0.230387
$$322$$ 0 0
$$323$$ 6825.00 1.17571
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 0 0
$$327$$ 126.000 0.0213083
$$328$$ 0 0
$$329$$ 312.000 0.0522830
$$330$$ 0 0
$$331$$ −3007.00 −0.499334 −0.249667 0.968332i $$-0.580321\pi$$
−0.249667 + 0.968332i $$0.580321\pi$$
$$332$$ 0 0
$$333$$ 4732.00 0.778715
$$334$$ 0 0
$$335$$ 0 0
$$336$$ 0 0
$$337$$ 83.0000 0.0134163 0.00670816 0.999978i $$-0.497865\pi$$
0.00670816 + 0.999978i $$0.497865\pi$$
$$338$$ 0 0
$$339$$ 183.000 0.0293192
$$340$$ 0 0
$$341$$ 4370.00 0.693985
$$342$$ 0 0
$$343$$ 3900.00 0.613936
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ 6189.00 0.957472 0.478736 0.877959i $$-0.341095\pi$$
0.478736 + 0.877959i $$0.341095\pi$$
$$348$$ 0 0
$$349$$ −5362.00 −0.822411 −0.411205 0.911543i $$-0.634892\pi$$
−0.411205 + 0.911543i $$0.634892\pi$$
$$350$$ 0 0
$$351$$ −636.000 −0.0967156
$$352$$ 0 0
$$353$$ −1690.00 −0.254815 −0.127407 0.991850i $$-0.540666\pi$$
−0.127407 + 0.991850i $$0.540666\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0 0
$$357$$ 450.000 0.0667130
$$358$$ 0 0
$$359$$ 1638.00 0.240809 0.120404 0.992725i $$-0.461581\pi$$
0.120404 + 0.992725i $$0.461581\pi$$
$$360$$ 0 0
$$361$$ 1422.00 0.207319
$$362$$ 0 0
$$363$$ −970.000 −0.140253
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 0 0
$$367$$ −7580.00 −1.07813 −0.539064 0.842265i $$-0.681222\pi$$
−0.539064 + 0.842265i $$0.681222\pi$$
$$368$$ 0 0
$$369$$ −3042.00 −0.429160
$$370$$ 0 0
$$371$$ 2412.00 0.337533
$$372$$ 0 0
$$373$$ −5630.00 −0.781529 −0.390765 0.920491i $$-0.627789\pi$$
−0.390765 + 0.920491i $$0.627789\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ −3264.00 −0.445901
$$378$$ 0 0
$$379$$ −4385.00 −0.594307 −0.297153 0.954830i $$-0.596037\pi$$
−0.297153 + 0.954830i $$0.596037\pi$$
$$380$$ 0 0
$$381$$ −902.000 −0.121288
$$382$$ 0 0
$$383$$ 12558.0 1.67541 0.837707 0.546119i $$-0.183896\pi$$
0.837707 + 0.546119i $$0.183896\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ −9672.00 −1.27043
$$388$$ 0 0
$$389$$ −6570.00 −0.856330 −0.428165 0.903701i $$-0.640840\pi$$
−0.428165 + 0.903701i $$0.640840\pi$$
$$390$$ 0 0
$$391$$ −13050.0 −1.68789
$$392$$ 0 0
$$393$$ 2068.00 0.265437
$$394$$ 0 0
$$395$$ 0 0
$$396$$ 0 0
$$397$$ 1268.00 0.160300 0.0801500 0.996783i $$-0.474460\pi$$
0.0801500 + 0.996783i $$0.474460\pi$$
$$398$$ 0 0
$$399$$ 546.000 0.0685067
$$400$$ 0 0
$$401$$ −6299.00 −0.784432 −0.392216 0.919873i $$-0.628291\pi$$
−0.392216 + 0.919873i $$0.628291\pi$$
$$402$$ 0 0
$$403$$ −2760.00 −0.341155
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 3458.00 0.421147
$$408$$ 0 0
$$409$$ −13459.0 −1.62715 −0.813575 0.581459i $$-0.802482\pi$$
−0.813575 + 0.581459i $$0.802482\pi$$
$$410$$ 0 0
$$411$$ −1339.00 −0.160701
$$412$$ 0 0
$$413$$ −1872.00 −0.223039
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 0 0
$$417$$ −2939.00 −0.345140
$$418$$ 0 0
$$419$$ −9567.00 −1.11546 −0.557731 0.830022i $$-0.688328\pi$$
−0.557731 + 0.830022i $$0.688328\pi$$
$$420$$ 0 0
$$421$$ 2708.00 0.313491 0.156746 0.987639i $$-0.449900\pi$$
0.156746 + 0.987639i $$0.449900\pi$$
$$422$$ 0 0
$$423$$ 1352.00 0.155405
$$424$$ 0 0
$$425$$ 0 0
$$426$$ 0 0
$$427$$ −1020.00 −0.115600
$$428$$ 0 0
$$429$$ −228.000 −0.0256595
$$430$$ 0 0
$$431$$ 5126.00 0.572879 0.286439 0.958098i $$-0.407528\pi$$
0.286439 + 0.958098i $$0.407528\pi$$
$$432$$ 0 0
$$433$$ 11445.0 1.27023 0.635117 0.772416i $$-0.280952\pi$$
0.635117 + 0.772416i $$0.280952\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ −15834.0 −1.73328
$$438$$ 0 0
$$439$$ −5096.00 −0.554029 −0.277015 0.960866i $$-0.589345\pi$$
−0.277015 + 0.960866i $$0.589345\pi$$
$$440$$ 0 0
$$441$$ 7982.00 0.861894
$$442$$ 0 0
$$443$$ 13247.0 1.42073 0.710366 0.703833i $$-0.248530\pi$$
0.710366 + 0.703833i $$0.248530\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 0 0
$$447$$ −208.000 −0.0220091
$$448$$ 0 0
$$449$$ 7449.00 0.782940 0.391470 0.920191i $$-0.371967\pi$$
0.391470 + 0.920191i $$0.371967\pi$$
$$450$$ 0 0
$$451$$ −2223.00 −0.232100
$$452$$ 0 0
$$453$$ −2678.00 −0.277756
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 3497.00 0.357949 0.178975 0.983854i $$-0.442722\pi$$
0.178975 + 0.983854i $$0.442722\pi$$
$$458$$ 0 0
$$459$$ 3975.00 0.404220
$$460$$ 0 0
$$461$$ −13108.0 −1.32430 −0.662148 0.749373i $$-0.730355\pi$$
−0.662148 + 0.749373i $$0.730355\pi$$
$$462$$ 0 0
$$463$$ −5428.00 −0.544839 −0.272420 0.962179i $$-0.587824\pi$$
−0.272420 + 0.962179i $$0.587824\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ −1516.00 −0.150219 −0.0751093 0.997175i $$-0.523931\pi$$
−0.0751093 + 0.997175i $$0.523931\pi$$
$$468$$ 0 0
$$469$$ −4578.00 −0.450730
$$470$$ 0 0
$$471$$ 1482.00 0.144983
$$472$$ 0 0
$$473$$ −7068.00 −0.687076
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 0 0
$$477$$ 10452.0 1.00328
$$478$$ 0 0
$$479$$ −14762.0 −1.40813 −0.704064 0.710137i $$-0.748633\pi$$
−0.704064 + 0.710137i $$0.748633\pi$$
$$480$$ 0 0
$$481$$ −2184.00 −0.207031
$$482$$ 0 0
$$483$$ −1044.00 −0.0983512
$$484$$ 0 0
$$485$$ 0 0
$$486$$ 0 0
$$487$$ −3926.00 −0.365306 −0.182653 0.983177i $$-0.558468\pi$$
−0.182653 + 0.983177i $$0.558468\pi$$
$$488$$ 0 0
$$489$$ −1469.00 −0.135850
$$490$$ 0 0
$$491$$ 996.000 0.0915455 0.0457728 0.998952i $$-0.485425\pi$$
0.0457728 + 0.998952i $$0.485425\pi$$
$$492$$ 0 0
$$493$$ 20400.0 1.86363
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 312.000 0.0281592
$$498$$ 0 0
$$499$$ 7804.00 0.700110 0.350055 0.936729i $$-0.386163\pi$$
0.350055 + 0.936729i $$0.386163\pi$$
$$500$$ 0 0
$$501$$ −4004.00 −0.357057
$$502$$ 0 0
$$503$$ −16732.0 −1.48319 −0.741593 0.670850i $$-0.765930\pi$$
−0.741593 + 0.670850i $$0.765930\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 0 0
$$507$$ −2053.00 −0.179836
$$508$$ 0 0
$$509$$ 10426.0 0.907906 0.453953 0.891026i $$-0.350013\pi$$
0.453953 + 0.891026i $$0.350013\pi$$
$$510$$ 0 0
$$511$$ 5886.00 0.509552
$$512$$ 0 0
$$513$$ 4823.00 0.415089
$$514$$ 0 0
$$515$$ 0 0
$$516$$ 0 0
$$517$$ 988.000 0.0840468
$$518$$ 0 0
$$519$$ 3224.00 0.272674
$$520$$ 0 0
$$521$$ −2235.00 −0.187941 −0.0939704 0.995575i $$-0.529956\pi$$
−0.0939704 + 0.995575i $$0.529956\pi$$
$$522$$ 0 0
$$523$$ −10855.0 −0.907564 −0.453782 0.891113i $$-0.649926\pi$$
−0.453782 + 0.891113i $$0.649926\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 17250.0 1.42585
$$528$$ 0 0
$$529$$ 18109.0 1.48837
$$530$$ 0 0
$$531$$ −8112.00 −0.662958
$$532$$ 0 0
$$533$$ 1404.00 0.114098
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 0 0
$$537$$ −4191.00 −0.336788
$$538$$ 0 0
$$539$$ 5833.00 0.466132
$$540$$ 0 0
$$541$$ 10608.0 0.843019 0.421510 0.906824i $$-0.361500\pi$$
0.421510 + 0.906824i $$0.361500\pi$$
$$542$$ 0 0
$$543$$ −3718.00 −0.293839
$$544$$ 0 0
$$545$$ 0 0
$$546$$ 0 0
$$547$$ −11583.0 −0.905399 −0.452700 0.891663i $$-0.649539\pi$$
−0.452700 + 0.891663i $$0.649539\pi$$
$$548$$ 0 0
$$549$$ −4420.00 −0.343608
$$550$$ 0 0
$$551$$ 24752.0 1.91374
$$552$$ 0 0
$$553$$ −6324.00 −0.486300
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ 17244.0 1.31176 0.655881 0.754864i $$-0.272297\pi$$
0.655881 + 0.754864i $$0.272297\pi$$
$$558$$ 0 0
$$559$$ 4464.00 0.337759
$$560$$ 0 0
$$561$$ 1425.00 0.107243
$$562$$ 0 0
$$563$$ −18416.0 −1.37858 −0.689291 0.724484i $$-0.742078\pi$$
−0.689291 + 0.724484i $$0.742078\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 0 0
$$567$$ −3894.00 −0.288417
$$568$$ 0 0
$$569$$ 11913.0 0.877713 0.438857 0.898557i $$-0.355384\pi$$
0.438857 + 0.898557i $$0.355384\pi$$
$$570$$ 0 0
$$571$$ 3900.00 0.285832 0.142916 0.989735i $$-0.454352\pi$$
0.142916 + 0.989735i $$0.454352\pi$$
$$572$$ 0 0
$$573$$ 870.000 0.0634289
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 0 0
$$577$$ 24899.0 1.79646 0.898231 0.439523i $$-0.144853\pi$$
0.898231 + 0.439523i $$0.144853\pi$$
$$578$$ 0 0
$$579$$ 2197.00 0.157693
$$580$$ 0 0
$$581$$ −2106.00 −0.150381
$$582$$ 0 0
$$583$$ 7638.00 0.542596
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ 1751.00 0.123120 0.0615601 0.998103i $$-0.480392\pi$$
0.0615601 + 0.998103i $$0.480392\pi$$
$$588$$ 0 0
$$589$$ 20930.0 1.46419
$$590$$ 0 0
$$591$$ −2314.00 −0.161058
$$592$$ 0 0
$$593$$ 10887.0 0.753922 0.376961 0.926229i $$-0.376969\pi$$
0.376961 + 0.926229i $$0.376969\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ −252.000 −0.0172758
$$598$$ 0 0
$$599$$ −14650.0 −0.999303 −0.499652 0.866226i $$-0.666539\pi$$
−0.499652 + 0.866226i $$0.666539\pi$$
$$600$$ 0 0
$$601$$ −4237.00 −0.287572 −0.143786 0.989609i $$-0.545928\pi$$
−0.143786 + 0.989609i $$0.545928\pi$$
$$602$$ 0 0
$$603$$ −19838.0 −1.33974
$$604$$ 0 0
$$605$$ 0 0
$$606$$ 0 0
$$607$$ 11440.0 0.764968 0.382484 0.923962i $$-0.375069\pi$$
0.382484 + 0.923962i $$0.375069\pi$$
$$608$$ 0 0
$$609$$ 1632.00 0.108591
$$610$$ 0 0
$$611$$ −624.000 −0.0413164
$$612$$ 0 0
$$613$$ −19370.0 −1.27626 −0.638130 0.769929i $$-0.720292\pi$$
−0.638130 + 0.769929i $$0.720292\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ −21346.0 −1.39280 −0.696400 0.717654i $$-0.745216\pi$$
−0.696400 + 0.717654i $$0.745216\pi$$
$$618$$ 0 0
$$619$$ 7436.00 0.482840 0.241420 0.970421i $$-0.422387\pi$$
0.241420 + 0.970421i $$0.422387\pi$$
$$620$$ 0 0
$$621$$ −9222.00 −0.595920
$$622$$ 0 0
$$623$$ −4794.00 −0.308295
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 0 0
$$627$$ 1729.00 0.110127
$$628$$ 0 0
$$629$$ 13650.0 0.865280
$$630$$ 0 0
$$631$$ 22490.0 1.41888 0.709440 0.704766i $$-0.248948\pi$$
0.709440 + 0.704766i $$0.248948\pi$$
$$632$$ 0 0
$$633$$ −741.000 −0.0465278
$$634$$ 0 0
$$635$$ 0 0
$$636$$ 0 0
$$637$$ −3684.00 −0.229145
$$638$$ 0 0
$$639$$ 1352.00 0.0837000
$$640$$ 0 0
$$641$$ −16086.0 −0.991199 −0.495600 0.868551i $$-0.665052\pi$$
−0.495600 + 0.868551i $$0.665052\pi$$
$$642$$ 0 0
$$643$$ −2396.00 −0.146950 −0.0734751 0.997297i $$-0.523409\pi$$
−0.0734751 + 0.997297i $$0.523409\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 23244.0 1.41239 0.706195 0.708018i $$-0.250410\pi$$
0.706195 + 0.708018i $$0.250410\pi$$
$$648$$ 0 0
$$649$$ −5928.00 −0.358543
$$650$$ 0 0
$$651$$ 1380.00 0.0830821
$$652$$ 0 0
$$653$$ 13598.0 0.814902 0.407451 0.913227i $$-0.366418\pi$$
0.407451 + 0.913227i $$0.366418\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 0 0
$$657$$ 25506.0 1.51459
$$658$$ 0 0
$$659$$ 9751.00 0.576396 0.288198 0.957571i $$-0.406944\pi$$
0.288198 + 0.957571i $$0.406944\pi$$
$$660$$ 0 0
$$661$$ 19104.0 1.12414 0.562072 0.827088i $$-0.310004\pi$$
0.562072 + 0.827088i $$0.310004\pi$$
$$662$$ 0 0
$$663$$ −900.000 −0.0527196
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ −47328.0 −2.74745
$$668$$ 0 0
$$669$$ −5092.00 −0.294272
$$670$$ 0 0
$$671$$ −3230.00 −0.185831
$$672$$ 0 0
$$673$$ 25402.0 1.45494 0.727470 0.686139i $$-0.240696\pi$$
0.727470 + 0.686139i $$0.240696\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ −2224.00 −0.126256 −0.0631280 0.998005i $$-0.520108\pi$$
−0.0631280 + 0.998005i $$0.520108\pi$$
$$678$$ 0 0
$$679$$ −5772.00 −0.326228
$$680$$ 0 0
$$681$$ 5876.00 0.330644
$$682$$ 0 0
$$683$$ 8281.00 0.463929 0.231965 0.972724i $$-0.425485\pi$$
0.231965 + 0.972724i $$0.425485\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 0 0
$$687$$ −604.000 −0.0335430
$$688$$ 0 0
$$689$$ −4824.00 −0.266734
$$690$$ 0 0
$$691$$ 481.000 0.0264806 0.0132403 0.999912i $$-0.495785\pi$$
0.0132403 + 0.999912i $$0.495785\pi$$
$$692$$ 0 0
$$693$$ −2964.00 −0.162472
$$694$$ 0 0
$$695$$ 0 0
$$696$$ 0 0
$$697$$ −8775.00 −0.476868
$$698$$ 0 0
$$699$$ 278.000 0.0150428
$$700$$ 0 0
$$701$$ −16788.0 −0.904528 −0.452264 0.891884i $$-0.649383\pi$$
−0.452264 + 0.891884i $$0.649383\pi$$
$$702$$ 0 0
$$703$$ 16562.0 0.888546
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ −2916.00 −0.155117
$$708$$ 0 0
$$709$$ −23452.0 −1.24225 −0.621127 0.783710i $$-0.713325\pi$$
−0.621127 + 0.783710i $$0.713325\pi$$
$$710$$ 0 0
$$711$$ −27404.0 −1.44547
$$712$$ 0 0
$$713$$ −40020.0 −2.10205
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ 2496.00 0.130007
$$718$$ 0 0
$$719$$ −20886.0 −1.08333 −0.541666 0.840594i $$-0.682206\pi$$
−0.541666 + 0.840594i $$0.682206\pi$$
$$720$$ 0 0
$$721$$ 7128.00 0.368184
$$722$$ 0 0
$$723$$ −2567.00 −0.132044
$$724$$ 0 0
$$725$$ 0 0
$$726$$ 0 0
$$727$$ 22576.0 1.15172 0.575858 0.817550i $$-0.304668\pi$$
0.575858 + 0.817550i $$0.304668\pi$$
$$728$$ 0 0
$$729$$ −15443.0 −0.784586
$$730$$ 0 0
$$731$$ −27900.0 −1.41165
$$732$$ 0 0
$$733$$ 35308.0 1.77917 0.889584 0.456771i $$-0.150994\pi$$
0.889584 + 0.456771i $$0.150994\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ −14497.0 −0.724564
$$738$$ 0 0
$$739$$ 188.000 0.00935818 0.00467909 0.999989i $$-0.498511\pi$$
0.00467909 + 0.999989i $$0.498511\pi$$
$$740$$ 0 0
$$741$$ −1092.00 −0.0541371
$$742$$ 0 0
$$743$$ 29870.0 1.47486 0.737432 0.675421i $$-0.236038\pi$$
0.737432 + 0.675421i $$0.236038\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 0 0
$$747$$ −9126.00 −0.446992
$$748$$ 0 0
$$749$$ −7950.00 −0.387833
$$750$$ 0 0
$$751$$ 22784.0 1.10706 0.553529 0.832830i $$-0.313281\pi$$
0.553529 + 0.832830i $$0.313281\pi$$
$$752$$ 0 0
$$753$$ 5395.00 0.261095
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 0 0
$$757$$ −28496.0 −1.36817 −0.684085 0.729402i $$-0.739798\pi$$
−0.684085 + 0.729402i $$0.739798\pi$$
$$758$$ 0 0
$$759$$ −3306.00 −0.158103
$$760$$ 0 0
$$761$$ 7397.00 0.352354 0.176177 0.984359i $$-0.443627\pi$$
0.176177 + 0.984359i $$0.443627\pi$$
$$762$$ 0 0
$$763$$ −756.000 −0.0358703
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ 3744.00 0.176256
$$768$$ 0 0
$$769$$ −8883.00 −0.416553 −0.208276 0.978070i $$-0.566785\pi$$
−0.208276 + 0.978070i $$0.566785\pi$$
$$770$$ 0 0
$$771$$ −1490.00 −0.0695993
$$772$$ 0 0
$$773$$ −33960.0 −1.58015 −0.790075 0.613010i $$-0.789959\pi$$
−0.790075 + 0.613010i $$0.789959\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 0 0
$$777$$ 1092.00 0.0504186
$$778$$ 0 0
$$779$$ −10647.0 −0.489690
$$780$$ 0 0
$$781$$ 988.000 0.0452669
$$782$$ 0 0
$$783$$ 14416.0 0.657964
$$784$$ 0 0
$$785$$ 0 0
$$786$$ 0 0
$$787$$ −32084.0 −1.45320 −0.726602 0.687059i $$-0.758901\pi$$
−0.726602 + 0.687059i $$0.758901\pi$$
$$788$$ 0 0
$$789$$ −3330.00 −0.150255
$$790$$ 0 0
$$791$$ −1098.00 −0.0493557
$$792$$ 0 0
$$793$$ 2040.00 0.0913525
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ −4766.00 −0.211820 −0.105910 0.994376i $$-0.533776\pi$$
−0.105910 + 0.994376i $$0.533776\pi$$
$$798$$ 0 0
$$799$$ 3900.00 0.172681
$$800$$ 0 0
$$801$$ −20774.0 −0.916371
$$802$$ 0 0
$$803$$ 18639.0 0.819123
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ 6096.00 0.265910
$$808$$ 0 0
$$809$$ −31278.0 −1.35930 −0.679651 0.733535i $$-0.737869\pi$$
−0.679651 + 0.733535i $$0.737869\pi$$
$$810$$ 0 0
$$811$$ −29956.0 −1.29704 −0.648519 0.761199i $$-0.724611\pi$$
−0.648519 + 0.761199i $$0.724611\pi$$
$$812$$ 0 0
$$813$$ −6006.00 −0.259089
$$814$$ 0 0
$$815$$ 0 0
$$816$$ 0 0
$$817$$ −33852.0 −1.44961
$$818$$ 0 0
$$819$$ 1872.00 0.0798693
$$820$$ 0 0
$$821$$ 14642.0 0.622423 0.311212 0.950341i $$-0.399265\pi$$
0.311212 + 0.950341i $$0.399265\pi$$
$$822$$ 0 0
$$823$$ 20844.0 0.882839 0.441419 0.897301i $$-0.354475\pi$$
0.441419 + 0.897301i $$0.354475\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 23751.0 0.998674 0.499337 0.866408i $$-0.333577\pi$$
0.499337 + 0.866408i $$0.333577\pi$$
$$828$$ 0 0
$$829$$ 11380.0 0.476772 0.238386 0.971171i $$-0.423382\pi$$
0.238386 + 0.971171i $$0.423382\pi$$
$$830$$ 0 0
$$831$$ 6976.00 0.291209
$$832$$ 0 0
$$833$$ 23025.0 0.957706
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 0 0
$$837$$ 12190.0 0.503403
$$838$$ 0 0
$$839$$ −29744.0 −1.22393 −0.611965 0.790885i $$-0.709621\pi$$
−0.611965 + 0.790885i $$0.709621\pi$$
$$840$$ 0 0
$$841$$ 49595.0 2.03350
$$842$$ 0 0
$$843$$ 3998.00 0.163343
$$844$$ 0 0
$$845$$ 0 0
$$846$$ 0 0
$$847$$ 5820.00 0.236101
$$848$$ 0 0
$$849$$ −13.0000 −0.000525511 0
$$850$$ 0 0
$$851$$ −31668.0 −1.27563
$$852$$ 0 0
$$853$$ −37726.0 −1.51432 −0.757159 0.653230i $$-0.773413\pi$$
−0.757159 + 0.653230i $$0.773413\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ 5429.00 0.216396 0.108198 0.994129i $$-0.465492\pi$$
0.108198 + 0.994129i $$0.465492\pi$$
$$858$$ 0 0
$$859$$ 32149.0 1.27696 0.638481 0.769638i $$-0.279563\pi$$
0.638481 + 0.769638i $$0.279563\pi$$
$$860$$ 0 0
$$861$$ −702.000 −0.0277864
$$862$$ 0 0
$$863$$ −29176.0 −1.15083 −0.575413 0.817863i $$-0.695159\pi$$
−0.575413 + 0.817863i $$0.695159\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 0 0
$$867$$ 712.000 0.0278902
$$868$$ 0 0
$$869$$ −20026.0 −0.781744
$$870$$ 0 0
$$871$$ 9156.00 0.356187
$$872$$ 0 0
$$873$$ −25012.0 −0.969677
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 20068.0 0.772689 0.386344 0.922355i $$-0.373738\pi$$
0.386344 + 0.922355i $$0.373738\pi$$
$$878$$ 0 0
$$879$$ −4466.00 −0.171370
$$880$$ 0 0
$$881$$ −36850.0 −1.40920 −0.704602 0.709603i $$-0.748874\pi$$
−0.704602 + 0.709603i $$0.748874\pi$$
$$882$$ 0 0
$$883$$ 30025.0 1.14431 0.572153 0.820147i $$-0.306108\pi$$
0.572153 + 0.820147i $$0.306108\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ 156.000 0.00590526 0.00295263 0.999996i $$-0.499060\pi$$
0.00295263 + 0.999996i $$0.499060\pi$$
$$888$$ 0 0
$$889$$ 5412.00 0.204176
$$890$$ 0 0
$$891$$ −12331.0 −0.463641
$$892$$ 0 0
$$893$$ 4732.00 0.177324
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 0 0
$$897$$ 2088.00 0.0777216
$$898$$ 0 0
$$899$$ 62560.0 2.32090
$$900$$ 0 0
$$901$$ 30150.0 1.11481
$$902$$ 0 0
$$903$$ −2232.00 −0.0822550
$$904$$ 0 0
$$905$$ 0 0
$$906$$ 0 0
$$907$$ −356.000 −0.0130328 −0.00651642 0.999979i $$-0.502074\pi$$
−0.00651642 + 0.999979i $$0.502074\pi$$
$$908$$ 0 0
$$909$$ −12636.0 −0.461067
$$910$$ 0 0
$$911$$ 8748.00 0.318149 0.159075 0.987267i $$-0.449149\pi$$
0.159075 + 0.987267i $$0.449149\pi$$
$$912$$ 0 0
$$913$$ −6669.00 −0.241743
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ −12408.0 −0.446836
$$918$$ 0 0
$$919$$ 36974.0 1.32716 0.663580 0.748105i $$-0.269036\pi$$
0.663580 + 0.748105i $$0.269036\pi$$
$$920$$ 0 0
$$921$$ 9041.00 0.323465
$$922$$ 0 0
$$923$$ −624.000 −0.0222527
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 0 0
$$927$$ 30888.0 1.09439
$$928$$ 0 0
$$929$$ 44382.0 1.56741 0.783706 0.621132i $$-0.213327\pi$$
0.783706 + 0.621132i $$0.213327\pi$$
$$930$$ 0 0
$$931$$ 27937.0 0.983457
$$932$$ 0 0
$$933$$ −346.000 −0.0121410
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ −2445.00 −0.0852451 −0.0426226 0.999091i $$-0.513571\pi$$
−0.0426226 + 0.999091i $$0.513571\pi$$
$$938$$ 0 0
$$939$$ −10646.0 −0.369988
$$940$$ 0 0
$$941$$ −7076.00 −0.245134 −0.122567 0.992460i $$-0.539113\pi$$
−0.122567 + 0.992460i $$0.539113\pi$$
$$942$$ 0 0
$$943$$ 20358.0 0.703020
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 1560.00 0.0535303 0.0267651 0.999642i $$-0.491479\pi$$
0.0267651 + 0.999642i $$0.491479\pi$$
$$948$$ 0 0
$$949$$ −11772.0 −0.402672
$$950$$ 0 0
$$951$$ −8116.00 −0.276740
$$952$$ 0 0
$$953$$ −35087.0 −1.19263 −0.596317 0.802749i $$-0.703370\pi$$
−0.596317 + 0.802749i $$0.703370\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 0 0
$$957$$ 5168.00 0.174564
$$958$$ 0 0
$$959$$ 8034.00 0.270523
$$960$$ 0 0
$$961$$ 23109.0 0.775704
$$962$$ 0 0
$$963$$ −34450.0 −1.15279
$$964$$ 0 0
$$965$$ 0 0
$$966$$ 0 0
$$967$$ −9360.00 −0.311269 −0.155635 0.987815i $$-0.549742\pi$$
−0.155635 + 0.987815i $$0.549742\pi$$
$$968$$ 0 0
$$969$$ 6825.00 0.226265
$$970$$ 0 0
$$971$$ 22269.0 0.735990 0.367995 0.929828i $$-0.380044\pi$$
0.367995 + 0.929828i $$0.380044\pi$$
$$972$$ 0 0
$$973$$ 17634.0 0.581007
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ 37249.0 1.21976 0.609878 0.792496i $$-0.291219\pi$$
0.609878 + 0.792496i $$0.291219\pi$$
$$978$$ 0 0
$$979$$ −15181.0 −0.495594
$$980$$ 0 0
$$981$$ −3276.00 −0.106620
$$982$$ 0 0
$$983$$ −17602.0 −0.571126 −0.285563 0.958360i $$-0.592181\pi$$
−0.285563 + 0.958360i $$0.592181\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 0 0
$$987$$ 312.000 0.0100619
$$988$$ 0 0
$$989$$ 64728.0 2.08112
$$990$$ 0 0
$$991$$ −26402.0 −0.846304 −0.423152 0.906059i $$-0.639076\pi$$
−0.423152 + 0.906059i $$0.639076\pi$$
$$992$$ 0 0
$$993$$ −3007.00 −0.0960969
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 0 0
$$997$$ 16836.0 0.534806 0.267403 0.963585i $$-0.413835\pi$$
0.267403 + 0.963585i $$0.413835\pi$$
$$998$$ 0 0
$$999$$ 9646.00 0.305491
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 200.4.a.f.1.1 yes 1
3.2 odd 2 1800.4.a.l.1.1 1
4.3 odd 2 400.4.a.j.1.1 1
5.2 odd 4 200.4.c.g.49.1 2
5.3 odd 4 200.4.c.g.49.2 2
5.4 even 2 200.4.a.e.1.1 1
8.3 odd 2 1600.4.a.be.1.1 1
8.5 even 2 1600.4.a.w.1.1 1
15.2 even 4 1800.4.f.p.649.1 2
15.8 even 4 1800.4.f.p.649.2 2
15.14 odd 2 1800.4.a.w.1.1 1
20.3 even 4 400.4.c.m.49.1 2
20.7 even 4 400.4.c.m.49.2 2
20.19 odd 2 400.4.a.k.1.1 1
40.19 odd 2 1600.4.a.v.1.1 1
40.29 even 2 1600.4.a.bf.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
200.4.a.e.1.1 1 5.4 even 2
200.4.a.f.1.1 yes 1 1.1 even 1 trivial
200.4.c.g.49.1 2 5.2 odd 4
200.4.c.g.49.2 2 5.3 odd 4
400.4.a.j.1.1 1 4.3 odd 2
400.4.a.k.1.1 1 20.19 odd 2
400.4.c.m.49.1 2 20.3 even 4
400.4.c.m.49.2 2 20.7 even 4
1600.4.a.v.1.1 1 40.19 odd 2
1600.4.a.w.1.1 1 8.5 even 2
1600.4.a.be.1.1 1 8.3 odd 2
1600.4.a.bf.1.1 1 40.29 even 2
1800.4.a.l.1.1 1 3.2 odd 2
1800.4.a.w.1.1 1 15.14 odd 2
1800.4.f.p.649.1 2 15.2 even 4
1800.4.f.p.649.2 2 15.8 even 4