# Properties

 Label 400.4.a.t.1.1 Level $400$ Weight $4$ Character 400.1 Self dual yes Analytic conductor $23.601$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [400,4,Mod(1,400)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(400, 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("400.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$400 = 2^{4} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 400.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: $$23.6007640023$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 200) Fricke sign: $$+1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 400.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+9.00000 q^{3} +26.0000 q^{7} +54.0000 q^{9} +O(q^{10})$$ $$q+9.00000 q^{3} +26.0000 q^{7} +54.0000 q^{9} +59.0000 q^{11} -28.0000 q^{13} -5.00000 q^{17} -109.000 q^{19} +234.000 q^{21} -194.000 q^{23} +243.000 q^{27} -32.0000 q^{29} -10.0000 q^{31} +531.000 q^{33} +198.000 q^{37} -252.000 q^{39} +117.000 q^{41} +388.000 q^{43} -68.0000 q^{47} +333.000 q^{49} -45.0000 q^{51} +18.0000 q^{53} -981.000 q^{57} -392.000 q^{59} -710.000 q^{61} +1404.00 q^{63} -253.000 q^{67} -1746.00 q^{69} +612.000 q^{71} +549.000 q^{73} +1534.00 q^{77} -414.000 q^{79} +729.000 q^{81} -121.000 q^{83} -288.000 q^{87} -81.0000 q^{89} -728.000 q^{91} -90.0000 q^{93} +1502.00 q^{97} +3186.00 q^{99} +O(q^{100})$$

## 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
$$3$$ 9.00000 1.73205 0.866025 0.500000i $$-0.166667\pi$$
0.866025 + 0.500000i $$0.166667\pi$$
$$4$$ 0 0
$$5$$ 0 0
$$6$$ 0 0
$$7$$ 26.0000 1.40387 0.701934 0.712242i $$-0.252320\pi$$
0.701934 + 0.712242i $$0.252320\pi$$
$$8$$ 0 0
$$9$$ 54.0000 2.00000
$$10$$ 0 0
$$11$$ 59.0000 1.61720 0.808599 0.588361i $$-0.200226\pi$$
0.808599 + 0.588361i $$0.200226\pi$$
$$12$$ 0 0
$$13$$ −28.0000 −0.597369 −0.298685 0.954352i $$-0.596548\pi$$
−0.298685 + 0.954352i $$0.596548\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ −5.00000 −0.0713340 −0.0356670 0.999364i $$-0.511356\pi$$
−0.0356670 + 0.999364i $$0.511356\pi$$
$$18$$ 0 0
$$19$$ −109.000 −1.31612 −0.658061 0.752965i $$-0.728623\pi$$
−0.658061 + 0.752965i $$0.728623\pi$$
$$20$$ 0 0
$$21$$ 234.000 2.43157
$$22$$ 0 0
$$23$$ −194.000 −1.75877 −0.879387 0.476108i $$-0.842047\pi$$
−0.879387 + 0.476108i $$0.842047\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ 0 0
$$27$$ 243.000 1.73205
$$28$$ 0 0
$$29$$ −32.0000 −0.204905 −0.102453 0.994738i $$-0.532669\pi$$
−0.102453 + 0.994738i $$0.532669\pi$$
$$30$$ 0 0
$$31$$ −10.0000 −0.0579372 −0.0289686 0.999580i $$-0.509222\pi$$
−0.0289686 + 0.999580i $$0.509222\pi$$
$$32$$ 0 0
$$33$$ 531.000 2.80107
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 198.000 0.879757 0.439878 0.898057i $$-0.355022\pi$$
0.439878 + 0.898057i $$0.355022\pi$$
$$38$$ 0 0
$$39$$ −252.000 −1.03467
$$40$$ 0 0
$$41$$ 117.000 0.445667 0.222833 0.974857i $$-0.428469\pi$$
0.222833 + 0.974857i $$0.428469\pi$$
$$42$$ 0 0
$$43$$ 388.000 1.37603 0.688017 0.725695i $$-0.258482\pi$$
0.688017 + 0.725695i $$0.258482\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ −68.0000 −0.211039 −0.105519 0.994417i $$-0.533650\pi$$
−0.105519 + 0.994417i $$0.533650\pi$$
$$48$$ 0 0
$$49$$ 333.000 0.970845
$$50$$ 0 0
$$51$$ −45.0000 −0.123554
$$52$$ 0 0
$$53$$ 18.0000 0.0466508 0.0233254 0.999728i $$-0.492575\pi$$
0.0233254 + 0.999728i $$0.492575\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ −981.000 −2.27959
$$58$$ 0 0
$$59$$ −392.000 −0.864984 −0.432492 0.901638i $$-0.642366\pi$$
−0.432492 + 0.901638i $$0.642366\pi$$
$$60$$ 0 0
$$61$$ −710.000 −1.49027 −0.745133 0.666916i $$-0.767614\pi$$
−0.745133 + 0.666916i $$0.767614\pi$$
$$62$$ 0 0
$$63$$ 1404.00 2.80774
$$64$$ 0 0
$$65$$ 0 0
$$66$$ 0 0
$$67$$ −253.000 −0.461326 −0.230663 0.973034i $$-0.574090\pi$$
−0.230663 + 0.973034i $$0.574090\pi$$
$$68$$ 0 0
$$69$$ −1746.00 −3.04629
$$70$$ 0 0
$$71$$ 612.000 1.02297 0.511486 0.859292i $$-0.329095\pi$$
0.511486 + 0.859292i $$0.329095\pi$$
$$72$$ 0 0
$$73$$ 549.000 0.880214 0.440107 0.897945i $$-0.354941\pi$$
0.440107 + 0.897945i $$0.354941\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ 1534.00 2.27033
$$78$$ 0 0
$$79$$ −414.000 −0.589603 −0.294802 0.955559i $$-0.595254\pi$$
−0.294802 + 0.955559i $$0.595254\pi$$
$$80$$ 0 0
$$81$$ 729.000 1.00000
$$82$$ 0 0
$$83$$ −121.000 −0.160018 −0.0800089 0.996794i $$-0.525495\pi$$
−0.0800089 + 0.996794i $$0.525495\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 0 0
$$87$$ −288.000 −0.354906
$$88$$ 0 0
$$89$$ −81.0000 −0.0964717 −0.0482359 0.998836i $$-0.515360\pi$$
−0.0482359 + 0.998836i $$0.515360\pi$$
$$90$$ 0 0
$$91$$ −728.000 −0.838628
$$92$$ 0 0
$$93$$ −90.0000 −0.100350
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 0 0
$$97$$ 1502.00 1.57222 0.786108 0.618089i $$-0.212093\pi$$
0.786108 + 0.618089i $$0.212093\pi$$
$$98$$ 0 0
$$99$$ 3186.00 3.23439
$$100$$ 0 0
$$101$$ −234.000 −0.230533 −0.115267 0.993335i $$-0.536772\pi$$
−0.115267 + 0.993335i $$0.536772\pi$$
$$102$$ 0 0
$$103$$ −1172.00 −1.12117 −0.560585 0.828097i $$-0.689424\pi$$
−0.560585 + 0.828097i $$0.689424\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 1125.00 1.01643 0.508214 0.861231i $$-0.330306\pi$$
0.508214 + 0.861231i $$0.330306\pi$$
$$108$$ 0 0
$$109$$ −1234.00 −1.08436 −0.542182 0.840261i $$-0.682402\pi$$
−0.542182 + 0.840261i $$0.682402\pi$$
$$110$$ 0 0
$$111$$ 1782.00 1.52378
$$112$$ 0 0
$$113$$ −567.000 −0.472025 −0.236013 0.971750i $$-0.575841\pi$$
−0.236013 + 0.971750i $$0.575841\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 0 0
$$117$$ −1512.00 −1.19474
$$118$$ 0 0
$$119$$ −130.000 −0.100144
$$120$$ 0 0
$$121$$ 2150.00 1.61533
$$122$$ 0 0
$$123$$ 1053.00 0.771917
$$124$$ 0 0
$$125$$ 0 0
$$126$$ 0 0
$$127$$ −2358.00 −1.64755 −0.823774 0.566918i $$-0.808136\pi$$
−0.823774 + 0.566918i $$0.808136\pi$$
$$128$$ 0 0
$$129$$ 3492.00 2.38336
$$130$$ 0 0
$$131$$ 1692.00 1.12848 0.564239 0.825611i $$-0.309169\pi$$
0.564239 + 0.825611i $$0.309169\pi$$
$$132$$ 0 0
$$133$$ −2834.00 −1.84766
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ −229.000 −0.142809 −0.0714043 0.997447i $$-0.522748\pi$$
−0.0714043 + 0.997447i $$0.522748\pi$$
$$138$$ 0 0
$$139$$ −2781.00 −1.69699 −0.848494 0.529205i $$-0.822490\pi$$
−0.848494 + 0.529205i $$0.822490\pi$$
$$140$$ 0 0
$$141$$ −612.000 −0.365530
$$142$$ 0 0
$$143$$ −1652.00 −0.966064
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 0 0
$$147$$ 2997.00 1.68155
$$148$$ 0 0
$$149$$ 1472.00 0.809335 0.404668 0.914464i $$-0.367387\pi$$
0.404668 + 0.914464i $$0.367387\pi$$
$$150$$ 0 0
$$151$$ −1322.00 −0.712469 −0.356235 0.934397i $$-0.615940\pi$$
−0.356235 + 0.934397i $$0.615940\pi$$
$$152$$ 0 0
$$153$$ −270.000 −0.142668
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ −298.000 −0.151484 −0.0757420 0.997127i $$-0.524133\pi$$
−0.0757420 + 0.997127i $$0.524133\pi$$
$$158$$ 0 0
$$159$$ 162.000 0.0808015
$$160$$ 0 0
$$161$$ −5044.00 −2.46909
$$162$$ 0 0
$$163$$ −341.000 −0.163860 −0.0819300 0.996638i $$-0.526108\pi$$
−0.0819300 + 0.996638i $$0.526108\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 684.000 0.316943 0.158472 0.987364i $$-0.449343\pi$$
0.158472 + 0.987364i $$0.449343\pi$$
$$168$$ 0 0
$$169$$ −1413.00 −0.643150
$$170$$ 0 0
$$171$$ −5886.00 −2.63224
$$172$$ 0 0
$$173$$ 2344.00 1.03012 0.515061 0.857154i $$-0.327769\pi$$
0.515061 + 0.857154i $$0.327769\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ −3528.00 −1.49820
$$178$$ 0 0
$$179$$ 1111.00 0.463911 0.231955 0.972726i $$-0.425488\pi$$
0.231955 + 0.972726i $$0.425488\pi$$
$$180$$ 0 0
$$181$$ 2042.00 0.838567 0.419284 0.907855i $$-0.362281\pi$$
0.419284 + 0.907855i $$0.362281\pi$$
$$182$$ 0 0
$$183$$ −6390.00 −2.58122
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 0 0
$$187$$ −295.000 −0.115361
$$188$$ 0 0
$$189$$ 6318.00 2.43157
$$190$$ 0 0
$$191$$ −5270.00 −1.99646 −0.998230 0.0594735i $$-0.981058\pi$$
−0.998230 + 0.0594735i $$0.981058\pi$$
$$192$$ 0 0
$$193$$ −613.000 −0.228625 −0.114313 0.993445i $$-0.536467\pi$$
−0.114313 + 0.993445i $$0.536467\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ −1174.00 −0.424589 −0.212295 0.977206i $$-0.568094\pi$$
−0.212295 + 0.977206i $$0.568094\pi$$
$$198$$ 0 0
$$199$$ −3428.00 −1.22113 −0.610564 0.791967i $$-0.709057\pi$$
−0.610564 + 0.791967i $$0.709057\pi$$
$$200$$ 0 0
$$201$$ −2277.00 −0.799041
$$202$$ 0 0
$$203$$ −832.000 −0.287660
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 0 0
$$207$$ −10476.0 −3.51755
$$208$$ 0 0
$$209$$ −6431.00 −2.12843
$$210$$ 0 0
$$211$$ −2339.00 −0.763144 −0.381572 0.924339i $$-0.624617\pi$$
−0.381572 + 0.924339i $$0.624617\pi$$
$$212$$ 0 0
$$213$$ 5508.00 1.77184
$$214$$ 0 0
$$215$$ 0 0
$$216$$ 0 0
$$217$$ −260.000 −0.0813362
$$218$$ 0 0
$$219$$ 4941.00 1.52457
$$220$$ 0 0
$$221$$ 140.000 0.0426128
$$222$$ 0 0
$$223$$ 3932.00 1.18075 0.590373 0.807131i $$-0.298981\pi$$
0.590373 + 0.807131i $$0.298981\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ 6084.00 1.77890 0.889448 0.457037i $$-0.151089\pi$$
0.889448 + 0.457037i $$0.151089\pi$$
$$228$$ 0 0
$$229$$ 4996.00 1.44168 0.720841 0.693101i $$-0.243756\pi$$
0.720841 + 0.693101i $$0.243756\pi$$
$$230$$ 0 0
$$231$$ 13806.0 3.93233
$$232$$ 0 0
$$233$$ −3222.00 −0.905924 −0.452962 0.891530i $$-0.649633\pi$$
−0.452962 + 0.891530i $$0.649633\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 0 0
$$237$$ −3726.00 −1.02122
$$238$$ 0 0
$$239$$ −2736.00 −0.740490 −0.370245 0.928934i $$-0.620726\pi$$
−0.370245 + 0.928934i $$0.620726\pi$$
$$240$$ 0 0
$$241$$ 1673.00 0.447168 0.223584 0.974685i $$-0.428224\pi$$
0.223584 + 0.974685i $$0.428224\pi$$
$$242$$ 0 0
$$243$$ 0 0
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 3052.00 0.786211
$$248$$ 0 0
$$249$$ −1089.00 −0.277159
$$250$$ 0 0
$$251$$ −5355.00 −1.34663 −0.673316 0.739355i $$-0.735131\pi$$
−0.673316 + 0.739355i $$0.735131\pi$$
$$252$$ 0 0
$$253$$ −11446.0 −2.84428
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ 5490.00 1.33252 0.666258 0.745721i $$-0.267895\pi$$
0.666258 + 0.745721i $$0.267895\pi$$
$$258$$ 0 0
$$259$$ 5148.00 1.23506
$$260$$ 0 0
$$261$$ −1728.00 −0.409810
$$262$$ 0 0
$$263$$ 3150.00 0.738545 0.369272 0.929321i $$-0.379607\pi$$
0.369272 + 0.929321i $$0.379607\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 0 0
$$267$$ −729.000 −0.167094
$$268$$ 0 0
$$269$$ 176.000 0.0398919 0.0199459 0.999801i $$-0.493651\pi$$
0.0199459 + 0.999801i $$0.493651\pi$$
$$270$$ 0 0
$$271$$ −2394.00 −0.536624 −0.268312 0.963332i $$-0.586466\pi$$
−0.268312 + 0.963332i $$0.586466\pi$$
$$272$$ 0 0
$$273$$ −6552.00 −1.45255
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 0 0
$$277$$ 6256.00 1.35699 0.678496 0.734604i $$-0.262632\pi$$
0.678496 + 0.734604i $$0.262632\pi$$
$$278$$ 0 0
$$279$$ −540.000 −0.115874
$$280$$ 0 0
$$281$$ −4802.00 −1.01944 −0.509721 0.860340i $$-0.670251\pi$$
−0.509721 + 0.860340i $$0.670251\pi$$
$$282$$ 0 0
$$283$$ 2123.00 0.445934 0.222967 0.974826i $$-0.428426\pi$$
0.222967 + 0.974826i $$0.428426\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 3042.00 0.625657
$$288$$ 0 0
$$289$$ −4888.00 −0.994911
$$290$$ 0 0
$$291$$ 13518.0 2.72316
$$292$$ 0 0
$$293$$ 8834.00 1.76139 0.880696 0.473682i $$-0.157075\pi$$
0.880696 + 0.473682i $$0.157075\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 0 0
$$297$$ 14337.0 2.80107
$$298$$ 0 0
$$299$$ 5432.00 1.05064
$$300$$ 0 0
$$301$$ 10088.0 1.93177
$$302$$ 0 0
$$303$$ −2106.00 −0.399296
$$304$$ 0 0
$$305$$ 0 0
$$306$$ 0 0
$$307$$ 1369.00 0.254505 0.127252 0.991870i $$-0.459384\pi$$
0.127252 + 0.991870i $$0.459384\pi$$
$$308$$ 0 0
$$309$$ −10548.0 −1.94192
$$310$$ 0 0
$$311$$ 10426.0 1.90098 0.950489 0.310758i $$-0.100583\pi$$
0.950489 + 0.310758i $$0.100583\pi$$
$$312$$ 0 0
$$313$$ 3574.00 0.645413 0.322707 0.946499i $$-0.395407\pi$$
0.322707 + 0.946499i $$0.395407\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ −9036.00 −1.60099 −0.800493 0.599343i $$-0.795429\pi$$
−0.800493 + 0.599343i $$0.795429\pi$$
$$318$$ 0 0
$$319$$ −1888.00 −0.331372
$$320$$ 0 0
$$321$$ 10125.0 1.76051
$$322$$ 0 0
$$323$$ 545.000 0.0938842
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 0 0
$$327$$ −11106.0 −1.87817
$$328$$ 0 0
$$329$$ −1768.00 −0.296271
$$330$$ 0 0
$$331$$ −10233.0 −1.69926 −0.849632 0.527376i $$-0.823176\pi$$
−0.849632 + 0.527376i $$0.823176\pi$$
$$332$$ 0 0
$$333$$ 10692.0 1.75951
$$334$$ 0 0
$$335$$ 0 0
$$336$$ 0 0
$$337$$ −4627.00 −0.747919 −0.373960 0.927445i $$-0.622000\pi$$
−0.373960 + 0.927445i $$0.622000\pi$$
$$338$$ 0 0
$$339$$ −5103.00 −0.817572
$$340$$ 0 0
$$341$$ −590.000 −0.0936959
$$342$$ 0 0
$$343$$ −260.000 −0.0409291
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ 4901.00 0.758212 0.379106 0.925353i $$-0.376232\pi$$
0.379106 + 0.925353i $$0.376232\pi$$
$$348$$ 0 0
$$349$$ −4482.00 −0.687438 −0.343719 0.939072i $$-0.611687\pi$$
−0.343719 + 0.939072i $$0.611687\pi$$
$$350$$ 0 0
$$351$$ −6804.00 −1.03467
$$352$$ 0 0
$$353$$ 1210.00 0.182441 0.0912207 0.995831i $$-0.470923\pi$$
0.0912207 + 0.995831i $$0.470923\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0 0
$$357$$ −1170.00 −0.173454
$$358$$ 0 0
$$359$$ 9882.00 1.45279 0.726396 0.687277i $$-0.241194\pi$$
0.726396 + 0.687277i $$0.241194\pi$$
$$360$$ 0 0
$$361$$ 5022.00 0.732177
$$362$$ 0 0
$$363$$ 19350.0 2.79783
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 0 0
$$367$$ −11260.0 −1.60155 −0.800773 0.598968i $$-0.795578\pi$$
−0.800773 + 0.598968i $$0.795578\pi$$
$$368$$ 0 0
$$369$$ 6318.00 0.891333
$$370$$ 0 0
$$371$$ 468.000 0.0654915
$$372$$ 0 0
$$373$$ 3230.00 0.448373 0.224186 0.974546i $$-0.428028\pi$$
0.224186 + 0.974546i $$0.428028\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 896.000 0.122404
$$378$$ 0 0
$$379$$ −11575.0 −1.56878 −0.784390 0.620268i $$-0.787024\pi$$
−0.784390 + 0.620268i $$0.787024\pi$$
$$380$$ 0 0
$$381$$ −21222.0 −2.85364
$$382$$ 0 0
$$383$$ −18.0000 −0.00240145 −0.00120073 0.999999i $$-0.500382\pi$$
−0.00120073 + 0.999999i $$0.500382\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ 20952.0 2.75207
$$388$$ 0 0
$$389$$ 10710.0 1.39593 0.697967 0.716130i $$-0.254088\pi$$
0.697967 + 0.716130i $$0.254088\pi$$
$$390$$ 0 0
$$391$$ 970.000 0.125460
$$392$$ 0 0
$$393$$ 15228.0 1.95458
$$394$$ 0 0
$$395$$ 0 0
$$396$$ 0 0
$$397$$ 3788.00 0.478877 0.239439 0.970912i $$-0.423037\pi$$
0.239439 + 0.970912i $$0.423037\pi$$
$$398$$ 0 0
$$399$$ −25506.0 −3.20024
$$400$$ 0 0
$$401$$ −10539.0 −1.31245 −0.656225 0.754565i $$-0.727848\pi$$
−0.656225 + 0.754565i $$0.727848\pi$$
$$402$$ 0 0
$$403$$ 280.000 0.0346099
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 11682.0 1.42274
$$408$$ 0 0
$$409$$ 5581.00 0.674725 0.337363 0.941375i $$-0.390465\pi$$
0.337363 + 0.941375i $$0.390465\pi$$
$$410$$ 0 0
$$411$$ −2061.00 −0.247352
$$412$$ 0 0
$$413$$ −10192.0 −1.21432
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 0 0
$$417$$ −25029.0 −2.93927
$$418$$ 0 0
$$419$$ −5193.00 −0.605476 −0.302738 0.953074i $$-0.597901\pi$$
−0.302738 + 0.953074i $$0.597901\pi$$
$$420$$ 0 0
$$421$$ 4788.00 0.554282 0.277141 0.960829i $$-0.410613\pi$$
0.277141 + 0.960829i $$0.410613\pi$$
$$422$$ 0 0
$$423$$ −3672.00 −0.422077
$$424$$ 0 0
$$425$$ 0 0
$$426$$ 0 0
$$427$$ −18460.0 −2.09214
$$428$$ 0 0
$$429$$ −14868.0 −1.67327
$$430$$ 0 0
$$431$$ −8006.00 −0.894746 −0.447373 0.894348i $$-0.647640\pi$$
−0.447373 + 0.894348i $$0.647640\pi$$
$$432$$ 0 0
$$433$$ 2395.00 0.265811 0.132906 0.991129i $$-0.457569\pi$$
0.132906 + 0.991129i $$0.457569\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ 21146.0 2.31476
$$438$$ 0 0
$$439$$ −1864.00 −0.202651 −0.101326 0.994853i $$-0.532308\pi$$
−0.101326 + 0.994853i $$0.532308\pi$$
$$440$$ 0 0
$$441$$ 17982.0 1.94169
$$442$$ 0 0
$$443$$ 5463.00 0.585903 0.292951 0.956127i $$-0.405363\pi$$
0.292951 + 0.956127i $$0.405363\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 0 0
$$447$$ 13248.0 1.40181
$$448$$ 0 0
$$449$$ 12969.0 1.36313 0.681565 0.731758i $$-0.261300\pi$$
0.681565 + 0.731758i $$0.261300\pi$$
$$450$$ 0 0
$$451$$ 6903.00 0.720731
$$452$$ 0 0
$$453$$ −11898.0 −1.23403
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ −18313.0 −1.87450 −0.937249 0.348659i $$-0.886637\pi$$
−0.937249 + 0.348659i $$0.886637\pi$$
$$458$$ 0 0
$$459$$ −1215.00 −0.123554
$$460$$ 0 0
$$461$$ 12492.0 1.26206 0.631031 0.775758i $$-0.282632\pi$$
0.631031 + 0.775758i $$0.282632\pi$$
$$462$$ 0 0
$$463$$ 4428.00 0.444464 0.222232 0.974994i $$-0.428666\pi$$
0.222232 + 0.974994i $$0.428666\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ −1084.00 −0.107412 −0.0537061 0.998557i $$-0.517103\pi$$
−0.0537061 + 0.998557i $$0.517103\pi$$
$$468$$ 0 0
$$469$$ −6578.00 −0.647641
$$470$$ 0 0
$$471$$ −2682.00 −0.262378
$$472$$ 0 0
$$473$$ 22892.0 2.22532
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 0 0
$$477$$ 972.000 0.0933015
$$478$$ 0 0
$$479$$ 13082.0 1.24787 0.623937 0.781474i $$-0.285532\pi$$
0.623937 + 0.781474i $$0.285532\pi$$
$$480$$ 0 0
$$481$$ −5544.00 −0.525540
$$482$$ 0 0
$$483$$ −45396.0 −4.27658
$$484$$ 0 0
$$485$$ 0 0
$$486$$ 0 0
$$487$$ −3014.00 −0.280446 −0.140223 0.990120i $$-0.544782\pi$$
−0.140223 + 0.990120i $$0.544782\pi$$
$$488$$ 0 0
$$489$$ −3069.00 −0.283814
$$490$$ 0 0
$$491$$ 3564.00 0.327579 0.163789 0.986495i $$-0.447628\pi$$
0.163789 + 0.986495i $$0.447628\pi$$
$$492$$ 0 0
$$493$$ 160.000 0.0146167
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 15912.0 1.43612
$$498$$ 0 0
$$499$$ 15796.0 1.41709 0.708543 0.705667i $$-0.249353\pi$$
0.708543 + 0.705667i $$0.249353\pi$$
$$500$$ 0 0
$$501$$ 6156.00 0.548962
$$502$$ 0 0
$$503$$ −10908.0 −0.966926 −0.483463 0.875365i $$-0.660621\pi$$
−0.483463 + 0.875365i $$0.660621\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 0 0
$$507$$ −12717.0 −1.11397
$$508$$ 0 0
$$509$$ 21946.0 1.91108 0.955540 0.294863i $$-0.0952739\pi$$
0.955540 + 0.294863i $$0.0952739\pi$$
$$510$$ 0 0
$$511$$ 14274.0 1.23570
$$512$$ 0 0
$$513$$ −26487.0 −2.27959
$$514$$ 0 0
$$515$$ 0 0
$$516$$ 0 0
$$517$$ −4012.00 −0.341291
$$518$$ 0 0
$$519$$ 21096.0 1.78422
$$520$$ 0 0
$$521$$ −6395.00 −0.537754 −0.268877 0.963174i $$-0.586653\pi$$
−0.268877 + 0.963174i $$0.586653\pi$$
$$522$$ 0 0
$$523$$ −5615.00 −0.469459 −0.234729 0.972061i $$-0.575420\pi$$
−0.234729 + 0.972061i $$0.575420\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 50.0000 0.00413289
$$528$$ 0 0
$$529$$ 25469.0 2.09329
$$530$$ 0 0
$$531$$ −21168.0 −1.72997
$$532$$ 0 0
$$533$$ −3276.00 −0.266228
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 0 0
$$537$$ 9999.00 0.803517
$$538$$ 0 0
$$539$$ 19647.0 1.57005
$$540$$ 0 0
$$541$$ −4112.00 −0.326781 −0.163391 0.986561i $$-0.552243\pi$$
−0.163391 + 0.986561i $$0.552243\pi$$
$$542$$ 0 0
$$543$$ 18378.0 1.45244
$$544$$ 0 0
$$545$$ 0 0
$$546$$ 0 0
$$547$$ −2167.00 −0.169386 −0.0846931 0.996407i $$-0.526991\pi$$
−0.0846931 + 0.996407i $$0.526991\pi$$
$$548$$ 0 0
$$549$$ −38340.0 −2.98053
$$550$$ 0 0
$$551$$ 3488.00 0.269680
$$552$$ 0 0
$$553$$ −10764.0 −0.827725
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ 19444.0 1.47912 0.739559 0.673092i $$-0.235034\pi$$
0.739559 + 0.673092i $$0.235034\pi$$
$$558$$ 0 0
$$559$$ −10864.0 −0.822000
$$560$$ 0 0
$$561$$ −2655.00 −0.199811
$$562$$ 0 0
$$563$$ 20416.0 1.52830 0.764149 0.645040i $$-0.223159\pi$$
0.764149 + 0.645040i $$0.223159\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 0 0
$$567$$ 18954.0 1.40387
$$568$$ 0 0
$$569$$ −3127.00 −0.230388 −0.115194 0.993343i $$-0.536749\pi$$
−0.115194 + 0.993343i $$0.536749\pi$$
$$570$$ 0 0
$$571$$ 22580.0 1.65489 0.827446 0.561545i $$-0.189793\pi$$
0.827446 + 0.561545i $$0.189793\pi$$
$$572$$ 0 0
$$573$$ −47430.0 −3.45797
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 0 0
$$577$$ 829.000 0.0598123 0.0299062 0.999553i $$-0.490479\pi$$
0.0299062 + 0.999553i $$0.490479\pi$$
$$578$$ 0 0
$$579$$ −5517.00 −0.395991
$$580$$ 0 0
$$581$$ −3146.00 −0.224644
$$582$$ 0 0
$$583$$ 1062.00 0.0754435
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ 7119.00 0.500567 0.250283 0.968173i $$-0.419476\pi$$
0.250283 + 0.968173i $$0.419476\pi$$
$$588$$ 0 0
$$589$$ 1090.00 0.0762524
$$590$$ 0 0
$$591$$ −10566.0 −0.735410
$$592$$ 0 0
$$593$$ 8217.00 0.569025 0.284512 0.958672i $$-0.408168\pi$$
0.284512 + 0.958672i $$0.408168\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ −30852.0 −2.11506
$$598$$ 0 0
$$599$$ 90.0000 0.00613907 0.00306953 0.999995i $$-0.499023\pi$$
0.00306953 + 0.999995i $$0.499023\pi$$
$$600$$ 0 0
$$601$$ −17117.0 −1.16176 −0.580879 0.813990i $$-0.697291\pi$$
−0.580879 + 0.813990i $$0.697291\pi$$
$$602$$ 0 0
$$603$$ −13662.0 −0.922653
$$604$$ 0 0
$$605$$ 0 0
$$606$$ 0 0
$$607$$ −15120.0 −1.01104 −0.505520 0.862815i $$-0.668699\pi$$
−0.505520 + 0.862815i $$0.668699\pi$$
$$608$$ 0 0
$$609$$ −7488.00 −0.498241
$$610$$ 0 0
$$611$$ 1904.00 0.126068
$$612$$ 0 0
$$613$$ 6570.00 0.432887 0.216444 0.976295i $$-0.430554\pi$$
0.216444 + 0.976295i $$0.430554\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ −18846.0 −1.22968 −0.614839 0.788653i $$-0.710779\pi$$
−0.614839 + 0.788653i $$0.710779\pi$$
$$618$$ 0 0
$$619$$ −16316.0 −1.05944 −0.529722 0.848172i $$-0.677704\pi$$
−0.529722 + 0.848172i $$0.677704\pi$$
$$620$$ 0 0
$$621$$ −47142.0 −3.04629
$$622$$ 0 0
$$623$$ −2106.00 −0.135434
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 0 0
$$627$$ −57879.0 −3.68655
$$628$$ 0 0
$$629$$ −990.000 −0.0627566
$$630$$ 0 0
$$631$$ −20170.0 −1.27251 −0.636256 0.771478i $$-0.719518\pi$$
−0.636256 + 0.771478i $$0.719518\pi$$
$$632$$ 0 0
$$633$$ −21051.0 −1.32180
$$634$$ 0 0
$$635$$ 0 0
$$636$$ 0 0
$$637$$ −9324.00 −0.579953
$$638$$ 0 0
$$639$$ 33048.0 2.04594
$$640$$ 0 0
$$641$$ −12726.0 −0.784160 −0.392080 0.919931i $$-0.628244\pi$$
−0.392080 + 0.919931i $$0.628244\pi$$
$$642$$ 0 0
$$643$$ 2196.00 0.134684 0.0673420 0.997730i $$-0.478548\pi$$
0.0673420 + 0.997730i $$0.478548\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ −16884.0 −1.02593 −0.512966 0.858409i $$-0.671453\pi$$
−0.512966 + 0.858409i $$0.671453\pi$$
$$648$$ 0 0
$$649$$ −23128.0 −1.39885
$$650$$ 0 0
$$651$$ −2340.00 −0.140878
$$652$$ 0 0
$$653$$ 4018.00 0.240791 0.120395 0.992726i $$-0.461584\pi$$
0.120395 + 0.992726i $$0.461584\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 0 0
$$657$$ 29646.0 1.76043
$$658$$ 0 0
$$659$$ −19071.0 −1.12732 −0.563658 0.826009i $$-0.690606\pi$$
−0.563658 + 0.826009i $$0.690606\pi$$
$$660$$ 0 0
$$661$$ 17424.0 1.02529 0.512644 0.858601i $$-0.328666\pi$$
0.512644 + 0.858601i $$0.328666\pi$$
$$662$$ 0 0
$$663$$ 1260.00 0.0738075
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 6208.00 0.360382
$$668$$ 0 0
$$669$$ 35388.0 2.04511
$$670$$ 0 0
$$671$$ −41890.0 −2.41005
$$672$$ 0 0
$$673$$ 5382.00 0.308263 0.154131 0.988050i $$-0.450742\pi$$
0.154131 + 0.988050i $$0.450742\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ 4496.00 0.255237 0.127618 0.991823i $$-0.459267\pi$$
0.127618 + 0.991823i $$0.459267\pi$$
$$678$$ 0 0
$$679$$ 39052.0 2.20718
$$680$$ 0 0
$$681$$ 54756.0 3.08114
$$682$$ 0 0
$$683$$ 3249.00 0.182020 0.0910099 0.995850i $$-0.470991\pi$$
0.0910099 + 0.995850i $$0.470991\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 0 0
$$687$$ 44964.0 2.49706
$$688$$ 0 0
$$689$$ −504.000 −0.0278677
$$690$$ 0 0
$$691$$ 13399.0 0.737658 0.368829 0.929497i $$-0.379759\pi$$
0.368829 + 0.929497i $$0.379759\pi$$
$$692$$ 0 0
$$693$$ 82836.0 4.54066
$$694$$ 0 0
$$695$$ 0 0
$$696$$ 0 0
$$697$$ −585.000 −0.0317912
$$698$$ 0 0
$$699$$ −28998.0 −1.56911
$$700$$ 0 0
$$701$$ −18148.0 −0.977804 −0.488902 0.872339i $$-0.662602\pi$$
−0.488902 + 0.872339i $$0.662602\pi$$
$$702$$ 0 0
$$703$$ −21582.0 −1.15787
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ −6084.00 −0.323638
$$708$$ 0 0
$$709$$ 4868.00 0.257858 0.128929 0.991654i $$-0.458846\pi$$
0.128929 + 0.991654i $$0.458846\pi$$
$$710$$ 0 0
$$711$$ −22356.0 −1.17921
$$712$$ 0 0
$$713$$ 1940.00 0.101898
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ −24624.0 −1.28257
$$718$$ 0 0
$$719$$ 17366.0 0.900755 0.450377 0.892838i $$-0.351289\pi$$
0.450377 + 0.892838i $$0.351289\pi$$
$$720$$ 0 0
$$721$$ −30472.0 −1.57398
$$722$$ 0 0
$$723$$ 15057.0 0.774517
$$724$$ 0 0
$$725$$ 0 0
$$726$$ 0 0
$$727$$ 21824.0 1.11335 0.556676 0.830729i $$-0.312076\pi$$
0.556676 + 0.830729i $$0.312076\pi$$
$$728$$ 0 0
$$729$$ −19683.0 −1.00000
$$730$$ 0 0
$$731$$ −1940.00 −0.0981580
$$732$$ 0 0
$$733$$ 31428.0 1.58366 0.791828 0.610744i $$-0.209130\pi$$
0.791828 + 0.610744i $$0.209130\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ −14927.0 −0.746056
$$738$$ 0 0
$$739$$ 14292.0 0.711420 0.355710 0.934596i $$-0.384239\pi$$
0.355710 + 0.934596i $$0.384239\pi$$
$$740$$ 0 0
$$741$$ 27468.0 1.36176
$$742$$ 0 0
$$743$$ 13950.0 0.688797 0.344398 0.938824i $$-0.388083\pi$$
0.344398 + 0.938824i $$0.388083\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 0 0
$$747$$ −6534.00 −0.320036
$$748$$ 0 0
$$749$$ 29250.0 1.42693
$$750$$ 0 0
$$751$$ 38736.0 1.88215 0.941076 0.338194i $$-0.109816\pi$$
0.941076 + 0.338194i $$0.109816\pi$$
$$752$$ 0 0
$$753$$ −48195.0 −2.33243
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 0 0
$$757$$ 3664.00 0.175919 0.0879593 0.996124i $$-0.471965\pi$$
0.0879593 + 0.996124i $$0.471965\pi$$
$$758$$ 0 0
$$759$$ −103014. −4.92644
$$760$$ 0 0
$$761$$ 19557.0 0.931591 0.465795 0.884892i $$-0.345768\pi$$
0.465795 + 0.884892i $$0.345768\pi$$
$$762$$ 0 0
$$763$$ −32084.0 −1.52231
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ 10976.0 0.516715
$$768$$ 0 0
$$769$$ −13283.0 −0.622883 −0.311442 0.950265i $$-0.600812\pi$$
−0.311442 + 0.950265i $$0.600812\pi$$
$$770$$ 0 0
$$771$$ 49410.0 2.30799
$$772$$ 0 0
$$773$$ −24840.0 −1.15580 −0.577900 0.816108i $$-0.696127\pi$$
−0.577900 + 0.816108i $$0.696127\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 0 0
$$777$$ 46332.0 2.13919
$$778$$ 0 0
$$779$$ −12753.0 −0.586552
$$780$$ 0 0
$$781$$ 36108.0 1.65435
$$782$$ 0 0
$$783$$ −7776.00 −0.354906
$$784$$ 0 0
$$785$$ 0 0
$$786$$ 0 0
$$787$$ 18044.0 0.817280 0.408640 0.912696i $$-0.366003\pi$$
0.408640 + 0.912696i $$0.366003\pi$$
$$788$$ 0 0
$$789$$ 28350.0 1.27920
$$790$$ 0 0
$$791$$ −14742.0 −0.662661
$$792$$ 0 0
$$793$$ 19880.0 0.890239
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ 6174.00 0.274397 0.137198 0.990544i $$-0.456190\pi$$
0.137198 + 0.990544i $$0.456190\pi$$
$$798$$ 0 0
$$799$$ 340.000 0.0150542
$$800$$ 0 0
$$801$$ −4374.00 −0.192943
$$802$$ 0 0
$$803$$ 32391.0 1.42348
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ 1584.00 0.0690947
$$808$$ 0 0
$$809$$ −1998.00 −0.0868306 −0.0434153 0.999057i $$-0.513824\pi$$
−0.0434153 + 0.999057i $$0.513824\pi$$
$$810$$ 0 0
$$811$$ 7156.00 0.309841 0.154921 0.987927i $$-0.450488\pi$$
0.154921 + 0.987927i $$0.450488\pi$$
$$812$$ 0 0
$$813$$ −21546.0 −0.929460
$$814$$ 0 0
$$815$$ 0 0
$$816$$ 0 0
$$817$$ −42292.0 −1.81103
$$818$$ 0 0
$$819$$ −39312.0 −1.67726
$$820$$ 0 0
$$821$$ 27922.0 1.18695 0.593474 0.804853i $$-0.297756\pi$$
0.593474 + 0.804853i $$0.297756\pi$$
$$822$$ 0 0
$$823$$ 22636.0 0.958738 0.479369 0.877613i $$-0.340866\pi$$
0.479369 + 0.877613i $$0.340866\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 26559.0 1.11674 0.558372 0.829591i $$-0.311426\pi$$
0.558372 + 0.829591i $$0.311426\pi$$
$$828$$ 0 0
$$829$$ 12580.0 0.527046 0.263523 0.964653i $$-0.415115\pi$$
0.263523 + 0.964653i $$0.415115\pi$$
$$830$$ 0 0
$$831$$ 56304.0 2.35038
$$832$$ 0 0
$$833$$ −1665.00 −0.0692543
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 0 0
$$837$$ −2430.00 −0.100350
$$838$$ 0 0
$$839$$ 11344.0 0.466792 0.233396 0.972382i $$-0.425016\pi$$
0.233396 + 0.972382i $$0.425016\pi$$
$$840$$ 0 0
$$841$$ −23365.0 −0.958014
$$842$$ 0 0
$$843$$ −43218.0 −1.76573
$$844$$ 0 0
$$845$$ 0 0
$$846$$ 0 0
$$847$$ 55900.0 2.26771
$$848$$ 0 0
$$849$$ 19107.0 0.772380
$$850$$ 0 0
$$851$$ −38412.0 −1.54729
$$852$$ 0 0
$$853$$ −14786.0 −0.593509 −0.296754 0.954954i $$-0.595904\pi$$
−0.296754 + 0.954954i $$0.595904\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ 29259.0 1.16624 0.583120 0.812386i $$-0.301832\pi$$
0.583120 + 0.812386i $$0.301832\pi$$
$$858$$ 0 0
$$859$$ 13651.0 0.542219 0.271109 0.962549i $$-0.412609\pi$$
0.271109 + 0.962549i $$0.412609\pi$$
$$860$$ 0 0
$$861$$ 27378.0 1.08367
$$862$$ 0 0
$$863$$ 29016.0 1.14451 0.572257 0.820074i $$-0.306068\pi$$
0.572257 + 0.820074i $$0.306068\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 0 0
$$867$$ −43992.0 −1.72324
$$868$$ 0 0
$$869$$ −24426.0 −0.953504
$$870$$ 0 0
$$871$$ 7084.00 0.275582
$$872$$ 0 0
$$873$$ 81108.0 3.14443
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ −21412.0 −0.824438 −0.412219 0.911085i $$-0.635246\pi$$
−0.412219 + 0.911085i $$0.635246\pi$$
$$878$$ 0 0
$$879$$ 79506.0 3.05082
$$880$$ 0 0
$$881$$ −1170.00 −0.0447427 −0.0223713 0.999750i $$-0.507122\pi$$
−0.0223713 + 0.999750i $$0.507122\pi$$
$$882$$ 0 0
$$883$$ −12655.0 −0.482304 −0.241152 0.970487i $$-0.577525\pi$$
−0.241152 + 0.970487i $$0.577525\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ 32764.0 1.24026 0.620128 0.784500i $$-0.287081\pi$$
0.620128 + 0.784500i $$0.287081\pi$$
$$888$$ 0 0
$$889$$ −61308.0 −2.31294
$$890$$ 0 0
$$891$$ 43011.0 1.61720
$$892$$ 0 0
$$893$$ 7412.00 0.277753
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 0 0
$$897$$ 48888.0 1.81976
$$898$$ 0 0
$$899$$ 320.000 0.0118716
$$900$$ 0 0
$$901$$ −90.0000 −0.00332779
$$902$$ 0 0
$$903$$ 90792.0 3.34592
$$904$$ 0 0
$$905$$ 0 0
$$906$$ 0 0
$$907$$ −29844.0 −1.09256 −0.546281 0.837602i $$-0.683957\pi$$
−0.546281 + 0.837602i $$0.683957\pi$$
$$908$$ 0 0
$$909$$ −12636.0 −0.461067
$$910$$ 0 0
$$911$$ −15628.0 −0.568363 −0.284182 0.958770i $$-0.591722\pi$$
−0.284182 + 0.958770i $$0.591722\pi$$
$$912$$ 0 0
$$913$$ −7139.00 −0.258780
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ 43992.0 1.58424
$$918$$ 0 0
$$919$$ −42974.0 −1.54253 −0.771263 0.636517i $$-0.780375\pi$$
−0.771263 + 0.636517i $$0.780375\pi$$
$$920$$ 0 0
$$921$$ 12321.0 0.440815
$$922$$ 0 0
$$923$$ −17136.0 −0.611092
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 0 0
$$927$$ −63288.0 −2.24234
$$928$$ 0 0
$$929$$ 13342.0 0.471191 0.235596 0.971851i $$-0.424296\pi$$
0.235596 + 0.971851i $$0.424296\pi$$
$$930$$ 0 0
$$931$$ −36297.0 −1.27775
$$932$$ 0 0
$$933$$ 93834.0 3.29259
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ 3005.00 0.104770 0.0523848 0.998627i $$-0.483318\pi$$
0.0523848 + 0.998627i $$0.483318\pi$$
$$938$$ 0 0
$$939$$ 32166.0 1.11789
$$940$$ 0 0
$$941$$ 16204.0 0.561355 0.280678 0.959802i $$-0.409441\pi$$
0.280678 + 0.959802i $$0.409441\pi$$
$$942$$ 0 0
$$943$$ −22698.0 −0.783827
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ −30200.0 −1.03629 −0.518146 0.855292i $$-0.673378\pi$$
−0.518146 + 0.855292i $$0.673378\pi$$
$$948$$ 0 0
$$949$$ −15372.0 −0.525813
$$950$$ 0 0
$$951$$ −81324.0 −2.77299
$$952$$ 0 0
$$953$$ 29583.0 1.00555 0.502774 0.864418i $$-0.332313\pi$$
0.502774 + 0.864418i $$0.332313\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 0 0
$$957$$ −16992.0 −0.573953
$$958$$ 0 0
$$959$$ −5954.00 −0.200485
$$960$$ 0 0
$$961$$ −29691.0 −0.996643
$$962$$ 0 0
$$963$$ 60750.0 2.03286
$$964$$ 0 0
$$965$$ 0 0
$$966$$ 0 0
$$967$$ −6480.00 −0.215494 −0.107747 0.994178i $$-0.534364\pi$$
−0.107747 + 0.994178i $$0.534364\pi$$
$$968$$ 0 0
$$969$$ 4905.00 0.162612
$$970$$ 0 0
$$971$$ 40171.0 1.32765 0.663825 0.747888i $$-0.268932\pi$$
0.663825 + 0.747888i $$0.268932\pi$$
$$972$$ 0 0
$$973$$ −72306.0 −2.38235
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ −50801.0 −1.66353 −0.831765 0.555129i $$-0.812669\pi$$
−0.831765 + 0.555129i $$0.812669\pi$$
$$978$$ 0 0
$$979$$ −4779.00 −0.156014
$$980$$ 0 0
$$981$$ −66636.0 −2.16873
$$982$$ 0 0
$$983$$ −58338.0 −1.89287 −0.946436 0.322891i $$-0.895345\pi$$
−0.946436 + 0.322891i $$0.895345\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 0 0
$$987$$ −15912.0 −0.513156
$$988$$ 0 0
$$989$$ −75272.0 −2.42013
$$990$$ 0 0
$$991$$ 51202.0 1.64126 0.820628 0.571462i $$-0.193624\pi$$
0.820628 + 0.571462i $$0.193624\pi$$
$$992$$ 0 0
$$993$$ −92097.0 −2.94321
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 0 0
$$997$$ −1764.00 −0.0560345 −0.0280173 0.999607i $$-0.508919\pi$$
−0.0280173 + 0.999607i $$0.508919\pi$$
$$998$$ 0 0
$$999$$ 48114.0 1.52378
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 400.4.a.t.1.1 1
4.3 odd 2 200.4.a.b.1.1 1
5.2 odd 4 400.4.c.b.49.1 2
5.3 odd 4 400.4.c.b.49.2 2
5.4 even 2 400.4.a.a.1.1 1
8.3 odd 2 1600.4.a.bz.1.1 1
8.5 even 2 1600.4.a.b.1.1 1
12.11 even 2 1800.4.a.c.1.1 1
20.3 even 4 200.4.c.b.49.1 2
20.7 even 4 200.4.c.b.49.2 2
20.19 odd 2 200.4.a.j.1.1 yes 1
40.19 odd 2 1600.4.a.c.1.1 1
40.29 even 2 1600.4.a.by.1.1 1
60.23 odd 4 1800.4.f.w.649.2 2
60.47 odd 4 1800.4.f.w.649.1 2
60.59 even 2 1800.4.a.bh.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
200.4.a.b.1.1 1 4.3 odd 2
200.4.a.j.1.1 yes 1 20.19 odd 2
200.4.c.b.49.1 2 20.3 even 4
200.4.c.b.49.2 2 20.7 even 4
400.4.a.a.1.1 1 5.4 even 2
400.4.a.t.1.1 1 1.1 even 1 trivial
400.4.c.b.49.1 2 5.2 odd 4
400.4.c.b.49.2 2 5.3 odd 4
1600.4.a.b.1.1 1 8.5 even 2
1600.4.a.c.1.1 1 40.19 odd 2
1600.4.a.by.1.1 1 40.29 even 2
1600.4.a.bz.1.1 1 8.3 odd 2
1800.4.a.c.1.1 1 12.11 even 2
1800.4.a.bh.1.1 1 60.59 even 2
1800.4.f.w.649.1 2 60.47 odd 4
1800.4.f.w.649.2 2 60.23 odd 4