# Properties

 Label 1600.4.a.by.1.1 Level $1600$ Weight $4$ Character 1600.1 Self dual yes Analytic conductor $94.403$ 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] = [1600,4,Mod(1,1600)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1600, 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("1600.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1600 = 2^{6} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 1600.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: $$94.4030560092$$ 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 $$\chi$$ $$=$$ 1600.1

## $q$-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 $$-0.747680\pi$$
−0.701934 + 0.712242i $$0.747680\pi$$
$$8$$ 0 0
$$9$$ 54.0000 2.00000
$$10$$ 0 0
$$11$$ −59.0000 −1.61720 −0.808599 0.588361i $$-0.799774\pi$$
−0.808599 + 0.588361i $$0.799774\pi$$
$$12$$ 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 $$-0.488644\pi$$
0.0356670 + 0.999364i $$0.488644\pi$$
$$18$$ 0 0
$$19$$ 109.000 1.31612 0.658061 0.752965i $$-0.271377\pi$$
0.658061 + 0.752965i $$0.271377\pi$$
$$20$$ 0 0
$$21$$ −234.000 −2.43157
$$22$$ 0 0
$$23$$ 194.000 1.75877 0.879387 0.476108i $$-0.157953\pi$$
0.879387 + 0.476108i $$0.157953\pi$$
$$24$$ 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 $$-0.467331\pi$$
0.102453 + 0.994738i $$0.467331\pi$$
$$30$$ 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 $$-0.466350\pi$$
0.105519 + 0.994417i $$0.466350\pi$$
$$48$$ 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 $$-0.357634\pi$$
0.432492 + 0.901638i $$0.357634\pi$$
$$60$$ 0 0
$$61$$ 710.000 1.49027 0.745133 0.666916i $$-0.232386\pi$$
0.745133 + 0.666916i $$0.232386\pi$$
$$62$$ 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 $$-0.645059\pi$$
−0.440107 + 0.897945i $$0.645059\pi$$
$$74$$ 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 $$-0.787907\pi$$
−0.786108 + 0.618089i $$0.787907\pi$$
$$98$$ 0 0
$$99$$ −3186.00 −3.23439
$$100$$ 0 0
$$101$$ 234.000 0.230533 0.115267 0.993335i $$-0.463228\pi$$
0.115267 + 0.993335i $$0.463228\pi$$
$$102$$ 0 0
$$103$$ 1172.00 1.12117 0.560585 0.828097i $$-0.310576\pi$$
0.560585 + 0.828097i $$0.310576\pi$$
$$104$$ 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 $$-0.317598\pi$$
0.542182 + 0.840261i $$0.317598\pi$$
$$110$$ 0 0
$$111$$ 1782.00 1.52378
$$112$$ 0 0
$$113$$ 567.000 0.472025 0.236013 0.971750i $$-0.424159\pi$$
0.236013 + 0.971750i $$0.424159\pi$$
$$114$$ 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 $$-0.191864\pi$$
0.823774 + 0.566918i $$0.191864\pi$$
$$128$$ 0 0
$$129$$ 3492.00 2.38336
$$130$$ 0 0
$$131$$ −1692.00 −1.12848 −0.564239 0.825611i $$-0.690831\pi$$
−0.564239 + 0.825611i $$0.690831\pi$$
$$132$$ 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 $$-0.477252\pi$$
0.0714043 + 0.997447i $$0.477252\pi$$
$$138$$ 0 0
$$139$$ 2781.00 1.69699 0.848494 0.529205i $$-0.177510\pi$$
0.848494 + 0.529205i $$0.177510\pi$$
$$140$$ 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 $$-0.632613\pi$$
−0.404668 + 0.914464i $$0.632613\pi$$
$$150$$ 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 $$-0.550657\pi$$
−0.158472 + 0.987364i $$0.550657\pi$$
$$168$$ 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 $$-0.574512\pi$$
−0.231955 + 0.972726i $$0.574512\pi$$
$$180$$ 0 0
$$181$$ −2042.00 −0.838567 −0.419284 0.907855i $$-0.637719\pi$$
−0.419284 + 0.907855i $$0.637719\pi$$
$$182$$ 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 $$-0.463533\pi$$
0.114313 + 0.993445i $$0.463533\pi$$
$$194$$ 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 $$-0.375383\pi$$
0.381572 + 0.924339i $$0.375383\pi$$
$$212$$ 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 $$-0.701019\pi$$
−0.590373 + 0.807131i $$0.701019\pi$$
$$224$$ 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 $$-0.756244\pi$$
−0.720841 + 0.693101i $$0.756244\pi$$
$$230$$ 0 0
$$231$$ 13806.0 3.93233
$$232$$ 0 0
$$233$$ 3222.00 0.905924 0.452962 0.891530i $$-0.350367\pi$$
0.452962 + 0.891530i $$0.350367\pi$$
$$234$$ 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 $$-0.264869\pi$$
0.673316 + 0.739355i $$0.264869\pi$$
$$252$$ 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 $$-0.732105\pi$$
−0.666258 + 0.745721i $$0.732105\pi$$
$$258$$ 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 $$-0.620393\pi$$
−0.369272 + 0.929321i $$0.620393\pi$$
$$264$$ 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 $$-0.506349\pi$$
−0.0199459 + 0.999801i $$0.506349\pi$$
$$270$$ 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 $$-0.604593\pi$$
−0.322707 + 0.946499i $$0.604593\pi$$
$$314$$ 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 $$-0.176824\pi$$
0.849632 + 0.527376i $$0.176824\pi$$
$$332$$ 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 $$-0.378000\pi$$
0.373960 + 0.927445i $$0.378000\pi$$
$$338$$ 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 $$-0.388313\pi$$
0.343719 + 0.939072i $$0.388313\pi$$
$$350$$ 0 0
$$351$$ −6804.00 −1.03467
$$352$$ 0 0
$$353$$ −1210.00 −0.182441 −0.0912207 0.995831i $$-0.529077\pi$$
−0.0912207 + 0.995831i $$0.529077\pi$$
$$354$$ 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 $$-0.204422\pi$$
0.800773 + 0.598968i $$0.204422\pi$$
$$368$$ 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 $$-0.212976\pi$$
0.784390 + 0.620268i $$0.212976\pi$$
$$380$$ 0 0
$$381$$ 21222.0 2.85364
$$382$$ 0 0
$$383$$ 18.0000 0.00240145 0.00120073 0.999999i $$-0.499618\pi$$
0.00120073 + 0.999999i $$0.499618\pi$$
$$384$$ 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 $$-0.745912\pi$$
−0.697967 + 0.716130i $$0.745912\pi$$
$$390$$ 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 $$-0.402099\pi$$
0.302738 + 0.953074i $$0.402099\pi$$
$$420$$ 0 0
$$421$$ −4788.00 −0.554282 −0.277141 0.960829i $$-0.589387\pi$$
−0.277141 + 0.960829i $$0.589387\pi$$
$$422$$ 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 $$-0.542431\pi$$
−0.132906 + 0.991129i $$0.542431\pi$$
$$434$$ 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 $$-0.113363\pi$$
0.937249 + 0.348659i $$0.113363\pi$$
$$458$$ 0 0
$$459$$ 1215.00 0.123554
$$460$$ 0 0
$$461$$ −12492.0 −1.26206 −0.631031 0.775758i $$-0.717368\pi$$
−0.631031 + 0.775758i $$0.717368\pi$$
$$462$$ 0 0
$$463$$ −4428.00 −0.444464 −0.222232 0.974994i $$-0.571334\pi$$
−0.222232 + 0.974994i $$0.571334\pi$$
$$464$$ 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 $$-0.455218\pi$$
0.140223 + 0.990120i $$0.455218\pi$$
$$488$$ 0 0
$$489$$ −3069.00 −0.283814
$$490$$ 0 0
$$491$$ −3564.00 −0.327579 −0.163789 0.986495i $$-0.552372\pi$$
−0.163789 + 0.986495i $$0.552372\pi$$
$$492$$ 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 $$-0.750647\pi$$
−0.708543 + 0.705667i $$0.750647\pi$$
$$500$$ 0 0
$$501$$ −6156.00 −0.548962
$$502$$ 0 0
$$503$$ 10908.0 0.966926 0.483463 0.875365i $$-0.339379\pi$$
0.483463 + 0.875365i $$0.339379\pi$$
$$504$$ 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 $$-0.904726\pi$$
−0.955540 + 0.294863i $$0.904726\pi$$
$$510$$ 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 $$-0.447757\pi$$
0.163391 + 0.986561i $$0.447757\pi$$
$$542$$ 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 $$-0.810207\pi$$
−0.827446 + 0.561545i $$0.810207\pi$$
$$572$$ 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 $$-0.509521\pi$$
−0.0299062 + 0.999553i $$0.509521\pi$$
$$578$$ 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 $$-0.591832\pi$$
−0.284512 + 0.958672i $$0.591832\pi$$
$$594$$ 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 $$-0.331301\pi$$
0.505520 + 0.862815i $$0.331301\pi$$
$$608$$ 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 $$-0.289221\pi$$
0.614839 + 0.788653i $$0.289221\pi$$
$$618$$ 0 0
$$619$$ 16316.0 1.05944 0.529722 0.848172i $$-0.322296\pi$$
0.529722 + 0.848172i $$0.322296\pi$$
$$620$$ 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 $$-0.328547\pi$$
0.512966 + 0.858409i $$0.328547\pi$$
$$648$$ 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 $$-0.309394\pi$$
0.563658 + 0.826009i $$0.309394\pi$$
$$660$$ 0 0
$$661$$ −17424.0 −1.02529 −0.512644 0.858601i $$-0.671334\pi$$
−0.512644 + 0.858601i $$0.671334\pi$$
$$662$$ 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 $$-0.549258\pi$$
−0.154131 + 0.988050i $$0.549258\pi$$
$$674$$ 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 $$-0.620241\pi$$
−0.368829 + 0.929497i $$0.620241\pi$$
$$692$$ 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 $$-0.337398\pi$$
0.488902 + 0.872339i $$0.337398\pi$$
$$702$$ 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 $$-0.541154\pi$$
−0.128929 + 0.991654i $$0.541154\pi$$
$$710$$ 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 $$-0.687924\pi$$
−0.556676 + 0.830729i $$0.687924\pi$$
$$728$$ 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 $$-0.615761\pi$$
−0.355710 + 0.934596i $$0.615761\pi$$
$$740$$ 0 0
$$741$$ −27468.0 −1.36176
$$742$$ 0 0
$$743$$ −13950.0 −0.688797 −0.344398 0.938824i $$-0.611917\pi$$
−0.344398 + 0.938824i $$0.611917\pi$$
$$744$$ 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 $$-0.549512\pi$$
−0.154921 + 0.987927i $$0.549512\pi$$
$$812$$ 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 $$-0.702244\pi$$
−0.593474 + 0.804853i $$0.702244\pi$$
$$822$$ 0 0
$$823$$ −22636.0 −0.958738 −0.479369 0.877613i $$-0.659134\pi$$
−0.479369 + 0.877613i $$0.659134\pi$$
$$824$$ 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 $$-0.584885\pi$$
−0.263523 + 0.964653i $$0.584885\pi$$
$$830$$ 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 $$-0.698168\pi$$
−0.583120 + 0.812386i $$0.698168\pi$$
$$858$$ 0 0
$$859$$ −13651.0 −0.542219 −0.271109 0.962549i $$-0.587391\pi$$
−0.271109 + 0.962549i $$0.587391\pi$$
$$860$$ 0 0
$$861$$ −27378.0 −1.08367
$$862$$ 0 0
$$863$$ −29016.0 −1.14451 −0.572257 0.820074i $$-0.693932\pi$$
−0.572257 + 0.820074i $$0.693932\pi$$
$$864$$ 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 $$-0.712919\pi$$
−0.620128 + 0.784500i $$0.712919\pi$$
$$888$$ 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 $$-0.516682\pi$$
−0.0523848 + 0.998627i $$0.516682\pi$$
$$938$$ 0 0
$$939$$ −32166.0 −1.11789
$$940$$ 0 0
$$941$$ −16204.0 −0.561355 −0.280678 0.959802i $$-0.590559\pi$$
−0.280678 + 0.959802i $$0.590559\pi$$
$$942$$ 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 $$-0.667687\pi$$
−0.502774 + 0.864418i $$0.667687\pi$$
$$954$$ 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 $$-0.465636\pi$$
0.107747 + 0.994178i $$0.465636\pi$$
$$968$$ 0 0
$$969$$ 4905.00 0.162612
$$970$$ 0 0
$$971$$ −40171.0 −1.32765 −0.663825 0.747888i $$-0.731068\pi$$
−0.663825 + 0.747888i $$0.731068\pi$$
$$972$$ 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 $$-0.187331\pi$$
0.831765 + 0.555129i $$0.187331\pi$$
$$978$$ 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 $$-0.104655\pi$$
0.946436 + 0.322891i $$0.104655\pi$$
$$984$$ 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 1600.4.a.by.1.1 1
4.3 odd 2 1600.4.a.c.1.1 1
5.4 even 2 1600.4.a.b.1.1 1
8.3 odd 2 200.4.a.j.1.1 yes 1
8.5 even 2 400.4.a.a.1.1 1
20.19 odd 2 1600.4.a.bz.1.1 1
24.11 even 2 1800.4.a.bh.1.1 1
40.3 even 4 200.4.c.b.49.2 2
40.13 odd 4 400.4.c.b.49.1 2
40.19 odd 2 200.4.a.b.1.1 1
40.27 even 4 200.4.c.b.49.1 2
40.29 even 2 400.4.a.t.1.1 1
40.37 odd 4 400.4.c.b.49.2 2
120.59 even 2 1800.4.a.c.1.1 1
120.83 odd 4 1800.4.f.w.649.1 2
120.107 odd 4 1800.4.f.w.649.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
200.4.a.b.1.1 1 40.19 odd 2
200.4.a.j.1.1 yes 1 8.3 odd 2
200.4.c.b.49.1 2 40.27 even 4
200.4.c.b.49.2 2 40.3 even 4
400.4.a.a.1.1 1 8.5 even 2
400.4.a.t.1.1 1 40.29 even 2
400.4.c.b.49.1 2 40.13 odd 4
400.4.c.b.49.2 2 40.37 odd 4
1600.4.a.b.1.1 1 5.4 even 2
1600.4.a.c.1.1 1 4.3 odd 2
1600.4.a.by.1.1 1 1.1 even 1 trivial
1600.4.a.bz.1.1 1 20.19 odd 2
1800.4.a.c.1.1 1 120.59 even 2
1800.4.a.bh.1.1 1 24.11 even 2
1800.4.f.w.649.1 2 120.83 odd 4
1800.4.f.w.649.2 2 120.107 odd 4