# Properties

 Label 1600.4.a.bq.1.1 Level $1600$ Weight $4$ Character 1600.1 Self dual yes Analytic conductor $94.403$ 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] = [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: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 800) 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)

 $$f(q)$$ $$=$$ $$q+5.00000 q^{3} +10.0000 q^{7} -2.00000 q^{9} +O(q^{10})$$ $$q+5.00000 q^{3} +10.0000 q^{7} -2.00000 q^{9} +15.0000 q^{11} -8.00000 q^{13} -21.0000 q^{17} -105.000 q^{19} +50.0000 q^{21} +10.0000 q^{23} -145.000 q^{27} +20.0000 q^{29} -230.000 q^{31} +75.0000 q^{33} +54.0000 q^{37} -40.0000 q^{39} -195.000 q^{41} -300.000 q^{43} +480.000 q^{47} -243.000 q^{49} -105.000 q^{51} -322.000 q^{53} -525.000 q^{57} -560.000 q^{59} +730.000 q^{61} -20.0000 q^{63} +255.000 q^{67} +50.0000 q^{69} -40.0000 q^{71} +317.000 q^{73} +150.000 q^{77} -830.000 q^{79} -671.000 q^{81} +75.0000 q^{83} +100.000 q^{87} -705.000 q^{89} -80.0000 q^{91} -1150.00 q^{93} -1434.00 q^{97} -30.0000 q^{99} +O(q^{100})$$

## 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
$$3$$ 5.00000 0.962250 0.481125 0.876652i $$-0.340228\pi$$
0.481125 + 0.876652i $$0.340228\pi$$
$$4$$ 0 0
$$5$$ 0 0
$$6$$ 0 0
$$7$$ 10.0000 0.539949 0.269975 0.962867i $$-0.412985\pi$$
0.269975 + 0.962867i $$0.412985\pi$$
$$8$$ 0 0
$$9$$ −2.00000 −0.0740741
$$10$$ 0 0
$$11$$ 15.0000 0.411152 0.205576 0.978641i $$-0.434093\pi$$
0.205576 + 0.978641i $$0.434093\pi$$
$$12$$ 0 0
$$13$$ −8.00000 −0.170677 −0.0853385 0.996352i $$-0.527197\pi$$
−0.0853385 + 0.996352i $$0.527197\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ −21.0000 −0.299603 −0.149801 0.988716i $$-0.547863\pi$$
−0.149801 + 0.988716i $$0.547863\pi$$
$$18$$ 0 0
$$19$$ −105.000 −1.26782 −0.633912 0.773405i $$-0.718552\pi$$
−0.633912 + 0.773405i $$0.718552\pi$$
$$20$$ 0 0
$$21$$ 50.0000 0.519566
$$22$$ 0 0
$$23$$ 10.0000 0.0906584 0.0453292 0.998972i $$-0.485566\pi$$
0.0453292 + 0.998972i $$0.485566\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ 0 0
$$27$$ −145.000 −1.03353
$$28$$ 0 0
$$29$$ 20.0000 0.128066 0.0640329 0.997948i $$-0.479604\pi$$
0.0640329 + 0.997948i $$0.479604\pi$$
$$30$$ 0 0
$$31$$ −230.000 −1.33256 −0.666278 0.745704i $$-0.732113\pi$$
−0.666278 + 0.745704i $$0.732113\pi$$
$$32$$ 0 0
$$33$$ 75.0000 0.395631
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 54.0000 0.239934 0.119967 0.992778i $$-0.461721\pi$$
0.119967 + 0.992778i $$0.461721\pi$$
$$38$$ 0 0
$$39$$ −40.0000 −0.164234
$$40$$ 0 0
$$41$$ −195.000 −0.742778 −0.371389 0.928477i $$-0.621118\pi$$
−0.371389 + 0.928477i $$0.621118\pi$$
$$42$$ 0 0
$$43$$ −300.000 −1.06394 −0.531972 0.846762i $$-0.678549\pi$$
−0.531972 + 0.846762i $$0.678549\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ 480.000 1.48969 0.744843 0.667240i $$-0.232525\pi$$
0.744843 + 0.667240i $$0.232525\pi$$
$$48$$ 0 0
$$49$$ −243.000 −0.708455
$$50$$ 0 0
$$51$$ −105.000 −0.288293
$$52$$ 0 0
$$53$$ −322.000 −0.834530 −0.417265 0.908785i $$-0.637011\pi$$
−0.417265 + 0.908785i $$0.637011\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ −525.000 −1.21996
$$58$$ 0 0
$$59$$ −560.000 −1.23569 −0.617846 0.786299i $$-0.711994\pi$$
−0.617846 + 0.786299i $$0.711994\pi$$
$$60$$ 0 0
$$61$$ 730.000 1.53224 0.766122 0.642695i $$-0.222184\pi$$
0.766122 + 0.642695i $$0.222184\pi$$
$$62$$ 0 0
$$63$$ −20.0000 −0.0399962
$$64$$ 0 0
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 255.000 0.464973 0.232487 0.972600i $$-0.425314\pi$$
0.232487 + 0.972600i $$0.425314\pi$$
$$68$$ 0 0
$$69$$ 50.0000 0.0872361
$$70$$ 0 0
$$71$$ −40.0000 −0.0668609 −0.0334305 0.999441i $$-0.510643\pi$$
−0.0334305 + 0.999441i $$0.510643\pi$$
$$72$$ 0 0
$$73$$ 317.000 0.508247 0.254124 0.967172i $$-0.418213\pi$$
0.254124 + 0.967172i $$0.418213\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ 150.000 0.222001
$$78$$ 0 0
$$79$$ −830.000 −1.18205 −0.591027 0.806652i $$-0.701277\pi$$
−0.591027 + 0.806652i $$0.701277\pi$$
$$80$$ 0 0
$$81$$ −671.000 −0.920439
$$82$$ 0 0
$$83$$ 75.0000 0.0991846 0.0495923 0.998770i $$-0.484208\pi$$
0.0495923 + 0.998770i $$0.484208\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 0 0
$$87$$ 100.000 0.123231
$$88$$ 0 0
$$89$$ −705.000 −0.839661 −0.419831 0.907602i $$-0.637911\pi$$
−0.419831 + 0.907602i $$0.637911\pi$$
$$90$$ 0 0
$$91$$ −80.0000 −0.0921569
$$92$$ 0 0
$$93$$ −1150.00 −1.28225
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 0 0
$$97$$ −1434.00 −1.50104 −0.750519 0.660849i $$-0.770196\pi$$
−0.750519 + 0.660849i $$0.770196\pi$$
$$98$$ 0 0
$$99$$ −30.0000 −0.0304557
$$100$$ 0 0
$$101$$ 1902.00 1.87382 0.936911 0.349567i $$-0.113671\pi$$
0.936911 + 0.349567i $$0.113671\pi$$
$$102$$ 0 0
$$103$$ −1480.00 −1.41581 −0.707906 0.706306i $$-0.750360\pi$$
−0.707906 + 0.706306i $$0.750360\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 1945.00 1.75729 0.878646 0.477474i $$-0.158447\pi$$
0.878646 + 0.477474i $$0.158447\pi$$
$$108$$ 0 0
$$109$$ 246.000 0.216170 0.108085 0.994142i $$-0.465528\pi$$
0.108085 + 0.994142i $$0.465528\pi$$
$$110$$ 0 0
$$111$$ 270.000 0.230876
$$112$$ 0 0
$$113$$ 753.000 0.626870 0.313435 0.949610i $$-0.398520\pi$$
0.313435 + 0.949610i $$0.398520\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 0 0
$$117$$ 16.0000 0.0126427
$$118$$ 0 0
$$119$$ −210.000 −0.161770
$$120$$ 0 0
$$121$$ −1106.00 −0.830954
$$122$$ 0 0
$$123$$ −975.000 −0.714738
$$124$$ 0 0
$$125$$ 0 0
$$126$$ 0 0
$$127$$ −1490.00 −1.04107 −0.520536 0.853840i $$-0.674268\pi$$
−0.520536 + 0.853840i $$0.674268\pi$$
$$128$$ 0 0
$$129$$ −1500.00 −1.02378
$$130$$ 0 0
$$131$$ −780.000 −0.520221 −0.260110 0.965579i $$-0.583759\pi$$
−0.260110 + 0.965579i $$0.583759\pi$$
$$132$$ 0 0
$$133$$ −1050.00 −0.684561
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ −2749.00 −1.71433 −0.857164 0.515044i $$-0.827776\pi$$
−0.857164 + 0.515044i $$0.827776\pi$$
$$138$$ 0 0
$$139$$ 735.000 0.448503 0.224251 0.974531i $$-0.428006\pi$$
0.224251 + 0.974531i $$0.428006\pi$$
$$140$$ 0 0
$$141$$ 2400.00 1.43345
$$142$$ 0 0
$$143$$ −120.000 −0.0701742
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 0 0
$$147$$ −1215.00 −0.681711
$$148$$ 0 0
$$149$$ −836.000 −0.459650 −0.229825 0.973232i $$-0.573815\pi$$
−0.229825 + 0.973232i $$0.573815\pi$$
$$150$$ 0 0
$$151$$ 1790.00 0.964690 0.482345 0.875981i $$-0.339785\pi$$
0.482345 + 0.875981i $$0.339785\pi$$
$$152$$ 0 0
$$153$$ 42.0000 0.0221928
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ −1374.00 −0.698453 −0.349227 0.937038i $$-0.613556\pi$$
−0.349227 + 0.937038i $$0.613556\pi$$
$$158$$ 0 0
$$159$$ −1610.00 −0.803027
$$160$$ 0 0
$$161$$ 100.000 0.0489510
$$162$$ 0 0
$$163$$ 1895.00 0.910600 0.455300 0.890338i $$-0.349532\pi$$
0.455300 + 0.890338i $$0.349532\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 720.000 0.333624 0.166812 0.985989i $$-0.446653\pi$$
0.166812 + 0.985989i $$0.446653\pi$$
$$168$$ 0 0
$$169$$ −2133.00 −0.970869
$$170$$ 0 0
$$171$$ 210.000 0.0939129
$$172$$ 0 0
$$173$$ −2512.00 −1.10395 −0.551976 0.833860i $$-0.686126\pi$$
−0.551976 + 0.833860i $$0.686126\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ −2800.00 −1.18904
$$178$$ 0 0
$$179$$ −165.000 −0.0688976 −0.0344488 0.999406i $$-0.510968\pi$$
−0.0344488 + 0.999406i $$0.510968\pi$$
$$180$$ 0 0
$$181$$ 3158.00 1.29686 0.648432 0.761273i $$-0.275425\pi$$
0.648432 + 0.761273i $$0.275425\pi$$
$$182$$ 0 0
$$183$$ 3650.00 1.47440
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 0 0
$$187$$ −315.000 −0.123182
$$188$$ 0 0
$$189$$ −1450.00 −0.558053
$$190$$ 0 0
$$191$$ −3290.00 −1.24637 −0.623183 0.782076i $$-0.714161\pi$$
−0.623183 + 0.782076i $$0.714161\pi$$
$$192$$ 0 0
$$193$$ −197.000 −0.0734734 −0.0367367 0.999325i $$-0.511696\pi$$
−0.0367367 + 0.999325i $$0.511696\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ −1746.00 −0.631459 −0.315729 0.948849i $$-0.602249\pi$$
−0.315729 + 0.948849i $$0.602249\pi$$
$$198$$ 0 0
$$199$$ 4660.00 1.65999 0.829997 0.557768i $$-0.188342\pi$$
0.829997 + 0.557768i $$0.188342\pi$$
$$200$$ 0 0
$$201$$ 1275.00 0.447421
$$202$$ 0 0
$$203$$ 200.000 0.0691490
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 0 0
$$207$$ −20.0000 −0.00671544
$$208$$ 0 0
$$209$$ −1575.00 −0.521268
$$210$$ 0 0
$$211$$ 265.000 0.0864614 0.0432307 0.999065i $$-0.486235\pi$$
0.0432307 + 0.999065i $$0.486235\pi$$
$$212$$ 0 0
$$213$$ −200.000 −0.0643370
$$214$$ 0 0
$$215$$ 0 0
$$216$$ 0 0
$$217$$ −2300.00 −0.719512
$$218$$ 0 0
$$219$$ 1585.00 0.489061
$$220$$ 0 0
$$221$$ 168.000 0.0511353
$$222$$ 0 0
$$223$$ 1060.00 0.318309 0.159154 0.987254i $$-0.449123\pi$$
0.159154 + 0.987254i $$0.449123\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ −4660.00 −1.36253 −0.681267 0.732035i $$-0.738571\pi$$
−0.681267 + 0.732035i $$0.738571\pi$$
$$228$$ 0 0
$$229$$ −1660.00 −0.479021 −0.239511 0.970894i $$-0.576987\pi$$
−0.239511 + 0.970894i $$0.576987\pi$$
$$230$$ 0 0
$$231$$ 750.000 0.213621
$$232$$ 0 0
$$233$$ −3462.00 −0.973404 −0.486702 0.873568i $$-0.661800\pi$$
−0.486702 + 0.873568i $$0.661800\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 0 0
$$237$$ −4150.00 −1.13743
$$238$$ 0 0
$$239$$ 4020.00 1.08800 0.544000 0.839085i $$-0.316909\pi$$
0.544000 + 0.839085i $$0.316909\pi$$
$$240$$ 0 0
$$241$$ 3985.00 1.06513 0.532565 0.846389i $$-0.321228\pi$$
0.532565 + 0.846389i $$0.321228\pi$$
$$242$$ 0 0
$$243$$ 560.000 0.147835
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 840.000 0.216388
$$248$$ 0 0
$$249$$ 375.000 0.0954404
$$250$$ 0 0
$$251$$ 6625.00 1.66600 0.833001 0.553272i $$-0.186621\pi$$
0.833001 + 0.553272i $$0.186621\pi$$
$$252$$ 0 0
$$253$$ 150.000 0.0372744
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ −2246.00 −0.545143 −0.272571 0.962136i $$-0.587874\pi$$
−0.272571 + 0.962136i $$0.587874\pi$$
$$258$$ 0 0
$$259$$ 540.000 0.129552
$$260$$ 0 0
$$261$$ −40.0000 −0.00948635
$$262$$ 0 0
$$263$$ −3950.00 −0.926112 −0.463056 0.886329i $$-0.653247\pi$$
−0.463056 + 0.886329i $$0.653247\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 0 0
$$267$$ −3525.00 −0.807964
$$268$$ 0 0
$$269$$ −2656.00 −0.602004 −0.301002 0.953623i $$-0.597321\pi$$
−0.301002 + 0.953623i $$0.597321\pi$$
$$270$$ 0 0
$$271$$ −3110.00 −0.697118 −0.348559 0.937287i $$-0.613329\pi$$
−0.348559 + 0.937287i $$0.613329\pi$$
$$272$$ 0 0
$$273$$ −400.000 −0.0886780
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 0 0
$$277$$ −6884.00 −1.49321 −0.746606 0.665267i $$-0.768318\pi$$
−0.746606 + 0.665267i $$0.768318\pi$$
$$278$$ 0 0
$$279$$ 460.000 0.0987078
$$280$$ 0 0
$$281$$ 4630.00 0.982928 0.491464 0.870898i $$-0.336462\pi$$
0.491464 + 0.870898i $$0.336462\pi$$
$$282$$ 0 0
$$283$$ 215.000 0.0451605 0.0225803 0.999745i $$-0.492812\pi$$
0.0225803 + 0.999745i $$0.492812\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ −1950.00 −0.401062
$$288$$ 0 0
$$289$$ −4472.00 −0.910238
$$290$$ 0 0
$$291$$ −7170.00 −1.44437
$$292$$ 0 0
$$293$$ 1602.00 0.319419 0.159710 0.987164i $$-0.448944\pi$$
0.159710 + 0.987164i $$0.448944\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 0 0
$$297$$ −2175.00 −0.424937
$$298$$ 0 0
$$299$$ −80.0000 −0.0154733
$$300$$ 0 0
$$301$$ −3000.00 −0.574475
$$302$$ 0 0
$$303$$ 9510.00 1.80309
$$304$$ 0 0
$$305$$ 0 0
$$306$$ 0 0
$$307$$ 1885.00 0.350432 0.175216 0.984530i $$-0.443938\pi$$
0.175216 + 0.984530i $$0.443938\pi$$
$$308$$ 0 0
$$309$$ −7400.00 −1.36237
$$310$$ 0 0
$$311$$ 9250.00 1.68656 0.843279 0.537476i $$-0.180622\pi$$
0.843279 + 0.537476i $$0.180622\pi$$
$$312$$ 0 0
$$313$$ −8162.00 −1.47394 −0.736970 0.675925i $$-0.763744\pi$$
−0.736970 + 0.675925i $$0.763744\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ −6924.00 −1.22678 −0.613392 0.789779i $$-0.710195\pi$$
−0.613392 + 0.789779i $$0.710195\pi$$
$$318$$ 0 0
$$319$$ 300.000 0.0526545
$$320$$ 0 0
$$321$$ 9725.00 1.69096
$$322$$ 0 0
$$323$$ 2205.00 0.379844
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 0 0
$$327$$ 1230.00 0.208010
$$328$$ 0 0
$$329$$ 4800.00 0.804354
$$330$$ 0 0
$$331$$ 8075.00 1.34091 0.670456 0.741949i $$-0.266098\pi$$
0.670456 + 0.741949i $$0.266098\pi$$
$$332$$ 0 0
$$333$$ −108.000 −0.0177729
$$334$$ 0 0
$$335$$ 0 0
$$336$$ 0 0
$$337$$ 4741.00 0.766346 0.383173 0.923677i $$-0.374831\pi$$
0.383173 + 0.923677i $$0.374831\pi$$
$$338$$ 0 0
$$339$$ 3765.00 0.603206
$$340$$ 0 0
$$341$$ −3450.00 −0.547883
$$342$$ 0 0
$$343$$ −5860.00 −0.922479
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ 8705.00 1.34671 0.673356 0.739319i $$-0.264852\pi$$
0.673356 + 0.739319i $$0.264852\pi$$
$$348$$ 0 0
$$349$$ 1470.00 0.225465 0.112733 0.993625i $$-0.464040\pi$$
0.112733 + 0.993625i $$0.464040\pi$$
$$350$$ 0 0
$$351$$ 1160.00 0.176399
$$352$$ 0 0
$$353$$ −1998.00 −0.301254 −0.150627 0.988591i $$-0.548129\pi$$
−0.150627 + 0.988591i $$0.548129\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0 0
$$357$$ −1050.00 −0.155664
$$358$$ 0 0
$$359$$ −8190.00 −1.20404 −0.602022 0.798480i $$-0.705638\pi$$
−0.602022 + 0.798480i $$0.705638\pi$$
$$360$$ 0 0
$$361$$ 4166.00 0.607377
$$362$$ 0 0
$$363$$ −5530.00 −0.799586
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 0 0
$$367$$ −5340.00 −0.759525 −0.379763 0.925084i $$-0.623994\pi$$
−0.379763 + 0.925084i $$0.623994\pi$$
$$368$$ 0 0
$$369$$ 390.000 0.0550206
$$370$$ 0 0
$$371$$ −3220.00 −0.450604
$$372$$ 0 0
$$373$$ 9378.00 1.30181 0.650904 0.759160i $$-0.274390\pi$$
0.650904 + 0.759160i $$0.274390\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ −160.000 −0.0218579
$$378$$ 0 0
$$379$$ 4045.00 0.548226 0.274113 0.961697i $$-0.411616\pi$$
0.274113 + 0.961697i $$0.411616\pi$$
$$380$$ 0 0
$$381$$ −7450.00 −1.00177
$$382$$ 0 0
$$383$$ −8090.00 −1.07932 −0.539660 0.841883i $$-0.681447\pi$$
−0.539660 + 0.841883i $$0.681447\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ 600.000 0.0788106
$$388$$ 0 0
$$389$$ −6574.00 −0.856851 −0.428425 0.903577i $$-0.640932\pi$$
−0.428425 + 0.903577i $$0.640932\pi$$
$$390$$ 0 0
$$391$$ −210.000 −0.0271615
$$392$$ 0 0
$$393$$ −3900.00 −0.500583
$$394$$ 0 0
$$395$$ 0 0
$$396$$ 0 0
$$397$$ −7036.00 −0.889488 −0.444744 0.895658i $$-0.646705\pi$$
−0.444744 + 0.895658i $$0.646705\pi$$
$$398$$ 0 0
$$399$$ −5250.00 −0.658719
$$400$$ 0 0
$$401$$ 8277.00 1.03076 0.515379 0.856963i $$-0.327651\pi$$
0.515379 + 0.856963i $$0.327651\pi$$
$$402$$ 0 0
$$403$$ 1840.00 0.227437
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 810.000 0.0986492
$$408$$ 0 0
$$409$$ −2179.00 −0.263434 −0.131717 0.991287i $$-0.542049\pi$$
−0.131717 + 0.991287i $$0.542049\pi$$
$$410$$ 0 0
$$411$$ −13745.0 −1.64961
$$412$$ 0 0
$$413$$ −5600.00 −0.667211
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 0 0
$$417$$ 3675.00 0.431572
$$418$$ 0 0
$$419$$ −12245.0 −1.42770 −0.713851 0.700297i $$-0.753051\pi$$
−0.713851 + 0.700297i $$0.753051\pi$$
$$420$$ 0 0
$$421$$ 660.000 0.0764048 0.0382024 0.999270i $$-0.487837\pi$$
0.0382024 + 0.999270i $$0.487837\pi$$
$$422$$ 0 0
$$423$$ −960.000 −0.110347
$$424$$ 0 0
$$425$$ 0 0
$$426$$ 0 0
$$427$$ 7300.00 0.827334
$$428$$ 0 0
$$429$$ −600.000 −0.0675251
$$430$$ 0 0
$$431$$ −7470.00 −0.834843 −0.417421 0.908713i $$-0.637066\pi$$
−0.417421 + 0.908713i $$0.637066\pi$$
$$432$$ 0 0
$$433$$ −1173.00 −0.130187 −0.0650933 0.997879i $$-0.520734\pi$$
−0.0650933 + 0.997879i $$0.520734\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ −1050.00 −0.114939
$$438$$ 0 0
$$439$$ 5660.00 0.615346 0.307673 0.951492i $$-0.400450\pi$$
0.307673 + 0.951492i $$0.400450\pi$$
$$440$$ 0 0
$$441$$ 486.000 0.0524781
$$442$$ 0 0
$$443$$ 1115.00 0.119583 0.0597915 0.998211i $$-0.480956\pi$$
0.0597915 + 0.998211i $$0.480956\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 0 0
$$447$$ −4180.00 −0.442298
$$448$$ 0 0
$$449$$ 4089.00 0.429781 0.214891 0.976638i $$-0.431060\pi$$
0.214891 + 0.976638i $$0.431060\pi$$
$$450$$ 0 0
$$451$$ −2925.00 −0.305394
$$452$$ 0 0
$$453$$ 8950.00 0.928273
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 4519.00 0.462560 0.231280 0.972887i $$-0.425709\pi$$
0.231280 + 0.972887i $$0.425709\pi$$
$$458$$ 0 0
$$459$$ 3045.00 0.309648
$$460$$ 0 0
$$461$$ −10068.0 −1.01717 −0.508583 0.861013i $$-0.669830\pi$$
−0.508583 + 0.861013i $$0.669830\pi$$
$$462$$ 0 0
$$463$$ −1460.00 −0.146548 −0.0732742 0.997312i $$-0.523345\pi$$
−0.0732742 + 0.997312i $$0.523345\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ 4740.00 0.469681 0.234841 0.972034i $$-0.424543\pi$$
0.234841 + 0.972034i $$0.424543\pi$$
$$468$$ 0 0
$$469$$ 2550.00 0.251062
$$470$$ 0 0
$$471$$ −6870.00 −0.672087
$$472$$ 0 0
$$473$$ −4500.00 −0.437442
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 0 0
$$477$$ 644.000 0.0618171
$$478$$ 0 0
$$479$$ −8610.00 −0.821296 −0.410648 0.911794i $$-0.634698\pi$$
−0.410648 + 0.911794i $$0.634698\pi$$
$$480$$ 0 0
$$481$$ −432.000 −0.0409512
$$482$$ 0 0
$$483$$ 500.000 0.0471031
$$484$$ 0 0
$$485$$ 0 0
$$486$$ 0 0
$$487$$ 17710.0 1.64788 0.823939 0.566678i $$-0.191772\pi$$
0.823939 + 0.566678i $$0.191772\pi$$
$$488$$ 0 0
$$489$$ 9475.00 0.876226
$$490$$ 0 0
$$491$$ 8660.00 0.795968 0.397984 0.917392i $$-0.369710\pi$$
0.397984 + 0.917392i $$0.369710\pi$$
$$492$$ 0 0
$$493$$ −420.000 −0.0383689
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ −400.000 −0.0361015
$$498$$ 0 0
$$499$$ −17300.0 −1.55201 −0.776006 0.630725i $$-0.782758\pi$$
−0.776006 + 0.630725i $$0.782758\pi$$
$$500$$ 0 0
$$501$$ 3600.00 0.321030
$$502$$ 0 0
$$503$$ −1860.00 −0.164877 −0.0824387 0.996596i $$-0.526271\pi$$
−0.0824387 + 0.996596i $$0.526271\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 0 0
$$507$$ −10665.0 −0.934219
$$508$$ 0 0
$$509$$ 5870.00 0.511165 0.255583 0.966787i $$-0.417733\pi$$
0.255583 + 0.966787i $$0.417733\pi$$
$$510$$ 0 0
$$511$$ 3170.00 0.274428
$$512$$ 0 0
$$513$$ 15225.0 1.31033
$$514$$ 0 0
$$515$$ 0 0
$$516$$ 0 0
$$517$$ 7200.00 0.612487
$$518$$ 0 0
$$519$$ −12560.0 −1.06228
$$520$$ 0 0
$$521$$ −12187.0 −1.02480 −0.512401 0.858746i $$-0.671244\pi$$
−0.512401 + 0.858746i $$0.671244\pi$$
$$522$$ 0 0
$$523$$ −5315.00 −0.444376 −0.222188 0.975004i $$-0.571320\pi$$
−0.222188 + 0.975004i $$0.571320\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 4830.00 0.399237
$$528$$ 0 0
$$529$$ −12067.0 −0.991781
$$530$$ 0 0
$$531$$ 1120.00 0.0915327
$$532$$ 0 0
$$533$$ 1560.00 0.126775
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 0 0
$$537$$ −825.000 −0.0662968
$$538$$ 0 0
$$539$$ −3645.00 −0.291282
$$540$$ 0 0
$$541$$ 1672.00 0.132874 0.0664371 0.997791i $$-0.478837\pi$$
0.0664371 + 0.997791i $$0.478837\pi$$
$$542$$ 0 0
$$543$$ 15790.0 1.24791
$$544$$ 0 0
$$545$$ 0 0
$$546$$ 0 0
$$547$$ −10595.0 −0.828171 −0.414085 0.910238i $$-0.635899\pi$$
−0.414085 + 0.910238i $$0.635899\pi$$
$$548$$ 0 0
$$549$$ −1460.00 −0.113500
$$550$$ 0 0
$$551$$ −2100.00 −0.162365
$$552$$ 0 0
$$553$$ −8300.00 −0.638249
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ 15264.0 1.16114 0.580571 0.814209i $$-0.302829\pi$$
0.580571 + 0.814209i $$0.302829\pi$$
$$558$$ 0 0
$$559$$ 2400.00 0.181591
$$560$$ 0 0
$$561$$ −1575.00 −0.118532
$$562$$ 0 0
$$563$$ −15400.0 −1.15281 −0.576406 0.817164i $$-0.695545\pi$$
−0.576406 + 0.817164i $$0.695545\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 0 0
$$567$$ −6710.00 −0.496990
$$568$$ 0 0
$$569$$ 14569.0 1.07340 0.536700 0.843773i $$-0.319671\pi$$
0.536700 + 0.843773i $$0.319671\pi$$
$$570$$ 0 0
$$571$$ 7780.00 0.570198 0.285099 0.958498i $$-0.407974\pi$$
0.285099 + 0.958498i $$0.407974\pi$$
$$572$$ 0 0
$$573$$ −16450.0 −1.19932
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 0 0
$$577$$ 19829.0 1.43066 0.715331 0.698786i $$-0.246276\pi$$
0.715331 + 0.698786i $$0.246276\pi$$
$$578$$ 0 0
$$579$$ −985.000 −0.0706998
$$580$$ 0 0
$$581$$ 750.000 0.0535546
$$582$$ 0 0
$$583$$ −4830.00 −0.343119
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ 795.000 0.0558998 0.0279499 0.999609i $$-0.491102\pi$$
0.0279499 + 0.999609i $$0.491102\pi$$
$$588$$ 0 0
$$589$$ 24150.0 1.68945
$$590$$ 0 0
$$591$$ −8730.00 −0.607621
$$592$$ 0 0
$$593$$ 21457.0 1.48589 0.742946 0.669352i $$-0.233428\pi$$
0.742946 + 0.669352i $$0.233428\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ 23300.0 1.59733
$$598$$ 0 0
$$599$$ 26570.0 1.81239 0.906194 0.422862i $$-0.138974\pi$$
0.906194 + 0.422862i $$0.138974\pi$$
$$600$$ 0 0
$$601$$ −25245.0 −1.71342 −0.856710 0.515799i $$-0.827495\pi$$
−0.856710 + 0.515799i $$0.827495\pi$$
$$602$$ 0 0
$$603$$ −510.000 −0.0344425
$$604$$ 0 0
$$605$$ 0 0
$$606$$ 0 0
$$607$$ 14040.0 0.938824 0.469412 0.882979i $$-0.344466\pi$$
0.469412 + 0.882979i $$0.344466\pi$$
$$608$$ 0 0
$$609$$ 1000.00 0.0665387
$$610$$ 0 0
$$611$$ −3840.00 −0.254255
$$612$$ 0 0
$$613$$ 19502.0 1.28496 0.642478 0.766304i $$-0.277906\pi$$
0.642478 + 0.766304i $$0.277906\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ −6286.00 −0.410154 −0.205077 0.978746i $$-0.565744\pi$$
−0.205077 + 0.978746i $$0.565744\pi$$
$$618$$ 0 0
$$619$$ −2420.00 −0.157137 −0.0785687 0.996909i $$-0.525035\pi$$
−0.0785687 + 0.996909i $$0.525035\pi$$
$$620$$ 0 0
$$621$$ −1450.00 −0.0936981
$$622$$ 0 0
$$623$$ −7050.00 −0.453374
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 0 0
$$627$$ −7875.00 −0.501590
$$628$$ 0 0
$$629$$ −1134.00 −0.0718848
$$630$$ 0 0
$$631$$ 2290.00 0.144475 0.0722373 0.997387i $$-0.476986\pi$$
0.0722373 + 0.997387i $$0.476986\pi$$
$$632$$ 0 0
$$633$$ 1325.00 0.0831975
$$634$$ 0 0
$$635$$ 0 0
$$636$$ 0 0
$$637$$ 1944.00 0.120917
$$638$$ 0 0
$$639$$ 80.0000 0.00495266
$$640$$ 0 0
$$641$$ −15150.0 −0.933524 −0.466762 0.884383i $$-0.654580\pi$$
−0.466762 + 0.884383i $$0.654580\pi$$
$$642$$ 0 0
$$643$$ 7860.00 0.482066 0.241033 0.970517i $$-0.422514\pi$$
0.241033 + 0.970517i $$0.422514\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 2420.00 0.147048 0.0735240 0.997293i $$-0.476575\pi$$
0.0735240 + 0.997293i $$0.476575\pi$$
$$648$$ 0 0
$$649$$ −8400.00 −0.508057
$$650$$ 0 0
$$651$$ −11500.0 −0.692351
$$652$$ 0 0
$$653$$ −20462.0 −1.22625 −0.613124 0.789987i $$-0.710087\pi$$
−0.613124 + 0.789987i $$0.710087\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 0 0
$$657$$ −634.000 −0.0376479
$$658$$ 0 0
$$659$$ 32205.0 1.90369 0.951843 0.306587i $$-0.0991870\pi$$
0.951843 + 0.306587i $$0.0991870\pi$$
$$660$$ 0 0
$$661$$ 27100.0 1.59466 0.797328 0.603546i $$-0.206246\pi$$
0.797328 + 0.603546i $$0.206246\pi$$
$$662$$ 0 0
$$663$$ 840.000 0.0492050
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 200.000 0.0116102
$$668$$ 0 0
$$669$$ 5300.00 0.306293
$$670$$ 0 0
$$671$$ 10950.0 0.629985
$$672$$ 0 0
$$673$$ 24182.0 1.38506 0.692532 0.721387i $$-0.256495\pi$$
0.692532 + 0.721387i $$0.256495\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ 25704.0 1.45921 0.729605 0.683869i $$-0.239704\pi$$
0.729605 + 0.683869i $$0.239704\pi$$
$$678$$ 0 0
$$679$$ −14340.0 −0.810484
$$680$$ 0 0
$$681$$ −23300.0 −1.31110
$$682$$ 0 0
$$683$$ 6525.00 0.365552 0.182776 0.983155i $$-0.441492\pi$$
0.182776 + 0.983155i $$0.441492\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 0 0
$$687$$ −8300.00 −0.460939
$$688$$ 0 0
$$689$$ 2576.00 0.142435
$$690$$ 0 0
$$691$$ 28955.0 1.59407 0.797033 0.603935i $$-0.206401\pi$$
0.797033 + 0.603935i $$0.206401\pi$$
$$692$$ 0 0
$$693$$ −300.000 −0.0164445
$$694$$ 0 0
$$695$$ 0 0
$$696$$ 0 0
$$697$$ 4095.00 0.222538
$$698$$ 0 0
$$699$$ −17310.0 −0.936659
$$700$$ 0 0
$$701$$ −9720.00 −0.523708 −0.261854 0.965107i $$-0.584334\pi$$
−0.261854 + 0.965107i $$0.584334\pi$$
$$702$$ 0 0
$$703$$ −5670.00 −0.304194
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ 19020.0 1.01177
$$708$$ 0 0
$$709$$ 25140.0 1.33167 0.665834 0.746100i $$-0.268076\pi$$
0.665834 + 0.746100i $$0.268076\pi$$
$$710$$ 0 0
$$711$$ 1660.00 0.0875596
$$712$$ 0 0
$$713$$ −2300.00 −0.120807
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ 20100.0 1.04693
$$718$$ 0 0
$$719$$ 30030.0 1.55762 0.778811 0.627259i $$-0.215823\pi$$
0.778811 + 0.627259i $$0.215823\pi$$
$$720$$ 0 0
$$721$$ −14800.0 −0.764467
$$722$$ 0 0
$$723$$ 19925.0 1.02492
$$724$$ 0 0
$$725$$ 0 0
$$726$$ 0 0
$$727$$ −36100.0 −1.84164 −0.920822 0.389984i $$-0.872481\pi$$
−0.920822 + 0.389984i $$0.872481\pi$$
$$728$$ 0 0
$$729$$ 20917.0 1.06269
$$730$$ 0 0
$$731$$ 6300.00 0.318760
$$732$$ 0 0
$$733$$ 5368.00 0.270493 0.135247 0.990812i $$-0.456817\pi$$
0.135247 + 0.990812i $$0.456817\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 3825.00 0.191175
$$738$$ 0 0
$$739$$ 25540.0 1.27132 0.635659 0.771970i $$-0.280728\pi$$
0.635659 + 0.771970i $$0.280728\pi$$
$$740$$ 0 0
$$741$$ 4200.00 0.208220
$$742$$ 0 0
$$743$$ 18730.0 0.924814 0.462407 0.886668i $$-0.346986\pi$$
0.462407 + 0.886668i $$0.346986\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 0 0
$$747$$ −150.000 −0.00734701
$$748$$ 0 0
$$749$$ 19450.0 0.948849
$$750$$ 0 0
$$751$$ 29940.0 1.45476 0.727381 0.686234i $$-0.240737\pi$$
0.727381 + 0.686234i $$0.240737\pi$$
$$752$$ 0 0
$$753$$ 33125.0 1.60311
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 0 0
$$757$$ 16216.0 0.778574 0.389287 0.921117i $$-0.372721\pi$$
0.389287 + 0.921117i $$0.372721\pi$$
$$758$$ 0 0
$$759$$ 750.000 0.0358673
$$760$$ 0 0
$$761$$ −25843.0 −1.23102 −0.615511 0.788128i $$-0.711050\pi$$
−0.615511 + 0.788128i $$0.711050\pi$$
$$762$$ 0 0
$$763$$ 2460.00 0.116721
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ 4480.00 0.210904
$$768$$ 0 0
$$769$$ −27475.0 −1.28839 −0.644196 0.764860i $$-0.722808\pi$$
−0.644196 + 0.764860i $$0.722808\pi$$
$$770$$ 0 0
$$771$$ −11230.0 −0.524564
$$772$$ 0 0
$$773$$ −3948.00 −0.183699 −0.0918497 0.995773i $$-0.529278\pi$$
−0.0918497 + 0.995773i $$0.529278\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 0 0
$$777$$ 2700.00 0.124661
$$778$$ 0 0
$$779$$ 20475.0 0.941711
$$780$$ 0 0
$$781$$ −600.000 −0.0274900
$$782$$ 0 0
$$783$$ −2900.00 −0.132360
$$784$$ 0 0
$$785$$ 0 0
$$786$$ 0 0
$$787$$ 22100.0 1.00099 0.500496 0.865739i $$-0.333151\pi$$
0.500496 + 0.865739i $$0.333151\pi$$
$$788$$ 0 0
$$789$$ −19750.0 −0.891152
$$790$$ 0 0
$$791$$ 7530.00 0.338478
$$792$$ 0 0
$$793$$ −5840.00 −0.261519
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ 16026.0 0.712259 0.356129 0.934437i $$-0.384096\pi$$
0.356129 + 0.934437i $$0.384096\pi$$
$$798$$ 0 0
$$799$$ −10080.0 −0.446314
$$800$$ 0 0
$$801$$ 1410.00 0.0621971
$$802$$ 0 0
$$803$$ 4755.00 0.208967
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ −13280.0 −0.579279
$$808$$ 0 0
$$809$$ 35770.0 1.55452 0.777260 0.629180i $$-0.216609\pi$$
0.777260 + 0.629180i $$0.216609\pi$$
$$810$$ 0 0
$$811$$ −30620.0 −1.32579 −0.662894 0.748714i $$-0.730672\pi$$
−0.662894 + 0.748714i $$0.730672\pi$$
$$812$$ 0 0
$$813$$ −15550.0 −0.670802
$$814$$ 0 0
$$815$$ 0 0
$$816$$ 0 0
$$817$$ 31500.0 1.34889
$$818$$ 0 0
$$819$$ 160.000 0.00682644
$$820$$ 0 0
$$821$$ 9150.00 0.388961 0.194481 0.980906i $$-0.437698\pi$$
0.194481 + 0.980906i $$0.437698\pi$$
$$822$$ 0 0
$$823$$ 28940.0 1.22574 0.612871 0.790183i $$-0.290015\pi$$
0.612871 + 0.790183i $$0.290015\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ −25765.0 −1.08336 −0.541679 0.840586i $$-0.682211\pi$$
−0.541679 + 0.840586i $$0.682211\pi$$
$$828$$ 0 0
$$829$$ 41584.0 1.74219 0.871093 0.491118i $$-0.163412\pi$$
0.871093 + 0.491118i $$0.163412\pi$$
$$830$$ 0 0
$$831$$ −34420.0 −1.43684
$$832$$ 0 0
$$833$$ 5103.00 0.212255
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 0 0
$$837$$ 33350.0 1.37723
$$838$$ 0 0
$$839$$ −20320.0 −0.836143 −0.418072 0.908414i $$-0.637294\pi$$
−0.418072 + 0.908414i $$0.637294\pi$$
$$840$$ 0 0
$$841$$ −23989.0 −0.983599
$$842$$ 0 0
$$843$$ 23150.0 0.945822
$$844$$ 0 0
$$845$$ 0 0
$$846$$ 0 0
$$847$$ −11060.0 −0.448673
$$848$$ 0 0
$$849$$ 1075.00 0.0434557
$$850$$ 0 0
$$851$$ 540.000 0.0217520
$$852$$ 0 0
$$853$$ 38882.0 1.56072 0.780360 0.625330i $$-0.215036\pi$$
0.780360 + 0.625330i $$0.215036\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ −15269.0 −0.608610 −0.304305 0.952575i $$-0.598424\pi$$
−0.304305 + 0.952575i $$0.598424\pi$$
$$858$$ 0 0
$$859$$ 19375.0 0.769577 0.384788 0.923005i $$-0.374274\pi$$
0.384788 + 0.923005i $$0.374274\pi$$
$$860$$ 0 0
$$861$$ −9750.00 −0.385922
$$862$$ 0 0
$$863$$ −22900.0 −0.903274 −0.451637 0.892202i $$-0.649160\pi$$
−0.451637 + 0.892202i $$0.649160\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 0 0
$$867$$ −22360.0 −0.875877
$$868$$ 0 0
$$869$$ −12450.0 −0.486004
$$870$$ 0 0
$$871$$ −2040.00 −0.0793602
$$872$$ 0 0
$$873$$ 2868.00 0.111188
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ −23444.0 −0.902677 −0.451338 0.892353i $$-0.649053\pi$$
−0.451338 + 0.892353i $$0.649053\pi$$
$$878$$ 0 0
$$879$$ 8010.00 0.307361
$$880$$ 0 0
$$881$$ 3750.00 0.143406 0.0717030 0.997426i $$-0.477157\pi$$
0.0717030 + 0.997426i $$0.477157\pi$$
$$882$$ 0 0
$$883$$ −37595.0 −1.43281 −0.716406 0.697684i $$-0.754214\pi$$
−0.716406 + 0.697684i $$0.754214\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ −11420.0 −0.432295 −0.216148 0.976361i $$-0.569349\pi$$
−0.216148 + 0.976361i $$0.569349\pi$$
$$888$$ 0 0
$$889$$ −14900.0 −0.562126
$$890$$ 0 0
$$891$$ −10065.0 −0.378440
$$892$$ 0 0
$$893$$ −50400.0 −1.88866
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 0 0
$$897$$ −400.000 −0.0148892
$$898$$ 0 0
$$899$$ −4600.00 −0.170655
$$900$$ 0 0
$$901$$ 6762.00 0.250028
$$902$$ 0 0
$$903$$ −15000.0 −0.552789
$$904$$ 0 0
$$905$$ 0 0
$$906$$ 0 0
$$907$$ 2620.00 0.0959158 0.0479579 0.998849i $$-0.484729\pi$$
0.0479579 + 0.998849i $$0.484729\pi$$
$$908$$ 0 0
$$909$$ −3804.00 −0.138802
$$910$$ 0 0
$$911$$ 46500.0 1.69112 0.845562 0.533877i $$-0.179266\pi$$
0.845562 + 0.533877i $$0.179266\pi$$
$$912$$ 0 0
$$913$$ 1125.00 0.0407799
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ −7800.00 −0.280893
$$918$$ 0 0
$$919$$ 45190.0 1.62207 0.811034 0.584999i $$-0.198905\pi$$
0.811034 + 0.584999i $$0.198905\pi$$
$$920$$ 0 0
$$921$$ 9425.00 0.337203
$$922$$ 0 0
$$923$$ 320.000 0.0114116
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 0 0
$$927$$ 2960.00 0.104875
$$928$$ 0 0
$$929$$ 15166.0 0.535609 0.267804 0.963473i $$-0.413702\pi$$
0.267804 + 0.963473i $$0.413702\pi$$
$$930$$ 0 0
$$931$$ 25515.0 0.898196
$$932$$ 0 0
$$933$$ 46250.0 1.62289
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ 23789.0 0.829405 0.414703 0.909957i $$-0.363886\pi$$
0.414703 + 0.909957i $$0.363886\pi$$
$$938$$ 0 0
$$939$$ −40810.0 −1.41830
$$940$$ 0 0
$$941$$ −47472.0 −1.64457 −0.822286 0.569074i $$-0.807302\pi$$
−0.822286 + 0.569074i $$0.807302\pi$$
$$942$$ 0 0
$$943$$ −1950.00 −0.0673391
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ −19760.0 −0.678050 −0.339025 0.940777i $$-0.610097\pi$$
−0.339025 + 0.940777i $$0.610097\pi$$
$$948$$ 0 0
$$949$$ −2536.00 −0.0867461
$$950$$ 0 0
$$951$$ −34620.0 −1.18047
$$952$$ 0 0
$$953$$ −36337.0 −1.23512 −0.617561 0.786523i $$-0.711879\pi$$
−0.617561 + 0.786523i $$0.711879\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 0 0
$$957$$ 1500.00 0.0506668
$$958$$ 0 0
$$959$$ −27490.0 −0.925650
$$960$$ 0 0
$$961$$ 23109.0 0.775704
$$962$$ 0 0
$$963$$ −3890.00 −0.130170
$$964$$ 0 0
$$965$$ 0 0
$$966$$ 0 0
$$967$$ 33880.0 1.12669 0.563344 0.826222i $$-0.309515\pi$$
0.563344 + 0.826222i $$0.309515\pi$$
$$968$$ 0 0
$$969$$ 11025.0 0.365505
$$970$$ 0 0
$$971$$ 17175.0 0.567633 0.283817 0.958879i $$-0.408399\pi$$
0.283817 + 0.958879i $$0.408399\pi$$
$$972$$ 0 0
$$973$$ 7350.00 0.242169
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ 13311.0 0.435882 0.217941 0.975962i $$-0.430066\pi$$
0.217941 + 0.975962i $$0.430066\pi$$
$$978$$ 0 0
$$979$$ −10575.0 −0.345228
$$980$$ 0 0
$$981$$ −492.000 −0.0160126
$$982$$ 0 0
$$983$$ 53110.0 1.72324 0.861621 0.507553i $$-0.169450\pi$$
0.861621 + 0.507553i $$0.169450\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 0 0
$$987$$ 24000.0 0.773990
$$988$$ 0 0
$$989$$ −3000.00 −0.0964555
$$990$$ 0 0
$$991$$ −8990.00 −0.288170 −0.144085 0.989565i $$-0.546024\pi$$
−0.144085 + 0.989565i $$0.546024\pi$$
$$992$$ 0 0
$$993$$ 40375.0 1.29029
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 0 0
$$997$$ 1236.00 0.0392623 0.0196311 0.999807i $$-0.493751\pi$$
0.0196311 + 0.999807i $$0.493751\pi$$
$$998$$ 0 0
$$999$$ −7830.00 −0.247978
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.bq.1.1 1
4.3 odd 2 1600.4.a.k.1.1 1
5.4 even 2 1600.4.a.l.1.1 1
8.3 odd 2 800.4.a.j.1.1 yes 1
8.5 even 2 800.4.a.b.1.1 1
20.19 odd 2 1600.4.a.bp.1.1 1
40.3 even 4 800.4.c.d.449.2 2
40.13 odd 4 800.4.c.c.449.1 2
40.19 odd 2 800.4.a.c.1.1 yes 1
40.27 even 4 800.4.c.d.449.1 2
40.29 even 2 800.4.a.i.1.1 yes 1
40.37 odd 4 800.4.c.c.449.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
800.4.a.b.1.1 1 8.5 even 2
800.4.a.c.1.1 yes 1 40.19 odd 2
800.4.a.i.1.1 yes 1 40.29 even 2
800.4.a.j.1.1 yes 1 8.3 odd 2
800.4.c.c.449.1 2 40.13 odd 4
800.4.c.c.449.2 2 40.37 odd 4
800.4.c.d.449.1 2 40.27 even 4
800.4.c.d.449.2 2 40.3 even 4
1600.4.a.k.1.1 1 4.3 odd 2
1600.4.a.l.1.1 1 5.4 even 2
1600.4.a.bp.1.1 1 20.19 odd 2
1600.4.a.bq.1.1 1 1.1 even 1 trivial