# Properties

 Label 1600.4.a.bt.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 25) 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)

 $$f(q)$$ $$=$$ $$q+7.00000 q^{3} -6.00000 q^{7} +22.0000 q^{9} +O(q^{10})$$ $$q+7.00000 q^{3} -6.00000 q^{7} +22.0000 q^{9} +43.0000 q^{11} -28.0000 q^{13} -91.0000 q^{17} +35.0000 q^{19} -42.0000 q^{21} -162.000 q^{23} -35.0000 q^{27} -160.000 q^{29} +42.0000 q^{31} +301.000 q^{33} -314.000 q^{37} -196.000 q^{39} -203.000 q^{41} +92.0000 q^{43} -196.000 q^{47} -307.000 q^{49} -637.000 q^{51} +82.0000 q^{53} +245.000 q^{57} +280.000 q^{59} +518.000 q^{61} -132.000 q^{63} +141.000 q^{67} -1134.00 q^{69} +412.000 q^{71} +763.000 q^{73} -258.000 q^{77} +510.000 q^{79} -839.000 q^{81} +777.000 q^{83} -1120.00 q^{87} -945.000 q^{89} +168.000 q^{91} +294.000 q^{93} -1246.00 q^{97} +946.000 q^{99} +O(q^{100})$$

## Coefficient data

For each $$n$$ we display the coefficients of the $$q$$-expansion $$a_n$$, the Satake parameters $$\alpha_p$$, and the Satake angles $$\theta_p = \textrm{Arg}(\alpha_p)$$.

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0 0
$$3$$ 7.00000 1.34715 0.673575 0.739119i $$-0.264758\pi$$
0.673575 + 0.739119i $$0.264758\pi$$
$$4$$ 0 0
$$5$$ 0 0
$$6$$ 0 0
$$7$$ −6.00000 −0.323970 −0.161985 0.986793i $$-0.551790\pi$$
−0.161985 + 0.986793i $$0.551790\pi$$
$$8$$ 0 0
$$9$$ 22.0000 0.814815
$$10$$ 0 0
$$11$$ 43.0000 1.17864 0.589318 0.807901i $$-0.299397\pi$$
0.589318 + 0.807901i $$0.299397\pi$$
$$12$$ 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
$$17$$ −91.0000 −1.29828 −0.649139 0.760669i $$-0.724871\pi$$
−0.649139 + 0.760669i $$0.724871\pi$$
$$18$$ 0 0
$$19$$ 35.0000 0.422608 0.211304 0.977420i $$-0.432229\pi$$
0.211304 + 0.977420i $$0.432229\pi$$
$$20$$ 0 0
$$21$$ −42.0000 −0.436436
$$22$$ 0 0
$$23$$ −162.000 −1.46867 −0.734333 0.678789i $$-0.762505\pi$$
−0.734333 + 0.678789i $$0.762505\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ 0 0
$$27$$ −35.0000 −0.249472
$$28$$ 0 0
$$29$$ −160.000 −1.02453 −0.512263 0.858829i $$-0.671193\pi$$
−0.512263 + 0.858829i $$0.671193\pi$$
$$30$$ 0 0
$$31$$ 42.0000 0.243336 0.121668 0.992571i $$-0.461176\pi$$
0.121668 + 0.992571i $$0.461176\pi$$
$$32$$ 0 0
$$33$$ 301.000 1.58780
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ −314.000 −1.39517 −0.697585 0.716502i $$-0.745742\pi$$
−0.697585 + 0.716502i $$0.745742\pi$$
$$38$$ 0 0
$$39$$ −196.000 −0.804747
$$40$$ 0 0
$$41$$ −203.000 −0.773251 −0.386625 0.922237i $$-0.626359\pi$$
−0.386625 + 0.922237i $$0.626359\pi$$
$$42$$ 0 0
$$43$$ 92.0000 0.326276 0.163138 0.986603i $$-0.447838\pi$$
0.163138 + 0.986603i $$0.447838\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ −196.000 −0.608288 −0.304144 0.952626i $$-0.598370\pi$$
−0.304144 + 0.952626i $$0.598370\pi$$
$$48$$ 0 0
$$49$$ −307.000 −0.895044
$$50$$ 0 0
$$51$$ −637.000 −1.74898
$$52$$ 0 0
$$53$$ 82.0000 0.212520 0.106260 0.994338i $$-0.466112\pi$$
0.106260 + 0.994338i $$0.466112\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ 245.000 0.569317
$$58$$ 0 0
$$59$$ 280.000 0.617846 0.308923 0.951087i $$-0.400032\pi$$
0.308923 + 0.951087i $$0.400032\pi$$
$$60$$ 0 0
$$61$$ 518.000 1.08726 0.543632 0.839324i $$-0.317049\pi$$
0.543632 + 0.839324i $$0.317049\pi$$
$$62$$ 0 0
$$63$$ −132.000 −0.263975
$$64$$ 0 0
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 141.000 0.257103 0.128551 0.991703i $$-0.458967\pi$$
0.128551 + 0.991703i $$0.458967\pi$$
$$68$$ 0 0
$$69$$ −1134.00 −1.97852
$$70$$ 0 0
$$71$$ 412.000 0.688668 0.344334 0.938847i $$-0.388105\pi$$
0.344334 + 0.938847i $$0.388105\pi$$
$$72$$ 0 0
$$73$$ 763.000 1.22332 0.611660 0.791121i $$-0.290502\pi$$
0.611660 + 0.791121i $$0.290502\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ −258.000 −0.381842
$$78$$ 0 0
$$79$$ 510.000 0.726323 0.363161 0.931726i $$-0.381697\pi$$
0.363161 + 0.931726i $$0.381697\pi$$
$$80$$ 0 0
$$81$$ −839.000 −1.15089
$$82$$ 0 0
$$83$$ 777.000 1.02755 0.513776 0.857924i $$-0.328246\pi$$
0.513776 + 0.857924i $$0.328246\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 0 0
$$87$$ −1120.00 −1.38019
$$88$$ 0 0
$$89$$ −945.000 −1.12550 −0.562752 0.826626i $$-0.690257\pi$$
−0.562752 + 0.826626i $$0.690257\pi$$
$$90$$ 0 0
$$91$$ 168.000 0.193530
$$92$$ 0 0
$$93$$ 294.000 0.327811
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 0 0
$$97$$ −1246.00 −1.30425 −0.652124 0.758112i $$-0.726122\pi$$
−0.652124 + 0.758112i $$0.726122\pi$$
$$98$$ 0 0
$$99$$ 946.000 0.960369
$$100$$ 0 0
$$101$$ −1302.00 −1.28271 −0.641356 0.767244i $$-0.721628\pi$$
−0.641356 + 0.767244i $$0.721628\pi$$
$$102$$ 0 0
$$103$$ −532.000 −0.508927 −0.254464 0.967082i $$-0.581899\pi$$
−0.254464 + 0.967082i $$0.581899\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ −1269.00 −1.14653 −0.573266 0.819370i $$-0.694324\pi$$
−0.573266 + 0.819370i $$0.694324\pi$$
$$108$$ 0 0
$$109$$ −1070.00 −0.940251 −0.470126 0.882599i $$-0.655791\pi$$
−0.470126 + 0.882599i $$0.655791\pi$$
$$110$$ 0 0
$$111$$ −2198.00 −1.87950
$$112$$ 0 0
$$113$$ 503.000 0.418746 0.209373 0.977836i $$-0.432858\pi$$
0.209373 + 0.977836i $$0.432858\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 0 0
$$117$$ −616.000 −0.486745
$$118$$ 0 0
$$119$$ 546.000 0.420603
$$120$$ 0 0
$$121$$ 518.000 0.389181
$$122$$ 0 0
$$123$$ −1421.00 −1.04169
$$124$$ 0 0
$$125$$ 0 0
$$126$$ 0 0
$$127$$ 874.000 0.610669 0.305334 0.952245i $$-0.401232\pi$$
0.305334 + 0.952245i $$0.401232\pi$$
$$128$$ 0 0
$$129$$ 644.000 0.439543
$$130$$ 0 0
$$131$$ −1092.00 −0.728309 −0.364155 0.931339i $$-0.618642\pi$$
−0.364155 + 0.931339i $$0.618642\pi$$
$$132$$ 0 0
$$133$$ −210.000 −0.136912
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ −411.000 −0.256307 −0.128154 0.991754i $$-0.540905\pi$$
−0.128154 + 0.991754i $$0.540905\pi$$
$$138$$ 0 0
$$139$$ 595.000 0.363074 0.181537 0.983384i $$-0.441893\pi$$
0.181537 + 0.983384i $$0.441893\pi$$
$$140$$ 0 0
$$141$$ −1372.00 −0.819456
$$142$$ 0 0
$$143$$ −1204.00 −0.704081
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 0 0
$$147$$ −2149.00 −1.20576
$$148$$ 0 0
$$149$$ 3200.00 1.75942 0.879712 0.475507i $$-0.157735\pi$$
0.879712 + 0.475507i $$0.157735\pi$$
$$150$$ 0 0
$$151$$ 202.000 0.108864 0.0544322 0.998517i $$-0.482665\pi$$
0.0544322 + 0.998517i $$0.482665\pi$$
$$152$$ 0 0
$$153$$ −2002.00 −1.05786
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ 406.000 0.206384 0.103192 0.994661i $$-0.467094\pi$$
0.103192 + 0.994661i $$0.467094\pi$$
$$158$$ 0 0
$$159$$ 574.000 0.286297
$$160$$ 0 0
$$161$$ 972.000 0.475803
$$162$$ 0 0
$$163$$ −3803.00 −1.82745 −0.913724 0.406336i $$-0.866806\pi$$
−0.913724 + 0.406336i $$0.866806\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ −4116.00 −1.90722 −0.953610 0.301046i $$-0.902664\pi$$
−0.953610 + 0.301046i $$0.902664\pi$$
$$168$$ 0 0
$$169$$ −1413.00 −0.643150
$$170$$ 0 0
$$171$$ 770.000 0.344347
$$172$$ 0 0
$$173$$ 1512.00 0.664481 0.332241 0.943195i $$-0.392195\pi$$
0.332241 + 0.943195i $$0.392195\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ 1960.00 0.832331
$$178$$ 0 0
$$179$$ −2585.00 −1.07940 −0.539698 0.841859i $$-0.681462\pi$$
−0.539698 + 0.841859i $$0.681462\pi$$
$$180$$ 0 0
$$181$$ 2758.00 1.13260 0.566300 0.824199i $$-0.308374\pi$$
0.566300 + 0.824199i $$0.308374\pi$$
$$182$$ 0 0
$$183$$ 3626.00 1.46471
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 0 0
$$187$$ −3913.00 −1.53020
$$188$$ 0 0
$$189$$ 210.000 0.0808214
$$190$$ 0 0
$$191$$ −2378.00 −0.900869 −0.450435 0.892809i $$-0.648731\pi$$
−0.450435 + 0.892809i $$0.648731\pi$$
$$192$$ 0 0
$$193$$ −3067.00 −1.14387 −0.571937 0.820298i $$-0.693808\pi$$
−0.571937 + 0.820298i $$0.693808\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 2346.00 0.848455 0.424227 0.905556i $$-0.360546\pi$$
0.424227 + 0.905556i $$0.360546\pi$$
$$198$$ 0 0
$$199$$ 4900.00 1.74549 0.872743 0.488180i $$-0.162339\pi$$
0.872743 + 0.488180i $$0.162339\pi$$
$$200$$ 0 0
$$201$$ 987.000 0.346356
$$202$$ 0 0
$$203$$ 960.000 0.331915
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 0 0
$$207$$ −3564.00 −1.19669
$$208$$ 0 0
$$209$$ 1505.00 0.498101
$$210$$ 0 0
$$211$$ −4307.00 −1.40524 −0.702621 0.711564i $$-0.747987\pi$$
−0.702621 + 0.711564i $$0.747987\pi$$
$$212$$ 0 0
$$213$$ 2884.00 0.927739
$$214$$ 0 0
$$215$$ 0 0
$$216$$ 0 0
$$217$$ −252.000 −0.0788335
$$218$$ 0 0
$$219$$ 5341.00 1.64800
$$220$$ 0 0
$$221$$ 2548.00 0.775552
$$222$$ 0 0
$$223$$ −2212.00 −0.664244 −0.332122 0.943236i $$-0.607765\pi$$
−0.332122 + 0.943236i $$0.607765\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ 476.000 0.139177 0.0695886 0.997576i $$-0.477831\pi$$
0.0695886 + 0.997576i $$0.477831\pi$$
$$228$$ 0 0
$$229$$ 2940.00 0.848387 0.424194 0.905572i $$-0.360558\pi$$
0.424194 + 0.905572i $$0.360558\pi$$
$$230$$ 0 0
$$231$$ −1806.00 −0.514399
$$232$$ 0 0
$$233$$ −1002.00 −0.281730 −0.140865 0.990029i $$-0.544988\pi$$
−0.140865 + 0.990029i $$0.544988\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 0 0
$$237$$ 3570.00 0.978466
$$238$$ 0 0
$$239$$ 2480.00 0.671204 0.335602 0.942004i $$-0.391060\pi$$
0.335602 + 0.942004i $$0.391060\pi$$
$$240$$ 0 0
$$241$$ 1897.00 0.507039 0.253520 0.967330i $$-0.418412\pi$$
0.253520 + 0.967330i $$0.418412\pi$$
$$242$$ 0 0
$$243$$ −4928.00 −1.30095
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ −980.000 −0.252453
$$248$$ 0 0
$$249$$ 5439.00 1.38427
$$250$$ 0 0
$$251$$ 2373.00 0.596743 0.298371 0.954450i $$-0.403557\pi$$
0.298371 + 0.954450i $$0.403557\pi$$
$$252$$ 0 0
$$253$$ −6966.00 −1.73102
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ 4494.00 1.09077 0.545385 0.838185i $$-0.316383\pi$$
0.545385 + 0.838185i $$0.316383\pi$$
$$258$$ 0 0
$$259$$ 1884.00 0.451993
$$260$$ 0 0
$$261$$ −3520.00 −0.834799
$$262$$ 0 0
$$263$$ −722.000 −0.169279 −0.0846396 0.996412i $$-0.526974\pi$$
−0.0846396 + 0.996412i $$0.526974\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 0 0
$$267$$ −6615.00 −1.51622
$$268$$ 0 0
$$269$$ 6160.00 1.39621 0.698107 0.715993i $$-0.254026\pi$$
0.698107 + 0.715993i $$0.254026\pi$$
$$270$$ 0 0
$$271$$ −7238.00 −1.62243 −0.811213 0.584751i $$-0.801192\pi$$
−0.811213 + 0.584751i $$0.801192\pi$$
$$272$$ 0 0
$$273$$ 1176.00 0.260713
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 0 0
$$277$$ 1776.00 0.385233 0.192616 0.981274i $$-0.438303\pi$$
0.192616 + 0.981274i $$0.438303\pi$$
$$278$$ 0 0
$$279$$ 924.000 0.198274
$$280$$ 0 0
$$281$$ 4542.00 0.964246 0.482123 0.876104i $$-0.339866\pi$$
0.482123 + 0.876104i $$0.339866\pi$$
$$282$$ 0 0
$$283$$ 7077.00 1.48652 0.743258 0.669005i $$-0.233280\pi$$
0.743258 + 0.669005i $$0.233280\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 1218.00 0.250510
$$288$$ 0 0
$$289$$ 3368.00 0.685528
$$290$$ 0 0
$$291$$ −8722.00 −1.75702
$$292$$ 0 0
$$293$$ −4158.00 −0.829054 −0.414527 0.910037i $$-0.636053\pi$$
−0.414527 + 0.910037i $$0.636053\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 0 0
$$297$$ −1505.00 −0.294037
$$298$$ 0 0
$$299$$ 4536.00 0.877337
$$300$$ 0 0
$$301$$ −552.000 −0.105703
$$302$$ 0 0
$$303$$ −9114.00 −1.72801
$$304$$ 0 0
$$305$$ 0 0
$$306$$ 0 0
$$307$$ −2569.00 −0.477591 −0.238796 0.971070i $$-0.576753\pi$$
−0.238796 + 0.971070i $$0.576753\pi$$
$$308$$ 0 0
$$309$$ −3724.00 −0.685602
$$310$$ 0 0
$$311$$ 2982.00 0.543710 0.271855 0.962338i $$-0.412363\pi$$
0.271855 + 0.962338i $$0.412363\pi$$
$$312$$ 0 0
$$313$$ −2422.00 −0.437379 −0.218689 0.975795i $$-0.570178\pi$$
−0.218689 + 0.975795i $$0.570178\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ −9484.00 −1.68036 −0.840181 0.542307i $$-0.817551\pi$$
−0.840181 + 0.542307i $$0.817551\pi$$
$$318$$ 0 0
$$319$$ −6880.00 −1.20754
$$320$$ 0 0
$$321$$ −8883.00 −1.54455
$$322$$ 0 0
$$323$$ −3185.00 −0.548663
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 0 0
$$327$$ −7490.00 −1.26666
$$328$$ 0 0
$$329$$ 1176.00 0.197067
$$330$$ 0 0
$$331$$ 183.000 0.0303885 0.0151942 0.999885i $$-0.495163\pi$$
0.0151942 + 0.999885i $$0.495163\pi$$
$$332$$ 0 0
$$333$$ −6908.00 −1.13681
$$334$$ 0 0
$$335$$ 0 0
$$336$$ 0 0
$$337$$ −2861.00 −0.462459 −0.231229 0.972899i $$-0.574275\pi$$
−0.231229 + 0.972899i $$0.574275\pi$$
$$338$$ 0 0
$$339$$ 3521.00 0.564113
$$340$$ 0 0
$$341$$ 1806.00 0.286805
$$342$$ 0 0
$$343$$ 3900.00 0.613936
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ −629.000 −0.0973098 −0.0486549 0.998816i $$-0.515493\pi$$
−0.0486549 + 0.998816i $$0.515493\pi$$
$$348$$ 0 0
$$349$$ −5950.00 −0.912597 −0.456298 0.889827i $$-0.650825\pi$$
−0.456298 + 0.889827i $$0.650825\pi$$
$$350$$ 0 0
$$351$$ 980.000 0.149027
$$352$$ 0 0
$$353$$ 11718.0 1.76682 0.883408 0.468604i $$-0.155243\pi$$
0.883408 + 0.468604i $$0.155243\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0 0
$$357$$ 3822.00 0.566615
$$358$$ 0 0
$$359$$ 8070.00 1.18640 0.593201 0.805054i $$-0.297864\pi$$
0.593201 + 0.805054i $$0.297864\pi$$
$$360$$ 0 0
$$361$$ −5634.00 −0.821403
$$362$$ 0 0
$$363$$ 3626.00 0.524286
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 0 0
$$367$$ −8316.00 −1.18281 −0.591406 0.806374i $$-0.701427\pi$$
−0.591406 + 0.806374i $$0.701427\pi$$
$$368$$ 0 0
$$369$$ −4466.00 −0.630056
$$370$$ 0 0
$$371$$ −492.000 −0.0688500
$$372$$ 0 0
$$373$$ 12062.0 1.67439 0.837194 0.546906i $$-0.184195\pi$$
0.837194 + 0.546906i $$0.184195\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 4480.00 0.612021
$$378$$ 0 0
$$379$$ −1735.00 −0.235148 −0.117574 0.993064i $$-0.537512\pi$$
−0.117574 + 0.993064i $$0.537512\pi$$
$$380$$ 0 0
$$381$$ 6118.00 0.822663
$$382$$ 0 0
$$383$$ −7602.00 −1.01421 −0.507107 0.861883i $$-0.669285\pi$$
−0.507107 + 0.861883i $$0.669285\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ 2024.00 0.265855
$$388$$ 0 0
$$389$$ −3030.00 −0.394928 −0.197464 0.980310i $$-0.563271\pi$$
−0.197464 + 0.980310i $$0.563271\pi$$
$$390$$ 0 0
$$391$$ 14742.0 1.90674
$$392$$ 0 0
$$393$$ −7644.00 −0.981142
$$394$$ 0 0
$$395$$ 0 0
$$396$$ 0 0
$$397$$ −1204.00 −0.152209 −0.0761046 0.997100i $$-0.524248\pi$$
−0.0761046 + 0.997100i $$0.524248\pi$$
$$398$$ 0 0
$$399$$ −1470.00 −0.184441
$$400$$ 0 0
$$401$$ 1077.00 0.134122 0.0670609 0.997749i $$-0.478638\pi$$
0.0670609 + 0.997749i $$0.478638\pi$$
$$402$$ 0 0
$$403$$ −1176.00 −0.145362
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ −13502.0 −1.64440
$$408$$ 0 0
$$409$$ −3955.00 −0.478147 −0.239074 0.971001i $$-0.576844\pi$$
−0.239074 + 0.971001i $$0.576844\pi$$
$$410$$ 0 0
$$411$$ −2877.00 −0.345285
$$412$$ 0 0
$$413$$ −1680.00 −0.200163
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 0 0
$$417$$ 4165.00 0.489115
$$418$$ 0 0
$$419$$ −6265.00 −0.730466 −0.365233 0.930916i $$-0.619011\pi$$
−0.365233 + 0.930916i $$0.619011\pi$$
$$420$$ 0 0
$$421$$ 3788.00 0.438517 0.219259 0.975667i $$-0.429636\pi$$
0.219259 + 0.975667i $$0.429636\pi$$
$$422$$ 0 0
$$423$$ −4312.00 −0.495642
$$424$$ 0 0
$$425$$ 0 0
$$426$$ 0 0
$$427$$ −3108.00 −0.352240
$$428$$ 0 0
$$429$$ −8428.00 −0.948503
$$430$$ 0 0
$$431$$ −15258.0 −1.70523 −0.852613 0.522544i $$-0.824983\pi$$
−0.852613 + 0.522544i $$0.824983\pi$$
$$432$$ 0 0
$$433$$ 13573.0 1.50641 0.753206 0.657784i $$-0.228506\pi$$
0.753206 + 0.657784i $$0.228506\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ −5670.00 −0.620670
$$438$$ 0 0
$$439$$ −8120.00 −0.882794 −0.441397 0.897312i $$-0.645517\pi$$
−0.441397 + 0.897312i $$0.645517\pi$$
$$440$$ 0 0
$$441$$ −6754.00 −0.729295
$$442$$ 0 0
$$443$$ −6183.00 −0.663122 −0.331561 0.943434i $$-0.607575\pi$$
−0.331561 + 0.943434i $$0.607575\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 0 0
$$447$$ 22400.0 2.37021
$$448$$ 0 0
$$449$$ −1975.00 −0.207586 −0.103793 0.994599i $$-0.533098\pi$$
−0.103793 + 0.994599i $$0.533098\pi$$
$$450$$ 0 0
$$451$$ −8729.00 −0.911380
$$452$$ 0 0
$$453$$ 1414.00 0.146657
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ −11831.0 −1.21101 −0.605504 0.795842i $$-0.707029\pi$$
−0.605504 + 0.795842i $$0.707029\pi$$
$$458$$ 0 0
$$459$$ 3185.00 0.323885
$$460$$ 0 0
$$461$$ −1932.00 −0.195189 −0.0975946 0.995226i $$-0.531115\pi$$
−0.0975946 + 0.995226i $$0.531115\pi$$
$$462$$ 0 0
$$463$$ 9228.00 0.926267 0.463133 0.886289i $$-0.346725\pi$$
0.463133 + 0.886289i $$0.346725\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ 13916.0 1.37892 0.689460 0.724324i $$-0.257848\pi$$
0.689460 + 0.724324i $$0.257848\pi$$
$$468$$ 0 0
$$469$$ −846.000 −0.0832935
$$470$$ 0 0
$$471$$ 2842.00 0.278031
$$472$$ 0 0
$$473$$ 3956.00 0.384560
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 0 0
$$477$$ 1804.00 0.173165
$$478$$ 0 0
$$479$$ 2310.00 0.220348 0.110174 0.993912i $$-0.464859\pi$$
0.110174 + 0.993912i $$0.464859\pi$$
$$480$$ 0 0
$$481$$ 8792.00 0.833432
$$482$$ 0 0
$$483$$ 6804.00 0.640979
$$484$$ 0 0
$$485$$ 0 0
$$486$$ 0 0
$$487$$ 17114.0 1.59242 0.796211 0.605019i $$-0.206835\pi$$
0.796211 + 0.605019i $$0.206835\pi$$
$$488$$ 0 0
$$489$$ −26621.0 −2.46185
$$490$$ 0 0
$$491$$ 17228.0 1.58348 0.791740 0.610858i $$-0.209175\pi$$
0.791740 + 0.610858i $$0.209175\pi$$
$$492$$ 0 0
$$493$$ 14560.0 1.33012
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ −2472.00 −0.223107
$$498$$ 0 0
$$499$$ 12500.0 1.12140 0.560698 0.828020i $$-0.310533\pi$$
0.560698 + 0.828020i $$0.310533\pi$$
$$500$$ 0 0
$$501$$ −28812.0 −2.56931
$$502$$ 0 0
$$503$$ 868.000 0.0769428 0.0384714 0.999260i $$-0.487751\pi$$
0.0384714 + 0.999260i $$0.487751\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 0 0
$$507$$ −9891.00 −0.866420
$$508$$ 0 0
$$509$$ −13370.0 −1.16427 −0.582136 0.813091i $$-0.697783\pi$$
−0.582136 + 0.813091i $$0.697783\pi$$
$$510$$ 0 0
$$511$$ −4578.00 −0.396319
$$512$$ 0 0
$$513$$ −1225.00 −0.105429
$$514$$ 0 0
$$515$$ 0 0
$$516$$ 0 0
$$517$$ −8428.00 −0.716950
$$518$$ 0 0
$$519$$ 10584.0 0.895156
$$520$$ 0 0
$$521$$ 21637.0 1.81945 0.909726 0.415210i $$-0.136292\pi$$
0.909726 + 0.415210i $$0.136292\pi$$
$$522$$ 0 0
$$523$$ 287.000 0.0239955 0.0119977 0.999928i $$-0.496181\pi$$
0.0119977 + 0.999928i $$0.496181\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ −3822.00 −0.315918
$$528$$ 0 0
$$529$$ 14077.0 1.15698
$$530$$ 0 0
$$531$$ 6160.00 0.503430
$$532$$ 0 0
$$533$$ 5684.00 0.461916
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 0 0
$$537$$ −18095.0 −1.45411
$$538$$ 0 0
$$539$$ −13201.0 −1.05493
$$540$$ 0 0
$$541$$ 5328.00 0.423417 0.211709 0.977333i $$-0.432097\pi$$
0.211709 + 0.977333i $$0.432097\pi$$
$$542$$ 0 0
$$543$$ 19306.0 1.52578
$$544$$ 0 0
$$545$$ 0 0
$$546$$ 0 0
$$547$$ 71.0000 0.00554980 0.00277490 0.999996i $$-0.499117\pi$$
0.00277490 + 0.999996i $$0.499117\pi$$
$$548$$ 0 0
$$549$$ 11396.0 0.885919
$$550$$ 0 0
$$551$$ −5600.00 −0.432973
$$552$$ 0 0
$$553$$ −3060.00 −0.235306
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ −18444.0 −1.40305 −0.701524 0.712646i $$-0.747497\pi$$
−0.701524 + 0.712646i $$0.747497\pi$$
$$558$$ 0 0
$$559$$ −2576.00 −0.194907
$$560$$ 0 0
$$561$$ −27391.0 −2.06141
$$562$$ 0 0
$$563$$ 672.000 0.0503045 0.0251522 0.999684i $$-0.491993\pi$$
0.0251522 + 0.999684i $$0.491993\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 0 0
$$567$$ 5034.00 0.372854
$$568$$ 0 0
$$569$$ −10935.0 −0.805657 −0.402829 0.915275i $$-0.631973\pi$$
−0.402829 + 0.915275i $$0.631973\pi$$
$$570$$ 0 0
$$571$$ 13588.0 0.995867 0.497934 0.867215i $$-0.334092\pi$$
0.497934 + 0.867215i $$0.334092\pi$$
$$572$$ 0 0
$$573$$ −16646.0 −1.21361
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 0 0
$$577$$ −8701.00 −0.627777 −0.313889 0.949460i $$-0.601632\pi$$
−0.313889 + 0.949460i $$0.601632\pi$$
$$578$$ 0 0
$$579$$ −21469.0 −1.54097
$$580$$ 0 0
$$581$$ −4662.00 −0.332896
$$582$$ 0 0
$$583$$ 3526.00 0.250484
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ 11361.0 0.798839 0.399420 0.916768i $$-0.369212\pi$$
0.399420 + 0.916768i $$0.369212\pi$$
$$588$$ 0 0
$$589$$ 1470.00 0.102836
$$590$$ 0 0
$$591$$ 16422.0 1.14300
$$592$$ 0 0
$$593$$ −11417.0 −0.790624 −0.395312 0.918547i $$-0.629364\pi$$
−0.395312 + 0.918547i $$0.629364\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ 34300.0 2.35143
$$598$$ 0 0
$$599$$ −21050.0 −1.43586 −0.717930 0.696116i $$-0.754910\pi$$
−0.717930 + 0.696116i $$0.754910\pi$$
$$600$$ 0 0
$$601$$ 7427.00 0.504083 0.252041 0.967716i $$-0.418898\pi$$
0.252041 + 0.967716i $$0.418898\pi$$
$$602$$ 0 0
$$603$$ 3102.00 0.209491
$$604$$ 0 0
$$605$$ 0 0
$$606$$ 0 0
$$607$$ 4144.00 0.277100 0.138550 0.990355i $$-0.455756\pi$$
0.138550 + 0.990355i $$0.455756\pi$$
$$608$$ 0 0
$$609$$ 6720.00 0.447140
$$610$$ 0 0
$$611$$ 5488.00 0.363373
$$612$$ 0 0
$$613$$ 30122.0 1.98469 0.992346 0.123489i $$-0.0394084\pi$$
0.992346 + 0.123489i $$0.0394084\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 11934.0 0.778679 0.389339 0.921094i $$-0.372703\pi$$
0.389339 + 0.921094i $$0.372703\pi$$
$$618$$ 0 0
$$619$$ −8540.00 −0.554526 −0.277263 0.960794i $$-0.589427\pi$$
−0.277263 + 0.960794i $$0.589427\pi$$
$$620$$ 0 0
$$621$$ 5670.00 0.366392
$$622$$ 0 0
$$623$$ 5670.00 0.364629
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 0 0
$$627$$ 10535.0 0.671017
$$628$$ 0 0
$$629$$ 28574.0 1.81132
$$630$$ 0 0
$$631$$ −3158.00 −0.199236 −0.0996181 0.995026i $$-0.531762\pi$$
−0.0996181 + 0.995026i $$0.531762\pi$$
$$632$$ 0 0
$$633$$ −30149.0 −1.89307
$$634$$ 0 0
$$635$$ 0 0
$$636$$ 0 0
$$637$$ 8596.00 0.534672
$$638$$ 0 0
$$639$$ 9064.00 0.561137
$$640$$ 0 0
$$641$$ −4278.00 −0.263605 −0.131803 0.991276i $$-0.542076\pi$$
−0.131803 + 0.991276i $$0.542076\pi$$
$$642$$ 0 0
$$643$$ −11508.0 −0.705803 −0.352901 0.935661i $$-0.614805\pi$$
−0.352901 + 0.935661i $$0.614805\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 8204.00 0.498505 0.249252 0.968439i $$-0.419815\pi$$
0.249252 + 0.968439i $$0.419815\pi$$
$$648$$ 0 0
$$649$$ 12040.0 0.728215
$$650$$ 0 0
$$651$$ −1764.00 −0.106201
$$652$$ 0 0
$$653$$ −5518.00 −0.330683 −0.165342 0.986236i $$-0.552873\pi$$
−0.165342 + 0.986236i $$0.552873\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 0 0
$$657$$ 16786.0 0.996780
$$658$$ 0 0
$$659$$ −13295.0 −0.785887 −0.392944 0.919563i $$-0.628543\pi$$
−0.392944 + 0.919563i $$0.628543\pi$$
$$660$$ 0 0
$$661$$ 9968.00 0.586551 0.293276 0.956028i $$-0.405255\pi$$
0.293276 + 0.956028i $$0.405255\pi$$
$$662$$ 0 0
$$663$$ 17836.0 1.04479
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 25920.0 1.50469
$$668$$ 0 0
$$669$$ −15484.0 −0.894837
$$670$$ 0 0
$$671$$ 22274.0 1.28149
$$672$$ 0 0
$$673$$ 15738.0 0.901419 0.450710 0.892671i $$-0.351171\pi$$
0.450710 + 0.892671i $$0.351171\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ −19824.0 −1.12540 −0.562702 0.826660i $$-0.690238\pi$$
−0.562702 + 0.826660i $$0.690238\pi$$
$$678$$ 0 0
$$679$$ 7476.00 0.422537
$$680$$ 0 0
$$681$$ 3332.00 0.187493
$$682$$ 0 0
$$683$$ −11073.0 −0.620346 −0.310173 0.950680i $$-0.600387\pi$$
−0.310173 + 0.950680i $$0.600387\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 0 0
$$687$$ 20580.0 1.14291
$$688$$ 0 0
$$689$$ −2296.00 −0.126953
$$690$$ 0 0
$$691$$ 6503.00 0.358011 0.179006 0.983848i $$-0.442712\pi$$
0.179006 + 0.983848i $$0.442712\pi$$
$$692$$ 0 0
$$693$$ −5676.00 −0.311130
$$694$$ 0 0
$$695$$ 0 0
$$696$$ 0 0
$$697$$ 18473.0 1.00389
$$698$$ 0 0
$$699$$ −7014.00 −0.379533
$$700$$ 0 0
$$701$$ 10148.0 0.546768 0.273384 0.961905i $$-0.411857\pi$$
0.273384 + 0.961905i $$0.411857\pi$$
$$702$$ 0 0
$$703$$ −10990.0 −0.589610
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ 7812.00 0.415559
$$708$$ 0 0
$$709$$ 9980.00 0.528641 0.264321 0.964435i $$-0.414852\pi$$
0.264321 + 0.964435i $$0.414852\pi$$
$$710$$ 0 0
$$711$$ 11220.0 0.591818
$$712$$ 0 0
$$713$$ −6804.00 −0.357380
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ 17360.0 0.904214
$$718$$ 0 0
$$719$$ −27510.0 −1.42691 −0.713456 0.700700i $$-0.752871\pi$$
−0.713456 + 0.700700i $$0.752871\pi$$
$$720$$ 0 0
$$721$$ 3192.00 0.164877
$$722$$ 0 0
$$723$$ 13279.0 0.683059
$$724$$ 0 0
$$725$$ 0 0
$$726$$ 0 0
$$727$$ 17024.0 0.868480 0.434240 0.900797i $$-0.357017\pi$$
0.434240 + 0.900797i $$0.357017\pi$$
$$728$$ 0 0
$$729$$ −11843.0 −0.601687
$$730$$ 0 0
$$731$$ −8372.00 −0.423597
$$732$$ 0 0
$$733$$ −34748.0 −1.75095 −0.875475 0.483263i $$-0.839451\pi$$
−0.875475 + 0.483263i $$0.839451\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 6063.00 0.303030
$$738$$ 0 0
$$739$$ 12020.0 0.598326 0.299163 0.954202i $$-0.403293\pi$$
0.299163 + 0.954202i $$0.403293\pi$$
$$740$$ 0 0
$$741$$ −6860.00 −0.340092
$$742$$ 0 0
$$743$$ −28642.0 −1.41423 −0.707115 0.707098i $$-0.750004\pi$$
−0.707115 + 0.707098i $$0.750004\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 0 0
$$747$$ 17094.0 0.837265
$$748$$ 0 0
$$749$$ 7614.00 0.371441
$$750$$ 0 0
$$751$$ 8752.00 0.425253 0.212627 0.977134i $$-0.431798\pi$$
0.212627 + 0.977134i $$0.431798\pi$$
$$752$$ 0 0
$$753$$ 16611.0 0.803902
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 0 0
$$757$$ 10256.0 0.492418 0.246209 0.969217i $$-0.420815\pi$$
0.246209 + 0.969217i $$0.420815\pi$$
$$758$$ 0 0
$$759$$ −48762.0 −2.33195
$$760$$ 0 0
$$761$$ 33957.0 1.61753 0.808765 0.588132i $$-0.200136\pi$$
0.808765 + 0.588132i $$0.200136\pi$$
$$762$$ 0 0
$$763$$ 6420.00 0.304613
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ −7840.00 −0.369082
$$768$$ 0 0
$$769$$ 27965.0 1.31137 0.655685 0.755034i $$-0.272380\pi$$
0.655685 + 0.755034i $$0.272380\pi$$
$$770$$ 0 0
$$771$$ 31458.0 1.46943
$$772$$ 0 0
$$773$$ 9912.00 0.461203 0.230601 0.973048i $$-0.425931\pi$$
0.230601 + 0.973048i $$0.425931\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 0 0
$$777$$ 13188.0 0.608902
$$778$$ 0 0
$$779$$ −7105.00 −0.326782
$$780$$ 0 0
$$781$$ 17716.0 0.811688
$$782$$ 0 0
$$783$$ 5600.00 0.255591
$$784$$ 0 0
$$785$$ 0 0
$$786$$ 0 0
$$787$$ −25564.0 −1.15789 −0.578944 0.815367i $$-0.696535\pi$$
−0.578944 + 0.815367i $$0.696535\pi$$
$$788$$ 0 0
$$789$$ −5054.00 −0.228045
$$790$$ 0 0
$$791$$ −3018.00 −0.135661
$$792$$ 0 0
$$793$$ −14504.0 −0.649498
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ 12446.0 0.553149 0.276575 0.960992i $$-0.410801\pi$$
0.276575 + 0.960992i $$0.410801\pi$$
$$798$$ 0 0
$$799$$ 17836.0 0.789728
$$800$$ 0 0
$$801$$ −20790.0 −0.917077
$$802$$ 0 0
$$803$$ 32809.0 1.44185
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ 43120.0 1.88091
$$808$$ 0 0
$$809$$ 33970.0 1.47629 0.738147 0.674640i $$-0.235701\pi$$
0.738147 + 0.674640i $$0.235701\pi$$
$$810$$ 0 0
$$811$$ −18732.0 −0.811060 −0.405530 0.914082i $$-0.632913\pi$$
−0.405530 + 0.914082i $$0.632913\pi$$
$$812$$ 0 0
$$813$$ −50666.0 −2.18565
$$814$$ 0 0
$$815$$ 0 0
$$816$$ 0 0
$$817$$ 3220.00 0.137887
$$818$$ 0 0
$$819$$ 3696.00 0.157691
$$820$$ 0 0
$$821$$ −6162.00 −0.261943 −0.130972 0.991386i $$-0.541810\pi$$
−0.130972 + 0.991386i $$0.541810\pi$$
$$822$$ 0 0
$$823$$ 25388.0 1.07530 0.537649 0.843169i $$-0.319313\pi$$
0.537649 + 0.843169i $$0.319313\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 25201.0 1.05964 0.529821 0.848109i $$-0.322259\pi$$
0.529821 + 0.848109i $$0.322259\pi$$
$$828$$ 0 0
$$829$$ 19740.0 0.827019 0.413509 0.910500i $$-0.364303\pi$$
0.413509 + 0.910500i $$0.364303\pi$$
$$830$$ 0 0
$$831$$ 12432.0 0.518967
$$832$$ 0 0
$$833$$ 27937.0 1.16202
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 0 0
$$837$$ −1470.00 −0.0607057
$$838$$ 0 0
$$839$$ 29680.0 1.22130 0.610648 0.791902i $$-0.290909\pi$$
0.610648 + 0.791902i $$0.290909\pi$$
$$840$$ 0 0
$$841$$ 1211.00 0.0496535
$$842$$ 0 0
$$843$$ 31794.0 1.29898
$$844$$ 0 0
$$845$$ 0 0
$$846$$ 0 0
$$847$$ −3108.00 −0.126083
$$848$$ 0 0
$$849$$ 49539.0 2.00256
$$850$$ 0 0
$$851$$ 50868.0 2.04904
$$852$$ 0 0
$$853$$ −1218.00 −0.0488904 −0.0244452 0.999701i $$-0.507782\pi$$
−0.0244452 + 0.999701i $$0.507782\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ −38731.0 −1.54379 −0.771894 0.635752i $$-0.780690\pi$$
−0.771894 + 0.635752i $$0.780690\pi$$
$$858$$ 0 0
$$859$$ 23555.0 0.935607 0.467803 0.883833i $$-0.345046\pi$$
0.467803 + 0.883833i $$0.345046\pi$$
$$860$$ 0 0
$$861$$ 8526.00 0.337474
$$862$$ 0 0
$$863$$ −24872.0 −0.981058 −0.490529 0.871425i $$-0.663196\pi$$
−0.490529 + 0.871425i $$0.663196\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 0 0
$$867$$ 23576.0 0.923510
$$868$$ 0 0
$$869$$ 21930.0 0.856069
$$870$$ 0 0
$$871$$ −3948.00 −0.153585
$$872$$ 0 0
$$873$$ −27412.0 −1.06272
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ −17124.0 −0.659335 −0.329667 0.944097i $$-0.606937\pi$$
−0.329667 + 0.944097i $$0.606937\pi$$
$$878$$ 0 0
$$879$$ −29106.0 −1.11686
$$880$$ 0 0
$$881$$ −658.000 −0.0251630 −0.0125815 0.999921i $$-0.504005\pi$$
−0.0125815 + 0.999921i $$0.504005\pi$$
$$882$$ 0 0
$$883$$ 33727.0 1.28540 0.642698 0.766120i $$-0.277815\pi$$
0.642698 + 0.766120i $$0.277815\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ −36036.0 −1.36412 −0.682058 0.731298i $$-0.738915\pi$$
−0.682058 + 0.731298i $$0.738915\pi$$
$$888$$ 0 0
$$889$$ −5244.00 −0.197838
$$890$$ 0 0
$$891$$ −36077.0 −1.35648
$$892$$ 0 0
$$893$$ −6860.00 −0.257067
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 0 0
$$897$$ 31752.0 1.18190
$$898$$ 0 0
$$899$$ −6720.00 −0.249304
$$900$$ 0 0
$$901$$ −7462.00 −0.275910
$$902$$ 0 0
$$903$$ −3864.00 −0.142399
$$904$$ 0 0
$$905$$ 0 0
$$906$$ 0 0
$$907$$ 39156.0 1.43347 0.716733 0.697348i $$-0.245637\pi$$
0.716733 + 0.697348i $$0.245637\pi$$
$$908$$ 0 0
$$909$$ −28644.0 −1.04517
$$910$$ 0 0
$$911$$ 43532.0 1.58318 0.791591 0.611051i $$-0.209253\pi$$
0.791591 + 0.611051i $$0.209253\pi$$
$$912$$ 0 0
$$913$$ 33411.0 1.21111
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ 6552.00 0.235950
$$918$$ 0 0
$$919$$ −28610.0 −1.02694 −0.513469 0.858108i $$-0.671640\pi$$
−0.513469 + 0.858108i $$0.671640\pi$$
$$920$$ 0 0
$$921$$ −17983.0 −0.643388
$$922$$ 0 0
$$923$$ −11536.0 −0.411389
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 0 0
$$927$$ −11704.0 −0.414682
$$928$$ 0 0
$$929$$ −24290.0 −0.857835 −0.428918 0.903344i $$-0.641105\pi$$
−0.428918 + 0.903344i $$0.641105\pi$$
$$930$$ 0 0
$$931$$ −10745.0 −0.378253
$$932$$ 0 0
$$933$$ 20874.0 0.732459
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ −34461.0 −1.20149 −0.600743 0.799442i $$-0.705128\pi$$
−0.600743 + 0.799442i $$0.705128\pi$$
$$938$$ 0 0
$$939$$ −16954.0 −0.589215
$$940$$ 0 0
$$941$$ 40628.0 1.40748 0.703738 0.710460i $$-0.251513\pi$$
0.703738 + 0.710460i $$0.251513\pi$$
$$942$$ 0 0
$$943$$ 32886.0 1.13565
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ −20904.0 −0.717306 −0.358653 0.933471i $$-0.616764\pi$$
−0.358653 + 0.933471i $$0.616764\pi$$
$$948$$ 0 0
$$949$$ −21364.0 −0.730774
$$950$$ 0 0
$$951$$ −66388.0 −2.26370
$$952$$ 0 0
$$953$$ −1807.00 −0.0614213 −0.0307106 0.999528i $$-0.509777\pi$$
−0.0307106 + 0.999528i $$0.509777\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 0 0
$$957$$ −48160.0 −1.62674
$$958$$ 0 0
$$959$$ 2466.00 0.0830358
$$960$$ 0 0
$$961$$ −28027.0 −0.940787
$$962$$ 0 0
$$963$$ −27918.0 −0.934211
$$964$$ 0 0
$$965$$ 0 0
$$966$$ 0 0
$$967$$ 57584.0 1.91497 0.957485 0.288482i $$-0.0931505\pi$$
0.957485 + 0.288482i $$0.0931505\pi$$
$$968$$ 0 0
$$969$$ −22295.0 −0.739132
$$970$$ 0 0
$$971$$ −27237.0 −0.900182 −0.450091 0.892983i $$-0.648608\pi$$
−0.450091 + 0.892983i $$0.648608\pi$$
$$972$$ 0 0
$$973$$ −3570.00 −0.117625
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ 13649.0 0.446950 0.223475 0.974710i $$-0.428260\pi$$
0.223475 + 0.974710i $$0.428260\pi$$
$$978$$ 0 0
$$979$$ −40635.0 −1.32656
$$980$$ 0 0
$$981$$ −23540.0 −0.766131
$$982$$ 0 0
$$983$$ −16002.0 −0.519211 −0.259606 0.965715i $$-0.583593\pi$$
−0.259606 + 0.965715i $$0.583593\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 0 0
$$987$$ 8232.00 0.265479
$$988$$ 0 0
$$989$$ −14904.0 −0.479191
$$990$$ 0 0
$$991$$ 37022.0 1.18672 0.593362 0.804936i $$-0.297800\pi$$
0.593362 + 0.804936i $$0.297800\pi$$
$$992$$ 0 0
$$993$$ 1281.00 0.0409379
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 0 0
$$997$$ 18396.0 0.584360 0.292180 0.956363i $$-0.405619\pi$$
0.292180 + 0.956363i $$0.405619\pi$$
$$998$$ 0 0
$$999$$ 10990.0 0.348056
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.bt.1.1 1
4.3 odd 2 1600.4.a.h.1.1 1
5.4 even 2 1600.4.a.i.1.1 1
8.3 odd 2 400.4.a.s.1.1 1
8.5 even 2 25.4.a.a.1.1 1
20.19 odd 2 1600.4.a.bs.1.1 1
24.5 odd 2 225.4.a.e.1.1 1
40.3 even 4 400.4.c.e.49.2 2
40.13 odd 4 25.4.b.b.24.2 2
40.19 odd 2 400.4.a.c.1.1 1
40.27 even 4 400.4.c.e.49.1 2
40.29 even 2 25.4.a.b.1.1 yes 1
40.37 odd 4 25.4.b.b.24.1 2
56.13 odd 2 1225.4.a.h.1.1 1
120.29 odd 2 225.4.a.c.1.1 1
120.53 even 4 225.4.b.f.199.1 2
120.77 even 4 225.4.b.f.199.2 2
280.69 odd 2 1225.4.a.i.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
25.4.a.a.1.1 1 8.5 even 2
25.4.a.b.1.1 yes 1 40.29 even 2
25.4.b.b.24.1 2 40.37 odd 4
25.4.b.b.24.2 2 40.13 odd 4
225.4.a.c.1.1 1 120.29 odd 2
225.4.a.e.1.1 1 24.5 odd 2
225.4.b.f.199.1 2 120.53 even 4
225.4.b.f.199.2 2 120.77 even 4
400.4.a.c.1.1 1 40.19 odd 2
400.4.a.s.1.1 1 8.3 odd 2
400.4.c.e.49.1 2 40.27 even 4
400.4.c.e.49.2 2 40.3 even 4
1225.4.a.h.1.1 1 56.13 odd 2
1225.4.a.i.1.1 1 280.69 odd 2
1600.4.a.h.1.1 1 4.3 odd 2
1600.4.a.i.1.1 1 5.4 even 2
1600.4.a.bs.1.1 1 20.19 odd 2
1600.4.a.bt.1.1 1 1.1 even 1 trivial