# Properties

 Label 1472.4.a.c.1.1 Level $1472$ Weight $4$ Character 1472.1 Self dual yes Analytic conductor $86.851$ 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] = [1472,4,Mod(1,1472)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1472, 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("1472.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1472 = 2^{6} \cdot 23$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 1472.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: $$86.8508115285$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 23) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 1472.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-5.00000 q^{3} +6.00000 q^{5} +8.00000 q^{7} -2.00000 q^{9} +O(q^{10})$$ $$q-5.00000 q^{3} +6.00000 q^{5} +8.00000 q^{7} -2.00000 q^{9} +34.0000 q^{11} +57.0000 q^{13} -30.0000 q^{15} -80.0000 q^{17} -70.0000 q^{19} -40.0000 q^{21} -23.0000 q^{23} -89.0000 q^{25} +145.000 q^{27} -245.000 q^{29} -103.000 q^{31} -170.000 q^{33} +48.0000 q^{35} +298.000 q^{37} -285.000 q^{39} +95.0000 q^{41} +88.0000 q^{43} -12.0000 q^{45} +357.000 q^{47} -279.000 q^{49} +400.000 q^{51} +414.000 q^{53} +204.000 q^{55} +350.000 q^{57} -408.000 q^{59} -822.000 q^{61} -16.0000 q^{63} +342.000 q^{65} +926.000 q^{67} +115.000 q^{69} -335.000 q^{71} -899.000 q^{73} +445.000 q^{75} +272.000 q^{77} +1322.00 q^{79} -671.000 q^{81} -36.0000 q^{83} -480.000 q^{85} +1225.00 q^{87} -460.000 q^{89} +456.000 q^{91} +515.000 q^{93} -420.000 q^{95} -964.000 q^{97} -68.0000 q^{99} +O(q^{100})$$

## 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 $$-0.659772\pi$$
−0.481125 + 0.876652i $$0.659772\pi$$
$$4$$ 0 0
$$5$$ 6.00000 0.536656 0.268328 0.963328i $$-0.413529\pi$$
0.268328 + 0.963328i $$0.413529\pi$$
$$6$$ 0 0
$$7$$ 8.00000 0.431959 0.215980 0.976398i $$-0.430705\pi$$
0.215980 + 0.976398i $$0.430705\pi$$
$$8$$ 0 0
$$9$$ −2.00000 −0.0740741
$$10$$ 0 0
$$11$$ 34.0000 0.931944 0.465972 0.884799i $$-0.345705\pi$$
0.465972 + 0.884799i $$0.345705\pi$$
$$12$$ 0 0
$$13$$ 57.0000 1.21607 0.608037 0.793909i $$-0.291957\pi$$
0.608037 + 0.793909i $$0.291957\pi$$
$$14$$ 0 0
$$15$$ −30.0000 −0.516398
$$16$$ 0 0
$$17$$ −80.0000 −1.14134 −0.570672 0.821178i $$-0.693317\pi$$
−0.570672 + 0.821178i $$0.693317\pi$$
$$18$$ 0 0
$$19$$ −70.0000 −0.845216 −0.422608 0.906313i $$-0.638885\pi$$
−0.422608 + 0.906313i $$0.638885\pi$$
$$20$$ 0 0
$$21$$ −40.0000 −0.415653
$$22$$ 0 0
$$23$$ −23.0000 −0.208514
$$24$$ 0 0
$$25$$ −89.0000 −0.712000
$$26$$ 0 0
$$27$$ 145.000 1.03353
$$28$$ 0 0
$$29$$ −245.000 −1.56881 −0.784403 0.620252i $$-0.787030\pi$$
−0.784403 + 0.620252i $$0.787030\pi$$
$$30$$ 0 0
$$31$$ −103.000 −0.596753 −0.298377 0.954448i $$-0.596445\pi$$
−0.298377 + 0.954448i $$0.596445\pi$$
$$32$$ 0 0
$$33$$ −170.000 −0.896764
$$34$$ 0 0
$$35$$ 48.0000 0.231814
$$36$$ 0 0
$$37$$ 298.000 1.32408 0.662039 0.749469i $$-0.269691\pi$$
0.662039 + 0.749469i $$0.269691\pi$$
$$38$$ 0 0
$$39$$ −285.000 −1.17017
$$40$$ 0 0
$$41$$ 95.0000 0.361866 0.180933 0.983495i $$-0.442088\pi$$
0.180933 + 0.983495i $$0.442088\pi$$
$$42$$ 0 0
$$43$$ 88.0000 0.312090 0.156045 0.987750i $$-0.450125\pi$$
0.156045 + 0.987750i $$0.450125\pi$$
$$44$$ 0 0
$$45$$ −12.0000 −0.0397523
$$46$$ 0 0
$$47$$ 357.000 1.10795 0.553977 0.832532i $$-0.313110\pi$$
0.553977 + 0.832532i $$0.313110\pi$$
$$48$$ 0 0
$$49$$ −279.000 −0.813411
$$50$$ 0 0
$$51$$ 400.000 1.09826
$$52$$ 0 0
$$53$$ 414.000 1.07297 0.536484 0.843911i $$-0.319752\pi$$
0.536484 + 0.843911i $$0.319752\pi$$
$$54$$ 0 0
$$55$$ 204.000 0.500134
$$56$$ 0 0
$$57$$ 350.000 0.813309
$$58$$ 0 0
$$59$$ −408.000 −0.900289 −0.450145 0.892956i $$-0.648628\pi$$
−0.450145 + 0.892956i $$0.648628\pi$$
$$60$$ 0 0
$$61$$ −822.000 −1.72535 −0.862675 0.505759i $$-0.831212\pi$$
−0.862675 + 0.505759i $$0.831212\pi$$
$$62$$ 0 0
$$63$$ −16.0000 −0.0319970
$$64$$ 0 0
$$65$$ 342.000 0.652614
$$66$$ 0 0
$$67$$ 926.000 1.68849 0.844246 0.535957i $$-0.180049\pi$$
0.844246 + 0.535957i $$0.180049\pi$$
$$68$$ 0 0
$$69$$ 115.000 0.200643
$$70$$ 0 0
$$71$$ −335.000 −0.559960 −0.279980 0.960006i $$-0.590328\pi$$
−0.279980 + 0.960006i $$0.590328\pi$$
$$72$$ 0 0
$$73$$ −899.000 −1.44137 −0.720685 0.693263i $$-0.756173\pi$$
−0.720685 + 0.693263i $$0.756173\pi$$
$$74$$ 0 0
$$75$$ 445.000 0.685122
$$76$$ 0 0
$$77$$ 272.000 0.402562
$$78$$ 0 0
$$79$$ 1322.00 1.88274 0.941371 0.337373i $$-0.109538\pi$$
0.941371 + 0.337373i $$0.109538\pi$$
$$80$$ 0 0
$$81$$ −671.000 −0.920439
$$82$$ 0 0
$$83$$ −36.0000 −0.0476086 −0.0238043 0.999717i $$-0.507578\pi$$
−0.0238043 + 0.999717i $$0.507578\pi$$
$$84$$ 0 0
$$85$$ −480.000 −0.612510
$$86$$ 0 0
$$87$$ 1225.00 1.50958
$$88$$ 0 0
$$89$$ −460.000 −0.547864 −0.273932 0.961749i $$-0.588324\pi$$
−0.273932 + 0.961749i $$0.588324\pi$$
$$90$$ 0 0
$$91$$ 456.000 0.525294
$$92$$ 0 0
$$93$$ 515.000 0.574226
$$94$$ 0 0
$$95$$ −420.000 −0.453590
$$96$$ 0 0
$$97$$ −964.000 −1.00907 −0.504533 0.863393i $$-0.668335\pi$$
−0.504533 + 0.863393i $$0.668335\pi$$
$$98$$ 0 0
$$99$$ −68.0000 −0.0690329
$$100$$ 0 0
$$101$$ 310.000 0.305407 0.152704 0.988272i $$-0.451202\pi$$
0.152704 + 0.988272i $$0.451202\pi$$
$$102$$ 0 0
$$103$$ −1044.00 −0.998722 −0.499361 0.866394i $$-0.666432\pi$$
−0.499361 + 0.866394i $$0.666432\pi$$
$$104$$ 0 0
$$105$$ −240.000 −0.223063
$$106$$ 0 0
$$107$$ 414.000 0.374046 0.187023 0.982356i $$-0.440116\pi$$
0.187023 + 0.982356i $$0.440116\pi$$
$$108$$ 0 0
$$109$$ −704.000 −0.618633 −0.309316 0.950959i $$-0.600100\pi$$
−0.309316 + 0.950959i $$0.600100\pi$$
$$110$$ 0 0
$$111$$ −1490.00 −1.27409
$$112$$ 0 0
$$113$$ 952.000 0.792537 0.396268 0.918135i $$-0.370305\pi$$
0.396268 + 0.918135i $$0.370305\pi$$
$$114$$ 0 0
$$115$$ −138.000 −0.111901
$$116$$ 0 0
$$117$$ −114.000 −0.0900795
$$118$$ 0 0
$$119$$ −640.000 −0.493014
$$120$$ 0 0
$$121$$ −175.000 −0.131480
$$122$$ 0 0
$$123$$ −475.000 −0.348206
$$124$$ 0 0
$$125$$ −1284.00 −0.918756
$$126$$ 0 0
$$127$$ −261.000 −0.182362 −0.0911811 0.995834i $$-0.529064\pi$$
−0.0911811 + 0.995834i $$0.529064\pi$$
$$128$$ 0 0
$$129$$ −440.000 −0.300309
$$130$$ 0 0
$$131$$ −1441.00 −0.961074 −0.480537 0.876974i $$-0.659558\pi$$
−0.480537 + 0.876974i $$0.659558\pi$$
$$132$$ 0 0
$$133$$ −560.000 −0.365099
$$134$$ 0 0
$$135$$ 870.000 0.554649
$$136$$ 0 0
$$137$$ 1556.00 0.970351 0.485175 0.874417i $$-0.338756\pi$$
0.485175 + 0.874417i $$0.338756\pi$$
$$138$$ 0 0
$$139$$ 25.0000 0.0152552 0.00762760 0.999971i $$-0.497572\pi$$
0.00762760 + 0.999971i $$0.497572\pi$$
$$140$$ 0 0
$$141$$ −1785.00 −1.06613
$$142$$ 0 0
$$143$$ 1938.00 1.13331
$$144$$ 0 0
$$145$$ −1470.00 −0.841909
$$146$$ 0 0
$$147$$ 1395.00 0.782705
$$148$$ 0 0
$$149$$ −822.000 −0.451952 −0.225976 0.974133i $$-0.572557\pi$$
−0.225976 + 0.974133i $$0.572557\pi$$
$$150$$ 0 0
$$151$$ 1489.00 0.802471 0.401235 0.915975i $$-0.368581\pi$$
0.401235 + 0.915975i $$0.368581\pi$$
$$152$$ 0 0
$$153$$ 160.000 0.0845440
$$154$$ 0 0
$$155$$ −618.000 −0.320251
$$156$$ 0 0
$$157$$ 632.000 0.321268 0.160634 0.987014i $$-0.448646\pi$$
0.160634 + 0.987014i $$0.448646\pi$$
$$158$$ 0 0
$$159$$ −2070.00 −1.03246
$$160$$ 0 0
$$161$$ −184.000 −0.0900698
$$162$$ 0 0
$$163$$ −3043.00 −1.46225 −0.731123 0.682245i $$-0.761004\pi$$
−0.731123 + 0.682245i $$0.761004\pi$$
$$164$$ 0 0
$$165$$ −1020.00 −0.481254
$$166$$ 0 0
$$167$$ 2224.00 1.03053 0.515264 0.857031i $$-0.327694\pi$$
0.515264 + 0.857031i $$0.327694\pi$$
$$168$$ 0 0
$$169$$ 1052.00 0.478835
$$170$$ 0 0
$$171$$ 140.000 0.0626086
$$172$$ 0 0
$$173$$ −3230.00 −1.41949 −0.709747 0.704457i $$-0.751191\pi$$
−0.709747 + 0.704457i $$0.751191\pi$$
$$174$$ 0 0
$$175$$ −712.000 −0.307555
$$176$$ 0 0
$$177$$ 2040.00 0.866304
$$178$$ 0 0
$$179$$ 369.000 0.154080 0.0770401 0.997028i $$-0.475453\pi$$
0.0770401 + 0.997028i $$0.475453\pi$$
$$180$$ 0 0
$$181$$ 1370.00 0.562604 0.281302 0.959619i $$-0.409234\pi$$
0.281302 + 0.959619i $$0.409234\pi$$
$$182$$ 0 0
$$183$$ 4110.00 1.66022
$$184$$ 0 0
$$185$$ 1788.00 0.710575
$$186$$ 0 0
$$187$$ −2720.00 −1.06367
$$188$$ 0 0
$$189$$ 1160.00 0.446442
$$190$$ 0 0
$$191$$ −4410.00 −1.67066 −0.835331 0.549747i $$-0.814724\pi$$
−0.835331 + 0.549747i $$0.814724\pi$$
$$192$$ 0 0
$$193$$ −135.000 −0.0503498 −0.0251749 0.999683i $$-0.508014\pi$$
−0.0251749 + 0.999683i $$0.508014\pi$$
$$194$$ 0 0
$$195$$ −1710.00 −0.627978
$$196$$ 0 0
$$197$$ −1221.00 −0.441587 −0.220794 0.975321i $$-0.570865\pi$$
−0.220794 + 0.975321i $$0.570865\pi$$
$$198$$ 0 0
$$199$$ 1098.00 0.391131 0.195566 0.980691i $$-0.437346\pi$$
0.195566 + 0.980691i $$0.437346\pi$$
$$200$$ 0 0
$$201$$ −4630.00 −1.62475
$$202$$ 0 0
$$203$$ −1960.00 −0.677660
$$204$$ 0 0
$$205$$ 570.000 0.194198
$$206$$ 0 0
$$207$$ 46.0000 0.0154455
$$208$$ 0 0
$$209$$ −2380.00 −0.787694
$$210$$ 0 0
$$211$$ −3676.00 −1.19937 −0.599683 0.800238i $$-0.704707\pi$$
−0.599683 + 0.800238i $$0.704707\pi$$
$$212$$ 0 0
$$213$$ 1675.00 0.538822
$$214$$ 0 0
$$215$$ 528.000 0.167485
$$216$$ 0 0
$$217$$ −824.000 −0.257773
$$218$$ 0 0
$$219$$ 4495.00 1.38696
$$220$$ 0 0
$$221$$ −4560.00 −1.38796
$$222$$ 0 0
$$223$$ −1656.00 −0.497282 −0.248641 0.968596i $$-0.579984\pi$$
−0.248641 + 0.968596i $$0.579984\pi$$
$$224$$ 0 0
$$225$$ 178.000 0.0527407
$$226$$ 0 0
$$227$$ 2940.00 0.859624 0.429812 0.902918i $$-0.358580\pi$$
0.429812 + 0.902918i $$0.358580\pi$$
$$228$$ 0 0
$$229$$ −3612.00 −1.04230 −0.521152 0.853464i $$-0.674498\pi$$
−0.521152 + 0.853464i $$0.674498\pi$$
$$230$$ 0 0
$$231$$ −1360.00 −0.387366
$$232$$ 0 0
$$233$$ −4325.00 −1.21605 −0.608026 0.793917i $$-0.708038\pi$$
−0.608026 + 0.793917i $$0.708038\pi$$
$$234$$ 0 0
$$235$$ 2142.00 0.594590
$$236$$ 0 0
$$237$$ −6610.00 −1.81167
$$238$$ 0 0
$$239$$ −2735.00 −0.740219 −0.370110 0.928988i $$-0.620680\pi$$
−0.370110 + 0.928988i $$0.620680\pi$$
$$240$$ 0 0
$$241$$ −6710.00 −1.79348 −0.896741 0.442556i $$-0.854072\pi$$
−0.896741 + 0.442556i $$0.854072\pi$$
$$242$$ 0 0
$$243$$ −560.000 −0.147835
$$244$$ 0 0
$$245$$ −1674.00 −0.436522
$$246$$ 0 0
$$247$$ −3990.00 −1.02784
$$248$$ 0 0
$$249$$ 180.000 0.0458114
$$250$$ 0 0
$$251$$ −6948.00 −1.74723 −0.873613 0.486621i $$-0.838229\pi$$
−0.873613 + 0.486621i $$0.838229\pi$$
$$252$$ 0 0
$$253$$ −782.000 −0.194324
$$254$$ 0 0
$$255$$ 2400.00 0.589388
$$256$$ 0 0
$$257$$ −4929.00 −1.19635 −0.598176 0.801365i $$-0.704108\pi$$
−0.598176 + 0.801365i $$0.704108\pi$$
$$258$$ 0 0
$$259$$ 2384.00 0.571948
$$260$$ 0 0
$$261$$ 490.000 0.116208
$$262$$ 0 0
$$263$$ −6138.00 −1.43911 −0.719554 0.694437i $$-0.755654\pi$$
−0.719554 + 0.694437i $$0.755654\pi$$
$$264$$ 0 0
$$265$$ 2484.00 0.575815
$$266$$ 0 0
$$267$$ 2300.00 0.527182
$$268$$ 0 0
$$269$$ 2063.00 0.467596 0.233798 0.972285i $$-0.424885\pi$$
0.233798 + 0.972285i $$0.424885\pi$$
$$270$$ 0 0
$$271$$ 1064.00 0.238500 0.119250 0.992864i $$-0.461951\pi$$
0.119250 + 0.992864i $$0.461951\pi$$
$$272$$ 0 0
$$273$$ −2280.00 −0.505465
$$274$$ 0 0
$$275$$ −3026.00 −0.663544
$$276$$ 0 0
$$277$$ −5729.00 −1.24268 −0.621340 0.783541i $$-0.713411\pi$$
−0.621340 + 0.783541i $$0.713411\pi$$
$$278$$ 0 0
$$279$$ 206.000 0.0442039
$$280$$ 0 0
$$281$$ −960.000 −0.203804 −0.101902 0.994794i $$-0.532493\pi$$
−0.101902 + 0.994794i $$0.532493\pi$$
$$282$$ 0 0
$$283$$ −114.000 −0.0239456 −0.0119728 0.999928i $$-0.503811\pi$$
−0.0119728 + 0.999928i $$0.503811\pi$$
$$284$$ 0 0
$$285$$ 2100.00 0.436468
$$286$$ 0 0
$$287$$ 760.000 0.156311
$$288$$ 0 0
$$289$$ 1487.00 0.302666
$$290$$ 0 0
$$291$$ 4820.00 0.970974
$$292$$ 0 0
$$293$$ 7048.00 1.40529 0.702643 0.711543i $$-0.252003\pi$$
0.702643 + 0.711543i $$0.252003\pi$$
$$294$$ 0 0
$$295$$ −2448.00 −0.483146
$$296$$ 0 0
$$297$$ 4930.00 0.963191
$$298$$ 0 0
$$299$$ −1311.00 −0.253569
$$300$$ 0 0
$$301$$ 704.000 0.134810
$$302$$ 0 0
$$303$$ −1550.00 −0.293878
$$304$$ 0 0
$$305$$ −4932.00 −0.925920
$$306$$ 0 0
$$307$$ 3872.00 0.719826 0.359913 0.932986i $$-0.382806\pi$$
0.359913 + 0.932986i $$0.382806\pi$$
$$308$$ 0 0
$$309$$ 5220.00 0.961021
$$310$$ 0 0
$$311$$ 4977.00 0.907459 0.453730 0.891139i $$-0.350093\pi$$
0.453730 + 0.891139i $$0.350093\pi$$
$$312$$ 0 0
$$313$$ −2536.00 −0.457965 −0.228983 0.973430i $$-0.573540\pi$$
−0.228983 + 0.973430i $$0.573540\pi$$
$$314$$ 0 0
$$315$$ −96.0000 −0.0171714
$$316$$ 0 0
$$317$$ −1434.00 −0.254074 −0.127037 0.991898i $$-0.540547\pi$$
−0.127037 + 0.991898i $$0.540547\pi$$
$$318$$ 0 0
$$319$$ −8330.00 −1.46204
$$320$$ 0 0
$$321$$ −2070.00 −0.359926
$$322$$ 0 0
$$323$$ 5600.00 0.964682
$$324$$ 0 0
$$325$$ −5073.00 −0.865844
$$326$$ 0 0
$$327$$ 3520.00 0.595280
$$328$$ 0 0
$$329$$ 2856.00 0.478591
$$330$$ 0 0
$$331$$ 5469.00 0.908167 0.454084 0.890959i $$-0.349967\pi$$
0.454084 + 0.890959i $$0.349967\pi$$
$$332$$ 0 0
$$333$$ −596.000 −0.0980799
$$334$$ 0 0
$$335$$ 5556.00 0.906139
$$336$$ 0 0
$$337$$ −7796.00 −1.26016 −0.630082 0.776529i $$-0.716979\pi$$
−0.630082 + 0.776529i $$0.716979\pi$$
$$338$$ 0 0
$$339$$ −4760.00 −0.762619
$$340$$ 0 0
$$341$$ −3502.00 −0.556141
$$342$$ 0 0
$$343$$ −4976.00 −0.783320
$$344$$ 0 0
$$345$$ 690.000 0.107676
$$346$$ 0 0
$$347$$ −10068.0 −1.55758 −0.778788 0.627288i $$-0.784165\pi$$
−0.778788 + 0.627288i $$0.784165\pi$$
$$348$$ 0 0
$$349$$ 7495.00 1.14956 0.574782 0.818306i $$-0.305087\pi$$
0.574782 + 0.818306i $$0.305087\pi$$
$$350$$ 0 0
$$351$$ 8265.00 1.25685
$$352$$ 0 0
$$353$$ 10617.0 1.60081 0.800405 0.599460i $$-0.204618\pi$$
0.800405 + 0.599460i $$0.204618\pi$$
$$354$$ 0 0
$$355$$ −2010.00 −0.300506
$$356$$ 0 0
$$357$$ 3200.00 0.474403
$$358$$ 0 0
$$359$$ −2522.00 −0.370769 −0.185384 0.982666i $$-0.559353\pi$$
−0.185384 + 0.982666i $$0.559353\pi$$
$$360$$ 0 0
$$361$$ −1959.00 −0.285610
$$362$$ 0 0
$$363$$ 875.000 0.126517
$$364$$ 0 0
$$365$$ −5394.00 −0.773520
$$366$$ 0 0
$$367$$ −7204.00 −1.02465 −0.512324 0.858792i $$-0.671215\pi$$
−0.512324 + 0.858792i $$0.671215\pi$$
$$368$$ 0 0
$$369$$ −190.000 −0.0268049
$$370$$ 0 0
$$371$$ 3312.00 0.463478
$$372$$ 0 0
$$373$$ 13310.0 1.84763 0.923815 0.382840i $$-0.125054\pi$$
0.923815 + 0.382840i $$0.125054\pi$$
$$374$$ 0 0
$$375$$ 6420.00 0.884073
$$376$$ 0 0
$$377$$ −13965.0 −1.90778
$$378$$ 0 0
$$379$$ 12952.0 1.75541 0.877704 0.479203i $$-0.159074\pi$$
0.877704 + 0.479203i $$0.159074\pi$$
$$380$$ 0 0
$$381$$ 1305.00 0.175478
$$382$$ 0 0
$$383$$ 2812.00 0.375161 0.187580 0.982249i $$-0.439936\pi$$
0.187580 + 0.982249i $$0.439936\pi$$
$$384$$ 0 0
$$385$$ 1632.00 0.216037
$$386$$ 0 0
$$387$$ −176.000 −0.0231178
$$388$$ 0 0
$$389$$ −1264.00 −0.164749 −0.0823745 0.996601i $$-0.526250\pi$$
−0.0823745 + 0.996601i $$0.526250\pi$$
$$390$$ 0 0
$$391$$ 1840.00 0.237987
$$392$$ 0 0
$$393$$ 7205.00 0.924794
$$394$$ 0 0
$$395$$ 7932.00 1.01039
$$396$$ 0 0
$$397$$ −7119.00 −0.899981 −0.449990 0.893033i $$-0.648573\pi$$
−0.449990 + 0.893033i $$0.648573\pi$$
$$398$$ 0 0
$$399$$ 2800.00 0.351317
$$400$$ 0 0
$$401$$ 4262.00 0.530758 0.265379 0.964144i $$-0.414503\pi$$
0.265379 + 0.964144i $$0.414503\pi$$
$$402$$ 0 0
$$403$$ −5871.00 −0.725696
$$404$$ 0 0
$$405$$ −4026.00 −0.493959
$$406$$ 0 0
$$407$$ 10132.0 1.23397
$$408$$ 0 0
$$409$$ 229.000 0.0276854 0.0138427 0.999904i $$-0.495594\pi$$
0.0138427 + 0.999904i $$0.495594\pi$$
$$410$$ 0 0
$$411$$ −7780.00 −0.933720
$$412$$ 0 0
$$413$$ −3264.00 −0.388888
$$414$$ 0 0
$$415$$ −216.000 −0.0255495
$$416$$ 0 0
$$417$$ −125.000 −0.0146793
$$418$$ 0 0
$$419$$ 15776.0 1.83940 0.919699 0.392623i $$-0.128432\pi$$
0.919699 + 0.392623i $$0.128432\pi$$
$$420$$ 0 0
$$421$$ 8728.00 1.01040 0.505198 0.863003i $$-0.331419\pi$$
0.505198 + 0.863003i $$0.331419\pi$$
$$422$$ 0 0
$$423$$ −714.000 −0.0820706
$$424$$ 0 0
$$425$$ 7120.00 0.812637
$$426$$ 0 0
$$427$$ −6576.00 −0.745281
$$428$$ 0 0
$$429$$ −9690.00 −1.09053
$$430$$ 0 0
$$431$$ 2928.00 0.327232 0.163616 0.986524i $$-0.447684\pi$$
0.163616 + 0.986524i $$0.447684\pi$$
$$432$$ 0 0
$$433$$ −5314.00 −0.589780 −0.294890 0.955531i $$-0.595283\pi$$
−0.294890 + 0.955531i $$0.595283\pi$$
$$434$$ 0 0
$$435$$ 7350.00 0.810128
$$436$$ 0 0
$$437$$ 1610.00 0.176240
$$438$$ 0 0
$$439$$ −2585.00 −0.281037 −0.140519 0.990078i $$-0.544877\pi$$
−0.140519 + 0.990078i $$0.544877\pi$$
$$440$$ 0 0
$$441$$ 558.000 0.0602527
$$442$$ 0 0
$$443$$ −2997.00 −0.321426 −0.160713 0.987001i $$-0.551379\pi$$
−0.160713 + 0.987001i $$0.551379\pi$$
$$444$$ 0 0
$$445$$ −2760.00 −0.294015
$$446$$ 0 0
$$447$$ 4110.00 0.434891
$$448$$ 0 0
$$449$$ −16562.0 −1.74078 −0.870389 0.492365i $$-0.836132\pi$$
−0.870389 + 0.492365i $$0.836132\pi$$
$$450$$ 0 0
$$451$$ 3230.00 0.337239
$$452$$ 0 0
$$453$$ −7445.00 −0.772178
$$454$$ 0 0
$$455$$ 2736.00 0.281903
$$456$$ 0 0
$$457$$ 3924.00 0.401656 0.200828 0.979626i $$-0.435637\pi$$
0.200828 + 0.979626i $$0.435637\pi$$
$$458$$ 0 0
$$459$$ −11600.0 −1.17961
$$460$$ 0 0
$$461$$ 4543.00 0.458977 0.229489 0.973311i $$-0.426295\pi$$
0.229489 + 0.973311i $$0.426295\pi$$
$$462$$ 0 0
$$463$$ −9616.00 −0.965213 −0.482606 0.875837i $$-0.660310\pi$$
−0.482606 + 0.875837i $$0.660310\pi$$
$$464$$ 0 0
$$465$$ 3090.00 0.308162
$$466$$ 0 0
$$467$$ 7826.00 0.775469 0.387735 0.921771i $$-0.373258\pi$$
0.387735 + 0.921771i $$0.373258\pi$$
$$468$$ 0 0
$$469$$ 7408.00 0.729360
$$470$$ 0 0
$$471$$ −3160.00 −0.309140
$$472$$ 0 0
$$473$$ 2992.00 0.290851
$$474$$ 0 0
$$475$$ 6230.00 0.601794
$$476$$ 0 0
$$477$$ −828.000 −0.0794791
$$478$$ 0 0
$$479$$ −11404.0 −1.08781 −0.543906 0.839146i $$-0.683055\pi$$
−0.543906 + 0.839146i $$0.683055\pi$$
$$480$$ 0 0
$$481$$ 16986.0 1.61018
$$482$$ 0 0
$$483$$ 920.000 0.0866697
$$484$$ 0 0
$$485$$ −5784.00 −0.541521
$$486$$ 0 0
$$487$$ 9267.00 0.862275 0.431137 0.902286i $$-0.358112\pi$$
0.431137 + 0.902286i $$0.358112\pi$$
$$488$$ 0 0
$$489$$ 15215.0 1.40705
$$490$$ 0 0
$$491$$ −18191.0 −1.67199 −0.835996 0.548735i $$-0.815110\pi$$
−0.835996 + 0.548735i $$0.815110\pi$$
$$492$$ 0 0
$$493$$ 19600.0 1.79055
$$494$$ 0 0
$$495$$ −408.000 −0.0370469
$$496$$ 0 0
$$497$$ −2680.00 −0.241880
$$498$$ 0 0
$$499$$ 19315.0 1.73278 0.866391 0.499366i $$-0.166434\pi$$
0.866391 + 0.499366i $$0.166434\pi$$
$$500$$ 0 0
$$501$$ −11120.0 −0.991627
$$502$$ 0 0
$$503$$ −8422.00 −0.746557 −0.373279 0.927719i $$-0.621766\pi$$
−0.373279 + 0.927719i $$0.621766\pi$$
$$504$$ 0 0
$$505$$ 1860.00 0.163899
$$506$$ 0 0
$$507$$ −5260.00 −0.460759
$$508$$ 0 0
$$509$$ 863.000 0.0751509 0.0375754 0.999294i $$-0.488037\pi$$
0.0375754 + 0.999294i $$0.488037\pi$$
$$510$$ 0 0
$$511$$ −7192.00 −0.622613
$$512$$ 0 0
$$513$$ −10150.0 −0.873554
$$514$$ 0 0
$$515$$ −6264.00 −0.535971
$$516$$ 0 0
$$517$$ 12138.0 1.03255
$$518$$ 0 0
$$519$$ 16150.0 1.36591
$$520$$ 0 0
$$521$$ 19260.0 1.61957 0.809785 0.586727i $$-0.199584\pi$$
0.809785 + 0.586727i $$0.199584\pi$$
$$522$$ 0 0
$$523$$ −11740.0 −0.981557 −0.490779 0.871284i $$-0.663288\pi$$
−0.490779 + 0.871284i $$0.663288\pi$$
$$524$$ 0 0
$$525$$ 3560.00 0.295945
$$526$$ 0 0
$$527$$ 8240.00 0.681101
$$528$$ 0 0
$$529$$ 529.000 0.0434783
$$530$$ 0 0
$$531$$ 816.000 0.0666881
$$532$$ 0 0
$$533$$ 5415.00 0.440056
$$534$$ 0 0
$$535$$ 2484.00 0.200734
$$536$$ 0 0
$$537$$ −1845.00 −0.148264
$$538$$ 0 0
$$539$$ −9486.00 −0.758054
$$540$$ 0 0
$$541$$ −17741.0 −1.40988 −0.704940 0.709267i $$-0.749026\pi$$
−0.704940 + 0.709267i $$0.749026\pi$$
$$542$$ 0 0
$$543$$ −6850.00 −0.541366
$$544$$ 0 0
$$545$$ −4224.00 −0.331993
$$546$$ 0 0
$$547$$ −6571.00 −0.513630 −0.256815 0.966461i $$-0.582673\pi$$
−0.256815 + 0.966461i $$0.582673\pi$$
$$548$$ 0 0
$$549$$ 1644.00 0.127804
$$550$$ 0 0
$$551$$ 17150.0 1.32598
$$552$$ 0 0
$$553$$ 10576.0 0.813268
$$554$$ 0 0
$$555$$ −8940.00 −0.683751
$$556$$ 0 0
$$557$$ 1372.00 0.104369 0.0521845 0.998637i $$-0.483382\pi$$
0.0521845 + 0.998637i $$0.483382\pi$$
$$558$$ 0 0
$$559$$ 5016.00 0.379524
$$560$$ 0 0
$$561$$ 13600.0 1.02352
$$562$$ 0 0
$$563$$ 4332.00 0.324284 0.162142 0.986767i $$-0.448160\pi$$
0.162142 + 0.986767i $$0.448160\pi$$
$$564$$ 0 0
$$565$$ 5712.00 0.425320
$$566$$ 0 0
$$567$$ −5368.00 −0.397592
$$568$$ 0 0
$$569$$ −3546.00 −0.261258 −0.130629 0.991431i $$-0.541700\pi$$
−0.130629 + 0.991431i $$0.541700\pi$$
$$570$$ 0 0
$$571$$ −6160.00 −0.451468 −0.225734 0.974189i $$-0.572478\pi$$
−0.225734 + 0.974189i $$0.572478\pi$$
$$572$$ 0 0
$$573$$ 22050.0 1.60760
$$574$$ 0 0
$$575$$ 2047.00 0.148462
$$576$$ 0 0
$$577$$ 2953.00 0.213059 0.106529 0.994310i $$-0.466026\pi$$
0.106529 + 0.994310i $$0.466026\pi$$
$$578$$ 0 0
$$579$$ 675.000 0.0484491
$$580$$ 0 0
$$581$$ −288.000 −0.0205650
$$582$$ 0 0
$$583$$ 14076.0 0.999946
$$584$$ 0 0
$$585$$ −684.000 −0.0483417
$$586$$ 0 0
$$587$$ −2949.00 −0.207356 −0.103678 0.994611i $$-0.533061\pi$$
−0.103678 + 0.994611i $$0.533061\pi$$
$$588$$ 0 0
$$589$$ 7210.00 0.504385
$$590$$ 0 0
$$591$$ 6105.00 0.424917
$$592$$ 0 0
$$593$$ 16390.0 1.13500 0.567501 0.823372i $$-0.307910\pi$$
0.567501 + 0.823372i $$0.307910\pi$$
$$594$$ 0 0
$$595$$ −3840.00 −0.264579
$$596$$ 0 0
$$597$$ −5490.00 −0.376366
$$598$$ 0 0
$$599$$ 12920.0 0.881297 0.440648 0.897680i $$-0.354749\pi$$
0.440648 + 0.897680i $$0.354749\pi$$
$$600$$ 0 0
$$601$$ −13835.0 −0.939004 −0.469502 0.882931i $$-0.655567\pi$$
−0.469502 + 0.882931i $$0.655567\pi$$
$$602$$ 0 0
$$603$$ −1852.00 −0.125073
$$604$$ 0 0
$$605$$ −1050.00 −0.0705596
$$606$$ 0 0
$$607$$ −6004.00 −0.401474 −0.200737 0.979645i $$-0.564334\pi$$
−0.200737 + 0.979645i $$0.564334\pi$$
$$608$$ 0 0
$$609$$ 9800.00 0.652079
$$610$$ 0 0
$$611$$ 20349.0 1.34735
$$612$$ 0 0
$$613$$ 16416.0 1.08162 0.540812 0.841143i $$-0.318117\pi$$
0.540812 + 0.841143i $$0.318117\pi$$
$$614$$ 0 0
$$615$$ −2850.00 −0.186867
$$616$$ 0 0
$$617$$ 3786.00 0.247032 0.123516 0.992343i $$-0.460583\pi$$
0.123516 + 0.992343i $$0.460583\pi$$
$$618$$ 0 0
$$619$$ 15824.0 1.02750 0.513748 0.857941i $$-0.328257\pi$$
0.513748 + 0.857941i $$0.328257\pi$$
$$620$$ 0 0
$$621$$ −3335.00 −0.215506
$$622$$ 0 0
$$623$$ −3680.00 −0.236655
$$624$$ 0 0
$$625$$ 3421.00 0.218944
$$626$$ 0 0
$$627$$ 11900.0 0.757959
$$628$$ 0 0
$$629$$ −23840.0 −1.51123
$$630$$ 0 0
$$631$$ −17852.0 −1.12627 −0.563135 0.826365i $$-0.690405\pi$$
−0.563135 + 0.826365i $$0.690405\pi$$
$$632$$ 0 0
$$633$$ 18380.0 1.15409
$$634$$ 0 0
$$635$$ −1566.00 −0.0978658
$$636$$ 0 0
$$637$$ −15903.0 −0.989168
$$638$$ 0 0
$$639$$ 670.000 0.0414785
$$640$$ 0 0
$$641$$ 10324.0 0.636152 0.318076 0.948065i $$-0.396963\pi$$
0.318076 + 0.948065i $$0.396963\pi$$
$$642$$ 0 0
$$643$$ −14702.0 −0.901696 −0.450848 0.892601i $$-0.648878\pi$$
−0.450848 + 0.892601i $$0.648878\pi$$
$$644$$ 0 0
$$645$$ −2640.00 −0.161163
$$646$$ 0 0
$$647$$ −11939.0 −0.725457 −0.362728 0.931895i $$-0.618155\pi$$
−0.362728 + 0.931895i $$0.618155\pi$$
$$648$$ 0 0
$$649$$ −13872.0 −0.839019
$$650$$ 0 0
$$651$$ 4120.00 0.248042
$$652$$ 0 0
$$653$$ −6159.00 −0.369097 −0.184548 0.982823i $$-0.559082\pi$$
−0.184548 + 0.982823i $$0.559082\pi$$
$$654$$ 0 0
$$655$$ −8646.00 −0.515767
$$656$$ 0 0
$$657$$ 1798.00 0.106768
$$658$$ 0 0
$$659$$ −21692.0 −1.28225 −0.641123 0.767438i $$-0.721531\pi$$
−0.641123 + 0.767438i $$0.721531\pi$$
$$660$$ 0 0
$$661$$ −16502.0 −0.971034 −0.485517 0.874227i $$-0.661369\pi$$
−0.485517 + 0.874227i $$0.661369\pi$$
$$662$$ 0 0
$$663$$ 22800.0 1.33556
$$664$$ 0 0
$$665$$ −3360.00 −0.195933
$$666$$ 0 0
$$667$$ 5635.00 0.327119
$$668$$ 0 0
$$669$$ 8280.00 0.478510
$$670$$ 0 0
$$671$$ −27948.0 −1.60793
$$672$$ 0 0
$$673$$ −27733.0 −1.58845 −0.794226 0.607622i $$-0.792124\pi$$
−0.794226 + 0.607622i $$0.792124\pi$$
$$674$$ 0 0
$$675$$ −12905.0 −0.735872
$$676$$ 0 0
$$677$$ 8814.00 0.500369 0.250184 0.968198i $$-0.419509\pi$$
0.250184 + 0.968198i $$0.419509\pi$$
$$678$$ 0 0
$$679$$ −7712.00 −0.435875
$$680$$ 0 0
$$681$$ −14700.0 −0.827174
$$682$$ 0 0
$$683$$ −22999.0 −1.28848 −0.644240 0.764823i $$-0.722826\pi$$
−0.644240 + 0.764823i $$0.722826\pi$$
$$684$$ 0 0
$$685$$ 9336.00 0.520745
$$686$$ 0 0
$$687$$ 18060.0 1.00296
$$688$$ 0 0
$$689$$ 23598.0 1.30481
$$690$$ 0 0
$$691$$ −12140.0 −0.668346 −0.334173 0.942512i $$-0.608457\pi$$
−0.334173 + 0.942512i $$0.608457\pi$$
$$692$$ 0 0
$$693$$ −544.000 −0.0298194
$$694$$ 0 0
$$695$$ 150.000 0.00818680
$$696$$ 0 0
$$697$$ −7600.00 −0.413014
$$698$$ 0 0
$$699$$ 21625.0 1.17015
$$700$$ 0 0
$$701$$ 20024.0 1.07888 0.539441 0.842024i $$-0.318636\pi$$
0.539441 + 0.842024i $$0.318636\pi$$
$$702$$ 0 0
$$703$$ −20860.0 −1.11913
$$704$$ 0 0
$$705$$ −10710.0 −0.572145
$$706$$ 0 0
$$707$$ 2480.00 0.131924
$$708$$ 0 0
$$709$$ 4956.00 0.262520 0.131260 0.991348i $$-0.458098\pi$$
0.131260 + 0.991348i $$0.458098\pi$$
$$710$$ 0 0
$$711$$ −2644.00 −0.139462
$$712$$ 0 0
$$713$$ 2369.00 0.124432
$$714$$ 0 0
$$715$$ 11628.0 0.608199
$$716$$ 0 0
$$717$$ 13675.0 0.712276
$$718$$ 0 0
$$719$$ −2760.00 −0.143158 −0.0715790 0.997435i $$-0.522804\pi$$
−0.0715790 + 0.997435i $$0.522804\pi$$
$$720$$ 0 0
$$721$$ −8352.00 −0.431407
$$722$$ 0 0
$$723$$ 33550.0 1.72578
$$724$$ 0 0
$$725$$ 21805.0 1.11699
$$726$$ 0 0
$$727$$ −7746.00 −0.395163 −0.197581 0.980287i $$-0.563309\pi$$
−0.197581 + 0.980287i $$0.563309\pi$$
$$728$$ 0 0
$$729$$ 20917.0 1.06269
$$730$$ 0 0
$$731$$ −7040.00 −0.356202
$$732$$ 0 0
$$733$$ 11976.0 0.603470 0.301735 0.953392i $$-0.402434\pi$$
0.301735 + 0.953392i $$0.402434\pi$$
$$734$$ 0 0
$$735$$ 8370.00 0.420044
$$736$$ 0 0
$$737$$ 31484.0 1.57358
$$738$$ 0 0
$$739$$ 15057.0 0.749500 0.374750 0.927126i $$-0.377728\pi$$
0.374750 + 0.927126i $$0.377728\pi$$
$$740$$ 0 0
$$741$$ 19950.0 0.989044
$$742$$ 0 0
$$743$$ −18532.0 −0.915038 −0.457519 0.889200i $$-0.651262\pi$$
−0.457519 + 0.889200i $$0.651262\pi$$
$$744$$ 0 0
$$745$$ −4932.00 −0.242543
$$746$$ 0 0
$$747$$ 72.0000 0.00352656
$$748$$ 0 0
$$749$$ 3312.00 0.161573
$$750$$ 0 0
$$751$$ 192.000 0.00932913 0.00466457 0.999989i $$-0.498515\pi$$
0.00466457 + 0.999989i $$0.498515\pi$$
$$752$$ 0 0
$$753$$ 34740.0 1.68127
$$754$$ 0 0
$$755$$ 8934.00 0.430651
$$756$$ 0 0
$$757$$ 9830.00 0.471965 0.235982 0.971757i $$-0.424169\pi$$
0.235982 + 0.971757i $$0.424169\pi$$
$$758$$ 0 0
$$759$$ 3910.00 0.186988
$$760$$ 0 0
$$761$$ −30219.0 −1.43947 −0.719736 0.694248i $$-0.755737\pi$$
−0.719736 + 0.694248i $$0.755737\pi$$
$$762$$ 0 0
$$763$$ −5632.00 −0.267224
$$764$$ 0 0
$$765$$ 960.000 0.0453711
$$766$$ 0 0
$$767$$ −23256.0 −1.09482
$$768$$ 0 0
$$769$$ 1122.00 0.0526142 0.0263071 0.999654i $$-0.491625\pi$$
0.0263071 + 0.999654i $$0.491625\pi$$
$$770$$ 0 0
$$771$$ 24645.0 1.15119
$$772$$ 0 0
$$773$$ −19300.0 −0.898024 −0.449012 0.893526i $$-0.648224\pi$$
−0.449012 + 0.893526i $$0.648224\pi$$
$$774$$ 0 0
$$775$$ 9167.00 0.424888
$$776$$ 0 0
$$777$$ −11920.0 −0.550357
$$778$$ 0 0
$$779$$ −6650.00 −0.305855
$$780$$ 0 0
$$781$$ −11390.0 −0.521852
$$782$$ 0 0
$$783$$ −35525.0 −1.62140
$$784$$ 0 0
$$785$$ 3792.00 0.172411
$$786$$ 0 0
$$787$$ −19396.0 −0.878517 −0.439258 0.898361i $$-0.644759\pi$$
−0.439258 + 0.898361i $$0.644759\pi$$
$$788$$ 0 0
$$789$$ 30690.0 1.38478
$$790$$ 0 0
$$791$$ 7616.00 0.342344
$$792$$ 0 0
$$793$$ −46854.0 −2.09815
$$794$$ 0 0
$$795$$ −12420.0 −0.554078
$$796$$ 0 0
$$797$$ 39034.0 1.73482 0.867412 0.497590i $$-0.165782\pi$$
0.867412 + 0.497590i $$0.165782\pi$$
$$798$$ 0 0
$$799$$ −28560.0 −1.26456
$$800$$ 0 0
$$801$$ 920.000 0.0405825
$$802$$ 0 0
$$803$$ −30566.0 −1.34328
$$804$$ 0 0
$$805$$ −1104.00 −0.0483365
$$806$$ 0 0
$$807$$ −10315.0 −0.449944
$$808$$ 0 0
$$809$$ −10310.0 −0.448060 −0.224030 0.974582i $$-0.571921\pi$$
−0.224030 + 0.974582i $$0.571921\pi$$
$$810$$ 0 0
$$811$$ −40693.0 −1.76193 −0.880965 0.473182i $$-0.843105\pi$$
−0.880965 + 0.473182i $$0.843105\pi$$
$$812$$ 0 0
$$813$$ −5320.00 −0.229496
$$814$$ 0 0
$$815$$ −18258.0 −0.784724
$$816$$ 0 0
$$817$$ −6160.00 −0.263784
$$818$$ 0 0
$$819$$ −912.000 −0.0389107
$$820$$ 0 0
$$821$$ 13934.0 0.592326 0.296163 0.955137i $$-0.404293\pi$$
0.296163 + 0.955137i $$0.404293\pi$$
$$822$$ 0 0
$$823$$ −6175.00 −0.261539 −0.130770 0.991413i $$-0.541745\pi$$
−0.130770 + 0.991413i $$0.541745\pi$$
$$824$$ 0 0
$$825$$ 15130.0 0.638496
$$826$$ 0 0
$$827$$ 28664.0 1.20525 0.602627 0.798023i $$-0.294121\pi$$
0.602627 + 0.798023i $$0.294121\pi$$
$$828$$ 0 0
$$829$$ 39590.0 1.65865 0.829323 0.558770i $$-0.188726\pi$$
0.829323 + 0.558770i $$0.188726\pi$$
$$830$$ 0 0
$$831$$ 28645.0 1.19577
$$832$$ 0 0
$$833$$ 22320.0 0.928382
$$834$$ 0 0
$$835$$ 13344.0 0.553040
$$836$$ 0 0
$$837$$ −14935.0 −0.616761
$$838$$ 0 0
$$839$$ 14316.0 0.589086 0.294543 0.955638i $$-0.404833\pi$$
0.294543 + 0.955638i $$0.404833\pi$$
$$840$$ 0 0
$$841$$ 35636.0 1.46115
$$842$$ 0 0
$$843$$ 4800.00 0.196110
$$844$$ 0 0
$$845$$ 6312.00 0.256970
$$846$$ 0 0
$$847$$ −1400.00 −0.0567941
$$848$$ 0 0
$$849$$ 570.000 0.0230416
$$850$$ 0 0
$$851$$ −6854.00 −0.276089
$$852$$ 0 0
$$853$$ −28366.0 −1.13861 −0.569304 0.822127i $$-0.692787\pi$$
−0.569304 + 0.822127i $$0.692787\pi$$
$$854$$ 0 0
$$855$$ 840.000 0.0335993
$$856$$ 0 0
$$857$$ 19283.0 0.768605 0.384303 0.923207i $$-0.374442\pi$$
0.384303 + 0.923207i $$0.374442\pi$$
$$858$$ 0 0
$$859$$ −26101.0 −1.03673 −0.518367 0.855158i $$-0.673460\pi$$
−0.518367 + 0.855158i $$0.673460\pi$$
$$860$$ 0 0
$$861$$ −3800.00 −0.150411
$$862$$ 0 0
$$863$$ −973.000 −0.0383793 −0.0191896 0.999816i $$-0.506109\pi$$
−0.0191896 + 0.999816i $$0.506109\pi$$
$$864$$ 0 0
$$865$$ −19380.0 −0.761780
$$866$$ 0 0
$$867$$ −7435.00 −0.291241
$$868$$ 0 0
$$869$$ 44948.0 1.75461
$$870$$ 0 0
$$871$$ 52782.0 2.05333
$$872$$ 0 0
$$873$$ 1928.00 0.0747456
$$874$$ 0 0
$$875$$ −10272.0 −0.396865
$$876$$ 0 0
$$877$$ −5694.00 −0.219239 −0.109620 0.993974i $$-0.534963\pi$$
−0.109620 + 0.993974i $$0.534963\pi$$
$$878$$ 0 0
$$879$$ −35240.0 −1.35224
$$880$$ 0 0
$$881$$ 45960.0 1.75758 0.878792 0.477205i $$-0.158350\pi$$
0.878792 + 0.477205i $$0.158350\pi$$
$$882$$ 0 0
$$883$$ 17188.0 0.655065 0.327532 0.944840i $$-0.393783\pi$$
0.327532 + 0.944840i $$0.393783\pi$$
$$884$$ 0 0
$$885$$ 12240.0 0.464907
$$886$$ 0 0
$$887$$ −8451.00 −0.319906 −0.159953 0.987125i $$-0.551134\pi$$
−0.159953 + 0.987125i $$0.551134\pi$$
$$888$$ 0 0
$$889$$ −2088.00 −0.0787731
$$890$$ 0 0
$$891$$ −22814.0 −0.857798
$$892$$ 0 0
$$893$$ −24990.0 −0.936460
$$894$$ 0 0
$$895$$ 2214.00 0.0826881
$$896$$ 0 0
$$897$$ 6555.00 0.243997
$$898$$ 0 0
$$899$$ 25235.0 0.936190
$$900$$ 0 0
$$901$$ −33120.0 −1.22463
$$902$$ 0 0
$$903$$ −3520.00 −0.129721
$$904$$ 0 0
$$905$$ 8220.00 0.301925
$$906$$ 0 0
$$907$$ 32774.0 1.19983 0.599913 0.800065i $$-0.295202\pi$$
0.599913 + 0.800065i $$0.295202\pi$$
$$908$$ 0 0
$$909$$ −620.000 −0.0226228
$$910$$ 0 0
$$911$$ 23690.0 0.861564 0.430782 0.902456i $$-0.358238\pi$$
0.430782 + 0.902456i $$0.358238\pi$$
$$912$$ 0 0
$$913$$ −1224.00 −0.0443686
$$914$$ 0 0
$$915$$ 24660.0 0.890967
$$916$$ 0 0
$$917$$ −11528.0 −0.415145
$$918$$ 0 0
$$919$$ 30044.0 1.07841 0.539206 0.842174i $$-0.318725\pi$$
0.539206 + 0.842174i $$0.318725\pi$$
$$920$$ 0 0
$$921$$ −19360.0 −0.692653
$$922$$ 0 0
$$923$$ −19095.0 −0.680953
$$924$$ 0 0
$$925$$ −26522.0 −0.942744
$$926$$ 0 0
$$927$$ 2088.00 0.0739794
$$928$$ 0 0
$$929$$ −39705.0 −1.40224 −0.701119 0.713044i $$-0.747316\pi$$
−0.701119 + 0.713044i $$0.747316\pi$$
$$930$$ 0 0
$$931$$ 19530.0 0.687508
$$932$$ 0 0
$$933$$ −24885.0 −0.873203
$$934$$ 0 0
$$935$$ −16320.0 −0.570825
$$936$$ 0 0
$$937$$ 17422.0 0.607419 0.303710 0.952765i $$-0.401775\pi$$
0.303710 + 0.952765i $$0.401775\pi$$
$$938$$ 0 0
$$939$$ 12680.0 0.440677
$$940$$ 0 0
$$941$$ 25292.0 0.876191 0.438095 0.898928i $$-0.355653\pi$$
0.438095 + 0.898928i $$0.355653\pi$$
$$942$$ 0 0
$$943$$ −2185.00 −0.0754543
$$944$$ 0 0
$$945$$ 6960.00 0.239586
$$946$$ 0 0
$$947$$ 33211.0 1.13961 0.569806 0.821779i $$-0.307018\pi$$
0.569806 + 0.821779i $$0.307018\pi$$
$$948$$ 0 0
$$949$$ −51243.0 −1.75281
$$950$$ 0 0
$$951$$ 7170.00 0.244483
$$952$$ 0 0
$$953$$ −14154.0 −0.481105 −0.240552 0.970636i $$-0.577329\pi$$
−0.240552 + 0.970636i $$0.577329\pi$$
$$954$$ 0 0
$$955$$ −26460.0 −0.896571
$$956$$ 0 0
$$957$$ 41650.0 1.40685
$$958$$ 0 0
$$959$$ 12448.0 0.419152
$$960$$ 0 0
$$961$$ −19182.0 −0.643886
$$962$$ 0 0
$$963$$ −828.000 −0.0277071
$$964$$ 0 0
$$965$$ −810.000 −0.0270205
$$966$$ 0 0
$$967$$ 46343.0 1.54115 0.770574 0.637350i $$-0.219970\pi$$
0.770574 + 0.637350i $$0.219970\pi$$
$$968$$ 0 0
$$969$$ −28000.0 −0.928266
$$970$$ 0 0
$$971$$ 11710.0 0.387015 0.193508 0.981099i $$-0.438014\pi$$
0.193508 + 0.981099i $$0.438014\pi$$
$$972$$ 0 0
$$973$$ 200.000 0.00658963
$$974$$ 0 0
$$975$$ 25365.0 0.833159
$$976$$ 0 0
$$977$$ 47854.0 1.56703 0.783513 0.621375i $$-0.213426\pi$$
0.783513 + 0.621375i $$0.213426\pi$$
$$978$$ 0 0
$$979$$ −15640.0 −0.510579
$$980$$ 0 0
$$981$$ 1408.00 0.0458246
$$982$$ 0 0
$$983$$ 22078.0 0.716357 0.358178 0.933653i $$-0.383398\pi$$
0.358178 + 0.933653i $$0.383398\pi$$
$$984$$ 0 0
$$985$$ −7326.00 −0.236980
$$986$$ 0 0
$$987$$ −14280.0 −0.460524
$$988$$ 0 0
$$989$$ −2024.00 −0.0650753
$$990$$ 0 0
$$991$$ 4288.00 0.137450 0.0687249 0.997636i $$-0.478107\pi$$
0.0687249 + 0.997636i $$0.478107\pi$$
$$992$$ 0 0
$$993$$ −27345.0 −0.873885
$$994$$ 0 0
$$995$$ 6588.00 0.209903
$$996$$ 0 0
$$997$$ −28966.0 −0.920123 −0.460061 0.887887i $$-0.652173\pi$$
−0.460061 + 0.887887i $$0.652173\pi$$
$$998$$ 0 0
$$999$$ 43210.0 1.36847
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 1472.4.a.c.1.1 1
4.3 odd 2 1472.4.a.h.1.1 1
8.3 odd 2 23.4.a.a.1.1 1
8.5 even 2 368.4.a.d.1.1 1
24.11 even 2 207.4.a.a.1.1 1
40.3 even 4 575.4.b.b.24.2 2
40.19 odd 2 575.4.a.g.1.1 1
40.27 even 4 575.4.b.b.24.1 2
56.27 even 2 1127.4.a.a.1.1 1
184.91 even 2 529.4.a.a.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
23.4.a.a.1.1 1 8.3 odd 2
207.4.a.a.1.1 1 24.11 even 2
368.4.a.d.1.1 1 8.5 even 2
529.4.a.a.1.1 1 184.91 even 2
575.4.a.g.1.1 1 40.19 odd 2
575.4.b.b.24.1 2 40.27 even 4
575.4.b.b.24.2 2 40.3 even 4
1127.4.a.a.1.1 1 56.27 even 2
1472.4.a.c.1.1 1 1.1 even 1 trivial
1472.4.a.h.1.1 1 4.3 odd 2