# Properties

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

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(400, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("400.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$400 = 2^{4} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 400.a (trivial)

## Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 Self dual: yes Analytic conductor: $$23.6007640023$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 10) Fricke sign: $$+1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

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

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+2.00000 q^{3} +26.0000 q^{7} -23.0000 q^{9} +O(q^{10})$$ $$q+2.00000 q^{3} +26.0000 q^{7} -23.0000 q^{9} +28.0000 q^{11} -12.0000 q^{13} +64.0000 q^{17} +60.0000 q^{19} +52.0000 q^{21} -58.0000 q^{23} -100.000 q^{27} +90.0000 q^{29} +128.000 q^{31} +56.0000 q^{33} -236.000 q^{37} -24.0000 q^{39} +242.000 q^{41} +362.000 q^{43} +226.000 q^{47} +333.000 q^{49} +128.000 q^{51} +108.000 q^{53} +120.000 q^{57} +20.0000 q^{59} +542.000 q^{61} -598.000 q^{63} -434.000 q^{67} -116.000 q^{69} +1128.00 q^{71} -632.000 q^{73} +728.000 q^{77} +720.000 q^{79} +421.000 q^{81} -478.000 q^{83} +180.000 q^{87} -490.000 q^{89} -312.000 q^{91} +256.000 q^{93} -1456.00 q^{97} -644.000 q^{99} +O(q^{100})$$

## 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
$$3$$ 2.00000 0.384900 0.192450 0.981307i $$-0.438357\pi$$
0.192450 + 0.981307i $$0.438357\pi$$
$$4$$ 0 0
$$5$$ 0 0
$$6$$ 0 0
$$7$$ 26.0000 1.40387 0.701934 0.712242i $$-0.252320\pi$$
0.701934 + 0.712242i $$0.252320\pi$$
$$8$$ 0 0
$$9$$ −23.0000 −0.851852
$$10$$ 0 0
$$11$$ 28.0000 0.767483 0.383742 0.923440i $$-0.374635\pi$$
0.383742 + 0.923440i $$0.374635\pi$$
$$12$$ 0 0
$$13$$ −12.0000 −0.256015 −0.128008 0.991773i $$-0.540858\pi$$
−0.128008 + 0.991773i $$0.540858\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ 64.0000 0.913075 0.456538 0.889704i $$-0.349089\pi$$
0.456538 + 0.889704i $$0.349089\pi$$
$$18$$ 0 0
$$19$$ 60.0000 0.724471 0.362235 0.932087i $$-0.382014\pi$$
0.362235 + 0.932087i $$0.382014\pi$$
$$20$$ 0 0
$$21$$ 52.0000 0.540349
$$22$$ 0 0
$$23$$ −58.0000 −0.525819 −0.262909 0.964821i $$-0.584682\pi$$
−0.262909 + 0.964821i $$0.584682\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ 0 0
$$27$$ −100.000 −0.712778
$$28$$ 0 0
$$29$$ 90.0000 0.576296 0.288148 0.957586i $$-0.406961\pi$$
0.288148 + 0.957586i $$0.406961\pi$$
$$30$$ 0 0
$$31$$ 128.000 0.741596 0.370798 0.928714i $$-0.379084\pi$$
0.370798 + 0.928714i $$0.379084\pi$$
$$32$$ 0 0
$$33$$ 56.0000 0.295405
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ −236.000 −1.04860 −0.524299 0.851534i $$-0.675673\pi$$
−0.524299 + 0.851534i $$0.675673\pi$$
$$38$$ 0 0
$$39$$ −24.0000 −0.0985404
$$40$$ 0 0
$$41$$ 242.000 0.921806 0.460903 0.887450i $$-0.347526\pi$$
0.460903 + 0.887450i $$0.347526\pi$$
$$42$$ 0 0
$$43$$ 362.000 1.28383 0.641913 0.766778i $$-0.278141\pi$$
0.641913 + 0.766778i $$0.278141\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ 226.000 0.701393 0.350697 0.936489i $$-0.385945\pi$$
0.350697 + 0.936489i $$0.385945\pi$$
$$48$$ 0 0
$$49$$ 333.000 0.970845
$$50$$ 0 0
$$51$$ 128.000 0.351443
$$52$$ 0 0
$$53$$ 108.000 0.279905 0.139952 0.990158i $$-0.455305\pi$$
0.139952 + 0.990158i $$0.455305\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ 120.000 0.278849
$$58$$ 0 0
$$59$$ 20.0000 0.0441318 0.0220659 0.999757i $$-0.492976\pi$$
0.0220659 + 0.999757i $$0.492976\pi$$
$$60$$ 0 0
$$61$$ 542.000 1.13764 0.568820 0.822462i $$-0.307400\pi$$
0.568820 + 0.822462i $$0.307400\pi$$
$$62$$ 0 0
$$63$$ −598.000 −1.19589
$$64$$ 0 0
$$65$$ 0 0
$$66$$ 0 0
$$67$$ −434.000 −0.791366 −0.395683 0.918387i $$-0.629492\pi$$
−0.395683 + 0.918387i $$0.629492\pi$$
$$68$$ 0 0
$$69$$ −116.000 −0.202388
$$70$$ 0 0
$$71$$ 1128.00 1.88548 0.942739 0.333531i $$-0.108240\pi$$
0.942739 + 0.333531i $$0.108240\pi$$
$$72$$ 0 0
$$73$$ −632.000 −1.01329 −0.506644 0.862155i $$-0.669114\pi$$
−0.506644 + 0.862155i $$0.669114\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ 728.000 1.07745
$$78$$ 0 0
$$79$$ 720.000 1.02540 0.512698 0.858569i $$-0.328646\pi$$
0.512698 + 0.858569i $$0.328646\pi$$
$$80$$ 0 0
$$81$$ 421.000 0.577503
$$82$$ 0 0
$$83$$ −478.000 −0.632136 −0.316068 0.948736i $$-0.602363\pi$$
−0.316068 + 0.948736i $$0.602363\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 0 0
$$87$$ 180.000 0.221816
$$88$$ 0 0
$$89$$ −490.000 −0.583594 −0.291797 0.956480i $$-0.594253\pi$$
−0.291797 + 0.956480i $$0.594253\pi$$
$$90$$ 0 0
$$91$$ −312.000 −0.359412
$$92$$ 0 0
$$93$$ 256.000 0.285440
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 0 0
$$97$$ −1456.00 −1.52407 −0.762033 0.647538i $$-0.775799\pi$$
−0.762033 + 0.647538i $$0.775799\pi$$
$$98$$ 0 0
$$99$$ −644.000 −0.653782
$$100$$ 0 0
$$101$$ −578.000 −0.569437 −0.284719 0.958611i $$-0.591900\pi$$
−0.284719 + 0.958611i $$0.591900\pi$$
$$102$$ 0 0
$$103$$ 1462.00 1.39859 0.699297 0.714831i $$-0.253497\pi$$
0.699297 + 0.714831i $$0.253497\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 966.000 0.872773 0.436387 0.899759i $$-0.356258\pi$$
0.436387 + 0.899759i $$0.356258\pi$$
$$108$$ 0 0
$$109$$ 370.000 0.325134 0.162567 0.986698i $$-0.448023\pi$$
0.162567 + 0.986698i $$0.448023\pi$$
$$110$$ 0 0
$$111$$ −472.000 −0.403606
$$112$$ 0 0
$$113$$ 528.000 0.439558 0.219779 0.975550i $$-0.429466\pi$$
0.219779 + 0.975550i $$0.429466\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 0 0
$$117$$ 276.000 0.218087
$$118$$ 0 0
$$119$$ 1664.00 1.28184
$$120$$ 0 0
$$121$$ −547.000 −0.410969
$$122$$ 0 0
$$123$$ 484.000 0.354803
$$124$$ 0 0
$$125$$ 0 0
$$126$$ 0 0
$$127$$ −1534.00 −1.07181 −0.535907 0.844277i $$-0.680030\pi$$
−0.535907 + 0.844277i $$0.680030\pi$$
$$128$$ 0 0
$$129$$ 724.000 0.494145
$$130$$ 0 0
$$131$$ −12.0000 −0.00800340 −0.00400170 0.999992i $$-0.501274\pi$$
−0.00400170 + 0.999992i $$0.501274\pi$$
$$132$$ 0 0
$$133$$ 1560.00 1.01706
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ 1224.00 0.763309 0.381655 0.924305i $$-0.375354\pi$$
0.381655 + 0.924305i $$0.375354\pi$$
$$138$$ 0 0
$$139$$ −3100.00 −1.89164 −0.945822 0.324685i $$-0.894742\pi$$
−0.945822 + 0.324685i $$0.894742\pi$$
$$140$$ 0 0
$$141$$ 452.000 0.269966
$$142$$ 0 0
$$143$$ −336.000 −0.196488
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 0 0
$$147$$ 666.000 0.373679
$$148$$ 0 0
$$149$$ 250.000 0.137455 0.0687275 0.997635i $$-0.478106\pi$$
0.0687275 + 0.997635i $$0.478106\pi$$
$$150$$ 0 0
$$151$$ −2152.00 −1.15978 −0.579892 0.814694i $$-0.696905\pi$$
−0.579892 + 0.814694i $$0.696905\pi$$
$$152$$ 0 0
$$153$$ −1472.00 −0.777805
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ 524.000 0.266368 0.133184 0.991091i $$-0.457480\pi$$
0.133184 + 0.991091i $$0.457480\pi$$
$$158$$ 0 0
$$159$$ 216.000 0.107735
$$160$$ 0 0
$$161$$ −1508.00 −0.738180
$$162$$ 0 0
$$163$$ −3518.00 −1.69050 −0.845249 0.534373i $$-0.820548\pi$$
−0.845249 + 0.534373i $$0.820548\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ −534.000 −0.247438 −0.123719 0.992317i $$-0.539482\pi$$
−0.123719 + 0.992317i $$0.539482\pi$$
$$168$$ 0 0
$$169$$ −2053.00 −0.934456
$$170$$ 0 0
$$171$$ −1380.00 −0.617142
$$172$$ 0 0
$$173$$ −4252.00 −1.86863 −0.934317 0.356444i $$-0.883989\pi$$
−0.934317 + 0.356444i $$0.883989\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ 40.0000 0.0169864
$$178$$ 0 0
$$179$$ −2500.00 −1.04390 −0.521952 0.852975i $$-0.674796\pi$$
−0.521952 + 0.852975i $$0.674796\pi$$
$$180$$ 0 0
$$181$$ −2578.00 −1.05868 −0.529340 0.848410i $$-0.677561\pi$$
−0.529340 + 0.848410i $$0.677561\pi$$
$$182$$ 0 0
$$183$$ 1084.00 0.437878
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 0 0
$$187$$ 1792.00 0.700770
$$188$$ 0 0
$$189$$ −2600.00 −1.00065
$$190$$ 0 0
$$191$$ 768.000 0.290945 0.145473 0.989362i $$-0.453530\pi$$
0.145473 + 0.989362i $$0.453530\pi$$
$$192$$ 0 0
$$193$$ 2608.00 0.972684 0.486342 0.873769i $$-0.338331\pi$$
0.486342 + 0.873769i $$0.338331\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ −5116.00 −1.85025 −0.925127 0.379659i $$-0.876041\pi$$
−0.925127 + 0.379659i $$0.876041\pi$$
$$198$$ 0 0
$$199$$ 3480.00 1.23965 0.619826 0.784739i $$-0.287203\pi$$
0.619826 + 0.784739i $$0.287203\pi$$
$$200$$ 0 0
$$201$$ −868.000 −0.304597
$$202$$ 0 0
$$203$$ 2340.00 0.809043
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 0 0
$$207$$ 1334.00 0.447920
$$208$$ 0 0
$$209$$ 1680.00 0.556019
$$210$$ 0 0
$$211$$ −3132.00 −1.02188 −0.510938 0.859618i $$-0.670702\pi$$
−0.510938 + 0.859618i $$0.670702\pi$$
$$212$$ 0 0
$$213$$ 2256.00 0.725721
$$214$$ 0 0
$$215$$ 0 0
$$216$$ 0 0
$$217$$ 3328.00 1.04110
$$218$$ 0 0
$$219$$ −1264.00 −0.390015
$$220$$ 0 0
$$221$$ −768.000 −0.233761
$$222$$ 0 0
$$223$$ 62.0000 0.0186181 0.00930903 0.999957i $$-0.497037\pi$$
0.00930903 + 0.999957i $$0.497037\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ −5314.00 −1.55376 −0.776878 0.629651i $$-0.783198\pi$$
−0.776878 + 0.629651i $$0.783198\pi$$
$$228$$ 0 0
$$229$$ −190.000 −0.0548277 −0.0274139 0.999624i $$-0.508727\pi$$
−0.0274139 + 0.999624i $$0.508727\pi$$
$$230$$ 0 0
$$231$$ 1456.00 0.414709
$$232$$ 0 0
$$233$$ 2408.00 0.677053 0.338526 0.940957i $$-0.390072\pi$$
0.338526 + 0.940957i $$0.390072\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 0 0
$$237$$ 1440.00 0.394675
$$238$$ 0 0
$$239$$ 5680.00 1.53727 0.768637 0.639685i $$-0.220935\pi$$
0.768637 + 0.639685i $$0.220935\pi$$
$$240$$ 0 0
$$241$$ −278.000 −0.0743052 −0.0371526 0.999310i $$-0.511829\pi$$
−0.0371526 + 0.999310i $$0.511829\pi$$
$$242$$ 0 0
$$243$$ 3542.00 0.935059
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ −720.000 −0.185476
$$248$$ 0 0
$$249$$ −956.000 −0.243309
$$250$$ 0 0
$$251$$ −3252.00 −0.817787 −0.408893 0.912582i $$-0.634085\pi$$
−0.408893 + 0.912582i $$0.634085\pi$$
$$252$$ 0 0
$$253$$ −1624.00 −0.403557
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ −1536.00 −0.372813 −0.186407 0.982473i $$-0.559684\pi$$
−0.186407 + 0.982473i $$0.559684\pi$$
$$258$$ 0 0
$$259$$ −6136.00 −1.47209
$$260$$ 0 0
$$261$$ −2070.00 −0.490919
$$262$$ 0 0
$$263$$ −4858.00 −1.13900 −0.569500 0.821991i $$-0.692863\pi$$
−0.569500 + 0.821991i $$0.692863\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 0 0
$$267$$ −980.000 −0.224626
$$268$$ 0 0
$$269$$ 2610.00 0.591578 0.295789 0.955253i $$-0.404417\pi$$
0.295789 + 0.955253i $$0.404417\pi$$
$$270$$ 0 0
$$271$$ 5168.00 1.15843 0.579213 0.815176i $$-0.303360\pi$$
0.579213 + 0.815176i $$0.303360\pi$$
$$272$$ 0 0
$$273$$ −624.000 −0.138338
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 0 0
$$277$$ 1924.00 0.417336 0.208668 0.977987i $$-0.433087\pi$$
0.208668 + 0.977987i $$0.433087\pi$$
$$278$$ 0 0
$$279$$ −2944.00 −0.631730
$$280$$ 0 0
$$281$$ 3042.00 0.645803 0.322901 0.946433i $$-0.395342\pi$$
0.322901 + 0.946433i $$0.395342\pi$$
$$282$$ 0 0
$$283$$ −1718.00 −0.360864 −0.180432 0.983587i $$-0.557750\pi$$
−0.180432 + 0.983587i $$0.557750\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 6292.00 1.29409
$$288$$ 0 0
$$289$$ −817.000 −0.166294
$$290$$ 0 0
$$291$$ −2912.00 −0.586613
$$292$$ 0 0
$$293$$ −2292.00 −0.456997 −0.228498 0.973544i $$-0.573382\pi$$
−0.228498 + 0.973544i $$0.573382\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 0 0
$$297$$ −2800.00 −0.547045
$$298$$ 0 0
$$299$$ 696.000 0.134618
$$300$$ 0 0
$$301$$ 9412.00 1.80232
$$302$$ 0 0
$$303$$ −1156.00 −0.219176
$$304$$ 0 0
$$305$$ 0 0
$$306$$ 0 0
$$307$$ 5406.00 1.00501 0.502503 0.864576i $$-0.332413\pi$$
0.502503 + 0.864576i $$0.332413\pi$$
$$308$$ 0 0
$$309$$ 2924.00 0.538319
$$310$$ 0 0
$$311$$ 5688.00 1.03710 0.518548 0.855048i $$-0.326473\pi$$
0.518548 + 0.855048i $$0.326473\pi$$
$$312$$ 0 0
$$313$$ −7352.00 −1.32767 −0.663833 0.747881i $$-0.731072\pi$$
−0.663833 + 0.747881i $$0.731072\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ 3484.00 0.617290 0.308645 0.951177i $$-0.400124\pi$$
0.308645 + 0.951177i $$0.400124\pi$$
$$318$$ 0 0
$$319$$ 2520.00 0.442298
$$320$$ 0 0
$$321$$ 1932.00 0.335931
$$322$$ 0 0
$$323$$ 3840.00 0.661496
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 0 0
$$327$$ 740.000 0.125144
$$328$$ 0 0
$$329$$ 5876.00 0.984664
$$330$$ 0 0
$$331$$ 7868.00 1.30654 0.653269 0.757125i $$-0.273397\pi$$
0.653269 + 0.757125i $$0.273397\pi$$
$$332$$ 0 0
$$333$$ 5428.00 0.893251
$$334$$ 0 0
$$335$$ 0 0
$$336$$ 0 0
$$337$$ −656.000 −0.106037 −0.0530187 0.998594i $$-0.516884\pi$$
−0.0530187 + 0.998594i $$0.516884\pi$$
$$338$$ 0 0
$$339$$ 1056.00 0.169186
$$340$$ 0 0
$$341$$ 3584.00 0.569163
$$342$$ 0 0
$$343$$ −260.000 −0.0409291
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ −5754.00 −0.890176 −0.445088 0.895487i $$-0.646828\pi$$
−0.445088 + 0.895487i $$0.646828\pi$$
$$348$$ 0 0
$$349$$ −3110.00 −0.477004 −0.238502 0.971142i $$-0.576656\pi$$
−0.238502 + 0.971142i $$0.576656\pi$$
$$350$$ 0 0
$$351$$ 1200.00 0.182482
$$352$$ 0 0
$$353$$ 7808.00 1.17727 0.588637 0.808397i $$-0.299665\pi$$
0.588637 + 0.808397i $$0.299665\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0 0
$$357$$ 3328.00 0.493379
$$358$$ 0 0
$$359$$ 9240.00 1.35841 0.679204 0.733949i $$-0.262325\pi$$
0.679204 + 0.733949i $$0.262325\pi$$
$$360$$ 0 0
$$361$$ −3259.00 −0.475142
$$362$$ 0 0
$$363$$ −1094.00 −0.158182
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 0 0
$$367$$ −3214.00 −0.457137 −0.228569 0.973528i $$-0.573405\pi$$
−0.228569 + 0.973528i $$0.573405\pi$$
$$368$$ 0 0
$$369$$ −5566.00 −0.785242
$$370$$ 0 0
$$371$$ 2808.00 0.392949
$$372$$ 0 0
$$373$$ 348.000 0.0483077 0.0241538 0.999708i $$-0.492311\pi$$
0.0241538 + 0.999708i $$0.492311\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ −1080.00 −0.147541
$$378$$ 0 0
$$379$$ −4940.00 −0.669527 −0.334764 0.942302i $$-0.608656\pi$$
−0.334764 + 0.942302i $$0.608656\pi$$
$$380$$ 0 0
$$381$$ −3068.00 −0.412542
$$382$$ 0 0
$$383$$ 6142.00 0.819430 0.409715 0.912214i $$-0.365628\pi$$
0.409715 + 0.912214i $$0.365628\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ −8326.00 −1.09363
$$388$$ 0 0
$$389$$ 3050.00 0.397535 0.198768 0.980047i $$-0.436306\pi$$
0.198768 + 0.980047i $$0.436306\pi$$
$$390$$ 0 0
$$391$$ −3712.00 −0.480112
$$392$$ 0 0
$$393$$ −24.0000 −0.00308051
$$394$$ 0 0
$$395$$ 0 0
$$396$$ 0 0
$$397$$ −5396.00 −0.682160 −0.341080 0.940034i $$-0.610793\pi$$
−0.341080 + 0.940034i $$0.610793\pi$$
$$398$$ 0 0
$$399$$ 3120.00 0.391467
$$400$$ 0 0
$$401$$ 14482.0 1.80348 0.901741 0.432276i $$-0.142289\pi$$
0.901741 + 0.432276i $$0.142289\pi$$
$$402$$ 0 0
$$403$$ −1536.00 −0.189860
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ −6608.00 −0.804782
$$408$$ 0 0
$$409$$ −1090.00 −0.131778 −0.0658888 0.997827i $$-0.520988\pi$$
−0.0658888 + 0.997827i $$0.520988\pi$$
$$410$$ 0 0
$$411$$ 2448.00 0.293798
$$412$$ 0 0
$$413$$ 520.000 0.0619553
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 0 0
$$417$$ −6200.00 −0.728094
$$418$$ 0 0
$$419$$ 7180.00 0.837150 0.418575 0.908182i $$-0.362530\pi$$
0.418575 + 0.908182i $$0.362530\pi$$
$$420$$ 0 0
$$421$$ −8138.00 −0.942095 −0.471047 0.882108i $$-0.656124\pi$$
−0.471047 + 0.882108i $$0.656124\pi$$
$$422$$ 0 0
$$423$$ −5198.00 −0.597483
$$424$$ 0 0
$$425$$ 0 0
$$426$$ 0 0
$$427$$ 14092.0 1.59710
$$428$$ 0 0
$$429$$ −672.000 −0.0756281
$$430$$ 0 0
$$431$$ 208.000 0.0232460 0.0116230 0.999932i $$-0.496300\pi$$
0.0116230 + 0.999932i $$0.496300\pi$$
$$432$$ 0 0
$$433$$ −12992.0 −1.44193 −0.720965 0.692971i $$-0.756301\pi$$
−0.720965 + 0.692971i $$0.756301\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ −3480.00 −0.380940
$$438$$ 0 0
$$439$$ −1080.00 −0.117416 −0.0587080 0.998275i $$-0.518698\pi$$
−0.0587080 + 0.998275i $$0.518698\pi$$
$$440$$ 0 0
$$441$$ −7659.00 −0.827017
$$442$$ 0 0
$$443$$ −9078.00 −0.973609 −0.486805 0.873511i $$-0.661838\pi$$
−0.486805 + 0.873511i $$0.661838\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 0 0
$$447$$ 500.000 0.0529065
$$448$$ 0 0
$$449$$ 14310.0 1.50408 0.752039 0.659119i $$-0.229071\pi$$
0.752039 + 0.659119i $$0.229071\pi$$
$$450$$ 0 0
$$451$$ 6776.00 0.707471
$$452$$ 0 0
$$453$$ −4304.00 −0.446401
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 2344.00 0.239929 0.119965 0.992778i $$-0.461722\pi$$
0.119965 + 0.992778i $$0.461722\pi$$
$$458$$ 0 0
$$459$$ −6400.00 −0.650820
$$460$$ 0 0
$$461$$ 11382.0 1.14992 0.574959 0.818182i $$-0.305018\pi$$
0.574959 + 0.818182i $$0.305018\pi$$
$$462$$ 0 0
$$463$$ 16062.0 1.61223 0.806117 0.591756i $$-0.201565\pi$$
0.806117 + 0.591756i $$0.201565\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ 17166.0 1.70096 0.850479 0.526008i $$-0.176312\pi$$
0.850479 + 0.526008i $$0.176312\pi$$
$$468$$ 0 0
$$469$$ −11284.0 −1.11097
$$470$$ 0 0
$$471$$ 1048.00 0.102525
$$472$$ 0 0
$$473$$ 10136.0 0.985315
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 0 0
$$477$$ −2484.00 −0.238437
$$478$$ 0 0
$$479$$ −7520.00 −0.717323 −0.358661 0.933468i $$-0.616767\pi$$
−0.358661 + 0.933468i $$0.616767\pi$$
$$480$$ 0 0
$$481$$ 2832.00 0.268458
$$482$$ 0 0
$$483$$ −3016.00 −0.284126
$$484$$ 0 0
$$485$$ 0 0
$$486$$ 0 0
$$487$$ −11814.0 −1.09927 −0.549634 0.835406i $$-0.685233\pi$$
−0.549634 + 0.835406i $$0.685233\pi$$
$$488$$ 0 0
$$489$$ −7036.00 −0.650673
$$490$$ 0 0
$$491$$ −14052.0 −1.29156 −0.645782 0.763522i $$-0.723468\pi$$
−0.645782 + 0.763522i $$0.723468\pi$$
$$492$$ 0 0
$$493$$ 5760.00 0.526202
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 29328.0 2.64696
$$498$$ 0 0
$$499$$ −7620.00 −0.683603 −0.341802 0.939772i $$-0.611037\pi$$
−0.341802 + 0.939772i $$0.611037\pi$$
$$500$$ 0 0
$$501$$ −1068.00 −0.0952390
$$502$$ 0 0
$$503$$ −1818.00 −0.161154 −0.0805772 0.996748i $$-0.525676\pi$$
−0.0805772 + 0.996748i $$0.525676\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 0 0
$$507$$ −4106.00 −0.359672
$$508$$ 0 0
$$509$$ 17850.0 1.55440 0.777198 0.629256i $$-0.216640\pi$$
0.777198 + 0.629256i $$0.216640\pi$$
$$510$$ 0 0
$$511$$ −16432.0 −1.42252
$$512$$ 0 0
$$513$$ −6000.00 −0.516387
$$514$$ 0 0
$$515$$ 0 0
$$516$$ 0 0
$$517$$ 6328.00 0.538308
$$518$$ 0 0
$$519$$ −8504.00 −0.719237
$$520$$ 0 0
$$521$$ −19238.0 −1.61772 −0.808860 0.588001i $$-0.799915\pi$$
−0.808860 + 0.588001i $$0.799915\pi$$
$$522$$ 0 0
$$523$$ −6278.00 −0.524891 −0.262445 0.964947i $$-0.584529\pi$$
−0.262445 + 0.964947i $$0.584529\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 8192.00 0.677133
$$528$$ 0 0
$$529$$ −8803.00 −0.723514
$$530$$ 0 0
$$531$$ −460.000 −0.0375938
$$532$$ 0 0
$$533$$ −2904.00 −0.235997
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 0 0
$$537$$ −5000.00 −0.401799
$$538$$ 0 0
$$539$$ 9324.00 0.745108
$$540$$ 0 0
$$541$$ −9818.00 −0.780238 −0.390119 0.920764i $$-0.627566\pi$$
−0.390119 + 0.920764i $$0.627566\pi$$
$$542$$ 0 0
$$543$$ −5156.00 −0.407486
$$544$$ 0 0
$$545$$ 0 0
$$546$$ 0 0
$$547$$ −12514.0 −0.978172 −0.489086 0.872236i $$-0.662670\pi$$
−0.489086 + 0.872236i $$0.662670\pi$$
$$548$$ 0 0
$$549$$ −12466.0 −0.969100
$$550$$ 0 0
$$551$$ 5400.00 0.417509
$$552$$ 0 0
$$553$$ 18720.0 1.43952
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ −10596.0 −0.806045 −0.403022 0.915190i $$-0.632040\pi$$
−0.403022 + 0.915190i $$0.632040\pi$$
$$558$$ 0 0
$$559$$ −4344.00 −0.328679
$$560$$ 0 0
$$561$$ 3584.00 0.269727
$$562$$ 0 0
$$563$$ 14002.0 1.04816 0.524080 0.851669i $$-0.324409\pi$$
0.524080 + 0.851669i $$0.324409\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 0 0
$$567$$ 10946.0 0.810739
$$568$$ 0 0
$$569$$ −7330.00 −0.540052 −0.270026 0.962853i $$-0.587032\pi$$
−0.270026 + 0.962853i $$0.587032\pi$$
$$570$$ 0 0
$$571$$ −5812.00 −0.425963 −0.212981 0.977056i $$-0.568317\pi$$
−0.212981 + 0.977056i $$0.568317\pi$$
$$572$$ 0 0
$$573$$ 1536.00 0.111985
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 0 0
$$577$$ −16736.0 −1.20750 −0.603751 0.797173i $$-0.706328\pi$$
−0.603751 + 0.797173i $$0.706328\pi$$
$$578$$ 0 0
$$579$$ 5216.00 0.374386
$$580$$ 0 0
$$581$$ −12428.0 −0.887436
$$582$$ 0 0
$$583$$ 3024.00 0.214822
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ −7434.00 −0.522716 −0.261358 0.965242i $$-0.584170\pi$$
−0.261358 + 0.965242i $$0.584170\pi$$
$$588$$ 0 0
$$589$$ 7680.00 0.537265
$$590$$ 0 0
$$591$$ −10232.0 −0.712163
$$592$$ 0 0
$$593$$ −25872.0 −1.79163 −0.895814 0.444429i $$-0.853407\pi$$
−0.895814 + 0.444429i $$0.853407\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ 6960.00 0.477142
$$598$$ 0 0
$$599$$ 3720.00 0.253748 0.126874 0.991919i $$-0.459506\pi$$
0.126874 + 0.991919i $$0.459506\pi$$
$$600$$ 0 0
$$601$$ −12958.0 −0.879481 −0.439740 0.898125i $$-0.644930\pi$$
−0.439740 + 0.898125i $$0.644930\pi$$
$$602$$ 0 0
$$603$$ 9982.00 0.674127
$$604$$ 0 0
$$605$$ 0 0
$$606$$ 0 0
$$607$$ −7214.00 −0.482384 −0.241192 0.970477i $$-0.577538\pi$$
−0.241192 + 0.970477i $$0.577538\pi$$
$$608$$ 0 0
$$609$$ 4680.00 0.311401
$$610$$ 0 0
$$611$$ −2712.00 −0.179568
$$612$$ 0 0
$$613$$ 4828.00 0.318109 0.159055 0.987270i $$-0.449155\pi$$
0.159055 + 0.987270i $$0.449155\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ −27656.0 −1.80452 −0.902260 0.431193i $$-0.858093\pi$$
−0.902260 + 0.431193i $$0.858093\pi$$
$$618$$ 0 0
$$619$$ 21220.0 1.37787 0.688937 0.724821i $$-0.258078\pi$$
0.688937 + 0.724821i $$0.258078\pi$$
$$620$$ 0 0
$$621$$ 5800.00 0.374792
$$622$$ 0 0
$$623$$ −12740.0 −0.819289
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 0 0
$$627$$ 3360.00 0.214012
$$628$$ 0 0
$$629$$ −15104.0 −0.957450
$$630$$ 0 0
$$631$$ −17672.0 −1.11491 −0.557457 0.830206i $$-0.688223\pi$$
−0.557457 + 0.830206i $$0.688223\pi$$
$$632$$ 0 0
$$633$$ −6264.00 −0.393320
$$634$$ 0 0
$$635$$ 0 0
$$636$$ 0 0
$$637$$ −3996.00 −0.248551
$$638$$ 0 0
$$639$$ −25944.0 −1.60615
$$640$$ 0 0
$$641$$ 7322.00 0.451173 0.225586 0.974223i $$-0.427570\pi$$
0.225586 + 0.974223i $$0.427570\pi$$
$$642$$ 0 0
$$643$$ −8238.00 −0.505249 −0.252624 0.967564i $$-0.581294\pi$$
−0.252624 + 0.967564i $$0.581294\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 6426.00 0.390467 0.195233 0.980757i $$-0.437454\pi$$
0.195233 + 0.980757i $$0.437454\pi$$
$$648$$ 0 0
$$649$$ 560.000 0.0338705
$$650$$ 0 0
$$651$$ 6656.00 0.400721
$$652$$ 0 0
$$653$$ 5908.00 0.354055 0.177027 0.984206i $$-0.443352\pi$$
0.177027 + 0.984206i $$0.443352\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 0 0
$$657$$ 14536.0 0.863171
$$658$$ 0 0
$$659$$ 26780.0 1.58301 0.791503 0.611166i $$-0.209299\pi$$
0.791503 + 0.611166i $$0.209299\pi$$
$$660$$ 0 0
$$661$$ −24538.0 −1.44390 −0.721950 0.691945i $$-0.756754\pi$$
−0.721950 + 0.691945i $$0.756754\pi$$
$$662$$ 0 0
$$663$$ −1536.00 −0.0899748
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ −5220.00 −0.303027
$$668$$ 0 0
$$669$$ 124.000 0.00716609
$$670$$ 0 0
$$671$$ 15176.0 0.873119
$$672$$ 0 0
$$673$$ 28848.0 1.65232 0.826158 0.563439i $$-0.190522\pi$$
0.826158 + 0.563439i $$0.190522\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ 26884.0 1.52620 0.763099 0.646282i $$-0.223677\pi$$
0.763099 + 0.646282i $$0.223677\pi$$
$$678$$ 0 0
$$679$$ −37856.0 −2.13959
$$680$$ 0 0
$$681$$ −10628.0 −0.598041
$$682$$ 0 0
$$683$$ 14282.0 0.800125 0.400063 0.916488i $$-0.368988\pi$$
0.400063 + 0.916488i $$0.368988\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 0 0
$$687$$ −380.000 −0.0211032
$$688$$ 0 0
$$689$$ −1296.00 −0.0716599
$$690$$ 0 0
$$691$$ 3428.00 0.188723 0.0943613 0.995538i $$-0.469919\pi$$
0.0943613 + 0.995538i $$0.469919\pi$$
$$692$$ 0 0
$$693$$ −16744.0 −0.917824
$$694$$ 0 0
$$695$$ 0 0
$$696$$ 0 0
$$697$$ 15488.0 0.841678
$$698$$ 0 0
$$699$$ 4816.00 0.260598
$$700$$ 0 0
$$701$$ 26942.0 1.45162 0.725810 0.687895i $$-0.241465\pi$$
0.725810 + 0.687895i $$0.241465\pi$$
$$702$$ 0 0
$$703$$ −14160.0 −0.759679
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ −15028.0 −0.799415
$$708$$ 0 0
$$709$$ −1950.00 −0.103292 −0.0516458 0.998665i $$-0.516447\pi$$
−0.0516458 + 0.998665i $$0.516447\pi$$
$$710$$ 0 0
$$711$$ −16560.0 −0.873486
$$712$$ 0 0
$$713$$ −7424.00 −0.389945
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ 11360.0 0.591697
$$718$$ 0 0
$$719$$ −12080.0 −0.626576 −0.313288 0.949658i $$-0.601430\pi$$
−0.313288 + 0.949658i $$0.601430\pi$$
$$720$$ 0 0
$$721$$ 38012.0 1.96344
$$722$$ 0 0
$$723$$ −556.000 −0.0286001
$$724$$ 0 0
$$725$$ 0 0
$$726$$ 0 0
$$727$$ 17226.0 0.878785 0.439393 0.898295i $$-0.355194\pi$$
0.439393 + 0.898295i $$0.355194\pi$$
$$728$$ 0 0
$$729$$ −4283.00 −0.217599
$$730$$ 0 0
$$731$$ 23168.0 1.17223
$$732$$ 0 0
$$733$$ 788.000 0.0397073 0.0198536 0.999803i $$-0.493680\pi$$
0.0198536 + 0.999803i $$0.493680\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ −12152.0 −0.607360
$$738$$ 0 0
$$739$$ 2060.00 0.102542 0.0512709 0.998685i $$-0.483673\pi$$
0.0512709 + 0.998685i $$0.483673\pi$$
$$740$$ 0 0
$$741$$ −1440.00 −0.0713896
$$742$$ 0 0
$$743$$ −3258.00 −0.160867 −0.0804337 0.996760i $$-0.525631\pi$$
−0.0804337 + 0.996760i $$0.525631\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 0 0
$$747$$ 10994.0 0.538487
$$748$$ 0 0
$$749$$ 25116.0 1.22526
$$750$$ 0 0
$$751$$ 4528.00 0.220012 0.110006 0.993931i $$-0.464913\pi$$
0.110006 + 0.993931i $$0.464913\pi$$
$$752$$ 0 0
$$753$$ −6504.00 −0.314766
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 0 0
$$757$$ −18236.0 −0.875560 −0.437780 0.899082i $$-0.644235\pi$$
−0.437780 + 0.899082i $$0.644235\pi$$
$$758$$ 0 0
$$759$$ −3248.00 −0.155329
$$760$$ 0 0
$$761$$ −18678.0 −0.889720 −0.444860 0.895600i $$-0.646747\pi$$
−0.444860 + 0.895600i $$0.646747\pi$$
$$762$$ 0 0
$$763$$ 9620.00 0.456445
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ −240.000 −0.0112984
$$768$$ 0 0
$$769$$ 27390.0 1.28441 0.642203 0.766534i $$-0.278020\pi$$
0.642203 + 0.766534i $$0.278020\pi$$
$$770$$ 0 0
$$771$$ −3072.00 −0.143496
$$772$$ 0 0
$$773$$ −9252.00 −0.430493 −0.215247 0.976560i $$-0.569056\pi$$
−0.215247 + 0.976560i $$0.569056\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 0 0
$$777$$ −12272.0 −0.566609
$$778$$ 0 0
$$779$$ 14520.0 0.667822
$$780$$ 0 0
$$781$$ 31584.0 1.44707
$$782$$ 0 0
$$783$$ −9000.00 −0.410771
$$784$$ 0 0
$$785$$ 0 0
$$786$$ 0 0
$$787$$ 5726.00 0.259352 0.129676 0.991556i $$-0.458606\pi$$
0.129676 + 0.991556i $$0.458606\pi$$
$$788$$ 0 0
$$789$$ −9716.00 −0.438401
$$790$$ 0 0
$$791$$ 13728.0 0.617082
$$792$$ 0 0
$$793$$ −6504.00 −0.291253
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ −27236.0 −1.21048 −0.605238 0.796045i $$-0.706922\pi$$
−0.605238 + 0.796045i $$0.706922\pi$$
$$798$$ 0 0
$$799$$ 14464.0 0.640425
$$800$$ 0 0
$$801$$ 11270.0 0.497136
$$802$$ 0 0
$$803$$ −17696.0 −0.777682
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ 5220.00 0.227699
$$808$$ 0 0
$$809$$ 10950.0 0.475873 0.237937 0.971281i $$-0.423529\pi$$
0.237937 + 0.971281i $$0.423529\pi$$
$$810$$ 0 0
$$811$$ 8828.00 0.382236 0.191118 0.981567i $$-0.438789\pi$$
0.191118 + 0.981567i $$0.438789\pi$$
$$812$$ 0 0
$$813$$ 10336.0 0.445879
$$814$$ 0 0
$$815$$ 0 0
$$816$$ 0 0
$$817$$ 21720.0 0.930094
$$818$$ 0 0
$$819$$ 7176.00 0.306166
$$820$$ 0 0
$$821$$ −16058.0 −0.682616 −0.341308 0.939951i $$-0.610870\pi$$
−0.341308 + 0.939951i $$0.610870\pi$$
$$822$$ 0 0
$$823$$ 41862.0 1.77305 0.886523 0.462684i $$-0.153113\pi$$
0.886523 + 0.462684i $$0.153113\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ −12154.0 −0.511047 −0.255524 0.966803i $$-0.582248\pi$$
−0.255524 + 0.966803i $$0.582248\pi$$
$$828$$ 0 0
$$829$$ −15390.0 −0.644773 −0.322386 0.946608i $$-0.604485\pi$$
−0.322386 + 0.946608i $$0.604485\pi$$
$$830$$ 0 0
$$831$$ 3848.00 0.160633
$$832$$ 0 0
$$833$$ 21312.0 0.886455
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 0 0
$$837$$ −12800.0 −0.528593
$$838$$ 0 0
$$839$$ 4280.00 0.176117 0.0880584 0.996115i $$-0.471934\pi$$
0.0880584 + 0.996115i $$0.471934\pi$$
$$840$$ 0 0
$$841$$ −16289.0 −0.667883
$$842$$ 0 0
$$843$$ 6084.00 0.248570
$$844$$ 0 0
$$845$$ 0 0
$$846$$ 0 0
$$847$$ −14222.0 −0.576947
$$848$$ 0 0
$$849$$ −3436.00 −0.138897
$$850$$ 0 0
$$851$$ 13688.0 0.551373
$$852$$ 0 0
$$853$$ −14452.0 −0.580102 −0.290051 0.957011i $$-0.593672\pi$$
−0.290051 + 0.957011i $$0.593672\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ 22584.0 0.900181 0.450090 0.892983i $$-0.351392\pi$$
0.450090 + 0.892983i $$0.351392\pi$$
$$858$$ 0 0
$$859$$ 26740.0 1.06212 0.531058 0.847336i $$-0.321795\pi$$
0.531058 + 0.847336i $$0.321795\pi$$
$$860$$ 0 0
$$861$$ 12584.0 0.498097
$$862$$ 0 0
$$863$$ −498.000 −0.0196432 −0.00982162 0.999952i $$-0.503126\pi$$
−0.00982162 + 0.999952i $$0.503126\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 0 0
$$867$$ −1634.00 −0.0640064
$$868$$ 0 0
$$869$$ 20160.0 0.786975
$$870$$ 0 0
$$871$$ 5208.00 0.202602
$$872$$ 0 0
$$873$$ 33488.0 1.29828
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 13244.0 0.509941 0.254970 0.966949i $$-0.417934\pi$$
0.254970 + 0.966949i $$0.417934\pi$$
$$878$$ 0 0
$$879$$ −4584.00 −0.175898
$$880$$ 0 0
$$881$$ 40842.0 1.56186 0.780932 0.624616i $$-0.214745\pi$$
0.780932 + 0.624616i $$0.214745\pi$$
$$882$$ 0 0
$$883$$ −12078.0 −0.460314 −0.230157 0.973154i $$-0.573924\pi$$
−0.230157 + 0.973154i $$0.573924\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ −18294.0 −0.692506 −0.346253 0.938141i $$-0.612546\pi$$
−0.346253 + 0.938141i $$0.612546\pi$$
$$888$$ 0 0
$$889$$ −39884.0 −1.50469
$$890$$ 0 0
$$891$$ 11788.0 0.443224
$$892$$ 0 0
$$893$$ 13560.0 0.508139
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 0 0
$$897$$ 1392.00 0.0518144
$$898$$ 0 0
$$899$$ 11520.0 0.427379
$$900$$ 0 0
$$901$$ 6912.00 0.255574
$$902$$ 0 0
$$903$$ 18824.0 0.693714
$$904$$ 0 0
$$905$$ 0 0
$$906$$ 0 0
$$907$$ 22566.0 0.826121 0.413060 0.910704i $$-0.364460\pi$$
0.413060 + 0.910704i $$0.364460\pi$$
$$908$$ 0 0
$$909$$ 13294.0 0.485076
$$910$$ 0 0
$$911$$ 6768.00 0.246140 0.123070 0.992398i $$-0.460726\pi$$
0.123070 + 0.992398i $$0.460726\pi$$
$$912$$ 0 0
$$913$$ −13384.0 −0.485154
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ −312.000 −0.0112357
$$918$$ 0 0
$$919$$ −22200.0 −0.796856 −0.398428 0.917200i $$-0.630444\pi$$
−0.398428 + 0.917200i $$0.630444\pi$$
$$920$$ 0 0
$$921$$ 10812.0 0.386827
$$922$$ 0 0
$$923$$ −13536.0 −0.482712
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 0 0
$$927$$ −33626.0 −1.19139
$$928$$ 0 0
$$929$$ −6330.00 −0.223553 −0.111776 0.993733i $$-0.535654\pi$$
−0.111776 + 0.993733i $$0.535654\pi$$
$$930$$ 0 0
$$931$$ 19980.0 0.703349
$$932$$ 0 0
$$933$$ 11376.0 0.399178
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ 19544.0 0.681403 0.340702 0.940172i $$-0.389335\pi$$
0.340702 + 0.940172i $$0.389335\pi$$
$$938$$ 0 0
$$939$$ −14704.0 −0.511019
$$940$$ 0 0
$$941$$ −9898.00 −0.342896 −0.171448 0.985193i $$-0.554845\pi$$
−0.171448 + 0.985193i $$0.554845\pi$$
$$942$$ 0 0
$$943$$ −14036.0 −0.484703
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 41406.0 1.42082 0.710409 0.703789i $$-0.248510\pi$$
0.710409 + 0.703789i $$0.248510\pi$$
$$948$$ 0 0
$$949$$ 7584.00 0.259417
$$950$$ 0 0
$$951$$ 6968.00 0.237595
$$952$$ 0 0
$$953$$ −25432.0 −0.864453 −0.432226 0.901765i $$-0.642272\pi$$
−0.432226 + 0.901765i $$0.642272\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 0 0
$$957$$ 5040.00 0.170240
$$958$$ 0 0
$$959$$ 31824.0 1.07159
$$960$$ 0 0
$$961$$ −13407.0 −0.450035
$$962$$ 0 0
$$963$$ −22218.0 −0.743474
$$964$$ 0 0
$$965$$ 0 0
$$966$$ 0 0
$$967$$ 12106.0 0.402588 0.201294 0.979531i $$-0.435485\pi$$
0.201294 + 0.979531i $$0.435485\pi$$
$$968$$ 0 0
$$969$$ 7680.00 0.254610
$$970$$ 0 0
$$971$$ −7812.00 −0.258186 −0.129093 0.991632i $$-0.541207\pi$$
−0.129093 + 0.991632i $$0.541207\pi$$
$$972$$ 0 0
$$973$$ −80600.0 −2.65562
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ −12576.0 −0.411814 −0.205907 0.978572i $$-0.566014\pi$$
−0.205907 + 0.978572i $$0.566014\pi$$
$$978$$ 0 0
$$979$$ −13720.0 −0.447899
$$980$$ 0 0
$$981$$ −8510.00 −0.276966
$$982$$ 0 0
$$983$$ 4342.00 0.140883 0.0704417 0.997516i $$-0.477559\pi$$
0.0704417 + 0.997516i $$0.477559\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 0 0
$$987$$ 11752.0 0.378997
$$988$$ 0 0
$$989$$ −20996.0 −0.675060
$$990$$ 0 0
$$991$$ −26272.0 −0.842137 −0.421068 0.907029i $$-0.638345\pi$$
−0.421068 + 0.907029i $$0.638345\pi$$
$$992$$ 0 0
$$993$$ 15736.0 0.502887
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 0 0
$$997$$ −44796.0 −1.42297 −0.711486 0.702700i $$-0.751978\pi$$
−0.711486 + 0.702700i $$0.751978\pi$$
$$998$$ 0 0
$$999$$ 23600.0 0.747418
Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000

## Twists

By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 400.4.a.n.1.1 1
4.3 odd 2 50.4.a.b.1.1 1
5.2 odd 4 80.4.c.a.49.1 2
5.3 odd 4 80.4.c.a.49.2 2
5.4 even 2 400.4.a.h.1.1 1
8.3 odd 2 1600.4.a.bh.1.1 1
8.5 even 2 1600.4.a.t.1.1 1
12.11 even 2 450.4.a.k.1.1 1
15.2 even 4 720.4.f.f.289.1 2
15.8 even 4 720.4.f.f.289.2 2
20.3 even 4 10.4.b.a.9.2 yes 2
20.7 even 4 10.4.b.a.9.1 2
20.19 odd 2 50.4.a.d.1.1 1
28.27 even 2 2450.4.a.o.1.1 1
40.3 even 4 320.4.c.d.129.2 2
40.13 odd 4 320.4.c.c.129.1 2
40.19 odd 2 1600.4.a.u.1.1 1
40.27 even 4 320.4.c.d.129.1 2
40.29 even 2 1600.4.a.bg.1.1 1
40.37 odd 4 320.4.c.c.129.2 2
60.23 odd 4 90.4.c.b.19.1 2
60.47 odd 4 90.4.c.b.19.2 2
60.59 even 2 450.4.a.j.1.1 1
140.27 odd 4 490.4.c.b.99.1 2
140.83 odd 4 490.4.c.b.99.2 2
140.139 even 2 2450.4.a.bb.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
10.4.b.a.9.1 2 20.7 even 4
10.4.b.a.9.2 yes 2 20.3 even 4
50.4.a.b.1.1 1 4.3 odd 2
50.4.a.d.1.1 1 20.19 odd 2
80.4.c.a.49.1 2 5.2 odd 4
80.4.c.a.49.2 2 5.3 odd 4
90.4.c.b.19.1 2 60.23 odd 4
90.4.c.b.19.2 2 60.47 odd 4
320.4.c.c.129.1 2 40.13 odd 4
320.4.c.c.129.2 2 40.37 odd 4
320.4.c.d.129.1 2 40.27 even 4
320.4.c.d.129.2 2 40.3 even 4
400.4.a.h.1.1 1 5.4 even 2
400.4.a.n.1.1 1 1.1 even 1 trivial
450.4.a.j.1.1 1 60.59 even 2
450.4.a.k.1.1 1 12.11 even 2
490.4.c.b.99.1 2 140.27 odd 4
490.4.c.b.99.2 2 140.83 odd 4
720.4.f.f.289.1 2 15.2 even 4
720.4.f.f.289.2 2 15.8 even 4
1600.4.a.t.1.1 1 8.5 even 2
1600.4.a.u.1.1 1 40.19 odd 2
1600.4.a.bg.1.1 1 40.29 even 2
1600.4.a.bh.1.1 1 8.3 odd 2
2450.4.a.o.1.1 1 28.27 even 2
2450.4.a.bb.1.1 1 140.139 even 2