Properties

 Label 450.4.a.k.1.1 Level $450$ Weight $4$ Character 450.1 Self dual yes Analytic conductor $26.551$ 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] = [450,4,Mod(1,450)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(450, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("450.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$450 = 2 \cdot 3^{2} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 450.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: $$26.5508595026$$ Analytic rank: $$1$$ 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 $$\chi$$ $$=$$ 450.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+2.00000 q^{2} +4.00000 q^{4} -26.0000 q^{7} +8.00000 q^{8} +O(q^{10})$$ $$q+2.00000 q^{2} +4.00000 q^{4} -26.0000 q^{7} +8.00000 q^{8} +28.0000 q^{11} -12.0000 q^{13} -52.0000 q^{14} +16.0000 q^{16} -64.0000 q^{17} -60.0000 q^{19} +56.0000 q^{22} -58.0000 q^{23} -24.0000 q^{26} -104.000 q^{28} -90.0000 q^{29} -128.000 q^{31} +32.0000 q^{32} -128.000 q^{34} -236.000 q^{37} -120.000 q^{38} -242.000 q^{41} -362.000 q^{43} +112.000 q^{44} -116.000 q^{46} +226.000 q^{47} +333.000 q^{49} -48.0000 q^{52} -108.000 q^{53} -208.000 q^{56} -180.000 q^{58} +20.0000 q^{59} +542.000 q^{61} -256.000 q^{62} +64.0000 q^{64} +434.000 q^{67} -256.000 q^{68} +1128.00 q^{71} -632.000 q^{73} -472.000 q^{74} -240.000 q^{76} -728.000 q^{77} -720.000 q^{79} -484.000 q^{82} -478.000 q^{83} -724.000 q^{86} +224.000 q^{88} +490.000 q^{89} +312.000 q^{91} -232.000 q^{92} +452.000 q^{94} -1456.00 q^{97} +666.000 q^{98} +O(q^{100})$$

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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 2.00000 0.707107
$$3$$ 0 0
$$4$$ 4.00000 0.500000
$$5$$ 0 0
$$6$$ 0 0
$$7$$ −26.0000 −1.40387 −0.701934 0.712242i $$-0.747680\pi$$
−0.701934 + 0.712242i $$0.747680\pi$$
$$8$$ 8.00000 0.353553
$$9$$ 0 0
$$10$$ 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 $$0.540858\pi$$
$$14$$ −52.0000 −0.992685
$$15$$ 0 0
$$16$$ 16.0000 0.250000
$$17$$ −64.0000 −0.913075 −0.456538 0.889704i $$-0.650911\pi$$
−0.456538 + 0.889704i $$0.650911\pi$$
$$18$$ 0 0
$$19$$ −60.0000 −0.724471 −0.362235 0.932087i $$-0.617986\pi$$
−0.362235 + 0.932087i $$0.617986\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 56.0000 0.542693
$$23$$ −58.0000 −0.525819 −0.262909 0.964821i $$-0.584682\pi$$
−0.262909 + 0.964821i $$0.584682\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ −24.0000 −0.181030
$$27$$ 0 0
$$28$$ −104.000 −0.701934
$$29$$ −90.0000 −0.576296 −0.288148 0.957586i $$-0.593039\pi$$
−0.288148 + 0.957586i $$0.593039\pi$$
$$30$$ 0 0
$$31$$ −128.000 −0.741596 −0.370798 0.928714i $$-0.620916\pi$$
−0.370798 + 0.928714i $$0.620916\pi$$
$$32$$ 32.0000 0.176777
$$33$$ 0 0
$$34$$ −128.000 −0.645642
$$35$$ 0 0
$$36$$ 0 0
$$37$$ −236.000 −1.04860 −0.524299 0.851534i $$-0.675673\pi$$
−0.524299 + 0.851534i $$0.675673\pi$$
$$38$$ −120.000 −0.512278
$$39$$ 0 0
$$40$$ 0 0
$$41$$ −242.000 −0.921806 −0.460903 0.887450i $$-0.652474\pi$$
−0.460903 + 0.887450i $$0.652474\pi$$
$$42$$ 0 0
$$43$$ −362.000 −1.28383 −0.641913 0.766778i $$-0.721859\pi$$
−0.641913 + 0.766778i $$0.721859\pi$$
$$44$$ 112.000 0.383742
$$45$$ 0 0
$$46$$ −116.000 −0.371810
$$47$$ 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
$$51$$ 0 0
$$52$$ −48.0000 −0.128008
$$53$$ −108.000 −0.279905 −0.139952 0.990158i $$-0.544695\pi$$
−0.139952 + 0.990158i $$0.544695\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ −208.000 −0.496342
$$57$$ 0 0
$$58$$ −180.000 −0.407503
$$59$$ 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 $$0.307400\pi$$
$$62$$ −256.000 −0.524388
$$63$$ 0 0
$$64$$ 64.0000 0.125000
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 434.000 0.791366 0.395683 0.918387i $$-0.370508\pi$$
0.395683 + 0.918387i $$0.370508\pi$$
$$68$$ −256.000 −0.456538
$$69$$ 0 0
$$70$$ 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 $$0.669114\pi$$
$$74$$ −472.000 −0.741471
$$75$$ 0 0
$$76$$ −240.000 −0.362235
$$77$$ −728.000 −1.07745
$$78$$ 0 0
$$79$$ −720.000 −1.02540 −0.512698 0.858569i $$-0.671354\pi$$
−0.512698 + 0.858569i $$0.671354\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ −484.000 −0.651815
$$83$$ −478.000 −0.632136 −0.316068 0.948736i $$-0.602363\pi$$
−0.316068 + 0.948736i $$0.602363\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ −724.000 −0.907801
$$87$$ 0 0
$$88$$ 224.000 0.271346
$$89$$ 490.000 0.583594 0.291797 0.956480i $$-0.405747\pi$$
0.291797 + 0.956480i $$0.405747\pi$$
$$90$$ 0 0
$$91$$ 312.000 0.359412
$$92$$ −232.000 −0.262909
$$93$$ 0 0
$$94$$ 452.000 0.495960
$$95$$ 0 0
$$96$$ 0 0
$$97$$ −1456.00 −1.52407 −0.762033 0.647538i $$-0.775799\pi$$
−0.762033 + 0.647538i $$0.775799\pi$$
$$98$$ 666.000 0.686491
$$99$$ 0 0
$$100$$ 0 0
$$101$$ 578.000 0.569437 0.284719 0.958611i $$-0.408100\pi$$
0.284719 + 0.958611i $$0.408100\pi$$
$$102$$ 0 0
$$103$$ −1462.00 −1.39859 −0.699297 0.714831i $$-0.746503\pi$$
−0.699297 + 0.714831i $$0.746503\pi$$
$$104$$ −96.0000 −0.0905151
$$105$$ 0 0
$$106$$ −216.000 −0.197922
$$107$$ 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
$$111$$ 0 0
$$112$$ −416.000 −0.350967
$$113$$ −528.000 −0.439558 −0.219779 0.975550i $$-0.570534\pi$$
−0.219779 + 0.975550i $$0.570534\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ −360.000 −0.288148
$$117$$ 0 0
$$118$$ 40.0000 0.0312059
$$119$$ 1664.00 1.28184
$$120$$ 0 0
$$121$$ −547.000 −0.410969
$$122$$ 1084.00 0.804432
$$123$$ 0 0
$$124$$ −512.000 −0.370798
$$125$$ 0 0
$$126$$ 0 0
$$127$$ 1534.00 1.07181 0.535907 0.844277i $$-0.319970\pi$$
0.535907 + 0.844277i $$0.319970\pi$$
$$128$$ 128.000 0.0883883
$$129$$ 0 0
$$130$$ 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
$$134$$ 868.000 0.559580
$$135$$ 0 0
$$136$$ −512.000 −0.322821
$$137$$ −1224.00 −0.763309 −0.381655 0.924305i $$-0.624646\pi$$
−0.381655 + 0.924305i $$0.624646\pi$$
$$138$$ 0 0
$$139$$ 3100.00 1.89164 0.945822 0.324685i $$-0.105258\pi$$
0.945822 + 0.324685i $$0.105258\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 2256.00 1.33323
$$143$$ −336.000 −0.196488
$$144$$ 0 0
$$145$$ 0 0
$$146$$ −1264.00 −0.716503
$$147$$ 0 0
$$148$$ −944.000 −0.524299
$$149$$ −250.000 −0.137455 −0.0687275 0.997635i $$-0.521894\pi$$
−0.0687275 + 0.997635i $$0.521894\pi$$
$$150$$ 0 0
$$151$$ 2152.00 1.15978 0.579892 0.814694i $$-0.303095\pi$$
0.579892 + 0.814694i $$0.303095\pi$$
$$152$$ −480.000 −0.256139
$$153$$ 0 0
$$154$$ −1456.00 −0.761869
$$155$$ 0 0
$$156$$ 0 0
$$157$$ 524.000 0.266368 0.133184 0.991091i $$-0.457480\pi$$
0.133184 + 0.991091i $$0.457480\pi$$
$$158$$ −1440.00 −0.725065
$$159$$ 0 0
$$160$$ 0 0
$$161$$ 1508.00 0.738180
$$162$$ 0 0
$$163$$ 3518.00 1.69050 0.845249 0.534373i $$-0.179452\pi$$
0.845249 + 0.534373i $$0.179452\pi$$
$$164$$ −968.000 −0.460903
$$165$$ 0 0
$$166$$ −956.000 −0.446988
$$167$$ −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
$$171$$ 0 0
$$172$$ −1448.00 −0.641913
$$173$$ 4252.00 1.86863 0.934317 0.356444i $$-0.116011\pi$$
0.934317 + 0.356444i $$0.116011\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 448.000 0.191871
$$177$$ 0 0
$$178$$ 980.000 0.412664
$$179$$ −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 $$0.677561\pi$$
$$182$$ 624.000 0.254143
$$183$$ 0 0
$$184$$ −464.000 −0.185905
$$185$$ 0 0
$$186$$ 0 0
$$187$$ −1792.00 −0.700770
$$188$$ 904.000 0.350697
$$189$$ 0 0
$$190$$ 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 $$0.338331\pi$$
$$194$$ −2912.00 −1.07768
$$195$$ 0 0
$$196$$ 1332.00 0.485423
$$197$$ 5116.00 1.85025 0.925127 0.379659i $$-0.123959\pi$$
0.925127 + 0.379659i $$0.123959\pi$$
$$198$$ 0 0
$$199$$ −3480.00 −1.23965 −0.619826 0.784739i $$-0.712797\pi$$
−0.619826 + 0.784739i $$0.712797\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 1156.00 0.402653
$$203$$ 2340.00 0.809043
$$204$$ 0 0
$$205$$ 0 0
$$206$$ −2924.00 −0.988955
$$207$$ 0 0
$$208$$ −192.000 −0.0640039
$$209$$ −1680.00 −0.556019
$$210$$ 0 0
$$211$$ 3132.00 1.02188 0.510938 0.859618i $$-0.329298\pi$$
0.510938 + 0.859618i $$0.329298\pi$$
$$212$$ −432.000 −0.139952
$$213$$ 0 0
$$214$$ 1932.00 0.617144
$$215$$ 0 0
$$216$$ 0 0
$$217$$ 3328.00 1.04110
$$218$$ 740.000 0.229904
$$219$$ 0 0
$$220$$ 0 0
$$221$$ 768.000 0.233761
$$222$$ 0 0
$$223$$ −62.0000 −0.0186181 −0.00930903 0.999957i $$-0.502963\pi$$
−0.00930903 + 0.999957i $$0.502963\pi$$
$$224$$ −832.000 −0.248171
$$225$$ 0 0
$$226$$ −1056.00 −0.310814
$$227$$ −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
$$231$$ 0 0
$$232$$ −720.000 −0.203751
$$233$$ −2408.00 −0.677053 −0.338526 0.940957i $$-0.609928\pi$$
−0.338526 + 0.940957i $$0.609928\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 80.0000 0.0220659
$$237$$ 0 0
$$238$$ 3328.00 0.906396
$$239$$ 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 $$0.511829\pi$$
$$242$$ −1094.00 −0.290599
$$243$$ 0 0
$$244$$ 2168.00 0.568820
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 720.000 0.185476
$$248$$ −1024.00 −0.262194
$$249$$ 0 0
$$250$$ 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
$$254$$ 3068.00 0.757888
$$255$$ 0 0
$$256$$ 256.000 0.0625000
$$257$$ 1536.00 0.372813 0.186407 0.982473i $$-0.440316\pi$$
0.186407 + 0.982473i $$0.440316\pi$$
$$258$$ 0 0
$$259$$ 6136.00 1.47209
$$260$$ 0 0
$$261$$ 0 0
$$262$$ −24.0000 −0.00565926
$$263$$ −4858.00 −1.13900 −0.569500 0.821991i $$-0.692863\pi$$
−0.569500 + 0.821991i $$0.692863\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 3120.00 0.719171
$$267$$ 0 0
$$268$$ 1736.00 0.395683
$$269$$ −2610.00 −0.591578 −0.295789 0.955253i $$-0.595583\pi$$
−0.295789 + 0.955253i $$0.595583\pi$$
$$270$$ 0 0
$$271$$ −5168.00 −1.15843 −0.579213 0.815176i $$-0.696640\pi$$
−0.579213 + 0.815176i $$0.696640\pi$$
$$272$$ −1024.00 −0.228269
$$273$$ 0 0
$$274$$ −2448.00 −0.539741
$$275$$ 0 0
$$276$$ 0 0
$$277$$ 1924.00 0.417336 0.208668 0.977987i $$-0.433087\pi$$
0.208668 + 0.977987i $$0.433087\pi$$
$$278$$ 6200.00 1.33759
$$279$$ 0 0
$$280$$ 0 0
$$281$$ −3042.00 −0.645803 −0.322901 0.946433i $$-0.604658\pi$$
−0.322901 + 0.946433i $$0.604658\pi$$
$$282$$ 0 0
$$283$$ 1718.00 0.360864 0.180432 0.983587i $$-0.442250\pi$$
0.180432 + 0.983587i $$0.442250\pi$$
$$284$$ 4512.00 0.942739
$$285$$ 0 0
$$286$$ −672.000 −0.138938
$$287$$ 6292.00 1.29409
$$288$$ 0 0
$$289$$ −817.000 −0.166294
$$290$$ 0 0
$$291$$ 0 0
$$292$$ −2528.00 −0.506644
$$293$$ 2292.00 0.456997 0.228498 0.973544i $$-0.426618\pi$$
0.228498 + 0.973544i $$0.426618\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ −1888.00 −0.370736
$$297$$ 0 0
$$298$$ −500.000 −0.0971954
$$299$$ 696.000 0.134618
$$300$$ 0 0
$$301$$ 9412.00 1.80232
$$302$$ 4304.00 0.820091
$$303$$ 0 0
$$304$$ −960.000 −0.181118
$$305$$ 0 0
$$306$$ 0 0
$$307$$ −5406.00 −1.00501 −0.502503 0.864576i $$-0.667587\pi$$
−0.502503 + 0.864576i $$0.667587\pi$$
$$308$$ −2912.00 −0.538723
$$309$$ 0 0
$$310$$ 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 $$0.731072\pi$$
$$314$$ 1048.00 0.188351
$$315$$ 0 0
$$316$$ −2880.00 −0.512698
$$317$$ −3484.00 −0.617290 −0.308645 0.951177i $$-0.599876\pi$$
−0.308645 + 0.951177i $$0.599876\pi$$
$$318$$ 0 0
$$319$$ −2520.00 −0.442298
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 3016.00 0.521972
$$323$$ 3840.00 0.661496
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 7036.00 1.19536
$$327$$ 0 0
$$328$$ −1936.00 −0.325908
$$329$$ −5876.00 −0.984664
$$330$$ 0 0
$$331$$ −7868.00 −1.30654 −0.653269 0.757125i $$-0.726603\pi$$
−0.653269 + 0.757125i $$0.726603\pi$$
$$332$$ −1912.00 −0.316068
$$333$$ 0 0
$$334$$ −1068.00 −0.174965
$$335$$ 0 0
$$336$$ 0 0
$$337$$ −656.000 −0.106037 −0.0530187 0.998594i $$-0.516884\pi$$
−0.0530187 + 0.998594i $$0.516884\pi$$
$$338$$ −4106.00 −0.660760
$$339$$ 0 0
$$340$$ 0 0
$$341$$ −3584.00 −0.569163
$$342$$ 0 0
$$343$$ 260.000 0.0409291
$$344$$ −2896.00 −0.453901
$$345$$ 0 0
$$346$$ 8504.00 1.32132
$$347$$ −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
$$351$$ 0 0
$$352$$ 896.000 0.135673
$$353$$ −7808.00 −1.17727 −0.588637 0.808397i $$-0.700335\pi$$
−0.588637 + 0.808397i $$0.700335\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 1960.00 0.291797
$$357$$ 0 0
$$358$$ −5000.00 −0.738151
$$359$$ 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
$$362$$ −5156.00 −0.748600
$$363$$ 0 0
$$364$$ 1248.00 0.179706
$$365$$ 0 0
$$366$$ 0 0
$$367$$ 3214.00 0.457137 0.228569 0.973528i $$-0.426595\pi$$
0.228569 + 0.973528i $$0.426595\pi$$
$$368$$ −928.000 −0.131455
$$369$$ 0 0
$$370$$ 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 $$0.492311\pi$$
$$374$$ −3584.00 −0.495519
$$375$$ 0 0
$$376$$ 1808.00 0.247980
$$377$$ 1080.00 0.147541
$$378$$ 0 0
$$379$$ 4940.00 0.669527 0.334764 0.942302i $$-0.391344\pi$$
0.334764 + 0.942302i $$0.391344\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 1536.00 0.205729
$$383$$ 6142.00 0.819430 0.409715 0.912214i $$-0.365628\pi$$
0.409715 + 0.912214i $$0.365628\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 5216.00 0.687791
$$387$$ 0 0
$$388$$ −5824.00 −0.762033
$$389$$ −3050.00 −0.397535 −0.198768 0.980047i $$-0.563694\pi$$
−0.198768 + 0.980047i $$0.563694\pi$$
$$390$$ 0 0
$$391$$ 3712.00 0.480112
$$392$$ 2664.00 0.343246
$$393$$ 0 0
$$394$$ 10232.0 1.30833
$$395$$ 0 0
$$396$$ 0 0
$$397$$ −5396.00 −0.682160 −0.341080 0.940034i $$-0.610793\pi$$
−0.341080 + 0.940034i $$0.610793\pi$$
$$398$$ −6960.00 −0.876566
$$399$$ 0 0
$$400$$ 0 0
$$401$$ −14482.0 −1.80348 −0.901741 0.432276i $$-0.857711\pi$$
−0.901741 + 0.432276i $$0.857711\pi$$
$$402$$ 0 0
$$403$$ 1536.00 0.189860
$$404$$ 2312.00 0.284719
$$405$$ 0 0
$$406$$ 4680.00 0.572080
$$407$$ −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
$$411$$ 0 0
$$412$$ −5848.00 −0.699297
$$413$$ −520.000 −0.0619553
$$414$$ 0 0
$$415$$ 0 0
$$416$$ −384.000 −0.0452576
$$417$$ 0 0
$$418$$ −3360.00 −0.393165
$$419$$ 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 $$0.656124\pi$$
$$422$$ 6264.00 0.722575
$$423$$ 0 0
$$424$$ −864.000 −0.0989612
$$425$$ 0 0
$$426$$ 0 0
$$427$$ −14092.0 −1.59710
$$428$$ 3864.00 0.436387
$$429$$ 0 0
$$430$$ 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 $$0.756301\pi$$
$$434$$ 6656.00 0.736171
$$435$$ 0 0
$$436$$ 1480.00 0.162567
$$437$$ 3480.00 0.380940
$$438$$ 0 0
$$439$$ 1080.00 0.117416 0.0587080 0.998275i $$-0.481302\pi$$
0.0587080 + 0.998275i $$0.481302\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 1536.00 0.165294
$$443$$ −9078.00 −0.973609 −0.486805 0.873511i $$-0.661838\pi$$
−0.486805 + 0.873511i $$0.661838\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ −124.000 −0.0131650
$$447$$ 0 0
$$448$$ −1664.00 −0.175484
$$449$$ −14310.0 −1.50408 −0.752039 0.659119i $$-0.770929\pi$$
−0.752039 + 0.659119i $$0.770929\pi$$
$$450$$ 0 0
$$451$$ −6776.00 −0.707471
$$452$$ −2112.00 −0.219779
$$453$$ 0 0
$$454$$ −10628.0 −1.09867
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 2344.00 0.239929 0.119965 0.992778i $$-0.461722\pi$$
0.119965 + 0.992778i $$0.461722\pi$$
$$458$$ −380.000 −0.0387691
$$459$$ 0 0
$$460$$ 0 0
$$461$$ −11382.0 −1.14992 −0.574959 0.818182i $$-0.694982\pi$$
−0.574959 + 0.818182i $$0.694982\pi$$
$$462$$ 0 0
$$463$$ −16062.0 −1.61223 −0.806117 0.591756i $$-0.798435\pi$$
−0.806117 + 0.591756i $$0.798435\pi$$
$$464$$ −1440.00 −0.144074
$$465$$ 0 0
$$466$$ −4816.00 −0.478749
$$467$$ 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
$$471$$ 0 0
$$472$$ 160.000 0.0156030
$$473$$ −10136.0 −0.985315
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 6656.00 0.640919
$$477$$ 0 0
$$478$$ 11360.0 1.08702
$$479$$ −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
$$482$$ −556.000 −0.0525417
$$483$$ 0 0
$$484$$ −2188.00 −0.205485
$$485$$ 0 0
$$486$$ 0 0
$$487$$ 11814.0 1.09927 0.549634 0.835406i $$-0.314767\pi$$
0.549634 + 0.835406i $$0.314767\pi$$
$$488$$ 4336.00 0.402216
$$489$$ 0 0
$$490$$ 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
$$494$$ 1440.00 0.131151
$$495$$ 0 0
$$496$$ −2048.00 −0.185399
$$497$$ −29328.0 −2.64696
$$498$$ 0 0
$$499$$ 7620.00 0.683603 0.341802 0.939772i $$-0.388963\pi$$
0.341802 + 0.939772i $$0.388963\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ −6504.00 −0.578262
$$503$$ −1818.00 −0.161154 −0.0805772 0.996748i $$-0.525676\pi$$
−0.0805772 + 0.996748i $$0.525676\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ −3248.00 −0.285358
$$507$$ 0 0
$$508$$ 6136.00 0.535907
$$509$$ −17850.0 −1.55440 −0.777198 0.629256i $$-0.783360\pi$$
−0.777198 + 0.629256i $$0.783360\pi$$
$$510$$ 0 0
$$511$$ 16432.0 1.42252
$$512$$ 512.000 0.0441942
$$513$$ 0 0
$$514$$ 3072.00 0.263619
$$515$$ 0 0
$$516$$ 0 0
$$517$$ 6328.00 0.538308
$$518$$ 12272.0 1.04093
$$519$$ 0 0
$$520$$ 0 0
$$521$$ 19238.0 1.61772 0.808860 0.588001i $$-0.200085\pi$$
0.808860 + 0.588001i $$0.200085\pi$$
$$522$$ 0 0
$$523$$ 6278.00 0.524891 0.262445 0.964947i $$-0.415471\pi$$
0.262445 + 0.964947i $$0.415471\pi$$
$$524$$ −48.0000 −0.00400170
$$525$$ 0 0
$$526$$ −9716.00 −0.805395
$$527$$ 8192.00 0.677133
$$528$$ 0 0
$$529$$ −8803.00 −0.723514
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 6240.00 0.508531
$$533$$ 2904.00 0.235997
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 3472.00 0.279790
$$537$$ 0 0
$$538$$ −5220.00 −0.418309
$$539$$ 9324.00 0.745108
$$540$$ 0 0
$$541$$ −9818.00 −0.780238 −0.390119 0.920764i $$-0.627566\pi$$
−0.390119 + 0.920764i $$0.627566\pi$$
$$542$$ −10336.0 −0.819131
$$543$$ 0 0
$$544$$ −2048.00 −0.161410
$$545$$ 0 0
$$546$$ 0 0
$$547$$ 12514.0 0.978172 0.489086 0.872236i $$-0.337330\pi$$
0.489086 + 0.872236i $$0.337330\pi$$
$$548$$ −4896.00 −0.381655
$$549$$ 0 0
$$550$$ 0 0
$$551$$ 5400.00 0.417509
$$552$$ 0 0
$$553$$ 18720.0 1.43952
$$554$$ 3848.00 0.295101
$$555$$ 0 0
$$556$$ 12400.0 0.945822
$$557$$ 10596.0 0.806045 0.403022 0.915190i $$-0.367960\pi$$
0.403022 + 0.915190i $$0.367960\pi$$
$$558$$ 0 0
$$559$$ 4344.00 0.328679
$$560$$ 0 0
$$561$$ 0 0
$$562$$ −6084.00 −0.456651
$$563$$ 14002.0 1.04816 0.524080 0.851669i $$-0.324409\pi$$
0.524080 + 0.851669i $$0.324409\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 3436.00 0.255169
$$567$$ 0 0
$$568$$ 9024.00 0.666617
$$569$$ 7330.00 0.540052 0.270026 0.962853i $$-0.412968\pi$$
0.270026 + 0.962853i $$0.412968\pi$$
$$570$$ 0 0
$$571$$ 5812.00 0.425963 0.212981 0.977056i $$-0.431683\pi$$
0.212981 + 0.977056i $$0.431683\pi$$
$$572$$ −1344.00 −0.0982438
$$573$$ 0 0
$$574$$ 12584.0 0.915063
$$575$$ 0 0
$$576$$ 0 0
$$577$$ −16736.0 −1.20750 −0.603751 0.797173i $$-0.706328\pi$$
−0.603751 + 0.797173i $$0.706328\pi$$
$$578$$ −1634.00 −0.117587
$$579$$ 0 0
$$580$$ 0 0
$$581$$ 12428.0 0.887436
$$582$$ 0 0
$$583$$ −3024.00 −0.214822
$$584$$ −5056.00 −0.358251
$$585$$ 0 0
$$586$$ 4584.00 0.323146
$$587$$ −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
$$591$$ 0 0
$$592$$ −3776.00 −0.262150
$$593$$ 25872.0 1.79163 0.895814 0.444429i $$-0.146593\pi$$
0.895814 + 0.444429i $$0.146593\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ −1000.00 −0.0687275
$$597$$ 0 0
$$598$$ 1392.00 0.0951892
$$599$$ 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 $$0.644930\pi$$
$$602$$ 18824.0 1.27443
$$603$$ 0 0
$$604$$ 8608.00 0.579892
$$605$$ 0 0
$$606$$ 0 0
$$607$$ 7214.00 0.482384 0.241192 0.970477i $$-0.422462\pi$$
0.241192 + 0.970477i $$0.422462\pi$$
$$608$$ −1920.00 −0.128070
$$609$$ 0 0
$$610$$ 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 $$0.449155\pi$$
$$614$$ −10812.0 −0.710646
$$615$$ 0 0
$$616$$ −5824.00 −0.380934
$$617$$ 27656.0 1.80452 0.902260 0.431193i $$-0.141907\pi$$
0.902260 + 0.431193i $$0.141907\pi$$
$$618$$ 0 0
$$619$$ −21220.0 −1.37787 −0.688937 0.724821i $$-0.741922\pi$$
−0.688937 + 0.724821i $$0.741922\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 11376.0 0.733338
$$623$$ −12740.0 −0.819289
$$624$$ 0 0
$$625$$ 0 0
$$626$$ −14704.0 −0.938802
$$627$$ 0 0
$$628$$ 2096.00 0.133184
$$629$$ 15104.0 0.957450
$$630$$ 0 0
$$631$$ 17672.0 1.11491 0.557457 0.830206i $$-0.311777\pi$$
0.557457 + 0.830206i $$0.311777\pi$$
$$632$$ −5760.00 −0.362532
$$633$$ 0 0
$$634$$ −6968.00 −0.436490
$$635$$ 0 0
$$636$$ 0 0
$$637$$ −3996.00 −0.248551
$$638$$ −5040.00 −0.312752
$$639$$ 0 0
$$640$$ 0 0
$$641$$ −7322.00 −0.451173 −0.225586 0.974223i $$-0.572430\pi$$
−0.225586 + 0.974223i $$0.572430\pi$$
$$642$$ 0 0
$$643$$ 8238.00 0.505249 0.252624 0.967564i $$-0.418706\pi$$
0.252624 + 0.967564i $$0.418706\pi$$
$$644$$ 6032.00 0.369090
$$645$$ 0 0
$$646$$ 7680.00 0.467749
$$647$$ 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
$$651$$ 0 0
$$652$$ 14072.0 0.845249
$$653$$ −5908.00 −0.354055 −0.177027 0.984206i $$-0.556648\pi$$
−0.177027 + 0.984206i $$0.556648\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ −3872.00 −0.230452
$$657$$ 0 0
$$658$$ −11752.0 −0.696262
$$659$$ 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 $$0.756754\pi$$
$$662$$ −15736.0 −0.923863
$$663$$ 0 0
$$664$$ −3824.00 −0.223494
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 5220.00 0.303027
$$668$$ −2136.00 −0.123719
$$669$$ 0 0
$$670$$ 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 $$0.190522\pi$$
$$674$$ −1312.00 −0.0749798
$$675$$ 0 0
$$676$$ −8212.00 −0.467228
$$677$$ −26884.0 −1.52620 −0.763099 0.646282i $$-0.776323\pi$$
−0.763099 + 0.646282i $$0.776323\pi$$
$$678$$ 0 0
$$679$$ 37856.0 2.13959
$$680$$ 0 0
$$681$$ 0 0
$$682$$ −7168.00 −0.402459
$$683$$ 14282.0 0.800125 0.400063 0.916488i $$-0.368988\pi$$
0.400063 + 0.916488i $$0.368988\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 520.000 0.0289412
$$687$$ 0 0
$$688$$ −5792.00 −0.320956
$$689$$ 1296.00 0.0716599
$$690$$ 0 0
$$691$$ −3428.00 −0.188723 −0.0943613 0.995538i $$-0.530081\pi$$
−0.0943613 + 0.995538i $$0.530081\pi$$
$$692$$ 17008.0 0.934317
$$693$$ 0 0
$$694$$ −11508.0 −0.629449
$$695$$ 0 0
$$696$$ 0 0
$$697$$ 15488.0 0.841678
$$698$$ −6220.00 −0.337293
$$699$$ 0 0
$$700$$ 0 0
$$701$$ −26942.0 −1.45162 −0.725810 0.687895i $$-0.758535\pi$$
−0.725810 + 0.687895i $$0.758535\pi$$
$$702$$ 0 0
$$703$$ 14160.0 0.759679
$$704$$ 1792.00 0.0959354
$$705$$ 0 0
$$706$$ −15616.0 −0.832459
$$707$$ −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
$$711$$ 0 0
$$712$$ 3920.00 0.206332
$$713$$ 7424.00 0.389945
$$714$$ 0 0
$$715$$ 0 0
$$716$$ −10000.0 −0.521952
$$717$$ 0 0
$$718$$ 18480.0 0.960540
$$719$$ −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
$$722$$ −6518.00 −0.335976
$$723$$ 0 0
$$724$$ −10312.0 −0.529340
$$725$$ 0 0
$$726$$ 0 0
$$727$$ −17226.0 −0.878785 −0.439393 0.898295i $$-0.644806\pi$$
−0.439393 + 0.898295i $$0.644806\pi$$
$$728$$ 2496.00 0.127071
$$729$$ 0 0
$$730$$ 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 $$0.493680\pi$$
$$734$$ 6428.00 0.323245
$$735$$ 0 0
$$736$$ −1856.00 −0.0929525
$$737$$ 12152.0 0.607360
$$738$$ 0 0
$$739$$ −2060.00 −0.102542 −0.0512709 0.998685i $$-0.516327\pi$$
−0.0512709 + 0.998685i $$0.516327\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 5616.00 0.277857
$$743$$ −3258.00 −0.160867 −0.0804337 0.996760i $$-0.525631\pi$$
−0.0804337 + 0.996760i $$0.525631\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 696.000 0.0341587
$$747$$ 0 0
$$748$$ −7168.00 −0.350385
$$749$$ −25116.0 −1.22526
$$750$$ 0 0
$$751$$ −4528.00 −0.220012 −0.110006 0.993931i $$-0.535087\pi$$
−0.110006 + 0.993931i $$0.535087\pi$$
$$752$$ 3616.00 0.175348
$$753$$ 0 0
$$754$$ 2160.00 0.104327
$$755$$ 0 0
$$756$$ 0 0
$$757$$ −18236.0 −0.875560 −0.437780 0.899082i $$-0.644235\pi$$
−0.437780 + 0.899082i $$0.644235\pi$$
$$758$$ 9880.00 0.473427
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 18678.0 0.889720 0.444860 0.895600i $$-0.353253\pi$$
0.444860 + 0.895600i $$0.353253\pi$$
$$762$$ 0 0
$$763$$ −9620.00 −0.456445
$$764$$ 3072.00 0.145473
$$765$$ 0 0
$$766$$ 12284.0 0.579424
$$767$$ −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
$$771$$ 0 0
$$772$$ 10432.0 0.486342
$$773$$ 9252.00 0.430493 0.215247 0.976560i $$-0.430944\pi$$
0.215247 + 0.976560i $$0.430944\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ −11648.0 −0.538839
$$777$$ 0 0
$$778$$ −6100.00 −0.281100
$$779$$ 14520.0 0.667822
$$780$$ 0 0
$$781$$ 31584.0 1.44707
$$782$$ 7424.00 0.339491
$$783$$ 0 0
$$784$$ 5328.00 0.242711
$$785$$ 0 0
$$786$$ 0 0
$$787$$ −5726.00 −0.259352 −0.129676 0.991556i $$-0.541394\pi$$
−0.129676 + 0.991556i $$0.541394\pi$$
$$788$$ 20464.0 0.925127
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 13728.0 0.617082
$$792$$ 0 0
$$793$$ −6504.00 −0.291253
$$794$$ −10792.0 −0.482360
$$795$$ 0 0
$$796$$ −13920.0 −0.619826
$$797$$ 27236.0 1.21048 0.605238 0.796045i $$-0.293078\pi$$
0.605238 + 0.796045i $$0.293078\pi$$
$$798$$ 0 0
$$799$$ −14464.0 −0.640425
$$800$$ 0 0
$$801$$ 0 0
$$802$$ −28964.0 −1.27525
$$803$$ −17696.0 −0.777682
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 3072.00 0.134251
$$807$$ 0 0
$$808$$ 4624.00 0.201326
$$809$$ −10950.0 −0.475873 −0.237937 0.971281i $$-0.576471\pi$$
−0.237937 + 0.971281i $$0.576471\pi$$
$$810$$ 0 0
$$811$$ −8828.00 −0.382236 −0.191118 0.981567i $$-0.561211\pi$$
−0.191118 + 0.981567i $$0.561211\pi$$
$$812$$ 9360.00 0.404522
$$813$$ 0 0
$$814$$ −13216.0 −0.569067
$$815$$ 0 0
$$816$$ 0 0
$$817$$ 21720.0 0.930094
$$818$$ −2180.00 −0.0931808
$$819$$ 0 0
$$820$$ 0 0
$$821$$ 16058.0 0.682616 0.341308 0.939951i $$-0.389130\pi$$
0.341308 + 0.939951i $$0.389130\pi$$
$$822$$ 0 0
$$823$$ −41862.0 −1.77305 −0.886523 0.462684i $$-0.846887\pi$$
−0.886523 + 0.462684i $$0.846887\pi$$
$$824$$ −11696.0 −0.494478
$$825$$ 0 0
$$826$$ −1040.00 −0.0438090
$$827$$ −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
$$831$$ 0 0
$$832$$ −768.000 −0.0320019
$$833$$ −21312.0 −0.886455
$$834$$ 0 0
$$835$$ 0 0
$$836$$ −6720.00 −0.278010
$$837$$ 0 0
$$838$$ 14360.0 0.591955
$$839$$ 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
$$842$$ −16276.0 −0.666162
$$843$$ 0 0
$$844$$ 12528.0 0.510938
$$845$$ 0 0
$$846$$ 0 0
$$847$$ 14222.0 0.576947
$$848$$ −1728.00 −0.0699761
$$849$$ 0 0
$$850$$ 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 $$0.593672\pi$$
$$854$$ −28184.0 −1.12932
$$855$$ 0 0
$$856$$ 7728.00 0.308572
$$857$$ −22584.0 −0.900181 −0.450090 0.892983i $$-0.648608\pi$$
−0.450090 + 0.892983i $$0.648608\pi$$
$$858$$ 0 0
$$859$$ −26740.0 −1.06212 −0.531058 0.847336i $$-0.678205\pi$$
−0.531058 + 0.847336i $$0.678205\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 416.000 0.0164374
$$863$$ −498.000 −0.0196432 −0.00982162 0.999952i $$-0.503126\pi$$
−0.00982162 + 0.999952i $$0.503126\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ −25984.0 −1.01960
$$867$$ 0 0
$$868$$ 13312.0 0.520552
$$869$$ −20160.0 −0.786975
$$870$$ 0 0
$$871$$ −5208.00 −0.202602
$$872$$ 2960.00 0.114952
$$873$$ 0 0
$$874$$ 6960.00 0.269366
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 13244.0 0.509941 0.254970 0.966949i $$-0.417934\pi$$
0.254970 + 0.966949i $$0.417934\pi$$
$$878$$ 2160.00 0.0830256
$$879$$ 0 0
$$880$$ 0 0
$$881$$ −40842.0 −1.56186 −0.780932 0.624616i $$-0.785255\pi$$
−0.780932 + 0.624616i $$0.785255\pi$$
$$882$$ 0 0
$$883$$ 12078.0 0.460314 0.230157 0.973154i $$-0.426076\pi$$
0.230157 + 0.973154i $$0.426076\pi$$
$$884$$ 3072.00 0.116881
$$885$$ 0 0
$$886$$ −18156.0 −0.688446
$$887$$ −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
$$891$$ 0 0
$$892$$ −248.000 −0.00930903
$$893$$ −13560.0 −0.508139
$$894$$ 0 0
$$895$$ 0 0
$$896$$ −3328.00 −0.124086
$$897$$ 0 0
$$898$$ −28620.0 −1.06354
$$899$$ 11520.0 0.427379
$$900$$ 0 0
$$901$$ 6912.00 0.255574
$$902$$ −13552.0 −0.500257
$$903$$ 0 0
$$904$$ −4224.00 −0.155407
$$905$$ 0 0
$$906$$ 0 0
$$907$$ −22566.0 −0.826121 −0.413060 0.910704i $$-0.635540\pi$$
−0.413060 + 0.910704i $$0.635540\pi$$
$$908$$ −21256.0 −0.776878
$$909$$ 0 0
$$910$$ 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
$$914$$ 4688.00 0.169656
$$915$$ 0 0
$$916$$ −760.000 −0.0274139
$$917$$ 312.000 0.0112357
$$918$$ 0 0
$$919$$ 22200.0 0.796856 0.398428 0.917200i $$-0.369556\pi$$
0.398428 + 0.917200i $$0.369556\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ −22764.0 −0.813115
$$923$$ −13536.0 −0.482712
$$924$$ 0 0
$$925$$ 0 0
$$926$$ −32124.0 −1.14002
$$927$$ 0 0
$$928$$ −2880.00 −0.101876
$$929$$ 6330.00 0.223553 0.111776 0.993733i $$-0.464346\pi$$
0.111776 + 0.993733i $$0.464346\pi$$
$$930$$ 0 0
$$931$$ −19980.0 −0.703349
$$932$$ −9632.00 −0.338526
$$933$$ 0 0
$$934$$ 34332.0 1.20276
$$935$$ 0 0
$$936$$ 0 0
$$937$$ 19544.0 0.681403 0.340702 0.940172i $$-0.389335\pi$$
0.340702 + 0.940172i $$0.389335\pi$$
$$938$$ −22568.0 −0.785577
$$939$$ 0 0
$$940$$ 0 0
$$941$$ 9898.00 0.342896 0.171448 0.985193i $$-0.445155\pi$$
0.171448 + 0.985193i $$0.445155\pi$$
$$942$$ 0 0
$$943$$ 14036.0 0.484703
$$944$$ 320.000 0.0110330
$$945$$ 0 0
$$946$$ −20272.0 −0.696723
$$947$$ 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
$$951$$ 0 0
$$952$$ 13312.0 0.453198
$$953$$ 25432.0 0.864453 0.432226 0.901765i $$-0.357728\pi$$
0.432226 + 0.901765i $$0.357728\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 22720.0 0.768637
$$957$$ 0 0
$$958$$ −15040.0 −0.507224
$$959$$ 31824.0 1.07159
$$960$$ 0 0
$$961$$ −13407.0 −0.450035
$$962$$ 5664.00 0.189828
$$963$$ 0 0
$$964$$ −1112.00 −0.0371526
$$965$$ 0 0
$$966$$ 0 0
$$967$$ −12106.0 −0.402588 −0.201294 0.979531i $$-0.564515\pi$$
−0.201294 + 0.979531i $$0.564515\pi$$
$$968$$ −4376.00 −0.145300
$$969$$ 0 0
$$970$$ 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
$$974$$ 23628.0 0.777300
$$975$$ 0 0
$$976$$ 8672.00 0.284410
$$977$$ 12576.0 0.411814 0.205907 0.978572i $$-0.433986\pi$$
0.205907 + 0.978572i $$0.433986\pi$$
$$978$$ 0 0
$$979$$ 13720.0 0.447899
$$980$$ 0 0
$$981$$ 0 0
$$982$$ −28104.0 −0.913274
$$983$$ 4342.00 0.140883 0.0704417 0.997516i $$-0.477559\pi$$
0.0704417 + 0.997516i $$0.477559\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 11520.0 0.372081
$$987$$ 0 0
$$988$$ 2880.00 0.0927379
$$989$$ 20996.0 0.675060
$$990$$ 0 0
$$991$$ 26272.0 0.842137 0.421068 0.907029i $$-0.361655\pi$$
0.421068 + 0.907029i $$0.361655\pi$$
$$992$$ −4096.00 −0.131097
$$993$$ 0 0
$$994$$ −58656.0 −1.87169
$$995$$ 0 0
$$996$$ 0 0
$$997$$ −44796.0 −1.42297 −0.711486 0.702700i $$-0.751978\pi$$
−0.711486 + 0.702700i $$0.751978\pi$$
$$998$$ 15240.0 0.483381
$$999$$ 0 0
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 450.4.a.k.1.1 1
3.2 odd 2 50.4.a.b.1.1 1
5.2 odd 4 90.4.c.b.19.2 2
5.3 odd 4 90.4.c.b.19.1 2
5.4 even 2 450.4.a.j.1.1 1
12.11 even 2 400.4.a.n.1.1 1
15.2 even 4 10.4.b.a.9.1 2
15.8 even 4 10.4.b.a.9.2 yes 2
15.14 odd 2 50.4.a.d.1.1 1
20.3 even 4 720.4.f.f.289.2 2
20.7 even 4 720.4.f.f.289.1 2
21.20 even 2 2450.4.a.o.1.1 1
24.5 odd 2 1600.4.a.bh.1.1 1
24.11 even 2 1600.4.a.t.1.1 1
60.23 odd 4 80.4.c.a.49.2 2
60.47 odd 4 80.4.c.a.49.1 2
60.59 even 2 400.4.a.h.1.1 1
105.62 odd 4 490.4.c.b.99.1 2
105.83 odd 4 490.4.c.b.99.2 2
105.104 even 2 2450.4.a.bb.1.1 1
120.29 odd 2 1600.4.a.u.1.1 1
120.53 even 4 320.4.c.d.129.2 2
120.59 even 2 1600.4.a.bg.1.1 1
120.77 even 4 320.4.c.d.129.1 2
120.83 odd 4 320.4.c.c.129.1 2
120.107 odd 4 320.4.c.c.129.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
10.4.b.a.9.1 2 15.2 even 4
10.4.b.a.9.2 yes 2 15.8 even 4
50.4.a.b.1.1 1 3.2 odd 2
50.4.a.d.1.1 1 15.14 odd 2
80.4.c.a.49.1 2 60.47 odd 4
80.4.c.a.49.2 2 60.23 odd 4
90.4.c.b.19.1 2 5.3 odd 4
90.4.c.b.19.2 2 5.2 odd 4
320.4.c.c.129.1 2 120.83 odd 4
320.4.c.c.129.2 2 120.107 odd 4
320.4.c.d.129.1 2 120.77 even 4
320.4.c.d.129.2 2 120.53 even 4
400.4.a.h.1.1 1 60.59 even 2
400.4.a.n.1.1 1 12.11 even 2
450.4.a.j.1.1 1 5.4 even 2
450.4.a.k.1.1 1 1.1 even 1 trivial
490.4.c.b.99.1 2 105.62 odd 4
490.4.c.b.99.2 2 105.83 odd 4
720.4.f.f.289.1 2 20.7 even 4
720.4.f.f.289.2 2 20.3 even 4
1600.4.a.t.1.1 1 24.11 even 2
1600.4.a.u.1.1 1 120.29 odd 2
1600.4.a.bg.1.1 1 120.59 even 2
1600.4.a.bh.1.1 1 24.5 odd 2
2450.4.a.o.1.1 1 21.20 even 2
2450.4.a.bb.1.1 1 105.104 even 2