# Properties

 Label 350.6.a.i.1.1 Level $350$ Weight $6$ Character 350.1 Self dual yes Analytic conductor $56.134$ 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] = [350,6,Mod(1,350)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(350, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("350.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$350 = 2 \cdot 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 350.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: $$56.1343369345$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 14) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 350.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+4.00000 q^{2} -10.0000 q^{3} +16.0000 q^{4} -40.0000 q^{6} -49.0000 q^{7} +64.0000 q^{8} -143.000 q^{9} +O(q^{10})$$ $$q+4.00000 q^{2} -10.0000 q^{3} +16.0000 q^{4} -40.0000 q^{6} -49.0000 q^{7} +64.0000 q^{8} -143.000 q^{9} -336.000 q^{11} -160.000 q^{12} -584.000 q^{13} -196.000 q^{14} +256.000 q^{16} +1458.00 q^{17} -572.000 q^{18} +470.000 q^{19} +490.000 q^{21} -1344.00 q^{22} +4200.00 q^{23} -640.000 q^{24} -2336.00 q^{26} +3860.00 q^{27} -784.000 q^{28} +4866.00 q^{29} -7372.00 q^{31} +1024.00 q^{32} +3360.00 q^{33} +5832.00 q^{34} -2288.00 q^{36} -14330.0 q^{37} +1880.00 q^{38} +5840.00 q^{39} +6222.00 q^{41} +1960.00 q^{42} -3704.00 q^{43} -5376.00 q^{44} +16800.0 q^{46} +1812.00 q^{47} -2560.00 q^{48} +2401.00 q^{49} -14580.0 q^{51} -9344.00 q^{52} +37242.0 q^{53} +15440.0 q^{54} -3136.00 q^{56} -4700.00 q^{57} +19464.0 q^{58} +34302.0 q^{59} +24476.0 q^{61} -29488.0 q^{62} +7007.00 q^{63} +4096.00 q^{64} +13440.0 q^{66} +17452.0 q^{67} +23328.0 q^{68} -42000.0 q^{69} +28224.0 q^{71} -9152.00 q^{72} -3602.00 q^{73} -57320.0 q^{74} +7520.00 q^{76} +16464.0 q^{77} +23360.0 q^{78} +42872.0 q^{79} -3851.00 q^{81} +24888.0 q^{82} +35202.0 q^{83} +7840.00 q^{84} -14816.0 q^{86} -48660.0 q^{87} -21504.0 q^{88} +26730.0 q^{89} +28616.0 q^{91} +67200.0 q^{92} +73720.0 q^{93} +7248.00 q^{94} -10240.0 q^{96} +16978.0 q^{97} +9604.00 q^{98} +48048.0 q^{99} +O(q^{100})$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 4.00000 0.707107
$$3$$ −10.0000 −0.641500 −0.320750 0.947164i $$-0.603935\pi$$
−0.320750 + 0.947164i $$0.603935\pi$$
$$4$$ 16.0000 0.500000
$$5$$ 0 0
$$6$$ −40.0000 −0.453609
$$7$$ −49.0000 −0.377964
$$8$$ 64.0000 0.353553
$$9$$ −143.000 −0.588477
$$10$$ 0 0
$$11$$ −336.000 −0.837255 −0.418627 0.908158i $$-0.637489\pi$$
−0.418627 + 0.908158i $$0.637489\pi$$
$$12$$ −160.000 −0.320750
$$13$$ −584.000 −0.958417 −0.479208 0.877701i $$-0.659076\pi$$
−0.479208 + 0.877701i $$0.659076\pi$$
$$14$$ −196.000 −0.267261
$$15$$ 0 0
$$16$$ 256.000 0.250000
$$17$$ 1458.00 1.22359 0.611794 0.791017i $$-0.290448\pi$$
0.611794 + 0.791017i $$0.290448\pi$$
$$18$$ −572.000 −0.416116
$$19$$ 470.000 0.298685 0.149343 0.988786i $$-0.452284\pi$$
0.149343 + 0.988786i $$0.452284\pi$$
$$20$$ 0 0
$$21$$ 490.000 0.242464
$$22$$ −1344.00 −0.592028
$$23$$ 4200.00 1.65550 0.827751 0.561096i $$-0.189620\pi$$
0.827751 + 0.561096i $$0.189620\pi$$
$$24$$ −640.000 −0.226805
$$25$$ 0 0
$$26$$ −2336.00 −0.677703
$$27$$ 3860.00 1.01901
$$28$$ −784.000 −0.188982
$$29$$ 4866.00 1.07443 0.537214 0.843446i $$-0.319477\pi$$
0.537214 + 0.843446i $$0.319477\pi$$
$$30$$ 0 0
$$31$$ −7372.00 −1.37778 −0.688892 0.724864i $$-0.741903\pi$$
−0.688892 + 0.724864i $$0.741903\pi$$
$$32$$ 1024.00 0.176777
$$33$$ 3360.00 0.537099
$$34$$ 5832.00 0.865207
$$35$$ 0 0
$$36$$ −2288.00 −0.294239
$$37$$ −14330.0 −1.72085 −0.860423 0.509581i $$-0.829800\pi$$
−0.860423 + 0.509581i $$0.829800\pi$$
$$38$$ 1880.00 0.211202
$$39$$ 5840.00 0.614825
$$40$$ 0 0
$$41$$ 6222.00 0.578057 0.289028 0.957321i $$-0.406668\pi$$
0.289028 + 0.957321i $$0.406668\pi$$
$$42$$ 1960.00 0.171448
$$43$$ −3704.00 −0.305492 −0.152746 0.988265i $$-0.548812\pi$$
−0.152746 + 0.988265i $$0.548812\pi$$
$$44$$ −5376.00 −0.418627
$$45$$ 0 0
$$46$$ 16800.0 1.17062
$$47$$ 1812.00 0.119650 0.0598251 0.998209i $$-0.480946\pi$$
0.0598251 + 0.998209i $$0.480946\pi$$
$$48$$ −2560.00 −0.160375
$$49$$ 2401.00 0.142857
$$50$$ 0 0
$$51$$ −14580.0 −0.784932
$$52$$ −9344.00 −0.479208
$$53$$ 37242.0 1.82114 0.910570 0.413355i $$-0.135643\pi$$
0.910570 + 0.413355i $$0.135643\pi$$
$$54$$ 15440.0 0.720548
$$55$$ 0 0
$$56$$ −3136.00 −0.133631
$$57$$ −4700.00 −0.191607
$$58$$ 19464.0 0.759735
$$59$$ 34302.0 1.28289 0.641445 0.767169i $$-0.278335\pi$$
0.641445 + 0.767169i $$0.278335\pi$$
$$60$$ 0 0
$$61$$ 24476.0 0.842201 0.421101 0.907014i $$-0.361644\pi$$
0.421101 + 0.907014i $$0.361644\pi$$
$$62$$ −29488.0 −0.974240
$$63$$ 7007.00 0.222424
$$64$$ 4096.00 0.125000
$$65$$ 0 0
$$66$$ 13440.0 0.379786
$$67$$ 17452.0 0.474961 0.237481 0.971392i $$-0.423678\pi$$
0.237481 + 0.971392i $$0.423678\pi$$
$$68$$ 23328.0 0.611794
$$69$$ −42000.0 −1.06201
$$70$$ 0 0
$$71$$ 28224.0 0.664466 0.332233 0.943197i $$-0.392198\pi$$
0.332233 + 0.943197i $$0.392198\pi$$
$$72$$ −9152.00 −0.208058
$$73$$ −3602.00 −0.0791109 −0.0395555 0.999217i $$-0.512594\pi$$
−0.0395555 + 0.999217i $$0.512594\pi$$
$$74$$ −57320.0 −1.21682
$$75$$ 0 0
$$76$$ 7520.00 0.149343
$$77$$ 16464.0 0.316453
$$78$$ 23360.0 0.434747
$$79$$ 42872.0 0.772869 0.386435 0.922317i $$-0.373706\pi$$
0.386435 + 0.922317i $$0.373706\pi$$
$$80$$ 0 0
$$81$$ −3851.00 −0.0652170
$$82$$ 24888.0 0.408748
$$83$$ 35202.0 0.560883 0.280441 0.959871i $$-0.409519\pi$$
0.280441 + 0.959871i $$0.409519\pi$$
$$84$$ 7840.00 0.121232
$$85$$ 0 0
$$86$$ −14816.0 −0.216015
$$87$$ −48660.0 −0.689246
$$88$$ −21504.0 −0.296014
$$89$$ 26730.0 0.357704 0.178852 0.983876i $$-0.442762\pi$$
0.178852 + 0.983876i $$0.442762\pi$$
$$90$$ 0 0
$$91$$ 28616.0 0.362248
$$92$$ 67200.0 0.827751
$$93$$ 73720.0 0.883849
$$94$$ 7248.00 0.0846055
$$95$$ 0 0
$$96$$ −10240.0 −0.113402
$$97$$ 16978.0 0.183213 0.0916067 0.995795i $$-0.470800\pi$$
0.0916067 + 0.995795i $$0.470800\pi$$
$$98$$ 9604.00 0.101015
$$99$$ 48048.0 0.492705
$$100$$ 0 0
$$101$$ 99204.0 0.967667 0.483833 0.875160i $$-0.339244\pi$$
0.483833 + 0.875160i $$0.339244\pi$$
$$102$$ −58320.0 −0.555031
$$103$$ 131644. 1.22267 0.611333 0.791373i $$-0.290634\pi$$
0.611333 + 0.791373i $$0.290634\pi$$
$$104$$ −37376.0 −0.338852
$$105$$ 0 0
$$106$$ 148968. 1.28774
$$107$$ −48852.0 −0.412499 −0.206250 0.978499i $$-0.566126\pi$$
−0.206250 + 0.978499i $$0.566126\pi$$
$$108$$ 61760.0 0.509504
$$109$$ −56374.0 −0.454478 −0.227239 0.973839i $$-0.572970\pi$$
−0.227239 + 0.973839i $$0.572970\pi$$
$$110$$ 0 0
$$111$$ 143300. 1.10392
$$112$$ −12544.0 −0.0944911
$$113$$ −8742.00 −0.0644043 −0.0322021 0.999481i $$-0.510252\pi$$
−0.0322021 + 0.999481i $$0.510252\pi$$
$$114$$ −18800.0 −0.135486
$$115$$ 0 0
$$116$$ 77856.0 0.537214
$$117$$ 83512.0 0.564007
$$118$$ 137208. 0.907140
$$119$$ −71442.0 −0.462473
$$120$$ 0 0
$$121$$ −48155.0 −0.299005
$$122$$ 97904.0 0.595526
$$123$$ −62220.0 −0.370823
$$124$$ −117952. −0.688892
$$125$$ 0 0
$$126$$ 28028.0 0.157277
$$127$$ −315992. −1.73847 −0.869234 0.494401i $$-0.835388\pi$$
−0.869234 + 0.494401i $$0.835388\pi$$
$$128$$ 16384.0 0.0883883
$$129$$ 37040.0 0.195973
$$130$$ 0 0
$$131$$ −24666.0 −0.125580 −0.0627900 0.998027i $$-0.520000\pi$$
−0.0627900 + 0.998027i $$0.520000\pi$$
$$132$$ 53760.0 0.268550
$$133$$ −23030.0 −0.112892
$$134$$ 69808.0 0.335848
$$135$$ 0 0
$$136$$ 93312.0 0.432604
$$137$$ −303234. −1.38031 −0.690155 0.723662i $$-0.742458\pi$$
−0.690155 + 0.723662i $$0.742458\pi$$
$$138$$ −168000. −0.750951
$$139$$ 250586. 1.10007 0.550034 0.835142i $$-0.314615\pi$$
0.550034 + 0.835142i $$0.314615\pi$$
$$140$$ 0 0
$$141$$ −18120.0 −0.0767557
$$142$$ 112896. 0.469848
$$143$$ 196224. 0.802439
$$144$$ −36608.0 −0.147119
$$145$$ 0 0
$$146$$ −14408.0 −0.0559399
$$147$$ −24010.0 −0.0916429
$$148$$ −229280. −0.860423
$$149$$ −60594.0 −0.223596 −0.111798 0.993731i $$-0.535661\pi$$
−0.111798 + 0.993731i $$0.535661\pi$$
$$150$$ 0 0
$$151$$ 124448. 0.444166 0.222083 0.975028i $$-0.428714\pi$$
0.222083 + 0.975028i $$0.428714\pi$$
$$152$$ 30080.0 0.105601
$$153$$ −208494. −0.720054
$$154$$ 65856.0 0.223766
$$155$$ 0 0
$$156$$ 93440.0 0.307412
$$157$$ −76040.0 −0.246203 −0.123101 0.992394i $$-0.539284\pi$$
−0.123101 + 0.992394i $$0.539284\pi$$
$$158$$ 171488. 0.546501
$$159$$ −372420. −1.16826
$$160$$ 0 0
$$161$$ −205800. −0.625721
$$162$$ −15404.0 −0.0461154
$$163$$ −124256. −0.366310 −0.183155 0.983084i $$-0.558631\pi$$
−0.183155 + 0.983084i $$0.558631\pi$$
$$164$$ 99552.0 0.289028
$$165$$ 0 0
$$166$$ 140808. 0.396604
$$167$$ 72420.0 0.200940 0.100470 0.994940i $$-0.467965\pi$$
0.100470 + 0.994940i $$0.467965\pi$$
$$168$$ 31360.0 0.0857241
$$169$$ −30237.0 −0.0814370
$$170$$ 0 0
$$171$$ −67210.0 −0.175770
$$172$$ −59264.0 −0.152746
$$173$$ 441552. 1.12167 0.560837 0.827926i $$-0.310479\pi$$
0.560837 + 0.827926i $$0.310479\pi$$
$$174$$ −194640. −0.487370
$$175$$ 0 0
$$176$$ −86016.0 −0.209314
$$177$$ −343020. −0.822974
$$178$$ 106920. 0.252935
$$179$$ −10692.0 −0.0249417 −0.0124709 0.999922i $$-0.503970\pi$$
−0.0124709 + 0.999922i $$0.503970\pi$$
$$180$$ 0 0
$$181$$ −546064. −1.23893 −0.619465 0.785024i $$-0.712651\pi$$
−0.619465 + 0.785024i $$0.712651\pi$$
$$182$$ 114464. 0.256148
$$183$$ −244760. −0.540272
$$184$$ 268800. 0.585308
$$185$$ 0 0
$$186$$ 294880. 0.624975
$$187$$ −489888. −1.02445
$$188$$ 28992.0 0.0598251
$$189$$ −189140. −0.385149
$$190$$ 0 0
$$191$$ −575976. −1.14241 −0.571204 0.820808i $$-0.693523\pi$$
−0.571204 + 0.820808i $$0.693523\pi$$
$$192$$ −40960.0 −0.0801875
$$193$$ 413938. 0.799912 0.399956 0.916534i $$-0.369025\pi$$
0.399956 + 0.916534i $$0.369025\pi$$
$$194$$ 67912.0 0.129551
$$195$$ 0 0
$$196$$ 38416.0 0.0714286
$$197$$ 494946. 0.908641 0.454320 0.890838i $$-0.349882\pi$$
0.454320 + 0.890838i $$0.349882\pi$$
$$198$$ 192192. 0.348395
$$199$$ 520364. 0.931482 0.465741 0.884921i $$-0.345788\pi$$
0.465741 + 0.884921i $$0.345788\pi$$
$$200$$ 0 0
$$201$$ −174520. −0.304688
$$202$$ 396816. 0.684244
$$203$$ −238434. −0.406095
$$204$$ −233280. −0.392466
$$205$$ 0 0
$$206$$ 526576. 0.864556
$$207$$ −600600. −0.974225
$$208$$ −149504. −0.239604
$$209$$ −157920. −0.250076
$$210$$ 0 0
$$211$$ 183284. 0.283412 0.141706 0.989909i $$-0.454741\pi$$
0.141706 + 0.989909i $$0.454741\pi$$
$$212$$ 595872. 0.910570
$$213$$ −282240. −0.426255
$$214$$ −195408. −0.291681
$$215$$ 0 0
$$216$$ 247040. 0.360274
$$217$$ 361228. 0.520753
$$218$$ −225496. −0.321364
$$219$$ 36020.0 0.0507497
$$220$$ 0 0
$$221$$ −851472. −1.17271
$$222$$ 573200. 0.780591
$$223$$ 1.27746e6 1.72023 0.860115 0.510100i $$-0.170392\pi$$
0.860115 + 0.510100i $$0.170392\pi$$
$$224$$ −50176.0 −0.0668153
$$225$$ 0 0
$$226$$ −34968.0 −0.0455407
$$227$$ 1.28764e6 1.65856 0.829279 0.558835i $$-0.188752\pi$$
0.829279 + 0.558835i $$0.188752\pi$$
$$228$$ −75200.0 −0.0958034
$$229$$ 350936. 0.442221 0.221110 0.975249i $$-0.429032\pi$$
0.221110 + 0.975249i $$0.429032\pi$$
$$230$$ 0 0
$$231$$ −164640. −0.203004
$$232$$ 311424. 0.379867
$$233$$ −836154. −1.00901 −0.504506 0.863408i $$-0.668325\pi$$
−0.504506 + 0.863408i $$0.668325\pi$$
$$234$$ 334048. 0.398813
$$235$$ 0 0
$$236$$ 548832. 0.641445
$$237$$ −428720. −0.495796
$$238$$ −285768. −0.327018
$$239$$ 774336. 0.876869 0.438434 0.898763i $$-0.355533\pi$$
0.438434 + 0.898763i $$0.355533\pi$$
$$240$$ 0 0
$$241$$ −1.15285e6 −1.27859 −0.639293 0.768963i $$-0.720773\pi$$
−0.639293 + 0.768963i $$0.720773\pi$$
$$242$$ −192620. −0.211428
$$243$$ −899470. −0.977172
$$244$$ 391616. 0.421101
$$245$$ 0 0
$$246$$ −248880. −0.262212
$$247$$ −274480. −0.286265
$$248$$ −471808. −0.487120
$$249$$ −352020. −0.359806
$$250$$ 0 0
$$251$$ 1.35801e6 1.36056 0.680282 0.732951i $$-0.261858\pi$$
0.680282 + 0.732951i $$0.261858\pi$$
$$252$$ 112112. 0.111212
$$253$$ −1.41120e6 −1.38608
$$254$$ −1.26397e6 −1.22928
$$255$$ 0 0
$$256$$ 65536.0 0.0625000
$$257$$ 317742. 0.300083 0.150042 0.988680i $$-0.452059\pi$$
0.150042 + 0.988680i $$0.452059\pi$$
$$258$$ 148160. 0.138574
$$259$$ 702170. 0.650418
$$260$$ 0 0
$$261$$ −695838. −0.632276
$$262$$ −98664.0 −0.0887985
$$263$$ −1.05101e6 −0.936951 −0.468475 0.883477i $$-0.655196\pi$$
−0.468475 + 0.883477i $$0.655196\pi$$
$$264$$ 215040. 0.189893
$$265$$ 0 0
$$266$$ −92120.0 −0.0798270
$$267$$ −267300. −0.229467
$$268$$ 279232. 0.237481
$$269$$ 1.18958e6 1.00234 0.501169 0.865349i $$-0.332903\pi$$
0.501169 + 0.865349i $$0.332903\pi$$
$$270$$ 0 0
$$271$$ −1.43008e6 −1.18287 −0.591435 0.806353i $$-0.701438\pi$$
−0.591435 + 0.806353i $$0.701438\pi$$
$$272$$ 373248. 0.305897
$$273$$ −286160. −0.232382
$$274$$ −1.21294e6 −0.976026
$$275$$ 0 0
$$276$$ −672000. −0.531003
$$277$$ −63302.0 −0.0495699 −0.0247849 0.999693i $$-0.507890\pi$$
−0.0247849 + 0.999693i $$0.507890\pi$$
$$278$$ 1.00234e6 0.777866
$$279$$ 1.05420e6 0.810795
$$280$$ 0 0
$$281$$ −496614. −0.375192 −0.187596 0.982246i $$-0.560070\pi$$
−0.187596 + 0.982246i $$0.560070\pi$$
$$282$$ −72480.0 −0.0542744
$$283$$ 1.15842e6 0.859803 0.429902 0.902876i $$-0.358548\pi$$
0.429902 + 0.902876i $$0.358548\pi$$
$$284$$ 451584. 0.332233
$$285$$ 0 0
$$286$$ 784896. 0.567410
$$287$$ −304878. −0.218485
$$288$$ −146432. −0.104029
$$289$$ 705907. 0.497168
$$290$$ 0 0
$$291$$ −169780. −0.117531
$$292$$ −57632.0 −0.0395555
$$293$$ −1.43886e6 −0.979151 −0.489575 0.871961i $$-0.662848\pi$$
−0.489575 + 0.871961i $$0.662848\pi$$
$$294$$ −96040.0 −0.0648013
$$295$$ 0 0
$$296$$ −917120. −0.608411
$$297$$ −1.29696e6 −0.853170
$$298$$ −242376. −0.158106
$$299$$ −2.45280e6 −1.58666
$$300$$ 0 0
$$301$$ 181496. 0.115465
$$302$$ 497792. 0.314073
$$303$$ −992040. −0.620758
$$304$$ 120320. 0.0746713
$$305$$ 0 0
$$306$$ −833976. −0.509155
$$307$$ 989098. 0.598954 0.299477 0.954104i $$-0.403188\pi$$
0.299477 + 0.954104i $$0.403188\pi$$
$$308$$ 263424. 0.158226
$$309$$ −1.31644e6 −0.784341
$$310$$ 0 0
$$311$$ −2.22050e6 −1.30182 −0.650909 0.759155i $$-0.725612\pi$$
−0.650909 + 0.759155i $$0.725612\pi$$
$$312$$ 373760. 0.217373
$$313$$ −2.33008e6 −1.34434 −0.672171 0.740396i $$-0.734638\pi$$
−0.672171 + 0.740396i $$0.734638\pi$$
$$314$$ −304160. −0.174092
$$315$$ 0 0
$$316$$ 685952. 0.386435
$$317$$ −427542. −0.238963 −0.119481 0.992836i $$-0.538123\pi$$
−0.119481 + 0.992836i $$0.538123\pi$$
$$318$$ −1.48968e6 −0.826086
$$319$$ −1.63498e6 −0.899569
$$320$$ 0 0
$$321$$ 488520. 0.264618
$$322$$ −823200. −0.442452
$$323$$ 685260. 0.365468
$$324$$ −61616.0 −0.0326085
$$325$$ 0 0
$$326$$ −497024. −0.259020
$$327$$ 563740. 0.291548
$$328$$ 398208. 0.204374
$$329$$ −88788.0 −0.0452235
$$330$$ 0 0
$$331$$ −396616. −0.198976 −0.0994879 0.995039i $$-0.531720\pi$$
−0.0994879 + 0.995039i $$0.531720\pi$$
$$332$$ 563232. 0.280441
$$333$$ 2.04919e6 1.01268
$$334$$ 289680. 0.142086
$$335$$ 0 0
$$336$$ 125440. 0.0606161
$$337$$ 3.21819e6 1.54361 0.771805 0.635860i $$-0.219354\pi$$
0.771805 + 0.635860i $$0.219354\pi$$
$$338$$ −120948. −0.0575847
$$339$$ 87420.0 0.0413154
$$340$$ 0 0
$$341$$ 2.47699e6 1.15356
$$342$$ −268840. −0.124288
$$343$$ −117649. −0.0539949
$$344$$ −237056. −0.108008
$$345$$ 0 0
$$346$$ 1.76621e6 0.793143
$$347$$ −2.78018e6 −1.23951 −0.619755 0.784796i $$-0.712768\pi$$
−0.619755 + 0.784796i $$0.712768\pi$$
$$348$$ −778560. −0.344623
$$349$$ −338800. −0.148895 −0.0744475 0.997225i $$-0.523719\pi$$
−0.0744475 + 0.997225i $$0.523719\pi$$
$$350$$ 0 0
$$351$$ −2.25424e6 −0.976635
$$352$$ −344064. −0.148007
$$353$$ 362046. 0.154642 0.0773209 0.997006i $$-0.475363\pi$$
0.0773209 + 0.997006i $$0.475363\pi$$
$$354$$ −1.37208e6 −0.581931
$$355$$ 0 0
$$356$$ 427680. 0.178852
$$357$$ 714420. 0.296676
$$358$$ −42768.0 −0.0176365
$$359$$ 876528. 0.358946 0.179473 0.983763i $$-0.442561\pi$$
0.179473 + 0.983763i $$0.442561\pi$$
$$360$$ 0 0
$$361$$ −2.25520e6 −0.910787
$$362$$ −2.18426e6 −0.876056
$$363$$ 481550. 0.191812
$$364$$ 457856. 0.181124
$$365$$ 0 0
$$366$$ −979040. −0.382030
$$367$$ −2.98062e6 −1.15516 −0.577578 0.816335i $$-0.696002\pi$$
−0.577578 + 0.816335i $$0.696002\pi$$
$$368$$ 1.07520e6 0.413875
$$369$$ −889746. −0.340173
$$370$$ 0 0
$$371$$ −1.82486e6 −0.688326
$$372$$ 1.17952e6 0.441924
$$373$$ −3.91441e6 −1.45678 −0.728391 0.685162i $$-0.759732\pi$$
−0.728391 + 0.685162i $$0.759732\pi$$
$$374$$ −1.95955e6 −0.724399
$$375$$ 0 0
$$376$$ 115968. 0.0423027
$$377$$ −2.84174e6 −1.02975
$$378$$ −756560. −0.272342
$$379$$ 3.60661e6 1.28974 0.644868 0.764294i $$-0.276912\pi$$
0.644868 + 0.764294i $$0.276912\pi$$
$$380$$ 0 0
$$381$$ 3.15992e6 1.11523
$$382$$ −2.30390e6 −0.807804
$$383$$ 2.66644e6 0.928826 0.464413 0.885619i $$-0.346265\pi$$
0.464413 + 0.885619i $$0.346265\pi$$
$$384$$ −163840. −0.0567012
$$385$$ 0 0
$$386$$ 1.65575e6 0.565623
$$387$$ 529672. 0.179775
$$388$$ 271648. 0.0916067
$$389$$ −213366. −0.0714910 −0.0357455 0.999361i $$-0.511381\pi$$
−0.0357455 + 0.999361i $$0.511381\pi$$
$$390$$ 0 0
$$391$$ 6.12360e6 2.02565
$$392$$ 153664. 0.0505076
$$393$$ 246660. 0.0805596
$$394$$ 1.97978e6 0.642506
$$395$$ 0 0
$$396$$ 768768. 0.246353
$$397$$ 4.09408e6 1.30371 0.651854 0.758345i $$-0.273992\pi$$
0.651854 + 0.758345i $$0.273992\pi$$
$$398$$ 2.08146e6 0.658657
$$399$$ 230300. 0.0724205
$$400$$ 0 0
$$401$$ 942366. 0.292657 0.146328 0.989236i $$-0.453254\pi$$
0.146328 + 0.989236i $$0.453254\pi$$
$$402$$ −698080. −0.215447
$$403$$ 4.30525e6 1.32049
$$404$$ 1.58726e6 0.483833
$$405$$ 0 0
$$406$$ −953736. −0.287153
$$407$$ 4.81488e6 1.44079
$$408$$ −933120. −0.277515
$$409$$ −4.84561e6 −1.43232 −0.716160 0.697936i $$-0.754102\pi$$
−0.716160 + 0.697936i $$0.754102\pi$$
$$410$$ 0 0
$$411$$ 3.03234e6 0.885469
$$412$$ 2.10630e6 0.611333
$$413$$ −1.68080e6 −0.484887
$$414$$ −2.40240e6 −0.688881
$$415$$ 0 0
$$416$$ −598016. −0.169426
$$417$$ −2.50586e6 −0.705694
$$418$$ −631680. −0.176830
$$419$$ −1.73485e6 −0.482754 −0.241377 0.970431i $$-0.577599\pi$$
−0.241377 + 0.970431i $$0.577599\pi$$
$$420$$ 0 0
$$421$$ −1.65145e6 −0.454109 −0.227055 0.973882i $$-0.572910\pi$$
−0.227055 + 0.973882i $$0.572910\pi$$
$$422$$ 733136. 0.200403
$$423$$ −259116. −0.0704115
$$424$$ 2.38349e6 0.643870
$$425$$ 0 0
$$426$$ −1.12896e6 −0.301408
$$427$$ −1.19932e6 −0.318322
$$428$$ −781632. −0.206250
$$429$$ −1.96224e6 −0.514765
$$430$$ 0 0
$$431$$ 4.14360e6 1.07445 0.537223 0.843440i $$-0.319473\pi$$
0.537223 + 0.843440i $$0.319473\pi$$
$$432$$ 988160. 0.254752
$$433$$ 3.03966e6 0.779121 0.389561 0.921001i $$-0.372627\pi$$
0.389561 + 0.921001i $$0.372627\pi$$
$$434$$ 1.44491e6 0.368228
$$435$$ 0 0
$$436$$ −901984. −0.227239
$$437$$ 1.97400e6 0.494474
$$438$$ 144080. 0.0358855
$$439$$ 2.54271e6 0.629703 0.314852 0.949141i $$-0.398045\pi$$
0.314852 + 0.949141i $$0.398045\pi$$
$$440$$ 0 0
$$441$$ −343343. −0.0840682
$$442$$ −3.40589e6 −0.829229
$$443$$ 2.43210e6 0.588806 0.294403 0.955681i $$-0.404879\pi$$
0.294403 + 0.955681i $$0.404879\pi$$
$$444$$ 2.29280e6 0.551961
$$445$$ 0 0
$$446$$ 5.10986e6 1.21639
$$447$$ 605940. 0.143437
$$448$$ −200704. −0.0472456
$$449$$ 1.82853e6 0.428042 0.214021 0.976829i $$-0.431344\pi$$
0.214021 + 0.976829i $$0.431344\pi$$
$$450$$ 0 0
$$451$$ −2.09059e6 −0.483981
$$452$$ −139872. −0.0322021
$$453$$ −1.24448e6 −0.284933
$$454$$ 5.15057e6 1.17278
$$455$$ 0 0
$$456$$ −300800. −0.0677432
$$457$$ −1.58063e6 −0.354030 −0.177015 0.984208i $$-0.556644\pi$$
−0.177015 + 0.984208i $$0.556644\pi$$
$$458$$ 1.40374e6 0.312697
$$459$$ 5.62788e6 1.24685
$$460$$ 0 0
$$461$$ 5.09604e6 1.11681 0.558407 0.829567i $$-0.311413\pi$$
0.558407 + 0.829567i $$0.311413\pi$$
$$462$$ −658560. −0.143546
$$463$$ 7.02338e6 1.52263 0.761313 0.648384i $$-0.224555\pi$$
0.761313 + 0.648384i $$0.224555\pi$$
$$464$$ 1.24570e6 0.268607
$$465$$ 0 0
$$466$$ −3.34462e6 −0.713479
$$467$$ 4.24845e6 0.901443 0.450722 0.892665i $$-0.351167\pi$$
0.450722 + 0.892665i $$0.351167\pi$$
$$468$$ 1.33619e6 0.282003
$$469$$ −855148. −0.179518
$$470$$ 0 0
$$471$$ 760400. 0.157939
$$472$$ 2.19533e6 0.453570
$$473$$ 1.24454e6 0.255775
$$474$$ −1.71488e6 −0.350581
$$475$$ 0 0
$$476$$ −1.14307e6 −0.231236
$$477$$ −5.32561e6 −1.07170
$$478$$ 3.09734e6 0.620040
$$479$$ 559284. 0.111377 0.0556883 0.998448i $$-0.482265\pi$$
0.0556883 + 0.998448i $$0.482265\pi$$
$$480$$ 0 0
$$481$$ 8.36872e6 1.64929
$$482$$ −4.61140e6 −0.904097
$$483$$ 2.05800e6 0.401400
$$484$$ −770480. −0.149502
$$485$$ 0 0
$$486$$ −3.59788e6 −0.690965
$$487$$ 1.32057e6 0.252312 0.126156 0.992010i $$-0.459736\pi$$
0.126156 + 0.992010i $$0.459736\pi$$
$$488$$ 1.56646e6 0.297763
$$489$$ 1.24256e6 0.234988
$$490$$ 0 0
$$491$$ 6.27193e6 1.17408 0.587040 0.809558i $$-0.300293\pi$$
0.587040 + 0.809558i $$0.300293\pi$$
$$492$$ −995520. −0.185412
$$493$$ 7.09463e6 1.31466
$$494$$ −1.09792e6 −0.202420
$$495$$ 0 0
$$496$$ −1.88723e6 −0.344446
$$497$$ −1.38298e6 −0.251144
$$498$$ −1.40808e6 −0.254422
$$499$$ −3.93785e6 −0.707959 −0.353979 0.935253i $$-0.615172\pi$$
−0.353979 + 0.935253i $$0.615172\pi$$
$$500$$ 0 0
$$501$$ −724200. −0.128903
$$502$$ 5.43204e6 0.962063
$$503$$ 7.59830e6 1.33905 0.669525 0.742790i $$-0.266498\pi$$
0.669525 + 0.742790i $$0.266498\pi$$
$$504$$ 448448. 0.0786386
$$505$$ 0 0
$$506$$ −5.64480e6 −0.980104
$$507$$ 302370. 0.0522419
$$508$$ −5.05587e6 −0.869234
$$509$$ −7.82664e6 −1.33900 −0.669501 0.742812i $$-0.733492\pi$$
−0.669501 + 0.742812i $$0.733492\pi$$
$$510$$ 0 0
$$511$$ 176498. 0.0299011
$$512$$ 262144. 0.0441942
$$513$$ 1.81420e6 0.304363
$$514$$ 1.27097e6 0.212191
$$515$$ 0 0
$$516$$ 592640. 0.0979866
$$517$$ −608832. −0.100178
$$518$$ 2.80868e6 0.459915
$$519$$ −4.41552e6 −0.719554
$$520$$ 0 0
$$521$$ 8.94454e6 1.44366 0.721828 0.692072i $$-0.243302\pi$$
0.721828 + 0.692072i $$0.243302\pi$$
$$522$$ −2.78335e6 −0.447087
$$523$$ −4.07481e6 −0.651407 −0.325704 0.945472i $$-0.605601\pi$$
−0.325704 + 0.945472i $$0.605601\pi$$
$$524$$ −394656. −0.0627900
$$525$$ 0 0
$$526$$ −4.20403e6 −0.662524
$$527$$ −1.07484e7 −1.68584
$$528$$ 860160. 0.134275
$$529$$ 1.12037e7 1.74069
$$530$$ 0 0
$$531$$ −4.90519e6 −0.754952
$$532$$ −368480. −0.0564462
$$533$$ −3.63365e6 −0.554019
$$534$$ −1.06920e6 −0.162258
$$535$$ 0 0
$$536$$ 1.11693e6 0.167924
$$537$$ 106920. 0.0160001
$$538$$ 4.75834e6 0.708760
$$539$$ −806736. −0.119608
$$540$$ 0 0
$$541$$ −1.18676e7 −1.74329 −0.871644 0.490140i $$-0.836946\pi$$
−0.871644 + 0.490140i $$0.836946\pi$$
$$542$$ −5.72032e6 −0.836416
$$543$$ 5.46064e6 0.794775
$$544$$ 1.49299e6 0.216302
$$545$$ 0 0
$$546$$ −1.14464e6 −0.164319
$$547$$ 5.37801e6 0.768516 0.384258 0.923226i $$-0.374457\pi$$
0.384258 + 0.923226i $$0.374457\pi$$
$$548$$ −4.85174e6 −0.690155
$$549$$ −3.50007e6 −0.495616
$$550$$ 0 0
$$551$$ 2.28702e6 0.320916
$$552$$ −2.68800e6 −0.375475
$$553$$ −2.10073e6 −0.292117
$$554$$ −253208. −0.0350512
$$555$$ 0 0
$$556$$ 4.00938e6 0.550034
$$557$$ 5.64878e6 0.771466 0.385733 0.922611i $$-0.373949\pi$$
0.385733 + 0.922611i $$0.373949\pi$$
$$558$$ 4.21678e6 0.573318
$$559$$ 2.16314e6 0.292789
$$560$$ 0 0
$$561$$ 4.89888e6 0.657188
$$562$$ −1.98646e6 −0.265301
$$563$$ −4.56407e6 −0.606850 −0.303425 0.952855i $$-0.598130\pi$$
−0.303425 + 0.952855i $$0.598130\pi$$
$$564$$ −289920. −0.0383778
$$565$$ 0 0
$$566$$ 4.63367e6 0.607973
$$567$$ 188699. 0.0246497
$$568$$ 1.80634e6 0.234924
$$569$$ 8.00165e6 1.03609 0.518047 0.855352i $$-0.326659\pi$$
0.518047 + 0.855352i $$0.326659\pi$$
$$570$$ 0 0
$$571$$ −1.37164e7 −1.76055 −0.880275 0.474464i $$-0.842642\pi$$
−0.880275 + 0.474464i $$0.842642\pi$$
$$572$$ 3.13958e6 0.401220
$$573$$ 5.75976e6 0.732855
$$574$$ −1.21951e6 −0.154492
$$575$$ 0 0
$$576$$ −585728. −0.0735597
$$577$$ −6.09797e6 −0.762510 −0.381255 0.924470i $$-0.624508\pi$$
−0.381255 + 0.924470i $$0.624508\pi$$
$$578$$ 2.82363e6 0.351551
$$579$$ −4.13938e6 −0.513144
$$580$$ 0 0
$$581$$ −1.72490e6 −0.211994
$$582$$ −679120. −0.0831073
$$583$$ −1.25133e7 −1.52476
$$584$$ −230528. −0.0279699
$$585$$ 0 0
$$586$$ −5.75544e6 −0.692364
$$587$$ 8.08462e6 0.968422 0.484211 0.874951i $$-0.339107\pi$$
0.484211 + 0.874951i $$0.339107\pi$$
$$588$$ −384160. −0.0458214
$$589$$ −3.46484e6 −0.411524
$$590$$ 0 0
$$591$$ −4.94946e6 −0.582893
$$592$$ −3.66848e6 −0.430211
$$593$$ −1.41575e6 −0.165330 −0.0826649 0.996577i $$-0.526343\pi$$
−0.0826649 + 0.996577i $$0.526343\pi$$
$$594$$ −5.18784e6 −0.603282
$$595$$ 0 0
$$596$$ −969504. −0.111798
$$597$$ −5.20364e6 −0.597546
$$598$$ −9.81120e6 −1.12194
$$599$$ 8.75460e6 0.996941 0.498470 0.866907i $$-0.333895\pi$$
0.498470 + 0.866907i $$0.333895\pi$$
$$600$$ 0 0
$$601$$ 8.70276e6 0.982813 0.491407 0.870930i $$-0.336483\pi$$
0.491407 + 0.870930i $$0.336483\pi$$
$$602$$ 725984. 0.0816462
$$603$$ −2.49564e6 −0.279504
$$604$$ 1.99117e6 0.222083
$$605$$ 0 0
$$606$$ −3.96816e6 −0.438942
$$607$$ 1.69578e7 1.86809 0.934045 0.357157i $$-0.116254\pi$$
0.934045 + 0.357157i $$0.116254\pi$$
$$608$$ 481280. 0.0528006
$$609$$ 2.38434e6 0.260510
$$610$$ 0 0
$$611$$ −1.05821e6 −0.114675
$$612$$ −3.33590e6 −0.360027
$$613$$ −1.76743e7 −1.89973 −0.949866 0.312658i $$-0.898780\pi$$
−0.949866 + 0.312658i $$0.898780\pi$$
$$614$$ 3.95639e6 0.423524
$$615$$ 0 0
$$616$$ 1.05370e6 0.111883
$$617$$ 9.70636e6 1.02646 0.513232 0.858250i $$-0.328448\pi$$
0.513232 + 0.858250i $$0.328448\pi$$
$$618$$ −5.26576e6 −0.554613
$$619$$ 1.48739e7 1.56027 0.780133 0.625613i $$-0.215151\pi$$
0.780133 + 0.625613i $$0.215151\pi$$
$$620$$ 0 0
$$621$$ 1.62120e7 1.68697
$$622$$ −8.88202e6 −0.920525
$$623$$ −1.30977e6 −0.135199
$$624$$ 1.49504e6 0.153706
$$625$$ 0 0
$$626$$ −9.32031e6 −0.950593
$$627$$ 1.57920e6 0.160424
$$628$$ −1.21664e6 −0.123101
$$629$$ −2.08931e7 −2.10561
$$630$$ 0 0
$$631$$ 1.26353e7 1.26331 0.631656 0.775248i $$-0.282375\pi$$
0.631656 + 0.775248i $$0.282375\pi$$
$$632$$ 2.74381e6 0.273251
$$633$$ −1.83284e6 −0.181809
$$634$$ −1.71017e6 −0.168972
$$635$$ 0 0
$$636$$ −5.95872e6 −0.584131
$$637$$ −1.40218e6 −0.136917
$$638$$ −6.53990e6 −0.636092
$$639$$ −4.03603e6 −0.391023
$$640$$ 0 0
$$641$$ 6.23398e6 0.599267 0.299634 0.954054i $$-0.403136\pi$$
0.299634 + 0.954054i $$0.403136\pi$$
$$642$$ 1.95408e6 0.187113
$$643$$ −1.06874e7 −1.01940 −0.509701 0.860352i $$-0.670244\pi$$
−0.509701 + 0.860352i $$0.670244\pi$$
$$644$$ −3.29280e6 −0.312860
$$645$$ 0 0
$$646$$ 2.74104e6 0.258425
$$647$$ −1.83258e7 −1.72109 −0.860544 0.509376i $$-0.829876\pi$$
−0.860544 + 0.509376i $$0.829876\pi$$
$$648$$ −246464. −0.0230577
$$649$$ −1.15255e7 −1.07411
$$650$$ 0 0
$$651$$ −3.61228e6 −0.334063
$$652$$ −1.98810e6 −0.183155
$$653$$ 7.28857e6 0.668897 0.334448 0.942414i $$-0.391450\pi$$
0.334448 + 0.942414i $$0.391450\pi$$
$$654$$ 2.25496e6 0.206155
$$655$$ 0 0
$$656$$ 1.59283e6 0.144514
$$657$$ 515086. 0.0465550
$$658$$ −355152. −0.0319779
$$659$$ 4.54337e6 0.407534 0.203767 0.979019i $$-0.434681\pi$$
0.203767 + 0.979019i $$0.434681\pi$$
$$660$$ 0 0
$$661$$ −2.10021e7 −1.86964 −0.934821 0.355120i $$-0.884440\pi$$
−0.934821 + 0.355120i $$0.884440\pi$$
$$662$$ −1.58646e6 −0.140697
$$663$$ 8.51472e6 0.752292
$$664$$ 2.25293e6 0.198302
$$665$$ 0 0
$$666$$ 8.19676e6 0.716072
$$667$$ 2.04372e7 1.77872
$$668$$ 1.15872e6 0.100470
$$669$$ −1.27746e7 −1.10353
$$670$$ 0 0
$$671$$ −8.22394e6 −0.705137
$$672$$ 501760. 0.0428620
$$673$$ −3.46923e6 −0.295253 −0.147627 0.989043i $$-0.547163\pi$$
−0.147627 + 0.989043i $$0.547163\pi$$
$$674$$ 1.28728e7 1.09150
$$675$$ 0 0
$$676$$ −483792. −0.0407185
$$677$$ 1.80916e7 1.51707 0.758536 0.651631i $$-0.225915\pi$$
0.758536 + 0.651631i $$0.225915\pi$$
$$678$$ 349680. 0.0292144
$$679$$ −831922. −0.0692481
$$680$$ 0 0
$$681$$ −1.28764e7 −1.06397
$$682$$ 9.90797e6 0.815687
$$683$$ −4.67752e6 −0.383675 −0.191838 0.981427i $$-0.561445\pi$$
−0.191838 + 0.981427i $$0.561445\pi$$
$$684$$ −1.07536e6 −0.0878848
$$685$$ 0 0
$$686$$ −470596. −0.0381802
$$687$$ −3.50936e6 −0.283685
$$688$$ −948224. −0.0763730
$$689$$ −2.17493e7 −1.74541
$$690$$ 0 0
$$691$$ 1.68960e7 1.34614 0.673069 0.739579i $$-0.264976\pi$$
0.673069 + 0.739579i $$0.264976\pi$$
$$692$$ 7.06483e6 0.560837
$$693$$ −2.35435e6 −0.186225
$$694$$ −1.11207e7 −0.876466
$$695$$ 0 0
$$696$$ −3.11424e6 −0.243685
$$697$$ 9.07168e6 0.707303
$$698$$ −1.35520e6 −0.105285
$$699$$ 8.36154e6 0.647282
$$700$$ 0 0
$$701$$ 2.40964e6 0.185207 0.0926035 0.995703i $$-0.470481\pi$$
0.0926035 + 0.995703i $$0.470481\pi$$
$$702$$ −9.01696e6 −0.690585
$$703$$ −6.73510e6 −0.513991
$$704$$ −1.37626e6 −0.104657
$$705$$ 0 0
$$706$$ 1.44818e6 0.109348
$$707$$ −4.86100e6 −0.365744
$$708$$ −5.48832e6 −0.411487
$$709$$ −5.77010e6 −0.431090 −0.215545 0.976494i $$-0.569153\pi$$
−0.215545 + 0.976494i $$0.569153\pi$$
$$710$$ 0 0
$$711$$ −6.13070e6 −0.454816
$$712$$ 1.71072e6 0.126468
$$713$$ −3.09624e7 −2.28092
$$714$$ 2.85768e6 0.209782
$$715$$ 0 0
$$716$$ −171072. −0.0124709
$$717$$ −7.74336e6 −0.562512
$$718$$ 3.50611e6 0.253813
$$719$$ −1.43716e7 −1.03677 −0.518385 0.855147i $$-0.673467\pi$$
−0.518385 + 0.855147i $$0.673467\pi$$
$$720$$ 0 0
$$721$$ −6.45056e6 −0.462124
$$722$$ −9.02080e6 −0.644024
$$723$$ 1.15285e7 0.820214
$$724$$ −8.73702e6 −0.619465
$$725$$ 0 0
$$726$$ 1.92620e6 0.135631
$$727$$ 1.40705e7 0.987353 0.493676 0.869646i $$-0.335653\pi$$
0.493676 + 0.869646i $$0.335653\pi$$
$$728$$ 1.83142e6 0.128074
$$729$$ 9.93049e6 0.692073
$$730$$ 0 0
$$731$$ −5.40043e6 −0.373796
$$732$$ −3.91616e6 −0.270136
$$733$$ 3.75000e6 0.257793 0.128897 0.991658i $$-0.458856\pi$$
0.128897 + 0.991658i $$0.458856\pi$$
$$734$$ −1.19225e7 −0.816819
$$735$$ 0 0
$$736$$ 4.30080e6 0.292654
$$737$$ −5.86387e6 −0.397664
$$738$$ −3.55898e6 −0.240539
$$739$$ 2.61318e7 1.76019 0.880093 0.474802i $$-0.157480\pi$$
0.880093 + 0.474802i $$0.157480\pi$$
$$740$$ 0 0
$$741$$ 2.74480e6 0.183639
$$742$$ −7.29943e6 −0.486720
$$743$$ 159072. 0.0105711 0.00528557 0.999986i $$-0.498318\pi$$
0.00528557 + 0.999986i $$0.498318\pi$$
$$744$$ 4.71808e6 0.312488
$$745$$ 0 0
$$746$$ −1.56577e7 −1.03010
$$747$$ −5.03389e6 −0.330067
$$748$$ −7.83821e6 −0.512227
$$749$$ 2.39375e6 0.155910
$$750$$ 0 0
$$751$$ −2.65311e7 −1.71654 −0.858272 0.513196i $$-0.828461\pi$$
−0.858272 + 0.513196i $$0.828461\pi$$
$$752$$ 463872. 0.0299126
$$753$$ −1.35801e7 −0.872802
$$754$$ −1.13670e7 −0.728143
$$755$$ 0 0
$$756$$ −3.02624e6 −0.192575
$$757$$ 1.52032e7 0.964260 0.482130 0.876100i $$-0.339863\pi$$
0.482130 + 0.876100i $$0.339863\pi$$
$$758$$ 1.44264e7 0.911981
$$759$$ 1.41120e7 0.889169
$$760$$ 0 0
$$761$$ 4.71380e6 0.295059 0.147530 0.989058i $$-0.452868\pi$$
0.147530 + 0.989058i $$0.452868\pi$$
$$762$$ 1.26397e7 0.788585
$$763$$ 2.76233e6 0.171776
$$764$$ −9.21562e6 −0.571204
$$765$$ 0 0
$$766$$ 1.06657e7 0.656779
$$767$$ −2.00324e7 −1.22954
$$768$$ −655360. −0.0400938
$$769$$ −1.58977e6 −0.0969434 −0.0484717 0.998825i $$-0.515435\pi$$
−0.0484717 + 0.998825i $$0.515435\pi$$
$$770$$ 0 0
$$771$$ −3.17742e6 −0.192504
$$772$$ 6.62301e6 0.399956
$$773$$ 9.69095e6 0.583334 0.291667 0.956520i $$-0.405790\pi$$
0.291667 + 0.956520i $$0.405790\pi$$
$$774$$ 2.11869e6 0.127120
$$775$$ 0 0
$$776$$ 1.08659e6 0.0647757
$$777$$ −7.02170e6 −0.417244
$$778$$ −853464. −0.0505518
$$779$$ 2.92434e6 0.172657
$$780$$ 0 0
$$781$$ −9.48326e6 −0.556327
$$782$$ 2.44944e7 1.43235
$$783$$ 1.87828e7 1.09485
$$784$$ 614656. 0.0357143
$$785$$ 0 0
$$786$$ 986640. 0.0569642
$$787$$ 1.57170e6 0.0904549 0.0452275 0.998977i $$-0.485599\pi$$
0.0452275 + 0.998977i $$0.485599\pi$$
$$788$$ 7.91914e6 0.454320
$$789$$ 1.05101e7 0.601054
$$790$$ 0 0
$$791$$ 428358. 0.0243425
$$792$$ 3.07507e6 0.174198
$$793$$ −1.42940e7 −0.807180
$$794$$ 1.63763e7 0.921860
$$795$$ 0 0
$$796$$ 8.32582e6 0.465741
$$797$$ 2.25298e6 0.125635 0.0628175 0.998025i $$-0.479991\pi$$
0.0628175 + 0.998025i $$0.479991\pi$$
$$798$$ 921200. 0.0512090
$$799$$ 2.64190e6 0.146403
$$800$$ 0 0
$$801$$ −3.82239e6 −0.210501
$$802$$ 3.76946e6 0.206940
$$803$$ 1.21027e6 0.0662360
$$804$$ −2.79232e6 −0.152344
$$805$$ 0 0
$$806$$ 1.72210e7 0.933728
$$807$$ −1.18958e7 −0.643000
$$808$$ 6.34906e6 0.342122
$$809$$ −2.37938e7 −1.27818 −0.639090 0.769132i $$-0.720689\pi$$
−0.639090 + 0.769132i $$0.720689\pi$$
$$810$$ 0 0
$$811$$ 5.32300e6 0.284187 0.142093 0.989853i $$-0.454617\pi$$
0.142093 + 0.989853i $$0.454617\pi$$
$$812$$ −3.81494e6 −0.203048
$$813$$ 1.43008e7 0.758812
$$814$$ 1.92595e7 1.01879
$$815$$ 0 0
$$816$$ −3.73248e6 −0.196233
$$817$$ −1.74088e6 −0.0912460
$$818$$ −1.93824e7 −1.01280
$$819$$ −4.09209e6 −0.213174
$$820$$ 0 0
$$821$$ 1.48802e7 0.770464 0.385232 0.922820i $$-0.374121\pi$$
0.385232 + 0.922820i $$0.374121\pi$$
$$822$$ 1.21294e7 0.626121
$$823$$ −2.00601e7 −1.03236 −0.516182 0.856479i $$-0.672647\pi$$
−0.516182 + 0.856479i $$0.672647\pi$$
$$824$$ 8.42522e6 0.432278
$$825$$ 0 0
$$826$$ −6.72319e6 −0.342867
$$827$$ −1.21539e7 −0.617949 −0.308975 0.951070i $$-0.599986\pi$$
−0.308975 + 0.951070i $$0.599986\pi$$
$$828$$ −9.60960e6 −0.487113
$$829$$ 3.21197e7 1.62325 0.811625 0.584179i $$-0.198583\pi$$
0.811625 + 0.584179i $$0.198583\pi$$
$$830$$ 0 0
$$831$$ 633020. 0.0317991
$$832$$ −2.39206e6 −0.119802
$$833$$ 3.50066e6 0.174798
$$834$$ −1.00234e7 −0.499001
$$835$$ 0 0
$$836$$ −2.52672e6 −0.125038
$$837$$ −2.84559e7 −1.40397
$$838$$ −6.93938e6 −0.341359
$$839$$ −1.01320e6 −0.0496922 −0.0248461 0.999691i $$-0.507910\pi$$
−0.0248461 + 0.999691i $$0.507910\pi$$
$$840$$ 0 0
$$841$$ 3.16681e6 0.154394
$$842$$ −6.60580e6 −0.321104
$$843$$ 4.96614e6 0.240686
$$844$$ 2.93254e6 0.141706
$$845$$ 0 0
$$846$$ −1.03646e6 −0.0497884
$$847$$ 2.35960e6 0.113013
$$848$$ 9.53395e6 0.455285
$$849$$ −1.15842e7 −0.551564
$$850$$ 0 0
$$851$$ −6.01860e7 −2.84886
$$852$$ −4.51584e6 −0.213128
$$853$$ −234824. −0.0110502 −0.00552510 0.999985i $$-0.501759\pi$$
−0.00552510 + 0.999985i $$0.501759\pi$$
$$854$$ −4.79730e6 −0.225088
$$855$$ 0 0
$$856$$ −3.12653e6 −0.145840
$$857$$ −2.83802e7 −1.31997 −0.659985 0.751279i $$-0.729437\pi$$
−0.659985 + 0.751279i $$0.729437\pi$$
$$858$$ −7.84896e6 −0.363994
$$859$$ 4.00081e7 1.84997 0.924986 0.380001i $$-0.124076\pi$$
0.924986 + 0.380001i $$0.124076\pi$$
$$860$$ 0 0
$$861$$ 3.04878e6 0.140158
$$862$$ 1.65744e7 0.759748
$$863$$ 2.08030e7 0.950823 0.475411 0.879764i $$-0.342299\pi$$
0.475411 + 0.879764i $$0.342299\pi$$
$$864$$ 3.95264e6 0.180137
$$865$$ 0 0
$$866$$ 1.21586e7 0.550922
$$867$$ −7.05907e6 −0.318933
$$868$$ 5.77965e6 0.260377
$$869$$ −1.44050e7 −0.647088
$$870$$ 0 0
$$871$$ −1.01920e7 −0.455211
$$872$$ −3.60794e6 −0.160682
$$873$$ −2.42785e6 −0.107817
$$874$$ 7.89600e6 0.349646
$$875$$ 0 0
$$876$$ 576320. 0.0253748
$$877$$ −3.03559e7 −1.33273 −0.666367 0.745624i $$-0.732152\pi$$
−0.666367 + 0.745624i $$0.732152\pi$$
$$878$$ 1.01708e7 0.445267
$$879$$ 1.43886e7 0.628125
$$880$$ 0 0
$$881$$ −2.58936e7 −1.12396 −0.561981 0.827150i $$-0.689961\pi$$
−0.561981 + 0.827150i $$0.689961\pi$$
$$882$$ −1.37337e6 −0.0594452
$$883$$ 1.88813e7 0.814950 0.407475 0.913216i $$-0.366409\pi$$
0.407475 + 0.913216i $$0.366409\pi$$
$$884$$ −1.36236e7 −0.586354
$$885$$ 0 0
$$886$$ 9.72840e6 0.416349
$$887$$ 2.34431e7 1.00048 0.500238 0.865888i $$-0.333246\pi$$
0.500238 + 0.865888i $$0.333246\pi$$
$$888$$ 9.17120e6 0.390296
$$889$$ 1.54836e7 0.657079
$$890$$ 0 0
$$891$$ 1.29394e6 0.0546033
$$892$$ 2.04394e7 0.860115
$$893$$ 851640. 0.0357378
$$894$$ 2.42376e6 0.101425
$$895$$ 0 0
$$896$$ −802816. −0.0334077
$$897$$ 2.45280e7 1.01784
$$898$$ 7.31412e6 0.302671
$$899$$ −3.58722e7 −1.48033
$$900$$ 0 0
$$901$$ 5.42988e7 2.22833
$$902$$ −8.36237e6 −0.342226
$$903$$ −1.81496e6 −0.0740709
$$904$$ −559488. −0.0227703
$$905$$ 0 0
$$906$$ −4.97792e6 −0.201478
$$907$$ 5.60873e6 0.226384 0.113192 0.993573i $$-0.463892\pi$$
0.113192 + 0.993573i $$0.463892\pi$$
$$908$$ 2.06023e7 0.829279
$$909$$ −1.41862e7 −0.569450
$$910$$ 0 0
$$911$$ 2.16215e7 0.863156 0.431578 0.902076i $$-0.357957\pi$$
0.431578 + 0.902076i $$0.357957\pi$$
$$912$$ −1.20320e6 −0.0479017
$$913$$ −1.18279e7 −0.469602
$$914$$ −6.32252e6 −0.250337
$$915$$ 0 0
$$916$$ 5.61498e6 0.221110
$$917$$ 1.20863e6 0.0474648
$$918$$ 2.25115e7 0.881654
$$919$$ 4.51695e7 1.76424 0.882119 0.471028i $$-0.156117\pi$$
0.882119 + 0.471028i $$0.156117\pi$$
$$920$$ 0 0
$$921$$ −9.89098e6 −0.384229
$$922$$ 2.03842e7 0.789706
$$923$$ −1.64828e7 −0.636835
$$924$$ −2.63424e6 −0.101502
$$925$$ 0 0
$$926$$ 2.80935e7 1.07666
$$927$$ −1.88251e7 −0.719512
$$928$$ 4.98278e6 0.189934
$$929$$ −2.28729e7 −0.869524 −0.434762 0.900545i $$-0.643168\pi$$
−0.434762 + 0.900545i $$0.643168\pi$$
$$930$$ 0 0
$$931$$ 1.12847e6 0.0426693
$$932$$ −1.33785e7 −0.504506
$$933$$ 2.22050e7 0.835117
$$934$$ 1.69938e7 0.637417
$$935$$ 0 0
$$936$$ 5.34477e6 0.199406
$$937$$ 1.79616e7 0.668336 0.334168 0.942514i $$-0.391545\pi$$
0.334168 + 0.942514i $$0.391545\pi$$
$$938$$ −3.42059e6 −0.126939
$$939$$ 2.33008e7 0.862395
$$940$$ 0 0
$$941$$ −1.79697e7 −0.661558 −0.330779 0.943708i $$-0.607311\pi$$
−0.330779 + 0.943708i $$0.607311\pi$$
$$942$$ 3.04160e6 0.111680
$$943$$ 2.61324e7 0.956974
$$944$$ 8.78131e6 0.320722
$$945$$ 0 0
$$946$$ 4.97818e6 0.180860
$$947$$ −4.32115e7 −1.56576 −0.782879 0.622174i $$-0.786250\pi$$
−0.782879 + 0.622174i $$0.786250\pi$$
$$948$$ −6.85952e6 −0.247898
$$949$$ 2.10357e6 0.0758213
$$950$$ 0 0
$$951$$ 4.27542e6 0.153295
$$952$$ −4.57229e6 −0.163509
$$953$$ 7.50965e6 0.267848 0.133924 0.990992i $$-0.457242\pi$$
0.133924 + 0.990992i $$0.457242\pi$$
$$954$$ −2.13024e7 −0.757806
$$955$$ 0 0
$$956$$ 1.23894e7 0.438434
$$957$$ 1.63498e7 0.577074
$$958$$ 2.23714e6 0.0787551
$$959$$ 1.48585e7 0.521708
$$960$$ 0 0
$$961$$ 2.57172e7 0.898288
$$962$$ 3.34749e7 1.16622
$$963$$ 6.98584e6 0.242746
$$964$$ −1.84456e7 −0.639293
$$965$$ 0 0
$$966$$ 8.23200e6 0.283833
$$967$$ 1.69305e7 0.582242 0.291121 0.956686i $$-0.405972\pi$$
0.291121 + 0.956686i $$0.405972\pi$$
$$968$$ −3.08192e6 −0.105714
$$969$$ −6.85260e6 −0.234448
$$970$$ 0 0
$$971$$ 2.86144e7 0.973949 0.486974 0.873416i $$-0.338101\pi$$
0.486974 + 0.873416i $$0.338101\pi$$
$$972$$ −1.43915e7 −0.488586
$$973$$ −1.22787e7 −0.415787
$$974$$ 5.28227e6 0.178412
$$975$$ 0 0
$$976$$ 6.26586e6 0.210550
$$977$$ −3.69445e7 −1.23826 −0.619132 0.785287i $$-0.712515\pi$$
−0.619132 + 0.785287i $$0.712515\pi$$
$$978$$ 4.97024e6 0.166161
$$979$$ −8.98128e6 −0.299489
$$980$$ 0 0
$$981$$ 8.06148e6 0.267450
$$982$$ 2.50877e7 0.830200
$$983$$ 3.88787e7 1.28330 0.641650 0.766998i $$-0.278250\pi$$
0.641650 + 0.766998i $$0.278250\pi$$
$$984$$ −3.98208e6 −0.131106
$$985$$ 0 0
$$986$$ 2.83785e7 0.929603
$$987$$ 887880. 0.0290109
$$988$$ −4.39168e6 −0.143133
$$989$$ −1.55568e7 −0.505743
$$990$$ 0 0
$$991$$ 2.49212e7 0.806092 0.403046 0.915180i $$-0.367951\pi$$
0.403046 + 0.915180i $$0.367951\pi$$
$$992$$ −7.54893e6 −0.243560
$$993$$ 3.96616e6 0.127643
$$994$$ −5.53190e6 −0.177586
$$995$$ 0 0
$$996$$ −5.63232e6 −0.179903
$$997$$ −1.01956e7 −0.324845 −0.162422 0.986721i $$-0.551931\pi$$
−0.162422 + 0.986721i $$0.551931\pi$$
$$998$$ −1.57514e7 −0.500603
$$999$$ −5.53138e7 −1.75356
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 350.6.a.i.1.1 1
5.2 odd 4 350.6.c.d.99.2 2
5.3 odd 4 350.6.c.d.99.1 2
5.4 even 2 14.6.a.a.1.1 1
15.14 odd 2 126.6.a.f.1.1 1
20.19 odd 2 112.6.a.c.1.1 1
35.4 even 6 98.6.c.c.79.1 2
35.9 even 6 98.6.c.c.67.1 2
35.19 odd 6 98.6.c.d.67.1 2
35.24 odd 6 98.6.c.d.79.1 2
35.34 odd 2 98.6.a.a.1.1 1
40.19 odd 2 448.6.a.l.1.1 1
40.29 even 2 448.6.a.e.1.1 1
60.59 even 2 1008.6.a.b.1.1 1
105.104 even 2 882.6.a.x.1.1 1
140.139 even 2 784.6.a.i.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
14.6.a.a.1.1 1 5.4 even 2
98.6.a.a.1.1 1 35.34 odd 2
98.6.c.c.67.1 2 35.9 even 6
98.6.c.c.79.1 2 35.4 even 6
98.6.c.d.67.1 2 35.19 odd 6
98.6.c.d.79.1 2 35.24 odd 6
112.6.a.c.1.1 1 20.19 odd 2
126.6.a.f.1.1 1 15.14 odd 2
350.6.a.i.1.1 1 1.1 even 1 trivial
350.6.c.d.99.1 2 5.3 odd 4
350.6.c.d.99.2 2 5.2 odd 4
448.6.a.e.1.1 1 40.29 even 2
448.6.a.l.1.1 1 40.19 odd 2
784.6.a.i.1.1 1 140.139 even 2
882.6.a.x.1.1 1 105.104 even 2
1008.6.a.b.1.1 1 60.59 even 2