# Properties

 Label 784.6.a.i.1.1 Level $784$ Weight $6$ Character 784.1 Self dual yes Analytic conductor $125.741$ 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] = [784,6,Mod(1,784)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(784, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("784.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$784 = 2^{4} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 784.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: $$125.740914733$$ Analytic rank: $$1$$ 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 $$\chi$$ $$=$$ 784.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+10.0000 q^{3} -84.0000 q^{5} -143.000 q^{9} +O(q^{10})$$ $$q+10.0000 q^{3} -84.0000 q^{5} -143.000 q^{9} +336.000 q^{11} -584.000 q^{13} -840.000 q^{15} +1458.00 q^{17} +470.000 q^{19} +4200.00 q^{23} +3931.00 q^{25} -3860.00 q^{27} +4866.00 q^{29} -7372.00 q^{31} +3360.00 q^{33} +14330.0 q^{37} -5840.00 q^{39} -6222.00 q^{41} -3704.00 q^{43} +12012.0 q^{45} -1812.00 q^{47} +14580.0 q^{51} -37242.0 q^{53} -28224.0 q^{55} +4700.00 q^{57} +34302.0 q^{59} -24476.0 q^{61} +49056.0 q^{65} +17452.0 q^{67} +42000.0 q^{69} -28224.0 q^{71} -3602.00 q^{73} +39310.0 q^{75} -42872.0 q^{79} -3851.00 q^{81} -35202.0 q^{83} -122472. q^{85} +48660.0 q^{87} -26730.0 q^{89} -73720.0 q^{93} -39480.0 q^{95} +16978.0 q^{97} -48048.0 q^{99} +O(q^{100})$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0 0
$$3$$ 10.0000 0.641500 0.320750 0.947164i $$-0.396065\pi$$
0.320750 + 0.947164i $$0.396065\pi$$
$$4$$ 0 0
$$5$$ −84.0000 −1.50264 −0.751319 0.659939i $$-0.770582\pi$$
−0.751319 + 0.659939i $$0.770582\pi$$
$$6$$ 0 0
$$7$$ 0 0
$$8$$ 0 0
$$9$$ −143.000 −0.588477
$$10$$ 0 0
$$11$$ 336.000 0.837255 0.418627 0.908158i $$-0.362511\pi$$
0.418627 + 0.908158i $$0.362511\pi$$
$$12$$ 0 0
$$13$$ −584.000 −0.958417 −0.479208 0.877701i $$-0.659076\pi$$
−0.479208 + 0.877701i $$0.659076\pi$$
$$14$$ 0 0
$$15$$ −840.000 −0.963943
$$16$$ 0 0
$$17$$ 1458.00 1.22359 0.611794 0.791017i $$-0.290448\pi$$
0.611794 + 0.791017i $$0.290448\pi$$
$$18$$ 0 0
$$19$$ 470.000 0.298685 0.149343 0.988786i $$-0.452284\pi$$
0.149343 + 0.988786i $$0.452284\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ 4200.00 1.65550 0.827751 0.561096i $$-0.189620\pi$$
0.827751 + 0.561096i $$0.189620\pi$$
$$24$$ 0 0
$$25$$ 3931.00 1.25792
$$26$$ 0 0
$$27$$ −3860.00 −1.01901
$$28$$ 0 0
$$29$$ 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 $$0.741903\pi$$
$$32$$ 0 0
$$33$$ 3360.00 0.537099
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 14330.0 1.72085 0.860423 0.509581i $$-0.170200\pi$$
0.860423 + 0.509581i $$0.170200\pi$$
$$38$$ 0 0
$$39$$ −5840.00 −0.614825
$$40$$ 0 0
$$41$$ −6222.00 −0.578057 −0.289028 0.957321i $$-0.593332\pi$$
−0.289028 + 0.957321i $$0.593332\pi$$
$$42$$ 0 0
$$43$$ −3704.00 −0.305492 −0.152746 0.988265i $$-0.548812\pi$$
−0.152746 + 0.988265i $$0.548812\pi$$
$$44$$ 0 0
$$45$$ 12012.0 0.884268
$$46$$ 0 0
$$47$$ −1812.00 −0.119650 −0.0598251 0.998209i $$-0.519054\pi$$
−0.0598251 + 0.998209i $$0.519054\pi$$
$$48$$ 0 0
$$49$$ 0 0
$$50$$ 0 0
$$51$$ 14580.0 0.784932
$$52$$ 0 0
$$53$$ −37242.0 −1.82114 −0.910570 0.413355i $$-0.864357\pi$$
−0.910570 + 0.413355i $$0.864357\pi$$
$$54$$ 0 0
$$55$$ −28224.0 −1.25809
$$56$$ 0 0
$$57$$ 4700.00 0.191607
$$58$$ 0 0
$$59$$ 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 $$-0.638356\pi$$
−0.421101 + 0.907014i $$0.638356\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ 49056.0 1.44015
$$66$$ 0 0
$$67$$ 17452.0 0.474961 0.237481 0.971392i $$-0.423678\pi$$
0.237481 + 0.971392i $$0.423678\pi$$
$$68$$ 0 0
$$69$$ 42000.0 1.06201
$$70$$ 0 0
$$71$$ −28224.0 −0.664466 −0.332233 0.943197i $$-0.607802\pi$$
−0.332233 + 0.943197i $$0.607802\pi$$
$$72$$ 0 0
$$73$$ −3602.00 −0.0791109 −0.0395555 0.999217i $$-0.512594\pi$$
−0.0395555 + 0.999217i $$0.512594\pi$$
$$74$$ 0 0
$$75$$ 39310.0 0.806956
$$76$$ 0 0
$$77$$ 0 0
$$78$$ 0 0
$$79$$ −42872.0 −0.772869 −0.386435 0.922317i $$-0.626294\pi$$
−0.386435 + 0.922317i $$0.626294\pi$$
$$80$$ 0 0
$$81$$ −3851.00 −0.0652170
$$82$$ 0 0
$$83$$ −35202.0 −0.560883 −0.280441 0.959871i $$-0.590481\pi$$
−0.280441 + 0.959871i $$0.590481\pi$$
$$84$$ 0 0
$$85$$ −122472. −1.83861
$$86$$ 0 0
$$87$$ 48660.0 0.689246
$$88$$ 0 0
$$89$$ −26730.0 −0.357704 −0.178852 0.983876i $$-0.557238\pi$$
−0.178852 + 0.983876i $$0.557238\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 0 0
$$93$$ −73720.0 −0.883849
$$94$$ 0 0
$$95$$ −39480.0 −0.448816
$$96$$ 0 0
$$97$$ 16978.0 0.183213 0.0916067 0.995795i $$-0.470800\pi$$
0.0916067 + 0.995795i $$0.470800\pi$$
$$98$$ 0 0
$$99$$ −48048.0 −0.492705
$$100$$ 0 0
$$101$$ −99204.0 −0.967667 −0.483833 0.875160i $$-0.660756\pi$$
−0.483833 + 0.875160i $$0.660756\pi$$
$$102$$ 0 0
$$103$$ −131644. −1.22267 −0.611333 0.791373i $$-0.709366\pi$$
−0.611333 + 0.791373i $$0.709366\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ −48852.0 −0.412499 −0.206250 0.978499i $$-0.566126\pi$$
−0.206250 + 0.978499i $$0.566126\pi$$
$$108$$ 0 0
$$109$$ −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
$$112$$ 0 0
$$113$$ 8742.00 0.0644043 0.0322021 0.999481i $$-0.489748\pi$$
0.0322021 + 0.999481i $$0.489748\pi$$
$$114$$ 0 0
$$115$$ −352800. −2.48762
$$116$$ 0 0
$$117$$ 83512.0 0.564007
$$118$$ 0 0
$$119$$ 0 0
$$120$$ 0 0
$$121$$ −48155.0 −0.299005
$$122$$ 0 0
$$123$$ −62220.0 −0.370823
$$124$$ 0 0
$$125$$ −67704.0 −0.387560
$$126$$ 0 0
$$127$$ −315992. −1.73847 −0.869234 0.494401i $$-0.835388\pi$$
−0.869234 + 0.494401i $$0.835388\pi$$
$$128$$ 0 0
$$129$$ −37040.0 −0.195973
$$130$$ 0 0
$$131$$ −24666.0 −0.125580 −0.0627900 0.998027i $$-0.520000\pi$$
−0.0627900 + 0.998027i $$0.520000\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ 0 0
$$135$$ 324240. 1.53120
$$136$$ 0 0
$$137$$ 303234. 1.38031 0.690155 0.723662i $$-0.257542\pi$$
0.690155 + 0.723662i $$0.257542\pi$$
$$138$$ 0 0
$$139$$ 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
$$142$$ 0 0
$$143$$ −196224. −0.802439
$$144$$ 0 0
$$145$$ −408744. −1.61448
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ −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 $$-0.571286\pi$$
−0.222083 + 0.975028i $$0.571286\pi$$
$$152$$ 0 0
$$153$$ −208494. −0.720054
$$154$$ 0 0
$$155$$ 619248. 2.07031
$$156$$ 0 0
$$157$$ −76040.0 −0.246203 −0.123101 0.992394i $$-0.539284\pi$$
−0.123101 + 0.992394i $$0.539284\pi$$
$$158$$ 0 0
$$159$$ −372420. −1.16826
$$160$$ 0 0
$$161$$ 0 0
$$162$$ 0 0
$$163$$ −124256. −0.366310 −0.183155 0.983084i $$-0.558631\pi$$
−0.183155 + 0.983084i $$0.558631\pi$$
$$164$$ 0 0
$$165$$ −282240. −0.807065
$$166$$ 0 0
$$167$$ −72420.0 −0.200940 −0.100470 0.994940i $$-0.532035\pi$$
−0.100470 + 0.994940i $$0.532035\pi$$
$$168$$ 0 0
$$169$$ −30237.0 −0.0814370
$$170$$ 0 0
$$171$$ −67210.0 −0.175770
$$172$$ 0 0
$$173$$ 441552. 1.12167 0.560837 0.827926i $$-0.310479\pi$$
0.560837 + 0.827926i $$0.310479\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ 343020. 0.822974
$$178$$ 0 0
$$179$$ 10692.0 0.0249417 0.0124709 0.999922i $$-0.496030\pi$$
0.0124709 + 0.999922i $$0.496030\pi$$
$$180$$ 0 0
$$181$$ 546064. 1.23893 0.619465 0.785024i $$-0.287349\pi$$
0.619465 + 0.785024i $$0.287349\pi$$
$$182$$ 0 0
$$183$$ −244760. −0.540272
$$184$$ 0 0
$$185$$ −1.20372e6 −2.58581
$$186$$ 0 0
$$187$$ 489888. 1.02445
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 575976. 1.14241 0.571204 0.820808i $$-0.306477\pi$$
0.571204 + 0.820808i $$0.306477\pi$$
$$192$$ 0 0
$$193$$ −413938. −0.799912 −0.399956 0.916534i $$-0.630975\pi$$
−0.399956 + 0.916534i $$0.630975\pi$$
$$194$$ 0 0
$$195$$ 490560. 0.923859
$$196$$ 0 0
$$197$$ −494946. −0.908641 −0.454320 0.890838i $$-0.650118\pi$$
−0.454320 + 0.890838i $$0.650118\pi$$
$$198$$ 0 0
$$199$$ 520364. 0.931482 0.465741 0.884921i $$-0.345788\pi$$
0.465741 + 0.884921i $$0.345788\pi$$
$$200$$ 0 0
$$201$$ 174520. 0.304688
$$202$$ 0 0
$$203$$ 0 0
$$204$$ 0 0
$$205$$ 522648. 0.868610
$$206$$ 0 0
$$207$$ −600600. −0.974225
$$208$$ 0 0
$$209$$ 157920. 0.250076
$$210$$ 0 0
$$211$$ −183284. −0.283412 −0.141706 0.989909i $$-0.545259\pi$$
−0.141706 + 0.989909i $$0.545259\pi$$
$$212$$ 0 0
$$213$$ −282240. −0.426255
$$214$$ 0 0
$$215$$ 311136. 0.459044
$$216$$ 0 0
$$217$$ 0 0
$$218$$ 0 0
$$219$$ −36020.0 −0.0507497
$$220$$ 0 0
$$221$$ −851472. −1.17271
$$222$$ 0 0
$$223$$ −1.27746e6 −1.72023 −0.860115 0.510100i $$-0.829608\pi$$
−0.860115 + 0.510100i $$0.829608\pi$$
$$224$$ 0 0
$$225$$ −562133. −0.740257
$$226$$ 0 0
$$227$$ −1.28764e6 −1.65856 −0.829279 0.558835i $$-0.811248\pi$$
−0.829279 + 0.558835i $$0.811248\pi$$
$$228$$ 0 0
$$229$$ −350936. −0.442221 −0.221110 0.975249i $$-0.570968\pi$$
−0.221110 + 0.975249i $$0.570968\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ 836154. 1.00901 0.504506 0.863408i $$-0.331675\pi$$
0.504506 + 0.863408i $$0.331675\pi$$
$$234$$ 0 0
$$235$$ 152208. 0.179791
$$236$$ 0 0
$$237$$ −428720. −0.495796
$$238$$ 0 0
$$239$$ −774336. −0.876869 −0.438434 0.898763i $$-0.644467\pi$$
−0.438434 + 0.898763i $$0.644467\pi$$
$$240$$ 0 0
$$241$$ 1.15285e6 1.27859 0.639293 0.768963i $$-0.279227\pi$$
0.639293 + 0.768963i $$0.279227\pi$$
$$242$$ 0 0
$$243$$ 899470. 0.977172
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ −274480. −0.286265
$$248$$ 0 0
$$249$$ −352020. −0.359806
$$250$$ 0 0
$$251$$ 1.35801e6 1.36056 0.680282 0.732951i $$-0.261858\pi$$
0.680282 + 0.732951i $$0.261858\pi$$
$$252$$ 0 0
$$253$$ 1.41120e6 1.38608
$$254$$ 0 0
$$255$$ −1.22472e6 −1.17947
$$256$$ 0 0
$$257$$ 317742. 0.300083 0.150042 0.988680i $$-0.452059\pi$$
0.150042 + 0.988680i $$0.452059\pi$$
$$258$$ 0 0
$$259$$ 0 0
$$260$$ 0 0
$$261$$ −695838. −0.632276
$$262$$ 0 0
$$263$$ −1.05101e6 −0.936951 −0.468475 0.883477i $$-0.655196\pi$$
−0.468475 + 0.883477i $$0.655196\pi$$
$$264$$ 0 0
$$265$$ 3.12833e6 2.73651
$$266$$ 0 0
$$267$$ −267300. −0.229467
$$268$$ 0 0
$$269$$ −1.18958e6 −1.00234 −0.501169 0.865349i $$-0.667097\pi$$
−0.501169 + 0.865349i $$0.667097\pi$$
$$270$$ 0 0
$$271$$ −1.43008e6 −1.18287 −0.591435 0.806353i $$-0.701438\pi$$
−0.591435 + 0.806353i $$0.701438\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ 1.32082e6 1.05320
$$276$$ 0 0
$$277$$ 63302.0 0.0495699 0.0247849 0.999693i $$-0.492110\pi$$
0.0247849 + 0.999693i $$0.492110\pi$$
$$278$$ 0 0
$$279$$ 1.05420e6 0.810795
$$280$$ 0 0
$$281$$ −496614. −0.375192 −0.187596 0.982246i $$-0.560070\pi$$
−0.187596 + 0.982246i $$0.560070\pi$$
$$282$$ 0 0
$$283$$ −1.15842e6 −0.859803 −0.429902 0.902876i $$-0.641452\pi$$
−0.429902 + 0.902876i $$0.641452\pi$$
$$284$$ 0 0
$$285$$ −394800. −0.287915
$$286$$ 0 0
$$287$$ 0 0
$$288$$ 0 0
$$289$$ 705907. 0.497168
$$290$$ 0 0
$$291$$ 169780. 0.117531
$$292$$ 0 0
$$293$$ −1.43886e6 −0.979151 −0.489575 0.871961i $$-0.662848\pi$$
−0.489575 + 0.871961i $$0.662848\pi$$
$$294$$ 0 0
$$295$$ −2.88137e6 −1.92772
$$296$$ 0 0
$$297$$ −1.29696e6 −0.853170
$$298$$ 0 0
$$299$$ −2.45280e6 −1.58666
$$300$$ 0 0
$$301$$ 0 0
$$302$$ 0 0
$$303$$ −992040. −0.620758
$$304$$ 0 0
$$305$$ 2.05598e6 1.26552
$$306$$ 0 0
$$307$$ −989098. −0.598954 −0.299477 0.954104i $$-0.596812\pi$$
−0.299477 + 0.954104i $$0.596812\pi$$
$$308$$ 0 0
$$309$$ −1.31644e6 −0.784341
$$310$$ 0 0
$$311$$ −2.22050e6 −1.30182 −0.650909 0.759155i $$-0.725612\pi$$
−0.650909 + 0.759155i $$0.725612\pi$$
$$312$$ 0 0
$$313$$ −2.33008e6 −1.34434 −0.672171 0.740396i $$-0.734638\pi$$
−0.672171 + 0.740396i $$0.734638\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ 427542. 0.238963 0.119481 0.992836i $$-0.461877\pi$$
0.119481 + 0.992836i $$0.461877\pi$$
$$318$$ 0 0
$$319$$ 1.63498e6 0.899569
$$320$$ 0 0
$$321$$ −488520. −0.264618
$$322$$ 0 0
$$323$$ 685260. 0.365468
$$324$$ 0 0
$$325$$ −2.29570e6 −1.20561
$$326$$ 0 0
$$327$$ −563740. −0.291548
$$328$$ 0 0
$$329$$ 0 0
$$330$$ 0 0
$$331$$ 396616. 0.198976 0.0994879 0.995039i $$-0.468280\pi$$
0.0994879 + 0.995039i $$0.468280\pi$$
$$332$$ 0 0
$$333$$ −2.04919e6 −1.01268
$$334$$ 0 0
$$335$$ −1.46597e6 −0.713695
$$336$$ 0 0
$$337$$ −3.21819e6 −1.54361 −0.771805 0.635860i $$-0.780646\pi$$
−0.771805 + 0.635860i $$0.780646\pi$$
$$338$$ 0 0
$$339$$ 87420.0 0.0413154
$$340$$ 0 0
$$341$$ −2.47699e6 −1.15356
$$342$$ 0 0
$$343$$ 0 0
$$344$$ 0 0
$$345$$ −3.52800e6 −1.59581
$$346$$ 0 0
$$347$$ −2.78018e6 −1.23951 −0.619755 0.784796i $$-0.712768\pi$$
−0.619755 + 0.784796i $$0.712768\pi$$
$$348$$ 0 0
$$349$$ 338800. 0.148895 0.0744475 0.997225i $$-0.476281\pi$$
0.0744475 + 0.997225i $$0.476281\pi$$
$$350$$ 0 0
$$351$$ 2.25424e6 0.976635
$$352$$ 0 0
$$353$$ 362046. 0.154642 0.0773209 0.997006i $$-0.475363\pi$$
0.0773209 + 0.997006i $$0.475363\pi$$
$$354$$ 0 0
$$355$$ 2.37082e6 0.998451
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ −876528. −0.358946 −0.179473 0.983763i $$-0.557439\pi$$
−0.179473 + 0.983763i $$0.557439\pi$$
$$360$$ 0 0
$$361$$ −2.25520e6 −0.910787
$$362$$ 0 0
$$363$$ −481550. −0.191812
$$364$$ 0 0
$$365$$ 302568. 0.118875
$$366$$ 0 0
$$367$$ 2.98062e6 1.15516 0.577578 0.816335i $$-0.303998\pi$$
0.577578 + 0.816335i $$0.303998\pi$$
$$368$$ 0 0
$$369$$ 889746. 0.340173
$$370$$ 0 0
$$371$$ 0 0
$$372$$ 0 0
$$373$$ 3.91441e6 1.45678 0.728391 0.685162i $$-0.240268\pi$$
0.728391 + 0.685162i $$0.240268\pi$$
$$374$$ 0 0
$$375$$ −677040. −0.248620
$$376$$ 0 0
$$377$$ −2.84174e6 −1.02975
$$378$$ 0 0
$$379$$ −3.60661e6 −1.28974 −0.644868 0.764294i $$-0.723088\pi$$
−0.644868 + 0.764294i $$0.723088\pi$$
$$380$$ 0 0
$$381$$ −3.15992e6 −1.11523
$$382$$ 0 0
$$383$$ −2.66644e6 −0.928826 −0.464413 0.885619i $$-0.653735\pi$$
−0.464413 + 0.885619i $$0.653735\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ 529672. 0.179775
$$388$$ 0 0
$$389$$ −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
$$392$$ 0 0
$$393$$ −246660. −0.0805596
$$394$$ 0 0
$$395$$ 3.60125e6 1.16134
$$396$$ 0 0
$$397$$ 4.09408e6 1.30371 0.651854 0.758345i $$-0.273992\pi$$
0.651854 + 0.758345i $$0.273992\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 942366. 0.292657 0.146328 0.989236i $$-0.453254\pi$$
0.146328 + 0.989236i $$0.453254\pi$$
$$402$$ 0 0
$$403$$ 4.30525e6 1.32049
$$404$$ 0 0
$$405$$ 323484. 0.0979976
$$406$$ 0 0
$$407$$ 4.81488e6 1.44079
$$408$$ 0 0
$$409$$ 4.84561e6 1.43232 0.716160 0.697936i $$-0.245898\pi$$
0.716160 + 0.697936i $$0.245898\pi$$
$$410$$ 0 0
$$411$$ 3.03234e6 0.885469
$$412$$ 0 0
$$413$$ 0 0
$$414$$ 0 0
$$415$$ 2.95697e6 0.842804
$$416$$ 0 0
$$417$$ 2.50586e6 0.705694
$$418$$ 0 0
$$419$$ −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 $$0.572910\pi$$
$$422$$ 0 0
$$423$$ 259116. 0.0704115
$$424$$ 0 0
$$425$$ 5.73140e6 1.53918
$$426$$ 0 0
$$427$$ 0 0
$$428$$ 0 0
$$429$$ −1.96224e6 −0.514765
$$430$$ 0 0
$$431$$ −4.14360e6 −1.07445 −0.537223 0.843440i $$-0.680527\pi$$
−0.537223 + 0.843440i $$0.680527\pi$$
$$432$$ 0 0
$$433$$ 3.03966e6 0.779121 0.389561 0.921001i $$-0.372627\pi$$
0.389561 + 0.921001i $$0.372627\pi$$
$$434$$ 0 0
$$435$$ −4.08744e6 −1.03569
$$436$$ 0 0
$$437$$ 1.97400e6 0.494474
$$438$$ 0 0
$$439$$ 2.54271e6 0.629703 0.314852 0.949141i $$-0.398045\pi$$
0.314852 + 0.949141i $$0.398045\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ 2.43210e6 0.588806 0.294403 0.955681i $$-0.404879\pi$$
0.294403 + 0.955681i $$0.404879\pi$$
$$444$$ 0 0
$$445$$ 2.24532e6 0.537500
$$446$$ 0 0
$$447$$ −605940. −0.143437
$$448$$ 0 0
$$449$$ 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
$$452$$ 0 0
$$453$$ −1.24448e6 −0.284933
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 1.58063e6 0.354030 0.177015 0.984208i $$-0.443356\pi$$
0.177015 + 0.984208i $$0.443356\pi$$
$$458$$ 0 0
$$459$$ −5.62788e6 −1.24685
$$460$$ 0 0
$$461$$ −5.09604e6 −1.11681 −0.558407 0.829567i $$-0.688587\pi$$
−0.558407 + 0.829567i $$0.688587\pi$$
$$462$$ 0 0
$$463$$ 7.02338e6 1.52263 0.761313 0.648384i $$-0.224555\pi$$
0.761313 + 0.648384i $$0.224555\pi$$
$$464$$ 0 0
$$465$$ 6.19248e6 1.32810
$$466$$ 0 0
$$467$$ −4.24845e6 −0.901443 −0.450722 0.892665i $$-0.648833\pi$$
−0.450722 + 0.892665i $$0.648833\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ 0 0
$$471$$ −760400. −0.157939
$$472$$ 0 0
$$473$$ −1.24454e6 −0.255775
$$474$$ 0 0
$$475$$ 1.84757e6 0.375722
$$476$$ 0 0
$$477$$ 5.32561e6 1.07170
$$478$$ 0 0
$$479$$ 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
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ −1.42615e6 −0.275303
$$486$$ 0 0
$$487$$ 1.32057e6 0.252312 0.126156 0.992010i $$-0.459736\pi$$
0.126156 + 0.992010i $$0.459736\pi$$
$$488$$ 0 0
$$489$$ −1.24256e6 −0.234988
$$490$$ 0 0
$$491$$ −6.27193e6 −1.17408 −0.587040 0.809558i $$-0.699707\pi$$
−0.587040 + 0.809558i $$0.699707\pi$$
$$492$$ 0 0
$$493$$ 7.09463e6 1.31466
$$494$$ 0 0
$$495$$ 4.03603e6 0.740358
$$496$$ 0 0
$$497$$ 0 0
$$498$$ 0 0
$$499$$ 3.93785e6 0.707959 0.353979 0.935253i $$-0.384828\pi$$
0.353979 + 0.935253i $$0.384828\pi$$
$$500$$ 0 0
$$501$$ −724200. −0.128903
$$502$$ 0 0
$$503$$ −7.59830e6 −1.33905 −0.669525 0.742790i $$-0.733502\pi$$
−0.669525 + 0.742790i $$0.733502\pi$$
$$504$$ 0 0
$$505$$ 8.33314e6 1.45405
$$506$$ 0 0
$$507$$ −302370. −0.0522419
$$508$$ 0 0
$$509$$ 7.82664e6 1.33900 0.669501 0.742812i $$-0.266508\pi$$
0.669501 + 0.742812i $$0.266508\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ 0 0
$$513$$ −1.81420e6 −0.304363
$$514$$ 0 0
$$515$$ 1.10581e7 1.83722
$$516$$ 0 0
$$517$$ −608832. −0.100178
$$518$$ 0 0
$$519$$ 4.41552e6 0.719554
$$520$$ 0 0
$$521$$ −8.94454e6 −1.44366 −0.721828 0.692072i $$-0.756698\pi$$
−0.721828 + 0.692072i $$0.756698\pi$$
$$522$$ 0 0
$$523$$ 4.07481e6 0.651407 0.325704 0.945472i $$-0.394399\pi$$
0.325704 + 0.945472i $$0.394399\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ −1.07484e7 −1.68584
$$528$$ 0 0
$$529$$ 1.12037e7 1.74069
$$530$$ 0 0
$$531$$ −4.90519e6 −0.754952
$$532$$ 0 0
$$533$$ 3.63365e6 0.554019
$$534$$ 0 0
$$535$$ 4.10357e6 0.619837
$$536$$ 0 0
$$537$$ 106920. 0.0160001
$$538$$ 0 0
$$539$$ 0 0
$$540$$ 0 0
$$541$$ −1.18676e7 −1.74329 −0.871644 0.490140i $$-0.836946\pi$$
−0.871644 + 0.490140i $$0.836946\pi$$
$$542$$ 0 0
$$543$$ 5.46064e6 0.794775
$$544$$ 0 0
$$545$$ 4.73542e6 0.682915
$$546$$ 0 0
$$547$$ 5.37801e6 0.768516 0.384258 0.923226i $$-0.374457\pi$$
0.384258 + 0.923226i $$0.374457\pi$$
$$548$$ 0 0
$$549$$ 3.50007e6 0.495616
$$550$$ 0 0
$$551$$ 2.28702e6 0.320916
$$552$$ 0 0
$$553$$ 0 0
$$554$$ 0 0
$$555$$ −1.20372e7 −1.65880
$$556$$ 0 0
$$557$$ −5.64878e6 −0.771466 −0.385733 0.922611i $$-0.626051\pi$$
−0.385733 + 0.922611i $$0.626051\pi$$
$$558$$ 0 0
$$559$$ 2.16314e6 0.292789
$$560$$ 0 0
$$561$$ 4.89888e6 0.657188
$$562$$ 0 0
$$563$$ 4.56407e6 0.606850 0.303425 0.952855i $$-0.401870\pi$$
0.303425 + 0.952855i $$0.401870\pi$$
$$564$$ 0 0
$$565$$ −734328. −0.0967763
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ 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 $$-0.157358\pi$$
0.880275 + 0.474464i $$0.157358\pi$$
$$572$$ 0 0
$$573$$ 5.75976e6 0.732855
$$574$$ 0 0
$$575$$ 1.65102e7 2.08249
$$576$$ 0 0
$$577$$ −6.09797e6 −0.762510 −0.381255 0.924470i $$-0.624508\pi$$
−0.381255 + 0.924470i $$0.624508\pi$$
$$578$$ 0 0
$$579$$ −4.13938e6 −0.513144
$$580$$ 0 0
$$581$$ 0 0
$$582$$ 0 0
$$583$$ −1.25133e7 −1.52476
$$584$$ 0 0
$$585$$ −7.01501e6 −0.847498
$$586$$ 0 0
$$587$$ −8.08462e6 −0.968422 −0.484211 0.874951i $$-0.660893\pi$$
−0.484211 + 0.874951i $$0.660893\pi$$
$$588$$ 0 0
$$589$$ −3.46484e6 −0.411524
$$590$$ 0 0
$$591$$ −4.94946e6 −0.582893
$$592$$ 0 0
$$593$$ −1.41575e6 −0.165330 −0.0826649 0.996577i $$-0.526343\pi$$
−0.0826649 + 0.996577i $$0.526343\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ 5.20364e6 0.597546
$$598$$ 0 0
$$599$$ −8.75460e6 −0.996941 −0.498470 0.866907i $$-0.666105\pi$$
−0.498470 + 0.866907i $$0.666105\pi$$
$$600$$ 0 0
$$601$$ −8.70276e6 −0.982813 −0.491407 0.870930i $$-0.663517\pi$$
−0.491407 + 0.870930i $$0.663517\pi$$
$$602$$ 0 0
$$603$$ −2.49564e6 −0.279504
$$604$$ 0 0
$$605$$ 4.04502e6 0.449296
$$606$$ 0 0
$$607$$ −1.69578e7 −1.86809 −0.934045 0.357157i $$-0.883746\pi$$
−0.934045 + 0.357157i $$0.883746\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 1.05821e6 0.114675
$$612$$ 0 0
$$613$$ 1.76743e7 1.89973 0.949866 0.312658i $$-0.101220\pi$$
0.949866 + 0.312658i $$0.101220\pi$$
$$614$$ 0 0
$$615$$ 5.22648e6 0.557213
$$616$$ 0 0
$$617$$ −9.70636e6 −1.02646 −0.513232 0.858250i $$-0.671552\pi$$
−0.513232 + 0.858250i $$0.671552\pi$$
$$618$$ 0 0
$$619$$ 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
$$622$$ 0 0
$$623$$ 0 0
$$624$$ 0 0
$$625$$ −6.59724e6 −0.675557
$$626$$ 0 0
$$627$$ 1.57920e6 0.160424
$$628$$ 0 0
$$629$$ 2.08931e7 2.10561
$$630$$ 0 0
$$631$$ −1.26353e7 −1.26331 −0.631656 0.775248i $$-0.717625\pi$$
−0.631656 + 0.775248i $$0.717625\pi$$
$$632$$ 0 0
$$633$$ −1.83284e6 −0.181809
$$634$$ 0 0
$$635$$ 2.65433e7 2.61229
$$636$$ 0 0
$$637$$ 0 0
$$638$$ 0 0
$$639$$ 4.03603e6 0.391023
$$640$$ 0 0
$$641$$ 6.23398e6 0.599267 0.299634 0.954054i $$-0.403136\pi$$
0.299634 + 0.954054i $$0.403136\pi$$
$$642$$ 0 0
$$643$$ 1.06874e7 1.01940 0.509701 0.860352i $$-0.329756\pi$$
0.509701 + 0.860352i $$0.329756\pi$$
$$644$$ 0 0
$$645$$ 3.11136e6 0.294477
$$646$$ 0 0
$$647$$ 1.83258e7 1.72109 0.860544 0.509376i $$-0.170124\pi$$
0.860544 + 0.509376i $$0.170124\pi$$
$$648$$ 0 0
$$649$$ 1.15255e7 1.07411
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ −7.28857e6 −0.668897 −0.334448 0.942414i $$-0.608550\pi$$
−0.334448 + 0.942414i $$0.608550\pi$$
$$654$$ 0 0
$$655$$ 2.07194e6 0.188701
$$656$$ 0 0
$$657$$ 515086. 0.0465550
$$658$$ 0 0
$$659$$ −4.54337e6 −0.407534 −0.203767 0.979019i $$-0.565319\pi$$
−0.203767 + 0.979019i $$0.565319\pi$$
$$660$$ 0 0
$$661$$ 2.10021e7 1.86964 0.934821 0.355120i $$-0.115560\pi$$
0.934821 + 0.355120i $$0.115560\pi$$
$$662$$ 0 0
$$663$$ −8.51472e6 −0.752292
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 2.04372e7 1.77872
$$668$$ 0 0
$$669$$ −1.27746e7 −1.10353
$$670$$ 0 0
$$671$$ −8.22394e6 −0.705137
$$672$$ 0 0
$$673$$ 3.46923e6 0.295253 0.147627 0.989043i $$-0.452837\pi$$
0.147627 + 0.989043i $$0.452837\pi$$
$$674$$ 0 0
$$675$$ −1.51737e7 −1.28183
$$676$$ 0 0
$$677$$ 1.80916e7 1.51707 0.758536 0.651631i $$-0.225915\pi$$
0.758536 + 0.651631i $$0.225915\pi$$
$$678$$ 0 0
$$679$$ 0 0
$$680$$ 0 0
$$681$$ −1.28764e7 −1.06397
$$682$$ 0 0
$$683$$ −4.67752e6 −0.383675 −0.191838 0.981427i $$-0.561445\pi$$
−0.191838 + 0.981427i $$0.561445\pi$$
$$684$$ 0 0
$$685$$ −2.54717e7 −2.07411
$$686$$ 0 0
$$687$$ −3.50936e6 −0.283685
$$688$$ 0 0
$$689$$ 2.17493e7 1.74541
$$690$$ 0 0
$$691$$ 1.68960e7 1.34614 0.673069 0.739579i $$-0.264976\pi$$
0.673069 + 0.739579i $$0.264976\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ −2.10492e7 −1.65300
$$696$$ 0 0
$$697$$ −9.07168e6 −0.707303
$$698$$ 0 0
$$699$$ 8.36154e6 0.647282
$$700$$ 0 0
$$701$$ 2.40964e6 0.185207 0.0926035 0.995703i $$-0.470481\pi$$
0.0926035 + 0.995703i $$0.470481\pi$$
$$702$$ 0 0
$$703$$ 6.73510e6 0.513991
$$704$$ 0 0
$$705$$ 1.52208e6 0.115336
$$706$$ 0 0
$$707$$ 0 0
$$708$$ 0 0
$$709$$ −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
$$712$$ 0 0
$$713$$ −3.09624e7 −2.28092
$$714$$ 0 0
$$715$$ 1.64828e7 1.20578
$$716$$ 0 0
$$717$$ −7.74336e6 −0.562512
$$718$$ 0 0
$$719$$ −1.43716e7 −1.03677 −0.518385 0.855147i $$-0.673467\pi$$
−0.518385 + 0.855147i $$0.673467\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ 0 0
$$723$$ 1.15285e7 0.820214
$$724$$ 0 0
$$725$$ 1.91282e7 1.35154
$$726$$ 0 0
$$727$$ −1.40705e7 −0.987353 −0.493676 0.869646i $$-0.664347\pi$$
−0.493676 + 0.869646i $$0.664347\pi$$
$$728$$ 0 0
$$729$$ 9.93049e6 0.692073
$$730$$ 0 0
$$731$$ −5.40043e6 −0.373796
$$732$$ 0 0
$$733$$ 3.75000e6 0.257793 0.128897 0.991658i $$-0.458856\pi$$
0.128897 + 0.991658i $$0.458856\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 5.86387e6 0.397664
$$738$$ 0 0
$$739$$ −2.61318e7 −1.76019 −0.880093 0.474802i $$-0.842520\pi$$
−0.880093 + 0.474802i $$0.842520\pi$$
$$740$$ 0 0
$$741$$ −2.74480e6 −0.183639
$$742$$ 0 0
$$743$$ 159072. 0.0105711 0.00528557 0.999986i $$-0.498318\pi$$
0.00528557 + 0.999986i $$0.498318\pi$$
$$744$$ 0 0
$$745$$ 5.08990e6 0.335984
$$746$$ 0 0
$$747$$ 5.03389e6 0.330067
$$748$$ 0 0
$$749$$ 0 0
$$750$$ 0 0
$$751$$ 2.65311e7 1.71654 0.858272 0.513196i $$-0.171539\pi$$
0.858272 + 0.513196i $$0.171539\pi$$
$$752$$ 0 0
$$753$$ 1.35801e7 0.872802
$$754$$ 0 0
$$755$$ 1.04536e7 0.667421
$$756$$ 0 0
$$757$$ −1.52032e7 −0.964260 −0.482130 0.876100i $$-0.660137\pi$$
−0.482130 + 0.876100i $$0.660137\pi$$
$$758$$ 0 0
$$759$$ 1.41120e7 0.889169
$$760$$ 0 0
$$761$$ −4.71380e6 −0.295059 −0.147530 0.989058i $$-0.547132\pi$$
−0.147530 + 0.989058i $$0.547132\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ 0 0
$$765$$ 1.75135e7 1.08198
$$766$$ 0 0
$$767$$ −2.00324e7 −1.22954
$$768$$ 0 0
$$769$$ 1.58977e6 0.0969434 0.0484717 0.998825i $$-0.484565\pi$$
0.0484717 + 0.998825i $$0.484565\pi$$
$$770$$ 0 0
$$771$$ 3.17742e6 0.192504
$$772$$ 0 0
$$773$$ 9.69095e6 0.583334 0.291667 0.956520i $$-0.405790\pi$$
0.291667 + 0.956520i $$0.405790\pi$$
$$774$$ 0 0
$$775$$ −2.89793e7 −1.73314
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ −2.92434e6 −0.172657
$$780$$ 0 0
$$781$$ −9.48326e6 −0.556327
$$782$$ 0 0
$$783$$ −1.87828e7 −1.09485
$$784$$ 0 0
$$785$$ 6.38736e6 0.369954
$$786$$ 0 0
$$787$$ −1.57170e6 −0.0904549 −0.0452275 0.998977i $$-0.514401\pi$$
−0.0452275 + 0.998977i $$0.514401\pi$$
$$788$$ 0 0
$$789$$ −1.05101e7 −0.601054
$$790$$ 0 0
$$791$$ 0 0
$$792$$ 0 0
$$793$$ 1.42940e7 0.807180
$$794$$ 0 0
$$795$$ 3.12833e7 1.75547
$$796$$ 0 0
$$797$$ 2.25298e6 0.125635 0.0628175 0.998025i $$-0.479991\pi$$
0.0628175 + 0.998025i $$0.479991\pi$$
$$798$$ 0 0
$$799$$ −2.64190e6 −0.146403
$$800$$ 0 0
$$801$$ 3.82239e6 0.210501
$$802$$ 0 0
$$803$$ −1.21027e6 −0.0662360
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ −1.18958e7 −0.643000
$$808$$ 0 0
$$809$$ −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 $$0.454617\pi$$
$$812$$ 0 0
$$813$$ −1.43008e7 −0.758812
$$814$$ 0 0
$$815$$ 1.04375e7 0.550431
$$816$$ 0 0
$$817$$ −1.74088e6 −0.0912460
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ 1.48802e7 0.770464 0.385232 0.922820i $$-0.374121\pi$$
0.385232 + 0.922820i $$0.374121\pi$$
$$822$$ 0 0
$$823$$ −2.00601e7 −1.03236 −0.516182 0.856479i $$-0.672647\pi$$
−0.516182 + 0.856479i $$0.672647\pi$$
$$824$$ 0 0
$$825$$ 1.32082e7 0.675628
$$826$$ 0 0
$$827$$ −1.21539e7 −0.617949 −0.308975 0.951070i $$-0.599986\pi$$
−0.308975 + 0.951070i $$0.599986\pi$$
$$828$$ 0 0
$$829$$ −3.21197e7 −1.62325 −0.811625 0.584179i $$-0.801417\pi$$
−0.811625 + 0.584179i $$0.801417\pi$$
$$830$$ 0 0
$$831$$ 633020. 0.0317991
$$832$$ 0 0
$$833$$ 0 0
$$834$$ 0 0
$$835$$ 6.08328e6 0.301941
$$836$$ 0 0
$$837$$ 2.84559e7 1.40397
$$838$$ 0 0
$$839$$ −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
$$842$$ 0 0
$$843$$ −4.96614e6 −0.240686
$$844$$ 0 0
$$845$$ 2.53991e6 0.122370
$$846$$ 0 0
$$847$$ 0 0
$$848$$ 0 0
$$849$$ −1.15842e7 −0.551564
$$850$$ 0 0
$$851$$ 6.01860e7 2.84886
$$852$$ 0 0
$$853$$ −234824. −0.0110502 −0.00552510 0.999985i $$-0.501759\pi$$
−0.00552510 + 0.999985i $$0.501759\pi$$
$$854$$ 0 0
$$855$$ 5.64564e6 0.264118
$$856$$ 0 0
$$857$$ −2.83802e7 −1.31997 −0.659985 0.751279i $$-0.729437\pi$$
−0.659985 + 0.751279i $$0.729437\pi$$
$$858$$ 0 0
$$859$$ 4.00081e7 1.84997 0.924986 0.380001i $$-0.124076\pi$$
0.924986 + 0.380001i $$0.124076\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ 2.08030e7 0.950823 0.475411 0.879764i $$-0.342299\pi$$
0.475411 + 0.879764i $$0.342299\pi$$
$$864$$ 0 0
$$865$$ −3.70904e7 −1.68547
$$866$$ 0 0
$$867$$ 7.05907e6 0.318933
$$868$$ 0 0
$$869$$ −1.44050e7 −0.647088
$$870$$ 0 0
$$871$$ −1.01920e7 −0.455211
$$872$$ 0 0
$$873$$ −2.42785e6 −0.107817
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 3.03559e7 1.33273 0.666367 0.745624i $$-0.267848\pi$$
0.666367 + 0.745624i $$0.267848\pi$$
$$878$$ 0 0
$$879$$ −1.43886e7 −0.628125
$$880$$ 0 0
$$881$$ 2.58936e7 1.12396 0.561981 0.827150i $$-0.310039\pi$$
0.561981 + 0.827150i $$0.310039\pi$$
$$882$$ 0 0
$$883$$ 1.88813e7 0.814950 0.407475 0.913216i $$-0.366409\pi$$
0.407475 + 0.913216i $$0.366409\pi$$
$$884$$ 0 0
$$885$$ −2.88137e7 −1.23663
$$886$$ 0 0
$$887$$ −2.34431e7 −1.00048 −0.500238 0.865888i $$-0.666754\pi$$
−0.500238 + 0.865888i $$0.666754\pi$$
$$888$$ 0 0
$$889$$ 0 0
$$890$$ 0 0
$$891$$ −1.29394e6 −0.0546033
$$892$$ 0 0
$$893$$ −851640. −0.0357378
$$894$$ 0 0
$$895$$ −898128. −0.0374784
$$896$$ 0 0
$$897$$ −2.45280e7 −1.01784
$$898$$ 0 0
$$899$$ −3.58722e7 −1.48033
$$900$$ 0 0
$$901$$ −5.42988e7 −2.22833
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ −4.58694e7 −1.86166
$$906$$ 0 0
$$907$$ 5.60873e6 0.226384 0.113192 0.993573i $$-0.463892\pi$$
0.113192 + 0.993573i $$0.463892\pi$$
$$908$$ 0 0
$$909$$ 1.41862e7 0.569450
$$910$$ 0 0
$$911$$ −2.16215e7 −0.863156 −0.431578 0.902076i $$-0.642043\pi$$
−0.431578 + 0.902076i $$0.642043\pi$$
$$912$$ 0 0
$$913$$ −1.18279e7 −0.469602
$$914$$ 0 0
$$915$$ 2.05598e7 0.811834
$$916$$ 0 0
$$917$$ 0 0
$$918$$ 0 0
$$919$$ −4.51695e7 −1.76424 −0.882119 0.471028i $$-0.843883\pi$$
−0.882119 + 0.471028i $$0.843883\pi$$
$$920$$ 0 0
$$921$$ −9.89098e6 −0.384229
$$922$$ 0 0
$$923$$ 1.64828e7 0.636835
$$924$$ 0 0
$$925$$ 5.63312e7 2.16469
$$926$$ 0 0
$$927$$ 1.88251e7 0.719512
$$928$$ 0 0
$$929$$ 2.28729e7 0.869524 0.434762 0.900545i $$-0.356832\pi$$
0.434762 + 0.900545i $$0.356832\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ 0 0
$$933$$ −2.22050e7 −0.835117
$$934$$ 0 0
$$935$$ −4.11506e7 −1.53938
$$936$$ 0 0
$$937$$ 1.79616e7 0.668336 0.334168 0.942514i $$-0.391545\pi$$
0.334168 + 0.942514i $$0.391545\pi$$
$$938$$ 0 0
$$939$$ −2.33008e7 −0.862395
$$940$$ 0 0
$$941$$ 1.79697e7 0.661558 0.330779 0.943708i $$-0.392689\pi$$
0.330779 + 0.943708i $$0.392689\pi$$
$$942$$ 0 0
$$943$$ −2.61324e7 −0.956974
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ −4.32115e7 −1.56576 −0.782879 0.622174i $$-0.786250\pi$$
−0.782879 + 0.622174i $$0.786250\pi$$
$$948$$ 0 0
$$949$$ 2.10357e6 0.0758213
$$950$$ 0 0
$$951$$ 4.27542e6 0.153295
$$952$$ 0 0
$$953$$ −7.50965e6 −0.267848 −0.133924 0.990992i $$-0.542758\pi$$
−0.133924 + 0.990992i $$0.542758\pi$$
$$954$$ 0 0
$$955$$ −4.83820e7 −1.71662
$$956$$ 0 0
$$957$$ 1.63498e7 0.577074
$$958$$ 0 0
$$959$$ 0 0
$$960$$ 0 0
$$961$$ 2.57172e7 0.898288
$$962$$ 0 0
$$963$$ 6.98584e6 0.242746
$$964$$ 0 0
$$965$$ 3.47708e7 1.20198
$$966$$ 0 0
$$967$$ 1.69305e7 0.582242 0.291121 0.956686i $$-0.405972\pi$$
0.291121 + 0.956686i $$0.405972\pi$$
$$968$$ 0 0
$$969$$ 6.85260e6 0.234448
$$970$$ 0 0
$$971$$ 2.86144e7 0.973949 0.486974 0.873416i $$-0.338101\pi$$
0.486974 + 0.873416i $$0.338101\pi$$
$$972$$ 0 0
$$973$$ 0 0
$$974$$ 0 0
$$975$$ −2.29570e7 −0.773400
$$976$$ 0 0
$$977$$ 3.69445e7 1.23826 0.619132 0.785287i $$-0.287485\pi$$
0.619132 + 0.785287i $$0.287485\pi$$
$$978$$ 0 0
$$979$$ −8.98128e6 −0.299489
$$980$$ 0 0
$$981$$ 8.06148e6 0.267450
$$982$$ 0 0
$$983$$ −3.88787e7 −1.28330 −0.641650 0.766998i $$-0.721750\pi$$
−0.641650 + 0.766998i $$0.721750\pi$$
$$984$$ 0 0
$$985$$ 4.15755e7 1.36536
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ −1.55568e7 −0.505743
$$990$$ 0 0
$$991$$ −2.49212e7 −0.806092 −0.403046 0.915180i $$-0.632049\pi$$
−0.403046 + 0.915180i $$0.632049\pi$$
$$992$$ 0 0
$$993$$ 3.96616e6 0.127643
$$994$$ 0 0
$$995$$ −4.37106e7 −1.39968
$$996$$ 0 0
$$997$$ −1.01956e7 −0.324845 −0.162422 0.986721i $$-0.551931\pi$$
−0.162422 + 0.986721i $$0.551931\pi$$
$$998$$ 0 0
$$999$$ −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 784.6.a.i.1.1 1
4.3 odd 2 98.6.a.a.1.1 1
7.6 odd 2 112.6.a.c.1.1 1
12.11 even 2 882.6.a.x.1.1 1
21.20 even 2 1008.6.a.b.1.1 1
28.3 even 6 98.6.c.c.79.1 2
28.11 odd 6 98.6.c.d.79.1 2
28.19 even 6 98.6.c.c.67.1 2
28.23 odd 6 98.6.c.d.67.1 2
28.27 even 2 14.6.a.a.1.1 1
56.13 odd 2 448.6.a.l.1.1 1
56.27 even 2 448.6.a.e.1.1 1
84.83 odd 2 126.6.a.f.1.1 1
140.27 odd 4 350.6.c.d.99.1 2
140.83 odd 4 350.6.c.d.99.2 2
140.139 even 2 350.6.a.i.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
14.6.a.a.1.1 1 28.27 even 2
98.6.a.a.1.1 1 4.3 odd 2
98.6.c.c.67.1 2 28.19 even 6
98.6.c.c.79.1 2 28.3 even 6
98.6.c.d.67.1 2 28.23 odd 6
98.6.c.d.79.1 2 28.11 odd 6
112.6.a.c.1.1 1 7.6 odd 2
126.6.a.f.1.1 1 84.83 odd 2
350.6.a.i.1.1 1 140.139 even 2
350.6.c.d.99.1 2 140.27 odd 4
350.6.c.d.99.2 2 140.83 odd 4
448.6.a.e.1.1 1 56.27 even 2
448.6.a.l.1.1 1 56.13 odd 2
784.6.a.i.1.1 1 1.1 even 1 trivial
882.6.a.x.1.1 1 12.11 even 2
1008.6.a.b.1.1 1 21.20 even 2