# Properties

 Label 441.6.a.z.1.2 Level $441$ Weight $6$ Character 441.1 Self dual yes Analytic conductor $70.729$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [441,6,Mod(1,441)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(441, 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("441.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$441 = 3^{2} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 441.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: $$70.7292645375$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\sqrt{2}, \sqrt{113})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - 2x^{3} - 59x^{2} + 60x + 674$$ x^4 - 2*x^3 - 59*x^2 + 60*x + 674 Coefficient ring: $$\Z[a_1, \ldots, a_{17}]$$ Coefficient ring index: $$7$$ Twist minimal: no (minimal twist has level 49) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$7.22929$$ of defining polynomial Character $$\chi$$ $$=$$ 441.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-2.81507 q^{2} -24.0754 q^{4} +45.9910 q^{5} +157.856 q^{8} +O(q^{10})$$ $$q-2.81507 q^{2} -24.0754 q^{4} +45.9910 q^{5} +157.856 q^{8} -129.468 q^{10} +551.781 q^{11} +1094.10 q^{13} +326.035 q^{16} +1180.71 q^{17} -1166.13 q^{19} -1107.25 q^{20} -1553.30 q^{22} -44.3851 q^{23} -1009.82 q^{25} -3079.97 q^{26} -3329.02 q^{29} +8784.01 q^{31} -5969.21 q^{32} -3323.80 q^{34} -2557.12 q^{37} +3282.73 q^{38} +7259.97 q^{40} +12761.3 q^{41} -96.7714 q^{43} -13284.3 q^{44} +124.947 q^{46} -7679.15 q^{47} +2842.73 q^{50} -26340.8 q^{52} +11953.3 q^{53} +25377.0 q^{55} +9371.43 q^{58} -9857.24 q^{59} -38517.9 q^{61} -24727.6 q^{62} +6370.65 q^{64} +50318.7 q^{65} -67548.9 q^{67} -28426.1 q^{68} +61374.6 q^{71} +1850.40 q^{73} +7198.49 q^{74} +28074.9 q^{76} -8.52913 q^{79} +14994.7 q^{80} -35923.9 q^{82} +95039.3 q^{83} +54302.3 q^{85} +272.419 q^{86} +87102.1 q^{88} -53605.6 q^{89} +1068.59 q^{92} +21617.4 q^{94} -53631.4 q^{95} +3110.79 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 10 q^{2} + 10 q^{4} + 270 q^{8}+O(q^{10})$$ 4 * q + 10 * q^2 + 10 * q^4 + 270 * q^8 $$4 q + 10 q^{2} + 10 q^{4} + 270 q^{8} + 1952 q^{11} - 1566 q^{16} + 3524 q^{22} + 7136 q^{23} + 2764 q^{25} + 3352 q^{29} - 27810 q^{32} - 9208 q^{37} + 20448 q^{43} - 1900 q^{44} + 56712 q^{46} + 43070 q^{50} + 102920 q^{53} + 96972 q^{58} - 40318 q^{64} + 63168 q^{65} - 22896 q^{67} + 153824 q^{71} - 17596 q^{74} - 90688 q^{79} + 272656 q^{85} + 161860 q^{86} + 154812 q^{88} + 212200 q^{92} - 108224 q^{95}+O(q^{100})$$ 4 * q + 10 * q^2 + 10 * q^4 + 270 * q^8 + 1952 * q^11 - 1566 * q^16 + 3524 * q^22 + 7136 * q^23 + 2764 * q^25 + 3352 * q^29 - 27810 * q^32 - 9208 * q^37 + 20448 * q^43 - 1900 * q^44 + 56712 * q^46 + 43070 * q^50 + 102920 * q^53 + 96972 * q^58 - 40318 * q^64 + 63168 * q^65 - 22896 * q^67 + 153824 * q^71 - 17596 * q^74 - 90688 * q^79 + 272656 * q^85 + 161860 * q^86 + 154812 * q^88 + 212200 * q^92 - 108224 * q^95

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −2.81507 −0.497639 −0.248820 0.968550i $$-0.580043\pi$$
−0.248820 + 0.968550i $$0.580043\pi$$
$$3$$ 0 0
$$4$$ −24.0754 −0.752355
$$5$$ 45.9910 0.822713 0.411356 0.911475i $$-0.365055\pi$$
0.411356 + 0.911475i $$0.365055\pi$$
$$6$$ 0 0
$$7$$ 0 0
$$8$$ 157.856 0.872041
$$9$$ 0 0
$$10$$ −129.468 −0.409414
$$11$$ 551.781 1.37494 0.687472 0.726211i $$-0.258720\pi$$
0.687472 + 0.726211i $$0.258720\pi$$
$$12$$ 0 0
$$13$$ 1094.10 1.79555 0.897776 0.440453i $$-0.145182\pi$$
0.897776 + 0.440453i $$0.145182\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 326.035 0.318393
$$17$$ 1180.71 0.990883 0.495442 0.868641i $$-0.335006\pi$$
0.495442 + 0.868641i $$0.335006\pi$$
$$18$$ 0 0
$$19$$ −1166.13 −0.741074 −0.370537 0.928818i $$-0.620826\pi$$
−0.370537 + 0.928818i $$0.620826\pi$$
$$20$$ −1107.25 −0.618972
$$21$$ 0 0
$$22$$ −1553.30 −0.684226
$$23$$ −44.3851 −0.0174951 −0.00874757 0.999962i $$-0.502784\pi$$
−0.00874757 + 0.999962i $$0.502784\pi$$
$$24$$ 0 0
$$25$$ −1009.82 −0.323143
$$26$$ −3079.97 −0.893537
$$27$$ 0 0
$$28$$ 0 0
$$29$$ −3329.02 −0.735057 −0.367529 0.930012i $$-0.619796\pi$$
−0.367529 + 0.930012i $$0.619796\pi$$
$$30$$ 0 0
$$31$$ 8784.01 1.64168 0.820841 0.571157i $$-0.193505\pi$$
0.820841 + 0.571157i $$0.193505\pi$$
$$32$$ −5969.21 −1.03049
$$33$$ 0 0
$$34$$ −3323.80 −0.493102
$$35$$ 0 0
$$36$$ 0 0
$$37$$ −2557.12 −0.307077 −0.153539 0.988143i $$-0.549067\pi$$
−0.153539 + 0.988143i $$0.549067\pi$$
$$38$$ 3282.73 0.368787
$$39$$ 0 0
$$40$$ 7259.97 0.717439
$$41$$ 12761.3 1.18559 0.592794 0.805354i $$-0.298025\pi$$
0.592794 + 0.805354i $$0.298025\pi$$
$$42$$ 0 0
$$43$$ −96.7714 −0.00798135 −0.00399067 0.999992i $$-0.501270\pi$$
−0.00399067 + 0.999992i $$0.501270\pi$$
$$44$$ −13284.3 −1.03445
$$45$$ 0 0
$$46$$ 124.947 0.00870627
$$47$$ −7679.15 −0.507071 −0.253535 0.967326i $$-0.581593\pi$$
−0.253535 + 0.967326i $$0.581593\pi$$
$$48$$ 0 0
$$49$$ 0 0
$$50$$ 2842.73 0.160809
$$51$$ 0 0
$$52$$ −26340.8 −1.35089
$$53$$ 11953.3 0.584520 0.292260 0.956339i $$-0.405593\pi$$
0.292260 + 0.956339i $$0.405593\pi$$
$$54$$ 0 0
$$55$$ 25377.0 1.13118
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 9371.43 0.365793
$$59$$ −9857.24 −0.368659 −0.184330 0.982864i $$-0.559011\pi$$
−0.184330 + 0.982864i $$0.559011\pi$$
$$60$$ 0 0
$$61$$ −38517.9 −1.32537 −0.662686 0.748897i $$-0.730584\pi$$
−0.662686 + 0.748897i $$0.730584\pi$$
$$62$$ −24727.6 −0.816965
$$63$$ 0 0
$$64$$ 6370.65 0.194417
$$65$$ 50318.7 1.47722
$$66$$ 0 0
$$67$$ −67548.9 −1.83836 −0.919182 0.393833i $$-0.871149\pi$$
−0.919182 + 0.393833i $$0.871149\pi$$
$$68$$ −28426.1 −0.745496
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 61374.6 1.44492 0.722458 0.691415i $$-0.243012\pi$$
0.722458 + 0.691415i $$0.243012\pi$$
$$72$$ 0 0
$$73$$ 1850.40 0.0406404 0.0203202 0.999794i $$-0.493531\pi$$
0.0203202 + 0.999794i $$0.493531\pi$$
$$74$$ 7198.49 0.152814
$$75$$ 0 0
$$76$$ 28074.9 0.557551
$$77$$ 0 0
$$78$$ 0 0
$$79$$ −8.52913 −0.000153758 0 −7.68788e−5 1.00000i $$-0.500024\pi$$
−7.68788e−5 1.00000i $$0.500024\pi$$
$$80$$ 14994.7 0.261946
$$81$$ 0 0
$$82$$ −35923.9 −0.589996
$$83$$ 95039.3 1.51429 0.757143 0.653249i $$-0.226595\pi$$
0.757143 + 0.653249i $$0.226595\pi$$
$$84$$ 0 0
$$85$$ 54302.3 0.815212
$$86$$ 272.419 0.00397183
$$87$$ 0 0
$$88$$ 87102.1 1.19901
$$89$$ −53605.6 −0.717357 −0.358678 0.933461i $$-0.616773\pi$$
−0.358678 + 0.933461i $$0.616773\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 1068.59 0.0131626
$$93$$ 0 0
$$94$$ 21617.4 0.252338
$$95$$ −53631.4 −0.609691
$$96$$ 0 0
$$97$$ 3110.79 0.0335693 0.0167846 0.999859i $$-0.494657\pi$$
0.0167846 + 0.999859i $$0.494657\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 24311.9 0.243119
$$101$$ −21835.9 −0.212994 −0.106497 0.994313i $$-0.533963\pi$$
−0.106497 + 0.994313i $$0.533963\pi$$
$$102$$ 0 0
$$103$$ 65341.4 0.606870 0.303435 0.952852i $$-0.401866\pi$$
0.303435 + 0.952852i $$0.401866\pi$$
$$104$$ 172710. 1.56579
$$105$$ 0 0
$$106$$ −33649.5 −0.290880
$$107$$ 108957. 0.920020 0.460010 0.887914i $$-0.347846\pi$$
0.460010 + 0.887914i $$0.347846\pi$$
$$108$$ 0 0
$$109$$ 86728.7 0.699192 0.349596 0.936901i $$-0.386319\pi$$
0.349596 + 0.936901i $$0.386319\pi$$
$$110$$ −71438.1 −0.562922
$$111$$ 0 0
$$112$$ 0 0
$$113$$ 101496. 0.747746 0.373873 0.927480i $$-0.378030\pi$$
0.373873 + 0.927480i $$0.378030\pi$$
$$114$$ 0 0
$$115$$ −2041.32 −0.0143935
$$116$$ 80147.3 0.553024
$$117$$ 0 0
$$118$$ 27748.9 0.183459
$$119$$ 0 0
$$120$$ 0 0
$$121$$ 143411. 0.890470
$$122$$ 108431. 0.659557
$$123$$ 0 0
$$124$$ −211478. −1.23513
$$125$$ −190165. −1.08857
$$126$$ 0 0
$$127$$ −3094.61 −0.0170253 −0.00851267 0.999964i $$-0.502710\pi$$
−0.00851267 + 0.999964i $$0.502710\pi$$
$$128$$ 173081. 0.933736
$$129$$ 0 0
$$130$$ −141651. −0.735124
$$131$$ −253431. −1.29027 −0.645136 0.764067i $$-0.723199\pi$$
−0.645136 + 0.764067i $$0.723199\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ 190155. 0.914842
$$135$$ 0 0
$$136$$ 186383. 0.864091
$$137$$ 97152.9 0.442236 0.221118 0.975247i $$-0.429029\pi$$
0.221118 + 0.975247i $$0.429029\pi$$
$$138$$ 0 0
$$139$$ 210308. 0.923249 0.461624 0.887076i $$-0.347267\pi$$
0.461624 + 0.887076i $$0.347267\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ −172774. −0.719047
$$143$$ 603702. 2.46878
$$144$$ 0 0
$$145$$ −153105. −0.604741
$$146$$ −5209.01 −0.0202243
$$147$$ 0 0
$$148$$ 61563.7 0.231031
$$149$$ −140406. −0.518109 −0.259055 0.965863i $$-0.583411\pi$$
−0.259055 + 0.965863i $$0.583411\pi$$
$$150$$ 0 0
$$151$$ 119696. 0.427205 0.213603 0.976921i $$-0.431480\pi$$
0.213603 + 0.976921i $$0.431480\pi$$
$$152$$ −184080. −0.646247
$$153$$ 0 0
$$154$$ 0 0
$$155$$ 403986. 1.35063
$$156$$ 0 0
$$157$$ 97616.9 0.316065 0.158032 0.987434i $$-0.449485\pi$$
0.158032 + 0.987434i $$0.449485\pi$$
$$158$$ 24.0101 7.65159e−5 0
$$159$$ 0 0
$$160$$ −274530. −0.847794
$$161$$ 0 0
$$162$$ 0 0
$$163$$ 182678. 0.538539 0.269270 0.963065i $$-0.413218\pi$$
0.269270 + 0.963065i $$0.413218\pi$$
$$164$$ −307232. −0.891984
$$165$$ 0 0
$$166$$ −267542. −0.753568
$$167$$ 451674. 1.25324 0.626619 0.779326i $$-0.284438\pi$$
0.626619 + 0.779326i $$0.284438\pi$$
$$168$$ 0 0
$$169$$ 825757. 2.22400
$$170$$ −152865. −0.405682
$$171$$ 0 0
$$172$$ 2329.81 0.00600481
$$173$$ −371647. −0.944095 −0.472047 0.881573i $$-0.656485\pi$$
−0.472047 + 0.881573i $$0.656485\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 179900. 0.437773
$$177$$ 0 0
$$178$$ 150904. 0.356985
$$179$$ −85003.4 −0.198291 −0.0991457 0.995073i $$-0.531611\pi$$
−0.0991457 + 0.995073i $$0.531611\pi$$
$$180$$ 0 0
$$181$$ −379442. −0.860892 −0.430446 0.902616i $$-0.641644\pi$$
−0.430446 + 0.902616i $$0.641644\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ −7006.46 −0.0152565
$$185$$ −117605. −0.252636
$$186$$ 0 0
$$187$$ 651496. 1.36241
$$188$$ 184878. 0.381497
$$189$$ 0 0
$$190$$ 150976. 0.303406
$$191$$ 922196. 1.82911 0.914555 0.404462i $$-0.132541\pi$$
0.914555 + 0.404462i $$0.132541\pi$$
$$192$$ 0 0
$$193$$ 505107. 0.976090 0.488045 0.872818i $$-0.337710\pi$$
0.488045 + 0.872818i $$0.337710\pi$$
$$194$$ −8757.11 −0.0167054
$$195$$ 0 0
$$196$$ 0 0
$$197$$ −251505. −0.461723 −0.230861 0.972987i $$-0.574154\pi$$
−0.230861 + 0.972987i $$0.574154\pi$$
$$198$$ 0 0
$$199$$ −208033. −0.372392 −0.186196 0.982513i $$-0.559616\pi$$
−0.186196 + 0.982513i $$0.559616\pi$$
$$200$$ −159407. −0.281794
$$201$$ 0 0
$$202$$ 61469.7 0.105994
$$203$$ 0 0
$$204$$ 0 0
$$205$$ 586904. 0.975399
$$206$$ −183941. −0.302002
$$207$$ 0 0
$$208$$ 356714. 0.571692
$$209$$ −643446. −1.01893
$$210$$ 0 0
$$211$$ −640577. −0.990525 −0.495262 0.868744i $$-0.664928\pi$$
−0.495262 + 0.868744i $$0.664928\pi$$
$$212$$ −287781. −0.439767
$$213$$ 0 0
$$214$$ −306723. −0.457838
$$215$$ −4450.62 −0.00656636
$$216$$ 0 0
$$217$$ 0 0
$$218$$ −244148. −0.347945
$$219$$ 0 0
$$220$$ −610960. −0.851052
$$221$$ 1.29182e6 1.77918
$$222$$ 0 0
$$223$$ 390135. 0.525354 0.262677 0.964884i $$-0.415395\pi$$
0.262677 + 0.964884i $$0.415395\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ −285720. −0.372108
$$227$$ −291353. −0.375279 −0.187639 0.982238i $$-0.560084\pi$$
−0.187639 + 0.982238i $$0.560084\pi$$
$$228$$ 0 0
$$229$$ −1.23040e6 −1.55045 −0.775227 0.631682i $$-0.782365\pi$$
−0.775227 + 0.631682i $$0.782365\pi$$
$$230$$ 5746.45 0.00716276
$$231$$ 0 0
$$232$$ −525506. −0.641000
$$233$$ −114279. −0.137903 −0.0689517 0.997620i $$-0.521965\pi$$
−0.0689517 + 0.997620i $$0.521965\pi$$
$$234$$ 0 0
$$235$$ −353172. −0.417174
$$236$$ 237317. 0.277363
$$237$$ 0 0
$$238$$ 0 0
$$239$$ 1.14782e6 1.29981 0.649906 0.760014i $$-0.274808\pi$$
0.649906 + 0.760014i $$0.274808\pi$$
$$240$$ 0 0
$$241$$ 812708. 0.901346 0.450673 0.892689i $$-0.351184\pi$$
0.450673 + 0.892689i $$0.351184\pi$$
$$242$$ −403713. −0.443133
$$243$$ 0 0
$$244$$ 927332. 0.997150
$$245$$ 0 0
$$246$$ 0 0
$$247$$ −1.27586e6 −1.33064
$$248$$ 1.38661e6 1.43161
$$249$$ 0 0
$$250$$ 535328. 0.541714
$$251$$ 406772. 0.407537 0.203768 0.979019i $$-0.434681\pi$$
0.203768 + 0.979019i $$0.434681\pi$$
$$252$$ 0 0
$$253$$ −24490.8 −0.0240548
$$254$$ 8711.54 0.00847248
$$255$$ 0 0
$$256$$ −691096. −0.659081
$$257$$ 1.69712e6 1.60281 0.801403 0.598125i $$-0.204087\pi$$
0.801403 + 0.598125i $$0.204087\pi$$
$$258$$ 0 0
$$259$$ 0 0
$$260$$ −1.21144e6 −1.11140
$$261$$ 0 0
$$262$$ 713427. 0.642090
$$263$$ −205694. −0.183372 −0.0916859 0.995788i $$-0.529226\pi$$
−0.0916859 + 0.995788i $$0.529226\pi$$
$$264$$ 0 0
$$265$$ 549746. 0.480892
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 1.62627e6 1.38310
$$269$$ −1.73425e6 −1.46127 −0.730635 0.682769i $$-0.760776\pi$$
−0.730635 + 0.682769i $$0.760776\pi$$
$$270$$ 0 0
$$271$$ 369042. 0.305248 0.152624 0.988284i $$-0.451228\pi$$
0.152624 + 0.988284i $$0.451228\pi$$
$$272$$ 384954. 0.315491
$$273$$ 0 0
$$274$$ −273492. −0.220074
$$275$$ −557201. −0.444304
$$276$$ 0 0
$$277$$ 1.22767e6 0.961351 0.480676 0.876899i $$-0.340391\pi$$
0.480676 + 0.876899i $$0.340391\pi$$
$$278$$ −592032. −0.459445
$$279$$ 0 0
$$280$$ 0 0
$$281$$ −2.00671e6 −1.51607 −0.758035 0.652214i $$-0.773841\pi$$
−0.758035 + 0.652214i $$0.773841\pi$$
$$282$$ 0 0
$$283$$ −1.78581e6 −1.32547 −0.662734 0.748855i $$-0.730604\pi$$
−0.662734 + 0.748855i $$0.730604\pi$$
$$284$$ −1.47762e6 −1.08709
$$285$$ 0 0
$$286$$ −1.69947e6 −1.22856
$$287$$ 0 0
$$288$$ 0 0
$$289$$ −25770.8 −0.0181503
$$290$$ 431002. 0.300943
$$291$$ 0 0
$$292$$ −44549.0 −0.0305760
$$293$$ −853248. −0.580639 −0.290319 0.956930i $$-0.593762\pi$$
−0.290319 + 0.956930i $$0.593762\pi$$
$$294$$ 0 0
$$295$$ −453345. −0.303301
$$296$$ −403658. −0.267784
$$297$$ 0 0
$$298$$ 395254. 0.257831
$$299$$ −48561.6 −0.0314134
$$300$$ 0 0
$$301$$ 0 0
$$302$$ −336952. −0.212594
$$303$$ 0 0
$$304$$ −380198. −0.235953
$$305$$ −1.77148e6 −1.09040
$$306$$ 0 0
$$307$$ −1.96068e6 −1.18730 −0.593652 0.804722i $$-0.702314\pi$$
−0.593652 + 0.804722i $$0.702314\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ −1.13725e6 −0.672128
$$311$$ −863604. −0.506307 −0.253153 0.967426i $$-0.581468\pi$$
−0.253153 + 0.967426i $$0.581468\pi$$
$$312$$ 0 0
$$313$$ 1.10047e6 0.634918 0.317459 0.948272i $$-0.397170\pi$$
0.317459 + 0.948272i $$0.397170\pi$$
$$314$$ −274799. −0.157286
$$315$$ 0 0
$$316$$ 205.342 0.000115680 0
$$317$$ −1.49591e6 −0.836097 −0.418048 0.908425i $$-0.637286\pi$$
−0.418048 + 0.908425i $$0.637286\pi$$
$$318$$ 0 0
$$319$$ −1.83689e6 −1.01066
$$320$$ 292993. 0.159949
$$321$$ 0 0
$$322$$ 0 0
$$323$$ −1.37686e6 −0.734318
$$324$$ 0 0
$$325$$ −1.10485e6 −0.580221
$$326$$ −514252. −0.267998
$$327$$ 0 0
$$328$$ 2.01445e6 1.03388
$$329$$ 0 0
$$330$$ 0 0
$$331$$ −2.74015e6 −1.37469 −0.687345 0.726331i $$-0.741224\pi$$
−0.687345 + 0.726331i $$0.741224\pi$$
$$332$$ −2.28810e6 −1.13928
$$333$$ 0 0
$$334$$ −1.27149e6 −0.623661
$$335$$ −3.10665e6 −1.51245
$$336$$ 0 0
$$337$$ −2.31353e6 −1.10968 −0.554842 0.831956i $$-0.687221\pi$$
−0.554842 + 0.831956i $$0.687221\pi$$
$$338$$ −2.32457e6 −1.10675
$$339$$ 0 0
$$340$$ −1.30735e6 −0.613329
$$341$$ 4.84685e6 2.25722
$$342$$ 0 0
$$343$$ 0 0
$$344$$ −15276.0 −0.00696006
$$345$$ 0 0
$$346$$ 1.04621e6 0.469819
$$347$$ 3.05926e6 1.36393 0.681966 0.731384i $$-0.261125\pi$$
0.681966 + 0.731384i $$0.261125\pi$$
$$348$$ 0 0
$$349$$ −210232. −0.0923921 −0.0461961 0.998932i $$-0.514710\pi$$
−0.0461961 + 0.998932i $$0.514710\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ −3.29370e6 −1.41686
$$353$$ 3.76790e6 1.60939 0.804697 0.593686i $$-0.202328\pi$$
0.804697 + 0.593686i $$0.202328\pi$$
$$354$$ 0 0
$$355$$ 2.82268e6 1.18875
$$356$$ 1.29057e6 0.539707
$$357$$ 0 0
$$358$$ 239291. 0.0986776
$$359$$ −1.00722e6 −0.412465 −0.206232 0.978503i $$-0.566120\pi$$
−0.206232 + 0.978503i $$0.566120\pi$$
$$360$$ 0 0
$$361$$ −1.11625e6 −0.450809
$$362$$ 1.06816e6 0.428414
$$363$$ 0 0
$$364$$ 0 0
$$365$$ 85101.8 0.0334354
$$366$$ 0 0
$$367$$ −1.52650e6 −0.591603 −0.295802 0.955249i $$-0.595587\pi$$
−0.295802 + 0.955249i $$0.595587\pi$$
$$368$$ −14471.1 −0.00557034
$$369$$ 0 0
$$370$$ 331066. 0.125722
$$371$$ 0 0
$$372$$ 0 0
$$373$$ 4.86297e6 1.80980 0.904898 0.425629i $$-0.139947\pi$$
0.904898 + 0.425629i $$0.139947\pi$$
$$374$$ −1.83401e6 −0.677988
$$375$$ 0 0
$$376$$ −1.21220e6 −0.442186
$$377$$ −3.64227e6 −1.31983
$$378$$ 0 0
$$379$$ 630878. 0.225604 0.112802 0.993617i $$-0.464017\pi$$
0.112802 + 0.993617i $$0.464017\pi$$
$$380$$ 1.29119e6 0.458704
$$381$$ 0 0
$$382$$ −2.59605e6 −0.910237
$$383$$ −565644. −0.197036 −0.0985182 0.995135i $$-0.531410\pi$$
−0.0985182 + 0.995135i $$0.531410\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ −1.42191e6 −0.485741
$$387$$ 0 0
$$388$$ −74893.5 −0.0252560
$$389$$ −592212. −0.198428 −0.0992140 0.995066i $$-0.531633\pi$$
−0.0992140 + 0.995066i $$0.531633\pi$$
$$390$$ 0 0
$$391$$ −52406.1 −0.0173356
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 708005. 0.229771
$$395$$ −392.264 −0.000126498 0
$$396$$ 0 0
$$397$$ 1.34312e6 0.427698 0.213849 0.976867i $$-0.431400\pi$$
0.213849 + 0.976867i $$0.431400\pi$$
$$398$$ 585629. 0.185317
$$399$$ 0 0
$$400$$ −329238. −0.102887
$$401$$ 3.68716e6 1.14507 0.572534 0.819881i $$-0.305960\pi$$
0.572534 + 0.819881i $$0.305960\pi$$
$$402$$ 0 0
$$403$$ 9.61057e6 2.94772
$$404$$ 525707. 0.160247
$$405$$ 0 0
$$406$$ 0 0
$$407$$ −1.41097e6 −0.422214
$$408$$ 0 0
$$409$$ 1.45630e6 0.430470 0.215235 0.976562i $$-0.430948\pi$$
0.215235 + 0.976562i $$0.430948\pi$$
$$410$$ −1.65218e6 −0.485397
$$411$$ 0 0
$$412$$ −1.57312e6 −0.456582
$$413$$ 0 0
$$414$$ 0 0
$$415$$ 4.37096e6 1.24582
$$416$$ −6.53090e6 −1.85029
$$417$$ 0 0
$$418$$ 1.81135e6 0.507062
$$419$$ 2.92192e6 0.813080 0.406540 0.913633i $$-0.366735\pi$$
0.406540 + 0.913633i $$0.366735\pi$$
$$420$$ 0 0
$$421$$ 2.01999e6 0.555450 0.277725 0.960661i $$-0.410420\pi$$
0.277725 + 0.960661i $$0.410420\pi$$
$$422$$ 1.80327e6 0.492924
$$423$$ 0 0
$$424$$ 1.88691e6 0.509725
$$425$$ −1.19231e6 −0.320197
$$426$$ 0 0
$$427$$ 0 0
$$428$$ −2.62319e6 −0.692182
$$429$$ 0 0
$$430$$ 12528.8 0.00326768
$$431$$ 5.43800e6 1.41009 0.705043 0.709164i $$-0.250928\pi$$
0.705043 + 0.709164i $$0.250928\pi$$
$$432$$ 0 0
$$433$$ −3.77335e6 −0.967179 −0.483590 0.875295i $$-0.660667\pi$$
−0.483590 + 0.875295i $$0.660667\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ −2.08802e6 −0.526041
$$437$$ 51758.6 0.0129652
$$438$$ 0 0
$$439$$ 2.35150e6 0.582350 0.291175 0.956670i $$-0.405954\pi$$
0.291175 + 0.956670i $$0.405954\pi$$
$$440$$ 4.00591e6 0.986439
$$441$$ 0 0
$$442$$ −3.63656e6 −0.885391
$$443$$ −4.80377e6 −1.16298 −0.581491 0.813553i $$-0.697530\pi$$
−0.581491 + 0.813553i $$0.697530\pi$$
$$444$$ 0 0
$$445$$ −2.46538e6 −0.590179
$$446$$ −1.09826e6 −0.261437
$$447$$ 0 0
$$448$$ 0 0
$$449$$ 2.76805e6 0.647975 0.323987 0.946061i $$-0.394976\pi$$
0.323987 + 0.946061i $$0.394976\pi$$
$$450$$ 0 0
$$451$$ 7.04142e6 1.63012
$$452$$ −2.44356e6 −0.562571
$$453$$ 0 0
$$454$$ 820179. 0.186754
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 241566. 0.0541061 0.0270530 0.999634i $$-0.491388\pi$$
0.0270530 + 0.999634i $$0.491388\pi$$
$$458$$ 3.46368e6 0.771567
$$459$$ 0 0
$$460$$ 49145.4 0.0108290
$$461$$ −990579. −0.217088 −0.108544 0.994092i $$-0.534619\pi$$
−0.108544 + 0.994092i $$0.534619\pi$$
$$462$$ 0 0
$$463$$ 6.20488e6 1.34518 0.672591 0.740014i $$-0.265181\pi$$
0.672591 + 0.740014i $$0.265181\pi$$
$$464$$ −1.08538e6 −0.234037
$$465$$ 0 0
$$466$$ 321702. 0.0686261
$$467$$ 6.88497e6 1.46086 0.730432 0.682985i $$-0.239319\pi$$
0.730432 + 0.682985i $$0.239319\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ 994206. 0.207602
$$471$$ 0 0
$$472$$ −1.55603e6 −0.321486
$$473$$ −53396.6 −0.0109739
$$474$$ 0 0
$$475$$ 1.17758e6 0.239473
$$476$$ 0 0
$$477$$ 0 0
$$478$$ −3.23121e6 −0.646838
$$479$$ 5.41288e6 1.07793 0.538963 0.842329i $$-0.318816\pi$$
0.538963 + 0.842329i $$0.318816\pi$$
$$480$$ 0 0
$$481$$ −2.79774e6 −0.551373
$$482$$ −2.28783e6 −0.448545
$$483$$ 0 0
$$484$$ −3.45268e6 −0.669950
$$485$$ 143069. 0.0276179
$$486$$ 0 0
$$487$$ −3.01043e6 −0.575182 −0.287591 0.957753i $$-0.592854\pi$$
−0.287591 + 0.957753i $$0.592854\pi$$
$$488$$ −6.08029e6 −1.15578
$$489$$ 0 0
$$490$$ 0 0
$$491$$ 7.24498e6 1.35623 0.678115 0.734956i $$-0.262797\pi$$
0.678115 + 0.734956i $$0.262797\pi$$
$$492$$ 0 0
$$493$$ −3.93062e6 −0.728356
$$494$$ 3.59163e6 0.662177
$$495$$ 0 0
$$496$$ 2.86389e6 0.522700
$$497$$ 0 0
$$498$$ 0 0
$$499$$ −5.12788e6 −0.921906 −0.460953 0.887425i $$-0.652492\pi$$
−0.460953 + 0.887425i $$0.652492\pi$$
$$500$$ 4.57829e6 0.818989
$$501$$ 0 0
$$502$$ −1.14509e6 −0.202806
$$503$$ −1.05978e7 −1.86766 −0.933830 0.357718i $$-0.883555\pi$$
−0.933830 + 0.357718i $$0.883555\pi$$
$$504$$ 0 0
$$505$$ −1.00426e6 −0.175233
$$506$$ 68943.5 0.0119706
$$507$$ 0 0
$$508$$ 74503.8 0.0128091
$$509$$ −8.78840e6 −1.50354 −0.751770 0.659425i $$-0.770800\pi$$
−0.751770 + 0.659425i $$0.770800\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ −3.59310e6 −0.605752
$$513$$ 0 0
$$514$$ −4.77753e6 −0.797619
$$515$$ 3.00512e6 0.499280
$$516$$ 0 0
$$517$$ −4.23721e6 −0.697194
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 7.94312e6 1.28820
$$521$$ −162133. −0.0261684 −0.0130842 0.999914i $$-0.504165\pi$$
−0.0130842 + 0.999914i $$0.504165\pi$$
$$522$$ 0 0
$$523$$ −7.14844e6 −1.14277 −0.571383 0.820684i $$-0.693593\pi$$
−0.571383 + 0.820684i $$0.693593\pi$$
$$524$$ 6.10144e6 0.970743
$$525$$ 0 0
$$526$$ 579044. 0.0912530
$$527$$ 1.03714e7 1.62671
$$528$$ 0 0
$$529$$ −6.43437e6 −0.999694
$$530$$ −1.54758e6 −0.239311
$$531$$ 0 0
$$532$$ 0 0
$$533$$ 1.39621e7 2.12879
$$534$$ 0 0
$$535$$ 5.01106e6 0.756912
$$536$$ −1.06630e7 −1.60313
$$537$$ 0 0
$$538$$ 4.88203e6 0.727185
$$539$$ 0 0
$$540$$ 0 0
$$541$$ −4.83604e6 −0.710390 −0.355195 0.934792i $$-0.615586\pi$$
−0.355195 + 0.934792i $$0.615586\pi$$
$$542$$ −1.03888e6 −0.151903
$$543$$ 0 0
$$544$$ −7.04793e6 −1.02109
$$545$$ 3.98874e6 0.575234
$$546$$ 0 0
$$547$$ 9.98777e6 1.42725 0.713626 0.700527i $$-0.247052\pi$$
0.713626 + 0.700527i $$0.247052\pi$$
$$548$$ −2.33899e6 −0.332719
$$549$$ 0 0
$$550$$ 1.56856e6 0.221103
$$551$$ 3.88205e6 0.544732
$$552$$ 0 0
$$553$$ 0 0
$$554$$ −3.45598e6 −0.478406
$$555$$ 0 0
$$556$$ −5.06324e6 −0.694611
$$557$$ −1.74619e6 −0.238481 −0.119241 0.992865i $$-0.538046\pi$$
−0.119241 + 0.992865i $$0.538046\pi$$
$$558$$ 0 0
$$559$$ −105877. −0.0143309
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 5.64904e6 0.754456
$$563$$ 755218. 0.100416 0.0502078 0.998739i $$-0.484012\pi$$
0.0502078 + 0.998739i $$0.484012\pi$$
$$564$$ 0 0
$$565$$ 4.66792e6 0.615181
$$566$$ 5.02719e6 0.659605
$$567$$ 0 0
$$568$$ 9.68836e6 1.26003
$$569$$ 4.39534e6 0.569131 0.284565 0.958657i $$-0.408151\pi$$
0.284565 + 0.958657i $$0.408151\pi$$
$$570$$ 0 0
$$571$$ 1.16104e7 1.49024 0.745121 0.666930i $$-0.232392\pi$$
0.745121 + 0.666930i $$0.232392\pi$$
$$572$$ −1.45344e7 −1.85740
$$573$$ 0 0
$$574$$ 0 0
$$575$$ 44821.1 0.00565344
$$576$$ 0 0
$$577$$ 1.06643e7 1.33350 0.666748 0.745283i $$-0.267686\pi$$
0.666748 + 0.745283i $$0.267686\pi$$
$$578$$ 72546.6 0.00903228
$$579$$ 0 0
$$580$$ 3.68606e6 0.454980
$$581$$ 0 0
$$582$$ 0 0
$$583$$ 6.59562e6 0.803682
$$584$$ 292097. 0.0354401
$$585$$ 0 0
$$586$$ 2.40195e6 0.288949
$$587$$ 1.39482e7 1.67079 0.835396 0.549648i $$-0.185238\pi$$
0.835396 + 0.549648i $$0.185238\pi$$
$$588$$ 0 0
$$589$$ −1.02433e7 −1.21661
$$590$$ 1.27620e6 0.150934
$$591$$ 0 0
$$592$$ −833711. −0.0977713
$$593$$ −1.17933e7 −1.37720 −0.688600 0.725142i $$-0.741774\pi$$
−0.688600 + 0.725142i $$0.741774\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 3.38033e6 0.389802
$$597$$ 0 0
$$598$$ 136705. 0.0156326
$$599$$ 4.38057e6 0.498843 0.249421 0.968395i $$-0.419760\pi$$
0.249421 + 0.968395i $$0.419760\pi$$
$$600$$ 0 0
$$601$$ 688570. 0.0777610 0.0388805 0.999244i $$-0.487621\pi$$
0.0388805 + 0.999244i $$0.487621\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ −2.88172e6 −0.321410
$$605$$ 6.59563e6 0.732601
$$606$$ 0 0
$$607$$ −9.37319e6 −1.03256 −0.516281 0.856420i $$-0.672684\pi$$
−0.516281 + 0.856420i $$0.672684\pi$$
$$608$$ 6.96085e6 0.763666
$$609$$ 0 0
$$610$$ 4.98684e6 0.542626
$$611$$ −8.40174e6 −0.910472
$$612$$ 0 0
$$613$$ −2.16685e6 −0.232904 −0.116452 0.993196i $$-0.537152\pi$$
−0.116452 + 0.993196i $$0.537152\pi$$
$$614$$ 5.51947e6 0.590849
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 5.07951e6 0.537166 0.268583 0.963256i $$-0.413445\pi$$
0.268583 + 0.963256i $$0.413445\pi$$
$$618$$ 0 0
$$619$$ −2.19034e6 −0.229766 −0.114883 0.993379i $$-0.536649\pi$$
−0.114883 + 0.993379i $$0.536649\pi$$
$$620$$ −9.72611e6 −1.01616
$$621$$ 0 0
$$622$$ 2.43111e6 0.251958
$$623$$ 0 0
$$624$$ 0 0
$$625$$ −5.59018e6 −0.572435
$$626$$ −3.09790e6 −0.315960
$$627$$ 0 0
$$628$$ −2.35016e6 −0.237793
$$629$$ −3.01923e6 −0.304278
$$630$$ 0 0
$$631$$ −7.18693e6 −0.718572 −0.359286 0.933228i $$-0.616980\pi$$
−0.359286 + 0.933228i $$0.616980\pi$$
$$632$$ −1346.38 −0.000134083 0
$$633$$ 0 0
$$634$$ 4.21109e6 0.416075
$$635$$ −142324. −0.0140070
$$636$$ 0 0
$$637$$ 0 0
$$638$$ 5.17097e6 0.502945
$$639$$ 0 0
$$640$$ 7.96017e6 0.768197
$$641$$ 1.76500e7 1.69668 0.848340 0.529452i $$-0.177602\pi$$
0.848340 + 0.529452i $$0.177602\pi$$
$$642$$ 0 0
$$643$$ 898309. 0.0856837 0.0428419 0.999082i $$-0.486359\pi$$
0.0428419 + 0.999082i $$0.486359\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 3.87597e6 0.365425
$$647$$ 1.38642e6 0.130207 0.0651035 0.997879i $$-0.479262\pi$$
0.0651035 + 0.997879i $$0.479262\pi$$
$$648$$ 0 0
$$649$$ −5.43904e6 −0.506886
$$650$$ 3.11022e6 0.288741
$$651$$ 0 0
$$652$$ −4.39804e6 −0.405173
$$653$$ 1.75425e7 1.60994 0.804968 0.593318i $$-0.202182\pi$$
0.804968 + 0.593318i $$0.202182\pi$$
$$654$$ 0 0
$$655$$ −1.16556e7 −1.06152
$$656$$ 4.16062e6 0.377484
$$657$$ 0 0
$$658$$ 0 0
$$659$$ −9.87522e6 −0.885795 −0.442898 0.896572i $$-0.646050\pi$$
−0.442898 + 0.896572i $$0.646050\pi$$
$$660$$ 0 0
$$661$$ −8.06792e6 −0.718221 −0.359110 0.933295i $$-0.616920\pi$$
−0.359110 + 0.933295i $$0.616920\pi$$
$$662$$ 7.71373e6 0.684100
$$663$$ 0 0
$$664$$ 1.50025e7 1.32052
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 147759. 0.0128599
$$668$$ −1.08742e7 −0.942880
$$669$$ 0 0
$$670$$ 8.74544e6 0.752652
$$671$$ −2.12534e7 −1.82231
$$672$$ 0 0
$$673$$ −1.12772e7 −0.959762 −0.479881 0.877334i $$-0.659320\pi$$
−0.479881 + 0.877334i $$0.659320\pi$$
$$674$$ 6.51274e6 0.552223
$$675$$ 0 0
$$676$$ −1.98804e7 −1.67324
$$677$$ −5.20372e6 −0.436357 −0.218179 0.975909i $$-0.570012\pi$$
−0.218179 + 0.975909i $$0.570012\pi$$
$$678$$ 0 0
$$679$$ 0 0
$$680$$ 8.57196e6 0.710899
$$681$$ 0 0
$$682$$ −1.36442e7 −1.12328
$$683$$ −6.05915e6 −0.497004 −0.248502 0.968631i $$-0.579938\pi$$
−0.248502 + 0.968631i $$0.579938\pi$$
$$684$$ 0 0
$$685$$ 4.46816e6 0.363833
$$686$$ 0 0
$$687$$ 0 0
$$688$$ −31550.9 −0.00254121
$$689$$ 1.30781e7 1.04954
$$690$$ 0 0
$$691$$ 7.36498e6 0.586781 0.293391 0.955993i $$-0.405216\pi$$
0.293391 + 0.955993i $$0.405216\pi$$
$$692$$ 8.94754e6 0.710295
$$693$$ 0 0
$$694$$ −8.61204e6 −0.678746
$$695$$ 9.67228e6 0.759569
$$696$$ 0 0
$$697$$ 1.50674e7 1.17478
$$698$$ 591818. 0.0459779
$$699$$ 0 0
$$700$$ 0 0
$$701$$ −7.80919e6 −0.600221 −0.300110 0.953904i $$-0.597023\pi$$
−0.300110 + 0.953904i $$0.597023\pi$$
$$702$$ 0 0
$$703$$ 2.98193e6 0.227567
$$704$$ 3.51520e6 0.267312
$$705$$ 0 0
$$706$$ −1.06069e7 −0.800898
$$707$$ 0 0
$$708$$ 0 0
$$709$$ 1.75650e7 1.31230 0.656150 0.754631i $$-0.272184\pi$$
0.656150 + 0.754631i $$0.272184\pi$$
$$710$$ −7.94606e6 −0.591569
$$711$$ 0 0
$$712$$ −8.46198e6 −0.625564
$$713$$ −389879. −0.0287214
$$714$$ 0 0
$$715$$ 2.77649e7 2.03110
$$716$$ 2.04649e6 0.149186
$$717$$ 0 0
$$718$$ 2.83539e6 0.205259
$$719$$ 8.09220e6 0.583773 0.291887 0.956453i $$-0.405717\pi$$
0.291887 + 0.956453i $$0.405717\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ 3.14232e6 0.224341
$$723$$ 0 0
$$724$$ 9.13520e6 0.647697
$$725$$ 3.36172e6 0.237529
$$726$$ 0 0
$$727$$ 1.51986e7 1.06652 0.533258 0.845952i $$-0.320967\pi$$
0.533258 + 0.845952i $$0.320967\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ −239568. −0.0166388
$$731$$ −114259. −0.00790858
$$732$$ 0 0
$$733$$ 5.83402e6 0.401059 0.200530 0.979688i $$-0.435734\pi$$
0.200530 + 0.979688i $$0.435734\pi$$
$$734$$ 4.29720e6 0.294405
$$735$$ 0 0
$$736$$ 264944. 0.0180285
$$737$$ −3.72722e7 −2.52765
$$738$$ 0 0
$$739$$ 6.47719e6 0.436290 0.218145 0.975916i $$-0.429999\pi$$
0.218145 + 0.975916i $$0.429999\pi$$
$$740$$ 2.83138e6 0.190072
$$741$$ 0 0
$$742$$ 0 0
$$743$$ 1.50899e7 1.00280 0.501401 0.865215i $$-0.332818\pi$$
0.501401 + 0.865215i $$0.332818\pi$$
$$744$$ 0 0
$$745$$ −6.45744e6 −0.426255
$$746$$ −1.36896e7 −0.900626
$$747$$ 0 0
$$748$$ −1.56850e7 −1.02502
$$749$$ 0 0
$$750$$ 0 0
$$751$$ −2.13997e6 −0.138455 −0.0692273 0.997601i $$-0.522053\pi$$
−0.0692273 + 0.997601i $$0.522053\pi$$
$$752$$ −2.50367e6 −0.161448
$$753$$ 0 0
$$754$$ 1.02533e7 0.656801
$$755$$ 5.50494e6 0.351467
$$756$$ 0 0
$$757$$ 2.10943e7 1.33791 0.668954 0.743304i $$-0.266742\pi$$
0.668954 + 0.743304i $$0.266742\pi$$
$$758$$ −1.77597e6 −0.112270
$$759$$ 0 0
$$760$$ −8.46605e6 −0.531675
$$761$$ −9.79958e6 −0.613403 −0.306701 0.951806i $$-0.599225\pi$$
−0.306701 + 0.951806i $$0.599225\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ −2.22022e7 −1.37614
$$765$$ 0 0
$$766$$ 1.59233e6 0.0980530
$$767$$ −1.07848e7 −0.661947
$$768$$ 0 0
$$769$$ −3.23493e7 −1.97265 −0.986323 0.164825i $$-0.947294\pi$$
−0.986323 + 0.164825i $$0.947294\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ −1.21606e7 −0.734366
$$773$$ −9.19713e6 −0.553609 −0.276805 0.960926i $$-0.589276\pi$$
−0.276805 + 0.960926i $$0.589276\pi$$
$$774$$ 0 0
$$775$$ −8.87030e6 −0.530499
$$776$$ 491058. 0.0292738
$$777$$ 0 0
$$778$$ 1.66712e6 0.0987456
$$779$$ −1.48812e7 −0.878609
$$780$$ 0 0
$$781$$ 3.38653e7 1.98668
$$782$$ 147527. 0.00862690
$$783$$ 0 0
$$784$$ 0 0
$$785$$ 4.48950e6 0.260030
$$786$$ 0 0
$$787$$ 1.46950e7 0.845731 0.422866 0.906192i $$-0.361024\pi$$
0.422866 + 0.906192i $$0.361024\pi$$
$$788$$ 6.05508e6 0.347379
$$789$$ 0 0
$$790$$ 1104.25 6.29506e−5 0
$$791$$ 0 0
$$792$$ 0 0
$$793$$ −4.21423e7 −2.37977
$$794$$ −3.78097e6 −0.212839
$$795$$ 0 0
$$796$$ 5.00847e6 0.280171
$$797$$ −2.33344e7 −1.30122 −0.650610 0.759412i $$-0.725487\pi$$
−0.650610 + 0.759412i $$0.725487\pi$$
$$798$$ 0 0
$$799$$ −9.06688e6 −0.502448
$$800$$ 6.02785e6 0.332995
$$801$$ 0 0
$$802$$ −1.03796e7 −0.569831
$$803$$ 1.02101e6 0.0558783
$$804$$ 0 0
$$805$$ 0 0
$$806$$ −2.70545e7 −1.46690
$$807$$ 0 0
$$808$$ −3.44693e6 −0.185740
$$809$$ −1.69301e6 −0.0909468 −0.0454734 0.998966i $$-0.514480\pi$$
−0.0454734 + 0.998966i $$0.514480\pi$$
$$810$$ 0 0
$$811$$ −2.12400e7 −1.13397 −0.566987 0.823727i $$-0.691891\pi$$
−0.566987 + 0.823727i $$0.691891\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ 3.97199e6 0.210110
$$815$$ 8.40155e6 0.443063
$$816$$ 0 0
$$817$$ 112848. 0.00591477
$$818$$ −4.09960e6 −0.214219
$$819$$ 0 0
$$820$$ −1.41299e7 −0.733847
$$821$$ −8.73550e6 −0.452304 −0.226152 0.974092i $$-0.572615\pi$$
−0.226152 + 0.974092i $$0.572615\pi$$
$$822$$ 0 0
$$823$$ −3.27964e7 −1.68782 −0.843910 0.536485i $$-0.819752\pi$$
−0.843910 + 0.536485i $$0.819752\pi$$
$$824$$ 1.03146e7 0.529215
$$825$$ 0 0
$$826$$ 0 0
$$827$$ −1.31248e7 −0.667311 −0.333656 0.942695i $$-0.608282\pi$$
−0.333656 + 0.942695i $$0.608282\pi$$
$$828$$ 0 0
$$829$$ 2.05402e7 1.03805 0.519026 0.854759i $$-0.326295\pi$$
0.519026 + 0.854759i $$0.326295\pi$$
$$830$$ −1.23046e7 −0.619970
$$831$$ 0 0
$$832$$ 6.97012e6 0.349085
$$833$$ 0 0
$$834$$ 0 0
$$835$$ 2.07729e7 1.03106
$$836$$ 1.54912e7 0.766601
$$837$$ 0 0
$$838$$ −8.22542e6 −0.404621
$$839$$ −2.83736e7 −1.39159 −0.695793 0.718243i $$-0.744947\pi$$
−0.695793 + 0.718243i $$0.744947\pi$$
$$840$$ 0 0
$$841$$ −9.42879e6 −0.459691
$$842$$ −5.68643e6 −0.276414
$$843$$ 0 0
$$844$$ 1.54221e7 0.745226
$$845$$ 3.79774e7 1.82972
$$846$$ 0 0
$$847$$ 0 0
$$848$$ 3.89720e6 0.186107
$$849$$ 0 0
$$850$$ 3.35645e6 0.159343
$$851$$ 113498. 0.00537236
$$852$$ 0 0
$$853$$ 1.02093e7 0.480424 0.240212 0.970720i $$-0.422783\pi$$
0.240212 + 0.970720i $$0.422783\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 1.71996e7 0.802295
$$857$$ 8.33206e6 0.387525 0.193763 0.981048i $$-0.437931\pi$$
0.193763 + 0.981048i $$0.437931\pi$$
$$858$$ 0 0
$$859$$ 3.12766e7 1.44623 0.723113 0.690729i $$-0.242710\pi$$
0.723113 + 0.690729i $$0.242710\pi$$
$$860$$ 107150. 0.00494023
$$861$$ 0 0
$$862$$ −1.53084e7 −0.701714
$$863$$ 3.73573e7 1.70745 0.853726 0.520722i $$-0.174337\pi$$
0.853726 + 0.520722i $$0.174337\pi$$
$$864$$ 0 0
$$865$$ −1.70924e7 −0.776719
$$866$$ 1.06222e7 0.481306
$$867$$ 0 0
$$868$$ 0 0
$$869$$ −4706.21 −0.000211408 0
$$870$$ 0 0
$$871$$ −7.39051e7 −3.30088
$$872$$ 1.36907e7 0.609724
$$873$$ 0 0
$$874$$ −145704. −0.00645199
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 38996.7 0.00171210 0.000856049 1.00000i $$-0.499728\pi$$
0.000856049 1.00000i $$0.499728\pi$$
$$878$$ −6.61965e6 −0.289800
$$879$$ 0 0
$$880$$ 8.27378e6 0.360162
$$881$$ −3.15554e7 −1.36973 −0.684864 0.728671i $$-0.740138\pi$$
−0.684864 + 0.728671i $$0.740138\pi$$
$$882$$ 0 0
$$883$$ −3.42253e7 −1.47722 −0.738611 0.674132i $$-0.764518\pi$$
−0.738611 + 0.674132i $$0.764518\pi$$
$$884$$ −3.11010e7 −1.33858
$$885$$ 0 0
$$886$$ 1.35230e7 0.578745
$$887$$ 2.69886e7 1.15178 0.575892 0.817526i $$-0.304655\pi$$
0.575892 + 0.817526i $$0.304655\pi$$
$$888$$ 0 0
$$889$$ 0 0
$$890$$ 6.94022e6 0.293696
$$891$$ 0 0
$$892$$ −9.39263e6 −0.395253
$$893$$ 8.95486e6 0.375777
$$894$$ 0 0
$$895$$ −3.90940e6 −0.163137
$$896$$ 0 0
$$897$$ 0 0
$$898$$ −7.79226e6 −0.322458
$$899$$ −2.92421e7 −1.20673
$$900$$ 0 0
$$901$$ 1.41135e7 0.579191
$$902$$ −1.98221e7 −0.811211
$$903$$ 0 0
$$904$$ 1.60218e7 0.652065
$$905$$ −1.74509e7 −0.708267
$$906$$ 0 0
$$907$$ 1.92103e7 0.775381 0.387690 0.921790i $$-0.373273\pi$$
0.387690 + 0.921790i $$0.373273\pi$$
$$908$$ 7.01442e6 0.282343
$$909$$ 0 0
$$910$$ 0 0
$$911$$ −2.86013e7 −1.14180 −0.570899 0.821020i $$-0.693405\pi$$
−0.570899 + 0.821020i $$0.693405\pi$$
$$912$$ 0 0
$$913$$ 5.24408e7 2.08206
$$914$$ −680027. −0.0269253
$$915$$ 0 0
$$916$$ 2.96224e7 1.16649
$$917$$ 0 0
$$918$$ 0 0
$$919$$ −4.21754e7 −1.64729 −0.823645 0.567106i $$-0.808063\pi$$
−0.823645 + 0.567106i $$0.808063\pi$$
$$920$$ −322235. −0.0125517
$$921$$ 0 0
$$922$$ 2.78855e6 0.108032
$$923$$ 6.71498e7 2.59442
$$924$$ 0 0
$$925$$ 2.58224e6 0.0992299
$$926$$ −1.74672e7 −0.669416
$$927$$ 0 0
$$928$$ 1.98716e7 0.757466
$$929$$ 3.01886e7 1.14763 0.573817 0.818983i $$-0.305462\pi$$
0.573817 + 0.818983i $$0.305462\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ 2.75130e6 0.103752
$$933$$ 0 0
$$934$$ −1.93817e7 −0.726984
$$935$$ 2.99630e7 1.12087
$$936$$ 0 0
$$937$$ −3.64068e6 −0.135467 −0.0677335 0.997703i $$-0.521577\pi$$
−0.0677335 + 0.997703i $$0.521577\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ 8.50275e6 0.313863
$$941$$ −1.88601e7 −0.694336 −0.347168 0.937803i $$-0.612857\pi$$
−0.347168 + 0.937803i $$0.612857\pi$$
$$942$$ 0 0
$$943$$ −566410. −0.0207420
$$944$$ −3.21380e6 −0.117379
$$945$$ 0 0
$$946$$ 150315. 0.00546104
$$947$$ 1.82172e7 0.660094 0.330047 0.943965i $$-0.392935\pi$$
0.330047 + 0.943965i $$0.392935\pi$$
$$948$$ 0 0
$$949$$ 2.02452e6 0.0729720
$$950$$ −3.31498e6 −0.119171
$$951$$ 0 0
$$952$$ 0 0
$$953$$ −2.76898e7 −0.987616 −0.493808 0.869571i $$-0.664396\pi$$
−0.493808 + 0.869571i $$0.664396\pi$$
$$954$$ 0 0
$$955$$ 4.24127e7 1.50483
$$956$$ −2.76343e7 −0.977921
$$957$$ 0 0
$$958$$ −1.52376e7 −0.536419
$$959$$ 0 0
$$960$$ 0 0
$$961$$ 4.85298e7 1.69512
$$962$$ 7.87585e6 0.274385
$$963$$ 0 0
$$964$$ −1.95662e7 −0.678133
$$965$$ 2.32304e7 0.803042
$$966$$ 0 0
$$967$$ −2.44768e7 −0.841761 −0.420881 0.907116i $$-0.638279\pi$$
−0.420881 + 0.907116i $$0.638279\pi$$
$$968$$ 2.26383e7 0.776526
$$969$$ 0 0
$$970$$ −402749. −0.0137437
$$971$$ −9.50151e6 −0.323403 −0.161702 0.986840i $$-0.551698\pi$$
−0.161702 + 0.986840i $$0.551698\pi$$
$$972$$ 0 0
$$973$$ 0 0
$$974$$ 8.47457e6 0.286233
$$975$$ 0 0
$$976$$ −1.25582e7 −0.421990
$$977$$ −4.69012e7 −1.57198 −0.785991 0.618238i $$-0.787847\pi$$
−0.785991 + 0.618238i $$0.787847\pi$$
$$978$$ 0 0
$$979$$ −2.95785e7 −0.986325
$$980$$ 0 0
$$981$$ 0 0
$$982$$ −2.03951e7 −0.674914
$$983$$ −2.35382e7 −0.776945 −0.388473 0.921460i $$-0.626997\pi$$
−0.388473 + 0.921460i $$0.626997\pi$$
$$984$$ 0 0
$$985$$ −1.15670e7 −0.379865
$$986$$ 1.10650e7 0.362458
$$987$$ 0 0
$$988$$ 3.07167e7 1.00111
$$989$$ 4295.21 0.000139635 0
$$990$$ 0 0
$$991$$ −2.64104e7 −0.854261 −0.427130 0.904190i $$-0.640475\pi$$
−0.427130 + 0.904190i $$0.640475\pi$$
$$992$$ −5.24336e7 −1.69173
$$993$$ 0 0
$$994$$ 0 0
$$995$$ −9.56766e6 −0.306371
$$996$$ 0 0
$$997$$ −1.95164e7 −0.621815 −0.310907 0.950440i $$-0.600633\pi$$
−0.310907 + 0.950440i $$0.600633\pi$$
$$998$$ 1.44354e7 0.458777
$$999$$ 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 441.6.a.z.1.2 4
3.2 odd 2 49.6.a.g.1.4 yes 4
7.6 odd 2 inner 441.6.a.z.1.1 4
12.11 even 2 784.6.a.bf.1.2 4
21.2 odd 6 49.6.c.h.18.1 8
21.5 even 6 49.6.c.h.18.2 8
21.11 odd 6 49.6.c.h.30.1 8
21.17 even 6 49.6.c.h.30.2 8
21.20 even 2 49.6.a.g.1.3 4
84.83 odd 2 784.6.a.bf.1.3 4

By twisted newform
Twist Min Dim Char Parity Ord Type
49.6.a.g.1.3 4 21.20 even 2
49.6.a.g.1.4 yes 4 3.2 odd 2
49.6.c.h.18.1 8 21.2 odd 6
49.6.c.h.18.2 8 21.5 even 6
49.6.c.h.30.1 8 21.11 odd 6
49.6.c.h.30.2 8 21.17 even 6
441.6.a.z.1.1 4 7.6 odd 2 inner
441.6.a.z.1.2 4 1.1 even 1 trivial
784.6.a.bf.1.2 4 12.11 even 2
784.6.a.bf.1.3 4 84.83 odd 2