# Properties

 Label 75.22.a.a.1.1 Level $75$ Weight $22$ Character 75.1 Self dual yes Analytic conductor $209.608$ 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] = [75,22,Mod(1,75)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(75, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 22, names="a")

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

chi := DirichletCharacter("75.1");

S:= CuspForms(chi, 22);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$75 = 3 \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$22$$ Character orbit: $$[\chi]$$ $$=$$ 75.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: $$209.608008215$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 3) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 75.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-1728.00 q^{2} +59049.0 q^{3} +888832. q^{4} -1.02037e8 q^{6} -5.38430e8 q^{7} +2.08798e9 q^{8} +3.48678e9 q^{9} +O(q^{10})$$ $$q-1728.00 q^{2} +59049.0 q^{3} +888832. q^{4} -1.02037e8 q^{6} -5.38430e8 q^{7} +2.08798e9 q^{8} +3.48678e9 q^{9} -6.41130e10 q^{11} +5.24846e10 q^{12} +1.30980e11 q^{13} +9.30407e11 q^{14} -5.47204e12 q^{16} -8.24203e12 q^{17} -6.02516e12 q^{18} +1.34921e13 q^{19} -3.17937e13 q^{21} +1.10787e14 q^{22} +2.33185e14 q^{23} +1.23293e14 q^{24} -2.26334e14 q^{26} +2.05891e14 q^{27} -4.78574e14 q^{28} -2.02456e15 q^{29} -6.86919e15 q^{31} +5.07688e15 q^{32} -3.78581e15 q^{33} +1.42422e16 q^{34} +3.09917e15 q^{36} -3.44400e15 q^{37} -2.33144e16 q^{38} +7.73424e15 q^{39} -2.18424e16 q^{41} +5.49396e16 q^{42} +7.17928e16 q^{43} -5.69857e16 q^{44} -4.02943e17 q^{46} -2.83545e17 q^{47} -3.23118e17 q^{48} -2.68639e17 q^{49} -4.86684e17 q^{51} +1.16419e17 q^{52} +2.17229e18 q^{53} -3.55780e17 q^{54} -1.12423e18 q^{56} +7.96695e17 q^{57} +3.49844e18 q^{58} +1.53483e18 q^{59} +4.31159e18 q^{61} +1.18700e19 q^{62} -1.87739e18 q^{63} +2.70285e18 q^{64} +6.54188e18 q^{66} -9.24391e18 q^{67} -7.32578e18 q^{68} +1.37693e19 q^{69} -2.03874e19 q^{71} +7.28033e18 q^{72} -1.66178e19 q^{73} +5.95123e18 q^{74} +1.19922e19 q^{76} +3.45204e19 q^{77} -1.33648e19 q^{78} +6.79403e19 q^{79} +1.21577e19 q^{81} +3.77437e19 q^{82} -3.95037e19 q^{83} -2.82593e19 q^{84} -1.24058e20 q^{86} -1.19548e20 q^{87} -1.33867e20 q^{88} +4.16117e19 q^{89} -7.05236e19 q^{91} +2.07262e20 q^{92} -4.05619e20 q^{93} +4.89965e20 q^{94} +2.99785e20 q^{96} -5.71815e19 q^{97} +4.64209e20 q^{98} -2.23548e20 q^{99} +O(q^{100})$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −1728.00 −1.19324 −0.596621 0.802523i $$-0.703491\pi$$
−0.596621 + 0.802523i $$0.703491\pi$$
$$3$$ 59049.0 0.577350
$$4$$ 888832. 0.423828
$$5$$ 0 0
$$6$$ −1.02037e8 −0.688919
$$7$$ −5.38430e8 −0.720443 −0.360222 0.932867i $$-0.617299\pi$$
−0.360222 + 0.932867i $$0.617299\pi$$
$$8$$ 2.08798e9 0.687513
$$9$$ 3.48678e9 0.333333
$$10$$ 0 0
$$11$$ −6.41130e10 −0.745286 −0.372643 0.927975i $$-0.621548\pi$$
−0.372643 + 0.927975i $$0.621548\pi$$
$$12$$ 5.24846e10 0.244697
$$13$$ 1.30980e11 0.263512 0.131756 0.991282i $$-0.457938\pi$$
0.131756 + 0.991282i $$0.457938\pi$$
$$14$$ 9.30407e11 0.859663
$$15$$ 0 0
$$16$$ −5.47204e12 −1.24420
$$17$$ −8.24203e12 −0.991563 −0.495782 0.868447i $$-0.665118\pi$$
−0.495782 + 0.868447i $$0.665118\pi$$
$$18$$ −6.02516e12 −0.397748
$$19$$ 1.34921e13 0.504855 0.252428 0.967616i $$-0.418771\pi$$
0.252428 + 0.967616i $$0.418771\pi$$
$$20$$ 0 0
$$21$$ −3.17937e13 −0.415948
$$22$$ 1.10787e14 0.889308
$$23$$ 2.33185e14 1.17370 0.586851 0.809695i $$-0.300367\pi$$
0.586851 + 0.809695i $$0.300367\pi$$
$$24$$ 1.23293e14 0.396936
$$25$$ 0 0
$$26$$ −2.26334e14 −0.314434
$$27$$ 2.05891e14 0.192450
$$28$$ −4.78574e14 −0.305344
$$29$$ −2.02456e15 −0.893618 −0.446809 0.894629i $$-0.647440\pi$$
−0.446809 + 0.894629i $$0.647440\pi$$
$$30$$ 0 0
$$31$$ −6.86919e15 −1.50525 −0.752624 0.658451i $$-0.771212\pi$$
−0.752624 + 0.658451i $$0.771212\pi$$
$$32$$ 5.07688e15 0.797117
$$33$$ −3.78581e15 −0.430291
$$34$$ 1.42422e16 1.18318
$$35$$ 0 0
$$36$$ 3.09917e15 0.141276
$$37$$ −3.44400e15 −0.117746 −0.0588728 0.998265i $$-0.518751\pi$$
−0.0588728 + 0.998265i $$0.518751\pi$$
$$38$$ −2.33144e16 −0.602415
$$39$$ 7.73424e15 0.152139
$$40$$ 0 0
$$41$$ −2.18424e16 −0.254138 −0.127069 0.991894i $$-0.540557\pi$$
−0.127069 + 0.991894i $$0.540557\pi$$
$$42$$ 5.49396e16 0.496327
$$43$$ 7.17928e16 0.506597 0.253298 0.967388i $$-0.418485\pi$$
0.253298 + 0.967388i $$0.418485\pi$$
$$44$$ −5.69857e16 −0.315873
$$45$$ 0 0
$$46$$ −4.02943e17 −1.40051
$$47$$ −2.83545e17 −0.786310 −0.393155 0.919472i $$-0.628617\pi$$
−0.393155 + 0.919472i $$0.628617\pi$$
$$48$$ −3.23118e17 −0.718338
$$49$$ −2.68639e17 −0.480962
$$50$$ 0 0
$$51$$ −4.86684e17 −0.572479
$$52$$ 1.16419e17 0.111684
$$53$$ 2.17229e18 1.70616 0.853081 0.521779i $$-0.174731\pi$$
0.853081 + 0.521779i $$0.174731\pi$$
$$54$$ −3.55780e17 −0.229640
$$55$$ 0 0
$$56$$ −1.12423e18 −0.495314
$$57$$ 7.96695e17 0.291478
$$58$$ 3.49844e18 1.06630
$$59$$ 1.53483e18 0.390944 0.195472 0.980709i $$-0.437376\pi$$
0.195472 + 0.980709i $$0.437376\pi$$
$$60$$ 0 0
$$61$$ 4.31159e18 0.773881 0.386940 0.922105i $$-0.373532\pi$$
0.386940 + 0.922105i $$0.373532\pi$$
$$62$$ 1.18700e19 1.79613
$$63$$ −1.87739e18 −0.240148
$$64$$ 2.70285e18 0.293044
$$65$$ 0 0
$$66$$ 6.54188e18 0.513442
$$67$$ −9.24391e18 −0.619541 −0.309771 0.950811i $$-0.600252\pi$$
−0.309771 + 0.950811i $$0.600252\pi$$
$$68$$ −7.32578e18 −0.420252
$$69$$ 1.37693e19 0.677637
$$70$$ 0 0
$$71$$ −2.03874e19 −0.743273 −0.371636 0.928378i $$-0.621203\pi$$
−0.371636 + 0.928378i $$0.621203\pi$$
$$72$$ 7.28033e18 0.229171
$$73$$ −1.66178e19 −0.452566 −0.226283 0.974062i $$-0.572657\pi$$
−0.226283 + 0.974062i $$0.572657\pi$$
$$74$$ 5.95123e18 0.140499
$$75$$ 0 0
$$76$$ 1.19922e19 0.213972
$$77$$ 3.45204e19 0.536936
$$78$$ −1.33648e19 −0.181538
$$79$$ 6.79403e19 0.807315 0.403658 0.914910i $$-0.367739\pi$$
0.403658 + 0.914910i $$0.367739\pi$$
$$80$$ 0 0
$$81$$ 1.21577e19 0.111111
$$82$$ 3.77437e19 0.303249
$$83$$ −3.95037e19 −0.279459 −0.139730 0.990190i $$-0.544623\pi$$
−0.139730 + 0.990190i $$0.544623\pi$$
$$84$$ −2.82593e19 −0.176290
$$85$$ 0 0
$$86$$ −1.24058e20 −0.604493
$$87$$ −1.19548e20 −0.515931
$$88$$ −1.33867e20 −0.512394
$$89$$ 4.16117e19 0.141456 0.0707278 0.997496i $$-0.477468\pi$$
0.0707278 + 0.997496i $$0.477468\pi$$
$$90$$ 0 0
$$91$$ −7.05236e19 −0.189845
$$92$$ 2.07262e20 0.497448
$$93$$ −4.05619e20 −0.869055
$$94$$ 4.89965e20 0.938259
$$95$$ 0 0
$$96$$ 2.99785e20 0.460216
$$97$$ −5.71815e19 −0.0787322 −0.0393661 0.999225i $$-0.512534\pi$$
−0.0393661 + 0.999225i $$0.512534\pi$$
$$98$$ 4.64209e20 0.573904
$$99$$ −2.23548e20 −0.248429
$$100$$ 0 0
$$101$$ 4.32417e20 0.389518 0.194759 0.980851i $$-0.437607\pi$$
0.194759 + 0.980851i $$0.437607\pi$$
$$102$$ 8.40989e20 0.683107
$$103$$ −1.84123e21 −1.34995 −0.674974 0.737841i $$-0.735845\pi$$
−0.674974 + 0.737841i $$0.735845\pi$$
$$104$$ 2.73483e20 0.181168
$$105$$ 0 0
$$106$$ −3.75371e21 −2.03587
$$107$$ 2.43805e21 1.19815 0.599077 0.800691i $$-0.295534\pi$$
0.599077 + 0.800691i $$0.295534\pi$$
$$108$$ 1.83003e20 0.0815658
$$109$$ −4.13676e21 −1.67372 −0.836859 0.547418i $$-0.815611\pi$$
−0.836859 + 0.547418i $$0.815611\pi$$
$$110$$ 0 0
$$111$$ −2.03365e20 −0.0679805
$$112$$ 2.94631e21 0.896374
$$113$$ −3.47910e21 −0.964146 −0.482073 0.876131i $$-0.660116\pi$$
−0.482073 + 0.876131i $$0.660116\pi$$
$$114$$ −1.37669e21 −0.347804
$$115$$ 0 0
$$116$$ −1.79950e21 −0.378741
$$117$$ 4.56699e20 0.0878373
$$118$$ −2.65219e21 −0.466491
$$119$$ 4.43775e21 0.714365
$$120$$ 0 0
$$121$$ −3.28977e21 −0.444548
$$122$$ −7.45043e21 −0.923428
$$123$$ −1.28977e21 −0.146727
$$124$$ −6.10556e21 −0.637966
$$125$$ 0 0
$$126$$ 3.24413e21 0.286554
$$127$$ −1.37141e21 −0.111488 −0.0557438 0.998445i $$-0.517753\pi$$
−0.0557438 + 0.998445i $$0.517753\pi$$
$$128$$ −1.53175e22 −1.14679
$$129$$ 4.23929e21 0.292484
$$130$$ 0 0
$$131$$ −2.45276e22 −1.43981 −0.719907 0.694071i $$-0.755815\pi$$
−0.719907 + 0.694071i $$0.755815\pi$$
$$132$$ −3.36495e21 −0.182370
$$133$$ −7.26455e21 −0.363719
$$134$$ 1.59735e22 0.739263
$$135$$ 0 0
$$136$$ −1.72092e22 −0.681713
$$137$$ −1.02835e22 −0.377204 −0.188602 0.982054i $$-0.560396\pi$$
−0.188602 + 0.982054i $$0.560396\pi$$
$$138$$ −2.37934e22 −0.808586
$$139$$ 8.70692e21 0.274289 0.137145 0.990551i $$-0.456207\pi$$
0.137145 + 0.990551i $$0.456207\pi$$
$$140$$ 0 0
$$141$$ −1.67430e22 −0.453977
$$142$$ 3.52294e22 0.886905
$$143$$ −8.39753e21 −0.196392
$$144$$ −1.90798e22 −0.414733
$$145$$ 0 0
$$146$$ 2.87155e22 0.540022
$$147$$ −1.58629e22 −0.277683
$$148$$ −3.06114e21 −0.0499039
$$149$$ −9.03997e22 −1.37313 −0.686564 0.727069i $$-0.740882\pi$$
−0.686564 + 0.727069i $$0.740882\pi$$
$$150$$ 0 0
$$151$$ −4.75206e22 −0.627514 −0.313757 0.949503i $$-0.601588\pi$$
−0.313757 + 0.949503i $$0.601588\pi$$
$$152$$ 2.81712e22 0.347094
$$153$$ −2.87382e22 −0.330521
$$154$$ −5.96512e22 −0.640695
$$155$$ 0 0
$$156$$ 6.87444e21 0.0644806
$$157$$ 1.50901e23 1.32356 0.661781 0.749697i $$-0.269801\pi$$
0.661781 + 0.749697i $$0.269801\pi$$
$$158$$ −1.17401e23 −0.963323
$$159$$ 1.28271e23 0.985053
$$160$$ 0 0
$$161$$ −1.25554e23 −0.845586
$$162$$ −2.10084e22 −0.132583
$$163$$ 4.83503e22 0.286042 0.143021 0.989720i $$-0.454318\pi$$
0.143021 + 0.989720i $$0.454318\pi$$
$$164$$ −1.94142e22 −0.107711
$$165$$ 0 0
$$166$$ 6.82624e22 0.333462
$$167$$ −4.78731e20 −0.00219568 −0.00109784 0.999999i $$-0.500349\pi$$
−0.00109784 + 0.999999i $$0.500349\pi$$
$$168$$ −6.63846e22 −0.285970
$$169$$ −2.29909e23 −0.930562
$$170$$ 0 0
$$171$$ 4.70440e22 0.168285
$$172$$ 6.38118e22 0.214710
$$173$$ 1.61804e23 0.512277 0.256139 0.966640i $$-0.417550\pi$$
0.256139 + 0.966640i $$0.417550\pi$$
$$174$$ 2.06580e23 0.615631
$$175$$ 0 0
$$176$$ 3.50829e23 0.927284
$$177$$ 9.06303e22 0.225712
$$178$$ −7.19050e22 −0.168791
$$179$$ −8.76377e22 −0.193970 −0.0969849 0.995286i $$-0.530920\pi$$
−0.0969849 + 0.995286i $$0.530920\pi$$
$$180$$ 0 0
$$181$$ 9.36624e22 0.184476 0.0922381 0.995737i $$-0.470598\pi$$
0.0922381 + 0.995737i $$0.470598\pi$$
$$182$$ 1.21865e23 0.226532
$$183$$ 2.54595e23 0.446800
$$184$$ 4.86885e23 0.806936
$$185$$ 0 0
$$186$$ 7.00910e23 1.03699
$$187$$ 5.28422e23 0.738999
$$188$$ −2.52024e23 −0.333260
$$189$$ −1.10858e23 −0.138649
$$190$$ 0 0
$$191$$ 1.20858e24 1.35340 0.676699 0.736260i $$-0.263410\pi$$
0.676699 + 0.736260i $$0.263410\pi$$
$$192$$ 1.59601e23 0.169189
$$193$$ 1.78822e24 1.79502 0.897509 0.440997i $$-0.145375\pi$$
0.897509 + 0.440997i $$0.145375\pi$$
$$194$$ 9.88096e22 0.0939466
$$195$$ 0 0
$$196$$ −2.38775e23 −0.203845
$$197$$ −1.90963e24 −1.54545 −0.772723 0.634743i $$-0.781106\pi$$
−0.772723 + 0.634743i $$0.781106\pi$$
$$198$$ 3.86292e23 0.296436
$$199$$ 1.44254e24 1.04995 0.524977 0.851116i $$-0.324074\pi$$
0.524977 + 0.851116i $$0.324074\pi$$
$$200$$ 0 0
$$201$$ −5.45844e23 −0.357692
$$202$$ −7.47216e23 −0.464790
$$203$$ 1.09008e24 0.643801
$$204$$ −4.32580e23 −0.242633
$$205$$ 0 0
$$206$$ 3.18165e24 1.61082
$$207$$ 8.13065e23 0.391234
$$208$$ −7.16728e23 −0.327861
$$209$$ −8.65020e23 −0.376262
$$210$$ 0 0
$$211$$ 3.98848e24 1.56979 0.784895 0.619629i $$-0.212717\pi$$
0.784895 + 0.619629i $$0.212717\pi$$
$$212$$ 1.93080e24 0.723119
$$213$$ −1.20385e24 −0.429129
$$214$$ −4.21295e24 −1.42969
$$215$$ 0 0
$$216$$ 4.29896e23 0.132312
$$217$$ 3.69858e24 1.08445
$$218$$ 7.14833e24 1.99715
$$219$$ −9.81262e23 −0.261289
$$220$$ 0 0
$$221$$ −1.07954e24 −0.261289
$$222$$ 3.51414e23 0.0811172
$$223$$ 4.62963e24 1.01940 0.509700 0.860352i $$-0.329756\pi$$
0.509700 + 0.860352i $$0.329756\pi$$
$$224$$ −2.73354e24 −0.574278
$$225$$ 0 0
$$226$$ 6.01188e24 1.15046
$$227$$ 3.43010e24 0.626664 0.313332 0.949644i $$-0.398555\pi$$
0.313332 + 0.949644i $$0.398555\pi$$
$$228$$ 7.08128e23 0.123537
$$229$$ 8.11792e23 0.135261 0.0676304 0.997710i $$-0.478456\pi$$
0.0676304 + 0.997710i $$0.478456\pi$$
$$230$$ 0 0
$$231$$ 2.03839e24 0.310000
$$232$$ −4.22724e24 −0.614374
$$233$$ −8.22188e23 −0.114218 −0.0571089 0.998368i $$-0.518188\pi$$
−0.0571089 + 0.998368i $$0.518188\pi$$
$$234$$ −7.89177e23 −0.104811
$$235$$ 0 0
$$236$$ 1.36421e24 0.165693
$$237$$ 4.01181e24 0.466104
$$238$$ −7.66844e24 −0.852411
$$239$$ −8.85525e24 −0.941940 −0.470970 0.882149i $$-0.656096\pi$$
−0.470970 + 0.882149i $$0.656096\pi$$
$$240$$ 0 0
$$241$$ 7.46934e24 0.727953 0.363977 0.931408i $$-0.381419\pi$$
0.363977 + 0.931408i $$0.381419\pi$$
$$242$$ 5.68472e24 0.530454
$$243$$ 7.17898e23 0.0641500
$$244$$ 3.83228e24 0.327992
$$245$$ 0 0
$$246$$ 2.22873e24 0.175081
$$247$$ 1.76720e24 0.133035
$$248$$ −1.43427e25 −1.03488
$$249$$ −2.33266e24 −0.161346
$$250$$ 0 0
$$251$$ 9.46474e23 0.0601914 0.0300957 0.999547i $$-0.490419\pi$$
0.0300957 + 0.999547i $$0.490419\pi$$
$$252$$ −1.66868e24 −0.101781
$$253$$ −1.49502e25 −0.874744
$$254$$ 2.36979e24 0.133032
$$255$$ 0 0
$$256$$ 2.08004e25 1.07535
$$257$$ 1.91825e25 0.951936 0.475968 0.879463i $$-0.342098\pi$$
0.475968 + 0.879463i $$0.342098\pi$$
$$258$$ −7.32550e24 −0.349004
$$259$$ 1.85435e24 0.0848290
$$260$$ 0 0
$$261$$ −7.05921e24 −0.297873
$$262$$ 4.23837e25 1.71805
$$263$$ −8.88429e23 −0.0346009 −0.0173004 0.999850i $$-0.505507\pi$$
−0.0173004 + 0.999850i $$0.505507\pi$$
$$264$$ −7.90469e24 −0.295831
$$265$$ 0 0
$$266$$ 1.25531e25 0.434006
$$267$$ 2.45713e24 0.0816694
$$268$$ −8.21628e24 −0.262579
$$269$$ −2.13847e25 −0.657211 −0.328605 0.944467i $$-0.606579\pi$$
−0.328605 + 0.944467i $$0.606579\pi$$
$$270$$ 0 0
$$271$$ −1.56435e25 −0.444791 −0.222395 0.974957i $$-0.571388\pi$$
−0.222395 + 0.974957i $$0.571388\pi$$
$$272$$ 4.51007e25 1.23370
$$273$$ −4.16435e24 −0.109607
$$274$$ 1.77699e25 0.450095
$$275$$ 0 0
$$276$$ 1.22386e25 0.287202
$$277$$ 8.04973e25 1.81863 0.909313 0.416112i $$-0.136608\pi$$
0.909313 + 0.416112i $$0.136608\pi$$
$$278$$ −1.50456e25 −0.327294
$$279$$ −2.39514e25 −0.501749
$$280$$ 0 0
$$281$$ 8.33171e25 1.61926 0.809632 0.586938i $$-0.199667\pi$$
0.809632 + 0.586938i $$0.199667\pi$$
$$282$$ 2.89320e25 0.541704
$$283$$ 4.46130e24 0.0804829 0.0402415 0.999190i $$-0.487187\pi$$
0.0402415 + 0.999190i $$0.487187\pi$$
$$284$$ −1.81209e25 −0.315020
$$285$$ 0 0
$$286$$ 1.45109e25 0.234343
$$287$$ 1.17606e25 0.183092
$$288$$ 1.77020e25 0.265706
$$289$$ −1.16088e24 −0.0168020
$$290$$ 0 0
$$291$$ −3.37651e24 −0.0454560
$$292$$ −1.47704e25 −0.191810
$$293$$ −9.67128e25 −1.21164 −0.605820 0.795602i $$-0.707155\pi$$
−0.605820 + 0.795602i $$0.707155\pi$$
$$294$$ 2.74111e25 0.331344
$$295$$ 0 0
$$296$$ −7.19099e24 −0.0809516
$$297$$ −1.32003e25 −0.143430
$$298$$ 1.56211e26 1.63848
$$299$$ 3.05426e25 0.309284
$$300$$ 0 0
$$301$$ −3.86554e25 −0.364974
$$302$$ 8.21155e25 0.748777
$$303$$ 2.55338e25 0.224889
$$304$$ −7.38293e25 −0.628140
$$305$$ 0 0
$$306$$ 4.96596e25 0.394392
$$307$$ −1.68163e26 −1.29056 −0.645278 0.763948i $$-0.723258\pi$$
−0.645278 + 0.763948i $$0.723258\pi$$
$$308$$ 3.06828e25 0.227569
$$309$$ −1.08723e26 −0.779393
$$310$$ 0 0
$$311$$ 2.30370e26 1.54327 0.771636 0.636065i $$-0.219439\pi$$
0.771636 + 0.636065i $$0.219439\pi$$
$$312$$ 1.61489e25 0.104597
$$313$$ 2.79658e26 1.75151 0.875753 0.482759i $$-0.160365\pi$$
0.875753 + 0.482759i $$0.160365\pi$$
$$314$$ −2.60756e26 −1.57933
$$315$$ 0 0
$$316$$ 6.03875e25 0.342163
$$317$$ −2.98501e25 −0.163615 −0.0818075 0.996648i $$-0.526069\pi$$
−0.0818075 + 0.996648i $$0.526069\pi$$
$$318$$ −2.21653e26 −1.17541
$$319$$ 1.29801e26 0.666002
$$320$$ 0 0
$$321$$ 1.43964e26 0.691755
$$322$$ 2.16957e26 1.00899
$$323$$ −1.11202e26 −0.500596
$$324$$ 1.08061e25 0.0470920
$$325$$ 0 0
$$326$$ −8.35494e25 −0.341317
$$327$$ −2.44272e26 −0.966322
$$328$$ −4.56064e25 −0.174723
$$329$$ 1.52669e26 0.566492
$$330$$ 0 0
$$331$$ 2.55594e26 0.889933 0.444967 0.895547i $$-0.353216\pi$$
0.444967 + 0.895547i $$0.353216\pi$$
$$332$$ −3.51122e25 −0.118443
$$333$$ −1.20085e25 −0.0392485
$$334$$ 8.27248e23 0.00261998
$$335$$ 0 0
$$336$$ 1.73977e26 0.517522
$$337$$ 4.91931e25 0.141837 0.0709187 0.997482i $$-0.477407\pi$$
0.0709187 + 0.997482i $$0.477407\pi$$
$$338$$ 3.97282e26 1.11039
$$339$$ −2.05437e26 −0.556650
$$340$$ 0 0
$$341$$ 4.40405e26 1.12184
$$342$$ −8.12921e25 −0.200805
$$343$$ 4.45381e26 1.06695
$$344$$ 1.49902e26 0.348292
$$345$$ 0 0
$$346$$ −2.79597e26 −0.611271
$$347$$ −2.98136e26 −0.632345 −0.316173 0.948702i $$-0.602398\pi$$
−0.316173 + 0.948702i $$0.602398\pi$$
$$348$$ −1.06258e26 −0.218666
$$349$$ 7.72834e26 1.54319 0.771595 0.636115i $$-0.219459\pi$$
0.771595 + 0.636115i $$0.219459\pi$$
$$350$$ 0 0
$$351$$ 2.69676e25 0.0507129
$$352$$ −3.25494e26 −0.594081
$$353$$ 7.30755e26 1.29461 0.647303 0.762233i $$-0.275897\pi$$
0.647303 + 0.762233i $$0.275897\pi$$
$$354$$ −1.56609e26 −0.269329
$$355$$ 0 0
$$356$$ 3.69858e25 0.0599529
$$357$$ 2.62045e26 0.412439
$$358$$ 1.51438e26 0.231453
$$359$$ −1.58936e25 −0.0235901 −0.0117951 0.999930i $$-0.503755\pi$$
−0.0117951 + 0.999930i $$0.503755\pi$$
$$360$$ 0 0
$$361$$ −5.32173e26 −0.745121
$$362$$ −1.61849e26 −0.220125
$$363$$ −1.94258e26 −0.256660
$$364$$ −6.26836e25 −0.0804618
$$365$$ 0 0
$$366$$ −4.39940e26 −0.533141
$$367$$ 1.40734e27 1.65732 0.828660 0.559752i $$-0.189104\pi$$
0.828660 + 0.559752i $$0.189104\pi$$
$$368$$ −1.27600e27 −1.46032
$$369$$ −7.61598e25 −0.0847127
$$370$$ 0 0
$$371$$ −1.16962e27 −1.22919
$$372$$ −3.60527e26 −0.368330
$$373$$ 9.30077e26 0.923797 0.461898 0.886933i $$-0.347169\pi$$
0.461898 + 0.886933i $$0.347169\pi$$
$$374$$ −9.13112e26 −0.881805
$$375$$ 0 0
$$376$$ −5.92035e26 −0.540599
$$377$$ −2.65177e26 −0.235479
$$378$$ 1.91562e26 0.165442
$$379$$ 2.18541e27 1.83578 0.917892 0.396830i $$-0.129890\pi$$
0.917892 + 0.396830i $$0.129890\pi$$
$$380$$ 0 0
$$381$$ −8.09801e25 −0.0643674
$$382$$ −2.08843e27 −1.61493
$$383$$ −2.10347e27 −1.58252 −0.791258 0.611482i $$-0.790574\pi$$
−0.791258 + 0.611482i $$0.790574\pi$$
$$384$$ −9.04484e26 −0.662099
$$385$$ 0 0
$$386$$ −3.09004e27 −2.14189
$$387$$ 2.50326e26 0.168866
$$388$$ −5.08247e25 −0.0333689
$$389$$ −2.97815e26 −0.190316 −0.0951582 0.995462i $$-0.530336\pi$$
−0.0951582 + 0.995462i $$0.530336\pi$$
$$390$$ 0 0
$$391$$ −1.92192e27 −1.16380
$$392$$ −5.60912e26 −0.330667
$$393$$ −1.44833e27 −0.831277
$$394$$ 3.29984e27 1.84409
$$395$$ 0 0
$$396$$ −1.98697e26 −0.105291
$$397$$ −6.36504e26 −0.328474 −0.164237 0.986421i $$-0.552516\pi$$
−0.164237 + 0.986421i $$0.552516\pi$$
$$398$$ −2.49271e27 −1.25285
$$399$$ −4.28964e26 −0.209993
$$400$$ 0 0
$$401$$ 2.43888e27 1.13286 0.566428 0.824111i $$-0.308325\pi$$
0.566428 + 0.824111i $$0.308325\pi$$
$$402$$ 9.43218e26 0.426814
$$403$$ −8.99728e26 −0.396651
$$404$$ 3.84346e26 0.165089
$$405$$ 0 0
$$406$$ −1.88367e27 −0.768211
$$407$$ 2.20805e26 0.0877542
$$408$$ −1.01618e27 −0.393587
$$409$$ 5.48032e26 0.206876 0.103438 0.994636i $$-0.467016\pi$$
0.103438 + 0.994636i $$0.467016\pi$$
$$410$$ 0 0
$$411$$ −6.07232e26 −0.217779
$$412$$ −1.63654e27 −0.572146
$$413$$ −8.26399e26 −0.281653
$$414$$ −1.40498e27 −0.466837
$$415$$ 0 0
$$416$$ 6.64970e26 0.210050
$$417$$ 5.14135e26 0.158361
$$418$$ 1.49475e27 0.448971
$$419$$ −6.08246e27 −1.78169 −0.890844 0.454309i $$-0.849886\pi$$
−0.890844 + 0.454309i $$0.849886\pi$$
$$420$$ 0 0
$$421$$ −4.05990e27 −1.13124 −0.565618 0.824667i $$-0.691362\pi$$
−0.565618 + 0.824667i $$0.691362\pi$$
$$422$$ −6.89209e27 −1.87314
$$423$$ −9.88659e26 −0.262103
$$424$$ 4.53568e27 1.17301
$$425$$ 0 0
$$426$$ 2.08026e27 0.512055
$$427$$ −2.32149e27 −0.557537
$$428$$ 2.16702e27 0.507811
$$429$$ −4.95866e26 −0.113387
$$430$$ 0 0
$$431$$ −7.87214e27 −1.71428 −0.857140 0.515084i $$-0.827761\pi$$
−0.857140 + 0.515084i $$0.827761\pi$$
$$432$$ −1.12664e27 −0.239446
$$433$$ 1.73785e27 0.360486 0.180243 0.983622i $$-0.442312\pi$$
0.180243 + 0.983622i $$0.442312\pi$$
$$434$$ −6.39115e27 −1.29401
$$435$$ 0 0
$$436$$ −3.67689e27 −0.709369
$$437$$ 3.14615e27 0.592550
$$438$$ 1.69562e27 0.311782
$$439$$ 8.37416e27 1.50336 0.751681 0.659526i $$-0.229243\pi$$
0.751681 + 0.659526i $$0.229243\pi$$
$$440$$ 0 0
$$441$$ −9.36687e26 −0.160321
$$442$$ 1.86545e27 0.311781
$$443$$ 3.30286e25 0.00539077 0.00269539 0.999996i $$-0.499142\pi$$
0.00269539 + 0.999996i $$0.499142\pi$$
$$444$$ −1.80757e26 −0.0288120
$$445$$ 0 0
$$446$$ −7.99999e27 −1.21639
$$447$$ −5.33801e27 −0.792776
$$448$$ −1.45530e27 −0.211121
$$449$$ 5.21713e27 0.739341 0.369670 0.929163i $$-0.379471\pi$$
0.369670 + 0.929163i $$0.379471\pi$$
$$450$$ 0 0
$$451$$ 1.40038e27 0.189406
$$452$$ −3.09233e27 −0.408632
$$453$$ −2.80604e27 −0.362296
$$454$$ −5.92721e27 −0.747763
$$455$$ 0 0
$$456$$ 1.66348e27 0.200395
$$457$$ −2.15211e26 −0.0253363 −0.0126682 0.999920i $$-0.504033\pi$$
−0.0126682 + 0.999920i $$0.504033\pi$$
$$458$$ −1.40278e27 −0.161399
$$459$$ −1.69696e27 −0.190826
$$460$$ 0 0
$$461$$ 1.68699e28 1.81239 0.906197 0.422855i $$-0.138972\pi$$
0.906197 + 0.422855i $$0.138972\pi$$
$$462$$ −3.52234e27 −0.369906
$$463$$ 1.90352e28 1.95415 0.977074 0.212898i $$-0.0682901\pi$$
0.977074 + 0.212898i $$0.0682901\pi$$
$$464$$ 1.10785e28 1.11184
$$465$$ 0 0
$$466$$ 1.42074e27 0.136290
$$467$$ 1.21027e28 1.13515 0.567576 0.823321i $$-0.307881\pi$$
0.567576 + 0.823321i $$0.307881\pi$$
$$468$$ 4.05929e26 0.0372279
$$469$$ 4.97720e27 0.446344
$$470$$ 0 0
$$471$$ 8.91053e27 0.764159
$$472$$ 3.20469e27 0.268779
$$473$$ −4.60286e27 −0.377559
$$474$$ −6.93240e27 −0.556175
$$475$$ 0 0
$$476$$ 3.94442e27 0.302768
$$477$$ 7.57429e27 0.568721
$$478$$ 1.53019e28 1.12396
$$479$$ 6.95253e27 0.499597 0.249798 0.968298i $$-0.419636\pi$$
0.249798 + 0.968298i $$0.419636\pi$$
$$480$$ 0 0
$$481$$ −4.51095e26 −0.0310274
$$482$$ −1.29070e28 −0.868625
$$483$$ −7.41382e27 −0.488199
$$484$$ −2.92405e27 −0.188412
$$485$$ 0 0
$$486$$ −1.24053e27 −0.0765466
$$487$$ 1.06412e28 0.642596 0.321298 0.946978i $$-0.395881\pi$$
0.321298 + 0.946978i $$0.395881\pi$$
$$488$$ 9.00250e27 0.532053
$$489$$ 2.85504e27 0.165146
$$490$$ 0 0
$$491$$ 1.68064e28 0.931361 0.465681 0.884953i $$-0.345810\pi$$
0.465681 + 0.884953i $$0.345810\pi$$
$$492$$ −1.14639e27 −0.0621869
$$493$$ 1.66865e28 0.886079
$$494$$ −3.05372e27 −0.158743
$$495$$ 0 0
$$496$$ 3.75885e28 1.87283
$$497$$ 1.09772e28 0.535486
$$498$$ 4.03083e27 0.192525
$$499$$ −5.12285e27 −0.239583 −0.119792 0.992799i $$-0.538223\pi$$
−0.119792 + 0.992799i $$0.538223\pi$$
$$500$$ 0 0
$$501$$ −2.82686e25 −0.00126768
$$502$$ −1.63551e27 −0.0718229
$$503$$ −1.99606e28 −0.858442 −0.429221 0.903200i $$-0.641212\pi$$
−0.429221 + 0.903200i $$0.641212\pi$$
$$504$$ −3.91994e27 −0.165105
$$505$$ 0 0
$$506$$ 2.58339e28 1.04378
$$507$$ −1.35759e28 −0.537260
$$508$$ −1.21895e27 −0.0472516
$$509$$ −2.57966e27 −0.0979550 −0.0489775 0.998800i $$-0.515596\pi$$
−0.0489775 + 0.998800i $$0.515596\pi$$
$$510$$ 0 0
$$511$$ 8.94749e27 0.326048
$$512$$ −3.81989e27 −0.136369
$$513$$ 2.77790e27 0.0971594
$$514$$ −3.31474e28 −1.13589
$$515$$ 0 0
$$516$$ 3.76802e27 0.123963
$$517$$ 1.81789e28 0.586026
$$518$$ −3.20432e27 −0.101222
$$519$$ 9.55436e27 0.295763
$$520$$ 0 0
$$521$$ −2.61230e28 −0.776652 −0.388326 0.921522i $$-0.626947\pi$$
−0.388326 + 0.921522i $$0.626947\pi$$
$$522$$ 1.21983e28 0.355435
$$523$$ −6.70750e28 −1.91555 −0.957774 0.287523i $$-0.907168\pi$$
−0.957774 + 0.287523i $$0.907168\pi$$
$$524$$ −2.18009e28 −0.610233
$$525$$ 0 0
$$526$$ 1.53520e27 0.0412873
$$527$$ 5.66161e28 1.49255
$$528$$ 2.07161e28 0.535367
$$529$$ 1.49036e28 0.377577
$$530$$ 0 0
$$531$$ 5.35163e27 0.130315
$$532$$ −6.45696e27 −0.154155
$$533$$ −2.86092e27 −0.0669684
$$534$$ −4.24592e27 −0.0974514
$$535$$ 0 0
$$536$$ −1.93011e28 −0.425943
$$537$$ −5.17492e27 −0.111988
$$538$$ 3.69528e28 0.784212
$$539$$ 1.72233e28 0.358454
$$540$$ 0 0
$$541$$ −2.15196e28 −0.430787 −0.215394 0.976527i $$-0.569103\pi$$
−0.215394 + 0.976527i $$0.569103\pi$$
$$542$$ 2.70319e28 0.530743
$$543$$ 5.53067e27 0.106507
$$544$$ −4.18438e28 −0.790392
$$545$$ 0 0
$$546$$ 7.19599e27 0.130788
$$547$$ 7.46789e28 1.33147 0.665734 0.746189i $$-0.268118\pi$$
0.665734 + 0.746189i $$0.268118\pi$$
$$548$$ −9.14032e27 −0.159870
$$549$$ 1.50336e28 0.257960
$$550$$ 0 0
$$551$$ −2.73156e28 −0.451148
$$552$$ 2.87500e28 0.465884
$$553$$ −3.65811e28 −0.581625
$$554$$ −1.39099e29 −2.17006
$$555$$ 0 0
$$556$$ 7.73899e27 0.116252
$$557$$ 7.95166e28 1.17214 0.586068 0.810262i $$-0.300675\pi$$
0.586068 + 0.810262i $$0.300675\pi$$
$$558$$ 4.13880e28 0.598709
$$559$$ 9.40343e27 0.133494
$$560$$ 0 0
$$561$$ 3.12028e28 0.426661
$$562$$ −1.43972e29 −1.93217
$$563$$ −5.46305e28 −0.719609 −0.359805 0.933028i $$-0.617157\pi$$
−0.359805 + 0.933028i $$0.617157\pi$$
$$564$$ −1.48817e28 −0.192408
$$565$$ 0 0
$$566$$ −7.70912e27 −0.0960356
$$567$$ −6.54605e27 −0.0800492
$$568$$ −4.25683e28 −0.511010
$$569$$ 9.43478e28 1.11187 0.555933 0.831227i $$-0.312361\pi$$
0.555933 + 0.831227i $$0.312361\pi$$
$$570$$ 0 0
$$571$$ 8.05027e28 0.914390 0.457195 0.889367i $$-0.348854\pi$$
0.457195 + 0.889367i $$0.348854\pi$$
$$572$$ −7.46400e27 −0.0832364
$$573$$ 7.13655e28 0.781384
$$574$$ −2.03223e28 −0.218473
$$575$$ 0 0
$$576$$ 9.42426e27 0.0976812
$$577$$ 1.67132e28 0.170104 0.0850519 0.996377i $$-0.472894\pi$$
0.0850519 + 0.996377i $$0.472894\pi$$
$$578$$ 2.00600e27 0.0200488
$$579$$ 1.05592e29 1.03635
$$580$$ 0 0
$$581$$ 2.12700e28 0.201334
$$582$$ 5.83461e27 0.0542401
$$583$$ −1.39272e29 −1.27158
$$584$$ −3.46975e28 −0.311145
$$585$$ 0 0
$$586$$ 1.67120e29 1.44578
$$587$$ −3.15730e28 −0.268297 −0.134149 0.990961i $$-0.542830\pi$$
−0.134149 + 0.990961i $$0.542830\pi$$
$$588$$ −1.40994e28 −0.117690
$$589$$ −9.26799e28 −0.759932
$$590$$ 0 0
$$591$$ −1.12762e29 −0.892264
$$592$$ 1.88457e28 0.146499
$$593$$ 5.48493e27 0.0418887 0.0209443 0.999781i $$-0.493333\pi$$
0.0209443 + 0.999781i $$0.493333\pi$$
$$594$$ 2.28101e28 0.171147
$$595$$ 0 0
$$596$$ −8.03502e28 −0.581971
$$597$$ 8.51805e28 0.606191
$$598$$ −5.27776e28 −0.369051
$$599$$ 1.25621e29 0.863136 0.431568 0.902080i $$-0.357960\pi$$
0.431568 + 0.902080i $$0.357960\pi$$
$$600$$ 0 0
$$601$$ 3.99325e28 0.264938 0.132469 0.991187i $$-0.457709\pi$$
0.132469 + 0.991187i $$0.457709\pi$$
$$602$$ 6.67965e28 0.435503
$$603$$ −3.22315e28 −0.206514
$$604$$ −4.22378e28 −0.265958
$$605$$ 0 0
$$606$$ −4.41223e28 −0.268347
$$607$$ 2.46990e29 1.47638 0.738189 0.674594i $$-0.235681\pi$$
0.738189 + 0.674594i $$0.235681\pi$$
$$608$$ 6.84978e28 0.402429
$$609$$ 6.43684e28 0.371699
$$610$$ 0 0
$$611$$ −3.71387e28 −0.207202
$$612$$ −2.55434e28 −0.140084
$$613$$ −2.63911e28 −0.142273 −0.0711364 0.997467i $$-0.522663\pi$$
−0.0711364 + 0.997467i $$0.522663\pi$$
$$614$$ 2.90585e29 1.53995
$$615$$ 0 0
$$616$$ 7.20777e28 0.369151
$$617$$ −3.09820e29 −1.55997 −0.779984 0.625800i $$-0.784773\pi$$
−0.779984 + 0.625800i $$0.784773\pi$$
$$618$$ 1.87873e29 0.930005
$$619$$ −2.50758e29 −1.22040 −0.610202 0.792246i $$-0.708912\pi$$
−0.610202 + 0.792246i $$0.708912\pi$$
$$620$$ 0 0
$$621$$ 4.80107e28 0.225879
$$622$$ −3.98080e29 −1.84150
$$623$$ −2.24050e28 −0.101911
$$624$$ −4.23221e28 −0.189291
$$625$$ 0 0
$$626$$ −4.83249e29 −2.08997
$$627$$ −5.10785e28 −0.217235
$$628$$ 1.34125e29 0.560963
$$629$$ 2.83855e28 0.116752
$$630$$ 0 0
$$631$$ −4.32770e28 −0.172167 −0.0860833 0.996288i $$-0.527435\pi$$
−0.0860833 + 0.996288i $$0.527435\pi$$
$$632$$ 1.41858e29 0.555039
$$633$$ 2.35516e29 0.906318
$$634$$ 5.15809e28 0.195232
$$635$$ 0 0
$$636$$ 1.14012e29 0.417493
$$637$$ −3.51864e28 −0.126739
$$638$$ −2.24296e29 −0.794702
$$639$$ −7.10863e28 −0.247758
$$640$$ 0 0
$$641$$ −8.73381e28 −0.294574 −0.147287 0.989094i $$-0.547054\pi$$
−0.147287 + 0.989094i $$0.547054\pi$$
$$642$$ −2.48770e29 −0.825431
$$643$$ 4.72013e29 1.54077 0.770386 0.637578i $$-0.220063\pi$$
0.770386 + 0.637578i $$0.220063\pi$$
$$644$$ −1.11596e29 −0.358383
$$645$$ 0 0
$$646$$ 1.92158e29 0.597332
$$647$$ −1.26799e28 −0.0387812 −0.0193906 0.999812i $$-0.506173\pi$$
−0.0193906 + 0.999812i $$0.506173\pi$$
$$648$$ 2.53849e28 0.0763903
$$649$$ −9.84027e28 −0.291365
$$650$$ 0 0
$$651$$ 2.18397e29 0.626105
$$652$$ 4.29753e28 0.121233
$$653$$ 2.76226e29 0.766790 0.383395 0.923584i $$-0.374755\pi$$
0.383395 + 0.923584i $$0.374755\pi$$
$$654$$ 4.22102e29 1.15306
$$655$$ 0 0
$$656$$ 1.19523e29 0.316198
$$657$$ −5.79425e28 −0.150855
$$658$$ −2.63812e29 −0.675962
$$659$$ 6.46511e28 0.163034 0.0815172 0.996672i $$-0.474023\pi$$
0.0815172 + 0.996672i $$0.474023\pi$$
$$660$$ 0 0
$$661$$ −2.27730e29 −0.556295 −0.278147 0.960538i $$-0.589720\pi$$
−0.278147 + 0.960538i $$0.589720\pi$$
$$662$$ −4.41667e29 −1.06191
$$663$$ −6.37459e28 −0.150855
$$664$$ −8.24829e28 −0.192132
$$665$$ 0 0
$$666$$ 2.07507e28 0.0468330
$$667$$ −4.72097e29 −1.04884
$$668$$ −4.25512e26 −0.000930591 0
$$669$$ 2.73375e29 0.588551
$$670$$ 0 0
$$671$$ −2.76429e29 −0.576763
$$672$$ −1.61413e29 −0.331559
$$673$$ 3.79243e29 0.766936 0.383468 0.923554i $$-0.374730\pi$$
0.383468 + 0.923554i $$0.374730\pi$$
$$674$$ −8.50057e28 −0.169246
$$675$$ 0 0
$$676$$ −2.04350e29 −0.394398
$$677$$ 3.39717e29 0.645559 0.322780 0.946474i $$-0.395383\pi$$
0.322780 + 0.946474i $$0.395383\pi$$
$$678$$ 3.54995e29 0.664218
$$679$$ 3.07882e28 0.0567220
$$680$$ 0 0
$$681$$ 2.02544e29 0.361805
$$682$$ −7.61020e29 −1.33863
$$683$$ −4.54742e29 −0.787677 −0.393838 0.919180i $$-0.628853\pi$$
−0.393838 + 0.919180i $$0.628853\pi$$
$$684$$ 4.18143e28 0.0713239
$$685$$ 0 0
$$686$$ −7.69619e29 −1.27313
$$687$$ 4.79355e28 0.0780929
$$688$$ −3.92853e29 −0.630306
$$689$$ 2.84526e29 0.449594
$$690$$ 0 0
$$691$$ −3.71838e29 −0.569947 −0.284974 0.958535i $$-0.591985\pi$$
−0.284974 + 0.958535i $$0.591985\pi$$
$$692$$ 1.43817e29 0.217118
$$693$$ 1.20365e29 0.178979
$$694$$ 5.15178e29 0.754542
$$695$$ 0 0
$$696$$ −2.49614e29 −0.354709
$$697$$ 1.80026e29 0.251994
$$698$$ −1.33546e30 −1.84140
$$699$$ −4.85494e28 −0.0659437
$$700$$ 0 0
$$701$$ −9.37969e29 −1.23637 −0.618186 0.786032i $$-0.712132\pi$$
−0.618186 + 0.786032i $$0.712132\pi$$
$$702$$ −4.66001e28 −0.0605128
$$703$$ −4.64668e28 −0.0594445
$$704$$ −1.73288e29 −0.218401
$$705$$ 0 0
$$706$$ −1.26275e30 −1.54478
$$707$$ −2.32826e29 −0.280626
$$708$$ 8.05551e28 0.0956629
$$709$$ −7.54578e29 −0.882914 −0.441457 0.897282i $$-0.645538\pi$$
−0.441457 + 0.897282i $$0.645538\pi$$
$$710$$ 0 0
$$711$$ 2.36893e29 0.269105
$$712$$ 8.68842e28 0.0972525
$$713$$ −1.60179e30 −1.76671
$$714$$ −4.52814e29 −0.492140
$$715$$ 0 0
$$716$$ −7.78952e28 −0.0822098
$$717$$ −5.22894e29 −0.543829
$$718$$ 2.74641e28 0.0281488
$$719$$ 1.22754e30 1.23989 0.619944 0.784646i $$-0.287155\pi$$
0.619944 + 0.784646i $$0.287155\pi$$
$$720$$ 0 0
$$721$$ 9.91374e29 0.972561
$$722$$ 9.19594e29 0.889111
$$723$$ 4.41057e29 0.420284
$$724$$ 8.32502e28 0.0781862
$$725$$ 0 0
$$726$$ 3.35677e29 0.306258
$$727$$ −9.04407e29 −0.813303 −0.406652 0.913583i $$-0.633304\pi$$
−0.406652 + 0.913583i $$0.633304\pi$$
$$728$$ −1.47252e29 −0.130521
$$729$$ 4.23912e28 0.0370370
$$730$$ 0 0
$$731$$ −5.91719e29 −0.502323
$$732$$ 2.26292e29 0.189367
$$733$$ 1.38874e30 1.14559 0.572795 0.819699i $$-0.305859\pi$$
0.572795 + 0.819699i $$0.305859\pi$$
$$734$$ −2.43189e30 −1.97759
$$735$$ 0 0
$$736$$ 1.18385e30 0.935578
$$737$$ 5.92655e29 0.461736
$$738$$ 1.31604e29 0.101083
$$739$$ 2.11506e30 1.60161 0.800804 0.598927i $$-0.204406\pi$$
0.800804 + 0.598927i $$0.204406\pi$$
$$740$$ 0 0
$$741$$ 1.04351e29 0.0768080
$$742$$ 2.02111e30 1.46673
$$743$$ −2.26805e30 −1.62282 −0.811411 0.584476i $$-0.801300\pi$$
−0.811411 + 0.584476i $$0.801300\pi$$
$$744$$ −8.46923e29 −0.597487
$$745$$ 0 0
$$746$$ −1.60717e30 −1.10231
$$747$$ −1.37741e29 −0.0931530
$$748$$ 4.69678e29 0.313208
$$749$$ −1.31272e30 −0.863202
$$750$$ 0 0
$$751$$ −6.98349e29 −0.446533 −0.223266 0.974757i $$-0.571672\pi$$
−0.223266 + 0.974757i $$0.571672\pi$$
$$752$$ 1.55157e30 0.978326
$$753$$ 5.58883e28 0.0347515
$$754$$ 4.58226e29 0.280984
$$755$$ 0 0
$$756$$ −9.85341e28 −0.0587635
$$757$$ 3.79778e29 0.223369 0.111685 0.993744i $$-0.464375\pi$$
0.111685 + 0.993744i $$0.464375\pi$$
$$758$$ −3.77639e30 −2.19054
$$759$$ −8.82794e29 −0.505034
$$760$$ 0 0
$$761$$ −7.50371e29 −0.417577 −0.208789 0.977961i $$-0.566952\pi$$
−0.208789 + 0.977961i $$0.566952\pi$$
$$762$$ 1.39934e29 0.0768060
$$763$$ 2.22736e30 1.20582
$$764$$ 1.07423e30 0.573608
$$765$$ 0 0
$$766$$ 3.63479e30 1.88833
$$767$$ 2.01032e29 0.103018
$$768$$ 1.22824e30 0.620856
$$769$$ 2.16884e29 0.108144 0.0540719 0.998537i $$-0.482780\pi$$
0.0540719 + 0.998537i $$0.482780\pi$$
$$770$$ 0 0
$$771$$ 1.13271e30 0.549600
$$772$$ 1.58943e30 0.760779
$$773$$ 3.08128e30 1.45495 0.727473 0.686136i $$-0.240695\pi$$
0.727473 + 0.686136i $$0.240695\pi$$
$$774$$ −4.32563e29 −0.201498
$$775$$ 0 0
$$776$$ −1.19394e29 −0.0541294
$$777$$ 1.09498e29 0.0489761
$$778$$ 5.14625e29 0.227094
$$779$$ −2.94700e29 −0.128303
$$780$$ 0 0
$$781$$ 1.30710e30 0.553951
$$782$$ 3.32107e30 1.38870
$$783$$ −4.16839e29 −0.171977
$$784$$ 1.47000e30 0.598412
$$785$$ 0 0
$$786$$ 2.50271e30 0.991915
$$787$$ −1.46460e30 −0.572775 −0.286387 0.958114i $$-0.592454\pi$$
−0.286387 + 0.958114i $$0.592454\pi$$
$$788$$ −1.69734e30 −0.655004
$$789$$ −5.24608e28 −0.0199768
$$790$$ 0 0
$$791$$ 1.87325e30 0.694612
$$792$$ −4.66764e29 −0.170798
$$793$$ 5.64732e29 0.203927
$$794$$ 1.09988e30 0.391949
$$795$$ 0 0
$$796$$ 1.28217e30 0.445000
$$797$$ −1.27731e30 −0.437505 −0.218752 0.975780i $$-0.570199\pi$$
−0.218752 + 0.975780i $$0.570199\pi$$
$$798$$ 7.41250e29 0.250573
$$799$$ 2.33698e30 0.779677
$$800$$ 0 0
$$801$$ 1.45091e29 0.0471519
$$802$$ −4.21439e30 −1.35177
$$803$$ 1.06541e30 0.337292
$$804$$ −4.85163e29 −0.151600
$$805$$ 0 0
$$806$$ 1.55473e30 0.473301
$$807$$ −1.26275e30 −0.379441
$$808$$ 9.02876e29 0.267799
$$809$$ −4.24975e30 −1.24424 −0.622120 0.782922i $$-0.713728\pi$$
−0.622120 + 0.782922i $$0.713728\pi$$
$$810$$ 0 0
$$811$$ 2.05863e30 0.587298 0.293649 0.955913i $$-0.405130\pi$$
0.293649 + 0.955913i $$0.405130\pi$$
$$812$$ 9.68902e29 0.272861
$$813$$ −9.23732e29 −0.256800
$$814$$ −3.81551e29 −0.104712
$$815$$ 0 0
$$816$$ 2.66315e30 0.712278
$$817$$ 9.68636e29 0.255758
$$818$$ −9.46999e29 −0.246854
$$819$$ −2.45901e29 −0.0632818
$$820$$ 0 0
$$821$$ −1.80717e29 −0.0453309 −0.0226655 0.999743i $$-0.507215\pi$$
−0.0226655 + 0.999743i $$0.507215\pi$$
$$822$$ 1.04930e30 0.259863
$$823$$ 6.23532e30 1.52462 0.762308 0.647214i $$-0.224066\pi$$
0.762308 + 0.647214i $$0.224066\pi$$
$$824$$ −3.84445e30 −0.928107
$$825$$ 0 0
$$826$$ 1.42802e30 0.336080
$$827$$ −3.37179e30 −0.783524 −0.391762 0.920067i $$-0.628134\pi$$
−0.391762 + 0.920067i $$0.628134\pi$$
$$828$$ 7.22678e29 0.165816
$$829$$ −2.70711e29 −0.0613315 −0.0306658 0.999530i $$-0.509763\pi$$
−0.0306658 + 0.999530i $$0.509763\pi$$
$$830$$ 0 0
$$831$$ 4.75328e30 1.04998
$$832$$ 3.54020e29 0.0772205
$$833$$ 2.21413e30 0.476904
$$834$$ −8.88425e29 −0.188963
$$835$$ 0 0
$$836$$ −7.68857e29 −0.159470
$$837$$ −1.41431e30 −0.289685
$$838$$ 1.05105e31 2.12599
$$839$$ −7.45457e30 −1.48909 −0.744547 0.667570i $$-0.767334\pi$$
−0.744547 + 0.667570i $$0.767334\pi$$
$$840$$ 0 0
$$841$$ −1.03399e30 −0.201446
$$842$$ 7.01551e30 1.34984
$$843$$ 4.91979e30 0.934882
$$844$$ 3.54509e30 0.665321
$$845$$ 0 0
$$846$$ 1.70840e30 0.312753
$$847$$ 1.77131e30 0.320272
$$848$$ −1.18868e31 −2.12280
$$849$$ 2.63435e29 0.0464668
$$850$$ 0 0
$$851$$ −8.03088e29 −0.138198
$$852$$ −1.07002e30 −0.181877
$$853$$ 6.10653e30 1.02525 0.512625 0.858613i $$-0.328673\pi$$
0.512625 + 0.858613i $$0.328673\pi$$
$$854$$ 4.01153e30 0.665277
$$855$$ 0 0
$$856$$ 5.09059e30 0.823746
$$857$$ 4.08307e30 0.652662 0.326331 0.945256i $$-0.394188\pi$$
0.326331 + 0.945256i $$0.394188\pi$$
$$858$$ 8.56856e29 0.135298
$$859$$ 5.27189e30 0.822316 0.411158 0.911564i $$-0.365125\pi$$
0.411158 + 0.911564i $$0.365125\pi$$
$$860$$ 0 0
$$861$$ 6.94452e29 0.105708
$$862$$ 1.36031e31 2.04555
$$863$$ −1.05537e31 −1.56780 −0.783902 0.620884i $$-0.786774\pi$$
−0.783902 + 0.620884i $$0.786774\pi$$
$$864$$ 1.04528e30 0.153405
$$865$$ 0 0
$$866$$ −3.00300e30 −0.430147
$$867$$ −6.85488e28 −0.00970062
$$868$$ 3.28742e30 0.459619
$$869$$ −4.35586e30 −0.601681
$$870$$ 0 0
$$871$$ −1.21077e30 −0.163256
$$872$$ −8.63747e30 −1.15070
$$873$$ −1.99379e29 −0.0262441
$$874$$ −5.43655e30 −0.707056
$$875$$ 0 0
$$876$$ −8.72177e29 −0.110742
$$877$$ −7.16069e30 −0.898378 −0.449189 0.893437i $$-0.648287\pi$$
−0.449189 + 0.893437i $$0.648287\pi$$
$$878$$ −1.44706e31 −1.79388
$$879$$ −5.71079e30 −0.699541
$$880$$ 0 0
$$881$$ −1.05943e31 −1.26714 −0.633568 0.773687i $$-0.718410\pi$$
−0.633568 + 0.773687i $$0.718410\pi$$
$$882$$ 1.61860e30 0.191301
$$883$$ −6.60744e30 −0.771695 −0.385848 0.922562i $$-0.626091\pi$$
−0.385848 + 0.922562i $$0.626091\pi$$
$$884$$ −9.59531e29 −0.110742
$$885$$ 0 0
$$886$$ −5.70734e28 −0.00643250
$$887$$ 2.74467e30 0.305698 0.152849 0.988250i $$-0.451155\pi$$
0.152849 + 0.988250i $$0.451155\pi$$
$$888$$ −4.24621e29 −0.0467374
$$889$$ 7.38406e29 0.0803205
$$890$$ 0 0
$$891$$ −7.79465e29 −0.0828096
$$892$$ 4.11496e30 0.432051
$$893$$ −3.82561e30 −0.396973
$$894$$ 9.22409e30 0.945974
$$895$$ 0 0
$$896$$ 8.24741e30 0.826196
$$897$$ 1.80351e30 0.178565
$$898$$ −9.01519e30 −0.882213
$$899$$ 1.39071e31 1.34512
$$900$$ 0 0
$$901$$ −1.79040e31 −1.69177
$$902$$ −2.41986e30 −0.226007
$$903$$ −2.28256e30 −0.210718
$$904$$ −7.26427e30 −0.662863
$$905$$ 0 0
$$906$$ 4.84884e30 0.432307
$$907$$ −5.45568e30 −0.480809 −0.240404 0.970673i $$-0.577280\pi$$
−0.240404 + 0.970673i $$0.577280\pi$$
$$908$$ 3.04878e30 0.265598
$$909$$ 1.50774e30 0.129839
$$910$$ 0 0
$$911$$ −8.75739e30 −0.736940 −0.368470 0.929640i $$-0.620118\pi$$
−0.368470 + 0.929640i $$0.620118\pi$$
$$912$$ −4.35955e30 −0.362657
$$913$$ 2.53270e30 0.208277
$$914$$ 3.71884e29 0.0302324
$$915$$ 0 0
$$916$$ 7.21547e29 0.0573274
$$917$$ 1.32064e31 1.03730
$$918$$ 2.93235e30 0.227702
$$919$$ 1.24295e31 0.954206 0.477103 0.878847i $$-0.341687\pi$$
0.477103 + 0.878847i $$0.341687\pi$$
$$920$$ 0 0
$$921$$ −9.92984e30 −0.745102
$$922$$ −2.91512e31 −2.16263
$$923$$ −2.67034e30 −0.195861
$$924$$ 1.81179e30 0.131387
$$925$$ 0 0
$$926$$ −3.28929e31 −2.33177
$$927$$ −6.41997e30 −0.449983
$$928$$ −1.02785e31 −0.712319
$$929$$ 2.24310e31 1.53703 0.768517 0.639829i $$-0.220995\pi$$
0.768517 + 0.639829i $$0.220995\pi$$
$$930$$ 0 0
$$931$$ −3.62451e30 −0.242816
$$932$$ −7.30787e29 −0.0484087
$$933$$ 1.36031e31 0.891008
$$934$$ −2.09134e31 −1.35451
$$935$$ 0 0
$$936$$ 9.53578e29 0.0603893
$$937$$ 1.10052e31 0.689176 0.344588 0.938754i $$-0.388019\pi$$
0.344588 + 0.938754i $$0.388019\pi$$
$$938$$ −8.60060e30 −0.532597
$$939$$ 1.65135e31 1.01123
$$940$$ 0 0
$$941$$ −2.38036e31 −1.42545 −0.712725 0.701444i $$-0.752539\pi$$
−0.712725 + 0.701444i $$0.752539\pi$$
$$942$$ −1.53974e31 −0.911828
$$943$$ −5.09332e30 −0.298283
$$944$$ −8.39866e30 −0.486412
$$945$$ 0 0
$$946$$ 7.95373e30 0.450520
$$947$$ −8.09762e30 −0.453610 −0.226805 0.973940i $$-0.572828\pi$$
−0.226805 + 0.973940i $$0.572828\pi$$
$$948$$ 3.56582e30 0.197548
$$949$$ −2.17660e30 −0.119257
$$950$$ 0 0
$$951$$ −1.76262e30 −0.0944631
$$952$$ 9.26593e30 0.491135
$$953$$ 3.42232e30 0.179410 0.0897048 0.995968i $$-0.471408\pi$$
0.0897048 + 0.995968i $$0.471408\pi$$
$$954$$ −1.30884e31 −0.678622
$$955$$ 0 0
$$956$$ −7.87083e30 −0.399220
$$957$$ 7.66461e30 0.384516
$$958$$ −1.20140e31 −0.596140
$$959$$ 5.53695e30 0.271754
$$960$$ 0 0
$$961$$ 2.63603e31 1.26577
$$962$$ 7.79493e29 0.0370232
$$963$$ 8.50095e30 0.399385
$$964$$ 6.63899e30 0.308527
$$965$$ 0 0
$$966$$ 1.28111e31 0.582540
$$967$$ −1.33121e31 −0.598780 −0.299390 0.954131i $$-0.596783\pi$$
−0.299390 + 0.954131i $$0.596783\pi$$
$$968$$ −6.86896e30 −0.305633
$$969$$ −6.56638e30 −0.289019
$$970$$ 0 0
$$971$$ 9.71774e30 0.418565 0.209283 0.977855i $$-0.432887\pi$$
0.209283 + 0.977855i $$0.432887\pi$$
$$972$$ 6.38091e29 0.0271886
$$973$$ −4.68806e30 −0.197610
$$974$$ −1.83881e31 −0.766773
$$975$$ 0 0
$$976$$ −2.35932e31 −0.962861
$$977$$ 1.40637e31 0.567816 0.283908 0.958851i $$-0.408369\pi$$
0.283908 + 0.958851i $$0.408369\pi$$
$$978$$ −4.93351e30 −0.197060
$$979$$ −2.66785e30 −0.105425
$$980$$ 0 0
$$981$$ −1.44240e31 −0.557906
$$982$$ −2.90414e31 −1.11134
$$983$$ 4.87291e31 1.84492 0.922458 0.386098i $$-0.126177\pi$$
0.922458 + 0.386098i $$0.126177\pi$$
$$984$$ −2.69301e30 −0.100877
$$985$$ 0 0
$$986$$ −2.88343e31 −1.05731
$$987$$ 9.01495e30 0.327064
$$988$$ 1.57074e30 0.0563841
$$989$$ 1.67410e31 0.594594
$$990$$ 0 0
$$991$$ 1.17847e30 0.0409773 0.0204887 0.999790i $$-0.493478\pi$$
0.0204887 + 0.999790i $$0.493478\pi$$
$$992$$ −3.48741e31 −1.19986
$$993$$ 1.50926e31 0.513803
$$994$$ −1.89685e31 −0.638965
$$995$$ 0 0
$$996$$ −2.07334e30 −0.0683829
$$997$$ −5.05438e31 −1.64956 −0.824781 0.565453i $$-0.808701\pi$$
−0.824781 + 0.565453i $$0.808701\pi$$
$$998$$ 8.85229e30 0.285881
$$999$$ −7.09089e29 −0.0226602
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 75.22.a.a.1.1 1
5.2 odd 4 75.22.b.b.49.1 2
5.3 odd 4 75.22.b.b.49.2 2
5.4 even 2 3.22.a.b.1.1 1
15.14 odd 2 9.22.a.a.1.1 1
20.19 odd 2 48.22.a.d.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
3.22.a.b.1.1 1 5.4 even 2
9.22.a.a.1.1 1 15.14 odd 2
48.22.a.d.1.1 1 20.19 odd 2
75.22.a.a.1.1 1 1.1 even 1 trivial
75.22.b.b.49.1 2 5.2 odd 4
75.22.b.b.49.2 2 5.3 odd 4