# Properties

 Label 75.22.b.b.49.2 Level $75$ Weight $22$ Character 75.49 Analytic conductor $209.608$ Analytic rank $0$ Dimension $2$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [75,22,Mod(49,75)]

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, 1]))

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

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

chi := DirichletCharacter("75.49");

S:= CuspForms(chi, 22);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$75 = 3 \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$22$$ Character orbit: $$[\chi]$$ $$=$$ 75.b (of order $$2$$, degree $$1$$, not minimal)

## 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: no Analytic conductor: $$209.608008215$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(i)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 3) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 49.2 Root $$1.00000i$$ of defining polynomial Character $$\chi$$ $$=$$ 75.49 Dual form 75.22.b.b.49.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+1728.00i q^{2} +59049.0i q^{3} -888832. q^{4} -1.02037e8 q^{6} +5.38430e8i q^{7} +2.08798e9i q^{8} -3.48678e9 q^{9} +O(q^{10})$$ $$q+1728.00i q^{2} +59049.0i q^{3} -888832. q^{4} -1.02037e8 q^{6} +5.38430e8i q^{7} +2.08798e9i q^{8} -3.48678e9 q^{9} -6.41130e10 q^{11} -5.24846e10i q^{12} +1.30980e11i q^{13} -9.30407e11 q^{14} -5.47204e12 q^{16} +8.24203e12i q^{17} -6.02516e12i q^{18} -1.34921e13 q^{19} -3.17937e13 q^{21} -1.10787e14i q^{22} +2.33185e14i q^{23} -1.23293e14 q^{24} -2.26334e14 q^{26} -2.05891e14i q^{27} -4.78574e14i q^{28} +2.02456e15 q^{29} -6.86919e15 q^{31} -5.07688e15i q^{32} -3.78581e15i q^{33} -1.42422e16 q^{34} +3.09917e15 q^{36} +3.44400e15i q^{37} -2.33144e16i q^{38} -7.73424e15 q^{39} -2.18424e16 q^{41} -5.49396e16i q^{42} +7.17928e16i q^{43} +5.69857e16 q^{44} -4.02943e17 q^{46} +2.83545e17i q^{47} -3.23118e17i q^{48} +2.68639e17 q^{49} -4.86684e17 q^{51} -1.16419e17i q^{52} +2.17229e18i q^{53} +3.55780e17 q^{54} -1.12423e18 q^{56} -7.96695e17i q^{57} +3.49844e18i q^{58} -1.53483e18 q^{59} +4.31159e18 q^{61} -1.18700e19i q^{62} -1.87739e18i q^{63} -2.70285e18 q^{64} +6.54188e18 q^{66} +9.24391e18i q^{67} -7.32578e18i q^{68} -1.37693e19 q^{69} -2.03874e19 q^{71} -7.28033e18i q^{72} -1.66178e19i q^{73} -5.95123e18 q^{74} +1.19922e19 q^{76} -3.45204e19i q^{77} -1.33648e19i q^{78} -6.79403e19 q^{79} +1.21577e19 q^{81} -3.77437e19i q^{82} -3.95037e19i q^{83} +2.82593e19 q^{84} -1.24058e20 q^{86} +1.19548e20i q^{87} -1.33867e20i q^{88} -4.16117e19 q^{89} -7.05236e19 q^{91} -2.07262e20i q^{92} -4.05619e20i q^{93} -4.89965e20 q^{94} +2.99785e20 q^{96} +5.71815e19i q^{97} +4.64209e20i q^{98} +2.23548e20 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 1777664 q^{4} - 204073344 q^{6} - 6973568802 q^{9}+O(q^{10})$$ 2 * q - 1777664 * q^4 - 204073344 * q^6 - 6973568802 * q^9 $$2 q - 1777664 q^{4} - 204073344 q^{6} - 6973568802 q^{9} - 128226080376 q^{11} - 1860813416448 q^{14} - 10944079986688 q^{16} - 26984203506040 q^{19} - 63587483465184 q^{21} - 246585903022080 q^{24} - 452667253199616 q^{26} + 40\!\cdots\!40 q^{29}+ \cdots + 44\!\cdots\!76 q^{99}+O(q^{100})$$ 2 * q - 1777664 * q^4 - 204073344 * q^6 - 6973568802 * q^9 - 128226080376 * q^11 - 1860813416448 * q^14 - 10944079986688 * q^16 - 26984203506040 * q^19 - 63587483465184 * q^21 - 246585903022080 * q^24 - 452667253199616 * q^26 + 4049124062247540 * q^29 - 13738389977403536 * q^31 - 28484454724823808 * q^34 + 6198331105419264 * q^36 - 15468488792930628 * q^39 - 43684806169250316 * q^41 + 113971443472760832 * q^44 - 805886758119545856 * q^46 + 537278411880734286 * q^49 - 973367226299838564 * q^51 + 711559752519106944 * q^54 - 2248458067362447360 * q^56 - 3069662953438136520 * q^59 + 8623179041595252124 * q^61 - 5405701776399663104 * q^64 + 13083762505171548672 * q^66 - 27538661562616356048 * q^69 - 40774722512809521456 * q^71 - 11902457457887516928 * q^74 + 23984423570680545280 * q^76 - 135880609491015255760 * q^79 + 24315330918113857602 * q^81 + 56518590103326425088 * q^84 - 248115974909646260736 * q^86 - 83223352373679388980 * q^89 - 141047188789442493376 * q^91 - 979930550310873919488 * q^94 + 599569380248886706176 * q^96 + 447096696856409014776 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/75\mathbb{Z}\right)^\times$$.

 $$n$$ $$26$$ $$52$$ $$\chi(n)$$ $$1$$ $$-1$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 1728.00i 1.19324i 0.802523 + 0.596621i $$0.203491\pi$$
−0.802523 + 0.596621i $$0.796509\pi$$
$$3$$ 59049.0i 0.577350i
$$4$$ −888832. −0.423828
$$5$$ 0 0
$$6$$ −1.02037e8 −0.688919
$$7$$ 5.38430e8i 0.720443i 0.932867 + 0.360222i $$0.117299\pi$$
−0.932867 + 0.360222i $$0.882701\pi$$
$$8$$ 2.08798e9i 0.687513i
$$9$$ −3.48678e9 −0.333333
$$10$$ 0 0
$$11$$ −6.41130e10 −0.745286 −0.372643 0.927975i $$-0.621548\pi$$
−0.372643 + 0.927975i $$0.621548\pi$$
$$12$$ − 5.24846e10i − 0.244697i
$$13$$ 1.30980e11i 0.263512i 0.991282 + 0.131756i $$0.0420615\pi$$
−0.991282 + 0.131756i $$0.957938\pi$$
$$14$$ −9.30407e11 −0.859663
$$15$$ 0 0
$$16$$ −5.47204e12 −1.24420
$$17$$ 8.24203e12i 0.991563i 0.868447 + 0.495782i $$0.165118\pi$$
−0.868447 + 0.495782i $$0.834882\pi$$
$$18$$ − 6.02516e12i − 0.397748i
$$19$$ −1.34921e13 −0.504855 −0.252428 0.967616i $$-0.581229\pi$$
−0.252428 + 0.967616i $$0.581229\pi$$
$$20$$ 0 0
$$21$$ −3.17937e13 −0.415948
$$22$$ − 1.10787e14i − 0.889308i
$$23$$ 2.33185e14i 1.17370i 0.809695 + 0.586851i $$0.199633\pi$$
−0.809695 + 0.586851i $$0.800367\pi$$
$$24$$ −1.23293e14 −0.396936
$$25$$ 0 0
$$26$$ −2.26334e14 −0.314434
$$27$$ − 2.05891e14i − 0.192450i
$$28$$ − 4.78574e14i − 0.305344i
$$29$$ 2.02456e15 0.893618 0.446809 0.894629i $$-0.352560\pi$$
0.446809 + 0.894629i $$0.352560\pi$$
$$30$$ 0 0
$$31$$ −6.86919e15 −1.50525 −0.752624 0.658451i $$-0.771212\pi$$
−0.752624 + 0.658451i $$0.771212\pi$$
$$32$$ − 5.07688e15i − 0.797117i
$$33$$ − 3.78581e15i − 0.430291i
$$34$$ −1.42422e16 −1.18318
$$35$$ 0 0
$$36$$ 3.09917e15 0.141276
$$37$$ 3.44400e15i 0.117746i 0.998265 + 0.0588728i $$0.0187506\pi$$
−0.998265 + 0.0588728i $$0.981249\pi$$
$$38$$ − 2.33144e16i − 0.602415i
$$39$$ −7.73424e15 −0.152139
$$40$$ 0 0
$$41$$ −2.18424e16 −0.254138 −0.127069 0.991894i $$-0.540557\pi$$
−0.127069 + 0.991894i $$0.540557\pi$$
$$42$$ − 5.49396e16i − 0.496327i
$$43$$ 7.17928e16i 0.506597i 0.967388 + 0.253298i $$0.0815154\pi$$
−0.967388 + 0.253298i $$0.918485\pi$$
$$44$$ 5.69857e16 0.315873
$$45$$ 0 0
$$46$$ −4.02943e17 −1.40051
$$47$$ 2.83545e17i 0.786310i 0.919472 + 0.393155i $$0.128617\pi$$
−0.919472 + 0.393155i $$0.871383\pi$$
$$48$$ − 3.23118e17i − 0.718338i
$$49$$ 2.68639e17 0.480962
$$50$$ 0 0
$$51$$ −4.86684e17 −0.572479
$$52$$ − 1.16419e17i − 0.111684i
$$53$$ 2.17229e18i 1.70616i 0.521779 + 0.853081i $$0.325269\pi$$
−0.521779 + 0.853081i $$0.674731\pi$$
$$54$$ 3.55780e17 0.229640
$$55$$ 0 0
$$56$$ −1.12423e18 −0.495314
$$57$$ − 7.96695e17i − 0.291478i
$$58$$ 3.49844e18i 1.06630i
$$59$$ −1.53483e18 −0.390944 −0.195472 0.980709i $$-0.562624\pi$$
−0.195472 + 0.980709i $$0.562624\pi$$
$$60$$ 0 0
$$61$$ 4.31159e18 0.773881 0.386940 0.922105i $$-0.373532\pi$$
0.386940 + 0.922105i $$0.373532\pi$$
$$62$$ − 1.18700e19i − 1.79613i
$$63$$ − 1.87739e18i − 0.240148i
$$64$$ −2.70285e18 −0.293044
$$65$$ 0 0
$$66$$ 6.54188e18 0.513442
$$67$$ 9.24391e18i 0.619541i 0.950811 + 0.309771i $$0.100252\pi$$
−0.950811 + 0.309771i $$0.899748\pi$$
$$68$$ − 7.32578e18i − 0.420252i
$$69$$ −1.37693e19 −0.677637
$$70$$ 0 0
$$71$$ −2.03874e19 −0.743273 −0.371636 0.928378i $$-0.621203\pi$$
−0.371636 + 0.928378i $$0.621203\pi$$
$$72$$ − 7.28033e18i − 0.229171i
$$73$$ − 1.66178e19i − 0.452566i −0.974062 0.226283i $$-0.927343\pi$$
0.974062 0.226283i $$-0.0726575\pi$$
$$74$$ −5.95123e18 −0.140499
$$75$$ 0 0
$$76$$ 1.19922e19 0.213972
$$77$$ − 3.45204e19i − 0.536936i
$$78$$ − 1.33648e19i − 0.181538i
$$79$$ −6.79403e19 −0.807315 −0.403658 0.914910i $$-0.632261\pi$$
−0.403658 + 0.914910i $$0.632261\pi$$
$$80$$ 0 0
$$81$$ 1.21577e19 0.111111
$$82$$ − 3.77437e19i − 0.303249i
$$83$$ − 3.95037e19i − 0.279459i −0.990190 0.139730i $$-0.955377\pi$$
0.990190 0.139730i $$-0.0446233\pi$$
$$84$$ 2.82593e19 0.176290
$$85$$ 0 0
$$86$$ −1.24058e20 −0.604493
$$87$$ 1.19548e20i 0.515931i
$$88$$ − 1.33867e20i − 0.512394i
$$89$$ −4.16117e19 −0.141456 −0.0707278 0.997496i $$-0.522532\pi$$
−0.0707278 + 0.997496i $$0.522532\pi$$
$$90$$ 0 0
$$91$$ −7.05236e19 −0.189845
$$92$$ − 2.07262e20i − 0.497448i
$$93$$ − 4.05619e20i − 0.869055i
$$94$$ −4.89965e20 −0.938259
$$95$$ 0 0
$$96$$ 2.99785e20 0.460216
$$97$$ 5.71815e19i 0.0787322i 0.999225 + 0.0393661i $$0.0125339\pi$$
−0.999225 + 0.0393661i $$0.987466\pi$$
$$98$$ 4.64209e20i 0.573904i
$$99$$ 2.23548e20 0.248429
$$100$$ 0 0
$$101$$ 4.32417e20 0.389518 0.194759 0.980851i $$-0.437607\pi$$
0.194759 + 0.980851i $$0.437607\pi$$
$$102$$ − 8.40989e20i − 0.683107i
$$103$$ − 1.84123e21i − 1.34995i −0.737841 0.674974i $$-0.764155\pi$$
0.737841 0.674974i $$-0.235845\pi$$
$$104$$ −2.73483e20 −0.181168
$$105$$ 0 0
$$106$$ −3.75371e21 −2.03587
$$107$$ − 2.43805e21i − 1.19815i −0.800691 0.599077i $$-0.795534\pi$$
0.800691 0.599077i $$-0.204466\pi$$
$$108$$ 1.83003e20i 0.0815658i
$$109$$ 4.13676e21 1.67372 0.836859 0.547418i $$-0.184389\pi$$
0.836859 + 0.547418i $$0.184389\pi$$
$$110$$ 0 0
$$111$$ −2.03365e20 −0.0679805
$$112$$ − 2.94631e21i − 0.896374i
$$113$$ − 3.47910e21i − 0.964146i −0.876131 0.482073i $$-0.839884\pi$$
0.876131 0.482073i $$-0.160116\pi$$
$$114$$ 1.37669e21 0.347804
$$115$$ 0 0
$$116$$ −1.79950e21 −0.378741
$$117$$ − 4.56699e20i − 0.0878373i
$$118$$ − 2.65219e21i − 0.466491i
$$119$$ −4.43775e21 −0.714365
$$120$$ 0 0
$$121$$ −3.28977e21 −0.444548
$$122$$ 7.45043e21i 0.923428i
$$123$$ − 1.28977e21i − 0.146727i
$$124$$ 6.10556e21 0.637966
$$125$$ 0 0
$$126$$ 3.24413e21 0.286554
$$127$$ 1.37141e21i 0.111488i 0.998445 + 0.0557438i $$0.0177530\pi$$
−0.998445 + 0.0557438i $$0.982247\pi$$
$$128$$ − 1.53175e22i − 1.14679i
$$129$$ −4.23929e21 −0.292484
$$130$$ 0 0
$$131$$ −2.45276e22 −1.43981 −0.719907 0.694071i $$-0.755815\pi$$
−0.719907 + 0.694071i $$0.755815\pi$$
$$132$$ 3.36495e21i 0.182370i
$$133$$ − 7.26455e21i − 0.363719i
$$134$$ −1.59735e22 −0.739263
$$135$$ 0 0
$$136$$ −1.72092e22 −0.681713
$$137$$ 1.02835e22i 0.377204i 0.982054 + 0.188602i $$0.0603955\pi$$
−0.982054 + 0.188602i $$0.939604\pi$$
$$138$$ − 2.37934e22i − 0.808586i
$$139$$ −8.70692e21 −0.274289 −0.137145 0.990551i $$-0.543793\pi$$
−0.137145 + 0.990551i $$0.543793\pi$$
$$140$$ 0 0
$$141$$ −1.67430e22 −0.453977
$$142$$ − 3.52294e22i − 0.886905i
$$143$$ − 8.39753e21i − 0.196392i
$$144$$ 1.90798e22 0.414733
$$145$$ 0 0
$$146$$ 2.87155e22 0.540022
$$147$$ 1.58629e22i 0.277683i
$$148$$ − 3.06114e21i − 0.0499039i
$$149$$ 9.03997e22 1.37313 0.686564 0.727069i $$-0.259118\pi$$
0.686564 + 0.727069i $$0.259118\pi$$
$$150$$ 0 0
$$151$$ −4.75206e22 −0.627514 −0.313757 0.949503i $$-0.601588\pi$$
−0.313757 + 0.949503i $$0.601588\pi$$
$$152$$ − 2.81712e22i − 0.347094i
$$153$$ − 2.87382e22i − 0.330521i
$$154$$ 5.96512e22 0.640695
$$155$$ 0 0
$$156$$ 6.87444e21 0.0644806
$$157$$ − 1.50901e23i − 1.32356i −0.749697 0.661781i $$-0.769801\pi$$
0.749697 0.661781i $$-0.230199\pi$$
$$158$$ − 1.17401e23i − 0.963323i
$$159$$ −1.28271e23 −0.985053
$$160$$ 0 0
$$161$$ −1.25554e23 −0.845586
$$162$$ 2.10084e22i 0.132583i
$$163$$ 4.83503e22i 0.286042i 0.989720 + 0.143021i $$0.0456816\pi$$
−0.989720 + 0.143021i $$0.954318\pi$$
$$164$$ 1.94142e22 0.107711
$$165$$ 0 0
$$166$$ 6.82624e22 0.333462
$$167$$ 4.78731e20i 0.00219568i 0.999999 + 0.00109784i $$0.000349453\pi$$
−0.999999 + 0.00109784i $$0.999651\pi$$
$$168$$ − 6.63846e22i − 0.285970i
$$169$$ 2.29909e23 0.930562
$$170$$ 0 0
$$171$$ 4.70440e22 0.168285
$$172$$ − 6.38118e22i − 0.214710i
$$173$$ 1.61804e23i 0.512277i 0.966640 + 0.256139i $$0.0824504\pi$$
−0.966640 + 0.256139i $$0.917550\pi$$
$$174$$ −2.06580e23 −0.615631
$$175$$ 0 0
$$176$$ 3.50829e23 0.927284
$$177$$ − 9.06303e22i − 0.225712i
$$178$$ − 7.19050e22i − 0.168791i
$$179$$ 8.76377e22 0.193970 0.0969849 0.995286i $$-0.469080\pi$$
0.0969849 + 0.995286i $$0.469080\pi$$
$$180$$ 0 0
$$181$$ 9.36624e22 0.184476 0.0922381 0.995737i $$-0.470598\pi$$
0.0922381 + 0.995737i $$0.470598\pi$$
$$182$$ − 1.21865e23i − 0.226532i
$$183$$ 2.54595e23i 0.446800i
$$184$$ −4.86885e23 −0.806936
$$185$$ 0 0
$$186$$ 7.00910e23 1.03699
$$187$$ − 5.28422e23i − 0.738999i
$$188$$ − 2.52024e23i − 0.333260i
$$189$$ 1.10858e23 0.138649
$$190$$ 0 0
$$191$$ 1.20858e24 1.35340 0.676699 0.736260i $$-0.263410\pi$$
0.676699 + 0.736260i $$0.263410\pi$$
$$192$$ − 1.59601e23i − 0.169189i
$$193$$ 1.78822e24i 1.79502i 0.440997 + 0.897509i $$0.354625\pi$$
−0.440997 + 0.897509i $$0.645375\pi$$
$$194$$ −9.88096e22 −0.0939466
$$195$$ 0 0
$$196$$ −2.38775e23 −0.203845
$$197$$ 1.90963e24i 1.54545i 0.634743 + 0.772723i $$0.281106\pi$$
−0.634743 + 0.772723i $$0.718894\pi$$
$$198$$ 3.86292e23i 0.296436i
$$199$$ −1.44254e24 −1.04995 −0.524977 0.851116i $$-0.675926\pi$$
−0.524977 + 0.851116i $$0.675926\pi$$
$$200$$ 0 0
$$201$$ −5.45844e23 −0.357692
$$202$$ 7.47216e23i 0.464790i
$$203$$ 1.09008e24i 0.643801i
$$204$$ 4.32580e23 0.242633
$$205$$ 0 0
$$206$$ 3.18165e24 1.61082
$$207$$ − 8.13065e23i − 0.391234i
$$208$$ − 7.16728e23i − 0.327861i
$$209$$ 8.65020e23 0.376262
$$210$$ 0 0
$$211$$ 3.98848e24 1.56979 0.784895 0.619629i $$-0.212717\pi$$
0.784895 + 0.619629i $$0.212717\pi$$
$$212$$ − 1.93080e24i − 0.723119i
$$213$$ − 1.20385e24i − 0.429129i
$$214$$ 4.21295e24 1.42969
$$215$$ 0 0
$$216$$ 4.29896e23 0.132312
$$217$$ − 3.69858e24i − 1.08445i
$$218$$ 7.14833e24i 1.99715i
$$219$$ 9.81262e23 0.261289
$$220$$ 0 0
$$221$$ −1.07954e24 −0.261289
$$222$$ − 3.51414e23i − 0.0811172i
$$223$$ 4.62963e24i 1.01940i 0.860352 + 0.509700i $$0.170244\pi$$
−0.860352 + 0.509700i $$0.829756\pi$$
$$224$$ 2.73354e24 0.574278
$$225$$ 0 0
$$226$$ 6.01188e24 1.15046
$$227$$ − 3.43010e24i − 0.626664i −0.949644 0.313332i $$-0.898555\pi$$
0.949644 0.313332i $$-0.101445\pi$$
$$228$$ 7.08128e23i 0.123537i
$$229$$ −8.11792e23 −0.135261 −0.0676304 0.997710i $$-0.521544\pi$$
−0.0676304 + 0.997710i $$0.521544\pi$$
$$230$$ 0 0
$$231$$ 2.03839e24 0.310000
$$232$$ 4.22724e24i 0.614374i
$$233$$ − 8.22188e23i − 0.114218i −0.998368 0.0571089i $$-0.981812\pi$$
0.998368 0.0571089i $$-0.0181882\pi$$
$$234$$ 7.89177e23 0.104811
$$235$$ 0 0
$$236$$ 1.36421e24 0.165693
$$237$$ − 4.01181e24i − 0.466104i
$$238$$ − 7.66844e24i − 0.852411i
$$239$$ 8.85525e24 0.941940 0.470970 0.882149i $$-0.343904\pi$$
0.470970 + 0.882149i $$0.343904\pi$$
$$240$$ 0 0
$$241$$ 7.46934e24 0.727953 0.363977 0.931408i $$-0.381419\pi$$
0.363977 + 0.931408i $$0.381419\pi$$
$$242$$ − 5.68472e24i − 0.530454i
$$243$$ 7.17898e23i 0.0641500i
$$244$$ −3.83228e24 −0.327992
$$245$$ 0 0
$$246$$ 2.22873e24 0.175081
$$247$$ − 1.76720e24i − 0.133035i
$$248$$ − 1.43427e25i − 1.03488i
$$249$$ 2.33266e24 0.161346
$$250$$ 0 0
$$251$$ 9.46474e23 0.0601914 0.0300957 0.999547i $$-0.490419\pi$$
0.0300957 + 0.999547i $$0.490419\pi$$
$$252$$ 1.66868e24i 0.101781i
$$253$$ − 1.49502e25i − 0.874744i
$$254$$ −2.36979e24 −0.133032
$$255$$ 0 0
$$256$$ 2.08004e25 1.07535
$$257$$ − 1.91825e25i − 0.951936i −0.879463 0.475968i $$-0.842098\pi$$
0.879463 0.475968i $$-0.157902\pi$$
$$258$$ − 7.32550e24i − 0.349004i
$$259$$ −1.85435e24 −0.0848290
$$260$$ 0 0
$$261$$ −7.05921e24 −0.297873
$$262$$ − 4.23837e25i − 1.71805i
$$263$$ − 8.88429e23i − 0.0346009i −0.999850 0.0173004i $$-0.994493\pi$$
0.999850 0.0173004i $$-0.00550718\pi$$
$$264$$ 7.90469e24 0.295831
$$265$$ 0 0
$$266$$ 1.25531e25 0.434006
$$267$$ − 2.45713e24i − 0.0816694i
$$268$$ − 8.21628e24i − 0.262579i
$$269$$ 2.13847e25 0.657211 0.328605 0.944467i $$-0.393421\pi$$
0.328605 + 0.944467i $$0.393421\pi$$
$$270$$ 0 0
$$271$$ −1.56435e25 −0.444791 −0.222395 0.974957i $$-0.571388\pi$$
−0.222395 + 0.974957i $$0.571388\pi$$
$$272$$ − 4.51007e25i − 1.23370i
$$273$$ − 4.16435e24i − 0.109607i
$$274$$ −1.77699e25 −0.450095
$$275$$ 0 0
$$276$$ 1.22386e25 0.287202
$$277$$ − 8.04973e25i − 1.81863i −0.416112 0.909313i $$-0.636608\pi$$
0.416112 0.909313i $$-0.363392\pi$$
$$278$$ − 1.50456e25i − 0.327294i
$$279$$ 2.39514e25 0.501749
$$280$$ 0 0
$$281$$ 8.33171e25 1.61926 0.809632 0.586938i $$-0.199667\pi$$
0.809632 + 0.586938i $$0.199667\pi$$
$$282$$ − 2.89320e25i − 0.541704i
$$283$$ 4.46130e24i 0.0804829i 0.999190 + 0.0402415i $$0.0128127\pi$$
−0.999190 + 0.0402415i $$0.987187\pi$$
$$284$$ 1.81209e25 0.315020
$$285$$ 0 0
$$286$$ 1.45109e25 0.234343
$$287$$ − 1.17606e25i − 0.183092i
$$288$$ 1.77020e25i 0.265706i
$$289$$ 1.16088e24 0.0168020
$$290$$ 0 0
$$291$$ −3.37651e24 −0.0454560
$$292$$ 1.47704e25i 0.191810i
$$293$$ − 9.67128e25i − 1.21164i −0.795602 0.605820i $$-0.792845\pi$$
0.795602 0.605820i $$-0.207155\pi$$
$$294$$ −2.74111e25 −0.331344
$$295$$ 0 0
$$296$$ −7.19099e24 −0.0809516
$$297$$ 1.32003e25i 0.143430i
$$298$$ 1.56211e26i 1.63848i
$$299$$ −3.05426e25 −0.309284
$$300$$ 0 0
$$301$$ −3.86554e25 −0.364974
$$302$$ − 8.21155e25i − 0.748777i
$$303$$ 2.55338e25i 0.224889i
$$304$$ 7.38293e25 0.628140
$$305$$ 0 0
$$306$$ 4.96596e25 0.394392
$$307$$ 1.68163e26i 1.29056i 0.763948 + 0.645278i $$0.223258\pi$$
−0.763948 + 0.645278i $$0.776742\pi$$
$$308$$ 3.06828e25i 0.227569i
$$309$$ 1.08723e26 0.779393
$$310$$ 0 0
$$311$$ 2.30370e26 1.54327 0.771636 0.636065i $$-0.219439\pi$$
0.771636 + 0.636065i $$0.219439\pi$$
$$312$$ − 1.61489e25i − 0.104597i
$$313$$ 2.79658e26i 1.75151i 0.482759 + 0.875753i $$0.339635\pi$$
−0.482759 + 0.875753i $$0.660365\pi$$
$$314$$ 2.60756e26 1.57933
$$315$$ 0 0
$$316$$ 6.03875e25 0.342163
$$317$$ 2.98501e25i 0.163615i 0.996648 + 0.0818075i $$0.0260693\pi$$
−0.996648 + 0.0818075i $$0.973931\pi$$
$$318$$ − 2.21653e26i − 1.17541i
$$319$$ −1.29801e26 −0.666002
$$320$$ 0 0
$$321$$ 1.43964e26 0.691755
$$322$$ − 2.16957e26i − 1.00899i
$$323$$ − 1.11202e26i − 0.500596i
$$324$$ −1.08061e25 −0.0470920
$$325$$ 0 0
$$326$$ −8.35494e25 −0.341317
$$327$$ 2.44272e26i 0.966322i
$$328$$ − 4.56064e25i − 0.174723i
$$329$$ −1.52669e26 −0.566492
$$330$$ 0 0
$$331$$ 2.55594e26 0.889933 0.444967 0.895547i $$-0.353216\pi$$
0.444967 + 0.895547i $$0.353216\pi$$
$$332$$ 3.51122e25i 0.118443i
$$333$$ − 1.20085e25i − 0.0392485i
$$334$$ −8.27248e23 −0.00261998
$$335$$ 0 0
$$336$$ 1.73977e26 0.517522
$$337$$ − 4.91931e25i − 0.141837i −0.997482 0.0709187i $$-0.977407\pi$$
0.997482 0.0709187i $$-0.0225931\pi$$
$$338$$ 3.97282e26i 1.11039i
$$339$$ 2.05437e26 0.556650
$$340$$ 0 0
$$341$$ 4.40405e26 1.12184
$$342$$ 8.12921e25i 0.200805i
$$343$$ 4.45381e26i 1.06695i
$$344$$ −1.49902e26 −0.348292
$$345$$ 0 0
$$346$$ −2.79597e26 −0.611271
$$347$$ 2.98136e26i 0.632345i 0.948702 + 0.316173i $$0.102398\pi$$
−0.948702 + 0.316173i $$0.897602\pi$$
$$348$$ − 1.06258e26i − 0.218666i
$$349$$ −7.72834e26 −1.54319 −0.771595 0.636115i $$-0.780541\pi$$
−0.771595 + 0.636115i $$0.780541\pi$$
$$350$$ 0 0
$$351$$ 2.69676e25 0.0507129
$$352$$ 3.25494e26i 0.594081i
$$353$$ 7.30755e26i 1.29461i 0.762233 + 0.647303i $$0.224103\pi$$
−0.762233 + 0.647303i $$0.775897\pi$$
$$354$$ 1.56609e26 0.269329
$$355$$ 0 0
$$356$$ 3.69858e25 0.0599529
$$357$$ − 2.62045e26i − 0.412439i
$$358$$ 1.51438e26i 0.231453i
$$359$$ 1.58936e25 0.0235901 0.0117951 0.999930i $$-0.496245\pi$$
0.0117951 + 0.999930i $$0.496245\pi$$
$$360$$ 0 0
$$361$$ −5.32173e26 −0.745121
$$362$$ 1.61849e26i 0.220125i
$$363$$ − 1.94258e26i − 0.256660i
$$364$$ 6.26836e25 0.0804618
$$365$$ 0 0
$$366$$ −4.39940e26 −0.533141
$$367$$ − 1.40734e27i − 1.65732i −0.559752 0.828660i $$-0.689104\pi$$
0.559752 0.828660i $$-0.310896\pi$$
$$368$$ − 1.27600e27i − 1.46032i
$$369$$ 7.61598e25 0.0847127
$$370$$ 0 0
$$371$$ −1.16962e27 −1.22919
$$372$$ 3.60527e26i 0.368330i
$$373$$ 9.30077e26i 0.923797i 0.886933 + 0.461898i $$0.152831\pi$$
−0.886933 + 0.461898i $$0.847169\pi$$
$$374$$ 9.13112e26 0.881805
$$375$$ 0 0
$$376$$ −5.92035e26 −0.540599
$$377$$ 2.65177e26i 0.235479i
$$378$$ 1.91562e26i 0.165442i
$$379$$ −2.18541e27 −1.83578 −0.917892 0.396830i $$-0.870110\pi$$
−0.917892 + 0.396830i $$0.870110\pi$$
$$380$$ 0 0
$$381$$ −8.09801e25 −0.0643674
$$382$$ 2.08843e27i 1.61493i
$$383$$ − 2.10347e27i − 1.58252i −0.611482 0.791258i $$-0.709426\pi$$
0.611482 0.791258i $$-0.290574\pi$$
$$384$$ 9.04484e26 0.662099
$$385$$ 0 0
$$386$$ −3.09004e27 −2.14189
$$387$$ − 2.50326e26i − 0.168866i
$$388$$ − 5.08247e25i − 0.0333689i
$$389$$ 2.97815e26 0.190316 0.0951582 0.995462i $$-0.469664\pi$$
0.0951582 + 0.995462i $$0.469664\pi$$
$$390$$ 0 0
$$391$$ −1.92192e27 −1.16380
$$392$$ 5.60912e26i 0.330667i
$$393$$ − 1.44833e27i − 0.831277i
$$394$$ −3.29984e27 −1.84409
$$395$$ 0 0
$$396$$ −1.98697e26 −0.105291
$$397$$ 6.36504e26i 0.328474i 0.986421 + 0.164237i $$0.0525162\pi$$
−0.986421 + 0.164237i $$0.947484\pi$$
$$398$$ − 2.49271e27i − 1.25285i
$$399$$ 4.28964e26 0.209993
$$400$$ 0 0
$$401$$ 2.43888e27 1.13286 0.566428 0.824111i $$-0.308325\pi$$
0.566428 + 0.824111i $$0.308325\pi$$
$$402$$ − 9.43218e26i − 0.426814i
$$403$$ − 8.99728e26i − 0.396651i
$$404$$ −3.84346e26 −0.165089
$$405$$ 0 0
$$406$$ −1.88367e27 −0.768211
$$407$$ − 2.20805e26i − 0.0877542i
$$408$$ − 1.01618e27i − 0.393587i
$$409$$ −5.48032e26 −0.206876 −0.103438 0.994636i $$-0.532984\pi$$
−0.103438 + 0.994636i $$0.532984\pi$$
$$410$$ 0 0
$$411$$ −6.07232e26 −0.217779
$$412$$ 1.63654e27i 0.572146i
$$413$$ − 8.26399e26i − 0.281653i
$$414$$ 1.40498e27 0.466837
$$415$$ 0 0
$$416$$ 6.64970e26 0.210050
$$417$$ − 5.14135e26i − 0.158361i
$$418$$ 1.49475e27i 0.448971i
$$419$$ 6.08246e27 1.78169 0.890844 0.454309i $$-0.150114\pi$$
0.890844 + 0.454309i $$0.150114\pi$$
$$420$$ 0 0
$$421$$ −4.05990e27 −1.13124 −0.565618 0.824667i $$-0.691362\pi$$
−0.565618 + 0.824667i $$0.691362\pi$$
$$422$$ 6.89209e27i 1.87314i
$$423$$ − 9.88659e26i − 0.262103i
$$424$$ −4.53568e27 −1.17301
$$425$$ 0 0
$$426$$ 2.08026e27 0.512055
$$427$$ 2.32149e27i 0.557537i
$$428$$ 2.16702e27i 0.507811i
$$429$$ 4.95866e26 0.113387
$$430$$ 0 0
$$431$$ −7.87214e27 −1.71428 −0.857140 0.515084i $$-0.827761\pi$$
−0.857140 + 0.515084i $$0.827761\pi$$
$$432$$ 1.12664e27i 0.239446i
$$433$$ 1.73785e27i 0.360486i 0.983622 + 0.180243i $$0.0576884\pi$$
−0.983622 + 0.180243i $$0.942312\pi$$
$$434$$ 6.39115e27 1.29401
$$435$$ 0 0
$$436$$ −3.67689e27 −0.709369
$$437$$ − 3.14615e27i − 0.592550i
$$438$$ 1.69562e27i 0.311782i
$$439$$ −8.37416e27 −1.50336 −0.751681 0.659526i $$-0.770757\pi$$
−0.751681 + 0.659526i $$0.770757\pi$$
$$440$$ 0 0
$$441$$ −9.36687e26 −0.160321
$$442$$ − 1.86545e27i − 0.311781i
$$443$$ 3.30286e25i 0.00539077i 0.999996 + 0.00269539i $$0.000857969\pi$$
−0.999996 + 0.00269539i $$0.999142\pi$$
$$444$$ 1.80757e26 0.0288120
$$445$$ 0 0
$$446$$ −7.99999e27 −1.21639
$$447$$ 5.33801e27i 0.792776i
$$448$$ − 1.45530e27i − 0.211121i
$$449$$ −5.21713e27 −0.739341 −0.369670 0.929163i $$-0.620529\pi$$
−0.369670 + 0.929163i $$0.620529\pi$$
$$450$$ 0 0
$$451$$ 1.40038e27 0.189406
$$452$$ 3.09233e27i 0.408632i
$$453$$ − 2.80604e27i − 0.362296i
$$454$$ 5.92721e27 0.747763
$$455$$ 0 0
$$456$$ 1.66348e27 0.200395
$$457$$ 2.15211e26i 0.0253363i 0.999920 + 0.0126682i $$0.00403251\pi$$
−0.999920 + 0.0126682i $$0.995967\pi$$
$$458$$ − 1.40278e27i − 0.161399i
$$459$$ 1.69696e27 0.190826
$$460$$ 0 0
$$461$$ 1.68699e28 1.81239 0.906197 0.422855i $$-0.138972\pi$$
0.906197 + 0.422855i $$0.138972\pi$$
$$462$$ 3.52234e27i 0.369906i
$$463$$ 1.90352e28i 1.95415i 0.212898 + 0.977074i $$0.431710\pi$$
−0.212898 + 0.977074i $$0.568290\pi$$
$$464$$ −1.10785e28 −1.11184
$$465$$ 0 0
$$466$$ 1.42074e27 0.136290
$$467$$ − 1.21027e28i − 1.13515i −0.823321 0.567576i $$-0.807881\pi$$
0.823321 0.567576i $$-0.192119\pi$$
$$468$$ 4.05929e26i 0.0372279i
$$469$$ −4.97720e27 −0.446344
$$470$$ 0 0
$$471$$ 8.91053e27 0.764159
$$472$$ − 3.20469e27i − 0.268779i
$$473$$ − 4.60286e27i − 0.377559i
$$474$$ 6.93240e27 0.556175
$$475$$ 0 0
$$476$$ 3.94442e27 0.302768
$$477$$ − 7.57429e27i − 0.568721i
$$478$$ 1.53019e28i 1.12396i
$$479$$ −6.95253e27 −0.499597 −0.249798 0.968298i $$-0.580364\pi$$
−0.249798 + 0.968298i $$0.580364\pi$$
$$480$$ 0 0
$$481$$ −4.51095e26 −0.0310274
$$482$$ 1.29070e28i 0.868625i
$$483$$ − 7.41382e27i − 0.488199i
$$484$$ 2.92405e27 0.188412
$$485$$ 0 0
$$486$$ −1.24053e27 −0.0765466
$$487$$ − 1.06412e28i − 0.642596i −0.946978 0.321298i $$-0.895881\pi$$
0.946978 0.321298i $$-0.104119\pi$$
$$488$$ 9.00250e27i 0.532053i
$$489$$ −2.85504e27 −0.165146
$$490$$ 0 0
$$491$$ 1.68064e28 0.931361 0.465681 0.884953i $$-0.345810\pi$$
0.465681 + 0.884953i $$0.345810\pi$$
$$492$$ 1.14639e27i 0.0621869i
$$493$$ 1.66865e28i 0.886079i
$$494$$ 3.05372e27 0.158743
$$495$$ 0 0
$$496$$ 3.75885e28 1.87283
$$497$$ − 1.09772e28i − 0.535486i
$$498$$ 4.03083e27i 0.192525i
$$499$$ 5.12285e27 0.239583 0.119792 0.992799i $$-0.461777\pi$$
0.119792 + 0.992799i $$0.461777\pi$$
$$500$$ 0 0
$$501$$ −2.82686e25 −0.00126768
$$502$$ 1.63551e27i 0.0718229i
$$503$$ − 1.99606e28i − 0.858442i −0.903200 0.429221i $$-0.858788\pi$$
0.903200 0.429221i $$-0.141212\pi$$
$$504$$ 3.91994e27 0.165105
$$505$$ 0 0
$$506$$ 2.58339e28 1.04378
$$507$$ 1.35759e28i 0.537260i
$$508$$ − 1.21895e27i − 0.0472516i
$$509$$ 2.57966e27 0.0979550 0.0489775 0.998800i $$-0.484404\pi$$
0.0489775 + 0.998800i $$0.484404\pi$$
$$510$$ 0 0
$$511$$ 8.94749e27 0.326048
$$512$$ 3.81989e27i 0.136369i
$$513$$ 2.77790e27i 0.0971594i
$$514$$ 3.31474e28 1.13589
$$515$$ 0 0
$$516$$ 3.76802e27 0.123963
$$517$$ − 1.81789e28i − 0.586026i
$$518$$ − 3.20432e27i − 0.101222i
$$519$$ −9.55436e27 −0.295763
$$520$$ 0 0
$$521$$ −2.61230e28 −0.776652 −0.388326 0.921522i $$-0.626947\pi$$
−0.388326 + 0.921522i $$0.626947\pi$$
$$522$$ − 1.21983e28i − 0.355435i
$$523$$ − 6.70750e28i − 1.91555i −0.287523 0.957774i $$-0.592832\pi$$
0.287523 0.957774i $$-0.407168\pi$$
$$524$$ 2.18009e28 0.610233
$$525$$ 0 0
$$526$$ 1.53520e27 0.0412873
$$527$$ − 5.66161e28i − 1.49255i
$$528$$ 2.07161e28i 0.535367i
$$529$$ −1.49036e28 −0.377577
$$530$$ 0 0
$$531$$ 5.35163e27 0.130315
$$532$$ 6.45696e27i 0.154155i
$$533$$ − 2.86092e27i − 0.0669684i
$$534$$ 4.24592e27 0.0974514
$$535$$ 0 0
$$536$$ −1.93011e28 −0.425943
$$537$$ 5.17492e27i 0.111988i
$$538$$ 3.69528e28i 0.784212i
$$539$$ −1.72233e28 −0.358454
$$540$$ 0 0
$$541$$ −2.15196e28 −0.430787 −0.215394 0.976527i $$-0.569103\pi$$
−0.215394 + 0.976527i $$0.569103\pi$$
$$542$$ − 2.70319e28i − 0.530743i
$$543$$ 5.53067e27i 0.106507i
$$544$$ 4.18438e28 0.790392
$$545$$ 0 0
$$546$$ 7.19599e27 0.130788
$$547$$ − 7.46789e28i − 1.33147i −0.746189 0.665734i $$-0.768118\pi$$
0.746189 0.665734i $$-0.231882\pi$$
$$548$$ − 9.14032e27i − 0.159870i
$$549$$ −1.50336e28 −0.257960
$$550$$ 0 0
$$551$$ −2.73156e28 −0.451148
$$552$$ − 2.87500e28i − 0.465884i
$$553$$ − 3.65811e28i − 0.581625i
$$554$$ 1.39099e29 2.17006
$$555$$ 0 0
$$556$$ 7.73899e27 0.116252
$$557$$ − 7.95166e28i − 1.17214i −0.810262 0.586068i $$-0.800675\pi$$
0.810262 0.586068i $$-0.199325\pi$$
$$558$$ 4.13880e28i 0.598709i
$$559$$ −9.40343e27 −0.133494
$$560$$ 0 0
$$561$$ 3.12028e28 0.426661
$$562$$ 1.43972e29i 1.93217i
$$563$$ − 5.46305e28i − 0.719609i −0.933028 0.359805i $$-0.882843\pi$$
0.933028 0.359805i $$-0.117157\pi$$
$$564$$ 1.48817e28 0.192408
$$565$$ 0 0
$$566$$ −7.70912e27 −0.0960356
$$567$$ 6.54605e27i 0.0800492i
$$568$$ − 4.25683e28i − 0.511010i
$$569$$ −9.43478e28 −1.11187 −0.555933 0.831227i $$-0.687639\pi$$
−0.555933 + 0.831227i $$0.687639\pi$$
$$570$$ 0 0
$$571$$ 8.05027e28 0.914390 0.457195 0.889367i $$-0.348854\pi$$
0.457195 + 0.889367i $$0.348854\pi$$
$$572$$ 7.46400e27i 0.0832364i
$$573$$ 7.13655e28i 0.781384i
$$574$$ 2.03223e28 0.218473
$$575$$ 0 0
$$576$$ 9.42426e27 0.0976812
$$577$$ − 1.67132e28i − 0.170104i −0.996377 0.0850519i $$-0.972894\pi$$
0.996377 0.0850519i $$-0.0271056\pi$$
$$578$$ 2.00600e27i 0.0200488i
$$579$$ −1.05592e29 −1.03635
$$580$$ 0 0
$$581$$ 2.12700e28 0.201334
$$582$$ − 5.83461e27i − 0.0542401i
$$583$$ − 1.39272e29i − 1.27158i
$$584$$ 3.46975e28 0.311145
$$585$$ 0 0
$$586$$ 1.67120e29 1.44578
$$587$$ 3.15730e28i 0.268297i 0.990961 + 0.134149i $$0.0428300\pi$$
−0.990961 + 0.134149i $$0.957170\pi$$
$$588$$ − 1.40994e28i − 0.117690i
$$589$$ 9.26799e28 0.759932
$$590$$ 0 0
$$591$$ −1.12762e29 −0.892264
$$592$$ − 1.88457e28i − 0.146499i
$$593$$ 5.48493e27i 0.0418887i 0.999781 + 0.0209443i $$0.00666728\pi$$
−0.999781 + 0.0209443i $$0.993333\pi$$
$$594$$ −2.28101e28 −0.171147
$$595$$ 0 0
$$596$$ −8.03502e28 −0.581971
$$597$$ − 8.51805e28i − 0.606191i
$$598$$ − 5.27776e28i − 0.369051i
$$599$$ −1.25621e29 −0.863136 −0.431568 0.902080i $$-0.642040\pi$$
−0.431568 + 0.902080i $$0.642040\pi$$
$$600$$ 0 0
$$601$$ 3.99325e28 0.264938 0.132469 0.991187i $$-0.457709\pi$$
0.132469 + 0.991187i $$0.457709\pi$$
$$602$$ − 6.67965e28i − 0.435503i
$$603$$ − 3.22315e28i − 0.206514i
$$604$$ 4.22378e28 0.265958
$$605$$ 0 0
$$606$$ −4.41223e28 −0.268347
$$607$$ − 2.46990e29i − 1.47638i −0.674594 0.738189i $$-0.735681\pi$$
0.674594 0.738189i $$-0.264319\pi$$
$$608$$ 6.84978e28i 0.402429i
$$609$$ −6.43684e28 −0.371699
$$610$$ 0 0
$$611$$ −3.71387e28 −0.207202
$$612$$ 2.55434e28i 0.140084i
$$613$$ − 2.63911e28i − 0.142273i −0.997467 0.0711364i $$-0.977337\pi$$
0.997467 0.0711364i $$-0.0226626\pi$$
$$614$$ −2.90585e29 −1.53995
$$615$$ 0 0
$$616$$ 7.20777e28 0.369151
$$617$$ 3.09820e29i 1.55997i 0.625800 + 0.779984i $$0.284773\pi$$
−0.625800 + 0.779984i $$0.715227\pi$$
$$618$$ 1.87873e29i 0.930005i
$$619$$ 2.50758e29 1.22040 0.610202 0.792246i $$-0.291088\pi$$
0.610202 + 0.792246i $$0.291088\pi$$
$$620$$ 0 0
$$621$$ 4.80107e28 0.225879
$$622$$ 3.98080e29i 1.84150i
$$623$$ − 2.24050e28i − 0.101911i
$$624$$ 4.23221e28 0.189291
$$625$$ 0 0
$$626$$ −4.83249e29 −2.08997
$$627$$ 5.10785e28i 0.217235i
$$628$$ 1.34125e29i 0.560963i
$$629$$ −2.83855e28 −0.116752
$$630$$ 0 0
$$631$$ −4.32770e28 −0.172167 −0.0860833 0.996288i $$-0.527435\pi$$
−0.0860833 + 0.996288i $$0.527435\pi$$
$$632$$ − 1.41858e29i − 0.555039i
$$633$$ 2.35516e29i 0.906318i
$$634$$ −5.15809e28 −0.195232
$$635$$ 0 0
$$636$$ 1.14012e29 0.417493
$$637$$ 3.51864e28i 0.126739i
$$638$$ − 2.24296e29i − 0.794702i
$$639$$ 7.10863e28 0.247758
$$640$$ 0 0
$$641$$ −8.73381e28 −0.294574 −0.147287 0.989094i $$-0.547054\pi$$
−0.147287 + 0.989094i $$0.547054\pi$$
$$642$$ 2.48770e29i 0.825431i
$$643$$ 4.72013e29i 1.54077i 0.637578 + 0.770386i $$0.279937\pi$$
−0.637578 + 0.770386i $$0.720063\pi$$
$$644$$ 1.11596e29 0.358383
$$645$$ 0 0
$$646$$ 1.92158e29 0.597332
$$647$$ 1.26799e28i 0.0387812i 0.999812 + 0.0193906i $$0.00617261\pi$$
−0.999812 + 0.0193906i $$0.993827\pi$$
$$648$$ 2.53849e28i 0.0763903i
$$649$$ 9.84027e28 0.291365
$$650$$ 0 0
$$651$$ 2.18397e29 0.626105
$$652$$ − 4.29753e28i − 0.121233i
$$653$$ 2.76226e29i 0.766790i 0.923584 + 0.383395i $$0.125245\pi$$
−0.923584 + 0.383395i $$0.874755\pi$$
$$654$$ −4.22102e29 −1.15306
$$655$$ 0 0
$$656$$ 1.19523e29 0.316198
$$657$$ 5.79425e28i 0.150855i
$$658$$ − 2.63812e29i − 0.675962i
$$659$$ −6.46511e28 −0.163034 −0.0815172 0.996672i $$-0.525977\pi$$
−0.0815172 + 0.996672i $$0.525977\pi$$
$$660$$ 0 0
$$661$$ −2.27730e29 −0.556295 −0.278147 0.960538i $$-0.589720\pi$$
−0.278147 + 0.960538i $$0.589720\pi$$
$$662$$ 4.41667e29i 1.06191i
$$663$$ − 6.37459e28i − 0.150855i
$$664$$ 8.24829e28 0.192132
$$665$$ 0 0
$$666$$ 2.07507e28 0.0468330
$$667$$ 4.72097e29i 1.04884i
$$668$$ − 4.25512e26i 0 0.000930591i
$$669$$ −2.73375e29 −0.588551
$$670$$ 0 0
$$671$$ −2.76429e29 −0.576763
$$672$$ 1.61413e29i 0.331559i
$$673$$ 3.79243e29i 0.766936i 0.923554 + 0.383468i $$0.125270\pi$$
−0.923554 + 0.383468i $$0.874730\pi$$
$$674$$ 8.50057e28 0.169246
$$675$$ 0 0
$$676$$ −2.04350e29 −0.394398
$$677$$ − 3.39717e29i − 0.645559i −0.946474 0.322780i $$-0.895383\pi$$
0.946474 0.322780i $$-0.104617\pi$$
$$678$$ 3.54995e29i 0.664218i
$$679$$ −3.07882e28 −0.0567220
$$680$$ 0 0
$$681$$ 2.02544e29 0.361805
$$682$$ 7.61020e29i 1.33863i
$$683$$ − 4.54742e29i − 0.787677i −0.919180 0.393838i $$-0.871147\pi$$
0.919180 0.393838i $$-0.128853\pi$$
$$684$$ −4.18143e28 −0.0713239
$$685$$ 0 0
$$686$$ −7.69619e29 −1.27313
$$687$$ − 4.79355e28i − 0.0780929i
$$688$$ − 3.92853e29i − 0.630306i
$$689$$ −2.84526e29 −0.449594
$$690$$ 0 0
$$691$$ −3.71838e29 −0.569947 −0.284974 0.958535i $$-0.591985\pi$$
−0.284974 + 0.958535i $$0.591985\pi$$
$$692$$ − 1.43817e29i − 0.217118i
$$693$$ 1.20365e29i 0.178979i
$$694$$ −5.15178e29 −0.754542
$$695$$ 0 0
$$696$$ −2.49614e29 −0.354709
$$697$$ − 1.80026e29i − 0.251994i
$$698$$ − 1.33546e30i − 1.84140i
$$699$$ 4.85494e28 0.0659437
$$700$$ 0 0
$$701$$ −9.37969e29 −1.23637 −0.618186 0.786032i $$-0.712132\pi$$
−0.618186 + 0.786032i $$0.712132\pi$$
$$702$$ 4.66001e28i 0.0605128i
$$703$$ − 4.64668e28i − 0.0594445i
$$704$$ 1.73288e29 0.218401
$$705$$ 0 0
$$706$$ −1.26275e30 −1.54478
$$707$$ 2.32826e29i 0.280626i
$$708$$ 8.05551e28i 0.0956629i
$$709$$ 7.54578e29 0.882914 0.441457 0.897282i $$-0.354462\pi$$
0.441457 + 0.897282i $$0.354462\pi$$
$$710$$ 0 0
$$711$$ 2.36893e29 0.269105
$$712$$ − 8.68842e28i − 0.0972525i
$$713$$ − 1.60179e30i − 1.76671i
$$714$$ 4.52814e29 0.492140
$$715$$ 0 0
$$716$$ −7.78952e28 −0.0822098
$$717$$ 5.22894e29i 0.543829i
$$718$$ 2.74641e28i 0.0281488i
$$719$$ −1.22754e30 −1.23989 −0.619944 0.784646i $$-0.712845\pi$$
−0.619944 + 0.784646i $$0.712845\pi$$
$$720$$ 0 0
$$721$$ 9.91374e29 0.972561
$$722$$ − 9.19594e29i − 0.889111i
$$723$$ 4.41057e29i 0.420284i
$$724$$ −8.32502e28 −0.0781862
$$725$$ 0 0
$$726$$ 3.35677e29 0.306258
$$727$$ 9.04407e29i 0.813303i 0.913583 + 0.406652i $$0.133304\pi$$
−0.913583 + 0.406652i $$0.866696\pi$$
$$728$$ − 1.47252e29i − 0.130521i
$$729$$ −4.23912e28 −0.0370370
$$730$$ 0 0
$$731$$ −5.91719e29 −0.502323
$$732$$ − 2.26292e29i − 0.189367i
$$733$$ 1.38874e30i 1.14559i 0.819699 + 0.572795i $$0.194141\pi$$
−0.819699 + 0.572795i $$0.805859\pi$$
$$734$$ 2.43189e30 1.97759
$$735$$ 0 0
$$736$$ 1.18385e30 0.935578
$$737$$ − 5.92655e29i − 0.461736i
$$738$$ 1.31604e29i 0.101083i
$$739$$ −2.11506e30 −1.60161 −0.800804 0.598927i $$-0.795594\pi$$
−0.800804 + 0.598927i $$0.795594\pi$$
$$740$$ 0 0
$$741$$ 1.04351e29 0.0768080
$$742$$ − 2.02111e30i − 1.46673i
$$743$$ − 2.26805e30i − 1.62282i −0.584476 0.811411i $$-0.698700\pi$$
0.584476 0.811411i $$-0.301300\pi$$
$$744$$ 8.46923e29 0.597487
$$745$$ 0 0
$$746$$ −1.60717e30 −1.10231
$$747$$ 1.37741e29i 0.0931530i
$$748$$ 4.69678e29i 0.313208i
$$749$$ 1.31272e30 0.863202
$$750$$ 0 0
$$751$$ −6.98349e29 −0.446533 −0.223266 0.974757i $$-0.571672\pi$$
−0.223266 + 0.974757i $$0.571672\pi$$
$$752$$ − 1.55157e30i − 0.978326i
$$753$$ 5.58883e28i 0.0347515i
$$754$$ −4.58226e29 −0.280984
$$755$$ 0 0
$$756$$ −9.85341e28 −0.0587635
$$757$$ − 3.79778e29i − 0.223369i −0.993744 0.111685i $$-0.964375\pi$$
0.993744 0.111685i $$-0.0356247\pi$$
$$758$$ − 3.77639e30i − 2.19054i
$$759$$ 8.82794e29 0.505034
$$760$$ 0 0
$$761$$ −7.50371e29 −0.417577 −0.208789 0.977961i $$-0.566952\pi$$
−0.208789 + 0.977961i $$0.566952\pi$$
$$762$$ − 1.39934e29i − 0.0768060i
$$763$$ 2.22736e30i 1.20582i
$$764$$ −1.07423e30 −0.573608
$$765$$ 0 0
$$766$$ 3.63479e30 1.88833
$$767$$ − 2.01032e29i − 0.103018i
$$768$$ 1.22824e30i 0.620856i
$$769$$ −2.16884e29 −0.108144 −0.0540719 0.998537i $$-0.517220\pi$$
−0.0540719 + 0.998537i $$0.517220\pi$$
$$770$$ 0 0
$$771$$ 1.13271e30 0.549600
$$772$$ − 1.58943e30i − 0.760779i
$$773$$ 3.08128e30i 1.45495i 0.686136 + 0.727473i $$0.259305\pi$$
−0.686136 + 0.727473i $$0.740695\pi$$
$$774$$ 4.32563e29 0.201498
$$775$$ 0 0
$$776$$ −1.19394e29 −0.0541294
$$777$$ − 1.09498e29i − 0.0489761i
$$778$$ 5.14625e29i 0.227094i
$$779$$ 2.94700e29 0.128303
$$780$$ 0 0
$$781$$ 1.30710e30 0.553951
$$782$$ − 3.32107e30i − 1.38870i
$$783$$ − 4.16839e29i − 0.171977i
$$784$$ −1.47000e30 −0.598412
$$785$$ 0 0
$$786$$ 2.50271e30 0.991915
$$787$$ 1.46460e30i 0.572775i 0.958114 + 0.286387i $$0.0924544\pi$$
−0.958114 + 0.286387i $$0.907546\pi$$
$$788$$ − 1.69734e30i − 0.655004i
$$789$$ 5.24608e28 0.0199768
$$790$$ 0 0
$$791$$ 1.87325e30 0.694612
$$792$$ 4.66764e29i 0.170798i
$$793$$ 5.64732e29i 0.203927i
$$794$$ −1.09988e30 −0.391949
$$795$$ 0 0
$$796$$ 1.28217e30 0.445000
$$797$$ 1.27731e30i 0.437505i 0.975780 + 0.218752i $$0.0701987\pi$$
−0.975780 + 0.218752i $$0.929801\pi$$
$$798$$ 7.41250e29i 0.250573i
$$799$$ −2.33698e30 −0.779677
$$800$$ 0 0
$$801$$ 1.45091e29 0.0471519
$$802$$ 4.21439e30i 1.35177i
$$803$$ 1.06541e30i 0.337292i
$$804$$ 4.85163e29 0.151600
$$805$$ 0 0
$$806$$ 1.55473e30 0.473301
$$807$$ 1.26275e30i 0.379441i
$$808$$ 9.02876e29i 0.267799i
$$809$$ 4.24975e30 1.24424 0.622120 0.782922i $$-0.286272\pi$$
0.622120 + 0.782922i $$0.286272\pi$$
$$810$$ 0 0
$$811$$ 2.05863e30 0.587298 0.293649 0.955913i $$-0.405130\pi$$
0.293649 + 0.955913i $$0.405130\pi$$
$$812$$ − 9.68902e29i − 0.272861i
$$813$$ − 9.23732e29i − 0.256800i
$$814$$ 3.81551e29 0.104712
$$815$$ 0 0
$$816$$ 2.66315e30 0.712278
$$817$$ − 9.68636e29i − 0.255758i
$$818$$ − 9.46999e29i − 0.246854i
$$819$$ 2.45901e29 0.0632818
$$820$$ 0 0
$$821$$ −1.80717e29 −0.0453309 −0.0226655 0.999743i $$-0.507215\pi$$
−0.0226655 + 0.999743i $$0.507215\pi$$
$$822$$ − 1.04930e30i − 0.259863i
$$823$$ 6.23532e30i 1.52462i 0.647214 + 0.762308i $$0.275934\pi$$
−0.647214 + 0.762308i $$0.724066\pi$$
$$824$$ 3.84445e30 0.928107
$$825$$ 0 0
$$826$$ 1.42802e30 0.336080
$$827$$ 3.37179e30i 0.783524i 0.920067 + 0.391762i $$0.128134\pi$$
−0.920067 + 0.391762i $$0.871866\pi$$
$$828$$ 7.22678e29i 0.165816i
$$829$$ 2.70711e29 0.0613315 0.0306658 0.999530i $$-0.490237\pi$$
0.0306658 + 0.999530i $$0.490237\pi$$
$$830$$ 0 0
$$831$$ 4.75328e30 1.04998
$$832$$ − 3.54020e29i − 0.0772205i
$$833$$ 2.21413e30i 0.476904i
$$834$$ 8.88425e29 0.188963
$$835$$ 0 0
$$836$$ −7.68857e29 −0.159470
$$837$$ 1.41431e30i 0.289685i
$$838$$ 1.05105e31i 2.12599i
$$839$$ 7.45457e30 1.48909 0.744547 0.667570i $$-0.232666\pi$$
0.744547 + 0.667570i $$0.232666\pi$$
$$840$$ 0 0
$$841$$ −1.03399e30 −0.201446
$$842$$ − 7.01551e30i − 1.34984i
$$843$$ 4.91979e30i 0.934882i
$$844$$ −3.54509e30 −0.665321
$$845$$ 0 0
$$846$$ 1.70840e30 0.312753
$$847$$ − 1.77131e30i − 0.320272i
$$848$$ − 1.18868e31i − 2.12280i
$$849$$ −2.63435e29 −0.0464668
$$850$$ 0 0
$$851$$ −8.03088e29 −0.138198
$$852$$ 1.07002e30i 0.181877i
$$853$$ 6.10653e30i 1.02525i 0.858613 + 0.512625i $$0.171327\pi$$
−0.858613 + 0.512625i $$0.828673\pi$$
$$854$$ −4.01153e30 −0.665277
$$855$$ 0 0
$$856$$ 5.09059e30 0.823746
$$857$$ − 4.08307e30i − 0.652662i −0.945256 0.326331i $$-0.894188\pi$$
0.945256 0.326331i $$-0.105812\pi$$
$$858$$ 8.56856e29i 0.135298i
$$859$$ −5.27189e30 −0.822316 −0.411158 0.911564i $$-0.634875\pi$$
−0.411158 + 0.911564i $$0.634875\pi$$
$$860$$ 0 0
$$861$$ 6.94452e29 0.105708
$$862$$ − 1.36031e31i − 2.04555i
$$863$$ − 1.05537e31i − 1.56780i −0.620884 0.783902i $$-0.713226\pi$$
0.620884 0.783902i $$-0.286774\pi$$
$$864$$ −1.04528e30 −0.153405
$$865$$ 0 0
$$866$$ −3.00300e30 −0.430147
$$867$$ 6.85488e28i 0.00970062i
$$868$$ 3.28742e30i 0.459619i
$$869$$ 4.35586e30 0.601681
$$870$$ 0 0
$$871$$ −1.21077e30 −0.163256
$$872$$ 8.63747e30i 1.15070i
$$873$$ − 1.99379e29i − 0.0262441i
$$874$$ 5.43655e30 0.707056
$$875$$ 0 0
$$876$$ −8.72177e29 −0.110742
$$877$$ 7.16069e30i 0.898378i 0.893437 + 0.449189i $$0.148287\pi$$
−0.893437 + 0.449189i $$0.851713\pi$$
$$878$$ − 1.44706e31i − 1.79388i
$$879$$ 5.71079e30 0.699541
$$880$$ 0 0
$$881$$ −1.05943e31 −1.26714 −0.633568 0.773687i $$-0.718410\pi$$
−0.633568 + 0.773687i $$0.718410\pi$$
$$882$$ − 1.61860e30i − 0.191301i
$$883$$ − 6.60744e30i − 0.771695i −0.922562 0.385848i $$-0.873909\pi$$
0.922562 0.385848i $$-0.126091\pi$$
$$884$$ 9.59531e29 0.110742
$$885$$ 0 0
$$886$$ −5.70734e28 −0.00643250
$$887$$ − 2.74467e30i − 0.305698i −0.988250 0.152849i $$-0.951155\pi$$
0.988250 0.152849i $$-0.0488447\pi$$
$$888$$ − 4.24621e29i − 0.0467374i
$$889$$ −7.38406e29 −0.0803205
$$890$$ 0 0
$$891$$ −7.79465e29 −0.0828096
$$892$$ − 4.11496e30i − 0.432051i
$$893$$ − 3.82561e30i − 0.396973i
$$894$$ −9.22409e30 −0.945974
$$895$$ 0 0
$$896$$ 8.24741e30 0.826196
$$897$$ − 1.80351e30i − 0.178565i
$$898$$ − 9.01519e30i − 0.882213i
$$899$$ −1.39071e31 −1.34512
$$900$$ 0 0
$$901$$ −1.79040e31 −1.69177
$$902$$ 2.41986e30i 0.226007i
$$903$$ − 2.28256e30i − 0.210718i
$$904$$ 7.26427e30 0.662863
$$905$$ 0 0
$$906$$ 4.84884e30 0.432307
$$907$$ 5.45568e30i 0.480809i 0.970673 + 0.240404i $$0.0772800\pi$$
−0.970673 + 0.240404i $$0.922720\pi$$
$$908$$ 3.04878e30i 0.265598i
$$909$$ −1.50774e30 −0.129839
$$910$$ 0 0
$$911$$ −8.75739e30 −0.736940 −0.368470 0.929640i $$-0.620118\pi$$
−0.368470 + 0.929640i $$0.620118\pi$$
$$912$$ 4.35955e30i 0.362657i
$$913$$ 2.53270e30i 0.208277i
$$914$$ −3.71884e29 −0.0302324
$$915$$ 0 0
$$916$$ 7.21547e29 0.0573274
$$917$$ − 1.32064e31i − 1.03730i
$$918$$ 2.93235e30i 0.227702i
$$919$$ −1.24295e31 −0.954206 −0.477103 0.878847i $$-0.658313\pi$$
−0.477103 + 0.878847i $$0.658313\pi$$
$$920$$ 0 0
$$921$$ −9.92984e30 −0.745102
$$922$$ 2.91512e31i 2.16263i
$$923$$ − 2.67034e30i − 0.195861i
$$924$$ −1.81179e30 −0.131387
$$925$$ 0 0
$$926$$ −3.28929e31 −2.33177
$$927$$ 6.41997e30i 0.449983i
$$928$$ − 1.02785e31i − 0.712319i
$$929$$ −2.24310e31 −1.53703 −0.768517 0.639829i $$-0.779005\pi$$
−0.768517 + 0.639829i $$0.779005\pi$$
$$930$$ 0 0
$$931$$ −3.62451e30 −0.242816
$$932$$ 7.30787e29i 0.0484087i
$$933$$ 1.36031e31i 0.891008i
$$934$$ 2.09134e31 1.35451
$$935$$ 0 0
$$936$$ 9.53578e29 0.0603893
$$937$$ − 1.10052e31i − 0.689176i −0.938754 0.344588i $$-0.888019\pi$$
0.938754 0.344588i $$-0.111981\pi$$
$$938$$ − 8.60060e30i − 0.532597i
$$939$$ −1.65135e31 −1.01123
$$940$$ 0 0
$$941$$ −2.38036e31 −1.42545 −0.712725 0.701444i $$-0.752539\pi$$
−0.712725 + 0.701444i $$0.752539\pi$$
$$942$$ 1.53974e31i 0.911828i
$$943$$ − 5.09332e30i − 0.298283i
$$944$$ 8.39866e30 0.486412
$$945$$ 0 0
$$946$$ 7.95373e30 0.450520
$$947$$ 8.09762e30i 0.453610i 0.973940 + 0.226805i $$0.0728280\pi$$
−0.973940 + 0.226805i $$0.927172\pi$$
$$948$$ 3.56582e30i 0.197548i
$$949$$ 2.17660e30 0.119257
$$950$$ 0 0
$$951$$ −1.76262e30 −0.0944631
$$952$$ − 9.26593e30i − 0.491135i
$$953$$ 3.42232e30i 0.179410i 0.995968 + 0.0897048i $$0.0285924\pi$$
−0.995968 + 0.0897048i $$0.971408\pi$$
$$954$$ 1.30884e31 0.678622
$$955$$ 0 0
$$956$$ −7.87083e30 −0.399220
$$957$$ − 7.66461e30i − 0.384516i
$$958$$ − 1.20140e31i − 0.596140i
$$959$$ −5.53695e30 −0.271754
$$960$$ 0 0
$$961$$ 2.63603e31 1.26577
$$962$$ − 7.79493e29i − 0.0370232i
$$963$$ 8.50095e30i 0.399385i
$$964$$ −6.63899e30 −0.308527
$$965$$ 0 0
$$966$$ 1.28111e31 0.582540
$$967$$ 1.33121e31i 0.598780i 0.954131 + 0.299390i $$0.0967831\pi$$
−0.954131 + 0.299390i $$0.903217\pi$$
$$968$$ − 6.86896e30i − 0.305633i
$$969$$ 6.56638e30 0.289019
$$970$$ 0 0
$$971$$ 9.71774e30 0.418565 0.209283 0.977855i $$-0.432887\pi$$
0.209283 + 0.977855i $$0.432887\pi$$
$$972$$ − 6.38091e29i − 0.0271886i
$$973$$ − 4.68806e30i − 0.197610i
$$974$$ 1.83881e31 0.766773
$$975$$ 0 0
$$976$$ −2.35932e31 −0.962861
$$977$$ − 1.40637e31i − 0.567816i −0.958851 0.283908i $$-0.908369\pi$$
0.958851 0.283908i $$-0.0916311\pi$$
$$978$$ − 4.93351e30i − 0.197060i
$$979$$ 2.66785e30 0.105425
$$980$$ 0 0
$$981$$ −1.44240e31 −0.557906
$$982$$ 2.90414e31i 1.11134i
$$983$$ 4.87291e31i 1.84492i 0.386098 + 0.922458i $$0.373823\pi$$
−0.386098 + 0.922458i $$0.626177\pi$$
$$984$$ 2.69301e30 0.100877
$$985$$ 0 0
$$986$$ −2.88343e31 −1.05731
$$987$$ − 9.01495e30i − 0.327064i
$$988$$ 1.57074e30i 0.0563841i
$$989$$ −1.67410e31 −0.594594
$$990$$ 0 0
$$991$$ 1.17847e30 0.0409773 0.0204887 0.999790i $$-0.493478\pi$$
0.0204887 + 0.999790i $$0.493478\pi$$
$$992$$ 3.48741e31i 1.19986i
$$993$$ 1.50926e31i 0.513803i
$$994$$ 1.89685e31 0.638965
$$995$$ 0 0
$$996$$ −2.07334e30 −0.0683829
$$997$$ 5.05438e31i 1.64956i 0.565453 + 0.824781i $$0.308701\pi$$
−0.565453 + 0.824781i $$0.691299\pi$$
$$998$$ 8.85229e30i 0.285881i
$$999$$ 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.b.b.49.2 2
5.2 odd 4 75.22.a.a.1.1 1
5.3 odd 4 3.22.a.b.1.1 1
5.4 even 2 inner 75.22.b.b.49.1 2
15.8 even 4 9.22.a.a.1.1 1
20.3 even 4 48.22.a.d.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
3.22.a.b.1.1 1 5.3 odd 4
9.22.a.a.1.1 1 15.8 even 4
48.22.a.d.1.1 1 20.3 even 4
75.22.a.a.1.1 1 5.2 odd 4
75.22.b.b.49.1 2 5.4 even 2 inner
75.22.b.b.49.2 2 1.1 even 1 trivial