LMFDB - Embedded newform 99.3.b.a.89.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [99,3,Mod(89,99)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(99, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0]))

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("99.89");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 99 = 3^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	99.b (of order $2$, degree $1$, 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:	$2.69755461717$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.65306824704.6
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 4x^{7} - 2x^{6} + 20x^{5} + x^{4} - 40x^{3} + 36x^{2} - 12x + 27 $ x^8 - 4x^7 - 2x^6 + 20x^5 + x^4 - 40x^3 + 36x^2 - 12x + 27
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{6}\cdot 3^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			89.4
Root			$-0.136233 + 0.707107i$ of defining polynomial
Character	$\chi$	$=$	99.89
Dual form			99.3.b.a.89.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.192662i q^{2} +3.96288 q^{4} +7.62670i q^{5} -2.45638 q^{7} -1.53415i q^{8} +O(q^{10})$	\(q-0.192662i q^{2} +3.96288 q^{4} +7.62670i q^{5} -2.45638 q^{7} -1.53415i q^{8} +1.46938 q^{10} +3.31662i q^{11} +20.4775 q^{13} +0.473253i q^{14} +15.5560 q^{16} -14.2972i q^{17} -25.2249 q^{19} +30.2237i q^{20} +0.638988 q^{22} -7.64508i q^{23} -33.1666 q^{25} -3.94524i q^{26} -9.73436 q^{28} -43.4065i q^{29} -33.4706 q^{31} -9.13363i q^{32} -2.75454 q^{34} -18.7341i q^{35} +24.5717 q^{37} +4.85988i q^{38} +11.7005 q^{40} -21.8024i q^{41} -17.2744 q^{43} +13.1434i q^{44} -1.47292 q^{46} +76.8928i q^{47} -42.9662 q^{49} +6.38995i q^{50} +81.1500 q^{52} -3.42175i q^{53} -25.2949 q^{55} +3.76845i q^{56} -8.36280 q^{58} -68.7090i q^{59} +31.6154 q^{61} +6.44852i q^{62} +60.4641 q^{64} +156.176i q^{65} +30.0905 q^{67} -56.6583i q^{68} -3.60936 q^{70} -102.248i q^{71} +131.058 q^{73} -4.73404i q^{74} -99.9632 q^{76} -8.14691i q^{77} -10.6412 q^{79} +118.641i q^{80} -4.20050 q^{82} +126.454i q^{83} +109.041 q^{85} +3.32813i q^{86} +5.08819 q^{88} +114.555i q^{89} -50.3007 q^{91} -30.2965i q^{92} +14.8143 q^{94} -192.383i q^{95} -109.469 q^{97} +8.27796i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 16 q^{4} + 16 q^{7}+O(q^{10}) $ 8 * q - 16 * q^4 + 16 * q^7	$ 8 q - 16 q^{4} + 16 q^{7} - 48 q^{10} - 8 q^{13} + 104 q^{16} + 40 q^{19} - 112 q^{25} - 32 q^{28} - 56 q^{31} - 216 q^{34} + 136 q^{37} + 432 q^{40} - 104 q^{43} + 24 q^{46} - 96 q^{49} + 280 q^{52} - 432 q^{58} - 8 q^{61} - 592 q^{64} + 112 q^{67} + 168 q^{70} + 448 q^{73} - 344 q^{76} + 448 q^{79} + 504 q^{82} + 48 q^{85} - 264 q^{88} - 544 q^{91} + 360 q^{94} - 152 q^{97}+O(q^{100}) $ 8 * q - 16 * q^4 + 16 * q^7 - 48 * q^10 - 8 * q^13 + 104 * q^16 + 40 * q^19 - 112 * q^25 - 32 * q^28 - 56 * q^31 - 216 * q^34 + 136 * q^37 + 432 * q^40 - 104 * q^43 + 24 * q^46 - 96 * q^49 + 280 * q^52 - 432 * q^58 - 8 * q^61 - 592 * q^64 + 112 * q^67 + 168 * q^70 + 448 * q^73 - 344 * q^76 + 448 * q^79 + 504 * q^82 + 48 * q^85 - 264 * q^88 - 544 * q^91 + 360 * q^94 - 152 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/99\mathbb{Z}\right)^\times$.

$n$	$46$	$56$
$\chi(n)$	$1$	$-1$

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 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	− 0.192662i	− 0.0963311i	−0.998839	−	0.0481656i	$-0.984662\pi$
$2$	− 0.192662i	− 0.0963311i	0.998839	−	0.0481656i	$-0.0153375\pi$
$3$	0	0
$3$	0	0
$4$	3.96288	0.990720
$5$	7.62670i	1.52534i	0.646787	+	0.762670i	$0.276112\pi$
$5$	7.62670i	1.52534i	−0.646787	+	0.762670i	$0.723888\pi$
$6$	0	0
$7$	−2.45638	−0.350912	−0.175456	−	0.984487i	$-0.556140\pi$
$7$	−2.45638	−0.350912	−0.175456	+	0.984487i	$0.556140\pi$
$8$	− 1.53415i	− 0.191768i
$9$	0	0
$10$	1.46938	0.146938
$11$	3.31662i	0.301511i
$11$	3.31662i	0.301511i
$12$	0	0
$13$	20.4775	1.57519	0.787597	−	0.616191i	$-0.211325\pi$
$13$	20.4775	1.57519	0.787597	+	0.616191i	$0.211325\pi$
$14$	0.473253i	0.0338038i
$15$	0	0
$16$	15.5560	0.972247
$17$	− 14.2972i	− 0.841015i	−0.907289	−	0.420507i	$-0.861852\pi$
$17$	− 14.2972i	− 0.841015i	0.907289	−	0.420507i	$-0.138148\pi$
$18$	0	0
$19$	−25.2249	−1.32762	−0.663812	−	0.747899i	$-0.731063\pi$
$19$	−25.2249	−1.32762	−0.663812	+	0.747899i	$0.731063\pi$
$20$	30.2237i	1.51119i
$21$	0	0
$22$	0.638988	0.0290449
$23$	− 7.64508i	− 0.332395i	−0.986093	−	0.166197i	$-0.946851\pi$
$23$	− 7.64508i	− 0.332395i	0.986093	−	0.166197i	$-0.0531489\pi$
$24$	0	0
$25$	−33.1666	−1.32666
$26$	− 3.94524i	− 0.151740i
$27$	0	0
$28$	−9.73436	−0.347656
$29$	− 43.4065i	− 1.49678i	−0.663261	−	0.748388i	$-0.730828\pi$
$29$	− 43.4065i	− 1.49678i	0.663261	−	0.748388i	$-0.269172\pi$
$30$	0	0
$31$	−33.4706	−1.07970	−0.539848	−	0.841762i	$-0.681518\pi$
$31$	−33.4706	−1.07970	−0.539848	+	0.841762i	$0.681518\pi$
$32$	− 9.13363i	− 0.285426i
$33$	0	0
$34$	−2.75454	−0.0810159
$35$	− 18.7341i	− 0.535261i
$36$	0	0
$37$	24.5717	0.664100	0.332050	−	0.943262i	$-0.392260\pi$
$37$	24.5717	0.664100	0.332050	+	0.943262i	$0.392260\pi$
$38$	4.85988i	0.127892i
$39$	0	0
$40$	11.7005	0.292512
$41$	− 21.8024i	− 0.531766i	−0.964005	−	0.265883i	$-0.914336\pi$
$41$	− 21.8024i	− 0.531766i	0.964005	−	0.265883i	$-0.0856635\pi$
$42$	0	0
$43$	−17.2744	−0.401731	−0.200866	−	0.979619i	$-0.564375\pi$
$43$	−17.2744	−0.401731	−0.200866	+	0.979619i	$0.564375\pi$
$44$	13.1434i	0.298713i
$45$	0	0
$46$	−1.47292	−0.0320200
$47$	76.8928i	1.63602i	0.575207	+	0.818008i	$0.304922\pi$
$47$	76.8928i	1.63602i	−0.575207	+	0.818008i	$0.695078\pi$
$48$	0	0
$49$	−42.9662	−0.876861
$50$	6.38995i	0.127799i
$51$	0	0
$52$	81.1500	1.56058
$53$	− 3.42175i	− 0.0645613i	−0.999479	−	0.0322806i	$-0.989723\pi$
$53$	− 3.42175i	− 0.0645613i	0.999479	−	0.0322806i	$-0.0102770\pi$
$54$	0	0
$55$	−25.2949	−0.459908
$56$	3.76845i	0.0672938i
$57$	0	0
$58$	−8.36280	−0.144186
$59$	− 68.7090i	− 1.16456i	−0.812989	−	0.582279i	$-0.802161\pi$
$59$	− 68.7090i	− 1.16456i	0.812989	−	0.582279i	$-0.197839\pi$
$60$	0	0
$61$	31.6154	0.518285	0.259143	−	0.965839i	$-0.416560\pi$
$61$	31.6154	0.518285	0.259143	+	0.965839i	$0.416560\pi$
$62$	6.44852i	0.104008i
$63$	0	0
$64$	60.4641	0.944752
$65$	156.176i	2.40271i
$66$	0	0
$67$	30.0905	0.449112	0.224556	−	0.974461i	$-0.427907\pi$
$67$	30.0905	0.449112	0.224556	+	0.974461i	$0.427907\pi$
$68$	− 56.6583i	− 0.833210i
$69$	0	0
$70$	−3.60936	−0.0515622
$71$	− 102.248i	− 1.44011i	−0.693915	−	0.720057i	$-0.744116\pi$
$71$	− 102.248i	− 1.44011i	0.693915	−	0.720057i	$-0.255884\pi$
$72$	0	0
$73$	131.058	1.79531	0.897655	−	0.440698i	$-0.145269\pi$
$73$	131.058	1.79531	0.897655	+	0.440698i	$0.145269\pi$
$74$	− 4.73404i	− 0.0639735i
$75$	0	0
$76$	−99.9632	−1.31531
$77$	− 8.14691i	− 0.105804i
$78$	0	0
$79$	−10.6412	−0.134698	−0.0673491	−	0.997729i	$-0.521454\pi$
$79$	−10.6412	−0.134698	−0.0673491	+	0.997729i	$0.521454\pi$
$80$	118.641i	1.48301i
$81$	0	0
$82$	−4.20050	−0.0512256
$83$	126.454i	1.52355i	0.647843	+	0.761774i	$0.275671\pi$
$83$	126.454i	1.52355i	−0.647843	+	0.761774i	$0.724329\pi$
$84$	0	0
$85$	109.041	1.28283
$86$	3.32813i	0.0386992i
$87$	0	0
$88$	5.08819	0.0578203
$89$	114.555i	1.28713i	0.765390	+	0.643566i	$0.222546\pi$
$89$	114.555i	1.28713i	−0.765390	+	0.643566i	$0.777454\pi$
$90$	0	0
$91$	−50.3007	−0.552755
$92$	− 30.2965i	− 0.329310i
$93$	0	0
$94$	14.8143	0.157599
$95$	− 192.383i	− 2.02508i
$96$	0	0
$97$	−109.469	−1.12854	−0.564272	−	0.825589i	$-0.690843\pi$
$97$	−109.469	−1.12854	−0.564272	+	0.825589i	$0.690843\pi$
$98$	8.27796i	0.0844690i
$99$	0	0
$100$	−131.435	−1.31435
$101$	− 25.4237i	− 0.251720i	−0.992048	−	0.125860i	$-0.959831\pi$
$101$	− 25.4237i	− 0.251720i	0.992048	−	0.125860i	$-0.0401690\pi$
$102$	0	0
$103$	58.9485	0.572316	0.286158	−	0.958182i	$-0.407622\pi$
$103$	58.9485	0.572316	0.286158	+	0.958182i	$0.407622\pi$
$104$	− 31.4155i	− 0.302072i
$105$	0	0
$106$	−0.659241	−0.00621926
$107$	− 193.507i	− 1.80848i	−0.427028	−	0.904238i	$-0.640439\pi$
$107$	− 193.507i	− 1.80848i	0.427028	−	0.904238i	$-0.359561\pi$
$108$	0	0
$109$	−85.6390	−0.785679	−0.392839	−	0.919607i	$-0.628507\pi$
$109$	−85.6390	−0.785679	−0.392839	+	0.919607i	$0.628507\pi$
$110$	4.87337i	0.0443034i
$111$	0	0
$112$	−38.2114	−0.341173
$113$	60.6031i	0.536310i	0.963376	+	0.268155i	$0.0864140\pi$
$113$	60.6031i	0.536310i	−0.963376	+	0.268155i	$0.913586\pi$
$114$	0	0
$115$	58.3068	0.507015
$116$	− 172.015i	− 1.48289i
$117$	0	0
$118$	−13.2376	−0.112183
$119$	35.1195i	0.295122i
$120$	0	0
$121$	−11.0000	−0.0909091
$122$	− 6.09110i	− 0.0499270i
$123$	0	0
$124$	−132.640	−1.06968
$125$	− 62.2844i	− 0.498275i
$126$	0	0
$127$	−154.958	−1.22014	−0.610069	−	0.792348i	$-0.708858\pi$
$127$	−154.958	−1.22014	−0.610069	+	0.792348i	$0.708858\pi$
$128$	− 48.1837i	− 0.376435i
$129$	0	0
$130$	30.0892	0.231455
$131$	208.528i	1.59182i	0.605415	+	0.795910i	$0.293007\pi$
$131$	208.528i	1.59182i	−0.605415	+	0.795910i	$0.706993\pi$
$132$	0	0
$133$	61.9620	0.465880
$134$	− 5.79731i	− 0.0432635i
$135$	0	0
$136$	−21.9341	−0.161280
$137$	197.914i	1.44463i	0.691565	+	0.722315i	$0.256922\pi$
$137$	197.914i	1.44463i	−0.691565	+	0.722315i	$0.743078\pi$
$138$	0	0
$139$	18.9225	0.136133	0.0680665	−	0.997681i	$-0.478317\pi$
$139$	18.9225	0.136133	0.0680665	+	0.997681i	$0.478317\pi$
$140$	− 74.2411i	− 0.530294i
$141$	0	0
$142$	−19.6994	−0.138728
$143$	67.9163i	0.474939i
$144$	0	0
$145$	331.049	2.28309
$146$	− 25.2499i	− 0.172944i
$147$	0	0
$148$	97.3747	0.657937
$149$	40.7220i	0.273302i	0.990619	+	0.136651i	$0.0436339\pi$
$149$	40.7220i	0.273302i	−0.990619	+	0.136651i	$0.956366\pi$
$150$	0	0
$151$	−177.672	−1.17663	−0.588316	−	0.808631i	$-0.700209\pi$
$151$	−177.672	−1.17663	−0.588316	+	0.808631i	$0.700209\pi$
$152$	38.6986i	0.254596i
$153$	0	0
$154$	−1.56960	−0.0101922
$155$	− 255.270i	− 1.64690i
$156$	0	0
$157$	−115.968	−0.738649	−0.369324	−	0.929301i	$-0.620411\pi$
$157$	−115.968	−0.738649	−0.369324	+	0.929301i	$0.620411\pi$
$158$	2.05015i	0.0129756i
$159$	0	0
$160$	69.6595	0.435372
$161$	18.7793i	0.116641i
$162$	0	0
$163$	10.2020	0.0625892	0.0312946	−	0.999510i	$-0.490037\pi$
$163$	10.2020	0.0625892	0.0312946	+	0.999510i	$0.490037\pi$
$164$	− 86.4004i	− 0.526832i
$165$	0	0
$166$	24.3630	0.146765
$167$	− 47.0263i	− 0.281595i	−0.990038	−	0.140797i	$-0.955033\pi$
$167$	− 47.0263i	− 0.281595i	0.990038	−	0.140797i	$-0.0449666\pi$
$168$	0	0
$169$	250.329	1.48124
$170$	− 21.0081i	− 0.123577i
$171$	0	0
$172$	−68.4565	−0.398003
$173$	166.304i	0.961292i	0.876915	+	0.480646i	$0.159598\pi$
$173$	166.304i	0.961292i	−0.876915	+	0.480646i	$0.840402\pi$
$174$	0	0
$175$	81.4700	0.465543
$176$	51.5933i	0.293144i
$177$	0	0
$178$	22.0704	0.123991
$179$	156.259i	0.872957i	0.899715	+	0.436479i	$0.143775\pi$
$179$	156.259i	0.872957i	−0.899715	+	0.436479i	$0.856225\pi$
$180$	0	0
$181$	86.0399	0.475359	0.237679	−	0.971344i	$-0.423613\pi$
$181$	86.0399	0.475359	0.237679	+	0.971344i	$0.423613\pi$
$182$	9.69104i	0.0532475i
$183$	0	0
$184$	−11.7287	−0.0637428
$185$	187.401i	1.01298i
$186$	0	0
$187$	47.4186	0.253575
$188$	304.717i	1.62083i
$189$	0	0
$190$	−37.0649	−0.195078
$191$	14.8142i	0.0775615i	0.999248	+	0.0387807i	$0.0123474\pi$
$191$	14.8142i	0.0775615i	−0.999248	+	0.0387807i	$0.987653\pi$
$192$	0	0
$193$	80.0888	0.414968	0.207484	−	0.978238i	$-0.433473\pi$
$193$	80.0888	0.414968	0.207484	+	0.978238i	$0.433473\pi$
$194$	21.0905i	0.108714i
$195$	0	0
$196$	−170.270	−0.868724
$197$	− 245.361i	− 1.24549i	−0.782426	−	0.622743i	$-0.786018\pi$
$197$	− 245.361i	− 1.24549i	0.782426	−	0.622743i	$-0.213982\pi$
$198$	0	0
$199$	67.4910	0.339151	0.169575	−	0.985517i	$-0.445760\pi$
$199$	67.4910	0.339151	0.169575	+	0.985517i	$0.445760\pi$
$200$	50.8824i	0.254412i
$201$	0	0
$202$	−4.89819	−0.0242485
$203$	106.623i	0.525237i
$204$	0	0
$205$	166.281	0.811125
$206$	− 11.3572i	− 0.0551318i
$207$	0	0
$208$	318.547	1.53148
$209$	− 83.6614i	− 0.400294i
$210$	0	0
$211$	−351.934	−1.66793	−0.833967	−	0.551814i	$-0.813936\pi$
$211$	−351.934	−1.66793	−0.833967	+	0.551814i	$0.813936\pi$
$212$	− 13.5600i	− 0.0639622i
$213$	0	0
$214$	−37.2815	−0.174213
$215$	− 131.747i	− 0.612777i
$216$	0	0
$217$	82.2166	0.378878
$218$	16.4994i	0.0756853i
$219$	0	0
$220$	−100.241	−0.455640
$221$	− 292.772i	− 1.32476i
$222$	0	0
$223$	274.780	1.23220	0.616098	−	0.787669i	$-0.288712\pi$
$223$	274.780	1.23220	0.616098	+	0.787669i	$0.288712\pi$
$224$	22.4357i	0.100159i
$225$	0	0
$226$	11.6759	0.0516634
$227$	− 205.054i	− 0.903323i	−0.892189	−	0.451662i	$-0.850831\pi$
$227$	− 205.054i	− 0.903323i	0.892189	−	0.451662i	$-0.149169\pi$
$228$	0	0
$229$	−60.3823	−0.263678	−0.131839	−	0.991271i	$-0.542088\pi$
$229$	−60.3823	−0.263678	−0.131839	+	0.991271i	$0.542088\pi$
$230$	− 11.2335i	− 0.0488413i
$231$	0	0
$232$	−66.5920	−0.287034
$233$	− 60.9371i	− 0.261533i	−0.991413	−	0.130766i	$-0.958256\pi$
$233$	− 60.9371i	− 0.261533i	0.991413	−	0.130766i	$-0.0417438\pi$
$234$	0	0
$235$	−586.438	−2.49548
$236$	− 272.285i	− 1.15375i
$237$	0	0
$238$	6.76621	0.0284295
$239$	371.303i	1.55357i	0.629767	+	0.776784i	$0.283150\pi$
$239$	371.303i	1.55357i	−0.629767	+	0.776784i	$0.716850\pi$
$240$	0	0
$241$	−323.523	−1.34242	−0.671211	−	0.741267i	$-0.734225\pi$
$241$	−323.523	−1.34242	−0.671211	+	0.741267i	$0.734225\pi$
$242$	2.11928i	0.00875737i
$243$	0	0
$244$	125.288	0.513476
$245$	− 327.690i	− 1.33751i
$246$	0	0
$247$	−516.543	−2.09127
$248$	51.3488i	0.207051i
$249$	0	0
$250$	−11.9999	−0.0479994
$251$	− 160.951i	− 0.641241i	−0.947208	−	0.320620i	$-0.896109\pi$
$251$	− 160.951i	− 0.641241i	0.947208	−	0.320620i	$-0.103891\pi$
$252$	0	0
$253$	25.3559	0.100221
$254$	29.8545i	0.117537i
$255$	0	0
$256$	232.573	0.908489
$257$	22.6233i	0.0880284i	0.999031	+	0.0440142i	$0.0140147\pi$
$257$	22.6233i	0.0880284i	−0.999031	+	0.0440142i	$0.985985\pi$
$258$	0	0
$259$	−60.3575	−0.233041
$260$	618.907i	2.38041i
$261$	0	0
$262$	40.1755	0.153342
$263$	− 39.0014i	− 0.148294i	−0.997247	−	0.0741472i	$-0.976377\pi$
$263$	− 39.0014i	− 0.148294i	0.997247	−	0.0741472i	$-0.0236235\pi$
$264$	0	0
$265$	26.0967	0.0984780
$266$	− 11.9377i	− 0.0448787i
$267$	0	0
$268$	119.245	0.444945
$269$	− 79.9847i	− 0.297341i	−0.988887	−	0.148670i	$-0.952501\pi$
$269$	− 79.9847i	− 0.297341i	0.988887	−	0.148670i	$-0.0474993\pi$
$270$	0	0
$271$	−203.564	−0.751158	−0.375579	−	0.926790i	$-0.622556\pi$
$271$	−203.564	−0.751158	−0.375579	+	0.926790i	$0.622556\pi$
$272$	− 222.407i	− 0.817674i
$273$	0	0
$274$	38.1306	0.139163
$275$	− 110.001i	− 0.400004i
$276$	0	0
$277$	−443.367	−1.60060	−0.800302	−	0.599598i	$-0.795327\pi$
$277$	−443.367	−1.60060	−0.800302	+	0.599598i	$0.795327\pi$
$278$	− 3.64565i	− 0.0131139i
$279$	0	0
$280$	−28.7409	−0.102646
$281$	258.628i	0.920385i	0.887819	+	0.460192i	$0.152220\pi$
$281$	258.628i	0.920385i	−0.887819	+	0.460192i	$0.847780\pi$
$282$	0	0
$283$	189.227	0.668648	0.334324	−	0.942458i	$-0.391492\pi$
$283$	189.227	0.668648	0.334324	+	0.942458i	$0.391492\pi$
$284$	− 405.197i	− 1.42675i
$285$	0	0
$286$	13.0849	0.0457514
$287$	53.5551i	0.186603i
$288$	0	0
$289$	84.5887	0.292694
$290$	− 63.7806i	− 0.219933i
$291$	0	0
$292$	519.366	1.77865
$293$	349.594i	1.19315i	0.802556	+	0.596576i	$0.203473\pi$
$293$	349.594i	1.19315i	−0.802556	+	0.596576i	$0.796527\pi$
$294$	0	0
$295$	524.023	1.77635
$296$	− 37.6966i	− 0.127353i
$297$	0	0
$298$	7.84560	0.0263275
$299$	− 156.552i	− 0.523586i
$300$	0	0
$301$	42.4327	0.140972
$302$	34.2306i	0.113346i
$303$	0	0
$304$	−392.397	−1.29078
$305$	241.121i	0.790562i
$306$	0	0
$307$	105.879	0.344883	0.172442	−	0.985020i	$-0.444834\pi$
$307$	105.879	0.344883	0.172442	+	0.985020i	$0.444834\pi$
$308$	− 32.2852i	− 0.104822i
$309$	0	0
$310$	−49.1809	−0.158648
$311$	− 423.614i	− 1.36210i	−0.732236	−	0.681051i	$-0.761523\pi$
$311$	− 423.614i	− 1.36210i	0.732236	−	0.681051i	$-0.238477\pi$
$312$	0	0
$313$	213.445	0.681934	0.340967	−	0.940075i	$-0.389245\pi$
$313$	213.445	0.681934	0.340967	+	0.940075i	$0.389245\pi$
$314$	22.3426i	0.0711548i
$315$	0	0
$316$	−42.1696	−0.133448
$317$	− 211.598i	− 0.667503i	−0.942661	−	0.333751i	$-0.891685\pi$
$317$	− 211.598i	− 0.667503i	0.942661	−	0.333751i	$-0.108315\pi$
$318$	0	0
$319$	143.963	0.451295
$320$	461.142i	1.44107i
$321$	0	0
$322$	3.61805	0.0112362
$323$	360.646i	1.11655i
$324$	0	0
$325$	−679.170	−2.08975
$326$	− 1.96555i	− 0.00602929i
$327$	0	0
$328$	−33.4481	−0.101976
$329$	− 188.878i	− 0.574098i
$330$	0	0
$331$	47.5307	0.143597	0.0717987	−	0.997419i	$-0.477126\pi$
$331$	47.5307	0.143597	0.0717987	+	0.997419i	$0.477126\pi$
$332$	501.124i	1.50941i
$333$	0	0
$334$	−9.06020	−0.0271263
$335$	229.492i	0.685049i
$336$	0	0
$337$	121.159	0.359524	0.179762	−	0.983710i	$-0.442467\pi$
$337$	121.159	0.359524	0.179762	+	0.983710i	$0.442467\pi$
$338$	− 48.2289i	− 0.142689i
$339$	0	0
$340$	432.116	1.27093
$341$	− 111.009i	− 0.325541i
$342$	0	0
$343$	225.904	0.658613
$344$	26.5015i	0.0770393i
$345$	0	0
$346$	32.0404	0.0926023
$347$	192.577i	0.554978i	0.960729	+	0.277489i	$0.0895023\pi$
$347$	192.577i	0.554978i	−0.960729	+	0.277489i	$0.910498\pi$
$348$	0	0
$349$	−261.888	−0.750396	−0.375198	−	0.926945i	$-0.622425\pi$
$349$	−261.888	−0.750396	−0.375198	+	0.926945i	$0.622425\pi$
$350$	− 15.6962i	− 0.0448462i
$351$	0	0
$352$	30.2928	0.0860592
$353$	− 424.807i	− 1.20342i	−0.798715	−	0.601709i	$-0.794487\pi$
$353$	− 424.807i	− 1.20342i	0.798715	−	0.601709i	$-0.205513\pi$
$354$	0	0
$355$	779.816	2.19667
$356$	453.967i	1.27519i
$357$	0	0
$358$	30.1053	0.0840930
$359$	− 176.407i	− 0.491384i	−0.969348	−	0.245692i	$-0.920985\pi$
$359$	− 176.407i	− 0.491384i	0.969348	−	0.245692i	$-0.0790153\pi$
$360$	0	0
$361$	275.294	0.762588
$362$	− 16.5766i	− 0.0457918i
$363$	0	0
$364$	−199.336	−0.547625
$365$	999.538i	2.73846i
$366$	0	0
$367$	−20.0741	−0.0546979	−0.0273489	−	0.999626i	$-0.508707\pi$
$367$	−20.0741	−0.0546979	−0.0273489	+	0.999626i	$0.508707\pi$
$368$	− 118.926i	− 0.323170i
$369$	0	0
$370$	36.1051	0.0975813
$371$	8.40513i	0.0226553i
$372$	0	0
$373$	−78.8064	−0.211277	−0.105639	−	0.994405i	$-0.533689\pi$
$373$	−78.8064	−0.211277	−0.105639	+	0.994405i	$0.533689\pi$
$374$	− 9.13577i	− 0.0244272i
$375$	0	0
$376$	117.965	0.313736
$377$	− 888.858i	− 2.35771i
$378$	0	0
$379$	601.156	1.58616	0.793082	−	0.609116i	$-0.208475\pi$
$379$	601.156	1.58616	0.793082	+	0.609116i	$0.208475\pi$
$380$	− 762.390i	− 2.00629i
$381$	0	0
$382$	2.85414	0.00747158
$383$	154.201i	0.402614i	0.979528	+	0.201307i	$0.0645188\pi$
$383$	154.201i	0.402614i	−0.979528	+	0.201307i	$0.935481\pi$
$384$	0	0
$385$	62.1340	0.161387
$386$	− 15.4301i	− 0.0399743i
$387$	0	0
$388$	−433.812	−1.11807
$389$	182.994i	0.470422i	0.971944	+	0.235211i	$0.0755781\pi$
$389$	182.994i	0.470422i	−0.971944	+	0.235211i	$0.924422\pi$
$390$	0	0
$391$	−109.304	−0.279549
$392$	65.9164i	0.168154i
$393$	0	0
$394$	−47.2718	−0.119979
$395$	− 81.1569i	− 0.205461i
$396$	0	0
$397$	111.625	0.281171	0.140586	−	0.990069i	$-0.455102\pi$
$397$	111.625	0.281171	0.140586	+	0.990069i	$0.455102\pi$
$398$	− 13.0030i	− 0.0326708i
$399$	0	0
$400$	−515.938	−1.28985
$401$	225.641i	0.562695i	0.959606	+	0.281348i	$0.0907814\pi$
$401$	225.641i	0.562695i	−0.959606	+	0.281348i	$0.909219\pi$
$402$	0	0
$403$	−685.395	−1.70073
$404$	− 100.751i	− 0.249384i
$405$	0	0
$406$	20.5422	0.0505967
$407$	81.4951i	0.200234i
$408$	0	0
$409$	−305.427	−0.746764	−0.373382	−	0.927678i	$-0.621802\pi$
$409$	−305.427	−0.746764	−0.373382	+	0.927678i	$0.621802\pi$
$410$	− 32.0360i	− 0.0781365i
$411$	0	0
$412$	233.606	0.567005
$413$	168.776i	0.408658i
$414$	0	0
$415$	−964.431	−2.32393
$416$	− 187.034i	− 0.449601i
$417$	0	0
$418$	−16.1184	−0.0385608
$419$	− 163.669i	− 0.390618i	−0.980742	−	0.195309i	$-0.937429\pi$
$419$	− 163.669i	− 0.390618i	0.980742	−	0.195309i	$-0.0625709\pi$
$420$	0	0
$421$	−330.809	−0.785770	−0.392885	−	0.919588i	$-0.628523\pi$
$421$	−330.809	−0.785770	−0.392885	+	0.919588i	$0.628523\pi$
$422$	67.8044i	0.160674i
$423$	0	0
$424$	−5.24946	−0.0123808
$425$	474.191i	1.11574i
$426$	0	0
$427$	−77.6596	−0.181873
$428$	− 766.845i	− 1.79169i
$429$	0	0
$430$	−25.3827	−0.0590295
$431$	− 339.558i	− 0.787837i	−0.919145	−	0.393919i	$-0.871119\pi$
$431$	− 339.558i	− 0.787837i	0.919145	−	0.393919i	$-0.128881\pi$
$432$	0	0
$433$	604.853	1.39689	0.698444	−	0.715665i	$-0.253876\pi$
$433$	604.853	1.39689	0.698444	+	0.715665i	$0.253876\pi$
$434$	− 15.8400i	− 0.0364978i
$435$	0	0
$436$	−339.377	−0.778388
$437$	192.846i	0.441296i
$438$	0	0
$439$	319.471	0.727724	0.363862	−	0.931453i	$-0.381458\pi$
$439$	319.471	0.727724	0.363862	+	0.931453i	$0.381458\pi$
$440$	38.8061i	0.0881957i
$441$	0	0
$442$	−56.4061	−0.127616
$443$	− 129.981i	− 0.293411i	−0.989180	−	0.146705i	$-0.953133\pi$
$443$	− 129.981i	− 0.293411i	0.989180	−	0.146705i	$-0.0468669\pi$
$444$	0	0
$445$	−873.675	−1.96332
$446$	− 52.9397i	− 0.118699i
$447$	0	0
$448$	−148.523	−0.331525
$449$	− 92.5002i	− 0.206014i	−0.994681	−	0.103007i	$-0.967154\pi$
$449$	− 92.5002i	− 0.206014i	0.994681	−	0.103007i	$-0.0328464\pi$
$450$	0	0
$451$	72.3104	0.160334
$452$	240.163i	0.531333i
$453$	0	0
$454$	−39.5062	−0.0870181
$455$	− 383.628i	− 0.843139i
$456$	0	0
$457$	136.208	0.298049	0.149024	−	0.988834i	$-0.452387\pi$
$457$	136.208	0.298049	0.149024	+	0.988834i	$0.452387\pi$
$458$	11.6334i	0.0254004i
$459$	0	0
$460$	231.063	0.502310
$461$	− 326.820i	− 0.708937i	−0.935068	−	0.354468i	$-0.884662\pi$
$461$	− 326.820i	− 0.708937i	0.935068	−	0.354468i	$-0.115338\pi$
$462$	0	0
$463$	547.369	1.18222	0.591112	−	0.806590i	$-0.298689\pi$
$463$	547.369	1.18222	0.591112	+	0.806590i	$0.298689\pi$
$464$	− 675.230i	− 1.45524i
$465$	0	0
$466$	−11.7403	−0.0251937
$467$	603.588i	1.29248i	0.763135	+	0.646239i	$0.223659\pi$
$467$	603.588i	1.29248i	−0.763135	+	0.646239i	$0.776341\pi$
$468$	0	0
$469$	−73.9139	−0.157599
$470$	112.985i	0.240393i
$471$	0	0
$472$	−105.410	−0.223325
$473$	− 57.2928i	− 0.121126i
$474$	0	0
$475$	836.624	1.76131
$476$	139.175i	0.292384i
$477$	0	0
$478$	71.5360	0.149657
$479$	457.810i	0.955762i	0.878425	+	0.477881i	$0.158595\pi$
$479$	457.810i	0.955762i	−0.878425	+	0.477881i	$0.841405\pi$
$480$	0	0
$481$	503.167	1.04609
$482$	62.3308i	0.129317i
$483$	0	0
$484$	−43.5917	−0.0900655
$485$	− 834.886i	− 1.72141i
$486$	0	0
$487$	665.399	1.36632	0.683161	−	0.730268i	$-0.260605\pi$
$487$	665.399	1.36632	0.683161	+	0.730268i	$0.260605\pi$
$488$	− 48.5027i	− 0.0993907i
$489$	0	0
$490$	−63.1335	−0.128844
$491$	− 52.3694i	− 0.106659i	−0.998577	−	0.0533293i	$-0.983017\pi$
$491$	− 52.3694i	− 0.106659i	0.998577	−	0.0533293i	$-0.0169833\pi$
$492$	0	0
$493$	−620.594	−1.25881
$494$	99.5183i	0.201454i
$495$	0	0
$496$	−520.667	−1.04973
$497$	251.161i	0.505354i
$498$	0	0
$499$	−489.865	−0.981694	−0.490847	−	0.871246i	$-0.663313\pi$
$499$	−489.865	−0.981694	−0.490847	+	0.871246i	$0.663313\pi$
$500$	− 246.826i	− 0.493651i
$501$	0	0
$502$	−31.0093	−0.0617714
$503$	759.327i	1.50960i	0.655957	+	0.754798i	$0.272265\pi$
$503$	759.327i	1.50960i	−0.655957	+	0.754798i	$0.727735\pi$
$504$	0	0
$505$	193.899	0.383959
$506$	− 4.88512i	− 0.00965438i
$507$	0	0
$508$	−614.079	−1.20882
$509$	362.954i	0.713073i	0.934281	+	0.356536i	$0.116042\pi$
$509$	362.954i	0.713073i	−0.934281	+	0.356536i	$0.883958\pi$
$510$	0	0
$511$	−321.928	−0.629996
$512$	− 237.543i	− 0.463951i
$513$	0	0
$514$	4.35865	0.00847987
$515$	449.583i	0.872976i
$516$	0	0
$517$	−255.024	−0.493278
$518$	11.6286i	0.0224491i
$519$	0	0
$520$	239.597	0.460763
$521$	− 184.301i	− 0.353746i	−0.984234	−	0.176873i	$-0.943402\pi$
$521$	− 184.301i	− 0.353746i	0.984234	−	0.176873i	$-0.0565981\pi$
$522$	0	0
$523$	461.562	0.882528	0.441264	−	0.897377i	$-0.354530\pi$
$523$	461.562	0.882528	0.441264	+	0.897377i	$0.354530\pi$
$524$	826.373i	1.57705i
$525$	0	0
$526$	−7.51410	−0.0142854
$527$	478.537i	0.908040i
$528$	0	0
$529$	470.553	0.889514
$530$	− 5.02784i	− 0.00948649i
$531$	0	0
$532$	245.548	0.461556
$533$	− 446.459i	− 0.837635i
$534$	0	0
$535$	1475.82	2.75854
$536$	− 46.1633i	− 0.0861255i
$537$	0	0
$538$	−15.4100	−0.0286432
$539$	− 142.503i	− 0.264383i
$540$	0	0
$541$	−200.369	−0.370367	−0.185184	−	0.982704i	$-0.559288\pi$
$541$	−200.369	−0.370367	−0.185184	+	0.982704i	$0.559288\pi$
$542$	39.2190i	0.0723598i
$543$	0	0
$544$	−130.586	−0.240047
$545$	− 653.143i	− 1.19843i
$546$	0	0
$547$	394.746	0.721657	0.360828	−	0.932632i	$-0.382494\pi$
$547$	394.746	0.721657	0.360828	+	0.932632i	$0.382494\pi$
$548$	784.311i	1.43122i
$549$	0	0
$550$	−21.1931	−0.0385329
$551$	1094.92i	1.98716i
$552$	0	0
$553$	26.1388	0.0472672
$554$	85.4201i	0.154188i
$555$	0	0
$556$	74.9876	0.134870
$557$	− 1055.65i	− 1.89524i	−0.319397	−	0.947621i	$-0.603480\pi$
$557$	− 1055.65i	− 1.89524i	0.319397	−	0.947621i	$-0.396520\pi$
$558$	0	0
$559$	−353.738	−0.632804
$560$	− 291.427i	− 0.520406i
$561$	0	0
$562$	49.8279	0.0886617
$563$	− 610.342i	− 1.08409i	−0.840350	−	0.542044i	$-0.817650\pi$
$563$	− 610.342i	− 1.08409i	0.840350	−	0.542044i	$-0.182350\pi$
$564$	0	0
$565$	−462.202	−0.818056
$566$	− 36.4570i	− 0.0644116i
$567$	0	0
$568$	−156.864	−0.276168
$569$	125.347i	0.220294i	0.993915	+	0.110147i	$0.0351321\pi$
$569$	125.347i	0.220294i	−0.993915	+	0.110147i	$0.964868\pi$
$570$	0	0
$571$	465.273	0.814838	0.407419	−	0.913241i	$-0.366429\pi$
$571$	465.273	0.814838	0.407419	+	0.913241i	$0.366429\pi$
$572$	269.144i	0.470532i
$573$	0	0
$574$	10.3180	0.0179757
$575$	253.561i	0.440976i
$576$	0	0
$577$	47.9960	0.0831820	0.0415910	−	0.999135i	$-0.486757\pi$
$577$	47.9960	0.0831820	0.0415910	+	0.999135i	$0.486757\pi$
$578$	− 16.2970i	− 0.0281956i
$579$	0	0
$580$	1311.91	2.26191
$581$	− 310.621i	− 0.534631i
$582$	0	0
$583$	11.3487	0.0194660
$584$	− 201.062i	− 0.344284i
$585$	0	0
$586$	67.3535	0.114938
$587$	811.870i	1.38308i	0.722337	+	0.691542i	$0.243068\pi$
$587$	811.870i	1.38308i	−0.722337	+	0.691542i	$0.756932\pi$
$588$	0	0
$589$	844.291	1.43343
$590$	− 100.959i	− 0.171118i
$591$	0	0
$592$	382.236	0.645669
$593$	746.682i	1.25916i	0.776936	+	0.629580i	$0.216773\pi$
$593$	746.682i	1.25916i	−0.776936	+	0.629580i	$0.783227\pi$
$594$	0	0
$595$	−267.846	−0.450162
$596$	161.377i	0.270766i
$597$	0	0
$598$	−30.1617	−0.0504376
$599$	− 699.995i	− 1.16861i	−0.811536	−	0.584303i	$-0.801368\pi$
$599$	− 699.995i	− 1.16861i	0.811536	−	0.584303i	$-0.198632\pi$
$600$	0	0
$601$	387.630	0.644976	0.322488	−	0.946574i	$-0.395481\pi$
$601$	387.630	0.644976	0.322488	+	0.946574i	$0.395481\pi$
$602$	− 8.17517i	− 0.0135800i
$603$	0	0
$604$	−704.091	−1.16571
$605$	− 83.8937i	− 0.138667i
$606$	0	0
$607$	−635.808	−1.04746	−0.523730	−	0.851884i	$-0.675460\pi$
$607$	−635.808	−1.04746	−0.523730	+	0.851884i	$0.675460\pi$
$608$	230.395i	0.378939i
$609$	0	0
$610$	46.4550	0.0761557
$611$	1574.57i	2.57704i
$612$	0	0
$613$	−996.122	−1.62499	−0.812497	−	0.582965i	$-0.801893\pi$
$613$	−996.122	−1.62499	−0.812497	+	0.582965i	$0.801893\pi$
$614$	− 20.3989i	− 0.0332230i
$615$	0	0
$616$	−12.4985	−0.0202898
$617$	− 2.79534i	− 0.00453054i	−0.999997	−	0.00226527i	$-0.999279\pi$
$617$	− 2.79534i	− 0.00453054i	0.999997	−	0.00226527i	$-0.000721059\pi$
$618$	0	0
$619$	−889.865	−1.43759	−0.718793	−	0.695224i	$-0.755305\pi$
$619$	−889.865	−1.43759	−0.718793	+	0.695224i	$0.755305\pi$
$620$	− 1011.61i	− 1.63162i
$621$	0	0
$622$	−81.6143	−0.131213
$623$	− 281.391i	− 0.451670i
$624$	0	0
$625$	−354.141	−0.566625
$626$	− 41.1229i	− 0.0656915i
$627$	0	0
$628$	−459.567	−0.731794
$629$	− 351.308i	− 0.558518i
$630$	0	0
$631$	973.037	1.54205	0.771027	−	0.636802i	$-0.219743\pi$
$631$	973.037	1.54205	0.771027	+	0.636802i	$0.219743\pi$
$632$	16.3251i	0.0258308i
$633$	0	0
$634$	−40.7670	−0.0643013
$635$	− 1181.82i	− 1.86113i
$636$	0	0
$637$	−879.841	−1.38123
$638$	− 27.7363i	− 0.0434738i
$639$	0	0
$640$	367.483	0.574192
$641$	909.627i	1.41907i	0.704668	+	0.709537i	$0.251096\pi$
$641$	909.627i	1.41907i	−0.704668	+	0.709537i	$0.748904\pi$
$642$	0	0
$643$	−816.993	−1.27059	−0.635297	−	0.772268i	$-0.719123\pi$
$643$	−816.993	−1.27059	−0.635297	+	0.772268i	$0.719123\pi$
$644$	74.4200i	0.115559i
$645$	0	0
$646$	69.4829	0.107559
$647$	− 975.225i	− 1.50730i	−0.657274	−	0.753651i	$-0.728291\pi$
$647$	− 975.225i	− 1.50730i	0.657274	−	0.753651i	$-0.271709\pi$
$648$	0	0
$649$	227.882	0.351128
$650$	130.850i	0.201308i
$651$	0	0
$652$	40.4295	0.0620084
$653$	− 1212.65i	− 1.85705i	−0.371272	−	0.928524i	$-0.621078\pi$
$653$	− 1212.65i	− 1.85705i	0.371272	−	0.928524i	$-0.378922\pi$
$654$	0	0
$655$	−1590.38	−2.42807
$656$	− 339.157i	− 0.517008i
$657$	0	0
$658$	−36.3897	−0.0553035
$659$	541.200i	0.821245i	0.911805	+	0.410622i	$0.134689\pi$
$659$	541.200i	0.821245i	−0.911805	+	0.410622i	$0.865311\pi$
$660$	0	0
$661$	48.0390	0.0726763	0.0363381	−	0.999340i	$-0.488431\pi$
$661$	48.0390	0.0726763	0.0363381	+	0.999340i	$0.488431\pi$
$662$	− 9.15737i	− 0.0138329i
$663$	0	0
$664$	194.000	0.292168
$665$	472.566i	0.710625i
$666$	0	0
$667$	−331.846	−0.497521
$668$	− 186.360i	− 0.278982i
$669$	0	0
$670$	44.2143	0.0659916
$671$	104.856i	0.156269i
$672$	0	0
$673$	662.951	0.985068	0.492534	−	0.870293i	$-0.336071\pi$
$673$	662.951	0.985068	0.492534	+	0.870293i	$0.336071\pi$
$674$	− 23.3429i	− 0.0346333i
$675$	0	0
$676$	992.023	1.46749
$677$	− 797.622i	− 1.17817i	−0.808070	−	0.589086i	$-0.799488\pi$
$677$	− 797.622i	− 1.17817i	0.808070	−	0.589086i	$-0.200512\pi$
$678$	0	0
$679$	268.897	0.396020
$680$	− 167.285i	− 0.246007i
$681$	0	0
$682$	−21.3873	−0.0313597
$683$	− 816.752i	− 1.19583i	−0.801559	−	0.597915i	$-0.795996\pi$
$683$	− 816.752i	− 1.19583i	0.801559	−	0.597915i	$-0.204004\pi$
$684$	0	0
$685$	−1509.43	−2.20355
$686$	− 43.5232i	− 0.0634449i
$687$	0	0
$688$	−268.720	−0.390582
$689$	− 70.0689i	− 0.101697i
$690$	0	0
$691$	697.023	1.00872	0.504358	−	0.863495i	$-0.331729\pi$
$691$	697.023	1.00872	0.504358	+	0.863495i	$0.331729\pi$
$692$	659.041i	0.952372i
$693$	0	0
$694$	37.1024	0.0534617
$695$	144.316i	0.207649i
$696$	0	0
$697$	−311.715	−0.447223
$698$	50.4560i	0.0722865i
$699$	0	0
$700$	322.856	0.461223
$701$	− 1184.54i	− 1.68979i	−0.534934	−	0.844894i	$-0.679664\pi$
$701$	− 1184.54i	− 1.68979i	0.534934	−	0.844894i	$-0.320336\pi$
$702$	0	0
$703$	−619.818	−0.881675
$704$	200.537i	0.284853i
$705$	0	0
$706$	−81.8442	−0.115927
$707$	62.4505i	0.0883317i
$708$	0	0
$709$	1189.88	1.67825	0.839126	−	0.543937i	$-0.183067\pi$
$709$	1189.88	1.67825	0.839126	+	0.543937i	$0.183067\pi$
$710$	− 150.241i	− 0.211607i
$711$	0	0
$712$	175.744	0.246831
$713$	255.885i	0.358885i
$714$	0	0
$715$	−517.977	−0.724444
$716$	619.237i	0.864857i
$717$	0	0
$718$	−33.9870	−0.0473356
$719$	219.672i	0.305524i	0.988263	+	0.152762i	$0.0488168\pi$
$719$	219.672i	0.305524i	−0.988263	+	0.152762i	$0.951183\pi$
$720$	0	0
$721$	−144.800	−0.200832
$722$	− 53.0388i	− 0.0734610i
$723$	0	0
$724$	340.966	0.470947
$725$	1439.65i	1.98572i
$726$	0	0
$727$	−391.450	−0.538445	−0.269223	−	0.963078i	$-0.586767\pi$
$727$	−391.450	−0.538445	−0.269223	+	0.963078i	$0.586767\pi$
$728$	77.1686i	0.106001i
$729$	0	0
$730$	192.573	0.263799
$731$	246.977i	0.337862i
$732$	0	0
$733$	−161.674	−0.220564	−0.110282	−	0.993900i	$-0.535175\pi$
$733$	−161.674	−0.220564	−0.110282	+	0.993900i	$0.535175\pi$
$734$	3.86753i	0.00526911i
$735$	0	0
$736$	−69.8273	−0.0948741
$737$	99.7990i	0.135412i
$738$	0	0
$739$	908.158	1.22890	0.614451	−	0.788955i	$-0.289378\pi$
$739$	908.158	1.22890	0.614451	+	0.788955i	$0.289378\pi$
$740$	742.648i	1.00358i
$741$	0	0
$742$	1.61935	0.00218241
$743$	− 424.124i	− 0.570826i	−0.958405	−	0.285413i	$-0.907869\pi$
$743$	− 424.124i	− 0.570826i	0.958405	−	0.285413i	$-0.0921307\pi$
$744$	0	0
$745$	−310.575	−0.416879
$746$	15.1830i	0.0203526i
$747$	0	0
$748$	187.914	0.251222
$749$	475.328i	0.634616i
$750$	0	0
$751$	−40.8598	−0.0544072	−0.0272036	−	0.999630i	$-0.508660\pi$
$751$	−40.8598	−0.0544072	−0.0272036	+	0.999630i	$0.508660\pi$
$752$	1196.14i	1.59061i
$753$	0	0
$754$	−171.249	−0.227121
$755$	− 1355.05i	− 1.79477i
$756$	0	0
$757$	−786.700	−1.03923	−0.519617	−	0.854400i	$-0.673925\pi$
$757$	−786.700	−1.03923	−0.519617	+	0.854400i	$0.673925\pi$
$758$	− 115.820i	− 0.152797i
$759$	0	0
$760$	−295.143	−0.388346
$761$	591.917i	0.777815i	0.921277	+	0.388908i	$0.127147\pi$
$761$	591.917i	0.777815i	−0.921277	+	0.388908i	$0.872853\pi$
$762$	0	0
$763$	210.362	0.275704
$764$	58.7071i	0.0768417i
$765$	0	0
$766$	29.7087	0.0387842
$767$	− 1406.99i	− 1.83441i
$768$	0	0
$769$	−662.841	−0.861952	−0.430976	−	0.902363i	$-0.641831\pi$
$769$	−662.841	−0.861952	−0.430976	+	0.902363i	$0.641831\pi$
$770$	− 11.9709i	− 0.0155466i
$771$	0	0
$772$	317.382	0.411117
$773$	194.400i	0.251487i	0.992063	+	0.125744i	$0.0401317\pi$
$773$	194.400i	0.251487i	−0.992063	+	0.125744i	$0.959868\pi$
$774$	0	0
$775$	1110.11	1.43239
$776$	167.941i	0.216419i
$777$	0	0
$778$	35.2561	0.0453163
$779$	549.963i	0.705986i
$780$	0	0
$781$	339.119	0.434211
$782$	21.0587i	0.0269293i
$783$	0	0
$784$	−668.380	−0.852525
$785$	− 884.452i	− 1.12669i
$786$	0	0
$787$	590.745	0.750629	0.375315	−	0.926898i	$-0.377535\pi$
$787$	590.745	0.750629	0.375315	+	0.926898i	$0.377535\pi$
$788$	− 972.336i	− 1.23393i
$789$	0	0
$790$	−15.6359	−0.0197922
$791$	− 148.864i	− 0.188198i
$792$	0	0
$793$	647.405	0.816400
$794$	− 21.5059i	− 0.0270855i
$795$	0	0
$796$	267.459	0.336003
$797$	− 44.3027i	− 0.0555868i	−0.999614	−	0.0277934i	$-0.991152\pi$
$797$	− 44.3027i	− 0.0555868i	0.999614	−	0.0277934i	$-0.00884805\pi$
$798$	0	0
$799$	1099.36	1.37591
$800$	302.932i	0.378665i
$801$	0	0
$802$	43.4725	0.0542051
$803$	434.669i	0.541307i
$804$	0	0
$805$	−143.224	−0.177918
$806$	132.050i	0.163833i
$807$	0	0
$808$	−39.0037	−0.0482720
$809$	642.524i	0.794220i	0.917771	+	0.397110i	$0.129987\pi$
$809$	642.524i	0.794220i	−0.917771	+	0.397110i	$0.870013\pi$
$810$	0	0
$811$	771.414	0.951189	0.475594	−	0.879665i	$-0.342233\pi$
$811$	771.414	0.951189	0.475594	+	0.879665i	$0.342233\pi$
$812$	422.535i	0.520363i
$813$	0	0
$814$	15.7010	0.0192887
$815$	77.8080i	0.0954699i
$816$	0	0
$817$	435.745	0.533348
$818$	58.8442i	0.0719366i
$819$	0	0
$820$	658.950	0.803598
$821$	− 313.109i	− 0.381375i	−0.981651	−	0.190687i	$-0.938928\pi$
$821$	− 313.109i	− 0.381375i	0.981651	−	0.190687i	$-0.0610717\pi$
$822$	0	0
$823$	−606.931	−0.737462	−0.368731	−	0.929536i	$-0.620208\pi$
$823$	−606.931	−0.737462	−0.368731	+	0.929536i	$0.620208\pi$
$824$	− 90.4356i	− 0.109752i
$825$	0	0
$826$	32.5167	0.0393665
$827$	− 297.721i	− 0.360002i	−0.983666	−	0.180001i	$-0.942390\pi$
$827$	− 297.721i	− 0.360002i	0.983666	−	0.180001i	$-0.0576101\pi$
$828$	0	0
$829$	1378.56	1.66292	0.831458	−	0.555588i	$-0.187507\pi$
$829$	1378.56	1.66292	0.831458	+	0.555588i	$0.187507\pi$
$830$	185.809i	0.223867i
$831$	0	0
$832$	1238.15	1.48817
$833$	614.298i	0.737453i
$834$	0	0
$835$	358.656	0.429528
$836$	− 331.540i	− 0.396579i
$837$	0	0
$838$	−31.5328	−0.0376286
$839$	567.982i	0.676975i	0.940971	+	0.338488i	$0.109915\pi$
$839$	567.982i	0.676975i	−0.940971	+	0.338488i	$0.890085\pi$
$840$	0	0
$841$	−1043.13	−1.24034
$842$	63.7344i	0.0756941i
$843$	0	0
$844$	−1394.67	−1.65246
$845$	1909.18i	2.25939i
$846$	0	0
$847$	27.0202	0.0319011
$848$	− 53.2285i	− 0.0627695i
$849$	0	0
$850$	91.3588	0.107481
$851$	− 187.853i	− 0.220743i
$852$	0	0
$853$	31.4201	0.0368348	0.0184174	−	0.999830i	$-0.494137\pi$
$853$	31.4201	0.0368348	0.0184174	+	0.999830i	$0.494137\pi$
$854$	14.9621i	0.0175200i
$855$	0	0
$856$	−296.868	−0.346808
$857$	478.358i	0.558178i	0.960265	+	0.279089i	$0.0900324\pi$
$857$	478.358i	0.558178i	−0.960265	+	0.279089i	$0.909968\pi$
$858$	0	0
$859$	−630.421	−0.733902	−0.366951	−	0.930240i	$-0.619598\pi$
$859$	−630.421	−0.733902	−0.366951	+	0.930240i	$0.619598\pi$
$860$	− 522.098i	− 0.607090i
$861$	0	0
$862$	−65.4200	−0.0758932
$863$	− 164.632i	− 0.190768i	−0.995441	−	0.0953838i	$-0.969592\pi$
$863$	− 164.632i	− 0.190768i	0.995441	−	0.0953838i	$-0.0304078\pi$
$864$	0	0
$865$	−1268.35	−1.46630
$866$	− 116.532i	− 0.134564i
$867$	0	0
$868$	325.815	0.375363
$869$	− 35.2927i	− 0.0406130i
$870$	0	0
$871$	616.179	0.707439
$872$	131.383i	0.150668i
$873$	0	0
$874$	37.1542	0.0425105
$875$	152.994i	0.174851i
$876$	0	0
$877$	1455.14	1.65922	0.829612	−	0.558341i	$-0.188562\pi$
$877$	1455.14	1.65922	0.829612	+	0.558341i	$0.188562\pi$
$878$	− 61.5500i	− 0.0701025i
$879$	0	0
$880$	−393.487	−0.447144
$881$	377.222i	0.428174i	0.976815	+	0.214087i	$0.0686777\pi$
$881$	377.222i	0.428174i	−0.976815	+	0.214087i	$0.931322\pi$
$882$	0	0
$883$	−1561.28	−1.76816	−0.884078	−	0.467339i	$-0.845213\pi$
$883$	−1561.28	−1.76816	−0.884078	+	0.467339i	$0.845213\pi$
$884$	− 1160.22i	− 1.31247i
$885$	0	0
$886$	−25.0424	−0.0282646
$887$	813.843i	0.917524i	0.888559	+	0.458762i	$0.151707\pi$
$887$	813.843i	0.917524i	−0.888559	+	0.458762i	$0.848293\pi$
$888$	0	0
$889$	380.636	0.428161
$890$	168.324i	0.189128i
$891$	0	0
$892$	1088.92	1.22076
$893$	− 1939.61i	− 2.17202i
$894$	0	0
$895$	−1191.74	−1.33156
$896$	118.358i	0.132096i
$897$	0	0
$898$	−17.8213	−0.0198455
$899$	1452.84i	1.61606i
$900$	0	0
$901$	−48.9216	−0.0542970
$902$	− 13.9315i	− 0.0154451i
$903$	0	0
$904$	92.9740	0.102847
$905$	656.201i	0.725084i
$906$	0	0
$907$	−486.113	−0.535957	−0.267979	−	0.963425i	$-0.586356\pi$
$907$	−486.113	−0.535957	−0.267979	+	0.963425i	$0.586356\pi$
$908$	− 812.606i	− 0.894941i
$909$	0	0
$910$	−73.9107	−0.0812205
$911$	− 868.058i	− 0.952863i	−0.879212	−	0.476432i	$-0.841930\pi$
$911$	− 868.058i	− 0.952863i	0.879212	−	0.476432i	$-0.158070\pi$
$912$	0	0
$913$	−419.402	−0.459367
$914$	− 26.2422i	− 0.0287114i
$915$	0	0
$916$	−239.288	−0.261231
$917$	− 512.226i	− 0.558589i
$918$	0	0
$919$	−828.692	−0.901732	−0.450866	−	0.892592i	$-0.648885\pi$
$919$	−828.692	−0.901732	−0.450866	+	0.892592i	$0.648885\pi$
$920$	− 89.4511i	− 0.0972295i
$921$	0	0
$922$	−62.9658	−0.0682926
$923$	− 2093.79i	− 2.26846i
$924$	0	0
$925$	−814.960	−0.881038
$926$	− 105.457i	− 0.113885i
$927$	0	0
$928$	−396.459	−0.427219
$929$	− 1250.65i	− 1.34623i	−0.739539	−	0.673114i	$-0.764956\pi$
$929$	− 1250.65i	− 1.34623i	0.739539	−	0.673114i	$-0.235044\pi$
$930$	0	0
$931$	1083.82	1.16414
$932$	− 241.487i	− 0.259106i
$933$	0	0
$934$	116.289	0.124506
$935$	361.648i	0.386789i
$936$	0	0
$937$	−628.988	−0.671278	−0.335639	−	0.941991i	$-0.608952\pi$
$937$	−628.988	−0.671278	−0.335639	+	0.941991i	$0.608952\pi$
$938$	14.2404i	0.0151817i
$939$	0	0
$940$	−2323.99	−2.47233
$941$	− 633.442i	− 0.673159i	−0.941655	−	0.336579i	$-0.890730\pi$
$941$	− 633.442i	− 0.673159i	0.941655	−	0.336579i	$-0.109270\pi$
$942$	0	0
$943$	−166.681	−0.176756
$944$	− 1068.83i	− 1.13224i
$945$	0	0
$946$	−11.0382	−0.0116682
$947$	− 929.570i	− 0.981594i	−0.871274	−	0.490797i	$-0.836706\pi$
$947$	− 929.570i	− 0.981594i	0.871274	−	0.490797i	$-0.163294\pi$
$948$	0	0
$949$	2683.74	2.82796
$950$	− 161.186i	− 0.169669i
$951$	0	0
$952$	53.8785	0.0565951
$953$	1189.77i	1.24845i	0.781244	+	0.624225i	$0.214585\pi$
$953$	1189.77i	1.24845i	−0.781244	+	0.624225i	$0.785415\pi$
$954$	0	0
$955$	−112.984	−0.118308
$956$	1471.43i	1.53915i
$957$	0	0
$958$	88.2027	0.0920696
$959$	− 486.153i	− 0.506938i
$960$	0	0
$961$	159.280	0.165744
$962$	− 96.9413i	− 0.100771i
$963$	0	0
$964$	−1282.09	−1.32996
$965$	610.813i	0.632967i
$966$	0	0
$967$	−920.308	−0.951714	−0.475857	−	0.879523i	$-0.657862\pi$
$967$	−920.308	−0.951714	−0.475857	+	0.879523i	$0.657862\pi$
$968$	16.8756i	0.0174335i
$969$	0	0
$970$	−160.851	−0.165826
$971$	295.006i	0.303816i	0.988395	+	0.151908i	$0.0485418\pi$
$971$	295.006i	0.303816i	−0.988395	+	0.151908i	$0.951458\pi$
$972$	0	0
$973$	−46.4809	−0.0477708
$974$	− 128.197i	− 0.131619i
$975$	0	0
$976$	491.808	0.503902
$977$	1373.57i	1.40591i	0.711235	+	0.702955i	$0.248136\pi$
$977$	1373.57i	1.40591i	−0.711235	+	0.702955i	$0.751864\pi$
$978$	0	0
$979$	−379.935	−0.388085
$980$	− 1298.60i	− 1.32510i
$981$	0	0
$982$	−10.0896	−0.0102745
$983$	944.469i	0.960803i	0.877049	+	0.480401i	$0.159509\pi$
$983$	944.469i	0.960803i	−0.877049	+	0.480401i	$0.840491\pi$
$984$	0	0
$985$	1871.30	1.89979
$986$	119.565i	0.121263i
$987$	0	0
$988$	−2047.00	−2.07186
$989$	132.064i	0.133533i
$990$	0	0
$991$	357.852	0.361102	0.180551	−	0.983566i	$-0.442212\pi$
$991$	357.852	0.361102	0.180551	+	0.983566i	$0.442212\pi$
$992$	305.708i	0.308173i
$993$	0	0
$994$	48.3892	0.0486813
$995$	514.734i	0.517320i
$996$	0	0
$997$	−197.472	−0.198066	−0.0990330	−	0.995084i	$-0.531575\pi$
$997$	−197.472	−0.198066	−0.0990330	+	0.995084i	$0.531575\pi$
$998$	94.3786i	0.0945677i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	99.3.b.a.89.4	✓	8
3.2	odd	2	inner	99.3.b.a.89.5	yes	8
4.3	odd	2		1584.3.i.b.881.8		8
11.10	odd	2		1089.3.b.g.485.5		8
12.11	even	2		1584.3.i.b.881.1		8
33.32	even	2		1089.3.b.g.485.4		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
99.3.b.a.89.4	✓	8	1.1	even	1	trivial
99.3.b.a.89.5	yes	8	3.2	odd	2	inner
1089.3.b.g.485.4		8	33.32	even	2
1089.3.b.g.485.5		8	11.10	odd	2
1584.3.i.b.881.1		8	12.11	even	2
1584.3.i.b.881.8		8	4.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 99 = 3^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	99.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-0.192662i q^{2} +3.96288 q^{4} +7.62670i q^{5} -2.45638 q^{7} -1.53415i q^{8} +O(q^{10})\)	\(q-0.192662i q^{2} +3.96288 q^{4} +7.62670i q^{5} -2.45638 q^{7} -1.53415i q^{8} +1.46938 q^{10} +3.31662i q^{11} +20.4775 q^{13} +0.473253i q^{14} +15.5560 q^{16} -14.2972i q^{17} -25.2249 q^{19} +30.2237i q^{20} +0.638988 q^{22} -7.64508i q^{23} -33.1666 q^{25} -3.94524i q^{26} -9.73436 q^{28} -43.4065i q^{29} -33.4706 q^{31} -9.13363i q^{32} -2.75454 q^{34} -18.7341i q^{35} +24.5717 q^{37} +4.85988i q^{38} +11.7005 q^{40} -21.8024i q^{41} -17.2744 q^{43} +13.1434i q^{44} -1.47292 q^{46} +76.8928i q^{47} -42.9662 q^{49} +6.38995i q^{50} +81.1500 q^{52} -3.42175i q^{53} -25.2949 q^{55} +3.76845i q^{56} -8.36280 q^{58} -68.7090i q^{59} +31.6154 q^{61} +6.44852i q^{62} +60.4641 q^{64} +156.176i q^{65} +30.0905 q^{67} -56.6583i q^{68} -3.60936 q^{70} -102.248i q^{71} +131.058 q^{73} -4.73404i q^{74} -99.9632 q^{76} -8.14691i q^{77} -10.6412 q^{79} +118.641i q^{80} -4.20050 q^{82} +126.454i q^{83} +109.041 q^{85} +3.32813i q^{86} +5.08819 q^{88} +114.555i q^{89} -50.3007 q^{91} -30.2965i q^{92} +14.8143 q^{94} -192.383i q^{95} -109.469 q^{97} +8.27796i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 16 q^{4} + 16 q^{7}+O(q^{10}) \) 8 * q - 16 * q^4 + 16 * q^7	\( 8 q - 16 q^{4} + 16 q^{7} - 48 q^{10} - 8 q^{13} + 104 q^{16} + 40 q^{19} - 112 q^{25} - 32 q^{28} - 56 q^{31} - 216 q^{34} + 136 q^{37} + 432 q^{40} - 104 q^{43} + 24 q^{46} - 96 q^{49} + 280 q^{52} - 432 q^{58} - 8 q^{61} - 592 q^{64} + 112 q^{67} + 168 q^{70} + 448 q^{73} - 344 q^{76} + 448 q^{79} + 504 q^{82} + 48 q^{85} - 264 q^{88} - 544 q^{91} + 360 q^{94} - 152 q^{97}+O(q^{100}) \) 8 * q - 16 * q^4 + 16 * q^7 - 48 * q^10 - 8 * q^13 + 104 * q^16 + 40 * q^19 - 112 * q^25 - 32 * q^28 - 56 * q^31 - 216 * q^34 + 136 * q^37 + 432 * q^40 - 104 * q^43 + 24 * q^46 - 96 * q^49 + 280 * q^52 - 432 * q^58 - 8 * q^61 - 592 * q^64 + 112 * q^67 + 168 * q^70 + 448 * q^73 - 344 * q^76 + 448 * q^79 + 504 * q^82 + 48 * q^85 - 264 * q^88 - 544 * q^91 + 360 * q^94 - 152 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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