LMFDB - Embedded newform 99.3.b.a.89.6

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			89.6
Root			$1.13623 + 0.707107i$ of defining polynomial
Character	$\chi$	$=$	99.89
Dual form			99.3.b.a.89.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.60688i q^{2} +1.41795 q^{4} -6.21249i q^{5} +11.1468 q^{7} +8.70597i q^{8} +O(q^{10})$	\(q+1.60688i q^{2} +1.41795 q^{4} -6.21249i q^{5} +11.1468 q^{7} +8.70597i q^{8} +9.98270 q^{10} +3.31662i q^{11} -17.7871 q^{13} +17.9115i q^{14} -8.31762 q^{16} +8.53964i q^{17} +16.4632 q^{19} -8.80900i q^{20} -5.32940 q^{22} -25.0834i q^{23} -13.5950 q^{25} -28.5817i q^{26} +15.8056 q^{28} +9.46540i q^{29} -46.1952 q^{31} +21.4585i q^{32} -13.7221 q^{34} -69.2494i q^{35} -37.4758 q^{37} +26.4543i q^{38} +54.0858 q^{40} -51.8375i q^{41} -55.6297 q^{43} +4.70281i q^{44} +40.3058 q^{46} +37.8600i q^{47} +75.2512 q^{49} -21.8455i q^{50} -25.2212 q^{52} +58.5761i q^{53} +20.6045 q^{55} +97.0437i q^{56} -15.2097 q^{58} -44.6296i q^{59} -10.1633 q^{61} -74.2300i q^{62} -67.7516 q^{64} +110.502i q^{65} -58.3755 q^{67} +12.1088i q^{68} +111.275 q^{70} +75.1766i q^{71} +112.274 q^{73} -60.2190i q^{74} +23.3440 q^{76} +36.9698i q^{77} +52.2849 q^{79} +51.6731i q^{80} +83.2963 q^{82} +18.0976i q^{83} +53.0524 q^{85} -89.3901i q^{86} -28.8744 q^{88} -136.779i q^{89} -198.269 q^{91} -35.5669i q^{92} -60.8364 q^{94} -102.278i q^{95} +127.754 q^{97} +120.919i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	1.60688i	0.803438i	0.915763	+	0.401719i	$0.131587\pi$
$2$	1.60688i	0.803438i	−0.915763	+	0.401719i	$0.868413\pi$
$3$	0	0
$3$	0	0
$4$	1.41795	0.354488
$5$	− 6.21249i	− 1.24250i	−0.783613	−	0.621249i	$-0.786626\pi$
$5$	− 6.21249i	− 1.24250i	0.783613	−	0.621249i	$-0.213374\pi$
$6$	0	0
$7$	11.1468	1.59240	0.796200	−	0.605033i	$-0.206840\pi$
$7$	11.1468	1.59240	0.796200	+	0.605033i	$0.206840\pi$
$8$	8.70597i	1.08825i
$9$	0	0
$10$	9.98270	0.998270
$11$	3.31662i	0.301511i
$11$	3.31662i	0.301511i
$12$	0	0
$13$	−17.7871	−1.36824	−0.684119	−	0.729370i	$-0.739813\pi$
$13$	−17.7871	−1.36824	−0.684119	+	0.729370i	$0.739813\pi$
$14$	17.9115i	1.27939i
$15$	0	0
$16$	−8.31762	−0.519851
$17$	8.53964i	0.502332i	0.967944	+	0.251166i	$0.0808139\pi$
$17$	8.53964i	0.502332i	−0.967944	+	0.251166i	$0.919186\pi$
$18$	0	0
$19$	16.4632	0.866485	0.433242	−	0.901277i	$-0.357369\pi$
$19$	16.4632	0.866485	0.433242	+	0.901277i	$0.357369\pi$
$20$	− 8.80900i	− 0.440450i
$21$	0	0
$22$	−5.32940	−0.242246
$23$	− 25.0834i	− 1.09058i	−0.838247	−	0.545290i	$-0.816419\pi$
$23$	− 25.0834i	− 1.09058i	0.838247	−	0.545290i	$-0.183581\pi$
$24$	0	0
$25$	−13.5950	−0.543802
$26$	− 28.5817i	− 1.09929i
$27$	0	0
$28$	15.8056	0.564486
$29$	9.46540i	0.326393i	0.986594	+	0.163197i	$0.0521805\pi$
$29$	9.46540i	0.326393i	−0.986594	+	0.163197i	$0.947820\pi$
$30$	0	0
$31$	−46.1952	−1.49017	−0.745084	−	0.666970i	$-0.767591\pi$
$31$	−46.1952	−1.49017	−0.745084	+	0.666970i	$0.767591\pi$
$32$	21.4585i	0.670579i
$33$	0	0
$34$	−13.7221	−0.403592
$35$	− 69.2494i	− 1.97855i
$36$	0	0
$37$	−37.4758	−1.01286	−0.506430	−	0.862281i	$-0.669035\pi$
$37$	−37.4758	−1.01286	−0.506430	+	0.862281i	$0.669035\pi$
$38$	26.4543i	0.696167i
$39$	0	0
$40$	54.0858	1.35214
$41$	− 51.8375i	− 1.26433i	−0.774835	−	0.632164i	$-0.782167\pi$
$41$	− 51.8375i	− 1.26433i	0.774835	−	0.632164i	$-0.217833\pi$
$42$	0	0
$43$	−55.6297	−1.29371	−0.646857	−	0.762611i	$-0.723917\pi$
$43$	−55.6297	−1.29371	−0.646857	+	0.762611i	$0.723917\pi$
$44$	4.70281i	0.106882i
$45$	0	0
$46$	40.3058	0.876214
$47$	37.8600i	0.805533i	0.915303	+	0.402766i	$0.131951\pi$
$47$	37.8600i	0.805533i	−0.915303	+	0.402766i	$0.868049\pi$
$48$	0	0
$49$	75.2512	1.53574
$50$	− 21.8455i	− 0.436911i
$51$	0	0
$52$	−25.2212	−0.485024
$53$	58.5761i	1.10521i	0.833444	+	0.552604i	$0.186366\pi$
$53$	58.5761i	1.10521i	−0.833444	+	0.552604i	$0.813634\pi$
$54$	0	0
$55$	20.6045	0.374627
$56$	97.0437i	1.73292i
$57$	0	0
$58$	−15.2097	−0.262237
$59$	− 44.6296i	− 0.756435i	−0.925717	−	0.378217i	$-0.876537\pi$
$59$	− 44.6296i	− 0.756435i	0.925717	−	0.378217i	$-0.123463\pi$
$60$	0	0
$61$	−10.1633	−0.166612	−0.0833060	−	0.996524i	$-0.526548\pi$
$61$	−10.1633	−0.166612	−0.0833060	+	0.996524i	$0.526548\pi$
$62$	− 74.2300i	− 1.19726i
$63$	0	0
$64$	−67.7516	−1.05862
$65$	110.502i	1.70003i
$66$	0	0
$67$	−58.3755	−0.871276	−0.435638	−	0.900122i	$-0.643477\pi$
$67$	−58.3755	−0.871276	−0.435638	+	0.900122i	$0.643477\pi$
$68$	12.1088i	0.178070i
$69$	0	0
$70$	111.275	1.58965
$71$	75.1766i	1.05882i	0.848365	+	0.529412i	$0.177588\pi$
$71$	75.1766i	1.05882i	−0.848365	+	0.529412i	$0.822412\pi$
$72$	0	0
$73$	112.274	1.53800	0.769000	−	0.639249i	$-0.220755\pi$
$73$	112.274	1.53800	0.769000	+	0.639249i	$0.220755\pi$
$74$	− 60.2190i	− 0.813771i
$75$	0	0
$76$	23.3440	0.307158
$77$	36.9698i	0.480127i
$78$	0	0
$79$	52.2849	0.661834	0.330917	−	0.943660i	$-0.392642\pi$
$79$	52.2849	0.661834	0.330917	+	0.943660i	$0.392642\pi$
$80$	51.6731i	0.645914i
$81$	0	0
$82$	83.2963	1.01581
$83$	18.0976i	0.218043i	0.994039	+	0.109022i	$0.0347717\pi$
$83$	18.0976i	0.218043i	−0.994039	+	0.109022i	$0.965228\pi$
$84$	0	0
$85$	53.0524	0.624146
$86$	− 89.3901i	− 1.03942i
$87$	0	0
$88$	−28.8744	−0.328119
$89$	− 136.779i	− 1.53684i	−0.639943	−	0.768422i	$-0.721042\pi$
$89$	− 136.779i	− 1.53684i	0.639943	−	0.768422i	$-0.278958\pi$
$90$	0	0
$91$	−198.269	−2.17878
$92$	− 35.5669i	− 0.386597i
$93$	0	0
$94$	−60.8364	−0.647196
$95$	− 102.278i	− 1.07661i
$96$	0	0
$97$	127.754	1.31705	0.658525	−	0.752559i	$-0.271181\pi$
$97$	127.754	1.31705	0.658525	+	0.752559i	$0.271181\pi$
$98$	120.919i	1.23387i
$99$	0	0
$100$	−19.2771	−0.192771
$101$	5.72551i	0.0566882i	0.999598	+	0.0283441i	$0.00902342\pi$
$101$	5.72551i	0.0566882i	−0.999598	+	0.0283441i	$0.990977\pi$
$102$	0	0
$103$	27.6215	0.268170	0.134085	−	0.990970i	$-0.457191\pi$
$103$	27.6215	0.268170	0.134085	+	0.990970i	$0.457191\pi$
$104$	− 154.854i	− 1.48898i
$105$	0	0
$106$	−94.1245	−0.887967
$107$	− 47.9169i	− 0.447821i	−0.974610	−	0.223911i	$-0.928118\pi$
$107$	− 47.9169i	− 0.447821i	0.974610	−	0.223911i	$-0.0718824\pi$
$108$	0	0
$109$	−79.6706	−0.730923	−0.365461	−	0.930826i	$-0.619089\pi$
$109$	−79.6706	−0.730923	−0.365461	+	0.930826i	$0.619089\pi$
$110$	33.1089i	0.300990i
$111$	0	0
$112$	−92.7148	−0.827811
$113$	214.762i	1.90055i	0.311415	+	0.950274i	$0.399197\pi$
$113$	214.762i	1.90055i	−0.311415	+	0.950274i	$0.600803\pi$
$114$	0	0
$115$	−155.830	−1.35504
$116$	13.4215i	0.115702i
$117$	0	0
$118$	71.7143	0.607748
$119$	95.1896i	0.799913i
$120$	0	0
$121$	−11.0000	−0.0909091
$122$	− 16.3312i	− 0.133862i
$123$	0	0
$124$	−65.5025	−0.528246
$125$	− 70.8532i	− 0.566826i
$126$	0	0
$127$	−15.6836	−0.123493	−0.0617465	−	0.998092i	$-0.519667\pi$
$127$	−15.6836	−0.123493	−0.0617465	+	0.998092i	$0.519667\pi$
$128$	− 23.0344i	− 0.179956i
$129$	0	0
$130$	−177.563	−1.36587
$131$	− 78.9245i	− 0.602477i	−0.953549	−	0.301239i	$-0.902600\pi$
$131$	− 78.9245i	− 0.602477i	0.953549	−	0.301239i	$-0.0974001\pi$
$132$	0	0
$133$	183.512	1.37979
$134$	− 93.8022i	− 0.700016i
$135$	0	0
$136$	−74.3458	−0.546661
$137$	− 135.084i	− 0.986017i	−0.870025	−	0.493008i	$-0.835897\pi$
$137$	− 135.084i	− 0.986017i	0.870025	−	0.493008i	$-0.164103\pi$
$138$	0	0
$139$	181.554	1.30615	0.653073	−	0.757295i	$-0.273480\pi$
$139$	181.554	1.30615	0.653073	+	0.757295i	$0.273480\pi$
$140$	− 98.1922i	− 0.701373i
$141$	0	0
$142$	−120.799	−0.850700
$143$	− 58.9932i	− 0.412540i
$144$	0	0
$145$	58.8037	0.405543
$146$	180.410i	1.23569i
$147$	0	0
$148$	−53.1389	−0.359047
$149$	123.629i	0.829725i	0.909884	+	0.414862i	$0.136170\pi$
$149$	123.629i	0.829725i	−0.909884	+	0.414862i	$0.863830\pi$
$150$	0	0
$151$	67.4578	0.446740	0.223370	−	0.974734i	$-0.428294\pi$
$151$	67.4578	0.446740	0.223370	+	0.974734i	$0.428294\pi$
$152$	143.328i	0.942949i
$153$	0	0
$154$	−59.4058	−0.385752
$155$	286.987i	1.85153i
$156$	0	0
$157$	168.395	1.07258	0.536291	−	0.844033i	$-0.319825\pi$
$157$	168.395	1.07258	0.536291	+	0.844033i	$0.319825\pi$
$158$	84.0154i	0.531743i
$159$	0	0
$160$	133.311	0.833193
$161$	− 279.599i	− 1.73664i
$162$	0	0
$163$	128.845	0.790458	0.395229	−	0.918583i	$-0.370665\pi$
$163$	128.845	0.790458	0.395229	+	0.918583i	$0.370665\pi$
$164$	− 73.5029i	− 0.448189i
$165$	0	0
$166$	−29.0806	−0.175184
$167$	− 51.8678i	− 0.310586i	−0.987868	−	0.155293i	$-0.950368\pi$
$167$	− 51.8678i	− 0.310586i	0.987868	−	0.155293i	$-0.0496322\pi$
$168$	0	0
$169$	147.381	0.872077
$170$	85.2486i	0.501463i
$171$	0	0
$172$	−78.8802	−0.458606
$173$	231.087i	1.33576i	0.744268	+	0.667881i	$0.232799\pi$
$173$	231.087i	1.33576i	−0.744268	+	0.667881i	$0.767201\pi$
$174$	0	0
$175$	−151.541	−0.865950
$176$	− 27.5864i	− 0.156741i
$177$	0	0
$178$	219.787	1.23476
$179$	− 294.651i	− 1.64609i	−0.567973	−	0.823047i	$-0.692272\pi$
$179$	− 294.651i	− 1.64609i	0.567973	−	0.823047i	$-0.307728\pi$
$180$	0	0
$181$	−157.848	−0.872090	−0.436045	−	0.899925i	$-0.643621\pi$
$181$	−157.848	−0.872090	−0.436045	+	0.899925i	$0.643621\pi$
$182$	− 318.594i	− 1.75052i
$183$	0	0
$184$	218.375	1.18682
$185$	232.818i	1.25848i
$186$	0	0
$187$	−28.3228	−0.151459
$188$	53.6837i	0.285551i
$189$	0	0
$190$	164.347	0.864986
$191$	− 215.226i	− 1.12684i	−0.826172	−	0.563419i	$-0.809486\pi$
$191$	− 215.226i	− 1.12684i	0.826172	−	0.563419i	$-0.190514\pi$
$192$	0	0
$193$	239.813	1.24255	0.621277	−	0.783591i	$-0.286614\pi$
$193$	239.813	1.24255	0.621277	+	0.783591i	$0.286614\pi$
$194$	205.284i	1.05817i
$195$	0	0
$196$	106.702	0.544400
$197$	298.396i	1.51470i	0.653010	+	0.757349i	$0.273506\pi$
$197$	298.396i	1.51470i	−0.653010	+	0.757349i	$0.726494\pi$
$198$	0	0
$199$	−66.9651	−0.336508	−0.168254	−	0.985744i	$-0.553813\pi$
$199$	−66.9651	−0.336508	−0.168254	+	0.985744i	$0.553813\pi$
$200$	− 118.358i	− 0.591790i
$201$	0	0
$202$	−9.20019	−0.0455455
$203$	105.509i	0.519748i
$204$	0	0
$205$	−322.040	−1.57093
$206$	44.3843i	0.215458i
$207$	0	0
$208$	147.946	0.711280
$209$	54.6023i	0.261255i
$210$	0	0
$211$	58.8876	0.279088	0.139544	−	0.990216i	$-0.455436\pi$
$211$	58.8876	0.279088	0.139544	+	0.990216i	$0.455436\pi$
$212$	83.0580i	0.391783i
$213$	0	0
$214$	76.9965	0.359797
$215$	345.599i	1.60744i
$216$	0	0
$217$	−514.929	−2.37295
$218$	− 128.021i	− 0.587251i
$219$	0	0
$220$	29.2162	0.132801
$221$	− 151.895i	− 0.687310i
$222$	0	0
$223$	−216.873	−0.972525	−0.486263	−	0.873813i	$-0.661640\pi$
$223$	−216.873	−0.972525	−0.486263	+	0.873813i	$0.661640\pi$
$224$	239.194i	1.06783i
$225$	0	0
$226$	−345.096	−1.52697
$227$	− 126.476i	− 0.557163i	−0.960413	−	0.278582i	$-0.910136\pi$
$227$	− 126.476i	− 0.557163i	0.960413	−	0.278582i	$-0.0898643\pi$
$228$	0	0
$229$	29.4781	0.128726	0.0643628	−	0.997927i	$-0.479499\pi$
$229$	29.4781	0.128726	0.0643628	+	0.997927i	$0.479499\pi$
$230$	− 250.400i	− 1.08869i
$231$	0	0
$232$	−82.4055	−0.355196
$233$	236.199i	1.01373i	0.862026	+	0.506864i	$0.169196\pi$
$233$	236.199i	1.01373i	−0.862026	+	0.506864i	$0.830804\pi$
$234$	0	0
$235$	235.205	1.00087
$236$	− 63.2826i	− 0.268147i
$237$	0	0
$238$	−152.958	−0.642680
$239$	− 119.069i	− 0.498197i	−0.968478	−	0.249098i	$-0.919866\pi$
$239$	− 119.069i	− 0.498197i	0.968478	−	0.249098i	$-0.0801342\pi$
$240$	0	0
$241$	102.143	0.423828	0.211914	−	0.977288i	$-0.432030\pi$
$241$	102.143	0.423828	0.211914	+	0.977288i	$0.432030\pi$
$242$	− 17.6756i	− 0.0730398i
$243$	0	0
$244$	−14.4111	−0.0590619
$245$	− 467.497i	− 1.90815i
$246$	0	0
$247$	−292.833	−1.18556
$248$	− 402.174i	− 1.62167i
$249$	0	0
$250$	113.852	0.455409
$251$	− 264.723i	− 1.05467i	−0.849656	−	0.527337i	$-0.823190\pi$
$251$	− 264.723i	− 1.05467i	0.849656	−	0.527337i	$-0.176810\pi$
$252$	0	0
$253$	83.1921	0.328822
$254$	− 25.2016i	− 0.0992190i
$255$	0	0
$256$	−233.993	−0.914036
$257$	299.407i	1.16501i	0.812828	+	0.582504i	$0.197927\pi$
$257$	299.407i	1.16501i	−0.812828	+	0.582504i	$0.802073\pi$
$258$	0	0
$259$	−417.736	−1.61288
$260$	156.687i	0.602641i
$261$	0	0
$262$	126.822	0.484053
$263$	− 345.564i	− 1.31393i	−0.753921	−	0.656966i	$-0.771840\pi$
$263$	− 345.564i	− 1.31393i	0.753921	−	0.656966i	$-0.228160\pi$
$264$	0	0
$265$	363.903	1.37322
$266$	294.881i	1.10858i
$267$	0	0
$268$	−82.7736	−0.308857
$269$	− 158.607i	− 0.589618i	−0.955556	−	0.294809i	$-0.904744\pi$
$269$	− 158.607i	− 0.589618i	0.955556	−	0.294809i	$-0.0952560\pi$
$270$	0	0
$271$	121.065	0.446734	0.223367	−	0.974734i	$-0.428295\pi$
$271$	121.065	0.446734	0.223367	+	0.974734i	$0.428295\pi$
$272$	− 71.0294i	− 0.261138i
$273$	0	0
$274$	217.064	0.792203
$275$	− 45.0897i	− 0.163962i
$276$	0	0
$277$	−221.505	−0.799656	−0.399828	−	0.916590i	$-0.630930\pi$
$277$	−221.505	−0.799656	−0.399828	+	0.916590i	$0.630930\pi$
$278$	291.735i	1.04941i
$279$	0	0
$280$	602.883	2.15315
$281$	− 26.5964i	− 0.0946489i	−0.998880	−	0.0473245i	$-0.984931\pi$
$281$	− 26.5964i	− 0.0946489i	0.998880	−	0.0473245i	$-0.0150695\pi$
$282$	0	0
$283$	−216.323	−0.764393	−0.382197	−	0.924081i	$-0.624832\pi$
$283$	−216.323	−0.764393	−0.382197	+	0.924081i	$0.624832\pi$
$284$	106.597i	0.375340i
$285$	0	0
$286$	94.7947	0.331450
$287$	− 577.822i	− 2.01332i
$288$	0	0
$289$	216.075	0.747663
$290$	94.4902i	0.325828i
$291$	0	0
$292$	159.199	0.545202
$293$	249.421i	0.851266i	0.904896	+	0.425633i	$0.139949\pi$
$293$	249.421i	0.851266i	−0.904896	+	0.425633i	$0.860051\pi$
$294$	0	0
$295$	−277.261	−0.939869
$296$	− 326.264i	− 1.10224i
$297$	0	0
$298$	−198.656	−0.666632
$299$	446.160i	1.49217i
$300$	0	0
$301$	−620.093	−2.06011
$302$	108.396i	0.358928i
$303$	0	0
$304$	−136.935	−0.450443
$305$	63.1396i	0.207015i
$306$	0	0
$307$	211.978	0.690483	0.345242	−	0.938514i	$-0.387797\pi$
$307$	211.978	0.690483	0.345242	+	0.938514i	$0.387797\pi$
$308$	52.4213i	0.170199i
$309$	0	0
$310$	−461.153	−1.48759
$311$	− 278.236i	− 0.894649i	−0.894372	−	0.447324i	$-0.852377\pi$
$311$	− 278.236i	− 0.894649i	0.894372	−	0.447324i	$-0.147623\pi$
$312$	0	0
$313$	378.259	1.20850	0.604248	−	0.796796i	$-0.293474\pi$
$313$	378.259	1.20850	0.604248	+	0.796796i	$0.293474\pi$
$314$	270.590i	0.861753i
$315$	0	0
$316$	74.1374	0.234612
$317$	− 446.608i	− 1.40886i	−0.709774	−	0.704429i	$-0.751203\pi$
$317$	− 446.608i	− 1.40886i	0.709774	−	0.704429i	$-0.248797\pi$
$318$	0	0
$319$	−31.3932	−0.0984112
$320$	420.906i	1.31533i
$321$	0	0
$322$	449.281	1.39528
$323$	140.590i	0.435263i
$324$	0	0
$325$	241.816	0.744051
$326$	207.037i	0.635084i
$327$	0	0
$328$	451.295	1.37590
$329$	422.018i	1.28273i
$330$	0	0
$331$	80.3709	0.242812	0.121406	−	0.992603i	$-0.461260\pi$
$331$	80.3709	0.242812	0.121406	+	0.992603i	$0.461260\pi$
$332$	25.6615i	0.0772935i
$333$	0	0
$334$	83.3452	0.249536
$335$	362.657i	1.08256i
$336$	0	0
$337$	171.695	0.509482	0.254741	−	0.967009i	$-0.418010\pi$
$337$	171.695	0.509482	0.254741	+	0.967009i	$0.418010\pi$
$338$	236.823i	0.700660i
$339$	0	0
$340$	75.2257	0.221252
$341$	− 153.212i	− 0.449303i
$342$	0	0
$343$	292.616	0.853109
$344$	− 484.311i	− 1.40788i
$345$	0	0
$346$	−371.328	−1.07320
$347$	466.748i	1.34510i	0.740053	+	0.672548i	$0.234800\pi$
$347$	466.748i	1.34510i	−0.740053	+	0.672548i	$0.765200\pi$
$348$	0	0
$349$	−298.940	−0.856561	−0.428280	−	0.903646i	$-0.640880\pi$
$349$	−298.940	−0.856561	−0.428280	+	0.903646i	$0.640880\pi$
$350$	− 243.508i	− 0.695737i
$351$	0	0
$352$	−71.1698	−0.202187
$353$	− 43.9061i	− 0.124380i	−0.998064	−	0.0621899i	$-0.980192\pi$
$353$	− 43.9061i	− 0.124380i	0.998064	−	0.0621899i	$-0.0198084\pi$
$354$	0	0
$355$	467.034	1.31559
$356$	− 193.946i	− 0.544792i
$357$	0	0
$358$	473.467	1.32253
$359$	657.031i	1.83017i	0.403261	+	0.915085i	$0.367877\pi$
$359$	657.031i	1.83017i	−0.403261	+	0.915085i	$0.632123\pi$
$360$	0	0
$361$	−89.9627	−0.249204
$362$	− 253.642i	− 0.700670i
$363$	0	0
$364$	−281.136	−0.772352
$365$	− 697.501i	− 1.91096i
$366$	0	0
$367$	−397.449	−1.08297	−0.541484	−	0.840711i	$-0.682137\pi$
$367$	−397.449	−1.08297	−0.541484	+	0.840711i	$0.682137\pi$
$368$	208.634i	0.566939i
$369$	0	0
$370$	−374.110	−1.01111
$371$	652.936i	1.75993i
$372$	0	0
$373$	−11.2157	−0.0300688	−0.0150344	−	0.999887i	$-0.504786\pi$
$373$	−11.2157	−0.0300688	−0.0150344	+	0.999887i	$0.504786\pi$
$374$	− 45.5112i	− 0.121688i
$375$	0	0
$376$	−329.609	−0.876618
$377$	− 168.362i	− 0.446584i
$378$	0	0
$379$	−99.5442	−0.262650	−0.131325	−	0.991339i	$-0.541923\pi$
$379$	−99.5442	−0.262650	−0.131325	+	0.991339i	$0.541923\pi$
$380$	− 145.024i	− 0.381643i
$381$	0	0
$382$	345.841	0.905344
$383$	− 164.100i	− 0.428461i	−0.976783	−	0.214230i	$-0.931276\pi$
$383$	− 164.100i	− 0.428461i	0.976783	−	0.214230i	$-0.0687243\pi$
$384$	0	0
$385$	229.674	0.596557
$386$	385.349i	0.998315i
$387$	0	0
$388$	181.148	0.466878
$389$	− 553.313i	− 1.42240i	−0.702991	−	0.711199i	$-0.748152\pi$
$389$	− 553.313i	− 1.42240i	0.702991	−	0.711199i	$-0.251848\pi$
$390$	0	0
$391$	214.203	0.547833
$392$	655.135i	1.67126i
$393$	0	0
$394$	−479.485	−1.21697
$395$	− 324.820i	− 0.822328i
$396$	0	0
$397$	18.8569	0.0474984	0.0237492	−	0.999718i	$-0.492440\pi$
$397$	18.8569	0.0474984	0.0237492	+	0.999718i	$0.492440\pi$
$398$	− 107.605i	− 0.270363i
$399$	0	0
$400$	113.078	0.282696
$401$	254.177i	0.633858i	0.948449	+	0.316929i	$0.102652\pi$
$401$	254.177i	0.633858i	−0.948449	+	0.316929i	$0.897348\pi$
$402$	0	0
$403$	821.680	2.03891
$404$	8.11849i	0.0200953i
$405$	0	0
$406$	−169.540	−0.417585
$407$	− 124.293i	− 0.305389i
$408$	0	0
$409$	−533.620	−1.30469	−0.652347	−	0.757920i	$-0.726216\pi$
$409$	−533.620	−1.30469	−0.652347	+	0.757920i	$0.726216\pi$
$410$	− 517.478i	− 1.26214i
$411$	0	0
$412$	39.1659	0.0950628
$413$	− 497.478i	− 1.20455i
$414$	0	0
$415$	112.431	0.270918
$416$	− 381.685i	− 0.917512i
$417$	0	0
$418$	−87.7391	−0.209902
$419$	− 345.851i	− 0.825420i	−0.910862	−	0.412710i	$-0.864582\pi$
$419$	− 345.851i	− 0.825420i	0.910862	−	0.412710i	$-0.135418\pi$
$420$	0	0
$421$	410.003	0.973880	0.486940	−	0.873436i	$-0.338113\pi$
$421$	410.003	0.973880	0.486940	+	0.873436i	$0.338113\pi$
$422$	94.6250i	0.224230i
$423$	0	0
$424$	−509.962	−1.20274
$425$	− 116.097i	− 0.273169i
$426$	0	0
$427$	−113.289	−0.265313
$428$	− 67.9438i	− 0.158747i
$429$	0	0
$430$	−555.335	−1.29148
$431$	− 101.979i	− 0.236610i	−0.992977	−	0.118305i	$-0.962254\pi$
$431$	− 101.979i	− 0.236610i	0.992977	−	0.118305i	$-0.0377461\pi$
$432$	0	0
$433$	−378.091	−0.873189	−0.436595	−	0.899658i	$-0.643816\pi$
$433$	−378.091	−0.873189	−0.436595	+	0.899658i	$0.643816\pi$
$434$	− 827.427i	− 1.90651i
$435$	0	0
$436$	−112.969	−0.259103
$437$	− 412.953i	− 0.944971i
$438$	0	0
$439$	594.777	1.35484	0.677422	−	0.735594i	$-0.263097\pi$
$439$	594.777	1.35484	0.677422	+	0.735594i	$0.263097\pi$
$440$	179.382i	0.407687i
$441$	0	0
$442$	244.077	0.552210
$443$	− 47.2023i	− 0.106551i	−0.998580	−	0.0532757i	$-0.983034\pi$
$443$	− 47.2023i	− 0.106551i	0.998580	−	0.0532757i	$-0.0169662\pi$
$444$	0	0
$445$	−849.739	−1.90953
$446$	− 348.488i	− 0.781364i
$447$	0	0
$448$	−755.214	−1.68575
$449$	608.886i	1.35609i	0.735019	+	0.678047i	$0.237173\pi$
$449$	608.886i	1.35609i	−0.735019	+	0.678047i	$0.762827\pi$
$450$	0	0
$451$	171.925	0.381209
$452$	304.522i	0.673721i
$453$	0	0
$454$	203.231	0.447646
$455$	1231.75i	2.70713i
$456$	0	0
$457$	−479.540	−1.04932	−0.524661	−	0.851311i	$-0.675808\pi$
$457$	−479.540	−1.04932	−0.524661	+	0.851311i	$0.675808\pi$
$458$	47.3677i	0.103423i
$459$	0	0
$460$	−220.959	−0.480346
$461$	372.082i	0.807119i	0.914953	+	0.403559i	$0.132227\pi$
$461$	372.082i	0.807119i	−0.914953	+	0.403559i	$0.867773\pi$
$462$	0	0
$463$	418.247	0.903342	0.451671	−	0.892185i	$-0.350828\pi$
$463$	418.247	0.903342	0.451671	+	0.892185i	$0.350828\pi$
$464$	− 78.7296i	− 0.169676i
$465$	0	0
$466$	−379.542	−0.814468
$467$	118.057i	0.252799i	0.991979	+	0.126400i	$0.0403422\pi$
$467$	118.057i	0.252799i	−0.991979	+	0.126400i	$0.959658\pi$
$468$	0	0
$469$	−650.700	−1.38742
$470$	377.946i	0.804139i
$471$	0	0
$472$	388.544	0.823187
$473$	− 184.503i	− 0.390070i
$474$	0	0
$475$	−223.818	−0.471196
$476$	134.974i	0.283559i
$477$	0	0
$478$	191.329	0.400270
$479$	692.647i	1.44603i	0.690833	+	0.723014i	$0.257244\pi$
$479$	692.647i	1.44603i	−0.690833	+	0.723014i	$0.742756\pi$
$480$	0	0
$481$	666.587	1.38584
$482$	164.131i	0.340520i
$483$	0	0
$484$	−15.5975	−0.0322261
$485$	− 793.669i	− 1.63643i
$486$	0	0
$487$	−813.256	−1.66993	−0.834965	−	0.550303i	$-0.814512\pi$
$487$	−813.256	−1.66993	−0.834965	+	0.550303i	$0.814512\pi$
$488$	− 88.4817i	− 0.181315i
$489$	0	0
$490$	751.210	1.53308
$491$	− 265.228i	− 0.540179i	−0.962835	−	0.270089i	$-0.912947\pi$
$491$	− 265.228i	− 0.540179i	0.962835	−	0.270089i	$-0.0870532\pi$
$492$	0	0
$493$	−80.8311	−0.163958
$494$	− 470.546i	− 0.952522i
$495$	0	0
$496$	384.234	0.774666
$497$	837.978i	1.68607i
$498$	0	0
$499$	492.715	0.987406	0.493703	−	0.869631i	$-0.335643\pi$
$499$	492.715	0.987406	0.493703	+	0.869631i	$0.335643\pi$
$500$	− 100.466i	− 0.200933i
$501$	0	0
$502$	425.377	0.847365
$503$	88.4886i	0.175922i	0.996124	+	0.0879608i	$0.0280350\pi$
$503$	88.4886i	0.175922i	−0.996124	+	0.0879608i	$0.971965\pi$
$504$	0	0
$505$	35.5697	0.0704350
$506$	133.679i	0.264188i
$507$	0	0
$508$	−22.2386	−0.0437767
$509$	261.110i	0.512986i	0.966546	+	0.256493i	$0.0825671\pi$
$509$	261.110i	0.512986i	−0.966546	+	0.256493i	$0.917433\pi$
$510$	0	0
$511$	1251.50	2.44911
$512$	− 468.135i	− 0.914327i
$513$	0	0
$514$	−481.110	−0.936012
$515$	− 171.598i	− 0.333200i
$516$	0	0
$517$	−125.568	−0.242877
$518$	− 671.250i	− 1.29585i
$519$	0	0
$520$	−962.029	−1.85006
$521$	− 221.574i	− 0.425287i	−0.977130	−	0.212643i	$-0.931793\pi$
$521$	− 221.574i	− 0.425287i	0.977130	−	0.212643i	$-0.0682072\pi$
$522$	0	0
$523$	−444.427	−0.849765	−0.424882	−	0.905249i	$-0.639685\pi$
$523$	−444.427	−0.849765	−0.424882	+	0.905249i	$0.639685\pi$
$524$	− 111.911i	− 0.213571i
$525$	0	0
$526$	555.278	1.05566
$527$	− 394.491i	− 0.748559i
$528$	0	0
$529$	−100.174	−0.189366
$530$	584.747i	1.10330i
$531$	0	0
$532$	260.211	0.489119
$533$	922.038i	1.72990i
$534$	0	0
$535$	−297.683	−0.556417
$536$	− 508.216i	− 0.948163i
$537$	0	0
$538$	254.862	0.473721
$539$	249.580i	0.463042i
$540$	0	0
$541$	508.057	0.939107	0.469553	−	0.882904i	$-0.344415\pi$
$541$	508.057	0.939107	0.469553	+	0.882904i	$0.344415\pi$
$542$	194.536i	0.358923i
$543$	0	0
$544$	−183.248	−0.336853
$545$	494.953i	0.908170i
$546$	0	0
$547$	44.8213	0.0819403	0.0409701	−	0.999160i	$-0.486955\pi$
$547$	44.8213	0.0819403	0.0409701	+	0.999160i	$0.486955\pi$
$548$	− 191.543i	− 0.349531i
$549$	0	0
$550$	72.4535	0.131734
$551$	155.831i	0.282815i
$552$	0	0
$553$	582.810	1.05391
$554$	− 355.931i	− 0.642474i
$555$	0	0
$556$	257.435	0.463012
$557$	− 89.6474i	− 0.160947i	−0.996757	−	0.0804734i	$-0.974357\pi$
$557$	− 89.6474i	− 0.160947i	0.996757	−	0.0804734i	$-0.0256432\pi$
$558$	0	0
$559$	989.492	1.77011
$560$	575.990i	1.02855i
$561$	0	0
$562$	42.7370	0.0760445
$563$	− 312.733i	− 0.555476i	−0.960657	−	0.277738i	$-0.910415\pi$
$563$	− 312.733i	− 0.555476i	0.960657	−	0.277738i	$-0.0895847\pi$
$564$	0	0
$565$	1334.21	2.36143
$566$	− 347.605i	− 0.614142i
$567$	0	0
$568$	−654.485	−1.15226
$569$	484.269i	0.851088i	0.904938	+	0.425544i	$0.139917\pi$
$569$	484.269i	0.851088i	−0.904938	+	0.425544i	$0.860083\pi$
$570$	0	0
$571$	−1049.98	−1.83884	−0.919422	−	0.393272i	$-0.871343\pi$
$571$	−1049.98	−1.83884	−0.919422	+	0.393272i	$0.871343\pi$
$572$	− 83.6494i	− 0.146240i
$573$	0	0
$574$	928.488	1.61757
$575$	341.009i	0.593060i
$576$	0	0
$577$	309.817	0.536945	0.268473	−	0.963287i	$-0.413481\pi$
$577$	309.817	0.536945	0.268473	+	0.963287i	$0.413481\pi$
$578$	347.205i	0.600701i
$579$	0	0
$580$	83.3807	0.143760
$581$	201.730i	0.347212i
$582$	0	0
$583$	−194.275	−0.333233
$584$	977.454i	1.67372i
$585$	0	0
$586$	−400.789	−0.683940
$587$	− 258.506i	− 0.440385i	−0.975456	−	0.220192i	$-0.929332\pi$
$587$	− 258.506i	− 0.440385i	0.975456	−	0.220192i	$-0.0706685\pi$
$588$	0	0
$589$	−760.522	−1.29121
$590$	− 445.524i	− 0.755126i
$591$	0	0
$592$	311.710	0.526537
$593$	− 601.720i	− 1.01470i	−0.861739	−	0.507352i	$-0.830624\pi$
$593$	− 601.720i	− 1.01470i	0.861739	−	0.507352i	$-0.169376\pi$
$594$	0	0
$595$	591.365	0.993890
$596$	175.300i	0.294127i
$597$	0	0
$598$	−716.924	−1.19887
$599$	− 168.322i	− 0.281005i	−0.990080	−	0.140502i	$-0.955128\pi$
$599$	− 168.322i	− 0.281005i	0.990080	−	0.140502i	$-0.0448718\pi$
$600$	0	0
$601$	−263.203	−0.437942	−0.218971	−	0.975731i	$-0.570270\pi$
$601$	−263.203	−0.437942	−0.218971	+	0.975731i	$0.570270\pi$
$602$	− 996.413i	− 1.65517i
$603$	0	0
$604$	95.6518	0.158364
$605$	68.3374i	0.112954i
$606$	0	0
$607$	139.029	0.229044	0.114522	−	0.993421i	$-0.463466\pi$
$607$	139.029	0.229044	0.114522	+	0.993421i	$0.463466\pi$
$608$	353.276i	0.581046i
$609$	0	0
$610$	−101.458	−0.166324
$611$	− 673.421i	− 1.10216i
$612$	0	0
$613$	−724.937	−1.18260	−0.591302	−	0.806450i	$-0.701386\pi$
$613$	−724.937	−1.18260	−0.591302	+	0.806450i	$0.701386\pi$
$614$	340.623i	0.554760i
$615$	0	0
$616$	−321.858	−0.522496
$617$	323.005i	0.523509i	0.965135	+	0.261754i	$0.0843011\pi$
$617$	323.005i	0.523509i	−0.965135	+	0.261754i	$0.915699\pi$
$618$	0	0
$619$	448.155	0.723999	0.362000	−	0.932178i	$-0.382094\pi$
$619$	448.155	0.723999	0.362000	+	0.932178i	$0.382094\pi$
$620$	406.934i	0.656345i
$621$	0	0
$622$	447.090	0.718795
$623$	− 1524.65i	− 2.44727i
$624$	0	0
$625$	−780.051	−1.24808
$626$	607.816i	0.970952i
$627$	0	0
$628$	238.776	0.380217
$629$	− 320.030i	− 0.508792i
$630$	0	0
$631$	12.9974	0.0205982	0.0102991	−	0.999947i	$-0.496722\pi$
$631$	12.9974	0.0205982	0.0102991	+	0.999947i	$0.496722\pi$
$632$	455.191i	0.720239i
$633$	0	0
$634$	717.644	1.13193
$635$	97.4343i	0.153440i
$636$	0	0
$637$	−1338.50	−2.10126
$638$	− 50.4449i	− 0.0790673i
$639$	0	0
$640$	−143.101	−0.223595
$641$	− 132.212i	− 0.206260i	−0.994668	−	0.103130i	$-0.967114\pi$
$641$	− 132.212i	− 0.206260i	0.994668	−	0.103130i	$-0.0328857\pi$
$642$	0	0
$643$	−1129.29	−1.75628	−0.878139	−	0.478405i	$-0.841215\pi$
$643$	−1129.29	−1.75628	−0.878139	+	0.478405i	$0.841215\pi$
$644$	− 396.458i	− 0.615617i
$645$	0	0
$646$	−225.910	−0.349707
$647$	351.154i	0.542741i	0.962475	+	0.271371i	$0.0874769\pi$
$647$	351.154i	0.542741i	−0.962475	+	0.271371i	$0.912523\pi$
$648$	0	0
$649$	148.020	0.228074
$650$	388.569i	0.597798i
$651$	0	0
$652$	182.695	0.280207
$653$	− 1091.70i	− 1.67182i	−0.548866	−	0.835910i	$-0.684940\pi$
$653$	− 1091.70i	− 1.67182i	0.548866	−	0.835910i	$-0.315060\pi$
$654$	0	0
$655$	−490.318	−0.748577
$656$	431.164i	0.657262i
$657$	0	0
$658$	−678.131	−1.03059
$659$	1111.39i	1.68648i	0.537534	+	0.843242i	$0.319356\pi$
$659$	1111.39i	1.68648i	−0.537534	+	0.843242i	$0.680644\pi$
$660$	0	0
$661$	−496.270	−0.750786	−0.375393	−	0.926866i	$-0.622492\pi$
$661$	−496.270	−0.750786	−0.375393	+	0.926866i	$0.622492\pi$
$662$	129.146i	0.195085i
$663$	0	0
$664$	−157.557	−0.237285
$665$	− 1140.07i	− 1.71439i
$666$	0	0
$667$	237.424	0.355958
$668$	− 73.5460i	− 0.110099i
$669$	0	0
$670$	−582.745	−0.869769
$671$	− 33.7080i	− 0.0502354i
$672$	0	0
$673$	−27.3883	−0.0406959	−0.0203479	−	0.999793i	$-0.506477\pi$
$673$	−27.3883	−0.0406959	−0.0203479	+	0.999793i	$0.506477\pi$
$674$	275.893i	0.409337i
$675$	0	0
$676$	208.979	0.309141
$677$	− 381.717i	− 0.563836i	−0.959439	−	0.281918i	$-0.909029\pi$
$677$	− 381.717i	− 0.563836i	0.959439	−	0.281918i	$-0.0909706\pi$
$678$	0	0
$679$	1424.05	2.09727
$680$	461.873i	0.679225i
$681$	0	0
$682$	246.193	0.360987
$683$	867.261i	1.26978i	0.772602	+	0.634891i	$0.218955\pi$
$683$	867.261i	1.26978i	−0.772602	+	0.634891i	$0.781045\pi$
$684$	0	0
$685$	−839.210	−1.22512
$686$	470.198i	0.685420i
$687$	0	0
$688$	462.707	0.672539
$689$	− 1041.90i	− 1.51219i
$690$	0	0
$691$	463.252	0.670408	0.335204	−	0.942146i	$-0.391195\pi$
$691$	463.252	0.670408	0.335204	+	0.942146i	$0.391195\pi$
$692$	327.670i	0.473511i
$693$	0	0
$694$	−750.007	−1.08070
$695$	− 1127.90i	− 1.62288i
$696$	0	0
$697$	442.673	0.635112
$698$	− 480.359i	− 0.688193i
$699$	0	0
$700$	−214.878	−0.306968
$701$	− 577.756i	− 0.824189i	−0.911141	−	0.412094i	$-0.864797\pi$
$701$	− 577.756i	− 0.824189i	0.911141	−	0.412094i	$-0.135203\pi$
$702$	0	0
$703$	−616.973	−0.877628
$704$	− 224.707i	− 0.319186i
$705$	0	0
$706$	70.5516	0.0999315
$707$	63.8211i	0.0902703i
$708$	0	0
$709$	−870.594	−1.22792	−0.613959	−	0.789338i	$-0.710424\pi$
$709$	−870.594	−1.22792	−0.613959	+	0.789338i	$0.710424\pi$
$710$	750.465i	1.05699i
$711$	0	0
$712$	1190.80	1.67247
$713$	1158.73i	1.62515i
$714$	0	0
$715$	−366.494	−0.512580
$716$	− 417.800i	− 0.583520i
$717$	0	0
$718$	−1055.77	−1.47043
$719$	533.694i	0.742272i	0.928579	+	0.371136i	$0.121032\pi$
$719$	533.694i	0.742272i	−0.928579	+	0.371136i	$0.878968\pi$
$720$	0	0
$721$	307.891	0.427033
$722$	− 144.559i	− 0.200220i
$723$	0	0
$724$	−223.821	−0.309145
$725$	− 128.682i	− 0.177493i
$726$	0	0
$727$	−749.734	−1.03127	−0.515636	−	0.856808i	$-0.672444\pi$
$727$	−749.734	−1.03127	−0.515636	+	0.856808i	$0.672444\pi$
$728$	− 1726.13i	− 2.37105i
$729$	0	0
$730$	1120.80	1.53534
$731$	− 475.058i	− 0.649874i
$732$	0	0
$733$	461.131	0.629101	0.314550	−	0.949241i	$-0.398146\pi$
$733$	461.131	0.629101	0.314550	+	0.949241i	$0.398146\pi$
$734$	− 638.651i	− 0.870097i
$735$	0	0
$736$	538.251	0.731320
$737$	− 193.610i	− 0.262700i
$738$	0	0
$739$	730.982	0.989150	0.494575	−	0.869135i	$-0.335324\pi$
$739$	730.982	0.989150	0.494575	+	0.869135i	$0.335324\pi$
$740$	330.125i	0.446115i
$741$	0	0
$742$	−1049.19	−1.41400
$743$	147.952i	0.199128i	0.995031	+	0.0995642i	$0.0317449\pi$
$743$	147.952i	0.199128i	−0.995031	+	0.0995642i	$0.968255\pi$
$744$	0	0
$745$	768.044	1.03093
$746$	− 18.0222i	− 0.0241585i
$747$	0	0
$748$	−40.1603	−0.0536902
$749$	− 534.120i	− 0.713111i
$750$	0	0
$751$	1020.61	1.35901	0.679503	−	0.733673i	$-0.262195\pi$
$751$	1020.61	1.35901	0.679503	+	0.733673i	$0.262195\pi$
$752$	− 314.905i	− 0.418757i
$753$	0	0
$754$	270.537	0.358802
$755$	− 419.081i	− 0.555074i
$756$	0	0
$757$	157.707	0.208332	0.104166	−	0.994560i	$-0.466783\pi$
$757$	157.707	0.208332	0.104166	+	0.994560i	$0.466783\pi$
$758$	− 159.955i	− 0.211023i
$759$	0	0
$760$	890.426	1.17161
$761$	727.440i	0.955900i	0.878387	+	0.477950i	$0.158620\pi$
$761$	727.440i	0.955900i	−0.878387	+	0.477950i	$0.841380\pi$
$762$	0	0
$763$	−888.072	−1.16392
$764$	− 305.180i	− 0.399450i
$765$	0	0
$766$	263.689	0.344242
$767$	793.832i	1.03498i
$768$	0	0
$769$	1249.12	1.62434	0.812172	−	0.583418i	$-0.198285\pi$
$769$	1249.12	1.62434	0.812172	+	0.583418i	$0.198285\pi$
$770$	369.058i	0.479296i
$771$	0	0
$772$	340.043	0.440470
$773$	1360.02i	1.75941i	0.475521	+	0.879704i	$0.342259\pi$
$773$	1360.02i	1.75941i	−0.475521	+	0.879704i	$0.657741\pi$
$774$	0	0
$775$	628.026	0.810356
$776$	1112.22i	1.43327i
$777$	0	0
$778$	889.105	1.14281
$779$	− 853.411i	− 1.09552i
$780$	0	0
$781$	−249.332	−0.319248
$782$	344.197i	0.440150i
$783$	0	0
$784$	−625.910	−0.798355
$785$	− 1046.15i	− 1.33268i
$786$	0	0
$787$	−20.7010	−0.0263037	−0.0131518	−	0.999914i	$-0.504186\pi$
$787$	−20.7010	−0.0263037	−0.0131518	+	0.999914i	$0.504186\pi$
$788$	423.110i	0.536942i
$789$	0	0
$790$	521.945	0.660689
$791$	2393.91i	3.02643i
$792$	0	0
$793$	180.776	0.227965
$794$	30.3006i	0.0381620i
$795$	0	0
$796$	−94.9532	−0.119288
$797$	514.624i	0.645701i	0.946450	+	0.322851i	$0.104641\pi$
$797$	514.624i	0.645701i	−0.946450	+	0.322851i	$0.895359\pi$
$798$	0	0
$799$	−323.311	−0.404645
$800$	− 291.729i	− 0.364662i
$801$	0	0
$802$	−408.431	−0.509265
$803$	372.371i	0.463724i
$804$	0	0
$805$	−1737.01	−2.15777
$806$	1320.34i	1.63814i
$807$	0	0
$808$	−49.8462	−0.0616908
$809$	− 983.864i	− 1.21615i	−0.793880	−	0.608074i	$-0.791942\pi$
$809$	− 983.864i	− 1.21615i	0.793880	−	0.608074i	$-0.208058\pi$
$810$	0	0
$811$	−1460.35	−1.80068	−0.900340	−	0.435186i	$-0.856683\pi$
$811$	−1460.35	−1.80068	−0.900340	+	0.435186i	$0.856683\pi$
$812$	149.606i	0.184244i
$813$	0	0
$814$	199.724	0.245361
$815$	− 800.446i	− 0.982142i
$816$	0	0
$817$	−915.844	−1.12098
$818$	− 857.461i	− 1.04824i
$819$	0	0
$820$	−456.636	−0.556873
$821$	477.870i	0.582058i	0.956714	+	0.291029i	$0.0939977\pi$
$821$	477.870i	0.582058i	−0.956714	+	0.291029i	$0.906002\pi$
$822$	0	0
$823$	−987.820	−1.20027	−0.600134	−	0.799900i	$-0.704886\pi$
$823$	−987.820	−1.20027	−0.600134	+	0.799900i	$0.704886\pi$
$824$	240.472i	0.291835i
$825$	0	0
$826$	799.385	0.967778
$827$	109.527i	0.132439i	0.997805	+	0.0662193i	$0.0210937\pi$
$827$	109.527i	0.132439i	−0.997805	+	0.0662193i	$0.978906\pi$
$828$	0	0
$829$	1512.47	1.82445	0.912225	−	0.409689i	$-0.134363\pi$
$829$	1512.47	1.82445	0.912225	+	0.409689i	$0.134363\pi$
$830$	180.663i	0.217666i
$831$	0	0
$832$	1205.11	1.44844
$833$	642.618i	0.771450i
$834$	0	0
$835$	−322.229	−0.385902
$836$	77.4233i	0.0926116i
$837$	0	0
$838$	555.740	0.663174
$839$	497.928i	0.593478i	0.954959	+	0.296739i	$0.0958992\pi$
$839$	497.928i	0.593478i	−0.954959	+	0.296739i	$0.904101\pi$
$840$	0	0
$841$	751.406	0.893468
$842$	658.824i	0.782452i
$843$	0	0
$844$	83.4997	0.0989332
$845$	− 915.604i	− 1.08355i
$846$	0	0
$847$	−122.615	−0.144764
$848$	− 487.213i	− 0.574544i
$849$	0	0
$850$	186.553	0.219474
$851$	940.020i	1.10461i
$852$	0	0
$853$	−234.494	−0.274905	−0.137452	−	0.990508i	$-0.543891\pi$
$853$	−234.494	−0.274905	−0.137452	+	0.990508i	$0.543891\pi$
$854$	− 182.041i	− 0.213163i
$855$	0	0
$856$	417.163	0.487340
$857$	− 473.810i	− 0.552870i	−0.961033	−	0.276435i	$-0.910847\pi$
$857$	− 473.810i	− 0.552870i	0.961033	−	0.276435i	$-0.0891531\pi$
$858$	0	0
$859$	−82.9051	−0.0965135	−0.0482568	−	0.998835i	$-0.515367\pi$
$859$	−82.9051	−0.0965135	−0.0482568	+	0.998835i	$0.515367\pi$
$860$	490.042i	0.569817i
$861$	0	0
$862$	163.868	0.190102
$863$	232.504i	0.269414i	0.990886	+	0.134707i	$0.0430092\pi$
$863$	232.504i	0.269414i	−0.990886	+	0.134707i	$0.956991\pi$
$864$	0	0
$865$	1435.63	1.65968
$866$	− 607.545i	− 0.701553i
$867$	0	0
$868$	−730.144	−0.841180
$869$	173.409i	0.199551i
$870$	0	0
$871$	1038.33	1.19211
$872$	− 693.610i	− 0.795424i
$873$	0	0
$874$	663.563	0.759226
$875$	− 789.786i	− 0.902613i
$876$	0	0
$877$	594.018	0.677329	0.338665	−	0.940907i	$-0.390025\pi$
$877$	594.018	0.677329	0.338665	+	0.940907i	$0.390025\pi$
$878$	955.732i	1.08853i
$879$	0	0
$880$	−171.380	−0.194750
$881$	976.976i	1.10894i	0.832204	+	0.554470i	$0.187079\pi$
$881$	976.976i	1.10894i	−0.832204	+	0.554470i	$0.812921\pi$
$882$	0	0
$883$	250.334	0.283503	0.141752	−	0.989902i	$-0.454727\pi$
$883$	250.334	0.283503	0.141752	+	0.989902i	$0.454727\pi$
$884$	− 215.380i	− 0.243643i
$885$	0	0
$886$	75.8482	0.0856075
$887$	− 766.566i	− 0.864223i	−0.901820	−	0.432112i	$-0.857769\pi$
$887$	− 766.566i	− 0.864223i	0.901820	−	0.432112i	$-0.142231\pi$
$888$	0	0
$889$	−174.822	−0.196650
$890$	− 1365.43i	− 1.53419i
$891$	0	0
$892$	−307.515	−0.344748
$893$	623.298i	0.697982i
$894$	0	0
$895$	−1830.52	−2.04527
$896$	− 256.760i	− 0.286562i
$897$	0	0
$898$	−978.404	−1.08954
$899$	− 437.256i	− 0.486381i
$900$	0	0
$901$	−500.218	−0.555181
$902$	276.263i	0.306278i
$903$	0	0
$904$	−1869.71	−2.06827
$905$	980.631i	1.08357i
$906$	0	0
$907$	1053.93	1.16199	0.580996	−	0.813907i	$-0.302663\pi$
$907$	1053.93	1.16199	0.580996	+	0.813907i	$0.302663\pi$
$908$	− 179.337i	− 0.197508i
$909$	0	0
$910$	−1979.26	−2.17501
$911$	− 643.333i	− 0.706183i	−0.935589	−	0.353092i	$-0.885130\pi$
$911$	− 643.333i	− 0.706183i	0.935589	−	0.353092i	$-0.114870\pi$
$912$	0	0
$913$	−60.0229	−0.0657424
$914$	− 770.561i	− 0.843065i
$915$	0	0
$916$	41.7985	0.0456316
$917$	− 879.756i	− 0.959385i
$918$	0	0
$919$	−33.6201	−0.0365834	−0.0182917	−	0.999833i	$-0.505823\pi$
$919$	−33.6201	−0.0365834	−0.0182917	+	0.999833i	$0.505823\pi$
$920$	− 1356.65i	− 1.47462i
$921$	0	0
$922$	−597.889	−0.648470
$923$	− 1337.17i	− 1.44873i
$924$	0	0
$925$	509.486	0.550795
$926$	672.071i	0.725779i
$927$	0	0
$928$	−203.113	−0.218872
$929$	576.473i	0.620530i	0.950650	+	0.310265i	$0.100418\pi$
$929$	576.473i	0.620530i	−0.950650	+	0.310265i	$0.899582\pi$
$930$	0	0
$931$	1238.88	1.33069
$932$	334.918i	0.359354i
$933$	0	0
$934$	−189.703	−0.203108
$935$	175.955i	0.188187i
$936$	0	0
$937$	46.1378	0.0492399	0.0246200	−	0.999697i	$-0.492162\pi$
$937$	46.1378	0.0492399	0.0246200	+	0.999697i	$0.492162\pi$
$938$	− 1045.59i	− 1.11471i
$939$	0	0
$940$	333.509	0.354797
$941$	863.034i	0.917146i	0.888657	+	0.458573i	$0.151639\pi$
$941$	863.034i	0.917146i	−0.888657	+	0.458573i	$0.848361\pi$
$942$	0	0
$943$	−1300.26	−1.37885
$944$	371.212i	0.393233i
$945$	0	0
$946$	296.473	0.313397
$947$	676.033i	0.713868i	0.934130	+	0.356934i	$0.116178\pi$
$947$	676.033i	0.713868i	−0.934130	+	0.356934i	$0.883822\pi$
$948$	0	0
$949$	−1997.03	−2.10435
$950$	− 359.648i	− 0.378577i
$951$	0	0
$952$	−828.718	−0.870502
$953$	972.843i	1.02082i	0.859931	+	0.510411i	$0.170507\pi$
$953$	972.843i	1.02082i	−0.859931	+	0.510411i	$0.829493\pi$
$954$	0	0
$955$	−1337.09	−1.40009
$956$	− 168.834i	− 0.176605i
$957$	0	0
$958$	−1113.00	−1.16179
$959$	− 1505.76i	− 1.57013i
$960$	0	0
$961$	1173.00	1.22060
$962$	1071.12i	1.11343i
$963$	0	0
$964$	144.833	0.150242
$965$	− 1489.84i	− 1.54387i
$966$	0	0
$967$	1715.71	1.77426	0.887128	−	0.461523i	$-0.152697\pi$
$967$	1715.71	1.77426	0.887128	+	0.461523i	$0.152697\pi$
$968$	− 95.7657i	− 0.0989315i
$969$	0	0
$970$	1275.33	1.31477
$971$	− 729.785i	− 0.751580i	−0.926705	−	0.375790i	$-0.877371\pi$
$971$	− 729.785i	− 0.751580i	0.926705	−	0.375790i	$-0.122629\pi$
$972$	0	0
$973$	2023.75	2.07991
$974$	− 1306.80i	− 1.34169i
$975$	0	0
$976$	84.5347	0.0866134
$977$	− 584.651i	− 0.598414i	−0.954188	−	0.299207i	$-0.903278\pi$
$977$	− 584.651i	− 0.598414i	0.954188	−	0.299207i	$-0.0967222\pi$
$978$	0	0
$979$	453.645	0.463376
$980$	− 662.888i	− 0.676416i
$981$	0	0
$982$	426.188	0.434000
$983$	− 76.3328i	− 0.0776529i	−0.999246	−	0.0388265i	$-0.987638\pi$
$983$	− 76.3328i	− 0.0776529i	0.999246	−	0.0388265i	$-0.0123620\pi$
$984$	0	0
$985$	1853.78	1.88201
$986$	− 129.885i	− 0.131730i
$987$	0	0
$988$	−415.222	−0.420266
$989$	1395.38i	1.41090i
$990$	0	0
$991$	−1492.66	−1.50622	−0.753109	−	0.657895i	$-0.771447\pi$
$991$	−1492.66	−1.50622	−0.753109	+	0.657895i	$0.771447\pi$
$992$	− 991.281i	− 0.999275i
$993$	0	0
$994$	−1346.53	−1.35465
$995$	416.020i	0.418111i
$996$	0	0
$997$	−1492.32	−1.49681	−0.748405	−	0.663242i	$-0.769180\pi$
$997$	−1492.32	−1.49681	−0.748405	+	0.663242i	$0.769180\pi$
$998$	791.732i	0.793319i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	99.3.b.a.89.6	yes	8
3.2	odd	2	inner	99.3.b.a.89.3	✓	8
4.3	odd	2		1584.3.i.b.881.2		8
11.10	odd	2		1089.3.b.g.485.3		8
12.11	even	2		1584.3.i.b.881.7		8
33.32	even	2		1089.3.b.g.485.6		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
99.3.b.a.89.3	✓	8	3.2	odd	2	inner
99.3.b.a.89.6	yes	8	1.1	even	1	trivial
1089.3.b.g.485.3		8	11.10	odd	2
1089.3.b.g.485.6		8	33.32	even	2
1584.3.i.b.881.2		8	4.3	odd	2
1584.3.i.b.881.7		8	12.11	even	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)

\(f(q)\)	\(=\)	\(q+1.60688i q^{2} +1.41795 q^{4} -6.21249i q^{5} +11.1468 q^{7} +8.70597i q^{8} +O(q^{10})\)	\(q+1.60688i q^{2} +1.41795 q^{4} -6.21249i q^{5} +11.1468 q^{7} +8.70597i q^{8} +9.98270 q^{10} +3.31662i q^{11} -17.7871 q^{13} +17.9115i q^{14} -8.31762 q^{16} +8.53964i q^{17} +16.4632 q^{19} -8.80900i q^{20} -5.32940 q^{22} -25.0834i q^{23} -13.5950 q^{25} -28.5817i q^{26} +15.8056 q^{28} +9.46540i q^{29} -46.1952 q^{31} +21.4585i q^{32} -13.7221 q^{34} -69.2494i q^{35} -37.4758 q^{37} +26.4543i q^{38} +54.0858 q^{40} -51.8375i q^{41} -55.6297 q^{43} +4.70281i q^{44} +40.3058 q^{46} +37.8600i q^{47} +75.2512 q^{49} -21.8455i q^{50} -25.2212 q^{52} +58.5761i q^{53} +20.6045 q^{55} +97.0437i q^{56} -15.2097 q^{58} -44.6296i q^{59} -10.1633 q^{61} -74.2300i q^{62} -67.7516 q^{64} +110.502i q^{65} -58.3755 q^{67} +12.1088i q^{68} +111.275 q^{70} +75.1766i q^{71} +112.274 q^{73} -60.2190i q^{74} +23.3440 q^{76} +36.9698i q^{77} +52.2849 q^{79} +51.6731i q^{80} +83.2963 q^{82} +18.0976i q^{83} +53.0524 q^{85} -89.3901i q^{86} -28.8744 q^{88} -136.779i q^{89} -198.269 q^{91} -35.5669i q^{92} -60.8364 q^{94} -102.278i q^{95} +127.754 q^{97} +120.919i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 16 q^{4} + 16 q^{7}+O(q^{10}) \) 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

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Database

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