LMFDB - Embedded newform 1089.3.b.g.485.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1089,3,Mod(485,1089)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1089, 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("1089.485");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1089 = 3^{2} \cdot 11^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1089.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:	$29.6731007888$
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:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{6}\cdot 3^{3} $
Twist minimal:	no (minimal twist has level 99)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			485.6
Root			$1.13623 + 0.707107i$ of defining polynomial
Character	$\chi$	$=$	1089.485
Dual form			1089.3.b.g.485.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.60688i q^{2} +1.41795 q^{4} +6.21249i q^{5} -11.1468 q^{7} +8.70597i q^{8} +O(q^{10})$	$q+1.60688i q^{2} +1.41795 q^{4} +6.21249i q^{5} -11.1468 q^{7} +8.70597i q^{8} -9.98270 q^{10} +17.7871 q^{13} -17.9115i q^{14} -8.31762 q^{16} +8.53964i q^{17} -16.4632 q^{19} +8.80900i q^{20} +25.0834i q^{23} -13.5950 q^{25} +28.5817i q^{26} -15.8056 q^{28} +9.46540i q^{29} -46.1952 q^{31} +21.4585i q^{32} -13.7221 q^{34} -69.2494i q^{35} -37.4758 q^{37} -26.4543i q^{38} -54.0858 q^{40} -51.8375i q^{41} +55.6297 q^{43} -40.3058 q^{46} -37.8600i q^{47} +75.2512 q^{49} -21.8455i q^{50} +25.2212 q^{52} -58.5761i q^{53} -97.0437i q^{56} -15.2097 q^{58} +44.6296i q^{59} +10.1633 q^{61} -74.2300i q^{62} -67.7516 q^{64} +110.502i q^{65} -58.3755 q^{67} +12.1088i q^{68} +111.275 q^{70} -75.1766i q^{71} -112.274 q^{73} -60.2190i q^{74} -23.3440 q^{76} -52.2849 q^{79} -51.6731i q^{80} +83.2963 q^{82} +18.0976i q^{83} -53.0524 q^{85} +89.3901i q^{86} +136.779i q^{89} -198.269 q^{91} +35.5669i q^{92} +60.8364 q^{94} -102.278i q^{95} +127.754 q^{97} +120.919i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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} - 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 - 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 $\left(\mathbb{Z}/1089\mathbb{Z}\right)^\times$.

$n$	$244$	$848$
$\chi(n)$	$1$	$-1$

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	$0.213374\pi$
$5$	6.21249i	1.24250i	−0.783613	+	0.621249i	$0.786626\pi$
$6$	0	0
$7$	−11.1468	−1.59240	−0.796200	−	0.605033i	$-0.793160\pi$
$7$	−11.1468	−1.59240	−0.796200	+	0.605033i	$0.793160\pi$
$8$	8.70597i	1.08825i
$9$	0	0
$10$	−9.98270	−0.998270
$11$	0	0
$11$	0	0
$12$	0	0
$13$	17.7871	1.36824	0.684119	−	0.729370i	$-0.260187\pi$
$13$	17.7871	1.36824	0.684119	+	0.729370i	$0.260187\pi$
$14$	− 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	$-0.642631\pi$
$19$	−16.4632	−0.866485	−0.433242	+	0.901277i	$0.642631\pi$
$20$	8.80900i	0.440450i
$21$	0	0
$22$	0	0
$23$	25.0834i	1.09058i	0.838247	+	0.545290i	$0.183581\pi$
$23$	25.0834i	1.09058i	−0.838247	+	0.545290i	$0.816419\pi$
$24$	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	$-0.276083\pi$
$43$	55.6297	1.29371	0.646857	+	0.762611i	$0.276083\pi$
$44$	0	0
$45$	0	0
$46$	−40.3058	−0.876214
$47$	− 37.8600i	− 0.805533i	−0.915303	−	0.402766i	$-0.868049\pi$
$47$	− 37.8600i	− 0.805533i	0.915303	−	0.402766i	$-0.131951\pi$
$48$	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	$-0.813634\pi$
$53$	− 58.5761i	− 1.10521i	0.833444	−	0.552604i	$-0.186366\pi$
$54$	0	0
$55$	0	0
$56$	− 97.0437i	− 1.73292i
$57$	0	0
$58$	−15.2097	−0.262237
$59$	44.6296i	0.756435i	0.925717	+	0.378217i	$0.123463\pi$
$59$	44.6296i	0.756435i	−0.925717	+	0.378217i	$0.876537\pi$
$60$	0	0
$61$	10.1633	0.166612	0.0833060	−	0.996524i	$-0.473452\pi$
$61$	10.1633	0.166612	0.0833060	+	0.996524i	$0.473452\pi$
$62$	− 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	$-0.822412\pi$
$71$	− 75.1766i	− 1.05882i	0.848365	−	0.529412i	$-0.177588\pi$
$72$	0	0
$73$	−112.274	−1.53800	−0.769000	−	0.639249i	$-0.779245\pi$
$73$	−112.274	−1.53800	−0.769000	+	0.639249i	$0.779245\pi$
$74$	− 60.2190i	− 0.813771i
$75$	0	0
$76$	−23.3440	−0.307158
$77$	0	0
$78$	0	0
$79$	−52.2849	−0.661834	−0.330917	−	0.943660i	$-0.607358\pi$
$79$	−52.2849	−0.661834	−0.330917	+	0.943660i	$0.607358\pi$
$80$	− 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
$88$	0	0
$89$	136.779i	1.53684i	0.639943	+	0.768422i	$0.278958\pi$
$89$	136.779i	1.53684i	−0.639943	+	0.768422i	$0.721042\pi$
$90$	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	$-0.380911\pi$
$109$	79.6706	0.730923	0.365461	+	0.930826i	$0.380911\pi$
$110$	0	0
$111$	0	0
$112$	92.7148	0.827811
$113$	− 214.762i	− 1.90055i	−0.311415	−	0.950274i	$-0.600803\pi$
$113$	− 214.762i	− 1.90055i	0.311415	−	0.950274i	$-0.399197\pi$
$114$	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
$121$	0	0
$122$	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	$-0.480333\pi$
$127$	15.6836	0.123493	0.0617465	+	0.998092i	$0.480333\pi$
$128$	− 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	$0.164103\pi$
$137$	135.084i	0.986017i	−0.870025	+	0.493008i	$0.835897\pi$
$138$	0	0
$139$	−181.554	−1.30615	−0.653073	−	0.757295i	$-0.726520\pi$
$139$	−181.554	−1.30615	−0.653073	+	0.757295i	$0.726520\pi$
$140$	− 98.1922i	− 0.701373i
$141$	0	0
$142$	120.799	0.850700
$143$	0	0
$144$	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	$-0.571706\pi$
$151$	−67.4578	−0.446740	−0.223370	+	0.974734i	$0.571706\pi$
$152$	− 143.328i	− 0.942949i
$153$	0	0
$154$	0	0
$155$	− 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
$176$	0	0
$177$	0	0
$178$	−219.787	−1.23476
$179$	294.651i	1.64609i	0.567973	+	0.823047i	$0.307728\pi$
$179$	294.651i	1.64609i	−0.567973	+	0.823047i	$0.692272\pi$
$180$	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
$187$	0	0
$188$	− 53.6837i	− 0.285551i
$189$	0	0
$190$	164.347	0.864986
$191$	215.226i	1.12684i	0.826172	+	0.563419i	$0.190514\pi$
$191$	215.226i	1.12684i	−0.826172	+	0.563419i	$0.809486\pi$
$192$	0	0
$193$	−239.813	−1.24255	−0.621277	−	0.783591i	$-0.713386\pi$
$193$	−239.813	−1.24255	−0.621277	+	0.783591i	$0.713386\pi$
$194$	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
$209$	0	0
$210$	0	0
$211$	−58.8876	−0.279088	−0.139544	−	0.990216i	$-0.544564\pi$
$211$	−58.8876	−0.279088	−0.139544	+	0.990216i	$0.544564\pi$
$212$	− 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
$220$	0	0
$221$	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	$-0.567970\pi$
$241$	−102.143	−0.423828	−0.211914	+	0.977288i	$0.567970\pi$
$242$	0	0
$243$	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	$0.176810\pi$
$251$	264.723i	1.05467i	−0.849656	+	0.527337i	$0.823190\pi$
$252$	0	0
$253$	0	0
$254$	25.2016i	0.0992190i
$255$	0	0
$256$	−233.993	−0.914036
$257$	− 299.407i	− 1.16501i	−0.812828	−	0.582504i	$-0.802073\pi$
$257$	− 299.407i	− 1.16501i	0.812828	−	0.582504i	$-0.197927\pi$
$258$	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	$0.0952560\pi$
$269$	158.607i	0.589618i	−0.955556	+	0.294809i	$0.904744\pi$
$270$	0	0
$271$	−121.065	−0.446734	−0.223367	−	0.974734i	$-0.571705\pi$
$271$	−121.065	−0.446734	−0.223367	+	0.974734i	$0.571705\pi$
$272$	− 71.0294i	− 0.261138i
$273$	0	0
$274$	−217.064	−0.792203
$275$	0	0
$276$	0	0
$277$	221.505	0.799656	0.399828	−	0.916590i	$-0.369070\pi$
$277$	221.505	0.799656	0.399828	+	0.916590i	$0.369070\pi$
$278$	− 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	$-0.375168\pi$
$283$	216.323	0.764393	0.382197	+	0.924081i	$0.375168\pi$
$284$	− 106.597i	− 0.375340i
$285$	0	0
$286$	0	0
$287$	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	$-0.612203\pi$
$307$	−211.978	−0.690483	−0.345242	+	0.938514i	$0.612203\pi$
$308$	0	0
$309$	0	0
$310$	461.153	1.48759
$311$	278.236i	0.894649i	0.894372	+	0.447324i	$0.147623\pi$
$311$	278.236i	0.894649i	−0.894372	+	0.447324i	$0.852377\pi$
$312$	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	$0.248797\pi$
$317$	446.608i	1.40886i	−0.709774	+	0.704429i	$0.751203\pi$
$318$	0	0
$319$	0	0
$320$	− 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	$-0.581990\pi$
$337$	−171.695	−0.509482	−0.254741	+	0.967009i	$0.581990\pi$
$338$	236.823i	0.700660i
$339$	0	0
$340$	−75.2257	−0.221252
$341$	0	0
$342$	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	$-0.359120\pi$
$349$	298.940	0.856561	0.428280	+	0.903646i	$0.359120\pi$
$350$	243.508i	0.695737i
$351$	0	0
$352$	0	0
$353$	43.9061i	0.124380i	0.998064	+	0.0621899i	$0.0198084\pi$
$353$	43.9061i	0.124380i	−0.998064	+	0.0621899i	$0.980192\pi$
$354$	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	$-0.495214\pi$
$373$	11.2157	0.0300688	0.0150344	+	0.999887i	$0.495214\pi$
$374$	0	0
$375$	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	$0.0687243\pi$
$383$	164.100i	0.428461i	−0.976783	+	0.214230i	$0.931276\pi$
$384$	0	0
$385$	0	0
$386$	− 385.349i	− 0.998315i
$387$	0	0
$388$	181.148	0.466878
$389$	553.313i	1.42240i	0.702991	+	0.711199i	$0.251848\pi$
$389$	553.313i	1.42240i	−0.702991	+	0.711199i	$0.748152\pi$
$390$	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	$-0.897348\pi$
$401$	− 254.177i	− 0.633858i	0.948449	−	0.316929i	$-0.102652\pi$
$402$	0	0
$403$	−821.680	−2.03891
$404$	8.11849i	0.0200953i
$405$	0	0
$406$	169.540	0.417585
$407$	0	0
$408$	0	0
$409$	533.620	1.30469	0.652347	−	0.757920i	$-0.273784\pi$
$409$	533.620	1.30469	0.652347	+	0.757920i	$0.273784\pi$
$410$	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
$418$	0	0
$419$	345.851i	0.825420i	0.910862	+	0.412710i	$0.135418\pi$
$419$	345.851i	0.825420i	−0.910862	+	0.412710i	$0.864582\pi$
$420$	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	$-0.736903\pi$
$439$	−594.777	−1.35484	−0.677422	+	0.735594i	$0.736903\pi$
$440$	0	0
$441$	0	0
$442$	−244.077	−0.552210
$443$	47.2023i	0.106551i	0.998580	+	0.0532757i	$0.0169662\pi$
$443$	47.2023i	0.106551i	−0.998580	+	0.0532757i	$0.983034\pi$
$444$	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	$-0.762827\pi$
$449$	− 608.886i	− 1.35609i	0.735019	−	0.678047i	$-0.237173\pi$
$450$	0	0
$451$	0	0
$452$	− 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	$-0.324192\pi$
$457$	479.540	1.04932	0.524661	+	0.851311i	$0.324192\pi$
$458$	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	$-0.959658\pi$
$467$	− 118.057i	− 0.252799i	0.991979	−	0.126400i	$-0.0403422\pi$
$468$	0	0
$469$	650.700	1.38742
$470$	377.946i	0.804139i
$471$	0	0
$472$	−388.544	−0.823187
$473$	0	0
$474$	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
$484$	0	0
$485$	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
$506$	0	0
$507$	0	0
$508$	22.2386	0.0437767
$509$	− 261.110i	− 0.512986i	−0.966546	−	0.256493i	$-0.917433\pi$
$509$	− 261.110i	− 0.512986i	0.966546	−	0.256493i	$-0.0825671\pi$
$510$	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
$517$	0	0
$518$	671.250i	1.29585i
$519$	0	0
$520$	−962.029	−1.85006
$521$	221.574i	0.425287i	0.977130	+	0.212643i	$0.0682072\pi$
$521$	221.574i	0.425287i	−0.977130	+	0.212643i	$0.931793\pi$
$522$	0	0
$523$	444.427	0.849765	0.424882	−	0.905249i	$-0.360315\pi$
$523$	444.427	0.849765	0.424882	+	0.905249i	$0.360315\pi$
$524$	− 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
$539$	0	0
$540$	0	0
$541$	−508.057	−0.939107	−0.469553	−	0.882904i	$-0.655585\pi$
$541$	−508.057	−0.939107	−0.469553	+	0.882904i	$0.655585\pi$
$542$	− 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	$-0.513045\pi$
$547$	−44.8213	−0.0819403	−0.0409701	+	0.999160i	$0.513045\pi$
$548$	191.543i	0.349531i
$549$	0	0
$550$	0	0
$551$	− 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	$-0.128657\pi$
$571$	1049.98	1.83884	0.919422	+	0.393272i	$0.128657\pi$
$572$	0	0
$573$	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
$583$	0	0
$584$	− 977.454i	− 1.67372i
$585$	0	0
$586$	−400.789	−0.683940
$587$	258.506i	0.440385i	0.975456	+	0.220192i	$0.0706685\pi$
$587$	258.506i	0.440385i	−0.975456	+	0.220192i	$0.929332\pi$
$588$	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	$0.0448718\pi$
$599$	168.322i	0.281005i	−0.990080	+	0.140502i	$0.955128\pi$
$600$	0	0
$601$	263.203	0.437942	0.218971	−	0.975731i	$-0.429730\pi$
$601$	263.203	0.437942	0.218971	+	0.975731i	$0.429730\pi$
$602$	− 996.413i	− 1.65517i
$603$	0	0
$604$	−95.6518	−0.158364
$605$	0	0
$606$	0	0
$607$	−139.029	−0.229044	−0.114522	−	0.993421i	$-0.536534\pi$
$607$	−139.029	−0.229044	−0.114522	+	0.993421i	$0.536534\pi$
$608$	− 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	$-0.298614\pi$
$613$	724.937	1.18260	0.591302	+	0.806450i	$0.298614\pi$
$614$	− 340.623i	− 0.554760i
$615$	0	0
$616$	0	0
$617$	− 323.005i	− 0.523509i	−0.965135	−	0.261754i	$-0.915699\pi$
$617$	− 323.005i	− 0.523509i	0.965135	−	0.261754i	$-0.0843011\pi$
$618$	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
$638$	0	0
$639$	0	0
$640$	143.101	0.223595
$641$	132.212i	0.206260i	0.994668	+	0.103130i	$0.0328857\pi$
$641$	132.212i	0.206260i	−0.994668	+	0.103130i	$0.967114\pi$
$642$	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	$-0.912523\pi$
$647$	− 351.154i	− 0.542741i	0.962475	−	0.271371i	$-0.0874769\pi$
$648$	0	0
$649$	0	0
$650$	− 388.569i	− 0.597798i
$651$	0	0
$652$	182.695	0.280207
$653$	1091.70i	1.67182i	0.548866	+	0.835910i	$0.315060\pi$
$653$	1091.70i	1.67182i	−0.548866	+	0.835910i	$0.684940\pi$
$654$	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
$671$	0	0
$672$	0	0
$673$	27.3883	0.0406959	0.0203479	−	0.999793i	$-0.493523\pi$
$673$	27.3883	0.0406959	0.0203479	+	0.999793i	$0.493523\pi$
$674$	− 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
$682$	0	0
$683$	− 867.261i	− 1.26978i	−0.772602	−	0.634891i	$-0.781045\pi$
$683$	− 867.261i	− 1.26978i	0.772602	−	0.634891i	$-0.218955\pi$
$684$	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
$704$	0	0
$705$	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
$715$	0	0
$716$	417.800i	0.583520i
$717$	0	0
$718$	−1055.77	−1.47043
$719$	− 533.694i	− 0.742272i	−0.928579	−	0.371136i	$-0.878968\pi$
$719$	− 533.694i	− 0.742272i	0.928579	−	0.371136i	$-0.121032\pi$
$720$	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	$-0.601854\pi$
$733$	−461.131	−0.629101	−0.314550	+	0.949241i	$0.601854\pi$
$734$	− 638.651i	− 0.870097i
$735$	0	0
$736$	−538.251	−0.731320
$737$	0	0
$738$	0	0
$739$	−730.982	−0.989150	−0.494575	−	0.869135i	$-0.664676\pi$
$739$	−730.982	−0.989150	−0.494575	+	0.869135i	$0.664676\pi$
$740$	− 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
$748$	0	0
$749$	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	$-0.801715\pi$
$769$	−1249.12	−1.62434	−0.812172	+	0.583418i	$0.801715\pi$
$770$	0	0
$771$	0	0
$772$	−340.043	−0.440470
$773$	− 1360.02i	− 1.75941i	−0.475521	−	0.879704i	$-0.657741\pi$
$773$	− 1360.02i	− 1.75941i	0.475521	−	0.879704i	$-0.342259\pi$
$774$	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
$781$	0	0
$782$	− 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	$-0.495814\pi$
$787$	20.7010	0.0263037	0.0131518	+	0.999914i	$0.495814\pi$
$788$	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	$-0.895359\pi$
$797$	− 514.624i	− 0.645701i	0.946450	−	0.322851i	$-0.104641\pi$
$798$	0	0
$799$	323.311	0.404645
$800$	− 291.729i	− 0.364662i
$801$	0	0
$802$	408.431	0.509265
$803$	0	0
$804$	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	$-0.143317\pi$
$811$	1460.35	1.80068	0.900340	+	0.435186i	$0.143317\pi$
$812$	− 149.606i	− 0.184244i
$813$	0	0
$814$	0	0
$815$	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
$836$	0	0
$837$	0	0
$838$	−555.740	−0.663174
$839$	− 497.928i	− 0.593478i	−0.954959	−	0.296739i	$-0.904101\pi$
$839$	− 497.928i	− 0.593478i	0.954959	−	0.296739i	$-0.0958992\pi$
$840$	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
$847$	0	0
$848$	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	$-0.456109\pi$
$853$	234.494	0.274905	0.137452	+	0.990508i	$0.456109\pi$
$854$	− 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	$-0.956991\pi$
$863$	− 232.504i	− 0.269414i	0.990886	−	0.134707i	$-0.0430092\pi$
$864$	0	0
$865$	−1435.63	−1.65968
$866$	− 607.545i	− 0.701553i
$867$	0	0
$868$	730.144	0.841180
$869$	0	0
$870$	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	$-0.609975\pi$
$877$	−594.018	−0.677329	−0.338665	+	0.940907i	$0.609975\pi$
$878$	− 955.732i	− 1.08853i
$879$	0	0
$880$	0	0
$881$	− 976.976i	− 1.10894i	−0.832204	−	0.554470i	$-0.812921\pi$
$881$	− 976.976i	− 1.10894i	0.832204	−	0.554470i	$-0.187079\pi$
$882$	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
$902$	0	0
$903$	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	$0.114870\pi$
$911$	643.333i	0.706183i	−0.935589	+	0.353092i	$0.885130\pi$
$912$	0	0
$913$	0	0
$914$	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	$-0.494177\pi$
$919$	33.6201	0.0365834	0.0182917	+	0.999833i	$0.494177\pi$
$920$	− 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	$-0.899582\pi$
$929$	− 576.473i	− 0.620530i	0.950650	−	0.310265i	$-0.100418\pi$
$930$	0	0
$931$	−1238.88	−1.33069
$932$	334.918i	0.359354i
$933$	0	0
$934$	189.703	0.203108
$935$	0	0
$936$	0	0
$937$	−46.1378	−0.0492399	−0.0246200	−	0.999697i	$-0.507838\pi$
$937$	−46.1378	−0.0492399	−0.0246200	+	0.999697i	$0.507838\pi$
$938$	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
$946$	0	0
$947$	− 676.033i	− 0.713868i	−0.934130	−	0.356934i	$-0.883822\pi$
$947$	− 676.033i	− 0.713868i	0.934130	−	0.356934i	$-0.116178\pi$
$948$	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	$-0.847303\pi$
$967$	−1715.71	−1.77426	−0.887128	+	0.461523i	$0.847303\pi$
$968$	0	0
$969$	0	0
$970$	−1275.33	−1.31477
$971$	729.785i	0.751580i	0.926705	+	0.375790i	$0.122629\pi$
$971$	729.785i	0.751580i	−0.926705	+	0.375790i	$0.877371\pi$
$972$	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	$0.0967222\pi$
$977$	584.651i	0.598414i	−0.954188	+	0.299207i	$0.903278\pi$
$978$	0	0
$979$	0	0
$980$	662.888i	0.676416i
$981$	0	0
$982$	426.188	0.434000
$983$	76.3328i	0.0776529i	0.999246	+	0.0388265i	$0.0123620\pi$
$983$	76.3328i	0.0776529i	−0.999246	+	0.0388265i	$0.987638\pi$
$984$	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	$-0.230820\pi$
$997$	1492.32	1.49681	0.748405	+	0.663242i	$0.230820\pi$
$998$	791.732i	0.793319i
$999$	0	0

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
99.3.b.a.89.3	✓	8	11.10	odd	2
99.3.b.a.89.6	yes	8	33.32	even	2
1089.3.b.g.485.3		8	3.2	odd	2	inner
1089.3.b.g.485.6		8	1.1	even	1	trivial
1584.3.i.b.881.2		8	132.131	odd	2
1584.3.i.b.881.7		8	44.43	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:	\( N \)	\(=\)	\( 1089 = 3^{2} \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1089.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.60688i q^{2} +1.41795 q^{4} +6.21249i q^{5} -11.1468 q^{7} +8.70597i q^{8} +O(q^{10})\)	\(q+1.60688i q^{2} +1.41795 q^{4} +6.21249i q^{5} -11.1468 q^{7} +8.70597i q^{8} -9.98270 q^{10} +17.7871 q^{13} -17.9115i q^{14} -8.31762 q^{16} +8.53964i q^{17} -16.4632 q^{19} +8.80900i q^{20} +25.0834i q^{23} -13.5950 q^{25} +28.5817i q^{26} -15.8056 q^{28} +9.46540i q^{29} -46.1952 q^{31} +21.4585i q^{32} -13.7221 q^{34} -69.2494i q^{35} -37.4758 q^{37} -26.4543i q^{38} -54.0858 q^{40} -51.8375i q^{41} +55.6297 q^{43} -40.3058 q^{46} -37.8600i q^{47} +75.2512 q^{49} -21.8455i q^{50} +25.2212 q^{52} -58.5761i q^{53} -97.0437i q^{56} -15.2097 q^{58} +44.6296i q^{59} +10.1633 q^{61} -74.2300i q^{62} -67.7516 q^{64} +110.502i q^{65} -58.3755 q^{67} +12.1088i q^{68} +111.275 q^{70} -75.1766i q^{71} -112.274 q^{73} -60.2190i q^{74} -23.3440 q^{76} -52.2849 q^{79} -51.6731i q^{80} +83.2963 q^{82} +18.0976i q^{83} -53.0524 q^{85} +89.3901i q^{86} +136.779i q^{89} -198.269 q^{91} +35.5669i q^{92} +60.8364 q^{94} -102.278i q^{95} +127.754 q^{97} +120.919i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 16 q^{4} - 16 q^{7}+O(q^{10}) \) 8 * q - 16 * q^4 - 16 * q^7	\( 8 q - 16 q^{4} - 16 q^{7} + 48 q^{10} + 8 q^{13} + 104 q^{16} - 40 q^{19} - 112 q^{25} + 32 q^{28} - 56 q^{31} - 216 q^{34} + 136 q^{37} - 432 q^{40} + 104 q^{43} - 24 q^{46} - 96 q^{49} - 280 q^{52} - 432 q^{58} + 8 q^{61} - 592 q^{64} + 112 q^{67} + 168 q^{70} - 448 q^{73} + 344 q^{76} - 448 q^{79} + 504 q^{82} - 48 q^{85} - 544 q^{91} - 360 q^{94} - 152 q^{97}+O(q^{100}) \) 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 - 544 * q^91 - 360 * q^94 - 152 * q^97

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