LMFDB - Embedded newform 1089.3.c.c.604.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1089,3,Mod(604,1089)]

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([0, 1]))

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

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

chi := DirichletCharacter("1089.604");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1089 = 3^{2} \cdot 11^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1089.c (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:	$4$
Coefficient field:	$\Q(\sqrt{-2}, \sqrt{3})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 4x^{2} + 1 $ x^4 + 4*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 363)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			604.4
Root			$1.93185i$ of defining polynomial
Character	$\chi$	$=$	1089.604
Dual form			1089.3.c.c.604.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.86370i q^{2} -10.9282 q^{4} +5.46410 q^{5} +1.93185i q^{7} -26.7685i q^{8} +O(q^{10})$	$q+3.86370i q^{2} -10.9282 q^{4} +5.46410 q^{5} +1.93185i q^{7} -26.7685i q^{8} +21.1117i q^{10} -10.4543i q^{13} -7.46410 q^{14} +59.7128 q^{16} -1.31268i q^{17} -4.65874i q^{19} -59.7128 q^{20} +28.6410 q^{23} +4.85641 q^{25} +40.3923 q^{26} -21.1117i q^{28} -40.2271i q^{29} +26.2679 q^{31} +123.639i q^{32} +5.07180 q^{34} +10.5558i q^{35} +70.2295 q^{37} +18.0000 q^{38} -146.266i q^{40} +24.9754i q^{41} +38.5355i q^{43} +110.660i q^{46} -67.2820 q^{47} +45.2679 q^{49} +18.7637i q^{50} +114.247i q^{52} +65.4256 q^{53} +51.7128 q^{56} +155.426 q^{58} -1.32051 q^{59} +58.2230i q^{61} +101.492i q^{62} -238.851 q^{64} -57.1233i q^{65} -0.162831 q^{67} +14.3452i q^{68} -40.7846 q^{70} +25.7513 q^{71} +92.6917i q^{73} +271.346i q^{74} +50.9117i q^{76} -88.5506i q^{79} +326.277 q^{80} -96.4974 q^{82} -152.755i q^{83} -7.17260i q^{85} -148.890 q^{86} -15.2154 q^{89} +20.1962 q^{91} -312.995 q^{92} -259.958i q^{94} -25.4558i q^{95} +68.9230 q^{97} +174.902i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 16 q^{4} + 8 q^{5}+O(q^{10}) $ 4 * q - 16 * q^4 + 8 * q^5	$ 4 q - 16 q^{4} + 8 q^{5} - 16 q^{14} + 128 q^{16} - 128 q^{20} - 24 q^{23} - 36 q^{25} + 120 q^{26} + 112 q^{31} + 48 q^{34} + 80 q^{37} + 72 q^{38} + 8 q^{47} + 188 q^{49} + 40 q^{53} + 96 q^{56} + 400 q^{58} + 64 q^{59} - 512 q^{64} - 160 q^{67} - 80 q^{70} + 200 q^{71} + 640 q^{80} - 192 q^{82} - 360 q^{86} - 144 q^{89} + 60 q^{91} - 864 q^{92} - 140 q^{97}+O(q^{100}) $ 4 * q - 16 * q^4 + 8 * q^5 - 16 * q^14 + 128 * q^16 - 128 * q^20 - 24 * q^23 - 36 * q^25 + 120 * q^26 + 112 * q^31 + 48 * q^34 + 80 * q^37 + 72 * q^38 + 8 * q^47 + 188 * q^49 + 40 * q^53 + 96 * q^56 + 400 * q^58 + 64 * q^59 - 512 * q^64 - 160 * q^67 - 80 * q^70 + 200 * q^71 + 640 * q^80 - 192 * q^82 - 360 * q^86 - 144 * q^89 + 60 * q^91 - 864 * q^92 - 140 * 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 $

$2$	3.86370i	1.93185i	0.258819	+	0.965926i	$0.416667\pi$
$2$	3.86370i	1.93185i	−0.258819	+	0.965926i	$0.583333\pi$
$3$	0	0
$3$	0	0
$4$	−10.9282	−2.73205
$5$	5.46410	1.09282	0.546410	−	0.837518i	$-0.315994\pi$
$5$	5.46410	1.09282	0.546410	+	0.837518i	$0.315994\pi$
$6$	0	0
$7$	1.93185i	0.275979i	0.990434	+	0.137989i	$0.0440640\pi$
$7$	1.93185i	0.275979i	−0.990434	+	0.137989i	$0.955936\pi$
$8$	− 26.7685i	− 3.34607i
$9$	0	0
$10$	21.1117i	2.11117i
$11$	0	0
$11$	0	0
$12$	0	0
$13$	− 10.4543i	− 0.804177i	−0.915601	−	0.402088i	$-0.868284\pi$
$13$	− 10.4543i	− 0.804177i	0.915601	−	0.402088i	$-0.131716\pi$
$14$	−7.46410	−0.533150
$15$	0	0
$16$	59.7128	3.73205
$17$	− 1.31268i	− 0.0772163i	−0.999254	−	0.0386082i	$-0.987708\pi$
$17$	− 1.31268i	− 0.0772163i	0.999254	−	0.0386082i	$-0.0122924\pi$
$18$	0	0
$19$	− 4.65874i	− 0.245197i	−0.992456	−	0.122598i	$-0.960877\pi$
$19$	− 4.65874i	− 0.245197i	0.992456	−	0.122598i	$-0.0391227\pi$
$20$	−59.7128	−2.98564
$21$	0	0
$22$	0	0
$23$	28.6410	1.24526	0.622631	−	0.782516i	$-0.286064\pi$
$23$	28.6410	1.24526	0.622631	+	0.782516i	$0.286064\pi$
$24$	0	0
$25$	4.85641	0.194256
$26$	40.3923	1.55355
$27$	0	0
$28$	− 21.1117i	− 0.753988i
$29$	− 40.2271i	− 1.38714i	−0.720388	−	0.693571i	$-0.756036\pi$
$29$	− 40.2271i	− 1.38714i	0.720388	−	0.693571i	$-0.243964\pi$
$30$	0	0
$31$	26.2679	0.847353	0.423677	−	0.905814i	$-0.360739\pi$
$31$	26.2679	0.847353	0.423677	+	0.905814i	$0.360739\pi$
$32$	123.639i	3.86370i
$33$	0	0
$34$	5.07180	0.149170
$35$	10.5558i	0.301595i
$36$	0	0
$37$	70.2295	1.89809	0.949047	−	0.315135i	$-0.102050\pi$
$37$	70.2295	1.89809	0.949047	+	0.315135i	$0.102050\pi$
$38$	18.0000	0.473684
$39$	0	0
$40$	− 146.266i	− 3.65665i
$41$	24.9754i	0.609155i	0.952487	+	0.304578i	$0.0985153\pi$
$41$	24.9754i	0.609155i	−0.952487	+	0.304578i	$0.901485\pi$
$42$	0	0
$43$	38.5355i	0.896174i	0.893990	+	0.448087i	$0.147895\pi$
$43$	38.5355i	0.896174i	−0.893990	+	0.448087i	$0.852105\pi$
$44$	0	0
$45$	0	0
$46$	110.660i	2.40566i
$47$	−67.2820	−1.43153	−0.715766	−	0.698340i	$-0.753922\pi$
$47$	−67.2820	−1.43153	−0.715766	+	0.698340i	$0.753922\pi$
$48$	0	0
$49$	45.2679	0.923836
$50$	18.7637i	0.375274i
$51$	0	0
$52$	114.247i	2.19705i
$53$	65.4256	1.23445	0.617223	−	0.786788i	$-0.288258\pi$
$53$	65.4256	1.23445	0.617223	+	0.786788i	$0.288258\pi$
$54$	0	0
$55$	0	0
$56$	51.7128	0.923443
$57$	0	0
$58$	155.426	2.67975
$59$	−1.32051	−0.0223815	−0.0111907	−	0.999937i	$-0.503562\pi$
$59$	−1.32051	−0.0223815	−0.0111907	+	0.999937i	$0.503562\pi$
$60$	0	0
$61$	58.2230i	0.954475i	0.878774	+	0.477238i	$0.158362\pi$
$61$	58.2230i	0.954475i	−0.878774	+	0.477238i	$0.841638\pi$
$62$	101.492i	1.63696i
$63$	0	0
$64$	−238.851	−3.73205
$65$	− 57.1233i	− 0.878821i
$66$	0	0
$67$	−0.162831	−0.00243032	−0.00121516	−	0.999999i	$-0.500387\pi$
$67$	−0.162831	−0.00243032	−0.00121516	+	0.999999i	$0.500387\pi$
$68$	14.3452i	0.210959i
$69$	0	0
$70$	−40.7846	−0.582637
$71$	25.7513	0.362694	0.181347	−	0.983419i	$-0.441954\pi$
$71$	25.7513	0.362694	0.181347	+	0.983419i	$0.441954\pi$
$72$	0	0
$73$	92.6917i	1.26975i	0.772615	+	0.634875i	$0.218948\pi$
$73$	92.6917i	1.26975i	−0.772615	+	0.634875i	$0.781052\pi$
$74$	271.346i	3.66684i
$75$	0	0
$76$	50.9117i	0.669891i
$77$	0	0
$78$	0	0
$79$	− 88.5506i	− 1.12089i	−0.828190	−	0.560447i	$-0.810629\pi$
$79$	− 88.5506i	− 1.12089i	0.828190	−	0.560447i	$-0.189371\pi$
$80$	326.277	4.07846
$81$	0	0
$82$	−96.4974	−1.17680
$83$	− 152.755i	− 1.84042i	−0.391423	−	0.920211i	$-0.628017\pi$
$83$	− 152.755i	− 1.84042i	0.391423	−	0.920211i	$-0.371983\pi$
$84$	0	0
$85$	− 7.17260i	− 0.0843836i
$86$	−148.890	−1.73128
$87$	0	0
$88$	0	0
$89$	−15.2154	−0.170959	−0.0854797	−	0.996340i	$-0.527242\pi$
$89$	−15.2154	−0.170959	−0.0854797	+	0.996340i	$0.527242\pi$
$90$	0	0
$91$	20.1962	0.221936
$92$	−312.995	−3.40212
$93$	0	0
$94$	− 259.958i	− 2.76551i
$95$	− 25.4558i	− 0.267956i
$96$	0	0
$97$	68.9230	0.710547	0.355273	−	0.934762i	$-0.384388\pi$
$97$	68.9230	0.710547	0.355273	+	0.934762i	$0.384388\pi$
$98$	174.902i	1.78471i
$99$	0	0
$100$	−53.0718	−0.530718
$101$	9.77804i	0.0968123i	0.998828	+	0.0484062i	$0.0154142\pi$
$101$	9.77804i	0.0968123i	−0.998828	+	0.0484062i	$0.984586\pi$
$102$	0	0
$103$	−102.138	−0.991635	−0.495818	−	0.868427i	$-0.665132\pi$
$103$	−102.138	−0.991635	−0.495818	+	0.868427i	$0.665132\pi$
$104$	−279.846	−2.69083
$105$	0	0
$106$	252.785i	2.38477i
$107$	− 21.8895i	− 0.204574i	−0.994755	−	0.102287i	$-0.967384\pi$
$107$	− 21.8895i	− 0.204574i	0.994755	−	0.102287i	$-0.0326161\pi$
$108$	0	0
$109$	48.4450i	0.444449i	0.974996	+	0.222225i	$0.0713318\pi$
$109$	48.4450i	0.444449i	−0.974996	+	0.222225i	$0.928668\pi$
$110$	0	0
$111$	0	0
$112$	115.356i	1.02997i
$113$	35.6462	0.315453	0.157726	−	0.987483i	$-0.449584\pi$
$113$	35.6462	0.315453	0.157726	+	0.987483i	$0.449584\pi$
$114$	0	0
$115$	156.497	1.36085
$116$	439.610i	3.78974i
$117$	0	0
$118$	− 5.10205i	− 0.0432377i
$119$	2.53590	0.0213101
$120$	0	0
$121$	0	0
$122$	−224.956	−1.84390
$123$	0	0
$124$	−287.061	−2.31501
$125$	−110.067	−0.880533
$126$	0	0
$127$	134.766i	1.06115i	0.847637	+	0.530576i	$0.178024\pi$
$127$	134.766i	1.06115i	−0.847637	+	0.530576i	$0.821976\pi$
$128$	− 428.296i	− 3.34607i
$129$	0	0
$130$	220.708	1.69775
$131$	− 11.9428i	− 0.0911667i	−0.998961	−	0.0455834i	$-0.985485\pi$
$131$	− 11.9428i	− 0.0911667i	0.998961	−	0.0455834i	$-0.0145147\pi$
$132$	0	0
$133$	9.00000	0.0676692
$134$	− 0.629132i	− 0.00469502i
$135$	0	0
$136$	−35.1384	−0.258371
$137$	62.7461	0.458001	0.229000	−	0.973426i	$-0.426454\pi$
$137$	62.7461	0.458001	0.229000	+	0.973426i	$0.426454\pi$
$138$	0	0
$139$	− 0.101536i	0 0.000730475i	−1.00000		0.000365237i	$-0.999884\pi$
$139$	− 0.101536i	0 0.000730475i	1.00000		0.000365237i	$-0.000116259\pi$
$140$	− 115.356i	− 0.823974i
$141$	0	0
$142$	99.4953i	0.700671i
$143$	0	0
$144$	0	0
$145$	− 219.805i	− 1.51590i
$146$	−358.133	−2.45297
$147$	0	0
$148$	−767.482	−5.18569
$149$	− 64.9994i	− 0.436238i	−0.975922	−	0.218119i	$-0.930008\pi$
$149$	− 64.9994i	− 0.436238i	0.975922	−	0.218119i	$-0.0699920\pi$
$150$	0	0
$151$	227.322i	1.50544i	0.658338	+	0.752722i	$0.271260\pi$
$151$	227.322i	1.50544i	−0.658338	+	0.752722i	$0.728740\pi$
$152$	−124.708	−0.820445
$153$	0	0
$154$	0	0
$155$	143.531	0.926005
$156$	0	0
$157$	−193.329	−1.23140	−0.615699	−	0.787982i	$-0.711126\pi$
$157$	−193.329	−1.23140	−0.615699	+	0.787982i	$0.711126\pi$
$158$	342.133	2.16540
$159$	0	0
$160$	675.573i	4.22233i
$161$	55.3302i	0.343666i
$162$	0	0
$163$	167.497	1.02759	0.513796	−	0.857913i	$-0.328239\pi$
$163$	167.497	1.02759	0.513796	+	0.857913i	$0.328239\pi$
$164$	− 272.936i	− 1.66424i
$165$	0	0
$166$	590.200	3.55542
$167$	− 281.550i	− 1.68593i	−0.537970	−	0.842964i	$-0.680809\pi$
$167$	− 281.550i	− 1.68593i	0.537970	−	0.842964i	$-0.319191\pi$
$168$	0	0
$169$	59.7077	0.353300
$170$	27.7128	0.163017
$171$	0	0
$172$	− 421.124i	− 2.44839i
$173$	− 170.057i	− 0.982991i	−0.870880	−	0.491495i	$-0.836451\pi$
$173$	− 170.057i	− 0.982991i	0.870880	−	0.491495i	$-0.163549\pi$
$174$	0	0
$175$	9.38186i	0.0536106i
$176$	0	0
$177$	0	0
$178$	− 58.7878i	− 0.330268i
$179$	196.354	1.09695	0.548474	−	0.836167i	$-0.315209\pi$
$179$	196.354	1.09695	0.548474	+	0.836167i	$0.315209\pi$
$180$	0	0
$181$	196.636	1.08639	0.543193	−	0.839608i	$-0.317215\pi$
$181$	196.636	1.08639	0.543193	+	0.839608i	$0.317215\pi$
$182$	78.0319i	0.428747i
$183$	0	0
$184$	− 766.678i	− 4.16673i
$185$	383.741	2.07428
$186$	0	0
$187$	0	0
$188$	735.272	3.91102
$189$	0	0
$190$	98.3538	0.517652
$191$	−61.7795	−0.323453	−0.161726	−	0.986836i	$-0.551706\pi$
$191$	−61.7795	−0.323453	−0.161726	+	0.986836i	$0.551706\pi$
$192$	0	0
$193$	− 325.624i	− 1.68717i	−0.536997	−	0.843584i	$-0.680441\pi$
$193$	− 325.624i	− 1.68717i	0.536997	−	0.843584i	$-0.319559\pi$
$194$	266.298i	1.37267i
$195$	0	0
$196$	−494.697	−2.52397
$197$	127.170i	0.645535i	0.946478	+	0.322767i	$0.104613\pi$
$197$	127.170i	0.645535i	−0.946478	+	0.322767i	$0.895387\pi$
$198$	0	0
$199$	−26.2679	−0.132000	−0.0659999	−	0.997820i	$-0.521024\pi$
$199$	−26.2679	−0.132000	−0.0659999	+	0.997820i	$0.521024\pi$
$200$	− 129.999i	− 0.649994i
$201$	0	0
$202$	−37.7795	−0.187027
$203$	77.7128	0.382822
$204$	0	0
$205$	136.468i	0.665697i
$206$	− 394.633i	− 1.91569i
$207$	0	0
$208$	− 624.256i	− 3.00123i
$209$	0	0
$210$	0	0
$211$	95.6217i	0.453183i	0.973990	+	0.226592i	$0.0727583\pi$
$211$	95.6217i	0.453183i	−0.973990	+	0.226592i	$0.927242\pi$
$212$	−714.985	−3.37257
$213$	0	0
$214$	84.5744	0.395207
$215$	210.562i	0.979358i
$216$	0	0
$217$	50.7458i	0.233852i
$218$	−187.177	−0.858610
$219$	0	0
$220$	0	0
$221$	−13.7231	−0.0620956
$222$	0	0
$223$	−326.138	−1.46250	−0.731252	−	0.682107i	$-0.761064\pi$
$223$	−326.138	−1.46250	−0.731252	+	0.682107i	$0.761064\pi$
$224$	−238.851	−1.06630
$225$	0	0
$226$	137.726i	0.609408i
$227$	− 25.1984i	− 0.111006i	−0.998459	−	0.0555030i	$-0.982324\pi$
$227$	− 25.1984i	− 0.111006i	0.998459	−	0.0555030i	$-0.0176762\pi$
$228$	0	0
$229$	172.210	0.752010	0.376005	−	0.926618i	$-0.377298\pi$
$229$	172.210	0.752010	0.376005	+	0.926618i	$0.377298\pi$
$230$	604.660i	2.62895i
$231$	0	0
$232$	−1076.82	−4.64147
$233$	56.6827i	0.243273i	0.992575	+	0.121637i	$0.0388143\pi$
$233$	56.6827i	0.243273i	−0.992575	+	0.121637i	$0.961186\pi$
$234$	0	0
$235$	−367.636	−1.56441
$236$	14.4308	0.0611474
$237$	0	0
$238$	9.79796i	0.0411679i
$239$	258.759i	1.08267i	0.840805	+	0.541337i	$0.182082\pi$
$239$	258.759i	1.08267i	−0.840805	+	0.541337i	$0.817918\pi$
$240$	0	0
$241$	− 340.350i	− 1.41224i	−0.708091	−	0.706121i	$-0.750443\pi$
$241$	− 340.350i	− 1.41224i	0.708091	−	0.706121i	$-0.249557\pi$
$242$	0	0
$243$	0	0
$244$	− 636.273i	− 2.60767i
$245$	247.349	1.00959
$246$	0	0
$247$	−48.7039	−0.197182
$248$	− 703.154i	− 2.83530i
$249$	0	0
$250$	− 425.265i	− 1.70106i
$251$	83.7025	0.333476	0.166738	−	0.986001i	$-0.446677\pi$
$251$	83.7025	0.333476	0.166738	+	0.986001i	$0.446677\pi$
$252$	0	0
$253$	0	0
$254$	−520.697	−2.04999
$255$	0	0
$256$	699.405	2.73205
$257$	32.8231	0.127716	0.0638581	−	0.997959i	$-0.479659\pi$
$257$	32.8231	0.127716	0.0638581	+	0.997959i	$0.479659\pi$
$258$	0	0
$259$	135.673i	0.523834i
$260$	624.256i	2.40098i
$261$	0	0
$262$	46.1436	0.176121
$263$	138.167i	0.525349i	0.964884	+	0.262675i	$0.0846046\pi$
$263$	138.167i	0.525349i	−0.964884	+	0.262675i	$0.915395\pi$
$264$	0	0
$265$	357.492	1.34903
$266$	34.7733i	0.130727i
$267$	0	0
$268$	1.77945	0.00663976
$269$	70.6307	0.262568	0.131284	−	0.991345i	$-0.458090\pi$
$269$	70.6307	0.262568	0.131284	+	0.991345i	$0.458090\pi$
$270$	0	0
$271$	148.329i	0.547340i	0.961824	+	0.273670i	$0.0882376\pi$
$271$	148.329i	0.547340i	−0.961824	+	0.273670i	$0.911762\pi$
$272$	− 78.3837i	− 0.288175i
$273$	0	0
$274$	242.432i	0.884790i
$275$	0	0
$276$	0	0
$277$	− 154.288i	− 0.556996i	−0.960437	−	0.278498i	$-0.910163\pi$
$277$	− 154.288i	− 0.556996i	0.960437	−	0.278498i	$-0.0898366\pi$
$278$	0.392305	0.00141117
$279$	0	0
$280$	282.564	1.00916
$281$	113.028i	0.402236i	0.979567	+	0.201118i	$0.0644574\pi$
$281$	113.028i	0.402236i	−0.979567	+	0.201118i	$0.935543\pi$
$282$	0	0
$283$	550.538i	1.94536i	0.232143	+	0.972682i	$0.425426\pi$
$283$	550.538i	1.94536i	−0.232143	+	0.972682i	$0.574574\pi$
$284$	−281.415	−0.990899
$285$	0	0
$286$	0	0
$287$	−48.2487	−0.168114
$288$	0	0
$289$	287.277	0.994038
$290$	849.261	2.92849
$291$	0	0
$292$	− 1012.95i	− 3.46902i
$293$	− 340.670i	− 1.16269i	−0.813656	−	0.581347i	$-0.802526\pi$
$293$	− 340.670i	− 1.16269i	0.813656	−	0.581347i	$-0.197474\pi$
$294$	0	0
$295$	−7.21539	−0.0244590
$296$	− 1879.94i	− 6.35115i
$297$	0	0
$298$	251.138	0.842746
$299$	− 299.422i	− 1.00141i
$300$	0	0
$301$	−74.4449	−0.247325
$302$	−878.305	−2.90829
$303$	0	0
$304$	− 278.187i	− 0.915088i
$305$	318.136i	1.04307i
$306$	0	0
$307$	481.653i	1.56890i	0.620191	+	0.784451i	$0.287055\pi$
$307$	481.653i	1.56890i	−0.620191	+	0.784451i	$0.712945\pi$
$308$	0	0
$309$	0	0
$310$	554.560i	1.78890i
$311$	219.636	0.706225	0.353112	−	0.935581i	$-0.385123\pi$
$311$	219.636	0.706225	0.353112	+	0.935581i	$0.385123\pi$
$312$	0	0
$313$	−16.3435	−0.0522157	−0.0261079	−	0.999659i	$-0.508311\pi$
$313$	−16.3435	−0.0522157	−0.0261079	+	0.999659i	$0.508311\pi$
$314$	− 746.968i	− 2.37888i
$315$	0	0
$316$	967.699i	3.06234i
$317$	526.277	1.66018	0.830090	−	0.557630i	$-0.188289\pi$
$317$	526.277	1.66018	0.830090	+	0.557630i	$0.188289\pi$
$318$	0	0
$319$	0	0
$320$	−1305.11	−4.07846
$321$	0	0
$322$	−213.779	−0.663911
$323$	−6.11543	−0.0189332
$324$	0	0
$325$	− 50.7703i	− 0.156216i
$326$	647.160i	1.98515i
$327$	0	0
$328$	668.554	2.03827
$329$	− 129.979i	− 0.395073i
$330$	0	0
$331$	138.138	0.417337	0.208668	−	0.977986i	$-0.433087\pi$
$331$	138.138	0.417337	0.208668	+	0.977986i	$0.433087\pi$
$332$	1669.34i	5.02812i
$333$	0	0
$334$	1087.83	3.25696
$335$	−0.889727	−0.00265590
$336$	0	0
$337$	− 449.005i	− 1.33236i	−0.745792	−	0.666179i	$-0.767929\pi$
$337$	− 449.005i	− 1.33236i	0.745792	−	0.666179i	$-0.232071\pi$
$338$	230.693i	0.682523i
$339$	0	0
$340$	78.3837i	0.230540i
$341$	0	0
$342$	0	0
$343$	182.112i	0.530938i
$344$	1031.54	2.99866
$345$	0	0
$346$	657.051	1.89899
$347$	497.497i	1.43371i	0.697223	+	0.716854i	$0.254419\pi$
$347$	497.497i	1.43371i	−0.697223	+	0.716854i	$0.745581\pi$
$348$	0	0
$349$	473.811i	1.35763i	0.734311	+	0.678813i	$0.237505\pi$
$349$	473.811i	1.35763i	−0.734311	+	0.678813i	$0.762495\pi$
$350$	−36.2487	−0.103568
$351$	0	0
$352$	0	0
$353$	−322.956	−0.914891	−0.457445	−	0.889238i	$-0.651235\pi$
$353$	−322.956	−0.914891	−0.457445	+	0.889238i	$0.651235\pi$
$354$	0	0
$355$	140.708	0.396360
$356$	166.277	0.467070
$357$	0	0
$358$	758.653i	2.11914i
$359$	− 56.7570i	− 0.158098i	−0.996871	−	0.0790488i	$-0.974812\pi$
$359$	− 56.7570i	− 0.158098i	0.996871	−	0.0790488i	$-0.0251883\pi$
$360$	0	0
$361$	339.296	0.939878
$362$	759.743i	2.09874i
$363$	0	0
$364$	−220.708	−0.606340
$365$	506.477i	1.38761i
$366$	0	0
$367$	7.93336	0.0216168	0.0108084	−	0.999942i	$-0.496560\pi$
$367$	7.93336	0.0216168	0.0108084	+	0.999942i	$0.496560\pi$
$368$	1710.24	4.64738
$369$	0	0
$370$	1482.66i	4.00719i
$371$	126.393i	0.340681i
$372$	0	0
$373$	− 439.390i	− 1.17799i	−0.808137	−	0.588994i	$-0.799524\pi$
$373$	− 439.390i	− 1.17799i	0.808137	−	0.588994i	$-0.200476\pi$
$374$	0	0
$375$	0	0
$376$	1801.04i	4.79000i
$377$	−420.546	−1.11551
$378$	0	0
$379$	32.1230	0.0847572	0.0423786	−	0.999102i	$-0.486506\pi$
$379$	32.1230	0.0847572	0.0423786	+	0.999102i	$0.486506\pi$
$380$	278.187i	0.732070i
$381$	0	0
$382$	− 238.697i	− 0.624863i
$383$	−430.918	−1.12511	−0.562556	−	0.826759i	$-0.690182\pi$
$383$	−430.918	−1.12511	−0.562556	+	0.826759i	$0.690182\pi$
$384$	0	0
$385$	0	0
$386$	1258.11	3.25936
$387$	0	0
$388$	−753.205	−1.94125
$389$	−392.459	−1.00889	−0.504446	−	0.863443i	$-0.668303\pi$
$389$	−392.459	−1.00889	−0.504446	+	0.863443i	$0.668303\pi$
$390$	0	0
$391$	− 37.5964i	− 0.0961545i
$392$	− 1211.76i	− 3.09121i
$393$	0	0
$394$	−491.349	−1.24708
$395$	− 483.850i	− 1.22494i
$396$	0	0
$397$	−364.580	−0.918336	−0.459168	−	0.888349i	$-0.651852\pi$
$397$	−364.580	−0.918336	−0.459168	+	0.888349i	$0.651852\pi$
$398$	− 101.492i	− 0.255004i
$399$	0	0
$400$	289.990	0.724974
$401$	65.1487	0.162466	0.0812329	−	0.996695i	$-0.474114\pi$
$401$	65.1487	0.162466	0.0812329	+	0.996695i	$0.474114\pi$
$402$	0	0
$403$	− 274.613i	− 0.681422i
$404$	− 106.856i	− 0.264496i
$405$	0	0
$406$	300.259i	0.739555i
$407$	0	0
$408$	0	0
$409$	− 315.942i	− 0.772474i	−0.922400	−	0.386237i	$-0.873775\pi$
$409$	− 315.942i	− 0.772474i	0.922400	−	0.386237i	$-0.126225\pi$
$410$	−527.272	−1.28603
$411$	0	0
$412$	1116.19	2.70920
$413$	− 2.55103i	− 0.00617682i
$414$	0	0
$415$	− 834.669i	− 2.01125i
$416$	1292.55	3.10710
$417$	0	0
$418$	0	0
$419$	−87.5204	−0.208879	−0.104440	−	0.994531i	$-0.533305\pi$
$419$	−87.5204	−0.208879	−0.104440	+	0.994531i	$0.533305\pi$
$420$	0	0
$421$	655.979	1.55815	0.779073	−	0.626933i	$-0.215690\pi$
$421$	655.979	1.55815	0.779073	+	0.626933i	$0.215690\pi$
$422$	−369.454	−0.875483
$423$	0	0
$424$	− 1751.35i	− 4.13054i
$425$	− 6.37490i	− 0.0149998i
$426$	0	0
$427$	−112.478	−0.263415
$428$	239.212i	0.558908i
$429$	0	0
$430$	−813.549	−1.89197
$431$	− 689.681i	− 1.60019i	−0.599875	−	0.800094i	$-0.704783\pi$
$431$	− 689.681i	− 1.60019i	0.599875	−	0.800094i	$-0.295217\pi$
$432$	0	0
$433$	392.072	0.905478	0.452739	−	0.891643i	$-0.350447\pi$
$433$	392.072	0.905478	0.452739	+	0.891643i	$0.350447\pi$
$434$	−196.067	−0.451766
$435$	0	0
$436$	− 529.416i	− 1.21426i
$437$	− 133.431i	− 0.305334i
$438$	0	0
$439$	475.693i	1.08358i	0.840513	+	0.541792i	$0.182254\pi$
$439$	475.693i	1.08358i	−0.840513	+	0.541792i	$0.817746\pi$
$440$	0	0
$441$	0	0
$442$	− 53.0221i	− 0.119959i
$443$	298.382	0.673549	0.336774	−	0.941585i	$-0.390664\pi$
$443$	298.382	0.673549	0.336774	+	0.941585i	$0.390664\pi$
$444$	0	0
$445$	−83.1384	−0.186828
$446$	− 1260.10i	− 2.82534i
$447$	0	0
$448$	− 461.425i	− 1.02997i
$449$	−705.808	−1.57195	−0.785977	−	0.618255i	$-0.787840\pi$
$449$	−705.808	−1.57195	−0.785977	+	0.618255i	$0.787840\pi$
$450$	0	0
$451$	0	0
$452$	−389.549	−0.861833
$453$	0	0
$454$	97.3590	0.214447
$455$	110.354	0.242536
$456$	0	0
$457$	− 427.720i	− 0.935929i	−0.883747	−	0.467965i	$-0.844987\pi$
$457$	− 427.720i	− 0.935929i	0.883747	−	0.467965i	$-0.155013\pi$
$458$	665.369i	1.45277i
$459$	0	0
$460$	−1710.24	−3.71790
$461$	192.487i	0.417542i	0.977965	+	0.208771i	$0.0669464\pi$
$461$	192.487i	0.417542i	−0.977965	+	0.208771i	$0.933054\pi$
$462$	0	0
$463$	−307.226	−0.663554	−0.331777	−	0.943358i	$-0.607648\pi$
$463$	−307.226	−0.663554	−0.331777	+	0.943358i	$0.607648\pi$
$464$	− 2402.07i	− 5.17688i
$465$	0	0
$466$	−219.005	−0.469968
$467$	821.031	1.75810	0.879048	−	0.476734i	$-0.158179\pi$
$467$	821.031	1.75810	0.879048	+	0.476734i	$0.158179\pi$
$468$	0	0
$469$	− 0.314566i	0 0.000670717i
$470$	− 1420.44i	− 3.02220i
$471$	0	0
$472$	35.3480i	0.0748899i
$473$	0	0
$474$	0	0
$475$	− 22.6247i	− 0.0476310i
$476$	−27.7128	−0.0582202
$477$	0	0
$478$	−999.769	−2.09157
$479$	− 466.250i	− 0.973382i	−0.873574	−	0.486691i	$-0.838204\pi$
$479$	− 466.250i	− 0.973382i	0.873574	−	0.486691i	$-0.161796\pi$
$480$	0	0
$481$	− 734.200i	− 1.52640i
$482$	1315.01	2.72824
$483$	0	0
$484$	0	0
$485$	376.603	0.776500
$486$	0	0
$487$	179.215	0.367999	0.183999	−	0.982926i	$-0.441096\pi$
$487$	179.215	0.367999	0.183999	+	0.982926i	$0.441096\pi$
$488$	1558.54	3.19374
$489$	0	0
$490$	955.682i	1.95037i
$491$	− 394.915i	− 0.804308i	−0.915572	−	0.402154i	$-0.868262\pi$
$491$	− 394.915i	− 0.804308i	0.915572	−	0.402154i	$-0.131738\pi$
$492$	0	0
$493$	−52.8052	−0.107110
$494$	− 188.177i	− 0.380926i
$495$	0	0
$496$	1568.53	3.16237
$497$	49.7477i	0.100096i
$498$	0	0
$499$	−589.347	−1.18106	−0.590528	−	0.807017i	$-0.701081\pi$
$499$	−589.347	−1.18106	−0.590528	+	0.807017i	$0.701081\pi$
$500$	1202.83	2.40566
$501$	0	0
$502$	323.402i	0.644226i
$503$	361.672i	0.719031i	0.933139	+	0.359515i	$0.117058\pi$
$503$	361.672i	0.719031i	−0.933139	+	0.359515i	$0.882942\pi$
$504$	0	0
$505$	53.4282i	0.105798i
$506$	0	0
$507$	0	0
$508$	− 1472.75i	− 2.89912i
$509$	−810.449	−1.59224	−0.796119	−	0.605141i	$-0.793117\pi$
$509$	−810.449	−1.59224	−0.796119	+	0.605141i	$0.793117\pi$
$510$	0	0
$511$	−179.067	−0.350424
$512$	989.108i	1.93185i
$513$	0	0
$514$	126.819i	0.246729i
$515$	−558.095	−1.08368
$516$	0	0
$517$	0	0
$518$	−524.200	−1.01197
$519$	0	0
$520$	−1529.11	−2.94059
$521$	−652.018	−1.25147	−0.625737	−	0.780034i	$-0.715202\pi$
$521$	−652.018	−1.25147	−0.625737	+	0.780034i	$0.715202\pi$
$522$	0	0
$523$	802.699i	1.53480i	0.641170	+	0.767399i	$0.278449\pi$
$523$	802.699i	1.53480i	−0.641170	+	0.767399i	$0.721551\pi$
$524$	130.514i	0.249072i
$525$	0	0
$526$	−533.836	−1.01490
$527$	− 34.4813i	− 0.0654295i
$528$	0	0
$529$	291.308	0.550676
$530$	1381.24i	2.60612i
$531$	0	0
$532$	−98.3538	−0.184876
$533$	261.100	0.489869
$534$	0	0
$535$	− 119.606i	− 0.223563i
$536$	4.35876i	0.00813201i
$537$	0	0
$538$	272.896i	0.507242i
$539$	0	0
$540$	0	0
$541$	38.3324i	0.0708548i	0.999372	+	0.0354274i	$0.0112792\pi$
$541$	38.3324i	0.0708548i	−0.999372	+	0.0354274i	$0.988721\pi$
$542$	−573.100	−1.05738
$543$	0	0
$544$	162.297	0.298341
$545$	264.708i	0.485703i
$546$	0	0
$547$	− 843.593i	− 1.54222i	−0.636704	−	0.771109i	$-0.719703\pi$
$547$	− 843.593i	− 1.54222i	0.636704	−	0.771109i	$-0.280297\pi$
$548$	−685.703	−1.25128
$549$	0	0
$550$	0	0
$551$	−187.408	−0.340123
$552$	0	0
$553$	171.067	0.309343
$554$	596.123	1.07603
$555$	0	0
$556$	1.10961i	0.00199569i
$557$	− 493.756i	− 0.886457i	−0.896409	−	0.443228i	$-0.853833\pi$
$557$	− 493.756i	− 0.886457i	0.896409	−	0.443228i	$-0.146167\pi$
$558$	0	0
$559$	402.862	0.720683
$560$	630.319i	1.12557i
$561$	0	0
$562$	−436.708	−0.777060
$563$	− 346.678i	− 0.615769i	−0.951424	−	0.307885i	$-0.900379\pi$
$563$	− 346.678i	− 0.615769i	0.951424	−	0.307885i	$-0.0996211\pi$
$564$	0	0
$565$	194.774	0.344733
$566$	−2127.12	−3.75815
$567$	0	0
$568$	− 689.324i	− 1.21360i
$569$	− 645.179i	− 1.13388i	−0.823758	−	0.566941i	$-0.808127\pi$
$569$	− 645.179i	− 1.13388i	0.823758	−	0.566941i	$-0.191873\pi$
$570$	0	0
$571$	− 43.7019i	− 0.0765358i	−0.999268	−	0.0382679i	$-0.987816\pi$
$571$	− 43.7019i	− 0.0765358i	0.999268	−	0.0382679i	$-0.0121840\pi$
$572$	0	0
$573$	0	0
$574$	− 186.419i	− 0.324771i
$575$	139.092	0.241900
$576$	0	0
$577$	−166.976	−0.289386	−0.144693	−	0.989477i	$-0.546219\pi$
$577$	−166.976	−0.289386	−0.144693	+	0.989477i	$0.546219\pi$
$578$	1109.95i	1.92033i
$579$	0	0
$580$	2402.07i	4.14151i
$581$	295.100	0.507917
$582$	0	0
$583$	0	0
$584$	2481.22	4.24866
$585$	0	0
$586$	1316.25	2.24615
$587$	−519.195	−0.884489	−0.442244	−	0.896895i	$-0.645818\pi$
$587$	−519.195	−0.884489	−0.442244	+	0.896895i	$0.645818\pi$
$588$	0	0
$589$	− 122.376i	− 0.207768i
$590$	− 27.8781i	− 0.0472511i
$591$	0	0
$592$	4193.60	7.08378
$593$	− 393.295i	− 0.663229i	−0.943415	−	0.331614i	$-0.892407\pi$
$593$	− 393.295i	− 0.663229i	0.943415	−	0.331614i	$-0.107593\pi$
$594$	0	0
$595$	13.8564	0.0232881
$596$	710.327i	1.19182i
$597$	0	0
$598$	1156.88	1.93458
$599$	−578.936	−0.966504	−0.483252	−	0.875481i	$-0.660544\pi$
$599$	−578.936	−0.966504	−0.483252	+	0.875481i	$0.660544\pi$
$600$	0	0
$601$	− 277.573i	− 0.461852i	−0.972971	−	0.230926i	$-0.925825\pi$
$601$	− 277.573i	− 0.461852i	0.972971	−	0.230926i	$-0.0741755\pi$
$602$	− 287.633i	− 0.477795i
$603$	0	0
$604$	− 2484.22i	− 4.11295i
$605$	0	0
$606$	0	0
$607$	964.135i	1.58836i	0.607682	+	0.794180i	$0.292099\pi$
$607$	964.135i	1.58836i	−0.607682	+	0.794180i	$0.707901\pi$
$608$	576.000	0.947368
$609$	0	0
$610$	−1229.18	−2.01506
$611$	703.386i	1.15121i
$612$	0	0
$613$	− 212.127i	− 0.346048i	−0.984918	−	0.173024i	$-0.944646\pi$
$613$	− 212.127i	− 0.346048i	0.984918	−	0.173024i	$-0.0553538\pi$
$614$	−1860.96	−3.03089
$615$	0	0
$616$	0	0
$617$	−304.231	−0.493081	−0.246540	−	0.969133i	$-0.579294\pi$
$617$	−304.231	−0.493081	−0.246540	+	0.969133i	$0.579294\pi$
$618$	0	0
$619$	161.026	0.260139	0.130069	−	0.991505i	$-0.458480\pi$
$619$	161.026	0.260139	0.130069	+	0.991505i	$0.458480\pi$
$620$	−1568.53	−2.52989
$621$	0	0
$622$	848.608i	1.36432i
$623$	− 29.3939i	− 0.0471812i
$624$	0	0
$625$	−722.825	−1.15652
$626$	− 63.1465i	− 0.100873i
$627$	0	0
$628$	2112.74	3.36424
$629$	− 92.1887i	− 0.146564i
$630$	0	0
$631$	448.862	0.711350	0.355675	−	0.934610i	$-0.384251\pi$
$631$	448.862	0.711350	0.355675	+	0.934610i	$0.384251\pi$
$632$	−2370.37	−3.75058
$633$	0	0
$634$	2033.38i	3.20722i
$635$	736.377i	1.15965i
$636$	0	0
$637$	− 473.245i	− 0.742927i
$638$	0	0
$639$	0	0
$640$	− 2340.25i	− 3.65665i
$641$	441.800	0.689236	0.344618	−	0.938743i	$-0.388009\pi$
$641$	441.800	0.689236	0.344618	+	0.938743i	$0.388009\pi$
$642$	0	0
$643$	248.636	0.386681	0.193340	−	0.981132i	$-0.438068\pi$
$643$	248.636	0.386681	0.193340	+	0.981132i	$0.438068\pi$
$644$	− 604.660i	− 0.938912i
$645$	0	0
$646$	− 23.6282i	− 0.0365762i
$647$	−272.000	−0.420402	−0.210201	−	0.977658i	$-0.567412\pi$
$647$	−272.000	−0.420402	−0.210201	+	0.977658i	$0.567412\pi$
$648$	0	0
$649$	0	0
$650$	196.161	0.301787
$651$	0	0
$652$	−1830.45	−2.80743
$653$	733.797	1.12373	0.561866	−	0.827228i	$-0.310084\pi$
$653$	733.797	1.12373	0.561866	+	0.827228i	$0.310084\pi$
$654$	0	0
$655$	− 65.2569i	− 0.0996288i
$656$	1491.35i	2.27340i
$657$	0	0
$658$	502.200	0.763222
$659$	− 1007.21i	− 1.52839i	−0.644986	−	0.764194i	$-0.723137\pi$
$659$	− 1007.21i	− 1.52839i	0.644986	−	0.764194i	$-0.276863\pi$
$660$	0	0
$661$	−331.299	−0.501208	−0.250604	−	0.968090i	$-0.580629\pi$
$661$	−331.299	−0.501208	−0.250604	+	0.968090i	$0.580629\pi$
$662$	533.726i	0.806233i
$663$	0	0
$664$	−4089.03	−6.15817
$665$	49.1769	0.0739502
$666$	0	0
$667$	− 1152.15i	− 1.72735i
$668$	3076.84i	4.60604i
$669$	0	0
$670$	− 3.43764i	− 0.00513081i
$671$	0	0
$672$	0	0
$673$	− 117.159i	− 0.174085i	−0.996205	−	0.0870426i	$-0.972258\pi$
$673$	− 117.159i	− 0.174085i	0.996205	−	0.0870426i	$-0.0277416\pi$
$674$	1734.82	2.57392
$675$	0	0
$676$	−652.497	−0.965233
$677$	− 851.030i	− 1.25706i	−0.777785	−	0.628530i	$-0.783657\pi$
$677$	− 851.030i	− 1.25706i	0.777785	−	0.628530i	$-0.216343\pi$
$678$	0	0
$679$	133.149i	0.196096i
$680$	−192.000	−0.282353
$681$	0	0
$682$	0	0
$683$	−871.654	−1.27621	−0.638107	−	0.769948i	$-0.720282\pi$
$683$	−871.654	−1.27621	−0.638107	+	0.769948i	$0.720282\pi$
$684$	0	0
$685$	342.851	0.500513
$686$	−703.626	−1.02569
$687$	0	0
$688$	2301.06i	3.34457i
$689$	− 683.979i	− 0.992713i
$690$	0	0
$691$	−217.000	−0.314038	−0.157019	−	0.987596i	$-0.550188\pi$
$691$	−217.000	−0.314038	−0.157019	+	0.987596i	$0.550188\pi$
$692$	1858.42i	2.68558i
$693$	0	0
$694$	−1922.18	−2.76971
$695$	− 0.554803i	0 0.000798277i
$696$	0	0
$697$	32.7846	0.0470367
$698$	−1830.67	−2.62273
$699$	0	0
$700$	− 102.527i	− 0.146467i
$701$	− 1091.52i	− 1.55709i	−0.627588	−	0.778545i	$-0.715958\pi$
$701$	− 1091.52i	− 1.55709i	0.627588	−	0.778545i	$-0.284042\pi$
$702$	0	0
$703$	− 327.181i	− 0.465407i
$704$	0	0
$705$	0	0
$706$	− 1247.81i	− 1.76743i
$707$	−18.8897	−0.0267181
$708$	0	0
$709$	−748.092	−1.05514	−0.527568	−	0.849513i	$-0.676896\pi$
$709$	−748.092	−1.05514	−0.527568	+	0.849513i	$0.676896\pi$
$710$	543.653i	0.765708i
$711$	0	0
$712$	407.294i	0.572041i
$713$	752.341	1.05518
$714$	0	0
$715$	0	0
$716$	−2145.79	−2.99692
$717$	0	0
$718$	219.292	0.305421
$719$	1219.21	1.69570	0.847848	−	0.530240i	$-0.177898\pi$
$719$	1219.21	1.69570	0.847848	+	0.530240i	$0.177898\pi$
$720$	0	0
$721$	− 197.316i	− 0.273670i
$722$	1310.94i	1.81571i
$723$	0	0
$724$	−2148.88	−2.96806
$725$	− 195.359i	− 0.269461i
$726$	0	0
$727$	−404.708	−0.556682	−0.278341	−	0.960482i	$-0.589784\pi$
$727$	−404.708	−0.556682	−0.278341	+	0.960482i	$0.589784\pi$
$728$	− 540.621i	− 0.742611i
$729$	0	0
$730$	−1956.88	−2.68065
$731$	50.5847	0.0691993
$732$	0	0
$733$	386.566i	0.527375i	0.964608	+	0.263688i	$0.0849388\pi$
$733$	386.566i	0.527375i	−0.964608	+	0.263688i	$0.915061\pi$
$734$	30.6521i	0.0417604i
$735$	0	0
$736$	3541.13i	4.81132i
$737$	0	0
$738$	0	0
$739$	− 73.7973i	− 0.0998610i	−0.998753	−	0.0499305i	$-0.984100\pi$
$739$	− 73.7973i	− 0.0998610i	0.998753	−	0.0499305i	$-0.0159000\pi$
$740$	−4193.60	−5.66703
$741$	0	0
$742$	−488.344	−0.658145
$743$	652.634i	0.878377i	0.898395	+	0.439188i	$0.144734\pi$
$743$	652.634i	0.878377i	−0.898395	+	0.439188i	$0.855266\pi$
$744$	0	0
$745$	− 355.163i	− 0.476729i
$746$	1697.67	2.27570
$747$	0	0
$748$	0	0
$749$	42.2872	0.0564582
$750$	0	0
$751$	−1047.76	−1.39515	−0.697577	−	0.716510i	$-0.745738\pi$
$751$	−1047.76	−1.39515	−0.697577	+	0.716510i	$0.745738\pi$
$752$	−4017.60	−5.34255
$753$	0	0
$754$	− 1624.87i	− 2.15499i
$755$	1242.11i	1.64518i
$756$	0	0
$757$	−1347.53	−1.78009	−0.890046	−	0.455871i	$-0.849328\pi$
$757$	−1347.53	−1.78009	−0.890046	+	0.455871i	$0.849328\pi$
$758$	124.114i	0.163738i
$759$	0	0
$760$	−681.415	−0.896599
$761$	989.300i	1.30000i	0.759934	+	0.650000i	$0.225231\pi$
$761$	989.300i	1.30000i	−0.759934	+	0.650000i	$0.774769\pi$
$762$	0	0
$763$	−93.5885	−0.122659
$764$	675.138	0.883689
$765$	0	0
$766$	− 1664.94i	− 2.17355i
$767$	13.8050i	0.0179987i
$768$	0	0
$769$	− 178.156i	− 0.231672i	−0.993268	−	0.115836i	$-0.963045\pi$
$769$	− 178.156i	− 0.231672i	0.993268	−	0.115836i	$-0.0369547\pi$
$770$	0	0
$771$	0	0
$772$	3558.48i	4.60943i
$773$	1032.22	1.33534	0.667670	−	0.744457i	$-0.267292\pi$
$773$	1032.22	1.33534	0.667670	+	0.744457i	$0.267292\pi$
$774$	0	0
$775$	127.568	0.164604
$776$	− 1844.97i	− 2.37754i
$777$	0	0
$778$	− 1516.34i	− 1.94903i
$779$	116.354	0.149363
$780$	0	0
$781$	0	0
$782$	145.261	0.185756
$783$	0	0
$784$	2703.08	3.44780
$785$	−1056.37	−1.34570
$786$	0	0
$787$	1235.66i	1.57009i	0.619436	+	0.785047i	$0.287361\pi$
$787$	1235.66i	1.57009i	−0.619436	+	0.785047i	$0.712639\pi$
$788$	− 1389.74i	− 1.76363i
$789$	0	0
$790$	1869.45	2.36639
$791$	68.8631i	0.0870583i
$792$	0	0
$793$	608.681	0.767567
$794$	− 1408.63i	− 1.77409i
$795$	0	0
$796$	287.061	0.360630
$797$	137.713	0.172789	0.0863945	−	0.996261i	$-0.472465\pi$
$797$	137.713	0.172789	0.0863945	+	0.996261i	$0.472465\pi$
$798$	0	0
$799$	88.3196i	0.110538i
$800$	600.439i	0.750549i
$801$	0	0
$802$	251.715i	0.313860i
$803$	0	0
$804$	0	0
$805$	302.330i	0.375565i
$806$	1061.02	1.31641
$807$	0	0
$808$	261.744	0.323940
$809$	395.965i	0.489450i	0.969593	+	0.244725i	$0.0786977\pi$
$809$	395.965i	0.489450i	−0.969593	+	0.244725i	$0.921302\pi$
$810$	0	0
$811$	89.0636i	0.109820i	0.998491	+	0.0549098i	$0.0174871\pi$
$811$	89.0636i	0.109820i	−0.998491	+	0.0549098i	$0.982513\pi$
$812$	−849.261	−1.04589
$813$	0	0
$814$	0	0
$815$	915.223	1.12297
$816$	0	0
$817$	179.527	0.219739
$818$	1220.70	1.49230
$819$	0	0
$820$	− 1491.35i	− 1.81872i
$821$	− 848.802i	− 1.03386i	−0.856027	−	0.516932i	$-0.827074\pi$
$821$	− 848.802i	− 1.03386i	0.856027	−	0.516932i	$-0.172926\pi$
$822$	0	0
$823$	−982.923	−1.19432	−0.597159	−	0.802123i	$-0.703704\pi$
$823$	−982.923	−1.19432	−0.597159	+	0.802123i	$0.703704\pi$
$824$	2734.10i	3.31808i
$825$	0	0
$826$	9.85641	0.0119327
$827$	656.661i	0.794028i	0.917813	+	0.397014i	$0.129954\pi$
$827$	656.661i	0.794028i	−0.917813	+	0.397014i	$0.870046\pi$
$828$	0	0
$829$	−262.867	−0.317089	−0.158544	−	0.987352i	$-0.550680\pi$
$829$	−262.867	−0.317089	−0.158544	+	0.987352i	$0.550680\pi$
$830$	3224.91	3.88544
$831$	0	0
$832$	2497.02i	3.00123i
$833$	− 59.4222i	− 0.0713352i
$834$	0	0
$835$	− 1538.42i	− 1.84242i
$836$	0	0
$837$	0	0
$838$	− 338.153i	− 0.403524i
$839$	26.9488	0.0321202	0.0160601	−	0.999871i	$-0.494888\pi$
$839$	26.9488	0.0321202	0.0160601	+	0.999871i	$0.494888\pi$
$840$	0	0
$841$	−777.221	−0.924162
$842$	2534.51i	3.01011i
$843$	0	0
$844$	− 1044.97i	− 1.23812i
$845$	326.249	0.386093
$846$	0	0
$847$	0	0
$848$	3906.75	4.60701
$849$	0	0
$850$	24.6307	0.0289773
$851$	2011.44	2.36362
$852$	0	0
$853$	− 797.944i	− 0.935456i	−0.883873	−	0.467728i	$-0.845073\pi$
$853$	− 797.944i	− 0.935456i	0.883873	−	0.467728i	$-0.154927\pi$
$854$	− 434.582i	− 0.508879i
$855$	0	0
$856$	−585.948	−0.684519
$857$	1113.40i	1.29918i	0.760286	+	0.649589i	$0.225059\pi$
$857$	1113.40i	1.29918i	−0.760286	+	0.649589i	$0.774941\pi$
$858$	0	0
$859$	−1187.27	−1.38215	−0.691075	−	0.722783i	$-0.742863\pi$
$859$	−1187.27	−1.38215	−0.691075	+	0.722783i	$0.742863\pi$
$860$	− 2301.06i	− 2.67565i
$861$	0	0
$862$	2664.72	3.09133
$863$	−366.351	−0.424509	−0.212254	−	0.977214i	$-0.568081\pi$
$863$	−366.351	−0.424509	−0.212254	+	0.977214i	$0.568081\pi$
$864$	0	0
$865$	− 929.211i	− 1.07423i
$866$	1514.85i	1.74925i
$867$	0	0
$868$	− 554.560i	− 0.638894i
$869$	0	0
$870$	0	0
$871$	1.70229i	0.00195441i
$872$	1296.80	1.48716
$873$	0	0
$874$	515.538	0.589861
$875$	− 212.632i	− 0.243008i
$876$	0	0
$877$	1043.66i	1.19003i	0.803715	+	0.595015i	$0.202854\pi$
$877$	1043.66i	1.19003i	−0.803715	+	0.595015i	$0.797146\pi$
$878$	−1837.94	−2.09332
$879$	0	0
$880$	0	0
$881$	−292.831	−0.332384	−0.166192	−	0.986093i	$-0.553147\pi$
$881$	−292.831	−0.332384	−0.166192	+	0.986093i	$0.553147\pi$
$882$	0	0
$883$	−1285.48	−1.45581	−0.727905	−	0.685678i	$-0.759506\pi$
$883$	−1285.48	−1.45581	−0.727905	+	0.685678i	$0.759506\pi$
$884$	149.969	0.169648
$885$	0	0
$886$	1152.86i	1.30120i
$887$	393.171i	0.443260i	0.975131	+	0.221630i	$0.0711377\pi$
$887$	393.171i	0.443260i	−0.975131	+	0.221630i	$0.928862\pi$
$888$	0	0
$889$	−260.349	−0.292856
$890$	− 321.222i	− 0.360924i
$891$	0	0
$892$	3564.11	3.99564
$893$	313.450i	0.351007i
$894$	0	0
$895$	1072.90	1.19877
$896$	827.405	0.923443
$897$	0	0
$898$	− 2727.03i	− 3.03678i
$899$	− 1056.68i	− 1.17540i
$900$	0	0
$901$	− 85.8828i	− 0.0953194i
$902$	0	0
$903$	0	0
$904$	− 954.195i	− 1.05553i
$905$	1074.44	1.18722
$906$	0	0
$907$	885.231	0.975999	0.487999	−	0.872844i	$-0.337727\pi$
$907$	885.231	0.975999	0.487999	+	0.872844i	$0.337727\pi$
$908$	275.373i	0.303274i
$909$	0	0
$910$	426.374i	0.468543i
$911$	−1441.04	−1.58182	−0.790910	−	0.611932i	$-0.790392\pi$
$911$	−1441.04	−1.58182	−0.790910	+	0.611932i	$0.790392\pi$
$912$	0	0
$913$	0	0
$914$	1652.58	1.80808
$915$	0	0
$916$	−1881.95	−2.05453
$917$	23.0718	0.0251601
$918$	0	0
$919$	− 345.405i	− 0.375848i	−0.982184	−	0.187924i	$-0.939824\pi$
$919$	− 345.405i	− 0.375848i	0.982184	−	0.187924i	$-0.0601759\pi$
$920$	− 4189.20i	− 4.55348i
$921$	0	0
$922$	−743.713	−0.806630
$923$	− 269.212i	− 0.291670i
$924$	0	0
$925$	341.063	0.368717
$926$	− 1187.03i	− 1.28189i
$927$	0	0
$928$	4973.62	5.35950
$929$	−1641.54	−1.76699	−0.883497	−	0.468436i	$-0.844818\pi$
$929$	−1641.54	−1.76699	−0.883497	+	0.468436i	$0.844818\pi$
$930$	0	0
$931$	− 210.892i	− 0.226522i
$932$	− 619.440i	− 0.664635i
$933$	0	0
$934$	3172.22i	3.39638i
$935$	0	0
$936$	0	0
$937$	1052.25i	1.12300i	0.827478	+	0.561498i	$0.189775\pi$
$937$	1052.25i	1.12300i	−0.827478	+	0.561498i	$0.810225\pi$
$938$	1.21539	0.00129573
$939$	0	0
$940$	4017.60	4.27404
$941$	393.715i	0.418401i	0.977873	+	0.209201i	$0.0670861\pi$
$941$	393.715i	0.418401i	−0.977873	+	0.209201i	$0.932914\pi$
$942$	0	0
$943$	715.320i	0.758558i
$944$	−78.8513	−0.0835289
$945$	0	0
$946$	0	0
$947$	−306.428	−0.323578	−0.161789	−	0.986825i	$-0.551726\pi$
$947$	−306.428	−0.323578	−0.161789	+	0.986825i	$0.551726\pi$
$948$	0	0
$949$	969.027	1.02110
$950$	87.4153	0.0920161
$951$	0	0
$952$	− 67.8823i	− 0.0713049i
$953$	− 25.7624i	− 0.0270330i	−0.999909	−	0.0135165i	$-0.995697\pi$
$953$	− 25.7624i	− 0.0270330i	0.999909	−	0.0135165i	$-0.00430256\pi$
$954$	0	0
$955$	−337.569	−0.353476
$956$	− 2827.77i	− 2.95792i
$957$	0	0
$958$	1801.45	1.88043
$959$	121.216i	0.126399i
$960$	0	0
$961$	−270.995	−0.281993
$962$	2836.73	2.94878
$963$	0	0
$964$	3719.42i	3.85832i
$965$	− 1779.24i	− 1.84377i
$966$	0	0
$967$	− 364.297i	− 0.376729i	−0.982099	−	0.188365i	$-0.939681\pi$
$967$	− 364.297i	− 0.376729i	0.982099	−	0.188365i	$-0.0603186\pi$
$968$	0	0
$969$	0	0
$970$	1455.08i	1.50008i
$971$	−1152.47	−1.18689	−0.593443	−	0.804876i	$-0.702232\pi$
$971$	−1152.47	−1.18689	−0.593443	+	0.804876i	$0.702232\pi$
$972$	0	0
$973$	0.196152	0.000201596 0
$974$	692.435i	0.710919i
$975$	0	0
$976$	3476.66i	3.56215i
$977$	1867.89	1.91187	0.955933	−	0.293587i	$-0.0948489\pi$
$977$	1867.89	1.91187	0.955933	+	0.293587i	$0.0948489\pi$
$978$	0	0
$979$	0	0
$980$	−2703.08	−2.75824
$981$	0	0
$982$	1525.84	1.55380
$983$	−866.267	−0.881248	−0.440624	−	0.897692i	$-0.645243\pi$
$983$	−866.267	−0.881248	−0.440624	+	0.897692i	$0.645243\pi$
$984$	0	0
$985$	694.872i	0.705454i
$986$	− 204.024i	− 0.206921i
$987$	0	0
$988$	532.246	0.538710
$989$	1103.70i	1.11597i
$990$	0	0
$991$	−566.851	−0.571999	−0.286000	−	0.958230i	$-0.592326\pi$
$991$	−566.851	−0.571999	−0.286000	+	0.958230i	$0.592326\pi$
$992$	3247.73i	3.27392i
$993$	0	0
$994$	−192.210	−0.193370
$995$	−143.531	−0.144252
$996$	0	0
$997$	− 117.138i	− 0.117490i	−0.998273	−	0.0587450i	$-0.981290\pi$
$997$	− 117.138i	− 0.117490i	0.998273	−	0.0587450i	$-0.0187099\pi$
$998$	− 2277.06i	− 2.28163i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1089.3.c.c.604.4		4
3.2	odd	2		363.3.c.a.241.1	✓	4
11.10	odd	2	inner	1089.3.c.c.604.1		4
33.2	even	10		363.3.g.d.40.4		16
33.5	odd	10		363.3.g.d.118.4		16
33.8	even	10		363.3.g.d.112.4		16
33.14	odd	10		363.3.g.d.112.1		16
33.17	even	10		363.3.g.d.118.1		16
33.20	odd	10		363.3.g.d.40.1		16
33.26	odd	10		363.3.g.d.94.4		16
33.29	even	10		363.3.g.d.94.1		16
33.32	even	2		363.3.c.a.241.4	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
363.3.c.a.241.1	✓	4	3.2	odd	2
363.3.c.a.241.4	yes	4	33.32	even	2
363.3.g.d.40.1		16	33.20	odd	10
363.3.g.d.40.4		16	33.2	even	10
363.3.g.d.94.1		16	33.29	even	10
363.3.g.d.94.4		16	33.26	odd	10
363.3.g.d.112.1		16	33.14	odd	10
363.3.g.d.112.4		16	33.8	even	10
363.3.g.d.118.1		16	33.17	even	10
363.3.g.d.118.4		16	33.5	odd	10
1089.3.c.c.604.1		4	11.10	odd	2	inner
1089.3.c.c.604.4		4	1.1	even	1	trivial

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.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+3.86370i q^{2} -10.9282 q^{4} +5.46410 q^{5} +1.93185i q^{7} -26.7685i q^{8} +O(q^{10})\)	\(q+3.86370i q^{2} -10.9282 q^{4} +5.46410 q^{5} +1.93185i q^{7} -26.7685i q^{8} +21.1117i q^{10} -10.4543i q^{13} -7.46410 q^{14} +59.7128 q^{16} -1.31268i q^{17} -4.65874i q^{19} -59.7128 q^{20} +28.6410 q^{23} +4.85641 q^{25} +40.3923 q^{26} -21.1117i q^{28} -40.2271i q^{29} +26.2679 q^{31} +123.639i q^{32} +5.07180 q^{34} +10.5558i q^{35} +70.2295 q^{37} +18.0000 q^{38} -146.266i q^{40} +24.9754i q^{41} +38.5355i q^{43} +110.660i q^{46} -67.2820 q^{47} +45.2679 q^{49} +18.7637i q^{50} +114.247i q^{52} +65.4256 q^{53} +51.7128 q^{56} +155.426 q^{58} -1.32051 q^{59} +58.2230i q^{61} +101.492i q^{62} -238.851 q^{64} -57.1233i q^{65} -0.162831 q^{67} +14.3452i q^{68} -40.7846 q^{70} +25.7513 q^{71} +92.6917i q^{73} +271.346i q^{74} +50.9117i q^{76} -88.5506i q^{79} +326.277 q^{80} -96.4974 q^{82} -152.755i q^{83} -7.17260i q^{85} -148.890 q^{86} -15.2154 q^{89} +20.1962 q^{91} -312.995 q^{92} -259.958i q^{94} -25.4558i q^{95} +68.9230 q^{97} +174.902i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 16 q^{4} + 8 q^{5}+O(q^{10}) \) 4 * q - 16 * q^4 + 8 * q^5	\( 4 q - 16 q^{4} + 8 q^{5} - 16 q^{14} + 128 q^{16} - 128 q^{20} - 24 q^{23} - 36 q^{25} + 120 q^{26} + 112 q^{31} + 48 q^{34} + 80 q^{37} + 72 q^{38} + 8 q^{47} + 188 q^{49} + 40 q^{53} + 96 q^{56} + 400 q^{58} + 64 q^{59} - 512 q^{64} - 160 q^{67} - 80 q^{70} + 200 q^{71} + 640 q^{80} - 192 q^{82} - 360 q^{86} - 144 q^{89} + 60 q^{91} - 864 q^{92} - 140 q^{97}+O(q^{100}) \) 4 * q - 16 * q^4 + 8 * q^5 - 16 * q^14 + 128 * q^16 - 128 * q^20 - 24 * q^23 - 36 * q^25 + 120 * q^26 + 112 * q^31 + 48 * q^34 + 80 * q^37 + 72 * q^38 + 8 * q^47 + 188 * q^49 + 40 * q^53 + 96 * q^56 + 400 * q^58 + 64 * q^59 - 512 * q^64 - 160 * q^67 - 80 * q^70 + 200 * q^71 + 640 * q^80 - 192 * q^82 - 360 * q^86 - 144 * q^89 + 60 * q^91 - 864 * q^92 - 140 * q^97

Introduction

L-functions

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