LMFDB - Embedded newform 1089.2.d.g.1088.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1089,2,Mod(1088,1089)]

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, 1]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("1089.1088");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1089 = 3^{2} \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1089.d (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:	$8.69570878012$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} + 2x^{14} - 16x^{12} - 72x^{10} + 26x^{8} + 360x^{6} + 725x^{4} + 1000x^{2} + 625 $ x^16 + 2x^14 - 16x^12 - 72x^10 + 26x^8 + 360x^6 + 725x^4 + 1000*x^2 + 625
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{8} $
Twist minimal:	no (minimal twist has level 99)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1088.4
Root			$0.556839 - 1.81878i$ of defining polynomial
Character	$\chi$	$=$	1089.1088
Dual form			1089.2.d.g.1088.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.35080 q^{2} +3.52626 q^{4} +2.24814i q^{5} -4.05107i q^{7} -3.58792 q^{8} +O(q^{10})$	$q-2.35080 q^{2} +3.52626 q^{4} +2.24814i q^{5} -4.05107i q^{7} -3.58792 q^{8} -5.28492i q^{10} +4.65266i q^{13} +9.52326i q^{14} +1.38197 q^{16} +0.0762914 q^{17} -2.39581i q^{19} +7.92750i q^{20} +3.22717i q^{23} -0.0541194 q^{25} -10.9375i q^{26} -14.2851i q^{28} +1.83922 q^{29} +1.67476 q^{31} +3.92711 q^{32} -0.179346 q^{34} +9.10737 q^{35} +7.26691 q^{37} +5.63207i q^{38} -8.06613i q^{40} -8.44210 q^{41} +4.28086i q^{43} -7.58644i q^{46} -6.21587i q^{47} -9.41120 q^{49} +0.127224 q^{50} +16.4065i q^{52} -1.22899i q^{53} +14.5349i q^{56} -4.32363 q^{58} +0.580290i q^{59} +3.78221i q^{61} -3.93702 q^{62} -11.9958 q^{64} -10.4598 q^{65} +12.9984 q^{67} +0.269023 q^{68} -21.4096 q^{70} +1.12047i q^{71} +13.3068i q^{73} -17.0831 q^{74} -8.44824i q^{76} +0.659861i q^{79} +3.10685i q^{80} +19.8457 q^{82} +10.2111 q^{83} +0.171513i q^{85} -10.0634i q^{86} -6.58983i q^{89} +18.8483 q^{91} +11.3798i q^{92} +14.6123i q^{94} +5.38611 q^{95} +16.4446 q^{97} +22.1238 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 16 q^{4}+O(q^{10}) $ 16 * q + 16 * q^4	$ 16 q + 16 q^{4} + 40 q^{16} - 32 q^{25} + 16 q^{31} + 40 q^{34} + 8 q^{37} + 16 q^{49} + 32 q^{58} - 104 q^{64} + 96 q^{67} - 64 q^{70} + 88 q^{82} + 48 q^{91}+O(q^{100}) $ 16 * q + 16 * q^4 + 40 * q^16 - 32 * q^25 + 16 * q^31 + 40 * q^34 + 8 * q^37 + 16 * q^49 + 32 * q^58 - 104 * q^64 + 96 * q^67 - 64 * q^70 + 88 * q^82 + 48 * q^91

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 $

$2$	−2.35080	−1.66227	−0.831133	−	0.556074i	$-0.812307\pi$
$2$	−2.35080	−1.66227	−0.831133	+	0.556074i	$0.812307\pi$
$3$	0	0
$3$	0	0
$4$	3.52626	1.76313
$5$	2.24814i	1.00540i	0.864462	+	0.502699i	$0.167659\pi$
$5$	2.24814i	1.00540i	−0.864462	+	0.502699i	$0.832341\pi$
$6$	0	0
$7$	− 4.05107i	− 1.53116i	−0.643339	−	0.765581i	$-0.722452\pi$
$7$	− 4.05107i	− 1.53116i	0.643339	−	0.765581i	$-0.277548\pi$
$8$	−3.58792	−1.26852
$9$	0	0
$10$	− 5.28492i	− 1.67124i
$11$	0	0
$11$	0	0
$12$	0	0
$13$	4.65266i	1.29042i	0.764007	+	0.645208i	$0.223229\pi$
$13$	4.65266i	1.29042i	−0.764007	+	0.645208i	$0.776771\pi$
$14$	9.52326i	2.54520i
$15$	0	0
$16$	1.38197	0.345492
$17$	0.0762914	0.0185034	0.00925169	−	0.999957i	$-0.497055\pi$
$17$	0.0762914	0.0185034	0.00925169	+	0.999957i	$0.497055\pi$
$18$	0	0
$19$	− 2.39581i	− 0.549636i	−0.961496	−	0.274818i	$-0.911382\pi$
$19$	− 2.39581i	− 0.549636i	0.961496	−	0.274818i	$-0.0886177\pi$
$20$	7.92750i	1.77264i
$21$	0	0
$22$	0	0
$23$	3.22717i	0.672912i	0.941699	+	0.336456i	$0.109228\pi$
$23$	3.22717i	0.672912i	−0.941699	+	0.336456i	$0.890772\pi$
$24$	0	0
$25$	−0.0541194	−0.0108239
$26$	− 10.9375i	− 2.14502i
$27$	0	0
$28$	− 14.2851i	− 2.69963i
$29$	1.83922	0.341534	0.170767	−	0.985311i	$-0.445375\pi$
$29$	1.83922	0.341534	0.170767	+	0.985311i	$0.445375\pi$
$30$	0	0
$31$	1.67476	0.300795	0.150398	−	0.988626i	$-0.451945\pi$
$31$	1.67476	0.300795	0.150398	+	0.988626i	$0.451945\pi$
$32$	3.92711	0.694222
$33$	0	0
$34$	−0.179346	−0.0307575
$35$	9.10737	1.53943
$36$	0	0
$37$	7.26691	1.19467	0.597337	−	0.801991i	$-0.296226\pi$
$37$	7.26691	1.19467	0.597337	+	0.801991i	$0.296226\pi$
$38$	5.63207i	0.913642i
$39$	0	0
$40$	− 8.06613i	− 1.27537i
$41$	−8.44210	−1.31843	−0.659217	−	0.751953i	$-0.729112\pi$
$41$	−8.44210	−1.31843	−0.659217	+	0.751953i	$0.729112\pi$
$42$	0	0
$43$	4.28086i	0.652825i	0.945227	+	0.326413i	$0.105840\pi$
$43$	4.28086i	0.652825i	−0.945227	+	0.326413i	$0.894160\pi$
$44$	0	0
$45$	0	0
$46$	− 7.58644i	− 1.11856i
$47$	− 6.21587i	− 0.906678i	−0.891338	−	0.453339i	$-0.850233\pi$
$47$	− 6.21587i	− 0.906678i	0.891338	−	0.453339i	$-0.149767\pi$
$48$	0	0
$49$	−9.41120	−1.34446
$50$	0.127224	0.0179922
$51$	0	0
$52$	16.4065i	2.27517i
$53$	− 1.22899i	− 0.168815i	−0.996431	−	0.0844074i	$-0.973100\pi$
$53$	− 1.22899i	− 0.168815i	0.996431	−	0.0844074i	$-0.0268997\pi$
$54$	0	0
$55$	0	0
$56$	14.5349i	1.94231i
$57$	0	0
$58$	−4.32363	−0.567721
$59$	0.580290i	0.0755474i	0.999286	+	0.0377737i	$0.0120266\pi$
$59$	0.580290i	0.0755474i	−0.999286	+	0.0377737i	$0.987973\pi$
$60$	0	0
$61$	3.78221i	0.484263i	0.970243	+	0.242131i	$0.0778465\pi$
$61$	3.78221i	0.484263i	−0.970243	+	0.242131i	$0.922153\pi$
$62$	−3.93702	−0.500001
$63$	0	0
$64$	−11.9958	−1.49947
$65$	−10.4598	−1.29738
$66$	0	0
$67$	12.9984	1.58801	0.794003	−	0.607914i	$-0.207993\pi$
$67$	12.9984	1.58801	0.794003	+	0.607914i	$0.207993\pi$
$68$	0.269023	0.0326238
$69$	0	0
$70$	−21.4096	−2.55894
$71$	1.12047i	0.132976i	0.997787	+	0.0664878i	$0.0211793\pi$
$71$	1.12047i	0.132976i	−0.997787	+	0.0664878i	$0.978821\pi$
$72$	0	0
$73$	13.3068i	1.55744i	0.627371	+	0.778720i	$0.284131\pi$
$73$	13.3068i	1.55744i	−0.627371	+	0.778720i	$0.715869\pi$
$74$	−17.0831	−1.98586
$75$	0	0
$76$	− 8.44824i	− 0.969079i
$77$	0	0
$78$	0	0
$79$	0.659861i	0.0742402i	0.999311	+	0.0371201i	$0.0118184\pi$
$79$	0.659861i	0.0742402i	−0.999311	+	0.0371201i	$0.988182\pi$
$80$	3.10685i	0.347356i
$81$	0	0
$82$	19.8457	2.19159
$83$	10.2111	1.12082	0.560409	−	0.828216i	$-0.310644\pi$
$83$	10.2111	1.12082	0.560409	+	0.828216i	$0.310644\pi$
$84$	0	0
$85$	0.171513i	0.0186032i
$86$	− 10.0634i	− 1.08517i
$87$	0	0
$88$	0	0
$89$	− 6.58983i	− 0.698520i	−0.937026	−	0.349260i	$-0.886433\pi$
$89$	− 6.58983i	− 0.698520i	0.937026	−	0.349260i	$-0.113567\pi$
$90$	0	0
$91$	18.8483	1.97584
$92$	11.3798i	1.18643i
$93$	0	0
$94$	14.6123i	1.50714i
$95$	5.38611	0.552603
$96$	0	0
$97$	16.4446	1.66970	0.834851	−	0.550477i	$-0.185554\pi$
$97$	16.4446	1.66970	0.834851	+	0.550477i	$0.185554\pi$
$98$	22.1238	2.23485
$99$	0	0
$100$	−0.190839	−0.0190839
$101$	−1.72190	−0.171335	−0.0856675	−	0.996324i	$-0.527302\pi$
$101$	−1.72190	−0.171335	−0.0856675	+	0.996324i	$0.527302\pi$
$102$	0	0
$103$	−3.23086	−0.318346	−0.159173	−	0.987251i	$-0.550883\pi$
$103$	−3.23086	−0.318346	−0.159173	+	0.987251i	$0.550883\pi$
$104$	− 16.6934i	− 1.63692i
$105$	0	0
$106$	2.88911i	0.280615i
$107$	14.8184	1.43255	0.716276	−	0.697817i	$-0.245845\pi$
$107$	14.8184	1.43255	0.716276	+	0.697817i	$0.245845\pi$
$108$	0	0
$109$	14.9258i	1.42963i	0.699313	+	0.714815i	$0.253489\pi$
$109$	14.9258i	1.42963i	−0.699313	+	0.714815i	$0.746511\pi$
$110$	0	0
$111$	0	0
$112$	− 5.59845i	− 0.529004i
$113$	11.2465i	1.05799i	0.848626	+	0.528993i	$0.177430\pi$
$113$	11.2465i	1.05799i	−0.848626	+	0.528993i	$0.822570\pi$
$114$	0	0
$115$	−7.25513	−0.676544
$116$	6.48556	0.602169
$117$	0	0
$118$	− 1.36415i	− 0.125580i
$119$	− 0.309062i	− 0.0283317i
$120$	0	0
$121$	0	0
$122$	− 8.89122i	− 0.804973i
$123$	0	0
$124$	5.90562	0.530340
$125$	11.1190i	0.994515i
$126$	0	0
$127$	6.06148i	0.537869i	0.963158	+	0.268935i	$0.0866715\pi$
$127$	6.06148i	0.537869i	−0.963158	+	0.268935i	$0.913328\pi$
$128$	20.3455	1.79830
$129$	0	0
$130$	24.5889	2.15659
$131$	7.15083	0.624771	0.312385	−	0.949955i	$-0.398872\pi$
$131$	7.15083	0.624771	0.312385	+	0.949955i	$0.398872\pi$
$132$	0	0
$133$	−9.70560	−0.841582
$134$	−30.5566	−2.63969
$135$	0	0
$136$	−0.273727	−0.0234719
$137$	10.6225i	0.907538i	0.891119	+	0.453769i	$0.149921\pi$
$137$	10.6225i	0.907538i	−0.891119	+	0.453769i	$0.850079\pi$
$138$	0	0
$139$	− 0.0530764i	− 0.00450188i	−0.999997	−	0.00225094i	$-0.999284\pi$
$139$	− 0.0530764i	− 0.00450188i	0.999997	−	0.00225094i	$-0.000716497\pi$
$140$	32.1149	2.71421
$141$	0	0
$142$	− 2.63400i	− 0.221041i
$143$	0	0
$144$	0	0
$145$	4.13482i	0.343378i
$146$	− 31.2816i	− 2.58888i
$147$	0	0
$148$	25.6250	2.10636
$149$	20.3949	1.67082	0.835409	−	0.549628i	$-0.185231\pi$
$149$	20.3949	1.67082	0.835409	+	0.549628i	$0.185231\pi$
$150$	0	0
$151$	− 1.18918i	− 0.0967737i	−0.998829	−	0.0483869i	$-0.984592\pi$
$151$	− 1.18918i	− 0.0967737i	0.998829	−	0.0483869i	$-0.0154080\pi$
$152$	8.59597i	0.697225i
$153$	0	0
$154$	0	0
$155$	3.76508i	0.302419i
$156$	0	0
$157$	−1.73309	−0.138315	−0.0691577	−	0.997606i	$-0.522031\pi$
$157$	−1.73309	−0.138315	−0.0691577	+	0.997606i	$0.522031\pi$
$158$	− 1.55120i	− 0.123407i
$159$	0	0
$160$	8.82869i	0.697969i
$161$	13.0735	1.03034
$162$	0	0
$163$	−1.00260	−0.0785299	−0.0392650	−	0.999229i	$-0.512502\pi$
$163$	−1.00260	−0.0785299	−0.0392650	+	0.999229i	$0.512502\pi$
$164$	−29.7690	−2.32457
$165$	0	0
$166$	−24.0044	−1.86310
$167$	13.5182	1.04607	0.523033	−	0.852312i	$-0.324800\pi$
$167$	13.5182	1.04607	0.523033	+	0.852312i	$0.324800\pi$
$168$	0	0
$169$	−8.64727	−0.665175
$170$	− 0.403194i	− 0.0309235i
$171$	0	0
$172$	15.0954i	1.15101i
$173$	−7.52294	−0.571959	−0.285979	−	0.958236i	$-0.592319\pi$
$173$	−7.52294	−0.571959	−0.285979	+	0.958236i	$0.592319\pi$
$174$	0	0
$175$	0.219242i	0.0165731i
$176$	0	0
$177$	0	0
$178$	15.4914i	1.16113i
$179$	12.5003i	0.934317i	0.884173	+	0.467159i	$0.154722\pi$
$179$	12.5003i	0.934317i	−0.884173	+	0.467159i	$0.845278\pi$
$180$	0	0
$181$	−18.0010	−1.33800	−0.669002	−	0.743261i	$-0.733278\pi$
$181$	−18.0010	−1.33800	−0.669002	+	0.743261i	$0.733278\pi$
$182$	−44.3085	−3.28437
$183$	0	0
$184$	− 11.5788i	− 0.853604i
$185$	16.3370i	1.20112i
$186$	0	0
$187$	0	0
$188$	− 21.9187i	− 1.59859i
$189$	0	0
$190$	−12.6617	−0.918573
$191$	21.6570i	1.56704i	0.621364	+	0.783522i	$0.286579\pi$
$191$	21.6570i	1.56704i	−0.621364	+	0.783522i	$0.713421\pi$
$192$	0	0
$193$	5.18693i	0.373363i	0.982420	+	0.186682i	$0.0597733\pi$
$193$	5.18693i	0.373363i	−0.982420	+	0.186682i	$0.940227\pi$
$194$	−38.6581	−2.77549
$195$	0	0
$196$	−33.1863	−2.37045
$197$	5.14679	0.366693	0.183347	−	0.983048i	$-0.441307\pi$
$197$	5.14679	0.366693	0.183347	+	0.983048i	$0.441307\pi$
$198$	0	0
$199$	24.2595	1.71971	0.859855	−	0.510539i	$-0.170554\pi$
$199$	24.2595	1.71971	0.859855	+	0.510539i	$0.170554\pi$
$200$	0.194176	0.0137303
$201$	0	0
$202$	4.04783	0.284804
$203$	− 7.45081i	− 0.522945i
$204$	0	0
$205$	− 18.9790i	− 1.32555i
$206$	7.59511	0.529176
$207$	0	0
$208$	6.42982i	0.445828i
$209$	0	0
$210$	0	0
$211$	− 18.7436i	− 1.29036i	−0.764029	−	0.645182i	$-0.776782\pi$
$211$	− 18.7436i	− 1.29036i	0.764029	−	0.645182i	$-0.223218\pi$
$212$	− 4.33373i	− 0.297642i
$213$	0	0
$214$	−34.8352	−2.38128
$215$	−9.62396	−0.656349
$216$	0	0
$217$	− 6.78456i	− 0.460566i
$218$	− 35.0875i	− 2.37643i
$219$	0	0
$220$	0	0
$221$	0.354958i	0.0238771i
$222$	0	0
$223$	−5.48860	−0.367544	−0.183772	−	0.982969i	$-0.558831\pi$
$223$	−5.48860	−0.367544	−0.183772	+	0.982969i	$0.558831\pi$
$224$	− 15.9090i	− 1.06297i
$225$	0	0
$226$	− 26.4383i	− 1.75865i
$227$	−13.7638	−0.913536	−0.456768	−	0.889586i	$-0.650993\pi$
$227$	−13.7638	−0.913536	−0.456768	+	0.889586i	$0.650993\pi$
$228$	0	0
$229$	−21.1326	−1.39648	−0.698242	−	0.715861i	$-0.746034\pi$
$229$	−21.1326	−1.39648	−0.698242	+	0.715861i	$0.746034\pi$
$230$	17.0554	1.12460
$231$	0	0
$232$	−6.59897	−0.433244
$233$	14.9655	0.980420	0.490210	−	0.871604i	$-0.336920\pi$
$233$	14.9655	0.980420	0.490210	+	0.871604i	$0.336920\pi$
$234$	0	0
$235$	13.9741	0.911571
$236$	2.04625i	0.133200i
$237$	0	0
$238$	0.726543i	0.0470948i
$239$	5.40500	0.349620	0.174810	−	0.984602i	$-0.444069\pi$
$239$	5.40500	0.349620	0.174810	+	0.984602i	$0.444069\pi$
$240$	0	0
$241$	11.7091i	0.754252i	0.926162	+	0.377126i	$0.123088\pi$
$241$	11.7091i	0.754252i	−0.926162	+	0.377126i	$0.876912\pi$
$242$	0	0
$243$	0	0
$244$	13.3370i	0.853817i
$245$	− 21.1577i	− 1.35171i
$246$	0	0
$247$	11.1469	0.709260
$248$	−6.00889	−0.381565
$249$	0	0
$250$	− 26.1386i	− 1.65315i
$251$	− 22.5981i	− 1.42638i	−0.700972	−	0.713189i	$-0.747250\pi$
$251$	− 22.5981i	− 1.42638i	0.700972	−	0.713189i	$-0.252750\pi$
$252$	0	0
$253$	0	0
$254$	− 14.2493i	− 0.894081i
$255$	0	0
$256$	−23.8365	−1.48978
$257$	− 27.5164i	− 1.71642i	−0.513295	−	0.858212i	$-0.671575\pi$
$257$	− 27.5164i	− 1.71642i	0.513295	−	0.858212i	$-0.328425\pi$
$258$	0	0
$259$	− 29.4388i	− 1.82924i
$260$	−36.8840	−2.28745
$261$	0	0
$262$	−16.8102	−1.03853
$263$	−32.0386	−1.97559	−0.987793	−	0.155773i	$-0.950213\pi$
$263$	−32.0386	−1.97559	−0.987793	+	0.155773i	$0.950213\pi$
$264$	0	0
$265$	2.76294	0.169726
$266$	22.8159	1.39893
$267$	0	0
$268$	45.8356	2.79986
$269$	− 27.0412i	− 1.64873i	−0.566058	−	0.824365i	$-0.691532\pi$
$269$	− 27.0412i	− 1.64873i	0.566058	−	0.824365i	$-0.308468\pi$
$270$	0	0
$271$	14.8811i	0.903963i	0.892027	+	0.451982i	$0.149283\pi$
$271$	14.8811i	0.903963i	−0.892027	+	0.451982i	$0.850717\pi$
$272$	0.105432	0.00639276
$273$	0	0
$274$	− 24.9713i	− 1.50857i
$275$	0	0
$276$	0	0
$277$	− 7.87985i	− 0.473454i	−0.971576	−	0.236727i	$-0.923925\pi$
$277$	− 7.87985i	− 0.473454i	0.971576	−	0.236727i	$-0.0760747\pi$
$278$	0.124772i	0.00748332i
$279$	0	0
$280$	−32.6765	−1.95279
$281$	−21.2297	−1.26646	−0.633229	−	0.773965i	$-0.718271\pi$
$281$	−21.2297	−1.26646	−0.633229	+	0.773965i	$0.718271\pi$
$282$	0	0
$283$	4.29511i	0.255318i	0.991818	+	0.127659i	$0.0407463\pi$
$283$	4.29511i	0.255318i	−0.991818	+	0.127659i	$0.959254\pi$
$284$	3.95107i	0.234453i
$285$	0	0
$286$	0	0
$287$	34.1996i	2.01874i
$288$	0	0
$289$	−16.9942	−0.999658
$290$	− 9.72012i	− 0.570785i
$291$	0	0
$292$	46.9231i	2.74597i
$293$	19.0955	1.11557	0.557787	−	0.829984i	$-0.311651\pi$
$293$	19.0955	1.11557	0.557787	+	0.829984i	$0.311651\pi$
$294$	0	0
$295$	−1.30457	−0.0759551
$296$	−26.0731	−1.51547
$297$	0	0
$298$	−47.9444	−2.77735
$299$	−15.0150	−0.868337
$300$	0	0
$301$	17.3421	0.999581
$302$	2.79551i	0.160864i
$303$	0	0
$304$	− 3.31093i	− 0.189895i
$305$	−8.50293	−0.486877
$306$	0	0
$307$	1.86240i	0.106293i	0.998587	+	0.0531463i	$0.0169250\pi$
$307$	1.86240i	0.106293i	−0.998587	+	0.0531463i	$0.983075\pi$
$308$	0	0
$309$	0	0
$310$	− 8.85095i	− 0.502700i
$311$	8.95461i	0.507769i	0.967235	+	0.253885i	$0.0817084\pi$
$311$	8.95461i	0.507769i	−0.967235	+	0.253885i	$0.918292\pi$
$312$	0	0
$313$	17.1369	0.968633	0.484316	−	0.874893i	$-0.339068\pi$
$313$	17.1369	0.968633	0.484316	+	0.874893i	$0.339068\pi$
$314$	4.07414	0.229917
$315$	0	0
$316$	2.32684i	0.130895i
$317$	27.3042i	1.53356i	0.641912	+	0.766778i	$0.278141\pi$
$317$	27.3042i	1.53356i	−0.641912	+	0.766778i	$0.721859\pi$
$318$	0	0
$319$	0	0
$320$	− 26.9682i	− 1.50757i
$321$	0	0
$322$	−30.7332	−1.71270
$323$	− 0.182780i	− 0.0101701i
$324$	0	0
$325$	− 0.251799i	− 0.0139673i
$326$	2.35692	0.130538
$327$	0	0
$328$	30.2896	1.67246
$329$	−25.1809	−1.38827
$330$	0	0
$331$	−5.87217	−0.322764	−0.161382	−	0.986892i	$-0.551595\pi$
$331$	−5.87217	−0.322764	−0.161382	+	0.986892i	$0.551595\pi$
$332$	36.0071	1.97615
$333$	0	0
$334$	−31.7785	−1.73884
$335$	29.2222i	1.59658i
$336$	0	0
$337$	− 16.5708i	− 0.902666i	−0.892356	−	0.451333i	$-0.850949\pi$
$337$	− 16.5708i	− 0.902666i	0.892356	−	0.451333i	$-0.149051\pi$
$338$	20.3280	1.10570
$339$	0	0
$340$	0.604800i	0.0327999i
$341$	0	0
$342$	0	0
$343$	9.76796i	0.527420i
$344$	− 15.3594i	− 0.828122i
$345$	0	0
$346$	17.6849	0.950748
$347$	−2.51817	−0.135183	−0.0675913	−	0.997713i	$-0.521531\pi$
$347$	−2.51817	−0.135183	−0.0675913	+	0.997713i	$0.521531\pi$
$348$	0	0
$349$	− 8.72945i	− 0.467277i	−0.972324	−	0.233638i	$-0.924937\pi$
$349$	− 8.72945i	− 0.467277i	0.972324	−	0.233638i	$-0.0750632\pi$
$350$	− 0.515393i	− 0.0275489i
$351$	0	0
$352$	0	0
$353$	0.536500i	0.0285550i	0.999898	+	0.0142775i	$0.00454483\pi$
$353$	0.536500i	0.0285550i	−0.999898	+	0.0142775i	$0.995455\pi$
$354$	0	0
$355$	−2.51897	−0.133693
$356$	− 23.2374i	− 1.23158i
$357$	0	0
$358$	− 29.3857i	− 1.55308i
$359$	−15.8058	−0.834197	−0.417099	−	0.908861i	$-0.636953\pi$
$359$	−15.8058	−0.834197	−0.417099	+	0.908861i	$0.636953\pi$
$360$	0	0
$361$	13.2601	0.697900
$362$	42.3167	2.22412
$363$	0	0
$364$	66.4639	3.48365
$365$	−29.9155	−1.56585
$366$	0	0
$367$	28.9332	1.51030	0.755152	−	0.655550i	$-0.227563\pi$
$367$	28.9332	1.51030	0.755152	+	0.655550i	$0.227563\pi$
$368$	4.45985i	0.232486i
$369$	0	0
$370$	− 38.4050i	− 1.99658i
$371$	−4.97873	−0.258483
$372$	0	0
$373$	− 6.60935i	− 0.342219i	−0.985252	−	0.171109i	$-0.945265\pi$
$373$	− 6.60935i	− 0.342219i	0.985252	−	0.171109i	$-0.0547352\pi$
$374$	0	0
$375$	0	0
$376$	22.3020i	1.15014i
$377$	8.55727i	0.440722i
$378$	0	0
$379$	1.54854	0.0795429	0.0397715	−	0.999209i	$-0.487337\pi$
$379$	1.54854	0.0795429	0.0397715	+	0.999209i	$0.487337\pi$
$380$	18.9928	0.974310
$381$	0	0
$382$	− 50.9112i	− 2.60484i
$383$	13.5078i	0.690217i	0.938563	+	0.345109i	$0.112158\pi$
$383$	13.5078i	0.690217i	−0.938563	+	0.345109i	$0.887842\pi$
$384$	0	0
$385$	0	0
$386$	− 12.1934i	− 0.620629i
$387$	0	0
$388$	57.9880	2.94390
$389$	14.9795i	0.759489i	0.925091	+	0.379745i	$0.123988\pi$
$389$	14.9795i	0.759489i	−0.925091	+	0.379745i	$0.876012\pi$
$390$	0	0
$391$	0.246206i	0.0124512i
$392$	33.7666	1.70547
$393$	0	0
$394$	−12.0991	−0.609542
$395$	−1.48346	−0.0746409
$396$	0	0
$397$	−20.7132	−1.03956	−0.519782	−	0.854299i	$-0.673987\pi$
$397$	−20.7132	−1.03956	−0.519782	+	0.854299i	$0.673987\pi$
$398$	−57.0292	−2.85861
$399$	0	0
$400$	−0.0747911	−0.00373956
$401$	− 27.0448i	− 1.35055i	−0.737565	−	0.675276i	$-0.764025\pi$
$401$	− 27.0448i	− 1.35055i	0.737565	−	0.675276i	$-0.235975\pi$
$402$	0	0
$403$	7.79208i	0.388151i
$404$	−6.07185	−0.302086
$405$	0	0
$406$	17.5154i	0.869273i
$407$	0	0
$408$	0	0
$409$	− 6.10550i	− 0.301898i	−0.988542	−	0.150949i	$-0.951767\pi$
$409$	− 6.10550i	− 0.301898i	0.988542	−	0.150949i	$-0.0482329\pi$
$410$	44.6158i	2.20342i
$411$	0	0
$412$	−11.3928	−0.561285
$413$	2.35080	0.115675
$414$	0	0
$415$	22.9561i	1.12687i
$416$	18.2715i	0.895836i
$417$	0	0
$418$	0	0
$419$	− 35.6619i	− 1.74220i	−0.491110	−	0.871098i	$-0.663409\pi$
$419$	− 35.6619i	− 1.74220i	0.491110	−	0.871098i	$-0.336591\pi$
$420$	0	0
$421$	14.3076	0.697312	0.348656	−	0.937251i	$-0.386638\pi$
$421$	14.3076	0.697312	0.348656	+	0.937251i	$0.386638\pi$
$422$	44.0625i	2.14493i
$423$	0	0
$424$	4.40952i	0.214145i
$425$	−0.00412884	−0.000200278 0
$426$	0	0
$427$	15.3220	0.741485
$428$	52.2536	2.52577
$429$	0	0
$430$	22.6240	1.09103
$431$	8.70321	0.419219	0.209609	−	0.977785i	$-0.432781\pi$
$431$	8.70321	0.419219	0.209609	+	0.977785i	$0.432781\pi$
$432$	0	0
$433$	−37.5533	−1.80469	−0.902347	−	0.431009i	$-0.858158\pi$
$433$	−37.5533	−1.80469	−0.902347	+	0.431009i	$0.858158\pi$
$434$	15.9491i	0.765583i
$435$	0	0
$436$	52.6321i	2.52062i
$437$	7.73170	0.369857
$438$	0	0
$439$	− 32.4669i	− 1.54956i	−0.632232	−	0.774779i	$-0.717861\pi$
$439$	− 32.4669i	− 1.54956i	0.632232	−	0.774779i	$-0.282139\pi$
$440$	0	0
$441$	0	0
$442$	− 0.834435i	− 0.0396900i
$443$	18.8215i	0.894235i	0.894475	+	0.447117i	$0.147549\pi$
$443$	18.8215i	0.894235i	−0.894475	+	0.447117i	$0.852451\pi$
$444$	0	0
$445$	14.8148	0.702290
$446$	12.9026	0.610955
$447$	0	0
$448$	48.5958i	2.29594i
$449$	− 7.22187i	− 0.340821i	−0.985373	−	0.170410i	$-0.945491\pi$
$449$	− 7.22187i	− 0.340821i	0.985373	−	0.170410i	$-0.0545094\pi$
$450$	0	0
$451$	0	0
$452$	39.6582i	1.86536i
$453$	0	0
$454$	32.3559	1.51854
$455$	42.3735i	1.98650i
$456$	0	0
$457$	− 4.22219i	− 0.197506i	−0.995112	−	0.0987529i	$-0.968515\pi$
$457$	− 4.22219i	− 0.197506i	0.995112	−	0.0987529i	$-0.0314853\pi$
$458$	49.6786	2.32133
$459$	0	0
$460$	−25.5834	−1.19283
$461$	14.1922	0.660997	0.330499	−	0.943806i	$-0.392783\pi$
$461$	14.1922	0.660997	0.330499	+	0.943806i	$0.392783\pi$
$462$	0	0
$463$	−40.3784	−1.87654	−0.938271	−	0.345902i	$-0.887573\pi$
$463$	−40.3784	−1.87654	−0.938271	+	0.345902i	$0.887573\pi$
$464$	2.54174	0.117997
$465$	0	0
$466$	−35.1808	−1.62972
$467$	11.3007i	0.522932i	0.965213	+	0.261466i	$0.0842060\pi$
$467$	11.3007i	0.522932i	−0.965213	+	0.261466i	$0.915794\pi$
$468$	0	0
$469$	− 52.6574i	− 2.43149i
$470$	−32.8504	−1.51527
$471$	0	0
$472$	− 2.08203i	− 0.0958334i
$473$	0	0
$474$	0	0
$475$	0.129660i	0.00594920i
$476$	− 1.08983i	− 0.0499523i
$477$	0	0
$478$	−12.7061	−0.581162
$479$	22.8983	1.04625	0.523126	−	0.852256i	$-0.324766\pi$
$479$	22.8983	1.04625	0.523126	+	0.852256i	$0.324766\pi$
$480$	0	0
$481$	33.8105i	1.54163i
$482$	− 27.5258i	− 1.25377i
$483$	0	0
$484$	0	0
$485$	36.9698i	1.67871i
$486$	0	0
$487$	−7.54210	−0.341765	−0.170883	−	0.985291i	$-0.554662\pi$
$487$	−7.54210	−0.341765	−0.170883	+	0.985291i	$0.554662\pi$
$488$	− 13.5703i	− 0.614298i
$489$	0	0
$490$	49.7374i	2.24691i
$491$	−38.1902	−1.72350	−0.861750	−	0.507334i	$-0.830631\pi$
$491$	−38.1902	−1.72350	−0.861750	+	0.507334i	$0.830631\pi$
$492$	0	0
$493$	0.140317	0.00631954
$494$	−26.2041	−1.17898
$495$	0	0
$496$	2.31446	0.103922
$497$	4.53911	0.203607
$498$	0	0
$499$	−8.05909	−0.360774	−0.180387	−	0.983596i	$-0.557735\pi$
$499$	−8.05909	−0.360774	−0.180387	+	0.983596i	$0.557735\pi$
$500$	39.2085i	1.75346i
$501$	0	0
$502$	53.1235i	2.37102i
$503$	26.0620	1.16205	0.581023	−	0.813887i	$-0.302653\pi$
$503$	26.0620	1.16205	0.581023	+	0.813887i	$0.302653\pi$
$504$	0	0
$505$	− 3.87106i	− 0.172260i
$506$	0	0
$507$	0	0
$508$	21.3743i	0.948332i
$509$	− 27.9633i	− 1.23945i	−0.784819	−	0.619726i	$-0.787244\pi$
$509$	− 27.9633i	− 1.23945i	0.784819	−	0.619726i	$-0.212756\pi$
$510$	0	0
$511$	53.9068	2.38469
$512$	15.3439	0.678111
$513$	0	0
$514$	64.6855i	2.85315i
$515$	− 7.26342i	− 0.320065i
$516$	0	0
$517$	0	0
$518$	69.2047i	3.04068i
$519$	0	0
$520$	37.5290	1.64576
$521$	− 32.2913i	− 1.41471i	−0.706859	−	0.707355i	$-0.749888\pi$
$521$	− 32.2913i	− 1.41471i	0.706859	−	0.707355i	$-0.250112\pi$
$522$	0	0
$523$	− 17.3600i	− 0.759101i	−0.925171	−	0.379550i	$-0.876079\pi$
$523$	− 17.3600i	− 0.759101i	0.925171	−	0.379550i	$-0.123921\pi$
$524$	25.2156	1.10155
$525$	0	0
$526$	75.3163	3.28395
$527$	0.127769	0.00556572
$528$	0	0
$529$	12.5853	0.547189
$530$	−6.49511	−0.282130
$531$	0	0
$532$	−34.2244	−1.48382
$533$	− 39.2782i	− 1.70133i
$534$	0	0
$535$	33.3139i	1.44029i
$536$	−46.6372	−2.01442
$537$	0	0
$538$	63.5684i	2.74063i
$539$	0	0
$540$	0	0
$541$	− 10.0763i	− 0.433214i	−0.976259	−	0.216607i	$-0.930501\pi$
$541$	− 10.0763i	− 0.433214i	0.976259	−	0.216607i	$-0.0694990\pi$
$542$	− 34.9825i	− 1.50263i
$543$	0	0
$544$	0.299605	0.0128455
$545$	−33.5552	−1.43735
$546$	0	0
$547$	− 23.8597i	− 1.02017i	−0.860125	−	0.510084i	$-0.829614\pi$
$547$	− 23.8597i	− 1.02017i	0.860125	−	0.510084i	$-0.170386\pi$
$548$	37.4575i	1.60010i
$549$	0	0
$550$	0	0
$551$	− 4.40642i	− 0.187720i
$552$	0	0
$553$	2.67315	0.113674
$554$	18.5239i	0.787007i
$555$	0	0
$556$	− 0.187161i	− 0.00793739i
$557$	39.1556	1.65908	0.829538	−	0.558450i	$-0.188604\pi$
$557$	39.1556	1.65908	0.829538	+	0.558450i	$0.188604\pi$
$558$	0	0
$559$	−19.9174	−0.842416
$560$	12.5861	0.531859
$561$	0	0
$562$	49.9068	2.10519
$563$	5.02067	0.211596	0.105798	−	0.994388i	$-0.466260\pi$
$563$	5.02067	0.211596	0.105798	+	0.994388i	$0.466260\pi$
$564$	0	0
$565$	−25.2838	−1.06370
$566$	− 10.0969i	− 0.424406i
$567$	0	0
$568$	− 4.02016i	− 0.168682i
$569$	−9.50917	−0.398645	−0.199323	−	0.979934i	$-0.563874\pi$
$569$	−9.50917	−0.398645	−0.199323	+	0.979934i	$0.563874\pi$
$570$	0	0
$571$	39.8291i	1.66680i	0.552673	+	0.833398i	$0.313608\pi$
$571$	39.8291i	1.66680i	−0.552673	+	0.833398i	$0.686392\pi$
$572$	0	0
$573$	0	0
$574$	− 80.3963i	− 3.35568i
$575$	− 0.174653i	− 0.00728352i
$576$	0	0
$577$	−13.0577	−0.543600	−0.271800	−	0.962354i	$-0.587619\pi$
$577$	−13.0577	−0.543600	−0.271800	+	0.962354i	$0.587619\pi$
$578$	39.9499	1.66170
$579$	0	0
$580$	14.5804i	0.605419i
$581$	− 41.3661i	− 1.71616i
$582$	0	0
$583$	0	0
$584$	− 47.7437i	− 1.97565i
$585$	0	0
$586$	−44.8898	−1.85438
$587$	− 23.7515i	− 0.980328i	−0.871630	−	0.490164i	$-0.836937\pi$
$587$	− 23.7515i	− 0.980328i	0.871630	−	0.490164i	$-0.163063\pi$
$588$	0	0
$589$	− 4.01240i	− 0.165328i
$590$	3.06679	0.126258
$591$	0	0
$592$	10.0426	0.412749
$593$	37.1489	1.52552	0.762761	−	0.646680i	$-0.223843\pi$
$593$	37.1489	1.52552	0.762761	+	0.646680i	$0.223843\pi$
$594$	0	0
$595$	0.694814	0.0284846
$596$	71.9178	2.94587
$597$	0	0
$598$	35.2971	1.44341
$599$	− 2.38579i	− 0.0974805i	−0.998811	−	0.0487403i	$-0.984479\pi$
$599$	− 2.38579i	− 0.0974805i	0.998811	−	0.0487403i	$-0.0155207\pi$
$600$	0	0
$601$	6.79386i	0.277127i	0.990354	+	0.138564i	$0.0442485\pi$
$601$	6.79386i	0.277127i	−0.990354	+	0.138564i	$0.955751\pi$
$602$	−40.7678	−1.66157
$603$	0	0
$604$	− 4.19334i	− 0.170624i
$605$	0	0
$606$	0	0
$607$	− 40.4963i	− 1.64370i	−0.569707	−	0.821848i	$-0.692943\pi$
$607$	− 40.4963i	− 1.64370i	0.569707	−	0.821848i	$-0.307057\pi$
$608$	− 9.40862i	− 0.381570i
$609$	0	0
$610$	19.9887	0.809318
$611$	28.9203	1.16999
$612$	0	0
$613$	− 0.202577i	− 0.00818201i	−0.999992	−	0.00409100i	$-0.998698\pi$
$613$	− 0.202577i	− 0.00818201i	0.999992	−	0.00409100i	$-0.00130221\pi$
$614$	− 4.37812i	− 0.176686i
$615$	0	0
$616$	0	0
$617$	− 8.08426i	− 0.325460i	−0.986671	−	0.162730i	$-0.947970\pi$
$617$	− 8.08426i	− 0.325460i	0.986671	−	0.162730i	$-0.0520299\pi$
$618$	0	0
$619$	20.9866	0.843523	0.421762	−	0.906707i	$-0.361412\pi$
$619$	20.9866	0.843523	0.421762	+	0.906707i	$0.361412\pi$
$620$	13.2766i	0.533203i
$621$	0	0
$622$	− 21.0505i	− 0.844048i
$623$	−26.6959	−1.06955
$624$	0	0
$625$	−25.2677	−1.01071
$626$	−40.2853	−1.61012
$627$	0	0
$628$	−6.11131	−0.243868
$629$	0.554403	0.0221055
$630$	0	0
$631$	−24.2601	−0.965779	−0.482890	−	0.875681i	$-0.660413\pi$
$631$	−24.2601	−0.965779	−0.482890	+	0.875681i	$0.660413\pi$
$632$	− 2.36753i	− 0.0941753i
$633$	0	0
$634$	− 64.1867i	− 2.54918i
$635$	−13.6270	−0.540772
$636$	0	0
$637$	− 43.7871i	− 1.73491i
$638$	0	0
$639$	0	0
$640$	45.7394i	1.80801i
$641$	16.1391i	0.637457i	0.947846	+	0.318728i	$0.103256\pi$
$641$	16.1391i	0.637457i	−0.947846	+	0.318728i	$0.896744\pi$
$642$	0	0
$643$	12.5840	0.496263	0.248132	−	0.968726i	$-0.420183\pi$
$643$	12.5840	0.496263	0.248132	+	0.968726i	$0.420183\pi$
$644$	46.1006	1.81662
$645$	0	0
$646$	0.429678i	0.0169055i
$647$	30.2507i	1.18928i	0.803993	+	0.594638i	$0.202705\pi$
$647$	30.2507i	1.18928i	−0.803993	+	0.594638i	$0.797295\pi$
$648$	0	0
$649$	0	0
$650$	0.591929i	0.0232174i
$651$	0	0
$652$	−3.53543	−0.138458
$653$	− 19.5926i	− 0.766716i	−0.923600	−	0.383358i	$-0.874768\pi$
$653$	− 19.5926i	− 0.766716i	0.923600	−	0.383358i	$-0.125232\pi$
$654$	0	0
$655$	16.0760i	0.628143i
$656$	−11.6667	−0.455508
$657$	0	0
$658$	59.1953	2.30767
$659$	1.34943	0.0525664	0.0262832	−	0.999655i	$-0.491633\pi$
$659$	1.34943	0.0525664	0.0262832	+	0.999655i	$0.491633\pi$
$660$	0	0
$661$	28.7859	1.11964	0.559821	−	0.828614i	$-0.310870\pi$
$661$	28.7859	1.11964	0.559821	+	0.828614i	$0.310870\pi$
$662$	13.8043	0.536519
$663$	0	0
$664$	−36.6368	−1.42178
$665$	− 21.8195i	− 0.846125i
$666$	0	0
$667$	5.93548i	0.229823i
$668$	47.6685	1.84435
$669$	0	0
$670$	− 68.6954i	− 2.65394i
$671$	0	0
$672$	0	0
$673$	21.7509i	0.838436i	0.907886	+	0.419218i	$0.137696\pi$
$673$	21.7509i	0.838436i	−0.907886	+	0.419218i	$0.862304\pi$
$674$	38.9545i	1.50047i
$675$	0	0
$676$	−30.4925	−1.17279
$677$	−18.9054	−0.726592	−0.363296	−	0.931674i	$-0.618349\pi$
$677$	−18.9054	−0.726592	−0.363296	+	0.931674i	$0.618349\pi$
$678$	0	0
$679$	− 66.6185i	− 2.55658i
$680$	− 0.615376i	− 0.0235986i
$681$	0	0
$682$	0	0
$683$	33.7466i	1.29128i	0.763642	+	0.645640i	$0.223409\pi$
$683$	33.7466i	1.29128i	−0.763642	+	0.645640i	$0.776591\pi$
$684$	0	0
$685$	−23.8807	−0.912436
$686$	− 22.9625i	− 0.876713i
$687$	0	0
$688$	5.91600i	0.225546i
$689$	5.71808	0.217841
$690$	0	0
$691$	−27.5329	−1.04740	−0.523701	−	0.851902i	$-0.675449\pi$
$691$	−27.5329	−1.04740	−0.523701	+	0.851902i	$0.675449\pi$
$692$	−26.5278	−1.00844
$693$	0	0
$694$	5.91971	0.224709
$695$	0.119323	0.00452618
$696$	0	0
$697$	−0.644059	−0.0243955
$698$	20.5212i	0.776738i
$699$	0	0
$700$	0.773102i	0.0292205i
$701$	10.4287	0.393886	0.196943	−	0.980415i	$-0.436899\pi$
$701$	10.4287	0.393886	0.196943	+	0.980415i	$0.436899\pi$
$702$	0	0
$703$	− 17.4101i	− 0.656636i
$704$	0	0
$705$	0	0
$706$	− 1.26120i	− 0.0474661i
$707$	6.97553i	0.262342i
$708$	0	0
$709$	−20.0467	−0.752869	−0.376435	−	0.926443i	$-0.622850\pi$
$709$	−20.0467	−0.752869	−0.376435	+	0.926443i	$0.622850\pi$
$710$	5.92160	0.222234
$711$	0	0
$712$	23.6438i	0.886088i
$713$	5.40473i	0.202409i
$714$	0	0
$715$	0	0
$716$	44.0793i	1.64732i
$717$	0	0
$718$	37.1562	1.38666
$719$	− 10.2376i	− 0.381799i	−0.981610	−	0.190899i	$-0.938860\pi$
$719$	− 10.2376i	− 0.381799i	0.981610	−	0.190899i	$-0.0611404\pi$
$720$	0	0
$721$	13.0885i	0.487440i
$722$	−31.1718	−1.16010
$723$	0	0
$724$	−63.4761	−2.35907
$725$	−0.0995374	−0.00369673
$726$	0	0
$727$	−14.4456	−0.535759	−0.267880	−	0.963452i	$-0.586323\pi$
$727$	−14.4456	−0.535759	−0.267880	+	0.963452i	$0.586323\pi$
$728$	−67.6261	−2.50639
$729$	0	0
$730$	70.3252	2.60285
$731$	0.326593i	0.0120795i
$732$	0	0
$733$	7.30917i	0.269970i	0.990848	+	0.134985i	$0.0430987\pi$
$733$	7.30917i	0.269970i	−0.990848	+	0.134985i	$0.956901\pi$
$734$	−68.0162	−2.51053
$735$	0	0
$736$	12.6735i	0.467151i
$737$	0	0
$738$	0	0
$739$	− 13.1337i	− 0.483131i	−0.970385	−	0.241565i	$-0.922339\pi$
$739$	− 13.1337i	− 0.483131i	0.970385	−	0.241565i	$-0.0776609\pi$
$740$	57.6085i	2.11773i
$741$	0	0
$742$	11.7040	0.429667
$743$	−35.3724	−1.29769	−0.648844	−	0.760921i	$-0.724747\pi$
$743$	−35.3724	−1.29769	−0.648844	+	0.760921i	$0.724747\pi$
$744$	0	0
$745$	45.8506i	1.67984i
$746$	15.5372i	0.568859i
$747$	0	0
$748$	0	0
$749$	− 60.0306i	− 2.19347i
$750$	0	0
$751$	−23.0686	−0.841785	−0.420892	−	0.907111i	$-0.638283\pi$
$751$	−23.0686	−0.841785	−0.420892	+	0.907111i	$0.638283\pi$
$752$	− 8.59012i	− 0.313249i
$753$	0	0
$754$	− 20.1164i	− 0.732597i
$755$	2.67343	0.0972960
$756$	0	0
$757$	9.79363	0.355956	0.177978	−	0.984034i	$-0.443044\pi$
$757$	9.79363	0.355956	0.177978	+	0.984034i	$0.443044\pi$
$758$	−3.64030	−0.132222
$759$	0	0
$760$	−19.3249	−0.700988
$761$	−48.4251	−1.75541	−0.877705	−	0.479201i	$-0.840926\pi$
$761$	−48.4251	−1.75541	−0.877705	+	0.479201i	$0.840926\pi$
$762$	0	0
$763$	60.4655	2.18900
$764$	76.3680i	2.76290i
$765$	0	0
$766$	− 31.7542i	− 1.14732i
$767$	−2.69989	−0.0974876
$768$	0	0
$769$	− 33.4223i	− 1.20524i	−0.798029	−	0.602619i	$-0.794124\pi$
$769$	− 33.4223i	− 1.20524i	0.798029	−	0.602619i	$-0.205876\pi$
$770$	0	0
$771$	0	0
$772$	18.2904i	0.658287i
$773$	− 6.51470i	− 0.234317i	−0.993113	−	0.117159i	$-0.962621\pi$
$773$	− 6.51470i	− 0.234317i	0.993113	−	0.117159i	$-0.0373786\pi$
$774$	0	0
$775$	−0.0906368	−0.00325577
$776$	−59.0021	−2.11805
$777$	0	0
$778$	− 35.2137i	− 1.26247i
$779$	20.2257i	0.724659i
$780$	0	0
$781$	0	0
$782$	− 0.578780i	− 0.0206971i
$783$	0	0
$784$	−13.0060	−0.464499
$785$	− 3.89622i	− 0.139062i
$786$	0	0
$787$	− 28.9940i	− 1.03352i	−0.856129	−	0.516761i	$-0.827137\pi$
$787$	− 28.9940i	− 1.03352i	0.856129	−	0.516761i	$-0.172863\pi$
$788$	18.1489	0.646527
$789$	0	0
$790$	3.48731	0.124073
$791$	45.5606	1.61995
$792$	0	0
$793$	−17.5974	−0.624901
$794$	48.6925	1.72803
$795$	0	0
$796$	85.5451	3.03207
$797$	27.0503i	0.958170i	0.877768	+	0.479085i	$0.159032\pi$
$797$	27.0503i	0.958170i	−0.877768	+	0.479085i	$0.840968\pi$
$798$	0	0
$799$	− 0.474217i	− 0.0167766i
$800$	−0.212533	−0.00751418
$801$	0	0
$802$	63.5768i	2.24498i
$803$	0	0
$804$	0	0
$805$	29.3911i	1.03590i
$806$	− 18.3176i	− 0.645210i
$807$	0	0
$808$	6.17803	0.217342
$809$	−41.8697	−1.47206	−0.736031	−	0.676948i	$-0.763302\pi$
$809$	−41.8697	−1.47206	−0.736031	+	0.676948i	$0.763302\pi$
$810$	0	0
$811$	12.7916i	0.449174i	0.974454	+	0.224587i	$0.0721033\pi$
$811$	12.7916i	0.449174i	−0.974454	+	0.224587i	$0.927897\pi$
$812$	− 26.2735i	− 0.922018i
$813$	0	0
$814$	0	0
$815$	− 2.25399i	− 0.0789538i
$816$	0	0
$817$	10.2561	0.358816
$818$	14.3528i	0.501834i
$819$	0	0
$820$	− 66.9248i	− 2.33711i
$821$	11.8623	0.413998	0.206999	−	0.978341i	$-0.433630\pi$
$821$	11.8623	0.413998	0.206999	+	0.978341i	$0.433630\pi$
$822$	0	0
$823$	26.2761	0.915927	0.457963	−	0.888971i	$-0.348579\pi$
$823$	26.2761	0.915927	0.457963	+	0.888971i	$0.348579\pi$
$824$	11.5921	0.403829
$825$	0	0
$826$	−5.52626	−0.192283
$827$	−25.8883	−0.900224	−0.450112	−	0.892972i	$-0.648616\pi$
$827$	−25.8883	−0.900224	−0.450112	+	0.892972i	$0.648616\pi$
$828$	0	0
$829$	17.3291	0.601863	0.300932	−	0.953646i	$-0.402702\pi$
$829$	17.3291	0.601863	0.300932	+	0.953646i	$0.402702\pi$
$830$	− 53.9651i	− 1.87315i
$831$	0	0
$832$	− 55.8124i	− 1.93495i
$833$	−0.717993	−0.0248770
$834$	0	0
$835$	30.3907i	1.05171i
$836$	0	0
$837$	0	0
$838$	83.8338i	2.89599i
$839$	31.0509i	1.07200i	0.844219	+	0.535999i	$0.180065\pi$
$839$	31.0509i	1.07200i	−0.844219	+	0.535999i	$0.819935\pi$
$840$	0	0
$841$	−25.6173	−0.883354
$842$	−33.6344	−1.15912
$843$	0	0
$844$	− 66.0948i	− 2.27508i
$845$	− 19.4402i	− 0.668765i
$846$	0	0
$847$	0	0
$848$	− 1.69842i	− 0.0583241i
$849$	0	0
$850$	0.00970608	0.000332916 0
$851$	23.4516i	0.803910i
$852$	0	0
$853$	− 0.319368i	− 0.0109350i	−0.999985	−	0.00546748i	$-0.998260\pi$
$853$	− 0.319368i	− 0.0109350i	0.999985	−	0.00546748i	$-0.00174036\pi$
$854$	−36.0190	−1.23254
$855$	0	0
$856$	−53.1674	−1.81722
$857$	−13.2476	−0.452528	−0.226264	−	0.974066i	$-0.572651\pi$
$857$	−13.2476	−0.452528	−0.226264	+	0.974066i	$0.572651\pi$
$858$	0	0
$859$	−8.58152	−0.292798	−0.146399	−	0.989226i	$-0.546768\pi$
$859$	−8.58152	−0.292798	−0.146399	+	0.989226i	$0.546768\pi$
$860$	−33.9365	−1.15723
$861$	0	0
$862$	−20.4595	−0.696853
$863$	53.7180i	1.82858i	0.405057	+	0.914292i	$0.367252\pi$
$863$	53.7180i	1.82858i	−0.405057	+	0.914292i	$0.632748\pi$
$864$	0	0
$865$	− 16.9126i	− 0.575046i
$866$	88.2802	2.99988
$867$	0	0
$868$	− 23.9241i	− 0.812037i
$869$	0	0
$870$	0	0
$871$	60.4771i	2.04919i
$872$	− 53.5525i	− 1.81352i
$873$	0	0
$874$	−18.1757	−0.614801
$875$	45.0440	1.52276
$876$	0	0
$877$	44.0154i	1.48630i	0.669127	+	0.743148i	$0.266668\pi$
$877$	44.0154i	1.48630i	−0.669127	+	0.743148i	$0.733332\pi$
$878$	76.3231i	2.57578i
$879$	0	0
$880$	0	0
$881$	7.44194i	0.250725i	0.992111	+	0.125363i	$0.0400095\pi$
$881$	7.44194i	0.250725i	−0.992111	+	0.125363i	$0.959991\pi$
$882$	0	0
$883$	−9.38632	−0.315875	−0.157937	−	0.987449i	$-0.550484\pi$
$883$	−9.38632	−0.315875	−0.157937	+	0.987449i	$0.550484\pi$
$884$	1.25167i	0.0420983i
$885$	0	0
$886$	− 44.2455i	− 1.48646i
$887$	−24.1847	−0.812043	−0.406022	−	0.913864i	$-0.633084\pi$
$887$	−24.1847	−0.812043	−0.406022	+	0.913864i	$0.633084\pi$
$888$	0	0
$889$	24.5555	0.823565
$890$	−34.8267	−1.16739
$891$	0	0
$892$	−19.3542	−0.648026
$893$	−14.8920	−0.498343
$894$	0	0
$895$	−28.1024	−0.939360
$896$	− 82.4210i	− 2.75349i
$897$	0	0
$898$	16.9772i	0.566535i
$899$	3.08024	0.102732
$900$	0	0
$901$	− 0.0937613i	− 0.00312364i
$902$	0	0
$903$	0	0
$904$	− 40.3517i	− 1.34208i
$905$	− 40.4687i	− 1.34523i
$906$	0	0
$907$	−35.8342	−1.18985	−0.594927	−	0.803780i	$-0.702819\pi$
$907$	−35.8342	−1.18985	−0.594927	+	0.803780i	$0.702819\pi$
$908$	−48.5347	−1.61068
$909$	0	0
$910$	− 99.6116i	− 3.30209i
$911$	− 8.75932i	− 0.290209i	−0.989416	−	0.145105i	$-0.953648\pi$
$911$	− 8.75932i	− 0.290209i	0.989416	−	0.145105i	$-0.0463519\pi$
$912$	0	0
$913$	0	0
$914$	9.92552i	0.328307i
$915$	0	0
$916$	−74.5191	−2.46218
$917$	− 28.9685i	− 0.956625i
$918$	0	0
$919$	41.5197i	1.36961i	0.728726	+	0.684805i	$0.240113\pi$
$919$	41.5197i	1.36961i	−0.728726	+	0.684805i	$0.759887\pi$
$920$	26.0308	0.858211
$921$	0	0
$922$	−33.3630	−1.09875
$923$	−5.21318	−0.171594
$924$	0	0
$925$	−0.393281	−0.0129310
$926$	94.9214	3.11931
$927$	0	0
$928$	7.22283	0.237101
$929$	54.1884i	1.77786i	0.458038	+	0.888932i	$0.348552\pi$
$929$	54.1884i	1.77786i	−0.458038	+	0.888932i	$0.651448\pi$
$930$	0	0
$931$	22.5474i	0.738963i
$932$	52.7721	1.72861
$933$	0	0
$934$	− 26.5656i	− 0.869253i
$935$	0	0
$936$	0	0
$937$	− 17.3958i	− 0.568295i	−0.958781	−	0.284147i	$-0.908290\pi$
$937$	− 17.3958i	− 0.568295i	0.958781	−	0.284147i	$-0.0917105\pi$
$938$	123.787i	4.04179i
$939$	0	0
$940$	49.2763	1.60722
$941$	21.5104	0.701219	0.350610	−	0.936522i	$-0.385974\pi$
$941$	21.5104	0.701219	0.350610	+	0.936522i	$0.385974\pi$
$942$	0	0
$943$	− 27.2441i	− 0.887191i
$944$	0.801941i	0.0261010i
$945$	0	0
$946$	0	0
$947$	− 44.7887i	− 1.45544i	−0.685875	−	0.727719i	$-0.740580\pi$
$947$	− 44.7887i	− 1.45544i	0.685875	−	0.727719i	$-0.259420\pi$
$948$	0	0
$949$	−61.9120	−2.00975
$950$	− 0.304804i	− 0.00988914i
$951$	0	0
$952$	1.10889i	0.0359393i
$953$	10.0337	0.325023	0.162511	−	0.986707i	$-0.448041\pi$
$953$	10.0337	0.325023	0.162511	+	0.986707i	$0.448041\pi$
$954$	0	0
$955$	−48.6879	−1.57550
$956$	19.0594	0.616425
$957$	0	0
$958$	−53.8294	−1.73915
$959$	43.0324	1.38959
$960$	0	0
$961$	−28.1952	−0.909522
$962$	− 79.4817i	− 2.56259i
$963$	0	0
$964$	41.2894i	1.32984i
$965$	−11.6609	−0.375378
$966$	0	0
$967$	− 4.32167i	− 0.138976i	−0.997583	−	0.0694878i	$-0.977863\pi$
$967$	− 4.32167i	− 0.138976i	0.997583	−	0.0694878i	$-0.0221365\pi$
$968$	0	0
$969$	0	0
$970$	− 86.9086i	− 2.79047i
$971$	− 57.3923i	− 1.84181i	−0.389790	−	0.920904i	$-0.627452\pi$
$971$	− 57.3923i	− 1.84181i	0.389790	−	0.920904i	$-0.372548\pi$
$972$	0	0
$973$	−0.215016	−0.00689311
$974$	17.7300	0.568105
$975$	0	0
$976$	5.22689i	0.167309i
$977$	3.45017i	0.110381i	0.998476	+	0.0551903i	$0.0175766\pi$
$977$	3.45017i	0.110381i	−0.998476	+	0.0551903i	$0.982423\pi$
$978$	0	0
$979$	0	0
$980$	− 74.6073i	− 2.38324i
$981$	0	0
$982$	89.7775	2.86491
$983$	1.03468i	0.0330010i	0.999864	+	0.0165005i	$0.00525252\pi$
$983$	1.03468i	0.0330010i	−0.999864	+	0.0165005i	$0.994747\pi$
$984$	0	0
$985$	11.5707i	0.368673i
$986$	−0.329856	−0.0105048
$987$	0	0
$988$	39.3068	1.25052
$989$	−13.8151	−0.439294
$990$	0	0
$991$	9.78910	0.310961	0.155481	−	0.987839i	$-0.450307\pi$
$991$	9.78910	0.310961	0.155481	+	0.987839i	$0.450307\pi$
$992$	6.57696	0.208819
$993$	0	0
$994$	−10.6705	−0.338449
$995$	54.5386i	1.72899i
$996$	0	0
$997$	− 11.6146i	− 0.367838i	−0.982941	−	0.183919i	$-0.941122\pi$
$997$	− 11.6146i	− 0.367838i	0.982941	−	0.183919i	$-0.0588785\pi$
$998$	18.9453	0.599703
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1089.2.d.g.1088.4		16
3.2	odd	2	inner	1089.2.d.g.1088.13		16
11.3	even	5		99.2.j.a.35.1	yes	16
11.7	odd	10		99.2.j.a.17.4	yes	16
11.10	odd	2	inner	1089.2.d.g.1088.14		16
33.14	odd	10		99.2.j.a.35.4	yes	16
33.29	even	10		99.2.j.a.17.1	✓	16
33.32	even	2	inner	1089.2.d.g.1088.3		16
44.3	odd	10		1584.2.cd.c.1025.1		16
44.7	even	10		1584.2.cd.c.17.4		16
99.7	odd	30		891.2.u.c.215.4		32
99.14	odd	30		891.2.u.c.431.4		32
99.25	even	15		891.2.u.c.134.4		32
99.29	even	30		891.2.u.c.215.1		32
99.40	odd	30		891.2.u.c.512.1		32
99.47	odd	30		891.2.u.c.134.1		32
99.58	even	15		891.2.u.c.431.1		32
99.95	even	30		891.2.u.c.512.4		32
132.47	even	10		1584.2.cd.c.1025.4		16
132.95	odd	10		1584.2.cd.c.17.1		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
99.2.j.a.17.1	✓	16	33.29	even	10
99.2.j.a.17.4	yes	16	11.7	odd	10
99.2.j.a.35.1	yes	16	11.3	even	5
99.2.j.a.35.4	yes	16	33.14	odd	10
891.2.u.c.134.1		32	99.47	odd	30
891.2.u.c.134.4		32	99.25	even	15
891.2.u.c.215.1		32	99.29	even	30
891.2.u.c.215.4		32	99.7	odd	30
891.2.u.c.431.1		32	99.58	even	15
891.2.u.c.431.4		32	99.14	odd	30
891.2.u.c.512.1		32	99.40	odd	30
891.2.u.c.512.4		32	99.95	even	30
1089.2.d.g.1088.3		16	33.32	even	2	inner
1089.2.d.g.1088.4		16	1.1	even	1	trivial
1089.2.d.g.1088.13		16	3.2	odd	2	inner
1089.2.d.g.1088.14		16	11.10	odd	2	inner
1584.2.cd.c.17.1		16	132.95	odd	10
1584.2.cd.c.17.4		16	44.7	even	10
1584.2.cd.c.1025.1		16	44.3	odd	10
1584.2.cd.c.1025.4		16	132.47	even	10

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:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1089.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-2.35080 q^{2} +3.52626 q^{4} +2.24814i q^{5} -4.05107i q^{7} -3.58792 q^{8} +O(q^{10})\)	\(q-2.35080 q^{2} +3.52626 q^{4} +2.24814i q^{5} -4.05107i q^{7} -3.58792 q^{8} -5.28492i q^{10} +4.65266i q^{13} +9.52326i q^{14} +1.38197 q^{16} +0.0762914 q^{17} -2.39581i q^{19} +7.92750i q^{20} +3.22717i q^{23} -0.0541194 q^{25} -10.9375i q^{26} -14.2851i q^{28} +1.83922 q^{29} +1.67476 q^{31} +3.92711 q^{32} -0.179346 q^{34} +9.10737 q^{35} +7.26691 q^{37} +5.63207i q^{38} -8.06613i q^{40} -8.44210 q^{41} +4.28086i q^{43} -7.58644i q^{46} -6.21587i q^{47} -9.41120 q^{49} +0.127224 q^{50} +16.4065i q^{52} -1.22899i q^{53} +14.5349i q^{56} -4.32363 q^{58} +0.580290i q^{59} +3.78221i q^{61} -3.93702 q^{62} -11.9958 q^{64} -10.4598 q^{65} +12.9984 q^{67} +0.269023 q^{68} -21.4096 q^{70} +1.12047i q^{71} +13.3068i q^{73} -17.0831 q^{74} -8.44824i q^{76} +0.659861i q^{79} +3.10685i q^{80} +19.8457 q^{82} +10.2111 q^{83} +0.171513i q^{85} -10.0634i q^{86} -6.58983i q^{89} +18.8483 q^{91} +11.3798i q^{92} +14.6123i q^{94} +5.38611 q^{95} +16.4446 q^{97} +22.1238 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 16 q^{4}+O(q^{10}) \) 16 * q + 16 * q^4	\( 16 q + 16 q^{4} + 40 q^{16} - 32 q^{25} + 16 q^{31} + 40 q^{34} + 8 q^{37} + 16 q^{49} + 32 q^{58} - 104 q^{64} + 96 q^{67} - 64 q^{70} + 88 q^{82} + 48 q^{91}+O(q^{100}) \) 16 * q + 16 * q^4 + 40 * q^16 - 32 * q^25 + 16 * q^31 + 40 * q^34 + 8 * q^37 + 16 * q^49 + 32 * q^58 - 104 * q^64 + 96 * q^67 - 64 * q^70 + 88 * q^82 + 48 * q^91

Introduction

L-functions

Modular forms

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Database

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