LMFDB - Embedded newform 3267.2.a.t.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [3267,2,Mod(1,3267)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3267, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("3267.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 3267 = 3^{3} \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3267.a (trivial)

Newform invariants

comment:select newform

sage:traces = [3,-1,0,5,-2,0,-7,-3,0,4,0,0,-4,13,0,5,-1,0,-2,-22,0,0,5,0,5, -2,0,-15,-3,0,4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(31)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$26.0871263404$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.564.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 5x + 3 $ x^3 - x^2 - 5*x + 3
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 297)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-2.08613$ of defining polynomial
Character	$\chi$	$=$	3267.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+2.08613 q^{2} +2.35194 q^{4} -1.35194 q^{5} +0.0861302 q^{7} +0.734191 q^{8} -2.82032 q^{10} -0.648061 q^{13} +0.179679 q^{14} -3.17226 q^{16} -3.43807 q^{17} -4.82032 q^{19} -3.17968 q^{20} +5.82032 q^{23} -3.17226 q^{25} -1.35194 q^{26} +0.202573 q^{28} -4.79001 q^{29} +0.648061 q^{31} -8.08613 q^{32} -7.17226 q^{34} -0.116443 q^{35} +11.4003 q^{37} -10.0558 q^{38} -0.992582 q^{40} -8.96227 q^{41} -9.61033 q^{43} +12.1419 q^{46} +3.87614 q^{47} -6.99258 q^{49} -6.61775 q^{50} -1.52420 q^{52} -7.46838 q^{53} +0.0632360 q^{56} -9.99258 q^{58} -6.69646 q^{59} +12.2281 q^{61} +1.35194 q^{62} -10.5242 q^{64} +0.876139 q^{65} -4.99258 q^{67} -8.08613 q^{68} -0.242915 q^{70} -7.46838 q^{71} +4.87614 q^{73} +23.7826 q^{74} -11.3371 q^{76} -10.8458 q^{79} +4.28870 q^{80} -18.6965 q^{82} -1.94418 q^{83} +4.64806 q^{85} -20.0484 q^{86} -5.04840 q^{89} -0.0558176 q^{91} +13.6890 q^{92} +8.08613 q^{94} +6.51678 q^{95} +15.5726 q^{97} -14.5874 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - q^{2} + 5 q^{4} - 2 q^{5} - 7 q^{7} - 3 q^{8} + 4 q^{10} - 4 q^{13} + 13 q^{14} + 5 q^{16} - q^{17} - 2 q^{19} - 22 q^{20} + 5 q^{23} + 5 q^{25} - 2 q^{26} - 15 q^{28} - 3 q^{29} + 4 q^{31} - 17 q^{32}+ \cdots - 48 q^{98}+O(q^{100}) $ 3 * q - q^2 + 5 * q^4 - 2 * q^5 - 7 * q^7 - 3 * q^8 + 4 * q^10 - 4 * q^13 + 13 * q^14 + 5 * q^16 - q^17 - 2 * q^19 - 22 * q^20 + 5 * q^23 + 5 * q^25 - 2 * q^26 - 15 * q^28 - 3 * q^29 + 4 * q^31 - 17 * q^32 - 7 * q^34 + 8 * q^35 - q^37 - 24 * q^38 + 24 * q^40 - q^41 - 5 * q^43 + 23 * q^46 - 7 * q^47 + 6 * q^49 - 23 * q^50 + 12 * q^52 - 12 * q^53 + 21 * q^56 - 3 * q^58 + 11 * q^59 + 16 * q^61 + 2 * q^62 - 15 * q^64 - 16 * q^65 + 12 * q^67 - 17 * q^68 - 34 * q^70 - 12 * q^71 - 4 * q^73 + 33 * q^74 + 22 * q^76 - 15 * q^79 - 10 * q^80 - 25 * q^82 - 12 * q^83 + 16 * q^85 - 27 * q^86 + 18 * q^89 + 6 * q^91 - 17 * q^92 + 17 * q^94 - 24 * q^95 - 3 * q^97 - 48 * q^98

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 $

$2$	2.08613	1.47512	0.737558	−	0.675283i	$-0.235979\pi$
$2$	2.08613	1.47512	0.737558	+	0.675283i	$0.235979\pi$
$3$	0	0
$3$	0	0
$4$	2.35194	1.17597
$5$	−1.35194	−0.604606	−0.302303	−	0.953212i	$-0.597755\pi$
$5$	−1.35194	−0.604606	−0.302303	+	0.953212i	$0.597755\pi$
$6$	0	0
$7$	0.0861302	0.0325542	0.0162771	−	0.999868i	$-0.494819\pi$
$7$	0.0861302	0.0325542	0.0162771	+	0.999868i	$0.494819\pi$
$8$	0.734191	0.259576
$9$	0	0
$10$	−2.82032	−0.891864
$11$	0	0
$11$	0	0
$12$	0	0
$13$	−0.648061	−0.179740	−0.0898699	−	0.995954i	$-0.528645\pi$
$13$	−0.648061	−0.179740	−0.0898699	+	0.995954i	$0.528645\pi$
$14$	0.179679	0.0480212
$15$	0	0
$16$	−3.17226	−0.793065
$17$	−3.43807	−0.833854	−0.416927	−	0.908940i	$-0.636893\pi$
$17$	−3.43807	−0.833854	−0.416927	+	0.908940i	$0.636893\pi$
$18$	0	0
$19$	−4.82032	−1.10586	−0.552929	−	0.833229i	$-0.686490\pi$
$19$	−4.82032	−1.10586	−0.552929	+	0.833229i	$0.686490\pi$
$20$	−3.17968	−0.710998
$21$	0	0
$22$	0	0
$23$	5.82032	1.21362	0.606810	−	0.794847i	$-0.292449\pi$
$23$	5.82032	1.21362	0.606810	+	0.794847i	$0.292449\pi$
$24$	0	0
$25$	−3.17226	−0.634452
$26$	−1.35194	−0.265137
$27$	0	0
$28$	0.202573	0.0382827
$29$	−4.79001	−0.889482	−0.444741	−	0.895659i	$-0.646704\pi$
$29$	−4.79001	−0.889482	−0.444741	+	0.895659i	$0.646704\pi$
$30$	0	0
$31$	0.648061	0.116395	0.0581976	−	0.998305i	$-0.481465\pi$
$31$	0.648061	0.116395	0.0581976	+	0.998305i	$0.481465\pi$
$32$	−8.08613	−1.42944
$33$	0	0
$34$	−7.17226	−1.23003
$35$	−0.116443	−0.0196824
$36$	0	0
$37$	11.4003	1.87420	0.937102	−	0.349056i	$-0.113498\pi$
$37$	11.4003	1.87420	0.937102	+	0.349056i	$0.113498\pi$
$38$	−10.0558	−1.63127
$39$	0	0
$40$	−0.992582	−0.156941
$41$	−8.96227	−1.39967	−0.699836	−	0.714304i	$-0.746743\pi$
$41$	−8.96227	−1.39967	−0.699836	+	0.714304i	$0.746743\pi$
$42$	0	0
$43$	−9.61033	−1.46556	−0.732781	−	0.680465i	$-0.761778\pi$
$43$	−9.61033	−1.46556	−0.732781	+	0.680465i	$0.761778\pi$
$44$	0	0
$45$	0	0
$46$	12.1419	1.79023
$47$	3.87614	0.565393	0.282696	−	0.959209i	$-0.408771\pi$
$47$	3.87614	0.565393	0.282696	+	0.959209i	$0.408771\pi$
$48$	0	0
$49$	−6.99258	−0.998940
$50$	−6.61775	−0.935891
$51$	0	0
$52$	−1.52420	−0.211368
$53$	−7.46838	−1.02586	−0.512931	−	0.858430i	$-0.671440\pi$
$53$	−7.46838	−1.02586	−0.512931	+	0.858430i	$0.671440\pi$
$54$	0	0
$55$	0	0
$56$	0.0632360	0.00845027
$57$	0	0
$58$	−9.99258	−1.31209
$59$	−6.69646	−0.871805	−0.435902	−	0.899994i	$-0.643571\pi$
$59$	−6.69646	−0.871805	−0.435902	+	0.899994i	$0.643571\pi$
$60$	0	0
$61$	12.2281	1.56564	0.782822	−	0.622245i	$-0.213779\pi$
$61$	12.2281	1.56564	0.782822	+	0.622245i	$0.213779\pi$
$62$	1.35194	0.171696
$63$	0	0
$64$	−10.5242	−1.31552
$65$	0.876139	0.108672
$66$	0	0
$67$	−4.99258	−0.609941	−0.304970	−	0.952362i	$-0.598647\pi$
$67$	−4.99258	−0.609941	−0.304970	+	0.952362i	$0.598647\pi$
$68$	−8.08613	−0.980587
$69$	0	0
$70$	−0.242915	−0.0290339
$71$	−7.46838	−0.886334	−0.443167	−	0.896439i	$-0.646145\pi$
$71$	−7.46838	−0.886334	−0.443167	+	0.896439i	$0.646145\pi$
$72$	0	0
$73$	4.87614	0.570709	0.285354	−	0.958422i	$-0.407889\pi$
$73$	4.87614	0.570709	0.285354	+	0.958422i	$0.407889\pi$
$74$	23.7826	2.76467
$75$	0	0
$76$	−11.3371	−1.30045
$77$	0	0
$78$	0	0
$79$	−10.8458	−1.22025	−0.610125	−	0.792305i	$-0.708881\pi$
$79$	−10.8458	−1.22025	−0.610125	+	0.792305i	$0.708881\pi$
$80$	4.28870	0.479492
$81$	0	0
$82$	−18.6965	−2.06468
$83$	−1.94418	−0.213402	−0.106701	−	0.994291i	$-0.534029\pi$
$83$	−1.94418	−0.213402	−0.106701	+	0.994291i	$0.534029\pi$
$84$	0	0
$85$	4.64806	0.504153
$86$	−20.0484	−2.16187
$87$	0	0
$88$	0	0
$89$	−5.04840	−0.535129	−0.267565	−	0.963540i	$-0.586219\pi$
$89$	−5.04840	−0.535129	−0.267565	+	0.963540i	$0.586219\pi$
$90$	0	0
$91$	−0.0558176	−0.00585127
$92$	13.6890	1.42718
$93$	0	0
$94$	8.08613	0.834021
$95$	6.51678	0.668608
$96$	0	0
$97$	15.5726	1.58116	0.790579	−	0.612360i	$-0.209780\pi$
$97$	15.5726	1.58116	0.790579	+	0.612360i	$0.209780\pi$
$98$	−14.5874	−1.47355
$99$	0	0
$100$	−7.46096	−0.746096
$101$	−2.56193	−0.254922	−0.127461	−	0.991844i	$-0.540683\pi$
$101$	−2.56193	−0.254922	−0.127461	+	0.991844i	$0.540683\pi$
$102$	0	0
$103$	14.9926	1.47726	0.738631	−	0.674109i	$-0.235472\pi$
$103$	14.9926	1.47726	0.738631	+	0.674109i	$0.235472\pi$
$104$	−0.475800	−0.0466561
$105$	0	0
$106$	−15.5800	−1.51327
$107$	6.47580	0.626039	0.313020	−	0.949747i	$-0.398659\pi$
$107$	6.47580	0.626039	0.313020	+	0.949747i	$0.398659\pi$
$108$	0	0
$109$	−0.172260	−0.0164996	−0.00824978	−	0.999966i	$-0.502626\pi$
$109$	−0.172260	−0.0164996	−0.00824978	+	0.999966i	$0.502626\pi$
$110$	0	0
$111$	0	0
$112$	−0.273227	−0.0258176
$113$	−17.5242	−1.64854	−0.824269	−	0.566198i	$-0.808414\pi$
$113$	−17.5242	−1.64854	−0.824269	+	0.566198i	$0.808414\pi$
$114$	0	0
$115$	−7.86872	−0.733762
$116$	−11.2658	−1.04600
$117$	0	0
$118$	−13.9697	−1.28601
$119$	−0.296122	−0.0271454
$120$	0	0
$121$	0	0
$122$	25.5094	2.30951
$123$	0	0
$124$	1.52420	0.136877
$125$	11.0484	0.988199
$126$	0	0
$127$	−8.47580	−0.752106	−0.376053	−	0.926598i	$-0.622719\pi$
$127$	−8.47580	−0.752106	−0.376053	+	0.926598i	$0.622719\pi$
$128$	−5.78259	−0.511114
$129$	0	0
$130$	1.82774	0.160303
$131$	−1.46838	−0.128293	−0.0641466	−	0.997940i	$-0.520433\pi$
$131$	−1.46838	−0.128293	−0.0641466	+	0.997940i	$0.520433\pi$
$132$	0	0
$133$	−0.415175	−0.0360003
$134$	−10.4152	−0.899734
$135$	0	0
$136$	−2.52420	−0.216448
$137$	12.8761	1.10008	0.550041	−	0.835137i	$-0.314612\pi$
$137$	12.8761	1.10008	0.550041	+	0.835137i	$0.314612\pi$
$138$	0	0
$139$	13.0787	1.10932	0.554661	−	0.832076i	$-0.312848\pi$
$139$	13.0787	1.10932	0.554661	+	0.832076i	$0.312848\pi$
$140$	−0.273866	−0.0231459
$141$	0	0
$142$	−15.5800	−1.30745
$143$	0	0
$144$	0	0
$145$	6.47580	0.537786
$146$	10.1723	0.841862
$147$	0	0
$148$	26.8129	2.20401
$149$	8.22808	0.674070	0.337035	−	0.941492i	$-0.390576\pi$
$149$	8.22808	0.674070	0.337035	+	0.941492i	$0.390576\pi$
$150$	0	0
$151$	−3.94418	−0.320973	−0.160487	−	0.987038i	$-0.551306\pi$
$151$	−3.94418	−0.320973	−0.160487	+	0.987038i	$0.551306\pi$
$152$	−3.53904	−0.287054
$153$	0	0
$154$	0	0
$155$	−0.876139	−0.0703732
$156$	0	0
$157$	−5.53162	−0.441471	−0.220736	−	0.975334i	$-0.570846\pi$
$157$	−5.53162	−0.441471	−0.220736	+	0.975334i	$0.570846\pi$
$158$	−22.6258	−1.80001
$159$	0	0
$160$	10.9320	0.864247
$161$	0.501305	0.0395084
$162$	0	0
$163$	7.64064	0.598461	0.299231	−	0.954181i	$-0.403270\pi$
$163$	7.64064	0.598461	0.299231	+	0.954181i	$0.403270\pi$
$164$	−21.0787	−1.64597
$165$	0	0
$166$	−4.05582	−0.314792
$167$	−21.1042	−1.63309	−0.816547	−	0.577279i	$-0.804114\pi$
$167$	−21.1042	−1.63309	−0.816547	+	0.577279i	$0.804114\pi$
$168$	0	0
$169$	−12.5800	−0.967694
$170$	9.69646	0.743685
$171$	0	0
$172$	−22.6029	−1.72346
$173$	23.0894	1.75545	0.877727	−	0.479162i	$-0.159059\pi$
$173$	23.0894	1.75545	0.877727	+	0.479162i	$0.159059\pi$
$174$	0	0
$175$	−0.273227	−0.0206541
$176$	0	0
$177$	0	0
$178$	−10.5316	−0.789378
$179$	−21.5168	−1.60824	−0.804120	−	0.594467i	$-0.797363\pi$
$179$	−21.5168	−1.60824	−0.804120	+	0.594467i	$0.797363\pi$
$180$	0	0
$181$	−13.2329	−0.983593	−0.491796	−	0.870710i	$-0.663660\pi$
$181$	−13.2329	−0.983593	−0.491796	+	0.870710i	$0.663660\pi$
$182$	−0.116443	−0.00863131
$183$	0	0
$184$	4.27323	0.315027
$185$	−15.4126	−1.13315
$186$	0	0
$187$	0	0
$188$	9.11644	0.664885
$189$	0	0
$190$	13.5949	0.986274
$191$	21.8761	1.58290	0.791451	−	0.611233i	$-0.209326\pi$
$191$	21.8761	1.58290	0.791451	+	0.611233i	$0.209326\pi$
$192$	0	0
$193$	6.82032	0.490937	0.245469	−	0.969405i	$-0.421058\pi$
$193$	6.82032	0.490937	0.245469	+	0.969405i	$0.421058\pi$
$194$	32.4865	2.33239
$195$	0	0
$196$	−16.4461	−1.17472
$197$	−6.85063	−0.488087	−0.244044	−	0.969764i	$-0.578474\pi$
$197$	−6.85063	−0.488087	−0.244044	+	0.969764i	$0.578474\pi$
$198$	0	0
$199$	17.3371	1.22899	0.614497	−	0.788919i	$-0.289359\pi$
$199$	17.3371	1.22899	0.614497	+	0.788919i	$0.289359\pi$
$200$	−2.32905	−0.164688
$201$	0	0
$202$	−5.34452	−0.376039
$203$	−0.412564	−0.0289563
$204$	0	0
$205$	12.1164	0.846249
$206$	31.2765	2.17914
$207$	0	0
$208$	2.05582	0.142545
$209$	0	0
$210$	0	0
$211$	−24.0713	−1.65714	−0.828568	−	0.559888i	$-0.810844\pi$
$211$	−24.0713	−1.65714	−0.828568	+	0.559888i	$0.810844\pi$
$212$	−17.5652	−1.20638
$213$	0	0
$214$	13.5094	0.923481
$215$	12.9926	0.886087
$216$	0	0
$217$	0.0558176	0.00378915
$218$	−0.359358	−0.0243388
$219$	0	0
$220$	0	0
$221$	2.22808	0.149877
$222$	0	0
$223$	−2.28870	−0.153263	−0.0766315	−	0.997059i	$-0.524416\pi$
$223$	−2.28870	−0.153263	−0.0766315	+	0.997059i	$0.524416\pi$
$224$	−0.696460	−0.0465342
$225$	0	0
$226$	−36.5578	−2.43179
$227$	−24.9926	−1.65882	−0.829408	−	0.558643i	$-0.811322\pi$
$227$	−24.9926	−1.65882	−0.829408	+	0.558643i	$0.811322\pi$
$228$	0	0
$229$	−2.06804	−0.136660	−0.0683301	−	0.997663i	$-0.521767\pi$
$229$	−2.06804	−0.136660	−0.0683301	+	0.997663i	$0.521767\pi$
$230$	−16.4152	−1.08238
$231$	0	0
$232$	−3.51678	−0.230888
$233$	0.992582	0.0650262	0.0325131	−	0.999471i	$-0.489649\pi$
$233$	0.992582	0.0650262	0.0325131	+	0.999471i	$0.489649\pi$
$234$	0	0
$235$	−5.24030	−0.341840
$236$	−15.7497	−1.02522
$237$	0	0
$238$	−0.617748	−0.0400427
$239$	−10.5726	−0.683885	−0.341942	−	0.939721i	$-0.611085\pi$
$239$	−10.5726	−0.683885	−0.341942	+	0.939721i	$0.611085\pi$
$240$	0	0
$241$	29.3929	1.89336	0.946682	−	0.322169i	$-0.104412\pi$
$241$	29.3929	1.89336	0.946682	+	0.322169i	$0.104412\pi$
$242$	0	0
$243$	0	0
$244$	28.7597	1.84115
$245$	9.45355	0.603965
$246$	0	0
$247$	3.12386	0.198767
$248$	0.475800	0.0302134
$249$	0	0
$250$	23.0484	1.45771
$251$	−3.35936	−0.212041	−0.106020	−	0.994364i	$-0.533811\pi$
$251$	−3.35936	−0.212041	−0.106020	+	0.994364i	$0.533811\pi$
$252$	0	0
$253$	0	0
$254$	−17.6816	−1.10944
$255$	0	0
$256$	8.98516	0.561573
$257$	−9.46357	−0.590322	−0.295161	−	0.955448i	$-0.595373\pi$
$257$	−9.46357	−0.590322	−0.295161	+	0.955448i	$0.595373\pi$
$258$	0	0
$259$	0.981913	0.0610131
$260$	2.06063	0.127795
$261$	0	0
$262$	−3.06324	−0.189247
$263$	22.6890	1.39907	0.699533	−	0.714600i	$-0.253391\pi$
$263$	22.6890	1.39907	0.699533	+	0.714600i	$0.253391\pi$
$264$	0	0
$265$	10.0968	0.620241
$266$	−0.866110	−0.0531046
$267$	0	0
$268$	−11.7422	−0.717272
$269$	4.64806	0.283397	0.141699	−	0.989910i	$-0.454744\pi$
$269$	4.64806	0.283397	0.141699	+	0.989910i	$0.454744\pi$
$270$	0	0
$271$	13.3626	0.811721	0.405860	−	0.913935i	$-0.366972\pi$
$271$	13.3626	0.811721	0.405860	+	0.913935i	$0.366972\pi$
$272$	10.9065	0.661301
$273$	0	0
$274$	26.8613	1.62275
$275$	0	0
$276$	0	0
$277$	9.40776	0.565257	0.282629	−	0.959229i	$-0.408794\pi$
$277$	9.40776	0.565257	0.282629	+	0.959229i	$0.408794\pi$
$278$	27.2839	1.63638
$279$	0	0
$280$	−0.0854912	−0.00510908
$281$	−28.8310	−1.71991	−0.859956	−	0.510368i	$-0.829509\pi$
$281$	−28.8310	−1.71991	−0.859956	+	0.510368i	$0.829509\pi$
$282$	0	0
$283$	17.3519	1.03147	0.515733	−	0.856749i	$-0.327520\pi$
$283$	17.3519	1.03147	0.515733	+	0.856749i	$0.327520\pi$
$284$	−17.5652	−1.04230
$285$	0	0
$286$	0	0
$287$	−0.771922	−0.0455651
$288$	0	0
$289$	−5.17968	−0.304687
$290$	13.5094	0.793297
$291$	0	0
$292$	11.4684	0.671136
$293$	−5.52420	−0.322727	−0.161364	−	0.986895i	$-0.551589\pi$
$293$	−5.52420	−0.322727	−0.161364	+	0.986895i	$0.551589\pi$
$294$	0	0
$295$	9.05321	0.527098
$296$	8.37003	0.486498
$297$	0	0
$298$	17.1648	0.994333
$299$	−3.77192	−0.218136
$300$	0	0
$301$	−0.827740	−0.0477101
$302$	−8.22808	−0.473473
$303$	0	0
$304$	15.2913	0.877017
$305$	−16.5316	−0.946598
$306$	0	0
$307$	24.2436	1.38365	0.691826	−	0.722064i	$-0.256806\pi$
$307$	24.2436	1.38365	0.691826	+	0.722064i	$0.256806\pi$
$308$	0	0
$309$	0	0
$310$	−1.82774	−0.103809
$311$	−1.05582	−0.0598699	−0.0299350	−	0.999552i	$-0.509530\pi$
$311$	−1.05582	−0.0598699	−0.0299350	+	0.999552i	$0.509530\pi$
$312$	0	0
$313$	13.0484	0.737539	0.368770	−	0.929521i	$-0.379779\pi$
$313$	13.0484	0.737539	0.368770	+	0.929521i	$0.379779\pi$
$314$	−11.5397	−0.651222
$315$	0	0
$316$	−25.5087	−1.43498
$317$	14.2691	0.801430	0.400715	−	0.916203i	$-0.368762\pi$
$317$	14.2691	0.801430	0.400715	+	0.916203i	$0.368762\pi$
$318$	0	0
$319$	0	0
$320$	14.2281	0.795374
$321$	0	0
$322$	1.04579	0.0582795
$323$	16.5726	0.922124
$324$	0	0
$325$	2.05582	0.114036
$326$	15.9394	0.882800
$327$	0	0
$328$	−6.58002	−0.363321
$329$	0.333853	0.0184059
$330$	0	0
$331$	7.80810	0.429172	0.214586	−	0.976705i	$-0.431160\pi$
$331$	7.80810	0.429172	0.214586	+	0.976705i	$0.431160\pi$
$332$	−4.57260	−0.250954
$333$	0	0
$334$	−44.0261	−2.40900
$335$	6.74967	0.368774
$336$	0	0
$337$	20.5726	1.12066	0.560330	−	0.828269i	$-0.310674\pi$
$337$	20.5726	1.12066	0.560330	+	0.828269i	$0.310674\pi$
$338$	−26.2436	−1.42746
$339$	0	0
$340$	10.9320	0.592869
$341$	0	0
$342$	0	0
$343$	−1.20518	−0.0650738
$344$	−7.05582	−0.380424
$345$	0	0
$346$	48.1675	2.58950
$347$	−2.06063	−0.110620	−0.0553101	−	0.998469i	$-0.517615\pi$
$347$	−2.06063	−0.110620	−0.0553101	+	0.998469i	$0.517615\pi$
$348$	0	0
$349$	−3.99519	−0.213858	−0.106929	−	0.994267i	$-0.534102\pi$
$349$	−3.99519	−0.213858	−0.106929	+	0.994267i	$0.534102\pi$
$350$	−0.569988	−0.0304671
$351$	0	0
$352$	0	0
$353$	−9.58002	−0.509893	−0.254946	−	0.966955i	$-0.582058\pi$
$353$	−9.58002	−0.509893	−0.254946	+	0.966955i	$0.582058\pi$
$354$	0	0
$355$	10.0968	0.535882
$356$	−11.8735	−0.629296
$357$	0	0
$358$	−44.8868	−2.37234
$359$	−20.2281	−1.06760	−0.533799	−	0.845612i	$-0.679236\pi$
$359$	−20.2281	−1.06760	−0.533799	+	0.845612i	$0.679236\pi$
$360$	0	0
$361$	4.23550	0.222921
$362$	−27.6055	−1.45091
$363$	0	0
$364$	−0.131280	−0.00688092
$365$	−6.59224	−0.345054
$366$	0	0
$367$	29.2207	1.52531	0.762653	−	0.646808i	$-0.223896\pi$
$367$	29.2207	1.52531	0.762653	+	0.646808i	$0.223896\pi$
$368$	−18.4636	−0.962480
$369$	0	0
$370$	−32.1526	−1.67153
$371$	−0.643253	−0.0333960
$372$	0	0
$373$	12.5120	0.647845	0.323923	−	0.946084i	$-0.394998\pi$
$373$	12.5120	0.647845	0.323923	+	0.946084i	$0.394998\pi$
$374$	0	0
$375$	0	0
$376$	2.84583	0.146762
$377$	3.10422	0.159875
$378$	0	0
$379$	−13.7719	−0.707416	−0.353708	−	0.935356i	$-0.615079\pi$
$379$	−13.7719	−0.707416	−0.353708	+	0.935356i	$0.615079\pi$
$380$	15.3271	0.786262
$381$	0	0
$382$	45.6365	2.33497
$383$	−9.22066	−0.471154	−0.235577	−	0.971856i	$-0.575698\pi$
$383$	−9.22066	−0.471154	−0.235577	+	0.971856i	$0.575698\pi$
$384$	0	0
$385$	0	0
$386$	14.2281	0.724190
$387$	0	0
$388$	36.6258	1.85939
$389$	−17.0484	−0.864388	−0.432194	−	0.901781i	$-0.642260\pi$
$389$	−17.0484	−0.864388	−0.432194	+	0.901781i	$0.642260\pi$
$390$	0	0
$391$	−20.0107	−1.01198
$392$	−5.13389	−0.259301
$393$	0	0
$394$	−14.2913	−0.719986
$395$	14.6629	0.737770
$396$	0	0
$397$	−17.3929	−0.872926	−0.436463	−	0.899722i	$-0.643769\pi$
$397$	−17.3929	−0.872926	−0.436463	+	0.899722i	$0.643769\pi$
$398$	36.1675	1.81291
$399$	0	0
$400$	10.0632	0.503162
$401$	−12.2839	−0.613428	−0.306714	−	0.951802i	$-0.599230\pi$
$401$	−12.2839	−0.613428	−0.306714	+	0.951802i	$0.599230\pi$
$402$	0	0
$403$	−0.419983	−0.0209208
$404$	−6.02551	−0.299780
$405$	0	0
$406$	−0.860663	−0.0427140
$407$	0	0
$408$	0	0
$409$	−20.2329	−1.00045	−0.500226	−	0.865895i	$-0.666750\pi$
$409$	−20.2329	−1.00045	−0.500226	+	0.865895i	$0.666750\pi$
$410$	25.2765	1.24832
$411$	0	0
$412$	35.2616	1.73722
$413$	−0.576767	−0.0283809
$414$	0	0
$415$	2.62842	0.129024
$416$	5.24030	0.256927
$417$	0	0
$418$	0	0
$419$	15.8129	0.772511	0.386255	−	0.922392i	$-0.373768\pi$
$419$	15.8129	0.772511	0.386255	+	0.922392i	$0.373768\pi$
$420$	0	0
$421$	15.1723	0.739451	0.369725	−	0.929141i	$-0.379452\pi$
$421$	15.1723	0.739451	0.369725	+	0.929141i	$0.379452\pi$
$422$	−50.2159	−2.44447
$423$	0	0
$424$	−5.48322	−0.266289
$425$	10.9065	0.529041
$426$	0	0
$427$	1.05321	0.0509682
$428$	15.2307	0.736203
$429$	0	0
$430$	27.1042	1.30708
$431$	32.1526	1.54874	0.774369	−	0.632735i	$-0.218068\pi$
$431$	32.1526	1.54874	0.774369	+	0.632735i	$0.218068\pi$
$432$	0	0
$433$	−26.0436	−1.25158	−0.625788	−	0.779994i	$-0.715222\pi$
$433$	−26.0436	−1.25158	−0.625788	+	0.779994i	$0.715222\pi$
$434$	0.116443	0.00558943
$435$	0	0
$436$	−0.405146	−0.0194030
$437$	−28.0558	−1.34209
$438$	0	0
$439$	11.6103	0.554131	0.277065	−	0.960851i	$-0.410638\pi$
$439$	11.6103	0.554131	0.277065	+	0.960851i	$0.410638\pi$
$440$	0	0
$441$	0	0
$442$	4.64806	0.221086
$443$	−0.232886	−0.0110647	−0.00553236	−	0.999985i	$-0.501761\pi$
$443$	−0.232886	−0.0110647	−0.00553236	+	0.999985i	$0.501761\pi$
$444$	0	0
$445$	6.82513	0.323542
$446$	−4.77453	−0.226081
$447$	0	0
$448$	−0.906451	−0.0428258
$449$	9.64545	0.455197	0.227598	−	0.973755i	$-0.426913\pi$
$449$	9.64545	0.455197	0.227598	+	0.973755i	$0.426913\pi$
$450$	0	0
$451$	0	0
$452$	−41.2159	−1.93863
$453$	0	0
$454$	−52.1378	−2.44695
$455$	0.0754620	0.00353771
$456$	0	0
$457$	−20.9171	−0.978462	−0.489231	−	0.872154i	$-0.662722\pi$
$457$	−20.9171	−0.978462	−0.489231	+	0.872154i	$0.662722\pi$
$458$	−4.31421	−0.201590
$459$	0	0
$460$	−18.5068	−0.862882
$461$	27.9293	1.30080	0.650400	−	0.759592i	$-0.274601\pi$
$461$	27.9293	1.30080	0.650400	+	0.759592i	$0.274601\pi$
$462$	0	0
$463$	37.6406	1.74931	0.874655	−	0.484747i	$-0.161088\pi$
$463$	37.6406	1.74931	0.874655	+	0.484747i	$0.161088\pi$
$464$	15.1952	0.705417
$465$	0	0
$466$	2.07065	0.0959212
$467$	−30.6210	−1.41697	−0.708485	−	0.705725i	$-0.750621\pi$
$467$	−30.6210	−1.41697	−0.708485	+	0.705725i	$0.750621\pi$
$468$	0	0
$469$	−0.430012	−0.0198561
$470$	−10.9320	−0.504254
$471$	0	0
$472$	−4.91648	−0.226299
$473$	0	0
$474$	0	0
$475$	15.2913	0.701614
$476$	−0.696460	−0.0319222
$477$	0	0
$478$	−22.0558	−1.00881
$479$	7.54384	0.344687	0.172344	−	0.985037i	$-0.444866\pi$
$479$	7.54384	0.344687	0.172344	+	0.985037i	$0.444866\pi$
$480$	0	0
$481$	−7.38811	−0.336869
$482$	61.3175	2.79293
$483$	0	0
$484$	0	0
$485$	−21.0532	−0.955977
$486$	0	0
$487$	43.0484	1.95071	0.975355	−	0.220643i	$-0.0708156\pi$
$487$	43.0484	1.95071	0.975355	+	0.220643i	$0.0708156\pi$
$488$	8.97774	0.406403
$489$	0	0
$490$	19.7213	0.890919
$491$	−28.1723	−1.27140	−0.635698	−	0.771938i	$-0.719288\pi$
$491$	−28.1723	−1.27140	−0.635698	+	0.771938i	$0.719288\pi$
$492$	0	0
$493$	16.4684	0.741699
$494$	6.51678	0.293204
$495$	0	0
$496$	−2.05582	−0.0923089
$497$	−0.643253	−0.0288539
$498$	0	0
$499$	−25.6210	−1.14695	−0.573477	−	0.819222i	$-0.694406\pi$
$499$	−25.6210	−1.14695	−0.573477	+	0.819222i	$0.694406\pi$
$500$	25.9852	1.16209
$501$	0	0
$502$	−7.00806	−0.312785
$503$	−8.06063	−0.359406	−0.179703	−	0.983721i	$-0.557514\pi$
$503$	−8.06063	−0.359406	−0.179703	+	0.983721i	$0.557514\pi$
$504$	0	0
$505$	3.46357	0.154127
$506$	0	0
$507$	0	0
$508$	−19.9346	−0.884453
$509$	−29.3977	−1.30303	−0.651516	−	0.758635i	$-0.725867\pi$
$509$	−29.3977	−1.30303	−0.651516	+	0.758635i	$0.725867\pi$
$510$	0	0
$511$	0.419983	0.0185789
$512$	30.3094	1.33950
$513$	0	0
$514$	−19.7422	−0.870793
$515$	−20.2691	−0.893161
$516$	0	0
$517$	0	0
$518$	2.04840	0.0900015
$519$	0	0
$520$	0.643253	0.0282085
$521$	42.6842	1.87003	0.935015	−	0.354608i	$-0.115386\pi$
$521$	42.6842	1.87003	0.935015	+	0.354608i	$0.115386\pi$
$522$	0	0
$523$	20.8310	0.910876	0.455438	−	0.890268i	$-0.349483\pi$
$523$	20.8310	0.910876	0.455438	+	0.890268i	$0.349483\pi$
$524$	−3.45355	−0.150869
$525$	0	0
$526$	47.3323	2.06379
$527$	−2.22808	−0.0970566
$528$	0	0
$529$	10.8761	0.472876
$530$	21.0632	0.914929
$531$	0	0
$532$	−0.976467	−0.0423352
$533$	5.80810	0.251577
$534$	0	0
$535$	−8.75489	−0.378507
$536$	−3.66551	−0.158326
$537$	0	0
$538$	9.69646	0.418044
$539$	0	0
$540$	0	0
$541$	−10.2691	−0.441501	−0.220751	−	0.975330i	$-0.570851\pi$
$541$	−10.2691	−0.441501	−0.220751	+	0.975330i	$0.570851\pi$
$542$	27.8761	1.19738
$543$	0	0
$544$	27.8007	1.19194
$545$	0.232886	0.00997572
$546$	0	0
$547$	2.31421	0.0989484	0.0494742	−	0.998775i	$-0.484245\pi$
$547$	2.31421	0.0989484	0.0494742	+	0.998775i	$0.484245\pi$
$548$	30.2839	1.29366
$549$	0	0
$550$	0	0
$551$	23.0894	0.983641
$552$	0	0
$553$	−0.934153	−0.0397242
$554$	19.6258	0.833821
$555$	0	0
$556$	30.7603	1.30453
$557$	−6.50131	−0.275469	−0.137735	−	0.990469i	$-0.543982\pi$
$557$	−6.50131	−0.275469	−0.137735	+	0.990469i	$0.543982\pi$
$558$	0	0
$559$	6.22808	0.263420
$560$	0.369387	0.0156094
$561$	0	0
$562$	−60.1452	−2.53707
$563$	1.70127	0.0716999	0.0358499	−	0.999357i	$-0.488586\pi$
$563$	1.70127	0.0716999	0.0358499	+	0.999357i	$0.488586\pi$
$564$	0	0
$565$	23.6917	0.996715
$566$	36.1984	1.52153
$567$	0	0
$568$	−5.48322	−0.230071
$569$	−16.8809	−0.707686	−0.353843	−	0.935305i	$-0.615125\pi$
$569$	−16.8809	−0.707686	−0.353843	+	0.935305i	$0.615125\pi$
$570$	0	0
$571$	−25.9500	−1.08598	−0.542988	−	0.839741i	$-0.682707\pi$
$571$	−25.9500	−1.08598	−0.542988	+	0.839741i	$0.682707\pi$
$572$	0	0
$573$	0	0
$574$	−1.61033	−0.0672139
$575$	−18.4636	−0.769984
$576$	0	0
$577$	−12.6406	−0.526237	−0.263118	−	0.964764i	$-0.584751\pi$
$577$	−12.6406	−0.526237	−0.263118	+	0.964764i	$0.584751\pi$
$578$	−10.8055	−0.449449
$579$	0	0
$580$	15.2307	0.632420
$581$	−0.167453	−0.00694711
$582$	0	0
$583$	0	0
$584$	3.58002	0.148142
$585$	0	0
$586$	−11.5242	−0.476060
$587$	−38.9655	−1.60828	−0.804140	−	0.594441i	$-0.797374\pi$
$587$	−38.9655	−1.60828	−0.804140	+	0.594441i	$0.797374\pi$
$588$	0	0
$589$	−3.12386	−0.128716
$590$	18.8862	0.777531
$591$	0	0
$592$	−36.1648	−1.48637
$593$	−14.1271	−0.580131	−0.290065	−	0.957007i	$-0.593677\pi$
$593$	−14.1271	−0.580131	−0.290065	+	0.957007i	$0.593677\pi$
$594$	0	0
$595$	0.400338	0.0164123
$596$	19.3519	0.792686
$597$	0	0
$598$	−7.86872	−0.321776
$599$	20.1648	0.823913	0.411957	−	0.911203i	$-0.364846\pi$
$599$	20.1648	0.823913	0.411957	+	0.911203i	$0.364846\pi$
$600$	0	0
$601$	34.0410	1.38856	0.694280	−	0.719705i	$-0.255723\pi$
$601$	34.0410	1.38856	0.694280	+	0.719705i	$0.255723\pi$
$602$	−1.72677	−0.0703780
$603$	0	0
$604$	−9.27648	−0.377455
$605$	0	0
$606$	0	0
$607$	2.57260	0.104419	0.0522093	−	0.998636i	$-0.483374\pi$
$607$	2.57260	0.104419	0.0522093	+	0.998636i	$0.483374\pi$
$608$	38.9777	1.58076
$609$	0	0
$610$	−34.4871	−1.39634
$611$	−2.51197	−0.101624
$612$	0	0
$613$	−44.0410	−1.77880	−0.889399	−	0.457131i	$-0.848877\pi$
$613$	−44.0410	−1.77880	−0.889399	+	0.457131i	$0.848877\pi$
$614$	50.5752	2.04105
$615$	0	0
$616$	0	0
$617$	−33.1042	−1.33273	−0.666363	−	0.745628i	$-0.732150\pi$
$617$	−33.1042	−1.33273	−0.666363	+	0.745628i	$0.732150\pi$
$618$	0	0
$619$	−29.6358	−1.19116	−0.595582	−	0.803294i	$-0.703079\pi$
$619$	−29.6358	−1.19116	−0.595582	+	0.803294i	$0.703079\pi$
$620$	−2.06063	−0.0827567
$621$	0	0
$622$	−2.20257	−0.0883151
$623$	−0.434820	−0.0174207
$624$	0	0
$625$	0.924538	0.0369815
$626$	27.2207	1.08796
$627$	0	0
$628$	−13.0100	−0.519157
$629$	−39.1952	−1.56281
$630$	0	0
$631$	−8.93196	−0.355576	−0.177788	−	0.984069i	$-0.556894\pi$
$631$	−8.93196	−0.355576	−0.177788	+	0.984069i	$0.556894\pi$
$632$	−7.96291	−0.316747
$633$	0	0
$634$	29.7671	1.18220
$635$	11.4588	0.454727
$636$	0	0
$637$	4.53162	0.179549
$638$	0	0
$639$	0	0
$640$	7.81771	0.309022
$641$	−39.1042	−1.54452	−0.772262	−	0.635304i	$-0.780875\pi$
$641$	−39.1042	−1.54452	−0.772262	+	0.635304i	$0.780875\pi$
$642$	0	0
$643$	−24.6284	−0.971250	−0.485625	−	0.874167i	$-0.661408\pi$
$643$	−24.6284	−0.971250	−0.485625	+	0.874167i	$0.661408\pi$
$644$	1.17904	0.0464607
$645$	0	0
$646$	34.5726	1.36024
$647$	−20.5774	−0.808981	−0.404491	−	0.914542i	$-0.632551\pi$
$647$	−20.5774	−0.808981	−0.404491	+	0.914542i	$0.632551\pi$
$648$	0	0
$649$	0	0
$650$	4.28870	0.168217
$651$	0	0
$652$	17.9703	0.703772
$653$	4.68904	0.183496	0.0917482	−	0.995782i	$-0.470755\pi$
$653$	4.68904	0.183496	0.0917482	+	0.995782i	$0.470755\pi$
$654$	0	0
$655$	1.98516	0.0775667
$656$	28.4307	1.11003
$657$	0	0
$658$	0.696460	0.0271508
$659$	40.2887	1.56943	0.784713	−	0.619860i	$-0.212811\pi$
$659$	40.2887	1.56943	0.784713	+	0.619860i	$0.212811\pi$
$660$	0	0
$661$	−9.98777	−0.388479	−0.194240	−	0.980954i	$-0.562224\pi$
$661$	−9.98777	−0.388479	−0.194240	+	0.980954i	$0.562224\pi$
$662$	16.2887	0.633078
$663$	0	0
$664$	−1.42740	−0.0553939
$665$	0.561292	0.0217660
$666$	0	0
$667$	−27.8794	−1.07949
$668$	−49.6358	−1.92047
$669$	0	0
$670$	14.0807	0.543984
$671$	0	0
$672$	0	0
$673$	6.58744	0.253927	0.126963	−	0.991907i	$-0.459477\pi$
$673$	6.58744	0.253927	0.126963	+	0.991907i	$0.459477\pi$
$674$	42.9171	1.65311
$675$	0	0
$676$	−29.5874	−1.13798
$677$	18.9016	0.726449	0.363225	−	0.931702i	$-0.381676\pi$
$677$	18.9016	0.726449	0.363225	+	0.931702i	$0.381676\pi$
$678$	0	0
$679$	1.34127	0.0514733
$680$	3.41256	0.130866
$681$	0	0
$682$	0	0
$683$	−9.15742	−0.350399	−0.175200	−	0.984533i	$-0.556057\pi$
$683$	−9.15742	−0.350399	−0.175200	+	0.984533i	$0.556057\pi$
$684$	0	0
$685$	−17.4078	−0.665116
$686$	−2.51417	−0.0959915
$687$	0	0
$688$	30.4865	1.16229
$689$	4.83997	0.184388
$690$	0	0
$691$	−14.4051	−0.547998	−0.273999	−	0.961730i	$-0.588346\pi$
$691$	−14.4051	−0.547998	−0.273999	+	0.961730i	$0.588346\pi$
$692$	54.3048	2.06436
$693$	0	0
$694$	−4.29873	−0.163178
$695$	−17.6816	−0.670702
$696$	0	0
$697$	30.8129	1.16712
$698$	−8.33449	−0.315465
$699$	0	0
$700$	−0.642614	−0.0242885
$701$	−42.0155	−1.58690	−0.793451	−	0.608634i	$-0.791718\pi$
$701$	−42.0155	−1.58690	−0.793451	+	0.608634i	$0.791718\pi$
$702$	0	0
$703$	−54.9533	−2.07260
$704$	0	0
$705$	0	0
$706$	−19.9852	−0.752152
$707$	−0.220660	−0.00829876
$708$	0	0
$709$	−12.0894	−0.454026	−0.227013	−	0.973892i	$-0.572896\pi$
$709$	−12.0894	−0.454026	−0.227013	+	0.973892i	$0.572896\pi$
$710$	21.0632	0.790489
$711$	0	0
$712$	−3.70649	−0.138907
$713$	3.77192	0.141260
$714$	0	0
$715$	0	0
$716$	−50.6062	−1.89124
$717$	0	0
$718$	−42.1984	−1.57483
$719$	5.48322	0.204490	0.102245	−	0.994759i	$-0.467398\pi$
$719$	5.48322	0.204490	0.102245	+	0.994759i	$0.467398\pi$
$720$	0	0
$721$	1.29131	0.0480910
$722$	8.83580	0.328834
$723$	0	0
$724$	−31.1229	−1.15668
$725$	15.1952	0.564334
$726$	0	0
$727$	−21.6816	−0.804127	−0.402063	−	0.915612i	$-0.631707\pi$
$727$	−21.6816	−0.804127	−0.402063	+	0.915612i	$0.631707\pi$
$728$	−0.0409808	−0.00151885
$729$	0	0
$730$	−13.7523	−0.508995
$731$	33.0410	1.22206
$732$	0	0
$733$	−26.9171	−0.994206	−0.497103	−	0.867691i	$-0.665603\pi$
$733$	−26.9171	−0.994206	−0.497103	+	0.867691i	$0.665603\pi$
$734$	60.9581	2.25000
$735$	0	0
$736$	−47.0639	−1.73480
$737$	0	0
$738$	0	0
$739$	2.62256	0.0964723	0.0482361	−	0.998836i	$-0.484640\pi$
$739$	2.62256	0.0964723	0.0482361	+	0.998836i	$0.484640\pi$
$740$	−36.2494	−1.33255
$741$	0	0
$742$	−1.34191	−0.0492631
$743$	43.3929	1.59193	0.795966	−	0.605341i	$-0.206963\pi$
$743$	43.3929	1.59193	0.795966	+	0.605341i	$0.206963\pi$
$744$	0	0
$745$	−11.1239	−0.407547
$746$	26.1016	0.955648
$747$	0	0
$748$	0	0
$749$	0.557762	0.0203802
$750$	0	0
$751$	−43.0288	−1.57014	−0.785071	−	0.619406i	$-0.787373\pi$
$751$	−43.0288	−1.57014	−0.785071	+	0.619406i	$0.787373\pi$
$752$	−12.2961	−0.448393
$753$	0	0
$754$	6.47580	0.235835
$755$	5.33229	0.194062
$756$	0	0
$757$	39.0968	1.42100	0.710499	−	0.703699i	$-0.248469\pi$
$757$	39.0968	1.42100	0.710499	+	0.703699i	$0.248469\pi$
$758$	−28.7300	−1.04352
$759$	0	0
$760$	4.78456	0.173554
$761$	23.0074	0.834018	0.417009	−	0.908902i	$-0.363078\pi$
$761$	23.0074	0.834018	0.417009	+	0.908902i	$0.363078\pi$
$762$	0	0
$763$	−0.0148368	−0.000537129 0
$764$	51.4513	1.86144
$765$	0	0
$766$	−19.2355	−0.695007
$767$	4.33971	0.156698
$768$	0	0
$769$	−47.1452	−1.70010	−0.850050	−	0.526703i	$-0.823428\pi$
$769$	−47.1452	−1.70010	−0.850050	+	0.526703i	$0.823428\pi$
$770$	0	0
$771$	0	0
$772$	16.0410	0.577328
$773$	−40.2643	−1.44820	−0.724102	−	0.689693i	$-0.757746\pi$
$773$	−40.2643	−1.44820	−0.724102	+	0.689693i	$0.757746\pi$
$774$	0	0
$775$	−2.05582	−0.0738471
$776$	11.4333	0.410430
$777$	0	0
$778$	−35.5652	−1.27507
$779$	43.2010	1.54784
$780$	0	0
$781$	0	0
$782$	−41.7449	−1.49279
$783$	0	0
$784$	22.1823	0.792225
$785$	7.47841	0.266916
$786$	0	0
$787$	18.6938	0.666364	0.333182	−	0.942863i	$-0.391878\pi$
$787$	18.6938	0.666364	0.333182	+	0.942863i	$0.391878\pi$
$788$	−16.1123	−0.573976
$789$	0	0
$790$	30.5887	1.08830
$791$	−1.50936	−0.0536668
$792$	0	0
$793$	−7.92454	−0.281409
$794$	−36.2839	−1.28767
$795$	0	0
$796$	40.7758	1.44526
$797$	14.8203	0.524963	0.262481	−	0.964937i	$-0.415459\pi$
$797$	14.8203	0.524963	0.262481	+	0.964937i	$0.415459\pi$
$798$	0	0
$799$	−13.3264	−0.471455
$800$	25.6513	0.906911
$801$	0	0
$802$	−25.6258	−0.904879
$803$	0	0
$804$	0	0
$805$	−0.677734	−0.0238870
$806$	−0.876139	−0.0308607
$807$	0	0
$808$	−1.88095	−0.0661715
$809$	8.88681	0.312443	0.156222	−	0.987722i	$-0.450069\pi$
$809$	8.88681	0.312443	0.156222	+	0.987722i	$0.450069\pi$
$810$	0	0
$811$	16.6177	0.583528	0.291764	−	0.956490i	$-0.405758\pi$
$811$	16.6177	0.583528	0.291764	+	0.956490i	$0.405758\pi$
$812$	−0.970326	−0.0340518
$813$	0	0
$814$	0	0
$815$	−10.3297	−0.361833
$816$	0	0
$817$	46.3249	1.62070
$818$	−42.2084	−1.47578
$819$	0	0
$820$	28.4971	0.995163
$821$	24.7852	0.865009	0.432505	−	0.901632i	$-0.357630\pi$
$821$	24.7852	0.865009	0.432505	+	0.901632i	$0.357630\pi$
$822$	0	0
$823$	−35.3175	−1.23109	−0.615545	−	0.788102i	$-0.711064\pi$
$823$	−35.3175	−1.23109	−0.615545	+	0.788102i	$0.711064\pi$
$824$	11.0074	0.383462
$825$	0	0
$826$	−1.20321	−0.0418651
$827$	−39.6720	−1.37953	−0.689765	−	0.724033i	$-0.742286\pi$
$827$	−39.6720	−1.37953	−0.689765	+	0.724033i	$0.742286\pi$
$828$	0	0
$829$	28.8081	1.00055	0.500273	−	0.865868i	$-0.333233\pi$
$829$	28.8081	1.00055	0.500273	+	0.865868i	$0.333233\pi$
$830$	5.48322	0.190325
$831$	0	0
$832$	6.82032	0.236452
$833$	24.0410	0.832971
$834$	0	0
$835$	28.5316	0.987377
$836$	0	0
$837$	0	0
$838$	32.9878	1.13954
$839$	36.7784	1.26973	0.634866	−	0.772622i	$-0.281055\pi$
$839$	36.7784	1.26973	0.634866	+	0.772622i	$0.281055\pi$
$840$	0	0
$841$	−6.05582	−0.208821
$842$	31.6513	1.09078
$843$	0	0
$844$	−56.6142	−1.94874
$845$	17.0074	0.585073
$846$	0	0
$847$	0	0
$848$	23.6917	0.813575
$849$	0	0
$850$	22.7523	0.780397
$851$	66.3536	2.27457
$852$	0	0
$853$	−0.531618	−0.0182023	−0.00910113	−	0.999959i	$-0.502897\pi$
$853$	−0.531618	−0.0182023	−0.00910113	+	0.999959i	$0.502897\pi$
$854$	2.19713	0.0751841
$855$	0	0
$856$	4.75447	0.162505
$857$	26.5274	0.906160	0.453080	−	0.891470i	$-0.350325\pi$
$857$	26.5274	0.906160	0.453080	+	0.891470i	$0.350325\pi$
$858$	0	0
$859$	−8.64806	−0.295068	−0.147534	−	0.989057i	$-0.547134\pi$
$859$	−8.64806	−0.295068	−0.147534	+	0.989057i	$0.547134\pi$
$860$	30.5578	1.04201
$861$	0	0
$862$	67.0745	2.28457
$863$	−2.95902	−0.100726	−0.0503631	−	0.998731i	$-0.516038\pi$
$863$	−2.95902	−0.100726	−0.0503631	+	0.998731i	$0.516038\pi$
$864$	0	0
$865$	−31.2154	−1.06136
$866$	−54.3303	−1.84622
$867$	0	0
$868$	0.131280	0.00445592
$869$	0	0
$870$	0	0
$871$	3.23550	0.109631
$872$	−0.126472	−0.00428288
$873$	0	0
$874$	−58.5281	−1.97974
$875$	0.951601	0.0321700
$876$	0	0
$877$	47.1910	1.59353	0.796763	−	0.604292i	$-0.206544\pi$
$877$	47.1910	1.59353	0.796763	+	0.604292i	$0.206544\pi$
$878$	24.2207	0.817408
$879$	0	0
$880$	0	0
$881$	22.8055	0.768336	0.384168	−	0.923263i	$-0.374488\pi$
$881$	22.8055	0.768336	0.384168	+	0.923263i	$0.374488\pi$
$882$	0	0
$883$	−34.9581	−1.17643	−0.588217	−	0.808703i	$-0.700170\pi$
$883$	−34.9581	−1.17643	−0.588217	+	0.808703i	$0.700170\pi$
$884$	5.24030	0.176250
$885$	0	0
$886$	−0.485830	−0.0163218
$887$	−42.2739	−1.41942	−0.709709	−	0.704495i	$-0.751173\pi$
$887$	−42.2739	−1.41942	−0.709709	+	0.704495i	$0.751173\pi$
$888$	0	0
$889$	−0.730022	−0.0244842
$890$	14.2381	0.477262
$891$	0	0
$892$	−5.38289	−0.180233
$893$	−18.6842	−0.625244
$894$	0	0
$895$	29.0894	0.972351
$896$	−0.498056	−0.0166389
$897$	0	0
$898$	20.1217	0.671469
$899$	−3.10422	−0.103531
$900$	0	0
$901$	25.6768	0.855419
$902$	0	0
$903$	0	0
$904$	−12.8661	−0.427920
$905$	17.8901	0.594686
$906$	0	0
$907$	7.75709	0.257570	0.128785	−	0.991673i	$-0.458892\pi$
$907$	7.75709	0.257570	0.128785	+	0.991673i	$0.458892\pi$
$908$	−58.7810	−1.95072
$909$	0	0
$910$	0.157424	0.00521854
$911$	−14.1749	−0.469634	−0.234817	−	0.972040i	$-0.575449\pi$
$911$	−14.1749	−0.469634	−0.234817	+	0.972040i	$0.575449\pi$
$912$	0	0
$913$	0	0
$914$	−43.6358	−1.44335
$915$	0	0
$916$	−4.86391	−0.160708
$917$	−0.126472	−0.00417647
$918$	0	0
$919$	−27.0591	−0.892596	−0.446298	−	0.894884i	$-0.647258\pi$
$919$	−27.0591	−0.892596	−0.446298	+	0.894884i	$0.647258\pi$
$920$	−5.77714	−0.190467
$921$	0	0
$922$	58.2643	1.91883
$923$	4.83997	0.159309
$924$	0	0
$925$	−36.1648	−1.18909
$926$	78.5233	2.58044
$927$	0	0
$928$	38.7326	1.27146
$929$	48.3903	1.58763	0.793817	−	0.608156i	$-0.208091\pi$
$929$	48.3903	1.58763	0.793817	+	0.608156i	$0.208091\pi$
$930$	0	0
$931$	33.7065	1.10469
$932$	2.33449	0.0764688
$933$	0	0
$934$	−63.8794	−2.09020
$935$	0	0
$936$	0	0
$937$	10.1675	0.332156	0.166078	−	0.986113i	$-0.446890\pi$
$937$	10.1675	0.332156	0.166078	+	0.986113i	$0.446890\pi$
$938$	−0.897061	−0.0292901
$939$	0	0
$940$	−12.3249	−0.401993
$941$	−56.4520	−1.84028	−0.920141	−	0.391587i	$-0.871926\pi$
$941$	−56.4520	−1.84028	−0.920141	+	0.391587i	$0.871926\pi$
$942$	0	0
$943$	−52.1633	−1.69867
$944$	21.2429	0.691398
$945$	0	0
$946$	0	0
$947$	49.6768	1.61428	0.807140	−	0.590360i	$-0.201014\pi$
$947$	49.6768	1.61428	0.807140	+	0.590360i	$0.201014\pi$
$948$	0	0
$949$	−3.16003	−0.102579
$950$	31.8997	1.03496
$951$	0	0
$952$	−0.217410	−0.00704629
$953$	−16.3897	−0.530913	−0.265457	−	0.964123i	$-0.585523\pi$
$953$	−16.3897	−0.530913	−0.265457	+	0.964123i	$0.585523\pi$
$954$	0	0
$955$	−29.5752	−0.957031
$956$	−24.8661	−0.804227
$957$	0	0
$958$	15.7374	0.508454
$959$	1.10902	0.0358123
$960$	0	0
$961$	−30.5800	−0.986452
$962$	−15.4126	−0.496921
$963$	0	0
$964$	69.1304	2.22654
$965$	−9.22066	−0.296824
$966$	0	0
$967$	−14.8097	−0.476246	−0.238123	−	0.971235i	$-0.576532\pi$
$967$	−14.8097	−0.476246	−0.238123	+	0.971235i	$0.576532\pi$
$968$	0	0
$969$	0	0
$970$	−43.9197	−1.41018
$971$	2.47099	0.0792979	0.0396490	−	0.999214i	$-0.487376\pi$
$971$	2.47099	0.0792979	0.0396490	+	0.999214i	$0.487376\pi$
$972$	0	0
$973$	1.12647	0.0361130
$974$	89.8046	2.87752
$975$	0	0
$976$	−38.7906	−1.24166
$977$	8.93676	0.285912	0.142956	−	0.989729i	$-0.454339\pi$
$977$	8.93676	0.285912	0.142956	+	0.989729i	$0.454339\pi$
$978$	0	0
$979$	0	0
$980$	22.2342	0.710244
$981$	0	0
$982$	−58.7710	−1.87546
$983$	−47.1304	−1.50323	−0.751613	−	0.659605i	$-0.770724\pi$
$983$	−47.1304	−1.50323	−0.751613	+	0.659605i	$0.770724\pi$
$984$	0	0
$985$	9.26164	0.295100
$986$	34.3552	1.09409
$987$	0	0
$988$	7.34713	0.233743
$989$	−55.9352	−1.77864
$990$	0	0
$991$	−17.1845	−0.545883	−0.272942	−	0.962031i	$-0.587997\pi$
$991$	−17.1845	−0.545883	−0.272942	+	0.962031i	$0.587997\pi$
$992$	−5.24030	−0.166380
$993$	0	0
$994$	−1.34191	−0.0425628
$995$	−23.4387	−0.743057
$996$	0	0
$997$	−56.7252	−1.79651	−0.898253	−	0.439479i	$-0.855163\pi$
$997$	−56.7252	−1.79651	−0.898253	+	0.439479i	$0.855163\pi$
$998$	−53.4487	−1.69189
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3267.2.a.t.1.3		3
3.2	odd	2		3267.2.a.w.1.1		3
11.10	odd	2		297.2.a.h.1.1	yes	3
33.32	even	2		297.2.a.g.1.3	✓	3
44.43	even	2		4752.2.a.bg.1.2		3
55.54	odd	2		7425.2.a.bm.1.3		3
99.32	even	6		891.2.e.t.298.1		6
99.43	odd	6		891.2.e.q.595.3		6
99.65	even	6		891.2.e.t.595.1		6
99.76	odd	6		891.2.e.q.298.3		6
132.131	odd	2		4752.2.a.bo.1.2		3
165.164	even	2		7425.2.a.bn.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
297.2.a.g.1.3	✓	3	33.32	even	2
297.2.a.h.1.1	yes	3	11.10	odd	2
891.2.e.q.298.3		6	99.76	odd	6
891.2.e.q.595.3		6	99.43	odd	6
891.2.e.t.298.1		6	99.32	even	6
891.2.e.t.595.1		6	99.65	even	6
3267.2.a.t.1.3		3	1.1	even	1	trivial
3267.2.a.w.1.1		3	3.2	odd	2
4752.2.a.bg.1.2		3	44.43	even	2
4752.2.a.bo.1.2		3	132.131	odd	2
7425.2.a.bm.1.3		3	55.54	odd	2
7425.2.a.bn.1.1		3	165.164	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 3267 = 3^{3} \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3267.a (trivial)

\(f(q)\)	\(=\)	\(q+2.08613 q^{2} +2.35194 q^{4} -1.35194 q^{5} +0.0861302 q^{7} +0.734191 q^{8} -2.82032 q^{10} -0.648061 q^{13} +0.179679 q^{14} -3.17226 q^{16} -3.43807 q^{17} -4.82032 q^{19} -3.17968 q^{20} +5.82032 q^{23} -3.17226 q^{25} -1.35194 q^{26} +0.202573 q^{28} -4.79001 q^{29} +0.648061 q^{31} -8.08613 q^{32} -7.17226 q^{34} -0.116443 q^{35} +11.4003 q^{37} -10.0558 q^{38} -0.992582 q^{40} -8.96227 q^{41} -9.61033 q^{43} +12.1419 q^{46} +3.87614 q^{47} -6.99258 q^{49} -6.61775 q^{50} -1.52420 q^{52} -7.46838 q^{53} +0.0632360 q^{56} -9.99258 q^{58} -6.69646 q^{59} +12.2281 q^{61} +1.35194 q^{62} -10.5242 q^{64} +0.876139 q^{65} -4.99258 q^{67} -8.08613 q^{68} -0.242915 q^{70} -7.46838 q^{71} +4.87614 q^{73} +23.7826 q^{74} -11.3371 q^{76} -10.8458 q^{79} +4.28870 q^{80} -18.6965 q^{82} -1.94418 q^{83} +4.64806 q^{85} -20.0484 q^{86} -5.04840 q^{89} -0.0558176 q^{91} +13.6890 q^{92} +8.08613 q^{94} +6.51678 q^{95} +15.5726 q^{97} -14.5874 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - q^{2} + 5 q^{4} - 2 q^{5} - 7 q^{7} - 3 q^{8} + 4 q^{10} - 4 q^{13} + 13 q^{14} + 5 q^{16} - q^{17} - 2 q^{19} - 22 q^{20} + 5 q^{23} + 5 q^{25} - 2 q^{26} - 15 q^{28} - 3 q^{29} + 4 q^{31} - 17 q^{32}+ \cdots - 48 q^{98}+O(q^{100}) \) 3 * q - q^2 + 5 * q^4 - 2 * q^5 - 7 * q^7 - 3 * q^8 + 4 * q^10 - 4 * q^13 + 13 * q^14 + 5 * q^16 - q^17 - 2 * q^19 - 22 * q^20 + 5 * q^23 + 5 * q^25 - 2 * q^26 - 15 * q^28 - 3 * q^29 + 4 * q^31 - 17 * q^32 - 7 * q^34 + 8 * q^35 - q^37 - 24 * q^38 + 24 * q^40 - q^41 - 5 * q^43 + 23 * q^46 - 7 * q^47 + 6 * q^49 - 23 * q^50 + 12 * q^52 - 12 * q^53 + 21 * q^56 - 3 * q^58 + 11 * q^59 + 16 * q^61 + 2 * q^62 - 15 * q^64 - 16 * q^65 + 12 * q^67 - 17 * q^68 - 34 * q^70 - 12 * q^71 - 4 * q^73 + 33 * q^74 + 22 * q^76 - 15 * q^79 - 10 * q^80 - 25 * q^82 - 12 * q^83 + 16 * q^85 - 27 * q^86 + 18 * q^89 + 6 * q^91 - 17 * q^92 + 17 * q^94 - 24 * q^95 - 3 * q^97 - 48 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more