LMFDB - Embedded newform 3344.2.a.z.1.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3344,2,Mod(1,3344)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3344, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))

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

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

chi := DirichletCharacter("3344.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3344 = 2^{4} \cdot 11 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3344.a (trivial)

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:	yes
Analytic conductor:	$26.7019744359$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.6.106392688.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} - 9x^{4} + 12x^{3} + 25x^{2} - 10x - 16 $ x^6 - 2x^5 - 9x^4 + 12x^3 + 25x^2 - 10*x - 16
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1672)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$-2.14822$ of defining polynomial
Character	$\chi$	$=$	3344.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.14822 q^{3} +0.677107 q^{5} +1.67711 q^{7} +6.91132 q^{9} +O(q^{10})$	$q+3.14822 q^{3} +0.677107 q^{5} +1.67711 q^{7} +6.91132 q^{9} +1.00000 q^{11} -4.16289 q^{13} +2.13169 q^{15} +6.86889 q^{17} -1.00000 q^{19} +5.27991 q^{21} -2.14077 q^{23} -4.54153 q^{25} +12.3137 q^{27} +2.01005 q^{29} +2.13169 q^{31} +3.14822 q^{33} +1.13558 q^{35} +2.81297 q^{37} -13.1057 q^{39} +10.9829 q^{41} -3.39032 q^{43} +4.67970 q^{45} -1.22974 q^{47} -4.18731 q^{49} +21.6248 q^{51} -11.7292 q^{53} +0.677107 q^{55} -3.14822 q^{57} +8.37650 q^{59} +9.52619 q^{61} +11.5910 q^{63} -2.81872 q^{65} +9.64636 q^{67} -6.73963 q^{69} -12.6011 q^{71} -14.1743 q^{73} -14.2977 q^{75} +1.67711 q^{77} +11.2387 q^{79} +18.0324 q^{81} +7.75976 q^{83} +4.65098 q^{85} +6.32808 q^{87} -4.08041 q^{89} -6.98160 q^{91} +6.71102 q^{93} -0.677107 q^{95} -12.4487 q^{97} +6.91132 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 4 q^{3} - 3 q^{5} + 3 q^{7} + 6 q^{9}+O(q^{10}) $ 6 * q + 4 * q^3 - 3 * q^5 + 3 * q^7 + 6 * q^9	$ 6 q + 4 q^{3} - 3 q^{5} + 3 q^{7} + 6 q^{9} + 6 q^{11} - 3 q^{13} + q^{15} - q^{17} - 6 q^{19} + 5 q^{21} + 4 q^{23} + 5 q^{25} + 16 q^{27} - 7 q^{29} + q^{31} + 4 q^{33} + 32 q^{35} + 5 q^{37} + 13 q^{39} - 6 q^{41} + 16 q^{43} + 4 q^{47} - 7 q^{49} + 35 q^{51} - 11 q^{53} - 3 q^{55} - 4 q^{57} + 18 q^{59} + 16 q^{61} + 6 q^{63} + 10 q^{65} + 18 q^{67} - 2 q^{69} + 7 q^{71} - 21 q^{73} + 23 q^{75} + 3 q^{77} + 22 q^{79} - 10 q^{81} + 16 q^{83} + 3 q^{87} - 13 q^{89} + 7 q^{91} - 9 q^{93} + 3 q^{95} - 15 q^{97} + 6 q^{99}+O(q^{100}) $ 6 * q + 4 * q^3 - 3 * q^5 + 3 * q^7 + 6 * q^9 + 6 * q^11 - 3 * q^13 + q^15 - q^17 - 6 * q^19 + 5 * q^21 + 4 * q^23 + 5 * q^25 + 16 * q^27 - 7 * q^29 + q^31 + 4 * q^33 + 32 * q^35 + 5 * q^37 + 13 * q^39 - 6 * q^41 + 16 * q^43 + 4 * q^47 - 7 * q^49 + 35 * q^51 - 11 * q^53 - 3 * q^55 - 4 * q^57 + 18 * q^59 + 16 * q^61 + 6 * q^63 + 10 * q^65 + 18 * q^67 - 2 * q^69 + 7 * q^71 - 21 * q^73 + 23 * q^75 + 3 * q^77 + 22 * q^79 - 10 * q^81 + 16 * q^83 + 3 * q^87 - 13 * q^89 + 7 * q^91 - 9 * q^93 + 3 * q^95 - 15 * q^97 + 6 * q^99

Display 100 coefficients 10 coefficients

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 $

$2$	0	0
$2$	0	0
$3$	3.14822	1.81763	0.908814	−	0.417201i	$-0.136989\pi$
$3$	3.14822	1.81763	0.908814	+	0.417201i	$0.136989\pi$
$4$	0	0
$5$	0.677107	0.302812	0.151406	−	0.988472i	$-0.451620\pi$
$5$	0.677107	0.302812	0.151406	+	0.988472i	$0.451620\pi$
$6$	0	0
$7$	1.67711	0.633887	0.316943	−	0.948444i	$-0.397343\pi$
$7$	1.67711	0.633887	0.316943	+	0.948444i	$0.397343\pi$
$8$	0	0
$9$	6.91132	2.30377
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	−4.16289	−1.15458	−0.577288	−	0.816540i	$-0.695889\pi$
$13$	−4.16289	−1.15458	−0.577288	+	0.816540i	$0.695889\pi$
$14$	0	0
$15$	2.13169	0.550399
$16$	0	0
$17$	6.86889	1.66595	0.832976	−	0.553310i	$-0.186635\pi$
$17$	6.86889	1.66595	0.832976	+	0.553310i	$0.186635\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	5.27991	1.15217
$22$	0	0
$23$	−2.14077	−0.446382	−0.223191	−	0.974775i	$-0.571647\pi$
$23$	−2.14077	−0.446382	−0.223191	+	0.974775i	$0.571647\pi$
$24$	0	0
$25$	−4.54153	−0.908305
$26$	0	0
$27$	12.3137	2.36977
$28$	0	0
$29$	2.01005	0.373257	0.186628	−	0.982431i	$-0.440244\pi$
$29$	2.01005	0.373257	0.186628	+	0.982431i	$0.440244\pi$
$30$	0	0
$31$	2.13169	0.382862	0.191431	−	0.981506i	$-0.438687\pi$
$31$	2.13169	0.382862	0.191431	+	0.981506i	$0.438687\pi$
$32$	0	0
$33$	3.14822	0.548036
$34$	0	0
$35$	1.13558	0.191948
$36$	0	0
$37$	2.81297	0.462450	0.231225	−	0.972900i	$-0.425727\pi$
$37$	2.81297	0.462450	0.231225	+	0.972900i	$0.425727\pi$
$38$	0	0
$39$	−13.1057	−2.09859
$40$	0	0
$41$	10.9829	1.71524	0.857622	−	0.514280i	$-0.171941\pi$
$41$	10.9829	1.71524	0.857622	+	0.514280i	$0.171941\pi$
$42$	0	0
$43$	−3.39032	−0.517019	−0.258510	−	0.966009i	$-0.583231\pi$
$43$	−3.39032	−0.517019	−0.258510	+	0.966009i	$0.583231\pi$
$44$	0	0
$45$	4.67970	0.697609
$46$	0	0
$47$	−1.22974	−0.179376	−0.0896878	−	0.995970i	$-0.528587\pi$
$47$	−1.22974	−0.179376	−0.0896878	+	0.995970i	$0.528587\pi$
$48$	0	0
$49$	−4.18731	−0.598187
$50$	0	0
$51$	21.6248	3.02808
$52$	0	0
$53$	−11.7292	−1.61113	−0.805564	−	0.592508i	$-0.798138\pi$
$53$	−11.7292	−1.61113	−0.805564	+	0.592508i	$0.798138\pi$
$54$	0	0
$55$	0.677107	0.0913011
$56$	0	0
$57$	−3.14822	−0.416993
$58$	0	0
$59$	8.37650	1.09053	0.545264	−	0.838264i	$-0.316429\pi$
$59$	8.37650	1.09053	0.545264	+	0.838264i	$0.316429\pi$
$60$	0	0
$61$	9.52619	1.21970	0.609852	−	0.792516i	$-0.291229\pi$
$61$	9.52619	1.21970	0.609852	+	0.792516i	$0.291229\pi$
$62$	0	0
$63$	11.5910	1.46033
$64$	0	0
$65$	−2.81872	−0.349619
$66$	0	0
$67$	9.64636	1.17849	0.589245	−	0.807954i	$-0.299425\pi$
$67$	9.64636	1.17849	0.589245	+	0.807954i	$0.299425\pi$
$68$	0	0
$69$	−6.73963	−0.811356
$70$	0	0
$71$	−12.6011	−1.49547	−0.747736	−	0.663996i	$-0.768859\pi$
$71$	−12.6011	−1.49547	−0.747736	+	0.663996i	$0.768859\pi$
$72$	0	0
$73$	−14.1743	−1.65897	−0.829485	−	0.558528i	$-0.811366\pi$
$73$	−14.1743	−1.65897	−0.829485	+	0.558528i	$0.811366\pi$
$74$	0	0
$75$	−14.2977	−1.65096
$76$	0	0
$77$	1.67711	0.191124
$78$	0	0
$79$	11.2387	1.26445	0.632225	−	0.774785i	$-0.282142\pi$
$79$	11.2387	1.26445	0.632225	+	0.774785i	$0.282142\pi$
$80$	0	0
$81$	18.0324	2.00360
$82$	0	0
$83$	7.75976	0.851744	0.425872	−	0.904783i	$-0.359967\pi$
$83$	7.75976	0.851744	0.425872	+	0.904783i	$0.359967\pi$
$84$	0	0
$85$	4.65098	0.504469
$86$	0	0
$87$	6.32808	0.678442
$88$	0	0
$89$	−4.08041	−0.432523	−0.216261	−	0.976336i	$-0.569386\pi$
$89$	−4.08041	−0.432523	−0.216261	+	0.976336i	$0.569386\pi$
$90$	0	0
$91$	−6.98160	−0.731871
$92$	0	0
$93$	6.71102	0.695901
$94$	0	0
$95$	−0.677107	−0.0694697
$96$	0	0
$97$	−12.4487	−1.26398	−0.631988	−	0.774978i	$-0.717761\pi$
$97$	−12.4487	−1.26398	−0.631988	+	0.774978i	$0.717761\pi$
$98$	0	0
$99$	6.91132	0.694614
$100$	0	0
$101$	11.5645	1.15071	0.575353	−	0.817905i	$-0.304865\pi$
$101$	11.5645	1.15071	0.575353	+	0.817905i	$0.304865\pi$
$102$	0	0
$103$	−10.0938	−0.994569	−0.497284	−	0.867588i	$-0.665669\pi$
$103$	−10.0938	−0.994569	−0.497284	+	0.867588i	$0.665669\pi$
$104$	0	0
$105$	3.57506	0.348891
$106$	0	0
$107$	14.3080	1.38320	0.691601	−	0.722280i	$-0.256906\pi$
$107$	14.3080	1.38320	0.691601	+	0.722280i	$0.256906\pi$
$108$	0	0
$109$	−20.0639	−1.92177	−0.960887	−	0.276940i	$-0.910680\pi$
$109$	−20.0639	−1.92177	−0.960887	+	0.276940i	$0.910680\pi$
$110$	0	0
$111$	8.85586	0.840561
$112$	0	0
$113$	−14.7940	−1.39171	−0.695853	−	0.718185i	$-0.744973\pi$
$113$	−14.7940	−1.39171	−0.695853	+	0.718185i	$0.744973\pi$
$114$	0	0
$115$	−1.44953	−0.135170
$116$	0	0
$117$	−28.7710	−2.65988
$118$	0	0
$119$	11.5199	1.05602
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	34.5767	3.11768
$124$	0	0
$125$	−6.46064	−0.577857
$126$	0	0
$127$	−10.5225	−0.933718	−0.466859	−	0.884332i	$-0.654614\pi$
$127$	−10.5225	−0.933718	−0.466859	+	0.884332i	$0.654614\pi$
$128$	0	0
$129$	−10.6735	−0.939749
$130$	0	0
$131$	9.76083	0.852808	0.426404	−	0.904533i	$-0.359780\pi$
$131$	9.76083	0.852808	0.426404	+	0.904533i	$0.359780\pi$
$132$	0	0
$133$	−1.67711	−0.145424
$134$	0	0
$135$	8.33770	0.717595
$136$	0	0
$137$	1.72444	0.147329	0.0736644	−	0.997283i	$-0.476531\pi$
$137$	1.72444	0.147329	0.0736644	+	0.997283i	$0.476531\pi$
$138$	0	0
$139$	2.73416	0.231908	0.115954	−	0.993255i	$-0.463007\pi$
$139$	2.73416	0.231908	0.115954	+	0.993255i	$0.463007\pi$
$140$	0	0
$141$	−3.87149	−0.326038
$142$	0	0
$143$	−4.16289	−0.348118
$144$	0	0
$145$	1.36102	0.113026
$146$	0	0
$147$	−13.1826	−1.08728
$148$	0	0
$149$	−18.6670	−1.52926	−0.764632	−	0.644468i	$-0.777079\pi$
$149$	−18.6670	−1.52926	−0.764632	+	0.644468i	$0.777079\pi$
$150$	0	0
$151$	10.5284	0.856791	0.428395	−	0.903591i	$-0.359079\pi$
$151$	10.5284	0.856791	0.428395	+	0.903591i	$0.359079\pi$
$152$	0	0
$153$	47.4731	3.83797
$154$	0	0
$155$	1.44338	0.115935
$156$	0	0
$157$	−11.6625	−0.930765	−0.465383	−	0.885110i	$-0.654083\pi$
$157$	−11.6625	−0.930765	−0.465383	+	0.885110i	$0.654083\pi$
$158$	0	0
$159$	−36.9261	−2.92843
$160$	0	0
$161$	−3.59030	−0.282956
$162$	0	0
$163$	−14.9612	−1.17185	−0.585926	−	0.810364i	$-0.699269\pi$
$163$	−14.9612	−1.17185	−0.585926	+	0.810364i	$0.699269\pi$
$164$	0	0
$165$	2.13169	0.165951
$166$	0	0
$167$	10.5989	0.820164	0.410082	−	0.912049i	$-0.365500\pi$
$167$	10.5989	0.820164	0.410082	+	0.912049i	$0.365500\pi$
$168$	0	0
$169$	4.32961	0.333047
$170$	0	0
$171$	−6.91132	−0.528522
$172$	0	0
$173$	3.76338	0.286124	0.143062	−	0.989714i	$-0.454305\pi$
$173$	3.76338	0.286124	0.143062	+	0.989714i	$0.454305\pi$
$174$	0	0
$175$	−7.61663	−0.575763
$176$	0	0
$177$	26.3711	1.98217
$178$	0	0
$179$	3.50407	0.261907	0.130953	−	0.991389i	$-0.458196\pi$
$179$	3.50407	0.261907	0.130953	+	0.991389i	$0.458196\pi$
$180$	0	0
$181$	25.3354	1.88317	0.941584	−	0.336778i	$-0.109337\pi$
$181$	25.3354	1.88317	0.941584	+	0.336778i	$0.109337\pi$
$182$	0	0
$183$	29.9906	2.21697
$184$	0	0
$185$	1.90468	0.140035
$186$	0	0
$187$	6.86889	0.502303
$188$	0	0
$189$	20.6514	1.50217
$190$	0	0
$191$	−4.57468	−0.331012	−0.165506	−	0.986209i	$-0.552926\pi$
$191$	−4.57468	−0.331012	−0.165506	+	0.986209i	$0.552926\pi$
$192$	0	0
$193$	1.55348	0.111822	0.0559111	−	0.998436i	$-0.482194\pi$
$193$	1.55348	0.111822	0.0559111	+	0.998436i	$0.482194\pi$
$194$	0	0
$195$	−8.87396	−0.635478
$196$	0	0
$197$	−14.8166	−1.05564	−0.527821	−	0.849356i	$-0.676991\pi$
$197$	−14.8166	−1.05564	−0.527821	+	0.849356i	$0.676991\pi$
$198$	0	0
$199$	8.99201	0.637427	0.318713	−	0.947851i	$-0.396749\pi$
$199$	8.99201	0.637427	0.318713	+	0.947851i	$0.396749\pi$
$200$	0	0
$201$	30.3689	2.14206
$202$	0	0
$203$	3.37107	0.236602
$204$	0	0
$205$	7.43662	0.519396
$206$	0	0
$207$	−14.7955	−1.02836
$208$	0	0
$209$	−1.00000	−0.0691714
$210$	0	0
$211$	−2.56841	−0.176817	−0.0884083	−	0.996084i	$-0.528178\pi$
$211$	−2.56841	−0.176817	−0.0884083	+	0.996084i	$0.528178\pi$
$212$	0	0
$213$	−39.6710	−2.71821
$214$	0	0
$215$	−2.29561	−0.156559
$216$	0	0
$217$	3.57506	0.242691
$218$	0	0
$219$	−44.6237	−3.01539
$220$	0	0
$221$	−28.5944	−1.92347
$222$	0	0
$223$	11.2774	0.755188	0.377594	−	0.925971i	$-0.376752\pi$
$223$	11.2774	0.755188	0.377594	+	0.925971i	$0.376752\pi$
$224$	0	0
$225$	−31.3879	−2.09253
$226$	0	0
$227$	−4.50002	−0.298677	−0.149339	−	0.988786i	$-0.547714\pi$
$227$	−4.50002	−0.298677	−0.149339	+	0.988786i	$0.547714\pi$
$228$	0	0
$229$	17.9898	1.18880	0.594401	−	0.804169i	$-0.297389\pi$
$229$	17.9898	1.18880	0.594401	+	0.804169i	$0.297389\pi$
$230$	0	0
$231$	5.27991	0.347393
$232$	0	0
$233$	−9.65321	−0.632403	−0.316201	−	0.948692i	$-0.602408\pi$
$233$	−9.65321	−0.632403	−0.316201	+	0.948692i	$0.602408\pi$
$234$	0	0
$235$	−0.832664	−0.0543170
$236$	0	0
$237$	35.3819	2.29830
$238$	0	0
$239$	−5.36583	−0.347087	−0.173543	−	0.984826i	$-0.555522\pi$
$239$	−5.36583	−0.347087	−0.173543	+	0.984826i	$0.555522\pi$
$240$	0	0
$241$	2.30762	0.148647	0.0743236	−	0.997234i	$-0.476320\pi$
$241$	2.30762	0.148647	0.0743236	+	0.997234i	$0.476320\pi$
$242$	0	0
$243$	19.8288	1.27202
$244$	0	0
$245$	−2.83526	−0.181138
$246$	0	0
$247$	4.16289	0.264878
$248$	0	0
$249$	24.4295	1.54815
$250$	0	0
$251$	−11.1322	−0.702659	−0.351330	−	0.936252i	$-0.614270\pi$
$251$	−11.1322	−0.702659	−0.351330	+	0.936252i	$0.614270\pi$
$252$	0	0
$253$	−2.14077	−0.134589
$254$	0	0
$255$	14.6423	0.916937
$256$	0	0
$257$	8.10554	0.505609	0.252805	−	0.967517i	$-0.418647\pi$
$257$	8.10554	0.505609	0.252805	+	0.967517i	$0.418647\pi$
$258$	0	0
$259$	4.71765	0.293141
$260$	0	0
$261$	13.8921	0.859898
$262$	0	0
$263$	−26.5787	−1.63891	−0.819455	−	0.573143i	$-0.805724\pi$
$263$	−26.5787	−1.63891	−0.819455	+	0.573143i	$0.805724\pi$
$264$	0	0
$265$	−7.94192	−0.487868
$266$	0	0
$267$	−12.8460	−0.786165
$268$	0	0
$269$	−4.21563	−0.257032	−0.128516	−	0.991707i	$-0.541021\pi$
$269$	−4.21563	−0.257032	−0.128516	+	0.991707i	$0.541021\pi$
$270$	0	0
$271$	−0.870973	−0.0529079	−0.0264539	−	0.999650i	$-0.508422\pi$
$271$	−0.870973	−0.0529079	−0.0264539	+	0.999650i	$0.508422\pi$
$272$	0	0
$273$	−21.9797	−1.33027
$274$	0	0
$275$	−4.54153	−0.273864
$276$	0	0
$277$	−17.0705	−1.02567	−0.512835	−	0.858487i	$-0.671405\pi$
$277$	−17.0705	−1.02567	−0.512835	+	0.858487i	$0.671405\pi$
$278$	0	0
$279$	14.7328	0.882027
$280$	0	0
$281$	−3.82059	−0.227917	−0.113959	−	0.993486i	$-0.536353\pi$
$281$	−3.82059	−0.227917	−0.113959	+	0.993486i	$0.536353\pi$
$282$	0	0
$283$	17.7958	1.05785	0.528926	−	0.848668i	$-0.322595\pi$
$283$	17.7958	1.05785	0.528926	+	0.848668i	$0.322595\pi$
$284$	0	0
$285$	−2.13169	−0.126270
$286$	0	0
$287$	18.4195	1.08727
$288$	0	0
$289$	30.1817	1.77539
$290$	0	0
$291$	−39.1914	−2.29744
$292$	0	0
$293$	−1.68332	−0.0983403	−0.0491702	−	0.998790i	$-0.515658\pi$
$293$	−1.68332	−0.0983403	−0.0491702	+	0.998790i	$0.515658\pi$
$294$	0	0
$295$	5.67179	0.330224
$296$	0	0
$297$	12.3137	0.714514
$298$	0	0
$299$	8.91179	0.515382
$300$	0	0
$301$	−5.68593	−0.327732
$302$	0	0
$303$	36.4075	2.09156
$304$	0	0
$305$	6.45025	0.369340
$306$	0	0
$307$	−7.93624	−0.452945	−0.226473	−	0.974018i	$-0.572719\pi$
$307$	−7.93624	−0.452945	−0.226473	+	0.974018i	$0.572719\pi$
$308$	0	0
$309$	−31.7775	−1.80776
$310$	0	0
$311$	−29.8589	−1.69314	−0.846570	−	0.532277i	$-0.821337\pi$
$311$	−29.8589	−1.69314	−0.846570	+	0.532277i	$0.821337\pi$
$312$	0	0
$313$	18.8447	1.06516	0.532582	−	0.846378i	$-0.321222\pi$
$313$	18.8447	1.06516	0.532582	+	0.846378i	$0.321222\pi$
$314$	0	0
$315$	7.84836	0.442205
$316$	0	0
$317$	5.77789	0.324518	0.162259	−	0.986748i	$-0.448122\pi$
$317$	5.77789	0.324518	0.162259	+	0.986748i	$0.448122\pi$
$318$	0	0
$319$	2.01005	0.112541
$320$	0	0
$321$	45.0447	2.51415
$322$	0	0
$323$	−6.86889	−0.382195
$324$	0	0
$325$	18.9059	1.04871
$326$	0	0
$327$	−63.1657	−3.49307
$328$	0	0
$329$	−2.06240	−0.113704
$330$	0	0
$331$	−10.4610	−0.574991	−0.287496	−	0.957782i	$-0.592823\pi$
$331$	−10.4610	−0.574991	−0.287496	+	0.957782i	$0.592823\pi$
$332$	0	0
$333$	19.4413	1.06538
$334$	0	0
$335$	6.53162	0.356861
$336$	0	0
$337$	−26.5246	−1.44489	−0.722444	−	0.691430i	$-0.756981\pi$
$337$	−26.5246	−1.44489	−0.722444	+	0.691430i	$0.756981\pi$
$338$	0	0
$339$	−46.5749	−2.52960
$340$	0	0
$341$	2.13169	0.115437
$342$	0	0
$343$	−18.7623	−1.01307
$344$	0	0
$345$	−4.56345	−0.245688
$346$	0	0
$347$	9.94655	0.533959	0.266979	−	0.963702i	$-0.413974\pi$
$347$	9.94655	0.533959	0.266979	+	0.963702i	$0.413974\pi$
$348$	0	0
$349$	1.51832	0.0812738	0.0406369	−	0.999174i	$-0.487061\pi$
$349$	1.51832	0.0812738	0.0406369	+	0.999174i	$0.487061\pi$
$350$	0	0
$351$	−51.2605	−2.73609
$352$	0	0
$353$	−34.4107	−1.83150	−0.915749	−	0.401750i	$-0.868402\pi$
$353$	−34.4107	−1.83150	−0.915749	+	0.401750i	$0.868402\pi$
$354$	0	0
$355$	−8.53227	−0.452846
$356$	0	0
$357$	36.2671	1.91946
$358$	0	0
$359$	−1.31071	−0.0691769	−0.0345884	−	0.999402i	$-0.511012\pi$
$359$	−1.31071	−0.0691769	−0.0345884	+	0.999402i	$0.511012\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	3.14822	0.165239
$364$	0	0
$365$	−9.59749	−0.502355
$366$	0	0
$367$	14.2993	0.746415	0.373207	−	0.927748i	$-0.378258\pi$
$367$	14.2993	0.746415	0.373207	+	0.927748i	$0.378258\pi$
$368$	0	0
$369$	75.9065	3.95153
$370$	0	0
$371$	−19.6711	−1.02127
$372$	0	0
$373$	−20.7508	−1.07443	−0.537217	−	0.843444i	$-0.680525\pi$
$373$	−20.7508	−1.07443	−0.537217	+	0.843444i	$0.680525\pi$
$374$	0	0
$375$	−20.3395	−1.05033
$376$	0	0
$377$	−8.36760	−0.430953
$378$	0	0
$379$	9.68867	0.497673	0.248837	−	0.968545i	$-0.419952\pi$
$379$	9.68867	0.497673	0.248837	+	0.968545i	$0.419952\pi$
$380$	0	0
$381$	−33.1271	−1.69715
$382$	0	0
$383$	36.1499	1.84717	0.923587	−	0.383390i	$-0.125243\pi$
$383$	36.1499	1.84717	0.923587	+	0.383390i	$0.125243\pi$
$384$	0	0
$385$	1.13558	0.0578746
$386$	0	0
$387$	−23.4316	−1.19109
$388$	0	0
$389$	−28.0423	−1.42180	−0.710900	−	0.703293i	$-0.751712\pi$
$389$	−28.0423	−1.42180	−0.710900	+	0.703293i	$0.751712\pi$
$390$	0	0
$391$	−14.7047	−0.743650
$392$	0	0
$393$	30.7293	1.55009
$394$	0	0
$395$	7.60979	0.382890
$396$	0	0
$397$	7.80222	0.391582	0.195791	−	0.980646i	$-0.437273\pi$
$397$	7.80222	0.391582	0.195791	+	0.980646i	$0.437273\pi$
$398$	0	0
$399$	−5.27991	−0.264326
$400$	0	0
$401$	−21.9629	−1.09677	−0.548387	−	0.836225i	$-0.684758\pi$
$401$	−21.9629	−1.09677	−0.548387	+	0.836225i	$0.684758\pi$
$402$	0	0
$403$	−8.87396	−0.442043
$404$	0	0
$405$	12.2098	0.606712
$406$	0	0
$407$	2.81297	0.139434
$408$	0	0
$409$	1.85738	0.0918414	0.0459207	−	0.998945i	$-0.485378\pi$
$409$	1.85738	0.0918414	0.0459207	+	0.998945i	$0.485378\pi$
$410$	0	0
$411$	5.42892	0.267789
$412$	0	0
$413$	14.0483	0.691271
$414$	0	0
$415$	5.25419	0.257918
$416$	0	0
$417$	8.60774	0.421523
$418$	0	0
$419$	−3.01873	−0.147474	−0.0737372	−	0.997278i	$-0.523493\pi$
$419$	−3.01873	−0.147474	−0.0737372	+	0.997278i	$0.523493\pi$
$420$	0	0
$421$	4.54661	0.221588	0.110794	−	0.993843i	$-0.464661\pi$
$421$	4.54661	0.221588	0.110794	+	0.993843i	$0.464661\pi$
$422$	0	0
$423$	−8.49910	−0.413240
$424$	0	0
$425$	−31.1953	−1.51319
$426$	0	0
$427$	15.9764	0.773154
$428$	0	0
$429$	−13.1057	−0.632749
$430$	0	0
$431$	12.1446	0.584984	0.292492	−	0.956268i	$-0.405516\pi$
$431$	12.1446	0.584984	0.292492	+	0.956268i	$0.405516\pi$
$432$	0	0
$433$	−4.70207	−0.225967	−0.112983	−	0.993597i	$-0.536041\pi$
$433$	−4.70207	−0.225967	−0.112983	+	0.993597i	$0.536041\pi$
$434$	0	0
$435$	4.28479	0.205440
$436$	0	0
$437$	2.14077	0.102407
$438$	0	0
$439$	−26.9200	−1.28482	−0.642411	−	0.766360i	$-0.722066\pi$
$439$	−26.9200	−1.28482	−0.642411	+	0.766360i	$0.722066\pi$
$440$	0	0
$441$	−28.9398	−1.37809
$442$	0	0
$443$	−4.33483	−0.205954	−0.102977	−	0.994684i	$-0.532837\pi$
$443$	−4.33483	−0.205954	−0.102977	+	0.994684i	$0.532837\pi$
$444$	0	0
$445$	−2.76288	−0.130973
$446$	0	0
$447$	−58.7680	−2.77963
$448$	0	0
$449$	10.4154	0.491533	0.245767	−	0.969329i	$-0.420960\pi$
$449$	10.4154	0.491533	0.245767	+	0.969329i	$0.420960\pi$
$450$	0	0
$451$	10.9829	0.517166
$452$	0	0
$453$	33.1458	1.55733
$454$	0	0
$455$	−4.72729	−0.221619
$456$	0	0
$457$	−10.4225	−0.487543	−0.243771	−	0.969833i	$-0.578385\pi$
$457$	−10.4225	−0.487543	−0.243771	+	0.969833i	$0.578385\pi$
$458$	0	0
$459$	84.5815	3.94793
$460$	0	0
$461$	34.1110	1.58871	0.794355	−	0.607454i	$-0.207809\pi$
$461$	34.1110	1.58871	0.794355	+	0.607454i	$0.207809\pi$
$462$	0	0
$463$	−28.7298	−1.33519	−0.667594	−	0.744525i	$-0.732676\pi$
$463$	−28.7298	−1.33519	−0.667594	+	0.744525i	$0.732676\pi$
$464$	0	0
$465$	4.54408	0.210727
$466$	0	0
$467$	−25.6777	−1.18822	−0.594111	−	0.804383i	$-0.702496\pi$
$467$	−25.6777	−1.18822	−0.594111	+	0.804383i	$0.702496\pi$
$468$	0	0
$469$	16.1780	0.747030
$470$	0	0
$471$	−36.7160	−1.69178
$472$	0	0
$473$	−3.39032	−0.155887
$474$	0	0
$475$	4.54153	0.208380
$476$	0	0
$477$	−81.0642	−3.71167
$478$	0	0
$479$	25.4486	1.16278	0.581389	−	0.813626i	$-0.302509\pi$
$479$	25.4486	1.16278	0.581389	+	0.813626i	$0.302509\pi$
$480$	0	0
$481$	−11.7101	−0.533934
$482$	0	0
$483$	−11.3031	−0.514308
$484$	0	0
$485$	−8.42912	−0.382747
$486$	0	0
$487$	−6.53376	−0.296073	−0.148036	−	0.988982i	$-0.547295\pi$
$487$	−6.53376	−0.296073	−0.148036	+	0.988982i	$0.547295\pi$
$488$	0	0
$489$	−47.1013	−2.12999
$490$	0	0
$491$	42.5368	1.91966	0.959831	−	0.280580i	$-0.0905269\pi$
$491$	42.5368	1.91966	0.959831	+	0.280580i	$0.0905269\pi$
$492$	0	0
$493$	13.8068	0.621827
$494$	0	0
$495$	4.67970	0.210337
$496$	0	0
$497$	−21.1333	−0.947960
$498$	0	0
$499$	43.6620	1.95458	0.977289	−	0.211912i	$-0.0679690\pi$
$499$	43.6620	1.95458	0.977289	+	0.211912i	$0.0679690\pi$
$500$	0	0
$501$	33.3676	1.49075
$502$	0	0
$503$	14.1374	0.630354	0.315177	−	0.949033i	$-0.397936\pi$
$503$	14.1374	0.630354	0.315177	+	0.949033i	$0.397936\pi$
$504$	0	0
$505$	7.83037	0.348447
$506$	0	0
$507$	13.6306	0.605356
$508$	0	0
$509$	6.20694	0.275118	0.137559	−	0.990494i	$-0.456074\pi$
$509$	6.20694	0.275118	0.137559	+	0.990494i	$0.456074\pi$
$510$	0	0
$511$	−23.7717	−1.05160
$512$	0	0
$513$	−12.3137	−0.543663
$514$	0	0
$515$	−6.83456	−0.301167
$516$	0	0
$517$	−1.22974	−0.0540838
$518$	0	0
$519$	11.8480	0.520067
$520$	0	0
$521$	28.7876	1.26121	0.630605	−	0.776104i	$-0.282807\pi$
$521$	28.7876	1.26121	0.630605	+	0.776104i	$0.282807\pi$
$522$	0	0
$523$	29.1445	1.27440	0.637199	−	0.770699i	$-0.280093\pi$
$523$	29.1445	1.27440	0.637199	+	0.770699i	$0.280093\pi$
$524$	0	0
$525$	−23.9788	−1.04652
$526$	0	0
$527$	14.6423	0.637829
$528$	0	0
$529$	−18.4171	−0.800743
$530$	0	0
$531$	57.8927	2.51233
$532$	0	0
$533$	−45.7207	−1.98038
$534$	0	0
$535$	9.68802	0.418850
$536$	0	0
$537$	11.0316	0.476049
$538$	0	0
$539$	−4.18731	−0.180360
$540$	0	0
$541$	23.3632	1.00446	0.502230	−	0.864734i	$-0.332513\pi$
$541$	23.3632	1.00446	0.502230	+	0.864734i	$0.332513\pi$
$542$	0	0
$543$	79.7616	3.42290
$544$	0	0
$545$	−13.5854	−0.581935
$546$	0	0
$547$	39.8726	1.70483	0.852416	−	0.522865i	$-0.175137\pi$
$547$	39.8726	1.70483	0.852416	+	0.522865i	$0.175137\pi$
$548$	0	0
$549$	65.8385	2.80992
$550$	0	0
$551$	−2.01005	−0.0856309
$552$	0	0
$553$	18.8485	0.801519
$554$	0	0
$555$	5.99637	0.254532
$556$	0	0
$557$	−16.8257	−0.712929	−0.356465	−	0.934309i	$-0.616018\pi$
$557$	−16.8257	−0.712929	−0.356465	+	0.934309i	$0.616018\pi$
$558$	0	0
$559$	14.1135	0.596938
$560$	0	0
$561$	21.6248	0.913000
$562$	0	0
$563$	−8.68841	−0.366173	−0.183086	−	0.983097i	$-0.558609\pi$
$563$	−8.68841	−0.366173	−0.183086	+	0.983097i	$0.558609\pi$
$564$	0	0
$565$	−10.0171	−0.421424
$566$	0	0
$567$	30.2422	1.27005
$568$	0	0
$569$	20.7590	0.870262	0.435131	−	0.900367i	$-0.356702\pi$
$569$	20.7590	0.870262	0.435131	+	0.900367i	$0.356702\pi$
$570$	0	0
$571$	28.5889	1.19641	0.598203	−	0.801344i	$-0.295881\pi$
$571$	28.5889	1.19641	0.598203	+	0.801344i	$0.295881\pi$
$572$	0	0
$573$	−14.4021	−0.601657
$574$	0	0
$575$	9.72237	0.405451
$576$	0	0
$577$	8.97270	0.373538	0.186769	−	0.982404i	$-0.440198\pi$
$577$	8.97270	0.373538	0.186769	+	0.982404i	$0.440198\pi$
$578$	0	0
$579$	4.89071	0.203251
$580$	0	0
$581$	13.0139	0.539909
$582$	0	0
$583$	−11.7292	−0.485774
$584$	0	0
$585$	−19.4811	−0.805443
$586$	0	0
$587$	−1.59725	−0.0659254	−0.0329627	−	0.999457i	$-0.510494\pi$
$587$	−1.59725	−0.0659254	−0.0329627	+	0.999457i	$0.510494\pi$
$588$	0	0
$589$	−2.13169	−0.0878346
$590$	0	0
$591$	−46.6461	−1.91876
$592$	0	0
$593$	−0.933009	−0.0383141	−0.0191571	−	0.999816i	$-0.506098\pi$
$593$	−0.933009	−0.0383141	−0.0191571	+	0.999816i	$0.506098\pi$
$594$	0	0
$595$	7.80018	0.319776
$596$	0	0
$597$	28.3089	1.15860
$598$	0	0
$599$	0.280302	0.0114528	0.00572641	−	0.999984i	$-0.498177\pi$
$599$	0.280302	0.0114528	0.00572641	+	0.999984i	$0.498177\pi$
$600$	0	0
$601$	−15.5729	−0.635230	−0.317615	−	0.948220i	$-0.602882\pi$
$601$	−15.5729	−0.635230	−0.317615	+	0.948220i	$0.602882\pi$
$602$	0	0
$603$	66.6691	2.71497
$604$	0	0
$605$	0.677107	0.0275283
$606$	0	0
$607$	0.305109	0.0123840	0.00619199	−	0.999981i	$-0.498029\pi$
$607$	0.305109	0.0123840	0.00619199	+	0.999981i	$0.498029\pi$
$608$	0	0
$609$	10.6129	0.430055
$610$	0	0
$611$	5.11925	0.207103
$612$	0	0
$613$	29.6347	1.19693	0.598467	−	0.801148i	$-0.295777\pi$
$613$	29.6347	1.19693	0.598467	+	0.801148i	$0.295777\pi$
$614$	0	0
$615$	23.4121	0.944069
$616$	0	0
$617$	−11.8931	−0.478798	−0.239399	−	0.970921i	$-0.576950\pi$
$617$	−11.8931	−0.478798	−0.239399	+	0.970921i	$0.576950\pi$
$618$	0	0
$619$	9.11269	0.366270	0.183135	−	0.983088i	$-0.441375\pi$
$619$	9.11269	0.366270	0.183135	+	0.983088i	$0.441375\pi$
$620$	0	0
$621$	−26.3608	−1.05782
$622$	0	0
$623$	−6.84329	−0.274170
$624$	0	0
$625$	18.3331	0.733324
$626$	0	0
$627$	−3.14822	−0.125728
$628$	0	0
$629$	19.3220	0.770418
$630$	0	0
$631$	−27.0964	−1.07869	−0.539346	−	0.842084i	$-0.681329\pi$
$631$	−27.0964	−1.07869	−0.539346	+	0.842084i	$0.681329\pi$
$632$	0	0
$633$	−8.08593	−0.321387
$634$	0	0
$635$	−7.12483	−0.282740
$636$	0	0
$637$	17.4313	0.690653
$638$	0	0
$639$	−87.0900	−3.44523
$640$	0	0
$641$	−31.8022	−1.25611	−0.628056	−	0.778168i	$-0.716149\pi$
$641$	−31.8022	−1.25611	−0.628056	+	0.778168i	$0.716149\pi$
$642$	0	0
$643$	34.3861	1.35606	0.678028	−	0.735036i	$-0.262835\pi$
$643$	34.3861	1.35606	0.678028	+	0.735036i	$0.262835\pi$
$644$	0	0
$645$	−7.22710	−0.284567
$646$	0	0
$647$	−31.0285	−1.21986	−0.609928	−	0.792457i	$-0.708802\pi$
$647$	−31.0285	−1.21986	−0.609928	+	0.792457i	$0.708802\pi$
$648$	0	0
$649$	8.37650	0.328806
$650$	0	0
$651$	11.2551	0.441122
$652$	0	0
$653$	−31.9348	−1.24971	−0.624853	−	0.780743i	$-0.714841\pi$
$653$	−31.9348	−1.24971	−0.624853	+	0.780743i	$0.714841\pi$
$654$	0	0
$655$	6.60913	0.258240
$656$	0	0
$657$	−97.9628	−3.82189
$658$	0	0
$659$	−45.5268	−1.77347	−0.886736	−	0.462277i	$-0.847033\pi$
$659$	−45.5268	−1.77347	−0.886736	+	0.462277i	$0.847033\pi$
$660$	0	0
$661$	−29.2167	−1.13640	−0.568200	−	0.822891i	$-0.692360\pi$
$661$	−29.2167	−1.13640	−0.568200	+	0.822891i	$0.692360\pi$
$662$	0	0
$663$	−90.0216	−3.49615
$664$	0	0
$665$	−1.13558	−0.0440359
$666$	0	0
$667$	−4.30305	−0.166615
$668$	0	0
$669$	35.5036	1.37265
$670$	0	0
$671$	9.52619	0.367754
$672$	0	0
$673$	45.9862	1.77264	0.886319	−	0.463076i	$-0.153254\pi$
$673$	45.9862	1.77264	0.886319	+	0.463076i	$0.153254\pi$
$674$	0	0
$675$	−55.9230	−2.15248
$676$	0	0
$677$	−16.9360	−0.650904	−0.325452	−	0.945559i	$-0.605516\pi$
$677$	−16.9360	−0.650904	−0.325452	+	0.945559i	$0.605516\pi$
$678$	0	0
$679$	−20.8779	−0.801218
$680$	0	0
$681$	−14.1671	−0.542884
$682$	0	0
$683$	−1.67863	−0.0642308	−0.0321154	−	0.999484i	$-0.510224\pi$
$683$	−1.67863	−0.0642308	−0.0321154	+	0.999484i	$0.510224\pi$
$684$	0	0
$685$	1.16763	0.0446129
$686$	0	0
$687$	56.6361	2.16080
$688$	0	0
$689$	48.8273	1.86017
$690$	0	0
$691$	−41.2185	−1.56803	−0.784013	−	0.620744i	$-0.786830\pi$
$691$	−41.2185	−1.56803	−0.784013	+	0.620744i	$0.786830\pi$
$692$	0	0
$693$	11.5910	0.440306
$694$	0	0
$695$	1.85132	0.0702244
$696$	0	0
$697$	75.4405	2.85751
$698$	0	0
$699$	−30.3905	−1.14947
$700$	0	0
$701$	−26.4831	−1.00025	−0.500126	−	0.865953i	$-0.666713\pi$
$701$	−26.4831	−1.00025	−0.500126	+	0.865953i	$0.666713\pi$
$702$	0	0
$703$	−2.81297	−0.106093
$704$	0	0
$705$	−2.62141	−0.0987281
$706$	0	0
$707$	19.3948	0.729418
$708$	0	0
$709$	36.8689	1.38464	0.692319	−	0.721591i	$-0.256589\pi$
$709$	36.8689	1.38464	0.692319	+	0.721591i	$0.256589\pi$
$710$	0	0
$711$	77.6741	2.91301
$712$	0	0
$713$	−4.56345	−0.170903
$714$	0	0
$715$	−2.81872	−0.105414
$716$	0	0
$717$	−16.8928	−0.630875
$718$	0	0
$719$	19.7419	0.736250	0.368125	−	0.929776i	$-0.380000\pi$
$719$	19.7419	0.736250	0.368125	+	0.929776i	$0.380000\pi$
$720$	0	0
$721$	−16.9283	−0.630444
$722$	0	0
$723$	7.26492	0.270185
$724$	0	0
$725$	−9.12869	−0.339031
$726$	0	0
$727$	−33.7540	−1.25187	−0.625934	−	0.779876i	$-0.715282\pi$
$727$	−33.7540	−1.25187	−0.625934	+	0.779876i	$0.715282\pi$
$728$	0	0
$729$	8.32841	0.308460
$730$	0	0
$731$	−23.2878	−0.861329
$732$	0	0
$733$	−35.0242	−1.29365	−0.646825	−	0.762639i	$-0.723904\pi$
$733$	−35.0242	−1.29365	−0.646825	+	0.762639i	$0.723904\pi$
$734$	0	0
$735$	−8.92603	−0.329242
$736$	0	0
$737$	9.64636	0.355328
$738$	0	0
$739$	8.99477	0.330878	0.165439	−	0.986220i	$-0.447096\pi$
$739$	8.99477	0.330878	0.165439	+	0.986220i	$0.447096\pi$
$740$	0	0
$741$	13.1057	0.481450
$742$	0	0
$743$	−17.8359	−0.654337	−0.327168	−	0.944966i	$-0.606094\pi$
$743$	−17.8359	−0.654337	−0.327168	+	0.944966i	$0.606094\pi$
$744$	0	0
$745$	−12.6396	−0.463079
$746$	0	0
$747$	53.6301	1.96222
$748$	0	0
$749$	23.9960	0.876794
$750$	0	0
$751$	10.8155	0.394662	0.197331	−	0.980337i	$-0.436773\pi$
$751$	10.8155	0.394662	0.197331	+	0.980337i	$0.436773\pi$
$752$	0	0
$753$	−35.0467	−1.27717
$754$	0	0
$755$	7.12887	0.259446
$756$	0	0
$757$	29.0949	1.05747	0.528737	−	0.848786i	$-0.322666\pi$
$757$	29.0949	1.05747	0.528737	+	0.848786i	$0.322666\pi$
$758$	0	0
$759$	−6.73963	−0.244633
$760$	0	0
$761$	−24.8989	−0.902586	−0.451293	−	0.892376i	$-0.649037\pi$
$761$	−24.8989	−0.902586	−0.451293	+	0.892376i	$0.649037\pi$
$762$	0	0
$763$	−33.6493	−1.21819
$764$	0	0
$765$	32.1444	1.16218
$766$	0	0
$767$	−34.8704	−1.25910
$768$	0	0
$769$	−22.4656	−0.810129	−0.405064	−	0.914288i	$-0.632751\pi$
$769$	−22.4656	−0.810129	−0.405064	+	0.914288i	$0.632751\pi$
$770$	0	0
$771$	25.5180	0.919010
$772$	0	0
$773$	−0.491907	−0.0176927	−0.00884633	−	0.999961i	$-0.502816\pi$
$773$	−0.491907	−0.0176927	−0.00884633	+	0.999961i	$0.502816\pi$
$774$	0	0
$775$	−9.68110	−0.347756
$776$	0	0
$777$	14.8522	0.532821
$778$	0	0
$779$	−10.9829	−0.393504
$780$	0	0
$781$	−12.6011	−0.450902
$782$	0	0
$783$	24.7511	0.884534
$784$	0	0
$785$	−7.89673	−0.281846
$786$	0	0
$787$	−30.7601	−1.09648	−0.548239	−	0.836322i	$-0.684702\pi$
$787$	−30.7601	−1.09648	−0.548239	+	0.836322i	$0.684702\pi$
$788$	0	0
$789$	−83.6756	−2.97893
$790$	0	0
$791$	−24.8112	−0.882184
$792$	0	0
$793$	−39.6564	−1.40824
$794$	0	0
$795$	−25.0030	−0.886763
$796$	0	0
$797$	−46.8176	−1.65836	−0.829182	−	0.558978i	$-0.811193\pi$
$797$	−46.8176	−1.65836	−0.829182	+	0.558978i	$0.811193\pi$
$798$	0	0
$799$	−8.44693	−0.298831
$800$	0	0
$801$	−28.2010	−0.996434
$802$	0	0
$803$	−14.1743	−0.500199
$804$	0	0
$805$	−2.43102	−0.0856822
$806$	0	0
$807$	−13.2718	−0.467188
$808$	0	0
$809$	−15.3194	−0.538601	−0.269300	−	0.963056i	$-0.586792\pi$
$809$	−15.3194	−0.538601	−0.269300	+	0.963056i	$0.586792\pi$
$810$	0	0
$811$	−21.0369	−0.738706	−0.369353	−	0.929289i	$-0.620421\pi$
$811$	−21.0369	−0.738706	−0.369353	+	0.929289i	$0.620421\pi$
$812$	0	0
$813$	−2.74202	−0.0961668
$814$	0	0
$815$	−10.1303	−0.354851
$816$	0	0
$817$	3.39032	0.118612
$818$	0	0
$819$	−48.2521	−1.68606
$820$	0	0
$821$	−49.3342	−1.72178	−0.860888	−	0.508794i	$-0.830092\pi$
$821$	−49.3342	−1.72178	−0.860888	+	0.508794i	$0.830092\pi$
$822$	0	0
$823$	−44.7400	−1.55954	−0.779769	−	0.626067i	$-0.784664\pi$
$823$	−44.7400	−1.55954	−0.779769	+	0.626067i	$0.784664\pi$
$824$	0	0
$825$	−14.2977	−0.497784
$826$	0	0
$827$	42.7710	1.48729	0.743646	−	0.668573i	$-0.233095\pi$
$827$	42.7710	1.48729	0.743646	+	0.668573i	$0.233095\pi$
$828$	0	0
$829$	48.6840	1.69086	0.845432	−	0.534083i	$-0.179343\pi$
$829$	48.6840	1.69086	0.845432	+	0.534083i	$0.179343\pi$
$830$	0	0
$831$	−53.7419	−1.86429
$832$	0	0
$833$	−28.7622	−0.996551
$834$	0	0
$835$	7.17656	0.248355
$836$	0	0
$837$	26.2489	0.907296
$838$	0	0
$839$	50.1921	1.73282	0.866412	−	0.499330i	$-0.166421\pi$
$839$	50.1921	1.73282	0.866412	+	0.499330i	$0.166421\pi$
$840$	0	0
$841$	−24.9597	−0.860680
$842$	0	0
$843$	−12.0281	−0.414268
$844$	0	0
$845$	2.93161	0.100851
$846$	0	0
$847$	1.67711	0.0576261
$848$	0	0
$849$	56.0253	1.92278
$850$	0	0
$851$	−6.02193	−0.206429
$852$	0	0
$853$	−25.6101	−0.876874	−0.438437	−	0.898762i	$-0.644468\pi$
$853$	−25.6101	−0.876874	−0.438437	+	0.898762i	$0.644468\pi$
$854$	0	0
$855$	−4.67970	−0.160042
$856$	0	0
$857$	−0.780477	−0.0266606	−0.0133303	−	0.999911i	$-0.504243\pi$
$857$	−0.780477	−0.0266606	−0.0133303	+	0.999911i	$0.504243\pi$
$858$	0	0
$859$	−18.5597	−0.633247	−0.316624	−	0.948551i	$-0.602549\pi$
$859$	−18.5597	−0.633247	−0.316624	+	0.948551i	$0.602549\pi$
$860$	0	0
$861$	57.9889	1.97625
$862$	0	0
$863$	−20.5115	−0.698220	−0.349110	−	0.937082i	$-0.613516\pi$
$863$	−20.5115	−0.698220	−0.349110	+	0.937082i	$0.613516\pi$
$864$	0	0
$865$	2.54821	0.0866417
$866$	0	0
$867$	95.0187	3.22700
$868$	0	0
$869$	11.2387	0.381246
$870$	0	0
$871$	−40.1567	−1.36066
$872$	0	0
$873$	−86.0371	−2.91191
$874$	0	0
$875$	−10.8352	−0.366296
$876$	0	0
$877$	14.4072	0.486497	0.243248	−	0.969964i	$-0.421787\pi$
$877$	14.4072	0.486497	0.243248	+	0.969964i	$0.421787\pi$
$878$	0	0
$879$	−5.29945	−0.178746
$880$	0	0
$881$	33.3794	1.12458	0.562291	−	0.826940i	$-0.309920\pi$
$881$	33.3794	1.12458	0.562291	+	0.826940i	$0.309920\pi$
$882$	0	0
$883$	−35.6404	−1.19940	−0.599698	−	0.800226i	$-0.704713\pi$
$883$	−35.6404	−1.19940	−0.599698	+	0.800226i	$0.704713\pi$
$884$	0	0
$885$	17.8561	0.600225
$886$	0	0
$887$	12.6762	0.425626	0.212813	−	0.977093i	$-0.431737\pi$
$887$	12.6762	0.425626	0.212813	+	0.977093i	$0.431737\pi$
$888$	0	0
$889$	−17.6473	−0.591871
$890$	0	0
$891$	18.0324	0.604107
$892$	0	0
$893$	1.22974	0.0411516
$894$	0	0
$895$	2.37263	0.0793083
$896$	0	0
$897$	28.0563	0.936773
$898$	0	0
$899$	4.28479	0.142906
$900$	0	0
$901$	−80.5666	−2.68406
$902$	0	0
$903$	−17.9006	−0.595694
$904$	0	0
$905$	17.1548	0.570245
$906$	0	0
$907$	−36.6608	−1.21730	−0.608651	−	0.793438i	$-0.708289\pi$
$907$	−36.6608	−1.21730	−0.608651	+	0.793438i	$0.708289\pi$
$908$	0	0
$909$	79.9256	2.65097
$910$	0	0
$911$	47.6952	1.58021	0.790106	−	0.612970i	$-0.210026\pi$
$911$	47.6952	1.58021	0.790106	+	0.612970i	$0.210026\pi$
$912$	0	0
$913$	7.75976	0.256810
$914$	0	0
$915$	20.3068	0.671323
$916$	0	0
$917$	16.3700	0.540584
$918$	0	0
$919$	29.7694	0.982001	0.491001	−	0.871159i	$-0.336631\pi$
$919$	29.7694	0.982001	0.491001	+	0.871159i	$0.336631\pi$
$920$	0	0
$921$	−24.9851	−0.823286
$922$	0	0
$923$	52.4568	1.72664
$924$	0	0
$925$	−12.7752	−0.420045
$926$	0	0
$927$	−69.7613	−2.29126
$928$	0	0
$929$	−50.6180	−1.66072	−0.830362	−	0.557225i	$-0.811866\pi$
$929$	−50.6180	−1.66072	−0.830362	+	0.557225i	$0.811866\pi$
$930$	0	0
$931$	4.18731	0.137234
$932$	0	0
$933$	−94.0024	−3.07750
$934$	0	0
$935$	4.65098	0.152103
$936$	0	0
$937$	16.1084	0.526238	0.263119	−	0.964763i	$-0.415249\pi$
$937$	16.1084	0.526238	0.263119	+	0.964763i	$0.415249\pi$
$938$	0	0
$939$	59.3273	1.93607
$940$	0	0
$941$	16.8109	0.548018	0.274009	−	0.961727i	$-0.411650\pi$
$941$	16.8109	0.548018	0.274009	+	0.961727i	$0.411650\pi$
$942$	0	0
$943$	−23.5119	−0.765654
$944$	0	0
$945$	13.9832	0.454874
$946$	0	0
$947$	−36.1350	−1.17423	−0.587114	−	0.809504i	$-0.699736\pi$
$947$	−36.1350	−1.17423	−0.587114	+	0.809504i	$0.699736\pi$
$948$	0	0
$949$	59.0058	1.91541
$950$	0	0
$951$	18.1901	0.589854
$952$	0	0
$953$	−29.1171	−0.943197	−0.471598	−	0.881813i	$-0.656323\pi$
$953$	−29.1171	−0.943197	−0.471598	+	0.881813i	$0.656323\pi$
$954$	0	0
$955$	−3.09755	−0.100234
$956$	0	0
$957$	6.32808	0.204558
$958$	0	0
$959$	2.89207	0.0933898
$960$	0	0
$961$	−26.4559	−0.853417
$962$	0	0
$963$	98.8868	3.18658
$964$	0	0
$965$	1.05187	0.0338610
$966$	0	0
$967$	−10.1650	−0.326883	−0.163442	−	0.986553i	$-0.552260\pi$
$967$	−10.1650	−0.326883	−0.163442	+	0.986553i	$0.552260\pi$
$968$	0	0
$969$	−21.6248	−0.694689
$970$	0	0
$971$	−15.9729	−0.512594	−0.256297	−	0.966598i	$-0.582503\pi$
$971$	−15.9729	−0.512594	−0.256297	+	0.966598i	$0.582503\pi$
$972$	0	0
$973$	4.58547	0.147003
$974$	0	0
$975$	59.5199	1.90616
$976$	0	0
$977$	42.7941	1.36910	0.684552	−	0.728964i	$-0.259998\pi$
$977$	42.7941	1.36910	0.684552	+	0.728964i	$0.259998\pi$
$978$	0	0
$979$	−4.08041	−0.130410
$980$	0	0
$981$	−138.668	−4.42733
$982$	0	0
$983$	−13.8981	−0.443280	−0.221640	−	0.975129i	$-0.571141\pi$
$983$	−13.8981	−0.443280	−0.221640	+	0.975129i	$0.571141\pi$
$984$	0	0
$985$	−10.0325	−0.319661
$986$	0	0
$987$	−6.49290	−0.206671
$988$	0	0
$989$	7.25790	0.230788
$990$	0	0
$991$	49.5286	1.57333	0.786664	−	0.617381i	$-0.211806\pi$
$991$	49.5286	1.57333	0.786664	+	0.617381i	$0.211806\pi$
$992$	0	0
$993$	−32.9337	−1.04512
$994$	0	0
$995$	6.08855	0.193020
$996$	0	0
$997$	51.6810	1.63675	0.818377	−	0.574682i	$-0.194874\pi$
$997$	51.6810	1.63675	0.818377	+	0.574682i	$0.194874\pi$
$998$	0	0
$999$	34.6381	1.09590

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3344.2.a.z.1.6		6
4.3	odd	2		1672.2.a.i.1.1	✓	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1672.2.a.i.1.1	✓	6	4.3	odd	2
3344.2.a.z.1.6		6	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:	\( N \)	\(=\)	\( 3344 = 2^{4} \cdot 11 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3344.a (trivial)

\(f(q)\)	\(=\)	\(q+3.14822 q^{3} +0.677107 q^{5} +1.67711 q^{7} +6.91132 q^{9} +O(q^{10})\)	\(q+3.14822 q^{3} +0.677107 q^{5} +1.67711 q^{7} +6.91132 q^{9} +1.00000 q^{11} -4.16289 q^{13} +2.13169 q^{15} +6.86889 q^{17} -1.00000 q^{19} +5.27991 q^{21} -2.14077 q^{23} -4.54153 q^{25} +12.3137 q^{27} +2.01005 q^{29} +2.13169 q^{31} +3.14822 q^{33} +1.13558 q^{35} +2.81297 q^{37} -13.1057 q^{39} +10.9829 q^{41} -3.39032 q^{43} +4.67970 q^{45} -1.22974 q^{47} -4.18731 q^{49} +21.6248 q^{51} -11.7292 q^{53} +0.677107 q^{55} -3.14822 q^{57} +8.37650 q^{59} +9.52619 q^{61} +11.5910 q^{63} -2.81872 q^{65} +9.64636 q^{67} -6.73963 q^{69} -12.6011 q^{71} -14.1743 q^{73} -14.2977 q^{75} +1.67711 q^{77} +11.2387 q^{79} +18.0324 q^{81} +7.75976 q^{83} +4.65098 q^{85} +6.32808 q^{87} -4.08041 q^{89} -6.98160 q^{91} +6.71102 q^{93} -0.677107 q^{95} -12.4487 q^{97} +6.91132 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 4 q^{3} - 3 q^{5} + 3 q^{7} + 6 q^{9}+O(q^{10}) \) 6 * q + 4 * q^3 - 3 * q^5 + 3 * q^7 + 6 * q^9	\( 6 q + 4 q^{3} - 3 q^{5} + 3 q^{7} + 6 q^{9} + 6 q^{11} - 3 q^{13} + q^{15} - q^{17} - 6 q^{19} + 5 q^{21} + 4 q^{23} + 5 q^{25} + 16 q^{27} - 7 q^{29} + q^{31} + 4 q^{33} + 32 q^{35} + 5 q^{37} + 13 q^{39} - 6 q^{41} + 16 q^{43} + 4 q^{47} - 7 q^{49} + 35 q^{51} - 11 q^{53} - 3 q^{55} - 4 q^{57} + 18 q^{59} + 16 q^{61} + 6 q^{63} + 10 q^{65} + 18 q^{67} - 2 q^{69} + 7 q^{71} - 21 q^{73} + 23 q^{75} + 3 q^{77} + 22 q^{79} - 10 q^{81} + 16 q^{83} + 3 q^{87} - 13 q^{89} + 7 q^{91} - 9 q^{93} + 3 q^{95} - 15 q^{97} + 6 q^{99}+O(q^{100}) \) 6 * q + 4 * q^3 - 3 * q^5 + 3 * q^7 + 6 * q^9 + 6 * q^11 - 3 * q^13 + q^15 - q^17 - 6 * q^19 + 5 * q^21 + 4 * q^23 + 5 * q^25 + 16 * q^27 - 7 * q^29 + q^31 + 4 * q^33 + 32 * q^35 + 5 * q^37 + 13 * q^39 - 6 * q^41 + 16 * q^43 + 4 * q^47 - 7 * q^49 + 35 * q^51 - 11 * q^53 - 3 * q^55 - 4 * q^57 + 18 * q^59 + 16 * q^61 + 6 * q^63 + 10 * q^65 + 18 * q^67 - 2 * q^69 + 7 * q^71 - 21 * q^73 + 23 * q^75 + 3 * q^77 + 22 * q^79 - 10 * q^81 + 16 * q^83 + 3 * q^87 - 13 * q^89 + 7 * q^91 - 9 * q^93 + 3 * q^95 - 15 * q^97 + 6 * q^99

Introduction

L-functions

Modular forms

Varieties

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Groups

Database

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