LMFDB - Embedded newform 8016.2.a.be.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8016,2,Mod(1,8016)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8016, 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("8016.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8016 = 2^{4} \cdot 3 \cdot 167 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8016.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:	$64.0080822603$
Analytic rank:	$0$
Dimension:	$11$
Coefficient field:	$\mathbb{Q}[x]/(x^{11} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{11} - x^{10} - 33 x^{9} + 22 x^{8} + 417 x^{7} - 151 x^{6} - 2470 x^{5} + 272 x^{4} + 6584 x^{3} + \cdots + 242 $ x^11 - x^10 - 33x^9 + 22x^8 + 417x^7 - 151x^6 - 2470x^5 + 272x^4 + 6584x^3 + 292x^2 - 5687*x + 242
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 4008)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$3.54432$ of defining polynomial
Character	$\chi$	$=$	8016.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} -2.54432 q^{5} -0.525270 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} -2.54432 q^{5} -0.525270 q^{7} +1.00000 q^{9} -5.59056 q^{11} +1.57904 q^{13} -2.54432 q^{15} +4.20935 q^{17} -3.78855 q^{19} -0.525270 q^{21} +2.45151 q^{23} +1.47356 q^{25} +1.00000 q^{27} -4.24006 q^{29} +5.76018 q^{31} -5.59056 q^{33} +1.33645 q^{35} -2.67489 q^{37} +1.57904 q^{39} +10.7672 q^{41} -8.36680 q^{43} -2.54432 q^{45} -4.77944 q^{47} -6.72409 q^{49} +4.20935 q^{51} -2.34459 q^{53} +14.2242 q^{55} -3.78855 q^{57} +5.25946 q^{59} -5.02359 q^{61} -0.525270 q^{63} -4.01758 q^{65} -8.91781 q^{67} +2.45151 q^{69} +5.10616 q^{71} -2.16373 q^{73} +1.47356 q^{75} +2.93655 q^{77} +2.52659 q^{79} +1.00000 q^{81} +4.01520 q^{83} -10.7099 q^{85} -4.24006 q^{87} +16.2275 q^{89} -0.829422 q^{91} +5.76018 q^{93} +9.63928 q^{95} -16.4673 q^{97} -5.59056 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 11 q + 11 q^{3} + 10 q^{5} + q^{7} + 11 q^{9}+O(q^{10}) $ 11 * q + 11 * q^3 + 10 * q^5 + q^7 + 11 * q^9	$ 11 q + 11 q^{3} + 10 q^{5} + q^{7} + 11 q^{9} + q^{11} + 10 q^{13} + 10 q^{15} + 17 q^{17} - 2 q^{19} + q^{21} + 3 q^{23} + 21 q^{25} + 11 q^{27} + 17 q^{29} + 15 q^{31} + q^{33} - 11 q^{35} + 4 q^{37} + 10 q^{39} + 16 q^{41} - 10 q^{43} + 10 q^{45} + 16 q^{47} + 22 q^{49} + 17 q^{51} + 42 q^{53} + 5 q^{55} - 2 q^{57} + 2 q^{59} + 12 q^{61} + q^{63} + 10 q^{65} + q^{67} + 3 q^{69} + 9 q^{71} + 24 q^{73} + 21 q^{75} + 22 q^{77} + 30 q^{79} + 11 q^{81} - 16 q^{83} + 25 q^{85} + 17 q^{87} + 37 q^{89} - q^{91} + 15 q^{93} - 5 q^{95} + 4 q^{97} + q^{99}+O(q^{100}) $ 11 * q + 11 * q^3 + 10 * q^5 + q^7 + 11 * q^9 + q^11 + 10 * q^13 + 10 * q^15 + 17 * q^17 - 2 * q^19 + q^21 + 3 * q^23 + 21 * q^25 + 11 * q^27 + 17 * q^29 + 15 * q^31 + q^33 - 11 * q^35 + 4 * q^37 + 10 * q^39 + 16 * q^41 - 10 * q^43 + 10 * q^45 + 16 * q^47 + 22 * q^49 + 17 * q^51 + 42 * q^53 + 5 * q^55 - 2 * q^57 + 2 * q^59 + 12 * q^61 + q^63 + 10 * q^65 + q^67 + 3 * q^69 + 9 * q^71 + 24 * q^73 + 21 * q^75 + 22 * q^77 + 30 * q^79 + 11 * q^81 - 16 * q^83 + 25 * q^85 + 17 * q^87 + 37 * q^89 - q^91 + 15 * q^93 - 5 * q^95 + 4 * q^97 + 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
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0	0
$5$	−2.54432	−1.13785	−0.568927	−	0.822388i	$-0.692641\pi$
$5$	−2.54432	−1.13785	−0.568927	+	0.822388i	$0.692641\pi$
$6$	0	0
$7$	−0.525270	−0.198533	−0.0992667	−	0.995061i	$-0.531650\pi$
$7$	−0.525270	−0.198533	−0.0992667	+	0.995061i	$0.531650\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−5.59056	−1.68562	−0.842809	−	0.538213i	$-0.819100\pi$
$11$	−5.59056	−1.68562	−0.842809	+	0.538213i	$0.819100\pi$
$12$	0	0
$13$	1.57904	0.437947	0.218974	−	0.975731i	$-0.429729\pi$
$13$	1.57904	0.437947	0.218974	+	0.975731i	$0.429729\pi$
$14$	0	0
$15$	−2.54432	−0.656940
$16$	0	0
$17$	4.20935	1.02092	0.510459	−	0.859902i	$-0.329476\pi$
$17$	4.20935	1.02092	0.510459	+	0.859902i	$0.329476\pi$
$18$	0	0
$19$	−3.78855	−0.869153	−0.434577	−	0.900635i	$-0.643102\pi$
$19$	−3.78855	−0.869153	−0.434577	+	0.900635i	$0.643102\pi$
$20$	0	0
$21$	−0.525270	−0.114623
$22$	0	0
$23$	2.45151	0.511174	0.255587	−	0.966786i	$-0.417731\pi$
$23$	2.45151	0.511174	0.255587	+	0.966786i	$0.417731\pi$
$24$	0	0
$25$	1.47356	0.294711
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	−4.24006	−0.787359	−0.393679	−	0.919248i	$-0.628798\pi$
$29$	−4.24006	−0.787359	−0.393679	+	0.919248i	$0.628798\pi$
$30$	0	0
$31$	5.76018	1.03456	0.517279	−	0.855817i	$-0.326945\pi$
$31$	5.76018	1.03456	0.517279	+	0.855817i	$0.326945\pi$
$32$	0	0
$33$	−5.59056	−0.973192
$34$	0	0
$35$	1.33645	0.225902
$36$	0	0
$37$	−2.67489	−0.439749	−0.219874	−	0.975528i	$-0.570565\pi$
$37$	−2.67489	−0.439749	−0.219874	+	0.975528i	$0.570565\pi$
$38$	0	0
$39$	1.57904	0.252849
$40$	0	0
$41$	10.7672	1.68155	0.840774	−	0.541387i	$-0.182101\pi$
$41$	10.7672	1.68155	0.840774	+	0.541387i	$0.182101\pi$
$42$	0	0
$43$	−8.36680	−1.27593	−0.637963	−	0.770067i	$-0.720223\pi$
$43$	−8.36680	−1.27593	−0.637963	+	0.770067i	$0.720223\pi$
$44$	0	0
$45$	−2.54432	−0.379285
$46$	0	0
$47$	−4.77944	−0.697153	−0.348577	−	0.937280i	$-0.613335\pi$
$47$	−4.77944	−0.697153	−0.348577	+	0.937280i	$0.613335\pi$
$48$	0	0
$49$	−6.72409	−0.960585
$50$	0	0
$51$	4.20935	0.589427
$52$	0	0
$53$	−2.34459	−0.322054	−0.161027	−	0.986950i	$-0.551481\pi$
$53$	−2.34459	−0.322054	−0.161027	+	0.986950i	$0.551481\pi$
$54$	0	0
$55$	14.2242	1.91799
$56$	0	0
$57$	−3.78855	−0.501806
$58$	0	0
$59$	5.25946	0.684723	0.342362	−	0.939568i	$-0.388773\pi$
$59$	5.25946	0.684723	0.342362	+	0.939568i	$0.388773\pi$
$60$	0	0
$61$	−5.02359	−0.643205	−0.321602	−	0.946875i	$-0.604221\pi$
$61$	−5.02359	−0.643205	−0.321602	+	0.946875i	$0.604221\pi$
$62$	0	0
$63$	−0.525270	−0.0661778
$64$	0	0
$65$	−4.01758	−0.498320
$66$	0	0
$67$	−8.91781	−1.08948	−0.544742	−	0.838604i	$-0.683372\pi$
$67$	−8.91781	−1.08948	−0.544742	+	0.838604i	$0.683372\pi$
$68$	0	0
$69$	2.45151	0.295127
$70$	0	0
$71$	5.10616	0.605990	0.302995	−	0.952992i	$-0.402013\pi$
$71$	5.10616	0.605990	0.302995	+	0.952992i	$0.402013\pi$
$72$	0	0
$73$	−2.16373	−0.253246	−0.126623	−	0.991951i	$-0.540414\pi$
$73$	−2.16373	−0.253246	−0.126623	+	0.991951i	$0.540414\pi$
$74$	0	0
$75$	1.47356	0.170151
$76$	0	0
$77$	2.93655	0.334651
$78$	0	0
$79$	2.52659	0.284263	0.142132	−	0.989848i	$-0.454604\pi$
$79$	2.52659	0.284263	0.142132	+	0.989848i	$0.454604\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	4.01520	0.440726	0.220363	−	0.975418i	$-0.429276\pi$
$83$	4.01520	0.440726	0.220363	+	0.975418i	$0.429276\pi$
$84$	0	0
$85$	−10.7099	−1.16165
$86$	0	0
$87$	−4.24006	−0.454582
$88$	0	0
$89$	16.2275	1.72011	0.860055	−	0.510201i	$-0.170429\pi$
$89$	16.2275	1.72011	0.860055	+	0.510201i	$0.170429\pi$
$90$	0	0
$91$	−0.829422	−0.0869471
$92$	0	0
$93$	5.76018	0.597303
$94$	0	0
$95$	9.63928	0.988969
$96$	0	0
$97$	−16.4673	−1.67201	−0.836003	−	0.548725i	$-0.815113\pi$
$97$	−16.4673	−1.67201	−0.836003	+	0.548725i	$0.815113\pi$
$98$	0	0
$99$	−5.59056	−0.561873
$100$	0	0
$101$	−7.51102	−0.747374	−0.373687	−	0.927555i	$-0.621907\pi$
$101$	−7.51102	−0.747374	−0.373687	+	0.927555i	$0.621907\pi$
$102$	0	0
$103$	−1.49723	−0.147527	−0.0737634	−	0.997276i	$-0.523501\pi$
$103$	−1.49723	−0.147527	−0.0737634	+	0.997276i	$0.523501\pi$
$104$	0	0
$105$	1.33645	0.130425
$106$	0	0
$107$	2.29357	0.221728	0.110864	−	0.993836i	$-0.464638\pi$
$107$	2.29357	0.221728	0.110864	+	0.993836i	$0.464638\pi$
$108$	0	0
$109$	13.3290	1.27669	0.638344	−	0.769751i	$-0.279620\pi$
$109$	13.3290	1.27669	0.638344	+	0.769751i	$0.279620\pi$
$110$	0	0
$111$	−2.67489	−0.253889
$112$	0	0
$113$	18.9368	1.78142	0.890711	−	0.454570i	$-0.150207\pi$
$113$	18.9368	1.78142	0.890711	+	0.454570i	$0.150207\pi$
$114$	0	0
$115$	−6.23741	−0.581642
$116$	0	0
$117$	1.57904	0.145982
$118$	0	0
$119$	−2.21104	−0.202686
$120$	0	0
$121$	20.2544	1.84131
$122$	0	0
$123$	10.7672	0.970842
$124$	0	0
$125$	8.97240	0.802516
$126$	0	0
$127$	0.286281	0.0254034	0.0127017	−	0.999919i	$-0.495957\pi$
$127$	0.286281	0.0254034	0.0127017	+	0.999919i	$0.495957\pi$
$128$	0	0
$129$	−8.36680	−0.736656
$130$	0	0
$131$	−12.5000	−1.09213	−0.546064	−	0.837744i	$-0.683874\pi$
$131$	−12.5000	−1.09213	−0.546064	+	0.837744i	$0.683874\pi$
$132$	0	0
$133$	1.99001	0.172556
$134$	0	0
$135$	−2.54432	−0.218980
$136$	0	0
$137$	12.4508	1.06374	0.531872	−	0.846825i	$-0.321489\pi$
$137$	12.4508	1.06374	0.531872	+	0.846825i	$0.321489\pi$
$138$	0	0
$139$	−13.8333	−1.17333	−0.586663	−	0.809831i	$-0.699559\pi$
$139$	−13.8333	−1.17333	−0.586663	+	0.809831i	$0.699559\pi$
$140$	0	0
$141$	−4.77944	−0.402502
$142$	0	0
$143$	−8.82773	−0.738212
$144$	0	0
$145$	10.7881	0.895899
$146$	0	0
$147$	−6.72409	−0.554594
$148$	0	0
$149$	17.9274	1.46867	0.734335	−	0.678787i	$-0.237494\pi$
$149$	17.9274	1.46867	0.734335	+	0.678787i	$0.237494\pi$
$150$	0	0
$151$	5.27238	0.429061	0.214530	−	0.976717i	$-0.431178\pi$
$151$	5.27238	0.429061	0.214530	+	0.976717i	$0.431178\pi$
$152$	0	0
$153$	4.20935	0.340306
$154$	0	0
$155$	−14.6557	−1.17718
$156$	0	0
$157$	23.2595	1.85631	0.928155	−	0.372195i	$-0.121395\pi$
$157$	23.2595	1.85631	0.928155	+	0.372195i	$0.121395\pi$
$158$	0	0
$159$	−2.34459	−0.185938
$160$	0	0
$161$	−1.28770	−0.101485
$162$	0	0
$163$	21.8496	1.71139	0.855696	−	0.517478i	$-0.173129\pi$
$163$	21.8496	1.71139	0.855696	+	0.517478i	$0.173129\pi$
$164$	0	0
$165$	14.2242	1.10735
$166$	0	0
$167$	−1.00000	−0.0773823
$167$	−1.00000	−0.0773823
$168$	0	0
$169$	−10.5066	−0.808202
$170$	0	0
$171$	−3.78855	−0.289718
$172$	0	0
$173$	10.0266	0.762310	0.381155	−	0.924511i	$-0.375526\pi$
$173$	10.0266	0.762310	0.381155	+	0.924511i	$0.375526\pi$
$174$	0	0
$175$	−0.774014	−0.0585100
$176$	0	0
$177$	5.25946	0.395325
$178$	0	0
$179$	−0.201597	−0.0150681	−0.00753404	−	0.999972i	$-0.502398\pi$
$179$	−0.201597	−0.0150681	−0.00753404	+	0.999972i	$0.502398\pi$
$180$	0	0
$181$	11.7308	0.871946	0.435973	−	0.899960i	$-0.356404\pi$
$181$	11.7308	0.871946	0.435973	+	0.899960i	$0.356404\pi$
$182$	0	0
$183$	−5.02359	−0.371354
$184$	0	0
$185$	6.80576	0.500370
$186$	0	0
$187$	−23.5326	−1.72088
$188$	0	0
$189$	−0.525270	−0.0382078
$190$	0	0
$191$	25.0393	1.81178	0.905891	−	0.423511i	$-0.139203\pi$
$191$	25.0393	1.81178	0.905891	+	0.423511i	$0.139203\pi$
$192$	0	0
$193$	−10.7710	−0.775315	−0.387658	−	0.921803i	$-0.626716\pi$
$193$	−10.7710	−0.775315	−0.387658	+	0.921803i	$0.626716\pi$
$194$	0	0
$195$	−4.01758	−0.287705
$196$	0	0
$197$	11.1738	0.796102	0.398051	−	0.917363i	$-0.369687\pi$
$197$	11.1738	0.796102	0.398051	+	0.917363i	$0.369687\pi$
$198$	0	0
$199$	23.0987	1.63742	0.818710	−	0.574207i	$-0.194690\pi$
$199$	23.0987	1.63742	0.818710	+	0.574207i	$0.194690\pi$
$200$	0	0
$201$	−8.91781	−0.629014
$202$	0	0
$203$	2.22717	0.156317
$204$	0	0
$205$	−27.3951	−1.91335
$206$	0	0
$207$	2.45151	0.170391
$208$	0	0
$209$	21.1801	1.46506
$210$	0	0
$211$	8.65386	0.595756	0.297878	−	0.954604i	$-0.403721\pi$
$211$	8.65386	0.595756	0.297878	+	0.954604i	$0.403721\pi$
$212$	0	0
$213$	5.10616	0.349868
$214$	0	0
$215$	21.2878	1.45182
$216$	0	0
$217$	−3.02565	−0.205394
$218$	0	0
$219$	−2.16373	−0.146212
$220$	0	0
$221$	6.64673	0.447108
$222$	0	0
$223$	−3.47630	−0.232791	−0.116395	−	0.993203i	$-0.537134\pi$
$223$	−3.47630	−0.232791	−0.116395	+	0.993203i	$0.537134\pi$
$224$	0	0
$225$	1.47356	0.0982370
$226$	0	0
$227$	−4.99391	−0.331458	−0.165729	−	0.986171i	$-0.552998\pi$
$227$	−4.99391	−0.331458	−0.165729	+	0.986171i	$0.552998\pi$
$228$	0	0
$229$	−0.724364	−0.0478673	−0.0239337	−	0.999714i	$-0.507619\pi$
$229$	−0.724364	−0.0478673	−0.0239337	+	0.999714i	$0.507619\pi$
$230$	0	0
$231$	2.93655	0.193211
$232$	0	0
$233$	3.07905	0.201716	0.100858	−	0.994901i	$-0.467841\pi$
$233$	3.07905	0.201716	0.100858	+	0.994901i	$0.467841\pi$
$234$	0	0
$235$	12.1604	0.793258
$236$	0	0
$237$	2.52659	0.164120
$238$	0	0
$239$	−23.7125	−1.53383	−0.766916	−	0.641748i	$-0.778210\pi$
$239$	−23.7125	−1.53383	−0.766916	+	0.641748i	$0.778210\pi$
$240$	0	0
$241$	14.1304	0.910219	0.455109	−	0.890436i	$-0.349600\pi$
$241$	14.1304	0.910219	0.455109	+	0.890436i	$0.349600\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	17.1082	1.09300
$246$	0	0
$247$	−5.98227	−0.380643
$248$	0	0
$249$	4.01520	0.254453
$250$	0	0
$251$	−0.523099	−0.0330177	−0.0165089	−	0.999864i	$-0.505255\pi$
$251$	−0.523099	−0.0330177	−0.0165089	+	0.999864i	$0.505255\pi$
$252$	0	0
$253$	−13.7053	−0.861645
$254$	0	0
$255$	−10.7099	−0.670682
$256$	0	0
$257$	19.5504	1.21952	0.609761	−	0.792586i	$-0.291266\pi$
$257$	19.5504	1.21952	0.609761	+	0.792586i	$0.291266\pi$
$258$	0	0
$259$	1.40504	0.0873048
$260$	0	0
$261$	−4.24006	−0.262453
$262$	0	0
$263$	2.94567	0.181638	0.0908189	−	0.995867i	$-0.471052\pi$
$263$	2.94567	0.181638	0.0908189	+	0.995867i	$0.471052\pi$
$264$	0	0
$265$	5.96539	0.366451
$266$	0	0
$267$	16.2275	0.993106
$268$	0	0
$269$	9.57622	0.583872	0.291936	−	0.956438i	$-0.405701\pi$
$269$	9.57622	0.583872	0.291936	+	0.956438i	$0.405701\pi$
$270$	0	0
$271$	−24.1059	−1.46433	−0.732164	−	0.681128i	$-0.761490\pi$
$271$	−24.1059	−1.46433	−0.732164	+	0.681128i	$0.761490\pi$
$272$	0	0
$273$	−0.829422	−0.0501989
$274$	0	0
$275$	−8.23800	−0.496770
$276$	0	0
$277$	−2.19302	−0.131766	−0.0658828	−	0.997827i	$-0.520986\pi$
$277$	−2.19302	−0.131766	−0.0658828	+	0.997827i	$0.520986\pi$
$278$	0	0
$279$	5.76018	0.344853
$280$	0	0
$281$	6.78076	0.404506	0.202253	−	0.979333i	$-0.435174\pi$
$281$	6.78076	0.404506	0.202253	+	0.979333i	$0.435174\pi$
$282$	0	0
$283$	4.02725	0.239395	0.119697	−	0.992810i	$-0.461808\pi$
$283$	4.02725	0.239395	0.119697	+	0.992810i	$0.461808\pi$
$284$	0	0
$285$	9.63928	0.570982
$286$	0	0
$287$	−5.65566	−0.333843
$288$	0	0
$289$	0.718627	0.0422722
$290$	0	0
$291$	−16.4673	−0.965333
$292$	0	0
$293$	22.8461	1.33468	0.667342	−	0.744751i	$-0.267432\pi$
$293$	22.8461	1.33468	0.667342	+	0.744751i	$0.267432\pi$
$294$	0	0
$295$	−13.3817	−0.779115
$296$	0	0
$297$	−5.59056	−0.324397
$298$	0	0
$299$	3.87103	0.223867
$300$	0	0
$301$	4.39483	0.253314
$302$	0	0
$303$	−7.51102	−0.431497
$304$	0	0
$305$	12.7816	0.731873
$306$	0	0
$307$	−2.74814	−0.156845	−0.0784223	−	0.996920i	$-0.524988\pi$
$307$	−2.74814	−0.156845	−0.0784223	+	0.996920i	$0.524988\pi$
$308$	0	0
$309$	−1.49723	−0.0851746
$310$	0	0
$311$	3.22030	0.182606	0.0913032	−	0.995823i	$-0.470897\pi$
$311$	3.22030	0.182606	0.0913032	+	0.995823i	$0.470897\pi$
$312$	0	0
$313$	−30.2753	−1.71126	−0.855629	−	0.517589i	$-0.826830\pi$
$313$	−30.2753	−1.71126	−0.855629	+	0.517589i	$0.826830\pi$
$314$	0	0
$315$	1.33645	0.0753006
$316$	0	0
$317$	13.1079	0.736214	0.368107	−	0.929783i	$-0.380006\pi$
$317$	13.1079	0.736214	0.368107	+	0.929783i	$0.380006\pi$
$318$	0	0
$319$	23.7043	1.32719
$320$	0	0
$321$	2.29357	0.128015
$322$	0	0
$323$	−15.9473	−0.887333
$324$	0	0
$325$	2.32680	0.129068
$326$	0	0
$327$	13.3290	0.737096
$328$	0	0
$329$	2.51050	0.138408
$330$	0	0
$331$	22.4120	1.23188	0.615938	−	0.787795i	$-0.288777\pi$
$331$	22.4120	1.23188	0.615938	+	0.787795i	$0.288777\pi$
$332$	0	0
$333$	−2.67489	−0.146583
$334$	0	0
$335$	22.6898	1.23967
$336$	0	0
$337$	1.50311	0.0818797	0.0409398	−	0.999162i	$-0.486965\pi$
$337$	1.50311	0.0818797	0.0409398	+	0.999162i	$0.486965\pi$
$338$	0	0
$339$	18.9368	1.02850
$340$	0	0
$341$	−32.2026	−1.74387
$342$	0	0
$343$	7.20885	0.389241
$344$	0	0
$345$	−6.23741	−0.335811
$346$	0	0
$347$	−29.4505	−1.58099	−0.790493	−	0.612471i	$-0.790175\pi$
$347$	−29.4505	−1.58099	−0.790493	+	0.612471i	$0.790175\pi$
$348$	0	0
$349$	5.58173	0.298783	0.149392	−	0.988778i	$-0.452268\pi$
$349$	5.58173	0.298783	0.149392	+	0.988778i	$0.452268\pi$
$350$	0	0
$351$	1.57904	0.0842829
$352$	0	0
$353$	−9.27363	−0.493586	−0.246793	−	0.969068i	$-0.579377\pi$
$353$	−9.27363	−0.493586	−0.246793	+	0.969068i	$0.579377\pi$
$354$	0	0
$355$	−12.9917	−0.689528
$356$	0	0
$357$	−2.21104	−0.117021
$358$	0	0
$359$	32.7886	1.73052	0.865258	−	0.501326i	$-0.167154\pi$
$359$	32.7886	1.73052	0.865258	+	0.501326i	$0.167154\pi$
$360$	0	0
$361$	−4.64689	−0.244573
$362$	0	0
$363$	20.2544	1.06308
$364$	0	0
$365$	5.50523	0.288157
$366$	0	0
$367$	−4.93563	−0.257638	−0.128819	−	0.991668i	$-0.541119\pi$
$367$	−4.93563	−0.257638	−0.128819	+	0.991668i	$0.541119\pi$
$368$	0	0
$369$	10.7672	0.560516
$370$	0	0
$371$	1.23154	0.0639385
$372$	0	0
$373$	32.9187	1.70446	0.852232	−	0.523165i	$-0.175249\pi$
$373$	32.9187	1.70446	0.852232	+	0.523165i	$0.175249\pi$
$374$	0	0
$375$	8.97240	0.463333
$376$	0	0
$377$	−6.69522	−0.344821
$378$	0	0
$379$	0.820431	0.0421427	0.0210714	−	0.999778i	$-0.493292\pi$
$379$	0.820431	0.0421427	0.0210714	+	0.999778i	$0.493292\pi$
$380$	0	0
$381$	0.286281	0.0146666
$382$	0	0
$383$	1.99971	0.102180	0.0510901	−	0.998694i	$-0.483730\pi$
$383$	1.99971	0.102180	0.0510901	+	0.998694i	$0.483730\pi$
$384$	0	0
$385$	−7.47153	−0.380784
$386$	0	0
$387$	−8.36680	−0.425309
$388$	0	0
$389$	−17.8177	−0.903395	−0.451698	−	0.892171i	$-0.649181\pi$
$389$	−17.8177	−0.903395	−0.451698	+	0.892171i	$0.649181\pi$
$390$	0	0
$391$	10.3192	0.521867
$392$	0	0
$393$	−12.5000	−0.630540
$394$	0	0
$395$	−6.42845	−0.323450
$396$	0	0
$397$	32.8025	1.64631	0.823156	−	0.567816i	$-0.192211\pi$
$397$	32.8025	1.64631	0.823156	+	0.567816i	$0.192211\pi$
$398$	0	0
$399$	1.99001	0.0996252
$400$	0	0
$401$	−35.4145	−1.76852	−0.884258	−	0.466998i	$-0.845336\pi$
$401$	−35.4145	−1.76852	−0.884258	+	0.466998i	$0.845336\pi$
$402$	0	0
$403$	9.09555	0.453082
$404$	0	0
$405$	−2.54432	−0.126428
$406$	0	0
$407$	14.9541	0.741248
$408$	0	0
$409$	−26.5720	−1.31390	−0.656950	−	0.753934i	$-0.728154\pi$
$409$	−26.5720	−1.31390	−0.656950	+	0.753934i	$0.728154\pi$
$410$	0	0
$411$	12.4508	0.614153
$412$	0	0
$413$	−2.76264	−0.135940
$414$	0	0
$415$	−10.2160	−0.501482
$416$	0	0
$417$	−13.8333	−0.677420
$418$	0	0
$419$	−26.4006	−1.28975	−0.644876	−	0.764287i	$-0.723091\pi$
$419$	−26.4006	−1.28975	−0.644876	+	0.764287i	$0.723091\pi$
$420$	0	0
$421$	30.2066	1.47218	0.736090	−	0.676884i	$-0.236670\pi$
$421$	30.2066	1.47218	0.736090	+	0.676884i	$0.236670\pi$
$422$	0	0
$423$	−4.77944	−0.232384
$424$	0	0
$425$	6.20271	0.300876
$426$	0	0
$427$	2.63874	0.127698
$428$	0	0
$429$	−8.82773	−0.426207
$430$	0	0
$431$	26.0759	1.25603	0.628015	−	0.778201i	$-0.283868\pi$
$431$	26.0759	1.25603	0.628015	+	0.778201i	$0.283868\pi$
$432$	0	0
$433$	22.3412	1.07365	0.536825	−	0.843694i	$-0.319624\pi$
$433$	22.3412	1.07365	0.536825	+	0.843694i	$0.319624\pi$
$434$	0	0
$435$	10.7881	0.517248
$436$	0	0
$437$	−9.28766	−0.444289
$438$	0	0
$439$	−2.40123	−0.114604	−0.0573021	−	0.998357i	$-0.518250\pi$
$439$	−2.40123	−0.114604	−0.0573021	+	0.998357i	$0.518250\pi$
$440$	0	0
$441$	−6.72409	−0.320195
$442$	0	0
$443$	6.06561	0.288186	0.144093	−	0.989564i	$-0.453974\pi$
$443$	6.06561	0.288186	0.144093	+	0.989564i	$0.453974\pi$
$444$	0	0
$445$	−41.2879	−1.95723
$446$	0	0
$447$	17.9274	0.847937
$448$	0	0
$449$	40.4576	1.90931	0.954656	−	0.297713i	$-0.0962237\pi$
$449$	40.4576	1.90931	0.954656	+	0.297713i	$0.0962237\pi$
$450$	0	0
$451$	−60.1945	−2.83445
$452$	0	0
$453$	5.27238	0.247718
$454$	0	0
$455$	2.11031	0.0989331
$456$	0	0
$457$	4.88373	0.228451	0.114226	−	0.993455i	$-0.463561\pi$
$457$	4.88373	0.228451	0.114226	+	0.993455i	$0.463561\pi$
$458$	0	0
$459$	4.20935	0.196476
$460$	0	0
$461$	−12.2825	−0.572051	−0.286026	−	0.958222i	$-0.592334\pi$
$461$	−12.2825	−0.572051	−0.286026	+	0.958222i	$0.592334\pi$
$462$	0	0
$463$	−20.0073	−0.929818	−0.464909	−	0.885359i	$-0.653913\pi$
$463$	−20.0073	−0.929818	−0.464909	+	0.885359i	$0.653913\pi$
$464$	0	0
$465$	−14.6557	−0.679643
$466$	0	0
$467$	−7.24777	−0.335387	−0.167693	−	0.985839i	$-0.553632\pi$
$467$	−7.24777	−0.335387	−0.167693	+	0.985839i	$0.553632\pi$
$468$	0	0
$469$	4.68426	0.216299
$470$	0	0
$471$	23.2595	1.07174
$472$	0	0
$473$	46.7752	2.15072
$474$	0	0
$475$	−5.58264	−0.256149
$476$	0	0
$477$	−2.34459	−0.107351
$478$	0	0
$479$	0.0521512	0.00238285	0.00119143	−	0.999999i	$-0.499621\pi$
$479$	0.0521512	0.00238285	0.00119143	+	0.999999i	$0.499621\pi$
$480$	0	0
$481$	−4.22375	−0.192587
$482$	0	0
$483$	−1.28770	−0.0585925
$484$	0	0
$485$	41.8982	1.90250
$486$	0	0
$487$	−6.29237	−0.285135	−0.142567	−	0.989785i	$-0.545536\pi$
$487$	−6.29237	−0.285135	−0.142567	+	0.989785i	$0.545536\pi$
$488$	0	0
$489$	21.8496	0.988073
$490$	0	0
$491$	−27.3974	−1.23643	−0.618214	−	0.786010i	$-0.712144\pi$
$491$	−27.3974	−1.23643	−0.618214	+	0.786010i	$0.712144\pi$
$492$	0	0
$493$	−17.8479	−0.803828
$494$	0	0
$495$	14.2242	0.639329
$496$	0	0
$497$	−2.68211	−0.120309
$498$	0	0
$499$	32.7237	1.46492	0.732458	−	0.680813i	$-0.238373\pi$
$499$	32.7237	1.46492	0.732458	+	0.680813i	$0.238373\pi$
$500$	0	0
$501$	−1.00000	−0.0446767
$502$	0	0
$503$	−1.12421	−0.0501261	−0.0250630	−	0.999686i	$-0.507979\pi$
$503$	−1.12421	−0.0501261	−0.0250630	+	0.999686i	$0.507979\pi$
$504$	0	0
$505$	19.1104	0.850402
$506$	0	0
$507$	−10.5066	−0.466616
$508$	0	0
$509$	−4.83949	−0.214506	−0.107253	−	0.994232i	$-0.534206\pi$
$509$	−4.83949	−0.214506	−0.107253	+	0.994232i	$0.534206\pi$
$510$	0	0
$511$	1.13654	0.0502778
$512$	0	0
$513$	−3.78855	−0.167269
$514$	0	0
$515$	3.80944	0.167864
$516$	0	0
$517$	26.7198	1.17513
$518$	0	0
$519$	10.0266	0.440120
$520$	0	0
$521$	−36.5789	−1.60255	−0.801275	−	0.598296i	$-0.795844\pi$
$521$	−36.5789	−1.60255	−0.801275	+	0.598296i	$0.795844\pi$
$522$	0	0
$523$	23.8955	1.04488	0.522439	−	0.852677i	$-0.325022\pi$
$523$	23.8955	1.04488	0.522439	+	0.852677i	$0.325022\pi$
$524$	0	0
$525$	−0.774014	−0.0337807
$526$	0	0
$527$	24.2466	1.05620
$528$	0	0
$529$	−16.9901	−0.738701
$530$	0	0
$531$	5.25946	0.228241
$532$	0	0
$533$	17.0018	0.736429
$534$	0	0
$535$	−5.83558	−0.252294
$536$	0	0
$537$	−0.201597	−0.00869956
$538$	0	0
$539$	37.5915	1.61918
$540$	0	0
$541$	−31.0286	−1.33402	−0.667012	−	0.745047i	$-0.732427\pi$
$541$	−31.0286	−1.33402	−0.667012	+	0.745047i	$0.732427\pi$
$542$	0	0
$543$	11.7308	0.503418
$544$	0	0
$545$	−33.9132	−1.45268
$546$	0	0
$547$	−3.82578	−0.163578	−0.0817892	−	0.996650i	$-0.526063\pi$
$547$	−3.82578	−0.163578	−0.0817892	+	0.996650i	$0.526063\pi$
$548$	0	0
$549$	−5.02359	−0.214402
$550$	0	0
$551$	16.0637	0.684335
$552$	0	0
$553$	−1.32714	−0.0564358
$554$	0	0
$555$	6.80576	0.288889
$556$	0	0
$557$	24.6847	1.04592	0.522962	−	0.852356i	$-0.324827\pi$
$557$	24.6847	1.04592	0.522962	+	0.852356i	$0.324827\pi$
$558$	0	0
$559$	−13.2115	−0.558788
$560$	0	0
$561$	−23.5326	−0.993549
$562$	0	0
$563$	−17.9996	−0.758592	−0.379296	−	0.925275i	$-0.623834\pi$
$563$	−17.9996	−0.758592	−0.379296	+	0.925275i	$0.623834\pi$
$564$	0	0
$565$	−48.1812	−2.02700
$566$	0	0
$567$	−0.525270	−0.0220593
$568$	0	0
$569$	−21.9387	−0.919718	−0.459859	−	0.887992i	$-0.652100\pi$
$569$	−21.9387	−0.919718	−0.459859	+	0.887992i	$0.652100\pi$
$570$	0	0
$571$	19.4601	0.814378	0.407189	−	0.913344i	$-0.366509\pi$
$571$	19.4601	0.814378	0.407189	+	0.913344i	$0.366509\pi$
$572$	0	0
$573$	25.0393	1.04603
$574$	0	0
$575$	3.61243	0.150649
$576$	0	0
$577$	8.17908	0.340500	0.170250	−	0.985401i	$-0.445543\pi$
$577$	8.17908	0.340500	0.170250	+	0.985401i	$0.445543\pi$
$578$	0	0
$579$	−10.7710	−0.447628
$580$	0	0
$581$	−2.10907	−0.0874988
$582$	0	0
$583$	13.1076	0.542861
$584$	0	0
$585$	−4.01758	−0.166107
$586$	0	0
$587$	−7.61306	−0.314225	−0.157112	−	0.987581i	$-0.550218\pi$
$587$	−7.61306	−0.314225	−0.157112	+	0.987581i	$0.550218\pi$
$588$	0	0
$589$	−21.8227	−0.899190
$590$	0	0
$591$	11.1738	0.459629
$592$	0	0
$593$	−21.7438	−0.892912	−0.446456	−	0.894806i	$-0.647314\pi$
$593$	−21.7438	−0.892912	−0.446456	+	0.894806i	$0.647314\pi$
$594$	0	0
$595$	5.62560	0.230627
$596$	0	0
$597$	23.0987	0.945365
$598$	0	0
$599$	−39.3506	−1.60782	−0.803911	−	0.594750i	$-0.797251\pi$
$599$	−39.3506	−1.60782	−0.803911	+	0.594750i	$0.797251\pi$
$600$	0	0
$601$	18.4627	0.753109	0.376555	−	0.926394i	$-0.377109\pi$
$601$	18.4627	0.753109	0.376555	+	0.926394i	$0.377109\pi$
$602$	0	0
$603$	−8.91781	−0.363161
$604$	0	0
$605$	−51.5336	−2.09514
$606$	0	0
$607$	6.84061	0.277652	0.138826	−	0.990317i	$-0.455667\pi$
$607$	6.84061	0.277652	0.138826	+	0.990317i	$0.455667\pi$
$608$	0	0
$609$	2.22717	0.0902496
$610$	0	0
$611$	−7.54693	−0.305316
$612$	0	0
$613$	46.6604	1.88460	0.942298	−	0.334776i	$-0.108661\pi$
$613$	46.6604	1.88460	0.942298	+	0.334776i	$0.108661\pi$
$614$	0	0
$615$	−27.3951	−1.10468
$616$	0	0
$617$	19.4605	0.783452	0.391726	−	0.920082i	$-0.371878\pi$
$617$	19.4605	0.783452	0.391726	+	0.920082i	$0.371878\pi$
$618$	0	0
$619$	−22.6463	−0.910232	−0.455116	−	0.890432i	$-0.650402\pi$
$619$	−22.6463	−0.910232	−0.455116	+	0.890432i	$0.650402\pi$
$620$	0	0
$621$	2.45151	0.0983756
$622$	0	0
$623$	−8.52381	−0.341499
$624$	0	0
$625$	−30.1964	−1.20786
$626$	0	0
$627$	21.1801	0.845853
$628$	0	0
$629$	−11.2595	−0.448947
$630$	0	0
$631$	−41.4338	−1.64945	−0.824727	−	0.565531i	$-0.808671\pi$
$631$	−41.4338	−1.64945	−0.824727	+	0.565531i	$0.808671\pi$
$632$	0	0
$633$	8.65386	0.343960
$634$	0	0
$635$	−0.728391	−0.0289053
$636$	0	0
$637$	−10.6176	−0.420685
$638$	0	0
$639$	5.10616	0.201997
$640$	0	0
$641$	−5.85111	−0.231105	−0.115552	−	0.993301i	$-0.536864\pi$
$641$	−5.85111	−0.231105	−0.115552	+	0.993301i	$0.536864\pi$
$642$	0	0
$643$	−1.42749	−0.0562947	−0.0281473	−	0.999604i	$-0.508961\pi$
$643$	−1.42749	−0.0562947	−0.0281473	+	0.999604i	$0.508961\pi$
$644$	0	0
$645$	21.2878	0.838207
$646$	0	0
$647$	10.5140	0.413349	0.206675	−	0.978410i	$-0.433736\pi$
$647$	10.5140	0.413349	0.206675	+	0.978410i	$0.433736\pi$
$648$	0	0
$649$	−29.4033	−1.15418
$650$	0	0
$651$	−3.02565	−0.118584
$652$	0	0
$653$	44.1836	1.72904	0.864519	−	0.502601i	$-0.167623\pi$
$653$	44.1836	1.72904	0.864519	+	0.502601i	$0.167623\pi$
$654$	0	0
$655$	31.8039	1.24268
$656$	0	0
$657$	−2.16373	−0.0844153
$658$	0	0
$659$	9.39910	0.366137	0.183069	−	0.983100i	$-0.441397\pi$
$659$	9.39910	0.366137	0.183069	+	0.983100i	$0.441397\pi$
$660$	0	0
$661$	47.0872	1.83148	0.915739	−	0.401774i	$-0.131606\pi$
$661$	47.0872	1.83148	0.915739	+	0.401774i	$0.131606\pi$
$662$	0	0
$663$	6.64673	0.258138
$664$	0	0
$665$	−5.06322	−0.196343
$666$	0	0
$667$	−10.3945	−0.402478
$668$	0	0
$669$	−3.47630	−0.134402
$670$	0	0
$671$	28.0847	1.08420
$672$	0	0
$673$	−8.95242	−0.345091	−0.172545	−	0.985002i	$-0.555199\pi$
$673$	−8.95242	−0.345091	−0.172545	+	0.985002i	$0.555199\pi$
$674$	0	0
$675$	1.47356	0.0567172
$676$	0	0
$677$	−8.00580	−0.307688	−0.153844	−	0.988095i	$-0.549165\pi$
$677$	−8.00580	−0.307688	−0.153844	+	0.988095i	$0.549165\pi$
$678$	0	0
$679$	8.64980	0.331949
$680$	0	0
$681$	−4.99391	−0.191367
$682$	0	0
$683$	−3.64312	−0.139400	−0.0697001	−	0.997568i	$-0.522204\pi$
$683$	−3.64312	−0.139400	−0.0697001	+	0.997568i	$0.522204\pi$
$684$	0	0
$685$	−31.6788	−1.21038
$686$	0	0
$687$	−0.724364	−0.0276362
$688$	0	0
$689$	−3.70220	−0.141043
$690$	0	0
$691$	−18.2229	−0.693230	−0.346615	−	0.938008i	$-0.612669\pi$
$691$	−18.2229	−0.693230	−0.346615	+	0.938008i	$0.612669\pi$
$692$	0	0
$693$	2.93655	0.111550
$694$	0	0
$695$	35.1964	1.33507
$696$	0	0
$697$	45.3227	1.71672
$698$	0	0
$699$	3.07905	0.116461
$700$	0	0
$701$	−18.3239	−0.692085	−0.346043	−	0.938219i	$-0.612475\pi$
$701$	−18.3239	−0.692085	−0.346043	+	0.938219i	$0.612475\pi$
$702$	0	0
$703$	10.1339	0.382209
$704$	0	0
$705$	12.1604	0.457988
$706$	0	0
$707$	3.94531	0.148379
$708$	0	0
$709$	33.1399	1.24460	0.622298	−	0.782781i	$-0.286199\pi$
$709$	33.1399	1.24460	0.622298	+	0.782781i	$0.286199\pi$
$710$	0	0
$711$	2.52659	0.0947545
$712$	0	0
$713$	14.1211	0.528840
$714$	0	0
$715$	22.4605	0.839977
$716$	0	0
$717$	−23.7125	−0.885558
$718$	0	0
$719$	−10.6679	−0.397847	−0.198924	−	0.980015i	$-0.563745\pi$
$719$	−10.6679	−0.397847	−0.198924	+	0.980015i	$0.563745\pi$
$720$	0	0
$721$	0.786451	0.0292890
$722$	0	0
$723$	14.1304	0.525515
$724$	0	0
$725$	−6.24796	−0.232043
$726$	0	0
$727$	16.3733	0.607254	0.303627	−	0.952791i	$-0.401802\pi$
$727$	16.3733	0.607254	0.303627	+	0.952791i	$0.401802\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−35.2188	−1.30261
$732$	0	0
$733$	10.1931	0.376492	0.188246	−	0.982122i	$-0.439720\pi$
$733$	10.1931	0.376492	0.188246	+	0.982122i	$0.439720\pi$
$734$	0	0
$735$	17.1082	0.631047
$736$	0	0
$737$	49.8556	1.83645
$738$	0	0
$739$	−14.0639	−0.517349	−0.258675	−	0.965964i	$-0.583286\pi$
$739$	−14.0639	−0.517349	−0.258675	+	0.965964i	$0.583286\pi$
$740$	0	0
$741$	−5.98227	−0.219764
$742$	0	0
$743$	8.93010	0.327614	0.163807	−	0.986492i	$-0.447623\pi$
$743$	8.93010	0.327614	0.163807	+	0.986492i	$0.447623\pi$
$744$	0	0
$745$	−45.6130	−1.67113
$746$	0	0
$747$	4.01520	0.146909
$748$	0	0
$749$	−1.20475	−0.0440204
$750$	0	0
$751$	16.7755	0.612146	0.306073	−	0.952008i	$-0.400985\pi$
$751$	16.7755	0.612146	0.306073	+	0.952008i	$0.400985\pi$
$752$	0	0
$753$	−0.523099	−0.0190628
$754$	0	0
$755$	−13.4146	−0.488208
$756$	0	0
$757$	−28.3838	−1.03163	−0.515814	−	0.856701i	$-0.672510\pi$
$757$	−28.3838	−1.03163	−0.515814	+	0.856701i	$0.672510\pi$
$758$	0	0
$759$	−13.7053	−0.497471
$760$	0	0
$761$	41.5928	1.50774	0.753869	−	0.657025i	$-0.228185\pi$
$761$	41.5928	1.50774	0.753869	+	0.657025i	$0.228185\pi$
$762$	0	0
$763$	−7.00133	−0.253465
$764$	0	0
$765$	−10.7099	−0.387218
$766$	0	0
$767$	8.30490	0.299873
$768$	0	0
$769$	−13.5540	−0.488770	−0.244385	−	0.969678i	$-0.578586\pi$
$769$	−13.5540	−0.488770	−0.244385	+	0.969678i	$0.578586\pi$
$770$	0	0
$771$	19.5504	0.704091
$772$	0	0
$773$	−36.6981	−1.31994	−0.659970	−	0.751292i	$-0.729431\pi$
$773$	−36.6981	−1.31994	−0.659970	+	0.751292i	$0.729431\pi$
$774$	0	0
$775$	8.48794	0.304896
$776$	0	0
$777$	1.40504	0.0504054
$778$	0	0
$779$	−40.7919	−1.46152
$780$	0	0
$781$	−28.5463	−1.02147
$782$	0	0
$783$	−4.24006	−0.151527
$784$	0	0
$785$	−59.1795	−2.11221
$786$	0	0
$787$	−20.9838	−0.747993	−0.373996	−	0.927430i	$-0.622013\pi$
$787$	−20.9838	−0.747993	−0.373996	+	0.927430i	$0.622013\pi$
$788$	0	0
$789$	2.94567	0.104869
$790$	0	0
$791$	−9.94692	−0.353672
$792$	0	0
$793$	−7.93245	−0.281690
$794$	0	0
$795$	5.96539	0.211570
$796$	0	0
$797$	8.30024	0.294010	0.147005	−	0.989136i	$-0.453037\pi$
$797$	8.30024	0.294010	0.147005	+	0.989136i	$0.453037\pi$
$798$	0	0
$799$	−20.1183	−0.711736
$800$	0	0
$801$	16.2275	0.573370
$802$	0	0
$803$	12.0965	0.426876
$804$	0	0
$805$	3.27632	0.115475
$806$	0	0
$807$	9.57622	0.337099
$808$	0	0
$809$	−40.2664	−1.41569	−0.707845	−	0.706368i	$-0.750333\pi$
$809$	−40.2664	−1.41569	−0.707845	+	0.706368i	$0.750333\pi$
$810$	0	0
$811$	11.0881	0.389357	0.194679	−	0.980867i	$-0.437634\pi$
$811$	11.0881	0.389357	0.194679	+	0.980867i	$0.437634\pi$
$812$	0	0
$813$	−24.1059	−0.845430
$814$	0	0
$815$	−55.5923	−1.94731
$816$	0	0
$817$	31.6981	1.10897
$818$	0	0
$819$	−0.829422	−0.0289824
$820$	0	0
$821$	40.0645	1.39826	0.699130	−	0.714995i	$-0.253571\pi$
$821$	40.0645	1.39826	0.699130	+	0.714995i	$0.253571\pi$
$822$	0	0
$823$	30.7172	1.07073	0.535366	−	0.844620i	$-0.320174\pi$
$823$	30.7172	1.07073	0.535366	+	0.844620i	$0.320174\pi$
$824$	0	0
$825$	−8.23800	−0.286810
$826$	0	0
$827$	26.1112	0.907977	0.453988	−	0.891008i	$-0.350001\pi$
$827$	26.1112	0.907977	0.453988	+	0.891008i	$0.350001\pi$
$828$	0	0
$829$	−3.46592	−0.120376	−0.0601882	−	0.998187i	$-0.519170\pi$
$829$	−3.46592	−0.120376	−0.0601882	+	0.998187i	$0.519170\pi$
$830$	0	0
$831$	−2.19302	−0.0760749
$832$	0	0
$833$	−28.3041	−0.980677
$834$	0	0
$835$	2.54432	0.0880498
$836$	0	0
$837$	5.76018	0.199101
$838$	0	0
$839$	−24.7905	−0.855864	−0.427932	−	0.903811i	$-0.640758\pi$
$839$	−24.7905	−0.855864	−0.427932	+	0.903811i	$0.640758\pi$
$840$	0	0
$841$	−11.0219	−0.380066
$842$	0	0
$843$	6.78076	0.233542
$844$	0	0
$845$	26.7322	0.919616
$846$	0	0
$847$	−10.6390	−0.365561
$848$	0	0
$849$	4.02725	0.138215
$850$	0	0
$851$	−6.55750	−0.224788
$852$	0	0
$853$	−35.4429	−1.21354	−0.606772	−	0.794876i	$-0.707536\pi$
$853$	−35.4429	−1.21354	−0.606772	+	0.794876i	$0.707536\pi$
$854$	0	0
$855$	9.63928	0.329656
$856$	0	0
$857$	−19.6459	−0.671092	−0.335546	−	0.942024i	$-0.608921\pi$
$857$	−19.6459	−0.671092	−0.335546	+	0.942024i	$0.608921\pi$
$858$	0	0
$859$	−4.37538	−0.149286	−0.0746430	−	0.997210i	$-0.523782\pi$
$859$	−4.37538	−0.149286	−0.0746430	+	0.997210i	$0.523782\pi$
$860$	0	0
$861$	−5.65566	−0.192744
$862$	0	0
$863$	10.7669	0.366510	0.183255	−	0.983065i	$-0.441337\pi$
$863$	10.7669	0.366510	0.183255	+	0.983065i	$0.441337\pi$
$864$	0	0
$865$	−25.5109	−0.867397
$866$	0	0
$867$	0.718627	0.0244059
$868$	0	0
$869$	−14.1251	−0.479160
$870$	0	0
$871$	−14.0816	−0.477136
$872$	0	0
$873$	−16.4673	−0.557335
$874$	0	0
$875$	−4.71293	−0.159326
$876$	0	0
$877$	−36.0855	−1.21852	−0.609261	−	0.792970i	$-0.708534\pi$
$877$	−36.0855	−1.21852	−0.609261	+	0.792970i	$0.708534\pi$
$878$	0	0
$879$	22.8461	0.770580
$880$	0	0
$881$	−10.9935	−0.370379	−0.185190	−	0.982703i	$-0.559290\pi$
$881$	−10.9935	−0.370379	−0.185190	+	0.982703i	$0.559290\pi$
$882$	0	0
$883$	53.8415	1.81191	0.905956	−	0.423372i	$-0.139154\pi$
$883$	53.8415	1.81191	0.905956	+	0.423372i	$0.139154\pi$
$884$	0	0
$885$	−13.3817	−0.449822
$886$	0	0
$887$	−50.4057	−1.69246	−0.846229	−	0.532820i	$-0.821132\pi$
$887$	−50.4057	−1.69246	−0.846229	+	0.532820i	$0.821132\pi$
$888$	0	0
$889$	−0.150375	−0.00504342
$890$	0	0
$891$	−5.59056	−0.187291
$892$	0	0
$893$	18.1072	0.605933
$894$	0	0
$895$	0.512927	0.0171453
$896$	0	0
$897$	3.87103	0.129250
$898$	0	0
$899$	−24.4235	−0.814569
$900$	0	0
$901$	−9.86921	−0.328791
$902$	0	0
$903$	4.39483	0.146251
$904$	0	0
$905$	−29.8470	−0.992147
$906$	0	0
$907$	52.2807	1.73595	0.867976	−	0.496607i	$-0.165421\pi$
$907$	52.2807	1.73595	0.867976	+	0.496607i	$0.165421\pi$
$908$	0	0
$909$	−7.51102	−0.249125
$910$	0	0
$911$	−46.9594	−1.55584	−0.777918	−	0.628366i	$-0.783724\pi$
$911$	−46.9594	−1.55584	−0.777918	+	0.628366i	$0.783724\pi$
$912$	0	0
$913$	−22.4473	−0.742896
$914$	0	0
$915$	12.7816	0.422547
$916$	0	0
$917$	6.56586	0.216824
$918$	0	0
$919$	40.3643	1.33150	0.665748	−	0.746177i	$-0.268113\pi$
$919$	40.3643	1.33150	0.665748	+	0.746177i	$0.268113\pi$
$920$	0	0
$921$	−2.74814	−0.0905543
$922$	0	0
$923$	8.06283	0.265391
$924$	0	0
$925$	−3.94159	−0.129599
$926$	0	0
$927$	−1.49723	−0.0491756
$928$	0	0
$929$	24.0090	0.787710	0.393855	−	0.919173i	$-0.371141\pi$
$929$	24.0090	0.787710	0.393855	+	0.919173i	$0.371141\pi$
$930$	0	0
$931$	25.4746	0.834895
$932$	0	0
$933$	3.22030	0.105428
$934$	0	0
$935$	59.8745	1.95811
$936$	0	0
$937$	20.1221	0.657359	0.328680	−	0.944442i	$-0.393396\pi$
$937$	20.1221	0.657359	0.328680	+	0.944442i	$0.393396\pi$
$938$	0	0
$939$	−30.2753	−0.987996
$940$	0	0
$941$	−26.2819	−0.856766	−0.428383	−	0.903597i	$-0.640917\pi$
$941$	−26.2819	−0.856766	−0.428383	+	0.903597i	$0.640917\pi$
$942$	0	0
$943$	26.3958	0.859564
$944$	0	0
$945$	1.33645	0.0434748
$946$	0	0
$947$	6.14635	0.199730	0.0998648	−	0.995001i	$-0.468159\pi$
$947$	6.14635	0.199730	0.0998648	+	0.995001i	$0.468159\pi$
$948$	0	0
$949$	−3.41662	−0.110908
$950$	0	0
$951$	13.1079	0.425054
$952$	0	0
$953$	−31.2709	−1.01296	−0.506481	−	0.862251i	$-0.669054\pi$
$953$	−31.2709	−1.01296	−0.506481	+	0.862251i	$0.669054\pi$
$954$	0	0
$955$	−63.7080	−2.06154
$956$	0	0
$957$	23.7043	0.766251
$958$	0	0
$959$	−6.54003	−0.211189
$960$	0	0
$961$	2.17965	0.0703114
$962$	0	0
$963$	2.29357	0.0739094
$964$	0	0
$965$	27.4049	0.882195
$966$	0	0
$967$	11.5240	0.370585	0.185293	−	0.982683i	$-0.440677\pi$
$967$	11.5240	0.370585	0.185293	+	0.982683i	$0.440677\pi$
$968$	0	0
$969$	−15.9473	−0.512302
$970$	0	0
$971$	5.19367	0.166673	0.0833365	−	0.996521i	$-0.473442\pi$
$971$	5.19367	0.166673	0.0833365	+	0.996521i	$0.473442\pi$
$972$	0	0
$973$	7.26622	0.232944
$974$	0	0
$975$	2.32680	0.0745173
$976$	0	0
$977$	23.7262	0.759067	0.379534	−	0.925178i	$-0.376084\pi$
$977$	23.7262	0.759067	0.379534	+	0.925178i	$0.376084\pi$
$978$	0	0
$979$	−90.7208	−2.89945
$980$	0	0
$981$	13.3290	0.425562
$982$	0	0
$983$	15.9426	0.508490	0.254245	−	0.967140i	$-0.418173\pi$
$983$	15.9426	0.508490	0.254245	+	0.967140i	$0.418173\pi$
$984$	0	0
$985$	−28.4297	−0.905847
$986$	0	0
$987$	2.51050	0.0799100
$988$	0	0
$989$	−20.5113	−0.652221
$990$	0	0
$991$	40.7291	1.29380	0.646902	−	0.762573i	$-0.276064\pi$
$991$	40.7291	1.29380	0.646902	+	0.762573i	$0.276064\pi$
$992$	0	0
$993$	22.4120	0.711223
$994$	0	0
$995$	−58.7703	−1.86314
$996$	0	0
$997$	45.1173	1.42888	0.714440	−	0.699697i	$-0.246681\pi$
$997$	45.1173	1.42888	0.714440	+	0.699697i	$0.246681\pi$
$998$	0	0
$999$	−2.67489	−0.0846297

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8016.2.a.be.1.1		11
4.3	odd	2		4008.2.a.k.1.1	✓	11

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4008.2.a.k.1.1	✓	11	4.3	odd	2
8016.2.a.be.1.1		11	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:	\( N \)	\(=\)	\( 8016 = 2^{4} \cdot 3 \cdot 167 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8016.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -2.54432 q^{5} -0.525270 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} -2.54432 q^{5} -0.525270 q^{7} +1.00000 q^{9} -5.59056 q^{11} +1.57904 q^{13} -2.54432 q^{15} +4.20935 q^{17} -3.78855 q^{19} -0.525270 q^{21} +2.45151 q^{23} +1.47356 q^{25} +1.00000 q^{27} -4.24006 q^{29} +5.76018 q^{31} -5.59056 q^{33} +1.33645 q^{35} -2.67489 q^{37} +1.57904 q^{39} +10.7672 q^{41} -8.36680 q^{43} -2.54432 q^{45} -4.77944 q^{47} -6.72409 q^{49} +4.20935 q^{51} -2.34459 q^{53} +14.2242 q^{55} -3.78855 q^{57} +5.25946 q^{59} -5.02359 q^{61} -0.525270 q^{63} -4.01758 q^{65} -8.91781 q^{67} +2.45151 q^{69} +5.10616 q^{71} -2.16373 q^{73} +1.47356 q^{75} +2.93655 q^{77} +2.52659 q^{79} +1.00000 q^{81} +4.01520 q^{83} -10.7099 q^{85} -4.24006 q^{87} +16.2275 q^{89} -0.829422 q^{91} +5.76018 q^{93} +9.63928 q^{95} -16.4673 q^{97} -5.59056 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 11 q + 11 q^{3} + 10 q^{5} + q^{7} + 11 q^{9}+O(q^{10}) \) 11 * q + 11 * q^3 + 10 * q^5 + q^7 + 11 * q^9	\( 11 q + 11 q^{3} + 10 q^{5} + q^{7} + 11 q^{9} + q^{11} + 10 q^{13} + 10 q^{15} + 17 q^{17} - 2 q^{19} + q^{21} + 3 q^{23} + 21 q^{25} + 11 q^{27} + 17 q^{29} + 15 q^{31} + q^{33} - 11 q^{35} + 4 q^{37} + 10 q^{39} + 16 q^{41} - 10 q^{43} + 10 q^{45} + 16 q^{47} + 22 q^{49} + 17 q^{51} + 42 q^{53} + 5 q^{55} - 2 q^{57} + 2 q^{59} + 12 q^{61} + q^{63} + 10 q^{65} + q^{67} + 3 q^{69} + 9 q^{71} + 24 q^{73} + 21 q^{75} + 22 q^{77} + 30 q^{79} + 11 q^{81} - 16 q^{83} + 25 q^{85} + 17 q^{87} + 37 q^{89} - q^{91} + 15 q^{93} - 5 q^{95} + 4 q^{97} + q^{99}+O(q^{100}) \) 11 * q + 11 * q^3 + 10 * q^5 + q^7 + 11 * q^9 + q^11 + 10 * q^13 + 10 * q^15 + 17 * q^17 - 2 * q^19 + q^21 + 3 * q^23 + 21 * q^25 + 11 * q^27 + 17 * q^29 + 15 * q^31 + q^33 - 11 * q^35 + 4 * q^37 + 10 * q^39 + 16 * q^41 - 10 * q^43 + 10 * q^45 + 16 * q^47 + 22 * q^49 + 17 * q^51 + 42 * q^53 + 5 * q^55 - 2 * q^57 + 2 * q^59 + 12 * q^61 + q^63 + 10 * q^65 + q^67 + 3 * q^69 + 9 * q^71 + 24 * q^73 + 21 * q^75 + 22 * q^77 + 30 * q^79 + 11 * q^81 - 16 * q^83 + 25 * q^85 + 17 * q^87 + 37 * q^89 - q^91 + 15 * q^93 - 5 * q^95 + 4 * q^97 + q^99

Introduction

L-functions

Modular forms

Varieties

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Database

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