LMFDB - Embedded newform 3016.2.a.g.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3016,2,Mod(1,3016)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3016, 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("3016.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3016 = 2^{3} \cdot 13 \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3016.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:	$24.0828812496$
Analytic rank:	$0$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - 2x^{9} - 21x^{8} + 40x^{7} + 138x^{6} - 243x^{5} - 318x^{4} + 448x^{3} + 312x^{2} - 240x - 128 $ x^10 - 2x^9 - 21x^8 + 40x^7 + 138x^6 - 243x^5 - 318x^4 + 448x^3 + 312x^2 - 240*x - 128
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-1.30047$ of defining polynomial
Character	$\chi$	$=$	3016.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.30047 q^{3} -2.99209 q^{5} +3.88835 q^{7} -1.30878 q^{9} +O(q^{10})$	$q-1.30047 q^{3} -2.99209 q^{5} +3.88835 q^{7} -1.30878 q^{9} +4.33056 q^{11} +1.00000 q^{13} +3.89112 q^{15} -2.80070 q^{17} -3.12346 q^{19} -5.05668 q^{21} +8.23426 q^{23} +3.95262 q^{25} +5.60343 q^{27} -1.00000 q^{29} -6.16033 q^{31} -5.63176 q^{33} -11.6343 q^{35} -5.85123 q^{37} -1.30047 q^{39} -0.970544 q^{41} +1.37440 q^{43} +3.91600 q^{45} +8.52811 q^{47} +8.11929 q^{49} +3.64223 q^{51} -5.70400 q^{53} -12.9574 q^{55} +4.06197 q^{57} +8.14158 q^{59} -4.13734 q^{61} -5.08901 q^{63} -2.99209 q^{65} -13.8596 q^{67} -10.7084 q^{69} -5.50177 q^{71} +10.1918 q^{73} -5.14026 q^{75} +16.8388 q^{77} +9.36127 q^{79} -3.36074 q^{81} +6.01882 q^{83} +8.37997 q^{85} +1.30047 q^{87} +4.27746 q^{89} +3.88835 q^{91} +8.01131 q^{93} +9.34569 q^{95} +12.7370 q^{97} -5.66776 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q + 2 q^{3} + 5 q^{5} - 2 q^{7} + 16 q^{9}+O(q^{10}) $ 10 * q + 2 * q^3 + 5 * q^5 - 2 * q^7 + 16 * q^9	$ 10 q + 2 q^{3} + 5 q^{5} - 2 q^{7} + 16 q^{9} + 4 q^{11} + 10 q^{13} + 8 q^{15} + 10 q^{17} - q^{19} + q^{21} + 23 q^{23} + 25 q^{25} + 2 q^{27} - 10 q^{29} - 13 q^{31} + 15 q^{33} - 12 q^{35} - 7 q^{37} + 2 q^{39} + 16 q^{41} - 12 q^{43} + 55 q^{45} + 11 q^{47} - 25 q^{51} + 11 q^{53} + 22 q^{55} - 6 q^{57} - 11 q^{59} + 34 q^{61} + 37 q^{63} + 5 q^{65} - 23 q^{67} + 2 q^{69} - 4 q^{71} + 39 q^{73} + 11 q^{75} + 32 q^{77} + 5 q^{79} + 38 q^{81} + 6 q^{83} + 45 q^{85} - 2 q^{87} - 24 q^{89} - 2 q^{91} + 13 q^{93} + 33 q^{95} + 19 q^{97}+O(q^{100}) $ 10 * q + 2 * q^3 + 5 * q^5 - 2 * q^7 + 16 * q^9 + 4 * q^11 + 10 * q^13 + 8 * q^15 + 10 * q^17 - q^19 + q^21 + 23 * q^23 + 25 * q^25 + 2 * q^27 - 10 * q^29 - 13 * q^31 + 15 * q^33 - 12 * q^35 - 7 * q^37 + 2 * q^39 + 16 * q^41 - 12 * q^43 + 55 * q^45 + 11 * q^47 - 25 * q^51 + 11 * q^53 + 22 * q^55 - 6 * q^57 - 11 * q^59 + 34 * q^61 + 37 * q^63 + 5 * q^65 - 23 * q^67 + 2 * q^69 - 4 * q^71 + 39 * q^73 + 11 * q^75 + 32 * q^77 + 5 * q^79 + 38 * q^81 + 6 * q^83 + 45 * q^85 - 2 * q^87 - 24 * q^89 - 2 * q^91 + 13 * q^93 + 33 * q^95 + 19 * q^97

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
$3$	−1.30047	−0.750826	−0.375413	−	0.926858i	$-0.622499\pi$
$3$	−1.30047	−0.750826	−0.375413	+	0.926858i	$0.622499\pi$
$4$	0	0
$5$	−2.99209	−1.33810	−0.669052	−	0.743215i	$-0.733300\pi$
$5$	−2.99209	−1.33810	−0.669052	+	0.743215i	$0.733300\pi$
$6$	0	0
$7$	3.88835	1.46966	0.734830	−	0.678252i	$-0.237262\pi$
$7$	3.88835	1.46966	0.734830	+	0.678252i	$0.237262\pi$
$8$	0	0
$9$	−1.30878	−0.436261
$10$	0	0
$11$	4.33056	1.30571	0.652857	−	0.757481i	$-0.273570\pi$
$11$	4.33056	1.30571	0.652857	+	0.757481i	$0.273570\pi$
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	0	0
$15$	3.89112	1.00468
$16$	0	0
$17$	−2.80070	−0.679271	−0.339635	−	0.940557i	$-0.610304\pi$
$17$	−2.80070	−0.679271	−0.339635	+	0.940557i	$0.610304\pi$
$18$	0	0
$19$	−3.12346	−0.716572	−0.358286	−	0.933612i	$-0.616639\pi$
$19$	−3.12346	−0.716572	−0.358286	+	0.933612i	$0.616639\pi$
$20$	0	0
$21$	−5.05668	−1.10346
$22$	0	0
$23$	8.23426	1.71696	0.858481	−	0.512845i	$-0.171408\pi$
$23$	8.23426	1.71696	0.858481	+	0.512845i	$0.171408\pi$
$24$	0	0
$25$	3.95262	0.790524
$26$	0	0
$27$	5.60343	1.07838
$28$	0	0
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	0	0
$31$	−6.16033	−1.10643	−0.553214	−	0.833039i	$-0.686599\pi$
$31$	−6.16033	−1.10643	−0.553214	+	0.833039i	$0.686599\pi$
$32$	0	0
$33$	−5.63176	−0.980363
$34$	0	0
$35$	−11.6343	−1.96656
$36$	0	0
$37$	−5.85123	−0.961936	−0.480968	−	0.876738i	$-0.659715\pi$
$37$	−5.85123	−0.961936	−0.480968	+	0.876738i	$0.659715\pi$
$38$	0	0
$39$	−1.30047	−0.208242
$40$	0	0
$41$	−0.970544	−0.151573	−0.0757867	−	0.997124i	$-0.524147\pi$
$41$	−0.970544	−0.151573	−0.0757867	+	0.997124i	$0.524147\pi$
$42$	0	0
$43$	1.37440	0.209594	0.104797	−	0.994494i	$-0.466581\pi$
$43$	1.37440	0.209594	0.104797	+	0.994494i	$0.466581\pi$
$44$	0	0
$45$	3.91600	0.583762
$46$	0	0
$47$	8.52811	1.24395	0.621976	−	0.783036i	$-0.286330\pi$
$47$	8.52811	1.24395	0.621976	+	0.783036i	$0.286330\pi$
$48$	0	0
$49$	8.11929	1.15990
$50$	0	0
$51$	3.64223	0.510014
$52$	0	0
$53$	−5.70400	−0.783505	−0.391752	−	0.920071i	$-0.628131\pi$
$53$	−5.70400	−0.783505	−0.391752	+	0.920071i	$0.628131\pi$
$54$	0	0
$55$	−12.9574	−1.74718
$56$	0	0
$57$	4.06197	0.538020
$58$	0	0
$59$	8.14158	1.05994	0.529972	−	0.848015i	$-0.322202\pi$
$59$	8.14158	1.05994	0.529972	+	0.848015i	$0.322202\pi$
$60$	0	0
$61$	−4.13734	−0.529732	−0.264866	−	0.964285i	$-0.585328\pi$
$61$	−4.13734	−0.529732	−0.264866	+	0.964285i	$0.585328\pi$
$62$	0	0
$63$	−5.08901	−0.641155
$64$	0	0
$65$	−2.99209	−0.371123
$66$	0	0
$67$	−13.8596	−1.69322	−0.846608	−	0.532216i	$-0.821359\pi$
$67$	−13.8596	−1.69322	−0.846608	+	0.532216i	$0.821359\pi$
$68$	0	0
$69$	−10.7084	−1.28914
$70$	0	0
$71$	−5.50177	−0.652940	−0.326470	−	0.945208i	$-0.605859\pi$
$71$	−5.50177	−0.652940	−0.326470	+	0.945208i	$0.605859\pi$
$72$	0	0
$73$	10.1918	1.19286	0.596429	−	0.802666i	$-0.296586\pi$
$73$	10.1918	1.19286	0.596429	+	0.802666i	$0.296586\pi$
$74$	0	0
$75$	−5.14026	−0.593546
$76$	0	0
$77$	16.8388	1.91895
$78$	0	0
$79$	9.36127	1.05322	0.526612	−	0.850106i	$-0.323462\pi$
$79$	9.36127	1.05322	0.526612	+	0.850106i	$0.323462\pi$
$80$	0	0
$81$	−3.36074	−0.373416
$82$	0	0
$83$	6.01882	0.660651	0.330326	−	0.943867i	$-0.392841\pi$
$83$	6.01882	0.660651	0.330326	+	0.943867i	$0.392841\pi$
$84$	0	0
$85$	8.37997	0.908935
$86$	0	0
$87$	1.30047	0.139425
$88$	0	0
$89$	4.27746	0.453410	0.226705	−	0.973963i	$-0.427205\pi$
$89$	4.27746	0.453410	0.226705	+	0.973963i	$0.427205\pi$
$90$	0	0
$91$	3.88835	0.407610
$92$	0	0
$93$	8.01131	0.830734
$94$	0	0
$95$	9.34569	0.958848
$96$	0	0
$97$	12.7370	1.29325	0.646625	−	0.762808i	$-0.276180\pi$
$97$	12.7370	1.29325	0.646625	+	0.762808i	$0.276180\pi$
$98$	0	0
$99$	−5.66776	−0.569632
$100$	0	0
$101$	−10.8390	−1.07852	−0.539260	−	0.842140i	$-0.681296\pi$
$101$	−10.8390	−1.07852	−0.539260	+	0.842140i	$0.681296\pi$
$102$	0	0
$103$	−4.19821	−0.413662	−0.206831	−	0.978377i	$-0.566315\pi$
$103$	−4.19821	−0.413662	−0.206831	+	0.978377i	$0.566315\pi$
$104$	0	0
$105$	15.1301	1.47654
$106$	0	0
$107$	2.30995	0.223311	0.111656	−	0.993747i	$-0.464385\pi$
$107$	2.30995	0.223311	0.111656	+	0.993747i	$0.464385\pi$
$108$	0	0
$109$	5.21859	0.499851	0.249925	−	0.968265i	$-0.419594\pi$
$109$	5.21859	0.499851	0.249925	+	0.968265i	$0.419594\pi$
$110$	0	0
$111$	7.60933	0.722246
$112$	0	0
$113$	6.11327	0.575088	0.287544	−	0.957767i	$-0.407161\pi$
$113$	6.11327	0.575088	0.287544	+	0.957767i	$0.407161\pi$
$114$	0	0
$115$	−24.6377	−2.29748
$116$	0	0
$117$	−1.30878	−0.120997
$118$	0	0
$119$	−10.8901	−0.998296
$120$	0	0
$121$	7.75377	0.704888
$122$	0	0
$123$	1.26216	0.113805
$124$	0	0
$125$	3.13386	0.280301
$126$	0	0
$127$	20.0522	1.77934	0.889672	−	0.456600i	$-0.150933\pi$
$127$	20.0522	1.77934	0.889672	+	0.456600i	$0.150933\pi$
$128$	0	0
$129$	−1.78737	−0.157369
$130$	0	0
$131$	11.6118	1.01452	0.507262	−	0.861792i	$-0.330657\pi$
$131$	11.6118	1.01452	0.507262	+	0.861792i	$0.330657\pi$
$132$	0	0
$133$	−12.1451	−1.05312
$134$	0	0
$135$	−16.7660	−1.44299
$136$	0	0
$137$	9.70371	0.829044	0.414522	−	0.910039i	$-0.363949\pi$
$137$	9.70371	0.829044	0.414522	+	0.910039i	$0.363949\pi$
$138$	0	0
$139$	−16.7298	−1.41901	−0.709503	−	0.704702i	$-0.751081\pi$
$139$	−16.7298	−1.41901	−0.709503	+	0.704702i	$0.751081\pi$
$140$	0	0
$141$	−11.0905	−0.933992
$142$	0	0
$143$	4.33056	0.362140
$144$	0	0
$145$	2.99209	0.248480
$146$	0	0
$147$	−10.5589	−0.870882
$148$	0	0
$149$	18.0019	1.47477	0.737387	−	0.675471i	$-0.236060\pi$
$149$	18.0019	1.47477	0.737387	+	0.675471i	$0.236060\pi$
$150$	0	0
$151$	−6.80650	−0.553905	−0.276953	−	0.960884i	$-0.589325\pi$
$151$	−6.80650	−0.553905	−0.276953	+	0.960884i	$0.589325\pi$
$152$	0	0
$153$	3.66551	0.296339
$154$	0	0
$155$	18.4323	1.48052
$156$	0	0
$157$	5.56587	0.444205	0.222102	−	0.975023i	$-0.428708\pi$
$157$	5.56587	0.444205	0.222102	+	0.975023i	$0.428708\pi$
$158$	0	0
$159$	7.41787	0.588276
$160$	0	0
$161$	32.0177	2.52335
$162$	0	0
$163$	14.6552	1.14788	0.573942	−	0.818896i	$-0.305413\pi$
$163$	14.6552	1.14788	0.573942	+	0.818896i	$0.305413\pi$
$164$	0	0
$165$	16.8507	1.31183
$166$	0	0
$167$	−4.81303	−0.372444	−0.186222	−	0.982508i	$-0.559624\pi$
$167$	−4.81303	−0.372444	−0.186222	+	0.982508i	$0.559624\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	4.08793	0.312612
$172$	0	0
$173$	12.1476	0.923568	0.461784	−	0.886992i	$-0.347210\pi$
$173$	12.1476	0.923568	0.461784	+	0.886992i	$0.347210\pi$
$174$	0	0
$175$	15.3692	1.16180
$176$	0	0
$177$	−10.5879	−0.795833
$178$	0	0
$179$	22.3518	1.67065	0.835325	−	0.549756i	$-0.185279\pi$
$179$	22.3518	1.67065	0.835325	+	0.549756i	$0.185279\pi$
$180$	0	0
$181$	−1.14611	−0.0851897	−0.0425949	−	0.999092i	$-0.513562\pi$
$181$	−1.14611	−0.0851897	−0.0425949	+	0.999092i	$0.513562\pi$
$182$	0	0
$183$	5.38047	0.397736
$184$	0	0
$185$	17.5074	1.28717
$186$	0	0
$187$	−12.1286	−0.886933
$188$	0	0
$189$	21.7881	1.58485
$190$	0	0
$191$	5.02219	0.363393	0.181696	−	0.983355i	$-0.441841\pi$
$191$	5.02219	0.363393	0.181696	+	0.983355i	$0.441841\pi$
$192$	0	0
$193$	10.6575	0.767145	0.383572	−	0.923511i	$-0.374694\pi$
$193$	10.6575	0.767145	0.383572	+	0.923511i	$0.374694\pi$
$194$	0	0
$195$	3.89112	0.278649
$196$	0	0
$197$	17.4075	1.24023	0.620117	−	0.784509i	$-0.287085\pi$
$197$	17.4075	1.24023	0.620117	+	0.784509i	$0.287085\pi$
$198$	0	0
$199$	10.8130	0.766514	0.383257	−	0.923642i	$-0.374802\pi$
$199$	10.8130	0.766514	0.383257	+	0.923642i	$0.374802\pi$
$200$	0	0
$201$	18.0239	1.27131
$202$	0	0
$203$	−3.88835	−0.272909
$204$	0	0
$205$	2.90396	0.202821
$206$	0	0
$207$	−10.7769	−0.749043
$208$	0	0
$209$	−13.5264	−0.935637
$210$	0	0
$211$	2.66832	0.183695	0.0918473	−	0.995773i	$-0.470723\pi$
$211$	2.66832	0.183695	0.0918473	+	0.995773i	$0.470723\pi$
$212$	0	0
$213$	7.15487	0.490244
$214$	0	0
$215$	−4.11234	−0.280459
$216$	0	0
$217$	−23.9535	−1.62607
$218$	0	0
$219$	−13.2541	−0.895629
$220$	0	0
$221$	−2.80070	−0.188396
$222$	0	0
$223$	7.09236	0.474940	0.237470	−	0.971395i	$-0.423682\pi$
$223$	7.09236	0.474940	0.237470	+	0.971395i	$0.423682\pi$
$224$	0	0
$225$	−5.17312	−0.344875
$226$	0	0
$227$	−17.5149	−1.16250	−0.581251	−	0.813724i	$-0.697436\pi$
$227$	−17.5149	−1.16250	−0.581251	+	0.813724i	$0.697436\pi$
$228$	0	0
$229$	5.02281	0.331917	0.165958	−	0.986133i	$-0.446928\pi$
$229$	5.02281	0.331917	0.165958	+	0.986133i	$0.446928\pi$
$230$	0	0
$231$	−21.8983	−1.44080
$232$	0	0
$233$	−0.906087	−0.0593597	−0.0296799	−	0.999559i	$-0.509449\pi$
$233$	−0.906087	−0.0593597	−0.0296799	+	0.999559i	$0.509449\pi$
$234$	0	0
$235$	−25.5169	−1.66454
$236$	0	0
$237$	−12.1740	−0.790788
$238$	0	0
$239$	18.8535	1.21953	0.609765	−	0.792582i	$-0.291264\pi$
$239$	18.8535	1.21953	0.609765	+	0.792582i	$0.291264\pi$
$240$	0	0
$241$	−7.55528	−0.486678	−0.243339	−	0.969941i	$-0.578243\pi$
$241$	−7.55528	−0.486678	−0.243339	+	0.969941i	$0.578243\pi$
$242$	0	0
$243$	−12.4398	−0.798011
$244$	0	0
$245$	−24.2937	−1.55207
$246$	0	0
$247$	−3.12346	−0.198741
$248$	0	0
$249$	−7.82728	−0.496034
$250$	0	0
$251$	−21.3974	−1.35059	−0.675295	−	0.737548i	$-0.735984\pi$
$251$	−21.3974	−1.35059	−0.675295	+	0.737548i	$0.735984\pi$
$252$	0	0
$253$	35.6590	2.24186
$254$	0	0
$255$	−10.8979	−0.682452
$256$	0	0
$257$	24.4266	1.52369	0.761845	−	0.647759i	$-0.224294\pi$
$257$	24.4266	1.52369	0.761845	+	0.647759i	$0.224294\pi$
$258$	0	0
$259$	−22.7516	−1.41372
$260$	0	0
$261$	1.30878	0.0810116
$262$	0	0
$263$	1.93073	0.119054	0.0595269	−	0.998227i	$-0.481041\pi$
$263$	1.93073	0.119054	0.0595269	+	0.998227i	$0.481041\pi$
$264$	0	0
$265$	17.0669	1.04841
$266$	0	0
$267$	−5.56270	−0.340432
$268$	0	0
$269$	−10.9359	−0.666774	−0.333387	−	0.942790i	$-0.608192\pi$
$269$	−10.9359	−0.666774	−0.333387	+	0.942790i	$0.608192\pi$
$270$	0	0
$271$	7.56548	0.459570	0.229785	−	0.973241i	$-0.426198\pi$
$271$	7.56548	0.459570	0.229785	+	0.973241i	$0.426198\pi$
$272$	0	0
$273$	−5.05668	−0.306044
$274$	0	0
$275$	17.1171	1.03220
$276$	0	0
$277$	0.351575	0.0211241	0.0105621	−	0.999944i	$-0.496638\pi$
$277$	0.351575	0.0211241	0.0105621	+	0.999944i	$0.496638\pi$
$278$	0	0
$279$	8.06253	0.482691
$280$	0	0
$281$	−18.0136	−1.07460	−0.537301	−	0.843391i	$-0.680556\pi$
$281$	−18.0136	−1.07460	−0.537301	+	0.843391i	$0.680556\pi$
$282$	0	0
$283$	−17.0492	−1.01347	−0.506735	−	0.862102i	$-0.669148\pi$
$283$	−17.0492	−1.01347	−0.506735	+	0.862102i	$0.669148\pi$
$284$	0	0
$285$	−12.1538	−0.719928
$286$	0	0
$287$	−3.77382	−0.222761
$288$	0	0
$289$	−9.15606	−0.538592
$290$	0	0
$291$	−16.5641	−0.971006
$292$	0	0
$293$	3.88341	0.226871	0.113436	−	0.993545i	$-0.463814\pi$
$293$	3.88341	0.226871	0.113436	+	0.993545i	$0.463814\pi$
$294$	0	0
$295$	−24.3604	−1.41832
$296$	0	0
$297$	24.2660	1.40806
$298$	0	0
$299$	8.23426	0.476200
$300$	0	0
$301$	5.34416	0.308032
$302$	0	0
$303$	14.0958	0.809780
$304$	0	0
$305$	12.3793	0.708836
$306$	0	0
$307$	−4.83724	−0.276076	−0.138038	−	0.990427i	$-0.544080\pi$
$307$	−4.83724	−0.276076	−0.138038	+	0.990427i	$0.544080\pi$
$308$	0	0
$309$	5.45964	0.310588
$310$	0	0
$311$	15.9411	0.903940	0.451970	−	0.892033i	$-0.350721\pi$
$311$	15.9411	0.903940	0.451970	+	0.892033i	$0.350721\pi$
$312$	0	0
$313$	−11.8058	−0.667303	−0.333652	−	0.942696i	$-0.608281\pi$
$313$	−11.8058	−0.667303	−0.333652	+	0.942696i	$0.608281\pi$
$314$	0	0
$315$	15.2268	0.857932
$316$	0	0
$317$	−14.2859	−0.802378	−0.401189	−	0.915995i	$-0.631403\pi$
$317$	−14.2859	−0.802378	−0.401189	+	0.915995i	$0.631403\pi$
$318$	0	0
$319$	−4.33056	−0.242465
$320$	0	0
$321$	−3.00402	−0.167668
$322$	0	0
$323$	8.74790	0.486746
$324$	0	0
$325$	3.95262	0.219252
$326$	0	0
$327$	−6.78662	−0.375301
$328$	0	0
$329$	33.1603	1.82819
$330$	0	0
$331$	−0.794698	−0.0436806	−0.0218403	−	0.999761i	$-0.506953\pi$
$331$	−0.794698	−0.0436806	−0.0218403	+	0.999761i	$0.506953\pi$
$332$	0	0
$333$	7.65798	0.419655
$334$	0	0
$335$	41.4691	2.26570
$336$	0	0
$337$	2.36993	0.129098	0.0645491	−	0.997915i	$-0.479439\pi$
$337$	2.36993	0.129098	0.0645491	+	0.997915i	$0.479439\pi$
$338$	0	0
$339$	−7.95012	−0.431791
$340$	0	0
$341$	−26.6777	−1.44468
$342$	0	0
$343$	4.35220	0.234997
$344$	0	0
$345$	32.0405	1.72500
$346$	0	0
$347$	2.04878	0.109984	0.0549921	−	0.998487i	$-0.482487\pi$
$347$	2.04878	0.109984	0.0549921	+	0.998487i	$0.482487\pi$
$348$	0	0
$349$	−17.8462	−0.955288	−0.477644	−	0.878554i	$-0.658509\pi$
$349$	−17.8462	−0.955288	−0.477644	+	0.878554i	$0.658509\pi$
$350$	0	0
$351$	5.60343	0.299089
$352$	0	0
$353$	−4.40092	−0.234237	−0.117119	−	0.993118i	$-0.537366\pi$
$353$	−4.40092	−0.234237	−0.117119	+	0.993118i	$0.537366\pi$
$354$	0	0
$355$	16.4618	0.873701
$356$	0	0
$357$	14.1623	0.749547
$358$	0	0
$359$	35.7341	1.88597	0.942986	−	0.332832i	$-0.108004\pi$
$359$	35.7341	1.88597	0.942986	+	0.332832i	$0.108004\pi$
$360$	0	0
$361$	−9.24397	−0.486525
$362$	0	0
$363$	−10.0835	−0.529248
$364$	0	0
$365$	−30.4948	−1.59617
$366$	0	0
$367$	−23.5888	−1.23133	−0.615663	−	0.788009i	$-0.711112\pi$
$367$	−23.5888	−1.23133	−0.615663	+	0.788009i	$0.711112\pi$
$368$	0	0
$369$	1.27023	0.0661255
$370$	0	0
$371$	−22.1792	−1.15149
$372$	0	0
$373$	−6.48105	−0.335576	−0.167788	−	0.985823i	$-0.553662\pi$
$373$	−6.48105	−0.335576	−0.167788	+	0.985823i	$0.553662\pi$
$374$	0	0
$375$	−4.07548	−0.210457
$376$	0	0
$377$	−1.00000	−0.0515026
$378$	0	0
$379$	35.6514	1.83129	0.915644	−	0.401991i	$-0.131682\pi$
$379$	35.6514	1.83129	0.915644	+	0.401991i	$0.131682\pi$
$380$	0	0
$381$	−26.0772	−1.33598
$382$	0	0
$383$	25.2723	1.29135	0.645676	−	0.763612i	$-0.276576\pi$
$383$	25.2723	1.29135	0.645676	+	0.763612i	$0.276576\pi$
$384$	0	0
$385$	−50.3831	−2.56776
$386$	0	0
$387$	−1.79879	−0.0914378
$388$	0	0
$389$	22.4529	1.13841	0.569205	−	0.822196i	$-0.307251\pi$
$389$	22.4529	1.13841	0.569205	+	0.822196i	$0.307251\pi$
$390$	0	0
$391$	−23.0617	−1.16628
$392$	0	0
$393$	−15.1007	−0.761731
$394$	0	0
$395$	−28.0098	−1.40932
$396$	0	0
$397$	14.2811	0.716748	0.358374	−	0.933578i	$-0.383331\pi$
$397$	14.2811	0.716748	0.358374	+	0.933578i	$0.383331\pi$
$398$	0	0
$399$	15.7944	0.790707
$400$	0	0
$401$	36.0208	1.79879	0.899397	−	0.437133i	$-0.144006\pi$
$401$	36.0208	1.79879	0.899397	+	0.437133i	$0.144006\pi$
$402$	0	0
$403$	−6.16033	−0.306868
$404$	0	0
$405$	10.0557	0.499670
$406$	0	0
$407$	−25.3391	−1.25601
$408$	0	0
$409$	−33.2324	−1.64324	−0.821619	−	0.570037i	$-0.806929\pi$
$409$	−33.2324	−1.64324	−0.821619	+	0.570037i	$0.806929\pi$
$410$	0	0
$411$	−12.6194	−0.622468
$412$	0	0
$413$	31.6574	1.55776
$414$	0	0
$415$	−18.0089	−0.884021
$416$	0	0
$417$	21.7566	1.06543
$418$	0	0
$419$	−2.82676	−0.138096	−0.0690482	−	0.997613i	$-0.521996\pi$
$419$	−2.82676	−0.138096	−0.0690482	+	0.997613i	$0.521996\pi$
$420$	0	0
$421$	−32.8459	−1.60081	−0.800405	−	0.599459i	$-0.795382\pi$
$421$	−32.8459	−1.60081	−0.800405	+	0.599459i	$0.795382\pi$
$422$	0	0
$423$	−11.1614	−0.542688
$424$	0	0
$425$	−11.0701	−0.536980
$426$	0	0
$427$	−16.0874	−0.778525
$428$	0	0
$429$	−5.63176	−0.271904
$430$	0	0
$431$	−4.47538	−0.215571	−0.107786	−	0.994174i	$-0.534376\pi$
$431$	−4.47538	−0.215571	−0.107786	+	0.994174i	$0.534376\pi$
$432$	0	0
$433$	−17.9383	−0.862062	−0.431031	−	0.902337i	$-0.641850\pi$
$433$	−17.9383	−0.862062	−0.431031	+	0.902337i	$0.641850\pi$
$434$	0	0
$435$	−3.89112	−0.186565
$436$	0	0
$437$	−25.7194	−1.23033
$438$	0	0
$439$	16.9549	0.809211	0.404606	−	0.914491i	$-0.367409\pi$
$439$	16.9549	0.809211	0.404606	+	0.914491i	$0.367409\pi$
$440$	0	0
$441$	−10.6264	−0.506018
$442$	0	0
$443$	28.7342	1.36520	0.682601	−	0.730791i	$-0.260849\pi$
$443$	28.7342	1.36520	0.682601	+	0.730791i	$0.260849\pi$
$444$	0	0
$445$	−12.7986	−0.606710
$446$	0	0
$447$	−23.4109	−1.10730
$448$	0	0
$449$	−1.23544	−0.0583042	−0.0291521	−	0.999575i	$-0.509281\pi$
$449$	−1.23544	−0.0583042	−0.0291521	+	0.999575i	$0.509281\pi$
$450$	0	0
$451$	−4.20300	−0.197912
$452$	0	0
$453$	8.85164	0.415886
$454$	0	0
$455$	−11.6343	−0.545425
$456$	0	0
$457$	6.53271	0.305587	0.152794	−	0.988258i	$-0.451173\pi$
$457$	6.53271	0.305587	0.152794	+	0.988258i	$0.451173\pi$
$458$	0	0
$459$	−15.6936	−0.732513
$460$	0	0
$461$	−42.6047	−1.98430	−0.992149	−	0.125061i	$-0.960087\pi$
$461$	−42.6047	−1.98430	−0.992149	+	0.125061i	$0.960087\pi$
$462$	0	0
$463$	17.4386	0.810440	0.405220	−	0.914219i	$-0.367195\pi$
$463$	17.4386	0.810440	0.405220	+	0.914219i	$0.367195\pi$
$464$	0	0
$465$	−23.9706	−1.11161
$466$	0	0
$467$	2.27674	0.105355	0.0526775	−	0.998612i	$-0.483224\pi$
$467$	2.27674	0.105355	0.0526775	+	0.998612i	$0.483224\pi$
$468$	0	0
$469$	−53.8909	−2.48845
$470$	0	0
$471$	−7.23823	−0.333520
$472$	0	0
$473$	5.95193	0.273670
$474$	0	0
$475$	−12.3459	−0.566467
$476$	0	0
$477$	7.46529	0.341812
$478$	0	0
$479$	−42.2711	−1.93142	−0.965708	−	0.259632i	$-0.916399\pi$
$479$	−42.2711	−1.93142	−0.965708	+	0.259632i	$0.916399\pi$
$480$	0	0
$481$	−5.85123	−0.266793
$482$	0	0
$483$	−41.6380	−1.89460
$484$	0	0
$485$	−38.1104	−1.73051
$486$	0	0
$487$	17.0451	0.772389	0.386195	−	0.922417i	$-0.373789\pi$
$487$	17.0451	0.772389	0.386195	+	0.922417i	$0.373789\pi$
$488$	0	0
$489$	−19.0586	−0.861861
$490$	0	0
$491$	15.9653	0.720505	0.360252	−	0.932855i	$-0.382691\pi$
$491$	15.9653	0.720505	0.360252	+	0.932855i	$0.382691\pi$
$492$	0	0
$493$	2.80070	0.126137
$494$	0	0
$495$	16.9585	0.762227
$496$	0	0
$497$	−21.3928	−0.959599
$498$	0	0
$499$	10.4074	0.465901	0.232950	−	0.972489i	$-0.425162\pi$
$499$	10.4074	0.465901	0.232950	+	0.972489i	$0.425162\pi$
$500$	0	0
$501$	6.25920	0.279640
$502$	0	0
$503$	41.4942	1.85014	0.925068	−	0.379801i	$-0.124007\pi$
$503$	41.4942	1.85014	0.925068	+	0.379801i	$0.124007\pi$
$504$	0	0
$505$	32.4312	1.44317
$506$	0	0
$507$	−1.30047	−0.0577558
$508$	0	0
$509$	32.0385	1.42008	0.710040	−	0.704161i	$-0.248677\pi$
$509$	32.0385	1.42008	0.710040	+	0.704161i	$0.248677\pi$
$510$	0	0
$511$	39.6293	1.75310
$512$	0	0
$513$	−17.5021	−0.772738
$514$	0	0
$515$	12.5614	0.553523
$516$	0	0
$517$	36.9315	1.62425
$518$	0	0
$519$	−15.7976	−0.693439
$520$	0	0
$521$	20.4343	0.895242	0.447621	−	0.894223i	$-0.352271\pi$
$521$	20.4343	0.895242	0.447621	+	0.894223i	$0.352271\pi$
$522$	0	0
$523$	−28.9043	−1.26390	−0.631948	−	0.775011i	$-0.717744\pi$
$523$	−28.9043	−1.26390	−0.631948	+	0.775011i	$0.717744\pi$
$524$	0	0
$525$	−19.9871	−0.872310
$526$	0	0
$527$	17.2533	0.751564
$528$	0	0
$529$	44.8031	1.94796
$530$	0	0
$531$	−10.6556	−0.462412
$532$	0	0
$533$	−0.970544	−0.0420389
$534$	0	0
$535$	−6.91158	−0.298814
$536$	0	0
$537$	−29.0678	−1.25437
$538$	0	0
$539$	35.1611	1.51450
$540$	0	0
$541$	−19.3017	−0.829844	−0.414922	−	0.909857i	$-0.636191\pi$
$541$	−19.3017	−0.829844	−0.414922	+	0.909857i	$0.636191\pi$
$542$	0	0
$543$	1.49048	0.0639626
$544$	0	0
$545$	−15.6145	−0.668853
$546$	0	0
$547$	−41.5603	−1.77699	−0.888494	−	0.458887i	$-0.848248\pi$
$547$	−41.5603	−1.77699	−0.888494	+	0.458887i	$0.848248\pi$
$548$	0	0
$549$	5.41487	0.231101
$550$	0	0
$551$	3.12346	0.133064
$552$	0	0
$553$	36.3999	1.54788
$554$	0	0
$555$	−22.7678	−0.966441
$556$	0	0
$557$	34.7866	1.47395	0.736977	−	0.675917i	$-0.236252\pi$
$557$	34.7866	1.47395	0.736977	+	0.675917i	$0.236252\pi$
$558$	0	0
$559$	1.37440	0.0581310
$560$	0	0
$561$	15.7729	0.665932
$562$	0	0
$563$	−34.9558	−1.47321	−0.736605	−	0.676323i	$-0.763572\pi$
$563$	−34.9558	−1.47321	−0.736605	+	0.676323i	$0.763572\pi$
$564$	0	0
$565$	−18.2915	−0.769528
$566$	0	0
$567$	−13.0678	−0.548794
$568$	0	0
$569$	8.73995	0.366398	0.183199	−	0.983076i	$-0.441355\pi$
$569$	8.73995	0.366398	0.183199	+	0.983076i	$0.441355\pi$
$570$	0	0
$571$	28.1979	1.18005	0.590024	−	0.807386i	$-0.299118\pi$
$571$	28.1979	1.18005	0.590024	+	0.807386i	$0.299118\pi$
$572$	0	0
$573$	−6.53120	−0.272845
$574$	0	0
$575$	32.5469	1.35730
$576$	0	0
$577$	−14.4510	−0.601605	−0.300802	−	0.953686i	$-0.597254\pi$
$577$	−14.4510	−0.601605	−0.300802	+	0.953686i	$0.597254\pi$
$578$	0	0
$579$	−13.8598	−0.575992
$580$	0	0
$581$	23.4033	0.970932
$582$	0	0
$583$	−24.7015	−1.02303
$584$	0	0
$585$	3.91600	0.161907
$586$	0	0
$587$	−32.7506	−1.35176	−0.675881	−	0.737011i	$-0.736237\pi$
$587$	−32.7506	−1.35176	−0.675881	+	0.737011i	$0.736237\pi$
$588$	0	0
$589$	19.2416	0.792835
$590$	0	0
$591$	−22.6379	−0.931200
$592$	0	0
$593$	−40.5064	−1.66340	−0.831699	−	0.555226i	$-0.812632\pi$
$593$	−40.5064	−1.66340	−0.831699	+	0.555226i	$0.812632\pi$
$594$	0	0
$595$	32.5843	1.33582
$596$	0	0
$597$	−14.0620	−0.575518
$598$	0	0
$599$	17.4968	0.714898	0.357449	−	0.933933i	$-0.383647\pi$
$599$	17.4968	0.714898	0.357449	+	0.933933i	$0.383647\pi$
$600$	0	0
$601$	11.7778	0.480425	0.240213	−	0.970720i	$-0.422783\pi$
$601$	11.7778	0.480425	0.240213	+	0.970720i	$0.422783\pi$
$602$	0	0
$603$	18.1392	0.738684
$604$	0	0
$605$	−23.2000	−0.943214
$606$	0	0
$607$	34.7204	1.40926	0.704629	−	0.709576i	$-0.251113\pi$
$607$	34.7204	1.40926	0.704629	+	0.709576i	$0.251113\pi$
$608$	0	0
$609$	5.05668	0.204907
$610$	0	0
$611$	8.52811	0.345010
$612$	0	0
$613$	−35.5975	−1.43777	−0.718885	−	0.695129i	$-0.755347\pi$
$613$	−35.5975	−1.43777	−0.718885	+	0.695129i	$0.755347\pi$
$614$	0	0
$615$	−3.77650	−0.152283
$616$	0	0
$617$	−6.58140	−0.264957	−0.132479	−	0.991186i	$-0.542294\pi$
$617$	−6.58140	−0.264957	−0.132479	+	0.991186i	$0.542294\pi$
$618$	0	0
$619$	−8.43699	−0.339111	−0.169556	−	0.985521i	$-0.554233\pi$
$619$	−8.43699	−0.339111	−0.169556	+	0.985521i	$0.554233\pi$
$620$	0	0
$621$	46.1402	1.85154
$622$	0	0
$623$	16.6323	0.666358
$624$	0	0
$625$	−29.1399	−1.16560
$626$	0	0
$627$	17.5906	0.702501
$628$	0	0
$629$	16.3876	0.653414
$630$	0	0
$631$	47.6744	1.89789	0.948943	−	0.315447i	$-0.102155\pi$
$631$	47.6744	1.89789	0.948943	+	0.315447i	$0.102155\pi$
$632$	0	0
$633$	−3.47006	−0.137923
$634$	0	0
$635$	−59.9980	−2.38095
$636$	0	0
$637$	8.11929	0.321698
$638$	0	0
$639$	7.20061	0.284852
$640$	0	0
$641$	−27.4957	−1.08602	−0.543008	−	0.839728i	$-0.682715\pi$
$641$	−27.4957	−1.08602	−0.543008	+	0.839728i	$0.682715\pi$
$642$	0	0
$643$	−6.27330	−0.247395	−0.123697	−	0.992320i	$-0.539475\pi$
$643$	−6.27330	−0.247395	−0.123697	+	0.992320i	$0.539475\pi$
$644$	0	0
$645$	5.34797	0.210576
$646$	0	0
$647$	−19.1243	−0.751852	−0.375926	−	0.926650i	$-0.622675\pi$
$647$	−19.1243	−0.751852	−0.375926	+	0.926650i	$0.622675\pi$
$648$	0	0
$649$	35.2576	1.38398
$650$	0	0
$651$	31.1508	1.22090
$652$	0	0
$653$	−12.0579	−0.471862	−0.235931	−	0.971770i	$-0.575814\pi$
$653$	−12.0579	−0.471862	−0.235931	+	0.971770i	$0.575814\pi$
$654$	0	0
$655$	−34.7435	−1.35754
$656$	0	0
$657$	−13.3388	−0.520397
$658$	0	0
$659$	35.0700	1.36613	0.683067	−	0.730356i	$-0.260646\pi$
$659$	35.0700	1.36613	0.683067	+	0.730356i	$0.260646\pi$
$660$	0	0
$661$	−18.0070	−0.700391	−0.350196	−	0.936677i	$-0.613885\pi$
$661$	−18.0070	−0.700391	−0.350196	+	0.936677i	$0.613885\pi$
$662$	0	0
$663$	3.64223	0.141452
$664$	0	0
$665$	36.3394	1.40918
$666$	0	0
$667$	−8.23426	−0.318832
$668$	0	0
$669$	−9.22339	−0.356597
$670$	0	0
$671$	−17.9170	−0.691678
$672$	0	0
$673$	−4.19099	−0.161551	−0.0807755	−	0.996732i	$-0.525740\pi$
$673$	−4.19099	−0.161551	−0.0807755	+	0.996732i	$0.525740\pi$
$674$	0	0
$675$	22.1482	0.852487
$676$	0	0
$677$	44.5103	1.71067	0.855335	−	0.518075i	$-0.173351\pi$
$677$	44.5103	1.71067	0.855335	+	0.518075i	$0.173351\pi$
$678$	0	0
$679$	49.5261	1.90064
$680$	0	0
$681$	22.7775	0.872836
$682$	0	0
$683$	−38.2691	−1.46432	−0.732162	−	0.681130i	$-0.761489\pi$
$683$	−38.2691	−1.46432	−0.732162	+	0.681130i	$0.761489\pi$
$684$	0	0
$685$	−29.0344	−1.10935
$686$	0	0
$687$	−6.53201	−0.249212
$688$	0	0
$689$	−5.70400	−0.217305
$690$	0	0
$691$	−7.76159	−0.295265	−0.147632	−	0.989042i	$-0.547165\pi$
$691$	−7.76159	−0.295265	−0.147632	+	0.989042i	$0.547165\pi$
$692$	0	0
$693$	−22.0383	−0.837164
$694$	0	0
$695$	50.0572	1.89878
$696$	0	0
$697$	2.71821	0.102959
$698$	0	0
$699$	1.17834	0.0445688
$700$	0	0
$701$	−17.3627	−0.655781	−0.327890	−	0.944716i	$-0.606338\pi$
$701$	−17.3627	−0.655781	−0.327890	+	0.944716i	$0.606338\pi$
$702$	0	0
$703$	18.2761	0.689296
$704$	0	0
$705$	33.1839	1.24978
$706$	0	0
$707$	−42.1458	−1.58506
$708$	0	0
$709$	20.6472	0.775422	0.387711	−	0.921781i	$-0.373266\pi$
$709$	20.6472	0.775422	0.387711	+	0.921781i	$0.373266\pi$
$710$	0	0
$711$	−12.2519	−0.459481
$712$	0	0
$713$	−50.7258	−1.89970
$714$	0	0
$715$	−12.9574	−0.484581
$716$	0	0
$717$	−24.5183	−0.915655
$718$	0	0
$719$	10.1544	0.378696	0.189348	−	0.981910i	$-0.439363\pi$
$719$	10.1544	0.378696	0.189348	+	0.981910i	$0.439363\pi$
$720$	0	0
$721$	−16.3241	−0.607942
$722$	0	0
$723$	9.82540	0.365411
$724$	0	0
$725$	−3.95262	−0.146797
$726$	0	0
$727$	0.756229	0.0280470	0.0140235	−	0.999902i	$-0.495536\pi$
$727$	0.756229	0.0280470	0.0140235	+	0.999902i	$0.495536\pi$
$728$	0	0
$729$	26.2597	0.972583
$730$	0	0
$731$	−3.84929	−0.142371
$732$	0	0
$733$	17.9903	0.664488	0.332244	−	0.943194i	$-0.392194\pi$
$733$	17.9903	0.664488	0.332244	+	0.943194i	$0.392194\pi$
$734$	0	0
$735$	31.5932	1.16533
$736$	0	0
$737$	−60.0198	−2.21086
$738$	0	0
$739$	23.1295	0.850831	0.425416	−	0.904998i	$-0.360128\pi$
$739$	23.1295	0.850831	0.425416	+	0.904998i	$0.360128\pi$
$740$	0	0
$741$	4.06197	0.149220
$742$	0	0
$743$	−1.95873	−0.0718589	−0.0359295	−	0.999354i	$-0.511439\pi$
$743$	−1.95873	−0.0718589	−0.0359295	+	0.999354i	$0.511439\pi$
$744$	0	0
$745$	−53.8634	−1.97340
$746$	0	0
$747$	−7.87732	−0.288216
$748$	0	0
$749$	8.98190	0.328192
$750$	0	0
$751$	−26.8488	−0.979728	−0.489864	−	0.871799i	$-0.662954\pi$
$751$	−26.8488	−0.979728	−0.489864	+	0.871799i	$0.662954\pi$
$752$	0	0
$753$	27.8266	1.01406
$754$	0	0
$755$	20.3657	0.741183
$756$	0	0
$757$	25.2360	0.917218	0.458609	−	0.888638i	$-0.348348\pi$
$757$	25.2360	0.917218	0.458609	+	0.888638i	$0.348348\pi$
$758$	0	0
$759$	−46.3734	−1.68325
$760$	0	0
$761$	−36.4389	−1.32091	−0.660455	−	0.750866i	$-0.729637\pi$
$761$	−36.4389	−1.32091	−0.660455	+	0.750866i	$0.729637\pi$
$762$	0	0
$763$	20.2917	0.734610
$764$	0	0
$765$	−10.9676	−0.396533
$766$	0	0
$767$	8.14158	0.293976
$768$	0	0
$769$	39.8090	1.43555	0.717775	−	0.696275i	$-0.245161\pi$
$769$	39.8090	1.43555	0.717775	+	0.696275i	$0.245161\pi$
$770$	0	0
$771$	−31.7660	−1.14403
$772$	0	0
$773$	−7.06507	−0.254113	−0.127056	−	0.991895i	$-0.540553\pi$
$773$	−7.06507	−0.254113	−0.127056	+	0.991895i	$0.540553\pi$
$774$	0	0
$775$	−24.3494	−0.874658
$776$	0	0
$777$	29.5878	1.06146
$778$	0	0
$779$	3.03146	0.108613
$780$	0	0
$781$	−23.8257	−0.852552
$782$	0	0
$783$	−5.60343	−0.200250
$784$	0	0
$785$	−16.6536	−0.594392
$786$	0	0
$787$	−31.4967	−1.12274	−0.561368	−	0.827566i	$-0.689725\pi$
$787$	−31.4967	−1.12274	−0.561368	+	0.827566i	$0.689725\pi$
$788$	0	0
$789$	−2.51085	−0.0893886
$790$	0	0
$791$	23.7706	0.845184
$792$	0	0
$793$	−4.13734	−0.146921
$794$	0	0
$795$	−22.1950	−0.787174
$796$	0	0
$797$	10.8030	0.382661	0.191331	−	0.981526i	$-0.438720\pi$
$797$	10.8030	0.382661	0.191331	+	0.981526i	$0.438720\pi$
$798$	0	0
$799$	−23.8847	−0.844980
$800$	0	0
$801$	−5.59827	−0.197805
$802$	0	0
$803$	44.1362	1.55753
$804$	0	0
$805$	−95.8000	−3.37651
$806$	0	0
$807$	14.2218	0.500631
$808$	0	0
$809$	−0.168060	−0.00590868	−0.00295434	−	0.999996i	$-0.500940\pi$
$809$	−0.168060	−0.00590868	−0.00295434	+	0.999996i	$0.500940\pi$
$810$	0	0
$811$	−39.9834	−1.40401	−0.702003	−	0.712174i	$-0.747711\pi$
$811$	−39.9834	−1.40401	−0.702003	+	0.712174i	$0.747711\pi$
$812$	0	0
$813$	−9.83867	−0.345057
$814$	0	0
$815$	−43.8497	−1.53599
$816$	0	0
$817$	−4.29290	−0.150189
$818$	0	0
$819$	−5.08901	−0.177824
$820$	0	0
$821$	47.1434	1.64531	0.822657	−	0.568538i	$-0.192491\pi$
$821$	47.1434	1.64531	0.822657	+	0.568538i	$0.192491\pi$
$822$	0	0
$823$	−11.9094	−0.415136	−0.207568	−	0.978221i	$-0.566555\pi$
$823$	−11.9094	−0.415136	−0.207568	+	0.978221i	$0.566555\pi$
$824$	0	0
$825$	−22.2602	−0.775001
$826$	0	0
$827$	30.7580	1.06956	0.534780	−	0.844991i	$-0.320394\pi$
$827$	30.7580	1.06956	0.534780	+	0.844991i	$0.320394\pi$
$828$	0	0
$829$	21.8967	0.760505	0.380253	−	0.924883i	$-0.375837\pi$
$829$	21.8967	0.760505	0.380253	+	0.924883i	$0.375837\pi$
$830$	0	0
$831$	−0.457212	−0.0158605
$832$	0	0
$833$	−22.7397	−0.787885
$834$	0	0
$835$	14.4010	0.498369
$836$	0	0
$837$	−34.5190	−1.19315
$838$	0	0
$839$	−41.7157	−1.44018	−0.720092	−	0.693878i	$-0.755901\pi$
$839$	−41.7157	−1.44018	−0.720092	+	0.693878i	$0.755901\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	0	0
$843$	23.4261	0.806839
$844$	0	0
$845$	−2.99209	−0.102931
$846$	0	0
$847$	30.1494	1.03594
$848$	0	0
$849$	22.1719	0.760939
$850$	0	0
$851$	−48.1805	−1.65161
$852$	0	0
$853$	9.17228	0.314053	0.157026	−	0.987594i	$-0.449809\pi$
$853$	9.17228	0.314053	0.157026	+	0.987594i	$0.449809\pi$
$854$	0	0
$855$	−12.2315	−0.418308
$856$	0	0
$857$	−2.78321	−0.0950728	−0.0475364	−	0.998870i	$-0.515137\pi$
$857$	−2.78321	−0.0950728	−0.0475364	+	0.998870i	$0.515137\pi$
$858$	0	0
$859$	56.3020	1.92100	0.960500	−	0.278279i	$-0.0897641\pi$
$859$	56.3020	1.92100	0.960500	+	0.278279i	$0.0897641\pi$
$860$	0	0
$861$	4.90773	0.167255
$862$	0	0
$863$	−20.0596	−0.682835	−0.341418	−	0.939912i	$-0.610907\pi$
$863$	−20.0596	−0.682835	−0.341418	+	0.939912i	$0.610907\pi$
$864$	0	0
$865$	−36.3469	−1.23583
$866$	0	0
$867$	11.9072	0.404388
$868$	0	0
$869$	40.5395	1.37521
$870$	0	0
$871$	−13.8596	−0.469614
$872$	0	0
$873$	−16.6700	−0.564195
$874$	0	0
$875$	12.1855	0.411947
$876$	0	0
$877$	8.01992	0.270814	0.135407	−	0.990790i	$-0.456766\pi$
$877$	8.01992	0.270814	0.135407	+	0.990790i	$0.456766\pi$
$878$	0	0
$879$	−5.05025	−0.170341
$880$	0	0
$881$	51.7885	1.74480	0.872399	−	0.488795i	$-0.162563\pi$
$881$	51.7885	1.74480	0.872399	+	0.488795i	$0.162563\pi$
$882$	0	0
$883$	39.6127	1.33307	0.666537	−	0.745472i	$-0.267776\pi$
$883$	39.6127	1.33307	0.666537	+	0.745472i	$0.267776\pi$
$884$	0	0
$885$	31.6799	1.06491
$886$	0	0
$887$	−36.6851	−1.23177	−0.615883	−	0.787838i	$-0.711201\pi$
$887$	−36.6851	−1.23177	−0.615883	+	0.787838i	$0.711201\pi$
$888$	0	0
$889$	77.9700	2.61503
$890$	0	0
$891$	−14.5539	−0.487574
$892$	0	0
$893$	−26.6372	−0.891381
$894$	0	0
$895$	−66.8786	−2.23551
$896$	0	0
$897$	−10.7084	−0.357543
$898$	0	0
$899$	6.16033	0.205458
$900$	0	0
$901$	15.9752	0.532212
$902$	0	0
$903$	−6.94991	−0.231279
$904$	0	0
$905$	3.42927	0.113993
$906$	0	0
$907$	48.8626	1.62245	0.811227	−	0.584731i	$-0.198800\pi$
$907$	48.8626	1.62245	0.811227	+	0.584731i	$0.198800\pi$
$908$	0	0
$909$	14.1859	0.470516
$910$	0	0
$911$	−25.3034	−0.838339	−0.419169	−	0.907908i	$-0.637679\pi$
$911$	−25.3034	−0.838339	−0.419169	+	0.907908i	$0.637679\pi$
$912$	0	0
$913$	26.0649	0.862621
$914$	0	0
$915$	−16.0989	−0.532213
$916$	0	0
$917$	45.1507	1.49101
$918$	0	0
$919$	56.6127	1.86748	0.933741	−	0.357950i	$-0.116524\pi$
$919$	56.6127	1.86748	0.933741	+	0.357950i	$0.116524\pi$
$920$	0	0
$921$	6.29067	0.207285
$922$	0	0
$923$	−5.50177	−0.181093
$924$	0	0
$925$	−23.1277	−0.760433
$926$	0	0
$927$	5.49455	0.180465
$928$	0	0
$929$	−31.6773	−1.03930	−0.519649	−	0.854380i	$-0.673937\pi$
$929$	−31.6773	−1.03930	−0.519649	+	0.854380i	$0.673937\pi$
$930$	0	0
$931$	−25.3603	−0.831151
$932$	0	0
$933$	−20.7310	−0.678701
$934$	0	0
$935$	36.2900	1.18681
$936$	0	0
$937$	27.9744	0.913883	0.456942	−	0.889497i	$-0.348945\pi$
$937$	27.9744	0.913883	0.456942	+	0.889497i	$0.348945\pi$
$938$	0	0
$939$	15.3531	0.501028
$940$	0	0
$941$	−21.7117	−0.707782	−0.353891	−	0.935287i	$-0.615142\pi$
$941$	−21.7117	−0.707782	−0.353891	+	0.935287i	$0.615142\pi$
$942$	0	0
$943$	−7.99171	−0.260246
$944$	0	0
$945$	−65.1921	−2.12070
$946$	0	0
$947$	−47.2894	−1.53670	−0.768350	−	0.640030i	$-0.778922\pi$
$947$	−47.2894	−1.53670	−0.768350	+	0.640030i	$0.778922\pi$
$948$	0	0
$949$	10.1918	0.330839
$950$	0	0
$951$	18.5784	0.602446
$952$	0	0
$953$	23.5207	0.761912	0.380956	−	0.924593i	$-0.375595\pi$
$953$	23.5207	0.761912	0.380956	+	0.924593i	$0.375595\pi$
$954$	0	0
$955$	−15.0269	−0.486258
$956$	0	0
$957$	5.63176	0.182049
$958$	0	0
$959$	37.7315	1.21841
$960$	0	0
$961$	6.94966	0.224182
$962$	0	0
$963$	−3.02322	−0.0974219
$964$	0	0
$965$	−31.8883	−1.02652
$966$	0	0
$967$	−52.3417	−1.68320	−0.841598	−	0.540105i	$-0.818385\pi$
$967$	−52.3417	−1.68320	−0.841598	+	0.540105i	$0.818385\pi$
$968$	0	0
$969$	−11.3764	−0.365461
$970$	0	0
$971$	−10.2292	−0.328270	−0.164135	−	0.986438i	$-0.552483\pi$
$971$	−10.2292	−0.328270	−0.164135	+	0.986438i	$0.552483\pi$
$972$	0	0
$973$	−65.0515	−2.08546
$974$	0	0
$975$	−5.14026	−0.164620
$976$	0	0
$977$	15.7862	0.505045	0.252523	−	0.967591i	$-0.418740\pi$
$977$	15.7862	0.505045	0.252523	+	0.967591i	$0.418740\pi$
$978$	0	0
$979$	18.5238	0.592024
$980$	0	0
$981$	−6.83000	−0.218065
$982$	0	0
$983$	2.13143	0.0679821	0.0339910	−	0.999422i	$-0.489178\pi$
$983$	2.13143	0.0679821	0.0339910	+	0.999422i	$0.489178\pi$
$984$	0	0
$985$	−52.0849	−1.65956
$986$	0	0
$987$	−43.1239	−1.37265
$988$	0	0
$989$	11.3172	0.359866
$990$	0	0
$991$	−31.0502	−0.986343	−0.493172	−	0.869932i	$-0.664163\pi$
$991$	−31.0502	−0.986343	−0.493172	+	0.869932i	$0.664163\pi$
$992$	0	0
$993$	1.03348	0.0327965
$994$	0	0
$995$	−32.3535	−1.02568
$996$	0	0
$997$	17.8906	0.566600	0.283300	−	0.959031i	$-0.408571\pi$
$997$	17.8906	0.566600	0.283300	+	0.959031i	$0.408571\pi$
$998$	0	0
$999$	−32.7870	−1.03733

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3016.2.a.g.1.3	✓	10
4.3	odd	2		6032.2.a.ba.1.8		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3016.2.a.g.1.3	✓	10	1.1	even	1	trivial
6032.2.a.ba.1.8		10	4.3	odd	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 3016 = 2^{3} \cdot 13 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3016.a (trivial)

\(f(q)\)	\(=\)	\(q-1.30047 q^{3} -2.99209 q^{5} +3.88835 q^{7} -1.30878 q^{9} +O(q^{10})\)	\(q-1.30047 q^{3} -2.99209 q^{5} +3.88835 q^{7} -1.30878 q^{9} +4.33056 q^{11} +1.00000 q^{13} +3.89112 q^{15} -2.80070 q^{17} -3.12346 q^{19} -5.05668 q^{21} +8.23426 q^{23} +3.95262 q^{25} +5.60343 q^{27} -1.00000 q^{29} -6.16033 q^{31} -5.63176 q^{33} -11.6343 q^{35} -5.85123 q^{37} -1.30047 q^{39} -0.970544 q^{41} +1.37440 q^{43} +3.91600 q^{45} +8.52811 q^{47} +8.11929 q^{49} +3.64223 q^{51} -5.70400 q^{53} -12.9574 q^{55} +4.06197 q^{57} +8.14158 q^{59} -4.13734 q^{61} -5.08901 q^{63} -2.99209 q^{65} -13.8596 q^{67} -10.7084 q^{69} -5.50177 q^{71} +10.1918 q^{73} -5.14026 q^{75} +16.8388 q^{77} +9.36127 q^{79} -3.36074 q^{81} +6.01882 q^{83} +8.37997 q^{85} +1.30047 q^{87} +4.27746 q^{89} +3.88835 q^{91} +8.01131 q^{93} +9.34569 q^{95} +12.7370 q^{97} -5.66776 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q + 2 q^{3} + 5 q^{5} - 2 q^{7} + 16 q^{9}+O(q^{10}) \) 10 * q + 2 * q^3 + 5 * q^5 - 2 * q^7 + 16 * q^9	\( 10 q + 2 q^{3} + 5 q^{5} - 2 q^{7} + 16 q^{9} + 4 q^{11} + 10 q^{13} + 8 q^{15} + 10 q^{17} - q^{19} + q^{21} + 23 q^{23} + 25 q^{25} + 2 q^{27} - 10 q^{29} - 13 q^{31} + 15 q^{33} - 12 q^{35} - 7 q^{37} + 2 q^{39} + 16 q^{41} - 12 q^{43} + 55 q^{45} + 11 q^{47} - 25 q^{51} + 11 q^{53} + 22 q^{55} - 6 q^{57} - 11 q^{59} + 34 q^{61} + 37 q^{63} + 5 q^{65} - 23 q^{67} + 2 q^{69} - 4 q^{71} + 39 q^{73} + 11 q^{75} + 32 q^{77} + 5 q^{79} + 38 q^{81} + 6 q^{83} + 45 q^{85} - 2 q^{87} - 24 q^{89} - 2 q^{91} + 13 q^{93} + 33 q^{95} + 19 q^{97}+O(q^{100}) \) 10 * q + 2 * q^3 + 5 * q^5 - 2 * q^7 + 16 * q^9 + 4 * q^11 + 10 * q^13 + 8 * q^15 + 10 * q^17 - q^19 + q^21 + 23 * q^23 + 25 * q^25 + 2 * q^27 - 10 * q^29 - 13 * q^31 + 15 * q^33 - 12 * q^35 - 7 * q^37 + 2 * q^39 + 16 * q^41 - 12 * q^43 + 55 * q^45 + 11 * q^47 - 25 * q^51 + 11 * q^53 + 22 * q^55 - 6 * q^57 - 11 * q^59 + 34 * q^61 + 37 * q^63 + 5 * q^65 - 23 * q^67 + 2 * q^69 - 4 * q^71 + 39 * q^73 + 11 * q^75 + 32 * q^77 + 5 * q^79 + 38 * q^81 + 6 * q^83 + 45 * q^85 - 2 * q^87 - 24 * q^89 - 2 * q^91 + 13 * q^93 + 33 * q^95 + 19 * q^97

Introduction

L-functions

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