LMFDB - Embedded newform 8008.2.a.z.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8008,2,Mod(1,8008)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8008, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("8008.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8008 = 2^{3} \cdot 7 \cdot 11 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8008.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:	$63.9442019386$
Analytic rank:	$0$
Dimension:	$15$
Coefficient field:	$\mathbb{Q}[x]/(x^{15} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{15} - x^{14} - 35 x^{13} + 32 x^{12} + 477 x^{11} - 392 x^{10} - 3236 x^{9} + 2330 x^{8} + \cdots + 2560 $ x^15 - x^14 - 35x^13 + 32x^12 + 477x^11 - 392x^10 - 3236x^9 + 2330x^8 + 11690x^7 - 7119x^6 - 22246x^5 + 11137x^4 + 20034x^3 - 8392x^2 - 6016*x + 2560
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$2.38300$ of defining polynomial
Character	$\chi$	$=$	8008.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.38300 q^{3} -4.12953 q^{5} -1.00000 q^{7} +2.67869 q^{9} +O(q^{10})$	$q-2.38300 q^{3} -4.12953 q^{5} -1.00000 q^{7} +2.67869 q^{9} -1.00000 q^{11} +1.00000 q^{13} +9.84066 q^{15} +1.31963 q^{17} -7.60564 q^{19} +2.38300 q^{21} +3.10483 q^{23} +12.0530 q^{25} +0.765692 q^{27} +2.76325 q^{29} +1.80742 q^{31} +2.38300 q^{33} +4.12953 q^{35} -7.02943 q^{37} -2.38300 q^{39} -2.48260 q^{41} -6.37587 q^{43} -11.0617 q^{45} -13.1687 q^{47} +1.00000 q^{49} -3.14467 q^{51} -3.26165 q^{53} +4.12953 q^{55} +18.1242 q^{57} -6.73839 q^{59} -12.9102 q^{61} -2.67869 q^{63} -4.12953 q^{65} +0.367821 q^{67} -7.39880 q^{69} +9.78486 q^{71} -10.6481 q^{73} -28.7223 q^{75} +1.00000 q^{77} -13.2011 q^{79} -9.86070 q^{81} -9.27258 q^{83} -5.44943 q^{85} -6.58483 q^{87} -3.12241 q^{89} -1.00000 q^{91} -4.30708 q^{93} +31.4077 q^{95} +15.8493 q^{97} -2.67869 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 15 q - q^{3} + 4 q^{5} - 15 q^{7} + 26 q^{9}+O(q^{10}) $ 15 * q - q^3 + 4 * q^5 - 15 * q^7 + 26 * q^9	$ 15 q - q^{3} + 4 q^{5} - 15 q^{7} + 26 q^{9} - 15 q^{11} + 15 q^{13} - 6 q^{15} + 8 q^{17} - 17 q^{19} + q^{21} + 7 q^{23} + 33 q^{25} - 4 q^{27} + 14 q^{29} - 4 q^{31} + q^{33} - 4 q^{35} + 3 q^{37} - q^{39} - 13 q^{43} + 20 q^{45} + 6 q^{47} + 15 q^{49} + 8 q^{51} + 38 q^{53} - 4 q^{55} + 24 q^{57} - 18 q^{59} + 23 q^{61} - 26 q^{63} + 4 q^{65} - 8 q^{67} + 43 q^{69} - 12 q^{71} + 11 q^{73} + 12 q^{75} + 15 q^{77} - q^{79} + 51 q^{81} - 16 q^{83} + 13 q^{85} - 25 q^{87} + 28 q^{89} - 15 q^{91} - 14 q^{93} + 49 q^{95} + 30 q^{97} - 26 q^{99}+O(q^{100}) $ 15 * q - q^3 + 4 * q^5 - 15 * q^7 + 26 * q^9 - 15 * q^11 + 15 * q^13 - 6 * q^15 + 8 * q^17 - 17 * q^19 + q^21 + 7 * q^23 + 33 * q^25 - 4 * q^27 + 14 * q^29 - 4 * q^31 + q^33 - 4 * q^35 + 3 * q^37 - q^39 - 13 * q^43 + 20 * q^45 + 6 * q^47 + 15 * q^49 + 8 * q^51 + 38 * q^53 - 4 * q^55 + 24 * q^57 - 18 * q^59 + 23 * q^61 - 26 * q^63 + 4 * q^65 - 8 * q^67 + 43 * q^69 - 12 * q^71 + 11 * q^73 + 12 * q^75 + 15 * q^77 - q^79 + 51 * q^81 - 16 * q^83 + 13 * q^85 - 25 * q^87 + 28 * q^89 - 15 * q^91 - 14 * q^93 + 49 * q^95 + 30 * q^97 - 26 * 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
$3$	−2.38300	−1.37583	−0.687913	−	0.725793i	$-0.741473\pi$
$3$	−2.38300	−1.37583	−0.687913	+	0.725793i	$0.741473\pi$
$4$	0	0
$5$	−4.12953	−1.84678	−0.923390	−	0.383863i	$-0.874594\pi$
$5$	−4.12953	−1.84678	−0.923390	+	0.383863i	$0.874594\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	2.67869	0.892895
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	0	0
$15$	9.84066	2.54085
$16$	0	0
$17$	1.31963	0.320056	0.160028	−	0.987112i	$-0.448841\pi$
$17$	1.31963	0.320056	0.160028	+	0.987112i	$0.448841\pi$
$18$	0	0
$19$	−7.60564	−1.74485	−0.872426	−	0.488746i	$-0.837455\pi$
$19$	−7.60564	−1.74485	−0.872426	+	0.488746i	$0.837455\pi$
$20$	0	0
$21$	2.38300	0.520013
$22$	0	0
$23$	3.10483	0.647401	0.323700	−	0.946160i	$-0.395073\pi$
$23$	3.10483	0.647401	0.323700	+	0.946160i	$0.395073\pi$
$24$	0	0
$25$	12.0530	2.41060
$26$	0	0
$27$	0.765692	0.147357
$28$	0	0
$29$	2.76325	0.513123	0.256562	−	0.966528i	$-0.417410\pi$
$29$	2.76325	0.513123	0.256562	+	0.966528i	$0.417410\pi$
$30$	0	0
$31$	1.80742	0.324622	0.162311	−	0.986740i	$-0.448105\pi$
$31$	1.80742	0.324622	0.162311	+	0.986740i	$0.448105\pi$
$32$	0	0
$33$	2.38300	0.414827
$34$	0	0
$35$	4.12953	0.698017
$36$	0	0
$37$	−7.02943	−1.15563	−0.577816	−	0.816167i	$-0.696095\pi$
$37$	−7.02943	−1.15563	−0.577816	+	0.816167i	$0.696095\pi$
$38$	0	0
$39$	−2.38300	−0.381585
$40$	0	0
$41$	−2.48260	−0.387717	−0.193858	−	0.981030i	$-0.562100\pi$
$41$	−2.48260	−0.387717	−0.193858	+	0.981030i	$0.562100\pi$
$42$	0	0
$43$	−6.37587	−0.972311	−0.486155	−	0.873872i	$-0.661601\pi$
$43$	−6.37587	−0.972311	−0.486155	+	0.873872i	$0.661601\pi$
$44$	0	0
$45$	−11.0617	−1.64898
$46$	0	0
$47$	−13.1687	−1.92085	−0.960424	−	0.278542i	$-0.910149\pi$
$47$	−13.1687	−1.92085	−0.960424	+	0.278542i	$0.910149\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−3.14467	−0.440342
$52$	0	0
$53$	−3.26165	−0.448023	−0.224011	−	0.974587i	$-0.571915\pi$
$53$	−3.26165	−0.448023	−0.224011	+	0.974587i	$0.571915\pi$
$54$	0	0
$55$	4.12953	0.556825
$56$	0	0
$57$	18.1242	2.40061
$58$	0	0
$59$	−6.73839	−0.877263	−0.438632	−	0.898667i	$-0.644537\pi$
$59$	−6.73839	−0.877263	−0.438632	+	0.898667i	$0.644537\pi$
$60$	0	0
$61$	−12.9102	−1.65299	−0.826493	−	0.562947i	$-0.809668\pi$
$61$	−12.9102	−1.65299	−0.826493	+	0.562947i	$0.809668\pi$
$62$	0	0
$63$	−2.67869	−0.337483
$64$	0	0
$65$	−4.12953	−0.512205
$66$	0	0
$67$	0.367821	0.0449365	0.0224682	−	0.999748i	$-0.492848\pi$
$67$	0.367821	0.0449365	0.0224682	+	0.999748i	$0.492848\pi$
$68$	0	0
$69$	−7.39880	−0.890711
$70$	0	0
$71$	9.78486	1.16125	0.580625	−	0.814171i	$-0.302808\pi$
$71$	9.78486	1.16125	0.580625	+	0.814171i	$0.302808\pi$
$72$	0	0
$73$	−10.6481	−1.24626	−0.623132	−	0.782117i	$-0.714140\pi$
$73$	−10.6481	−1.24626	−0.623132	+	0.782117i	$0.714140\pi$
$74$	0	0
$75$	−28.7223	−3.31656
$76$	0	0
$77$	1.00000	0.113961
$78$	0	0
$79$	−13.2011	−1.48524	−0.742621	−	0.669712i	$-0.766418\pi$
$79$	−13.2011	−1.48524	−0.742621	+	0.669712i	$0.766418\pi$
$80$	0	0
$81$	−9.86070	−1.09563
$82$	0	0
$83$	−9.27258	−1.01780	−0.508899	−	0.860826i	$-0.669947\pi$
$83$	−9.27258	−1.01780	−0.508899	+	0.860826i	$0.669947\pi$
$84$	0	0
$85$	−5.44943	−0.591074
$86$	0	0
$87$	−6.58483	−0.705968
$88$	0	0
$89$	−3.12241	−0.330975	−0.165487	−	0.986212i	$-0.552920\pi$
$89$	−3.12241	−0.330975	−0.165487	+	0.986212i	$0.552920\pi$
$90$	0	0
$91$	−1.00000	−0.104828
$92$	0	0
$93$	−4.30708	−0.446624
$94$	0	0
$95$	31.4077	3.22236
$96$	0	0
$97$	15.8493	1.60926	0.804628	−	0.593779i	$-0.202365\pi$
$97$	15.8493	1.60926	0.804628	+	0.593779i	$0.202365\pi$
$98$	0	0
$99$	−2.67869	−0.269218
$100$	0	0
$101$	−12.2140	−1.21534	−0.607669	−	0.794190i	$-0.707895\pi$
$101$	−12.2140	−1.21534	−0.607669	+	0.794190i	$0.707895\pi$
$102$	0	0
$103$	8.51121	0.838634	0.419317	−	0.907840i	$-0.362270\pi$
$103$	8.51121	0.838634	0.419317	+	0.907840i	$0.362270\pi$
$104$	0	0
$105$	−9.84066	−0.960350
$106$	0	0
$107$	−11.1982	−1.08257	−0.541283	−	0.840840i	$-0.682061\pi$
$107$	−11.1982	−1.08257	−0.541283	+	0.840840i	$0.682061\pi$
$108$	0	0
$109$	12.3455	1.18248	0.591242	−	0.806494i	$-0.298638\pi$
$109$	12.3455	1.18248	0.591242	+	0.806494i	$0.298638\pi$
$110$	0	0
$111$	16.7511	1.58995
$112$	0	0
$113$	−6.24125	−0.587127	−0.293564	−	0.955940i	$-0.594841\pi$
$113$	−6.24125	−0.587127	−0.293564	+	0.955940i	$0.594841\pi$
$114$	0	0
$115$	−12.8215	−1.19561
$116$	0	0
$117$	2.67869	0.247645
$118$	0	0
$119$	−1.31963	−0.120970
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	5.91603	0.533431
$124$	0	0
$125$	−29.1255	−2.60506
$126$	0	0
$127$	−9.91700	−0.879991	−0.439996	−	0.898000i	$-0.645020\pi$
$127$	−9.91700	−0.879991	−0.439996	+	0.898000i	$0.645020\pi$
$128$	0	0
$129$	15.1937	1.33773
$130$	0	0
$131$	14.1696	1.23800	0.619001	−	0.785390i	$-0.287538\pi$
$131$	14.1696	1.23800	0.619001	+	0.785390i	$0.287538\pi$
$132$	0	0
$133$	7.60564	0.659492
$134$	0	0
$135$	−3.16194	−0.272137
$136$	0	0
$137$	−9.40961	−0.803918	−0.401959	−	0.915658i	$-0.631670\pi$
$137$	−9.40961	−0.803918	−0.401959	+	0.915658i	$0.631670\pi$
$138$	0	0
$139$	−18.7698	−1.59204	−0.796018	−	0.605273i	$-0.793064\pi$
$139$	−18.7698	−1.59204	−0.796018	+	0.605273i	$0.793064\pi$
$140$	0	0
$141$	31.3809	2.64275
$142$	0	0
$143$	−1.00000	−0.0836242
$144$	0	0
$145$	−11.4109	−0.947626
$146$	0	0
$147$	−2.38300	−0.196546
$148$	0	0
$149$	8.60016	0.704552	0.352276	−	0.935896i	$-0.385408\pi$
$149$	8.60016	0.704552	0.352276	+	0.935896i	$0.385408\pi$
$150$	0	0
$151$	−8.59577	−0.699514	−0.349757	−	0.936841i	$-0.613736\pi$
$151$	−8.59577	−0.699514	−0.349757	+	0.936841i	$0.613736\pi$
$152$	0	0
$153$	3.53486	0.285777
$154$	0	0
$155$	−7.46379	−0.599506
$156$	0	0
$157$	13.2549	1.05786	0.528929	−	0.848666i	$-0.322594\pi$
$157$	13.2549	1.05786	0.528929	+	0.848666i	$0.322594\pi$
$158$	0	0
$159$	7.77252	0.616401
$160$	0	0
$161$	−3.10483	−0.244695
$162$	0	0
$163$	5.45835	0.427531	0.213765	−	0.976885i	$-0.431427\pi$
$163$	5.45835	0.427531	0.213765	+	0.976885i	$0.431427\pi$
$164$	0	0
$165$	−9.84066	−0.766094
$166$	0	0
$167$	−8.29995	−0.642270	−0.321135	−	0.947033i	$-0.604064\pi$
$167$	−8.29995	−0.642270	−0.321135	+	0.947033i	$0.604064\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	−20.3731	−1.55797
$172$	0	0
$173$	7.59488	0.577428	0.288714	−	0.957415i	$-0.406772\pi$
$173$	7.59488	0.577428	0.288714	+	0.957415i	$0.406772\pi$
$174$	0	0
$175$	−12.0530	−0.911120
$176$	0	0
$177$	16.0576	1.20696
$178$	0	0
$179$	20.9143	1.56321	0.781606	−	0.623773i	$-0.214401\pi$
$179$	20.9143	1.56321	0.781606	+	0.623773i	$0.214401\pi$
$180$	0	0
$181$	−19.3279	−1.43663	−0.718316	−	0.695717i	$-0.755087\pi$
$181$	−19.3279	−1.43663	−0.718316	+	0.695717i	$0.755087\pi$
$182$	0	0
$183$	30.7651	2.27422
$184$	0	0
$185$	29.0282	2.13420
$186$	0	0
$187$	−1.31963	−0.0965006
$188$	0	0
$189$	−0.765692	−0.0556959
$190$	0	0
$191$	−10.7889	−0.780659	−0.390330	−	0.920675i	$-0.627639\pi$
$191$	−10.7889	−0.780659	−0.390330	+	0.920675i	$0.627639\pi$
$192$	0	0
$193$	−9.02658	−0.649748	−0.324874	−	0.945757i	$-0.605322\pi$
$193$	−9.02658	−0.649748	−0.324874	+	0.945757i	$0.605322\pi$
$194$	0	0
$195$	9.84066	0.704704
$196$	0	0
$197$	2.70507	0.192728	0.0963642	−	0.995346i	$-0.469279\pi$
$197$	2.70507	0.192728	0.0963642	+	0.995346i	$0.469279\pi$
$198$	0	0
$199$	25.4749	1.80587	0.902934	−	0.429780i	$-0.141409\pi$
$199$	25.4749	1.80587	0.902934	+	0.429780i	$0.141409\pi$
$200$	0	0
$201$	−0.876517	−0.0618248
$202$	0	0
$203$	−2.76325	−0.193942
$204$	0	0
$205$	10.2520	0.716028
$206$	0	0
$207$	8.31685	0.578061
$208$	0	0
$209$	7.60564	0.526093
$210$	0	0
$211$	−13.0462	−0.898137	−0.449068	−	0.893497i	$-0.648244\pi$
$211$	−13.0462	−0.898137	−0.449068	+	0.893497i	$0.648244\pi$
$212$	0	0
$213$	−23.3173	−1.59768
$214$	0	0
$215$	26.3293	1.79564
$216$	0	0
$217$	−1.80742	−0.122696
$218$	0	0
$219$	25.3744	1.71464
$220$	0	0
$221$	1.31963	0.0887677
$222$	0	0
$223$	−28.0218	−1.87648	−0.938239	−	0.345987i	$-0.887544\pi$
$223$	−28.0218	−1.87648	−0.938239	+	0.345987i	$0.887544\pi$
$224$	0	0
$225$	32.2862	2.15241
$226$	0	0
$227$	−21.5709	−1.43171	−0.715855	−	0.698249i	$-0.753963\pi$
$227$	−21.5709	−1.43171	−0.715855	+	0.698249i	$0.753963\pi$
$228$	0	0
$229$	−2.53360	−0.167425	−0.0837125	−	0.996490i	$-0.526678\pi$
$229$	−2.53360	−0.167425	−0.0837125	+	0.996490i	$0.526678\pi$
$230$	0	0
$231$	−2.38300	−0.156790
$232$	0	0
$233$	−21.0656	−1.38005	−0.690027	−	0.723783i	$-0.742401\pi$
$233$	−21.0656	−1.38005	−0.690027	+	0.723783i	$0.742401\pi$
$234$	0	0
$235$	54.3804	3.54738
$236$	0	0
$237$	31.4583	2.04343
$238$	0	0
$239$	−29.3680	−1.89966	−0.949828	−	0.312773i	$-0.898742\pi$
$239$	−29.3680	−1.89966	−0.949828	+	0.312773i	$0.898742\pi$
$240$	0	0
$241$	−11.8695	−0.764583	−0.382292	−	0.924042i	$-0.624865\pi$
$241$	−11.8695	−0.764583	−0.382292	+	0.924042i	$0.624865\pi$
$242$	0	0
$243$	21.2010	1.36004
$244$	0	0
$245$	−4.12953	−0.263826
$246$	0	0
$247$	−7.60564	−0.483935
$248$	0	0
$249$	22.0966	1.40031
$250$	0	0
$251$	8.09614	0.511024	0.255512	−	0.966806i	$-0.417756\pi$
$251$	8.09614	0.511024	0.255512	+	0.966806i	$0.417756\pi$
$252$	0	0
$253$	−3.10483	−0.195199
$254$	0	0
$255$	12.9860	0.813214
$256$	0	0
$257$	21.2832	1.32761	0.663804	−	0.747907i	$-0.268941\pi$
$257$	21.2832	1.32761	0.663804	+	0.747907i	$0.268941\pi$
$258$	0	0
$259$	7.02943	0.436788
$260$	0	0
$261$	7.40189	0.458165
$262$	0	0
$263$	−16.2926	−1.00465	−0.502324	−	0.864680i	$-0.667521\pi$
$263$	−16.2926	−1.00465	−0.502324	+	0.864680i	$0.667521\pi$
$264$	0	0
$265$	13.4691	0.827399
$266$	0	0
$267$	7.44070	0.455364
$268$	0	0
$269$	−2.84260	−0.173316	−0.0866581	−	0.996238i	$-0.527619\pi$
$269$	−2.84260	−0.173316	−0.0866581	+	0.996238i	$0.527619\pi$
$270$	0	0
$271$	−26.6277	−1.61752	−0.808758	−	0.588142i	$-0.799860\pi$
$271$	−26.6277	−1.61752	−0.808758	+	0.588142i	$0.799860\pi$
$272$	0	0
$273$	2.38300	0.144226
$274$	0	0
$275$	−12.0530	−0.726823
$276$	0	0
$277$	1.98116	0.119037	0.0595183	−	0.998227i	$-0.481044\pi$
$277$	1.98116	0.119037	0.0595183	+	0.998227i	$0.481044\pi$
$278$	0	0
$279$	4.84151	0.289854
$280$	0	0
$281$	4.59238	0.273959	0.136979	−	0.990574i	$-0.456261\pi$
$281$	4.59238	0.273959	0.136979	+	0.990574i	$0.456261\pi$
$282$	0	0
$283$	−25.1266	−1.49362	−0.746810	−	0.665037i	$-0.768416\pi$
$283$	−25.1266	−1.49362	−0.746810	+	0.665037i	$0.768416\pi$
$284$	0	0
$285$	−74.8445	−4.43340
$286$	0	0
$287$	2.48260	0.146543
$288$	0	0
$289$	−15.2586	−0.897564
$290$	0	0
$291$	−37.7690	−2.21406
$292$	0	0
$293$	13.6889	0.799717	0.399858	−	0.916577i	$-0.369059\pi$
$293$	13.6889	0.799717	0.399858	+	0.916577i	$0.369059\pi$
$294$	0	0
$295$	27.8263	1.62011
$296$	0	0
$297$	−0.765692	−0.0444299
$298$	0	0
$299$	3.10483	0.179557
$300$	0	0
$301$	6.37587	0.367499
$302$	0	0
$303$	29.1059	1.67209
$304$	0	0
$305$	53.3132	3.05270
$306$	0	0
$307$	−24.0130	−1.37050	−0.685248	−	0.728310i	$-0.740306\pi$
$307$	−24.0130	−1.37050	−0.685248	+	0.728310i	$0.740306\pi$
$308$	0	0
$309$	−20.2822	−1.15381
$310$	0	0
$311$	2.74936	0.155902	0.0779508	−	0.996957i	$-0.475162\pi$
$311$	2.74936	0.155902	0.0779508	+	0.996957i	$0.475162\pi$
$312$	0	0
$313$	14.5595	0.822952	0.411476	−	0.911421i	$-0.365013\pi$
$313$	14.5595	0.822952	0.411476	+	0.911421i	$0.365013\pi$
$314$	0	0
$315$	11.0617	0.623256
$316$	0	0
$317$	1.70097	0.0955360	0.0477680	−	0.998858i	$-0.484789\pi$
$317$	1.70097	0.0955360	0.0477680	+	0.998858i	$0.484789\pi$
$318$	0	0
$319$	−2.76325	−0.154712
$320$	0	0
$321$	26.6852	1.48942
$322$	0	0
$323$	−10.0366	−0.558451
$324$	0	0
$325$	12.0530	0.668580
$326$	0	0
$327$	−29.4193	−1.62689
$328$	0	0
$329$	13.1687	0.726012
$330$	0	0
$331$	−8.37913	−0.460559	−0.230279	−	0.973125i	$-0.573964\pi$
$331$	−8.37913	−0.460559	−0.230279	+	0.973125i	$0.573964\pi$
$332$	0	0
$333$	−18.8296	−1.03186
$334$	0	0
$335$	−1.51893	−0.0829878
$336$	0	0
$337$	31.9631	1.74114	0.870570	−	0.492045i	$-0.163751\pi$
$337$	31.9631	1.74114	0.870570	+	0.492045i	$0.163751\pi$
$338$	0	0
$339$	14.8729	0.807784
$340$	0	0
$341$	−1.80742	−0.0978773
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	30.5535	1.64495
$346$	0	0
$347$	−2.29261	−0.123074	−0.0615370	−	0.998105i	$-0.519600\pi$
$347$	−2.29261	−0.123074	−0.0615370	+	0.998105i	$0.519600\pi$
$348$	0	0
$349$	5.30273	0.283849	0.141924	−	0.989878i	$-0.454671\pi$
$349$	5.30273	0.283849	0.141924	+	0.989878i	$0.454671\pi$
$350$	0	0
$351$	0.765692	0.0408696
$352$	0	0
$353$	7.28992	0.388003	0.194002	−	0.981001i	$-0.437853\pi$
$353$	7.28992	0.388003	0.194002	+	0.981001i	$0.437853\pi$
$354$	0	0
$355$	−40.4069	−2.14457
$356$	0	0
$357$	3.14467	0.166433
$358$	0	0
$359$	−10.6492	−0.562046	−0.281023	−	0.959701i	$-0.590674\pi$
$359$	−10.6492	−0.562046	−0.281023	+	0.959701i	$0.590674\pi$
$360$	0	0
$361$	38.8457	2.04451
$362$	0	0
$363$	−2.38300	−0.125075
$364$	0	0
$365$	43.9716	2.30158
$366$	0	0
$367$	19.8805	1.03776	0.518878	−	0.854848i	$-0.326350\pi$
$367$	19.8805	1.03776	0.518878	+	0.854848i	$0.326350\pi$
$368$	0	0
$369$	−6.65010	−0.346190
$370$	0	0
$371$	3.26165	0.169337
$372$	0	0
$373$	14.2762	0.739192	0.369596	−	0.929193i	$-0.379496\pi$
$373$	14.2762	0.739192	0.369596	+	0.929193i	$0.379496\pi$
$374$	0	0
$375$	69.4061	3.58411
$376$	0	0
$377$	2.76325	0.142315
$378$	0	0
$379$	1.61778	0.0830996	0.0415498	−	0.999136i	$-0.486770\pi$
$379$	1.61778	0.0830996	0.0415498	+	0.999136i	$0.486770\pi$
$380$	0	0
$381$	23.6322	1.21071
$382$	0	0
$383$	31.9602	1.63309	0.816544	−	0.577284i	$-0.195887\pi$
$383$	31.9602	1.63309	0.816544	+	0.577284i	$0.195887\pi$
$384$	0	0
$385$	−4.12953	−0.210460
$386$	0	0
$387$	−17.0790	−0.868172
$388$	0	0
$389$	16.1620	0.819445	0.409722	−	0.912210i	$-0.365626\pi$
$389$	16.1620	0.819445	0.409722	+	0.912210i	$0.365626\pi$
$390$	0	0
$391$	4.09721	0.207205
$392$	0	0
$393$	−33.7661	−1.70327
$394$	0	0
$395$	54.5144	2.74292
$396$	0	0
$397$	−4.25533	−0.213569	−0.106784	−	0.994282i	$-0.534055\pi$
$397$	−4.25533	−0.213569	−0.106784	+	0.994282i	$0.534055\pi$
$398$	0	0
$399$	−18.1242	−0.907346
$400$	0	0
$401$	−3.67307	−0.183425	−0.0917123	−	0.995786i	$-0.529234\pi$
$401$	−3.67307	−0.183425	−0.0917123	+	0.995786i	$0.529234\pi$
$402$	0	0
$403$	1.80742	0.0900340
$404$	0	0
$405$	40.7200	2.02339
$406$	0	0
$407$	7.02943	0.348436
$408$	0	0
$409$	−36.4619	−1.80292	−0.901462	−	0.432859i	$-0.857505\pi$
$409$	−36.4619	−1.80292	−0.901462	+	0.432859i	$0.857505\pi$
$410$	0	0
$411$	22.4231	1.10605
$412$	0	0
$413$	6.73839	0.331574
$414$	0	0
$415$	38.2914	1.87965
$416$	0	0
$417$	44.7285	2.19036
$418$	0	0
$419$	16.0477	0.783981	0.391991	−	0.919969i	$-0.371787\pi$
$419$	16.0477	0.783981	0.391991	+	0.919969i	$0.371787\pi$
$420$	0	0
$421$	14.8661	0.724528	0.362264	−	0.932076i	$-0.382004\pi$
$421$	14.8661	0.724528	0.362264	+	0.932076i	$0.382004\pi$
$422$	0	0
$423$	−35.2747	−1.71512
$424$	0	0
$425$	15.9054	0.771527
$426$	0	0
$427$	12.9102	0.624770
$428$	0	0
$429$	2.38300	0.115052
$430$	0	0
$431$	35.2548	1.69817	0.849083	−	0.528259i	$-0.177155\pi$
$431$	35.2548	1.69817	0.849083	+	0.528259i	$0.177155\pi$
$432$	0	0
$433$	23.2659	1.11809	0.559044	−	0.829138i	$-0.311168\pi$
$433$	23.2659	1.11809	0.559044	+	0.829138i	$0.311168\pi$
$434$	0	0
$435$	27.1922	1.30377
$436$	0	0
$437$	−23.6142	−1.12962
$438$	0	0
$439$	−20.3967	−0.973479	−0.486740	−	0.873547i	$-0.661814\pi$
$439$	−20.3967	−0.973479	−0.486740	+	0.873547i	$0.661814\pi$
$440$	0	0
$441$	2.67869	0.127556
$442$	0	0
$443$	22.6525	1.07625	0.538125	−	0.842865i	$-0.319133\pi$
$443$	22.6525	1.07625	0.538125	+	0.842865i	$0.319133\pi$
$444$	0	0
$445$	12.8941	0.611238
$446$	0	0
$447$	−20.4942	−0.969341
$448$	0	0
$449$	28.3603	1.33841	0.669203	−	0.743080i	$-0.266636\pi$
$449$	28.3603	1.33841	0.669203	+	0.743080i	$0.266636\pi$
$450$	0	0
$451$	2.48260	0.116901
$452$	0	0
$453$	20.4837	0.962409
$454$	0	0
$455$	4.12953	0.193595
$456$	0	0
$457$	18.3825	0.859897	0.429949	−	0.902853i	$-0.358532\pi$
$457$	18.3825	0.859897	0.429949	+	0.902853i	$0.358532\pi$
$458$	0	0
$459$	1.01043	0.0471627
$460$	0	0
$461$	19.4658	0.906611	0.453306	−	0.891355i	$-0.350245\pi$
$461$	19.4658	0.906611	0.453306	+	0.891355i	$0.350245\pi$
$462$	0	0
$463$	31.7150	1.47392	0.736962	−	0.675935i	$-0.236260\pi$
$463$	31.7150	1.47392	0.736962	+	0.675935i	$0.236260\pi$
$464$	0	0
$465$	17.7862	0.824816
$466$	0	0
$467$	19.7391	0.913415	0.456708	−	0.889617i	$-0.349029\pi$
$467$	19.7391	0.913415	0.456708	+	0.889617i	$0.349029\pi$
$468$	0	0
$469$	−0.367821	−0.0169844
$470$	0	0
$471$	−31.5865	−1.45543
$472$	0	0
$473$	6.37587	0.293163
$474$	0	0
$475$	−91.6707	−4.20614
$476$	0	0
$477$	−8.73695	−0.400037
$478$	0	0
$479$	19.8051	0.904919	0.452459	−	0.891785i	$-0.350547\pi$
$479$	19.8051	0.904919	0.452459	+	0.891785i	$0.350547\pi$
$480$	0	0
$481$	−7.02943	−0.320514
$482$	0	0
$483$	7.39880	0.336657
$484$	0	0
$485$	−65.4502	−2.97194
$486$	0	0
$487$	−28.5640	−1.29436	−0.647179	−	0.762338i	$-0.724052\pi$
$487$	−28.5640	−1.29436	−0.647179	+	0.762338i	$0.724052\pi$
$488$	0	0
$489$	−13.0072	−0.588208
$490$	0	0
$491$	4.33540	0.195654	0.0978270	−	0.995203i	$-0.468811\pi$
$491$	4.33540	0.195654	0.0978270	+	0.995203i	$0.468811\pi$
$492$	0	0
$493$	3.64646	0.164228
$494$	0	0
$495$	11.0617	0.497187
$496$	0	0
$497$	−9.78486	−0.438911
$498$	0	0
$499$	−37.4890	−1.67824	−0.839119	−	0.543948i	$-0.816929\pi$
$499$	−37.4890	−1.67824	−0.839119	+	0.543948i	$0.816929\pi$
$500$	0	0
$501$	19.7788	0.883651
$502$	0	0
$503$	−33.3876	−1.48868	−0.744339	−	0.667802i	$-0.767235\pi$
$503$	−33.3876	−1.48868	−0.744339	+	0.667802i	$0.767235\pi$
$504$	0	0
$505$	50.4380	2.24446
$506$	0	0
$507$	−2.38300	−0.105833
$508$	0	0
$509$	−17.5923	−0.779765	−0.389883	−	0.920865i	$-0.627484\pi$
$509$	−17.5923	−0.779765	−0.389883	+	0.920865i	$0.627484\pi$
$510$	0	0
$511$	10.6481	0.471044
$512$	0	0
$513$	−5.82357	−0.257117
$514$	0	0
$515$	−35.1473	−1.54877
$516$	0	0
$517$	13.1687	0.579157
$518$	0	0
$519$	−18.0986	−0.794440
$520$	0	0
$521$	−34.3912	−1.50670	−0.753352	−	0.657617i	$-0.771564\pi$
$521$	−34.3912	−1.50670	−0.753352	+	0.657617i	$0.771564\pi$
$522$	0	0
$523$	−12.3505	−0.540050	−0.270025	−	0.962853i	$-0.587032\pi$
$523$	−12.3505	−0.540050	−0.270025	+	0.962853i	$0.587032\pi$
$524$	0	0
$525$	28.7223	1.25354
$526$	0	0
$527$	2.38512	0.103897
$528$	0	0
$529$	−13.3601	−0.580872
$530$	0	0
$531$	−18.0500	−0.783304
$532$	0	0
$533$	−2.48260	−0.107533
$534$	0	0
$535$	46.2431	1.99926
$536$	0	0
$537$	−49.8389	−2.15071
$538$	0	0
$539$	−1.00000	−0.0430730
$540$	0	0
$541$	−27.0080	−1.16116	−0.580582	−	0.814202i	$-0.697175\pi$
$541$	−27.0080	−1.16116	−0.580582	+	0.814202i	$0.697175\pi$
$542$	0	0
$543$	46.0584	1.97655
$544$	0	0
$545$	−50.9811	−2.18379
$546$	0	0
$547$	−21.8043	−0.932283	−0.466142	−	0.884710i	$-0.654356\pi$
$547$	−21.8043	−0.932283	−0.466142	+	0.884710i	$0.654356\pi$
$548$	0	0
$549$	−34.5825	−1.47594
$550$	0	0
$551$	−21.0163	−0.895324
$552$	0	0
$553$	13.2011	0.561369
$554$	0	0
$555$	−69.1742	−2.93628
$556$	0	0
$557$	42.2458	1.79001	0.895006	−	0.446054i	$-0.147171\pi$
$557$	42.2458	1.79001	0.895006	+	0.446054i	$0.147171\pi$
$558$	0	0
$559$	−6.37587	−0.269671
$560$	0	0
$561$	3.14467	0.132768
$562$	0	0
$563$	6.61243	0.278681	0.139340	−	0.990245i	$-0.455502\pi$
$563$	6.61243	0.278681	0.139340	+	0.990245i	$0.455502\pi$
$564$	0	0
$565$	25.7734	1.08429
$566$	0	0
$567$	9.86070	0.414110
$568$	0	0
$569$	27.8366	1.16697	0.583485	−	0.812124i	$-0.301689\pi$
$569$	27.8366	1.16697	0.583485	+	0.812124i	$0.301689\pi$
$570$	0	0
$571$	23.8527	0.998203	0.499102	−	0.866543i	$-0.333663\pi$
$571$	23.8527	0.998203	0.499102	+	0.866543i	$0.333663\pi$
$572$	0	0
$573$	25.7100	1.07405
$574$	0	0
$575$	37.4224	1.56062
$576$	0	0
$577$	25.8409	1.07577	0.537886	−	0.843018i	$-0.319223\pi$
$577$	25.8409	1.07577	0.537886	+	0.843018i	$0.319223\pi$
$578$	0	0
$579$	21.5103	0.893939
$580$	0	0
$581$	9.27258	0.384692
$582$	0	0
$583$	3.26165	0.135084
$584$	0	0
$585$	−11.0617	−0.457345
$586$	0	0
$587$	28.3623	1.17064	0.585319	−	0.810803i	$-0.300969\pi$
$587$	28.3623	1.17064	0.585319	+	0.810803i	$0.300969\pi$
$588$	0	0
$589$	−13.7466	−0.566418
$590$	0	0
$591$	−6.44618	−0.265161
$592$	0	0
$593$	9.61815	0.394970	0.197485	−	0.980306i	$-0.436723\pi$
$593$	9.61815	0.394970	0.197485	+	0.980306i	$0.436723\pi$
$594$	0	0
$595$	5.44943	0.223405
$596$	0	0
$597$	−60.7067	−2.48456
$598$	0	0
$599$	39.1503	1.59964	0.799820	−	0.600241i	$-0.204928\pi$
$599$	39.1503	1.59964	0.799820	+	0.600241i	$0.204928\pi$
$600$	0	0
$601$	22.4511	0.915801	0.457901	−	0.889003i	$-0.348602\pi$
$601$	22.4511	0.915801	0.457901	+	0.889003i	$0.348602\pi$
$602$	0	0
$603$	0.985277	0.0401236
$604$	0	0
$605$	−4.12953	−0.167889
$606$	0	0
$607$	−14.8149	−0.601318	−0.300659	−	0.953732i	$-0.597207\pi$
$607$	−14.8149	−0.601318	−0.300659	+	0.953732i	$0.597207\pi$
$608$	0	0
$609$	6.58483	0.266831
$610$	0	0
$611$	−13.1687	−0.532747
$612$	0	0
$613$	4.38049	0.176926	0.0884632	−	0.996079i	$-0.471804\pi$
$613$	4.38049	0.176926	0.0884632	+	0.996079i	$0.471804\pi$
$614$	0	0
$615$	−24.4304	−0.985129
$616$	0	0
$617$	−31.7412	−1.27785	−0.638926	−	0.769268i	$-0.720621\pi$
$617$	−31.7412	−1.27785	−0.638926	+	0.769268i	$0.720621\pi$
$618$	0	0
$619$	−23.4688	−0.943293	−0.471646	−	0.881788i	$-0.656340\pi$
$619$	−23.4688	−0.943293	−0.471646	+	0.881788i	$0.656340\pi$
$620$	0	0
$621$	2.37734	0.0953993
$622$	0	0
$623$	3.12241	0.125097
$624$	0	0
$625$	60.0096	2.40038
$626$	0	0
$627$	−18.1242	−0.723812
$628$	0	0
$629$	−9.27622	−0.369867
$630$	0	0
$631$	9.11944	0.363039	0.181520	−	0.983387i	$-0.441898\pi$
$631$	9.11944	0.363039	0.181520	+	0.983387i	$0.441898\pi$
$632$	0	0
$633$	31.0891	1.23568
$634$	0	0
$635$	40.9525	1.62515
$636$	0	0
$637$	1.00000	0.0396214
$638$	0	0
$639$	26.2106	1.03687
$640$	0	0
$641$	−28.3898	−1.12133	−0.560664	−	0.828043i	$-0.689454\pi$
$641$	−28.3898	−1.12133	−0.560664	+	0.828043i	$0.689454\pi$
$642$	0	0
$643$	−46.8478	−1.84750	−0.923748	−	0.383000i	$-0.874891\pi$
$643$	−46.8478	−1.84750	−0.923748	+	0.383000i	$0.874891\pi$
$644$	0	0
$645$	−62.7428	−2.47049
$646$	0	0
$647$	10.4983	0.412729	0.206364	−	0.978475i	$-0.433837\pi$
$647$	10.4983	0.412729	0.206364	+	0.978475i	$0.433837\pi$
$648$	0	0
$649$	6.73839	0.264505
$650$	0	0
$651$	4.30708	0.168808
$652$	0	0
$653$	27.2132	1.06493	0.532467	−	0.846451i	$-0.321265\pi$
$653$	27.2132	1.06493	0.532467	+	0.846451i	$0.321265\pi$
$654$	0	0
$655$	−58.5137	−2.28632
$656$	0	0
$657$	−28.5229	−1.11278
$658$	0	0
$659$	−30.9420	−1.20533	−0.602664	−	0.797995i	$-0.705894\pi$
$659$	−30.9420	−1.20533	−0.602664	+	0.797995i	$0.705894\pi$
$660$	0	0
$661$	−12.0855	−0.470071	−0.235036	−	0.971987i	$-0.575521\pi$
$661$	−12.0855	−0.470071	−0.235036	+	0.971987i	$0.575521\pi$
$662$	0	0
$663$	−3.14467	−0.122129
$664$	0	0
$665$	−31.4077	−1.21794
$666$	0	0
$667$	8.57942	0.332196
$668$	0	0
$669$	66.7759	2.58171
$670$	0	0
$671$	12.9102	0.498394
$672$	0	0
$673$	−36.8691	−1.42120	−0.710599	−	0.703597i	$-0.751576\pi$
$673$	−36.8691	−1.42120	−0.710599	+	0.703597i	$0.751576\pi$
$674$	0	0
$675$	9.22887	0.355219
$676$	0	0
$677$	10.7715	0.413984	0.206992	−	0.978343i	$-0.433633\pi$
$677$	10.7715	0.413984	0.206992	+	0.978343i	$0.433633\pi$
$678$	0	0
$679$	−15.8493	−0.608242
$680$	0	0
$681$	51.4034	1.96978
$682$	0	0
$683$	−17.8073	−0.681376	−0.340688	−	0.940176i	$-0.610660\pi$
$683$	−17.8073	−0.681376	−0.340688	+	0.940176i	$0.610660\pi$
$684$	0	0
$685$	38.8573	1.48466
$686$	0	0
$687$	6.03756	0.230347
$688$	0	0
$689$	−3.26165	−0.124259
$690$	0	0
$691$	−45.5635	−1.73332	−0.866659	−	0.498901i	$-0.833737\pi$
$691$	−45.5635	−1.73332	−0.866659	+	0.498901i	$0.833737\pi$
$692$	0	0
$693$	2.67869	0.101755
$694$	0	0
$695$	77.5105	2.94014
$696$	0	0
$697$	−3.27610	−0.124091
$698$	0	0
$699$	50.1993	1.89871
$700$	0	0
$701$	−9.32725	−0.352285	−0.176143	−	0.984365i	$-0.556362\pi$
$701$	−9.32725	−0.352285	−0.176143	+	0.984365i	$0.556362\pi$
$702$	0	0
$703$	53.4633	2.01641
$704$	0	0
$705$	−129.588	−4.88058
$706$	0	0
$707$	12.2140	0.459355
$708$	0	0
$709$	16.0470	0.602656	0.301328	−	0.953521i	$-0.402570\pi$
$709$	16.0470	0.602656	0.301328	+	0.953521i	$0.402570\pi$
$710$	0	0
$711$	−35.3617	−1.32617
$712$	0	0
$713$	5.61173	0.210161
$714$	0	0
$715$	4.12953	0.154436
$716$	0	0
$717$	69.9838	2.61359
$718$	0	0
$719$	7.43191	0.277163	0.138582	−	0.990351i	$-0.455746\pi$
$719$	7.43191	0.277163	0.138582	+	0.990351i	$0.455746\pi$
$720$	0	0
$721$	−8.51121	−0.316974
$722$	0	0
$723$	28.2851	1.05193
$724$	0	0
$725$	33.3055	1.23693
$726$	0	0
$727$	23.6021	0.875352	0.437676	−	0.899133i	$-0.355802\pi$
$727$	23.6021	0.875352	0.437676	+	0.899133i	$0.355802\pi$
$728$	0	0
$729$	−20.9398	−0.775548
$730$	0	0
$731$	−8.41376	−0.311194
$732$	0	0
$733$	−11.4296	−0.422161	−0.211081	−	0.977469i	$-0.567698\pi$
$733$	−11.4296	−0.422161	−0.211081	+	0.977469i	$0.567698\pi$
$734$	0	0
$735$	9.84066	0.362978
$736$	0	0
$737$	−0.367821	−0.0135489
$738$	0	0
$739$	−38.7671	−1.42607	−0.713035	−	0.701128i	$-0.752680\pi$
$739$	−38.7671	−1.42607	−0.713035	+	0.701128i	$0.752680\pi$
$740$	0	0
$741$	18.1242	0.665810
$742$	0	0
$743$	−21.5813	−0.791741	−0.395871	−	0.918306i	$-0.629557\pi$
$743$	−21.5813	−0.791741	−0.395871	+	0.918306i	$0.629557\pi$
$744$	0	0
$745$	−35.5146	−1.30115
$746$	0	0
$747$	−24.8383	−0.908787
$748$	0	0
$749$	11.1982	0.409172
$750$	0	0
$751$	−32.4851	−1.18540	−0.592699	−	0.805424i	$-0.701938\pi$
$751$	−32.4851	−1.18540	−0.592699	+	0.805424i	$0.701938\pi$
$752$	0	0
$753$	−19.2931	−0.703079
$754$	0	0
$755$	35.4964	1.29185
$756$	0	0
$757$	−7.89858	−0.287079	−0.143539	−	0.989645i	$-0.545848\pi$
$757$	−7.89858	−0.287079	−0.143539	+	0.989645i	$0.545848\pi$
$758$	0	0
$759$	7.39880	0.268559
$760$	0	0
$761$	−3.52347	−0.127726	−0.0638628	−	0.997959i	$-0.520342\pi$
$761$	−3.52347	−0.127726	−0.0638628	+	0.997959i	$0.520342\pi$
$762$	0	0
$763$	−12.3455	−0.446937
$764$	0	0
$765$	−14.5973	−0.527767
$766$	0	0
$767$	−6.73839	−0.243309
$768$	0	0
$769$	13.7748	0.496733	0.248366	−	0.968666i	$-0.420106\pi$
$769$	13.7748	0.496733	0.248366	+	0.968666i	$0.420106\pi$
$770$	0	0
$771$	−50.7178	−1.82656
$772$	0	0
$773$	−9.56538	−0.344043	−0.172021	−	0.985093i	$-0.555030\pi$
$773$	−9.56538	−0.344043	−0.172021	+	0.985093i	$0.555030\pi$
$774$	0	0
$775$	21.7848	0.782534
$776$	0	0
$777$	−16.7511	−0.600943
$778$	0	0
$779$	18.8817	0.676509
$780$	0	0
$781$	−9.78486	−0.350130
$782$	0	0
$783$	2.11580	0.0756125
$784$	0	0
$785$	−54.7365	−1.95363
$786$	0	0
$787$	−13.7142	−0.488858	−0.244429	−	0.969667i	$-0.578600\pi$
$787$	−13.7142	−0.488858	−0.244429	+	0.969667i	$0.578600\pi$
$788$	0	0
$789$	38.8253	1.38222
$790$	0	0
$791$	6.24125	0.221913
$792$	0	0
$793$	−12.9102	−0.458456
$794$	0	0
$795$	−32.0968	−1.13836
$796$	0	0
$797$	14.0085	0.496208	0.248104	−	0.968733i	$-0.420193\pi$
$797$	14.0085	0.496208	0.248104	+	0.968733i	$0.420193\pi$
$798$	0	0
$799$	−17.3777	−0.614780
$800$	0	0
$801$	−8.36396	−0.295526
$802$	0	0
$803$	10.6481	0.375763
$804$	0	0
$805$	12.8215	0.451897
$806$	0	0
$807$	6.77391	0.238453
$808$	0	0
$809$	−9.88445	−0.347519	−0.173759	−	0.984788i	$-0.555591\pi$
$809$	−9.88445	−0.347519	−0.173759	+	0.984788i	$0.555591\pi$
$810$	0	0
$811$	−11.1730	−0.392337	−0.196168	−	0.980570i	$-0.562850\pi$
$811$	−11.1730	−0.392337	−0.196168	+	0.980570i	$0.562850\pi$
$812$	0	0
$813$	63.4537	2.22542
$814$	0	0
$815$	−22.5404	−0.789555
$816$	0	0
$817$	48.4925	1.69654
$818$	0	0
$819$	−2.67869	−0.0936009
$820$	0	0
$821$	22.5226	0.786046	0.393023	−	0.919529i	$-0.371429\pi$
$821$	22.5226	0.786046	0.393023	+	0.919529i	$0.371429\pi$
$822$	0	0
$823$	−9.76449	−0.340369	−0.170184	−	0.985412i	$-0.554436\pi$
$823$	−9.76449	−0.340369	−0.170184	+	0.985412i	$0.554436\pi$
$824$	0	0
$825$	28.7223	0.999981
$826$	0	0
$827$	20.7005	0.719827	0.359914	−	0.932986i	$-0.382806\pi$
$827$	20.7005	0.719827	0.359914	+	0.932986i	$0.382806\pi$
$828$	0	0
$829$	−12.8243	−0.445408	−0.222704	−	0.974886i	$-0.571488\pi$
$829$	−12.8243	−0.445408	−0.222704	+	0.974886i	$0.571488\pi$
$830$	0	0
$831$	−4.72111	−0.163774
$832$	0	0
$833$	1.31963	0.0457223
$834$	0	0
$835$	34.2749	1.18613
$836$	0	0
$837$	1.38393	0.0478355
$838$	0	0
$839$	−25.6033	−0.883925	−0.441962	−	0.897034i	$-0.645718\pi$
$839$	−25.6033	−0.883925	−0.441962	+	0.897034i	$0.645718\pi$
$840$	0	0
$841$	−21.3644	−0.736705
$842$	0	0
$843$	−10.9436	−0.376919
$844$	0	0
$845$	−4.12953	−0.142060
$846$	0	0
$847$	−1.00000	−0.0343604
$848$	0	0
$849$	59.8766	2.05496
$850$	0	0
$851$	−21.8252	−0.748157
$852$	0	0
$853$	33.2906	1.13985	0.569925	−	0.821697i	$-0.306972\pi$
$853$	33.2906	1.13985	0.569925	+	0.821697i	$0.306972\pi$
$854$	0	0
$855$	84.1313	2.87723
$856$	0	0
$857$	7.53823	0.257501	0.128750	−	0.991677i	$-0.458903\pi$
$857$	7.53823	0.257501	0.128750	+	0.991677i	$0.458903\pi$
$858$	0	0
$859$	−40.2738	−1.37413	−0.687063	−	0.726598i	$-0.741100\pi$
$859$	−40.2738	−1.37413	−0.687063	+	0.726598i	$0.741100\pi$
$860$	0	0
$861$	−5.91603	−0.201618
$862$	0	0
$863$	−48.7883	−1.66077	−0.830386	−	0.557188i	$-0.811880\pi$
$863$	−48.7883	−1.66077	−0.830386	+	0.557188i	$0.811880\pi$
$864$	0	0
$865$	−31.3633	−1.06638
$866$	0	0
$867$	36.3612	1.23489
$868$	0	0
$869$	13.2011	0.447817
$870$	0	0
$871$	0.367821	0.0124631
$872$	0	0
$873$	42.4554	1.43690
$874$	0	0
$875$	29.1255	0.984622
$876$	0	0
$877$	−46.3037	−1.56357	−0.781783	−	0.623551i	$-0.785690\pi$
$877$	−46.3037	−1.56357	−0.781783	+	0.623551i	$0.785690\pi$
$878$	0	0
$879$	−32.6208	−1.10027
$880$	0	0
$881$	−30.1314	−1.01515	−0.507577	−	0.861606i	$-0.669459\pi$
$881$	−30.1314	−1.01515	−0.507577	+	0.861606i	$0.669459\pi$
$882$	0	0
$883$	−25.8259	−0.869112	−0.434556	−	0.900645i	$-0.643095\pi$
$883$	−25.8259	−0.869112	−0.434556	+	0.900645i	$0.643095\pi$
$884$	0	0
$885$	−66.3102	−2.22899
$886$	0	0
$887$	5.72015	0.192064	0.0960320	−	0.995378i	$-0.469385\pi$
$887$	5.72015	0.192064	0.0960320	+	0.995378i	$0.469385\pi$
$888$	0	0
$889$	9.91700	0.332605
$890$	0	0
$891$	9.86070	0.330346
$892$	0	0
$893$	100.156	3.35160
$894$	0	0
$895$	−86.3663	−2.88691
$896$	0	0
$897$	−7.39880	−0.247039
$898$	0	0
$899$	4.99436	0.166571
$900$	0	0
$901$	−4.30416	−0.143392
$902$	0	0
$903$	−15.1937	−0.505614
$904$	0	0
$905$	79.8151	2.65314
$906$	0	0
$907$	16.8955	0.561005	0.280502	−	0.959853i	$-0.409499\pi$
$907$	16.8955	0.561005	0.280502	+	0.959853i	$0.409499\pi$
$908$	0	0
$909$	−32.7175	−1.08517
$910$	0	0
$911$	33.4662	1.10879	0.554393	−	0.832255i	$-0.312950\pi$
$911$	33.4662	1.10879	0.554393	+	0.832255i	$0.312950\pi$
$912$	0	0
$913$	9.27258	0.306878
$914$	0	0
$915$	−127.045	−4.19999
$916$	0	0
$917$	−14.1696	−0.467921
$918$	0	0
$919$	−6.11858	−0.201833	−0.100917	−	0.994895i	$-0.532178\pi$
$919$	−6.11858	−0.201833	−0.100917	+	0.994895i	$0.532178\pi$
$920$	0	0
$921$	57.2230	1.88556
$922$	0	0
$923$	9.78486	0.322073
$924$	0	0
$925$	−84.7257	−2.78576
$926$	0	0
$927$	22.7989	0.748812
$928$	0	0
$929$	20.7544	0.680930	0.340465	−	0.940257i	$-0.389416\pi$
$929$	20.7544	0.680930	0.340465	+	0.940257i	$0.389416\pi$
$930$	0	0
$931$	−7.60564	−0.249265
$932$	0	0
$933$	−6.55171	−0.214493
$934$	0	0
$935$	5.44943	0.178215
$936$	0	0
$937$	−15.9243	−0.520225	−0.260112	−	0.965578i	$-0.583760\pi$
$937$	−15.9243	−0.520225	−0.260112	+	0.965578i	$0.583760\pi$
$938$	0	0
$939$	−34.6953	−1.13224
$940$	0	0
$941$	13.7973	0.449781	0.224890	−	0.974384i	$-0.427798\pi$
$941$	13.7973	0.449781	0.224890	+	0.974384i	$0.427798\pi$
$942$	0	0
$943$	−7.70804	−0.251008
$944$	0	0
$945$	3.16194	0.102858
$946$	0	0
$947$	32.4473	1.05440	0.527199	−	0.849742i	$-0.323242\pi$
$947$	32.4473	1.05440	0.527199	+	0.849742i	$0.323242\pi$
$948$	0	0
$949$	−10.6481	−0.345652
$950$	0	0
$951$	−4.05341	−0.131441
$952$	0	0
$953$	−22.8503	−0.740193	−0.370096	−	0.928993i	$-0.620675\pi$
$953$	−22.8503	−0.740193	−0.370096	+	0.928993i	$0.620675\pi$
$954$	0	0
$955$	44.5531	1.44171
$956$	0	0
$957$	6.58483	0.212857
$958$	0	0
$959$	9.40961	0.303852
$960$	0	0
$961$	−27.7332	−0.894620
$962$	0	0
$963$	−29.9964	−0.966619
$964$	0	0
$965$	37.2755	1.19994
$966$	0	0
$967$	19.2282	0.618336	0.309168	−	0.951007i	$-0.399949\pi$
$967$	19.2282	0.618336	0.309168	+	0.951007i	$0.399949\pi$
$968$	0	0
$969$	23.9172	0.768331
$970$	0	0
$971$	−47.4207	−1.52180	−0.760901	−	0.648868i	$-0.775243\pi$
$971$	−47.4207	−1.52180	−0.760901	+	0.648868i	$0.775243\pi$
$972$	0	0
$973$	18.7698	0.601733
$974$	0	0
$975$	−28.7223	−0.919849
$976$	0	0
$977$	30.7258	0.983005	0.491502	−	0.870876i	$-0.336448\pi$
$977$	30.7258	0.983005	0.491502	+	0.870876i	$0.336448\pi$
$978$	0	0
$979$	3.12241	0.0997927
$980$	0	0
$981$	33.0697	1.05584
$982$	0	0
$983$	54.3116	1.73227	0.866135	−	0.499810i	$-0.166597\pi$
$983$	54.3116	1.73227	0.866135	+	0.499810i	$0.166597\pi$
$984$	0	0
$985$	−11.1707	−0.355927
$986$	0	0
$987$	−31.3809	−0.998866
$988$	0	0
$989$	−19.7960	−0.629475
$990$	0	0
$991$	−52.5008	−1.66774	−0.833871	−	0.551960i	$-0.813880\pi$
$991$	−52.5008	−1.66774	−0.833871	+	0.551960i	$0.813880\pi$
$992$	0	0
$993$	19.9675	0.633648
$994$	0	0
$995$	−105.199	−3.33504
$996$	0	0
$997$	13.7985	0.437002	0.218501	−	0.975837i	$-0.429883\pi$
$997$	13.7985	0.437002	0.218501	+	0.975837i	$0.429883\pi$
$998$	0	0
$999$	−5.38238	−0.170291

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8008.2.a.z.1.3	✓	15

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8008.2.a.z.1.3	✓	15	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:	\( N \)	\(=\)	\( 8008 = 2^{3} \cdot 7 \cdot 11 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8008.a (trivial)

\(f(q)\)	\(=\)	\(q-2.38300 q^{3} -4.12953 q^{5} -1.00000 q^{7} +2.67869 q^{9} +O(q^{10})\)	\(q-2.38300 q^{3} -4.12953 q^{5} -1.00000 q^{7} +2.67869 q^{9} -1.00000 q^{11} +1.00000 q^{13} +9.84066 q^{15} +1.31963 q^{17} -7.60564 q^{19} +2.38300 q^{21} +3.10483 q^{23} +12.0530 q^{25} +0.765692 q^{27} +2.76325 q^{29} +1.80742 q^{31} +2.38300 q^{33} +4.12953 q^{35} -7.02943 q^{37} -2.38300 q^{39} -2.48260 q^{41} -6.37587 q^{43} -11.0617 q^{45} -13.1687 q^{47} +1.00000 q^{49} -3.14467 q^{51} -3.26165 q^{53} +4.12953 q^{55} +18.1242 q^{57} -6.73839 q^{59} -12.9102 q^{61} -2.67869 q^{63} -4.12953 q^{65} +0.367821 q^{67} -7.39880 q^{69} +9.78486 q^{71} -10.6481 q^{73} -28.7223 q^{75} +1.00000 q^{77} -13.2011 q^{79} -9.86070 q^{81} -9.27258 q^{83} -5.44943 q^{85} -6.58483 q^{87} -3.12241 q^{89} -1.00000 q^{91} -4.30708 q^{93} +31.4077 q^{95} +15.8493 q^{97} -2.67869 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 15 q - q^{3} + 4 q^{5} - 15 q^{7} + 26 q^{9}+O(q^{10}) \) 15 * q - q^3 + 4 * q^5 - 15 * q^7 + 26 * q^9	\( 15 q - q^{3} + 4 q^{5} - 15 q^{7} + 26 q^{9} - 15 q^{11} + 15 q^{13} - 6 q^{15} + 8 q^{17} - 17 q^{19} + q^{21} + 7 q^{23} + 33 q^{25} - 4 q^{27} + 14 q^{29} - 4 q^{31} + q^{33} - 4 q^{35} + 3 q^{37} - q^{39} - 13 q^{43} + 20 q^{45} + 6 q^{47} + 15 q^{49} + 8 q^{51} + 38 q^{53} - 4 q^{55} + 24 q^{57} - 18 q^{59} + 23 q^{61} - 26 q^{63} + 4 q^{65} - 8 q^{67} + 43 q^{69} - 12 q^{71} + 11 q^{73} + 12 q^{75} + 15 q^{77} - q^{79} + 51 q^{81} - 16 q^{83} + 13 q^{85} - 25 q^{87} + 28 q^{89} - 15 q^{91} - 14 q^{93} + 49 q^{95} + 30 q^{97} - 26 q^{99}+O(q^{100}) \) 15 * q - q^3 + 4 * q^5 - 15 * q^7 + 26 * q^9 - 15 * q^11 + 15 * q^13 - 6 * q^15 + 8 * q^17 - 17 * q^19 + q^21 + 7 * q^23 + 33 * q^25 - 4 * q^27 + 14 * q^29 - 4 * q^31 + q^33 - 4 * q^35 + 3 * q^37 - q^39 - 13 * q^43 + 20 * q^45 + 6 * q^47 + 15 * q^49 + 8 * q^51 + 38 * q^53 - 4 * q^55 + 24 * q^57 - 18 * q^59 + 23 * q^61 - 26 * q^63 + 4 * q^65 - 8 * q^67 + 43 * q^69 - 12 * q^71 + 11 * q^73 + 12 * q^75 + 15 * q^77 - q^79 + 51 * q^81 - 16 * q^83 + 13 * q^85 - 25 * q^87 + 28 * q^89 - 15 * q^91 - 14 * q^93 + 49 * q^95 + 30 * q^97 - 26 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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