LMFDB - Embedded newform 8008.2.a.v.1.8

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:	$1$
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} - 2 x^{10} - 19 x^{9} + 33 x^{8} + 120 x^{7} - 178 x^{6} - 296 x^{5} + 380 x^{4} + 280 x^{3} + \cdots + 64 $ x^11 - 2x^10 - 19x^9 + 33x^8 + 120x^7 - 178x^6 - 296x^5 + 380x^4 + 280x^3 - 295x^2 - 87x + 64
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
Root			$-0.979305$ 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)

$f(q)$	$=$	$q+0.979305 q^{3} +2.33286 q^{5} -1.00000 q^{7} -2.04096 q^{9} +O(q^{10})$	$q+0.979305 q^{3} +2.33286 q^{5} -1.00000 q^{7} -2.04096 q^{9} +1.00000 q^{11} -1.00000 q^{13} +2.28458 q^{15} -2.60681 q^{17} +2.89553 q^{19} -0.979305 q^{21} +3.81847 q^{23} +0.442245 q^{25} -4.93664 q^{27} -3.10856 q^{29} -7.32534 q^{31} +0.979305 q^{33} -2.33286 q^{35} -1.46215 q^{37} -0.979305 q^{39} -0.286060 q^{41} -9.21304 q^{43} -4.76128 q^{45} +3.81847 q^{47} +1.00000 q^{49} -2.55286 q^{51} -11.8763 q^{53} +2.33286 q^{55} +2.83561 q^{57} -9.80563 q^{59} +1.26111 q^{61} +2.04096 q^{63} -2.33286 q^{65} -4.93124 q^{67} +3.73944 q^{69} +16.2180 q^{71} +0.183169 q^{73} +0.433093 q^{75} -1.00000 q^{77} -15.7288 q^{79} +1.28841 q^{81} +15.3290 q^{83} -6.08133 q^{85} -3.04423 q^{87} -10.7996 q^{89} +1.00000 q^{91} -7.17374 q^{93} +6.75487 q^{95} +4.70388 q^{97} -2.04096 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 11 q - 2 q^{3} + 2 q^{5} - 11 q^{7} + 9 q^{9}+O(q^{10}) $ 11 * q - 2 * q^3 + 2 * q^5 - 11 * q^7 + 9 * q^9	$ 11 q - 2 q^{3} + 2 q^{5} - 11 q^{7} + 9 q^{9} + 11 q^{11} - 11 q^{13} - 7 q^{15} - 4 q^{17} - 16 q^{19} + 2 q^{21} - 3 q^{23} + 11 q^{25} - 11 q^{27} + q^{29} + 14 q^{31} - 2 q^{33} - 2 q^{35} - 8 q^{37} + 2 q^{39} + 4 q^{41} - 30 q^{43} + 13 q^{45} - 3 q^{47} + 11 q^{49} - 14 q^{51} - 5 q^{53} + 2 q^{55} - 22 q^{57} + 11 q^{59} + 15 q^{61} - 9 q^{63} - 2 q^{65} - 41 q^{67} + 12 q^{69} + q^{71} - 8 q^{73} - 24 q^{75} - 11 q^{77} - 26 q^{79} + 19 q^{81} - 31 q^{83} - 27 q^{85} - 25 q^{87} + 2 q^{89} + 11 q^{91} - 37 q^{93} - 6 q^{97} + 9 q^{99}+O(q^{100}) $ 11 * q - 2 * q^3 + 2 * q^5 - 11 * q^7 + 9 * q^9 + 11 * q^11 - 11 * q^13 - 7 * q^15 - 4 * q^17 - 16 * q^19 + 2 * q^21 - 3 * q^23 + 11 * q^25 - 11 * q^27 + q^29 + 14 * q^31 - 2 * q^33 - 2 * q^35 - 8 * q^37 + 2 * q^39 + 4 * q^41 - 30 * q^43 + 13 * q^45 - 3 * q^47 + 11 * q^49 - 14 * q^51 - 5 * q^53 + 2 * q^55 - 22 * q^57 + 11 * q^59 + 15 * q^61 - 9 * q^63 - 2 * q^65 - 41 * q^67 + 12 * q^69 + q^71 - 8 * q^73 - 24 * q^75 - 11 * q^77 - 26 * q^79 + 19 * q^81 - 31 * q^83 - 27 * q^85 - 25 * q^87 + 2 * q^89 + 11 * q^91 - 37 * q^93 - 6 * q^97 + 9 * 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
$3$	0.979305	0.565402	0.282701	−	0.959208i	$-0.408770\pi$
$3$	0.979305	0.565402	0.282701	+	0.959208i	$0.408770\pi$
$4$	0	0
$5$	2.33286	1.04329	0.521644	−	0.853163i	$-0.325319\pi$
$5$	2.33286	1.04329	0.521644	+	0.853163i	$0.325319\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	−2.04096	−0.680320
$10$	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
$15$	2.28458	0.589877
$16$	0	0
$17$	−2.60681	−0.632244	−0.316122	−	0.948719i	$-0.602381\pi$
$17$	−2.60681	−0.632244	−0.316122	+	0.948719i	$0.602381\pi$
$18$	0	0
$19$	2.89553	0.664280	0.332140	−	0.943230i	$-0.392229\pi$
$19$	2.89553	0.664280	0.332140	+	0.943230i	$0.392229\pi$
$20$	0	0
$21$	−0.979305	−0.213702
$22$	0	0
$23$	3.81847	0.796205	0.398103	−	0.917341i	$-0.369669\pi$
$23$	3.81847	0.796205	0.398103	+	0.917341i	$0.369669\pi$
$24$	0	0
$25$	0.442245	0.0884490
$26$	0	0
$27$	−4.93664	−0.950057
$28$	0	0
$29$	−3.10856	−0.577245	−0.288623	−	0.957443i	$-0.593197\pi$
$29$	−3.10856	−0.577245	−0.288623	+	0.957443i	$0.593197\pi$
$30$	0	0
$31$	−7.32534	−1.31567	−0.657835	−	0.753162i	$-0.728527\pi$
$31$	−7.32534	−1.31567	−0.657835	+	0.753162i	$0.728527\pi$
$32$	0	0
$33$	0.979305	0.170475
$34$	0	0
$35$	−2.33286	−0.394326
$36$	0	0
$37$	−1.46215	−0.240376	−0.120188	−	0.992751i	$-0.538350\pi$
$37$	−1.46215	−0.240376	−0.120188	+	0.992751i	$0.538350\pi$
$38$	0	0
$39$	−0.979305	−0.156814
$40$	0	0
$41$	−0.286060	−0.0446750	−0.0223375	−	0.999750i	$-0.507111\pi$
$41$	−0.286060	−0.0446750	−0.0223375	+	0.999750i	$0.507111\pi$
$42$	0	0
$43$	−9.21304	−1.40497	−0.702487	−	0.711696i	$-0.747927\pi$
$43$	−9.21304	−1.40497	−0.702487	+	0.711696i	$0.747927\pi$
$44$	0	0
$45$	−4.76128	−0.709770
$46$	0	0
$47$	3.81847	0.556981	0.278490	−	0.960439i	$-0.410166\pi$
$47$	3.81847	0.556981	0.278490	+	0.960439i	$0.410166\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−2.55286	−0.357472
$52$	0	0
$53$	−11.8763	−1.63133	−0.815666	−	0.578523i	$-0.803629\pi$
$53$	−11.8763	−1.63133	−0.815666	+	0.578523i	$0.803629\pi$
$54$	0	0
$55$	2.33286	0.314563
$56$	0	0
$57$	2.83561	0.375585
$58$	0	0
$59$	−9.80563	−1.27658	−0.638292	−	0.769794i	$-0.720359\pi$
$59$	−9.80563	−1.27658	−0.638292	+	0.769794i	$0.720359\pi$
$60$	0	0
$61$	1.26111	0.161469	0.0807343	−	0.996736i	$-0.474273\pi$
$61$	1.26111	0.161469	0.0807343	+	0.996736i	$0.474273\pi$
$62$	0	0
$63$	2.04096	0.257137
$64$	0	0
$65$	−2.33286	−0.289356
$66$	0	0
$67$	−4.93124	−0.602447	−0.301224	−	0.953554i	$-0.597395\pi$
$67$	−4.93124	−0.602447	−0.301224	+	0.953554i	$0.597395\pi$
$68$	0	0
$69$	3.73944	0.450176
$70$	0	0
$71$	16.2180	1.92472	0.962360	−	0.271780i	$-0.0876122\pi$
$71$	16.2180	1.92472	0.962360	+	0.271780i	$0.0876122\pi$
$72$	0	0
$73$	0.183169	0.0214383	0.0107191	−	0.999943i	$-0.496588\pi$
$73$	0.183169	0.0214383	0.0107191	+	0.999943i	$0.496588\pi$
$74$	0	0
$75$	0.433093	0.0500092
$76$	0	0
$77$	−1.00000	−0.113961
$78$	0	0
$79$	−15.7288	−1.76963	−0.884813	−	0.465947i	$-0.845714\pi$
$79$	−15.7288	−1.76963	−0.884813	+	0.465947i	$0.845714\pi$
$80$	0	0
$81$	1.28841	0.143156
$82$	0	0
$83$	15.3290	1.68257	0.841286	−	0.540591i	$-0.181799\pi$
$83$	15.3290	1.68257	0.841286	+	0.540591i	$0.181799\pi$
$84$	0	0
$85$	−6.08133	−0.659612
$86$	0	0
$87$	−3.04423	−0.326376
$88$	0	0
$89$	−10.7996	−1.14476	−0.572379	−	0.819989i	$-0.693979\pi$
$89$	−10.7996	−1.14476	−0.572379	+	0.819989i	$0.693979\pi$
$90$	0	0
$91$	1.00000	0.104828
$92$	0	0
$93$	−7.17374	−0.743882
$94$	0	0
$95$	6.75487	0.693035
$96$	0	0
$97$	4.70388	0.477606	0.238803	−	0.971068i	$-0.423245\pi$
$97$	4.70388	0.477606	0.238803	+	0.971068i	$0.423245\pi$
$98$	0	0
$99$	−2.04096	−0.205124
$100$	0	0
$101$	−10.6825	−1.06294	−0.531472	−	0.847076i	$-0.678361\pi$
$101$	−10.6825	−1.06294	−0.531472	+	0.847076i	$0.678361\pi$
$102$	0	0
$103$	−13.1224	−1.29299	−0.646494	−	0.762919i	$-0.723766\pi$
$103$	−13.1224	−1.29299	−0.646494	+	0.762919i	$0.723766\pi$
$104$	0	0
$105$	−2.28458	−0.222953
$106$	0	0
$107$	−2.16009	−0.208824	−0.104412	−	0.994534i	$-0.533296\pi$
$107$	−2.16009	−0.208824	−0.104412	+	0.994534i	$0.533296\pi$
$108$	0	0
$109$	13.2998	1.27389	0.636945	−	0.770909i	$-0.280198\pi$
$109$	13.2998	1.27389	0.636945	+	0.770909i	$0.280198\pi$
$110$	0	0
$111$	−1.43189	−0.135909
$112$	0	0
$113$	12.1758	1.14540	0.572701	−	0.819764i	$-0.305895\pi$
$113$	12.1758	1.14540	0.572701	+	0.819764i	$0.305895\pi$
$114$	0	0
$115$	8.90795	0.830671
$116$	0	0
$117$	2.04096	0.188687
$118$	0	0
$119$	2.60681	0.238966
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	−0.280140	−0.0252593
$124$	0	0
$125$	−10.6326	−0.951010
$126$	0	0
$127$	−1.59695	−0.141707	−0.0708533	−	0.997487i	$-0.522572\pi$
$127$	−1.59695	−0.141707	−0.0708533	+	0.997487i	$0.522572\pi$
$128$	0	0
$129$	−9.02237	−0.794376
$130$	0	0
$131$	5.72718	0.500386	0.250193	−	0.968196i	$-0.419506\pi$
$131$	5.72718	0.500386	0.250193	+	0.968196i	$0.419506\pi$
$132$	0	0
$133$	−2.89553	−0.251074
$134$	0	0
$135$	−11.5165	−0.991182
$136$	0	0
$137$	13.7036	1.17078	0.585390	−	0.810752i	$-0.300941\pi$
$137$	13.7036	1.17078	0.585390	+	0.810752i	$0.300941\pi$
$138$	0	0
$139$	−6.16193	−0.522648	−0.261324	−	0.965251i	$-0.584159\pi$
$139$	−6.16193	−0.522648	−0.261324	+	0.965251i	$0.584159\pi$
$140$	0	0
$141$	3.73944	0.314918
$142$	0	0
$143$	−1.00000	−0.0836242
$144$	0	0
$145$	−7.25185	−0.602233
$146$	0	0
$147$	0.979305	0.0807717
$148$	0	0
$149$	3.66700	0.300413	0.150206	−	0.988655i	$-0.452006\pi$
$149$	3.66700	0.300413	0.150206	+	0.988655i	$0.452006\pi$
$150$	0	0
$151$	−16.4743	−1.34066	−0.670331	−	0.742062i	$-0.733848\pi$
$151$	−16.4743	−1.34066	−0.670331	+	0.742062i	$0.733848\pi$
$152$	0	0
$153$	5.32040	0.430129
$154$	0	0
$155$	−17.0890	−1.37262
$156$	0	0
$157$	−0.898098	−0.0716760	−0.0358380	−	0.999358i	$-0.511410\pi$
$157$	−0.898098	−0.0716760	−0.0358380	+	0.999358i	$0.511410\pi$
$158$	0	0
$159$	−11.6305	−0.922359
$160$	0	0
$161$	−3.81847	−0.300937
$162$	0	0
$163$	6.05949	0.474616	0.237308	−	0.971435i	$-0.423735\pi$
$163$	6.05949	0.474616	0.237308	+	0.971435i	$0.423735\pi$
$164$	0	0
$165$	2.28458	0.177855
$166$	0	0
$167$	−2.96978	−0.229809	−0.114904	−	0.993377i	$-0.536656\pi$
$167$	−2.96978	−0.229809	−0.114904	+	0.993377i	$0.536656\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	−5.90966	−0.451923
$172$	0	0
$173$	−4.79498	−0.364556	−0.182278	−	0.983247i	$-0.558347\pi$
$173$	−4.79498	−0.364556	−0.182278	+	0.983247i	$0.558347\pi$
$174$	0	0
$175$	−0.442245	−0.0334306
$176$	0	0
$177$	−9.60271	−0.721784
$178$	0	0
$179$	−15.6819	−1.17212	−0.586060	−	0.810268i	$-0.699322\pi$
$179$	−15.6819	−1.17212	−0.586060	+	0.810268i	$0.699322\pi$
$180$	0	0
$181$	21.5030	1.59830	0.799152	−	0.601129i	$-0.205282\pi$
$181$	21.5030	1.59830	0.799152	+	0.601129i	$0.205282\pi$
$182$	0	0
$183$	1.23501	0.0912946
$184$	0	0
$185$	−3.41100	−0.250782
$186$	0	0
$187$	−2.60681	−0.190629
$188$	0	0
$189$	4.93664	0.359088
$190$	0	0
$191$	−12.9627	−0.937951	−0.468975	−	0.883211i	$-0.655377\pi$
$191$	−12.9627	−0.937951	−0.468975	+	0.883211i	$0.655377\pi$
$192$	0	0
$193$	−10.8808	−0.783214	−0.391607	−	0.920133i	$-0.628081\pi$
$193$	−10.8808	−0.783214	−0.391607	+	0.920133i	$0.628081\pi$
$194$	0	0
$195$	−2.28458	−0.163602
$196$	0	0
$197$	−14.8956	−1.06127	−0.530635	−	0.847600i	$-0.678046\pi$
$197$	−14.8956	−1.06127	−0.530635	+	0.847600i	$0.678046\pi$
$198$	0	0
$199$	1.66656	0.118139	0.0590696	−	0.998254i	$-0.481187\pi$
$199$	1.66656	0.118139	0.0590696	+	0.998254i	$0.481187\pi$
$200$	0	0
$201$	−4.82919	−0.340625
$202$	0	0
$203$	3.10856	0.218178
$204$	0	0
$205$	−0.667338	−0.0466089
$206$	0	0
$207$	−7.79334	−0.541675
$208$	0	0
$209$	2.89553	0.200288
$210$	0	0
$211$	−16.7614	−1.15390	−0.576950	−	0.816780i	$-0.695757\pi$
$211$	−16.7614	−1.15390	−0.576950	+	0.816780i	$0.695757\pi$
$212$	0	0
$213$	15.8823	1.08824
$214$	0	0
$215$	−21.4927	−1.46579
$216$	0	0
$217$	7.32534	0.497276
$218$	0	0
$219$	0.179378	0.0121212
$220$	0	0
$221$	2.60681	0.175353
$222$	0	0
$223$	−4.79512	−0.321105	−0.160552	−	0.987027i	$-0.551328\pi$
$223$	−4.79512	−0.321105	−0.160552	+	0.987027i	$0.551328\pi$
$224$	0	0
$225$	−0.902605	−0.0601736
$226$	0	0
$227$	−11.5168	−0.764397	−0.382199	−	0.924080i	$-0.624833\pi$
$227$	−11.5168	−0.764397	−0.382199	+	0.924080i	$0.624833\pi$
$228$	0	0
$229$	14.1605	0.935754	0.467877	−	0.883794i	$-0.345019\pi$
$229$	14.1605	0.935754	0.467877	+	0.883794i	$0.345019\pi$
$230$	0	0
$231$	−0.979305	−0.0644335
$232$	0	0
$233$	−17.8496	−1.16936	−0.584682	−	0.811263i	$-0.698781\pi$
$233$	−17.8496	−1.16936	−0.584682	+	0.811263i	$0.698781\pi$
$234$	0	0
$235$	8.90795	0.581091
$236$	0	0
$237$	−15.4033	−1.00055
$238$	0	0
$239$	−25.0009	−1.61717	−0.808587	−	0.588377i	$-0.799767\pi$
$239$	−25.0009	−1.61717	−0.808587	+	0.588377i	$0.799767\pi$
$240$	0	0
$241$	−3.85076	−0.248049	−0.124025	−	0.992279i	$-0.539580\pi$
$241$	−3.85076	−0.248049	−0.124025	+	0.992279i	$0.539580\pi$
$242$	0	0
$243$	16.0717	1.03100
$244$	0	0
$245$	2.33286	0.149041
$246$	0	0
$247$	−2.89553	−0.184238
$248$	0	0
$249$	15.0117	0.951329
$250$	0	0
$251$	15.6235	0.986144	0.493072	−	0.869989i	$-0.335874\pi$
$251$	15.6235	0.986144	0.493072	+	0.869989i	$0.335874\pi$
$252$	0	0
$253$	3.81847	0.240065
$254$	0	0
$255$	−5.95547	−0.372946
$256$	0	0
$257$	3.79105	0.236479	0.118239	−	0.992985i	$-0.462275\pi$
$257$	3.79105	0.236479	0.118239	+	0.992985i	$0.462275\pi$
$258$	0	0
$259$	1.46215	0.0908537
$260$	0	0
$261$	6.34445	0.392712
$262$	0	0
$263$	12.8208	0.790563	0.395282	−	0.918560i	$-0.370647\pi$
$263$	12.8208	0.790563	0.395282	+	0.918560i	$0.370647\pi$
$264$	0	0
$265$	−27.7057	−1.70195
$266$	0	0
$267$	−10.5761	−0.647249
$268$	0	0
$269$	17.2682	1.05286	0.526430	−	0.850218i	$-0.323530\pi$
$269$	17.2682	1.05286	0.526430	+	0.850218i	$0.323530\pi$
$270$	0	0
$271$	6.99630	0.424995	0.212497	−	0.977162i	$-0.431840\pi$
$271$	6.99630	0.424995	0.212497	+	0.977162i	$0.431840\pi$
$272$	0	0
$273$	0.979305	0.0592702
$274$	0	0
$275$	0.442245	0.0266684
$276$	0	0
$277$	21.5108	1.29246	0.646228	−	0.763144i	$-0.276345\pi$
$277$	21.5108	1.29246	0.646228	+	0.763144i	$0.276345\pi$
$278$	0	0
$279$	14.9507	0.895077
$280$	0	0
$281$	13.7124	0.818013	0.409007	−	0.912531i	$-0.365875\pi$
$281$	13.7124	0.818013	0.409007	+	0.912531i	$0.365875\pi$
$282$	0	0
$283$	−26.2520	−1.56052	−0.780259	−	0.625457i	$-0.784913\pi$
$283$	−26.2520	−1.56052	−0.780259	+	0.625457i	$0.784913\pi$
$284$	0	0
$285$	6.61508	0.391844
$286$	0	0
$287$	0.286060	0.0168856
$288$	0	0
$289$	−10.2045	−0.600267
$290$	0	0
$291$	4.60653	0.270040
$292$	0	0
$293$	−2.63462	−0.153916	−0.0769581	−	0.997034i	$-0.524521\pi$
$293$	−2.63462	−0.153916	−0.0769581	+	0.997034i	$0.524521\pi$
$294$	0	0
$295$	−22.8752	−1.33185
$296$	0	0
$297$	−4.93664	−0.286453
$298$	0	0
$299$	−3.81847	−0.220828
$300$	0	0
$301$	9.21304	0.531031
$302$	0	0
$303$	−10.4614	−0.600991
$304$	0	0
$305$	2.94199	0.168458
$306$	0	0
$307$	−15.6740	−0.894565	−0.447282	−	0.894393i	$-0.647608\pi$
$307$	−15.6740	−0.894565	−0.447282	+	0.894393i	$0.647608\pi$
$308$	0	0
$309$	−12.8508	−0.731058
$310$	0	0
$311$	0.412463	0.0233886	0.0116943	−	0.999932i	$-0.496277\pi$
$311$	0.412463	0.0233886	0.0116943	+	0.999932i	$0.496277\pi$
$312$	0	0
$313$	−11.1735	−0.631563	−0.315782	−	0.948832i	$-0.602267\pi$
$313$	−11.1735	−0.631563	−0.315782	+	0.948832i	$0.602267\pi$
$314$	0	0
$315$	4.76128	0.268268
$316$	0	0
$317$	18.4553	1.03655	0.518276	−	0.855214i	$-0.326574\pi$
$317$	18.4553	1.03655	0.518276	+	0.855214i	$0.326574\pi$
$318$	0	0
$319$	−3.10856	−0.174046
$320$	0	0
$321$	−2.11539	−0.118070
$322$	0	0
$323$	−7.54810	−0.419987
$324$	0	0
$325$	−0.442245	−0.0245313
$326$	0	0
$327$	13.0246	0.720260
$328$	0	0
$329$	−3.81847	−0.210519
$330$	0	0
$331$	−17.1894	−0.944815	−0.472407	−	0.881380i	$-0.656615\pi$
$331$	−17.1894	−0.944815	−0.472407	+	0.881380i	$0.656615\pi$
$332$	0	0
$333$	2.98420	0.163533
$334$	0	0
$335$	−11.5039	−0.628526
$336$	0	0
$337$	−21.5406	−1.17339	−0.586697	−	0.809807i	$-0.699572\pi$
$337$	−21.5406	−1.17339	−0.586697	+	0.809807i	$0.699572\pi$
$338$	0	0
$339$	11.9238	0.647613
$340$	0	0
$341$	−7.32534	−0.396689
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	8.72361	0.469663
$346$	0	0
$347$	7.68411	0.412505	0.206252	−	0.978499i	$-0.433873\pi$
$347$	7.68411	0.412505	0.206252	+	0.978499i	$0.433873\pi$
$348$	0	0
$349$	−11.8842	−0.636145	−0.318073	−	0.948066i	$-0.603036\pi$
$349$	−11.8842	−0.636145	−0.318073	+	0.948066i	$0.603036\pi$
$350$	0	0
$351$	4.93664	0.263498
$352$	0	0
$353$	4.68606	0.249414	0.124707	−	0.992194i	$-0.460201\pi$
$353$	4.68606	0.249414	0.124707	+	0.992194i	$0.460201\pi$
$354$	0	0
$355$	37.8343	2.00804
$356$	0	0
$357$	2.55286	0.135112
$358$	0	0
$359$	14.1521	0.746918	0.373459	−	0.927647i	$-0.378172\pi$
$359$	14.1521	0.746918	0.373459	+	0.927647i	$0.378172\pi$
$360$	0	0
$361$	−10.6159	−0.558732
$362$	0	0
$363$	0.979305	0.0514002
$364$	0	0
$365$	0.427307	0.0223663
$366$	0	0
$367$	−16.9661	−0.885625	−0.442813	−	0.896614i	$-0.646019\pi$
$367$	−16.9661	−0.885625	−0.442813	+	0.896614i	$0.646019\pi$
$368$	0	0
$369$	0.583837	0.0303933
$370$	0	0
$371$	11.8763	0.616586
$372$	0	0
$373$	5.89452	0.305206	0.152603	−	0.988288i	$-0.451234\pi$
$373$	5.89452	0.305206	0.152603	+	0.988288i	$0.451234\pi$
$374$	0	0
$375$	−10.4126	−0.537703
$376$	0	0
$377$	3.10856	0.160099
$378$	0	0
$379$	0.354818	0.0182258	0.00911289	−	0.999958i	$-0.497099\pi$
$379$	0.354818	0.0182258	0.00911289	+	0.999958i	$0.497099\pi$
$380$	0	0
$381$	−1.56390	−0.0801212
$382$	0	0
$383$	−25.1046	−1.28279	−0.641393	−	0.767213i	$-0.721643\pi$
$383$	−25.1046	−1.28279	−0.641393	+	0.767213i	$0.721643\pi$
$384$	0	0
$385$	−2.33286	−0.118894
$386$	0	0
$387$	18.8034	0.955833
$388$	0	0
$389$	17.4783	0.886187	0.443094	−	0.896475i	$-0.353881\pi$
$389$	17.4783	0.886187	0.443094	+	0.896475i	$0.353881\pi$
$390$	0	0
$391$	−9.95401	−0.503396
$392$	0	0
$393$	5.60866	0.282919
$394$	0	0
$395$	−36.6931	−1.84623
$396$	0	0
$397$	22.1899	1.11368	0.556840	−	0.830620i	$-0.312014\pi$
$397$	22.1899	1.11368	0.556840	+	0.830620i	$0.312014\pi$
$398$	0	0
$399$	−2.83561	−0.141958
$400$	0	0
$401$	36.8152	1.83846	0.919230	−	0.393720i	$-0.128812\pi$
$401$	36.8152	1.83846	0.919230	+	0.393720i	$0.128812\pi$
$402$	0	0
$403$	7.32534	0.364901
$404$	0	0
$405$	3.00568	0.149353
$406$	0	0
$407$	−1.46215	−0.0724762
$408$	0	0
$409$	−4.59380	−0.227149	−0.113574	−	0.993529i	$-0.536230\pi$
$409$	−4.59380	−0.227149	−0.113574	+	0.993529i	$0.536230\pi$
$410$	0	0
$411$	13.4200	0.661961
$412$	0	0
$413$	9.80563	0.482504
$414$	0	0
$415$	35.7603	1.75541
$416$	0	0
$417$	−6.03441	−0.295506
$418$	0	0
$419$	35.4748	1.73306	0.866528	−	0.499129i	$-0.166347\pi$
$419$	35.4748	1.73306	0.866528	+	0.499129i	$0.166347\pi$
$420$	0	0
$421$	−10.7158	−0.522254	−0.261127	−	0.965304i	$-0.584094\pi$
$421$	−10.7158	−0.522254	−0.261127	+	0.965304i	$0.584094\pi$
$422$	0	0
$423$	−7.79334	−0.378925
$424$	0	0
$425$	−1.15285	−0.0559214
$426$	0	0
$427$	−1.26111	−0.0610294
$428$	0	0
$429$	−0.979305	−0.0472813
$430$	0	0
$431$	26.3325	1.26839	0.634197	−	0.773172i	$-0.281331\pi$
$431$	26.3325	1.26839	0.634197	+	0.773172i	$0.281331\pi$
$432$	0	0
$433$	−20.8589	−1.00242	−0.501208	−	0.865327i	$-0.667111\pi$
$433$	−20.8589	−1.00242	−0.501208	+	0.865327i	$0.667111\pi$
$434$	0	0
$435$	−7.10177	−0.340504
$436$	0	0
$437$	11.0565	0.528903
$438$	0	0
$439$	−12.7517	−0.608606	−0.304303	−	0.952575i	$-0.598424\pi$
$439$	−12.7517	−0.608606	−0.304303	+	0.952575i	$0.598424\pi$
$440$	0	0
$441$	−2.04096	−0.0971886
$442$	0	0
$443$	−11.9295	−0.566789	−0.283395	−	0.959003i	$-0.591461\pi$
$443$	−11.9295	−0.566789	−0.283395	+	0.959003i	$0.591461\pi$
$444$	0	0
$445$	−25.1940	−1.19431
$446$	0	0
$447$	3.59112	0.169854
$448$	0	0
$449$	0.651619	0.0307518	0.0153759	−	0.999882i	$-0.495106\pi$
$449$	0.651619	0.0307518	0.0153759	+	0.999882i	$0.495106\pi$
$450$	0	0
$451$	−0.286060	−0.0134700
$452$	0	0
$453$	−16.1334	−0.758013
$454$	0	0
$455$	2.33286	0.109366
$456$	0	0
$457$	34.3642	1.60749	0.803745	−	0.594974i	$-0.202838\pi$
$457$	34.3642	1.60749	0.803745	+	0.594974i	$0.202838\pi$
$458$	0	0
$459$	12.8689	0.600668
$460$	0	0
$461$	−10.1979	−0.474966	−0.237483	−	0.971392i	$-0.576322\pi$
$461$	−10.1979	−0.474966	−0.237483	+	0.971392i	$0.576322\pi$
$462$	0	0
$463$	29.6391	1.37745	0.688723	−	0.725025i	$-0.258172\pi$
$463$	29.6391	1.37745	0.688723	+	0.725025i	$0.258172\pi$
$464$	0	0
$465$	−16.7353	−0.776083
$466$	0	0
$467$	5.97904	0.276677	0.138338	−	0.990385i	$-0.455824\pi$
$467$	5.97904	0.276677	0.138338	+	0.990385i	$0.455824\pi$
$468$	0	0
$469$	4.93124	0.227704
$470$	0	0
$471$	−0.879512	−0.0405258
$472$	0	0
$473$	−9.21304	−0.423616
$474$	0	0
$475$	1.28053	0.0587549
$476$	0	0
$477$	24.2390	1.10983
$478$	0	0
$479$	−30.3925	−1.38867	−0.694334	−	0.719653i	$-0.744301\pi$
$479$	−30.3925	−1.38867	−0.694334	+	0.719653i	$0.744301\pi$
$480$	0	0
$481$	1.46215	0.0666684
$482$	0	0
$483$	−3.73944	−0.170151
$484$	0	0
$485$	10.9735	0.498281
$486$	0	0
$487$	32.6920	1.48141	0.740707	−	0.671828i	$-0.234490\pi$
$487$	32.6920	1.48141	0.740707	+	0.671828i	$0.234490\pi$
$488$	0	0
$489$	5.93409	0.268349
$490$	0	0
$491$	30.6634	1.38382	0.691911	−	0.721983i	$-0.256769\pi$
$491$	30.6634	1.38382	0.691911	+	0.721983i	$0.256769\pi$
$492$	0	0
$493$	8.10343	0.364960
$494$	0	0
$495$	−4.76128	−0.214004
$496$	0	0
$497$	−16.2180	−0.727475
$498$	0	0
$499$	26.0432	1.16586	0.582928	−	0.812524i	$-0.301907\pi$
$499$	26.0432	1.16586	0.582928	+	0.812524i	$0.301907\pi$
$500$	0	0
$501$	−2.90832	−0.129934
$502$	0	0
$503$	−36.0360	−1.60677	−0.803384	−	0.595462i	$-0.796969\pi$
$503$	−36.0360	−1.60677	−0.803384	+	0.595462i	$0.796969\pi$
$504$	0	0
$505$	−24.9207	−1.10896
$506$	0	0
$507$	0.979305	0.0434925
$508$	0	0
$509$	−21.9584	−0.973290	−0.486645	−	0.873600i	$-0.661780\pi$
$509$	−21.9584	−0.973290	−0.486645	+	0.873600i	$0.661780\pi$
$510$	0	0
$511$	−0.183169	−0.00810291
$512$	0	0
$513$	−14.2942	−0.631104
$514$	0	0
$515$	−30.6127	−1.34896
$516$	0	0
$517$	3.81847	0.167936
$518$	0	0
$519$	−4.69575	−0.206120
$520$	0	0
$521$	−19.5117	−0.854825	−0.427412	−	0.904057i	$-0.640575\pi$
$521$	−19.5117	−0.854825	−0.427412	+	0.904057i	$0.640575\pi$
$522$	0	0
$523$	−27.8683	−1.21859	−0.609297	−	0.792942i	$-0.708548\pi$
$523$	−27.8683	−1.21859	−0.609297	+	0.792942i	$0.708548\pi$
$524$	0	0
$525$	−0.433093	−0.0189017
$526$	0	0
$527$	19.0958	0.831824
$528$	0	0
$529$	−8.41931	−0.366057
$530$	0	0
$531$	20.0129	0.868487
$532$	0	0
$533$	0.286060	0.0123906
$534$	0	0
$535$	−5.03920	−0.217864
$536$	0	0
$537$	−15.3574	−0.662719
$538$	0	0
$539$	1.00000	0.0430730
$540$	0	0
$541$	−15.9951	−0.687684	−0.343842	−	0.939027i	$-0.611728\pi$
$541$	−15.9951	−0.687684	−0.343842	+	0.939027i	$0.611728\pi$
$542$	0	0
$543$	21.0580	0.903685
$544$	0	0
$545$	31.0266	1.32903
$546$	0	0
$547$	−39.3080	−1.68069	−0.840345	−	0.542053i	$-0.817647\pi$
$547$	−39.3080	−1.68069	−0.840345	+	0.542053i	$0.817647\pi$
$548$	0	0
$549$	−2.57388	−0.109850
$550$	0	0
$551$	−9.00093	−0.383453
$552$	0	0
$553$	15.7288	0.668856
$554$	0	0
$555$	−3.34041	−0.141792
$556$	0	0
$557$	−37.9317	−1.60722	−0.803608	−	0.595159i	$-0.797089\pi$
$557$	−37.9317	−1.60722	−0.803608	+	0.595159i	$0.797089\pi$
$558$	0	0
$559$	9.21304	0.389670
$560$	0	0
$561$	−2.55286	−0.107782
$562$	0	0
$563$	−34.9013	−1.47091	−0.735457	−	0.677571i	$-0.763032\pi$
$563$	−34.9013	−1.47091	−0.735457	+	0.677571i	$0.763032\pi$
$564$	0	0
$565$	28.4045	1.19498
$566$	0	0
$567$	−1.28841	−0.0541080
$568$	0	0
$569$	23.3620	0.979384	0.489692	−	0.871895i	$-0.337109\pi$
$569$	23.3620	0.979384	0.489692	+	0.871895i	$0.337109\pi$
$570$	0	0
$571$	15.0498	0.629815	0.314907	−	0.949122i	$-0.398027\pi$
$571$	15.0498	0.629815	0.314907	+	0.949122i	$0.398027\pi$
$572$	0	0
$573$	−12.6945	−0.530319
$574$	0	0
$575$	1.68870	0.0704235
$576$	0	0
$577$	17.8152	0.741656	0.370828	−	0.928702i	$-0.379074\pi$
$577$	17.8152	0.741656	0.370828	+	0.928702i	$0.379074\pi$
$578$	0	0
$579$	−10.6556	−0.442831
$580$	0	0
$581$	−15.3290	−0.635952
$582$	0	0
$583$	−11.8763	−0.491865
$584$	0	0
$585$	4.76128	0.196855
$586$	0	0
$587$	28.4425	1.17395	0.586974	−	0.809606i	$-0.300319\pi$
$587$	28.4425	1.17395	0.586974	+	0.809606i	$0.300319\pi$
$588$	0	0
$589$	−21.2107	−0.873973
$590$	0	0
$591$	−14.5874	−0.600044
$592$	0	0
$593$	34.3548	1.41078	0.705392	−	0.708818i	$-0.250771\pi$
$593$	34.3548	1.41078	0.705392	+	0.708818i	$0.250771\pi$
$594$	0	0
$595$	6.08133	0.249310
$596$	0	0
$597$	1.63207	0.0667961
$598$	0	0
$599$	−1.93947	−0.0792447	−0.0396223	−	0.999215i	$-0.512615\pi$
$599$	−1.93947	−0.0792447	−0.0396223	+	0.999215i	$0.512615\pi$
$600$	0	0
$601$	−19.9317	−0.813032	−0.406516	−	0.913644i	$-0.633256\pi$
$601$	−19.9317	−0.813032	−0.406516	+	0.913644i	$0.633256\pi$
$602$	0	0
$603$	10.0645	0.409857
$604$	0	0
$605$	2.33286	0.0948443
$606$	0	0
$607$	−2.52805	−0.102611	−0.0513053	−	0.998683i	$-0.516338\pi$
$607$	−2.52805	−0.102611	−0.0513053	+	0.998683i	$0.516338\pi$
$608$	0	0
$609$	3.04423	0.123358
$610$	0	0
$611$	−3.81847	−0.154479
$612$	0	0
$613$	13.3609	0.539641	0.269821	−	0.962911i	$-0.413036\pi$
$613$	13.3609	0.539641	0.269821	+	0.962911i	$0.413036\pi$
$614$	0	0
$615$	−0.653527	−0.0263528
$616$	0	0
$617$	7.81741	0.314717	0.157359	−	0.987542i	$-0.449702\pi$
$617$	7.81741	0.314717	0.157359	+	0.987542i	$0.449702\pi$
$618$	0	0
$619$	23.8668	0.959288	0.479644	−	0.877463i	$-0.340766\pi$
$619$	23.8668	0.959288	0.479644	+	0.877463i	$0.340766\pi$
$620$	0	0
$621$	−18.8504	−0.756440
$622$	0	0
$623$	10.7996	0.432678
$624$	0	0
$625$	−27.0156	−1.08063
$626$	0	0
$627$	2.83561	0.113243
$628$	0	0
$629$	3.81155	0.151977
$630$	0	0
$631$	−17.6974	−0.704523	−0.352262	−	0.935902i	$-0.614587\pi$
$631$	−17.6974	−0.704523	−0.352262	+	0.935902i	$0.614587\pi$
$632$	0	0
$633$	−16.4145	−0.652417
$634$	0	0
$635$	−3.72547	−0.147841
$636$	0	0
$637$	−1.00000	−0.0396214
$638$	0	0
$639$	−33.1003	−1.30943
$640$	0	0
$641$	33.6388	1.32865	0.664327	−	0.747442i	$-0.268718\pi$
$641$	33.6388	1.32865	0.664327	+	0.747442i	$0.268718\pi$
$642$	0	0
$643$	18.5558	0.731770	0.365885	−	0.930660i	$-0.380766\pi$
$643$	18.5558	0.731770	0.365885	+	0.930660i	$0.380766\pi$
$644$	0	0
$645$	−21.0480	−0.828762
$646$	0	0
$647$	31.2113	1.22704	0.613521	−	0.789678i	$-0.289752\pi$
$647$	31.2113	1.22704	0.613521	+	0.789678i	$0.289752\pi$
$648$	0	0
$649$	−9.80563	−0.384905
$650$	0	0
$651$	7.17374	0.281161
$652$	0	0
$653$	−40.4228	−1.58186	−0.790932	−	0.611903i	$-0.790404\pi$
$653$	−40.4228	−1.58186	−0.790932	+	0.611903i	$0.790404\pi$
$654$	0	0
$655$	13.3607	0.522046
$656$	0	0
$657$	−0.373840	−0.0145849
$658$	0	0
$659$	−4.56349	−0.177768	−0.0888841	−	0.996042i	$-0.528330\pi$
$659$	−4.56349	−0.177768	−0.0888841	+	0.996042i	$0.528330\pi$
$660$	0	0
$661$	−37.3024	−1.45090	−0.725448	−	0.688277i	$-0.758367\pi$
$661$	−37.3024	−1.45090	−0.725448	+	0.688277i	$0.758367\pi$
$662$	0	0
$663$	2.55286	0.0991449
$664$	0	0
$665$	−6.75487	−0.261943
$666$	0	0
$667$	−11.8699	−0.459606
$668$	0	0
$669$	−4.69588	−0.181553
$670$	0	0
$671$	1.26111	0.0486846
$672$	0	0
$673$	38.2640	1.47497	0.737484	−	0.675365i	$-0.236014\pi$
$673$	38.2640	1.47497	0.737484	+	0.675365i	$0.236014\pi$
$674$	0	0
$675$	−2.18320	−0.0840315
$676$	0	0
$677$	25.4686	0.978839	0.489419	−	0.872049i	$-0.337209\pi$
$677$	25.4686	0.978839	0.489419	+	0.872049i	$0.337209\pi$
$678$	0	0
$679$	−4.70388	−0.180518
$680$	0	0
$681$	−11.2785	−0.432192
$682$	0	0
$683$	−12.6773	−0.485082	−0.242541	−	0.970141i	$-0.577981\pi$
$683$	−12.6773	−0.485082	−0.242541	+	0.970141i	$0.577981\pi$
$684$	0	0
$685$	31.9687	1.22146
$686$	0	0
$687$	13.8675	0.529077
$688$	0	0
$689$	11.8763	0.452450
$690$	0	0
$691$	−33.9122	−1.29008	−0.645041	−	0.764148i	$-0.723160\pi$
$691$	−33.9122	−1.29008	−0.645041	+	0.764148i	$0.723160\pi$
$692$	0	0
$693$	2.04096	0.0775297
$694$	0	0
$695$	−14.3749	−0.545272
$696$	0	0
$697$	0.745703	0.0282455
$698$	0	0
$699$	−17.4802	−0.661161
$700$	0	0
$701$	43.5469	1.64474	0.822371	−	0.568951i	$-0.192651\pi$
$701$	43.5469	1.64474	0.822371	+	0.568951i	$0.192651\pi$
$702$	0	0
$703$	−4.23371	−0.159677
$704$	0	0
$705$	8.72361	0.328550
$706$	0	0
$707$	10.6825	0.401755
$708$	0	0
$709$	44.5607	1.67351	0.836757	−	0.547574i	$-0.184449\pi$
$709$	44.5607	1.67351	0.836757	+	0.547574i	$0.184449\pi$
$710$	0	0
$711$	32.1018	1.20391
$712$	0	0
$713$	−27.9716	−1.04754
$714$	0	0
$715$	−2.33286	−0.0872441
$716$	0	0
$717$	−24.4835	−0.914353
$718$	0	0
$719$	−34.7805	−1.29709	−0.648546	−	0.761175i	$-0.724623\pi$
$719$	−34.7805	−1.29709	−0.648546	+	0.761175i	$0.724623\pi$
$720$	0	0
$721$	13.1224	0.488704
$722$	0	0
$723$	−3.77107	−0.140248
$724$	0	0
$725$	−1.37475	−0.0510568
$726$	0	0
$727$	28.7891	1.06773	0.533864	−	0.845571i	$-0.320740\pi$
$727$	28.7891	1.06773	0.533864	+	0.845571i	$0.320740\pi$
$728$	0	0
$729$	11.8738	0.439772
$730$	0	0
$731$	24.0166	0.888287
$732$	0	0
$733$	7.49094	0.276684	0.138342	−	0.990385i	$-0.455823\pi$
$733$	7.49094	0.276684	0.138342	+	0.990385i	$0.455823\pi$
$734$	0	0
$735$	2.28458	0.0842681
$736$	0	0
$737$	−4.93124	−0.181645
$738$	0	0
$739$	−34.5073	−1.26937	−0.634686	−	0.772770i	$-0.718871\pi$
$739$	−34.5073	−1.26937	−0.634686	+	0.772770i	$0.718871\pi$
$740$	0	0
$741$	−2.83561	−0.104169
$742$	0	0
$743$	−50.7560	−1.86206	−0.931028	−	0.364948i	$-0.881087\pi$
$743$	−50.7560	−1.86206	−0.931028	+	0.364948i	$0.881087\pi$
$744$	0	0
$745$	8.55462	0.313417
$746$	0	0
$747$	−31.2858	−1.14469
$748$	0	0
$749$	2.16009	0.0789281
$750$	0	0
$751$	−22.7913	−0.831666	−0.415833	−	0.909441i	$-0.636510\pi$
$751$	−22.7913	−0.831666	−0.415833	+	0.909441i	$0.636510\pi$
$752$	0	0
$753$	15.3001	0.557568
$754$	0	0
$755$	−38.4323	−1.39870
$756$	0	0
$757$	−7.70078	−0.279889	−0.139945	−	0.990159i	$-0.544692\pi$
$757$	−7.70078	−0.279889	−0.139945	+	0.990159i	$0.544692\pi$
$758$	0	0
$759$	3.73944	0.135733
$760$	0	0
$761$	0.329560	0.0119465	0.00597327	−	0.999982i	$-0.498099\pi$
$761$	0.329560	0.0119465	0.00597327	+	0.999982i	$0.498099\pi$
$762$	0	0
$763$	−13.2998	−0.481485
$764$	0	0
$765$	12.4118	0.448748
$766$	0	0
$767$	9.80563	0.354061
$768$	0	0
$769$	24.9152	0.898465	0.449233	−	0.893415i	$-0.351697\pi$
$769$	24.9152	0.898465	0.449233	+	0.893415i	$0.351697\pi$
$770$	0	0
$771$	3.71259	0.133706
$772$	0	0
$773$	36.0350	1.29609	0.648044	−	0.761603i	$-0.275587\pi$
$773$	36.0350	1.29609	0.648044	+	0.761603i	$0.275587\pi$
$774$	0	0
$775$	−3.23959	−0.116370
$776$	0	0
$777$	1.43189	0.0513689
$778$	0	0
$779$	−0.828294	−0.0296767
$780$	0	0
$781$	16.2180	0.580325
$782$	0	0
$783$	15.3458	0.548416
$784$	0	0
$785$	−2.09514	−0.0747787
$786$	0	0
$787$	−37.0281	−1.31991	−0.659955	−	0.751305i	$-0.729425\pi$
$787$	−37.0281	−1.31991	−0.659955	+	0.751305i	$0.729425\pi$
$788$	0	0
$789$	12.5555	0.446986
$790$	0	0
$791$	−12.1758	−0.432922
$792$	0	0
$793$	−1.26111	−0.0447833
$794$	0	0
$795$	−27.1324	−0.962285
$796$	0	0
$797$	−24.8666	−0.880820	−0.440410	−	0.897797i	$-0.645167\pi$
$797$	−24.8666	−0.880820	−0.440410	+	0.897797i	$0.645167\pi$
$798$	0	0
$799$	−9.95401	−0.352148
$800$	0	0
$801$	22.0416	0.778803
$802$	0	0
$803$	0.183169	0.00646388
$804$	0	0
$805$	−8.90795	−0.313964
$806$	0	0
$807$	16.9108	0.595290
$808$	0	0
$809$	−38.9534	−1.36953	−0.684764	−	0.728765i	$-0.740095\pi$
$809$	−38.9534	−1.36953	−0.684764	+	0.728765i	$0.740095\pi$
$810$	0	0
$811$	−4.20179	−0.147545	−0.0737724	−	0.997275i	$-0.523504\pi$
$811$	−4.20179	−0.147545	−0.0737724	+	0.997275i	$0.523504\pi$
$812$	0	0
$813$	6.85151	0.240293
$814$	0	0
$815$	14.1359	0.495161
$816$	0	0
$817$	−26.6766	−0.933297
$818$	0	0
$819$	−2.04096	−0.0713170
$820$	0	0
$821$	24.8090	0.865842	0.432921	−	0.901432i	$-0.357483\pi$
$821$	24.8090	0.865842	0.432921	+	0.901432i	$0.357483\pi$
$822$	0	0
$823$	14.1090	0.491809	0.245905	−	0.969294i	$-0.420915\pi$
$823$	14.1090	0.491809	0.245905	+	0.969294i	$0.420915\pi$
$824$	0	0
$825$	0.433093	0.0150784
$826$	0	0
$827$	37.9753	1.32053	0.660266	−	0.751032i	$-0.270444\pi$
$827$	37.9753	1.32053	0.660266	+	0.751032i	$0.270444\pi$
$828$	0	0
$829$	10.8367	0.376374	0.188187	−	0.982133i	$-0.439739\pi$
$829$	10.8367	0.376374	0.188187	+	0.982133i	$0.439739\pi$
$830$	0	0
$831$	21.0656	0.730757
$832$	0	0
$833$	−2.60681	−0.0903206
$834$	0	0
$835$	−6.92809	−0.239757
$836$	0	0
$837$	36.1626	1.24996
$838$	0	0
$839$	47.9990	1.65711	0.828555	−	0.559908i	$-0.189163\pi$
$839$	47.9990	1.65711	0.828555	+	0.559908i	$0.189163\pi$
$840$	0	0
$841$	−19.3368	−0.666788
$842$	0	0
$843$	13.4286	0.462506
$844$	0	0
$845$	2.33286	0.0802529
$846$	0	0
$847$	−1.00000	−0.0343604
$848$	0	0
$849$	−25.7087	−0.882320
$850$	0	0
$851$	−5.58318	−0.191389
$852$	0	0
$853$	−34.6149	−1.18519	−0.592595	−	0.805500i	$-0.701897\pi$
$853$	−34.6149	−1.18519	−0.592595	+	0.805500i	$0.701897\pi$
$854$	0	0
$855$	−13.7864	−0.471486
$856$	0	0
$857$	−26.3956	−0.901656	−0.450828	−	0.892611i	$-0.648871\pi$
$857$	−26.3956	−0.901656	−0.450828	+	0.892611i	$0.648871\pi$
$858$	0	0
$859$	−28.1458	−0.960323	−0.480162	−	0.877180i	$-0.659422\pi$
$859$	−28.1458	−0.960323	−0.480162	+	0.877180i	$0.659422\pi$
$860$	0	0
$861$	0.280140	0.00954714
$862$	0	0
$863$	−1.08423	−0.0369077	−0.0184539	−	0.999830i	$-0.505874\pi$
$863$	−1.08423	−0.0369077	−0.0184539	+	0.999830i	$0.505874\pi$
$864$	0	0
$865$	−11.1860	−0.380336
$866$	0	0
$867$	−9.99336	−0.339392
$868$	0	0
$869$	−15.7288	−0.533562
$870$	0	0
$871$	4.93124	0.167089
$872$	0	0
$873$	−9.60043	−0.324925
$874$	0	0
$875$	10.6326	0.359448
$876$	0	0
$877$	15.7169	0.530723	0.265362	−	0.964149i	$-0.414509\pi$
$877$	15.7169	0.530723	0.265362	+	0.964149i	$0.414509\pi$
$878$	0	0
$879$	−2.58010	−0.0870245
$880$	0	0
$881$	−33.9150	−1.14262	−0.571312	−	0.820733i	$-0.693566\pi$
$881$	−33.9150	−1.14262	−0.571312	+	0.820733i	$0.693566\pi$
$882$	0	0
$883$	−5.38111	−0.181089	−0.0905444	−	0.995892i	$-0.528861\pi$
$883$	−5.38111	−0.181089	−0.0905444	+	0.995892i	$0.528861\pi$
$884$	0	0
$885$	−22.4018	−0.753028
$886$	0	0
$887$	18.0514	0.606107	0.303053	−	0.952974i	$-0.401994\pi$
$887$	18.0514	0.606107	0.303053	+	0.952974i	$0.401994\pi$
$888$	0	0
$889$	1.59695	0.0535600
$890$	0	0
$891$	1.28841	0.0431633
$892$	0	0
$893$	11.0565	0.369991
$894$	0	0
$895$	−36.5837	−1.22286
$896$	0	0
$897$	−3.73944	−0.124856
$898$	0	0
$899$	22.7713	0.759464
$900$	0	0
$901$	30.9592	1.03140
$902$	0	0
$903$	9.02237	0.300246
$904$	0	0
$905$	50.1635	1.66749
$906$	0	0
$907$	17.4949	0.580909	0.290454	−	0.956889i	$-0.406194\pi$
$907$	17.4949	0.580909	0.290454	+	0.956889i	$0.406194\pi$
$908$	0	0
$909$	21.8025	0.723143
$910$	0	0
$911$	−43.7260	−1.44871	−0.724353	−	0.689429i	$-0.757861\pi$
$911$	−43.7260	−1.44871	−0.724353	+	0.689429i	$0.757861\pi$
$912$	0	0
$913$	15.3290	0.507314
$914$	0	0
$915$	2.88111	0.0952466
$916$	0	0
$917$	−5.72718	−0.189128
$918$	0	0
$919$	−5.76629	−0.190212	−0.0951061	−	0.995467i	$-0.530319\pi$
$919$	−5.76629	−0.190212	−0.0951061	+	0.995467i	$0.530319\pi$
$920$	0	0
$921$	−15.3497	−0.505789
$922$	0	0
$923$	−16.2180	−0.533821
$924$	0	0
$925$	−0.646629	−0.0212610
$926$	0	0
$927$	26.7823	0.879646
$928$	0	0
$929$	0.993416	0.0325929	0.0162965	−	0.999867i	$-0.494812\pi$
$929$	0.993416	0.0325929	0.0162965	+	0.999867i	$0.494812\pi$
$930$	0	0
$931$	2.89553	0.0948972
$932$	0	0
$933$	0.403927	0.0132240
$934$	0	0
$935$	−6.08133	−0.198881
$936$	0	0
$937$	7.98084	0.260723	0.130361	−	0.991467i	$-0.458386\pi$
$937$	7.98084	0.260723	0.130361	+	0.991467i	$0.458386\pi$
$938$	0	0
$939$	−10.9423	−0.357087
$940$	0	0
$941$	2.46084	0.0802212	0.0401106	−	0.999195i	$-0.487229\pi$
$941$	2.46084	0.0802212	0.0401106	+	0.999195i	$0.487229\pi$
$942$	0	0
$943$	−1.09231	−0.0355705
$944$	0	0
$945$	11.5165	0.374632
$946$	0	0
$947$	−22.5354	−0.732301	−0.366150	−	0.930556i	$-0.619324\pi$
$947$	−22.5354	−0.732301	−0.366150	+	0.930556i	$0.619324\pi$
$948$	0	0
$949$	−0.183169	−0.00594591
$950$	0	0
$951$	18.0733	0.586068
$952$	0	0
$953$	−0.811014	−0.0262713	−0.0131357	−	0.999914i	$-0.504181\pi$
$953$	−0.811014	−0.0262713	−0.0131357	+	0.999914i	$0.504181\pi$
$954$	0	0
$955$	−30.2403	−0.978552
$956$	0	0
$957$	−3.04423	−0.0984060
$958$	0	0
$959$	−13.7036	−0.442513
$960$	0	0
$961$	22.6606	0.730986
$962$	0	0
$963$	4.40867	0.142067
$964$	0	0
$965$	−25.3833	−0.817118
$966$	0	0
$967$	49.2075	1.58241	0.791203	−	0.611554i	$-0.209455\pi$
$967$	49.2075	1.58241	0.791203	+	0.611554i	$0.209455\pi$
$968$	0	0
$969$	−7.39189	−0.237462
$970$	0	0
$971$	19.9554	0.640398	0.320199	−	0.947350i	$-0.396250\pi$
$971$	19.9554	0.640398	0.320199	+	0.947350i	$0.396250\pi$
$972$	0	0
$973$	6.16193	0.197542
$974$	0	0
$975$	−0.433093	−0.0138701
$976$	0	0
$977$	−29.0330	−0.928849	−0.464424	−	0.885613i	$-0.653739\pi$
$977$	−29.0330	−0.928849	−0.464424	+	0.885613i	$0.653739\pi$
$978$	0	0
$979$	−10.7996	−0.345158
$980$	0	0
$981$	−27.1444	−0.866653
$982$	0	0
$983$	−0.451059	−0.0143866	−0.00719328	−	0.999974i	$-0.502290\pi$
$983$	−0.451059	−0.0143866	−0.00719328	+	0.999974i	$0.502290\pi$
$984$	0	0
$985$	−34.7495	−1.10721
$986$	0	0
$987$	−3.73944	−0.119028
$988$	0	0
$989$	−35.1797	−1.11865
$990$	0	0
$991$	−12.7744	−0.405791	−0.202896	−	0.979200i	$-0.565035\pi$
$991$	−12.7744	−0.405791	−0.202896	+	0.979200i	$0.565035\pi$
$992$	0	0
$993$	−16.8337	−0.534200
$994$	0	0
$995$	3.88785	0.123253
$996$	0	0
$997$	14.5482	0.460745	0.230373	−	0.973102i	$-0.426006\pi$
$997$	14.5482	0.460745	0.230373	+	0.973102i	$0.426006\pi$
$998$	0	0
$999$	7.21812	0.228371

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8008.2.a.v.1.8	✓	11

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8008.2.a.v.1.8	✓	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:	\( N \)	\(=\)	\( 8008 = 2^{3} \cdot 7 \cdot 11 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8008.a (trivial)

\(f(q)\)	\(=\)	\(q+0.979305 q^{3} +2.33286 q^{5} -1.00000 q^{7} -2.04096 q^{9} +O(q^{10})\)	\(q+0.979305 q^{3} +2.33286 q^{5} -1.00000 q^{7} -2.04096 q^{9} +1.00000 q^{11} -1.00000 q^{13} +2.28458 q^{15} -2.60681 q^{17} +2.89553 q^{19} -0.979305 q^{21} +3.81847 q^{23} +0.442245 q^{25} -4.93664 q^{27} -3.10856 q^{29} -7.32534 q^{31} +0.979305 q^{33} -2.33286 q^{35} -1.46215 q^{37} -0.979305 q^{39} -0.286060 q^{41} -9.21304 q^{43} -4.76128 q^{45} +3.81847 q^{47} +1.00000 q^{49} -2.55286 q^{51} -11.8763 q^{53} +2.33286 q^{55} +2.83561 q^{57} -9.80563 q^{59} +1.26111 q^{61} +2.04096 q^{63} -2.33286 q^{65} -4.93124 q^{67} +3.73944 q^{69} +16.2180 q^{71} +0.183169 q^{73} +0.433093 q^{75} -1.00000 q^{77} -15.7288 q^{79} +1.28841 q^{81} +15.3290 q^{83} -6.08133 q^{85} -3.04423 q^{87} -10.7996 q^{89} +1.00000 q^{91} -7.17374 q^{93} +6.75487 q^{95} +4.70388 q^{97} -2.04096 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 11 q - 2 q^{3} + 2 q^{5} - 11 q^{7} + 9 q^{9}+O(q^{10}) \) 11 * q - 2 * q^3 + 2 * q^5 - 11 * q^7 + 9 * q^9	\( 11 q - 2 q^{3} + 2 q^{5} - 11 q^{7} + 9 q^{9} + 11 q^{11} - 11 q^{13} - 7 q^{15} - 4 q^{17} - 16 q^{19} + 2 q^{21} - 3 q^{23} + 11 q^{25} - 11 q^{27} + q^{29} + 14 q^{31} - 2 q^{33} - 2 q^{35} - 8 q^{37} + 2 q^{39} + 4 q^{41} - 30 q^{43} + 13 q^{45} - 3 q^{47} + 11 q^{49} - 14 q^{51} - 5 q^{53} + 2 q^{55} - 22 q^{57} + 11 q^{59} + 15 q^{61} - 9 q^{63} - 2 q^{65} - 41 q^{67} + 12 q^{69} + q^{71} - 8 q^{73} - 24 q^{75} - 11 q^{77} - 26 q^{79} + 19 q^{81} - 31 q^{83} - 27 q^{85} - 25 q^{87} + 2 q^{89} + 11 q^{91} - 37 q^{93} - 6 q^{97} + 9 q^{99}+O(q^{100}) \) 11 * q - 2 * q^3 + 2 * q^5 - 11 * q^7 + 9 * q^9 + 11 * q^11 - 11 * q^13 - 7 * q^15 - 4 * q^17 - 16 * q^19 + 2 * q^21 - 3 * q^23 + 11 * q^25 - 11 * q^27 + q^29 + 14 * q^31 - 2 * q^33 - 2 * q^35 - 8 * q^37 + 2 * q^39 + 4 * q^41 - 30 * q^43 + 13 * q^45 - 3 * q^47 + 11 * q^49 - 14 * q^51 - 5 * q^53 + 2 * q^55 - 22 * q^57 + 11 * q^59 + 15 * q^61 - 9 * q^63 - 2 * q^65 - 41 * q^67 + 12 * q^69 + q^71 - 8 * q^73 - 24 * q^75 - 11 * q^77 - 26 * q^79 + 19 * q^81 - 31 * q^83 - 27 * q^85 - 25 * q^87 + 2 * q^89 + 11 * q^91 - 37 * q^93 - 6 * q^97 + 9 * q^99

Introduction

L-functions

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Database

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