LMFDB - Embedded newform 8008.2.a.n.1.9

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:	$9$
Coefficient field:	$\mathbb{Q}[x]/(x^{9} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{9} - 3x^{8} - 15x^{7} + 45x^{6} + 64x^{5} - 201x^{4} - 63x^{3} + 282x^{2} + 3x - 116 $ x^9 - 3x^8 - 15x^7 + 45x^6 + 64x^5 - 201x^4 - 63x^3 + 282x^2 + 3x - 116
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.9
Root			$-2.82774$ 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)

$f(q)$	$=$	$q+2.82774 q^{3} +0.233208 q^{5} -1.00000 q^{7} +4.99612 q^{9} +O(q^{10})$	$q+2.82774 q^{3} +0.233208 q^{5} -1.00000 q^{7} +4.99612 q^{9} +1.00000 q^{11} -1.00000 q^{13} +0.659452 q^{15} -1.37647 q^{17} -2.67708 q^{19} -2.82774 q^{21} -3.07457 q^{23} -4.94561 q^{25} +5.64452 q^{27} +9.22098 q^{29} +6.01875 q^{31} +2.82774 q^{33} -0.233208 q^{35} -4.26255 q^{37} -2.82774 q^{39} +8.07364 q^{41} +6.43767 q^{43} +1.16514 q^{45} +11.0424 q^{47} +1.00000 q^{49} -3.89232 q^{51} +3.87764 q^{53} +0.233208 q^{55} -7.57010 q^{57} -9.61739 q^{59} +10.7387 q^{61} -4.99612 q^{63} -0.233208 q^{65} +11.2232 q^{67} -8.69409 q^{69} +0.633784 q^{71} +6.61461 q^{73} -13.9849 q^{75} -1.00000 q^{77} +1.86994 q^{79} +0.972871 q^{81} +11.0882 q^{83} -0.321005 q^{85} +26.0746 q^{87} +12.4318 q^{89} +1.00000 q^{91} +17.0195 q^{93} -0.624318 q^{95} +5.54876 q^{97} +4.99612 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q - 3 q^{3} + q^{5} - 9 q^{7} + 12 q^{9}+O(q^{10}) $ 9 * q - 3 * q^3 + q^5 - 9 * q^7 + 12 * q^9	$ 9 q - 3 q^{3} + q^{5} - 9 q^{7} + 12 q^{9} + 9 q^{11} - 9 q^{13} + 11 q^{15} + 7 q^{17} - 17 q^{19} + 3 q^{21} + 11 q^{23} + 18 q^{25} - 9 q^{27} + 9 q^{29} - 8 q^{31} - 3 q^{33} - q^{35} + 2 q^{37} + 3 q^{39} + 18 q^{41} + 7 q^{43} + 5 q^{45} + 15 q^{47} + 9 q^{49} - 7 q^{51} - 4 q^{53} + q^{55} + 22 q^{57} - 23 q^{59} + 12 q^{61} - 12 q^{63} - q^{65} - 16 q^{67} - 32 q^{69} - 6 q^{71} + 4 q^{73} - 14 q^{75} - 9 q^{77} + 21 q^{79} + 5 q^{81} - 16 q^{83} + 53 q^{85} + 41 q^{87} + 5 q^{89} + 9 q^{91} + 29 q^{93} + 19 q^{95} + 18 q^{97} + 12 q^{99}+O(q^{100}) $ 9 * q - 3 * q^3 + q^5 - 9 * q^7 + 12 * q^9 + 9 * q^11 - 9 * q^13 + 11 * q^15 + 7 * q^17 - 17 * q^19 + 3 * q^21 + 11 * q^23 + 18 * q^25 - 9 * q^27 + 9 * q^29 - 8 * q^31 - 3 * q^33 - q^35 + 2 * q^37 + 3 * q^39 + 18 * q^41 + 7 * q^43 + 5 * q^45 + 15 * q^47 + 9 * q^49 - 7 * q^51 - 4 * q^53 + q^55 + 22 * q^57 - 23 * q^59 + 12 * q^61 - 12 * q^63 - q^65 - 16 * q^67 - 32 * q^69 - 6 * q^71 + 4 * q^73 - 14 * q^75 - 9 * q^77 + 21 * q^79 + 5 * q^81 - 16 * q^83 + 53 * q^85 + 41 * q^87 + 5 * q^89 + 9 * q^91 + 29 * q^93 + 19 * q^95 + 18 * q^97 + 12 * 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
$3$	2.82774	1.63260	0.816299	−	0.577630i	$-0.196022\pi$
$3$	2.82774	1.63260	0.816299	+	0.577630i	$0.196022\pi$
$4$	0	0
$5$	0.233208	0.104294	0.0521469	−	0.998639i	$-0.483394\pi$
$5$	0.233208	0.104294	0.0521469	+	0.998639i	$0.483394\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	4.99612	1.66537
$10$	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
$15$	0.659452	0.170270
$16$	0	0
$17$	−1.37647	−0.333844	−0.166922	−	0.985970i	$-0.553383\pi$
$17$	−1.37647	−0.333844	−0.166922	+	0.985970i	$0.553383\pi$
$18$	0	0
$19$	−2.67708	−0.614165	−0.307083	−	0.951683i	$-0.599353\pi$
$19$	−2.67708	−0.614165	−0.307083	+	0.951683i	$0.599353\pi$
$20$	0	0
$21$	−2.82774	−0.617064
$22$	0	0
$23$	−3.07457	−0.641093	−0.320546	−	0.947233i	$-0.603866\pi$
$23$	−3.07457	−0.641093	−0.320546	+	0.947233i	$0.603866\pi$
$24$	0	0
$25$	−4.94561	−0.989123
$26$	0	0
$27$	5.64452	1.08629
$28$	0	0
$29$	9.22098	1.71229	0.856147	−	0.516733i	$-0.172852\pi$
$29$	9.22098	1.71229	0.856147	+	0.516733i	$0.172852\pi$
$30$	0	0
$31$	6.01875	1.08100	0.540500	−	0.841344i	$-0.318235\pi$
$31$	6.01875	1.08100	0.540500	+	0.841344i	$0.318235\pi$
$32$	0	0
$33$	2.82774	0.492247
$34$	0	0
$35$	−0.233208	−0.0394194
$36$	0	0
$37$	−4.26255	−0.700759	−0.350379	−	0.936608i	$-0.613947\pi$
$37$	−4.26255	−0.700759	−0.350379	+	0.936608i	$0.613947\pi$
$38$	0	0
$39$	−2.82774	−0.452801
$40$	0	0
$41$	8.07364	1.26089	0.630445	−	0.776234i	$-0.282872\pi$
$41$	8.07364	1.26089	0.630445	+	0.776234i	$0.282872\pi$
$42$	0	0
$43$	6.43767	0.981735	0.490868	−	0.871234i	$-0.336680\pi$
$43$	6.43767	0.981735	0.490868	+	0.871234i	$0.336680\pi$
$44$	0	0
$45$	1.16514	0.173688
$46$	0	0
$47$	11.0424	1.61070	0.805350	−	0.592799i	$-0.201977\pi$
$47$	11.0424	1.61070	0.805350	+	0.592799i	$0.201977\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−3.89232	−0.545033
$52$	0	0
$53$	3.87764	0.532635	0.266318	−	0.963885i	$-0.414193\pi$
$53$	3.87764	0.532635	0.266318	+	0.963885i	$0.414193\pi$
$54$	0	0
$55$	0.233208	0.0314458
$56$	0	0
$57$	−7.57010	−1.00268
$58$	0	0
$59$	−9.61739	−1.25208	−0.626039	−	0.779792i	$-0.715325\pi$
$59$	−9.61739	−1.25208	−0.626039	+	0.779792i	$0.715325\pi$
$60$	0	0
$61$	10.7387	1.37495	0.687475	−	0.726208i	$-0.258719\pi$
$61$	10.7387	1.37495	0.687475	+	0.726208i	$0.258719\pi$
$62$	0	0
$63$	−4.99612	−0.629452
$64$	0	0
$65$	−0.233208	−0.0289259
$66$	0	0
$67$	11.2232	1.37113	0.685566	−	0.728011i	$-0.259555\pi$
$67$	11.2232	1.37113	0.685566	+	0.728011i	$0.259555\pi$
$68$	0	0
$69$	−8.69409	−1.04665
$70$	0	0
$71$	0.633784	0.0752163	0.0376082	−	0.999293i	$-0.488026\pi$
$71$	0.633784	0.0752163	0.0376082	+	0.999293i	$0.488026\pi$
$72$	0	0
$73$	6.61461	0.774182	0.387091	−	0.922042i	$-0.373480\pi$
$73$	6.61461	0.774182	0.387091	+	0.922042i	$0.373480\pi$
$74$	0	0
$75$	−13.9849	−1.61484
$76$	0	0
$77$	−1.00000	−0.113961
$78$	0	0
$79$	1.86994	0.210384	0.105192	−	0.994452i	$-0.466454\pi$
$79$	1.86994	0.210384	0.105192	+	0.994452i	$0.466454\pi$
$80$	0	0
$81$	0.972871	0.108097
$82$	0	0
$83$	11.0882	1.21709	0.608543	−	0.793521i	$-0.291754\pi$
$83$	11.0882	1.21709	0.608543	+	0.793521i	$0.291754\pi$
$84$	0	0
$85$	−0.321005	−0.0348179
$86$	0	0
$87$	26.0746	2.79549
$88$	0	0
$89$	12.4318	1.31777	0.658886	−	0.752243i	$-0.271028\pi$
$89$	12.4318	1.31777	0.658886	+	0.752243i	$0.271028\pi$
$90$	0	0
$91$	1.00000	0.104828
$92$	0	0
$93$	17.0195	1.76484
$94$	0	0
$95$	−0.624318	−0.0640537
$96$	0	0
$97$	5.54876	0.563391	0.281695	−	0.959504i	$-0.409103\pi$
$97$	5.54876	0.563391	0.281695	+	0.959504i	$0.409103\pi$
$98$	0	0
$99$	4.99612	0.502129
$100$	0	0
$101$	3.29342	0.327707	0.163854	−	0.986485i	$-0.447608\pi$
$101$	3.29342	0.327707	0.163854	+	0.986485i	$0.447608\pi$
$102$	0	0
$103$	4.71506	0.464589	0.232294	−	0.972646i	$-0.425377\pi$
$103$	4.71506	0.464589	0.232294	+	0.972646i	$0.425377\pi$
$104$	0	0
$105$	−0.659452	−0.0643560
$106$	0	0
$107$	−6.23341	−0.602607	−0.301303	−	0.953528i	$-0.597422\pi$
$107$	−6.23341	−0.602607	−0.301303	+	0.953528i	$0.597422\pi$
$108$	0	0
$109$	17.1305	1.64080	0.820401	−	0.571789i	$-0.193750\pi$
$109$	17.1305	1.64080	0.820401	+	0.571789i	$0.193750\pi$
$110$	0	0
$111$	−12.0534	−1.14406
$112$	0	0
$113$	−13.0752	−1.23001	−0.615005	−	0.788523i	$-0.710846\pi$
$113$	−13.0752	−1.23001	−0.615005	+	0.788523i	$0.710846\pi$
$114$	0	0
$115$	−0.717015	−0.0668620
$116$	0	0
$117$	−4.99612	−0.461892
$118$	0	0
$119$	1.37647	0.126181
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	22.8302	2.05853
$124$	0	0
$125$	−2.31940	−0.207453
$126$	0	0
$127$	9.93787	0.881844	0.440922	−	0.897546i	$-0.354652\pi$
$127$	9.93787	0.881844	0.440922	+	0.897546i	$0.354652\pi$
$128$	0	0
$129$	18.2041	1.60278
$130$	0	0
$131$	−16.4134	−1.43405	−0.717023	−	0.697050i	$-0.754496\pi$
$131$	−16.4134	−1.43405	−0.717023	+	0.697050i	$0.754496\pi$
$132$	0	0
$133$	2.67708	0.232133
$134$	0	0
$135$	1.31635	0.113293
$136$	0	0
$137$	−1.35339	−0.115628	−0.0578140	−	0.998327i	$-0.518413\pi$
$137$	−1.35339	−0.115628	−0.0578140	+	0.998327i	$0.518413\pi$
$138$	0	0
$139$	18.9706	1.60907	0.804533	−	0.593908i	$-0.202416\pi$
$139$	18.9706	1.60907	0.804533	+	0.593908i	$0.202416\pi$
$140$	0	0
$141$	31.2251	2.62963
$142$	0	0
$143$	−1.00000	−0.0836242
$144$	0	0
$145$	2.15041	0.178582
$146$	0	0
$147$	2.82774	0.233228
$148$	0	0
$149$	−11.4974	−0.941901	−0.470950	−	0.882160i	$-0.656089\pi$
$149$	−11.4974	−0.941901	−0.470950	+	0.882160i	$0.656089\pi$
$150$	0	0
$151$	5.60453	0.456090	0.228045	−	0.973651i	$-0.426767\pi$
$151$	5.60453	0.456090	0.228045	+	0.973651i	$0.426767\pi$
$152$	0	0
$153$	−6.87704	−0.555975
$154$	0	0
$155$	1.40362	0.112742
$156$	0	0
$157$	−1.67781	−0.133904	−0.0669521	−	0.997756i	$-0.521327\pi$
$157$	−1.67781	−0.133904	−0.0669521	+	0.997756i	$0.521327\pi$
$158$	0	0
$159$	10.9650	0.869579
$160$	0	0
$161$	3.07457	0.242310
$162$	0	0
$163$	−12.6136	−0.987971	−0.493986	−	0.869470i	$-0.664460\pi$
$163$	−12.6136	−0.987971	−0.493986	+	0.869470i	$0.664460\pi$
$164$	0	0
$165$	0.659452	0.0513383
$166$	0	0
$167$	−22.0678	−1.70765	−0.853827	−	0.520557i	$-0.825724\pi$
$167$	−22.0678	−1.70765	−0.853827	+	0.520557i	$0.825724\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	−13.3750	−1.02282
$172$	0	0
$173$	6.88776	0.523667	0.261833	−	0.965113i	$-0.415673\pi$
$173$	6.88776	0.523667	0.261833	+	0.965113i	$0.415673\pi$
$174$	0	0
$175$	4.94561	0.373853
$176$	0	0
$177$	−27.1955	−2.04414
$178$	0	0
$179$	−10.9773	−0.820482	−0.410241	−	0.911977i	$-0.634555\pi$
$179$	−10.9773	−0.820482	−0.410241	+	0.911977i	$0.634555\pi$
$180$	0	0
$181$	0.875823	0.0650994	0.0325497	−	0.999470i	$-0.489637\pi$
$181$	0.875823	0.0650994	0.0325497	+	0.999470i	$0.489637\pi$
$182$	0	0
$183$	30.3663	2.24474
$184$	0	0
$185$	−0.994061	−0.0730848
$186$	0	0
$187$	−1.37647	−0.100658
$188$	0	0
$189$	−5.64452	−0.410578
$190$	0	0
$191$	−1.48144	−0.107193	−0.0535966	−	0.998563i	$-0.517068\pi$
$191$	−1.48144	−0.107193	−0.0535966	+	0.998563i	$0.517068\pi$
$192$	0	0
$193$	−17.5835	−1.26569	−0.632845	−	0.774278i	$-0.718113\pi$
$193$	−17.5835	−1.26569	−0.632845	+	0.774278i	$0.718113\pi$
$194$	0	0
$195$	−0.659452	−0.0472244
$196$	0	0
$197$	−5.60141	−0.399084	−0.199542	−	0.979889i	$-0.563945\pi$
$197$	−5.60141	−0.399084	−0.199542	+	0.979889i	$0.563945\pi$
$198$	0	0
$199$	−15.2498	−1.08103	−0.540513	−	0.841336i	$-0.681770\pi$
$199$	−15.2498	−1.08103	−0.540513	+	0.841336i	$0.681770\pi$
$200$	0	0
$201$	31.7363	2.23851
$202$	0	0
$203$	−9.22098	−0.647186
$204$	0	0
$205$	1.88284	0.131503
$206$	0	0
$207$	−15.3609	−1.06766
$208$	0	0
$209$	−2.67708	−0.185178
$210$	0	0
$211$	−7.62190	−0.524713	−0.262357	−	0.964971i	$-0.584500\pi$
$211$	−7.62190	−0.524713	−0.262357	+	0.964971i	$0.584500\pi$
$212$	0	0
$213$	1.79218	0.122798
$214$	0	0
$215$	1.50132	0.102389
$216$	0	0
$217$	−6.01875	−0.408580
$218$	0	0
$219$	18.7044	1.26393
$220$	0	0
$221$	1.37647	0.0925917
$222$	0	0
$223$	−5.72478	−0.383360	−0.191680	−	0.981457i	$-0.561394\pi$
$223$	−5.72478	−0.383360	−0.191680	+	0.981457i	$0.561394\pi$
$224$	0	0
$225$	−24.7089	−1.64726
$226$	0	0
$227$	13.2484	0.879327	0.439663	−	0.898163i	$-0.355098\pi$
$227$	13.2484	0.879327	0.439663	+	0.898163i	$0.355098\pi$
$228$	0	0
$229$	−5.53650	−0.365862	−0.182931	−	0.983126i	$-0.558559\pi$
$229$	−5.53650	−0.365862	−0.182931	+	0.983126i	$0.558559\pi$
$230$	0	0
$231$	−2.82774	−0.186052
$232$	0	0
$233$	−8.78101	−0.575263	−0.287632	−	0.957741i	$-0.592868\pi$
$233$	−8.78101	−0.575263	−0.287632	+	0.957741i	$0.592868\pi$
$234$	0	0
$235$	2.57518	0.167986
$236$	0	0
$237$	5.28770	0.343473
$238$	0	0
$239$	8.36273	0.540940	0.270470	−	0.962728i	$-0.412821\pi$
$239$	8.36273	0.540940	0.270470	+	0.962728i	$0.412821\pi$
$240$	0	0
$241$	−30.5523	−1.96805	−0.984023	−	0.178040i	$-0.943024\pi$
$241$	−30.5523	−1.96805	−0.984023	+	0.178040i	$0.943024\pi$
$242$	0	0
$243$	−14.1825	−0.909810
$244$	0	0
$245$	0.233208	0.0148991
$246$	0	0
$247$	2.67708	0.170339
$248$	0	0
$249$	31.3545	1.98701
$250$	0	0
$251$	−17.0290	−1.07486	−0.537429	−	0.843309i	$-0.680605\pi$
$251$	−17.0290	−1.07486	−0.537429	+	0.843309i	$0.680605\pi$
$252$	0	0
$253$	−3.07457	−0.193297
$254$	0	0
$255$	−0.907720	−0.0568436
$256$	0	0
$257$	−2.20347	−0.137449	−0.0687244	−	0.997636i	$-0.521893\pi$
$257$	−2.20347	−0.137449	−0.0687244	+	0.997636i	$0.521893\pi$
$258$	0	0
$259$	4.26255	0.264862
$260$	0	0
$261$	46.0692	2.85161
$262$	0	0
$263$	22.3070	1.37551	0.687755	−	0.725943i	$-0.258596\pi$
$263$	22.3070	1.37551	0.687755	+	0.725943i	$0.258596\pi$
$264$	0	0
$265$	0.904298	0.0555506
$266$	0	0
$267$	35.1540	2.15139
$268$	0	0
$269$	−16.6954	−1.01794	−0.508968	−	0.860785i	$-0.669973\pi$
$269$	−16.6954	−1.01794	−0.508968	+	0.860785i	$0.669973\pi$
$270$	0	0
$271$	16.0899	0.977394	0.488697	−	0.872454i	$-0.337472\pi$
$271$	16.0899	0.977394	0.488697	+	0.872454i	$0.337472\pi$
$272$	0	0
$273$	2.82774	0.171143
$274$	0	0
$275$	−4.94561	−0.298232
$276$	0	0
$277$	6.30137	0.378613	0.189306	−	0.981918i	$-0.439376\pi$
$277$	6.30137	0.378613	0.189306	+	0.981918i	$0.439376\pi$
$278$	0	0
$279$	30.0704	1.80027
$280$	0	0
$281$	18.7296	1.11731	0.558656	−	0.829400i	$-0.311317\pi$
$281$	18.7296	1.11731	0.558656	+	0.829400i	$0.311317\pi$
$282$	0	0
$283$	−28.6955	−1.70577	−0.852886	−	0.522098i	$-0.825150\pi$
$283$	−28.6955	−1.70577	−0.852886	+	0.522098i	$0.825150\pi$
$284$	0	0
$285$	−1.76541	−0.104574
$286$	0	0
$287$	−8.07364	−0.476572
$288$	0	0
$289$	−15.1053	−0.888548
$290$	0	0
$291$	15.6905	0.919791
$292$	0	0
$293$	3.45232	0.201687	0.100843	−	0.994902i	$-0.467846\pi$
$293$	3.45232	0.201687	0.100843	+	0.994902i	$0.467846\pi$
$294$	0	0
$295$	−2.24285	−0.130584
$296$	0	0
$297$	5.64452	0.327528
$298$	0	0
$299$	3.07457	0.177807
$300$	0	0
$301$	−6.43767	−0.371061
$302$	0	0
$303$	9.31294	0.535014
$304$	0	0
$305$	2.50435	0.143399
$306$	0	0
$307$	22.1992	1.26697	0.633487	−	0.773753i	$-0.281623\pi$
$307$	22.1992	1.26697	0.633487	+	0.773753i	$0.281623\pi$
$308$	0	0
$309$	13.3330	0.758486
$310$	0	0
$311$	−24.4459	−1.38620	−0.693099	−	0.720843i	$-0.743755\pi$
$311$	−24.4459	−1.38620	−0.693099	+	0.720843i	$0.743755\pi$
$312$	0	0
$313$	6.66325	0.376629	0.188314	−	0.982109i	$-0.439698\pi$
$313$	6.66325	0.376629	0.188314	+	0.982109i	$0.439698\pi$
$314$	0	0
$315$	−1.16514	−0.0656480
$316$	0	0
$317$	23.3545	1.31172	0.655860	−	0.754883i	$-0.272306\pi$
$317$	23.3545	1.31172	0.655860	+	0.754883i	$0.272306\pi$
$318$	0	0
$319$	9.22098	0.516276
$320$	0	0
$321$	−17.6265	−0.983815
$322$	0	0
$323$	3.68494	0.205036
$324$	0	0
$325$	4.94561	0.274333
$326$	0	0
$327$	48.4405	2.67877
$328$	0	0
$329$	−11.0424	−0.608788
$330$	0	0
$331$	−12.5712	−0.690976	−0.345488	−	0.938423i	$-0.612287\pi$
$331$	−12.5712	−0.690976	−0.345488	+	0.938423i	$0.612287\pi$
$332$	0	0
$333$	−21.2962	−1.16703
$334$	0	0
$335$	2.61734	0.143001
$336$	0	0
$337$	−8.04391	−0.438179	−0.219090	−	0.975705i	$-0.570309\pi$
$337$	−8.04391	−0.438179	−0.219090	+	0.975705i	$0.570309\pi$
$338$	0	0
$339$	−36.9732	−2.00811
$340$	0	0
$341$	6.01875	0.325934
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	−2.02753	−0.109159
$346$	0	0
$347$	9.68601	0.519972	0.259986	−	0.965612i	$-0.416282\pi$
$347$	9.68601	0.519972	0.259986	+	0.965612i	$0.416282\pi$
$348$	0	0
$349$	−22.8823	−1.22486	−0.612432	−	0.790523i	$-0.709809\pi$
$349$	−22.8823	−1.22486	−0.612432	+	0.790523i	$0.709809\pi$
$350$	0	0
$351$	−5.64452	−0.301282
$352$	0	0
$353$	−17.6747	−0.940729	−0.470364	−	0.882472i	$-0.655877\pi$
$353$	−17.6747	−0.940729	−0.470364	+	0.882472i	$0.655877\pi$
$354$	0	0
$355$	0.147804	0.00784460
$356$	0	0
$357$	3.89232	0.206003
$358$	0	0
$359$	11.8212	0.623901	0.311950	−	0.950098i	$-0.399018\pi$
$359$	11.8212	0.623901	0.311950	+	0.950098i	$0.399018\pi$
$360$	0	0
$361$	−11.8332	−0.622801
$362$	0	0
$363$	2.82774	0.148418
$364$	0	0
$365$	1.54258	0.0807424
$366$	0	0
$367$	−26.8779	−1.40301	−0.701507	−	0.712662i	$-0.747489\pi$
$367$	−26.8779	−1.40301	−0.701507	+	0.712662i	$0.747489\pi$
$368$	0	0
$369$	40.3369	2.09985
$370$	0	0
$371$	−3.87764	−0.201317
$372$	0	0
$373$	−17.9892	−0.931445	−0.465723	−	0.884931i	$-0.654206\pi$
$373$	−17.9892	−0.931445	−0.465723	+	0.884931i	$0.654206\pi$
$374$	0	0
$375$	−6.55866	−0.338688
$376$	0	0
$377$	−9.22098	−0.474905
$378$	0	0
$379$	0.474924	0.0243952	0.0121976	−	0.999926i	$-0.496117\pi$
$379$	0.474924	0.0243952	0.0121976	+	0.999926i	$0.496117\pi$
$380$	0	0
$381$	28.1017	1.43970
$382$	0	0
$383$	10.4478	0.533857	0.266929	−	0.963716i	$-0.413991\pi$
$383$	10.4478	0.533857	0.266929	+	0.963716i	$0.413991\pi$
$384$	0	0
$385$	−0.233208	−0.0118854
$386$	0	0
$387$	32.1634	1.63496
$388$	0	0
$389$	−6.69516	−0.339458	−0.169729	−	0.985491i	$-0.554289\pi$
$389$	−6.69516	−0.339458	−0.169729	+	0.985491i	$0.554289\pi$
$390$	0	0
$391$	4.23207	0.214025
$392$	0	0
$393$	−46.4129	−2.34122
$394$	0	0
$395$	0.436085	0.0219418
$396$	0	0
$397$	30.9425	1.55296	0.776480	−	0.630142i	$-0.217003\pi$
$397$	30.9425	1.55296	0.776480	+	0.630142i	$0.217003\pi$
$398$	0	0
$399$	7.57010	0.378979
$400$	0	0
$401$	28.2888	1.41267	0.706337	−	0.707875i	$-0.250346\pi$
$401$	28.2888	1.41267	0.706337	+	0.707875i	$0.250346\pi$
$402$	0	0
$403$	−6.01875	−0.299815
$404$	0	0
$405$	0.226881	0.0112738
$406$	0	0
$407$	−4.26255	−0.211287
$408$	0	0
$409$	−15.5923	−0.770988	−0.385494	−	0.922710i	$-0.625969\pi$
$409$	−15.5923	−0.770988	−0.385494	+	0.922710i	$0.625969\pi$
$410$	0	0
$411$	−3.82704	−0.188774
$412$	0	0
$413$	9.61739	0.473241
$414$	0	0
$415$	2.58586	0.126935
$416$	0	0
$417$	53.6440	2.62696
$418$	0	0
$419$	−7.66727	−0.374571	−0.187285	−	0.982306i	$-0.559969\pi$
$419$	−7.66727	−0.374571	−0.187285	+	0.982306i	$0.559969\pi$
$420$	0	0
$421$	11.7765	0.573951	0.286976	−	0.957938i	$-0.407350\pi$
$421$	11.7765	0.573951	0.286976	+	0.957938i	$0.407350\pi$
$422$	0	0
$423$	55.1692	2.68242
$424$	0	0
$425$	6.80751	0.330213
$426$	0	0
$427$	−10.7387	−0.519683
$428$	0	0
$429$	−2.82774	−0.136525
$430$	0	0
$431$	−2.78161	−0.133985	−0.0669926	−	0.997753i	$-0.521340\pi$
$431$	−2.78161	−0.133985	−0.0669926	+	0.997753i	$0.521340\pi$
$432$	0	0
$433$	37.5833	1.80614	0.903068	−	0.429497i	$-0.141309\pi$
$433$	37.5833	1.80614	0.903068	+	0.429497i	$0.141309\pi$
$434$	0	0
$435$	6.08080	0.291552
$436$	0	0
$437$	8.23089	0.393737
$438$	0	0
$439$	14.4168	0.688078	0.344039	−	0.938955i	$-0.388205\pi$
$439$	14.4168	0.688078	0.344039	+	0.938955i	$0.388205\pi$
$440$	0	0
$441$	4.99612	0.237911
$442$	0	0
$443$	36.1621	1.71811	0.859056	−	0.511882i	$-0.171052\pi$
$443$	36.1621	1.71811	0.859056	+	0.511882i	$0.171052\pi$
$444$	0	0
$445$	2.89921	0.137436
$446$	0	0
$447$	−32.5116	−1.53774
$448$	0	0
$449$	−29.9928	−1.41545	−0.707724	−	0.706489i	$-0.750278\pi$
$449$	−29.9928	−1.41545	−0.707724	+	0.706489i	$0.750278\pi$
$450$	0	0
$451$	8.07364	0.380173
$452$	0	0
$453$	15.8481	0.744611
$454$	0	0
$455$	0.233208	0.0109330
$456$	0	0
$457$	−1.98949	−0.0930646	−0.0465323	−	0.998917i	$-0.514817\pi$
$457$	−1.98949	−0.0930646	−0.0465323	+	0.998917i	$0.514817\pi$
$458$	0	0
$459$	−7.76954	−0.362651
$460$	0	0
$461$	−12.9126	−0.601401	−0.300700	−	0.953719i	$-0.597220\pi$
$461$	−12.9126	−0.601401	−0.300700	+	0.953719i	$0.597220\pi$
$462$	0	0
$463$	35.4548	1.64773	0.823863	−	0.566789i	$-0.191814\pi$
$463$	35.4548	1.64773	0.823863	+	0.566789i	$0.191814\pi$
$464$	0	0
$465$	3.96908	0.184062
$466$	0	0
$467$	−36.1342	−1.67209	−0.836046	−	0.548659i	$-0.815138\pi$
$467$	−36.1342	−1.67209	−0.836046	+	0.548659i	$0.815138\pi$
$468$	0	0
$469$	−11.2232	−0.518239
$470$	0	0
$471$	−4.74442	−0.218612
$472$	0	0
$473$	6.43767	0.296004
$474$	0	0
$475$	13.2398	0.607485
$476$	0	0
$477$	19.3732	0.887037
$478$	0	0
$479$	28.2792	1.29211	0.646055	−	0.763291i	$-0.276418\pi$
$479$	28.2792	1.29211	0.646055	+	0.763291i	$0.276418\pi$
$480$	0	0
$481$	4.26255	0.194355
$482$	0	0
$483$	8.69409	0.395595
$484$	0	0
$485$	1.29402	0.0587582
$486$	0	0
$487$	−20.9278	−0.948330	−0.474165	−	0.880436i	$-0.657250\pi$
$487$	−20.9278	−0.948330	−0.474165	+	0.880436i	$0.657250\pi$
$488$	0	0
$489$	−35.6679	−1.61296
$490$	0	0
$491$	−19.1424	−0.863884	−0.431942	−	0.901901i	$-0.642171\pi$
$491$	−19.1424	−0.863884	−0.431942	+	0.901901i	$0.642171\pi$
$492$	0	0
$493$	−12.6925	−0.571639
$494$	0	0
$495$	1.16514	0.0523690
$496$	0	0
$497$	−0.633784	−0.0284291
$498$	0	0
$499$	−33.6064	−1.50443	−0.752214	−	0.658918i	$-0.771014\pi$
$499$	−33.6064	−1.50443	−0.752214	+	0.658918i	$0.771014\pi$
$500$	0	0
$501$	−62.4019	−2.78791
$502$	0	0
$503$	14.5290	0.647814	0.323907	−	0.946089i	$-0.395003\pi$
$503$	14.5290	0.647814	0.323907	+	0.946089i	$0.395003\pi$
$504$	0	0
$505$	0.768052	0.0341779
$506$	0	0
$507$	2.82774	0.125584
$508$	0	0
$509$	27.6413	1.22518	0.612590	−	0.790401i	$-0.290128\pi$
$509$	27.6413	1.22518	0.612590	+	0.790401i	$0.290128\pi$
$510$	0	0
$511$	−6.61461	−0.292613
$512$	0	0
$513$	−15.1109	−0.667160
$514$	0	0
$515$	1.09959	0.0484537
$516$	0	0
$517$	11.0424	0.485645
$518$	0	0
$519$	19.4768	0.854937
$520$	0	0
$521$	2.07965	0.0911111	0.0455555	−	0.998962i	$-0.485494\pi$
$521$	2.07965	0.0911111	0.0455555	+	0.998962i	$0.485494\pi$
$522$	0	0
$523$	−18.3341	−0.801696	−0.400848	−	0.916145i	$-0.631284\pi$
$523$	−18.3341	−0.801696	−0.400848	+	0.916145i	$0.631284\pi$
$524$	0	0
$525$	13.9849	0.610352
$526$	0	0
$527$	−8.28466	−0.360886
$528$	0	0
$529$	−13.5470	−0.589000
$530$	0	0
$531$	−48.0496	−2.08518
$532$	0	0
$533$	−8.07364	−0.349708
$534$	0	0
$535$	−1.45368	−0.0628482
$536$	0	0
$537$	−31.0410	−1.33952
$538$	0	0
$539$	1.00000	0.0430730
$540$	0	0
$541$	16.1030	0.692321	0.346161	−	0.938175i	$-0.387485\pi$
$541$	16.1030	0.692321	0.346161	+	0.938175i	$0.387485\pi$
$542$	0	0
$543$	2.47660	0.106281
$544$	0	0
$545$	3.99497	0.171126
$546$	0	0
$547$	−0.467630	−0.0199944	−0.00999720	−	0.999950i	$-0.503182\pi$
$547$	−0.467630	−0.0199944	−0.00999720	+	0.999950i	$0.503182\pi$
$548$	0	0
$549$	53.6519	2.28981
$550$	0	0
$551$	−24.6854	−1.05163
$552$	0	0
$553$	−1.86994	−0.0795179
$554$	0	0
$555$	−2.81095	−0.119318
$556$	0	0
$557$	−42.4841	−1.80011	−0.900055	−	0.435777i	$-0.856474\pi$
$557$	−42.4841	−1.80011	−0.900055	+	0.435777i	$0.856474\pi$
$558$	0	0
$559$	−6.43767	−0.272284
$560$	0	0
$561$	−3.89232	−0.164334
$562$	0	0
$563$	8.27832	0.348890	0.174445	−	0.984667i	$-0.444187\pi$
$563$	8.27832	0.348890	0.174445	+	0.984667i	$0.444187\pi$
$564$	0	0
$565$	−3.04924	−0.128282
$566$	0	0
$567$	−0.972871	−0.0408567
$568$	0	0
$569$	−5.10154	−0.213868	−0.106934	−	0.994266i	$-0.534103\pi$
$569$	−5.10154	−0.213868	−0.106934	+	0.994266i	$0.534103\pi$
$570$	0	0
$571$	15.3622	0.642887	0.321443	−	0.946929i	$-0.395832\pi$
$571$	15.3622	0.642887	0.321443	+	0.946929i	$0.395832\pi$
$572$	0	0
$573$	−4.18913	−0.175003
$574$	0	0
$575$	15.2056	0.634119
$576$	0	0
$577$	−7.19740	−0.299632	−0.149816	−	0.988714i	$-0.547868\pi$
$577$	−7.19740	−0.299632	−0.149816	+	0.988714i	$0.547868\pi$
$578$	0	0
$579$	−49.7217	−2.06636
$580$	0	0
$581$	−11.0882	−0.460015
$582$	0	0
$583$	3.87764	0.160596
$584$	0	0
$585$	−1.16514	−0.0481725
$586$	0	0
$587$	22.6709	0.935729	0.467864	−	0.883800i	$-0.345024\pi$
$587$	22.6709	0.935729	0.467864	+	0.883800i	$0.345024\pi$
$588$	0	0
$589$	−16.1127	−0.663913
$590$	0	0
$591$	−15.8393	−0.651544
$592$	0	0
$593$	34.6355	1.42231	0.711154	−	0.703036i	$-0.248173\pi$
$593$	34.6355	1.42231	0.711154	+	0.703036i	$0.248173\pi$
$594$	0	0
$595$	0.321005	0.0131599
$596$	0	0
$597$	−43.1224	−1.76488
$598$	0	0
$599$	−18.3020	−0.747800	−0.373900	−	0.927469i	$-0.621980\pi$
$599$	−18.3020	−0.747800	−0.373900	+	0.927469i	$0.621980\pi$
$600$	0	0
$601$	−5.23738	−0.213637	−0.106819	−	0.994279i	$-0.534066\pi$
$601$	−5.23738	−0.213637	−0.106819	+	0.994279i	$0.534066\pi$
$602$	0	0
$603$	56.0724	2.28345
$604$	0	0
$605$	0.233208	0.00948126
$606$	0	0
$607$	−26.3505	−1.06954	−0.534768	−	0.844999i	$-0.679601\pi$
$607$	−26.3505	−1.06954	−0.534768	+	0.844999i	$0.679601\pi$
$608$	0	0
$609$	−26.0746	−1.05659
$610$	0	0
$611$	−11.0424	−0.446728
$612$	0	0
$613$	47.4507	1.91652	0.958259	−	0.285903i	$-0.0922936\pi$
$613$	47.4507	1.91652	0.958259	+	0.285903i	$0.0922936\pi$
$614$	0	0
$615$	5.32418	0.214692
$616$	0	0
$617$	−15.2072	−0.612220	−0.306110	−	0.951996i	$-0.599027\pi$
$617$	−15.2072	−0.612220	−0.306110	+	0.951996i	$0.599027\pi$
$618$	0	0
$619$	−30.5023	−1.22599	−0.612995	−	0.790086i	$-0.710036\pi$
$619$	−30.5023	−1.22599	−0.612995	+	0.790086i	$0.710036\pi$
$620$	0	0
$621$	−17.3545	−0.696411
$622$	0	0
$623$	−12.4318	−0.498071
$624$	0	0
$625$	24.1872	0.967487
$626$	0	0
$627$	−7.57010	−0.302321
$628$	0	0
$629$	5.86729	0.233944
$630$	0	0
$631$	0.914236	0.0363952	0.0181976	−	0.999834i	$-0.494207\pi$
$631$	0.914236	0.0363952	0.0181976	+	0.999834i	$0.494207\pi$
$632$	0	0
$633$	−21.5528	−0.856645
$634$	0	0
$635$	2.31759	0.0919709
$636$	0	0
$637$	−1.00000	−0.0396214
$638$	0	0
$639$	3.16646	0.125263
$640$	0	0
$641$	−7.74078	−0.305742	−0.152871	−	0.988246i	$-0.548852\pi$
$641$	−7.74078	−0.305742	−0.152871	+	0.988246i	$0.548852\pi$
$642$	0	0
$643$	−8.11887	−0.320177	−0.160089	−	0.987103i	$-0.551178\pi$
$643$	−8.11887	−0.320177	−0.160089	+	0.987103i	$0.551178\pi$
$644$	0	0
$645$	4.24534	0.167160
$646$	0	0
$647$	−7.99447	−0.314295	−0.157148	−	0.987575i	$-0.550230\pi$
$647$	−7.99447	−0.314295	−0.157148	+	0.987575i	$0.550230\pi$
$648$	0	0
$649$	−9.61739	−0.377516
$650$	0	0
$651$	−17.0195	−0.667046
$652$	0	0
$653$	21.8267	0.854146	0.427073	−	0.904217i	$-0.359545\pi$
$653$	21.8267	0.854146	0.427073	+	0.904217i	$0.359545\pi$
$654$	0	0
$655$	−3.82774	−0.149562
$656$	0	0
$657$	33.0474	1.28930
$658$	0	0
$659$	24.7032	0.962301	0.481150	−	0.876638i	$-0.340219\pi$
$659$	24.7032	0.962301	0.481150	+	0.876638i	$0.340219\pi$
$660$	0	0
$661$	−40.6473	−1.58100	−0.790499	−	0.612463i	$-0.790179\pi$
$661$	−40.6473	−1.58100	−0.790499	+	0.612463i	$0.790179\pi$
$662$	0	0
$663$	3.89232	0.151165
$664$	0	0
$665$	0.624318	0.0242100
$666$	0	0
$667$	−28.3506	−1.09774
$668$	0	0
$669$	−16.1882	−0.625872
$670$	0	0
$671$	10.7387	0.414563
$672$	0	0
$673$	21.6599	0.834929	0.417464	−	0.908693i	$-0.362919\pi$
$673$	21.6599	0.834929	0.417464	+	0.908693i	$0.362919\pi$
$674$	0	0
$675$	−27.9156	−1.07447
$676$	0	0
$677$	−1.68620	−0.0648059	−0.0324029	−	0.999475i	$-0.510316\pi$
$677$	−1.68620	−0.0648059	−0.0324029	+	0.999475i	$0.510316\pi$
$678$	0	0
$679$	−5.54876	−0.212942
$680$	0	0
$681$	37.4630	1.43559
$682$	0	0
$683$	19.7272	0.754839	0.377420	−	0.926042i	$-0.376811\pi$
$683$	19.7272	0.754839	0.377420	+	0.926042i	$0.376811\pi$
$684$	0	0
$685$	−0.315622	−0.0120593
$686$	0	0
$687$	−15.6558	−0.597305
$688$	0	0
$689$	−3.87764	−0.147726
$690$	0	0
$691$	−1.36535	−0.0519405	−0.0259703	−	0.999663i	$-0.508268\pi$
$691$	−1.36535	−0.0519405	−0.0259703	+	0.999663i	$0.508268\pi$
$692$	0	0
$693$	−4.99612	−0.189787
$694$	0	0
$695$	4.42410	0.167816
$696$	0	0
$697$	−11.1132	−0.420941
$698$	0	0
$699$	−24.8304	−0.939173
$700$	0	0
$701$	23.7869	0.898418	0.449209	−	0.893427i	$-0.351706\pi$
$701$	23.7869	0.898418	0.449209	+	0.893427i	$0.351706\pi$
$702$	0	0
$703$	11.4112	0.430382
$704$	0	0
$705$	7.28194	0.274254
$706$	0	0
$707$	−3.29342	−0.123862
$708$	0	0
$709$	20.1134	0.755373	0.377687	−	0.925934i	$-0.376720\pi$
$709$	20.1134	0.755373	0.377687	+	0.925934i	$0.376720\pi$
$710$	0	0
$711$	9.34244	0.350369
$712$	0	0
$713$	−18.5051	−0.693021
$714$	0	0
$715$	−0.233208	−0.00872149
$716$	0	0
$717$	23.6477	0.883138
$718$	0	0
$719$	−0.628754	−0.0234485	−0.0117243	−	0.999931i	$-0.503732\pi$
$719$	−0.628754	−0.0234485	−0.0117243	+	0.999931i	$0.503732\pi$
$720$	0	0
$721$	−4.71506	−0.175598
$722$	0	0
$723$	−86.3940	−3.21303
$724$	0	0
$725$	−45.6034	−1.69367
$726$	0	0
$727$	7.70045	0.285594	0.142797	−	0.989752i	$-0.454390\pi$
$727$	7.70045	0.285594	0.142797	+	0.989752i	$0.454390\pi$
$728$	0	0
$729$	−43.0231	−1.59345
$730$	0	0
$731$	−8.86129	−0.327747
$732$	0	0
$733$	46.7135	1.72540	0.862702	−	0.505713i	$-0.168771\pi$
$733$	46.7135	1.72540	0.862702	+	0.505713i	$0.168771\pi$
$734$	0	0
$735$	0.659452	0.0243243
$736$	0	0
$737$	11.2232	0.413412
$738$	0	0
$739$	−17.6767	−0.650247	−0.325123	−	0.945672i	$-0.605406\pi$
$739$	−17.6767	−0.650247	−0.325123	+	0.945672i	$0.605406\pi$
$740$	0	0
$741$	7.57010	0.278095
$742$	0	0
$743$	−18.7809	−0.689004	−0.344502	−	0.938786i	$-0.611952\pi$
$743$	−18.7809	−0.689004	−0.344502	+	0.938786i	$0.611952\pi$
$744$	0	0
$745$	−2.68128	−0.0982345
$746$	0	0
$747$	55.3979	2.02690
$748$	0	0
$749$	6.23341	0.227764
$750$	0	0
$751$	−44.2451	−1.61453	−0.807264	−	0.590191i	$-0.799052\pi$
$751$	−44.2451	−1.61453	−0.807264	+	0.590191i	$0.799052\pi$
$752$	0	0
$753$	−48.1535	−1.75481
$754$	0	0
$755$	1.30702	0.0475674
$756$	0	0
$757$	−24.6289	−0.895153	−0.447576	−	0.894246i	$-0.647713\pi$
$757$	−24.6289	−0.895153	−0.447576	+	0.894246i	$0.647713\pi$
$758$	0	0
$759$	−8.69409	−0.315576
$760$	0	0
$761$	−0.473922	−0.0171796	−0.00858982	−	0.999963i	$-0.502734\pi$
$761$	−0.473922	−0.0171796	−0.00858982	+	0.999963i	$0.502734\pi$
$762$	0	0
$763$	−17.1305	−0.620165
$764$	0	0
$765$	−1.60378	−0.0579848
$766$	0	0
$767$	9.61739	0.347264
$768$	0	0
$769$	−22.5935	−0.814743	−0.407372	−	0.913262i	$-0.633555\pi$
$769$	−22.5935	−0.814743	−0.407372	+	0.913262i	$0.633555\pi$
$770$	0	0
$771$	−6.23084	−0.224398
$772$	0	0
$773$	52.3031	1.88121	0.940605	−	0.339502i	$-0.110258\pi$
$773$	52.3031	1.88121	0.940605	+	0.339502i	$0.110258\pi$
$774$	0	0
$775$	−29.7664	−1.06924
$776$	0	0
$777$	12.0534	0.432413
$778$	0	0
$779$	−21.6138	−0.774395
$780$	0	0
$781$	0.633784	0.0226786
$782$	0	0
$783$	52.0480	1.86004
$784$	0	0
$785$	−0.391280	−0.0139654
$786$	0	0
$787$	−53.8006	−1.91778	−0.958892	−	0.283770i	$-0.908415\pi$
$787$	−53.8006	−1.91778	−0.958892	+	0.283770i	$0.908415\pi$
$788$	0	0
$789$	63.0785	2.24565
$790$	0	0
$791$	13.0752	0.464900
$792$	0	0
$793$	−10.7387	−0.381343
$794$	0	0
$795$	2.55712	0.0906918
$796$	0	0
$797$	21.8313	0.773305	0.386653	−	0.922225i	$-0.373631\pi$
$797$	21.8313	0.773305	0.386653	+	0.922225i	$0.373631\pi$
$798$	0	0
$799$	−15.1996	−0.537723
$800$	0	0
$801$	62.1110	2.19458
$802$	0	0
$803$	6.61461	0.233425
$804$	0	0
$805$	0.717015	0.0252715
$806$	0	0
$807$	−47.2102	−1.66188
$808$	0	0
$809$	−15.7343	−0.553189	−0.276595	−	0.960987i	$-0.589206\pi$
$809$	−15.7343	−0.553189	−0.276595	+	0.960987i	$0.589206\pi$
$810$	0	0
$811$	−36.4679	−1.28056	−0.640280	−	0.768142i	$-0.721182\pi$
$811$	−36.4679	−1.28056	−0.640280	+	0.768142i	$0.721182\pi$
$812$	0	0
$813$	45.4982	1.59569
$814$	0	0
$815$	−2.94159	−0.103039
$816$	0	0
$817$	−17.2342	−0.602948
$818$	0	0
$819$	4.99612	0.174579
$820$	0	0
$821$	21.3255	0.744264	0.372132	−	0.928180i	$-0.378627\pi$
$821$	21.3255	0.744264	0.372132	+	0.928180i	$0.378627\pi$
$822$	0	0
$823$	−5.13206	−0.178892	−0.0894462	−	0.995992i	$-0.528510\pi$
$823$	−5.13206	−0.178892	−0.0894462	+	0.995992i	$0.528510\pi$
$824$	0	0
$825$	−13.9849	−0.486892
$826$	0	0
$827$	20.9148	0.727280	0.363640	−	0.931540i	$-0.381534\pi$
$827$	20.9148	0.727280	0.363640	+	0.931540i	$0.381534\pi$
$828$	0	0
$829$	42.1026	1.46228	0.731141	−	0.682226i	$-0.238988\pi$
$829$	42.1026	1.46228	0.731141	+	0.682226i	$0.238988\pi$
$830$	0	0
$831$	17.8186	0.618122
$832$	0	0
$833$	−1.37647	−0.0476920
$834$	0	0
$835$	−5.14638	−0.178098
$836$	0	0
$837$	33.9730	1.17428
$838$	0	0
$839$	−9.91855	−0.342426	−0.171213	−	0.985234i	$-0.554769\pi$
$839$	−9.91855	−0.342426	−0.171213	+	0.985234i	$0.554769\pi$
$840$	0	0
$841$	56.0265	1.93195
$842$	0	0
$843$	52.9623	1.82412
$844$	0	0
$845$	0.233208	0.00802261
$846$	0	0
$847$	−1.00000	−0.0343604
$848$	0	0
$849$	−81.1435	−2.78484
$850$	0	0
$851$	13.1055	0.449251
$852$	0	0
$853$	15.4273	0.528221	0.264111	−	0.964492i	$-0.414922\pi$
$853$	15.4273	0.528221	0.264111	+	0.964492i	$0.414922\pi$
$854$	0	0
$855$	−3.11917	−0.106673
$856$	0	0
$857$	32.0757	1.09568	0.547842	−	0.836582i	$-0.315449\pi$
$857$	32.0757	1.09568	0.547842	+	0.836582i	$0.315449\pi$
$858$	0	0
$859$	55.6842	1.89992	0.949961	−	0.312368i	$-0.101122\pi$
$859$	55.6842	1.89992	0.949961	+	0.312368i	$0.101122\pi$
$860$	0	0
$861$	−22.8302	−0.778050
$862$	0	0
$863$	−13.7556	−0.468245	−0.234122	−	0.972207i	$-0.575222\pi$
$863$	−13.7556	−0.468245	−0.234122	+	0.972207i	$0.575222\pi$
$864$	0	0
$865$	1.60628	0.0546152
$866$	0	0
$867$	−42.7139	−1.45064
$868$	0	0
$869$	1.86994	0.0634333
$870$	0	0
$871$	−11.2232	−0.380283
$872$	0	0
$873$	27.7223	0.938257
$874$	0	0
$875$	2.31940	0.0784100
$876$	0	0
$877$	31.4276	1.06123	0.530617	−	0.847612i	$-0.321960\pi$
$877$	31.4276	1.06123	0.530617	+	0.847612i	$0.321960\pi$
$878$	0	0
$879$	9.76227	0.329273
$880$	0	0
$881$	−0.0935359	−0.00315131	−0.00157565	−	0.999999i	$-0.500502\pi$
$881$	−0.0935359	−0.00315131	−0.00157565	+	0.999999i	$0.500502\pi$
$882$	0	0
$883$	1.12055	0.0377094	0.0188547	−	0.999822i	$-0.493998\pi$
$883$	1.12055	0.0377094	0.0188547	+	0.999822i	$0.493998\pi$
$884$	0	0
$885$	−6.34221	−0.213191
$886$	0	0
$887$	−7.49853	−0.251776	−0.125888	−	0.992044i	$-0.540178\pi$
$887$	−7.49853	−0.251776	−0.125888	+	0.992044i	$0.540178\pi$
$888$	0	0
$889$	−9.93787	−0.333306
$890$	0	0
$891$	0.972871	0.0325924
$892$	0	0
$893$	−29.5615	−0.989237
$894$	0	0
$895$	−2.56000	−0.0855713
$896$	0	0
$897$	8.69409	0.290287
$898$	0	0
$899$	55.4988	1.85099
$900$	0	0
$901$	−5.33748	−0.177817
$902$	0	0
$903$	−18.2041	−0.605793
$904$	0	0
$905$	0.204249	0.00678947
$906$	0	0
$907$	−55.2841	−1.83568	−0.917839	−	0.396953i	$-0.870068\pi$
$907$	−55.2841	−1.83568	−0.917839	+	0.396953i	$0.870068\pi$
$908$	0	0
$909$	16.4543	0.545755
$910$	0	0
$911$	−32.9859	−1.09287	−0.546436	−	0.837501i	$-0.684016\pi$
$911$	−32.9859	−1.09287	−0.546436	+	0.837501i	$0.684016\pi$
$912$	0	0
$913$	11.0882	0.366965
$914$	0	0
$915$	7.08167	0.234113
$916$	0	0
$917$	16.4134	0.542018
$918$	0	0
$919$	−32.9500	−1.08692	−0.543460	−	0.839435i	$-0.682886\pi$
$919$	−32.9500	−1.08692	−0.543460	+	0.839435i	$0.682886\pi$
$920$	0	0
$921$	62.7736	2.06846
$922$	0	0
$923$	−0.633784	−0.0208613
$924$	0	0
$925$	21.0809	0.693136
$926$	0	0
$927$	23.5570	0.773714
$928$	0	0
$929$	−42.5624	−1.39643	−0.698213	−	0.715890i	$-0.746021\pi$
$929$	−42.5624	−1.39643	−0.698213	+	0.715890i	$0.746021\pi$
$930$	0	0
$931$	−2.67708	−0.0877379
$932$	0	0
$933$	−69.1266	−2.26310
$934$	0	0
$935$	−0.321005	−0.0104980
$936$	0	0
$937$	19.3789	0.633082	0.316541	−	0.948579i	$-0.397479\pi$
$937$	19.3789	0.633082	0.316541	+	0.948579i	$0.397479\pi$
$938$	0	0
$939$	18.8419	0.614883
$940$	0	0
$941$	−37.8923	−1.23525	−0.617627	−	0.786471i	$-0.711906\pi$
$941$	−37.8923	−1.23525	−0.617627	+	0.786471i	$0.711906\pi$
$942$	0	0
$943$	−24.8230	−0.808348
$944$	0	0
$945$	−1.31635	−0.0428208
$946$	0	0
$947$	−25.5405	−0.829956	−0.414978	−	0.909831i	$-0.636211\pi$
$947$	−25.5405	−0.829956	−0.414978	+	0.909831i	$0.636211\pi$
$948$	0	0
$949$	−6.61461	−0.214719
$950$	0	0
$951$	66.0405	2.14151
$952$	0	0
$953$	29.0985	0.942592	0.471296	−	0.881975i	$-0.343786\pi$
$953$	29.0985	0.942592	0.471296	+	0.881975i	$0.343786\pi$
$954$	0	0
$955$	−0.345484	−0.0111796
$956$	0	0
$957$	26.0746	0.842871
$958$	0	0
$959$	1.35339	0.0437033
$960$	0	0
$961$	5.22539	0.168561
$962$	0	0
$963$	−31.1429	−1.00357
$964$	0	0
$965$	−4.10062	−0.132004
$966$	0	0
$967$	46.9373	1.50940	0.754700	−	0.656070i	$-0.227782\pi$
$967$	46.9373	1.50940	0.754700	+	0.656070i	$0.227782\pi$
$968$	0	0
$969$	10.4201	0.334740
$970$	0	0
$971$	−1.51495	−0.0486171	−0.0243085	−	0.999705i	$-0.507738\pi$
$971$	−1.51495	−0.0486171	−0.0243085	+	0.999705i	$0.507738\pi$
$972$	0	0
$973$	−18.9706	−0.608170
$974$	0	0
$975$	13.9849	0.447876
$976$	0	0
$977$	−31.2958	−1.00124	−0.500621	−	0.865667i	$-0.666895\pi$
$977$	−31.2958	−1.00124	−0.500621	+	0.865667i	$0.666895\pi$
$978$	0	0
$979$	12.4318	0.397323
$980$	0	0
$981$	85.5859	2.73255
$982$	0	0
$983$	25.9289	0.827004	0.413502	−	0.910503i	$-0.364305\pi$
$983$	25.9289	0.827004	0.413502	+	0.910503i	$0.364305\pi$
$984$	0	0
$985$	−1.30630	−0.0416220
$986$	0	0
$987$	−31.2251	−0.993905
$988$	0	0
$989$	−19.7931	−0.629383
$990$	0	0
$991$	24.6397	0.782707	0.391354	−	0.920240i	$-0.372007\pi$
$991$	24.6397	0.782707	0.391354	+	0.920240i	$0.372007\pi$
$992$	0	0
$993$	−35.5481	−1.12809
$994$	0	0
$995$	−3.55637	−0.112744
$996$	0	0
$997$	1.64604	0.0521305	0.0260653	−	0.999660i	$-0.491702\pi$
$997$	1.64604	0.0521305	0.0260653	+	0.999660i	$0.491702\pi$
$998$	0	0
$999$	−24.0600	−0.761226

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8008.2.a.n.1.9	✓	9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8008.2.a.n.1.9	✓	9	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)

\(f(q)\)	\(=\)	\(q+2.82774 q^{3} +0.233208 q^{5} -1.00000 q^{7} +4.99612 q^{9} +O(q^{10})\)	\(q+2.82774 q^{3} +0.233208 q^{5} -1.00000 q^{7} +4.99612 q^{9} +1.00000 q^{11} -1.00000 q^{13} +0.659452 q^{15} -1.37647 q^{17} -2.67708 q^{19} -2.82774 q^{21} -3.07457 q^{23} -4.94561 q^{25} +5.64452 q^{27} +9.22098 q^{29} +6.01875 q^{31} +2.82774 q^{33} -0.233208 q^{35} -4.26255 q^{37} -2.82774 q^{39} +8.07364 q^{41} +6.43767 q^{43} +1.16514 q^{45} +11.0424 q^{47} +1.00000 q^{49} -3.89232 q^{51} +3.87764 q^{53} +0.233208 q^{55} -7.57010 q^{57} -9.61739 q^{59} +10.7387 q^{61} -4.99612 q^{63} -0.233208 q^{65} +11.2232 q^{67} -8.69409 q^{69} +0.633784 q^{71} +6.61461 q^{73} -13.9849 q^{75} -1.00000 q^{77} +1.86994 q^{79} +0.972871 q^{81} +11.0882 q^{83} -0.321005 q^{85} +26.0746 q^{87} +12.4318 q^{89} +1.00000 q^{91} +17.0195 q^{93} -0.624318 q^{95} +5.54876 q^{97} +4.99612 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q - 3 q^{3} + q^{5} - 9 q^{7} + 12 q^{9}+O(q^{10}) \) 9 * q - 3 * q^3 + q^5 - 9 * q^7 + 12 * q^9	\( 9 q - 3 q^{3} + q^{5} - 9 q^{7} + 12 q^{9} + 9 q^{11} - 9 q^{13} + 11 q^{15} + 7 q^{17} - 17 q^{19} + 3 q^{21} + 11 q^{23} + 18 q^{25} - 9 q^{27} + 9 q^{29} - 8 q^{31} - 3 q^{33} - q^{35} + 2 q^{37} + 3 q^{39} + 18 q^{41} + 7 q^{43} + 5 q^{45} + 15 q^{47} + 9 q^{49} - 7 q^{51} - 4 q^{53} + q^{55} + 22 q^{57} - 23 q^{59} + 12 q^{61} - 12 q^{63} - q^{65} - 16 q^{67} - 32 q^{69} - 6 q^{71} + 4 q^{73} - 14 q^{75} - 9 q^{77} + 21 q^{79} + 5 q^{81} - 16 q^{83} + 53 q^{85} + 41 q^{87} + 5 q^{89} + 9 q^{91} + 29 q^{93} + 19 q^{95} + 18 q^{97} + 12 q^{99}+O(q^{100}) \) 9 * q - 3 * q^3 + q^5 - 9 * q^7 + 12 * q^9 + 9 * q^11 - 9 * q^13 + 11 * q^15 + 7 * q^17 - 17 * q^19 + 3 * q^21 + 11 * q^23 + 18 * q^25 - 9 * q^27 + 9 * q^29 - 8 * q^31 - 3 * q^33 - q^35 + 2 * q^37 + 3 * q^39 + 18 * q^41 + 7 * q^43 + 5 * q^45 + 15 * q^47 + 9 * q^49 - 7 * q^51 - 4 * q^53 + q^55 + 22 * q^57 - 23 * q^59 + 12 * q^61 - 12 * q^63 - q^65 - 16 * q^67 - 32 * q^69 - 6 * q^71 + 4 * q^73 - 14 * q^75 - 9 * q^77 + 21 * q^79 + 5 * q^81 - 16 * q^83 + 53 * q^85 + 41 * q^87 + 5 * q^89 + 9 * q^91 + 29 * q^93 + 19 * q^95 + 18 * q^97 + 12 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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