LMFDB - Embedded newform 8008.2.a.o.1.4

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:	$9$
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} - 2x^{8} - 12x^{7} + 20x^{6} + 47x^{5} - 55x^{4} - 68x^{3} + 37x^{2} + 21x - 9 $ x^9 - 2x^8 - 12x^7 + 20x^6 + 47x^5 - 55x^4 - 68x^3 + 37x^2 + 21x - 9
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$0.559729$ 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)

$f(q)$	$=$	$q-0.559729 q^{3} +0.0677230 q^{5} +1.00000 q^{7} -2.68670 q^{9} +O(q^{10})$	$q-0.559729 q^{3} +0.0677230 q^{5} +1.00000 q^{7} -2.68670 q^{9} -1.00000 q^{11} +1.00000 q^{13} -0.0379065 q^{15} +5.99740 q^{17} +3.72835 q^{19} -0.559729 q^{21} -5.88978 q^{23} -4.99541 q^{25} +3.18301 q^{27} +3.26141 q^{29} +3.14312 q^{31} +0.559729 q^{33} +0.0677230 q^{35} -6.81776 q^{37} -0.559729 q^{39} -7.71736 q^{41} -7.44015 q^{43} -0.181952 q^{45} -8.91390 q^{47} +1.00000 q^{49} -3.35692 q^{51} -5.08944 q^{53} -0.0677230 q^{55} -2.08687 q^{57} +9.41662 q^{59} +3.81485 q^{61} -2.68670 q^{63} +0.0677230 q^{65} -7.92425 q^{67} +3.29668 q^{69} +14.7308 q^{71} -4.61482 q^{73} +2.79608 q^{75} -1.00000 q^{77} +10.9215 q^{79} +6.27849 q^{81} +5.90384 q^{83} +0.406162 q^{85} -1.82550 q^{87} +11.7275 q^{89} +1.00000 q^{91} -1.75930 q^{93} +0.252495 q^{95} +8.39662 q^{97} +2.68670 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q - 2 q^{3} - 4 q^{5} + 9 q^{7} + q^{9}+O(q^{10}) $ 9 * q - 2 * q^3 - 4 * q^5 + 9 * q^7 + q^9	$ 9 q - 2 q^{3} - 4 q^{5} + 9 q^{7} + q^{9} - 9 q^{11} + 9 q^{13} - 4 q^{15} - 9 q^{17} + 9 q^{19} - 2 q^{21} - 10 q^{23} - q^{25} - 8 q^{27} - 13 q^{29} + 7 q^{31} + 2 q^{33} - 4 q^{35} - 15 q^{37} - 2 q^{39} - 18 q^{41} + 7 q^{43} + 9 q^{45} - 11 q^{47} + 9 q^{49} + q^{51} - 16 q^{53} + 4 q^{55} - 26 q^{57} - 2 q^{59} + 2 q^{61} + q^{63} - 4 q^{65} + 13 q^{67} - 3 q^{69} - q^{71} - 6 q^{73} - 4 q^{75} - 9 q^{77} - 21 q^{79} - 23 q^{81} - 30 q^{83} + q^{85} + q^{87} - 36 q^{89} + 9 q^{91} - 22 q^{93} - 23 q^{95} - 5 q^{97} - q^{99}+O(q^{100}) $ 9 * q - 2 * q^3 - 4 * q^5 + 9 * q^7 + q^9 - 9 * q^11 + 9 * q^13 - 4 * q^15 - 9 * q^17 + 9 * q^19 - 2 * q^21 - 10 * q^23 - q^25 - 8 * q^27 - 13 * q^29 + 7 * q^31 + 2 * q^33 - 4 * q^35 - 15 * q^37 - 2 * q^39 - 18 * q^41 + 7 * q^43 + 9 * q^45 - 11 * q^47 + 9 * q^49 + q^51 - 16 * q^53 + 4 * q^55 - 26 * q^57 - 2 * q^59 + 2 * q^61 + q^63 - 4 * q^65 + 13 * q^67 - 3 * q^69 - q^71 - 6 * q^73 - 4 * q^75 - 9 * q^77 - 21 * q^79 - 23 * q^81 - 30 * q^83 + q^85 + q^87 - 36 * q^89 + 9 * q^91 - 22 * q^93 - 23 * q^95 - 5 * q^97 - q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	−0.559729	−0.323160	−0.161580	−	0.986860i	$-0.551659\pi$
$3$	−0.559729	−0.323160	−0.161580	+	0.986860i	$0.551659\pi$
$4$	0	0
$5$	0.0677230	0.0302867	0.0151433	−	0.999885i	$-0.495180\pi$
$5$	0.0677230	0.0302867	0.0151433	+	0.999885i	$0.495180\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	−2.68670	−0.895568
$10$	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
$15$	−0.0379065	−0.00978742
$16$	0	0
$17$	5.99740	1.45458	0.727291	−	0.686329i	$-0.240779\pi$
$17$	5.99740	1.45458	0.727291	+	0.686329i	$0.240779\pi$
$18$	0	0
$19$	3.72835	0.855343	0.427672	−	0.903934i	$-0.359334\pi$
$19$	3.72835	0.855343	0.427672	+	0.903934i	$0.359334\pi$
$20$	0	0
$21$	−0.559729	−0.122143
$22$	0	0
$23$	−5.88978	−1.22810	−0.614052	−	0.789265i	$-0.710462\pi$
$23$	−5.88978	−1.22810	−0.614052	+	0.789265i	$0.710462\pi$
$24$	0	0
$25$	−4.99541	−0.999083
$26$	0	0
$27$	3.18301	0.612571
$28$	0	0
$29$	3.26141	0.605628	0.302814	−	0.953050i	$-0.402074\pi$
$29$	3.26141	0.605628	0.302814	+	0.953050i	$0.402074\pi$
$30$	0	0
$31$	3.14312	0.564521	0.282261	−	0.959338i	$-0.408916\pi$
$31$	3.14312	0.564521	0.282261	+	0.959338i	$0.408916\pi$
$32$	0	0
$33$	0.559729	0.0974363
$34$	0	0
$35$	0.0677230	0.0114473
$36$	0	0
$37$	−6.81776	−1.12083	−0.560417	−	0.828211i	$-0.689359\pi$
$37$	−6.81776	−1.12083	−0.560417	+	0.828211i	$0.689359\pi$
$38$	0	0
$39$	−0.559729	−0.0896284
$40$	0	0
$41$	−7.71736	−1.20525	−0.602625	−	0.798025i	$-0.705878\pi$
$41$	−7.71736	−1.20525	−0.602625	+	0.798025i	$0.705878\pi$
$42$	0	0
$43$	−7.44015	−1.13461	−0.567306	−	0.823507i	$-0.692014\pi$
$43$	−7.44015	−1.13461	−0.567306	+	0.823507i	$0.692014\pi$
$44$	0	0
$45$	−0.181952	−0.0271238
$46$	0	0
$47$	−8.91390	−1.30023	−0.650113	−	0.759837i	$-0.725278\pi$
$47$	−8.91390	−1.30023	−0.650113	+	0.759837i	$0.725278\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−3.35692	−0.470062
$52$	0	0
$53$	−5.08944	−0.699088	−0.349544	−	0.936920i	$-0.613664\pi$
$53$	−5.08944	−0.699088	−0.349544	+	0.936920i	$0.613664\pi$
$54$	0	0
$55$	−0.0677230	−0.00913177
$56$	0	0
$57$	−2.08687	−0.276412
$58$	0	0
$59$	9.41662	1.22594	0.612970	−	0.790106i	$-0.289975\pi$
$59$	9.41662	1.22594	0.612970	+	0.790106i	$0.289975\pi$
$60$	0	0
$61$	3.81485	0.488441	0.244221	−	0.969720i	$-0.421468\pi$
$61$	3.81485	0.488441	0.244221	+	0.969720i	$0.421468\pi$
$62$	0	0
$63$	−2.68670	−0.338493
$64$	0	0
$65$	0.0677230	0.00840001
$66$	0	0
$67$	−7.92425	−0.968101	−0.484050	−	0.875040i	$-0.660835\pi$
$67$	−7.92425	−0.968101	−0.484050	+	0.875040i	$0.660835\pi$
$68$	0	0
$69$	3.29668	0.396874
$70$	0	0
$71$	14.7308	1.74822	0.874111	−	0.485725i	$-0.161444\pi$
$71$	14.7308	1.74822	0.874111	+	0.485725i	$0.161444\pi$
$72$	0	0
$73$	−4.61482	−0.540124	−0.270062	−	0.962843i	$-0.587044\pi$
$73$	−4.61482	−0.540124	−0.270062	+	0.962843i	$0.587044\pi$
$74$	0	0
$75$	2.79608	0.322863
$76$	0	0
$77$	−1.00000	−0.113961
$78$	0	0
$79$	10.9215	1.22877	0.614383	−	0.789008i	$-0.289405\pi$
$79$	10.9215	1.22877	0.614383	+	0.789008i	$0.289405\pi$
$80$	0	0
$81$	6.27849	0.697610
$82$	0	0
$83$	5.90384	0.648030	0.324015	−	0.946052i	$-0.394967\pi$
$83$	5.90384	0.648030	0.324015	+	0.946052i	$0.394967\pi$
$84$	0	0
$85$	0.406162	0.0440544
$86$	0	0
$87$	−1.82550	−0.195715
$88$	0	0
$89$	11.7275	1.24311	0.621556	−	0.783370i	$-0.286501\pi$
$89$	11.7275	1.24311	0.621556	+	0.783370i	$0.286501\pi$
$90$	0	0
$91$	1.00000	0.104828
$92$	0	0
$93$	−1.75930	−0.182430
$94$	0	0
$95$	0.252495	0.0259055
$96$	0	0
$97$	8.39662	0.852547	0.426274	−	0.904594i	$-0.359826\pi$
$97$	8.39662	0.852547	0.426274	+	0.904594i	$0.359826\pi$
$98$	0	0
$99$	2.68670	0.270024
$100$	0	0
$101$	−2.93667	−0.292210	−0.146105	−	0.989269i	$-0.546674\pi$
$101$	−2.93667	−0.292210	−0.146105	+	0.989269i	$0.546674\pi$
$102$	0	0
$103$	−1.13245	−0.111584	−0.0557918	−	0.998442i	$-0.517768\pi$
$103$	−1.13245	−0.111584	−0.0557918	+	0.998442i	$0.517768\pi$
$104$	0	0
$105$	−0.0379065	−0.00369930
$106$	0	0
$107$	−13.8679	−1.34066	−0.670331	−	0.742062i	$-0.733848\pi$
$107$	−13.8679	−1.34066	−0.670331	+	0.742062i	$0.733848\pi$
$108$	0	0
$109$	−11.4726	−1.09887	−0.549436	−	0.835536i	$-0.685157\pi$
$109$	−11.4726	−1.09887	−0.549436	+	0.835536i	$0.685157\pi$
$110$	0	0
$111$	3.81610	0.362208
$112$	0	0
$113$	5.15434	0.484879	0.242440	−	0.970166i	$-0.422052\pi$
$113$	5.15434	0.484879	0.242440	+	0.970166i	$0.422052\pi$
$114$	0	0
$115$	−0.398874	−0.0371952
$116$	0	0
$117$	−2.68670	−0.248386
$118$	0	0
$119$	5.99740	0.549781
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	4.31963	0.389488
$124$	0	0
$125$	−0.676919	−0.0605455
$126$	0	0
$127$	−4.23839	−0.376097	−0.188048	−	0.982160i	$-0.560216\pi$
$127$	−4.23839	−0.376097	−0.188048	+	0.982160i	$0.560216\pi$
$128$	0	0
$129$	4.16446	0.366661
$130$	0	0
$131$	4.05064	0.353906	0.176953	−	0.984219i	$-0.443376\pi$
$131$	4.05064	0.353906	0.176953	+	0.984219i	$0.443376\pi$
$132$	0	0
$133$	3.72835	0.323289
$134$	0	0
$135$	0.215563	0.0185527
$136$	0	0
$137$	−17.9390	−1.53263	−0.766317	−	0.642462i	$-0.777913\pi$
$137$	−17.9390	−1.53263	−0.766317	+	0.642462i	$0.777913\pi$
$138$	0	0
$139$	−0.672660	−0.0570543	−0.0285271	−	0.999593i	$-0.509082\pi$
$139$	−0.672660	−0.0570543	−0.0285271	+	0.999593i	$0.509082\pi$
$140$	0	0
$141$	4.98937	0.420181
$142$	0	0
$143$	−1.00000	−0.0836242
$144$	0	0
$145$	0.220872	0.0183425
$146$	0	0
$147$	−0.559729	−0.0461657
$148$	0	0
$149$	−16.2523	−1.33144	−0.665722	−	0.746200i	$-0.731876\pi$
$149$	−16.2523	−1.33144	−0.665722	+	0.746200i	$0.731876\pi$
$150$	0	0
$151$	−19.5829	−1.59363	−0.796816	−	0.604223i	$-0.793484\pi$
$151$	−19.5829	−1.59363	−0.796816	+	0.604223i	$0.793484\pi$
$152$	0	0
$153$	−16.1132	−1.30268
$154$	0	0
$155$	0.212862	0.0170975
$156$	0	0
$157$	7.57796	0.604787	0.302394	−	0.953183i	$-0.402214\pi$
$157$	7.57796	0.604787	0.302394	+	0.953183i	$0.402214\pi$
$158$	0	0
$159$	2.84871	0.225917
$160$	0	0
$161$	−5.88978	−0.464180
$162$	0	0
$163$	3.52539	0.276130	0.138065	−	0.990423i	$-0.455912\pi$
$163$	3.52539	0.276130	0.138065	+	0.990423i	$0.455912\pi$
$164$	0	0
$165$	0.0379065	0.00295102
$166$	0	0
$167$	2.09660	0.162240	0.0811199	−	0.996704i	$-0.474150\pi$
$167$	2.09660	0.162240	0.0811199	+	0.996704i	$0.474150\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	−10.0170	−0.766018
$172$	0	0
$173$	−17.9656	−1.36590	−0.682949	−	0.730466i	$-0.739303\pi$
$173$	−17.9656	−1.36590	−0.682949	+	0.730466i	$0.739303\pi$
$174$	0	0
$175$	−4.99541	−0.377618
$176$	0	0
$177$	−5.27076	−0.396174
$178$	0	0
$179$	−22.4605	−1.67878	−0.839390	−	0.543530i	$-0.817087\pi$
$179$	−22.4605	−1.67878	−0.839390	+	0.543530i	$0.817087\pi$
$180$	0	0
$181$	9.81854	0.729806	0.364903	−	0.931046i	$-0.381102\pi$
$181$	9.81854	0.729806	0.364903	+	0.931046i	$0.381102\pi$
$182$	0	0
$183$	−2.13528	−0.157845
$184$	0	0
$185$	−0.461720	−0.0339463
$186$	0	0
$187$	−5.99740	−0.438573
$188$	0	0
$189$	3.18301	0.231530
$190$	0	0
$191$	−7.68151	−0.555815	−0.277907	−	0.960608i	$-0.589641\pi$
$191$	−7.68151	−0.555815	−0.277907	+	0.960608i	$0.589641\pi$
$192$	0	0
$193$	4.49874	0.323826	0.161913	−	0.986805i	$-0.448234\pi$
$193$	4.49874	0.323826	0.161913	+	0.986805i	$0.448234\pi$
$194$	0	0
$195$	−0.0379065	−0.00271454
$196$	0	0
$197$	−23.3937	−1.66673	−0.833367	−	0.552720i	$-0.813590\pi$
$197$	−23.3937	−1.66673	−0.833367	+	0.552720i	$0.813590\pi$
$198$	0	0
$199$	7.94472	0.563187	0.281593	−	0.959534i	$-0.409137\pi$
$199$	7.94472	0.563187	0.281593	+	0.959534i	$0.409137\pi$
$200$	0	0
$201$	4.43543	0.312851
$202$	0	0
$203$	3.26141	0.228906
$204$	0	0
$205$	−0.522643	−0.0365030
$206$	0	0
$207$	15.8241	1.09985
$208$	0	0
$209$	−3.72835	−0.257896
$210$	0	0
$211$	−11.7063	−0.805893	−0.402947	−	0.915223i	$-0.632014\pi$
$211$	−11.7063	−0.805893	−0.402947	+	0.915223i	$0.632014\pi$
$212$	0	0
$213$	−8.24525	−0.564955
$214$	0	0
$215$	−0.503869	−0.0343636
$216$	0	0
$217$	3.14312	0.213369
$218$	0	0
$219$	2.58305	0.174546
$220$	0	0
$221$	5.99740	0.403429
$222$	0	0
$223$	−7.20553	−0.482518	−0.241259	−	0.970461i	$-0.577560\pi$
$223$	−7.20553	−0.482518	−0.241259	+	0.970461i	$0.577560\pi$
$224$	0	0
$225$	13.4212	0.894746
$226$	0	0
$227$	27.9570	1.85557	0.927784	−	0.373118i	$-0.121711\pi$
$227$	27.9570	1.85557	0.927784	+	0.373118i	$0.121711\pi$
$228$	0	0
$229$	2.80719	0.185504	0.0927522	−	0.995689i	$-0.470434\pi$
$229$	2.80719	0.185504	0.0927522	+	0.995689i	$0.470434\pi$
$230$	0	0
$231$	0.559729	0.0368275
$232$	0	0
$233$	−24.6588	−1.61545	−0.807725	−	0.589559i	$-0.799301\pi$
$233$	−24.6588	−1.61545	−0.807725	+	0.589559i	$0.799301\pi$
$234$	0	0
$235$	−0.603676	−0.0393795
$236$	0	0
$237$	−6.11308	−0.397087
$238$	0	0
$239$	3.32613	0.215149	0.107575	−	0.994197i	$-0.465692\pi$
$239$	3.32613	0.215149	0.107575	+	0.994197i	$0.465692\pi$
$240$	0	0
$241$	−2.82777	−0.182153	−0.0910763	−	0.995844i	$-0.529031\pi$
$241$	−2.82777	−0.182153	−0.0910763	+	0.995844i	$0.529031\pi$
$242$	0	0
$243$	−13.0633	−0.838010
$244$	0	0
$245$	0.0677230	0.00432666
$246$	0	0
$247$	3.72835	0.237230
$248$	0	0
$249$	−3.30455	−0.209417
$250$	0	0
$251$	7.53681	0.475719	0.237860	−	0.971300i	$-0.423554\pi$
$251$	7.53681	0.475719	0.237860	+	0.971300i	$0.423554\pi$
$252$	0	0
$253$	5.88978	0.370288
$254$	0	0
$255$	−0.227341	−0.0142366
$256$	0	0
$257$	−5.49785	−0.342947	−0.171473	−	0.985189i	$-0.554853\pi$
$257$	−5.49785	−0.342947	−0.171473	+	0.985189i	$0.554853\pi$
$258$	0	0
$259$	−6.81776	−0.423635
$260$	0	0
$261$	−8.76244	−0.542381
$262$	0	0
$263$	9.56226	0.589634	0.294817	−	0.955554i	$-0.404741\pi$
$263$	9.56226	0.589634	0.294817	+	0.955554i	$0.404741\pi$
$264$	0	0
$265$	−0.344672	−0.0211730
$266$	0	0
$267$	−6.56422	−0.401724
$268$	0	0
$269$	16.5264	1.00763	0.503817	−	0.863810i	$-0.331929\pi$
$269$	16.5264	1.00763	0.503817	+	0.863810i	$0.331929\pi$
$270$	0	0
$271$	−15.9241	−0.967318	−0.483659	−	0.875257i	$-0.660692\pi$
$271$	−15.9241	−0.967318	−0.483659	+	0.875257i	$0.660692\pi$
$272$	0	0
$273$	−0.559729	−0.0338763
$274$	0	0
$275$	4.99541	0.301235
$276$	0	0
$277$	2.24395	0.134826	0.0674129	−	0.997725i	$-0.478526\pi$
$277$	2.24395	0.134826	0.0674129	+	0.997725i	$0.478526\pi$
$278$	0	0
$279$	−8.44464	−0.505567
$280$	0	0
$281$	−2.98659	−0.178165	−0.0890826	−	0.996024i	$-0.528394\pi$
$281$	−2.98659	−0.178165	−0.0890826	+	0.996024i	$0.528394\pi$
$282$	0	0
$283$	19.1529	1.13852	0.569260	−	0.822158i	$-0.307230\pi$
$283$	19.1529	1.13852	0.569260	+	0.822158i	$0.307230\pi$
$284$	0	0
$285$	−0.141329	−0.00837160
$286$	0	0
$287$	−7.71736	−0.455541
$288$	0	0
$289$	18.9688	1.11581
$290$	0	0
$291$	−4.69983	−0.275509
$292$	0	0
$293$	−20.0959	−1.17401	−0.587006	−	0.809583i	$-0.699694\pi$
$293$	−20.0959	−1.17401	−0.587006	+	0.809583i	$0.699694\pi$
$294$	0	0
$295$	0.637722	0.0371296
$296$	0	0
$297$	−3.18301	−0.184697
$298$	0	0
$299$	−5.88978	−0.340615
$300$	0	0
$301$	−7.44015	−0.428843
$302$	0	0
$303$	1.64374	0.0944305
$304$	0	0
$305$	0.258353	0.0147933
$306$	0	0
$307$	−9.37376	−0.534989	−0.267494	−	0.963559i	$-0.586196\pi$
$307$	−9.37376	−0.534989	−0.267494	+	0.963559i	$0.586196\pi$
$308$	0	0
$309$	0.633865	0.0360593
$310$	0	0
$311$	−32.7099	−1.85481	−0.927404	−	0.374062i	$-0.877965\pi$
$311$	−32.7099	−1.85481	−0.927404	+	0.374062i	$0.877965\pi$
$312$	0	0
$313$	−27.9948	−1.58236	−0.791179	−	0.611585i	$-0.790532\pi$
$313$	−27.9948	−1.58236	−0.791179	+	0.611585i	$0.790532\pi$
$314$	0	0
$315$	−0.181952	−0.0102518
$316$	0	0
$317$	−31.1588	−1.75005	−0.875026	−	0.484076i	$-0.839156\pi$
$317$	−31.1588	−1.75005	−0.875026	+	0.484076i	$0.839156\pi$
$318$	0	0
$319$	−3.26141	−0.182604
$320$	0	0
$321$	7.76228	0.433248
$322$	0	0
$323$	22.3604	1.24417
$324$	0	0
$325$	−4.99541	−0.277096
$326$	0	0
$327$	6.42153	0.355111
$328$	0	0
$329$	−8.91390	−0.491439
$330$	0	0
$331$	29.9779	1.64774	0.823868	−	0.566782i	$-0.191812\pi$
$331$	29.9779	1.64774	0.823868	+	0.566782i	$0.191812\pi$
$332$	0	0
$333$	18.3173	1.00378
$334$	0	0
$335$	−0.536654	−0.0293205
$336$	0	0
$337$	−21.1528	−1.15227	−0.576133	−	0.817356i	$-0.695439\pi$
$337$	−21.1528	−1.15227	−0.576133	+	0.817356i	$0.695439\pi$
$338$	0	0
$339$	−2.88503	−0.156693
$340$	0	0
$341$	−3.14312	−0.170210
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	0.223261	0.0120200
$346$	0	0
$347$	−27.2318	−1.46188	−0.730941	−	0.682441i	$-0.760919\pi$
$347$	−27.2318	−1.46188	−0.730941	+	0.682441i	$0.760919\pi$
$348$	0	0
$349$	26.5098	1.41904	0.709518	−	0.704688i	$-0.248913\pi$
$349$	26.5098	1.41904	0.709518	+	0.704688i	$0.248913\pi$
$350$	0	0
$351$	3.18301	0.169897
$352$	0	0
$353$	2.18554	0.116324	0.0581622	−	0.998307i	$-0.481476\pi$
$353$	2.18554	0.116324	0.0581622	+	0.998307i	$0.481476\pi$
$354$	0	0
$355$	0.997613	0.0529478
$356$	0	0
$357$	−3.35692	−0.177667
$358$	0	0
$359$	1.04556	0.0551825	0.0275913	−	0.999619i	$-0.491216\pi$
$359$	1.04556	0.0551825	0.0275913	+	0.999619i	$0.491216\pi$
$360$	0	0
$361$	−5.09937	−0.268388
$362$	0	0
$363$	−0.559729	−0.0293781
$364$	0	0
$365$	−0.312529	−0.0163585
$366$	0	0
$367$	10.9854	0.573436	0.286718	−	0.958015i	$-0.407436\pi$
$367$	10.9854	0.573436	0.286718	+	0.958015i	$0.407436\pi$
$368$	0	0
$369$	20.7343	1.07938
$370$	0	0
$371$	−5.08944	−0.264231
$372$	0	0
$373$	3.83457	0.198546	0.0992732	−	0.995060i	$-0.468348\pi$
$373$	3.83457	0.198546	0.0992732	+	0.995060i	$0.468348\pi$
$374$	0	0
$375$	0.378891	0.0195659
$376$	0	0
$377$	3.26141	0.167971
$378$	0	0
$379$	−19.5933	−1.00644	−0.503219	−	0.864159i	$-0.667851\pi$
$379$	−19.5933	−1.00644	−0.503219	+	0.864159i	$0.667851\pi$
$380$	0	0
$381$	2.37235	0.121539
$382$	0	0
$383$	−15.3887	−0.786325	−0.393162	−	0.919469i	$-0.628619\pi$
$383$	−15.3887	−0.786325	−0.393162	+	0.919469i	$0.628619\pi$
$384$	0	0
$385$	−0.0677230	−0.00345148
$386$	0	0
$387$	19.9895	1.01612
$388$	0	0
$389$	21.5189	1.09105	0.545526	−	0.838094i	$-0.316330\pi$
$389$	21.5189	1.09105	0.545526	+	0.838094i	$0.316330\pi$
$390$	0	0
$391$	−35.3234	−1.78638
$392$	0	0
$393$	−2.26726	−0.114368
$394$	0	0
$395$	0.739637	0.0372152
$396$	0	0
$397$	−38.9058	−1.95263	−0.976314	−	0.216358i	$-0.930582\pi$
$397$	−38.9058	−1.95263	−0.976314	+	0.216358i	$0.930582\pi$
$398$	0	0
$399$	−2.08687	−0.104474
$400$	0	0
$401$	−18.7239	−0.935026	−0.467513	−	0.883986i	$-0.654850\pi$
$401$	−18.7239	−0.935026	−0.467513	+	0.883986i	$0.654850\pi$
$402$	0	0
$403$	3.14312	0.156570
$404$	0	0
$405$	0.425198	0.0211283
$406$	0	0
$407$	6.81776	0.337944
$408$	0	0
$409$	−30.9317	−1.52947	−0.764736	−	0.644343i	$-0.777131\pi$
$409$	−30.9317	−1.52947	−0.764736	+	0.644343i	$0.777131\pi$
$410$	0	0
$411$	10.0410	0.495286
$412$	0	0
$413$	9.41662	0.463362
$414$	0	0
$415$	0.399825	0.0196267
$416$	0	0
$417$	0.376507	0.0184376
$418$	0	0
$419$	26.9846	1.31828	0.659142	−	0.752018i	$-0.270920\pi$
$419$	26.9846	1.31828	0.659142	+	0.752018i	$0.270920\pi$
$420$	0	0
$421$	6.02731	0.293753	0.146876	−	0.989155i	$-0.453078\pi$
$421$	6.02731	0.293753	0.146876	+	0.989155i	$0.453078\pi$
$422$	0	0
$423$	23.9490	1.16444
$424$	0	0
$425$	−29.9595	−1.45325
$426$	0	0
$427$	3.81485	0.184613
$428$	0	0
$429$	0.559729	0.0270240
$430$	0	0
$431$	14.6053	0.703515	0.351757	−	0.936091i	$-0.385584\pi$
$431$	14.6053	0.703515	0.351757	+	0.936091i	$0.385584\pi$
$432$	0	0
$433$	−27.8074	−1.33634	−0.668170	−	0.744008i	$-0.732922\pi$
$433$	−27.8074	−1.33634	−0.668170	+	0.744008i	$0.732922\pi$
$434$	0	0
$435$	−0.123629	−0.00592754
$436$	0	0
$437$	−21.9592	−1.05045
$438$	0	0
$439$	6.88720	0.328708	0.164354	−	0.986401i	$-0.447446\pi$
$439$	6.88720	0.328708	0.164354	+	0.986401i	$0.447446\pi$
$440$	0	0
$441$	−2.68670	−0.127938
$442$	0	0
$443$	29.0522	1.38031	0.690155	−	0.723662i	$-0.257542\pi$
$443$	29.0522	1.38031	0.690155	+	0.723662i	$0.257542\pi$
$444$	0	0
$445$	0.794221	0.0376497
$446$	0	0
$447$	9.09691	0.430269
$448$	0	0
$449$	−13.3528	−0.630159	−0.315080	−	0.949065i	$-0.602031\pi$
$449$	−13.3528	−0.630159	−0.315080	+	0.949065i	$0.602031\pi$
$450$	0	0
$451$	7.71736	0.363396
$452$	0	0
$453$	10.9611	0.514997
$454$	0	0
$455$	0.0677230	0.00317490
$456$	0	0
$457$	−34.1995	−1.59979	−0.799893	−	0.600142i	$-0.795111\pi$
$457$	−34.1995	−1.59979	−0.799893	+	0.600142i	$0.795111\pi$
$458$	0	0
$459$	19.0898	0.891035
$460$	0	0
$461$	−12.1878	−0.567642	−0.283821	−	0.958877i	$-0.591602\pi$
$461$	−12.1878	−0.567642	−0.283821	+	0.958877i	$0.591602\pi$
$462$	0	0
$463$	30.8823	1.43522	0.717610	−	0.696445i	$-0.245236\pi$
$463$	30.8823	1.43522	0.717610	+	0.696445i	$0.245236\pi$
$464$	0	0
$465$	−0.119145	−0.00552521
$466$	0	0
$467$	13.0774	0.605148	0.302574	−	0.953126i	$-0.402154\pi$
$467$	13.0774	0.605148	0.302574	+	0.953126i	$0.402154\pi$
$468$	0	0
$469$	−7.92425	−0.365908
$470$	0	0
$471$	−4.24161	−0.195443
$472$	0	0
$473$	7.44015	0.342098
$474$	0	0
$475$	−18.6247	−0.854559
$476$	0	0
$477$	13.6738	0.626081
$478$	0	0
$479$	−6.94634	−0.317386	−0.158693	−	0.987328i	$-0.550728\pi$
$479$	−6.94634	−0.317386	−0.158693	+	0.987328i	$0.550728\pi$
$480$	0	0
$481$	−6.81776	−0.310863
$482$	0	0
$483$	3.29668	0.150004
$484$	0	0
$485$	0.568644	0.0258208
$486$	0	0
$487$	−4.34711	−0.196986	−0.0984931	−	0.995138i	$-0.531402\pi$
$487$	−4.34711	−0.196986	−0.0984931	+	0.995138i	$0.531402\pi$
$488$	0	0
$489$	−1.97326	−0.0892340
$490$	0	0
$491$	−31.7114	−1.43111	−0.715557	−	0.698555i	$-0.753827\pi$
$491$	−31.7114	−1.43111	−0.715557	+	0.698555i	$0.753827\pi$
$492$	0	0
$493$	19.5600	0.880937
$494$	0	0
$495$	0.181952	0.00817812
$496$	0	0
$497$	14.7308	0.660766
$498$	0	0
$499$	−6.81967	−0.305290	−0.152645	−	0.988281i	$-0.548779\pi$
$499$	−6.81967	−0.305290	−0.152645	+	0.988281i	$0.548779\pi$
$500$	0	0
$501$	−1.17353	−0.0524293
$502$	0	0
$503$	14.6880	0.654906	0.327453	−	0.944867i	$-0.393810\pi$
$503$	14.6880	0.654906	0.327453	+	0.944867i	$0.393810\pi$
$504$	0	0
$505$	−0.198880	−0.00885006
$506$	0	0
$507$	−0.559729	−0.0248584
$508$	0	0
$509$	−31.9160	−1.41465	−0.707325	−	0.706889i	$-0.750098\pi$
$509$	−31.9160	−1.41465	−0.707325	+	0.706889i	$0.750098\pi$
$510$	0	0
$511$	−4.61482	−0.204148
$512$	0	0
$513$	11.8674	0.523958
$514$	0	0
$515$	−0.0766929	−0.00337949
$516$	0	0
$517$	8.91390	0.392033
$518$	0	0
$519$	10.0558	0.441403
$520$	0	0
$521$	−35.4113	−1.55140	−0.775699	−	0.631103i	$-0.782602\pi$
$521$	−35.4113	−1.55140	−0.775699	+	0.631103i	$0.782602\pi$
$522$	0	0
$523$	35.9759	1.57312	0.786558	−	0.617516i	$-0.211861\pi$
$523$	35.9759	1.57312	0.786558	+	0.617516i	$0.211861\pi$
$524$	0	0
$525$	2.79608	0.122031
$526$	0	0
$527$	18.8506	0.821143
$528$	0	0
$529$	11.6896	0.508242
$530$	0	0
$531$	−25.2997	−1.09791
$532$	0	0
$533$	−7.71736	−0.334276
$534$	0	0
$535$	−0.939178	−0.0406042
$536$	0	0
$537$	12.5718	0.542514
$538$	0	0
$539$	−1.00000	−0.0430730
$540$	0	0
$541$	19.8508	0.853454	0.426727	−	0.904380i	$-0.359666\pi$
$541$	19.8508	0.853454	0.426727	+	0.904380i	$0.359666\pi$
$542$	0	0
$543$	−5.49572	−0.235844
$544$	0	0
$545$	−0.776957	−0.0332812
$546$	0	0
$547$	−9.13375	−0.390531	−0.195266	−	0.980750i	$-0.562557\pi$
$547$	−9.13375	−0.390531	−0.195266	+	0.980750i	$0.562557\pi$
$548$	0	0
$549$	−10.2494	−0.437432
$550$	0	0
$551$	12.1597	0.518020
$552$	0	0
$553$	10.9215	0.464430
$554$	0	0
$555$	0.258438	0.0109701
$556$	0	0
$557$	25.2021	1.06785	0.533924	−	0.845533i	$-0.320717\pi$
$557$	25.2021	1.06785	0.533924	+	0.845533i	$0.320717\pi$
$558$	0	0
$559$	−7.44015	−0.314685
$560$	0	0
$561$	3.35692	0.141729
$562$	0	0
$563$	43.3037	1.82503	0.912516	−	0.409041i	$-0.134137\pi$
$563$	43.3037	1.82503	0.912516	+	0.409041i	$0.134137\pi$
$564$	0	0
$565$	0.349067	0.0146854
$566$	0	0
$567$	6.27849	0.263672
$568$	0	0
$569$	−26.9199	−1.12854	−0.564271	−	0.825590i	$-0.690843\pi$
$569$	−26.9199	−1.12854	−0.564271	+	0.825590i	$0.690843\pi$
$570$	0	0
$571$	23.1542	0.968973	0.484486	−	0.874799i	$-0.339006\pi$
$571$	23.1542	0.968973	0.484486	+	0.874799i	$0.339006\pi$
$572$	0	0
$573$	4.29956	0.179617
$574$	0	0
$575$	29.4219	1.22698
$576$	0	0
$577$	−12.2078	−0.508216	−0.254108	−	0.967176i	$-0.581782\pi$
$577$	−12.2078	−0.508216	−0.254108	+	0.967176i	$0.581782\pi$
$578$	0	0
$579$	−2.51807	−0.104648
$580$	0	0
$581$	5.90384	0.244932
$582$	0	0
$583$	5.08944	0.210783
$584$	0	0
$585$	−0.181952	−0.00752277
$586$	0	0
$587$	8.37754	0.345778	0.172889	−	0.984941i	$-0.444690\pi$
$587$	8.37754	0.345778	0.172889	+	0.984941i	$0.444690\pi$
$588$	0	0
$589$	11.7187	0.482859
$590$	0	0
$591$	13.0941	0.538621
$592$	0	0
$593$	−5.59888	−0.229918	−0.114959	−	0.993370i	$-0.536674\pi$
$593$	−5.59888	−0.229918	−0.114959	+	0.993370i	$0.536674\pi$
$594$	0	0
$595$	0.406162	0.0166510
$596$	0	0
$597$	−4.44689	−0.181999
$598$	0	0
$599$	−12.4406	−0.508310	−0.254155	−	0.967163i	$-0.581797\pi$
$599$	−12.4406	−0.508310	−0.254155	+	0.967163i	$0.581797\pi$
$600$	0	0
$601$	−27.8637	−1.13658	−0.568291	−	0.822827i	$-0.692395\pi$
$601$	−27.8637	−1.13658	−0.568291	+	0.822827i	$0.692395\pi$
$602$	0	0
$603$	21.2901	0.867000
$604$	0	0
$605$	0.0677230	0.00275333
$606$	0	0
$607$	−25.8119	−1.04767	−0.523837	−	0.851819i	$-0.675500\pi$
$607$	−25.8119	−1.04767	−0.523837	+	0.851819i	$0.675500\pi$
$608$	0	0
$609$	−1.82550	−0.0739732
$610$	0	0
$611$	−8.91390	−0.360618
$612$	0	0
$613$	2.29366	0.0926400	0.0463200	−	0.998927i	$-0.485251\pi$
$613$	2.29366	0.0926400	0.0463200	+	0.998927i	$0.485251\pi$
$614$	0	0
$615$	0.292538	0.0117963
$616$	0	0
$617$	−7.52515	−0.302951	−0.151475	−	0.988461i	$-0.548402\pi$
$617$	−7.52515	−0.302951	−0.151475	+	0.988461i	$0.548402\pi$
$618$	0	0
$619$	−31.5568	−1.26838	−0.634188	−	0.773179i	$-0.718666\pi$
$619$	−31.5568	−1.26838	−0.634188	+	0.773179i	$0.718666\pi$
$620$	0	0
$621$	−18.7473	−0.752301
$622$	0	0
$623$	11.7275	0.469852
$624$	0	0
$625$	24.9312	0.997249
$626$	0	0
$627$	2.08687	0.0833415
$628$	0	0
$629$	−40.8888	−1.63035
$630$	0	0
$631$	32.9840	1.31307	0.656536	−	0.754295i	$-0.272021\pi$
$631$	32.9840	1.31307	0.656536	+	0.754295i	$0.272021\pi$
$632$	0	0
$633$	6.55234	0.260432
$634$	0	0
$635$	−0.287037	−0.0113907
$636$	0	0
$637$	1.00000	0.0396214
$638$	0	0
$639$	−39.5773	−1.56565
$640$	0	0
$641$	−32.4356	−1.28113	−0.640565	−	0.767904i	$-0.721300\pi$
$641$	−32.4356	−1.28113	−0.640565	+	0.767904i	$0.721300\pi$
$642$	0	0
$643$	−25.4901	−1.00523	−0.502615	−	0.864510i	$-0.667629\pi$
$643$	−25.4901	−1.00523	−0.502615	+	0.864510i	$0.667629\pi$
$644$	0	0
$645$	0.282030	0.0111049
$646$	0	0
$647$	−28.6608	−1.12677	−0.563386	−	0.826194i	$-0.690502\pi$
$647$	−28.6608	−1.12677	−0.563386	+	0.826194i	$0.690502\pi$
$648$	0	0
$649$	−9.41662	−0.369635
$650$	0	0
$651$	−1.75930	−0.0689522
$652$	0	0
$653$	−18.8725	−0.738537	−0.369268	−	0.929323i	$-0.620392\pi$
$653$	−18.8725	−0.738537	−0.369268	+	0.929323i	$0.620392\pi$
$654$	0	0
$655$	0.274322	0.0107186
$656$	0	0
$657$	12.3986	0.483717
$658$	0	0
$659$	−36.3820	−1.41724	−0.708621	−	0.705589i	$-0.750682\pi$
$659$	−36.3820	−1.41724	−0.708621	+	0.705589i	$0.750682\pi$
$660$	0	0
$661$	−4.62919	−0.180055	−0.0900274	−	0.995939i	$-0.528695\pi$
$661$	−4.62919	−0.180055	−0.0900274	+	0.995939i	$0.528695\pi$
$662$	0	0
$663$	−3.35692	−0.130372
$664$	0	0
$665$	0.252495	0.00979135
$666$	0	0
$667$	−19.2090	−0.743775
$668$	0	0
$669$	4.03315	0.155930
$670$	0	0
$671$	−3.81485	−0.147271
$672$	0	0
$673$	45.0681	1.73725	0.868623	−	0.495474i	$-0.165006\pi$
$673$	45.0681	1.73725	0.868623	+	0.495474i	$0.165006\pi$
$674$	0	0
$675$	−15.9005	−0.612009
$676$	0	0
$677$	−23.0202	−0.884738	−0.442369	−	0.896833i	$-0.645862\pi$
$677$	−23.0202	−0.884738	−0.442369	+	0.896833i	$0.645862\pi$
$678$	0	0
$679$	8.39662	0.322233
$680$	0	0
$681$	−15.6483	−0.599645
$682$	0	0
$683$	−27.2504	−1.04271	−0.521353	−	0.853341i	$-0.674573\pi$
$683$	−27.2504	−1.04271	−0.521353	+	0.853341i	$0.674573\pi$
$684$	0	0
$685$	−1.21489	−0.0464184
$686$	0	0
$687$	−1.57127	−0.0599475
$688$	0	0
$689$	−5.08944	−0.193892
$690$	0	0
$691$	−9.25937	−0.352243	−0.176122	−	0.984368i	$-0.556355\pi$
$691$	−9.25937	−0.352243	−0.176122	+	0.984368i	$0.556355\pi$
$692$	0	0
$693$	2.68670	0.102059
$694$	0	0
$695$	−0.0455546	−0.00172798
$696$	0	0
$697$	−46.2841	−1.75313
$698$	0	0
$699$	13.8022	0.522048
$700$	0	0
$701$	−6.13999	−0.231904	−0.115952	−	0.993255i	$-0.536992\pi$
$701$	−6.13999	−0.231904	−0.115952	+	0.993255i	$0.536992\pi$
$702$	0	0
$703$	−25.4190	−0.958697
$704$	0	0
$705$	0.337895	0.0127259
$706$	0	0
$707$	−2.93667	−0.110445
$708$	0	0
$709$	35.6369	1.33837	0.669186	−	0.743095i	$-0.266643\pi$
$709$	35.6369	1.33837	0.669186	+	0.743095i	$0.266643\pi$
$710$	0	0
$711$	−29.3428	−1.10044
$712$	0	0
$713$	−18.5123	−0.693291
$714$	0	0
$715$	−0.0677230	−0.00253270
$716$	0	0
$717$	−1.86173	−0.0695276
$718$	0	0
$719$	−4.58229	−0.170891	−0.0854453	−	0.996343i	$-0.527231\pi$
$719$	−4.58229	−0.170891	−0.0854453	+	0.996343i	$0.527231\pi$
$720$	0	0
$721$	−1.13245	−0.0421746
$722$	0	0
$723$	1.58278	0.0588643
$724$	0	0
$725$	−16.2921	−0.605073
$726$	0	0
$727$	24.7591	0.918264	0.459132	−	0.888368i	$-0.348161\pi$
$727$	24.7591	0.918264	0.459132	+	0.888368i	$0.348161\pi$
$728$	0	0
$729$	−11.5236	−0.426799
$730$	0	0
$731$	−44.6215	−1.65039
$732$	0	0
$733$	−18.7928	−0.694126	−0.347063	−	0.937842i	$-0.612821\pi$
$733$	−18.7928	−0.694126	−0.347063	+	0.937842i	$0.612821\pi$
$734$	0	0
$735$	−0.0379065	−0.00139820
$736$	0	0
$737$	7.92425	0.291893
$738$	0	0
$739$	27.3711	1.00686	0.503430	−	0.864036i	$-0.332071\pi$
$739$	27.3711	1.00686	0.503430	+	0.864036i	$0.332071\pi$
$740$	0	0
$741$	−2.08687	−0.0766630
$742$	0	0
$743$	−23.2912	−0.854473	−0.427236	−	0.904140i	$-0.640513\pi$
$743$	−23.2912	−0.854473	−0.427236	+	0.904140i	$0.640513\pi$
$744$	0	0
$745$	−1.10066	−0.0403250
$746$	0	0
$747$	−15.8619	−0.580355
$748$	0	0
$749$	−13.8679	−0.506723
$750$	0	0
$751$	42.0393	1.53404	0.767019	−	0.641625i	$-0.221739\pi$
$751$	42.0393	1.53404	0.767019	+	0.641625i	$0.221739\pi$
$752$	0	0
$753$	−4.21857	−0.153733
$754$	0	0
$755$	−1.32621	−0.0482658
$756$	0	0
$757$	−9.42982	−0.342733	−0.171366	−	0.985207i	$-0.554818\pi$
$757$	−9.42982	−0.342733	−0.171366	+	0.985207i	$0.554818\pi$
$758$	0	0
$759$	−3.29668	−0.119662
$760$	0	0
$761$	−6.56000	−0.237800	−0.118900	−	0.992906i	$-0.537937\pi$
$761$	−6.56000	−0.237800	−0.118900	+	0.992906i	$0.537937\pi$
$762$	0	0
$763$	−11.4726	−0.415335
$764$	0	0
$765$	−1.09124	−0.0394537
$766$	0	0
$767$	9.41662	0.340015
$768$	0	0
$769$	−4.52941	−0.163335	−0.0816673	−	0.996660i	$-0.526024\pi$
$769$	−4.52941	−0.163335	−0.0816673	+	0.996660i	$0.526024\pi$
$770$	0	0
$771$	3.07731	0.110826
$772$	0	0
$773$	32.8214	1.18050	0.590251	−	0.807220i	$-0.299029\pi$
$773$	32.8214	1.18050	0.590251	+	0.807220i	$0.299029\pi$
$774$	0	0
$775$	−15.7012	−0.564003
$776$	0	0
$777$	3.81610	0.136902
$778$	0	0
$779$	−28.7731	−1.03090
$780$	0	0
$781$	−14.7308	−0.527109
$782$	0	0
$783$	10.3811	0.370990
$784$	0	0
$785$	0.513203	0.0183170
$786$	0	0
$787$	35.0492	1.24937	0.624685	−	0.780877i	$-0.285228\pi$
$787$	35.0492	1.24937	0.624685	+	0.780877i	$0.285228\pi$
$788$	0	0
$789$	−5.35227	−0.190546
$790$	0	0
$791$	5.15434	0.183267
$792$	0	0
$793$	3.81485	0.135469
$794$	0	0
$795$	0.192923	0.00684227
$796$	0	0
$797$	−6.76785	−0.239730	−0.119865	−	0.992790i	$-0.538246\pi$
$797$	−6.76785	−0.239730	−0.119865	+	0.992790i	$0.538246\pi$
$798$	0	0
$799$	−53.4602	−1.89129
$800$	0	0
$801$	−31.5083	−1.11329
$802$	0	0
$803$	4.61482	0.162853
$804$	0	0
$805$	−0.398874	−0.0140585
$806$	0	0
$807$	−9.25032	−0.325627
$808$	0	0
$809$	39.9210	1.40355	0.701774	−	0.712399i	$-0.252391\pi$
$809$	39.9210	1.40355	0.701774	+	0.712399i	$0.252391\pi$
$810$	0	0
$811$	11.1134	0.390243	0.195121	−	0.980779i	$-0.437490\pi$
$811$	11.1134	0.390243	0.195121	+	0.980779i	$0.437490\pi$
$812$	0	0
$813$	8.91316	0.312598
$814$	0	0
$815$	0.238750	0.00836304
$816$	0	0
$817$	−27.7395	−0.970482
$818$	0	0
$819$	−2.68670	−0.0938810
$820$	0	0
$821$	−21.1493	−0.738117	−0.369058	−	0.929406i	$-0.620320\pi$
$821$	−21.1493	−0.738117	−0.369058	+	0.929406i	$0.620320\pi$
$822$	0	0
$823$	−3.91154	−0.136348	−0.0681738	−	0.997673i	$-0.521717\pi$
$823$	−3.91154	−0.136348	−0.0681738	+	0.997673i	$0.521717\pi$
$824$	0	0
$825$	−2.79608	−0.0973469
$826$	0	0
$827$	25.0037	0.869463	0.434732	−	0.900560i	$-0.356843\pi$
$827$	25.0037	0.869463	0.434732	+	0.900560i	$0.356843\pi$
$828$	0	0
$829$	17.3484	0.602535	0.301267	−	0.953540i	$-0.402590\pi$
$829$	17.3484	0.602535	0.301267	+	0.953540i	$0.402590\pi$
$830$	0	0
$831$	−1.25600	−0.0435703
$832$	0	0
$833$	5.99740	0.207798
$834$	0	0
$835$	0.141988	0.00491370
$836$	0	0
$837$	10.0046	0.345809
$838$	0	0
$839$	−30.4509	−1.05128	−0.525641	−	0.850706i	$-0.676175\pi$
$839$	−30.4509	−1.05128	−0.525641	+	0.850706i	$0.676175\pi$
$840$	0	0
$841$	−18.3632	−0.633214
$842$	0	0
$843$	1.67168	0.0575758
$844$	0	0
$845$	0.0677230	0.00232974
$846$	0	0
$847$	1.00000	0.0343604
$848$	0	0
$849$	−10.7204	−0.367924
$850$	0	0
$851$	40.1552	1.37650
$852$	0	0
$853$	−13.3141	−0.455867	−0.227933	−	0.973677i	$-0.573197\pi$
$853$	−13.3141	−0.455867	−0.227933	+	0.973677i	$0.573197\pi$
$854$	0	0
$855$	−0.678380	−0.0232001
$856$	0	0
$857$	46.4971	1.58831	0.794155	−	0.607716i	$-0.207914\pi$
$857$	46.4971	1.58831	0.794155	+	0.607716i	$0.207914\pi$
$858$	0	0
$859$	3.05818	0.104344	0.0521719	−	0.998638i	$-0.483386\pi$
$859$	3.05818	0.104344	0.0521719	+	0.998638i	$0.483386\pi$
$860$	0	0
$861$	4.31963	0.147213
$862$	0	0
$863$	19.7544	0.672447	0.336223	−	0.941782i	$-0.390850\pi$
$863$	19.7544	0.672447	0.336223	+	0.941782i	$0.390850\pi$
$864$	0	0
$865$	−1.21668	−0.0413684
$866$	0	0
$867$	−10.6174	−0.360585
$868$	0	0
$869$	−10.9215	−0.370487
$870$	0	0
$871$	−7.92425	−0.268503
$872$	0	0
$873$	−22.5592	−0.763514
$874$	0	0
$875$	−0.676919	−0.0228841
$876$	0	0
$877$	54.8557	1.85234	0.926172	−	0.377100i	$-0.123079\pi$
$877$	54.8557	1.85234	0.926172	+	0.377100i	$0.123079\pi$
$878$	0	0
$879$	11.2482	0.379393
$880$	0	0
$881$	50.2294	1.69227	0.846136	−	0.532967i	$-0.178923\pi$
$881$	50.2294	1.69227	0.846136	+	0.532967i	$0.178923\pi$
$882$	0	0
$883$	−24.3228	−0.818527	−0.409264	−	0.912416i	$-0.634214\pi$
$883$	−24.3228	−0.818527	−0.409264	+	0.912416i	$0.634214\pi$
$884$	0	0
$885$	−0.356952	−0.0119988
$886$	0	0
$887$	−42.7244	−1.43454	−0.717272	−	0.696793i	$-0.754609\pi$
$887$	−42.7244	−1.43454	−0.717272	+	0.696793i	$0.754609\pi$
$888$	0	0
$889$	−4.23839	−0.142151
$890$	0	0
$891$	−6.27849	−0.210337
$892$	0	0
$893$	−33.2342	−1.11214
$894$	0	0
$895$	−1.52109	−0.0508446
$896$	0	0
$897$	3.29668	0.110073
$898$	0	0
$899$	10.2510	0.341890
$900$	0	0
$901$	−30.5234	−1.01688
$902$	0	0
$903$	4.16446	0.138585
$904$	0	0
$905$	0.664941	0.0221034
$906$	0	0
$907$	23.8736	0.792709	0.396355	−	0.918097i	$-0.370275\pi$
$907$	23.8736	0.792709	0.396355	+	0.918097i	$0.370275\pi$
$908$	0	0
$909$	7.88998	0.261694
$910$	0	0
$911$	40.7706	1.35079	0.675395	−	0.737456i	$-0.263973\pi$
$911$	40.7706	1.35079	0.675395	+	0.737456i	$0.263973\pi$
$912$	0	0
$913$	−5.90384	−0.195388
$914$	0	0
$915$	−0.144608	−0.00478058
$916$	0	0
$917$	4.05064	0.133764
$918$	0	0
$919$	−5.97228	−0.197007	−0.0985036	−	0.995137i	$-0.531406\pi$
$919$	−5.97228	−0.197007	−0.0985036	+	0.995137i	$0.531406\pi$
$920$	0	0
$921$	5.24676	0.172887
$922$	0	0
$923$	14.7308	0.484870
$924$	0	0
$925$	34.0576	1.11981
$926$	0	0
$927$	3.04256	0.0999307
$928$	0	0
$929$	−45.0363	−1.47759	−0.738797	−	0.673928i	$-0.764606\pi$
$929$	−45.0363	−1.47759	−0.738797	+	0.673928i	$0.764606\pi$
$930$	0	0
$931$	3.72835	0.122192
$932$	0	0
$933$	18.3087	0.599399
$934$	0	0
$935$	−0.406162	−0.0132829
$936$	0	0
$937$	−32.3538	−1.05695	−0.528476	−	0.848948i	$-0.677236\pi$
$937$	−32.3538	−1.05695	−0.528476	+	0.848948i	$0.677236\pi$
$938$	0	0
$939$	15.6695	0.511354
$940$	0	0
$941$	32.9427	1.07390	0.536951	−	0.843614i	$-0.319576\pi$
$941$	32.9427	1.07390	0.536951	+	0.843614i	$0.319576\pi$
$942$	0	0
$943$	45.4536	1.48017
$944$	0	0
$945$	0.215563	0.00701227
$946$	0	0
$947$	0.689703	0.0224123	0.0112062	−	0.999937i	$-0.496433\pi$
$947$	0.689703	0.0224123	0.0112062	+	0.999937i	$0.496433\pi$
$948$	0	0
$949$	−4.61482	−0.149803
$950$	0	0
$951$	17.4405	0.565546
$952$	0	0
$953$	33.4341	1.08304	0.541519	−	0.840689i	$-0.317849\pi$
$953$	33.4341	1.08304	0.541519	+	0.840689i	$0.317849\pi$
$954$	0	0
$955$	−0.520215	−0.0168338
$956$	0	0
$957$	1.82550	0.0590102
$958$	0	0
$959$	−17.9390	−0.579282
$960$	0	0
$961$	−21.1208	−0.681316
$962$	0	0
$963$	37.2590	1.20065
$964$	0	0
$965$	0.304668	0.00980762
$966$	0	0
$967$	−21.0089	−0.675600	−0.337800	−	0.941218i	$-0.609683\pi$
$967$	−21.0089	−0.675600	−0.337800	+	0.941218i	$0.609683\pi$
$968$	0	0
$969$	−12.5158	−0.402065
$970$	0	0
$971$	54.3874	1.74538	0.872688	−	0.488279i	$-0.162375\pi$
$971$	54.3874	1.74538	0.872688	+	0.488279i	$0.162375\pi$
$972$	0	0
$973$	−0.672660	−0.0215645
$974$	0	0
$975$	2.79608	0.0895461
$976$	0	0
$977$	1.96966	0.0630149	0.0315075	−	0.999504i	$-0.489969\pi$
$977$	1.96966	0.0630149	0.0315075	+	0.999504i	$0.489969\pi$
$978$	0	0
$979$	−11.7275	−0.374812
$980$	0	0
$981$	30.8234	0.984115
$982$	0	0
$983$	−43.2341	−1.37895	−0.689477	−	0.724308i	$-0.742160\pi$
$983$	−43.2341	−1.37895	−0.689477	+	0.724308i	$0.742160\pi$
$984$	0	0
$985$	−1.58429	−0.0504798
$986$	0	0
$987$	4.98937	0.158813
$988$	0	0
$989$	43.8209	1.39342
$990$	0	0
$991$	4.72198	0.149999	0.0749994	−	0.997184i	$-0.476105\pi$
$991$	4.72198	0.149999	0.0749994	+	0.997184i	$0.476105\pi$
$992$	0	0
$993$	−16.7795	−0.532481
$994$	0	0
$995$	0.538041	0.0170570
$996$	0	0
$997$	39.1894	1.24114	0.620570	−	0.784151i	$-0.286901\pi$
$997$	39.1894	1.24114	0.620570	+	0.784151i	$0.286901\pi$
$998$	0	0
$999$	−21.7010	−0.686590

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8008.2.a.o.1.4	✓	9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8008.2.a.o.1.4	✓	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)

\(f(q)\)	\(=\)	\(q-0.559729 q^{3} +0.0677230 q^{5} +1.00000 q^{7} -2.68670 q^{9} +O(q^{10})\)	\(q-0.559729 q^{3} +0.0677230 q^{5} +1.00000 q^{7} -2.68670 q^{9} -1.00000 q^{11} +1.00000 q^{13} -0.0379065 q^{15} +5.99740 q^{17} +3.72835 q^{19} -0.559729 q^{21} -5.88978 q^{23} -4.99541 q^{25} +3.18301 q^{27} +3.26141 q^{29} +3.14312 q^{31} +0.559729 q^{33} +0.0677230 q^{35} -6.81776 q^{37} -0.559729 q^{39} -7.71736 q^{41} -7.44015 q^{43} -0.181952 q^{45} -8.91390 q^{47} +1.00000 q^{49} -3.35692 q^{51} -5.08944 q^{53} -0.0677230 q^{55} -2.08687 q^{57} +9.41662 q^{59} +3.81485 q^{61} -2.68670 q^{63} +0.0677230 q^{65} -7.92425 q^{67} +3.29668 q^{69} +14.7308 q^{71} -4.61482 q^{73} +2.79608 q^{75} -1.00000 q^{77} +10.9215 q^{79} +6.27849 q^{81} +5.90384 q^{83} +0.406162 q^{85} -1.82550 q^{87} +11.7275 q^{89} +1.00000 q^{91} -1.75930 q^{93} +0.252495 q^{95} +8.39662 q^{97} +2.68670 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q - 2 q^{3} - 4 q^{5} + 9 q^{7} + q^{9}+O(q^{10}) \) 9 * q - 2 * q^3 - 4 * q^5 + 9 * q^7 + q^9	\( 9 q - 2 q^{3} - 4 q^{5} + 9 q^{7} + q^{9} - 9 q^{11} + 9 q^{13} - 4 q^{15} - 9 q^{17} + 9 q^{19} - 2 q^{21} - 10 q^{23} - q^{25} - 8 q^{27} - 13 q^{29} + 7 q^{31} + 2 q^{33} - 4 q^{35} - 15 q^{37} - 2 q^{39} - 18 q^{41} + 7 q^{43} + 9 q^{45} - 11 q^{47} + 9 q^{49} + q^{51} - 16 q^{53} + 4 q^{55} - 26 q^{57} - 2 q^{59} + 2 q^{61} + q^{63} - 4 q^{65} + 13 q^{67} - 3 q^{69} - q^{71} - 6 q^{73} - 4 q^{75} - 9 q^{77} - 21 q^{79} - 23 q^{81} - 30 q^{83} + q^{85} + q^{87} - 36 q^{89} + 9 q^{91} - 22 q^{93} - 23 q^{95} - 5 q^{97} - q^{99}+O(q^{100}) \) 9 * q - 2 * q^3 - 4 * q^5 + 9 * q^7 + q^9 - 9 * q^11 + 9 * q^13 - 4 * q^15 - 9 * q^17 + 9 * q^19 - 2 * q^21 - 10 * q^23 - q^25 - 8 * q^27 - 13 * q^29 + 7 * q^31 + 2 * q^33 - 4 * q^35 - 15 * q^37 - 2 * q^39 - 18 * q^41 + 7 * q^43 + 9 * q^45 - 11 * q^47 + 9 * q^49 + q^51 - 16 * q^53 + 4 * q^55 - 26 * q^57 - 2 * q^59 + 2 * q^61 + q^63 - 4 * q^65 + 13 * q^67 - 3 * q^69 - q^71 - 6 * q^73 - 4 * q^75 - 9 * q^77 - 21 * q^79 - 23 * q^81 - 30 * q^83 + q^85 + q^87 - 36 * q^89 + 9 * q^91 - 22 * q^93 - 23 * q^95 - 5 * q^97 - q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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