LMFDB - Embedded newform 8008.2.a.w.1.1

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:	$11$
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} - 3 x^{10} - 19 x^{9} + 55 x^{8} + 128 x^{7} - 361 x^{6} - 343 x^{5} + 1012 x^{4} + 215 x^{3} + \cdots + 160 $ x^11 - 3x^10 - 19x^9 + 55x^8 + 128x^7 - 361x^6 - 343x^5 + 1012x^4 + 215x^3 - 1090x^2 + 240x + 160
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-2.60864$ 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)

$f(q)$	$=$	$q-2.60864 q^{3} -4.23324 q^{5} -1.00000 q^{7} +3.80501 q^{9} +O(q^{10})$	$q-2.60864 q^{3} -4.23324 q^{5} -1.00000 q^{7} +3.80501 q^{9} +1.00000 q^{11} +1.00000 q^{13} +11.0430 q^{15} +1.40113 q^{17} +5.23220 q^{19} +2.60864 q^{21} -0.274156 q^{23} +12.9203 q^{25} -2.09998 q^{27} +6.21898 q^{29} -1.08819 q^{31} -2.60864 q^{33} +4.23324 q^{35} -2.68509 q^{37} -2.60864 q^{39} -7.30001 q^{41} +6.45488 q^{43} -16.1075 q^{45} -8.44880 q^{47} +1.00000 q^{49} -3.65505 q^{51} +11.3550 q^{53} -4.23324 q^{55} -13.6489 q^{57} +8.72494 q^{59} -11.3147 q^{61} -3.80501 q^{63} -4.23324 q^{65} -12.2114 q^{67} +0.715175 q^{69} -7.37487 q^{71} -1.64748 q^{73} -33.7045 q^{75} -1.00000 q^{77} +3.59160 q^{79} -5.93693 q^{81} +14.5699 q^{83} -5.93133 q^{85} -16.2231 q^{87} +0.400823 q^{89} -1.00000 q^{91} +2.83870 q^{93} -22.1491 q^{95} -19.4305 q^{97} +3.80501 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 11 q + 3 q^{3} - 2 q^{5} - 11 q^{7} + 14 q^{9}+O(q^{10}) $ 11 * q + 3 * q^3 - 2 * q^5 - 11 * q^7 + 14 * q^9	$ 11 q + 3 q^{3} - 2 q^{5} - 11 q^{7} + 14 q^{9} + 11 q^{11} + 11 q^{13} + 7 q^{15} + 9 q^{17} + 20 q^{19} - 3 q^{21} + 12 q^{23} + 13 q^{25} + 15 q^{27} + 8 q^{29} + 7 q^{31} + 3 q^{33} + 2 q^{35} - 10 q^{37} + 3 q^{39} - 2 q^{41} + 24 q^{43} - 6 q^{45} + 2 q^{47} + 11 q^{49} + 17 q^{51} + 3 q^{53} - 2 q^{55} - 16 q^{57} + q^{59} - 22 q^{61} - 14 q^{63} - 2 q^{65} + 14 q^{67} - 22 q^{69} + 6 q^{71} + 3 q^{73} - 11 q^{77} + 8 q^{79} - 9 q^{81} + 29 q^{83} - 9 q^{85} + 19 q^{87} + 20 q^{89} - 11 q^{91} - q^{93} + 18 q^{95} - 25 q^{97} + 14 q^{99}+O(q^{100}) $ 11 * q + 3 * q^3 - 2 * q^5 - 11 * q^7 + 14 * q^9 + 11 * q^11 + 11 * q^13 + 7 * q^15 + 9 * q^17 + 20 * q^19 - 3 * q^21 + 12 * q^23 + 13 * q^25 + 15 * q^27 + 8 * q^29 + 7 * q^31 + 3 * q^33 + 2 * q^35 - 10 * q^37 + 3 * q^39 - 2 * q^41 + 24 * q^43 - 6 * q^45 + 2 * q^47 + 11 * q^49 + 17 * q^51 + 3 * q^53 - 2 * q^55 - 16 * q^57 + q^59 - 22 * q^61 - 14 * q^63 - 2 * q^65 + 14 * q^67 - 22 * q^69 + 6 * q^71 + 3 * q^73 - 11 * q^77 + 8 * q^79 - 9 * q^81 + 29 * q^83 - 9 * q^85 + 19 * q^87 + 20 * q^89 - 11 * q^91 - q^93 + 18 * q^95 - 25 * q^97 + 14 * 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
$3$	−2.60864	−1.50610	−0.753050	−	0.657963i	$-0.771418\pi$
$3$	−2.60864	−1.50610	−0.753050	+	0.657963i	$0.771418\pi$
$4$	0	0
$5$	−4.23324	−1.89316	−0.946581	−	0.322466i	$-0.895488\pi$
$5$	−4.23324	−1.89316	−0.946581	+	0.322466i	$0.895488\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	3.80501	1.26834
$10$	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
$15$	11.0430	2.85129
$16$	0	0
$17$	1.40113	0.339824	0.169912	−	0.985459i	$-0.445652\pi$
$17$	1.40113	0.339824	0.169912	+	0.985459i	$0.445652\pi$
$18$	0	0
$19$	5.23220	1.20035	0.600174	−	0.799869i	$-0.295098\pi$
$19$	5.23220	1.20035	0.600174	+	0.799869i	$0.295098\pi$
$20$	0	0
$21$	2.60864	0.569252
$22$	0	0
$23$	−0.274156	−0.0571655	−0.0285827	−	0.999591i	$-0.509099\pi$
$23$	−0.274156	−0.0571655	−0.0285827	+	0.999591i	$0.509099\pi$
$24$	0	0
$25$	12.9203	2.58406
$26$	0	0
$27$	−2.09998	−0.404142
$28$	0	0
$29$	6.21898	1.15484	0.577418	−	0.816449i	$-0.304060\pi$
$29$	6.21898	1.15484	0.577418	+	0.816449i	$0.304060\pi$
$30$	0	0
$31$	−1.08819	−0.195445	−0.0977223	−	0.995214i	$-0.531156\pi$
$31$	−1.08819	−0.195445	−0.0977223	+	0.995214i	$0.531156\pi$
$32$	0	0
$33$	−2.60864	−0.454106
$34$	0	0
$35$	4.23324	0.715548
$36$	0	0
$37$	−2.68509	−0.441427	−0.220713	−	0.975339i	$-0.570839\pi$
$37$	−2.68509	−0.441427	−0.220713	+	0.975339i	$0.570839\pi$
$38$	0	0
$39$	−2.60864	−0.417717
$40$	0	0
$41$	−7.30001	−1.14007	−0.570035	−	0.821620i	$-0.693070\pi$
$41$	−7.30001	−1.14007	−0.570035	+	0.821620i	$0.693070\pi$
$42$	0	0
$43$	6.45488	0.984360	0.492180	−	0.870493i	$-0.336200\pi$
$43$	6.45488	0.984360	0.492180	+	0.870493i	$0.336200\pi$
$44$	0	0
$45$	−16.1075	−2.40117
$46$	0	0
$47$	−8.44880	−1.23238	−0.616192	−	0.787596i	$-0.711325\pi$
$47$	−8.44880	−1.23238	−0.616192	+	0.787596i	$0.711325\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−3.65505	−0.511809
$52$	0	0
$53$	11.3550	1.55974	0.779868	−	0.625944i	$-0.215286\pi$
$53$	11.3550	1.55974	0.779868	+	0.625944i	$0.215286\pi$
$54$	0	0
$55$	−4.23324	−0.570810
$56$	0	0
$57$	−13.6489	−1.80784
$58$	0	0
$59$	8.72494	1.13589	0.567945	−	0.823066i	$-0.307739\pi$
$59$	8.72494	1.13589	0.567945	+	0.823066i	$0.307739\pi$
$60$	0	0
$61$	−11.3147	−1.44869	−0.724347	−	0.689435i	$-0.757859\pi$
$61$	−11.3147	−1.44869	−0.724347	+	0.689435i	$0.757859\pi$
$62$	0	0
$63$	−3.80501	−0.479386
$64$	0	0
$65$	−4.23324	−0.525069
$66$	0	0
$67$	−12.2114	−1.49186	−0.745931	−	0.666023i	$-0.767995\pi$
$67$	−12.2114	−1.49186	−0.745931	+	0.666023i	$0.767995\pi$
$68$	0	0
$69$	0.715175	0.0860969
$70$	0	0
$71$	−7.37487	−0.875235	−0.437618	−	0.899161i	$-0.644178\pi$
$71$	−7.37487	−0.875235	−0.437618	+	0.899161i	$0.644178\pi$
$72$	0	0
$73$	−1.64748	−0.192823	−0.0964117	−	0.995342i	$-0.530737\pi$
$73$	−1.64748	−0.192823	−0.0964117	+	0.995342i	$0.530737\pi$
$74$	0	0
$75$	−33.7045	−3.89186
$76$	0	0
$77$	−1.00000	−0.113961
$78$	0	0
$79$	3.59160	0.404087	0.202043	−	0.979377i	$-0.435242\pi$
$79$	3.59160	0.404087	0.202043	+	0.979377i	$0.435242\pi$
$80$	0	0
$81$	−5.93693	−0.659659
$82$	0	0
$83$	14.5699	1.59925	0.799626	−	0.600499i	$-0.205031\pi$
$83$	14.5699	1.59925	0.799626	+	0.600499i	$0.205031\pi$
$84$	0	0
$85$	−5.93133	−0.643343
$86$	0	0
$87$	−16.2231	−1.73930
$88$	0	0
$89$	0.400823	0.0424872	0.0212436	−	0.999774i	$-0.493237\pi$
$89$	0.400823	0.0424872	0.0212436	+	0.999774i	$0.493237\pi$
$90$	0	0
$91$	−1.00000	−0.104828
$92$	0	0
$93$	2.83870	0.294359
$94$	0	0
$95$	−22.1491	−2.27245
$96$	0	0
$97$	−19.4305	−1.97287	−0.986435	−	0.164152i	$-0.947511\pi$
$97$	−19.4305	−1.97287	−0.986435	+	0.164152i	$0.947511\pi$
$98$	0	0
$99$	3.80501	0.382418
$100$	0	0
$101$	1.81498	0.180597	0.0902985	−	0.995915i	$-0.471218\pi$
$101$	1.81498	0.180597	0.0902985	+	0.995915i	$0.471218\pi$
$102$	0	0
$103$	3.55278	0.350066	0.175033	−	0.984563i	$-0.443997\pi$
$103$	3.55278	0.350066	0.175033	+	0.984563i	$0.443997\pi$
$104$	0	0
$105$	−11.0430	−1.07769
$106$	0	0
$107$	11.7127	1.13231	0.566157	−	0.824297i	$-0.308430\pi$
$107$	11.7127	1.13231	0.566157	+	0.824297i	$0.308430\pi$
$108$	0	0
$109$	−5.20875	−0.498907	−0.249454	−	0.968387i	$-0.580251\pi$
$109$	−5.20875	−0.498907	−0.249454	+	0.968387i	$0.580251\pi$
$110$	0	0
$111$	7.00445	0.664833
$112$	0	0
$113$	13.2500	1.24645	0.623226	−	0.782042i	$-0.285822\pi$
$113$	13.2500	1.24645	0.623226	+	0.782042i	$0.285822\pi$
$114$	0	0
$115$	1.16057	0.108224
$116$	0	0
$117$	3.80501	0.351773
$118$	0	0
$119$	−1.40113	−0.128442
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	19.0431	1.71706
$124$	0	0
$125$	−33.5286	−2.99889
$126$	0	0
$127$	1.88008	0.166830	0.0834152	−	0.996515i	$-0.473417\pi$
$127$	1.88008	0.166830	0.0834152	+	0.996515i	$0.473417\pi$
$128$	0	0
$129$	−16.8385	−1.48254
$130$	0	0
$131$	−0.107903	−0.00942753	−0.00471376	−	0.999989i	$-0.501500\pi$
$131$	−0.107903	−0.00942753	−0.00471376	+	0.999989i	$0.501500\pi$
$132$	0	0
$133$	−5.23220	−0.453689
$134$	0	0
$135$	8.88973	0.765106
$136$	0	0
$137$	15.4405	1.31917	0.659586	−	0.751629i	$-0.270732\pi$
$137$	15.4405	1.31917	0.659586	+	0.751629i	$0.270732\pi$
$138$	0	0
$139$	−11.9735	−1.01558	−0.507788	−	0.861482i	$-0.669537\pi$
$139$	−11.9735	−1.01558	−0.507788	+	0.861482i	$0.669537\pi$
$140$	0	0
$141$	22.0399	1.85609
$142$	0	0
$143$	1.00000	0.0836242
$144$	0	0
$145$	−26.3264	−2.18629
$146$	0	0
$147$	−2.60864	−0.215157
$148$	0	0
$149$	−12.0736	−0.989106	−0.494553	−	0.869147i	$-0.664668\pi$
$149$	−12.0736	−0.989106	−0.494553	+	0.869147i	$0.664668\pi$
$150$	0	0
$151$	−16.2032	−1.31860	−0.659300	−	0.751880i	$-0.729147\pi$
$151$	−16.2032	−1.31860	−0.659300	+	0.751880i	$0.729147\pi$
$152$	0	0
$153$	5.33132	0.431012
$154$	0	0
$155$	4.60657	0.370008
$156$	0	0
$157$	16.6850	1.33161	0.665805	−	0.746125i	$-0.268088\pi$
$157$	16.6850	1.33161	0.665805	+	0.746125i	$0.268088\pi$
$158$	0	0
$159$	−29.6212	−2.34912
$160$	0	0
$161$	0.274156	0.0216065
$162$	0	0
$163$	−11.7138	−0.917494	−0.458747	−	0.888567i	$-0.651702\pi$
$163$	−11.7138	−0.917494	−0.458747	+	0.888567i	$0.651702\pi$
$164$	0	0
$165$	11.0430	0.859697
$166$	0	0
$167$	5.70202	0.441235	0.220618	−	0.975360i	$-0.429193\pi$
$167$	5.70202	0.441235	0.220618	+	0.975360i	$0.429193\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	19.9086	1.52245
$172$	0	0
$173$	13.3128	1.01215	0.506077	−	0.862488i	$-0.331095\pi$
$173$	13.3128	1.01215	0.506077	+	0.862488i	$0.331095\pi$
$174$	0	0
$175$	−12.9203	−0.976684
$176$	0	0
$177$	−22.7602	−1.71076
$178$	0	0
$179$	20.9215	1.56375	0.781874	−	0.623436i	$-0.214264\pi$
$179$	20.9215	1.56375	0.781874	+	0.623436i	$0.214264\pi$
$180$	0	0
$181$	−9.44619	−0.702130	−0.351065	−	0.936351i	$-0.614180\pi$
$181$	−9.44619	−0.702130	−0.351065	+	0.936351i	$0.614180\pi$
$182$	0	0
$183$	29.5159	2.18188
$184$	0	0
$185$	11.3666	0.835693
$186$	0	0
$187$	1.40113	0.102461
$188$	0	0
$189$	2.09998	0.152751
$190$	0	0
$191$	10.7309	0.776457	0.388229	−	0.921563i	$-0.373087\pi$
$191$	10.7309	0.776457	0.388229	+	0.921563i	$0.373087\pi$
$192$	0	0
$193$	−2.18699	−0.157423	−0.0787114	−	0.996897i	$-0.525081\pi$
$193$	−2.18699	−0.157423	−0.0787114	+	0.996897i	$0.525081\pi$
$194$	0	0
$195$	11.0430	0.790806
$196$	0	0
$197$	−0.0933139	−0.00664834	−0.00332417	−	0.999994i	$-0.501058\pi$
$197$	−0.0933139	−0.00664834	−0.00332417	+	0.999994i	$0.501058\pi$
$198$	0	0
$199$	−18.4710	−1.30937	−0.654687	−	0.755900i	$-0.727200\pi$
$199$	−18.4710	−1.30937	−0.654687	+	0.755900i	$0.727200\pi$
$200$	0	0
$201$	31.8552	2.24689
$202$	0	0
$203$	−6.21898	−0.436487
$204$	0	0
$205$	30.9027	2.15834
$206$	0	0
$207$	−1.04317	−0.0725051
$208$	0	0
$209$	5.23220	0.361919
$210$	0	0
$211$	−4.77947	−0.329032	−0.164516	−	0.986374i	$-0.552606\pi$
$211$	−4.77947	−0.329032	−0.164516	+	0.986374i	$0.552606\pi$
$212$	0	0
$213$	19.2384	1.31819
$214$	0	0
$215$	−27.3251	−1.86355
$216$	0	0
$217$	1.08819	0.0738711
$218$	0	0
$219$	4.29769	0.290411
$220$	0	0
$221$	1.40113	0.0942503
$222$	0	0
$223$	23.8621	1.59792	0.798961	−	0.601382i	$-0.205383\pi$
$223$	23.8621	1.59792	0.798961	+	0.601382i	$0.205383\pi$
$224$	0	0
$225$	49.1619	3.27746
$226$	0	0
$227$	16.8737	1.11995	0.559974	−	0.828511i	$-0.310811\pi$
$227$	16.8737	1.11995	0.559974	+	0.828511i	$0.310811\pi$
$228$	0	0
$229$	−27.9791	−1.84891	−0.924454	−	0.381293i	$-0.875479\pi$
$229$	−27.9791	−1.84891	−0.924454	+	0.381293i	$0.875479\pi$
$230$	0	0
$231$	2.60864	0.171636
$232$	0	0
$233$	3.14143	0.205802	0.102901	−	0.994692i	$-0.467188\pi$
$233$	3.14143	0.205802	0.102901	+	0.994692i	$0.467188\pi$
$234$	0	0
$235$	35.7658	2.33310
$236$	0	0
$237$	−9.36920	−0.608595
$238$	0	0
$239$	−3.21199	−0.207766	−0.103883	−	0.994590i	$-0.533127\pi$
$239$	−3.21199	−0.207766	−0.103883	+	0.994590i	$0.533127\pi$
$240$	0	0
$241$	−21.6758	−1.39626	−0.698131	−	0.715970i	$-0.745985\pi$
$241$	−21.6758	−1.39626	−0.698131	+	0.715970i	$0.745985\pi$
$242$	0	0
$243$	21.7873	1.39765
$244$	0	0
$245$	−4.23324	−0.270452
$246$	0	0
$247$	5.23220	0.332917
$248$	0	0
$249$	−38.0076	−2.40863
$250$	0	0
$251$	−4.83888	−0.305428	−0.152714	−	0.988270i	$-0.548801\pi$
$251$	−4.83888	−0.305428	−0.152714	+	0.988270i	$0.548801\pi$
$252$	0	0
$253$	−0.274156	−0.0172360
$254$	0	0
$255$	15.4727	0.968938
$256$	0	0
$257$	−25.9269	−1.61728	−0.808639	−	0.588305i	$-0.799795\pi$
$257$	−25.9269	−1.61728	−0.808639	+	0.588305i	$0.799795\pi$
$258$	0	0
$259$	2.68509	0.166844
$260$	0	0
$261$	23.6633	1.46472
$262$	0	0
$263$	11.6514	0.718454	0.359227	−	0.933250i	$-0.383040\pi$
$263$	11.6514	0.718454	0.359227	+	0.933250i	$0.383040\pi$
$264$	0	0
$265$	−48.0686	−2.95283
$266$	0	0
$267$	−1.04560	−0.0639900
$268$	0	0
$269$	−18.7238	−1.14161	−0.570806	−	0.821085i	$-0.693369\pi$
$269$	−18.7238	−1.14161	−0.570806	+	0.821085i	$0.693369\pi$
$270$	0	0
$271$	15.1264	0.918861	0.459431	−	0.888214i	$-0.348054\pi$
$271$	15.1264	0.918861	0.459431	+	0.888214i	$0.348054\pi$
$272$	0	0
$273$	2.60864	0.157882
$274$	0	0
$275$	12.9203	0.779124
$276$	0	0
$277$	14.9411	0.897722	0.448861	−	0.893602i	$-0.351830\pi$
$277$	14.9411	0.897722	0.448861	+	0.893602i	$0.351830\pi$
$278$	0	0
$279$	−4.14057	−0.247889
$280$	0	0
$281$	1.08542	0.0647508	0.0323754	−	0.999476i	$-0.489693\pi$
$281$	1.08542	0.0647508	0.0323754	+	0.999476i	$0.489693\pi$
$282$	0	0
$283$	23.2747	1.38354	0.691770	−	0.722118i	$-0.256831\pi$
$283$	23.2747	1.38354	0.691770	+	0.722118i	$0.256831\pi$
$284$	0	0
$285$	57.7792	3.42254
$286$	0	0
$287$	7.30001	0.430906
$288$	0	0
$289$	−15.0368	−0.884519
$290$	0	0
$291$	50.6872	2.97134
$292$	0	0
$293$	−0.337922	−0.0197416	−0.00987081	−	0.999951i	$-0.503142\pi$
$293$	−0.337922	−0.0197416	−0.00987081	+	0.999951i	$0.503142\pi$
$294$	0	0
$295$	−36.9348	−2.15043
$296$	0	0
$297$	−2.09998	−0.121853
$298$	0	0
$299$	−0.274156	−0.0158548
$300$	0	0
$301$	−6.45488	−0.372053
$302$	0	0
$303$	−4.73463	−0.271997
$304$	0	0
$305$	47.8977	2.74261
$306$	0	0
$307$	12.4325	0.709559	0.354780	−	0.934950i	$-0.384556\pi$
$307$	12.4325	0.709559	0.354780	+	0.934950i	$0.384556\pi$
$308$	0	0
$309$	−9.26793	−0.527234
$310$	0	0
$311$	−7.24242	−0.410680	−0.205340	−	0.978691i	$-0.565830\pi$
$311$	−7.24242	−0.410680	−0.205340	+	0.978691i	$0.565830\pi$
$312$	0	0
$313$	−13.5508	−0.765936	−0.382968	−	0.923762i	$-0.625098\pi$
$313$	−13.5508	−0.765936	−0.382968	+	0.923762i	$0.625098\pi$
$314$	0	0
$315$	16.1075	0.907556
$316$	0	0
$317$	−31.8318	−1.78785	−0.893926	−	0.448214i	$-0.852060\pi$
$317$	−31.8318	−1.78785	−0.893926	+	0.448214i	$0.852060\pi$
$318$	0	0
$319$	6.21898	0.348196
$320$	0	0
$321$	−30.5544	−1.70538
$322$	0	0
$323$	7.33100	0.407908
$324$	0	0
$325$	12.9203	0.716690
$326$	0	0
$327$	13.5878	0.751404
$328$	0	0
$329$	8.44880	0.465797
$330$	0	0
$331$	12.4174	0.682522	0.341261	−	0.939969i	$-0.389146\pi$
$331$	12.4174	0.682522	0.341261	+	0.939969i	$0.389146\pi$
$332$	0	0
$333$	−10.2168	−0.559878
$334$	0	0
$335$	51.6939	2.82434
$336$	0	0
$337$	−19.2536	−1.04881	−0.524405	−	0.851469i	$-0.675712\pi$
$337$	−19.2536	−1.04881	−0.524405	+	0.851469i	$0.675712\pi$
$338$	0	0
$339$	−34.5644	−1.87728
$340$	0	0
$341$	−1.08819	−0.0589287
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	−3.02751	−0.162995
$346$	0	0
$347$	14.7354	0.791039	0.395519	−	0.918458i	$-0.370565\pi$
$347$	14.7354	0.791039	0.395519	+	0.918458i	$0.370565\pi$
$348$	0	0
$349$	18.4423	0.987195	0.493598	−	0.869690i	$-0.335682\pi$
$349$	18.4423	0.987195	0.493598	+	0.869690i	$0.335682\pi$
$350$	0	0
$351$	−2.09998	−0.112089
$352$	0	0
$353$	24.3021	1.29347	0.646736	−	0.762714i	$-0.276133\pi$
$353$	24.3021	1.29347	0.646736	+	0.762714i	$0.276133\pi$
$354$	0	0
$355$	31.2196	1.65696
$356$	0	0
$357$	3.65505	0.193446
$358$	0	0
$359$	−14.4760	−0.764012	−0.382006	−	0.924160i	$-0.624767\pi$
$359$	−14.4760	−0.764012	−0.382006	+	0.924160i	$0.624767\pi$
$360$	0	0
$361$	8.37589	0.440836
$362$	0	0
$363$	−2.60864	−0.136918
$364$	0	0
$365$	6.97419	0.365046
$366$	0	0
$367$	11.6270	0.606924	0.303462	−	0.952844i	$-0.401857\pi$
$367$	11.6270	0.606924	0.303462	+	0.952844i	$0.401857\pi$
$368$	0	0
$369$	−27.7766	−1.44599
$370$	0	0
$371$	−11.3550	−0.589525
$372$	0	0
$373$	−23.4923	−1.21639	−0.608194	−	0.793789i	$-0.708106\pi$
$373$	−23.4923	−1.21639	−0.608194	+	0.793789i	$0.708106\pi$
$374$	0	0
$375$	87.4641	4.51663
$376$	0	0
$377$	6.21898	0.320294
$378$	0	0
$379$	22.3227	1.14664	0.573319	−	0.819332i	$-0.305656\pi$
$379$	22.3227	1.14664	0.573319	+	0.819332i	$0.305656\pi$
$380$	0	0
$381$	−4.90447	−0.251263
$382$	0	0
$383$	8.92748	0.456173	0.228086	−	0.973641i	$-0.426753\pi$
$383$	8.92748	0.456173	0.228086	+	0.973641i	$0.426753\pi$
$384$	0	0
$385$	4.23324	0.215746
$386$	0	0
$387$	24.5609	1.24850
$388$	0	0
$389$	11.6391	0.590125	0.295063	−	0.955478i	$-0.404659\pi$
$389$	11.6391	0.590125	0.295063	+	0.955478i	$0.404659\pi$
$390$	0	0
$391$	−0.384129	−0.0194262
$392$	0	0
$393$	0.281480	0.0141988
$394$	0	0
$395$	−15.2041	−0.765002
$396$	0	0
$397$	22.6512	1.13683	0.568415	−	0.822742i	$-0.307557\pi$
$397$	22.6512	1.13683	0.568415	+	0.822742i	$0.307557\pi$
$398$	0	0
$399$	13.6489	0.683301
$400$	0	0
$401$	9.21116	0.459983	0.229992	−	0.973193i	$-0.426130\pi$
$401$	9.21116	0.459983	0.229992	+	0.973193i	$0.426130\pi$
$402$	0	0
$403$	−1.08819	−0.0542066
$404$	0	0
$405$	25.1324	1.24884
$406$	0	0
$407$	−2.68509	−0.133095
$408$	0	0
$409$	27.0869	1.33936	0.669680	−	0.742649i	$-0.266431\pi$
$409$	27.0869	1.33936	0.669680	+	0.742649i	$0.266431\pi$
$410$	0	0
$411$	−40.2787	−1.98680
$412$	0	0
$413$	−8.72494	−0.429326
$414$	0	0
$415$	−61.6778	−3.02764
$416$	0	0
$417$	31.2345	1.52956
$418$	0	0
$419$	18.2586	0.891991	0.445996	−	0.895035i	$-0.352850\pi$
$419$	18.2586	0.891991	0.445996	+	0.895035i	$0.352850\pi$
$420$	0	0
$421$	−22.2622	−1.08499	−0.542496	−	0.840058i	$-0.682521\pi$
$421$	−22.2622	−1.08499	−0.542496	+	0.840058i	$0.682521\pi$
$422$	0	0
$423$	−32.1478	−1.56308
$424$	0	0
$425$	18.1031	0.878128
$426$	0	0
$427$	11.3147	0.547555
$428$	0	0
$429$	−2.60864	−0.125946
$430$	0	0
$431$	12.7557	0.614423	0.307211	−	0.951641i	$-0.400604\pi$
$431$	12.7557	0.614423	0.307211	+	0.951641i	$0.400604\pi$
$432$	0	0
$433$	−19.7888	−0.950990	−0.475495	−	0.879718i	$-0.657731\pi$
$433$	−19.7888	−0.950990	−0.475495	+	0.879718i	$0.657731\pi$
$434$	0	0
$435$	68.6762	3.29277
$436$	0	0
$437$	−1.43444	−0.0686185
$438$	0	0
$439$	−16.0732	−0.767130	−0.383565	−	0.923514i	$-0.625304\pi$
$439$	−16.0732	−0.767130	−0.383565	+	0.923514i	$0.625304\pi$
$440$	0	0
$441$	3.80501	0.181191
$442$	0	0
$443$	−6.32182	−0.300359	−0.150179	−	0.988659i	$-0.547985\pi$
$443$	−6.32182	−0.300359	−0.150179	+	0.988659i	$0.547985\pi$
$444$	0	0
$445$	−1.69678	−0.0804352
$446$	0	0
$447$	31.4956	1.48969
$448$	0	0
$449$	3.46064	0.163318	0.0816588	−	0.996660i	$-0.473978\pi$
$449$	3.46064	0.163318	0.0816588	+	0.996660i	$0.473978\pi$
$450$	0	0
$451$	−7.30001	−0.343744
$452$	0	0
$453$	42.2684	1.98594
$454$	0	0
$455$	4.23324	0.198457
$456$	0	0
$457$	−22.4301	−1.04924	−0.524618	−	0.851338i	$-0.675792\pi$
$457$	−22.4301	−1.04924	−0.524618	+	0.851338i	$0.675792\pi$
$458$	0	0
$459$	−2.94235	−0.137337
$460$	0	0
$461$	−7.88326	−0.367160	−0.183580	−	0.983005i	$-0.558769\pi$
$461$	−7.88326	−0.367160	−0.183580	+	0.983005i	$0.558769\pi$
$462$	0	0
$463$	−2.67510	−0.124323	−0.0621613	−	0.998066i	$-0.519799\pi$
$463$	−2.67510	−0.124323	−0.0621613	+	0.998066i	$0.519799\pi$
$464$	0	0
$465$	−12.0169	−0.557269
$466$	0	0
$467$	−24.8172	−1.14840	−0.574202	−	0.818713i	$-0.694688\pi$
$467$	−24.8172	−1.14840	−0.574202	+	0.818713i	$0.694688\pi$
$468$	0	0
$469$	12.2114	0.563871
$470$	0	0
$471$	−43.5253	−2.00554
$472$	0	0
$473$	6.45488	0.296796
$474$	0	0
$475$	67.6017	3.10178
$476$	0	0
$477$	43.2061	1.97827
$478$	0	0
$479$	−2.35109	−0.107424	−0.0537121	−	0.998556i	$-0.517105\pi$
$479$	−2.35109	−0.107424	−0.0537121	+	0.998556i	$0.517105\pi$
$480$	0	0
$481$	−2.68509	−0.122430
$482$	0	0
$483$	−0.715175	−0.0325416
$484$	0	0
$485$	82.2540	3.73496
$486$	0	0
$487$	12.3638	0.560256	0.280128	−	0.959963i	$-0.409623\pi$
$487$	12.3638	0.560256	0.280128	+	0.959963i	$0.409623\pi$
$488$	0	0
$489$	30.5570	1.38184
$490$	0	0
$491$	−35.8734	−1.61895	−0.809473	−	0.587158i	$-0.800247\pi$
$491$	−35.8734	−1.61895	−0.809473	+	0.587158i	$0.800247\pi$
$492$	0	0
$493$	8.71361	0.392441
$494$	0	0
$495$	−16.1075	−0.723979
$496$	0	0
$497$	7.37487	0.330808
$498$	0	0
$499$	30.5969	1.36971	0.684853	−	0.728681i	$-0.259866\pi$
$499$	30.5969	1.36971	0.684853	+	0.728681i	$0.259866\pi$
$500$	0	0
$501$	−14.8745	−0.664545
$502$	0	0
$503$	6.57749	0.293276	0.146638	−	0.989190i	$-0.453155\pi$
$503$	6.57749	0.293276	0.146638	+	0.989190i	$0.453155\pi$
$504$	0	0
$505$	−7.68324	−0.341900
$506$	0	0
$507$	−2.60864	−0.115854
$508$	0	0
$509$	33.9723	1.50579	0.752897	−	0.658138i	$-0.228656\pi$
$509$	33.9723	1.50579	0.752897	+	0.658138i	$0.228656\pi$
$510$	0	0
$511$	1.64748	0.0728804
$512$	0	0
$513$	−10.9875	−0.485111
$514$	0	0
$515$	−15.0398	−0.662731
$516$	0	0
$517$	−8.44880	−0.371578
$518$	0	0
$519$	−34.7283	−1.52440
$520$	0	0
$521$	−12.6166	−0.552745	−0.276373	−	0.961051i	$-0.589132\pi$
$521$	−12.6166	−0.552745	−0.276373	+	0.961051i	$0.589132\pi$
$522$	0	0
$523$	9.85278	0.430832	0.215416	−	0.976522i	$-0.430889\pi$
$523$	9.85278	0.430832	0.215416	+	0.976522i	$0.430889\pi$
$524$	0	0
$525$	33.7045	1.47098
$526$	0	0
$527$	−1.52470	−0.0664168
$528$	0	0
$529$	−22.9248	−0.996732
$530$	0	0
$531$	33.1985	1.44069
$532$	0	0
$533$	−7.30001	−0.316199
$534$	0	0
$535$	−49.5829	−2.14365
$536$	0	0
$537$	−54.5768	−2.35516
$538$	0	0
$539$	1.00000	0.0430730
$540$	0	0
$541$	24.9585	1.07305	0.536524	−	0.843885i	$-0.319737\pi$
$541$	24.9585	1.07305	0.536524	+	0.843885i	$0.319737\pi$
$542$	0	0
$543$	24.6417	1.05748
$544$	0	0
$545$	22.0499	0.944513
$546$	0	0
$547$	42.9468	1.83627	0.918135	−	0.396267i	$-0.129694\pi$
$547$	42.9468	1.83627	0.918135	+	0.396267i	$0.129694\pi$
$548$	0	0
$549$	−43.0524	−1.83743
$550$	0	0
$551$	32.5389	1.38620
$552$	0	0
$553$	−3.59160	−0.152730
$554$	0	0
$555$	−29.6515	−1.25864
$556$	0	0
$557$	−28.6132	−1.21238	−0.606190	−	0.795320i	$-0.707303\pi$
$557$	−28.6132	−1.21238	−0.606190	+	0.795320i	$0.707303\pi$
$558$	0	0
$559$	6.45488	0.273012
$560$	0	0
$561$	−3.65505	−0.154316
$562$	0	0
$563$	−1.74675	−0.0736167	−0.0368083	−	0.999322i	$-0.511719\pi$
$563$	−1.74675	−0.0736167	−0.0368083	+	0.999322i	$0.511719\pi$
$564$	0	0
$565$	−56.0903	−2.35974
$566$	0	0
$567$	5.93693	0.249328
$568$	0	0
$569$	−28.9628	−1.21418	−0.607092	−	0.794632i	$-0.707664\pi$
$569$	−28.9628	−1.21418	−0.607092	+	0.794632i	$0.707664\pi$
$570$	0	0
$571$	32.6492	1.36633	0.683163	−	0.730266i	$-0.260604\pi$
$571$	32.6492	1.36633	0.683163	+	0.730266i	$0.260604\pi$
$572$	0	0
$573$	−27.9930	−1.16942
$574$	0	0
$575$	−3.54218	−0.147719
$576$	0	0
$577$	40.0961	1.66922	0.834611	−	0.550839i	$-0.185692\pi$
$577$	40.0961	1.66922	0.834611	+	0.550839i	$0.185692\pi$
$578$	0	0
$579$	5.70506	0.237094
$580$	0	0
$581$	−14.5699	−0.604460
$582$	0	0
$583$	11.3550	0.470278
$584$	0	0
$585$	−16.1075	−0.665964
$586$	0	0
$587$	−2.80858	−0.115923	−0.0579613	−	0.998319i	$-0.518460\pi$
$587$	−2.80858	−0.115923	−0.0579613	+	0.998319i	$0.518460\pi$
$588$	0	0
$589$	−5.69362	−0.234602
$590$	0	0
$591$	0.243422	0.0100131
$592$	0	0
$593$	−40.2590	−1.65324	−0.826620	−	0.562761i	$-0.809739\pi$
$593$	−40.2590	−1.65324	−0.826620	+	0.562761i	$0.809739\pi$
$594$	0	0
$595$	5.93133	0.243161
$596$	0	0
$597$	48.1842	1.97205
$598$	0	0
$599$	40.6411	1.66055	0.830276	−	0.557353i	$-0.188183\pi$
$599$	40.6411	1.66055	0.830276	+	0.557353i	$0.188183\pi$
$600$	0	0
$601$	−14.1375	−0.576682	−0.288341	−	0.957528i	$-0.593104\pi$
$601$	−14.1375	−0.576682	−0.288341	+	0.957528i	$0.593104\pi$
$602$	0	0
$603$	−46.4646	−1.89218
$604$	0	0
$605$	−4.23324	−0.172106
$606$	0	0
$607$	−11.9496	−0.485018	−0.242509	−	0.970149i	$-0.577970\pi$
$607$	−11.9496	−0.485018	−0.242509	+	0.970149i	$0.577970\pi$
$608$	0	0
$609$	16.2231	0.657392
$610$	0	0
$611$	−8.44880	−0.341802
$612$	0	0
$613$	10.7642	0.434762	0.217381	−	0.976087i	$-0.430249\pi$
$613$	10.7642	0.434762	0.217381	+	0.976087i	$0.430249\pi$
$614$	0	0
$615$	−80.6141	−3.25067
$616$	0	0
$617$	29.6035	1.19179	0.595895	−	0.803062i	$-0.296797\pi$
$617$	29.6035	1.19179	0.595895	+	0.803062i	$0.296797\pi$
$618$	0	0
$619$	−8.23480	−0.330985	−0.165492	−	0.986211i	$-0.552921\pi$
$619$	−8.23480	−0.330985	−0.165492	+	0.986211i	$0.552921\pi$
$620$	0	0
$621$	0.575723	0.0231030
$622$	0	0
$623$	−0.400823	−0.0160587
$624$	0	0
$625$	77.3330	3.09332
$626$	0	0
$627$	−13.6489	−0.545086
$628$	0	0
$629$	−3.76217	−0.150008
$630$	0	0
$631$	21.4708	0.854738	0.427369	−	0.904077i	$-0.359441\pi$
$631$	21.4708	0.854738	0.427369	+	0.904077i	$0.359441\pi$
$632$	0	0
$633$	12.4679	0.495556
$634$	0	0
$635$	−7.95885	−0.315837
$636$	0	0
$637$	1.00000	0.0396214
$638$	0	0
$639$	−28.0614	−1.11009
$640$	0	0
$641$	47.5979	1.88001	0.940003	−	0.341167i	$-0.110822\pi$
$641$	47.5979	1.88001	0.940003	+	0.341167i	$0.110822\pi$
$642$	0	0
$643$	−13.5169	−0.533055	−0.266528	−	0.963827i	$-0.585876\pi$
$643$	−13.5169	−0.533055	−0.266528	+	0.963827i	$0.585876\pi$
$644$	0	0
$645$	71.2813	2.80670
$646$	0	0
$647$	−12.1073	−0.475988	−0.237994	−	0.971267i	$-0.576490\pi$
$647$	−12.1073	−0.475988	−0.237994	+	0.971267i	$0.576490\pi$
$648$	0	0
$649$	8.72494	0.342484
$650$	0	0
$651$	−2.83870	−0.111257
$652$	0	0
$653$	−6.92823	−0.271123	−0.135561	−	0.990769i	$-0.543284\pi$
$653$	−6.92823	−0.271123	−0.135561	+	0.990769i	$0.543284\pi$
$654$	0	0
$655$	0.456779	0.0178478
$656$	0	0
$657$	−6.26869	−0.244565
$658$	0	0
$659$	−20.9067	−0.814409	−0.407205	−	0.913337i	$-0.633496\pi$
$659$	−20.9067	−0.814409	−0.407205	+	0.913337i	$0.633496\pi$
$660$	0	0
$661$	−10.0284	−0.390058	−0.195029	−	0.980797i	$-0.562480\pi$
$661$	−10.0284	−0.390058	−0.195029	+	0.980797i	$0.562480\pi$
$662$	0	0
$663$	−3.65505	−0.141950
$664$	0	0
$665$	22.1491	0.858907
$666$	0	0
$667$	−1.70497	−0.0660167
$668$	0	0
$669$	−62.2476	−2.40663
$670$	0	0
$671$	−11.3147	−0.436798
$672$	0	0
$673$	37.8756	1.46000	0.729998	−	0.683449i	$-0.239521\pi$
$673$	37.8756	1.46000	0.729998	+	0.683449i	$0.239521\pi$
$674$	0	0
$675$	−27.1324	−1.04433
$676$	0	0
$677$	41.1723	1.58238	0.791190	−	0.611570i	$-0.209462\pi$
$677$	41.1723	1.58238	0.791190	+	0.611570i	$0.209462\pi$
$678$	0	0
$679$	19.4305	0.745675
$680$	0	0
$681$	−44.0174	−1.68675
$682$	0	0
$683$	−25.4470	−0.973702	−0.486851	−	0.873485i	$-0.661854\pi$
$683$	−25.4470	−0.973702	−0.486851	+	0.873485i	$0.661854\pi$
$684$	0	0
$685$	−65.3634	−2.49741
$686$	0	0
$687$	72.9873	2.78464
$688$	0	0
$689$	11.3550	0.432593
$690$	0	0
$691$	11.9083	0.453012	0.226506	−	0.974010i	$-0.427270\pi$
$691$	11.9083	0.453012	0.226506	+	0.974010i	$0.427270\pi$
$692$	0	0
$693$	−3.80501	−0.144540
$694$	0	0
$695$	50.6866	1.92265
$696$	0	0
$697$	−10.2283	−0.387424
$698$	0	0
$699$	−8.19487	−0.309958
$700$	0	0
$701$	−12.6570	−0.478048	−0.239024	−	0.971014i	$-0.576828\pi$
$701$	−12.6570	−0.478048	−0.239024	+	0.971014i	$0.576828\pi$
$702$	0	0
$703$	−14.0489	−0.529866
$704$	0	0
$705$	−93.3001	−3.51389
$706$	0	0
$707$	−1.81498	−0.0682593
$708$	0	0
$709$	47.3254	1.77734	0.888672	−	0.458543i	$-0.151629\pi$
$709$	47.3254	1.77734	0.888672	+	0.458543i	$0.151629\pi$
$710$	0	0
$711$	13.6661	0.512518
$712$	0	0
$713$	0.298334	0.0111727
$714$	0	0
$715$	−4.23324	−0.158314
$716$	0	0
$717$	8.37893	0.312917
$718$	0	0
$719$	−43.8336	−1.63472	−0.817358	−	0.576130i	$-0.804562\pi$
$719$	−43.8336	−1.63472	−0.817358	+	0.576130i	$0.804562\pi$
$720$	0	0
$721$	−3.55278	−0.132312
$722$	0	0
$723$	56.5445	2.10291
$724$	0	0
$725$	80.3512	2.98417
$726$	0	0
$727$	−6.36554	−0.236085	−0.118042	−	0.993009i	$-0.537662\pi$
$727$	−6.36554	−0.236085	−0.118042	+	0.993009i	$0.537662\pi$
$728$	0	0
$729$	−39.0244	−1.44535
$730$	0	0
$731$	9.04414	0.334510
$732$	0	0
$733$	−51.5147	−1.90274	−0.951369	−	0.308055i	$-0.900322\pi$
$733$	−51.5147	−1.90274	−0.951369	+	0.308055i	$0.900322\pi$
$734$	0	0
$735$	11.0430	0.407327
$736$	0	0
$737$	−12.2114	−0.449814
$738$	0	0
$739$	13.6559	0.502340	0.251170	−	0.967943i	$-0.419185\pi$
$739$	13.6559	0.502340	0.251170	+	0.967943i	$0.419185\pi$
$740$	0	0
$741$	−13.6489	−0.501406
$742$	0	0
$743$	−53.6217	−1.96719	−0.983594	−	0.180397i	$-0.942262\pi$
$743$	−53.6217	−1.96719	−0.983594	+	0.180397i	$0.942262\pi$
$744$	0	0
$745$	51.1103	1.87254
$746$	0	0
$747$	55.4385	2.02839
$748$	0	0
$749$	−11.7127	−0.427975
$750$	0	0
$751$	36.6968	1.33909	0.669543	−	0.742773i	$-0.266490\pi$
$751$	36.6968	1.33909	0.669543	+	0.742773i	$0.266490\pi$
$752$	0	0
$753$	12.6229	0.460004
$754$	0	0
$755$	68.5922	2.49633
$756$	0	0
$757$	−7.61695	−0.276843	−0.138421	−	0.990373i	$-0.544203\pi$
$757$	−7.61695	−0.276843	−0.138421	+	0.990373i	$0.544203\pi$
$758$	0	0
$759$	0.715175	0.0259592
$760$	0	0
$761$	4.53319	0.164328	0.0821640	−	0.996619i	$-0.473817\pi$
$761$	4.53319	0.164328	0.0821640	+	0.996619i	$0.473817\pi$
$762$	0	0
$763$	5.20875	0.188569
$764$	0	0
$765$	−22.5688	−0.815975
$766$	0	0
$767$	8.72494	0.315039
$768$	0	0
$769$	41.8819	1.51030	0.755150	−	0.655553i	$-0.227564\pi$
$769$	41.8819	1.51030	0.755150	+	0.655553i	$0.227564\pi$
$770$	0	0
$771$	67.6341	2.43578
$772$	0	0
$773$	−6.79558	−0.244420	−0.122210	−	0.992504i	$-0.538998\pi$
$773$	−6.79558	−0.244420	−0.122210	+	0.992504i	$0.538998\pi$
$774$	0	0
$775$	−14.0598	−0.505041
$776$	0	0
$777$	−7.00445	−0.251283
$778$	0	0
$779$	−38.1951	−1.36848
$780$	0	0
$781$	−7.37487	−0.263893
$782$	0	0
$783$	−13.0597	−0.466717
$784$	0	0
$785$	−70.6318	−2.52096
$786$	0	0
$787$	−12.7651	−0.455028	−0.227514	−	0.973775i	$-0.573060\pi$
$787$	−12.7651	−0.455028	−0.227514	+	0.973775i	$0.573060\pi$
$788$	0	0
$789$	−30.3942	−1.08206
$790$	0	0
$791$	−13.2500	−0.471115
$792$	0	0
$793$	−11.3147	−0.401796
$794$	0	0
$795$	125.394	4.44726
$796$	0	0
$797$	−24.8727	−0.881035	−0.440518	−	0.897744i	$-0.645205\pi$
$797$	−24.8727	−0.881035	−0.440518	+	0.897744i	$0.645205\pi$
$798$	0	0
$799$	−11.8379	−0.418794
$800$	0	0
$801$	1.52514	0.0538881
$802$	0	0
$803$	−1.64748	−0.0581384
$804$	0	0
$805$	−1.16057	−0.0409046
$806$	0	0
$807$	48.8438	1.71938
$808$	0	0
$809$	4.60106	0.161765	0.0808823	−	0.996724i	$-0.474226\pi$
$809$	4.60106	0.161765	0.0808823	+	0.996724i	$0.474226\pi$
$810$	0	0
$811$	−39.9867	−1.40412	−0.702061	−	0.712117i	$-0.747737\pi$
$811$	−39.9867	−1.40412	−0.702061	+	0.712117i	$0.747737\pi$
$812$	0	0
$813$	−39.4593	−1.38390
$814$	0	0
$815$	49.5872	1.73696
$816$	0	0
$817$	33.7732	1.18158
$818$	0	0
$819$	−3.80501	−0.132958
$820$	0	0
$821$	−16.5520	−0.577670	−0.288835	−	0.957379i	$-0.593268\pi$
$821$	−16.5520	−0.577670	−0.288835	+	0.957379i	$0.593268\pi$
$822$	0	0
$823$	−23.3445	−0.813738	−0.406869	−	0.913487i	$-0.633379\pi$
$823$	−23.3445	−0.813738	−0.406869	+	0.913487i	$0.633379\pi$
$824$	0	0
$825$	−33.7045	−1.17344
$826$	0	0
$827$	32.5511	1.13191	0.565957	−	0.824435i	$-0.308507\pi$
$827$	32.5511	1.13191	0.565957	+	0.824435i	$0.308507\pi$
$828$	0	0
$829$	−20.3887	−0.708128	−0.354064	−	0.935221i	$-0.615200\pi$
$829$	−20.3887	−0.708128	−0.354064	+	0.935221i	$0.615200\pi$
$830$	0	0
$831$	−38.9759	−1.35206
$832$	0	0
$833$	1.40113	0.0485463
$834$	0	0
$835$	−24.1380	−0.835330
$836$	0	0
$837$	2.28518	0.0789873
$838$	0	0
$839$	13.5508	0.467825	0.233913	−	0.972258i	$-0.424847\pi$
$839$	13.5508	0.467825	0.233913	+	0.972258i	$0.424847\pi$
$840$	0	0
$841$	9.67568	0.333644
$842$	0	0
$843$	−2.83148	−0.0975212
$844$	0	0
$845$	−4.23324	−0.145628
$846$	0	0
$847$	−1.00000	−0.0343604
$848$	0	0
$849$	−60.7154	−2.08375
$850$	0	0
$851$	0.736135	0.0252344
$852$	0	0
$853$	−36.3946	−1.24613	−0.623064	−	0.782171i	$-0.714112\pi$
$853$	−36.3946	−1.24613	−0.623064	+	0.782171i	$0.714112\pi$
$854$	0	0
$855$	−84.2777	−2.88224
$856$	0	0
$857$	4.90107	0.167417	0.0837086	−	0.996490i	$-0.473324\pi$
$857$	4.90107	0.167417	0.0837086	+	0.996490i	$0.473324\pi$
$858$	0	0
$859$	33.6457	1.14797	0.573987	−	0.818864i	$-0.305396\pi$
$859$	33.6457	1.14797	0.573987	+	0.818864i	$0.305396\pi$
$860$	0	0
$861$	−19.0431	−0.648988
$862$	0	0
$863$	−15.6605	−0.533088	−0.266544	−	0.963823i	$-0.585882\pi$
$863$	−15.6605	−0.533088	−0.266544	+	0.963823i	$0.585882\pi$
$864$	0	0
$865$	−56.3563	−1.91617
$866$	0	0
$867$	39.2257	1.33217
$868$	0	0
$869$	3.59160	0.121837
$870$	0	0
$871$	−12.2114	−0.413768
$872$	0	0
$873$	−73.9333	−2.50226
$874$	0	0
$875$	33.5286	1.13347
$876$	0	0
$877$	−52.8168	−1.78350	−0.891748	−	0.452532i	$-0.850521\pi$
$877$	−52.8168	−1.78350	−0.891748	+	0.452532i	$0.850521\pi$
$878$	0	0
$879$	0.881518	0.0297328
$880$	0	0
$881$	−7.95857	−0.268131	−0.134065	−	0.990972i	$-0.542803\pi$
$881$	−7.95857	−0.268131	−0.134065	+	0.990972i	$0.542803\pi$
$882$	0	0
$883$	19.9764	0.672260	0.336130	−	0.941816i	$-0.390882\pi$
$883$	19.9764	0.672260	0.336130	+	0.941816i	$0.390882\pi$
$884$	0	0
$885$	96.3496	3.23875
$886$	0	0
$887$	21.4876	0.721483	0.360742	−	0.932666i	$-0.382524\pi$
$887$	21.4876	0.721483	0.360742	+	0.932666i	$0.382524\pi$
$888$	0	0
$889$	−1.88008	−0.0630560
$890$	0	0
$891$	−5.93693	−0.198895
$892$	0	0
$893$	−44.2058	−1.47929
$894$	0	0
$895$	−88.5659	−2.96043
$896$	0	0
$897$	0.715175	0.0238790
$898$	0	0
$899$	−6.76742	−0.225706
$900$	0	0
$901$	15.9099	0.530036
$902$	0	0
$903$	16.8385	0.560349
$904$	0	0
$905$	39.9880	1.32925
$906$	0	0
$907$	−20.4920	−0.680426	−0.340213	−	0.940348i	$-0.610499\pi$
$907$	−20.4920	−0.680426	−0.340213	+	0.940348i	$0.610499\pi$
$908$	0	0
$909$	6.90601	0.229058
$910$	0	0
$911$	−41.1095	−1.36202	−0.681009	−	0.732275i	$-0.738459\pi$
$911$	−41.1095	−1.36202	−0.681009	+	0.732275i	$0.738459\pi$
$912$	0	0
$913$	14.5699	0.482192
$914$	0	0
$915$	−124.948	−4.13065
$916$	0	0
$917$	0.107903	0.00356327
$918$	0	0
$919$	33.0881	1.09148	0.545738	−	0.837956i	$-0.316250\pi$
$919$	33.0881	1.09148	0.545738	+	0.837956i	$0.316250\pi$
$920$	0	0
$921$	−32.4319	−1.06867
$922$	0	0
$923$	−7.37487	−0.242747
$924$	0	0
$925$	−34.6923	−1.14067
$926$	0	0
$927$	13.5184	0.444001
$928$	0	0
$929$	42.0036	1.37809	0.689046	−	0.724717i	$-0.258030\pi$
$929$	42.0036	1.37809	0.689046	+	0.724717i	$0.258030\pi$
$930$	0	0
$931$	5.23220	0.171478
$932$	0	0
$933$	18.8929	0.618525
$934$	0	0
$935$	−5.93133	−0.193975
$936$	0	0
$937$	22.4350	0.732919	0.366459	−	0.930434i	$-0.380570\pi$
$937$	22.4350	0.732919	0.366459	+	0.930434i	$0.380570\pi$
$938$	0	0
$939$	35.3492	1.15358
$940$	0	0
$941$	17.0370	0.555391	0.277696	−	0.960669i	$-0.410429\pi$
$941$	17.0370	0.555391	0.277696	+	0.960669i	$0.410429\pi$
$942$	0	0
$943$	2.00134	0.0651727
$944$	0	0
$945$	−8.88973	−0.289183
$946$	0	0
$947$	38.5119	1.25147	0.625734	−	0.780036i	$-0.284799\pi$
$947$	38.5119	1.25147	0.625734	+	0.780036i	$0.284799\pi$
$948$	0	0
$949$	−1.64748	−0.0534796
$950$	0	0
$951$	83.0378	2.69268
$952$	0	0
$953$	43.7149	1.41606	0.708032	−	0.706180i	$-0.249583\pi$
$953$	43.7149	1.41606	0.708032	+	0.706180i	$0.249583\pi$
$954$	0	0
$955$	−45.4263	−1.46996
$956$	0	0
$957$	−16.2231	−0.524418
$958$	0	0
$959$	−15.4405	−0.498600
$960$	0	0
$961$	−29.8158	−0.961801
$962$	0	0
$963$	44.5671	1.43616
$964$	0	0
$965$	9.25804	0.298027
$966$	0	0
$967$	−32.3289	−1.03963	−0.519813	−	0.854280i	$-0.673998\pi$
$967$	−32.3289	−1.03963	−0.519813	+	0.854280i	$0.673998\pi$
$968$	0	0
$969$	−19.1239	−0.614350
$970$	0	0
$971$	−27.9945	−0.898386	−0.449193	−	0.893435i	$-0.648288\pi$
$971$	−27.9945	−0.898386	−0.449193	+	0.893435i	$0.648288\pi$
$972$	0	0
$973$	11.9735	0.383852
$974$	0	0
$975$	−33.7045	−1.07941
$976$	0	0
$977$	50.1446	1.60427	0.802134	−	0.597144i	$-0.203698\pi$
$977$	50.1446	1.60427	0.802134	+	0.597144i	$0.203698\pi$
$978$	0	0
$979$	0.400823	0.0128104
$980$	0	0
$981$	−19.8193	−0.632783
$982$	0	0
$983$	39.1300	1.24805	0.624027	−	0.781403i	$-0.285495\pi$
$983$	39.1300	1.24805	0.624027	+	0.781403i	$0.285495\pi$
$984$	0	0
$985$	0.395020	0.0125864
$986$	0	0
$987$	−22.0399	−0.701537
$988$	0	0
$989$	−1.76964	−0.0562714
$990$	0	0
$991$	−36.5443	−1.16087	−0.580434	−	0.814307i	$-0.697117\pi$
$991$	−36.5443	−1.16087	−0.580434	+	0.814307i	$0.697117\pi$
$992$	0	0
$993$	−32.3925	−1.02795
$994$	0	0
$995$	78.1922	2.47886
$996$	0	0
$997$	−51.0538	−1.61689	−0.808444	−	0.588572i	$-0.799690\pi$
$997$	−51.0538	−1.61689	−0.808444	+	0.588572i	$0.799690\pi$
$998$	0	0
$999$	5.63865	0.178399

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8008.2.a.w.1.1	✓	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)

\(f(q)\)	\(=\)	\(q-2.60864 q^{3} -4.23324 q^{5} -1.00000 q^{7} +3.80501 q^{9} +O(q^{10})\)	\(q-2.60864 q^{3} -4.23324 q^{5} -1.00000 q^{7} +3.80501 q^{9} +1.00000 q^{11} +1.00000 q^{13} +11.0430 q^{15} +1.40113 q^{17} +5.23220 q^{19} +2.60864 q^{21} -0.274156 q^{23} +12.9203 q^{25} -2.09998 q^{27} +6.21898 q^{29} -1.08819 q^{31} -2.60864 q^{33} +4.23324 q^{35} -2.68509 q^{37} -2.60864 q^{39} -7.30001 q^{41} +6.45488 q^{43} -16.1075 q^{45} -8.44880 q^{47} +1.00000 q^{49} -3.65505 q^{51} +11.3550 q^{53} -4.23324 q^{55} -13.6489 q^{57} +8.72494 q^{59} -11.3147 q^{61} -3.80501 q^{63} -4.23324 q^{65} -12.2114 q^{67} +0.715175 q^{69} -7.37487 q^{71} -1.64748 q^{73} -33.7045 q^{75} -1.00000 q^{77} +3.59160 q^{79} -5.93693 q^{81} +14.5699 q^{83} -5.93133 q^{85} -16.2231 q^{87} +0.400823 q^{89} -1.00000 q^{91} +2.83870 q^{93} -22.1491 q^{95} -19.4305 q^{97} +3.80501 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 11 q + 3 q^{3} - 2 q^{5} - 11 q^{7} + 14 q^{9}+O(q^{10}) \) 11 * q + 3 * q^3 - 2 * q^5 - 11 * q^7 + 14 * q^9	\( 11 q + 3 q^{3} - 2 q^{5} - 11 q^{7} + 14 q^{9} + 11 q^{11} + 11 q^{13} + 7 q^{15} + 9 q^{17} + 20 q^{19} - 3 q^{21} + 12 q^{23} + 13 q^{25} + 15 q^{27} + 8 q^{29} + 7 q^{31} + 3 q^{33} + 2 q^{35} - 10 q^{37} + 3 q^{39} - 2 q^{41} + 24 q^{43} - 6 q^{45} + 2 q^{47} + 11 q^{49} + 17 q^{51} + 3 q^{53} - 2 q^{55} - 16 q^{57} + q^{59} - 22 q^{61} - 14 q^{63} - 2 q^{65} + 14 q^{67} - 22 q^{69} + 6 q^{71} + 3 q^{73} - 11 q^{77} + 8 q^{79} - 9 q^{81} + 29 q^{83} - 9 q^{85} + 19 q^{87} + 20 q^{89} - 11 q^{91} - q^{93} + 18 q^{95} - 25 q^{97} + 14 q^{99}+O(q^{100}) \) 11 * q + 3 * q^3 - 2 * q^5 - 11 * q^7 + 14 * q^9 + 11 * q^11 + 11 * q^13 + 7 * q^15 + 9 * q^17 + 20 * q^19 - 3 * q^21 + 12 * q^23 + 13 * q^25 + 15 * q^27 + 8 * q^29 + 7 * q^31 + 3 * q^33 + 2 * q^35 - 10 * q^37 + 3 * q^39 - 2 * q^41 + 24 * q^43 - 6 * q^45 + 2 * q^47 + 11 * q^49 + 17 * q^51 + 3 * q^53 - 2 * q^55 - 16 * q^57 + q^59 - 22 * q^61 - 14 * q^63 - 2 * q^65 + 14 * q^67 - 22 * q^69 + 6 * q^71 + 3 * q^73 - 11 * q^77 + 8 * q^79 - 9 * q^81 + 29 * q^83 - 9 * q^85 + 19 * q^87 + 20 * q^89 - 11 * q^91 - q^93 + 18 * q^95 - 25 * q^97 + 14 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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