LMFDB - Embedded newform 8008.2.a.w.1.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

chi = DirichletCharacter(H, H._module([0, 0, 0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("8008.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8008 = 2^{3} \cdot 7 \cdot 11 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8008.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$63.9442019386$
Analytic rank:	$0$
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			1.8
Root			$1.36523$ 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)

$f(q)$	$=$	$q+1.36523 q^{3} -2.50606 q^{5} -1.00000 q^{7} -1.13614 q^{9} +O(q^{10})$	$q+1.36523 q^{3} -2.50606 q^{5} -1.00000 q^{7} -1.13614 q^{9} +1.00000 q^{11} +1.00000 q^{13} -3.42134 q^{15} -6.95100 q^{17} -2.13943 q^{19} -1.36523 q^{21} +3.66368 q^{23} +1.28032 q^{25} -5.64679 q^{27} +7.07852 q^{29} +3.87339 q^{31} +1.36523 q^{33} +2.50606 q^{35} +1.84900 q^{37} +1.36523 q^{39} +2.04538 q^{41} +3.91369 q^{43} +2.84724 q^{45} -7.44426 q^{47} +1.00000 q^{49} -9.48971 q^{51} -3.73212 q^{53} -2.50606 q^{55} -2.92081 q^{57} -10.1471 q^{59} -9.48106 q^{61} +1.13614 q^{63} -2.50606 q^{65} +2.87619 q^{67} +5.00176 q^{69} +2.08744 q^{71} +2.95403 q^{73} +1.74793 q^{75} -1.00000 q^{77} -13.2191 q^{79} -4.30074 q^{81} -3.25465 q^{83} +17.4196 q^{85} +9.66382 q^{87} +14.5678 q^{89} -1.00000 q^{91} +5.28807 q^{93} +5.36153 q^{95} -2.32609 q^{97} -1.13614 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	1.36523	0.788216	0.394108	−	0.919064i	$-0.371054\pi$
$3$	1.36523	0.788216	0.394108	+	0.919064i	$0.371054\pi$
$4$	0	0
$5$	−2.50606	−1.12074	−0.560371	−	0.828242i	$-0.689341\pi$
$5$	−2.50606	−1.12074	−0.560371	+	0.828242i	$0.689341\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	−1.13614	−0.378715
$10$	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
$15$	−3.42134	−0.883387
$16$	0	0
$17$	−6.95100	−1.68586	−0.842932	−	0.538020i	$-0.819172\pi$
$17$	−6.95100	−1.68586	−0.842932	+	0.538020i	$0.819172\pi$
$18$	0	0
$19$	−2.13943	−0.490819	−0.245409	−	0.969420i	$-0.578922\pi$
$19$	−2.13943	−0.490819	−0.245409	+	0.969420i	$0.578922\pi$
$20$	0	0
$21$	−1.36523	−0.297918
$22$	0	0
$23$	3.66368	0.763929	0.381965	−	0.924177i	$-0.375248\pi$
$23$	3.66368	0.763929	0.381965	+	0.924177i	$0.375248\pi$
$24$	0	0
$25$	1.28032	0.256063
$26$	0	0
$27$	−5.64679	−1.08673
$28$	0	0
$29$	7.07852	1.31445	0.657224	−	0.753695i	$-0.271730\pi$
$29$	7.07852	1.31445	0.657224	+	0.753695i	$0.271730\pi$
$30$	0	0
$31$	3.87339	0.695681	0.347840	−	0.937554i	$-0.386915\pi$
$31$	3.87339	0.695681	0.347840	+	0.937554i	$0.386915\pi$
$32$	0	0
$33$	1.36523	0.237656
$34$	0	0
$35$	2.50606	0.423601
$36$	0	0
$37$	1.84900	0.303974	0.151987	−	0.988383i	$-0.451433\pi$
$37$	1.84900	0.303974	0.151987	+	0.988383i	$0.451433\pi$
$38$	0	0
$39$	1.36523	0.218612
$40$	0	0
$41$	2.04538	0.319435	0.159718	−	0.987163i	$-0.448942\pi$
$41$	2.04538	0.319435	0.159718	+	0.987163i	$0.448942\pi$
$42$	0	0
$43$	3.91369	0.596832	0.298416	−	0.954436i	$-0.403542\pi$
$43$	3.91369	0.596832	0.298416	+	0.954436i	$0.403542\pi$
$44$	0	0
$45$	2.84724	0.424442
$46$	0	0
$47$	−7.44426	−1.08586	−0.542928	−	0.839779i	$-0.682684\pi$
$47$	−7.44426	−1.08586	−0.542928	+	0.839779i	$0.682684\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−9.48971	−1.32883
$52$	0	0
$53$	−3.73212	−0.512646	−0.256323	−	0.966591i	$-0.582511\pi$
$53$	−3.73212	−0.512646	−0.256323	+	0.966591i	$0.582511\pi$
$54$	0	0
$55$	−2.50606	−0.337916
$56$	0	0
$57$	−2.92081	−0.386871
$58$	0	0
$59$	−10.1471	−1.32104	−0.660519	−	0.750810i	$-0.729664\pi$
$59$	−10.1471	−1.32104	−0.660519	+	0.750810i	$0.729664\pi$
$60$	0	0
$61$	−9.48106	−1.21392	−0.606962	−	0.794730i	$-0.707612\pi$
$61$	−9.48106	−1.21392	−0.606962	+	0.794730i	$0.707612\pi$
$62$	0	0
$63$	1.13614	0.143141
$64$	0	0
$65$	−2.50606	−0.310838
$66$	0	0
$67$	2.87619	0.351383	0.175691	−	0.984445i	$-0.443784\pi$
$67$	2.87619	0.351383	0.175691	+	0.984445i	$0.443784\pi$
$68$	0	0
$69$	5.00176	0.602142
$70$	0	0
$71$	2.08744	0.247734	0.123867	−	0.992299i	$-0.460470\pi$
$71$	2.08744	0.247734	0.123867	+	0.992299i	$0.460470\pi$
$72$	0	0
$73$	2.95403	0.345743	0.172872	−	0.984944i	$-0.444695\pi$
$73$	2.95403	0.345743	0.172872	+	0.984944i	$0.444695\pi$
$74$	0	0
$75$	1.74793	0.201833
$76$	0	0
$77$	−1.00000	−0.113961
$78$	0	0
$79$	−13.2191	−1.48727	−0.743634	−	0.668587i	$-0.766899\pi$
$79$	−13.2191	−1.48727	−0.743634	+	0.668587i	$0.766899\pi$
$80$	0	0
$81$	−4.30074	−0.477860
$82$	0	0
$83$	−3.25465	−0.357244	−0.178622	−	0.983918i	$-0.557164\pi$
$83$	−3.25465	−0.357244	−0.178622	+	0.983918i	$0.557164\pi$
$84$	0	0
$85$	17.4196	1.88942
$86$	0	0
$87$	9.66382	1.03607
$88$	0	0
$89$	14.5678	1.54418	0.772090	−	0.635514i	$-0.219212\pi$
$89$	14.5678	1.54418	0.772090	+	0.635514i	$0.219212\pi$
$90$	0	0
$91$	−1.00000	−0.104828
$92$	0	0
$93$	5.28807	0.548347
$94$	0	0
$95$	5.36153	0.550081
$96$	0	0
$97$	−2.32609	−0.236178	−0.118089	−	0.993003i	$-0.537677\pi$
$97$	−2.32609	−0.236178	−0.118089	+	0.993003i	$0.537677\pi$
$98$	0	0
$99$	−1.13614	−0.114187
$100$	0	0
$101$	−4.10069	−0.408034	−0.204017	−	0.978967i	$-0.565400\pi$
$101$	−4.10069	−0.408034	−0.204017	+	0.978967i	$0.565400\pi$
$102$	0	0
$103$	1.02451	0.100948	0.0504739	−	0.998725i	$-0.483927\pi$
$103$	1.02451	0.100948	0.0504739	+	0.998725i	$0.483927\pi$
$104$	0	0
$105$	3.42134	0.333889
$106$	0	0
$107$	17.1404	1.65702	0.828510	−	0.559974i	$-0.189189\pi$
$107$	17.1404	1.65702	0.828510	+	0.559974i	$0.189189\pi$
$108$	0	0
$109$	−4.00804	−0.383900	−0.191950	−	0.981405i	$-0.561481\pi$
$109$	−4.00804	−0.383900	−0.191950	+	0.981405i	$0.561481\pi$
$110$	0	0
$111$	2.52431	0.239597
$112$	0	0
$113$	19.5131	1.83564	0.917820	−	0.396998i	$-0.129948\pi$
$113$	19.5131	1.83564	0.917820	+	0.396998i	$0.129948\pi$
$114$	0	0
$115$	−9.18138	−0.856168
$116$	0	0
$117$	−1.13614	−0.105037
$118$	0	0
$119$	6.95100	0.637197
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	2.79242	0.251784
$124$	0	0
$125$	9.32174	0.833761
$126$	0	0
$127$	2.28715	0.202951	0.101476	−	0.994838i	$-0.467644\pi$
$127$	2.28715	0.202951	0.101476	+	0.994838i	$0.467644\pi$
$128$	0	0
$129$	5.34309	0.470433
$130$	0	0
$131$	17.6134	1.53889	0.769443	−	0.638716i	$-0.220534\pi$
$131$	17.6134	1.53889	0.769443	+	0.638716i	$0.220534\pi$
$132$	0	0
$133$	2.13943	0.185512
$134$	0	0
$135$	14.1512	1.21794
$136$	0	0
$137$	19.5011	1.66609	0.833047	−	0.553202i	$-0.186594\pi$
$137$	19.5011	1.66609	0.833047	+	0.553202i	$0.186594\pi$
$138$	0	0
$139$	11.6059	0.984401	0.492201	−	0.870482i	$-0.336193\pi$
$139$	11.6059	0.984401	0.492201	+	0.870482i	$0.336193\pi$
$140$	0	0
$141$	−10.1631	−0.855890
$142$	0	0
$143$	1.00000	0.0836242
$144$	0	0
$145$	−17.7392	−1.47316
$146$	0	0
$147$	1.36523	0.112602
$148$	0	0
$149$	23.4002	1.91702	0.958509	−	0.285062i	$-0.0920140\pi$
$149$	23.4002	1.91702	0.958509	+	0.285062i	$0.0920140\pi$
$150$	0	0
$151$	16.2095	1.31911	0.659554	−	0.751657i	$-0.270745\pi$
$151$	16.2095	1.31911	0.659554	+	0.751657i	$0.270745\pi$
$152$	0	0
$153$	7.89734	0.638462
$154$	0	0
$155$	−9.70693	−0.779679
$156$	0	0
$157$	−22.1910	−1.77104	−0.885518	−	0.464605i	$-0.846196\pi$
$157$	−22.1910	−1.77104	−0.885518	+	0.464605i	$0.846196\pi$
$158$	0	0
$159$	−5.09520	−0.404076
$160$	0	0
$161$	−3.66368	−0.288738
$162$	0	0
$163$	−10.8479	−0.849675	−0.424838	−	0.905270i	$-0.639669\pi$
$163$	−10.8479	−0.849675	−0.424838	+	0.905270i	$0.639669\pi$
$164$	0	0
$165$	−3.42134	−0.266351
$166$	0	0
$167$	−7.59111	−0.587418	−0.293709	−	0.955895i	$-0.594890\pi$
$167$	−7.59111	−0.587418	−0.293709	+	0.955895i	$0.594890\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	2.43070	0.185880
$172$	0	0
$173$	20.9005	1.58903	0.794517	−	0.607242i	$-0.207724\pi$
$173$	20.9005	1.58903	0.794517	+	0.607242i	$0.207724\pi$
$174$	0	0
$175$	−1.28032	−0.0967828
$176$	0	0
$177$	−13.8531	−1.04126
$178$	0	0
$179$	−9.95225	−0.743866	−0.371933	−	0.928260i	$-0.621305\pi$
$179$	−9.95225	−0.743866	−0.371933	+	0.928260i	$0.621305\pi$
$180$	0	0
$181$	20.3309	1.51118	0.755591	−	0.655044i	$-0.227350\pi$
$181$	20.3309	1.51118	0.755591	+	0.655044i	$0.227350\pi$
$182$	0	0
$183$	−12.9438	−0.956836
$184$	0	0
$185$	−4.63369	−0.340676
$186$	0	0
$187$	−6.95100	−0.508307
$188$	0	0
$189$	5.64679	0.410744
$190$	0	0
$191$	20.4594	1.48039	0.740194	−	0.672393i	$-0.234734\pi$
$191$	20.4594	1.48039	0.740194	+	0.672393i	$0.234734\pi$
$192$	0	0
$193$	25.9273	1.86629	0.933143	−	0.359504i	$-0.117054\pi$
$193$	25.9273	1.86629	0.933143	+	0.359504i	$0.117054\pi$
$194$	0	0
$195$	−3.42134	−0.245008
$196$	0	0
$197$	−7.13128	−0.508083	−0.254041	−	0.967193i	$-0.581760\pi$
$197$	−7.13128	−0.508083	−0.254041	+	0.967193i	$0.581760\pi$
$198$	0	0
$199$	12.6834	0.899106	0.449553	−	0.893254i	$-0.351583\pi$
$199$	12.6834	0.899106	0.449553	+	0.893254i	$0.351583\pi$
$200$	0	0
$201$	3.92667	0.276966
$202$	0	0
$203$	−7.07852	−0.496815
$204$	0	0
$205$	−5.12584	−0.358004
$206$	0	0
$207$	−4.16246	−0.289311
$208$	0	0
$209$	−2.13943	−0.147987
$210$	0	0
$211$	4.21190	0.289959	0.144980	−	0.989435i	$-0.453688\pi$
$211$	4.21190	0.289959	0.144980	+	0.989435i	$0.453688\pi$
$212$	0	0
$213$	2.84984	0.195268
$214$	0	0
$215$	−9.80792	−0.668895
$216$	0	0
$217$	−3.87339	−0.262943
$218$	0	0
$219$	4.03293	0.272520
$220$	0	0
$221$	−6.95100	−0.467575
$222$	0	0
$223$	−8.08545	−0.541442	−0.270721	−	0.962658i	$-0.587262\pi$
$223$	−8.08545	−0.541442	−0.270721	+	0.962658i	$0.587262\pi$
$224$	0	0
$225$	−1.45462	−0.0969749
$226$	0	0
$227$	−13.9030	−0.922778	−0.461389	−	0.887198i	$-0.652649\pi$
$227$	−13.9030	−0.922778	−0.461389	+	0.887198i	$0.652649\pi$
$228$	0	0
$229$	−14.6021	−0.964931	−0.482465	−	0.875915i	$-0.660259\pi$
$229$	−14.6021	−0.964931	−0.482465	+	0.875915i	$0.660259\pi$
$230$	0	0
$231$	−1.36523	−0.0898256
$232$	0	0
$233$	−5.29291	−0.346750	−0.173375	−	0.984856i	$-0.555467\pi$
$233$	−5.29291	−0.346750	−0.173375	+	0.984856i	$0.555467\pi$
$234$	0	0
$235$	18.6557	1.21696
$236$	0	0
$237$	−18.0472	−1.17229
$238$	0	0
$239$	−10.4853	−0.678238	−0.339119	−	0.940743i	$-0.610129\pi$
$239$	−10.4853	−0.678238	−0.339119	+	0.940743i	$0.610129\pi$
$240$	0	0
$241$	−26.5716	−1.71163	−0.855815	−	0.517283i	$-0.826944\pi$
$241$	−26.5716	−1.71163	−0.855815	+	0.517283i	$0.826944\pi$
$242$	0	0
$243$	11.0689	0.710068
$244$	0	0
$245$	−2.50606	−0.160106
$246$	0	0
$247$	−2.13943	−0.136129
$248$	0	0
$249$	−4.44334	−0.281585
$250$	0	0
$251$	3.59865	0.227145	0.113572	−	0.993530i	$-0.463771\pi$
$251$	3.59865	0.227145	0.113572	+	0.993530i	$0.463771\pi$
$252$	0	0
$253$	3.66368	0.230333
$254$	0	0
$255$	23.7818	1.48927
$256$	0	0
$257$	4.76016	0.296931	0.148465	−	0.988918i	$-0.452567\pi$
$257$	4.76016	0.296931	0.148465	+	0.988918i	$0.452567\pi$
$258$	0	0
$259$	−1.84900	−0.114891
$260$	0	0
$261$	−8.04223	−0.497801
$262$	0	0
$263$	−5.17281	−0.318969	−0.159485	−	0.987200i	$-0.550983\pi$
$263$	−5.17281	−0.318969	−0.159485	+	0.987200i	$0.550983\pi$
$264$	0	0
$265$	9.35290	0.574544
$266$	0	0
$267$	19.8884	1.21715
$268$	0	0
$269$	2.76915	0.168838	0.0844189	−	0.996430i	$-0.473097\pi$
$269$	2.76915	0.168838	0.0844189	+	0.996430i	$0.473097\pi$
$270$	0	0
$271$	0.461384	0.0280271	0.0140136	−	0.999902i	$-0.495539\pi$
$271$	0.461384	0.0280271	0.0140136	+	0.999902i	$0.495539\pi$
$272$	0	0
$273$	−1.36523	−0.0826275
$274$	0	0
$275$	1.28032	0.0772059
$276$	0	0
$277$	2.67521	0.160738	0.0803688	−	0.996765i	$-0.474390\pi$
$277$	2.67521	0.160738	0.0803688	+	0.996765i	$0.474390\pi$
$278$	0	0
$279$	−4.40073	−0.263465
$280$	0	0
$281$	4.67238	0.278731	0.139365	−	0.990241i	$-0.455494\pi$
$281$	4.67238	0.278731	0.139365	+	0.990241i	$0.455494\pi$
$282$	0	0
$283$	19.8954	1.18266	0.591330	−	0.806430i	$-0.298603\pi$
$283$	19.8954	1.18266	0.591330	+	0.806430i	$0.298603\pi$
$284$	0	0
$285$	7.31972	0.433583
$286$	0	0
$287$	−2.04538	−0.120735
$288$	0	0
$289$	31.3163	1.84214
$290$	0	0
$291$	−3.17564	−0.186160
$292$	0	0
$293$	−1.59278	−0.0930515	−0.0465257	−	0.998917i	$-0.514815\pi$
$293$	−1.59278	−0.0930515	−0.0465257	+	0.998917i	$0.514815\pi$
$294$	0	0
$295$	25.4292	1.48054
$296$	0	0
$297$	−5.64679	−0.327660
$298$	0	0
$299$	3.66368	0.211876
$300$	0	0
$301$	−3.91369	−0.225581
$302$	0	0
$303$	−5.59839	−0.321619
$304$	0	0
$305$	23.7601	1.36050
$306$	0	0
$307$	26.2170	1.49628	0.748142	−	0.663539i	$-0.230946\pi$
$307$	26.2170	1.49628	0.748142	+	0.663539i	$0.230946\pi$
$308$	0	0
$309$	1.39869	0.0795687
$310$	0	0
$311$	7.35890	0.417285	0.208643	−	0.977992i	$-0.433095\pi$
$311$	7.35890	0.417285	0.208643	+	0.977992i	$0.433095\pi$
$312$	0	0
$313$	11.1622	0.630926	0.315463	−	0.948938i	$-0.397840\pi$
$313$	11.1622	0.630926	0.315463	+	0.948938i	$0.397840\pi$
$314$	0	0
$315$	−2.84724	−0.160424
$316$	0	0
$317$	−0.157771	−0.00886131	−0.00443065	−	0.999990i	$-0.501410\pi$
$317$	−0.157771	−0.00886131	−0.00443065	+	0.999990i	$0.501410\pi$
$318$	0	0
$319$	7.07852	0.396321
$320$	0	0
$321$	23.4005	1.30609
$322$	0	0
$323$	14.8712	0.827454
$324$	0	0
$325$	1.28032	0.0710191
$326$	0	0
$327$	−5.47190	−0.302597
$328$	0	0
$329$	7.44426	0.410415
$330$	0	0
$331$	−20.6192	−1.13333	−0.566666	−	0.823948i	$-0.691767\pi$
$331$	−20.6192	−1.13333	−0.566666	+	0.823948i	$0.691767\pi$
$332$	0	0
$333$	−2.10073	−0.115119
$334$	0	0
$335$	−7.20790	−0.393810
$336$	0	0
$337$	28.9988	1.57966	0.789832	−	0.613324i	$-0.210168\pi$
$337$	28.9988	1.57966	0.789832	+	0.613324i	$0.210168\pi$
$338$	0	0
$339$	26.6399	1.44688
$340$	0	0
$341$	3.87339	0.209756
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	−12.5347	−0.674845
$346$	0	0
$347$	−12.2083	−0.655373	−0.327687	−	0.944786i	$-0.606269\pi$
$347$	−12.2083	−0.655373	−0.327687	+	0.944786i	$0.606269\pi$
$348$	0	0
$349$	−9.23734	−0.494463	−0.247232	−	0.968956i	$-0.579521\pi$
$349$	−9.23734	−0.494463	−0.247232	+	0.968956i	$0.579521\pi$
$350$	0	0
$351$	−5.64679	−0.301403
$352$	0	0
$353$	−3.80828	−0.202694	−0.101347	−	0.994851i	$-0.532315\pi$
$353$	−3.80828	−0.202694	−0.101347	+	0.994851i	$0.532315\pi$
$354$	0	0
$355$	−5.23125	−0.277646
$356$	0	0
$357$	9.48971	0.502249
$358$	0	0
$359$	−4.77484	−0.252007	−0.126003	−	0.992030i	$-0.540215\pi$
$359$	−4.77484	−0.252007	−0.126003	+	0.992030i	$0.540215\pi$
$360$	0	0
$361$	−14.4228	−0.759097
$362$	0	0
$363$	1.36523	0.0716560
$364$	0	0
$365$	−7.40297	−0.387489
$366$	0	0
$367$	−1.69589	−0.0885249	−0.0442625	−	0.999020i	$-0.514094\pi$
$367$	−1.69589	−0.0885249	−0.0442625	+	0.999020i	$0.514094\pi$
$368$	0	0
$369$	−2.32385	−0.120975
$370$	0	0
$371$	3.73212	0.193762
$372$	0	0
$373$	7.23577	0.374654	0.187327	−	0.982298i	$-0.440018\pi$
$373$	7.23577	0.374654	0.187327	+	0.982298i	$0.440018\pi$
$374$	0	0
$375$	12.7263	0.657184
$376$	0	0
$377$	7.07852	0.364563
$378$	0	0
$379$	1.66902	0.0857318	0.0428659	−	0.999081i	$-0.486351\pi$
$379$	1.66902	0.0857318	0.0428659	+	0.999081i	$0.486351\pi$
$380$	0	0
$381$	3.12248	0.159970
$382$	0	0
$383$	−4.53441	−0.231698	−0.115849	−	0.993267i	$-0.536959\pi$
$383$	−4.53441	−0.231698	−0.115849	+	0.993267i	$0.536959\pi$
$384$	0	0
$385$	2.50606	0.127720
$386$	0	0
$387$	−4.44652	−0.226029
$388$	0	0
$389$	−11.5731	−0.586781	−0.293390	−	0.955993i	$-0.594784\pi$
$389$	−11.5731	−0.586781	−0.293390	+	0.955993i	$0.594784\pi$
$390$	0	0
$391$	−25.4662	−1.28788
$392$	0	0
$393$	24.0463	1.21298
$394$	0	0
$395$	33.1279	1.66684
$396$	0	0
$397$	−0.925549	−0.0464520	−0.0232260	−	0.999730i	$-0.507394\pi$
$397$	−0.925549	−0.0464520	−0.0232260	+	0.999730i	$0.507394\pi$
$398$	0	0
$399$	2.92081	0.146224
$400$	0	0
$401$	−0.841999	−0.0420474	−0.0210237	−	0.999779i	$-0.506693\pi$
$401$	−0.841999	−0.0420474	−0.0210237	+	0.999779i	$0.506693\pi$
$402$	0	0
$403$	3.87339	0.192947
$404$	0	0
$405$	10.7779	0.535558
$406$	0	0
$407$	1.84900	0.0916515
$408$	0	0
$409$	−36.8116	−1.82021	−0.910107	−	0.414373i	$-0.864001\pi$
$409$	−36.8116	−1.82021	−0.910107	+	0.414373i	$0.864001\pi$
$410$	0	0
$411$	26.6236	1.31324
$412$	0	0
$413$	10.1471	0.499305
$414$	0	0
$415$	8.15633	0.400378
$416$	0	0
$417$	15.8448	0.775921
$418$	0	0
$419$	19.3190	0.943797	0.471898	−	0.881653i	$-0.343569\pi$
$419$	19.3190	0.943797	0.471898	+	0.881653i	$0.343569\pi$
$420$	0	0
$421$	−11.0823	−0.540118	−0.270059	−	0.962844i	$-0.587043\pi$
$421$	−11.0823	−0.540118	−0.270059	+	0.962844i	$0.587043\pi$
$422$	0	0
$423$	8.45775	0.411230
$424$	0	0
$425$	−8.89947	−0.431688
$426$	0	0
$427$	9.48106	0.458821
$428$	0	0
$429$	1.36523	0.0659140
$430$	0	0
$431$	−30.5689	−1.47245	−0.736226	−	0.676736i	$-0.763394\pi$
$431$	−30.5689	−1.47245	−0.736226	+	0.676736i	$0.763394\pi$
$432$	0	0
$433$	41.2742	1.98351	0.991756	−	0.128143i	$-0.0409016\pi$
$433$	41.2742	1.98351	0.991756	+	0.128143i	$0.0409016\pi$
$434$	0	0
$435$	−24.2181	−1.16117
$436$	0	0
$437$	−7.83817	−0.374951
$438$	0	0
$439$	18.3897	0.877694	0.438847	−	0.898562i	$-0.355387\pi$
$439$	18.3897	0.877694	0.438847	+	0.898562i	$0.355387\pi$
$440$	0	0
$441$	−1.13614	−0.0541021
$442$	0	0
$443$	5.91161	0.280869	0.140435	−	0.990090i	$-0.455150\pi$
$443$	5.91161	0.280869	0.140435	+	0.990090i	$0.455150\pi$
$444$	0	0
$445$	−36.5076	−1.73063
$446$	0	0
$447$	31.9467	1.51103
$448$	0	0
$449$	25.1616	1.18745	0.593725	−	0.804668i	$-0.297657\pi$
$449$	25.1616	1.18745	0.593725	+	0.804668i	$0.297657\pi$
$450$	0	0
$451$	2.04538	0.0963133
$452$	0	0
$453$	22.1297	1.03974
$454$	0	0
$455$	2.50606	0.117486
$456$	0	0
$457$	−16.5321	−0.773340	−0.386670	−	0.922218i	$-0.626375\pi$
$457$	−16.5321	−0.773340	−0.386670	+	0.922218i	$0.626375\pi$
$458$	0	0
$459$	39.2508	1.83207
$460$	0	0
$461$	12.3658	0.575930	0.287965	−	0.957641i	$-0.407021\pi$
$461$	12.3658	0.575930	0.287965	+	0.957641i	$0.407021\pi$
$462$	0	0
$463$	23.6445	1.09885	0.549426	−	0.835543i	$-0.314847\pi$
$463$	23.6445	1.09885	0.549426	+	0.835543i	$0.314847\pi$
$464$	0	0
$465$	−13.2522	−0.614556
$466$	0	0
$467$	−18.1152	−0.838274	−0.419137	−	0.907923i	$-0.637667\pi$
$467$	−18.1152	−0.838274	−0.419137	+	0.907923i	$0.637667\pi$
$468$	0	0
$469$	−2.87619	−0.132810
$470$	0	0
$471$	−30.2959	−1.39596
$472$	0	0
$473$	3.91369	0.179952
$474$	0	0
$475$	−2.73914	−0.125681
$476$	0	0
$477$	4.24023	0.194147
$478$	0	0
$479$	19.8014	0.904748	0.452374	−	0.891828i	$-0.350577\pi$
$479$	19.8014	0.904748	0.452374	+	0.891828i	$0.350577\pi$
$480$	0	0
$481$	1.84900	0.0843071
$482$	0	0
$483$	−5.00176	−0.227588
$484$	0	0
$485$	5.82930	0.264695
$486$	0	0
$487$	−31.6554	−1.43444	−0.717222	−	0.696844i	$-0.754587\pi$
$487$	−31.6554	−1.43444	−0.717222	+	0.696844i	$0.754587\pi$
$488$	0	0
$489$	−14.8099	−0.669728
$490$	0	0
$491$	−10.6317	−0.479801	−0.239900	−	0.970797i	$-0.577115\pi$
$491$	−10.6317	−0.479801	−0.239900	+	0.970797i	$0.577115\pi$
$492$	0	0
$493$	−49.2028	−2.21598
$494$	0	0
$495$	2.84724	0.127974
$496$	0	0
$497$	−2.08744	−0.0936346
$498$	0	0
$499$	−24.8459	−1.11226	−0.556128	−	0.831097i	$-0.687713\pi$
$499$	−24.8459	−1.11226	−0.556128	+	0.831097i	$0.687713\pi$
$500$	0	0
$501$	−10.3636	−0.463013
$502$	0	0
$503$	27.5413	1.22801	0.614003	−	0.789303i	$-0.289558\pi$
$503$	27.5413	1.22801	0.614003	+	0.789303i	$0.289558\pi$
$504$	0	0
$505$	10.2766	0.457301
$506$	0	0
$507$	1.36523	0.0606320
$508$	0	0
$509$	−14.6001	−0.647137	−0.323568	−	0.946205i	$-0.604883\pi$
$509$	−14.6001	−0.647137	−0.323568	+	0.946205i	$0.604883\pi$
$510$	0	0
$511$	−2.95403	−0.130679
$512$	0	0
$513$	12.0809	0.533385
$514$	0	0
$515$	−2.56747	−0.113136
$516$	0	0
$517$	−7.44426	−0.327398
$518$	0	0
$519$	28.5340	1.25250
$520$	0	0
$521$	22.0372	0.965468	0.482734	−	0.875767i	$-0.339644\pi$
$521$	22.0372	0.965468	0.482734	+	0.875767i	$0.339644\pi$
$522$	0	0
$523$	−5.25870	−0.229947	−0.114973	−	0.993369i	$-0.536678\pi$
$523$	−5.25870	−0.229947	−0.114973	+	0.993369i	$0.536678\pi$
$524$	0	0
$525$	−1.74793	−0.0762858
$526$	0	0
$527$	−26.9239	−1.17282
$528$	0	0
$529$	−9.57748	−0.416412
$530$	0	0
$531$	11.5286	0.500296
$532$	0	0
$533$	2.04538	0.0885953
$534$	0	0
$535$	−42.9547	−1.85709
$536$	0	0
$537$	−13.5871	−0.586327
$538$	0	0
$539$	1.00000	0.0430730
$540$	0	0
$541$	−11.5048	−0.494629	−0.247314	−	0.968935i	$-0.579548\pi$
$541$	−11.5048	−0.494629	−0.247314	+	0.968935i	$0.579548\pi$
$542$	0	0
$543$	27.7563	1.19114
$544$	0	0
$545$	10.0444	0.430253
$546$	0	0
$547$	24.5985	1.05176	0.525878	−	0.850560i	$-0.323737\pi$
$547$	24.5985	1.05176	0.525878	+	0.850560i	$0.323737\pi$
$548$	0	0
$549$	10.7719	0.459731
$550$	0	0
$551$	−15.1440	−0.645156
$552$	0	0
$553$	13.2191	0.562134
$554$	0	0
$555$	−6.32606	−0.268526
$556$	0	0
$557$	10.0876	0.427425	0.213713	−	0.976897i	$-0.431444\pi$
$557$	10.0876	0.427425	0.213713	+	0.976897i	$0.431444\pi$
$558$	0	0
$559$	3.91369	0.165531
$560$	0	0
$561$	−9.48971	−0.400656
$562$	0	0
$563$	19.2430	0.810997	0.405498	−	0.914096i	$-0.367098\pi$
$563$	19.2430	0.810997	0.405498	+	0.914096i	$0.367098\pi$
$564$	0	0
$565$	−48.9009	−2.05728
$566$	0	0
$567$	4.30074	0.180614
$568$	0	0
$569$	−5.90004	−0.247342	−0.123671	−	0.992323i	$-0.539467\pi$
$569$	−5.90004	−0.247342	−0.123671	+	0.992323i	$0.539467\pi$
$570$	0	0
$571$	3.34319	0.139908	0.0699540	−	0.997550i	$-0.477715\pi$
$571$	3.34319	0.139908	0.0699540	+	0.997550i	$0.477715\pi$
$572$	0	0
$573$	27.9318	1.16687
$574$	0	0
$575$	4.69066	0.195614
$576$	0	0
$577$	−26.8351	−1.11716	−0.558581	−	0.829450i	$-0.688654\pi$
$577$	−26.8351	−1.11716	−0.558581	+	0.829450i	$0.688654\pi$
$578$	0	0
$579$	35.3967	1.47104
$580$	0	0
$581$	3.25465	0.135025
$582$	0	0
$583$	−3.73212	−0.154569
$584$	0	0
$585$	2.84724	0.117719
$586$	0	0
$587$	13.7173	0.566175	0.283088	−	0.959094i	$-0.408641\pi$
$587$	13.7173	0.566175	0.283088	+	0.959094i	$0.408641\pi$
$588$	0	0
$589$	−8.28684	−0.341453
$590$	0	0
$591$	−9.73584	−0.400479
$592$	0	0
$593$	12.9093	0.530123	0.265061	−	0.964232i	$-0.414608\pi$
$593$	12.9093	0.530123	0.265061	+	0.964232i	$0.414608\pi$
$594$	0	0
$595$	−17.4196	−0.714133
$596$	0	0
$597$	17.3158	0.708690
$598$	0	0
$599$	22.2955	0.910970	0.455485	−	0.890243i	$-0.349466\pi$
$599$	22.2955	0.910970	0.455485	+	0.890243i	$0.349466\pi$
$600$	0	0
$601$	−40.7634	−1.66278	−0.831388	−	0.555693i	$-0.812453\pi$
$601$	−40.7634	−1.66278	−0.831388	+	0.555693i	$0.812453\pi$
$602$	0	0
$603$	−3.26777	−0.133074
$604$	0	0
$605$	−2.50606	−0.101886
$606$	0	0
$607$	30.6094	1.24240	0.621200	−	0.783652i	$-0.286646\pi$
$607$	30.6094	1.24240	0.621200	+	0.783652i	$0.286646\pi$
$608$	0	0
$609$	−9.66382	−0.391598
$610$	0	0
$611$	−7.44426	−0.301162
$612$	0	0
$613$	24.4188	0.986267	0.493134	−	0.869954i	$-0.335851\pi$
$613$	24.4188	0.986267	0.493134	+	0.869954i	$0.335851\pi$
$614$	0	0
$615$	−6.99796	−0.282185
$616$	0	0
$617$	27.5479	1.10904	0.554519	−	0.832171i	$-0.312902\pi$
$617$	27.5479	1.10904	0.554519	+	0.832171i	$0.312902\pi$
$618$	0	0
$619$	11.5735	0.465178	0.232589	−	0.972575i	$-0.425280\pi$
$619$	11.5735	0.465178	0.232589	+	0.972575i	$0.425280\pi$
$620$	0	0
$621$	−20.6880	−0.830181
$622$	0	0
$623$	−14.5678	−0.583645
$624$	0	0
$625$	−29.7624	−1.19049
$626$	0	0
$627$	−2.92081	−0.116646
$628$	0	0
$629$	−12.8524	−0.512458
$630$	0	0
$631$	−29.9830	−1.19360	−0.596802	−	0.802389i	$-0.703562\pi$
$631$	−29.9830	−1.19360	−0.596802	+	0.802389i	$0.703562\pi$
$632$	0	0
$633$	5.75022	0.228551
$634$	0	0
$635$	−5.73171	−0.227456
$636$	0	0
$637$	1.00000	0.0396214
$638$	0	0
$639$	−2.37164	−0.0938205
$640$	0	0
$641$	−39.3612	−1.55468	−0.777338	−	0.629083i	$-0.783430\pi$
$641$	−39.3612	−1.55468	−0.777338	+	0.629083i	$0.783430\pi$
$642$	0	0
$643$	32.3990	1.27769	0.638846	−	0.769335i	$-0.279412\pi$
$643$	32.3990	1.27769	0.638846	+	0.769335i	$0.279412\pi$
$644$	0	0
$645$	−13.3901	−0.527234
$646$	0	0
$647$	−17.1826	−0.675516	−0.337758	−	0.941233i	$-0.609669\pi$
$647$	−17.1826	−0.675516	−0.337758	+	0.941233i	$0.609669\pi$
$648$	0	0
$649$	−10.1471	−0.398308
$650$	0	0
$651$	−5.28807	−0.207256
$652$	0	0
$653$	−9.40166	−0.367915	−0.183958	−	0.982934i	$-0.558891\pi$
$653$	−9.40166	−0.367915	−0.183958	+	0.982934i	$0.558891\pi$
$654$	0	0
$655$	−44.1400	−1.72469
$656$	0	0
$657$	−3.35621	−0.130938
$658$	0	0
$659$	29.3062	1.14161	0.570803	−	0.821087i	$-0.306632\pi$
$659$	29.3062	1.14161	0.570803	+	0.821087i	$0.306632\pi$
$660$	0	0
$661$	−2.76318	−0.107475	−0.0537376	−	0.998555i	$-0.517113\pi$
$661$	−2.76318	−0.107475	−0.0537376	+	0.998555i	$0.517113\pi$
$662$	0	0
$663$	−9.48971	−0.368550
$664$	0	0
$665$	−5.36153	−0.207911
$666$	0	0
$667$	25.9334	1.00415
$668$	0	0
$669$	−11.0385	−0.426773
$670$	0	0
$671$	−9.48106	−0.366012
$672$	0	0
$673$	35.4195	1.36532	0.682661	−	0.730735i	$-0.260822\pi$
$673$	35.4195	1.36532	0.682661	+	0.730735i	$0.260822\pi$
$674$	0	0
$675$	−7.22968	−0.278270
$676$	0	0
$677$	16.3492	0.628350	0.314175	−	0.949365i	$-0.398272\pi$
$677$	16.3492	0.628350	0.314175	+	0.949365i	$0.398272\pi$
$678$	0	0
$679$	2.32609	0.0892670
$680$	0	0
$681$	−18.9809	−0.727349
$682$	0	0
$683$	32.8225	1.25592	0.627959	−	0.778246i	$-0.283891\pi$
$683$	32.8225	1.25592	0.627959	+	0.778246i	$0.283891\pi$
$684$	0	0
$685$	−48.8709	−1.86726
$686$	0	0
$687$	−19.9352	−0.760574
$688$	0	0
$689$	−3.73212	−0.142182
$690$	0	0
$691$	41.4186	1.57564	0.787820	−	0.615906i	$-0.211210\pi$
$691$	41.4186	1.57564	0.787820	+	0.615906i	$0.211210\pi$
$692$	0	0
$693$	1.13614	0.0431586
$694$	0	0
$695$	−29.0851	−1.10326
$696$	0	0
$697$	−14.2174	−0.538524
$698$	0	0
$699$	−7.22605	−0.273314
$700$	0	0
$701$	−29.2130	−1.10336	−0.551681	−	0.834055i	$-0.686013\pi$
$701$	−29.2130	−1.10336	−0.551681	+	0.834055i	$0.686013\pi$
$702$	0	0
$703$	−3.95580	−0.149196
$704$	0	0
$705$	25.4694	0.959232
$706$	0	0
$707$	4.10069	0.154222
$708$	0	0
$709$	−13.1309	−0.493142	−0.246571	−	0.969125i	$-0.579304\pi$
$709$	−13.1309	−0.493142	−0.246571	+	0.969125i	$0.579304\pi$
$710$	0	0
$711$	15.0188	0.563250
$712$	0	0
$713$	14.1908	0.531451
$714$	0	0
$715$	−2.50606	−0.0937212
$716$	0	0
$717$	−14.3149	−0.534598
$718$	0	0
$719$	−33.0469	−1.23244	−0.616221	−	0.787573i	$-0.711337\pi$
$719$	−33.0469	−1.23244	−0.616221	+	0.787573i	$0.711337\pi$
$720$	0	0
$721$	−1.02451	−0.0381547
$722$	0	0
$723$	−36.2764	−1.34913
$724$	0	0
$725$	9.06275	0.336582
$726$	0	0
$727$	−34.1675	−1.26720	−0.633601	−	0.773660i	$-0.718424\pi$
$727$	−34.1675	−1.26720	−0.633601	+	0.773660i	$0.718424\pi$
$728$	0	0
$729$	28.0138	1.03755
$730$	0	0
$731$	−27.2040	−1.00618
$732$	0	0
$733$	−50.4856	−1.86473	−0.932365	−	0.361519i	$-0.882258\pi$
$733$	−50.4856	−1.86473	−0.932365	+	0.361519i	$0.882258\pi$
$734$	0	0
$735$	−3.42134	−0.126198
$736$	0	0
$737$	2.87619	0.105946
$738$	0	0
$739$	9.57891	0.352366	0.176183	−	0.984357i	$-0.443625\pi$
$739$	9.57891	0.352366	0.176183	+	0.984357i	$0.443625\pi$
$740$	0	0
$741$	−2.92081	−0.107299
$742$	0	0
$743$	44.9691	1.64976	0.824879	−	0.565310i	$-0.191243\pi$
$743$	44.9691	1.64976	0.824879	+	0.565310i	$0.191243\pi$
$744$	0	0
$745$	−58.6422	−2.14848
$746$	0	0
$747$	3.69775	0.135294
$748$	0	0
$749$	−17.1404	−0.626295
$750$	0	0
$751$	−4.46629	−0.162977	−0.0814886	−	0.996674i	$-0.525967\pi$
$751$	−4.46629	−0.162977	−0.0814886	+	0.996674i	$0.525967\pi$
$752$	0	0
$753$	4.91299	0.179039
$754$	0	0
$755$	−40.6218	−1.47838
$756$	0	0
$757$	0.888507	0.0322933	0.0161467	−	0.999870i	$-0.494860\pi$
$757$	0.888507	0.0322933	0.0161467	+	0.999870i	$0.494860\pi$
$758$	0	0
$759$	5.00176	0.181552
$760$	0	0
$761$	31.8251	1.15366	0.576829	−	0.816865i	$-0.304290\pi$
$761$	31.8251	1.15366	0.576829	+	0.816865i	$0.304290\pi$
$762$	0	0
$763$	4.00804	0.145101
$764$	0	0
$765$	−19.7912	−0.715551
$766$	0	0
$767$	−10.1471	−0.366390
$768$	0	0
$769$	−19.6341	−0.708024	−0.354012	−	0.935241i	$-0.615183\pi$
$769$	−19.6341	−0.708024	−0.354012	+	0.935241i	$0.615183\pi$
$770$	0	0
$771$	6.49872	0.234046
$772$	0	0
$773$	−16.3467	−0.587951	−0.293976	−	0.955813i	$-0.594978\pi$
$773$	−16.3467	−0.587951	−0.293976	+	0.955813i	$0.594978\pi$
$774$	0	0
$775$	4.95916	0.178138
$776$	0	0
$777$	−2.52431	−0.0905591
$778$	0	0
$779$	−4.37595	−0.156785
$780$	0	0
$781$	2.08744	0.0746946
$782$	0	0
$783$	−39.9710	−1.42845
$784$	0	0
$785$	55.6119	1.98488
$786$	0	0
$787$	32.1605	1.14640	0.573199	−	0.819416i	$-0.305702\pi$
$787$	32.1605	1.14640	0.573199	+	0.819416i	$0.305702\pi$
$788$	0	0
$789$	−7.06209	−0.251417
$790$	0	0
$791$	−19.5131	−0.693806
$792$	0	0
$793$	−9.48106	−0.336682
$794$	0	0
$795$	12.7689	0.452865
$796$	0	0
$797$	−5.05707	−0.179131	−0.0895653	−	0.995981i	$-0.528548\pi$
$797$	−5.05707	−0.179131	−0.0895653	+	0.995981i	$0.528548\pi$
$798$	0	0
$799$	51.7450	1.83061
$800$	0	0
$801$	−16.5511	−0.584804
$802$	0	0
$803$	2.95403	0.104245
$804$	0	0
$805$	9.18138	0.323601
$806$	0	0
$807$	3.78052	0.133081
$808$	0	0
$809$	−49.2727	−1.73234	−0.866169	−	0.499752i	$-0.833425\pi$
$809$	−49.2727	−1.73234	−0.866169	+	0.499752i	$0.833425\pi$
$810$	0	0
$811$	−13.6325	−0.478703	−0.239351	−	0.970933i	$-0.576935\pi$
$811$	−13.6325	−0.478703	−0.239351	+	0.970933i	$0.576935\pi$
$812$	0	0
$813$	0.629896	0.0220914
$814$	0	0
$815$	27.1855	0.952267
$816$	0	0
$817$	−8.37306	−0.292936
$818$	0	0
$819$	1.13614	0.0397001
$820$	0	0
$821$	12.7035	0.443355	0.221678	−	0.975120i	$-0.428847\pi$
$821$	12.7035	0.443355	0.221678	+	0.975120i	$0.428847\pi$
$822$	0	0
$823$	−31.4946	−1.09783	−0.548916	−	0.835877i	$-0.684959\pi$
$823$	−31.4946	−1.09783	−0.548916	+	0.835877i	$0.684959\pi$
$824$	0	0
$825$	1.74793	0.0608550
$826$	0	0
$827$	−6.72413	−0.233821	−0.116911	−	0.993142i	$-0.537299\pi$
$827$	−6.72413	−0.233821	−0.116911	+	0.993142i	$0.537299\pi$
$828$	0	0
$829$	21.4684	0.745628	0.372814	−	0.927906i	$-0.378393\pi$
$829$	21.4684	0.745628	0.372814	+	0.927906i	$0.378393\pi$
$830$	0	0
$831$	3.65228	0.126696
$832$	0	0
$833$	−6.95100	−0.240838
$834$	0	0
$835$	19.0238	0.658344
$836$	0	0
$837$	−21.8722	−0.756014
$838$	0	0
$839$	5.05923	0.174664	0.0873320	−	0.996179i	$-0.472166\pi$
$839$	5.05923	0.174664	0.0873320	+	0.996179i	$0.472166\pi$
$840$	0	0
$841$	21.1055	0.727776
$842$	0	0
$843$	6.37887	0.219700
$844$	0	0
$845$	−2.50606	−0.0862109
$846$	0	0
$847$	−1.00000	−0.0343604
$848$	0	0
$849$	27.1618	0.932192
$850$	0	0
$851$	6.77413	0.232214
$852$	0	0
$853$	−37.1607	−1.27236	−0.636179	−	0.771542i	$-0.719486\pi$
$853$	−37.1607	−1.27236	−0.636179	+	0.771542i	$0.719486\pi$
$854$	0	0
$855$	−6.09147	−0.208324
$856$	0	0
$857$	18.4170	0.629113	0.314556	−	0.949239i	$-0.398144\pi$
$857$	18.4170	0.629113	0.314556	+	0.949239i	$0.398144\pi$
$858$	0	0
$859$	31.9287	1.08939	0.544696	−	0.838633i	$-0.316645\pi$
$859$	31.9287	1.08939	0.544696	+	0.838633i	$0.316645\pi$
$860$	0	0
$861$	−2.79242	−0.0951654
$862$	0	0
$863$	−33.2911	−1.13324	−0.566621	−	0.823978i	$-0.691750\pi$
$863$	−33.2911	−1.13324	−0.566621	+	0.823978i	$0.691750\pi$
$864$	0	0
$865$	−52.3778	−1.78090
$866$	0	0
$867$	42.7540	1.45200
$868$	0	0
$869$	−13.2191	−0.448428
$870$	0	0
$871$	2.87619	0.0974561
$872$	0	0
$873$	2.64277	0.0894442
$874$	0	0
$875$	−9.32174	−0.315132
$876$	0	0
$877$	−42.3655	−1.43058	−0.715291	−	0.698826i	$-0.753706\pi$
$877$	−42.3655	−1.43058	−0.715291	+	0.698826i	$0.753706\pi$
$878$	0	0
$879$	−2.17452	−0.0733447
$880$	0	0
$881$	−13.8694	−0.467271	−0.233635	−	0.972324i	$-0.575062\pi$
$881$	−13.8694	−0.467271	−0.233635	+	0.972324i	$0.575062\pi$
$882$	0	0
$883$	33.3508	1.12234	0.561171	−	0.827700i	$-0.310351\pi$
$883$	33.3508	1.12234	0.561171	+	0.827700i	$0.310351\pi$
$884$	0	0
$885$	34.7167	1.16699
$886$	0	0
$887$	−19.1447	−0.642815	−0.321407	−	0.946941i	$-0.604156\pi$
$887$	−19.1447	−0.642815	−0.321407	+	0.946941i	$0.604156\pi$
$888$	0	0
$889$	−2.28715	−0.0767084
$890$	0	0
$891$	−4.30074	−0.144080
$892$	0	0
$893$	15.9265	0.532959
$894$	0	0
$895$	24.9409	0.833682
$896$	0	0
$897$	5.00176	0.167004
$898$	0	0
$899$	27.4179	0.914437
$900$	0	0
$901$	25.9419	0.864251
$902$	0	0
$903$	−5.34309	−0.177807
$904$	0	0
$905$	−50.9503	−1.69365
$906$	0	0
$907$	51.8340	1.72112	0.860560	−	0.509349i	$-0.170114\pi$
$907$	51.8340	1.72112	0.860560	+	0.509349i	$0.170114\pi$
$908$	0	0
$909$	4.65898	0.154528
$910$	0	0
$911$	35.7836	1.18556	0.592782	−	0.805363i	$-0.298029\pi$
$911$	35.7836	1.18556	0.592782	+	0.805363i	$0.298029\pi$
$912$	0	0
$913$	−3.25465	−0.107713
$914$	0	0
$915$	32.4380	1.07237
$916$	0	0
$917$	−17.6134	−0.581644
$918$	0	0
$919$	−34.4832	−1.13749	−0.568747	−	0.822512i	$-0.692572\pi$
$919$	−34.4832	−1.13749	−0.568747	+	0.822512i	$0.692572\pi$
$920$	0	0
$921$	35.7923	1.17940
$922$	0	0
$923$	2.08744	0.0687090
$924$	0	0
$925$	2.36730	0.0778364
$926$	0	0
$927$	−1.16399	−0.0382304
$928$	0	0
$929$	17.7595	0.582671	0.291335	−	0.956621i	$-0.405900\pi$
$929$	17.7595	0.582671	0.291335	+	0.956621i	$0.405900\pi$
$930$	0	0
$931$	−2.13943	−0.0701170
$932$	0	0
$933$	10.0466	0.328911
$934$	0	0
$935$	17.4196	0.569681
$936$	0	0
$937$	−21.2309	−0.693582	−0.346791	−	0.937942i	$-0.612729\pi$
$937$	−21.2309	−0.693582	−0.346791	+	0.937942i	$0.612729\pi$
$938$	0	0
$939$	15.2390	0.497306
$940$	0	0
$941$	26.5606	0.865851	0.432926	−	0.901430i	$-0.357481\pi$
$941$	26.5606	0.865851	0.432926	+	0.901430i	$0.357481\pi$
$942$	0	0
$943$	7.49362	0.244026
$944$	0	0
$945$	−14.1512	−0.460338
$946$	0	0
$947$	20.3575	0.661531	0.330765	−	0.943713i	$-0.392693\pi$
$947$	20.3575	0.661531	0.330765	+	0.943713i	$0.392693\pi$
$948$	0	0
$949$	2.95403	0.0958919
$950$	0	0
$951$	−0.215394	−0.00698463
$952$	0	0
$953$	−28.8455	−0.934397	−0.467199	−	0.884152i	$-0.654737\pi$
$953$	−28.8455	−0.934397	−0.467199	+	0.884152i	$0.654737\pi$
$954$	0	0
$955$	−51.2723	−1.65913
$956$	0	0
$957$	9.66382	0.312387
$958$	0	0
$959$	−19.5011	−0.629725
$960$	0	0
$961$	−15.9969	−0.516028
$962$	0	0
$963$	−19.4739	−0.627538
$964$	0	0
$965$	−64.9752	−2.09163
$966$	0	0
$967$	23.0280	0.740531	0.370266	−	0.928926i	$-0.379267\pi$
$967$	23.0280	0.740531	0.370266	+	0.928926i	$0.379267\pi$
$968$	0	0
$969$	20.3026	0.652213
$970$	0	0
$971$	56.7466	1.82109	0.910543	−	0.413415i	$-0.135664\pi$
$971$	56.7466	1.82109	0.910543	+	0.413415i	$0.135664\pi$
$972$	0	0
$973$	−11.6059	−0.372069
$974$	0	0
$975$	1.74793	0.0559785
$976$	0	0
$977$	−26.5919	−0.850750	−0.425375	−	0.905017i	$-0.639858\pi$
$977$	−26.5919	−0.850750	−0.425375	+	0.905017i	$0.639858\pi$
$978$	0	0
$979$	14.5678	0.465588
$980$	0	0
$981$	4.55371	0.145389
$982$	0	0
$983$	28.3572	0.904454	0.452227	−	0.891903i	$-0.350630\pi$
$983$	28.3572	0.904454	0.452227	+	0.891903i	$0.350630\pi$
$984$	0	0
$985$	17.8714	0.569430
$986$	0	0
$987$	10.1631	0.323496
$988$	0	0
$989$	14.3385	0.455937
$990$	0	0
$991$	7.41346	0.235496	0.117748	−	0.993043i	$-0.462432\pi$
$991$	7.41346	0.235496	0.117748	+	0.993043i	$0.462432\pi$
$992$	0	0
$993$	−28.1499	−0.893310
$994$	0	0
$995$	−31.7854	−1.00767
$996$	0	0
$997$	−5.39225	−0.170774	−0.0853871	−	0.996348i	$-0.527213\pi$
$997$	−5.39225	−0.170774	−0.0853871	+	0.996348i	$0.527213\pi$
$998$	0	0
$999$	−10.4409	−0.330336

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8008.2.a.w.1.8	✓	11	1.1	even	1	trivial

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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)

\(f(q)\)	\(=\)	\(q+1.36523 q^{3} -2.50606 q^{5} -1.00000 q^{7} -1.13614 q^{9} +O(q^{10})\)	\(q+1.36523 q^{3} -2.50606 q^{5} -1.00000 q^{7} -1.13614 q^{9} +1.00000 q^{11} +1.00000 q^{13} -3.42134 q^{15} -6.95100 q^{17} -2.13943 q^{19} -1.36523 q^{21} +3.66368 q^{23} +1.28032 q^{25} -5.64679 q^{27} +7.07852 q^{29} +3.87339 q^{31} +1.36523 q^{33} +2.50606 q^{35} +1.84900 q^{37} +1.36523 q^{39} +2.04538 q^{41} +3.91369 q^{43} +2.84724 q^{45} -7.44426 q^{47} +1.00000 q^{49} -9.48971 q^{51} -3.73212 q^{53} -2.50606 q^{55} -2.92081 q^{57} -10.1471 q^{59} -9.48106 q^{61} +1.13614 q^{63} -2.50606 q^{65} +2.87619 q^{67} +5.00176 q^{69} +2.08744 q^{71} +2.95403 q^{73} +1.74793 q^{75} -1.00000 q^{77} -13.2191 q^{79} -4.30074 q^{81} -3.25465 q^{83} +17.4196 q^{85} +9.66382 q^{87} +14.5678 q^{89} -1.00000 q^{91} +5.28807 q^{93} +5.36153 q^{95} -2.32609 q^{97} -1.13614 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 11 q + 3 q^{3} - 2 q^{5} - 11 q^{7} + 14 q^{9}+O(q^{10}) \) 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

Properties

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