LMFDB - Embedded newform 8008.2.a.l.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:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 2x^{7} - 9x^{6} + 12x^{5} + 24x^{4} - 10x^{3} - 18x^{2} - 2x + 1 $ x^8 - 2x^7 - 9x^6 + 12x^5 + 24x^4 - 10x^3 - 18x^2 - 2*x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-0.405405$ 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)

$f(q)$	$=$	$q-0.867389 q^{3} -3.42583 q^{5} +1.00000 q^{7} -2.24764 q^{9} +O(q^{10})$	$q-0.867389 q^{3} -3.42583 q^{5} +1.00000 q^{7} -2.24764 q^{9} +1.00000 q^{11} +1.00000 q^{13} +2.97152 q^{15} +0.909259 q^{17} -7.30405 q^{19} -0.867389 q^{21} +2.42356 q^{23} +6.73629 q^{25} +4.55174 q^{27} +8.71455 q^{29} -7.50953 q^{31} -0.867389 q^{33} -3.42583 q^{35} +1.77295 q^{37} -0.867389 q^{39} -10.1812 q^{41} +9.94047 q^{43} +7.70001 q^{45} +8.73735 q^{47} +1.00000 q^{49} -0.788681 q^{51} -11.5902 q^{53} -3.42583 q^{55} +6.33545 q^{57} +3.50987 q^{59} -7.45653 q^{61} -2.24764 q^{63} -3.42583 q^{65} +1.60849 q^{67} -2.10217 q^{69} +16.6917 q^{71} +10.3230 q^{73} -5.84298 q^{75} +1.00000 q^{77} -4.33168 q^{79} +2.79478 q^{81} +2.06354 q^{83} -3.11496 q^{85} -7.55891 q^{87} +10.1262 q^{89} +1.00000 q^{91} +6.51368 q^{93} +25.0224 q^{95} +7.60929 q^{97} -2.24764 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 5 q^{3} - 7 q^{5} + 8 q^{7} + 7 q^{9}+O(q^{10}) $ 8 * q - 5 * q^3 - 7 * q^5 + 8 * q^7 + 7 * q^9	$ 8 q - 5 q^{3} - 7 q^{5} + 8 q^{7} + 7 q^{9} + 8 q^{11} + 8 q^{13} - 5 q^{15} + 3 q^{17} - 19 q^{19} - 5 q^{21} + q^{23} + 15 q^{25} - 11 q^{27} - 21 q^{29} + 2 q^{31} - 5 q^{33} - 7 q^{35} - 12 q^{37} - 5 q^{39} - 6 q^{41} - 19 q^{43} - 17 q^{45} - 7 q^{47} + 8 q^{49} - 19 q^{51} - 26 q^{53} - 7 q^{55} + 2 q^{57} - 17 q^{59} + 7 q^{63} - 7 q^{65} - 24 q^{67} + 20 q^{69} + 2 q^{71} + 2 q^{73} + 18 q^{75} + 8 q^{77} + 7 q^{79} - 4 q^{81} - 6 q^{83} + 19 q^{85} - 13 q^{87} + 13 q^{89} + 8 q^{91} + 13 q^{93} + 21 q^{95} + 18 q^{97} + 7 q^{99}+O(q^{100}) $ 8 * q - 5 * q^3 - 7 * q^5 + 8 * q^7 + 7 * q^9 + 8 * q^11 + 8 * q^13 - 5 * q^15 + 3 * q^17 - 19 * q^19 - 5 * q^21 + q^23 + 15 * q^25 - 11 * q^27 - 21 * q^29 + 2 * q^31 - 5 * q^33 - 7 * q^35 - 12 * q^37 - 5 * q^39 - 6 * q^41 - 19 * q^43 - 17 * q^45 - 7 * q^47 + 8 * q^49 - 19 * q^51 - 26 * q^53 - 7 * q^55 + 2 * q^57 - 17 * q^59 + 7 * q^63 - 7 * q^65 - 24 * q^67 + 20 * q^69 + 2 * q^71 + 2 * q^73 + 18 * q^75 + 8 * q^77 + 7 * q^79 - 4 * q^81 - 6 * q^83 + 19 * q^85 - 13 * q^87 + 13 * q^89 + 8 * q^91 + 13 * q^93 + 21 * q^95 + 18 * q^97 + 7 * 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
$3$	−0.867389	−0.500787	−0.250394	−	0.968144i	$-0.580560\pi$
$3$	−0.867389	−0.500787	−0.250394	+	0.968144i	$0.580560\pi$
$4$	0	0
$5$	−3.42583	−1.53208	−0.766038	−	0.642795i	$-0.777775\pi$
$5$	−3.42583	−1.53208	−0.766038	+	0.642795i	$0.777775\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	−2.24764	−0.749212
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	0	0
$15$	2.97152	0.767244
$16$	0	0
$17$	0.909259	0.220528	0.110264	−	0.993902i	$-0.464830\pi$
$17$	0.909259	0.220528	0.110264	+	0.993902i	$0.464830\pi$
$18$	0	0
$19$	−7.30405	−1.67566	−0.837831	−	0.545929i	$-0.816177\pi$
$19$	−7.30405	−1.67566	−0.837831	+	0.545929i	$0.816177\pi$
$20$	0	0
$21$	−0.867389	−0.189280
$22$	0	0
$23$	2.42356	0.505348	0.252674	−	0.967552i	$-0.418690\pi$
$23$	2.42356	0.505348	0.252674	+	0.967552i	$0.418690\pi$
$24$	0	0
$25$	6.73629	1.34726
$26$	0	0
$27$	4.55174	0.875983
$28$	0	0
$29$	8.71455	1.61825	0.809126	−	0.587635i	$-0.199941\pi$
$29$	8.71455	1.61825	0.809126	+	0.587635i	$0.199941\pi$
$30$	0	0
$31$	−7.50953	−1.34875	−0.674376	−	0.738388i	$-0.735587\pi$
$31$	−7.50953	−1.34875	−0.674376	+	0.738388i	$0.735587\pi$
$32$	0	0
$33$	−0.867389	−0.150993
$34$	0	0
$35$	−3.42583	−0.579070
$36$	0	0
$37$	1.77295	0.291471	0.145735	−	0.989324i	$-0.453445\pi$
$37$	1.77295	0.291471	0.145735	+	0.989324i	$0.453445\pi$
$38$	0	0
$39$	−0.867389	−0.138893
$40$	0	0
$41$	−10.1812	−1.59003	−0.795017	−	0.606587i	$-0.792538\pi$
$41$	−10.1812	−1.59003	−0.795017	+	0.606587i	$0.792538\pi$
$42$	0	0
$43$	9.94047	1.51591	0.757954	−	0.652308i	$-0.226199\pi$
$43$	9.94047	1.51591	0.757954	+	0.652308i	$0.226199\pi$
$44$	0	0
$45$	7.70001	1.14785
$46$	0	0
$47$	8.73735	1.27447	0.637237	−	0.770668i	$-0.280077\pi$
$47$	8.73735	1.27447	0.637237	+	0.770668i	$0.280077\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−0.788681	−0.110437
$52$	0	0
$53$	−11.5902	−1.59204	−0.796018	−	0.605272i	$-0.793064\pi$
$53$	−11.5902	−1.59204	−0.796018	+	0.605272i	$0.793064\pi$
$54$	0	0
$55$	−3.42583	−0.461938
$56$	0	0
$57$	6.33545	0.839151
$58$	0	0
$59$	3.50987	0.456946	0.228473	−	0.973550i	$-0.426627\pi$
$59$	3.50987	0.456946	0.228473	+	0.973550i	$0.426627\pi$
$60$	0	0
$61$	−7.45653	−0.954711	−0.477356	−	0.878710i	$-0.658405\pi$
$61$	−7.45653	−0.954711	−0.477356	+	0.878710i	$0.658405\pi$
$62$	0	0
$63$	−2.24764	−0.283176
$64$	0	0
$65$	−3.42583	−0.424922
$66$	0	0
$67$	1.60849	0.196508	0.0982541	−	0.995161i	$-0.468674\pi$
$67$	1.60849	0.196508	0.0982541	+	0.995161i	$0.468674\pi$
$68$	0	0
$69$	−2.10217	−0.253072
$70$	0	0
$71$	16.6917	1.98095	0.990473	−	0.137709i	$-0.0439738\pi$
$71$	16.6917	1.98095	0.990473	+	0.137709i	$0.0439738\pi$
$72$	0	0
$73$	10.3230	1.20822	0.604109	−	0.796902i	$-0.293529\pi$
$73$	10.3230	1.20822	0.604109	+	0.796902i	$0.293529\pi$
$74$	0	0
$75$	−5.84298	−0.674690
$76$	0	0
$77$	1.00000	0.113961
$78$	0	0
$79$	−4.33168	−0.487352	−0.243676	−	0.969857i	$-0.578353\pi$
$79$	−4.33168	−0.487352	−0.243676	+	0.969857i	$0.578353\pi$
$80$	0	0
$81$	2.79478	0.310531
$82$	0	0
$83$	2.06354	0.226502	0.113251	−	0.993566i	$-0.463874\pi$
$83$	2.06354	0.226502	0.113251	+	0.993566i	$0.463874\pi$
$84$	0	0
$85$	−3.11496	−0.337865
$86$	0	0
$87$	−7.55891	−0.810400
$88$	0	0
$89$	10.1262	1.07338	0.536688	−	0.843781i	$-0.319675\pi$
$89$	10.1262	1.07338	0.536688	+	0.843781i	$0.319675\pi$
$90$	0	0
$91$	1.00000	0.104828
$92$	0	0
$93$	6.51368	0.675438
$94$	0	0
$95$	25.0224	2.56724
$96$	0	0
$97$	7.60929	0.772607	0.386303	−	0.922372i	$-0.373752\pi$
$97$	7.60929	0.772607	0.386303	+	0.922372i	$0.373752\pi$
$98$	0	0
$99$	−2.24764	−0.225896
$100$	0	0
$101$	−9.98874	−0.993916	−0.496958	−	0.867774i	$-0.665550\pi$
$101$	−9.98874	−0.993916	−0.496958	+	0.867774i	$0.665550\pi$
$102$	0	0
$103$	−3.69030	−0.363617	−0.181808	−	0.983334i	$-0.558195\pi$
$103$	−3.69030	−0.363617	−0.181808	+	0.983334i	$0.558195\pi$
$104$	0	0
$105$	2.97152	0.289991
$106$	0	0
$107$	−3.62081	−0.350037	−0.175019	−	0.984565i	$-0.555999\pi$
$107$	−3.62081	−0.350037	−0.175019	+	0.984565i	$0.555999\pi$
$108$	0	0
$109$	6.78259	0.649654	0.324827	−	0.945773i	$-0.394694\pi$
$109$	6.78259	0.649654	0.324827	+	0.945773i	$0.394694\pi$
$110$	0	0
$111$	−1.53783	−0.145965
$112$	0	0
$113$	−3.73068	−0.350953	−0.175476	−	0.984484i	$-0.556147\pi$
$113$	−3.73068	−0.350953	−0.175476	+	0.984484i	$0.556147\pi$
$114$	0	0
$115$	−8.30270	−0.774231
$116$	0	0
$117$	−2.24764	−0.207794
$118$	0	0
$119$	0.909259	0.0833517
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	8.83105	0.796269
$124$	0	0
$125$	−5.94823	−0.532026
$126$	0	0
$127$	1.46792	0.130257	0.0651283	−	0.997877i	$-0.479254\pi$
$127$	1.46792	0.130257	0.0651283	+	0.997877i	$0.479254\pi$
$128$	0	0
$129$	−8.62225	−0.759147
$130$	0	0
$131$	−7.47741	−0.653304	−0.326652	−	0.945145i	$-0.605921\pi$
$131$	−7.47741	−0.653304	−0.326652	+	0.945145i	$0.605921\pi$
$132$	0	0
$133$	−7.30405	−0.633341
$134$	0	0
$135$	−15.5935	−1.34207
$136$	0	0
$137$	−10.7072	−0.914776	−0.457388	−	0.889267i	$-0.651215\pi$
$137$	−10.7072	−0.914776	−0.457388	+	0.889267i	$0.651215\pi$
$138$	0	0
$139$	4.93720	0.418768	0.209384	−	0.977834i	$-0.432854\pi$
$139$	4.93720	0.418768	0.209384	+	0.977834i	$0.432854\pi$
$140$	0	0
$141$	−7.57868	−0.638240
$142$	0	0
$143$	1.00000	0.0836242
$144$	0	0
$145$	−29.8545	−2.47929
$146$	0	0
$147$	−0.867389	−0.0715410
$148$	0	0
$149$	−19.8785	−1.62851	−0.814255	−	0.580507i	$-0.802854\pi$
$149$	−19.8785	−1.62851	−0.814255	+	0.580507i	$0.802854\pi$
$150$	0	0
$151$	−1.98286	−0.161363	−0.0806814	−	0.996740i	$-0.525710\pi$
$151$	−1.98286	−0.161363	−0.0806814	+	0.996740i	$0.525710\pi$
$152$	0	0
$153$	−2.04368	−0.165222
$154$	0	0
$155$	25.7264	2.06639
$156$	0	0
$157$	4.15912	0.331934	0.165967	−	0.986131i	$-0.446925\pi$
$157$	4.15912	0.331934	0.165967	+	0.986131i	$0.446925\pi$
$158$	0	0
$159$	10.0532	0.797272
$160$	0	0
$161$	2.42356	0.191003
$162$	0	0
$163$	16.3000	1.27671	0.638357	−	0.769741i	$-0.279614\pi$
$163$	16.3000	1.27671	0.638357	+	0.769741i	$0.279614\pi$
$164$	0	0
$165$	2.97152	0.231333
$166$	0	0
$167$	6.91211	0.534875	0.267438	−	0.963575i	$-0.413823\pi$
$167$	6.91211	0.534875	0.267438	+	0.963575i	$0.413823\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	16.4168	1.25543
$172$	0	0
$173$	−18.0163	−1.36975	−0.684875	−	0.728660i	$-0.740143\pi$
$173$	−18.0163	−1.36975	−0.684875	+	0.728660i	$0.740143\pi$
$174$	0	0
$175$	6.73629	0.509216
$176$	0	0
$177$	−3.04442	−0.228833
$178$	0	0
$179$	−1.81957	−0.136001	−0.0680004	−	0.997685i	$-0.521662\pi$
$179$	−1.81957	−0.136001	−0.0680004	+	0.997685i	$0.521662\pi$
$180$	0	0
$181$	9.89301	0.735341	0.367671	−	0.929956i	$-0.380155\pi$
$181$	9.89301	0.735341	0.367671	+	0.929956i	$0.380155\pi$
$182$	0	0
$183$	6.46771	0.478107
$184$	0	0
$185$	−6.07381	−0.446555
$186$	0	0
$187$	0.909259	0.0664916
$188$	0	0
$189$	4.55174	0.331091
$190$	0	0
$191$	−27.2842	−1.97421	−0.987106	−	0.160066i	$-0.948829\pi$
$191$	−27.2842	−1.97421	−0.987106	+	0.160066i	$0.948829\pi$
$192$	0	0
$193$	18.6515	1.34256	0.671282	−	0.741202i	$-0.265744\pi$
$193$	18.6515	1.34256	0.671282	+	0.741202i	$0.265744\pi$
$194$	0	0
$195$	2.97152	0.212795
$196$	0	0
$197$	−3.21615	−0.229141	−0.114571	−	0.993415i	$-0.536549\pi$
$197$	−3.21615	−0.229141	−0.114571	+	0.993415i	$0.536549\pi$
$198$	0	0
$199$	−22.9553	−1.62726	−0.813629	−	0.581384i	$-0.802511\pi$
$199$	−22.9553	−1.62726	−0.813629	+	0.581384i	$0.802511\pi$
$200$	0	0
$201$	−1.39519	−0.0984088
$202$	0	0
$203$	8.71455	0.611642
$204$	0	0
$205$	34.8790	2.43605
$206$	0	0
$207$	−5.44729	−0.378612
$208$	0	0
$209$	−7.30405	−0.505231
$210$	0	0
$211$	−10.7700	−0.741436	−0.370718	−	0.928745i	$-0.620888\pi$
$211$	−10.7700	−0.741436	−0.370718	+	0.928745i	$0.620888\pi$
$212$	0	0
$213$	−14.4782	−0.992032
$214$	0	0
$215$	−34.0543	−2.32249
$216$	0	0
$217$	−7.50953	−0.509780
$218$	0	0
$219$	−8.95407	−0.605060
$220$	0	0
$221$	0.909259	0.0611634
$222$	0	0
$223$	26.5963	1.78102	0.890508	−	0.454967i	$-0.150349\pi$
$223$	26.5963	1.78102	0.890508	+	0.454967i	$0.150349\pi$
$224$	0	0
$225$	−15.1407	−1.00938
$226$	0	0
$227$	−23.0232	−1.52810	−0.764051	−	0.645155i	$-0.776793\pi$
$227$	−23.0232	−1.52810	−0.764051	+	0.645155i	$0.776793\pi$
$228$	0	0
$229$	16.3899	1.08307	0.541536	−	0.840677i	$-0.317843\pi$
$229$	16.3899	1.08307	0.541536	+	0.840677i	$0.317843\pi$
$230$	0	0
$231$	−0.867389	−0.0570700
$232$	0	0
$233$	20.5382	1.34550	0.672752	−	0.739868i	$-0.265112\pi$
$233$	20.5382	1.34550	0.672752	+	0.739868i	$0.265112\pi$
$234$	0	0
$235$	−29.9327	−1.95259
$236$	0	0
$237$	3.75725	0.244060
$238$	0	0
$239$	7.90111	0.511081	0.255540	−	0.966798i	$-0.417747\pi$
$239$	7.90111	0.511081	0.255540	+	0.966798i	$0.417747\pi$
$240$	0	0
$241$	−16.4307	−1.05839	−0.529197	−	0.848499i	$-0.677507\pi$
$241$	−16.4307	−1.05839	−0.529197	+	0.848499i	$0.677507\pi$
$242$	0	0
$243$	−16.0794	−1.03149
$244$	0	0
$245$	−3.42583	−0.218868
$246$	0	0
$247$	−7.30405	−0.464745
$248$	0	0
$249$	−1.78989	−0.113430
$250$	0	0
$251$	−30.7620	−1.94168	−0.970839	−	0.239732i	$-0.922941\pi$
$251$	−30.7620	−1.94168	−0.970839	+	0.239732i	$0.922941\pi$
$252$	0	0
$253$	2.42356	0.152368
$254$	0	0
$255$	2.70189	0.169199
$256$	0	0
$257$	23.0017	1.43481	0.717404	−	0.696657i	$-0.245330\pi$
$257$	23.0017	1.43481	0.717404	+	0.696657i	$0.245330\pi$
$258$	0	0
$259$	1.77295	0.110166
$260$	0	0
$261$	−19.5871	−1.21241
$262$	0	0
$263$	5.96523	0.367832	0.183916	−	0.982942i	$-0.441123\pi$
$263$	5.96523	0.367832	0.183916	+	0.982942i	$0.441123\pi$
$264$	0	0
$265$	39.7060	2.43912
$266$	0	0
$267$	−8.78337	−0.537533
$268$	0	0
$269$	−16.9669	−1.03449	−0.517244	−	0.855838i	$-0.673042\pi$
$269$	−16.9669	−1.03449	−0.517244	+	0.855838i	$0.673042\pi$
$270$	0	0
$271$	−12.3148	−0.748071	−0.374036	−	0.927414i	$-0.622026\pi$
$271$	−12.3148	−0.748071	−0.374036	+	0.927414i	$0.622026\pi$
$272$	0	0
$273$	−0.867389	−0.0524968
$274$	0	0
$275$	6.73629	0.406214
$276$	0	0
$277$	−13.0742	−0.785554	−0.392777	−	0.919634i	$-0.628486\pi$
$277$	−13.0742	−0.785554	−0.392777	+	0.919634i	$0.628486\pi$
$278$	0	0
$279$	16.8787	1.01050
$280$	0	0
$281$	17.1597	1.02366	0.511830	−	0.859087i	$-0.328968\pi$
$281$	17.1597	1.02366	0.511830	+	0.859087i	$0.328968\pi$
$282$	0	0
$283$	−6.64306	−0.394889	−0.197444	−	0.980314i	$-0.563264\pi$
$283$	−6.64306	−0.394889	−0.197444	+	0.980314i	$0.563264\pi$
$284$	0	0
$285$	−21.7042	−1.28564
$286$	0	0
$287$	−10.1812	−0.600977
$288$	0	0
$289$	−16.1732	−0.951368
$290$	0	0
$291$	−6.60022	−0.386912
$292$	0	0
$293$	3.74092	0.218547	0.109274	−	0.994012i	$-0.465148\pi$
$293$	3.74092	0.218547	0.109274	+	0.994012i	$0.465148\pi$
$294$	0	0
$295$	−12.0242	−0.700077
$296$	0	0
$297$	4.55174	0.264119
$298$	0	0
$299$	2.42356	0.140158
$300$	0	0
$301$	9.94047	0.572959
$302$	0	0
$303$	8.66412	0.497741
$304$	0	0
$305$	25.5448	1.46269
$306$	0	0
$307$	1.48349	0.0846673	0.0423337	−	0.999104i	$-0.486521\pi$
$307$	1.48349	0.0846673	0.0423337	+	0.999104i	$0.486521\pi$
$308$	0	0
$309$	3.20093	0.182095
$310$	0	0
$311$	−13.9879	−0.793181	−0.396591	−	0.917996i	$-0.629807\pi$
$311$	−13.9879	−0.793181	−0.396591	+	0.917996i	$0.629807\pi$
$312$	0	0
$313$	−33.0829	−1.86996	−0.934979	−	0.354704i	$-0.884582\pi$
$313$	−33.0829	−1.86996	−0.934979	+	0.354704i	$0.884582\pi$
$314$	0	0
$315$	7.70001	0.433847
$316$	0	0
$317$	23.8898	1.34178	0.670891	−	0.741556i	$-0.265912\pi$
$317$	23.8898	1.34178	0.670891	+	0.741556i	$0.265912\pi$
$318$	0	0
$319$	8.71455	0.487921
$320$	0	0
$321$	3.14065	0.175294
$322$	0	0
$323$	−6.64127	−0.369530
$324$	0	0
$325$	6.73629	0.373662
$326$	0	0
$327$	−5.88314	−0.325338
$328$	0	0
$329$	8.73735	0.481706
$330$	0	0
$331$	−26.6066	−1.46243	−0.731217	−	0.682145i	$-0.761047\pi$
$331$	−26.6066	−1.46243	−0.731217	+	0.682145i	$0.761047\pi$
$332$	0	0
$333$	−3.98494	−0.218373
$334$	0	0
$335$	−5.51040	−0.301065
$336$	0	0
$337$	−1.86210	−0.101435	−0.0507174	−	0.998713i	$-0.516151\pi$
$337$	−1.86210	−0.101435	−0.0507174	+	0.998713i	$0.516151\pi$
$338$	0	0
$339$	3.23595	0.175753
$340$	0	0
$341$	−7.50953	−0.406664
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	7.20167	0.387725
$346$	0	0
$347$	−25.1182	−1.34841	−0.674207	−	0.738543i	$-0.735514\pi$
$347$	−25.1182	−1.34841	−0.674207	+	0.738543i	$0.735514\pi$
$348$	0	0
$349$	−31.7793	−1.70111	−0.850553	−	0.525889i	$-0.823733\pi$
$349$	−31.7793	−1.70111	−0.850553	+	0.525889i	$0.823733\pi$
$350$	0	0
$351$	4.55174	0.242954
$352$	0	0
$353$	−5.68546	−0.302606	−0.151303	−	0.988487i	$-0.548347\pi$
$353$	−5.68546	−0.302606	−0.151303	+	0.988487i	$0.548347\pi$
$354$	0	0
$355$	−57.1830	−3.03496
$356$	0	0
$357$	−0.788681	−0.0417414
$358$	0	0
$359$	−19.5556	−1.03211	−0.516053	−	0.856557i	$-0.672599\pi$
$359$	−19.5556	−1.03211	−0.516053	+	0.856557i	$0.672599\pi$
$360$	0	0
$361$	34.3491	1.80785
$362$	0	0
$363$	−0.867389	−0.0455261
$364$	0	0
$365$	−35.3649	−1.85108
$366$	0	0
$367$	7.73949	0.403998	0.201999	−	0.979386i	$-0.435256\pi$
$367$	7.73949	0.403998	0.201999	+	0.979386i	$0.435256\pi$
$368$	0	0
$369$	22.8836	1.19127
$370$	0	0
$371$	−11.5902	−0.601733
$372$	0	0
$373$	11.0111	0.570134	0.285067	−	0.958508i	$-0.407984\pi$
$373$	11.0111	0.570134	0.285067	+	0.958508i	$0.407984\pi$
$374$	0	0
$375$	5.15943	0.266432
$376$	0	0
$377$	8.71455	0.448822
$378$	0	0
$379$	−31.2500	−1.60520	−0.802602	−	0.596515i	$-0.796552\pi$
$379$	−31.2500	−1.60520	−0.802602	+	0.596515i	$0.796552\pi$
$380$	0	0
$381$	−1.27325	−0.0652308
$382$	0	0
$383$	12.0347	0.614942	0.307471	−	0.951557i	$-0.400517\pi$
$383$	12.0347	0.614942	0.307471	+	0.951557i	$0.400517\pi$
$384$	0	0
$385$	−3.42583	−0.174596
$386$	0	0
$387$	−22.3426	−1.13574
$388$	0	0
$389$	−21.9664	−1.11374	−0.556871	−	0.830599i	$-0.687998\pi$
$389$	−21.9664	−1.11374	−0.556871	+	0.830599i	$0.687998\pi$
$390$	0	0
$391$	2.20365	0.111443
$392$	0	0
$393$	6.48582	0.327166
$394$	0	0
$395$	14.8396	0.746660
$396$	0	0
$397$	28.4284	1.42678	0.713390	−	0.700767i	$-0.247159\pi$
$397$	28.4284	1.42678	0.713390	+	0.700767i	$0.247159\pi$
$398$	0	0
$399$	6.33545	0.317169
$400$	0	0
$401$	−0.131204	−0.00655199	−0.00327600	−	0.999995i	$-0.501043\pi$
$401$	−0.131204	−0.00655199	−0.00327600	+	0.999995i	$0.501043\pi$
$402$	0	0
$403$	−7.50953	−0.374076
$404$	0	0
$405$	−9.57443	−0.475757
$406$	0	0
$407$	1.77295	0.0878817
$408$	0	0
$409$	20.6271	1.01995	0.509973	−	0.860191i	$-0.329656\pi$
$409$	20.6271	1.01995	0.509973	+	0.860191i	$0.329656\pi$
$410$	0	0
$411$	9.28729	0.458108
$412$	0	0
$413$	3.50987	0.172710
$414$	0	0
$415$	−7.06931	−0.347019
$416$	0	0
$417$	−4.28247	−0.209713
$418$	0	0
$419$	−32.5082	−1.58813	−0.794064	−	0.607834i	$-0.792039\pi$
$419$	−32.5082	−1.58813	−0.794064	+	0.607834i	$0.792039\pi$
$420$	0	0
$421$	9.04250	0.440704	0.220352	−	0.975420i	$-0.429279\pi$
$421$	9.04250	0.440704	0.220352	+	0.975420i	$0.429279\pi$
$422$	0	0
$423$	−19.6384	−0.954851
$424$	0	0
$425$	6.12503	0.297108
$426$	0	0
$427$	−7.45653	−0.360847
$428$	0	0
$429$	−0.867389	−0.0418779
$430$	0	0
$431$	−19.5955	−0.943880	−0.471940	−	0.881631i	$-0.656446\pi$
$431$	−19.5955	−0.943880	−0.471940	+	0.881631i	$0.656446\pi$
$432$	0	0
$433$	−4.92299	−0.236584	−0.118292	−	0.992979i	$-0.537742\pi$
$433$	−4.92299	−0.236584	−0.118292	+	0.992979i	$0.537742\pi$
$434$	0	0
$435$	25.8955	1.24159
$436$	0	0
$437$	−17.7018	−0.846792
$438$	0	0
$439$	6.02126	0.287379	0.143689	−	0.989623i	$-0.454103\pi$
$439$	6.02126	0.287379	0.143689	+	0.989623i	$0.454103\pi$
$440$	0	0
$441$	−2.24764	−0.107030
$442$	0	0
$443$	−8.94157	−0.424827	−0.212413	−	0.977180i	$-0.568132\pi$
$443$	−8.94157	−0.424827	−0.212413	+	0.977180i	$0.568132\pi$
$444$	0	0
$445$	−34.6907	−1.64450
$446$	0	0
$447$	17.2424	0.815537
$448$	0	0
$449$	9.11032	0.429943	0.214971	−	0.976620i	$-0.431034\pi$
$449$	9.11032	0.429943	0.214971	+	0.976620i	$0.431034\pi$
$450$	0	0
$451$	−10.1812	−0.479413
$452$	0	0
$453$	1.71991	0.0808084
$454$	0	0
$455$	−3.42583	−0.160605
$456$	0	0
$457$	−27.7125	−1.29634	−0.648169	−	0.761497i	$-0.724465\pi$
$457$	−27.7125	−1.29634	−0.648169	+	0.761497i	$0.724465\pi$
$458$	0	0
$459$	4.13871	0.193179
$460$	0	0
$461$	4.69992	0.218897	0.109448	−	0.993992i	$-0.465092\pi$
$461$	4.69992	0.218897	0.109448	+	0.993992i	$0.465092\pi$
$462$	0	0
$463$	10.8472	0.504113	0.252057	−	0.967713i	$-0.418893\pi$
$463$	10.8472	0.504113	0.252057	+	0.967713i	$0.418893\pi$
$464$	0	0
$465$	−22.3148	−1.03482
$466$	0	0
$467$	−5.70150	−0.263834	−0.131917	−	0.991261i	$-0.542113\pi$
$467$	−5.70150	−0.263834	−0.131917	+	0.991261i	$0.542113\pi$
$468$	0	0
$469$	1.60849	0.0742731
$470$	0	0
$471$	−3.60758	−0.166228
$472$	0	0
$473$	9.94047	0.457063
$474$	0	0
$475$	−49.2022	−2.25755
$476$	0	0
$477$	26.0506	1.19277
$478$	0	0
$479$	−14.7463	−0.673775	−0.336888	−	0.941545i	$-0.609374\pi$
$479$	−14.7463	−0.673775	−0.336888	+	0.941545i	$0.609374\pi$
$480$	0	0
$481$	1.77295	0.0808394
$482$	0	0
$483$	−2.10217	−0.0956521
$484$	0	0
$485$	−26.0681	−1.18369
$486$	0	0
$487$	−12.3416	−0.559252	−0.279626	−	0.960109i	$-0.590210\pi$
$487$	−12.3416	−0.559252	−0.279626	+	0.960109i	$0.590210\pi$
$488$	0	0
$489$	−14.1384	−0.639362
$490$	0	0
$491$	−22.0297	−0.994186	−0.497093	−	0.867697i	$-0.665599\pi$
$491$	−22.0297	−0.994186	−0.497093	+	0.867697i	$0.665599\pi$
$492$	0	0
$493$	7.92379	0.356869
$494$	0	0
$495$	7.70001	0.346090
$496$	0	0
$497$	16.6917	0.748727
$498$	0	0
$499$	−32.4841	−1.45419	−0.727094	−	0.686538i	$-0.759129\pi$
$499$	−32.4841	−1.45419	−0.727094	+	0.686538i	$0.759129\pi$
$500$	0	0
$501$	−5.99549	−0.267859
$502$	0	0
$503$	14.3855	0.641418	0.320709	−	0.947178i	$-0.396079\pi$
$503$	14.3855	0.641418	0.320709	+	0.947178i	$0.396079\pi$
$504$	0	0
$505$	34.2197	1.52276
$506$	0	0
$507$	−0.867389	−0.0385221
$508$	0	0
$509$	−13.7739	−0.610518	−0.305259	−	0.952269i	$-0.598743\pi$
$509$	−13.7739	−0.610518	−0.305259	+	0.952269i	$0.598743\pi$
$510$	0	0
$511$	10.3230	0.456663
$512$	0	0
$513$	−33.2461	−1.46785
$514$	0	0
$515$	12.6423	0.557088
$516$	0	0
$517$	8.73735	0.384268
$518$	0	0
$519$	15.6271	0.685954
$520$	0	0
$521$	−35.1454	−1.53975	−0.769875	−	0.638195i	$-0.779681\pi$
$521$	−35.1454	−1.53975	−0.769875	+	0.638195i	$0.779681\pi$
$522$	0	0
$523$	23.0708	1.00882	0.504409	−	0.863465i	$-0.331710\pi$
$523$	23.0708	1.00882	0.504409	+	0.863465i	$0.331710\pi$
$524$	0	0
$525$	−5.84298	−0.255009
$526$	0	0
$527$	−6.82811	−0.297437
$528$	0	0
$529$	−17.1263	−0.744624
$530$	0	0
$531$	−7.88892	−0.342350
$532$	0	0
$533$	−10.1812	−0.440996
$534$	0	0
$535$	12.4043	0.536284
$536$	0	0
$537$	1.57827	0.0681075
$538$	0	0
$539$	1.00000	0.0430730
$540$	0	0
$541$	−0.824455	−0.0354461	−0.0177230	−	0.999843i	$-0.505642\pi$
$541$	−0.824455	−0.0354461	−0.0177230	+	0.999843i	$0.505642\pi$
$542$	0	0
$543$	−8.58109	−0.368250
$544$	0	0
$545$	−23.2360	−0.995319
$546$	0	0
$547$	−2.22571	−0.0951643	−0.0475821	−	0.998867i	$-0.515152\pi$
$547$	−2.22571	−0.0951643	−0.0475821	+	0.998867i	$0.515152\pi$
$548$	0	0
$549$	16.7596	0.715281
$550$	0	0
$551$	−63.6515	−2.71164
$552$	0	0
$553$	−4.33168	−0.184202
$554$	0	0
$555$	5.26835	0.223629
$556$	0	0
$557$	−16.3352	−0.692144	−0.346072	−	0.938208i	$-0.612485\pi$
$557$	−16.3352	−0.692144	−0.346072	+	0.938208i	$0.612485\pi$
$558$	0	0
$559$	9.94047	0.420437
$560$	0	0
$561$	−0.788681	−0.0332982
$562$	0	0
$563$	14.7377	0.621121	0.310561	−	0.950554i	$-0.399483\pi$
$563$	14.7377	0.621121	0.310561	+	0.950554i	$0.399483\pi$
$564$	0	0
$565$	12.7807	0.537686
$566$	0	0
$567$	2.79478	0.117370
$568$	0	0
$569$	−31.8702	−1.33607	−0.668034	−	0.744131i	$-0.732864\pi$
$569$	−31.8702	−1.33607	−0.668034	+	0.744131i	$0.732864\pi$
$570$	0	0
$571$	22.9263	0.959437	0.479719	−	0.877422i	$-0.340739\pi$
$571$	22.9263	0.959437	0.479719	+	0.877422i	$0.340739\pi$
$572$	0	0
$573$	23.6660	0.988660
$574$	0	0
$575$	16.3258	0.680834
$576$	0	0
$577$	46.9569	1.95484	0.977420	−	0.211306i	$-0.0677718\pi$
$577$	46.9569	1.95484	0.977420	+	0.211306i	$0.0677718\pi$
$578$	0	0
$579$	−16.1781	−0.672339
$580$	0	0
$581$	2.06354	0.0856099
$582$	0	0
$583$	−11.5902	−0.480017
$584$	0	0
$585$	7.70001	0.318356
$586$	0	0
$587$	−44.1204	−1.82104	−0.910522	−	0.413460i	$-0.864320\pi$
$587$	−44.1204	−1.82104	−0.910522	+	0.413460i	$0.864320\pi$
$588$	0	0
$589$	54.8499	2.26005
$590$	0	0
$591$	2.78965	0.114751
$592$	0	0
$593$	27.5657	1.13199	0.565994	−	0.824409i	$-0.308493\pi$
$593$	27.5657	1.13199	0.565994	+	0.824409i	$0.308493\pi$
$594$	0	0
$595$	−3.11496	−0.127701
$596$	0	0
$597$	19.9112	0.814910
$598$	0	0
$599$	0.716096	0.0292589	0.0146294	−	0.999893i	$-0.495343\pi$
$599$	0.716096	0.0292589	0.0146294	+	0.999893i	$0.495343\pi$
$600$	0	0
$601$	25.4663	1.03879	0.519395	−	0.854534i	$-0.326157\pi$
$601$	25.4663	1.03879	0.519395	+	0.854534i	$0.326157\pi$
$602$	0	0
$603$	−3.61530	−0.147226
$604$	0	0
$605$	−3.42583	−0.139280
$606$	0	0
$607$	38.7807	1.57406	0.787030	−	0.616914i	$-0.211617\pi$
$607$	38.7807	1.57406	0.787030	+	0.616914i	$0.211617\pi$
$608$	0	0
$609$	−7.55891	−0.306302
$610$	0	0
$611$	8.73735	0.353475
$612$	0	0
$613$	4.90078	0.197940	0.0989702	−	0.995090i	$-0.468445\pi$
$613$	4.90078	0.197940	0.0989702	+	0.995090i	$0.468445\pi$
$614$	0	0
$615$	−30.2537	−1.21994
$616$	0	0
$617$	−1.23198	−0.0495976	−0.0247988	−	0.999692i	$-0.507895\pi$
$617$	−1.23198	−0.0495976	−0.0247988	+	0.999692i	$0.507895\pi$
$618$	0	0
$619$	1.17625	0.0472777	0.0236388	−	0.999721i	$-0.492475\pi$
$619$	1.17625	0.0472777	0.0236388	+	0.999721i	$0.492475\pi$
$620$	0	0
$621$	11.0314	0.442676
$622$	0	0
$623$	10.1262	0.405698
$624$	0	0
$625$	−13.3038	−0.532153
$626$	0	0
$627$	6.33545	0.253013
$628$	0	0
$629$	1.61207	0.0642774
$630$	0	0
$631$	33.4335	1.33097	0.665483	−	0.746413i	$-0.268226\pi$
$631$	33.4335	1.33097	0.665483	+	0.746413i	$0.268226\pi$
$632$	0	0
$633$	9.34176	0.371302
$634$	0	0
$635$	−5.02883	−0.199563
$636$	0	0
$637$	1.00000	0.0396214
$638$	0	0
$639$	−37.5170	−1.48415
$640$	0	0
$641$	−11.2177	−0.443072	−0.221536	−	0.975152i	$-0.571107\pi$
$641$	−11.2177	−0.443072	−0.221536	+	0.975152i	$0.571107\pi$
$642$	0	0
$643$	−38.3894	−1.51393	−0.756965	−	0.653456i	$-0.773319\pi$
$643$	−38.3894	−1.51393	−0.756965	+	0.653456i	$0.773319\pi$
$644$	0	0
$645$	29.5384	1.16307
$646$	0	0
$647$	42.1996	1.65904	0.829519	−	0.558479i	$-0.188615\pi$
$647$	42.1996	1.65904	0.829519	+	0.558479i	$0.188615\pi$
$648$	0	0
$649$	3.50987	0.137775
$650$	0	0
$651$	6.51368	0.255291
$652$	0	0
$653$	47.0711	1.84204	0.921018	−	0.389520i	$-0.127359\pi$
$653$	47.0711	1.84204	0.921018	+	0.389520i	$0.127359\pi$
$654$	0	0
$655$	25.6163	1.00091
$656$	0	0
$657$	−23.2024	−0.905211
$658$	0	0
$659$	−32.7415	−1.27543	−0.637714	−	0.770273i	$-0.720120\pi$
$659$	−32.7415	−1.27543	−0.637714	+	0.770273i	$0.720120\pi$
$660$	0	0
$661$	34.8229	1.35445	0.677226	−	0.735775i	$-0.263182\pi$
$661$	34.8229	1.35445	0.677226	+	0.735775i	$0.263182\pi$
$662$	0	0
$663$	−0.788681	−0.0306298
$664$	0	0
$665$	25.0224	0.970327
$666$	0	0
$667$	21.1203	0.817779
$668$	0	0
$669$	−23.0693	−0.891911
$670$	0	0
$671$	−7.45653	−0.287856
$672$	0	0
$673$	−33.4918	−1.29101	−0.645507	−	0.763754i	$-0.723354\pi$
$673$	−33.4918	−1.29101	−0.645507	+	0.763754i	$0.723354\pi$
$674$	0	0
$675$	30.6619	1.18018
$676$	0	0
$677$	−14.6552	−0.563245	−0.281623	−	0.959525i	$-0.590873\pi$
$677$	−14.6552	−0.563245	−0.281623	+	0.959525i	$0.590873\pi$
$678$	0	0
$679$	7.60929	0.292018
$680$	0	0
$681$	19.9701	0.765255
$682$	0	0
$683$	14.4897	0.554432	0.277216	−	0.960808i	$-0.410588\pi$
$683$	14.4897	0.554432	0.277216	+	0.960808i	$0.410588\pi$
$684$	0	0
$685$	36.6809	1.40151
$686$	0	0
$687$	−14.2164	−0.542389
$688$	0	0
$689$	−11.5902	−0.441552
$690$	0	0
$691$	10.4303	0.396787	0.198393	−	0.980122i	$-0.436428\pi$
$691$	10.4303	0.396787	0.198393	+	0.980122i	$0.436428\pi$
$692$	0	0
$693$	−2.24764	−0.0853806
$694$	0	0
$695$	−16.9140	−0.641584
$696$	0	0
$697$	−9.25734	−0.350647
$698$	0	0
$699$	−17.8146	−0.673812
$700$	0	0
$701$	−18.4783	−0.697915	−0.348958	−	0.937139i	$-0.613464\pi$
$701$	−18.4783	−0.697915	−0.348958	+	0.937139i	$0.613464\pi$
$702$	0	0
$703$	−12.9497	−0.488406
$704$	0	0
$705$	25.9633	0.977833
$706$	0	0
$707$	−9.98874	−0.375665
$708$	0	0
$709$	7.82630	0.293923	0.146961	−	0.989142i	$-0.453051\pi$
$709$	7.82630	0.293923	0.146961	+	0.989142i	$0.453051\pi$
$710$	0	0
$711$	9.73603	0.365130
$712$	0	0
$713$	−18.1998	−0.681588
$714$	0	0
$715$	−3.42583	−0.128119
$716$	0	0
$717$	−6.85334	−0.255943
$718$	0	0
$719$	−14.8722	−0.554640	−0.277320	−	0.960778i	$-0.589446\pi$
$719$	−14.8722	−0.554640	−0.277320	+	0.960778i	$0.589446\pi$
$720$	0	0
$721$	−3.69030	−0.137434
$722$	0	0
$723$	14.2518	0.530031
$724$	0	0
$725$	58.7038	2.18020
$726$	0	0
$727$	−1.91835	−0.0711476	−0.0355738	−	0.999367i	$-0.511326\pi$
$727$	−1.91835	−0.0711476	−0.0355738	+	0.999367i	$0.511326\pi$
$728$	0	0
$729$	5.56275	0.206028
$730$	0	0
$731$	9.03846	0.334300
$732$	0	0
$733$	−28.8629	−1.06607	−0.533037	−	0.846092i	$-0.678949\pi$
$733$	−28.8629	−1.06607	−0.533037	+	0.846092i	$0.678949\pi$
$734$	0	0
$735$	2.97152	0.109606
$736$	0	0
$737$	1.60849	0.0592494
$738$	0	0
$739$	5.33825	0.196371	0.0981854	−	0.995168i	$-0.468696\pi$
$739$	5.33825	0.196371	0.0981854	+	0.995168i	$0.468696\pi$
$740$	0	0
$741$	6.33545	0.232739
$742$	0	0
$743$	2.23808	0.0821070	0.0410535	−	0.999157i	$-0.486929\pi$
$743$	2.23808	0.0821070	0.0410535	+	0.999157i	$0.486929\pi$
$744$	0	0
$745$	68.1003	2.49500
$746$	0	0
$747$	−4.63808	−0.169698
$748$	0	0
$749$	−3.62081	−0.132302
$750$	0	0
$751$	49.9912	1.82421	0.912103	−	0.409962i	$-0.134458\pi$
$751$	49.9912	1.82421	0.912103	+	0.409962i	$0.134458\pi$
$752$	0	0
$753$	26.6826	0.972368
$754$	0	0
$755$	6.79293	0.247220
$756$	0	0
$757$	−11.3987	−0.414293	−0.207147	−	0.978310i	$-0.566418\pi$
$757$	−11.3987	−0.414293	−0.207147	+	0.978310i	$0.566418\pi$
$758$	0	0
$759$	−2.10217	−0.0763040
$760$	0	0
$761$	52.5718	1.90573	0.952864	−	0.303399i	$-0.0981215\pi$
$761$	52.5718	1.90573	0.952864	+	0.303399i	$0.0981215\pi$
$762$	0	0
$763$	6.78259	0.245546
$764$	0	0
$765$	7.00131	0.253133
$766$	0	0
$767$	3.50987	0.126734
$768$	0	0
$769$	7.86379	0.283576	0.141788	−	0.989897i	$-0.454715\pi$
$769$	7.86379	0.283576	0.141788	+	0.989897i	$0.454715\pi$
$770$	0	0
$771$	−19.9514	−0.718534
$772$	0	0
$773$	−20.1738	−0.725601	−0.362800	−	0.931867i	$-0.618179\pi$
$773$	−20.1738	−0.725601	−0.362800	+	0.931867i	$0.618179\pi$
$774$	0	0
$775$	−50.5864	−1.81712
$776$	0	0
$777$	−1.53783	−0.0551695
$778$	0	0
$779$	74.3639	2.66436
$780$	0	0
$781$	16.6917	0.597278
$782$	0	0
$783$	39.6664	1.41756
$784$	0	0
$785$	−14.2484	−0.508549
$786$	0	0
$787$	−35.9802	−1.28255	−0.641277	−	0.767309i	$-0.721595\pi$
$787$	−35.9802	−1.28255	−0.641277	+	0.767309i	$0.721595\pi$
$788$	0	0
$789$	−5.17417	−0.184205
$790$	0	0
$791$	−3.73068	−0.132648
$792$	0	0
$793$	−7.45653	−0.264789
$794$	0	0
$795$	−34.4406	−1.22148
$796$	0	0
$797$	−48.5740	−1.72058	−0.860289	−	0.509807i	$-0.829717\pi$
$797$	−48.5740	−1.72058	−0.860289	+	0.509807i	$0.829717\pi$
$798$	0	0
$799$	7.94452	0.281057
$800$	0	0
$801$	−22.7601	−0.804187
$802$	0	0
$803$	10.3230	0.364291
$804$	0	0
$805$	−8.30270	−0.292632
$806$	0	0
$807$	14.7169	0.518058
$808$	0	0
$809$	−10.6830	−0.375595	−0.187797	−	0.982208i	$-0.560135\pi$
$809$	−10.6830	−0.375595	−0.187797	+	0.982208i	$0.560135\pi$
$810$	0	0
$811$	18.1166	0.636160	0.318080	−	0.948064i	$-0.396962\pi$
$811$	18.1166	0.636160	0.318080	+	0.948064i	$0.396962\pi$
$812$	0	0
$813$	10.6817	0.374625
$814$	0	0
$815$	−55.8409	−1.95602
$816$	0	0
$817$	−72.6056	−2.54015
$818$	0	0
$819$	−2.24764	−0.0785388
$820$	0	0
$821$	12.3637	0.431498	0.215749	−	0.976449i	$-0.430781\pi$
$821$	12.3637	0.431498	0.215749	+	0.976449i	$0.430781\pi$
$822$	0	0
$823$	20.3849	0.710573	0.355287	−	0.934757i	$-0.384383\pi$
$823$	20.3849	0.710573	0.355287	+	0.934757i	$0.384383\pi$
$824$	0	0
$825$	−5.84298	−0.203427
$826$	0	0
$827$	−27.7046	−0.963383	−0.481692	−	0.876341i	$-0.659977\pi$
$827$	−27.7046	−0.963383	−0.481692	+	0.876341i	$0.659977\pi$
$828$	0	0
$829$	−1.72155	−0.0597919	−0.0298960	−	0.999553i	$-0.509518\pi$
$829$	−1.72155	−0.0597919	−0.0298960	+	0.999553i	$0.509518\pi$
$830$	0	0
$831$	11.3404	0.393395
$832$	0	0
$833$	0.909259	0.0315040
$834$	0	0
$835$	−23.6797	−0.819470
$836$	0	0
$837$	−34.1814	−1.18148
$838$	0	0
$839$	−48.3893	−1.67059	−0.835293	−	0.549806i	$-0.814702\pi$
$839$	−48.3893	−1.67059	−0.835293	+	0.549806i	$0.814702\pi$
$840$	0	0
$841$	46.9434	1.61874
$842$	0	0
$843$	−14.8841	−0.512636
$844$	0	0
$845$	−3.42583	−0.117852
$846$	0	0
$847$	1.00000	0.0343604
$848$	0	0
$849$	5.76212	0.197755
$850$	0	0
$851$	4.29685	0.147294
$852$	0	0
$853$	−5.66959	−0.194123	−0.0970615	−	0.995278i	$-0.530944\pi$
$853$	−5.66959	−0.194123	−0.0970615	+	0.995278i	$0.530944\pi$
$854$	0	0
$855$	−56.2412	−1.92341
$856$	0	0
$857$	−30.4254	−1.03931	−0.519656	−	0.854375i	$-0.673940\pi$
$857$	−30.4254	−1.03931	−0.519656	+	0.854375i	$0.673940\pi$
$858$	0	0
$859$	−49.7599	−1.69779	−0.848893	−	0.528565i	$-0.822730\pi$
$859$	−49.7599	−1.69779	−0.848893	+	0.528565i	$0.822730\pi$
$860$	0	0
$861$	8.83105	0.300961
$862$	0	0
$863$	−11.5042	−0.391607	−0.195804	−	0.980643i	$-0.562731\pi$
$863$	−11.5042	−0.391607	−0.195804	+	0.980643i	$0.562731\pi$
$864$	0	0
$865$	61.7206	2.09856
$866$	0	0
$867$	14.0285	0.476433
$868$	0	0
$869$	−4.33168	−0.146942
$870$	0	0
$871$	1.60849	0.0545015
$872$	0	0
$873$	−17.1029	−0.578846
$874$	0	0
$875$	−5.94823	−0.201087
$876$	0	0
$877$	46.6066	1.57379	0.786897	−	0.617084i	$-0.211686\pi$
$877$	46.6066	1.57379	0.786897	+	0.617084i	$0.211686\pi$
$878$	0	0
$879$	−3.24484	−0.109446
$880$	0	0
$881$	−25.6361	−0.863701	−0.431850	−	0.901945i	$-0.642139\pi$
$881$	−25.6361	−0.863701	−0.431850	+	0.901945i	$0.642139\pi$
$882$	0	0
$883$	−31.7196	−1.06745	−0.533724	−	0.845659i	$-0.679208\pi$
$883$	−31.7196	−1.06745	−0.533724	+	0.845659i	$0.679208\pi$
$884$	0	0
$885$	10.4297	0.350590
$886$	0	0
$887$	−30.5155	−1.02461	−0.512304	−	0.858804i	$-0.671208\pi$
$887$	−30.5155	−1.02461	−0.512304	+	0.858804i	$0.671208\pi$
$888$	0	0
$889$	1.46792	0.0492323
$890$	0	0
$891$	2.79478	0.0936286
$892$	0	0
$893$	−63.8180	−2.13559
$894$	0	0
$895$	6.23352	0.208364
$896$	0	0
$897$	−2.10217	−0.0701894
$898$	0	0
$899$	−65.4422	−2.18262
$900$	0	0
$901$	−10.5385	−0.351088
$902$	0	0
$903$	−8.62225	−0.286931
$904$	0	0
$905$	−33.8917	−1.12660
$906$	0	0
$907$	32.8552	1.09094	0.545470	−	0.838131i	$-0.316351\pi$
$907$	32.8552	1.09094	0.545470	+	0.838131i	$0.316351\pi$
$908$	0	0
$909$	22.4510	0.744654
$910$	0	0
$911$	−28.1405	−0.932335	−0.466167	−	0.884697i	$-0.654366\pi$
$911$	−28.1405	−0.932335	−0.466167	+	0.884697i	$0.654366\pi$
$912$	0	0
$913$	2.06354	0.0682930
$914$	0	0
$915$	−22.1573	−0.732497
$916$	0	0
$917$	−7.47741	−0.246926
$918$	0	0
$919$	15.7955	0.521046	0.260523	−	0.965468i	$-0.416105\pi$
$919$	15.7955	0.521046	0.260523	+	0.965468i	$0.416105\pi$
$920$	0	0
$921$	−1.28676	−0.0424003
$922$	0	0
$923$	16.6917	0.549415
$924$	0	0
$925$	11.9431	0.392686
$926$	0	0
$927$	8.29446	0.272426
$928$	0	0
$929$	−49.4784	−1.62333	−0.811667	−	0.584121i	$-0.801439\pi$
$929$	−49.4784	−1.62333	−0.811667	+	0.584121i	$0.801439\pi$
$930$	0	0
$931$	−7.30405	−0.239380
$932$	0	0
$933$	12.1330	0.397215
$934$	0	0
$935$	−3.11496	−0.101870
$936$	0	0
$937$	−31.7528	−1.03732	−0.518659	−	0.854981i	$-0.673569\pi$
$937$	−31.7528	−1.03732	−0.518659	+	0.854981i	$0.673569\pi$
$938$	0	0
$939$	28.6958	0.936451
$940$	0	0
$941$	18.7322	0.610652	0.305326	−	0.952248i	$-0.401235\pi$
$941$	18.7322	0.610652	0.305326	+	0.952248i	$0.401235\pi$
$942$	0	0
$943$	−24.6747	−0.803520
$944$	0	0
$945$	−15.5935	−0.507256
$946$	0	0
$947$	−57.7445	−1.87645	−0.938223	−	0.346032i	$-0.887529\pi$
$947$	−57.7445	−1.87645	−0.938223	+	0.346032i	$0.887529\pi$
$948$	0	0
$949$	10.3230	0.335099
$950$	0	0
$951$	−20.7217	−0.671948
$952$	0	0
$953$	44.3505	1.43665	0.718326	−	0.695706i	$-0.244909\pi$
$953$	44.3505	1.43665	0.718326	+	0.695706i	$0.244909\pi$
$954$	0	0
$955$	93.4708	3.02464
$956$	0	0
$957$	−7.55891	−0.244345
$958$	0	0
$959$	−10.7072	−0.345753
$960$	0	0
$961$	25.3930	0.819130
$962$	0	0
$963$	8.13827	0.262252
$964$	0	0
$965$	−63.8968	−2.05691
$966$	0	0
$967$	0.387769	0.0124698	0.00623490	−	0.999981i	$-0.498015\pi$
$967$	0.387769	0.0124698	0.00623490	+	0.999981i	$0.498015\pi$
$968$	0	0
$969$	5.76056	0.185056
$970$	0	0
$971$	−18.1039	−0.580983	−0.290492	−	0.956878i	$-0.593819\pi$
$971$	−18.1039	−0.580983	−0.290492	+	0.956878i	$0.593819\pi$
$972$	0	0
$973$	4.93720	0.158279
$974$	0	0
$975$	−5.84298	−0.187125
$976$	0	0
$977$	38.8671	1.24347	0.621735	−	0.783228i	$-0.286428\pi$
$977$	38.8671	1.24347	0.621735	+	0.783228i	$0.286428\pi$
$978$	0	0
$979$	10.1262	0.323635
$980$	0	0
$981$	−15.2448	−0.486729
$982$	0	0
$983$	25.3806	0.809516	0.404758	−	0.914424i	$-0.367356\pi$
$983$	25.3806	0.809516	0.404758	+	0.914424i	$0.367356\pi$
$984$	0	0
$985$	11.0180	0.351062
$986$	0	0
$987$	−7.57868	−0.241232
$988$	0	0
$989$	24.0913	0.766060
$990$	0	0
$991$	32.4897	1.03207	0.516035	−	0.856567i	$-0.327407\pi$
$991$	32.4897	1.03207	0.516035	+	0.856567i	$0.327407\pi$
$992$	0	0
$993$	23.0783	0.732368
$994$	0	0
$995$	78.6409	2.49308
$996$	0	0
$997$	33.9469	1.07511	0.537554	−	0.843229i	$-0.319348\pi$
$997$	33.9469	1.07511	0.537554	+	0.843229i	$0.319348\pi$
$998$	0	0
$999$	8.07000	0.255323

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8008.2.a.l.1.4	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8008.2.a.l.1.4	✓	8	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)

\(f(q)\)	\(=\)	\(q-0.867389 q^{3} -3.42583 q^{5} +1.00000 q^{7} -2.24764 q^{9} +O(q^{10})\)	\(q-0.867389 q^{3} -3.42583 q^{5} +1.00000 q^{7} -2.24764 q^{9} +1.00000 q^{11} +1.00000 q^{13} +2.97152 q^{15} +0.909259 q^{17} -7.30405 q^{19} -0.867389 q^{21} +2.42356 q^{23} +6.73629 q^{25} +4.55174 q^{27} +8.71455 q^{29} -7.50953 q^{31} -0.867389 q^{33} -3.42583 q^{35} +1.77295 q^{37} -0.867389 q^{39} -10.1812 q^{41} +9.94047 q^{43} +7.70001 q^{45} +8.73735 q^{47} +1.00000 q^{49} -0.788681 q^{51} -11.5902 q^{53} -3.42583 q^{55} +6.33545 q^{57} +3.50987 q^{59} -7.45653 q^{61} -2.24764 q^{63} -3.42583 q^{65} +1.60849 q^{67} -2.10217 q^{69} +16.6917 q^{71} +10.3230 q^{73} -5.84298 q^{75} +1.00000 q^{77} -4.33168 q^{79} +2.79478 q^{81} +2.06354 q^{83} -3.11496 q^{85} -7.55891 q^{87} +10.1262 q^{89} +1.00000 q^{91} +6.51368 q^{93} +25.0224 q^{95} +7.60929 q^{97} -2.24764 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 5 q^{3} - 7 q^{5} + 8 q^{7} + 7 q^{9}+O(q^{10}) \) 8 * q - 5 * q^3 - 7 * q^5 + 8 * q^7 + 7 * q^9	\( 8 q - 5 q^{3} - 7 q^{5} + 8 q^{7} + 7 q^{9} + 8 q^{11} + 8 q^{13} - 5 q^{15} + 3 q^{17} - 19 q^{19} - 5 q^{21} + q^{23} + 15 q^{25} - 11 q^{27} - 21 q^{29} + 2 q^{31} - 5 q^{33} - 7 q^{35} - 12 q^{37} - 5 q^{39} - 6 q^{41} - 19 q^{43} - 17 q^{45} - 7 q^{47} + 8 q^{49} - 19 q^{51} - 26 q^{53} - 7 q^{55} + 2 q^{57} - 17 q^{59} + 7 q^{63} - 7 q^{65} - 24 q^{67} + 20 q^{69} + 2 q^{71} + 2 q^{73} + 18 q^{75} + 8 q^{77} + 7 q^{79} - 4 q^{81} - 6 q^{83} + 19 q^{85} - 13 q^{87} + 13 q^{89} + 8 q^{91} + 13 q^{93} + 21 q^{95} + 18 q^{97} + 7 q^{99}+O(q^{100}) \) 8 * q - 5 * q^3 - 7 * q^5 + 8 * q^7 + 7 * q^9 + 8 * q^11 + 8 * q^13 - 5 * q^15 + 3 * q^17 - 19 * q^19 - 5 * q^21 + q^23 + 15 * q^25 - 11 * q^27 - 21 * q^29 + 2 * q^31 - 5 * q^33 - 7 * q^35 - 12 * q^37 - 5 * q^39 - 6 * q^41 - 19 * q^43 - 17 * q^45 - 7 * q^47 + 8 * q^49 - 19 * q^51 - 26 * q^53 - 7 * q^55 + 2 * q^57 - 17 * q^59 + 7 * q^63 - 7 * q^65 - 24 * q^67 + 20 * q^69 + 2 * q^71 + 2 * q^73 + 18 * q^75 + 8 * q^77 + 7 * q^79 - 4 * q^81 - 6 * q^83 + 19 * q^85 - 13 * q^87 + 13 * q^89 + 8 * q^91 + 13 * q^93 + 21 * q^95 + 18 * q^97 + 7 * q^99

Introduction

L-functions

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Varieties

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Database

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